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+{"text":"\\section{\\label{}}\n\\section{INTRODUCTION}\nOPERA~\\cite{OPERA} is a long baseline neutrino experiment located\n in the Gran Sasso underground laboratory (LNGS) in Italy. The collaboration is composed\nof about 200 physicists coming from 36 institutions in 13 different countries. \nThe experiment is a massive hybrid detector\n with nuclear emulsions used as very precise tracking devices and electronic detectors \nto locate the neutrino interaction events in the emulsions.\n It is designed to primarily search\nfor $\\nu_{\\tau}$ appearance in the CERN high energy $\\nu_{\\mu}$ beam CNGS~\\cite{CNGS} at \n730 km from the neutrino source,\nin order to establish unambiguously  the origin of  the neutrino\n oscillations observed at the \"atmospheric\" $\\Delta m^{2}$ scale. The preferred hypothesis \nto describe this phenomenon being  $\\nu_{\\mu} \\rightarrow \\nu_{\\tau}$ oscillation.\nCombining all the present known neutrino data \n the best fit values of a global three flavour \nanalysis of neutrino oscillations~\\cite{Fogli2008} give for \n $\\nu_{\\mu} \\rightarrow \\nu_{\\tau}$ oscillation parameters \n $\\Delta m^{2}=2.39$x$10^{-3}$$ \\mathrm{eV}^{2}$  and $\\mathrm{sin}^{2}2\\theta$=0.995.\nThe range of allowed values at 3 $\\sigma$  is  \n2.06x$10^{-3} < \\Delta m^{2} <$ 2.81x$10^{-3}$$ \\mathrm{eV}^{2}$.\nIn addition to the dominant $\\nu_{\\mu}\\rightarrow\\nu_{\\tau}$ oscillation in $\\nu_{\\mu}$ beam, it is possible that a \nsub-leading $\\nu_{\\mu}\\rightarrow\\nu_{e}$ transition occurs as well.\nThis process will also be investigated by OPERA profiting from its \nexcellent electron identification capabilities \nto asses a possible improvement on the knowledge of the third yet unknown mixing angle $\\theta_{13}$.\n\nThe $\\nu_{\\tau}$ direct appearance search is based on the observation\nof events produced by  charged current interaction (CC) with the\n$\\tau$ decaying in leptonic and hadronic modes.\nIn order to directly observe the $\\tau$ kinematics,\nthe principle of the OPERA experiment is to observe the $\\tau$ trajectories \nand the decay products in emulsion films composed of\ntwo thin emulsion layers (44 $\\mu$m thick) put on either side of a plastic base\n (205 $\\mu$m thick). The detector concept which is described in the next section combines\nmicrometer tracking resolution, large target mass together with  good lepton identification. \nThis concept allows to reject efficiently the main topological background coming from charm production in \n$\\nu_\\mu$ charged current interactions.\n\n\n\\section{DETECTOR OVERVIEW}\n\nThe OPERA detector is installed in the Hall C of the Gran Sasso underground laboratory.\nFigure~\\ref{fig:opera} shows a recent picture of the detector which is 20 m long with a \ncross section of about 8x9 $\\mathrm{m}^{2}$ and composed \nof two identical parts called super modules (SM). Each SM has a target section and\na muon spectrometer. \n\\begin{figure*}[htb]\n\\vspace{-0.3cm}\n\\begin{center}\n    \\includegraphics[width=17cm]{opera_det.eps}\n\\end{center}\n\\vspace{-0.8cm}\n\\caption{View of the OPERA detector in Hall C of the Gran Sasso Underground Laboratory in May 2007.}\n\\label{fig:opera}\n\\vspace{0cm}\n\\end{figure*}\n\nThe spectrometer  allows a determination of the charge and momentum\nof muons going through by measuring their curvature in  a \ndipolar magnet made of 990 tons of iron, and providing 1.53 Tesla transverse to the \nneutrino beam axis. Each spectrometer is  equipped with six vertical planes of \ndrift tubes as precision tracker together with 22 planes (8x8 $\\mathrm{m}^{2}$) \nof RPC bakelite chambers reaching a spatial resolution of $\\sim$1 cm and an efficiency of 96\\%.\nThe precision tracker planes are composed of 4 staggered layers of 168 aluminium tubes,\n8 m long  with 38 mm  outer  diameter. The spatial resolution\nof this detector is better than 500 $\\mu$m.\nThe physics performance of the complete spectrometer should reduce the charge confusion \nto less than 0.3\\% and gives a momentum resolution better than 20\\% for momentum less than\n50 GeV. The muon identification efficiency reaches 95\\% adding  the target\ntracker information for the cases where  the  muons stop inside the target. \n\nThe target section is composed of 31 vertical light supporting steel structures, called walls,\ninterleaved with double layered planes of \n 6.6 m long scintillator strips in the two transverse\ndirections. The main goals\nof this electronic detector are to provide a trigger for the\nneutrino interactions, an efficient event pattern recognition together with the magnetic spectrometer\nallowing a clear classification of the $\\nu$ interactions\n and a precise localisation of the event.\nThe electronic target tracker spatial resolution\nreaches $\\sim$0.8 cm and has an efficiency of 99\\%.\\\\\nThe walls contain the basic target detector units, called ECC brick, sketched in Fig.~\\ref{fig:brick}\nwhich are obtained by stacking 56 lead plates with 57 emulsion films. This structure provides\nmany advantages like a massive target coupled to a very precise tracker, as well as a standalone\ndetector to measure electromagnetic showers and charged particle momentum using the multiple\ncoulomb scattering in the lead. The ECC concept has been\nalready succesfully used for the direct $\\nu_\\tau$ observation perfomed in 2000 by the DONUT \nexperiment~\\cite{donut}.\n\\begin{figure*}[htb]\n\\vspace{-0.3cm}\n\\begin{center}\n    \\includegraphics[width=6cm]{brick_emul3.eps}\n    \\includegraphics[width=6cm]{brique.eps}\n\\end{center}\n\\vspace{-0.8cm}\n\\caption{a)Schematic structure of an ECC cell. The $\\tau$ decay kink is\nreconstructed by using four track segments in the emulsion films. b) Picture of an assembled brick. \n Each brick weights about 8.6 kg and has a thickness of 10 radiation length $X_{o}$.          }\n\\label{fig:brick}\n\\vspace{0cm}\n\\end{figure*}\n\n Behind each brick, an  emulsion film doublet, called Changeable Sheet (CS) is attached in \n a separate enveloppe. The CS can be detached from the brick for analysis\n  to confirm and locate the tracks produced in neutrino interactions.\n\nBy the time of this conference, 146500  bricks (1.25 kton of target) assembled\n underground at an average rate of about 700 bricks\/day by\na dedicated fully automated Brick Assembly Machine (BAM) with precise robotics  were\ninstalled in the support steel structures from the sides of the walls\nusing two automated manipulator systems (BMS) running on each side of the experiment. \\\\\nWhen a candidate brick has been located by the electronic detectors, \nthe brick is removed using the BMS and the changeable\nsheet is detached and developped. The film is then scanned to \nsearch for the tracks originating from the neutrino interaction. If none\nare found then the brick is left untouched and another\none is removed. When a neutrino \n event is confirmed the brick is exposed to cosmics\nto collect enough alignment tracks before going to the development. \nAfter development the emulsions are sent to the scanning laboratories hosting\nautomated optical  microscopes in Europe and Japan, each region using\na different technology~\\cite{scan1,scan2}. This step is the \nstart of the detailed analysis consisting of finding the neutrino vertex\nand looking for a decay kink topology in the vertex region.  \n\n\n\\section{THE CNGS BEAM STARTUP}\n The CNGS neutrino beam~\\cite{CNGS} is a high energy $\\nu_\\mu$ beam optimised\nto maximise the $\\nu_\\tau$ charged current interactions at Gran Sasso produced\nby oscillation mechanism at the atmospheric $\\Delta m^{2}$.\nThe mean neutrino energy is about 17 GeV with a  contamination of 2.4\\% $\\overline{\\nu}_{\\mu}$,\n 0.9\\% $\\nu_{e}$ and less than 0.06\\% of $\\overline{\\nu}_{e}$.\nUsing the CERN  SPS accelerator in a shared mode with fixed target experiment together with LHC, \n4.5x$10^{19}$ protons on target (pot) per year should normally be delivered,\n assuming 200 days of operation.\nThe number of charged current and neutral current interactions expected in the Gran Sasso\nlaboratory from $\\nu_\\mu$\nare then about 2900 \/kton\/year and 875 \/kton\/year respectively.\nIf the $\\nu_{\\mu}\\rightarrow\\nu_{\\tau}$ oscillation hypothesis is confirmed, \nthe number of $\\tau$'s produced via\ncharged current interaction at the Gran Sasso should be\nof the order of 14 \/kton\/year \nfor  $\\Delta m^{2}=$2.5x$10^{-3}$$ \\mathrm{eV}^{2}$ at full mixing. \n\nA first CNGS short run took place in August 2006. The OPERA target was empty at that time\nbut the electronic detectors were taking data.\nDuring this run, 319 events correlated in time with the beam and coming from neutrino \ninteractions in the surrounding rock and inside the detector have been recorded. \n The delivered intensity correponded to  $7.6\\mathrm{x}10^{17}$ pot, with a \npeak intensity of $1.7$x$10^{13}$ pot per extraction corresponding to 70\\% of the expected nominal value.\n The reconstructed zenith angle distribution from penetrating muon tracks was showing a clear\npeak centered around $3.4^{o}$ as expected for neutrinos originating from CERN.\nDetails and results can  be found in Ref~\\cite{cngs2006}.\n\n\\section{FIRST NEUTRINO EVENTS AND DETECTOR PERFORMANCES}\nA second CNGS physics run took place in October\n2007 with a total of  $8.24\\mathrm{x}10^{17}$ pot delivered and 369 reconstructed beam related events.\nSimilar selection criteria to the 2006 analysis~\\cite{cngs2006},\n based on GPS timing systems and synchronisation between OPERA and CNGS, have been used\nto select events compatible with the CNGS proton extraction time window. \n The OPERA target was filled with 80\\% of the first\nsupermodule corresponding to a total target mass of 0.5 kton. \nAmong the selected beam events, 38 were recorded and reconstructed inside the OPERA target\nfor 31.5$\\pm$6 expected.\nAmong them, 29 were classified as Charged Current (CC) and 9\nas Neutral Current (NC) in agreement with expectation. \nFor each event the electronic detector hits were used to find the most probable\nbrick where the neutrino interaction may have occured. \nThe left part of Figure~\\ref{fig:cngs_event} shows an event display of the first neutrino \ninteraction located in the OPERA detector. The black dots represent hits in the \nelectronic detector. The event is a charged current event with a clear muon track traversing\nboth target and spectrometer sections over more than 18 m. The right part of the figure\nshows the result of the detailed analysis of the emulsions after scanning the identified \nbrick where a clear reconstructed\ninteraction vertex is visible with two photon conversions compatible with \na $\\pi^{o}$ decay.\n\n\\begin{figure*}[htb]\n\\vspace{-0.3cm}\n\\begin{center}\n    \\includegraphics[width=10cm]{cngs_event.eps}\n    \\includegraphics[width=7cm]{vertex.eps}\n\n\\end{center}\n\\vspace{-0.8cm}\n\\caption{a) Charged current neutrino interaction recorded in OPERA. The event display \nshows the hits left in the electronic detectors). b) Emulsion reconstruction of the \nneutrino interaction vertex in the corresponding target brick.}\n\\label{fig:cngs_event}\n\\vspace{0cm}\n\\end{figure*}\n\nThe extensive study of the recorded events have confirmed the OPERA\nperformances and the validity of the methods and algorithms used which, for example, give\nimpact parameter resolution of the order of a few microns,  particle momentum estimation, \nshower detection for e\/$\\pi$ separation.\n Figure~\\ref{fig:charm2007} shows the longitudinal and \ntransverse views of another reconstructed event vertex where a clear decay topology similar to what is expected\nfrom a $\\tau$ decay is visible. However, the presence of a prompt muon attached to the primary vertex and the momentum\nbalance in the transverse plane is in favour of a $\\nu_\\mu^{CC}$ interaction producing a charm particle. \n \n\\begin{figure*}[htb]\n\\vspace{-0.3cm}\n\\begin{center}\n    \\includegraphics[width=9cm]{charm2007_v2.eps}\n\\end{center}\n\\vspace{-0.8cm}\n\\caption{Longitudinal and transverse view of a reconstructed neutrino interaction vertex with a charm\ndecay candidate topology.}\n\\label{fig:charm2007}\n\\vspace{0cm}\n\\end{figure*}\n\n\n\\section{CONCLUSIONS}\n\nThe OPERA detector is completed and is now massive with 1.25 kton of lead-emulsion target offering\na huge and precise tracking device. \nWith the cosmic data taking and the first CNGS neutrino runs in 2006 and 2007,\nthe design goals and detector perfomances\nwere reached and the first levels of the reconstruction software and analysis tools were validated.\nThe observation in 2007 of 38 neutrino events in the target bricks, the \nlocalization and reconstruction of neutrino vertex in emulsions was an important phase which\nsuccesfully validated the OPERA detector concept.\\\\\nHaving now the full OPERA target, the next important step is the 2008 CNGS neutrino run which\nstarted already  in June.\nIt is expected to have about $2.28$x$10^{19}$ pot in 123 days of SPS running assuming a nominal \nintensity of $2$x$10^{13}$ pot\/extraction. This intensity, when reached, should lead\n to about 20 neutrino interactions\/day  in the target and eventually the  \nobservation of the first $\\tau$ event candidate.\\\\\nIn 5 years of CNGS running at 4.5x$10^{19}$ pot per year, OPERA should be able \nto observe 10 to 15 $\\nu_\\tau$ events after oscillation at\nfull mixing in the range $2.5$x$10^{-3} < \\Delta m^{2} <$ 3x$10^{-3}$ $\\mathrm{eV}^{2}$,\nwith a total background less than 0.76 events.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} The combination of control theory techniques with quantum mechanics (see, e.g., \\cite{AltaTico}, \\cite{Mikobook}, \\cite{Optrev})  has generated a rich set  of control algorithms for quantum mechanical systems modeled by the \nSchr\\\"odinger (operator) equation \n\\be{Scrod11}\n\\dot X=AX+ \\sum_{j=1}^m B_j u_jX, \\qquad X(0)={\\bf 1}. \n\\end{equation}    \nHere we assume that we have a finite dimensional model, $A$ and $ B_j$ are matrices in $\\mathfrak{su}(n)$ for each $j=1,...,m$, $X$ is the unitary propagator, which is  equal to the identity ${\\bf 1}$ at time zero, and $u_j$ are the controls. These are usually electromagnetic fields, constant in space but possibly time-varying, which are the output of an appropriately engineered  pulse-shaping device. Many of the proposed algorithms in the literature involve control functions which are only piecewise continuous and in fact have `jumps' at certain points of the interval of control. For example, control algorithms  based on {\\it Lie group decompositions} (see, e.g., \\cite{Rama1}) involve `switches' between different Hamiltonians; Algorithms based on \n{\\it optimal control}, even if they produce smooth control functions, often require a jump at the beginning of the control interval in order for  the control to achieve the prescribed value in norm (assuming a bound in norm of the optimal control as in \\cite{Bosca1}). Beside the practical  problem of generating (almost) instantaneous switches with pulse shapers, such discontinuities introduce undesired high frequency \ncomponents in the dynamics of the controlled system. For these reasons, it is important  to have algorithms which produce {\\it smooth} control functions whose values at the beginning and the end of the control interval are equal to zero. \n\nThis paper describes a method to design control functions without discontinuities in order to drive the state of a class of quantum systems of the form (\\ref{Scrod11}) to an arbitrary final configuration. Our main example of application will be the simultaneous control of two quantum bits in zero field NMR, a system which was also considered in \\cite{Xinhua} in the context of optimal control. As compared to that paper, we abandon here the requirement of time optimality (under the requirement of bounded norm for the control) but introduce a novel \nmethod which will allow us more flexibility in the control design.  The result is a control algorithm that does not present discontinuities, with the control equal to zero at the beginning and at the end of the control interval.  \n\nThe paper is organized in two main sections each of which divided into several subsections. In Section \\ref{GenTheor} we describe the class of systems we consider and the general theory underlying our method. We also present two simple examples of quantum systems where the theory applies. In Section \\ref{App2QB} we detail the application to the system of two spin $\\frac{1}{2}$ particles in zero field NMR above mentioned. This section includes a description of the model as well as the explicit numerical treatment of a control problem: the independent control of the two spin $\\frac{1}{2}$ particles to two different types of Hadamard gates. \n\n\n\n\\section{General Theory}\\label{GenTheor}\n\n\\subsection{Class of systems considered}\\label{Classo} \n\nConsider the class of control systems (\\ref{Scrod11}) with $A$ and $ B_j$, $j=1,...,m$, in $\\mathfrak{su}(n)$ and let ${\\cal L}$ denote the Lie algebra generated by $\\{A, B_1,...,B_m\\}$. We assume that ${\\cal L}$ is semi-simple, which implies, since ${\\cal L} \\subseteq \\mathfrak{su}(n)$, that the associated Lie group $e^{\\cal L}$ is compact. The Lie algebra ${\\cal L}$ is called, in quantum control theory, the {\\it dynamical Lie algebra} associated to the system (\\ref{Scrod11}).  Since $e^{\\cal L}$ is compact, the Lie group $e^{\\cal L}$ is the set of states for (\\ref{Scrod11}) reachable by changing the control \\cite{Suss}. In particular if ${\\cal L}=\\mathfrak{su}(n)$, the system is said to be {\\it controllable} because every special unitary matrix can be obtained with appropriate control. These are known facts in quantum control theory (see, e.g., \\cite{Mikobook}). We assume that ${\\cal L}$ has a (vector space) decomposition ${\\cal L}={\\cal K} \\oplus {\\cal P}$, such that $[{\\cal K}, {\\cal K}] \\subseteq {\\cal K}$, i.e., ${\\cal K}$ is a {\\it Lie subalgebra of ${\\cal L}$}, which we also assume to be semisimple so that $e^{\\cal K}$ is compact. Moreover $[{\\cal K}, {\\cal P}] \\subseteq {\\cal P}$. A special case is when in addition $[{\\cal P}, {\\cal P} ] \\subseteq {\\cal K}$ in which case the decomposition ${\\cal L}={\\cal K} \\oplus {\\cal P}$ defines a symmetric space of $e^{\\cal L}$ \\cite{Helgason}. We assume, in the model (\\ref{Scrod11}), that such a decomposition exists so that $A \\in {\\cal K}$ and $\\{B_1,...,B_m\\}$ forms a basis for ${\\cal P}$. \n\n\nUnder such circumstances, we can reduce ourselves to the case $A=0$ in (\\ref{Scrod11}), i.e., to systems of the form \n\\be{Scro}\n\\dot U= \\sum_{k=1}^m \\hat u_k B_k U, \\qquad U(0)={\\bf 1}.   \n\\end{equation}   \nTo see this, assume that for any fixed interval $[0,t_f]$ and any desired final condition $ U_f$, we are able to find a control $\\hat u_k$ steering the state $U$ in (\\ref{Scro}) from the identity ${\\bf 1}$ to $U_f$. Let $a_{kj}=a_{kj}(t)$, $k,j=1,...,m$ the coefficients forming an $m \\times m$ orthogonal matrix, so that, for any $j=1,...,m$,  \n\\be{transforB}\ne^{-At} B_j e^{At}=\\sum_{k=1}^m a_{kj}(t) B_k. \n\\end{equation} \nLet $X_f$ be  the desired final condition for (\\ref{Scrod11}) and $\\hat u_k$ be the control steering the state $U$ of system (\\ref{Scro}) from the identity ${\\bf 1}$ to $e^{-At_f} X_f$, in time $t_f$. Then the control $u_j$ obtained inverting \n\\be{tobeinverted}\n\\hat u_k(t):=\\sum_{j=1}^m a_{kj}(t)u_j(t), \n\\end{equation}\nsteers the state $X$ of (\\ref{Scrod11}) from the identity to $X_f$. This follows from the fact that, if $U=U(t)$ is the solution of (\\ref{Scro}) with the control $\\hat u_k$, and final condition $e^{-At_f} X_f$,  then $X=e^{At}U$ is a solution of   (\\ref{Scrod11}), with the controls $u_j$ given by (\\ref{tobeinverted}) and therefore  the final condition at $t_f$ is $X_f$. Notice that the transformation (\\ref{tobeinverted}) does not modify the smoothness properties of the control, neither does it modify the fact that the control is zero at the beginning and at the end of the control interval (or at any other point). \nTherefore in the following we shall deal with {\\it driftless systems} of the form (\\ref{Scro}) with the Lie algebraic ${\\cal L}={\\cal K} \\oplus {\\cal P}$, structure above described. In particular $\\{B_1,...,B_m\\}$ forms a basis for ${\\cal P}$.  \n\n\\subsection{Symmetry reduction}\n \nThe compact Lie group $e^{\\cal K}$ can be seen as a Lie \ntransformation group which acts on $e^{\\cal L}$  via conjugation $X \\in e^{\\cal L}  \\rightarrow KXK^{-1}$, $ K \\in e^{\\cal K}$. Moreover this is  a {\\it group of symmetries} for system (\\ref{Scro}) \nin the sense that  $K B_j K^{-1} \\in {\\cal P}$ for each $j$, for every $K \\in e^{\\cal K}$. In particular let $K B_j K^{-1} :=\\sum_{k=1}^m a_{kj} B_k$ for an orthogonal matrix $\\{a_{kj}\\}$ depending on $K \\in e^{\\cal K}$ (cf. (\\ref{transforB})).  If $U=U(t)$ is a trajectory corresponding to a control $\\hat u_k$, $KUK^{-1}$ is the trajectory \ncorresponding to controls $u_k:=\\sum_{j=1}^ma_{kj} \\hat u_j$, as it is easily seen from (\\ref{Scro}) and \n$$\nK\\dot U K^{-1}=\\sum_{j=1}^m \\hat u_j KB_j K^{-1} KUK^{-1}= \\sum_{j=1}^m \\hat u_j \\left( \\sum_{k=1}^m a_{kj} B_k \\right) KUK^{-1}=$$\n$$=\\sum_{k=1}^m \\left( \\sum_{j=1}^m a_{kj} \\hat u_j \\right) B_k KUK^{-1}=\\sum_{k=1}^m u_k B_k KUK^{-1}. \n$$  \nThis suggests to treat the control problem on the {\\it quotient space} $e^{\\cal L}\/e^{\\cal K}$ corresponding \nto the above action of $e^{\\cal K}$ on $e^{\\cal L}$.  \n    \n \nFrom the theory of Lie transformation groups (see, e.g., \\cite{Bredon}) we know that the quotient space $e^{\\cal L}\/e^{\\cal K}$ has the structure of a {\\it stratified space} where each stratum corresponds to an {\\it orbit type}, i.e., a set of points in $e^{\\cal L}$ which have conjugate isotropy groups. The stratum corresponding to the smallest possible isotropy group, $K_{min}$, is known  to be a connected manifold which is {\\it open and dense} in $e^{\\cal L}\/e^{\\cal K}$. We denote it here by $e^{\\cal L}_{reg}\/e^{\\cal K}$, where $reg$ stands for the {\\it regular} part.  Its preimage in $e^{\\cal L}$, $e^{\\cal L}_{reg}$, under the natural projection $\\pi \\, : \\, e^{\\cal L} \\rightarrow  e^{\\cal L}\/e^{\\cal K}$ is open and dense in $e^{\\cal L}$ \nas well. This is called the {\\it regular part} of   $e^{\\cal L}\/e^{\\cal K}$, (resp. $e^{\\cal L}$). The complementary set in $e^{\\cal L}\/e^{\\cal K}$, (resp. $e^{\\cal L}$) is called the {\\it singular part}. The dimension of $e^{\\cal L}_{reg}\/e^{\\cal K}$ as a manifold is \n\\be{dimensio}\n\\dim (e^{\\cal L}_{reg}\/e^{\\cal K}) =\\dim (e^{\\cal L})-\\dim (e^{\\cal K})+\\dim K_{min}=\\dim ({\\cal L})-\\dim ({\\cal K})+(\\dim K_{min}), \n\\end{equation}   \nwhere $\\dim ( K_{min})$ is the dimension of the minimal isotropy group as a Lie group.\\footnote{More discussion on these basic facts in the theory of Lie transformation groups can be found in \\cite{NOIJDCS} and references therein.}   In particular, if \n$K_{min}$ is a {\\it discrete  Lie group}, i.e., it has dimension zero, the right hand side of (\\ref{dimensio}) is the dimension of the subspace ${\\cal P}$.  This is verified  for instance in $K-P$ problems (cf., e.g., \\cite{conBenECC})  when $e^{\\cal L}=SU(n)$. We shall assume this to be the case in the following. \n\nAccording to a result in \\cite{conBenECC}, under the assumption that the minimal isotropy group $K_{min}$ is discrete,   the restriction of $\\pi_*$ to $R_{x*} {\\cal P}$ is an isomorphism onto $T_{\\pi(x)} \\left( e^{\\cal L}_{reg}\/{e^{\\cal K}} \\right)$ for each point $x$ in the regular part, $e^{\\cal L}_{reg}$.  Here, as it is often done, we have identified the Lie algebra ${\\cal L}$ with the tangent space of $e^{\\cal L}$ at the identity ${\\bf 1}$, and therefore ${\\cal P}$ is  identified with a subspace of the tangent space at $\\bf{1}$. The map $R_x$ denotes the {\\it right translation} by $x$ so that $R_{x*}{\\cal P}$ is a subspace (with the same dimension) of the tangent space at $x$, $T_x {e^{\\cal L}}$.\\footnote{Recall that for a map $f \\, : \\, M \\rightarrow N$ for two manifolds $M$ and $N$, $f_*$ denotes the {\\it differential} (also called {\\it push-forward}) $f_* \\, : \\, T_xM \\rightarrow  T_{f(x)}N$ between two tangent spaces. When we want to emphasize the point $x$ we write $f_*|_x$.  } In Appendix B, we show that in given coordinates the determinant of the restriction of  $\\pi_*$ to $R_{x*}{\\cal P}$ is invariant under the action of $e^{\\cal K}$. The above isomorphism result says that in the regular part $\\det \\pi_* \\not=0$. In this situation, for every regular point $U \\in e^{\\cal L}$, for every tangent vector $V \\in  T_{\\pi(U)}(e^{\\cal L}_{reg}\/e^{\\cal K})$, we can find a tangent vector $\\pi_*^{-1} V \\in R_{U*} {\\cal P}$. Such a tangent vector is {\\it horizontal} for system (\\ref{Scro})  which means  that it  can be written as a linear combination of the available vector fields $\\{  B_k U \\}$ in (\\ref{Scro}). If $\\Gamma=\\Gamma(t)$ is a curve entirely contained in $e^{\\cal L}_{reg}\/e^{\\cal K}$ and $U=U(t)$ a curve in $e^{\\cal L}_{reg}$ such that $\\pi(U(t))=\\Gamma(t)$ for every $t$, i.e., $U$ is a `{\\it lift}' of $\\Gamma$, then $\\pi_*|_{U}^{-1} \\dot \\Gamma$ is a horizontal tangent vector at $U(t)$ for every $t$. If $\\Gamma$ joins two points $\\Gamma_0$ and $\\Gamma_1$ in  $e^{\\cal L}_{reg}\/e^{\\cal K}$, in the interval $[t_0,t_1]$,  and $U_0$ is such that $\\pi(U_0)=\\Gamma_0$, then the solution of the differential system \n\\be{diffsys}\n\\dot U=\\pi_*|_{U}^{-1} \\dot \\Gamma, \\qquad U(t_0)=U_0,  \n\\end{equation} \nis such that $\\pi(U(t_1))=\\Gamma_1$. Therefore, once we prescribe an arbitrary trajectory $\\Gamma$ to move in the quotient space between two given orbits $\\Gamma_0$ and $\\Gamma_1$  in the regular part,  the control specified by \n\\be{deficontr}\n\\pi_*|_{U}^{-1} \\dot \\Gamma=\\sum_{j=1}^m u_j B_j U\n\\end{equation}\n will allow us to move between two states $U_0$ and $U_1$ such that $\\pi(U_0)=\\Gamma_0$ and $\\pi(U_1)=\\Gamma_1$. \n \n \n \\subsection{Methodology for Control} \n The above treatment suggests a general methodology to design control laws \n  for  systems of the form (\\ref{Scro}) described in subsection \\ref{Classo}. In fact, \n   given the freedom in the choice of the trajectory $\\Gamma=\\Gamma(t)$ above mentioned,  \n   we can design such controls satisfying various requirements and in particular  without discontinuity. Such a  methodology can be summarized as follows. \n   \n First of all we need to obtain a geometric description of the orbit space $e^{\\cal L}\/e^{\\cal K}$, and in particular of its regular part  $e^{\\cal L}_{reg}\/e^{\\cal K}$, and verify that the minimal isotropy group, which is the isotropy group of the elements in $e^{\\cal L}_{reg}$, is discrete so that the right hand side of \n (\\ref{dimensio}) is equal to $\\dim {\\cal P}$. This is a weak assumption, easily verified in the examples that will follow and that can be proven true in several cases \\cite{conBenECC}, \\cite{BenTesi}. Then one chooses\n  coordinates for the manifold $e^{\\cal L}_{reg}\/e^{\\cal K}$. These are   expressed in terms of the original coordinates in \n $e^{\\cal L}$ or, more commonly, in terms of the entries of the matrices in $e^{\\cal L}$. Such coordinates are a complete set of {\\it independent invariants} with respect to the (conjugacy) action of the group $e^{\\cal K}$. The word `complete' here means that the knowledge of their values uniquely determines the {\\it orbit}, i.e., a point in $e^{\\cal L}_{reg}\/e^{\\cal K}$. There are $m=\\dim({\\cal P})$ of them, as this is the dimension of $e^{\\cal L}_{reg}\/e^{\\cal K}$ (cf. (\\ref{dimensio})).  Once we have coordinates $\\{x^1,...,x^m\\}$, the tangent vectors $\\left \\{ \\frac{\\partial}{\\partial x^1},...,\\frac{\\partial}{\\partial x^m}\\right\\}$ at every regular point in the quotient space determine a basis of the tangent space of  $e^{\\cal L}_{reg}\/e^{\\cal K}$. For any trajectory $\\Gamma$ in \n $e^{\\cal L}_{reg}\/e^{\\cal K}$,\n  we can write the tangent vector $\\dot \\Gamma(t)$ as  $\\dot \\Gamma(t)=\\sum_{j=1}^m \\dot x^{j}\\frac{\\partial}{\\partial x^j}$, for some functions $\\dot x^j$. With this choice of coordinates, one then needs to calculate, for every regular point $U$,  the  matrix for $\\pi_*|_U$ as restricted to $R_{U*} {\\cal P}$  and its inverse $\\pi_*|_U^{-1}$. This allows us to implement \n formula (\\ref{deficontr}) to obtain the control from a given prescribed trajectory in the orbit space. \n \n \n We remark that there is an issue concerning the fact that our initial condition which is the identity ${\\bf 1}$ in (\\ref{Scro}) (and possibly the final desired condition)  is  not in the regular part of $e^{\\cal L}$.  In fact the whole Lie group $e^{\\cal K}$ is the isotropy group of the identity. If we take for $\\Gamma$ a trajectory which starts from the class corresponding to the identity, the matrix corresponding to $\\pi_*$ may become  singular as $t \\rightarrow 0$ and therefore it will be impossible to derive the control directly from formula (\\ref{deficontr}). This problem can be overcome by applying a {\\it preliminary control} which takes the state of system (\\ref{Scro}) out of the singular part and into  the regular part of $e^{\\cal L}$.  To avoid discontinuities, such a control is chosen to be zero at the beginning and at the end of the control interval. It takes the system to a point $U_0$ with $\\pi(U_0)=\\Gamma_0 \\in e^{\\cal L}_{reg}\/e^{\\cal K}$. Then we choose the trajectory in the quotient space $\\Gamma$ in the regular part of the quotient space which joins $\\Gamma_0$ and $\\Gamma_1$ where $\\Gamma_1$ is the orbit of the desired final condition. The control obtained through (\\ref{deficontr}) will steer system (\\ref{Scro}) to a state $\\hat U_f$ in the same orbit as the desired final condition. Therefore if $U_f$ is the desired final condition we will have \n $\\pi(\\hat U_f)=\\pi(U_f)=\\Gamma_1$.  Notice that we also want $\\dot \\Gamma\\to 0$ at both the initial and final point so that the control is zero and can be {\\it concatenated} continuously with the preliminary control above described. \n \n \n \n \n It is possible that the desired final condition $U_f$ is also in the singular part of $e^{\\cal L}$. This problem can be tackled in two ways. We can recall that the regular part is open and dense and therefore we can always drive to a state in the regular part arbitrarily close to the desired $U_f$. This means that our algorithm  will only give an approximate control, but which will steer the system arbitrarily close to $U_f$. Alternatively we can select a regular element $S \\in e^{\\cal L}$ and such that $U_f S^{-1}$ is also regular.\\footnote{Such an element $S\\in e^{\\mathcal{L}}_{\\text{reg}}$ always exists for any $U\\in e^{\\cal L}$, by the following argument: Assume that it does not exist. Then for every regular $S$, $US$ is singular. Therefore, by indicating by $L_U$ the left translation by $U$ we have,  $L_U(e^{\\mathcal{L}}_{\\text{reg}})\\subseteq e^{\\mathcal{L}}_{\\text{sing}}$. Then by applying the unique bi-invariant Haar measure $\\mu$ on $e^{\\cal L}$ with $\\mu(e^{\\cal L})=1$ implies $\\mu(e^{\\mathcal{L}}_{\\text{reg}})=\\mu(L_U(e^{\\mathcal{L}}_{\\text{reg}}))\\leq\\mu(e^{\\mathcal{L}}_{\\text{sing}})$. On the other hand, $\\mu(e^{\\mathcal{L}}_{\\text{sing}})=0$ since $\\mu$ must also correspond to the Riemannian volume of the bi-invariant Killing metric (normalized if necessary) and each stratum in $e^{\\mathcal{L}}_{\\text{sing}}$ has dimension strictly less than dimension of $e^{\\cal L}$ and thus has volume $0$ and therefore invariant measure $0$. But $\\mu(e^{\\mathcal{L}}_{\\text{reg}})=\\mu(e^{\\mathcal{L}})-\\mu(e^{\\mathcal{L}}_{\\text{sing}})=\\mu(e^{\\mathcal{L}})=1$. This is a contradiction.} Then we find two controls: $u_1$ driving $U$ in (\\ref{Scro}) from the identity to $S$ in (\\ref{Scro}) and $u_2$ driving $U$ in (\\ref{Scro}) from the identity to $U_fS^{-1}$ in (\\ref{Scro}). In particular, because of the right invariance of system (\\ref{Scro}), $u_2$ also drives $S$ to $U_f$. \n Therefore, the concatenation of $u_1$ (first) and $u_2$ (second) will drive to the desired final configuration. Therefore in the following, for simplicity, we shall assume that the final desired state is in the regular part. \n \n \n The (concatenated) control $\\hat u_j$  obtained from the tangent vector $\\dot \\Gamma$ at every time $t$ for a trajectory on the quotient space $\\Gamma$ (cf. (\\ref{deficontr})) will drive the state $U$ of (\\ref{Scro}) from the identity ${\\bf 1}$ only to a state $\\hat U_f$ which is in the same orbit as the desired final state $U_f$. \n  There exists $K \\in e^{\\cal K}$ such \n that $K\\hat U_f K^{-1}=U_f$. Once such a  $K$ is found it will determine through \n $\\sum_{j=1}^m KB_j \\hat u_jK^{-1}=  \\sum_{k=1}^m B_k  u_k$ the actual control $\\{u_k\\}$ to apply. We remark that this tranformation does not modify the smoothness properties of the control, nor the fact that it is zero at some point (in particular at the beginning and at the end of the control interval).    \n\n \n\\subsection{Examples}\n \n\\subsubsection{Control of a single  spin $\\frac{1}{2}$ particle}\\label{Twol}\n\nConsider the Schr\\\"odinger operator equation (\\ref{Scro}) in the form \n\\be{Scro1}\n\\dot U=\\begin{pmatrix} 0 & \\alpha(t) \\cr \n-\\alpha^*(t) & 0\n\\end{pmatrix} U, \\qquad U(0)={\\bf 1}_2, \n\\end{equation}\nwith $U$ in $SU(2)$. The complex-valued function $\\alpha$ is a \ncomplex control field representing the $x$ and $y$ \ncomponents of an electro-magnetic field. \nThe dynamical Lie algebra ${\\cal L}$ is $\\mathfrak{su}(2)$ and it has a decomposition \n$\\mathfrak{su}(2)={\\cal K} \\oplus {\\cal P}$ with ${\\cal K}$ diagonal and ${\\cal P}$ \nanti-diagonal matrices.  \n The one-dimensional \nLie group of {\\it diagonal} matrices in $SU(2)=e^{\\cal L}$ is a symmetry group $e^{\\cal K}$ \nfor the above system and the structure of the quotient space $SU(2)\/e^{\\cal K}$ is that of \na closed unit disc \\cite{NoiAutomat}. The entry $u_{1,1}$ of \n$U \\in SU(2)$, which is a complex number with absolute value $\\leq 1$, \ndetermines the orbit of the matrix $U$. The regular part \nof $SU(2)$ corresponds to  matrices with $|u_{1,1}| < 1$, i.e., the interior \nof the unit disc $\\mathring D$, in the complex plane. The singular part is \nthe boundary of the unit disc. Denote by $z$ the (complex) coordinate in the \ninterior of the complex unit disc. This corresponds  to two {\\it real} coordinates invariant under the action of $e^{\\cal K}$ \n (conjugation by diagonal matrices). Let $\\Gamma=\\Gamma(t)$ be a desired trajectory inside \n the unit disc, which we denote by $z_d$ in the chosen coordinates.  The tangent \nvector $\\dot \\Gamma$ in (\\ref{diffsys}) is given in complex coordinates by $\\dot{z_d} \\frac{\\partial}{\\partial z}$.\\footnote{This means $\\dot x_d \\frac{\\partial}{\\partial x_d} +\\dot y_d \\frac{\\partial}{\\partial y_d}$ where $x_d$ and $y_d$ are the real and imaginary parts of $z_d$.} In the coordinates on $SU(2)$ used in equation (\\ref{Scro}) the corresponding tangent vector for $\\dot U$ is given by (cf. (\\ref{Scro1})) $\\begin{pmatrix} 0 & \\alpha \\cr -\\alpha & 0 \\end{pmatrix} U$ and the value of the control $\\alpha$ is obtained by solving (\\ref{diffsys}) which gives \n\\be{diffebis}\n\\dot z_d=\\frac{d}{dt}|_{t=0} z \\left( e^{\\begin{pmatrix}0 & \\alpha \\cr -\\alpha^* & 0  \\end{pmatrix}t} U \\right), \n\\end{equation}\nwhere $z(P)$ denotes the the $(1,1)$ entry of the matrix $P$. Equation (\\ref{diffebis}) gives $\\alpha=\\frac{\\dot z_d}{u_{21}}$, which,  \nas expected from the above recalled  isomorphism theorem of \\cite{conBenECC},  gives a one-to-one correspondence  between $\\alpha$ and $\\dot z_d$ as long as $U$ is in the \nregular part of $SU(2)$, i.e., it is not diagonal, i.e., $u_{2,1} \\not=0$. \n\n\n\n\n  \n\\subsection{Control of a three level system in the $\\Lambda$ configuration}\\label{Lambda}\nConsider a three level quantum system where the controls couple level $|1\\rangle $ to level $|2\\rangle $ and level $|1 \\rangle $ to level $|3\\rangle$ but not level $|2\\rangle$ and $|3\\rangle$ directly. Assuming that $|1\\rangle$ is the highest energy level, the energy level diagram takes the so-called $\\Lambda$ configuration (see, e.g., \\cite{Lambda1}). The Schr\\\"odinger operator equation (\\ref{Scro}) is such that \n\\be{lambdascro}\n\\sum_{j=1}^m u_j B_j =\\sum_{j=1}^4 u_j B_j=\n\\begin{pmatrix}0 & \\alpha & \\beta \\cr -\\alpha^* & 0 & 0 \\cr \n- \\beta^* & 0 & 0 \\end{pmatrix},  \n\\end{equation}\nwith the complex control functions  $\\alpha$ and $\\beta$. Such a system admits a group of symmetries given by $e^{\\cal K}=S(U(1) \\times U(2))$, i.e., block diagonal matrices in $SU(3)$ with one block of dimension $1 \\times 1$ and one block of dimension $2 \\times 2$, and determinant equal to $1$. The Lie subalgebra ${\\cal K}$ consists of matrices in $\\mathfrak{su}(3)$ with a block diagonal structure with one block of dimension $1 \\times 1$ and one block of dimension $2 \\times 2$. The complementary subspace ${\\cal P}$ is spanned by \nantidiagonal matrices in $\\mathfrak{su}(2)$ with the same partition of rows and columns.   Such a system  was studied in \\cite{ADS} in the context of optimal control and a description of the orbit space $SU(3)\/e^{\\cal K}$ was given. \nIt was shown  that the regular part $SU(3)_{reg}\/e^{\\cal K}$ is \nhomeomorphic to the product of two open unit discs \n$\\mathring D_1 \\times \\mathring D_2$ in the complex plane. Up to a similarity transformation in  $e^{\\cal K}=S(U(1) \\times U(2))$, a matrix $U$ in $SU(3)$ \ncan be written as   \n\\be{canonicalformSU3}\nU=\\begin{pmatrix} z_1 & \\sqrt{1-|z_1|^2} & 0 \\cr \n-\\sqrt{1-|z_1|^2} & z_1^* & 0 \\cr 0 & 0 & 1 \\end{pmatrix}\n\\begin{pmatrix} 1 & 0 & 0 \\cr 0 & z_2 & w \\cr 0 & - w^* & z_2^* \\end{pmatrix},   \n\\end{equation}\nfor complex numbers $z_1,$ $z_2$ and $w$,  where $|z_1|\\leq 1$ and $|z_2| \\leq 1$. Strict inequalities  hold  if and only if $U$ is in the regular part in which case $z_1$ and $z_2$ can be taken as the coordinates in $SU(3)_{reg}\/e^{\\cal K}$. An alternative set of (complex) coordinates is given by $(z_1,T)$  where $T$ is the trace of the (lower) $2 \\times 2$ block of the element $U \\in SU(3)$ which is invariant (along with $z_1$) under the conjugation  action of elements in $e^{\\cal K}=S(U(1) \\times U(2))$. The  coordinates $(z_1,T)$ are related to the coordinates $(z_1,z_2)$ by  (from (\\ref{canonicalformSU3})) $T=z_1^*z_2+z_2^*$ which is inverted as $z_2=\\frac{T^*-z_1 T}{1-|z_1|^2}$. A desired trajectory $\\Gamma$ in $SU(3)_{reg}\/S(U(1)\\times U(2))$ is written in these coordinates as $(z_{1d},T_d):=  (z_{1d}(t),T_d(t))$ . The associated tangent vector $\\dot \\Gamma$ in (\\ref{diffsys})  is \n $\\dot z_{1d}(t) \\frac{\\partial}{\\partial z_{1d}}+\\dot T_{d} \\frac{\\partial}{\\partial T}$. By applying $\\pi_*|_{U}\\dot U$ in (\\ref{diffsys}) to $z_1$ and $T$ with the restriction  that $\\dot U$ is of the form   $\\begin{pmatrix}0 & \\alpha & \\beta \\cr -\\alpha^* & 0 & 0 \\cr \n- \\beta^* & 0 & 0 \\end{pmatrix}U$, we obtain two equations for $\\alpha$ and $\\beta$, \n$$\n\\alpha u_{2,1}+\\beta u_{3,1}=\\dot z_{1d}, \\qquad -\\alpha u_{1,2}^*-\\beta u_{1,3}^*=\\dot T_{d}^*.  \n$$ \nThese are solved, using $\\hat D:=u_{1,3}^*u_{2,1}-u_{3,1} u_{1,2}^*$ by \n\\be{alphabetasol}\n\\alpha=\\frac{u_{1,3}^* \\dot z_{1,d}+ u_{3,1} \\dot T_d^*}{\\hat D}, \\qquad \n\\beta=-\\frac{u_{1,2}^* \\dot z_{1,d}+ u_{2,1} \\dot T_d^*}{\\hat D}. \n\\end{equation}\n\nThe  quantity $\\hat D$ is different from zero if and only if the  matrix $U$ is in the regular part of $SU(3)$. This can be shown  in two steps: First one shows that $\\hat D$ is invariant under the action of $S(U(1) \\times U(2))$ by writing a matrix in  \n$S(U(1) \\times U(2))$ with an Euler-type decomposition \nas $F_1 R F_2$ with $F_1$ and $F_2$ diagonal and $R$ of the form \n$\nR=\\begin{pmatrix} 1 & 0 & 0 \\cr \n0 & \\cos(\\theta) & \\sin (\\theta) \\cr \n0 & -\\sin(\\theta) & \\cos(\\theta) \n\\end{pmatrix}, \n$\nand verifying that conjugation by each factor in $F_1 R F_2$  leaves $\\hat D$ unchanged. The second step is to verify that $\\hat D$ for the matrix $X$ in the form \n(\\ref{canonicalformSU3}) is different from zero if and only if $|z_1| \\not=1$ and \n$|z_2| \\not=1$. This gives a quick  \ntest to check whether a matrix is in the regular part, i.e., if its isotropy group is the smallest possible one, which, in this case,  is the group of scalar matrices in $SU(3)$. \n{This fact also follows from the result in Appendix B which shows in general that $\\det(\\pi_*)$, with $\\pi_* :\\, R_{p *}{\\cal P} \\rightarrow T_{\\pi(p)} e^{\\cal L}_{reg}\/e^{\\cal K}$ is  invariant under the action of $e^{\\cal K}$. }\n\nAs always, we have the problem that the initial point ${\\bf 1}$ is  in the singular part of the orbit space decomposition and therefore $\\hat D$ in (\\ref{alphabetasol}) is zero at time $0$. As suggested in the previous section, we can however apply a preliminary control to \nsteer away from the singular part.  \n\n\n\n\n \n \n\n\n    \n\n\\section{Simultaneous control of two independent spin $\\frac{1}{2}$ particles}\\label{App2QB}\n\n\\subsection{The model}\nThe dynamics of two spin $\\frac{1}{2}$ particles with different gyromagnetic ratios in zero field NMR can be described by the Schr\\\"odinger equation (\\ref{Scro}) (after appropriate normalization) where \n\\be{Hamilto}\n\\sum_{k=1}^m \\hat u_k B_k:=\\sum_{x,y,z} u_{x,y,z}(t)( i\\sigma_{x,y,z} \\otimes {\\bf 1}+ \\gamma\n i {\\bf 1} \\otimes \\sigma_{x,y,z}).   \n\\end{equation}  \nHere $u_{x,y,z}$ are the controls representing the $x,y,z$ components of the electromagnetic field, and $\\sigma_{x,y,z}$ are the Pauli matrices defined as \n\\be{Pauli}\n\\sigma_x:=  \\begin{pmatrix} 0 & 1 \\cr 1 & 0 \\end{pmatrix}, \\qquad\n\\sigma_y:=  \\begin{pmatrix} 0 & -i \\cr i & 0 \\end{pmatrix}, \\qquad\n\\sigma_z:=  \\begin{pmatrix} 1 & 0 \\cr 0 & -1 \\end{pmatrix}.  \\qquad\n\\end{equation}\nThe parameter $\\gamma$ is the ratio of the two gyromagnetic ratios and \nwe shall assume that  $|\\gamma|\\not=1$. Under this assumption,  the dynamical Lie algebra ${\\cal  L}$ for system (\\ref{Scro}), (\\ref{Hamilto}) is the $6-$dimensional Lie algebra spanned by $\\{\\sigma_1 \\otimes {\\bf 1} + {\\bf 1} \\otimes \\sigma_2 \\, | \\, \\sigma_1, \\sigma_2 \\in \\mathfrak{su}(2)\\}$.\\footnote{This Lie algebra is isomorphic to $\\mathfrak{so}(4)$.}   The corresponding Lie group $e^{\\cal L}$, which is the set of reachable states for system  (\\ref{Scro}), (\\ref{Hamilto}), is $\\{X_1 \\otimes X_2 \\, | \\, X_1, X_2 \\in SU(2)\\}$, i.e. the tensor product $SU(2)\\otimes SU(2)$. It is convenient to slightly relax the description of the state space and look at system (\\ref{Scro}), (\\ref{Hamilto}) as a system on the Lie group $SU(2) \\times SU(2)$, i.e., the Cartesian direct product of $SU(2)$ with itself, and the dynamical equations (\\ref{Scro}), (\\ref{Hamilto}) \nreplaced by\n\\be{reple}\n\\dot U= \\sigma(t)U, \\quad U(0)={\\bf 1}, \\qquad \\dot V= \\gamma \\sigma(t)V, \\quad V(0)={\\bf 1},  \n\\end{equation} \nwith $\\sigma(t):=\\sum_{x,y,z} i u_{x,y,z}(t) \\sigma_{x,y,z}$. \nThe controls that drive system (\\ref{reple}) to $(\\pm U_f, \\pm V_f)$ drive system (\\ref{Scro}), (\\ref{Hamilto}) to the state $U_f \\otimes V_f$. Therefore we shall focus on the steering problem for system (\\ref{reple}) which consists of steering one spin to $U_f$ and the other to $V_f$, simultaneously. Since $|\\gamma| \\not=1$, the dynamical Lie algebra associated with (\\ref{reple}) is spanned by the pairs $(\\sigma_1, \\sigma_2)$ with $\\sigma_1$ and $\\sigma_2$ in $\\mathfrak{su}(2)$. Such a Lie algebra can be written as ${\\cal K} \\oplus {\\cal P}$ with ${\\cal K}$ spanned by elements  of the form \n$(\\sigma, \\sigma)$ with $\\sigma \\in \\mathfrak{su}(2)$ and ${\\cal P}$ spanned by elements of the \nform $(\\sigma, \\gamma \\sigma)$ with $\\sigma \\in \\mathfrak{su}(2)$. At every $p \\in G=SU(2) \\times SU(2)$, the vector fields in (\\ref{reple}) belong to $R_{p*} {\\cal P}$. \n\n\\subsection{Symmetries and the the structure of the quotient space}  \n\nThe Lie group $SU(2)$ acts on $G:=SU(2) \\times SU(2)$ by {\\it simultaneous conjugation}, that is, for $K \\in SU(2)$, $(U_f, V_f) \\rightarrow (K U_f K^{-1}, K V_f K^{-1})$ and this \nis a group of symmetries for system (\\ref{reple}) in that  if $\\sigma=\\sigma(t)$ is the control steering to $(U_f,V_f)$, then $K\\sigma K^{-1}$ is the control steering to $(KU_f K^{-1}, KV_f K^{-1})$. The quotient space of \n$SU(2) \\times SU(2)$ under this action, $(SU(2) \\times SU(2))\/SU(2)$ was described in \\cite{Xinhua} as follows. \n\nConsider \na pair $(U_f, V_f)$ and let $\\phi \\in [0,\\pi]$ be a real number so that the two eigenvalues of $U_f$ are $e^{i\\phi}$ \nand $e^{-i\\phi}$. If $0< \\phi < \\pi$ then $U_f \\not= \\pm {\\bf 1}$ and there exists a unitary \nmatrix $S$ such that $SU_f S^{\\dagger}:=D_f$ is diagonal. Therefore the matrix $(U_f, V_f)$ is in the same orbit as $(D_f, SV_f S^\\dagger)$. The parameter $\\phi$ determines the orbit, along with the $(1,1)-$entry of $SV_f S^\\dagger$, which does not depend on the choice of $S$.\\footnote{All the possible diagonalizing matrices differ by a diagonal factor that does not affect the $(1,1)$ entry of  $SV_f S^\\dagger$.} Such a $(1,1)$-entry has absolute value $\\leq 1$ and therefore it is an element of the unit disk in the complex plane. The orbits corresponding to the values of $ 0< \\phi < \\pi$ (for the eigenvalue of the first matrix) are in one-to-one correspondence with the points  of a solid cylinder with height  equal to $\\phi$. When $\\phi=0$ (or $\\phi=\\pi$), the matrix $U_f$ is $\\pm$ identity  and therefore the equivalence class  is determined by the eigenvalue of the matrices $V_f$, which are $e^{\\pm i \\psi}$ for $\\psi\\in [0, \\pi]$. In the geometric description, the solid cylinder degenerates at the two ends to become a segment $[0,\\pi]$. The regular part of the orbit space $G_{reg}$ is represented by points in the interior of the solid cylinder. Such points \ncorrespond to pairs \n$$\n\\left( \\begin{pmatrix} e^{i \\phi} & 0 \\cr 0 & e^{-i\\phi} \\end{pmatrix},  \\begin{pmatrix} z & w \\cr -w^* & z^* \\end{pmatrix}\\right)\n$$ \nwith $\\phi \\in (0,\\pi)$ and $|z|<1$. For these pairs,  the isotropy group is the discrete group $\\{ \\pm {\\bf 1} \\}$. In general points that are in the singular part correspond to pairs of matrices $(U_f, V_f)$ which can be simultaneously diagonalized. Therefore \nthe condition that they commute \n\\be{commut34}\nU_f V_f=V_fU_f, \n\\end{equation} \nis necessary and sufficient for a pair $(U_f,V_f)$ to belong to the singular part. \n\n\nAssume that $p$ is a regular point in $G_{reg}$ for this problem and $\\pi$ is the natural projection $\\pi: G_{reg} \\rightarrow G_{reg}\/SU(2)$.  Then from the theory in the previous section, the differential $\\pi_*|_{p}$ is an isomorphism from $R_{p*}{\\cal P}$ to $T_{\\pi(p)} G_{reg}\/SU(2)$. Let us choose a basis for ${\\cal P}$ given by $(i\\sigma_{x,y,z}, \\gamma i\\sigma_{x,y,z})$. To choose the three coordinates in $G_{reg}\/SU(2)$, we consider a general element $p$ in $SU(2) \\times SU(2)$ written as \n\\be{puntop}\np:=(U_f,V_f):=\\left( \\begin{pmatrix} x & y \\cr -y^* & x^* \\end{pmatrix}, \\begin{pmatrix} z & w \\cr \n- w^* & z^* \\end{pmatrix}\\right). \n\\end{equation} \nFor a complex number $q$ we shall denoted by $q_R:=\\texttt{Re}(q)$ and $q_I:=\\texttt{Im}(q)$. Notice that in (\\ref{puntop}) we have \n$$\nx_R^2+x_I^2+y_R^2+y_I^2=z_R^2+z_I^2+w_R^2+w_I^2=1. \n$$ \nCoordinates in $G_{reg}\/SU(2)$ must be independent invariant functions of $(U_f,V_f)$ in (\\ref{puntop}). We choose  \n\\be{ccordinates}\nx^1:=x_R, \\qquad x^2:=z_R, \\qquad x^3:=x_I z_I+w_R y_R+ w_I y_I.\n\\end{equation}\nIt is a direct verification to check that at any point $p \\in SU(2) \\times SU(2)$, $x^1$, $x^2$ and $x^3$ are unchanged by the (double conjugation) action of $SU(2)$, i.e., they are invariant. We remark also that we can consider two unit \nvectors $\\vec V:=(x_R,x_I, y_R, y_I)$, and  $\\vec W:=(z_R,z_I, w_R, w_I)$, and, if we do that, \n $x^3=\\vec V \\cdot \\vec W-x_Rz_R$. \n\n\\subsection{Choice of invariants}\nWe pause a moment to detail how the invariant coordinates in (\\ref{ccordinates})\n were chosen. We do this  because the method can be used for other examples. We consider the vectors $\\vec V:=[x_R, x_I, y_R, y_I]^T$ and  \n $\\vec W:=[z_R, z_I, w_R, w_I]^T$ and the adjoint action of $SU(2)$ on $SU(2) \\times SU(2)$ which gives a linear action on $\\vec Q:=[\\vec V^T, \\vec W^T]^T$. We are looking for functions $f=f(\\vec V, \\vec W)$ invariant under this action.  Given that every element of $SU(2)$ can be written according to Euler's decomposition as $e^{i \\sigma_z \\alpha} e^{i \\sigma_y \\theta}  e^{i \\sigma_z \\beta}$, for real parameters $\\alpha, \\beta$ and $\\theta$, it is enough that $f$ is invariant  with respect to transformations of the form \n$e^{i \\sigma_z \\beta}$ and $e^{i \\sigma_y \\theta}$, for general real $\\beta$ and $\\theta$, in order for $f$ to be invariant with respect to all of $SU(2)$. If $X_z:=X_z(\\beta):=e^{i \\sigma_z \\beta}$ then $Ad_{X_z}$ acting on  $[\\vec V^T, \\vec W^T]^T$ is \n\\be{adXz}\nAd_{X_z(\\beta)}:=\\begin{pmatrix} 1 & 0  & 0 & 0 \\cr 0 & 1 & 0 & 0 \\cr \n0 & 0 & \\cos(\\beta) & - \\sin (\\beta) \\cr \n0 & 0 & \\sin(\\beta) & \\cos(\\beta) \n\\end{pmatrix}. \n\\end{equation}\nIf $X_y:=X_y(\\theta):=e^{i \\sigma_y \\theta}$ then $Ad_{X_y}$ acting on  $[\\vec V^T, \\vec W^T]^T$ is \n\\be{adXy}\nAd_{X_y(\\theta)}:=\\begin{pmatrix} 1 & 0  & 0 & 0 \\cr 0 & \\cos(\\theta)  & 0 & \\cos(\\theta) \\cr \n0  & 0  & 1 & 0 \\cr \n0 & -\\sin(\\theta) & 0 & \\cos(\\theta) \n\\end{pmatrix}. \n\\end{equation}\nWe first look for {\\it linear invariants}, i.e., invariant functions $f$ of the form \n$f(\\vec V, \\vec W):=\\vec a^{\\, T} \\vec V + \\vec b^{\\, T} \\vec W$. From the condition\n$$\n\\vec a^{\\, T} \\vec V + \\vec b^{\\, T} \\vec W=\\vec a^{\\, T} Ad_{X}\\vec V + \\vec b^{\\, T} Ad_{X}\\vec W, \n$$ \nwhere $X$ may be equal to $X_z(\\beta)$ or $X_{y}(\\theta)$, with arbitrary $\\beta$ and $\\theta$, we find that the last three components of $\\vec a$ and $\\vec b$ must be zero. Therefore all linear invariants $f$ must be of the form \n$f=a_1 x_R+b_1 z_R$, from which we get the invariant $x_R$ and $z_R$ in (\\ref{ccordinates}). \n\n\n\nWe then try to find {\\it quadratic invariants} and therefore an $8 \\times 8$ symmetric matrix $P$ so that\n$f(\\vec V, \\vec W)=[\\vec V^T, \\vec W^T] P [\\vec V^T, \\vec W^T]^T$ and  \n$$\n[\\vec V^T, \\vec W^T] P [\\vec V^T, \\vec W^T]^T=[\\vec V^T, \\vec W^T] \\begin{pmatrix}Ad_X^T & 0 \\cr 0 & Ad_X^T \\end{pmatrix}P   \\begin{pmatrix}Ad_X & 0 \\cr 0 & Ad_X \\end{pmatrix} [\\vec V^T, \\vec W^T]^T, \n$$\nfor $X=X_z(\\beta)$ and $X=X_{y}(\\theta)$ as defined in (\\ref{adXz}) and (\\ref{adXy}) for every $\\beta$ and $\\theta$ (and for every $\\vec V$ and $\\vec W$). This leads to the condition \n$$\n\\begin{pmatrix} Ad_X & 0 \\cr \n0 & Ad_X \\end{pmatrix} P=P \\begin{pmatrix} Ad_X & 0 \\cr \n0 & Ad_X \\end{pmatrix}.   \n$$ \nFrom this, we find that the matrix $P$ must be of the form \n$$\nP=\\begin{pmatrix} e & 0 & 0 & 0& d & 0& 0 & 0\\cr \n 0 & c & 0 & 0 &0 & g & 0 & 0 \\cr \n 0 & 0 & c & 0 & \n 0 & 0 & g & 0 \\cr \n 0 & 0 & 0 & c & 0 & 0 & 0 & g \\cr \n d & 0 & 0 & 0 & r & 0 & 0 & 0\\cr \n 0 & g & 0 & 0 & 0 & h & 0 & 0 \\cr \n 0 & 0 & g & 0 & 0 & 0 & h&  0 \\cr \n 0 & 0 & 0 & g & 0 & 0 & 0 & h    \\end{pmatrix}. \n$$\nIt follows that all quadratic invariants must be of the form \n$$\nf=ex_R^2+2dx_R z_R + r z_R^2+c(x_I^2+y_R^2+y_I^2) + h (z_I^2+w_R^2+w_I^2)+ 2g(x_I z_I+w_R y_R+ w_I y_I). \n$$\nBecause of $x_R^2+x_I^2+y_R^2+y_I^2=z_R^2+z_I^2+w_R^2+w_I^2=1$, all terms can be  written in terms of the (linear) invariant $x_R$ and $z_R$ except the last one which we choose as the third coordinate in (\\ref{ccordinates}). \n\n\\subsection{Algorithm for control}\\label{AlgoCon}\n\nAt the point $\\pi(p) \\in G_{reg}\/SU(2)$, the tangent vectors \n$\\frac{\\partial}{\\partial x^j},$ $j=1,2,3$ span $T_{\\pi(p)} G_{reg}\/SU(2)$, so that a general tangent vector at $\\pi(p)$ can be written as $a_1\\frac{\\partial}{\\partial x^1}+a_2 \\frac{\\partial}{\\partial x^2}+ a_3 \\frac{\\partial}{\\partial x^3}$. We calculate the matrix of the isomorphism $\\pi_*|_{p}$  mapping \nthe coordinates $\\alpha_x, \\alpha_y, \\alpha_z,$ in $R_{p*} (\\sigma, \\gamma \\sigma):=R_{p*}(\\alpha_x(i \\sigma_x, \\gamma i \\sigma_x) +\\alpha_y (i \\sigma_y, \\gamma i \\sigma_y)+ \\alpha_z (i \\sigma_z, \\gamma i \\sigma_z) ) \\in R_{p*} {\\cal P}$ to $(a_1,a_2,a_3)$, (cf. \n(\\ref{diffsys})). Denote this matrix by $\\Pi_{j,l}:=\\Pi_{j,l}(p)$ with $j=1,2,3$ and $l=x,y,z$. \nWe have \n$$\n\\Pi_{j,l}(p)= \\pi_{*}|_p R_{p*} (i \\sigma_l, i \\gamma \\sigma_l) x^j. \n$$\nFor the sake of illustration, let us calculate  $\\Pi_{1,x}(p)$. This is given by (recall $p$ is defined in (\\ref{puntop})) \n$$\n\\Pi_{1,x}(p):=\\pi_{p*}R_{p*} (i \\sigma_x, i \\gamma \\sigma_x) x^1=R_{p*} (i \\sigma_x, i \\gamma \\sigma_x) (x^1 \\circ \\pi)=\\frac{d}{dt}|_{t=0} x^1\\circ \\pi \\left( e^{i \\sigma_x t} \\begin{pmatrix} x & y \\cr -y^* & x^*\\end{pmatrix}\\, , \\,e^{i \\gamma \\sigma_x t} \\begin{pmatrix} z &  w \\cr -w^* & w^*\\end{pmatrix}  \\right). \n$$\nThis simplifies because $x^1 \\circ \\pi $ does not depend on the second factor. Therefore the  $\\Pi_{1,x}(p)$ entry is the derivative at $t=0$ of the real part of the $(1,1)$ entry of the matrix \n$$\ne^{i \\sigma_x t} \\begin{pmatrix} x &  y \\cr -y^* & x^*\\end{pmatrix}  \n= \n\\begin{pmatrix} \\cos(t) & i \\sin(t) \\cr i \\sin(t) & \\cos(t) \\end{pmatrix} \\begin{pmatrix} x &  y \\cr -y^* & x^*\\end{pmatrix}.\n$$ \nThis leads to  the result \n$$\n\\Pi_{1,x}(p)=-y_I. \n$$\n\n\nThe quantities \n\\be{D1D2D3}\n\\Delta_1:=z_I y_R- x_I w_R, \\qquad \\Delta_2:=z_I y_I-x_Iw_I, \\qquad \\Delta_3:=w_Ry_I-w_Iy_R, \n\\end{equation}\nappear routinely in calculations that follow. \n\n\n\n\nSimilar calculations to the ones above for $\\Pi_{1,x}(p)$  lead to the full matrix $\\Pi(p)$,  which is given by \n{\n\\be{PiofP}\n\\Pi(p):=\n\\begin{pmatrix} - y_I & -y_R & -x_I \\cr \n- \\gamma w_I & - \\gamma w_R & - \\gamma z_I \\cr \n(\\gamma -1) \\Delta_1 +\\gamma z_R y_I \n+w_I x_R \n& (1-\\gamma) \\Delta_2  \n+ w_R x_R \n+ \\gamma z_R y_R \n &\n(\\gamma-1) \\Delta_3 \n+x_R z_I \n+ \\gamma z_R x_I\n\\end{pmatrix}\n\\end{equation} }\nThe determinant of this matrix is different from zero if and only if $p$ is in the regular part \nand it is another invariant under the action of $SU(2)$ on $SU(2) \\times SU(2)$ (cf. Appendix B). It can be explicitly  computed as \n\\be{determinante}\n\\det (\\Pi (p))=\\gamma(\\gamma-1)(\\Delta_1^2+\\Delta_2^2+\\Delta_3^2), \n\\end{equation}\nwhich can be seen to be equal to zero if and only if  condition (\\ref{commut34}) is verified. The invariant \n$\\Delta:=\\Delta_1^2+\\Delta_2^2+\\Delta_3^2$ can be expressed in terms of the (minimal) invariants $x_R$, $z_R$ and $x^3$ in (\\ref{ccordinates}) as\\footnote{This can be seen by expanding the left hand side using the definitions of $\\Delta_{1,2,3}$ (\\ref{D1D2D3}) and the right hand side using the definition of $x^3$ (\\ref{ccordinates}), so that (\\ref{delt}) reduces to $y_I^2w_R^2+w_I^2y_R^2+y_R^2 z_I^2+x_I^2 w_R^2+x_I^2 w_I^2+y_I^2 z_I^2=(1-x_R^2)(1-z_R^2)-z_I^2x_I^2-y_R^2 w_R^2-y_I^2 w_I^2$, and writing $(1-x_R^2)=x_I^2+y_R^2+y_I^2$ and  $(1-z_R^2)=z_I^2+w_R^2+w_I^2$, we obtain an identity.}\n\\be{delt}\n\\Delta=\\Delta_1^2+\\Delta_2^2+\\Delta_3^2=(1-x_R^2)(1-z_R^2)-(x^3)^2. \n\\end{equation}\n\n\n\n\n\nWhen we design a control law, the components $a_1,a_2,a_3$ of the tangent vector  at every time $t$ in the tangent space at $\\pi(p(t))$ are given by the derivatives $\\dot x^1$, $\\dot x^2$, $\\dot x^3$ of the desired trajectory in the quotient space. The corresponding components,  $\\alpha_x$, $\\alpha_y$ and $\\alpha_z$, of the tangent vector in $R_{p(t)*}{\\cal P}$ give  the appropriate control functions $(u_x, u_y, u_z)$. The matrix $\\Pi(p)$ in (\\ref{PiofP}) gives the map from the control to trajectories. Since we want to specify trajectories and compute the corresponding controls, we need the inverse of the matrix $\\Pi(p)$ (cf. (\\ref{deficontr})). This is found from (\\ref{PiofP}) to be \n\\be{Piinverse}\n\\det(\\Pi(p)) \\Pi^{-1}(p):=\n\\begin{pmatrix}\n\\gamma(\\gamma -1)(-w_R \\Delta_3 - z_I \\Delta_2) + \\gamma^2z_R \\Delta_1 &\n(\\gamma -1)(x_I \\Delta_2 + y_R \\Delta_3)+ x_R \\Delta_1 & \\gamma \\Delta_1 \\cr \n\\gamma (\\gamma -1)(w_I \\Delta_3 - z_I \\Delta_1)- \\gamma^2 z_R \\Delta_2 & \n(\\gamma -1) (x_I \\Delta_1 - y_I \\Delta_3) - x_R \\Delta_2 & \n- \\gamma \\Delta_2 \\cr \n\\gamma ( \\gamma-1) ( w_I \\Delta_2 + w_R \\Delta_1) + \\gamma^2 z_R \\Delta_3 \n& \n-(\\gamma-1)(y_I \\Delta_2 +y_R \\Delta_1) + x_R \\Delta_3 \n& \n\\gamma \\Delta_3\n\\end{pmatrix}. \n\\end{equation}\n\n\nWe remark that $\\Pi^{-1}(p)$ is not defined if we are in the singular part of the space $G=SU(2) \\times SU(2)$ as the determinant of $\\Pi$ is zero there. This is in particular true at the beginning as the initial point $p \\in SU(2) \\times SU(2)$ is the identity. In order  to follow a  prescribed trajectory in the quotient space $G_{reg}\/SU(2)$, we need therefore to apply a preliminary control to drive the state to an arbitrary point in $G_{reg}$ and after that we \nshall apply the control  corresponding to a prescribed trajectory in the quotient space. \n\n\n\nThe preliminary control in an interval $[0,T_1]$ to move the state from the singular part of the quotient space has to involve at least two different directions in the tangent space. In other terms, if we use $\\sigma(t)=u(t) \\sigma$ for some function $u=u(t)$ and a constant matrix $\\sigma \\in \\mathfrak{su}(2)$ we remain in the singular part. To see this, notice that if \n$d:=\\int_0^{T_1} u(s)ds$, then the solution of  (\\ref{reple}) will be $(U_f, V_f)=(e^{d \\sigma},  e^{\\gamma d \\sigma})$, a pair that satisfies the condition (\\ref{commut34}). Therefore the simplest control strategy of moving in one direction only will not work if we want to move the state from the singular part. Furthermore, we want $u_x(0)=u_y(0)=u_z(0)=0$ and $u_x(T_1)=u_y(T_1)=u_z(T_1)=0$ to avoid discontinuities at the initial time $t=0$ and at the time of concatenation with the second portion  of the control. We propose to prescribe a trajectory \nfor $U=U(t)$ in  (\\ref{reple}) and, from that trajectory, to  derive  the control to be used in the equation for $V=V(t)$ in (\\ref{reple}). We choose a smooth function $\\delta:=\\delta(t)$ such that \n$\\delta(0)=0$ and $\\delta(T_1)=\\delta_0 \\not=0 $, and $\\dot \\delta(0)=\\dot \\delta(T_1)=0$. \nWe also choose a smooth function $\\epsilon:=\\epsilon(t)$, with $\\epsilon(0)=\\epsilon_0 \\not=0 $ and $\\epsilon(T_1)=0$, and $\\dot \\epsilon(0)=\\dot \\epsilon(T_1)=0$. We choose for $U=U(t)$ in (\\ref{reple}) \n\\be{Uoft}\nU(t)=\\begin{pmatrix} \\cos(\\delta(t)) & e^{i \\epsilon(t)} \\sin(\\delta(t)) \\cr \n-e^{-i\\epsilon(t)} \\sin(\\delta(t)) & \\cos(\\delta(t)) \\end{pmatrix}, \n\\end{equation}\nwhich at time $T_1$ gives \n\\be{Poil}\nU(T_1)=\\begin{pmatrix} \\cos(\\delta_0) &  \\sin(\\delta_0) \\cr \n-\\sin(\\delta_0)   & \\cos(\\delta_0) \\end{pmatrix}. \n\\end{equation}\nThe corresponding control $\\sigma$ is $\\sigma(t)=\\dot U U^\\dagger$,  which is \n\\be{controlSS}\n\\sigma(t)=\\begin{pmatrix} i \\dot \\epsilon \\sin^2(\\delta) & \\dot \\delta e^{i \\epsilon}+ \\frac{i}{2} \\, \\dot \\epsilon  \\,\n{\\sin(2 \\delta)}\\, e^{i\\epsilon}  \\cr \n- \\dot \\delta e^{-i \\epsilon}+ \\frac{i}{2} \\dot \\epsilon \\,  {\\sin(2 \\delta)} \\, e^{-i\\epsilon} & -i \\dot \\epsilon \\sin^2(\\delta) \\end{pmatrix}.\n\\end{equation}\nPlacing this in the second equation of (\\ref{reple}) and integrating numerically we obtain the values for $V(T_1)$,  the second component of $(U,V)$, and therefore  the values of $(z_R,z_I,w_R,w_I)$. Using these values and the expression for $(x_R,x_I,z_R,z_I)$ in (\\ref{Poil}), and using the formula  for $\\Delta$  given in  \n(\\ref{delt}), we obtain \n\\be{accor5}\n\\Delta=\\Delta_1^2+\\Delta_2^2+\\Delta_3^2=\\sin^2(\\delta_0)(1-z_R^2(T_1)-w_R^2(T_1))=\\sin^2(\\delta_0)(z^2_I(T_1)+w^2_I(T_1)),\n\\end{equation}\nwhich has to be different from zero in order for the state to be in the regular part. \n\n\nThe second portion  of the control depends on the trajectory followed, $(x^1,x^2,x^3)=(x^1(t),x^2(t),x^3(t))$, and it is obtained by multiplying by $\\Pi^{-1}$ in (\\ref{Piinverse}) $(\\dot x^1,\\dot x^2, \\dot x^3)$. The trajectory $(x^1,x^2,x^3)$ is almost completely arbitrary. However it has to satisfy certain conditions which we now discuss.  Let us denote the interval where the second part of the control is used by $[0,T_2]$. The initial condition $(x^1(0),x^2(0),x^3(0))$ has to agree with the one given by the previous interval of  control. The final condition  $(x^1(T_2),x^2(T_2),x^3(T_2))$ has to agree with the orbit of the  desired final condition. Moreover, care has to be taken to make sure that the trajectory is such that $\\Delta$  in (\\ref{delt}) is never zero because this would create a singularity in $\\Pi^{-1}(p)$. Furthermore \nwe need  $\\dot x^1(0)=\\dot x^2(0)=\\dot x^3(0)=0$, which gives $\\sigma(0)=0$, \n to ensure continuity  with the  control in the previous interval, and we also choose $\\dot x^1(T_2)=\\dot x^2(T_2)=\\dot x^3(T_2)=0$ to ensure that the control is switched off at the end of the procedure. Finally, the functions $(x^1, x^2, x^3)$ have to be representative of a possible trajectory for special unitary matrices. This means that,  with $\\vec V:=(x_R, x_I, y_R, y_I)^T$ and $\\vec W:=(z_R, z_I, w_R, w_I)^T$, \n $\\|\\vec V (t)\\|^2=\\|\\vec W(t)\\|^2=1$, at every $t$. Therefore $|x_R(t)| < 1$ at every $t$, $|z_R(t)| < 1$, at every $t$ (to avoid singularity), and from the Schwartz inequality $|\\vec V \\cdot \\vec W |\\leq 1$ we also must have $|x_R z_R + x^3|\\leq 1$ and therefore \n\\be{condx3} \n -1-x_R z_R \\leq  x^3 \\leq  1 - x_R z_R. \n\\end{equation}  \n \n \nOnce the functions $(\\dot x^1, \\dot x^2,\\dot x^3)$ are chosen, the system to integrate numerically is (\\ref{reple}) with $(u_x,u_y,u_z)$ given by $\\Pi^{-1}(p)(\\dot x^1, \\dot x^2,\\dot x^3)^T$. By deriving $(u_x,u_y,u_z)$ using the explicit expression of $\\Pi^{-1}$ given in (\\ref{Piinverse}) and replacing into (\\ref{reple}), it is possible to obtain a  simplified system of differential equations for $(x_R,x_I,...,z_R,z_I,...,w_I)$ without implementing the preliminary step of calculating the control. We found this system to be more stable in numerical integration with MATLAB and report it in Appendix A for future use. \n\n\n\n\\subsection{Numerical example: Driving to two different Hadamard  gates} \n\nWe now apply the above technique to  a specific numerical example: The problem is  to drive \nthe system (\\ref{Hamilto}) so that the first spin performs the Hadamard-type  gate \n\\be{Hada1}\nH_1:=\\begin{pmatrix} \\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\cr \n\\frac{-1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}}\\end{pmatrix}\n\\end{equation}\nand the second spin performs the Hadamard gate\n\\be{Hada2}\nH_2:=\\begin{pmatrix} \\frac{1}{\\sqrt{2}} & \\frac{-i}{\\sqrt{2}} \\cr \n\\frac{-i}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}}\\end{pmatrix}. \n\\end{equation}\nWe want to  drive system (\\ref{reple}) to $(U_f, V_f)=(H_1, H_2)$. The orbit of the desired final condition is characterized by the invariant coordinates \n\\be{finalorbit}\nx^1=x_R=\\frac{1}{\\sqrt{2}}, \\qquad x^2=z_R=\\frac{1}{\\sqrt{2}}, \\qquad x^3=x_I z_I+ y_R w_R+ y_I w_I=0. \n\\end{equation}\nWe take  a physical value for the ratio $\\gamma$ between the two gyromagnetic ratios. In particular we will choose \n  $\\gamma\\approx \\frac{1}{0.2514}$ which corresponds to the Hydrogen-Carbon ($^{1} H-^{13} C$) system also considered in  \\cite{Xinhua}. \n\nWe first consider the control that moves the state away from the singular part, in a time  interval $[0,1]$. We choose \n$\\sigma$ in (\\ref{controlSS}) with  the functions \n$\\delta$ and $\\epsilon$ as follows:  \n\\be{deltaandepsilon}\n\\delta=6 \\delta_0 \\left( \\frac{t^2}{2} -\\frac{t^3}{3} \\right), \\qquad \\epsilon=\\epsilon_0 + 6 \\epsilon_0\n\\left( \\frac{t^3}{3} -\\frac{t^2}{2} \\right). \n\\end{equation}\nWith these functions $\\delta$ and $\\epsilon$, $\\sigma$  satisfies all the requirements described above. \nFrom (\\ref{deltaandepsilon}) and (\\ref{controlSS}) we obtain the controls $u_x,u_y,u_z$ which replaced into (\\ref{reple}) give the dynamics in the interval $[0,T_1]=[0,1]$. Numerical integration with the values of the parameters $\\delta_0=0.5$ and $\\epsilon_0=1$, gives the following conditions at time $T_1=1$ (cf. (\\ref{Poil}) \n\\be{condizioni}\nU(1)=\\begin{pmatrix} \\cos(0.5) & \\sin(0.5) \\cr \n-\\sin(0.5) & \\cos(0.5) \\end{pmatrix}, \\qquad V(1) \\approx \\begin{pmatrix} -0.3472+i0.7769 & -0.5123 -i 0.1157\\cr \n0.5123-i0.1157  &  -0.3472-i0.7769 \\end{pmatrix}. \n\\end{equation}\nThe value of $\\Delta$ is, according to (\\ref{accor5}), \n$\\Delta \\approx \\sin^2(0.5)\\left( (0.7769 )^2+(0.1157)^2 \\right) \\not= 0$, as desired. \n\n\n\nThe values of the variables to be used as initial conditions in the integration in the subsequent interval of the procedure are \n$\nx_R(1)=x^1(1)=\\cos(0.5), \\quad x_I(1)=0, \\quad y_R(1)=\\sin(0.5), \\quad y_I(1)=0, \\quad \nz_R(1)=x^2(1)=-0.3472 , \\quad z_I(1)=0.7769 , \\quad w_R(1)=-0.5123, \\quad w_I(1)=-0.1157, \n$ \nand $x^3(1)=x_I(1) z_I(1)+y_R(1) w_R(1)+y_I(1) w_I(1)=\\sin(0.5) \\times (-0.5123)\\approx -0.2456.$ For the subsequent \n interval $[0,T_2]$ we choose the trajectory $x^1(t),x^2(t),x^3(t)$ in the quotient space as follows: \n$T_2=10$ and the trajectory in the interval $[0,T_2]$ is \n\\be{x1t}\nx^1(t)=-\\frac{1}{500} \\left( \\frac{1}{\\sqrt{2}}-\\cos(0.5)\\right)t^3+\\frac{3}{100} \\left( \\frac{1}{\\sqrt{2}} - \\cos(0.5) \\right)t^2+ \\cos(0.1);\n\\end{equation}\n\\be{x2t}\nx^2(t)=-\\frac{1}{500} \\left( \\frac{1}{\\sqrt{2}}+0.3472 \\right) t^3+\\frac{3}{100} \\left( \\frac{1}{\\sqrt{2}}+\n0.3472 \\right)t^2-0.3472; \n\\end{equation}  \n\\be{x3t}\nx^3(t):=\\frac{-0.2456}{500} t^3-\\frac{3 \\times (-0.2456)}{100} t^2-0.2456,  \n\\end{equation}\nwhich are easily seen to satisfy the conditions at the endpoints. Moreover by plotting $x^1$ and $x^2$ we see that \n$|x^1(t)|\\leq 1$ and $|x^2(t)|\\leq 1$ for every $t \\in [0,10]$ (Figure \\ref{Fig1}). \nBy plotting $x^3$ vs $1-x^1 x^2$ and $-1-x^1x^2$ (Figure \\ref{Fig2}) we find that $-1-x^1 x^2(t) \\leq x^3(t) \\leq 1-x^1 x^2(t)$ for every $t \\in [0,10]$ as required from condition (\\ref{condx3}). \nBy plotting $\\Delta=\\Delta(t)$ in $[0,10]$ we know that $\\Delta(t) \\not=0$ for every $t \\in [0,10]$ (Figure \\ref{Fig3}). Therefore the whole trajectory is in the regular part. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth, height=0.4\\textheight]{Fig1}\n\\vspace*{-16mm}\n\\caption{Prescribed $x^1$ and $x^2$ variables in the (second) interval $[0,10]$.}\n\\label{Fig1}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth, height=.4\\textheight]{Fig2}\n\\vspace*{-16mm}\n\\caption{$x^3=x^3(t)$ vs $-1-x^1(t)x^2(t)$ and $ 1-x^1(t)x^2(t)$ in the (second) interval $[0,10]$.}\n\\label{Fig2}\n\\end{figure}\n\n\n\n\n \n \n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth, height=.4\\textheight]{Fig3}\n\\vspace*{-16mm}\n\\caption{$\\Delta=\\Delta(t)=\\Delta_1^2(t)+\\Delta_2^2(t)+\\Delta_3^2(t)$ in the (second) interval $[0,10]$.}\n\\label{Fig3}\n\\end{figure}\n\nThe {\\it full} trajectory, in the union  of the two intervals, and with the concatenation \nof the two controls is depicted in Figure \\ref{Fig4}. Let us denote the full control by $\\hat \\sigma=\\hat \\sigma(t)=\nu_x i\\sigma_x+ u_y i \\sigma_y+ u_z i \\sigma_z$. The final condition $(\\hat U_f, \\hat V_f)$ is given by \n\\be{Finalprel}\n\\hat U_f=\\begin{pmatrix}  0.7071-0.2795i  & 0.5913+0.2685i \\cr \n-0.5913+0.2685i & 0.7071+0.2795i \\end{pmatrix}, \n\\qquad \\hat V_f=\\begin{pmatrix}  0.7071+0.2708i  &  0.3718-0.5369i \\cr \n -0.3718-0.5369i & 0.7071-0.2708i \\end{pmatrix}.  \n\\end{equation} \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth, height=.4\\textheight]{BeforeConjugation}\n\\vspace*{-16mm}\n\\caption{Trajectory under the preliminary control in the total interval $[0,11]$.}\n\\label{Fig4}\n\\end{figure}\nThis condition, as expected, is in the same orbit as the desired final condition $(H_1,H_2)$  in (\\ref{Hada1}), (\\ref{Hada2}), that is, there exists a matrix $K \\in SU(2)$ such that\n$K\\hat U_fK^\\dagger=H_1$ and $K\\hat V_fK^\\dagger=H_2$. The matrix $K$ solving these equations is found to be \n$$\nK=\\begin{pmatrix}  0.1485-0.2460i &  -0.2444+0.9260i \\cr \n 0.2444+0.9260i & 0.1485+0.2460i\\end{pmatrix}. \n$$\nIn particular, to find $K$ one can diagonalize $\\hat U_f$ and $H_1$, i.e., $\\hat U_f=P \\Lambda P^\\dagger$ and $H_1=R\\Lambda R^\\dagger$ for a diagonal matrix $\\Lambda$, so that, from $KP\\Lambda P^\\dagger K^\\dagger=R \\Lambda R^\\dagger$,  we find that $R^\\dagger K P=D$, for $D$, a diagonal matrix. This matrix is found by solving $DP^\\dagger \\hat V_f P=R^ \\dagger H_2 R D$. The control $K \\hat \\sigma K^\\dagger$ steers then to the desired final condition. The resulting trajectory leading to the  desired final condition (\\ref{Hada1}), (\\ref{Hada2}) is given in Figure \\ref{Fig5}.   \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth, height=.4\\textheight]{AfterConjugation}\n\\vspace*{-16mm}\n\\caption{Trajectory under control algorithm leading to the desired final condition (\\ref{Hada1}) and (\\ref{Hada2}).}\n\\label{Fig5}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\n The notion of the complexity $c(M)$ of a compact $3$-manifold $M$\nwas introduced in \\cite{Matveev-1990}. The complexity is defined as\nthe minimal possible number of true vertices of an almost simple\nspine of $M$. If $M$ is closed and irreducible and $c(M)>0$, then\n$c(M)$ is the minimal number of tetrahedra needed to obtain $M$ by\ngluing together their faces. The problem of calculating the\ncomplexity $c(M)$ is very difficult. The exact values of the\ncomplexity are presently known only for certain infinite series of\nirreducible boundary irreducible $3$-manifolds \\cite{FMP,\nAnisov-2005, Jaco-Rubinstein-Tillmann-2009}. In addition, this\nproblem is solved for all closed orientable irreducible manifolds up\nto complexity $12$ (see \\cite{Matveev-2005}). Note that the table\ngiven in \\cite{Matveev-2005} contains $36833$ manifolds and is only\navailable in electronic form \\cite{Atlas}.\n\n The task of finding an upper bound for the complexity of a manifold\n$M$ does not present any particular difficulties. To do that it\nsuffices to construct an almost simple spine $P$ of $M$. The number\nof true vertices of $P$ will serve as an upper bound for the\ncomplexity. It is known \\cite[2.1.2]{Matveev-2003} that an almost\nsimple spine can be easily constructed from practically any\nrepresentation of a manifold. The rather large number of manifolds\nin \\cite{Atlas} gives rise to a new task of finding potentially\nsharp upper bounds for the complexity, i.e. upper bounds that would\nyield the exact value of the complexity for all manifolds from the\ntable \\cite{Atlas}. An important result in this direction was\nobtained by Martelli and Petronio \\cite{Martelli-Petronio-2004}.\nThey found a potentially sharp upper bound for the complexity of all\nclosed orientable Seifert manifolds. Similar results for infinite\nfamilies of graph manifolds can be found in\n\\cite{Fominykh-Ovchinnikov-2005, Fominykh-2008}.\n\n An upper bound $h(r\/s, t\/u, v\/w)$ for the complexity of hyperbolic\nmanifolds obtained by surgeries on the link $6^3_1$ (in Rolfsen's\nnotation \\cite{Rolfsen-1976}) with rational parameters $(r\/s, t\/u,\nv\/w)$ is given by Martelli and Petronio in\n\\cite{Martelli-Petronio-2004}. It turns out that the bound is not\nsharp for a large number of manifolds, as the following two examples\nshow. First, the value of $h$ is equal to $10$ only for $13$ of $24$\nmanifolds of complexity $10$ obtained by surgeries on $6^3_1$ (see\n\\cite{Martelli-Petronio-2004}). Second, on analyzing the table\n\\cite{Atlas} we noticed that the bound is not sharp for $44$ of $46$\nmanifolds of the type $6^3_1(1, 2, v\/w)$ with complexity less or\nequal to $12$. Denote by $4_1(p\/q)$ the closed orientable\n$3$-manifold obtained from the figure eight knot $4_1$ by\n$p\/q$-surgery. Since the manifolds $4_1(p\/q)$ and $6^3_1(1, 2,\np\/q+1)$ are homeomorphic, a potentially sharp upper bound for the\ncomplexity of such manifolds become important.\n\n The following theorem is the main result of the paper. To give an\nexact formulation, we need to introduce a certain\n$\\mathbb{N}$-valued function $\\omega(p\/q)$ on the set of\nnon-negative rational numbers. Let $p\\geqslant 0$, $q\\geqslant 1$ be\nrelatively prime integers, let $[p\/q]$ be the integer part of $p\/q$,\nand let $rem(p,q)$ be the remainder of the division of $p$ by $q$.\nAs in \\cite{Matveev-2003}, we denote by $S(p,q)$ the sum of all\npartial quotients in the expansion of $p\/q$ as a regular continued\nfraction. Now we define:\n $$\\omega(p\/q) = a(p\/q) + \\max\\{[p\/q]-3, 0\\} + S(rem(p,q),q),$$\nwhere\n $$a(p\/q) = \\left\\\n\\begin{array}{ll}\n    6, & \\hbox{if } p\/q=4, \\\\\n    7, & \\hbox{if } p\/q\\in \\mathbb{Z} \\hbox{ and } p\/q\\neq 4,\\\\\n    8, & \\hbox{if } p\/q\\not\\in \\mathbb{Z}.\\\\\n\\end{array\n\\right.$$\n\n\\begin{theorem}\n For any two relatively prime integers $p\\geqslant 0$ and\n$q\\geqslant 1$ we have the inequality $c(4_1(p\/q))\\leqslant\n\\omega(p\/q)$. Moreover, if $\\omega(p\/q)\\leqslant 12$, then\n$c(4_1(p\/q)) = \\omega(p\/q)$.\n\\end{theorem}\n\n Note that the restrictions $p\\geqslant 0$ and $q\\geqslant 1$ in the\nabove theorem are inessential, since the knot $4_1$ is equivalent to\nits mirror image, which implies $4_1(-p\/q)$ is homeomorphic to\n$4_1(p\/q)$.\n\n\n\\section{Preliminaries}\n\n In this section we recall some known definitions and facts that\nwill be used in the paper.\n\n\\subsection{Theta-curves on a torus}\n\n By a theta-curve $\\theta\\subset T$ on a torus $T$ we mean a graph\nthat is homeomorphic to a circle with a diameter and such that\n$T\\setminus \\theta$ is an open disc. It is well known\n\\cite{Martelli-Petronio-2004, Anisov-1994} that any two theta-curves\non $T$ can be transformed into each other by isotopies and by a\nsequence of flips (see Fig.~\\ref{flip-transformation}). Let us endow\nthe set $\\Theta(T)$ of theta-curves on $T$ with the distance\nfunction $d$ defining for given $\\theta, \\theta'\\in \\Theta(T)$ the\ndistance $d(\\theta, \\theta')$ between them as the minimal number of\nflips required to transform $\\theta$ into $\\theta'$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{flip.eps}\n\\caption{A flip-transformation} \\label{flip-transformation}\n\\end{figure}\n\n For calculating the distance between two theta-curves on a torus\nwe use the classical ideal triangulation $\\mathbb{F}$ (Farey\ntesselation) of the hyperbolic plane $\\mathbb{H}^2$. If we view the\nhyperbolic plane $\\mathbb{H}^2$ as the upper half plane of\n$\\mathbb{C}$ bounded by the circle $\\partial \\mathbb{H}^2 =\n\\mathbb{R}\\cup \\{\\infty\\}$, then the triangulation $\\mathbb{F}$ has\nvertices at the points of $\\mathbb{Q}\\cup \\{1\/0\\}\\subset \\partial\n\\mathbb{H}^2$, where $1\/0=\\infty$, and its edges are all the\ngeodesics in $\\mathbb{H}^2$ with endpoints the pairs $a\/b$, $c\/d$\nsuch that $ad-bc=\\pm 1$. For convenience, the images of the\nhyperbolic plane $\\mathbb{H}^2$ and of the triangulation\n$\\mathbb{F}$ under the mapping $z\\to (z-i)\/(z+i)$ are shown in\nFig.~\\ref{triangulation}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.5]{circle.eps}\n\\caption{The ideal Farey triangulation of the hyperbolic plane}\n\\label{triangulation}\n\\end{figure}\n\n Fix some coordinate system $(\\mu, \\lambda)$ on a torus $T$. We now\nconstruct a map $\\Psi_{\\mu, \\lambda}$ from $\\Theta(T)$ to the set of\ntriangles of $\\mathbb{F}$. To do that we consider the map\n$\\psi_{\\mu, \\lambda}$ that assigns to each nontrivial simple closed\ncurve $\\mu^{\\alpha}\\lambda^{\\beta}$ on $T$ the point $\\alpha\/\n\\beta\\in \\partial\\mathbb{H}^2$. Note that each theta-curve $\\theta$\non $T$ contains three nontrivial simple closed curves $\\ell_1$,\n$\\ell_2$, $\\ell_3$, that are formed by the pairs of edges of\n$\\theta$. Since the intersection index of every two curves $\\ell_i$,\n$\\ell_j$, $i\\neq j$, is equal to $\\pm 1$, the points $\\psi_{\\mu,\n\\lambda}(\\ell_1)$, $\\psi_{\\mu, \\lambda}(\\ell_2)$, $\\psi_{\\mu,\n\\lambda}(\\ell_3)$ are the vertices of a triangle $\\triangle$ of the\nFarey triangulation, and we define $\\Psi_{\\mu, \\lambda}(\\theta)$ to\nbe $\\triangle$.\n\n Denote by $\\Sigma$ the graph dual to the triangulation\n$\\mathbb{F}$. This graph is a tree because the triangulation is\nideal. We now define the distance between any two triangles of\n$\\mathbb{F}$ to be the number of edges of the only simple path in\n$\\Sigma$ that joins the corresponding vertices of the dual graph.\nThe key observation used for the practical calculations is that for\nany coordinate system $(\\mu, \\lambda)$ on $T$ the distance between\nany two theta-curves $\\theta$, $\\theta'$ is equal to the distance\nbetween the triangles $\\Psi_{\\mu, \\lambda}(\\theta)$, $\\Psi_{\\mu,\n\\lambda}(\\theta')$ of the Farey triangulation. The reason is that if\n$\\theta'$ is obtained from $\\theta$ via a flip, the corresponding\ntriangles have a common edge.\n\n\\subsection{Simple and special spines}\n\n A compact polyhedron $P$, following Matveev \\cite{Matveev-2003}, is\ncalled simple if the link of each point $x\\in P$ is homeomorphic to\none of the following $1$-dimensional polyhedra:\n\\begin{itemize}\n    \\item[(a)] a circle (the point $x$ is then called nonsingular);\n    \\item[(b)] a circle with a diameter (then $x$ is a triple point);\n    \\item[(c)] a circle with three radii (then $x$ is a true vertex).\n\\end{itemize}\n\n The components of the set of nonsingular points are said to\nbe the $2$-components of $P$, while the components of the set of\ntriple points are said to be the triple lines of $P$. A simple\npolyhedron is special if each of its triple lines is an open\n$1$-cell and each of its $2$-components is an open $2$-cell.\n\n A subpolyhedron $P$ of a $3$-manifold $M$ is a spine of $M$ if\n$\\partial M\\neq\\emptyset$ and the manifold $M\\setminus P$ is\nhomeomorphic to $\\partial M \\times (0,1]$, or $\\partial M=\\emptyset$\nand $M\\setminus P$ is an open ball. A spine of a $3$-manifold is\ncalled simple or special if it is a simple or special polyhedron,\nrespectively.\n\n\n\n\\subsection{Relative spines}\n\n A manifold with boundary pattern, following Johannson\n\\cite{Johannson-1979}, is a $3$-manifold $M$ with a fixed graph\n$\\Gamma \\subset \\partial M$ that does not have any isolated\nvertices. A manifold $M$ with boundary pattern $\\Gamma$ can be\nconveniently viewed as a pair $(M, \\Gamma)$. The case $\\Gamma =\n\\emptyset$ is also allowed.\n\n\\begin{definition}\n Let $(M, \\Gamma)$ be a $3$-manifold with boundary pattern. Then a\nsubpolyhedron $P\\subset M$ is called a relative spine of $(M,\n\\Gamma)$ if the following holds:\n \\begin{enumerate}\n \\item $M\\setminus P$ is an open ball;\n \\item $\\partial M \\subset P$;\n \\item $\\partial M \\cap Cl(P\\setminus\\partial M) = \\Gamma$.\n \\end{enumerate}\n\\end{definition}\n\nA relative spine is simple if it is a simple polyhedron.\nObviously, if $M$ is closed, then any relative spine of $(M,\n\\emptyset)$ is a spine of $M$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{2blocks.eps}\n\\caption{Examples of simple relative spines} \\label{blocks}\n\\end{figure}\n\n\\begin{example}\n Let $V$ be a solid torus with a meridian $m$. Choose a\nsimple closed curve $\\ell$ on $\\partial V$ that intersects $m$ twice\nin the same direction. Note that $\\ell$ decomposes $m$ into two\narcs. Consider a theta-curve $\\theta_V\\subset \\partial V$ consisting\nof $\\ell$ and an arc (denote it by $\\gamma$) of $m$. Then the\nmanifold $(V, \\theta_V)$ has a simple relative spine without\ninterior true vertices. This spine is the union of $\\partial V$, a\nM\\\"{o}bius strip inside V, and a part of meridional disc bounded by\n$\\gamma$ (Figure~\\ref{blocks}a).\n\n Note that among the three nontrivial simple closed curves contained in\n$\\theta_V$, none is isotopic to the meridian $m$ of $V$. On the\nother hand, applying the flip to $\\theta_V$ along $\\gamma$, we get a\ntheta-curve $\\theta_m\\subset \\partial V$ containing $m$.\n\\end{example}\n\n\\begin{example}\n Let $\\theta$, $\\theta'$ be two theta-curves on a torus $T$ such that\n$\\theta'$ is obtained from $\\theta$ by exactly one flip. Then the\nmanifold\n $$(T\\times [0, 1], (\\theta\\times \\{0\\})\\cup (\\theta'\\times \\{1\\}))$$\nhas a simple relative spine $R$ with one interior true vertex (in\nFigure~\\ref{blocks}b the torus $T$ is represented as a square with\nthe sides identified). Note that $R$ satisfies the following\nconditions:\n \\begin{itemize}\n \\item[(1)] for each $t\\in [0, 1\/2)$ a theta-curve $\\theta_t$, where\n $R\\cap (T\\times \\{t\\}) = \\theta_t\\times \\{t\\}$, is isotopic to\n $\\theta$;\n \\item[(2)] for each $t\\in (1\/2, 1]$ the theta-curve $\\theta_t$ is isotopic to\n $\\theta'$;\n \\item[(3)] $R\\cap (T\\times \\{1\/2\\})$ is a wedge of two circles.\n \\end{itemize}\n\\end{example}\n\n\\subsection{Assembling of manifolds with boundary patterns}\n\n Denote by $\\mathscr{T}$ the class of all manifolds $(M,\n\\Gamma)$ such that any component $T$ of $\\partial M$ is a torus and\n$T \\cap \\Gamma$ is a theta-curve. Let $(M, \\Gamma)$ and $(M',\n\\Gamma')$ be two manifolds in $\\mathscr{T}$ with nonempty\nboundaries. Choose two tori $T\\subseteq \\partial M$, $T'\\subseteq\n\\partial M'$ and a homeomorphism $\\varphi: T\\to T'$ taking the theta-curve\n$\\theta = T\\cap \\Gamma$ to the theta-curve $\\theta' = T'\\cap\n\\Gamma'$. Then we can construct a new manifold $(W, \\Xi)\\in\n\\mathscr{T}$, where $W = M\\cup_\\varphi M'$, and $\\Xi =\n(\\Gamma\\setminus \\theta)\\cup (\\Gamma'\\setminus \\theta')$. In this\ncase we say that the manifold $(W, \\Xi)$ is obtained assembling $(M,\n\\Gamma)$ and $(M', \\Gamma')$ \\cite{Martelli-Petronio-2001}.\n\n Note that if manifolds $(M, \\Gamma)$ and $(M', \\Gamma')$ have\nsimple relative spines denoted $P$ and $P'$ respectively, with $v$\nand $v'$ interior true vertices, then the manifold $(W, \\Xi)$ has a\nsimple relative spine $R$ with $v + v'$ interior true vertices.\nIndeed, $R$ can be obtained by gluing $P$ and $P'$ along $\\varphi$\nand removing the open disc in $P\\cup_\\varphi P'$ that is obtained by\nidentifying $T\\setminus \\theta$ with $T'\\setminus \\theta'$.\n\n To prove the main theorem of the paper we generalize the notion of the assembling\nby removing the restriction $\\varphi(\\theta) = \\theta'$.\n\n\\begin{lemma}\n \\label{assembling}\n Let $(M, \\Gamma)$ and $(M', \\Gamma')$ be two manifolds in\n$\\mathscr{T}$ with nonempty boundaries that admit simple relative\nspines with $v$ and $v'$ interior true vertices respectively. Then\nfor any homeomorphism $\\varphi: T\\to T'$ of a torus $T\\subseteq\n\\partial M$ onto a torus $T'\\subseteq \\partial M'$ there exists a simple\nrelative spine of a manifold $(W, \\Upsilon)$, where $W =\nM\\cup_\\varphi M'$ and $\\Upsilon = (\\Gamma\\setminus \\theta)\\cup\n(\\Gamma'\\setminus \\theta')$, with $v + v' + d(\\varphi(\\theta),\n\\theta')$ interior true vertices.\n\\end{lemma}\n\n\\begin{proof}\n First, by induction on the number $n = d(\\varphi(\\theta), \\theta')$\nwe prove that there exists a simple relative spine of the manifold\n $$(M'', \\Gamma'') = (T'\\times [0, 1], (\\varphi(\\theta)\\times \\{0\\})\\cup\n (\\theta'\\times \\{1\\}))$$\nwith $n$ interior true vertices. If $n=0$, i.e. the theta-curve\n$\\varphi(\\theta)$ is isotopic to the theta-curve $\\theta'$, the\ndesired spine is isotopic to the polyhedron\n$(\\varphi(\\theta)\\times [0, 1])\\cup \\partial M''$. Suppose that $n\n> 0$. As has already been alluded to in the beginning of the\nsection 1.1, there exists a sequence $\\{\\theta_i\\}_{i=0}^n$ of\npairwise distinct theta-curves on the torus $T'$ such that\n$\\theta_0 = \\varphi(\\theta)$, $\\theta_n = \\theta'$, and $\\theta_i$\nis obtained from $\\theta_{i-1}$ by a flip, for $i=1\\ldots n$. The\ninduction assumption implies that the manifold\n \\begin{equation}\n \\label{m1}\n (T'\\times [0, 1\/2], (\\theta_0\\times \\{0\\})\\cup\n (\\theta_{n-1}\\times \\{1\/2\\}))\n \\end{equation}\nhas a simple relative spine with $n-1$ interior true vertices.\nFurthermore, the simple relative spine of the manifold\n \\begin{equation}\n \\label{m2}\n (T'\\times [1\/2, 1], (\\theta_{n-1}\\times \\{1\/2\\})\\cup\n (\\theta_n\\times \\{1\\}))\n \\end{equation}\nwith one interior true vertex is described in the Example 2. Then\nthe desired spine of the manifold $(M'', \\Gamma'')$ is obtained by\nassembling the manifolds (\\ref{m1}) and (\\ref{m2}) along the\nidentity map on $T'\\times \\{1\/2\\}$.\n\n Now, note that the consecutive assemblings of the manifolds $(M,\n\\Gamma)$, $(M'', \\Gamma'')$ and $(M', \\Gamma')$ along natural\nhomeomorphisms that take each point $x\\in T$ to the point\n$(\\varphi(x), 0)\\in T'\\times \\{0\\}$, and each point $(y, 1)\\in\nT'\\times \\{1\\}$ to the point $y\\in T'$, yield the manifold $(W,\n\\Upsilon)$ and its simple relative spine with $v + v' +\nd(\\varphi(\\theta), \\theta')$ interior true vertices.\n\\end{proof}\n\n\n\\section{Relative spines of the figure eight knot complement}\n\n In this section we construct some simple relative spines of\nthe figure eight knot complement $E(4_1)$. Let us fix a canonical\ncoordinate system on the boundary torus $\\partial E(4_1)$ consisting\nof oriented closed curves $\\mu$, $\\lambda$ such that the meridian\n$\\mu$ generates $H_1(E(4_1); \\mathbb{Z})$ and the longitude\n$\\lambda$ bounds a surface in $E(4_1)$. This system determines the\nmap $\\Psi_{\\mu, \\lambda}$ from $\\Theta(T)$ to the set of triangles\nof the Farey triangulation. Denote by $\\triangle^{(i)}$ the triangle\nof $\\mathbb{F}$ with the vertices at $i$, $i+1$, and $\\infty$.\n\n\\begin{proposition}\n \\label{spine}\n For any $i\\in\\{ 0, 1, 2, 3\\}$ there exists a theta-curve\n$\\theta^{(i)}$ on the torus $\\partial E(4_1)$ such that the\nmanifold $(E(4_1), \\theta^{(i)})$ has a simple relative spine with\n$10$ interior true vertices and $\\Psi_{\\mu, \\lambda}(\\theta^{(i)})\n= \\triangle^{(i)}$.\n\\end{proposition}\n\n\\begin{proof}\n Step 1. Let $P$ be a special spine of an arbitrary compact orientable\n$3$-manifold $M$ whose boundary is a torus, and let $\\theta$ be a\ntheta-curve on $\\partial M$. We begin the proof by describing a\nmethod for constructing a simple relative spine $R(P, \\theta)$ of\nthe manifold $M$.\n\n By Theorem 1.1.7 \\cite{Matveev-2003}, $M$ can be identified with\nthe mapping cylinder of a local embedding $f:\\partial M\\to P$.\nDenote by $f_{|\\theta}:\\theta\\to P$ the restriction to $\\theta$ of\nthe map $f$. Then the union $R(P, \\theta)$ of the mapping cylinder\nof $f_{|\\theta}$ and of $\\partial M$ is a relative spine of $M$,\nsince $\\partial M \\subset R(P, \\theta)$, $\\partial M \\cap Cl(R(P,\n\\theta)\\setminus\\partial M) = \\theta$, and $M\\setminus R(P, \\theta)$\nis homeomorphic to the direct product of the open disc $\\partial\nM\\setminus \\theta$ with an interval. In general, $R(P, \\theta)$ just\nconstructed is not necessarily a simple polyhedron. This can be\ndealt with by introducing the notion of general position. We say\nthat a theta-curve $\\theta\\subset \\partial M$ is in general position\nwith respect to the map $f$, if the image $f(\\theta)$ satisfies the\nfollowing conditions.\n\\begin{enumerate}\n    \\item $f(\\theta)$ contains no true vertices of $P$.\n    \\item For any intersection point $x$ of $f(\\theta)$ with\n    the triple lines of $P$ there exists a neighborhood $U(x)\\subset P$\n    such that the intersection $U(x)\\cap f(\\theta)$ is an arc\n    meeting the set of the triple lines of $P$ transversally exactly at $x$.\n    \\item  For any intersection point $x$ of the set $f(\\theta)$ with\n    the $2$-components of $P$ its inverse image $f^{-1}_{|\\theta}(x)$ consists of at most\n    two points of $\\theta$. Moreover, if $f^{-1}_{|\\theta}(x)$ consists of exactly\n    two points, then there exists a neighborhood $U(x)\\subset P$ such that\n    the inverse image $f^{-1}_{|\\theta}(U(x)\\cap f(\\theta))$ of the\n    intersection $U(x)\\cap f(\\theta)$ is the disjoint union of two\n    arcs $\\gamma_1$, $\\gamma_2$ of $\\theta$, and the images $f(\\gamma_1)$, $f(\\gamma_2)$\n    intersect each other transversally at exactly one point $x$.\n    Such a point $x$ is called the self-intersection point of the image $f(\\theta)$\n    of $\\theta$.\n\\end{enumerate}\n\nObviously, if a theta-curve $\\theta$ is in general position with\nrespect to the map $f$, then the relative spine $R(P, \\theta)$ of\nthe manifold $M$ is simple.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{P_2.eps}\n\\caption{A minimal spine of the complement of the figure eight knot}\n\\label{minspine}\n\\end{figure}\n\n Step 2. We consider now the minimal special spine $P$ of the manifold\n$M = E(4_1)$ shown in Figure \\ref{minspine} (see\n\\cite[2.4.2]{Matveev-2003}). To construct the theta-curves\n$\\theta^{(i)}$, $i\\in\\{ 0, 1, 2, 3\\}$, we need to describe certain\ncell decompositions of the torus $T=\\partial M$ and of its universal\ncovering $\\tilde{T}$. The local embedding $f:T\\to P$ determines a\ncell decomposition of $T$ as follows.\n\\begin{enumerate}\n    \\item The inverse image $f^{-1}(C)$ of every open $k$-dimensional\n    cell $C$ of $P$ consists of two open $2$-cells if $k=2$,\n    three open arcs if $k=1$, and four points if $k=0$.\n    \\item The restriction of $f$ to each of these cells\n    is a homeomorphism onto the corresponding cell of $P$.\n\\end{enumerate}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.7]{decompositions.eps}\n\\caption{Cell decompositions of $\\tilde{T}$ (left) and $T$ (right)}\n\\label{decomposition}\n\\end{figure}\n\n Construct the universal covering $\\tilde{T}$ of $T$. It can\nbe presented as a plane decomposed into hexagons, see Fig.\n\\ref{decomposition}a. The group of covering translations is\nisomorphic to the group $\\pi_1(T) = H_1(T; \\mathbb{Z})$. We choose a\nbasis $\\tilde{\\mu}$, $\\tilde{\\lambda}$ as shown in Fig.\n\\ref{decomposition}a. It is easy to see that the corresponding\nelements of $\\pi_1(T)$ (which can be also viewed as oriented loops)\nform the canonical coordinate system $(\\mu, \\lambda)$ on $T$. If we\nfactor this covering by the translations $\\tilde{\\mu}$,\n$\\tilde{\\lambda}$, we recover $T$. If we additionally identify the\nhexagons marked by the letter $A$ with respect to the composition of\nthe symmetry in the dotted diagonal of the hexagon and the\ntranslation by $-\\tilde{\\mu} + \\tilde{\\lambda}\/2$, and do the same\nfor the hexagons marked by the letter $B$, we obtain $P$. The torus\n$T$ is shown in Fig. \\ref{decomposition}b as a polygon $D$ composed\nof four hexagons. Each side of $D$ is identified with some other one\nvia the translation along one of the three vectors $\\tilde{\\mu}$,\n$-2\\tilde{\\mu}+\\tilde{\\lambda}$, and $-\\tilde{\\mu}+\\tilde{\\lambda}$.\nThe spine $P$ can be presented as the union of two hexagons, see\nFig. \\ref{theta0} (right). The edges of the hexagons are decorated\nwith four different patterns. To recover $P$, one should identify\nthe edges having the same pattern.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4]{theta0_2.eps}\n\\caption{The theta-curve $\\theta^{(0)}$} \\label{theta0}\n\\end{figure}\n\n Step 3. Now for each $i\\in\\{ 0, 1, 2, 3\\}$ we exhibit a theta-curve\n$\\theta^{(i)}\\subset \\partial M$ such that the simple relative\nspine $R(P, \\theta^{(i)})$ of $M$ has $10$ interior true vertices\nand $\\Psi_{\\mu, \\lambda}(\\theta^{(i)}) = \\triangle^{(i)}$.\n\n Consider the wedge of the three arcs on $\\tilde{T}$, see Fig.\n\\ref{theta0} (left). The projections of the arcs onto $T$ yield a\ntheta-curve that we denoted by $\\theta^{(0)}$. It can be checked\ndirectly that $\\theta^{(0)}$ is in general position with respect to\nthe map $f$, and $\\Psi_{\\mu, \\lambda}(\\theta^{(0)}) =\n\\triangle^{(0)}$. It remains to note that the set of the interior\ntrue vertices of $R(P, \\theta^{(0)})$ consists of (a) the two true\nvertices of the special polyhedron $P$, (b) the images under $f$ of\nthe two vertices of $\\theta^{(0)}$, (c) the five intersection points\nof the set $f(\\theta^{(0)})$ with the triple lines of $P$, see Fig.\n\\ref{theta0} (left), and (d) one self-intersection point of the\nimage $f(\\theta^{(0)})$ of $\\theta^{(0)}$ (shown in Fig.\n\\ref{theta0} (right) as a fat gray dot).\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4]{theta1_2.eps}\n\\caption{The theta-curve $\\theta^{(1)}$} \\label{theta1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4]{theta2_2.eps}\n\\caption{The theta-curve $\\theta^{(2)}$} \\label{theta2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4]{theta3_2.eps}\n\\caption{The theta-curve $\\theta^{(3)}$} \\label{theta3}\n\\end{figure}\n\n The theta-curves $\\theta^{(1)}$, $\\theta^{(2)}$, $\\theta^{(3)}$\nsatisfying the conclusion of the Proposition are shown in Fig.\n\\ref{theta1}, \\ref{theta2}, \\ref{theta3}. We point out that among\nthe $10$ interior true vertices of $R(P, \\theta^{(3)})$ there are\n$6$ intersection points of the set $f(\\theta^{(3)})$ with the triple\nlines of $P$, see Fig. \\ref{theta3} (left), while there are no\nself-intersection points of the image $f(\\theta^{(3)})$ of\n$\\theta^{(3)}$, see Fig. \\ref{theta3} (right).\n\\end{proof}\n\n\n\\section{Proof of the main theorem}\n\n Let $p\\geqslant 0$ and $q\\geqslant 1$ be two relatively prime integers.\nTo prove the inequality $c(4_1(p\/q))\\leqslant \\omega(p\/q)$ it\nsuffices to construct a simple spine of the manifold $4_1(p\/q)$ with\n$\\omega(p\/q)$ true vertices.\n\n Thurston \\cite{Thurston-1978} proved that the manifold $4_1(p\/q)$ is\nhyperbolic except for $p\/q\\in \\{0, 1, 2, 3, 4, \\infty\\}$. The case\n$p\/q=\\infty$ does not satisfy the assumptions of the Theorem. In\neach of the five remaining cases the non-hyperbolic manifold\n$4_1(p\/q)$ has complexity $7$ and $\\omega(p\/q)=7$.\n\n Let us construct a simple spine of the hyperbolic manifold\n$4_1(p\/q)$. Recall that the meridian $m$ and the theta-curve\n$\\theta_m$ on the boundary of $(V, \\theta_V)$ were fixed in Example\n1. Let $(\\mu, \\lambda)$ be the canonical coordinate system on the\nboundary torus $\\partial E(4_1)$ of the figure eight knot complement\n$E(4_1)$. Among all homeomorphisms $\\partial V \\to\n\\partial E(4_1)$ that take $m$ to the curve $\\mu^p\\lambda^q$, we\nchoose a homeomorphism $\\varphi$ such that the distance between the\ntheta-curves $\\varphi(\\theta_m)$ and $\\theta^{(0)}$ be as small as\npossible. For convenience denote by $z$ the number $\\min\\{[p\/q],\n3\\}$. By the Proposition, the manifold $(E(4_1), \\theta^{(z)})$ has\na simple relative spine with $10$ interior true vertices. Since\n$4_1(p\/q) = V\\cup_\\varphi E(4_1)$, it follows from Lemma that the\nmanifold $(4_1(p\/q), \\emptyset)$ has a simple relative spine\n$Q_{p\/q}$ with $10 + d(\\varphi(\\theta_V), \\theta^{(z)})$ interior\ntrue vertices. Moreover, $Q_{p\/q}$ is a spine of $4_1(p\/q)$, since\n$\\partial 4_1(p\/q) = \\emptyset$.\n\n Now let us prove that $d(\\varphi(\\theta_V), \\theta^{(z)}) = -2 +\n\\max\\{[p\/q]-3, 0\\} + S(rem(p,q),q)$. Recall that for each $i\\in\\{ 0,\n1, 2, 3\\}$ the map $\\Psi_{\\mu, \\lambda}$ takes $\\theta^{(i)}$ to the\ntriangle $\\triangle^{(i)}$ of the Farey triangulation with the\nvertices at $i$, $i+1$, and $\\infty$. Denote by $\\triangle_V$ and\n$\\triangle_m$ the triangles $\\Psi_{\\mu, \\lambda}(\\varphi(\\theta_V))$\nand $\\Psi_{\\mu, \\lambda}(\\varphi(\\theta_m))$, respectively. Since\nthe distance between theta-curves on $\\partial E(4_1)$ is equal to\nthe distance between the corresponding triangles of $\\mathbb{F}$, it\nis sufficient to find $d(\\triangle_V, \\triangle^{(z)})$.\n\n The choice of $\\varphi$ guarantees us that $\\triangle_m$ is\nthe closest triangle to $\\triangle^{(0)}$ among all the triangles\nwith a vertex at $p\/q$. This implies (see \\cite[Proposition\n4.3]{Martelli-Petronio-2004} and \\cite[Lemma 2]{Fominykh-2008}) that\n$d(\\triangle_m, \\triangle^{(0)}) = S(p,q)-1$. Since the theta-curve\n$\\theta_V$ is obtained from $\\theta_m$ by exactly one flip and\n$\\theta_V$ does not contain the meridian $m$, the triangle\n$\\triangle_V$ has a common edge with $\\triangle_m$ and $p\/q$ is not\na vertex of $\\triangle_V$. Hence, $d(\\triangle_V, \\triangle^{(0)}) =\nS(p,q)-2$. Analyzing the Farey triangulation, we can notice that\n$d(\\triangle_V, \\triangle^{(z)}) = d(\\triangle_V, \\triangle^{(0)}) -\nd(\\triangle^{(z)}, \\triangle^{(0)})$. Taking into account that\n$d(\\triangle^{(z)}, \\triangle^{(0)})=z$, $S(p,q) = [p\/q] +\nS(rem(p,q),q)$ and $[p\/q] - \\min\\{[p\/q], 3\\} = \\max\\{[p\/q]-3, 0\\}$\nwe get the equality $d(\\varphi(\\theta_V), \\theta^{(z)}) =\nd(\\triangle_V, \\triangle^{(z)}) = -2 + \\max\\{[p\/q]-3, 0\\} +\nS(rem(p,q),q)$.\n\n Note that if $p\/q\\not\\in \\mathbb{Z}$, then $Q_{p\/q}$ is the desired\nspine, since it contains $\\omega(p\/q)$ true vertices. On the other\nhand, if $p\/q\\in \\mathbb{Z}$, the spine $Q_{p\/q}$ contains\n$\\omega(p\/q)+1$ true vertices. In this case $Q_{p\/q}$ can be\ntransformed into another simple spine $Q_{p\/q}'$ of $4_1(p\/q)$ by a\nsequence of moves along boundary curves of length $4$ (similar\narguments can be found in \\cite[page 81]{Matveev-2003}). The spine\n$Q_{p\/q}'$ has the same number of true vertices but possesses a\nboundary curve of length $3$, hence it can be simplified. The result\nis a new spine of $4_1(p\/q)$ with $\\omega(p\/q)$ true vertices.\n\n To conclude the proof of the theorem, it remains to note that the\ntable \\cite{Atlas}contains $46$ hyperbolic manifolds of the type\n$4_1(p\/q)$ satisfying $\\omega(p\/q)\\leqslant 12$. For each of them\nour upper bound is sharp.\n\n\n\\footnotesize\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgment}\n\nThe authors greatly appreciate the financial support from the Rail Manufacturing Cooperative Research Centre (funded jointly by participating rail organizations and the Australian Federal Government's Business-Cooperative Research Centres Program) through Project R3.7.3 - Rail infrastructure defect detection through video analytics.\n\n\\small\n\n\\section{Conclusion}\n\nWe propose a Poisson Transfer Network (PTN) to tackle the semi-supervised few-shot problem, aiming to explore the value of unlabeled novel-class data from two aspects. We propose to employ the Poisson learning model to capture the relations between the few labeled and unlabeled data, which results in a more stable and informative classifier than previous semi-supervised few-shot models. Moreover, we propose to adopt the unsupervised contrastive learning to improve the generality of the embedding on novel classes, which avoids the possible over-fitting problem when training with few labeled samples. \nIntegrating the two modules, the proposed PTN can fully explore the unlabeled auxiliary information boosting the performance of few-shot learning.\nExtensive experiments indicate that PTN outperforms state-of-the-art few-shot and semi-supervised few-shot methods.\n\n\n\\section{Experiments}\n\\subsection{Datasets}\nWe evaluate the proposed PTN on two few-shot benchmark datasets: miniImageNet and tieredImageNet. The miniImageNet dataset \\cite{vinyals2016matching} is a subset of the ImageNet, consisting of 100 classes, and each class contains 600 images of size 84$\\times$84. We follow the standard split of 64 base, 16 validation , and 20 test classes  \\cite{vinyals2016matching,tian2020rethinking}. The tieredImageNet \\cite{ren2018meta} is another subset but with 608 classes instead. We follow the standard split of 351 base, 97 validation, and 160 test classes for the experiments \\cite{ren2018meta,liu2018learning}. We resize the images from tieredImageNet to 84$\\times$84 pixels, and randomly select $C$ classes from the novel class to construct the few-shot task. Within each class, $K$ examples are selected as the labeled data, and $V$ examples from the rest as queries. The extra $N$ unlabeled samples are selected from the $C$ classes or rest novel classes. We set $C=5, K= \\{1,5\\}, V=15$ and study different sizes of $N$. We run 600 few-shot tasks and report the mean accuracy with the 95\\% confidence interval.\n\n\\subsection{Implementation Details}\nSame as previous works~\\cite{rusu2018meta,dhillon2019baseline,liu2019prototype,tian2020rethinking,yu2020transmatch}, we adopt the wide residual network (WRN-28-10) \\cite{zagoruyko2016wide} as the backbone of our base model $W_{\\phi} \\circ f_{\\theta_0}$, and we follow the protocals in \\cite{tian2020rethinking,yu2020transmatch} fusing the base and validation classes to train the base model from scratch. We set the batch size to 64 with SGD learning rate as 0.05 and weight decay as $5e^{-4}$. We reduce the learning rate by 0.1 after 60 and 80 epochs. The base model is trained for 100 epochs.\n\n\n\\begin{table*}[t]\n\\centering\n\\resizebox{2.0\\columnwidth}{!}{%\n\\begin{tabular}{lcccc}\n\\hline\n\\multicolumn{1}{c}{\\multirow{2}{*}{Methods}}                              & \\multirow{2}{*}{Type}  & \\multirow{2}{*}{Backbone} & \\multicolumn{2}{c}{miniImageNet}                                                    \\\\ \\cline{4-5} \n\\multicolumn{1}{c}{}                                                      &                        &                           & 1-shot                                   & 5-shot                                   \\\\ \\hline\nPrototypical-Net \\cite{snell2017prototypical}            & Metric, Meta           & ConvNet-256               & 49.42$\\pm$0.78                           & 68.20$\\pm$0.66                           \\\\\nRelation Network \\cite{sung2018learning}            & Metric, Meta           & ConvNet-64                & 50.44$\\pm$0.82                           & 65.32$\\pm$0.70                           \\\\\nTADAM \\cite{oreshkin2018tadam}                      & Metric, Meta           & ResNet-12                 & 58.50$\\pm$0.30                           & 76.70$\\pm$0.30\n                          \\\\\nDPGN \\cite{yang2020dpgn}                            & Metric, Meta       & ResNet-12                 & 67.77$\\pm$0.32          & 84.60$\\pm$0.43   \n                          \\\\\nRFS \\cite{tian2020rethinking}                       & Metric, Transfer       & ResNet-12                 & 64.82$\\pm$0.60                           & 82.14$\\pm$0.43\n                         \\\\ \\hline\nMAML \\cite{finn2017model}                           & Optimization, Meta     & ConvNet-64                & 48.70$\\pm$1.84                           & 63.11$\\pm$0.92                           \\\\\nSNAIL \\cite{mishra2018simple}                       & Optimization, Meta     & ResNet-12                 & 55.71$\\pm$0.99                           & 68.88$\\pm$0.92                           \\\\\nLEO \\cite{rusu2018meta}                             & Optimization, Meta     & WRN-28-10                 & 61.76$\\pm$0.08                           & 77.59$\\pm$0.12                           \\\\\nMetaOptNet \\cite{lee2019meta}                       & Optimization, Meta     & ResNet-12                 & 64.09$\\pm$0.62                           & 80.00$\\pm$0.45                           \\\\ \\hline\nTPN \\cite{liu2018learning}                          & Transductive, Meta     & ConvNet-64                & 55.51$\\pm$0.86                           & 69.86$\\pm$0.65                           \\\\\nBD-CSPN \\cite{liu2019prototype}                     & Transductive, Meta     & WRN-28-10                 & 70.31$\\pm$0.93                           & 81.89$\\pm$0.60                           \\\\\nTransductive Fine-tuning \\cite{dhillon2019baseline} & Transductive, Transfer & WRN-28-10                 & 65.73$\\pm$0.68                           & 78.40$\\pm$0.52                           \\\\\nLaplacianShot \\cite{ziko2020laplacian}              & Transductive, Transfer & DenseNet                  & 75.57$\\pm$0.19                           & 84.72$\\pm$0.13                           \\\\ \\hline\nMasked Soft k-Means \\cite{ren2018meta}              & Semi, Meta             & ConvNet-128               & 50.41$\\pm$0.31                           & 64.39$\\pm$0.24                           \\\\\nTPN-semi \\cite{liu2018learning}                     & Semi, Meta             & ConvNet-64                & 52.78$\\pm$0.27                           & 66.42$\\pm$0.21                           \\\\\nLST \\cite{li2019learning}                           & Semi, Meta             & ResNet-12                 & 70.10$\\pm$1.90                           & 78.70$\\pm$0.80                           \\\\ \\hline\nTransMatch \\cite{yu2020transmatch}                  & Semi, Transfer         & WRN-28-10                 & 62.93$\\pm$1.11                          & 82.24$\\pm$0.59                           \\\\\nDPN (Ours)                                                                      & Semi, Transfer         & WRN-28-10                 & \\multicolumn{1}{l}{79.67$\\pm$1.06}       & \\multicolumn{1}{l}{86.30$\\pm$0.95}       \\\\\nPTN (Ours)                                                                       & Semi, Transfer         & WRN-28-10                 & \\textbf{82.66$\\pm$0.97} & \\textbf{88.43$\\pm$0.67} \\\\ \\hline \\hline\n\\multicolumn{1}{c}{\\multirow{2}{*}{Methods}}                              & \\multirow{2}{*}{Type}  & \\multirow{2}{*}{Backbone} & \\multicolumn{2}{c}{tieredImageNet}                                                  \\\\ \\cline{4-5} \n\\multicolumn{1}{c}{}                                                      &                        &                           & 1-shot                                   & 5-shot                                   \\\\ \\hline\nPrototypical-Net \\cite{snell2017prototypical}            & Metric, Meta           & ConvNet-256               & 53.31$\\pm$0.89                           & 72.69$\\pm$0.74                           \\\\\nRelation Network \\cite{sung2018learning}            & Metric, Meta           & ConvNet-64                & 54.48$\\pm$0.93                           & 71.32$\\pm$0.78                           \\\\\nDPGN \\cite{yang2020dpgn}                            & Metric, Meta       & ResNet-12                 & 72.45$\\pm$0.51          & 87.24$\\pm$0.39   \n                          \\\\\nRFS \\cite{tian2020rethinking}                       & Metric, Transfer       & ResNet-12                 & 71.52$\\pm$0.69                           & 86.03$\\pm$0.49                           \\\\ \\hline\nMAML \\cite{finn2017model}                           & Optimization, Meta     & ConvNet-64                & 51.67$\\pm$1.81                           & 70.30$\\pm$1.75                           \\\\\nLEO \\cite{rusu2018meta}                             & Optimization, Meta     & WRN-28-10                 & 66.33$\\pm$0.05                           & 81.44$\\pm$0.09                           \\\\\nMetaOptNet \\cite{lee2019meta}                       & Optimization, Meta     & ResNet-12                 & 65.81$\\pm$0.74                           & 81.75$\\pm$0.53                           \\\\ \\hline\nTPN \\cite{liu2018learning}                          & Transductive, Meta     & ConvNet-64                & 59.91$\\pm$0.94                           & 73.30$\\pm$0.75                           \\\\\nBD-CSPN \\cite{liu2019prototype}                     & Transductive, Meta     & WRN-28-10                 & 78.74$\\pm$0.95                           & 86.92$\\pm$0.63                           \\\\\nTransductive Fine-tuning \\cite{dhillon2019baseline} & Transductive, Transfer & WRN-28-10                 & 73.34$\\pm$0.71                           & 85.50$\\pm$0.50                           \\\\\nLaplacianShot \\cite{ziko2020laplacian}              & Transductive, Transfer & DenseNet                  & 80.30$\\pm$0.22                           & 87.93$\\pm$0.15                           \\\\ \\hline\nMasked Soft k-Means \\cite{ren2018meta}              & Semi, Meta             & ConvNet-128               & 52.39$\\pm$0.44                           & 69.88$\\pm$0.20                           \\\\\nTPN-semi \\cite{liu2018learning}                     & Semi, Meta             & ConvNet-64                & 55.74$\\pm$0.29                           & 71.01$\\pm$0.23                           \\\\\nLST \\cite{li2019learning}                           & Semi, Meta             & ResNet-12                 & 77.70$\\pm$1.60                           & 85.20$\\pm$0.80                           \\\\ \\hline\nDPN (Ours)                                                                       & Semi, Transfer         & WRN-28-10                 & \\multicolumn{1}{l}{82.18$\\pm$1.06}       & \\multicolumn{1}{l}{88.02$\\pm$0.72}       \\\\\nPTN (Ours)                                                                      & Semi, Transfer         & WRN-28-10                 & \\textbf{84.70$\\pm$1.14} & \\textbf{89.14$\\pm$0.71} \\\\ \\hline\n\\end{tabular} }\n\\caption{The 5-way, 1-shot and 5-shot classification accuracy (\\%) on the two datasets with 95\\% confidence interval. Tne best results are in bold. The upper and lower parts of the table show the results on miniImageNet and tieredImageNet, respectively.}\n\\label{Res1}\n\\end{table*}\n\nIn unsupervised embedding transfer, the data augmentation $T$ is defined same as \\cite{lee2019meta,tian2020rethinking}. For fair comparisons against TransMatch~\\cite{yu2020transmatch}, we also augment each labeled image 10 times by random transformations and generate the prototypes of each class as labeled samples. We apply SGD optimizer with a momentum of 0.9. The learning rate is initialized as $1e^{-3}$, and the cosine learning rate scheduler is used for 10 epochs. We set the batch size to 80 with $\\lambda = 1$ in Eq. (\\ref{UT}).\nFor Poisson inference, we construct the graph by connecting each sample to its $K$-nearest neighbors with Gaussian weights. We set $K = 30$ and the weight matrix $W$ is summarized with $w_{ii} = 0$, which accelerates the convergence of the iteration in Algorithm \\ref{algorithm} without change the solution of the Equation \\ref{PO}. We set the max $tp = 100$ in step 7 of Algorithm \\ref{algorithm} by referring to the stop constraint discussed in the Proposed Algorithm section. \nWe set hyper-parameters $\\mu = 1.5, M_1 = 20, M_2 = 40$ and $M_3 = 100$ empirically. Moreover, we set ${\\varphi} = 10, \\upsilon_{\\alpha}=0.5, \\upsilon_{\\sigma}=1.0$.\n\\subsection{Experimental Results}\n\\subsubsection{Comparison with the State-Of-The-Art}\nIn our experiments, we group the compared methods into five categories, and the experimental results on two datasets are summarized in Table \\ref{Res1}. \nWith the auxiliary unlabeled data available, our proposed PTN outperforms the metric-based and optimization-based few-shot models by large margins, indicating that the proposed model effectively utilizes the unlabeled information for assisting few-shot recognition.\nBy integrating the unsupervised embedding transfer and PoissonMBO classifier, PTN achieves superior performance over both transductive and existing SSFSL approaches. Specifically, under the 5-way-1-shot setting, the classification accuracies are 81.57\\% vs. 63.02\\%~TransMatch \\cite{yu2020transmatch}, 84.70\\% vs. 80.30\\%~LaplacianShot \\cite{ziko2020laplacian} on miniImageNet and tieredImageNet, respectively; under the 5-way-5-shot setting, the classification accuracies are 88.43\\% vs. 78.70\\%~LST \\cite{li2019learning}, 89.14\\% vs. 81.89\\%~BD-CSPN \\cite{liu2019prototype} on miniImageNet and tieredImageNet, respectively.\nThese results demonstrate the superiority of PTN for SSFSL tasks.\n\\subsubsection{Different Extra Unlabeled Samples}\n\\begin{table}[]\n\\centering\n\\begin{tabular}{@{}cccc@{}}\n\\toprule\nMethods & Num\\_U          & 1-shot         & 5-shot         \\\\ \\midrule\n~~PTN$^*$    & \\multicolumn{1}{c}{0}   & 76.20$\\pm$0.82 & 84.25$\\pm$0.61 \\\\\nPTN    & \\multicolumn{1}{c}{0}   & 77.01$\\pm$0.94 & 85.32$\\pm$0.68 \\\\\nPTN    & \\multicolumn{1}{c}{20}  & 77.20$\\pm$0.92 & 85.93$\\pm$0.82 \\\\\nPTN    & \\multicolumn{1}{c}{50}  & 79.92$\\pm$1.06 & 86.09$\\pm$0.75 \\\\\nPTN    & \\multicolumn{1}{c}{100} & 81.57$\\pm$0.94 & 87.17$\\pm$0.58 \\\\\nPTN    & \\multicolumn{1}{c}{200} & \\textbf{82.66$\\pm$0.97} & \\textbf{88.43$\\pm$0.76} \\\\ \\bottomrule\n\\end{tabular}\n\\caption{The 5-way, 1-shot and 5-shot classification accuracy (\\%) with different number of extra unlabeled samples on miniImageNet. PTN$^*$ denotes that we adopt PTN as the transductive model without fine-tune embedding. Best results are in bold.}\n\\label{Res2}\n\\end{table}\n\nWe show the results of using different numbers of extra unlabeled instances in Table \\ref{Res2}. For Num\\_U = 0, PTN$^*$ can be viewed as the transductive model without extra unlabeled data, where we treat query samples as the unlabeled data, and we do not fine-tune the embedding with query labels for fair comparisons. Contrary to PTN$^*$, the proposed PTN model utilize the query samples to fine-tune the embedding when Num\\_U=0.\nIt can be observed that our PTN model achieves better performances with more extra unlabeled samples, which indicates the effectiveness of PTN in mining the unlabeled auxiliary information for the few-shot problem.\n\n\n\\subsubsection{Results with Distractor Classes}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.95\\linewidth]{.\/images\/Few-shot.pdf}\n\\end{center}\n\\caption{The 5-way, 1-shot and 5-shot classification accuracy (\\%) with different number of extra unlabeled samples on miniImageNet. w\/D means with distractor classes.}\n\\label{dist}\n\\end{figure}\n\nInspired by \\cite{ren2018meta,liu2018learning,yu2020transmatch}, we further investigate the influence of distractor classes, where the extra unlabeled data are collected from classes with no overlaps to labeled support samples. We follow the settings in \\cite{ren2018meta,liu2018learning}. As shown in Figure \\ref{dist}, even with distractor class data, the proposed PTN still outperforms other SSFSL methods by a large margin, which indicates the robustness of the proposed PTN in dealing with distracted unlabeled data.\n\n\\subsection{Ablation Study}\n\\begin{table}[t]\n\\begin{minipage}{16.5cm}\n\\begin{tabular}{@{}lcc@{}}\n\\toprule\nMethods                                            & 1-shot         & 5-shot         \\\\ \\midrule\nTransMatch                                         & 62.93$\\pm$1.11 & 82.24$\\pm$0.59 \\\\\nLabel Propagation (LP)                  & 74.04$\\pm$1.00 & 82.60$\\pm$0.68 \\\\\nPoissonMBO                              & 79.67$\\pm$1.02 & 86.30$\\pm$0.65 \\\\\nDPN                         & 80.00$\\pm$0.83 & 87.17$\\pm$0.51 \\\\\nUnsup Trans+LP  \\footnote{Unsup Trans means Unsupervised Embedding Transfer.} & 75.65$\\pm$1.06 & 84.46$\\pm$0.68 \\\\\nUnsup Trans+PoissonMBO        & 80.73$\\pm$1.11 & 87.41$\\pm$0.63 \\\\\nUnsup Trans+PTN \\footnote{PTN consists of Unsup Trans and DPN.}    & \\textbf{82.66$\\pm$0.97} & \\textbf{88.43$\\pm$0.76} \\\\ \\bottomrule\n\\end{tabular}\n\\end{minipage}\n\\caption{Ablation studies about the proposed PTN, all methods are based on a pretrained embedding with 200 extra unlabeled samples each class on miniImageNet for 5-way, 1-shot and 5-shot classification (\\%). Best results are in bold.}\n\\label{aby}\n\\end{table}\n\nWe analyze different components of the PTN and summarize the results in Table ~\\ref{aby}. All compared approaches are based on the pre-trained WRN-28-10 embedding. \n\nFirst of all, we investigate the graph propagation component (classifier).\nIt can be observed that graph-based models such as Label Propagation~\\cite{zhou2004learning} and PoissonMBO~\\cite{calder2020poisson} outperform the inductive model TransMatch~\\cite{yu2020transmatch}, which is consistent with previous researches~\\cite{zhu2005semi,liu2018learning,ziko2020laplacian}. Compared to directly applying PoissonMBO on few-shot tasks, the proposed DPN \\textit{\\textbf{(without Unsupervised Embedding Transfer)}} achieves better performance, which indicates it is necessary to perform the feature calibration to eliminate the cross-class biases between support and query data distributions before \nlabel inference.\n\nFor investigating the proposed unsupervised embedding transfer in representation learning, we observe that all the graph-based models achieve clear improvement after incorporating the proposed transfer module. For instance, the Label Propagation obtains 1.61\\%, 1.86\\% performance gains on 5-way-1-shot, and 5-way-5-shot minImageNet classification. These results indicate the effectiveness of the proposed unsupervised embedding transfer. \nFinally, by integrating the unsupervised embedding transfer and graph propagation classifier, the PTN model achieves the best performances compared against all other approaches in Table \\ref{aby}. \n\n\\subsection{Inference Time}\nWe conduct inference time experiments to investigate the computation efficiency of the proposed Poisson Transfer Network (PTN) on the \\textit{mini}ImageneNet~\\cite{vinyals2016matching} dataset. Same as \\cite{ziko2020laplacian}, we compute the average inference time required for each 5-shot task. The results are shown in Table~\\ref{time}. Compared with inductive models, the proposed PTN costs more time due to the graph-based Poisson inference. However, our model achieves better classification performance than inductive ones and other transductive models, with affordable inference time.  \n\n\\begin{table}[h]\n\\caption{Average inference time (in seconds) for the 5-shot tasks in \\textit{mini}ImageneNet dataset.}\n\\begin{tabular}{@{}lc@{}}\n\\toprule\nMethods                & Inference Time \\\\ \\midrule\nSimpleShot~\\cite{wang2019simpleshot}             &      0.009          \\\\\nLaplacianShot~\\cite{ziko2020laplacian}          &     0.012         \\\\\nTransductive fine-tune~\\cite{dhillon2019baseline} &    20.7            \\\\\nPTN(Ours)              &      13.68          \\\\ \\bottomrule\n\\end{tabular}\n\n\\label{time}\n\\end{table}\n\n\\begin{table*}[t!]\n\\centering\n\\caption{Accuracy with various extra unlabeled samples for different semi-supervised few-shot methods on the \\textit{mini}ImageNet dataset. All results are averaged over 600 episodes with 95\\% confidence intervals. The best results are in bold.}\n\\begin{tabular}{@{}lccccc@{}}\n\\toprule\n                  & \\multicolumn{5}{c}{\\textit{mini}ImageNet  5-way-1-shot}                                                        \\\\ \\midrule\n                  & 0                       & 20                      & 50                      & 100                     & 200                     \\\\ \\midrule\nTransMatch~\\cite{yu2020transmatch}        & -                       & 58.43$\\pm$0.93          & 61.21$\\pm$1.03          & 63.02$\\pm$1.07          & 62.93$\\pm$1.11          \\\\\nLabel Propagation~\\cite{zhou2004learning} & 69.74$\\pm$0.72& 71.80$\\pm$1.02& 72.97$\\pm$1.06 & 73.35$\\pm$1.05                   & 74.04$\\pm$1.00                   \\\\\nPoissonMBO~\\cite{calder2020poisson}       & 74.79$\\pm$1.06 & 76.01$\\pm$0.99 & 76.67$\\pm$1.02 & 78.28$\\pm$1.02                   &   79.67$\\pm$1.02        \\\\\nDPN (Ours)        & 75.85$\\pm$0.97 & 76.10$\\pm$1.06 & 77.01$\\pm$0.92& 79.55$\\pm$1.13 & 80.00$\\pm$0.83                   \\\\\nPTN (Ours)        & \\textbf{77.01$\\pm$0.94} & \\textbf{77.20$\\pm$0.92} & \\textbf{79.92$\\pm$1.06} & \\textbf{81.57$\\pm$0.94} & \\textbf{82.66$\\pm$0.97} \\\\ \\midrule\n                  & \\multicolumn{5}{c}{\\textit{mini}ImageNet  5-way-5-shot}                                                        \\\\ \\midrule\n                  & 0                       & 20                      & 50                      & 100                     & 200                     \\\\ \\midrule\nTransMatch        & -                       & 76.43$\\pm$0.61          & 79.30$\\pm$0.59          & 81.19$\\pm$0.59          & 82.24$\\pm$0.59          \\\\\nLabel Propagation & 75.50$\\pm$0.60 & 78.47$\\pm$0.60 & 80.40$\\pm$0.61 & 81.65$\\pm$0.59 & 82.60$\\pm$0.68                   \\\\\nPoissonMBO        & 83.89$\\pm$0.66 & 84.43$\\pm$0.67  & 84.94$\\pm$0.82 & 85.51$\\pm$0.81 & 86.30$\\pm$0.65                 \\\\\nDPN (Ours)        & 84.74$\\pm$0.63 & 85.04$\\pm$0.66 & 85.36$\\pm$0.60  & 86.09$\\pm$0.63 & 87.17$\\pm$0.51                   \\\\\nPTN (Ours)        & \\textbf{85.32$\\pm$0.68} & \\textbf{85.93$\\pm$0.82} & \\textbf{86.09$\\pm$0.75} & \\textbf{87.17$\\pm$0.58} & \\textbf{88.43$\\pm$0.76} \\\\ \\bottomrule\n\\label{unlab}\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Results with Different Extra Unlabeled}\nWe conduct further experiments to investigate the current semi-supervised few-shot methods in mining the value of the unlabeled data. All approaches are based on a pre-trained WRN-28-10~\\cite{zagoruyko2016wide} model for fair comparisons. As indicated in Table \\ref{unlab}, with more unlabeled samples, all the models achieve higher classification performances. However, our proposed PTN model achieves the highest performance among the compared methods, which validates the superior capacity of the proposed model in using the extra unlabeled information for boosting few-shot methods.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Semi-supervised comparison on the \\textit{mini}ImageNet dataset.}\n\\begin{threeparttable}\n\\begin{tabular}{@{}lcccc@{}}\n\\toprule\nMethods             & 1-shot         & 5-shot         & \\multicolumn{1}{l}{1-shot w\/D} & \\multicolumn{1}{l}{5-shot w\/D} \\\\ \\midrule\nSoft K-Means~\\cite{ren2018meta} & 50.09$\\pm$0.45 & 64.59$\\pm$0.28 & 48.70$\\pm$0.32 & 63.55$\\pm$0.28 \\\\\nSoft K-Means+Cluster~\\cite{ren2018meta} & 49.03$\\pm$0.24 & 63.08$\\pm$0.18 & 48.86$\\pm$0.32 & 61.27$\\pm$0.24 \\\\\nMasked Soft k-Means~\\cite{ren2018meta} & 50.41$\\pm$0.31 & 64.39$\\pm$0.24 & 49.04$\\pm$0.31                 & 62.96$\\pm$0.14                 \\\\\nTPN-semi~\\cite{liu2018learning}            & 52.78$\\pm$0.27 & 66.42$\\pm$0.21 & 50.43$\\pm$0.84                 & 64.95$\\pm$0.73                 \\\\\nTransMatch~\\cite{yu2020transmatch}          & 63.02$\\pm$1.07 & 81.19$\\pm$0.59 & 62.32$\\pm$1.04                 & 80.28$\\pm$0.62                 \\\\ \\midrule\nPTN (Ours)                & \\textbf{82.66$\\pm$0.97} & \\textbf{88.43$\\pm$0.67} & \\textbf{81.92$\\pm$1.02} & \\textbf{87.59$\\pm$0.61} \\\\ \\bottomrule\n\\end{tabular}\n\\begin{tablenotes}\n\\item[$\\divideontimes$]``w\/D\" means with distraction classification. In this setting, many extra unlabeled samples are from the distraction classes, which is different from the support labeled classes. All results are averaged over 600 episodes with 95\\% confidence intervals. The best results are in bold. \n\\end{tablenotes}\n\\end{threeparttable}\n\\label{mini}\n\\end{table*}\n\n\\begin{table*}[!hpbt]\n\\centering\n\\caption{Semi-supervised comparison on the \\textit{tiered}ImageNet dataset.}\n\\begin{threeparttable}\n\\begin{tabular}{@{}lcccc@{}}\n\\toprule\nMethods             & 1-shot         & 5-shot         & \\multicolumn{1}{l}{1-shot w\/D} \n& \\multicolumn{1}{l}{5-shot w\/D} \n\\\\ \\midrule\nSoft K-Means~\\cite{ren2018meta} & 51.52$\\pm$0.36 & 70.25$\\pm$0.31 & 49.88$\\pm$0.52 & 68.32$\\pm$0.22 \\\\\nSoft K-Means+Cluster~\\cite{ren2018meta} & 51.85$\\pm$0.25 & 69.42$\\pm$0.17 & 51.36$\\pm$0.31 & 67.56$\\pm$0.10 \\\\\nMasked Soft k-Means~\\cite{ren2018meta} & 52.39$\\pm$0.44 & 69.88$\\pm$0.20 & 51.38$\\pm$0.38& 69.08$\\pm$0.25                 \\\\\nTPN-semi~\\cite{liu2018learning}            & 55.74$\\pm$0.29 & 71.01$\\pm$0.23 & 53.45$\\pm$0.93& 69.93$\\pm$0.80                 \\\\ \\midrule\nPTN (Ours)                & \\textbf{84.70$\\pm$1.14} & \\textbf{89.14$\\pm$0.71} & \\textbf{83.84$\\pm$1.07}                 & \\textbf{88.06$\\pm$0.62} \\\\ \\bottomrule\n\\end{tabular}\n\\begin{tablenotes}\n\\item[$\\divideontimes$]``w\/D\" means with distraction classification. In this setting, many extra unlabeled samples are from the distraction classes, which is different from the support labeled classes. All results are averaged over 600 episodes with 95\\% confidence intervals. The best results are in bold. \n\\end{tablenotes}\n\\end{threeparttable}\n\\label{tiered}\n\\end{table*}\n\n\n\\subsection{Results with Distractor Classification}\nWe report the results of the proposed PTN on both \\textit{mini}ImageNet~\\cite{vinyals2016matching} and \\textit{tiered}ImageneNet~\\cite{ren2018meta} datasets under different settings in Table~\\ref{mini} and Table~\\ref{tiered}, respectively. It can be observed that the classification results of all semi-supervised few-shot models are degraded due to the distractor classes. However, the proposed PTN model still outperforms other semi-supervised few-shot methods with a large margin. This also indicates the superiority of the proposed PTN model in dealing with the semi-supervised few-shot classification tasks over previous approaches.\n\n\n\n\\section{Introduction}\n\\noindent\nFew-shot learning \\cite{miller2000learning,fei2006one,vinyals2016matching} aims to learn a model that generalizes well with a few instances of each novel class.\nIn general, a few-shot learner is firstly trained on a substantial annotated dataset, also noted as the base-class set, and then adapted to unseen novel classes with a few labeled instances.\nDuring the evaluation, a set of few-shot tasks are fed to the learner, where each task consists of a few support (labeled) samples and a certain number of query (unlabeled) data.\nThis research topic has been proved immensely appealing in the past few years, as a large number of few-shot learning methods are proposed from various perspectives. Mainstream methods can be roughly grouped into two categories. The first one is learning from episodes \\cite{vinyals2016matching}, also known as meta-learning, which adopts the base-class data to create a set of episodes. Each episode is a few-shot learning task, with support and query samples that simulate the evaluation procedure.\nThe second type is the transfer-learning based method, which focuses on learning a decent classifier by transferring the domain knowledge from a model pre-trained on the large base-class set~\\cite{chen2018closer,qiao2018few}. This paradigm decouples the few-shot learning progress into representation learning and classification, and has shown favorable performance against meta-learning methods in recent works \\cite{tian2020rethinking,ziko2020laplacian}. Our method shares somewhat similar motivation with transfer-learning based methods and proposes to utilize the extra unlabeled novel-class data and a pre-trained embedding to tackle the few-shot problem.\n\nCompared with collecting labeled novel-class data, it is much easier to obtain abundant unlabeled data from these classes. Therefore, semi-supervised few-shot learning (SSFSL)~\\cite{ren2018meta,liu2018learning,li2019learning,yu2020transmatch} is proposed to combine the auxiliary information from labeled base-class data and extra unlabeled novel-class data to enhance the performance of few-shot learners.\nThe core challenge in SSFSL is how to fully explore the auxiliary information from these unlabeled.\nPrevious SSFSL works indicate that graph-based models~\\cite{liu2018learning,ziko2020laplacian} can learn a better classifier than inductive ones~\\cite{ren2018meta,li2019learning,yu2020transmatch}, since these methods directly model the relationship between the labeled and unlabeled samples during the inference.\nHowever, current graph-based models adopt the Laplace learning~\\cite{zhu2003semi} to conduct label propagation, the solutions of Laplace learning develop localized spikes near the labeled samples but are almost constant far from the labeled samples, \\textit{i.e.,} label values are not propagated well, especially with few labeled samples. Therefore, these models suffer from the underdeveloped message-passing capacity for the labels.\nOn the other hand, most SSFSL methods adapt the feature embedding pre-trained on base-class data (meta- or transfer- pre-trained) as the novel-class embedding. This may lead to the embedding degeneration problem, as the pre-trained model is designed for the base-class recognition, it tends to learn the embedding that represents only base-class information, and lose information that might be useful outside base classes.\n\nTo address the above issues, we propose a novel transfer-learning based SSFSL method, named Poisson Transfer Network (PTN). Specifically, \\textbf{\\textit{to improve the capacity of graph-based SSFSL models in message passing}}, we propose to revise the Poisson model tailored for few-shot problems by incorporating the query feature calibration and the Poisson MBO model. Poisson learning~\\cite{calder2020poisson} has been provably more stable and informative than traditional Laplace learning in low label rate semi-supervised problems. However, directly employing Poisson MBO for SSFSL may suffer from the cross-class bias due to the data distribution drift between the support and query data. Therefore, we improve the Poisson MBO model by explicitly eliminating the cross-class bias before label inference. \n\\textbf{\\textit{To tackle the novel-class embedding degeneration problem}}, we propose to transfer the pre-trained base-class embedding to the novel-class embedding by adopting unsupervised contrastive training \\cite{he2020momentum,chen2020simple} on the extra unlabeled novel-class data. Constraining the distances between the augmented positive pairs, while pushing the negative ones distant, the proposed transfer scheme captures the novel-class distribution implicitly. This strategy effectively avoids the possible overfitting of retraining feature embedding on the few labeled instances.\n\nBy integrating the Poisson learning and the novel-class specific embedding, the proposed PTN model can fully explore the auxiliary information of extra unlabeled data for SSFSL tasks.\nThe contributions are summarized as follows:\n\\begin{itemize}\n\\item We propose a Poisson learning based model to improve the capacity of mining the relations between the labeled and unlabeled data for graph-based SSFSL.\n\n\\item We propose to adapt unsupervised contrastive learning in the representation learning with extra unlabeled data to improve the generality of the pre-trained base-class embedding for novel-class recognition. \n\n\\item \nExtensive experiments are conducted on two benchmark datasets to investigate the effectiveness of PTN, and PTN achieves state-of-the-art performance.\n\\end{itemize}\n\\section{Methodology}\n\\subsection{Problem Definition}\\label{def}\nIn the standard few-shot learning, there exists a labeled support set $S$ of $C$ different classes, ${ S } = \\left\\{ \\left({x} _ { s } ,  {y} _ { s } \\right) \\right\\} _ { s = 1 } ^ { K \\times C }$, where $x_s$ is the labeled sample and $y_s$ denote its label. We use the standard basis vector $\\mathbf{e}_{i} \\in \\mathbb{R}^{C}$ represent the $i$-th class, \\textit{i.e.}, $y_s \\in \\left\\{\\mathbf{e}_{1}, \\mathbf{e}_{2}, \\ldots, \\mathbf{e}_{C}\\right\\}$. Given an unlabeled query sample $x_q$ from the query set ${ Q } = \\left\\{  {x} _ { q } \\right\\} _ { q=1 } ^ { V }$, the goal is to assign the query to one of the $C$ support classes. The labeled support set and unlabeled query set share the same label space, and the novel-class dataset $\\mathcal{D}_{novel}$ is thus defined as $\\mathcal{D}_{novel} = S \\cup Q$. If $S$ contains $K$ labeled samples for each of $C$ categories, the task is noted as a $C$-way-$K$-shot problem.\nIt is far from obtaining an ideal classifier with the limited annotated $S$. Therefore, few-shot models usually utilize a fully annotated dataset, which has similar data distribution but disjoint label space with $\\mathcal{D}_{novel}$ as an auxiliary dataset $\\mathcal{D}_{base}$, noted as the base-class set.\n\nFor the semi-supervised few-shot learning (SSFSL), we have an extra unlabeled support set ${ U } = \\left\\{  {x} _ { u } \\right\\} _ { u = 1 } ^ { N }$. These additional $N$ unlabeled samples are usually from each of the $C$ support classes in standard-setting, or other novel-class under distractor classification settings. Then the new novel-class dataset $\\mathcal{D}_{novel}$ is defined as $\\mathcal{D}_{novel} = S \\cup Q \\cup U$. The goal of SSFSL is maximizing the value of the extra unlabeled data to improve the few-shot methods.\n\nFor a clear understanding, the details of proposed PTN are introduced as follows: we first introduce the proposed Representation Learning, and then we illustrate the proposed Poisson learning model for label inference.\n\n\\subsection{Representation Leaning}\nThe representation learning aims to learn a well-generalized novel-class embedding through Feature Embedding Pre-training and Unsupervised Embedding Transfer. \n\\subsubsection{Feature Embedding Pre-training}\nOn the left side of Figure \\ref{framework}, the first part of PTN is the feature embedding pre-training.\n\nBy employing the cross-entropy loss between predictions and ground-truth labels in $\\mathcal{D}_{base}$, we train the base encoder $f_{\\theta_0}$ in a fully-supervised way, which is the same as \\cite{chen2018closer,yu2020transmatch,tian2020rethinking}. \nThis stage can generate powerful embedding for the downstream few-shot learner.\n\n\\subsubsection{Unsupervised Embedding Transfer} \\label{UET}\nDirectly employ the pre-trained base-class embedding for the novel-class  may suffer from the degeneration problem. However, retraining the base-class embedding with the limited labeled instances is easy to lead to overfitting. How can we train a novel-class embedding to represent things beyond labels when our only supervision is the limited labels? Our solution is unsupervised contrastive learning.\nUnsupervised learning, especially Contrastive learning~\\cite{he2020momentum,chen2020simple}, recently has shown great potential in representation learning for various downstream vision tasks, and most of these works training a model from scratch. \nHowever, unsupervised pre-trained models perform worse than fully-supervised pre-trained models. \nUnlike previous works, we propose to adopt contrastive learning to retrain the pre-trained embedding with the unlabeled novel data. In this way, we can learn a decent novel-class embedding by integrating the fully-supervised pre-trained scheme with unsupervised contrastive fine-tuning. \n\nSpecifically, for a minibatch of $n$ examples from the unlabeled novel-class subset ${ U_i = \\{x_u\\}_{u=1}^n }$, randomly sampling two data augmentation operators $t,t'\\in {T}$, we can generate a new feature set $Z = \\{ Z_t = \\{f_{\\theta_0} \\circ t(x_u)\\}_{u=1}^n \\} \\cup  \\{ Z_{t'} =  \\{f_{\\theta_0} \\circ t'(x_u)\\}_{u=1}^n \\}$, resulting in $n$ pairs of feature points. We treat each feature pair from the same raw data input as the positive pair, and the other $2(n-1)$ feature points as negative samples. Then the contrastive loss for the minibatch is defined as\n\\begin{equation}\n\\ell_{cont}=- \\sum_{i,j = 1}^{n} \\log \\frac{\\exp \\left(\\operatorname{cosine}\\left({z}_{i}, {z}_{j}\\right) \/ \\tau\\right)}{\\sum_{k \\neq i} \\exp \\left(\\operatorname{cosine}\\left({z}_{i}, {z}_{k}\\right) \/ \\tau\\right)},\n\\label{NCE}\n\\end{equation}\n\nwhere $z_i,z_j$ denote a positive feature pair from $Z$, $\\tau$ is a temperature parameter, and $\\operatorname{cosine}(\\cdot)$ represents the consine similarity. Then, we adopt a Kullback-Leibler divergence ($\\ell_{KL}$) between two feature subset $Z_t$ and $Z_{t'}$ as the regulation term. Therefore, the final unsupervised embedding transfer loss $\\ell_{UT}$ is defined as \n\\begin{equation}\n\\ell_{UT} = \\ell_{cont} + \\lambda \\ell_{KL}(Z_t ~\\|~ Z_{t'}).\n\\label{UT}\n\\end{equation}\nBy training the extra unlabeled data with this loss, we can learn a robust novel-class embedding $f_{\\theta}$ from $f_{\\theta_0}$.\n\n\\subsection{Poisson Label Inference}\nPrevious studies \\cite{zhu2003semi,zhou2004learning,zhu2005semi,liu2018learning,ziko2020laplacian} indicate that the graph-based few-shot classifier has shown superior performance against inductive ones. Therefore, we propose constructing the classifier with a graph-based Poisson model, which adopts different optimizing strategy with representation learning. \nPoisson model~\\cite{calder2020poisson} has been proved superior over traditional Laplace-based graph models~\\cite{zhu2003semi,zhou2004learning} both theoretically and experimentally, especially for the low label rate semi-supervised problem. \nHowever, directly applying this model to the few-shot task will suffer from a cross-class bias challenge, caused by the data distribution bias between support data (including labeled support and unlabeled support data) and query data. \n\nTherefore, we revise this powerful model by eliminating the support-query bias as the classifier. We explicitly propose a query feature calibration strategy before the final Poisson label inference. \nIt is worth noticing that the proposed graph-based classifier can be directly appended to the pre-trained embedding without adopting the unsupervised embedding transfer training. We dob this baseline model as \\textit{Decoupled Poisson Network} (\\textit{DPN}).  \n\n\\subsubsection{Query Feature Calibration}\nThe support-query data distribution bias, also referred to as the cross-class bias~\\cite{liu2019prototype}, is one of the reasons for the degeneracy of the few-shot learner. In this paper, we propose a simple but effective method to eliminate this distribution bias for Poisson graph inference. For a SSFSL task, we fuse the labeled support set $S$ and the extra unlabeled set $U$ as the final support set $ B = S \\cup U$. We denote the normalized embedded support feature set and query feature set as $Z_b = \\{z_b\\}$ and $Z_q = \\{z_q\\}$,\nthe cross-class bias is defined as \n\\begin{equation} \\\n\\begin{split}\n& \\Delta_{\\text {cross}}=\\mathbb{E}_{z_{b} \\sim p_{\\mathcal{B}}}\\left[z_{b}\\right]-\\mathbb{E}_{z_{q} \\sim p_{\\mathcal{Q}}}\\left[z_{q}\\right] \\\\\n& \\quad\\quad~ = \\frac{1}{|\\mathcal{B}|} \\sum_{b=1}^{|\\mathcal{B}|} {z}_{b}-\\frac{1}{|\\mathcal{Q}|} \\sum_{q=1}^{|\\mathcal{Q}|} {z}_{q}.\n\\end{split}\n\\label{QFR}\n\\end{equation}\nWe then add the bias $\\Delta_{cross}$ to query features. To such a degree, support-query bias is somewhat eliminated. After that, a Poisson MBO model is adopted to infer the query label.\n\n\\subsubsection{The Poisson Merriman\u2013Bence\u2013Osher Model}\nWe denote the embedded feature set as $Z_{novel} = Z_b \\cup Z_q = \\{z_1, z_2, \\dots, z_m\\}$ ($m=K\\times C + N + V)$, where the first $K \\times C$ feature points belong to the labeled support set, the last $V$ feature points belong to the query set, and the remaining $N$ points denote the unlabeled support set. We build a graph with the feature points as the vertices, and the edge weight $w_{ij}$ is the similarity between feature \npoint $z_i$ and $z_j$, defined as $w_{i j}=\\exp \\left(-4\\left|z_{i}-z_{j}\\right|^{2} \/ d_{K}\\left(z_{i}\\right)^{2}\\right)$, where $d_{K}\\left(z_{i}\\right)^{2}$ is the distance between $z_i$ and its $K$-th nearest neighbor. We set $w_{ij} \\ge 0$ and $w_{ij} = w_{ji}$. Correspondingly, we define the weight matrix as $W=[w_{ij}]$, the degree matrix as $D=\\operatorname{diag}([d_i=\\sum_{j=1}^{m}w_{ij}])$, and the unnormalized Laplacian as $L = D - W$. \nAs the first $K\\times C$ feature points have the ground-truth label, we use $\\bar y = \\frac{1}{K \\times C} \\sum_{s=1}^{K\\times C} y_s $ to denote the average label vector, and we let indicator $\\mathbb{I}_{ij} = 1 $ if $i=j$, else $\\mathbb{I}_{ij} = 0 $. The goal of this model is to learn a classifier $g: z \\rightarrow \\mathbb{R}^{C}$.\nBy solving the Poisson equation:\n\\begin{equation}\n\\begin{split}\n& L  g\\left(z_{i}\\right)=\\sum_{j=1}^{K\\times C}\\left(y_{j}-\\bar{y}\\right) \\mathbb{I}_{ij} \\quad \\text { for } i=1, \\ldots, m, \n\\end{split}\n\\label{PO}\n\\end{equation}\nsatisfying $\\sum_{i=1}^{m} \\sum_{k=1}^{m}w_{ik} g\\left(z_{i}\\right)=0$, we can then result in the label prediction function $g(z_i)=(g_1(z_i),g_2(z_i),\\dots,g_C(z_i))$. The predict label $\\hat{y_i}$ of vertex $z_i$ is then determined as $\\hat{y_i} = {\\arg \\max_{j \\in\\{1, \\ldots, C\\}} }\\left\\{g_{j}(x_i)\\right\\}$. Let $G$ denote the set of $ m \\times  C $ matrix, which is the prediction label matrix of the all data. We concatenate the support label to form a label matrix $Y = [y_s] \\in \\mathbb{R}^{C \\times (\nK\\times C)} $. Let $A = [Y - \\bar y, \\mathbf{0}^{C \\times (m-K\\times C)}]$ denotes the initial label of all the data, in which all unlabeled data's label is zero.  The query label of Eq. (\\ref{PO}) can be determined by\n\\begin{equation}\n    G^{tp+1} = G^{tp} + D^{-1} ( A^T - LG^{tp}),\n    \\label{POSO}\n\\end{equation}\nwhere $G^{tp}$ denotes the predicted labels of all data at the timestamp $tp$.\nWe can get a stable classifier $g$ with a certain number of iteration using Eq. (\\ref{POSO}). After that, we adopt a graph-cut method to improve the inference performance by incrementally adjusting the classifier's decision boundary. The graph-cut problem is defined as\n\\begin{equation}\n\\min_{g: Z\\rightarrow H \\atop(g)_{z}=o}\\left\\{ g^T L g -\\mu \\sum_{i=1}^{K \\times  C}\\left(y_{i}-\\bar{y}\\right) \\cdot g\\left(z_{i}\\right)\\right\\},\n\\label{POMBO}\n\\end{equation}\nwhere $H = \\{ \\mathbf{e}_{1}, \\mathbf{e}_{2}, \\ldots, \\mathbf{e}_{C} \\}$ denotes the annotated samples' label set, $(g)_z = \\frac{1}{m}\\sum_{i=1}^m g(z_i)$ is the fraction of vertices to each of $C$ classes, and $o =[o_1,o_2,\\dots,o_C]^T \\in \\mathbb{R}^{C}$ is the piror knowledge of the class size distribution that $o_i$ is the fraction of data belonging to class $i$. With the constraint $(g)_z = o$, we can encode the prior \nknowledge into the Poisson Model.\n$g^T L g = \\frac{1}{2} \\sum_{i, j=1}^{m} w_{i j}(g(i)-g(j))^{2}$, this term is the graph-cut energy of the classification given by $g=[g(z_1),g(z_2),\\dots,g(z_m)]^T$, widely used in semi-supervised graph models~\\cite{zhu2003semi,zhu2005semi,zhou2004learning}.\n\nIn Eq. (\\ref{POMBO}), the solution will get discrete values, which is hard to solve.\nTo relax this problem, we use the Merriman-Bence-Osher (MBO) scheme~\\cite{garcia2014multiclass} by replacing the graph-cut energy with the Ginzburg-Landau approximation:\n\n\\begin{equation}\n\\begin{split}\n& \\min _{g\\in \\mathrm{SP}\\{Z\\rightarrow \\mathbb{R}^C\\} \\atop(g)_{z}=o}\\left\\{ \\mathrm{GL}_{\\tau'} (g) -\\mu \\sum_{i=1}^{K \\times  C}\\left(y_{i}-\\bar{y}\\right) \\cdot g\\left(z_{i}\\right)\\right\\},\\\\\n& \\mathrm{GL}_{\\tau'}(g)= g^T L g  +\\frac{1}{\\tau'} \\sum_{i=1}^{m} \\prod_{j=1}^{C}\\left|g\\left(z_{i}\\right)-\\mathbf{e}_{j}\\right|^{2}.\n\\end{split}\n\\label{GLPOMBO}\n\\end{equation}\n\nIn Eq. (\\ref{GLPOMBO}), $\\mathrm{SP}\\{Z\\rightarrow \\mathbb{R}^C\\}$ represents the space of projections $g: Z\\rightarrow \\mathbb{R}^C$, which allow the classifier $g$ to take on any real values, instead of the discrete value from $H$ in Eq. (\\ref{POMBO}). More importantly, this leads to a more efficiently computation of the Poisson model. The Eq. (\\ref{GLPOMBO}) can be efficiently solved with alternates \ngradient decent strategy, as shown in lines 9-20 of Algorithm \\ref{algorithm}. \n\n\\begin{algorithm}[t]\n\\caption{PTN for SSFSL}\\label{algorithm}\n\\SetKwData{Left}{left}\\SetKwData{This}{this}\\SetKwData{Up}{up}\n  \\SetKwFunction{Union}{Union}\\SetKwFunction{FindCompress}{FindCompress}\n  \\SetKwInOut{Input}{Input}\\SetKwInOut{Output}{Output}\n  \\SetKwProg{PoissonMBO}{$PoissonMBO$}{}{$G\\leftarrow G[m-V:m,:]$;}\n  \\Input{$\\mathcal{D}_{base}$, $\\mathcal{D}_{novel}=S\\cup U \\cup Q$,\\\\ $o$, $\\mu$, $M_{1}, M_{2}, M_{3}$}\n  \\Output{Query samples' label prediction $G$}\n\nTrain a base model $\\mathbf{W}_{\\phi} \\circ f_{\\theta_0} (x)$ with all samples and labels from $\\mathcal{D}_{base}$;\n\nApply unsupervised embedding transfer method to fine-tune the $f_{\\theta_0}$ with novel unlabeled data $U$ by using $\\ell_{UT}$ in Eq. (\\ref{UT}), and result in $f_{\\theta}$;\n\nApply $f_{\\theta}$ to extract features on $D_{novel}$ as $Z_{novel}$;\n\nApply query feature calibration using Eq. (\\ref{QFR});\n\nCompute $W, D, L, A$ according to $Z_{novel}$, $G \\leftarrow  \\mathbf{0}^{m \\times C}$\n\n\\PoissonMBO{}{\n\nUpdate $G$ uisng Eq. (\\ref{POSO}) with given steps\n\n$\\mathrm{d}_{mx} \\leftarrow 1 \/ \\max _{1 \\leq i \\leq m} {D}_{i i}$, $G \\leftarrow \\mu G$\n\n\\For{ $i=1$ \\KwTo $M_{1} $}{\n\\For{ $j=1$ \\KwTo $M_{2} $}{${G} \\leftarrow {G}-\\mathrm{d}_{mx}\\left({L} {G}-\\mu {A}^{T}\\right)$}\n$r \\leftarrow \\textbf{ones}(1,C)$\\\\\n\\For{ $j=1$ \\KwTo $M_3$}{$\\hat{o} \\leftarrow \\frac{1}{n} \\mathbf{1}^{T} \\mathbf{P r o j}_{H}({G} \\cdot \\operatorname{diag}({r}))$\\\\\n${r} \\leftarrow \\max \\left(\\min \\left({r}+{\\varphi} \\cdot ({o}-\\hat{o}), \\upsilon_{\\alpha}\\right), \\upsilon_{\\sigma }\\right)$\n}\n$G \\leftarrow \\mathbf{P r o j}_{H}({G} \\cdot \\operatorname{diag}({r}))$ }\n}\n\\end{algorithm}\n\\subsection{Proposed Algorithm}\nThe overall proposed algorithm is summarized in Algorithm \\ref{algorithm}. Inputting the base-class set $\\mathcal{D}_{base}$, novel-class set $\\mathcal{D}_{novel}$, prior classes' distribution $o$, and other parameters, PTN will predict the query samples' label $G \\in \\mathbb{R}^{V \\times C}$. The query label $\\hat{y}_q$ is then determined as $\\hat{y}_q = \\arg \\max _{1 \\leq j \\leq C} G_{qj}$. \nMore specifically, once the encoder $f_\\theta$ is learned using the base set $\\mathcal{D}_{base}$, we employ the proposed unsupervised embedding transfer method in step 2 in Algorithm \\ref{algorithm}. After that, we build the graph with the feature set $Z_{novel}$ and compute the related matrices $W, D, L, A$ in step 3-5. In the label inference stage in steps 6-20, we first apply Poisson model to robust propagate the labels in step 7, and then solve the graph-cut problem by using MBO scheme in several steps of gradient-descent to boost the classification performance. The stop condition in step 7 follow the constraint: $\\left\\|\\mathbf{sp}_{tp}-{W} \\mathbf{1} \/\\left(\\mathbf{1}^{T} {W} \\mathbf{1}\\right)\\right\\|_{\\infty} \\leq 1 \/ m$, where $\\mathbf{1}$ is a all-ones \ncolumn vector, $ \\mathbf{sp}_{tp}={W} {D}^{-1} \\mathbf{sp}_{tp-1}$, $\\mathbf{sp_{0}}$ is a $m$-column vector with ones in the first $K\\times C$ positions and zeros elsewhere. \nSteps 9-19 are aimed to solve the graph-cut problem in Eq. (\\ref{GLPOMBO}), \nTo solve the problem, we first divide the Eq. (\\ref{GLPOMBO}) into $E_1 = g^TLg-\\mu \\sum_{i=1}^{K \\times  C}\\left(y_{i}-\\bar{y}\\right) \\cdot g\\left(z_{i}\\right)$ and $E_2=\\frac{1}{\\tau'} \\sum_{i=1}^{m} \\prod_{j=1}^{C}\\left|g\\left(z_{i}\\right)-\\mathbf{e}_{j}\\right|^{2}$, and then \nemploying the gradient decent alternative on these two energy functions. \nSteps 10-12 are used to optimize the $E_1$.\nWe optimize the $E_2$ in steps 14-17, $\\mathbf{Proj}_{H}: \\mathbb{R}^{C} \\rightarrow H$ is the closet point projection, $r =[r_1,\\dots,r_C]^T$ ($r_i > 0$), ${\\varphi}$ is the time step, and $\\upsilon_{\\alpha}, \\upsilon_{\\sigma }$ are the clipping values, \nBy adopting the gradient descent scheme in steps 14-17, the vector $r$ is generated that also satisfies the constraint $(g)_z = o$ in Eq.(\\ref{GLPOMBO}). After obtaining the PoissonMBO's solution $G$, the query samples' label prediction matrix is resolved by step 20.  \n\nThe main inference complexity of PTN is $\\mathcal{O}(M_1 M_2 E)$\n, where $E$ is the number of edges in the graph. As a graphed-based model, PTN's inference complexity is heavier than inductive models. However, previous studies \\cite{liu2018learning,calder2020poisson} indicate that this complexity is affordable for few-shot tasks since the data scale is not very big. Moreover, we do not claim that our model is the final solution for SSFSL. We aim to design a new method to make full use of the extra unlabeled information. We report inference time comparison experiments in Table~\\ref{time}. The average inference time of PTN is 13.68s.\n\n\n\n\\section{Related Work}\n\\subsection{Few-Shot Learning} \nAs a representative of the learning methods with limited samples, \\textit{e.g.,} weakly supervised learning~\\cite{lan2017robust,zhang2018adversarial}, semi-supervised learning~\\cite{zhu2003semi,calder2019properly}, \nfew-shot learning can be roughly grouped into two categories: meta-learning models and transfer-learning models. Meta-learning models adopt the episode training mechanism~\\cite{vinyals2016matching}, of which metric-based models optimize the transferable embedding of both auxiliary and target data, and queries are identified according to the embedding distances~\\cite{sung2018learning,li2019distribution,Simon_2020_CVPR,zhang2020sgone}. Meanwhile, meta-optimization models~\\cite{finn2017model,rusu2018meta} target at designing optimization-centered algorithms to adapt the knowledge from meta-training to meta-testing. \nInstead of separating base classes into a set of few-shot tasks, transfer-learning methods~\\cite{qiao2018few,gidaris2018dynamic,chen2018closer,qi2018low} utilize all base classes to pre-train the few-shot model, which is then adapted to novel-class recognition. \nMost recently, Tian \\textit{et al.} \\cite{tian2020rethinking} decouple the learning procedure into the base-class embedding pre-training and novel-class classifier learning. By adopting multivariate logistic regression and knowledge distillation, the proposed model outperforms the meta-learning approaches. \nOur proposed method is inspired by the transfer-learning framework, where we adapt this framework to the semi-supervised few-shot learning by exploring both unlabeled novel-class data and base-class data to boost the performance of few-shot tasks.\n\n\\subsection{Semi-Supervised Few-shot Learning (SSFSL)}\nSSFSL aims to leverage the extra unlabeled novel-class data to improve the few-shot learning. \\citeauthor{ren2018meta} \\cite{ren2018meta} propose a meta-learning based framework by extending the prototypical network \\cite{snell2017prototypical} with unlabeled data to refine class prototypes. LST \\cite{li2019learning} re-trains the base model using the unlabeled data with generated pseudo labels. During the evaluation, it dynamically adds the unlabeled sample with high prediction confidence into testing. In \\cite{yu2020transmatch}, TransMatch proposes to initialize the novel-class classifier with the pre-trained feature imprinting, and then employs MixMatch \\cite{berthelot2019mixmatch} to fine-tune the whole model with both labeled and unlabeled data.\nAs closely related research to SSFSL, the transductive few-shot approaches~\\cite{liu2018learning,kim2019edge,ziko2020laplacian} also attempt to utilize unlabeled data to improve the performance of the few-shot learning. These methods adopt the entire query set as the unlabeled data and perform inference on all query samples together. For instance, TPN \\cite{liu2018learning} employs graph-based transductive inference to address the few-shot problem, and a semi-supervised extension model is also presented in their work.\n\nUnlike the above approaches, in this paper, we adopt the transfer-learning framework and propose to fully explore the extra unlabeled information in both classifier learning and embedding learning with different learning strategies. \n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\linewidth]{.\/images\/flowchart.pdf}\n\\end{center}\n   \\caption{The overview of the proposed PTN. We first pre-train a feature embedding $f_{\\theta_0}$ from the base-class set using standard cross-entropy loss. This embedding is then fine-tuned with the external novel-class unlabeled data by adopting unsupervised transferring loss $\\ell_{UT}$ to generate $f_{\\theta}$. Finally, we revise a graph model named PoissonMBO to conduct the query label inference.}\n\\label{framework}\n\\end{figure*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nDynamics of liquid drops is familiar in daily life: we observe rain drops\nrolling on a new umbrella, honey dripping off from a spoon, and oil droplets\nfloating on the surface of vegetable soup and so on. Such everyday phenomena\nare in fact important not only in physical sciences\n\\cite{RichardClanetQuere2002,DoshiCohenZhangSiegelHowellBasaranNagel2003,CouderProtiereFortBoudaoud2005,RistenpartBirdBelmonteDollarStone2009,PhysRevLett.79.1265,BirdDeCourbinStone2010,EtienneWedge2014}\nbut also in a variety of practical issues such as ink-jet printing\n\\cite{Calvert2001}, microfluidics manipulations\n\\cite{SquiresQuake2005,MathileTabeling2014}, and emulsification, formation of\nspray and foams \\cite{DynamicsDroplets,PhysicsFoams,Sylvie}. From such\nphenomena familiar to everybody, researchers have successfully extracted a\nnumber of scaling laws representing the essential physics \\cite{CapilaryText},\nwhich include scaling laws associated with the lifetime of a bubble in viscous\nliquid \\cite{DebregeasGennesBrochard-Wyart1998,EriOkumura2007} and contact\ndynamics of a drop to another drop\n\\cite{AartsLekkerkerkerGuoWegdamBonn2005,YokotaPNAS2011} or to a solid plate\n\\cite{BirdMandreStone2008,DavidCoalPRE}. Here, we report on a crossover of two\nscaling regimes experimentally revealed for viscous friction acting on a fluid\ndrop in a confined space. In particular, we study the descending motion (due\nto gravity) of an oil droplet surrounded by another immiscible oil in a\nHele-Shaw cell. The friction law thus revealed is nonlinear and replaces the\nwell-known Stokes' law in the Hele-Shaw cell geometry.\n\nA closely related topic of the rising bubble in a Hele-Shaw cell is\ntheoretically discussed by Taylor and Saffman in a pioneering paper\n\\cite{TAYLORSAFFMAN1959} in 1958 (earlier than the Bretherton's paper on\nbubbles in tubes \\cite{Bretherton,Clanet2004}). The solution of Taylor and\nSaffman was further discussed by Tanveer \\cite{Tanveer1986}. There are many\nother theoretical works on fluid drops in the Hele-Shaw cell geometry, notably\nin the context of the topological transition associated with droplet breakup\n\\cite{Eggers1997,ConstantinDupontGoldsteinKadanoffShelleyZhou1993,GoldsteinPesciShelley1995,Howell1999}%\n. \\ As for experimental studies, a number of researchers have investigated the\nrising motion of a bubble in a Hele-Shaw cell\n\\cite{Maxworthy1986,Kopf-SillHomsy1988,MaruvadaPark1996}. However, unlike the\npresent study, systematic and quantitative studies in a constant velocity\nregime have mostly concerned with the case in which there is a forced flow in\nthe outer fluid phase and most of the studies have been performed with the\ncell strongly inclined nearly to a horizontal position (one of a few examples\nof the case with the cell set in the upright position but with external flow\n\\cite{HeleShawPetroleum2010} demonstrates relevance of the present work to\nimportant problems in petroleum industry, such as the suction of crude oil\nfrom the well).\\ \n\nOne of the features of the present study compared with most of previous ones\non the dynamics of fluid drops in a Hele-Shaw cell is that in the present case\nthe existence of a thin liquid film surrounding a fluid drop plays a crucial\nrole: In many previous works, the existence of such thin films is not\nconsidered. In this respect, the present problem is closely related to the\ndynamics governed by thin film dissipation such as the imbibition of textured\nsurfaces\n\\cite{StoneNM2007,IshinoReyssatEPL2007,ObaraPRER2012,TaniPlosOne2014,TaniSR2015,DominicVellaImbibition2016}%\n. In this sense, our problem is quasi two-dimensional, although the geometry\nof the Hele-Shaw cell is often associated with a purely two-dimensional problem.\n\n\\section*{Experiment}\n\nWe fabricated a Hele-Shaw cell of thickness $D$\n\\cite{EriOkumura2007,EriOkumura2010,YokotaPNAS2011} and filled the cell with\nolive oil (150-00276, Wako; kinematic viscosity $\\nu_{ex}=60$ cS and density\n$\\rho_{ex}=910$ kg\/m$^{3}$). This oil plays a role of an external surrounding\nliquid for a drop of poly(dimethylsiloxane) (PDMS) to be inserted at the top\nof the cell using a syringe (SS-01T, Termo). We observe the inserted drop\ngoing down in the cell, as illustrated in Fig. \\ref{Fig1}(a), because of the\ndensity difference $\\Delta\\rho=\\rho_{in}-\\rho_{ex}>0$. The drop density\n$\\rho_{in}$ depends on its kinematic viscosity $\\nu_{in}$ only slightly (see\nthe details for Methods). The drop size is characterized by the cell thickness\n$D$ and the width $R_{T}$, i.e., the size in the direction transverse to that\nof gravity (see Fig. \\ref{Fig1}(b)), which is slightly smaller than the size\nin the longitudinal direction, $R_{L}$. As shown in Fig. \\ref{Fig1}(c), a thin\nfilm of olive oil exists between a cell plate and the surface of the drop. We\ncan think of two limiting cases for the distribution of liquid flow: (1)\nInternal Regime: The velocity gradient is predominantly created in the\ninternal side of the droplet as in the left illustration. (2) External Regime:\nThe gradient is predominantly exists in the external side of the droplet as in\nthe right.\n\nThe width and height of the cell are 10 cm and 40 cm, respectively, and are\nmuch larger than the drop size to remove any finite size effects in the\ndirection of width and height. The cell is made of acrylic plates of thickness\n5 mm, to avoid thinning deformation of the cell due to the effect of capillary\nadhesion \\cite{CapilaryText}.\n\nWe took snapshots of the descending drop at a regular time interval using a\ndigital camera (Lumix DMC-G3, Panasonic) and a camera controller (PS1,\nEtsumi). The obtained data were analyzed with the software, Image J, to obtain\nthe position as a function of time to determine the descending velocity of the\ndrop. Some examples are shown in Fig. \\ref{Fig1}(d). This plot show the\nfollowing facts. (1) The descending motion can be characterized by a\nwell-defined constant velocity (to guarantee a long stationary regime, the\ncell height is made significantly larger (40 cm) than the drop size; because\nof a small density difference, the constant-velocity regime starts after a\nlong transient regime). (2) The descending velocity is dependent on the\nkinematic viscosity of the internal liquid of the drop $\\nu_{in}$ for the\nthinner cell ($D=0.7$ mm) as predicted in the previous study\n\\cite{EriSoftMat2011}, which is not the case for the thicker cell ($D=1.5$\nmm); These examples clearly demonstrate the existence of a novel scaling\nregime different from the one discussed in the previous study\n\\cite{EriSoftMat2011}.\n\nIn the present study, the dependence of the descending velocity on the drop\nsize is negligible. In the previous study \\cite{EriSoftMat2011}, it was found\nthat the descending speed of drops is dependent on $R_{T}$ for $R_{T}\/D<10$ if\na glycerol drop goes down in PDMS oil. However, in the present combination\n(i.e., a PDMS drop going down in olive oil), we do not observe a significant\ndependence on $R_{T}$ in our data even for fairly small drops, whereas $R_{T}$\nis in the range $1.31<R_{T}\/D<15.8$ in the present study (the size dependence\nin the previous study may be caused by the polarity of the glycerol aqueous\nsolution: We expect that if the liquid is polar, the drop may subject to\nelectrostatic (attractive) force from the acrylic cell plates and this effect\ntends to make the drop less mobile).\\ The data analysis below neglects any\npossible small dependences of the velocity on the drop size.\n\n\\section*{Theory}\n\nAt the level of scaling laws, the characteristic energy scales are given as\nfollows. The gravitational energy gain for the descending drop per unit time\nis expressed as%\n\n\\begin{equation}\n\\mathcal{\\dot{E}}=\\alpha\\Delta\\rho gR_{T}R_{L}DV, \\label{eq3}%\n\\end{equation}\nwhere $\\alpha$ is a numerical coefficient. The viscous dissipation per unit\ntime in the internal regime discussed above (Fig. \\ref{Fig1}(c) left) is\nwritten as%\n\n\\begin{equation}\nT\\dot{S}_{in}=(\\alpha\/k_{in})\\eta_{in}(V\/D)^{2}R_{T}R_{L}D, \\label{eq1}%\n\\end{equation}\nwhere $\\alpha\/k_{in}$ is a numerical coefficient. Strictly speaking, because\nof the existence of the thin film of thickness $h$ (Fig. \\ref{Fig1}(c)), the\nvelocity gradient $V\/D$ in the above expression should be replaced with\n$V\/(D-2h)$, which is not essential, however, because the relation $D\\gg h$ is\nwell satisfied in the present study (see the next section). The viscous\ndissipation per unit time in the external regime discussed above (Fig.\n\\ref{Fig1}(c) right) is given as%\n\n\\begin{equation}\nT\\dot{S}_{ex}=(\\alpha\/k_{ex})\\eta_{ex}(V\/h)^{2}R_{T}R_{L}h, \\label{eq2}%\n\\end{equation}\nwhere $\\alpha\/k_{ex}$ is a numerical coefficient.\n\nIn the internal regime the velocity is given by the balance between $T\\dot\n{S}_{in}$ and $\\mathcal{\\dot{E}}$,%\n\n\\begin{equation}\nV_{in}=k_{in}\\Delta\\rho gD^{2}\/\\eta_{in}, \\label{eq4}%\n\\end{equation}\nwhereas in the external regime the velocity is given by the balance between\n$T\\dot{S}_{ex}$ and $\\mathcal{\\dot{E}}$,%\n\n\\begin{equation}\nV_{ex}=k_{ex}\\Delta\\rho gDh\/\\eta_{ex}. \\label{eq5}%\n\\end{equation}\n\n\nThe thickness of the thin film formed between the drop and cell plates may be\ngiven by the law of Landau, Levich and Derjaguin (LLD),%\n\n\\begin{equation}\nh=k_{LL}Ca^{2\/3}\\kappa^{-1}, \\label{eq6}%\n\\end{equation}\nwhere the numerical coefficient is of the order of unity \\cite{CapilaryText}\n($k_{LL}=0.94$, in the original papers \\cite{LandauLevich,Derjaguin1943}).\nHere the capillary length $\\kappa^{-1}$ is defined as $\\kappa^{-1}%\n=\\sqrt{\\gamma\/(\\Delta\\rho g)}$, which is smaller than the cell thickness $D$\n(If otherwise, the length $\\kappa^{-1}$ is replaced with $D$ with the\ncoefficient $k_{LL}=1.337$ \\cite{ParkHomsyHeleShaw1984}). The capillary number\n$Ca$ is defined as\n\\begin{equation}\nCa=\\eta_{ex}V_{ex}\/\\gamma\\label{eq7}%\n\\end{equation}\n\n\nRemoving $h$ from Eqs. (\\ref{eq5}) and (\\ref{eq6}), we obtain another\nexpression for the velocity in the external regime,%\n\\begin{equation}\n\\eta_{ex}V_{ex}\/\\gamma=(k_{1}D\/\\kappa^{-1})^{3} \\label{eq8}%\n\\end{equation}\nwith $k_{1}=k_{ex}k_{LL}$. Removing $V_{ex}$ from Eqs. (\\ref{eq5}) and\n(\\ref{eq6}), we obtain an expression for the thickness of the thin film,%\n\\begin{equation}\nh\/\\kappa^{-1}=(k_{2}D\/\\kappa^{-1})^{2} \\label{eq9}%\n\\end{equation}\nwith $k_{2}=k_{LL}^{3\/2}k_{ex}=k_{LL}^{1\/2}k_{1}$.\n\nThe condition for the internal regime is given by $T\\dot{S}_{in}<T\\dot{S}%\n_{ex}$, which leads to the equation, $\\eta_{ex}\/\\eta_{in}>k_{3}D\/\\kappa^{-1}$.\nIn other words, the phase boundary between the internal and external regimes\nis given by\n\\begin{equation}\n\\eta_{ex}\/\\eta_{in}=k_{3}D\/\\kappa^{-1}\\label{eq10}%\n\\end{equation}\nwith $k_{3}=k_{1}^{3}\/k_{in}$. This means that the phase boundary between the\ninternal and external regime is a straight line with the slope $k_{3}$ on the\nplot of $\\eta_{ex}\/\\eta_{in}$ as a function of $D\/\\kappa^{-1}$.\n\n\\section*{Experiment and theory}\n\nThe experimental data for the descending velocity of drops $V$ are plotted as\na function of $\\Delta\\rho gD^{2}\/\\eta_{in}$ in Fig. \\ref{Fig2}(a). In view of\nEq. (\\ref{eq4}), the data points in the internal regime would be on a straight\nline of slope 1. This is almost true: there is a series of data well on the\ndashed line of slope close to one. Naturally, there is a slight deviation from\nthe theory: the slope of the straight dashed line obtained by a numerical\nfitting is in fact $1.24\\pm0.06$, a value slightly larger than one, but the\ncoefficient corresponding $k_{in}$ is $0.150\\pm0.015$, the order of magnitude\nof which is consistent with the scaling arguments.\n\nSome detailed remarks for the above arguments are as follows. (1) Even in the\nprevious study \\cite{EriSoftMat2011} in which the internal scaling regime was\nconfirmed for the first time, the scaling regime described by Eq. (\\ref{eq4})\nwas shown with some deviations, similarly to the present case (whereas another\nscaling regime first established in the previous paper \\cite{EriSoftMat2011}\nis almost perfectly demonstrated). (2) We note here that the data represented\nby the red filled circle and red filled inverse triangle are exceptional ones\nand their seemingly strange behavior will be explained in Discussion. (3) We\nhave confirmed that even if we replace $D$ with $D-2h$ in the analysis (by\nusing the thickness $h$ estimated from Eq. (\\ref{eq6})) when $D$ is used as a\nlength scale characterizing the viscous gradient  (i.e., when $D$ is used in\nthe expression $V\/D$ in Eq. (2)), any visible differences are not introduced\ninto the plots given in Fig. \\ref{Fig2} (This correction could be motivated by\nconsidering the existence of thin films surrounding the drops as in Fig.\n\\ref{Fig1}(c) as mentioned above).\n\nIn Fig. \\ref{Fig2}(b), it is shown that some of the data we obtained clearly\nsatisfy Eq. (\\ref{eq8}), which describes the external regime. In Fig.\n\\ref{Fig2}(b), we collected the data points that are off the dashed line of\nslope close to one in Fig. \\ref{Fig2}(a) and that are thus ruled out from the\ninternal regime. The data thus selected and plotted in Fig. \\ref{Fig2}(b) are\nalmost on the straight line of slope 3 in accordance with Eq. (\\ref{eq8}). The\nstraight line is obtained by a numerical fitting with fixing the slope to 3.0;\nas a result of this fitting, the coefficient is given as $k_{1}=0.167\\pm\n0.003$, the order of magnitude of which is consistent with the scaling arguments.\n\nWe confirm this scaling law in Eq. (\\ref{eq8}) also in Fig. \\ref{Fig2}(a). In\nthe light of Eq. (\\ref{eq8}), the data in the external regime for a given $D$\nshould take almost the same values, because $\\eta_{ex}$ and $\\gamma$ are both\nconstant and $\\kappa^{-1}$ is almost constant (note that $\\Delta\\rho$ is\nalmost constant) in the present study. In fact, in Fig. \\ref{Fig2}(a), the\ndata points for a fixed $D$ that are off the dashed line, which data are shown\nto be in the external regime in Fig. \\ref{Fig2}(b), take almost a constant\nvalue, that is, they are located almost on a horizontal line. This fact also\nconfirms that the data in question are independent of $\\eta_{in}$, that is,\nthey are certainly not in the internal regime. Strictly speaking, the data\nlabeled as a given $D$ can have slightly different measured values of $D$ (see\nMethods), which is the main reason the data for a \"given\" $D$ that are off the\ndashed line in Fig. \\ref{Fig2}(a) slightly deviate from the straight\nhorizontal line corresponding the $D$ value.\n\nThe scaling law in Eq. (\\ref{eq8}) can be confirmed in Fig. \\ref{Fig2}(a) in a\nstill another way. The open marks of the same shape, say diamond, but with\ndifferent colors (that are the data for a given $\\nu_{in}$ but with different\n$D$) are almost on a straight line of a slope close to one (This slope may\nseem to be slightly larger than one, which may be because of the uncertainty\non the cell spacing $D$ as already mentioned\\ in the last sentence of the\nparagraph just above this one, or because the exponent 3 in Eq. (\\ref{eq8})\nmay be in fact slightly larger than 3 in a more complete theory beyond the\npresent arguments at the level of scaling laws). For a such series of data,\nthe velocity $V$ scales with $D^{3}$ according to Eq. (\\ref{eq8}), thus when\nplotted as a function of $D^{2}$ as in Fig. \\ref{Fig2}(a), the quantity\nlinearly scales with $D$, as \\ reasonably well confirmed.\n\nThe phase diagram based on Eq. (\\ref{eq10}) is shown in Fig. \\ref{Fig2}(c), in\nwhich\\ we plot all the data (except for the special data mentioned above), to\ndemonstrate further consistency of the present arguments. As expected from Eq.\n(\\ref{eq10}), we can indeed draw a straight line of slope 1 on Fig.\n\\ref{Fig2}(c), which divides the internal and external regimes; Above the\nstraight line of slope 1 in Fig. \\ref{Fig2}(c) lie the data in the internal\nregime described by Eq. (\\ref{eq4}), i.e., the data on the straight dashed\nline in Fig. \\ref{Fig2}(a); Below the straight line in Fig. \\ref{Fig2}(c) lie\nthe data in the external regime described by Eq. (\\ref{eq8}), i.e., the data\non the straight line in Fig. \\ref{Fig2}(b). The coefficient $k_{3}$ of Eq.\n(\\ref{eq10}), i.e., the line dividing two regimes shown in Fig. \\ref{Fig2}(c),\nis $k_{3}=0.017$,\\ the order of magnitude of which is consistent with the\nscaling arguments in a profound\\ sense: The numerical coefficient, $k_{in}$,\n$k_{1}$, and $k_{3}$, are predicted to satisfy the relation $k_{3}=k_{1}%\n^{3}\/k_{in}$, and this relation is satisfied at a quantitative level in the\npresent analysis (0.017 vs $(0.167)^{3}\/0.15\\simeq0.031$). This quantitative\nagreement is indeed quite satisfactory, if we consider slight deviations of\nthe data from the predicted theory. For example, the value $0.15$ used in the\nestimation in the parentheses is not the value of $k_{in}$ itself (the precise\ndefinition of $k_{in}$ is the coefficient appearing in Eq. (\\ref{eq4}),\n$V_{in}=k_{in}\\Delta\\rho gD^{2}\/\\eta_{in}$, but the value of $k_{in}$, 0.15,\nused in the above is in fact the value of the coefficient $k_{in}^{\\prime}$\nappearing in the relation $V_{in}=(k_{in}^{\\prime}\\Delta\\rho gD^{2}\/\\eta\n_{in})^{\\alpha}$ obtained when the data corresponding to the internal regime\nin Fig. \\ref{Fig2}(a) are numerically fitted by this relation with $\\alpha$\ndetermined to be not equal to one but close to 1.24, as mentioned in the first\nparagraph in Experiment and Theory).\\ In addition, the exponent in (\\ref{eq8})\nmight also be slightly deviated from 3 as suggested in the paragraph just\nabove this one.\n\nThe crossover from the internal to external regime can explicitly be seen in\nthe data for $D=1.0$ mm (red data) in Fig. \\ref{Fig2}(a). As $\\eta_{in}$\ndecreases from the left-most data for $\\nu_{in}=30000$ cS (red open diamonds)\nto the data for $\\nu_{in}=5000$ cS (red open inverse triangle), the velocity\nis independent of $\\nu_{in}$, which reveals that the three data on the\nhorizontal line are in the external regime. However, the data for $\\nu\n_{in}=1000$ cS and $\\nu_{in}=500$ cS are on the straight dashed line with a\nslope close to one, which confirms that these two data are in the internal\nregime. Since the phase boundary expressed by Eq. (\\ref{eq10}) is obtained\nalso by equating $V_{in}$ and $V_{ex}$ in Eqs. (\\ref{eq4}) and (\\ref{eq8}),\nthe crossover between the two regimes occurs in Fig. \\ref{Fig2}(a) near at the\ncross point between the horizontal line connecting the data on the external\nregime for a given $D$ and the straight dashed line of a slope close to one\nrepresenting the internal regime.\n\nThe behavior of the data close to the crossover points are quite intriguing.\nThe data for $D=2.0$ mm and 3.0 mm at $\\nu_{in}=1000$ cS (green filled square\nand purple filled square) are located at the position close to the phase\nboundary in Fig. \\ref{Fig2}(c) (and the data have already been confirmed to be\nin the internal regime in Fig. \\ref{Fig2}(a): in this plot, these data points\nare reasonably well on the dashed line).\\ We have confirmed that, when these\ntwo data are plotted in Fig. \\ref{Fig2}(b), they are nearly on the straight\nline of slope close to 3 in Fig. \\ref{Fig2}(b). The two points can be\ndescribed by both Eqs. (\\ref{eq4}) and (\\ref{eq8}), which is reasonable\nbecause they are nearly on the phase boundary. However, this is not always the\ncase. The data for $D=0.7$ mm and $\\nu_{in}=5000$ cS (black filled inverse\ntriangle) and for $D=1.5$ mm and $\\nu_{in}=3000$ cS (blue open triangle) are\nalso positioned close to the phase boundary in Fig. \\ref{Fig2}(c). However,\nthe former is rather in the internal regime and the latter rather in the\nexternal regime. This is in a sense logical because the blue open triangle is\nrather away from the crossover point for $D=1.5$ mm in Fig. \\ref{Fig2}(a) but\nthis is not the case for black filled inverse triangle. In general, how\nquickly the crossover occurs seems to be a subtle problem.\n\n\\section*{Discussion}\n\nThe direct measurement of the thickness $h$ supports the above analysis. We\nused a laser distance sensor\\ (ZS-HLDS2+ZS-HLDC11+Smart Monitor Zero Pro.,\nOmron), as illustrated in Fig. \\ref{Fig3}(a). The measurement is extremely\ndelicate and difficult, because we have six reflective planes I to VI with\nsignificantly different strengths of reflection where the target two\nreflections II and III are the smallest and the second smallest among them\n(see Fig. \\ref{Fig3}(b)). The six surfaces are the front and back surfaces of\nthe front cell plate (interface I and II), the front and back interfaces\nbetween olive oil and the PDMS drop (interface III and IV), and the front and\nback surfaces of the back cell plate (interface V and VI). To determine $h$,\nwe need to detect reflection from interface II and III, where the\nreflection\\ from II is small compared with that of III (see Fig.\n\\ref{Fig3}(b)) and significantly small compared with that of I, because the\nrefractive index of olive oil is $n_{olive}=1.47$, that of acrylic plate is\n$n_{acr}=1.491$, that of PDMS oil is $n_{PDMS}=1.403$ and that of air is\n$n_{air}=1$. Furthermore, the object (the descending drop) is moving. In spite\nof these experimental difficulties, we obtained a reasonably good correlation\nbetween the measured thickness and the experimentally\\ obtained value as shown\nin Fig. \\ref{Fig3}(c), by virtue of various efforts (for example, in the\nscreen shot Fig. \\ref{Fig3}(b), the two target peaks are intensionally\npositioned off-center because the precision of measurement becomes the maximum\nwhen the reflection angle is the largest). Here, the slope of the line\nobtained by a numerical fitting is $0.749\\pm0.027$ (the slope here is not the\nexponent but the coefficient for the linear relationship), the order of\nmagnitude of which is consistent with the scaling argument.\n\nExceptional data mentioned above reveal an intriguing phenomenon. In Fig.\n\\ref{Fig2}(a), the data for $D=1$ mm and for $\\nu_{in}=10000$ cS are\nrepresented by two different marks, the red filled circle and the red open\ncircle, with the former described by the internal regime and the latter by the\nexternal regime. The data for $D=1$ mm and for $\\nu_{in}=5000$ cS are also\ncategorized into two filled and open symbols. The experimental difference in\nacquiring these two different types (filled and open symbols) of data obtained\nfor identical drop viscosity and cell spacing\\ is that, when the drop goes\ndown on the same path multiple times in the same cell, the first drop is in\nthe external regime (open marks) whereas the drop going down after the first\none is always in the internal regime (filled marks). This apparently\nmysterious effect is quite reproducible and is understood by considering a\npossibility of mixing of olive oil and PDMS at the surface of the drops. For\nthe first drop, such a mixing effect is negligible and the drop is governed by\nthe dynamics of the external regime. However, after the first one, because of\nthe mixing effect, the viscosity of the thin film surrounding the drop\nincreases (because $\\nu_{in}\\gg\\nu_{ex}$)\\ so that making a velocity gradient\nin the external thin film is no longer favored in terms of energy and instead\nthe velocity gradient inside the drop is favored to realize the dynamics in\nthe internal regime. Because of this reason, the red filled circles\\ and the\nred filled inverse triangles\\ are not shown in the phase diagram given in Fig.\n\\ref{Fig2}(c). This seemingly mysterious behavior tends to be suppressed if\nthe viscosity is too small (because the \"external\" viscosity does not get\nsufficiently viscous), or too large (because the mixing is not sufficiently\neffective). This is why we observed this phenomenon only for the two values of viscosity.\n\nThe present study suggests that Stokes' drag friction $F=6\\pi\\eta_{ex}VR$ for\na solid sphere of radius $R$ surrounded by a viscous liquid of viscosity\n$\\eta_{ex}$ is replaced in the external regime of the Hele-Shaw cell geometry\nby\n\\begin{equation}\nF_{ex}\\simeq\\eta_{ex}VR_{T}R_{L}\/h\\simeq\\eta_{ex}Ca^{-2\/3}VR_{T}R_{L}%\n\/\\kappa^{-1}.\\label{eq11}%\n\\end{equation}\nThis expression possesses a nonlinear dependence on the velocity $V$ due to\nthe extra $V$ dependence contained in the capillary number $Ca$. This is\nstrikingly different from the two other expressions for the drag force:\n$F_{in}\\simeq\\eta_{in}VR_{T}R_{L}\/D$ and $F_{bubble}\\simeq\\eta_{ex}VR_{T}%\n^{2}\/D$, which are both linear in velocity; The former corresponds to the\ninternal regime in the present study, whereas the latter corresponds to the\ncase in which the dominant dissipation is the one associated with the velocity\ngradient $V\/D$ in the surrounding external liquid \\cite{EriSoftMat2011}. The\nviscous friction forces including the nonlinear friction in Eq. (\\ref{eq11})\nare relevant to the dynamics of emulsion, foam, antifoam and soft gels\n\\cite{Sylvie,AnnLaureSylvieSM2009,DominiqueMicrogravity2015}, in particular,\nnonlinear rheology of such systems\n\\cite{DenkovSoftMat2009,DurianPRL10,CloitreNP2011}.\n\nWe intentionally used several times the phrase, \\textquotedblleft the order of\nmagnitude of which is consistent with the scaling argument,\\textquotedblright%\n\\ which may be vague compared with an expression like, \\textquotedblleft being\nof order one further supports the scaling argument.\\textquotedblright\\ The\nreason we used the seemingly vague expression is that whether a coefficient\nfor a scaling law is of the order of one or not is in fact a subtle issue.\nDepending on the problem or on the definition of the coefficient, the orders\nof magnitude can be fairly larger or smaller than one. An example of such a\ncase can be given by exploiting the relation, $k_{3}=k_{1}^{3}\/k_{in}$, given\nabove: The three coefficients, $k_{1},k_{3}$, and $k_{in}$ are all\ncoefficients for some scaling laws so that, for example, $k_{1}$ and $k_{in}$\ncan be 5 and 1, respectively, but this example implies $k_{3}$ is much larger\nthan one ($k_{3}=5^{3}$).\\ \n\nIn the present study, the consistency of the whole scaling arguments is\nchecked in several ways, which clearly deepens our physical understanding. For\nexample, a new scaling regime is demonstrated through a clear data collapse\n(Fig. \\ref{Fig2}(b)), and the crossover of this regime to another is shown\n(Fig. \\ref{Fig2}(a)), which is completed by the phase diagram (Fig.\n\\ref{Fig2}(c)) and a separate measurements on thin-film thickness (Fig.\n\\ref{Fig3}(c)). In addition, data arrangements in the crossover diagram (Fig.\n\\ref{Fig2}(a)) are interpreted from various viewpoints, confirming the\nconsistency of the arguments.\n\n\\section*{Conclusion}\n\nIn summary, we show in Fig. \\ref{Fig2}(b) the existence of a novel scaling\nregime for the descending velocity of a drop surrounded by thin external fluid\nin the Hele-Shaw cell, in which regime the viscous dissipation in the thin\nfilm is essential. This regime corresponds to a nonlinear form of viscous drag\nfriction. In this regime, the thickness of the film is determined by the law\nof LLD, as directly confirmed in Fig. \\ref{Fig3}(c). The crossover between\nthis regime and another regime in which the viscous dissipation in the\ninternal side of the drop governs the dynamics is shown in Fig. \\ref{Fig2}(a).\nThe phase boundary between the two regimes are given in Fig. \\ref{Fig2}(c).\n\nThere are some other scaling regimes for the viscous drag friction in the\nHele-Shaw cell geometry with the existence of thin films surrounding a fluid\ndrop. For example, the dissipation associated with the velocity gradient $V\/D$\nin the internal drop liquid has been revealed to be important for a rising\nbubble in the Hele-Shaw cell \\cite{EriSoftMat2011}. The dissipation associated\nwith the dynamic meniscus (in the context of LLD theory\n\\cite{LandauLevich,Derjaguin1943,CapilaryText}) formed in the external\nthin-film has been found to be important in a non Hele-Shaw cell geometry\n\\cite{PascalEPL2002}. In addition, the present external regime will give\nanother scaling law if the capillary length $\\kappa^{-1}$ is, unlike in the\npresent study, larger than the cell thickness $D$.\n\nConfirmation of such other regimes for viscous drag friction in the Hele-Shaw\ncell geometry, as well as crossovers among various scaling regimes would be\nexplored in future studies. The simple friction laws for confined fluid drops\nand the crossover between them revealed in the present study (and in future\nstudies) are relevant to fundamental issues including rheology of foam and\nemulsion, as well as applications such as in microfluidics.\n\n\\section*{Methods}\n\nThe density of PDMS oil $\\rho_{in}$ slightly depends on viscosity: (1) 970\nkg\/m$^{3}$ for the kinematic viscosities $\\nu_{in}=500,1000,$ and $3000$ cS\n(SN-4, SN-5, and SN-6, As One). (2) 975 kg\/m$^{3}$ for $\\nu_{in}=5000$ and\n$10000$ cS (SN-7 and SN-8, As One). (3) 976 kg\/m$^{3}$ for $\\nu_{in}=30000$ cS\n(KF-96H, ShinEtsu).\n\nThe cell thickness $D$ is controlled by spaces, and is directly measured using\nthe laser distance sensor\\ (ZS-HLDS5, Omron) for most of the cells. In all the\nfigures of the present study, for simplicity, the cell thickness $D$ is\nrepresented by an approximate value, which is slightly different from measured\nvalues. For some of the data the measurement of $D$ was not performed and in\nsuch a case an approximate value of $D$ is used, instead of measured values,\nto plot the data points, which does not cause serious difficulties in\nanalyzing and interpreting the data. This is because the difference between\nthe $D$ value used for labeling and the measured value of the cell thickness\nis rather small.\n\nThe interfacial tension between PDMS and olive oil was measured by using\npendant drop tensiometry. It is recently discussed that measured values for\npendant drops are dependent on Bond number and Worthington number, with both\nscaling with $B=\\Delta\\rho gR_{0}^{2}\/\\gamma$ ($R_{0}$: the drop radius at the\napex of the pendant drop) when the drop size is of the same order of magnitude\nas the needle diameter, and that the measured value approach the correct value\nas $B$ approaches one \\cite{PendantDrop2015} (one could expect that the\nexperimental precision will be optimized when the drop is most \"swelled,\" that\nis, when the droplet is on the verge of detaching off from the needle tip due\nto gravity, that is, when $B=1$).\\ We measured the value of tension as a\nfunction of $B$ by using the software, OpenDrop, developed by Michael Neeson,\nJoe Berry and Rico Tabor. We extrapolated the data thus obtained to the value\nat $B=1$ to have a pragmatic\\ value, $\\gamma=0.78$ mN\/m, because it was\nexperimentally difficult to approach $B=1$. This is possibly because the\ntension is significantly small, which might lead to an extra error in the measurement.\n\nEven though the measurement of the interfacial tension contains an extra error\nand our analysis numerically depends on the measured value, this does not\nbring any uncertainties in the present arguments at the level of scaling laws.\nWe explain this by an example. Introducing the experimentally measured value\nof surface tension $\\gamma_{m}$, we define a numerical coefficient $\\beta$ as\n$\\gamma=\\beta^{2}\\gamma_{m}$ and the corresponding capillary length\n$\\kappa^{-1}=\\beta\\kappa_{m}^{-1}$. With these \"measured\" quantities, Eq.\n(\\ref{eq8}) can be expressed as $\\eta_{ex}V_{ex}\/\\gamma_{m}=k_{1,m}%\n^{3}(D\/\\kappa_{m}^{-1})^{3}$ with $k_{1,m}^{3}=k_{1}^{3}\/\\beta$. By noting\nthat the values of the interfacial tension and capillary number used in Fig.\n\\ref{Fig2}(c) that experimentally confirms the relation Eq. (\\ref{eq8}) are in\nfact not $\\gamma$ and $\\kappa^{-1}$ but $\\gamma_{m}$ and $\\kappa_{m}^{-1}$,\nrespectively, the coefficient we determined from Fig. \\ref{Fig2}(c) is in fact\nnot $k_{1}$ but $k_{1,m}$. However, since Eq. (\\ref{eq10}) can be expressed as\n$\\eta_{ex}\/\\eta_{in}=k_{3,m}D\/\\kappa_{m}^{-1}$ with $k_{3,m}=k_{3}\/\\beta$, the\nphase boundary line $\\eta_{ex}\/\\eta_{in}=k_{3,m}D\/\\kappa_{m}^{-1}$ on the\n$(\\eta_{ex}\/\\eta_{in},D\/\\kappa_{m}^{-1})$ space and the line $\\eta_{ex}%\n\/\\eta_{in}=k_{3}D\/\\kappa^{-1}$ on the $(\\eta_{ex}\/\\eta_{in},D\/\\kappa^{-1})$\nspace have the same physical meaning. From these reasons, a special care is\nneeded when one compares the numerical coefficient obtained experimentally in\nthe present study with more sophisticated experiments or calculations.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\subsection{\\@startsection{subsection}{1}%\n  \\z@{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}%\n  {\\normalfont\\bfseries\\centering}\n\\makeatother\n\n\\setcounter{tocdepth}{3\n\\let\\oldtocsection=\\tocsection\n\\let\\oldtocsubsection=\\tocsubsection\n\\let\\oldtocsubsubsection=\\tocsubsubsection\n\\renewcommand{\\tocsection}[2]{\\hspace{0em}\\oldtocsection{#1}{#2}}\n\\renewcommand{\\tocsubsection}[2]{\\hspace{1em}\\oldtocsubsection{#1}{#2}}\n\\renewcommand{\\tocsubsubsection}[2]{\\hspace{2em}\\oldtocsubsubsection{#1}{#2}}\n\n\n\\setlength{\\textwidth}{\\paperwidth}\n\\addtolength{\\textwidth}{-2.5in}\n\\calclayout\n\n\\begin{document}\n\\title[]{Adams-type maps are not stable under composition}\n\n\\author{Robert Burklund}\n\\address{Department of Mathematics, MIT, Cambridge, MA, USA}\n\\email{burklund@mit.edu}\n\n\\author{Ishan Levy}\n\\address{Department of Mathematics, MIT, Cambridge, MA, USA}\n\\email{ishanl@mit.edu}\n\n\\author{Piotr Pstr\\k{a}gowski}\n\\address{Department of Mathematics, Harvard, Cambridge, MA, USA}\n\\email{pstragowski.piotr@gmail.com}\n\n\\begin{abstract}\nWe give a simple counterexample to the plausible conjecture that Adams-type maps of ring spectra are stable under composition. We then show that over a field, this failure is quite extreme, as any map of $\\mathbb{E}_{\\infty}$-algebras is a transfinite composition of Adams-type maps. \n\\end{abstract}\n\n\\maketitle \n\n\n\n\nThe Adams spectral sequence is a fundamental tool in stable homotopy theory which, given a map $A \\to B$ of ring spectra, lets us compute the homotopy groups of $A$ in terms of information living over $B$. \nUnfortunately, the $E_{2}$-page of the Adams spectral sequence can be difficult to identify in general. \nThe standard additional assumption which makes the $E_2$-page computable in terms of homological algebra is that the map $A \\to B$ is flat in one of the senses below.\n\n\\begin{definition}\nWe say a map $A \\rightarrow B$ of $\\mathbb{E}_{1}$-rings is \n\\begin{enumerate}\n    \\item \\emph{Adams-type} if we can write $B \\simeq \\varinjlim B_{\\alpha}$ as a filtered colimit of perfect $A$-modules with the property that $\\pi_{*}(B \\otimes_{A} B_{\\alpha})$ is projective as a $\\pi_*B$-module, \n    \\item \\emph{descent-flat}\\footnote{In older literature, what we call descent-flat is often simply referred to as \\emph{flat}. We avoid the latter term as it is also often used to refer to maps such that $\\pi_*A \\rightarrow \\pi_*B$ is flat, a much stronger condition than the one we work with.} if $\\pi_{*}(B \\otimes_{A} B)$ is flat as a left $\\pi_*B$-module. \n\\end{enumerate}\n\\end{definition}\n\nIf $A \\rightarrow B$ is descent-flat, then for any pair $M$, $N$ of $A$-modules such that $\\pi_{*}(B \\otimes_{A} N)$ is projective $\\pi_*B$-module, the associated Adams spectral sequence computing homotopy classes of $A$-module maps from $M$ to $N$ has signature\n\\[\n\\Ext^{s, t}_{(\\pi_*B,\\, \\pi_*(B \\otimes_A B))} \\left(\\pi_{*}(B \\otimes_{A} N),\\, \\pi_{*}(B \\otimes_{A} M)\\right) \\Longrightarrow [N,\\, M]_{A}.\n\\]\nIf $A \\rightarrow B$ is furthermore Adams-type, then one can construct an Adams spectral sequence with the above signature with no projectivity assumption on $\\pi_{*}(B \\otimes_{A} N)$, as in the work of Devinatz \\cite[\\S 1]{dev_morava}.\n\nAs both of the above notions are a form of flatness, it is natural to expect that they are stable under composition, and the authors of this note have spent considerable amount of time trying to show that this is indeed the case. To our surprise, this is very far from being true; our first result is the following simple counterexample. \n\n\\begin{theorem}\n\\label{theorem:main_theorem}\nBoth of the maps $\\mathbb{S} \\rightarrow \\mathrm{MU}$ and $\\mathrm{MU} \\rightarrow \\mathbb{Z}$ are Adams-type, but the composite $\\mathbb{S} \\rightarrow \\mathbb{Z}$ is not even descent-flat. In particular, neither Adams-type nor descent-flat maps of $\\mathbb{E}_{\\infty}$-ring spectra are stable under composition.\n\\end{theorem}\n\n\\begin{proof}\nIt is well-known that $\\mathbb{S} \\rightarrow \\mathrm{MU}$ is Adams-type \\cite[13.4(iv)]{adams1995stable}. Similarly, $\\mathbb{Z}_{*} \\mathbb{Z}$ is known to contain torsion elements in positive degrees and so is not flat over $\\mathbb{Z}_{*}$ \\cite{cartan1955seminaire\n, hence $\\mathbb{S} \\rightarrow \\mathbb{Z}$ is not descent-flat. \n\nTo see that $\\mathrm{MU} \\to \\mathbb{Z}$ is Adams-type, note that if we write $\\mathrm{MU}_{*} \\simeq \\mathbb{Z}[b_{1}, b_{2}, \\ldots]$ for some choice of generators $b_{i}$, then \n\\[\n\\mathbb{Z} = \\colim_{n} \\ \\bigotimes^{\\mathrm{MU}}_{1 \\leq i \\leq n}\\mathrm{MU}\/b_i\n\\]\nin $\\mathrm{MU}$-modules. We claim this is the needed expression of $\\mathbb{Z}$ as a filtered colimit. To see this, note that we have an equivalence \n\\[\n\\mathbb{Z} \\otimes_{\\mathrm{MU}} \\mathrm{MU}\/b_{i} \\simeq \\mathbb{Z}\/b_{i} \\simeq \\mathbb{Z} \\oplus \\Sigma^{|b_{i}|+1} \\mathbb{Z}\n\\]\nas $b_{i}$ vanishes in $\\mathbb{Z}$ and hence for each $n$ we have \n\\[\n\\mathbb{Z} \\otimes_{\\mathrm{MU}} \\bigotimes^{\\mathrm{MU}}_{1 \\leq i \\leq n}\\mathrm{MU}\/b_i \\simeq \\bigotimes^{\\mathbb{Z}}_{1 \\leq i \\leq n} \\mathbb{Z} \\oplus \\Sigma^{|b_{i}|+1} \\mathbb{Z}\n\\]\nwhich is a perfect $\\mathbb{Z}$-module whose homotopy groups form a finitely generated free abelian group, as needed. \n\\end{proof}\n\n\n\n\nThe counterexample of \\cref{theorem:main_theorem} was contrary to our expectations, and led us to ask which maps of ring spectra can be written as compositions (perhaps infinite) of Adams-type maps. In \\cref{theorem:a_map_of_commutative_k_algebras_a_tf_composite_of_adams_type_maps} below we show that, surprisingly, at least for $\\mathbb{E}_{\\infty}$-algebras over a field, \\emph{any} map of ring spectra has this property.\n\n\n\n\n\\begin{lemma}\\label{lem:adamstypestability}\nLet $R \\to R'$ be a map of $\\mathbb{E}_{2}$-algebras. If $A$ is an $\\mathbb{E}_{1}$-$R$-algebra such that the unit $R \\rightarrow A$ is of Adams-type, then so is $R' \\rightarrow R' \\otimes_{R} A$. \n\\end{lemma}\n\n\\begin{proof}\n    We start with an observation about the Adams-type condition:\n    If $B \\to C$ is a map of $\\mathbb{E}_1$-algebras, and $M$ is a $B$-module, then \n    $\\pi_*(C \\otimes_B M)$ is projective as a $\\pi_*C$-module iff the $C$-module $C \\otimes_B M$ is a retract of a sum suspensions of copies of $C$. We call a $B$-module which satisfies this condition $C$-projective.\n    \n    Now we prove the lemma.\n    Write $A$ as a filtered colimit $A \\simeq \\colim A_{\\alpha}$ of $A$-projective $R$-modules.\n    Then $R' \\otimes_{A} R \\simeq \\colim R' \\otimes_{R} A_{\\alpha}$ is a filtered colimit with the required property, as \n    \\[ (R' \\otimes_R A) \\otimes_{R'} (R' \\otimes_R A_\\alpha) \\simeq R' \\otimes_R (A \\otimes_R A_\\alpha) \\]\n    which implies that $R' \\otimes_R A_\\alpha$ is $(R' \\otimes_R A)$-projective.\n\\end{proof}\n    \n   \n   \n   \n\n   \n   \n   \n   \n    \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \n   \n\n\\begin{corollary} \\label{cor:po-at}\nAdams-type maps of $\\mathbb{E}_{\\infty}$-ring spectra are stable under pushouts. \n\\end{corollary}\n\n\n\n\\begin{lemma}\n\\label{lemma:adams_type_if_source_or_target_is_a_field}\nLet $A \\rightarrow B$ be a map of $\\mathbb{E}_{1}$-rings such that either $\\pi_{*}A$ or $\\pi_{*}B$ is a graded skew-field (that is, every non-zero homogeneous element is invertible). Then $A \\rightarrow B$ is Adams-type. \n\\end{lemma}\n\n\\begin{proof}\nSuppose that the first condition holds. Then, every $A$-module is a direct sum of shifts of $A$, and hence for any way to write $B \\simeq \\varinjlim B_{\\alpha}$ as a filtered colimit of perfect $A$-modules, $B \\otimes_{A} B_{\\alpha}$ is a finite direct sum of shifts of $B$ and hence $\\pi_{*}(B \\otimes_{A} B_{\\alpha})$ is a free $B_{*}$-module. \n\nSimilarly, if $\\pi_{*} B$ is a graded skew-field, then the perfect $B$-modules $B \\otimes_{A} B_{\\alpha}$ are also finite sums of shifts of $B$, as needed.\n\\end{proof}\n\n\n\n\\begin{theorem}\n\\label{theorem:a_map_of_commutative_k_algebras_a_tf_composite_of_adams_type_maps}\nLet $k$ be a field and $A \\rightarrow B$ a map of $\\mathbb{E}_{\\infty}$-$k$-algebras. Then any $A \\rightarrow B$ can be factored as transfinite composite of Adams-type maps.\n\\end{theorem}\n\n\\begin{proof}\nLet $k \\{ x_{n} \\}$ denote the free $\\mathbb{E}_{\\infty}$-$k$-algebra on a variable of degree $| x_{n} | = n$.  Let $\\mathcal{G}$ denote the set of maps $i_{n}: k \\rightarrow k \\{ x_{n} \\}$ together with the maps $p_{n}: k \\{ x_{n} \\} \\rightarrow k$ determined by $x_{n} \\mapsto 0$. As a consequence of \\cref{lemma:adams_type_if_source_or_target_is_a_field}, all of the $i_{n}$ and $p_{n}$ are Adams-type, as either the source or target is a field. By the small object argument, the given map $A \\rightarrow B$ can be factored as \n\\[\nA \\rightarrow B^{\\prime} \\rightarrow B\n\\]\nwhere $B^{\\prime} \\rightarrow B$ has the right lifting property with respect to the maps in $\\mathcal{G}$, and $A \\rightarrow B^{\\prime}$ is a transfinite composition of maps $A_{\\alpha} \\to A_{\\alpha+1}$ which fit into pushout squares \n\\begin{center}\n\\begin{tikzcd}\n\tS \\ar[r,\"f\"]\\ar[d]& \\ar[d]T \\\\\n\tA_{\\alpha}\\ar[r] & A_{\\alpha+1}\\arrow[ul, phantom, \"\\ulcorner\", very near start]\n\\end{tikzcd}\n\\end{center}\nwhere $f \\in \\mathcal{G}$. Each such map is Adams-type by \\cref{cor:po-at}, and it follows that $A \\rightarrow B^{\\prime}$ is a transfinite composite of such maps. On the other hand, one may observe  that the right lifting property with respect to $i_n$ implies surjectivity on $\\pi_n(-)$ and the right lifting property with respect to $p_n$ implies injectivity on $\\pi_n(-)$, therefore  $B^{\\prime} \\rightarrow B$ is an equivalence.\n    \\qedhere\n\\end{proof}\n\n\\begin{remark}\nBy using a more careful argument, where we also allow pushouts along maps $k\\{S\\} \\to k$ and $k \\to k\\{S\\}$ for $S$ a (graded) set of generators, one can show that in the context of \\cref{theorem:a_map_of_commutative_k_algebras_a_tf_composite_of_adams_type_maps} the given map factors as an $\\omega$-indexed composite of Adams-type maps. \n\\end{remark}\n\n\nWe believe the above two results show that the somewhat unexplored theory of Adams-type ring spectra still has a few surprises up its sleeve. To further emphasize this point, we share with the reader a few natural questions which we believe are open.\n\n\\begin{enumerate}\n    \\item Does there exist a descent-flat map $A \\rightarrow B$ which is not Adams-type\\footnote{If we had to guess, we expect that a descent-flat map of ring spectra which is not Adams-type does indeed exist, although we couldn't find one. A curious variant of this question (which we also do not know the answer to) would be to ask if exists a descent-flat map $A \\rightarrow B$ such that the associated homology theory $\\pi_{*}(B \\otimes_{A} -): \\Mod(A) \\rightarrow \\euscr{C}\\mathrm{omod}_{\\pi_{*}(B \\otimes_{A} B)}$ is adapted in the sense of \\cite[Definition 2.19]{patchkoria2021adams}, which is not Adams-type? In other words, does possessing a modified Adams spectral sequence based on $\\pi_{*}(B \\otimes_{A} B)$-comodules characterize Adams-type maps?}?\n    \\item Is every map of $\\mathbb{E}_{1}$-algebras (over the sphere) a transfinite composite of Adams-type maps? Is it true for every map of $\\mathbb{E}_{\\infty}$-algebras? \n    \\item Let $A \\rightarrow B$ be an Adams-type map and $M$ be an $A$-module such that $\\pi_{*}(B \\otimes_{A} M)$ is a flat $B_{*}$-module. Can we write $M \\simeq \\varinjlim M_{\\alpha}$ as a filtered colimit of perfect $A$-modules such that $\\pi_{*}(B \\otimes_{A} M_{\\alpha})$ is a projective $B_{*}$-module?\n\\end{enumerate}\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\nNowadays commercial quantum computers,\nwhich strive for near-term intermediate-scale quantum advantages~\\cite{PreskillNISQ,PreskillSupremacy},\nare accessed by the cloud~\\cite{Devitt}\nraising the problem of a client\nClara (C)\nnot trusting the remote server(s).\nRisk mitigation strategies include blind quantum computing (BQC)\nfor weakly quantum clients\n(using either single-server prepare-and-send\nor receive-and-measure protocols)~\\cite{Childs, BFK09, Fitzsimons, FK17,BKB+12, GKK19}\nand for purely classical clients\n(using multi-server entanglement-based protocols)~\\cite{GKK19, HZM+17,CCKW21} and quantum homomorphic encryption~\\cite{BJ15,OTF18,TFB+20,OTFR20}\nwhereby~C delegates quantum computation to one or more remote servers, who\nare denied key information about the computation~\\cite{Fitzsimons}.\nBuilding on successful experimental factorization\nof the odd semiprime\n(odd integer~$N=pq$ for~$p,q\\in\\mathbb{P}$ and~$p\\neq q$)\n$N=15$~\\cite{HZM+17},\nwe devise a protocol for C\nto delegate secure factorization of $N=21$~\\cite{Martin}\nto two remote quantum servers\ncalled Alice (A) \\&\\ Bob (B).\n\nOur approach extends the\nBQC factorization of~15\nin two ways~\\cite{HZM+17}.\nFirst we increase~$N$ from~15 to\nthe next odd semiprime number~21.\nSecond,\nwe choose a harder base $a$~\\cite{SSV13}\nfor modular exponentiation (modexp) $f(x):=a^x \\bmod N$\n(where $\\gcd(a,N)=1$ with~$\\gcd$ denoting greatest common divisor).\nThe period~$r$ of~$f(x)$\nyields a solution~$p=\\gcd(a^{\\nicefrac{r}2}+1, N)$ when\nthe following two conditions are simultaneously met:\n(i) either~$r$ is even, or~$r$ is odd and~$a$ is a perfect square,\nand (ii)~$a^{\\nicefrac{r}2}\\not\\equiv-1\\bmod{N}$.\nPeriod finding is sped up\nsubexponentially by quantum computing~\\cite{Shors}.\nFor~$N=15$ with~$a=11$,\n$r=2$ was achieved experimentally~\\cite{HZM+17};\nin contrast, we treat the hard case~$N=21$ with~$a=4$, for which~$r=3$.\nThis harder~$a$ requires incorporating a non-Clifford operator,\nfor which we employ the controlled-controlled-not (C$^2$NOT or Toffoli) gate~\\cite{Shi03}.\n\nThe remainder of our paper is orgainsed as follows:\nIn Sec.~\\ref{sec:Background}, we summarily recall\nsome relevant prerequisites to our work\nand point to comprehensive resources on key topics.\nIn Sec.~\\ref{sec:Approach}, we describe our methodology to construct\na blind quantum factorization scheme for given~$N$ and~$a$.\nIn Sec.~\\ref{sec:Results} we present a formal algorithm,\nalong with a function library, to design blind quantum factorization\ncircuits for arbitrary~$N$ and~$a$\nand also present the resulting circuits for two cases of~$N=21, a=4$.\nWe finish with a discussion on the significance of our results\nin Sec.~\\ref{sec:Disc} and a conclusion in Sec.~\\ref{sec:Conc}.\n\n\n\\section{Background}\n\\label{sec:Background}\n\nIn this section we briefly review\nthe state-of-the-art in BQC,\nblind quantum factorization,\nand some other key concepts\nthat are fundamental to our work.\nBQC is a quantum cryptographic protocol that\nallows clients with limited or no quantum hardware\nto outsource a comptuation to remote quantum server(s)\nwithout revealing information about the computation itself to the server(s)~\\cite{BFK09}.\nSeveral BQC protocols have already been developed\nand demonstrated for weakly quantum clients~\\cite{Childs, BFK09, Fitzsimons, FK17},\nbut a purely classical client communicating only classically\nwith a single quantum server might not be able to achieve secure BQC~\\cite{Aaronson}.\nNevertheless, this obstacle is overcome\nif multiple servers sharing non-local resources are employed~\\cite{RUV}.\nA brief overview of verifiable BQC can be found in Ref.~\\cite{GKK19}.\n\n\nSecure BQC for completely classical clients, thus,\nwarrants the remote and classical leveraging of\nquantum-advantageous algorithms,\nlike Shor's factorization~\\cite{Shors}\nor Grover's search~\\cite{Grover},\nwhich serve as prime candidates for delegation~\\cite{BKB+12, HZM+17}.\nDelegated Shor's factorization is known to be feasible\nin the measurement-based quantum computation model~\\cite{BFK09}\nand has been demonstrated experimentally for $N=15$\nin the quantum circuit model~\\cite{HZM+17}.\nThe approach in both these works comprises C\ndelegating the quantum period-finding subroutine,\nwhich computes the period~$r>1$ of~$f(x)$\nfor a given odd semiprime~$N=pq$ with unknown~$p$ \\&~$q$,\nof Shor's algorithm to remote server(s).\nFrom the outputs returned by the servers to her,\nC classically computes the factors as\n\\begin{equation}\n\tp,q=\\gcd(a^{\\nicefrac{r}2}\\pm 1, N).\n\\end{equation}\n\nA proof-of-principle implementation of BQC for a completely classical client\nhas been demonstrated in Ref.~\\cite{HZM+17},\nwherein Shor's algorithm~\\cite{Shors} is executed for factorizing\n$N=15$ using verifiable BQC based on the RUV protocol~\\cite{RUV}.\nThis blind quantum factorization was performed for the choice of base~$a=11$,\nwhich results in~$r=2$, thus\nmaking the experimental demonstration sufficiently challenging for a proof-of-concept\nbut not as realistic as, for example, the case~$a=7$ and~$r=4$ would be.\nThis is because~$r=2$ implies that the quantum period-finding circuit\nhas reduced to a classical coin-toss experiment~\\cite{SM09}---an anomaly\nthat can be rationalised as the choice of base~$a=11$\n(implicitly) assuming pre-knowledge of the factors~\\cite{SM09}.\n\nA pre-knowledgeless factorization scheme would have to\nchoose a random base~$a$ from some set of allowed bases.\nWithout any prior ansatz,\nsuch a choice would yield a hard base with high probability;\na hard base implies a period~$r>2$\nand a period-finding circuit requiring the multi-qubit Toffoli gate,\nwhich is a non-Clifford operator.\nIntroducing\na non-Clifford operator\nbrings\nthe quantum resource called ``magic''\ninto play~\\cite{MagicState}.\n``Magic'' enables quantum circuits to violate conditions for efficient classical simulatability~\\cite{Gottesman1, Gottesman2}\nso its inclusion is important for\nscaling considerations concerning BQC factorization's quantum advantage.\nTo this end,\nwe now succinctly summarize the Clifford hierarchy of unitary operators.\n\nThe~$n$-qubit Pauli group is\n\\begin{equation}\n\\bm{C}^{(1)}_n := \\{\\pm1, \\pm\\text{i}\\}\\times \\{ I, X, Y, Z \\}^{\\otimes n},\n\\end{equation}\nqubits being two-level systems spanned by logical states~$\\ket0,\\ket1\\in\\mathscr{H}_2$,\nand~$\\mathscr{H}_d$ a $d$-dimensional Hilbert space.\nLogical states are~$Z$-eigenstates\nand comprise our computational basis,\nand\n\\begin{equation}\nX,Y,Z\\in\\mathcal{U}(\\mathscr{H}_2)\n\\end{equation}\nare the single-qubit Pauli operators,\nwith~$\\mathcal{U}(\\mathscr{H}_d)$ the group of unitary operators on~$\\mathscr{H}_d$.\nThe $n$-qubit Clifford group $\\bm{C}^{(2)}_n$\nis the normalizer\nof the Pauli group, i.e.,\n\\begin{equation}\n\\bm{C}^{(2)}_n:=\\{ u\\in \\mathcal{U}(\\mathscr{H}_{2^n}) ;  u \\,\\bm{C}^{(1)}_n u^\\dagger\\subseteq\\bm{C}^{(1)}_n \\}.\n\\end{equation}\nThis group is generated by the Hadamard, phase (phase-shift by~$\\pi\/2$) and controlled-not (CNOT) gates.\n\nThe Pauli and Clifford groups constitute\nthe first two levels of the Clifford hierarchy,\nand the subsequent levels~$\\bm{C}^{(k>2)}_n$  are\ndefined recursively by~\\cite{Gottesman3}\n\\begin{equation}\n\\bm{C}_n^{(k)} := \\{ u\\in \\mathcal{U}(\\mathscr{H}_{2^n}); u\\,\\bm{C}^{(1)}_n u^\\dagger\n\\subseteq\\bm{C}_n^{(k-1)}\\}.\n\\end{equation}\nConjugation with~$\\bm{C}^{(2)}_n$ maps~$\\bm{C}^{(1)}_n$ into itself,\nso Clifford operators can be blindly delegated\nusing one-time Pauli pads~\\cite{Childs}.\nHowever, $\\bm{C}^{(2)}_n$ does not constitute a universal gate-set.\nMoreover, stabilizer quantum circuits,\nwhich comprise only Clifford operators\nand computational basis measurements\n(corresponding to a projective-valued measure\n$\\ket{\\epsilon}\\bra{\\epsilon}$\nfor $\\ket\\epsilon$\na computational basis state)~\\cite{Clifford2},\ncan be simulated\nefficiently (polynomial-time)\nclassically~\\cite{Gottesman1, Gottesman2}.\nThe experimental blind quantum factorization of~15\nrequires only a stabilizer circuit in its simplest form~\\cite{HZM+17}.\n\nIn contrast, the C$^2$NOT gate\nalong with~$\\bm{C}^{(2)}_n$\nconstitutes a universal gate-set~\\cite{BMPRV00,Unweyling}.\nA circuit comprising both Clifford and C$^2$NOT gates\nalso circumvents the simulatability theorem~\\cite{Gottesman1, Gottesman2}.\nHowever, the inclusion of ``magic''\nentails significant resource costs~\\cite{Shi03, BJ15}.\nImportantly, for our protocol,\nonly one of A \\&\\ B needs ``magic''\nwhereas the other executes a stabilizer circuit,\nthereby simplifying the scheme for experimental realization.\n\n\\section{Approach}\n\\label{sec:Approach}\n\nIn this section we explain our approach\nto solving the blind quantum factorization of~21.\nFirst we describe the setup for blind quantum factorization,\nnamely the classical client, the bipartite quantum server\nand their collective resources.\nNext we describe our mathematical representation\nof the computation circuit~$\\mathcal{C}$ to be delegated to the servers\nand~$\\mathcal{C}$'s associated representations.\nOur scheme for blind quantum factorization\nrelies upon computation by teleportation\non maximally entangled states, as identified in~\\cite{Gottesman3};\nin spirit, this is similar to the RUV protocol~\\cite{RUV}\nbut we highlight some important distinctions in Sec.~\\ref{sec:Disc}.\nNext we establish the mathematical backbone\nof our scheme---a procedure to obtain two blind circuits, one for each server,\nfrom the circuit the classical client wishes to execute---via\nLemma~\\ref{lemma:concat=simult} and Fact~\\ref{fact:transpose=reverse}.\nWe conclude this section\nby reviewing the full procedure to obtain the blind quantum circuits\nfrom the input quantum circuit.\n\nSimilar to the BQC factorization of~15,\nwhich we summarize in Fig.~\\ref{fig:schemefor15}~\\cite{HZM+17},\nour scheme is based on the\nReichardt-Unger-Vazirani (RUV) protocol~\\cite{RUV}.\nThe RUV protocol is a multi-round, two-server BQC scheme for a classical client\n(as single-server BQC is not secure for classical clients~\\cite{Aaronson, Morimae}).\nIn each round of our protocol, servers~A \\&~B\nreceive~$n$ copies of the entangled two-qubit pair\n$\\ket{\\Phi}:=\\ket{00}+\\ket{11}\\in\\mathscr{H}_{4}$\n(for~$\\ket{00}\\equiv\\ket0\\otimes\\ket0$,\nand implied normalization employed throughout)\nfrom a periodic source of entanglement, Deborah (D).\nEach server receives one qubit from each copy of~$\\ket\\Phi$\nand, thus, A \\&\\ B collectively share the resource\n\\begin{equation}\n\\label{eq:sharedresource}\n\\ket{\\Phi}^{\\otimes n}\n=(\\mathds1\\otimes\\mathds1 + X\\otimes X)^{\\otimes n}\n\\ket{00}^{\\otimes n}\n\\in\\mathscr{H}_{4^n}\n\\end{equation}\nbut have no other means for communicating~\\cite{RUV}.\nWe index the~$2n$ qubits in~$\\ket\\Phi^{\\otimes n}$\nas shown in Fig.~\\ref{fig:schemefor15} for~$n=3$.\n\n\nTo delegate an~$n$-qubit quantum circuit~$\\mathcal{C}$\nin the RUV protocol,\nC instructs each server to either compute,\nby executing a quantum circuit~($\\mathcal{A}$ for A,~$\\mathcal{B}$ for B),\nor perform the measurement part of a Clauser-Horne-Shimony-Holt (CHSH) test~\\cite{CHSH}.\nThere are, thus, four distinct subprotocols:\nA~\\&~B could both compute (computational subprotocol),\nor both measure (CHSH subprotocol),\nor else one computes while the other measures\n(two tomography subprotocols)~\\cite{RUV}.\nWhen both compute,\nA~\\&~B report to C their~$i^\\text{th}$ $Z$-measurement outcomes\n\\begin{equation}\n\\{(a_i,b_i);a_i,b_i\\in\\{0,1\\},i\\in[n]:=\\{1, \\dots, n\\}\\}.\n\\end{equation}\nFrom their combined outcomes, C recovers the output from~$\\mathcal{C}$\nwhereas the output from either~$\\mathcal{A}$ or~$\\mathcal{B}$\nalone yields no information about~$\\mathcal{C}$ except depth,\nthereby blinding~A~\\&~B.\n\n\\begin{figure}\n\\begin{tikzpicture}\n\t\\node [anchor=south west,inner sep=0] (image) at (0,0) {\\includegraphics[width=6 cm]{Fig1Fact15.eps}};\n\t\\begin{scope}[x={(image.south east)},y={(image.north west)}]\n\t\t\\draw [thick, red,rounded corners=3] (-0.01, -0.03) rectangle (0.4, 0.47);\n\t\t\\draw [thick, red,rounded corners=3] (0.59, -0.03) rectangle (1.01, 0.47);\n\t\t\\node [black] at (-0.06, 0.24) {$\\mathcal{A}$};\n\t\t\\node [black] at (1.06, 0.24) {$\\mathcal{B}$};\n\t\t\\node [black] at (0.0173, 0.419) {\\scriptsize 1};\n\t\t\\node [black] at (0.017, 0.27) {\\scriptsize 2};\n\t\t\\node [black] at (0.017, 0.122) {\\scriptsize 3};\n\t\t\\node [black] at (0.983, 0.419) {\\scriptsize 4};\n\t\t\\node [black] at (0.983, 0.27) {\\scriptsize 5};\n\t\t\\node [black] at (0.983, 0.122) {\\scriptsize 6};\n\t\\end{scope}%\n\\end{tikzpicture}%\n\\caption{%\n\tBQC scheme for factorizing~15.\n\tQuantum servers A \\&\\ B\n\tjointly compute circuits~$\\mathcal{A}$ \\&~$\\mathcal{B}$\n\t(rounded rectangles), respectively,\n\ton state~$\\ket{\\Phi}^{\\otimes 3}$\n\tsupplied by entanglement source D,\n\tand report outcomes to classical client C.\n\tEach of $\\mathcal{A}$ \\&~$\\mathcal{B}$ involves\n\ta CNOT and a Hadamard gate ($H$), and~$Z$-basis measurements.\n\tSolid and dotted arrows represent\n\tclassical and quantum communication,\n\trespectively,\n\tarrowheads indicate directionality,\n\tand numbers represent indices for qubits.}%\n\\label{fig:schemefor15}%\n\\end{figure}\n\nNow we discuss the underlying primitive gates\nfor~$\\mathcal{C}$ over~$n$ qubits.\nEach computational cycle,\nwith execution by circuit component~$\\mathcal{C}_\\nu$,\nallows one or more of the following primitive gates\noperating in parallel:\n\\begin{enumerate}[(i)]\n\\item single-qubit Hadamard~($H$),\n\\item single-qubit NOT~($X$),\n\\item two-qubit controlled-rotation~(CR$^k$),\nand\n\\item multi-controlled NOT~(C$^l$NOT,\nwhere~$l\\in[n-1]$ denotes the number of controls),\nalso known as the multi-controlled Toffoli gate.\n\\end{enumerate}\nOur~CR$^k$ gates are restricted to rotations\n$\\mathrm{R}^k \\textcolor{red}{:} \\, 0\\leq k<n$\nthat impart a phase of~$\\nicefrac{\\pi}{2^k}$ on~$\\ket1$\nand zero phase on~$\\ket0$.\nOur choice of primitives is natural for Shor factorization~\\cite{Shors},\nand these primitives are composites of the ``standard set''\nof CNOT,~$H$ and the~R$^2$ gate called~$T$,\nas described in the Appendix A~\\cite{BMPRV00,BBCD+95,MDM05,GKMR14,SM09}.\n\nIn Appendix B, we show that\nthere are at most\n\\begin{equation}\n\\left ( \\frac{1133233}{8!}  \\right) \\, n^{\\nicefrac{n}{2}} \\, n!\n\\end{equation}\nallowed circuit components over~$n$ qubits,\nso\nwe can label~$\\mathcal{C}_\\nu$ by a bit string~$\\bm{B}(\\mathcal{C}_\\nu)$\nwith size at most\n\\begin{equation}\n\\left\\lceil\\log \\{(\\nicefrac{1133233}{8!}) n^{\\nicefrac{n}{2}} n! \\}\\right\\rceil \\, .\n\\end{equation}\nThe depth~$d$ circuit~$\\mathcal{C}$\nis then a composition of~$d$ circuit components\n\\begin{equation}%\n\\label{eq:cyclecomposition}%\n\\mathcal{C}%\n=\\bigcircle_{\\nu=1}^d \\mathcal{C}_{\\nu}%\n:= \\mathcal{C}_d \\circ \\cdots \\circ \\mathcal{C}_1  ,%\n\\end{equation}%\nwhere~$\\mathcal{C}_\\nu$ is the circuit component\nfor the~$\\nu^\\text{th}$ computational cycle and~$\\circ$ denotes composition.\nCorrespondingly,~$\\mathcal{C}$ is represented\nby the bit string\n\\begin{equation}\n\\bm{B}(\\mathcal{C})=\\Vert_{\\nu=1}^d \\bm{B}(\\mathcal{C}_\\nu),\n\\end{equation}\nwhere~$\\Vert$ denotes concatenation of bit strings.\nWe also represent each\ncircuit component~$\\mathcal{C}_\\nu$\nby unitary operator~$G_\\nu \\in \\mathcal{U}(\\mathscr{H}_{2^n})$\nand represent~$\\mathcal C$\nby unitary operator~$G\\in \\mathcal{U}(\\mathscr{H}_{2^n})$,\nso that\n\\begin{equation}\nG=G_d \\, G_{d-1} \\cdots  G_1 \\, .\n\\end{equation}\n\nWe now focus on RUV-based blind quantum factorization\nof odd semiprime~$N$,\nwherein factorization circuit~$\\mathcal{C}$\nacts on two computational registers\nof sizes~$t$ and~$L$~\\cite{Shors, NielsenChuang}.\nThus, now~$n=t+L$, and the registers are collectively\ninitialized to~$\\ket{0}^{\\otimes n}$.\nFor uncompiled factorization circuits,\n$t\\geq2\\lceil\\log{N}\\rceil+1$\nand~$L\\geq\\lceil\\log{N}\\rceil$,\nbut we employ compilation so these bounds do not apply~\\cite{SSV13}.\nTo convert~$\\mathcal{C}$ into\nblind circuits~$\\mathcal{A}$ \\&~$\\mathcal{B}$,\nwe employ a two-step procedure:\nfirst we partition~$\\mathcal C$\ninto a first-stage circuit~$\\mathcal{C}_<$\nand a second-stage circuit~$\\mathcal{C}_>$;\nthen we convert the sequential computation~$\\mathcal{C}_>\\circ\\, \\mathcal{C}_<$\ninto a bipartite computation~$\\mathcal{A}\\otimes\\mathcal{B}$\non~$\\ket\\Phi^{\\otimes n}$.\n\nIn our scheme we require\n$\\mathcal{A}$ to be a stabilizer circuit,\nand~$\\mathcal{B}$ to have the minimum possible~$d$.\nThus, before partitioning~$\\mathcal{C}$,\nwe first minimize reduced depth~$d_>$\n(defined to be\nthe number of cycles including\nthe first non-Clifford cycle and\nthen all subsequent cycles, whether Clifford or not)\nover all circuits that are permutations of the cycles of~$\\mathcal{C}$\nand are~$\\mathcal{C}$-equivalent (i.e., map input to the same output as~$\\mathcal{C}$).\nIn case the minimum reduced depth~$d^*_>$ is not achieved uniquely,\nwe choose an optimal circuit~$\\mathcal{C}^*$.\nThen,~$\\mathcal{C}_<$ is\nthe composition of the first~$d-d_>^*$ cycles in~$\\mathcal{C}^*$\nand~$\\mathcal{C}_>$ is\nthe composition of the last~$d^*_>$ cycles in~$\\mathcal{C}^*$.\nThus,\nfor\n\\begin{equation}\n\\mathcal{C}^* = \\bigcircle_{\\nu=1}^d \\mathcal{C}^*_\\nu \\, ,\n\\end{equation}\nwe partition as\n\\begin{equation}\n\\label{eq:circuitpartition}\n\\mathcal{C}_< := \\bigcircle_{\\nu=1}^{d-d^*_>} \\mathcal{C}^*_\\nu\n\\quad  \\&\\ \\quad\n\\mathcal{C}_> := \\bigcircle_{\\nu=d-d^*_>+1}^{d} \\mathcal{C}^*_\\nu  \\, ,\n\\end{equation}\nso that\n\\begin{equation}\n\\mathcal{C}^* = \\mathcal{C}_>\\circ\\mathcal{C}_< .\n\\end{equation}\nThe bit strings~$\\bm{B}(\\mathcal{C}_<)$\n\\&~$\\bm{B}(\\mathcal{C}_>)$\nrepresenting~$\\mathcal{C}_<$ \\&~$\\mathcal{C}_>$,\nrespectively,\nare determined by first permuting the component bit strings of~$\\bm{B}(\\mathcal{C})$\nand then partitioning into $\\bm{B}(\\mathcal{C}_<)\\Vert\\bm{B}(\\mathcal{C}_>)$\nfollowing Eq.~\\eqref{eq:circuitpartition};\nwe denote this operation by the bit-string function \\textsc{part}.\n\nTo establish the second step in our procedure,\nwe first introduce some notation,\nprove a lemma and state a fact.\nBelow we denote transposition in the computational basis by~${}^\\top$.\nFor~$x=(x_{n}\\cdots x_1)\\in\\{0,1\\}^n$, we define\n\\begin{equation}\n\\label{eq:Xx}\nX^x:=X_{n}^{x_{n}} \\otimes \\cdots \\otimes  X_1^{x_1} \\in\\mathcal{U}(\\mathscr{H}_{2^n}) ,\n\\end{equation}\nwhere the subscripts below~$X$\nindicate the index of the qubit being targeted.\nThus, a computational basis state is\n\\begin{equation}\n\\ket{x}=X^x\\ket0^{\\otimes n}\\in \\mathscr{H}_{2^n} \\, .\n\\end{equation}\n\\begin{lemma}\n\\label{lemma:concat=simult}\nFor any~$x\\in\\{0,1\\}^n$\nand unitary operators~$G_<, G_>, G_A, G_B \\in\\mathcal{U}(\\mathscr{H}_{2^n})$,\nthe mapping\n\\begin{equation}\n\t\\label{eq:mapforparallelization}\n\tX^x G_<^\\top \\mapsto G_A  \\, , \\quad X^x G_>\\mapsto G_B\n\\end{equation}\nleads to the equality\n\\begin{equation}\n\t\\label{eq:equationforparallelization}\n\tX^x G_> G_< X^x\\ket0^{\\otimes n} = \\left( \\bra0^{\\otimes n} G_A  \\otimes G_B\\right)\\ket\\Phi^{\\otimes n} .\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFrom the ``ricochet'' property~\\cite{Wilde17},\n\\begin{align}\n\t\\label{eq:ricochetproperty}\n\t&(\\ket0^{\\otimes n} \\bra0^{\\otimes n} G_A \\otimes G_B) \\left(\\mathds1\\otimes\\mathds1 + X\\otimes X\\right)^{\\otimes n} \\nonumber\\\\\n\t=&\\left [ \\mathds 1 \\otimes \\left (G_B G_A^\\top \\ket0^{\\otimes n} \\bra0^{\\otimes n}  \\right) \\right] \\left(\\mathds1\\otimes\\mathds1 + X\\otimes X\\right)^{\\otimes n} ,\n\\end{align}\nso assign\n$G_A\\gets X^x G_<^\\top$ and\n$G_B\\gets X^x G_>$.\n\\end{proof}\n\\begin{fact}\n\\label{fact:transpose=reverse}\nAs each of our primitive gates admits\na symmetric matrix representation in the computational basis,\nthe operator $G_<^\\top$ represents a circuit~$\\mathcal{C}_<^\\top$\nthat consists of the components of~$\\mathcal{C}_<$ executed in reverse order, i.e.,\n\\begin{equation}\n\t\\label{eq:transposedcircuit}\n\t\\mathcal{C}_<^\\top := \\bigcircle_{\\nu=d-d^*_>}^{1} \\mathcal{C}^*_\\nu \\, .\n\\end{equation}\nWe denote the operation\nof obtaining~$\\bm{B}(\\mathcal{C}_<^\\top)$\nby reversing the order of components\nin~$\\bm{B}(\\mathcal{C}_<)$\nby the bit-string function~\\textsc{rev}.\n\\end{fact}\n\n\nWe now explain how we use Lemma~\\ref{lemma:concat=simult} and Fact~\\ref{fact:transpose=reverse}\nto convert~$\\mathcal{C}_>\\circ\\, \\mathcal{C}_<$\ninto~$\\mathcal{A}\\otimes\\mathcal{B}$.\nLet~$G_<, G_>, G_A$ and~$G_B$ be unitary operators\nrepresenting~$\\mathcal{C}_<, \\mathcal{C}_>, \\mathcal{A}$ and~$\\mathcal{B}$, respectively,\nso that Map~\\eqref{eq:mapforparallelization} implies\n\\begin{equation}\n\\mathcal{A}=X^x \\circ \\mathcal{C}^\\top_< \\quad \\text{\\&} \\quad \\mathcal{B}= X^x \\circ \\mathcal{C}_> \\, .\n\\end{equation}\nAlso, consider any~$x\\in\\{0,1\\}^{t+L}$\nwith~$x_i=0$ for all~$i\\in[t]$.\nThen,\n\\begin{equation}\nX^xG_>G_< X^x =G_>G_<=G,\n\\end{equation}\nbecause the second register of~$\\mathcal{C}$\nis operated on by only C$^l$NOTs~\\cite{Shors},\nand Eq.~\\eqref{eq:equationforparallelization} simplifies to\n\\begin{equation}\n\\label{eq:unitaryparallelization}\nG \\ket 0^{\\otimes n} = \\bra{x} G_<^\\top \\otimes X^x G_> \\ket\\Phi^{\\otimes n}.%\n\\end{equation}\nAs the output from only the first register of~$\\mathcal{C}^*$\nis used to compute~$r$ \\cite{Shors},\nthe~$X^x$ in Eq.~\\eqref{eq:unitaryparallelization} can be ignored.\nFinally, we have\n\\begin{equation}\n\\mathcal{A}=\\mathcal{C}_<^\\top \\quad \\text{\\&} \\quad \\mathcal{B}=\\mathcal{C}_> \\,,\n\\end{equation}\nand B's outcomes~$\\{b_i; i\\in[t]\\}$ are identical\nto the output from the first register of~$\\mathcal{C}$\nwhenever A reports~$a_i=0$ for all~$i\\in[t]$.\nThis completes our description of\nthe procedure to obtain~$\\mathcal{A}~\\&~\\mathcal{B}$\nfrom~$\\mathcal{C}$.\n\n\\section{Results}\n\\label{sec:Results}\n\nIn this section we present our results, which are two-fold.\nFirstly, we present a scalable algorithm to design circuits~$\\mathcal{A}~\\&~\\mathcal{B}$\nfor blind-quantumly factorizing arbitrary odd semiprime~$N$ using arbitrary base~$a$.\nSecondly, we present the outputs of this algorithm,\ni.e., the factorization circuits~$\\mathcal{A}~\\&~\\mathcal{B}$,\nalong with the rest of the blind factorization scheme,\nfor two cases corresponding to~$N=21$ and~$a=4$.\nOur algorithm requires  certain standard functions so we start by specifying a function library,\nthough we don't reproduce the corresponding algorithms here as they are standard in literature.\n\n\n\nWe now introduce our concept\nfor the function library~\\textsc{funcLib}\nfor designing circuits~$\\mathcal{A}$\n\\&~$\\mathcal{B}$.~\\textsc{funcLib}\ncomprises four functions,\nwith two of them~(\\textsc{part} \\&~\\textsc{rev}) already discussed and\nthe other two well established\nin literature on factorization~\\cite{BCDP, Shors, SSV13}.\nThe integer function\n\\begin{equation}\n\\label{eq:maximumdepthshorcircuitfunction}\n\\textsc{maxDep}(N, a) = 96 \\lfloor\\log{a}\\rfloor \\lfloor\\log{N}\\rfloor^2\n\\end{equation}\nyields an upper bound\non factorization-circuit depth, given~$N$ and~$a$,\nbased on complexity arguments for scaling modular exponentiation~\\cite{BCDP}.\nThe bit-string function~\\textsc{ShorCir}\nreturns a bit string representing the compiled factorization circuit,\ngiven~$N, a, t$ and~$n$~\\cite{Shors,SSV13}.\nOur procedure for designing\ncircuits~$\\mathcal{A}$ \\&~$\\mathcal{B}$\nis described in Alg.~\\ref{algorithm:circuitdesign},\nwhere we employ `type' notation USINT for nonnegative integers and\nBIN for bit strings (with~[\\,] denoting array size).\n\n\\begin{algorithm}[H]\n\\caption{Parallelizing Factorization}\n\\label{algorithm:circuitdesign}\n\\begin{algorithmic}[1]\n\t\\Require\n\t\\Statex USINT \\textsc{num}\n\t\\Comment $N=pq$,\\; $p\\neq q$,\\; $p,q\\in\\mathbb{P}\\setminus\\{2\\}$ \\;\n\t\\Statex USINT \\textsc{base}\n\t\\Comment Base~$a$: $a<N, \\, \\gcd(a, N) = 1$ \\;\n\t\\Statex USINT \\textsc{siz1}\n\t\\Comment First-register size of Shor circuit\\;\n\t\\Statex USINT \\textsc{siz2}\n\t\\Comment Second-register size of Shor circuit\\;\n\t\\Ensure\n\t\\Statex BIN[\\;] \\textsc{circDesA}, \\textsc{circDesB}\n\t\\Procedure{\\textsc{cirDesign}}{\\textsc{num}, \\textsc{base}, \\textsc{siz1}, \\textsc{siz2}}\n\t\\State Import \\textsc{funcLib}\n\t\\Comment For functions\\;\n\t\\State USINT \\textsc{depth}, \\textsc{sizeCom}, \\textsc{sizCir}\n\t\\State $\\textsc{depth}\\gets \\textsc{maxDep}(\\textsc{num}, \\textsc{base})$\n\t\\Comment Eq.~\\eqref{eq:maximumdepthshorcircuitfunction}\n\t\\State $\\textsc{siz2}\\gets\\textsc{siz1}+\\textsc{siz2}$\n\t\\Comment Replace by total register size\n\t\\State $\\textsc{sizeCom} \\gets\\left\\lceil\\log  ( \\frac{133233}{8!}* \\textsc{siz2}**{\\frac{\\textsc{siz2}}{2}} \\,* \\textsc{siz2}!)\\right\\rceil$\n\t\\State $\\textsc{sizCir}\\gets \\textsc{sizeCom} * \\textsc{depth}$\n\t\\Comment Space-time product\n\t\\State BIN[\\textsc{sizCir}] \\textsc{cirDes}, \\textsc{cirDesL}, \\textsc{cirDesG}, \\textsc{cirDesA}, \\textsc{cirDesB}\n\t\\State $\\textsc{cirDes}\\gets$~\\Call{\\textsc{ShorCir}}{\\textsc{num}, \\textsc{base}, \\textsc{siz1},  \\textsc{siz2}}\n\t\\State $\\textsc{cirDesL}\\|\\textsc{cirDesG} \\gets \\textsc{part}(\\textsc{cirDes})$\n\t\\Comment Optimal partition\n\t\\State $\\textsc{cirDesA}\\gets\\textsc{rev}(\\textsc{cirDesL})$\n\t\\Comment Reverse order\n\t\\State $\\textsc{cirDesB}\\gets\\textsc{cirDesG}$\n\t\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nNext we describe the computational subprotocol for\nour scheme.\nPrior to executing the subprotocol,\nC runs Alg.~\\ref{algorithm:circuitdesign}\nto design~$\\mathcal{A}$ \\&\\ $\\mathcal{B}$\nand sends the output bit strings to the servers.\nThen, in every instance of the subprotocol,\nC instructs A \\&\\ B to execute their circuits\nand report measurement outcomes.\nIf~$a_i=0$ for all~$i\\in[t]$,\nwhich occurs with probability~$2^{-t}$,\nC computes a candidate for~$r$ by\nclassically processing~$\\{b_i; i\\in[t]\\}$.\nFor completeness, we describe this classical processing\nin Appendix~C.\nThe exponentially small probability\ncould be improved by instructing B to\nPauli-correct before computing,\nbut doing so blindly would entail higher space requirements~\\cite{RUV, Gottesman3};\nin this work we instead focus on~$N$\nwith~$t$ sufficiently small for~$2^{-t}$ to be a feasible probability.\n\nThis concludes our discussion of the computation subprotocol,\nwhereas the three other RUV subprotocols\nare standard so we describe them\nin Appendix D.\nIn our full multi-round protocol\nfor blind quantum factorization\n(summarized in Appendix D),\nC runs one of the four sub-protocols at random,\ninteracting with the servers as required,\nuntil the correct~$r$ is found.\nAlthough tomographic verification via\nthe stabilizer framework~\\cite{RUV, HZM+17}\nis inapplicable here\ndue to our inclusion of\nnon-Clifford operators~\\cite{RUVProof, JKMW01},\nC can classically verify~$r$\nin~$\\operatorname{polylog}N$ time\nby checking~$\\gcd(p,N)=p\\in\\mathbb{P}$~\\cite{GKK19}.\n\\begin{figure}\n\\begin{tikzpicture}\n\t\\node[anchor=south west,inner sep=0] (image) at (0,0) {\\includegraphics[width=0.8 \\columnwidth]{Fig2Fact21t2.eps}};\n\t\\begin{scope}[\n\t\tx={(image.south east)},\n\t\ty={(image.north west)}\n\t\t]\n\t\t\\node [black] at (0.71, 1.08) {QFT$^{\\dagger}$};\n\t\t\\node [black] at (0.34, 1.07) {modexp};\n\t\t\\node [black] at (0.07, 1.07) {QFT};\n\t\t\\draw[red,thick, densely dashed] (0.41,0.06) -- (0.41,0.92);\n\t\\end{scope}%\n\\end{tikzpicture}%\n\\caption{%\n\tOptimally permuted~$\\mathcal{C}^*$ for~$N=21,a=4$, $t=2$ and $L=3$.\n\tUnlabeled input states are~$\\ket0$\n\tand measurements are in the $Z$ basis.\n\tQuantum Fourier transform (QFT) is performed using Hadamard~($H$) gates,\n\tmodexp using CNOT and C$^2$NOT gates,\n\tand inverse QFT~($\\text{QFT}^\\dagger$) using a controlled-phase gate~($S$) and Hadamard gates.\n\tThe vertical dashed line (red) indicates the optimal partition of the circuit.}%\n\\label{fig:factorization21with5qubits}\n\\end{figure}\n\\begin{figure\n\\begin{tikzpicture}%\n\t\\node[anchor=south west,inner sep=0] (image) at (0,0) {\\includegraphics[width=0.98\\columnwidth]{Fig3Fact21t3.eps}};\n\t\\begin{scope}[\n\t\tx={(image.south east)},\n\t\ty={(image.north west)}\n\t\t]\n\t\t\\node [black] at (0.048, 1.05) {\\footnotesize QFT};\n\t\t\\node [black] at (0.411, 1.05) {\\footnotesize modexp};\n\t\t\\node [black] at (0.852, 0.748) {\\footnotesize QFT$^{\\dagger}$};\n\t\t\\draw[red,thick, densely dashed] (0.410,0.067) -- (0.410,0.94);\n\t\\end{scope}%\n\\end{tikzpicture}%\n\\caption{%\n\tOptimally permuted~$\\mathcal{C}^*$ for~$N=21,a=4$, $t=3$ and $L=3$.\n\tUnlabeled input states are~$\\ket0$\n\tand measurements are in the $Z$ basis.\n\tQFT is performed using Hadamard~($H$) gates,\n\tmodexp using NOT, CNOT, C$^2$NOT, and C$^3$NOT gates.\n\tThe inverse QFT (QFT$^{\\dagger}$) is abridged as a 3-qubit gate for readability.\n\tThe vertical dashed line (red) indicates the optimal partition of the circuit.}%\n\\label{fig:factorization21with6qubits}\n\\end{figure}\n\nWe now present circuits~$\\mathcal{C}^*$,~$\\mathcal{A}$ and~$\\mathcal{B}$\nspecifically for~$N=21$ and~$a=4$\nwith two cases considered,\nnamely~$t=2$ and~$t=3$,\nboth with~$L=3$.\nThe five-qubit~($t=2$) and six-qubit~($t=3$)\noptimally permuted circuits~$\\mathcal{C}^*$\nare shown in Figs.~\\ref{fig:factorization21with5qubits}\nand~\\ref{fig:factorization21with6qubits},\nrespectively,\nwhere a vertical dashed line indicates the partition\ninto a first\nand a second stage.\nFor both cases,\nthe output from the first register\nof~$\\mathcal{C}^*$\nis shown in Fig.~\\ref{fig:outputsall} in Appendix C.\nFor~$t=2$,\nthe blind circuits~$\\mathcal{A}$ \\&~$\\mathcal{B}$\nare shown in Fig.~\\ref{fig:blindfact21}.\nFor~$t=3$,\nwe do not show~$\\mathcal{A}$ \\&~$\\mathcal{B}$ explicitly\nas they could be similarly obtained\nfrom~$\\mathcal{C}^*$ in Fig.~\\ref{fig:factorization21with6qubits}:\n$\\mathcal{A}$ is the first stage performed in reverse order\nwhereas~$\\mathcal{B}$ is the second stage.\nBoth~$\\mathcal{A}$ \\&~$\\mathcal{B}$ terminate with~$Z$ measurements of all qubits.\n\n\\section{Discussion}\n\\label{sec:Disc}\n\\begin{comment}\n{\\color{purple}\nOur Alg.~\\ref{algorithm:circuitdesign} is scalable\nin the following sense:\nthe expected number of runs of the computational subprotocol\nfor C to factor~$N$ goes like~$2^t$,\nwhich for uncompiled circuits grows as~$2^{2 \\lceil \\log N \\rceil} \\approx N^2$\nwhereas sub-polynomial complexity classical factorization algorithms exist.\nThe catch here is that we use compiled circuits, not uncompiled,\nwhere~$t$ can be freely chosen, even to be lower than~$  \\log N $,\nso our scheme is not necessarily polynomial-time in~$N$.\n}\n\n{\\color{purple}\nNext, our scheme ensures~$\\mathcal{A}$ is a stabilizer circuit;\nthis implies not only that~$A$ is not resource-intensive to implement,\nbut also that~$\\mathcal{A}$ can be implemented, e.g.,\nin an error-corrected, fault-tolerant fashion~\\cite{Gottesman3}.\nA being a stabilizer circuit also implies\nC can efficiently tomographically check\nwhether A performs~$\\mathcal{A}$ honestly\nby simply collating B's reported outcomes~$\\{b_i\\}$.\n}\n\n{\\color{purple}\nOn the other hand,~$\\mathcal{B}$ is expected to include non-Clifford operations,\nand is thus resource intensive (to extent of being experimentally infeasible at present).\nInterestingly, the problem of verifiability of the servers in this case was left as an open problem in Ref.~\\cite{RUV}.\nhowever, for us there exists an easy fix:\nC tomographically ensures A's honesty,\nand, as factorization admits classical~$\\operatorname{polylog} N$ verification,\nif~$r$ is not verified after~$2^t$ trials she declares B dishonest.\n}\n\\end{comment}\n\nIn this section, we discuss the significance\nsome implications of our results.\nOur Alg.~\\ref{algorithm:circuitdesign} designs\nscalable blind quantum factorization circuits\nfor arbitrary~$N$ and~$a$,\nwith expected number of runs growing as~$2^t$\n(note that we use compiled circuits,\nso~$t$ is not restricted by the~$2 \\lceil \\log N \\rceil+1$ bound).\nAlg.~\\ref{algorithm:circuitdesign}, via \\textsc{Part}, also ensures\nthat~$\\mathcal{A}$ is always a stabilizer circuit,\nand thus not resource-intensive to implement.\nMoreover,~$\\mathcal{A}$ can be implemented fault-tolerantly straightforwardly~\\cite{Gottesman3}\nand affords complete tomographic verification via just CHSH measurements~\\cite{RUV}.\n\nOn the other hand, we expect~$\\mathcal{B}$ to include non-Clifford operations\nand be resource intensve (potentially experimentally infeasible at present).\nThis can also be seen in the circuits~$\\mathcal{C}^*$,~$\\mathcal{A}$ and~$\\mathcal{B}$\nfor~$t\\in\\{2,3\\}$.\nFor~$t=2$,~$\\mathcal{C}^*$ has~$d=8$,~$d^*_>=5$\nand incorporates three non-Clifford gates\n(one~CR$^1$ and two~C$^2$NOTs);\n$\\mathcal{A}$ \\&~$\\mathcal{B}$ thus\nhave depths of three \\& five, respectively.\nFor~$t=3$,~$\\mathcal{C}^*$\nhas~$d=17$,~$d^*_>=13$ and incorporates eight non-Clifford gates\n(one~CR$^2$, two~CR$^1$s,\nthree~C$^2$NOTs and two~C$^3$NOTs);\n$\\mathcal{A}$ \\&~$\\mathcal{B}$ thus\nhave depths of four \\& thirteen, respectively.\nIn both cases,~$\\mathcal{A}$\nis conveniently a stabilizer circuit\nwhereas~$\\mathcal{B}$ incorporates\nall non-Clifford gates in~$\\mathcal{C}^*$.\n\nFor~$t=2$,\ndespite following Shor's algorithm~\\cite{Martin},\n$\\mathcal{C}^*$ never yields the correct period\ndue to insufficient space in the first register;\nhence, $\\mathcal{A}$ \\&~$\\mathcal{B}$\nfail to factorize~21.\nHowever, the output from~$\\mathcal{C}^*$\nis sufficient to establish a proof-of-concept as in Ref.~\\cite{Martin}.\nFurther, photonic implementations of~$\\mathcal{A}$ \\&~$\\mathcal{B}$,\nwhich entail scaling up from\none C$^2$NOT to two~\\cite{Lanyon}\nand from three EPR pairs to five~\\cite{HZM+17},\nare more feasible\nin this case compared to~$t=3$.\nFor~$t=3$, $\\mathcal{C}^*$\ndelivers the correct period\nwith probability~0.47\nso~$\\mathcal{A}$ \\&~$\\mathcal{B}$\nsucceed in factorizing with probability~0.058,\nbut~$\\mathcal{B}$ requires significant resources.\n\nAnother advantage of our blind quantum factorization scheme\nis the low qubit count required to factorize~$N$.\nOur scheme requires, at worst,~$\\mathrm{O}(\\log N)$ qubits to guarantee factorization\ncompared to the~$\\mathrm{O}((\\log N)^2)$ qubits\na literal adaptation of the RUV protocol to quantum factorization would require.\nFinally, we remedy RUV's open problem of tomographic verifiability\nof non-Clifford computations in the context of factorization\nby declaring B to be dishonest if C finds A honest\n(which requires only CHSH measurments)\nbut still does not recieve the correct period\n(which can be checked efficiently classically)\nfrom B.\n\n\n\\begin{figure}\n\\begin{tikzpicture}\n\t\\node[anchor=south west,inner sep=0] (image) at (0,0) {\\includegraphics[width=0.8\\columnwidth]{Fig5BlindFact21.eps}};\n\t\\begin{scope}[%\n\t\tx={(image.south east)},\n\t\ty={(image.north west)}\n\t\t]\n\t\t\\node [black] at (-0.06, 0.5) {\\footnotesize $\\mathcal{A}$};\n\t\t\\node [black] at (1.06, 0.5) {\\footnotesize $\\mathcal{B}$};\n\t\t\\draw [thick,red,rounded corners=3] (-0.02, -0.04) rectangle (0.454, 1.04);\n\t\t\\draw [thick,red,rounded corners=3] (0.501, -0.04) rectangle (1.02, 1.04);\n\t\\end{scope}%\n\\end{tikzpicture}%\n\\caption{%\n\tBlind circuits~$\\mathcal{A}$ \\&~$\\mathcal{B}$\n\tfor $N=21, a=4$, $t=2$ and $L=3$.\n\t$\\mathcal{A}$ \\&~$\\mathcal{B}$ (rounded rectangles)\n\teach act on input registers initialized to one half of the bipartite state~$\\ket\\Phi^{\\otimes 5}$ (red dots).\n\t$\\mathcal{A}$ comprises NOT, CNOT, and Hadamard~($H$) gates\n\twhereas~$\\mathcal{B}$ comprises C$^2$NOT, controlled-phase~($S$) and Hadamard gates.\n\tAll measurements are in the~$Z$ basis.}\n\\label{fig:blindfact21}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:Conc}\n\nHere\nwe have developed a BQC scheme\nfor factorizing~21.\nOur multi-round protocol is accessible\nto a classical client,\nwho communicates with\ntwo remote quantum servers.\nThe servers send\nthe client~$Z$-measurement results\nfor each round.\nBy processing these data,\nthe client determines candidates\nfor factors of~21\nor verifies honesty of the servers,\nall while concealing the actual task.\nOur choice of hard~$a$ implies that\nservers employ non-Clifford gates,\nwhich is a non-trivial requirement\nunseen for $N=15$~\\cite{HZM+17};\nour non-Clifford analysis\nestablishes a foundation for\nfuture BQC factorization protocols.\nFinally, our protocol for~$t=2$ motivates\na challenging but feasible photonic experiment\nthat would set a milestone towards\nsecure quantum computation\nfor classical clients.\n\n\\begin{acknowledgments}\nAritra Das thanks and acknowledges financial support from the Shastri Indo-Canadian Institute through their Shastri Research Student Fellowship program\nand from the Australian Research Council Centre of Excellence CE170100012.\nAritra Das is also grateful to the Indian Institute of Technology Kanpur,\nwhere a majority of this work was undertaken.\nFinally, Aritra Das and Barry C. Sanders acknowledge the traditional owners of the land on which this work was undertaken at the University of Calgary: the Treaty 7 First Nations.\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev4-2}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter*{Streszczenie}\n\nCelem niniejszej pracy doktorskiej jest badanie zbior\u00f3w $\\K$-prostokre\u015blnych w geometrii afinicznej. \n\nPierwszy rozdzia\u0142 jest po\u015bwi\u0119cony w\u0142asno\u015bci $\\K$-prostokre\u015blno\u015bci oraz wa\\-run\\-kom r\u00f3w\\-no\\-wa\u017c\\-nym.\nNiech $\\K$ b\u0119dzie cia\u0142em algebraicznie domkni\u0119tym. Krzyw\u0105 $\\Gamma$ nazywamy parametryczn\u0105 je\u017celi jest obrazem cia\u0142a $\\K$ po\\-przez odwzorowanie\nregularne. Definicj\u0105 \\ref{k-uniruleddef} wprowadzamy stopie\u0144 $\\K$-pros\\-to\\-kre\u015bl\\-no\u015bci dla rozmaito\u015bci afinicznych. Rozmaito\u015b\u0107 afiniczna $X$ \nma stopie\u0144 $\\K$-pros\\-to\\-kre\u015bl\\-no\u015bci co najwy\u017cej $d$ je\u017celi jest pokryta krzywymi pa\\-ra\\-me\\-try\\-cznymi stopnia co najwy\u017cej $d$. \nW Propozycji \\ref{k-uniruledofdeg} podajemy dwa warunki r\u00f3w\\-no\\-wa\u017c\\-ne. W pierwszym z nich \u017c\u0105damy tylko aby g\u0119sty zbi\u00f3r by\u0142 pokryty \ntakimi krzywymi. Kolejny warunek r\u00f3wnowa\u017cny zapewnia istnienie dominuj\u0105cego, generycznie sko\u0144czonego odwzorowania wielomianowego o stopniu ze wzgl\u0119du \nna pierwsz\u0105 wsp\u00f3\u0142\\-rz\u0119d\\-n\u0105 co najwy\u017cej $d$ z cylindra $\\K\\times W$ w rozmaito\u015b\u0107 $X$. Wprowadzamy now\u0105 Definicj\u0119 \\ref{k-uniruleddef} rozmaito\u015bci \n$\\K$-prostokre\u015blnych, kt\u00f3ra ma t\u0105 zalet\u0119 w stosunku do poprzedniej, \u017ce dzi\u0119ki Propozycji \\ref{indep} nie zale\u017cy od cia\u0142a podstawowego. W sekcji \n\\ref{rr} przedstawiamy analogiczne  wyniki dla cia\u0142a liczb rzeczywistych.\n\nTwierdzenie \\ref{Sfkuni} m\u00f3wi, \u017ce je\u017celi $f:X\\rightarrow Y$ jest generycznie sko\u0144\\-czo\\-nym odwzorowaniem regularnym pomi\u0119dzy rozmaito\u015bciami \nafi\\-nicz\\-ny\\-mi oraz rozmaito\u015b\u0107 $X$ jest $\\K$-prostokre\u015blna, to zbi\u00f3r $\\s_f$ punkt\u00f3w niew\u0142a\u015bciwo\u015bci odwzorowania $f$ jest r\u00f3wnie\u017c $\\K$-prostokre\u015blny. \nW drugim rozdziale zajmujemy si\u0119 ograniczeniem od g\u00f3ry stopnia $\\K$-prostokre\u015blno\u015bci zbioru $\\s_f$ w zale\u017cno\u015bci od stopnia odwzorowania $f$ oraz \nstopnia $\\K$-pro\\-sto\\-kre\u015bl\\-no\\-\u015bci rozmaito\u015bci $X$. Dla cia\u0142a liczb zespolonych $\\C$ przedstawiamy kompletne wyniki Twierdzenia \\ref{cn} i \\ref{multc}. \nPierwsze z nich daje ograniczenie przez iloczyn stopnia odwzorowania $f$ oraz stopnia $\\C$-prostokre\u015blno\u015bci rozmaito\u015bci $X$. Natomiast drugie m\u00f3wi, \n\u017ce dla $f:\\C^n\\rightarrow\\C^m$ stopnia $d$ stopie\u0144 $\\C$-prostokre\u015blno\u015bci $\\s_f$ wynosi co najwy\u017cej $d-1$. Przed\\-sta\\-wia\\-my r\u00f3wnie\u017c wyniki dla dowolnego \ncia\u0142a algebraicznie domkni\u0119tego $\\K$ - Twierdzenia \\ref{kn}, \\ref{kw}, \\ref{kkw} oraz dla cia\u0142a liczb rzeczywistych $\\R$ - Twier\\-dze\\-nia \n\\ref{rn1}, \\ref{rxw1}, \\ref{multc1}.\n\nW ostatnim rozdziale dowodzimy $\\K$-prostokre\u015blno\u015bci zbior\u00f3w zwi\u0105\\-za\\-nych ze zbiorem punkt\u00f3w sta\u0142ych dzia\u0142ania grupy algebraicznej. \nDok\u0142adnie w Twierdzeniu \\ref{glowne} dowodzimy, \u017ce je\u017celi nietrywialna, sp\u00f3jna grupa unipotentna dzia\u0142a efektywnie na rozmaito\u015bci afinicznej, \nto zbi\u00f3r pun\\-kt\u00f3w sta\u0142ych tego dzia\u0142ania jest $\\K$-prostokre\u015blny. W szczeg\u00f3lno\u015bci nie zawiera izolowanych punkt\u00f3w. \nNatomiast w Twierdzeniu \\ref{glowne1} pokazujemy, \u017ce gdy niesko\u0144czona grupa algebraiczna dzia\u0142a efektywnie na przestrzeni afinicznej $\\K^n$, to ka\u017cda hiperpowierzchnia \nzawarta w zbiorze punkt\u00f3w sta\u0142ych jest $\\K$-prostokre\u015blna.\n\n\\pagebreak\n\\addcontentsline{toc}{chapter}{Abstract}\n\\chapter*{Abstract}\n\nThe main goal of this thesis is to study $\\K$-uniruled sets that appear in affine geometry. \n\nThe first Chapter discusses the property of $\\K$-uniruledness and its equi\\-valent conditions.\nLet $\\K$ be an algebraically closed field. A curve $\\Gamma$ is called parametric if it is \nan image of the field $\\K$ under a regular map. In \\ref{k-uniruleddef} we define the degree of $\\K$-uniruledness \nfor an affine variety $X$. The degree of $\\K$-uniruledness \nof an affine variety $X$ is at most $d$ if $X$ is covered by parametric curves of degree at most $d$. \nIn Proposition \\ref{k-uniruledofdeg} we prove two equivalent conditions. In the first one we require that only a\ndense subset of the variety $X$ is covered with such curves. Another equivalent condition asserts the existence of an \naffine variety $W$ with $\\dim W=\\dim X-1$, and a dominant polynomial map $\\phi:\\K\\times W\\rightarrow X$ of degree in the first coordinate\nat most $d$. We state a new Definition \\ref{k-uniruleddef} according to which $\\K$-uniruled \nvarieties are those with a finite degree of $\\K$-uniruledness. One of the advantages of the new definition is that by \nProposition \\ref{indep} it does not depend on the base field. In section \\ref{rr} we present analogous results for the field of real \nnumbers.\n\nTheorem \\ref{Sfkuni} asserts that if $f:X\\rightarrow Y$ is a generically finite regular mapping between affine \nvarieties and $X$ is $\\K$-uniruled, then the set $\\s_f$ of points at which $f$ is not proper is also $\\K$-uniruled. \nIn the second Chapter we bound from above the degree of $\\K$-uniruledness of the set $\\s_f$ in terms of degree of \na mapping $f$ and degree of $\\K$-uniruledness of a variety $X$. For the field of complex numbers $\\C$ we get \nthe best possible results Theorems \\ref{cn} and \\ref{multc}. The first one gives a bound by the product of the degree \nof $f$ and degree $\\C$-uniruledness of a variety $X$. The second theorem bounds by $d-1$ degree of $\\C$-uniruledness \nof the set $\\s_f$ for a mapping $f:\\C^n\\rightarrow\\C^m$ of degree $d$. We also present results for an arbitrary \nalgebraically closed field $\\K$ - Theorems \\ref{kn}, \\ref{kw}, \\ref{kkw} and for the field of real numbers $\\R$ - Theorems \\ref{rn1}, \\ref{rxw1}, \\ref{multc1}.\n\nIn the last Chapter we show that some sets associated with the set of fixed points of an algebraic group action are $\\K$-uniruled. \nIn Theorem \\ref{glowne} we prove that if nontrivial connected unipotent group  acts effectively on an affine variety then \nthe set of fixed points of this action is a $\\K$-uniruled variety. In particular there are no isolated fixed points. \nIn Theorem \\ref{glowne1}, on the other hand, we show that if an arbitrary infinite algebraic group acts effectively on an affine space $\\K^n$, \nthen every hypersurface contained in the set of fixed points is $\\K$-uniruled.\n\n\\pagebreak\n\\addcontentsline{toc}{chapter}{Acknowledgements}\n\\chapter*{Acknowledgements}\n\nFirst and foremost I would like to thank my supervisor Zbigniew Jelonek for his constant and enormous help during my work on this thesis. Most of the\nresults presented here come from our two joint papers. It was a great pleasure and opportunity to learn from him. Throughout the thesis he provided\nme with encouragement and useful ideas. I am sincerely grateful for his effort.\n\nI would also like to express my gratitude to professors Rosa-Maria Mir\u00f3-Roig, Jaros\u0142aw Wi\u015bniewski and Mikhail Zaidenberg for many interesting and \nfruitful discussions. I thank also S\u0142awomir Cynk with whom I had written my master thesis, and who introduced me to algebraic geometry.\n\nI acknowledge the administrative help of Anna Poczma\u0144ska.\n\nWriting this thesis would not be possible without the constant support of my family. I thank a lot my mum for her love and motivation.\n\nI thank my friends: Piotr Achinger, Agnieszka Bodzenta, Maria Donten-Bury, Micha\u0142 Farnik, Wojciech Lubawski, Mateusz Micha\u0142ek and Karol Palka \nfor interesting discussions, questions, answers and a great working atmosphere.\n\nI acknowledge a support of Polish Ministry of Science and Higher Education grant N N201 413139.\n\\pagebreak\n\n\\chapter{$\\K$-uniruled varieties}\n\nThis Chapter can be treated as a preliminary to the rest of the thesis.\n\nWe are going to remind an old, well known definition of $\\K$-uniruled varieties and introduce our new one. The relation between both definitions will \nbe given. Additionally for $\\K$-uniruled varieties we introduce a degree of $\\K$-uniruledness.  \n\nWe end the Chapter by stating a few open problems concerning bounding the degree of $\\K$-uniruledness. \n\nIn the whole Chapter, unless stated otherwise, $\\K$ is assumed to be an arbitrary algebraically closed field.\n\n\\section{Introduction}\n\nWe begin with a presentation of some necessary preliminaries.\n\n\\begin{proposition}\\label{birrat} Let $\\Gamma \\subset \\K^m$ be an affine curve. The following conditions are equivalent:\n\\begin{enumerate}\n\\item there exists a regular dominant map $\\varphi:\\K \\rightarrow \\Gamma$,\n\\item there exists a regular birational map $\\varphi: \\K \\rightarrow \\Gamma$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nIt is an immediate consequence of L\\\"{u}roth's Theorem \\ref{luroth}. \n\\end{proof}\n\n\\begin{theorem}[L\\\"{u}roth]\\label{luroth}\nSuppose $\\Le$ and $\\M$ are arbitrary fields (not necessarily algebraically closed), and $\\xi$ is transcendental over $\\Le$, such that \n$$\\Le\\subsetneq\\M\\subset\\Le(\\xi).$$ Then $\\M=\\Le(\\eta)$ for some $\\eta$ in $\\Le(\\xi)$.\n\\end{theorem}\n\n\\begin{proof}(see \\cite{pi})\nSuppose $\\eta\\in\\Le(\\xi)\\setminus\\Le$, then \n$$\\eta=\\frac{f(\\xi)}{g(\\xi)}$$\nfor some polynomials $f,g\\in\\Le[X]$, at least one of which is of positive degree. Moreover we may assume that $f$ and $g$ have no common factor in $\\Le[X]$ of positive degree. Since $\\eta\\notin\\Le$, we have\n$$\\deg(f(X)-g(X)\\eta)=\\max(\\deg(f),\\deg(g)).$$\nClearly $f(\\xi)-g(\\xi)\\eta=0$, so $\\xi$ is algebraic over $\\Le(\\eta)$, hence $\\eta$ is transcendental over $\\Le$. As a consequence $f(X)-g(X)\\eta$ is irreducible in $\\Le[X]$, and\n$$\\max(\\deg(f),\\deg(g))=[\\Le(\\xi):\\Le(\\eta)].$$\n\nChoosing $\\eta$ from $\\M\\setminus\\Le$ shows that $\\Le(\\xi)$ is algebraic of finite degree over $\\M$. Denote $[\\Le(\\xi):\\M]=n$, then the minimal polynomial of $\\xi$ over $\\M$ can be written as\n$$\\frac{p(\\xi)}{q(\\xi)}\\sum_{i=0}^{n}f_i(\\xi)\\xi^i=0,$$\nwhere $$p,q,f_i\\in\\Le[X],\\text{ }\\frac{p(\\xi)}{q(\\xi)}f_i(\\xi)\\in\\M,$$\nand polynomials $f_i$ with no common factor of positive degree. We have $f_n(\\xi)\\neq 0$ and since $\\xi$ is not algebraic over $\\Le$\n$$\\eta:=\\frac{f_j(\\xi)}{f_n(\\xi)}\\notin\\Le$$\nfor some $j$. Clearly $\\eta\\in\\M$. Assume equation \n$$\\eta=\\frac{f(\\xi)}{g(\\xi)}$$\nholds as in the first part of the proof. Since $\\xi$ is a root of $f(X)-g(X)\\eta\\in\\M[X]$, this polynomial is divisible in $\\M[X]$ by $\\frac{p(\\xi)}{q(\\xi)}\\sum_{i=0}^{n}f_i(\\xi)X^i$. Therefore \n$$f(X)g(\\xi)-g(X)f(\\xi)=r(X,\\xi)\\sum_{i=0}^{n}f_i(\\xi)X^i,\\;\\;\\;\\;\\;\\;\\;\\;(\\star)$$\nfor some polynomial $r\\in\\Le[X,Y]$. \n\nSay the degree  of $f(X)g(\\xi)-g(X)f(\\xi)$ in $\\xi$ is $m$. This is the same as the degree in $X$, so $n\\leq m$. Additionally\n$$m\\leq\\max(\\deg(f),\\deg(g))\\leq\\max(\\deg(f_j),\\deg(f_n))\\leq\\max(\\deg(f_0),\\dots).$$\nBy $(\\star)$, there must be equalities, and $r$ must be constant in $\\xi$. But now $f(X)$ and $g(X)$ are both divisible by $r(X)$, hence $r$ is also constant in $X$. We get that $n=m$, and $\\M=\\Le(\\eta).$  \n\\end{proof}\n\n\\begin{definition}\nAn affine curve $\\Gamma\\subset\\K^m$ is called a \\textup{parametric curve}, if equivalent conditions from Proposition \\ref{birrat} hold.\n\\end{definition}\n\nWe are ready to present the formerly known definition of $\\K$-uniruled varieties, which was introduced in \\cite{st}.\n\n\\begin{definition}\nAn affine variety $X\\subset\\K^m$ is called \\textup{$\\K$-uniruled}, if for every point $x\\in X$ there exists a parametric curve $l_x\\subset X$ passing through $x$. In other words $X$ is covered by parametric curves.\n\\end{definition}\n\nA parametric curve is an image of the field, which is irreducible, so it is also irreducible. Hence each parametric curve on a variety is contained in some irreducible component. So a variety is $\\K$-uniruled if and only if all its irreducible components are $\\K$-uniruled.\n\nThe following characterization of components of $\\K$-uniruled varieties is known \\cite{st}.\n\n\\begin{proposition}\\label{k-uniruled}\nLet $\\K$ be an uncountable field and let $X\\subset \\K^m$ be an irreducible affine variety. The following conditions  are equivalent:\n\\begin{enumerate}\n\\item $X$ is $\\K$-uniruled,\n\\item there exists an open, non-empty subset $U\\subset X$, such that for every point $x\\in U$ there exists a parametric curve $l_x\\subset X$ passing through $x$, \n\\item there exists an affine variety $W$ with $\\dim W = \\dim X-1$, and a regular dominant map $\\phi: \\K\\times W\\rightarrow X$. \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nImplication $(1)\\Rightarrow (2)$ is trivial. \n\nTo prove implication $(2)\\Rightarrow (3)$ let us define\n$$S_d=\\{\\varphi:\\K\\rightarrow\\K^m\\text{ such that }\\varphi(\\K)\\subset X\\text{ and }\\deg\\varphi=d\\}.$$ \nEach $\\varphi=(\\varphi_1,\\dots,\\varphi_m)$, where $\\varphi_i(t)=\\sum_{j=0}^{d}a_{j}^{i} t^{j}$, corresponds to a point \n$$(a^1_0,\\dots,a^m_d)\\in(\\K^{d+1})^m\\setminus(\\K^d\\times\\{0\\})^m.$$ \nConditions $\\varphi(\\K)\\subset X$ are polynomial equations, so $S_d$ is a quasiprojective variety. Consider the morphism: \n$$F_d:\\K\\times S_d\\ni (t,\\varphi)\\rightarrow \\varphi(t)\\in X.$$ \nLet us denote the image by $X_d:=F_d(\\K\\times S_d)$. We know that $U\\subset\\bigcup_{d\\in N}X_d$, since for every point $x\\in U$ there is a parametric curve $l_x\\subset X$ passing through $x$. Let $\\overline{X_d}$ be the closure of $X_d$ in $X$. From Baire's Theorem \\ref{baire} for the Zariski topology there exists $d$ such that $\\overline{X_d}=X$. Now the map \n$$F_d:\\K\\times S_d:\\rightarrow X$$\nis dominant, so when we restrict it to some irreducible component $Y$ (suppose $Y\\subset \\K^M$)  of $S_d$ the map $\\Phi=F_d\\vert_{\\K\\times Y}$ is \nstill dominant. Suppose $\\dim X=n$ and $\\dim(Y)=s$, on an open subset of $X$ fibers of the map $\\Phi$ are of dimension $s+1-n$. Let $x$ be one of \nsuch points. From the construction of the set $S_d$ we know that the fiber $F=\\Phi^{-1}(x)$ does not contain any line of the type $\\K\\times\\{y\\}$, \nso in particular the image $F'$ of the fiber $F$ under projection $\\K\\times Y\\rightarrow Y$ has the same dimension. For general linear subspace \n$L\\subset \\K^M$ of dimension $M+n-s-1$ the dimension of $L\\cap F'$ equals to $0$. Let us fix such $L$, and let $R$ be any irreducible component of \n$L\\cap Y$ intersecting $F'$. Now mapping the\n$$\\Phi\\vert_{\\K\\times R}:\\K\\times R\\rightarrow X$$\nconfirms the assertion, since it has one fiber of dimension $0$ (at $x$), the dimension of $R$ is $n-1$ (at most $n-1$, because of the $0$ dimensional fiber, \nat least $n-1$ because of the small dimension of $L$), so as a consequence it is dominant.   \n\nTo prove implication $(3)\\Rightarrow (1)$ we note that for \n$$\\phi: \\K\\times W\\ni (t,w)\\to \\phi(t,w)\\in X$$ \nthere exists $d\\in\\N$ such that $\\deg_t \\phi \\leq d$. Next we use implication $(3)\\Rightarrow (1)$ from Proposition \\ref{k-uniruledofdeg}, which gives us condition $(1)$. \n\\end{proof}\n\nBaire's Theorem reads as follows.\n\n\\begin{theorem}[Baire]\\label{baire}\nLet $\\K$ be an uncountable field, let $X\\subset\\K^m$ be an irreducible affine variety, and let $X_d$ for $d\\in\\N$ be its closed subsets. If $\\bigcup_{d\\in N}X_d$ contains a non-empty open subset $U$ of $X$, then $X_d=X$ for some $d$.  \n\\end{theorem}\n\n\\begin{proof}\nWe are going to prove it by induction on the dimension of $X$. If $\\dim X=0$, then $X$ is just a point, and the assertion is clearly trivial. \n\nWhen $\\dim X>0$, then there exists a regular function $f:\\K^m\\rightarrow \\K$ non-constant on $X$ (one of coordinates is good). \n\nAssume that the assertion is false, which means that all $X_d$ are of a lower dimension than $X$, and consider sets \n$$X^c=X\\cap\\{f=c\\}$$ \nfor $c\\in\\K$, which are of pure codimension one. When $U\\cap X^c\\neq\\emptyset$, then it means that one of irreducible \ncomponents $R$ of $X^c$ satisfies the conditions of the theorem with sets $R\\cap X_d$ for $d\\in\\N$. From the inductive \nassumption we know that $R$ equals to $R\\cap X_d$ for some $d$, so to one of irreducible components of $X_d$ (they are of the same dimension). \nBut there are only countably many irreducible components of $X_d$ for some $d\\in\\N$, and uncountably many $c\\in\\K$ for \nwhich $U\\cap X^c\\neq\\emptyset$, since open set $U$ intersects almost all $X_c$ (all, except the finite number). The contradiction \nshows that the hypothesis was false.\n\\end{proof}\n\n\\section{The degree of $\\K$-uniruledness}\n\nFor an uncountable field $\\K$ there is a nice characterization of $\\K$-uniruled varieties, namely Proposition \\ref{k-uniruled}. However to work over an arbitrary algebraically closed field we need a refined version of the definition. We introduce it in our papers \\cite{jela}, \\cite{jela2}. It coincides with the older one for uncountable fields.\n\nMoreover with the new definition of $\\K$-uniruledness we are able to measure the degree of $\\K$-uniruledness of $\\K$-uniruled varieties.\n\n\\begin{definition}\nAn affine curve $\\Gamma\\subset \\K^m$ is called a \\textup{parametric curve of degree at most $d$}, if there exists a polynomial dominant map $f:\\K\\rightarrow \\Gamma$ of degree at most $d$ (by degree of $f=(f_1,\\dots,f_m)$ we mean $\\deg f:=\\max_i \\deg  f_i$).\n\\end{definition}\n\nNow we prove the following:\n\n\\begin{proposition}\\label{k-uniruledofdeg}\nLet $X\\subset \\K^m$ be an irreducible affine variety of dimension $n$, and let $d$ be an integer. The following conditions  are equivalent:\n\\begin{enumerate}\n\\item for every point $x\\in X$ there exists a parametric curve $l_x\\subset X$ of degree at most $d$ passing through $x$, \n\\item there exists a dense in the Zariski topology subset $U\\subset X$, such that for every point $x\\in U$ there exists a parametric curve $l_x\\subset X$ of degree at most $d$ passing through $x$, \n\\item there exists an affine variety $W$ with $\\dim W = \\dim X-1$, and a dominant polynomial map $\\phi: \\K\\times W\\ni (t,w)\\to \\phi(t,w)\\in X$, such that $\\deg_t \\phi \\leq d$. \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nImplication $(1)\\Rightarrow (2)$ is obvious. \n\nWe prove $(2)\\Rightarrow (1)$. Suppose that \n$$X=\\{x\\in \\K^m:f_1(x)=0,\\dots,f_r(x)=0\\}.$$\nFor $a=(a_1,\\dots,a_m)\\in \\K^m$ and $b=(b_{1,1}:\\dots:b_{d,m})\\in\\p^M$, where $M=dm-1$, let $$\\varphi_{a,b}:\\K\\ni t\\rightarrow (a_1+b_{1,1}t+\\dots+b_{1,d}^dt^d,\\dots,a_m+b_{m,1}t+\\dots+b_{m,d}^dt^d)\\in\\K^m$$\nbe a parametric curve of degree at most $d$. Consider a variety and a projection\n$$\\K^m\\times\\p^M\\supset V=\\{(a,b)\\in \\K^m\\times\\p^M:\\forall_{t,i}\\;f_i(\\varphi_{a,b}(t))=0\\}\\ni(a,b)\\rightarrow a\\in \\K^m.$$\nThe definition of the set $V$ says that parametric curves $\\varphi_{a,b}$ are contained in $X$. Hence the image of the projection is contained in $X$ and contains dense set $U$, since through every point of $U$ passes a parametric curve of degree at most $d$. But since $\\p^M$ is complete and $V$ is closed we have that the image is closed, and hence it is the whole set $X$. \n\nLet us prove $(2)\\Rightarrow (3)$. For some affine chart $V_j=V\\cap \\{b_j=1\\}$ the above map is dominant. We consider the following dominant mapping\n$$\\Phi: \\K\\times V_j\\ni (t,\\phi)\\rightarrow \\phi(t)\\in X.$$\nAfter replacing $V_j$ by one of its irreducible components $Y\\subset \\K^M$ the map remains dominant. Suppose $\\dim X=n$ and $\\dim(Y)=s$, on an open subset of $X$ fibers of the map $\\Phi'=\\Phi\\vert_{k\\times Y}$ are of dimension $s+1-n$. Let $x$ be one of such points. From the construction of the set $V$ we know that the fiber $F=\\Phi'^{-1}(x)$ does not contain any line of the type $\\K\\times\\{y\\}$, so in particular the image $F'$ of the fiber $F$ under projection $\\K\\times Y\\rightarrow Y$ has the same dimension. For general linear subspace $L\\subset \\K^M$ of dimension $M+n-s-1$ the dimension of $L\\cap F'$ equals to $0$. Let us fix such $L$, and let $R$ be any irreducible component of $L\\cap Y$ intersecting $F'$. Now mapping \n$$\\Phi'\\vert_{\\K\\times R}:\\K\\times R\\rightarrow X$$ \nconfirms the assertion, since it has one fiber of dimension $0$ (at $x$), dimension of $R$ is $n-1$ (at most $n-1$, because of the $0$ dimensional fiber, at least $n-1$ because of the small dimension of $L$), so as a consequence it is dominant.   \n\nTo prove the implication $(3)\\Rightarrow (2)$ it is enough to notice that for each $w\\in W$ the map \n$$\\phi_w: \\K\\ni t\\rightarrow\\phi(t,w)\\in X$$\nis a parametric curve of degree at most $d$ or it is constant. Image of $\\phi$ contains an open dense subset, so after excluding points with \ninfinite preimages (closed set, at most of codimension one) we get an open set $U$ with required properties. \n\\end{proof}\n\nWe are ready to define the degree of $\\K$-uniruledness - a parameter which measures degree of parametric curves covering a variety.\n\n\\begin{definition}\\label{k-uniruleddef}\nWe say that an affine variety $X$ has \\textup{degree of $\\K$-uni\\-ruled\\-ness at most $d$}, if all its irreducible components satisfy the above conditions. A \\textup{degree of $\\K$-uniruledness} is the minimum $d$ for which it has degree of $\\K$-uni\\-ruled\\-ness at most $d$. An affine variety is called \\textup{$\\K$-uniruled}, if it is $\\K$-uniruled of some degree.\n\\end{definition}\n\nThe condition (3) from Propositions \\ref{k-uniruled} and \\ref{k-uniruledofdeg} is clearly the same, so both definitions of \n$\\K$-uniruled varieties coincide for an uncountable field $\\K$. From now on we are going to use only the second - Definition \\ref{k-uniruleddef}.\n\n\\begin{example}\nLet $X\\subset \\K^n$ be a hypersurface of degree $d<n$. It is well known that $X$ is covered by affine lines, so its degree of $\\K$-uniruledness is one.\n\\end{example}\n\n\\begin{proof}\nLet $f\\in\\K[x_1,\\dots,x_n]$ be an equation of $X$. We have to show that through every point of $X$ passes a line. Without the loss of generality \nit is enough to assume that $O=(0,\\dots,0)\\in X$ and show that through $O$ passes a line (it is because we can make a linear change of coordinates \nand the degree of $f$ stays unchanged). A parametrization of such line $L$ is \n$$\\K\\ni t\\rightarrow (a_1t,\\dots,a_nt)\\in\\K^n,$$ \nfor $(a_1,\\dots,a_n)\\in\\p^{n-1}$. Line $L$ is contained in variety $X$ if and only if $f(a_1t,\\dots,a_nt)=0$ for all $t\\in\\K$. It happens if and only if $f_i(a_1,\\dots,a_n)=0$ for $i=1,\\dots,d$ (where $f_i$ is the part of $f$ of degree $i$). Now we use the fact that $d$ equations define a variety in $\\p^{n-1}$ of dimension at least $n-1-d$, which for $d<n$ is greater or equal to zero. It means that there exists $(a_1,\\dots,a_n)\\in\\p^{n-1}$ for which the corresponding line is contained in $X$.\n\\end{proof}\n\n\\begin{proposition}\\label{indep} The degree of $\\K$-uniruledness of a variety does not depend on the base field. Namely let $\\K\\subset\\Le$ be a field extension, and let $f_1,\\dots,f_r\\in\\K[x_1,\\dots,x_m]$. Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item $\\K X:=\\{x\\in\\K^m:f_1(x)=0,\\dots,f_r(x)=0\\}\\subset\\K^m$ has degree of $\\K$-uniruledness at most $d$,\n\\item $\\Le X:=\\{x\\in\\Le^m:f_1(x)=0,\\dots,f_r(x)=0\\}\\subset\\Le^m$ has degree of $\\Le$-uniruledness at most $d$\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nImplication $(1)\\Rightarrow (2)$. Observe that the set $\\K X$ is dense in $\\Le X$. \nIndeed let $g\\in\\Le[x_1,\\dots,x_m]$ and $g\\equiv 0$ on $\\K X$. Let $\\mathcal{B}$ be any basis of vector space $\\Le$ over $\\K$. Now $g$ has a decomposition\n$$g=b_1g_1+\\dots+b_kg_k\\text{ for }g_1,\\dots,g_k\\in\\K[x]\\text{ and distinct }b_1,\\dots,b_k\\in\\mathcal{B}.$$\nNow $g\\equiv 0$ on $\\K X$, so also for each $i$ there is $g_i\\equiv 0$ on $\\K X$. Hence for each $i$ $g_i$ belongs to the radical of the ideal $(f_1,\\dots,f_r)$ in $\\K[x]$. It implies that for each $i$ polynomial $g_i$ belongs to the radical of the ideal $(f_1,\\dots,f_r)$ in $\\Le[x]$, and as a consequence $g$ also belongs. So $g\\equiv 0$ on $\\Le X$, and it follows that $\\K X$ is dense in $\\Le X$. Of course any $\\K$ parametric curve in $\\K X$ extends to a $\\Le$ parametric curve in $\\Le X$. From the assumption and Proposition \\ref{k-uniruledofdeg} condition $(2)$ we get that $X\\subset\\Le^m$ has degree of $\\Le$-uniruledness at most $d$.\n\nImplication $(2)\\Rightarrow (1)$. As in the proof of Proposition \\ref{k-uniruledofdeg} for \n$a=(a_1,\\dots,a_m)\\in \\Le^m$ and $b=(b_{1,1}:\\dots:b_{d,m})\\in\\p\\Le^M$, where $M=dm-1$, let \n$$\\varphi_{a,b}:\\Le\\ni t\\rightarrow (a_1+b_{1,1}t+\\dots+b_{1,d}^dt^d,\\dots,a_m+b_{m,1}t+\\dots+b_{m,d}^dt^d)\\in\\Le^m$$\nbe a parametric curve of degree at most $d$. We consider a variety and a projection\n$$\\Le^m\\times\\p\\Le^M\\supset V=\\{(a,b)\\in\\Le^m\\times\\p\\Le^M:\\forall_{t,i}\\;f_i(\\varphi_{a,b}(t))=0\\}\\ni(a,b)\\xrightarrow{p} a\\in \\Le^m.$$\nFrom our assumption the image of the projection is the whole $\\Le X$. But the above variety $V$ and also projection $p$ are defined over the \nsmaller field $\\K$. The anologous argument to the one from opposite implication shows that over $\\K$ the image of the projection is also the \nwhole variety $\\K X$, which gives condition $(1)$. Indeed if for some $g\\in\\K[x]$ $g\\equiv 0$ on $p(\\K V)$ then $g\\circ p$ belongs to the radical \nof the ideal $(h_1,\\dots,h_s)$ in $\\K[x]$, where $h_i$ are equations defining $V$. So $g\\circ p$ belongs to the radical of the ideal \n$(h_1,\\dots,h_s)$ in $\\Le[x]$ and $g\\equiv 0$ on $p(\\Le V)$. It follows that $g$ belongs to the radical of the ideal $(f_1,\\dots,f_r)$ in $\\Le[x]$ \nwhich implies that $g$ belongs to the radical of the ideal $(f_1,\\dots,f_r)$ in $\\K[x]$.\n\\end{proof}\n\n\\section{A real field case}\\label{rr}\n\nFor the whole section we assume that the base field is $\\R$. \n\nLet us specify that by a \\textit{real parametric curve of degree at most $d$} in a semialgebraic set $X\\subset \\R^n$ we mean the image of a real polynomial mapping $f:\\R\\to X$ of degree at most $d$. Thus in general a real parametric curve does not have to be algebraic, but only semialgebraic. The real counterpart of Proposition \\ref{k-uniruledofdeg} is:\n\n\\begin{proposition}\\label{real}\nLet $X\\subset \\R^m$ be a closed irreducible semialgebraic set of dimension at least two, and let $d$ be an integer. The following conditions are equivalent:\n\\begin{enumerate}\n\\item for every point $x\\in X$ there exists a parametric curve $l_x\\subset X$ of degree at most $d$ passing through $x$,\n\\item there exists a dense in the classical topology subset $U\\subset X$, such that for every point $x\\in U$ there exists a parametric curve $l_x\\subset X$ of degree at most $d$ passing through $x$,\n\\item for every polynomial mapping $f:X\\to\\R^n$, and every sequence $(x_k)_{k\\in\\N}\\subset X$ such that $f(x_k)\\to a\\in \\R^n$ there exists a semialgebraic curve $W$ and a generically finite polynomial map $\\phi:\\R\\times W\\ni (t,w)\\to \\phi(t,w)\\in X$ such that $\\deg_t \\phi \\leq d$, \nand there exists a sequence $(y_k)_{k\\in\\N}\\in\\R\\times W$ such that $f(\\phi(y_k))\\to a$. Moreover, if $x_k\\to\\infty$ then also $\\phi(y_k)\\to\\infty$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nFirst we prove implication $(2)\\Rightarrow (1)$. Suppose that \n$$X=\\{x\\in\\R^m:f_1(x)=0,\\dots,f_r(x)=0\\;g_1(x)\\geq 0,\\dots, g_s(x)\\geq 0\\}.$$ \nFor $a=(a_1,\\dots,a_m)\\in\\R^m$ and $b=(b_{1,1},\\dots,b_{d,m})\\in\\R^M$, where $M=dm$, let $$\\varphi_{a,b}:\\R\\ni t\\rightarrow (a_1+b_{1,1}t+\\dots+b_{1,d}^dt^d,\\dots,a_m+b_{m,1}t+\\dots+b_{m,d}^dt^d)\\in\\R^m$$ \nbe a parametric curve of degree at most $d$. If there exists a parametric curve of degree at most $d$ passing through $a$, then after reparametrization we can assume that it is $\\varphi_{a,b}$ with $\\sum_{i,j} b_{i,j}^2=1$. This means that $b\\in S_{M-1}$, where $S_{M-1}$ denotes the unit sphere in $\\R^M$ with the center in the origin. Consider a semialgebraic set\n$$V=\\{(a,b)\\in\\R^m\\times S_{M-1}:\\forall_{t,i}\\;f_i(\\varphi_{a,b}(t))=0, \\forall_{t,j}\\;g_j(\\varphi_{a,b}(t))\\geq 0\\}.$$\nThe definition of the set $V$ says that parametric curves $\\varphi_{a,b}(t)$ are contained in $X$. It is easy to see that $V$ is closed. For any $a\\in X$, by the assumption there is a sequence of points $a_k\\to a$, such that for every $k$ there is a parametric curve $\\varphi_{a_k,b_k}\\in V$. We can assume that $\\Vert a_k\\Vert<\\Vert a\\Vert+1$ for all $k$. The set $V$ is closed, the sequence $(a_k,b_k)_{k\\in\\N}\\subset V$ is bounded, so there exists a subsequence $(a_{k_r},b_{k_r})_{r\\in\\N}$ which converges to $(a,b)\\in V$. Now parametric curve $\\varphi_{a,b}\\subset X$ of degree at most $d$ passes through $a$.\n\nTo prove $(1)\\Rightarrow (3)$ consider a semialgebraic set $V$ as above. We have a surjective mapping $$\\Phi: \\R\\times V\\ni (t,\\varphi_{a,b})\\to \\varphi_{a,b}(t)\\in X.$$ Let $f:X\\to\\R^n$ be a polynomial mapping,  \nand let $f(x_k)\\to a\\in \\R^n$ for a sequence $(x_k)_{k\\in\\N}\\subset X$. Put $g=f\\circ\\Phi$. Hence there exists a sequence $z_k\\in \\R\\times V$ such that $g(z_k)\\to a$. Due to the real Curve Selection Lemma analogous to Lemma \\ref{curve} there is a semialgebraic curve $W_1\\subset \\R\\times V$ such that $a\\in \\overline{g(W_1)}$. Take $W_2=p_2(W_1)$, where $p_2: \\R\\times V \\to V$ is a projection. If $W_2$ is a curve then let $W:=W_2$, if it is a point we take as $W$ any\nsemialgebraic curve in $V$ containing a point $p_2(W_1)$. Now $(W,\\Phi\\vert_{\\R\\times W})$ is a good pair.\n\nFinally to prove $(3)\\Rightarrow (2)$ it is enough to take $f$ being an identity in the third condition.\n\\end{proof}\n\nNote that the first two conditions are the same as for general algebraically closed field in Proposition \\ref{k-uniruledofdeg}.\n\n\\begin{definition}\\label{r-uniruleddef}\nWe say that a closed semialgebraic set $X$ has \\textup{degree of $\\R$-uni\\-ruled\\-ness at most $d$}, if all its irreducible components satisfy the above conditions. A \\textup{degree of $\\R$-uniruledness} is the minimum $d$ for which it has degree of $\\R$-uni\\-ruled\\-ness at most $d$. A closed semialgebraic set is called \\textup{$\\R$-uniruled}, if it is $\\R$-uniruled of some degree.\n\\end{definition}\n\n\\begin{example}\nLet $X=\\{ (x,y)\\in\\R^2 : x\\geq 0, y\\ge 0\\}$. It is easy to check that the  degree of $\\R$-uniruledness of $X$ is two. It has a ruling with curves $l_a$ of the form $l_a=\\{(a, t^2):t\\ge 0\\}$ for  $a\\ge 0$.\n\\end{example}\n\n\\section{Remarks}\n\nThe statement of Proposition \\ref{k-uniruled} suggests a question.\n\n\\begin{question}\nDoes equivalence from Proposition \\ref{k-uniruled} hold also for countable fields?\n\\end{question}\n\nHowever it is known that there exist K3 surfaces, which are not $\\C$-uniruled, but their dense subset is covered by parametric curves. In this sense \nconditions $(1)$ and $(2)$ from  Proposition \\ref{k-uniruledofdeg} with no upper bound on the degree of parametric curves are not equivalent.\n\n\\pagebreak\n\n\\chapter[Bounding degree of $\\K$-uniruledness]{Bounding degree of $\\K$-uniruledness of the non-properness set of a polynomial mapping}\n\nIn this Chapter we are going to recall the definition, motivation and some known facts about the set $\\s_f$ of points at which regular mapping $f$ is not proper. It is known that if $f:X\\rightarrow Y$ is a generically finite regular dominant map between affine varieties then the set $\\s_f$ is a hypersurface in $Y$ or it is empty. If $X$ is additionally $\\K$-uniruled then the set $\\s_f$ is also $\\K$-uniruled. The main goal of this Chapter is to bound from above the degree of $\\K$-uniruledness of the set $\\s_f$ in terms of the degree of the map $f$ and the degree of $\\K$-uniruledness of $X$. We prove:\n\n\\begin{theoremm}\nLet $\\K=\\C$, or $\\R$, and let $f:\\K^n\\rightarrow \\K^m$ be a generically\nfinite polynomial mapping of degree $d$. Then the set $\\s_f$ has degree of $\\K$-uniruledness at most $d-1$ or it is empty.\n\\end{theoremm}\n\n\\begin{theoremm}\nLet $X$ be an affine variety with degree of $\\C$-uniruledness at most $d_1$ and let\n$f:X\\rightarrow\\C^m$ be a generically finite polynomial mapping of degree\n$d_2$. Then the set $\\s_f$ has degree of $\\C$-uniruledness at most $d_1d_2$ or it is empty.\n\\end{theoremm}\n\n\\begin{theoremm}\nLet $X$ be a closed semialgebraic set with degree of $\\R$-uni\\-ruled\\-ness at most $d_1$, and let \n$f:X\\rightarrow\\R^m$ be a generically finite polynomial mapping of degree $d_2$. \nThen the set $\\s_f$ has degree of $\\R$-uniruledness at most $2d_1d_2$ or it is empty.\n\\end{theoremm}\n\n\\begin{theoremm}\nLet $\\K$ be an arbitrary algebraically closed field, and let\n$f:\\K^n\\rightarrow\\K^m$ be a generically finite polynomial mapping of degree\n$d$. Then the set $\\s_f$ has degree\nof $\\K$-uniruledness at most $d$ or it is empty.\n\\end{theoremm}\n\n\\section{The set $\\s_f$ of points at which regular map $f$ is not proper}\n\nWe begin with some well-known facts concerning proper maps between topological spaces. It will provide a motivation for the definition of proper maps between affine varieties over arbitrary field.\n\n\\begin{definition}\nLet $X,Y$ be topological spaces, and let $f:X\\rightarrow Y$ be a continuous map. We say that $f$ is \\textup{proper} if for every compact set $K\\subset Y$ the set $f^{-1}(K)$ is also compact. \n\\end{definition}\n\n\\begin{definition}\nLet $X,Y$ be topological spaces, and let $f:X\\rightarrow Y$ be a continuous map. We say that $f$ is \\textup{proper at a point $y\\in\\overline{f(X)}$} if there exists an open neighborhood $U$ of $y$, such that $f\\vert_{f^{-1}(U)}:f^{-1}(U)\\rightarrow U$ is a proper map. \n\\end{definition}\n\n\\begin{proposition}\\label{topeq}\nLet $X$ be a topological space, $Y$ be a locally compact topological space, and let $f:X\\rightarrow Y$ be a continuous map. Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $f$ is proper at every point of $\\overline{f(X)}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nOne implication is obvious. Namely if $f$ is proper then it is also proper at every point, we can take an open neighborhood $U=Y$. \n\nTo prove the other implication suppose that $f$ is proper at every point. Let $K\\subset Y$ be a compact set, so $K':=K\\cap\\overline{f(X)}$ is also \ncompact. For any $y\\in K'$ there exists an open neighborhood $U_y$ such that $f\\vert_{f^{-1}(U_y)}$ is proper. From local compactness we have that \nfor any $y\\in K'$ there exists an open neighborhood $V_y$ such that $\\overline{V_y}$ is compact subset of $U_y$. The family of sets $\\{V_y\\}_{y\\in K'}$ forms an open cover of compact set $K'$. Hence there are $y_1,\\dots,y_k\\in K'$ such that $K'\\subset V_{y_1}\\cup\\dots\\cup V_{y_k}$. Therefore:\n$$f^{-1}(K)=f^{-1}(K')=(f\\vert_{f^{-1}(U_1)})^{-1}(K'\\cap\\overline{V_1})\\cup\\dots\\cup (f\\vert_{f^{-1}(U_k)})^{-1}(K'\\cap\\overline{V_k}).$$ \nThe right hand side is a finite sum of compact sets, so $f^{-1}(K)$ is also compact.\n\\end{proof}\n\nIf $X,Y$ are affine varieties over $\\C$, then there exists a purely algebraic condition for a regular map $f:X\\rightarrow Y$ to be proper. \n\n\\begin{proposition}\nLet $X,Y$ be affine varieties over $\\C$, and let $f:X\\rightarrow Y$ be a regular map. Then $f$ is proper in the classical topology if and only if $f^*:\\C[Y]\\ni h\\rightarrow h\\circ f\\in\\C[X]$ is an integral extension of rings.\n\\end{proposition}\n\n\nIt an easy consequence of \\cite{twwi} and the fact that monic polynomials with bound\\-ed coefficients have bounded roots.\n\nMotivated by the above equivalence we are going to extend the definition of proper maps to maps between affine varieties over \narbitrary algebraically closed field. \n\nFor maps between affine varieties the properties of being finite and of being proper coincide, hence we define finite maps. \n\n\\begin{definition}\\label{properdef}\nLet $X,Y$ be affine varieties, and let $f:X\\rightarrow Y$ be a regular, dominant map. We say that $f$ is \\textup{finite} if the \ninduced ring extension $f^*:\\K[Y]\\ni h\\rightarrow h\\circ f\\in\\K[X]$ is an integral extension of rings.\n\\end{definition}\n\nFinally we are ready to introduce the set $\\s_f$. Let $f:X\\rightarrow Y$ be a generically finite regular map of affine varieties. A measure of non-properness of $f$ is the set $\\s_f$ of points, at which the map $f$ is not proper. \n\n\\begin{definition}\nLet $X,Y$ be affine varieties, and let $f:X\\rightarrow Y$ be a regular map. We say that $f$ is \\textup{proper} at a point $y\\in\\overline{f(X)}$, \nif there exists a Zariski open neighborhood $U$ of $y$ such that $f^{-1}(U)$ is also affine, and \n$$f\\vert_{f^{-1}(U)}:f^{-1}(U)\\rightarrow U$$ is a finite map. We denote the set of points from $\\overline{f(X)}$ at which $f$ is not proper by $\\s_f$.\n\\end{definition}\n\nThe set $\\s_f$ was first introduced by Jelonek in \\cite{je93}, and it is the main object of study of the papers \\cite{je99, je01, st, st2}. The set $\\s_f$ indicates how far map $f$ is from a proper mapping, and therefore it is important in affine algebraic geometry as well as in applied mathematics (\\cite{ha, je03, je05, sa}).\n\nBeing finite is a local property. We have the following equivalences in the spirit of Proposition \\ref{topeq}. \n\n\\begin{proposition}[\\cite{sh}, page $62$, Theorem $5$] \nLet $X,Y$ be affine varieties, and let $f:X\\rightarrow Y$ be a regular map. The following conditions are equivalent:\n\\begin{enumerate}\n\\item $f$ is finite,\n\\item for every open set $U\\subset Y$ such that $f^{-1}(U)$ is affine the restriction $f\\vert_{f^{-1}(U)}:f^{-1}(U)\\rightarrow U$ is finite,\n\\item for every $y\\in\\overline{f(X)}$ the map $f$ is proper at $y$.\n\\end{enumerate}\n\\end{proposition}\n\nThe above Proposition enables us to extend Definition \\ref{properdef} of finite maps.\n\n\\begin{definition}\nLet $X,Y$ be quasiprojective varieties, and let $f:X\\rightarrow Y$ be a regular map. We say that $f$ is \\textup{finite} if for every point $y\\in Y$ exists an affine neighborhood $U$ such that $f^{-1}(U)$ is affine and \n$$f\\vert_{f^{-1}(U)}: f^{-1}(U)\\rightarrow U$$\nis a finite map between affine varieties.\n\\end{definition}\n\nThe following properties of the set of points, at which a map is not proper, will be frequently used.\n\n\\begin{proposition}\\label{zlozenie}\nLet $X,Y,Z$ be affine varieties, and let $f:X\\rightarrow Y$, $g:Y\\rightarrow Z$ be regular dominant maps. Then $\\s_g\\subset \\s_{g\\circ f}$.\n\\end{proposition}\n\n\\begin{proof}\nThe assertion is equivalent to the fact that complements satisfy $Z\\setminus\\s_{g\\circ f}\\subset Z\\setminus \\s_g$. Let $z\\in Z\\setminus \\s_{g\\circ f}$, then there exists an open set $z\\in U$ such that the composition\n$$(g\\circ f)^{-1}(U)\\xrightarrow{f}g^{-1}(U)\\xrightarrow{g}U$$\nis finite. So by the definition the composition\n$$\\K[U]\\xrightarrow{g^*}\\K[g^{-1}(U)]\\xrightarrow{f^*}\\K[(g\\circ f)^{-1}(U)]$$\nis an integral extension of rings. Of course then \n$$\\K[U]\\xrightarrow{g^*}\\K[g^{-1}(U)]$$\nis also an integral extension of rings. Hence the map\n$$g^{-1}(U)\\xrightarrow{g}U$$\nis finite and $z\\in Z\\setminus \\s_g$. \n\\end{proof}\n\n\\begin{corollary}\\label{isoo}\nIf $f:X\\rightarrow Y$ is an isomorphism between affine varieties, then $f$ is finite.\n\\end{corollary}\n\n\\begin{proof}\nThere exists $g:Y\\rightarrow X$ such that $f\\circ g=id_Y$. Then by Proposition \\ref{zlozenie} $\\s_{f}\\subset \\s_{id_Y}=\\emptyset$, so $f$ is finite.\n\\end{proof}\n\n\\begin{proposition}\\label{zlozeniesk}\nLet $X,Y,Z$ be affine varieties, $g:Y\\rightarrow Z$ be a regular dominant map, and let $f:X\\rightarrow Y$ be a finite map. Then $\\s_g=\\s_{g\\circ f}$.\n\\end{proposition}\n\n\\begin{proof}\nInclusion $\\s_g\\subset\\s_{g\\circ f}$ follows from Proposition \\ref{zlozenie}.\n\nTo prove the other inclusion let $z\\in Z\\setminus\\s_g$, then there exists an open set $z\\in U$ such that \n$$g^{-1}(U)\\xrightarrow{g}U$$\nis finite. So by the definition \n$$\\K[U]\\xrightarrow{g^*}\\K[g^{-1}(U)]$$\nis an integral extension of rings. From the assumption we know that $f$ is finite, so\n$$\\K[g^{-1}(U)]\\xrightarrow{f^*}\\K[(g\\circ f)^{-1}(U)]$$\nis also an integral extension of rings. Now the composition \n$$\\K[U]\\xrightarrow{g^*}\\K[g^{-1}(U)]\\xrightarrow{f^*}\\K[(g\\circ f)^{-1}(U)]$$\nis also an integral extension of rings. Hence the map\n$$(g\\circ f)^{-1}(U)\\xrightarrow{g\\circ f}U$$\nis finite and $z\\in Z\\setminus\\s_{g\\circ f}$. \n\\end{proof}\n\n\\begin{proposition}[\\cite{je00}]\\label{graph}\nLet $f:X\\rightarrow Y$ be a generically finite mapping between affine varieties $X\\subset\\K^n$ and $Y\\subset\\K^m$. Let \n$$\\graph(f)=\\{(x,y)\\in X\\times Y:y_i=f_i(x)\\}$$ \nand let $\\overline{\\graph(f)}$ be its closure in $\\p^n\\times\\K^m$. Then there is an equality $$\\s_f=p_2(\\overline{\\graph(f)}\\setminus\\graph(f)),$$ \nwhere $p_2$ denotes the projection $\\p^n\\times\\K^m\\rightarrow\\K^m$.\n\\end{proposition}\n\n\\begin{proof}\nObserve that morphism \n$$i:X\\ni x\\rightarrow (x,f(x))\\in\\graph(f)$$\nis finite and that $f=p_2\\circ i$. Now due to Proposition \\ref{zlozeniesk} $\\s_f=\\s_{p_2}$. So it is enough to show that $\\s_{p_2}=p_2(\\overline{\\graph(f)}\\setminus\\graph(f))$.\n\nWe will show the inclusion $\\subset$. Suppose \n$$y\\notin p_2(\\overline{\\graph(f)}\\setminus\\graph(f)).$$\nThen there exists an open neighborhood $U$ of $y$, which is disjoint from $$p_2(\\overline{\\graph(f)}\\setminus\\graph(f)).$$ \nSince the projection $p_2$ is proper, the mapping\n$$p_2\\vert_{p_2^{-1}(U)}: p_2^{-1}(U)\\rightarrow U$$ \nis also proper. The map $f\\vert_{f^{-1}(U)}$ has finite fibers, hence also the map $p_2\\vert_{p_2^{-1}(U)}$ has finite fibers. Due to \\cite{ii} (page $142$, Theorem $2.27$) the map $p_2\\vert_{p_2^{-1}(U)}$ is finite, and consequently $y\\notin \\s_{p_2}$. \n\nThe opposite inclusion is also true, however since we will not use it, we do not prove it.\n\\end{proof}\n\nThe following property is well-known as curve selection lemma:\n\n\\begin{lemma}\\label{curve}\nLet $X,Y$ be affine varieties, let $f:X\\rightarrow Y$ be a regular dominant map, and let $y\\in \\s_f$. Then there exists an affine curve $\\Gamma\\subset X$, such that $y\\in \\s_{f\\vert_{\\Gamma}}$.\n\\end{lemma}\n\n\\begin{proof}\nIt is enough to show that if $X,Y$ are affine varieties, $\\dim X=n>1$, $f:X\\rightarrow Y$ is a regular dominant map, and let $y\\in \\s_f$, then there exists an affine variety $X'\\subset X$, such that $$\\dim X'=\\dim X-1\\text{ and }y\\in \\s_{f\\vert_{X'}}$$ Then the assertion follows from the induction.\n\nSuppose $X\\subset\\K^m$. Without the loss of generality we can assume $X$ is irreducible. From Proposition \\ref{graph} we know that \n$$y\\in \\s_f=p_2(\\overline{\\graph(f)}\\setminus \\graph(f)).$$ \nSo there exists\n$$x\\in\\overline{\\graph(f)}\\setminus\\graph(f)=\\overline{\\graph(f)}\\cap (\\p^m\\setminus\\K^m),$$\nsuch that $(x,y)\\in\\overline{\\graph(f)}$.\nConsider irreducible hypersurfaces $H$ in \\newline $\\overline{\\graph(f)}$ passing through $(x,y)$ and different \nfrom $\\overline{\\graph(f)}\\setminus\\graph(f)$. Take any such $H$, now $X'$ equal to projection of $H\\cap\\K^m$ to $X$ satisfies our requirements.\n\\end{proof}\n\nTwo most important properties of the set $\\s_f$ are the following:\n\n\\begin{theorem}[\\cite{je99}, page $5$, Theorem $3.8$]\\label{hiper}\nLet $X,Y$ be affine varieties, and let $f:X\\rightarrow Y$ be a regular generically finite map. Then the set $\\s_f$ is a hypersurface in $\\overline{f(X)}$ or it is empty. \n\\end{theorem}\n\n\nThe second property is that if $X$ is additionally $\\K$-uniruled, then the set $\\s_f$ is also $\\K$-uniruled. We are going to prove it. First we will consider the case of surfaces.\n\nLet $X$ be a smooth projective surface and let $D=\\sum_{i=1}^n D_i$ be a simple normal crossing divisor on $X$ (here we consider only reduced divisors). Let $\\graph(D)$ be a graph of $D$, with vertices $D_i$, and an edge between $D_i$ and $D_j$ for each point of intersection of $D_i$ and $D_j$. \n\n\\begin{definition}\nWe say that a simple normal crossing divisor $D$ on a smooth surface $X$ is a tree if $\\graph(D)$ is a tree (it is connected and acyclic).\n\\end{definition}\n\nThe following fact is obvious.\n\n\\begin{proposition}\\label{acykl}\nLet $X$ be a smooth projective surface and let $D\\subset X$ be a divisor which is a tree. If $D', D''\\subset D$ are connected divisors without common components, then $D'$ and $D''$ have at most one common point.\n\\end{proposition}\n\n\\begin{theorem}\\label{Sfsurface}\nLet $\\Gamma$ be an affine curve, and let $f:\\Gamma\\times\\K\\rightarrow\\K^m$ be a generically finite mapping. Then the set $S_{f}$ is $\\K$-uniruled or it is empty.\n\\end{theorem}\n\n\\begin{proof}\nTake an affine normalization $\\nu:\\Gamma^{\\nu}\\rightarrow\\Gamma$ (\\cite{sh}) of the curve $\\Gamma$. From Proposition \\ref{zlozeniesk} we get that $\\s_f=\\s_{\\nu\\circ f}$, because normalization is a finite mapping. Hence we can assume that the curve $\\Gamma$ is smooth. \n\nLet $\\overline{\\Gamma}$ be a smooth completion of $\\Gamma$, and let us denote $\\overline{\\Gamma}\\setminus\\Gamma=\\{a_1,\\dots,a_l\\}$. Let $X=\\Gamma\\times\\K$ and $\\overline{X}= \\overline{\\Gamma}\\times\\p^1$ be a projective closure of $X$. The divisor \n$$D=\\overline{X}\\setminus X=\\overline{\\Gamma}\\times\\infty+\\sum_{i=1}^l \\{a_i\\}\\times\\p^1$$\nis a tree. We can resolve points of indeterminacy of the rational map $f:\\overline{X}\\dashrightarrow\\p^m$.\n\\vspace{5mm}\n\n\\begin{center}\n\\begin{picture}(240,160)(-40,40)\n\\put(-20,117.5){\\makebox(0,0)[l]{$\\pi\\left\\{\\rule{0mm}{2.7cm}\\right.$}}\n\\put(0,205){\\makebox(0,0)[tl]{$\\overline{X}_m$}}\n\\put(0,153){\\makebox(0,0)[tl]{$\\overline{X}_{m-1}$}}\n\\put(4,105){\\makebox(0,0)[tl]{$\\vdots$}}\n\\put(0,40){\\makebox(0,0)[tl]{$\\overline{X}$}}\n\\put(170,40){\\makebox(0,0)[tl]{$\\p^m$}}\n\\put(80,50){\\makebox(0,0)[tl]{$f$}}\n\\put(100,140){\\makebox(0,0)[tl]{$f'$}}\n\\put(10,70){\\makebox(0,0)[tl]{$\\pi_1$}}\n\\put(10,130){\\makebox(0,0)[tl]{$\\pi_{m-1}$}}\n\\put(10,180){\\makebox(0,0)[tl]{$\\pi_m$}}\n\\put(5,190){\\vector(0,-1){30}} \\put(5,140){\\vector(0,-1){30}}\n\\put(5,80){\\vector(0,-1){33}}\n\\multiput(20,35)(8,0){17}{\\line(1,0){5}}\n\\put(157,35){\\vector(1,0){10}} \\put(20,200){\\vector(1,-1){150}}\n\\end{picture}\n\\end{center}\n\\vspace{5mm}\n \nDenote the proper transform of $\\overline{\\Gamma}\\times\\infty$ by $\\pi^*(\\overline{\\Gamma}\\times\\infty)$. It is an easy observation that $f'(\\pi^*(\\overline{\\Gamma}\\times\\infty))\\subset H_\\infty$, where $H_\\infty$ is the hyperplane at infinity of $\\p^m$. \n\nNote that the divisor $D'=\\pi^*(D)$ is a tree, since each $\\pi_i$ is a blow up. Moreover irreducible components of $D'$ are $\\pi^*(\\overline{\\Gamma}\\times\\infty)$ and $\\p^1$'s. The curve $C=f'^{-1}(H_\\infty)\\subset D'$ is a complement of a semi-affine variety $f'^{-1}(\\K^m)$ hence it is connected (for details see \\cite{je99}, Lemma~$4.5$). \n\nNow $\\s_f=f'(D'\\setminus C)$.\nLet $R\\subset \\s_f$ be an irreducible component. From Proposition \\ref{hiper} we know that $R$ is a curve. So there is an irreducible curve $Z\\subset D'$, such that $R=f'(Z\\setminus C)$. By Proposition \\ref{acykl} we have that $Z$ has at most one common point with $C$. Of course it has at least one, since every morphism from irreducible projective variety to an affine variety is constant. So $R=f'(Z\\setminus C)=f'(\\K)$. This completes the proof due to Proposition \\ref{k-uniruledofdeg} and the fact that $\\s_f$ consists only of finite number of parametric curves.\n\\end{proof}\n\n\\begin{theorem}[\\cite{je10}, page $3673$, Theorem $4.1$]\\label{Sfkuni}\nLet $X,Y$ be affine varieties, and let $f:X\\rightarrow Y$ be a regular dominant map. If $X$ is additionally $\\K$-uniruled, then the set $\\s_f$ is also $\\K$-uniruled or it is empty.\n\\end{theorem}\n\n\\begin{proof}\nDue to Proposition \\ref{k-uniruledofdeg} there exists a regular dominant map $g:\\K\\times W\\rightarrow X$. From Proposition \\ref{zlozenie} we have \nthat $\\s_f\\subset \\s_{g\\circ f}$, and from Proposition \\ref{hiper} both sets are of the same pure dimension. Therefore it is enough to prove the assertion for $f:\\K\\times W\\rightarrow Y$. \n\nLet $y\\in \\s_f$. Using Lemma \\ref{curve} we get that there is a curve $\\Gamma\\subset\\K\\times W$ such that $f\\vert_{\\Gamma}$ is not proper at $y$. Let $p_2:\\K\\times W\\rightarrow W$ be a projection, and $\\Gamma'=p_2(\\Gamma)$. Of course $\\Gamma'$ is still a curve, because otherwise $\\Gamma=\\K$ and $f\\vert_{\\Gamma}$ would be finite. We have that $\\Gamma\\subset\\K\\times\\Gamma'$, so \n$$y\\in \\s_{f\\vert_{\\Gamma}}\\subset \\s_{f\\vert_{\\K\\times\\Gamma'}}\\subset \\s_f.$$ \nFrom Proposition \\ref{Sfsurface} we get that there exists a parametric curve contained in $\\s_f$ passing through $y$. By Propositions \\ref{k-uniruled} and \\ref{k-uniruledofdeg} this completes the proof for uncountable fields $\\K$. For countable fields we extend $X,Y,f$ to an uncountable field $\\K\\subset\\Le$ and use Proposition \\ref{indep}.\n\\end{proof}\n\nTo motivate one conjecture we will need one more property of the set $\\s_f$. It describes its degree as a hypersurface.\n\n\\begin{theorem}[\\cite{je93}, page $264$, Theorem $15$]\\label{hiperc}\nLet $f:\\C^n\\rightarrow\\C^n$ be a dominant polynomial  map. Then the degree of a hypersurface $\\s_f$ is at most $$\\frac{\\prod_i\\deg f_i-\\mu(f)}{\\min_i\\deg f_i},$$ where $\\mu(f)$ denotes the multiplicity of $f$.\n\\end{theorem}\n\n\n\\section{A complex field case}\\label{c}\n\nFor the whole section we asume that the base field is $\\C$. \n\nThe condition that a map is finite at a point $y$ is equivalent to the fact that it is locally\nproper in the classical topology sense (there exists an open neighborhood\n$U$ of $y$ such that $f^{-1}(\\overline{U})$ is compact). This\ncharacterization gives the following:\n\n\\begin{proposition}[\\cite{je93}, page $260$, Proposition $3$]\\label{topo}\nLet $f:X\\rightarrow Y$ be a generically finite mapping between\naffine varieties over $\\C$. Then $y\\in \\s_f$ if and only if there exists a\nsequence $(x_n)_{n\\in\\N}$, such that $\\vert x_n\\vert\\rightarrow \\infty$ and\n$f(x_n)\\rightarrow y$.\n\\end{proposition}\n\nWe will need a classical theorem of complex analysis.\n\n\n\n\\begin{theorem}[Rouch\\'{e} \\cite{ru}]\\label{rouche}\nLet $f,g:\\C\\rightarrow\\C$ be holomorphic functions. Suppose that $\\vert g(z)\\vert<\\vert f(z)\\vert$ for every $z\\in\\partial D$, where $D$ is a region. Then $f$ and $f+g$ have the same number of zeros (counted with multiplicities) inside $D$. \n\\end{theorem}\n\nWe are ready to prove the following:\n\n\\begin{theorem}\\label{cn}\nLet  $f:\\C^n\\rightarrow \\C^m$ be a generically finite polynomial mapping of\ndegree $d$. Then the set $\\s_f$ has degree of $\\C$-uniruledness at most $d-1$ or it is empty.\n\\end{theorem}\n\n\\begin{proof}  Let $y\\in \\s_f$, by an affine transformation we can assume that \n$$y=O=(0,0,\\dots,0)\\in \\C^m.$$\nFrom the same reason we can assume that $O\\notin f^{-1}(\\s_f)$.\n\nDue to Proposition \\ref{topo} there exists a sequence of points\n$\\lvert x_k\\rvert\\to\\infty$ such that $f(x_k)\\to O$. Let us consider lines\n$$L_k(t)=tO+(1-t)x_k=(1-t)x_k,\\;\\; t\\in\\C.$$\nDenote $l_k(t)=f(L_k(t))$.\nInfinite fibers cover only a subset of codimension at least $2$ in $\\C^n$, and due to Proposition \\ref{hiper} the codimension of $\\s_f$ equals to $1$. Thus by Proposition \\ref{k-uniruledofdeg} we can assume that $\\deg l_k>0$. Note also that $\\deg l_k\\leq d$. Each curve $l_k$ is given by $m$ polynomials of one variable: \n$$l_k(t)=(\\sum_{i=0}^d a^1_i(k)\nt^i,\\dots,\\sum_{i=0}^d a^m_i(k) t^i).$$ \nHence we can associate $l_k$ with a point\n$$(a^1_0(k),\\dots,a^1_d(k);\\ldots;a^m_0(k),\\dots,a^m_d(k))\\in \\C^N.$$\n\nFor each $i$ when $k\\to \\infty$, then $a^i_0(k)\\to 0$. So we\ncan change the parametrization of $l_k$ by putting $t\\rightarrow\n\\lambda_k t$, in such a way that $\\Vert l_k\\Vert=1,$ for $k\\gg 0$\n(we consider here $l_k$ as an element of $\\C^N$ with the Euclidean\nnorm). \n\nNow, since unit sphere is compact, there exists a subsequence $(l_{k_r})_{r\\in\\N}$ of $(l_k)_{k\\in\\N}$, which is convergent to a polynomial mapping \n$$l : \\C\\rightarrow \\C^m,$$ with $l (0) = O$. Moreover, $l$ is non-constant, because $\\Vert l \\Vert\n= 1$, and $l(0)= O$. \n\nChoosing a subsequence we can also assume that the limit\n$$\\textit{lim}_{k\\rightarrow \\infty}\\lambda_k=\\lambda$$\nexists in the compactification of the field $\\C$. Consider two cases:\n\\begin{enumerate}\n\\item $\\lambda\\in\\C$ is finite. If $k\\to\\infty$, then $L_k(\\lambda_kt)=(1-\\lambda_kt)x_k\\to\\infty$ for $t\\not=\\lambda^{-1}$.\n\\item $\\lambda=\\infty$. If $k\\to\\infty$, then $\\Vert L_k(\\lambda_k t)\\Vert \\geq (\\vert\\lambda_kt\\vert-1)\\Vert x_k\\Vert$, and hence $\\Vert L_k(\\lambda_k t)\\Vert \\to\n \\infty$ for every $t\\neq 0$.\n\\end{enumerate}\n\nOn the other hand if $k\\to\\infty$\n$$f(L_k(\\lambda_k t))=l_k(\\lambda_kt)\\to l(t),$$ \nso using once more Proposition \\ref{topo} this means that the curve $l$ is contained in the set $\\s_f$. As a consequence the set $\\s_f$ has \ndegree of $\\C$-uniruled\\-ness at most $d$.\n\nTo complete the proof we should show that $\\deg l<d$. It is enough to prove that $\\deg l<\\deg l_k$ for some $k$. Suppose the contrary, what is $\\deg  l=\\deg  l_k$ for all $k$.\nLet \n$$l(t)=(l_1(t),\\dots,l_m(t))\\text{ and }l_k(t)=(l_1^k(t),\\dots,l_m^k(t)).$$ We can assume that a component\n$l_1(t)$ has the maximum degree. Denote $a=(a_1,\\dots,a_m)=f(O)$. All roots of\nthe polynomial $l_1(t)-a_1$ are contained in the interior of some disc \n$$D=\\{ t\\in \\C : \\vert t\\vert<R\\}.$$ \nLet \n$$\\epsilon =\\inf\\{ \\vert l_1(t)-a_1\\vert: t\\in\\partial D\\}.$$ \nFor $k\\gg 0$ we have \n$$\\vert (l_1-a_1)-(l_1^k-a_1)\\vert_D<\\epsilon.$$ \n\nConsequently by Rouch\\'{e} Theorem \\ref{rouche} these polynomials\nhave the same number of  zeros (counted with multiplicities) in $D$. In particular\nzeros of $l_1^k-a_1$ are bounded. All curves $L_k$ pass through $O$, so all $l_k$ pass through $a=f(O)$. These means that there exists a sequence \n$t_k$ such that $l_k(t_k)=a$. We have just showed that $|t_k|<R$, since $t_k$ is a root of the polynomial $l_1^k-a_1$. \n\nSo we can assume that the sequence $t_k$ converges to some $t_0$. When we pass to the limit we get $l(t_0)=a$, this is a contradiction, since $a=f(O)\\not\\in \\s_f$.\n\\end{proof}\n\nIn a similar way we can prove the following generalization of\nTheorem \\ref{cn}:\n\n\\begin{theorem}\\label{cxw}\nLet  $X=\\C\\times W\\subset \\C\\times \\C^n$ be an affine cylinder and\nlet \n$$f:\\C\\times W\\ni (t,w)\\rightarrow (f_1(t,w),\\dots,f_m(t,w))\\in\n\\C^m$$ \nbe a generically finite polynomial mapping. Assume that for every $i$ there is\n$\\deg _tf_i\\leq d$. Then the set $\\s_f$ has degree of $\\C$-uniruledness at most $d$ or it is empty.\n\\end{theorem}\n\n\\begin{proof} As before for $y\\in \\s_f$ because of an affine transformation we can assume that \n$$y=O=(0,0,...,0)\\in \\C^m.$$ \n\nDue to Proposition \\ref{topo} there exists a sequence of points $(a_k,w_k)_{k\\in\\N}\\subset\\C\\times W$, such that $$\\vert(a_k,w_k)\\vert\\to\\infty\\text{ and }f(a_k,w_k)\\to y.$$ \nLet us consider lines \n$$L_k(t)=((1-t)a_k, w_k),\\;\\; t\\in\\C.$$ \nDenote $l_k(t)=f(L_k(t))$. \n\nInfinite fibers cover only a subset of codimension at least $2$ in $\\overline{f(\\C\\times W)}$, while due to Proposition \\ref{hiper} the codimension of $\\s_f$ equals to $1$. Thus by Proposition \\ref{k-uniruledofdeg} we can assume that $\\deg l_k>0$. Note also that $\\deg l_k\\leq d$.  Each curve $l_k$ is given by $m$ polynomials of one\nvariable: \n$$l_k(t)=(\\sum_{i=0}^d a^1_i(k) t^i,\\dots,\\sum_{i=0}^d\na^m_i(k) t^i).$$\nAs before we can associate $l_k$ with a point\n$$(a^1_0(k),\\dots,a^1_d(k);\\ldots;a^m_0(k),\\dots,a^m_d(k))\\in \\C^N.$$\n\nFor each $i$ when $k\\to \\infty$, then $a^i_0(k)\\to 0$. Thus we\ncan change the parametrization of $l_k$ by $t\\rightarrow\n\\lambda_k t$, in such a way that $\\Vert l_k\\Vert=1,$ for $k\\gg 0$\n(we consider $l_k$ as an element of  $\\C^N$ with the Euclidean\nnorm). \n\nCompactness of unit sphere implies that there exists a\nsubsequence $(l_{k_r})_{r\\in\\N}$ of $(l_k)_{k\\in\\N}$, which is convergent to a\npolynomial mapping $$l : \\C\\rightarrow \\C^m,$$ with $l(0) = O$.\nMoreover, $l$ is non-constant, because $\\Vert l \\Vert = 1$ and\n$l(0)= O$. \n\nWe can also assume that the limit\n$$\\textit{lim}_{k\\rightarrow \\infty}\\lambda_k=\\lambda$$ \nexists in the compactification of the field $\\C$. \nWe consider two cases:\n\\begin{enumerate}\n \\item $\\lambda$ is finite. If $k\\to\\infty$, then $L_k(\\lambda_kt)=((1-\\lambda_kt)a_k,w_k)\\to\\infty$ for $t\\not=\\lambda^{-1}$.\n\\item $\\lambda=\\infty$. If $k\\to\\infty$, then $\\Vert L_k(\\lambda_k t)\\Vert \\geq \\max((\\vert\\lambda_kt\\vert-1)\\vert a_k\\vert,\\Vert w_k\\Vert)$, and $\\Vert L_k(\\lambda_k t)\\Vert \\to \\infty$ for every $t\\neq 0$.\n\\end{enumerate}\n\nBut on the other hand \n$$f(L_k(\\lambda_k t))=l_k(\\lambda_kt)\\to l(t),$$ so using once more Proposition \\ref{topo} we get that the curve $l$ is contained in the set $\\s_f$. Thus the set $\\s_f$ has degree of $\\C$-uniruledness at most $d$.\n\\end{proof}\n\n\\begin{corollary}\\label{wn}\nLet  $f=(f_1,...,f_m) :\\C^n\\rightarrow \\C^m$ be a generically\nfinite polynomial mapping with $d=\\min_j\\max_i\\deg _{x_j}f_i$. Then\nthe set $\\s_f$ has degree of $\\C$-uniruledness at most $d$ or it is empty.\n\\end{corollary}\n\n\\begin{theorem}\\label{multc}\nLet $X$ be an affine variety with degree of $\\C$-uniruledness at\nmost $d_1$ and let $f:X\\rightarrow \\C^m$ be a\ngenerically finite polynomial mapping of degree $d_2$. Then the set $\\s_f$ has degree of $\\C$-uniruledness at\nmost $d_1d_2$ or it is empty.\n\\end{theorem}\n\n\\begin{proof}\nBy Definition \\ref{k-uniruleddef} there exists an affine variety\n$W$ with $\\dim W = \\dim X-1$, and a dominant polynomial map \n$$\\phi:\\C\\times W\\rightarrow X$$\n of degree in the  first coordinate at\nmost $d_1$. Equality $\\dim  \\C\\times W = \\dim  X$ implies that\n$\\phi$ is generically finite, hence also generically finite is the map\n$$f\\circ\\phi:\\C\\times W\\rightarrow \\C^m.$$ \nIt is of degree in the  first coordinate at\nmost $d_1d_2$. Due to Theorem \\ref{cxw} the set $\\s_{f\\circ\\phi}$ \nhas degree of $\\C$-uniruledness at most $d_1d_2$.\nBy Proposition \\ref{zlozenie} there is an inclusion $\\s_f\\subset\\s_{f\\circ\\phi}$. From Theorem\n\\ref{hiper} we know that if not empty, then both sets are of pure\ndimension $\\dim X-1$, so components of $\\s_f$ are components of\n$\\s_{f\\circ\\phi}$. This implies the assertion.\n\\end{proof}\n\n\\begin{example}\nConsider the following polynomial mapping \n$$f: \\C^n\\ni (x_1,\\ldots, x_n)\\to (x_1, x_1x_2,\\ldots,x_1x_n)\\in \\C^n.$$ \nWe have $\\deg f=2$ and $\\s_f=\\{x\\in\\C^n:x_1=0\\}$. The set $\\s_f$ has degree of $\\C$-uniruledness equal to one. This shows that in general Theorems \\ref{cn}, \\ref{cxw} and Corollary \\ref{wn} can not be improved.\n\\end{example}\n\n\\begin{example}\nConsider the following polynomial mapping \n$$f: \\C^2\\ni (x,y)\\to (x+(xy)^d, xy)\\in \\C^2.$$ \nWe have $\\deg f =2d$ and $\\s_f=\\{(s,t)\\in\\C^2:s=t^d\\}$. The set $\\s_f$ has degree of $\\C$-uniruledness equal to $d$. This shows that in general Corollary\n\\ref{wn} can not be improved.\n\\end{example}\n\n\\begin{example}\nFor $n>2$ let $X=\\{ x\\in \\C^n : x_1x_2=1\\}$, and \n$$f:X\\ni (x_1,\\ldots, x_n)\\to (x_2,\\ldots, x_n)\\in\\C^{n-1}.$$ The variety $X$ has degree of $\\C$-uniruledness equal to one, moreover we have $\\deg f=1$ and $\\s_f=\\{x\\in\\C^{n-1}:x_1=0\\}$. So the set $\\s_f$ has degree of $\\C$-uniruledness one. This shows that in general Theorems \\ref{cxw}, \\ref{multc} can not be improved.\n\\end{example}\n\n\\begin{remark}\nLet us note that by Proposition \\ref{indep} all results from this\nsection remain true for arbitrary algebraically closed field of\ncharacteristic zero.\n\\end{remark}\n\n\\section{A real field case}\\label{r}\n\nFor the whole section we assume that the base field is $\\R$. \n\nLet $X\\subset\\R^m$ be a closed semialgebraic set, and let $f: X\\to\\R^n$ be a polynomial mapping. \nAs in the complex case we say that $f$ is not proper at a point $y\\in\\R^n$ if there is no neighborhood $U$ of $y$ such that $f^{-1}(\\overline{U})$ \nis compact. The set of all points $y\\in\\overline{f(X)}$ at which the mapping $f$ is not proper we denote as before by $\\s_f$. This set is also closed and semialgebraic \\cite{je02}. The results of \\cite{je02} can be generalized as follows:\n\n\\begin{theorem}\\label{rn1}\nLet $f:\\R^n\\rightarrow\\R^m$ be a generically finite polynomial mapping of degree $d$. Then the set $\\s_f$ has degree of $\\R$-uniruledness at most $d-1$ or it is empty.\n\\end{theorem}\n\n\\begin{theorem}\\label{rxw1}\nLet $X=\\R\\times W\\subset \\R\\times \\R^n$ be a closed semialgebraic cylinder and let \n$$f:\\R\\times W\\ni (t,w)\\rightarrow (f_1(t,w),\\dots,f_m(t,w))\\in \\R^m$$ \nbe a generically finite polynomial mapping. Assume that for every $i$ there is $\\deg _tf_i\\leq d$. Then the set $\\s_f$ has degree of $\\R$-uniruledness at most $d$ or it is empty.\n\\end{theorem}\n\n\\begin{corollary}\\label{rwn1}\nLet  $f=(f_1,...,f_m) :\\R^n\\rightarrow \\R^m$ be a generically finite polynomial mapping with $d=\\min_j\\max_i\\deg _{x_j}f_i$. Then the set $\\s_f$ has degree of $\\R$-uniruledness at most $d$ or it is empty.\n\\end{corollary}\n\nProofs of these facts are exactly the same as in the complex case (see section \\ref{c}). To prove a real analog of Theorem \\ref{multc} we need some new ideas. \n\n\n\n\n\n\n\\begin{theorem}\\label{multc1}\nLet  $X$ be a closed semialgebraic set with degree of $\\R$-uni\\-ruled\\-ness at most $d_1$, and let $f:X\\rightarrow\\R^m$ be a generically finite polynomial mapping of degree $d_2$. Then the set $\\s_f$ has degree of $\\R$-uniruledness at most $2d_1d_2$ or it is empty.\n\\end{theorem}\n\n\\begin{proof}\nLet $a\\in \\s_f$ and let $(x_k)_{k\\in\\N}\\subset X$ be a sequence of points such that $f(x_k)\\to a$ and $x_k\\to\\infty$. \nBy Proposition \\ref{real} there exists a semialgebraic curve $W$ and a generically finite polynomial map \n$$\\phi:\\R\\times W \\ni (t,w)\\to \\phi(t,w)\\in X,$$ \nsuch that $\\deg_t\\phi \\leq d_1$, and there exists a sequence $(y_k)_{k\\in\\N}\\subset\\R\\times W$\nsuch that $f(\\phi(y_k))\\to a$ and $\\phi(y_k)\\to\\infty$. In particular $a\\in\\s_{f\\circ\\phi}$. Let $\\Gamma$ be a Zariski closure of $W$. We can assume that $\\Gamma$ and its complexification $\\Gamma^c$ are smooth. Denote $Z:=\\R\\times\\Gamma$. We have the induced mapping $\\phi: Z \\to X$. Hence we have also the induced complex mapping $\\phi^c: Z^c:=\\C\\times\\Gamma^c\\to X^c$, where $Z^c,X^c$ denote the complexification of $Z$ and $X$ respectively.\n\nLet $\\overline{\\Gamma^c}$ be a smooth completion of $\\Gamma^c$ and let us denote $\\overline{\\Gamma^c}\\setminus\\Gamma^c=\\{ a_1,..., a_l\\}.$ Now $\\p^1\\C\\times\\overline{\\Gamma^c}$ is a projective closure of $Z^c$. The divisor \n$$D=\\overline{Z^c}\\setminus Z^c=\\infty\\times\\overline{\\Gamma^c}+\\sum_{i=1}^l \\p^1\\C\\times\\{a_i\\}$$ \nis a tree. The mapping $\\phi$ induces a rational mapping $\\phi: \\overline{Z^c}\\dashrightarrow\\overline{X^c}$, where $\\overline{X^c}$ denotes the\nprojective closure of $X^c.$ We can resolve points of indeterminacy of this mapping:\n\n\\vspace{3mm} \n\\begin{center}\\begin{picture}(240,160)(-40,40)\n\\put(-20,117.5){\\makebox(0,0)[l]{$\\pi\\left\\{\\rule{0mm}{2.7cm}\\right.$}}\n\\put(0,205){\\makebox(0,0)[tl]{$\\overline{Z^c}_m$}}\n\\put(0,153){\\makebox(0,0)[tl]{$\\overline{Z^c}_{m-1}$}}\n\\put(4,105){\\makebox(0,0)[tl]{$\\vdots$}}\n\\put(0,40){\\makebox(0,0)[tl]{$\\overline{Z^c}$}}\n\\put(170,40){\\makebox(0,0)[tl]{$\\overline{X^c}$}}\n\\put(80,50){\\makebox(0,0)[tl]{$\\phi$}}\n\\put(100,140){\\makebox(0,0)[tl]{$\\phi '$}}\n\\put(10,70){\\makebox(0,0)[tl]{$\\pi_1$}}\n\\put(10,130){\\makebox(0,0)[tl]{$\\pi_{m-1}$}}\n\\put(10,180){\\makebox(0,0)[tl]{$\\pi_m$}}\n\\put(5,190){\\vector(0,-1){30}} \\put(5,140){\\vector(0,-1){30}}\n\\put(5,80){\\vector(0,-1){33}}\n\\multiput(20,35)(8,0){17}{\\line(1,0){5}}\n\\put(157,35){\\vector(1,0){10}} \\put(20,200){\\vector(1,-1){150}}\n\\end{picture}\n\\end{center}\n\n\\vspace{3mm} \nObserve that $H:=\\pi^{-1}(\\overline{Z})$ has a structure of a real variety and $\\R\\times\\Gamma\\subset H$. Denote $Q:=\\overline{Z_m}\\cap \\phi'^{-1}(X^c).$ Then the mapping $\\phi': Q\\to X^c$ is proper. Moreover, $Q=\\overline{Z_m}\\setminus\n\\phi'^{-1}(\\overline{X^c}\\setminus X^c).$ The divisor\n$D_1=\\phi'^{-1}(\\overline{X^c}\\setminus X^c)$ is connected  as a\ncomplement of a semi-affine variety $\\phi'^{-1}(X^c)$  (for\ndetails see \\cite{je99}, page $8$, Lemma~$4.5$). Note that  the divisor\n$D'=\\pi^*(D)$ is a tree. Hence the divisor $D_1\\subset D'$ is also\na tree.\n\nWe can consider the mapping $f'=f\\circ\\phi': \\R\\times \\Gamma\\to \\R^n$ as the regular mapping $f': Q\\to \\C^n.$ This mapping\ninduces a rational mapping from $H^c=\\overline{Z_m^c}$ to $\\p^n\\C$. As\nbefore we can resolve its points of indeterminacy:\n\\vspace{3mm} \n\n\\begin{center}\\begin{picture}(240,160)(-40,40)\n\\put(-20,117.5){\\makebox(0,0)[l]{$\\psi\\left\\{\\rule{0mm}{2.7cm}\\right.$}}\n\\put(0,205){\\makebox(0,0)[tl]{$\\overline{H^c_k}$}}\n\\put(0,153){\\makebox(0,0)[tl]{$\\overline{H^c_{k-1}}$}}\n\\put(4,105){\\makebox(0,0)[tl]{$\\vdots$}}\n\\put(0,40){\\makebox(0,0)[tl]{$\\overline{H^c}$}}\n\\put(170,40){\\makebox(0,0)[tl]{${\\p^n(\\C)}$}}\n\\put(80,50){\\makebox(0,0)[tl]{$f'$}}\n\\put(100,140){\\makebox(0,0)[tl]{$F$}}\n\\put(10,70){\\makebox(0,0)[tl]{$\\rho_1$}}\n\\put(10,130){\\makebox(0,0)[tl]{$\\rho_{k-1}$}}\n\\put(10,180){\\makebox(0,0)[tl]{$\\rho_k$}}\n\\put(5,190){\\vector(0,-1){30}} \\put(5,140){\\vector(0,-1){30}}\n\\put(5,80){\\vector(0,-1){33}}\n\\multiput(20,35)(8,0){17}{\\line(1,0){5}}\n\\put(157,35){\\vector(1,0){10}} \\put(20,200){\\vector(1,-1){150}}\n\\end{picture}\n\\end{center}\n\n\\vspace{3mm} \nNote that  the divisor $D_1'=\\psi^*(D_1)$ is a tree.\nDenote a proper transform of\n$\\infty\\times\\overline{\\Gamma}$ by $\\infty'\\times\\overline{\\Gamma}$. \nIt is an easy observation that\n$F(\\infty'\\times\\overline{\\Gamma})\\subset \\pi_\\infty$, where\n$\\pi_\\infty$ denotes the hyperplane at infinity of $\\p^n\\C$.\nNow $\\s_{f'}= F(D_1\\setminus F^{-1}(\\pi_\\infty)).$ The curve\n$L=F^{-1}(\\pi_\\infty)$ is connected (the same argument as above). Now\nby Proposition \\ref{acykl} we have that every irreducible curve\n$l\\subset D_1$ (we have $l\\cong \\p^1\\C$)\nwhich does not belong to $L$ has at most one common point with\n$L$. \n\nLet $R$ be an irreducible component of $\\s_{f'}$. Hence $R$\nis a curve. There is a curve $l\\subset D_1,$ which has exactly one\ncommon point with $L$, such that $R=F(l\\setminus L).$ If $l$ is\ngiven by blowing up of a real point, then $L$ also has a real\ncommon point  with $l$ (since the conjugate point also is a\ncommon point of $l$ and $L$). When we restrict to the real model\n$l^r$ of $l$ we have  $l^r\\setminus L\\cong \\R.$ Hence if we\nrestrict our consideration only to the real points and to the set\n$Q^r:=\\overline{\\R\\times W}\\subset Q$ (we consider here a closure in the\neuclidian topology) we see that the set $\\s$ of\nnon-proper points of the mapping $f'|_{Q^r}$ is a union of\nparametric curves $F(l^r\\setminus L), \\ l\\in D_1,\n\\psi(l)\\in \\overline{Q^r}$, where the last closure is the closure in a real\nprojective space. Of course $a\\in \\s\\subset\\s_f$.\nSimilarly the set $\\s_{f\\circ\\phi}$\n is a union of parametric curves $F(l^r\\setminus L), \\ l\\in \\psi^*(D'), \\pi(\\psi(l))\\subset\n\\overline{\\R\\times W}\\subset\\overline{Z}$. Hence we can say that a \"irreducible\ncomponent\" of the set of non-proper points of $f'|_{Q^r}$ is also a\n\"irreducible\" component of $\\s_{f\\circ\\phi}$. Now we can finish the\nproof by Theorem \\ref{cxw} and the following Lemma \\ref{xx}.\n\\end{proof}\n\n\\begin{lemma}\\label{xx}\nLet $\\psi:\\R\\to\\R^n$ be a parametric curve. If there exists a parametric curve $\\phi: \\R\\to \\R^n$ of degree at most $d$ with $\\psi(\\R)\\subset \\phi(\\R)$, then $\\psi(R)$ has degree of $\\R$-uniruledness at most $2d$.\n\\end{lemma}\n\n\\begin{proof}\nIndeed, let $\\phi(t)=(\\phi_1(t),\\dots,\\phi_n(t))$ and consider a\nfield \n$$\\Le=\\R(\\phi_1,\\dots,\\phi_n).$$ \nBy the L\\\"{u}roth Theorem \\ref{luroth} there\nexists a rational function $g(t)\\in \\R(t)$ such that $\\Le=\\R(g(t)).$\nIn particular $\\phi_i(t)=f_i(g(t)).$ In fact we have two induced\nmappings \n$$\\overline{f} : \\p^1\\R\\to \\p^n\\R\\text{ and }\n\\overline{g}: \\p^1\\R\\to \\p^1\\R.$$ \nMoreover,\n$\\overline{f}\\circ \\overline{g}=\\overline{\\phi}.$ Let $A_\\infty$\ndenote the unique point at infinity of a Zariski closure of\n$\\psi(\\R)$ and let $\\infty=\\overline{f}^{-1}(A_\\infty).$ This implies\nthat $\\#\\overline{g}^{-1}(\\infty)=1$ and we can assume that\n$\\overline{g}^{-1}(\\infty)=\\infty$, i.e., $g\\in \\R[t].$ Similarly\n$f_i\\in \\R[t].$ Now if deg $g=1$ then $f:\\R\\to \\R^n$ covers the whole\n$\\phi(R)$, because the image of $f$ is open and closed in\n$\\phi(R).$ Otherwise we can compose $f$ with suitable polynomial\nof degree two to obtain the whole $\\phi(R)$ in the image.\n\nIn the same way we can compose $f$ with suitable polynomial of degree one or two to obtain whole\n$\\psi(\\R)$ in the image. In any case $\\psi(\\R)$ has a parametrization of degree bounded by\n$2\\deg f\\leq 2d$.\n\\end{proof}\n\nTheorem \\ref{multc1} gives the following:\n\n\\begin{corollary}\\label{multc3}\nLet $X$ be a closed semialgebraic set which is $\\R$-uniruled and let $f:X\\rightarrow \\R^m$ be a generically\nfinite polynomial mapping. Then every connected component of the set $\\s_f$ is unbounded.\n\\end{corollary}\n\n\\section{A field of positive characteristic case}\\label{positive}\n\nIn this section we assume $\\K$ to be an arbitrary algebraically closed field. We begin with a useful lemma.\n\n\\begin{lemma}\\label{tool}\nLet $A\\subset\\K^n$ be an affine set, and let $f$ be a regular function on $\\K^n$ not equal to\n$0$ on any component of $A$. Suppose that for each $c\\in \\K^*$ the set \n$$A_c:=A\\cap\\{x\\in\\K^n:f(x)=c\\}$$\nhas degree of $\\K$-uniruledness at most $d$. Then $A_0$ also has degree of $\\K$-uniruledness at most $d$.\n\\end{lemma}\n\\begin{proof}\nSuppose that \n$$A=\\{x\\in\\K^n:g_1(x)=0,\\dots,g_r(x)=0\\}.$$ \nFor $a=(a_1,\\dots,a_n)\\in\\K^n$ and $b=(b_{1,1}:\\dots:b_{d,n})\\in\\p^{dn-1}$, let\n$$\\varphi_{a,b}:\\K\\ni t\\rightarrow (a_1+b_{1,1}t+\\dots+b_{1,d}^dt^d,\\dots,a_n+b_{n,1}t+\\dots+b_{n,d}^dt^d)\\in\\K^n$$ \nbe a parametric curve of degree at most $d$. Let us consider a variety and a projection\n$$\\K^n\\times\\p^{dn-1}\\supset V=\\{(a,b)\\in\\K^n\\times\\p^{dn-1}:\\forall_{t,i}\\;g_i(\\varphi_{a,b}(t))=0\\;\\text{and}$$\n$$\\forall_{t_1,t_2}\\;f(\\varphi_{a,b}(t_1))=f(\\varphi_{a,b}(t_2))\\}\\ni(a,b)\\rightarrow a\\in\\K^n.$$\nThe definition of the set $V$ says that parametric curves $\\varphi_{a,b}$ are contained in $A$ and $f$ is constant on them. Hence $V$ is closed and the image of the projection is contained in $A$. Moreover the image contains all $A_c$ for $c\\in \\K^*$, since they are filled with\nparametric curves of degree at most $d$. But since $\\p^{dn-1}$ is complete and $V$ is closed we have that the image is closed, and hence it is the whole set $A$. In particular $A_0$ is contained in the image, so it is filled with parametric curves of degree at most $d$.\n\\end{proof}\n\nWe are ready to prove an analog of Theorem \\ref{cn} for general fields.\n\n\\begin{theorem}\\label{kn}\nLet $\\K$ be an arbitrary algebraically closed field, and let\n$f:\\K^n\\rightarrow\\K^m$ be a generically finite polynomial mapping of degree\n$d$. Then the set $\\s_f$ has degree of $\\K$-uniruledness at most $d$ or it is empty.\n\\end{theorem}\n\\begin{proof}\nIf $n=1$ then the map is proper and $\\s_f$ is empty. \n\nWe consider the case $n\\geq 2$. Due to Proposition \\ref{graph} $$\\s_f=p_2(\\overline{\\graph(f)}\\setminus\\graph(f)),$$\nso it is enough to prove that the set\n$$\\overline{\\graph(f)}\\setminus\\graph(f)$$ \nis filled with parametric curves of degree at most $d$. It is because we take images of curves under projection $p_2$ which is of degree $1$, \nand both sets are of the same dimension, so only on a codimension $1$ subvariety images of curves can become points. \n\nDenote the coordinates in $\\p^n\\times\\K^m$ by $(x_{0} : \\dots : x_n;x_{n+1},\\dots,x_{n+m})$. Let us take an arbitrary point $$z\\in\\overline{\\graph(f)}\\setminus\\graph(f)=\\overline{\\graph(f)}\\cap\\{x_0=0\\}.$$ \nThere exists $1\\leq i\\leq n$ such that $z_i\\neq 0$. Consider sets \n$$A:=\\overline{\\graph(f)}\\cap\\{x_i\\neq 0\\}\\text{ and }A_c:=\\overline{\\graph(f)}\\cap\\{x_i\\neq 0\\}\\cap\\{x_0=cx_i\\}\\text{ for }c\\in\\K.$$ \nThe set $A$ is affine and sets $A_c$ satisfy\nassumptions of Lemma \\ref{tool} for $f=\\frac{x_0}{x_i}$. Sets $A_c$ for $c\\neq 0$ are filled with parametric\ncurves of degree at most $d$, because we can take $j\\neq 0,i$ and curves\n$$\\K\\ni t\\rightarrow (c:a_1:\\dots:a_{i-1}:1:\\dots:a_{j-1}:t:a_{j+1}:\\dots)\\xrightarrow{f} A_c\\subset\\graph(f).$$\nHence the set \n$$A_0=\\overline{\\graph(f)}\\cap\\{x_i\\neq 0\\}\\cap\\{x_0=0\\}$$ \nis also filled with such curves and we get that though $z$ passes a parametric curve of degree at most $d$ in $$\\overline{\\graph(f)}\\cap\\{x_0=0\\},$$ \nthis finishes the proof.\n\\end{proof}\n\nBy looking carefully at the proof of the above theorem we get the following slightly more general result:\n\n\\begin{theorem}\\label{kw}\nLet $\\K$ be an arbitrary algebraically closed field, and let $f:\\K\\times W\\rightarrow\\K^m$ be a generically finite polynomial mapping of degree $d$. Then the set $\\s_f$ is empty or all its components have degree of $\\K$-uniruledness at most $d$, except possibly components of $p_2(\\overline{\\graph(f)}\\cap \\{(0:1:0:\\dots)\\}\\times \\K^m)$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\K\\times W\\subset\\K\\times\\K^n$. As before it is enough to show that \n$$\\overline{\\graph(f)}\\setminus\\graph(f)$$ \nis filled with parametric curves of degree at most $d$. Let us take an arbitrary point $$z\\in\\overline{\\graph(f)}\\setminus\\graph(f)=\\overline{\\graph(f)}\\cap\\{x_0=0\\}.$$ \n\nIf there exists $i\\neq 1$ such that $z_i\\neq 0$ then the idea from the proof of Theorem \\ref{kn} works.\nWe apply Lemma \\ref{tool} to the set \n$$A:=\\overline{\\graph(f)}\\cap\\{x_i\\neq 0\\}\\text{ and }f=\\frac{x_0}{x_i}.$$ \nSets $A_c$ for $c\\neq 0$ are filled with parametric\ncurves of degree at most $d$\n$$\\K\\ni t\\rightarrow (c:t:a_2:\\dots:a_{i-1}:1:a_{i+1}\\dots:a_n)\\xrightarrow{f} A_c\\subset\\graph(f).$$\n\nOtherwise $z=(0,1,0\\dots,0)$. So for every component $C$ of $\\s_f$ which is different from components of $p_2(\\overline{\\graph(f)}\\cap \\{(0:1:0:\\dots)\\}\\times \\K^m)$ its open dense subset is covered by parametric curves of degree at most $d$, thus by Proposition \\ref{k-uniruledofdeg} component $C$ has degree of $\\K$-uniruledness at most $d$.\n\\end{proof}\n\nWhen applying Theorem \\ref{kw} to two different directions we get:\n\n\\begin{corollary}\\label{kkw}\nLet $\\K$ be an arbitrary algebraically closed field, and let $f:\\K^2\\times W\\rightarrow\\K^m$ be a generically finite polynomial mapping of degree $d$. Then the set $\\s_f$ has degree of $\\K$-uniruledness at most $d$ or it is empty.\n\\end{corollary}\n\n\\begin{proof}\nApply Theorem \\ref{kw} to two directions. As previously it is enough to show that any component $C$ of  $$\\overline{\\graph(f)}\\setminus\\graph(f)$$  \nhas degree of $\\K$-uniruledness at most $d$. From the first direction we get that for any component of $$\\overline{\\graph(f)}\\setminus\\graph(f)\\setminus\\overline{\\graph(f)}\\cap \\{(0:1:0:0:\\dots)\\}\\times \\K^m,$$ \nhas degree of $\\K$-uniruledness at most $d$,\nwhile from the second direction we get it for any component of \n$$\\overline{\\graph(f)}\\setminus\\graph(f)\\setminus\\overline{\\graph(f)}\\cap \\{(0:0:1:0:\\dots)\\}\\times \\K^m.$$\nSince $C$ is a component of at least one of them the assertion follows.\n\\end{proof}\n\n\\section{Remarks}\n\nWhen we consider all generically finite mappings $\\C^n\\rightarrow\\C^n$ of degree at most $D$, then by Theorem \\ref{Sfkuni} hypersurfaces $\\s_f$ are all $\\C$-uniruled and by Theorem \\ref{hiperc} their degree is bounded. It happens that they all are also of bounded degree of $\\C$-uniruledness (namely by $D-1$, this follows from Theorem \\ref{cn}). It is reasonable to ask the following question:\n\n\\begin{question}\nDoes for every $n$ and $d$ exist a universal constant $D=D(n,d)$, such that all $\\K$-uniruled hypersurfaces in $\\K^n$ of degree at \nmost $d$ have degree of $\\K$-uniruledness at most $D(n,d)$?\n\\end{question}\n\nWe can ask an even stronger question:\n\n\\begin{question}\\label{22}\nDoes for every $n$ and $d$ there exist a universal constant $D=D(n,d)$, such that if a hypersurface in $\\K^n$ of degree at most $d$ \ncontains a parametric curve passing through $O=(0,\\dots,0)$, then it also contains such a parametric curve of degree at most $D(n,d)$?\n\\end{question}\n\nWe can show the following equivalent condition to the affirmative answer to Question \\ref{22}. Let us define\n$$K_{n,d,D}=\\{a=(a_1,\\dots,a_s)\\in\\K^{\\binom{n+d}{d}}:\\text{ such that hypersurface in }\\K^n\\text{ defined}$$ $$\\text{by }f_a=a_1+a_2x_1+\\dots+a_sx_1^d\\cdots x_n^d=0\\text{ contains a parametric curve}$$ \n$$\\text{passing through }O\\text{ of degree at most }D\\},$$\nand let\n$$K_{n,d}=\\{(a_1,\\dots,a_s)\\in\\K^{\\binom{n+d}{d}}:\\text{ such that hypersurface in }\\K^n\\text{ defined by}$$ $$f_a=a_1+a_2x_1+\\dots+a_sx_1^d\\cdots x_n^d=0\\text{ contains a parametric curve}$$\n$$\\text{passing through }O\\}.$$\n\n\\begin{proposition}\nThe set $K_{n,d,D}$ is closed.  \n\\end{proposition}\n\n\\begin{proof}\nFor $b=(b_{1,1}:\\dots:b_{D,n})\\in\\p^M$, where $M=Dn-1$, let \n$$\\varphi_{b}:\\K\\ni t\\rightarrow (b_{1,1}t+\\dots+b_{1,D}^Dt^D,\\dots,b_{n,1}t+\\dots+b_{n,D}^Dt^D)\\in\\K^n$$\nbe a parametric curve of degree at most $D$ passing through $O$. Consider a variety and a projection\n$$\\K^{\\binom{n+d}{d}}\\times\\p^M\\supset V=\\{(a,b):\\forall_{t}\\;f_a(\\varphi_{b}(t))=0\\}\\ni(a,b)\\rightarrow a\\in \\K^{\\binom{n+d}{d}}.$$\nBy the definition the set $V$ is closed and it consists of pairs $(a,b)$ corresponding to pairs $(f_a,\\varphi_b)$ of a hypersurface and a parametric curve passing through $O$ contained in it. Hence the image equals to $K_{n,d,D}$. But since $\\p^M$ is complete and $V$ is closed the image is also closed. \n\\end{proof}\n\n\\begin{proposition}\nLet $\\K$ be an uncountable field. Then the following conditions are equivalent:\n\\begin{enumerate}\n\\item the set $K_{n,d}$ is closed,\n\\item the answer to Question \\ref{22} is positive.  \n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nObviously $K_{n,d}=\\bigcup_{D\\in\\N}K_{n,d,D}$. \nImplication $(1)\\Rightarrow(2)$ follows from Baire's Theorem \\ref{baire}, the opposite one is trivial.\n\\end{proof}\n\nAffirmative answers in the case of $\\K=\\C$ from section \\ref{c}\nand $\\K=\\R$ from section \\ref{r} lead to the following natural conjecture:\n\n\\begin{conjecture}\nLet $\\K$ be an arbitrary algebraically closed field, and let\n$f:\\K^n\\rightarrow \\K^m$ be a generically finite polynomial mapping of degree\n$d$. Then the set $\\s_f$ has degree of $\\K$-uniruledness at most $d-1$ or it is empty.\n\\end{conjecture}\n\nTheorems \\ref{multc} and \\ref{multc1} suggests the following two conjectures:\n\n\\begin{conjecture}\nLet $\\K$ be an arbitrary algebraically closed field, $X$ be an affine variety with degree of $\\K$-uniruledness at most $d_1$ and let $f:X\\rightarrow\\K^m$ be a generically finite polynomial mapping of degree $d_2$. Then the set $\\s_f$ has degree of $\\K$-uniruledness at most $d_1d_2$ or it is empty.\n\\end{conjecture}\n\n\\begin{conjecture}\nLet $X$ be a closed semialgebraic set with degree of $\\R$-uni\\-ruled\\-ness at most $d_1$, and let $f:X\\rightarrow\\R^m$ be a generically finite polynomial mapping of degree $d_2$. Then the set $\\s_f$ has degree of $\\R$-uniruledness at most $d_1d_2$ or it is empty.\n\\end{conjecture}\n\n\\pagebreak\n\n\\chapter{The set of fixed points of group actions}\n\nIn the whole Chapter, unless stated otherwise, $\\K$ is assumed to be an arbitrary algebraically closed field.\nWe are going to remind the definition and a characterization of unipotent groups. The main goal of this Chapter is to show that under some conditions the set of fixed points $\\Fix(\\G)$ of action of an algebraic group $\\G$ is $\\K$-uniruled. \n\nIn particular we prove:\n\n\\begin{theoremm}\nLet $\\G$ be a nontrivial connected unipotent algebraic group which acts effectively on an affine variety $X$. Then the set of fixed points of $\\G$ is a $\\K$-uniruled variety.\n\\end{theoremm}\n\n\\begin{theoremm}\nLet $\\G$ be an infinite connected algebraic group which acts effectively on $\\K^n$, for $n\\geq 2$. Assume that an irreducible hypersurface $H$ is contained in the set of fixed points of $\\G$. Then $H$ is $\\K$- uniruled.\n\\end{theoremm}\n\n\\section{Introduction}\n\nWe will use the following definition of unipotent groups.\n\n\\begin{definition}\\label{series}\nAn algebraic group $\\G$ is \\textup{unipotent} if there exists a series of normal algebraic subgroups\n$$0=\\G_0\\vartriangleleft \\G_1\\vartriangleleft\\dots\\vartriangleleft \\G_r=\\G,$$\nsuch that $\\G_i\/\\G_{i-1}\\cong \\G_a=(\\K,+,0)$.\n\\end{definition}\n\nLet $\\G$ be a connected unipotent algebraic group, which acts effectively on a variety $X$. The set\nof fixed points of this action was intensively studied (see \\cite{bi, caso, gr, ho}). \nIn particular Bia\\l ynicki-Birula proved that if $X$ is an affine variety,\nthen $\\G$ has no isolated fixed points.\n\nHere we also consider the case when $X$ is an affine variety. We\ngeneralize the result of Bia\\l ynicki-Birula by proving, that\nthe set $\\Fix(\\G)$ of fixed points of $\\G$ is in fact a $\\K$-uniruled\nvariety. So through every point $x\\in\\Fix(\\G)$ passes a\nparametric curve.\n\nWe also show that if an arbitrary infinite connected algebraic group $\\G$\nacts effectively on $\\K^n$ and the set of fixed points contains a\nhypersurface $H$ then this hypersurface is $\\K$-uniruled. It was known before for the case $\\K=\\C$ \\cite{je03}.\n\n\\section{The set of fixed points of a unipotent group}\n\nWe begin with a quite natural observation:\n\n\\begin{lemma}\\label{ginv}\nLet algebraic group $\\G$ act on affine varieties $X,Y$ and let $f:X\\rightarrow Y$ be a generically finite $\\G$-invariant mapping (for any $x\\in X,g\\in\\G$ there is $gf(x)=f(gx)$). Then the set $\\s_f$ of points at which map $f$ is not proper is also $\\G$-invariant (for any $y\\in \\s_f,g\\in\\G$ there is $gy\\in \\s_f$). \n\\end{lemma}\n\n\\begin{proof}\nIt is enough to show that the complement of the set $\\s_f$ is\n$\\G$-invariant.\n\nSuppose that $f$ is proper at $y\\in Y$. This means that\nthere exists an open neighborhood $U$ of $y$ such that the mapping\n$$f\\vert_{f^{-1}(U)}:f^{-1}(U)\\to U$$\nis finite. We have the following diagram:\n\n\\begin{center}\n\\begin{picture}(240,160)(-40,40)\n\\put(140,160){\\makebox(0,0)[tl]{$f^{-1}(gU)=gf^{-1}(U)$}}\n\\put(20,160){\\makebox(0,0)[tl]{$f^{-1}(U)$}}\n\\put(30,40){\\makebox(0,0)[tl]{$U$}}\n\\put(180,40){\\makebox(0,0)[tl]{$gU$}}\n\\put(190,100){\\makebox(0,0)[tl]{$f\\vert_{f^{-1}(gU)}$}}\n\\put(40,100){\\makebox(0,0)[tl]{$f\\vert_{f^{-1}(U)}$}}\n\\put(95,50){\\makebox(0,0)[tl]{$g$}}\n\\put(95,170){\\makebox(0,0)[tl]{$g^{-1}$}}\n\\put(33,145){\\vector(0,-1){100}} \\put(45,35){\\vector(1,0){125}}\n\\put(135,155){\\vector(-1,0){75}} \\put(183,145){\\vector(0,-1){100}}\n\\end{picture}\n\\end{center}\n\\vspace{3mm}\n\nHorizontal mappings are isomorphisms, hence by Corollary \\ref{isoo} they are finite. The map $f\\vert_{f^{-1}(gU)}$ is a composition of finite maps, so it is also finite. Hence $gy\\notin \\s_f$.\n\\end{proof}\n\nThe aim of this section is to prove the following:\n\n\\begin{theorem}\\label{glowne}\nLet $\\G$ be a nontrivial connected unipotent algebraic group which acts effectively on an affine variety $X$. Then the set of fixed points of $\\G$ is a $\\K$-uniruled variety.\n\\end{theorem}\n\n\\begin{proof} By the induction on $\\dim\\G$ we can easily reduce the general case to the case of $\\G=\\G_a$. Indeed let \n$$0=\\G_0\\vartriangleleft \\G_1\\vartriangleleft\\dots\\vartriangleleft \\G_r=\\G$$\nbe a normal series like in Definition \\ref{series}. Now $\\Fix(\\G)\\subset\\Fix(\\G_{r-1})$. The set $\\Fix(\\G_{r-1})$ is invariant under $\\G$ action, moreover $\\G_{r-1}$ acts trivially on it. So the group $\\G\/\\G_{r-1}\\cong \\G_a$ acts on $\\Fix(\\G_{r-1})$. If this action is trivial, then $\\Fix(\\G)=\\Fix(\\G_{r-1})$ and the assertion follows from the inductive assumption. Otherwise $\\G_a$ acts effectively on $\\Fix(\\G_{r-1})$ so the set of fixed points $\\Fix(\\G)$ is $\\K$-uniruled.\n\nFirst assume that the field $\\K$ is uncountable. Take a point $a\\in\\Fix(\\G)$. By Propositions \\ref{k-uniruled} and \\ref{k-uniruledofdeg} it is enough to prove, that\nthere exists a parametric curve $S\\subset\\Fix(\\G)$ passing through $a$.\nLet $L$ be an irreducible curve in $X$ passing through $a$, which is\nnot contained in any orbit of $\\G$ and which is not contained in\n$\\Fix(\\G)$. \n\nConsider a surface $Y=L\\times\\G$. There is natural $\\G$\naction on $Y$. For $h\\in\\G$ and $y=(l,g)\\in Y$ we put\n$h(y)=(l,hg)\\in Y$. Take a mapping\n$$\\Phi : L\\times G\\ni (x, g)\\to gx\\in X.$$ \nIt is a generically finite polynomial mapping. \n\nObserve that it is $\\G-$invariant,\n$\\Phi(gy)=g\\Phi(y)$. Due to Lemma \\ref{ginv} this implies that the set $\\s_\\Phi$ of points\nat which the mapping $\\Phi$ is not finite is also $\\G$-invariant.\n\nTheorem \\ref{Sfsurface} gives that the set $\\s_\\Phi$ is $\\K$-uniruled. Let $\\s_\\Phi=S_1\\cup\\dots\\cup S_k$ be the decomposition of $\\s_\\Phi$ \ninto parametric curves. Since the set $\\s_\\Phi$ is $\\G$-invariant, we have that each curve $S_i$ is also\n$\\G$-invariant. \n\nNote that the point $a$ belongs to $\\s_\\Phi$,\nbecause the fiber over $a$ has an infinite number of points. We can\nassume that $a\\in S_1$. We want to show that $S_1\\subset\\Fix(\\G)$. Let $x\\in S_1$, assume $x\\notin\\Fix(\\G)$, then $\\G x=S_1$ and $a$ would be in the orbit\nof $x$, this is a contradiction with $a\\in\\Fix(\\G)$. Hence $a\\in S_1\\subset\\Fix(\\G)$, which ends the proof of an uncountable field case.\n\nNow assume that the field $\\K$ is countable.\nLet $\\Le$ be an uncountable algebraically closed extension of $\\K$. \nThen the group $\\Le\\G$ acts on $\\Le X$ and $\\Fix(\\Le\\G)=\\Le\\Fix(\\G)$.\nBy the first part of our proof the variety\n$\\Fix(\\Le\\G)$ is $\\Le$-uniruled, so due to Proposition \\ref{indep} the set $\\Fix(\\G)$ is $\\K$-uniruled.\n\\end{proof}\n\nWe get a following corollary:\n\n\\begin{corollary}[Bia\\l ynicki-Birula \\cite{bi}]\nLet $\\G$ be a nontrivial connected unipotent  group which acts\neffectively on an affine variety $X$. Then $\\G$ has no isolated fixed points.\n\\end{corollary}\n\n\\section{Hypersurface contained in the set\\newline of fixed points}\n\nWe generalize the main result of \\cite{je03} from $\\C$ to an arbitrary algebraically closed field.\n\n\\begin{theorem}\\label{glowne1}\nLet $\\G$ be an infinite connected algebraic group which acts\neffectively on $\\K^n$, for $n\\geq 2$. Assume that an irreducible\nhypersurface $H$ is contained in the set $\\Fix(\\G)$ of fixed points of $G$.\nThen $H$ is $\\K$-uniruled.\n\\end{theorem}\n\n\\begin{proof} Since $\\G$ acts effectively on the affine space $\\K^n$, then by the Chevalley Theorem (\\cite{sh}, page $190$, Theorem C) we can assume that the group $\\G$ is affine. \nIn particular it contains either the subgroup $\\G_m=(\\K^*,\\cdot,1)$ or the subgroup $\\G_a=(\\K,+,0)$ (see \\cite{ii}). Thus we can assume that the group $\\G$ is either $\\G_m$ or $\\G_a$.\n\nAs before we can assume that the field $\\K$ is uncountable. Take a\npoint $a \\in H$. By Propositions \\ref{k-uniruled} and \\ref{k-uniruledofdeg} it is enough to\nprove, that there exists a parametric curve $S\\subset H$ passing\nthrough $a$. Let $L$ be a line in $\\K^n$ going through $a$ such\nthat the set $L\\cap\\Fix(\\G)$ is finite. Denote $L\\cap H=\\{a, a_1,\\dots,a_m\\}$. \n\nConsider a mapping\n$$\\phi:L\\times\\G\\ni (x,g)\\to gx\\in\\K^n.$$ \nObserve that $\\phi(L\\times\\G)$ is a union of disjoint orbits of $\\G$. This\nimplies that $\\phi(L\\times\\G)\\cap H=\\{ a, a_1,\\dots,a_m\\}$. Take\n$X=\\overline{\\phi(L\\times\\G)}$. Note that $X\\cap H$ is a union of\ncurves. This means that there exists a curve $S\\subset X\\cap H$, which\ncontains the point $a$. However $S\\subset \\overline{X\\setminus\n\\phi(L\\times\\G)}.$ This implies that $S\\subset\\s_\\phi$ and we\nconclude by Theorem \\ref{Sfsurface}.\n\\end{proof}\n\n\\section{A real field case}\n\nWe give a real counterpart of Theorem \\ref{glowne}.\n\n\\begin{theorem}\\label{glowne2}\nLet $\\G$ be a real nontrivial connected unipotent algebraic group, which acts effectively and polynomially on a closed semialgebraic $\\R$-uniruled set $X$. Then the set $\\Fix(\\G)$ of fixed points of $\\G$ is also $\\R$-uniruled. In particular it does not contain isolated points.\n\\end{theorem}\n\n\\begin{proof} By the induction on $\\dim\\G$ we\ncan easily reduce the problem to the case of $\\G=\\G_a$. \n\nAssume that $\\G=\\G_a$. Let $D$ be the degree of $\\R$-uniruledness\nof $X$. Take a point $a\\in\\Fix(\\G)$. Let \n$$\\phi:\\G\\times X\\ni\n(g,x)\\to \\phi(g,x)\\in X$$ \nbe a polynomial action of $\\G$ on $X$.\nThis action also induces a polynomial action of the\ncomplexification $\\G^c=(\\C,+,0)$ of $\\G$ on $X^c$. We will denote\nthis action by $\\overline{\\phi}$. Assume that $\\deg_g\\phi\\le d$.\nBy Definition \\ref{r-uniruleddef} we have to prove, that\nthere exists a parametric curve $S\\subset\\Fix(\\G)$ passing through\n$a$ of degree bounded by max$(d,D)$. Let $L$ be a parametric\ncurve in $X$ passing through $a$. If it is contained in\n$\\Fix(\\G)$, then the assertion is true. Otherwise consider a closed semialgebraic\nsurface $Y=L\\times\\G$. There is a natural $\\G$ action on $Y$: \n$$\\text{for }h\\in \\G\\text{ and }y=(l,g)\\in Y\\text{ we put }h(y)=(l,hg)\\in Y.$$ \nTake a generically finite polynomial mapping\n$$\\Phi:L\\times\\G\\ni (x, g)\\to \\phi(g,x)\\in X.$$ \n\nObserve that it is $\\G$-invariant, which means\n$\\Phi(gy)=g\\Phi(y)$. Lemma \\ref{ginv} implies that the set $\\s_\\Phi$ of points\nat which the mapping $\\Phi$ is not finite is also $\\G$-invariant.\n\nLet $\\s_\\Phi=S_1\\cup\\dots\\cup S_k$ be a decomposition of $\\s_\\Phi$ into irreducible components. Due to Theorem \\ref{multc1} each $\\s_i$ is a parametric curve. Since\nthe set $\\s_\\Phi$ is $\\G-$invariant each $S_i$ is also $\\G$-invariant. \n\nNote that the point $a$ belongs to $\\s_\\Phi$, because the fiber over $a$ has infinite number of points. We can assume that $a\\in S_1$. Let us note that the point $a$ is also a fixed point for $\\G^c$. We want to show that $S_1\\subset\\Fix(\\G)$. Let $x\\in S_1$. The set $S_1^c$ is also $\\G^c$-invariant and if $x\\not\\in\\Fix(\\G)$ then $\\G^cx=S_1^c$ and $a$ would be in the orbit of $x$, which is a contradiction. Hence $S_1\\subset\\Fix(\\G)$ completes the proof.\n\\end{proof}\n\n\\begin{corollary}\nLet $\\G$ be a real nontrivial connected unipotent group which acts effectively and polynomially on a closed semialgebraic set $X$. If the set $\\Fix(\\G)$ of fixed points of $\\G$ is nowhere dense in $X$, then it is $\\R$-uniruled.\n\\end{corollary}\n\n\\begin{corollary}\nLet $\\G$ be a real nontrivial connected unipotent group which acts effectively and polynomially on a connected Nash submanifold $X\\subset\\R^n$. Then the set $\\Fix(\\G)$ is $\\R$-uniruled.\n\\end{corollary}\n\n\\section{Remarks}\n\nTo finish we state:\n\n\\begin{conjecture} Let $\\G$ be an algebraic group, which\nacts effectively on $\\K^n$. If $R$ is an irreducible component of\nthe set $\\Fix(\\G)$ of fixed points of $\\G$, then $R$ is a $\\K$-uniruled variety or it is a point.\n\\end{conjecture}\n\\pagebreak\n\\clearpage\n\\addcontentsline{toc}{chapter}{Bibliography}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf Introduction}\n\nThere is an intriguing parallel between the $D=5$ pure Yang-Mills theory\nand the $D=4$ chiral lagrangian theory of mesons. We first summarize \nfeatures of the $D=5$ Yang-Mills theory.\n\nThe pure $SU(N)$ Yang-Mills $D=5$ gauge theory supports a \ntopological soliton, unique to $D=5$ \\cite{deser1}. \nThis soliton is simply the instanton, an $SU(2) $ configuration,\nlifted to a time slice in $D=5$, associated\nwith the nontrivial homotopy class  $\\Pi_3(SU(2))$, and\nit carries a conserved topological current \\cite{topo1,topo2}.\nThe theory actually has two conserved\ntopological currents, built out of the gauge fields: firstly an \nadjoint representation current\n(only for groups possessing $d$-symbols, e.g., SU(3) and higher), and secondly \na singlet current (present for all groups). The adjoint current controls \ntransitions between the various ways in which the instantonic soliton\ncan be embedded into the gauge group (e.g., a pure ``I-spin'' embedding can\nflip to a ``U-spin'' or ``V-spin'' embedding\nfor $SU(3)$). The singlet current is identically conserved, \nand yields the topological charge of the soliton.\n\nEach of these currents is topological, and cannot be derived by Noetherian variation\nof the gauge kinetic term action. The theory \nmust therefore be supplemented with an additional \nChern-Simons term. The Chern-Simons term that\ngenerates the adjoint current is known as\nthe ``second Chern character'',\n(the $D=5$ generalization of the Deser-Jackiw-Schonfeld-Siegel-Templeton\nmass term of $D=3$ \\cite{schonfeld,deser}; see also \\cite{niemi}). \nUnder variation \nof the gauge fields, the second Chern character generates\nthe adjoint current as a source term in the equation\nof motion of the gauge field.  \nWhile not manifestly gauge invariant, under small gauge\ntransformations (those continuously connected\nto the identity), the action containing\nthe second Chern character is invariant.  By contrast, \nfor topologically nontrivial transformations,\nthe action shifts by an additive numerical factor, and\nthe coefficient of the Chern character is necessarily quantized so \nthe path integral is invariant (\\ie, with the proper \ncoefficient, this shift in the action is then $2\\pi N$) \\cite{deser,Wu}.\n\nThe singlet current has no associated Chern-Simons term\nbuilt out of the gauge fields alone. We presently propose to \nintroduce a ``dual variable,'' a vector\npotential associated with the instantonic soliton. This\nallows us to write a new $U(1)$-gauge\ninvariant topological term which is analogous to the second Chern character\nand which generates the singlet current. \n\nOn the other hand, chiral theories of mesons in $D=4$ based on flavor \n$SU(N)_L\\times SU(N)_R$ also possess remarkable, and quite similar, \ntopological properties.  The theories support \nthe skyrmion solution, which is an $SU(2) $ configuration\nand a stable topological object \n(whose core is stabilized when the ``Skyrme'' term is added). \nThe skyrmion also reflects the nontrivial $\\Pi_3(SU(2))$, and \nit carries a conserved\n(modulo anomalies) singlet current, the Goldstone-Wilczek current\n\\cite{goldstone}, which is interpreted as baryon number.\nThe chiral theory also contains adjoint representation \ntopological currents, conserved modulo anomalies. \nThese latter currents again exist only for groups with $d$-symbols, and \ngovern transitions\nof the embeddings of the skyrmion in the diagonal subgroup $SU(N)$. \nA connection between the instantonic soliton of $D=5$ and the skyrmion \nof $D=4$, through compactification, and a matching \nof their $U(1)$ currents was discussed in \nsome detail in ref.(\\cite{topo2}). In fact, the full form of the \nGoldstone-Wilczek current can be easily inferred from this matching.\n\nThe adjoint topological currents in $D=4$ chiral\ntheories derive from \nthe Wess-Zumino-Witten term. \nRemarkably, the WZW term is neither \nmanifestly chirally nor gauge invariant; yet it possesses both symmetries \nfor small transformations---those\nthat are continuously connected to the identity. The overall invariance \nof the path integral \nunder large topological chiral and gauge transformations leads again to \nquantization conditions on the WZW term coefficient \\cite{Witten}.\nThe singlet Goldstone-Wilczek current has no corresponding WZW term, but\nas will be discussed below, and elaborated\nin a companion paper, \\cite{hill}, \nsuch a term can be written, provided the $\\sigma$ and $\\eta'$\nmesons are incorporated into the theory. \nThis, in turn, leads to a new singlet axial\nvector topological current. \n\nThese topologically interesting aspects of $D=4$ chiral lagrangians\nhave long been known to\nfollow from the structure of the theory in one higher dimension.  \nIn the case of a $D=4$, $SU(N)_L\\times SU(N)_R$ nonlinear $\\sigma$-model, \ndescribed by an\n$N\\times N$ unitary matrix field, Witten has shown that the WZW term\ncan be obtained by promoting the global theory\nof mesons to $D=5$, where a certain manifestly chirally invariant\nChern-Simons term occurs, built out of the mesons. One \nthen compactifies the fifth-dimension \nterm, back to $D=4$. This results in the global \nWess-Zumino term of $D=4$ \\cite{Witten}.  \nAt this stage, by performing\ngauge transformations upon the global object, one can \ninfer how to introduce the gauge fields to compensate\nthe local changes in the WZ term.\nThis leads to the full Wess-Zumino-Witten term\nfor the gauged chiral lagrangian, which contains the \nfull anomaly structure of the theory. \nThe WZW term plays another crucial role, that of locking\nthe parity of the pion to the parity of space \\cite{Witten}.\nCertain allowed transitions, \nsuch as $K+\\bar{K}\\rightarrow 3 \\pi$, which would be\nabsent without the WZW term, now occur\nwith a topologically quantized amplitude. \n\nA conceptual drawback of \nthis procedure is that local gauge invariance\nis induced into a non-gauge invariant object, {\\em a posteriori}.  Local\ngauge invariance, however, is a more fundamental symmetry than the\nchiral symmetry which it breaks, and it is thus preferable to rely on \na procedure in which local gauge invariance\nis present at the outset.   \nThen, upon compactification, one would\nrequire that the meson fields appear with their proper kinetic terms \nand gauging.   \nImplementing such a procedure for compactification, we would expect that\nthe second Chern character  of the $D=5$ pure Yang-Mills theory\nmorphs into the $D=4$ WZW term.\nThis approach, at least,\nmay shed some light on the interplay on the interrelationship\nof the various symmetries and topology. \n\nIndeed, there exists in principle such a procedure, \nthe latticization of extra dimensions, \\cite{wang},\nor ``dimensional deconstruction\" \\cite{harvard}. \nThe approach latticizes only the extra dimensions, yielding\nthe effective kinetic and interaction terms,\nwhile keeping the $D=4$ subspace\nin the continuum. This  is related to the earlier\n``transverse lattice'' of Bardeen, Pearson and Rabinovici \\cite{bardeen}.\nFor extra dimensional theories, this is a powerful tool, leading to a \ncontinuum $D=4$ effective description of a theory\nthat originated as a pure Yang-Mills theory \nin higher dimensions, with emphasis\non maximal manifest gauge invariance. \nIn this approach, one derives\nthe gauge invariant effective lagrangian for a theory in $D=4$\nthat is defined by compactification of a theory in higher dimensions.\nStarting with pure Yang-Mills in $D=5$, one can thus engineer a $D=4$ gauged\nchiral lagrangian. The mesons then appear in the compactified\ntheory, packaged into exponential chiral fields,\nwhich are the Wilson links associated with the latticization.\n\nIn the present work, we study the deconstruction of\nthe $D=5$ Yang-Mills theory, supplemented with the \nsecond Chern character, and a new singlet auxiliary term.\nWe begin with a discussion of\nthe physical basis of orbifold boundary conditions, and\nconsideration of the topological\naspects of a gauged chiral lagrangian in $D=4$\nthrough the pure Yang-Mills theory in $D=5$ without mesons. \nPresently, we will show how to derive the WZW\nterm for a gauged chiral \nlagrangian in $D=4$, by \nmatching of the vector potentials and the field strengths of\nthe $D=5$ Yang-Mills theory onto the relevant operators\nin the deconstructed effective lagrangian. \nWe begin in the next section, after a review of\n$D=5$ Yang-Mills and the Chern-Simons terms, with a simple\nheuristic discussion that readily yields the WZW \nterm on orbifold compactification to $D=4$. \nWe will also\nanticipate how a new WZW-like term arises from\nthe singlet auxiliary  term in the parent $D=5$ \nYang-Mills theory. This new WZW term generates the\nsinglet Goldstone-Wilczek current, and a new $U(1)$ axial current\nfor the skyrmion.\n\nWe then study the general issue of\ntopological deformation of $D=5$ Yang-Mills into \n$D=4$ chiral theories in more detail. \nFirst, we note that there is\na key element we must address that\nis missing in a naive deconstruction, and\nwhich is essential to the propagation of topology\nfrom one geometrical dimension to another. This\nis the consistency of {\\em the Bianchi \nidentities}. The ordinary $D=4$ Bianchi \nidentities are always automatically \nsatisfied by the\nspecification that the field strength tensor is a commutator \nof covariant derivatives,\nsince the $D=4$ relations are simply the \nJacobi identity for antisymmetrized nested\ncommutators. However, we'll find that \nthere is an additional nontrivial Bianchi \nconstraint involving the ``lattice hopping derivative.'' \nThis is seen to fail for the first plausible definition\nof the hopping derivative in $D=5$.\n\nWe thus formulate the Bianchi identities on the \ndeconstruction lattice, and we find that\nwe are able to satisfy them \nwith a modified definition\nof the $D=4$ covariant derivative. The ordinary\nderivative is modified by the addition\nof  a vector combination of chiral currents\nwith a special coefficient of $1\/2$.  The\nformalism automatically\nimplements the ``magnetic superconductivity,''\nor confinement phase on the orbifold branes, $G_{4\\mu} =0$.\n\nThe Bianchi-consistent modification implies that the effective  \nfield strength, $G_{\\mu\\nu}$, is modified by the  addition \nof terms involving the commutators of chiral currents of the mesons.\nThis term occurs with a fixed coefficient.\nThe gauge action is therefore modified as well, and there now appear \nin the classical action two Skyrme terms.  The usual Skyrme term\nis generated by the current commutator terms in the field strength\ntensor and now has a fixed coefficient. \nMoreover, a new Skyrme term that involves the gauge\nfield, is also present. We thus conjecture\nthat this modified theory may tighten the link between the \ninstantonic soliton in $D=5$ and\nthe skyrmion in the deconstructed theory in $D=4$. \nWith these terms,\nthere may exist a ``self-dual,'' and even an analytic\nskyrmion solution, matching the instantonic soliton \nat large distances.\n\nGiven the new Bianchi-consistent action and field \nstrength, the pure Yang-Mills \nsecond Chern character (CS2) term again goes into the \nWZW term. The resulting WZW term is consistent with\nWitten's minimal coefficient, but is larger by\na factor of $5$. Thus, we infer that the dimensionality\nof the space-time, $D=5$, appears as the\nindex of the WZW term in the Bianchi-compliant theory, \nwhere we normally would install\n$N_c=3$, the number of colors of QCD.\n\n\n\n\\section{The D=5 Pure Yang-Mills Theory and Heuristic Derivation of the\n$D=4$\nWess-Zumino-Witten Term}\n\n\\vskip 0.15in\n\\noindent{\\bf (i) Preliminaries}\n\\vskip 0.15in\n\n\nWe start with an $SU(N)$ Yang-Mills gauge theory\nin $D=5$. The theory relies on vector potentials, $A_A^a(x)$ \nand coordinates $x^A$, where $(A=0,1,2,3,4)$,\nand where $x^\\mu$ and $(\\mu = 0,1,2,3)$ refers to the\nusual space-time dimensions. When we say ``fifth component\nof a vector, $x^A$'' we mean, of course, $x^4$.\n\nThe covariant derivative is \n\\beq\nD_A = \\partial_A - iA_A~, \\qquad \\qquad  A_A = A_A^aQ^a ,\n\\eeq\nwhere $Q^a$ is an abstract operator that takes on\nthe values of $Q^a=\\lambda^a\/2$ in the adjoint representation.\nThe field strength then is \n\\beq\nG_{AB} = i[D_A,D_B]\n=\\partial_A A_B-\\partial_B A_A - i[A_A,A_B] .\n\\eeq\nThis theory has the standard kinetic term:\n\\beq\n\\label{start}\n{\\cal{L}} = -\\frac{1}{2\\tilde{g}^2}\\Tr(G_{AB}G^{AB})  ,\n\\eeq\nwhere $1\/\\tilde{g}_2^2$ is the coupling with dimensions of mass. With this\nnormalization, gauge fields have the canonical dimensionality \nwith respect to $D=4$, \\ie, $[A_A]= M^{1}$, and $[G_{AB}]= M^2$. \n\nThe theory possesses two identically conserved Chern-Simons \ncurrents of the form:\n\\bea\n\\label{cur1}\nJ_A & = & \\epsilon_{ABCDE}\\Tr(G^{BC}G^{DE}),\n\\eea\n\\bea\n\\label{current2}\nJ^a_A & = & \\epsilon_{ABCDE}\n\\Tr \\Bigl ( \\frac{\\lambda^a}{2}\\{G^{BC},G^{DE}\\}\\Bigr ).\n\\eea\nThe second current requires that $SU(N)$ possess a $d$-symbol, \nhence $N\\geq 3$; and it is further {\\em covariantly}\nconserved, \n\\beq\n[D^A,~J^a_A~ Q^a]=0. \n\\eeq\nThese topological  \ncurrents do not arise from eq. (\\ref{start}) \nunder local Noetherian variation of the fields. \n\nThe adjoint currents can be derived from an\naction containing the ``second Chern character.''\nThe second Chern character, which we'll abbreviate as CS2, \nis derived by ascending to \n$D=6$ and considering the generalization of the Pontryagin index\n(a $D=6$ generalization of the $\\theta$-term),\n\\beq\n{\\cal{L}}_0 =\\epsilon_{ABCDEF}\\Tr G^{AB}G^{CD}G^{EF} .\n\\eeq\nThis can be written as a total\ndivergence, \n\\beq\n\\frac{1}{8}{\\cal{L}}_0 \n=\n-\\partial^F\\epsilon_{ABCDEF}\n\\Tr\\Bigl (A_A \\partial_B A_C \\partial_D A_E   \n- \\frac{3i}{2}A_A A_BA_C \\partial_D A_E \n- \\frac{3}{5}A_A A_B A_C A_D A_E\\Bigr ). \n\\eeq\nFormally, compactifying the sixth\ndimension and integrating ${\\cal{L}}_0$ over \nthe boundary in $x^5$ leads to ${\\cal{L}}_1$,\nthe second Chern character as an element\nof the $D=5$ Lagrangian, \n\\bea\n\\label{CSterm0}\n{\\cal{L}}_1 & = & c\\epsilon^{ABCDE}\n\\Tr \\Bigl (A_A \\partial_B A_C \\partial_D A_E   \n - \\frac{3i}{2}A_A A_BA_C \\partial_D A_E \n- \\frac{3}{5}A_A A_B A_C A_D A_E \\Bigr ) .\n\\eea\nThis can be rewritten in a convenient\nform involving gauge covariant field\nstrengths,\n\\bea\n\\label{CS2}\n{\\cal{L}}_1 \n& = &\n\\frac{c}{4}\\epsilon^{ABCDE}\n\\Tr \\Bigl (A_A G_{BC}G_{DE}   \n + i A_A A_B A_C G_{DE}\n- \\frac{2}{5}A_A A_B A_C A_D A_E \\Bigr ) ,\n\\eea\nhence, for pure gauge configurations all but the last term vanish. \nThe second Chern characters can be constructed in any odd\ndimension from a general algorithm \\cite{Wu}.\n\nWhile not manifestly gauge invariant, it is\nstraightforward to verify that CS2 is indeed gauge\ninvariant for gauge transformations continuously\nconnected to the identity. By contrast, for topologically\nnontrivial gauge transformations, the action shifts by a constant. \nHence, the coefficient $c$ must be chosen\nfor effective invariance so that the action shifts by $2\\pi N$: \nthe path integral is then invariant. \nIt can be shown that this factor is:\n\\beq\nc = \\frac{1}{48\\pi^2} ~.\n\\eeq\nThe\nvariation of the action with respect to the gauge field $A^a$ indeed \ngenerates the current of eq.(\\ref{current2}) as a source for the\nequation of motion.\n\n\\newpage\n\\vskip 0.15in\n\\noindent{\\bf (ii) Heuristic Derivation of $D=4$\nWess-Zumino-Witten Term}\n\\vskip 0.15in\n\nConsider orbifold compactification of the $D=5$ Yang-Mills theory\nto a $D=4$ theory.\nOrbifold compactification is usually specified mathematically\nfollowing Horava and Witten \\cite{horava}, such as\n``compactification on $S_1\/Z_2$.'' One thus\nconsiders an interval $0\\leq x^4\\leq 2a$, \nclassifies basis functions as even\n$P=(+)$ or odd $P=(-)$ under reflection about $x^4=a$, and under\ncompactification, demands that the $P=(+)$ basis functions \nare assigned to the \n$D=4$ vector potentials, $A_\\mu^A$, and the $P=(-)$ basis functions\nto the $A_4^A$ vector potentials. \nOrbifolding is the basis of many\nmodels of low energy extra dimensions, but we prefer a more\nphysical statement on orbifold compactification. \n\nAlternatively, we can consider two branes to be located at\n$x^4=0$ and $x^4=a$. Each brane $i$ has a normal vector \n$\\eta_i^A$; \\eg, for brane ``L'' we have $\\eta_L=(0,0,0,0,1)$\nand for brane ``R'' we have $\\eta_{R} = (0,0,0,0,-1)$. The orbifold\nboundary conditions can be viewed as a special gauge choice\nfor the boundary condition applied on each brane of: \n\\beq\n \\left. \\eta_{A} ~G^{AB} \\right|_{L,R} = 0.\n\\eeq\nThis boundary condition is manifestly gauge invariant. \nFor the $\\eta_i$ defined above, we see that $G_{04}=0$,\nhence the normal component\nof the chromoelectric field strength is zero. Moreover,\nthe ``parallel'' magnetic field $G_{\\mu 4} $ where $\\mu \\neq 0$\nis also zero. These boundary conditions on the\nbranes are dual to those of an electric superconductor, and\nthey thus correspond to a magnetic superconductor. A magnetic\nsuperconductor would form electric flux tubes between electric charges\n(quarks) in the medium, hence a magnetic superconducting phase is\na confinement phase.\n\nWe thus consider the orbifold compactification as a kind of parallel\nplate magnetic superconducting capacitor (it can likewise be \nviewed as a magnetic superconducting\nJosephson junction). Spanning the gap\nbetween the plates,  is a Wilson line:\n\\beq\nU = P\\exp\\left( -i\\int_0^{a} dx^A A_A \\right) = \\exp(i\\tilde{\\pi}),\n\\eeq\nwhere, upon compactification, we view the Wilson line as a\nchiral field of mesons, as indicated, with $\\tilde{\\pi} =\n\\pi^a\\lambda^a\/f_\\pi $, where $f_\\pi = 95$ MeV. \n\nIn the superconducting boundary brane, or\ncapacitor plate regions (we'll refer to these generically\nas the ``end-zones''), we \ncan perform local gauge transformations. If the gauge group is $SU(N)$,\nthen there exist gauge transformations $V_L$ ($V_R$) that are\nconstant over the entire left-hand (right-hand) end-zone. These\ncan be identified as global $SU(N)_L$ ($SU(N)_R$) transformations. Under\nthese transformations we see that $U$ transforms as \n\\beq\nU \\rightarrow V_L U V^\\dagger_R ~, \n\\eeq\nand the theory under compactification becomes a gauged $SU(N)_L\\times\nSU(N)_R$ chiral lagrangian. The gauge fields should be viewed\nas left-- and right-- handed combinations of\nthe normal vector and axial vector mesons of QCD, and\nthey should be supplemented with additional Higgs fields\nto acquire masses. We thus do not pass to a unitary gauge\nin which the $A_4$ modes are eaten by gauge fields to acquire masses.\n\n\nIn the end-zones, we have\nthe magnetic\nsuperconducting phase. Here we hypothesize that the vector potentials\nare determined by ``London currents,'' the chiral currents\nbuilt out of the Wilson line:\n\\beq\n\\label{london}\nA_{A}(\\makebox{L end-zone}) = iU[\\partial_A, U^\\dagger] \\equiv i\\alpha_A  ~,\n\\qquad\nA_{A}(\\makebox{R end-zone}) = iU^\\dagger[\\partial_A, U] \\equiv i\\beta_A ~.\n\\eeq\nLondon currents are generated by the magnetic condensate kinetic term (\\eg,\nanalogous to a Higgs field), that locks the vector\npotential to the Nambu-Goldstone boson, \\eg, in our\npresent case $A_A(L)= \\partial_A\\tilde{\\pi} + ...$ \nand $A_A(R)= -\\partial_A\\tilde{\\pi} + ...$ in the endzones.\nThe particular definitions given in eq.\\ (\\ref{london})\nare pure gauges, and thus the gauge field\nstrength vanishes (\\eg, using form notation,\n$d\\alpha = -\\alpha^2$, and   $(1\/2)G = dA-iA^2 = 0$ when $A=i\\alpha$).\n\nWe now seek the low energy effective \ntheory. We substitute the London current vector\npotentials into the Chern-Simons term of eq.(\\ref{CS2})\nto obtain the $D=4$ effective topological\n lagrangian:\n\\beq\n \\left(\\frac{1}{2\\times 5}\\right) \\frac{i}{48\\pi^2}\\epsilon_{ABCDE}\\left(\n \\Tr\\alpha^A\\alpha^B\\alpha^C\\alpha^D\\alpha^E +  \n \\Tr\\beta^A\\beta^B\\beta^C\\beta^D\\beta^E\\right).\n\\eeq\nwhere the $\\alpha$ ($\\beta$) terms reside on the left (right)\nend-zone.\nTo leading order in the expansion\nin pions, we can write, \n\\bea\n\\epsilon_{ABCDE}\\Tr\\alpha^A\\alpha^B\\alpha^C\\alpha^D\\alpha^E & = &\ni\\epsilon_{ABCDE}\\partial_A\\Tr\\tilde{\\pi}\\alpha^B\\alpha^C\\alpha^D\\alpha^E + ...,\n\\nonumber \\\\\n\\epsilon_{ABCDE}\\Tr\\beta^A\\beta^B\\beta^C\\beta^D\\beta^E & = &\n-i\\epsilon_{ABCDE}\\partial_A\\Tr\\tilde{\\pi}\\beta^B\\beta^C\\beta^D\\beta^E + ...,\n\\eea\nThus, when we integrate \n$x^4$ over\nthe gap between the end-zones, $\\int_0^a dx^4 $,\nwe arrive at the effective lagrangian, \n\\beq\n\\label{WZ}\n \\frac{1}{240\\pi^2}\\epsilon_{\\mu\\nu\\rho\\sigma}\\left(\n \\Tr\\tilde{\\pi}\\alpha^\\mu\\alpha^\\nu\\alpha^\\rho\\alpha^\\sigma \\right).\n\\eeq\nEq.(\\ref{WZ}) is the precise structure of the leading\npiece of the Wess-Zumino term in an expansion \nin pions with\nWitten's normalization. \n\nA few comments are in order. Note that the expression\nis hermitian---and it can be written as either\n$\\Tr(\\pi \\alpha^4)$  or $\\Tr(\\pi \\beta^4)$. Witten's derivation \ninvolves compactification on a disk, and the WZW term\nresides on the periodic boundary of the disk, while\nthe present approach has used the orbifold configuration. \nWitten writes in an expansion in pions $(2\/15\\pi^2F_\\pi^5)\\Tr(A\\partial_\\mu A\n\\partial_\\nu A\\partial_\\rho A\\partial_\\sigma A) + ...$\nwith $A=\\pi^a\\lambda^a$ and $F_\\pi = 2f_\\pi$, which is consistent\nwith eq.(\\ref{WZ}). We note that the $\\alpha$ terms \nin the above derivation received a minus sign upon integrating\nfrom $0$ to $a$\n(which canceled against $i^2$) since the left end-zone\nresides at the lower limit of the integral; the $\\beta$\nterms received a positive sign. In\nthe $D=4$  theory the currents $\\alpha(x_\\mu)$ and $\\beta(x_\\mu)$ \nare viewed as residing\nat a common point in $D=4$ space-time, and we then have\nidentities such as: \n\\beq\n\\epsilon_{ABCDE}\\Tr\\tilde{\\pi}\\beta^B\\beta^C\\beta^D\\beta^E\n=\\epsilon_{ABCDE}\\Tr U\\tilde{\\pi}U^\\dagger\\alpha^B\\alpha^C\\alpha^D\\alpha^E,\n\\eeq\nand $U\\tilde{\\pi}U^\\dagger =\\tilde{\\pi}$, and \nwe use $U\\beta U^\\dagger = -\\alpha$\nand $U\\tilde{\\pi} U^\\dagger$ to bring the two terms into the\ncommon form.\n\n\nWith covariant London currents, \\eg,\n$\\alpha_A \\rightarrow U[D_A, U^\\dagger]$, the expression\n becomes fully gauge invariant. The field strength is then\n nonzero, and other\noperators like $\\Tr(\\pi \\alpha^2 G)$, $\\Tr(\\pi \\alpha G \\alpha)$, \\etc,\nnow appear. This expression can be integrated by parts\ninto the full Wess-Zumino-Witten term \nwhich will be developed\nin greater detail elsewhere \\cite{hill}.\n\nThe present ``derivation'' is only meant to be heuristic, and\nis not well-defined (\\eg, operators like $ U[D_A, U^\\dagger]$\nhave path dependence). Nonetheless, there\nare many alternative ways to proceed to formalize the deformation\ntheory from $D=5$ pure Yang-Mills into $D=4$ chiral lagrangians. \nIn the subsequent sections we'll\nbe led to a particular and well-defined deformation \nof the $D=5$ Yang-Mills theory\ninto a $D=4$ chiral lagrangian in which, \\eg, $A_\\mu(L)\\rightarrow\nA_\\mu(L)+i\\half U[D_\\mu, U^\\dagger]$. \n\n\\vskip 0.15in\n\\noindent{\\bf (iii) Singlet Auxiliary Chern-Simons Term\nand a New Singlet WZW Term}\n\\vskip 0.15in\n\nWe presently turn to the singlet topological current,\nand we'll merely anticipate some results that follow for\nthe compactification and deconstruction, using the\ntechniques of the next section. \n  \nThe singlet Chern-Simons current can be\ngenerated by an additional modification \nof the Lagrangian of the form\n(CS1):\n\\beq\n\\label{CS1}\n{\\cal{L}}_2 = {c'}\\epsilon_{ABCDE}V^A\\Tr(G^{BC}G^{DE}),\n\\eeq\nwhere $V^A$ is a singlet auxiliary vector field. Since it is identically \nconserved, the CS singlet current couples to this vector field \nin ${\\cal{L}}_2$ compatibly with a simple abelian gauge-invariance, \n$\\delta V^A= \\partial_A \\sigma$. \nIf the vector field $V$ is endowed with kinetic terms, the singlet current \nis also generated as a source in the corresponding Maxwell equations\nof motion for $V$.  \nNote that the singlet current cannot be derived\nfrom CS2, as the Chern-Simons term of eq.(\\ref{CSterm0}) only exists\nin $SU(N)$ for $N\\geq 3$ and does not occur, e.g., in $SU(2)$ Yang-Mills, while\nthe singlet current is always present. One might argue\nthat in $SU(2)$ the form of the current can be inferred, {\\em a posteriori},\ne.g., by considering the $\\lambda^8$\ncomponent in $SU(3)$ of the adjoint current, and setting coset fields\nto zero to descend to $SU(2)$. The singlet current cannot\narise from direct variation of CS2, eq.(\\ref{CSterm0}), and \neq.(\\ref{CS1}) (CS1) is required to generate it {\\em a priori}. \n\nThe appearance of $V_A$ is linked to the instantonic soliton \n\\cite{topo1,topo2}, the 't Hooft \ninstanton lifted to a static ``monopole'' configuration in $D=5$.\nThis object has  a mass of $8\\pi^2\/\\tilde{g}^2$, and it \ndescends to the \nskyrmion, characterized by the Goldstone-Wilczek current \\cite{goldstone},\nin $D=4$\n\\cite{topo2}.                  \n$V_A$ can be interpreted as an effective field associated with the \ninstantonic soliton. \nThe choice of $V_A$ is dictated by the degrees of freedom in the theory. \nWe must generate a conserved current, hence the\nvariation $\\delta V^A= \\partial_A \\sigma $, \\ie, we have\nno complex fields to draw upon. However, the instantonic\nsoliton must be described as a massive excitation, hence we cannot use\na Nambu-Goldstone field $\\sigma$ by itself. We may thus infer that the\ninstantonic soliton is associated with a massive $U(1)$ gauge field. \n\n\nMaking use of the chiral deconstruction techniques\ndiscussed in the present paper, we can deconstruct\neq.(\\ref{CS1}) to obtain a new auxiliary WZW term that generates\nthe Goldstone-Wilczek current. The field $V^A$ is\ndecomposed into $V^A_L+V^B_R$ where $V_L$ ($V_R$) has support\nin the $L$ ($R$) end-zone.\nThe $x^4$ integrated zero modes of $V^A$\nare then defined in terms of $\\sigma$ and $\\eta'$ fields\nof a chiral theory of mesons:\n\\bea\n\\int dx^4 V_{4R} & = & a(\\sigma + \\eta') ,\n\\quad\n\\int dx^4 V_{4L}  = a(\\sigma - \\eta') , \\nonumber \\\\\n\\int dx^4 V_{\\mu R} & = & af^{-1}\\partial_\\mu(\\sigma - \\eta' ),\n\\qquad\n\\int dx^4 V_{\\mu L} = af^{-1}\\partial_\\mu(\\sigma + \\eta'),\n\\eea\nwhere the choices are consistent with\nparities, and the Noether variations\nthat we would make for the original $V^A$\nto generate the currents (here we have set\nthe decay constant of $\\sigma$ and $\\eta'$\nto unity). Note that $\\sigma$ and $\\eta'$\ncan be viewed as glueballs, physical objects\nin the end-zone phases, even though the\ntheory is quarkless.\nWe find  that CS1, using methods\ndeveloped in the next section, deconstructs \nto terms containing the following form:\n\\bea\n\\label{CS2GW04}\n{\\cal{L}}_2 \n& \\rightarrow & -\\frac{ac'}{2} \\eta'\\epsilon_{\\mu\\nu\\rho\\sigma}\n\\Tr\\left(G_L^{\\mu\\nu}G_L^{\\rho\\sigma} \n+ G_R^{\\mu\\nu}G_R^{\\rho\\sigma}\n\\right) \n+i\\frac{ac'}{2f} \\partial^\\mu (\\eta') \\epsilon_{\\mu\\nu\\rho\\sigma}\n\\Tr\\left(\\alpha^\\nu  G_L^{\\rho\\sigma}\n-\\alpha^\\nu  \\overline{G}_R^{\\rho\\sigma}\\right)  \\cr\n&& \\cr\n&&\n-\\frac{2ac'}{f}\\partial^\\mu \\sigma \\epsilon_{\\mu\\nu\\rho\\sigma}\n\\Tr\\left(\\frac{3 i}{2} \\alpha^\\nu  G_L^{\\rho\\sigma}\n+\\frac{3i}{2} \\alpha^\\nu  \\overline{G}_R^{\\rho\\sigma}\n+ \\alpha^\\nu \\alpha^\\rho \\alpha^\\nu \\right)\n\\cr\n&&\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad\n+\\;\\;\\frac{3a}{2}c'\\sigma \\epsilon_{\\mu\\nu\\rho\\sigma}\\Tr\\left(\nG_{L\\mu\\nu}G_{L\\rho\\sigma} - G_{R\\mu\\nu} {G}_{R\\rho\\sigma} \n\\right),  \\cr\n&&\n\\eea\nwhere $\\alpha = U[D, U^\\dagger]$ and $\\beta=U^\\dagger[D, U]$,\nand $\\overline{G}_R^{\\rho\\sigma} = U{G}_R^{\\rho\\sigma}U^\\dagger$.\nThis is a new WZW-like term\nthat correctly generates the full Goldstone-Wilczek current,\n\\cite{topo2,goldstone} \nwith the correct normalization of a unit of baryon number for the\nskyrmion, provided $c' = 1\/48\\pi^2$ (identifying $c=c'$\nof the second Chern character), \n\\be\n\\label{eqb}\n{Q}^\\mu = \\frac{1}{24\\pi^2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\Tr \\left(  \n\\alpha_\\nu \\alpha_\\rho \\alpha_\\sigma  \n+\\frac{3i}{2}(\n G_{L\\nu\\rho} \\alpha_\\sigma + \\overline{G}_{R\\nu\\rho} \\alpha_\\sigma)\\right).\n\\ee\nThus, by constructing the Noether equation\nof motion of the $\\sigma$ meson,\nwe generate\nthe full conservation equation of the GW current, including its anomaly,\n\\beq\n\\partial_\\mu {Q}^\\mu = \n-\\frac{1}{32\\pi^2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\Tr \\left(  \nG_{L\\mu\\nu}G_{L\\rho\\sigma}  - {G}_{R\\mu\\nu}{G}_{R\\rho\\sigma} \\right).\n\\eeq\nThis shows that the singlet topological Chern-Simons current matches\nthe full GW current under compactification. \n\nMoreover, by Noether variation of\nthe $\\eta'$, we obtain a ``$U(1)$ axial current,'' \n\\be\n\\label{cur3}\nQ_5^\\mu  = \\frac{Z}{32\\pi^2}\\epsilon^{\\mu\\nu\\rho\\sigma}\\Tr \\left(  \nG_{L\\nu\\rho} \\alpha_\\sigma - \\overline{G}_{R\\nu\\rho} \\alpha_\\sigma  \n )\\right),\n\\ee\nThis actually has an indeterminate normalization\n$Z$. Its divergence equation likewise follows from\nthe $\\eta'$ equation of motion:\n\\bea\n\\label{eqb}\n\\partial_\\mu {Q}_5^\\mu  & = & \\frac{Z}{32\\pi^2}\n\\epsilon^{\\mu\\nu\\rho\\sigma}\n\\Tr\\left[\niG_{L\\mu\\nu }\\alpha_\\rho\\alpha_\\sigma\n+i\\overline{G}_{R\\mu\\nu }\\alpha_\\rho\\alpha_\\sigma\n+G_{L\\mu\\nu }G_{L\\rho\\sigma} + {G}_{R\\mu\\nu }{G}_{R\\rho\\sigma}\n\\right],\n\\eea\nand $Z=1$ is thus favored by matching to the $U(1)$ axial anomaly.\nThe last two terms are the correct form of an\naxial current anomaly, while the first terms on\nthe {\\em rhs} are \nanalogous to the Skyrme terms. The first two terms\nform a pseudoscalar and can be\ninterpreted as $2im\\bar{\\psi}\\gamma^5\\psi$\nin the axial current divergence of a massive nucleon.\nThis current, to our knowledge, has not been\npreviously discussed in the literature.\nThe details of this derivation will be \npresented elsewhere, \\cite{hill}.\n\n\\vskip 0.15in\n\\noindent\n\\section{Deconstruction and Bianchi Identities}\n\\vskip 0.15in\n\nThe heuristic argument presented in section II\nsuggests that a direct morphing of \nthe Chern-Simons terms  of $D=5$ Yang-Mills into $D=4$ chiral\nlagrangians is possible and meaningful. We expect that\nthere are many possible deformations of the parent theory\nin $D=5$, through deconstruction, that can yield various\nchiral theories in $D=4$. These deformations may or may not\nexploit the full geometrical and topological matching. We \nwould expect that\nan integral multiple of\nthe minimal coefficient of Witten, $1\/240\\pi^2$, will always obtain in\na consistent theory. \n\nThe heuristic argument indeed gave the ``minimal\ncoefficient'' of the WZ term of Witten.  We will now turn to a more\nliteral interpretation of dimensional deconstruction of\npure $D=5$ Yang-Mills which pays\ncloser attention to the details of topological mapping---in particular,\nto the definition of motion (``hopping'') in the fifth dimension\nand to the Bianchi identities.  Remarkably, the present construction\nyields the WZW term with a coefficient that is of the form $N\/240\\pi^2$,\nwhere the index $N=D=5$, is the dimensionality of the parent space-time. \n\n\n\\vskip 0.15in\n\\noindent{\\bf (i) Preliminaries}\n\\vskip 0.15in\n\nWe presently consider the \ncompactification of the $SU(N)$ Yang-Mills theory in $D=5$\non the interval $0\\leq x^4 \\leq a$. First, construct\na coarse-grained lattice of the $x^4$ dimension\nwith $2$ slices.  \nOn each slice lies a copy of the gauge group with hermitian generators \n$Q_i^a$. The covariant derivative is a sum over all slices with\nthe appropriate abstract charge assigned to each gauge field,\n\\beq\nD_\\mu = \\partial_\\mu -i A_{L\\mu}^a Q_L^a - i A_{R\\mu}^a Q_R^a,\n\\eeq\nwhere we use the notation ``left,'' $L$ (``right,'' $R$) \nfor brane 1 (2).\nThe generators $Q_L$ and $Q_R$ act on\nthe given slice, and the \nslices are connected from $L$ to $R$ \nby a unitary Wilson link $U$,\nconnecting the 1st to the 2nd slice (while the link $U^\\dagger$ \nconnects slice 2 to slice 1). Thus,\n\\beq\n[Q_L, Q_R ] = 0 .\n\\eeq\nThe hermitian field\nstrength tensor is,\n\\beq\nG_{\\mu\\nu} = i[D_\\mu, D_\\nu] = G_{L\\mu\\nu}^a Q^a_L + G_{R\\mu\\nu}^a Q^a_R ,\n\\eeq\nand also resolves into $L$ and $R$ operator components. \n\n\n\\vskip 0.15in\n\\noindent{\\bf (ii) Matrix Formalism}\n\\vskip 0.15in\n\nWe choose to define a ``left-handed derivative,'' \n$D_{L\\mu} = \\partial_\\mu -i A_{L\\mu}^a Q_L^a$ , \nso that $G_{L\\mu\\nu}^a Q_L^a = i[D_{L\\mu}, D_{L,\\nu}]$;  \nand, respectively, a ``right-handed derivative,'' \n$D_{R\\mu} = \\partial_\\mu -i A_{R\\mu}^a Q_R^a$,\nso that $G_{R\\mu\\nu}^a Q_R^a = i[D_{R\\mu}, D_{R,\\nu}]$\nfor the right-handed fields. $D_L$ applies to fields\non the left-hand lattice slice, while $D_R$\napplies on the right-hand slice. We further \nrequire $[D_{L\\mu}, D_{R\\nu }] =0$, which\ndoes not hold, naively;  however, we can still implement\nthis construction as a $2\\times 2$ matrix representation.\n\nOperators are defined as left-handed and right-handed \nin the chirality matrix format,\n\\beq\n{\\cal{O}} \n= \\left( \\begin{array}{cc} \nO^L & 0 \\\\\n0 & O^R \\\\\n\\end{array} \\right).\n\\eeq\nHence, the matrix covariant derivative can be defined\nas\n\\beq\n{\\cal{D}}_\\mu \n= \\left( \\begin{array}{cc} \nD_\\mu^L & 0 \\\\\n0 & D_\\mu^R \\\\\n\\end{array} \\right).\n\\eeq\nThe commutator, then, yields the field strengths residing\non their respective lattice slices:\n\\beq\n{\\cal{G}}_{\\mu\\nu} = i[{\\cal{D}}_\\mu, {\\cal{D}}_\\nu]  \n= \\left( \\begin{array}{cc} \nG^L_{\\mu\\nu} & 0 \\\\\n0 & G^R_{\\mu\\nu} \\\\\n\\end{array} \\right).\n\\eeq\nThe gauge transformations in this space are thus,\n\\beq\n{\\cal O} \\rightarrow {\\cal V O V}^\\dagger ,\n\\qquad\n{\\cal V} = \\left( \\begin{array}{cc} \nV_L & 0 \\\\\n0 & V_R \\\\\n\\end{array} \\right).\n\\eeq\nLattice link fields are off-diagonal matrices,\n\\beq\n{\\cal{U}} = \\left( \\begin{array}{cc} \n0 & U \\\\\n0 & 0 \\\\\n\\end{array} \\right), \n\\qquad\n{\\cal{U}}^\\dagger = \\left( \\begin{array}{cc} \n0 & 0 \\\\\nU^\\dagger & 0 \\\\\n\\end{array} \\right).\n\\eeq\nNote, then, \n\\beq\n{\\cal{U}}^\\dagger {\\cal{U}} = \\left( \\begin{array}{cc} \n0 & 0 \\\\\n0 & 1 \\\\\n\\end{array} \\right),\n\\qquad\n{\\cal{U}} {\\cal{U}}^\\dagger = \\left( \\begin{array}{cc} \n1 & 0 \\\\\n0 & 0 \\\\\n\\end{array} \\right),\n\\eeq\nso that \n\\beq\n{\\cal{U}} {\\cal{U}}^\\dagger + {\\cal{U}}^\\dagger {\\cal{U}} = \\openone ~,\n\\qquad\n{\\cal{U}} {\\cal{U}}^\\dagger - {\\cal{U}}^\\dagger {\\cal{U}} = \\sigma_z .\n\\eeq\n\nThe lattice Wilson links transform as bifundamentals,\n\\beq\n{\\cal{U}} \\rightarrow {\\cal  V U V}^\\dagger = \\left( \\begin{array}{cc} \n0 & V_L U V_R^\\dagger \\\\\n0 & 0 \\\\\n\\end{array} \\right),\n\\qquad\n{\\cal{U}}^\\dagger \\rightarrow {\\cal V {U}}^\\dagger {\\cal V}^\\dagger \n= \\left( \\begin{array}{cc} \n0 & 0 \\\\\nV_R U^\\dagger V_L^\\dagger & 0 \\\\\n\\end{array} \\right).\n\\eeq\nThe commutators of operators with link fields are:\n\\bea\n\\label{comm}\n[{\\cal{O}}, {\\cal{U}}] & = & \\left( \\begin{array}{cc} \n0 & O_LU-UO_R \\\\\n0 & 0 \\\\\n\\end{array} \\right),\n\\qquad\n[{\\cal{O}}, {\\cal{U}}^\\dagger]  =  \\left( \\begin{array}{cc} \n0 & 0 \\\\\nU^\\dagger O_L - O_R U^\\dagger & 0 \\\\\n\\end{array} \\right).\n\\eea\nThe abstract charge is defined as\n\\beq\n{\\cal Q}^a = \\left( \\begin{array}{cc} \nQ^a_L & 0 \\\\\n0 & Q^a_R \\\\\n\\end{array} \\right).\n\\eeq\n\nThus, define the $Q^a$ as having commutators on the $U$'s:\n\\beq\nT^a \\equiv  \\frac{\\lambda^a}{2} ,\n\\qquad\n[Q_L, U] = T^aU ,\\qquad \n[Q_R, U] =  -UT^a .\n\\eeq\nWe often enconter these charges sandwiched between $U$ and\n$U^\\dagger$ matrices. We thus see that, e.g.,\n\\bea\nU^\\dagger Q_L^a U & = & U^\\dagger T^a U  + Q_L^a .\n\\eea\n\nThe structure of eq. (\\ref{comm}) allows \ncovariant differentiation to be written as a commutation\nrelation, and takes the following form on $U$, \n\\beq\n[D_\\mu , U] = \\partial_\\mu U -iA_{L\\mu}^a\\frac{\\lambda^a}{2}U\n+ iA_{R\\mu}^aU\\frac{\\lambda^a}{2} .\n\\eeq\nThis corresponds to the chirality matrix commutator, \n\\beq\n[{\\cal{D}}_\\mu, {\\cal {U}}]\n= \\left( \\begin{array}{cc} \n0 & D_{L\\mu}U-UD_{R\\mu} \\\\\n0 & 0 \\\\\n\\end{array} \\right).\n\\eeq\n\nFrom the link field $U$, we may thus form left-handed (right-invariant), \nand right-handed \n(left-invariant) chiral currents, respectively (non-matrix),\n\\beq\n\\label{alpha}\n\\alpha_{\\mu} \\equiv U [D_\\mu , U^\\dagger],  \n\\qquad \\qquad \n\\beta_{\\mu} \\equiv  U^\\dagger [D_\\mu , U] .\n\\eeq\nMore explicitly, \n\\bea\n\\alpha_\\mu & = & \n U(\\partial_\\mu - iA^a_{R\\mu}\\frac{\\lambda^a}{2} )U^\\dagger \n + iA^a_{L\\mu}\\frac{\\lambda^a}{2} \n = U(D_{R\\mu}U^\\dagger -U^\\dagger D_{L\\mu}),\n\\cr\n\\beta_\\mu & = &  U^\\dagger(\\partial_\\mu - iA^a_{L\\mu}\\frac{\\lambda^a}{2} )U \n + iA^a_{R\\mu}\\frac{\\lambda^a}{2} \n = U^\\dagger(D_{L\\mu}U -U D_{R\\mu}),\n\\eea\nwhere the action of the derivatives follows Leibniz's rule, \n$~D_L U = [D_L,U] + UD_L$; likewise, \n$D_R U^\\dagger = [D_R,U^\\dagger] + U^\\dagger D_R$.\n\nIn the chiral matrix representation, these amount to\n\\beq\n{\\hat{\\alpha}}_{\\mu} = {\\cal{U}}[{\\cal{D}}_\\mu, {\\cal{U}}^\\dagger]  \n= \\left( \\begin{array}{cc} \n\\alpha_{\\mu} & 0 \\\\\n0 & 0 \\\\\n\\end{array} \\right),\n\\qquad\n{\\hat{\\beta}}_{\\mu} = {\\cal{U}}^\\dagger[{\\cal{D}}_\\mu, {\\cal{U}}]  \n= \\left( \\begin{array}{cc} \n0 & 0 \\\\\n0 & \\beta_{\\mu} \\\\\n\\end{array} \\right).\n\\eeq\n\n\nFinally, it is useful to define the hermitian link chiral matrix:\n\\beq\n{\\cal{U}}_+ \\equiv  {\\cal{U}} + {\\cal{U}}^\\dagger \n= \\left( \\begin{array}{cc} \n0 & U \\\\\nU^\\dagger & 0 \\\\\n\\end{array} \\right).\n\\eeq\nThus,\n\\beq\n{\\cal{U}}_+ {\\cal{U}}_+ = \\openone ,\n \\eeq\nand one sees that \n\\beq\n\\label{central}\n{\\cal{A}}_\\mu \\equiv {\\hat{\\alpha}}_{\\mu}+ {\\hat{\\beta}}_{\\mu}      = \n{\\cal {U}}[{\\cal {D}}_\\mu, {\\cal {U}}^\\dagger]  + {\\cal {U}}^\\dagger\n[{\\cal{D}}_\\mu, {\\cal {U}}]\n=  {\\cal{U}}_+[{\\cal D}_\\mu, {\\cal{U}}_+ ]\n=\n\\left( \\begin{array}{cc} \n\\alpha_{\\mu} & 0 \\\\\n0 & \\beta_\\mu \\\\\n\\end{array} \\right).\n\\qquad\n\\eeq\nA useful set of relationships that recur throughout,\nespecially in computing current divergences,\nis \n\\bea\n\\label{alpha1}\n[ D_\\mu, \\alpha_\\nu ] - [D_\\nu, \\alpha_\\mu]\n& = &\n-[\\alpha_\\mu, \\alpha_\\nu] -iU[G_{\\mu\\nu}, U^\\dagger], \\cr\n[D_\\mu, \\beta_\\nu] - [D_\\nu, \\beta_\\mu]\n& = &\n-[\\beta_\\mu, \\beta_\\nu] -iU^\\dagger[G_{\\mu\\nu}, U] ,\n\\eea\nwith the correspondence in the chirality matrix representation:\n\\bea\n\\label{curvature}\n[{\\cal{D}}_\\mu, {\\cal{A}}_\\nu]\n-[{\\cal{D}}_\\nu, {\\cal{A}}_\\mu]\n& = & - [{\\cal{A}}_\\mu, {\\cal{A}}_\\mu] -\ni {\\cal{U}}_+ [{\\cal{G}}_{\\mu\\nu}, {\\cal{U}}_+ ]\n\\cr &&\\cr\n& = & \\left( \\begin{array}{cc} \niG^L_{\\mu\\nu} -iU G^R_{\\mu\\nu}U^\\dagger -[\\alpha_{\\mu},\\alpha_\\nu] & 0 \\\\\n0 & \niG^R_{\\mu\\nu} -iU^\\dagger  G^L_{\\mu\\nu}U-[\\beta_{\\mu},\\beta_\\nu] \\\\\n\\end{array} \\right).\n\\eea\n\n\nHow should ${\\cal D}_4$ be defined in D=4? \n${\\cal D}_4$ is some kind of a lattice \nderivative, or ``brane-hop'', in the $x^4$ direction. In general, hopping \non an $N$-slice lattice works through the Wilson link, $U_i$, fields,\nwhich are identified with the continuum $A^4$ through:\n\\beq\nU_i = P \\exp \\left( -i\\int_{x^4_i}^{x^4_{i+1}} A_4 dx^4 \\right ) .\n\\eeq \nConsider a field $\\psi_i(x)$ on the $i$th slice, where $\\psi_i(x)\n\\rightarrow V_i(x) \\psi_i (x)$ under local gauge transformations\n$V_i(x)$ of the local gauge symmetry group on the $i$th slice.\nTo define a covariant derivative in the $x^4$ direction, \none seeks a difference like $(\\psi_{i+1}(x) - \\psi_i(x))\/a$, for lattice \nspacing $a$. But since this has a mixed gauge symmetry, \none is led to define the covariant difference \n$(U_{i}\\psi_{i+1}(x) - \\psi_i(x))\/a$.\nThe link now pulls the first term back from the slice\n$i+1$ to $i$, where the $i$-covariant difference can be computed,\ninvariant under $i+1$ transformations. A vanishing covariant difference \nthus amounts to link-gauge transformation. For adjoint quantities, both sides \nof the corresponding operator need such adjustment.\n\nThe hopping derivative in deconstruction must handle left and right in \na manner consistent with parity. One possibility would be to define an \n{\\em off diagonal} (antihermitian) hopping derivative as a commutator,\nand thus traceless,\n\\beq\n[{\\cal  D}^4, {\\cal  O}] \\equiv -\\frac{1}{a} [{\\cal{U}}_+,{\\cal{O}}]=\n-\\frac{1}{a} \\left( \\begin{array}{cc} \n0 & -UO_R + O_L U \\\\\nO_RU^\\dagger - U^\\dagger O_L & \n0\n\\end{array} \\right).\n\\eeq\nWith this definition, the hopping derivative obeys Leibniz's chain rule \nof differentiation, as a commutator, and so the Bianchi identities in $D=4$ \nare automatically satisfied, so there is no need for modification of the \ntheory. Thus, an orbifold\ncompactification solves the Bianchi identity with the usual spectrum.\nIn addition, by Leibniz's rule, ${\\cal O \\psi}$ hop-transforms \nexactly like ${\\cal \\psi}$.  This may be at the root of a deficiency,\nhowever. Being off diagonal, this ${\\cal D}_4$ maps operators from one\nrepresentation into another. For example,\nit maps an adjoint representation under $SU(N)_L$, \\ie, $(N^2-1, 0)$,\ninto  a bifundamental under $SU(N)_L\\times SU(N)_R$, \\ie $(N,\\bar{N})$.\nA covariant derivative then which does not faithfully map a given\nrepresentation into itself is unsatisfactory. \n\nMoreover, if it \nis applied to fermions, one immediately encounters the fermion doubling \nproblem. The remedy\nto this is the addition of a Wilson term, which is a \ncontinuum second derivative.\nIf we generalize the Wilson term to the case of\nhigher representations, such as adjoint\noperators, we are led to the diagonal definition, below \\cite{leib}.\nThe Wilson term projects out the unwanted fermionic doublers, and \npermits the appearance of anomalies consistently with topology. \nMultiplication of the above ${\\cal D}_4$ by $-\\sigma_z {\\cal{U}}_+ $  \non the left, however, leads to a different, {\\em diagonal hopping derivative} \ndefined below.\n\nA preferred definition, and the one we will be using presently is a \n{\\em diagonal hopping derivative},\n\\beq\n{\\cal D}^4 ({\\cal O}) \\equiv \n\\frac{1}{a}\\left({\\cal{U}}[{\\cal{O}},{\\cal{U}}^\\dagger]\n-{\\cal{U}}^\\dagger[{\\cal{O}},{\\cal{U}}]\\right)\n=\n\\left( \\begin{array}{cc} \nUO_RU^\\dagger - O_L  & 0 \\\\\n0 & O_R - U^\\dagger O_L U\n\\end{array} \\right),\n\\eeq\nwhere $ a$ is the spacing between neighboring slices.\n\nNote the second term pushes the previous slice fields forward,\nas the first term pulls the subsequent slice fields back, hence\na relative sign difference, which is commensurate \nwith parity:\n Under parity, $L\\leftrightarrow R$,\n$U\\leftrightarrow U^\\dagger$,\nand ${\\cal D}^4 \\rightarrow -{\\cal D}^4$, and hence the definitions are\nparity invariant.  It is important to note, however, that, like lattice \nderivatives, this derivative does not obey the Leibniz rule of differentiation,\nand so {\\em cannot} be written as a commutator \n$[{\\cal D}^4 , {\\cal A} {\\cal B}]= \n[{\\cal D}^4 , {\\cal A}] {\\cal B}+ {\\cal A} [{\\cal D}^4 ,  {\\cal B}]$.\n\nWe thus define the coset field strength as a transform, not a commutator,\nthrough the diagonal hopping derivative:\n\\bea\n\\label{trivium}\n{\\cal G}_{4\\mu} & =& -{\\cal G}_{\\mu 4} \\equiv i{\\cal D}_4 ( {\\cal D}_\\mu)  \n\\nonumber \\\\               & = & \n \\frac{i}{a}\\left({\\cal{U}} [{\\cal D}_\\mu, {\\cal {U}}^\\dagger]\n- {\\cal{U}}^\\dagger[{\\cal D}_\\mu, {\\cal{U}}] \\right)\n=\\frac{i}{a}(\\hat\\alpha_\\mu - \\hat\\beta_\\mu) \n= \\frac{i}{a}\\left( \\begin{array}{cc} \n\\alpha_\\mu & 0 \\\\\n0 & -\\beta_\\mu \\\\\n\\end{array} \\right).\n\\eea\n\nThe conventional deconstructed\nlagrangian in the chirality matrix formalism can then be written,\n\\bea\n\\label{lag2}\n{\\cal{L}} & = & -\\frac{1}{2{g}^2}\\left(  \n\\Tr {\\cal{G}}_{\\mu\\nu}{\\cal{G}}^{\\mu\\nu}\n-\\Tr {\\cal{G}}_{4\\nu}{\\cal{G}}^{4\\nu}\n\\right)\n\\cr\n& = & -\\frac{1}{2{g}^2}  \\Tr G_{L\\mu\\nu}G^{L\\mu\\nu}\n-\\frac{1}{2{g}^2}  \\Tr G_{R\\mu\\nu}G^{R\\mu\\nu}\n- \\frac{1}{8} f_\\pi^2~ \\Bigl (\\Tr (\\alpha_\\mu)^2  + \\Tr (\\beta_\\mu)^2 \\Bigr ), \n\\eea\nwhere we identify $1\/g^2 = a\/\\tilde{g}^2$, and \n$f^2_\\pi = 4\/ a\\tilde{g}^2 = 1\/a^2{g}^2$.\n\nIt could be interpreted as a gauged chiral lagrangian\nwith external vector fields, $A^\\mu _L$ and $A^\\mu _R$. We may wish to\nassign the octet of vector mesons, including the\n$\\rho$ to a vector combination of the fields, and the axial\nvector mesons to the axial vector combination. To do\nthis in detail would require additional Higgs fields to give\nmasses to the vector ($\\rho$ ) and axial vector ($A_1$)\ncombinations. Once these combinations have acquired with longitudinal degrees\nof freedom, then one cannot eliminate the mesons by gauge transformations. \n\n\nAs an effective fundamental theory, this represents a massless zero mode \ntogether with a massive KK mode. To see this, pass to unitary gauge to remove \nthe spinless mesons altogether, i.e., note that \n$\\Tr (\\alpha_\\mu)^2 = \\Tr (\\beta_\\mu)^2$, and introduce a ``St\\\"{u}ckelberg'' \nfield, \n\\beq\nV_\\mu \\equiv -i\\alpha_\\mu\/g .\n\\eeq\nThe corresponding field strength, by eq. (\\ref{alpha1}) is:\n\\bea\nF^V_{\\mu\\nu} & = & [D_\\mu, V_\\nu] - [D_\\nu, V_\\mu] - i[V_\\mu, V_\\nu]\n\\cr && \\cr\n& = & -\\frac{1}{g}U[G_{\\mu\\nu}, U^\\dagger] = \n\\frac{1}{g}G^a_{L\\mu\\nu}\\frac{\\lambda^a}{2}\n-\\frac{1}{g}UG^a_{R\\mu\\nu}\\frac{\\lambda^a}{2}U^\\dagger   .  \n\\eea\nFurther, the orthogonal zero-mode field strength is likewise right-invariant, \n\\bea\nF^0_{\\mu\\nu} & = &  \\frac{1}{g}\n\\left(UG^a_{R\\mu\\nu}\\frac{\\lambda^a}{2}U^\\dagger +\nG^a_{L\\mu\\nu}\\frac{\\lambda^a}{2}\\right). \n\\eea  \nThus, the effective lagrangian takes the form,\n\\beq\n\\label{lag3}\n{\\cal{L}} = -\\frac{1}{2}  \\Tr F^0_{\\mu\\nu}F^{0\\mu\\nu}\n-\\frac{1}{2}  \\Tr F^V_{\\mu\\nu}F^{V\\mu\\nu}\n- \\frac{1}{4}g^2f_\\pi^2~ \\Tr V_{\\mu}V_{\\mu}, \n\\eeq\ndescribing a massless zero mode and massive KK mode\nof mass $gf_\\pi\/\\sqrt{2}$.  (The spinless mesons have been\nabsorbed into the longitudinal components of $V_\\mu$.) \n\nNote that one can always perform\na left gauge transformation on these fields, \n$D_R \\rightarrow U D_R U^\\dagger = D_R^\\prime$\nleading to $G^a_{R\\mu\\nu}{}^\\prime =\nUG^a_{R\\mu\\nu}\\frac{\\lambda^a}{2}U^\\dagger$,\nhence  $gF^V_{\\mu\\nu}  = G^a_{L\\mu\\nu}\\frac{\\lambda^a}{2}\n-G^a_{R\\mu\\nu}{}^\\prime\\frac{\\lambda^a}{2} $; thus \n$gF^0_{\\mu\\nu}  =  G^a_{R\\mu\\nu}{}^\\prime\\frac{\\lambda^a}{2} +\nG^a_{L\\mu\\nu}\\frac{\\lambda^a}{2}$. With these field\nredefinitions, evidently only one linearly\nrealized symmetry transforms\nall fields, the vectorial symmetry,\n$O \\rightarrow VOV^\\dagger $, where $V= V_L$.\n\n\\vskip 0.15in\n\\noindent{\\bf (iii) Bianchi Identities}\n\\vskip 0.15in\n\nThe {\\em Bianchi identities} in $D=5$ are just the Jacobi identities for \ncovariant derivatives,  \n\\beq\n\\epsilon_{ABCDE}[D^C, G^{DE}] = i \\epsilon_{ABCDE}[D^C, [D^D , D^E]] =0.\n\\eeq\nConsistency in $D=4$ requires:\n\\beq\n\\label{one}\n\\epsilon_{\\mu\\nu\\rho\\sigma} [{\\cal D}^\\nu, {\\cal G}^{\\rho\\sigma}]=  \ni\\epsilon_{\\mu\\nu\\rho\\sigma} [{\\cal D}^\\nu, [{\\cal D}^{\\rho}, \n{\\cal D}^{\\sigma}]] \n= 0, \n\\eeq\nas well as,\n\\beq \n\\label{two}\n\\epsilon_{\\mu\\nu\\rho\\sigma}{\\cal D}^4 ( {\\cal G}^{\\mu\\nu} )  =  \n\\epsilon_{\\mu\\nu\\rho\\sigma}\\left([{\\cal D}^{\\mu}, \n{\\cal G}^{4\\nu}]-[{\\cal D}^{\\nu}, {\\cal G}^{4\\mu}]\\right). \n\\eeq\nEq. (\\ref{one}) holds automatically in the $D=4$ theory,\nas ${\\cal G}_{\\mu\\nu}$ is defined as a commutator of \ncovariant derivatives, for {\\em any choice of ${\\cal D}_\\mu$}. \n\nThe off-diagonal hopping derivative\nsatisfies the coset identity eq. (\\ref{two}), \nwhile the diagonal hopping derivative, not being a commutator,  does not,\nin general: the Bianchi relation implies\na nontrivial constraint. Consider the \ndiagonal hopping on the {\\em lhs} of eq.(\\ref{two}),\n\\beq\n\\label{bianchilhs}\n{\\cal{D}}^4 ({\\cal {G}}^{\\mu\\nu} )=  \\frac{1}{a}\\left(\n{\\cal{U}}[ {\\cal{G}}_{\\mu\\nu}, {\\cal{U}}^\\dagger ]- \n{\\cal{U}}^\\dagger [{\\cal{G}}_{\\mu\\nu}, {\\cal{U}}]\\right),\n\\eeq\nand compare to the {\\em rhs} of eq.(\\ref{two}),\n\\bea\n\\label{bianchirhs10}\ni[{\\cal D}_\\mu , {\\cal D}_4 ( {\\cal{D}}_\\nu) ] \n - i[{\\cal{D}}_\\nu ,  {\\cal{D}}_4 ( {\\cal{D}}_\\mu) ] & = & \n\\frac{i}{a}\\left( [{\\cal{D}}_\\mu, {\\cal{U}} \n[{\\cal{D}}_\\nu, {\\cal{U}}^\\dagger]]\n-[{\\cal{D}}_\\mu, {\\cal{U}}^\\dagger [{\\cal{D}}_\\nu, {\\cal{U}}]] - \n(\\mu\\leftrightarrow \\nu) \\right)\n\\nonumber \\\\\n& = & \\; \\frac{i}{a}\\left( {\\cal{U}}[[{\\cal{D}}_{\\mu}, {\\cal{D}}_{\\nu}], \n{\\cal{U}}^\\dagger ]- {\\cal{U}}^\\dagger[[{\\cal{D}}_{\\mu}, \n{\\cal{D}}_{\\nu}], {\\cal{U}} ]\\right.   \\nonumber \\\\\n& & \\left.\n+[{\\cal{D}}_{[\\mu}, {\\cal{U}}][{\\cal{D}}_{\\nu]}, {\\cal{U}}^\\dagger] \n- [{\\cal{D}}_{[\\mu}, {\\cal{U}}^\\dagger][{\\cal{D}}_{\\nu]}, {\\cal{U}}] \\right)\n\\nonumber \\\\\n& = &\n{\\cal{D}}_4 ({\\cal{G}}^{\\mu\\nu}) \n+\\frac{i}{a} \\left(-  [\\hat{\\alpha}_\\mu, \\hat{\\alpha}_\\nu] \n+ [\\hat{\\beta}_\\mu, \\hat {\\beta}_\\nu]\\right).\n\\eea\n\nThe first term of the {\\em rhs} is consistent,\nbut the last term is an unwanted nonvanishing current commutator.\nThis term is nonzero, and is the current algebra \nof the chiral theory. Thus, the Bianchi identity fails given\nthe presence of this term.\n\nNonetheless, the constraint, eq.(\\ref{two}), can be satisfied if we\nconsider a {\\em modified covariant derivative}.\nWe find that the desired modification takes the form,\n\\beq\n\\label{newderiv}\n\\fbox{ ${\\cal{D}}'_\\mu \\equiv {\\cal{D}}_\\mu + \\frac{1}{2} {\\cal{A}}_\\mu$} ~.\n\\eeq\t\t\nThe Bianchi identities of eq.(\\ref{one}) thus remain automatic in the \n$D=4$ subspace, since the gauge field strengths \nare defined, as usual, by commutators of ${\\cal D}'_\\mu$.\nThe Bianchi constraint, eq. (\\ref{two}), \nnow\nrequires the vanishing of the following expression,\nwith the modified derivative: \n\\beq\n\\epsilon^{\\mu\\nu\\rho\\sigma} \n\\left(  [{\\cal{D}}'_\\mu, U][{\\cal{D}}'_\\nu, U^\\dagger  ]  \n- [{\\cal{D}}'_\\mu, U^\\dagger] [{\\cal{D}}'_\\nu , U] \\right)\n  =   0.\n\\eeq\nTo see the vanishing of this constraint, we first note:\n\\beq\n{\\cal{U}}_+ [{\\cal{A}}_{\\mu},{\\cal{U}}_+] = -2{\\cal{A}}_{\\mu},\n\\eeq\nhence, \n\\beq\n\\{ {\\cal{U}}_+ ,  {\\cal{A}}_{\\mu} \\}=0 ,\n\\eeq\nand so, by eqn. (\\ref{central}),\n\\beq\n[ {\\cal{D}}'_{\\mu}, {\\cal{U}}_+] =  0.\n\\eeq\nIt is evident that this, in fact, resolves into the two components,\n\\beq\n[ {\\cal{D}}'_{\\mu}, {\\cal{U}}] =  0,  \\qquad \\qquad \n[ {\\cal{D}}'_{\\mu}, {\\cal{U}}^\\dagger ] =  0,\n\\eeq\nand the Bianchi constraint is therefore satisfied.\nWe remark that one can derive the same result without recourse to the\nmatrix representation by careful analysis, where, allowing\nan arbitrary factor $w$ in the current part\nof  eq.(\\ref{newderiv}), one obtains the unwanted current commutators\nof eq.(\\ref{bianchirhs10}), multiplied by a factor of $(1-4w+4w^2)$.\nThe Bianchi constraint is thus satisfied with the new \ncovariant derivative\nof eq.(\\ref{newderiv}) with the special coefficient of $w=1\/2$.\nThe matrix formulation both streamlines and automates this derivation. \n\n \nObserve that the field strength ${\\cal G}'_{4\\mu}$ of (\\ref{trivium}) \nmanifestly vanishes for this hopping-flat modified derivative,\n\\beq\n{\\cal G}'_{4\\mu} = i{\\cal D}_4 ( {\\cal D}'_\\mu)=0.  \n\\eeq\n(Actually, by ${\\cal{D}}_4 ({\\cal{G}}'_{\\mu\\nu})=0$, each of the three terms\nin the respective coset identity eq. (\\ref{two}) vanishes separately \nfor modified covariant derivatives.) \n\nThe rest of the field strength tensor, by (\\ref{curvature}),\nreduces to \n\\beq\n{\\cal{G}}'_{\\mu\\nu} \n= i[{\\cal{D}}'_\\mu, {\\cal{D}}'_\\nu]  \n=\\half ({\\cal{G}}_{\\mu\\nu}+  {\\cal{U}}_+ {\\cal{G}}_{\\mu\\nu}{\\cal{U}}_+ \n-\\frac{i}{2}[{\\cal{A}}_\\mu, {\\cal{A}}_\\mu] ),\n\\eeq\nso that \n\\beq\n{\\cal{G}}'_{\\mu\\nu} =\n\\half \\left( \\begin{array}{cc} \nG^L_{\\mu\\nu}+ U G^R_{\\mu\\nu} U^\\dagger \n- \\frac{i}{2}[\\alpha_\\mu,\\alpha_\\nu] & 0 \\\\\n0 & G^R_{\\mu\\nu} + U^\\dagger G^L_{\\mu\\nu} U\n-\\frac{i}{2}[\\beta_\\mu,\\beta_\\nu]\\\\\n\\end{array} \\right).\n\\eeq\nEvidently, the right-slice amounts to the gauge-transformed image \ntheory of the left-slice,  \n\\beq\n{\\cal{G}}'_{\\mu\\nu} \n= \\half \\left( \\begin{array}{cc} \nG^L_{\\mu\\nu}+ U G^R_{\\mu\\nu} U^\\dagger \n- \\frac{i}{2}[\\alpha_\\mu,\\alpha_\\nu] & 0 \\\\\n0 & U^\\dagger (U G^R_{\\mu\\nu} U^\\dagger \n + G^L_{\\mu\\nu} \n-\\frac{i}{2}[\\alpha_\\mu,\\alpha_\\nu])U\\\\\n\\end{array} \\right),\n\\eeq\nFor gauge-invariant combinations, this ``hop-invariant'' setup effectively \ndoubles up the theory. The effective field strength appearing here \nis simply the (hop-symmetric) \nzero-mode combination encountered previously,\n\\beq\nF_{\\mu\\nu}^0 = G^L_{\\mu\\nu}+  U G^R_{\\mu\\nu} U^\\dagger ,\n\\eeq\nwhereas the orthogonal KK-mode combination is absent. \n\nIn effect, the diagonal hopping derivative\nBianchi-compatible theory of the two-slice orbifold\ncontains only one propagating gauge field, together\nwith the spinless mesons. This does not mean that there is no\nKK mode, but that the simplest hop-symmetric deconstruction truncates\nthe spectrum on the propagating zero mode. To obtain the second KK mode \nwould require that we start with $N=3$ branes, and\nwe would expect that, for any $N$,  the Bianchi-improved theory would \ndescribe the zero mode and $N-2$ KK modes. \n\nNote that the chiral feature of an orbifold is still present, \\ie, we may treat\n$F_{\\mu\\nu}^0$ as any combination of left-hand or right-hand\ngauging. For example, we may gauge only the left-hand side\nof the meson fields, whence setting $ G^{R}_{\\mu\\nu} =0$, so that \n$F^0_{\\mu\\nu}= G^L_{\\mu\\nu}$; or, else, we may choose to gauge\nisospin, $G^L_{\\mu\\nu} = G^{R}_{\\mu\\nu} $, so\n$F^0_{\\mu\\nu}= 2G^L_{\\mu\\nu}$ (which rescales the coupling constant). \n\nIn the simplifying case that we set the right-hand \nYang-Mills fields to zero (\\ie, we retain only a single\n$SU(N)_L$ gauge group), we end up with a pure left-handed chiral theory:\n\\beq\n{\\cal{G}}'_{\\mu\\nu} = \\half \\left( \\begin{array}{cc} \nG^L_{\\mu\\nu} \n- \\frac{i}{2}[\\alpha_\\mu,\\alpha_\\nu] & 0 \\\\\n0 &  U^\\dagger \\left( G^L_{\\mu\\nu} \n-\\frac{i}{2}[\\alpha_\\mu,\\alpha_\\nu]\\right)U\\\\\n\\end{array} \\right).\n\\eeq\nThe resulting gauge action is then, \n\\bea\n\\label{skyrmeterm}\n-\\frac{1}{2\\tilde{g}^2}\\Tr {\\cal{G}}'_{\\mu\\nu}{\\cal{G}}'^{\\mu\\nu}\n& = & -\\frac{1}{4\\tilde{g}^2} \n\\left(\n\\Tr G^L_{\\mu\\nu} G^{L\\mu\\nu} \n-i\\Tr (G^{L\\mu\\nu} \n[\\alpha_\\mu, \\alpha_\\nu])  \n- \\frac{1}{4}\\Tr[\\alpha_\\mu, \\alpha_\\nu][\\alpha^{\\mu}, \\alpha^{\\nu}]\n \\right) . \n\\eea\n\nThe resulting theory has several interesting properties \nevident at this point. The last term, $ \\Tr([\\alpha,\\alpha]^2)$, \nis the Skyrme term required for\nthe stability of the core of the Skyrmion solution. \nIt is normally a puzzle to understand how these terms are\ngenerated in a deconstructed theory, since they are needed classically,\nbecause the skyrmion core is not an entirely short-distance structure.\nThe Bianchi identities have fixed the coefficient of the \nSkyrme terms to definite values.\nWhile one  could always add other contributions to the\nSkyrme terms by hand, their appearance here reflects self-consistency with\nthe parent $D=5$ theory, which admits stable large instantonic solitons,\nwhich, in turn, carry the current that matches to the Skyrmionic current. \n\nWe note that the new cross-term of the form $G^L [\\alpha,\\alpha] $, \nwhich is  allowed by the presence of the\ngauge field. This term has significant effects\nupon the mass of the skyrmion, and bounds related to those\nof magnetic monopoles arise \\cite{briyahe}.\n\nWe are thus led to speculate that this Bianchi-consistent theory, \nwith these fixed Skyrme terms, points to a more intricate relationship\nbetween the instantonic soliton and the skyrmion. Perhaps we could \nnow find a skyrmion solution that is ``self-dual,''  matching\nthe self-duality of the instantonic soliton in $D=5$, which, in turn, is\na consequence of the self-duality of the instanton. \n\nIn non-matrix notation, the modified derivative reads,\n\\beq\nD'_\\mu = \\partial_\\mu -i (A_{L\\mu}+ \\frac{i}{2} \\alpha_\\mu )\\cdot \nQ_L - i (A_{R\\mu}+ \\frac{i}{2} \\beta_\\mu )\\cdot Q_R ,\n\\eeq\nand hence,\n\\beq\n\\label{nonmatrixprimeder} \nD'_{L\\mu}  = \\half ( D_{L\\mu}+ U   D_{R\\mu} U^\\dagger)\n=  \\partial_\\mu - i A_{\\mu L} +\\half \\alpha_\\mu,\\quad \nD'_{R\\mu}  = \\half ( D_{R\\mu}+ U^\\dagger D_{L\\mu} U) \n=  \\partial_\\mu - i A_{\\mu R} +\\half \\beta_\\mu .\n\\eeq\nEffectively, the gauge fields are augmented by the meson currents \n$\\alpha_\\mu$ and $\\beta_\\mu$. In the limit of vanishing gauge fields, \nthe effective primed gauge fields are still non-trivial, \n\\beq\n\\fbox{ $A'_{\\mu L} \\rightarrow \\frac{i}{2}  U \\partial_\\mu U^\\dagger, \\qquad   \n\\qquad  \nA'_{\\mu R} \\rightarrow \\frac{i}{2}   U^\\dagger  \\partial_\\mu U  $} ~,\n\\label{trans}\n\\eeq\t\nreminiscent of the London equation inside a superconducting medium.\nSince they are not pure gauges, because of the coefficient of $1\/2$, they \nyield nonvanishing primed field strengths, and hence the Skyrme term \nexhibited above. \n\n\nTo summarize, the deconstruction prescription we have been led to \nis based on the diagonal hopping derivative \n${\\cal D}_4$; the Bianchi-consistent hopping-flat \nmodified covariant derivatives, ${\\cal D}'_\\mu$;\nand the corresponding field strengths, ${\\cal{G}}'_{\\mu \\nu}$.\nHaving rejected the nonvanishing ${\\cal{G}}_{\\mu 4}$, in favor \nof its vanishing primed counterpart, we have forfeited the meson currents' \nkinetic term, in the naive chiral lagrangian above. To recover them, we might, \nfor instance, supplement the lagrangian with a term of the form:\n\\beq\n\\sim \\frac{f_\\pi^2 }{8}\\Tr{\\cal{A_\\mu}}{\\cal{A_\\mu}},\n\\eeq\nor somehow match ${\\cal {A_\\mu}}\\mapsto {\\cal G}_{4\\mu} $. \nThis is equivalent to\ndefining ${\\cal G}_{4\\mu} $ as an off-diagonal operator using\nthe off-diagonal hopping derivative. Another possibility,\nmore consistent with Wilson fermions, is a hybrid\nhopping derivative that is a combination of the off-diagonal Leibnitz form\nand the diagonal form discussed above (see \\cite{leib}; this\nhappens automatically with supersymmetric \ndeconstruction in which hopping terms \nare defined as superpotentials).\nWe will never\nneed this operator in the derivation of the usual\nWZW term in the subsequent section, so these ambiguities\nare irrelevant. We will need the fact, however, \nthat the diagonal ${\\cal G}'_{4\\mu}=0$.\n\nUltimately, such prescriptions codify a number of implicit choices \nof brane configurations and phenomenological outcomes. Unlike the off-diagonal \nantihermitian hopping derivative, the hermitian diagonal one \npreserves topological structures \nassociated with chirality (\\eg, anomalies). \n\n\\section{Derivation of the WZW Term in the Bianchi Theory}\n\n\nThe CS2 lagrangian may be written in a form more suitable for \nsubsequent considerations. Specifically, we start by separating the $A_4$ \ncomponent,\n\\bea\n\\label{CStermr}\n{\\cal{L}}_1 & = & {\\cal{L}}_{1a} + {\\cal{L}}_{1b}, \\cr\n{\\cal{L}}_{1a} & = & \\frac{c}{4}\\epsilon^{\\mu\\nu\\rho\\sigma}\n\\Tr(A_4 G_{\\mu\\nu}G_{\\rho\\sigma}  ~, \n + i A_4 A_\\mu A_\\nu G_{\\rho\\sigma}\n + i A_4 A_\\mu G_{\\nu\\rho}A_\\sigma + i A_A G_{\\mu\\nu}A_\\rho A_\\sigma\n \\cr\n& & \\;\\;\\;\\;\\; \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n- {2}A_4 A_\\mu A_\\nu A_\\rho A_\\sigma )~, \n\\cr\n{\\cal{L}}_{1b} & = &\\frac{c}{2}\\epsilon^{\\mu\\nu\\rho\\sigma}\n\\Tr(A_\\mu G_{\\nu\\rho}G_{\\sigma4} \n + A_\\mu G_{\\nu 4 }G_{\\rho\\sigma }\n + i A_\\mu A_\\nu A_{\\rho}G_{\\sigma4} ) .\n\\eea\nThis helps re-express ${\\cal{L}}_{1a}$ as a lower CS covariant current \ndivergence plus an anomaly term,\n\\bea\n\\label{CSterm1a}\n{\\cal{L}}_{1a} & = &  -\\frac{c}{2} \\Tr(A_4 ~[D_\\mu, K^{\\mu}]) +\n\\frac{3c}{4}\\epsilon^{\\mu\\nu\\rho\\sigma}\\Tr(A_4 \nG_{\\mu\\nu}G_{\\rho\\sigma}), \n\\eea\nwhere \n\\bea\nK^{\\mu} \\equiv  \\epsilon^{\\mu\\nu\\rho\\sigma}\n\\left(iA_\\nu A_\\rho A_\\sigma + G_{\\nu\\rho}A_\\sigma\n+ A_\\nu G_{\\rho\\sigma}\\right).\n \\eea\nLikewise, since $G_{\\mu 4} = [D_\\mu , A_4] -\\partial_4 A_\\mu$, \nthe second term can be written as\n\\beq\n\\label{CSterm01b}\n{\\cal{L}}_{1b} = -\\frac{c}{2}\\Tr(([D_\\mu , A_4] -\\partial_4 A_\\mu) K^{\\mu}). \n\\eeq\nThe combined CS2 is then \n\\bea\n\\label{CS11}\n{\\cal{L}}_{1} = \n \\frac{c}{2}\\Tr ((\\partial_4A_\\mu)K^\\mu)\n+\\frac{3c}{4}\\epsilon^{\\mu\\nu\\rho\\sigma}\\Tr(A_4 \nG_{\\mu\\nu}G_{\\rho\\sigma}),\n\\eea\nwhere some total divergences have been discarded.\n \n\nOur problem is the interpretation of\nthe first term above. This problem is obviated when\n$G_{\\mu 4}=0$, whence we use eq.(\\ref{CSterm1a})\nfor the full lagrangian. We then need to\ninterpret $[D_\\mu, A_4]$.\nConsider the definition\nof the Wilson line, which we identify with\nthe chiral field of mesons:\n\\beq\nU = \\exp (-i\\int A_4 dx^4) = \\exp (i\\tilde{\\pi}) ,\n\\eeq\nwhere, for a zero-mode $A_4$ we can neglect path-ordering.\nWe can then write, upon\nexpanding the $U$'s to second order (this is\nthe order necessary for for consistent WZW terms---see below):\n\\beq\n\\alpha_\\mu = U [D_\\mu, U^\\dagger]\n= -i [D_\\mu, \\tilde{\\pi} ] - \\half \n(\\tilde{\\pi} [D_\\mu, \\tilde{\\pi} ] - [D_\\mu, \\tilde{\\pi} ]\\tilde{\\pi} ) \n+ O(\\tilde{\\pi}^3), \n\\eeq\n\\beq\n\\alpha_\\mu = U [D_\\mu, U^\\dagger]\n= -i [D_\\mu, \\int dx^4 A_4] - \\half \n(\\tilde{\\pi} [D_\\mu, \\int dx^4 A_4] - [D_\\mu, \\int dx^4 A_4]\\tilde{\\pi} ) \n+ ...\n\\eeq\nWe invert this to make the identification \n\\beq\n\\label{res1}\n[D_\\mu, \\int dx^4  A_4] = i\\alpha_\\mu \n- \\frac{1}{2}\n(\\tilde{\\pi} \\alpha_\\mu - \\alpha_\\mu \\tilde{\\pi} ) \n+ ...  .\n\\eeq\n\nWe now impose the condition that, from our Bianchi-improved theory,\n$G_{4\\mu} = 0$, equivalently, $\\partial_4A_\\mu = [D_\\mu, A_4]$, and we\nsubstitute eq.(\\ref{res1}) into the expression eq.(\\ref{CS11}).\nThe full lagrangian upon integrating over\n$x^4$ becomes, \n\\bea\n\\label{CSterm1a}\n{\\cal{L}}_{1} & = &  i\\frac{c}{2}\\Tr(\\alpha_\\mu K^{\\mu}) -\n\\frac{c}{4}\\Tr(\\tilde{\\pi}\\alpha_\\mu K^{\\mu}-\\tilde{\\pi}K^{\\mu}\\alpha_\\mu ) \n+\\frac{3c}{4}\\epsilon^{\\mu\\nu\\rho\\sigma}\\Tr(\\tilde{\\pi} \nG_{\\mu\\nu}G_{\\rho\\sigma}) +... ,\n\\eea \n\nWe can now check that we recover the Wess-Zumino term.\nTurn off the gauge fields, but make the deconstructive\nreplacement, with the modified vector potential\nand field strength summarized in (\\ref{trans}), with the primes omitted, \n\\beq\nA_\\mu \\rightarrow i\\frac{\\alpha_\\mu}{2},\n\\qquad \\makebox{hence,}\n\\qquad G_{\\mu\\nu}\\rightarrow -\\frac{i}{2}[\\alpha_\\mu,\\alpha_\\nu],\n\\qquad K_\\mu \\rightarrow\n\\frac{5}{8}\\epsilon_{\\mu\\nu\\rho\\sigma}\\alpha^\\nu\\alpha^\\rho\\alpha^\\sigma .\n\\eeq\n\nOwing to the vector potential which \nis no longer a pure gauge (due to the factor of $1\/2$), \nthe $G_{\\mu\\nu}$ terms\nare now non-negligible and active in our expression for the \nsecond Chern character,\nand this modifies the WZW term's overall coefficient from\nthe heuristic argument result in which $G_{\\mu\\nu}=0$.\nThe CS term thus becomes on the left end-zone (the $(11)$\nmatrix element contribution to the trace):\n\\beq\n{\\cal{L}}_{1L}  =  \n -\\frac{c}{2}\\epsilon_{\\mu\\nu\\rho\\sigma}\\Tr(\\tilde{\\pi} \n \\alpha^\\mu \\alpha^\\nu \\alpha^\\rho \\alpha^\\sigma ) +...  .\n\\eeq\nFrom the right end-zone, we likewise get the result:\n\\beq\n{\\cal{L}}_{1R}  =  \n -\\frac{c}{2}\\epsilon_{\\mu\\nu\\rho\\sigma}\\Tr(U^\\dagger\\tilde{\\pi}  U  \n \\beta^\\mu \\beta^\\nu \\beta^\\rho \\beta^\\sigma )  +... , \n\\eeq\nwhich is equivalent, since\n$\\tilde{\\pi} = U\\tilde{\\pi}U^\\dagger$.\n\nThus, combining, we obtain the Wess-Zumino term\nfor the Bianchi-consistent theory:\n\\beq\n{\\cal{L}}_{1}  =  \n -\\frac{N}{240\\pi^2}\\epsilon_{\\mu\\nu\\rho\\sigma}\\Tr(\\tilde{\\pi}\n \\alpha^\\mu \\alpha^\\nu \\alpha^\\rho \\alpha^\\sigma ) +... ,\n\\eeq\nwhere the ``index'' $N$ is given by the dimensionality\nof the parent theory space-time:\n\\beq\nN = D = 5 . \n\\eeq\nEvidently, the ``'t Hooft matching'' of our Bianchi\nimproved theory intrinsically identifies\n$D=5$, reflected in the value of\nthis index. We have no deeper interpretation for\nthis result at present.\n\nParenthetically, we may suggest care in manipulating the \nWZ term. For example, we could write (using forms,\nand $d\\alpha = -\\alpha^2$ when the vector\npotential is ignored):\n\\beq\n\\tr(\\tilde{\\pi}   \\alpha^4) = \\tr(\\tilde{\\pi}   d\\alpha d\\alpha)\n= \\tr(d\\tilde{\\pi}   \\alpha^3 ).\n\\eeq\nBy naively replacing $d\\tilde{\\pi}   \\rightarrow i\\alpha$, we get \nzero for the {\\em rhs} by cyclicity of the\n$\\epsilon$-symbol, $\\Tr (\\alpha^4) =0$; so we would only access the vanishing, leading \npart of the WZ term: zero! Of course, at the next order in the \nexpansion in pions, we recover the properly modified covariant derivative,\n\\beq\n\\label{picur}\nd\\tilde{\\pi}    \\rightarrow i\\alpha\n-(1\/2)[\\tilde{\\pi}   , \\alpha]   ,\n\\eeq\nand hence consistency for the WZ term, to leading non-trivial order. \n\nThe higher orders for the WZ term have been discussed mathematically in, \n\\eg, \\cite{BCZ}.  \nBeyond leading order, however, the WZ term is not universal in form, as \nan expansion in pions. Indeed, the expansion of unitary chiral fields, such as \n$U=\\exp(i\\tilde{\\pi})$, as a power series\nin $\\tilde{\\pi}$ is non-universal beyond the second\norder. (This owes to the fact that pion fields\nare ``coordinates'', which parameterize the unitary manifold\nsatisfying $U^\\dagger U =1$. We could equally well have\nchosen, \\eg, $U= (1+i\\tilde{\\pi} )\/\\sqrt{1 + \\pi_a\\pi^a\/f_\\pi^2}$. Upon\ncomparing expansions of both $U$s, it is evident that universality \nis lost at $O(\\tilde{\\pi}^3)$.) Physically, there is no general way to lock\nthe coefficients of higher order terms to lower order\nones without additional constraints. Imposing\nthe equations of motion, however, does lock\nthe higher order terms to the universal lower order ones\n(one must use an expansion in pions in the kinetic term\nas well as in the WZ term when the equations of motion\nare implemented). The actual on-mass-shell matrix\nelements are thus universal.\nConsequently, the form of the WZ term is\nuniversal only at the fifth order in $\\Tr(\\pi\\alpha^4)$, since\nat the next order we pick up nonuniversal terms from expansions\nof $\\alpha$. Moreover, there is no way to insure\nthe self-consistency beyond this order off-mass shell.\n\n\\vskip .1in\n\\noindent\n\\section{Conclusions}\n\\vskip .1in\n\nWe have initiated the discussion as to how the Chern-Simons terms\nof a $D=5 $ pure Yang-Mills theory can be deformed into\nthe Wess-Zumino-Witten terms of gauged chiral\nlagrangians of $D=4$. \n\nAdjoint currents in $D=5$ are controlled by the second Chern character. \nThis in\nturn becomes the WZW term in $D=4$. The minimal coefficient of Witten for the\nWess-Zumino term follows from the simplest case of pure gauge\nvector potentials generated by London currents in the orbifold \nmagnetic superconducting end-zones, as shown in our heuristic argument. \n\nSinglet currents follow from introduction of a singlet\n$U(1)$ vector potential in $D=5$ Yang-Mills,\nwhich is a dual variable describing the\ninstantonic soliton that uniquely\noccurs there. We summarize how this morphs into a new\nWZW term in $D=4$, involving the $\\sigma$ and $\\eta'$ fields,\nand which generates the corresponding chiral current equations of\nmotion. A new $U(1)$ axial current, associated with the $\\eta'$, \nhas also been identified.\nThese results are a consequence of the present\napproach, and may have application to skyrmion physics.\n\nWe then embark upon a formal discussion of the latticization\nof the extra (fifth) dimension, and study hopping\nderivatives and the Bianchi identities. The coset Bianchi\nidentity is shown to fail in the case of the diagonal \nhopping derivative in the fifth dimension, \nthe most natural definition for a lattice gauge\ntheory. We find, however, that the coset Bianchi identity can\nbe rescued if the basic $D=4$ covariant derivative is modified by\nthe addition of a chiral vector current with the special\ncoefficient of $1\/2$. \n\nThis result has intriguing implications. For one, it converts the\norbifold compactification into an effective periodic compactification. It also\nprovides a Skyrme term in the effective action that must match the topology of\nthe instantonic soliton to the skyrmion. We conjecture that with\nthe fixed coefficient of the Skyrme term provided by the theory,\nthe matching may be quite powerful, leading perhaps to an analytic skyrmion\nsolution and some form of ``self-duality.'' \n\nWe finally examine the WZW term implied by the Bianchi-consistent theory. Again\nwe obtain the WZW term, but now with a coefficient that has an index\nof $N=D=5$. Many other issues are raised and future lines to explore \nare suggested by the\npresent work.\n\n\n\\vskip .2in\n\\noindent\n{\\bf Acknowledgments}\n\\vskip .1in\nWe thank Nima Arkani-Hamed,\nBill Bardeen, and Lisa Randall for some helpful discussions.\nOne of us (CTH) thanks the University\nof Minnesota's Frontiers II Workshop, and Harvard University\nfor its hospitality during the final\ncompletion of the present work.\nThis work is supported in part by\nthe US Department of Energy, High Energy Physics Division,\nContract W-31-109-ENG-38, and \ngrant DE-AC02-76CHO3000.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe fundamental paradigm of solar system astronomy prior to the time of \nCopernicus was that the Earth was at the center of the solar system.  Also, \ncelestial bodies were assumed to move along perfect circles.  This led to the \nsystem of deferents and epicycles.  One prime motivation for the use of epicycles\nwas to account for retrograde motion. Copernicus' great book {\\em On the \nRevolutions of the Heavenly Spheres} (1543) asserted that the Sun was \nphysically and truly at the center of the solar system, and that this provided \na much simpler explanation for the retrograde motion of the planets. However, \nCopernicus retained circular motion.  Also, he retained the notion of epicycles \nbecause they were needed to account for variations of distance of the planets \nfrom the Sun \\citep{Ging93}.\n\nIn ancient Greek astronomy, the {\\em direction} towards the Moon, Sun, \nor a planet was more important than the implied distance to it.  \nPtolemy's model of the motion of the Moon implied that its distance from \nthe Earth varied nearly a factor of two.  Naked eye observations by this \nauthor have demonstrated that without a telescope one can show that the \nangular size of the Moon varies in a regular fashion, implying that \nMoon's distance varies in a regular fashion \\citep{Kri10}. The implied \neccentricity of the Moon's orbit was $\\approx$0.04.  (The true eccentricity\nof the Moon's orbit is 0.055, but its orbit is anything but a simple ellipse,\nowing to the gravitational force of the Sun.\\footnote[2]{\\citet{Lah12} derived\na value of the Earth's orbital eccentricity of 0.017 $\\pm$ 0.001, which compares\nextremely well with the official modern value of 0.0167.  This was \naccomplished by determining the variation of the\nequation of time (difference of apparent solar time and mean solar time)\nover the course of the year using observations of the length of the\nshadow of a {\\em gnomon}.  It was also necessary to know the obliquity of the\necliptic, which is directly obtained from such observations on the first\nday of summer and the first day of winter.})  The point here is \nthat epicycles in Ptolemaic astronomy were a geometrical device to \nexplain retrograde motion (in the case of the planets) or to determine \nthe direction towards the Moon.  In Copernicus' model, the use of an \nepicycle implies a realistic, physical variation of distance.\n\nIn 1609 Johannes Kepler published the original versions of his first two \nlaws of planetary motion: 1) the orbit of a planet is an ellipse, with \nthe Sun at one focus; and 2) what we now call the law of areas, that the \nradius vector of a planet sweeps out equal areas in equal times.  The \nSecond Law can be stated as follows: \n\n\\begin{equation}\nr^2 \\rm{d}\\theta \\; = \\; h \\; , \n\\end{equation}\n\n\n\\noindent\nwhere $r$ is the \ndistance between a planet and the Sun, d$\\theta$ is an angular increment \nin radians, and $h$ is a constant unique to each planet.\n\nNewton's breakthroughs in mathematics and mechanics led to the \nrealization that Kepler's First Law needed correction. The very center \nof the Sun is not at the focus of a planetary orbit. A planet orbits the \n{\\em center of mass} of the planet-Sun system, and the Sun orbits that \ncenter of mass too \\citep[][chapter 2]{Car_Ost07}. This idea, of course, \nhas led to the discovery of many extra-solar planets via the radial \nvelocity method.\n\nIn the autumn of 2015 Mars was nicely situated in the constellation Leo before sunrise.\nWe began a sequence of observations of Mars\nusing a simple cross staff (Fig. \\ref{f1}).\\footnote[3]{A pattern for \nmaking the cross staff can be obtained from this link: \nhttps:\/\/sites.google.com\/a\/uw.edu\/introductory-astronomy-clearinghouse\/labs-exercises\/measuring-angular-sizes-and-distances. \nThe reader should note that when printed out the scale may look like inches, but \nthe scale is, in fact, somewhat different.}\n\nSay the full width of the cross staff is $d$, and suppose at a linear distance \n$D$ down the ruler the angular separation of two celestial objects exactly matches \nthe width of the cross staff.  Then the angular separation of the two objects will be \n\n\\begin{equation}\n\\theta \\; = \\; 2 ~\\rm{tan}^{-1} \\left(\\frac{d}{2D}\\right) \\; .\n\\end{equation}\n\nIf an observer can measure the angular separation of a planet and two \nstars of known celestial coordinates, there are two solutions for the \nposition of the planet, one on each side of the great circle arc joining \nthe two stars.  If the planet is close to being on the great circle arc \nbetween the two stars, perhaps no solution results, given errors of \nmeasurement.  If the positions of a planet and the two stars form a \nspherical triangle with reasonably equal sides, this\nis ideal, and the planetary position can be \ndetermined as accurately as possible.  Because we are using a handheld \nnaked eye instrument, it is advised to use three to five reference \nstars.  We assume a system of accurate stellar coordinates of bright \nstars along the zodiac.  We adopt the J2000.0 coordinates of such stars \nfrom the SIMBAD database.\n\nUnderstanding Johannes Kepler's efforts to discover the elliptical \nnature of Mars' orbit requires serious effort.  A good place to start is \nan article by \\citet{Ging89}.  To make a long story shorter, Kepler \nbelieved that an ovoid would fit the (then) most accurate data available \n(obtained by Tycho and his assistants). Kepler found anomalies \namounting to 8\\arcmin ~between the measured ecliptic longitudes and his model. \nSince he believed that Tycho's data were good to $\\pm$2\\arcmin ~or \nbetter, he concluded that there was a problem with the {\\em model}.  \nThis led him to conclude that his {\\em approximation} to the ovoid \n(namely, an ellipse) was the true orbital shape.\n\nWe wondered if it were possible to demonstrate from simple naked eye \nobservations that the orbit of Mars is indeed an ellipse.  Or, requiring \nless rigor, are the positions of Mars consistent with an elliptical \norbit? Here we present results based on nearly a year of observations.\nA full blown orbital determination for the planet Mars is beyond the\nscope of the present paper.  That would involve \nsimultaneously solving for all six orbital elements.  We are only trying \nto show that a data set obtained with simple equipment can be fit with \nan ellipse of eccentricity $\\approx$ 0.093.  Other values of the \neccentricity can be shown to give ecliptic longitudes that differ from \nthe observational data by 1.5 deg, far larger systematic \ndifferences than the internal random errors of the observations.\n\n\\section {Data Acquisition}\n\nIn Table \\ref{data} we give various data relating to Mars.  For each \nJulian Date we give the ``true'' right ascension ($\\alpha$) and declination \n($\\delta$) of the planet, obtained using an algorithm by \n\\citet{vanF_Pul79}.  These coordinates are accurate to $\\pm$1\\arcmin.\nNote that these coordinates will correspond to the equinox of date in the years\n2015 or 2016.  To convert these coordinates to ecliptic latitude ($\\beta$) and\nlongitude ($\\lambda$) we need the following formulas from spherical\ntrigonometry \\citep[][p. 40]{Smart77}:\n\n\\begin{equation}\n\\rm{sin}(\\beta) \\; = \\; \\rm{sin}(\\delta) ~\\rm{cos}(\\epsilon) - \n\\rm{cos}(\\delta) ~\\rm{sin}(\\alpha) ~\\rm{sin}(\\epsilon) \\; ;\n\\end{equation}\n\n\\begin{equation}\n\\rm{sin}(\\lambda) \\; = \\; \\frac{\\rm{cos}(\\delta) ~\\rm{sin}(\\alpha) ~\\rm{cos}(\\epsilon) + \n\\rm{sin}(\\delta) ~\\rm{sin}(\\epsilon)}{\\rm{cos}(\\beta)} \\; ;\n\\end{equation}\n\n\\begin{equation}\n\\rm{cos}(\\lambda) \\; = \\; \\frac{\\rm{cos}(\\delta) ~\\rm{cos}(\\alpha)}{\\rm{cos}(\\beta)} \\; .\n\\end{equation}\n\n\\noindent\n$\\epsilon$ is the obliquity of the ecliptic, 23\\arcdeg ~26\\arcmin ~21\\arcsec.406\nfor the year 2000.  Using the {\\sc atan2} function in FORTRAN or Python \nwith arguments sin($\\lambda$) and cos($\\lambda$), we obtain the ecliptic \nlongitude in the correct quadrant.\n\nTable \\ref{data} also gives the observed right ascension, declination, \necliptic longitude, and ecliptic latitude of Mars derived from the cross \nstaff measurements, along with the number of reference stars used and the \nvalue for each date of the Sun's ecliptic longitude. The values of the \nSun's longitude were calculated using the second method of \\citet[][p. \n80]{Meeus88}. This method uses the Sun's mean anomaly, but apparently does \n{\\em not} use the Earth's orbital eccentricity.  On occasion we desired \none more sufficiently bright reference star and used the {\\em derived} \nposition of Saturn as that reference position.\n\nConsider two celestial objects with equatorial coordinates ($\\alpha _1$, $\\delta _1$) and\n($\\alpha _2$, $\\delta _2$). The angular separation ($\\theta$) between two objects is:\n\n\\begin{equation}\n\\rm{cos} (\\theta) \\; = \\; \\rm{sin}(\\delta _1) ~\\rm{sin}(\\delta _2) +\n\\rm{cos}(\\delta _1) ~\\rm{cos}(\\delta _2) ~\\rm{cos}(\\alpha _1 - \\alpha _2) \\; .\n\\end{equation}\n\nNext consider a spherical quadrilateral that is bounded by starting and ending \nright ascensions, and starting and ending declinations.  The quadrilateral is \ndivided into a grid, given a nominal increment in each coordinate of 0.01 deg.  \nWe used a computer program of our devising that uses the coordinates of two \nreference stars and the measured angular distance of a planet from each of these \nstars to determine the coordinates of the planet.  If the coordinates of the \nstars are J2000.0 coordinates, then the derived right ascension and declination \nof the planet also have J2000.0 coordinates.\n\nThe derived ecliptic coordinates of Mars are shown in Figure \\ref{f2}.  The \nsolid line in the plot shows the locus of ``true'' positions from \n\\citet{vanF_Pul79}.  The earliest observations are of lesser quality.  On \nthese occasions we only used two reference stars, and was getting used to a new \nobserving procedure.  On almost all subsequent occasions we used more than two \nreference stars.  Two of these first three observations are easily shown to be \noutliers. Our first three observations will thus be excluded from subsequent \nanalysis.\n\nAnother thing to note about Figure \\ref{f2} is that Mars was north of \nthe ecliptic until JD $\\approx$ 2,457,504 (April 25, 2016) and then was \nlocated south of the ecliptic.  In other words, the orbit of Mars is \ninclined to the ecliptic.  The modern value of the orbital inclination \nis 1.850 deg \\citep[][on p. 295]{Tho_etal00}.  For our purposes here we will\nassume that Mars' orbit is coplanar with the ecliptic.\n\nGiven that most positions of Mars listed in Table \\ref{data} were \nderived from angular separations with respect to three to five reference \nstars, almost all derived right ascensions and declinations have \neasy-to-calculate internal random errors. These are sometimes as small \nas $\\pm$0.01 deg (which we do not really believe). On one occasion (JD \n2457399.9840) the internal random error for right ascension was \n$\\pm$0.23 deg and the internal random error of declination was $\\pm$0.47 \ndeg.  Typical internal random errors for right ascension and declination \nare $\\sigma _{\\alpha} \\approx \\sigma _{\\delta} \\approx \\pm~0.10$ deg.\n\nSince we have ``true'' positions of Mars from \\citet{vanF_Pul79} we can \nmake a direct estimate of the accuracy of our observations.  The easiest \nway to do this is to precess the ecliptic longitudes from column 4 in \nTable \\ref{data} to equinox J2000 by subtracting 50.25 arc seconds per year \ntimes the number of years from JD 2,457,543.5 (January 0.0, 2000) to the \ndate of observation.  The ecliptic latitudes require no precession \ncorrection.  The subsequent differences of ecliptic longitude and \nlatitude (``observed'' {\\em minus} ``true'') give mean differences of $-0.062 \n\\pm 0.016$ deg in ecliptic longitude and $-0.037 \\pm0.027$ deg in \necliptic latitude.  The standard deviations of the distributions of \ndifferences are: $\\sigma _{\\lambda} = \\pm$0.10 deg and $\\sigma _{\\beta} \n= \\pm$0.17 deg.  The square root of the sum of squares of those errors \nis $\\pm$0.20 deg, or $\\pm$12\\arcmin.  This is the accuracy of the \nposition of a bright object (such as a planet) obtainable with our \nsimple cross staff.\n\n\\section{Fitting the Data}\n\nIn Book 5, Chapter 19, of {\\em On the Revolutions of the Heavenly Spheres} \n\\citet{Cop1543} derived the perigee, mean, and apogee distances of Mars.  \nHe obtained the values 1.374, 1.520, and 1.649 AU, respectively.\nThus, Copernicus knew the amount\nby which Mars' distance from the Sun varies, and his mean distance\nis very close to the modern value of the semi-major axis size of\nMars' orbit (1.5237 AU).  Further discussion of Copernicus' model\ncan be found in Appendix \\ref{sec:comparison}.\n\nLet us consider the elliptical orbit of Mars.  The equation of an\nellipse is:\n\n\\begin{equation}\nr \\; = \\; \\frac{a(1 \\; - \\; e)}{1 \\; + \\; e \\; \\rm{cos} \\; \\theta} \\; ,\n\\end{equation}\n\n\\noindent\nwhere r is the Mars-Sun distance, $a$ is the semi-major axis of\nthe ellipse, $e$ is the eccentricity, and angle $\\theta$ = 0 when Mars is \nat perihelion.\n\nThe velocity along the orbit \\citep[][Eq. 2.36]{Car_Ost07} is\n\n\\begin{equation}\nv^2 \\; = \\; G (M_{\\odot} + M_{Mars}) \\left(\\frac{2}{r} \\; - \\; \\frac{1}{a} \\right) \\; .\n\\end{equation}\n\n\\noindent\nSince the mass of Mars is $\\approx$ 3.23 $\\times$ 10$^{-7}$ M$_{\\odot}$\n\\citep[][p. 295]{Tho_etal00}, for our purposes here we shall ignore it.\n\nWe wish to calculate the position of Mars at increments of one Earth day\nstarting at the moment of its perihelion.  At perihelion r = r$_{min}$ = $a (1 - e)$.\nUsing the known semi-major axis size and eccentricity of Mars' orbit,\nwe can calculate the maximum velocity at perihelion with Equation 8.\nOn perihelion day traveling at velocity\n$v_{max}$ Mars moves 0.6349 degrees along its orbit as viewed from the Sun.\nThis allows us to calculate the constant $h$ for Mars using Equation 1.\nThen, by alternating use of Equations 7 and 1 we can calculate $r$ and $\\theta$\nfor Mars each day along its orbit.  The X-Y coordinates are obtained simply:\n$X = r$ cos($\\theta$) and $Y = r$ sin($\\theta$).\n\nFor the Earth let us adopt a circular orbit.  (The eccentricity of the Earth's\norbit is actually about 0.0167.)  On average, the Earth moves 360.0\/365.2422 = \n0.985647 degrees per day with respect to the Sun.  This gives us another \nset of X-Y coordinates in the same coordinate system, with\nthe Sun at the origin and the +X-axis in the direction of Mars' perihelion.\n\nConsider Figure \\ref{f3}.  According to the Solar Systems Dynamic Group, Horizons \nOn-Line Ephemeris System at Jet Propulsion Laboratory, the previous perihelion of \nMars occurred on Julian Date 2,457,003.8524 (December 12, 2014, at 08:27:29 UT).  \nThe next perihelion occurs on Julian Date 2,457,691.0507 (October 29, 2016, at \n13:13:04 UT).  For the moment we will take the perihelion dates as given.  They \nare not directly observable, but the date of opposition of Mars essentially is.  \nMars' ecliptic longitude differs from that of the Sun by 180 deg near the \nmid-time of its retrograde motion. From our data we find opposition to have \noccurred on May 22, 2016, at 2 hours UT.  According to the {\\em Astronomical \nAlmanac} for 2016 opposition occurred on May 22nd at 11 hours UT.  The agreement \nis reasonably good.\n\nMars' latest opposition occurred 526.9 days after the perihelion of December 12, \n2014.  Call it 527 days.  The X-Y coordinates of the day-by-day position of Mars \nin our coordinate system give $\\theta$ = 265.545 deg on the day of opposition.  \nSince the Earth moves 0.985647 deg per day along its orbit, the 269th pair of \nEarth coordinates gives an angle $\\theta$ most closely matching that of Mars \n(265.139 vs. 265.545 deg, in fact).  Given the index $i$ of the Earth's \ncoordinates, the corresponding index of Mars' coordinates for the same day is \nequal to $i$ + (527 $-$ 269).\n\nWe wrote a simple program in Python that calculates the X-Y coordinates of Mars\nfor each day starting at perihelion, and X-Y coordinates of the Earth.  With \nan appropriate offset of the indices of the two sets of coordinates we can\nobtain the direction toward Mars {\\em from the Earth} for any given date.  \nThen we can find an arithmetic offset between days-since-Mars-perihelion and Julian \nDate, and we can find another arithmetic offset to transform the angles obtained\nto ecliptic longitude.  This is just an example of shifting a template in\ntwo coordinates to minimize the sum of squares of deviations from the\ndata to the offset template.  This produces a goodness of fit parameter.\nThe square root of the goodness of fit parameter divided by the number of\ndata points minus 2 gives the RMS scatter of the fit of the template\nto the data.  Varying the model parameters allows us to search for an\neven better fit to the data.\n\nOur program lets us a generate a template of ecliptic longitude values\nvs. time.  Using the official values of Mars' perihelion date, orbital\neccentricity, and orbit size given in Table \\ref{summary}, we can adjust\nthe X and Y axis values to minimize a goodness of fit parameter and \nmake a plot that fits the data well enough to eye.  The goodness of\nfit parameter is the sum of squares of differences between a template\nand the data, in other words, like $\\chi^2$ minimization, but with\nequal weights for all the points.\n\nWe wondered if a better fit to our data might be obtained with different fit \nparameters.  In other words, we start by deriving the date of perihelion, using \nour data.  In the top panel of Figure \\ref{f4} we show the goodness of fit \nparameter for a range of dates with respect to October 29.5. On the basis of our \ndata, we find the best fit is obtained for a perihelion date of October 25.4. The \ngoodness of fit parameter doubles, compared to the value at the minimum, for \nperihelion dates October 21.5 and October 29.3. Our perihelion date is therefore \nOctober 25.4 $\\pm$ 3.9, 2016 (UT). This is the equivalent of December 8.5, 2014 \n(the date of the previous perihelion).  In what follows that is our effective\nperihelion date, allowing us to work {\\em forward} in time toward the \nsubsequent perihelion on October 25.4, 2016.\n\nHow robust is the eccentricity?  In the middle panel of Figure \\ref{f4} we show \nthe goodness of fit parameter for a range of eccentricities ranging from 0.043 to \n0.133.  The minimum of the sum of squares of residuals occurs for $e$ = 0.086,\nwith an uncertainty of $\\pm$0.010.\n\nFinally, we tried a number of values of the orbital semi-major axis size,\nranging from 1.50 to 1.55.  The goodness of fit parameter is minimized for\n$a$ = 1.526 AU.  Using our determination of the time of perihelion, the eccentricity, and a slight\nadjustment to the mean distance from the Sun, we obtain a\nsum of squares of residuals of 1.111.  There were 42 data points\nused for the analysis, and 2 constraints (adjusting time to truncated Julian \nDate and adjusting angle to ecliptic longitude).  The best fit to our data\ngives an RMS residual of $\\pm 10^{\\prime}.0$ for the fit of the ecliptic longitudes\nto the best model.  This is only slightly better than the average difference\nbetween the data and the ``true'' values ($\\pm$12\\arcmin). \nIgnoring the eccentricity of the Earth's orbit was not a serious mistake.\n\nIn Figure \\ref{f5} we plot the observed ecliptic longitudes of Mars (for equinox\nJ2000) vs. the Julian Date.  Low order polynomial fits to the two stationary\npoints indicate that retrograde motion lasted for 74.1 days.  The range of\necliptic longitude between the stationary points was\n15.23 $\\pm$ 0.42 deg.  We also show three fits of the data\nfor values of the orbital eccentricity of 0.053, 0.086, and 0.123.  Even if this\nfigure is displayed in color, it is difficult to see that the middle locus is\nbetter than the other two.  In Figure \\ref{f6} we show the differentials\n(data {\\em minus} model).  Figure \\ref{f6} clearly shows that the\neccentricity cannot be as small as 0.053 or as large as 0.123.  In both\ncases the model differs from the data by 1.0 to 1.5 degrees 190 days before\nopposition and 70 days after opposition.  Given that the individual positions\nof Mars are good to $\\pm$0.2 deg on average, values of\nthe eccentricity of 0.053 or 0.123 are strongly rejected by the data.  \n\n\\section{Conclusions}\n\nIn Table \\ref{summary} we summarize the values derived from our simple dataset\nbased on naked eye observations using a handheld cross staff.  For\ncomparison we also give the official (``true'') values based on a much\nmore sophisticated orbital determination using the most modern methods.  \nAdmittedly, in science we almost never have the ``true'' values for comparison.  \nTypically, we can only calculate our internal random errors and estimate sources \nof systematic error.\n\nOur value of the orbital eccentricity of Mars (0.086 $\\pm$ 0.010) compares \nreasonably well the official modern value (0.0934).  While our data are not \naccurate enough to {\\em prove} that Mars' orbit is an ellipse, if we fit the data \nwith an ellipse, the eccentricity must be near 0.09.  Thus, a dataset based on \nnaked eye observations can be shown to be in strong agreement with Kepler's First \nLaw.\n\n\\acknowledgments\n\nWe made use of the SIMBAD database, operated at CDS, Strasbourg, France.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nPropulsion of lightweight spacecraft to sub-relativistic speeds that greatly exceed $30 ~\\rm km\\,{\\rm s}^{-1}$ (or $10^{-4}c$) attainable by chemical rockets is a promising frontier for space exploration (\\citealt{Turyshev:2020to}; \\citealt{2020arXiv200703530F}). \nRecently, \\cite{2017ApJ...834L..20C} proposed a novel method to measure the mass of planets via interferometry by an array of relativistic spacecraft, envisioned by the Breakthrough Starshot initiative\\footnote{https:\/\/breakthroughinitiatives.org\/Initiative\/3}. \\cite{2018AcAau.152..370P} suggested a precursor that will launch slower spacecraft at $v\\sim 0.01c$ to explore the Solar system. \\cite{Witten:2020tc} proposed to use a sub-relativistic spacecraft of speeds $v\\gtrsim 0.001c$ to indirectly probe Planet 9 through its gravitational influence on the spacecraft trajectory. \\cite{2020ApJ...895L..35H} calculated the turbulent noise associated with drag and magnetic forces from the interstellar medium (ISM) at these high speeds. \n\nAs relativistic spacecraft moves through the solar system, it is heated by its interaction with gas particles as well as solar wind and radiation (\\citealt{2017ApJ...837....5H}). Here, we study the detectability of the hot surface of a relativistic spacecraft from its long-wavelength thermal emission. In particular, we aim to derive the minimum size and speed of the spacecraft that could be detected with a high-sensitivity telescope.\n\nThe structure of the paper is as follows. In Section \\ref{sec:model}, we describe the model of calculations, including spacecraft heating, cooling, and its thermal emission. In Section \\ref{sec:results}, we present our numerical results of the surface temperature and emission flux, and discuss the minimum sizes of objects detectable by telescopes. In Section \\ref{sec:discuss}, we discuss implications of our findings.\n\n\n\n\\section{Model}\\label{sec:model}\nLet us first evaluate the equilibrium temperature of the spacecraft due to collisional and radiative heating by absorption of interstellar starlight and solar radiation. Collisional heating is dominated by H and He which carry most of the gas mass. The rate of collisional heating to the spacecraft surface moving at a speed $v$ can be written as,\n\\begin{eqnarray}\n\\frac{dE_{\\rm coll}}{dt} = \\frac{1.4n_{\\rm H}m_{\\rm H}v^{3}\\pi R^{2}}{2}\n\\simeq 9.86\\times 10^{4}\\left(\\frac{n_{\\rm H}}{1\\,{\\rm {cm}}^{-3}}\\right) \\left(\\frac{R}{1\\,{\\rm {cm}}}\\right)^{2}\\left(\\frac{v}{0.1c}\\right)^{3} \\,{\\rm {erg}}\\,{\\rm s}^{-1},\\label{eq:dEdt_coll}\n\\ena\nwhere $\\pi R^{2}$ is the frontal surface area (for a radius $R$) of the spacecraft, $n_{\\rm H}$ is the hydrogen density, and a factor of $1.4$ accounts for the $10\\%$ abundance of He relative to H. \n\nBeyond the heliopause, at a distance $D>D_{\\rm hp}=100\\,{\\rm {au}}$, from the Sun, the hydrogen density is $n_{\\rm H}=n_{\\rm ISM}$ with $n_{\\rm ISM}$ the ISM density. Inside the heliosphere, $n_{\\rm H}=n_{\\rm sw}+n_{\\rm neu}$ with $n_{\\rm neu}$ being the neutral hydrogen density and $n_{\\rm sw}$ being the proton density of the solar wind. We take $n_{\\rm neu}= 0.1\\,{\\rm {cm}}^{-3}$ (\\citealt{2009SSRv..146..235F}). The mean proton density of the solar wind at a distance $D$ is described by $n_{\\rm sw}=n_{0}(D\/1\\,{\\rm {au}})^{2}\\,{\\rm {cm}}^{-3}$ with $n_{0}\\approx 6\\,{\\rm {cm}}^{-3}$ being the density at $D=1\\,{\\rm {au}}$ (\\citealt{Venzmer:2018gy}). Note that heating by heavy elements and dust is negligible on average because it carries only a small fraction of the gas mass.\n\nThe heating by cosmic rays (CRs) of particle energy $E$ and flux $F_{\\rm CR}$ is given by,\n\\begin{eqnarray}\n\\frac{dE_{\\rm CR}}{dt} = F_{\\rm CR}E\\pi R^{2}\\simeq 0.02\\left(\\frac{E}{1\\rm GeV}\\right)\\left(\\frac{F_{\\rm CR}}{1\\,{\\rm {cm}}^{2}\\,{\\rm s}^{-1}}\\right)\\left(\\frac{R}{1\\,{\\rm {cm}}}\\right)^{2}\\,{\\rm {erg}}\\,{\\rm s}^{-1}.\\label{eq:dECR}\n\\ena\nThe CR's  measured near the Earth have $F_{\\rm CR}\\approx 1\\,{\\rm {cm}}^{2}\\,{\\rm s}^{-1}$ (\\citealt{1985A&A...144..147L}) and $E=1$ GeV, yielding subdominance of CR heating relative to collisional heating for $v\\gtrsim 10^{-3}$c.\n \nWe assume that the local radiation field has the same spectrum as the interstellar radiation field (ISRF) in the solar neighborhood \\citep{1983A&A...128..212M}, with a total radiation energy density of $u_{\\rm MMP}\\approx 8.64\\times 10^{-13} \\,{\\rm {erg}}\\,{\\rm {cm}}^{-3}$, where the local radiation field is expressed as a factor, $U$, times the energy density of the ISRF, $u_{\\rm MMP}$. The heating rate of the frontal surface by the background radiation is given by,\n\\begin{eqnarray}\n\\frac{dE_{\\rm abs}}{dt}=\\pi R^{2}cUu_{\\rm MMP}\\epsilon_{\\star},\\label{eq:dErad}\n\\ena\nwhere $\\epsilon_{\\star}$ is the surface absorption efficiency averaged over the starlight radiation spectrum.  \n\nThe local radiation field includes the averaged ISRF and the solar radiation, with\n\\begin{eqnarray}\nU=1+\\frac{L_{\\star}}{4\\pi D^{2}cu_{\\rm MMP}}=1+0.5\\times 10^{8}\\left(\\frac{L_{\\star}}{L_{\\odot}}\\right)\\left(\\frac{1\\,{\\rm {au}}}{D}\\right)^{2},\n\\ena\nimplying an enhancement by the Sun within $D=10^{4}\\,{\\rm {au}}$. \n\nThe surface of the spacecraft at a temperature $T_{s}$ also emits thermal radiation, which results in radiative cooling at a rate:\n\\begin{eqnarray}\n\\frac{dE_{\\rm emiss}}{dt}=\\int d\\nu \\pi R^{2}Q_{\\rm abs, \\nu}\\pi B_{\\nu}(T_{s})\n=\\pi R^{2}\\epsilon_{T} \\sigma T_{s}^{4},~~~~\n\\label{eq:dEcdt}\n\\ena\nwhere,\n\\begin{eqnarray}\n\\epsilon_{T}=\\frac{\\int d\\nu Q_{\\rm abs,\\nu}B_{\\nu}(T_{s})}{\\int d\\nu B_{\\nu}(T_{s})}, \\label{eq:Qabsavg}\n\\ena\nis the Planck-spectrum averaged absorption efficiency, and $Q_{\\rm abs}$ is the absorption efficiency of the spacecraft material.\n \nThe energy balance between radiative heating and cooling yields the surface equilibrium temperature,\n\\begin{eqnarray}\nT_{\\rm s}=\\left(\\frac{cUu_{\\rm MMP}+1.4 n_{\\rm H}m_{\\rm H}v^{3}\/2+F_{\\rm CR}E_{\\rm CR}}{\\sigma}\\right)^{1\/4}\\left(\\frac{\\epsilon_{\\star}}{\\epsilon_{T}}\\right)^{1\/4}.\\label{eq:Tsp}\n\\ena\n\nFor the ISM and $v\\gtrsim 10^{-3}c$, collisional heating dominates. Equation (\\ref{eq:Tsp}) yields the surface temperature,\n\\begin{eqnarray}\nT_{\\rm s}\\simeq \\left(\\frac{1.4 n_{\\rm H}m_{\\rm H}v^{3}}{2\\sigma} \\right)^{1\/4}\\left(\\frac{\\epsilon_{\\star}}{4\\epsilon_{T}}\\right)^{1\/4}\\simeq 272.6\\left(\\frac{n_{\\rm H}}{10\\,{\\rm {cm}}^{-3}}\\right)^{1\/4}\\left(\\frac{v}{0.1c}\\right)^{3\/4}\\left(\\frac{\\epsilon_{\\star}}{\\epsilon_{T}}\\right)^{1\/4}\\rm K,\\label{eq:Tsp_coll}\n\\ena\nimplying that the emission spectrum peaks at a wavelength,\n\\begin{eqnarray}\n\\lambda_{\\rm max}=\\frac{b}{T_{\\rm sp}}\\simeq 10.6\\left(\\frac{n_{\\rm H}}{10\\,{\\rm {cm}}^{-3}}\\right)^{-1\/4}\\left(\\frac{v}{0.1c}\\right)^{-3\/4}\\left(\\frac{\\epsilon_{\\star}}{\\epsilon_{T}}\\right)^{-1\/4}\\mum,\n\\ena \nwith $b$ is Wien's constant.\n\nWhen the spacecraft enters the Solar system, radiative heating dominates. Equation (\\ref{eq:Tsp}) yields the equilibrium temperature\n\\begin{eqnarray}\nT_{\\rm s}\\simeq 388.7\\left(\\frac{D}{1\\,{\\rm {au}}}\\right)^{-1\/2}\\left(\\frac{\\epsilon_{\\star}}{\\epsilon_{T}}\\right)^{1\/4}\\,{\\rm K},\\label{eq:Tsp_rad}\n\\ena\nimplying a peak wavelength\n\\begin{eqnarray}\n\\lambda_{\\rm max}=\\frac{b}{T_{\\rm s}}\\simeq 7.45\\left(\\frac{D}{1\\,{\\rm {au}}}\\right)^{1\/2}\\left(\\frac{\\epsilon_{\\star}}{\\epsilon_{T}}\\right)^{-1\/4}\\mum.\n\\ena \n\nEquations (\\ref{eq:Tsp_coll}) and (\\ref{eq:Tsp_rad}) imply that radiative heating by the Sun dominates over collisional heating at $D<10\\,{\\rm {au}}$ for a speed $v=0.1c$. \n\nThe bolometric radiation flux observed on Earth from the spacecraft is,\n\\begin{eqnarray}\nF=\\frac{1}{4}\\theta^{2}\\sigma T_{\\rm s}^4=\\frac{1}{4}\\left(\\frac{R}{D}\\right)^{2}\\sigma T_{\\rm s}^{4},\n\\ena\nwhere $\\theta=(R\/D)$ is the angular size of the spacecraft on the sky, namely its radius, $R$, divided by its distance, $D$. For a temperature of $T_{s}=100\\,{\\rm K}$, the bolometric flux is,\n\\begin{eqnarray}\nF\\simeq 6.3\\times 10^{-24} \\left(\\frac{R}{1\\rm m}\\frac{100\\,{\\rm {au}}}{D}\\right)^{2}\\left(\\frac{T_{\\rm s}}{100\\rm K}\\right)^{4}~\\frac{\\rm erg}{\\,{\\rm {cm}}^{2}\\,{\\rm s}}.\n\\ena\n\nThe spectral flux density is given by \n\\begin{eqnarray}\nF_{\\nu}=\\frac{1}{4}\\theta^{2}\\pi B_{\\nu}(T_{\\rm s})=\\frac{1}{4}\\left(\\frac{R}{D}\\right)^{2}\\frac{2\\pi h\\nu^{3}}{c^{2}}\\frac{1}{\\,{\\rm {exp}}(h\\nu\/kT_{\\rm s})-1}.\\label{eq:Sflux}\n\\ena\n\n\n\n\n\\section{Results}\\label{sec:results}\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{f1.pdf}\n\\includegraphics[width=0.5\\textwidth]{f2.pdf}\n\\caption{Left panel: Surface temperature of the spacecraft as a function of its speed relative to the speed of light, $v\/c$, at different heliocentric distances, $D$. Right panel: Spectral flux density for different spacecraft speeds, assuming the object radius $R=10$ m and $D=1,10, 100$ au. Horizontal black lines mark the sensitivity of various telescopes.}\n\\label{fig:Teq}\n\\end{figure}\n\nThe left panel of Figure \\ref{fig:Teq} shows the surface temperature as a function of spacecraft speed in units of the speed of light, ($v\/c$), for different heliocentric distances, and assuming $\\epsilon_{\\star}\/\\epsilon_{T}=1$. The temperature is dictated by radiative heating at low speeds. When the speed is sufficiently large, collisional heating becomes important and $T_{\\rm sp}$ increases with $v\/c$. For example, at $D=10$ au, the temperature increases with speed for $v\\gtrsim 0.02c$. At $D=1000$ au, the temperature at $v>0.005c$ is higher than that for $D=10, 100$ au due to the stronger collisional heating.\n\nThe right panel of Figure \\ref{fig:Teq} shows the spectral flux density from Equation (\\ref{eq:Sflux}) as a function of frequency (wavelength) for different spacecraft speeds, $v\/c$, assuming a spacecraft radius $R=10$ m. The frequency bands of several telescopes are shown, including ALMA (Atacama Large Millimeter\/submillimeter Array), JWST (James Webb Space Telescope), and LSST (Legacy Survey of Space and Time). The sensitivity is $S=8.7~\\mu$Jy at $\\lambda\\approx 870\\mum$ for ALMA\\footnote{https:\/\/almascience.eso.org\/proposing\/sensitivity-calculator}, $0.01~\\mu$Jy at $\\lambda\\sim 0.6-28.3\\mum$ with integration time of $10^{4}$ s for JWST\\footnote{https:\/\/www.stsci.edu\/files\/live\/sites\/www\/files\/home\/news\/newsletters\/documents\/2013-volume030-issue02.pdf} (\\citealt{Anonymous:vWxVySok}; \\citealt{2015PASP..127..686G}). LSST can reach the limiting magnitude of $m=25$ with two exposure times of 15 s (\\cite{2019ApJ...873..111I}), yielding a sensitivity of $S=3631\\times 10^{-m\/2.5}~\\rm Jy\\sim 0.36\\mu$J for the $\\sim 0.3-1\\mum$ wavelengths.\n\nWe find that only when a 10 meter-size objects enter the solar system, it is feasible to detect them using JWST. At a distance of 100 au, JWST can detect an object of size $\\sim 400$ m moving at $v\\gtrsim 0.1c$. For travel through the Solar wind near the Earth's orbit, i.e., at a distance of $D\\sim$ 1 au, we get detectability for objects larger than $50$ cm. At $D=10\\,{\\rm {au}}$, objects larger than $100$ m can be detected by JWST.\n\n\\section{Discussion}\\label{sec:discuss}\nMost sky surveys dismiss objects that move too fast on the sky. A sub-relativistic nearby object would move at an angular speed of $(v\/D)=2.48(v\/10^{-3}c)(10\\,{\\rm {au}}\/D)~{\\rm arcsec\/min}$, and this would limit the integration time within the field of view of telescopes. Hence the sensitivity will be lower than for steady point sources, unless a special tracking program is applied.\n\nThe situation improves for objects closer to the Sun, since the solar wind density increases inversely with the square of the distance from the Sun. The planned size of the transmitter or sails of Starshot spacecraft is a few meters and the goal of Starshot is to probe the habitable zone around a star, similar to the region around the Earth's orbit. Thus, the mid- and far-infrared special regimes is optimal for detecting analogous spacecraft in our vicinity.\n\n\n\\acknowledgments \nThis work is supported in part by a grant from the Breakthrough Prize Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nKendall's tau is a measure of association which evaluates every copula   $ C : [0,1]^d\\to[0,1] $   by a real number   $ \\kappa[C] $   \nsatisfying   $ - 1\/(2^{d-1}\\!-\\!1) \\leq \\kappa[C] \\leq 1 $. \n\n\\bigskip\nIn the present paper we study Kendall's tau for a class of copulas related to the order statistic   $ \\XXX_{:d} = (X_{1:d},\\dots,X_{d:d}) $   \nof a random vector   $ \\XXX = (X_1,\\dots,X_d) $   with identical univariate marginal distributions.  \nIn this case, \nevery copula   $ C $   for   $ \\XXX $   determines a copula   $ C_{:d} $   for   $ \\XXX_{:d} $; \nsee \nNavarro and Spizzichino [2010] for the special case where the distribution functions of the coordinates of   $ \\XXX $   are continuous and \nDietz et al.\\ [2016] for the general case. \nSince the construction of the order transform   $ C_{:d} $   of a copula   $ C $   \nis determined by a map transforming every random vector into its order statistic, \nwe shall study Kendall's tau for the order transform   $ C_{:d} $   of an arbitrary copula   $ C $. \n\n\\bigskip\nAs a general result, \nwe show that the order transform   $ C_{:d} $   of a copula   $ C $   satisfies   $ \\kappa[C] \\leq \\kappa[C_{:d}] $   \n(Theorem \n\\ref{t.Kendall}). \nWe also show that the inequality is strict when   $ C $   is the product copula \n(Theorem \n\\ref{t.product}), \nwhich corresponds to the case where the coordinates of   $ \\XXX $   are also independent. \nBy contrast, \nthe inequality becomes an equality when   $ C $   is the upper Fr{\\'e}chet--Hoeffding bound (and hence maximizes Kendall's tau) \nor a copula which is symmetric and minimizes Kendall's tau \n(Corollary \n\\ref{c.bounds} and Theorem \n\\ref{t.Kendall}). \nThis result is of interest \nsince the computation of   $ C_{:d} $   or   $ \\kappa[C_{:d}] $   may be tedious and is not needed in the case where   $ \\kappa[C] = \\kappa[C_{:d}] $   and \nsince there exist many symmetric copulas minimizing Kendall's tau; \nsee \nFuchs et al.\\ [2018]. \n\n\\bigskip\nThe major part of this paper is devoted to the order transform   $ \\Pi_{:d} $   of the product copula   $ \\Pi $.   \nWe first determine   $ \\kappa[\\Pi_{:d}] $   \n(Theorem \n\\ref{t.product}) and then present some identities for the computation of   $ \\kappa[\\varrho_K(\\Pi_{:d})] $,   \nwhere   $ \\varrho_K(\\Pi_{:d}) $   denotes the multivariate margin of   $ \\Pi_{:d} $   with respect to the coordinates \nin   $ K\\subseteq\\{1,\\dots,d\\} $   with   $ |K|\\geq2 $   \n(Theorem \n\\ref{t.product-K}). \nIn particular, \nwe obtain an explicit formula for Kendall's tau of   $ \\varrho_{\\{1,\\dots,k\\}}(\\Pi_{:d}) $   \n(Corollary \n\\ref{c.product-lowertail}), \nwhich is a copula for the lower   $ k $   coordinates of   $ \\XXX_{:d} $, \nand we show that, \ndue to a general reflection principle \n(Theorem \n\\ref{t.product-reflection}), \nthis formula is also valid for Kendall's tau of   $ \\varrho_{\\{d-k+1,\\dots,d\\}}(\\Pi_{:d}) $,   \nwhich is a copula for the upper   $ k $   coordinates of   $ \\XXX_{:d} $. \nWe thus extend certain results for the case   $ |K|=2 $; \nsee \nAv{\\'e}rous et al.\\ [2005] and \nNavarro and Balakrischnan [2010]. \n\n\\bigskip\nThis paper is organized as follows: \nSection \\ref{preliminaries} collects some definitions and results on copulas and related topics which will be needed in this paper. \nSection \\ref{kendall} provides a brief discussion of Kendall's tau and Kendall's distribution function. \nSection \\ref{ot-copula} starts with the definition of the order transform of the Euclidean space, \nwhich turns every random vector into its order statistic, \nand proceeds with the construction of the order transform of a copula.  \nIn Section \\ref{ot-kendall} we present some result of Kendall's tau for the order transform of an arbitrary copula and \nin Section \\ref{ot-product} we study Kendall's for the order transform of the product copula and its multivariate margins. \nSome auxiliary results needed in Section \\ref{ot-product} are established in the Appendix. \n\n\\bigskip\nThroughout this paper we shall use the following notation: \nLet   $ \\I := [0,1] $   and \nlet   $ \\leb $   denote the Lebesgue measure on   $ \\BB(\\R) $. \nFurthermore, \nlet   $ d\\geq2 $   be an integer, \nwhich will be kept fixed, \nand \nlet   $ \\leb^d $   denote the Lebesgue measure on   $ \\BB(\\R^d) $. \nWe denote \nby   $ \\eee_1,\\dots,\\eee_d $   the standard basic unit vectors in   $ \\R^d $, \nby   $ \\zero $   the vector in   $ \\R^d $   with all coordinates being equal to   $ 0 $   and \nby   $ \\eins $   the vector in   $ \\R^d $   with all coordinates being equal to   $ 1 $.\nFor   $ \\xxx,\\yyy\\in\\R^d $, \nwe write   $ \\xxx\\leq\\yyy $   if   $ x_k \\leq y_k $   holds for every   $ k\\in\\{1,\\dots,d\\} $. \nThen we have   $ \\I^d = [\\zero,\\eins] $. \nOn the collection of all real--valued maps on a set   $ S $, \nthe pointwise order   $ \\leq $   is defined by letting   $ g \\leq h $   if   $ g(s) \\leq h(s) $   holds for every   $ s \\in S $. \n\n\\bigskip\nDue to the central role of the order transform   $ T $   of the Euclidean space in the construction of the order transform of a copula   $ C $, \nthe symbol   $ C_{:d} $   used in the Abstract and in this Introduction will henceforth be replaced by   $ C_T $. \n\n\n\n\n\\section{Preliminaries}\n\\label{preliminaries}\n\nIn this section, \nwe recall some definitions and results on \ncopulas, \ncopula measures, \ngroups of transformations of copulas and \na biconvex form for copulas. \nFor further details we refer to \nFuchs [2014; 2016]. \n\n\n\\subsection*{Copulas}\n\nFor   $ K\\subseteq\\{1,...,d\\} $, \nwe consider the map   $ \\eeta_K : \\I^d\\times\\I^d \\to \\I^d $   given coordinatewise by\n\\begin{eqnarray*}\n        (\\eeta_K(\\uuu,\\vvv))_k                \n& := &  \\begin{cases}                         \n         u_k & k \\in \\{1,...,d\\} \\setminus K  \\\\\n         v_k & k \\in K                        \n        \\end{cases}                           \n\\end{eqnarray*}\nand for   $ k\\in\\{1,\\dots,d\\} $   we put   $ \\eeta_k := \\eeta_{\\{k\\}} $. \n\n\\bigskip\nA \n\\emph{copula} is a function   $ C: \\I^{d} \\to \\I $   satisfying the following conditions: \n\\begin{romlist}\n\\item   The inequality   $ \\sum_{K \\subseteq \\{1,...,d\\}} (-1)^{d-|K|}\\,C(\\eeta_K(\\uuu,\\vvv)) \\geq 0 $   \nholds for all   $ \\uuu,\\vvv\\in\\I^d $   such that   $ \\uuu\\leq\\vvv $. \n\\item   The identity   $ C(\\eeta_k(\\uuu,\\zero)) = 0   $   holds for every   $ k\\in\\{1,...,d\\} $   and every   $ \\uuu\\in\\I^d $. \n\\item   The identity   $ C(\\eeta_k(\\eins,\\uuu)) = u_k $   holds for every   $ k\\in\\{1,...,d\\} $   and every   $ \\uuu\\in\\I^d $. \n\\end{romlist}\nThis definition of a copula is in accordance with the literature;\nsee, \ne.\\,g., \nDurante and Sempi [2016] and \nNelsen [2006]. \nThe collection   $ \\CC $   of all copulas is convex.\n\n\\bigskip\nThe following copulas are of particular interest: \n\\begin{hylist}\n\\item   The \n\\emph{upper Fr{\\'e}chet--Hoeffding bound}   $ M $   given by \n$ M(\\uuu) := \\min\\{u_1,\\dots,u_d\\} $ \nis a copula and every copula   $ C $   satisfies   $ C \\leq M $. \n\\item   The \n\\emph{product copula}   $ \\Pi $   given by \n$ \\Pi(\\uuu) := \\prod_{k=1}^d u_k $ \nis a copula. \n\\item   In the case   $ d=2 $, \nthe \n\\emph{lower Fr{\\'e}chet--Hoeffding bound}   $ W $   given by \n$ W(\\uuu) := \\max\\{ \\sum_{k=1}^d u_k + 1 - d, 0 \\} $   \nis a copula and every copula   $ C $   satisfies   $ W \\leq C $. \n\\end{hylist}\n\n\n\\subsection*{A Group of Transformations of Copulas}\n\nLet   $ \\Phi $   denote the collection of all transformations   $ \\CC\\to\\CC $   \nand consider the composition   $ \\circ: \\Phi\\times\\Phi \\to \\Phi $   given by   $ (\\varphi_2\\circ\\varphi_1)(C) := \\varphi_2(\\varphi_1(C)) $   \nand the map   $ \\iota\\in\\Phi $   given by   $ \\iota(C) := C $. \nThen   $ (\\Phi,\\circ) $   is a semigroup with neutral element   $ \\iota $. \nFor   $ i,j,k\\in\\{1,...,d\\} $   with   $ i \\neq j $, \nwe define the maps   $ \\pi_{i,j},\\nu_k : \\CC\\to\\CC $   by letting\n\\begin{eqnarray*}\n        (\\pi_{i,j}(C))(\\uuu)                                      \n& := &  C(\\eeta_{\\{i,j\\}}(\\uuu,u_j\\,\\eee_i+u_i\\,\\eee_j))          \\\\*\n        (\\nu_k    (C))(\\uuu)                                      \n& := &  C(\\eeta_k(\\uuu,\\eins)) - C(\\eeta_k(\\uuu,\\eins\\!-\\!\\uuu))  \n\\end{eqnarray*}\nEach of these maps is an involution and there exists \n\\begin{hylist}\n\\item   a smallest subgroup   $ \\Gamma^\\pi $   of   $ \\Phi $   containing every   $ \\pi_{i,j} $, \n\\item   a smallest subgroup   $ \\Gamma^\\nu $   of   $ \\Phi $   containing every   $ \\nu_k $   and \n\\item   a smallest subgroup   $ \\Gamma     $   of   $ \\Phi $   containing   $ \\Gamma^\\pi\\cup\\Gamma^\\nu $. \n\\end{hylist}\nThe group   $ \\Gamma^\\nu $   is commutative. \nMoreover, \nthe \n\\emph{total reflection} \n\\begin{eqnarray*}\n        \\tau                              \n& := &  \\bcirc_{k\\in\\{1,...,d\\}} \\nu_{k}  \n\\end{eqnarray*}\ntransforms every copula into its survival copula and satisfies   $ \\tau(M)=M $, \nand we put   $ \\Gamma^\\tau := \\{\\iota,\\tau\\} $. \nThe total reflection is used in the definition of the \n\\emph{concordance order}   $ \\leq_c $   on   $ \\CC $   \nwhich is defined by letting   $ C \\leq_c D $  if and only if \n$ C \\leq D $   and   $ \\tau(C) \\leq \\tau(D) $. \n\n\\bigskip\nThe group   $ \\Gamma $    is a representation of the hyperoctahedral group, \nwhich also has a well--known geometric representation: \n\n\\bigskip\nLet   $ \\tilde\\Phi $   denote the collection of all transformations   $ \\I^d\\to\\I^d $   \nand consider the composition   $ \\diamond: \\I^d\\times\\I^d\\to\\I^d $   \ngiven by   $ (\\tilde\\varphi_2\\diamond\\tilde\\varphi_1)(\\uuu) := \\tilde\\varphi_2(\\tilde\\varphi_1(\\uuu)) $   \nand the map   $ \\tilde\\iota\\in\\Phi $   \ngiven by   $ \\tilde\\iota(\\uuu) := \\uuu $. \nThen   $ (\\tilde\\Phi,\\diamond) $ is a semigroup with neutral element   $ \\tilde\\iota $. \nFor   $ i,j,k\\in\\{1,...,d\\} $   with   $ i \\neq j $, \nwe define the maps   $ \\tilde\\pi_{i,j},\\tilde\\nu_k : \\I^d\\to\\I^d $   by letting\n\\begin{eqnarray*}\n        \\tilde\\pi_{i,j}(\\uuu)                           \n& := &  \\eeta_{\\{i,j\\}}(\\uuu,u_j\\,\\eee_i+u_i\\,\\eee_j))  \\\\*\n        \\tilde\\nu_k    (\\uuu)                           \n& := &  \\eeta_k(\\uuu,\\eins\\!-\\!\\uuu)                    \n\\end{eqnarray*}\nEach of these maps is an involution and there exists \n\\begin{hylist}\n\\item   a smallest subgroup   $ \\tilde\\Gamma^\\pi $   of   $ \\tilde\\Phi $   containing every   $ \\tilde\\pi_{i,j} $, \n\\item   a smallest subgroup   $ \\tilde\\Gamma^\\nu $   of   $ \\tilde\\Phi $   containing every   $ \\tilde\\nu_k $   and \n\\item   a smallest subgroup   $ \\tilde\\Gamma     $   of   $ \\tilde\\Phi $   containing   $ \\tilde\\Gamma^\\pi\\cup\\tilde\\Gamma^\\nu $. \n\\end{hylist}\nThe groups   $ \\Gamma $   and   $ \\tilde\\Gamma $   are related by an isomorphism   $ J : (\\Gamma,\\circ) \\to (\\tilde\\Gamma,\\diamond) $   \nsatisfying   $ J(\\pi_{i,j}) = \\tilde\\pi_{i,j} $   and   $ J(\\nu_k) = \\tilde\\nu_k $   for all   $ i,j,k\\in\\{1,...,d\\} $. \nFor   $ \\gamma\\in\\Gamma $   we put   $ \\tilde\\gamma := J(\\gamma) $, \nand for   $ \\tilde\\gamma\\in\\tilde\\Gamma $   we put   $ \\gamma := J^{-1}(\\tilde\\gamma) $. \nThen   $ \\pi(C) = C\\circ\\tilde\\pi $   holds for every   $ \\pi\\in\\Gamma^\\pi $   and every   $ C\\in\\CC $. \n\n\n\\subsection*{Copula Measures}\n\nSince every copula   $ C\\in\\CC $   has a unique extension to a distribution function   $ \\R^d\\to\\I $, \nthere exists a unique probability measure   $ Q^C : \\BB(\\I^d)\\to\\I $   satisfying   \n\\begin{eqnarray*}\n        Q^C[[\\zero,\\uuu]]  \n&  = &  C(\\uuu)            \n\\end{eqnarray*}\nfor every   $ \\uuu\\in\\I^d $. \nThe probability measure   $ Q^C $   is said to be the \n\\emph{copula measure} with respect to   $ C $. \nIt satisfies   $ Q^C[(\\uuu,\\vvv)] = Q^C[[\\uuu,\\vvv]] $   for all   $ \\uuu,\\vvv\\in\\I^d $   such that   $ \\uuu\\leq\\vvv $, \nand   $ Q^{\\gamma(C)} = (Q^C)_{\\tilde\\gamma} $   holds for every   $ \\gamma\\in\\Gamma $. \n\n\n\\subsection*{A Biconvex Form for Copulas}\n\nConsider the map   $ [.\\,,.]: \\CC\\times\\CC \\to \\R $   given by \n\\begin{eqnarray*}\n        [C,D]                            \n& := &  \\int_{\\I^d} C(\\uuu)\\,dQ^D(\\uuu)  \n\\end{eqnarray*}\nThe map   $ [.\\,,.] $   is in either argument linear with respect to convex combinations and is therefore called a \n\\emph{biconvex form}. \nMoreover, \nthe map   $ [.\\,,.] $   is in either argument monotone with respect to the concordance order. \nIt satisfies   $ 0 \\leq [C,D] \\leq 1\/2 $   for all   $ C,D\\in\\CC $, \nand the bounds are attained since   $ [M,M] = 1\/2 $   and since   $ [\\nu(M),\\nu(M)] = 0 $   holds for every   $ \\nu\\in\\Gamma^\\nu\\setminus\\Gamma^\\tau $. \nWe also note that   $ [\\tau(C),\\tau(D)] = [D,C] $   holds for all   $ C,D\\in\\CC $. \n\n\n\n\n\\section{Kendall's Tau and Kendall's Distribution Function}\n\\label{kendall}\n\nThe map   $ \\kappa : \\CC\\to\\R $   given by \n\\begin{eqnarray*}\n        \\kappa[C]                                                       \n& := &  \\frac{2^{d}}{2^{d-1}-1}\\,\\biggl( [C,C] - \\frac{1}{2^d} \\biggr)  \n\\;\\,=\\,\\;  \\frac{2^d\\,[C,C]-1}{2^{d-1}-1}\n\\end{eqnarray*}\nis called \n\\emph{Kendall's tau}; \nsee \nNelsen [2002]. \nKendall's tau satisfies \n\\begin{eqnarray*}\n        -\\,\\frac{1}{2^{d-1}-1}  \n&\\leq&  \\kappa[C]               \n\\;\\,\\leq\\,\\;  1                 \n\\end{eqnarray*}\nand the bounds are attained. \nThe following result is obvious from the properties of the biconvex form and will be tacitly used throughout this paper: \n\n\\bigskip\n\\begin{lemma}{}\n\\label{l.maxmin}\n\\begin{onelist}\n\\item   Kendall's tau is monotone with respect to the concordance order. \n\\item   A copula   $ C $   maximizes Kendall's tau if and only if   $ [C,C] = 1\/2 $. \n\\item   A copula   $ C $   minimizes Kendall's tau if and only if   $ [C,C] = 0   $. \n\\end{onelist}\n\\end{lemma}\n\n\\bigskip\nFor a copula   $ C $, \nthe function   $ K_C: \\R\\to\\I $   given by \n\\begin{eqnarray*}\n        K_C(t)                                        \n& := &  Q^C[\\{ \\uuu\\in\\I^d \\colon C(\\uuu) \\leq t \\}]  \n\\;\\,=\\,\\;  (Q^C)_C[[0,t]]                             \n\\end{eqnarray*}\nis said to be \n\\emph{Kendall's distribution function} with respect to   $ C $, \nwhich is well--known from the literature. \nIt is easy to see that   $ K_C $   is indeed a distribution function \nwhich satisfies   $ K_C(0) = 0 $   and   $ K_C(1) = 1 $   as well as   $ t \\leq  K_C(t) $   for every   $ t\\in\\I $. \nThe following result implies that Kendall's tau can be expressed in term of Kendall's distribution function: \n\n\\bigskip\n\\begin{lemma}{}\n\\label{distribution.l-Kendall-2}\nThe identity \n$$  [C,C]  =  \\int_{\\I} \\Bigl( 1 - K_C(t) \\Bigr) d\\leb(t)  $$\nholds for every copula   $ C $. \nIn particular, \n$ \\kappa[C] \\leq \\kappa[D] $   holds for any two copulas   $ C $   and   $ D $   satisfying   $ K_D \\leq K_C $. \n\\end{lemma}\n\n\\bigskip\nWe refer to Fuchs et al.\\ [2018] for further details on the results of this section. \nThe comparison of copulas via Kendall's distribution function was considered by \nCap{\\'e}ra{\\`a} et al.\\ [1997].  \n\n\n\n\n\\section{The Order Transform of a Copula} \n\\label{ot-copula}\n\nThroughout this section, \nconsider the map   $ T : \\overline\\R^d\\to\\overline\\R^d $   which is defined coordinatewise by letting \n\\begin{eqnarray*}\n        (T(\\xxx))_k                                                     \n& := &  \\min_{J\\subseteq\\{1,\\dots,d\\},\\;|J|=k} \\; \\max_{l \\in J}\\; x_l  \n\\end{eqnarray*}\nThen the coordinates of   $ T(\\xxx) $   satisfy   $ (T(\\xxx))_1 \\leq \\dots \\leq (T(\\xxx))_d $, \nand  $ \\xxx\\leq\\yyy $   implies   $ T(\\xxx) \\leq T(\\yyy) $. \nMoreover, \nthe map   $ T $   is measurable and satisfies   $ T^{-1}(\\I^d) = \\I^d $. \nThe map   $ T $   is called the \n\\emph{order transform} and associates with every random vector its \n\\emph{order statistic}. \n\n\\bigskip\nFor a distribution function   $ F : \\overline\\R^d\\to\\I $, \nlet   $ F_1,\\dots,F_d $   denote the univariate marginal distribution functions of   $ F $   and \nlet   $ F_1^\\leftarrow,\\dots,F_d^\\leftarrow $   denote the corresponding lower quantile functions. \nConsider also the maps   $ \\FFF : \\overline\\R^d\\to\\I^d $   and   $ \\FFF^\\leftarrow : \\I^d\\to\\overline\\R^d $   which are defined coordinatewise \nby letting \n$$   (\\FFF(\\xxx))_k             \n :=  F_k(x_k)                   \n\\qquad\\text{and}\\qquad          \n     (\\FFF^\\leftarrow(\\uuu))_k  \n :=  F_k^\\leftarrow(u_k)        \n$$\nFurthermore, \nlet   $ Q^F $   denote the distribution   $ \\BB(\\overline\\R^d) \\to \\I $   corresponding to   $ F $   and \nlet   $ F_T $   denote the distribution function corresponding to   $ (Q^F)_T $. \nThen we have   $ Q^{F_T} = (Q^F)_T $. \n\n\\bigskip\nFor the remainder of this section, \nconsider a fixed copula   $ C $   and let   $ H^C $   denote the distribution function extending   $ C $. \nThen we have \n\\begin{eqnarray*}\n        H^C           \n&  = &  C\\circ\\HHH^C  \n\\end{eqnarray*}\nSince the univariate marginal distribution functions of   $ H^C   $   are continuous, \nthose of   $ H^C_T := (H^C)_T $   are continuous as well. \nTherefore, \nthere exists a unique copula   $ C_T $   satisfying \n\\begin{eqnarray*}\n        H^C_T             \n&  = &  C_T\\circ\\HHH^C_T  \n\\end{eqnarray*}\nand hence   $ C_T = H^C_T \\circ (\\HHH^C_T)^\\leftarrow $. \nThe copula   $ C_T $   was introduced and studied by \nDietz et al.\\ [2016] and is called the \n\\emph{order transform} of the copula   $ C $. \n\n\\bigskip\nThe following theorem is due to \nDietz et al.\\ [2016; Theorem 5.2]: \n\n\\bigskip\n\\begin{theorem}{}\n\\label{equal.t}\nLet   $ F $   be a distribution function satisfying   $ F_1=\\ldots=F_n $.   \nIf   $ C $   is a copula for   $ F $, \nthen   $ C_T $   is a copula for   $ F_T $. \n\\end{theorem}\n\n\\bigskip\nWe complete this section with two technical results which will be needed later. \n\n\\bigskip\n\\begin{lemma}{}\n\\label{l-image}\n$ H^C_T $   satisfies   \n$ (Q^{H^C_T\\circ(\\HHH^C_T)^\\leftarrow})_{(\\HHH^C_T)^\\leftarrow} = Q^{H^C_T} $   and \n$ (Q^{C_T})_{C_T} = (Q^{H^C_T})_{H^C_T} $. \n\\end{lemma\n\n\\bigskip\n\\begin{proof}\nFor every   $ \\xxx\\in\\overline\\R^d $   we have \n\\begin{eqnarray*}\n        (Q^{H^C_T\\circ(\\HHH^C_T)^\\leftarrow})_{(\\HHH^C_T)^\\leftarrow}[[-\\iinfty,\\xxx]]                                                         \n&  = &   Q^{H^C_T\\circ(\\HHH^C_T)^\\leftarrow} \\Bigl[ \\Bigl\\{ \\uuu\\in\\I^d \\Bigm| (\\HHH^C_T)^{\\leftarrow}(\\uuu)\\leq         \\xxx  \\Bigr\\} \\Bigr]  \\\\\n&  = &   Q^{H^C_T\\circ(\\HHH^C_T)^\\leftarrow} \\Bigl[ \\Bigl\\{ \\uuu\\in\\I^d \\Bigm|                         \\uuu \\leq\\HHH^C_T(\\xxx) \\Bigr\\} \\Bigr]  \\\\[.5ex]\n&  = &  (   H^C_T\\circ(\\HHH^C_T)^\\leftarrow \\circ \\HHH^C_T )(\\xxx)                                                                             \\\\[1ex]\n&  = &      H^C_T                                           (\\xxx)                                                                             \\\\*[1ex]\n&  = &   Q^{H^C_T}                                                   [[-\\iinfty,\\xxx]]                                                         \n\\end{eqnarray*}\nThis yields the first identity. \nSince   $ C_T = H^C_T\\circ(\\HHH^C_T)^\\leftarrow $, \nthe second identity follows from the first. \n\\end{proof}\n\n\\bigskip\n\\begin{lemma}{}\n\\label{l.commute}\nLet   $ S : \\overline\\R^d\\to\\overline\\R^d $   be a measurable map satisfying   $ S(\\I^d)\\subseteq\\I^d $   and \n$$  ((\\HHH^C_T)^\\leftarrow \\circ S)(\\uuu)  =  (S \\circ (\\HHH^C_T)^\\leftarrow)(\\uuu)  $$\nfor every   $ \\uuu\\in\\I^d $. \nThen \n$$  \\int_{\\I^d}          (  C_T \\circ S)(\\uuu)\\,dQ^{  C_T}(\\uuu)  \n =  \\int_{\\overline\\R^d} (H^C_T \\circ S)(\\xxx)\\,dQ^{H^C_T}(\\xxx)  \n$\n\\end{lemma\n\n\\bigskip\n\\begin{proof}\nLemma \n\\ref{l-image} yields \n\\begin{eqnarray*}\n        \\int_{\\I^d}          (  C_T                             \\circ S)(\\uuu)\\,d Q^{  C_T}                          (\\uuu)                           \n&  = &  \\int_{\\I^d}          (H^C_T \\circ (\\HHH^C_T)^\\leftarrow \\circ S)(\\uuu)\\,d Q^{H^C_T\\circ(\\HHH^C_T)^\\leftarrow}(\\uuu)                           \\\\\n&  = &  \\int_{\\I^d}          (H^C_T \\circ S \\circ (\\HHH^C_T)^\\leftarrow)(\\uuu)\\,d Q^{H^C_T\\circ(\\HHH^C_T)^\\leftarrow}(\\uuu)                           \\\\\n&  = &  \\int_{\\overline\\R^d} (H^C_T \\circ S)(\\xxx)                            \\,d(Q^{H^C_T\\circ(\\HHH^C_T)^\\leftarrow})_{(\\HHH^C_T)^\\leftarrow}(\\xxx)  \\\\*\n&  = &  \\int_{\\overline\\R^d} (H^C_T \\circ S)(\\xxx)                            \\,d Q^{H^C_T}                                                   (\\xxx)  \n\\end{eqnarray*}\nas was to be shown. \n\\end{proof}\n\n\\bigskip\nThe previous results are remarkable since   $ C_T  $   and the restriction of   $ H^C_T $   to   $ \\I^d $   are usually distinct. \n\n\n\n\n\\section{The Order Transform and Kendall's Tau}\n\\label{ot-kendall}\n\nThroughout this section, \nwe consider a fixed copula   $ C $. \nThe discussion of Kendall's tau for the order transform   $ C_T $   of   $ C $   basically relies on the discussion of   $ [C_T,C_T] $. \n\n\\bigskip\nThe following result provides a comparison of the copulas   $ C $   and   $ C_T $   in terms of Kendall's distribution function: \n\n\\bigskip\n\\begin{theorem}{}\n\\label{t.order}\n$ C_T $   satisfies \n$ K_{C_T} \\leq K_C $. \nIn particular, \n$$  [C,C]  \\leq  [C_T,C_T]  $\n\\end{theorem\n\n\\bigskip\n\\begin{proof}\nLemma \n\\ref{l-image} yields \n$$  (Q^{C_T})_{C_T}            \n =  ( Q^{H^C_T} )_{H^C_T}      \n =  ((Q^{H^C})_T)_{H^C_T}      \n =  (Q^{H^C})_{H^C_T \\circ T}  \n$$\nand for every   $ \\uuu\\in\\I^d $   we have \n$$    C(\\uuu)\n  =   Q^{H^C} \\Bigl[ \\Bigl\\{ \\vvv\\in\\I^d \\Bigm|   \\vvv  \\leq   \\uuu  \\Bigr\\} \\Bigr]\n\\leq  Q^{H^C} \\Bigl[ \\Bigl\\{ \\vvv\\in\\I^d \\Bigm| T(\\vvv) \\leq T(\\uuu) \\Bigr\\} \\Bigr]\n  =   (H^C_T \\circ T)(\\uuu)\n$$\nFor every   $ t\\in\\I^d $, \nwe thus obtain \n\\begin{eqnarray*}\n        K_{C_T}(t)                                                                         \n&  = &  (Q^{C_T})_{C_T}[[0,t]]    \t                                                       \\\\[1ex]\n&  = &  (Q^{H^C})_{H^C_T \\circ T}[[0,t]]                                                   \\\\[.5ex]\n&  = &  Q^C \\Bigl[ \\Bigl\\{ \\uuu\\in\\I^d \\Bigm| (H^C_T \\circ T)(\\uuu) \\leq t \\Bigr\\} \\Bigr]  \\\\\n&\\leq&  Q^C \\Bigl[ \\Bigl\\{ \\uuu\\in\\I^d \\Bigm|              C (\\uuu) \\leq t \\Bigr\\} \\Bigr]  \\\\*[1ex]\n&  = &  K_C(t)                                                                             \n\\end{eqnarray*}\nwhich proves the first inequality. \nThe second inequality then follows from Lemma \n\\ref{distribution.l-Kendall-2}. \n\\end{proof}\n\n\\bigskip\nThe next result provides several useful representations of   $ [C_T,C_T] $: \n\n\\bigskip\n\\begin{theorem}{} \n\\label{t.biconvex-1}\n$ C_T $   satisfies \n\\begin{eqnarray*}\n        [C_T,C_T]                                                                           \n&  = &  \\int_{\\I^d} Q^C \\Bigl[                                                              \n        \\Bigl\\{ \\vvv\\in\\I^d \\Bigm| T(\\vvv) \\leq        T(\\uuu) \\Bigr\\} \\Bigr] \\,dQ^C(\\uuu)  \\\\\n&  = &  \\int_{\\I^d} Q^C\\biggl[ \\bigcup\\nolimits_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi}              \n        \\Bigl\\{ \\vvv\\in\\I^d \\Bigm|   \\vvv  \\leq\\tilde\\pi(\\uuu) \\Bigr\\} \\biggr]\\,dQ^C(\\uuu)  \\\\*\n&  = &  \\int_{\\I^d} H^C_T(T(\\uuu))                                            \\,dQ^C(\\uuu)  \n\\end{eqnarray*\n\\end{theorem\n\n\\bigskip\n\\begin{proof}\nLemma \n\\ref{l-image} yields \n\\begin{eqnarray*}\n        [C_T,C_T]                                                                                                                        \n&  = &  \\int_{\\I^d}               C_T(\\uuu)                                                                          \\,dQ^{  C_T}(\\uuu)  \\\\\n&  = &  \\int_{\\overline\\R^d}    H^C_T(\\xxx)                                                                          \\,dQ^{H^C_T}(\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d} Q^{H^C_T}\\Bigl[ \\Bigl\\{ \\sss\\in\\overline\\R^d \\Bigm|   \\sss  \\leq   \\xxx  \\Bigr\\} \\Bigr] \\,dQ^{H^C_T}(\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d} Q^{H^C}  \\Bigl[ \\Bigl\\{ \\yyy\\in\\overline\\R^d \\Bigm| T(\\yyy) \\leq   \\xxx  \\Bigr\\} \\Bigr] \\,dQ^{H^C_T}(\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d} Q^{H^C}  \\Bigl[ \\Bigl\\{ \\yyy\\in\\overline\\R^d \\Bigm| T(\\yyy) \\leq T(\\zzz) \\Bigr\\} \\Bigr] \\,dQ^{H^C}  (\\zzz)  \\\\*\n&  = &  \\int_{\\I^d}          Q^C      \\Bigl[ \\Bigl\\{ \\vvv\\in         \\I^d \\Bigm| T(\\vvv) \\leq T(\\uuu) \\Bigr\\} \\Bigr] \\,dQ^C      (\\uuu)  \n\\end{eqnarray*}\nMoreover, \nfor any   $ \\uuu,\\vvv\\in\\I^d $   there exist some   $ \\tilde\\pi_\\uuu, \\tilde\\pi_\\vvv \\in \\tilde\\Gamma^\\pi $   such that   \n$ T(\\uuu) = \\tilde\\pi_\\uuu(\\uuu) $   and \n$ T(\\vvv) = \\tilde\\pi_\\vvv(\\vvv) $.  \nSince   $ \\tilde\\Gamma^\\pi $   is a group, \nthis yields   $ T(\\vvv) \\leq T(\\uuu) $   if and only if there exists some   $ \\tilde\\pi\\in\\tilde\\Gamma^\\pi $   such that   $ \\vvv\\leq\\tilde\\pi(\\uuu) $. \nThis yields \n\\begin{eqnarray*}\n        [C_T,C_T]                                                                            \n&  = &  \\int_{\\I^d} Q^C\\Bigl[                                                                \n        \\Bigl\\{ \\vvv\\in\\I^d \\Bigm| T(\\vvv) \\leq         T(\\uuu) \\Bigr\\} \\Bigr] \\,dQ^C(\\uuu)  \\\\*\n&  = &  \\int_{\\I^d} Q^C\\biggl[ \\bigcup\\nolimits_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi}               \n        \\Bigl\\{ \\vvv\\in\\I^d \\Bigm|   \\vvv  \\leq \\tilde\\pi(\\uuu) \\Bigr\\} \\biggr]\\,dQ^C(\\uuu)  \n\\end{eqnarray*}\nFinally, \nwe have \n\\begin{eqnarray*}\n        [C_T,C_T]                                                                                               \n&  = &  \\int_{\\I^d} Q^C      \\Bigl[ \\Bigl\\{ \\vvv\\in\\I^d \\Bigm| T(\\vvv) \\leq T(\\uuu) \\Bigr\\} \\Bigr] \\,dQ^C(\\uuu)  \\\\\n&  = &  \\int_{\\I^d} Q^{H^C_T}\\Bigl[ \\Bigl\\{ \\www\\in\\I^d \\Bigm|   \\www  \\leq T(\\uuu) \\Bigr\\} \\Bigr] \\,dQ^C(\\uuu)  \\\\*\n&  = &  \\int_{\\I^d}    H^C_T (T(\\uuu))                                                             \\,dQ^C(\\uuu)  \n\\end{eqnarray*}\nwhich completes the proof. \n\\end{proof}\n\n\\bigskip\n\\begin{corollary}{}\n\\label{c.biconvex}\n\\begin{onelist}\n\\item   If   $ [C_T,C_T] = 0 $, \nthen $ [C,C] = 0 $.\n\\item   If   $ C $   is symmetric with   $ [C,C] = 0 $, \nthen   $ [C_T,C_T] = 0 $. \n\\end{onelist}\n\\end{corollary}\n\n\\bigskip\n\\begin{proof}\nBecause of Theorem \n\\ref{t.order} we have   $ [C,C] \\leq [C_T,C_T] $,   \nwhich yields (1). \nFrom Theorem \n\\ref{t.biconvex-1} we obtain \n\\begin{eqnarray*}\n        [C_T,C_T]                                                                                                                                              \n&  = &  \\int_{\\I^d} Q^C\\biggl[ \\bigcup\\nolimits_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} \\Bigl\\{ \\vvv\\in\\I^d \\Bigm| \\vvv\\leq\\tilde\\pi(\\uuu) \\Bigr\\} \\biggr]\\,dQ^C(\\uuu)  \\\\\n&\\leq&  \\sum_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} \\int_{\\I^d} Q^C\\Bigl[              \\Bigl\\{ \\vvv\\in\\I^d \\Bigm| \\vvv\\leq\\tilde\\pi(\\uuu) \\Bigr\\} \\Bigr ]\\,dQ^C(\\uuu)  \\\\\n&  = &  \\sum_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} \\int_{\\I^d}  (C \\circ \\tilde\\pi)(\\uuu)                                                               \\,dQ^C(\\uuu)  \\\\\n&  = &  \\sum_{      \\pi\\in      \\Gamma^\\pi} \\int_{\\I^d}  (\\pi(C))(\\uuu)                                                                          \\,dQ^C(\\uuu)  \\\\*\n&  = &  \\sum_{      \\pi\\in      \\Gamma^\\pi} [\\pi(C),C]                                                                                                         \n\\end{eqnarray*}\nIf   $ C $   is symmetric, \nthen the previous inequality becomes   $ [C_T,C_T] \\leq d!\\,[C,C] $   since   $ |\\Gamma^\\pi| = d! $\\,. \nThis yields (2). \n\\end{proof}\n\n\\bigskip\nThe following result resumes Theorem \n\\ref{t.order} and Corollary \n\\ref{c.biconvex} in terms of Kendall's tau: \n\n\\bigskip\n\\begin{theorem}{}\n\\label{t.Kendall}\nKendall's tau satisfies \n$$  \\kappa[C]  \\leq  \\kappa[C_T]  $$\nIn particular: \n\\begin{onelist}\n\\item   If   $ C_T $   minimizes Kendall's tau, \nthen   $ C $   minimizes Kendall's tau as well. \n\\item   If   $ C $   is symmetric and minimizes Kendall's tau, \nthen   $ C_T $   minimizes Kendall's tau as well. \n\\end{onelist}\n\\end{theorem}\n\\pagebreak\n\n\\bigskip\nWe note in passing that Theorem \n\\ref{t.Kendall} yields two results due to \nDietz et al.\\ [2016; Examples 4.2 and 4.3]: \n\n\\bigskip\n\\begin{corollary}{}\n\\label{c.bounds}\nThe upper Fr{\\'e}chet--Hoeffding bound satisfies   $ M_T=M $, \nand in the bivariate case \nthe lower Fr{\\'e}chet--Hoeffding bound satisfies   $ W_T=W $. \n\\end{corollary}\n\n\\bigskip\n\\begin{proof}\nIt has been shown by \nFuchs et al.\\ [2018; Theorems 3.2 and 3.3] \nthat the upper Fr{\\'e}chet--Hoeffding bound is the only copula maximizing Kendall's tau and \nthat in the bivariate case the lower Fr{\\'e}chet--Hoeffding bound is the only copula minimizing Kendall's tau. \nThe assertion follows. \n\\end{proof}\n\n\\bigskip\nTheorem \n\\ref{t.Kendall} provides conditions under which    $ \\kappa[C] = \\kappa[C_T] $. \nBelow we shall show that the product copula satisfies   $ \\kappa[\\Pi] < \\kappa[\\Pi_T] $; \nsee Theorem \n\\ref{t.product}. \n\n\\bigskip\nThe following example provides a copula   $ D $   for which   $ D \\not\\leq_c D_T $; \nit thus shows that Theorem \n\\ref{t.Kendall} cannot be obtained from the fact that Kendall's tau is monotone with respect to the concordance order   $ \\leq_c $: \n\n\\bigskip\n\\begin{example}{}\n\\label{e.bounds}\nAssume that   $ d=2 $   and consider the copula \n\\begin{eqnarray*}\n        D              \n& := &  \\frac{M+W}{2}  \n\\end{eqnarray*}\nLet   $ F_{(0,1)} $   denote the distribution function corresponding to the uniform distribution on   $ (0,1) $. \nAccording to \nDietz et al.\\ [2016; Examples 4.2 and 4.3], \nwe have \n\\begin{eqnarray*}\n        H^D_T(\\xxx)                                                                              \n&  = &  \\frac{1}{2}\\,H^M_T(\\xxx) + \\frac{1}{2}\\,H^W_T(\\xxx)                                      \\\\*\n&  = &  \\frac{1}{2} \\, F_{(0,1)}(\\min\\{x_1,x_2\\})                                                \n      + \\frac{1}{2} \\, \\Bigl( F_{(0,1)}(2x_1) + \\Bigl( 2\\,F_{(0,1)}(x_2)-1 \\Bigr)^+ -1 \\Bigr)^+  \n\\end{eqnarray*}\nand hence \n\\begin{eqnarray*}\n        \\HHH^D_T(\\xxx)                                                                         \n&  = &  \\leftmatrix{c}                                                                         \n\t\t \\dps \\frac{1}{2}\\,F_{(0,1)} (x_1) + \\frac{1}{2}\\,F_{(0,1)}(2x_1)                      \\\\[2ex]\n\t\t \\dps \\frac{1}{2}\\,F_{(0,1)} (x_2) + \\frac{1}{2}\\,\\Bigl( 2\\,F_{(0,1)}(x_2)-1 \\Bigr)^+  \n\t\t\\rightmatrix                                                                           \n\\end{eqnarray*}\nSince the univariate marginal distribution functions of   $ H^D_T $   are continuous and strictly increasing on   $ (0,1) $, \nwe obtain \n$$  D                             \n    \\leftmatrix{c}                \n       3\/8                        \\\\\n       4\/8                        \n      \\rightmatrix                \n =  \\frac{3}{16}                  \n >  \\frac{2}{16}                  \n =  (H^D_T\\circ(\\HHH^D_T )^{-1})  \n    \\leftmatrix{c}                \n     3\/8                          \\\\\n     4\/8                          \n    \\rightmatrix                  \n =  D_T                           \n    \\leftmatrix{c}                \n     3\/8                          \\\\\n     4\/8                          \n    \\rightmatrix   \n$$\nand hence   $ D \\not\\leq D_T $. \nSince   $ d=2 $, \nthe concordance order agrees with the pointwise order and we obtain   $ D \\not\\leq_c D_T $. \n\\end{example}\n\n\\bigskip\nThe following example shows that the map   $ C \\mapsto C_T $   is not order preserving with respect to the concordance order: \n\n\\bigskip\n\\begin{example}{}\n\\label{e.order}\nAssume that   $ d=2 $. \nFor every symmetric copula   $ C : \\I^2\\to\\I $, \nTheorem \n\\ref{t.biconvex-1} yields \n\\begin{eqnarray*}\n        [C_T,C_T]                                                                 \n&  = &  2\\,[C,C] - \\int_{\\I^2} C(\\min\\{u_1,u_2\\},\\min\\{u_1,u_2\\})\\,dQ^C(u_1,u_2)  \n\\end{eqnarray*}\nand for the computation of   $ [C,C] $   in the case where   $ C $   is a shuffle we refer to \nFuchs et al.\\ [2018; Theorem 5.1]. \nThe copula   $ A: \\I^2\\to\\I $   \n\\begin{center}\n\\begin{tikzpicture}[xscale=0.8,yscale=0.8] \n\\draw [thin]  (0,0) -- (0,4);\n\\draw [thin]  (1,0) -- (1,4);\n\\draw [thin]  (2,0) -- (2,4);\n\\draw [thin]  (3,0) -- (3,4);\n\\draw [thin]  (4,0) -- (4,4);\n\\draw [thin]  (0,0) -- (4,0);\n\\draw [thin]  (0,1) -- (4,1);\n\\draw [thin]  (0,2) -- (4,2);\n\\draw [thin]  (0,3) -- (4,3);\n\\draw [thin]  (0,4) -- (4,4);\n\\draw [thick] (0,2) -- (2,4);\n\\draw [thick] (2,0) -- (4,2);\n\\end{tikzpicture}\n\\end{center}\ndefined as the shuffle of   $ M $   with respect to the shuffling structure   $ \\{[\\aaa_i,\\bbb_i]\\}_{i\\in\\{1,\\dots,4\\}} $   with \n$$\\begin{array}{ccccccc}\n \\aaa_1 &=& (\\phantom{00}0,2\/4)  &&  \\bbb_1 &=& (1\/4,          3\/4)  \\\\\n \\aaa_2 &=& (1\/4          ,3\/4)  &&  \\bbb_2 &=& (2\/4,\\phantom{00}1)  \\\\\n \\aaa_3 &=& (2\/4,\\phantom{00}0)  &&  \\bbb_3 &=& (3\/4          ,1\/4)  \\\\\n \\aaa_4 &=& (3\/4,          1\/4)  &&  \\bbb_4 &=& (\\phantom{00}1,2\/4)  \n\\end{array}$$\nis symmetric and satisfies   $ [A_T,A_T] = 1\/2 $,   \nand the copula   $ B : \\I^2\\to\\I $   \n\\begin{center}\n\\begin{tikzpicture}[xscale=0.8,yscale=0.8] \n\\draw [thin]  (0,0) -- (0,4);\n\\draw [thin]  (1,0) -- (1,4);\n\\draw [thin]  (2,0) -- (2,4);\n\\draw [thin]  (3,0) -- (3,4);\n\\draw [thin]  (4,0) -- (4,4);\n\\draw [thin]  (0,0) -- (4,0);\n\\draw [thin]  (0,1) -- (4,1);\n\\draw [thin]  (0,2) -- (4,2);\n\\draw [thin]  (0,3) -- (4,3);\n\\draw [thin]  (0,4) -- (4,4);\n\\draw [thick] (0,0) -- (1,1);\n\\draw [thick] (1,3) -- (2,4);\n\\draw [thick] (2,2) -- (3,3);\n\\draw [thick] (3,1) -- (4,2);\n\\end{tikzpicture}\n\\end{center}\ndefined as the shuffle of   $ M $   with \n$$\\begin{array}{ccccccc}\n \\aaa_1 &=& (\\phantom{00}0,\\phantom{00}0)  &&  \\bbb_1 &=& (1\/4,          1\/4)  \\\\\n \\aaa_2 &=& (1\/4          ,          3\/4)  &&  \\bbb_2 &=& (2\/4,\\phantom{00}1)  \\\\\n \\aaa_3 &=& (2\/4          ,          2\/4)  &&  \\bbb_3 &=& (3\/4          ,3\/4)  \\\\\n \\aaa_4 &=& (3\/4          ,          1\/4)  &&  \\bbb_4 &=& (\\phantom{00}1,2\/4)  \n\\end{array}$$\nis symmetric as well and satisfies   $ [B_T,B_T] = 3\/8 $. \n\\\\\nIt it obvious that   $ A \\leq B $. \nOn the other hand, \nwe have   $ \\kappa[A_T] = 1 $, \nand it then follows from \nFuchs et al.\\ [2018; Theorem 3.2] \nthat   $ A_T = M $.\nThis yields   $ B_T \\leq M = A_T $, \nand from   $ [B_T,B_T] < [A_T,A_T] $   we obtain   $ A_T \\neq B_T $   and hence   $ A_T \\not\\leq B_T $. \nSince   $ d=2 $, \nthis yields   $ A \\leq_c B $   and   $ A_T \\not\\leq_c B_T $. \n\\end{example}\n\n\\bigskip\nReturning to the case of arbitrary dimension   $ d $, \nwe conclude this section with another representation of   $ [C_T,C_T] $   for a class of copulas \nwhich includes every copula whose copula measure is absolutely continuous with respect to Lebesgue measure. \nThis class includes the product copula, \nwhich will be studied in the following section. \n\n\\bigskip\nFor   $ \\tilde\\pi\\in\\tilde\\Gamma^\\pi $, \ndefine\n\\begin{eqnarray*}\n        A_{\\tilde\\pi}                                                                                                                       \n& := &  \\Bigl\\{ \\uuu \\in \\I^d \\Bigm|  (\\tilde\\pi(\\uuu))_i < (\\tilde\\pi(\\uuu))_{i+1} \\;\\text{\\rm for all}\\; i\\in\\{1,\\dots,d\\!-\\!1\\} \\Bigr\\}  \n\\end{eqnarray*}\nThen every    $ \\uuu \\in A_{\\tilde\\pi} $   satisfies   $ T(\\uuu) = \\tilde\\pi(\\uuu) $    \nand the family   $ \\fa{A}{{\\tilde\\pi}}{{\\tilde\\Gamma^\\pi}} $   is disjoint. \n\n\\bigskip\n\\begin{lemma}{} \n\\label{l-biconvex-2}\nAssume that   $ C $   satisfies   $ Q^C[ \\sum_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} A_{\\tilde\\pi} ] = 1 $.\nThen \n$$  [C_T,C_T]  =  \\sum_{\\pi\\in\\Gamma^\\pi} \\int_{T(\\I^d)} H^C_T (\\uuu) \\,dQ^{\\pi(C)}(\\uuu)  $$\nIn particular, \nif   $ C $   is also symmetric, \nthen \n$$  [C_T,C_T]  = d! \\int_{T(\\I^d)} H^C_T(\\uuu) \\,dQ^C(\\uuu)  $\n\\end{lemma\n\n\\bigskip\n\\begin{proof}{}\nSince   $ Q^C[ \\sum_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} A_{\\tilde\\pi} ] = 1 $, \nTheorem \n\\ref{t.biconvex-1} yields \n\\begin{eqnarray*} \n        [C_T,C_T]                                                                                                                 \n&  = &  \\int_{\\I^d}                                                                   H^C_T(        T(\\uuu)) \\,dQ^C       (\\uuu)  \\\\\n&  = &  \\int_{\\I^d} \\sum_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} \\chi_{A_{\\tilde\\pi}}(\\uuu) \\, H^C_T(        T(\\uuu)) \\,dQ^C       (\\uuu)  \\\\\n&  = &  \\sum_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} \\int_{\\I^d} \\chi_{A_{\\tilde\\pi}}(\\uuu) \\, H^C_T(\\tilde\\pi(\\uuu)) \\,dQ^C       (\\uuu)  \\\\*\n&  = &  \\sum_{\\pi\\in\\Gamma^\\pi} \\int_{T(\\I^d)}                                        H^C_T(          \\vvv ) \\,dQ^{\\pi(C)}(\\vvv)  \n\\end{eqnarray*}\nThis proves the assertion.\n\\end{proof}\n\n\n\n\n\\section{The Order Transform of the Product Copula}\n\\label{ot-product}\n\nIn the present section we determine Kendall's tau for the order transform   $ \\Pi_T $   of the product copula   $ \\Pi $   \nand for certain marginals of the order transform. \nTo this end, \nwe first recall a formula for the distribution function   $ H^\\Pi_T $; \nsee Dietz et al.\\ [2016; Example 5.3]: \n\n\\bigskip\n\\begin{lemma}{}\n\\label{l.product-identity}\nThe product copula   $ \\Pi $   satisfies \n$$  H^\\Pi_T(\\uuu)  =  d!\\,\\det\\Bigl[ (a_{i,j}(\\uuu))_{i,j\\in\\{1,\\dots,d\\}} \\Bigr]  $$\nfor every   $ \\uuu \\in T(\\I^d) $, \nwhere \n$$  a_{i,j}(\\uuu)                                                               \n:=  \\begin{cases}                                                               \n     \\dps \\frac{u_i^{j-i+1}}{(j\\!-\\!i\\!+\\!1)!}  &   \\text{if   $ i \\leq j+1 $}  \\\\[2ex]\n     0                                          &   \\text{else}                 \n    \\end{cases}                                                                 \n$$\nfor all   $ i,j\\in\\{1,\\dots,d\\} $. \n\\end{lemma\n\n\\bigskip\nWe also recall that the copula measure of the product copula is the Lebesgue measure, \nwhich means that Lemma \n\\ref{l-biconvex-2} applies to the product copula. \n\n\\bigskip\n\\begin{theorem}{}\n\\label{t.product}\nThe product copula satisfies  \n$$  [\\Pi,\\Pi]      \n =  \\frac{1}{2^d}  \n <  \\frac{1}{d+1}  \n =  [\\Pi_T,\\Pi_T]  \n$$\nand hence \n$$  \\kappa[\\Pi]                               \n =  0                                         \n <  \\frac{2^d - (d\\!+\\!1)}{(2^{d-1}-1)(d+1)}  \n =  \\kappa[\\Pi_T]                             \n$$\nIn particular, \n$ \\lim_{d\\to\\infty} \\kappa[\\Pi_T] = 0 $. \n\\end{theorem}\n\n\\bigskip\n\\begin{proof}{}\nSince the product copula is absolutely continuous and symmetric, \nLemma \n\\ref{l-biconvex-2} together with Lemma \n\\ref{l.product-identity} and Lemma \n\\ref{l.app-1} yields \n\\begin{eqnarray*} \n        [\\Pi_T,\\Pi_T]                                                                                    \n&  = &  d! \\int_{T(\\I^d)} H^\\Pi_T(\\uuu)                                                 \\,dQ^\\Pi (\\uuu)  \\\\\n&  = &  d! \\int_{T(\\I^d)} d! \\, \\det\\Bigl[ (a_{i,j}(\\uuu))_{i,j\\in\\{1,\\dots,d\\}} \\Bigr] \\,d\\leb^d(\\uuu)  \\\\\n&  = &  d!\\,d!\\,\\frac{1}{d!\\,(d+1)!}                                                                     \\\\*\n&  = &  \\frac{1}{d+1}                                                                                    \n\\end{eqnarray*}\nThe identity for   $ [\\Pi,\\Pi] $   is well known. \n\\end{proof}\n\n\\bigskip\nSimilar but more complicated identities can be obtained for certain margins of the order transform of the product copula. \nIn the sequel, \nwe use the canonical extensions \nof transformations of   $ \\I^d $   \nto transformations of   $ \\overline\\R^d $   without changing the notation. \nWe also need the following definitions: \n\n\\bigskip\nConsider   $ K\\subseteq\\{1,\\dots,d\\} $   such that   $ |K|\\geq2 $.   \nThen there exists a unique strictly increasing sequence   $ \\fa{k}{j}{\\{1,\\dots,|K|\\}} \\subseteq \\{1,\\dots,d\\} $   \nsuch that   $ K = \\fa{k}{j}{\\{1,\\dots,|K|\\}} $   and we denote \n\\begin{hylist}\n\\item   by   $ \\overline\\R^K $   a copy of   $ \\overline\\R^{|K|} $   with coordinates   $ k_1,\\dots,k_{|K|} $   instead of   $ 1,\\dots,|K| $, \n\\item   by   $ \\I^K $   the unit cube of   $ \\overline\\R^K $,   and \n\\item   by   $ \\CC^K $   the collection of all copulas   $ \\I^K\\to\\I $. \n\\end{hylist}\nWe also define a map   $ \\tilde\\varrho_K : \\I^K\\to\\I^d $   by letting \n\\begin{eqnarray*}\n        \\tilde\\varrho_K(\\vvv)                                                                         \n& := &  \\sum_{j \\in \\{1,\\dots,|K|\\}} v_j\\,\\eee_{k_j} + \\sum_{k \\in \\{1,\\dots,d\\} \\setminus K} \\eee_k  \n\\end{eqnarray*}\nand the map   $ \\varrho_K : \\CC^d\\to\\CC^K $   given by \n\\begin{eqnarray*}\n        \\varrho_K(C)             \n& := &  C \\circ \\tilde\\varrho_K  \n\\end{eqnarray*}\nThen   $ \\varrho_K $   associates indeed with every copula in   $ \\CC^d $   a copula in   $ \\CC^K $, \nwhich is called the \n\\emph{margin} of   $ C $   with respect to   $ K $. \nWe have the following result: \n\n\\bigskip\n\\begin{theorem}{}\n\\label{t.product-K}\n$ \\varrho_K(\\Pi_T) $   satisfies \n\\begin{eqnarray*}\n        \\Bigl[ \\varrho_K(\\Pi_T),\\varrho_K(\\Pi_T) \\Bigr]                            \n&  = &     \\int_{\\I^d}            \\Pi_T(\\eeta_K(\\eins,\\uuu)) \\,dQ^{  \\Pi_T}(\\uuu)  \\\\*\n&  = &     \\int_{\\overline\\R^d} H^\\Pi_T(\\eeta_K(\\eins,\\xxx)) \\,dQ^{H^\\Pi_T}(\\xxx)  \\\\*\n&  = &  d! \\int_{T(\\I^d)}       H^\\Pi_T(\\eeta_K(\\eins,\\uuu)) \\,dQ^{  \\Pi  }(\\uuu)  \n\\end{eqnarray*\n\\end{theorem\n\n\\bigskip\n\\begin{proof}\nConsider the map   $ \\eeta_{K,\\eins} : \\I^d\\to\\I^d $   given by   $ \\eeta_{K,\\eins}(\\uuu) := \\eeta_K(\\eins,\\uuu) $. \nThen we obtain \n\\begin{eqnarray*}\n        \\Bigl[ \\varrho_K(\\Pi_T),\\varrho_K(\\Pi_T) \\Bigr]                                                                                               \n&  = &  \\Bigl[ \\Pi_T\\circ\\tilde\\varrho_K,\\Pi_T\\circ\\tilde\\varrho_K \\Bigr]                                                                             \\\\[.5ex]\n&  = &  \\int_{\\I^K}                                        \\Pi_T(\\tilde\\varrho_K(\\vvv)) \\,d Q^{\\Pi_T \\circ \\tilde\\varrho_K}                   (\\vvv)  \\\\\n&  = &  \\int_{\\tilde\\varrho_K^{-1}(\\eeta_{K,\\eins}(\\I^d))} \\Pi_T(\\tilde\\varrho_K(\\vvv)) \\,d Q^{\\Pi_T \\circ \\tilde\\varrho_K}                   (\\vvv)  \\\\\n&  = &  \\int_{\\eeta_{K,\\eins}(\\I^d)}                       \\Pi_T(\\uuu)                  \\,d(Q^{\\Pi_T \\circ \\tilde\\varrho_K})_{\\tilde\\varrho_K}(\\uuu)  \\\\\n&  = &  \\int_{\\eeta_{K,\\eins}(\\I^d)}                       \\Pi_T(\\uuu)                  \\,d Q^{\\Pi_T}                                         (\\uuu)  \\\\\n&  = &  \\int_{\\eeta_{K,\\eins}(\\I^d)}                       \\Pi_T(\\uuu)                  \\,d(Q^{\\Pi_T})_{\\eeta_{K,\\eins}}                      (\\uuu)  \\\\\n&  = &  \\int_{\\I^d}                                        \\Pi_T(\\eeta_{K,\\eins}(\\uuu)) \\,d Q^{\\Pi_T}                                         (\\uuu)  \\\\*\n&  = &  \\int_{\\I^d}                                        \\Pi_T(\\eeta_K (\\eins, \\uuu)) \\,d Q^{\\Pi_T}                                         (\\uuu)  \n\\end{eqnarray*}\nwhich gives the first identity. \nThe second identity then follows from the first identity and Lemma \n\\ref{l.commute}. \nFurthermore, \nwe have \n\\begin{eqnarray*}\n        \\Bigl[ \\varrho_K(\\Pi_T),\\varrho_K(\\Pi_T) \\Bigr]                                                                        \n&  = &  \\int_{\\overline\\R^d}                                     H^\\Pi_T(\\eeta_K(\\eins,  \\xxx ))         \\,dQ^{H^\\Pi_T}(\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d}                                     H^\\Pi_T(\\eeta_K(\\eins,T(\\xxx)))         \\,dQ^{H^\\Pi}  (\\xxx)  \\\\\n&  = &  \\int_{\\I^d}                                              H^\\Pi_T(\\eeta_K(\\eins,T(\\uuu)))                               \n        \\Biggl( \\sum_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} \\chi_{A_{\\tilde\\pi}}(\\uuu) \\Biggr)                   \\,dQ^\\Pi      (\\uuu)  \\\\\n&  = &  \\sum_{\\tilde\\pi\\in\\tilde\\Gamma^\\pi} \\int_{A_{\\tilde\\pi}} H^\\Pi_T(\\eeta_K(\\eins,\\tilde\\pi(\\uuu))) \\,dQ^\\Pi      (\\uuu)  \\\\*\n&  = &  d! \\int_{T(\\I^d)}                                        H^\\Pi_T(\\eeta_K(\\eins,          \\uuu )) \\,dQ^\\Pi      (\\uuu)  \n\\end{eqnarray*}\nwhich gives the last identity. \n\\end{proof}\n\n\\bigskip\nFrom the previous result, \ncombined with Lemma \n\\ref{l.product-identity}, \nwe obtain explicit formulas for   $ [\\varrho_K(\\Pi_T),\\varrho_K(\\Pi_T)] $,   \nand hence for   $ \\kappa[\\varrho_K(\\Pi_T)] $,   \nin the case   $ K=\\{1,\\dots,k\\} $: \n\n\\bigskip\n\\begin{corollary}{}\n\\label{c.product-lowertail}\n$ \\varrho_{\\{1,\\dots,k\\}}(\\Pi_T) $   satisfies \n\\begin{eqnarray*}\n        \\Bigl[ \\varrho_{\\{1,\\dots,k\\}}(\\Pi_T),\\varrho_{\\{1,\\dots,k\\}}(\\Pi_T) \\Bigr]                                        \n&  = &  \\frac{1}{2} - \\frac{1}{4} \\sum_{h=2}^k \\frac{1}{2h-1}\\, \\binom{2h}{h} \\binom{2d+2-2h}{d+1-h} \\bigg\/ \\binom{2d}{d}  \\\\*\n&  = &  \\frac{1}{d+1} + \\frac{1}{2d} \\sum_{l=1}^{d-k} \\binom{d}{l-1} \\binom{d}{l} \\bigg\/ \\binom{2d-1}{2l-1}                \n\\end{eqnarray*}\nfor every   $ k\\in\\{2,\\dots,d\\} $. \nIn particular, \n\\begin{eqnarray*}\n        \\kappa[\\varrho_{\\{1,\\dots,k\\}}(\\Pi_T)]                                                                                 \n&  = &  1 - \\frac{2^{k-2}}{2^{k-1}-1} \\sum_{h=2}^k \\frac{1}{2h-1}\\, \\binom{2h}{h} \\binom{2d+2-2h}{d+1-h} \\bigg\/ \\binom{2d}{d}  \n\\end{eqnarray*}\nholds for every   $ k\\in\\{2,\\dots,d\\} $, \nthe sequence   $ \\{ \\kappa[\\varrho_{\\{1,\\dots,k\\}}(\\Pi_T)] \\}_{k\\in\\{1,\\dots,d\\}} $   is decreasing with \n\\begin{eqnarray*}\n        \\kappa[\\varrho_{\\{1,\\dots,d\\}}(\\Pi_T)]  \n&  = &  \\frac{1}{d+1}                           \n\\end{eqnarray*} \nand for every   $ k\\in\\{2,3,\\dots\\} $   the sequence   $ \\{ \\kappa[\\varrho_{\\{1,\\dots,k\\}}(\\Pi_T)] \\}_{d\\in\\{k,k+1,\\dots\\}} $   is increasing with \n\\begin{eqnarray*}\n        \\lim_{d\\to\\infty} \\kappa[\\varrho_{\\{1,\\dots,k\\}}(\\Pi_T)]                                        \n&  = &  1 - \\frac{2^{k-2}}{2^{k-1}-1} \\sum_{h=2}^k \\frac{1}{2h-1}\\, \\binom{2h}{h} \\,\\frac{1}{2^{2h-2}}  \n\\end{eqnarray*\n\\end{corollary\n\n\\bigskip\n\\begin{proof}\nSince   $ \\eeta_{\\{1,\\dots,k\\}}(\\eins,\\uuu) \\in T(\\I^d) $   holds for every   $ \\uuu \\in T(\\I^d) $, \nTheorem \n\\ref{t.product-K} together with Lemma \n\\ref{l.product-identity} and Corollary \n\\ref{c.app-1} yields \n\\begin{eqnarray*}\n        \\Bigl[ \\varrho_{\\{1,\\dots,k\\}}(\\Pi_T),\\varrho_{\\{1,\\dots,k\\}}(\\Pi_T) \\Bigr]                                                                           \n&  = &  d! \\int_{T(\\I^d)}                                        H^\\Pi_T(\\eeta_{\\{1,\\dots,k\\}}(\\eins,\\uuu))                             \\,dQ^\\Pi (\\uuu)       \\\\\n&  = &  d! \\int_{T(\\I^d)} d!                       \\,\\det\\Bigl[ (a_{i,j}(\\eeta_{\\{1,\\dots,k\\}}(\\eins,\\uuu)))_{i,j\\in\\{1,...,d\\}} \\Bigr] \\,d\\leb^d(\\uuu)       \\\\\n&  = &  (d!)^2 \\biggl( \\frac{1}{d!\\,(d\\!+\\!1)!} + \\frac{1}{(2d)!} \\sum_{l=1}^{d-k} \\frac{(2d\\!-\\!2l)!}{(d\\!-\\!l)!\\,(d\\!+\\!1\\!-\\!l)!} \\binom{2l-1}{l} \\biggr)  \\\\*\n&  = &  \\frac{1}{d+1}                      + \\frac{1}{2d} \\sum_{l=1}^{d-k} \\binom{d}{l-1} \\binom{d}{l} \\bigg\/ \\binom{2d-1}{2l-1}                              \n\\end{eqnarray*}\nwhich gives the second identity. \nOn the other hand, \nLemma \n\\ref{l.combi} yields \n\\begin{eqnarray*} \n\t    \\sum_{l=1}^{d-k} \\frac{(2d-2l)!}{(d-l)!\\,(d+1-l)!}\\,\\frac{(2l-1)!}{l! (l-1)!}             \n&  = &  \\frac{1}{4} \\sum_{l=1}^{d-k} \\binom{2d+2-2l}{d+1-l} \\binom{2l}{l} \\frac{1}{2d-2l+1}       \\\\\n&  = &  \\frac{1}{4} \\sum_{h=k+1}^d   \\binom{2h}{h} \\binom{2d+2-2h}{d+1-h} \\frac{1}{2h-1}   \t      \\\\\n&  = &  \\frac{1}{4} \\biggl(                      - \\binom{2d+2}{d+1}      \\frac{1}{2d+1}          \n                  - \\sum_{h=0}^k     \\binom{2h}{h} \\binom{2d+2-2h}{d+1-h} \\frac{1}{2h-1} \\biggr)  \\\\*\n&  = &  \\frac{1}{4} \\biggl( \\binom{2d}{d} \\frac{2(d-1)}{d+1}                                      \n\t\t\t\t  - \\sum_{h=2}^k     \\binom{2h}{h} \\binom{2d+2-2h}{d+1-h} \\frac{1}{2h-1} \\biggr)  \n\\end{eqnarray*}\nand hence \n\\begin{eqnarray*}  \n        \\Bigl[ \\varrho_{\\{1,\\dots,k\\}}(\\Pi_T),\\varrho_{\\{1,\\dots,k\\}}(\\Pi_T) \\Bigr]                              \n&  = &  \\frac{1}{d+1}                                                                                            \n      + \\frac{(d!)^2}{(2d)!} \\sum_{l=1}^{d-k} \\frac{(2d\\!-\\!2l)!}{(d\\!-\\!l)!\\,(d\\!+\\!1\\!-\\!l)!} \\binom{2l-1}{l}  \\\\\n&  = &  \\frac{1}{d+1}                                                                                            \n      + \\frac{1}{2}\\,\\frac{d-1}{d+1}                                                                             \n      - \\frac{1}{4}\\,\\frac{(d!)^2}{(2d)!} \\sum_{h=2}^k \\binom{2h}{h} \\binom{2d+2-2h}{d+1-h} \\frac{1}{2h-1}       \\\\*\n&  = &  \\frac{1}{2}                                                                                              \n      - \\frac{1}{4} \\sum_{h=2}^k \\frac{1}{2h-1}\\, \\binom{2h}{h} \\binom{2d+2-2h}{d+1-h} \\bigg\/ \\binom{2d}{d}      \n\\end{eqnarray*}\nwhich gives the first identity. \n\\end{proof}\n\n\\bigskip\nThe first identity of Corollary \n\\ref{c.product-lowertail} is suitable for small   $ k $   while the second is suitable for large  $ k $. \nFor   $ k=2 $, \nCorollary \n\\ref{c.product-lowertail} yields \n\\begin{eqnarray*}\n        \\kappa[\\varrho_{\\{1,2\\}}(\\Pi_T)]  \n&  = &  \\frac{d-1}{2d-1}                  \n\\end{eqnarray*}\nwhich is in accordance with the literature; \nsee \nAv{\\'e}rous et al.\\ [2005] and \nNavarro and Balakrishnan [2010]. \nIn particular, \nthe sequence   $ \\{ \\kappa[\\varrho_{\\{1,2\\}}(\\Pi_T)] \\}_{d\\in\\{2,3,\\dots\\}} $   is increasing \nwith   $ \\lim_{d\\to\\infty}\\kappa[\\varrho_{\\{1,2\\}}(\\Pi_T)] = 1\/2 $. \nWe illustrate Corollary \n\\ref{c.product-lowertail} by numerical examples for certain values of   $ d $   and   $ k\\in\\{2,\\dots,d\\} $: \n\n\\bigskip\n\\begin{examples}{}\n\\label{e.product-lowertail}\nFor   $ d\\in\\{2,\\dots,5\\} $   and   $ k\\in\\{2,\\dots,d\\} $, \nthe table \n$$\\begin{array}{|c|@{\\;}cccc|}\n\\hline\\rule{0ex}{2.5ex\n   & \\multicolumn{4}{c|}{d} \\\\\n k &    2    &    3    &    4    &    5    \\\\\n\\hline\\rule{0ex}{2.5ex\n 2 & 315\/945 & 378\/945 & 405\/945 & 420\/945 \\\\\n 3 &         & 315\/945 & 369\/945 & 395\/945 \\\\\n 4 &         &         & 297\/945 & 345\/945 \\\\\n 5 &         &         &         & 273\/945 \\\\\n\\hline\n\\end{array}$$\npresents the values of   $ \\kappa[\\varrho_{\\{1,\\dots,k\\}}(\\Pi_T)] $   for   $ d\\in\\{2,\\dots,5\\} $   and   $ k\\in\\{2,\\dots,d\\} $. \n\\end{examples}\n\n\\bigskip\nWe now introduce a general reflection principle which, \nwhen combined with Corollary \n\\ref{c.product-lowertail}, \nyields explicit formulas for   $ [\\varrho_K(\\Pi_T),\\varrho_K(\\Pi_T)] $, \nand hence for   $ \\kappa[\\varrho_K(\\Pi_T)] $,   \nin the case   $ K=\\{d-k+1,\\dots,d\\} $. \nWe shall need the following lemma: \n\n\\bigskip\n\\begin{lemma}{}\n\\label{l.product-taurho}\nThe identity \n$$  \\rho_K(\\tau(\\Pi_T))  =  \\tau(\\rho_K(\\Pi_T))  $$\nholds for every   $ K\\subseteq\\{1,\\dots,d\\} $   such that   $|K|\\geq2 $. \n\\end{lemma}\n\n\\bigskip\nIn the identity of Lemma \n\\ref{l.product-taurho}, \nthe transformation   $ \\tau $   acts on   $ \\CC^d $   on the left hand side and on   $ \\CC^K $   on the right hand side. \n\\pagebreak\n\n\\bigskip\n\\begin{proof}\nAccording to \nFuchs [2014; Theorem 4.1], \nevery copula   $ D $   satisfies \n\\begin{eqnarray*}\n        (\\tau(D))(\\vvv)                                                                  \n&  = &  \\sum_{L \\subseteq \\{1,...,d\\}} (-1)^{d-|L|} \\, D(\\eeta_L(\\eins\\!-\\!\\vvv,\\eins))  \n\\end{eqnarray*}\nFor every   $ \\uuu\\in\\I^K $   we obtain \n\\begin{eqnarray*}\n        (\\rho_K(\\tau(\\Pi_T)))(\\uuu)                                                                                                    \n&  = &  (\\tau(\\Pi_T))(\\tilde\\rho_K(\\uuu))                                                                                              \\\\[1ex]\n&  = &  \\sum_{L \\subseteq \\{1,...,d\\}} (-1)^{d-|L|}  \\, \\Pi_T(\\eeta_L                           (\\eins\\!-\\!\\tilde\\rho_K(\\uuu),\\eins))  \\\\\n&  = &  \\sum_{L \\subseteq \\{1,...,d\\}} (-1)^{d-|L|}  \\, \\Pi_T(\\eeta_{\\{1,\\dots,d\\} \\setminus L} (\\eins,\\eins\\!-\\!\\tilde\\rho_K(\\uuu)))  \\\\\n&  = &  \\sum_{J \\subseteq \\{1,...,d\\}} (-1)^{|J|}    \\, \\Pi_T(\\eeta_J                           (\\eins,\\eins\\!-\\!\\tilde\\rho_K(\\uuu)))  \\\\\n&  = &  \\sum_{J \\subseteq K}           (-1)^{|J|}    \\, \\Pi_T(\\eeta_J                           (\\eins,\\eins\\!-\\!\\tilde\\rho_K(\\uuu)))  \\\\\n&  = &  \\sum_{J \\subseteq K}           (-1)^{|J|}    \\, \\Pi_T(\\eeta_{\\{1,\\dots,d\\} \\setminus L} (\\eins\\!-\\!\\tilde\\rho_K(\\uuu),\\eins))  \\\\\n&  = &  \\sum_{L \\subseteq K}           (-1)^{|K|-|L|}\\, \\Pi_T(\\eeta_L              (\\tilde\\rho_K(\\eins\\!-\\!\\uuu),\\eins))               \\\\\n&  = &  \\sum_{L \\subseteq K}           (-1)^{|K|-|L|}\\, (\\Pi_T \\circ \\tilde\\rho_K) (\\eeta_L     (\\eins\\!-\\!\\uuu ,\\eins))               \\\\\n&  = &  \\sum_{L \\subseteq K}           (-1)^{|K|-|L|}\\, (\\rho_K(\\Pi_T))            (\\eeta_L     (\\eins\\!-\\!\\uuu ,\\eins))               \\\\*[1ex]\n&  = &  (\\tau(\\rho_K(\\Pi_T)))(\\uuu)                                                                                                    \n\\end{eqnarray*}\nNote that in the last three lines the transformation   $ \\eeta_L $   acts on   $ \\I^K $. \n\\end{proof}\n\n\\bigskip\nConsider now the map   $ b : \\{1,\\dots,d\\}\\to\\{1,\\dots,d\\} $   given by \n\\begin{eqnarray*}\n        b(i)   \n& := &  d-i+1  \n\\end{eqnarray*}\nand the linear map   $ B : \\overline\\R^d\\to\\overline\\R^d $   given by the $ (d \\times d) $--matrix with entries \n\\begin{eqnarray*}\n        b_{i,j}                         \n& := &  \\begin{cases}                   \n         1  &  \\text{if   $ i+j=d+1 $}  \\\\\n         0  &  \\text{else}              \n        \\end{cases}                     \n\\end{eqnarray*}\nWe can now state the announced reflection principle: \n\n\\bigskip\n\\begin{theorem}{}\n\\label{t.product-reflection}\nThe identity \n$$  \\Bigl[ \\varrho_{b(K)}(\\Pi_T) , \\varrho_{b(K)}(\\Pi_T) \\Bigr]  \n =  \\Bigl[ \\varrho_K(\\Pi_T) , \\varrho_K(\\Pi_T) \\Bigr]            \n$$\nholds for every   $ K\\subseteq\\{1,\\dots,d\\} $   such that   $ |K|\\geq2 $. \nIn particular, \n$$  \\kappa[\\varrho_{b(K)}(\\Pi_T)]  =  \\kappa[\\varrho_K(\\Pi_T)]  $$\nholds for every   $ K\\subseteq\\{1,\\dots,d\\} $   such that   $ |K|\\geq2 $. \n\\end{theorem}\n\\pagebreak\n\n\\bigskip\n\\begin{proof}\nUsing Lemma \n\\ref{l.product-taurho} and Theorem \n\\ref{t.product-K} we obtain \n\\begin{eqnarray*}\n        \\Bigl[      \\varrho_{b(K)}(\\Pi_T)  ,      \\varrho_{b(K)}(\\Pi_T)  \\Bigr]       \n&  = &  \\Bigl[ \\tau(\\varrho_{b(K)}(\\Pi_T)) , \\tau(\\varrho_{b(K)}(\\Pi_T)) \\Bigr]       \\\\\n&  = &  \\Bigl[ \\varrho_{b(K)}(\\tau(\\Pi_T)) , \\varrho_{b(K)}(\\tau(\\Pi_T)) \\Bigr]       \\\\*\n&  = &  \\int_{\\I^d} (\\tau(\\Pi_T))(\\eeta_{b(K)}(\\eins,\\uuu)) \\,dQ^{\\tau(\\Pi_T)}(\\uuu)  \n\\end{eqnarray*}\nand from Theorem \n\\ref{t.product-K} we also obtain \n\\begin{eqnarray*}\n        \\Bigl[ \\varrho_K(\\Pi_T) , \\varrho_K(\\Pi_T) \\Bigr]                       \n&  = &  \\int_{\\I^d}            \\Pi_T(\\eeta_K(\\eins,\\uuu)) \\,dQ^{  \\Pi_T}(\\uuu)  \\\\*\n&  = &  \\int_{\\overline\\R^d} H^\\Pi_T(\\eeta_K(\\eins,\\xxx)) \\,dQ^{H^\\Pi_T}(\\xxx)  \n\\end{eqnarray*}\nIt remains to show that the two integrals are identical. \nWe have   $ B \\circ T = \\tilde\\tau \\circ T \\circ \\tilde\\tau $   and hence   $ \\tilde\\tau \\circ T = B \\circ T \\circ \\tilde\\tau $, \nand we also have   \n$ Q^{H^\\Pi_{T \\circ \\tilde\\tau}} = (Q^{H^{\\Pi}})_{T \\circ \\tilde\\tau} = ((Q^{H^\\Pi})_{\\tilde\\tau})_T = (Q^{H^\\Pi})_T = Q^{H^\\Pi_T} $. \nFrom these identities and Lemma \n\\ref{l.commute} we obtain \n\\begin{eqnarray*}\n\\lefteqn{                                                                                                              \n \t\t\\int_{\\I^d}             \\Pi_T (\\eeta_L(\\eeta_{b(K)}(\\zero,  \\uuu ),\\eins))                                     \n        \\,dQ^{\\Pi_T}  (\\uuu)                                                                                          }\\\\*\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_T (\\eeta_L(\\eeta_{b(K)}(\\zero,  \\xxx ),\\eins))                                     \n        \\,dQ^{H^\\Pi_T}(\\xxx)                                                                                           \\\\\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_T (\\eeta_L(\\eeta_{b(K)}(\\zero,T(\\xxx)),\\eins))                                     \n        \\,dQ^{H^\\Pi}  (\\xxx)                                                                                           \\\\\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_T (\\eeta_L(\\eins - \\eeta_{b(K)}(\\eins,(        \\tilde\\tau \\circ T)(\\xxx)),\\eins))  \n        \\,dQ^{H^\\Pi}  (\\xxx)                                                                                           \\\\\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_T (\\eeta_L(\\eins - \\eeta_{b(K)}(\\eins,(B \\circ T \\circ \\tilde\\tau)(\\xxx)),\\eins))  \n        \\,dQ^{H^\\Pi}  (\\xxx)                                                                                           \\\\\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_T (\\eeta_L(\\eins - \\eeta_{b(K)}(\\eins,B(\\zzz))                           ,\\eins))  \n        \\,dQ^{H^\\Pi_{T \\circ \\tilde\\tau}}(\\zzz)                                                                        \\\\\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_T                            (\\eeta_L(\\eins - B \\circ \\eeta_K(\\eins,\\zzz),\\eins))  \n        \\,dQ^{H^\\Pi_T}                   (\\zzz)                                                                        \\\\\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_T                            ((B \\circ \\eeta_L)(\\eins-\\eeta_K(\\eins,\\zzz),\\eins))  \n        \\,dQ^{H^\\Pi_T}                   (\\zzz)                                                                        \\\\\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_{B \\circ T}                           (\\eeta_L (\\eins-\\eeta_K(\\eins,\\zzz),\\eins))  \n        \\,dQ^{H^\\Pi_T}                   (\\zzz)                                                                        \\\\*\n&  = &  \\int_{\\overline\\R^d}  H^\\Pi_{\\tilde\\tau \\circ T \\circ \\tilde\\tau} (\\eeta_L (\\eeta_K(\\zero,\\eins-\\zzz),\\eins))  \n        \\,dQ^{H^\\Pi_T}                   (\\zzz)                                                                        \n\\end{eqnarray*}\nUsing the first identity in the proof of Lemma \n\\ref{l.product-taurho} we thus obtain \n\\begin{eqnarray*}\n\\lefteqn{\n        \\Bigl[ \\varrho_{b(K)}(\\Pi_T),\\varrho_{b(K)}(\\Pi_T) \\Bigr]                                                                                }\\\\*\n&  = &  \\int_{\\I^d} (\\tau(\\Pi_T))(\\eeta_{b(K)}(\\eins,\\uuu))                                                             \\,dQ^{\\tau(\\Pi_T)}(\\uuu)  \\\\\n&  = &  \\int_{\\I^d} \\sum_{L\\subseteq\\{1,\\dots,d\\}} (-1)^{d-|L|}\\,                                                                                 \n        \\Pi_T            (\\eeta_L(\\eins-\\eeta_{b(K)}(\\eins,\\uuu),\\eins))                                                \\,dQ^{\\tau(\\Pi_T)}(\\uuu)  \\\\\n&  = &  \\sum_{L\\subseteq\\{1,\\dots,d\\}} (-1)^{d-|L|}\\,\\int_{\\I^d}                                                                                  \n        \\Pi_T            (\\eeta_L(\\eeta_{b(K)}(\\zero,\\tilde\\tau(\\uuu)),\\eins))                                          \\,dQ^{\\tau(\\Pi_T)}(\\uuu)  \\\\\n&  = &  \\sum_{L\\subseteq\\{1,\\dots,d\\}} (-1)^{d-|L|}\\,\\int_{\\I^d}                                                                                  \n        \\Pi_T            (\\eeta_L(\\eeta_{b(K)}(\\zero,\\uuu),\\eins))                                                      \\,dQ^{\\Pi_T}      (\\uuu)  \\\\\n&  = &  \\sum_{L\\subseteq\\{1,\\dots,d\\}} (-1)^{d-|L|}\\,\\int_{\\overline\\R^d}                                                                         \n        H^\\Pi_{\\tilde\\tau \\circ T \\circ \\tilde\\tau} (\\eeta_L(\\eeta_K(\\zero,\\eins\\!-\\!\\xxx),\\eins))                      \\,dQ^{H^\\Pi_T}    (\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d} \\sum_{L\\subseteq\\{1,\\dots,d\\}} (-1)^{d-|L|}\\,                                                                        \n                    Q^\\Pi_{\\tilde\\tau \\circ T \\circ \\tilde\\tau}                                                                                   \n        \\Bigl[ \\Bigl[ \\zero,                         \\eeta_L(\\eeta_K(\\zero,\\eins\\!-\\!\\xxx),\\eins)         \\Bigr] \\Bigr] \\,dQ^{H^\\Pi_T}    (\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d} Q^\\Pi_{\\tilde\\tau \\circ T \\circ \\tilde\\tau}                                                                          \n        \\Bigl[ \\Bigl[                                        \\eeta_K(\\zero,\\eins\\!-\\!\\xxx)        , \\eins \\Bigr] \\Bigr] \\,dQ^{H^\\Pi_T}    (\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d} Q^\\Pi_{                 T \\circ \\tilde\\tau}                                                                          \n        \\Bigl[ \\Bigl[                        \\zero , \\eins - \\eeta_K(\\zero,\\eins\\!-\\!\\xxx)                \\Bigr] \\Bigr] \\,dQ^{H^\\Pi_T}    (\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d} Q^\\Pi_T                                                                                                              \n        \\Bigl[ \\Bigl[                        \\zero ,         \\eeta_K(\\eins,          \\xxx)                \\Bigr] \\Bigr] \\,dQ^{H^\\Pi_T}    (\\xxx)  \\\\\n&  = &  \\int_{\\overline\\R^d} H^\\Pi_T(                        \\eeta_K(\\eins,          \\xxx))                             \\,dQ^{H^\\Pi_T}    (\\xxx)  \\\\*\n&  = &  \\Bigl[ \\varrho_K(\\Pi_T),\\varrho_K(\\Pi_T) \\Bigr]                                                                                           \n\\end{eqnarray*}\nas was to be shown. \n\\end{proof}\n\n\\bigskip\nTheorem \n\\ref{t.product-reflection} extends a result of \nAv{\\'e}rous et al.\\ [2005; Proposition 10] for   $ |K|=2 $. \nAs noted before, \ncombining Theorem \n\\ref{t.product-reflection} and Corollary  \n\\ref{c.product-lowertail} yields explicit formulas for   $ [\\varrho_K(\\Pi_T),\\varrho_K(\\Pi_T)] $,   \nand hence for   $ \\kappa[\\varrho_K(\\Pi_T)] $,   \nin the case   $ K=\\{d-k+1,\\dots,d\\} $. \n\n\\bigskip\nAs a final remark, \nwe note that Theorem \n\\ref{t.product-K} combined with Lemma \n\\ref{l.product-identity} provides a general tool to compute Kendall's tau for the margins of   $ \\Pi_T $   with respect to any   $ K\\subseteq\\{1,\\dots,d\\} $. \nWe illustrate this by the following example which is not covered by Corollary  \n\\ref{c.product-lowertail}: \n\n\\bigskip\n\\begin{example}{}\nAssume that   $ d=5 $   and consider   $ K=\\{1,2,3,5\\} $. \nSince \n\\linebreak\n  $ Q^\\Pi_T[ \\I^d \\setminus T(\\I^d) ] = 0 $, \nTheorem \n\\ref{t.product-K} together with Lemma \n\\ref{l.product-identity} yields \n\\begin{eqnarray*}\n        \\Bigl[ \\rho_{\\{1,2,3,5\\}}(\\Pi_T), \\rho_{\\{1,2,3,5\\}}(\\Pi_T) \\Bigr]  \n&  = &  5! \\int_{T(\\I^5)} H^\\Pi_T((u_1,u_2,u_3,  1,u_5)) \\,dQ^\\Pi(\\uuu)     \\\\\n&  = &  5! \\int_{T(\\I^5)} H^\\Pi_T((u_1,u_2,u_3,u_5,u_5)) \\,dQ^\\Pi(\\uuu)     \\\\\n&  = &  (5!)^2 \\int_{T(\\I^5)} \\det                                          \n\t\t\\left( \\begin{matrix}                                               \n\t\t  u_1 & u_1^2\/2! & u_1^3\/3! & u_1^4\/4! & u_1^5\/5!                   \\\\\n\t\t    1 &      u_2 & u_2^2\/2! & u_2^3\/3! & u_2^4\/4!                   \\\\\n\t\t\t0 &        1 &      u_3 & u_3^2\/2! & u_3^3\/3!                   \\\\\n\t\t\t0 &        0 &        1 &      u_5 & u_5^2\/2!                   \\\\\n\t\t\t0 &        0 &        0 &        1 &      u_5                   \n\t\t\\end{matrix} \\right)                               dQ^\\Pi(\\uuu)     \\\\\n&  = &  (5!)^2 \\frac{47}{3\\,628\\,800}                                       \\\\*\n&  = &  \\frac{47}{252}                                                      \n\\end{eqnarray*}\nWe thus obtain \n\\begin{eqnarray*}\n        \\kappa[\\rho_{\\{1,2,3,5\\}}(\\Pi_T)]                                               \n&  = &  \\frac{2^4\\,[\\rho_{\\{1,2,3,5\\}}(\\Pi_T),\\rho_{\\{1,2,3,5\\}}(\\Pi_T)]-1}{2^{4-1}-1}  \n\\,\\;=\\;\\, \\frac{125}{441}                                                               \n\\end{eqnarray*}\nand Theorem \n\\ref{t.product-reflection} then yields   $ \\kappa[\\rho_{\\{1,3,4,5\\}}(\\Pi_T)] = \\kappa[\\rho_{\\{1,2,3,5\\}}(\\Pi_T)] = 125\/441 $. \n\\end{example}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe suppression of  $J\/\\psi$ production in  relativistic  heavy ion\ncollisions is considered as   an important signature to identify the quark-gluon plasma\n\\cite{Matsui86}. The dissociation of $J\/\\psi$ in the quark-gluon\nplasma due to color screening can result in a reduction of its\nproduction. The interpretation of suppression requires the detailed knowledge of the\nexpected suppression due to the $J\/\\psi$ dissociation in the hadronic environment.\nThe  in-medium hadron properties\n   can affect  the productions  of the open-charmed mesons and  the $J\/\\psi$ in\nthe relativistic heavy ion collisions,   the higher charmonium states are\nconsidered as the major source of the $J\/\\psi$  \\cite{Jpsi-Source}. For example,\n the higher  charmonium states can decay to the $D\\bar{D}$, $D^*\\bar{D}^*$ pairs instead\nof decaying to the lowest  state $J\/\\psi$ in case of the  mass reductions\n   of the $D$, $D^*$, $\\bar{D}$, $\\bar{D}^*$ mesons  are large enough.\n We have to  disentangle  the color screening  versus the recombination of off-diagonal $\\bar{c}c$ (or $\\bar{b}b$) pairs in the hot dense medium  versus  cold nuclear matter  effects, such as nuclear absorption,\nshadowing and anti-shadowing, so as to draw a  definite conclusion on appearance  of the quark-gluon plasma \\cite{Recombine-cc,CNM}.\nThe upcoming  FAIR\n(Facility for Antiproton and Ion Research) project at GSI (Institute for Heavy Ion Research) in\nDarmstadt (Germany)\n provides the opportunity to study the in-medium properties of the charmoniums  or charmed hadrons for the first time.\n  The CBM (Compressed  Baryonic  Matter) collaboration intends to study the  properties of the\nhadrons in the nuclear matter \\cite{CBM}, while the $\\rm \\bar{P}ANDA$ (anti-Proton Annihilation at Darmstadt) collaboration will focus on the charm spectroscopy, and mass and width modifications of the charmed hadrons in the nuclear matter \\cite{PANDA}.\n  However,\n the in-medium mass modifications are not easy to access experimentally despite the interesting physics involved,  and they\nrequire more detailed theoretical studies.\nOn the other hand, the bottomonium states  are also sensitive to the color screening, the\n $\\Upsilon$ suppression in high energy heavy ion collisions can also be taken  as a\nsignature to identify the quark-gluon plasma \\cite{QGP-rev}.\nThe suppressions  on the\n $\\Upsilon$ production in ultra-relativistic heavy ion collisions will be studied in details  at the Relativistic Heavy Ion Collider (RHIC) and\n Large Hadron Collider (LHC).\n\n\n\n Extensive theoretical and experimental studies are required  to explore the  hadron properties in nuclear matter.\n The connection between the condensates and the nuclear density dependence of the in-medium hadron masses is\nnot straightforward.\nThe QCD sum rules provides  a powerful theoretical tool  in\n studying the in-medium hadronic properties \\cite{SVZ79,PRT85}, and has been applied extensively\n  to study the light-flavor hadrons and  charmonium states in the nuclear matter \\cite{C-parameter,Drukarev1991,Jpsi-etac}.\n   The\n  works on the heavy mesons and heavy baryons are  few, only the $D$,  $B$, $D_0$, $B_0$, $D^*$, $B^*$, $D_1$, $B_1$,  $\\Lambda_Q$,   $\\Sigma_Q$, $\\Xi_{QQ}$ and $\\Omega_{QQ}$  are studied with the QCD sum rules \\cite{Hay,WangHuang,Azi,Hil,Hilger2,WangH}.\nThe heavy mesons (heavy baryons) contain a  heavy quark and a light quark (two light quarks),\nthe existence of a light quark (two light quarks) in the heavy mesons (heavy baryons) leads to large\ndifference between the  mass-shifts of\n the heavy mesons (heavy baryons) and heavy quarkonia  in the nuclear matter.\nThe former have  large contributions  from the light-quark\ncondensates in the nuclear matter and the   modifications of the masses originate mainly from the modifications of the quark condensates,  while the latter\n  are dominated by the gluon condensates, and the mass modifications are mild \\cite{Jpsi-etac,Hay,WangHuang,Azi,Hil,Hilger2,WangH}.\n\n  In previous works \\cite{Hay,WangHuang,Azi,Hil,Hilger2},\n  the properties of the heavy mesons in the nuclear matter are studied with the QCD sum rules by taking the leading order approximation for the contributions of the   quark condensates. In this article, I take into account the next-to-leading order contributions of the quark condensates,  and study the properties of the scalar, pseudoscalar, vector and axialvector  heavy mesons in the nuclear matter with the QCD sum rules in a systematic way, and make predictions for the  modifications of the masses and decay constants of the heavy mesons in the nuclear matter. Measuring those effects is a long term physics\ngoal based on further theoretical studies on the reaction\ndynamics and on the exploration of the experimental\nability to identify more complicated\nprocesses \\cite{CBM,PANDA}.   Furthermore, I study the heavy-meson-nucleon scattering lengths as a byproduct. From the negative or positive sign of the scattering lengths, I can obtain the conclusion qualitatively that the interactions are  attractive or repulsive, which favor or disfavor  the\nformations of the heavy-meson-nucleon bound states. For example, the $\\Sigma_c(2800)$  and $\\Lambda_c(2940)$   can  be assigned to be the $S$-wave $DN$  state with $J^P ={\\frac{1}{2}}^-$ and  the $S$-wave $D^*N$  state with $J^P ={\\frac{3}{2}}^-$ respectively based on the QCD sum rules \\cite{ZhangDN}.\n\n\n\nThe article is arranged as follows:  I study in-medium properties of the heavy mesons\n with  the  QCD sum rules in Sec.2; in Sec.3, I present the numerical results and discussions; and Sec.4 is reserved for my\nconclusions.\n\n\\section{The properties of the heavy mesons in the nuclear matter with  QCD sum rules}\n\nI study the  scalar, pseudoscalar, vector and axialvector heavy  mesons in the nuclear matter with\nthe two-point correlation functions $\\Pi(q)$ and $\\Pi_{\\mu\\nu}(q)$, respectively. In the Fermi gas\napproximation for the nuclear matter, I divide the $\\Pi(q)$ and $\\Pi_{\\mu\\nu}(q)$\ninto  the vacuum part $\\Pi^0(q)$ and $\\Pi^0_{\\mu\\nu}(q)$  and the static one-nucleon part\n$\\Pi_N(q)$ and $\\Pi^N_{\\mu\\nu}(q)$,  and  expand  the\n$\\Pi_N(q)$ and $\\Pi^N_{\\mu\\nu}(q)$  up to the order ${\\mathcal{O}}(\\rho_N)$  at relatively low nuclear density  \\cite{Drukarev1991,Hay},\n\\begin{eqnarray}\n\\Pi(q) &=& i\\int d^{4}x\\ e^{iq \\cdot x} \\langle\nT\\left\\{J(x)J^{\\dag}(0)\\right\\} \\rangle_{\\rho_N}\\nonumber \\\\\n& =& \\Pi_{0}(q)+ \\frac{\\rho_N}{2m_N}T_{N}(q)\\, , \\nonumber \\\\\n \\Pi_{\\mu\\nu}(q) &=& i\\int d^{4}x\\ e^{iq \\cdot x} \\langle T\\left\\{J_\\mu(x)J_\\nu^{\\dag}(0)\\right\\} \\rangle_{\\rho_N} \\nonumber\\\\\n &=&\\Pi^{0}_{\\mu\\nu}(q)+ \\frac{\\rho_N}{2m_N}T^{N}_{\\mu\\nu}(q)\\, ,\n \\end{eqnarray}\nwhere the $\\rho_N$ is the density of the nuclear matter, and the forward scattering amplitudes $T_{N}(q)$ and $T^{N}_{\\mu\\nu}(q)$ are defined as\n\\begin{eqnarray}\nT_{N}(\\omega,\\mbox{\\boldmath $q$}\\,) &=&i\\int d^{4}x e^{iq\\cdot x}\\langle N(p)|\nT\\left\\{J(x)J^{\\dag}(0)\\right\\} |N(p) \\rangle\\, , \\nonumber\\\\\nT^{N}_{\\mu\\nu}(\\omega,\\mbox{\\boldmath $q$}\\,) &=&i\\int d^{4}x e^{iq\\cdot x}\\langle N(p)|\nT\\left\\{J_\\mu(x)J_\\nu^{\\dag}(0)\\right\\} |N(p) \\rangle\\, ,\n\\end{eqnarray}\n where the $J(x)$ and $J_\\mu(x)$ denote the isospin averaged currents $\\eta(x)$, $\\eta_5(x)$, $\\eta_\\mu(x)$ and $\\eta_{5\\mu}(x)$, respectively,\n\\begin{eqnarray}\n \\eta(x) &=&\\eta^\\dag(x) =\\frac{\\bar{c}(x) q(x)+\\bar{q}(x) c(x)}{2}\\, , \\nonumber\\\\\n  \\eta_{5}(x) &=&\\eta_{5}^\\dag(x) =\\frac{\\bar{c}(x)i \\gamma_5q(x)+\\bar{q}(x)i\\gamma_5 c(x)}{2}\\,,\\nonumber\\\\\n \\eta_\\mu(x) &=&\\eta_\\mu^\\dag(x) =\\frac{\\bar{c}(x)\\gamma_\\mu q(x)+\\bar{q}(x)\\gamma_\\mu c(x)}{2}\\, , \\nonumber\\\\\n  \\eta_{5\\mu}(x) &=&\\eta_{5\\mu}^\\dag(x) =\\frac{\\bar{c}(x)\\gamma_\\mu \\gamma_5q(x)+\\bar{q}(x)\\gamma_\\mu\\gamma_5 c(x)}{2}\\,,\n\\end{eqnarray}\n which interpolate the scalar, pseudoscalar, vector and axialvector mesons $D_0$, $D$, $D^*$ and $D_1$, respectively.\n I choose the isospin averaged currents since the $D_0$, $D$, $D^*$ and $D_1$ mesons are produced in pairs in the antiproton-nucleon\nannihilation processes.\nThe $q$ denotes the $u$ or $d$ quark, the $q^{\\mu}=(\\omega,\\mbox{\\boldmath $q$}\\,)$ is the four-momentum carried by\nthe  currents $J(x)$ and $J_\\mu(x)$, the $|N(p)\\rangle$ denotes the isospin and\nspin averaged static nucleon state with the four-momentum $p =\n(m_N,0)$, and  $\\langle N(\\mbox{\\boldmath $p$})|N(\\mbox{\\boldmath\n$p$}')\\rangle = (2\\pi)^{3} 2p_{0}\\delta^{3}(\\mbox{\\boldmath\n$p$}-\\mbox{\\boldmath $p$}')$ \\cite{Hay}.\n\nI can decompose the correlation functions $T^{N}_{\\mu\\nu}(\\omega,\\mbox{\\boldmath $q$}\\,)$ as\n\\begin{eqnarray}\nT^{N}_{\\mu\\nu}(\\omega,\\mbox{\\boldmath $q$}\\,) &=&T_{N}(\\omega,\\mbox{\\boldmath $q$}\\,)\\left(-g_{\\mu\\nu}+\\frac{q_\\mu q_\\nu}{q^2}\\right)+T^0_N(\\omega,\\mbox{\\boldmath $q$}\\,) q_\\mu q_\\nu\\nonumber\\\\\n&&+T^1_N(\\omega,\\mbox{\\boldmath $q$}\\,) \\left(q_\\mu u_\\nu+q_\\nu u_\\mu \\right)+ T^2_N(\\omega,\\mbox{\\boldmath $q$}\\,) u_\\mu u_\\nu\\, ,\n\\end{eqnarray}\naccording to Lorentz covariance, where the $T_{N}(\\omega,\\mbox{\\boldmath $q$}\\,)$ denotes the contributions of the vector and axialvector charmed mesons,\n and the $T^{0\/1\/2}_{N}(\\omega,\\mbox{\\boldmath $q$}\\,)$ are irrelevant in the present analysis.\n\n\nIn the limit $\\mbox{\\boldmath $q$}\\rightarrow {\\bf 0}$, the forward scattering amplitude\n$T_{N}(\\omega,\\mbox{\\boldmath $q$}\\,)$ can be related to the $DN$ ($D_0N$, $D^*N$ and $D_1N$)\nscattering $T$-matrix,\n\\begin{eqnarray}\n{{\\cal T}_{D\/D_0\/D^*\/D_1\\,N}}(m_{D\/D_0\/D^*\/D_1},0) =\n8\\pi(m_N+m_{D\/D_0\/D^*\/D_1})a_{D\/D_0\/D^*\/D_1} \\, ,\n\\end{eqnarray}\n where the  $a_{D\/D_0\/D^*\/D_1}$ are the $D\/D_0\/D^*\/D_1\\,N$\nscattering lengths.  I can parameterize  the\nphenomenological spectral densities $\\rho(\\omega,0)$ with three unknown   parameters $a,\\,b$ and $c$ near the pole positions of the charmed mesons $D$, $D_0$, $D^*$ and $D_1$ according to Ref.\\cite{Hay},\n\\begin{eqnarray}\n\\rho(\\omega,0) &=& -\\frac{1}{\\pi} \\mbox{Im} \\left[\\frac{{{\\cal T}_{D\/D_0N}}(\\omega,{\\bf 0})}{\\left(\\omega^{2}-\nm_{D\/D_0}^2+i\\varepsilon\\right)^{2}} \\right]\\frac{f_{D\/D_0}^2m_{D\/D_0}^4}{m_c^2}+ \\cdots \\,, \\nonumber \\\\\n&=& a\\,\\frac{d}{d\\omega^2}\\delta\\left(\\omega^{2}-m_{D\/D_0}^2\\right) +\nb\\,\\delta\\left(\\omega^{2}-m_{D\/D_0}^2\\right) + c\\,\\delta\\left(\\omega^{2}-s_{0}\\right)\\, ,\n\\end{eqnarray}\nfor the pseudoscalar and scalar currents $\\eta_5(x)$ and $\\eta(x)$,\n\\begin{eqnarray}\n\\rho(\\omega,0) &=& -\\frac{1}{\\pi} \\mbox{Im} \\left[\\frac{{{\\cal T}_{D^*\/D_1N}}(\\omega,{\\bf 0})}{\\left(\\omega^{2}-\nm_{D^*\/D_1}^2+i\\varepsilon\\right)^{2}} \\right]f_{D^*\/D_1}^2m_{D^*\/D_1}^2+ \\cdots \\,, \\nonumber\\\\\n&=& a\\,\\frac{d}{d\\omega^2}\\delta\\left(\\omega^{2}-m_{D^*\/D_1}^2\\right) +\nb\\,\\delta\\left(\\omega^{2}-m_{D^*\/D_1}^2\\right) + c\\,\\delta\\left(\\omega^{2}-s_{0}\\right)\\, ,\n\\end{eqnarray}\nfor the vector and axialvector currents $\\eta_\\mu(x)$ and $\\eta_{5\\mu}(x)$.\n\nNow the hadronic correlation functions $\\Pi(\\omega,0)$  and $\\Pi_{\\mu\\nu}(\\omega,0)$ at the phenomenological side can be written as\n\\begin{eqnarray}\n\\Pi(\\omega,0)&=&\\frac{\\left( f_{D\/D_0}+\\delta f_{D\/D_0}\\right)^2\\left( m_{D\/D_0}+\\delta m_{D\/D_0}\\right)^4}{m_c^2}\\frac{1}{\\left(m_{D\/D_0}+\\delta m_{D\/D_0}\\right)^2-\\omega^2}+\\cdots  \\nonumber\\\\\n&=&\\frac{ f_{D\/D_0}^2m_{D\/D_0}^4}{m_c^2}\\frac{1}{m_{D\/D_0}^2-\\omega^2}+\\cdots\\nonumber\\\\\n&&+\\frac{\\rho_N}{2m_N}\\left[\\frac{a}{\\left(m_{D\/D_0}^2-\\omega^2\\right)^2}+\\frac{b}{m_{D\/D_0}^2-\\omega^2} +\\cdots \\right]\\, ,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\Pi_{\\mu\\nu}(\\omega,0)&=&\\left( f_{D^*\/D_1}+\\delta f_{D^*\/D_1}\\right)^2\\left( m_{D^*\/D_1}+\\delta m_{D^*\/D_1}\\right)^2\\frac{1}{\\left(m_{D^*\/D_1}+\\delta m_{D^*\/D_1}\\right)^2-\\omega^2}\\nonumber\\\\\n&&\\left( -g_{\\mu\\nu}+\\frac{q_\\mu q_\\nu}{q^2}\\right) +\\cdots \\, ,\\nonumber\\\\\n&=&f_{D^*\/D_1}^2m_{D^*\/D_1}^2\\frac{1}{m_{D^*\/D_1}^2-\\omega^2}\\left( -g_{\\mu\\nu}+\\frac{q_\\mu q_\\nu}{q^2}\\right)+\\cdots\\nonumber\\\\\n&&+\\frac{\\rho_N}{2m_N}\\left[\\left(\\frac{a}{\\left(m_{D^*\/D_1}^2-\\omega^2\\right)^2}+\\frac{b}{m_{D^*\/D_1}^2-\\omega^2} +\\cdots\\right)\\left( -g_{\\mu\\nu}+\\frac{q_\\mu q_\\nu}{q^2}\\right)+\\cdots \\right] \\, .\\nonumber\\\\\n\\end{eqnarray}\n\n\n\n\n\n\nIn Eqs.(6-7), the first\nterm denotes the double-pole term, and corresponds  to the on-shell\neffect of the $T$-matrix,\n\\begin{eqnarray}\na&=&-8\\pi(m_N+m_{D\/D_0})a_{D\/D_0}\\frac{f_{D\/D_0}^2m_{D\/D_0}^4}{m_c^2}\\, ,\n\\end{eqnarray}\nfor the currents $\\eta_5(x)$ and $\\eta(x)$ and\n\\begin{eqnarray}\na&=&-8\\pi(m_N+m_{D^*\/D_1})a_{D^*\/D_1}f_{D^*\/D_1}^2m_{D^*\/D_1}^2\\, ,\n\\end{eqnarray}\nfor the currents  $\\eta_\\mu(x)$ and $\\eta_{5\\mu}(x)$;\nthe second term denotes  the single-pole term,\nand corresponds to the off-shell effect of the $T$-matrix;\n the\nthird term denotes   the continuum term or the  remaining effects,\nwhere the $s_{0}$ is the continuum threshold parameter.\nIn general, the continuum contributions are approximated by $\\rho_{QCD}(\\omega,0)\\theta(\\omega^2-s_0)$, where the $\\rho_{QCD}(\\omega,0)$ are the\nperturbative QCD spectral densities, and $\\theta(x)=1$ for $x\\geq 0$, else $\\theta(x)=0$. In this article, the QCD spectral densities are of the type $\\delta(\\omega^2-m_Q^2)$, which include both the ground state and continuum state contributions,  I have attributed  the excited state contributions to the continuum state contributions,  so the collective continuum state contributions can be approximated as $c\\,\\delta(\\omega^2-s_0)$, then I obtain\n the result $c\/\\left(s_0-\\omega^2\\right)$ in the hadronic representation, see Eq.(15). The doublet $\\left(D(2550), D(2600)\\right)$ or $\\left(D_J(2580), D^*_J (2650)\\right)$ is assigned  to be\n the first radial excited state of the doublet $(D,D^*)$ \\cite{WangHQET}. The single-pole contributions come from the doublet $\\left(D(2550), D(2600)\\right)$ or $\\left(D_J(2580), D^*_J (2650)\\right)$ are  of the form $1\/\\left(m_{D(2550)\/D(2600)}^2-\\omega^2\\right)$, so the approximation $c\/\\left(s_0-\\omega^2\\right)$ is reasonable.\n\nThen the shifts of the masses and decay  constants  of\nthe charmed-mesons can be approximated as\n\\begin{eqnarray}\n\\delta m_{D\/D_0\/D^*\/D_1} &=&2\\pi\\frac{m_{N}+m_{D\/D_0\/D^*\/D_1}}{m_Nm_{D\/D_0\/D^*\/D_1}}\\rho_N a_{D\/D_0\/D^*\/D_1}\\, ,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\delta f_{D\/D_0}&=&\\frac{m_c^2}{2f_{D\/D_0}m_{D\/D_0}^4}\\left(\\frac{b\\rho_N}{2m_N}-\\frac{4f_{D\/D_0}^2m_{D\/D_0}^3\\delta m_{D\/D_0}}{m_c^2} \\right) \\, , \\nonumber\\\\\n\\delta f_{D^*\/D_1}&=&\\frac{1}{2f_{D^*\/D_1}m_{D^*\/D_1}^2}\\left(\\frac{b\\rho_N}{2m_N}-2f_{D^*\/D_1}^2m_{D^*\/D_1}\\delta m_{D^*\/D_1} \\right) \\, .\n\\end{eqnarray}\n\n\n\n\nIn calculations, I have used the following definitions for the decay constants of the heavy mesons,\n\\begin{eqnarray}\n\\langle 0|\\eta(0)|D_0+\\bar{D}_0\\rangle &=&\\frac{f_{D_0}m^2_{D_0}}{m_c} \\,,\\nonumber\\\\\n\\langle 0|\\eta_5(0)|D+\\bar{D}\\rangle &=&\\frac{f_{D}m^2_{D}}{m_c} \\,,\\nonumber\\\\\n\\langle 0|\\eta_\\mu(0)|D^*+\\bar{D}^*\\rangle &=&f_{D^*}m_{D^*}\\epsilon_\\mu\\,,\\nonumber\\\\\n\\langle 0|\\eta_{5\\mu}(0)|D_1+\\bar{D}_1\\rangle &=&f_{D_1}m_{D_1}\\epsilon_{\\mu}\\,,\n\\end{eqnarray}\nwith summations of the polarization vectors  $\\sum_\\lambda \\epsilon_\\mu(\\lambda,q)\\epsilon^*_\\nu(\\lambda,q)=-g_{\\mu\\nu}+\\frac{q_\\mu q_\\nu}{q^2}$.\n\n\nIn the low energy limit $\\omega\\rightarrow 0$, the\n$T_{N}(\\omega,{\\bf 0})$ is equivalent to the Born term $T_{N}^{\\rm\nBorn}(\\omega,{\\bf 0})$. Now I take\ninto account the Born terms  at the phenomenological side,\n\\begin{eqnarray}\nT_{N}(\\omega^2)&=&T_{N}^{\\rm\nBorn}(\\omega^2)+\\frac{a}{\\left(m_{D\/D_0\/D^*\/D_1}^2-\\omega^2\\right)^2}+\\frac{b}{m_{D\/D_0\/D^*\/D_1}^2-\\omega^2}+\\frac{c}{s_0-\\omega^2}\n\\, ,\n\\end{eqnarray}\nwith the constraint\n\\begin{eqnarray}\n\\frac{a}{m_{D\/D_0\/D^*\/D_1}^4}+\\frac{b}{m_{D\/D_0\/D^*\/D_1}^2}+\\frac{c}{s_0}&=&0 \\, .\n\\end{eqnarray}\n\n\nThe contributions from the intermediate spin-$\\frac{3}{2}$ charmed\nbaryon states are zero in the soft-limit $q_\\mu \\to 0$\n\\cite{Wangzg}, and I only take into account the intermediate\nspin-$\\frac{1}{2}$ charmed  baryon states in calculating the Born terms,\n\\begin{eqnarray}\n\\left(D\/D_0\/D^*\/D_1\\right)^0(c\\bar{u})+p(uud)\\ \\mbox{or}\\ n(udd)&\\longrightarrow&\n\\Lambda_c^+,\\Sigma_c^+(cud)\\ \\mbox{or}\\ \\Sigma_c^0(cdd) \\, ,\\nonumber\\\\\n \\left(D\/D_0\/D^*\/D_1\\right)^+(c\\bar{d})+p(uud)\\ \\mbox{or}\\ n(udd)\n&\\longrightarrow& \\Sigma_c^{++}(cuu)\\ \\mbox{or}\\\n\\Lambda_c^+,\\Sigma_c^+(cud)\\, ,\n\\end{eqnarray}\nwhere   $M_{\\Lambda_c}=2.286\\,\\rm{GeV}$   and\n$M_{\\Sigma_c}=2.454\\,\\rm{GeV}$ \\cite{PDG}. I can take $M_H\\approx\n2.4\\,\\rm{GeV}$ as the average value, where the $H$ means either\n$\\Lambda_c^+$, $\\Sigma_c^+$, $\\Sigma_c^{++}$ or $\\Sigma_c^0$. In the case of the bottom baryons, I take the approximation $M_H=\\frac{M_{\\Sigma_b}+M_{\\Lambda_b}}{2}\\approx 5.7\\,\\rm{GeV}$ \\cite{PDG}. I write down the Feynman  diagrams and calculate the  Born terms directly,\nand obtain    the  results,\n\\begin{eqnarray}\nT_{N}^{\\rm\nBorn}(\\omega,{\\bf0})&=&\\frac{2m_N(m_H+m_N)}\n{\\left[\\omega^2-(m_H+m_N)^2\\right]\\left[\\omega^2-m_{D}^2\\right]^2}\\left(\\frac{f_{D}m_{D}^2g_{DNH}}{m_c}\\right)^2\\,,\n\\end{eqnarray}\nfor the current $\\eta_5(x)$,\n\\begin{eqnarray}\nT_{N}^{\\rm\nBorn}(\\omega,{\\bf0})&\\to&T_{N}^{\\rm\nBorn}(\\omega,{\\bf0})\\,\\left( {\\rm with}\\,\\,\\, m_N \\to -m_N\\, , \\,\\,\\,  D\\to D_0 \\right)\\,,\n\\end{eqnarray}\nfor the current $\\eta(x)$,\n\n\\begin{eqnarray}\nT_{N}^{\\rm\nBorn}(\\omega,{\\bf0})&=&\\frac{2m_N(m_H+m_N)}\n{\\left[\\omega^2-(m_H+m_N)^2\\right]\\left[\\omega^2-m_{D^*}^2\\right]^2}\\left(f_{D^*}m_{D^*}g_{D^*NH}\\right)^2\\,,\n\\end{eqnarray}\nfor the current $\\eta_\\mu(x)$,\n\\begin{eqnarray}\nT_{N}^{\\rm\nBorn}(\\omega,{\\bf0})&\\to&T_{N}^{\\rm\nBorn}(\\omega,{\\bf0})\\,\\left( {\\rm with}\\,\\,\\, m_N \\to -m_N\\, , \\,\\,\\,  D^*\\to D_1 \\right)\\,,\n\\end{eqnarray}\nfor the current $\\eta_{5\\mu}(x)$, where the $g_{D\/D_0\/D^*\/D_1NH}$ denote the  strong coupling constants\n$g_{D\/D_0\/D^*\/D_1N\\Lambda_c}$\n and  $g_{D\/D_0\/D^*\/D_1N\\Sigma_c}$.\nOn the other hand, there are no inelastic channels for the\n$(\\bar{D}\/\\bar{D}_0\/\\bar{D}^*\/\\bar{D}_1)^0  N$ and $(\\bar{D}\/\\bar{D}_0\/\\bar{D}^*\/\\bar{D}_1)^- N$ interactions, and $T_{N}^{\\rm\nBorn}(0)=0$.\nIn calculations, I have used the following definitions for  the hadronic coupling constants,\n \\begin{eqnarray}\n \\langle\\Lambda_c\/\\Sigma_c(p-q)|D(-q)N(p)\\rangle &=&g_{\\Lambda_c\/\\Sigma_cDN}\\overline{U}_{\\Lambda_c\/\\Sigma_c}(p-q) i\\gamma_5 U_N(p)\\,,\\nonumber\\\\\n \\langle\\Lambda_c\/\\Sigma_c(p-q)|D_0(-q)N(p)\\rangle &=& g_{\\Lambda_c\/\\Sigma_cD_0N}\\overline{U}_{\\Lambda_c\/\\Sigma_c}(p-q) U_N(p)\\,,\\nonumber\\\\\n \\langle\\Lambda_c\/\\Sigma_c(p-q)|D^*(-q)N(p)\\rangle &=&\\overline{U}_{\\Lambda_c\/\\Sigma_c}(p-q)\\left( g_{\\Lambda_c\/\\Sigma_cD^*N}\\!\\not\\!{\\epsilon}+i\\frac{g^T_{\\Lambda_c\/\\Sigma_cD^*N}}{M_N+M_{\\Lambda_c\/\\Sigma_c}}\\sigma^{\\alpha\\beta}\\epsilon_\\alpha q_\\beta\\right)U_N(p)\\,,\\nonumber\\\\\n \\langle\\Lambda_c\/\\Sigma_c(p-q)|D_1(-q)N(p)\\rangle &=&\\overline{U}_{\\Lambda_c\/\\Sigma_c}(p-q)\\left( g_{\\Lambda_c\/\\Sigma_cD_1N}\\!\\not\\!{\\epsilon}+i\\frac{g^T_{\\Lambda_c\/\\Sigma_cD_1N}}{M_N+M_{\\Lambda_c\/\\Sigma_c}}\\sigma^{\\alpha\\beta}\\epsilon_\\alpha q_\\beta\\right)\\gamma_5U_N(p)\\,,\\nonumber\\\\\n \\end{eqnarray}\n where the $U_N$ and $\\overline{U}_{\\Lambda_c\/\\Sigma_c}$ are the Dirac spinors of the nucleon and the charmed  baryons $\\Lambda_c\/\\Sigma_c$, respectively. In the limit $q_\\mu \\to 0$, the  strong coupling constants $g_{\\Lambda_c\/\\Sigma_cD^*N}^T$ and  $g_{\\Lambda_c\/\\Sigma_cD_1N}^T$ have no contributions.\n\n\n For example, near the thresholds, the $D^*N$  can translate  to the $DN$, $D^*N$, $\\pi\\Sigma_c$, $\\eta\\Lambda_c$,   etc, we can  take into account  the intermediate baryon-meson loops or the re-scattering effects  with the Bethe-Salpeter equation to obtain the full $D^*N \\to D^*N$ scattering amplitude, and   generate higher baryon states  dynamically \\cite{DN-BSE-Negative}. We can saturate the full $D^*N\\to D^*N$ scattering amplitude with the tree-level Feynman diagrams describing   the  exchanges of the higher resonances $\\Lambda_c(2595)$, $\\Sigma_c({\\frac{1}{2}}^-)$, etc.  While in other coupled-channels analysis,  the $\\Lambda_c(2595)$ emerges as a $DN$ quasi-bound state rather than a $D^*N$ quasi-bound state \\cite{DN-BSE-Negative}.     The translations  $D^*N $ to the ground  states $\\Lambda_c $ and $\\Sigma_c$ are favored in the phase-space, as   the $\\Lambda_c(2595)$ and $\\Sigma_c({\\frac{1}{2}}^-)$ with $J^P={\\frac{1}{2}}^-$  have the average mass $m_{H'}\\approx 2.7\\,\\rm{GeV}$  \\cite{PDG,Wang-Negative}. In fact,  $m_{H'}^2> s_0$, I can absorb the high resonances into the continuum states in case the high resonances do not dominate the\n QCD sum rules.\n In calculations, I observe that the mass-shift $\\delta m_{D^*}$ does not sensitive to contributions of the  ground  states $\\Lambda_c $ and $\\Sigma_c$,\n  the contributions from the spin-$\\frac{1}{2}$ higher resonances maybe even smaller.\nIn this article, I neglect the intermediate baryon-meson loops, their  effects are absorbed into continuum contributions.\n\n\nAt the low nuclear density,  the condensates $\\langle{\\cal {O}}\\rangle_{\\rho_N}$ in the nuclear matter can be approximated as\n\\begin{eqnarray}\n\\langle{\\cal{O}}\\rangle_{\\rho_N} &=&\\langle{\\cal{O}}\\rangle+\\frac{\\rho_N}{2m_N}\\langle {\\cal{O}}\\rangle_N \\, ,\n\\end{eqnarray}\nbased on the Fermi gas model, where the   $\\langle{\\cal{O}}\\rangle$ and $\\langle\n{\\cal{O}}\\rangle_N$ denote the vacuum condensates and nuclear matter induced condensates,  respectively \\cite{Drukarev1991}. I neglect the terms\nproportional to $p_F^4$, $p_F^5$, $p_F^6$, $\\cdots$\nat the normal nuclear matter with the saturation density\n$\\rho_N=\\rho_0=\\frac{2p_F^3}{3\\pi^2}$, as the Fermi momentum\n$p_F=0.27\\,\\rm{GeV}$ is a small quantity \\cite{Drukarev1991}.\n\nI carry out the operator product expansion to the  nuclear matter induced condensates $\\frac{\\rho_N}{2m_N}\\langle\n{\\cal{O}}\\rangle_N$ up to dimension-5\n at the large space-like  region in the nuclear matter,\nand take into account the one-loop corrections to the quark condensate $\\langle \\bar{q}q\\rangle_N$. I insert the following term\n\\begin{eqnarray}\n\\frac{1}{2!}\\,\\, ig_s \\int d^D y \\bar{\\psi}(y)\\gamma^\\mu \\psi(y)\\frac{\\lambda^a}{2}G^a_\\mu(y)\\,\\, ig_s \\int d^D z \\bar{\\psi}(z)\\gamma^\\nu \\psi(z)\\frac{\\lambda^b}{2}G^b_\\nu(z) \\, ,\n\\end{eqnarray}\n with the dimension  $D=4-2\\epsilon$, into the correlation functions $T_N(q)$ and $T^N_{\\mu\\nu}(q)$ firstly, where the $\\psi$ denotes the quark fields, the $G^a_\\mu$  denotes the gluon field, the $\\lambda^a$ denotes  the Gell-Mann matrix, then contract the quark fields with Wick theorem,  and extract the quark condensate $\\langle\\bar{q}{q}\\rangle_N$ according to the formula  $\\langle N|q_\\alpha^i q_\\beta^j|N\\rangle=-\\frac{1}{12}\\langle \\bar{q}q\\rangle_N\\delta_{ij}\\delta_{\\alpha\\beta}$ to obtain the perturbative corrections $\\alpha_s\\langle\\bar{q}{q}\\rangle_N$, where the $i$ and $j$ are color indexes and the $\\alpha$ and $\\beta$ are Dirac spinor  indexes. There are six Feynman diagrams make contributions, see Fig.1. Now I calculate  the first diagram explicitly for the current $\\eta_5(x)$ in Fig.1,\n \\begin{eqnarray}\n 2T^{(\\alpha_s,1)}_N(q^2) &=&-\\frac{{\\rm Tr}(\\frac{\\lambda^a}{2} \\frac{\\lambda^b}{2})\\langle \\bar{q}q\\rangle_Ng_s^2 \\mu^{2\\epsilon}}{12} \\frac{i}{(2\\pi)^D}\\int d^Dk{\\rm Tr}\\left\\{ i\\gamma_5 \\frac{i}{\\!\\not\\! {k}}\\gamma^\\alpha \\gamma^\\beta \\frac{i}{\\!\\not\\! {k}} i \\gamma_5 \\frac{i}{\\!\\not\\! {q}+\\!\\not\\! {k}-m_c} \\frac{-i\\delta_{ab}g_{\\alpha\\beta}}{k^2}\\right\\} \\nonumber\\\\\n &=&-\\frac{4Dm_c\\langle \\bar{q}q\\rangle_Ng_s^2 \\mu^{2\\epsilon}}{3(2\\pi)^D}i\\frac{\\partial}{\\partial t}\\frac{(-2\\pi i)^2}{2\\pi i} \\int_{m_c^2}^{\\infty} ds \\frac{\\int d^Dk \\delta \\left( k^2-t\\right)\\delta\\left( (k+q)^2-m_c^2\\right)}{s-q^2}\\mid_{t=0} \\nonumber\\\\\n  &=& -\\frac{Dm_c\\langle \\bar{q}q\\rangle_Ng_s^2 \\mu^{2\\epsilon}\\left[1+\\epsilon(\\log4\\pi-\\gamma_E) \\right]}{12\\pi^2}  \\int_{m_c^2}^{\\infty} ds \\frac{1}{s-q^2} \\frac{s+m_c^2}{s^{1-\\epsilon}(s-m_c^2)^{1+2\\epsilon}} \\, ,\n   \\end{eqnarray}\n   where I have used Cutkosky's rule to obtain the QCD spectral density. There exists  infrared divergence at the end point $s=m_c^2$.\n   It is difficult to carry out the integral over $s$, I can perform the Borel transform $B_{M^2}$ firstly, then carry out the  integral over $s$,\n \\begin{eqnarray}\n B_{M^2}2T^{(\\alpha_s,1)}_N(q^2)&=& -\\frac{Dm_c\\langle \\bar{q}q\\rangle_Ng_s^2 \\mu^{2\\epsilon}\\left[1+\\epsilon(\\log4\\pi-\\gamma_E) \\right]}{12\\pi^2M^2}  \\int_{m_c^2}^{\\infty} ds   \\frac{s+m_c^2}{s^{1-\\epsilon}(s-m_c^2)^{1+2\\epsilon}}\\exp\\left( -\\frac{s}{M^2}\\right) \\nonumber\\\\\n &=&\\frac{m_c\\langle \\bar{q}q\\rangle_Ng_s^2}{3\\pi^2M^2}\\exp\\left(-\\frac{m_c^2}{M^2} \\right)\\left( \\frac{1}{\\epsilon}-\\log4\\pi+\\gamma_E\\right)+\\frac{m_c\\langle \\bar{q}q\\rangle_Ng_s^2}{3\\pi^2M^2}\\Gamma\\left(0,\\frac{m_c^2}{M^2} \\right) \\nonumber\\\\\n &&-\\frac{m_c\\langle \\bar{q}q\\rangle_Ng_s^2}{6\\pi^2M^2}\\exp\\left(-\\frac{m_c^2}{M^2} \\right)+\\frac{m_c\\langle \\bar{q}q\\rangle_Ng_s^2}{3\\pi^2M^2}\\exp\\left(-\\frac{m_c^2}{M^2} \\right)\\log\\frac{m_c^2\\mu^2}{M^4} \\, ,\n \\end{eqnarray}\n where\n \\begin{eqnarray}\n\\Gamma(0,x)&=&e^{-x}\\int_0^\\infty dt \\frac{1}{t+x}e^{-t} \\, .\n\\end{eqnarray}\n Other diagrams are calculated analogously,   I regularize the divergences in $D=4-2\\epsilon$ dimension, then remove the ultraviolet divergences through renormalization and absorb the\n    infrared divergences into the quark condensate $\\langle \\bar{q}q\\rangle_N$.\n \\begin{figure}\n \\centering\n \\includegraphics[totalheight=6cm,width=14cm]{qq-aphs.eps}\n    \\caption{The perturbative $\\mathcal{O}(\\alpha_s)$ corrections to the quark condensate $\\langle\\bar{q}q\\rangle_N$. }\n\\end{figure}\n\n\nI calculate the contributions of other condensates at the tree level, the calculations are straightforward and  cumbersome. In calculations, I use the following formulas,\n\\begin{eqnarray}\n\\langle\nq_{\\alpha}(x)\\bar{q}_{\\beta}(0)\\rangle_N&=&-\\frac{1}{4}\\left[\\left(\\langle\\bar{q}q\\rangle_N+x^{\\mu}\\langle\\bar{q}D_{\\mu}q\\rangle_N\n+\\frac{1}{2}x^{\\mu}x^{\\nu}\\langle\\bar{q}D_{\\mu}D_{\\nu}q\\rangle_N\n+\\cdots\\right)\\delta_{\\alpha\\beta}\\right.\\nonumber \\\\\n&&\\left.+\\left(\\langle\\bar{q}\\gamma_{\\lambda}q\\rangle_N+x^{\\mu}\\langle\\bar{q}\n\\gamma_{\\lambda}D_{\\mu} q\\rangle_N\n+\\frac{1}{2}x^{\\mu}x^{\\nu}\\langle\\bar{q}\\gamma_{\\lambda}D_{\\mu}D_{\\nu}\nq\\rangle_N\n+\\cdots\\right)\\gamma^{\\lambda}_{\\alpha\\beta} \\right] \\, ,\n\\end{eqnarray}\n and\n\\begin{eqnarray}\n\\langle\ng_{s}q^i_{\\alpha}\\bar{q}^j_{\\beta}G_{\\mu\\nu}^{a}\\rangle_N&=&-\\frac{1}{96} \\frac{\\lambda^a_{ij}}{2}\\left\\{\\langle g_{s}\\bar{q}\\sigma Gq\\rangle_N\\left[\\sigma_{\\mu\\nu}+i(u_{\\mu}\\gamma_{\\nu}-u_{\\nu}\\gamma_{\\mu\n})\n\\!\\not\\! {u}\\right]_{\\alpha\\beta} +\\langle g_{s}\\bar{q}\\!\\not\\! {u}\\sigma Gq\\rangle_N\\right.\\nonumber\\\\\n&&\\left.\\left[\\sigma_{\\mu\\nu}\\!\\not\\!\n{u}+i(u_{\\mu}\\gamma_{\\nu}-u_{\\nu}\\gamma_{\\mu}\n)\\right]_{\\alpha\\beta}\n-4\\langle\\bar{q}u D u D q\\rangle_N\\left[\\sigma_{\\mu\\nu}+2i(u_{\\mu}\\gamma_{\\nu}-u_{\\nu}\\gamma_{\\mu}\n)\\!\\not\\! {u}\\right]_{\\alpha\\beta}\\right\\} \\, , \\nonumber\\\\\n\\end{eqnarray}\nwhere  $D_\\mu=\\partial_\\mu-ig_s\\frac{\\lambda^a}{2}G^a_\\mu$,\n\\begin{eqnarray} \\label{ }\n\\langle\\bar{q}\\gamma_{\\mu}q\\rangle_N&=&\\langle\\bar{q}\\!\\not\\!{u}q\\rangle_N\nu_{\\mu} \\, , \\nonumber \\\\\n\\langle\\bar{q}D_{\\mu}q\\rangle_N&=&\\langle\\bar{q}u D\nq\\rangle_N\nu_{\\mu}=0\\, , \\nonumber \\\\\n\\langle\\bar{q}\\gamma_{\\mu}D_{\\nu}q\\rangle_N&=&\\frac{4}{3}\\langle\\bar{q}\n\\!\\not\\! {u}u D q\\rangle_N\\left(u_{\\mu}u_{\\nu}-\\frac{1}{4}g_{\\mu\\nu}\\right) \\, , \\nonumber \\\\\n\\langle\\bar{q}D_{\\mu}D_{\\nu}q\\rangle_N&=&\\frac{4}{3}\\langle\\bar{q}\nu D u D q\\rangle_N\\left(u_{\\mu}u_{\\nu}-\\frac{1}{4}g_{\\mu\\nu}\\right)\n-\\frac{1}{6} \\langle\ng_{s}\\bar{q}\\sigma Gq\\rangle_N\\left(u_{\\mu}u_{\\nu}-g_{\\mu\\nu}\\right) \\, , \\nonumber \\\\\n\\langle\\bar{q}\\gamma_{\\lambda}D_{\\mu}D_{\\nu}q\\rangle_N&=&2\\langle\\bar{q}\n\\!\\not\\! {u}u  D u  D\nq\\rangle_N\\left[u_{\\lambda}u_{\\mu}u_{\\nu} -\\frac{1}{6}\n\\left(u_{\\lambda}g_{\\mu\\nu}+u_{\\mu}g_{\\lambda\\nu}+u_{\\nu}g_{\\lambda\\mu}\\right)\\right]\n\\nonumber\\\\\n&&-\\frac{1}{6} \\langle\ng_{s}\\bar{q}\\!\\not\\! {u}\\sigma Gq\\rangle_N(u_{\\lambda}u_{\\mu}u_{\\nu}-u_{\\lambda}g_{\\mu\\nu}) \\, ,\n\\end{eqnarray}\nand\n\\begin{equation}\n \\langle\nG_{\\alpha\\beta}^{a}G_{\\mu\\nu}^{b}\\rangle_N=\\frac{\\delta^{ab}}{96}\n\\langle\nGG\\rangle_N\\left(g_{\\alpha\\mu}g_{\\beta\\nu}-g_{\\alpha\\nu}g_{\\beta\\mu}\\right)+O\\left(\\langle\n\\textbf{E}^{2}+\\textbf{B}^{2}\\rangle_N\\right).\n\\end{equation}\n\n\n\nOnce analytical results at the level of quark-gluon degree's of  freedom are obtained,\n then I set  $\\omega^2=q^2$, and take the\nquark-hadron duality below the continuum threshold $s_0$, and perform the Borel transform  with respect\nto the variable $Q^2=-\\omega^2$, finally   obtain  the following QCD sum\nrules:\n\\begin{eqnarray}\n a\\, C_a+b\\, C_b &=&C_f \\, ,\n\\end{eqnarray}\n\\begin{eqnarray}\nC_a &=&\\frac{1}{M^2}\\exp\\left(-\\frac{m_{D}^2}{M^2}\\right)-\\frac{s_0}{m_{D}^4}\\exp\\left(-\\frac{s_0}{M^2}\\right) \\, ,\\nonumber\\\\\nC_b&=&\\exp\\left(-\\frac{m_{D}^2}{M^2}\\right)-\\frac{s_0}{m_{D}^2}\\exp\\left(-\\frac{s_0}{M^2}\\right) \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\nC_f&=& \\frac{2m_N(m_H+m_N)}{(m_H+m_N)^2-m_{D}^2}\\left(\\frac{f_{D}m_{D}^2g_{DNH}}{m_c}\\right)^2\\left\\{ \\left[\\frac{1}{M^2}-\\frac{1}{m_{D}^2-(m_H+m_N)^2}\\right] \\exp\\left(-\\frac{m_{D}^2}{M^2}\\right)\\right.\\nonumber\\\\\n&&\\left.+\\frac{1}{(m_H+m_N)^2-m_{D}^2}\\exp\\left(-\\frac{(m_H+m_N)^2}{M^2}\\right)\\right\\}-\\frac{m_c\\langle\\bar{q}q\\rangle_N}{2}\\left\\{1+\\frac{\\alpha_s}{\\pi} \\left[ 6-\\frac{4m_c^2}{3M^2} \\right.\\right.\\nonumber\\\\\n&&\\left.\\left.-\\frac{2}{3}\\left( 1-\\frac{m_c^2}{M^2}\\right)\\log\\frac{m_c^2}{\\mu^2}-2\\Gamma\\left(0,\\frac{m_c^2}{M^2}\\right)\\exp\\left( \\frac{m_c^2}{M^2}\\right) \\right]\\right\\}\\exp\\left(- \\frac{m_c^2}{M^2}\\right) \\nonumber\\\\\n&&+\\frac{1}{2}\\left\\{-2\\left(1-\\frac{m_c^2}{M^2}\\right)\\langle q^\\dag i D_0q\\rangle_N +\\frac{4m_c\n}{M^2}\\left(1-\\frac{m_c^2}{2M^2}\\right)\\langle \\bar{q} i D_0 i D_0q\\rangle_N+\\frac{1}{12}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle_N\\right\\} \\nonumber\\\\\n&&\\exp\\left(- \\frac{m_c^2}{M^2}\\right)\\, ,\n\\end{eqnarray}\n for the current $\\eta_5(x)$,\n \\begin{eqnarray}\n C_i &\\to& C_i\\left( {\\rm with}\\,\\,\\, m_N \\to -m_N\\, , \\,\\,\\,m_c \\to -m_c\\, , \\,\\,\\,D \\to D_0\\right)\\, ,\n \\end{eqnarray}\nfor the current $\\eta(x)$,\n\\begin{eqnarray}\nC_a &=&\\frac{1}{M^2}\\exp\\left(-\\frac{m_{D^*}^2}{M^2}\\right)-\\frac{s_0}{m_{D^*}^4}\\exp\\left(-\\frac{s_0}{M^2}\\right) \\, ,\\nonumber\\\\\nC_b&=&\\exp\\left(-\\frac{m_{D^*}^2}{M^2}\\right)-\\frac{s_0}{m_{D^*}^2}\\exp\\left(-\\frac{s_0}{M^2}\\right) \\, ,\n\\end{eqnarray}\n\\begin{eqnarray}\nC_f&=& \\frac{2m_N(m_H+m_N)}{(m_H+m_N)^2-m_{D^*}^2}\\left(f_{D^*}m_{D^*}g_{D^*NH}\\right)^2\\left\\{ \\left[\\frac{1}{M^2}-\\frac{1}{m_{D^*}^2-(m_H+m_N)^2}\\right] \\exp\\left(-\\frac{m_{D^*}^2}{M^2}\\right)\\right.\\nonumber\\\\\n&&\\left.+\\frac{1}{(m_H+m_N)^2-m_{D^*}^2}\\exp\\left(-\\frac{(m_H+m_N)^2}{M^2}\\right)\\right\\}-\\frac{m_c\\langle\\bar{q}q\\rangle_N}{2}\\left\\{1+\\frac{\\alpha_s}{\\pi} \\left[ \\frac{8}{3}-\\frac{4m_c^2}{3M^2} \\right.\\right.\\nonumber\\\\\n&&\\left.\\left.+\\frac{2}{3}\\left( 2+\\frac{m_c^2}{M^2}\\right)\\log\\frac{m_c^2}{\\mu^2}-\\frac{2m_c^2}{3M^2}\\Gamma\\left(0,\\frac{m_c^2}{M^2}\\right)\\exp\\left( \\frac{m_c^2}{M^2}\\right) \\right]\\right\\}\\exp\\left(- \\frac{m_c^2}{M^2}\\right) \\nonumber\\\\\n&&+\\frac{1}{2}\\left\\{-\\frac{4\\langle q^\\dag i D_0q\\rangle_N}{3} +\\frac{2m_c^2\\langle q^\\dag i D_0q\\rangle_N}{M^2}+\\frac{2m_c\\langle\\bar{q}g_s\\sigma Gq\\rangle_N}{3M^2}+\\frac{16m_c\\langle \\bar{q} i D_0 i D_0q\\rangle_N}{3M^2}\\right.\\nonumber\\\\\n&&\\left.-\\frac{2m_c^3\\langle \\bar{q} i D_0 i D_0q\\rangle_N}{M^4}-\\frac{1}{12}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle_N\\right\\}\\exp\\left(- \\frac{m_c^2}{M^2}\\right)\\, ,\n\\end{eqnarray}\n for the current $\\eta_\\mu(x)$,\n\\begin{eqnarray}\n C_i &\\to& C_i\\left( {\\rm with}\\,\\,\\, m_N \\to -m_N\\, , \\,\\,\\,m_c \\to -m_c\\, , \\,\\,\\,D^* \\to D_1\\right)\\, ,\n \\end{eqnarray}\nfor the current $\\eta_{5\\mu}(x)$, where $i=a,b,f$.\nIn this article, I neglect the contributions from the heavy quark condensates $\\langle \\bar{Q}Q\\rangle$, $\\langle \\bar{Q}Q\\rangle=-\\frac{1}{12\\pi m_Q}\\langle \\frac{\\alpha_s GG}{\\pi} \\rangle$ up to the order $\\mathcal{O}(\\alpha_s)$ (here I count the condensate $ \\langle \\frac{\\alpha_s GG}{\\pi} \\rangle$ as of the order $\\mathcal{O}(\\alpha_s)$), the heavy quark condensates have practically no effect on the polarization\nfunctions, for detailed discussions about this subject, one can consult Ref.\\cite{PRT85}.\nIn Ref.\\cite{QQcond}, Buchheim, Hilger and Kampfer study the contributions of the  condensates involve the heavy quarks in details, the results indicate that\nthose condensates are either suppressed by the heavy quark mass $m_Q$ or by the additional factor $\\frac{\\alpha_s}{4\\pi}$ (or $g_s^2\/(4\\pi)^2$).\nNeglecting  the in-medium effects on the heavy quark condensates cannot affect the predictions remarkably, as the main contributions come from the terms $\\langle \\bar{q}q\\rangle_N$.\n\n\nDifferentiate  above equation with respect to  $\\tau=\\frac{1}{M^2}$, then\neliminate the\n parameter $b$ ($a$), I can obtain the QCD sum rules for\n the parameter $a$ ($b$),\n \\begin{eqnarray}\n a&=&\\frac{C_f\\left(-\\frac{d}{d\\tau}\\right)C_b-C_b\\left(-\\frac{d}{d\\tau}\\right)C_f}{C_a\\left(-\\frac{d}{d\\tau}\\right)C_b-C_b\\left(-\\frac{d}{d\\tau}\\right)C_a}\\, , \\nonumber\\\\\n  b&=&\\frac{C_f\\left(-\\frac{d}{d\\tau}\\right)C_a-C_a\\left(-\\frac{d}{d\\tau}\\right)C_f}{C_b\\left(-\\frac{d}{d\\tau}\\right)C_a-C_a\\left(-\\frac{d}{d\\tau}\\right)C_b}\\, .\n \\end{eqnarray}\n With the simple replacements $m_c \\to m_b$, $D\/D_0\/D^*\/D_1 \\to B\/B_0\/B^*\/B_1$, $\\Lambda_c \\to \\Lambda_b$ and $\\Sigma_c \\to \\Sigma_b$,\n I can obtain the corresponding the QCD sum rules for  the bottom  mesons in the nuclear matter.\n\n\n\n\\section{Numerical results and discussions}\nAt the normal nuclear matter with the saturation density\n$\\rho_N=\\rho_0=\\frac{2p_F^3}{3\\pi^2}$, where  the Fermi momentum\n$p_F=0.27\\,\\rm{GeV}$ is a small quantity,\n  the condensates $\\langle{\\cal {O}}\\rangle_{\\rho_N}$ in the nuclear matter can be approximated as\n$\\langle{\\cal{O}}\\rangle_{\\rho_N} =\\langle{\\cal{O}}\\rangle+\\frac{\\rho_N}{2m_N}\\langle {\\cal{O}}\\rangle_N $,   the terms\nproportional to $p_F^4$, $p_F^5$, $p_F^6$, $\\cdots$ can be neglected safely,\nwhere the   $\\langle{\\cal{O}}\\rangle=\\langle0|{\\cal{O}}|0\\rangle$ and $\\langle\n{\\cal{O}}\\rangle_N=\\langle N|\n{\\cal{O}}|N\\rangle$ denote the vacuum condensates and nuclear matter induced condensates,  respectively \\cite{Drukarev1991}.\n\n\nThe  input parameters at the QCD side are taken as $\\rho_N=(0.11\\,\\rm{GeV})^3$,\n  $\\langle\\bar{q} q\\rangle_N={\\sigma_N \\over m_u+m_d } (2m_N)$,\n $\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle_N= - 0.65 \\,{\\rm {GeV}} (2m_N)$, $\\sigma_N=45\\,\\rm{MeV}$,\n $m_u+m_d=12\\,\\rm{MeV}$,\n$\\langle q^\\dagger iD_0 q\\rangle_N=0.18 \\,{\\rm{GeV}}(2m_N)$,\n$\\langle\\bar{q}g_s\\sigma G q\\rangle_N=3.0\\,{\\rm GeV}^2(2m_N) $,\n$\\langle \\bar{q} iD_0iD_0\nq\\rangle_N+{1\\over8}\\langle\\bar{q}g_s\\sigma G\nq\\rangle_N=0.3\\,{\\rm{GeV}}^2(2m_N)$,\n $m_N=0.94\\,\\rm{GeV}$ \\cite{C-parameter}, $m_c=(1.3\\pm0.1)\\,\\rm{GeV}$, $m_b=(4.7\\pm0.1)\\,\\rm{GeV}$, $\\alpha_s=0.45$ and $\\mu=1\\,\\rm{GeV}$.\n  If we take the normalization $\\langle N(\\mbox{\\boldmath $p$})|N(\\mbox{\\boldmath\n$p$}')\\rangle = (2\\pi)^{3} \\delta^{3}(\\mbox{\\boldmath\n$p$}-\\mbox{\\boldmath $p$}')$, then $\\langle{\\cal{O}}\\rangle_{\\rho_N} =\\langle{\\cal{O}}\\rangle+\\rho_N\\langle {\\cal{O}}\\rangle_N $, the unit $2m_N$ in the brackets in the values of  the condensates $\\langle\\bar{q} q\\rangle_N$, $\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle_N$, $\\cdots$ disappears.\nI choose the values of the nuclear matter  induced condensates determined in Ref.\\cite{C-parameter}, which are still widely used in the literatures.\nAlthough the values of some condensates are updated, those condensates are irrelevant to the present work. The updates focus  on\nthe four-quark condensate \\cite{update-4q}. In this article, I take into account the condensates up to dimension-5, the four-quark condensates have no contributions, the dominant  contributions come from the nuclear matter induced condensate $\\langle\\bar{q} q\\rangle_N$,  $\\langle\\bar{q} q\\rangle_N={\\sigma_N \\over m_u+m_d } (2m_N)$. The value $m_u+m_d=12\\,\\rm{MeV}$ is obtained from the famous Gell-Mann-Oakes-Renner relation at the energy scale $\\mu=1\\,\\rm{GeV}$, while the value $\\sigma_N=45\\,\\rm{MeV}$ is still widely used \\cite{update-4q}.\n\n\nThe parameters at the hadronic side are taken as\n$m_D=1.870\\,\\rm{GeV}$, $m_B=5.280\\,\\rm{GeV}$, $m_{D_0}=2.355\\,\\rm{GeV}$, $m_{B_0}=5.740\\,\\rm{}GeV$,\n $m_{D^*}=2.010\\,\\rm{GeV}$, $m_{B^*}=5.325\\,\\rm{GeV}$,  $m_{D_1}=2.420\\,\\rm{GeV}$, $m_{B_1}=5.750\\,\\rm{}GeV$,\n$f_{D}=0.210\\,\\rm{GeV}$, $f_B=0.190\\,\\rm{GeV}$,\n $f_{D_0}=0.334 \\frac{m_c}{m_{D_0}}\\,\\rm{GeV}$,\n$f_{B_0}=0.280\\frac{m_b}{m_{B_0}}\\,\\rm{GeV}$,\n$f_{D^*}=0.270\\,\\rm{GeV}$, $f_{B^*}=0.195\\,\\rm{GeV}$,\n $f_{D_1}=0.305\\,\\rm{GeV}$,\n$f_{B_1}=0.255\\,\\rm{GeV}$,\n $s^0_{D}=(6.2\\pm0.5)\\,\\rm{GeV}^2$, $s^0_{B}=(33.5\\pm1.0)\\,\\rm{GeV}^2$,\n$s^0_{D^*}=(6.5\\pm0.5)\\,\\rm{GeV}^2$, $s^0_{B^*}=(35.0\\pm1.0)\\,\\rm{GeV}^2$,\n$s^0_{D_0}=(8.0\\pm0.5)\\,\\rm{GeV}^2$, $s^0_{B_0}=(39.0\\pm1.0)\\,\\rm{GeV}^2$,\n$s^0_{D_1}=(8.5\\pm0.5)\\,\\rm{GeV}^2$ and $s^0_{B_1}=(39.0\\pm1.0)\\,\\rm{GeV}^2$, which are determined by the conventional two-point correlation functions\n using the QCD sum rules \\cite{WangHuang,WangJHEP}. I neglect the uncertainties of the decay constants to avoid double  counting as the main uncertainties of the decay constants originate from the uncertainties of the continuum threshold parameters $s_0$.\n\n\nThe value of the strong coupling constant $g_{DN\\Lambda_c}$ is $g_{\\Lambda_c DN}=6.74$ from\nthe  QCD sum rules \\cite{Nielsen98}, while the average value of the strong coupling constants $g_{\\Lambda_cDN}$ and $g_{\\Sigma_cDN}$ from the light-cone QCD sum rules is $\\frac{g_{\\Lambda_cDN}+g_{\\Sigma_cDN}}{2}=6.775$ \\cite{Khodjamirian1108}, those values are consistent with each other.\nThe average value of the strong coupling constants $g_{\\Lambda_c D^*N}$ and $g_{\\Sigma_c D^*N}$ from the light-cone QCD sum rules is  $\\frac{g_{\\Lambda_c D^*N}+g_{\\Sigma_cD^*N}}{2}=3.86$ \\cite{Khodjamirian1108}.  In this\narticle, I take the approximation\n$g_{DN\\Lambda_c}\\approx\ng_{DN\\Sigma_c}\\approx g_{BN\\Lambda_b}\\approx\ng_{BN\\Sigma_b}\\approx g_{D_0N\\Lambda_c}\\approx\ng_{D_0N\\Sigma_c}\\approx g_{B_0N\\Lambda_b}\\approx\ng_{B_0N\\Sigma_b}\\approx6.74$ and\n$g_{\\Lambda_cD^*N}\\approx g_{\\Sigma_cD^*N}\n\\approx g_{\\Lambda_cD_1N}\\approx g_{\\Sigma_cD_1N}\\approx g_{\\Lambda_bB^*N}\\approx g_{\\Sigma_bB^*N}\n\\approx g_{\\Lambda_bB_1N}\\approx g_{\\Sigma_bB_1N}\\approx3.86$.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[totalheight=5cm,width=7cm]{massPD.EPS}\n  \\includegraphics[totalheight=5cm,width=7cm]{massVD.EPS}\n  \\includegraphics[totalheight=5cm,width=7cm]{massSD.EPS}\n  \\includegraphics[totalheight=5cm,width=7cm]{massAD.EPS}\n   \\includegraphics[totalheight=5cm,width=7cm]{massPB.EPS}\n    \\includegraphics[totalheight=5cm,width=7cm]{massVB.EPS}\n    \\includegraphics[totalheight=5cm,width=7cm]{massSB.EPS}\n    \\includegraphics[totalheight=5cm,width=7cm]{massAB.EPS}\n \\caption{(Color online) The shifts of the masses  of the heavy mesons in the nuclear matter with variations  of the Borel parameter $M^2$, the I (II) denotes contributions up to  the next-to-leading order (leading order) are included. }\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[totalheight=5cm,width=7cm]{decayPD.EPS}\n  \\includegraphics[totalheight=5cm,width=7cm]{decayVD.EPS}\n  \\includegraphics[totalheight=5cm,width=7cm]{decaySD.EPS}\n  \\includegraphics[totalheight=5cm,width=7cm]{decayAD.EPS}\n  \\includegraphics[totalheight=5cm,width=7cm]{decayPB.EPS}\n  \\includegraphics[totalheight=5cm,width=7cm]{decayVB.EPS}\n   \\includegraphics[totalheight=5cm,width=7cm]{decaySB.EPS}\n    \\includegraphics[totalheight=5cm,width=7cm]{decayAB.EPS}\n \\caption{(Color online) The shifts of the decay constants  of the heavy mesons in the nuclear matter with variations  of the Borel parameter $M^2$, the I (II) denotes contributions up to  the next-to-leading order (leading order) are included. }\n\\end{figure}\n\nIn Figs.2-3, I plot the shifts of the masses and decay constants of the heavy mesons   in the nuclear matter with variations  of  the Borel\nparameter $M^2$, respectively. From the figures, I can see that there appear platforms. In this article, I choose the Borel parameters $M^2$ according to the criterion that the uncertainties originate from the Borel parameters $M^2$  are  negligible. The values of the Borel parameters $M^2$ are shown explicitly in Table 1.\nFrom Figs.2-3 and Table 1, I can see that the Borel parameters $M^2$ in the QCD sum rules for the mass-shift $\\delta m$ and decay-constant-shift $\\delta f$ of the same meson are different. It is not un-acceptable, as the mass-shift $\\delta m$ and decay-constant-shift $\\delta f$ come from different QCD sum rules, not coupled QCD sum rules, see Eq.(39), the platforms maybe appear in different places  in different QCD sum rules.\n\nI can obtain the shifts of the masses and decay constants of the heavy mesons in the nuclear matter in the Borel windows, which are shown explicitly in Table 2.\nFrom the Table 2, I can obtain the fractions of the shifts $\\frac{\\delta m_{D\/D^*\/D_0\/D_1}}{m_{D\/D^*\/D_0\/D_1}}\\leq 5\\%$, $\\frac{\\delta f_{D\/D^*\/D_0\/D_1}}{f_{D\/D^*\/D_0\/D_1}}\\leq 10\\%$, $\\frac{\\delta m_{B\/B^*\/B_0\/B_1}}{m_{B\/B^*\/B_0\/B_1}}= (5-15)\\%$ and $\\frac{\\delta f_{B\/B^*\/B_0\/B_1}}{f_{B\/B^*\/B_0\/B_1}}= (25-55)\\%$, which are shown explicitly in Table 3.\nIn calculations, I observe that the main contributions come from the terms $m_c\\langle\\bar{q}q\\rangle_N$ and   $m_b\\langle\\bar{q}q\\rangle_N$.\n From Table 3, I can see that the next-to-leading order corrections $\\alpha_s \\langle\\bar{q}q\\rangle_N$ are important. In the case of  the shifts $\\frac{\\delta m_{B^*\/B_1}}{m_{B^*\/B_1}}$ and $\\frac{\\delta f_{B^*\/B_1}}{f_{B^*\/B_1}}$, the next-to-leading order contributions $\\alpha_s \\langle\\bar{q}q\\rangle_N$ and the leading order contributions $\\langle\\bar{q}q\\rangle_N$ are almost equivalent. In this article, I choose the special energy scale $\\mu=1\\,\\rm{GeV}$. The logarithm $\\log \\frac{m_b^2}{\\mu^2}$ in the next-to-leading contributions is very large and enhances the next-to-leading  contributions greatly.  Although the nuclear matter induced condensates  evolve with the   renormalization group equation, their  evolving behaviors with the energy scales are not well known, as  this subject has not been studied in details yet at the present time. A larger energy scale $\\mu$ can lead to smaller logarithm $\\log \\frac{m_b^2}{\\mu^2}$ therefore  more reasonable predictions. In Table 4, I present the main uncertainties, which originate from the uncertainties of the heavy quark masses and the continuum threshold parameters.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n       &$\\delta m_{D}$ &$\\delta m_{D^*}$ &$\\delta m_{D_0}$  &$\\delta m_{D_1}$  &$\\delta m_{B}$  &$\\delta m_{B^*}$  &$\\delta m_{B_0}$  &$\\delta m_{B_1}$\\\\ \\hline\n $M^2$ &$4.4-5.4$      &$4.6-5.6$        &$6.0-7.0$         &$6.6-7.6$         &$29-33$         &$30-34$           &$32-36$           &$32-36$      \\\\ \\hline\\hline\n       &$\\delta f_{D}$ &$\\delta f_{D^*}$ &$\\delta f_{D_0}$  &$\\delta f_{D_1}$  &$\\delta f_{B}$  &$\\delta f_{B^*}$  &$\\delta f_{B_0}$  &$\\delta f_{B_1}$\\\\ \\hline\n $M^2$ &$1.9-2.9$      &$3.5-4.5$        &$4.3-5.3$         &$5.3-6.3$         &$25-29$         &$27-31$           &$30-34$           &$31-35$    \\\\ \\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The Borel parameters in the QCD sum rules for the shifts of the masses and decay constants of the heavy mesons in the nuclear matter, the unit is $\\rm{GeV}^2$.   }\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n           &$\\delta m_{D}$ &$\\delta m_{D^*}$ &$\\delta m_{D_0}$  &$\\delta m_{D_1}$  &$\\delta m_{B}$  &$\\delta m_{B^*}$  &$\\delta m_{B_0}$  &$\\delta m_{B_1}$\\\\ \\hline\n{\\rm NLO}  &$-72$          &$-102$           &$80$              &$97$              &$-473$          &$-687$            &$295$             &$522$   \\\\ \\hline\n{\\rm LO}   &$-47$          &$-70$            &$54$              &$66$              &$-329$          &$-340$            &$209$             &$260$   \\\\ \\hline\n\\cite{Hay} &$-48$          &                 &                  &                  &                &                  &                  &\\\\ \\hline\n\\cite{Hil} &$+45$          &                 &                  &                  & $+60$          &                  &                  &\\\\ \\hline\n\\cite{Azi} &$-46$          &                 &                  &                  & $-242$         &                  &                  &\\\\ \\hline\\hline\n           &$\\delta f_{D}$ &$\\delta f_{D^*}$ &$\\delta f_{D_0}$  &$\\delta f_{D_1}$  &$\\delta f_{B}$  &$\\delta f_{B^*}$  &$\\delta f_{B_0}$  &$\\delta f_{B_1}$\\\\ \\hline\n{\\rm NLO}  &$-6$           &$-26$            &$11$              &$31$              &$-71$           &$-111$            &$56$              &$134$  \\\\ \\hline\n{\\rm LO}   &$-4$           &$-18$            &$7$               &$21$              &$-48$           &$-55$             &$39$              &$67$  \\\\ \\hline\n\\cite{Azi} &$-2$           &                 &                  &                  & $-23$          &                  &                  &\\\\ \\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The shifts of the masses and decay constants of the heavy mesons in the nuclear matter, where the NLO (LO) denotes contributions up to  the next-to-leading order (leading order) are included, the unit is MeV.  }\n\\end{table}\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n           &$\\frac{\\delta m_{D}}{m_{D}}$   &$\\frac{\\delta m_{D^*}}{m_{D^*}}$  &$\\frac{\\delta m_{D_0}}{m_{D_0}}$   &$\\frac{\\delta m_{D_1}}{m_{D_1}}$\n           &$\\frac{\\delta m_{B}}{ m_{B}}$  &$\\frac{\\delta m_{B^*}}{m_{B^*}}$  &$\\frac{\\delta m_{B_0}}{m_{B_0}}$   &$\\frac{\\delta m_{B_1}}{m_{B_1}}$\\\\ \\hline\n{\\rm NLO}  &$-4\\%$         &$-5\\%$           &$3\\%$             &$4\\%$             &$-9\\%$          &$-13\\%$           &$5\\%$             &$9\\%$   \\\\ \\hline\n{\\rm LO}   &$-3\\%$         &$-3\\%$           &$2\\%$             &$3\\%$             &$-6\\%$          &$-6\\%$            &$4\\%$             &$5\\%$   \\\\  \\hline\\hline\n           &$\\frac{\\delta f_{D}}{f_{D}}$   &$\\frac{\\delta f_{D^*}}{f_{D^*}}$  &$\\frac{\\delta f_{D_0}}{f_{D_0}}$   &$\\frac{\\delta f_{D_1}}{f_{D_1}}$\n           &$\\frac{\\delta f_{B}}{f_{B}}$   &$\\frac{\\delta f_{B^*}}{f_{B^*}}$  &$\\frac{\\delta f_{B_0}}{f_{B_0}}$   &$\\frac{\\delta f_{B_1}}{f_{B_1}}$\\\\ \\hline\n{\\rm NLO}  &$-3\\%$         &$-10\\%$          &$6\\%$             &$10\\%$            &$-37\\%$         &$-57\\%$           &$24\\%$            &$53\\%$  \\\\ \\hline\n{\\rm LO}   &$-2\\%$         &$-7\\%$           &$4\\%$             &$7\\%$             &$-25\\%$         &$-28\\%$           &$17\\%$            &$26\\%$  \\\\ \\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The fractions of the  shifts of the masses and decay constants of the heavy mesons in the nuclear matter, where the NLO (LO) denotes contributions up to  the next-to-leading order (leading order) are included.  }\n\\end{table}\n\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n             &$\\delta (\\delta m_{D})$   &$\\delta(\\delta m_{D^*})$   &$\\delta(\\delta m_{D_0})$     &$\\delta(\\delta m_{D_1})$\n             &$\\delta(\\delta m_{B})$    &$\\delta(\\delta m_{B^*})$   &$\\delta(\\delta m_{B_0})$     &$\\delta(\\delta m_{B_1})$              \\\\ \\hline\n$\\delta m_Q$ &$\\pm14$                   &$\\pm4$                     &$\\pm26$                      &$\\pm6$\n             &$\\pm18$                   &$\\pm1$                     &$\\pm25$                      &$\\pm1$   \\\\ \\hline\n$\\delta s_0$ &$\\pm9$                    &$\\pm14$                    &$\\pm12$                      &$\\pm13$\n             &$\\pm65$                   &$\\pm80$                    &$\\pm39$                      &$\\pm70$   \\\\ \\hline \\hline\n             &$\\delta (\\delta f_{D}$)   &$\\delta (\\delta f_{D^*})$  &$\\delta (\\delta f_{D_0})$    &$\\delta (\\delta f_{D_1})$\n             &$\\delta (\\delta f_{B})$   &$\\delta (\\delta f_{B^*})$  &$\\delta (\\delta f_{B_0})$    &$\\delta (\\delta f_{B_1})$\\\\ \\hline\n$\\delta m_Q$ &$\\pm1$                    &$\\pm1$                     &$\\pm3$                       &$\\pm2$\n             &$\\pm2$                    &$\\pm1$                     &$\\pm3$                       &$\\pm0$   \\\\ \\hline\n$\\delta s_0$ &$\\pm1$                    &$\\pm7$                     &$\\pm4$                       &$\\pm8$\n             &$\\pm21$                   &$\\pm25$                    &$\\pm15$                      &$\\pm34$   \\\\ \\hline \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The uncertainties of the shifts of\n    the masses and decay constants of the heavy mesons in the nuclear matter originate from the uncertainties of the heavy quark masses and continuum threshold parameters, where the unit is MeV.  }\n\\end{table}\n\n\nThe mass-shifts of the negative (positive) parity  mesons are negative (positive), the decays of the high\ncharmonium  states to the $D\\bar{D}$ and $D^*\\bar{D}^*$ ($D_0\\bar{D}_0$ and $D_1\\bar{D}_1$) pairs are  enhanced (suppressed) in the phase space,\n and we  should take into account those effects carefully in studying the production of the  $J\/\\psi$ so as to identifying the quark-gluon plasmas.\n\n\n The  currents $\\bar{Q}  q$ and $\\bar{Q}i\\gamma_5 q$ (also $\\bar{Q}\\gamma_\\mu q$ and $\\bar{Q}\\gamma_\\mu\\gamma_5 q$) are mixed with each other under the  chiral transformation $q\\to e^{i\\alpha\\gamma_5}q$, the  currents $\\bar{Q}  q$, $\\bar{Q}i\\gamma_5 q$, $\\bar{Q}\\gamma_\\mu q$, $\\bar{Q}\\gamma_\\mu\\gamma_5 q$ are not conserved in the limit $m_q \\to 0$, it is better to take the doublets $(D,D_0)$ and $(D^*,D_1)$ as  the parity-doublets rather than  the chiral-doublets.\nThe  quark condensate $\\langle\\bar{q}q\\rangle_{\\rho_N}$ serves as the order parameter,\n  and  undergoes  reduction in the nuclear matter,\n    the chiral symmetry is partially  restored; however, there appear new medium-induced condensates, which also break the chiral symmetry. In this article, the $\\langle\n{\\cal{O}}\\rangle_N$  are companied by  the heavy quark masses $m_Q$, $m_Q^2$ or $m_Q^3$, the net effects cannot warrant that  the chiral symmetry is monotonously restored with the increase of the $\\rho_N$.\n When the $\\rho_N$ is large enough, the order parameter $\\langle\\bar{q}q\\rangle_{\\rho_N} \\to 0$, the chiral symmetry is restored, the Fermi  gas  approximation for the nuclear matter breaks down,  and the parity-doublets\n  maybe have degenerated  masses approximately. In this article, I study the parity-doublets  at the low $\\rho_N$, the mass breaking effects of the parity-doublets   maybe even larger, see Table 2.\n  We expect that smaller mass splitting of the parity-doublets at the high nuclear density is favored, however, larger  mass splitting of the parity-doublets at the lower nuclear density cannot be excluded.\n  In Refs.\\cite{Hil,Hilger2}, the mass center $\\overline{m}_P$ of the pseudoscalar mesons increases in the nuclear matter while the mass center $\\overline{m}_S$ of the scalar mesons  decreases in the nuclear matter,  the mass breaking effect  $\\overline{m}_S-\\overline{m}_P$ of the parity-doublets is  smaller than that in the vacuum.\n\n\n\nIn Table 5, I show the scattering lengths $a_{D}$,  $a_{D^*}$,  $a_{D_0}$,  $a_{D_1}$,  $a_{B}$,  $a_{B^*}$,   $a_{B_0}$,  $a_{B_1}$ explicitly, the  $a_{D}$,  $a_{D^*}$,    $a_{B}$,  $a_{B^*}$ are negative, which indicate the interactions $DN$,  $D^*N$,    $BN$,  $B^*N$ are attractive,  the  $a_{D_0}$,  $a_{D_1}$,     $a_{B_0}$,  $a_{B_1}$ are positive, which indicate the interactions $D_0N$,  $D_1N$,    $B_0N$,  $B_0N$ are repulsive. It is difficult (possible)  to form the $D_0N$,  $D_1N$,    $B_0N$,  $B_0N$ ($DN$,  $D^*N$,    $BN$,  $B^*N$) bound states. In Ref.\\cite{ZhangDN}, Zhang studies the $S$-wave $DN$ and $D^*N$ bound states with the QCD sum rules, the numerical results indicate that  the $\\Sigma_c(2800)$  and $\\Lambda_c(2940)$   can  be assigned to be the $S$-wave $DN$  state with $J^P ={\\frac{1}{2}}^-$ and  the $S$-wave $D^*N$  state with $J^P ={\\frac{3}{2}}^-$,  respectively.\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n           &$a_{D}$        &$a_{D^*}$        &$a_{D_0}$         &$a_{D_1}$         &$a_{B}$         &$a_{B^*}$         &$a_{B_0}$         &$a_{B_1}$\\\\ \\hline\n{\\rm NLO}  &$-1.1$         &$-1.5$           &$1.3$             &$1.6$             &$-8.9$          &$-12.9$           &$5.6$             &$9.9$   \\\\ \\hline\n{\\rm LO}   &$-0.7$         &$-1.1$           &$0.9$             &$1.1$             &$-6.2$          &$-6.4$            &$4.0$             &$5.0$   \\\\ \\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The heavy-meson-nucleon scattering lengths, where the NLO (LO) denotes contributions up to  the next-to-leading order (the leading order) are included, the unit is fm.  }\n\\end{table}\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[totalheight=6cm,width=8cm]{Vacuum-PD.EPS}\n   \\caption{(Color online) The contributions of the perturbative term (I) and quark condensate term (II) in the QCD sum rules for the $D$ mesons  in the vacuum. Furthermore, I  show  the  mass $m_D$ (III) and decay constant $f_D$ (IV) explicitly, which are normalized to be 1 at the value $M^2=1.5\\rm{GeV}^2$. }\n\\end{figure}\n\n\n\nIn the present work and Refs.\\cite{Hay,WangHuang,Azi},  the correlation functions  are  divided\ninto  a vacuum part   and a static one-nucleon part, and  the nuclear matter  induced effects are extracted  explicitly;\n  while in Refs.\\cite{Hil,Hilger2},   the pole terms of (or ground state contributions to)  the  hadronic spectral densities of the whole correlation functions are  parameterized as $\\Delta\\Pi(\\omega)= F_{+}\\delta(\\omega-m_{+})- F_{-}\\delta(\\omega+m_{-})$, where $m_{\\pm}=m\\pm\\Delta m$ and $F_{\\pm}=F\\pm\\Delta F$,\nand  QCD  sum rules for the   mass center $\\overline{m}$ and the mass splitting $\\Delta m$ are obtained.\nIn the leading order approximation, the present predictions of  the $\\delta m_{D}$ and $\\delta m_{B}$ are compatible with that of Refs.\\cite{Hay,Azi} and differ greatly from that of Refs.\\cite{Hil,Hilger2}, see Table 2.\nThe values obtained from the QCD sum rules depend heavily on the Borel windows, the values extracted from different Borel windows especially in different QCD sum rules maybe differ from each other greatly.\n\nIn Refs.\\cite{Hil,Hilger2}, the authors study the masses of the heavy mesons in the nuclear matter directly by  including both the vacuum part   and the static one-nucleon part in the QCD sum rules, then the continuum contributions are well approximated by $\\rho_{QCD}(\\omega^2)\\theta\\left(\\omega^2-{\\omega_0^{\\pm}}^2\\right)$, where the ${\\omega_0^{\\pm}}^2$ are the continuum threshold parameters,   it is one of the advantages of Refs.\\cite{Hil,Hilger2}. However, they define the moments $S_n(M^2)$ to study the mass-shifts,\n\\begin{eqnarray}\nS_n(M^2)&=&\\int_{\\omega_0^-}^{\\omega_0^+}d\\omega \\omega^n \\Delta\\Pi(\\omega) \\exp\\left( -\\frac{\\omega^2}{M^2}\\right)\\, ,\n\\end{eqnarray}\nthe odd moment $o=S_0(M^2)$ and the even moment $e=S_1(M^2)$, then obtain $\\frac{do}{d(1\/M^2)}=-S_2(M^2)$ and $\\frac{de}{d(1\/M^2)}=-S_3(M^2)$ by assuming the $F_{\\pm}$ and $m_{\\pm}$ are independent on the Borel parameters  at the phenomenological side.  In fact, $\\frac{do}{d(1\/M^2)}\\neq-S_2(M^2)$ and $\\frac{de}{d(1\/M^2)}\\neq-S_3(M^2)$ at the operator product expansion side according to the QCD spectral densities $\\Delta\\Pi(\\omega)$, which  depend  on the Borel parameters explicitly, the approximations $\\frac{do}{d(1\/M^2)}=-S_2(M^2)$ and $\\frac{de}{d(1\/M^2)}=-S_3(M^2)$ lead to undetermined uncertainties.\n\nIn Refs.\\cite{Hil,Hilger2}, the perturbative $\\mathcal{O}(\\alpha_s)$ corrections  to the  perturbative terms are taken into account. In the QCD sum rules for the pseudoscalar $D$  mesons  in the vacuum, if we take into account the perturbative  $\\mathcal{O}(\\alpha_s)$ corrections to the perturbative term and vacuum condensate term, the  two criteria (pole dominance and convergence of the operator product expansion) of the QCD sum rules leads  to the Borel window $M^2=(1.2-1.8)\\,\\rm{GeV}^2$, the resulting predictions\nof the mass $m_{D}$  and decay constant $f_{D}$  are consistent with the experimental data.  In Fig.4, I plot the contributions of the perturbative term   and quark condensate  term in the operator product expansion. From the figure, I can see that the main contributions come from the perturbative term, the quark condensate $\\langle \\bar{q}q\\rangle$ plays a less important role.\n\nThe modifications of the condensates in the nuclear matter are mild, for example, $\\langle\\bar{q}q\\rangle_{\\rho_N}\\approx 0.64 \\langle\\bar{q}q\\rangle $, while the perturbative contributions are not modified (or modified slightly by introducing a minor  splitting $\\Delta s_0$, ${\\omega_0^{\\pm}}^2=s_0\\pm\\Delta s_0$) by the nuclear matter.\nIf we turn on the in-medium effects, the contributions of  the quark condensate  are even smaller, the Borel windows are determined dominantly\nby the perturbative terms \\cite{Hil,Hilger2}.\nIf the     perturbative  $\\mathcal{O}(\\alpha_s^2)$ corrections to the perturbative terms are also included, the contributions of the perturbative are even larger \\cite{WangJHEP}, the QCD sum rules are  dominated by the perturbative terms, which are not (or slightly) affected by the nuclear matter.\nIt is not favored to extract the  mass-shifts in the nuclear matter, and impairs the predictive ability.\n\nIn the present work and Refs.\\cite{Hay,WangHuang,Azi},  the correlation functions   are divided into\n   the vacuum part   and the static one-nucleon part,  which are of the orders ${\\mathcal{O}}(0)$ and ${\\mathcal{O}}(\\rho_N)$, respectively. We can obtain  independent QCD sum rules from the two parts respectively. The QCD sum rules correspond to the orders ${\\mathcal{O}}(0)$ and ${\\mathcal{O}}(\\rho_N)$ respectively can   have quite different Borel parameters. In this article, I  separate the nuclear matter induced effects unambiguously, study the QCD sum rules correspond to the order ${\\mathcal{O}}(\\rho_N)$, and determine the Borel parameters by the criteria of the QCD sum rules.\n\n\n\nIn the conventional QCD sum rules, we usually choose the  Borel parameters $M^2$  to satisfy the  following three  criteria:\n\n$\\bf{1_\\cdot}$ Pole dominance at the phenomenological side;\n\n$\\bf{2_\\cdot}$ Convergence of the operator product expansion;\n\n$\\bf{3_\\cdot}$ Appearance of the Borel platforms.\n\n\nIn the present work and Refs.\\cite{Hay,WangHuang,Azi}, the nuclear matter  induced effects are extracted  explicitly, the resulting QCD sum rules are not contaminated by the contributions of the vacuum part, the Borel windows are determined completely by the nuclear matter  induced effects,  it  is the advantage.\nAs the QCD spectral densities are of the form $\\delta(\\omega^2-m_Q^2)$, we have to take the hadronic spectral densities to be the  form $\\delta(\\omega^2-m_{H}^2)$ and model the continuum contributions with the function $\\delta(\\omega^2-s_0)$, and determine the $s_0$ by some constraints, see Eq.(16), where the $H$ denotes  the ground state and excited state heavy mesons. In this article, I attribute the higher excited states to the continuum contributions, the $\\delta$-type hadronic spectral densities  make sense. So the pole dominance at the phenomenological side can be  released as the  continuum contributions are already taken into account.\nFurthermore, I expect that the couplings of\n the interpolating  currents  to the excited states are more weak than that to the ground states, the uncertainties originate from continuum contributions are very small. For example, the decay constants of the pseudoscalar mesons $\\pi(140)$ and $\\pi(1300)$ have the hierarchy $f_{\\pi(1300)}\\ll f_{\\pi(140)}$ from the Dyson-Schwinger equation \\cite{CDRoberts}, the lattice QCD \\cite{Latt-pion},  the QCD sum rules \\cite{QCDSR-pion}, etc, or from the experimental data \\cite{pion-exp}.\n\nIn the present work and Refs.\\cite{Hay,WangHuang,Azi}, large Borel parameters are chosen  to warrant the convergence of the operator product expansion and to obtain the Borel platforms, and small Borel parameters cannot lead to platforms. In the Borel windows, where the platforms appear, the main contributions come from terms $\\langle \\bar{q}q\\rangle_N$, the operator product expansion is well convergent. The criteria $\\bf{2}$ and $\\bf{3}$ can be satisfied.  The continuum contributions are not  suppressed efficiently for large Borel parameters compared to that for small Borel parameters.   In calculations, I  observe that the predictions are insensitive to the $s_0$, the uncertainties originate from the continuum threshold parameters $s_0$ are very small in almost all cases,  the large Borel parameters make sense. Furthermore, the    continuum contributions are already taken into account. On the other hand, from Eqs.(8-9) and Eqs.(12-13), we can see that the mass-shifts $\\delta m_{D\/D_0\/D^*\/D_1}$ and decay constant shifts $\\delta f_{D\/D_0\/D^*\/D_1}$ reduce to zero in the limit $\\rho_N \\to 0$, the QCD sum rules correspond to the nuclear matter induced effects decouple, their Borel parameters (irrespective of large or small) are also irrelevant to the ones in the QCD sum rules for the vacuum part of the correlation functions.\nSo the present predictions are sensible.\n\n\nThe predictions depend on the in-medium hadronic spectral functions \\cite{Kwon-2008}, for example, there are two generic prototypes of the  in-medium\nspectral functions for the $\\rho$ meson, they differ in details at the\nlow mass end of the spectrum. The Klingl-Kaise-Weise spectral function\n emphasizes the role of chiral in-medium $\\pi\\pi$ interactions \\cite{KKW-97}, while  the Rapp-Wambach spectral function focuses\non the role of nucleon-hole, $\\Delta(1232)$-hole and $N^*(1520)$-hole excitations \\cite{RW-99}. Both of the\nspectral functions account quite well for the low-mass enhancements observed in dilepton spectra from high-energy nuclear collisions. However, the QCD\nsum rules analysis of the lowest spectral moments reveals qualitative differences with respect to their Brown-Rho  scaling properties \\cite{Kwon-2008}. If the simple spectral densities $F\\delta(\\omega^2-M_{P\/V}^2)$ analogous to the ones in Refs.\\cite{Hil,Hilger2} are taken, where the $P$ denotes the pseudoscalar mesons $\\pi$, $\\eta_c$, the $V$ denotes the vector mesons $\\rho$, $\\omega$, $\\phi$, $J\/\\psi$, the $F$ denotes the constant pole residues, the in-medium  mass-shifts $\\delta M_{P\/V}$ are smaller  than zero qualitatively \\cite{Lee-Vector}. I expect that the S-wave mesons $q^{\\prime}\\bar{q}$, $c\\bar{q}$, $c\\bar{c}$ with the spin-parity $J^P=0^-$ (or $1^-$) have analogous in-medium mass-shifts, at least qualitatively. Further studies based on more sophisticated hadronic spectral densities are needed.\n\n\nIn fact, there are controversies about the mass-shifts of the $D$ and $B$ mesons in the nuclear matter, some  theoretical approaches indicate negative mass-shifts \\cite{DN-BSE-Negative}, while others indicate positive mass-shifts \\cite{Positive-BD}.  The different predictions  originate mainly from whether or not the   heavy pseudoscalar and heavy vector mesons are treated  on equal footing in the coupled-channel approaches. If we obtain   the meson-baryon interaction kernel  by   treating the heavy pseudoscalar and heavy vector mesons on equal footing as required by heavy quark symmetry, the  mass-shift $\\delta M_{D}$ is negative \\cite{DN-BSE-Negative}, which is consistent with the present work;\nfurthermore, the attractive D-nucleus interaction can lead to the formation  of $D$-nucleus bound states, which can be confronted to the experimental data in the future directly \\cite{D-Nuclei}.\n\n\n\nThe upcoming  FAIR  project at GSI\n provides the opportunity to study the in-medium properties of the charmoniums  or charmed hadrons for the first time, however,\nthe high mass of charmed hadrons requires a high momentum in the antiproton beam to produce them, the\nconditions for observing in-medium effects seem unfavorable, as  the hadrons  sensitive\nto the in-medium effects are either at rest or have a small momentum relative to the nuclear\nmedium. We have to find\nprocesses that would slow down the charmed hadrons inside the nuclear matter, but this\nrequires more detailed theoretical studies.\nFurther theoretical studies on the reaction\ndynamics and on the exploration of the experimental\nability  to identify more complicated\nprocesses are still needed.\n\n\n\n\n\\section{Conclusion}\n In this article, I divide  the two-point correlation functions of the  scalar, pseudoscalar, vector and axialvector currents in the nuclear matter\ninto  two parts, i.e. the vacuum part   and the static one-nucleon part, then study  the in-medium modifications of  the masses and decay constants by deriving\n QCD sum rules from the   static one-nucleon part of the two-point correlation functions. In the operator product expansion,\n I calculate the contributions of the nuclear matter induced condensates up to dimension 5,  especially I calculate the next-to-leading order contributions of the in-medium quark condensate and obtain concise expressions, which also have applications in studying the mesons properties in the vacuum.\n  In calculation, I observe that the next-to-leading order contributions of the in-medium quark condensate are very large and should be taken into account.\n\n All in all, I study the properties of the scalar, pseudoscalar, vector and axialvector heavy mesons  with the QCD sum rules in a systematic way,  and obtain    the shifts of the masses and decay constants in the nuclear matter.\n The numerical results indicate that the mass-shifts of the negative parity and positive parity heavy mesons are negative and positive, respectively. For the pseudoscalar   meson $D$, I obtain the prediction $\\delta M_{D}<0$, which is in contrast to the prediction in Refs.\\cite{Hil,Hilger2}, where the mass-shift is positive  $\\delta M_{D}>0$.   In Refs.\\cite{Hil,Hilger2}, the authors study the masses of the heavy mesons in the nuclear matter directly by  including both the vacuum part   and\n  static one-nucleon part with  the QCD sum rules, and   parameterize the spectral density of the whole correlation functions by a simple function\n  $\\Delta\\Pi(\\omega)= F_{+}\\delta(\\omega-m_{+})- F_{-}\\delta(\\omega+m_{-})$.\n  I discuss the differences between the QCD sum rules in the present work and that in Refs.\\cite{Hil,Hilger2} in details, and show why I prefer the present predictions.   In the present work and Refs.\\cite{Hay,WangHuang,Azi,Hil,Hilger2}, the finite widths of the mesons in the nuclear matter are neglected, further   studies based on the  more sophisticated hadronic spectral densities by including the finite widths are  needed.\n\n  As the masses of the heavy meson paries, such as the $D\\bar{D}$, $D^*\\bar{D}^*$, $D_0 \\bar{D}_0$, $D_1 \\bar{D}_1$ are modified in the nuclear environment,\n  we  should take into account those effects carefully in studying the production of the  $J\/\\psi$ (and $\\Upsilon$) so as to identifying the quark-gluon plasmas. Furthermore, I study the heavy-meson-nucleon scattering lengths as a byproduct,\nand obtain the conclusion qualitatively about the possible existence  of the  heavy-meson-nucleon bound states.\n\n\n\n\\section*{Acknowledgements}\nThis  work is supported by National Natural Science Foundation,\nGrant Numbers 11375063, and Natural Science Foundation of Hebei province, Grant Number A2014502017.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Background}\n\\label{sec:Bg}\nThis section first presents the definition of the image spam email and then briefly provides the concept of a convolutional neural network. Finally, we present the description of data augmentation that is a widely used technique~\\cite{Krizhevsky17:dnn} for improving the robustness of deep learning models. It increases the size of labeled training samples by leveraging task-specific data transformations that preserve class labels.\n\n\\subsection{Image Spam Email}\n\nSince image spam emails appeared in 2004, several studies were conducted to formally define image spam emails and construct models to detect image spam emails in academia. Klangpraphan et al.~\\cite{klangpraphant2010pimsi} observed that image spam emails contain an image-based link to a website, which looks like a text. Soranamageswari et al.~\\cite{soranamageswari2010statistical} introduced the definition of image spam email as spam email having at least one image containing spam content.  \n\n\\begin{figure}[!ht]\n\\centering\n\\begin{subfigure}[b]{0.49\\textwidth}{}\n\\centering\n\\includegraphics[width=\\textwidth,height=4.3cm]{pic\/spam_1.JPG}\n\\caption{Spam email containing a link}\n\\label{fig:y equals x}\n\\end{subfigure}\n\\hspace{0.15em}\n\\begin{subfigure}[b]{0.49\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{pic\/spam_2.JPG}\n\\caption{Spam email showing an advertisement}\n\\label{fig:three sin x}\n\\end{subfigure}\n\\caption{Examples of image spam emails.}\n\\label{fig:imagespam}\n\\end{figure}\n\nFigure \\ref{fig:imagespam} shows two examples of image spam emails. In Figure \\ref{fig:imagespam}(a), if a user clicks the ``Verify Email'' button, it tries to visit an attacker's website or download malware. In Figure \\ref{fig:imagespam}(b), the spam image shows unwanted advertisement information to email recipients. Basically, the goal of image spam emails is to hide the attacker's message into an image for circumventing text-based spam filters. Based on this observation, in this paper, we define the image spam email as spam email with images displaying unwanted text information.\n\n\\subsection{Convolutional Neural Network (CNN)}\n\nConvolutional neural network (CNN) is a kind of deep learning methods. Recently, in many classification tasks, CNN outperformed traditional machine learning methods. Therefore, it is widely believed that CNN has the potential to be used for security applications. \n\nCNN can automatically extract features of target objects from lower to higher levels by using convolutional and pooling layers. Convolutional layers play a role in extracting the features of the input. A convolutional layer consists of a set of filters and activation functions. A filter is a function to emphasize key features that are used to recognize target objects. The raw input data is converted into feature maps with filters, which becomes more clear after processing the activation functions. A pooling layer (or sub-sampling) reduces the number of features, which prevents overfitting caused by a high number of features and improve the learning rate. Finally, feature map layers are used as the input layer for the fully connected classifier. These are popularly applied to computer vision tasks such as object recognition~\\cite{bappy2016cnn}.\n\n\n\\subsection{Data Augmentation}\n\nIn a classification problem, it is widely known that the performance of classifiers deteriorates when an imbalanced training dataset is used. If the number of instances in the major class is significantly greater than that in the minor class, the classification performance on the major class will be higher, and vice versa.\n\nData augmentation is a popularly used method to solve the imbalance problem~\\cite{shorten2019survey}, which increases the number of instances in minority classes to balance between majority classes and minority classes. In the image domain, new samples are typically generated by applying the geometric transformations or adding noise to training samples. Figure \\ref{fig:mani} shows typically used image manipulation techniques such as flipping, rotation, and color transformation for image applications.\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{subfigure}[t]{.235\\linewidth}\n\\centering\n\\captionsetup{justification=centering}\n\\includegraphics[width=\\textwidth]{pic\/origin.JPG}\n\\caption{Original image}\n\\label{fig:D_spam}\n\\end{subfigure}\n\\hspace{0.05em}\n\\begin{subfigure}[t]{.235\\linewidth}\n\\centering\n\\captionsetup{justification=centering}\n\\includegraphics[width=\\textwidth]{pic\/flip.JPG}\n\\caption{Flipping}\n\\label{fig:D_spam2}\n\\end{subfigure}\n\\hspace{0.05em}\n\\begin{subfigure}[t]{.235\\linewidth}\n\\centering\n\\includegraphics[width=\\textwidth]{pic\/rotation.JPG}\n\\caption{Rotation}\n\\label{fig:D_ham}\n\\end{subfigure}\n\\hspace{0.05em}\n\\begin{subfigure}[t]{.235\\linewidth}\n\\centering\n\\captionsetup{justification=centering}\n\\includegraphics[width=\\textwidth]{pic\/color_transform.JPG}\n\\caption{Color transformation}\n\\label{fig:D_ham2}\n\\end{subfigure}\n\\caption{Examples of the image manipulation techniques.}\n\\label{fig:mani}\n\\end{figure}\n\nIn the image spam detection problem, however, the effects of such general data augmentation techniques would be limited because the typical ham and spam images are different from the samples generated from such augmentation techniques. Therefore, in this paper, we focus on developing data augmentation for image ham and spam emails.\n\n\n\n\\section{Conclusion}\n\\label{sec:CS}\n\nIn this paper, we proposed a new image spam email detection tool called DeepCapture. To overcome the performance degrade of existing models against entirely new and unseen datasets, we developed a classifier using CNN-XGBoost and data augmentation techniques tailored towards the image spam detection task. To show the feasibility of DeepCapture, we evaluate its performance with three publicly available datasets consisting of spam and non-spam image samples. The experimental results demonstrated that DeepCapture is capable of achieving 88\\% F1-score, which has 6\\% improvement over the best existing spam detection model, CNN-SVM~\\cite{shang2016image}, with an F1-score of 82\\%. Furthermore, DeepCapture outperforms other classifiers in cross data training scenarios to evaluate the performance of classifiers with the new and unseen dataset.\n\nFor future work, we plan to develop more sophisticated data augmentation methods to add a more real-like synthetic dataset effectively. In addition, we will increase the size of the dataset and examine any changes in detection accuracy. It would also be interesting to add the functionality of DeepCapture to an open-source project such as SpamAssassin.\n\n\n\n\n\n\n\n\\section{Evaluation}\n\\label{sec:eva}\nThis section presents the performance evaluation results of DeepCapture (presented in Section~\\ref{sec:architecture}) compared with state-of-the-art classification methods:  SVM~\\cite{annadatha2018image}, RSVM~\\cite{annadatha2018image} and CNN-SVM~\\cite{shang2016image}.\n\n\\subsection{Dataset}\n\n\nTo evaluate the performance of image spam email detection models, we use publicly available mixed datasets with two spam (``Personal spam'' and ``SpamArchive spam'') and two ham (``Personal ham'' and ``Normal image ham'') image datasets. Figure \\ref{fig:imagesample} shows examples of those datasets.\n \n\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{subfigure}[t]{.235\\linewidth}\n\\centering\n\\captionsetup{justification=centering}\n\\includegraphics[width=\\textwidth]{pic\/Dataset_spam_1.png}\n\\caption{Personal spam}\n\\label{fig:D_spam}\n\\end{subfigure}\n\\hspace{0.05em}\n\\begin{subfigure}[t]{.235\\linewidth}\n\\centering\n\\captionsetup{justification=centering}\n\\includegraphics[width=\\textwidth]{pic\/Dataset_spam_2.png}\n\\caption{SpamArchive spam}\n\\label{fig:D_spam2}\n\\end{subfigure}\n\\hspace{0.05em}\n\\begin{subfigure}[t]{.235\\linewidth}\n\\centering\n\\includegraphics[width=\\textwidth]{pic\/Dataset_ham_3.png}\n\\caption{Personal ham}\n\\label{fig:D_ham}\n\\end{subfigure}\n\\hspace{0.05em}\n\\begin{subfigure}[t]{.235\\linewidth}\n\\centering\n\\captionsetup{justification=centering}\n\\includegraphics[width=\\textwidth]{pic\/Dataset_ham_4.png}\n\\caption{Normal image ham}\n\\label{fig:D_ham2}\n\\end{subfigure}\n\\caption{Examples of image datasets.}\n\\label{fig:imagesample}\n\\end{figure}\n\n\n\nIn the ``Personal spam'' dataset, spam images were collected from 10 email accounts for one month, and ham images were collected from two email accounts for two years. The ``SpamArchive spam'' dataset~\\cite{dredze2007learning} was constructed with many anonymous users. ``Normal image ham'' dataset~\\cite{gao2008image} was collected from a photo-sharing website called ``Flickr'' (\\url{https:\/\/www.flickr.com\/}) and 20 scanned documents. From those datasets, we removed unnecessary image samples such as duplicated images, solid color background images, small and unknown images. In particular, since the ``SpamArchive spam'' dataset contains a lot of duplicated images such as the advertisement for a watch or corporate logo, we need to remove such duplicated images. After eliminating image samples that cannot be categorized as either normal ham or spam images, we were left with a dataset of 8,313 samples for experiments. The details of the dataset are presented in Table \\ref{tab:dataset}. In the final dataset, the number of spam images is 6,000, while the number of ham images is 2,313. The ratio of ham to spam is around 1:3.\n\n\\begin{table}[ht!]\n\\centering\n\\caption{Description of the datasets.}\n\\label{tab:dataset}\n\\vspace{2mm}\n\\begin{tabular}{ |p{1.5cm}||p{3cm}|p{2cm}|}\n\\Xhline{3\\arrayrulewidth}\n\\hline\n Category& Corpus &Total count\\\\\n \\hline\n Spam   & Personal spam    & 786\\\\\n        & SpamArchive spam  & 5,214\\\\\n        \\cline{2-3}\n        & Total & 6,000\\\\\n        \\hline\n Ham    & Personal ham    & 1,503\\\\\n        & Normal image ham  & 810 \\\\\n        \\cline{2-3}\n        & Total & 2,313\\\\\n\\hline\n\\Xhline{3\\arrayrulewidth}\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\subsection{Experiment setup}\n\nOur experiments were conducted using the Google Colab environment (\\url{https:\/\/colab.research.google.com\/}). It supports a GPU Nvidia Tesla K80 with 13GB of memory and an Intel(R) Xeon(R) CPU at 2.30GHz. We use Keras framework with the scikit-learn library in Python 3 to implement DeepCapture. \n\nFor classification, we randomly divided 8,313 samples into a training set (60\\%) and a testing set (40\\%) with similar class distributions.\n\nTo address the data imbalance issue and make the classifier more robust against new and unseen image datasets, we use the data augmentation (DA) techniques presented in Section~\\ref{sec:Data Augmentation in DeepCapture} to create additional image samples. Finally, we obtained 5,214 ham-like and 4,497 spam-like image samples with the images in the training set through our data augmentation techniques. Those image samples are used for training only.\n\n\n\\subsection{Classification results}\n\nTo evaluate the performance of classifiers, we use the four following metrics:\n\n\\begin{itemize}\n\\item \\textbf{Accuracy (Acc.)}: the proportion of correctly classified images;\n\\item \\textbf{Precision (Pre.)}: the proportion of images classified as spam that actually are spam;\n\\item \\textbf{Recall (Rec.)}: the proportion of spam images that were accurately classified;\n\\item \\textbf{F1-score (F1.)}: the harmonic mean of \\textit{precision} and \\textit{recall}.\n\\end{itemize}\n\nBecause the dataset used in our experiments is imbalanced, accuracy is not the best measure to evaluate the performance of classifiers. F1-score would be a more effective measure since it considers both precision and recall measures. Table \\ref{tab:evaluation} shows the performance of classifiers with\/without data augmentation techniques used for DeepCapture. DeepCapture produced the best results in all metrics except precision (accuracy: 85\\%, precision: 91\\%, recall: 85\\%, F1-score: 88\\%). The existing solutions (SVM~\\cite{annadatha2018image} with DA, RSVM~\\cite{annadatha2018image} with DA, and CNN-SVM~\\cite{shang2016image}) achieved high precision, but their recall was poor. Interestingly, traditional machine learning-based solutions (SVM~\\cite{annadatha2018image} and RSVM~\\cite{annadatha2018image}) failed to achieve a very low F1-score, less than 20\\%, without the training samples generated by the proposed data augmentation method. In contrast with those existing techniques, deep learning-based solutions (DeepCapture and CNN-SVM~\\cite{shang2016image}), achieved an F1-score of 85\\% and 82\\%, respectively, without data augmentation.\n\n\\begin{table}[!ht]\n\\centering\n\\caption{Performance of classifiers (DA represents ``Data Augmentation'').}\n\\vspace{2mm}\n\\label{tab:evaluation}\n\\begin{tabular}{|p{4.2cm}||p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|}\n\\Xhline{3\\arrayrulewidth}\n\\hline\n Model & Acc. & Pre. & Rec. & F1.\\\\\n\\hline\n\\textbf{DeepCapture} & \\textbf{85\\%} & \\textbf{91\\%} & \\textbf{85\\%} & \\textbf{88\\%}\\\\\nDeepCapture without DA & 81\\% & 90\\% & 81\\% & 85\\%\\\\\nSVM~\\cite{annadatha2018image} & 51\\% & 50\\% & 09\\% & 15\\%  \\\\ \nSVM~\\cite{annadatha2018image} with DA & 71\\% & 96\\% & 36\\%  & 52\\%  \\\\\nRSVM~\\cite{annadatha2018image} & 53\\% & 52\\% & 11\\% & 18\\%  \\\\\nRSVM~\\cite{annadatha2018image} with DA  & 73\\% & 98\\% & 42\\% & 59\\%  \\\\\nCNN-SVM~\\cite{shang2016image} & 76\\% & 99\\% & 71\\% & 82\\%  \\\\\nCNN-SVM~\\cite{shang2016image} with DA & 84\\% & 90\\% & 83\\% & 86\\%  \\\\\n\\hline\n\\Xhline{3\\arrayrulewidth}\n\\end{tabular}\n\\end{table}\n\nWe compare DeepCapture against existing solutions (SVM~\\cite{annadatha2018image}, RSVM~\\cite{annadatha2018image} and CNN-SVM~\\cite{shang2016image}) with respect to the training and testing times. Training time refers to the time taken to train a model with training samples. Testing time refers to the time taken to perform classification with all testing samples. Table~\\ref{tab:evaluation4} shows the training and testing times of all classifiers. DeepCapture took 300.27 seconds for training and 5.79 seconds for testing. CNN-based solutions such as DeepCapture and CNN-SVM outperformed SVM and RSVM with respect to the training time. However, DeepCapture produced the worst result with respect to the training time. We surmise that the testing time of XGBoost is relatively slower than other classifiers such as SVM and RSVM because XGBoost is an ensemble of multiple regression trees. For a single image, however, the average testing time of DeepCapture was only 0.0017 seconds. Hence, we believe that the testing time of DeepCapture would be practically acceptable.\n\n\\begin{table}[!ht]\n\\centering\n\\caption{Training and testing times (sec.) of classifiers.}\n\\vspace{2mm}\n\\label{tab:evaluation4}\n\\begin{tabular}{|p{4.5cm}||p{2cm}|p{2cm}|}\n\\Xhline{3\\arrayrulewidth}\n\\hline\n Model & Training time & Testing time\\\\\n\\hline\n\\textbf{DeepCapture} & \\textbf{300.27} &  \\textbf{5.79}\\\\\nSVM (Annadatha et al.~\\cite{annadatha2018image})  & 2000.00 & 0.01  \\\\\nRSVM (Annadatha et al.~\\cite{annadatha2018image}) & 2000.00 & 0.01  \\\\\nCNN-SVM (Shang et al.~\\cite{shang2016image}) & 320.24 &  0.03  \\\\\n\\hline\n\\Xhline{3\\arrayrulewidth}\n\\end{tabular}\n\\end{table}\n\nTo test the robustness of classifiers against new and unseen image spam emails, we evaluate the performance of DeepCapture with cross data training. For cross data training, we trained classifiers on ham and spam images collected from one specific source, and evaluated the performance of classifiers against a different unseen dataset.\n\nFor training, we used 6,024 samples collected from ``SpamArchive spam'' and ``Normal image ham'' datasets, while for testing, we used 2,289 samples collected from ``Personal spam'' and ``Personal ham'' datasets. To make classifiers more robust against the unseen dataset, we additionally created 5,190 ham-like and 786 spam-like image samples with the images in the training set through our data augmentation techniques. Those image samples are used for training only. Table~\\ref{tab:evaluation2} shows the evaluation results for the first cross data training scenario. DeepCapture achieved an F1-score of 72\\% and outperformed the other classifiers. Surprisingly, F1-scores of all classifiers, including DeepCapture itself, are less than 35\\% without the training samples created by data augmentation, indicating that our data augmentation techniques are necessary to process unseen and unexpected image samples. \n\n\\begin{table}[!ht]\n\\centering\n\\caption{Performance of classifiers with a cross data training scenario (training dataset: ``SpamArchive spam'' and ``Normal image ham'' datasets; and testing dataset: ``Personal spam'' and ``Personal ham'' datasets).}\n\\vspace{2mm}\n\\label{tab:evaluation2}\n\\begin{tabular}{|p{4.2cm}||p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|}\n\\Xhline{3\\arrayrulewidth}\n\\hline\n Model & Acc. & Pre. & Rec. & F1.\\\\\n\\hline\n\\textbf{DeepCapture} & \\textbf{71\\%} & \\textbf{81\\%} & \\textbf{71\\%} & \\textbf{72\\%}\\\\\nDeepCapture without DA & 36\\% & 37\\% & 34\\% & 35\\%\\\\\nSVM~\\cite{annadatha2018image} & 89\\% & 14\\% & 10\\% & 12\\%  \\\\ \nSVM~\\cite{annadatha2018image} with DA & 65\\% & 45\\% & 22\\%  & 29\\%  \\\\\nRSVM~\\cite{annadatha2018image} & 90\\% & 12\\% & 11\\% & 13\\%  \\\\\nRSVM~\\cite{annadatha2018image} with DA  & 69\\% & 58\\% & 27\\% & 30\\%  \\\\\nCNN-SVM~\\cite{shang2016image} & 35\\% & 35\\% & 34\\% & 35\\%  \\\\\nCNN-SVM~\\cite{shang2016image} with DA & 68\\% & 73\\% & 45\\% & 55\\%  \\\\\n\\hline\n\\Xhline{3\\arrayrulewidth}\n\\end{tabular}\n\\end{table}\n\n\n\n\n\n\n\n\n\nAs another cross data training scenario, we used 2,289 samples collected from ``Personal spam'' and ``Personal ham'' datasets for training while we used 6,024 samples collected from ``SpamArchive spam'' and ``Normal image ham'' datasets for testing. Again, to make classifiers more robust against the unseen dataset, we additionally created 4,497 ham-like and 5,214 spam-like image samples with the images in the training set through our data augmentation techniques. Those image samples are used for training only. Table~\\ref{tab:evaluation3} shows the evaluation results for the second cross data training scenario. DeepCapture and RSVM~\\cite{annadatha2018image} with DA achieved an F1-score of 76\\% and outperformed the other classifiers. F1-scores of all classifiers, including DeepCapture, are less than 40\\% without the training samples created by data augmentation. \n\nWe note that in the second cross data training scenario, RSVM~\\cite{annadatha2018image} with DA also produced the best classification results comparable with DeepCapture. We surmise that underlying dataset differences may explain this. In the ``Personal spam'' dataset, the ratio of spam to ham image samples is approximately 1.9:1 while in the ``SpamArchive spam'' dataset, the ratio of spam to ham image samples is approximately 6.4:1. These results demonstrate that the performance of RSVM~\\cite{annadatha2018image} with DA can significantly be affected by the class distribution of samples. In contrast, DeepCapture overall works well regardless of the imbalanced class distribution of samples.\n\n\n\\begin{table}[!ht]\n\\centering\n\\caption{Performance of classifiers with a cross data training scenario (training dataset: \"Personal spam\" and \"Personal ham\" datasets; and testing dataset: \"SpamArchive spam\" and \"Normal image ham\" datasets).}\n\\vspace{2mm}\n\\label{tab:evaluation3}\n\\begin{tabular}{|p{4.2cm}||p{1.2cm}|p{1.2cm}|p{1.2cm}|p{1.2cm}|}\n\\Xhline{3\\arrayrulewidth}\n\\hline\n Model & Acc. & Pre. & Rec. & F1.\\\\\n\\hline\n\\textbf{DeepCapture} & \\textbf{73\\%} & \\textbf{82\\%} & \\textbf{72\\%} & \\textbf{76\\%}\\\\\nDeepCapture without DA & 31\\% & 47\\% & 32\\% & 38\\%\\\\\nSVM~\\cite{annadatha2018image} & 74\\% & 61\\% & 14\\% & 22\\%  \\\\ \nSVM~\\cite{annadatha2018image} with DA & 60\\% & 84\\% & 52\\%  & 64\\%  \\\\\nRSVM~\\cite{annadatha2018image} & 82\\% & 71\\% & 22\\% & 34\\%  \\\\\nRSVM~\\cite{annadatha2018image} with DA  & 62\\% & 94\\% & 67\\% & 76\\%  \\\\\nCNN-SVM~\\cite{shang2016image} & 24\\% & 42\\% & 23\\% & 30\\%  \\\\\nCNN-SVM~\\cite{shang2016image} with DA & 64\\% & 69\\% & 47\\% & 56\\%  \\\\\n\\hline\n\\Xhline{3\\arrayrulewidth}\n\\end{tabular}\n\\end{table}\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\n\nImage-based spam emails (also referred to as ``image spam emails'') are designed to evade traditional text-based spam detection methods by replacing sentences or words contained in a spam email with images for expressing the same meaning~\\cite{ismail2019image}. As image spam emails become popular~\\cite{emailadvertisement}, several spam detection methods~\\cite{Fumera06:spam,kim2017analysis,kumar2017svm} have been proposed to detect image spam emails with statistical properties of image spam emails (e.g., the ratio of text contents in an email sample). However, these countermeasures have disadvantages due to a high processing cost for text recognition in images~\\cite{attar2013survey}. Recently, a convolutional neural network (CNN) model-based detection~\\cite{shang2016image} was presented to address this processing cost issue and improve the detection accuracy. The recent advance of deep learning technologies in the image domain would bring a new angle or approach to security applications. CNN has the potential to process raw data inputs (e.g., the input image itself) by extracting important (low-level) features in an automated manner~\\cite{Krizhevsky17:dnn}. However, we found that the detection accuracy of the existing CNN based image spam detection model~\\cite{shang2016image} could be degraded significantly against new and unseen image spam emails.\n\nTo overcome the limitation of existing image spam detectors against new and unseen datasets, we propose a new image spam email detection tool called DeepCapture. DeepCapture consists of two phases: (1) data augmentation to introduce new training samples and (2) classification using a CNN-XGBoost model. In this paper, we focus on developing new data augmentation techniques tailored for image spam training dataset and designing an effective CNN architecture capable of detecting images used for spam emails with the optimized configuration for number of layers, number of filters, filter size, activation function, a number of epochs and batch size.\n\n\n\nTo examine the feasibility of DeepCapture, we evaluate the performance of DeepCapture compared with existing image spam email detectors such as RSVM based detector~\\cite{annadatha2018image} and CNN-SVM based detector~\\cite{shang2016image}. In our experiments, we use a dataset consisting of 6,000 spam and 2,313 non-spam (hereinafter referred to as ham) image samples collected from real-world user emails. We also use our data augmentation techniques to balance the distribution of ham and spam samples and avoid performance degradation against new and unseen datasets. We evaluate the performance of the DeepCapture in two ways. First, we evaluate the performance of DeepCapture with\/without data augmentation. Second, we also evaluate the performance of DeepCapture via cross data training scenarios with\/without data augmentation. Our experimental results demonstrate that DeepCapture produced the best classification results in F1-score (88\\%) compared with existing solutions. Moreover, for two cross data training scenarios against unseen datasets, DeepCapture also produced the best F1-score results compared with other classifiers. The use of data augmentation techniques would be necessary for processing new and unseen datasets. In the cross data training scenarios, F1-scores of all classifiers are less than 40\\% without applying our data augmentation techniques.\n\n\nThis paper is constructed as follows: Section \\ref{sec:Bg} describes the background of image spam email, convolutional neural network and data augmentation. Section \\ref{sec:architecture} describes the model architecture of DeepCapture. Section \\ref{sec:eva} describes experiment setups and evaluation results of DeepCapture. Section \\ref{sec:Rw} describes the related work for image spam detection and we conclude in Section \\ref{sec:CS}.\n\\section{Related work}\n\\label{sec:Rw}\n\nTo avoid spam analysis and detection, spammers introduced the image spam technique to replace text spam messages with images. This strategy would be an effective technique to circumvent the text analysis of emails, which are commonly used in spam filters~\\cite{Biggio11:spam}. To detect image spam emails, several classification methods have been proposed~\\cite{Fumera06:spam,shang2016image,kim2017analysis,kumar2017svm,annadatha2018image,fatichah2019image}. However, the solutions offered so far exhibit several critical weaknesses. Existing detection techniques can be categorized into two approaches: (1) keyword-based analysis and (2) image classification.\n\n\n\n\\subsection{Keyword-based analysis} \n\nKeyword-based analysis is to extract texts from a given image and analyze them using a text-based spam filter. Several techniques~\\cite{Fumera06:spam,kim2017analysis,kumar2017svm} using keyword analysis were introduced. Also, this approach was deployed in real-world spam filters such as SpamAssassin (\\url{https:\/\/spamassassin.apache.org\/}). Unsurprisingly, the performance of this approach depends on the performance of optical character recognition (OCR). Sophisticated spammers can intentionally embed abnormal text characters into an image, which cannot be recognized by typical OCR programs but can still be interpreted by human victims. The performance of keyword-based spam detection methods could be degraded significantly against such image spam emails. Moreover, a high processing cost of OCR is always required for analyzing images. Therefore, in this paper, we propose an image spam detection method in the direction of establishing an image classifier to distinguish spam images from ham images.\n\n\n\\subsection{Image classification} \n\nTo address the high processing cost issue of keyword-based analysis, some researchers have tried to develop image spam detection methods using low-level features that are directly extracted from images. Annadatha et al.~\\cite{annadatha2018image} demonstrated that image spam emails could be detected with high accuracy using either Principal Component Analysis (PCA) or Support Vector Machines (SVM). To build a classifier, they manually selected 21 features (e.g., image color, object edges) that can be extracted from spam and ham images. Shang et al.~\\cite{shang2016image} proposed an alternative image classification method using a CNN model and an SVM classifier together, which is composed of 13 layers. The CNN model proceeds classification in the last fully connected layer. However, they use the output from the last fully connected layer as the input for the SVM classifier. In this paper, we develop a more compact CNN-XGBoost model consisting of 8 layers. Our evaluation results show that DeepCapture outperforms Shang et al.'s architecture in terms of detection accuracy. Fatichah et al.~\\cite{fatichah2019image} also discussed the possibility of CNN models to detect image spam. Unlike other previous studies, they focused on building CNN models to detect the image spam on Instagram (\\url{https:\/\/www.instagram.com\/}), a social photo-sharing service. They evaluated the performance of four pre-trained CNN models (3-layer, 5-layer, AlexNet, and VGG16) with 8,000 images collected from Instagram. They found that the VGG16 architecture achieves the best accuracy (about 0.84) compared with the other models. Since VGG16 is a pre-trained network and its performance is not advantageous, we do not directly compare DeepCapture with VGG16.\n\n\nWe note that the performances of previous methods have been evaluated on different data sets with different configurations. Therefore, we cannot directly compare their reported performances. In this paper, we needed to reimplement their models and used the publicly available datasets to compare the performance of DeepCature with those of the best existing models (SVM~\\cite{annadatha2018image}, RSVM~\\cite{annadatha2018image} and CNN-SVM~\\cite{shang2016image}).\n\n\n\\section*{Acknowledgement} \n\\label{sec:ack}\n\\begin{small}\nHyoungshick Kim is the corresponding author. This work has been supported in part by the Cyber Security Research Centre Limited whose activities are partially funded by the Australian Government's Cooperative Research Centres Programme and the NRF grant (No. 2017H1D8A2031628) and the ITRC Support Program (IITP-2019-\n2015-0-00403) funded by the Korea government. The authors would like to thank all the anonymous reviewers for their valuable feedback.\n\\end{small}\n\n\\section{Overview of DeepCapture}\n\\label{sec:architecture}\n\nWe designed DeepCapture using data augmentation and CNN to make it robust against new and unseen datasets. Figure \\ref{fig:model} shows an overview of  DeepCapture architecture.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=1\\columnwidth]{pic\/model2.JPG}\n\\caption{Overview of DeepCapture.}\n\\label{fig:model}\n\\end{figure}\n\nDeepCapture consists of two phases: (1) data augmentation to introduce new training samples and (2) classification using a CNN model.\n\n\n\n\\subsection{Data Augmentation in DeepCapture}\n\\label{sec:Data Augmentation in DeepCapture}\n\nTo address the class imbalance problem in image spam datasets and generalize the detection model, we introduce a new data augmentation method to create new ham and spam samples for training. The goal of data augmentation is to make augmented samples that are similar to real data.\n\nFor both ham and spam images, we commonly remove unnecessary images such as duplicate images, solid color background images, small and unknown images that cannot be recognized by human users. After removing unnecessary images, we apply different data augmentation methods to ham and spam images, respectively. \n\nFor ham images, we randomly choose an image among ham images and use an API to search images that are similar to the given image. For example, the Google Image Search API can be used to crawl the images similar to the ones we uploaded. For each uploaded image, $N$ (e.g., $N=100$) similar images can be obtained as ham-like images for training (see Figure~\\ref{fig:similar}). Those images would be regarded as additional ham images because those are also actually used images on other websites.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=1\\columnwidth]{pic\/similar.jpg}\n\\caption{Data augmentation process for ham images.}\n\\label{fig:similar}\n\\end{figure}\n\nFor spam images, we randomly choose two images among spam images and split each image in half from left to right (``left and right parts''). Next, we then combine the left part of an image with the right part of the other image. To combine parts from different images, we resize a part of an image so that its size is the same as the size of the part of another image (see Figure~\\ref{fig:cropflip}). Our key observation is that a spam image typically consists of the image and text parts. Therefore, it is essential to create augmented samples having both image and text parts. Our data augmentation techniques are designed to produce such image samples.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=1\\columnwidth]{pic\/generate_spam.JPG}\n\\caption{Data augmentation process for spam images.}\n\\label{fig:cropflip}\n\\end{figure}\n\n\\subsection{CNN-XGBoost Classification in DeepCapture}\n\nAs shown in Figure~\\ref{fig:model}, the architecture of DeepCapture is composed of eight layers. Given an input image, the input image is resized to 32x32 pixels. The first six layers are convolutional layers, and the remaining two layers are used for the XGBoost classifier to determine whether a given image is spam or not. \n\nAll convolutional layers use 3x3 kernel size and the Leaky ReLU function~\\cite{maas2013rectifier}, which is used as the activation function. The Leaky ReLU function has the advantage to solve the gradient saturation problem and improve convergence speed. Unlike ReLU, in which the negative value is totally dropped, Leaky ReLU assigns a relatively small positive gradient for negative inputs. We also apply the 2x2 max pooling to the 3rd and 6th layers, which selects the maximum value from the prior feature map. The use of max pooling reduces the dimension of feature parameters and can help extract key features from the feature map. We also use regularization techniques to prevent the overfitting problem. We specifically use both L2 regularization~\\cite{ng2004feature} and the dropout method~\\cite{srivastava2014dropout} as regularization techniques for DeepCapture. \n\n\n\n\nAfter extracting features from the input image through the convolutional layers, we use the XGBoost~\\cite{chen2016xgboost} classifier, which is a decision-tree-based ensemble machine learning algorithm that uses a gradient boosting framework. XGBoost builds a series of gradient boosted decision trees in a parallel manner and makes the final decision using a majority vote over those decision trees. In many situations, a CNN model typically uses the fully connected layers. However, for the image spam detection problem, we found that we can improve the detection accuracy if we replace the fully connected layers with a classifier such as XGBoost. We use the random search~\\cite{bergstra2012random} method for optimizing hyperparameters used in the XGBoost classifier.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Content Guidelines}\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\t%\n\n\n\t\n\t\\section{Introduction}\n\t\n\tOver the past decades, nanoelectromechanical systems (NEMS) have received considerable attention as versatile elements in mesoscopic devices. \n\tFor example, they are well-suited for sensor applications, and have been shown to act as highly sensitive mass \n\t\\cite{hanay2012single,chaste2012nanomechanical}, force \\cite{mamin2001sub,Rugar2004,moser2013ultrasensitive,Simonsen2019}, or infrared \\cite{piller2019nanoelectromechanical} detectors. \n\tEven more, they serve as amplifiers or oscillators. \n\tOn the other hand, nanomechanical systems lend themselves to the exploration of fundamental physical phenomena both in the classical \\cite{Yang2019,huber2020spectral,Karg2020} and quantum regimes\n\t\\cite{sudhir2017quantum,rossi2018measurement,mason2019continuous,Guo2019}, with the prospect of serving as hybrid transducer\n\t\\cite{Andrews2014} or as storage element \\cite{OConnell2010} in future quantum technologies. \n\t\n\tMany of the underlying devices are based on a one-dimensional string or two-dimensional membrane resonators  \\cite{Thompson2008,Wilson2009,Joeckel2014,Ftouni2015,Fink2016,Serra2016,Norte2016,Reinhardt2016,sudhir2017quantum,Tsaturyan2017NatNano-UltracoherentNanomechSoftclamping,Ghadimi2018StrainEngineering,rossi2018measurement,mason2019continuous,Simonsen2019,Guo2019,Yang2019,yuksel2019nonlinear,huber2020spectral,Karg2020}\n\tthat are both characterized by a strong intrinsic tensile stress in the device layer. Tensile stress in the device layer enables remarkably large mechanical quality factors as a result of a process commonly referred to as dissipation dilution \\cite{GonzalezSaulson1994BrownianMotionAnelasticWire,Unterreithmeier2010PRL-DampingNanomechRes,Yu2012PRL-ControlMaterialDampingHighQMembraneMicrores,Villanueva2014PRL-EvidenceSurfaceLossSiN}. \n\tThis process relies on the stress-induced increase of the stored vibrational energy of the resonator, while the dissipated energy remains largely unaffected. \n\t\n\tSeveral approaches have been pursued to suppress the limiting dissipation mechanism and to further enhance the potential of this class of nanomechanical resonators. For example, clamping losses can be reduced by means of mechanical impedance mismatch engineering \\cite{bib:Rieger2014} \n\tthat is exploited, e.g., in trampoline resonators \\cite{Kleckner2011,Reinhardt2016,Norte2016}. In addition, phononic bandgaps have been employed to reduce clamping losses \\cite{Alegre2011,yu2014phononic,Tsaturyan2014}. Recently, soft clamping has been presented as an innovative approach of boosting the quality factor \\cite{Tsaturyan2017NatNano-UltracoherentNanomechSoftclamping}. It is also based on phononic bandgap engineering, but additionally prevents mechanical strain at the interface between the resonator and its clamping points and thus enhances the dissipation dilution. \n\tSoft-clamped membranes have recently enabled measurement-based quantum control \\cite{rossi2018measurement} or measurements below the standard quantum limit \\cite{mason2019continuous}, and may find application in magnetic resonance force imaging \\cite{Simonsen2019}. \n\tFor string resonators, soft clamping can be beneficially combined with stress engineering \\cite{Ghadimi2018StrainEngineering,fedorov2019generalized}, which boosts the tensile stress to close to the mechanical limits and enables quality factors $Q$ of around $800$ million and a $Q \\times$ frequency product of more than $ 10^{15}\\,$Hz. \n\tThese developments pave the way towards an ever increasing sensitivity and the observation of mechanical quantum phenomena at room temperature.\n\t\n\tHere we describe a previously unappreciated aspect of nanomechanical string resonators: the one-dimensional tensile stress in the string is not solely determined by elastic material properties, but significantly depends on its length as well as other geometric parameters. This allows one to increase the tensile stress by approximately $50$\\,\\% by using shorter strings and thus boost the dissipation dilution. The observed length dependence of the tensile stress is material independent. We demonstrate length-dependent tensile stress for nanomechanical resonators fabricated from four different wafers featuring the three complementary, tensile-stressed device layers silicon nitride (SiN), silicon carbide (SiC), and indium gallium phosphide (InGaP). A simple elastic model is developed that captures the observed features. It describes the geometric reconstruction of the string resonator by a combination of two effects that determine the stress distribution in the device layer: the vertical release of the string leads to a deformation of the clamping structure, while the subsequent lateral release undercuts the clamping pads. Our results entail important insights for strain engineering of nanomechanical devices and may be exploited to boost the mechanical quality factor of string resonators.\n\t\n\t\n\t\\section{Experimental results}\n\t\n\tWe investigate nanomechanical string resonators fabricated from three distinct, strongly stressed material platforms, namely amorphous silicon nitride (SiN), crystalline silicon carbide (SiC), and crystalline indium gallium phosphide (InGaP). The $100$\\,nm thick amorphous stoichiometric SiN film is deposited by low-pressure chemical vapor deposition on top of two different substrates, a fused silica wafer (material denoted SiN-FS) and a sacrificial SiO$_2$ layer atop a silicon wafer (SiN-Si). The $110$\\,nm thick film of crystalline 3C-SiC is epitaxially grown on a Si(111) wafer (SiC). The III-V heterostructure hosts a $100$\\,nm thick In$_{0.415}$Ga$_{0.585}$P film epitaxially grown atop a sacrificial layer of Al$_y$Ga$_{1-y}$As on a GaAs wafer.\n\tSee Tab.~S.1. of the Supplemental Material \\cite{siCite} for the full details of all wafers.\n\t\n\tSeries of nanomechanical string resonators such as that depicted in Fig.~\\ref{fig:SEM} are defined in all four material systems. The length of the strings increases from $10$ to $110\\,\\mu$m in steps of $10\\,\\mu$m. \n\t%\n\t\\begin{figure}[t!]\n\t\t\\includegraphics[width=0.9\\linewidth]{SEMresonator.jpg\n\t\t\\caption{\n\t\t\t\\label{fig:SEM} \n\t\t\tScanning electron micrograph of a series of nano\\-string resonators with lengths increasing from $10$ to $110\\,\\mu$m in steps of $10\\,\\mu$m. \n\t\t}\n\t\\end{figure}\n\t%\n\tThe resonators are characterized using piezo actuation and interferometric detection. For all wafers, we record the resonance curves of the out-of-plane flexural modes in the linear response regime. The measurements are performed at room temperature and inside a vacuum chamber (pressure less than $10^{-3}$\\,mbar) to avoid gas damping.\n\t\n\tFor each resonator length we probe the fundamental out-of-plane mode as well as a series of up to $30$ higher-order modes, and determine the corresponding eigenfrequencies by Lorentzian fits, as shown for the case of SiN-FS in Fig.~\\ref{fig:ModeFreq}.\n\t%\n\t\\begin{figure}[t!]\n\t\t\\includegraphics[width=0.8\\linewidth]{nfOnFS\n\t\t\\caption{\n\t\t\t\\label{fig:ModeFreq} \n\t\t\tEigenfrequencies of the out-of-plane modes as a function of the mode number for SiN string resonators on fused silica (SiN-FS). The resonator lengths range from $10$ to $110\\,\\mu$m.\n\t\t\tFits of the eigenfrequencies using Eq.~(\\ref{eq:frequency}) are included as solid lines.\n\t\t}\n\t\\end{figure}\n\t%\n\tFollowing Euler-Bernoulli beam theory of a doubly clamped string with simply supported boundary conditions \\cite{Timoshenko1990-VibrationProblemsEngineering,Cleland2002foundations}, we can conveniently express the eigenfrequency of the $n$-th mode as\n\t%\n\t\\begin{equation}\n\t\tf_n = \\frac{n^2 \\pi}{2 L^2} \\sqrt{\\frac{E_1 h_1^2}{12\\rho}}\\sqrt{1+ \\frac{12 \\sigma L^2}{n^2 \\pi^2 E_1 h_1^2}} \\label{eq:frequency}\n\t\\end{equation}\n\t%\n\twhere $E_1$ is Young's modulus, $\\rho$ is the mass density, $h_1$ is the thickness of the resonator, and $\\sigma$ is the tensile stress of the string.\n\t\n\tThe resonance frequencies shown in Fig.\\,\\ref{fig:ModeFreq} are fitted with Eq.~(\\ref{eq:frequency}). To a very good approximation, the eigenfrequencies of all resonators scale linearly with the mode number as expected for stress-dominated nanostrings for which  \n\t$f_n \\approx (n\/2 L) \\sqrt{\\sigma \/ \\rho}$.\n\tThe fits of the eigenfrequencies as a function of mode number allow to extract the tensile stress of each nanostring resonator. The obtained values are shown as a function of the resonator length for all four materials in Fig.\\,\\ref{fig:sigL}. Clearly, the tensile stress is not constant, but decreases for increasing resonator length. The same qualitative behavior is observed in all four material systems.\n\t\n\t\\begin{figure}[t!]\n\t\t\\includegraphics[width=0.8\\linewidth]{sigL\n\t\t\\caption{\n\t\t\t\\label{fig:sigL} \n\t\t\tExperimentally determined tensile stress as a function of the length of the nanostring for all four material systems. \n\t\t\tFits of Eq.~(\\ref{eq:sigLM}) are included as solid lines. The obtained fit parameters are summarized in Tab.~\\ref{tab:disp}.\n\t\t\tThe shaded areas indicate the uncertainty resulting from measurement errors of the pedestal height $h_0$ and undercut $a_\\mathrm{uc}$.\n\t\t}\n\t\\end{figure}\n\t\n\t\n\t\\section{Elastic model} \n\t\n\tA length dependence of the tensile stress in a nanostring has not been reported in the literature. We have developed a simple theoretical model to quantify this previously unappreciated phenomenon. Our model is based on elastic theory. As such, it is material independent and can be applied to all material systems under investigation. The model assumes a prismatic string of length $L$, width $w$ and thickness $h_1$. Its cross-sectional area is $A_{\\textrm{s}} = w \\, h_1$. On both ends, the string is attached to a rectangular clamping structure. It consists of a clamping pad in the device layer with lithographic dimensions $2 a_x$ and $2 a_y$, as well as thickness $h_1$ (Fig.~\\ref{fig:toymodel}(a)), which is supported by a pedestal of height $h_0$ in the underlying sacrificial or substrate layer (Fig.~\\ref{fig:toymodel}(b) and (c)). As a result of the wet etching process required to suspend the nanostrings, the clamping pads exhibit a certain undercut $a_\\mathrm{uc}$, i.e. the width of the pad $2a_x$ (in $x$ direction, the width $a_y$ in $y$ direction may differ) is larger than that of the remaining pedestal $2 a_\\mathrm{p} = 2a_x - 2a_\\mathrm{uc}$. The cross-sectional area of the clamping pad (in $yz$-plane) is  $A_{\\textrm{p}} = 2a_y \\, h_1$. The geometric parameters of the four investigated samples are summarized in Tab.~\\ref{tab:ModelParameters}. The thickness of the device layer $h_1$ is obtained from the wafer growth protocol, whereas the height of the pedestal $h_0$ is determined by atomic force microscopy. The half-widths of the clamping pad $a_x$ and $a_y$, the undercut $ a_\\mathrm{uc} $ and the width of the nanostring $w$ have been extracted using electron beam microscopy. The elastic and material parameters of the samples \\cite{DosterUnpub,maluf2002introduction,li1987single,henisch2013silicon,Ioffe1999ShurEtAl-HandbookSeriesSemiconductorParametersVOL2} are listed in Tab.~S.2. in the Supplemental Material \\cite{siCite}.\n\t\n\t\\begin{figure}[ht]\n\t\t\\includegraphics[width=\\linewidth]{ToymodelSketch2\n\t\t\\caption{\n\t\t\t\\label{fig:toymodel} \n\t\t\tSample geometry and parameters of the model. \n\t\t\t(a) Lithographic dimensions of the nanostring and its clamping pads.  \n\t\t\t(b) Cross section through the clamping structure illustrating the shearing contraction of the pedestal following the vertical release of the structure.\n\t\t\t(c) Cross section through the clamping structure illustrating the lateral contraction of the undercut areas of the clamping pad following the horizontal release. Combining the vertical and horizontal releases leads to the string's length change of $ \\Delta L $. Areas supported by a pedestal are colored in dark blue, undercut areas of the clamping pads are indicated by a lighter color and the string is marked with the lightest blue. Dotted lines serve as guides to the eye.\n\t\t}\n\t\\end{figure}\n\t%\n\t\\begin{table}\n\t\t\\caption{\n\t\t\t\\label{tab:ModelParameters}\n\t\t\tGeometric parameters of the investigated samples.\n\t\t}\n\t\t\\begin{ruledtabular}\n\t\t\t\\begin{tabular}{crrrrrr}\n\t\t\t\t& \\multicolumn{1}{c}{\\textrm{$h_1$}} & \\multicolumn{1}{c}{\\textrm{$h_0$}} &  \n\t\t\t\t\\multicolumn{1}{c}{\\textrm{$2a_x$}} & \n\t\t\t\t\\multicolumn{1}{c}{\\textrm{$2a_y$}} &\n\t\t\t\t\\multicolumn{1}{c}{\\textrm{$a_\\mathrm{uc}$}} &\n\t\t\t\t\\multicolumn{1}{c}{\\textrm{$w$}} \\\\\n\t\t\t\t& \\multicolumn{1}{c}{\\textrm{(nm)}} & \\multicolumn{1}{c}{\\textrm{(nm)}} & \\multicolumn{1}{c}{\\textrm{($ \\mu $m)}} & \\multicolumn{1}{c}{\\textrm{($ \\mu $m)}} & \\multicolumn{1}{c}{\\textrm{(nm)}} & \\multicolumn{1}{c}{\\textrm{(nm)}}  \\\\\n\t\t\t\t\\hline\n\t\t\t\tSiN-FS\t& 100(5) & 460(20)\t& 13.7(2) & 13.6(2)\t& 570(100) \t& 420(25) \\\\\n\t\t\t\tSiN-Si\t& 100(5) & 365(20)\t& 14.1(2) & 15.0(2) & 410(150) \t& 340(25) \\\\\n\t\t\t\tSiC \t& 110(15)& 570(40) \t& 14.2(2) & 15.0(2) & 860(150) \t& 360(30) \\\\\n\t\t\t\tInGaP \t& 100(1) & 990(10)\t& 12.7(2) & 13.3(2) & 640(170)\t& 250(15) \\\\\n\t\t\t\\end{tabular}\n\t\t\\end{ruledtabular}\n\t\\end{table}\n\t\n\tAs we show in the following, the tensile stress in the device layer atop an unstressed sacrificial layer or substrate gives rise to a balance of forces that in turn leads to a length- and geometry-dependent change in the one-dimensional tensile stress of the nanostring. To quantify the contributing forces, we roughly follow the process sequence required to fabricate a freely suspended nanostring. \n\tFirst, we consider the vertical release of the nanostructure. It comprises all vertical etching contributions, notably the anisotropic dry etch employed for the pattern transfer following electron beam lithography. Note that the subsequent wet etching process releasing the nanostring may also have a vertical etching component, e.g., for the case of an isotropic wet etch. The vertical release penetrates both the device layer and the sacrificial or substrate layer and defines the height of the pedestal $h_0$.\n\n\tFollowing this vertical release, the tensile-stressed device layer will contract and induce a certain amount of shear in the pedestal (Fig.~\\ref{fig:toymodel}(b)). As a result, the tensile stress in the pad relaxes to a value $\\sigma_\\mathrm{p}$. \n\t\n\tSecond, the lateral release is considered. It accounts for the lateral etching during the nanostring release.\n\tAs a result of this lateral release, the two-dimensional stress in the nanostring relaxes in the direction perpendicular to the string. At the same time, the undercut parts of the tensile-stressed clamping pad contract, which applies additional stress on the nanostring (Fig.~\\ref{fig:toymodel}(c)). The combination of the described effects gives rise to the tensile stress experienced by the nanostring $\\sigma$ (see Eq.~(\\ref{eq:frequency}) and Fig.~\\ref{fig:sigL}). The model assumes a clear separation between the vertical and lateral release, which will be described in Secs.~\\ref{Pedestal} and~\\ref{Undercut}, respectively, and neglects geometric and elastic reconfigurations of the sheared pedestal and stressed clamping pad arising from the lateral releases, which we can safely assume to be small.\n\t\n\t\\subsection{Pedestal shear from vertical release}\\label{Pedestal}\n\t\n\tTo evaluate the shear of the pedestal induced by the vertical release of the structure, we first consider an isolated clamping structure and focus on its cross section along the $x$-$z$ direction as indicated in Fig.~\\ref{fig:toymodel}(b). \n\tThe resonator will be included at a later stage. \n\tFollowing the vertical release of the structure, the strong tensile stress in the device layer leads to a contraction of the clamping pad in order to minimize internal forces. \n\tThis contraction produces an increasing shear of the pedestal. \n\tThe reconfiguration of the clamping structure stops once equilibrium between the reduced tensile force and the counteracting shearing force is reached. \n\tThe shear stress $\\tau$ of such a shear-constrained material system can be expressed as \\cite{Taylor1962} \n\t\n\t\\begin{equation}\n\t\t\\tau = \\sigma_{\\mathrm{2D}} h_1  k  \\tanh\\left(k a_{x}\\right) , \\quad k = \\sqrt{\\frac{G_0}{h_0}  \\frac{1}{E_1 h_1}} \\label{eq:mSL:tau}\n\t\\end{equation}\n\t%\n\twhere $h_0$ and $h_1$ are the heights of the pedestal and the clamping pad, respectively, $G_0$ is the shear modulus of the pedestal, $E_1$ is Young's modulus of the clamping pad, and $\\sigma_{2D}$ is the initial two-dimensional stress in the device layer.\n\tThis results in the maximum contraction of the clamping pad $\\Delta p$ from its original half-width $a_\\mathrm{x}$:  \n\t%\n\t\\begin{equation}\n\t\t\\Delta p = \\frac{h_0}{G_0} \\tau = \\frac{\\sigma_{\\mathrm{2D}}}{E_1 k} \\tanh\\left(k a_{x}\\right) .\n\t\t\\label{umax}\n\t\\end{equation}\n\tIn consequence, the tensile stress in the clamping pad is reduced to $\\sigma_\\mathrm{p} = \\sigma_{2D} - E_1 \\frac{\\Delta p}{a_{x}} $ according to Hooke's law. Note that a similar model that also accounts for additional shear in the device layer is presented in Ref.~\\cite{suhir2008interfacial}. \n\t\n\t\n\tFor the sake of simplicity, we neglect the minute counterforce exerted by the presence of the resonator, which will lead to a slightly reduced contraction of the pad to which it is attached. Similarly, the effect of shear of the pedestal underneath the resonator will be ignored. \n\t\n\tAn experimental verification of the contraction of the clamping pad following the vertical release is discussed in the Supplemental Material \\cite{siCite}. \n\t\n\t\\subsection{Undercut of clamping pads from lateral release}\\label{Undercut}\n\t\n\tThe lateral release of the nanostrings results in an undercut of the clamping pads. More specifically, the width of the pedestal is reduced by $a_\\mathrm{uc}$ from either side such that the rim of the clamping pad gets freely suspended as shown in Fig.~\\ref{fig:toymodel}(c). This enables a relaxation of the tensile force in the undercut parts of the pads that gives rise to a contraction by an amount $ \\Delta c $. The resulting contracting force acting on the interface between the clamping pad and the nanostring can be expressed as\n\t%\n\t\\begin{equation}\n\t\tF_\\mathrm{c} = \\sigma_\\mathrm{p} A_\\mathrm{p} - E_1 \\frac{\\Delta c}{a_\\mathrm{uc}} A_\\mathrm{p},\n\t\\end{equation}\n\t%\n\twhere $\\sigma_\\mathrm{p}$ is the remaining tensile stress in the clamping pad following the vertical release, and $E_1 \\frac{\\Delta c}{a_\\mathrm{uc}}$ is its reduction in the undercut part of the clamping pad, again according to Hooke's law. \n\n\tNote that in the absence of the nanostring, the suspended part of the clamping pad fully relaxes such that $F_\\mathrm{c} = 0$.   \n\tIn the presence of the nanostring, however, the contracting force of the clamping pad is counteracted by a second force acting on the interface between the clamping pad and the nanostring which is associated with the elongation $\\Delta L$ of the nanostring \n\t\n\t\n\t\\begin{equation}\n\t\tF_\\mathrm{s} = \\sigma_\\infty A_\\mathrm{s} + E_1 \\frac{\\Delta L}{L} A_\\mathrm{s},\n\t\\end{equation}\n\twhere $\\sigma_\\infty$ is the one-dimensional stress of an infinitely long nanostring after the lateral release, and $E_1 \\frac{\\Delta L}{L} $ is its modification according to Hooke's law. \n\t\n\tThe equilibrium condition for the clamping pad - nanostring interface \n\t%\n\t\\begin{equation}\n\t\tF_c = F_s  \\label{eq:interfaceCondition}\n\t\\end{equation}\n\t%\n\tdetermines the final geometric reconfiguration of the clamping pad and the string, under the boundary condition that the total length of the compound between the centers of the clamping pads has to be conserved,\n\t%\n\t\\begin{equation}\n\t\t2 \\Delta p + 2 \\Delta c = \\Delta L . \\label{eq:conservLength}\n\t\\end{equation}\n\t%\n\t\n\tEquations (\\ref{eq:interfaceCondition}) and (\\ref{eq:conservLength}) form a second-order system of linear equations with the unknown parameters $ \\Delta L $ and $ \\Delta c $. The third unknown $\\Delta p$ is determined using Eq.~(\\ref{umax}). Solving for the elongation of the resonator yields\n\t%\n\t\\begin{equation}\n\t\t\\Delta L = 2 L \\, \\frac{\\left( A_\\mathrm{p} a_\\mathrm{uc} \\sigma_\\mathrm{p} + A_\\mathrm{p} E_1 \\Delta p - A_\\mathrm{s} a_\\mathrm{uc} \\sigma_\\infty \\right)}{E_1 \\left( 2 A_\\mathrm{s} a_\\mathrm{uc} + A_\\mathrm{p} L \\right)} .\n\t\t\\label{eq:delL}\n\t\\end{equation}\n\t%\n\t\n\tThis length change of the resonator directly translates into an additional strain $ \\varepsilon=\\Delta L\/L $, giving rise to a length-dependent stress $ \\sigma(L) $ of the doubly clamped string resonator via Hooke's law\n\t%\n\t\\begin{equation}\n\t\t\\sigma(L) = \\sigma_\\infty + E_1 \\frac{\\Delta L}{L}. \\label{eq:sigLM} \n\t\\end{equation}\n\t%\n\t\n\t\n\t\\section{Discussion}\n\t\n\tTo validate the theoretical model, we fit Eq.~(\\ref{eq:sigLM}) to the experimental data measured on all four material systems, using the geometric and material parameters specified in Tabs.~\\ref{tab:ModelParameters} and S.2 of the Supplemental Material \\cite{siCite}. The initial two-dimensional stress \n\t$\\sigma_\\mathrm{2D}$ can be calculated from the epitaxial lattice mismatch of the crystalline InGaP sample. The mismatch of the lattice constants of the In$_{1-x}$Ga$_x$P layer with respect to the underlying Al$_{y}$Ga$_{1-y}$As sacrificial layer induces an in-plane strain $\\varepsilon^\\parallel(x) = (a^\\parallel_\\mathrm{L} - a^\\infty_\\mathrm{L}(x) )\/a^\\infty_\\mathrm{L}(x)$, where $a^\\infty_\\mathrm{L}(x)$ is the lattice constant of In$_{1-x}$Ga$_x$P and $a^\\parallel_\\mathrm{L}$ is the in-plane lattice constant of the strained In$_{1-x}$Ga$_x$P film. This allows to compute the two-dimensional stress of the thin In$_{1-x}$Ga$_x$P layer $\\sigma_\\mathrm{2D} = \\varepsilon^\\parallel E_1 \/(1-\\nu_1)$ \\cite{Bueckle2018APL-StressControl}. The ratio $ E_1\/(1-\\nu_1) $ including the Poisson ratio $\\nu_1$ (see Tab.~S.2. in the Supplemental Material \\cite{siCite}) represents the biaxial modulus of the stressed thin film, which is required as no stress occurs in the $z$ direction normal to the substrate. The obtained $\\sigma_\\mathrm{2D}$ value of $0.95$\\,GPa has also been used as an input parameter for the model.  \n\tIn principle, the same argument can be made for SiC which is also an epitaxially grown crystalline thin-film material. However, the crystallization of 3C-SiC atop a Si wafer is more complex than that of the III-V heterostructures. SiC and Si feature the same crystal structure, but exhibit a lattice mismatch of approximately $20\\,\\%$. This implies a nontrivial, commensurate growth of the SiC film that strongly depends on growth conditions, such that the two-dimensional stress cannot be predicted from the crystal structure \\cite{pakula2004fabrication,zorman2008micro,iacopi2013evidence,Iacopi2013APL-QrientationDependentStressRelax3C-SiCFilms,Kermany2016JAP-FactorsAffectionFQproduct3CSiC,Romero2020}. \n\tHence, we set $ \\sigma_{\\mathrm{2D}} $ as an additional fit parameter for SiC. \n\tThe same applies for the amorphous thin-film materials SiN-FS and SiN-Si.\n\tThe one-dimensional stress  $ \\sigma_\\infty $ is employed as fit parameter for all material systems. \n\t\n\tThe results of the fits are included in Fig.~\\ref{fig:sigL} as solid lines. The shaded area represents the model's uncertainty arising from the error of the input parameters.\n\t%\n\tAs long as $A_\\textrm{s} \\ll A_\\textrm{p}$ and $a_\\textrm{uc} \\ll L$, the length dependence of Eq.~(\\ref{eq:sigLM}) can be approximated as $ \\sigma(L) \\propto 1\/L $. This holds true for all nanostrings under investigation, such that a $1\/L$ dependence of the stress can be assumed.  \n\t%\n\tWe find remarkable agreement between the model and the experimental data. This is particularly noteworthy for the case of the InGaP samples for which only one fit parameter, $ \\sigma_\\infty $, is employed, which, in the above approximation of small $A_\\textrm{s} a_\\textrm{uc}$ corresponds to a vertical offset and thus the limit $\\sigma(L\\rightarrow \\infty)$. \n\tAlso the results for SiN and SiC, which involve two fitting parameters, show good agreement between the model and the experimental data. Again, $ \\sigma_\\infty $ can be interpreted as the tensile stress of an infinitely long string, whereas the two-dimensional stress in the as-grown device layer $\\sigma_\\mathrm{2D}$ can, at least to some extent, be compared to literature values.\n\t\n\tIn Table~\\ref{tab:disp} we summarize the parameters obtained from the elastic model as well as the fit parameters for the case of the longest strings. \n\tThe as-grown two-dimensional stress in low-pressure chemical vapor deposition (LPCVD) grown stoichiometric SiN on silicon is found to depend on growth conditions, but has been reported to amount to $1.1$\\,GPa~\\cite{Ghadimi2018StrainEngineering,Bereyhi2019} and $1.4$\\,GPa~\\cite{Unterrreithmeier2009Nature-TransductionDielectricForces}, which is close to the value found here. \n\n\tThe same applies for high stress 3C-SiC(111), for which an as-grown two-dimensional stress of $1.3$\\,GPa has been reported~\\cite{iacopi2013evidence}, which is somewhat lower than our result. The growth of high stress SiN on a fused silica substrate is poorly characterized, and no comparison with the literature could be obtained. \n\tCertainly, all observed two-dimensional stress values \n\tare well within the yield strength of the respective material, which amounts to approximately $6-7$\\,GPa (or even $12$\\,GPa according to Ref.~\\cite{Yang2002}) for high stress LPCVD-deposited SiN \\cite{Kaushik2005,Ghadimi2018StrainEngineering,Bereyhi2019}, and $21$\\,GPa for SiC \\cite{Petersen1982}. \n\t\n\t\n\n\t\n\t\\begin{table}\n\t\t\\caption{\n\t\t\t\\label{tab:disp}\n\t\t\tParameters of the elastic model for the case of a long string, as well as two-dimensional and one-dimensional stress.  \n\t\t}\n\t\t\\begin{ruledtabular}\n\t\t\t\\begin{tabular}{cddddd}\n\t\t\t\t& \\multicolumn{1}{c}{\\textrm{$\\Delta p$}} & \\multicolumn{1}{c}{\\textrm{$\\Delta c$}} & \\multicolumn{1}{c}{\\textrm{$\\Delta L$}} & \n\t\t\t\t\\multicolumn{1}{c}{\\textrm{$\\sigma_\\mathrm{2D}$}} & \\multicolumn{1}{c}{\\textrm{$\\sigma_\\infty$}} \\\\\n\t\t\t\t& \\multicolumn{1}{c}{\\textrm{(nm)}} & \\multicolumn{1}{c}{\\textrm{(nm)}} & \\multicolumn{1}{c}{\\textrm{(nm)}} & \\multicolumn{1}{c}{\\textrm{(GPa)}} & \\multicolumn{1}{c}{\\textrm{(MPa)}} \\\\\n\t\t\t\t\\hline\n\t\t\t\tSiN-FS \t& 8 \t& 6  \t& 27 \t& 3.1\\footnotemark[1] & 1560\\footnotemark[1] \\\\\n\t\t\t\tSiN-Si \t& 3 \t& 2 & 8 & 1.2\\footnotemark[1] & 900\\footnotemark[1]  \\\\\n\t\t\t\tSiC  \t& 3 \t& 5 & 16\t& 2.5\\footnotemark[1] & 1110\\footnotemark[1] \\\\\n\t\t\t\tInGaP \t& 3 \t& 5 \t& 15 \t& 0.95\\footnotemark[2] & 540\\footnotemark[1] \\\\\n\t\t\t\\end{tabular}\n\t\t\\end{ruledtabular}\n\t\t\\footnotemark[1]{From fit}\n\t\t\\footnotemark[2]{Calculated with $\\sigma_\\mathrm{2D}=\\varepsilon^\\parallel E_1\/(1-\\nu_1)$ \\cite{Bueckle2018APL-StressControl}}\n\t\\end{table}\n\t\n\tA more general consideration of the length dependence of the tensile stress according to Eqs.~(\\ref{eq:delL}) and (\\ref{eq:sigLM}) reveals that two geometric parameters, the height of the pedestal $h_0$ and the undercut of the pedestal $a_\\textrm{uc}$, dominate the stress enhancement of short nanostrings. This suggests that maximum tensile stress can be achieved for short strings with large $h_0$ and $a_\\textrm{uc}$. However, it has to be noted that this limit can only be achieved for sufficiently large clamping pads avoiding a softening of the entire clamping structure under overly large undercuts, an unwanted side effect that is not accounted for in our model.\n\t\n\tFinally, we discuss the relation between $ \\sigma_\\infty $ and $ \\sigma_{\\mathrm{2D}} $. For a one-dimensional nanostring processed from a thin film under biaxial and isotropic stress, the one-dimensional stress follows from the initial two-dimensional stress according to  \n\t%\n\t\\begin{equation}\n\t\t\\sigma_{\\mathrm{1D}} = \\sigma_{\\mathrm{2D}} (1-\\nu_1) \\label{eq:sigma1D-2D}.\n\t\\end{equation} \n\t%\n\tFor the nanostrings under investigation, this simple picture does not hold, as the stress relaxation along the $y$ direction upon releasing the string assumes a more complicated stress configuration following the contraction of the device layer described in Sec.~\\ref{Pedestal}. Not only does the contraction of the clamps by an amount $\\Delta p$ reduced the two-dimensional stress in the clamping pads from $\\sigma_{\\mathrm{2D}}$ to $\\sigma_{\\mathrm{p}}$. A similar contraction also occurs along the $y$ direction of the string, such that the tensile stress in the string before the lateral release cannot be considered isotropic. \n\t\n\tAdditionally, we wish to note that high-resolution x-ray diffraction measurements performed on In$_{1-x}$Ga$_x$P wafers have shown a compositional variation in the direction normal to the substrate \\cite{Bueckle2018APL-StressControl}. This can furthermore lead to strain gradients inside the device layer. A similar observation has been made for 3C-SiC in Ref.~\\cite{Romero2020}. This suggests that a more thorough analysis of the length-dependent stress should assume a three-dimensional strain tensor accounting for a vertical strain gradient rather than a biaxial isotropic thin-film stress $\\sigma_{\\mathrm{2D}}$.\n\t\n\t\n\t\\section{Conclusion}\n\t\n\tIn conclusion, we report on the observation of a length dependence of the tensile stress in nanomechanical string resonators. This previously unappreciated effect is material independent, and experimentally observed on samples fabricated from four different wafers, featuring the three complementary material platforms amorphous silicon nitride, crystalline silicon carbide and crystalline indium gallium phosphide. We develop a simple elastic model that describes the observed $1\/L$ dependence of the tensile stress, and which allows us to explain the observed length dependence by a combination of the elastic reconfiguration of the device under the vertical and lateral releases of the one-dimensional nanostring. The one-dimensional tensile stress relaxes to a value considerably smaller than the initial two-dimensional stress value for long strings. For shorter strings, this value increases by approximately $50$\\,\\%. Besides the length, particularly the height of the supporting pedestal and the size of the undercut of the clamping pads influence the resulting stress. Thus, the geometric parameters of the nanostring allow us to control the tensile stress, and enable stress engineering of the quality factor of the device without the need for complex phononic metamaterial processing.\n\t\n\tData and analysis code are available onlschneg \\cite{zenodo}.\n\t\n\t\\begin{acknowledgments}\n\t\tWe thank Ralf Messmer for technical support with the AFM measurements.\n\t\tFinancial support from the European\n\t\tUnions Horizon 2020 programme for Research and Innovation\n\t\tunder Grant Agreement No. 732894 (FET Proactive HOT) and the German Federal Ministry of Education\n\t\tand Research through Contract No. 13N14777 funded within\n\t\tthe European QuantERA cofund project QuaSeRT is gratefully\n\t\tacknowledged. \n\t\tWe further acknowledge support from the Deutsche Forschungsgemeinschaft via\n\t\tthe Collaborative Research Center SFB 1432 and via project WE 4721\/1-1, as well as project QT-6 SPOC of the Baden-W\\\"urttemberg Foundation\n\t\t\n\t\tM.B. and Y.S.K. contributed equally to this work.\n\t\t\n\t\\end{acknowledgments}\n\t\n\t\n\t\n\\section{Wafers and material parameters}\t\n\t\\label{A}\n\t\n\tIn Tab. \\ref{tab:wafers} the growth parameters of the four wafers employed in this work are summarized, stating the thickness of the device layer, sacrificial layer (if the system has one), substrate, and the corresponding supplier. The two SiN wafers were grown by Low Pressure Chemical Vapor Deposition (LPCVD), the SiC in a two-stage Chemical Vapor Deposition process, and the InGaP using Metal-Organic Chemical Vapor Deposition (MOCVD).\n\t\n\t\n\t\\begin{table*}[h!]\n\t\t\\caption{\n\t\t\t\\label{tab:wafers}\n\t\t\tBasic parameters of the wafers on which the string resonators were fabricated.\n\t\t}\n\t\t\\begin{ruledtabular}\n\t\t\t\\begin{tabular}{lllll}\n\t\t\t\t& resonator\/device layer & sacrificial layer & substrate & source \\\\\n\t\t\t\t\\hline\n\t\t\t\tSiN-FS \t\t\t& 100\\,nm SiN\t& --- \t& SiO$_2$ & HSG-IMIT  \\\\\n\t\t\t\tSiN-Si \t\t\t& 100\\,nm SiN \t& 400\\,nm SiO$_2$ \t& Si(100) & HSG-IMIT  \\\\\n\t\t\t\tSiC \t\t\t& 110\\,nm 3C-SiC \t& --- \t& Si(111) & NOVASiC  \\\\\n\t\t\t\tInGaP \t& \\multicolumn{1}{r}{100\\,nm In$_{0.415}$Ga$_{0.585}$P} \t& \\multicolumn{1}{l}{1000\\,nm Al$_{0.85}$Ga$_{0.15}$As} \t& GaAs & CNRS  \\\\\n\t\t\t\\end{tabular}\n\t\t\\end{ruledtabular}\n\t\\end{table*}\n\t\n\t\\newpage\n\t\n\tAll material parameters employed in the theoretical calculations, i.e. the Young's modulus $ E $, the shear modulus $ G $, the Poisson ratio $ \\nu $, and the mass density $\\rho$ of the respective materials, are listed in Tab. \\ref{tab:ModelParameters}. \n\t\n\t\\begin{table}[h]\n\t\t\\caption{\n\t\t\t\\label{tab:ModelParameters}\n\t\t\tYoung's modulus, shear modulus and density of the materials used within the main text. All shear mouduli where calculated via $ G = \\frac{E}{2(1+\\nu)} $ . \n\t\t}\n\t\t\\begin{ruledtabular}\n\t\t\t\\begin{tabular}{crrrr}\n\t\t\t\t& \\multicolumn{1}{c}{Young's modulus $ E $} & \\multicolumn{1}{c}{Shear modulus $ G $}& \\multicolumn{1}{c}{Poisson's ratio $ \\nu $} &\\multicolumn{1}{c}{density $\\rho$} \\\\\n\t\t\t\t& \\multicolumn{1}{c}{\\textrm{(GPa)}} & \\multicolumn{1}{c}{\\textrm{(GPa)}} & & \\multicolumn{1}{c}{(g\/cm$ ^3 $)} \\\\\n\t\t\t\t\\hline\n\t\t\t\tSiN\t& 260 \\cite{DosterUnpub} & 104 & 0.25\\cite{maluf2002introduction} & 3.1 \\cite{maluf2002introduction}\t  \\\\\n\t\t\t\tSiO$_2$\t& 73 \\cite{maluf2002introduction} & 31 & 0.17 \\cite{maluf2002introduction} &2.2 \\cite{maluf2002introduction}\t  \\\\\n\t\t\t\tSi & 160 \\cite{maluf2002introduction} & 66 & 0.22 \\cite{maluf2002introduction} &2.4 \\cite{maluf2002introduction}\t  \\\\\n\t\t\t\tSiC\t& 419 \\cite{li1987single}\\footnotemark[1] &\t184 & 0.14 \\cite{maluf2002introduction} &3.166 \\cite{henisch2013silicon}\\\\\n\t\t\t\tInGaP\t& 124 \\cite{Ioffe1999ShurEtAl-HandbookSeriesSemiconductorParametersVOL2}\\footnotemark[1]  & 47 & 0.32\\cite{Ioffe1999ShurEtAl-HandbookSeriesSemiconductorParametersVOL2}\\footnotemark[2] & 4.418 \\cite{Ioffe1999ShurEtAl-HandbookSeriesSemiconductorParametersVOL2}\t  \\\\\n\t\t\t\tGaAs\t& 75  \\cite{maluf2002introduction} & 29 & 0.31 \\cite{Ioffe1999ShurEtAl-HandbookSeriesSemiconductorParametersVOL2} &5.3 \\cite{maluf2002introduction}\t  \\\\\n\t\t\t\\end{tabular}\n\t\t\\end{ruledtabular}\n\t\t\\footnotemark[1]{Determined by utilizing the elastic constants and the stiffness tensor as described in \\cite{Bueckle2018APL-StressControl}.}\n\t\t\\footnotemark[2]{Calculated with $ \\nu = \\frac{c_{12}}{c_{11}+c_{12}} $ where $ c_{ij} $ are the elastic constants.}\n\t\t\\vspace{-4mm}\n\t\\end{table}\n\t\n\t\n\t\\section{Finite Element Method Simulations}\n\t\\label{B}\n\t\n\tThe geometric reconfiguration of the pedestal, the clamping pad and the string was explored in more detail by finite element method (FEM) simulations to validate our theoretical considerations. To this end, the individual $10\\,\\mu$m long and $300$\\,nm wide SiN-FS string resonator held in place by two SiO$ _2 $ pedestals on a SiO$_2$ substrate shown in Fig.~\\ref{fig:femSimulationPedestal} was considered. The thickness of the device layer was set to $100$\\,nm, a $500$\\,nm undercut and a pedestal height of $1\\,\\mu$m were assumed, as well as an initial two-dimensional tensile stress of $2.9$\\,GPa. A perfectly matched layer was included to mimick an infinite substrate, but did not noticeably influence the result.\n\tA close look at Fig.~\\ref{fig:femSimulationPedestal} clearly reveals the shearing of the pedestal as well as the contraction of the clamping pad due to the stressed device layer. Also apparent is the resulting elongation of the string and the enhanced tensile stress in the string, which, for the case of the extremely short length of the simulated string, even exceeds the remaining tensile stress in the clamping pad. \n\tThese observations qualitatively support all assumptions of the elastic model put forward in the main text.\\\\\n\n\t\n\t\\begin{figure}[h!]\n\t\t\\includegraphics[width=0.99\\linewidth]{femSimulationPedestal}\n\t\t\\caption{ FEM simulations of a single string resonator with an initial stress of $2.9$\\,GPa. Furthermore, we set a thickness of $100$\\,nm for the device layer, a $500$\\,nm undercut and a pedestal height of $1\\,\\mu $m.\n\t\t\t\\label{fig:femSimulationPedestal} \n\t\t}\n\t\\end{figure}\n\t\n\t\\section{Measuring the pedestal contraction}\n\t\\label{C}\n\t\n\tTo further support our elastic model, we have experimentally quantified the shearing of the pedestal using the test structures discussed in the following.\n\tAn array of quadratic pedestals is fabricated on SiN-FS (see Fig.~\\ref{fig:MeasurePedCont}(a) and (c)), the material for which the biggest contraction is expected (c.f. Tab. \\RNum{2}). As shown in Fig.~\\ref{fig:MeasurePedCont}(a) and (b), the uncontracted width of a pedestal is $ 2 a $ and the pedestal-pedestal distance is $ d $. An anistropic ICP-RIE etch step (etching depth of around 350~nm) allows for a contraction of the pedestal by $ 2 \\Delta p $ to $ 2 a_{\\mathrm{con}} = 2 a - 2 \\Delta p$. Because the contraction is in the nanometer regime and the pedestal in the micrometer regime, we can not simply image the whole pedestal and directly measure $ 2a $ and $ 2a_{\\mathrm{con}}  $ and calculate the contraction $ 2 \\Delta p $, as this is beyond the resolution of our scanning electron microscopy. However, as indicated schematically in Fig. ~\\ref{fig:MeasurePedCont}(b), the separation of two closely-spaced pedestals of the test structure can be mapped out with a higher resolution. Comparison of their spacing before and after the contraction, $ d $ and $ \\tilde{d} $, respectively, indeed yields in increase of the gap as shown in Fig. ~\\ref{fig:MeasurePedCont}(d), which indicates a contraction of the clamping structure. For our sample chip we measure an average value of $ d = 793(6) $~nm and $ \\tilde{d} = 824 (7) $~nm (over the entire gap, not just the section close to the center shown in Fig. ~\\ref{fig:MeasurePedCont}(d)), corresponding to a contraction of  $ \\Delta p = \\frac{\\tilde{d}-d}{2} = 15(7)~ $nm. The theoretical value of $ \\Delta p = 8~ $nm is just within the uncertainty of the measured value.  \n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\includegraphics[width=0.9\\linewidth]{MeasurePedestalContraction}\n\t\t\\caption{ Array of pedestals (a) and a close up (b) including length annotations. Dashed lines and solid lines correspond to the pedestal before and after contraction, respectively. (c) SEM image of the array structure before it was etched. (d) SEM image of the gap between two pedestals before (left) and after (right) contraction.\n\t\t\t\\label{fig:MeasurePedCont} \n\t\t}\n\t\\end{figure}\n\t\n\t\n\t\\clearpage\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \n\nAlthough the use of low clinker ratio cements is increasing, the development of\nnew clinker types remains a reliable strategy to reduce greenhouse gas emissions\nand improve the properties of cement for concrete structures applications. In a\ncontext of durable design, considerable efforts are being made for a better\nunderstanding of original Portland cement (OPC) hydration. However, considering\nthe hydration of all clinker phases together would turn the study highly\ncomplex. Most of the time, the tricalcium silicate (\\ce{C3S}) received a\nparticular attention, due to its predominance in OPC clinker (about\n\\SI{50}{\\percent} to \\SI{70}{\\percent} by mass). \\ce{C3S} is the main phase\nresponsible for OPC setting and strength development. The reactivity and early\nhydration of tricalcium silicate (\\ce{C3S}) is a very relevant topic towards a more sustainable design of OPC. The hydration itself\nencompasses several processes such as dissolution, phase growth, diffusion and\ncomplexation \\cite{Bullard2011}. Atomistic simulation methods have shown good\ncapabilities in predicting the reactivity of mineral surfaces and the behavior\nof solid\/liquid interfaces. Towards a better understanding of the phases and processes occurring in cementitious systems, many atomistic force fields have been optimized \\cite{Mishra2017}.  Molecular dynamics (MD) and density\nfunctional theory (DFT), have already been used to compute surface energies and\ndetermine Wulff shapes for monoclinic \\ce{M3 C3S}\n\\cite{Manzano2015,Durgun2014,Mishra2013}. In recent studies, molecular and\ndissociative adsorption of single water molecule were investigated on multiple\nsurface planes of \\ce{M3 C3S} polymorph \\cite{Zhang2018b,Zhang2019,Qi2019}.\nZhang et al. have shown that the adsorption energy decreases with increasing\namount of adsorbed molecules. Reactive MD studies indicates that after\napproximately \\SI{0.3}{\\ns}, the structural properties of the surface are lost,\nmaking further hydration process independent of the crystallographic surface\nplane, and driven by proton hopping mechanisms towards the bulk\n\\cite{Manzano2015,Huang2015}. No correlation was found between water adsorption\nenergy and surface energy, when using static computational methods\n\\cite{Manzano2015}. However, the proton diffusion after the initial stage of\nhydration was related to the location of the valence band maximum (VBM), which\nis mainly constituted of oxygen 2p orbitals \\cite{Huang2015,Huang2014}. Previous\nDFT studies reveal that the local density of state of the VBM is close to the\noxygen anions for \\ce{C3S}, and close to oxygen in silicates for \\ce{C2S}\n\\cite{Durgun2012}. The higher reactivity of \\ce{C3S}\nwhen compared to \\ce{C2S} is explained by the difference in their electronic\nstructure, arising from the presence of oxygen anions in \\ce{C3S}. Calculations\nof a single water molecule sorption on a (100) surface of \\ce{T1 C3S}, shown\nthat chemisorption occured only in regions close to oxide ions. This behaviour\nwas associated to the higher degree of freedom of oxide ions when compared to\noxygen in silicate \\cite{Huang2015}.\n\nProton transfer (PT) frequency strongly depends on hydrogen bonds (HB)\nfluctuation due to thermal motion \\cite{Tocci2014}, and thus cannot be analyzed\nby a \\SI{0}{\\kelvin}, DFT investigation. Furthermore, such a phenomenon cannot\nbe captured considering a single water molecule adsorption. A previous\ncomputational study found structural changes, as well as a huge increase in PT\nrate from a solid\/water monolayer interface to a thicker water film\n\\cite{Tocci2014}. Towards a better understanding of the \\ce{C3S}\/water\ninterface, we performed an \\emph{ab initio} MD (AIMD) simulation, considering a\nwater film thick enough to account for fluctuation of the HB network. AIMD is a\npowerful tool that has been used extensively to investigate the structural and\ndynamical behavior of water\/oxide interfaces at the DFT level of theory\n\\cite{Bjoerneholm2016,Tocci2014,Ngouana-Wakou2017,Rimsza2016,Cimas2014,Laporte2015,Lee2015,Liu2008}.\nHowever, only few AIMD studies were conducted on cementitious materials\n\\cite{Churakov2009,Churakov2009a,Churakov2017}. As far as we know, this is the\nfirst time that the very early hydration stage of \\ce{C3S} is investigated using\nAIMD. In particular, the structure of water and the PT dynamics are analysed and\nquantified, and the results are compared with reactive molecular dynamics\ncalculations, performed for that purpose.\n\n\\section{Computational Methods}\n\nA simulation of the \\ce{C3S}\/water interface was performed on the symmetric,\nCa-rich, (040) plane (as in \\cite{Mishra2013}). The \\ce{M3 C3S} model employed\nwas refined from XRD analysis by Mumme et al. \\cite{Mumme1995}. The unit\ncell of 54 atoms was optimized, at the DFT level with the Quantum Espresso code,\nusing the PBE exchange-correlation functional \\cite{Perdew1996,Perdew1997} with\na Grimme D2 correction for van der Waals interactions \\cite{Grimme2006}. The\nkinetic energy cutoffs for wave functions and charge density were\n\\SI{45}{\\rydberg} and \\SI{405}{\\rydberg}, respectively. The Monkhorst-Pack\nmethod was used for the integration of the first Brillouin zone, with a\n\\num{3x3x3} k-point mesh. During the optimization process, the atoms were\nallowed to relax. In order to build a surface model, the optimized unit cell was\nconverted to an orthorhombic supercell of 162 atoms, with lattice parameters $a$\n= \\SI{12.28}{\\angstrom}, $b$ = \\SI{7.09}{\\angstrom} and $c$ =\n\\SI{25.59}{\\angstrom}. This transformation was performed with the Atomsk code,\nwhich searches linear combination of the unit cell vectors producing vectors\naligned with Cartesian axes \\cite{Hirel2015}. The optimized monoclinic cell and\ncorresponding orthorhombic supercell are represented in \\cref{fig:cells}.\n\n\\begin{figure}\n\\centering\n\\includegraphics{figure1}\n\\caption{\\ce{M3} monoclinic cell, optimized from the model by Mumme et al.\n\\cite{Mumme1995} (right). Corresponding orthorhombic supercell (left). Color\ncode: calcium cations in green, oxygen  anions and silicate oxygen in red,\nsilicon atoms in yellow.}\n\\label{fig:cells}\n\\end{figure}\n\nThe surface model was created from three orthorhombic supercells, with a\n\\SI{20}{\\angstrom} thick vacuum layer, thus resulting in a \\SI{12.28 x 25.59\nx 21.28}{\\angstrom} structure. The relaxation of the surface and the AIMD\nsimulation were performed with the CP2K code, using a PBE functional, and a\ncombination of Gaussian and plane wave basis functions (GPW), with Grimme D2\ncorrection. A \\SI{400}{\\rydberg} planewave cutoff was adopted, and the\nreciprocal space was sampled only at the $\\gamma$ point. To relax the surface, the periodicity was applied for in-plane directions, and removed in the\ndirection of the vacuum. The atoms of the surface were allowed to relax at\nthe DFT level, ensuring that almost no change occurs within the middle of\nthe slab. The interface model was created adding a \\SI{15}{\\angstrom} thick\nlayer of water (157 molecules), with a \\SI{15}{\\angstrom} vacuum region. The\nstructure of the \\ce{C3S}(040)\/water interface is depited in \\cref{fig:system}.\n\n\\begin{figure}\n\\centering\n\\includegraphics{figure2}\n\\caption{Structure of the investigated \\ce{C3S}(040)\/water interface.}\n\\label{fig:system}\n\\end{figure}\n\nWhile the atoms of the mineral surface were kept fixed, the water molecules were\nallowed to relax on the surface during a \\SI{2}{ns} classical MD run in NVT\nensemble at \\SI{300}{\\kelvin}, using the INTERFACE FF parameters for \\ce{C3S}\n\\cite{Mishra2013} and a SPC model for water \\cite{Berendsen1981}. In order to\nminimize the computational time, the bottom layer was removed so that the\nremaining slab was composed of two orthorhombic supercells (\\SI{\\sim\n14}{\\angstrom} thick) and during the AIMD run, the lower supercell, considered\nas the bulk, was fixed. Afterwards, a \\SI{18}{\\ps} AIMD run was performed within\nthe Born-Oppenheimer approximation, in the canonical ensemble, with a\nNose-Hoover thermostat, integrating the equation of motion with a \\SI{0.5}{\\fs}\ntimestep. Based on the evolution of the energy of the system, it was considered\nthat the equilibrium was reached after \\SI{6}{\\ps}, and the remaining\nsimulation time was used for analysis of equilibrium properties. A slightly higher temperature of\n\\SI{360}{\\kelvin} (compared to standard conditions) was used to balance the low\ndiffusivity of water using the PBE functional. Deuterium masses were used for\nprotons to minimize the vibrational frequency of nuclei. It deserves to highlight that such substitution could decrease the frequency of PT, because of the lower vibrational frequency of deuterium nuclei in comparison to hydrogen nuclei. However, this method has already been used in the literature to prevent from energy drifts and it is understood that its benefits outweigh losses\n\\cite{Leung2006,Tocci2014}. A reactive molecular dynamics simulation was\nperformed using the ReaxFF, with the current optimized set of parameters for\nCa\/O\/H\/Si elements \\cite{Fogarty2010,Manzano2012,Manzano2012a}. The simulation\nmethod and parameters, as well as the system size, are in accordance with\npreviously reported calculations \\cite{Manzano2015}.\n\n\\section{Results}\n\\subsection{Water structure}\n\nIn this article, Oi refers to oxygen anions, Os refers to oxygens in silicates,\nand Odw refers to oxygens resulting from the dissociation of water molecules.\nAt the very first steps of the simulation, three oxide ions Oi from equivalent\nsites are protonated. The hydration model for \\ce{C3S} proposed by\nPustovgar et al. considers that protonation of oxide ions occurs before\nprotonation of silicates \\cite{Pustovgar2017}. Although hydroxides are more\nstable than silanol groups, our simulation indicates that on the considered\n(040) Ca-rich surface, protonation of Oi only occurs on sites close to\nsilicates. The other superficial oxide ions are shielded by four\ncalcium cations, hindering any protonation reaction. The negatively charged region allows water molecules to form hydrogen bonds with oxide ions on one side and with oxygens in silicates on the other side, thus leading to protonation of Oi (see\n\\cref{fig:oxide_protonation}).\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figure3}\n\t\\caption{Atomic structure of the (040) surface. Hydrogen in hydroxide H-Oi are colorized in white. The snapshot shows the position of the water\n\tmolecule before protonation of the oxide ion.  The ions of the surface are\n\tin the same layer, except for the calcium indicated by blue circles, having\n\tperpendicular coordinate $z$ \\SI{\\sim 2}{\\angstrom} larger.}\n\t\\label{fig:oxide_protonation}\n\\end{figure}\n\nThe number of H-Os groups formed on silicates tends to stabilize after \\SI{\\sim\n1}{\\ps} whereas the number of H-Odw groups stabilize after \\SI{\\sim 0.25}{\\ps}\n(see \\cref{fig:ho-coverage}). Both hydroxyl groups fluctuate during the whole\nsimulation due to proton transfer. Conversely, the hydroxides H-Oi formed on\noxygen anions Oi are very stable and no backward PT occurs. From our simulation\nusing the ReaxFF, within the timescale of \\SI{18}{\\ps}, a steady state is\nreached very quickly as in the AIMD simulation. However, all hydroxyl groups\nformed in the ReaxFF simulation are very stable, and the currently developed set\nof parameters for Ca\/O\/H\/Si failed to describe the PT dynamics between water and\nOi\/Os atoms which is observed in our AIMD simulation. The complexity to optimize\nparameters for ReaxFF relies on the fact that the same set of parameters is\nemployed for each element \\cite{Senftle2016}. It means that the parameters are\nthe same independently of the environment. Other approaches based on the\nempirical valence bond model has succeeded in reproducing \\ce{OH-} solvation and\ntransport in water solutions \\cite{Ufimtsev2009}. The implementation of a PT\nmodel almost doubled the diffusion of \\ce{OH-} ions when compared to a classical\nmodel. The hydroxyl coverage of each hydroxyl type is close to the result of the\nAIMD simulation. Therefore, within the timescale of the simulation, the ReaxFF\nis representing the protonation state of the (040) surface in good agreement\nwith the AIMD simulation. The average total hydroxyl coverage over the last\n\\SI{12}{\\ps} of simulation is \\SI{5.36(37)}{\\hopernmsquared} for the AIMD\nsimulation, and \\SI{5.17(1)}{\\hopernmsquared} for the ReaxFF simulation. These\nvalues are also in agreement with previous investigation on \\ce{C3S} hydration,\nusing the ReaxFF \\cite{Manzano2015,Huang2015}.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics{figure4}\n\t\\caption{Number of hydroxyl groups formed at the surface on oxide ions\n\t(Oi-H), oxygen in silicates (Os-H), and from water dissociation (Odw-H) for\n\t(a) AIMD simulation, (b) ReaxFF simulation.}\n\t\\label{fig:ho-coverage}\n\\end{figure}\nThe atomic density profile of water oxygen and hydrogen atom, along the axis\nperpendicular to the surface, is reported in \\cref{fig:1d_density_snap}.\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics{figure5}\n\t\\caption{Atomic density profile of water molecules along the z axis for\n\tAIMD. The origin adopted is the average coordinate of the uppermost oxygen\n\tsilicate layer. The layering observed results from the hydrogen bond network\n\tcreated between the strong interaction between the water molecules and the\n\tionic surface.}\n\t\\label{fig:1d_density_snap}\n\\end{figure*}\nThe layered structure of the interfacial water results from the effect of\nexcluded volume, electrostatic force field, and hydrogen bonding network.\nPrevious investigations based on classical MD simulations showed that this\nlayering is lost with protonation of the surface \\cite{Claverie2019}. The\nclosest hydrogen's peak from the surface, is at the average position of oxygen\nin silicates Os ($z = 0$), and corresponds to chemisorbed H. The thickness of\nthe layered region is approximately the same as in our previous classical MD\ninvestigation: \\SIrange{\\sim 5}{6}{\\angstrom} \\cite{Claverie2019}. \n\nThe radial distribution functions (RDF) of H-Oi, H-Os and Ow-Ca pairs are\nplotted in \\cref{fig:rdf}. \n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics{figure6}\n\t\\caption{Radial distribution function H-Oi, H-Os, and Ow-Ca pairs employing AIMD, ReaxFF, and IFF \\cite{Claverie2019}. The inset shows a zoom out ($\\times 0.5$).}\n\t\\label{fig:rdf}\n\\end{figure}\nFor H-Oi and Ow-Ca pairs, sharp peaks raise at \\SI{\\sim 0.97}{\\angstrom} and\n\\SI{\\sim 2.50}{\\angstrom} respectively, indicating superficial hydroxides and\nthe first coordination shell of calcium cation. No second coordination shell is\nnoticed for both pairs. Yet for the H-Os pairs, two peaks stand at \\SI{\\sim\n1.02}{\\angstrom} and \\SI{\\sim 1.58}{\\angstrom}, corresponding to hydroxyl groups\nformed on silicates and H-bonds between water and oxygen in silicates. The RDF\nobtained in the ReaxFF simulation is very similar, with two main differences: in\none case, only one correlation peak is observed for the H- Os pair, suggesting\nan Os protonation, and the coordination peak for Ow-Ca pairs is split in two,\nindicating two different distances of correlated water molecules. RDF for dry\n\\ce{C3S}\/water interface, obtained from previous MD investigation\n\\cite{Claverie2019}, was plotted for comparative purpose. In such classical\nsimulation, no PT occurs and the first coordination peak correspond to H-bonds\nbetween water and superficial anions (r \\SI{\\sim 1.53}{\\angstrom}).\n\nThe orientation of water molecules in contact with the surface is a\ncharacteristic of the hydrophilic\/hydrophobic behavior of the surface. The\nprobability distribution of the angle $\\theta$ between the water dipole moment\nand the $z$ axis is depicted in \\cref{fig:angdis_water} a). Within the contact\nlayer ($z$ \\SI{< 1.6}{\\angstrom}), most of the water molecules have $\\theta$\n\\SIrange[range-phrase = --]{\\sim 20}{50}{\\degree} or $\\theta$\n\\SIrange[range-phrase = --]{\\sim 120}{160}{\\degree}, meaning that their dipole\nmoments point preferentially towards or against the surface. This feature is\ncharacteristic of hydrophilic surfaces \\cite{Deryagin1987}. Less probably, water\nmolecules forming a single hydrogen bond with silicates orient with their dipole\nmoment parallel to the surface.   Water molecules in the contact layer orient\naccording to the charge of superficial ionic species. Thus, two regions can be\ndistinguished: one where the water dipole is oriented upward and H atoms\ncoordinate with Os, and a second where the water dipole is oriented downward and\nOw atoms coordinate with calcium cations. These regions are mapped on the\nsurface in \\cref{fig:angdis_water} b) by collecting $\\theta$ and the $x$ and $y$\ncoordinates of water molecules within \\SI{3}{\\angstrom} from the surface, during\nthe whole simulation.  The effect of the topology of \\ce{C3S} surfaces on the\nstructure of water molecules has already been reported in a previous molecular\ndynamics investigation \\cite{Alex2017}.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figure7}\n\t\\caption{(a) Probability distribution of the angle $\\theta$ between the\n\twater dipole moment and the z axis, perpendicular to the surface. Lighter\n\tregions correspond to higher probabilities (b) Density mapping of $\\theta$\n\tfor water molecules within \\SI{3}{\\angstrom} from the surface. Color code:\n\tCa in green, Os and Oi in red, Si in yellow, H in hydroxide H-Oi in\n\twhite.}\n\t\\label{fig:angdis_water}\n\\end{figure}\nThe probability distribution of H-Os and H-Odw hydroxyl groups is plotted in\n\\cref{fig:map_oh}. H-Odw groups are principally located on Ca-rich, positively\ncharged regions, but also close to protonated silicates.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\columnwidth]{figure8}\n\\caption{Normalized probability distribution of H-Os (blue) and H-Odw (red) hydroxyl groups.}\n\\label{fig:map_oh}\n\\end{figure}\n\n\\subsection{Proton transfer analysis}\n\nThe frequency $\\nu$ of PT between water molecules and Os-H and Odw-H groups is\nreported in \\cref{fig:frequency}. The lifetime $\\tau$ is defined as\nthe time for a proton to return to the oxygen atoms to which it was initially\nbonded.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figure9}\n\t\\caption{Frequency $\\nu$ of PT as a function of its lifetime $\\tau$ (left). Evolution of the total number of hops within the simulation time (right).}\n\t\\label{fig:frequency}\n\\end{figure}\nThe inset plot shows the evolution of proton hops during the simulation. Hops\nbetween Ow and Os reach a plateau after \\SI{\\sim 8}{\\ps}. Conversely, hops\nwithin Odw-Ow pairs seems to increase steadily during the whole simulation and\nare about 5 times more frequent than within Os-Ow pairs during the first\n\\SI{8}{\\ps}. The typical lifetime of PT events is shorter than\n\\SI{100}{\\fs}, which corresponds to the time scale of Eigen-Zendel structure\ninterchanging in bulk water, obtained by femtosecond vibrational spectroscopy\n\\cite{Woutersen2006}. Lifetimes of approximately the same duration were reported\nfor PT at water\/ZnO($10\\bar{1}0$) interface \\cite{Tocci2014}. The longest\nlifetimes reported are \\SI{\\sim 0.2}{\\ps} for transfer between Ow and Os, and\n\\SI{\\sim 2}{\\ps} for transfer between Ow and Odw.\n\nThe free energy profiles of PT between Ow and Os and between Ow and Odw pairs\nwere obtained from a standard method as follows: $F = -k_BT \\log P$ (see \\cref{fig:free-energy}) \\cite{Tocci2014, Ufimtsev2009, Zhu2002}.\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics{figure10}\n\t\\caption{Free energy contour plot of PT between Ow and Os as a function of\n\tthe distance $d_{\\text{Oa-Ob}}$ between oxygen atoms, and of\n\t$\\delta_{\\text{a-b}} = d_{\\text{Oa-H}} - d_{\\text{Ob-H}}$. The red circles\n\tshows the local free energy minima, corresponding to the most stable\n\tconfigurations before and after the proton jump. The red square is the\n\tsaddle point, where the proton is equidistant from both oxygen atoms.}\n\t\\label{fig:free-energy}\n\\end{figure*}\n$P$ is a probability distribution function of the distance $d_{\\text{Oa-Ob}}$\nbetween two oxygen atoms Oa and Ob, and of $\\delta_{\\text{a-b}} =\nd_{\\text{Oa-H}} - d_{\\text{Ob-H}}$. In the case of PT between Ow and Os, the\nmolecular configuration of water forming an H-bond with Os is more stable than\nthe dissociative configuration. The free energy profile of PT between Ow and Odw\nsuggests that the molecular adsorption of water on the surface is more stable\nthan the dissociated form. This observation is contrary to previous DFT and\nreactive MD calculations on a single water molecule, where dissociative\nadsorption energies were generally larger than molecular adsorption energies,\nresulting in more stable configurations \\cite{Zhang2019, Zhang2018b,Qi2019,\nManzano2015}. This consolidates the idea that static calculations on single\nmolecule adsorption cannot describe accurately the properties of a solid\/water\ninterface. Tocci and Michaelides also reported considerable differences in PT\nrate between a monolayer and an thicker water film \\cite{Tocci2014}. These\ndifferences arise from the decrease of the free energy barrier of proton\ntransfers induced by H-bond fluctuations. The free energy barrier is \\SI{\\sim\n3.24}{\\kt} for PT from Ow to Os and \\SI{\\sim 2.24}{\\kt} from Os to Ow. The\nbarrier is \\SI{\\sim 3.30}{\\kt} for transfer from Ow to Odw and \\SI{\\sim\n3.60}{\\kt} for the reverse reaction. These results suggest that hydroxides\nformed by water dissociation are more stable than silanol groups. However, the\nenergy barrier for the creation of both hydroxyl groups is almost equal.\n\nElectron density difference analysis allows to map the distribution of electrons\ninvolved in PT, and more generally in the adsorption. This analysis was realized\nperforming static DFT calculation on the total system, as well as \\ce{C3S}, and\nwater independently on the system configuration at \\SI{5}{\\ps}. The electron\ndensity difference $\\Delta \\rho$ was calculated as follow:\n\\begin{equation}\n\\Delta \\rho = \\rho_{\\ce{C3S}\/\\ce{H2O}} - \\rho_{\\ce{C3S}} -\\rho_{\\ce{H2O}}\n\\end{equation}\nwhere $\\rho_{\\ce{C3S}\/\\ce{H2O}}$ corresponds to the electron density of the\ninterface system, $\\rho_{\\ce{C3S}}$ and $\\rho_{\\ce{H2O}}$ are the electron\ndensity of the \\ce{C3S} and water alone, respectively. A positive value of\n$\\Delta \\rho$ indicates a high electron density, while a\nnegative value points out an electron depletion region.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics{figure11}\n\t\\caption{Snapshots of the isosurface of electron density difference. Golden and cyan isosurface represent positive and negative $\\Delta \\rho$, respectively. a) A water molecule between a silicate and a silanol group.  b1) Electron density delocalization during a proton exchange between a molecular water and a hydroxide. b2) Electron abundance and depletion regions around an hydroxide created upon water dissociation.}\n\t\\label{fig:edensity}\n\\end{figure*}\n\nThe proton transfers occurring at the surface create an electron\ndelocalization, and thus high and low electron density regions. Water molecules\nact as charge carrier and the electron depletion or gain highly depends on the\nlocation of the molecule (see \\cref{fig:edensity} a and b1). High electron\ndensity is observed around the silicate oxygen close to the water molecule,\ncreating a depletion region on the silicon atom. A large depletion region is\nobserved around the silanol group (\\cref{fig:edensity} a). Charge depletion\nregions are observed around H of hydroxyl groups. Their magnitude increase in\nthis order: H-Oi $<$ H-Odw $<$ H-Os. In other words, the magnitude of the\ndepletion regions on H decreases as a function of the stability of the hydroxyl\ngroup. A large depletion region indicates a greater charge separation, and a\nmore ionic bond, while a small depletion region reveals a more covalent bond.\n\n\\section{Conclusion}\nThe very early hydration of the (040) surface of \\ce{C3S} was investigated\nthrough a \\SI{18}{\\ps} AIMD simulation. As a first observation, only 1\/3 of the\noxide ions on the surface were protonated during the whole simulation. The\nhydroxides formed are highly stable and no proton exchange was observed.\nAlthough the oxide ion is very unstable in water, we found that its environment\non the surface is an important factor for the creation of hydrogen bond with\nwater molecules and for protonation to occur. Thus, the pKa of hydroxide and\nsilicic acid in solution cannot predict accurately the protonation state of the\nsurface during the very early stage of hydration. The structure of water at\nthe interface, resulting from the formation of the hydrogen bond network, is\nvery similar to that of our previous classical molecular dynamics study, with a\nthickness of the layered region of approximately \\SIrange{5}{6}{\\angstrom} from\nthe surface. The (040) surface is composed of Ca-rich regions (positively\ncharged) and Si-rich regions (negatively charged). Water molecules in the\ncontact layer orient their dipole moment in accordance with the surface charge,\nmaking either H-bonds in Si-rich regions, or creating strong Ca-Ow interactions\nin Ca-rich regions. Energy barrier analysis suggests that the molecular\nadsorption of water on the \\ce{C3S} surface is more stable than dissociative\nadsorption. Based on proton transfer energy analysis, the hydroxyl groups formed\nwere classified in order of stability as follow: H-Oi $>$ H-Odw $>$ H-Os. From\nelectron density difference, high electron density, and depletion regions were\nobserved. These observations revealed that the magnitude of the electron\ndepletion region upon adsorption is smaller for more stable hydroxyl groups. \n\n\\section*{Acknowledgments}\nThe authors acknowledge Brazilian science agencies CAPES (PDSE process\nn\\si{\\degree}88881.188619\/2018-01) and CNPq for financial support, as well as\nHegoi Manzano and Gabriele Tocci for helpful discussions. The authors also thank\nthe anonymous reviewers for their careful reading of our manuscript and their\nmany insightful comments and suggestions.\n\t\n\\bibliographystyle{elsarticle-num}\n\\biboptions{sort&compress}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}}\n\\newcommand{\\subsec}[1]{\\subsection{#1}}\n\\newcommand{{\\cal O}}{{\\cal O}}\n\\newcommand{{\\cal N}}{{\\cal N}}\n\\newcommand{{\\cal A}}{{\\cal A}}\n\\newcommand{{\\cal N}}{{\\cal N}}\n\\def\\slashed#1{{\\ooalign{\\hfil\\hfil\/\\hfil\\cr $#1$}}}\n\\newcommand{\\gb}[1]{\\langle#1]}\n\\newcommand{\\rf}[1]{(\\ref{#1})}\n\\def{\\cal M}{{\\cal M}}\n\\def\\lambda{\\lambda}\n\\def\\tilde\\lambda{\\tilde\\lambda}\n\\def\\Bbb{CP}{\\Bbb{CP}}\n\\def\\Bbb{RP}{\\Bbb{RP}}\n\n\n\n\\begin{document}\n\\baselineskip=16pt\n\\pagestyle{plain}\n\\setcounter{page}{1}\n\n\\begin{titlepage}\n\n\\begin{flushright}\nhep-th\/0504194 \\\\\n\\end{flushright}\n\\vfil\n\n\\begin{center}\n{\\huge Lectures on Twistor Strings}\\\\ \\huge{ and} \\linebreak\n\\huge{Perturbative Yang-Mills Theory}\n\\end{center}\n\n\\vfil\n\\begin{center}\n{\\large Freddy Cachazo and Peter Svr\\v{c}ek\\footnote{On leave from\nPrinceton University.}\n}\\end{center}\n\n$$\\seqalign{\\span\\hfil$\\displaystyle{##}$\\, & \\sl\\span\\hbox{##}}{\n & Institute for Advanced Study,\n Princeton, NJ  08540, USA  \\cr\n}$$\n\\vfil\n\n\\begin{center}\n{\\large Abstract}\n\\end{center}\n\n\\noindent Recently, Witten proposed a topological string theory in\ntwistor space that is dual to a weakly coupled gauge theory. In\nthis lectures we will discuss aspects of the twistor string\ntheory. Along the way we will learn new things about Yang-Mills\nscattering amplitudes. The string theory sheds light on Yang-Mills\nperturbation theory and leads to new methods for computing\nYang-Mills scattering amplitudes\\footnote{These lecture notes are\nbased on lectures given by the second author at RTN Winter School\non Strings,\n Supergravity and Gauge Theories, 31\/1-4\/2 2005 SISSA, Trieste, Italy}.\n\n\\vfil\n\\begin{flushleft}\nApril, 2005\n\\end{flushleft}\n\\end{titlepage}\n\\newpage\n\\renewcommand{\\thefootnote}{\\arabic{footnote}}\n\\setcounter{footnote}{0}\n\\renewcommand{\\baselinestretch}{1.2} \n\\tableofcontents\n\\def\\overline{\\overline}\n\n\\section{Introduction}\n\nThe idea that a gauge theory should be dual to a string theory\ngoes back to 't Hooft \\cite{'tHooft:1973jz}.  't Hooft considered\n$U(N)$ gauge theory in the large $N$ limit while keeping\n$\\lambda=g_{YM}^2 N$ fixed. He observed that the perturbative\nexpansion of Yang-Mills can be reorganized in terms of Riemann\nsurfaces, which he interpreted as an evidence for a hypothetical\ndual string theory with string coupling $g_s\\sim 1\/N.$\n\nIn 1997, Maldacena proposed a concrete example of this duality\n\\cite{Maldacena:1997re}. He considered the maximally\nsupersymmetric Yang-Mills theory and conjectured that it is dual\nto type IIB string theory on $AdS_5\\times S^5.$ This duality led\nto many new insights from string theory about gauge theories and\nvice versa. At the moment, we have control over the duality only\nfor strongly coupled gauge theory. This corresponds to the limit\nof large radius of $AdS_5\\times S^5$ in which the string theory is\nwell described by supergravity. However, QCD is asymptotically\nfree, so we would also like to have a string theory description of\na weakly coupled gauge theory.\n\nIn weakly coupled field theories, the natural object to study is\nthe perturbative $S$ matrix. The perturbative expansion of the $S$\nmatrix is conventionally computed using Feynman rules. Starting\nfrom early studies of de Witt \\cite{DeWitt:1967uc}, it was\nobserved that scattering amplitudes show simplicity that is not\napparent from the Feynman rules. For example, the maximally\nhelicity violating (MHV) amplitudes can be expressed as  simple\nholomorphic functions.\n\nRecently, Witten proposed a string theory that is dual to a weakly\ncoupled ${\\cal N}=4$ gauge theory \\cite{Witten:2003nn}. The\nperturbative expansion of the gauge theory is related to\nD-instanton expansion of the string theory. The string theory in\nquestion is the topological open string B-model on a Calabi-Yau\nsupermanifold $\\Bbb{CP}^{3|4},$ which is a supersymmetric\ngeneralization of  Penrose's twistor space.\n\nAt tree level, evaluating the instanton contribution has led to\nnew insights about scattering amplitudes. `Disconnected'\ninstantons give the MHV diagram construction of  amplitudes in\nterms of Feynman diagrams with vertices that are suitable\noff-shell continuations of the MHV amplitudes\n\\cite{Cachazo:2004kj}. The `connected' instanton contributions\nexpress the amplitudes as integrals over the moduli space of\nholomorphic curves in twistor space \\cite{Roiban:2004yf}.\nSurprisingly, the MHV diagram construction and the connected\ninstanton integral can be related via localization on the moduli\nspace \\cite{Gukov:2004ei}.\n\nDespite the successes of the twistor string theory at tree level,\nthere are still many open questions. The most pressing issue is\nperhaps the closed strings that give ${\\cal N}=4$ conformal\nsupergravity \\cite{Berkovits:2004jj}. At tree level, it is\npossible to recover the Yang-Mills scattering amplitudes by\nextracting the single-trace amplitudes. At loop level, the single\ntrace gluon scattering amplitudes receive contributions from\ninternal supergravity states, so it would be difficult to extract\nthe Yang-Mills contribution to the gluon scattering amplitudes.\nSince, ${\\cal N}=4$ Yang-Mills theory is consistent without conformal\nsupergravity, it is likely that there exists a version of the\ntwistor string theory that is dual to pure Yang-Mills theory.\nIndeed, the MHV diagram construction that at tree level has been\nderived from twistor string theory seems to compute loop\namplitudes as well \\cite{Brandhuber:2004yw}.\n\nThe study of twistor structure of scattering amplitudes has\ninspired new developments in perturbative Yang-Mills theory\nitself. At tree level, this has led to recursion relations for\non-shell amplitudes \\cite{Britto:2004ap}. At one loop, unitarity\ntechniques \\cite{Bern:1994cg,Bern:1994zx} have been used to find\nnew ways of computing  ${\\cal N}=4$ \\cite{Britto:2004nc} and ${\\cal N}=1$\n\\cite{Britto:2005ha} Yang-Mills amplitudes.\n\nIn these lectures we will discuss aspects of twistor string\ntheory. Along the way we will learn lessons about Yang-Mills\nscattering amplitudes. The string theory sheds light on Yang-Mills\nperturbation theory and leads to new methods for computing\nYang-Mills scattering amplitudes. In the last section, we will\ndescribe further developments in perturbative Yang-Mills.\n\n\\section{Helicity Amplitudes}\n\\subsection{Spinors}\n\\label{spinors}\n\nRecall\\footnote{The sections $2-4$\n are based on lectures given by E. Witten at PITP, IAS Summer 2004}\n that the complexified Lorentz group is locally isomorphic\nto\n\\begin{equation}\nSO(3,1,\\Bbb C)\\cong Sl(2,\\Bbb C)\\times Sl(2,\\Bbb C),\n\\label{sothree}\n\\end{equation}\nhence the finite dimensional representations are classified as\n$(p,q)$ where $p$ and $q$ are integer or half-integer. The\nnegative and positive chirality spinors transform in the\nrepresentations $(1\/2,0)$ and $(0,1\/2)$ respectively. We write\ngenerically $\\lambda_a,a=1,2$ for a spinor transforming as\n$(1\/2,0)$ and $\\tilde\\lambda_{\\dot a}, \\dot a=1,2$ for a spinor\ntransforming as $(0,1\/2).$\n\nThe spinor indices of type $(1\/2,0)$ are raised and lowered using\nthe antisymmetric tensors $\\epsilon_{ab}$ and $\\epsilon^{ab}$\nobeying $\\epsilon_{12}=1$ and\n$\\epsilon^{ac}\\epsilon_{cb}=\\delta^a{}_b$ \\eqalign{raise}{\n\\lambda^a = \\epsilon^{ab}\\lambda_b\\qquad \\lambda_a =\n\\epsilon_{ab}\\lambda^b.} Given two spinors $\\lambda$ and\n$\\lambda',$ both of negative chirality, we can form the Lorentz\ninvariant product\n\\eqn{skew}{\\vev{\\lambda,\\lambda'}=\\epsilon_{ab}\\lambda^a\\lambda'{}^b.}\nIt follows that $\\vev{\\lambda,\\lambda'}=-\\vev{\\lambda',\\lambda}$,\nso the product is antisymmetric in its two variables. In\nparticular, $\\vev{\\lambda,\\lambda'}=0$ implies that $\\lambda$\nequals $\\lambda'$  up to a scaling $\\lambda^a=c \\lambda'^a.$\n\nSimilarly, we lower and raise the indices of positive chirality\nspinors with the antisymmetric tensor $\\epsilon_{\\dot a\\dot b}$\nand its inverse $\\epsilon^{\\dot a \\dot b}.$ For two spinors\n$\\tilde\\lambda$ and $\\tilde\\lambda',$ both of positive chirality\nwe define the antisymmetric product\n\\eqn{angled}{[\\tilde\\lambda,\\tilde\\lambda']=-[\\tilde\\lambda',\\tilde\\lambda]=\\epsilon_{\\dot\na\\dot b}\\tilde\\lambda^{\\dot a}\\tilde\\lambda'^{\\dot b}.}\n\nThe vector representation of $SO(3,1,\\Bbb C)$ is the $(1\/2,1\/2)$\nrepresentation. Thus a momentum vector $p_\\mu, \\mu=0,\\dots,3$ can\nbe represented as a bi-spinor $p_{a\\dot a}$ with one spinor index\n$a$ and $\\dot a$ of each chirality. The explicit mapping from\n$p_\\mu$ to $p_{a \\dot a}$ can be made using the chiral part of the\nDirac matrices.  In signature $+---,$ one can take the Dirac\nmatrices to be \\begin{equation}\\gamma^\\mu=\\begin{pmatrix}0 &\n\\sigma^\\mu \\cr \\overline\\sigma^\\mu &0\\end{pmatrix},\\end{equation} where\n$\\sigma^\\mu=(1,\\vec\\sigma), \\overline\\sigma^\\mu=(1,-\\vec\\sigma)$ with\n$\\vec\\sigma$ being the $2\\times 2$ Pauli matrices.  For any\nvector, the relation between  $p_\\mu,$ and $p_{a\\dot a}$ is\n\\eqn{transf}{ p_{a\\dot a}= p_\\mu \\sigma^\\mu_{a \\dot\na}=p_0+\\vec\\sigma\\cdot\\vec p.} It follows that, \\eqn{detr}{p_\\mu\np^\\mu= \\det(p_{a\\dot a}).} Hence, $p_\\mu$ is lightlike if the\ncorresponding determinant is zero. This is equivalent to the rank\nof the $2\\times2$ matrix $p_{a\\dot a}$ being less than or equal to\none. So $p^\\mu$ is lightlike precisely, when it can be written as\na product \\eqn{bispi}{p_{a\\dot a}=\\lambda_a\\tilde\\lambda_{\\dot a}}\nfor some  spinors $\\lambda_a$ and $\\tilde\\lambda_{\\dot a}.$ For a\ngiven null vector $p,$ the spinors $\\lambda$ and $\\tilde\\lambda$\nare unique up to a scaling\n\\eqn{scalm}{(\\lambda,\\tilde\\lambda)\\rightarrow\n(t\\lambda,t^{-1}\\tilde\\lambda)\\qquad t\\in \\Bbb C^\\ast.} There is\nno continuous way to pick $\\lambda$ as a function $p.$ In\nMinkowski signature, the $\\lambda$'s form the Hopf line bundle\nover the sphere $S^2$ of directions of the lightlike vector $p.$\n\nFor complex momenta, the spinors $\\lambda^a$ and\n$\\tilde\\lambda^{\\dot a}$ are independent complex variables, each\nof which parameterizes a copy of $\\Bbb  {CP}^1.$ Hence, the\ncomplex lightcone $p_\\mu p^\\mu=0$ is a complex cone over the\nconnected manifold $\\Bbb {CP}^1\\times \\Bbb {CP}^1.$\n\nFor real null momenta  in Minkowski signature $+---$, we can fix\nthe scaling up to a $Z_2$ by requiring $\\lambda^a$ and\n$\\tilde\\lambda^{\\dot a}$ to be complex conjugates\n\\eqn{fixe}{\\overline\\lambda^{\\dot a}=\\pm\\tilde\\lambda^{\\dot a}.} Hence,\nthe negative chirality spinors $\\lambda$ are conventionally called\n`holomorphic' and the positive chirality spinor\n`anti-holomorphic.' In \\rf{fixe} the $+$ sign is for a future\npointing null vector $p^\\mu,$ and $-$ is for a past pointing\n$p^\\mu.$\n\nOne can also consider other signatures. For example in the\nsignature $++--,$  the spinors $\\lambda$ and $\\tilde\\lambda$ are\nreal and independent. Indeed, with signature $++--,$ the Lorentz\ngroup is $SO(2,2),$ which is locally isomorphic to $Sl(2,\\Bbb\nR)\\times Sl(2,\\Bbb R).$ Hence, the spinor representations are\nreal.\n\nLet us remark, that if $p$ and $p'$ are two lightlike vectors\ngiven by $p_{a\\dot a}=\\lambda_a\\tilde\\lambda_{\\dot a}$ and\n$p'_{a\\dot a}=\\lambda'_a\\tilde\\lambda'_{\\dot a}$ then their scalar\nproduct can be expressed as\\footnote{This differs from the `-'\nsign convention used in the perturbative QCD literature.}\n\\eqn{product}{2 p\\cdot\np'=\\vev{\\lambda,\\lambda'}[\\tilde\\lambda,\\tilde\\lambda'].}\n\nGiven $p,$ the additional physical information in $\\lambda$ is\nequivalent to a choice of wavefunction of a helicity $-1\/2$\nmassless particle with momentum $p.$ To see this, we write the\nchiral Dirac equation for a negative chirality spinor $\\psi^a$\n\\eqn{chdir}{0=i\\sigma^\\mu_{a\\dot a} \\partial_\\mu \\psi^a.} A plane\nwave $\\psi^a=\\rho^a\\exp(ip\\cdot x)$ satisfies this equation if and\nonly if $p_{a\\dot a}\\rho^a=0.$ Writing  $p_{a\\dot a}=\\lambda_a\n\\tilde\\lambda_{\\dot a},$ we get $\\lambda_a\\rho^a=0,$ that is\n$\\rho^a= c\\cdot \\lambda^a$ for a constant $c.$ Hence the negative\nhelicity fermion has wavefunction \\eqn{negw}{\\psi^a=c\\lambda^a\n\\exp(ix_{a\\dot a} \\lambda^a\\tilde\\lambda^{\\dot a}).} Similarly,\n$\\tilde\\lambda$ defines a wavefunction for a helicity $+1\/2$\nfermion $\\psi^{\\dot a}=c \\tilde\\lambda^{\\dot a}\\exp(ix_{a\\dot a}\n\\lambda^a\\tilde\\lambda^{\\dot a}).$\n\nThere is an analogous description of wavefunctions of massless\nparticles of helicity $\\pm1.$ Usually, we describe massless gluons\nwith their momentum vector $p^\\mu$ and polarization vector\n$\\epsilon^\\mu.$ The polarization vector obeys the constraint\n\\eqn{decop}{p_\\mu\\ \\epsilon^\\mu=0} that represents the decoupling\nof longitudinal modes and it is subject to the gauge invariance\n\\eqn{gaugi}{\\epsilon^\\mu\\rightarrow\\epsilon^\\mu+wp^\\mu,} for any\nconstant $w.$ Suppose that instead of being given only a lightlike\nvector $p_{a\\dot a}$, one is also given a decomposition $p_{a\\dot\na}=\\lambda_a\\tilde\\lambda_{\\dot a}.$ Then we have enough\ninformation to determine the polarization vector up to a gauge\ntransformation once the helicity is specified. For a positive\nhelicity gluon, we take \\eqn{pospol}{\\epsilon_{a\\dot\na}^+=\\frac{\\mu_a\\tilde\\lambda_{\\dot a}}{\\vev{\\mu,\\lambda}},} where\n$\\mu$ is any negative chirality spinor that is not a multiple of\n$\\lambda.$ To get a negative helicity polarization vector, we take\n\\eqn{negpol}{\\epsilon_{a\\dot a}^-=\\frac{\\lambda_a\\tilde\\mu_{\\dot\na}}{[\\tilde\\lambda,\\tilde\\mu]},} where $\\tilde\\mu$ is any positive\nchirality spinor that is not a multiple of $\\tilde\\lambda.$ We\nwill explain the expression for the positive helicity vector. The\nnegative helicity case is similar.\n\nClearly, the constraint \\eqn{con}{p^\\mu\\ \\epsilon^+_\\mu=p^{a\\dot\na}\\epsilon^+_{a\\dot a}=0}  holds because $\\tilde\\lambda^{\\dot\na}\\tilde\\lambda_{\\dot a}=0.$ Moreover, $\\epsilon^+$ is also\nindependent of $\\mu^a$ up to a gauge transformation. To see this,\nnotice that $\\mu$ lives in a two dimensional space that is spanned\nwith $\\lambda$ and $\\mu.$ Hence, any change in $\\tilde\\mu$ is of\nthe form \\eqn{vam}{\\delta \\mu^a=\\alpha\\mu^a+\\beta\\lambda^a} for some\nparameters $\\alpha$ and $\\beta.$ The polarization vector\n\\rf{pospol} is invariant under the $\\alpha$ term, because this\nsimply rescales $\\mu$ and $\\epsilon^+_{a \\dot a}$ is invariant\nunder the rescaling of $\\mu.$ The $\\beta$ term amounts to a  gauge\ntransformation of the polarization vector\n\\eqn{poch}{\\delta\\epsilon^+_{a\\dot a}=\n\\beta{\\lambda_a\\tilde\\lambda_{\\dot a}\\over \\vev{\\mu,\\lambda}}.}\n\nUnder the scaling $(\\lambda,\\tilde\\lambda)\\rightarrow\n(t\\lambda,t^{-1}\\tilde\\lambda),\\, t\\in {\\Bbb C}^\\ast$ the\npolarization vectors scale like \\eqn{scal}{\\epsilon^-\\rightarrow\nt^{+2}\\epsilon^- \\qquad \\epsilon^+\\rightarrow t^{-2}\\epsilon^+.}\nThis could have been anticipated, since $\\tilde\\lambda_{\\dot a}$\ngives the wavefunction of a helicity $+1\/2$ particle  so a\nhelicity $+1$ polarization vector should scale like\n$\\tilde\\lambda^2.$ Similarly, the helicity $-1$ polarization\nvector scales like $\\lambda^2.$\n\nTo show  more directly that $\\epsilon^+$ describes a massless\nparticle of helicity $+1,$ we must show that the corresponding\nlinearized field strength $F_{\\mu\\nu}=\\partial_\\mu\nA_\\nu-\\partial_\\nu A_\\mu$ is anti-selfdual. Indeed, the field\nstrength written in a bispinor notation has the decomposition\n\\eqn{sef}{F_{a\\dot a b\\dot b}=\\epsilon_{ab}\\tilde f_{\\dot a\\dot\nb}+\\epsilon_{\\dot a \\dot b}f_{ab},} where $f_{ab}$ and $\\tilde\nf_{\\dot a \\dot b}$ are the selfdual and anti-selfdual parts of\n$F.$ Substituting $A_{a\\dot a}=\\epsilon_{a\\dot a}^+\\exp(ix_{a\\dot\na}\\lambda^a\\tilde\\lambda^{\\dot a})$ we find that $F_{a\\dot a b\\dot b}=\n\\epsilon_{ab} \\tilde\\lambda_{\\dot a}\\tilde\\lambda_{\\dot b}\\exp(ix_{a\\dot a}\n\\lambda^a\\tilde\\lambda^{\\dot a}).$\n\nSo far, we have seen that the wavefunction of a massless particle\nwith helicity $h$ scales under\n$(\\lambda,\\tilde\\lambda)\\rightarrow(t\\lambda,t^{-1}\\tilde\\lambda)$\nas $t^{-2h}$ if $|h|\\leq1.$ This is true for any $h,$ as can be\nseen from the following argument. Consider a massless particle\nmoving in the $\\vec n$ direction. Then a rotation by angle\n$\\theta$ around the $\\vec n$ axis acts on the spinors as\n\\eqn{rota}{(\\lambda,\\tilde\\lambda)\\rightarrow\n(e^{-i\\theta\/2}\\lambda,e^{+i\\theta\/2}\\tilde\\lambda).} Hence,\n$\\lambda,\\tilde\\lambda$ carry $-\\frac{1}{2}$ or $+\\frac{1}{2}$ units of\nangular momentum around the $\\vec n$ axis. Clearly, a massless\nparticle of helicity $h$ carries $h$ units of angular momentum\naround the $\\vec n$ axis. Hence the wavefunction of the particle\ngets transformed as $\\psi\\rightarrow e^{ih\\theta}\\psi$ under the\nrotation around $\\vec n$ axis, so it obeys the auxiliary condition\n\\eqn{scali}{\\left(\\lambda^a\\frac{\\partial}{\\partial\\lambda^a}-\\tilde\\lambda^{\\dot\na}\\frac{\\partial}{\\partial\\tilde\\lambda^{\\dot a}}\\right)\n\\psi(\\lambda,\\tilde\\lambda)=-2h\\psi(\\lambda,\\tilde\\lambda).}\nClearly, this constraint holds for wavefunctions of massless\nparticles of any spin. The spinors $\\lambda,\\tilde\\lambda$ give us\na convenient way of writing the wavefunction of massless particle\nof any spin, as we have seen in detail above for particles with\n$|h|\\leq 1.$\n\n\n\n\\subsection{Scattering Amplitudes}\n\nLet us consider scattering  of massless particles in four\ndimensions. Consider the situation with $n$ particles of momenta\n$p_1,p_2,\\dots, p_n.$ For scattering of scalar particles, the\ninitial and final states of the particles are completely\ndetermined by their momenta. The scattering amplitude is simply a\nfunction of the momenta $p_i,$\n\\eqn{ascal}{A_{scalar}(p_1,p_2,\\dots,p_n).} In fact, by Lorentz\ninvariance, it is a function of the Lorentz invariant products\n$p_i\\cdot p_j$ only.\n\nFor particles with spin, the scattering amplitude is a function of\nboth the momenta $p_i$ and the wavefunctions $\\psi_i$ of each\nparticle \\eqn{ascat}{A(p_1,\\psi_1;\\dots;p_n,\\psi_n).} Here, $A$ is\nlinear in each of the wavefunctions $\\psi_i.$ The description of\n$\\psi_i$ depends on the spin of the particle. As we have seen\nexplicitly above in the case of massless particles of spin $\\frac{1}{2}$\nor $1,$ the spinors $\\lambda,\\tilde\\lambda$ give a unified\ndescription of the wavefunctions of particles with spin. Hence, to\ndescribe the wavefunctions, we specify for each particle the\nhelicity $h_i$ and the spinors $\\lambda_i$ and $\\tilde\\lambda_i.$\nThe spinors determine the momenta $p_i=\\lambda_i\\tilde\\lambda_i$\nand the wavefunctions $\\psi_i(\\lambda_i,\\tilde\\lambda_i, h_i)$. So\nfor massless particles with spin, the scattering amplitude is a\nfunction of the spinors and helicities of the external particles\n\\eqn{scaf}{A(\\lambda_1,\\tilde\\lambda_1,h_1;\\dots;\\lambda_n,\\tilde\\lambda_n,h_n).}\nIn labelling the helicities we take all particles to be incoming.\nTo obtain an amplitude with incoming particles as well as outgoing\nparticles, we use crossing symmetry, that relates an incoming\nparticle of one helicity  to an outgoing particle of the opposite\nhelicity.\n\nIt follows from \\rf{scali} that the amplitude obeys the conditions\n\\eqn{scalia}{\\left(\\lambda^a_i\\frac{\\partial}{\\partial\\lambda^a_i}-\\tilde\\lambda_i^{\\dot\na}\\frac{\\partial}{\\partial\\tilde\\lambda^{\\dot a}_i}\\right)\nA(\\lambda_i,\\tilde\\lambda_i,h_i)=-2h_iA(\\lambda_i,\\tilde\\lambda_i,h_i)}\nfor each particle $i,$ with helicity $h_i.$ In summary, a general\nscattering amplitude of massless particles can be written as\n\\eqn{scatmas}{A=(2\\pi)^4 \\delta^4\\left(\\sum_i\n\\lambda_i^a\\tilde\\lambda^{\\dot a}_i\\right)\nA(\\lambda_i,\\tilde\\lambda_i,h_i),} where we have written\nexplicitly the delta function of momentum conservation.\n\n\\begin{figure}\n \\centering\n \\includegraphics[height=2.0in]{ampl_notation.eps}\n \\caption{A scattering amplitude of $n$ gluons in Yang-Mills theory.\n          Each gluon comes with the color factor $T_i,$ spinors\n          $\\lambda_i,\\tilde\\lambda_i$ and helicity label $h_i=\\pm1.$}\n \\label{ampl_notation}\n\\end{figure}\n\n\n\\subsection{Maximally Helicity Violating Amplitudes}\n\nTo make the discussion more concrete, we consider tree level\nscattering of $n$ gluons in Yang-Mills theory. These amplitudes\nare of phenomenological importance. The multijet production at LHC\nwill be dominated by tree level QCD scattering.\n\nConsider Yang-Mills theory with gauge group $U(N).$ Recall that\ntree level scattering amplitudes are planar and lead to single\ntrace interactions. In an index loop, the gluons are attached in a\ndefinite cyclic order, say $1,2,\\dots, n.$ Then the amplitude\ncomes with a group theory factor $\\mop{Tr} T_1T_2\\dots T_n.$ It is\nsufficient to give the amplitude with one cyclic order. The full\namplitude is obtained from this by summing over the cyclic\npermutations to achieve Bose symmetry\n\\eqn{boses}{A=g^{n-2}(2\\pi)^4\\delta^4\\left(\\sum_i^n\np_i\\right){\\cal A}(1,2,\\dots, n)\\mop{Tr} (T_1T_2\\dots T_n)\\, +\\,\npermutations.} Here, $g$ is the coupling constant of the gauge\ntheory. In the rest of the lecture notes, we will always consider\ngluons in the cyclic order $1,2,\\dots, n$ and we will omit the\ngroup theory factor and the delta function of momentum\nconservation in writing the formulas. Hence we will consider the\n`reduced color ordered amplitude' ${\\cal A}(1,2,\\dots, n).$\n\nThe scattering amplitude with $n$ incoming gluons of the same\nhelicity vanishes. So does the amplitude, for $n\\geq 3,$ with\n$n-1$ incoming gluons of one helicity and one of the opposite\nhelicity. The first nonzero amplitudes, the maximally helicity\nviolating (MHV) amplitudes, have $n-2$ gluons of one helicity and\ntwo gluons of the other helicity. Suppose that gluons $r,s$ have\nnegative helicity and the rest of gluons have positive helicity.\nThen the tree level amplitude, stripped of the momentum delta\nfunction and the group theory factor, is\n\\eqn{mhv}{{\\cal A}(r^-,s^-)=g^{n-2}\\frac{\\vev{\\lambda_r,\\lambda_s}^4}{\\prod_{k=1}^n\n\\vev{\\lambda_k,\\lambda_{k+1}}}.} The amplitude ${\\cal A}(r^+,s^+)$ with\ngluons $r,s$ of positive helicity and the rest of the gluons of\nnegative helicity follows from \\rf{mhv} by exchange\n$\\vev{}\\leftrightarrow [].$ Note, that the amplitude has the\ncorrect homogeneity in each variable. It is homogeneous of degree\n$-2$ in $\\lambda_i$ for positive helicity gluons; and of degree\n$-2$ for negative helicity gluons $i=r,s$ as required by the\nauxiliary condition \\rf{scalia}. The amplitude ${\\cal A}$ is sometimes\ncalled `holomorphic' because it depends on the `holomorphic'\nspinors $\\lambda_i$ only.\n\n\\section{Twistor Space}\n\n\\subsection{Conformal Invariance of Scattering Amplitudes}\n\nBefore discussing twistor space, let us show the conformal\ninvariance of the MHV tree level amplitude. Firstly, we need to\nconstruct representation of the conformal group generators in\nterms of the spinors $\\lambda,\\tilde\\lambda.$ We will consider the conformal\ngenerators for a single particle. The generators of the\n$n$-particle system are given by the sum of the generators over\nthe $n$ particles.\n\nSome of the generators are clear. The Lorentz generators are the\nfirst order differential operators\n\\eqalign{lorentz}{J_{ab}&=&\\frac{i}{2}\\left(\\lambda_a\n\\frac{\\partial}{\\partial\\lambda^b}+\\lambda_b\\frac{\\partial}{\\partial\\lambda^a}\n\\right)\\cr  \\tilde J_{\\dot a \\dot\nb}&=&\\frac{i}{2}\\left(\\tilde\\lambda_{\\dot\na}\\frac{\\partial}{\\partial\\tilde\\lambda^{\\dot\nb}}+\\tilde\\lambda_{\\dot\nb}\\frac{\\partial}{\\partial\\tilde\\lambda^{\\dot a}}\\right).} The\nmomentum operator is the multiplication operator\n\\eqn{momentum}{P_{a\\dot a}=\\lambda_a\\tilde\\lambda_{\\dot a}.} The\nremaining generators are the dilatation operator $D$ and the\ngenerator of special conformal transformation $K_{a\\dot a}.$ The\ncommutation relations of the dilatation operator are\n\\eqn{comd}{[D,P]=iP,\\qquad [D,K]=-iK,} so $P$ has dimension $+1$\nand $K$ has dimension $-1.$ We see from (\\ref{momentum}) that it\nis natural to take $\\lambda$ and $\\tilde\\lambda$ to have dimension\n$1\/2.$ Hence, a natural guess for the special conformal generator\nrespecting all the symmetries is \\eqn{kguess}{K_{a\\dot\na}=\\frac{\\partial^2}{\\partial\\lambda^a\\partial\\tilde\\lambda^{\\dot\na}}.} We find the dilatation operator $D$ from the closure of the\nconformal algebra.  The commutation relation\n\\eqn{comrel}{[K_{a\\dot a},P^{b\\dot b}]=-i\\left(\\delta_a{}^b\\tilde\nJ_{\\dot a}{}^{\\dot b}+\\delta_{\\dot a}{}^{\\dot\nb}J_{a}{}^b+\\delta_a{}^b\\delta_{\\dot a}{}^{\\dot b}D\\right)}\ndetermines the dilatation operator to be\n\\eqn{dilat}{D=\\frac{i}{2}\\left(\\lambda^a\\frac{\\partial}{\\partial\\lambda^a}\n+\\tilde\\lambda^{\\dot a}\\frac{\\partial}{\\partial\\tilde\\lambda^{\\dot\na}}+2\\right).}\n\nWe are now ready to verify that the MHV amplitude\n\\eqn{mhvdil}{{\\cal A}(r^-,s^-)=(2\\pi)^4\\delta^4\\left(\\sum_i\\lambda_i^a\\tilde\\lambda^{\\dot\na}_i\\right)\n\\frac{\\vev{\\lambda_r,\\lambda_s}^4}{\\prod_{k=1}^n\\vev{\\lambda_k,\\lambda_{k+1}}},}\nis invariant under the conformal group. The Lorentz generators are\nclearly symmetries of the amplitude. The momentum operator\nannihilates the amplitude thanks to the delta function of momentum\nconservation.\n\nIt remains to verify that the amplitude is annihilated by $D$ and\n$K.$ For simplicity, we will only consider the dilatation operator\n$D.$ The numerator contains the delta function of momentum\nconservation which has dimension $D=-4$ and the factor\n$\\vev{\\lambda_r,\\lambda_s}^4$ of dimension $4.$ Hence, $D$\ncommutes with the numerator. So we are left with the denominator\n\\eqn{denomi}{\\frac{1}{\\prod_{k=1}^n\\vev{\\lambda_k,\\lambda_{k+1}}}.}\nThis is annihilated by $D_k$ for each particle $k,$ since the $-2$\ncoming from the second power of $\\lambda_k$ in the denominator\ngets cancelled against the $+2$ from the definition of the\ndilatation operator \\rf{dilat}.\n\n\\subsection{Transform to Twistor Space}\n\\label{transform}\n\nWe have demonstrated conformal invariance of the MHV amplitude,\nhowever the representation of the conformal group that we have\nencountered above is quite exotic. The Lorentz generators are\nfirst order differential operators, but the momentum is a\nmultiplication operator and the special conformal generator is a\nsecond order differential operator.\n\nWe can put the action of the conformal group into a more standard\nform if we make the following transformation\n\\eqalign{four}{\\tilde\\lambda_{\\dot a}&\\rightarrow\ni\\frac{\\partial}{\\partial\\mu^{\\dot a}} \\cr\n\\frac{\\partial}{\\partial\\tilde\\lambda^{\\dot a}}&\\rightarrow\ni\\mu_{\\dot a}.}  Making this substitution we have arbitrarily\nchosen to Fourier transform $\\tilde\\lambda$ rather than $\\lambda.$ This choice\nbreaks the symmetry between positive and negative helicities. The\namplitude with $n_1$ positive helicity and $n_2$ negative helicity\ngluons has different description in twistor space from an\namplitude with $n_2$ positive helicity gluons and $n_1$ negative\nhelicity gluons.\n\nUpon making this substitution, all operators become first order.\nThe Lorentz generators take the form \\eqalign{lorentztwo}{J_{ab}\n&=&\\frac{i}{2}\\left(\\lambda_a\n\\frac{\\partial}{\\partial\\lambda^b}+\\lambda_b\\frac{\\partial}{\\partial\\lambda^a}\n\\right)\\cr  \\tilde J_{\\dot a \\dot b} &=&\\frac{i}{2}\\left(\\mu_{\\dot\na}\\frac{\\partial}{\\partial\\mu^{\\dot b}}+\\mu_{\\dot\nb}\\frac{\\partial}{\\partial\\mu^{\\dot a}}\\right).} The momentum and\nspecial conformal operators become \\eqalign{fir}{P_{a\\dot\na}&=&i\\lambda_a\\frac{\\partial}{\\partial\\mu^{\\dot a}} \\cr K_{a\\dot\na}&=&i\\mu_{\\dot a}\\frac{\\partial}{\\partial\\lambda^a}.} Finally,\nthe dilatation operator (\\ref{dilat}) becomes a {\\it homogeneous}\nfirst order operator\n\\eqn{dilafirst}{D=\\frac{i}{2}\\left(\\lambda^a{\\partial\\over\n\\partial\\lambda^a} -\\mu^{\\dot a}\\frac{\\partial}{\\partial\\mu^{\\dot\na}}\\right).}\n\nThis representation of the four dimensional conformal group is\neasy to explain. The conformal group of Minkowski space is\n$SO(4,2)$ which is the same as $SU(2,2).$ $SU(2,2),$ or its\ncomplexification $Sl(4,\\Bbb C),$ has an obvious four-dimensional\nrepresentation acting on \\eqn{zi}{Z^I=(\\lambda^a,\\mu^{\\dot a}).}\n$Z^I$ is called a twistor and the space $\\Bbb C^4$ spanned by\n$Z^I$ is called the twistor space. The action of $Sl(4,\\Bbb C)$ on\nthe $Z^I$ is generated by $15$ traceless matrices $\\Lambda^I{}_J,\\\nI,J=1,\\dots, 4,$ that correspond to the $15$ first order operators\n$J_{ab}, \\tilde J_{\\dot a \\dot b}, D, P_{a\\dot a}, K_{a\\dot a}.$\n\nIf we are in signature $++--,$ the conformal group is $SO(3,3)\n\\cong Sl(4,\\Bbb R).$ The twistor space is a copy of $\\Bbb R^4$ and\nwe can consider $\\lambda$ and $\\mu$ to be real. In the Euclidean\nsignature $++++$, the conformal group is $SO(5,1)\\cong SU^\\ast(4)$\nwhere $SU^\\ast(4)$ is the noncompact version of $SU(4),$ so we\nmust think of twistor space as a copy of $\\Bbb C^4.$\n\n\nFor signature $++--,$ where $\\tilde\\lambda$ is real, the\ntransformation from momentum space scattering amplitudes to\ntwistor space scattering amplitudes is made by a simple Fourier\ntransform that is familiar from quantum mechanics\n\\eqn{fouriert}{\\tilde {\\cal A}(\\lambda_i,\\mu_i)= \\int\\prod_{j=1}^n\n{d^2\\tilde\\lambda_j\\over (2\\pi)^2} \\exp(i[\\mu_j,\\tilde\\lambda_j])\n{\\cal A}(\\lambda_i,\\tilde\\lambda_i).} The same Fourier transform turns\na momentum space wavefunction $\\psi(\\lambda,\\tilde\\lambda)$ to a\ntwistor space wavefunction\n\\eqn{wavefo}{\\tilde\\psi(\\lambda,\\mu)=\\int {d^2\\tilde\\lambda\\over\n(2\\pi)^2} \\exp(i[\\mu,\\tilde\\lambda]) \\psi(\\lambda,\\tilde\\lambda).}\n\nRecall that the scattering amplitude of  massless particles obeys\nthe auxiliary condition\n\\eqn{sara}{\\left(\\lambda^a_i\\frac{\\partial}{\\partial\\lambda^a_i}-\\tilde\\lambda_i^{\\dot\na}\\frac{\\partial}{\\partial\\tilde\\lambda^{\\dot a}_i}\\right)\n{\\cal A}(\\lambda_i,\\tilde\\lambda_i,h_i)=-2h_i{\\cal A}(\\lambda_i,\\tilde\\lambda_i,h_i)}\nfor each particle $i,$ with helicity $h_i.$ In terms of\n$\\lambda_i$ and $\\mu_i,$ this becomes\n\\eqn{lica}{\\left(\\lambda^a_i\\frac{\\partial}{\\partial\\lambda^a_i}+\\mu_i^{\\dot\na}\\frac{\\partial}{\\partial\\mu^{\\dot a}_i}\\right)\n\\tilde{\\cal A}(\\lambda_i,\\mu_i,h_i)=-(2h_i+2)\\tilde{\\cal A}(\\lambda_i,\\mu_i,h_i).}\nThere is a similar condition for the twistor wavefunctions of\nparticles. The operator on the left hand side coincides with\n$Z^I{\\partial\\over\n\\partial Z^I}$ that generates the scaling of the twistor coordinates\n\\eqn{rescal}{Z^I\\rightarrow t Z^I, \\qquad t\\in \\Bbb C^\\ast.}\n\n\nSo the wavefunctions and scattering amplitudes have known behavior\nunder the $\\Bbb C^\\ast$ action $Z^I\\rightarrow t Z^I.$ Hence, we\ncan identify the sets of $Z^I$ that differ by the scaling\n$Z^I\\rightarrow t Z^I$ and throw away the point $Z^I=0.$ We get\nthe projective twistor space\\footnote{The twistor wavefunctions\n\\rf{wavefo} are regular only on the subset $\\Bbb{CP}'^{3|4}$ of\n$\\Bbb{CP}^{3|4}$ with $(\\lambda^1,\\lambda^2)\\neq(0,0),$ which is the\nprecise definition of the projective twistor space. In the rest of\nthe lecture notes, we do not distinguish between these two spaces,\nunless necessary.} $\\Bbb {CP}^3$ or $\\Bbb {RP}^3$ if $Z^I$ are\ncomplex or real-valued. The $Z^I$ are the homogeneous coordinates\non the projective twistor space. It follows from \\rf{lica} that,\nthe scattering amplitudes are homogeneous functions of degree\n$-2h_i-2$ in the twistor coordinates $Z_i^I$ of each particle\nparticle. In the complex case, this means that scattering\namplitudes are sections of the complex line bundle ${\\cal\nO}(-2h_i-2)$ over a $\\Bbb {CP}^3_i$ for each particle. For further\ndetails on twistor transform, see any standard textbook on twistor\ntheory, e.g. \\cite{Huggett:1986fs,Atiyah:1979iu}.\n\n\n\\subsection{Scattering Amplitudes in Twistor Space}\n\\label{twist}\n\nIn an $n$ gluon scattering process, after the Fourier transform\ninto twistor space, the external gluons are associated with points\n$P_i$ in the projective twistor space. The scattering amplitudes\nare functions of the twistors $P_i,$ that is, they are functions\ndefined on the product of $n$ copies of twistor space, one for\neach particle.\n\nLet us see what happens to the tree level MHV amplitude with $n-2$\ngluons of positive helicity and $2$ gluons of negative helicity,\nafter Fourier transform into twistor space. We work in $++--$\nsignature, for which the twistor space is a copy of $\\Bbb {RP}^3.$\nThe advantage of $++--$ signature is that the transform to twistor\nspace is an ordinary Fourier transform and the scattering\namplitudes are ordinary functions on a product of $\\Bbb {RP}^3$'s,\none for each particle. With other signatures, the twistor\ntransform involves $\\overline\\partial$-cohomology and other\nmathematical machinery.\n\nWe recall that the MHV amplitude with negative helicity gluons\n$r,s$ is\n\\eqn{mhvrecall}{A(\\lambda_i,\\tilde\\lambda_i)=(2\\pi)^4\\delta^4(\\sum_i\n\\lambda_i\\tilde\\lambda_i) f(\\lambda_i),} where\n\\eqn{fln}{f(\\lambda_i)=g^{n-2}{\\vev{\\lambda_r,\\lambda_s}^4\\over\n\\prod_k\\vev{\\lambda_k, \\lambda_{k+1}}}.}  The only property of\n$f(\\lambda_i),$ that we need is that it is a function of the\nholomorphic spinors $\\lambda_i$ only. It does not depend on the\nanti-holomorphic spinors $\\tilde\\lambda_i.$\n\nWe express the delta function of momentum conservation as an\nintegral \\eqn{deltan}{(2\\pi)^4\\delta^4(\\sum_i\n\\lambda_i^a\\tilde\\lambda_i^{\\dot a})=\\int d^4x^{a\\dot a}\n\\exp\\left(i x_{b\\dot b}\\sum_i \\lambda_i^b\\tilde\\lambda_i^{\\dot\nb}\\right).} Hence, we can rewrite the amplitude as \\eqn{mhvnew}{\nA(\\lambda_i,\\tilde\\lambda_i)=\\int d^4x \\exp\\left(i x_{b\\dot\nb}\\sum_i \\lambda_i^b\\tilde\\lambda_i^{\\dot b}\\right) f(\\lambda_i).}\nTo transform the amplitude into twistor space, we simply carry out\na Fourier transform with respect to all $\\tilde\\lambda$'s. Hence,\nthe twistor space amplitude is\n\\eqn{mhvint}{A(\\lambda_i,\\mu_i)=\\int\n{d^2\\tilde\\lambda_1\\over(2\\pi)^2}\\dots{d^2\\tilde\\lambda_n\\over\n(2\\pi)^2}\\exp\\left(i\\sum_{j=1}^n \\mu_{j\\dot\na}\\tilde\\lambda_j^{\\dot a}\\right)\\int d^4x\\, \\exp\\left(i x_{b\\dot\nb}\\sum_j \\lambda_j^b\\tilde\\lambda_j^{\\dot b}\\right) f(\\lambda_i).}\nThe only dependece on $\\tilde\\lambda_i$ is in the exponential factors. Hence\nthe integrals over $\\tilde\\lambda_j$ gives a product of delta\nfunctions with the result \\cite{Nair:1988bq}\n\\eqn{mhvtwistor}{A(\\lambda_i,\\mu_i)=\\int d^4x \\prod_{j=1}^n\n\\delta^2(\\mu_{j\\dot a}+x_{a\\dot a}\\lambda^a_j)f(\\lambda_i).} This\nequation has a simple geometrical interpretation. Pick some\n$x^{a\\dot a}$ and consider the equation \\eqn{inci}{\\mu_{\\dot\na}+x_{a\\dot a}\\lambda^a=0.} The solution set for $x=0$ is a $\\Bbb\n{RP}^1$ or $\\Bbb {CP}^1$ depending on whether the variables are\nreal or complex. This is true for any $x$ as the equation lets us\nsolve for $\\mu_{\\dot a}$ in terms of $\\lambda^a.$ So\n$(\\lambda^1,\\lambda^2)$ are the homogeneous coordinates on the\ncurve.\n\nIn real twistor space, which is appropriate for signature $++--,$\nthe curve $\\Bbb {RP}^1$ can be described more intuitively as a\nstraight line, see fig. \\ref{cp_one}. Indeed, throwing away the\nset $Z^1=0,$ we can describe the rest of $\\Bbb {RP}^3$ as a copy\nof $\\Bbb R^3$ with the coordinates $x_i=Z_i\/Z_1, i=2,3,4.$ The\nequations \\rf{inci} determine two planes whose intersection is the\nstraight line in question.\n\nIn complex twistor space, the genus zero curve $\\Bbb {CP}^1$ is\ntopologically a sphere $S^2.$ The $\\Bbb {CP}^1$  is an example of\na holomorphic curve in $\\Bbb {CP}^3.$ The simplest holomorphic\ncurves are defined by vanishing of a pair of homogeneous\npolynomials in the $Z^I$ \\eqalign{completeint}{f(Z^1,\\dots,\nZ^4)&=&0 \\cr g(Z^1,\\dots, Z^4)&=&0.} If $f$ is  homogeneous of\ndegree $d_1$ and $g$ is homogeneous of degree $d_2,$ the curve has\ndegree $d_1 d_2.$ The equations \\eqn{incide}{\\mu_{\\dot b}+x_{b\\dot\nb}\\lambda^b=0, \\quad \\dot b=1,2} are  both  linear, $d_1=d_2=1$.\nHence the degree of the $\\Bbb {CP}^1$ is $d=d_1d_2=1.$ Moreover,\nevery degree one genus zero curve in $\\Bbb {CP}^3$ is of the form\n\\rf{incide} for some $x^{b\\dot b}.$\n\nThe area of a holomorphic curve of degree $d,$ using the natural\nmetric on $\\Bbb {CP}^3,$ is $2\\pi d.$ So the curves we found with\n$d=1$ have the minimal area among nontrivial holomorphic curves.\nThey are associated with the minimal nonzero Yang-Mills tree\namplitudes, the MHV amplitudes.\n\nGoing back to the amplitude \\rf{mhvtwistor}, the\n$\\delta$-functions mean that the amplitude vanishes unless\n$\\mu_{j\\dot a}+x_{a\\dot a}\\lambda^a_j=0, j=1,\\dots n,$ that is,\nunless some curve of degree one determined by $x_{a\\dot a}$\ncontains all $n$ points $(\\lambda_j,\\mu_j).$ The result is that\nthe MHV amplitudes are supported on genus zero curves of degree\none. This is a consequence of the holomorphy of these amplitudes.\n\n\\begin{figure}\n \\centering\n \\includegraphics[height=1.5in]{cp_one.eps}\n \\caption{(a) In\n          complex twistor space $\\Bbb{CP}^3$, the MHV amplitude localizes to a $\\Bbb{CP}^1.$\n          (b) In the real case, the amplitude is associated to a real line in $\\Bbb R^3.$}\n \\label{cp_one}\n\\end{figure}\n\n\nThe general conjecture is that an $l$-loop amplitude with $p$\ngluons of positive helicity and $q$ gluons of negative helicity is\nsupported on a holomorphic curve in twistor space. The degree of\nthe curve is determined by  \\eqn{dec}{d=q-1+l.} The genus of the\ncurve is bounded by the number of the loops \\eqn{lops}{g\\leq l.}\nThe MHV amplitudes are a special case of this for $q=2,l=0.$\nIndeed the conjecture in this case gives that MHV amplitudes are\nsupported in twistor space on a genus zero curve of degree one.\n\nThe natural interpretation of this is that the curve is the\nworldsheet of a string. In some way of describing the perturbative\ngauge theory, the amplitudes arise from coupling of the gluons to\na string. In the next two sections we discuss a proposal for such\na string theory due to Witten \\cite{Witten:2003nn} in which the\nstrings in questions are D1-strings. There is an alternative\nversion of twistor string theory due to Berkovits\n\\cite{Berkovits:2004hg, Berkovits:2004tx}, discussed in section\n\\ref{berko}, in which the curves come from fundamental strings.\nThe Berkovits's twistor string theory seems to give an equivalent\ndescription of the scattering amplitudes. Further proposals\n\\cite{Neitzke:2004pf,Aganagic:2004yh,Bars:2004dg} have not yet\nbeen used for computing scattering amplitudes.\n\n\n\n\\section{Twistor String Theory}\n\nIn this section, we will describe a string theory that gives a\nnatural framework for understanding the twistor properties of\nscattering amplitudes discussed in previous section. This is a\ntopological string theory whose target space is a supersymmetric\nversion of the twistor space.\n\n\\subsection{Brief Review of Topological Strings}\n\nFirstly, let us consider an ${\\cal N}=2$ topological field theory\nin $D=2$ \\cite{Witten:1991zz}. The ${\\cal N}=2$ supersymmetry algebra\nhas two supersymmetry generators $Q_i, i=1,2$ that satisfy the\nanticommutation relations \\eqn{susyalg}{\\{Q_{\\alpha i}, Q_{\\beta\nj}\\}=\\delta_{ij} \\gamma^\\mu_{\\alpha\\beta}P_\\mu.} In two\ndimensions, the Lorentz group $SO(1,1)$ is generated by the\nLorentz boost $L.$ We diagonalize $L$ by going into the light-cone\nframe $P_\\pm=P_0\\pm P_1,$ \\eqalign{boostdiag}{[L,P_\\pm]&=&\\pm\nP_\\pm \\cr \\{L,Q_\\pm\\}&=&\\pm\\frac{1}{2} Q_\\pm.} The  commutation\nrelations of the ${\\cal N}=2$ supersymmetry algebra become\n\\eqalign{susydiag}{\\{Q_{+i},Q_{+j}\\}&=&\\delta_{ij} P_+\\cr\n\\{Q_{-i}, Q_{-j}\\}&=&\\delta_{ij} P_- \\cr \\{Q_{+i}, Q_{-j}\\}&=&0.}\nWe let \\eqn{brstcharge}{Q=Q_{+1}+iQ_{+2}+Q_{-1}\\pm iQ_{-2}} with\neither choice of sign.  It follows from \\rf{susydiag} that $Q$ is\nnilpotent \\eqn{br}{Q^2=0,} so we  would like to consider $Q$ as a\nBRST operator.\n\nHowever $Q$ (\\ref{brstcharge}) is not a scalar so this\nconstruction would violate Lorentz invariance.  There is a way out\nif the theory has left and right R-symmetries $R_+$ and $R_-.$\nUnder $R_+,$ the combination of supercharges $Q_{+1}\\pm iQ_{+2}$\nhas charge $\\pm1\/2$ and $Q_{-1}\\pm iQ_{-2}$ is neutral. For $R_-,$\nthe same is true with `left' and `right'  interchanged.\n\nHence, we can make $Q$ scalar if we modify the Lorentz generator\n$L$ to be \\eqn{lmod}{L'=L-\\frac{1}{2} R_+ \\mp\\frac{1}{2} R_-.} At a more\nfundamental level, this change in the Lorentz generator arises if\nwe replace the stress tensor $T_{\\mu\\nu}$ with \\eqn{tmunu}{\\tilde\nT_{\\mu\\nu}= T_{\\mu\\nu}-\\frac{1}{2}(\\partial_\\mu J^+_\\nu+\\partial_\\nu\nJ^+_\\mu) \\mp\\frac{1}{2}(\\partial_\\mu J^-_\\nu+\\partial_\\nu J^-_\\mu),}\nwhere $J^+_\\nu$ and $J^-_\\mu$ are the left and right R-symmetry\ncurrents. The substitution \\rf{tmunu} is usually referred to as\n`twisting' the stress tensor.\n\nWe give a new interpretation to the theory by taking $Q$ to be a\nBRST operator. A state $\\Psi$ is considered to be physical if it\nis annihilated by $Q$ \\eqn{qani}{Q\\Psi=0.} Two states $\\Psi$ and\n$\\Psi'$ are equivalent if \\eqn{psiequi}{\\Psi-\\Psi'= Q\\Phi,} for\nsome $\\Phi.$ Similarly, we take the physical operators to commute\nwith the BRST charge \\eqn{ocu}{[Q,{\\cal O}]=0.} Two operators are\nequivalent if they differ by an anticommutator of $Q,$\n\\eqn{eqvi}{{\\cal O}'\\sim {\\cal O}+ \\{Q,{\\cal V}\\},} for some\noperator ${\\cal V}.$\n\nThe theory with the stress tensor $\\tilde T_{\\mu\\nu}$ and BRST\noperator $Q$ is called a topological field theory. The basis for\nthe name is that one can use the supersymmetry  algebra to show\nthat the twisted stress tensor is BRST trivial\n\\eqn{trivial}{\\tilde T_{\\mu\\nu}=\\{Q,\\Lambda_{\\mu\\nu}\\}.} It\nfollows that in some sense the worldsheet metric is irrelevant.\nThe correlation function \\eqn{cor}{\\vev{{\\cal O}_1(x_1)\n{\\cal O}_2(x_2)\\dots{\\cal O}_n(x_n)}_\\Sigma} of physical operators ${\\cal O}_i$\nobeying $[Q,{\\cal O}_i]=0$ on a fixed Riemann surface $\\Sigma$ is\nindependent of metric on $\\Sigma.$ Indeed, varying the metric\n$g_{\\mu\\nu}\\rightarrow g_{\\mu\\nu}+\\delta g_{\\mu\\nu},$ the\ncorrelation function stays the same up to BRST trivial terms\n\\eqn{mvev}{\\vev{{\\cal O}_1(x_1)\\dots {\\cal O}_n(x_n)\\int_\\Sigma\n\\delta(\\sqrt{g} g^{\\mu\\nu})\\tilde T_{\\mu\\nu}}=\\vev{{\\cal O}_1(x_1)\\dots\n{\\cal O}_n(x_n)\\int_\\Sigma\\delta(\\sqrt{g}\ng^{\\mu\\nu})\\{Q,\\Lambda_{\\mu\\nu}\\}}=0.}\n\nMore importantly for us, we can also construct a topological\nstring theory in which one obtains the correlation functions by\nintegrating \\rf{cor} over the moduli of the Riemann surface\n$\\Sigma$ using $\\Lambda_{\\mu\\nu}$ where the antighost $b_{\\mu\\nu}$\nusually appears in the definition of the string measure.\n\nFor an ${\\cal N}=2$ supersymmetric  field theory in two dimensions with\nanomaly-free left and right R-symmetries we get two topological\nstring theories, depending on the choice of sign in\n(\\ref{brstcharge}). We would like to consider the case that the\n${\\cal N}=2$ model is a sigma model with a target space being a complex\nmanifold $X.$ In this case, the two R-symmetries exist\nclassically, so classically we can construct the two topological\nstring theories, called the A-model and the B-model. Quantum\nmechanically, however, there is an anomaly, and the B-model only\nexists if $X$ is a Calabi-Yau manifold.\n\n\\subsection{Open String B-model on a Super-Twistor Space}\n\nTo define open strings in the B-model, one needs BRST invariant\nboundary conditions. The simplest such conditions are Neumann\nboundary conditions \\cite{Witten:1992fb}. Putting in $N$ space\nfilling D5-branes gives $Gl(n,\\Bbb C)$ (whose compact real form is\n$U(N)$) gauge symmetry.  The zero modes of the D5-D5 strings give\na $(0,1)$ form gauge field $A=A_{\\overline i}dz^{\\overline i}$ in the target\nspace. The BRST operator acts as the $\\overline\\partial$ operator and\nthe string $\\ast$ product is just the wedge product. Hence, $A$ is\nsubject to the gauge invariance \\eqn{gaui}{\\delta\nA=Q\\epsilon=\\overline\\partial \\epsilon+[A,\\epsilon],} and the string\nfield theory action reduces to the action of the holomorphic\nChern-Simons theory \\cite{Witten:1992fb} \\eqn{hcsc}{S={1\\over\n2}\\int\\Omega\\wedge \\mop{Tr}\\left(A\\wedge \\overline\\partial A+{2\\over3}\nA\\wedge A\\wedge A\\right),} where $\\Omega$ is the Calabi-Yau volume\nform.\n\nWe would like to consider the open string B-model with target\nspace $\\Bbb {CP}^3,$ but we cannot, since $\\Bbb {CP}^3$ is not a\nCalabi-Yau manifold and the B-model is well defined only on a\nCalabi-Yau manifold. On a non-Calabi-Yau manifold, the R-symmetry\nthat we used to twist the stress tensor is anomalous. A way out is\nto introduce spacetime supersymmetry. Instead of $\\Bbb {CP}^3,$\nwhich has homogeneous coordinates $Z^I, I=1,\\dots, 4$ we consider\na supermanifold $\\Bbb {CP}^{3|N}$ with bosonic and fermionic\ncoordinates \\eqn{coordi}{Z^I,\\quad \\psi^A\\qquad I=1,\\dots, 4,\n\\quad A=1,\\dots, N,} with identification of two sets of\ncoordinates that differ by a scaling \\eqn{suca}{(Z^I,\\psi^A)\\cong\n(tZ^I, t\\psi^A) \\qquad t\\in \\Bbb C^\\ast.} The $\\Bbb {CP}^{3|N}$ is\na Calabi-Yau supermanifold if and only if the number of fermionic\ndimensions is $N=4.$ To see this, we construct the holomorphic\nmeasure on $\\Bbb {CP}^{3|4}.$ We start with the $(4|N)$ form on\n$\\Bbb C^{4|N}$ \\eqn{ozero}{\\Omega_0=dZ^1\\dots dZ^4d\\psi^1\\dots\nd\\psi^N} and study its behavior under the rescaling symmetry\n(\\ref{suca}). For this, recall that $d\\psi$ scales oppositely to\n$\\psi$ \\eqn{suop}{(dZ^I, d\\psi^A)\\rightarrow (t dZ^I, t^{-1}\nd\\psi^A).} It follows, that $\\Omega_0$ is $\\Bbb C^\\ast$ invariant\nif and only if $N=4.$  In this case we can divide by the $C^\\ast$\naction and get a Calabi-Yau measure on $\\Bbb {CP}^{3|4}$\n\\eqn{holf}{\\Omega={1\\over 4!}\\epsilon_{IJKL}Z^I dZ^JdZ^KdZ^L\n{1\\over 4!}\\epsilon_{ABCD} \\psi^A\\psi^B\\psi^C\\psi^D.}\n\nThe twistor space $\\Bbb {CP}^3$ has a natural $Sl(4,\\Bbb C)$ group\naction that acts as $Z^A\\rightarrow \\Lambda^A{}_B Z^B$ on the\nhomogeneous coordinates of $\\Bbb{CP}^3$. The real form $SU(2,2)$ of\n$Sl(4,\\Bbb C)$ is the conformal group of Minkowski space.\nSimilarly, the super-twistor space $\\Bbb {CP}^{3|N}$ has a natural\n$Sl(4|N,\\Bbb C)$ symmetry. The real form $SU(2,2|N)$ of this is\nthe superconformal symmetry group with $N$ supersymmetries.\n\nFor $N=4$, the superconformal group $SU(2,2|4)$ is the symmetry\ngroup of ${\\cal N}=4$ super-Yang-Mills theory.  In a sense, this is the\nsimplest non-abelian gauge theory in four dimensions. The ${\\cal N}=4$\nsuperconformal symmetry uniquely determines the states and\ninteractions of the gauge theory. In particular, the beta function\nof ${\\cal N}=4$ gauge theory vanishes.\n\nNow we know a new reason for ${\\cal N}=4$ to be special. The\ntopological B-model on $\\Bbb {CP}^{3|N}$ exists if and only if\n${\\cal N}=4.$  The B-model on $\\Bbb {CP}^{3|4}$ has a $SU(2,2|4)$\nsymmetry coming from the geometric transformations of the twistor\nspace. This is related via the twistor transform to the ${\\cal N}=4$\nsuperconformal symmetry.\n\nIn the topological B-model with space-filling branes on $\\Bbb\n{CP}^{3|4}$, the basic field is the holomorphic gauge field\n${\\cal A}={\\cal A}_{\\overline I} dZ^{\\overline I}$, \\eqn{abasic}{{\\cal A}(Z,\\overline Z,\n\\psi)= A(Z,\\overline Z)+\\psi^A \\xi_{A}(Z,\\overline Z)+{1\\over\n2!}\\psi^A\\psi^B \\psi_{AB}(Z,\\overline Z)+\\dots+{1\\over\n4!}\\epsilon_{ABCD}\\psi^A\\psi^B\\psi^C\\psi^D G(Z,\\overline Z).} The\naction is the same as \\rf{hcsc}, except that the gauge field ${\\cal A}$\nnow depends on $\\psi$ \\eqn{actsu}{S=\\frac{1}{2}\\int\n\\Omega\\wedge\\mop{Tr}\\left({\\cal A} \\overline\\partial {\\cal A}+\n{2\\over3}{\\cal A}\\wedge{\\cal A}\\wedge{\\cal A}\\right),} and the holomorphic three\nform is  (\\ref{holf}).  The classical equations of motions\nobtained from \\rf{actsu} are\n\\eqn{cseq}{\\overline\\partial{\\cal A}+{\\cal A}\\wedge{\\cal A}=0.} Linearizing the\nequations of motions around the trivial solutions ${\\cal A}=0,$ they\ntell us that \\eqn{tela}{\\overline\\partial \\Phi=0,} where $\\Phi$ is any\nof the components of ${\\cal A}.$ The gauge invariance reduces to\n$\\delta\\Phi=\\overline\\partial\\alpha.$ Hence for each component $\\Phi,$\nthe field $\\Phi$ defines an element of a cohomology group.\n\nThis action has the amazing property that its spectrum is the same\nas that of ${\\cal N}=4$ super Yang-Mills theory in Minkowski space. To\nsee this, we need to use that the elements of  $(0,1)$ cohomology\ngroups of degree $2h-2$ are related by twistor transform to\nhelicity $h$ free fields on Minkowski space.\n\nTo figure out the degrees of various components ${\\cal A}$, notice that\nthe action must be invariant under the $\\Bbb C^*$ action\n$Z^I\\rightarrow tZ^I.$ Since the holomorphic measure is also\ninvariant under the scaling, the only way that the action\n\\rf{actsu} is invariant is that the superfield ${\\cal A}$ is also\ninvariant, in other words, ${\\cal A}$ is of degree zero\n\\eqn{azero}{{\\cal A}\\in H^{0,1}(\\Bbb {CP}^{3|4},{\\cal O}(0)).} Looking back\nat the expansion (\\ref{abasic}) of the superfield, we identify the\ncomponents, via the twistor correspondence, with fields in\nMinkowski space of definite helicity. $A$ is is of degree zero,\njust like the superfield ${\\cal A}.$ Hence, it is related by twistor\ntransform to a field of helicity $+1.$ The field $G$ has degree\n$-4$ to off-set the degree $4$ coming the four $\\psi$, so it\ncorresponds to a field of helicity $-1.$ Continuing in this\nfashion, we obtain the complete spectrum of ${\\cal N}=4$ supersymmetric\nYang-Mills theory. The twistor fields\n$A,\\xi_A,\\phi_{AB},\\xi_{ABC},G$  describe, via twistor transform,\nparticles of helicities $1,+\\frac{1}{2},0,-\\frac{1}{2},-1$ respectively.\n\nThe fields also have the correct representations under the $SU(4)$\nR-symmetry group. This symmetry is realized in twistor space by\nthe natural geometric action on the fermionic coordinates\n$\\psi^A\\rightarrow \\Lambda^A{}_B \\psi^B.$ Hence, $\\psi^A$\ntransforms in the $\\bf 4$ of the $SU(4)_R.$ The holomorphic gauge\nsuperfield ${\\cal A}(Z,\\psi)$ is invariant under the R-symmetry, hence\nthe representations of the components of ${\\cal A}$ must be conjugate\nto the representations of the $\\psi$ factors that they multiply in\n\\rf{abasic}. Hence, $A, \\xi_{A}, \\phi_{AB},\\xi_{ABC}$ and $G$\ntransform in the $\\bf 1, \\overline 4, 6, 4, 1$ representation of\n$SU(4)_R$ respectively\\footnote{The construction of twistor string\nhas been generalized to theories with less supersymmetry or with\nproduct gauge groups, by orbifolding the fermionic directions of\nthe super-twistor space \\cite{Park:2004bw,Giombi:2004xv}.}\n\n\\subsection{ D-Instantons}\n\nThe action (\\ref{actsu}) also describes some of the interactions\nof ${\\cal N}=4$ super Yang-Mills, but not all. It cannot describe the\nfull interactions, because an extra $U(1)$ R-symmetry gets in the\nway. The fermionic coordinates $\\psi^A, A=1,\\dots, 4$ have an\nextra $U(1)_R$ besides the $SU(4)_R$ considered above. Indeed, the\nfull R-symmetry group in twistor space is\n\\eqn{rsym}{U(4)_R=SU(4)_R\\times U(1)_R,} where we take the extra\n$U(1)_R$,  which we call $S,$ to rotate the fermions by a common\nphase \\eqn{extrau}{S:\\ Z^I\\rightarrow Z^I,\\quad \\psi^A\\rightarrow\ne^{i\\theta} \\psi^A.} In the B-model, the extra $U(1)_R$ is\nanomalous, since it does not leave fixed the holomorphic measure\n$\\Omega\\sim d^3Z d\\psi^1\\dots d\\psi^4.$ Under the $S$\ntransformation, the holomorphic  measure transforms as\n$\\Omega\\rightarrow e^{-4i\\theta}\\Omega$, so it has charge $S=-4$,\nhence the $B$-model action has $S=-4.$\n\nHowever, as we have set things up so far, the anomaly of the\nB-model is too trivial to agree with the anomaly of ${\\cal N}=4$\nYang-Mills theory. With the normalization \\rf{extrau}, the $S$\ncharges of fields are given by their degrees. The ${\\cal N}=4$\nYang-Mills action is a sum of terms with $S=-4$ and $S=-8.$ For\nillustration, consider the positive and negative helicity gluons\nthat have $S$-charge $0$ and $-4$ respectively. The kinetic term\nand the $++-$ three-gluon vertex contribute to the $S=-4$ part of\nthe Yang-Mills action. The $--+$ and the $--++$ vertices\ncontribute to the $S=-8$ part.\n\nThe action of the open string B-model (\\ref{actsu}) has $S=-4$\ncoming from the anomaly of $S$ of the holomorphic measure\n$\\Omega$. To get the $S=-8$ piece of the Yang-Mills action, we\nneed to enrich the B-model with nonperturbative instanton\ncontributions.\n\n\nThe instantons in question are Euclidean D1-branes wrapped on\nholomorphic curves in $\\Bbb {CP}^{3|4}$ on which open string can\nend. The gauge theory amplitudes come from coupling of the open\nstrings to the D1-branes. The massless modes on the worldvolume of\na D-instanton are a $U(1)$ gauge field and the modes that describe\nthe motion of the instanton. In the following, we will study\nmostly tree level amplitudes. These get contributions from genus\nzero instantons on which the $U(1)$ line bundles have a fixed\ndegree $d=-1$. Hence the bundles do not have any discrete or\ncontinuous moduli, so we will ignore the $U(1)$ gauge field from\nnow on. The modes describing the motion of the D-instanton make up\nthe moduli space ${\\cal M}$ of holomorphic curves $C$ in the\ntwistor space. To construct scattering amplitudes we need to\nintegrate over ${\\cal M}.$\n\n\n\\section{Tree Level Amplitudes from Twistor String Theory}\n\n\\subsection{Basic Setup}\n\\label{basic}\n\nRecall that the interactions of  the full gauge theory come from\nEuclidean D1-brane instantons on which the open strings can end.\nThe open strings are described by the holomorphic gauge field\n${\\cal A}.$ To find the coupling of the open strings to the\nD-instantons, let us consider the effective action of the D1-D5\nand D5-D1 strings. Quantizing the zero modes of the D1-D5 strings\nleads to a fermionic zero form field $\\alpha^i$ living on the\nworldvolume of the D-instanton. $\\alpha^i$ transforms in the\nfundamental representation of the $Gl(n,\\Bbb C)$ gauge group\ncoming from the Chan-Paton factors. The D5-D1 strings are\ndescribed by a fermion $\\beta_i$ transforming in the\nantifundamental representation. The kinetic operator for the\ntopological strings is the BRST operator $Q,$ which acts as\n$\\overline\\partial$ on the low energy modes. So the effective action of\nthe D1-D5 strings is \\eqn{done}{S=\\int_C \\beta(\\overline\\partial+{\\cal\nA})\\alpha,} where $C$ is the holomorphic curve wrapped by the\nD-instanton. From this we read off the vertex operator for an open\nstring with wavefunction $\\phi={\\cal A}_{\\overline I}dZ^{\\overline I}$\n\\eqn{verterxop}{V_\\phi=\\int_C J\\wedge \\phi,} where $J_i{}^j=\n\\beta_i\\alpha^j dz$ is  a holomorphic current made from the free\nfermions $\\alpha^j,\\beta_i$. These currents generate a current\nalgebra on the worldvolume of the D-instanton.\n\nTo compute a scattering amplitude,  we evaluate the correlation\nfunction \\eqalign{conf}{{\\cal A} =\\int d{\\cal M} \\vev{\nV_{\\phi_1}V_{\\phi_2}\\dots V_{\\phi_n}} =\\int d{\\cal M} \\left\\langle\n\\int_C J_1\\phi_1\\dots \\int_C J_n\\phi_n \\right\\rangle.} We can\nthink of this as integrating out the fermions $\\alpha,\\beta$\nliving on the D-instanton. Hence, the generating function for\nscattering amplitudes is simply the integral of Dirac operator\nover moduli space of D-instantons \\eqn{dii}{\\int d{\\cal M}\n\\det(\\overline\\partial+ A).} Here,  $d{\\cal M} $ is the holomorphic\nmeasure on the moduli space of holomorphic curves of genus zero\nand degree $d.$ In topological B-model, the action is holomorphic\nfunction of the fields and all path integrals are contour\nintegral. Hence, the integral is actually over a\nmiddle-dimensional Lagrangian cycle in the moduli space. This\nintegral is a higher dimensional generalization of the familiar\ncontour integral from complex analysis.\n\nThe correlator of the currents on D1-instanton\\footnote{Here we\nwrite the single trace contribution to the correlator that\nreproduces the gauge theory scattering amplitude. As discussed in\nsection \\ref{closed}, the multitrace contributions correspond to\ngluon scattering processes with exchange of internal conformal\nsupergravity states.} \\eqn{curc}{\\langle J_1(z_1) J_2(z_2) \\dots\nJ_n(z_n)\\rangle={\\mop{Tr}(T_1 T_2\\dots T_n)dz_1 dz_2\\dots dz_n\\over\n(z_1-z_2)(z_2-z_3)\\dots (z_n-z_1)}+\\,permutations} follows from\nthe free fermion correlator on a sphere\n\\eqn{fref}{\\alpha^i(z)\\beta_j(z')\\sim {\\delta^i{}_j\\over z-z'}.}\n\n\n\\vskip .2in\\noindent{\\it Scattering Wavefunctions}\n\nWe would like to compute  the scattering amplitudes of plane waves\n$\\phi(x)=\\exp\\,(i \\, p\\cdot x)=\\exp\\,(i \\, \\pi^a \\tilde\\pi^{\\dot\na} x_{a \\dot a}).$ These are wavefunctions of external particles\nwith definite momentum ${p^{a \\dot a}=\\pi^a \\tilde\\pi^{\\dot a}}.$\nThe twistor wavefunctions corresponding to plane waves are\n\\eqn{twav}{\\phi(\\lambda,\\mu,\\psi)= \\overline\\delta(\\vev{\\lambda,\\pi})\n\\exp(i [{\\tilde\\pi},\\mu])g(\\psi),} where $g(\\psi)$ encodes the\ndependence on fermionic coordinates. For a positive helicity gluon\n$g(\\psi)=1$ and for a negative helicity gluon\n$g(\\psi)=\\psi^1\\psi^2\\psi^3\\psi^4.$ Here, we have introduced the\nholomorphic delta function \\eqn{hold}{\\overline\\delta(f)=\\overline\\partial\n\\overline f \\delta^2(f),} which is a closed $(0,1)$ form. We normalize\nit so that for any function $f(z)$, we have \\eqn{delf}{\\int dz\n\\,\\overline\\delta(z-a)f(z)=f(a).}\n\nThe idea of \\rf{twav} is that the delta function\n$\\delta(\\vev{\\lambda,\\pi})$ sets $\\lambda^a$ equal to  $\\pi^a.$\nThe Fourier transform of the exponential $\\exp(i\n[{\\tilde\\pi},\\mu])$ back into Minkowski space gives another delta\nfunction that sets $\\tilde\\lambda^{\\dot a}$ equal to\n$\\tilde\\pi^{\\dot a}.$ The twistor string computation with these\nwavefunctions gives directly momentum space scattering amplitudes.\n\nActually, the wavefunctions should be modified slightly so that\nthey are invariant under the scaling of the homogeneous\ncoordinates of $\\Bbb{CP}^{3|4}.$ From the basic properties of delta\nfunctions, it follows that $\\overline\\delta(\\vev{\\lambda,\\pi})$ is\nhomogeneous of degree $-1$ in both $\\lambda$ and $\\pi.$ Hence, for\npositive helicity gluons, the wavefunction is actually\n\\eqn{poh}{\\phi^+(\\lambda,\\mu)=\\overline\\delta(\\vev{\\lambda,\\pi})(\\lambda\/\\pi)\n\\exp \\big(i[{\\tilde\\pi},\\mu](\\pi\/\\lambda)\\big).} Here,\n$\\lambda\/\\pi$ is a well defined holomorphic function, since\n$\\lambda$ is a multiple of $\\pi$ on the support of the delta\nfunction. The power of $(\\lambda\/\\pi)$ was chosen, so that the\nwavefunction is homogeneous of degree zero in overall scaling of\n$\\lambda,\\mu,\\psi.$  Under the scaling\n\\eqn{scals}{(\\pi,\\tilde\\pi)\\rightarrow (t\\pi,t^{-1}\\tilde\\pi),}\nthe wavefunction is homogeneous of degree $-2$ as expected for a\npositive helicity gluon \\rf{scalia}. For negative helicity gluon,\nthe wavefunction is\n\\eqn{nev}{\\phi^-(\\lambda,\\mu)=\\overline\\delta(\\vev{\\lambda,\\pi})(\\pi\/\\lambda)^3\n\\exp\\big(i[\\tilde\\pi,\\mu](\\pi\/\\lambda)\\big)\n\\psi^1\\psi^2\\psi^3\\psi^4.} Under the scaling \\rf{scals}, the\nwavefunction is homogeneous of degree $+2$ as expected. For\nwavefunctions of particles with helicity $h,$ there are similar\nformulas with $2-2h$ factors of $\\psi.$\n\n\\vskip .2in\\noindent{\\it MHV Amplitudes}\n\nWe saw in section \\ref{twist} that MHV amplitudes, after Fourier\ntransform into twistor space, localize on genus zero degree one\ncurve $C$, that is, a linearly embedded copy of $\\Bbb{CP}^1.$ Here we\nwill evaluate the degree one instanton contribution and confirm\nthat it gives the MHV amplitude.\n\nConsider the moduli space of such curves. Each curve $C$ can be\ndescribed by the equations \\eqn{dege}{\\mu^{\\dot a}=x^{a {\\dot a}}\n\\lambda_a \\qquad \\psi^A=\\theta^{Aa} \\lambda_a,} where $\\lambda^a$\nare the homogeneous coordinates and $x^{a\\dot a}$ and\n$\\theta^{Aa}$ are the moduli of $C$.  The holomorphic measure on\nthe moduli space is \\eqn{meas}{d{\\cal M}=d^4x d^8\\theta.} Hence, the\nmoduli space has $4$ bosonic and $8$ fermionic dimensions.\n\nIn terms of the homogeneous coordinate $\\lambda^a$ the current\ncorrelator \\rf{curc} becomes\n\\eqn{cunu}{\\vev{J_1(\\pi_1)J_2(\\pi_2)\\dots\nJ_n(\\pi_n)}={\\prod_i\\vev{\\lambda_i,d\\lambda_i}\\over\n\\vev{\\lambda_1,\\lambda_2}\\vev{\\lambda_2,\\lambda_3}\\dots \\vev{\\lambda_n,\\lambda_1}},} which we\nfound by setting $z_i=\\lambda^2_i\/\\lambda^1_i$. We stripped away the color\nfactors and kept only the contribution of the term with\n$1,2,\\dots, n$ cyclic order. We multiply  this with the\nwavefunctions\n$\\psi_i(\\lambda,\\mu,\\psi)=\\overline\\delta(\\vev{\\lambda,\\pi_i})\\exp\\left(i[\\mu,\\tilde\\pi_i]\\right)\ng_i(\\psi)$ and integrate over the positions $\\lambda_i,\\tilde\\lambda_i$ of the\nvertex operators.  We perform the integral over the positions of\nthe vertex operators using the formula \\eqn{homl}{\\int_{\\Bbb{CP}^1}\n\\vev{\\lambda,d\\lambda}\\ \\overline\\delta(\\vev{\\lambda,\\pi})f(\\lambda)=f(\\pi),} where $f(\\lambda)$\nis a homogeneous function of $\\lambda^a$ of degree $-1.$ This is the\nhomogeneous version of definition of holomorphic delta function\n\\eqn{hdef}{\\int_{\\Bbb C} dz\\ \\overline\\delta(z-b)f(z)=f(b).} Hence,\neach wavefunction contributes a factor of \\eqn{wco}{\\int_C\n\\vev{\\lambda,d\\lambda}\\phi_i=\n\\exp\\left(i[\\tilde\\pi_i,\\mu_i]\\right)g_i(\\psi_i),} where\n$\\mu_i^{\\dot a}=x^{a\\dot\na}\\lambda_{ia},\\psi^A_i=\\theta^{aA}\\lambda_{ia}$ and the delta\nfunction sets $\\lambda^a_i=\\pi^a_i$ in the correlation function.\nSo the amplitude becomes \\eqn{ale}{{\\cal A}={1\\over \\prod_k\n\\vev{\\pi_k,\\pi_{k+1}}} \\int d^4x d^8\\theta\\ \\exp\\Big(i\\sum_k\n[\\tilde\\pi_k,\\mu_k]\\Big) \\prod_k g_k(\\psi_k).}\n\nThe fermionic part of the wavefunctions is $g_i=1$ for the\npositive helicity gluons and\n$g_i=\\psi^1_i\\psi^2_i\\psi^3_i\\psi^4_i$ for the negative helicity\ngluons. Since we are integrating over eight fermionic moduli\n$d^8\\theta,$ we get nonzero contribution to amplitudes with\nexactly two negative helicities $r^-,s^-.$ Setting\n$\\psi^A=\\theta^{Aa}\\pi_a,$ the integral over fermionic dimensions\nof the moduli space gives the numerator of the MHV amplitude\n\\eqn{fermi}{\\int d^8\\theta \\prod_{A=1}^4 \\psi_r^A\n\\prod_{B=1}^4\\psi_s^B= \\vev{r,s}^4.} Setting $\\mu^{\\dot\na}_i=x^{a\\dot a}\\pi_{ia},\\ i=1,\\dots, n,$ the integral over\nbosonic moduli gives the delta function of momentum conservation\n\\eqn{bosni}{\\int d^4x \\exp\\left(ix_{a\\dot\na}\\sum_i\\pi_i^a\\tilde\\pi^{\\dot a}_i\\right)=\\delta^4(\\sum_{i=1}^n\n\\pi_i^a\\tilde\\pi_i^{\\dot a}).}\n\nCollecting the various pieces, we get the familiar MHV amplitude\n\\eqn{mhva}{{\\cal A}(r^-,s^-)={\\vev{r,s}^4 \\over \\prod_{i=1}^n\n\\vev{i,i+1}}\\delta^4(\\sum_{i=1}^n \\pi_i\\tilde\\pi_i).}\n\n\n\n\\subsection{Higher Degree Instantons}\n\\label{highd}\n\n\\vskip .15in\\noindent{\\it Instanton Measure}\n\nHere we will construct the measure on the moduli space of genus\nzero degree $d$ curves. Such curves can be described as degree $d$\nmaps from an abstract $\\Bbb{CP}^1$ with homogeneous coordinates $(u,v)$\n\\eqalign{didi}{Z^I&=&P^I(u,v) \\cr \\psi^A&=&\\chi^A(u,v).} Here\n$P^I,\\chi^A$ are homogeneous polynomials of degree $d$ in $u,v.$\nThe space of homogeneous polynomials of degree $d$ is a linear\nspace of dimension $d+1,$ spanned by $u^d,u^{d-1}v,\\dots, v^d.$\nPicking a basis $b^\\alpha(u,v),\\alpha=1,\\dots,d+1$, we write\n\\eqalign{expa}{P^I&=&\\sum_\\alpha P_\\alpha^I\\ b^\\alpha\\cr \\psi^A&=&\n\\sum_\\alpha \\chi^A_\\alpha\\ b^\\alpha.} A natural measure is\n\\eqn{namo}{d{\\cal M}_0=\\prod_{\\alpha=1}^{d+1}\\prod_{A,I=1}^4\ndP^I_\\alpha\\ d\\chi^A_\\alpha.} This measure is invariant under a\ngeneral $Gl(d+1,\\Bbb C)$ transformation of the basis $b_\\alpha.$\nSince the number of bosonic and fermionic coordinates is the same,\nthe Jacobians cancel between fermionic and bosonic parts of the\nmeasure. The description \\rf{didi} is redundant, we need to divide\nby the $\\Bbb C^\\ast$ action that rescales $P^I$ and $\\chi^A$ by a\ncommon factor. This reduces the space of curves from $\\Bbb\nC^{4d+4|4d+4}$ to $\\Bbb{CP}^{4d+3|4d+4}.$ The curve $C$ also stays\ninvariant under an $Sl(2,\\Bbb C)$ transformation on $(u,v)$ so the\nactual moduli space of genus zero degree $d$ curves is\n\\eqn{modsp}{{\\cal M}=\\Bbb{CP}^{4d+3|4d+4}\/Sl(2,\\Bbb C).}  As $d{\\cal M}_0$\nis $Gl(2,\\Bbb C)$ invariant, it descends to a holomorphic measure\n\\eqn{holm}{d{\\cal M}={d{\\cal M}_0\\over Gl(2,\\Bbb C)}.} on ${\\cal M}.$ Hence,\n${\\cal M}$ is a Calabi-Yau supermanifold of dimension $(4d|4d+4).$\n\nWe can now understand why amplitudes with different helicities\ncome from holomorphic curves of different degrees. Integrating\nover the moduli space, the measure absorbs $4d+4$ fermion zero\nmodes. These come from the fermionic factors $g(\\psi)$ in the\nwavefunctions of the gluons \\rf{twav}. A positive helicity gluon\ndoes not contribute any zero modes while a negative helicity gluon\n$g^-(\\psi)=\\psi^1\\psi^2\\psi^3\\psi^4$ gives $4$ zero modes. Hence,\ninstantons of degree $d$ contribute to amplitudes with $d+1$\nnegative helicity gluons.\n\nAlternatively, we can get this from counting the $S$ charge\nanomaly. Wavefunctions of particles with different helicities\nviolate $S$ by different amount. The positive helicity gluons do\nnot violate $S$ while the negative helicity gluons violate $S$ by\n$-4$ units. So, the amplitudes with $p$ positive helicity gluons\nand $q$ negative helicity gluons violates the $S$ charge by $-4q$\nunits.\n\nIn the twistor string, there is a source of  violation of $S$ from\nthe instanton measure. Since the $S$ charge of $Z$ and $\\psi$ is\n$0$ and $1$ respectively, the charges of the coefficients\n$P^I_\\alpha, \\chi^A_\\alpha$ are $0,1.$ Hence, the differentials\n$dP^I_\\alpha, d\\chi^A_\\alpha$ have charges $0,-1$ and the $S$\ncharge of the $(4d|4d+4)$ dimensional measure $d{\\cal M}$ is $-4d-4.$\n\nSo an instanton can contribute to an amplitude with $q$ negative\nhelicity gluons if and only if \\eqn{reld}{d=q-1.} This is the\nfamiliar formula discussed at in subsection \\ref{twist}. For $l$\nloop amplitudes, this relation generalizes to $d=q-1+l.$\n\n \\vskip\n.15in\\noindent{\\it Evaluating the Instanton Contribution}\n\nHere we consider the connected instanton contribution along the\nlines of the calculation of the MHV amplitude. The amplitude is\n\\cite{Roiban:2004yf,Roiban:2004vt,Witten:2004cp}\n\\eqalign{wop}{{\\cal A}&=&\\int d{\\cal M}_d \\prod_i\\int_C {\\vev{u_i,du_i}\n\\over \\prod_k \\vev{u_k,u_{k+1}}}\\overline\\delta(\\vev{\\lambda(u_i),\\pi_i})\n\\exp\\left( i[\\mu(u_i),\\tilde\\pi_i]\\right) g_i(\\psi_i).} Here\n$d{\\cal M}_d$ is the measure on the moduli space of genus zero degree\n$d$ curves. Next comes the correlator of currents on the\nworldvolume of the D1-instanton and the wavefunctions in which we\nuse the parameterization $\\lambda^a_i(u_i)=P^a(u_i), \\mu^{\\dot\na}(u_i)=P^{\\dot a}(u_i).$\n\nThis is not really an integral. The integral over the $2d+2$\nparameters $P^{\\dot a}_\\alpha$ gives $2d+2$ delta functions\nbecause $P^{\\dot a}$ appear only in the exponential\n$\\exp\\left(\\sum_i P(u_i)_{\\dot a}\\tilde\\pi_i^{\\dot a}\\right)$.\nHence, we are left with an integral over $4d-(2d+2)+2n=2d+2n-2$\nbosonic variables. Here the $2n$ integrals come from the\nintegration over the positions of the vertex operators. Now there\nare $2n$ delta functions from the wavefunctions since each\nholomorphic delta function is really a product of two real delta\nfunctions $\\overline\\delta(z)=d\\overline z\\ \\delta^2(z)$, and $2d+2$ delta\nfunctions from the integral over the exponentials, which gives a\ntotal of $2d+2n+2$. There are four more delta functions than\nintegration variables. The four extra delta functions impose\nmomentum conservation. Hence, the delta functions localize the\nintegral to a sum of contributions from a finite number of points\non the moduli space.\n\n\n\\vskip .15in\\noindent{\\it Parity Invariance}\n\nIn the helicity formalism, the parity symmetry of Yang-Mills\nscattering amplitudes is apparent. The parity changes the signs of\nthe helicities of the gluons. The parity conjugate amplitude can\nbe obtained by simply exchanging $\\lambda_i$'s with $\\tilde\\lambda_i$'s.\n\nTo go to twistor space, one Fourier transforms with respect to\n$\\tilde\\lambda_i$, which breaks the symmetry between $\\lambda$ and $\\tilde\\lambda.$ Indeed,\nthe result \\rf{wop} for the scattering amplitude treats $\\lambda$ and\n$\\tilde\\lambda$ asymmetrically. An amplitude with $p$ positive helicities\nand $q$ negative helicities has contribution from instantons of\ndegree $q-1$, while the parity conjugate amplitude with $q$ gluons\nof positive helicity and $p$ gluons of negative helicity has\ncontribution from instantons of degree $p-1.$ To show that these\ntwo are related by an exchange of $\\lambda_i$ and $\\tilde\\lambda_i$ requires some\namount of work. We refer the interested reader to the original\nliterature \\cite{Roiban:2004yf,Roiban:2004vt,\nWitten:2004cp,Berkovits:2004tx}.\n\n\n\\vskip .15in\\noindent{\\it Localization on the Moduli Space}\n\nRecall that a tree level amplitude with $q$ negative helicity\ngluons and arbitrary number of positive helicity gluons receives\ncontribution from instantons wrapping holomorphic curves of degree\n$d=q-1.$ The degree $d$ instanton can consist of several disjoint\nlower degree instantons whose degrees add up to $d.$ For\ndisconnected scattering amplitudes the instantons are connected by\nopen strings.\n\\begin{figure}\n \\centering\n \\includegraphics[height=1.6in]{con_discon.eps}\n \\caption{An amplitude with tree negative helicity gluons has contribution\n          from two configurations: (a) Connected $d=2$ instanton. (b) Two disjoint\n          $d=1$ instantons. The dashed line represents an open string\n          connecting the instantons. }\n \\label{coni}\n\\end{figure}\nA priory, one expects that the amplitude receives contributions\nfrom all possible instanton configurations with total degree\n$q-1.$ So for example an amplitude with three negative helicity\ngluons has contribution from a  connected $d=2$ instanton and a\ncontribution from two disjoint $d=1$ instantons, fig. \\ref{coni}.\n\nWhat one actually finds is that the connected and disconnected\ninstanton contributions reproduce the whole amplitude {\\it\nseparately}. For example, in the case of amplitude with three\nnegative helicity gluons, it seems that there are two different\nways to compute the same amplitude. One can either evaluate it\nfrom the connected $d=2$ instantons\n\\cite{Roiban:2004yf,Roiban:2004vt}, see fig. \\ref{coni}(a).\nAlternatively, the amplitude comes from evaluating the\ncontribution of the two disjoint $d=1$ instantons\n\\cite{Cachazo:2004kj}, fig. \\ref{coni}(b).\n\nWe can explain the equality of various instanton contributions\nroughly as follows \\cite{Gukov:2004ei}.  Consider the connected\ncontribution. The amplitude is expressed as a `contour' integral\nover a middle-dimensional Lagrangian cycle in the moduli space of\ndegree two curves . The integrand comes from the correlation\nfunction on the worldvolume of the D-instanton and from the\nmeasure on the moduli space. It has poles in the region of the\nmoduli space, where the instanton degenerates to two intersecting\ninstantons of lower degrees $d_1+d_2=d$ , fig. \\ref{locali}.\nPicking a contour that encircles the pole, the integral localizes\nto an integral over the moduli space ${\\cal M}'$ of the intersecting\nlower degree curves.\n\nSimilarly, the  disconnected contribution has a pole when the two\nends of the propagator coincide. This comes from the pole of the\nopen string propagator \\eqn{opro}{\\overline\\partial G={\\overline\n\\delta}^3(Z'{}^I-Z^I) \\delta^4(\\psi'{}^A-\\psi^A).} Hence, the\nintegral over disjoint instantons also localizes on the moduli\nspace of intersecting instantons. It can be shown that the\nlocalized integrals coming from either connected or disconnected\ninstanton configurations agree \\cite{Gukov:2004ei} which explains\nwhy the connected and disconnected instanton calculations give the\nentire scattering amplitude separately.\n\n\n\\vskip .2in \\noindent{\\it Towards MHV Diagrams}\n\nStarting with a higher degree instanton contribution, successive\nlocalization reduces the integral to the moduli space of\nintersecting degree one curves. As we will review below, this\nintegral can be evaluated leading to a combinatorial prescription\nfor the scattering amplitudes \\cite{Cachazo:2004kj}. Indeed,\ndegree one instantons give MHV amplitudes, so the localization of\nthe moduli integral leads to a diagrammatic construction based on\na suitable generalization of the MHV amplitudes.\n\n\n\\begin{figure}\n \\centering\n  \\includegraphics[height=1.5in]{degree_two.eps}\n \\caption{Localization of the connected instanton contribution to next to MHV amplitude;\n  (a) the integral over the moduli space of connected degree two\n  curves, localizes to an integral over the degenerate\n  curves of (b), that is intersecting complex lines.\n  In the figure, we draw the real section of the curves.}\n \\label{locali}\n\\end{figure}\n\n\n\\subsection{MHV Diagrams}\n\nIn this subsection, we start with a motivation of the MHV diagrams\nconstruction of amplitudes from basic properties of twistor\ncorrespondence. We then go on to discuss simple examples and\nextensions to loop amplitudes. In the next subsection, we give a\nheuristic derivation of the MHV rules from twistor string theory.\n\nRecall that MHV scattering amplitudes are supported on $\\Bbb\n{CP}^1$'s in twistor space. Each such $\\Bbb {CP}^1$ can be\nassociated to a point $x^{a\\dot a}$ in Minkowski space\\footnote{We\nare being slightly imprecise here. The space of $\\Bbb {CP}^1$'s is\nactually a copy of the complexified Minkowski space $\\Bbb C^4.$\nThe Minkowski space $\\Bbb R^{3|1}$ corresponds to $\\Bbb {CP}^1$'s\nthat lie entirely in the 'null twistor space', defined by\nvanishing of the pseudo-hermitian norm $Q(\\lambda,\\mu)=i(\\lambda^a\n\\overline\\mu_a-\\overline\\lambda^{\\dot a}\\mu_{\\dot a}).$ Indeed, for a $\\Bbb\n{CP}^1$ corresponding to point a point in Minkowski space\n$x^{a\\dot a}$ is a hermitian matrix, hence it follows from\n\\rf{cpone} that $Q$ vanishes.} \\eqn{cpone}{\\mu_{\\dot a}+x_{a\\dot\na}\\lambda^a=0.} So, in a sense, we can think of MHV amplitudes as\nlocal interaction vertices \\cite{Cachazo:2004kj}. To take this\nanalogy further, we can try to build more complicated amplitudes\nfrom Feynman diagrams with vertices that are suitable off-shell\ncontinuations of the MHV amplitudes, fig. \\ref{localize}. MHV\namplitudes are functions of holomorphic spinors $\\lambda_i$ only.\nHence, to use them as vertices in Feynman diagrams, we need to\ndefine $\\lambda$ for internal off-shell momenta $p^2\\neq0.$\n\n\\begin{figure}\n \\centering\n \\includegraphics[height=1.5in]{localize.eps}\n \\caption{Two representations of a degree three MHV diagram. (a) In\n          Minkowski space, the MHV vertices are represented by points.\n          (b) In twistor space, each MHV vertex corresponds to a line.\n          The three lines pairwise intersect.}\n \\label{localize}\n\\end{figure}\n\nTo motivate the off-shell continuation, notice that for on-shell\nmomentum $p^{a\\dot a}=\\lambda^a\\tilde\\lambda^{\\dot a}$, we can\nextract the holomorphic spinors $\\lambda$ from the momentum $p$ by\npicking arbitrary anti-holomorphic spinor $\\eta^{\\dot a}$ and\ncontracting it with $p^{a\\dot a}.$ This gives $\\lambda^a$ up to a\nscalar factor \\eqn{lax}{\\lambda^a= {p^{a\\dot a}\\eta_{\\dot a}\\over\n[\\tilde\\lambda, \\eta]}.}  For off-shell momenta, this strategy\nalmost works except for the factor $[\\tilde\\lambda,\\eta]$ in the\ndenominator which depends on the undefined spinor $\\tilde\\lambda$.\nFortunately, $[\\tilde\\lambda, \\eta]$ scales out of Feynman\ndiagrams, so we take as our definition\n\\eqn{lambdaoffshell}{\\lambda^a= p^{a\\dot a}\\eta_{\\dot a}.} This is\nclearly well-defined for off-shell momentum. We complete the\ndefinition of the MHV rules, by taking $1\/k^2$ for the propagator\nconnecting the MHV vertices.\n\nConsider an MHV diagram with $v$ vertices. Each vertex gives two\nnegative helicity gluons. To make a connected tree level graph,\nthe vertices are connected with $v-1$ propagators. The propagators\nabsorb $v-1$ negative helicities, leaving $v+1$ negative helicity\nexternal gluons. Hence, to find all MHV graphs contributing to a\ngiven amplitude, draw all possible tree graphs of $v$ vertices and\n$v-1$ links assigning opposite helicities to the two ends of\ninternal lines. The external gluons are distributed among the\nvertices while preserving cyclic ordering. MHV graphs are those\nfor which each vertex has two negative helicity gluons emanating\nfrom it.\n\nFor further work on MHV vertices construction of tree-level gluon\namplitudes, see\n\\cite{Bena:2004ry,Kosower:2004yz,Wu:2004fb,Zhu:2004kr,Birthwright:2005ak}.\nMHV vertices have many generalizations; in particular, to\namplitudes with fermions and scalars\n\\cite{Georgiou:2004wu,Georgiou:2004by,Khoze:2004ba,Wu:2004jx,Su:2004ym},\nwith Higgses \\cite{Dixon:2004za,Badger:2004ty} and with\nelectroweak vector-boson currents \\cite{Bern:2004ba}. For an\nattempt to generalize MHV vertices to gravity, see\n\\cite{Giombi:2004ix,Nair:2005iv,Abe:2005se}.\n\n \\vskip .2in \\noindent{\\it Examples}\n\nHere we discuss concrete amplitudes to illustrate the method.\nConsider first the $+---$ gluon amplitude. This amplitude vanishes\nin Yang-Mills theory. It has contribution from two diagrams, see\nfig. \\ref{pmmm}.\n\n\\begin{figure}\n \\centering\n  \\includegraphics[height=1.8in]{pmmm.eps}\n \\caption{MHV diagrams contributing to the $+---$\n          amplitude, which is expected to vanish.}\n \\label{pmmm}\n\\end{figure}\n\nThe first of the two diagrams gives\n\\eqn{firstd}{{\\vev{2,\\lambda}^4 \\over \\vev{1,2} \\vev{2,\\lambda}\n\\vev{\\lambda, 1}} {1 \\over p^2} {\\vev{3,4}^4 \\over \\vev{3,4}\n\\vev{4,\\lambda} \\vev{\\lambda, 3}},} where we associate to the\ninternal momentum $p=p_1+p_2=-p_3-p_4$ the holomorphic spinor\n\\eqn{hof}{\\lambda^a=p^{a\\dot a}\\eta_{\\dot a}=(p_1+p_2)^{a {\\dot\na}}\\eta_{\\dot a}.} The second diagram can be obtained from the\nfirst by exchanging particles $2$ and $4$\n\\eqn{secd}{{\\vev{\\lambda', 4}^4\\over\\vev{1,\n\\lambda'}\\vev{\\lambda', 4}\\vev{4, 1}}{1\\over\np'^2}{\\vev{2,3}^4\\over\\vev{2,3}\\vev{3,\\lambda'}\\vev{\\lambda',2}},}\nwhere $\\lambda'^a=(p_1+p_4)^{a\\dot a}\\eta_{\\dot a}.$ Denoting\n$\\phi_i=\\lambda_i^{\\dot a}\\eta_{\\dot a},$ the first and second\ndiagrams  give respectively \\eqn{firg}{-{\\phi_1^3\\over\n\\phi_2\\phi_3\\phi_4}{\\vev{34}\\over [21]}-{\\phi^3_1\\over\n\\phi_2\\phi_3\\phi_4}{\\vev{32}\\over[41]}.} The sum of these\ncontributions vanishes, because momentum conservation implies\n$\\vev{32}[21]+\\vev{34}[41]=\\sum_i \\vev{3i}[i1]=0.$\n\nIt is easy to compute more complicated amplitudes. For example,\nthe $n$ gluon $---++\\dots++$ amplitude is a sum of $2(n-3)$ MHV\ndiagrams, which can be obtained from fig. \\ref{pmmm} by adding\nadditional $+$ helicities on the MHV vertices. The diagrams can be\nevaluated to give \\eqalign{nmhv}{ A&=&\\sum_{i=3}^{n-1} {\\vev{1\n\\lambda_{2,i}}^3 \\over \\vev{\\lambda_{2,i} i+1} \\vev{i+1 i+2} \\dots\n\\vev{n1}}{1 \\over q_{2i}^2} {\\vev{23}^3 \\over \\vev{\\lambda_{2,i}\n2} \\vev{34}\\dots \\vev{i\\lambda_{2,i}}} \\cr &+&\\sum_{i=4}^n\n{\\vev{12}^3 \\over \\vev{2\\lambda_{3,i}} \\vev{\\lambda_{3,i}i+1}\n\\dots \\vev{n1}} {1\\over q_{3i}^2} {\\vev{\\lambda_{3,i} 3}^3 \\over\n\\vev{3,4} \\dots \\vev{i-1 i}\\vev{i\\lambda_{3,i}},}} where\n$q_{ij}=p_i+p_{i+1}+\\dots+p_j$ and the corresponding spinor\n$\\lambda_{i,j}^a$ is defined in the usual way\n$\\lambda^a_{i,j}=q_{ij}^{a\\dot a}\\ \\eta_{\\dot a}.$\n\n\n\\vskip .2in\\noindent{\\it Loop Amplitudes}\n\nSimilarly, one can compute loop amplitudes using MHV diagrams.\nThis has been carried out for the one loop MHV amplitude in\n${\\cal N}=4$ \\cite{Brandhuber:2004yw} and ${\\cal N}=1$\n\\cite{Quigley:2004pw,Bedford:2004py} Yang-Mills theory, in\nagreement with the known answers.\n\n\\begin{figure}\n \\centering\n \\includegraphics[height=1.2in]{mhv_loop.eps}\n \\caption{Schematic representation of a hypothetical twistor string computation of\n          one-loop MHV amplitude. The picture shows a diagram in which the\n           negative helicity gluons $i^-,j^-$ are on the same MHV vertex.}\n \\label{mhv_loop}\n\\end{figure}\n\nThe expression for an MHV diagram  contributing to the one-loop\nMHV amplitude is just what one would expect for a one-loop Feynman\ndiagram with MHV vertices, fig. \\ref{mhv_loop}. There are two MHV\nvertices, each coming with two negative helicity gluons. The\nvertices are connected with two Feynman propagators that absorb\ntwo negative helicities, leaving two negative helicity external\ngluons \\eqn{ampli}{{\\cal A}^{loop}=\\sum_{{\\cal D}, h} \\int\n{d^{4-2\\epsilon} p\\over (2\\pi)^4} {\\cal A}_L(\\lambda_k,\n\\lambda_p,\\lambda_{p-p_L}){1\\over\np^2(p-p_L)^2}{\\cal A}_R(\\lambda_k,\\lambda_p,\\lambda_{p-p_L}).} The off-shell\nspinors  entering the MHV amplitudes ${\\cal A}_L, {\\cal A}_R$ are determined\nin terms of the momenta of the internal lines\n\\eqn{ofsh}{\\lambda_p^a=p^{a\\dot a}\\eta_{\\dot a},\\qquad\n\\lambda_{p-p_L}^a=(p-p_L)^{a\\dot a}\\eta_{\\dot a},} which is the same\nprescription as for tree level MHV diagrams. The sum in \\rf{ampli}\nis over partitions ${\\cal D}$ of the gluons among the two MHV\ndiagrams that preserve the cyclic order and over the helicities of\nthe internal particles\\footnote{Similarly, the double-trace\ncontribution to one-loop MHV amplitudes comes from Feynman\ndiagrams with double-trace MHV vertices\n\\cite{Luo:2004ss,Luo:2004nw}.}.\n\nThis calculation makes the twistor structure of one-loop MHV\namplitudes manifest. The two MHV vertices are supported on  lines\nin twistor space, so the amplitude is a sum of contributions, each\nof which is supported on a disjoint union two lines. In a\nhypothetical twistor string theory computation of the amplitude,\nthese two lines are connected by open string propagators, see fig.\n\\ref{loop_twistor}. This pictures agrees with studies of the\ntwistor structure using differential equations\n\\cite{Cachazo:2004zb}, after taking into account the holomorphic\nanomaly of the differential equations\n\\cite{Cachazo:2004by,Bena:2004xu}.\n\nFinally, we make a few remarks about the nonsupersymmetric\none-loop MHV amplitudes. The ${\\cal N}=0$ MHV amplitudes are sums of\ncut-constructible terms and  rational terms. The cut-constructible\nterms are correctly reproduced from MHV diagrams\n\\cite{Bedford:2004nh}.  The rational terms are single valued\nfunctions of the spinors, hence they are free of cuts in four\ndimensions. Their twistor structure suggests that they receive\ncontribution from diagrams in which, alongside with MHV vertices,\nthere are new one-loop vertices coming from one-loop all-plus\nhelicity amplitudes \\cite{Cachazo:2004zb}. However, a suitable\noff-shell continuation of the one-loop all-plus amplitude has not\nbeen found yet. There has been recent progress in computing the\nrational part of some one-loop QCD amplitudes using a\ngeneralization \\cite{Bern:2005hs} of the tree level recursion\nrelations reviewed in section \\ref{perturb}.\n\n\n\\begin{figure}\n \\centering\n  \\includegraphics[height=1.4in]{loop_twistor.eps}\n   \\caption{Twistor space structure of the one-loop MHV amplitude.\n     The two MHV vertices are represented by lines. In a hypothetical\n     twistor string computation of the amplitude, the lines are\n     connected by two twistor propagators to make a loop.}\n \\label{loop_twistor}\n\\end{figure}\n\n\n\\subsection{Heuristic Derivation of MHV Diagrams from Twistor String Theory}\n\nHere, we will make an analysis of the disconnected twistor\ndiagrams that contribute to tree level amplitudes\\footnote{For an\nattempt to derive MHV rules from ${\\cal N}=4$ superspace constraints,\nsee \\cite{Abe:2004ep}.}. We will evaluate the twistor string\namplitude corresponding the twistor contribution of fig.\n\\ref{thret} and show how it leads to the MHV diagrammatic rules of\nthe last subsection.\n\n\nThe physical field of the open string B-model is a $(0,1)$-form\n${\\cal A}$ with kinetic operator  $\\overline\\partial$ coming from the\nChern-Simons action. The twistor propagator for ${\\cal A}$ is a\n$(0,2)$-form on $\\Bbb {CP}^3\\times \\Bbb{CP}^3$ that is a\n$(0,1)$-form on each copy of $\\Bbb{CP}^3.$ The propagator obeys\nthe equation \\eqn{opror}{\\overline\\partial G={\\overline\n\\delta}^3(Z_2^I-Z_1^I) \\delta^4(\\psi_2^A-\\psi_1^A).} Here,\n$\\overline\\delta(z)=d{\\overline z} \\delta(z)\\delta({\\overline z})$ is the\nholomorphic delta function $(0,1)$-form.\n\nIn an axial gauge, the twistor propagator becomes\n\\eqn{tiwp}{G=\\overline\\delta(\\lambda_2^2-\\lambda_1^2)\\overline\\delta(\\mu_2^{\\dot\n1}-\\mu_1^{\\dot 1}){1\\over \\mu_2^{\\dot 2}-\\mu_1^{\\dot\n2}}\\prod_{A=1}^4(\\psi_2^A-\\psi_1^A),} where we set\n$\\lambda_1^1=\\lambda_2^1=1.$\n\nFor simplicity, we evaluate the contribution from two degree-one\ninstantons $C_1$ and $C_2$ connected by twistor propagator. This\nconfiguration contributes to amplitudes with three negative\nhelicity gluons. The instantons $C_i, i=1,2$ are described by the\nequations \\eqn{insto}{\\mu^{\\dot a}_k=x_i^{a {\\dot a}}\n\\lambda_{ka}, \\qquad \\psi_k^A=\\theta^{Aa}_i\\lambda_{ka}\\qquad\ni=1,2,\\ k=1,\\dots, n.} Here, $x^{a {\\dot a}}_i$ and\n$\\theta^{Aa}_i$ are the bosonic and fermionic moduli of $C_i$.\n\nWith our choice of gauge, the twistor propagator is supported on\npoints such that $\\lambda^a_1=\\lambda^a_2.$ Since $\\mu^{\\dot\na}_2-\\mu^{\\dot a}_1=y^{a\\dot a}\\lambda_a,$ where $y^{a\\dot\na}=x^{a\\dot a}_2-x^{a\\dot a}_1,$ the condition $\\mu^{\\dot\n1}_2-\\mu^{\\dot 1}_1=0$ implies $\\lambda^a=y^{a\\dot 1}.$ Hence, the\nbosonic part of the propagator gives $1\/(\\mu^{\\dot 2}_2-\\mu^{\\dot\n2}_1)=1\/y^2.$\n\n\\begin{figure}\n \\centering\n  \\includegraphics[height=1.2in]{instantons.eps}\n \\caption{Twistor string contribution to an amplitude with three negative\n helicity external gluons.\n Two disconnected degree one instantons are connected by an open string.}\n \\label{thret}\n\\end{figure}\n\nThe  correlators of the gluon vertex operators on $C_1$ and $C_2$\nand the integral over $\\theta^{Aa}_i$ give two MHV amplitudes\n${\\cal A}_L$ and ${\\cal A}_R$ as explained in the $d=1$ computation. So we\nare left with the integral \\eqn{dtwoi}{\\int d^4x_1 d^4x_2 {\\cal A}_L\n{1\\over (x_2-x_1)^2} {\\cal A}_R \\prod_{i \\in L} \\exp(i x_1 \\cdot p_i)\n\\prod_{j \\in R} \\exp(i x_2 \\cdot p_j),} where the integral is over\na suitably chosen  $4\\times4$ real dimensional `contour' in the\nmoduli space $\\Bbb C^4\\times \\Bbb C^4$ of two degree one curves.\nWe rewrite the exponential as \\eqn{expo}{\\exp(i y \\cdot P)\n\\prod_{j \\in L,R} \\exp(i x \\cdot p_i),} where $x\\equiv x_1$ and\n$P=\\sum_{i \\in R} p_i$ is momentum of the off-shell line\nconnecting the two vertices. The integral \\eqn{dem}{\\int d^4x\n\\prod_{i \\in L,R} \\exp(i x \\cdot p_i)=(2\\pi)^4\\delta^4(\\sum_i\np_i)} gives the delta function of momentum conservation. We are\nleft with \\eqn{abem}{A= \\int d^4 y{1 \\over y^2} \\exp(i y \\cdot P)\n{\\cal A}_L{\\cal A}_R.}  The integrand has a pole at $y^2=0,$ which is the\ncondition for the curves $C_1$ and $C_2$ to intersect. The space\n$y^2=0$ is the familiar conifold. It is a cone over $\\Bbb\n{CP}^1\\times \\Bbb {CP}^1$ so we parameterize it as \\eqn{cop}{y^{a\n{\\dot a}}=t \\lambda^a \\tilde\\lambda^{\\dot a}.} Here $\\lambda^a\\in\n{\\cal O}(1,0),\\, \\tilde\\lambda^{\\dot a} \\in {\\cal O}(0,1),$ hence\n$t\\in {\\cal O}(-1,-1)$ so that \\rf{cop} is well-defined. We choose\na contour that picks the residue at $y^2=0.$ The residue is the\nvolume form on the conifold \\eqn{refa}{ {\\rm Res} \\, {d^4 y \\over\ny^2} = t dt \\vev{\\lambda, d\\lambda} [\\tilde\\lambda,\nd\\tilde\\lambda].} Taking the residue, the integral becomes\n\\eqn{aboc}{ I= \\int t dt\\vev{\\lambda, d\\lambda} [\\tilde\\lambda,\nd\\tilde\\lambda] \\exp(i t P_{a\\dot a}\\lambda^a \\tilde\\lambda^{\\dot\na}){\\cal A}_L {\\cal A}_R,} where the MHV vertices depend on the holomorphic\nspinor $\\lambda$ only. We pick the contour $t\\in(-\\infty,\\infty)$\nand ${\\tilde\\lambda=\\overline\\lambda},$ that is, we integrate over the\nreal light-cone. For ${t\\in(0,\\pm\\infty)}$ we regulate the\nintegral with the prescription ${P=(p^0\\pm i\\epsilon,\\vec{p})},$\nso \\eqn{princ}{{\\int_{-\\infty}^\\infty t dt \\exp(i tP_{a\\dot\na}\\lambda^a \\tilde\\lambda^{\\dot a})=-{2\\over (P_{a\\dot a}\\lambda^a\n\\tilde\\lambda^{\\dot a})^2}}.} Hence we have \\eqn{next}{{I= \\int\n\\vev{\\lambda, d\\lambda} [\\tilde\\lambda, d\\tilde\\lambda] {1 \\over\n(P \\lambda \\tilde\\lambda)^2} {\\cal A}_L{\\cal A}_R(\\lambda)}.} To reduce the\nintegral \\rf{next} to a sum over MHV diagrams, we use the identity\n\\eqn{rein}{{ {[\\tilde\\lambda,d\\tilde\\lambda] \\over (P \\lambda\n\\tilde\\lambda )^2}= -{1\\over P\\lambda \\eta} \\, \\overline\\partial \\left(\n{[\\tilde\\lambda, \\eta] \\over P \\lambda \\tilde\\lambda } \\right)},} where\n$\\eta^{\\dot a}$ is an arbitrary positive helicity spinor to write\nthe integral as \\eqn{ipre}{{I=\\int \\vev{\\lambda ,d\\lambda} \\,\n{{\\cal A}_L {\\cal A}_R \\over (P\\lambda \\eta)} \\overline \\partial \\left(\n{[\\tilde\\lambda, \\eta] \\over (P \\lambda \\tilde\\lambda)} \\right)}.}\nNow we can integrate by parts. The $\\overline\\partial$ operator acting\non the holomorphic function on the left gives zero except for\ncontributions coming from  poles of the holomorphic function,\n$\\overline\\partial\\left(1\/z\\right)=\\overline\\delta(z).$ These evaluate to a\nsum over residues \\eqn{byp}{{I=\\sum {\\rm Res} \\, \\left(\n{{\\cal A}_L{\\cal A}_R \\over P\\lambda \\eta} \\right) {[\\tilde\\lambda, \\eta]\n\\over P \\lambda \\tilde\\lambda}}.} The residues of\n${1\/(P\\lambda\\eta)}$ are at \\eqn{pol}{\\lambda^a=P^{a\\dot\na}\\eta_{\\dot a}.} Substituting this back into \\rf{byp},\n$P\\lambda\\tilde\\lambda$ evaluates to $P^2 [\\tilde\\lambda,\\eta],$ so we\nhave \\eqn{mhve}{{I={1 \\over P^2} {\\cal A}_L{\\cal A}_R(\\lambda=P\\eta)}.} But\nthis is precisely the contribution from an MHV diagram. Summing\nover all cyclicly ordered partitions of the gluons among the two\ninstantons gives the sum over MHV diagrams contributing to the\nscattering amplitude.\n\nThere are additional additional poles in \\rf{byp} that come from\nthe MHV vertices ${\\cal A}_L{\\cal A}_R$ \\eqn{apo}{{{1 \\over\n\\prod_{\\alpha=1}^4 \\vev{\\lambda_\\alpha, \\lambda}}},} where\n$\\alpha$ runs over the four gluons  adjacent to the twistor line.\nThe poles are located at  ${\\lambda=\\lambda_\\alpha},$ which is the\ncondition of the twistor line to meet the gluon vertex operator.\nConsider the two diagrams, fig. \\ref{cancellation} in which the\nfunction ${\\cal A}_L{\\cal A}_R$ has a pole at $\\lambda=\\lambda_\\alpha.$ The\ngraphs differ by whether the gluon $\\alpha$ is on the left vertex\njust after the propagator or on the right vertex just before the\npropagaor. The reversed order of ${\\lambda}$ and\n${\\lambda_\\alpha}$ in the two diagrams changes the sign of the\nresidue. The rest of the residue \\rf{byp} stays the same after\ntaking ${\\lambda=\\lambda_\\alpha}.$ The off-shell momenta of the\ntwo diagrams differ by ${\\delta P=\\lambda_\\alpha\n\\tilde\\lambda_\\alpha},$ so the diagrams have the same value of the\ndenominators ${(P\\lambda_\\alpha\n\\tilde\\lambda_\\alpha)(P\\lambda_\\alpha \\eta)}.$ Hence, all poles at\n${\\lambda=\\lambda_\\alpha}$ get cancelled among pairs of diagrams.\n\n\\begin{figure}\n \\centering\n  \\includegraphics[height=1.6in]{cancellation.eps}\n \\caption{The graphs contributing to the pole at $\\lambda=\\lambda_\\alpha.$\n          The reversed order of $\\alpha$ and the internal line in the two graphs,\n          changes the sign of the residue of the pole.}\n \\label{cancellation}\n\\end{figure}\n\nThis derivation clearly generalizes to several disconnected degree\none instantons that contribute to a general tree level amplitude.\nAn amplitude with $d+1$ negative helicity gluons gets\ncontributions from diagrams with $d$ disconnected degree one\ninstantons. The evaluation of the twistor contributions leads to\nMHV diagrams with $d$ MHV vertices.\n\nLet us remark that the integral \\rf{next} could be taken as the\nstarting point in the study of MHV diagrams.  Since \\rf{princ} is\nclearly Lorentz invariant,\\footnote{The Lorentz invariance\nrequires some elaboration, because the choice of contour\n$\\overline\\lambda=\\tilde\\lambda,$ breaks the complexified Lorentz group\n$Sl(2,\\Bbb C)\\times Sl(2,\\Bbb C)$ to the diagonal $Sl(2,\\Bbb C),$\nthe real Minkowski group. It can be argued from the holomorphic\nproperties of the integral \\rf{next}, that it is invariant under\nthe full $Sl(2,\\Bbb C)\\times Sl(2,\\Bbb C)$ \\cite{Cachazo:2004kj}.}\nthe MHV diagram construction must be Lorentz invariant as well.\nAlthough separate MHV diagrams depend on the auxiliary spinor\n$\\eta,$ the sum of all diagrams contributing to a given amplitude\nis independent of the auxiliary spinor $\\eta^{\\dot a}$.\n\n\\vskip .2in \\noindent{\\it Loops in Twistor Space?}\n\nWe have just seen that the disconnected instanton contribution\nleads to tree level MHV diagrams. However, the MHV diagram\nconstruction seems to work for loop amplitudes as well, as\ndiscussed in previous subsection. Hence, one would like to\ngeneralize the twistor string derivation to higher genus instanton\nconfigurations, which contribute to loop amplitudes in Yang-Mills\ntheory. For example, the one-loop MHV amplitude should come from a\nconfiguration of two degree one instantons connected by two\ntwistor propagators to make a loop, fig. \\ref{loop_twistor}. An\nattempt to evaluate this contribution runs into difficulties: the\ntwo twistor propagators are both inserted at the same point\n$\\lambda^a=y^{a\\dot a}\\eta_{\\dot a}$ on the D-instanton\nworldvolume making the answer ill-defined. Some of these\ndifficulties are presumably related to the closed string sector of\nthe twistor string theory, that we will now review.\n\n\n\n\n\\section{Closed Strings}\n\\label{closed}\n\n\nThe closed strings of the topological B-model on supertwistor\nspace are related by twistor transform to ${\\cal N}=4$ conformal\nsupergravity \\cite{Berkovits:2004jj}. The conformal group is the\ngroup of linear transformations of the twistor space, so the\ntwistor string is manifestly conformally invariant. Hence it\nnecessarily leads to a conformal theory of gravity.\n\n\n\\subsection{Closed String Spectrum}\n\nLet us see how the closed strings are related to the conformal\nsupergravity fields. The most obvious closed string field is the\ndeformation of complex structure of $\\Bbb{CP}'^{3|4},$ the $'$ means\nthat we throw away the set $\\lambda^a=0.$ In this and the\nfollowing section, we parameterize $\\Bbb{CP}^{3|4}$ with homogeneous\ncoordinates $Z^I,\\ I=1,\\dots,8.$ Recall that the complex structure\nis conventionally defined in terms of the tensor field\n$j^A=J^A{}_B dZ^B$ obeying $J^2=-1.$ The indices $A,B$ can be both\nholomorphic or antiholomorphic. In local holomorphic coordinates\n$J^I{}_J=i$ and $J^{\\overline I}{}_{\\overline J}=-i$. The first order\nperturbations of the complex structure are described by a field\n$J^I{}_{\\overline J}$ and its complex conjugate $J^{\\overline I}{}_J$. From\n$J^I{}_{\\overline J}$ we form the vector valued $(0,1)$ form\n$j^I=J^I{}_{\\overline J} dZ^{\\overline J}$ with equations of motion\n\\eqn{ejej}{\\overline\\partial j^I=0} that express the integrability\ncondition on the deformed complex structure. $j^I$ is volume\npreserving $\\partial_I J^I_{\\overline J}=0$, since the holomorphic\nvolume $\\Omega$ is part of the definition of the\nB-model\\footnote{This extra condition is not understood from\nB-model perpective \\cite{Berkovits:2004jj}. One can guess it from\nanalogous condition in the Berkovits's open twistor string.},\n$j^I$ is subject to the gauge symmetry $j^I\\rightarrow\nj^I+\\epsilon \\overline\\partial \\kappa^I$, where $\\kappa^I$ is a volume\npreserving vector field.\n\n\nAccording to twistor transform \\cite{Penrose:1976jq}, volume\npreserving deformations of complex structure of twistor space are\nrelated to anti-selfdual perturbations of the spacetime.\nAnti-selfdual perturbations correspond to positive helicity\nconformal supergravitons. The ${\\cal N}=4$ positive helicity\nsupermultiplet contains fields going from the helicity $+2$\ngraviton to a complex scalar $\\overline C.$\n\nThe negative helicity graviton is part of a separate ${\\cal N}=4$\nsuperfield. It comes from an RR two form field\n\\eqn{bfield}{b=B_{\\overline I J}\\ d\\overline Z^{\\overline I}\\wedge dZ^J} that\ncouples to the D1-branes of the B-model via \\eqn{bcop}{\\int_C b,}\nwhere $C$ is the worldvolume of the D1-brane. The equations of\nmotion of $b$ are \\eqn{eomg}{\\overline\\partial b=0} and $b$ is subject\nto the gauge invariance $b\\rightarrow b+\\overline\\partial \\lambda.$ In\norder to relate $b$ to the fields of the Berkovits's open twistor\nstring that we discuss in next section, one needs to assume that\n$b$ is also invariant under the gauge transformation $B_{\\overline\nIJ}\\rightarrow B_{\\overline IJ}+\\partial_J\\ \\chi_{\\overline I}.$\n\n\\subsection{Conformal Supergravity}\n\nConformal supergravity in four dimensions has action\n\\eqn{weyla}{S\\sim\\int d^4x \\sqrt{-g}W_{abcd}W^{abcd},} where $W$\nis the Weyl tensor. This theory is generally considered\nunphysical. Expanding the action around flat space\n$g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$ leads to a fourth order\nkinetic operator $S\\sim\\int d^4x\\, h\n\\partial^4h$ for the fluctuations of the metric, and thus to\nlack of unitarity.\n\nWe can see a sign of the supergravity already in the tree level\nMHV amplitude calculation of section \\ref{basic}. There we found\nthat the single trace terms agree with the tree level MHV\namplitude in gauge theory. We remarked that the current algebra\ncorrelators give additional multi-trace contributions. These come\nfrom an exchange of an internal conformal supergravity state,\nwhich is a singlet under the gauge group\\footnote{These\nopen-closed string interactions can be used to study deformations\nof ${\\cal N}=4$ gauge theory by turning on closed string background\nfield \\cite{Kulaxizi:2004pa,Chiou:2005jn}.}. For example, the four\ngluon MHV amplitude has a contribution $\\mop{Tr}\\ T_1T_2 \\mop{Tr}\\ T_3T_4$\ncoming from an exchange of supergravity state in the\n$12\\rightarrow 34$ channel, fig. \\ref{graviton}. In twistor string\ntheory, this comes from the double trace contribution of the\ncurrent algebra on the worldvolume of the D-instanton\n\\eqn{twiam}{\\int_{{\\cal M}} d{\\cal M} \\left\\langle V_1 V_2\\right\\rangle\n\\left\\langle V_3 V_4\\right\\rangle.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[height=1.1in]{graviton.eps}\n \\caption{A double trace  $ \\mop{Tr} T_1T_2\\mop{Tr} T_3T_4$ contribution to\n          tree level four gluon scattering amplitude coming from\n          exchange of conformal supergravity particle,\n          which is represented by a dashed line.}\n \\label{graviton}\n\\end{figure}\n\nAt tree level, it is possible to recover the pure gauge theory\nscattering amplitudes by keeping the single-trace terms. However,\nat the loop level, the diagrams that include conformal\nsupergravity particles can generate single-trace interactions.\nHence the presence of conformal supergravity coming from the\nclosed strings puts an obstruction to computation of Yang-Mills\nloop amplitudes in the present formulation of twistor string\ntheory.\n\nIn twistor string theory, the conformal supergravitons have the\nsame coupling as gauge bosons, so it is not possible to remove the\nconformal supergravity states by going to weak coupling. Since,\nYang-Mills theory is consistent without conformal supergravity, it\nis likely that there is a version of the twistor string theory\nthat does not contain the conformal supergravity states.\n\n\\section{Berkovits's Open Twistor String}\n\\label{berko}\n\nHere we will describe the open string version of the twistor\nstring \\cite{Berkovits:2004hg}. In this string theory, both\nYang-Mills and conformal supergravity states come from open string\nvertex operators.\n\n\\subsection{The Spectrum}\n\nThe action of the open string theory is \\eqn{opac}{S=\\int\nd^2z\\left( Y_I\\overline\\nabla_{\\overline z} Z^I+\\overline Y_I \\nabla_z\\overline\nZ^I+S_C \\right).} For Euclidean signature of the worldsheet,\n$Z^I=(\\lambda^a,\\tilde\\lambda^{\\dot a},\\phi^A),\\ a,{\\dot a}=1,2,\\ A=1,\\dots,4,\\\nI=1,\\dots,8,$ are homogeneous coordinates on $\\Bbb{CP}^{3|4}$ and $\\overline\nZ^I$ are their complex conjugates. $Y_I$ and $\\overline Y_I$ are\nconjugates to $Z^I$ and $\\overline Z^I.$ Notice that $Z^I,I=5,\\dots, 8$\nwere denoted as $\\phi^A,A=1,\\dots, 4$ in previous sections. Before\ntwisting, $Z$ and $\\overline Z$ have conformal weight zero and $Y$ and\n$\\overline Y$ have conformal weight one. The covariant derivatives are\n\\eqn{covd}{\\nabla_z=\\partial_z-A_z\\qquad \\nabla_{\\overline\nz}=\\overline\\partial_{\\overline z}-A_{\\overline z},} where $A$ is a worldsheet\ngauge field that gauges the $Gl(1,\\Bbb C)$ symmetry\n$Z^I\\rightarrow tZ^I,\\ Y_I\\rightarrow t^{-1} Y_I.$ $S_C$ is the\naction of a  current algebra with central charge $+28$ which\ncancels $-26$ of the conformal ghosts and $-2$ of the $Gl(1,\\Bbb\nC)$ ghosts.  The open string boundary conditions are\n\\eqn{bondc}{Z^I=\\overline Z^I,\\quad Y_I=\\overline Y_I\\quad j_r=\\overline j_r,}\nwhere $j_r,\\ r=1,\\dots \\dim G$ are the currents of the current\nalgebra. On the boundary, $Z$ and $Y$ are real and the $Gl(1,\\Bbb\nC)$ gauge group is broken to the  group $Gl(1,\\Bbb R)$ of real\nscalings of $Z,Y.$\n\nThe physical open string vertex operators are described by\ndimension one fields that are neutral under $Gl(1)$ and primary\nwith respect to Virasoro and $Gl(1)$ generators\n\\eqn{genp}{T=Y_I\\partial Z^I+T_C,\\qquad J=Y_IZ^I.} The fields\ncorresponding to Yang-Mills states are\n\\eqn{vym}{V_\\phi=j_r\\phi^r(Z)} where $\\phi^r(Z)$ is a dimension\nzero $Gl(1,\\Bbb R)$ neutral function of $Z.$ That is, $\\phi$ is\nany function on $\\Bbb{RP}^{3|4}.$ $V_\\phi$ has clearly dimension one.\n$\\phi$ is related by twistor transform to gauge fields on\nspacetime with signature $++--.$\n\nThe vertex operators describing the conformal supergravity are\n\\eqn{vertg}{V_f=Y_If^I(Z),\\qquad V_g=\\partial Z^I g_I(Z).} These\nhave dimension one, since $Y_I$ and $\\partial Z^I$ have dimension\none. The $Gl(1)$ invariance requires that $f^I$ has $Gl(1)$ charge\n$+1$ and $g_I$ has $Gl(1)$ charge $-1.$ The vertex operators are\nprimary if \\eqn{pric}{\\partial_I f^I=0,\\qquad Z^Ig_I=0.}  We\nidentify two vertex operators that differ by null states\n\\eqn{din}{\\delta V_f=J_{-1}\\Lambda=Y_IZ^I\\Lambda,\\qquad \\delta\nV_\\phi=T_{-1}\\chi=\\partial Z^I\\partial_I\\chi.} Hence, $f^I$ and\n$g_I$ are subject to the gauge invariance \\eqn{fgga}{\\delta\nf^I=Z^I\\Lambda,\\qquad \\delta g_I=\\partial_I \\chi.}\n\nSince $f^I$ has $Gl(1)$ charge $+1,$ we can construct $Gl(1)$\nneutral the vector field \\eqn{ups}{\\Upsilon=f^I{\\partial\\over\n\\partial Z^I}.}  $\\Upsilon$ descends to\na vector field on on the real twistor space $\\Bbb R^{3|4}$ thanks\nto the gauge invariance $\\delta f^I=Z^I\\Lambda$ that kills the\nvertical part of the vector field along the $Gl(1)$ orbits\n$Z^I\\rightarrow tZ^I.$ The primary condition $\\partial_I f^I=0$\nimplies that $\\Upsilon$ preserves the volume measure $\\Omega\\sim Z\nd^7Z$ on $\\Bbb{RP}^{3|4}.$ Hence $\\Upsilon$ is a volume preserving\nvector field on $\\Bbb{RP}^{3|4}$. Similarly, we can summarize the\nconditions on $g$ by considering the $1$ form\n\\eqn{oneg}{\\Theta=g_I dZ^I.} The constraint $g_I Z^I=0$ means that\n$\\Theta$ annihilates the vertical vector field\n$Z^I\\partial\/\\partial Z^I,$ so it descends to a one form on\n$\\Bbb{RP}^{3|4}.$ The gauge invariance $\\delta g_I=\\partial_I \\chi$\nmeans that $\\Theta$ is actually an abelian gauge field on\n$\\Bbb{RP}^{3|4}.$\n\n\\vskip .2in\\noindent{\\it Comparison with B-model}\n\nRecall that that B-model is defined on $\\Bbb{CP}'^{3|4}$. The open\nstrings correspond to gauge fields in Minkowski space and the\nclosed strings correspond to conformal supergravity. On the other\nhand in the open twistor string both gauge theory and conformal\nsupergravity states come from the open string vertex operators.\nThe boundary of the worldsheet (and hence the vertex operators)\nlives in $\\Bbb{RP}'^{3|4}$. Hence the twistor fields are related by\ntwistor transform to fields on spacetime with signature $++--.$\n\nThe gauge field is described in B-model by a $(0,1)$ form ${\\cal A}$\nthat is an element of $H^1(\\Bbb{CP}'^{3|4},{\\cal O})$. This has equations of\nmotion $\\overline\\partial {\\cal A}=0$ and gauge invariance\n$\\delta{\\cal A}=\\overline\\partial\\epsilon,$ where $\\epsilon$ is a function\non $\\Bbb{CP}'^{3|4}$. In open string, the gauge field comes from a\nfunction $\\phi$ on $\\Bbb{RP}'^{3|4}.$ If $\\phi$ is real-analytic, we\ncan extend it to a complex neighborhood of $\\Bbb{RP}'^{3|4}$ in\n$\\Bbb{CP}'^{3|4}$. Then the relation between the two fields is\n\\cite{Atiyah:1979iu,Berkovits:2004jj,Witten:2003nn}\n\\eqn{reat}{{\\cal A}=\\phi\\ \\overline\\partial(\\theta\\left({\\rm Im}\\ z\\right))=\n{i\\over 2} \\phi\\ \\delta\\left({\\rm Im}\\ z\\right) d\\overline z,} where\n$z=\\lambda^2\/\\lambda^1$ and $\\theta(x)=1$ for $x\\geq0$ and $0$ for\n$x<0.$\n\nThe B-model closed string field giving deformation of complex\nstructure $j^I=J^I{}_{\\overline J} dZ^J$ is related to the open string\nvolume preserving vector field $\\Upsilon=f^I\\ {\\partial\/\\partial\nZ^I}$ as $j^I= f^I\\ \\overline\\partial(\\theta\\left({\\rm Im}\\ z\n\\right)).$ Similarly, the RR-two form $b=B_{I\\overline J}\\ dZ^I\\wedge\ndZ^{\\overline J}$ gets related to the abelian gauge field $\\Theta=g_I\\\ndZ^I$ of the open string by $b=\\Theta\\\n\\overline\\partial(\\theta\\left({\\rm Im}\\ z \\right)).$\n\nHence, we get the open twistor string wavefunction by considering\n$\\lambda^a,\\mu^{\\dot a}$ real and  by replacing holomorphic delta\nfunctions $\\overline\\delta(\\vev{\\lambda,\\pi})$ with real delta functions\n\\eqn{bwa}{\\phi(\\lambda,\\mu,\\psi)=\\delta(\\vev{\\lambda,\\pi})\n\\exp\\left(i[\\tilde\\pi,\\mu]\\right) g(\\psi).}\n\n\n\\subsection{ Tree Level Yang-Mills Amplitudes}\n\nA tree level $n$ gluon scattering amplitude has contribution from\nworldsheet $D$ of disk topology. The gluon vertex operators are\ninserted along the boundary of the worldsheet. Taking the disk to\nbe the upper half-plane ${\\rm Im}\\ z\\geq 0$, we insert the vertex\noperators at $z_i,\\ {\\rm Im}\\ z_i=0.$ Hence, the scattering\namplitude is \\eqn{scatr}{{\\cal A}=\\sum_d\\int d{\\cal M}  \\vev{\\int dz_1\nV_\\phi(z_1) \\dots \\int dz_n V_\\phi(z_n)},} where the sum is over\n$U(1)$ worldsheet instantons and $d{\\cal M}$ is the measure.\n\nIn two dimensions, the instanton number of a $Gl(1)$ gauge bundle\nis the degree of the line bundle. Recall that the  bundle ${\\cal O}(d)$\nof degree $d$ homogeneous functions has degree $d$. Hence, on a\nworldsheet with instanton number $d$, $Z^I$'s are sections of\n${\\cal O}(d).$ But this is just the parametric description of an\nalgebraic curve of degree $d$ discussed in section \\ref{highd}.\nWhile in B-model we summed over D-instantons, in the open twistor\nstring we are summing over  worldsheet instantons. Both\ndescription lead to the same curves in twistor space. The only\ndifference is that for B-model we consider holomorphic curves,\nwhile here we are interested in real algebraic curves.\n\nThe discussion of the real case is entirely analogous to the\nholomorphic case.  Each $Z^I$ has $d+1$ real zero modes that are\nlocal coordinates on the moduli space\n${\\cal M}=\\Bbb{RP}^{4d+3|4d+4}\/Sl(2,\\Bbb R).$ The measure is just the\nholomorphic measure \\rf{holm} restricted to real curves. The\nmoduli space of degree $d$ instantons has $4d+4$ fermionic\ndimensions. Since negative helicity gluon gives $4$ zero modes and\npositive helicity gluon gives no zero modes, a degree $d$\ninstanton contributes to amplitudes with $d+1$ negative helicity\ngluons. Parameterizing the disk using $z,\\ {\\rm Im} z\\geq 0,$ the\namplitude is the real version of \\rf{wop}\n\\eqalign{rwop}{{\\cal A}&=&\\int d{\\cal M}_d \\prod_i\\int_D {dz_i \\over \\prod_k\n(z_k-z_{k+1})} \\delta(\\vev{\\lambda(z_i),\\pi_i}) \\exp\\left(\ni[\\mu(z_i),\\tilde\\pi_i]\\right) g_i(\\psi_i).} In\n\\cite{Berkovits:2004tx}, a cubic open string field theory was\nconstructed for the Berkovits's twistor string theory. Since the\ntwistor string field theory gives the correct cubic\nsuper-Yang-Mills vertices, it provides further support that\n\\rf{rwop} correctly computes tree-level Yang-Mills amplitudes.\n\n\n\n\\section{Recent Results in Perturbative Yang-Mills}\n\\label{perturb}\n\nIn this part of the lecture we shift gears and concentrate on new\ntechniques for the calculation of scattering amplitudes in gauge\ntheory. We will discuss two main results: BCFW recursion relations\n\\cite{Britto:2004ap, Britto:2005fq} for tree amplitudes of gluons\nand quadruple cuts of ${\\cal N}=4$ one-loop amplitudes of gluons\n\\cite{Britto:2004nc}.\n\n\n\\subsection{BCFW Recursion Relations}\n\nWe have seen how tree-level amplitudes of gluons can be computed\nin a simple and systematic manner by using MHV diagrams. However,\nfrom the study of infrared divergencies of one-loop ${\\cal N}=4$\namplitudes of gluons, surprisingly simple and compact forms for\nmany tree amplitudes were found in \\cite{Bern:2004ky,\nRoiban:2004ix}. These miraculously simple formulas were given an\nexplanation when a set of recursion relations for amplitudes of\ngluons was conjectured in \\cite{Britto:2004ap}. The\nBritto-Cachazo-Feng-Witten (BCFW) recursion relations were later\nproven and extended in \\cite{Britto:2005fq}. Here we review the\nBCFW proof of the general set of recursion relations. The reason\nwe choose to spend more time in the proof than in recursion\nrelation itself is that the proof is constructive and the same\nmethod can and has been applied to many other problems from field\ntheory to perhaps string theory.\n\nConsider a tree-level amplitude ${\\cal A} (1,2,\\ldots ,n-1,n)$ of $n$\ncyclically ordered gluons, with any specified helicities. Denote\nthe momentum of the $i^{th}$ gluon by $p_i$ and the corresponding\nspinors by $\\lambda_i$ and $\\tilde\\lambda_i$. Thus, $p_i^{a\\dot a}\n=\\lambda_i^a\\tilde\\lambda_i^{\\dot a}$, as usual in these lectures.\n\nIn what follows, we single out two of the gluons for special\ntreatment.  Using the cyclic symmetry, without any loss of\ngenerality, we can take these to be the gluons $k$ and $n$. We\nintroduce a complex variable $z$, and let\n\\eqalign{deftwo}{ p_{k}(z) & = \\lambda_{k}(\\tilde\\lambda_{k} -\nz\\tilde\\lambda_n), \\cr p_n(z) & = (\\lambda_n +\nz\\lambda_{k})\\tilde\\lambda_n. }  We leave the momenta of the other\ngluons unchanged, so $p_s(z)=p_s $ for $s\\not= k,n$. In effect, we\nhave made the transformation\n\\eqn{thetrans}{\\tilde\\lambda_k\\to\\tilde\\lambda_k-z\\tilde\\lambda_n,\\quad\n~~\\lambda_n\\to\\lambda_n+z\\lambda_k,}\nwith $\\lambda_k$ and $\\tilde\\lambda_n$ fixed. Note that $p_k(z)$\nand $p_n(z)$ are on-shell for all $z$, and $p_k(z)+p_n(z)$ is\nindependent of $z$. As a result, we can define the following\nfunction of a complex variable $z$,\n\\eqn{defz}{{\\cal A}(z) = {\\cal A}(p_1,\\ldots ,p_{k-1}, p_{k}(z),\np_{k+1},\\ldots ,p_{n-1}, p_n(z)). }\nThe right hand side is a physical, on-shell amplitude for all $z$.\nMomentum is conserved and all momenta are on-shell.\n\n\nFor any $z\\not=0$, the deformation \\rf{deftwo} does not make sense\nfor real momenta in Minkowski space, as it does not respect the\nMinkowski space reality condition $\\tilde\\lambda = \\pm\n\\overline\\lambda$. However, \\rf{deftwo} makes perfect sense for complex\nmomenta or  (if $z$ is real) for real momenta in signature\n$++-\\,-$.   In any case, we think of ${\\cal A}(z)$ as an auxiliary\nfunction. In the end, all answers are given in terms of spinor\ninner products and are valid for any signature.\n\nHere we assume that the helicities $(h_k,h_n)$ are $(-,+)$. The\nproof can be extended to helicities $(+,+)$, or $(-,-)$ but we\nrefer the reader to \\cite{Britto:2005fq}.\n\nWe claim three facts about ${\\cal A}(z)$: $(1)$ It is a rational\nfunction. $(2)$ It only has simple poles. $(3)$ It vanishes for\n$z\\to \\infty$.\n\nThese three properties of ${\\cal A}(z)$ imply that it can be written as\nfollows\n\\eqn{ratio}{{\\cal A}(z) = \\sum_{p\\in \\{ {\\rm poles}\\} }\n\\frac{c_p}{z-z_p},}\nwhere $c_p$ is the residue at a given pole and the sum is over the\nwhole set of poles. It turns out that, as we will see below, $c_p$\nis proportional to the product of two physical amplitudes with\nfewer gluons than ${\\cal A}(z)$. Therefore, \\rf{ratio} provides a\nrecursion relation for amplitudes of gluons.\n\nLet us prove the three statements. $(1)$ This is easy. Note that\nthe original tree-level amplitude is a rational function of spinor\nproducts. Since the $z$ dependence enters only via the shift\n$\\tilde\\lambda_k \\to \\tilde\\lambda_k - z\\tilde\\lambda_n$ and\n$\\lambda_n \\to \\lambda_n + z \\lambda_k$, ${\\cal A}(z)$ is clearly\nrational in $z$.\n\n$(2)$ By definition, ${\\cal A}(z)$ is constructed out of Feynman\ndiagrams. The only singularities ${\\cal A}(z)$ can have come from\npropagators. Recall that ${\\cal A}(z)$ is color-ordered. This means\nthat all propagators are of the form $1\/P^2_{ij}$ where\n$P_{ij}=p_i+\\ldots + p_j$. Clearly, $P_{ij}$ is $z$ independent if\nboth $k,n\\in \\{ i,\\ldots, j\\}$ or if $k,n\\not\\in \\{ i,\\ldots,\nj\\}$. By momentum conservation it is enough to consider\npropagators for which $n\\in \\{ i,\\ldots, j\\}$ and $k \\not\\in \\{\ni,\\ldots, j\\}$. Since the shift of $p_n$ is by a null vector, one\nhas\n\\eqn{porpi}{ P_{ij}^2(z)  = P^2_{ij}(0) - z\n\\langle\\lambda_k|P_{ij}|\\tilde \\lambda_n],}\nwhere for any spinors $\\lambda,\\tilde\\lambda$ and vector $p$, we\ndefine $\\langle \\lambda|p|\\tilde\\lambda]=- p_{a\\dot\na}\\lambda^a\\tilde\\lambda^{\\dot a}$. Hence, the propagator\n$1\/P_{ij}(z)^2$ has only a single, simple pole, which is located\nat $z_{ij}= P_{ij}^2\/\\langle\\lambda_k|P_{ij}|\\tilde\\lambda_n]$.\n\n$(3)$ Recall that any Feynman diagram contributing to the\namplitude ${\\cal A}(z)$ is linear in the polarization vectors\n$\\epsilon_{a\\dot a}$ of the external gluons. Polarization vectors\nof gluons of negative and positive helicity and momentum $p_{a\\dot\na}= \\lambda_a \\tilde\\lambda_{\\dot a}$ can be written respectively\nas follows (see section \\ref{spinors} ),\n\\eqn{poli}{\\epsilon_{a\\dot a}^{-} = {\\lambda_a\\tilde\\mu_{\\dot\na}\\over [\\tilde\\lambda , \\tilde\\mu ] }, \\qquad \\epsilon_{a\\dot\na}^{+} = {\\mu_a \\tilde\\lambda_{\\dot a}\\over \\vev{\\mu, \\lambda}},}\nwhere $\\mu$ and $\\tilde\\mu$ are fixed reference spinors.\n\nOnly the polarization vectors of gluons $k$ and $n$ can depend on\n$z$. Consider the $k^{th}$ gluon first. Recall that $\\lambda_k$\ndoes not depend on $z$ and $\\tilde\\lambda_k(z)$ is linear in $z$.\nSince $h_k=-1$, it follows from \\rf{poli} that $\\epsilon^{-}_k$\ngoes as $1\/z$ as $z\\to \\infty$. A similar argument leads to\n$\\epsilon^{+}_n \\sim 1\/z$ as $z\\to \\infty$.\n\nThe remaining pieces in a Feynman diagram are the propagators and\nvertices. It is clear that the vanishing of ${\\cal A}(z)$ as $z\\to\n\\infty$ can only be spoiled by the momenta from the cubic\nvertices, since the quartic vertices have no momentum factors and\nthe propagators are either constant or vanish for $z\\to\\infty$.\n\nLet us now construct the most dangerous class of graphs and show\nthat they vanish precisely as $1\/z$. The $z$ dependence in a tree\ndiagram ``flows\" from the $k^{th}$ gluon to the $n^{th}$ gluon\nalong a unique path of propagators. Each such propagator\ncontributes a factor of $1\/z$.  If there are $r$ such propagators,\nthe number of cubic vertices through which the $z$-dependent\nmomentum flows is at most $r+1$. (If all vertices are cubic, then\n starting from the $k^{th}$ gluon, we\nfind a cubic vertex and then a propagator, and so on. The final\ncubic vertex is then joined to the $n^{th}$ gluon.) So the\nvertices and propagators give a factor that grows for large $z$ at\nmost linearly in $z$.\n\nAs the product of polarization vectors vanishes as $1\/z^2$, it\nfollows  that for this helicity configuration, ${\\cal A}(z)$ vanishes\nas $1\/z$ for $z\\to \\infty$.\n\nNow we can rewrite \\rf{ratio} more precisely as follows\n\\eqn{collo}{{\\cal A}(z)=\\sum_{i,j}{c_{ij}\\over z-z_{ij}},}\nwhere $c_{ij}$ is the residue of ${\\cal A}(z)$ at the pole $z=z_{ij}$.\nFrom the above discussion, the sum over $i$ and $j$ runs over all\npairs such that $n$ is in the range from $i $ to $j$ while $k$ is\nnot. At this point it is clear the smallest number of poles is\nachieved when $k$ and $n$ are adjacent, i.e., $k=n-1$. This is the\nchoice we make in the examples below.\n\nFinally, we have to compute the residues $c_{ij}$. To get a pole\nat $P_{ij}^2(z)=0$, a tree diagram must contain a propagator that\ndivides it into a ``left'' part containing all external gluons not\nin the range from $i$ to $j$, and a ``right'' part containing all\nexternal gluons that are in that range. The internal line\nconnecting the two parts of the diagram has momentum $P_{ij}(z)$,\nand we need to sum over the helicity $h=\\pm$ at, say, the left of\nthis line. (The helicity at the other end is opposite.)  The\ncontribution of such diagrams near $z=z_{ij}$ is $\\sum_h\n{\\cal A}_L^h(z){\\cal A}_R^{-h}(z)\/P_{ij}(z)^2$, where ${\\cal A}_L^h(z)$ and\n${\\cal A}_R^{-h}(z)$ are the amplitudes on the left and the right with\nindicated helicities. Since the denominator $P_{ij}(z)^2$ is\nlinear in $z$, to obtain the function $c_{ij}\/(z-z_{ij})$ that\nappears in \\rf{collo}, we must simply set $z$ equal to $z_{ij}$ in\nthe numerator. When we do this, the internal line becomes\non-shell, and the numerator becomes a product\n${\\cal A}_L^h(z_{ij}){\\cal A}_R^{-h}(z_{ij})$ of physical, on-shell\nscattering amplitudes. More precisely we have,\n\\eqn{preci}{{\\cal A}_L^h(z_{ij}) = {\\cal A}(p_{j+1},\\ldots\n,p_k(z_{ij}),\\ldots , p_{i-1}, P^{h}_{ij}(z_{ij})), \\quad\n{\\cal A}_R^{-h}(z_{ij}) = {\\cal A}(-P^{-h}_{ij}(z_{ij}),p_i,\\ldots,\np_n(z_{ij}),\\ldots , p_j). }\n\nThe formula \\rf{collo} for the function ${\\cal A}(z)$ therefore becomes\n\\eqn{gollo}{{\\cal A}(z)=\\sum_{i,j}\\sum_h{{\\cal A}_L^h(z_{ij}){\\cal A}_R^{-h}(z_{ij})\\over\nP_{ij}(z)^2}.} To get the physical scattering amplitude\n${\\cal A}(1,2,\\ldots, n-1,n)$, we set $z$ to zero in the denominator\nwithout touching the numerator. Hence,\n\\eqn{wollo}{{\\cal A}(1,2,\\ldots,\nn-1,n)=\\sum_{i,j}\\sum_h{{\\cal A}_L^h(z_{ij}){\\cal A}_R^{-h}(z_{ij})\\over\nP_{ij}^2}.}\nThis is the BCFW recursion relation\n\\cite{Britto:2004ap,Britto:2005fq}.\n\n\\subsubsection{Examples}\n\nLet us illustrate some of the compact formulas one can obtain\nusing the recursion relations \\rf{wollo}.\n\nConsider two of the six-gluon next-to-MHV amplitudes, for example,\namplitudes with three minus and three plus helicity gluons: $\n{\\cal A}(1^-,2^-,3^-,4^+,5^+,6^+)$ and ${\\cal A}(1^+,2^-,3^+,4^-,5^+,6^-)$.\nAs mentioned above, the recursion relations \\rf{wollo} have the\nsmallest number of terms when $k$ and $n$ are chosen to be\nadjacent gluons. In the first example we choose to shift $p_3$ and\n$p_4$, while in the second we shift $p_2$ and $p_3$. The results\nare the following:\n\\eqn{lepta}{ {\\cal A}(1^-,2^-,3^-,4^+,5^+,6^-) = {1\\over \\gb{ 5 | 3+4 |\n2}}\\left( {\\gb{ 1 | 2+3 | 4}^3 \\over [2~3][3~4]\\vev{5~6}\\vev{6~1}\nt_2^{[3]}} + {\\gb{ 3 | 4+5 | 6}^3 \\over\n[6~1][1~2]\\vev{3~4}\\vev{4~5} t_3^{[3]} }\\right).}\n\\eqalign{alos}{ {\\cal A}(1^+,2^-,3^+,4^-,5^+,6^-) &=& \\frac{ [1~3]^4\n\\vev{4~6}^4 }{ [1~2] [2~3] \\vev{4~5} \\vev{5~6} t_{1}^{[3]}\n\\gb{6|1+2|3} \\gb{4|2+3|1}} \\\\ &+& \\frac{ \\vev{2~6}^4 [3~5]^4 }{\n\\vev{6~1}\\vev{1~2} [3~4] [4~5] t_{3}^{[3]} \\gb{6|4+5|3}\n\\gb{2|3+4|5}} \\\\ &+& \\frac{ [1~5]^4 \\vev{2~4}^4 }{\n\\vev{2~3}\\vev{3~4} [5~6] [6~1] t_{2}^{[3]} \\gb{4|2+3|1}\n\\gb{2|3+4|5}},}\nwhere $t_i^{[r]} = p_i + \\ldots + p_{i+r-1}$.\n\nIt is interesting to observe that while \\rf{lepta} and \\rf{alos}\nare simpler than the amplitudes computed by Berends, Giele,\nMangano, Parke, Xu \\cite{Berends:1987me, Mangano:1987xk,\nBerends:1989hf, Mangano:1990by}; the former possess spurious\npoles, like $\\gb{ 5 | 3+4 | 2}$, while the latter only have\nphysical poles. One can use the recursion relations to find\nfurther simple formulas for tree-level gluon amplitudes\n\\cite{Britto:2005dg}.\n\nAlso note that the two-term form \\rf{lepta} was obtained in\n\\cite{Roiban:2004ix} as a collinear limit of a very compact form\nof the seven-gluon amplitude, which was originally obtained from\nthe infrared behavior of a one-loop ${\\cal N}=4$ amplitude\n\\cite{Bern:2004ky}.\n\nLet us also mention that many generalizations of the BCFW\nrecursion relations have been made, in particular, to include\namplitudes with fermions and scalars \\cite{Luo:2005rx,Luo:2005my}\nand to gravity amplitudes \\cite{Bedford:2005yy, Cachazo:2005ca}.\nThe recursion relations have also been generalized to amplitudes\nwith massive particles \\cite{Badger:2005zh}, and to some one-loop\namplitudes in QCD \\cite{Bern:2005hs}.\n\n\n\\subsection{One-Loop ${\\cal N}=4$ Amplitudes of Gluons and Quadruple Cuts}\n\nSupersymmetric amplitudes of gluons are very special. The main\nreason is that these amplitudes are four-dimensional\ncut-constructible. This means that a complete knowledge of their\nbranch cuts and discontinuities, when the dimensional\nregularization parameter is taken to zero, is enough to determine\nthe full amplitude. This is not true for non-supersymmetric\namplitudes. As an example consider the one-loop\n${\\cal A}(1^+,2^+,\\ldots , n^+)$ and ${\\cal A}(1^-,2^+,\\ldots , n^+)$\namplitudes. One can prove that both series of amplitudes are\nsingle valued functions of the kinematical invariants. This is\nenough to conclude that they vanish in any supersymmetric\ntheory\\footnote{This can also be derived using supersymmetric Ward\nidentities. For a nice review see \\cite{Dixon:1996wi}.}. In\ncontrast, in non-supersymmetric gauge theories, they are\ninteresting rational functions. These two series of amplitudes\nwere shown to be reproduced by a generalization of the BCFW\nrecursion relations in \\cite{Bern:2005hs}.\n\nHere we concentrate on ${\\cal N}=4$ one-loop amplitudes. The\nreason these amplitudes are special within the class of\nsupersymmetric amplitudes is that they can be expressed in terms\nof known scalar box integrals with coefficients that are rational\nfunctions of the spinor products.\n\nIn this part of the lectures, we explain a new technique that\nallows the computation of any given scalar box coefficient as the\nproduct of four tree-level amplitudes. Now recall that either by\nusing MHV diagrams or the BCFW recursion relations\\footnote{We\nalso need the corresponding generalizations to include fermions\nand scalars \\cite{Georgiou:2004by, Wu:2004fb, Wu:2004jx,\nGeorgiou:2004wu}.}, any tree-level amplitude can be easily\ncomputed. This implies that the new technique solves the problem\nof computing one-loop amplitudes of gluons in ${\\cal N}=4$ super\nYang-Mills.\n\n\n\\subsubsection{Review of The Unitarity-Based Method}\n\nOne of the most successful methods in the calculation of one-loop\namplitudes of gluons is the unitarity-based method\n\\cite{Bern:1994zx, Bern:1994cg}. This method was used to calculate\nall MHV amplitudes \\cite{Bern:1994zx} and all six-gluon\nnext-to-MHV amplitudes \\cite{Bern:1994cg} more than a decade ago.\nWe review the basic idea of the method focusing on the points that\nprepare the ground for the quadruple cut method.\n\nThe unitarity-based method can be described as a three-step\nprocedure: (1) Consider a given amplitude and use\nPassarino-Veltman or other reduction techniques \\cite{passarino}\nto find a set of basic integrals. In supersymmetric amplitudes of\ngluons, this means that any tensor Feynman integrals that enters\nin a Feynman diagram calculation can be reduced to a set of scalar\nintegrals, that is Feynman integrals in a scalar field theory with\na massless particle running in the loop, with rational\ncoefficients. In particular, for ${\\cal N}=4$ super Yang-Mills,\nonly scalar box integrals appear.\n\nScalar box integrals are defined as follows,\n\\eqn{ifou}{I_{(K_1,K_2,K_3,K_4)} = \\int d^4\\ell { 1\\over\n(\\ell^2+i\\epsilon) ((\\ell-K_1)^2 + i\\epsilon )((\\ell -K_1-K_2)^2 +\ni\\epsilon )((\\ell+K_4)^2 + i\\epsilon )}.}\nThis is really a function of only three momenta $K_1, K_2, K_3$,\nfor $K_4=-K_1-K_2-K_3$ by momentum conservation. This integral is\nUV finite but it has IR divergencies when at least one $K_i$ is\nnull, i.e., $K_i^2=0$. This implies that a regularization\nprocedure, like dimensional regularization, is required. The\nstructure of the IR singular terms is well understood\n\\cite{Catani:1998bh}. We do not discuss it here because it is not\nrelevant for the quadruple cut technique.\n\nIn a given amplitude, $K_i$ is the sum of consecutive momenta of\nexternal gluons. We discuss this in more detail below.\n\n(2) Consider a unitarity cut in a given channel, say the\n$s-$channel. Recall that this is defined by summing over all\nFeynman diagrams that contain two propagators whose momenta differ\nby $s$ and by cutting those two propagators. Cutting a propagator\n$1\/(P^2+i\\epsilon)$ means removing the principal part, i.e.,\nreplacing the propagator by $\\delta^{(+)} (P^2)$. When this is\ndone, the internal particles go on-shell and the sum over Feynman\ndiagrams produces two tree-level amplitudes while the integration\nover the internal momenta becomes an integration over the Lorentz\ninvariant phase space of two null vectors; this is known as a cut\nintegral. As an example, consider the cut in the\n$P_{ij}^2$-channel, see figure \\rf{uni}, the cut integral is given\nby \\cite{cuts}\n\\eqn{cutii}{C = \\int d\\mu ~ {\\cal A}^{\\rm tree} (\\ell_1,i,...,j,\n\\ell_2) ~ {\\cal A}^{\\rm tree}(-\\ell_2,j+1,..., i-1,-\\ell_1),}\nwhere $d\\mu$ is the Lorentz invariant phase space measure for\n$(\\ell_1,\\ell_2)$. The measure is explicitly given by\n\\eqn{mme}{ d\\mu = d^4\\ell_1 d^4\\ell_2 \\delta^{(+)} (\\ell_1^2)\n\\delta^{(+)} (\\ell_2^2) \\delta^{(4)}(\\ell_1 + \\ell_2 - P_{ij}),}\nwith $P_{ij}$ denoting the sum of the momenta of gluons from $i$\nto $j$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[height=1.7in]{GeneralCut.eps}\n\\caption{Unitarity cut in the $P_{ij}^2-$channel.\nThe blobs represent tree-level amplitudes in which the propagator\nlines are interpreted as external on-shell particles.} \\label{uni}\n\\end{figure}\n\n(3) Use reduction techniques to write the integrand of \\rf{cutii}\nas a sum of terms that contain a constant coefficient times two\npropagators. Once this is achieved, it is easy to construct a\nfunction of scalar box integrals with given coefficients that has\nsuch a cut. Then repeat this for all other cuts, remembering that\na given scalar box integral has cuts in several different\nchannels. This means that one should not just add the functions\nobtained from the study of each channel. Instead one has to\ncombine them while avoiding to overcount. Once a function with all\nthe correct discontinuities has been constructed, this must be the\nfinal answer for the amplitude. The reason is that supersymmetric\namplitudes are four-dimensional cut-constructible, as mentioned\nabove.\n\nUsing this technique, all MHV amplitudes and the six-gluon NMHV\namplitudes were computed more than ten years ago. More recently,\nthe seven-gluon NMHV amplitude with all minus helicity gluons\nadjacent was computed by using a combination of this method and\nthe holomorphic anomaly of unitarity cuts \\cite{Cachazo:2004by} in\n\\cite{Cachazo:2004dr, Britto:2004nj}. The same result as well as\nall other helicity configurations for the seven-gluon amplitude\nwere obtained by the unitarity-based method in \\cite{Bern:2004ky}.\n\nAt this point it is important to mention that the integrand in the\ncut integral \\rf{cutii} is complicated because in general there\nare many scalar box integrals sharing the same branch cut. The\nreduction techniques, though systematic, can lead to very large\nexpressions for the scalar box coefficients \\cite{Bern:2004ky}.\nThese large expressions can be shown to be equivalent to simple\nformulas obtained as educated guesses \\cite{Bern:2004ky}. This is\na hint that there must be a more direct method for computing such\ncoefficients.\n\nA related difficulty comes from the fact that a given scalar box\nintegral has many different branch cuts. This means that after its\ncoefficient has been computed from a given cut, one still has to\ndisentangle it from other unknown coefficients over and over again\nin the other cuts. This somehow reduces the efficiency of the\nmethod.\n\nOne way to improve the situation is by cutting three propagators\n\\cite{Bern:2004ky}\\cite{Bern:2004bt}. Note that triple cuts where\na single gluon in trapped in between two cut propagators vanish.\nThis would correspond to a cut in a one-particle channel. In the\nnext part of this lecture we will reconsider this issue.\n\nNote that the number of scalar boxes with a given triple cut is\nless than that with a given unitarity cut. However, in general one\nstill has to apply reduction techniques. A class of amplitudes for\nwhich triple cuts are very suitable are next-to-MHV (NMHV)\namplitudes \\cite{Bern:2004bt}. But, one might expect that this\nprocedure becomes cumbersome already for NNMHV amplitudes.\n\nIt turns out that there is a way of avoiding the reduction\ntechniques as well as the recalculation of known coefficients.\nThis is achieved by considering quadruple cuts\n\\cite{Britto:2004nc} which we now discuss.\n\n\\subsubsection{Quadruple Cuts}\n\nConsider a scalar one-loop Feynman integral, $I$. The integral $I$\nis a function of the kinematical invariants constructed out of the\nexternal momenta. In general, $I$ is a complicated multi-valued\nfunction with branch cuts that are like domain walls in the space\nof kinematical invariants, $\\Sigma$. As it is well known, cutting\ntwo propagators in the loop computes the imaginary part of the\nintegral in a certain region of $\\Sigma$. This imaginary part of\n$I$ can be thought of as the discontinuity of $I$ across the\nbranch cut of interest.\n\nNow consider unitarity cuts in several possible channels. One can\nask what is the discontinuity across the intersection of two or\nmore cuts. The answer is given by the union of the set of cut\npropagators! Of particular interest to us is the meaning of\ncutting all propagators in a one-loop integral; such a cut\nintegral computes the discontinuity across the singularity of\nhighest codimension, which is known as the leading singularity.\nFor a more extensive discussion and references see\n\\cite{Eden:1966}\\footnote{In \\cite{Eden:1966}, the arguments are\nmade for a massive scalar field theory. However, it turns out that\nthe relevant results for our discussion can be used in massless\ntheories with little modifications.}.\n\nAs mentioned above, ${\\cal N}=4$ one-loop amplitudes of gluons can\nbe written as a linear combination of scalar box integrals with\nrational coefficients. The scalar box integrals can be thought of\nas a ``basis of vectors\" in some sort of vector space. The idea is\nthat this basis is in some appropriate sense\northogonal\\footnote{To push the analogy even further, one can\nthink of the scalar box functions defined in \\cite{Bern:1994zx},\nwhich are scalar box integrals nicely normalized, as an\northonormal basis!} (In less supersymmetric theories there also\nare bubble and triangle integrals which break the orthogonality\ncondition).\n\nThe one-loop amplitude ${\\cal A}_n^{\\rm 1-loop}$ can now be interpreted\nas a general vector which can be written as a linear combination\nof the basis. All we need is the appropriate way of projecting the\n``vector\" ${\\cal A}_n^{\\rm 1-loop}$ onto a given vector $I$ in order to\ncompute the corresponding coefficient.\n\n{}From our discussion, it is clear that the natural way of doing\nthis is to consider ${\\cal A}_n^{\\rm 1-loop}$ in the region near the\nleading singularity of $I$, which is unique to $I$. The\ndiscontinuity of ${\\cal A}_n^{\\rm 1-loop}$ across such a singularity is\nthe coefficient of $I$, up to a normalization, which is the analog\nof the norm of $I$.\n\nLet us see how this works in practice. Recall that the scalar box\nintegral \\rf{ifou} is,\n\\eqn{ifout}{I_{(K_1,K_2,K_3)} = \\int d^4\\ell { 1\\over\n(\\ell^2+i\\epsilon) ((\\ell-K_1)^2 + i\\epsilon )((\\ell -K_1-K_2)^2 +\ni\\epsilon )((\\ell+K_4)^2 + i\\epsilon )}.}\nIn the expansion of ${\\cal A}^{\\rm 1-loop}_n$, each $K_i$ in \\rf{ifout}\nis the sum of the momenta of consecutive external gluons.\n\nThe discontinuity across the leading singularity $\\Delta_{LS}$ is\ncomputed by cutting all four propagators. This is called a\nquadruple cut:\n\\eqn{wpuvv}{\\Delta_{LS} I_{(K_1,K_2,K_3)} = \\int d^4\\ell\n~\\delta^{(+)}(\\ell^2) ~\\delta^{(+)}((\\ell-K_1)^2)\n~\\delta^{(+)}((\\ell -K_1-K_2)^2)~\\delta^{(+)}((\\ell+K_4)^2).}\n\nIn order to make the discussion more explicit we introduce\nnotation for the coefficients of $I$ in the expansion of\n${\\cal A}_n^{\\rm 1-loop}$ as follows:\n \\eqn{ampli}{ {\\cal A}_n^{\\rm 1-loop} =\n\\sum_{1<i<j<k<m<n} B_{ijkm} I_{(p_{i+1}+\\ldots + p_{j},\np_{j+1}+\\ldots +p_{k}, p_{k+1}+\\ldots +p_{m})}, }\nwhere the coefficients $B_{ijkm}$ are rational functions of the\nspinor products, as mentioned above.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[height=2.in]{quadruple.eps}\n\\caption{A quadruple cut diagram. Momenta in the cut propagators\nflows clockwise and external momenta are taken outgoing.  The\ntree-level amplitude ${\\cal A}^{\\rm tree}_1$, for example, has external\nmomenta $(-\\ell_1, i+1,...,j,\\ell_2)$.} \\label{druple}\n\\end{figure}\n\nBy cutting four given propagators in all possible Feynman diagrams\ncontributing to ${\\cal A}_n^{\\rm 1-loop}$ one finds the product of four\ntree-level amplitudes integrated over the Lorentz invariant phase\nspace of four null vectors.\n\nComparing the two sides of \\rf{ampli} we find\n\\eqn{kasa}{\\int d\\mu {\\cal A}^{\\rm tree}_1{\\cal A}^{\\rm tree}_2{\\cal A}^{\\rm\ntree}_3{\\cal A}^{\\rm tree}_4 = B_{ijkm} \\int d^4\\ell\n~\\delta^{(+)}(\\ell^2) ~\\delta^{(+)}((\\ell-K_1)^2)\n~\\delta^{(+)}((\\ell -K_1-K_2)^2)~\\delta^{(+)}((\\ell+K_4)^2)}\nwhere the measure is the same one both sides of the integrals,\n\\eqn{qeqi}{d\\mu = d^4\\ell ~\\delta^{(+)}(\\ell^2)\n~\\delta^{(+)}((\\ell-K_1)^2) ~\\delta^{(+)}((\\ell\n-K_1-K_2)^2)~\\delta^{(+)}((\\ell+K_4)^2),}\nand the tree-level amplitudes are defined as follows (see figure\n\\rf{druple})\n\\eqn{treeam}{\\begin{array}{rlrl} {\\cal A}_1^{\\rm tree} = &\n{\\cal A}(-\\ell_1,i+1,i+2,\\ldots ,j-1,j,\\ell_2 ), & \\quad {\\cal A}_2^{\\rm\ntree} = & {\\cal A}(-\\ell_2 , j+1,j+2,\\ldots ,k-1, k, \\ell_3), \\\\\n{\\cal A}_3^{\\rm tree} = & {\\cal A}(-\\ell_3, k+1,k+2,\\ldots , m-1,m,\\ell_4),\n& \\quad {\\cal A}_4^{\\rm tree} = & {\\cal A}(-\\ell_4, m+1,m+2,\\ldots ,i-1,i,\n\\ell_1).\n\\end{array} }\n\nIn general one might expect that four delta functions localize the\nintegral producing a Jacobian which is common to both sides of\n\\rf{kasa} and cancels out to give\n\\eqn{cool}{ B_{ijkl} = {1\\over |{\\cal S}|}\\sum_{\\cal S}{\\cal A}^{\\rm\ntree}_1{\\cal A}^{\\rm tree}_2{\\cal A}^{\\rm tree}_3{\\cal A}^{\\rm tree}_4. }\nHere ${\\cal S}$ is the set of solutions to the conditions imposed\nby the delta functions, and $|{\\cal S} |$ is the number of\nsolutions.\n\nThe derivation of the formula for the coefficients \\rf{cool}\nassumes that the Jacobian is a smooth function and that it does\nnot vanish for generic momenta of the external gluons and that it\nis the same for all solutions ${\\cal S}$ \\cite{Britto:2004nj}.\n\nIt turns out that both assumptions are not valid if at least one\nof the momenta $K_i$ in the box integral is null, i.e., if\n$K_i^2=0$ for some $i$. This is where the problem of defining a\ncut in a one-particle channel, which we mentioned in the\ndiscussion of triple cuts, comes back again.\n\nIt is not difficult to see that by using $\\delta^{(+)}(\\ell^2)$ to\nreduce the integration over arbitrary $\\ell$'s to those lying in\nthe future light-cone and by using that, say, $K_1^2=0$ one finds\nthat two of the three remaining delta functions are enough to\nlocalize the integral. The last delta function, that can be\nthought of as part of the Jacobian, imposes an extra constraint on\nthe external momenta beyond momentum conservation. Therefore, this\nmakes our two assumptions fail.\n\nThis problem of defining a cut in a one-particle channel is the\nfamiliar statement that a gluon cannot decay into two gluons. In\nother words, the tree-level amplitude ${\\cal A}_1^{\\rm tree}$ in\n\\rf{kasa} vanishes.\n\nIt turns out that the way out of both problems is the same.\nConsider a Wick rotation of \\rf{kasa} in to $--++$\nsignature\\footnote{In some sense it is even more natural to\ncomplexify all momenta.}. In this case one needs all four delta\nfunctions in order to localize the integral. The integration can\nbe done and produces a smooth and generically nonzero Jacobian.\nThe reason for this will be clear shortly.\n\nIt remains to see what happens to ${\\cal A}_1^{\\rm tree}$. If it is\nstill zero it would imply that all coefficients with $K_1^2=0$ are\nzero. This is known to be false in MHV and NMHV amplitudes.\n\nLet us look more closely at the tree-level amplitude ${\\cal A}_1$ and\nthe delta function containing only $K_1$. There are two cases,\n\\eqalign{cases}{{\\cal A}_1(K_1^+,\\ell^+,(\\ell-K_1)^-) &=& {[K_1,~\n\\ell]^3\\over [K_1,~\\ell-K_1][\\ell-K_1, ~\\ell]},\\\\\n{\\cal A}_2(K_1^-,\\ell^-,(\\ell-K_1)^+) &=& {\\vev{K_1,~ \\ell}^3\\over\n\\vev{K_1,~\\ell-K_1}\\vev{\\ell-K_1, ~\\ell}}.}\nThe delta function is given by $\\delta((\\ell-K_1)^2)$. This\nimplies that $\\vev{\\ell,~K_1}[\\ell,~K_1]=0$. As reviewed in\nsection 2, in Minkowski space with real momenta $\\lambda$ and\n$\\tilde\\lambda$ are complex but not independent, i.e.,\n$\\tilde\\lambda = \\pm\\overline\\lambda$ and therefore the only solution\nis $\\vev{\\ell,~K_1} = [\\ell,~K_1] = 0$. On the other hand, in\n$--++$ signature, $\\lambda$ and $\\tilde\\lambda$ are real and\nindependent, therefore we can have $\\vev{\\ell,~K_1} = 0$ while\n$[\\ell,~K_1] \\neq 0$ or vice versa. This is also the reason why\nfour delta functions are required to localize the integral\n\\rf{wpuvv}.\n\nThis explains how the problem is completely solved. When summing\nover the set of solutions ${\\cal S}$, one must take into account\nthe two possibilities, $\\vev{\\ell,~K_1} = 0$ or $[\\ell,~K_1] = 0$.\nOne of them makes ${\\cal A}_1^{\\rm tree}$ vanish while the other does\nnot. Actually, the presence of two solutions is important even in\nthe case when no $K_i$ is a null vector. The reason is that each\nsolution produces a function with a square root. However, the\ncoefficient must be a rational function. The resolution to this\nlittle puzzle is that by adding the two solutions, which differ\nonly in the branch the square root takes, one always produces a\nrational function.\n\nOne can easily see that in general there are only two solutions to\nthe delta function constraints. The two solutions can be found\nexplicitly in full generality; we refer the reader to\n\\cite{Britto:2004nj} for the actual formula. This implies that\n$|{\\cal S}| = 2$. Using this in \\rf{cool} we find a formula for\nall one-loop ${\\cal N} =4$ amplitude coefficients in terms of\ntree-level amplitudes,\n\\eqn{fine}{ B_{ijkl} = {1\\over 2}\\sum_{h,\\cal S}{\\cal A}^{\\rm\ntree}_1{\\cal A}^{\\rm tree}_2{\\cal A}^{\\rm tree}_3{\\cal A}^{\\rm tree}_4.}\nThe sum on the right hand side of \\rf{fine} is over the two\nsolutions ${\\cal S}$ and over all internal particles in the ${\\cal\nN}=4$ supermultiplet\\footnote{This method has been generalized to\none-loop amplitudes in ${\\cal N}=8$ supergravity in\n\\cite{Bern:2005bb,Bjerrum-Bohr:2005xx}.}.\n\n\\subsubsection{Examples}\n\nAs a simple example consider the coefficient of\n$I_{(3+4,5+6,7+1)}$ in ${\\cal A}(1^-,2^-,3^-,4^+,5^+,6^+,7^+)$.\n\nIn this case, only one internal helicity configuration gives a non\nzero contribution and it allows only gluons to run in the loop.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[height=2.in]{ThreeMass.eps}\n\\caption{$(a)$ Scalar Box Integral $I_{3+4,5+6,7+1}$. $(b)$\nQuadruple cut diagram of ${\\cal A}_7^{\\rm 1-loop}$ corresponding to\n$I_{3+4,5+6,7+1}$. Blobs represent tree-level amplitudes.}\n\\label{threemass}\n\\end{figure}\n\nUsing \\rf{fine} we find\n\\eqn{exco}{\n B_{3572} = \\frac{1}{2} { [\\ell_1 ~\\ell_4]^3\\over\n[\\ell_1~2][2~\\ell_4] }   { [4~\\ell_2]^3 \\over\n[\\ell_2~\\ell_1][\\ell_1~3][3~4] }   {[5~6]^3\\over [6~\\ell_3]\n[\\ell_3~\\ell_2] [\\ell_2 ~ 5] }   {[\\ell_3 ~ 7]^3 \\over [7~1] [1~\n\\ell_4 ] [\\ell_4 ~ \\ell_3 ] } }\n\nAfter solving the equations for $\\ell_i$ and plugging in the\nanswer in \\rf{exco} one finds a simple expression for $B_{3572}$\n\\cite{Britto:2004nj, Bern:2004ky}\n\\eqn{anwi}{-{ \\vev{1~2}^3 \\vev{2~3}^3 [5~6]^3 \\over \\vev{7~1}\n\\vev{3~4} \\gb{2 | 3+4| 5} \\gb{ 2| 7+1|6} ( \\vev{7~1} \\gb{ 2| 3+4\n|1} - t_2^{[3]} \\vev{7~2}) ( t_7^{[2]}\\vev{2~4} -\\vev{3~4} \\gb{\n2|7+1| 3})}.}\n\n\\pagebreak\n\\begin{center}\n{\\bf Acknowledgements}\n\\end{center}\n\nIt is pleasure to thank R. Britto, B. Feng for proof-reading parts\nof the manuscript and especially to E. Witten for giving us a\npermission to base sections 2-4 of the lecture notes on his\nlectures given at PITP, IAS Summer 2004. Work of F. Cachazo was\nsupported in part by the Martin A. and Helen Chooljian Membership\nat the Institute for Advanced Study and by DOE grant\nDE-FG02-90ER40542 and that of P. Svr\\v{c}ek in part by Princeton\nUniversity Centennial Fellowship and by NSF grants PHY-9802484 and\nPHY-0243680. Opinions and conclusions expressed here are those of\nthe authors and do not necessarily reflect the views of funding\nagencies.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Background}\n\nIn the seminal paper \\cite{SaZy} Salem and Zygmund studied the distribution of a random trigonometric polynomial \n$$\nQ_N(\\theta) = \\frac{1}{\\sqrt{N}} \\sum_{n=1}^N r_n e(n\\theta).\n$$\nHere the coefficients $r_n$ are taken to be independent random variables; for concreteness take $r_n$ independent and identically distributed taking the values $+1$ and $-1$ with probability $1\/2$ for all $n$. Among other results, Salem and Zygmund demonstrated that\n\\begin{enumerate}[(i)]\n\\item if $\\theta$ is drawn randomly at uniform from the interval $(0,1)$, then $Q_N(\\theta)$ converges in distribution (and in moments) to a standard complex Gaussian, and\n\\item almost surely, we have for sufficiently large $N$,\n$$\n\\max_{\\theta}|Q_N(\\theta)| \\asymp \\sqrt{\\log N}.\n$$ \n\\end{enumerate}\nIn fact Halasz \\cite{Ha} showed that the estimate in (ii) can be replaced with an asymptotic formula: almost surely $\\max_\\theta |Q_n(\\theta)| \\sim \\sqrt{\\log N}$. These results have stimulated a great deal of subsequent work, for instance \\cite{angst_poly,borwein_lockhart,kahane,weber00,weber06}.\n\nOur purpose in this note is to investigate what happens when independence between coefficients is replaced by a type of weak but non-local dependence. Instead of the i.i.d. sequence of Salem and Zygmund, we consider coefficients given by a random multiplicative function. In this case it is not clear that we should expect the same asymptotic behavior as found by Salem and Zygmund. \n\nRecall that a function $f: \\mathbb{N} \\rightarrow \\mathbb{C}$ is said to be \\emph{multiplicative} if it satisfies $f(mn) = f(m)f(n)$ for all coprime $m,n \\in \\mathbb{N}$; if this holds for all $m,n$ the function is said to be \\emph{completely multiplicative}. We will slightly abuse terminology and say that a function defined only on an interval $[1,N]$ is multiplicative or completely multiplicative if this relation holds for all $m,n \\in [1,N]$ with $mn \\in [1,N]$.\n\n\\subsection{Polynomials with non-random multiplicative coefficients}\n\nThe Fekete polynomials are perhaps the most famous example of a trigonometric polynomial for which the coefficients are completely multiplicative and take the value $\\pm 1$. For primes $p$ they are the $p-1$ degree polynomial defined by\n$$\nF_p(\\theta) = \\frac{1}{\\sqrt{p-1}}\\sum_{n=1}^{p-1} \\left(\\frac{n}{p}\\right) e(n\\theta),\n$$\nwhere $\\left(\\frac{\\cdot}{p}\\right)$ is the Legendre symbol. Recently, building upon work in \\cite{HoJe,BoCh}, G\\\"unther and Schmidt \\cite{GuSc} established the $p\\rightarrow\\infty$ limiting value of moments \n$$\n\\int_0^1 \\left| F_p(\\theta) \\right|^{2k}\\,d\\theta,\n$$\nfor all fixed integers $k$. The moments are \\emph{not} Gaussian. Earlier, Montgomery \\cite{Mo} has shown that $$\\log \\log p \\ll \\max_\\theta |F_p(\\theta)| \\ll \\log p,$$ and conjectured that \n$$\n\\max_\\theta |F_p(\\theta)| \\asymp \\log \\log p.\n$$\n\nBy contrast, if $\\lambda(n)$ is the Liouville function, then the polynomials defined by \n$$\nL_N(\\theta) = \\frac{1}{\\sqrt{N}} \\sum_{n=1}^N \\lambda(n) e(n\\theta),\n$$\narise naturally in the circle method and other parts of number theory (though it is perhaps more common to have the M\\\"obius function in place of the Liouville function, see for instance \\cite{davenport}). These polynomials also have coefficients which are completely multiplicative and take the value $\\pm 1$. Numerical evidence in this case suggests that $L_N(\\theta)$ does indeed tend to a Gaussian distribution as $N\\rightarrow\\infty$ (so that in particular the polynomial's $L^1$-norm, investigated in \\cite{balog_perelli,balog_rusza}, will be of constant order) and $\\max_\\theta |L_N(\\theta)| \\asymp \\sqrt{\\log N}$, though it would seem that both of these claims lie far outside the reach of what can now be proven.\n\n\\subsection{Main results}\n\nThe known and conjectured results for polynomials with non-random multiplicative coefficients that we have just described leave us wondering what behavior we should expect to be typical for random coefficients.\n\nIn order to investigate this, we consider random multiplicative functions of the following sort: if for each prime $p$, $X(p)$ is an independent random variable taking the values $+1$ and $-1$ with probability $1\/2$ each, and if for composite $n = p_1^{e_1}...p_k^{e_k}$ we set $X(n) = X(p_1)^{e_1}\\cdots X(p_k)^{e_k}$, we say that $\\{X(n)\\}_{n\\geq 1}$ is a \\emph{Rademacher random multiplicative function} (Some authors restrict $n$ to be squarefree, setting $X(n) = 0$ otherwise, but we will not follow this convention here). If instead $X(p)$ is uniformly distributed on the complex unit circle, and $X(n)$ is generated from $X(p)$ as before, we say that $\\{X(n)\\}_{n\\geq 1}$ is a \\emph{Steinhaus random multiplicative function}. \n\nWe define the random trigonometric polynomial\n$$\nP_N(\\theta) = \\frac{1}{\\sqrt{N}} \\sum_{n=1}^N X(n) e(n\\theta),\n$$\nand the normalized unit-circle-moments,\n\\begin{equation} \\label{eq:P_N_moments_normalized}\n\\mathfrak{m}_N^{(j,k)} = \\int_0^1 \\big( P_N(\\theta) \\big)^j \\big( \\overline{P_N(\\theta)} \\big)^k \\, d\\theta, \\quad\\quad  \\mathfrak{m}_N^{(k)} = \\int_0^1 \\big| P_N(\\theta) \\big|^{2k}\\, d\\theta.\n\\end{equation}\nNote that $\\mathfrak{m}_N^{(j,k)}$ and $\\mathfrak{m}_N^{(k)}$ are random variables, determined by the random function $X(n)$. From the definition we have $\\mathfrak{m}_N^{(j,k)} = \\overline{\\mathfrak{m}_N^{(k,j)}}$.\n\nRandom multiplicative functions have been extensively studied, starting with Wintner \\cite{wintner}. Some recent works are \\cite{heap,harper15,sarkozy,najnudel,harper20,bondarenko,aymone2020}.\n\nOur main result shows that $\\mathfrak{m}_N^{(j,k)}$ tend to Gaussian moments for relatively large $j$ and $k$, at the level of the mean and mean-square. Throughout this paper we will use the abbreviation $\\log_2 N = \\log \\log N$.\n\n\\begin{thm}\n\\label{thm:main_moments_Gaussian}\nFor $X(n)$ a Rademacher or Steinhaus random multiplicative function, we have\n\\begin{equation}\\label{eq:moment_mean_square_Gaussian}\n\\E\\, \\left| \\mathfrak{m}_N^{(j,k)} - k!\\, \\mathbf{1}_{jk} \\right|^2 \\ll \\frac{1}{N^{1\/15k}}\n\\end{equation}\nuniformly for $1 \\leq j \\leq k \\leq A\\, (\\log N \/ \\log_2 N)^{1\/3},$ where $A > 0$ is a small absolute constant.\n\\end{thm}\n\nAs a consequence of Theorem \\ref{thm:main_moments_Gaussian}, we will prove the following central limit theorem.\n\n\\begin{thm}\n\\label{thm:Gaussian_RMF}\nFor $X(n)$ drawn randomly as a Rademacher or Steinhaus random multiplicative function, define random trigonometric polynomials $P_N$ by\n$$\nP_N(\\theta) = \\frac{1}{\\sqrt{N}}\\sum_{n=1}^N X(n) e(n\\theta).\n$$\nThen almost surely,\n\\begin{equation}\n\\label{eq:as_CLT}\n\\lim_{N\\rightarrow\\infty} \\mathrm{meas}\\Big\\{ \\theta \\in [0,1]:\\, P_N(\\theta) \\in E\\Big\\} = \\frac{1}{\\pi} \\int_{E} e^{-x^2-y^2}\\, dxdy,\n\\end{equation}\nfor any rectangle $E \\subset \\mathbb{C}$.\n\\end{thm}\n\nHere ``almost surely\" means upon drawing $X(n)$, this event occurs with probability $1$. The measure in \\eqref{eq:as_CLT} is the Lebesgue measure on $[0,1]$, with $\\mathrm{meas}\\{[0,1]\\} = 1$.\n\n\nTheorem \\ref{thm:Gaussian_RMF} was first obtained by A. Harper \\cite{harper_com} using martingale methods, though this work was not published. The approach we give here is via moments.\n\nFurthermore, from the information on high moments in Theorems \\ref{thm:main_moments_Gaussian} and a few additional estimates we obtain information about the sup-norm of $P_N$. Recall that a sequence of events $E_n$ is said to occur \\emph{asymptotically almost surely} if $\\mathbb{P}(E_n) = 1-o(1)$ as $n\\rightarrow\\infty$.\n\n\\begin{thm}\\label{thm:supnorm}\nFor $X(n)$ a Rademacher or Steinhaus random multiplicative function, we have\n$$\n\\Big(\\frac{\\log N}{\\log_2 N}\\Big)^{1\/6} \\,\\leq \\, \\max_\\theta |P_N(\\theta)| \\,\\leq \\, \\exp\\big( 3\\sqrt{\\log N \\log_2 N}\\big),\n$$\nasymptotically almost surely as $N\\rightarrow\\infty$.\n\\end{thm}\n\nThis result allows us to see that $\\max_\\theta|P_N(\\theta)|$ will typically be at least as large as a power of $\\log N$, thus distinguishing conjectural behavior of Fekete polynomials from typical behavior of random multiplicative functions.\n\nA well-known conjecture of S\\'{a}rk\\\"{o}zy \\cite[Conj. 49]{sarkozy} is that if $f(n): \\N \\to \\{\\pm1\\}$ is \\emph{any} multiplicative function, then\n$$\n\\limsup_{N\\to\\infty} \\frac1{\\sqrt{N}} \\max_{\\theta} \\Big| \\sum_{n \\le N} f(n) e(n\\theta) \\Big| = +\\infty.\n$$\nThe lower bound in Theorem \\ref{thm:supnorm} shows that for a typical completely multiplicative function a rather stronger statement holds.\n\n\\subsection{Long tails}\n\nIn fact it seems likely that asymptotically almost surely,\n\\begin{equation}\\label{eq:supnorm_conj}\n\\max_\\theta |P_N(\\theta)| \\sim (\\log N)^{1\/2},\n\\end{equation}\nand it may even be that this is true almost surely as for independent coefficients. Nonetheless we expect the random variables $\\max_\\theta |P_N(\\theta)|$ to have rather longer tails than in the independent case, presenting a serious challenge in proving \\eqref{eq:supnorm_conj} via a moment or exponential moment method.\n\n\nIn fact, such long tails are already present in the computation of high moments. $\\E\\, \\mathfrak{m}_N^{(k)}$ is much larger than a Gaussian moment for $k$ larger than $(\\log N)^{1\/2+\\epsilon}$, and the standard deviation of $\\mathfrak{m}_N^{(k)}$ will be at least as large as the expected value, indicating that there is no concentration of this random variable in the mean-square sense.\n\n\\begin{thm}\n\\label{thm:main_moments_larger}\nFor $X(n)$ a Rademacher or Steinhaus random multiplicative function, we have for positive integers $k \\leq N^{1\/4}$,\n\\begin{equation}\\label{eq:moment_mean_larger}\n\\E\\, \\mathfrak{m}_N^{(k)} = e^{O(\\log N)} \\E\\, \\Big| \\frac{1}{\\sqrt{N}} \\sum_{n\\leq N} X(n)\\Big|^{2k}.\n\\end{equation}\n\nAs a consequence for any $\\varepsilon > 0$, there is a constant $B_\\varepsilon$ such that if $N$ is sufficiently large (depending on $\\varepsilon$) and $B_\\varepsilon (\\log N \/ \\log_2 N)^{1\/2} \\leq k \\leq N^{1\/4}$,\n\\begin{equation}\\label{eq:moment_mean_lowerbound}\n\\E\\, \\mathfrak{m}_N^{(k)} \\gg \\Big(\\E\\, \\Big| \\frac{1}{\\sqrt{N}} \\sum_{n\\leq N} X(n)\\Big|^{2k}\\Big)^{1-\\varepsilon}.\n\\end{equation}\n\nMoreover, there is an absolute constant $B$ such that for $B (\\log N \/ \\log_2 N)^{1\/2} \\leq k \\leq N^{1\/4}$,\n\\begin{equation}\\label{eq:moment_variance_larger}\n\\Var(\\mathfrak{m}_N^{(k)}) \\gg (\\E\\, \\mathfrak{m}_N^{(k)})^2.\n\\end{equation}\n\\end{thm}\n\n\\begin{rmk}\nThe implicit constants in this theorem are to be understood as absolute (independent of $k, N$ and $\\epsilon$).\n\\end{rmk}\n\nA considerable amount is known about the moments on the right hand side of \\eqref{eq:moment_mean_larger} and \\eqref{eq:moment_mean_lowerbound}. We recall the most relevant result in Theorem \\ref{thm:harper_highmoments} below.\n\nFrom Theorem \\ref{thm:harper_highmoments} the reader should check that the right hand side of \\eqref{eq:moment_mean_lowerbound} is considerably larger than the gaussian moment $k!$ for large $k$. Thus the asymptotic $\\E\\, \\mathfrak{m}_N^{(k)} \\sim k!$ implied by Theorem \\ref{thm:main_moments_Gaussian} for $k \\ll (\\log N\/\\log_2 N)^{1\/3}$ cannot be true for $k \\gg (\\log N\/\\log _2 N)^{1\/2}$.\n\nNote that while \\eqref{eq:moment_variance_larger} indicates that $\\mathfrak{m}_N^{(k)}$ will not concentrate around its mean value for such large $k$ in the mean-square sense, it is quite possible that $\\mathfrak{m}_N^{(k)}$ will concentrate around a median value in probability, even for $k$ of order $(\\log N)^{1\/2}$ or larger. The latter phenomenon would be far more subtle, and we are not able to address it here.\n\n\\begin{rmk}\nThe estimate \\eqref{eq:moment_mean_larger} is not useful for $k = o(\\log N \/ \\log_2 N)^{1\/2})$ as in this range the error term is larger than the main term. It seems possible that with more work the range of $k$ in which the moments of Theorem \\ref{thm:main_moments_Gaussian} are Gaussian in mean square can be extended to  $k = o( (\\log N\/\\log_2 N)^{1\/2})$, but Theorem \\ref{thm:main_moments_larger} tells us we can go no further. \n\\end{rmk}\n\n\\begin{rmk}\nAs we will discuss in section \\ref{sec:supremum}, unit-circle-moments would need to be well understood for $k \\approx \\log N$ in order to capture the true order of magnitude of $\\max_\\theta \\left| P_N(\\theta) \\right|$ (see also \\cite[Sec. 4.5]{green}). Theorem \\ref{thm:main_moments_larger} thus rules out such an approach, at least if \\eqref{eq:supnorm_conj} is true.\n\\end{rmk}\n\n\\begin{rmk}\nNote that $N^{-1\/2} \\sum_{n\\leq N} X(n) = P_N(0)$. Thus the average value of $2k$th moments on the unit circle are essentially controlled for large enough $k$ by the values of the polynomial at $\\theta = 0$.\n\\end{rmk}\n\n\\subsection{Notation}\nWe will write $f \\ll g$ or alternatively $f=O(g)$, if there exists a positive constant $C$ such that $|f|\\leq C |g|$. A subscript $f\\ll_t g$ may be added to emphasize the depence of the implicit constant $C$ on the parameter $t$. The symbol $\\tau_k$ is reserved for the $k$-fold divisor function and we will write $n = \\square$ to indicate that the integer $n$ is a perfect square.\n\n\\subsection{Acknowledgments}\nWe thank Adam Harper for explaining his previous work on Theorem \\ref{thm:Gaussian_RMF} to us, pointing us in the direction of \\cite{VW} for the computation of high-moments, and comments on a preprint version, Trevor Wooley for explaining some of the details of a point counting method to us, and Maxim Gerspach for pointing out a mistake in Theorem \\ref{thm:main_moments_larger} in an earlier draft. Likewise we thank Oleksiy Klurman and an anonymous referee for helpful comments and corrections.\nJ.B. is supported by ERC Advanced Grant 692616. The research of A.N. was funded by ISF Grant 1903\/18. B.R. received partial support from an NSERC grant and US NSF FRG grant 1854398.\n\n\\section{Method of proof}\n\n\\subsection{Point counting}\n\nWe will explicitly compute moments. Note that for $X(n)$ a Rademacher random multiplicative function, the value of\n\\begin{equation} \\label{eq:P_N_moments_avg}\nN^{\\tfrac12 (j+k)} \\cdot \\E \\int_0^1 \\big( P_N(\\theta) \\big)^j \\big( \\overline{P_N(\\theta)} \\big)^k\\, d\\theta\n\\end{equation}\nis equal to the count of integer solutions to the system of equations\n\\begin{equation} \\label{eq:fundsystem_squares}\n\\begin{split}\nm_1 \\cdots m_{j} &n_1 \\cdots n_{k}=\\square   \\\\\nm_1+\\cdots m_j&=n_1 +\\cdots +n_k,\n\\end{split}\n\\end{equation}\nwith $m_r, n_s \\in [1,N]$ for all $r,s$. We label this count by $\\mathcal{N}^{\\, \\square}_{j,k}(N)$. \n\nLikewise for $X(n)$ a Steinhaus random multiplicative function the value of \\eqref{eq:P_N_moments_avg} is given by the count of integer solutions to the system of equations\n\\begin{equation} \\label{eq:fundsystem_circles}\n\\begin{split}\nm_1 \\cdots m_{j} &=n_1 \\cdots n_{k}   \\\\\nm_1+\\cdots m_j&=n_1 +\\cdots +n_k,\n\\end{split}\n\\end{equation}\nwith $m_r, n_s \\in [1,N]$ for all $r,s$. We label this count by $\\mathcal{N}^{\\, \\bullet}_{j,k}(N)$.\n\nAn important subcollection of solutions to the systems \\eqref{eq:fundsystem_squares} and \\eqref{eq:fundsystem_circles} is given by the \\emph{diagonal} contributions,\nin which $\\{m_r\\}$ is a permutation of $\\{n_s\\}$.\nIn the notation of multisets, we define the count of diagonal contributions to be the count of integers solutions to\n$$\n\\{m_1,...,m_j\\} =  \\{n_1,...,n_k\\},\n$$\nwith $m_r, n_s \\in [1,N]$ for all $r,s$. We label this count by $\\mathcal{D}_{j,k}(N)$.\n\n\\begin{lem} \\label{lem:diag_count}\nFor $j,k \\leq N^{1\/2}$,\n$$\n\\mathcal{D}_{j,k}(N) = \\mathbf{1}_{jk} \\, k! \\,  N^k \\left( 1+ O(k^2\/N)\\right).\n$$\n\\end{lem}\n\n\\begin{proof}\nIf $j\\neq k$ obviously $\\mathcal{D}_{j,k}(N) = 0$. For $j=k$, we have\n$$\nk! N (N-1) \\cdots (N-k+1) \\leq \\mathcal{D}_{j,k}(N) \\leq k! N^k,\n$$\nwith the lower bound coming from matches of $m_1,...,m_k$ in which all $m_j$ are distinct. And a crude bound reveals\n\\begin{multline*}\nN(N-1)\\cdots (N-k+1) = N^k \\exp\\left( \\log(1-\\tfrac{1}{N}) + \\cdots + \\log(1 - \\tfrac{k-1}{N}) \\right) \\\\= N^k \\left( 1 + O(k^2\/N)\\right). \\qedhere\n\\end{multline*}\n\\end{proof}\n\nThese diagonal contributions will constitute the most important contribution to the counts \\eqref{eq:fundsystem_squares} and \\eqref{eq:fundsystem_circles}. We thus seek to bound \\emph{off-diagonal} counts,\n$$\n\\mathcal{E}^{\\, \\square}_{j,k}(N) = \\mathcal{N}^{\\, \\square}_{j,k}(N) - \\mathcal{D}_{j,k}(N),\n$$\nand\n$$\n\\mathcal{E}^{\\, \\bullet}_{j,k}(N) = \\mathcal{N}^{\\, \\bullet}_{j,k}(N) - \\mathcal{D}_{j,k}(N).\n$$\n\n\\subsection{A factorization of Vaughan and Wooley}\n\nOur main tool will be a variant of a device introduced by Vaughan and Wooley \\cite{VW}. Let $sq(\\nu)$ be the largest square divisor of a positive integer $\\nu$, so that $\\nu\/sq(\\nu)$ is squarefree.\n\n\\begin{lem}\\label{lem:triangular}\nIf for positive integers $\\nu_1,...,\\nu_\\ell$ we have \n$$\n\\nu_1\\cdots \\nu_\\ell = \\square,\n$$\nthen there exists an upper triangular array of $\\binom{\\ell+1}{2}$ positive integers\n\\begin{equation} \\label{eq:tri_array}\n\\begin{matrix}\nb_{11} & b_{12} & \\cdots & b_{1\\ell} \\\\\n & b_{22} & \\cdots & b_{2\\ell} \\\\\n &  & \\ddots & \\vdots \\\\\n & &  & b_{\\ell \\ell}\n\\end{matrix}\n\\end{equation}\nsuch that for $1 \\leq r \\leq \\ell$, we have $b_{rr}^2 = sq(\\nu_r)$, and $\\nu_r$ is the product of row $r$ and column $r$ in this array:\n$$\n\\nu_r = (b_{1r}\\cdots b_{rr}) (b_{rr}\\cdots b_{r\\ell}).\n$$\n\\end{lem}\n\n\\begin{proof}\nNote the numbers $\\nu_r\/sq(\\nu_r)$ will be squarefree for all $r$. We fix $\\ell$ and let $E = (\\nu_1\/sq(\\nu_1))\\cdots (\\nu_\\ell\/sq(\\nu_\\ell))$. Obviously such an array exists if $E=1$. We use induction on $E$, supposing we have proved the claim for all smaller values. We have $E = \\square$, and so if $E > 1$, we must have $\\gcd(\\nu_u\/sq(\\nu_u), \\nu_v\/sq(\\nu_v))~>~1$ for some distinct indices $u < v$. \nLet $c = \\gcd(\\nu_u\/sq(\\nu_u), \\nu_v\/sq(\\nu_v))$.\nNote that $c$ must be squarefree, and further note\n$$\n(\\nu_1\/sq(\\nu_1))\\cdots \\left(\\tfrac{\\nu_u\/sq(\\nu_u)}{c}\\right) \\cdots \\left(\\tfrac{\\nu_v\/sq(\\nu_v)}{c}\\right) \\cdots (\\nu_\\ell\/sq(\\nu_\\ell)) = \\square.\n$$\nBy inductive hypothesis an array $b_{rs}$ generates the values $\\nu_1,\\cdots, \\left(\\tfrac{\\nu_u}{c}\\right), \\cdots, \\left(\\tfrac{\\nu_v}{c}\\right), \\cdots, \\nu_\\ell$ with its row-times-column products, and thus one may check the array\n$$\n\\begin{cases}\nb_{rs} & \\textrm{if}\\; (r,s) \\neq (u,v) \\\\\nc \\, b_{uv} & \\textrm{if}\\; (r,s) = (u,v)\n\\end{cases}\n$$\ngenerates the values $\\nu_1,...,\\nu_\\ell$ with its row-times-column products, which completes the proof.\n\\end{proof}\n\nThe above lemma will allow us to consider moments for both Rademacher and Steinhaus random multiplicative functions. If we were treating the Steinhaus case alone, then it would be enough to use the original version of the device, that follows.\n\n\\begin{lem} [Vaughan-Wooley] \\label{lem:rectangular}\nIf for positive integers $m_1,...,m_j$ and $n_1,...,n_k$ we have\n$$\nm_1\\cdots m_j = n_1 \\cdots n_k,\n$$\nthen there exists a $j\\times k$ array of positive integers\n\\begin{equation} \\label{eq:rect_array}\n\\begin{matrix}\na_{11} & a_{12} & \\cdots & a_{1k} \\\\\na_{21} & a_{22} & \\cdots & a_{2k} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_{j1} & a_{j2} & \\cdots & a_{jk}\n\\end{matrix}\n\\end{equation}\nsuch that for $1 \\leq r \\leq j$ the product of row $r$ is equal to $m_r$:\n$$\nm_r = a_{r1}a_{r2}\\cdots a_{rk},\n$$\nand for $1 \\leq s \\leq k$ the product of column $s$ is equal to $n_s$:\n$$\nn_s = a_{1s}a_{2s}\\cdots a_{js}.\n$$\n\\end{lem}\n\nThis lemma can be proved using the same inductive procedure as above (see also the original proof in \\cite[Section 8]{VW}, and see also \\cite[Section 4]{granville}). \n\n\\subsection{A rough outline}\n\nWe carry out the details of bounding $\\mathcal{E}^{\\, \\square}_{j,k}(N)$ and $\\mathcal{E}^{\\, \\bullet}_{j,k}(N)$ later, but an approximation to the idea can be explained here. As observed by Vaughan and and Wooley, each non-diagonal solution to \\eqref{eq:fundsystem_circles} implies at least one `non-diagonal' solution to\n$$\n(a_{11}\\cdots a_{1k}) + \\cdots + (a_{j1}\\cdots a_{jk}) = (a_{11}\\cdots a_{j1}) + \\cdots +( a_{1k} \\cdots a_{jk}),\n$$\nin which at least one of the products on the left hand side has no matching product on the right hand side, which will be seen to force a solution to one of the $a_{rs}$ in terms of the others, e.g.\n$$\na_{11}(a_{12}\\cdots a_{1k} - a_{21}\\cdots a_{j1}) = (a_{12}\\cdots a_{j2} + \\cdots + a_{1k}\\cdots a_{jk}) - (a_{21}\\cdots a_{2k} + \\cdots + a_{j1}\\cdots a_{jk}),\n$$\nand this loss in degrees of freedom entails fewer non-diagonal solutions than diagonal. This is a slight oversimplification, but with only a little more care the argument can be made rigorous. A similar argument works for the system \\eqref{eq:fundsystem_squares}. Likewise we obtain the mean-square estimate in Theorem \\ref{thm:main_moments_Gaussian} by reducing the variance of $\\mathfrak{m}_N^{(j,k)}$ to a related point count.\n\nNote however the number of solutions to e.g. $m_1 = a_{11}\\cdots a_{1k}$ grows as $k$ grows, and this phenomenon is responsible for this argument breaking down as $k$ increases with $N$ in the form of Theorem \\ref{thm:main_moments_larger}. \n\nTheorem \\ref{thm:Gaussian_RMF} then follows from these moment estimates by using the moment method and a simple bound on $L^p$-norms of short-interval sums of random multiplicative functions to bootstrap from an asymptotically almost sure central limit theorem to an almost sure central limit theorem. Theorem \\ref{thm:supnorm} likewise follows from these moment estimates, using the fact that for degree $N$ trigonometric polynomials, the sup-norm is reasonably close to an $L^p$-norm, even for $p$ relatively small compared to $N$. \n\n\n\n\\section{Point counting}\n\n\\subsection{Preliminary results}\n\nWe state the following results regarding the $k$-fold divisor function for easy reference later.\n\n\\begin{lem}[Divisor sum bounds]\\label{lem:divisorsumbound}\nFor $N\\geq 3$ and $\\ell \\geq 1$,\n$$\n\\sum_{n \\leq N} \\tau_\\ell(n) \\le N (2\\log N)^{\\ell-1}.\n$$\n\\end{lem}\n\n\\begin{proof}\nWe have\n$$\n\\sum_{n\\leq N} \\tau_\\ell(n) = \\sum_{a_1\\cdots a_\\ell \\leq N} 1 \\le \\sum_{a_1\\cdots a_{\\ell-1} \\leq N} \\frac{N}{a_1\\cdots a_{\\ell-1}} \\leq N (1+ \\log N)^{\\ell-1}.\n$$\nAs $1+ \\log N \\leq 2\\log N$ for $N\\geq 3$ the result follows.\n\\end{proof}\n\nWe also note the well-known pointwise divisor bound: for all fixed $\\ell \\geq 1$ and $\\varepsilon > 0$, we have\n\\begin{equation}\\label{eq:divisor_bound}\n\\tau_\\ell(n) \\leq_{\\ell, \\varepsilon} n^\\varepsilon,\n\\end{equation}\nobtained by iterating the bound for $\\ell = 2$ (see e.g. \\cite[(2.20)]{montgomeryvaughan}).\n\nIn addition we have the crude estimates\n\\begin{equation}\\label{eq:div_jk_bound}\n\\tau_j(n)\\tau_k(n) \\leq \\tau_{jk}(n),\n\\end{equation}\n\\begin{equation}\\label{eq:div_nm_bound}\n\\tau_k(nm) \\leq \\tau_k(n)\\tau_k(m).\n\\end{equation}\nBoth are simple if slightly tedious to verify by setting $m = p^a$ and $n = p^b$ for a prime $p$, then using multiplicativity to extend to other values. \\eqref{eq:div_nm_bound} has appeared before as \\cite[(45)]{sandor}.\n\nThese estimates then imply the following bounds, for all $\\ell \\geq 1$\n\\begin{equation}\\label{eq:tau_squared}\n\\sum_{n\\leq N} \\tau_\\ell(n)^2 \\leq \\sum_{n\\leq N} \\tau_{\\ell^2}(n) \\leq N (2 \\log N)^{\\ell^2-1},\n\\end{equation}\n\\begin{equation}\\label{eq:tau_of_squares}\n\\sum_{n\\leq N} \\tau_\\ell(n^2) \\leq \\sum_{n\\leq N} \\tau_\\ell(n)^2 \\leq N (2\\log N)^{\\ell^2-1}.\n\\end{equation}\nA somewhat better estimate than \\eqref{eq:tau_squared} has appeared in \\cite[Theorem 1.7]{Nor} for a limited range of $\\ell$, but the estimate here will be sufficient for our purposes.\n\n\\subsection{The mean of moments: Gaussian range}\n\nWe compute counts which are related to $\\mathbb{E} \\, \\mathfrak{m}_N^{(j,k)}$. In the arguments that follow we label the collection of triangular arrays of positive integers \\eqref{eq:tri_array} by $T_\\ell$ (thus $T_\\ell$ is isomorphic to $\\mathbb{N}^{\\binom{\\ell+1}{2}}$) and for $\\mathbf{b} \\in T_\\ell$, we let\n$$\nb_r^\\ast = (b_{1r}\\cdots b_{rr})(b_{rr}\\cdots b_{r\\ell})\n$$\ndenote the product of the $r$th column and $r$th row.\n\n\\begin{lem}\\label{lem:square_offdiag}\nUniformly for all $1\\leq j\\leq k$ and $N\\geq 10$,\n$$\n\\Esquare_{j,k}(N) \\ll N^{(j+k)\/2} \\exp\\Big[ - \\frac{\\log N}{6k} + O(k^2 (\\log k + \\log_2 N))\\Big].\n$$\n\\end{lem}\n\n\\begin{proof}\nWe first prove an upper bound for $\\Fsquare_{j,k}(N)$ defined as follows: $\\Fsquare_{j,k}(N)$ is the count of $(m_1,...,m_j,n_1,...,n_k) \\in \\mathbb{N}^{j+k}$ such that $m_r, n_s \\in [1,N]$ for all $1\\leq r \\leq j$, $1 \\leq s \\leq k$, and\n\\begin{gather*}\nm_1 \\cdots m_{j} n_1 \\cdots n_{k}=\\square   \\\\\nm_1+\\cdots m_j=n_1 +\\cdots +n_k, \\\\\n\\textrm{the sets} \\; \\{m_1,...m_j\\} \\; \\textrm{and} \\; \\{n_1,...,n_k\\} \\; \\textrm{have no elements in common.}\n\\end{gather*}\n\nWe also define $\\Gsquare_{j,k}(N)$ to be the count of $\\mathbf{b} \\in T_{j+k}$ satisfying\n\\begin{enumerate}[(i)]\n\\item $b_1^\\ast,...,b_{j+k}^\\ast \\in [1,N]$,\n\\item $b_1^\\ast + \\cdots + b_j^\\ast = b_{j+1}^\\ast + \\cdots + b_{j+k}^\\ast$, and\n\\item the sets $\\{b_1^\\ast,...,b_j^\\ast\\}$ and $\\{b_{j+1}^\\ast,...,b_{j+k}^\\ast\\}$ have no elements in common.\n\\end{enumerate}\nFor notational reasons let $\\ell = j+k$. For indices $u \\leq v$ of the $\\ell \\times \\ell$ upper triangle, we likewise let $\\Gsquare_{j,k}(N;\\, u,v)$ be the count of $\\mathbf{b} \\in T_\\ell$ satisfying (i), (ii), (iii) above and in addition\n\\begin{enumerate}[(iv)]\n\\item $b_{uv}$ is a maximal entry of $\\mathbf{b}$.\n\\end{enumerate}\n\nBy Lemma \\ref{lem:triangular},\n\\begin{equation}\\label{eq:F_boundedby_G}\n\\Fsquare_{j,k}(N) \\leq \\Gsquare_{j,k}(N),\n\\end{equation}\nand plainly\n\\begin{equation}\\label{eq:G_boundedby_maximalG}\n\\Gsquare_{j,k}(N) \\leq \\sum_{1 \\leq u \\leq v \\leq \\ell} \\Gsquare_{j,k}(N;\\, u,v).\n\\end{equation}\n\nSo it will be sufficient to treat $\\Gsquare_{j,k}(N;\\, u,v)$. When $b_{uv}$ is the largest entry of $\\mathbf{b}$, then  $b_u^\\ast \\leq N$ and $b_v^\\ast \\leq N$ imply that either\n\\begin{equation}\\label{eq:bprod_small1}\n\\begin{split}\n(b_{1u}\\cdots  b_{uu}) (b_{uu} \\cdots \\widehat{b_{uv}} \\cdots b_{u\\ell}) &\\leq N^{1-1\/(\\ell+1)}\\\\ \n\\textrm{and}&  \\\\\n(b_{1v}\\cdots \\widehat{b_{uv}} \\cdots b_{vv}) (b_{vv} \\cdots b_{v\\ell}) &\\leq N^{1-1\/(\\ell+1)}\n\\end{split}\n\\quad (\\textrm{if}\\; u < v)\n\\end{equation}\nor\n\\begin{equation}\\label{eq:bprod_small2}\n(b_{1v}\\cdots \\widehat{b_{vv}}) (\\widehat{b_{vv}}\\cdots b_{v\\ell}) \\leq N^{1-2\/(\\ell+1)} \\quad (\\textrm{if}\\; u = v),\n\\end{equation}\nwhere $\\widehat{b_{uv}}$ or $\\widehat{b_{vv}}$ indicates these terms are excluded from the products above.\n\nFurthermore, if $1 \\leq u \\leq j$ and $j+1 \\leq v \\leq k$, and utilizing that by (ii)  $b_1^\\ast + \\cdots + b_j^\\ast = b_{j+1}^\\ast + \\cdots + b_{j+k}^\\ast$, we must have \n\\begin{align*}\n&b_{uv} \\left( (b_{1u}\\cdots  b_{uu}) (b_{uu} \\cdots \\widehat{b_{uv}} \\cdots b_{u\\ell}) - (b_{1v}\\cdots \\widehat{b_{uv}} \\cdots b_{vv}) (b_{vv} \\cdots b_{v\\ell})\\right) \\\\ \n&\\quad\\quad = b_u^\\ast - b_v^\\ast \n= \\sum_{\\substack{j+1 \\leq r \\leq j+k \\\\ r \\neq v}} b_r^\\ast - \\sum_{\\substack{1\\leq r \\leq j \\\\ r \\neq u}} b_r^\\ast.\n\\end{align*}\nBy (iii) the sets $\\{b_1^\\ast,...,b_j^\\ast\\}$ and $\\{b_{j+1}^\\ast,...,b_{j+k}^\\ast\\}$ are disjoint, and in particular $b_u^\\ast - b_v^\\ast \\ne 0$. Hence, if $1 \\leq u \\leq j$ and $j+1 \\leq v \\leq k$, for $\\mathbf{b}$ contributing to the count of $\\Gsquare_{j,k}(N;\\, u,v)$, we have that $b_{uv}$ is determined by the values of $b_{rs}$ for $(r,s) \\neq (u,v)$. An easier analysis shows $b_{uv}$ is determined by other values of $b_{rs}$ for $u,v$ not in this range as well. It thus follows for all $u\\leq v$,\n$$\n\\Gsquare_{j,k}(N;\\, u,v) \\leq \\#\\{ (b_{11},b_{12},..., \\widehat{b_{uv}},...,b_{\\ell\\ell}):\\, b_{11}^2 b_{12}^2 \\cdots \\widehat{ (b_{uv}^2)} \\cdots b_{\\ell\\ell}^2 \\leq N^{\\ell - 1\/(\\ell+1)}\\},\n$$\nsince multiplying together the relations $b_r^\\ast \\leq N$ for all $r \\neq u,v$ with the relations \\eqref{eq:bprod_small1} or \\eqref{eq:bprod_small2} for $r = u$ or $v$ yield the inequality inside the brackets above. The above count is just the sum of a divisor function, and applying Lemma \\ref{lem:divisorsumbound} we obtain,\n$$\n\\Gsquare_{j,k}(N; \\, u,v) \\ll N^{\\tfrac{\\ell}{2} - \\tfrac{1}{2(\\ell+1)}} (\\ell \\log N)^{\\binom{\\ell+1}{2}-1},\n$$\nfor all $u,v$. Thus from \\eqref{eq:G_boundedby_maximalG} and \\eqref{eq:F_boundedby_G},\n\\begin{align}\\label{eq:F_bound}\n\\notag \\Fsquare_{j,k}(N) &\\ll \\binom{\\ell+1}{2} N^{\\tfrac{\\ell}{2} - \\tfrac{1}{2(\\ell+1)}} (\\ell \\log N)^{\\binom{\\ell+1}{2}} \\\\\n\\notag & \\ll \\ell^{\\ell^2} N^{\\tfrac{\\ell}{2}- \\tfrac{1}{3\\ell}} (\\log N)^{\\ell^2} \\\\\n& \\ll (2k)^{4k^2} N^{\\tfrac{j+k}{2}- \\tfrac{1}{6k}} (\\log N)^{4k^2},\n\\end{align}\nas $\\ell = j+k \\leq 2k$.\n\nTurning finally to $\\Esquare_{j,k}(N)$ with $j\\leq k$, if $m_1,...m_j,n_1,...,n_k$ is a non-diagonal solution to the system \\eqref{eq:fundsystem_squares}, then either (i) $m_r = n_s$ for some $1\\leq r \\leq j$ and $1 \\leq s \\leq k$ and the remaining $m, n$ are a non-diagonal solution to such a system with $j$ and $k$ reduced by $1$, or the sets $\\{m_1,...,m_j\\}$ and $\\{n_1,...,n_k\\}$ have no elements in common. Hence\n$$\n\\Esquare_{j,k}(N) \\leq jk \\,N \\, \\Esquare_{j-1,k-1}(N) + \\Fsquare_{j,k}(N),\n$$\nand by inductively expanding this relation\n\\begin{multline*}\n\\Esquare_{j,k}(N) \\leq \\Fsquare_{j,k}(N) + jk\\, N \\, \\Fsquare_{j-1,k-1}(N) + j(j-1)k(k-1)\\, N^2 \\, \\Fsquare_{j-2,k-2}(N) \\\\\n+ \\cdots + j!\\, k(k-1)\\cdots(k-j+1)\\, N^{j-1} \\Fsquare_{1,k-j+1}(N),\n\\end{multline*}\nnoting that $\\Esquare_{1,k}(N) = \\Fsquare_{1,k}(N)$. Hence from \\eqref{eq:F_bound},\n\\begin{align*}\n\\Esquare_{j,k}(N) &\\ll j \\cdot j!\\, k! \\, (2k)^{4k^2} \\, N^{(j+k)\/2-1\/6k} (\\log N)^{4k^2} \\\\\n&\\ll  N^{(j+k)\/2} \\exp\\Big[ - \\frac{\\log N}{6k} + O(k^2 (\\log k + \\log_2 N))\\Big],\n\\end{align*}\nas claimed.\n\\end{proof}\n\nAs an immediate consequence, because any off-diagonal solution to \\eqref{eq:fundsystem_circles} also constitutes an off-diagonal solution to \\eqref{eq:fundsystem_squares}, we immediately obtain\n\n\\begin{lem}\\label{lem:circle_offdiag}\nUniformly for all $1\\leq j\\leq k$ and $N\\geq 10$,\n$$\n\\Ecircle_{j,k}(N) \\ll N^{(j+k)\/2} \\exp\\Big[ - \\frac{\\log N}{6k} + O(k^2 (\\log k + \\log_2 N))\\Big].\n$$\n\\end{lem}\n\nNote that if $A$ is a sufficiently small constant, for $k \\leq A\\, (\\log N \/ \\log_2 N)^{1\/3},$ we have $\\exp[ - \\log N\/6k + O(k^2 (\\log k + \\log_2 N))] \\ll 1\/N^{1\/7k}$. Thus, combining these upper-bounds for off-diagonal counts with Lemma \\ref{lem:diag_count}, we deduce\n\n\\begin{prop}\\label{prop:expectation_estimate}\nFor $X(n)$ a Rademacher or Steinhaus random multiplicative function, we have\n$$\n\\mathbb{E}\\, \\mathfrak{m}_N^{(j,k)} =  k!\\, \\mathbf{1}_{jk} + O\\Big(\\frac{1}{N^{1\/7k}}\\Big),\n$$\nuniformly for $1 \\leq j \\leq k \\leq A\\, (\\log N \/ \\log_2 N)^{1\/3},$ for a sufficiently small absolute constant $A > 0$.\n\\end{prop}\n\n\n\\subsection{The variance of moments: Gaussian range}\n\nWe define the count\n$$\n\\Vcircle_{j,k}(N)= \\sideset{}{^*}\\sum_{\\substack{ {\\bf m}, {\\bf m'} \\in [1,N]^j \\\\ {\\bf n}, {\\bf n'} \\in [1,N]^k}} 1\n$$\nwhere the asterisked sum above is restricted to tuples ${\\bf m},{ \\bf m'}, {\\bf n}, {\\bf n'}$ satisfying,\n\\begin{align}\\label{Vsys}\n\\begin{split}\nm_1 + \\cdots + m_j = n_1 + \\cdots + n_k&, \\quad m_1\\cdots m_j \\neq n_1 \\cdots n_k, \\\\\nm_1' + \\cdots + m_j' = n_1' + \\cdots + n_k'&,\\quad m_1'\\cdots m_j' \\neq n_1' \\cdots n_k',\\\\\nm_1\\cdots m_j m_1'\\cdots m_j' &= n_1\\cdots n_k n_1' \\cdots n_k'.\n\\end{split}\n\\end{align}\n\nThe reader can verify for $X(n)$ a Steinhaus random multiplicative function,\n\\begin{equation}\\label{eq:Steinhaus_variance}\nN^{j+k} \\cdot \\Var\\Big( \\int_0^1 (P_N(\\theta))^j (\\overline{P_N(\\theta)})^k\\, d\\theta \\Big) = \\Vcircle_{j,k}(N).\n\\end{equation}\n\n\\begin{cor}\n\\label{prop:count_twofold}\nUniformly for all $1 \\leq j \\leq k$ and $N \\geq 10$,\n\\begin{equation}\\label{Vbound}\n\\Vcircle_{j,k}(N) \\ll N^{j+k}\\exp\\Big[ - \\frac{\\log N}{12k} + O(k^2 (\\log k + \\log_2 N))\\Big].\n\\end{equation}\n\\end{cor}\n\n\n\n\\begin{proof}\nLet us first suppose that  $j \\neq k$. Since each solution $({\\bf m},{ \\bf m'}, {\\bf n}, {\\bf n'})$ to \\eqref{Vsys} may be combined to form a pair \n\\begin{equation}\\label{concat}\n{\\bf \\underline{m} }=(m_1,...,m_j,m_1',...,m_j'), \\ \\ {\\bf \\underline{n} }=(n_1,...,n_k,n_1',...,n_k'),\n\\end{equation} \nit follows from Lemmas \\ref{lem:diag_count} and \\ref{lem:circle_offdiag} that\n$$\n\\Vcircle_{j,k}(N) \\leq \\Ncircle_{2j,2k}(N) \\ll N^{j+k} \\exp\\Big[ - \\frac{\\log N}{12k} + O(k^2 (\\log k + \\log_2 N))\\Big],\n$$\nsince for $2j \\neq 2k$ all contributions to $\\Ncircle_{2j,2k}(N)$ are off-diagonal.\n\nWe may now focus on the case $j=k$, once again arguing inductively with $k$. Writing  $\\Vcircle_{k}(N) =\\Vcircle_{k,k}(N)$, we begin by observing that $\\Vcircle_{1}(N)=0$.\n\nWe make two simplifications. First, it will be sufficient to restrict our attention to solutions $({\\bf m},{ \\bf m'}, {\\bf n}, {\\bf n'})$ satisfying the following property: either $\\{m_1,...,m_k\\}$ and $\\{n_1,...,n_k\\}$ have no elements in common, or $\\{m'_1,...,m'_k\\}$ and $\\{n'_1,...,n'_k\\}$ have no elements in common. Indeed,  letting $\\Wcircle_k(N)$ be the set of solutions for which this is not the case, by reducing the system by removing two equal pairs ($m_{r_1} = n_{s_1}$) and ($m'_{r_2} = n'_{s_2}$) from the solution, we see\n$$\n\\Wcircle_k(N) \\leq k^2 N^2 \\Vcircle_{k-1}(N)\n$$\nand the quantity $\\Vcircle_{k-1}(N)$ we will deal with inductively.\n\nSecond, we may also assume that the tuple ${\\bf \\underline{m} }=(m_1,...m_K,m'_1,...,m_K')$ is a permutation of  ${\\bf \\underline{n} }=(n_1,...n_K,n'_1,...,n_K')$ since if $\\Ucircle_k(N)$ is the set of solutions for which this is not the case we have\n\\begin{equation}\\label{e2kbound}\n\\Ucircle_k(N) \\leq \\Ecircle_{2k,2k}(N) \\ll N^{2k} \\exp\\Big[ - \\frac{\\log N}{12k} + O(k^2 (\\log k + \\log_2 N))\\Big]\n\\end{equation} \nby Lemma \\ref{lem:circle_offdiag}.\n\nLet $\\widetilde{V}_{k}(N)$ denote the set of solutions satisfying both simplifications. Given any $({\\bf m},{ \\bf m'}, {\\bf n}, {\\bf n'}) \\in \\widetilde{V}_{k}(N)$, we see that ${\\bf m}$ has to be a permutation of ${\\bf n'}$ and similarly that ${ \\bf m'}$ must be a permutation of ${\\bf n}$. Thus\n\\begin{equation}\\label{eq:simplification_bound}\n|\\widetilde{V}_{k}(N)| \\leq \n(k !)^2 \\sum_{\\substack{ {\\bf m}, {\\bf m'} \\in [1,N]^K \\\\ m_1 + \\cdots + m_k = m'_1 + \\cdots + m'_k }} 1 \\leq k^{2k} N^{2k-1}.\n\\end{equation}\nHence\n$$\n\\Vcircle_k(N) \\ll k^{2k} N^{2k-1} + \\Ucircle_k(N) + k^2 N^2 \\Vcircle_{k-1}(N).\n$$\nExpanding inductively and using \\eqref{e2kbound} we verify the claim.\n\\end{proof}\n\nWe also have an analogous count for Rademacher random multiplicative functions. Define\n$$\n\\Vsquare_{j,k}(N) = \\sideset{}{^*}\\sum_{\\substack{ {\\bf m}, {\\bf m'} \\in [1,N]^j \\\\ {\\bf n}, {\\bf n'} \\in [1,N]^k} } 1,\n$$\nwhere here the asterisked sum is restricted to ${\\bf m},{ \\bf m'}, {\\bf n}, {\\bf n'}$ satisfying,\n\\begin{align}\\label{vsquaresys}\n\\begin{split}\nm_1 + \\cdots + m_j = n_1 + \\cdots + n_k&,   \\quad \\ m_1\\cdots m_j  n_1 \\cdots n_k  \\neq \\square,\\\\\nm_1' + \\cdots + m_j' = n_1' + \\cdots + n_k'&,    \\quad \\  m_1'\\cdots m_j' n_1' \\cdots n_k'  \\neq \\square,\\\\\nm_1\\cdots m_j m_1'\\cdots m_j' & n_1\\cdots n_k n_1' \\cdots n_k'  =  \\square.\n\\end{split}\n\\end{align}\nPlainly for Rademacher random multiplicative functions, \\eqref{eq:Steinhaus_variance} remains true with $\\Vsquare_{j,k}(N)$ in place of $\\Vcircle_{j,k}(N)$. \n\n\n\\begin{cor}\n\\label{prop:count_twofold_Sq}\nUniformly for all $1 \\leq j \\leq k$ and $N \\geq 10$, we have the estimate\n\\begin{equation}\\label{Vbound_square}\n\\Vsquare_{j,k}(N) \\ll N^{j+k}\\exp\\Big[ - \\frac{\\log N}{12k} + O(k^2 (\\log k + \\log_2 N))\\Big].\n\\end{equation}\n\n\\end{cor}\n\n\\begin{proof}\nAs in \\eqref{concat}, each solution $({\\bf m},{ \\bf m'}, {\\bf n}, {\\bf n'})$ to \\eqref{vsquaresys} may be concatenated to form a pair ${\\bf \\underline{m} }, {\\bf \\underline{n} }$. \nWhen $j \\neq k$ we may apply Lemma \\ref{lem:square_offdiag} to find that \n\\begin{equation}\\label{vsquarejk}\n\\Vsquare_{j,k}(N) \\leq \\mathcal{E}^{\\square}_{2j,2k}(N) \\ll N^{j+k}\\exp\\Big[ - \\frac{\\log N}{12k} + O(k^2 (\\log k + \\log_2 N))\\Big]. \n\\end{equation}\nWhen $j=k$, we consider two sets of solutions to \\eqref{vsquaresys}. First we count the number of solutions $({\\bf m},{ \\bf m'}, {\\bf n}, {\\bf n'})$ for which \n$m_1\\cdots m_j m_1'\\cdots m_j' = n_1\\cdots n_k n_1' \\cdots n_k'$. Invoking Corollary \\ref{prop:count_twofold} these contribute no more than $N^{j+k}\\exp[ - \\log N\/12k + O(k^2 (\\log k + \\log_2 N))]$ to $\\Vsquare_{k,k}(N)$. The remaining solutions are counted in precisely the same manner as \\eqref{vsquarejk} since we have removed all diagonal tuples $({\\bf \\underline{m} }, {\\bf \\underline{n} })$.\n\\end{proof}\n\nFrom Corollaries \\ref{prop:count_twofold} and \\ref{prop:count_twofold_Sq}, as we did in Proposition \\ref{prop:expectation_estimate}, we obtain the following estimate for the variance.\n\n\\begin{prop}\\label{prop:variance_estimate}\nFor $X(n)$ a Rademacher or Steinhaus random multiplicative function, we have\n$$\n\\Var(\\mathfrak{m}_N^{(j,k)}) =  O\\Big(\\frac{1}{N^{1\/15k}}\\Big),\n$$\nuniformly for $1 \\leq j \\leq k \\leq A\\, (\\log N \/ \\log_2 N)^{1\/3},$ for a sufficiently small absolute constant $A > 0$.\n\\end{prop}\n\n\\subsection{Moments of short interval polynomials}\n\nIn bounding the mean of moments of sums $\\sum_{n \\in [M,M+L]} X(n) e(n\\theta)$, we will use the following estimate.\n\n\\begin{lem}\\label{lem:shortintervals}\nFix $\\ell \\geq 1$ and $\\varepsilon > 0$. For any $N\\geq 1$ and any collection of intervals $I_1, ..., I_\\ell \\subset [1,2N]$, we have\n\\begin{equation}\\label{eq:count_shortint_Sq}\n\\sum_{\\substack{\\nu_1\\cdots \\nu_\\ell = \\square \\\\ \\nu_r \\in I_r}} 1 \\ll_{\\ell,\\varepsilon} N^\\varepsilon \\prod_{r=1}^\\ell (|I_r|^{1\/2}+1) \n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nNote that by Lemma \\ref{lem:triangular} it will be sufficient to show\n\\begin{equation}\\label{eq:triangular_shortint}\n\\sum_{\\substack{\\mathbf{b} \\in T_\\ell \\\\ b_r^\\ast \\in I_r}} 1 \\ll_{\\ell,\\epsilon} N^\\varepsilon \\prod_{r=1}^\\ell (|I_r|^{1\/2}+1).\n\\end{equation}\nWe will prove the above estimate by induction.\n\nFor $\\ell=1$ this is just the claim\n$$\n\\sum_{b_{11}^2 \\in I_1} 1 \\ll N^\\varepsilon (|I_1|^{1\/2}+1),\n$$\nbut for $I = (T-L,T]$ the left hand side is $ \\lfloor \\sqrt{T} \\rfloor -\\lfloor \\sqrt{T-L} \\rfloor \\ll L\/\\sqrt{T}+1 \\leq L^{1\/2}+1,$ as claimed.\n\nSuppose \\eqref{eq:triangular_shortint} has been verified for $\\ell-1$. Then for $\\ell$, suppose without loss of generality $|I_\\ell| \\leq |I_1|, \\, |I_2|,\\, ... , |I_{\\ell-1}|$. We have by inductive hypothesis\n$$\n\\sum_{\\substack{\\mathbf{b} \\in T_\\ell \\\\ b_r^\\ast \\in I_r}} 1 = \\sum_{\\beta_1 \\cdots \\beta_{\\ell-1} \\beta_\\ell^2 \\in I_\\ell} \\sum_{\\substack{\\mathbf{b} \\in T_{\\ell-1} \\\\ \\beta_r b_r^\\ast \\in I_r \\\\ \\forall\\, 1 \\leq r \\leq \\ell-1}} 1 \\ll \\sum_{\\beta_1 \\cdots \\beta_{\\ell-1} \\beta_\\ell^2 \\in I_\\ell} \\prod_{r=1}^{\\ell-1} \\Big(\\frac{|I_r|^{1\/2}}{\\beta_r^{1\/2}} + 1\\Big) N^\\varepsilon,\n$$\nwhere we have relabeled the right column of $T_\\ell$ by $\\beta_r$ in place of $b_{r \\ell}$.\n\nNote that the number of tuples satisfying $\\beta_1 \\cdots \\beta_{\\ell-1} \\beta_\\ell^2 \\in I_\\ell$ is $\\ll_{\\ell} (|I_\\ell|+1)N^\\varepsilon$ by the pointwise divisor bound \\eqref{eq:divisor_bound}. If we expand the product above, getting $2^{\\ell-1}$ terms, the contribution from any term in which $1$ appears instead of $|I_s|^{1\/2}\/\\beta_s^{1\/2}$ is thus\n$$\n\\ll (|I_\\ell|+1) N^\\varepsilon \\cdot \\prod_{\\substack{1 \\leq r \\leq \\ell-1 \\\\ r\\neq s}} (|I_r|^{1\/2}+1) N^\\varepsilon.\n$$\nAs $|I_\\ell| \\leq |I_s|$ and $\\varepsilon > 0$ is arbitrary, this yields an acceptable contribution to \\eqref{eq:triangular_shortint}.\n\nOn the other hand, in expanding the product, the only term not considered above is\n\\begin{equation}\\label{eq:induction_leftover}\n\\sum_{\\beta_1 \\cdots \\beta_{\\ell-1} \\beta_\\ell^2 \\in I_\\ell} \\prod_{r=1}^{\\ell-1} \\frac{|I_r|^{1\/2}}{\\beta_r^{1\/2}} N^\\varepsilon = \\sum_{\\delta\\beta_\\ell^2 \\in I_\\ell} \\frac{\\tau_{\\ell-1}(\\delta)}{\\delta^{1\/2}} \\prod_{r=1}^{\\ell-1} |I_r|^{1\/2} N^\\varepsilon.\n\\end{equation}\nIf we show\n\\begin{equation}\\label{eq:simple_bound}\n\\sum_{\\delta \\beta^2 \\in I} \\frac{1}{\\delta^{1\/2}} \\ll (|I|^{1\/2}+1) N^\\varepsilon,\n\\end{equation}\nfor all intervals $I \\subset [1,2N]$, we will be done, since then \\eqref{eq:induction_leftover} therefore yields an acceptable contribution to \\eqref{eq:triangular_shortint}, by the pointwise divisor bound.\n\nBut note that if $I \\subset [N, 2N]$, we can split the sum in \\eqref{eq:simple_bound} into two pieces: one in which $\\delta > |I|$, which yields a contribution no larger than the right hand side by the divisor bound, and a second in which $\\delta \\leq |I|$, which yields a contribution no more than\n$$\n\\sum_{\\delta \\leq |I|} \\frac{1}{\\delta^{1\/2}} \\sum_{b^2 \\in I\/\\delta} 1 \\ll \\sum_{\\delta \\leq |I|} \\frac{1}{\\delta^{1\/2}} \\Big( \\frac{|I|\/\\delta}{\\sqrt{N\/\\delta}} + 1\\Big) \\ll |I|^{1\/2} \\log N.\n$$\nThis yields \\eqref{eq:simple_bound} if $I \\subset [N,2N]$. But then summing dyadically (splitting $I$ up into pieces if necessary) gives \\eqref{eq:simple_bound} for all $I \\subset [1,2N]$.\n\\end{proof}\n\n\\section{Proofs of the main results}\n\n\\subsection{Gaussian moments and an almost sure central limit theorem}\n\n\\subsubsection{A proof of Theorem \\ref{thm:main_moments_Gaussian}}\n\nIt does not take much more work to prove Theorem~\\ref{thm:main_moments_Gaussian}. Indeed, using Propositions \\ref{prop:expectation_estimate} and \\ref{prop:variance_estimate}, note that in both the Rademacher and Steinhaus cases, we have from the triangle inequality,\n$$\n(\\E\\, |\\mathfrak{m}_N^{(j,k)} - k! \\mathbf{1}_{jk}|^2)^{1\/2} \\leq (\\E\\, |\\mathfrak{m}_N^{(j,k)} - \\E\\, \\mathfrak{m}_N^{(j,k)}|^2)^{1\/2} + (\\E\\, |k! \\mathbf{1}_{jk} - \\E\\, \\mathfrak{m}_N^{(j,k)}|^2)^{1\/2} \\ll \\frac{1}{N^{1\/30k}},\n$$\nwhich implies the theorem.\n\n\\subsubsection{From asymptotically almost sure to almost sure}\n\nTheorem \\ref{thm:main_moments_Gaussian} shows that asymptotically almost surely the fixed moments of the polynomials $P_N(\\theta)$ become Gaussian. In this section we will prove the almost sure central limit theorem, Theorem \\ref{thm:Gaussian_RMF}, by repeatedly applying Borel-Cantelli type arguments to this result. We begin with a lemma.\n\n\\begin{lem}\n\\label{lem:sqrt_cancel}\nAlmost surely for all $p> 0$, $\\varepsilon > 0$, \nand uniformly for $L \\le M$,\n\\begin{equation}\n\\label{eq:sqrt_cancel}\n\\int_0^1 \\Big|\\sum_{M < n \\leq M+L} X(n) e(n\\theta) \\Big|^p\\, d\\theta = O_{p,\\varepsilon}(L^{p\/2+\\varepsilon} M^\\varepsilon).\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nBy H\\\"older's inequality, if \\eqref{eq:sqrt_cancel} is true for some exponent $p$, it will be true for all smaller exponents $p$, and plainly if it is true for some $\\varepsilon$, it remains true for all larger $\\varepsilon$. Therefore by examining any countable collection of $p\\rightarrow\\infty$ and $\\varepsilon \\rightarrow 0$, it will be sufficient to prove that for any fixed $p$ and $\\varepsilon$, \\eqref{eq:sqrt_cancel} is true almost surely. \n\nNow, we note that Lemma \\ref{lem:shortintervals} immediately implies that for all $k\\geq 1$ and $L \\leq M$,\n\\begin{equation}\n\\label{eq:2k_expint}\n\\EE \\int_0^1 \\Big| \\sum_{M < n \\leq M+L} X(n) e(n\\theta)\\Big|^{2k}\\, d\\theta = O_{k,\\varepsilon}( L^k M^\\varepsilon)\n\\end{equation}\n\nFix $p$ and $\\varepsilon$, and let $\\mathcal{A}$ be the event that for infinitely many $M, L$,\n$$\n\\int_0^1 \\Big| \\sum_{M < n \\leq M+L} X(n) e(n\\theta)\\Big|^p \\, d\\theta \\geq L^{p\/2+\\varepsilon} M^\\varepsilon.\n$$\nTo prove the lemma, we need only to show that $\\mathcal{A}$ is null.\n\nBut on $\\mathcal{A}$, we see by H\\\"older that for all $t\\geq 1$,\n$$\n\\int_0^1 \\Big| \\sum_{M < n \\leq M+L} X(n) e(n\\theta)\\Big|^{pt} \\, d\\theta \\geq L^{pt\/2+\\varepsilon t} M^{\\varepsilon t},\n$$\nfor infinitely many $M, L$. We choose $t$ large enough that $\\varepsilon t \\geq 2+\\varepsilon$ and also take $pt$ an even integer $2k$. Then on the one hand from \\eqref{eq:2k_expint}\n$$\n\\EE \\sum_{M,L\\geq 1} \\frac{1}{L^{pt\/2 + \\varepsilon t} M^{\\varepsilon t}} \\int_0^1 \\Big| \\sum_{M < n \\leq M+L} X(n) e(n\\theta) \\Big|^{pt}\\, d\\theta \\leq \\sum_{M,L\\geq 1} \\frac{1}{L^{2} M^2} < +\\infty.\n$$\nBut on the other hand the sum here is infinite on $\\mathcal{A}$, so $\\mathcal{A}$ must be a null event.\n\\end{proof}\n\nWe now turn to the almost sure central limit theorem.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:Gaussian_RMF}]\nWe use the method of moments: we will show that for all $j,k \\geq 1$, almost surely\n\\begin{equation}\n\\label{eq:as_moments}\n\\mathfrak{m}_N^{(j,k)} = \\mathbf{1}_{jk}\\cdot k! + o(1),\n\\end{equation}\nas $N\\rightarrow\\infty$. Since these are the moments of a standard complex normal random variable, the theorem will follow (see e.g. \\cite[p. 388]{billingsley}).\n\nTo establish \\eqref{eq:as_moments}, fix arbitrary $1 \\leq j \\leq k$ and let $\\lambda \\in \\N$ be such that $(\\lambda+1)\/15k > 1$. Then by Proposition \\ref{prop:variance_estimate},\n$$\n\\sum_{M=1}^\\infty \\Var\\big(\\mathfrak{m}^{(j,k)}_{M^{\\lambda+1}}\\big) < +\\infty,\n$$\nand so almost surely,\n\\begin{equation}\n\\label{eq:as_moments_subseq}\n\\mathfrak{m}^{(j,k)}_{M^{\\lambda+1}} = \\EE \\mathfrak{m}^{(j,k)}_{M^{\\lambda+1}}+ o(1) = \\mathbf{1}_{jk}\\cdot k! + o(1),\n\\end{equation}\nas $M\\rightarrow\\infty$, with the last equation following from Proposition \\ref{prop:expectation_estimate}. Note that this implies in particular almost surely, for all $p > 0$,\n$$\n\\int_0^1 |P_{M^{\\lambda+1}}(\\theta)|^p \\, d\\theta = O_p(1).\n$$\n\nIt remains to pass from the sequence $\\{M^{\\lambda+1}\\}$ to the integers. We note that if we take $N$ with $M^{\\lambda+1} \\leq N < (M+1)^{\\lambda+1}$, then $N = M^{\\lambda+1} + L$, for $L = O(M^\\lambda)$.\n\nWrite\n$$\n\\sqrt{N} P_N(\\theta) = M^{(\\lambda+1)\/2} P_{M^{\\lambda+1}}(\\theta) + \\sum_{M^{\\lambda+1} < n \\leq N} X(n) e(n\\theta) \\eqqcolon F + f,\n$$\nand observe that for all $X$ and $\\theta$, by a binomial expansion,\n$$\n(F+f)^j \\overline{(F+f)}^k = F^j \\overline{F}^k + O_{j,k}\\Big(\\sum_{\\substack{a\\leq j, b\\leq k \\\\ (a,b) \\neq (0,0)}} |F|^{(j-a) + (k-b)} |f|^{a+b}\\Big).\n$$\nWe seek to integrate the error term in $\\theta$; note that for any $p_1 p_2 \\geq 0$,\n$$\n\\int_0^1 |F|^{p_1} |f|^{p_2} \\, d\\theta \n\\leq \\Big(\\int_0^1 |F|^{2p_1}\\, d\\theta \\Big)^{1\/2} \\Big(\\int_0^1 |f|^{2p_2}\\, d\\theta \\Big)^{1\/2}.\n$$\nBut from our almost sure computation of moments along the sequence $\\{M^{\\lambda+1}\\}$ we have that almost surely\n$$\n\\int_0^1 |F|^p\\, d\\theta = O_p(M^{p (\\lambda+1)\/2}) = O_p(N^{p\/2}),\n$$\nfor any $p > 0$. Furthermore, because $L = O(M^\\lambda)$ and $N > M^{\\lambda+1}$, applying Lemma \\ref{lem:sqrt_cancel} with $\\varepsilon < 1\/4(\\lambda + 1)$ we find that\n$$\n\\int_0^1 |f|^p\\, d\\theta = O_{p,\\lambda}(N^{2\\varepsilon} N^{(p\/2)(1-1\/(\\lambda+1))}) = o(N^{p\/2}),\n$$\nfor any $p > 0$.\n\nTherefore almost surely\n$$\n\\int_0^1 (F+f)^j \\overline{(F+f)}^k \\, d\\theta = \\int_0^1 F^j \\overline{F}^k\\, d\\theta + o(N^{(j+k)\/2}).\n$$\nBut dividing by $N^{(j+k)\/2}$  applying \\eqref{eq:as_moments_subseq}, we see that \\eqref{eq:as_moments} holds as claimed.\n\\end{proof}\n\n\\subsection{Estimates for the sup-norm}\\label{sec:supremum}\n\nBefore proving Theorem \\ref{thm:supnorm} we need the following simple lemma, which tells us the sup-norm of a degree $N$ polynomial is characterized by $L^p$ norms for $p$ of order at least $\\log N$, and gives less precise bounds in the case of smaller $p$.\n\n\\begin{lem}\\label{lem:Lp_to_sup}\nFor an arbitrary trigonometric polynomial $p_N(\\theta) = \\sum_{n\\leq N} a_n e(n\\theta)$ of degree $N$,\n\\begin{equation}\\label{eq:Lp_to_sup}\n\\Big(\\int_0^1 |p_N(\\theta)|^{2k}\\, d\\theta\\Big)^{1\/2k}\\leq \\max_\\theta|p_N(\\theta)| \\ll \\Big( N \\int_0^1 |p_N(\\theta)|^{2k}\\,d\\theta \\Big)^{1\/2k},\n\\end{equation}\nfor all $k\\geq 1$, where the implied constant is absolute\n\\end{lem}\n\n\\begin{proof}\nLet $H = \\max_\\theta |p_N(\\theta)|$. The lower bound in \\eqref{eq:Lp_to_sup} is obvious. For the upper bound, note that Bernstein's inequality (see \\cite[Ex 7.16]{katznelson}) implies $|p_N'(\\theta)| \\leq 2\\pi N H$. Thus if $|p_N(\\theta)|$ achieves its maximum for $\\theta = \\theta^\\ast$, then $|p_N(\\theta^\\ast + t)| \\geq H - 2\\pi NH |t|$, so\n$$\n|p_N(\\theta^\\ast + t)| \\geq H\/2, \\quad \\textrm{for} \\; |t| \\leq 1\/2\\pi N.\n$$\nThis immediately implies the upper bound in \\eqref{eq:Lp_to_sup}.\n\\end{proof}\n\nWe now turn to:\n\n\\begin{proof}[Proof of Theorem \\ref{thm:supnorm}]\nWe will prove the lower bound first. We may take $X(n)$ either a Rademacher or Steinhaus random multiplicative function. Note by Chebyshev's inequality,\n$$\n\\mathbb{P}\\left(|\\mathfrak{m}_N^{(k)}-k!| \\,\\geq\\, k!\/2 \\right) \\leq \\frac{\\E\\, \\left| \\mathfrak{m}_N^{(k)} - k! \\right|^2}{(k!)^2\/4} = o(1),\n$$\nfor any choice of $k \\leq A\\,(\\log N\/\\log_2 N)^{1\/3}$ for a small absolute constant $A$. Thus $\\mathfrak{m}_N^{(k)} \\geq k!\/2$ with probability $1-o(1)$, for any choice of $k$ in this range. Hence using the lower bound in Lemma \\ref{lem:Lp_to_sup},\n$$\nH \\coloneqq \\max_\\theta |P_N(\\theta)| \\geq (\\mathfrak{m}_N^{(k)})^{1\/2k} \\geq (k!\/2)^{1\/2k} \\gg \\sqrt{k},\n$$\nwith probability $1-o(1)$ for any choice of $k$ in this range. Using Stirling's formula and taking $k$ to be of order $(\\log N\/\\log_2 N)^{1\/3}$ yields the lower bound in the theorem.\n\nFor the upper bound, if $X(n)$ is a Rademacher random multiplicative function,\n\\begin{multline*}\n\\E\\, \\int_0^1 |P_N(\\theta)|^{2k}\\, d\\theta = \\frac{1}{N^k} \\Nsquare_{k,k}(N) \\leq \\frac{1}{N^k} \\sum_{\\substack{m_1\\cdots m_k n_1\\cdots n_k = \\square \\\\ m_r, n_s \\leq N}} 1 \\\\\n\\leq \\frac{1}{N^k} \\sum_{n \\leq N^k} \\tau_{2k}(n^2) \\leq (2k \\log N)^{4k^2-1},\n\\end{multline*}\nusing \\eqref{eq:tau_of_squares} in the final step. It is easy to see in the same way that if $X(n)$ is instead a Steinhaus random multiplicative function the same bound is satisfied (in fact a slightly better bound is satisfied)\n\nThus from Lemma \\ref{lem:Lp_to_sup}\n$$\n\\E\\, H^{2k} \\leq C^{2k} N (2k)^{4k^2-1} (\\log N)^{4k^2-1},\n$$\nHence for all $\\lambda > 0$ and all integers $k \\geq 1$,\n$$\n\\PP (H \\geq \\lambda) \\leq \\frac{C^{2k} N (2k)^{4k^2-1} (\\log N)^{4k^2-1}}{\\lambda^{2k}}.\n$$\nWe approximately optimize the right-hand side in $k$ by setting $k = \\lfloor \\log \\lambda \/ 6 \\log_2 N \\rfloor$, and then we set $\\lambda = \\exp(3 \\sqrt{\\log N \\log_2 N})$ to see that for such $\\lambda$, \n$$\n\\PP (H \\geq \\lambda) \\rightarrow 0.\n$$\nThis proves the claim.\n\\end{proof}\n\n\\subsection{Large mean and variance}\n\nIn this section we prove Theorem \\ref{thm:main_moments_larger}. We first recall a recent result estimating moments of sums of random multiplicative functions.\n\n\\begin{thm}[Harper]\\label{thm:harper_highmoments}\nThere is a small positive absolute constant $c$ such that the following holds. For $X(n)$ a Steinhaus random multiplicative function, for all $1 \\leq \\ell \\leq c \\log N \/ \\log_2 N$,\n\\begin{equation}\\label{eq:steinhaus_high_moments}\n\\mathbb{E} \\, \\Big| \\sum_{n\\leq N} X(n) \\Big|^{2\\ell} = N^\\ell \\exp(-\\ell^2 \\log \\ell - \\ell^2 \\log_2(2 \\ell) + O(\\ell^2)) (\\log N)^{(\\ell-1)^2}.\n\\end{equation}\nFor $X(n)$ a Rademacher random multiplicative function, for all $2 \\leq \\ell \\leq c \\log N \/ \\log_2 N$,\n\\begin{equation}\\label{eq:rad_high_moments}\n\\mathbb{E} \\, \\Big| \\sum_{n\\leq N} X(n) \\Big|^{2\\ell} = N^\\ell \\exp(-2 \\ell^2 \\log \\ell - 2\\ell^2 \\log_2(2 \\ell) + O(\\ell^2)) (\\log N)^{2 \\ell^2+O(\\ell)}.\n\\end{equation}\n\n\\begin{proof}\nThe estimate \\eqref{eq:steinhaus_high_moments} is just a restatement of of Harper \\cite[Thm. 1]{harper_high}. \\eqref{eq:rad_high_moments} is very nearly \\cite[Thm.2]{harper_high}, except the result there actually pertains to Rademacher random multiplicative functions supported on squarefree integers. However, in our setting, we may recover the estimate \\eqref{eq:rad_high_moments} by way of H\\\"older's inequality. Writing $\\flat$ to indicate a summation over squarefree integers, we see that\n\\begin{align*}\n\\Big| \\sum_{n\\leq N} X(n) \\Big|^{2\\ell} =&\\Big| \\sum_{r^2 \\leq N} \\sum_{n \\leq N\/r^2}^{\\flat} X(n) \\Big|^{2 \\ell} \n\\leq (\\log N)^{2 \\ell} \\max_{\\substack{D \\leq N^{1\/2}\\\\ \\text{dyadic} }}   \\Big| \\sum_{r \\in [D,2D]} \\sum_{n \\leq N\/r^2}^{\\flat} X(n) \\Big|^{2 \\ell}\\\\\n&\\leq  (\\log N)^{2 \\ell} \\sum_{\\substack{D \\leq N^{1\/2} \\\\ \\text{dyadic} }} D^{2 \\ell -1}   \\sum_{r \\in [D,2D]}  \\Big|\\sum_{n \\leq N\/r^2}^{\\flat} X(n) \\Big|^{2 \\ell}.\n\\end{align*}\nWe have used H\\\"older's inequality in the last step, and then replaced the maximum with a sum. Taking expectations, we obtain the desired upper bound. The matching lower bound follows upon noting that our $2\\ell$-th moments, expanded into lattice point counts, are larger than those in the squarefree Rademacher setting. \n\\end{proof}\n\\end{thm}\n\n\\begin{rmk}\nThese estimates hold for all $\\ell$, though we only need them for integer $\\ell$. Harper actually gives a formula for \\eqref{eq:rad_high_moments} in a slightly larger range $1\\leq \\ell \\leq c \\log N \/ \\log_2 N$, with a somewhat more complicated right hand side (involving a phase change at $\\ell = (1+\\sqrt{5})\/2 \\approx 1.618$), but we will not need this.\n\\end{rmk}\n\n\n\nWe now prove Theorem \\ref{thm:main_moments_larger}, treating the mean-value estimate \\eqref{eq:moment_mean_larger}, the lower bound \\eqref{eq:moment_mean_lowerbound}, and the variance estimate \\eqref{eq:moment_variance_larger}  separately.\n\n\n\\begin{proof}[Proof of \\eqref{eq:moment_mean_larger} of Theorem \\ref{thm:main_moments_larger}]\nWe first treat the case that $X(n)$ is a Steinhaus random multiplicative function. We will find upper and lower bounds for the quantity $\\E\\, \\mathfrak{m}_N^{(k)}$. \n\nFor the upper bound, note by examining the point counts corresponding to the left- and right- hand side, we have\n\\begin{equation}\\label{eq:add_to_mult_comp}\n\\E\\, \\int_0^1 |P_N(\\theta)|^{2k}\\, d\\theta \\leq \\E\\, \\Big| \\frac{1}{\\sqrt{N}} \\sum_{n\\leq N} X(n) \\Big|^{2k}.\n\\end{equation}\n\nFor the lower bound, note\n\n\\begin{align*}\n\\E\\, & \\int_0^1 |P_N(\\theta)|^{2k}\\, d\\theta  \\geq   \\E\\, \\int_{[0,N^{-3\/2}]}|P_N(\\theta)|^{2k}\\, d\\theta  \\\\\n&= \\frac1{N^k} \\int_{[0,N^{-3\/2}]}  \\sum_{\\substack{m_r, n_s \\in [1,N] \\\\ m_1\\cdots m_k = n_1\\cdots n_k}} e([(m_1+\\cdots+m_k) - (n_1+\\cdots+n_k)]\\theta)\\,d\\theta\\\\\n&\\geq \\frac{1}{N^{k + 3\/2}}  \\sum_{\\substack{m_r, n_s \\in [1,N] \\\\ m_1\\cdots m_k = n_1\\cdots n_k}} \\Big( 1 - O\\Big(\\frac{k}{N^{1\/2}}\\Big)\\Big)\n\\end{align*}\nusing a pointwise bound of the (positive) integrand in $\\theta$ to arrive at the last line. For $k \\leq N^{1\/4}$ and sufficiently large $N$ this gives\n\\begin{equation}\\label{eq:lower_add_to_mult_comp}\n\\E\\, \\int_0^1 |P_N(\\theta)|^{2k}\\, d\\theta \\gg \\frac{1}{N^{3\/2}} \\E\\, \\Big| \\frac{1}{\\sqrt{N}} \\sum_{n\\leq N} X(n) \\Big|^{2k}\n\\end{equation}\nBy possibly adjusting the implicit constant and using compactness it follows that this result is true for all $N$.\n\n\\eqref{eq:add_to_mult_comp} and \\eqref{eq:lower_add_to_mult_comp} then give \\eqref{eq:moment_mean_larger}.\n\nThe case for $X(n)$ a Rademacher random multiplicative function is the same, replacing a sum over $m_1\\cdots m_k = n_1 \\cdots n_k$ with a sum over $m_1\\cdots m_k n_1 \\cdots n_k = \\square$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\eqref{eq:moment_mean_lowerbound} of Theorem \\ref{thm:main_moments_larger}]\nLet $X(n)$ be either a Steinhaus or Rademacher random multiplicative function. We claim that for any $\\delta > 0$, that if $N$ is sufficiently large, $C_\\delta$ is a sufficiently large constant, and  $C_\\delta (\\log N\/\\log_2 N)^{1\/2} \\leq k$, then\n\\begin{equation}\\label{eq:log_lowerbound}\n\\log N \\leq \\delta \\log \\E\\, \\Big| \\frac{1}{\\sqrt{N}} \\sum_{n\\leq N} X(n)\\Big|^{2k}.\n\\end{equation}\n\nThis follows directly from Theorem \\ref{thm:harper_highmoments} if $k = k_0 = \\lceil C_\\delta (\\log N\/\\log_2 N)^{1\/2} \\rceil$ as long as $C_\\delta$ and $N$ are chosen large enough. But note for $k \\geq k_0$\n$$\n\\log \\E\\, \\Big| \\frac{1}{\\sqrt{N}} \\sum_{n\\leq N} X(n)\\Big|^{2k_0} \\leq \\frac{2k_0}{2k} \\log \\E\\, \\Big| \\frac{1}{\\sqrt{N}} \\sum_{n\\leq N} X(n)\\Big|^{2k} \\leq \\log \\E\\, \\Big| \\frac{1}{\\sqrt{N}} \\sum_{n\\leq N} X(n)\\Big|^{2k},\n$$\nusing H\\\"older's inequality to deduce the first inequality. This implies that \\eqref{eq:log_lowerbound} is true for all $k$ as claimed.\n\n\\eqref{eq:moment_mean_lowerbound} then follows from \\eqref{eq:moment_mean_larger} and \\eqref{eq:log_lowerbound}, choosing $\\delta$ based on $\\varepsilon$ and the implicit constant in \\eqref{eq:moment_mean_larger}.\n\\end{proof}\n\nIn order to treat the variance estimate \\eqref{eq:moment_variance_larger} we use the following lemma.\n\n\\begin{lem}\\label{lem:var_lowerbound}\nLet $X(n)$ be a Rademacher or Steinhaus random multiplicative function and suppose that $k\\leq N^{1\/4}$. We have the lower bound\n\\begin{equation}\\label{vartransfer}\nN^{2k} \\cdot \\Var(\\mathfrak{m}_N^{(k)})  \\geq N^{-3} \\ \\E \\Big|\\sum_{n \\leq N} X(n) \\Big|^{4k} - \\left( \\E \\Big|\\sum_{n \\leq N} X(n) \\Big|^{2k} \\right)^2.\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nWe treat the case that $X(n)$ is a Steinhaus random multiplicative function first. Since $\\E (\\mathfrak{m}_N^{(k)}) \\leq N^{-k}\\  \\E \\left( |\\sum_{n \\leq N} X(n)|^{2k} \\right) $, we first observe that\n\n\\begin{align}\\label{Varlower}\n\\Var(\\mathfrak{m}_N^{(k)}) &=\\E[(\\mathfrak{m}_N^{(k)})^2] - (\\E\\, \\mathfrak{m}_N^{(k)})^2 \\notag \\\\\n&\\geq   N^{-2k}\\sideset{}{^*}\\sum_{\\substack{ {\\bf m}, {\\bf m'} \\in [1,N]^k \\\\ {\\bf n}, {\\bf n'} \\in [1,N]^k}} 1 - N^{-2k} \\left( \\E  \\big|\\sum_{n \\leq N} X(n)\\big|^{2k}  \\right)^2\n\\end{align}\nwhere the asterisked sum here is restricted to tuples ${\\bf m},{ \\bf m'}, {\\bf n}, {\\bf n'}$ satisfying,\n\\begin{align}\\label{Vsys1}\n\\begin{split}\n\\sum_{j \\leq k} m_j =  \\sum_{j \\leq k} n_j, &\\qquad \\sum_{j \\leq k} m_j' =  \\sum_{j \\leq k} n_j' \\\\\nm_1\\cdots m_k m_1'\\cdots m_k' &= n_1\\cdots n_k n_1' \\cdots n_k'.\n\\end{split}\n\\end{align}\n\n\n\nTo treat the asterisked sum on the right-hand side of \\eqref{Varlower}, consider the random trigonometric polynomial\n$$\nF_k(\\alpha,\\alpha')= \\sum_{ {\\bf n}, {\\bf n'} \\in [1,N]^k} \nX(n_1 \\cdots n_k n_1' \\cdots n_k') \\, e\\Big(\\alpha \\sum_{j \\leq k} n_j +\\alpha' \\sum_{j \\leq k} n_j' \\Big)\n$$\nand observe that \n\n\\begin{align}\\label{starsumlower}\n\\sideset{}{^*}\\sum_{\\substack{ {\\bf m}, {\\bf m'} \\in [1,N]^k \\\\ {\\bf n}, {\\bf n'} \\in [1,N]^k}} 1  \n&= \\int_{[0,1]^2} \\E \\, | F_k(\\alpha,\\alpha')|^2 \\, d \\alpha \\, d \\alpha' \\notag \\\\\n&\\geq \\int_{[0,N^{-3\/2}]^2} \\E \\, | F_k(\\alpha,\\alpha')|^2 \\, d \\alpha \\, d \\alpha'.  \n\\end{align}\n\nFor any $|\\alpha|,|\\alpha'|\\leq N^{-3\/2}$ we gather that \n\n\n\\begin{align*}\n\\E \\, |F_k(\\alpha,\\alpha')|^2 &=\n\\sideset{}{^\\dagger}\\sum_{\\substack{ {\\bf n}, {\\bf n'}   \\in [1,N]^k  \\\\  {\\bf m}, {\\bf m'} \\in [1,N]^k }} \ne\\Big(\\alpha \\big(\\sum_{j \\leq k} n_j - \\sum_{j \\leq k} m_j \\big)+\\alpha' \\big(\\sum_{j \\leq k} n_j' - \\sum_{j \\leq k} m_j' \\big) \\Big)\\\\\n& \\geq \\sideset{}{^\\dagger}\\sum_{\\substack{ {\\bf n}, {\\bf n'}   \\in [1,N]^k  \\\\  {\\bf m}, {\\bf m'} \\in [1,N]^k }} \\left( 1+O\\Big( \\frac{k}{N^{1\/2} } \\Big) \\right)\n\\gg \\sideset{}{^\\dagger}\\sum_{\\substack{ {\\bf n}, {\\bf n'}   \\in [1,N]^k  \\\\  {\\bf m}, {\\bf m'} \\in [1,N]^k }} 1,\n\\end{align*}\nwhere the outer sums range over ${\\bf n}, {\\bf n'}, {\\bf m}, {\\bf m'} \\in [1,N]^k $ satisfying the multiplicative constraint in \\eqref{Vsys1}. Inserting this information into \\eqref{starsumlower} and then \\eqref{Varlower}, we get that \n \n$$\\sideset{}{^*}\\sum_{\\substack{ {\\bf m}, {\\bf m'} \\in [1,N]^k \\\\ {\\bf n}, {\\bf n'} \\in [1,N]^k}} 1  \\gg N^{-3} \\sideset{}{^\\dagger}\\sum_{\\substack{ {\\bf n}, {\\bf n'}   \\in [1,N]^k  \\\\  {\\bf m}, {\\bf m'} \\in [1,N]^k }} 1=N^{-3} \\ \\E \\Big( \\Big|\\sum_{n \\leq N} X(n)\\Big|^{4k} \\Big).$$\nCollecting all of the previous estimates, we find the lower bound \\eqref{vartransfer}.\n\nThe proof for $X(n)$ a Rademacher random variable carries through in the same manner. The multiplicative constraint in \\eqref{Vsys1} is replaced by the constraint \n$$\nm_1\\cdots m_k m_1'\\cdots m_k' n_1\\cdots n_k n_1' \\cdots n_k' = \\square.\n$$\n\\end{proof}\n\n\\begin{proof}[Proof of \\eqref{eq:moment_variance_larger} of Theorem \\ref{thm:main_moments_larger}]\nLemma \\ref{lem:var_lowerbound} shows that in the range $k \\leq N^{1\/4}$, for $X(n)$ either a Steinhaus or Rademacher random multiplicative function, we will have\n\\begin{equation}\\label{eq:var_upper_const1}\n\\Var(\\mathfrak{m}_N^{(k)}) \\geq (\\E\\, \\mathfrak{m}_N^{(k)})^2\n\\end{equation}\nif\n\\begin{equation}\\label{eq:moment_ratio}\n\\Big(\\E\\, \\Big| \\sum_{n\\leq N} X(n)\\Big|^{4k}\\Big)\\bigg\/\\Big(\\E\\,\\Big| \\sum_{n\\leq N} X(n) \\Big|^{2k}\\Big)^2 \\geq 2N^3.\n\\end{equation}\nBut for $k \\leq \\tfrac{c}{2}\\log N \/ \\log_2 N$ we may apply the estimates of Theorem \\ref{thm:harper_highmoments} to the numerator and denominator of the left-hand side of \\eqref{eq:moment_ratio}, and we see with a little computation that such an inequality is true as long as $k \\geq B (\\log N \/ \\log_2 N)^{1\/2}$ for a sufficiently large absolute constant $B$, in both the Steinhaus and Rademacher cases. \n\nThis proves \\eqref{eq:moment_ratio} for $B(\\log N \/\\log_2 N)^{1\/2} \\leq k \\leq \\tfrac{c}{2} \\log N \/ \\log_2 N$. But note that if for $p \\in (0,\\infty)$ we define\n$$\n\\phi(p) = \\log\\, \\E\\, \\Big| \\sum_{n \\leq N} X(n)\\Big|^{2p},\n$$\nthen $\\phi(p)$ is a convex function (see \\cite[Thm. 5.5.1]{garling}). It follows that $\\phi(2k) - 2 \\phi(k)$ is an increasing function in $k$. Therefore, fixing $N$, if \\eqref{eq:moment_ratio} is true for some value of $k$, it will remain true for all larger values of $k$. This implies \\eqref{eq:var_upper_const1} for all $B(\\log N \/\\log_2 N)^{1\/2} \\leq k \\leq N^{1\/4}$, completing the proof of Theorem \\ref{thm:main_moments_larger}.\n\\end{proof}\n\n\\begin{rmk}\nWe have made no attempt to optimize the upper limit $N^{1\/4}$ above. It may be that \\eqref{eq:moment_variance_larger} of Theorem \\ref{thm:main_moments_larger} holds for all $B(\\log N \/\\log_2 N)^{1\/2} \\leq k < \\infty$. \n\\end{rmk}\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{\\textbf{$\\#$ #1}}}\n\n\\newcommand{\\myrefeq}[1]{Eq.~\\eqref{#1}}\n\\newcommand{\\myreffig}[1]{Fig.~\\ref{#1}}\n\\newcommand{\\myreftab}[1]{Table~\\ref{#1}}\n\\newcommand{\\myrefsec}[1]{Sec.~\\ref{#1}}\n\\newcommand{\\myrefapp}[1]{Appendix~\\ref{#1}}\n\\newcommand{\\myrefalg}[1]{Alg.~\\ref{#1}}\n\n\\makeatletter\n\\renewcommand{\\ALG@name}{Alg.}\n\\makeatother\n\n\\newcommand{\\mathbf{v}}{\\mathbf{v}}\n\\newcommand{\\mathbf{w}}{\\mathbf{w}}\n\\newcommand{\\rho}{\\rho}\n\\renewcommand{\\vec}{\\mathbf}\n\\newcommand{\\mathbb{E}}{\\mathbb{E}}\n\\captionsetup[table]{labelformat=empty}\n\n\\begin{abstract}\nWe propose a temporally coherent generative model addressing \nthe super-resolution problem for fluid flows. \nOur work represents a first approach\nto synthesize four-dimensional physics fields with neural networks.\nBased on a conditional generative adversarial network\nthat is designed for the inference of three-dimensional volumetric data, \nour model generates consistent and detailed results \nby using a novel temporal discriminator, in addition to the commonly used spatial one.\nOur experiments show that the generator is able to\ninfer more realistic high-resolution details by using\nadditional physical quantities, such as low-resolution velocities or vorticities.\nBesides improvements in the training process and in the generated outputs, \nthese inputs offer means for artistic control as well.\nWe additionally employ a physics-aware data augmentation step,\nwhich is crucial to avoid overfitting and to reduce memory requirements.\nIn this way, our network learns to \ngenerate advected quantities with highly detailed, realistic, and temporally coherent features.\nOur method works instantaneously, using only a single time-step of low-resolution fluid data.\nWe demonstrate the abilities of our method using a variety of complex inputs and applications in\ntwo and three dimensions.\n\\end{abstract}\n\n\\begin{document}\n\\title{tempoGAN: A Temporally Coherent, Volumetric GAN for Super-resolution Fluid Flow}\n\\author{You Xie*}\n\\affiliation{\n\t\\institution{Technical University of Munich}}\n\\email{you.xie@tum.de}\n\\author{Erik Franz*}\n\\affiliation{\n\t\\institution{Technical University of Munich}}\n\\email{franzer@in.tum.de}\n\\author{Mengyu Chu*}\n\\affiliation{\n\t\\institution{Technical University of Munich}}\n\\email{mengyu.chu@tum.de}\n\\author{Nils Thuerey}\n\\affiliation{\n\t\\institution{Technical University of Munich}}\n\\email{nils.thuerey@tum.de}\n\\renewcommand\\shortauthors{Xie, Y., Franz, E., Chu, M., Thuerey, N.}\n\n\\begin{CCSXML}\n<ccs2012>\n<concept>\n<concept_id>10010147.10010257.10010293.10010294<\/concept_id>\n<concept_desc>Computing methodologies~Neural networks<\/concept_desc>\n<concept_significance>500<\/concept_significance>\n<\/concept>\n<concept>\n<concept_id>10010147.10010371.10010352.10010379<\/concept_id>\n<concept_desc>Computing methodologies~Physical simulation<\/concept_desc>\n<concept_significance>500<\/concept_significance>\n<\/concept>\n<\/ccs2012>\n\\end{CCSXML}\n\n\\ccsdesc[500]{Computing methodologies~Neural networks}\n\\ccsdesc[500]{Computing methodologies~Physical simulation}\n\n\\keywords{physics-based deep learning, generative models, computer animation, fluid simulation}\n\n\\thanks{\n(*) Similar amount of contributions.\n }\n\n\\begin{teaserfigure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{teaser7.jpg}\n\t\\caption{Our convolutional neural network learns to generate highly detailed,\n\tand temporally coherent features based on a low-resolution field containing a single time-step\n\tof density and velocity data. We introduce a novel discriminator that ensures\n\tthe synthesized details change smoothly over time.}\n\t\\label{fig:teaser}\n\\end{teaserfigure}\n\n\\maketitle\n\n\\section{Introduction} \\label{sec:intro}\n\nGenerative models were highly successful in the last years\nto represent and synthesize complex natural images \n\\cite{goodfellow2014generative}.\nThese works demonstrated that deep convolutional neural networks (CNNs) are able to\ncapture the distribution of, e.g., photos of human faces, and generate novel,\npreviously unseen versions that are virtually indistinguishable from the original\ninputs.  Likewise, similar algorithms were shown to be extremely successful at\ngenerating natural high-resolution images from a coarse input \\cite{karras2017growgan}. \nHowever, in their original\nform, these generative models do not take into account the temporal\nevolution of the data, which is crucial for realistic physical systems. In the\nfollowing, we will extend these methods to generate high-resolution volumetric\ndata sets of passively advected flow quantities, and ensuring temporal coherence \nis one of the core aspects that we will focus on below.\nWe will demonstrate that it is especially important to make the training process\naware of the underlying transport phenomena, such that the network can learn to\ngenerate stable and highly detailed solutions.\n\nCapturing the intricate details of turbulent flows has been a long-standing\nchallenge for numerical simulations. Resolving such details with discretized\nmodels induces enormous com\\-pu\\-tatio\\-nal costs and quickly becomes\ninfeasible for flows on human space and time scales. \nWhile algorithms to increase the apparent resolution\nof simulations can alleviate this problem \\cite{Kim:2008:wlt}, \nthey are typically based on procedural models that are only loosely inspired by\nthe underlying physics.\nIn contrast to all previous methods, our algorithm represents a\nphysically-based interpolation, that does not require any form\nof additional temporal data or quantities tracked over time. The\nsuper-resolution process is instantaneous, based on volumetric\ndata from a single frame of a fluid simulation.\nWe found that inference of \nhigh-resolution data in a fluid flow setting benefits from the availability\nof information about the flow. In our case, this takes the shape of\nadditional physical variables such as velocity and vorticity\nas inputs, which in turn yield means for artistic control.\nA particular challenge in the field of super-resolution flow is how to \nevaluate the quality of the generated output. As we are typically\ntargeting turbulent motions, a single coarse approximation can be associated\nwith a large variety of significantly different high-resolution versions.\nAs long as the output matches the correlated spatial \nand temporal distributions of the reference data, it represents a correct solution. To encode this requirement in the training\nprocess of a neural network, we employ so-called generative adversarial\nnetworks (GANs). These methods\ntrain a {\\em generator}, as well as a second network, the {\\em discriminator} \nthat learns to judge how closely the generated output matches the ground truth data.\nIn this way, we train a specialized, data-driven loss function\nalongside the generative network, while making sure it\nis differentiable and compatible with the training process.\nWe not only employ this adversarial approach for the\nsmoke density outputs, but we also train a specialized and novel adversarial\nloss function that learns to judge the temporal coherence of the\noutputs. \n\nWe additionally present best practices to set up a training pipeline for\nphysics-based GANs. E.g., we found it particularly useful to have physics-aware\ndata augmentation functionality in place. The large amounts of space-time data\nthat arise in the context of many physics problems quickly bring typical\nhardware environments to their limits. \nAs such, we found data augmentation crucial to avoid overfitting.\nWe also explored a variety of different variants\nfor setting up the networks as well as training them, \nand we will evaluate them in terms of their capabilities to learn\nhigh-resolution physics functions below.\n\nTo summarize, the main contributions of our work are:\n\\begin{itemize}\n{\\vspace{-2pt}\n\\item a novel temporal discriminator, to generate consistent and high\\-ly detailed results over time,\n\\item artistic control of the outputs, in the form of additional loss terms and\n\tan intentional entangling of the physical quantities used as inputs,\n\\item a physics aware data augmentation method,\n\\item and a thorough evaluation of adversarial training processes for physics functions.}\n\\end{itemize}\nTo the best of our knowledge, our approach is the first generative adversarial network\nfor four-dimensional functions,\nand we will demonstrate that it\nsuccessfully learns to infer solutions for flow transport processes from approximate solutions.\n\\revision{ A high level preview of the architecture we propose can\nbe found in \\myreffig{fig:tempoGANcoarse}.}\n\\begin{figure}[tb]\n\t\\centering \n\t\\includegraphics[width=\\linewidth]{coarseview.pdf} \\\\\n\t\\caption{ \\revision{This figure gives a high level overview of our approach: a generator on the \n\tleft, is guided during training by two discriminator networks (right), one of which\n\tfocuses on space ($D_s$), while the other one focuses on temporal aspects ($D_t$).\n\tAt runtime, both are discarded, and only the generator network is evaluated.} }\n\t\\label{fig:tempoGANcoarse}\n\\end{figure}\n\\section{Related Work}\nIn the area of computer vision, deep learning techniques have achiev\\-ed significant breakthroughs\nin numerous fields such as classification \\cite{krizhevsky2012imagenet}, object detection \\cite{girshick2014rich}, \nstyle transfer \\cite{luan2017deep}, novel view synthesis \\cite{flynn2016deepstereo}, and \nadditionally, in the area of content creation. For more in-depth reviews of neural networks\nand deep learning techniques, we refer the readers to corresponding books \\cite{bishop2006book,Goodfellow2016}. \n\nOne of the popular methods to generate content are so called {\\em generative adversarial networks} (GANs), \nintroduced by Goodfellow et al. \\cite{goodfellow2014generative}. They were shown \nto be particularly powerful at re-creating the distributions of complex data sets such as images\nof human faces.\nDepending on the kind of input data they take, GANs can be separated into unconditional and conditional ones. \nThe formers generate realistic data from samples of a synthetic data distribution\nlike Gaussian noise. The DC-GAN \\cite{RadfordMC15} is a good example of an unconditional\nGAN. It was designed for generic natural images, \nwhile the cycle-consistent GAN by Zhu et al.  \\shortcite{zhu2017cycle} was developed\nto translate between different classes of images.\nThe conditional GANs were introduced by Mirza and Osindero \\shortcite{mirza2014conditional}, and\nprovide the network with an input that is in some way related to the target function in order\nto control the generated output.\nTherefore, conditional variants are popular for transformation tasks,\nsuch as image translations problems \\cite{isola2016image} and \nsuper resolution problems \\cite{ledig2016photo}. \n\nIn the field of super-resolution techniques, researchers have explored different network architectures.\nE.g., convolutional net\\-works \\cite{dong2016image} were shown to be more effective than fully connected architectures.\nThese networks can be trained with smaller tiles and\n later on be applied to images of arbitrary sizes \\cite{isola2016image}.\nBatch normalization \\cite{lim2017enhanced} significantly improves results by removing value \nshifting in hidden layers, \nand networks employing so-called {\\em residual blocks}\n\\cite{kim2016accurate, lim2017enhanced} enable the training of deep networks \nwithout strongly vanishing gradients.\n\nIn term of loss functions, pixel-wise loss between the network output and the ground truth data \nused to be common, such as the $L_1$ and $L_2$ loss \\cite{dong2016image}.\nNowadays, using the adversarial structure of GANs, or using pre-trained\nnetworks, such as the VGG net \\cite{simonyan2014} often leads to higher perceptual qualities\n\\cite{mathieu2015deep, johnson2016perceptual}.\n\nOur method likewise uses residual blocks in conjunction with a conditional GAN architecture to\ninfer three-dimensional flow quantities. Here, we try to use standard architectures. \nWhile our generator is similar\nto approaches for image super-resolution \\cite{ledig2016photo}, \nwe show that loss terms and discriminators are crucial for high-quality outputs.\nWe also employ a fairly traditional GAN training, instead of recently proposed alternatives\n\\cite{arjovsky2017wasserstein,berthelot2017began}, which could potentially lead to additional gains in quality.\nBesides the super-resolution task, our work differs from many works in the GAN area\nwith its focus on temporal coherence, as we will demonstrate in more detail later on.\n\nWhile most works have focused on single images, several papers have addressed\ntemporal changes of data sets. One way to solve this problem is by \ndirectly incorporating the time axis, i.e., by using sequences of data as input and output.\nE.g., Saito et al. propose a temporal generator in their work \\cite{saito2017temporal},\nwhile Yu et al. \\cite{yu2017seqgan} proposed a sequence generator that learns a stochastic policy.\nIn these works, results need to be generated\nsequentially, while our algorithm processes individual frames independently, and in arbitrary order, if necessary.\nIn addition, such approaches would explode in terms of weights and computational resources for\ntypical four-dimensional fluid data sets.\n\nAn alternative here is to generate single frame data with additional\nloss terms to keep the results coherent over time.\nBhattacharjee etc. \\shortcite{bhattacharjee2017temporal} achieved improved coherence in their results for video frame prediction, \nby adding specially designed distance measures as a discontinuity penalty between nearby frames.\nFor video style transfer, a $L_2$ loss on warped nearby frames helped to \nalleviate temporal discontinuities, as shown by Ruder et al. \\shortcite{Ruder2016Artistic}. \nIn addition to a $L_2$ loss on nearby frames, Chen et al. \\shortcite{Chen2017ICCV} used neural networks to learn \nframe warping and frame combination in VGG feature space.\nSimilarly, Liu etc. \\cite{liu2017VideoSR} used neural networks to learn\nspatial alignment for low-resolution inputs, and adaptive aggregation for high-resolution outputs,\nwhich also improved the temporal coherence.\nDue to the three-dimensional data sets we are facing, we also adopt the single\nframe view. However, in contrast to all previous works, we propose the use\nof a temporal discriminator. We will show that relying on data-driven, learned\nloss functions in the form of a discriminator helps to improve results over manually\ndesigned losses. Once our networks are trained, this discriminator can be discarded.\nThus, unlike, e.g., aggregation methods, our approach does not influence runtime performance.\nWhile previous work shows that warping layers are useful in motion field \nlearning~\\cite{de2017deep, Chen2017ICCV}, our work targets the opposite direction:\nby providing our networks with velocities, warping layers \ncan likewise improve the training of temporally coherent content generation.\n\nMore recently, deep learning algorithms have begun to influence \ncomputer graphics algorithms.\nE.g., they were successfully used for efficient and noise-free renderings \\cite{bako2017kernel,chaitanya2017interactive}, \nthe illumination of volumes \\cite{kallweit2017deep}, \nfor modeling porous media \\cite{mosser2017porous}, and for character control \\cite{peng2017deeploco}.\nFirst works also exist that target numerical simulations. E.g., a conditional GAN\nwas used to compute solutions for smaller, two-dimensional advection-diffusion\nproblems \\cite{farimani2017, long2017pde}.\nOthers have demonstrated the inference of SPH forces with regression forests \\cite{ladicky2015data},\nproposed CNNs for fast pressure projections \\cite{tompson2016accelerating},\nlearned space-time deformations for interactive liquids \\cite{rtliquids2017} , and \nmodeled splash statistics with NNs \\cite{um2017mlflip}.\nCloser to our line of work, Chu et al. \\shortcite{chu2017cnnpatch} proposed\na method to look up pre-computed patches using CNN-based descriptors. Despite a similar goal, \ntheir methods still require additional Lagrangian tracking information, while our method\ndoes not require any modifications of a basic solver. In addition, our method does not use\nany stored data at runtime apart from the trained generator model.\n\nAs our method focuses on the conditional inference of\nhigh-re\\-so\\-lu\\-tion flow data sets, i.e. solutions of the {\\em Navier-Stokes} (NS) equations, we also give a brief overview of the\nrelated work here, with a particular focus on single-phase flows.\n\\revision{After the introduction of the stable fluids algorithm \\cite{Stam1999},} \n\\revision{a variety of extensions and variants have been developed over the years.\nE.g., more accurate Eulerian ad\\-vec\\-tion sche\\-mes \\cite{Kim05FlowFixer,Selle:2008:USM} \nare often employed, an alternative to which are Lagrangian versions \\cite{rasmussen2003smoke,magnus2011capturing}.\nWhile grids are more commonly used, particles can achieve non-dissipative results for which\na Eulerian grid would require a significant amount of refinement.\nFurthermore, procedural turbulence methods to increase apparent resolutions are popular \nextensions \\cite{Kim:2008:wlt,narain:2008:procTurb,schechter2008evolving}.\nIn contrast to our work, the different advection schemes and procedural\nturbulence methods require a calculation of the\nhigh-resolution transport of the density field over the full simulation sequence. \nAdditionally, Eulerian and Lagrangian representations can be advected in a parallelized \nfashion on a per frame basis, in line with the application of convolutions for NNs.\nOur method infers an instantaneous solution to the underlying advection problem\nbased only on a single snapshot of data, without having to compute a series of previous \ntime steps.}\n\nOur work also shares similarities in terms\nof goals with other phy\\-sics-based up-sampling algorithms \\cite{kavan2011physics},\nand due to this goal, is related to fluid control methods \\cite{McNamaraAdjointMethod,Pan:2013}.\nThese methods would work very well in conjunction with our approach, in order to \ngenerate a coarse input with the right shape and timing.\n\n\\section{Adversarial Loss Functions}\n\\label{sec:loss}\nBased on a set of low-resolution inputs, with corresponding\nhigh-resolution references, our goal is to train a CNN that produces\na temporally coherent, high-resolution solution with adversarial train\\-ing.\nWe will first very briefly summarize the basics of adversarial training,\nand then explain our extensions for temporal coherence and for \\revision{results control.} \n\\subsection{Generative Adversarial Networks}\n\\label{sec:ganbasic}\nGANs consist of two models, which are trained in conjunction:\nthe generator $G$ and the discriminator $D$. Both will be realized as convolutional neural networks in our case. \nFor regular ones, i.e., not conditional GANs,\nthe goal is to train a generator $G(x)$ that maps a simple data distribution, typically noise, $x$ to a complex desired output $y$, e.g., natural images.\nInstead of using a manually specified loss term to train the generator, another NN, the discriminator, is used as complex, learned loss function \\cite{goodfellow2014generative}.\nThis discriminator takes the form of a simple binary classifier, which is trained in a supervised manner \nto reject {\\em generated} data, i.e., it should return $D(G(x)) = 0$, and accept the {\\em real} data with $D(y) =1$.\nFor training, the loss for the discriminator is thus given by a sigmoid cross entropy\nfor the two classes ``generated'' and ``real'':\n\\revision{\n\\resizeEq{\n\\begin{split}\n\\mathcal{L}_D(D,G)= & \\mathbb{E}_{y\\sim p_{\\text{y}}(y)}[-\\log D(y)] +\\mathbb{E}_{x\\sim p_{\\text{x}}(x)}[-\\log (1-D(G(x)))] \\\\\n= & \\mathbb{E}_m [-\\log D(y_m)] + \\mathbb{E}_n [-\\log (1-D(G(x_n)))]\\ ,\n\\end{split} \n}{eq:disloss}\n}\nwhere $n$ is the number of drawn inputs $x$, while $m$ denotes the number of real data samples $y$.\n\\revision{Here we use the notation $y\\sim p_{\\text{y}}(y)$ for samples $y$ being drawn from a corresponding \nprobability data distribution $p_{\\text{y}}$, which will later on be represented by our numerical simulation framework.}\n\\revision{The continuous distribution $\\mathcal{L}_D(D,G)$ yields the average of discrete samples $y_n$ and $x_m$ in the second line of \\myrefeq{eq:disloss}.\nWe will omit the $y\\sim p_{\\text{y}}(y)$ and $x\\sim p_{\\text{x}}(x)$ subscripts of the sigmoid cross entropy,}\nand $n$ and $m$ subscripts of $D(y_m)$ and $G(x_n)$, for clarity below.\n\nIn contrast to the discriminator, the generator is trained to ``fool'' the discriminator into accepting its samples and thus to generate output that is close to the real data from $y$.\nIn practice, this means that the generator is trained to drive the discriminator result for its outputs to one.\nInstead of directly using the negative discriminator loss, GANs typically use\n\\revision{\n\\resizeEq{\n\\begin{split}\n\\mathcal{L}_G(D,G)= \\mathbb{E}_{x\\sim p_{\\text{x}}(x)}[-\\log (D(G(x)))] = \\mathbb{E}_n [-\\log (D(G(x)))]\n\\end{split}}{}\n}\nas the loss function for the generator, in order to reduce diminishing gradient problems \n\\cite{goodfellow2016nips}.\nAs $D$ is realized as a NN, it is guaranteed to be sufficiently differentiable as a loss function for $G$.\nIn practice, both discriminator and generator are trained in turns and will optimally reach an\nequilibrium state.\n\nAs we target a super-resolution problem, our goal is not to generate an arbitrary high-resolution output, but\none that corresponds to a low-resolution input, and hence we employ a {\\em conditional} GAN.\nIn terms of the dual optimization problem described above, this means that the input $x$ now represents\nthe low-resolution data set, and the discriminator is provided with $x$ in order to establish and \nensure the correct relationship\nbetween input and output, i.e., we now have $D(x,y)$ and $D(x,G(x))$ \\cite{mirza2014conditional}.\nFurthermore, previous work \\cite{zhao2015loss}\nhas shown that an\nadditional $L_1$ loss term with a small weight can be added to the generator to ensure that its output stays close \nto the ground truth $y$. This yields $\\lambda _{L_1}  \\mathbb{E}_n \\left \\| G(x)-{y}\\right \\|_{1}$, where $\\lambda _{L_1}$\ncontrols the strength of this term, and we use $\\mathbb{E}$ for consistency to denote the\nexpected value, in this discrete case being equivalent to an average.\n\n\\subsection{ Loss in Feature Spaces}\n\\label{sec:featureloss}\n\n\\revision{In order to further control the coupled, non-linear optimization process, the features\nof the underlying CNNs can be constrained. This is an important issue, as controlling\nthe training process of GANs is known as a difficult problem.\nHere, we extend previous work on feature space losses, which\nwere shown to improve realism in natural images~\\cite{dosovitskiy2016generating}, and\nwere also shown to help with mode collapse problems \\cite{salimans2016improved}.} \nTo achieve this goal, an $L_2$ loss over parts or the whole feature space of a neural network is introduced for the generator.\nI.e., the intermediate results of the generator network are constrained w.r.t. a set of intermediate reference data.\nWhile previous work typically makes use of manually selected layers\nof pre-trained networks, such as the VGG net, we propose to use features of the discriminator as constraints instead.\n\nThus, we incorporate a novel loss term of the form\n\\begin{equation}\n\\mathcal{L}_{f}=\\mathbb{E} _{n,j} \\lambda _{f}^{j}\\left \\| F^{j}(G(x))-F^{j}(y) \\right \\|_{2}^{2} \\ ,\n\\label{eq:featureloss}\n\\end{equation}\nwhere $j$ is a layer in our discriminator network, and $F^{j}$ denotes the activations of the corresponding layer.\nThe factor $\\lambda _{f}^{j}$ is a weighting term, which can be adjusted on a per layer basis, as we will discuss in \\myrefsec{sec:artcontrol}. \nIt is particularly important in this case that we can employ the discriminator here, as no suitable, pre-trained networks\nare available for three-dimensional flow problems.\n\n\\revision{Interestingly, these weights yields different and realistic results both for positive as well as negative choices for the weights.\nFor $\\lambda _{f} > 0$\nthese loss terms effectively encourage a minimization of the mean feature space distances of real and\ngenerated data sets, such that generated features resemble features of the reference. \nSurprisingly, we found that training runs with $\\lambda _{f} < 0$\nalso yield excellent, and often slightly better results.\nAs we are targeting conditional GANs, our networks\nare highly constrained by the inputs. \nOur explanation for this behavior is that a negative feature loss in this setting encourages the optimization\nto generate results that differ in terms of the features, but are still similar, ideally indistinguishable, in \nterms of their final output. This is possible as we are not targeting a single ground-truth result, but \nrather, we give the generator the freedom to generate any result that best fits the collection of inputs\nit receives. From our experience, this loss term drives the generator towards realistic detail,\nan example of which can be seen in \\myreffig{fig:2dlayerlossexample}. \nNote \nthat due to the non-linear nature of the optimization, linearly changing $\\lambda _{f}$ \nyields to models with significant differences in the generated small scale features.}\n\\begin{figure}[t]\n     \\centering \n\t \\begin{overpic}[width=0.24\\linewidth]{low_smallcut.png}\n\t \\put( 5,5){\\small \\color{white}{$a)$}}\n\t \\end{overpic}\n\t \\begin{overpic}[width=0.24\\linewidth]{l2_smallcut.png}\n\t \\put( 5,5){\\small \\color{white}{$b)$}}\n \t\\end{overpic}\n\t \\begin{overpic}[width=0.24\\linewidth]{test_13_smallcut.png}\n\t \\put( 5,5){\\small \\color{white}{$c)$}}\n\t \\end{overpic}\n\t \\begin{overpic}[width=0.24\\linewidth]{highreference_smallcut.png}\n\t\\put( 5,5){\\small \\color{white}{$d)$}}\n   \t\\end{overpic}\n     \\caption{From left to right: a) a sample, low-resolution input, b) a CNN output with naive $L_2$ loss (no GAN training), c)\n     our tempoGAN output, and d) the high-resolution reference. The $L_2$ version learns a smooth result\n     without small scale details, while our output in (c) surpasses the detail of the reference in certain regions.\n     } \\label{fig:2dlayerlossexample}\n\\end{figure}\n\n\\subsection{ Temporal Coherence}\n\\label{sec:tempo}\nWhile the GAN process described so far is highly successful at generating highly \ndetailed and realistic outputs for static frames, these details are particularly challenging\nin terms of their temporal coherence.\nSince both the generator and the discriminator work on every frame independently, subtle changes of the input $x$ can lead to outputs $G(x)$ with distinctly different \n details for higher spatial frequencies.\n\nWhen the ground truth data $y$ comes from a transport process, such as frame motion or flow motion, it\n typically exhibits a very high degree of temporal coherence, and a velocity field $v_y$ exists for which \n $y^{t} = \\mathcal{A}( y^{t-1}, v_y^{t-1} )$. Here, we denote the advection operator\n (also called warp or transport in other works) with\n$\\mathcal{A}$, and we assume without loss of generality\nthat the time step between frame $t$ and $t-1$ is equal to one. Discrete time steps\nwill be denoted by superscripts, i.e., for a function $y$ of space and time $y^t = y(\\vec{x},t)$ denotes\na full spatial sample at time $t$.\nSimilarly, in order to solve the temporal coherence problem, the relationship $G(x^{t}) = \\mathcal{A}( G(x^{t-1}), v_{G(x)}^{t-1})$ should hold, \nwhich assumes that we can compute a motion $v_{G(x)}$ based on the generator input $x$. \nWhile directly computing such a motion can be difficult and unnecessary for general GAN problems, \nwe can make use of the ground truth data for $y$ in our conditional setting. I.e., in the following,\nwe will use a velocity reference $v_y$ corresponding to the target $y$, and perform a \nspatial down-sampling to compute the velocity $v_x$ for input $x$.\n\nEquipped with $v_x$, one possibility to improve temporal coherence would be\nto add an $L_2$ loss term of the form:\n\\begin{equation}\\label{eq:l2VelSingleSide}\n\\mathcal{L}_{2,t} = \\| G(x^{t}) - \\mathcal{A}( G(x^{t-1}), v_{x}^{t-1})\\|_{2}^{2}\n\\end{equation}\nWe found that extending the forward-advection difference with backward-advection improves the results further, i.e., the \nfollowing $L_2$ loss is clearly preferable over \\myrefeq{eq:l2VelSingleSide}:\n\\resizeEq{\n\\begin{split}\n\\mathcal{L}_{2,t} = \\| G(x^{t}) - \\mathcal{A}( G(x^{t-1}), v_{x}^{t-1})\\|_{2}^{2} + \\| G(x^{t}) - \\mathcal{A}( G(x^{t+1}), -v_{x}^{t+1})\\|_{2}^{2} \\\n\\end{split}\n}{eq:l2VelDoubleSide}\n, where we align the next frame at $t+1$ by advecting with $-v_{x}^{t+1}$.\n\nWhile this $\\mathcal{L}_{2,t}$ based loss improves temporal coherence, our tests\nshow that its effect is relatively small. E.g., it can improve outlines, but leads to clearly unsatisfactory results,\nwhich are best seen in the accompanying video. One side effect of this loss term\nis that it can easily be minimized by simply reducing the values of $G(x)$. \nThis is visible, e.g., in the second column of \\myreffig{fig:tempoStill}, which contains noticeably less density\nthan the other versions and the ground truth. However, we do not want to drive the generator\ntowards darker outputs, but rather make it aware of how the data should change over time.\n\n\\begin{figure}[b]\n\t\\centering\n\t\\begin{overpic}[width=\\linewidth]{tempofinal.png}\n\t\t\\put( 1.2,1.1){\\small \\color{white}{None}}\n\t\t\\put( 21,2){\\small \\color{white}{$\\mathcal{L}_{2,t}$}}\n\t\t\\put( 41,2){\\small \\color{white}{$\\mathcal{L}_{ \\footnotesize{D_t}' }$}}\n\t\t\\put( 61,2){\\small \\color{white}{$\\mathcal{L}_{D_t}$}}\n\t\t\\put( 81,2){\\small \\color{white}{$y$}}\n\t\\end{overpic}\n\t\\caption{ A comparison of different approaches for temporal coherence.\n\t\tThe top two rows show the inferred densities, while the bottom two rows contain the \n\t\ttime derivative of the frame content computed with a finite difference between frame $t$ and $t+1$.\n\t\tPositive and negative values are color-coded with red and blue, respectively.\n\t\tFrom left to right: no temporal loss applied, $\\mathcal{L}_{2,t}$ loss applied, \n\t\t$\\mathcal{L}_{D_t'}$, i.e., applied without advection, $\\mathcal{L}_{D_t}$ \n\t\tapplied with advection (our full tempoGAN approach), and the ground-truth $y$.\n\t\tFrom left to right across the different versions, the derivatives become \n\t\tless jagged and less noisy, as well as more structured and narrow.\n\t\tThis means the temporal coherence is improved, esp. for the result\n\t\tfrom our algorithm ($\\mathcal{L}_{D_t}$).}\n\t\\label{fig:tempoStill}\n\\end{figure}\n\nInstead of manually encoding the allowed temporal changes, \nwe propose to use another discriminator $D_t$, that learns from the given data whose changes are admissible.\nIn this way, the original spatial discriminator, which we will denote as $D_s(x,G(x))$ from now on, guarantees that \nour generator learns to generate realistic details, while the new temporal discriminator $D_t$ mainly focuses on \ndriving $G(x)$ towards solutions that match the temporal evolution of the\nground-truth $y$.\n\nSpecifically, $D_t$ takes three frames as input. We will denote such sets of three frames with a tilde in the following.\nAs real data for the discriminator, the set $\\widetilde{Y}_{\\mathcal{A}}$ contains three consecutive and advected frames, thus\n$\\widetilde{Y}_{\\mathcal{A}} = \\{ \\mathcal{A}( y^{t-1}, v_x^{t-1} )$, $y^{t}$, $\\mathcal{A}( y^{t+1}, -v_x^{t+1} )\\}$. \nThe generated data set contains correspondingly advected samples from the generator:\n$\\widetilde{G}_{\\mathcal{A}}(${\\footnotesize{$\\widetilde{X}$}}$) = \\{ \\mathcal{A}( G(x^{t-1}), v_{x}^{t-1})$, $G(x^{t})$, $\\mathcal{A}( G(x^{t+1}), -v_{x}^{t+1}) \\}$. \n\nSimilar to our spatial discriminator $D_s$, the temporal discriminator  $D_t$ is trained as a\nbinary classifier on the two sets of data:\n\\resizeEq{\n\\begin{split}\n\\mathcal{L}_{D_t}(D_t,G)= \\mathbb{E}_m [-\\log D_t\\Big(\\widetilde{Y}_{\\mathcal{A}}\\Big) ]\n+ \\mathbb{E}_n [-\\log (1-D_t\\Big(\\widetilde{G}_{\\mathcal{A}}\\Big(\\widetilde{X}\\Big)\\Big))]  \\\n\\end{split}}{eq:tempod}\n, where set $\\widetilde{X}$ also contains three consecutive frames, i.e., $\\widetilde{X} =\\{x^{t-1},$ $ x^t, x^{t+1}\\} $.\nNote that unlike the spatial discriminator, $D_t$ is not a \nconditional discriminator. It does not ``see'' the conditional input $x$, and thus\n$D_t$ is forced to make its judgment purely based on the given sequence.\n\n\\begin{figure}[b]\n\t\\centering \n\t\\begin{overpic}[width=\\linewidth]{advectInput.png}\n\t\t\\put( 3.0,2.0){\\small \\color{black}{a) $\\widetilde{Y}$}}\n\t\t\\put(26.0,2){\\small \\color{black}{a) $\\widetilde{Y}_{\\mathcal{A}}$}}\n\t\t\\put(54.0,2){\\small \\color{white}{b) $\\widetilde{Y}$}}\n\t\t\\put(77.0,2){\\small \\color{white}{b) $\\widetilde{Y}_{\\mathcal{A}}$}}\n\t\\end{overpic}\n\t\\caption{ These images highlight data alignment due to advection. \n\t\tThree consecutive frames are encoded as R, G, B channels\n\t\tof a single image, thus, ideally a fully aligned image would only contain shades of grey.\n\t\tThe two rows contain front and top views in the top and bottom row, respectively.\n        We show two examples, a) and b). Each of them contains $\\widetilde{Y}$ left, \n        and $\\widetilde{Y}_{\\mathcal{A}}$ right.\n\t\tThe RGB channels are the three input frames, t-1, t, and t+1.\n        Compared with $\\widetilde{Y}$, $\\widetilde{Y}_{\\mathcal{A}}$ is significantly less saturated, \n\t\ti.e., better aligned. }\n\t\\label{fig:tempoData}\n\\end{figure}\n\nIn \\myreffig{fig:tempoStill}, we show a comparison of the different loss variants for improving temporal\ncoherence. The first column is generated with only the spatial discriminator, i.e., \nprovides a baseline for the improvements.\nThe second column shows the result using the $L_2$-based temporal loss $\\mathcal{L}_{2,t}$ from \\myrefeq{eq:l2VelDoubleSide}, \nwhile the fourth column shows the result using $D_t$ from \\myrefeq{eq:tempod}. The last column is the ground-truth data $y$.\nThe first two rows show the generated density fields. While $\\mathcal{L}_{2,t} $ reduces overall density content, the result \nwith $D_t$ is clearly closer to the ground truth.\nThe bottom two rows show time derivatives of the densities for frames t and t+1. \nAgain, the result from $D_t$ and the ground-truth $y$ match closely in terms of their time derivatives.\nThe large and jagged values of the first two rows indicate the undesirable temporal changes produced\nby the regular GAN and the $\\mathcal{L}_{2,t}$ loss.\n\nIn the third column of \\myreffig{fig:tempoStill}, we show a simpler variant of \nour temporal discriminator. Here, we employ the discriminator without aligning the\nset of inputs with advection operations, i.e.,\n\\resizeEq{\n\\begin{split}\n\\mathcal{L}_{D_t'}(D_t',G)=& \\mathbb{E}_m [-\\log D_t\\Big(\\widetilde{Y}\\Big)] + \\mathbb{E}_n \n[-\\log (1-D_t(\\widetilde{G}\\Big(\\widetilde{X}\\Big) ))]\n\\end{split}}{}\nwith $\\widetilde{Y} = \\{y^{t-1}, y^t, y^{t+1} \\}$ \nand $\\widetilde{G}\\Big(\\widetilde{X}\\Big) = \\{G(x^{t-1}), G(x^{t}), G(x^{t+1})\\}$.\n\n\\begin{figure*}[tb]\n     \\centering \n\t\\includegraphics[width=0.99\\linewidth]{outline1.pdf} \\\\\n     \\caption{ Here an overview of our tempoGAN architecture is shown. The three neural networks (blue boxes)\n     are trained in conjunction. The data flow between them is highlighted by the red and black arrows. \\revision{Note that $x$ and $y$ denote fluid data that contains velocity and\/or vorticity fields, as well as density depending on the chosen architecture (see \\myrefsec{sec:inputs}).} }\n     \\label{fig:nnArchResnet}\n\\end{figure*}\n\nThis version improves results compared to $\\mathcal{L}_{2,t}$, but does not \nreach the level of quality of $\\mathcal{L}_{D_t}$, as can be seen in \\myreffig{fig:tempoStill}.\nAdditionally, we found that  $\\mathcal{L}_{D_t}$\noften exhibits a faster convergence during the training process. This is an indication that the underlying\nneural networks have difficulties aligning and comparing the data by themselves when using  $\\mathcal{L}_{D_t'}$. \nThis intuition is illustrated in \\myreffig{fig:tempoData}, where\nwe show example content of the regular data sets $\\widetilde{Y}$ and the advected version\n$\\widetilde{Y}_{\\mathcal{A}}$ side by side. In this figure, the three chronological frames\nare visualized as red, green, and blue channels of the images. Thus, a pure gray-scale image\nwould mean perfect alignment, while increasing visibility of individual colors indicates un-aligned\nfeatures in the data. \\myreffig{fig:tempoData} shows\nthat, although not perfect, the advected one leads to clear improvements in terms of aligning the features of the data sets,\ndespite only using the approximated coarse velocity fields $v_x$. Our experiments show\nthat this alignment successfully improves the backpropagated gradients such that the\ngenerator learns to produce more coherent outputs.\nHowever, when no flow fields are available, $\\mathcal{L}_{D_t'}$ still represents a better choice than the simpler $\\mathcal{L}_{2,t}$ version.\nWe see this as another indicator of the power of adversarial training models. It seems\nto be preferable to let a neural network learn and judge the specifics of a data set, instead\nof manually specifying metrics, as we have demonstrated for \ndata sets of fluid flow motions above.\n\nIt is worth pointing out that our formulation for $D_t$ in \\myrefeq{eq:tempod} means\nthat the advection step is an inherent part of the generator training process. While\n$v_x$ can be pre-computed, it needs to be applied to the outputs of the generator during\ntraining. This in turn means that the advection needs to be tightly integrated into the training loop.\nThe results discussed in the previous paragraph indicate that if this is done correctly, the loss gradients of the\ntemporal discriminator are successfully passed through the advection steps to give the generator\nfeedback such that it can improve its results. In the general case, advection is a non-linear function, the discrete\napproximation for which we have abbreviated with $\\mathcal{A}(y^{t}, v_y^{t})$ above. Given a known \nflow field $v_y$ and time step, we can linearize this equation to yield a matrix $M y = \\mathcal{A}(y^{t}, v_y^{t}) = y^{t+1}$.\nE.g., for a first order approximation, $M$ would encode the Euler-step lookup of source positions\nand linear interpolation to compute the solution. While we have found first order scheme (i.e., semi-Lagrangian\nadvection) to work well, $M$ could likewise\nencode higher-order methods for advection.\n\nWe have implemented this process as an advection layer in our network training, which computes\nthe advection coefficients, and performs the matrix multiplication such that the \ndiscriminator receives the correct sets of inputs. When training the generator, the same \ncode is used, and the underlying NN framework can easily compute the necessary derivatives.\nIn this way, the generator actually receives three accumulated, and aligned gradients\nfrom the three input frames that were passed to $D_t$.\n\n\\subsection{Full Algorithm}\n\\label{sec:totalloss}\nWhile the previous sections have explained the different parts of our final loss function,\nwe summarize and discuss the combined loss in the following section.\nWe will refer to our full algorithm as {\\em tempoGAN}.\nThe resulting optimization problem that is solved with NN training consists\nof three coupled non-linear sub-problems: the generator, the conditional spatial discriminator,\nand the un-con\\-di\\-tio\\-nal temporal discriminator. The generator has to effectively minimize both discriminator losses, additional feature space constraints,\nand a $L_1$ regularization term. Thus, the loss functions can be summarized as:\n\\resizeEq{\n\\begin{split}\n\\mathcal{L}_{D_t}(D_t,G)= & -\\mathbb{E}_m [\\log D_t(\\widetilde{Y}_{\\mathcal{A}})] -\n \\mathbb{E}_n [\\log \\Big(1-D_t\\Big(\\widetilde{G}_{\\mathcal{A}}\\Big(\\widetilde{X}\\Big)\\Big)\\Big)] \\\\\n\\mathcal{L}_{D_s}(D_s,G)=& -\\mathbb{E}_m [\\log D_s(x, y)] - \\mathbb{E}_n [\\log (1-D_s(x, G(x)))] \\\\\n\\mathcal{L}_{G}(D_s,D_t,G)=& -\\mathbb{E}_n [\\log D_s(x, G(x)) ]\n- \\mathbb{E}_n [\\log D_t\\Big(\\widetilde{G}_{\\mathcal{A}}\\Big(\\widetilde{X}\\Big)\\Big)] \\\\\n& + \\mathbb{E} _{n,j} \\lambda _{f}^{j} \\left \\| F^{j}(G(x))-F^{j}({y}) \\right \\|_{2}^{2} + \\lambda _{L_1}  \\mathbb{E}_n \\left \\| G(x)-{y}\\right \\|_{1}\n\\end{split}}{eq:totalLoss}\n\nOur generator has to effectively compete against two powerful adversaries,\nwho, along the lines of  \"the enemy of my enemy is my friend\", \nimplicitly cooperate to expose the results of the generator. E.g., we have \nperformed tests without $D_s$, only using $D_t$, and the resulting generator\noutputs were smooth in time, but clearly less detailed than when using both discriminators.\n\nAmong the loss terms of the generator, the $L_1$ term has a relatively minor role to stabilize\nthe training by keeping the averaged output close to the target. However, due to the complex\noptimization problem, it is nonetheless helpful for successful training runs.\nThe feature space loss, on the other hand, directly influences the generated features.\nIn the adversarial setting the discriminator most likely learns distinct features \nthat only arise for the ground truth (positive features), or those that make it easy \nto identify generated versions, i.e., negative features that are only produced by the generator.\nThus, while training, the generator will receive gradients to make it produce more\nfeatures of the targets from $F(y)$, while the gradients from $F(G(x))$ will penalize\nthe generation of recognizable negative features. \n\nWhile positive values for $\\lambda_f$ reinforce this behavior, it is less clear\nwhy negative values can lead to even better results in certain cases. \n\\revision{Our explanation\nfor this behavior is that the negative weights drive the generator towards distinct features that\nhave to adhere to the positive and negative features detected by the discriminator,\nas explained above in \\myrefsec{sec:featureloss}, but at the same time differ from the average features in $y$.}\nThus, the generator cannot simply create different or no features,\nas the discriminator would easily detect this. Instead it needs to develop features\nthat are like the ones present in the outputs $y$, but\ndon't correspond to the average features in $F(y)$,\nwhich, e.g., leads to the fine detailed outputs shown in \\myreffig{fig:2dlayerlossexample}. \n\n\\begin{algorithm}\n\t\\revision{\n\t\t\\caption{tempoGAN training algorithm}\\label{alg:tempoGANalg}\t\t\n\t\t\\begin{algorithmic}[1]   \\small    \n\t\t\t\\For{number of training steps}\n\t\t\t\\For{$k_{D_{s}}$}\n\t\t\t\\State Compute data-augmented mini batch $x, y$\n\t\t\t\\State Update $D_{s}$ with $\\nabla_{D_{s}}[\\mathcal{L}_{D_s}(D_s,G)]$\n\t\t\t\\EndFor\n\t\t\t\\For{$k_{D_{t}}$}\n\t\t\t\\State Compute data-augmented mini batch $\\widetilde{X}, \\widetilde{Y}$\n\t\t\t\\State Compute advected frames $\\widetilde{Y}_{\\mathcal{A}}$ and $\\widetilde{G}_{\\mathcal{A}}\\Big(\\widetilde{X}\\Big)$\n\t\t\t\\State Update $D_{t}$ with $\\nabla_{D_{t}}[\\mathcal{L}_{D_t}(D_t,G)]$\n\t\t\t\\EndFor\n\t\t\t\\For{$k_{G}$}\n\t\t\t\\State Compute data-augmented mini batch $x, y, \\widetilde{X}$\n\t\t\t\\State Compute advected frames $\\widetilde{G}_{\\mathcal{A}}\\Big(\\widetilde{X}\\Big)$\n\t\t\t\\State Update $G$ with $\\nabla_{G}[\\mathcal{L}_{G}(D_s,D_t,G)]$\n\t\t\t\\EndFor\n\t\t\t\\EndFor\n\t\t\\end{algorithmic}\n\t}\n\\end{algorithm}\n\n\\section{Architecture and Training Data}\n\\label{sec:nnarch}\nWhile our loss function theoretically works with any realization of $G, D_s$ and $D_t$, their\nspecifics naturally have significant impact on performance and the quality of the generated outputs.\nA variety of network architectures has been proposed\nfor training generative models \\cite{goodfellow2014generative,RadfordMC15,berthelot2017began}, and\nin the following, we will focus on pure convolutional networks for the generator, i.e., networks without any\nfully connected layers. \nA fully convolutional network has the advantage that the trained network can\nbe applied to inputs of arbitrary sizes later on. \nWe have experimented with a large variety of generator architectures, and while many \nsimpler networks only yielded sub-optimal results,\nwe have achieved high quality results with generators based on the popular U-net \\cite{ronneberger2015u, isola2016image}, \nas well as with residual networks ({\\em res-nets}) \\cite{lim2017enhanced}.\nThe U-net concatenates activations from earlier layers to later layers \n(so called {\\em skip connections}) in order to allow the network to combine high- and low-level information, \nwhile the res-net\nprocesses the data using by multiple {\\em residual blocks}. Each of these residual blocks convolves the\ninputs without changing their spatial size, and the result of two convolutional layers is added to the original \nsignal as a ``residual'' correction. In the following, we will focus on the latter architecture, as it gave slightly \nsharper results in our tests. \n\nWe found the discriminator architecture to be less crucial. \nAs long as enough non-linearity is introduced over the course of several hidden layers,\nand there are enough weights, changing the connectivity of the discriminator did not significantly influence\nthe generated outputs. Thus, in the following, we will always use discriminators with four convolutional layers\nwith leaky ReLU activations~\\!\\footnote{With a leaky tangent of 0.2 for the negative half space.}\nfollowed by a fully connected layer to output the final score. \n\\revision{As suggested by Odena et al. \\shortcite{odena2016deconv}, we use the \nnearest-neighbor interpolation layers as the first two layers in our generator, instead of deconvolutional ones,\nand in the discriminator networks, the kernel size is divisible by the corresponding stride.\nAn overview of the architecture of our neural \nnetworks is shown in \\myreffig{fig:nnArchResnet}, while their details, such as layer configuration and activation \nfunctions, can be found in \\myrefapp{app:nnarch}.}\n\\subsection{Data Generation and Training} \\label{sec:datagen}\n\n\\revision{We use a randomized smoke simulation setup to generate the desired number of\ntraining samples. For this  \nwe employ a standard fluids solve \\cite{Stam1999} with MacCormack advection and MiC-preconditioned CG solver.}\nWe typically generate around 20 simulations,\nwith 120 frames of output per simulation. For each of these, we randomly \ninitialize a certain number of smoke inflow regions, another set of velocity inflows,\nand a randomized buoyancy force.  As inputs $x$,\nwe use a down-sampled version of the simulation data sets, typically by a factor of 4,\nwhile the full resolution data is used as ground truth $y$. \n\\revision{Note that this\nsetup is inherently {\\em multi-modal}: for a single low resolution configuration,\nan infinitely large number of correct high resolution exists. We do not explicitly\nsample the high resolution solution space, but the down-sampling\nin conjunction with data augmentation lead to ambiguous low- and high-resolution\npairs of input data.}\nTo prevent a large number of primarily empty samples, we discard inputs with\naverage smoke density of less than 0.02.\nDetails of the parametrization can be found in \\myrefapp{app:data}, and visualizations of the training data\nsets can be found in the supplemental video.\nIn \naddition, we show examples generated from a two-dimensional rising smoke \nsimulation with a different simulation setup than the one used for generating\nthe training data. It is, e.g., used in \\myreffig{fig:2dlayerlossexample}.\n\nWe use the same modalities for all training runs:\nwe employ the commonly used ADAM optimizer~\\!\\footnote{Parameterized with $\\beta=0.5$.} \nwith an initial learning rate of $2 \\cdot 10^{-4}$\nthat decays  to 1\/20th for second half of the training iterations. \n\\revision{All parameters were determined experimentally, details are given in \\myrefapp{app:data}.\nThe number of training iterations is typically on the order of 10k. \nWe use 20\\% of the data for testing and the remaining 80\\% for training.\nOur networks did not require any additional regularization such \nas dropout or weight decay. \n\\revision{The training procedure is summarized again in \\myrefalg{alg:tempoGANalg}. \nDue to the typically limited amount of GPU memory, especially for 3D data sets, we perform multiple training steps\nfor each of the components. \nDetail are listed in \\myrefapp{app:data}.\nIn \\myrefalg{alg:tempoGANalg}, we use $k_{D_{s}}$, $k_{D_{t}}$, and $k_{G}$ to denote the number training iterations for $D_{s}$, $D_{t}$, and $G$, respectively.\n}\\\\\n\nWhile the coupled non-linear optimization can yield different results\neven for runs with the same parameters due to the non-deterministic nature\nof parallelized operations, we found the results to be stable in terms of quality.\nIn particular, we did not find it necessary to change the weights of the different\ndiscriminator loss terms. However, if desired, $\\lambda_{f}$  can be used to influence\nthe learned details as described above.}\nFor training and running the trained networks, we use \nNvidia GeForce GTX 1080 Ti GPUs (each with 11GB Ram) and Intel Core i7-6850K CPUs,\nwhile we used the {\\em tensorflow} and {\\em mantaflow} software frameworks for\ndeep learning and fluid simulation implementations, respectively.\n\n\\subsection{  Input Fields }\n\\label{sec:inputs}\n\n\\begin{figure}[tb]\n    \\begin{overpic}[width=0.74 \\linewidth]{comp_0099_z1.png}\n\t\t\\put( 2,3){\\small \\color{black}{a)}}\n\t\t\\put(35,3){\\small \\color{black}{b)}} \n\t\t\\put(68,3){\\small \\color{black}{c)}}  \n\t\\end{overpic}\n    \\begin{overpic}[width=0.2466 \\linewidth]{t75_img_0099_z1.png}\n\t\t\\put(6 ,6){\\small \\color{black}{d)}} \n\t\\end{overpic}\n\t\\caption{ An illustration of different training results after 40k iterations with different input\n\t\tfields: a) $\\rho$, b) $\\rho+\\mathbf{v}$, c) $\\rho+\\mathbf{v}+\\mathbf{w}$,\n\t\tall with similar network sizes. Version d) with only $\\rho$ has 2x the number of weights.\n\t\tThe seams in the images show the size of the training patches. \n\t\tSupplemental physical fields lead to clear improvements in b) and c), that even additional \n\t\tweights cannot compensate for.}\n\t\\label{fig:densVelVort}\n\\end{figure}\n\\begin{figure}[tb]\n\t\\centering \n\t\\begin{overpic}[width=0.49\\linewidth]{t10_img_0140_noaug_z2.png}\n\t\t\\put( 3,3){\\small \\color{white}{a)}}\n\t\\end{overpic}\n\t\\begin{overpic}[width=0.49\\linewidth]{t05_img_0140_resn_z2.png}\n\t\t\\put( 3,3){\\small \\color{white}{b)}}\n\t\\end{overpic}\n\t\\caption{ \n\t\tAn identical GAN network trained with the same set of input data\n\t\tWhile version a) did not use data augmentation, leading to blurry results with streak-like artifacts,\n\t\tversion b), with data augmentation, produced sharp and detailed outputs.\n\t} \\label{fig:dataAugComp}\n\\end{figure}\n\nOn first sight, it might seem redundant and unnecessary to input flow velocity $\\mathbf{v}$ and vorticity\n$\\mathbf{w}$ in addition to the density $\\rho$. After all, we are only interested in the\nfinal output density, and many works on GANs exist, which demonstrate that\ndetailed images can be learned purely based on image content.\n\nHowever, over the course of numerous training runs, we noticed that giving the networks additional \ninformation about the underlying physics significantly improves convergence and quality of the inferred results. \nAn example is shown in \\myreffig{fig:densVelVort}. Here, we show how the training evolves\nfor three networks with identical size, structure and parameters, the only difference being the\ninput fields. From left to right, the networks receive $(\\rho)$, $(\\rho,\\mathbf{v})$, and $(\\rho,\\mathbf{v},\\mathbf{w})$.\nNote that these fields are only given to the \ngenerator, while the discriminator always only receives $(\\rho)$ as input. \nThe version with only density passed to the generator, $G(\\rho)$, fails to reconstruct smooth and detailed outputs. \nEven after 40000 iterations, the results exhibit strong grid artifacts and lack detailed structures. \nIn contrast, both versions with additional inputs start to yield higher quality outputs earlier during training.\nWhile adding $\\mathbf{v}$ is crucial, the addition of $\\mathbf{w}$ \nonly yields subtle improvements (most apparent at the top of the images in \\myreffig{fig:densVelVort}),\nwhich is why we will use $(\\rho,\\mathbf{v})$ to generate our final results below. \n\\newRachel{ The full training run comparison is in our supplemental video.} \n\nWe believe that the insight that auxiliary fields help improving training and inference quality is a surprising\nand important one. The networks do not get any explicit guidance on how to use the additional information.\nHowever, it clearly not only learns to use this information, but also benefits from having this\nsupporting information about the underlying physics processes. \nWhile larger networks can potentially alleviate the quality problems of\nthe den\\-si\\-ty-only version, as illustrated in \\myreffig{fig:densVelVort} d), \nwe believe it is highly preferable to instead construct and train smaller, \nphysics-aware networks. This not only shortens training times and accelerates convergence, but also makes evaluating\nthe trained model more efficient in the long run. The availability of physical inputs turned out\nto be a crucial addition in order to successfully realize high-dimensional GAN outputs for space-time data, \nwhich we will demonstrate in \\myrefsec{sec:results}. \n\\subsection{  Augmenting Physical Data}\n\\label{sec:dataaug}\nData augmentation turned out to be an important component\nof our pipeline due to the high dimensionality of our data sets and\nthe large amount of memory they require. \nWithout sufficient enough training data, the adversarial training yields undesirable results\ndue to overfitting.\nWhile data augmentation is common practice for natural images \\cite{dosovitskiy2016discriminative, krizhevsky2012imagenet},\nwe describe several aspects below that play a role for physical data sets.\n\nThe augmentation process allows us to train networks having millions of weights\nwith data sets that only contain a few hundred samples without overfitting. \nAt the same time, we can ensure that the trained networks respect the\ninvariants of the underlying physical problems, which is crucial\nfor the complex space-time data sets of flow fields that we are considering.\nE.g., we know from theory that solutions obey Galilean invariance, and we can make\nsure our networks are aware of this property not by providing large data sets, but instead\nby generating data with different inertial frames on the fly while training. \n\nIn order to minimize the necessary size of the training set without deteriorating the result quality,\nwe generate modified data sets at training time. We focus on spatial transformations, which\ntake the form of  $\\tilde{\\vec{x}}(\\vec{p}) = \\vec{x}( A \\vec{p} )$, where $\\vec{p}$ is a spatial position, and $A$ denotes\nan $4\\times4$ matrix. For applying augmentation, we distinguish three types of components of a data set: \\begin{itemize}\n\\item {\\em passive}: these components can be transformed in a straight forward manner as described above. An example of passive\n\tcomponents are the advected smoke fields $\\rho$, shown in many of our examples.\n\\item {\\em directional}: the content of these components needs to be transformed in conjunction with the augmentation. A good example\n\tis velocity, whose directions need to be adjusted for rotations and flips, i.e., $\\tilde{\\vec{v}}(\\vec{p}) = A_{3\\times3}\\vec{v}( A \\vec{p} )$, where \n\t$A_{3\\times3}$ is the upper left $3\\times3$ matrix of $A$.\n\\item {\\em derived}: finally, derived components would be invalid after applying augmentation, and thus need to be re-computed. \n\t\\revision{A good example are physical quantities such as vorticity,\n\twhich contain mixed derivatives that cannot be easily transformed into a new frame of reference. \n\tHowever, these quantities typically can be calculated anew from other fields after augmentation.}\n\\end{itemize}\n\nIf the data set contains quantities that cannot be computed from other augmented fields, this unfortunately\nmeans that augmentation cannot be applied easily. However, we believe that a large class of typical physics data\nsets can in practice be augmented as described here.\n\n\\begin{figure*}[tb]\n\t\\centering \n\t\\begin{overpic}[width=0.33\\linewidth]{3d_lowinput.png}\n\t\t\\put( 3,3){\\small \\color{blue1}{a)}}\n\t\\end{overpic}\n\t\\begin{overpic}[width=0.33\\linewidth]{3d_13.png}\n\t\t\\put( 3,3){\\small \\color{blue1}{b)}}\n\t\\end{overpic}\n\t\\begin{overpic}[width=0.33\\linewidth]{3d_highreference.png}\n\t\t\\put( 3,3){\\small \\color{blue1}{c)}}\n\t\\end{overpic} \n\t\\caption{ These images show our algorithm applied to a 3D volume. F.l.t.r.: \n\t\t\\revision{a). a coarse input volume (down-sampled from the reference c, rendered with cubic up-sampling),}\n\t\tb). our result, and c). the high resolution reference. As in 2D, our trained model generates\n\t\tsharp features and detailed sheets that are at least on par with the reference. }\n\t\\label{fig:plume3d}\n\\end{figure*}\n\nFor matrix $A$, we consider affine transformation matrices that contain combinations of randomized translations, uniform scaling, reflections,\nand rotations. Here, only those transformations are allowed that do not violate the physical model\nfor the data set. \\revision{While shearing and non-uniform scaling could easily be added, they violate the NS momentum equation \nand thus should not be used for flow data.}\nWe have used values in the range $[0.85, 1.15]$ for scaling, and rotations by $[-90, 90]$\ndegrees.\nWe typically do not load derived components into memory\nfor training, as they are re-computed after augmentation. Thus, they are computed on the fly for a training batch\nand discarded afterwards.\n\nThe outputs of our simulations typically have significantly larger size than \nthe input tiles that our networks receive.\nIn this way, we have many choices for choosing offsets, in order\nto train the networks for shift invariance. This also aligns with our\ngoal to train a network that will later on work for arbitrarily sized inputs.\nWe found it important to take special care at spatial boundaries of the tiles.\nWhile data could be extended by Dirichlet or periodic boundary conditions, it is important that\nthe data set boundaries after augmentation do not lie outside the original data set. We enforce this \nby choosing suitable translations after applying the other transformations. This ensures that all data sets\ncontain only valid content, and the network does not learn from potentially unphysical or unrepresentative \ndata near boundaries.\nWe also do not augment the time axis in the same way as the spatial axes. We found that the spatial transformations above applied to\nvelocity fields give enough variance in terms of temporal changes.\n\\begin{figure}[tb]\n\t\\centering \\includegraphics[width=\\linewidth]{greenwlt.png}\\\\     \n\t\\caption{ We apply our algorithm to a horizontal jet of smoke in this example. The inset\n\t\tshows the coarse input (rendered with cubic up-sampling), and the result\n\t\tof our algorithm. The diffuse streaks caused by procedural turbulence in the input\n\t\t(esp. near the inflow) are turned into detailed wisps of smoke by our algorithm. }\n\t\\label{fig:green}\n\\end{figure}\n\\begin{figure}[tb]\n\t\\centering \\includegraphics[width=\\linewidth]{house.png}\\\\   \n\t\\caption{Our algorithm generated a high-resolution volume around an obstacle\n\t\twith a final resolution of $1024 \\times 720 \\times 720$.\n\t\tThe inset shows the input volume. This scene is also shown in \\myreffig{fig:teaser}\n\t\twith a different visualization. }\n\t\\label{fig:house}\n\\end{figure}\nAn example of the huge difference that data augmentation can make is shown in \\myreffig{fig:dataAugComp}.\nHere we compare two runs with the same amount of training data (160 frames of \ndata), one with, the other one without data augmentation.\nWhile training a GAN directly with this data produces blurry results,\nthe network converges to a final state with significantly sharper results with data augmentation. \nThe possibility to successfully train networks with only a small amount\nof training data is what makes it possible to train networks for 3D+time data, as \nwe will demonstrate in \\myrefsec{sec:results}.\n\\section{Results and Applications}\n\\label{sec:results}\nIn the following, we will apply our method discussed so far\nto different data sets, and explore different application settings. Among others, we will discuss related topics\nsuch as art direction, training convergence, and performance.~\\footnote{We will make code and trained models available upon acceptance of our work.}\n\\subsection{3D Results}\nWe have primarily used the 2D rising plume example in the previous sections \nto ensure the different variants can be compared easily. \nIn  \\myreffig{fig:plume3d}, we demonstrate that these results directly extend\nto 3D. \\revision{We apply our method to a three-dimensional plume with resolution $64^3$, \nwhich in this case was generated by down-sampling a $256^3$ simulation such that we can compare our result to\nthis reference solution.\nFor this input data, the $256^3$ output produced by our tempoGAN exhibits small scale features that are at least\nas detailed as the ground truth reference.} The temporal coherence is especially important\nin this setting, which is best seen in the accompanying video.\n\nWe also apply our trained 3D model to two different inputs with higher resolutions. In both cases,\nwe use a regular simulation augmented with additional turbulence to generate an interesting\nset of inputs for our method. A first scene with $150\\times100\\times100$ is shown in  \\myreffig{fig:green}, \nwhere we generate a $600\\times400\\times400$\noutput with our method. The output closely resembles\nthe input volumes, but exhibits a large number of fine details.\n\\revision{Note that our networks were only trained with down-sampled inputs,\nbut our models generalize well to regular simulation inputs without re-sampling, as illustrated by this example.}\n\nOur method also has no problems with obstacles in the flow, as shown in \\myreffig{fig:house}.\nThis example has resolutions of $256 \\times 180 \\times 180$ and $1024 \\times 720 \\times 720$ for input and output volumes. \nThe small-scale features closely adhere to the input flow around the obstacle. Although the obstacle is\ncompletely filled with densities towards the end of the simulation, there are no leaking artifacts\nas our method is applied independently to each input volume in the sequence.\nWhen showing the low-resolution input, we always employ cubic up-sampling, in order\nto not make the input look unnecessarily bad.\n\n\\subsection{\\revision{Fine Tuning Results}}\n\\label{sec:artcontrol}\n\\revision{\nGANs have a reputation for being particularly hard to influence and control,\nand influencing the outcome of simulation results is an important topic for applications in computer graphics. \nIn contrast to procedural methods,\nregular GAN models typically lack intuitive control knobs to influence the generated results.}\nWhile we primarily rely on traditional guiding techniques to control the low-resolution input,\nour method offers different ways to adjust the details produced by our tempoGAN algorithm.\n\\begin{figure}[tb]\n\t\\centering \\begin{overpic}[width=0.99\\linewidth]{summary.png}\n\t\t\\put( 3,53){\\small \\color{white}{a)}}\n\t\t\\put( 53,53){\\small \\color{white}{b)}}\n\t\t\\put( 3,3){\\small \\color{white}{c)}}\n\t\t\\put( 53,3){\\small \\color{white}{d)}}\n\t\\end{overpic} \t\n\t\\caption{The  red\\&green images on the left of each pair represent the modified velocity inputs, while the corresponding result is shown on the right. For reference, pair a) shows the unmodified input velocity, and the regular output of our algorithm. }\n\t\\label{fig:differentvelocity}\n\\end{figure}\n\nA first control knob for fine-tuning the results is to modify the data fields of the conditional\ninputs. As described in \\myrefsec{sec:inputs}, our generator receives the velocity in addition to the density, \nand it internally builds tight relationships between the two. \nWe can use these entangled\ninputs to control the features produced in the outputs. To achieve this, we modify the velocity components\npassed to the generator with various procedural functions. \\myreffig{fig:differentvelocity} shows the results of original input and several modified velocity\n examples and the resulting density configurations. \n\\revision{We have also experimented with noise fields instead \\cite{mirza2014conditional},\n but found that the trained networks completely ignored these fields. Instead, the strongly correlated velocity fields\nnaturally provide a much more meaningful input for our networks, and as a consequence provide means for influencing the\nresults.}\n\nIn addition, \\myreffig{fig:zerovelocity} demonstrates that we can effectively suppress the generation\n of small scale details by setting all velocities to zero. Thus, the network learns a correlation between\n velocity magnitudes and amount of features. This is another indicator that the network learns to extract \n meaningful relationships from the data, as we expect turbulence and small-scale details to primarily \n form in regions with large velocities.\nThree-dimensional data can similarly be controlled, as illustrated in \\myreffig{fig:3ddifferentvelocity}. \n\nIn \\myrefsec{sec:featureloss}, we discussed the influence of the $\\lambda_{f}$ parameter for small scale features.\nFor situations where we might not have additional channels such as the velocity above, we can use $\\lambda_{f}$\nto globally let the network generate different features. However, as this only provides a uniform change that is encoded\nin the trained network, the resulting differences are more subtle than those from the velocity modifications above. Examples\nof different 2D and 3D outputs can be found in  \\myreffig{fig:styleLLCompare} and  \\myreffig{fig:3ddifferentstyle}, respectively.\n\\begin{figure}[t]\n    \\centering \n\t\\begin{overpic}[width=0.99\\linewidth]{styl_150th.png}\n\t\\end{overpic} \n    \\caption{\n\t\\revision{An illustration how the entangled inputs of density and velocity can be used to fine tune the results:\n\ton the left the velocities were scaled up by a factor of 2, while the right hand side was scaled by zero.\n\tThe network has learned a relationship between detail and velocities, leading\n\tto reduced details in regions where the velocity was set to zero. }}\n\\label{fig:zerovelocity}\n\\end{figure}\n\\begin{figure}[t]\n     \\centering \\begin{overpic}[width=0.49\\linewidth]{test_08_3.jpg}\n     \\put( 3,3){\\small \\color{blue1}{a)}}\n     \\end{overpic} \n     \\begin{overpic}[width=0.49\\linewidth]{test_08_1.png}\n     \t\\put( 3,3){\\small \\color{blue1}{b)}}\n     \\end{overpic} \n     \\centering \\begin{overpic}[width=0.49\\linewidth]{test_08_4.jpg}\n     \t\\put( 3,3){\\small \\color{blue1}{c)}}\n     \\end{overpic} \n     \\begin{overpic}[width=0.49\\linewidth]{test_08_2.png}\n     \t\\put( 3,3){\\small \\color{blue1}{d)}}\n     \\end{overpic} \n     \\caption{a) is the result of tempoGAN with velocity set to zero.\n     \tThe other\n     three examples were generated with modified velocity inputs to achieve more stylized outputs.}\n     \\label{fig:3ddifferentvelocity}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering \\begin{overpic}[width=0.49\\linewidth]{test_13_comp_4_small.png}\n\t\t\\put( 3,3){\\small \\color{white}{a)}}\n\t\\end{overpic} \n\t\\centering \\begin{overpic}[width=0.49\\linewidth]{test_9_comp_4_small.png}\n\t\t\\put( 3,3){\\small \\color{white}{b)}}\n\t\\end{overpic} \n\t\\caption{ A comparison of training runs with different feature loss weights: a) \n\t$\\lambda _{f}^{1,...,4}=-10^{-5}$ , \n\tb) \n\t$\\lambda _{f}^{1,4}=1\/3\\cdot{10^{-4}}$, $\\lambda _{f}^{2,3}=-1\/3\\cdot10^{-4}$. }\n\t\\label{fig:styleLLCompare}\n\\end{figure}\n\\begin{figure}[t]\n     \\centering \\begin{overpic}[width=0.32\\linewidth]{test_13_dark_cut.png}\n     \t     \t\\put( 3,3){\\small \\color{white}{a)}}\n     \\end{overpic} \n     \\begin{overpic}[width=0.32\\linewidth]{test_9_dark_cut.png}\n     \t     \t\\put( 3,3){\\small \\color{white}{b)}}\n     \\end{overpic} \n     \\begin{overpic}[width=0.32\\linewidth]{refeh_dark_cut.png}\n     \t     \t\\put( 3,3){\\small \\color{white}{c)}}\n     \\end{overpic} \n     \\caption{ A comparison of training runs with different feature loss weights in 3D:\n     a) with $\\lambda _{f}^{1,...,4}=-1\/3\\cdot10^{-6}$ ,\n     b) with $\\lambda _{f}^{1}=1\/3\\cdot10^{-6}$, $\\lambda _{f}^{2,3,4}=-1\/3\\cdot10^{-6}$. The latter yields a sharpened result.\n          Image c) shows the high resolution reference. }\n     \\label{fig:3ddifferentstyle}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering \\begin{overpic}[width=0.49\\linewidth]{crop_v13.png}    \n\t\t\\put( 3,3){\\small \\color{white}{a)}}\n\t\\end{overpic} \n\t\\begin{overpic}[width=0.49\\linewidth]{crop_wlt_2x_data_pos.png}    \n\t\t\\put( 3,3){\\small \\color{white}{b)}}\n\t\\end{overpic} \n\t\\caption{ Our regular model a) and one trained with wavelet turbulence data\n\t\tb). In contrast to the model trained with real simulation data, the wavelet\n\t\tturbulence model produces flat regions with sharper swirls, mimicking\n\t\tthe input data.\n\t}\n\t\\label{fig:waveletTurbulence}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering \\begin{overpic}[width=0.49\\linewidth]{plume_4x_cut.png}   \n\t\t\\put( 3,3){\\small \\color{white}{a)}}\n\t\\end{overpic}  \n\t\\begin{overpic}[width=0.49\\linewidth]{plume_8x_cut.png}  \n\t\t\\put( 3,3){\\small \\color{white}{b)}}\n\t\\end{overpic}   \n\t\\caption{a) is the network output after a single application. \n\tb) is the network recursively applied to a) with a scaling factor of 2, resulting in a total increase of $8\\times$.}\n\t\\label{fig:recursive}\n\\end{figure}\n\\subsection{Additional Variants}\n\nIn order to verify that our network can not only work with two- or three-dimensional\ndata from a Navier-Stokes solver, we generated a more synthetic data set by applying\nstrong wavelet turbulence to a $4\\times$ up-sampled input flow. We then trained our network\nwith down-sampled inputs, i.e., giving it the task to learn the output\nof the wavelet turbulence algorithm. Note that a key difference here is that wavelet\nturbulence normally requires a full high-resolution advection over time, while our method\ninfers high-resolution data sets purely based on low-resolution data from a single frame.\n\\begin{figure*}[t]\n     \\centering\n\t\t\\includegraphics[width=0.99\\linewidth, height = 3.3cm]{graphs.pdf}\n     \\caption{ Several discriminator loss functions over the course of the 40k training iterations.\n     a) $D_{s}$ (spatial discriminator) loss is shown in green without $D_{t}$, and orange with $D_{t}$.\n     b) Temporal discriminator loss in blue with only $D_{t}$, and in red for tempoGAN (i.e., with $D_{s}$, and feature loss).\n     c) Spatial discriminator loss is shown in green with $\\mathcal{L}_{f}$, and in dark blue without.\n     For each graph, the dark lines show smoothed curves. The full data is shown in a lighter color in the background.\n\t}\n     \\label{fig:graph-all}\n\\end{figure*}\n\nOur network successfully learns to generate structures similar \nto the wavelet turbulence outputs, shown in \\myreffig{fig:waveletTurbulence}. However, this data set \nturned out to be more difficult to learn than the original fluid simulation inputs.\nThe training runs required two times more training data than the regular simulation runs,\nand we used a feature loss of $\\lambda _{f}^{1,...,4}=10^{-5}$. \nWe assume that these more difficult training conditions are caused by the more \nchaotic nature of the procedural turbulence,\nand the less reasonable correlations between density and velocity inputs. Note that\ndespite using more wavelet turbulence input data, it is still a comparatively small data set.\n\nWe additionally were curious how well our network works when it is applied\nto a generated output, i.e., a recursive application. The result can be\nfound in \\myreffig{fig:recursive}, where we applied our network \nto its own output for an additional $2\\times$ upsampling. Thus, in total this led to an $8\\times$ increase of resolution.\nWhile the output is plausible, and clearly \ncontains even more fine features such as thin sheets, there is a tendency\nto amplify features generated during the first application.\n\\subsection{Training Progress}\n\nWith the training settings given in \\myrefapp{app:data}, our training\nruns typically converged to stable solutions of around $1\/2$ for the discriminator outputs after sigmoid activation. \nWhile this by itself does not guarantee that a desirable solution was found, it at least indicates convergence\ntowards one of the available local minima.\n\n\\revision{However, it is interesting how the discriminator loss changes \nin the presence of the temporal discriminator.\n\\myreffig{fig:graph-all} shows several graphs of discriminator losses over the course of a full training run.\nNote that we show the final loss outputs from \\myrefeq{eq:disloss} and \\myrefeq{eq:tempod} here.\nA large value means the discriminator does ``worse'', i.e.,\nit has more difficulty distinguishing real samples from the generated ones. \nCorrespondingly, lower values mean it can separate them more successfully.\nIn \\myreffig{fig:graph-all}a) it is visible that the spatial discriminator loss decreases when \nthe temporal discriminator is introduced. \nHere the graph only shows the spatial discriminator loss, and the discriminator itself\nis unchanged when the second discriminator is introduced. The training run corresponding to the\ngreen line is trained with only a spatial discriminator, and for the orange line with both spatial and temporal discriminators.\nOur interpretation\nof the lower loss for the spatial discriminator network is that the existence of a temporal discriminator in the optimization \nprevents the generator from using\nthe part of the solution space with detailed, but flickering outputs. Hence, the generator is driven to find a solution from the temporally\ncoherent ones, and as a consequence has a harder time, which in turn makes the job easier for the spatial discriminator. This manifests\nitself as a lower loss for the spatial discriminator, i.e. the lower orange curve in \\myreffig{fig:graph-all}a).}\n\nConversely, the existence of a spatial discriminator does not noticeably influence\nthe temporal discriminator, as shown in \\myreffig{fig:graph-all}b). This is also intuitive, as the spatial discriminator does not influence temporal changes. \n\\revision{We found that a generator trained only with $D_t$ typically produces fewer details\nthan a generator trained with both. In conjunction, our tests indicate that the two discriminators \nsuccessfully influence different aspects of the solution space,\nas intended.}\nLastly, \\myreffig{fig:graph-all}c) shows\nthat activating the negative feature loss from \\myrefsec{sec:featureloss} makes the task for\nthe generator slightly harder, resulting in a lowered spatial discriminator loss. \n\\subsection{Performance}\n\nTraining our two- and three-dimensional models\nis relatively expensive. Our full 2D runs typically\ntake around 14 hours to complete (1 GPU), while the 3D runs\ntook ca. 9 days using two GPUs. However, in practice, the state of the\nmodel after a quarter of this time is already indicative of\nthe final performance. The remainder of the time is typically\nspent fine-tuning the network.\n\nWhen using our trained network to generate high-resolution\noutputs in 3D, the limited memory of current GPUs poses a constraint\non the volumes that can be processed at once, as the intermediate\nlayers with their feature maps can take up significant amounts of memory. \nHowever, this does not pose a problem for generating larger final\nvolumes, as we can subdivide the input volumes, and process them piece by piece.\nWe generate tiles with a size of $136^3$ on one GPU, with a corresponding input of size $34^3$. \nOur 8 convolutional layers with a receptive field of 16 cells mean that up to four \ncells of an input could be influenced by a boundary. In practice, we found 3 input cells\nto be enough in terms of overlap. Generating a single $136^3$ output took ca. 2.2 seconds\non average. \nThus, generating a $256^3$ volume from an $64^3$ input took 17.9s on average.\n\\revision{Comparing the performance of our model with high resolution simulations\nis inherently difficult, due to the substantially different implementations and hardware platforms (CPU vs. GPU).\nHowever, for the example of \\myreffig{fig:green} we estimate that fluid simulation at the full resolution would\ntake ca. 31.5 minutes per frame of animation on average, while the evaluation of all volume\ntiles with our evaluation pipeline took ca. 3.9 minutes.} \n\n\\revision{\nThe cost for the trained model scales linearly with the number of cells in the volume,\nand in contrast to all previous methods for increasing the resolution of flow simulations,\nour method does not require any additional tracking information. It is also fully independent for all frames.\nThus, our method could ideally be applied on the fly before rendering a volume, after which the \nhigh resolution data could be discarded.\nAdditionally, due to GPU memory restrictions we currently evaluate our model in \nvolumetric tiles with 3 cells of overlap for the input. This overlap can potentially be reduced further,\nand become unnecessary when enough memory is available to process the full input volume at once.\n}\n\n\\subsection{Limitations and Discussion}\n\\label{sec:limitations}\n\nOne limitation of our approach is that the network encodes a fixed\nresolution difference for the generated details. While the initial up-sampling\nlayers can be stripped, and the network could thus be applied to inputs of any size,\nit will be interesting to explore different up-sampling factors beyond the factor\nof four which we have used throughout.\n\\revision{With our current implementation, our method can also be slower than, e.g., calculating\nthe advection for a high resolution grid. However, a high-res advection would typically not lead to\ndifferent dynamics than those contained in the input flow, and require a sequential solve\nfor the whole animation sequence.}\nOur networks have so far also focused\non buoyant smoke clouds. While obstacle interactions worked in our tests,\nwe assume that networks trained for larger data sets and with other types of interactions\ncould yield even better results.\n\nOur three-dimensional networks needed a long time to train, circa nine days\nfor our final model. Luckily, this is a one-time cost, and the network can be flexibly reused\nafterwards. However, if the synthesized small-scale features need to be fine-tuned,\nwhich we luckily did not find necessary for our work, the long runtimes could make this\na difficult process. \nThe feature loss weights clearly also are data dependent, e.g., we used\ndifferent settings for simulation and wavelet turbulence data.\nHere, it will be an interesting direction for future work to \ngive the network additional inputs for fine tuning the results beyond the velocity\nmodifications which discussed in \\myrefsec{sec:artcontrol}.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe have realized a first conditional GAN approach for four-di\\-men\\-sional \ndata sets, and\nwe have demonstrated that it is possible to train generators that preserve temporal coherence\nusing our novel time discriminator.\nThe network architecture of this temporal discriminator, which ensures\nthat the generator receives gradient information even for complex transport processes, makes it \npossible to robustly train networks for temporal evolutions. We have shown\nthat this discriminator improves the generation of stable details as well as the learning process itself. \nAt the same time, our fully convolutional networks can be applied to inputs of arbitrary size, and \nour approach  provides basic means for art direction of the generated outputs.\nWe also found it very promising to see that our CNNs are able to benefit from coherent, physical \ninformation even in complex 3D settings, which led to reduced network sizes.\n\nOverall, we believe that our contributions yield a robust and very general method \nfor generative models of physics problems, and for super-resolution flows in particular.\nIt will be highly interesting as future work to apply our tempoGAN to other physical problem settings,\nor even to non-physical data such as video streams.\n\n\\begin{acks}\nThis work was funded by the ERC Starting Grant {\\em realFlow} (StG-2015-637014). We would like to thank Wei He for helping with making the videos, and all members of the graphics labs of TUM, IST Austria and ETH Zurich for the thorough discussions.\n\\end{acks}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nThree interacting quantum systems may also be viewed as a bipartite system \ncoupled to a third, ``external\" system. The Raman coupled model, introduced some years ago \n\\cite{eber90} constitutes an example of a simple, analytically solvable model \ninvolving three quantum subsystems; a three level atom coupled to two \nmodes of the cavity quantized field. In the limit in which the excited atomic \nstate is well far-off resonance, a simpler, effective two-level \nHamiltonian may be derived, either by an adiabatic elimination of the upper atomic \nlevel \\cite{eber90} or via a unitary transformation \\cite{bose95}. In such a\nprocedure, energy (Stark) shifts arise and care must be taken, given that they may \nnot always be neglected \\cite{knig91}. In fact, the presence of the shifts\nnormally leads to a very different dynamics, e.g., from a non-periodic to a periodic \natomic inversion. Several features of the dynamics of such model have already been  \ninvestigated \\cite{eber90}; in particular, quantum entanglement and possible applications \nto quantum computation \\cite{agar04,garr04}. Here we are going to discuss a different aspect \nof that system; the influence of one of the fields (a cavity mode) on \nthe dynamics of the atom as well as on the bipartite entanglement between the atom and the \nother cavity mode. We are going to consider different field preparations, such as coherent and \nthermal states, and we will also compare our results to the case in which one of the modes is not quantized, \nbut treated as a classical field, instead. For simplicity, we do not take into account cavity losses or \natomic spontaneous decay. Our paper is organized as follows: in Sec. II we present the model and solution. \nIn Sec. III we discuss dynamical features with different preparations. In Sec. IV we present the \nevolution of bipartite entanglement, and in Sec. V we summarize our conclusions.\n\n\\section{The model and solution}\n\\subsection{Two quantized modes}\n\nWe consider a three level atom (levels 1,2,3) in interaction with two modes (mode 1, of frequency \n$\\omega_1$ and mode 2, of frequency $\\omega_2$) of the quantized field \nin a lambda configuration. Direct transitions between the lower levels 1 and 2 are forbidden. \nIf the upper level (level 3) is highly detuned from the fields (detuning \n$\\Delta$), the effective Hamiltonian may be written as \\cite{bose95}\n\\begin{equation}\nH_{eff}  =  \\sigma_{11}E_{1}+\\sigma_{22}E_{2}+\\hbar\\omega_{1}a_{1}^{\\dagger}a_{1}+\\hbar\\omega_{2}a_{2}^{\\dagger}a_{2}\n -\\hbar\\frac{g_{1}^{2}}{\\Delta}\\sigma_{11}a_{1}^{\\dagger}a_{1}-\\hbar\\frac{g_{2}^{2}}{\\Delta}\\sigma_{22}a_{2}^{\\dagger}a_{2}+\\label{heffective}\n -\\hbar\\frac{g_{1}g_{2}}{\\Delta}\\left(a_{1}^{\\dagger}a_{2}\\sigma_{12}+a_{2}^{\\dagger}a_{1}\\sigma_{21}\\right),\n\\end{equation}\nwhere $\\sigma_{12}$ and $\\sigma_{21}$ are the transition operators between levels 1 and 2, \n$a_1(a_1^\\dagger)$ is the annihilation (creation) operator of mode 1, \n$a_2(a_2^\\dagger)$ is the annihilation (creation) operator of mode 2, and $g_1(g_2)$ are \nthe couplings of the transitions $1-3$ and $2-3$, respectively. The effective\nHamiltonian is valid in the limit $g_1\/\\Delta(g_2\/\\Delta)\\ll 1$, but under certain conditions, \nexact similar Hamiltonians may also be obtained \\cite{wu96}. The Stark shift terms \n$-\\hbar \\frac{g_1^2}{\\Delta}\\sigma_{11}a_1^\\dagger a_1$ and $-\\hbar \\frac{g_2^2}{\\Delta}\\sigma_{22}a_2^\\dagger a_2$ \nare usually neglected \\cite{eber90,garr04}, but one should be very careful, given that their inclusion results in a \nRabi frequency depending linearly on the photon numbers $n_1$ and $n_2$. As a consequence, because of the shifts, \nthe dynamics of the Raman coupled model becomes basically periodic, with Rabi frequency\n\\begin{equation}\n\\Omega_{n_{1},n_{2}}=\\frac{\\left[g_{1}^{2}n_{1}+g_{2}^{2}\\left(n_{2}+1\\right)\\right]}{2\\Delta}\\label{rabifreq}\n\\end{equation} \nin contrast to the Rabi frequency, which is proportional to $\\sqrt{n_1(n_2+1)}$, if the Stark shifts are \nneglected \\cite{eber90}. \nAssuming an initial density operator of the product form,\n\\begin{eqnarray}\n\\rho\\left(0\\right)=\\sum_{n_{1}n_{2}m_{1}m_{2}}^{\\infty}\\rho_{n_{1}n_{2}m_{1}m_{2}}\\left|1;n_{1},n_{2}\\right\\rangle \\left\\langle 1;m_{1},m_{2}\\right|,\n\\label{initialdo}\n\\end{eqnarray}\ni.e., with the atom initially prepared in level 1, and the fields in generic states characterized by the coefficients \n$\\rho_{n_{1}n_{2} m_{1}m_{2}}$. The full time-dependent density operator for the tripartite system may be written as\n\n\\begin{eqnarray}\n\\rho\\left(t\\right) && = \\sum_{n_{1}n_{2}m_{1}m_{2}}^{\\infty}\\{ A_{n_{1}n_{2}m_{1}m_{2}}\\left|1;n_{1},n_{2}\\right\\rangle \\left\\langle 1;m_{1},m_{2}\\right|\n +  B_{n_{1}n_{2}m_{1}m_{2}}\\left|2;n_{1}-1,n_{2}+1\\right\\rangle \\left\\langle 2;m_{1}-1,m_{2}+1\\right|\\nonumber \\\\\n\\nonumber\\\\&& +  C_{n_{1}n_{2}m_{1}m_{2}}\\left|1;n_{1},n_{2}\\right\\rangle \\left\\langle 2;m_{1}-1,m_{2}+1\\right|+h.c.\\},\\nonumber \n\\label{densityop}\\\\\n\\end{eqnarray}\nwith coefficients\n\\begin{eqnarray}\nA_{n_{1}n_{2}m_{1}m_{2}}&=&\\rho_{n_{1}n_{2}m_{1}m_{2}}e^{i\\nu_{n_{1}n_{2}m_{1}m_{2}}t}k_{1,n_{1},n_{2}}\\, k_{1,m_{1},m_{2}}^{*},\\nonumber \\\\\nB_{n_{1}n_{2}m_{1}m_{2}}&=&\\rho_{n_{1}n_{2}m_{1}m_{2}}e^{i\\nu_{n_{1}n_{2}m_{1}m_{2}}t}k_{2,n_{1},n_{2}}\\, k_{2,m_{1},m_{2}}^{*},\\nonumber \\\\ \nC_{n_{1}n_{2}m_{1}m_{2}}&=&\\rho_{n_{1}n_{2}m_{1}m_{2}}e^{i\\nu_{n_{1}n_{2}m_{1}m_{2}}t}k_{1,n_{1},n_{2}}\\, k_{2,m_{1},m_{2}}^{*},\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\nu_{n_{1}n_{2}m_{1}m_{2}} = \\left(m_{1}-n_{1}\\right)\\left(\\omega_{1}-\\frac{g_{1}^{2}}{2\\Delta}\\right)+\n\\left(m_{2}-n_{2}\\right)\\left(\\omega_{2}-\\frac{g_{2}^{2}}{2\\Delta}\\right),\\nonumber\n\\end{equation}\nand\n\n\\begin{eqnarray}\nk_{1,n_{1},n_{2}}\\left(t\\right) & = & \\cos\\left(\\Omega_{n_{1},n_{2}}t\\right) \n+ i\\left[\\frac{n_{1}-r^{2}\\left(n_{2}+1\\right)}{n_{1}+r^{2}\\left(n_{2}+1\\right)}\\right]\\sin\\left(\\Omega_{n_{1},n_{2}}\\,t\\right),\\nonumber \\\\\nk_{2,n_{1},n_{2}}\\left(t\\right) & = & \\frac{2ri\\sqrt{n_{1}\\left(n_{2}+1\\right)}}{\\left[n_{1}+r^{2}\\left(n_{2}+1\\right)\\right]}\\sin\\left(\\Omega_{n_{1},n_{2}}\\,t\\right),\\nonumber \\label{coeff}\n\\end{eqnarray}\nwhere the Rabi frequency $\\Omega_{n_{1},n_{2}}$ is given in Eq. (\\ref{rabifreq}) and having defined $r\\equiv g_2\/g_1$.\n\n\\subsubsection{Atomic dynamics with different initial field preparations}\n\nThe atomic response to the fields may be characterized by the atomic population inversion as a function of time, or\n\\begin{equation}\n\\label{atomicinv}\nW\\left(t\\right)  =  2\\, \\mbox{Tr} \\left[ \\rho(t) |2\\rangle\\langle 2| \\right] - 1 \n =  8\\sum_{n_{1},n_{2}}^{\\infty}p_{n_1} p_{n_2}\n\\frac{r^{2}n_{1}\\left(n_{2}+1\\right)}{\\left[n_{1}+r^{2}\\left(n_{2}+1\\right)\\right]^{2}}\\sin^{2}\\left(\\Omega_{n_{1},n_{2}}t\\right) - 1,\\nonumber\n\\end{equation}\nwhere $p_{n_1} p_{n_2}= \\rho_{n_{1}n_{2}\\,;\\, n_{1}n_{2}}$ is the product of the photon number \ndistributions of the initial fields. The atomic inversion has\npeculiar features depending on the field statistics. For instance, in another well known model of optical \nresonance having a two level atom coupled to a single mode field, the Jaynes Cummings model (JCM), a field \ninitially in a Fock state leads to pure oscillations of the atomic inversion, while an initial coherent state \ncauses collapses and revivals \\cite{eber80} of the Rabi oscillations.\nOn the other hand, in the JCM, for a field initially in a thermal state, the structure of collapses and revivals \nbecomes highly desorganized \\cite{radm82} and look like random. In the Raman coupled model, in which an effective two \nlevel atom is coupled to two fields, rather than one, we expect of course a different behaviour. Perhaps the most \nstriking difference to the JCM is the periodicity of the atomic inversion; moreover, the ``revival'' \ntimes do not depend on the intensity of the fields, and the field statistics has only\nlittle effect on the pattern of collapses and revivals \\cite{knig91}. \nIn order to illustrate this, in Fig. (\\ref{figure1}) we have plots of the atomic inversion as a \nfunction of the scaled time $\\tau = g_1 t$ for modes 1 and 2 prepared in coherent states: \n$p_{n_i} = e^{-\\bar{n}_{i}}\\bar{n}_{i}^{n_{i}}\/n_{i}!$ [Fig. \\ref{figure1}(a)] and for mode 1 prepared in a coherent state and mode 2 prepared in a \nthermal state: $p_{n_1}  = e^{-\\bar{n}_{1}}\\bar{n}_{1}^{n_{1}}\/n_{1}!$ and $p_{n_2} = \\bar{n}_{2}^{n_{2}}\/\\left(\\bar{n}_{2}+1\\right)^{n_{2}+1}$\n[Fig. \\ref{figure1}(b)]. \nWe may note the periodical and well defined revivals occurring at the same times in both cases. Curioulsly, even in the \ncase of a thermal (chaotic) initial preparation, revivals are well defined, although\ntheir amplitude is slightly suppressed; probably an effect due to the broader thermal distribution. We also note that \nif one of the fields is prepared in a Fock state, if the other mode is prepared in a coherent state, for instance, collapses \nand revivals will still occur, due to the presence of the ``in phase\" and ``out of phase \nterms\" in Eq. (\\ref{atomicinv}). In Fig. (\\ref{figure2}) we have a plot of the atomic inversion for mode 1 prepared in a \n$N$ photons Fock state and mode 2 prepared in a coherent state, and we note again the pattern of periodic and regular \ncollapses and revivals. In what follows, this particular preparation (Fock-coherent) is going to be compared with a situation \ntermed ``partially classical\", having one of the fields, mode 2, treated classically rather than prepared in a coherent state \n(fully quantized case).\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics{fig1.pdf}\n\\end{center}\n\\caption{Atomic inversion as a function of the scaled time $\\tau = g_1 t$ for a) modes 1 and 2 \nprepared in coherent states and b) mode 1 prepared in a \ncoherent state and mode 2 prepared in a thermal state. \nIn both cases $\\overline{n}_1 = 10.5$ and $\\overline{n}_2 = 10.1$.\nWe have considered $g_1 = g_2$ and $r = 1.012$.}\n\\label{figure1}      \n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics{fig2.pdf}\n\\end{center}\n\\caption{Atomic inversion for mode 1 prepared in Fock state ($N = 5$) and mode 2 prepared in a coherent state ($\\overline{n} = 5$), with $r = 1.023$.}\n\\label{figure2}      \n\\end{figure}\n\n\\subsection{One quantized mode}\n\nNow we consider mode 2 as being a classical field of amplitude $\\Omega_L$;  the ``partially classical\" case, keeping mode 1 quantized. \nThe effetive Hamiltonian in the far-off resonance limit for the excited state is, \nin this case \\cite{mato01},\n\\begin{equation}\nH_{eff}^{\\prime}=\\frac{g^{2}a^{\\dagger}a}{\\Delta}\\sigma_{11}+\\frac{g^{2}r^{\\prime}{}^2}{\\Delta}\\sigma_{22}+\\lambda\\left(\\sigma_{12}a^{\\dagger}\n+\\sigma_{21}a\\right),\\label{semclasshamil}\n\\end{equation}\nwhere $g$ is the quantum field\/atom coupling constant, $\\lambda=g\\left|\\Omega_{L}^{*}\\right|\/\\Delta$ \nis the effective coupling constant and $r^{\\prime} = \\left|\\Omega_{L}^{*}\\right|\/g$. \nNote that apart from some shifts, the effective Hamiltonian is very similar to a JCM Hamiltonian. \nThis means that the atomic response to a field (mode 1) prepared in a Fock state is going to be in the form of pure Rabi oscillations, \nin contrast to the case where mode 2 is a quantum field prepared in the ``quasi-classical\" coherent state, which shows collapses and \nrevivals. This is another example showing that regarding the atomic response to a field, an intense quantum coherent state is not \nequivalent to a classical field.\n\n\\section{Entanglement}\n\nWe would like now to discuss the entanglement between the atom and one mode of the field \n(mode 1, for instance) considering the other mode as an ``external\" coupled\nsub-system. We then trace over the variables of mode 2, initially prepared in a coherent state, \nand examine the degree of entanglement between the atom and the remaining \nfield (mode 1) initially prepared, for the sake of simplicity, in a Fock state. In order to quantify \nentanglement, we use the negativity. As we have done in the case of \nthe atomic inversion, we will compare the results with the case in which mode 2 is considered to be a \nclassical field. For an initial state having the atom in level 1,\nmode 1 in a Fock state $|N\\rangle$ and mode 2 in a coherent state $|\\alpha\\rangle$, i.e., a product state \n$\\left|\\psi\\left(0\\right)\\right\\rangle =\\left|1\\right\\rangle \\otimes\\left|N\\right\\rangle \\otimes\\left|\\alpha\\right\\rangle$, \nthe joint atom-mode 1 density operator, obtained after tracing over mode 2 from Eq. (\\ref{densityop}), \n$\\tilde{\\rho}(t) = Tr_2[\\rho(t)]$ is given by\n\\begin{eqnarray}\n&&\\tilde{\\rho}\\left(t\\right)  =  \\sum_{n=0}^{\\infty}\\left|c_{n}\\right|^{2}k_{1,N,n}\\, k_{1,N,n}^{*}\\left|1;N\\right\\rangle \\left\\langle 1;N\\right| \n+\\sum_{n=0}^{\\infty}\\left|c_{n}\\right|^{2}\\, k_{2,N,n}\\, k_{2,N,n}^{*}\\left|2;N-1\\right\\rangle \\left\\langle 2;N-1\\right|\\nonumber \\\\\n&&+F(t)\\sum_{n=0}^{\\infty}c_{n+1}c_{n}^{*}\\, k_{1,N,n+1}\\, k_{2,N,n}^{*}\\left|1;N\\right\\rangle \\left\\langle 2;N-1\\right|\n+ h.c.,\\nonumber \\label{reduceddens}\n\\end{eqnarray}\nwith $F(t) = e^{-i\\left(\\omega_{2}-\\frac{g_{2}^{2}}{2\\Delta}\\right)t}$ and $c_{n}=e^{-\\bar{n}\/2}\\alpha^{n}\/\\sqrt{n!}$.\n\nWe may then calculate the negativity as a function of time\n\n\\begin{eqnarray}\n{\\cal {N}}\\left(t\\right) =2r\\sqrt{\\bar{n}N}\\left\\{ \\Bigg[\\sum_{n=0}^{\\infty}e^{-\\bar{n}}\\frac{\\bar{n}^{n}}{n!}\\left[\\frac{N-r^{2}\\left(n+2\\right)}{N+r^{2}\\left(n+2\\right)}\\right]\\right.\n\\frac{\\sin\\left(\\Omega_{N,n}t\\right)\\sin\\left(\\Omega_{N,n+1}t\\right)}{\\left[N+r^{2}\\left(n+1\\right)\\right]}\\Bigg]^{2}\\nonumber\\\\\n\\left.+\\left[\\sum_{n=0}^{\\infty}e^{-\\bar{n}}\\frac{\\bar{n}^{n}}{n!}\\:\\frac{\\sin\\left(\\Omega_{N,n}t\\right)\\cos\\left(\\Omega_{N,n+1}t\\right)}{\\left[N+r^{2}\\left(n+1\\right)\\right]}\\right]^{2}\\right\\} ^{1\/2},\n\\end{eqnarray}\nwhere $\\bar{n} = |\\alpha|^2$ and $\\Omega_{N,n}=\\left[g_{1}^{2}N+g_{2}^{2}\\left(n+1\\right)\\right]\/2\\Delta$. \nIt is also worth to compare the negativity to the linear entropy relative to the atomic state (obtained after tracing over mode 1, $\\rho_A(t) = \nTr_1[\\tilde{\\rho}(t)]$), which is defined as $\\zeta_A(t) = 1 - Tr[\\rho_A^2(t)]$ or\n\n\\begin{equation}\n\\zeta_{A}\\left(t\\right)  =\n2 \\sum_{n=0}^{\\infty}e^{-\\bar{n}}\\:\\frac{\\bar{n}^{n}}{n!}\\frac{4r^{2}N\\left(n+1\\right)}{\\left[N+r^{2}\\left(n+1\\right)\\right]^{2}}\\sin^{2}\\left(\\Omega_{N,n}t\\right) \n-2\\left\\{ \\sum_{n=0}^{\\infty}e^{-\\bar{n}}\\:\\frac{\\bar{n}^{n}}{n!}\\frac{4r^{2}N\\left(n+1\\right)}\n{\\left[N+r^{2}\\left(n+1\\right)\\right]^{2}}\\sin^{2}\\left(\\Omega_{N,n}t\\right)\\right\\} ^{2}.\\label{linearentro}\n\\end{equation}\n\nThe linear entropy may be used as a measure of the quantum state purity, as it it zero for a pure \nstate and $> 0$ for a mixed state. In Fig. (\\ref{figure3}) we have plotted the negativity (\\ref{figure3}a) and \nthe linear entropy (\\ref{figure3}b) as a function of the scaled time $g_1 t$. \nWe note that entanglement is maximum approximately in the middle of the collapse region, and the atom-mode 1 \nstate tends to become separable at the revival times themselves (see Fig. \\ref{figure2}). At the same time one may notice the \nsimilarities and differences between the negativity and the linear entropy; the maximum of entanglement coincides\nwith the maximum of mixedness, and separability approximately occurs at times in which the atom is close to a pure state. \nHowever, we should point out that there are time intervals of maximum mixedness which do not correspond to maximum entanglement; \nthis explicitly shows us the inadequacy of the linear entropy as a \nmeasure of entanglement (as expected), given that the state under consideration (atom-mode 1) is generally not a pure state. \nAgain, the evolution of entanglement in the ``partially classical\" case is going to be very different. We would like to remark \nthat in this case a bipartite quantum system is under the action of an external classical field, instead of a  \ntripartite system from which we have obtained a bipartite system by tracing over one of the subsystems. \nHaving mode 1 prepared in a $N$ photon Fock state (mode 2 being a classical field), we have the following \nexpressions for the atomic inversion (with the atom initially in state 1),\n\\begin{eqnarray}\nW^{\\prime}\\left(t\\right)=\\frac{8r^{\\prime2}N}{\\left(N+r^{\\prime2}\\right)}\\sin^{2}\\left[\\frac{\\left(N+r^{\\prime2}\\right)}{2r^{\\prime}}\\tau^{\\prime}\\right]-1, \\label{invsemiclass}\n\\end{eqnarray}\nand the negativity,\n\\begin{equation}\n{\\cal {N^{\\prime}}}\\left(t\\right) = \\frac{2r^{\\prime}\\sqrt{N}}{\\left(N+r^{\\prime2}\\right)}\n\\left\\{\\sin^{2}\\left[\\frac{\\left(N+r^{\\prime2}\\right)}{2r^{\\prime}}\\tau^{\\prime}\\right]\\right.\n\\left.-\\frac{4r^{\\prime2}N}{\\left(N+r^{\\prime2}\\right)^{2}}\\sin^{4}\\left[\\frac{\\left(N+r^{\\prime2}\\right)}{2r^{\\prime}}\\tau^{\\prime}\\right]\\right\\}^{1\/2},\n\\label{negatsemiclass} \n\\end{equation}\nwhere the scaled time is defined as $\\tau^{\\prime} = g\\left|\\Omega_{L}^{*}\\right|t\/\\Delta$ and $r^{\\prime} = \\left|\\Omega_{L}^{*}\\right|\/g$.\nCompared to the fully quantized situation; there are no collapses and revivals, as the atomic inversion \nand the negativity are now periodic functions of time. This is illustrated in Fig. (\\ref{figure4}), \nwhere we have plotted the atomic inversion in Eq. (\\ref{invsemiclass}) and the negativity in Eq. (\\ref{negatsemiclass}) \nas a function of the scaled time $\\tau^{\\prime}$. For a convenient choice of the parameter $r^{\\prime}$, the atomic population \ncompletely inverts and returns to its initial state at times in which atom-mode 1 are in a separable state, \na very different behaviour from what happens if mode 2 is a quantized field. \n\n\\section{Conclusions}\n\nWe have presented a study of the dynamics of three coupled quantum systems: one three level atom and two quantized cavity \nfields, focusing on some properties of the atomic system (population inversion) and the atom-mode 1 bipartite system \n(entanglement). The second field (mode 2) has been basically traced out and treated as an ``external\" \nsubsystem. Our study has been based on the Raman coupled model, involving an effective two level atom coupled to two \nelectromagnetic cavity fields. We have considered different preparations for mode 2: coherent and thermal states of the \nquantized field, as well as a classical field. In order to keep the consistency of the effective Hamiltonian, as has been \nalready pointed out \\cite{knig91,agar04} we have retained the Stark shift terms in the Raman Hamiltonian, which have the \nremarkable effect of keeping the dynamics periodic. Moreover, the periodic revival times will not depend on the statistics \nof the fields. This means that having one mode (or even two) prepared in the (highly mixed) thermal state will not change \nthe regular atomic response during the atom\/fields interaction, as we have shown explicitly here. This is a nice example of \ndynamical features robust against substantial variations in the field statistics.  We have addressed the issue of bipartite \nentanglement between the atom and mode 1, after tracing out mode 2: we have found that\nfor mode 1 initially prepared in a Fock state and mode 2 in a coherent state, the atom\/mode 1 reach maximum entanglement \nin the collapse region of the atomic inversion. We have also compared both the atomic response and \nentanglement in the case in which mode 2 is treated as a classical field (``partially classical\" case), rather than a \nquantum coherent ``quasi-classical\" field, which result in very different evolutions.\n\\\\\n\\begin{flushleft}\nWe thank Prof. C.J. Villas-B\\^oas for fruitful comments.\nThis work is supported by the Agencies CNPq (INCT of Quantum Information) and FAPESP (CePOF), Brazil.\n\\end{flushleft}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics{fig3.pdf}\n\\end{center}\n\\caption{Negativity (a) and linear entropy (b) as a function of the scaled time $g_1 t$\nfor mode 1 prepared in Fock state ($N = 5$) and mode 2 prepared in a coherent state ($\\overline{n} = 5$), with $r = 1.023$.}\n\\label{figure3}      \n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics{fig4.pdf}\n\\end{center}\n\\caption{Atomic inversion (red, dashed curve) and the Negativity (blue, continuous curve) \nas a function of the scaled time $\\tau^{\\prime}$ in the ``partially classical\" case, having mode 1 prepared in a Fock state ($N = 2$) and $r^\\prime = 1.41$.}\n\\label{figure4}      \n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nVector equilibrium problem is a unified model of several problems, for\ninstance, vector variational inequalities and vector optimization problems.\nFor further relevant information on this topic, the reader is referred to\nthe following recent publications available in our bibliography: [1-7],\n[9-12], [14-17].\n\nIn this paper, we will suppose that $X$\\textit{\\ }is a nonempty, convex and\ncompact set in a Hausdorff locally convex space $E$ , $A:X\\rightarrow 2^{X}$\nand $F:X\\times X\\times X\\rightarrow 2^{X}$ are correspondences and $C\\subset\nX$ is a nonempty closed convex cone with int$C\\neq \\emptyset $.\n\nWe consider the following generalized strong vector quasi-equilibrium\nproblem (in short, GSVQEOP):\\newline\n\nfind $x^{\\ast }\\in X$ such that $x^{\\ast }\\in \\overline{A}(x^{\\ast })$ and\neach $u\\in A(x^{\\ast })$ implies that $F(u,x^{\\ast },z)\\nsubseteq $int$C$\nfor each $z\\in A(x^{\\ast }).$\n\n\\section{Preliminary results}\n\nLet $X$, $Y$ be topological spaces and $T:X\\rightarrow 2^{Y}$ be a\ncorrespondence. $T$ is said to be \\textit{upper semicontinuous} if for each \nx\\in X$ and each open set $V$ in $Y$ with $T(x)\\subset V$, there exists an\nopen neighborhood $U$ of $x$ in $X$ such that $T(x)\\subset V$ for each $y\\in\nU$. $T$ is said to be \\textit{lower semicontinuous} if for each x$\\in X$ and\neach open set $V$ in $Y$ with $T(x)\\cap V\\neq \\emptyset $, there exists an\nopen neighborhood $U$ of $x$ in $X$ such that $T(y)\\cap V\\neq \\emptyset $\nfor each $y\\in U$. $T$ is said to have \\textit{open lower sections} if \nT^{-1}(y):=\\{x\\in X:y\\in T(x)\\}$ is open in $X$ for each $y\\in Y.$\n\nThe following lemma will be crucial in the proofs.\n\n\\begin{lemma}\n(Yannelis and Prabhakar, \\cite{yan}). \\textit{Let }$X$\\textit{\\ be a\nparacompact Hausdorff topological space and }$Y$\\textit{\\ be a Hausdorff\ntopological vector space. Let }$T:X\\rightarrow 2^{Y}$\\textit{\\ be a\ncorrespondence with nonempty convex values\\ and for each }$y\\in Y$\\textit{, \n$T^{-1}(y)$\\textit{\\ is open in }$X$\\textit{. Then, }$T$\\textit{\\ has a\ncontinuous selection that is, there exists a continuous function }\nf:X\\rightarrow Y$\\textit{\\ such that }$f(x)\\in T(x)$\\textit{\\ for each }\nx\\in X$\\textit{.\\medskip }\n\\end{lemma}\n\nThe correspondence $\\overline{T}$ is defined by $\\overline{T}(x):=\\{y\\in\nY:(x,y)\\in $cl$_{X\\times Y}$ Gr $T\\}$ (the set cl$_{X\\times Y}$ Gr $(T)$ is\ncalled the adherence of the graph of $T$)$.$ It is easy to see that cl \nT(x)\\subset \\overline{T}(x)$ for each $x\\in X.$\n\nIf $X$ and $Y$ are topological vector spaces, $K$ is a nonempty subset of \nX, $ $C$ is a nonempty closed convex cone and $T:K\\rightarrow 2^{Y}$ is a\ncorrespondence, then \\cite{luc}, $T$ is called \\textit{upper }$C$\\textit\n-continuous at} $x_{0}\\in K$ if, for any neighborhood $U$ of the origin in \nY,$ there is a neighborhood $V$ of $x_{0}$ such that, for all $x\\in V,$ \nT(x)\\subset T(x_{0})+U+C.$ $T$ is called \\textit{lower }$C$\\textit\n-continuous at} $x_{0}\\in K$ if, for any neighborhood $U$ of the origin in \nY,$ there is a neighborhood $V$ of $x_{0}$ such that, for all $x\\in V,$ \nT(x_{0})\\subset T(x)+U-C.$\n\nThe property of properly $C-$quasi-convexity for correspondences is\npresented below.\n\nLet $X$ be a nonempty convex subset of a topological vector space\\textit{\\ }\nE,$ $Y$ be a topological vector space, and $C$ be a pointed closed convex\ncone in $Z$ with its interior int$C\\neq \\emptyset .$ Let $T:X\\rightarrow\n2^{Y}$ be a correspondence with nonempty values. $T$ is said to be \\textit\nproperly }$C-$\\textit{quasi-convex on} $X$ (\\cite{long}), if for any \nx_{1},x_{2}\\in X$ and $\\lambda \\in \\lbrack 0,1],$ either $T(x_{1})\\subset\nT(\\lambda x_{1}+(1-\\lambda )x_{2})+C$ or $T(x_{2})\\subset T(\\lambda\nx_{1}+(1-\\lambda )x_{2})+C.\\medskip $\n\nIn order to establish our main theorems, we need to prove some auxiliary\nresults. The starting point is the following statement:\n\n\\begin{theorem}\nLet $X$ be \\textit{a nonempty, convex and compact set in a} \\textit\nHausdorff locally convex space }$E$ and let $\\mathcal{C\\ }$\\ be a nonempty\nsubset of $X\\times X.$\\textit{\\ Assume that t}he following conditions are\nfulfilled:\n\\end{theorem}\n\n\\textit{a) }$\\mathcal{C}^{-}(y)$ $=\\{x\\in X:(x,y)\\in C\\}$ \\textit{is open fo\n} \\textit{each } $y\\in X;$\n\n\\textit{b)} $\\mathcal{C}^{+}(x)=\\{y\\in X:(x,y)\\in C\\}\\mathit{\\ }$ \\textit{is\nconvex and nonempty for each} $x\\in X.$\n\n\\textit{Then, there exists }$x^{\\ast }\\in X$ such that $(x^{\\ast },x^{\\ast\n})\\in C$\\textit{.\\medskip }\n\n\\begin{proof}\nLet us define the correspondence $T:X\\rightarrow 2^{X},$ by\n\n$T(x)=\\mathcal{C}^{+}(x)$ for each $x\\in X.$\n\nThe correspondence $T$ is nonempty and convex valued and it has open lower\nsections$.$\n\nWe apply the Yannelis and Prabhakar's Lemma and we obtain that $T$\\textit{\\ \nhas a continuous selection $f:X\\rightarrow X.$\n\nAccording to the Tychonoff fixed point Theorem \\cite{is}, there exists \nx^{\\ast }\\in X$ such that $f(x^{\\ast })=x^{\\ast }.$ Hence, $x^{\\ast }\\in\nT(x^{\\ast })$ and obviously, $(x^{\\ast },x^{\\ast })\\in \\mathcal{C}.\\medskip\n\\medskip $\n\\end{proof}\n\nThe next two results are direct consequences of Theorem 1.\\medskip\n\n\\begin{theorem}\nLet $X$ be \\textit{a nonempty, convex and compact set in a} \\textit\nHausdorff locally convex space }$E$ , and let $A:X\\rightarrow 2^{X}$ and \nP:X\\times X\\rightarrow 2^{X}$ be correspondences such that the following\nconditions are fulfilled:\n\\end{theorem}\n\n\\textit{a)} $A$\\textit{\\ has nonempty, convex values and open lower sections\n}\n\n\\textit{b)} \\textit{the set }$\\{y\\in X:$ $A(x)\\cap P(x,y)=\\emptyset \\}\\cap\nA(x)$ \\ \\textit{is nonempty for each} $x\\in X;$\n\n\\textit{c) }$\\{y\\in X:$\\textit{\\ }$A(x)\\cap P(x,y)=\\emptyset \\}$\\textit{\\ is\nconvex for each }$x\\in X;$\n\n\\textit{d)} $\\{x\\in X:A(x)\\cap P(x,y)=\\emptyset \\}\\mathit{\\ }$ \\textit{is\nopen for each} $y\\in X.$\n\n\\textit{Then, there exists }$x^{\\ast }\\in $\\textit{\\ }$X$\\textit{\\ such that \n}$\\ x^{\\ast }\\in A(x^{\\ast })$ \\textit{and }$A(x^{\\ast })\\cap P(x^{\\ast\n},x^{\\ast })=\\emptyset .\\medskip $\n\n\\begin{proof}\nLet us define the set $\\mathcal{C}=\\{(x,y)\\in X\\times X:$ $A(x)\\cap\nP(x,y)=\\emptyset \\}\\cap $Gr$A.$\n\nThen, $\\mathcal{C}^{+}(x)=\\{y\\in X:$ $A(x)\\cap P(x,y)=\\emptyset \\}\\cap A(x)$\nfor each $\\ x\\in X$ and\n\n$\\mathcal{C}^{-}(y)=A^{-1}(y)\\cap \\{x\\in X:A(x)\\cap P(x,y)=\\emptyset \\}\\ \nfor each $x\\in X.$\n\nAssumption b) implies that $\\mathcal{C}$ is nonempty$.$ The set $\\mathcal{C\n^{-}(y)$ is open for each $y\\in X$ since Assumptions a) and d) hold.\n\nAccording to Assumptions b) and c), $A(x)\\cap $ $\\mathcal{C}^{+}(x)$ is\nnonempty and convex for each $x\\in X.$\n\nAll hypotheses of Theorem 1 are fulfilled, and then, there exists $x^{\\ast\n}\\in $\\ $X$\\ such that $\\ x^{\\ast }\\in A(x^{\\ast })$ and $A(x^{\\ast })\\cap\nP(x^{\\ast },x^{\\ast })=\\emptyset .$\n\\end{proof}\n\nWe establish the following result as a consequence of Theorem 2. It will be\nused in order to prove the existence of solutions for the considered vector\nquasi-equilibrium problem.\n\n\\begin{theorem}\nLet $X$ be \\textit{a nonempty, convex and compact set in a} \\textit\nHausdorff locally convex space }$E$ , and let $A:X\\rightarrow 2^{X}$, \nP:X\\times X\\rightarrow 2^{X}$ be correspondences such that the following\nconditions are fulfilled:\n\\end{theorem}\n\n\\textit{a)} $A$\\textit{\\ has nonempty, convex values and open lower sections\n}\n\n\\textit{b)} \\textit{the set }$\\{y\\in X:$ $\\overline{A}(x)\\cap\nP(x,y)=\\emptyset \\}\\cap A(x)$ \\ \\textit{is nonempty for each} $x\\in X;$\n\n\\textit{c) }$\\{y\\in X:$\\textit{\\ }$\\overline{A}(x)\\cap P(x,y)=\\emptyset \\}\n\\textit{\\ is convex for each }$x\\in X;$\n\n\\textit{d)} $\\{x\\in X:\\overline{A}(x)\\cap P(x,y)=\\emptyset \\}\\mathit{\\ }$ \n\\textit{is open for each} $y\\in X.$\n\n\\textit{Then, there exists }$x^{\\ast }\\in $\\textit{\\ }$X$\\ \\textit{\\ such\nthat }$\\ x^{\\ast }\\in A(x^{\\ast })$ \\textit{and }$A(x^{\\ast })\\cap P(x^{\\ast\n},x^{\\ast })=\\emptyset .\\medskip $\n\nWe note that, according to Theorem 2, there exists $x^{\\ast }\\in $\\textit{\\ \n$X$\\textit{\\ }such that $\\ x^{\\ast }\\in A(x^{\\ast })$ and\\textit{\\ }\n\\overline{A}(x^{\\ast })\\cap P(x^{\\ast },x^{\\ast })=\\emptyset .$ Obviously, \n\\overline{A}(x^{\\ast })\\cap P(x^{\\ast },x^{\\ast })=\\emptyset $ implies \nA(x^{\\ast })\\cap P(x^{\\ast },x^{\\ast })=\\emptyset .\\medskip $\n\nIf $A(x)=X$ for each $x\\in X$, Theorem 2 implies the following corollary.\n\n\\begin{corollary}\nLet $X$ be \\textit{a nonempty, convex and compact set in a} \\textit\nHausdorff locally convex space }$E$ , and let $P:X\\times X\\rightarrow 2^{X}$\nbe a correspondence such that the following conditions are fulfilled:\n\\end{corollary}\n\n\\textit{a) }$\\{y\\in X:$\\textit{\\ }$P(x,y)=\\emptyset \\}$\\textit{\\ is nonempty\nand convex for each }$x\\in X;$\n\n\\textit{b)} $\\{x\\in X:P(x,y)=\\emptyset \\}\\mathit{\\ }$ \\textit{is open for\neach} $y\\in X.$\n\n\\textit{Then, there exists }$x^{\\ast }\\in $\\textit{\\ }$X$\\textit{\\ such that \n}$\\ P(x^{\\ast },x^{\\ast })=\\emptyset .\\medskip $\n\nBy applying an approximation method of proof, we can prove Theorem 4.\n\n\\begin{theorem}\nLet $X$ be \\textit{a nonempty, convex and compact set in a} \\textit\nHausdorff locally convex space }$E$ and $\\mathcal{C\\ }$\\ be a nonempty,\nclosed subset of $X\\times X.$\\textit{\\ Assume that there exists a sequence }\n(G_{k})_{k\\in \\mathbb{N}^{\\ast }}$ \\textit{of subsets of }$X\\times X$ such\nthat \\textit{t}he following conditions are fulfilled:\n\\end{theorem}\n\n\\textit{a) }for each $k\\in \\mathbb{N}^{\\ast },$ $G_{k}^{-}(y)$ $=\\{x\\in\nX:(x,y)\\in G_{k}\\}$ \\textit{is open for} \\textit{each } $y\\in X;$\n\n\\textit{b)} for each $k\\in \\mathbb{N}^{\\ast },$ $G_{k}^{+}(x)=\\{y\\in\nX:(x,y)\\in G_{k}\\}\\mathit{\\ }$ \\textit{is convex and nonempty for each} \nx\\in X;$\n\n\\textit{c) }$G_{k}\\supseteq G_{k+1}$\\textit{\\ for each }$k\\in \\mathbb{N\n^{\\ast };$\n\n\\textit{d) for every open set }$G$\\textit{\\ with }$G\\supset \\mathcal{C},\n\\textit{\\ there exists }$k\\in \\mathbb{N}^{\\ast }$\\textit{\\ such that }\nG_{k}\\subseteq \\mathcal{C}.$\n\n\\textit{Then, there exists }$x^{\\ast }\\in X$\\textit{\\ such that }$(x^{\\ast\n},x^{\\ast })\\in \\mathcal{C}$\\textit{.\\medskip }\n\n\\begin{proof}\nFor each $k\\in \\mathbb{N}^{\\ast },$ we apply Theorem 1. Let $x^{k}\\in X$\nsuch that $(x^{k},x^{k})\\in G_{k}.$ Since $X$ is a compact set, we can\nconsider that the sequence $(x^{k})_{k}$ converges to some $x^{\\ast }\\in X.$\nWe claim that $(x^{\\ast },x^{\\ast })\\in \\mathcal{C}.$\n\nIndeed, let us suppose, by way of contradiction, that $(x^{\\ast },x^{\\ast\n})\\notin \\mathcal{C}.$ Since $\\mathcal{C}$\\textit{\\ }is nonempty and\ncompact, we can choose a neighborhood $V_{(x^{\\ast },x^{\\ast })}$ of \n(x^{\\ast },x^{\\ast })$ and an open set $G$ such that $G\\supset \\mathcal{C}$\nand $V_{(x^{\\ast },x^{\\ast })}\\cap G=\\emptyset .$ According to Assumptions\nd) and c), there exists $k_{1}\\in \\mathbb{N}^{\\ast }$ such that \nG_{k}\\subseteq G$ for each $k\\geq k_{1}.$ Since $V_{(x^{\\ast },x^{\\ast })}$\nis a neighborhood of $(x^{\\ast },x^{\\ast }),$ there exists $k_{2}\\in \\mathbb\nN}^{\\ast }$ such that $(x^{k},x^{k})\\in V_{(x^{\\ast },x^{\\ast })}$ for each \nk\\geq k_{2}.$ $\\ $Hence, for $k\\geq $max($k_{2},k_{1}),$ \n(x^{k},x^{k})\\notin G_{k},$ which is a contradiction.\n\nConsequently, $(x^{\\ast },x^{\\ast })\\in \\mathcal{C}.$\n\\end{proof}\n\nTheorem 5 is a consequence of Theorem 4 and it will be used in Section 3 in\norder to prove the existence of solutions for GSVQEP.\n\n\\begin{theorem}\n\\textit{Let }$X$\\textit{\\ be a nonempty, convex and compact set in a\nHausdorff locally convex space }$E$\\textit{\\ , and let }$A:X\\rightarrow\n2^{X} $\\textit{\\ and }$P:X\\times X\\rightarrow 2^{X}$\\textit{\\ be\ncorrespondences such that the following conditions are fulfilled:}\n\\end{theorem}\n\n\\textit{a) }$A$\\textit{\\ has nonempty, convex values and open lower sections\n}\n\n\\textit{b) the set }$U=\\{(x,y)\\in X\\times X:$\\textit{\\ }$\\overline{A}(x)\\cap\nP(x,y)=\\emptyset \\}$\\textit{\\ \\ is closed} \\textit{and} $U\\cap $Gr$\\overline\nA}$ \\textit{is nonempty;}\n\n\\textit{c) there exists a sequence }$(P_{k})_{k\\in N}$\\textit{\\ of\ncorrespondences, where, for each }$k\\in \\mathbb{N}^{\\ast },$\\textit{\\ }\nP_{k}:X\\times X\\rightarrow 2^{X}$\\textit{\\ and let }$U_{k}=\\{(x,y)\\in\nX\\times X:$\\textit{\\ }$A(x)\\cap P_{k}(x,y)=\\emptyset \\}$\\textit{. Assume\nthat:}\n\n\\textit{\\ }$\\ \\ \\ $\\textit{c1) }$U_{k}^{+}(x)=\\{y\\in X:$\\textit{\\ }\n\\overline{A}(x)\\cap P_{k}(x,y)=\\emptyset \\}$\\textit{\\ is convex for each }\nx\\in X$\\textit{\\ and }$U_{k}^{+}(x)\\cap A(x)\\neq \\emptyset $\\textit{\\ for\neach }$x\\in X;$\n\n\\textit{\\ \\ \\ \\ c2 ) }$U_{k}^{-}(y)=\\{x\\in X:\\overline{A}(x)\\cap\nP_{k}(x,y)=\\emptyset \\}\\ $\\textit{\\ is open for each }$y\\in X;$\n\n\\textit{\\ \\ \\ \\ c3) for each }$k\\in \\mathbb{N}^{\\ast },$\\textit{\\ }\nP_{k}(x,y)\\subseteq P_{k+1}(x,y)$\\textit{\\ for each }$(x,y)\\in X\\times X;$\n\n\\textit{\\ \\ \\ \\ \\ c4) for every open set }$G$\\textit{\\ with }$G\\supset U\\cap \n$Gr$\\overline{A},$\\textit{\\ there exists }$k\\in \\mathbb{N}^{\\ast }$\\textit{\\\nsuch that }$G\\supseteq U_{k}\\cap $Gr$A.$\n\n\\textit{Then, there exists }$x^{\\ast }\\in $\\textit{\\ }$X$\\textit{\\ such that \n}$\\ x^{\\ast }\\in \\overline{A}(x^{\\ast })$\\textit{\\ and }$A(x^{\\ast })\\cap\nP(x^{\\ast },x^{\\ast })=\\emptyset .\\medskip $\n\n\\begin{proof}\nLet us define $\\mathcal{C=}U\\cap $Gr$\\overline{A}.$ According to Assumptions\nb) and c), $\\mathcal{C\\ }$\\ is a nonempty and closed subset of $X\\times X.$\n\nFurther, \\textit{\\ }for each\\textit{\\ }$k\\in \\mathbb{N}^{\\ast },$ let us\ndefine $G_{k}=U_{k}\\cap $Gr$A\\subseteq X\\times X.$\n\nThen, for each $k\\in \\mathbb{N}^{\\ast },$ $G_{k}^{+}(x)$ $=\\{y\\in X:(x,y)\\in\nG_{k}\\}=U_{k}^{+}(x)\\cap A(x)$ is nonempty and convex for each\\textit{\\ }\nx\\in X,$ since Assumptions a) and c1) hold.\n\nFor each $k\\in \\mathbb{N}^{\\ast },$ $G_{k}^{-}(y)$ $=\\{x\\in X:(x,y)\\in\nG_{k}\\}=U_{k}^{-}(y)\\cap A^{-1}(y)$ is open for each\\textit{\\ }$y\\in X,$\nsince Assumptions a) and c2) hold.\n\nAssumption c3) implies that, \\textit{\\ }for each\\textit{\\ }$k\\in \\mathbb{N\n^{\\ast },$ $U_{k+1}\\subseteq U_{k}$ and then, $G_{k}\\supseteq G_{k+1}$ and\nAssumption c4) implies that for every open set $G$ with $G\\supset \\mathcal{C\n,$ there exists $k\\in \\mathbb{N}^{\\ast }$ such that $G_{k}\\subseteq \\mathcal\nC}.$\n\nAll hypotheses of Theorem 4 are verified. Therefore, there exists $x^{\\ast\n}\\in X$ such that $(x^{\\ast },x^{\\ast })\\in \\mathcal{C}$.\n\nConsequently, there exists $x^{\\ast }\\in X$ such that $\\ x^{\\ast }\\in \n\\overline{A}(x^{\\ast })$\\textit{\\ }and\\textit{\\ }$A(x^{\\ast })\\cap P(x^{\\ast\n},x^{\\ast })=\\emptyset .\\medskip $\n\\end{proof}\n\n\\section{Main results}\n\nThis section is devoted to the study of the existence of solutions for the\nconsidered generalized strong vector quasi-equilibrium problem. We derive\nour main results by using the auxiliary theorems concerning correspondences,\nwhich have been established in the previous section. This new approach to\nsolve GSVQEP is intended to provide new conditions under which the solutions\nexist.\\medskip\n\nThe first theorem states that GSVQEP has solutions if $F(\\cdot ,y,\\cdot )$\nis lower ($-C$)-semicontinuous for each $y\\in X$ and $F(u,\\cdot ,z)$\\ is\nproperly $C-$\\ quasi-convex for each $(u,z)\\in X\\times X.$\n\n\\begin{theorem}\n\\textit{Let }$F:X\\times X\\times X\\rightarrow 2^{X}$\\textit{\\ be a\ncorrespondence with nonempty values. Suppose that:}\n\\end{theorem}\n\n\\textit{a) }$A$\\textit{\\ has nonempty, convex values and open lower sections\n}\n\n\\textit{b) for each }$x\\in X,$\\textit{\\ there exists }$y\\in A(x)$\\textit{\\\nsuch that each }$u\\in \\overline{A}(x)$\\textit{\\ implies that }\nF(u,y,z)\\nsubseteq $\\textit{int}$C$\\textit{\\ for each }$z\\in \\overline{A}(x);\n$\n\n\\textit{c) }$F(\\cdot ,y,\\cdot ):$\\textit{\\ }$X\\times X\\rightarrow 2^{X}\n\\textit{\\ is lower (}$-C$\\textit{)-semicontinuous for each }$y\\in X;$\n\n\\textit{d) for each }$(u,z)\\in X\\times X,$\\textit{\\ }$F(u,\\cdot\n,z):X\\rightarrow 2^{X}$\\textit{\\ is properly }$C-$\\textit{\\ quasi-convex.}\n\n\\textit{Then, there exists }$x^{\\ast }\\in X$\\textit{\\ such that }$x^{\\ast\n}\\in A(x^{\\ast })$\\textit{\\ and each }$u\\in A(x^{\\ast })$\\textit{\\ implies\nthat }$F(u,x^{\\ast },z)\\nsubseteq $\\textit{int}$C$\\textit{\\ for each }$z\\in\nA(x^{\\ast }),$\\textit{\\ that is, }$x^{\\ast }$\\textit{\\ is a solution for\nGSVQEP.\\medskip }\n\n\\begin{proof}\nLet us define $P:X\\times X\\rightarrow 2^{X},$ by\n\n$P(x,y)=\\{u\\in X:$ $\\exists z\\in \\overline{A}(x)$ such that \nF(u,y,z)\\subseteq C\\}$ for each $(x,y)\\in X\\times X.$\n\nAssumption b) implies that the set $\\{y\\in X:$ $\\overline{A}(x)\\cap\nP(x,y)=\\emptyset \\}\\cap A(x)$ \\ is nonempty for each $x\\in X.$\n\nWe claim that the set $E(x)$ $\\ $is convex for each $x\\in X,$ where\\ \nE(x)=\\{y\\in X:$\\textit{\\ }$\\overline{A}(x)\\cap P(x,y)=\\emptyset \\}=$\n\n$=\\{y\\in X:$\\textit{\\ }each $u\\in \\overline{A}(x)$ implies that \nF(u,y,z)\\nsubseteq C$ for each $z\\in \\overline{A}(x)\\}.$\n\nIndeed, let us fix $x_{0}\\in X$ and let us consider $y_{1},y_{2}\\in\nE(x_{0}). $ This means that each $u\\in \\overline{A}(x_{0})$ implies that \nF(u,y_{1},z)\\nsubseteq C$ and $F(u,y_{2},z)\\nsubseteq C$ for each $z\\in \n\\overline{A}(x_{0}).$\n\nLet $y(\\lambda )=\\lambda y_{1}+(1-\\lambda )y_{2}$ be defined for each \n\\lambda \\in \\lbrack 0,1].$\n\nWe claim that $y(\\lambda )\\in E(x_{0})$ for each $\\lambda \\in \\lbrack 0,1].$\n\nSuppose, on the contrary, that there exist $\\lambda _{0}\\in \\lbrack 0,1]$ $,$\n$u^{\\prime }\\in \\overline{A}(x_{0})$ and $z^{\\prime }\\in \\overline{A}(x_{0})$\nsuch that $F(u^{\\prime },y(\\lambda _{0}),z^{\\prime })\\subseteq C.$ Since \n(F(u^{\\prime },\\cdot ,z^{\\prime }):X\\rightarrow 2^{X}$ is\\textit{\\ }properl\n\\textit{\\ }$C-$\\textit{\\ }quasi-convex$,$ we have that:\n\n$F(u^{\\prime },y_{1},z^{\\prime })\\subseteq F(u^{\\prime },y(\\lambda\n),z^{\\prime })+C$ or $F(u^{\\prime },y_{2},z^{\\prime })\\subseteq F(u^{\\prime\n},y(\\lambda ),z^{\\prime })+C$\n\nOn the other hand, it is true that $F(u^{\\prime },y(\\lambda ),z^{\\prime\n})\\subseteq C.$ We obtain that:\n\n$F(u^{\\prime },y_{j},z^{\\prime })\\subseteq C+C\\subseteq C$ for $j=1$ or for \nj=2$.\n\nThis contradicts the assumption that $y_{1},y_{2}\\in E(x_{0})$.\nConsequently, $E(x_{0})$ is convex and Assumption c) from Theorem 3 is\nfulfilled.\n\nNow, we will prove that $D(y)=\\{x\\in X:\\overline{A}(x)\\cap P(x,y)=\\emptyset\n\\}\\mathit{\\ }$ is open for each $y\\in X.$\n\nIn order to do this, we will show that $^{C}D(y)$ is closed for each $y\\in X,\n$ where $\\ ^{C}D(y)=\\{x\\in X:\\overline{A}(x)\\cap P(x,y)\\neq \\emptyset \\\n\\mathit{=}\\{x\\in X:$ there exist $u,z\\in \\overline{A}(x)$ such that \nF(u,y,z)\\subseteq C\\}.$\n\nLet $(x_{\\alpha })_{\\alpha \\in \\Lambda }$ be a net in $^{C}D(y)$ such that\nlim$_{\\alpha }x_{\\alpha }=x_{0}.$ Then, there exist $u_{\\alpha },z_{\\alpha\n}\\in \\overline{A}(x_{\\alpha })$ such that $F(u_{\\alpha },y,z_{\\alpha\n})\\subseteq C.$\n\nSince $X$ is a compact set, we can suppose that $(u_{\\alpha })_{\\alpha \\in\n\\Lambda },(z_{\\alpha })_{\\alpha \\in \\Lambda }$ are convergent nets and let\nlim$_{\\alpha }u_{\\alpha }=u_{0}$ and lim$_{\\alpha }z_{\\alpha }=z_{0}.$\n\nThe closedness of $\\overline{A}$ implies that $u_{0},z_{0}\\in \\overline{A\n(x_{0}).$\n\nNow, we claim that $F(u_{0},y,z_{0})\\subseteq C.$\n\nSince $F(u_{\\alpha },y,z_{\\alpha })\\subseteq C$ and $F(\\cdot ,y,\\cdot ):\n\\textit{\\ }$X\\times X\\rightarrow 2^{X}$\\textit{\\ }is lower ($-C\n)-semicontinuous, for each neighborhood $U$ of the origin in $X,$ there\nexists a subnet $(u_{\\beta },z_{\\beta })_{\\beta }$ of $(u_{\\alpha\n},z_{\\alpha })_{\\alpha }$ such that $F(u_{0},y,z_{0})\\subset F(u_{\\beta\n},y,z_{\\beta })+U+C.$ Hence, $F(u_{0},y,z_{0})\\subset U+C.$\n\nWe will show that $F(u_{0},y,z_{0})\\subset C.$ Suppose, by way of\ncontradiction, that there exists $t\\in F(u_{0},y,z_{0})\\cap ^{C}C.$ We note\nthat $B=C-t$ is a closed set which does not contain $0.$ It follows that \n^{C}B$ is open and contains $0.$ Since $X$ is locally convex, there exists a\nconvex neighborhood $U_{1}$ of origin such that $U_{1}\\subset X\\backslash B$\nand $U_{1}=-U_{1}$. Thus, $0\\notin B+U_{1}$ and then, $t\\notin C+U_{1},$\nwhich is a contradiction. Therefore, $F(u_{0},y,z_{0})\\subset C.$\n\nWe proved that there exist $u_{0},z_{0}\\in \\overline{A}(x_{0})$ such that \nF(u_{0},y,z_{0})\\subseteq C.$ It follows that $^{C}D(y)$ is closed. Then, \nD(y)$ is an open set and Assumption d) from Theorem 3 is fulfilled.\n\nConsequently, all conditions of Theorem 3 are verified, so that there exists \n$x^{\\ast }\\in $\\ $X$\\ such that $\\ x^{\\ast }\\in A(x^{\\ast })$ and $A(x^{\\ast\n})\\cap P(x^{\\ast },x^{\\ast })=\\emptyset .\\medskip $ Obviously, $x^{\\ast }\n\\textit{\\ }is a solution for GSVQEP.\\medskip\n\\end{proof}\n\nIn order to obtain a second result concerning the existence of solutions of\nGSVQEP, we use an approximation method and Theorem 5. We mention that this\nresult does not require convexity properties for the correspondence $F.$\n\n\\begin{theorem}\n\\textit{Let }$F:X\\times X\\times X\\rightarrow 2^{X}$\\textit{\\ be a\ncorrespondence. Suppose that:}\n\\end{theorem}\n\n\\textit{a) }$A$\\textit{\\ has nonempty, convex values and open lower\nsections; }$\\overline{A}$\\textit{\\ is lower semicontinuous;}\n\n\\textit{b) }$F$\\textit{\\ is upper semicontinuous with nonempty, closed\nvalues;}\n\n\\textit{c) }$U\\cap $Gr$\\overline{A}$\\textit{\\ is nonempty, where }$U=\\{\n\\textit{\\ }$(x,y)\\in X\\times X:u\\in \\overline{A}(x)$\\textit{\\ implies that }\nF(u,y,z)\\nsubseteq $\\textit{int}$C$\\textit{\\ for each }$z\\in \\overline{A\n(x)\\};$\n\n\\textit{d) there exists a sequence }$(F_{k})_{k\\in N}$\\textit{\\ of\ncorrespondences, such that, for each }$k\\in \\mathbb{N}^{\\ast },$\\textit{\\ }\nF_{k}:X\\times X\\times X\\rightarrow 2^{X}$\\textit{\\ and let }\nU_{k}=\\{(x,y)\\in X\\times X:u\\in \\overline{A}(x)$\\textit{\\ implies that }\nF_{k}(u,y,z)\\nsubseteq $\\textit{int}$C$\\textit{\\ for each }$z\\in \\overline{A\n(x)\\}$\\textit{. Assume that:}\n\n\\textit{d1) for each }$k\\in \\mathbb{N}^{\\ast }$\\textit{\\ and for each }$x\\in\nX,$\\textit{\\ there exists }$y\\in A(x)$\\textit{\\ such that each }$u\\in \n\\overline{A}(x)$\\textit{\\ implies that }$F_{k}(u,y,z)\\nsubseteq $\\textit{int\n$C$\\textit{\\ for each }$z\\in A(x);$\n\n\\textit{d2) for each }$k\\in \\mathbb{N}^{\\ast }$\\textit{\\ and for each }\n(u,z)\\in X\\times X,$\\textit{\\ }$F_{k}(u,\\cdot ,z):X\\rightarrow 2^{X}$\\textit\n\\ is properly }$C-$\\textit{\\ quasi-convex;}\n\n\\textit{d3) for each }$k\\in \\mathbb{N}^{\\ast }$ \\textit{and} \\textit{\\ for\neach }$y\\in X,$\\textit{\\ }$F_{k}(\\cdot ,y,\\cdot ):$\\textit{\\ }$X\\times\nX\\rightarrow 2^{X}$\\textit{\\ is lower (}$-C$\\textit{)-semicontinuous}$;$\n\n\\textit{d4) for each }$k\\in \\mathbb{N}^{\\ast },$\\textit{\\ for each }\n(x,y)\\in X\\times X,$\\textit{\\ and for each }$u\\in X$\\textit{\\ with the\nproperty that }$\\exists z\\in \\overline{A}(x)$\\textit{\\ such that }\nF_{k}(u,y,z)\\subseteq C,$\\textit{\\ there exists }$z^{\\prime }\\in \\overline{A\n(x)$\\textit{\\ such that }$F_{k+1}(u,y,z^{\\prime })\\subseteq C;$\n\n\\textit{d5) for every open set }$G$\\textit{\\ with }$G\\supset U\\cap $Gr\n\\overline{A},$\\textit{\\ there exists }$k\\in \\mathbb{N}^{\\ast }$\\textit{\\\nsuch that }$G\\supseteq U_{k}\\cap $\\textit{Gr}$A.$\n\n\\textit{Then, there exists }$x^{\\ast }\\in X$\\textit{\\ such that }$x^{\\ast\n}\\in \\overline{A}(x^{\\ast })$\\textit{\\ and each }$u\\in A(x^{\\ast })$\\textit\n\\ implies that }$F(u,x^{\\ast },z)\\nsubseteq $\\textit{int}$C$\\textit{\\ for\neach }$z\\in A(x^{\\ast }),$\\textit{\\ that is, }$x^{\\ast }$\\textit{\\ is a\nsolution for GSVQEP.\\medskip }\n\n\\begin{proof}\nLet us define $P:X\\times X\\rightarrow 2^{X},$ by\n\n$P(x,y)=\\{u\\in X:$ $\\exists z\\in \\overline{A}(x)$ such that \nF(u,y,z)\\subseteq C\\}$ for each $(x,y)\\in X\\times X.$\n\nWe claim that, $U=\\{$\\textit{\\ }$(x,y)\\in X\\times X:u\\in \\overline{A}(x)\n\\textit{\\ }implies that $F(u,y,z)\\nsubseteq $int$C$ for each $z\\in \\overline\nA}(x)\\}$ is closed$.$\n\nLet $(x^{0},y^{0})\\in $cl$U.$ Then, there exists $(x^{\\alpha },y^{\\alpha\n})_{\\alpha \\in \\Lambda }$ a net in $U$ such that $\\lim_{\\alpha }(x^{\\alpha\n},y^{\\alpha })=(x^{0},y^{0})\\in X\\times X.$ Let $u\\in \\overline{A}(x^{0})$\nand $z\\in \\overline{A}(x^{0}).$ Since $\\overline{A}$ is lower semicontinuous\nand $\\lim_{\\alpha }x^{\\alpha }=x^{0},$ there exists nets $(u^{\\alpha\n})_{\\alpha \\in \\Lambda }\\ $\\ and $(z^{\\alpha })_{\\alpha \\in \\Lambda }$ in $X$\nsuch that $u^{\\alpha },z^{\\alpha }\\in \\overline{A}(x^{\\alpha })$ for each \n\\alpha \\in \\Lambda $ and $\\lim_{\\alpha }u^{\\alpha }=u,$ $\\lim_{\\alpha\n}z^{\\alpha }=z.$ Since $(x^{\\alpha },y^{\\alpha })_{\\alpha \\in \\Lambda }$ is\na net in $U,$ $\\ $then$,$ for each $\\alpha \\in \\Lambda ,$ $F(u^{\\alpha\n},y^{\\alpha },z^{\\alpha })\\nsubseteq $int$C$ $,$ that is, $F(u^{\\alpha\n},y^{\\alpha },z^{\\alpha })\\cap W\\neq \\emptyset ,$ where $W=X\\backslash $int\nC,$ that is, there exists $(t^{\\alpha })_{\\alpha \\in \\Lambda }$ a net in $X$\nsuch that $t^{\\alpha }\\in F(u^{\\alpha },y^{\\alpha },z^{\\alpha })\\cap W$ for\neach $\\alpha \\in \\Lambda .$\n\nSince $X$ is compact, we can suppose that $\\lim_{\\alpha }t^{\\alpha }=t^{0}.$\nThe closedness of $W$ implies that $t^{0}\\in W.$ We invoke here the\nclosedness of $F$ and we conclude that $t^{0}\\in F(u,y^{0},z).$ Therefore, \nF(u,y^{0},z)\\cap W\\neq \\emptyset ,$ and, thus, $u\\in \\overline{A}(x^{0})$\nimplies $F(u,y^{0},z)\\nsubseteq $int$C$ for each $z\\in \\overline{A}(x^{0}).$\nHence, $U$ is closed.\n\nFor each $k\\in \\mathbb{N}^{\\ast },$ let us define $P_{k}:X\\times\nX\\rightarrow 2^{X},$ by\n\n$P_{k}(x,y)=\\{u\\in X:$ $\\exists z\\in \\overline{A}(x)$ such that \nF_{k}(u,y,z)\\subseteq C\\}$ for each $(x,y)\\in X\\times X$ and\n\n$U_{k}=\\{(x,y)\\in X\\times X:u\\in \\overline{A}(x)$ implies that \nF_{k}(u,y,z)\\nsubseteq $int$C$ for each $z\\in \\overline{A}(x)\\}=\\{(x,y)\\in\nX\\times X:$\\textit{\\ }$\\overline{A}(x)\\cap P_{k}(x,y)=\\emptyset \\}.$\n\n\\textit{\\ }Let\\textit{\\ }$k\\in \\mathbb{N}^{\\ast }.$\n\nAssumption d1) implies that $U_{k}^{+}(x)\\cap A(x)\\neq \\emptyset $\\textit{\\ \nfor each\\textit{\\ }$x\\in X$ and Assumption d2) implies that $U_{k}^{+}(x)$\nis convex \\textit{\\ }for each\\textit{\\ }$x\\in X$ (we use a similar proof\nwith the one of Theorem 6)\n\nSince $F_{k}(\\cdot ,y,\\cdot ):$\\textit{\\ }$X\\times X\\rightarrow 2^{X}\n\\textit{\\ }is lower ($-C$)-semicontinuous for each $y\\in X,$ by following an\nargument similar with the one from the proof of Theorem 6, we can prove that\n\n$U_{k}^{-}(y)=\\{x\\in X:\\overline{A}(x)\\cap P_{k}(x,y)=\\emptyset \\}\\ $ is\nopen for each\\textit{\\ }$y\\in X.$\n\nAssumption d4) implies that $P_{k}(x,y)\\subseteq P_{k+1}(x,y)$\\textit{\\ }for\neach\\textit{\\ }$(x,y)\\in X\\times X.$\n\nAll conditions of Theorem 5 are verified, so that there exists $x^{\\ast }\\in \n$\\ $X$\\ such that $\\ x^{\\ast }\\in \\overline{A}(x^{\\ast })$ and $A(x^{\\ast\n})\\cap P(x^{\\ast },x^{\\ast })=\\emptyset .\\medskip $ Obvioulsy, $x^{\\ast }\n\\textit{\\ }is a solution for GSVQEP.\\medskip\n\\end{proof}\n\n\\section{Concluding remarks}\n\nThis paper developed a framework for discussing the existence of solutions\nfor a generalized strong vector quasi-equilibrium problem. The results have\nbeen obtained under assumptions which are different than the existing ones\nin literature. An approximation technique of proof has been developed.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec-1}\n\n\\subsection{Background}\n\\label{Sec-1.1}\n\nMicroorganisms are ubiquitous in nature and manage critical ecosystem services, ranging from global nutrient cycling to human health. Microbes do not exist in a vacuum in nature, but instead are members of diverse communities known as microbiomes, wherein microbes may interact with other microbes of different species. These microbial interactions may be favorable or antagonistic and are crucial for the successful establishment and maintenance of a microbial community, and frequently result in important pathogenic or beneficial effect to the host or environment \\citep{braga2016microbial}. A thorough understanding of how microbe interact with one another is critical to uncovering the underlying role microorganisms play in the host or environment.\nHowever, it is a highly challenging biological task as it is estimated that only 1\\% of bacteria are cultivatable. The inability to culture the majority of microbial species has motivated the use of culture-independent methods for microbiome studies in different environments \\citep{faust2012microbial, tang2019microbial}. \n\nFortunately, recent innovations in in situ DNA sequencing provide opportunities to infer how microbes interact with one another or their environments. Modern studies on microbial interactions frequently rely on DNA sequencing techniques through the bioinformatic analysis of taxonomically diagnostic genetic markers (e.g., 16S rRNA) sequenced directly from a sample.  The counts of the taxonomically diagnostic genetic markers can be used to represent the abundance of microbial species, e.g., Operational Taxonomic Units (OTUs) or phylotypes (e.g., genera), in a sample. Here, the frequency with which a taxon's marker is observed in a sequence library represents its relative abundance in the community. When such abundance data are available from many communities, interactions among microbiota can be inferred through \\textcolor{blue}{statistical} correlation analysis \\citep{faust2012microbial}. \\textcolor{blue}{For example}, if the relative abundances of two microbial taxa are statistically correlated, then it is inferred that they interact on some level. This approach has been used to document interaction networks in the healthy human microbiome \\citep{faust2012microbial}, as well as free-living microbial communities \\citep{freilich2010large}, and has been useful for generating hypotheses of host-microbiome interaction \\citep{morgan2015associations}.\n\n\\subsection{Methodological Innovations}\n\\label{Sec-1.2}\n\nDespite the great potential of microbiome interaction networks as a tool to advance microbiome research, the power of this approach has been limited by the availability of effective statistical methods and computationally efficient estimation techniques. Microbial abundance data possess a few important features that pose tremendous challenges to standard statistical tools. First, the data are represented as compositional counts of the 16S rRNA sequences because the total count of sequences per sample is predetermined by how deeply the sequencing is conducted, a concept named sequencing depth. The counts only carry information about the relative abundances of the taxa instead of their absolute abundances. Second, the sequencing depth is always finite and often varies considerably across samples in a microbiome dataset. Thus, the observed relative abundance of a taxon in a sample is only an estimator of its true relative abundance with the variance depending on sample-specific sequencing depth, causing the ``heteroscedasticity'' issue \\citep{mcmurdie2014waste}. Third, the data are high-dimensional in nature. It is likely that the number of taxa is far more than the number of samples in any biological experiment.\n\n\nWhen such abundance data are available, one common strategy to resolve the interactions among microbial taxa is to use correlation-type analyses. For example, after the sample correlations are calculated between \\textcolor{blue}{the relative abundances of} each pair of microbial taxa, a threshold is then applied such that an interaction is deemed present if the sample correlation exceeds the threshold. More \\textcolor{blue}{recently developed methods} have started to \\textcolor{blue}{account for} the compositional feature and aim to construct sparse networks for the absolute abundances instead of relative abundances, \\textcolor{blue}{including} SparCC \\citep{friedman2012inferring}, CCLasso \\citep{fang2015cclasso}, and REBACCA \\citep{ban2015investigating}, \\textcolor{blue}{among others.}\n\n\\textcolor{blue}{The above-mentioned network methods based on marginal correlations} could lead to spurious correlations that are caused by confounding factors such as other \\textcolor{blue}{species} in the same community. \\textcolor{blue}{To eliminate the detection of spurious correlations}, interactions among taxa can be modeled through their conditional dependencies given the other taxa. The Gaussian graphical model \\textcolor{blue}{is a classical approach to modeling} the conditional dependency, \\textcolor{blue}{in which the conditional dependency is determined by the nonzero entries of the inverse covariance matrix of a multivariate normal distribution used to model the data.} Graphical lasso \\citep{yuan2007model, banerjee2008model, friedman2008sparse} and neighborhood selection \\citep{meinshausen2006high} are two commonly used methods to estimate sparse inverse covariance matrix under the \\textcolor{blue}{high-dimensional} Gaussian graphical model. However, \\textcolor{blue}{when applied to microbiome abundance data, its multivariate normality assumption was violated by either the count or the compositional features of the data.}\n\nSeveral methods have been proposed to infer microbial conditional dependence networks based on Gaussian graphical models, such as SPIEC-EASI \\citep{kurtz2015sparse}, gCoda \\citep{fang2017gcoda}, CD-trace \\citep{yuan2019compositional}, and SPRING \\citep{yoon2019microbial}. In order to transform the discrete counts to continuous variables and to remove the compositionality constraint, all these methods take the centered log-ratio transformation \\citep{aitchison1986statistical} on the observed counts as their first step. However, the centered log-ratio transformation suffers from an undefined inverse covariance matrix of the transformed data. To partially address this issue, these methods impose a sparsity assumption on the inverse covariance matrix and adding an $L_1$-norm penalty to their objective functions. Additionally, these methods ignore the heteroscedasticity issue as they simply treat the observed relative abundances as the truth. Ignoring the heteroscedasticity issue could impact downstream analysis including constructing microbial interaction networks \\citep{mcmurdie2014waste}.\n\nIn this article, we provide a new statistical tool to help unleash the full potential of microbiome interaction networks as a research tool in the microbiome field. We adopt the logistic normal multinomial distribution to model the compositional count data \\citep{aitchison1986statistical, billheimer2001statistical, xia2013logistic}. Compared to previous methods, this model accounts for the heteroscedasticity issue as the sequencing depth is treated as the number of trials in the multinomial distribution. Additionally, the additive log-ratio transformation applied to the multinomial probabilities results in a well-defined inverse covariance matrix in contrast to the centered log-ratio transformation. Based on this model, we develop an efficient algorithm that iterates between Newton-Raphson and graphical lasso for estimating a sparse inverse covariance matrix. We call this new approach ``compositional graphical lasso''. We establish the theoretical convergence of the algorithm and illustrate the advantage of compositional graphical lasso in comparison to current methods under a variety of simulation scenarios. We further apply the developed method to the data from the Zebrafish Parasite Infection study \\citep{gaulke2019longitudinal} (see Section \\ref{Sec-1.3}) to investigate how microbial interactions associate with parasite infection.\n\n\n\\subsection{Zebrafish Parasite Infection Study}\n\\label{Sec-1.3}\n\nHelminth parasites represent a significant threat to the health of human and animal populations, and there is a growing need for tools to treat, diagnose, and prevent these infections. A growing body of evidence points to the gut microbiome as an agent that interacts with parasites to influence their success in the gut. To clarify how the gut microbiome varies in accordance with parasitic infection dynamics, the Zebrafish Parasite Infection Study was a recent effort \\citep{gaulke2019longitudinal} conducted at Oregon State University, which assessed the association of an intestinal helminth of zebrafish, \\textit{Pseudocapillaria tomentosa}, and the gut microbiome of 210 4-month-old 5D line zebrafish. Among these fish, 105 were exposed to \\textit{P.\\ tomentosa} and the remaining 105 were unexposed controls. At each of the seven time points after exposure, a randomly selected group of 30 fish (15 exposed and 15 unexposed) were euthanized and fecal samples were collected. The parasite burden and tissue damage in \\textit{P.\\ tomentosa}-infected fish were also monitored over 12 weeks of infection.\n\nPrevious analyses \\citep{gaulke2019longitudinal} of the Zebrafish Parasite Infection Study data have revealed that parasite exposure, burden, and intestinal lesions were correlated with gut microbial diversity. They also identified individual taxa whose abundance associated with parasite burden, suggesting that gut microbiota may influence \\textit{P.\\ tomentosa} success. Numerous associations between taxon abundance, burden, and gut pathologic changes were also observed, indicating that the magnitude of microbiome disruption during infection varies with infection severity. However, it remains unclear how parasite success may disrupt or be modulated by the microbial interactions in the gut. Understanding how the microbial interactions associate with parasitic infection can help resolve potential drug or diagnostic test for parasitic infection.\n\n\\subsection{Benchmarking and Novel Biological Discoveries}\n\\label{Sec-1.4}\n\nTo evaluate the performance of our method and to benchmark it against previously available tools, we take advantage of a unique data resource provided by the \\textit{Tara} Oceans Project on ocean planktons. This data set is particularly suitable for method comparison because it enjoys an experimentally validated sub-network of plankton interactome that has served as a gold standard for method benchmarking. Compared to other methods, compositional graphical lasso performs better in reconstructing the microbial interactions that are validated by the literature. In addition, it performs the best in picking up the keystone taxa, which are those with an excessive number of interactions with other taxa in the literature, such as \\textit{Amoebophyra} \\citep{chambouvet2008control}, \\textit{Blastodinium} \\citep{skovgaard2012parasitic}, \\textit{Phaeocystis} \\citep{verity2007current}, and \\textit{Syndinium} \\citep{skovgaard2005phylogenetic}. All these genera are well-described keystone organisms in marine ecosystems. Finally, compositional graphical lasso affords an opportunity to resolve novel modulators of community composition. For example, as one of a few described genera within the syndinean dinoflagellates---an enigmatic lineage with abundant diversity in marine environmental clone libraries, \\textit{Euduboscquella} \\citep{bachvaroff2012molecular} ranked high in the degree distribution uniquely by compositional graphical lasso.\n\nTo investigate the role of the gut microbiome plays in the parasite infections, we apply compositional graphical lasso to the Zebrafish Parasite Infection Study data. {\\color{blue}Interestingly, compositional graphical lasso identifies changes in interaction degree between infected and uninfected individuals for three taxa, \\textit{Photobacterium}, \\textit{Gemmobacter}, and \\textit{Paucibacter}, which are inversely predicted by other methods. Further investigation of these method-specific taxa interaction changes reveals their biological plausibility, and provides insight into their relevance in the context of parasite-linked changes in the zebrafish gut microbiome. In particular, based on our observations, we speculate on the potential pathobiotic roles of \\textit{Photobacterium} and \\textit{Gemmobacter} in the zebrafish gut, and the potential probiotic role of \\textit{Paucibacter}. Future studies should seek to experimentally validate the ecological roles of \\textit{Photobacterium}, \\textit{Gemmobacter}, and \\textit{Paucibacter} in the zebrafish gut, including their impacts on the rest of the microbial community and their roles in infection induced tissue damage.}\n\n\n\n\n\\section{Compositional Graphical Lasso}\n\\label{Sec-2}\n\n\\subsection{Logistic Normal Multinomial Model}\n\\label{Sec-2.1}\n\nConsider a microbiome abundance dataset with $n$ independent samples, each of which is composed of the observed counts of $K + 1$ taxa, denoted by $\\mathbf{x}_i = (x_{i,1}, \\ldots, x_{i,K+1})'$ for the $i$-th sample, $i = 1,\\ldots,n$. Due to the compositional property of the data, the total count of all taxa for each sample $i$ is a fixed number, denoted by $M_i$. Naturally, a multinomial distribution is imposed on the observed counts:\n\\begin{equation} \n\\mathbf{x}_i | \\mathbf{p}_i \\sim \\text{Multinomial}(M_i; p_{i,1}, \\ldots, p_{i,K +1}), \\label{multinomial}\n\\end{equation}\nwhere $\\mathbf{p}_i = (p_{i,1}, \\ldots, p_{i,K+1})'$ are the multinomial probabilities for all taxa and $\\sum_{k=1}^{K+1} p_{i,k} = 1$.\n\nTo apply the additive log-ratio transformation \\citep{aitchison1986statistical} on the multinomial probabilities, we choose one taxon, without loss of generality the $(K+1)$-th taxon, as a reference to which all the other tax are compared. The transformed multinomial probabilities are given by\n\\begin{equation}\nz_{i,k} = \\log (\\frac{p_{i, k}}{p_{i, K + 1}}),\\ i = 1,\\ldots,n, \\ k = 1,\\ldots,K. \\label{log.ratio.transformation}\n\\end{equation}\nLet $\\mathbf{z}_i = (z_{i,1}, \\ldots, z_{i,K})'$ for $i = 1,\\ldots,n$, and further assume that they follow an i.i.d. multivariate normal distribution\n\\begin{equation}\n\\mathbf{z}_1, \\ldots, \\mathbf{z}_n \\stackrel{i.i.d.}{\\sim} N (\\boldsymbol\\mu, \\boldsymbol\\Sigma), \\label{logistic.normal}\n\\end{equation}\nwhere $\\boldsymbol\\mu$ is the mean and $\\boldsymbol\\Sigma$ is the covariance matrix. Let $\\boldsymbol\\Omega = \\boldsymbol\\Sigma^{-1}$ be the inverse covariance matrix or the precision matrix.\n\nThe above model given in (\\ref{multinomial})--(\\ref{logistic.normal}) is often referred to as the logistic normal multinomial model. In this model, a multinomial distribution is imposed on the compositional counts, which is the distribution of the observed data given the multinomial probabilities. In addition, to capture the variation of the multinomial probabilities across samples, we impose a logistic normal distribution on the multinomial probabilities as a prior distribution. We thereby obtain as our final model the logistic normal multinomial model, which is a hierarchical model with two levels.\n\nThe logistic normal multinomial model has a long history in modeling compositional count data and it has also been applied to analyze microbiome abundance data. For example, \\citet{xia2013logistic} proposed a penalized regression under this model to identify a subset of covariates that are associated with the taxon composition. Our objective is different from \\citet{xia2013logistic} as we aim to reveal the microbial interaction network by finding a sparse estimator of the inverse covariance matrix $\\boldsymbol{\\Omega}$ in (\\ref{logistic.normal}). It is also noteworthy that \\cite{jiang2020microbial} has the same objective as ours. However, \\cite{jiang2020microbial} did not make full use of the logistic normal multinomial model as it focused on correcting the bias of a naive estimator of the $\\boldsymbol{\\Sigma}$ that does not require the logistic normal part of the model. By contrast, we aim to find an estimator of $\\boldsymbol{\\Omega}$ directly based on the logistic normal multinomial model.\n\n\n\\subsection{Objective Function}\n\\label{Sec-2.2}\n\n{\\color{blue}From the logistic normal multinomial model in (\\ref{multinomial})--(\\ref{logistic.normal}), we aim to derive an objective function of $\\boldsymbol{\\Omega}$ to estimate the microbial interaction network. To this end, we take a two-step procedure similar to SPIEC-EASI \\citep{kurtz2015sparse}. In the first step, we find estimated values of $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$ based on the logistic normal multinomial model given $\\mu$ and $\\boldsymbol{\\Sigma}$; in the second step, we find an estimate of $\\boldsymbol{\\Omega}$ based on the estimated values of $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$.\n\nIn the first step, we consider the posterior distribution of $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$ given the data $\\mathbf{x}_1,\\ldots,\\mathbf{x}_n$ and find the maximum a posteriori (MAP) estimates of $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$. For $i=1,\\ldots,n$, the logarithm of the posterior density function of $\\mathbf{z}_i$ given $\\mathbf{x}_i$ is\n\\begin{align*} \n& \\log[f_{\\boldsymbol{\\mu},\\boldsymbol{\\Omega}}(\\mathbf{z}_i | \\mathbf{x}_i)] \\propto \\log[f_{\\boldsymbol{\\mu},\\boldsymbol{\\Omega}}(\\mathbf{x}_i, \\mathbf{z}_i)] \\\\\n\\propto{}& \\sum_{k=1}^{K+1} x_{i,k} \\log p_{i,k} + \\frac12 \\log[\\det(\\boldsymbol{\\Omega})] - \\frac12 (\\mathbf{z}_i - \\boldsymbol{\\mu})' \\boldsymbol{\\Omega} (\\mathbf{z}_i - \\boldsymbol{\\mu}) \\\\\n={} &\\sum_{k=1}^{K} x_{i,k} z_{i,k} - M_i \\log(\\sum_{k=1}^K e^{z_{i,k}} + 1) + \\frac12 \\log[\\det(\\boldsymbol{\\Omega})] - \\frac12 (\\mathbf{z}_i - \\boldsymbol{\\mu})' \\boldsymbol{\\Omega} (\\mathbf{z}_i - \\boldsymbol{\\mu}),\n\\end{align*}\nwhere $\\propto$ denotes that two quantities are equal up to a term not depending on $\\mathbf{z}_i$. In the above derivation, we ignored the marginal density function of $\\mathbf{x}_i$ that does not depend on $\\mathbf{z}_i$, which does not affect the estimation of $\\mathbf{z}_i$.}\n\nBy independence between all the samples, the logarithm of the posterior density function of $(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n)$ given the data $(\\mathbf{x}_1,\\ldots,\\mathbf{x}_n)$ can be written as (again, ignoring a term independent of $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$):\n\\begin{equation} \n\\sum_{i=1}^n \\left[\\sum_{k=1}^{K} x_{i,k} z_{i,k} - M_i \\log(\\sum_{k=1}^K e^{z_{i,k}} + 1)\\right] + \\frac{n}2 \\log[\\det(\\boldsymbol{\\Omega})] - \\frac12 \\sum_{i=1}^n (\\mathbf{z}_i - \\boldsymbol{\\mu})' \\boldsymbol{\\Omega} (\\mathbf{z}_i - \\boldsymbol{\\mu}). \\label{objective.function.1}\n\\end{equation}\nGiven the values of the multivariate normal parameters $\\boldsymbol{\\mu}$ and $\\boldsymbol{\\Omega}$, one can maximize (\\ref{objective.function.1}) with respect to $(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n)$. This leads to the MAP estimator $(\\hat\\mathbf{z}_1,\\ldots,\\hat\\mathbf{z}_n)$.\n\n{\\color{blue}In the second step, we find a sparse inverse covariance estimator of $\\boldsymbol{\\Omega}$ based on the estimate values $(\\hat\\mathbf{z}_1,\\ldots,\\hat\\mathbf{z}_n)$. Hereby, we use the graphical lasso estimator, which minimizes the $L_1$ penalized negative log-likelihood function as follows:\n\\begin{equation}\n-\\frac{1}2 \\log[\\det(\\boldsymbol{\\Omega})] + \\frac1{2n} \\sum_{i=1}^n (\\hat\\mathbf{z}_i - \\boldsymbol{\\mu})' \\boldsymbol{\\Omega} (\\hat\\mathbf{z}_i - \\boldsymbol{\\mu}) + \\lambda \\|\\boldsymbol{\\Omega}\\|_1. \\label{objective.function.2}\n\\end{equation}\n\nIt turns out that the above two-step procedure is equivalent to minimizing an overall objective function with respect to both $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$ and $(\\boldsymbol{\\mu},\\boldsymbol{\\Omega})$:\n\\begin{align}\n\\ell(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n,\\boldsymbol{\\mu},\\boldsymbol{\\Omega}) ={}& -\\frac1n\\sum_{i=1}^n \\left[\\sum_{k=1}^{K} x_{i,k} z_{i,k} - M_i \\log(\\sum_{k=1}^K e^{z_{i,k}} + 1)\\right] \\notag\\\\\n& -\\frac12 \\log[\\det(\\boldsymbol{\\Omega})] + \\frac{1}{2n} \\sum_{i=1}^n (\\mathbf{z}_i - \\boldsymbol{\\mu})' \\boldsymbol{\\Omega} (\\mathbf{z}_i - \\boldsymbol{\\mu}) + \\lambda \\|\\boldsymbol{\\Omega}\\|_1. \\label{objective.function}\n\\end{align}\nIn other words, $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$, $\\boldsymbol{\\mu}$, and $\\boldsymbol{\\Omega}$ are all treated as unknown parameters in the minimization of the objective function (\\ref{objective.function}).\n\nIt is noteworthy that similar ideas to the above two-step procedure have been used in existing microbial interaction network estimation methods. For example, SPIEC-EASI is also a two-step procedure. In its first step, the abundance counts are converted into their centered log-ratio transformed data; in its second step, either graphical lasso or neighborhood selection is applied to estimate a sparse inverse covariance matrix based on the centered log-ratio transformed data in the first step.\n\nAlthough both our method and SPIEC-EASI can be regarded as two-step procedures, we underline two important distinctions between them. First, our method is based on the additive log-ratio transformation and SPIEC-EASI uses the centered log-ratio transformation. As mentioned in the Introduction, the centered log-ratio transformation suffers from an undefined inverse covariance matrix of the transformed data while the additive log-ratio transformation does not. Second, in the first step, our method accounts for the sequencing depths [$M_i$'s in (\\ref{multinomial})] and further the uncertainty of the observed relative abundances, and thus addresses the heteroscedasticity issue. However, the heteroscedasticity issue is ignored in SPIEC-EASI.} \n\n\n\\subsection{Computational Algorithm}\n\\label{Sec-2.3}\n\nThe objective function (\\ref{objective.function})  naturally includes three sets of parameters $(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n)$, $\\boldsymbol{\\mu}$, and $\\boldsymbol{\\Omega}$, which motivates us to apply a block coordinate descent algorithm. A block coordinate descent algorithm minimizes the objective function iteratively for each set of parameters given current values of the other sets. Given the initial values $(\\mathbf{z}_1^{(0)},\\ldots,\\mathbf{z}_n^{(0)})$, $\\boldsymbol{\\mu}^{(0)}$, and $\\boldsymbol{\\Omega}^{(0)}$, a block coordinate algorithm repeats the following steps cyclically for iteration $t= 0,1,2,\\ldots$ until the algorithm converges.\n\\begin{enumerate}[nosep]\n\\item Given $\\boldsymbol{\\mu}^{(t)}$ and $\\boldsymbol{\\Omega}^{(t)}$, find $(\\mathbf{z}_1^{(t+1)},\\ldots,\\mathbf{z}_n^{(t+1)})$ that maximizes (\\ref{objective.function}).\n\\item Given $(\\mathbf{z}_1^{(t+1)},\\ldots,\\mathbf{z}_n^{(t+1)})$ and $\\boldsymbol{\\Omega}^{(t)}$, find $\\boldsymbol{\\mu}^{(t+1)}$ that maximizes (\\ref{objective.function}).\n\\item Given $(\\mathbf{z}_1^{(t+1)},\\ldots,\\mathbf{z}_n^{(t+1)})$ and $\\boldsymbol{\\mu}^{(t+1)}$, find $\\boldsymbol{\\Omega}^{(t+1)}$ that maximizes (\\ref{objective.function}).\n\\end{enumerate}\n\nBelow, we will present the details of this algorithm in each iteration. For the initial values $(\\mathbf{z}_1^{(0)},\\ldots,\\mathbf{z}_n^{(0)})$, we use their maximum likelihood estimators from the multinomial distribution, i.e.,\n\\[ {z}_{i,k}^{(0)} = \\log (\\frac{x_{i, k}}{x_{i, K + 1}}),\\ i = 1,\\ldots,n, \\ k = 1,\\ldots,K.\\] \nIf $x_{i, k} = 0$ for some $i$ and $k$, we add a small constant to it to evaluate the log ratio. For the initial value $\\boldsymbol{\\mu}^{(0)}$, we have a closed form minimizer of $\\boldsymbol{\\mu}$ for (\\ref{objective.function}) given the values of $(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n)$, which is $\\boldsymbol{\\mu} = \\bar{\\mathbf{z}} = \\frac1n \\sum_{i=1}^n \\mathbf{z}_i$. Therefore, we set the initial value as $\\boldsymbol{\\mu}^{(0)} = \\frac1n \\sum_{i=1}^n \\mathbf{z}_i^{(0)}$. Finally, for the initial value $\\boldsymbol{\\Omega}^{(0)}$, we use the estimate of the graphical lasso algorithm taking the sample covariance matrix computed from $\\mathbf{z}_1^{(0)},\\ldots,\\mathbf{z}_n^{(0)}$ as input.\n\nIn step 1, given $\\boldsymbol{\\mu}^{(t)}$ and $\\boldsymbol{\\Omega}^{(t)}$, minimizing the objective function (\\ref{objective.function}) with respect to $(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n)$ is equivalent to minimizing the following objective function with respect to each $\\mathbf{z}_i$ separately, for $i = 1,\\ldots,n$:\n\\begin{equation}\n\\ell_i^{(t)}(\\mathbf{z}_i) = \\frac12 (\\mathbf{z}_i - \\boldsymbol{\\mu}^{(t)})' \\boldsymbol{\\Omega}^{(t)} (\\mathbf{z}_i - \\boldsymbol{\\mu}^{(t)}) - \\left[\\sum_{k=1}^{K} x_{i,k} z_{i,k} - M_i \\log(\\sum_{k=1}^K e^{z_{i,k}} + 1)\\right]. \\label{objective.function.zi}\n\\end{equation}\nThe above objective function is a smooth and convex function in $\\mathbf{z}_i$ with the following positive definite Hessian matrix\n\\begin{equation*}\n\\boldsymbol{\\Omega}^{(t)} + \\frac{M_i}{\\left(\\sum_{k=1}^K e^{z_{i,k}} + 1\\right)^2} \\left\\{\\left(\\sum_{k=1}^K e^{z_{i,k}} + 1\\right) \\text{diag} (e^{\\mathbf{z}_i}) - (e^{\\mathbf{z}_i})(e^{\\mathbf{z}_i})'\\right\\},\n\\end{equation*}\nwhere $e^{\\mathbf{z}_i} = (e^{z_{i,1}}, \\ldots, e^{z_{i,K}})'$ and $\\text{diag}(e^{\\mathbf{z}_i})$ is the diagonal matrix with the diagonal elements $e^{z_{i,1}}, \\ldots, e^{z_{i,K}}$. Therefore, we apply the Newton-Raphson algorithm to find the minimizer numerically. In addition, we implement a line search procedure in each Newton-Raphson iteration following the Armijo rule \\citep{armijo1966minimization}. This procedure ensures sufficient decrease in the objective function at each iteration to prevent possible divergence of the algorithm.\n\nStep 2 is similar to the initialization step, in which $\\boldsymbol{\\mu}$ has a closed-form solution and is updated as $\\bar{\\mathbf{z}}^{(t+1)} = \\frac1n \\sum_{i=1}^n \\mathbf{z}_i^{(t+1)}$ from the current numerical values of $(\\mathbf{z}_1^{(t+1)},\\ldots,\\mathbf{z}_n^{(t+1)})$ that are computed from the Newton-Raphson algorithm in step 1.\n\nIn step 3, given $(\\mathbf{z}_1^{(t+1)},\\ldots,\\mathbf{z}_n^{(t+1)})$ and $\\boldsymbol{\\mu}^{(t+1)} = \\bar{\\mathbf{z}}^{(t+1)}$, the objective function for $\\boldsymbol{\\Omega}$ can be simplified as\n\\begin{align}\n\\ell^{(t)}(\\boldsymbol{\\Omega}) &= -\\frac12 \\log[\\det(\\boldsymbol{\\Omega})] + \\frac{1}{2n} \\sum_{i=1}^n (\\mathbf{z}_i^{(t+1)} - \\boldsymbol{\\mu}^{(t+1)})' \\boldsymbol{\\Omega} (\\mathbf{z}_i^{(t+1)} - \\boldsymbol{\\mu}^{(t+1)}) + \\lambda \\|\\boldsymbol{\\Omega}\\|_1, \\notag \\\\\n&= -\\frac12 \\log[\\det(\\boldsymbol{\\Omega})] + \\frac{1}{2} \\mathrm{tr} (\\mathbf{S}^{(t+1)} \\boldsymbol{\\Omega}) + \\lambda \\|\\boldsymbol{\\Omega}\\|_1, \\label{objective.function.omega} \n\\end{align}\nwhere $\\mathbf{S}^{(t+1)} = \\frac{1}{n} \\sum_{i=1}^n (\\mathbf{z}_i^{(t+1)} - \\bar{\\mathbf{z}}^{(t+1)}) (\\mathbf{z}_i^{(t+1)} - \\bar{\\mathbf{z}}^{(t+1)})'$. It is obvious that minimizing the objective function (\\ref{objective.function.omega}) becomes a graphical lasso problem \\citep{yuan2007model, banerjee2008model, friedman2008sparse}. It is well known that the graphical lasso objective function is a convex function in $\\boldsymbol{\\Omega}$ \\citep{banerjee2008model} and efficient algorithms have been developed for its optimization \\citep{friedman2008sparse}. In this paper, we implement this step using the graphical lasso algorithm included in the \\texttt{huge} \\citep{zhao2012huge} package in R.\n\nThe above block coordinate descent algorithm iterates between Newton-Raphson and graphical lasso and is designed specifically to optimize the objective function (\\ref{objective.function}) for compositional count data. Therefore, we name this algorithm the compositional graphical lasso algorithm, and the entire approach the compositional graphical lasso method including both the model and the algorithm for the analysis of microbiome abundance data.\n\n\\subsection{Theoretical Convergence}\n\\label{Sec-2.4}\n\nUnfortunately, the objective function (\\ref{objective.function}) is not necessarily a convex function jointly in $(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n)$, $\\boldsymbol{\\mu}$, and $\\boldsymbol{\\Omega}$. However, \nwe have shown that it is convex in each of the three subsets of its parameters. The convergence property of such an optimization problem has been studied in the literature. For example, \\cite{tseng2001convergence} studied the convergence property of a block coordinate descent method applied to minimize a nonconvex function with certain separability and regularity properties. We will establish the convergence property of the compositional graphical lasso algorithm following \\cite{tseng2001convergence}.\n\nRecall that our algorithm treats the three sets of parameters $(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n)$, $\\boldsymbol{\\mu}$, and $\\boldsymbol{\\Omega}$ as three blocks and optimizes for each block iteratively. In addition, as in \\cite{tseng2001convergence}, the objective function (\\ref{objective.function}) can be regarded as the sum of two parts, the first of which is an inseparable but differentiable function given by\n\\begin{equation}\n\\ell_0(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n,\\boldsymbol{\\mu},\\boldsymbol{\\Omega}) = \\frac{1}{2n} \\sum_{i=1}^n (\\mathbf{z}_i - \\boldsymbol{\\mu})' \\boldsymbol{\\Omega} (\\mathbf{z}_i - \\boldsymbol{\\mu}), \\label{objective.function.inseparable}\n\\end{equation}\nand the second of which is a sum of separable and differentiable functions given by $\\ell_1(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n) + \\ell_2(\\boldsymbol{\\Omega})$, where\n\\begin{align}\n\\ell_1(\\mathbf{z}_1,\\ldots,\\mathbf{z}_n) &= -\\frac1n\\sum_{i=1}^n \\left[\\sum_{k=1}^{K} x_{i,k} z_{i,k} - M_i \\log(\\sum_{k=1}^K e^{z_{i,k}} + 1)\\right], \\label{objective.function.separable.1}\\\\\n\\ell_2(\\boldsymbol{\\Omega}) &= -\\frac12 \\log[\\det(\\boldsymbol{\\Omega})] + \\lambda \\|\\boldsymbol{\\Omega}\\|_1. \\label{objective.function.separable.2}\n\\end{align} \n\\cite{tseng2001convergence} established the convergence property of a block coordinate descent algorithm under regularity conditions on $\\ell_0$, $\\ell_1$, and $\\ell_2$.\n\nTo present the main convergence property of the compositional graphical lasso algorithm, let's review the definition of a cluster point in real analysis. A cluster point of a set $\\mathcal{A} \\subset \\mathbb{R}^n$ is a real vector $\\mathbf{a} \\in \\mathbb{R}^n$ such that for every $\\delta > 0$, there exists a point $\\mathbf{x}$ in $\\mathcal{A}\\setminus \\{\\mathbf{a}\\}$ satisfying that $\\|\\mathbf{x} - \\mathbf{a}\\|_2 < \\delta$. Obviously, any limit point of the set $\\mathcal{A}$ is a cluster point. Furthermore, define a cluster point of the compositional graphical lasso algorithm to be a cluster point of the set $\\{(\\mathbf{z}_1^{(t)},\\ldots,\\mathbf{z}_n^{(t)},\\boldsymbol{\\mu}^{(t)},\\boldsymbol{\\Omega}^{(t)}): t = 0,1,2,\\ldots\\}$, which are minimizers found at each iteration $t$. Then, the following theorem presents a theoretical property for every cluster point of our algorithm as follows.\n\n\\begin{thm} \\label{Thm-1}\nAny cluster point of the compositional graphical lasso algorithm is a stationary point of the objective function (\\ref{objective.function}).\n\\end{thm}\n\nThe proof of Theorem \\ref{Thm-1} can be found in the supplementary materials. This theorem guarantees that a cluster point, usually a limit point, of the compositional graphical lasso algorithm, is at least a stationary point. It is noteworthy that there exists a global minimizer for the objective function in (\\ref{objective.function}) because we have proved the coerciveness of the objective function in the proof \\citep[Lemma 8.4]{calafiore2014optimization}. Therefore, to achieve global optimization in practice, one can run the algorithm multiple times starting with different initial values and choose the one solution that yields the smallest objective function among the multiple ones.\n\nIn addition, the values of the objective function at each iteration, i.e., \\\\\n$\\{\\ell(\\mathbf{z}_1^{(t)},\\ldots,\\mathbf{z}_n^{(t)},\\boldsymbol{\\mu}^{(t)},\\boldsymbol{\\Omega}^{(t)}): t = 0,1,2,\\ldots\\}$, will always converge. This is because that the objective function is bounded below (as $\\ell_0 + \\ell_2$ is bounded below as shown in the proof and $\\ell_1 > 0$ by definition)\nand our algorithm results in non-increasing objective function values between two iterations. Therefore, the values of objective function will always converge to a limit point. In practice, we have always observed the numerical convergence of both the minimizers and the values of the objective function after a certain number of iterations.\n\n\n\n\n\n\n\\subsection{Tuning Parameter Selection}\n\\label{Sec-2.5}\n\nThere is a large body of literature on the selection of a tuning parameter in the variable selection framework. Common approaches can be broadly categorized into three types: criteirion-based methods, prediction-based methods, and stability-based methods. Criterion-based methods such as Akaike information criterion (AIC) \\citep{akaike1974new} and Bayesian information criteria (BIC) \\citep{schwarz1978estimating} balance the model complexity and the goodness of fit, prediction-based methods such as cross validation \\citep{stone1974cross, geisser1975predictive} and generalized cross validation \\citep{golub1979generalized} aim to minimize the expected prediction error of the selected model on independent datasets. Stability-based methods such as stability selection \\citep{meinshausen2010stability} and the Stability Approach to Regularization Selection (StARS) \\citep{liu2010stability} select a model with high stability under subsampling or bootstrapping the original data.\n\nIn this work, we apply StARS to select the tuning parameter $\\lambda$ in our objective function (\\ref{objective.function}). In StARS, we draw $N$ subsamples without replacement from the original dataset with $n$ observations, each subsample of size $b$. For each value of the tuning parameter $\\lambda$, we obtain an estimate of $\\boldsymbol{\\Omega}$, i.e., a network for each subsample. Then, we measure the total instability of these resultant networks across the $N$ subsamples. The total instability of these networks is defined by averaging the instabilities of each edge across the $N$ subsamples over all possible edges, where the instability of each edge is estimated as the twice the sample variance of the Bernoulli indicator of whether this edge is selected or not in each of the $N$ subsamples.\n\nStarting from a large penalty which corresponds to the empty network, the instability of networks increases as $\\lambda$ decreases. StARS stops and selects the tuning parameter to be the minimum value of $\\lambda$'s with which the instability of the resultant networks is less than a threshold $\\beta > 0$. In principle, StARS selects a tuning parameter so that the resultant network is the densest among networks with a total instability less than a threshold $\\beta$ without violating some sparsity assumption. The selected network is the ``densest on the sparse side,\" as it starts with the empty network and stops when the instability first across the threshold.\n\n\\section{Simulation Studies}\n\\label{Sec-3}\n\n\\subsection{Simulation Settings}\n\\label{Sec-3.1}\n\nTo evaluate the performance of compositional graphical lasso, we conduct \\textcolor{blue}{simulation studies} and compare it with other network estimation methods.\n\nGiven that our goal is to estimate the true network, i.e., $\\boldsymbol{\\Omega}$ in (\\ref{logistic.normal}), we consider the following three types of true precision matrices $\\boldsymbol{\\Omega} = (\\omega_{kl})_{1\\le k,l \\le K}$, which are different in the pattern of edge distributions as well as the degree of connectedness.\n\n\\begin{enumerate}[nosep]\n\\item Chain network: $\\omega_{kk} = 1.5$, $\\omega_{kl} = 0.5$ if $|k - l| = 1$, and $\\omega_{kl} = 0$ if $|k - l| > 1$. A node is designed to be connected to its adjacent nodes, and the connectedness of nodes is balanced.\n\\item Random network: $\\omega_{kl} = 1$ with probability $3\/K$ for $k \\ne l$. A node is connected to all other nodes randomly with a fixed probability, set to be $3\/K$. Similar to the chain structure, the connectedness of nodes is balanced.\n\\item Hub network: All nodes are randomly split into $\\lceil K \/ 20 \\rceil$ disjoint groups, and a hub node $k$ is selected from each group. For any other node $l$ in the same group, $\\omega_{kl} = 1$. All the remaining entries of $\\boldsymbol{\\Omega}$ are zero. Here, nodes are partitioned into the same group at random, but is then designated to be connected to the hub node at certain. The degree of connectedness among nodes is extremely unbalanced in this case: the hub nodes are connected to all the other nodes in its group (around 20 nodes) and all the other nodes are only connected to the hub node in its group, i.e., just one node.\n\\end{enumerate}\n\nIn addition to the true network, we also consider two other factors that are expected to influence the result. The first factor is the sequencing depth, $M_i$, in the multinomial distribution (\\ref{multinomial}). \\textcolor{blue}{We simulate $M_i$ from uniform distributions, the detail of which will be discussed in the following subsections.} The second factor is the degree of variation in the logistic normal distribution (\\ref{logistic.normal}). For each of the three types of precision matrices, we consider an additional factor by multiplying a positive constant $c$ to $\\boldsymbol{\\Omega}$ so that the true precision matrix is $c\\boldsymbol{\\Omega}$. We choose $c = 1$ and $c = 1\/5$ separately and call the two settings low and high compositional variation, respectively.\n\nThe data are simulated following the logistic normal multinomial model in (\\ref{multinomial})--(\\ref{logistic.normal}). We first simulate $\\mathbf{z}_i \\sim N (\\boldsymbol{\\mu}, \\boldsymbol{\\Sigma})$ independently for $i = 1,\\ldots,n$; then, we perform the inverse log-ratio transformation (also know as the softmax transformation, the inverse transformation of (\\ref{log.ratio.transformation})) to obtain the multinomial probabilities $\\mathbf{p}_i$ for $i = 1,\\ldots,n$; last, we simulate multinomial counts $\\mathbf{x}_i$ from a multinomial distribution with sequencing depth $M_i$ and probabilities $\\mathbf{p}_i$. Throughout this simulation study, we fix $n = 100$ and $K = 200$.\n\nThe simulation results are based on 100 replicates. For each replicate, we apply compositional graphical lasso, neighborhood selection, and graphical lasso separately to obtain a sparse estimator of $\\boldsymbol{\\Omega}$. For neighborhood selection and graphical lasso, we first obtain an estimate of $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$ from the multinomial distribution via the additive log-ratio transformation\n\\begin{equation}\n\\tilde{z}_{i,k}= \\log (\\frac{x_{i, k}}{x_{i, K + 1}}),\\ i = 1,\\ldots,n, \\ k = 1,\\ldots,K, \\label{z.surrogates}\n\\end{equation}\nand then apply neighborhood selection and graphical lasso directly on the estimates $\\tilde\\mathbf{z}_1,\\ldots,\\tilde\\mathbf{z}_n$ by treating them as surrogates for their true counterparts, i.e., $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$. These methods are almost identical to SPIEC-EASI although we replaced the centered log-ratio transformation in SPIEC-EASI with additive log-ratio transformation for a fair comparison.\n\nTo compare the performance of the three methods in terms of network recovery, all three methods are applied with a sequence of tuning parameter values, and their true positive rates (TPR) and false positive rates (FPR) in terms of edge selection are recorded for each value of $\\lambda$. An ROC curve is plotted from the average TPR and the average FPR over the 100 replicates for each of the tuning parameters.\n\nIn addition, we apply StARS to select an optimal tuning parameter $\\lambda$. Following the recommendation in \\citet{liu2010stability}, we set the threshold for the total instability to be $\\beta = 0.05$, the size of each subsample $b = 7 \\sqrt{n}$, and the number of subsamples $N = 50$. Once the optimal tuning parameter is determined by StARS, we fit the whole dataset with the selected tuning parameter and evaluate the resultant network using three criteria: precision, recall, and F1 score, which are defined as\n\\begin{equation*}\n\\text{Precision} = \\frac{\\text{TP}}{\\text{TP} + \\text{FP}},\\quad \\text{Recall} = \\frac{\\text{TP}}{\\text{TP} + \\text{FN}}, \\quad \\text{F1} = \\frac{2 \\times \\text{Precision} \\times \\text{Recall}}{\\text{Precision} + \\text{Recall}},\n\\end{equation*}\nwhere TP, FP, and FN are numbers of true positives, false positives, and false negatives, respectively. \n\n\\subsection{\\textcolor{blue}{Simulation Results for Dense Data}}\n\\label{Sec-3.2}\n\n\\textcolor{blue}{In this subsection, we evaluate the performance of compositional graphical lasso on dense data, in which most of the simulated counts are nonzero. This corresponds to a simulation setting where the sequencing depths $M_i$'s are relatively larger than the number of taxa $K + 1$. Still, to evaluate the effect of the sequencing depth on the performance of network estimation methods, we simulate $M_i$ from two uniform distributions, Uniform$(20K, 40K)$ and Uniform$(100K, 200K)$, and call the two settings low and high sequencing depth in this subsection, respectively.}\n\nFigure \\ref{ROC} presents the ROC curves for compositional graphical lasso (Comp-gLASSO), neighborhood selection (MB), and graphical lasso (gLASSO), from which we can see that compositional graphical lasso dominates its competitors in terms of edge selection in all settings. In particular, the advantage of the compositional graphical lasso over neighborhood selection and graphical lasso is the most obvious when the compositional variation is high and the sequencing depth is low, no matter which type of network structure is considered. On the contrary, the three methods perform very similarly for all types of network structures when the compositional variation is low and the sequencing depth is high. The difference between compositional graphical lasso and the rest is intermediate for the other two settings when both compositional variation and sequencing depth are high or when both are low. Comparing graphical lasso and neighborhood selection, they tend to perform more similarly although graphical lasso seems to outperform neighborhood selection in some settings with a small margin.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{ROC_MB.pdf}\n\\caption{ROC curves for compositional graphical lasso (Comp-gLASSO), graphical lasso (gLASSO) and neighborhood selection (MB). Solid blue: Comp-gLASSO; dashed red: gLASSO; dotted black: MB. $\\mathbf{h\/l, h\/l}$: high\/low sequencing depth, high\/low compositional variation.}\n\\label{ROC}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{bar_graph_MB.pdf}\n\\caption{Recall, precision and F1 score for the network selected by StARS for compositional graphical lasso (Comp-gLASSO), graphical lasso (gLASSO) and neighborhood selection (MB). Red (left): Comp-gLASSO; green (middle): gLASSO; blue (right): MB. $\\mathbf{h\/l, h\/l}$: high\/low sequencing depth, high\/low compositional variation.}\n\\label{Recall.Precision.F1}\n\\end{figure}\n\nThe above observations agree with our expectation about how the two factors, compositional variation and sequencing depth, affect the comparison between the methods. Recall that neighborhood selection and graphical lasso replace the true values of $\\mathbf{z}_1,\\ldots,\\mathbf{z}_n$ by their estimates\/surrogates $\\tilde\\mathbf{z}_1,\\ldots,\\tilde\\mathbf{z}_n$ as in (\\ref{z.surrogates}) without taking into account the estimation accuracy or uncertainty of these surrogates. First, a higher sequencing depth leads to more accurate surrogates $\\tilde\\mathbf{z}_1,\\ldots,\\tilde\\mathbf{z}_n$; therefore, it is not surprising to see that the three methods perform more similarly when the sequencing depth is high. Second, a higher compositional variation results in a higher variation in $\\mathbf{z}_i$'s and further in $\\mathbf{p}_i$'s that are multinomial probabilities. Since neighborhood selection and graphical lasso ignore the multinomial component in the model, it is also not surprising to see that their performance is deteriorated by a high compositional variation.\n\nFigure \\ref{Recall.Precision.F1} presents the recall, precision, and F1 score from 50 replicates of the estimated network resulted from the tuning parameter selected by StARS. The first observation would be that the precisions of both compositional graphical lasso and graphical lasso are much worse than their recalls, whereas the precisions and recalls are more comparable for neighborhood selection. Interestingly, StARS results in a much more sparse network for neighborhood selection than the other methods under the same stability threshold, suggesting that fewer edges selected by neighborhood selection are stable enough (within the instability threshold). When it comes to method comparison, compositional graphical lasso has much higher recall than neighborhood selection in most settings, but have comparable or lower precision in most of the settings with high sequencing depth. The network from compositional graphical lasso has higher F1 score than the ones from neighborhood selection in most settings, except when sequencing depth is high and compositional variation is low for chain and hub networks. In addition, the network from compositional graphical lasso has higher or comparable precision, recall, and F1 score than the ones from graphical lasso in all settings. Similar to the observations from the ROC curves, the advantage of compositional graphical lasso is more obvious with a low sequencing depth or a high compositional variation.\n\n\\subsection{\\textcolor{blue}{Simulation Results for Sparse Data}}\n\\label{Sec-3.3}\n\n\\textcolor{blue}{In this subsection, we evaluate the performance of compositional graphical lasso on sparse data, in which a range of proportions of the compositional counts are simulated as zero. We only present the simulation results for the chain network, given that the simulation results for the other network types are similar. To examine how our method performs for different sparsity levels, we simulate $M_i$ from four uniform distributions, Uniform$(8K, 16K)$, Uniform$(4K, 8K)$, Uniform$(2K, 4K)$, and Uniform$(K, 2K)$, respectively. Note that the sparsity level of the simulated data also depends on the compositional variation (see Section \\ref{Sec-3.1}). In a typically simulated dataset with high compositional variation, the sparsity level is around 40\\%, 50\\%, 60\\%, and 70\\% with the four uniform distributions for $M_i$; in a typically simulated dataset with low compositional variation, the sparsity level is around 5\\%, 10\\%, 25\\%, and 40\\% with the four uniform distributions for $M_i$. In both cases, we refer to these four sparsity levels as 1, 2, 3, and 4, with 1 the least sparse and 4 the most sparse. Due to the limited space, we place the figures resulted from this simulation in the supplementary materials.}\n\n\\textcolor{blue}{Figure \\ref{ROC_MB_Sparse} presents the ROC curves for compositional graphical lasso (Comp-gLASSO), neighborhood selection (MB), and graphical lasso (gLASSO). Similar to Figure \\ref{ROC}, we can see that compositional graphical lasso dominates its competitors in terms of edge selection across all sparsity levels. As the sequencing depths for sparse data are relatively small compared to the ones used for dense data, the advantage of the compositional graphical lasso over neighborhood selection and graphical lasso is obviously and consistently observed in all simulation settings. In addition, graphical lasso and neighborhood selection tend to perform more similarly although graphical lasso seems to outperform neighborhood selection with a small margin. This result is consistent with that observed for dense data in Section \\ref{Sec-3.2}.}\n\n\\textcolor{blue}{Figure \\ref{Recall.Precision.F1.Sparse} presents the recall, precision, and F1 score from 50 replicates of the estimated network resulted from the tuning parameter selected by StARS. The first observation is that all methods perform worse for sparse data than for dense data from the comparison between Figure \\ref{Recall.Precision.F1.Sparse} and Figure \\ref{Recall.Precision.F1}; all methods perform worse as the sparsity level increases. Similar to Figure \\ref{Recall.Precision.F1}, Figure \\ref{Recall.Precision.F1.Sparse} implies that compositional graphical lasso outperforms graphical lasso and neighborhood selection, with a higher recall, precision, and F1 score consistently observed in most settings. The only exception is that, when data are extremely sparse with a low compositional variation, graphical lasso seems to outperform compositional graphical lasso by a very small margin as both methods are not working well. For sparse data, StARS results in an almost empty network for neighborhood selection, resulting in zero recall, precision, and F1 score for most settings. This agrees with our observation from the simulation results for dense data that neighborhood selection tends to produce a much sparser network than compositional graphical lasso and graphical lasso when its tuning parameter is selected by StARS.}\n\n\\section{Real Data}\n\\label{Sec-4}\n\n\\subsection{Benchmark Study: \\textit{Tara} Oceans Project}\n\\label{Sec-4.1}\nTo better understand the ocean, the largest ecosystem on the earth, the \\textit{Tara} Oceans Project aims to build the global ocean interactome that can be used to predict the dynamics and structure of ocean ecosystems. To achieve this, the \\textit{Tara} Oceans Consortium sampled both plankton and environmental data at 210 sites from the world's oceans using the 110-foot research schooner \\textit{Tara} during the \\textit{Tara} Oceans Expedition (2009-2013). The data collected was later processed using sequencing and imaging techniques. One unique advantage of the \\textit{Tara} Oceans Project is that it has generated a list of 91 genus-level marine planktonic interactions that have been validated in the literature \\citep{lima2015determinants}. Though this list only comprises interactions between microbes that represent a small fraction of the total marine eukaryotic diversity and is therefore far from complete, it could serve as partial ground truth for us to evaluate the interactions identified by different methods. Thus, our major goal is to use the \\textit{Tara} Oceans Project as a benchmark study in order to compare the performance of different methods in constructing the planktonic interactions.\n\nAs the partial ground truth is a list of genus-level interactions, we choose to analyze the genus-level abundance data, which are aggregated from the original OTU abundance data downloadable from the \\textit{Tara} Oceans Project data repository (\\url{http:\/\/taraoceans.sb-roscoff.fr\/EukDiv\/}). As a benchmark study, we only include the 81 genera that are involved in the list of gold-standard interactions in our analysis. In addition, we discard the samples with too few sequence reads (less than 100), resulting in 324 samples left in our analysis.\n\nSimilar to the simulation study, we apply compositional graphical lasso, graphical lasso, and neighborhood selection to estimate the interaction network among the 81 genera. We first pick the genus \\textit{Acrosphaera}, which has the largest average relative abundance among those genera not involved in the gold-standard list, and use this genus as the reference taxon for all three methods. Then, we apply each method with a sequence of 70 decreasing tuning parameter values, resulting in a sequence of interaction networks starting from an empty network. Finally, we apply StARS to find the optimal tuning parameter, in which the parameters $\\beta$, $b$, and $N$ are set the same as in the simulation study.\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.4\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{TARA_path_half.pdf}\n  \\caption{}\n  \\label{tara_path}\n\\end{subfigure}%\n\\begin{subfigure}{.4\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{TARA_degree_half.pdf}\n  \\caption{}\n  \\label{tara_degree}\n\\end{subfigure}\n\\caption{(a): Number of identified literature interactions versus number of edges of the estimated network from the \\textit{Tara} dataset. (b): The degree distribution of vertices from the networks selected by StARS. Solid red: compositional graphical lasso; dashed green: graphical lasso; dashed dotted blue: neighborhood selection.}\n\\label{network.P}\n\\end{figure}\n\nFirst, we compare the three methods in terms of how highly they rank the literature validated interactions amongst their top reported edges. Specifically, we start with a large value of the tuning parameter that results in an empty network, then decrease the tuning parameter so that the network becomes denser, and stop until the network has about 200 edges (out of a total of 3240 possible edges). At each value of the tuning parameter, we plot the number of literature validated interactions included in the network versus the total number of edges of the network, resulting in a step function for each method (Figure \\ref{tara_path}). From Figure \\ref{tara_path}, we observe that compositional graphical lasso identifies slightly more literature validated interactions than graphical lasso until the total number of edges arrives 175 and graphical lasso identifies one more literature validated interaction afterwards. Neighborhood selection selects much fewer literature validated interactions than either compositional graphical lasso or graphical lasso. These observations imply that compositional graphical lasso slightly outperforms graphical lasso in reconstructing the literature validated interactions, while its advantage over neighborhood selection is much more obvious.\n\nSecond, we compare the overall topologies of the three interaction networks with tuning parameters selected by StARS. We find that compositional graphical lasso, graphical lasso, and neighborhood selection identify 749, 921, and 190 edges, respectively, with the same instability threshold used in StARS. This agrees with our observation in the simulation study that the network from neighborhood selection is much sparser than those from compositional graphical lasso and graphical lasso. The degree distributions from the networks estimated by the three methods are shown in Figure \\ref{tara_degree}.\nThe center of the three degree distributions are ranked as neighborhood selection, compositional graphical lasso, and graphical lasso in the ascending order, which is also reflected in the densities of the three interaction networks.\n\nThird, we investigate the high-degree nodes, i.e., hub genera, in the interaction networks. High-degree nodes are often thought to represent taxa that elicit an effect on a large number of other members of their community, such as keystone taxa and generalistic parasites. It is observed that there are a few hub genera that have an excessive number of interactions with other genera in the literature, such as \\textit{Amoebophrya}, \\textit{Blastodinium}, and \\textit{Parvilucifera}, which are referred to as benchmark hub genera. Although the literature validated interactions are rather incomplete, it is still of interest to evaluate how well the three methods pick up those benchmark hub genera. Since the density of networks from the three methods are rather different, it is hard to compare the degrees of the hub genera from the three networks directly, but it is reasonable to compare the ranks of those degrees within each degree distribution. The method that generates lower ranks (degree of genera ranked in descending order) for those hubs in their degree distribution are believed to pick up the benchmark hub genera better. \n\nA list of 7 benchmark hubs (which has degree $\\geq 5$) along with their degrees from the incomplete network constructed from the literature is shown in Table \\ref{tab:degree rank of lit hubs}, followed by the corresponding ranks of those genera in the degree distributions from each of the three methods and their corresponding degrees in the parentheses. We can see that compositional graphical lasso generates lower ranks than graphical lasso for all 7 genera, while neighborhood selection generates lower ranks than compositional graphical lasso for 3 genera, and the opposite for the other 4 genera. Overall, compositional graphical lasso performs the best in picking up the benchmark hub genera among the three methods.\n\n\\begin{table}[htbp]\n\\centering\n\\begin{tabular}{lllll}\n  \\hline\n & Literature & Comp-gLASSO & gLASSO & MB \\\\ \n  \\hline\n  \\textit{Amoebophrya} & 1 (21) & 31 (19) & 47 (21) & 2 (9) \\\\ \n  \\textit{Blastodinium} & 2 (12) & 13 (23) & 26 (24) & 30 (5) \\\\ \n  \\textit{Parvilucifera} & 2 (12) & 46 (17) & 67 (19) & 58 (3) \\\\ \n  \\textit{Syndinium} & 4 (7) & 14 (23) & 27 (24) & 19 (6) \\\\ \n  \\textit{Vampyrophrya} & 4 (7) & 34 (19) & 60 (20) & 6 (8) \\\\ \n  \\textit{Phaeocystis} & 6 (6) & 1 (31) & 3 (29) & 17 (6) \\\\ \n  \\textit{Pirsonia} & 7 (5) & 64 (15) & 68 (19) & 33 (5) \\\\\n   \\hline\n\\end{tabular}\n\\caption{\\label{tab:degree rank of lit hubs} For the hub genera, their ranks in each degree distribution (in descending order) from the literature, compositional graphical lasso (Comp-gLASSO), graphical lasso (gLASSO) and neighborhood selection (MB). The numbers in the parentheses are the corresponding degrees of the genera.}\n\\end{table}\n\nWe find that compositional graphical lasso predicts a high degree for genera that are known to act as keystone species or parasitize other taxa, many of which are also high-degree in the literature validated network. For example, \\textit{Phaeocystis} elicits a high degree in both literature validated and compositional graphical lasso networks. This genus is a well-described keystone organism in some marine ecosystems \\citep{verity2007current}, where it causes large phytoplankton blooms and plays an important role in the global cycling of carbon and sulfur \\citep{verity1996organism}. Similarly, these two networks predict a high degree for taxa that are known parasites, such as \\textit{Blastodinium} \\citep{skovgaard2012parasitic}, \\textit{Amoebophyra} \\citep{chambouvet2008control}, and \\textit{Syndinium} \\citep{skovgaard2005phylogenetic}. In some instances, compositional graphical lasso uniquely reveals high-degree genera even when compared to the literature validated network, such as in the case of the parasite \\textit{Euduboscquella} \\citep{bachvaroff2012molecular}, which is ranked the 5th by compositional graphical lasso but only has 1 interaction in the literature. Given the fact that the literature validated network only captures a subset of interactions that exist in nature (i.e., those interactions that have been explicitly tested), we posit that compositional graphical lasso affords an opportunity to resolve novel modulators of community composition, such as keystone taxa and generalistic parasites, that follow-up experiments can validate.\n\n\\subsection{Zebrafish Parasite Infection Study}\n\\label{Sec-4.2}\nThe Zebrafish Parasite Infection Study, recently conducted at Oregon State University, used a zebrafish helminth intestinal infection model to resolve how the gut microbiome and parasite burden covary during infection \\citep{gaulke2019longitudinal}. This study quantified the infection burden of an intestinal helminth of zebrafish, \\textit{Pseudocapillaria tomentosa}, and the gut microbiome in 210 4-month-old zebrafish. Half of these fish were exposed to the parasite, \\textit{P.\\ tomentosa}, and the other half were unexposed. Given that not all exposed fish would be ultimately infected, parasite burden was measured more accurately as the total number of worms in their fecal samples. In addition, the same fecal samples were used to measure the abundance of the gut microbiome species via 16S rRNA sequencing, which resulted in gut microbiome data for 207 fish. Among these fish, 81 were infected after being exposed to \\textit{P.\\ tomentosa} and the other 126 fish were not infected, as indicated by the total number of worms. Our major goal is to evaluate how the microbial interaction network associates with successful parasite infection in the gut.\n\nSimilar to the \\textit{Tara} Oceans Project, we choose to analyze the genus-level abundance data. We discard those genera that have a nonzero abundance in less than 5\\% of the samples, resulting in 42 genera left in our analysis. In addition, we follow the same strategy to define a reference genus as in \\citet{jiang2020microbial}, i.e., combining those OTUs that do not have a genus-level taxonomic classification into a pseudo-genus and using it as the reference genus. The analysis is conducted in the same way as in the \\textit{Tara} Oceans Project, except that the methods (compositional graphical lasso, graphical lasso, and neighborhood selection) are applied separately for the uninfected and the infected fish, resulting in three interaction networks for each group of fish. Analyzing the uninfected and infected fish separately allows us to compare the microbial interaction networks between the two groups.\n\nFirst, we compare the overall topologies of the interaction networks with tuning parameters selected by StARS. We find that compositional graphical lasso, graphical lasso, and neighborhood selection identify 262, 312, and 78 edges, respectively, for uninfected fish, and 241, 259, and 60 edges, respectively, for infected fish. Comparing the two groups of fish, the interaction networks for the uninfected fish are slightly denser than the infected fish. The comparison among the three methods agrees with our observation in the simulation study and in the \\textit{Tara} Oceans Project that the network from neighborhood selection tends to be much sparser than those from compositional graphical lasso and graphical lasso. The degree distributions based on the three methods are shown in Figure \\ref{fish_degree}, which again suggests high similarity between the two groups of fish. Similar to the \\textit{Tara} Oceans Project, neighborhood selection results in the lowest median degree, followed by composition graphical lasso and graphical lasso, regardless of whether the fish are infected or not.\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.4\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{not_infected_degree.pdf}\n  \\caption{}\n  \\label{not_infected_degree}\n\\end{subfigure}%\n\\begin{subfigure}{.4\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{infected_degree.pdf}\n  \\caption{}\n  \\label{infected_degree}\n\\end{subfigure}\n\\caption{(a): The degree distribution of vertices from the networks selected by StARS for uninfected fish. (b): The degree distribution of vertices from the networks selected by StARS for infected fish. Solid red: compositional graphical lasso; dashed green: graphical lasso; dashed dotted blue: neighborhood selection.}\n\\label{fish_degree}\n\\end{figure}\n\nSecond, we further investigate the degree of each node in the interaction networks. This analysis helps identify those high-degree nodes, i.e., hub genera, which are indicative of keystone taxa in the microbial community. {\\color{blue}Due to the limited space, we refer to Table \\ref{fish_hubs} in the supplementary materials that presents all 42 genera with their degrees in each network in descending order.} Although the networks are similar in density between the two groups of fish, we have observed a substantial difference between the degrees of individual nodes. To see this we sort the nodes in each network based on their degrees and compare the top ten nodes between networks. Taking compositional graphical lasso as an example, only three out of the top ten nodes overlap between uninfected and infected fish: \\textit{Paicibacter}, \\textit{Photobacterium}, \\textit{Fusobacterium}. This suggests that the network structures between the uninfected and infected fish are more distinctive than what their overall topologies might appear. The hub genera identified by different methods are quite different from each other too. For the group of uninfected fish, only three out of the top ten nodes overlap between the networks from the three methods: \\textit{Aeromonas}, \\textit{Photobacterium}, and \\textit{Rheinheimera}; similarly, for the group of infected fish, only three out of the top ten nodes overlap between the networks from the three methods: \\textit{Fusobacterium}, \\textit{Paucibacter}, and \\textit{Yersinia}.\n\n{\\color{blue}Third, we compare the networks generated from uninfected and infected fish to clarify the relationship between infection and the interactions that exist in the gut microbiome, motivated by the hypothesis that intestinal dysbiosis is defined not only by changes in community composition, but also by how members of the community ecologically interact. In order to understand this relationship, we identify microbes whose interaction set changes between infected and uninfected hosts then compare these results across methods. One of the bacterial taxa that compositional graphical lasso identifies as increasing in interaction degree among infected individuals, while other methods estimate a decrease in detected interaction degree, is the genus \\textit{Gemmobacter}. Our earlier work \\citep{gaulke2019longitudinal} showed that the abundance of this taxon positively links to parasite exposure, indicating that it is more ecologically successful in infected versus uninfected intestines. The analysis from compositional graphical lasso indicates that this increase in \\textit{Gemmobacter} relative abundance in infected individuals is coincident with an increase in the ecological interactions between \\textit{Gemmobacter} and other members of the gut microbiome, as it interacts with a greater number of other taxa in infected individuals. While \\textit{Gemmobacter} is not a particularly well studied genus, prior work links it to dysbiotic diseases in a variety of hosts \\citep{bates2022microbiome, ni2020gut}. Our observation of an increase in its interaction with other taxa in infected individuals strengthens the hypothesis that \\textit{Gemmobacter} may act as an agent of dysbiosis, as its increase in its relative abundance not only links to a disease context, but its more integrated role in the microbial community (i.e., it potentially impacts a larger number of taxa in the gut).\n\nThe results from compositional graphical lasso also uniquely suggest that the \\textit{Paucibacter} genus in uninfected individuals displays a higher number of identified interactions relative to infected individuals. The other methods we evaluated find that \\textit{Paucibacter}'s interactions are relatively consistent between infected and uninfected individuals. The results from compositional graphical lasso are notable because the genus \\textit{Paucibacter} includes clades that have been identified as being conserved in the healthy zebrafish gut microbiome \\citep{sharpton2021phylogenetic}, presumably because these taxa are critical to the commensal microbial community. Based on our observations, namely that disruption to the broader microbiome by parasite infection appears to disrupt the set of interactions between \\textit{Paucibacter} and other taxa in the community, we hypothesize that \\textit{Paucibacter} is a keystone member of the healthy zebrafish gut microbiome and that parasite exposure induced dysbiosis occurs in part by disrupting \\textit{Paucibacter} in the gut. Collectively, our observations reveal patterns of interaction between infected and uninfected individuals that clarify the potential ecological role of \\textit{Gemmobacter} and \\textit{Paucibacter}, which future research should seek to empirically test.}\n\n{\\color{blue}In addition to these observations about \\textit{Gemmobacter} and \\textit{Paucibacter}, compositional graphical lasso also finds insightful patterns about \\textit{Photobacterium}, a genus which contains pathogens of marine fish \\citep{romalde2002photobacterium, osorio2018photobacterium}, though its role as a pathogen in zebrafish is less clear.} However, as prior work has cultured \\textit{Photobacterium} from the zebrafish gut \\citep{cantas2012culturable} and other studies have demonstrated that \\textit{Photobacterium} elicits toxicity to zebrafish cells in culture \\citep{osorio2018photobacterium}, observations collectively point to \\textit{Photobacterium} as a potential pathogen or pathobiont of zebrafish. Consistent with this speculation, prior work found that \\textit{Photobacterium} statistically explained variation in \\textit{P. tomentosa} worm burden as well as worm infection induced intestinal inflammation \\citep{gaulke2019longitudinal}. We sought to determine if our interaction network analysis could clarify the role of \\textit{Photobacterium} in the zebrafish gut microbiome, especially as it relates to parasite infection.\n\nInterestingly, \\textit{Photobacterium} is identified as an interaction hub (i.e., a relatively high degree node) in all networks assembled from all methods we evaluated. However, compositional graphical lasso uniquely reveals that the number of interactions linked to \\textit{Photobacterium} increases in infected fish (19 edges) as compared to uninfected fish (16 edges), whereas the other approaches observe the opposite pattern (graphical lasso: 11 versus 17 edges; neighborhood selection: 3 versus 5 edges). These differences in the \\textit{Photobacterium} subgraph across approaches also includes variation in the specific taxa that \\textit{Photobacterium} is inferred to interact with. {\\color{blue}To better understand changes in the set of \\textit{Photobacterium} interactions, we extracted the first-degree sub-graph connected to \\textit{Photobacterium} that each of the three methods produced for both the infected and uninfected host interaction networks. Compositional graphical lasso alone estimates eight interactions in the \\textit{Photobacterium}-specific infected host network, and six of these eight genera have previously been linked to parasite exposure, disease burden, or histopathology score.} Two of these parasite etiology-linked taxa are \\textit{Aeromonas} and \\textit{Gemmobacter}, genera which are notable because they contain microbes that elicit pathogenic or pathobiotic phenotypes. For example, the \\textit{Aeromonas} genus includes well-characterized and abundant opportunistic pathogens, such as \\textit{A. hydrophila} \\citep{saraceni2016establishment}, and the \\textit{Gemmobacter} genus, as noted above, has been shown to opportunistically increase abundance \\citep{huang2020exposure} in dysbiotic fish gut communities which display impeded overall gut function. {\\color{blue}Given that \\textit{Photobacterium} proliferates in the infected host gut, these observations suggest that \\textit{Photobacterium} may work alongside and even promote the growth of other pathobiotic taxa to disrupt the composition of the gut microbiome and induce dysbiosis upon infection.}\n\nBased on these collective observations, we hypothesize that \\textit{Photobacterium} is an intestinal pathobiont of the zebrafish gut and contributes to dysbiosis in infected fish. \\textit{Photobacterium} is a relatively prevalent genus in the zebrafish intestines, even in uninfected fish. However, \\textit{P. tomentosa} infection links to an increase in the relative abundance of \\textit{Photobacterium}, which also positively associates with intestinal hyperplasia \\citep{gaulke2019longitudinal}, possibly due to the cytotoxic effect of \\textit{Photobacterium}. Given that \\textit{Photobacterium} is also an interaction hub whose influence on the community increases in infected fish, at least according to compositional graphic lasso, infection induced changes in \\textit{Photobacterium} relative abundance may drive additional changes in the success of other taxa in the microbiome, including opportunistic pathogens and pathobionts like members of \\textit{Aeromonas} and \\textit{Gemmobacter}. Going forward, future studies should seek to experimentally validate our novel hypothesis about the pathobiotic role of \\textit{Photobacterium} in the zebrafish gut, including its impact on the rest of the microbial community and its role in infection induced tissue damage. If our hypothesis is accurate, \\textit{Photobacterium} may serve as an important model taxon for discerning how the gut microbiome and helminths interact to impact infection outcomes.\n\n{\\color{blue}Collectively, these results provide evidence for our hypothesis that in the case of intestinal helminth infection, dysbiosis may be defined by not only a change in the composition of the gut microbiome, but a restructuring in how key members of the microbial community interact with the rest of the microbiota.  Notably, these patterns were only observed by compositional graphical lasso, as the other methods did not observe unique evidence of such infection-associated inversions for \\textit{Gemmobacter}, \\textit{Paucibacter}, or \\textit{Photobacterium}.\n\nFinally, we visualize the six genus interaction networks, three for infected fish and three for uninfected fish (Figure \\ref{fish_StARS}). For a better visualization, we only keep the top $100$ edges for the networks resulted from compositional graphical lasso and graphical lasso as they are much denser than the ones from neighborhood selection. Their edges are ranked in the exactly same way as in the \\textit{Tara} Oceans Study, first by selection probability and then by edge weight (see Figure \\ref{tara_StARS} in the supplementary materials for details). For all networks, darker blue implies higher magnitude in the absolute value of partial correlation.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.8]{Fish_StARS_genus_level.pdf}\n\\caption{Inferred networks from each method with edges filtered by selection probability and ranked by edge weight separately for the groups of uninfected and infected fish. In each network, darker blue implies stronger (larger in absolute value) edge weight.}\n\\label{fish_StARS}\n\\end{figure}}\n\n\\section{Discussion}\n\\label{Sec-5}\n\nA growing body of work points to the gut microbiome as an agent to treat, diagnose, and prevent human diseases. The innovation of these utilities requires foundational knowledge about the pathobiotic {\\color{blue}or probiotic} role that the gut microbiome play in the development and progression of the diseases. While most studies have focused on investigating how the abundance of gut microbes covary with a disease, such as infection \\citep{gaulke2019longitudinal}, how the microbial interactions associate with the diseases is largely unknown. Understanding how the microbial interactions associate with a disease is critical in identifying microbes as a pathobiont {\\color{blue}or probiotic}, which are candidates for potential drug or diagnostic test.\n\nThis work focuses on gaining a better understanding of the underlying role that microorganisms play in their communities by constructing microbial interaction networks. We propose a novel method called compositional graphical lasso that simultaneously accounts for the following key features of microbiome abundance data:  (a) the data are compositional counts and only carry information about relative abundances of the taxa; (b) the observed relative abundance is subject to heteroscedasticity as its variance depends on the varying sequence depth across samples; (c) the data are high-dimensional as the number of taxa are often larger than the number of samples. We have demonstrated the advantages of our approach over previously proposed methods in simulations and on a benchmark dataset.\n\n\nWe apply compositional graphical lasso to the data from the Zebrafish Parasite Infection Study, which used a parasite infection model to identify gut microbiota that positively and negatively associate with infection burden \\citep{gaulke2019longitudinal}. {\\color{blue}Our approach identified method-specific changes in interaction degree between infected and uninfected individuals for three taxa, \\textit{Photobacterium}, \\textit{Gemmobacter}, and \\textit{Paucibacter}. Further investigation of these method-specific taxa interaction changes reveals their biological plausibility, and provides insight into their relevance in the context of parasite-linked changes in the zebrafish gut microbiome. In particular, we hypothesize that \\textit{Photobacterium} and \\textit{Gemmobacter} are pathobionts of zebrafish gut for parasitic infection and that \\textit{Paucibacter} is a probiotic. Future studies should seek to experimentally validate their ecological roles in the zebrafish gut, including their impacts on the rest of the microbial community and their roles in infection induced tissue damage.}\n\n\nIt is noteworthy that compositional graphical lasso requires one to choose a reference taxon. As a general rule of thumb, we recommend choosing a ``common'' taxon that has a high average relative abundance and relatively few zero counts across samples, as its true relative abundance serves as the denominator in the additive log-ratio transformation. In practice, we also suggest that a user choose a taxon that is not of direct interest to investigate, as the reference taxon will not be represented in the resultant network. {\\color{blue}It is also of interest to study how much the choice of the reference taxon may affect the estimated network, and whether some robustness could be guaranteed across different choices of the reference. Because the robustness against the choice of the reference is so important not just for compositional graphical lasso, but for many other network estimation methods as well, we have chosen to devote a major effort to this issue in a separate ongoing project. In this ongoing work, we have theoretically established the reference-invariance property of the inverse covariance matrix of a class of additive log-ratio transformation-based models, including the logistic normal multinomial model as a special case. However, as it is beyond the scope of this paper, this invariance property will be detailed in the future.}\n\n\\bibliographystyle{asa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nBlind decoding, also known as blind detection, requires the receiver of a set of bits to identify if said bits compose a codeword of a particular channel code. In 3GPP LTE\/LTE-Advanced standards blind detection is used by the user equipment (UE) to receive control information related to the downlink shared channel. The UE attempts the decoding of a set of candidates, to identify if one of the candidates holds its control information. \nBlind detection will be required in the $5^{\\text{th}}$ generation wireless communication standard (5G) as well: ongoing discussions are considering a substantial reduction of the time frame allocated to blind detection, from $16\\mu$s to $4\\mu$s. Blind detection must be performed very frequently, and given the high number of decoding attempts that must be performed in a limited time \\cite{3GPP_R8}, it can lead to large implementation costs and high energy consumption. Blind detection solutions for codes adopted in previous generation standards can be found in \\cite{Moosavi_GLOBECOM11,Xia_TSP14,Zhou_ENT13}.\n\nPolar codes are a class of capacity-achieving error correcting codes, introduced by Ar{\\i}kan in \\cite{arikan}. They are characterized by simple encoding and decoding algorithms, and have been selected for use in 5G \\cite{3gpp_polar}. In \\cite{arikan}, the successive-cancellation (SC) decoding algorithm has been proposed as well. It is optimal for infinite code lengths, but its error-correction performance degrades quickly at moderate and short code lengths. In its original formulation, it also suffers from long decoding latency. SC list (SCL) decoding has been proposed in \\cite{tal_list} to improve the error-correction performance of SC, at the cost of increased decoding latency. In \\cite{sarkis, hashemi_SSCL, xiong_symbol, hashemi_FSSCL}, a series of techniques has been proposed, aimed at improving the decoding speed of both SC and SCL without sacrificing error-correction performance.\n\nBlind detection of polar codes has been recently addressed in \\cite{Condo_COMML17}, where a blind detection scheme fitting within 3GPP LTE-A and future 5G requirements has been proposed. It is based on a two-step scheme: a first SC decoding phase helps selecting a set of candidates, subsequently decoded with SCL. An early stopping criterion for SCL is also proposed to reduce average latency. Another recent work on polar code blind detection \\cite{Giard_BD} detaches itself from 4G-5G standard requirements, and proposes a metric on which the outcome of the blind detection can be based.\n\nIn this work, we extend the blind detection scheme presented in \\cite{Condo_COMML17} and its early stopping criterion by considering SCL also in the first decoding phase, and provide improved detection accuracy results. We then propose an architecture to implement the blind detection scheme: it relies on an SCL decoder with tunable list size, that can be used for both the first and second decoding stages. The architecture is synthesized and implementation results are reported for various system parameters.\n\nThe rest of the paper is organized as follows. Section~\\ref{sec:prel} introduces background information on polar codes and blind detection. Section~\\ref{sec:blind} details the proposed blind detection scheme, and provides simulation results to evaluate its performance. The architecture of the blind detection system is detailed in Section~\\ref{sec:HW}, and implementation results are given in Section~\\ref{sec:impl}. Finally, Section~\\ref{sec:conc} draws the conclusion.\n\n\n\\section{Preliminaries} \\label{sec:prel}\n\n\\subsection{Polar Codes} \\label{sec:prel:PC}\n\n\nA polar code $\\mathcal{P}(N,K)$ is a linear block code of length $N=2^n$ and rate $K\/N$, and it can be expressed as the concatenation of two polar codes of length $N\/2$. This is due to the fact that the encoding process is represented by a modulo-$2$ matrix multiplication a\n\\begin{equation}\n\\mathbf{x} = \\mathbf{u} \\mathbf{G}^{\\otimes n}\\text{,}\n\\end{equation}\nwhere $\\mathbf{u} = \\{u_0,u_1,\\ldots,u_{N-1}\\}$ is the input vector, $\\mathbf{x} = \\{x_0,x_1,\\ldots,x_{N-1}\\}$ is the codeword, and the generator matrix $\\mathbf{G}^{\\otimes n}$ is the $n$-th Kronecker product of the polarizing matrix $\\mathbf{G}=\\bigl[\\begin{smallmatrix} 1&0\\\\ 1&1 \\end{smallmatrix} \\bigr]$. The polarization effect brought by polar codes allows to divide the $N$-bit input vector $\\mathbf{u}$ between reliable and unreliable bit-channels.\nThe $K$ information bits are assigned to the most reliable bit-channels of $\\mathbf{u}$, while the remaining $N-K$, called frozen bits, are set to a predefined value, usually $0$.\nCodeword $\\mathbf{x}$ is transmitted through the channel, and the decoder receives the logarithmic likelihood ratio (LLR) vector $\\mathbf{y} = \\{y_0,y_1,\\ldots,y_{N-1}\\}$.\n\nIn the seminal work on polar codes \\cite{arikan}, the SC decoder is proposed. The SC-based decoding process can be represented as a binary tree search, in which the tree is explored depth first, with priority given to the left branches. Fig.~\\ref{fig:tree} shows an example of SC decoding tree for $\\mathcal{P}(16,8)$, where nodes at stage $s$ contain $2^s$ bits. White leaf nodes are frozen bits, while black leaf nodes are information bits.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=1.8, thick]\n\n\\fill (0,0) circle [radius=.05];\n\n\\fill (-1,-.5) circle [radius=.05];\n\\fill (1,-.5) circle [radius=.05];\n\n\\fill (-1.5,-1) circle [radius=.05];\n\\fill  (-.5,-1) circle [radius=.05];\n\\fill  (.5,-1) circle [radius=.05];\n\\fill  (1.5,-1) circle [radius=.05];\n\n\\fill  (-1.75,-1.5) circle [radius=.05];\n\\fill  (-1.25,-1.5) circle [radius=.05];\n\\fill (-.75,-1.5) circle [radius=.05];\n\\fill (-.25,-1.5) circle [radius=.05];\n\\fill  (.25,-1.5) circle [radius=.05];\n\\fill  (.75,-1.5) circle [radius=.05];\n\\fill  (1.25,-1.5) circle [radius=.05];\n\\fill (1.75,-1.5) circle [radius=.05];\n\n\\draw (-1.875,-2) circle [radius=.05];\n\\draw  (-1.625,-2) circle [radius=.05];\n\\draw  (-1.375,-2) circle [radius=.05];\n\\draw  (-1.125,-2) circle [radius=.05];\n\\draw  (-.875,-2) circle [radius=.05];\n\\fill  (-.625,-2) circle [radius=.05];\n\\fill  (-.375,-2) circle [radius=.05];\n\\fill  (-.125,-2) circle [radius=.05];\n\\fill  (.125,-2) circle [radius=.05];\n\\fill  (.375,-2) circle [radius=.05];\n\\fill  (.625,-2) circle [radius=.05];\n\\fill  (.875,-2) circle [radius=.05];\n\\draw  (1.125,-2) circle [radius=.05];\n\\draw  (1.375,-2) circle [radius=.05];\n\\draw  (1.625,-2) circle [radius=.05];\n\\fill  (1.875,-2) circle [radius=.05];\n\n\\draw (0,-.05) -- (-1,-.45);\n\\draw (0,-.05) -- (1,-.45);\n\n\\draw (-1,-.55) -- (-1.5,-.95);\n\\draw (-1,-.55) -- (-.5,-.95);\n\\draw (1,-.55) -- (.5,-.95);\n\\draw (1,-.55) -- (1.5,-.95);\n\n\\draw  (-1.5,-1.05) -- (-1.75,-1.45);\n\\draw  (-1.5,-1.05) -- (-1.25,-1.45);\n\\draw (-.5,-1.05) -- (-.75,-1.45);\n\\draw (-.5,-1.05) -- (-.25,-1.45);\n\\draw  (.5,-1.05) -- (.25,-1.45);\n\\draw  (.5,-1.05) -- (.75,-1.45);\n\\draw  (1.5,-1.05) -- (1.25,-1.45);\n\\draw  (1.5,-1.05) -- (1.75,-1.45);\n\n\\draw  (-1.75,-1.55) -- (-1.875,-1.95);\n\\draw  (-1.75,-1.55) -- (-1.625,-1.95);\n\\draw  (-1.25,-1.55) -- (-1.375,-1.95);\n\\draw  (-1.25,-1.55) -- (-1.125,-1.95);\n\\draw  (-.75,-1.55) -- (-.875,-1.95);\n\\draw  (-.75,-1.55) -- (-.625,-1.95);\n\\draw (-.25,-1.55) -- (-.375,-1.95);\n\\draw (-.25,-1.55) -- (-.125,-1.95);\n\\draw  (.25,-1.55) -- (.125,-1.95);\n\\draw (.25,-1.55) -- (.375,-1.95);\n\\draw (.75,-1.55) -- (.625,-1.95);\n\\draw  (.75,-1.55) -- (.875,-1.95);\n\\draw  (1.25,-1.55) -- (1.125,-1.95);\n\\draw  (1.25,-1.55) -- (1.375,-1.95);\n\\draw  (1.75,-1.55) -- (1.625,-1.95);\n\\draw (1.75,-1.55) -- (1.875,-1.95);\n\n\\draw [very thin,gray,dashed] (-2,0) -- (2,0);\n\\draw [very thin,gray,dashed] (-2,-.5) -- (2,-.5);\n\\draw [very thin,gray,dashed] (-2,-1) -- (2,-1);\n\\draw [very thin,gray,dashed] (-2,-1.5) -- (2,-1.5);\n\\draw [very thin,gray,dashed] (-2,-2) -- (2,-2);\n\n\\node at (-2.3,0) {$s=4$};\n\\node at (-2.3,-.5) {$s=3$};\n\\node at (-2.3,-1) {$s=2$};\n\\node at (-2.3,-1.5) {$s=1$};\n\\node at (-2.3,-2) {$s=0$};\n\n\\end{tikzpicture}\n\\caption{Binary tree example for $\\mathcal{P}(16,8)$. White circles at $s=0$ are frozen bits, black circles at $s=0$ are information bits.}\n\\label{fig:tree}\n\\end{figure}\n\nFig.~\\ref{fig:MessagePassing} portrays the message passing among SC tree nodes. Parents pass LLR values $\\alpha$ to children, that send in return the hard bit estimates $\\beta$. \nThe left and right branch messages $\\alpha^\\text{l}$ and $\\alpha^\\text{r}$, in the hardware-friendly version of \\cite{leroux}, are computed as\n\\begin{align}\n\\alpha^{\\text{l}}_i = & \\text{sgn}(\\alpha_i)\\text{sgn}(\\alpha_{i+2^{s-1}})\\min(|\\alpha_i|,|\\alpha_{i+2^{s-1}}|) \\text{,} \\label{eq1} \\\\\n\\alpha^{\\text{r}}_i =& \\alpha_{i+2^{s-1}} + (1-2\\beta^\\text{l}_i)\\alpha_i \\text{,}\n\\label{eq2}\n\\end{align}\nwhile $\\beta$ is computed as\n\\begin{equation}\n\\beta_i =\n  \\begin{cases}\n    \\beta^\\text{l}_i\\oplus \\beta^\\text{r}_i, & \\text{if} \\quad i < 2^{s-1} \\text{,}\\\\\n    \\beta^\\text{r}_{i-2^{s-1}}, & \\text{otherwise},\n  \\end{cases}\n  \\label{eq3}\n\\end{equation}\nwhere $\\oplus$ denotes the bitwise XOR. The SC operations are scheduled according to the following order: each node receives $\\alpha$ first, then sends $\\alpha^\\text{l}$, receives $\\beta^\\text{l}$, sends $\\alpha^\\text{r}$, receives $\\beta^\\text{r}$, and finally sends $\\beta$.\nWhen a leaf node is reached, $\\beta_i$ is set as the estimated bit $\\hat{u}_i$:\n\\begin{equation}\n\\hat{u}_i =\n  \\begin{cases}\n    0 \\text{,} & \\text{if } i \\in \\mathcal{F} \\text{ or } \\alpha_{i}\\geq 0\\text{,}\\\\\n    1 \\text{,} & \\text{otherwise,}\n  \\end{cases} \\label{eq6}\n\\end{equation}\nwhere $\\mathcal{F}$ is the set of frozen bits.\n\nThe SC decoding process requires full tree exploration: however, in \\cite{alamdar, sarkis} it has been shown that it is possible to prune the tree by identifying patterns in the sequence of frozen and information bits, achieving substantial speed increments. This improved SC decoding is called fast simplified SC (Fast-SSC).\n\n\nSC decoding suffers from modest error-correction performance with moderate and short code lengths. To improve it, the SCL algorithm was proposed in \\cite{tal_list}. It is based on the same process as SC, but each time that a bit is estimated at a leaf node, both its possible values $0$ and $1$ are considered. A set of $L$ codeword candidates is stored, so that a bit estimation results in $2L$ new candidates, half of which must be discarded. To this purpose, a path metric (PM) is associated to each candidate and updated at every new estimate: the $L$ paths with the lowest PM survive. In the LLR-based SCL proposed in \\cite{balatsoukas}, the hardware-friendly formulation of the PM is\n\\begin{align}\n \\text{PM}_{{i}_l} =& \\begin{cases}\n    \\text{PM}_{{i-1}_l}, & \\text{if } \\hat{u}_{i_l} = \\frac{1}{2}\\left(1-\\text{sgn}\\left(\\alpha_{i_l}\\right)\\right)\\text{,}\\\\\n    \\text{PM}_{{i-1}_l} + |\\alpha_{i_l}|, & \\text{otherwise,}\n  \\end{cases} \\label{eq7}\n\\end{align} \nwhere $l$ is the path index and $\\hat{u}_{i_l}$ is the estimate of bit $i$ at path $l$.\nAs with SC decoding, SCL tree pruning techniques relying on the identification of frozen-information bit patterns have been proposed in \\cite{hashemi_SSCL,hashemi_FSSCL}, called simplified SCL (SSCL) and Fast-SSCL.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=.5]\n\n\\draw [very thin,gray,dashed] (-2,0) -- (2,0);\n\\draw [very thin,gray,dashed] (-2,-2) -- (2,-2);\n\\draw [very thin,gray,dashed] (-2,-4) -- (2,-4);\n\n\\node at (-3,0) {$s+1$};\n\\node at (-3,-2) {$s$};\n\\node at (-3,-4) {$s-1$};\n\n\\fill (0,0) circle [radius=.25];\n\\fill  (0,-2) circle [radius=.2];\n\\fill  (-1.5,-4) circle [radius=.15];\n\\fill  (1.5,-4) circle [radius=.15];\n\n\\draw [->,very thick] (-.1,-.4) -- (-.1,-1.7) node [left,midway,rotate=0] {$\\alpha$};\n\\draw [->,very thick] (.1,-1.7) -- (.1,-.4) node [right,midway,rotate=0] {$\\beta$};\n\n\\draw [->,very thick] (-.25,-2.2) -- (-1.45,-3.75) node [left,midway,rotate=0] {$\\alpha^{\\text{l}}$};\n\\draw [->,very thick] (-1.3,-3.85) -- (-.1,-2.3) node [right,near start,rotate=0] {$\\beta^{\\text{l}}$};\n\\draw [<-,very thick] (.25,-2.2) -- (1.45,-3.75) node [right,midway,rotate=0] {$\\beta^{\\text{r}}$};\n\\draw [<-,very thick] (1.3,-3.85) -- (.1,-2.3) node [left,near start,rotate=0] {$\\alpha^{\\text{r}}$};\n\n\\end{tikzpicture}\n\\caption{Message passing in tree graph representation of SC decoding.}\n\\label{fig:MessagePassing}\n\\end{figure}\n\n\n\n\\subsection{Blind Detection}\n\nThe physical downlink control channel (PDCCH) is used in 3GPP LTE\/LTE-Advanced to transmit the downlink control information (DCI) related to the downlink shared channel. The DCI carries information regarding the channel resource allocation, transport format and hybrid automatic repeat request, and allows the UE to receive, demodulate and decode.\n\nA cyclic redundancy check (CRC) is attached to the DCI payload before transmission. The CRC is masked according to an ID, like the radio network temporary identifier (RNTI), of the UE to which the transmission is directed, or according to one of the system-wide IDs. Finally, the DCI is encoded with a convolutional code. The UE is not aware of the format with which the DCI has been transmitted: it thus has to explore a combination of PDCCH locations, PDCCH formats, and DCI formats in the common search space (CSS) and UE-specific search space (UESSS) and attempt decoding to identify useful DCIs. This process is called blind decoding, or blind detection. For each PDCCH candidate in the search space, the UE performs channel decoding, and demasks the CRC with its ID. If no error is found in the CRC, the DCI is considered as carrying the UE control information.\n\n\nBased on LTE standard R8 \\cite{3GPP_R8}, the performance specifications for the blind detection process are the following:\n\\begin{itemize}\n \\item The DCI of PDCCH is from $8$ to $57$ bits plus $16$-bit CRC, masked by $16$-bit ID.\n \\item In UESSS, a maximum of $2$ DCI formats can be sent per transmission time interval (TTI) for $2$ potential frame lengths. Therefore, $16$ candidate locations in UESSS $\\rightarrow$ $32$ candidates.\n \\item In CSS, a maximum of $2$ DCI formats can be sent per TTI for $2$ potential frame lengths. Therefore, $6$ candidate locations in CSS $\\rightarrow$ $12$ candidates.\n \\item Code length could be between $72$ and $576$ bits.\n \\item Information length (including $16$-bit CRC) could be between $24$ and $73$ bits.\n \\item Target signal-to-noise ratio (SNR) is dependent on the targeted block error rate (BLER): $10^{-2}$.\n \\item There are two types of false-alarm scenarios: Type-1, when the UE ID is not transmitted but detected, and Type-2, when the UE ID is transmitted but another one is detected. The target false-alarm rate (FAR) is below $1.52\\times 10^{-5}$.\n \\item Missed detection occurs when UE ID is transmitted but not detected. The missed detection rate (MDR) is close to BLER curve.\n \\item The available time frame for blind detection is $16\\mu$s.\n\\end{itemize}\n\n\n\n\\section{Blind Detection Scheme} \\label{sec:blind}\n\nIn \\cite{Condo_COMML17}, polar codes have been considered within a blind detection framework, and a blind detection scheme has been proposed. Frozen bit positions are selected to instead transmit the RNTI. Fig.~\\ref{fig:scheme} shows the block diagram of the devised blind detection scheme. $C_1$ candidates are received at the same time: in this case, $C_1=44$. The $C_1$ candidates are decoded with the simple SC algorithm, and a PM is obtained for each candidate, equivalent to the LLR of the last decoded bit: thanks to the serial nature of SC decoding, the LLR of the last bit can be interpreted as a reliability measure on the decoding process.\nThe PMs are then sorted, to help the selection of the best candidates to forward to the following decoding phase. $C_2$ candidates are in fact selected to be decoded with the more powerful SCL decoding algorithm, that guarantees a better error-correction performance, at a higher implementation complexity. The $C_2$ candidates are chosen as:\n\\begin{enumerate}\n \\item All candidates whose ID, after the first phase, matches the one assigned to the UE. If more than $C_2$ are present, the ones with the highest PMs are selected.\n \\item If free slots among the $C_2$ remain, the candidates with the smallest PMs are selected. The candidates with large PMs have higher probability to be correctly decoded: if their ID does not match the one assigned to the UE, it is probably a different one. On the other hand, candidates with small PMs have a higher chance of being incorrectly decoded, and a transmission to the UE might be hiding among them.\n\\end{enumerate}\nAfter the SCL decoding phase, if one of the $C_2$ candidates matches the UE ID, it is selected, otherwise no selection is attempted.\n\n\\begin{figure}[t!]\n  \\centering\n  \\input{.\/Arch.tikz}\n  \\caption{Polar codes blind detection scheme.}\n  \\label{fig:scheme}\n\\end{figure}\n\nIn \\cite{Condo_COMML17}, an early stopping criterion has been proposed as well, to reduce the latency and energy expenditure of the second phase of the blind detection scheme, The first phase requires the full decoding of each candidate, to identify the $C_2$ codewords that will be sent to the second phase. In the second phase, however, all codewords whose ID does not match the UE ID will be discarded. Thus, as soon as the ID is shown to be different, the decoding can be interrupted. Since SC-based decoding algorithms estimate codeword bits sequentially, the ID evaluation can be performed every time an ID bit is estimated. In case the estimated bit is different from the UE ID bit, the decoding is stopped. \n\nThree methods of ID bits have been described in \\cite{Condo_COMML17} to choose the bits assigned to the ID:\n\\begin{itemize}\n \\item ID mode 1: the ID bits are the $16$ most reliable bits after the $K$ information bits. \n \\item ID mode 2: the ID bits are the $16$ most reliable bits, while the $K$ information bits are the most reliable bits after the $16$ ID bits.\n \\item ID mode 3: considering the order with which bits are decoded in SC-based algorithms, the ID bits are the first $16$ to be decoded among the $K+16$ most reliable bits.\n\\end{itemize}\nThe three techniques yield negligible differences in terms of error-correction performance, while ID mode 3 yields considerable advantages over mode 1 and mode 2 when early stopping is applied. In fact, since the ID bits are decoded earlier, the average percentage of estimated bits decreases, and the reduction in average latency is more substantial.\n\nIn this work, we generalize the blind detection scheme proposed in \\cite{Condo_COMML17}, by considering SCL also for the first decoding phase. In particular, we consider a list sizes $L_1\\ge1$ for the first decoding phase, and a list size $L_{\\max}>L_1$ for the second decoding phase. It should be noted that when $L_1=1$, the blind detection scheme reverts to that of \\cite{Condo_COMML17}.\n\n\n\n\\subsection{Simulation Results} \\label{sec:simres}\n\n\\begin{figure}[t!]\n  \\centering\n  \\input{.\/blerHW.tikz}\n    \\\\\n  \\ref{blerlegend}\n  \\caption{BLER curves with SCL when $L=8$.}\n  \\label{fig:BLER}\n\\end{figure}\n\n\\begin{figure}[t!]\n  \\centering\n    \\input{.\/MissedDetectionHW.tikz}\n      \\\\\n  \\ref{MDRlegend}\n  \\caption{Missed detection rates after the second decoding phase with $L_1=2$, $L_{\\max}=8$, and $C_2=5$. Transmissions include $C_1\/2$ cases of $N_1=256$ and $C_1\/2$ cases of $N_2=512$.}\n  \\label{fig:MD_SCL}\n\\end{figure}\n\n\n\nTo evaluate the effectiveness of the proposed blind detection scheme, simulations were performed. The BLER, MDR, and FAR have been measured on the additive white Gaussian noise (AWGN) channel, with binary phase-shift keying (BPSK) modulation, at the variation of different code parameters. We focused on polar codes with block lengths $N=\\{256, 512\\}$, since in \\cite{Condo_COMML17} it has been shown that they constitute the most critical cases in terms of speed. Four information lengths $K=\\{8, 16, 32, 57\\}$ have been considered, while the number of ID bits has been set to $16$. The 3GPP standardization committee has decided that information bits in polar codes must be assigned to the $K$ most reliable bit-channels \\cite{3gpp_polar_AH}: thus, the ID bits have been assigned according to ID mode 1. The ID values assigned to the $C_1$ candidates are randomly selected over $16$ bits. While different numbers of candidates passed to the second phase have been considered in \\cite{Condo_COMML17}, we have focused here on $C_2=5$, for which a good tradeoff between accuracy and latency is found. At the same time, we set $L_{\\max}=8$ and $L_1=2$: it is a representative case for which $L_{\\max}$ guarantees good error-correction performance, and at which SCL decoders can be implemented with reasonable complexity.\n\n\nFig.~\\ref{fig:BLER} plots the BLER curves for all the considered code lengths and rates. As expected, their error-correction performance improves as the code length increases and the code rate decreases. In Fig.~\\ref{fig:MD_SCL}, the first of the metrics specific to the blind detection problem, the MDR, is depicted. The MDR can be defined as the number of missed detections divided by the number of transmissions in which the UE ID was sent. The curves in Fig.~\\ref{fig:MD_SCL} have been obtained considering $C_1\/2$ candidates of length $N_1=256$, and $C_1\/2$ candidates of length $N_2=512$ in each transmission, with $K_1=K_2$ information bits. Together with the MDR, in Fig.~\\ref{fig:MD_SCL} the BLER curves relative to the aggregate transmissions are portrayed. It can be seen that the MDR curve is always lower than the relative BLER curve.\n\n\nThe FAR curves for the considered case study are portrayed in Fig.~\\ref{fig:FAR}. The system target FAR is equivalent to the FAR obtained with a $16$-bit CRC: in 5G, a CRC of at least $16$-bits long is foreseen. Here, we evaluate the additional contribution that the proposed blind detection scheme can bring in lowering the FAR on top of the CRC. It can be seen that the FAR is kept below the $10^{-4}$ threshold at SNR values for which the BLER is still very high, and decreases as the channel conditions improve. In the blind detection method presented in \\cite{Giard_BD}, the FAR increases as the MDR decreases. On the other hand, the proposed scheme allows to decrease both at the same time, thus avoiding performance limitations that could make it unappealing for 5G standard applications.\n\nThe impact of the devised early stopping criterion on the average number of estimated bits is shown in Fig.~\\ref{fig:avg_est}, for $K=32$ and $K=57$. These results consider each of the $C_2$ candidates separately, since the number of candidates of length $N_1$ and $N_2$ in the second phase depends on the PMs received from the first phase, and thus on channel SNR. The solid curves have been obtained in cases the UE ID was sent through the considered code, while the dashed curves in cases it was not sent through the code. \n\n\n\n\\begin{itemize}\n \\item For $N=256$ (curves with a circle marker), it is possible to observe the same behavior noted in \\cite{Condo_COMML17} for $N=128$ as well. In case the UE ID was sent, as the channel conditions improve, the number of estimated bits increases until stabilizing at a maximum average value. This phenomenon can be explained by the fact that when the SNR is low, it is more likely that the codeword carrying the UE ID is not selected to be among the $C_2$ candidates. Thus the decoders in the second phase easily encounter ID bits different from the UE ID early in the decoding process. As the channel conditions improve, the codeword with the UE ID falls among the $C_2$ candidates with rising probability. Consequently, the decoder tasked with its decoding does not interrupt the process, reaching $100\\%$ estimated bits, while the remaining $C_2-1$ decoders stop the decoding early, thus averaging the estimated bit percentage at a stable value ($67\\%$ for $K=32$ and $61\\%$ for $K=57$). The dashed curves show instead a stable value regardless of channel conditions: since among the $C_2$ candidates there is never one carrying the UE ID, all second phase decoders tend to stop the decoding early, at a percentage independent of the SNR, and mostly influenced by the position of bits assigned to the ID.\n\\item For $N=512$ (curves with a cross marker) a similar behavior to the $N=256$ case can be observed when the UE ID is not sent, with the average number of estimated bits stable at all the considered SNR values. On the other hand, when the UE ID is sent, the trend is different: at low SNR values, the percentage of estimated bits is very close to $100\\%$. As the SNR value increases, the average starts to decrease, until it settles on a stable value.\nThis behavior is due to the fact that at low SNR, it is very unlikely that a codeword with $N=512$ is among the $C_2$ second phase candidates if the UE ID is not matching: the longer code length and lower rate contribute to a higher decoding reliability during the first phase, that allows to screen out unlikely candidates better than the $N=256$ case.\n\\end{itemize}\n\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\input{.\/FalseAlarmHW.tikz}\n  \\\\\n  \\ref{FARlegend}\n  \\caption{False alarm rates after the second decoding phase with $L_1=2$, $L_{\\max}=8$, and $C_2=5$. Transmissions include $C_1\/2$ cases of $N_1=256$ and $C_1\/2$ cases of $N_2=512$.}\n  \\label{fig:FAR}\n\\end{figure}\n\n\\begin{figure}[t!]\n  \\centering\n  \\input{.\/AvgEstHW.tikz}\n  \\\\\n  \\ref{ESlegend}\n  \\caption{Average percentage of estimated bits during the second decoding phase with early stopping when $L_{\\max}=8$ and $C_2=5$.}\n  \\label{fig:avg_est}\n\\end{figure}\n\n\n\n\n\\section{Hardware Architecture} \\label{sec:HW}\nTo evaluate the implementation cost of the devised blind detection scheme, we designed a decoder architecture that supports it, portrayed in Fig.~\\ref{fig:BDarch}. An array of flexible list size SCL decoders handles both the first and second decoding phase. A dedicated module selects the $C_2$ candidates for the second phase according to the criteria described in Section~\\ref{sec:blind}.\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[scale=0.55]{.\/BDsys.pdf}\n  \\caption{Polar codes blind detection system architecture.}\n  \\label{fig:BDarch}\n\\end{figure}\n\n\\subsection{Flexible list size SCL decoder}\n\nWe based our SCL decoder architecture on that of \\cite{hashemi_SSCL_TCASI,hashemi_FSSCL}: the decoding process follows the one described in Section~\\ref{sec:prel:PC} for a list size $L_{\\max}$. Most of the datapath and memories are instantiated $L_{\\max}$ times: multiple candidates are stored at the same time, with the best candidate being selected at the end of the decoding. While in \\cite{hashemi_SSCL_TCASI,hashemi_FSSCL} the final candidate is selected according to a CRC check, in the proposed architecture no CRC is considered, and the validity of the final candidate is based on the matching ID and PM value.\n\nThe SC decoding tree is descended by computing (\\ref{eq1}) and (\\ref{eq2}) at each stage $s$, with priority being given to left branches. These calculations are performed by $L_{\\max}$ parallel sets of $P$ processing elements (PEs), with $P$ being a power of $2$. In the stages for which $2^s>2P$, the operations in (\\ref{eq1}) and (\\ref{eq2}) are performed over $2^s\/(2P)$ steps, while a single step is needed otherwise. Internal memories store the updated LLR values between stages.\n\nPEs get two LLR values as input, and concurrently compute both $\\alpha^{\\text{l}}$ and $\\alpha^{\\text{r}}$ according to (\\ref{eq1}) and (\\ref{eq2}), respectively. The correct output is selected depending on the index of the leaf node to be estimated. When a leaf node is reached, the decoder controller module identifies the leaf node as either an information bit or a frozen bit. If a frozen bit is found, the paths are not split, and the bit is estimated only as $0$, and the $L$ memories are updated with the same bit or LLR values. Instead, in case of an information bit, both $0$ and $1$ are considered, so that paths are split, and the PMs updated for the $2L$ candidates according to (\\ref{eq7}). Afterwards, the PMs are sorted, identifying the $L$ surviving paths.\n\nAll memories in the decoder are registers, enabling the internal LLR and $\\beta$ values to be read, updated by the PEs, and written back in a single clock cycle. At the same time, the paths are either updated or split and updated, and the new PMs computed. In the following clock cycle, in case the paths were split, the PMs are sorted and the surviving paths selected.\n\nCodes with different code lengths can be decoded by storing the appropriate memory offsets for every considered code in a dedicated memory.\n\nThis baseline decoder has been modified to better fit the needs of the proposed blind detection scheme. In order to maximize resource sharing, the SCL decoder has been sized for $L_{\\max}>L_1$, and the effective list size can be selected through a dedicated input.\nThe $L_{\\max}-L_1$ paths that are not used in the first decoding phase are used to decode up to $\\left\\lfloor(L_{\\max}-L_1)\/L_1\\right\\rfloor$ additional candidates at the same time. In order to exploit the unused paths, additional functional modules are necessary.\n\\begin{itemize}\n \\item The baseline decoder uses a single memory to store the channel LLR values, sharing it among the different paths. If different codewords have to be decoded at the same time, the channel memory needs to be instantiated not once, but $\\left\\lfloor L_{\\max}\/L_1\\right\\rfloor$ times.\n \\item The decoder relies on sorting and selection logic that identifies the surviving $L_{\\max}$ ones after paths are split. To support the parallel decoding of $\\left\\lfloor L_{\\max}\/L_1\\right\\rfloor$ candidates, as many sorting and selection modules targeting the selection of $L_1$ paths out of $2L_1$ are instantiated.\n\\end{itemize}\nIf $L_1=1$ is selected, the path splitting and PM sorting steps are bypassed, reverting decoders to the standard SC case. Since a single set of SCL decoders can handle both decoding phases, the total number of decoders is $N_{\\text{SCL}_{\\max}}$ (see Fig. \\ref{fig:BDarch}). However, the effective number of decoders for the first decoding phase is $N_{\\text{SCL}_1}=N_{\\text{SCL}_{\\max}}\\times \\left\\lfloor L_{\\max}\/L_1\\right\\rfloor$.\n\nThe early stopping technique described in Section~\\ref{sec:blind} has been also implemented. The decoder receives as input the position of the ID bits and the value of the UE ID: every time a bit in an ID position is estimated, the bit value is compared to the expected UE ID bit. All paths whose estimated bit does not match the UE ID bit are deactivated. This operation is performed after the $L$ surviving paths have been selected, in order not to force the survival of unlikely paths and increase the FAR. In case all paths have been deactivated, the decoding is stopped. The early stopping logic can be activated and deactivated by means of a dedicated control signal. Since the same hardware is used for both decoding phases, early stopping is enabled only during the second one.\n\n\n\n\\subsection{PM sorting and candidate selection}\n\n\\begin{figure*}[t!]\n  \\centering\n  \\includegraphics{.\/C2sel.pdf}\n  \\caption{PM sorting and candidate selection architecture.}\n  \\label{fig:C2sel}\n\\end{figure*}\n\nFig.~\\ref{fig:C2sel} depicts the architecture of the PM sorting and candidate selection block. It processes the output of the first decoding phase to select the $C_2$ candidates for the second phase, and selects the overall system output based on the results from the second phase. For each of the $N_{\\text{SCL}_1}$ first phase decoders, a PM and a flag signalling a UE ID match are received. They are stored every time the respective \\texttt{Valid} signal is risen by the decoder. The \\texttt{Valid} signal is also used as an enable for the PM and UE ID match register address counter, and for the counter keeping track of how many codewords had a matching UE ID after the first phase. When all the $C_1$ candidates have gone through the first decoding phase, a \\texttt{Valid} signal is issued to the sorter module, that receives as input all the stored PMs. The sorter module returns the $C_2$ minimum PMs in as many clock cycles: each PM is compared to all the others, and a single clock cycle is necessary to identify the minimum one, that is excluded from the subsequent comparison. When the $C_2$ minima have been found, the selector module considers how many candidates had a matching UE ID after the first phase, and selects the $C_2$ candidates for the second phase among them and those with the minimum PM values. The $C_2$ candidates are sent to the $N_{\\text{SCL}_{\\max}}$ decoders by means of a dedicated counter. Returning PMs and UE ID match flags are received and compared by another selector: when all $C_2$ candidates have been decoded, the selected codeword, if any, is output.\n\n\n\\section{Implementation Results} \\label{sec:impl}\n\nThe architecture proposed in Section~\\ref{sec:HW} has been described in VHDL and synthesized in TSMC 65~nm CMOS technology. Table~\\ref{tab:asic} reports the synthesis results for the architecture sized for a maximum code length $N_{\\max}=512$, a maximum list size $L_{\\max}=8$, $C_2=5$, and a target frequency $f=1$~GHz. Various $N_{\\text{SCL}_{\\max}}$ values have been considered, leading to different latencies and area occupations. Since during the first decoding phase $L_1=2$, the effective number of decoders $N_{\\text{SCL}_1}$ is equal to $4N_{\\text{SCL}_{\\max}}$, even if only $N_{\\text{SCL}_{\\max}}$ are physically instantiated.\nRegarding the area, the $N_{\\text{SCL}_{\\max}}$ SCL decoders contribute to the majority of the complexity, ranging from $97.8\\%$ when $N_{\\text{SCL}_{\\max}}=1$ to $99.7\\%$ when $N_{\\text{SCL}_{\\max}}=5$. The logic complexity of the PM sorting and candidate selection module remains almost unchanged at the variation of $N_{\\text{SCL}_{\\max}}$, being mainly affected by $C_1$ and $C_2$. Memories have been synthesized with registers only, without the use of RAM, and account for $36\\%$ of the total area occupation.\n\nThe worst case latency of the proposed blind detection system can be found as \n\\begin{align} \\label{eq:lat2}\n\\begin{split}\n T_{\\text{bd}}=&\\left\\lceil \\frac{C_1}{N_{\\text{SCL}_1}} \\right\\rceil \\left(\\frac{T^1_{\\text{SCL}}}{2}+\\frac{T^2_{\\text{SCL}}}{2}\\right)\\\\\n &+ T_{\\text{sort}} + \\left\\lceil \\frac{C_2}{N_{\\text{SCL}_{\\max}}} \\right\\rceil \\max\\left(T^1_{\\text{SCL}},T^2_{\\text{SCL}}\\right)~ \\text{,}\\\\\n\\end{split} \n\\end{align}\nwhere $T^1_{\\text{SCL}}$ and $T^2_{\\text{SCL}}$ are the SCL decoding latencies for codes of length $N_1$ and $N_2$, respectively, while $T_{\\text{sort}}$ is the number of time steps required to sort the PM of the first decoding phase and obtain the $C_2$ candidates out of the $C_1$ candidate locations. Also, it is worth remembering that for the proposed architecture, $N_{\\text{SCL}_1}=\\left\\lfloor L_{\\max}\/L_1\\right\\rfloor \\times N_{\\text{SCL}_{\\max}}$.\nThe SCL decoding latency can be found as \\cite{balatsoukas}\n\\begin{equation*}\nT^x_{\\text{SCL}}=2N_x+K_x+16-2 \\text{,}\n\\end{equation*}\nfor $x\\in\\{1,2\\}$.\nFrom the results presented in Table \\ref{tab:asic}, it is possible to see that even when considering the relatively old 65~nm technology node, the $16\\mu$s worst case latency target can be reached with a single SCL decoder running at a frequency of $1$~GHz, while $N_{\\text{SCL}_{\\max}}=5$ guarantees a worst case latency of $3.6\\mu$s, meeting the $4\\mu$s target as well.\n\nHowever, considering only the worst case latency is indeed an unrealistic scenario. To begin with, while there is no guarantee on how the $C_2$ candidates are distributed among $N_1$ and $N_2$, simulation results have shown that we can expect the $C_2$ candidates either to favor the shorter code length, or to be equally divided between $N_1$ and $N_2$ candidates. Thus, the factor $$\\left\\lceil\\frac{C_2}{N_{\\text{SCL}_{\\max}}} \\right\\rceil\\max\\left(T^1_{\\text{SCL}},T^2_{\\text{SCL}}\\right)$$ in (\\ref{eq:lat2}), that represents the contribution of the second decoding phase, could be better expressed as: \n\\begin{equation*}\n\\left\\lceil\\frac{\\left\\lceil C_2\/2\\right\\rceil}{N_{\\text{SCL}_{\\max}}} \\right\\rceil T^1_{\\text{SCL}} + \\left\\lceil\\frac{C_2-\\left\\lceil C_2\/2\\right\\rceil}{N_{\\text{SCL}_{\\max}}} \\right\\rceil T^2_{\\text{SCL}}~.\n\\end{equation*}\nNote that this is still a conservative assumption, since it entails the $C_2$ candidates equally divided among the two code lengths. We can refine this assumption by taking in account the effect of early stopping. We can approximate the latency reduction with a multiplicative factor $E^x$ associated to  $T^x_{\\text{SCL}}$. Consequently, the average latency of the blind detection system, for $N_{\\text{SCL}_{\\max}}<C_2$, can be computed as\n\\begin{align} \\label{eq:lat3}\n\\begin{split}\n T_{\\text{bd}}&=\\left\\lceil \\frac{C_1}{N_{\\text{SCL}_1}} \\right\\rceil \\left(\\frac{T^1_{\\text{SCL}}}{2}+\\frac{T^2_{\\text{SCL}}}{2}\\right) + T_{\\text{sort}}\\\\\n & + \\left\\lceil\\frac{\\left\\lceil C_2\/2\\right\\rceil}{N_{\\text{SCL}_{\\max}}} \\right\\rceil T^1_{\\text{SCL}}E^1 + \\left\\lceil\\frac{C_2-\\left\\lceil C_2\/2\\right\\rceil}{N_{\\text{SCL}_{\\max}}} \\right\\rceil T^2_{\\text{SCL}}E^2~ \\text{,}\\\\\n\\end{split} \n\\end{align}\nwhile for $N_{\\text{SCL}_{\\max}}\\ge C_2$ it becomes\n \\begin{align} \\label{eq:lat4}\n \\begin{split}\nT_{\\text{bd}}=\\left\\lceil \\frac{C_1}{N_{\\text{SCL}_1}} \\right\\rceil  & \\left(\\frac{T^1_{\\text{SCL}}}{2}+\\frac{T^2_{\\text{SCL}}}{2}\\right) + T_{\\text{sort}}\\\\\n  & + \\max\\left(T^1_{\\text{SCL}}E^1,T^2_{\\text{SCL}}E^2\\right)~ \\text{.}\\\\\n \\end{split}  \n \\end{align}\n\n\nConsidering the number of UEs connected to the shared channel, blind detection is dominated by instances in which a particular UE ID is not sent. Thus, we can set $E^x$ as the fraction of bits expressed by the dashed curves in Fig.~\\ref{fig:avg_est}. The average latency results in Table~\\ref{tab:asic} show substantial reduction with respect to the worst case latency case, within a more realistic framework. Even within the 65~nm technology node, with $N_{\\text{SCL}_{\\max}}\\ge 4$, the average latency is below $4\\mu$s. With the latest technology nodes, a substantially higher frequency will be easy to achieve, along with proportionally smaller area occupation. It is consequently safe to assume that the $4\\mu$s worst case latency target can be easily met for $N_{\\text{SCL}_{\\max}}\\ge3$, and the average latency with $N_{\\text{SCL}_{\\max}}\\ge2$.\n\n\\begin{table}[tbp!]\n  \\centering\n \n    \\caption{TSMC CMOS 65~nm blind detection scheme synthesis results for $L_{\\max}=8$, $P=64$, $C_2=5$, and $f=1$~GHz.}\n    \t\t\\setlength{\\extrarowheight}{1.8pt}\n  \\begin{tabular}{cccccc}\n\\toprule\n    \\multirow{2}{*}{$N_{\\text{SCL}_{\\max}}$} & Area & \\multicolumn{2}{c}{Worst case latency} & \\multicolumn{2}{c}{Average latency}\\\\\n    & [mm$^2$] & [clock cycles] & [$\\mu$s] & [clock cycles] & [$\\mu$s]\\\\\n\\cmidrule(lr){1-1}\n\\cmidrule(lr){2-2}\n\\cmidrule(lr){3-4}    \n\\cmidrule(lr){5-6}\n   \n    $1$ & $1.555$ & $14720$ & $14.7$ & $11843$ & $11.8$\\\\\n    $2$ & $3.086$ & $8330$ & $8.3$ & $6470$ & $6.5$\\\\\n    $3$ & $4.596$ & $5555$ & $5.6$ & $4541$ & $4.5$\\\\\n    $4$ & $6.117$ & $4710$ & $4.7$ & $3696$ & $3.7$\\\\\n    $5$ & $7.654$ & $3620$ & $3.6$ & $3451$ & $3.5$\\\\\n\\bottomrule\n\t\\end{tabular}\n  \\label{tab:asic}\n\\end{table}\n\n\n\\section{Conclusion} \\label{sec:conc}\nIn this work, we propose a polar codes blind detection scheme. The candidates go through a first, coarser decoding phase, that helps to select a few of them for a second, finer decoding phase. An early stopping criterion is proposed for the second phase, to reduce average latency. We evaluate the effectiveness of the blind detection scheme, and propose an architecture to implement it. It is based on an SCL decoder with tunable list size, that can be used for both decoding stages. The architecture is synthesized and implementation results are reported for various system parameters. The reported area occupation and latency, obtained in 65~nm CMOS technology, are able to meet 5G requirements, and are guaranteed to meet them with even less resource usage in the latest technology nodes.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{$\\epsilon_0$-modal sets}\\label{appendix:eps-modal-set}\n\n\n\\begin{definition}\\label{eps-modal-set} For $\\epsilon_0 \\ge 0$, connected set $M$ is an {\\bf $\\epsilon_0$-modal set} of $f$ if there exists $f_M > \\epsilon_0$ such that $\\sup_{x\\in M} f(x) = f_M$ and $M$ is a CC of the level set ${\\mathcal{X}^{f_M - \\epsilon_0}} := \\{ x : f(x) \\ge f_M - \\epsilon_0 \\}$.\n\\end{definition}\n\nWe require the following Assumption~\\ref{assumption2} on $\\epsilon_0$-modal sets. Note that under Assumption~\\ref{assumption-main} on modal-sets, Assumption~\\ref{assumption2} on $\\epsilon_0$-modal sets will hold for $\\epsilon_0$ sufficiently small.\n\n\\begin{assumption}\\label{assumption2}\nThe $\\epsilon_0$-modal sets are on the interior of $\\mathcal{X}$ and $f_M \\ge 2\\epsilon_0$ for all $\\epsilon_0$-modal sets $M$. \n\\end{assumption}\n\n\\begin{remark}\nSince each $\\epsilon_0$-modal set contains a modal-set, it follows that the number of $\\epsilon_0$-modal sets is finite.\n\\end{remark}\n\nThe following extends Proposition~\\ref{prop:main-assumptions} to show the additional properties of the regions around the $\\epsilon_0$-modal sets necessary in our analysis. The proof is in Appendix~\\ref{supportinglemmas}.\n\n\\begin{proposition}[Extends Proposition~\\ref{prop:main-assumptions}] \\label{prop:main-assumptions-general}\nFor any $\\epsilon_0$-modal set $M$, there exists $\\lambda_M, A_M, r_M, l_M, u_M, r_s, S_M$ such that the following holds. $A_M$ is a CC of $\\mathcal{X}^{\\lambda_M} := \\{x : f(x) \\ge \\lambda_M\\}$ containing $M$ which satisifies the following.\n\\begin{itemize} \n\\item \\emph{$A_M$ isolates $M$ by a valley}: $A_M$ does not intersect any other $\\epsilon_0$-modal sets and $A_M$ and $\\mathcal{X}^{\\lambda_M} \\backslash A_M$ are $r_s$-separated by $S_M$ with $r_s>0$ where $r_s$ does not depend on $M$. \n\n\\item \\emph{$A_M$ is full-dimensional}: $A_M$ contains an envelope $B(M, r_M)$ of $M$, with $r_M>0$.  \n\\item \\emph{$f$ is smooth around some maximum modal-set in $M$}: There exists modal-set $M_0 \\subseteq M$ such that $f$ has density $f_M$ on $M_0$ and $f_M - f(x) \\le u_M(d(x, M_0))$ for $x \\in B(M_0, r_M)$\n\\item \\emph{$f$ is both \\emph{smooth} and has \\emph{curvature} around $M$}: $u_M$ and $l_M$ are increasing continuous functions on $[0, r_M]$, $u_M(0) = l_M(0) = 0$ and $u_M(r), l_M(r) > 0$ for $r > 0$, and\n\\begin{align*}\nl_M(d(x, M)) \\le f_M -  \\epsilon_0 - f(x) \\le u_M(d(x, M))  \\forall x \\in B(M, r_M).\n\\end{align*}\n\n\\end{itemize}\n\\end{proposition}\n\n\n\n\\begin{algorithm}[tb]\n   \\caption{M-cores (estimating $\\epsilon_0$-modal-sets)}\n   \\label{alg:epsilonmodalset}\n\\begin{algorithmic}\n   \\STATE Initialize $\\widehat{\\mathcal{M}}:= \\emptyset$. Define $\\beta_k = 4\\frac{C_{\\delta, n}}{\\sqrt{k}}$. \n   \\STATE Sort the $X_i$'s in descending order of $f_k$ values. \n   \\FOR{$i=1$ {\\bfseries to} $n$}\n   \\STATE Define $\\lambda := f_k(X_i)$.\n   \\STATE Let $A$ be the CC of $G(\\lambda - 9\\beta_k \\lambda - \\epsilon_0 - \\tilde{\\epsilon})$ that contains $X_i$.\n   \\IF{$A$ is disjoint from all cluster-cores in $\\widehat{\\mathcal{M}}$}\n   \\STATE Add $\\widehat{M} := \\{ x \\in A : f_k(x) > \\lambda - \\beta_k \\lambda  - \\epsilon_0 \\}$ to\n    $\\widehat{\\mathcal{M}}$. \n    \\ENDIF\n   \\ENDFOR\n   \\STATE \\textbf{return} $\\widehat{\\mathcal{M}}$. \n\n\\end{algorithmic}\n\\end{algorithm}\n\n\nNext we give admissibility conditions for $\\epsilon_0$-modal sets. The only changes (compared to admissibility conditions for modal-sets) are the constant factors. In particular, when $\\epsilon_0=0$ and $\\tilde\\epsilon = 0$ it is the admissibility conditions for modal-sets. As discussed in the main text, a larger $\\tilde\\epsilon$ value will prune more aggressively at the cost of requiring a larger number of samples. Furthermore, it is implicit below that $\\tilde\\epsilon < l_M(\\min\\{r_M, r_s\\}\/2)$. This ensures that we don't prune too aggressively that the estimated $\\epsilon_0$-modal sets merge together. \n\n\n\\begin{definition} $k$ is {\\bf admissible} for an $\\epsilon_0$-modal set $M$ if (letting $u_M^{-1}, l_M^{-1}$ be the inverses of $u_M, l_M$)\n\\begin{align*}\n\\max \\left\\{ \\left(\\frac{24C_{\\delta, n} (\\sup_{x \\in \\mathcal{X}} f(x) + \\epsilon_0)}{l_M(\\min\\{r_M, r_s\\}\/2) - \\tilde\\epsilon}  \\right)^2, 2^{7 + d} C_{\\delta, n}^2 \\right\\} \\\\\n\\le k \\le \\frac{v_d \\cdot (f_M - \\epsilon_0)}{2^{2+2d}} \\left(u_M^{-1} \\left ( \\frac{C_{\\delta, n} (f_M - \\epsilon_0)}{2\\sqrt{k}}\\right) \\right)^d \\cdot n.\n\\end{align*}\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\\section{Supporting lemmas and propositions}\\label{supportinglemmas}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:main-assumptions-general}]\nLet $M$ be an $\\epsilon_0$-modal set with maximum density $f_M$ and minimum density $f_M - \\epsilon_0$ (i.e. $f_M - \\epsilon_0 \\le f(x) \\le f_M$ for $x\\in M$). \nDefine ${\\mathcal{X}^{\\lambda}} := \\{ x : f(x) \\ge \\lambda\\}$.\nLet $A_1,...,A_m$ be the CCs of ${\\mathcal{X}^{f_M - \\epsilon_0}}$ (there are a finite number of CCs since each CC contains at least one modal-set and the number of modal-sets is finite). \nDefine $r_{\\text{min}} :=  \\min_{A_i \\neq A_j} \\inf_{x \\in A_i, x' \\in A_j} |x - x'|$, which is the minimum distance between pairs of points in different CCs.\nNext, define the one-sided Hausdorff distance for closed sets $A, B$: $d_{H'}(A, B) := \\max_{x \\in A} \\min_{x \\in B} |x - y|$. Then consider\n$g(t) := d_{H'}({\\mathcal{X}^{f_M - \\epsilon_0 - t}}, {\\mathcal{X}^{f_M - \\epsilon_0}})$.\n\nSince $f$ is continuous and has a finite number of modal-sets, $g$ has a finite number of points of discontinuity (i.e. when $f_M - \\epsilon_0 - t$ is the density of some modal-set) and we have $g(t) \\rightarrow 0$ as $t \\rightarrow 0$. \nThus, there exists $0 < \\lambda_M < f_M - \\epsilon_0$ such that $g(f_M - \\epsilon_0 - \\lambda_M) < \\frac{1}{4}r_{\\text{min}}$ and there are no modal-sets or $\\epsilon_0$-modal sets with minimum density in $[\\lambda_M, f_M - \\epsilon_0)$.  \nFor each $A_i$, there exists exactly one CC of $\\mathcal{X}^{\\lambda_M}$, $A_i'$, such that\n$A_i \\subset A_i'$. Since $g(f_M - \\epsilon_0 - \\lambda_M) < \\frac{1}{4}r_{\\text{min}}$, it follows that $A_i' \\subseteq B(A_i, \\frac{1}{4}r_{\\text{min}})$. Thus, the $A_i'$'s are pairwise separated by distance at least $\\frac{1}{2}r_{\\text{min}}$. Moreover, there are no other CCs in ${\\mathcal{X}^{f_M - \\epsilon_0}}$ because there are no modal-sets with density in $[\\lambda_M, f_M - \\epsilon_0)$. \n\nThen, let $A_M$ be the CC of $\\mathcal{X}^{\\lambda_M}$ containing $M$. Then $A_M$ contains no other $\\epsilon_0$-modal sets and it is $\\frac{1}{5}r_{\\text{min}}$-separated by $\\mathcal{X}^{\\lambda_M} \\backslash M$ by some set $S_M$ (i.e. take $S_M := \\{x : d(x, A_M) = \\frac{1}{5} r_{\\text{min}}\\}$). Since there is a finite number of modal-sets, it suffices to take $r_s$ to be the minimum of the corresponding $\\frac{1}{5}r_{\\text{min}}$ for each $\\epsilon_0$-modal set. This resolves the first part of the proposition.\n\nLet $h(r) := \\inf_{x \\in B(M, r)} f(x)$. Since $f$ is continuous, $h$ is continuous and decreasing with $h(0) = f_M - \\epsilon_0 > \\lambda_M$. Take $r_M > 0$ sufficiently small so that $h(r_M) > \\lambda_M$. This resolves the second part of the proposition. \n\n\nTake $M_0$ to be some modal-set with density $f_M$ in $M$. One must exist since $M$ has local-maxima at level $f_M$.\nFor each $r$, let $u_M(r) := \\max\\{f_M - \\epsilon_0 - \\inf_{x \\in B(M, r)} f(x), f_M - \\inf_{x \\in B(M_0, r)} f(x) \\}$. Then, we have $f_M - f(x) \\le u_M(d(x, M_0))$ and $f_M - \\epsilon_0 - f(x) \\le u_M(d(x, M))$. Clearly $u_M$ is increasing on $[0, r_M]$ with $u_M(0) = 0$ and continuous since $f$ is continuous.  If $u_M$ is not strictly increasing then we can replace it with a strictly increasing continuous function while still having $u_M(r) \\rightarrow 0$ as $r \\rightarrow 0$ (i.e. by adding an appropriate strictly increasing continuous function). This resolves the third part of the proposition and the upper bound in the fourth part of the proposition. \n\nNow, define $g_M(t) := d({\\mathcal{X}^{f_M - \\epsilon_0 - t}} \\cap {A_M}, M)$ \nfor $t \\in  [0, \\frac{1}{2} (f_M - \\epsilon_0 - \\lambda_M)]$. \nThen, $g_M$ is continuous, $g_M(0) = 0$ and is strictly increasing. \nDefine $l_M$ to be the inverse of $g_M$. Clearly $l_M$ is continuous, strictly increasing, and $l_M(r) \\rightarrow 0$ as $r \\rightarrow 0$. From the definition of $g_M$, it follows that for $x \\in B(M, r_M)$, $f_M - \\epsilon_0- f(x) \\ge l_M(d(x, M))$\nas desired.\n\\end{proof}\n\nWe need the following result giving guarantees on the empirical balls.\n\\begin{lemma}[\\citep{CD10}] \\label{ball_bounds} \nPick $0 < \\delta < 1$. Assume that $k \\ge d \\log n$. Then with probability at least $1 - \\delta$, for every ball $B \\subset \\mathbb{R}^d$ we have\n\\begin{align*}\n\\mathcal{F}(B) \\ge C_{\\delta, n} \\frac{\\sqrt{d \\log n}}{n} &\\Rightarrow \\mathcal{F}_n(B) > 0\\\\\n\\mathcal{F}(B) \\ge \\frac{k}{n} + C_{\\delta, n} \\frac{\\sqrt{k}}{n} &\\Rightarrow \\mathcal{F}_n(B) \\ge \\frac{k}{n} \\\\\n\\mathcal{F}(B) \\le \\frac{k}{n} - C_{\\delta, n}\\frac{\\sqrt{k}}{n} &\\Rightarrow \\mathcal{F}_n(B) < \\frac{k}{n}.\n\\end{align*}\n\\end{lemma}\n\n\n\nLemma~\\ref{fk_bounds} of \\cite{dasgupta2014optimal} establish convergence rates for $f_k$. \n\n\\begin{definition}\\label{rhat} For $x \\in \\mathbb{R}^d$ and $\\epsilon > 0$, define \n$\\hat{r}(\\epsilon, x):=\\sup\\left\\{r : \\sup_{x' \\in B(x, r)} f(x') - f(x) \\le \\epsilon \\right\\}$  and\n$\\check{r}(\\epsilon, x):=\\sup\\left\\{r : \\sup_{x' \\in B(x, r)} f(x) - f(x') \\le \\epsilon \\right\\}$.\n\\end{definition}\n\n\\begin{lemma}[Bounds on $f_k$]\\label{fk_bounds}  Suppose that $\\frac{C_{\\delta, n}}{\\sqrt{k}} < \\frac{1}{2}$. Then the follow two statements each hold with probability at least $1 - \\delta$: \n\\begin{align*}\nf_k(x) < \\left(1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}} \\right)(f(x) + \\epsilon),\n\\end{align*}\nfor all $x\\in \\mathbb{R}^d$ and all $\\epsilon > 0$ provided $k$ satisfies $v_d\\cdot \\hat{r}(\\epsilon, x)^d \\cdot (f(x) + \\epsilon) \\ge \\frac{k}{n} - C_{\\delta, n}\\frac{\\sqrt{k}}{n}$.\n\\begin{align*}\nf_k(x) \\ge \\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}} \\right)(f(x) - \\epsilon),\n\\end{align*}\nfor all $x\\in \\mathbb{R}^d$ and all $\\epsilon > 0$  provided $k$ satisfies $v_d\\cdot \\check{r}(\\epsilon, x)^d \\cdot (f(x) - \\epsilon) \\ge \\frac{k}{n} + C_{\\delta, n}\\frac{\\sqrt{k}}{n}$. \n\\end{lemma}\n\n\n\\begin{lemma}[Extends Lemma~\\ref{r_n_upper_bound}] \\label{r_n_upper_bound_general}(Upper bound on $r_n$)  Let $M$ be an $\\epsilon_0$-modal set with maximum density $f_M$ and suppose that $k$ is admissible. With probability at least $1 - \\delta$,\n\\begin{align*}\nr_n(M) \\le  \\left(\\frac{2C_{\\delta, n} \\sqrt{d \\log n}}{n\\cdot v_d\\cdot (f_M - \\epsilon_0)}\\right)^{1\/d}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{r_n_upper_bound_general}]\nDefine $r_0 := \\left(\\frac{2C_{\\delta, n} \\sqrt{d \\log n}}{nv_d\\cdot (f_M - \\epsilon_0)}\\right)^{1\/d}$ and $r := (4k\/(nv_df_M))^{1\/d}$. Since $k$ is admissible, we have that $u_M(r_0) \\le u_M(r) \\le (f_M  - \\epsilon_0)\/ 2$. We have\n\\begin{align*}\n\\mathcal{F}(B(x, r_0)) &\\ge v_d{r_0}^d(f_M -\\epsilon_0 - u_M(r_0)) \\ge  v_d{r_0}^d (f_M - \\epsilon_0)\/2 \n= \\frac{C_{\\delta, n} \\sqrt{d\\log n}}{n}.\n\\end{align*}\nBy Lemma~\\ref{ball_bounds}, this implies that $\\mathcal{F}_n (B(x, r_0)) > 0$ with probability at least $1 - \\delta$ and therefore we have $r_n(x) \\le r_0$.\n\\end{proof}\n\n\n\n\n\n\\section{Isolation Results} \\label{appendix:isolation}\n\n\n\nThe following extends Lemma~\\ref{isolation} to handle more general\n$\\epsilon_0$-modal sets and pruning parameter $\\tilde\\epsilon$. \n\n\\begin{lemma}[Extends Lemma~\\ref{isolation}] (Isolation) \\label{isolation_general} \nLet $M$ be an $\\epsilon_0$-modal set and $k$ be admissible for $M$.  Suppose $0 \\le \\tilde\\epsilon < l_M(\\min\\{r_M, r_s\\}\/2)$ and let $\\hat{x}_M := \\argmax_{x \\in \\mathcal{X}_M \\cap X_{[n]}} f_k(x)$. Then the following holds\nwith probability at least $1-5\\delta$: when processing sample point $\\hat{x}_M$ in Algorithm~\\ref{alg:epsilonmodalset} we will add $\\widehat{M}$ to $\\widehat{\\mathcal{M}}$ where  \n$\\widehat{M}$ does not contain points outside of $\\mathcal{X}_M$.\n\\end{lemma}\n\n\\begin{proof}  Define  $\\widehat{f}_M := f_k(\\hat{x}_M)$, $\\lambda = \\widehat{f}_M$ and $\\bar{r} := \\min\\{r_M, r_s\\} \/ 2$.\nIt suffices to show that (\\rm{i}) $\\mathcal{X} \\backslash \\mathcal{X}_M$ and $B(M, \\bar{r})$ are disconnected in $G(\\lambda -  9\\beta_k \\lambda - \\epsilon_0 - \\tilde{\\epsilon})$ and (\\rm{ii}) $\\hat{x}_M \\in B(M, \\bar{r})$. \n\nIn order to show (\\rm{i}), we first show that $G(\\lambda - 9\\beta_k \\lambda - \\epsilon_0 - \\tilde{\\epsilon})$ contains no points from $B(S_M, r_s\/2)$ and no points from $\\mathcal{X}_M \\backslash B(M, \\bar{r})$. Then, all that will be left is showing that there are no edges between $B(M, \\bar{r})$ and $\\mathcal{X} \\backslash \\mathcal{X}_M$.\n   \n\nWe first prove bounds on $f_k$ that will help us show (\\rm{i}) and (\\rm{ii}). Let $\\bar{F} := f_M - \\epsilon_0 - l_M(\\bar{r}\/2)$. Then for all $x \\in \\mathcal{X}_M \\backslash B(M, \\bar{r})$, we have $\\hat{r}(\\bar{F} - f(x), x) \\ge \\bar{r}\/2$. Thus the conditions for Lemma~\\ref{fk_bounds} are satisfied by the admissibility of $k$ and hence $f_k(x) < \\left(1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}} \\right) \\bar{F}$. Now,\n\\begin{align*}\n\\sup_{x \\in \\mathcal{X}_M\\backslash B(M, \\bar{r})} f_k(x) &< (1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}}) \\bar{F} = (1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}}) (f_M - \\epsilon_0 - l_M(\\bar{r}\/2))\\\\\n &\\le (1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}})^3 \\widehat{f}_M - (1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}}) \\cdot (\\epsilon_0 + l_M(\\bar{r}\/2)) \\le \\lambda-  9\\beta_k \\lambda - \\epsilon_0 - \\tilde{\\epsilon},\n\\end{align*}\nwhere the second inequality holds by using Lemma~\\ref{fk_bounds} as follows. Choose $x \\in M_0$ and $\\epsilon = \\frac{C_{\\delta, n}}{2\\sqrt{k}} f_M$. Then $\\check{r}(\\epsilon, x) \\ge u^{-1}(\\epsilon)$. The conditions for Lemma~\\ref{fk_bounds} hold by the admissibility of $k$ and thus $\\widehat{f}_M \\ge f_k(x) \\ge (1 - C_{\\delta, n}\/\\sqrt{k})^2 f_M$. Furthermore it follows from Lemma~\\ref{fk_bounds} that $\\widehat{f}_M <  (1 + 2C_{\\delta, n}\/\\sqrt{k}) f_M$; \ncombine this admissibility of $k$ to obtain the last inequality. Finally, from the above, we also have $\\sup_{x \\in \\mathcal{X}_M\\backslash B(M, \\bar{r})} f_k(x) < \\widehat{f}_M$, implying (\\rm{ii}).\n\n\\noindent Next, if $x \\in B(S_M, r_s\/2)$, then $\\hat{r}(\\bar{F} - f(x) , x) \\ge \\bar{r}\/2$ and the same holds for $B(S_M, r_s\/2)$:\n\\begin{align*}\n\\sup_{x \\in B(S_M, r_s\/2)} f_k(x) < \n\\lambda- 9\\beta_k \\lambda - \\epsilon_0 -\\tilde{\\epsilon}.\n\\end{align*}\nThus, $G(\\lambda- 9\\beta_k \\lambda - \\epsilon_0 - \\tilde{\\epsilon})$ contains no point from $B(S_M, r_s\/2)$ and no point from $\\mathcal{X}_M \\backslash B(M, \\bar{r})$. \n\n\\noindent All that remains is showing that there is no edge between $B(M, \\bar{r})$ and $\\mathcal{X} \\backslash \\mathcal{X}_M$. It suffices to show that any such edge will have length less than $r_s$ since $B(S_M, r_s\/2)$ separates them by a width of $r_s$. We have for all $x \\in B(M, \\bar{r})$, \n\\begin{align*}\\mathcal{F}(B(x, \\bar{r})) \\ge v_d\\bar{r}^d\\inf_{x' \\in B(x, 2\\bar{r})} f(x') \\ge \\frac{k}{n} + C_{\\delta, n}\\frac{\\sqrt{k}}{n}.\n\\end{align*}\nThus by Lemma~\\ref{ball_bounds}, we have $r_k(x) \\le \\bar{r} < r_s$, establishing (\\rm{i}).\n\n\\end{proof}\n\n\n\n\\section{Integrality Results} \\label{appendix:integrality}\n\nThe goal is to show that the $\\widehat{M} \\in \\widehat{\\mathcal{M}}$ refered to above contains $B(M, r_n(M))$. We give a condition under which $B(M, r_n(M)) \\cap X_{[n]}$ would be connected in $G(\\lambda)$ for some $\\lambda$. It is adapted from arguments in Theorem V.2 in \\cite{CDKvL14}.\n\n\n\\begin{lemma}\\label{connectedness} (Connectedness)\nLet $M$ be an $\\epsilon_0$-modal set and $k$ be admissible for $M$. Then with probability at least $1 - \\delta$, $B(M, r_n(M)) \\cap X_{[n]}$ is connected in $G(\\lambda)$ if \n\\begin{align*}\n\\lambda \\le \\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)^2 (f_M - \\epsilon_0).\n\\end{align*}\n\\end{lemma}\n\n\n\\begin{proof}\nFor simplicity of notation, let $A := B(M, r_n(M))$. It suffices to prove the result for $\\lambda = (1 - \\frac{C_{\\delta, n}}{\\sqrt{k}})^2 (f_M - \\epsilon_0)$.\nDefine $r_{\\lambda} = (k\/(nv_d\\lambda))^{1\/d}$ and $r_o = (k\/(2nv_df_M))^{1\/d}$. \nFirst, we show that each $x \\in B(A, r_\\lambda)$, there is a sample point in $B(x, r_o)$. \nWe have for $x\\in B(A, r_\\lambda)$,\n\\begin{align*}\n\\mathcal{F}(B(x, r_o)) &\\ge v_d r_o^d \\inf_{x' \\in B(x, r_o + r_\\lambda )} f(x') \\ge v_d r_o^d(f_M - \\epsilon_0 - u_M(r_o + r_\\lambda + r_n(M)))  \\\\\n&\\ge v_d r_o^d (f_M - \\epsilon_0) \\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}}\\right) \\ge C_{\\delta, n} \\frac{\\sqrt{d \\log n}}{n}.\n\\end{align*}\nThus by Lemma~\\ref{ball_bounds} we have that with probability at least $1 - \\delta$, $B(x, r_o)$ contains a sample uniformly over $x \\in B(A, r_\\lambda)$. \n\n \\noindent Now, let $x$ and $x'$ be two points in $A\\cap X_{[n]}$. We now show that there exists $x = x_0,x_1,...,x_p = x'$ such that $||x_i - x_{i+1}|| < r_o$ and $x_i \\in B(A, r_o)$. Note that since $A$ is connected and the density in $B(A, r_o + r_\\lambda)$ is lower bounded by a positive quantity, then for arbitrary $\\gamma \\in (0, 1)$, we can choose $x = z_0, z_1,...,z_p = x'$ where $||z_{i+1} - z_i|| \\le \\gamma r_o$. Next, choose $\\gamma$ sufficiently small such that \n\\begin{align*}\nv_d\\left( \\frac{(1-\\gamma)r_o}{2}\\right)^d\\gamma \\ge \\frac{C_{\\delta, n}\\sqrt{d\\log n}}{n},\n\\end{align*}\nthen there exists a sample point $x_i$ in $B(z_i, (1-\\gamma)r_o\/2)$. Moreover we obtain that \n\\begin{align*}\n||x_{i+1} - x_i|| &\\le ||x_{i+1} - z_{i+1}|| + ||z_{i+1} - z_i|| + ||z_i -x_i|| \\le r_o.\n\\end{align*} \n\n\\noindent All that remains is to show $(x_i, x_{i+1}) \\in G(\\lambda)$. We see that $x_i \\in B(A, r_o)$. However, for each $x \\in B(A, r_o)$, we have \n\\begin{align*}\n\\mathcal{F}(B(x, r_\\lambda)) &\\ge v_dr_\\lambda^d \\inf_{x' \\in B(x, r_o + r_\\lambda)} f(x') \n\\ge v_d r_\\lambda^d (f_M - \\epsilon_0) \\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)\n\\ge \\frac{k}{n} + \\frac{C_{\\delta, n} \\sqrt{k}}{n}.\n\\end{align*}\n Thus $r_k(x_i) \\le r_\\lambda$ for all $i$. Therefore, $x_i \\in G(\\lambda)$ for all $x_i$. Finally, $||x_{i+1} - x_i|| \\le r_o \\le \\min \\{ r_k(x_i), r_k(x_{i+1})\\}$ and thus $(x_i, x_{i+1}) \\in G(\\lambda)$. Therefore, $A \\cap X_{[n]}$ is connected in $G(\\lambda)$, as desired. \n \\end{proof}\n\n\n\n\nThe following extends Lemma~\\ref{integrality} handle more general $\\epsilon_0$-modal sets. \n\\begin{lemma}[Extends Lemma~\\ref{integrality}] (Integrality) \\label{integrality_general} \nLet $M$ be an $\\epsilon_0$-modal set with density $f_M$, and suppose $k$ is admissible for $M$. Let $\\hat{x}_M := \\argmax_{x \\in \\mathcal{X}_M \\cap X_{[n]}} f_k(x)$. Then the following holds with probability at least $1 - 3\\delta$. When processing sample point $\\hat{x}_M$ in Algorithm~\\ref{alg:modalset}, if we add $\\widehat{M}$ to $\\widehat{\\mathcal{M}}$, then $B(M, r_n(M)) \\cap X_{[n]} \\subseteq \\widehat{M}$.\n\\end{lemma}\n\n\\begin{proof}  Define $\\widehat{f}_M := f_k(\\hat{x}_M)$ and $\\lambda := \\widehat{f}_M$. It suffices to show that $B(M, r_n(M)) \\cap X_{[n]}$ is connected in $G(\\lambda - 9\\beta_k \\lambda - \\tilde{\\epsilon})$. By Lemma~\\ref{connectedness}, $B(M, r_n(M)) \\cap X_{[n]}$ is connected in $G(\\lambda_0)$ when $\\lambda_0 \\le (1 - \\frac{C_{\\delta, n}}{\\sqrt{k}})^2 (f_M - \\epsilon_0) $. Indeed, we have that\n\\begin{align*}\n\\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)^2 (f_M  - \\epsilon_0)\n&\\ge \\widehat{f}_M \\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)^2 \/ \\left(1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}}\\right) -  \\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)^2 \\epsilon_0 \\\\\n&\\ge \\lambda -  \\beta_k \\lambda - \\epsilon_0 \\ge \\lambda -  9\\beta_k \\lambda - \\epsilon_0 -  \\tilde{\\epsilon},\n\\end{align*}\nwhere the first inequality follows from Lemma~\\ref{fk_bounds},\nas desired.\n\\end{proof}\n\n\n\\section{Theorem~\\ref{theo:main}} \\label{appendix:theomain}\n\nCombining the  isolation and integrality, we obtain the following extention of Corollary~\\ref{identification}.\n\n\n\\begin{corollary}[Extends Corollary~\\ref{identification}] (Identification) \\label{identification_general}\nSuppose we have the assumptions of Lemmas~\\ref{isolation_general} and~\\ref{integrality_general} for $\\epsilon_0$-modal set $M$. Define $\\widehat{f}_M := \\max_{x \\in \\mathcal{X}_M \\cap X_{[n]}} f_k(x)$ and $\\lambda := \\widehat{f}_M$. With probability at least $1 - 5\\delta$, there exists $\\widehat{M} \\in \\widehat{\\mathcal{M}}$ such that $B(M, r_n(M)) \\cap X_{[n]} \\subseteq  \\widehat{M} \\subseteq \\{ x \\in \\mathcal{X}_M \\cap X_{[n]} : f_k(x) \\ge \\lambda- \\beta_k \\lambda  - \\epsilon_0 \\}$\n\\end{corollary}\n\n\\begin{proof}\nBy Lemma~\\ref{isolation_general}, there exists $\\widehat{M} \\in  \\widehat{\\mathcal{M}}$ which contains only points in $\\mathcal{X}_M$ with maximum $f_k$ value of $\\widehat{f}_M$. Thus, we have $\\widehat{M} \\subseteq \\{ x \\in \\mathcal{X}_M \\cap X_{[n]} : f_k(x) \\ge \\widehat{f}_M - \\beta_k \\widehat{f}_M - \\epsilon_0 \\}$. By Lemma~\\ref{integrality_general}, $B(M, r_n(M)) \\cap X_{[n]} \\subseteq  \\widehat{M}$.\n\\end{proof}\n\n\nThe following extends Theorem~\\ref{theo:main} to handle more general $\\epsilon_0$-modal sets and pruning parameter $\\tilde{\\epsilon}$. \n\n\\begin{theorem}[Extends Theorem~\\ref{theo:main}] \\label{theo:main_general}\nLet $\\delta > 0$ and $M$ be an $\\epsilon_0$-modal set. Suppose $k$ is admissible for $M$ and $0 \\le \\tilde\\epsilon < l_M(\\min\\{r_M, r_s\\}\/2)$. Then with probability at least $1 - 6\\delta$, there exists $\\widehat{M} \\in \\widehat{\\mathcal{M}}$ such that \n \\begin{align*}\n d(M, \\widehat{M}) \\le l_M^{-1}\\left(\\frac{8C_{\\delta,n }}{\\sqrt{k}}f_M\\right), \\end{align*}\n which goes to $0$ as $C_{\\delta, n}\/\\sqrt{k} \\rightarrow 0$.\n\\end{theorem}\n\n\\begin{proof} Define $\\tilde{r} = l_M^{-1}\\left(\\frac{8C_{\\delta,n }}{\\sqrt{k}}f_M\\right)$. There are two directions to show: $\\max_{x \\in \\widehat{M}} d(x, M) \\le \\tilde{r}$ and $\\sup_{x \\in M} d(x, \\widehat{M}) \\le \\tilde{r}$ with probability at least $1 - \\delta$.\n\nWe first show $\\max_{x \\in \\widehat{M}} d(x, M) \\le \\tilde{r}$.\nBy Corollary~\\ref{identification_general} we have $\\widehat{M} \\in \\widehat{\\mathcal{M}}$ such that $\\widehat{M} \\subseteq \\{ x \\in \\mathcal{X}_M : f_k(x) \\ge \\widehat{f}_M - \\beta_k \\widehat{f}_M - \\epsilon_0 \\}$ where $\\widehat{f}_M := \\max_{x \\in \\mathcal{X}_M \\cap X_{[n]}} f_k(x)$. \nHence, it suffices to show \n\\begin{align}\\label{consistency_to_show}\n\\inf_{x \\in B(M_0, r_n(M))} f_k(x) \\ge \\sup_{\\mathcal{X}_M \\backslash B(M, \\tilde{r})} f_k(x) + \\beta_k \\widehat{f}_M + \\epsilon_0.\n\\end{align}\nDefine $r := (4\/f_Mv_d)^{1\/d}(k\/n)^{1\/d}$. For any $x \\in B(M_0, r + r_n(M))$, $f(x) \\ge f_M - u_M(r + r_n(M)) := \\check{F}$. Thus, for any $x \\in B(M_0, r_n(M))$ we can let $\\epsilon = f(x) - \\check{F}$ and thus $\\check{r}(\\epsilon, x) \\ge r$ and hence the conditions for Lemma~\\ref{fk_bounds} are satisfied. Therefore, with probability at least $1-\\delta$,\n\\begin{align}\\label{consistency_bound1}\n \\inf_{x \\in B(M_0, r_n(M))} f_k(x) \\ge \\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)(f_M - u_M(r +r_n(M))).\n\\end{align}\nFor any $x\\in \\mathcal{X}_M \\backslash B(M, \\tilde{r}\/2)$, $f(x) \\le f_M - \\epsilon_0 - l_M(\\tilde{r}\/2) := \\hat{F}$. Now, for any $x \\in \\mathcal{X} \\backslash B(M, \\tilde{r})$, let $\\epsilon := \\hat{F} - f(x)$. We have $\\hat{r}(\\epsilon, x) \\ge \\tilde{r}\/2 = l^{-1}_M (8C_{\\delta, n}\/\\sqrt{k})\/2 \\ge l^{-1}_M (u_M(2r)) \/ 2 \\ge r$ (since $l_M$ is increasing and $l_M \\le u_M$) and thus the conditions for Lemma~\\ref{fk_bounds} hold. Hence, with probability at least $1 - \\delta$,\n\\begin{align}\\label{consistency_bound2}\n\\sup_{x \\in \\mathcal{X}_M \\backslash B(M, \\tilde{r})} f_k(x) \\le \\left(1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)(f_M - \\epsilon_0 -  l_M(\\tilde{r})).\n\\end{align}\nThus, by (\\ref{consistency_bound1}) and (\\ref{consistency_bound2}) applied to (\\ref{consistency_to_show}) it suffices to show that\n\\begin{align}\\label{consistency_to_show_2}\n&\\left(1 - \\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)(f_M - u_M(r + r_n(M))) \n\\ge \\left(1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)(f_M - \\epsilon_0 - l_M(\\tilde{r})) + \\beta_k \\widehat{f}_M + \\epsilon_0,\n\\end{align}\nwhich holds when \n\\begin{align}\\label{consistency_to_show_3}\nl_M(\\tilde{r}) \\ge u_M(r + r_n(M)) + \\frac{3C_{\\delta, n}}{\\sqrt{k}}f_M + \\beta_k \\widehat{f}_M.\n\\end{align}\nThe admissibility of $k$ ensures that $r_n(M) \\le r \\le r_M\/2$ so that the regions of $\\mathcal{X}$ we are dealing with in this proof are confined within $B(M_0, r_M)$ and $B(M, r_M) \\backslash M$.\n\nBy the admissibility of $k$, $u_M(2r) \\le \\frac{C_{\\delta, n}}{2\\sqrt{k}}f_M$. This gives\n\\begin{align*}\nl_M(\\tilde{r}) &= \\frac{8C_{\\delta,n }}{\\sqrt{k}}f_M \\ge u_M(2r) + \\frac{15C_{\\delta,n }}{2\\sqrt{k}}f_M \n\\ge u_M(r + r_n(M)) + \\frac{3C_{\\delta, n}}{\\sqrt{k}} f_M + \\beta_k \\widehat{f}_M,\n\\end{align*}\nwhere the second inequality holds since $C_{\\delta, n}\/\\sqrt{k} < 1\/16$, $u$ is increasing, $r \\ge r_n(M)$, and $\\widehat{f}_M \\le \\left(1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}}\\right)f_M$ by Lemma~\\ref{fk_bounds}. Thus, showing (\\ref{consistency_to_show_3}), as desired.\n\nThis shows one direction of the Hausdorff bound. We now show the other direction, that $\\sup_{x \\in M} d(x, M) \\le \\tilde{r}$.\n\nIt suffices to show for each point $x \\in M$ that the distance to the closest sample point $r_n(x) \\le \\tilde{r}$ since $\\widehat{M}$ contains these sample points by Corollary~\\ref{identification_general}. However, by Lemma~\\ref{r_n_upper_bound_general} and the admissibility of $k$, $r_n(x) \\le \\tilde{r}$  as desired.\n\\end{proof}\n\n\n\n\n\n\\section{Theorem~\\ref{pruning}} \\label{appendix:pruning}\n\nWe need the following Lemma~\\ref{connectedness_pruning} which gives guarantees us that \ngiven points in separate CCs of the pruned graph, these points will also be in separate CCs of $f$ at a nearby level. \\cite{CDKvL14} gives a result for a different graph and the proof can be  adapted to give the same result for our graph (but slightly different assumptions on $k$).\n\n\\begin{lemma}[Separation of level sets under pruning, \\cite{CDKvL14}]  \\label{connectedness_pruning} \nFix $\\epsilon > 0$ and let $r(\\epsilon) := \\inf_{x \\in \\mathbb{R}^d} \\min \\{\\hat{r}(\\epsilon, x), \\check{r}(\\epsilon, x) \\}$. Define $\\Lambda := \\max_{x \\in \\mathbb{R}^d} f(x)$ and assume $\\tilde\\epsilon_0 \\ge 2 \\epsilon + \\beta_k(\\lambda_f + \\epsilon)$ and let $\\tilde{G}(\\lambda)$ be the graph with vertices in $G(\\lambda)$ and edges between pairs of vertices if they are connected in $G(\\lambda - \\tilde\\epsilon_0)$. Then the following holds with probability at least $1 - \\delta$.\\\\\n\nLet $\\tilde{A}_1$ and $\\tilde{A}_2$ denote two disconnected sets of points $\\tilde{G}(\\lambda)$. Define $\\lambda_f := \\inf_{x \\in \\tilde{A}_1 \\cup \\tilde{A}_2}f(x)$.  Then $\\tilde{A}_1$ and $\\tilde{A}_2$ are disconnected in the level set $\\{ x \\in \\mathcal{X} : f(x) \\ge \\lambda_f\\}$ if $k$ satisfies\n\\begin{align*}\nv_d (r(\\epsilon) \/ 2) ^d (\\lambda_f - \\epsilon) \\ge \\frac{k}{n} + C_{\\delta, n}\\frac{\\sqrt{k}}{n}\n\\end{align*}\nand \n\\begin{align*}\nk \\ge \\max \\{ 8 \\Lambda C_{\\delta, n}^2\/(\\lambda_f - \\epsilon), 2^{d + 7} C_{\\delta, n}^2 \\}.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nWe prove the contrapositive. Let $A$ be a CC of $\\{x \\in \\mathcal{X} : f(x) \\ge \\lambda_f \\}$ with $\\lambda_f = \\min_{x \\in A \\cap X_{[n]}} f(x)$. Then it suffices to show $A \\cap X_{[n]}$ is connected in $G(\\lambda')$ for $\\lambda' := \\min_{x \\in A \\cap X_{[n]}} f_k(x) - \\tilde\\epsilon_0$. \n\nWe first show $A \\cap X_{[n]}$ is connected in $G(\\lambda)$ for $\\lambda = (\\lambda_f - \\epsilon) \/ (1 + C_{\\delta, n}\/\\sqrt{k})$ and all that will remain is showing $\\lambda' \\le \\lambda$.\n\nDefine $r_o := (k\/(2nv_df_M))^{1\/d}$ and $r_\\lambda := (k\/(nv_d\\lambda))^{1\/d}$. Then from the first assumption on $k$, it follows that $r_\\lambda \\le r(\\epsilon)\/2$. Now for each $x \\in B(A, r_\\lambda)$, we have \n\\begin{align*}\n\\mathcal{F}(B(x, r_o)) \\ge v_d r_o^d \\inf_{x' \\in B(x, r_o + r_\\lambda )} f(x') \n\\ge v_d r_o^d(\\lambda_f - \\epsilon) \\ge C_{\\delta, n} \\frac{\\sqrt{d \\log n}}{n}.\n\\end{align*}\nThus, by Lemma~\\ref{ball_bounds} we have with probability at least $1-\\delta$ that $B(x, r_0)$ contains a sample point.\n\nNow, in the same way shown as in Lemma~\\ref{connectedness}, we have the following. If $x$ and $x'$ be two points in $A\\cap X_{[n]}$ then there exists $x = x_0,x_1,...,x_p = x'$ such that $||x_i - x_{i+1}|| < r_o$ and $x_i \\in B(A, r_o)$. \n\nNext is showing $(x_i, x_{i+1}) \\in G(\\lambda)$. We see that $x_i \\in B(A, r_o)$. However, for each $x \\in B(A, r_o)$, we have \n\\begin{align*}\n\\mathcal{F}(B(x, r_\\lambda)) &\\ge v_dr_\\lambda^d \\inf_{x' \\in B(x, r_o + r_\\lambda)} f(x') \n\\ge v_d r_\\lambda^d (\\lambda_f - \\epsilon) \\ge \\frac{k}{n} + \\frac{C_{\\delta, n} \\sqrt{k}}{n}.\n\\end{align*}\n Thus $r_k(x_i) \\le r_\\lambda$ for all $i$. Therefore, $x_i \\in G(\\lambda)$ for all $x_i$. Finally, $||x_{i+1} - x_i|| \\le r_o \\le \\min \\{ r_k(x_i), r_k(x_{i+1})\\}$ and thus $(x_i, x_{i+1}) \\in G(\\lambda)$. Therefore, $A \\cap X_{[n]}$ is connected in $G(\\lambda)$.\n\nAll that remains is showing $\\lambda' \\le \\lambda$. We have\n\\begin{align*}\n\\lambda' = \\min_{x \\in A \\cap X_{[n]}} f_k(x) - \\tilde\\epsilon_0 \n\\le \\left(1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}} \\right) (\\lambda_f + \\epsilon) - \\tilde\\epsilon_0\n\\le \\lambda,\n\\end{align*}\nwhere the first inequality holds by Lemma~\\ref{fk_bounds}, and the second inequality holds from the assumption on $\\tilde\\epsilon_0$, as desired. \n\n\\end{proof}\n\n\nWe state the pruning result for more general choices of $\\tilde{\\epsilon}$. Its proof is standard and given here for completion. (See e.g. \\cite{dasgupta2014optimal}). \n\n\\begin{theorem}[Extends Theorem~\\ref{pruning}] \\label{pruning_general}\nLet $0< \\delta < 1$ and $\\tilde\\epsilon \\ge 0$. There exists $\\lambda_0 = \\lambda_0(n, k)$ such that the following holds with probability at least $1 - \\delta$. All $\\epsilon_0$-modal set estimates in $\\widehat{\\mathcal{M}}$ chosen at level $\\lambda \\ge \\lambda_0$ can be injectively mapped to $\\epsilon_0$-modal sets $\\braces{M:  \\lambda_M \\geq \\min_{\\{x\\in \\mathcal{X}_{[n]} : f_k(x) \\ge \\lambda - \\beta_k \\lambda \\}} f(x)}$, provided $k$ is admissible for all such $M$.  \n\nIn particular, if $f$ is H\\\"older-continuous, (i.e. $||f(x) - f(x')|| \\le c||x - x'||^\\alpha $ for some $0 < \\alpha\\le 1$, $c > 0$) and $\\tilde\\epsilon = 0$, \nthen $\\lambda_0 \\to 0$ as $n\\to \\infty$, provided \n$C_1 \\log n \\le k \\le C_2 n^{2\\alpha \/ (2\\alpha + d)}$, for some $C_1, C_2$ independent of $n$. \n\\end{theorem}\n\n\\begin{proof}\nDefine $r(\\epsilon) := \\inf_{x \\in \\mathbb{R}^d} \\min \\{\\hat{r}(\\epsilon, x), \\check{r}(\\epsilon, x) \\}$. Since $f$ is uniformly continuous, it follows that $r(0) = 0$, $r$ is increasing, and $r(\\epsilon) > 0$ for $\\epsilon > 0$. \n\nThus, there exists $\\tilde{\\lambda}_{n,k,\\tilde\\epsilon} > 0$ such that \n\\begin{align*}\n\\tilde{\\lambda}_{n, k, \\tilde\\epsilon} = \\frac{k}{n\\cdot v_d \\cdot r((8\\beta_k \\tilde{\\lambda}_{n,k,\\tilde\\epsilon} + \\tilde{\\epsilon})\/3)}.\n\\end{align*}\nDefine\n\\begin{align*}\n\\lambda_0 := \\max \\{\\tilde\\lambda_{n,k,\\tilde\\epsilon}, 32\\beta_k \\sup_{x \\in \\mathcal{X}} f(x) + 4 \\tilde\\epsilon \\}.\n\\end{align*}\n\nLet us identify each estimated $\\epsilon_0$-modal set $\\widehat{M}$ with the point $\\hat{x}_M := \\max_{x \\in \\widehat{M}} f_k(x)$. Let us call these points modal-points. Then it suffices to show that there is an injection from modal-points to the $\\epsilon_0$-modal sets. \n\nDefine $G'(\\lambda)$ to be the graph with vertices in $G(\\lambda - \\beta_k \\lambda)$ and edges between vertices if they are in the same CC of $G(\\lambda - 9\\beta_k \\lambda - \\tilde\\epsilon)$ and $X_{[n]}^\\lambda := \\{ x : f_k(x) \\ge \\lambda \\}$.  \nLet $\\tilde{A}_{i, \\lambda} := \\tilde{A}_i \\cap X_{[n]}^\\lambda$ for $i = 1,...,m$ to be the vertices of the CCs of $G'(\\lambda)$ which do not contain any modal-points chosen thus far as part of estimated modal-sets. \n\nFix level $\\lambda > 0$ such that $\\lambda_f := \\inf_{x \\in X_{[n]}^\\lambda} f(x) \\ge \\lambda_0\/2$. Then the conditions are satisified for Lemma~\\ref{connectedness_pruning} with $\\epsilon = (8\\beta_k\\lambda + \\tilde\\epsilon)\/3$. Suppose that $\\tilde{A}_{1,\\lambda},...,\\tilde{A}_{m,\\lambda}$ are in ascending order according to $\\lambda_{i, f} := \\min_{x \\in \\tilde{A}_{i, \\lambda}} f(x)$. Starting with $i = 1$, by Lemma~\\ref{connectedness_pruning}, $\\mathcal{X}^{\\lambda_1, f}$ can be partitioned into disconneced subsets $A_1$ and $\\mathcal{X}^{\\lambda_1, f} \\backslash A_1$ containing respectively $\\tilde{A}_{1, \\lambda}$ and $\\cup_{i=2}^m \\tilde{A}_{i, \\lambda}$. Assign the modal-point $\\argmax_{x \\in \\tilde{A}_{1, \\lambda}} f_k(x)$ to any $\\epsilon_0$-modal set in $A_1$. Repeat the same argument successively for any $\\tilde{A}_{i, \\lambda}$ and $\\cup_{j=i+1}^m \\tilde{A}_{j, \\lambda}$ until all modal-points are assigned to distinct $\\epsilon_0$-modal sets in disjoint sets $A_i$. \n\nNow by Lemma~\\ref{connectedness_pruning}, $\\mathcal{X}^{\\lambda_f}$ can be partitioned into disconnected subsets $A$ and $\\mathcal{X}^{\\lambda_f} \\backslash A$ containing respectively $\\tilde{A}_\\lambda := \\cup_{i=1}^m \\tilde{A}_{i, \\lambda}$ and $\\mathcal{X}_{[n]}^{\\lambda_f} \\backslash \\tilde{A}_{\\lambda}$. Thus, the modal-points in $\\tilde{A}_\\lambda$ were assigned to $\\epsilon_0$-modal sets in $A$. \n\n\\noindent Now we repeat the argument for all $\\lambda' > \\lambda$ to show that the modal-points  in $X_{[n]}^\\lambda \\backslash \\tilde{A}_\\lambda$ can be assigned to distinct $\\epsilon_0$-modal sets in $\\mathcal{X}^\\lambda \\backslash A^\\lambda$. (We have $\\lambda'_f := \\min_{x \\in X_{[n]}^{\\lambda' - \\beta_k\\lambda'} }f(x) \\ge \\lambda_f$).\n\n\\noindent Finally, it remains to show that $\\lambda \\ge \\lambda_0$ implies $\\lambda_f \\ge \\lambda_0 \/2$. We have $\\lambda_0 \/ 4 \\ge 8\\beta_k\\lambda + \\tilde\\epsilon$, thus $r(\\lambda_0 \/ 4) \\ge r(8\\beta_k\\lambda + \\tilde\\epsilon)$. It follows that\n\\begin{align*}\nv_d(r(\\lambda_0\/4))^d\\cdot (\\lambda_0\/4) \\ge \\frac{k}{n} + C_{\\delta, n}\\frac{\\sqrt{k}}{n}.\n\\end{align*} \nHence, for all $x$ such that $f(x) \\le \\lambda_0 \/ 2$, we have\n\\begin{align*}\nf_k(x) \\le (1 + 2\\frac{C_{\\delta, n}}{\\sqrt{k}}) (f(x) + \\lambda_0 \/ 4) \\le \\lambda_0.\n\\end{align*}\n\nTo see the second part, suppose we have $C_1, C_2 > 0$ such that $C_1 \\log n \\le k \\le C_2 n^{2\\alpha \/ (2\\alpha + d)}$. This combined with the fact that $r(\\epsilon) \\ge (\\epsilon\/C)^{1\/\\alpha}$ implies $\\lambda_0 \\rightarrow 0$, as desired.\n\n\\end{proof}\n\n\n\n\n\\section{Point-Cloud Density} \\label{appendix:pointcloud}\nHere we formalize the fact that modal-sets can serve as good models for high-density structures in data, for instance \na low-dimensional structure $M$ $+$ noise. \n\n\\begin{lemma} (Point Cloud with Gaussian Noise)\nLet $M \\subseteq \\mathbb{R}^d$ be compact (with possibly multiple connected-components of differing dimension). Then there exists a density $f$ over $\\mathbb{R}^d$ such that the density is uniform in $M$ and has Gaussian decays around $M$ i.e.\n\\begin{align*}\nf(x) = \\frac{1}{Z} \\exp(-d(x, M)^2\/(2\\sigma^2)),\n\\end{align*}  \nwhere $\\sigma > 0$ and $Z > 0$ depends on $M, \\sigma$. Thus, the modal-sets of $f$ are the connected-components of $M$. \n\\end{lemma}\n\n\\begin{proof}\nSince $M$ is compact in $\\mathbb{R}^d$, it is bounded. Thus there exists $R > 0$ such that $M \\subseteq B(0, R)$. \nIt suffices to show that for any $\\sigma > 0$,\n\\begin{align*}\n\\int_{\\mathbb{R}^d} \\exp(-d(x, M)^2\/(2\\sigma^2)) dx < \\infty.\n\\end{align*}\nBy a scaling of $x$ by $\\sigma$, it suffices to show that \n\\begin{align*}\n\\int_{\\mathbb{R}^d} g(x) dx < \\infty,\n\\end{align*}\nwhere $g(x) := \\exp(-\\frac{1}{2} d(x, M)^2)$. Consider level sets $\\mathcal{X}^\\lambda := \\{ x \\in \\mathbb{R}^d : g(x) \\ge \\lambda \\}$. \nNote that $\\mathcal{X}^\\lambda \\subseteq B(M, \\sqrt{2 \\log(1\/\\lambda)})$ based on the decay in $g$ around $M$. Clearly the image of $g$ is $(0, 1]$ so consider partitioning this range into intervals $[1, 1\/2], [1\/2, 1\/3], ...$. Then it follows that \n\\begin{align*}\n\\int_{\\mathbb{R}^d} g(x) dx \n&\\le \\sum_{n=2}^\\infty \\text{Vol}(\\mathcal{X}^{1\/n}) \\left(\\frac{1}{n-1} - \\frac{1}{n}\\right)\n\\le \\sum_{n=2}^\\infty \\frac{\\text{Vol}(B(M, \\sqrt{2 \\log(n))}) }{(n-1)n}\\\\\n&\\le \\sum_{n=2}^\\infty \\frac{\\text{Vol}(B(0, R + \\sqrt{2 \\log(n))}) }{(n-1)n}\n= \\sum_{n=2}^\\infty \\frac{v_d(R + \\sqrt{2 \\log(n))})^d }{(n-1)n} \\\\\n&\\le v_d\\cdot 2^{d-1} \\sum_{n=2}^\\infty \\frac{R^d + (2 \\log(n))^{d\/2}}{(n-1)n} < \\infty,\n\\end{align*}\nwhere the last inequality holds by AM-GM. As desired.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Implementation} \\label{appendix:implementation}\n\nIn this section, we explain how to implement Algorithm~\\ref{alg:epsilonmodalset} (which supersedes Algorithm~\\ref{alg:modalset}) efficiently. Here we assume that for our sample $X_{[n]}$, we have the $k$-nearest neighbors for each sample point. In our implementation, we simply use kd-tree, although one could replace it with any method that can produce the $k$-nearest neighbors for all the sample points. In particular, one could use approximate $k$-NN methods if scale is an issue. \n\nThis section now concerns with what remains: constructing a data structure that maintains the CCs of the mutual $k$-NN graph as we traverse down the levels. At level $\\lambda$ in Algorithm~\\ref{alg:epsilonmodalset}, we must keep track of the mutual $k$-NN graph for points $x$ such that $f_k(x) \\ge \\lambda - 9\\beta_k \\lambda - \\epsilon_0 - \\tilde\\epsilon$. Thus as $\\lambda$ decreases, we add more vertices (and corresponding edges to the mutual $k$-nearest neighbors). Algorithm~\\ref{alg:epsilonmodalsetinterface} shows what functions this data structure must support. Namely, adding nodes and edges, getting CCs of nodes, and checking if a CC intersects with the current estimates of the $\\epsilon_0$-modal sets.\n\nWe implement this data structure as a disjoint-set forest data structure.  \nThe CCs can be represented as disjoint-sets of forests. Adding a node corresponds to making a set while adding an edge corresponds to a union operation. We can identify the verticies with the roots of the corresponding set's trees and thus getConnectedComponent and componentSeen can be implemented in a straightforward way.\n\nIn sum, the bulk of the time complexity is in preprocessing the data. This consists of obtaining the initial $k$-NN graph, i.e. distances to nearest neighbors; this one time operation is of worst-case order $O(n^2)$, similar to usual clustering procedures (e.g. Mean-Shift, K-Means, Spectral Clustering), but average case $O(nk \\log n)$. After this preprocessing step, the estimation procedure itself requires just $O(nk)$ operations, each with amortized $O(\\alpha(n))$ where $\\alpha$ is the inverse Ackermann function. Thus, the implementation provided in Algorithm~\\ref{alg:epsilonmodalsetimpl} is near-linear in $k$ and $n$. \n\n\n\n\n\\begin{algorithm}[tb]\n   \\caption{Interface for Mutual $k$-NN graph construction}\n      \\label{alg:epsilonmodalsetinterface}\n\\begin{algorithmic}\n\\STATE InitializeGraph()  \\hfill \/\/ Creates an empty graph\n\\STATE addNode(G, node)   \\hfill \/\/ Adds a node\n\\STATE addEdge(G, node1, node2) \\hfill \/\/ Adds an edge\n\\STATE getConnectedComponent(G, node)   \\hfill\/\/ Get the vertices in node's CC\n\\STATE componentSeen(G, node)       \\hfill   \/\/ checks whether node's CC intersects with the estimates. If not, then marks the component as seen.\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\begin{algorithm}[tb]\n   \\caption{Implementation of M-cores (Algorithm~\\ref{alg:epsilonmodalset})}\n   \\label{alg:epsilonmodalsetimpl}\n\\begin{algorithmic}\n\n\\STATE Let $\\text{kNNSet}(x)$ be the $k$-nearest neighbors of $x \\in X_{[n]}$. \n\\STATE $\\widehat{\\mathcal{M}} \\leftarrow \\{ \\}$\n\\STATE $G \\leftarrow \\text{InitializeGraph()}$ \n\\STATE Sort points in descending order of $f_k$ values\n\\STATE Let $p \\leftarrow 1$\n\\FOR{$i = 1,...,n$}\n\\STATE $\\lambda \\leftarrow f_k(X_i)$\n\\WHILE {$p < n$ and $f_k(X_p) \\ge \\lambda - 9\\beta_k \\lambda -\\epsilon_0 - \\tilde\\epsilon$}\n\\STATE addNode($G, X_p$)\n\\FOR{$x \\in \\text{kNNSet}(X_p) \\cap G$}\n\\STATE addEdge($G, x, X_p$)\n\\ENDFOR \n\\STATE $p \\leftarrow p + 1$\n\\ENDWHILE\n\\IF{not componentSeen($G, X_i$)}\n\\STATE $\\text{toAdd} \\leftarrow \\text{getConnectedComponent}(G, X_i)$\n\\STATE Delete all $x$ from $\\text{toAdd}$ where $f_k(x) < \\lambda - \\beta_k \\lambda$\n\\STATE $\\widehat{\\mathcal{M}} \\leftarrow \\widehat{\\mathcal{M}} + \\{\\text{toAdd}\\}$\n\\ENDIF\n\\ENDFOR\n\\RETURN{$\\widehat{\\mathcal{M}}$}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\subsection{Practical Setup} \n\nThe analysis prescribes a setting of $\\beta_k = O(1\/\\sqrt{k})$. Throughout the experiments we simply fix $\\beta_k = 2 \/ \\sqrt{k}$, and let our choice of $k$ be the essential parameter. \nAs we will see, M-cores yields competitive and stable performance for a wide-range of settings of $k$. The implementation can be done efficiently and is described in Appendix~\\ref{appendix:implementation}. \n\n\n\nWe will release an optimized Python\/C++  version of the code at \\cite{url}. \n\n\n\n\\subsection{Qualitative Experiments on General Structures} \n\\begin{figure}[h]\n\\centering \n\\includegraphics[width=0.155\\textwidth]{eye1_original}\n\\includegraphics[width=0.155\\textwidth]{eye1_filtered_new}\n\\includegraphics[width=0.155\\textwidth]{eye1_estimated_new}\n\\includegraphics[width=0.155\\textwidth]{eye2_original}\n\\includegraphics[width=0.155\\textwidth]{eye2_filtered_new}\n\\includegraphics[width=0.155\\textwidth]{eye2_estimated_new}\n\\caption{Diabetic Retinopathy: (Left 3 figures) An unhealthy eye, (Right 3 figures) A healthy eye. \nIn both cases, shown are (1) original image, (2) a filter applied to the image, (3) modal-sets (structures of capillaries)  estimated by M-cores on the corresponding filtered image. The unhealthy eye is characterized by a proliferation of \ndamaged blood capillaries, while a healthy eye has visually fewer capillaries. The analysis task is to automatically discover the higher number of capillary-structures in the unhealthy eye. M-cores discovers $29$ structures for unhealthy eye vs $6$ for healthy eye.\n}\n\\label{fig:eye}\n\\end{figure} \n\nWe start with a qualitative experiment highlighting the flexibility of the procedure in fitting \na large variety of high-density structures. For these experiments, we use $k = \\frac{1}{2} \\cdot \\log^2 n$, which is within the\ntheoretical range for admissible values of $k$ (see Theorem \\ref{theo:main} and Remark \\ref{kadmissible}). \n\nWe consider a medical imaging problem. Figure~\\ref{fig:eye} displays the procedure applied to the Diabetic Retinopathy detection problem \\cite{drd}. While this is by no means an end-to-end treatment of this detection problem, it gives a sense of M-cores' versatility in fitting \nreal-world patterns. In particular, M-cores automatically estimates a reasonable number of clusters, independent \nof shape, while pruning away (most importantly in the case of the healthy eye) false clusters due to noisy data. As a result, it correctly picks up a much larger number of clusters in the case of the unhealthy eye. \n\n\n\n\\subsection{Clustering applications}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=1\\textwidth]{real_data_examples_new.png}\n\\caption{Comparison on real datasets (along the rows) across different hyperparameter settings for each algorithm (along the columns). The hyperparameters being tuned are displayed at the bottom of the figure for each clustering algorithm. Scores: the blue line with triangular markers is Adjusted-Mutual-Information, and the dotted red line is Adjusted-Rand-Index. }\n\\label{fig:realworld}\n\\end{figure*} \nWe now evaluate the performance of M-cores on clustering applications, where for\n{\\bf clustering:} we assign every point $x_i\\in X_{[n]}$ to $\\argmin_{\\widehat{M} \\in \\widehat{\\mathcal{M}}} d(x_i, \\widehat M)$, i.e. to the closest estimated modal-set. \n\nWe compare M-cores to two common density-based clustering procedures, DBSCAN and Mean-Shift, as implemented in the \\textit{sci-kit-learn} package. Mean-Shift clusters data around point-modes, i.e. local-maxima of $f$, and is therefore most similar to M-cores in its objective. \n\n\n\n{\\bf Clustering scores.} We compute two established scores which evaluate a clustering against a labeled ground-truth. The \\emph{rand-index}-score is the $0$-$1$ accuracy in grouping pairs of points, (see e.g. \\cite{hubert}); the \\emph{mutual information}-score is the (information theoretic) mutual-information between the distributions induced by the clustering and \nthe ground-truth (each cluster is a mass-point of the distribution, see e.g. \\cite{vinh2010information}). For both scores we report the \\emph{adjusted} version, which adjusts the score so that a random clustering (with the same number of clusters as the ground-truth) scores near $0$ (see e.g. \\cite{hubert}, \\cite{vinh2010information}). \n\n{\\bf Datasets.} Phonemes \\cite{hastie2005elements}, and UCI datasets: Glass, Seeds, Iris, and Wearable Computing. They are described in the table below\n\\begin{center}\n    \\begin{tabular}{ | p{2cm}| p{0.7cm} | p{0.35cm} | p{0.7cm} | p{8cm} |}\n    \\hline\n    {\\small \\emph{Dataset}} & $n$ & $d$ & {\\small \\emph{Labels}} & {\\small \\emph{Description}}  \\\\ \\hline\n    {\\small Phonemes} & {\\small 4509} & {\\small 256} & {\\small 5} & {\\small Log-periodograms of spoken phonemes} \\\\ \\hline\n    {\\small Glass} & {\\small 214} & {\\small 7} & {\\small 6} & \\small{Properties of different types of glass} \\\\ \\hline\n    {\\small Seeds} & {\\small 210} & {\\small 7} & {\\small 3} & \\small{Geometric measurements of wheat kernels} \\\\ \\hline\n    {\\small Iris} & {\\small 150} & {\\small 4} & {\\small 3}  & \\small{Various measurements over species of flowers} \\\\ \\hline\n    {\\small Wearable} & {\\small 10000} & {\\small 12} & {\\small 5}  & \n    \\small{4 sensors on a human body, recording body posture and activity} \\\\ \\hline\n    \\end{tabular}\n\\end{center}\n\n{\\bf Results.} Figure~\\ref{fig:realworld} reports the performance of the procedures for each dataset. Rather than reporting the performance of the \nprocedures under \\emph{optimal-tuning}, we report their performance \\emph{over a range} of hyperparameter settings, \nmindful of the fact that optimal-tuning is hardly found in practice (this is a general problem in clustering given the lack of ground-truth to guide tuning). \n\n\nFor M-cores we vary the parameter $k$. For DBSCAN and Mean-Shift, we vary the main parameters, respectively \\emph{eps} (choice of level-set), and \\emph{bandwidth} (used in density estimation). \nM-cores yields competitive performance across the board, with stable scores over a large range of values of $k$ (relative to sample size). Such stable performance to large changes in $k$ is quite desirable, considering that proper tuning of hyperparameters remains a largely open problem in clustering. \n\\vspace{0.3cm}\n\n{\\bf Conclusion} \\\\\n\n\\vspace{-0.2cm} \n\nWe presented a theoretically-motivated procedure which can consistently estimate modal-sets, i.e. nontrivial high-density structures in data, under benign distributional conditions. This procedure is easily implemented and yields competitive and stable scores in clustering applications. \n\n\n\n\n\n\\subsection*{#1}}{\\qed\\medskip}\n\\theoremstyle{definition}\n\\newtheorem{discussion}{Discussion}\n\n\\newcommand{\\mathcal{G}}{\\mathcal{G}}\n\\renewcommand{\\P}{\\mathbb{P}}\n\\newcommand{\\mathbb{E}}{\\mathbb{E}}\n\n\\newcommand{\\hat{C}}{\\hat{C}}\n\\newcommand{\\check{C}}{\\check{C}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\\newcommand{\\mathcal{X}}{\\mathcal{X}}\n\\newcommand{\\mathcal{Y}}{\\mathcal{Y}}\n\\newcommand{X_{[n]}}{X_{[n]}}\n\\newcommand{\\mathbf{Y}}{\\mathbf{Y}}\n\\newcommand{\\mathcal{M}}{\\mathcal{M}}\n\\newcommand{\\Sh}[2]{\\mathcal{S}\\left(#1, #2\\right)}\n\\newcommand{\\mathcal{N}_r}{\\mathcal{N}_r}\n\\newcommand{\\mathcal{P}}{\\mathcal{P}}\n\\newcommand{\\mathcal{H}}{\\mathcal{H}}\n\\newcommand{\\mathcal{C}}{\\mathcal{C}}\n\\newcommand{\\mathcal{B}}{\\mathcal{B}}\n\\newcommand{\\mathcal{P}}{\\mathcal{P}}\n\\newcommand{\\widetilde{S}}{\\widetilde{S}}\n\\newcommand{\\mathbf{A}}{\\mathbf{A}}\n\\newcommand{\\A^{\\!\\!*}}{\\mathbf{A}^{\\!\\!*}}\n\\newcommand{\\tilde{R}}{\\tilde{R}}\n\\newcommand{\\bar{a}}{\\bar{a}}\n\\newcommand{\\widetilde{f}}{\\widetilde{f}}\n\\newcommand{\\mathcal{S}}{\\mathcal{S}}\n\\newcommand{\\mathcal{L}}{\\mathcal{L}}\n\\newcommand{\\mathcal{O}}{\\mathcal{O}}\n\\newcommand{\\mathbb{R}}{\\mathbb{R}}\n\\newcommand{\\mathbb{N}}{\\mathbb{N}}\n\\newcommand{\\mathbb{E}}{\\mathbb{E}}\n\\newcommand{\\ext}[2]{{\\overline{\\mathbb{E}}_{#1}}\\left[#2\\right]}\n\\newcommand{\\exf}[2]{{\\mathbb{E}_{#1}}\\left[#2\\right]}\n\\newcommand{\\ex}[1]{{\\mathbb{E}}\\left[#1\\right]}\n\\newcommand{{\\expectation} \\,}{{\\mathbb{E}} \\,}\n\\newcommand{\\expecf}[1]{{\\mathbb{E}_{#1}} \\,}\n\\newcommand{\\norm}[1]{\\left\\|#1\\right\\|}\n\\newcommand{\\abs}[1]{\\left|#1\\right|}\n\\newcommand{\\paren}[1]{\\left(#1\\right)}\n\\newcommand{\\pr}[1]{\\mathbb{P}\\left(#1\\right)}\n\\newcommand{\\prf}[2]{\\mathbb{P}_{#1}\\left(#2\\right)}\n\\newcommand{\\widetilde{\\sigma}}{\\widetilde{\\sigma}}\n\\newcommand{\\mathbf{\\top}}{\\mathbf{\\top}}\n\\newcommand{\\widetilde{x}}{\\widetilde{x}}\n\\newcommand{\\widetilde{X}}{\\widetilde{X}}\n\\newcommand{\\widetilde{m}}{\\widetilde{m}}\n\\newcommand{\\SigmaT_Z}{\\SigmaT_Z}\n\\newcommand{\\tilde{\\rho}}{\\tilde{\\rho}}\n\\newcommand{\\mathring{\\delta}}{\\mathring{\\delta}}\n\\newcommand{\\text{tr}}{\\text{tr}}\n\\newcommand{\\diameter}[1]{\\Delta_n\\left(#1\\right)}\n\\newcommand{\\diam}[1]{\\Delta_n^2\\left(#1\\right)}\n\\newcommand{\\diamA}[1]{\\Delta_{n,a}^2\\left(#1\\right)}\n\\newcommand{\\Delta^2_{\\X}}{\\Delta^2_{\\mathcal{X}}}\n\\newcommand{\\meanN}[1]{\\bar{#1}_n}\n\\newcommand{\\mean}[1]{\\bar{#1}}\n\\newcommand{\\text{err}}{\\text{err}}\n\\newcommand{\\ind}[1]{\\mathds{1}\\left\\{#1\\right\\}}\n\\newcommand{\\braces}[1]{\\left\\{#1\\right\\}}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\level}{level}\n\\DeclareMathOperator*{\\volume}{vol}\n\\DeclareMathOperator*{\\Expectation}{\\mathbb{E}}\n\\newcommand{\\lev}[1]{\\level\\left(#1\\right)}\n\\newcommand{\\vol}[1]{\\volume\\left(#1\\right)}\n\n\n\\title{Modal-set estimation with an application to clustering}\n\n\n\\author{\n  Heinrich Jiang \\thanks{Much of this work was done when this author was at Princeton University Mathematics Department.}\\\\\n  \\texttt{heinrich.jiang@gmail.com} \\\\\n  \\And\n  Samory Kpotufe\\\\\n  ORFE, Princeton University\\\\\n  \\texttt{samory@princeton.edu}\\\\\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\n{\\bf Abstract}. We present a first procedure that can estimate -- with statistical consistency guarantees -- any local-maxima of a density, under benign distributional conditions. The procedure estimates all such local maxima, or \\emph{modal-sets}, of any bounded shape or dimension, including usual point-modes. In practice, modal-sets can arise as dense low-dimensional structures in noisy data, and more generally serve to better model the rich variety of locally-high-density structures in data. \n\nThe procedure is then shown to be competitive on clustering applications, and moreover is quite stable to a wide range of settings of its tuning parameter. \n\\end{abstract}\n\n\n\n\\section{Introduction}\nMode estimation is a basic problem in data analysis. \n Modes, i.e. points of locally high density, serve as a measure of central tendency and are therefore important in unsupervised problems such as outlier detection, image or audio segmentation, and clustering in particular (as cluster cores). In the present work, we are interested in capturing a wider generality of \\emph{modes}, i.e. general structures (other than single-points) of locally high density, that can arise in modern data. \n\nFor example, application data in $\\mathbb{R}^d$ (e.g. speech, vision) are often well modeled as arising from \na lower-dimensional structure $M$ + noise. In other words, such data is densest on $M$, hence \nthe ambient density $f$ is more closely modeled as locally maximal at (or near) $M$, a nontrivial subset of $\\mathbb{R}^d$, \nrather than maximal only at single points in $\\mathbb{R}^d$. Such a situation is illustrated in Figure \\ref{fig:clustercores}. \n\nWe therefore extend the notion of \\emph{mode} to any connected \nsubset of $\\mathbb{R}^d$ where the unknown density $f$ is locally maximal; we refer to these as \\emph{modal-sets} of $f$.  A modal-set can be of any bounded shape and dimension, from $0$-dimensional (point modes), to full dimensional surfaces, and aim to capture the possibly rich variety of dense structures in data. \n\nOur main contribution is a procedure, M(odal)-cores, that consistently estimates all such modal-sets from data, of general shape and dimension, with minimal assumption on the unknown $f$. The procedure builds on recent developments in topological data analysis \n\\cite{SN10, CD10, KV11, RSNW12, balakrishnan2013cluster, CDKvL14, eldridge2015beyond}, and works by traversing certain $k$-NN graphs which encode level sets of a $k$-NN density estimate. We show that, if $f$ is continuous on compact support, the Hausdorff distance between any modal-set and its estimate vanishes as $n\\to \\infty$ (Theorem \\ref{theo:main}); the estimation rate for point-modes matches (up to $\\log n$) the known minimax rates. Furthermore, under mild additional smoothness condition on $f$ (H\\\"older continuity), \\emph{false} structures (due to empirical variability) are correctly identified and pruned. We know of no such general statistical guarantees in mode estimation. \n\nWhile there is often a gap between theoretical procedures and practical ones, the present procedure is easy to implement and yields competitive scores on clustering applications; here, as in \\emph{mode-based clustering}, clusters are simply defined as regions of high-density of the data, and the estimated modal-sets serve as the centers of these regions, i.e. as  \\emph{cluster-cores}. A welcome aspect of the resulting clustering procedure is its stability to tuning \nsettings of the parameter $k$ (from $k$-NN): it maintains high clustering scores (computed with knowledge of the ground-truth) over a wide range of settings of $k$, for various datasets. \n Such stability to tuning is of practical importance, since typically the ground-truth is unknown, so clustering procedures  come with tuning parameters that are hard to set in practice. Practitioners therefore use various rule-of-thumbs and can thus benefit from procedures that are less-sensitive to their hyperparameters. \n \n\n\nIn the next section we put our result in context with respect to previous work on mode estimation and density-based clustering in general. \n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure} \n\\centering \n\\includegraphics[width=0.25\\textwidth]{3d_2000_samples.jpg}\n\\includegraphics[width=0.25\\textwidth]{3d_density.jpg}\n\\includegraphics[width=0.25\\textwidth]{3d_estimated.jpg}\n\n\\label{fig:clustercores}\n\\caption{Main phase of M-cores. (Left) Points-cloud generated as three 1-dimensional rings + noise. (Middle) The 3 rings, and (Right) their estimate (as modal-sets) by M-cores.}\n\\end{figure}\n\n\\subsection*{Related Work} \n$\\bullet$ Much theoretical work on mode-estimation is concerned with understanding the statistical difficulty of the problem, and as such, often only considers the case of densities with single point-modes \\cite{parzen1962estimation, chernoff1964estimation, eddy1980optimum, devroye1979recursive, tsybakov1990recursive, abraham2004asymptotic}. \nThe more practical case of densities with multiple point-modes has received less attention in the theoretical literature. However there exist practical estimators, e.g., the popular \\emph{Mean-Shift} procedure (which doubles as a clustering procedure), which are however harder to analyze. Recently, \\cite{arias2013estimation} shows the consistency of a variant of Mean-Shift. Other recent work of \\cite{genovese2013nonparametric} derives a method for pruning false-modes obtained by mode-seeking procedures. Also recent, the work of \\cite{dasgupta2014optimal} shows that point-modes of a $k$-NN density estimate $f_k$ approximate the true modes of the unknown density $f$, assuming $f$ only has point-modes and bounded Hessian at the modes; their procedure, therefore operates on level-sets of $f_k$ (similar to ours), but fails in the presence of more general high-density structures such as modal-sets. To handle such general structures, we have to identify more appropriate level-sets to operate on, the main technical difficulty being that local-maxima of $f_k$ can be relatively far (in Hausdorff) from those of $f$, for instance single-point modes rather than more general modal-sets, due to data-variability. \nThe present procedure handles general structures, and is consistent under the much weaker conditions of continuity (of $f$) on a compact domain.\n\nA related line of work, which seeks more general structures than point-modes, is that of \\emph{ridge} estimation (see e.g. \\citep{ozertem2011locally, genovese2014nonparametric}). A ridge is typically defined as a lower-dimensional structure away from which the density curves (in some but not all directions), and can serve to capture various lower-dimensional patterns apparent in point clouds. In contrast, the modal-sets defined here can be full-dimensional and are always local maxima of the density. Also, unlike in ridge estimation, we do not require local differentiability of the unknown $f$, nor knowledge of the dimension of the structure, thus allowing a different but rich set of practical structures. \n\n$\\bullet$ A main application of the present work, and of mode-estimation in general, is \\emph{density-based clustering}.  Such clustering was formalized in early work of \\cite{carmichael1968finding, hartigan1975clustering, H81}, and can take various forms, each with their advantage. \n\nIn its hierarchical version, one is interested in estimating the connected components (CCs) of \\emph{all} level sets \n$\\braces{f \\ge \\lambda}_{\\lambda>0}$ of the unknown density $f$. Many recent works analyze approaches that consistently estimate such a hierarchy under quite general conditions, e.g. \\cite{SN10, CD10, KV11, RSNW12, balakrishnan2013cluster, CDKvL14, eldridge2015beyond}. \n\nIn the \\emph{flat} clustering version, one is interested in estimating the CCs of $\\braces{f \\ge \\lambda}$ for a single $\\lambda$, somehow appropriately chosen \n\\citep{RV09, SSN09, MHL09, RW10, S11, sriperumbudur2012consistency}. The popular DBSCAN procedure  \\citep{ester1996density} can be viewed as estimating such single level set. The main disadvantage here is in the ambiguity in the choice of $\\lambda$, especially when the levels $\\lambda$ of $f$ have different numbers of clusters (CCs). \n\n\nAnother common flat clustering approach, most related to the present work, is \\emph{mode-based} clustering. The approach clusters points to estimated modes of $f$, a fixed target, and therefore does away with the ambiguity in choosing an appropriate level $\\lambda$ of $f$ \\citep{fukunaga1975estimation, cheng1995mean, comaniciu2002mean, li2007nonparametric, chazal2013persistence}. As previously discussed, \nthese approaches are however hard to analyze in that mode-estimation is itself not an easy problem. Popular examples are extensions of $k$-Means to categorical data \\cite{chaturvedi2001k}, and the many variants of Mean-Shift which cluster points by gradient ascent to the closest mode. \nNotably, the recent work \\cite{wasserman2014feature} analyzes clustering error of Mean-Shift in a general high-dimensional setting with potentially irrelevant features. The main assumption is that $f$ only has point-modes. \n\n\n\n\n\n\\section{Overview of Results}\n\\label{sec:Overview}\n\\input{Overview.tex}\n\n\n\\section{Analysis Overview}\n\\label{sec:analysis}\n\\input{Analysis.tex}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\input{Experiments.tex}\n\n\n{\n\n\\subsection{Basic Setup and Definitions}\nWe have samples $X_{[n]} = \\{X_1,...,X_n\\}$ drawn i.i.d. from \na distribution $\\mathcal{F}$ over $\\mathbb{R}^d$ with density $f$. We let $\\mathcal{X}$ denote the support of $f$. \nOur main aim is to estimate all local maxima of $f$, or \\emph{modal-sets} of $f$, as we will soon define. \n\nWe first require the following notions of distance between sets. \n\\begin{definition}\\label{ball} \nFor $M\\subset \\mathcal{X}$, $x\\in \\mathcal{X}$, let $d(x, M) := \\inf_{x'\\in M} \\norm{x - x'}$. \nThe {\\bf Hausdorff} distance between $A, B \\subset \\mathcal{X}$ is defined as \n$d(A,B) := \\max \\{ \\sup_{x \\in A} d(x, B), \\sup_{y \\in B} d(y, A) \\}.$\n\\end{definition}\n\nA modal set, defined below, extends the notion of a point-mode to general subsets of $\\mathcal{X}$ \nwhere $f$ is locally maximal. These can arise for instance, as discussed earlier, in applications where high-dimensional \ndata might be modeled as a (disconnected) manifold $\\mathcal{M}$ + ambient noise, each connected component of which \ninduces a modal set of $f$ in ambient space $\\mathbb{R}^D$ (see e.g. Figure \\ref{fig:clustercores}). \n\n\\begin{definition}\\label{modal-set} For any $M \\subset \\mathcal{X}$ and $r>0$, define the \\emph{envelope} $B(M, r) := \\{x : d(x, M) \\leq r\\}$. A connected set $M$ is a {\\bf modal-set} of $f$ if  $\\forall x \\in M$, $f(x) = f_M$ for some fixed $f_M$, and there exist $r>0$ such that $f(x) < f_M$ for all $x \\in B(M, r) \\backslash M$. \n\\end{definition}\n\n\\begin{remark} \nThe above definition can be relaxed to \\emph{$\\epsilon_0$-modal sets}, i.e., to allow $f$ to vary by a small $\\epsilon_0$ on $M$. Our results extend easily to this more relaxed definition, with minimal changes to some constants. This is because the procedure operates on $f_k$, and therefore already needs to account for variations in $f_k$  \non $M$. This is described in Appendix~\\ref{appendix:eps-modal-set}. \n\\end{remark}\n\n\n\n\n\n\\subsection{Estimating Modal-sets}\nThe algorithm relies on nearest-neighbor density estimate $f_k$, defined as follows. \n\n\\begin{definition} \\label{kNNdensity} Let $r_k(x) := \\min \\{ r : |B(x, r) \\cap X_{[n]} | \\ge k \\}$. \nDefine the {\\bf $k$-NN density estimate} as\n$$f_k(x) := \\frac{k}{n\\cdot v_d\\cdot r_k(x)^d},\n\\text{where } v_d \\text{ is the volume of a unit sphere in } \\mathbb{R}^d.$$\n\\end{definition}\n\nFurthermore, we need an estimate of the level-sets of $f$; various recent work on cluster-tree estimation (see e.g. \\cite{CDKvL14}) have shown that such level sets are encoded by subgraphs of certain \\emph{modified} $k$-NN graphs. \nHere however, we directly use $k$-NN graphs, simplifying implementation details, but requiring a bit of side analysis.\n\n\\begin{definition}\n Let $G(\\lambda)$ denote the (mutual) {\\bf $k$-NN graph} with vertices $\\{ x \\in X_{[n]} : f_k(x) \\ge \\lambda\\}$ and an edge between $x$ and $x'$ iff $||x - x'|| \\le \\min \\{r_k(x), r_k(x') \\}$. \n \\end{definition}\n $G(\\lambda)$ can be viewed as approximating the $\\lambda$-level set of $f_k$, hence approximates the $\\lambda$-level set of $f$ (implicit in the connectedness result in Appendix~\\ref{appendix:integrality}).  \n \nAlgorithm \\ref{alg:modalset} (M-cores) estimates the modal-sets of the unknown $f$.\nIt is based on various insights described below. \nA basic idea, used for instance in point-mode estimation \\cite{dasgupta2014optimal}, is to proceed top-down on the level sets of $f_k$ (i.e. on $G(\\lambda), \\, \\lambda \\to 0$), and identify new modal-sets as they appear in separate CCs at a level $\\lambda$. \n\nHere we have to however be careful: the CCs of $G(\\lambda)$ (essentially modes of $f_k$) might be \nsingleton points (since $f_k$ might take unique values over samples $x\\in X_{[n]}$) while the modal-sets to be estimated might be of any dimension and shape. Fortunately, if a datapoint $x$, locally maximizes $f_k$, and belongs to some modal-set $M$ of $f$, then the rest of $M\\cap X_{[n]}$ must be at a nearby level; Algorithm \\ref{alg:modalset} therefore proceeds by checking a nearby level ($\\lambda - 9\\beta_k \\lambda$) from which it picks a specific set of points as an estimate of $M$. The main parameter here is $\\beta_k$ which is worked out explicitly in terms of $k$ and requires no a priori knowledge of distributional parameters. The confidence level $\\delta$ can be viewed in practice as fixed (e.g. $\\delta = 0.05$). The essential algorithmic parameter is therefore just $k$, which, as we will show, can be chosen over a wide range (w.r.t. $n$) while ensuring statistical consistency. \n\n\\begin{definition}Let $0< \\delta < 1$. Define $C_{\\delta, n} := 16\\log(2\/\\delta)\\sqrt{d \\log n}$, and define $\\beta_k = 4\\frac{C_{\\delta, n}}{\\sqrt{k}}$. \\end{definition}\n\nWe note that the above definition of $\\beta_k$ is somewhat conservative (needed towards theoretical guarantees), since the exact constants $C_{\\delta, n}$ turn out to have little effect in implementation. \n\nA further algorithmic difficulty is that a level $G(\\lambda)$ might have too many CCs w.r.t. the ground truth. For example, due to variability in the data, $f_k$ might have more modal-sets than $f$, inducing too many CCs at some level $G(\\lambda)$. Fortunately, it can be shown that the nearby level $\\lambda - 9\\beta_k \\lambda$ will likely have the right number of CCs. Such lookups down to lower-level act as a way of \\emph{pruning false modal-sets}, and trace back to earlier work \\citep{KV11} on pruning cluster-trees. Here, we need further care: \nwe run the risk of over-estimating a given $M$ if we look too far down (aggressive pruning), since \na CC at lower level might contain points \\emph{far outside} of a modal-set $M$. \nTherefore, the main difficulty here is in figuring out how far down to look and yet not over-estimate \\emph{any} $M$ (to ensure consistency). In particular our lookup \\emph{distance} of $9\\beta_k \\lambda$ is adapted to the level $\\lambda$ unlike in aggressive pruning. \n\nFinally, for clustering with M-cores, we can simply assign every data-point to the closest estimated modal-set (acting as cluster-cores).\n\n\\begin{algorithm}[tb]\n   \\caption{M-cores (estimating modal-sets).}\n   \\label{alg:modalset}\n\\begin{algorithmic}\n   \\STATE Initialize $\\widehat{\\mathcal{M}}:= \\emptyset$. Define $\\beta_k = 4\\frac{C_{\\delta, n}}{\\sqrt{k}}$.   \n   \\STATE Sort the $X_i$'s in decreasing order of $f_k$ values (i.e. $f_k(X_i) \\geq f_k(X_{i+1})$). \n   \\FOR{$i=1$ {\\bfseries to} $n$}\n   \\STATE Define $\\lambda := f_k(X_i)$.\n   \\STATE Let $A$ be the CC of $G(\\lambda - 9\\beta_k \\lambda)$ that contains $X_i$.  \\hfill (\\rm{i})\n   \\IF{$A$ is disjoint from all cluster-cores in $\\widehat{\\mathcal{M}}$}\n   \\STATE Add $\\widehat{M} := \\{ x \\in A : f_k(x) > \\lambda - \\beta_k \\lambda \\}$ to\n    $\\widehat{\\mathcal{M}}$. \n    \\ENDIF\n   \\ENDFOR\n   \\STATE \\textbf{return} $\\widehat{\\mathcal{M}}$. \\textit{     \/\/ Each $\\widehat M\\in \\widehat{\\mathcal{M}}$ is a cluster-core estimating \n   a modal-set of the unknown $f$.}\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\subsection{Consistency Results}\nOur consistency results rely on the following mild assumptions. \n\\begin{assumption} $f$ is continuous with compact support $\\mathcal{X}$. Furthermore $f$ has a finite number of modal-sets \nall in the interior of its support $\\mathcal{X}$. \n\\label{assumption-main}\n\\end{assumption}\n\n\nWe will express the convergence of the procedure explicitly in terms of quantities that characterize the behavior of $f$ at the boundary of every modal set. The first quantity has to do with how \\emph{salient} a modal-set, i.e whether it is sufficiently \\emph{separated} from other modal sets. We start with the following definition of \\emph{separation}. \n\n\\begin{definition}\\label{rsalient} \nTwo sets $A, A' \\subset \\mathcal{X}$ are {\\bf $r$-separated}, if there exists a set $S$ such that every path from $A$\n to $A'$ crosses $S$ and $\\sup_{x \\in B(S, r)} f(x) < \\inf_{x \\in A\\cup A'} f(x)$.\n\\end{definition}\nThe next quantities characterize the \\emph{change} in $f$ in a neighborhood of a modal set $M$. \nThe existence of a proper such neighborhood $A_M$, and appropriate functions $u_M$ and $l_M$ capturing smoothness and curvature, follow from the above assumptions on $f$. This is captured in the proposition below. \n\n\\begin{proposition}\n\\label{prop:main-assumptions}\nLet $M$ be a modal-set of $f$. Then there exists a CC $A_M$ of some level-set $\\mathcal{X}^{\\lambda_M} := \\{x : f(x) \\ge \\lambda_M\\}$, containing $M$, such that the following holds. \n\\begin{itemize} \n\\item \\emph{$A_M$ isolates $M$ by a valley}: $A_M$ does not intersect any other modal-set; and $A_M$ and $\\mathcal{X}^{\\lambda_M} \\backslash A_M$ are $r_s$-separated (by some $S_M$) for some $r_s>0$ independent of $M$. \n\\item \\emph{$A_M$ is full-dimensional}: $A_M$ contains an envelope $B(M, r_M)$ of $M$, for some $r_M>0$.  \n\\item \\emph{$f$ is both \\emph{smooth} and has \\emph{curvature} around $M$}: there exist functions $u_M$ and $l_M$, increasing and continuous on $[0, r_M]$, $u_M(0) = l_M(0) = 0$, such that $\\forall x \\in B(M, r_M)$, \n\\begin{align*}\nl_M(d(x, M)) \\le f_M -  f(x) \\le u_M(d(x, M)). \n\\end{align*}\n\n\\end{itemize}\n\\end{proposition}\n\n\n\nFinally, our consistency guarantees require the following admissibility condition on $k = k(n)$. \nThis condition results, roughly, from needing the density estimate $f_k$ to properly approximate the behavior \nof $f$ in the neighborhood of a modal-set $M$. In particular, we intuitively need $f_k$ values to be smaller for \npoints far from $M$ than for points close to $M$, and this should depend on the smoothness and curvature of $f$ \naround $M$ (as captured by $u_M$ and $l_M$). \n\n\\begin{definition} $k$ is {\\bf admissible} for a modal-set $M$ if (we let $u_M^{-1}$ denote the inverse of $u_M$):\n$$\\max \\left\\{ \\left(\\frac{24 \\sup_{x \\in \\mathcal{X}} f(x)}{l_M(\\min\\{r_M, r_s\\}\/2)} \\right)^2, 2^{7 + d}  \\right\\}\\cdot C_{\\delta, n}^2 \\le k \\le \\frac{v_d \\cdot f_M}{2^{2+2d}} \\left(u_M^{-1} \\left ( \nf_M\\frac{C_{\\delta, n}}{2\\sqrt{k}}\\right) \\right)^d \\cdot n.$$\n\\end{definition}\n\n\\begin{remark}\\label{kadmissible} The admissibility condition on $k$, although seemingly opaque, allows for a wide range of settings of $k$. For example, suppose $u_M(t) = c t^{\\alpha}$ for some $c, \\alpha > 0$. These are polynomial tail conditions common in mode estimation, following e.g. from H\\\"older assumptions on $f$. \nAdmissibility then (ignoring $\\log (1\/\\delta)$), is immediately seen to correspond to the wide range\n$$C_1\\cdot \\log n \\leq k \\leq C_2\\cdot n^{2\\alpha\/(2\\alpha + d)},$$ where $C_1, C_2$ are constants depending on $M$, but independent of $k$ and $n$. It's clear then that even the simple choice $k = \\Theta(log^2 n)$ is always admissible \\emph{for any} $M$ for $n$ sufficiently large.\n\\end{remark}\n\n\n{\\bf Main theorems.} We then have the following two main consistency results for Algorithm \\ref{alg:modalset}. Theorem \\ref{theo:main} states a rate (in terms of $l_M$ and $u_M$) at which \nany modal-set $M$ is approximated by some estimate in $\\widehat{\\mathcal{M}}$; Theorem \\ref{pruning} establishes \\emph{pruning} guarantees. \n\\begin{theorem} \n\\label{theo:main}\nLet $0< \\delta < 1$. The following holds with probability at least $1- 6\\delta$, simultaneously for all modal-sets $M$ of $f$. Suppose $k$ is admissible for $M$. Then there exists $\\widehat{M} \\in \\widehat{\\mathcal{M}}$ such that the following holds. Let $l_M^{-1}$ denote the inverse of $l_M$. \n \\begin{align*}\n d(M, \\widehat{M}) \\le l_M^{-1}\\left(\\frac{8C_{\\delta,n }}{\\sqrt{k}}f_M\\right), \n \\text{ which goes to } 0 \\text{ as } C_{\\delta, n}\/\\sqrt{k} \\rightarrow 0.\n \\end{align*}\n\\end{theorem}\nIf $k$ is admissible for all modal-sets $M$ of $f$, then $\\widehat{\\mathcal{M}}$ estimates all modal-sets of $f$ \nat the above rates. These rates can be instantiated under the settings in Remark~\\ref{kadmissible}: \nsuppose $l_M(t) = c_1 t^{\\alpha_1}$, $u_M(t) = c t^\\alpha$, $\\beta_1 \\geq \\beta$; then \nthe above bound becomes $d(M, \\widehat{M}) \\lesssim k^{-1\/2{\\alpha_1}}$ for admissible $k$. As in the remark, $k = \\Theta (\\log^2 n)$ is admissible, simultaneously for all $M$ (for $n$ sufficiently large), and therefore all modal-sets of $f$ are recovered at the above rate. \nIn particular, taking large $k = O(n^{2\\alpha\/(2\\alpha + d)})$ optimizes the rate to $O(n^{-\\alpha\/(2\\alpha_1\\alpha + \\alpha_1 d)})$. Note that for $\\alpha_1 = \\alpha = 2$, the resulting rate ($n^{-1\/(4+d)}$) is tight (see e.g. \\cite{tsybakov1990recursive} for matching lower-bounds in the case of point-modes $M = \\braces{x}$.).  \n\nFinally, Theorem~\\ref{pruning} (pruning guarantees) states that any estimated modal-set in $\\widehat{\\mathcal{M}}$, at a sufficiently high level (w.r.t. to $k$), corresponds to a \\emph{true} modal-set of $f$ at a similar level. Its proof consists of showing that if two sets of points are wrongly disconnected at level $\\lambda$, they remain connected at nearby level $\\lambda - 9\\beta_k \\lambda$ (so are reconnected by the procedure). The main technicality is the dependence of the nearby level on the empirical $\\lambda$; the proof is less involved and given in  Appendix~\\ref{appendix:pruning}. \n\n\\begin{theorem} \\label{pruning}\nLet $0< \\delta < 1$. There exists $\\lambda_0 = \\lambda_0(n, k)$ such that the following holds with probability at least $1 - \\delta$. All modal-set estimates in $\\widehat{\\mathcal{M}}$ chosen at level $\\lambda \\ge \\lambda_0$ can be injectively mapped to modal-sets $\\braces{M:  \\lambda_M \\geq \\min_{\\{x\\in \\mathcal{X}_{[n]} : f_k(x) \\ge \\lambda - \\beta_k \\lambda \\}} f(x)}$, provided $k$ is admissible for all such $M$. \n\nIn particular, if $f$ is H\\\"older-continuous, (i.e. $||f(x) - f(x')|| \\le c||x - x'||^\\alpha $ for some $0 < \\alpha\\le 1$, $c > 0$)\nthen $\\lambda_0 \\xrightarrow{n \\to \\infty} 0$, provided \n$C_1 \\log n \\le k \\le C_2 n^{2\\alpha \/ (2\\alpha + d)}$, for some $C_1, C_2$ independent $n$. \n\n\n\\end{theorem}\n\\begin{remark}\nThus with little additional smoothness ($\\alpha \\approx 0$) over uniform continuity of $f$, any estimate above level $\\lambda_0 \\to 0$ corresponds to a true modal-set of $f$. We note that these pruning guarantees can be strengthened as needed by implementing a more aggressive pruning: simply replace $G(\\lambda - 9\\beta_k \\lambda)$ in the procedure (on line (\\rm{i})) with $G(\\lambda - 9\\beta_k \\lambda - \\tilde\\epsilon)$ using a \\emph{pruning parameter} $\\tilde \\epsilon \\ge 0$. This allows $\\lambda_0 \\rightarrow 0$ faster. However the rates of Theorem \\ref{theo:main} (while maintained) then require a larger initial sample size $n$. \nThis is discussed in Appendix~\\ref{appendix:pruning}.  \n\\end{remark}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the framework of research and development of novel acceleration schemes and technology, the upgrade of the SPARC\\_LAB test facility \\cite{ferrario2013sparc_lab} at INFN-LNF is foreseen, based on a high gradient linac. High brightness electron bunches are fundamental for the successful development of plasma-based accelerators, for instance, whereas external injection schemes are considered, i.e. particle beam driven and laser driven plasma wakefield accelerators (PWFA and LWFA, respectively). Indeed, the ultimate beam brightness and its stability and reproducibility are strongly influenced by the RF-generated electron beam.\n\nIn this scenario the SPARC\\_LAB upgrade, named as EuPRAXIA@SPARC\\_LAB \\cite{ferrario_eaac2017}, might be one of the possible candidates to host EuPRAXIA (European Plasma Research Accelerator with eXcellence In Applications) \\cite{Walker:IPAC2017-TUOBB3}. EuPRAXIA is a design study in the framework of Horizon 2020 (INFRADEV-1-2014), funded to bring together for the first time novel acceleration schemes, based for instance on plasmas, modern lasers, the latest correction\/feedback technologies and large-scale user areas. Such a research infrastructure would achieve the required quantum leap in accelerator technology towards more compact and more cost-effective accelerators, opening new horizons for applications and research.\n\nThe preliminary EuPRAXIA@SPARC\\_LAB linac layout is based on an S-band Gun, three S-band TW structures and an X-band booster with a bunch compressor \\cite{ferrario_eaac2017}. The booster design has been driven by the need of a high accelerating gradient required to achieve a high facility compactness, which is one of the main goals of the EuPRAXIA project. The baseline technology chosen for the EuPRAXIA@SPARC\\_LAB booster is X-band. The total space allocated for the linac accelerating sections is $\\approx$25 m, corresponding to an active length of $\\approx$16 m taking into account the space required to accommodate beam diagnostics, magnetic elements, vacuum equipment and flanges. Two average accelerating gradient options are foreseen for the X-band linac: high gradient (HG) of 57 MV\/m and very high gradient (VHG) of 80 MV\/m, corresponding to double the power of the HG case. The RF linac layout is based on klystrons with SLEDs \\cite{farkas} that feed several TW accelerating structures. The operating mode is the $2\\pi\/3$ mode at 11.9942 GHz. The preliminary RF system parameters are summarized in Table \\ref{tab:RF_parameters} \\cite{cpi_website}.\n\n\\begin{table}[hbt]\n\\center{\n\\caption{RF system parameters.}\n  \\label{tab:RF_parameters}\n\\begin{tabular}{lcr}\n\\midrule\nFrequency & 11.9942 GHz\\\\\nPeak RF power & \\SI {50}{\\mega W}\\\\\nRF pulse length $t_k$ & \\SI {1.5}{\\micro s}\\\\\nUnloaded Q-factor $Q_0$ of SLED & 180000\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\nIn this paper we illustrate the preliminary RF design of the X-band booster. The single cell parameters have been calculated by electromagnetic (e.m.) simulations. On the basis of these results, the accelerating structure length and geometry have been optimized  by numerical studies. Finally, the basic RF power distribution layout has been designed. \n\n\\section{Single cell study}\\label{sec:single_cell}\n\nThe main single cell parameters (shunt impedance per unit length $R$, normalized group velocity $v_g\/c$, Q-factor $Q$, peak value of modified Poynting vector \\cite{poynting,poynting4} normalized to the average accelerating field $S_{c\\;max}\/E_{acc}^2$) as a function of the iris radius $a$ have been calculated with ANSYS Electronics Desktop \\cite{hfss_website}. The results are reported in Fig. \\ref{fig:cell_parameters}.\n\n\n\nAccording to beam dynamics calculations and single bunch beam break up limits, an average iris radius $\\langle a \\rangle$=3.2 mm has been taken into account (the corresponding parameters are given in Tab. \\ref{tab:a_3.2}) \\cite{vaccarezza_eaac2017}.\n\n\n\n\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[scale=1]{cell_parameters.pdf}\n\\caption{Single cell parameters as a function of the iris radius.}\n\\label{fig:cell_parameters}\n\\end{figure}\n\n\\begin{table}[hbt]\n\\center{\n\\caption{Single cell parameters for an iris radius of 3.2 mm.}\n  \\label{tab:a_3.2}\n\\begin{tabular}{lcr}\n\\midrule\niris radius $a$ [mm] & 3.2\\\\\niris thickness $t$ [mm]& 2.5\\\\\ncell radius $b$ [mm] & 10.139\\\\\ncell length $d$ [mm]& 8.332\\\\\n$R$ [M$\\Omega$\/m] & 93\\\\\n$v_g\/c$ [$\\%$] & 1.382\\\\\n$Q$ & 6396\\\\\n$S_{c\\;max}\/E_{acc}^2$ [A\/V] & $3.9 \\cdot 10^{-4}$\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\\section{Analytical optimization of structure effective shunt impedance}\n\nThe accelerating gradient distribution along the structure after one filling time $t_f$ is given by the formula \\cite{anal_grudiev}:\n\\begin{linenomath*}\n\\begin{equation}\\label{eq:G}\nG(z,t_f) = G_0[t_f-\\tau(z)] g(z),\n\\end{equation}\n\\end{linenomath*}\nwhere $z$ is the longitudinal position, $\\tau(z) = \\int_{0}^{z} \\frac{dz^\\prime}{v_g(z^\\prime)}$ is the signal time delay and $g(z)$ is defined as:\n\\begin{linenomath*}\n\\begin{equation}\ng(z)=\\sqrt{\\frac{v_g(0)}{v_g(z)}} \\sqrt{\\frac{R(z)Q(0)}{R(0)Q(z)}} e^{-\\frac{1}{2}\\int_{0}^{z} \\frac{\\omega}{v_g(z^\\prime)Q(z^\\prime)} dz^\\prime}.\n\\end{equation}\n\\end{linenomath*}\n$G_0$ is the gradient at the beginning of the structure given by:\n\\begin{linenomath*}\n\\begin{equation}\nG_0(t)=\\sqrt{\\frac{\\omega R(0) P_0(t)}{v_g(0) Q(0)}}.\n\\end{equation}\n\\end{linenomath*}\n$P_0(t)$ is the input RF power and, due to the SLED, is given by:\n\\begin{linenomath*}\n\\begin{equation}\nP_0(t)=P_k \\cdot k_{SLED}^2(t),\n\\end{equation}\n\\end{linenomath*}\nwhere $P_k$ is the power from the klystron and $k_{SLED}(t)$ is the SLED electric field gain factor \\cite{farkas}.\n\nIntegrating Eq. \\eqref{eq:G} along the structure length $L_s$, we obtain the accelerating voltage $V_a$.\n\nThe efficiency of the structure is given by the effective shunt impedance per unit length $R_s$ defined as \\cite{neal}:\n\\begin{linenomath*}\n\\begin{equation}\\label{eq:R_s}\nR_s = \\frac{V_a^2}{P_k L_s} = \\frac{V_a \\langle G \\rangle}{P_k} = \\frac{\\langle G \\rangle^2 L_s}{P_k},\n\\end{equation}\n\\end{linenomath*}\n where $\\langle G \\rangle = V_a\/L_s$ is the average accelerating gradient. \n\n\n\nFor a constant impedance (CI) structure $R_s$, as a function of the section attenuation $\\tau_s$ ($= \\frac{\\omega}{2 v_g Q}L_s$), is given by \\cite{leduff}:\n\\begin{linenomath*}\n\\begin{align}\\label{eq:Rs_CI}\nR_s =\\;&2 \\tau_s R \\left\\{ \\frac{1 - \\frac{2 Q_l}{Q_e}}{\\tau_s} \\left( 1 - e^{-\\tau_s} \\right) + \\right.  \\nonumber \\\\ &\\left. + \\frac{\\frac{2 Q_l}{Q_e} \\left[ 2 - e^{- \\left( \\frac{\\omega t_k}{2 Q_l} - \\tau_s \\frac{Q}{Q_l} \\right)} \\right]}{\\tau_s \\left( 1 - \\frac{Q_l}{Q_e} \\right)} \\left( e^{-\\tau_s \\frac{Q_l}{Q_e}} - e^{-\\tau_s} \\right) \\right\\}^2,\n\\end{align}\n\\end{linenomath*}\nwhere $Q_l = \\frac{Q_0 Q_e}{Q_0 + Q_e}$ is the loaded Q-factor of SLED (being $Q_e$ the external quality factor).\nFor a constant gradient (CG) structure $R_s$ is given by \\cite{farkas}:\n\\begin{linenomath*}\n\\begin{align}\\label{eq:Rs_CG}\nR_s =\\;&R \\frac{2 \\tau_s}{1+\\tau_s} \\left\\{ 1 - \\frac{2 Q_l}{Q_e} + \\frac{2 Q_l}{Q_e} \\left[ 2 - e^{-\\frac{\\omega t_k}{2 Q_l}} \\left( \\frac{1 + \\tau_s}{1 - \\tau_s} \\right)^{\\frac{Q}{2 Q_l}} \\right] \\cdot \\right.  \\nonumber \\\\ &\\left. \\cdot \\frac{1 - \\tau_s}{2 \\tau_s} \\frac{1}{1-Q\/2Q_l} \\left[ \\left( \\frac{1 + \\tau_s}{1 - \\tau_s} \\right)^{1 - \\frac{Q}{2 Q_l}} - 1 \\right] \\right\\}^2.\n\\end{align}\n\\end{linenomath*}\nFigure \\ref{fig:Rs_CI_CG} shows $R_s$ as a function of $\\tau_s$ for both structures, with the parameters of Tabs. \\ref{tab:RF_parameters} and  \\ref{tab:a_3.2}. In both cases the value of the external Q-factor $Q_e$ of the SLED has been chosen in order to maximize $R_s$. \n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[scale=1]{Rs_CI_CG.pdf}\n\\caption{Effective shunt impedance per unit length for the CI and CG structure.}\n\\label{fig:Rs_CI_CG}\n\\end{figure}\n\nThe accelerating gradient for a CI structure is given by \\cite{leduff}:\n\\begin{linenomath*}\n\\begin{align}\nG(z,t_f) = &\\sqrt{\\frac{\\omega}{v_g} \\frac{R}{Q} P_k}\\;e^{-\\tau_s\\frac{z}{L_s}} \\cdot \\nonumber \\\\ &\\cdot \\left\\{ 1 + \\frac{2 Q_l}{Q_e} \\left[ e^{- \\frac{\\omega (L_s - z)}{2 v_g Q_l}} \\left( 2 - e^{- \\frac{\\omega \\left(t_k - t_f\\right)}{2 Q_l}} \\right) - 1 \\right] \\right\\},\n\\end{align}\n\\end{linenomath*}\nwhile for a CG structure is given by \\cite{farkas}:\n\\begin{linenomath*}\n\\begin{align}\nG(z,t_f) = &\\sqrt{\\frac{2 \\tau_s}{1 + \\tau_s} \\frac{R}{L_s} P_k} \\left\\{ 1 + \\frac{2 Q_l}{Q_e} \\cdot \\right. \\nonumber \n\\\\ &\\left. \\cdot \\left[ \\left( \\frac{1 + \\tau_s \\left(1 -  \\frac{2z}{L_s}\\right)}{1 - \\tau_s} \\right)^{- \\frac{Q}{2 Q_l}} \\left( 2 - e^{-\\frac{\\omega \\left(t_k - t_f\\right)}{2 Q_l}} \\right) - 1 \\right] \\right\\}.\n\\end{align}\n\\end{linenomath*}\nThe previous formulas \\eqref{eq:Rs_CI},\\eqref{eq:Rs_CG} allow calculating the optimum $\\tau_s$ (=$\\tau_{s0}$) that maximize $R_s$. This fixes also the filling time of the structure, i.e. the compressed pulse length and allows to calculate the corresponding gradient profiles given in Figure \\ref{fig:G_CI_CG}.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[scale=1]{G_CI_CG.pdf}\n\\caption{Normalized gradient distribution along the structure for the CI and CG structure.}\n\\label{fig:G_CI_CG}\n\\end{figure}\n\nThe final optimum structures parameters are summarized in Table \\ref{tab:CI_vs_CG}.\n\n\\begin{table}[hbt]\n\\center{\n\\caption{CI and CG structure parameters (analytical study).}\n  \\label{tab:CI_vs_CG}\n\\begin{tabular}{lcr}\n\\toprule\n\\textbf{Parameter} & \\textbf{CI} & \\textbf{CG}\\\\\n\\midrule\n$R_s$ \\SI {}{[M\\ohm\/m]} & 343 & 344\\\\\nOptimal structure length $L_s$ [m] & 0.474 & 0.432\\\\\nFilling time $t_f$ [ns] & 114 & 118\\\\\nExternal Q-factor $Q_e$ of SLED & 20030 & 21170\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Numerical optimization}\n\nIn the analytical study, the CG solution is approximated because of the assumption of constant $R\/Q$ \\cite{lapostolle} along the structure. On the other hand in the CI case it is quite easy to verify that, in the VHG case, one exceeds the maximum value of $S_{c\\;max}$ that allows to have a Breakdown rate (BDR) lower than $10^{-6}$ bpp\/m \\cite{poynting,poynting4}. For these reasons we also performed a numerical study.\n\nTo this purpose we have considered a linear tapering of the irises as sketched in Figure \\ref{fig:tapering}, defined by the modulation angle $\\theta$. We have then calculated (by Eq. \\eqref{eq:G}) the gradient profile along the structure for different $\\theta$ and $L_s$. In the calculation we have used the polynomial fits of the single cell parameters illustrated in section \\ref{sec:single_cell}. From the gradient profiles we have finally calculated the effective shunt impedance per unit length and the peak value of the modified Poynting vector.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics*[width=180pt]{linear_tapering_2.png}\n\\caption{Sketch of the linear iris tapering.}\n\\label{fig:tapering}\n\\end{figure}\n\nFigure \\ref{fig:BFL_G} shows, as an example, the normalized gradient profile as a function of $\\theta$ for $L_s$ equal to 0.5 m. $R_s$ and $S_{c\\;max}$ (VHG case), as a function of $\\theta$ and for different $L_s$, are given in Figures \\ref{fig:BFL_Rs} and \\ref{fig:BFL_Sc}. In Figure \\ref{fig:BFL_Sc} we have also reported the maximum value of $S_{c\\;max}$ that, according to the scaling law given in \\cite{poynting}, allows having a BDR lower than $10^{-6}$ bpp\/m. From the plot it is quite easy to note that the 0.4 m and 0.5 m long structures have the same efficiency while the 0.667 m case is worse. Concerning $S_{c\\;max}$, the 0.4 m solution is better but requires, on the other hand, a larger number of structures per unit length. In conclusion the 0.5 m case with $\\theta$=\\SI{0.1}{\\degree} has been chosen as the design baseline for the X-band linac.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[scale=1]{BFL_G_theta_60cells_fl.pdf}\n\\caption{Normalized gradient after one filling time as function of the modulation angle (0.5 m case).}\n\\label{fig:BFL_G}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[scale=1]{BFL_Rs_48_60_80cells.pdf}\n\\caption{Effective shunt impedance per unit length as a function of the modulation angle for three structure lengths.}\n\\label{fig:BFL_Rs}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics*[scale=1]{BFL_Sc_max_48_60_80cells_80MVm.pdf}\n\\caption{Peak value of modified Poynting vector as a function of the modulation angle for three structure lengths (VHG case).}\n\\label{fig:BFL_Sc}\n\\end{figure}\n\n\n\\section{Linac basic layout}\n\nAccording to the results of numerical study, the TW X-band accelerating sections optimized for the EuPRAXIA@SPARC\\_LAB application are 0.5 m long and show an effective shunt impedance per unit length of \\SI {346}{\\mega\\ohm\/m}. Commercially available X-band klystrons \\cite{cpi_website} provide up to 50 MW peak power with \\SI {1.5}{\\micro s} long RF pulses. We have estimated the RF losses in the waveguide distribution system  of $\\approx$-7 dB and, as a consequence, $\\approx$40 MW available input power. The basic RF module of the EuPRAXIA@SPARC\\_LAB X-band linac can be conveniently composed by a group of 8 TW sections assembled on a single girder and powered by one (for HG) or two (for VHG) klystrons by means of one pulse compressor system and a waveguide network splitting and transporting the RF power to the input couplers of the sections. The sketch of the basic module is given in Fig. \\ref{fig:RF_layout} while the final main linac parameters are shown in Tab. \\ref{tab:final_parameters}.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=180pt]{RF_layout.png}\n\\caption{RF power distribution layout of a single module for the HG and VHG cases.}\n\\label{fig:RF_layout}\n\\end{figure}\n\n\\begin{table}[hbt]\n\\center{\n\\caption{X-band linac parameters.}\n  \\label{tab:final_parameters}\n\\begin{tabular}{lcr}\n\\midrule\nFrequency of operation [GHz] & \\multicolumn{2}{c}{11.9942}\\\\\nRF pulse length $t_f$ [ns] & \\multicolumn{2}{c}{129}\\\\\nUnloaded Q-factor $Q_0$ of SLED & \\multicolumn{2}{c}{180000}\\\\\nExternal Q-factor $Q_e$ of SLED & \\multicolumn{2}{c}{21800}\\\\\n$a$ first-last cell [mm] & \\multicolumn{2}{c}{$3.636 - 2.764$}\\\\\nStructure length $L_s$ [m] & \\multicolumn{2}{c}{0.5}\\\\\nActive length $L_t$ [m] & \\multicolumn{2}{c}{16}\\\\\nNo. of structures $N_s$ & \\multicolumn{2}{c}{32}\\\\\n$v_g\/c$ first-last cell [\\%] & \\multicolumn{2}{c}{$2.23 - 0.77$}\\\\\n$R_s$ \\SI {}{[\\mega\\ohm\/m]} & \\multicolumn{2}{c}{346}\\\\\n& HG & VHG\\\\\nAverage gradient $\\langle G \\rangle$ [MV\/m] & 57 & 80\\\\\nEnergy gain $W_{gain}$ [MeV] & 912 & 1280\\\\\nTotal Required RF power $P_{RF}$ [W] & 150 & 296\\\\\nNo. of klystrons $N_k$ & 4 & 8\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\n\\section{Conclusions}\nIn the paper we have illustrated the preliminary RF design of the EuPRAXIA@SPARC\\_LAB X-band linac. It has been done performing e.m. simulations of the single cell, and analytical and numerical optimization of the structure efficiency taking into account the available space, local field quantity (modified Poynting vector) minimization and available commercial klystrons. The final linac\nlayout and main parameters have been finally shown.\n\n\\section*{Acknowledgments}\n\nThis work was supported by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 653782.\n\n\n\n\n\n\\bibliographystyle{elsarticle-num}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Acknowledgements}\nThis manuscript has been authored under contract number\nDE-AC02-98CH10886 with the U.S.~Department of Energy.  Accordingly,\nthe U.S. Government retains a non-exclusive, royalty-free license to\npublish or reproduce the published form of this contribution, or allow\nothers to do so, for U.S.~Government purposes.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\nIntermediate band solar cells (IBSCs) \\cite{Luque_PRL_1997,Okasa_APR_2015} constitute an attractive approach to develop next-generation solar cells, since they offer the potential to significantly exceed the single-junction Shockley-Queisser (detailed balance) photovoltaic efficiency limit. \\cite{Shockley_JAP_1961} This is achieved via introduction of an intermediate band (IB) lying energetically within the band gap of a host matrix semiconductor. In the case that the IB is electrically isolated from both the valence band (VB) and conduction band (CB) of the host matrix semiconductor -- via gaps in the density of states -- carrier generation due to absorption of photons having energy less than the band gap $E_{g} = E_{\\scalebox{0.7}{\\textrm{CB}}} - E_{\\scalebox{0.7}{\\textrm{VB}}}$ of the host matrix semiconductor can occur via two-step photon absorption (TSPA). In a hole-based IBSC, TSPA proceeds via (i) an electron in the IB being promoted to the CB via absorption of a sub-band gap photon having energy $E_{\\scalebox{0.7}{\\textrm{H}}} = E_{\\scalebox{0.7}{\\textrm{CB}}} - E_{\\scalebox{0.7}{\\textrm{IB}}}$, and (ii) subsequent promotion of the resulting IB hole to the VB via the absorption of a second sub-band gap photon having energy $E_{\\scalebox{0.7}{\\textrm{L}}} = E_{\\scalebox{0.7}{\\textrm{IB}}} - E_{\\scalebox{0.7}{\\textrm{VB}}}$. The presence of the IB therefore enhances the photocurrent generated by a single-junction solar cell at fixed illumination, by allowing absorption of photons having energy $< E_{g}$, while the electrical isolation of the IB from the host matrix CB and VB ensures that the open-circuit voltage $V_{\\scalebox{0.7}{\\textrm{OC}}}$ is determined by $E_{g}$ rather than being limited by the sub-band gap energies $E_{\\scalebox{0.7}{\\textrm{L}}}$ and $E_{\\scalebox{0.7}{\\textrm{H}}}$. \\cite{Luque_PRL_1997,Luque_AM_2010}\n\nDue to their promise of high efficiency -- with a predicted detailed balance limit of 63.8\\% under concentrated illumination \\cite{Luque_AM_2010,Luque_PRL_1997,Okasa_APR_2015} -- IBSCs have attracted significant research interest. However, despite the simplicity of the underlying concepts, practical realisation of IBSCs has proved extremely challenging. \\cite{Marti_JPE_2013,Okasa_APR_2015} Practical efforts to realise IBSCs have to date centred primarily on two approaches to introduce an IB into the band gap of a host matrix semiconductor, relying on (i) a bound electron or hole ground state in a three-dimensional quantum confined heterostructure possessing a discrete density of states, e.g.~in a quantum dot (QD), or (ii) an IB formed via incorporation of a small concentration of a substitutional impurity, e.g.~in a highly-mismatched alloy (HMA) such as dilute nitride GaN$_{x}$As$_{y}$P$_{1-x-y}$. \\cite{Kudraweic_PRA_2014} The performance of QD-IBSCs has been limited by a combination of poor sub-band gap absorption, which limits short-circuit current density $J_{\\scalebox{0.7}{\\textrm{SC}}}$, and short radiative lifetimes $\\tau_{\\scalebox{0.7}{\\textrm{rad}}}$ for carriers occupying IB states, resulting in loss of carriers from the IB via radiative recombination and so reducing $V_{\\scalebox{0.7}{\\textrm{OC}}}$. Also, the presence of insufficiently large band offsets in QDs can lead to loss of carriers via thermionic emission (e.g.~from the IB to CB in an electron-based QD-IBSC). \\cite{Ramiro_IEEEJP_2015} The use of localised impurity states in HMA-IBSCs also presents fundamental issues, increasing losses associated with non-radiative (Shockley-Read-Hall) recombination of carriers occupying IB states at defect sites, thereby limiting carrier extraction and degrading overall efficiency.\n\n\n\nAs a result, research efforts to realise IBSCs have increasingly shifted away from conventional platforms such as type-I QDs, and towards novel materials and heterostructures whose electronic and optical properties offer the potential to overcome the aforementioned limitations. In particular, there is increasing interest in heterostructures having type-II band alignment, \\cite{Kechiantz_N_2007,Tayagaki_APL_2012,Kechiantz_PP_2015,Takeshi_APL_2016} due to their potential to suppress radiative losses as a consequence of the intrinsically large radiative lifetimes associated with their bound eigenstates (resulting from reduced electron-hole spatial overlaps), \\cite{Cuadra_PE_2002,Nishikawa_APL_2012} as well as reduced intra-band carrier relaxation via non-radiative (Auger) recombination. \\cite{Tomic_APL_2011,Tomic_APL_2013} Much of the work to date on type-II QD-IBSC systems has centred on InAs\/GaAs$_{1-x}$Sb$_{x}$ QDs -- an electron-based IBSC, in which the IB is formed by the lowest energy bound electron states in the InAs dot, and holes are localised in a GaAs$_{1-x}$Sb$_{x}$ quantum well buffer layer -- where increases in QD uniformity and number density have been reported in epitaxial growth of vertical QD stacks. \\cite{Ban_APL_2010,Liu_SEMSC_2012,Hatch_OE_2014,Cheng_SEMSC_2016} Experimental investigations have revealed increased $J_{\\scalebox{0.7}{\\textrm{SC}}}$ compared to conventional type-I InAs\/GaAs QDs, \\cite{Liu_SEMSC_2012} in line with the theoretically predicted increase in the radiative lifetime of IB states in these structures. \\cite{Tomic_APL_2013} While these promising results highlight the potential of type-II heterostructures as candidate IBSCs, the InAs\/GaAs(Sb) system suffers from a relatively low CB offset, leading to carrier leakage via thermionic emission from the IB at room temperature. \\cite{Cheng_SEMSC_2016} Furthermore, the sub-band gaps in InAs\/GaAs(Sb) QDs are far from the optimum values $E_{\\scalebox{0.7}{\\textrm{H}}} = 0.97$ eV and $E_{\\scalebox{0.7}{\\textrm{L}}} = 0.45$ eV required to maximise the calculated detailed balance efficiency for an IBSC based on a GaAs host matrix semiconductor. \\cite{Wang_IETO_2014}\n\nWhat is therefore required is to broaden investigations to additional type-II heterostructure systems, focusing on the ability to engineer the electronic properties so as to reliably tune the energy and character of the IB states in order to maximise overall efficiency. One such class of novel type-II heterostructures are GaAs$_{1-x}$Sb$_{x}$\/GaAs quantum rings (QRs). These hole-based IBSCs -- in which the IB is formed by the highest energy bound hole eigenstate in the QR -- have attracted increasing attention due not only to their type-II band alignment, but also to their large band offsets, which are expected to mitigate carrier loss via thermionic emission from the IB. Experimental analysis of prototype IBSCs based on type-II GaAs$_{1-x}$Sb$_{x}$\/GaAs vertical QR stacks has revealed several promising properties compared to conventional QD-IBSCs, including (i) enhanced TSPA and external quantum efficiency, \\cite{Wagener_JAP_2014_2,Shoji_AIPA_2017} (ii) reduced losses via radiative recombination, leading to improved carrier extraction and overall efficiency, \\cite{Hwang_JAP_2012} and (iii) recovery of $V_{\\scalebox{0.7}{\\textrm{OC}}}$ under concentrated illumination. \\cite{Tsai_OE_2014,Fujita_PP_2015,Montesdeoca_SEMSC_2018}\n\n\n\nDespite numerous and ongoing experimental investigations of type-II GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs for IBSC applications, there is little detailed information available in the literature regarding their electronic properties from a theoretical perspective. Here, we present a combined analytical and numerical analysis of the electronic properties of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs and demonstrate that minor changes in morphology, compatible with established epitaxial growth, can be exploited to tune the QR hole ground state (IB) to an optimum energy to maximise IBSC efficiency. Using an analytical analysis -- based on a solution of the time-independent Schr\\\"{o}dinger equation for a cylindrical QR of infinite potential depth -- we highlight that the QR geometry offers significant flexibility, compared to the conventional QD geometry, to engineer the valence band (VB) structure of GaAs$_{1-x}$Sb$_{x}$\/GaAs structures for IBSC applications. Our numerical calculations -- based on a multi-band \\textbf{k}$\\cdot$\\textbf{p} Hamiltonian, and including full strain and piezoelectric effects -- corroborate this finding. We highlight that type-II GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs are ideally suited to IBSC applications, due not only to their intrinsically large radiative lifetimes, but also due to their large VB offsets, which can be expected to mitigate thermionic emission of holes from the IB. Furthermore, the strong confinement of the highest energy hole (IB) states in these QRs is expected to mitigate the impact of miniband formation via electronic coupling between QRs in high-density stacks, which has been demonstrated to lead to a closing of the gap in the density of states between the IB and CB in electron-based InAs\/GaAs QD stacks. \\cite{Tomic_PRB_2010}\n\nWe undertake a numerical optimisation of the QR morphology, by varying the QR dimensions and alloy composition, to identify structures which allow to achieve optimum IB energy so as to maximise IBSC efficiency. Our results confirm the potential of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs for IBSC applications, and provide guidelines for the growth of suitable structures for prototype IBSCs.\n\n\n\nThe remainder of this paper is organised as follows. In Sec.~\\ref{sec:theoretical_model} we outline the theoretical models we have developed to calculate the electronic properties of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs, describing analytical and numerical approaches in Secs.~\\ref{sec:theoretical_model_analytical} and~\\ref{sec:theoretical_model_numerical} respectively. OWe present our results in Sec.~\\ref{sec:results}, beginning in Sec.~\\ref{sec:results_analytical} with an analysis of the QR ground state obtained from the analytical model. In Sec.~\\ref{sec:results_numerical} we describe the electronic structure of real GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs, based on numerical multi-band \\textbf{k}$\\cdot$\\textbf{p} calculations. Finally, in Sec.~\\ref{sec:conclusions} we summarise and conclude.\n\n\n\n\\section{Theoretical model}\n\\label{sec:theoretical_model}\n\nIn this section we describe the theoretical approaches we have applied to investigate the electronic properties of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs. We begin in Sec.~\\ref{sec:theoretical_model_analytical} with a description of the analytical solution of Schr\\\"{o}dinger's equation for a QR of infinite potential depth, and in Sec.~\\ref{sec:theoretical_model_numerical} describe numerical strain relaxation and multi-band \\textbf{k}$\\cdot$\\textbf{p} calculations for realistic QRs.\n\n\n\n\\subsection{Analytical: solution of Schr\\\"{o}dinger's equation}\n\\label{sec:theoretical_model_analytical}\n\n\nOur analytical analysis starts via the solution of the time-independent Schr\\\"{o}dinger equation for the eigenstates of an idealised, [001]-oriented cylindrical QR of infinite potential depth. The QR is taken to have inner and outer radii $a_{1}$ and $a_{2}$ in the plane perpendicular to the [001] direction, and height $h$ along [001]. A schematic illustration of the QR geometry is shown in Fig.~\\ref{fig:analytical}(a). Choosing the origin of a cylindrical polar coordinate system $( r, \\phi, z )$ to lie at the centre of the base of the QR, the potential -- which is axially symmetric about the [001], or $z$, direction and hence independent of $\\phi$ -- is\n\n\\begin{equation}\n\tV (r, h) = \\left\\{\n\t\t\\begin{array}{lr}\n\t\t\t0 \\; , & a_{1} < r < a_{2}~\\textrm{and}~0 < z < h \\\\\n\t\t\t\\infty, & \\textrm{otherwise}\n\t\t\\end{array}\n\t\t\\right. \\, .\n\t\\label{eq:quantum_ring_potential}\n\\end{equation}\n\n\n\nWe proceed via separation of variables by writing the QR eigenstates $\\psi_{lmn} ( r, \\phi, z ) = R_{lm} ( r ) \\Phi_{m} ( \\phi ) Z_{n} ( z )$, yielding three separable differential equations (one in each of the radial, polar and longitudinal coordinates $r$, $\\phi$ and $z$). The solution of the $\\phi$ and $z$ equations are as in the conventional cylindrical QD of infinite potential depth (corresponding here to $a_{1} = 0$). The solution in the polar ($\\phi$) direction is trivial due to the axial symmetry of the potential about the $z$ direction, with $\\Phi_{m} ( \\phi ) = \\frac{ 1 }{ \\sqrt{ 2 \\pi } } \\, e^{ i m \\phi }$ for integer $m$. Along the $z$ direction the eigenstates are those of the infinite square well\n\n\\begin{equation}\n    Z_{n} ( z ) = \\sqrt{ \\frac{ 2 }{ h } } \\sin \\left( \\frac{ n \\pi z }{ h } \\right) \\, .\n    \\label{eq:general_solution_longitudinal}\n\\end{equation}\n\nWithin the QR $V ( r, z ) = 0$ and the radial equation reduces to the (cylindrical) Bessel equation, the general solution of which is $R_{lm} ( r ) = A_{lm} J_{m} ( kr ) + B_{lm} Y_{m} ( kr )$, where $J_{m} ( kr )$ and $Y_{m} ( kr )$ are respectively the Bessel functions of the first and second kind. Here, $l$ is a positive integer which indexes the (discrete) allowed values $k_{lm}$ of the radial wave vector $k$. To this point, the general solution is identical to that of a cylindrical QD of infinite potential depth. The difference in the QR case arises due to the presence of the central barrier for $r \\leq a_{1}$. Due to the presence of the central barrier we seek radial wave functions satisfying the boundary conditions $R_{lm} ( a_{1} ) = 0$ and $R_{lm} ( a_{2} ) = 0$. Applying the first of these conditions allows us to solve for $B_{lm}$ in terms of $A_{lm}$, giving\n\n\\begin{equation}\n    R_{lm} ( r ) = A_{lm} \\left( J_{m} ( kr ) - \\frac{ J_{m} ( k a_{1} ) }{ Y_{m} ( k a_{1} ) } \\, Y_{m} ( kr ) \\right) \\, ,\n    \\label{eq:radial_wave_function}\n\\end{equation}\n\n\\noindent\nwhere the constant $A_{lm}$ can be determined via normalisation. Applying the second boundary condition then yields the transcendental equation\n\n\\begin{equation}\n    J_{m} ( k a_{1} ) \\, Y_{m} ( \\rho k a_{1} ) - J_{m} ( \\rho k a_{1} ) \\, Y_{m} ( k a_{1} ) = 0 \\, ,\n    \\label{eq:quantum_ring_transcendental}\n\\end{equation}\n\n\\noindent\nwhere we have defined $\\rho = \\frac{ a_{2} }{ a_{1} }$ as the ratio of the outer and inner QR radii (so that $\\rho k a_{1} = k a_{2}$). The wave vector $k_{lm}$ associated with the radial wave function $R_{lm} ( r )$ is then, for a given value of $m$, determined via the $l^{\\scalebox{0.7}{\\textrm{th}}}$ root $k_{lm} a_{1}$ of Eq.~\\eqref{eq:quantum_ring_transcendental}. Defining $a = a_{2} - a_{1}$ as the radial thickness of the QR, we note that Eq.~\\eqref{eq:radial_wave_function} reduces to $R_{lm} ( r ) = A_{lm} J_{m} ( kr )$ in the limit $a_{1} \\to 0$ ($a_{2} \\to a$), with Eq.~\\eqref{eq:quantum_ring_transcendental} correspondingly reducing to $J_{m} ( ka ) = 0$, yielding the well-known solution of Schr\\\"{o}dinger's equation for a cylindrical QD of radius $a$ as the QR is transformed into a QD via the removal of the central potential barrier.\n\nThe energies of the QR eigenstates are given by the sum of the in- and out-of-plane contributions\n\n\\begin{equation}\n    E_{lmn} = \\frac{ \\hbar^{2} }{ 2 m_{0} } \\left( \\frac{ k_{lm}^{2} }{ m_{\\perp}^{\\ast} } + \\frac{ \\pi^{2} n^{2} }{ m_{\\parallel}^{\\ast} h^{2} } \\right) \\, ,\n    \\label{eq:quantum_ring_ground_state_energy}\n\\end{equation}\n\n\\noindent\nwhere $m_{0}$ is the free electron mass, and where $m_{\\parallel}^{\\ast}$ and $m_{\\perp}^{\\ast}$ are respectively the (relative) effective mass parallel and perpendicular to the [001] growth direction. The QR ground state is obtained for quantum numbers $( l, m, n ) = ( 1, 0, 1 )$.\n\n\n\n\\begin{figure*}[ht!]\n\t\\includegraphics[width=1.00\\textwidth]{.\/figure_1.pdf}\n\t\\caption{(a) Schematic illustration of the QR geometry considered in this work, viewed side-on from the [100] direction (top) and top-down from the [001] direction (bottom). The cylindrical QR has inner and outer radii $a_{1}$ and $a_{2}$, radial thickness $a = a_{2} - a_{1}$, and height $h$ along the growth direction. (b) Variation of the ground state radial wave vector $k_{10}$, computed as the first root of Eq.~\\eqref{eq:quantum_ring_transcendental} for $m = 0$, as a function of the inner radius $a_{1}$ for a QR of fixed radial thickness $a$. As $a_{1}$ increases as a proportion of $a$, $k_{10}$ rapidly converges to a value of $\\frac{ \\pi }{ a }$ (dashed black line). The inset shows the Bessel functions $J_{0} (x)$ (solid blue line) and $Y_{0} (x)$ (dashed blue line), as well as the left-hand side of Eq.~\\eqref{eq:quantum_ring_transcendental} for $\\rho = 2$ (i.e.~$a_{1} = a$). (c) Contour map showing the confinement energy associated with the HH ground state in a GaSb\/GaAs QR having outer radius $a_{2} = 12$ nm, calculated via Eqs.~\\eqref{eq:quantum_ring_transcendental} and Eq.~\\eqref{eq:quantum_ring_ground_state_energy}, as a function of the QR inner radius $a_{1}$ and height $h$.}\n  \\label{fig:analytical}\n\\end{figure*}\n\n\n\n\\subsection{Numerical: multi-band \\textbf{k}$\\cdot$\\textbf{p} calculations}\n\\label{sec:theoretical_model_numerical}\n\nOur numerical analysis of the electronic properties of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs is based on multi-band \\textbf{k}$\\cdot$\\textbf{p} calculations. \\cite{Arkani_NUSOD_2019} We employ a supercell approach, by embedding a cylindrical GaAs$_{1-x}$Sb$_{x}$\/GaAs QR in a GaAs matrix, and relax the supercell by minimising the total elastic energy with respect to the components $\\epsilon_{ij}$ of the strain tensor to obtain the strain fields $\\epsilon_{ij} ( \\textbf{r} )$. The QR VB eigenstates are computed using a strain-dependent 6-band \\textbf{k}$\\cdot$\\textbf{p} Hamiltonian -- i.e.~the Luttinger-Kohn VB Hamiltonian, including Bik-Pikus strain-related terms -- which explicitly treats heavy-hole (HH), light-hole (LH) and spin-split-off (SO) VB states. The QR CB eigenstates are computed using a strain-dependent 1-band (effective mass) Hamiltonian. We note that the separate treatment of the CB and VB eigenstates has been chosen to circumvent the emergence of spurious solutions in full 8-band \\textbf{k}$\\cdot$\\textbf{p} calculations, which arise in the presence of large plane wave cut-off energies due to the strong inter-band coupling present in narrow-gap GaSb. Both the 6-band VB and 1-band CB calculations explicitly include the strain-induced piezoelectric potential, computed to second order for a given structure using the relaxed strain fields. The QR band offsets are computed firstly for an unstrained structure using the model solid theory -- assuming a natural (unstrained) VB offset of 0.58 eV between GaSb and GaAs \\cite{Wei_APL_1998,Hinuma_PRB_2014} -- with the position-dependent band edge energy (confining potential) profiles then computed via direct diagonalisation of the strain-dependent bulk 6- and 1-band \\textbf{k}$\\cdot$\\textbf{p} Hamiltonians at each real space grid point in the supercell.\n\nOur numerical calculations have been implemented using a plane wave (reciprocal space) approach, via the S\/Phi\/nX software library. \\cite{Marquardt_CPC_2010} We employ [001]-oriented supercells of size of 50 nm $\\times$ 50 nm $\\times$ 14 nm, and a plane wave cut-off energy equivalent to a real space resolution of 0.2 nm in each of the $x$, $y$ and $z$ directions. This choice of plane wave cut-off was validated by examining the convergence of the bound carrier eigenstate energies, which were found to vary by $<1$ meV with respect to further increases in the size of the plane wave basis set. All material parameters used in our calculations -- including lattice and elastic constants, band gaps, VB spin-orbit splitting energies and Luttinger parameters, electron effective masses, and CB and VB edge deformation potentials -- are taken from Ref.~\\onlinecite{Vurgaftman_JAP_2001}, with the exception of the first and second order piezoelectric coefficients, which are taken from Ref.~\\onlinecite{Bester_PRB_2006}. All calculations are performed at temperature $T = 300$ K.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nIn this section we present the results of our theoretical analysis, beginning in Sec.~\\ref{sec:results_analytical} with a description of trends in the electronic properties of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs based on the analytical treatment described in Sec.~\\ref{sec:theoretical_model_analytical}. Specifically, we describe (i) the rapid convergence of the ground state radial wave vector $k_{10}$ in a QR having fixed radial thickness $a = a_{2} - a_{1}$, allowing the ground state energy in a realistic QR to be easily and accurately estimated, and (ii) trends in the HH confinement energy in realistic GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs. In Sec.~\\ref{sec:results_numerical} we present the results of our numerical calculations, including the strain fields and band offsets (confining potentials) in relaxed QRs, as well as the electronic properties obtained from multi-band \\textbf{k}$\\cdot$\\textbf{p} calculations, allowing morphologies supporting optimised IBSC sub-band gaps to be identified.\n\n\n\n\\subsection{Analytical: ground state radial wave vector convergence and QR HH confinement energy}\n\\label{sec:results_analytical}\n\n\nWe begin by demonstrating a useful convergence property of the ground state radial wave vector $k_{10}$, obtained from the first ($l = 0$) root of Eq.~\\eqref{eq:quantum_ring_transcendental} for $m = 0$. Specifically, for a QR having radial thickness $a = a_{2} - a_{1}$, $k_{10}$ converges rapidly to $\\frac{ \\pi }{ a }$ with increasing inner radius $a_{1}$. This provides a useful approximation, $k_{10} \\approx \\frac{ \\pi }{ a }$, which can be used to estimate the radial contribution to the QR ground state energy (cf.~Eq.~\\eqref{eq:quantum_ring_ground_state_energy}).\n\nTo see that this is the case we note that, for large $x$, $J_{0} (x) \\approx \\sqrt{ \\frac{ 2 }{ \\pi x } } \\cos \\left( x - \\frac{ \\pi }{ 4 } \\right)$ and $Y_{0} (x) \\approx \\sqrt{ \\frac{ 2 }{ \\pi x } } \\sin \\left( x - \\frac{ \\pi }{ 4 } \\right)$, so that the $m = 0$ radial wave function can be written in the form\n\n\\begin{equation}\n    R_{l0} (r) \\approx C_{l0} \\sqrt{ \\frac{ 2 }{ \\pi k r } } \\cos \\left( kr + \\theta \\right) \\, ,\n    \\label{eq:radial_wave_function_limit}\n\\end{equation}\n\n\\noindent\nfor some phase $\\theta$ and normalisation constant $C_{l0}$. This approximate expression for $R_{l0} (r)$ satisfies the boundary conditions $R_{l0} ( a_{1} ) = 0$ and $R_{l0} ( a_{2} ) = 0$ for radial wave vectors $k_{l0} \\sim \\frac{ l \\pi }{ a }$. We therefore deduce that, for a QR of fixed radial thickness $a$, as the inner radius $a_{1}$ increases, the first root $k_{10}$ of Eq.~\\eqref{eq:quantum_ring_transcendental} should converge to $\\frac{ \\pi }{ a }$.\n\nTo verify that this is the case we have computed the first root $k_{10} a_{1}$ of Eq.~\\eqref{eq:quantum_ring_transcendental} as a function of $a_{1}$ for a QR having fixed thickness $a$ (= 11.5 nm). The results of this analysis are summarised in Fig.~\\ref{fig:analytical}(b), where the solid green line shows the calculated variation of $k_{10}$ (in units of $\\frac{ \\pi }{ a }$) as a function of $\\frac{ a_{1} }{ a }$ ($= \\frac{ 1 }{ \\rho - 1 }$). The inset to Fig.~\\ref{fig:analytical}(b) shows the functions $J_{0} (x)$ (solid blue line) and $Y_{0} (x)$ (dashed blue line), as well as the left-hand side of Eq.~\\eqref{eq:quantum_ring_transcendental} for $m = 0$ and $\\rho = 2$ ($a_{2} = 2 a_{1}$, solid red line). Examining Fig.~\\ref{fig:analytical}(b), we in fact note \\textit{rapid} convergence of $k_{10}$ to $\\frac{ \\pi }{ a }$ with increasing $\\frac{ a_{1} }{ a }$. For $a_{1} = 0$ -- i.e.~for a cylindrical QD of radius $a$ -- $k_{10}$ is given by the first root of $J_{0} ( ka ) = 0$. This solution, highlighted by the open green circles in Fig.~\\ref{fig:analytical}(b) and its inset, is $k_{10} a \\approx 2.4048$ (the first zero of $J_{0} (ka)$), so that $k_{10} \\approx 0.7655 \\times \\frac{ \\pi }{ a }$. As $\\frac{ a_{1} }{ a }$ increases due to the inclusion of the central potential barrier to form a QR, $k_{10}$ then rapidly approaches $\\frac{ \\pi }{ a }$. For example, for $\\frac{ a_{1} }{ a } = 1$ ($\\rho = 2$), which corresponds well to real QR dimensions, we compute $k_{10} \\approx 0.9941 \\times \\frac{ \\pi }{ a }$.\n\nGenerally, we find $k_{10} = \\frac{ \\pi }{ a }$ to be an excellent approximation to the ground state radial wave vector for $\\frac{ a_{1} }{ a } \\gtrsim 1$ -- i.e.~for $a_{1} \\gtrsim \\frac{ a_{2} }{ 2 }$ or, equivalently, $\\rho \\lesssim 2$. For $a = 11.5$ nm we note that $\\rho = 2$ corresponds to an outer QR radius $a_{2} = 2 a_{1} = 23$ nm, dimensions typical of epitaxially grown GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs, which will be analysed in further detail in Sec.~\\ref{sec:results_numerical}. This rapid convergence of $k_{10}$ then provides a simple and reliable approach to estimate the radial contribution to the QR ground state energy, circumventing the requirement to numerically compute the roots of Eq.~\\eqref{eq:quantum_ring_transcendental}, with the overall accuracy of the corresponding estimate of the ground state energy $E_{101}$ being better for small QR aspect ratios $\\frac{ h }{ 2 a_{2} }$ (where the radial component contributes a small proportion of the total confinement energy, cf.~Eq.~\\eqref{eq:quantum_ring_ground_state_energy}).\n\n\n\nWe have used Eq.~\\eqref{eq:quantum_ring_ground_state_energy} to estimate the confinement energy associated with the HH ground state in a GaSb\/GaAs QR. To do so we set $m_{\\parallel}^{\\ast} = ( \\gamma_{1} - 2 \\gamma_{2} )^{-1}$ and $m_{\\perp}^{\\ast} = ( \\gamma_{1} + 2 \\gamma_{2} )^{-1}$, which are the bulk HH VB edge effective masses admitted by the 6-band Luttinger-Kohn Hamiltonian, \\cite{Luttinger_PR_1955,Vurgaftman_JAP_2001} where $\\gamma_{1}$ and $\\gamma_{2}$ are the VB Luttinger parameters. Following Ref.~\\onlinecite{Vurgaftman_JAP_2001} we set $\\gamma_{1} = 13.4$ and $\\gamma_{2} = 4.7$ and respectively obtain $m_{\\parallel}^{\\ast} = 0.250$ and $m_{\\perp}^{\\ast} = 0.044$ for the (relative) HH effective masses parallel and perpendicular to the [001] direction in GaSb. The results of our calculations are summarised in Fig.~\\ref{fig:analytical}(c), in which the solid red lines are contours of constant HH ground state (confinement) energy as a function of inner radius $a_{1}$ and height $h$, for a QR having fixed outer radius $a_{2} = 12$ nm. We note from these results the flexibility offered by the QR geometry from the perspective of band structure engineering for IBSC applications: the HH confinement energy can readily be tuned across a broad range via relatively minor adjustments in QR morphology, allowing the IB energy to be tuned in a hole-based QR-IBSC. This provides distinct advantages compared to, e.g., equivalent GaAs$_{1-x}$Sb$_{x}$\/GaAs QDs, since the QR inner radius $a_{1}$ provides an additional parameter by which the electronic properties can be tuned. The realities of epitaxial growth do not in general allow $a_{1}$ to be fine tuned independently of the other QR dimensions $a_{2}$ and $h$, the relationships between which are in practice determined in large part by the (Stranski-Krastanov) strain relaxation mechanism that drives QR formation. However, it is generally observed that Stranski-Krastanov QR formation tends to fix $a_{2}$, with the inner radius $a_{1}$ then depending largely on growth rate. Correspondingly, QR heights are generally found to be in the range $h = 3 \\pm 2$ nm. \\cite{Khan_EMC_2016} Despite incomplete control over the precise morphology of individual QRs during epitaxial growth, we emphasise that GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs offer additional benefits for IBSC applications due to the nature of carrier localisation within these structures. To elucidate these properties requires a more detailed, quantitative analysis of the electronic properties, to which we now turn our attention.\n\n\n\n\\subsection{Numerical: strain, band offsets, carrier localisation and IBSC sub-band gaps}\n\\label{sec:results_numerical}\n\n\nWhile the analytical treatment of Secs.~\\ref{sec:theoretical_model_analytical} and~\\ref{sec:results_analytical} provides useful insight into some of the general features of the electronic properties of QRs, this approach neglects several key factors which play an important role in determining the nature of the electronic properties in a real semiconductor QR. Firstly, the large lattice mismatch which drives QR formation -- 7.2\\% in the case of GaSb\/GaAs -- produces large, strongly position dependent strain fields, which impact the electronic properties directly (as well as via the associated strain-induced piezoelectric potential). Secondly, the natural type-II band offsets between GaSb and GaAs produce confining potentials which are markedly different in nature for electrons and holes. Thirdly, the analytical treatment described above neglects band hybridisation effects, which can be expected to play a role in determining precise the nature of QR eigenstates in the presence of strain, quantum-confinement, spin-orbit coupling and narrow band gap, all of which are present in real GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs. In order to quantitatively understand the QR electronic structure we have therefore undertaken multi-band \\textbf{k}$\\cdot$\\textbf{p} calculations, based on the formalism described in Sec.~\\ref{sec:theoretical_model_numerical}. We provide here an overview of the results of these calculations, elucidating the electronic properties of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs and identifying optimised QR morphologies providing electronic properties well-suited to IBSC applications.\n\n\n\nFor our \\textbf{k}$\\cdot$\\textbf{p} calculations we begin by considering an exemplar GaSb\/GaAs QR having inner radius $a_{1} = 5$ nm, outer radius $a_{2} = 11.5$ nm and height $h = 3$ nm (dimensions typical of epitaxially grown QRs \\cite{Khan_EMC_2016}). The solid blue and red lines in Fig.~\\ref{fig:numerical}(a) respectively show linescans -- taken through the centre of the QR along the [100] direction -- of the calculated hydrostatic and biaxial components $\\epsilon_{\\scalebox{0.7}{\\textrm{hyd}}} = \\textrm{Tr}( \\epsilon )$ and $\\epsilon_{\\scalebox{0.7}{\\textrm{bia}}} = \\epsilon_{zz} - \\frac{1}{2} ( \\epsilon_{xx} + \\epsilon_{yy} )$ of the strain fields in the structure. The hydrostatic strain reaches values as low as $-10.5$\\% within the GaSb regions, reflecting that the QR is under significant compressive strain when grown epitaxially on GaAs. We note that strain relaxation in the structure acts to place the central GaAs region of the QR under a minor amount ($\\lesssim 1$\\%) of tensile strain, due to its being surrounded in the plane perpendicular to [001] by GaSb material having a larger lattice constant. The calculated biaxial strain resembles that associated with a cylindrical QD, \\cite{Andreev_JAP_1999} varying strongly at the interfaces with the surrounding barrier material, albeit with additional features due to the presence of the central GaAs barrier. We note that the strain-induced piezoelectric potential associated with this structure attains a maximum value of $\\approx 17$ meV throughout the calculational supercell, and has minimal impact on the electronic properties.\n\n\n\n\\begin{figure*}[ht!]\n\t\\includegraphics[width=1.00\\textwidth]{.\/figure_2.pdf}\n\t\\caption{(a) Linescan along the [100] in-plane direction, through the centre of the QR, showing the calculated hydrostatic (solid blue line) and biaxial (solid red line) components of the strain field for a GaSb\/GaAs QR having $a_{1} = 5$ nm, $a_{2} = 11.5$ nm and $h = 3$ nm. (b) Linescan along the [100] in-plane direction, through the centre of the QR, showing the calculated bulk band edge energies (band offsets) for the CB (solid blue line), and light-hole (LH; dashed green line) and heavy-hole (HH; solid red line) VBs, for the same QR as in (a). The dashed red line shows the energy of the highest energy bound hole state $h1$ -- i.e.~the intermediate band -- at energy $E_{\\protect\\scalebox{0.7}{\\textrm{IB}}}$, separated from the CB and VB edges by the sub-band gaps $E_{\\protect\\scalebox{0.7}{\\textrm{H}}}$ and $E_{\\protect\\scalebox{0.7}{\\textrm{L}}}$ respectively. (c) Side-on view, along the [010] direction, of the probability density associated with the lowest energy quasi-bound electron (green) and highest energy bound hole (orange) states in the same QR as in (a). (d) Top-down view, along the [001] direction, of the probability density associated with the same two states as in (c). (e) Lower energy sub-band gap $E_{\\protect\\scalebox{0.7}{\\textrm{L}}}$ as a function of QR height $h$ for GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs having inner and outer radii $a_{1} = 5$ nm and $a_{2} = 11.5$ nm, estimated based on Eqs.~\\eqref{eq:quantum_ring_transcendental} and~\\eqref{eq:quantum_ring_ground_state_energy}. (f) As in (e), but with the values of $E_{\\protect\\scalebox{0.7}{\\textrm{L}}}$ obtained from full numerical multi-band \\textbf{k}$\\cdot$\\textbf{p} calculations.}\n  \\label{fig:numerical}\n\\end{figure*}\n\n\n\nFigure~\\ref{fig:numerical}(b) shows a linescan of the band edge energies for the same GaSb\/GaAs QR, calculated using the strain fields shown (in part) in Fig.~\\ref{fig:numerical}(a). The band edge energy (confining potential) profiles are shown -- using solid blue, solid red and dashed green lines for the CB, HH VB and LH VB edges, respectively -- as a linescan along the [100] direction, through the centre of the QR. We firstly note the presence of large type-II band offsets in the QR. There exist strong confining potentials for holes within the GaSb region, reaching a maximum depth of $\\approx 820$ meV for HH states, while electrons are excluded from the GaSb region by a CB potential barrier having a maximum height of $\\approx 670$ meV. Due to the compressive strain in the GaSb region, the HH VB edge is pushed higher in energy than the LH VB edge, resulting in a larger confining potential for HH states and ensuring that the highest energy hole bound state -- i.e.~the IB in a hole-based GaAs$_{1-x}$Sb$_{x}$\/GaAs QR-IBSC -- is primarily HH-like. We therefore expect that hole states will be strongly confined within the GaSb region, experiencing strong annular localisation in the plane perpendicular to the [001] direction, as well as strong localisation along the [001] direction. We also note that the aforementioned minor tensile strain in the central barrier of the QR tends to reduce slightly the CB edge energy in the centre of the QR. This, combined with the presence of a confining potential for electrons in the plane perpendicular to the [001] direction, suggests the possibility of quasi-localised electron states residing in the central barrier region.\n\nThese expected trends in carrier localisation are verified by our calculated probability densities, shown in Figs.~\\ref{fig:numerical}(c) and~\\ref{fig:numerical}(d), which respectively show cross-sections through the centre of the QR of the calculated electron (green) and hole (orange) probability densities in the plane perpendicular to the [010] and [001] directions. We observe very strong localisation of the highest energy hole bound state within the QR, with only minimal penetration of the hole probability density into the GaSb region. We also note the presence of a resonant electron state within the central GaAs barrier of the QR, which our calculations identify as a consequence of the type-II band offsets  (cf.~Fig.~\\ref{fig:numerical}(b)). While this electron state is strongly localised in the plane of the QR, it is less strongly localised along [001], due to the absence of a strongly confining CB potential along that direction.\n\nWe note that the quasi-localised electron state for which the probability density is shown in Figs.~\\ref{fig:numerical}(c) and~\\ref{fig:numerical}(d) was not the lowest energy electron state identified in our calculation: it was in fact the sixth lowest energy electron state. The type-II band alignment in this GaSb\/GaAs QR results in a large range of delocalised electron states lying at and above the GaAs CB edge in energy, which are characterised by their probability density being excluded from the central region of the QR in the plane of the QR. (The precise number of delocalised states appearing in a given energy range in a numerical calculation depends on a combination of the supercell dimensions and the size of the plane wave basis set employed.) The quasi-bound electron state identified in our analysis lies only 31 meV above the GaAs CB edge in energy and, given its higher spatial overlap with the highest energy bound hole state compared to the delocalised CB edge states described above, transitions involving this state can be expected to contribute significantly to the band edge optical absorption. From the perspective of IBSC design and optimisation, the presence of this unusual quasi-bound electron state is particularly appealing. By varying the QR morphology the in-plane confinement of this electron state can be engineered so as to control the spatial overlap with the highest energy bound hole state. This overlap governs a key trade-off in an IBSC, between (i) the generation rate associated with carriers occupying IB states (i.e.~the generation of holes in the VB via absorption of photons having energy $\\geq E_{\\scalebox{0.7}{\\textrm{L}}}$), and (ii) the radiative lifetime for recombination involving electron and hole states (i.e.~recombination of quasi-bound CB electrons with bound IB holes via emission of photons having energy $> E_{\\scalebox{0.7}{\\textrm{H}}}$). Furthermore, the weak electron localisation along [001] suggests that photo-generated electrons can be readily collected from the QRs in a real IBSC structure, with hole extraction proceeding via excitation of holes from the IB to the GaAs VB via TSPA. Finally, the large VB offsets in these QRs result in very large ionisation energies of $\\approx 0.5$ eV for IB hole states. This ionisation energy is far in excess of the thermal energy in the range of temperatures relevant to IBSC operation, and so electrical leakage of carriers via thermionic emission from the IB should be effectively suppressed in these heterostructures.\n\n\n\nFor this GaSb\/GaAs QR we calculate that the highest energy bound hole state -- the energy of which is denoted by a horizontal dotted red line in Fig.~\\ref{fig:numerical}(b) -- lies 502 meV above the GaAs barrier VB edge, corresponding to IBSC sub-band gaps $E_{\\scalebox{0.6}{\\textrm{L}}} = 0.502$ eV and $E_{\\scalebox{0.6}{\\textrm{H}}} = E_{g} (\\textrm{GaAs}) - E_{\\scalebox{0.6}{\\textrm{L}}} = 0.922$ eV at temperature $T = 300$ K. We note that these sub-band gaps are -- in addition to being in good quantitative agreement with experimental measurements \\cite{Wagener_JAP_2014} -- close to the optimum values associated with an IBSC implemented using a GaAs host matrix. Specifically, for GaAs (which has $E_{g} = 1.42$ eV at $T = 300$ K) the optimum sub-band gaps $E_{\\scalebox{0.6}{\\textrm{L}}} = 0.45$ eV and $E_{\\scalebox{0.6}{\\textrm{H}}} = 0.97$ eV correspond to a detailed balance efficiency limit of $\\approx 58$\\% under concentrated illumination, close to the overall efficiency limit for an ideal IBSC. However, for a given host matrix band gap $E_{g}$ the detailed balance efficiency is a sensitive function of the sub-band gaps $E_{\\scalebox{0.6}{\\textrm{L}}}$ and $E_{\\scalebox{0.6}{\\textrm{H}}}$, reducing rapidly as the sub-band gaps are detuned from their optimum values. \\cite{Wang_IETO_2014} It is therefore desirable to engineer the band structure such that the sub-band gaps are as close as possible to their optimum energies. The calculated value of $E_{\\scalebox{0.6}{\\textrm{L}}}$ ($E_{\\scalebox{0.6}{\\textrm{H}}}$) for the exemplar GaSb\/GaAs QR considered above is only $\\approx 50$ meV higher (lower) than the optimum value, suggesting that minor changes in morphology will be sufficient to realise optimum sub-band gaps.\n\nIn order to identify optimised QR morphologies we therefore proceed by repeating our analysis as a function of (i) QR height $h$, and (ii) QR Sb composition $x$, for QRs having the same inner and outer radii $a_{1} = 5$ nm and $a_{2} = 11.5$ nm considered above. We note that we have chosen to vary the QR height rather than in-plane dimensions since (i) real GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs typically have low aspect ratios $\\frac{ h }{ 2a_{2} } \\approx 0.1$, (ii) the confinement energies in such low aspect ratio structures, and hence the sub-band gaps, are dominated by confinement along the [001] direction (cf.~Eq.~\\eqref{eq:quantum_ring_ground_state_energy}), and (iii) characterisation of epitaxially grown GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs has revealed significantly larger \\textit{relative} variations in $h$ than in $a_{1}$ or $a_{2}$. The results of these calculations are summarised in Figs.~\\ref{fig:numerical}(e) and~\\ref{fig:numerical}(f), which respectively show the sub-band gap energy $E_{\\scalebox{0.7}{\\textrm{L}}}$ calculated based on the analytical and numerical models. Results are shown for GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs having Sb compositions $x = 100$, 90, 80 and 70\\% using red, green, blue and orange closed circles, respectively.\n\nNote that since the analytical model assumes a confining potential of infinite depth, it is not capable of directly predicting $E_{\\scalebox{0.7}{\\textrm{L}}}$ (which is obtained from the full numerical calculations as the difference $E_{h1} - E_{\\scalebox{0.7}{\\textrm{VB}}}$ in energy between the highest energy bound hole state and host matrix VB edge). In order to estimate $E_{\\scalebox{0.7}{\\textrm{L}}}$ using the analytical model we have therefore (i) used Eqs.~\\eqref{eq:quantum_ring_transcendental} and~\\eqref{eq:quantum_ring_ground_state_energy} to calculate the confinement energy $\\Delta E_{h1}$ associated with the HH ground state in a QR of infinite potential depth, and (ii) used the (maximum) HH band offset $\\Delta E_{\\scalebox{0.7}{\\textrm{HH}}}$ extracted from a full numerical calculation to compute $E_{\\scalebox{0.7}{\\textrm{L}}} = \\Delta E_{\\scalebox{0.7}{\\textrm{HH}}} - \\Delta E_{h1}$. Due to its assumption of an infinitely deep confining potential, the analytical approach naturally overestimates $\\Delta E_{h1}$ -- in particular the contribution associated with confinement along [001] -- and hence underestimates $E_{\\scalebox{0.7}{\\textrm{L}}}$ for short QRs having $h \\lesssim 3$ nm. Nonetheless, we observe that the analytical model accurately captures the key trends observed in the results of the full numerical calculations, and provides reasonably good quantitative agreement with the results of the numerical calculations for $h \\gtrsim 3$ nm, across the full range of Sb compositions considered.\n\nExamining Fig.~\\ref{fig:numerical}(f), the results of our numerical calculations suggest that optimum IBSC sub-band gaps are obtained in short GaSb\/GaAs QRs having $h \\approx 2$ nm, which lies well within the range of heights observed in real structures. \\cite{Khan_EMC_2016} By reducing the QR Sb composition $x$ at fixed $h$ to form alloyed GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs, we find that $E_{\\scalebox{0.7}{\\textrm{L}}}$ increases by $\\approx 45$ -- 50 meV for each 10\\% reduction in $x$. Therefore, in QRs having reduced Sb composition -- due, e.g., to interfacial Sb-As intermixing \\cite{Timm_JVSTB_2008,Carrington_PB_2012} -- the QR height should be slightly increased in order to restore $E_{\\scalebox{0.7}{\\textrm{L}}}$ to its optimum energy. In practice, this constitutes reducing the confinement energy $\\Delta E_{h1}$ so as to maintain fixed hole ionisation energy in response to the reduction in $\\Delta E_{\\scalebox{0.7}{\\textrm{HH}}}$ associated with a reduction in $x$. For Sb compositions $x \\gtrsim 75$\\% our calculations indicate that an optimum sub-band gap $E_{\\scalebox{0.7}{\\textrm{L}}} = 0.45$ eV is maintained for an $\\approx 1$ nm increase in $h$ in response to each 10\\% reduction in $x$. For Sb compositions $x \\lesssim 75$\\% our calculations indicate that QR heights $h \\gtrsim 5$ nm are required to obtain optimised electronic properties. Such heights are outside the range typically obtained via epitaxial growth, suggesting that only epitaxial growth of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs having $x \\gtrsim 75$\\% is likely to produce heterostructures possessing electronic properties which are well-suited to IBSC applications.\n\nWe note, however, that our analysis of the electronic properties here has focused solely on idealised QR structures, possessing exact cylindrical shape and uniform alloy composition throughout the GaAs$_{1-x}$Sb$_{x}$ region. The presence of Sb-As intermixing at the ring-barrier interface in real GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs \\cite{Timm_JVSTB_2008,Carrington_PB_2012} results in a non-uniform alloy composition, which may modify the strain fields and confining potentials compared to those considered in our exploratory calculations here. Overall, we expect the resulting modifications of the electronic properties to be quantitative rather than qualitative in nature compared to those described above for ideal QRs. Of more importance from the perspective of designing real IBSC devices is to build upon our initial analysis here by undertaking theoretical investigations of the optical properties of, and radiative and non-radiative losses in, GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs and vertical QR stacks.\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\n\nIn summary, we have presented a theoretical analysis of the electronic properties of type-II GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs based on a combined analytical and numerical approach, and identified optimised combinations of QR morphology and alloy composition for the realisation of hole-based IBSCs -- in which the IB is formed by the highest energy bound hole state in the QR -- offering the maximum theoretical efficiency available via inclusion of an IB in a GaAs matrix.\n\n\n\nAnalytically, we presented the solution of the time-independent Schr\\\"{o}dinger equation for a cylindrical QR of infinite potential depth and derived a transcendental equation which must be satisfied by bound QR eigenstates. The relationship to the solution of the well-known problem of the cylindrical QD was described, and it was demonstrated that (i) the QR eigenstates evolve smoothly from those of the QD, and (ii) the convergence properties of the QR ground state allow the confinement energy to be estimated straightforwardly, and to high accuracy, for realistic QRs having dimensions typical of those achieved via epitaxial growth. Our analytical analysis demonstrated that type-II GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs offer significant benefits from the perspective of band structure engineering for IBSC applications, allowing the IB energy in a hole-based IBSC to be tuned across a broad range via changes in QR morphology.\n\nNumerically, we used multi-band \\textbf{k}$\\cdot$\\textbf{p} calculations -- including full strain and piezoelectric effects -- to analyse the electronic properties of GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs, both as a function of QR dimensions and Sb composition. We further demonstrated that the nature of the carrier confinement in these heterostructures is ideally suited to IBSC applications. Strong hole localisation, with large ionisation energies in excess of 0.4 eV, can be expected to mitigate carrier leakage from the IB via thermionic emission. Additionally, the interplay between strain relaxation and type-II band alignment in these QRs was demonstrated to give rise to electron states which, in the plane of the QR, are strongly localised in the central barrier of the QR. The unusual nature of the carrier localisation in these heterostructures suggests the potential to engineer the trade-off between the electron-hole overlap (which mediates carrier generation via optical absorption) and the radiative lifetime for photo-generated electron-hole pairs (which mediates carrier loss via radiative recombination).\n\nFor inner and outer radii $a_{1} = 5$ nm and $a_{2} = 11.5$ nm, typical of epitaxially grown QRs, our calculations indicate that an optimum IB sub-band gap $E_{\\scalebox{0.7}{\\textrm{L}}} = E_{\\scalebox{0.7}{\\textrm{IB}}} - E_{\\scalebox{0.7}{\\textrm{VB}}}$ ($E_{\\scalebox{0.7}{\\textrm{H}}} = E_{\\scalebox{0.7}{\\textrm{CB}}} - E_{\\scalebox{0.7}{\\textrm{IB}}}$) of 0.45 eV (0.97 eV) can be obtained in GaSb\/GaAs QRs having height $h \\approx 2$ nm. For GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs having reduced Sb compositions $x$, our calculations indicate that decreases in $x$ can be compensated by slight increases in $h$ -- by $\\approx 1$ nm for each 10\\% reduction in $x$ -- in order to maintain optimum sub-band gap energies. QRs grown to these specifications have a detailed balance efficiency limit of $\\approx 58$\\% under concentrated illumination, close to the overall limit of 63.8\\% for ideal IBSCs. Given the sensitivity of the theoretical IBSC efficiency to the IB energy, our analysis suggests that careful control of QR morphology provides a viable route to realising optimised heterostructures suitable for IBSC applications.\n\nOur initial calculations here, however, were guided by detailed balance efficiency limits which include a number of assumptions -- e.g.~optimium absorption spectra, infinite carrier mobilities, absence of non-radiative carrier losses, etc.~-- which are not reflective of the conditions in real quantum-confined heterostructure. Further theoretical work is therefore required to quantify, and identify pathways towards simultaneously optimising the optical properties and mitigating losses (both radiative and non-radiative) in real GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs and vertical QR stacks, in order to identify rigorously optimised heterostructures for IBSC applications and to quantify photovoltaic efficiencies that can be realistically achieved using this novel platform.\n\n\n\nOverall, our calculations highlight the suitability of type-II GaAs$_{1-x}$Sb$_{x}$\/GaAs QRs for applications in hole-based IBSCs, and provide initial information regarding optimised combinations of QR alloy composition and morphology to guide the growth and fabrication of prototype QR-IBSC devices.\n\n\n\n\\section*{Acknowledgements}\n\nThis work was supported by the European Commission via the Marie Sk\\l{}odowska-Curie Innovative Training Network PROMIS (project no.~641899), by the National University of Ireland (NUI; via the Post-Doctoral Fellowship in the Sciences, held by C.A.B.), and by Science Foundation Ireland (SFI; project no.~15\/IA\/3082). The authors thank Prof.~Anthony Krier and Dr.~Denise Montesdeoca (Lancaster University, U.K.) for useful discussions.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec:intro}}\n\nIt is a classical result in Newtonian mechanics that (uncharged) $n$-body configurations cannot be in equilibrium for $n>1$ if the bodies are separated by a plane. Indeed, since the Newtonian gravitational force is always attractive, it is clear that separated bodies  cannot be in balance. Interestingly, the situation might be rather different in the nonlinear theory of general relativity. If we consider \\emph{rotating} objects, then the effect of spin--spin repulsion might be able to compensate for the gravitational attraction. Therefore, in order to better understand the nature of the gravitational interaction in general relativity, it is an important question as to whether an equilibrium of (physically reasonable) rotating bodies is possible.\n\nProbably the simplest type of such an equilibrium configuration might be a configuration consisting of two aligned rotating black holes in vacuum, i.e.\\ a nonlinear superposition of two Kerr black holes. Especially since the discovery of a family of exact candidate solutions --- the \\emph{double-Kerr-NUT solution} \\cite{KramerNeugebauer1980,Neugebauer1980} --- such two-black-hole configurations have attracted great interest. It remained unclear, however, whether there is any choice of the parameters for which the double-Kerr-NUT solution does indeed describe regular spacetimes containing two black holes in equilibrium. Moreover, it was not a priori guaranteed that there could not be other candidate solutions that were not contained in this family. These questions were addressed in a series of papers \\cite{NeugebauerHennig2009,HennigNeugebauer2011,NeugebauerHennig2012,Chrusciel2011} with the following results. Firstly, by studying a boundary value problem for an asymptotically flat spacetime with two black hole event horizons, it turned out that any regular solution would necessarily be a member of the double-Kerr-NUT family. Secondly, it was shown that none of the candidate solutions can describe a regular equilibrium configuration, for at least one of the two black holes would necessarily violate a geometric inequality between horizon area and angular momentum \\cite{HennigAnsorgCederbaum2008}, which needs to be satisfied by physically reasonable black holes. Hence it turned out that stationary two-black-hole configurations in vacuum do not exist --- the spin--spin repulsion is not strong enough to compensate for the gravitational attraction.\n\nWhat happens if we study the more general situation of \\emph{electrovacuum} rather than vacuum solutions, i.e.\\ if we allow for electromagnetic fields and consider possible equilibrium configurations with charged bodies? In non-relativistic physics, it is easily possible to construct charged $n$-body configurations. One simply needs to choose sufficiently large charges such that the electromagnetic forces cancel the gravitational forces. In the context of \\emph{relativistic} two-black-hole configurations, however, there are upper limits for the charges (and rotation rates), since we otherwise obtain naked singularities instead of black holes. Hence it was not a priori guaranteed that there are any relativistic configurations where the combined spin--spin and charge--charge repulsions lead to stationary equilibrium.\n\nNevertheless, a class of static configurations is given by the well-known \\emph{Majumdar--Papapetrou solution} \\cite{Majumdar1947,Papapetrou1947}, which describes the superposition of an arbitrary number of extremal Reissner--Nordstr\\\"om black holes at arbitrary positions\\footnote{A remarkable extension of the Majumdar--Papapetrou solution in the cosmological setting (with cosmological constant $\\Lambda>0$) was constructed in \\cite{KastorTraschen1993}. This solution contains an arbitrary number of black holes at fixed coordinate positions. However, due to the cosmological expansion, the proper distances between the black holes vary, i.e.\\ it is not an equilibrium configuration in the context discussed here. \nAnother interesting configuration is the superpartner of the Majumdar--Papapetrou solution in $N=2$ supergravity, which was derived in \\cite{AichelburgEmbacher1986}. The black holes in this family of solutions carry angular momenta of a quantum mechanical nature, and the analysis of forces in \\cite{KastorTraschen1999} shows that static configurations can exist due to a balance of the gauge and gravitational spin--spin forces.}.\n\nHence general relativity does evidently permit equilibrium states with charged black holes. \nHowever, this example requires \\emph{extremal} black holes, i.e.\\ degenerate horizons with vanishing surface gravity $\\kappa$. According to the third law of black hole thermodynamics, however, it should not be possible by any procedure to reduce $\\kappa$ to zero by any finite sequence of operations. In line with this principle is a result by Thorne \\cite{Thorne1974}, who studied a black hole that swallows matter and radiation from an accretion disk. It turned out that the accreting matter can spin up the black hole up to a limiting state in which the ratio of the black hole rotation parameter and mass is about $0.998$ --- close to, but not quite at the extremal limit of $1$. Hence, while extremal black holes are certainly mathematically perfectly valid solutions to Einstein's field equations, they should rather be considered as limiting configurations that cannot exactly be realised in nature. Therefore, the above example for an equilibrium configuration of charged extremal black holes is most likely an unphysical idealisation, and the question remains as to whether equilibrium states with more realistic \\emph{non-extremal} black holes are possible.\n\nAs far as \\emph{static} solutions are concerned, this was answered in the negative. It was shown by Chru\\'sciel and Tod \\cite{ChruscielTod2007} that every static solution to the electrovacuum Einstein--Maxwell equations with disconnected horizons (i.e.\\ multiple black holes) can only contain degenerate horizons. Moreover, such solutions are necessarily locally diffeomorphic to an open subset of a Majumdar--Papapetrou spacetime.\n\nThe problem is considerably more complicated in the case of \\emph{rotating} black holes, i.e.\\ non-static solutions, and it is currently not known whether physically reasonable equilibrium configurations do exist. Nevertheless, some families of exact candidate solutions were constructed \\cite{ChamorroMankoSibgatullin1993,MankoMartinRuiz1994}. These were obtained by assuming that the solution (in terms of the Ernst potentials $\\mathcal E$ and $\\Phi$, see below) does have particular boundary values $\\mathcal E_+(\\zeta)$ and $\\Phi_+(\\zeta)$ on the upper part of the symmetry axis (above both black holes) in terms of a cylindrical coordinate $\\zeta$. The chosen boundary values already determine the solution uniquely \\cite{HauserErnst1981}, and the explicit solution in the entire spacetime was calculated by applying a particular technique from soliton theory (`Sibgatullin's integral method' \\cite{Sibgatullin1984,MankoSibgatullin1993}).\nA plausible form of the axis data $\\mathcal E_+$ and $\\Phi_+$ was obtained by starting from the Kerr--Newman data of a single black hole with mass $M$, rotation parameter $a$, and charge $Q$,\n\\begin{equation}\n \\mathcal E_+(\\zeta)=1-\\frac{2M}{\\zeta+M-\\mathrm{i} a}\\equiv\\frac{\\zeta-M-\\mathrm{i} a}{\\zeta+M-\\mathrm{i} a},\\quad\n \\Phi_+(\\zeta)=\\frac{Q}{\\zeta+M-\\mathrm{i} a}\n\\end{equation}\nand including additional terms to describe a second black hole. In \\cite{ChamorroMankoSibgatullin1993}, boundary data of the form\n\\begin{equation}\\fl\\label{eq:bdata1}\n \\mathcal E_+(\\zeta)=1-\\frac{2M_1}{\\zeta+\\zeta_1-\\mathrm{i} a_1}-\\frac{2M_2}{\\zeta+\\zeta_2-\\mathrm{i} a_2},\\quad\n \\Phi_+(\\zeta)=\\frac{Q_1}{\\zeta+\\zeta_1-\\mathrm{i} a_1}+\\frac{Q_2}{\\zeta+\\zeta_2-\\mathrm{i} a_2}\n\\end{equation}\nwere considered. On the other hand, the solution in \\cite{MankoMartinRuiz1994} was constructed from the  data\n\\begin{eqnarray}\n \\label{eq:bdata2a}\n \\mathcal E_+(\\zeta) &=&\\frac{(\\zeta+\\zeta_1-M_1-\\mathrm{i} a_1)(\\zeta+\\zeta_2-M_2-\\mathrm{i} a_2)}\n             {(\\zeta+\\zeta_1+M_1-\\mathrm{i} a_1)(\\zeta+\\zeta_2+M_2-\\mathrm{i} a_2)},\\\\\n \\label{eq:bdata2b}\n \\Phi_+(\\zeta)&=&\\frac{Q_1(\\zeta+\\zeta_2-\\mathrm{i} a_2)+Q_2(\\zeta+\\zeta_1-\\mathrm{i} a_1)}\n               {(\\zeta+\\zeta_1+M_1-\\mathrm{i} a_1)(\\zeta+\\zeta_2+M_2-\\mathrm{i} a_2)}.\n\\end{eqnarray}\nIn both cases, the boundary values of the potentials are rational functions of $\\zeta$ that depend on a number of free parameters.\n\nIn order to decide whether the candidate solutions  \\cite{ChamorroMankoSibgatullin1993,MankoMartinRuiz1994} contain any physically reasonable equilibrium configurations, it needs to be studied whether the parameters can be chosen such that all of the following requirements are satisfied:\n\\begin{enumerate}\n \\item the solutions have a vanishing NUT parameter (corresponding to the appropriate behaviour at infinity in an asymptotically flat spacetime),\n \\item there are no conical singularities on the symmetry axes, in particular, between the two black holes (which would correspond to `struts' that keep the two black holes apart),\n \\item the norm of the axial Killing vector vanishes on the symmetry axis, \n \\item there is no global magnetic charge,\n \\item the solutions are free of singularities off the symmetry axis.\n\\end{enumerate}\nUsing the conditions (i)--(iv), one can write down an algebraic system of equations for the parameters that ensures the correct behaviour at infinity and on the axis. Unfortunately, the equations are rather involved, which makes it very difficult to decide whether there are parameter values satisfying those conditions. However, even if the correct behaviour on the axis is obtained in a subset of the parameter space, the solutions would likely violate condition (v), and it is probably even more difficult to check regularity off the axis. Hence it is not clear whether the solution families \\cite{ChamorroMankoSibgatullin1993} and \\cite{MankoMartinRuiz1994} contain any physically acceptable equilibrium configurations.\n\nMoreover, the question remains as to whether the boundary data \\eref{eq:bdata1} or \\eref{eq:bdata2a} and \\eref{eq:bdata2b} do contain the axis potentials for actual equilibrium configurations with non-extremal rotating and charged black holes, if any exist, or whether data of some other form need to be considered. This is exactly the problem that we address in this paper. Generalising the considerations for one black hole in vacuum \\cite{Neugebauer2000} (which leads to a constructive uniqueness proof of the Kerr solution), two black holes in vacuum \\cite{NeugebauerMeinel,NeugebauerHennig2009,HennigNeugebauer2011,NeugebauerHennig2012}, or a single black hole in electrovacuum \\cite{Meinel2012} (which extends the constructive uniqueness proof to the Kerr--Newman solution), we study a boundary value problem for two aligned charged and rotating black holes. As a result, we will obtain the most general form of the axis potentials $\\mathcal E_+$ and $\\Phi_+$.\n\nThese considerations crucially rely on the fact that the Einstein--Maxwell equations in electrovacuum for axisymmetric and stationary spacetimes can be reformulated in terms of a linear matrix problem, as a consequence of which methods from soliton theory are applicable. Note that closely related techniques also work in the context of Gowdy-symmetric cosmological models (which have two spacelike Killing vectors rather than a spacelike and a timelike Killing vector) \\cite{HennigAnsorg2010,BeyerHennig2012,Hennig2016b,Hennig2019}. Yet another type of application is the investigation of the interior region of axisymmetric and stationary black holes \\cite{AnsorgHennig2009,HennigAnsorg2009}.\n\nThis paper is organised as follows. In Sec.~\\ref{sec:fieldeqns}, we recapitulate the Ernst formulation of the Einstein--Maxwell equations and the associated linear problem. Then we integrate the linear problem along the black hole horizons, symmetry axis and at infinity in Sec.~\\ref{sec:integration}. This will eventually allow us to obtain the general form of the axis data. Finally, in Sec.~\\ref{sec:discussion}, we summarise our results.\n\n\\section{Field equations\\label{sec:fieldeqns}}\n\\subsection{Ernst formulation}\n\nWe describe the exterior electrovacuum region of an axisymmetric and stationary spacetime containing two aligned rotating and charged black holes with Weyl--Lewis--Papapetrou coordinates $(\\rho,\\zeta,\\varphi,t)$. The line element can be written in the standard form\n\\begin{equation}\\label{eq:metric}\n \\mathrm{d} s^2=f^{-1\n          \\left[\\mathrm{e}^{2k}(\\mathrm{d}\\rho^2+\\mathrm{d}\\zeta^2)+\\rho^2\\,\\mathrm{d}\\varphi^2\\right]\n         -f(\\mathrm{d} t+a\\,\\mathrm{d}\\varphi)^2,\n\\end{equation}\nwhere the three metric functions $f$, $k$ and $a$ are functions of $\\rho$ and $\\zeta$ alone. The electromagnetic field can be given in terms of an electromagnetic 4-potential of the form $(A_\\mu)=[0,0,A_\\varphi(\\rho,\\zeta),A_t(\\rho,\\zeta)]$.\n\nIt is well-known that the corresponding Einstein--Maxwell equations can be written in a very elegant and concise form if we replace the metric functions and electromagnetic potential in terms of the two corresponding complex Ernst potentials $\\mathcal E(\\rho,\\zeta)$ and $\\Phi(\\rho,\\zeta)$ \\cite{Ernst1968b}. The resulting \\emph{Ernst equations} read\n\\begin{equation}\\label{eq:Ernst}\n f\\Delta\\mathcal E   = (\\nabla\\mathcal E+2\\bar\\Phi\\nabla\\Phi)\\cdot\\nabla\\mathcal E,\\quad\n f\\Delta\\Phi = (\\nabla\\mathcal E+2\\bar\\Phi\\nabla\\Phi)\\cdot\\nabla\\Phi,\n\\end{equation}\nwhere $\\Delta$ and $\\nabla$ refer to the Laplace and nabla operators in flat cylindrical coordinates $(\\rho,\\zeta,\\varphi)$, respectively, and a bar denotes complex conjugation. Note that the metric function $f$ is related to the Ernst potentials as follows,\n\\begin{equation}\n f=\\Re(\\mathcal E)+|\\Phi|^2.\n\\end{equation}\nHence, if we define $b=\\Im(\\mathcal E)$, then the Ernst potential $\\mathcal E$ can be expressed as\n\\begin{equation}\\label{eq:b}\n \\mathcal E=f-|\\Phi|^2+\\mathrm{i} b.\n\\end{equation}\nMore details about the Ernst formulation of the field equations and the relation to the metric and electromagnetic functions can be found in \\cite{Ernst1968b,Stephani}.\n\nIn our coordinates, the event horizon of a black hole is necessarily located at $\\rho=0$ and corresponds to an interval on the $\\zeta$-axis.\n\\begin{figure}\\centering\n \\includegraphics[width=4.5cm]{IntegrationPathX.pdf}\n \\caption{In Weyl--Lewis--Papapetrou coordinates, the event horizons ${\\mathcal H_1}$ and ${\\mathcal H_2}$ are located on the $\\zeta$-axis. The symmetry axis has the three parts $\\mathcal A_+$, $\\mathcal A_0$ and $\\mathcal A_-$. In Sec.~\\ref{sec:integration} below we will integrate the linear problem, which is equivalent to the field equations, along the dashed path. The part ${\\mathcal C}$ of this path is an infinitely large semicircle.}\n \\label{fig:IntegrationPath}\n\\end{figure}\nThe present situation of a (candidate) spacetime with two black holes is sketched in Fig.~\\ref{fig:IntegrationPath}. We denote the endpoints of the horizons by $A$, $B$, and $C$, $D$, and the corresponding $\\zeta$-intervals by $[K_B,K_A]$ and $[K_D,K_C]$, respectively.\n\n\\subsection{The linear problem}\nIt is a most remarkable property of the Ernst equations \\eref{eq:Ernst} that they belong to the class of integrable partial differential equations. They are equivalent to an associated \\emph{linear} matrix problem, and techniques from soliton theory, like the inverse scattering method, can be used to study properties of the solutions and to construct exact solutions.\n\nA linear problem (LP) for the electrovacuum Ernst equations was first found by Belinski \\cite{Belinski1979}, and a modified version was constructed by Neugebauer and Kramer \\cite{NeugebauerKramer}. Here we will use a minor reformulation of Neugebauer and Kramer's LP, which is due to Meinel \\cite{Meinel2012}. In order to state the LP, we first define the complex coordinates\n\\begin{equation}\n z=\\rho+\\mathrm{i}\\zeta,\\quad\\textrm{and}\\quad \\bar z=\\rho-\\mathrm{i}\\zeta,\n\\end{equation}\nand the function\n\\begin{equation}\\label{eq:lambda}\n \\lambda=\\sqrt{\\frac{K-\\mathrm{i}\\bar z}{K+\\mathrm{i} z}},\n\\end{equation}\nwhich depends on the complex coordinates and on an important additional degree of freedom, the \\emph{spectral parameter} $K$. \nDue to the square root, the complex function $\\lambda$ is defined on a two-sheeted Riemannian $K$-surface with branch points at $K_1=\\mathrm{i}\\bar z$ and $K_2=-\\mathrm{i} z$.\n\nThe LP is a system of equations for a $3\\times 3$ matrix function ${\\mathbf Y}={\\mathbf Y}(\\rho,\\zeta;K)$ and reads\n\\begin{eqnarray}\n\\label{eq:LP1}\n {\\mathbf Y}_{,z} &=& \\left[\n \\left(\\begin{array}{ccc}\n   B_1 & 0 & C_1\\\\ 0 & A_1 & 0\\\\ D_1 & 0 & 0\n \\end{array}\\right)\n +\\lambda\n \\left(\\begin{array}{ccc}\n  0 & B_1 & 0\\\\ A_1 & 0 & -C_1\\\\ 0 & D_1 & 0\n \\end{array}\\right)\\right]{\\mathbf Y},\\\\\n \\label{eq:LP2}\n {\\mathbf Y}_{,\\bar z} &=& \\left[\n \\left(\\begin{array}{ccc}\n   B_2 & 0 & C_2\\\\ 0 & A_2 & 0\\\\ D_2 & 0 & 0\n \\end{array}\\right)\n +\\frac{1}{\\lambda}\n \\left(\\begin{array}{ccc}\n  0 & B_2 & 0\\\\ A_2 & 0 & -C_2\\\\ 0 & D_2 & 0\n \\end{array}\\right)\\right]{\\mathbf Y}.\n\\end{eqnarray}\nThe matrix elements are given in terms of the Ernst potentials by\n\\begin{eqnarray}\n A_1 &= \\bar B_2 = \\frac{1}{2f}(\\mathcal E_{,z}+2\\bar\\Phi\\Phi_{,z}),\\quad\n C_1 &= f\\bar D_2 = \\Phi_{,z},\\\\\n A_2 &= \\bar B_1 = \\frac{1}{2f}(\\mathcal E_{,\\bar z}+2\\bar\\Phi\\Phi_{,\\bar z}),\\quad\n C_2 &= f\\bar D_1 = \\Phi_{,\\bar z}.\n\\end{eqnarray}\nNote that integrability of the LP \\eref{eq:LP1}, \\eref{eq:LP2} is ensured by virtue of the Ernst equations \\eref{eq:Ernst}, since the integrability condition ${\\mathbf Y}_{,z\\bar z}={\\mathbf Y}_{,\\bar z z}$ turns out to be equivalent to the Ernst equations.\n\nSince the LP  contains $\\lambda$, the matrix function ${\\mathbf Y}$ will in general also take on different values on the two Riemannian $K$-sheets. Only at the branch points the function values are unique. If some function ${\\mathbf Y}$ solves the LP on one sheet, then one can show that a particular solution on the other sheet is given by ${\\mathbf J}{\\mathbf Y}$ with\n\\begin{equation}\n {\\mathbf J}=\\mathrm{diag}(1,-1,1).\n\\end{equation}\nThe general solution on the other sheet can be obtained by multiplying ${\\mathbf J}{\\mathbf Y}$ on the right by a matrix that depends on $K$ only. However, only for a particular choice of this matrix, we obtain a solution that correctly connects to the solution on the first sheet through the branch cut. Hence the solutions on the two sheets are related via\n\\begin{equation}\\label{eq:sheets}\n {\\mathbf Y}|_{-\\lambda}={\\mathbf J}{\\mathbf Y}|_{\\lambda}{\\mathbf B}(K)\n\\end{equation}\nfor some $3\\times 3$ matrix ${\\mathbf B}$. It is possible to impose certain gauge conditions which  enforce that ${\\mathbf B}$ takes on a particular form. For example, the matrix obtained with conditions used in \\cite{Meinel2012} is given by ${\\mathbf B}=\\small\\left(\\begin{array}{ccc}0&1&0\\{\\mathbf 1}&0&0\\{\\mathbf 0}&0&1\\end{array}\\right)$. Here, however, we will demonstrate that the discussion can easily be done in full generality, i.e.\\ without referring to any particular gauge. Indeed, the final physical results will, of course, be independent of any gauge choice. The only property of the matrix ${\\mathbf B}$ that we will later use is that\n\\begin{equation}\\label{eq:B2}\n {\\mathbf B}^2={\\mathbf 1},\n\\end{equation}\nwhere ${\\mathbf 1}$ is the $3\\times 3$ identity matrix. This immediately follows by applying the transformation \\eref{eq:sheets} twice and using that this must lead back to the original solution ${\\mathbf Y}$.\n\nAn important ingredient of our construction of two-black-hole solutions is to study the LP not only in the coordinates introduced above, but also in certain rotating frames of reference. They can be introduced by a simple transformation of the coordinate $\\varphi$,\n\\begin{equation}\n \\tilde\\varphi=\\varphi-\\Omega t,\n\\end{equation}\nwhere $\\Omega$ is the angular velocity of the rotating frame. The other coordinates $\\rho$, $\\zeta$ and $t$ remain unchanged. In the following, we will consider the two particular frames that are co-rotating with either the first or the second black hole. If we denote the angular velocities of the two black holes by $\\Omega_1$ and $\\Omega_2$, then these frames correspond to choosing $\\Omega=\\Omega_{1\/2}$. Note that we always assume rotating black holes with $\\Omega_1\\neq0$ and $\\Omega_2\\neq0$.\n\nFortunately, we do not need to solve the LP both in the original and the rotating frame, in order to obtain the two solutions ${\\mathbf Y}$ and $\\tilde{\\mathbf Y}$. Instead, there is a simple relation between both solutions. The corresponding transformation in the vacuum case was given in \\cite{NeugebauerMeinel}, and the generalisation to electrovacuum was presented in \\cite{AnsorgHennig2009,HennigAnsorg2009}. In the present formulation of the LP, it reads \\cite{Meinel2012}\n\\begin{equation}\\fl\\label{eq:rotframe}\n \\tilde{\\mathbf Y}(\\rho,\\zeta; K)=\\left[\n  \\left(\\begin{array}{ccc}\n         c_- & 0   & 0\\\\\n         0   & c_+ & 0\\\\\n         0   & 0   & 1\n        \\end{array}\\right)\n  +\\mathrm{i}(K+\\mathrm{i} z)\\frac{\\Omega}{f}\n  \\left(\\begin{array}{ccc}\n         -1      & -\\lambda & 0\\\\\n         \\lambda & 1        & 0\\\\\n         0       & 0        & 0\n        \\end{array}\\right)\n\\right]{\\mathbf Y}(\\rho,\\zeta;K),\n\\end{equation}\nwhere\n\\begin{equation}\n c_{\\pm}=1+\\Omega\\left(a\\pm\\frac{\\rho}{f}\\right).\n\\end{equation}\nThe reason why using additional coordinate systems does actually add some new information is that the above transformation formula depends on the metric function $a$, which takes on specific boundary values at the symmetry axis and horizons (see next section). Considering not only ${\\mathbf Y}$ but also $\\tilde{\\mathbf Y}$ does therefore incorporate these boundary conditions into our calculations.\n\n\n\\section{Integration of the linear problem\\label{sec:integration}}\n\n\\subsection{Solutions on the axis parts and horizons}\n\nSimilarly to the study of two-black-hole configurations in vacuum and the other applications of the LP mentioned in the introduction, we intend to integrate the LP along the boundaries of the physical domain. The integration path consists of the event horizons ${\\mathcal H_1}$ and ${\\mathcal H_2}$ of the two black holes, the three parts $\\mathcal A_+$, $\\mathcal A_0$ and $\\mathcal A_-$ of the symmetry axis, and a semicircle ${\\mathcal C}$ in the limit of an infinite radius, cf.\\ Fig.~\\ref{fig:IntegrationPath}.\n\nIn the following discussion, we will make use of the well-known boundary values for the metric and Ernst potentials at symmetry axes and black hole (Killing) horizons in Weyl--Lewis--Papapetrou coordinates, as well as the behaviour at infinity,\n\\begin{eqnarray}\n \\mathcal A_+,\\ \\mathcal A_0,\\ \\mathcal A_-:\\quad && a=0,\\label{eq:condA}\\\\\n {\\mathcal H_1}:                  && a=-\\frac{1}{\\Omega_1},\\label{eq:condH1}\\\\\n {\\mathcal H_2}:                  && a=-\\frac{1}{\\Omega_2},\\label{eq:condH2}\\\\\n A,\\ B,\\ C,\\ D:        && f=0,\\label{eq:condf}\\\\\n {\\mathcal C}:                   && \\mathcal E\\to 1,\\quad \\Phi\\to 0.\\label{eq:condC}\n\\end{eqnarray}\nHere, $A$, $B$, $C$, $D$ refers to the endpoints of the horizons, see Fig.~\\ref{fig:IntegrationPath}.\n \n\nFirstly, we consider the LP anywhere on the $\\zeta$-axis, i.e.\\ at $\\rho=0$. According to \\eref{eq:lambda}, the function $\\lambda$ simplifies to $\\lambda=\\pm1$ on the two Riemannian sheets. As a consequence, the LP \\eref{eq:LP1}, \\eref{eq:LP2} also becomes particularly simple and reduces to an ODE. The general solution can easily be derived. In the sheet with $\\lambda=1$ it reads\n\\begin{equation}\n {\\mathbf Y}(0,\\zeta; K)={\\mathbf E}(\\zeta){\\mathbf C}(K),\\quad\n {\\mathbf E}:=\\left(\\begin{array}{ccc}\n             \\bar\\mathcal E+2|\\Phi|^2 & 1 & \\Phi\\\\\n             \\mathcal E & -1 & -\\Phi\\\\\n             2\\bar\\Phi & 0 & 1\n            \\end{array}\\right).\n\\end{equation}\nHence ${\\mathbf Y}$ depends on the boundary values of the Ernst potentials and on a $K$-dependent `integration constant', a $3\\times 3$ matrix ${\\mathbf C}$. The solution in the other sheet with $\\lambda=-1$ is readily obtained from \\eref{eq:sheets}.\n\nUsing \\eref{eq:rotframe}, we can also construct the solution in the frame that rotates with angular velocity $\\Omega$. The result is\n\\begin{equation}\\fl\n \\tilde{\\mathbf Y} = \\left[\\left(\\begin{array}{ccc}\n                    1+\\Omega a & 0 & 0\\\\\n                    0 & 1+\\Omega a & 0\\\\\n                    0 & 0 & 1\n                   \\end{array}\\right){\\mathbf E}\n        +2\\mathrm{i}\\Omega(K-\\zeta)\\left(\\begin{array}{ccc}\n                                   -1 & 0 & 0\\\\\n                                   1 & 0 & 0\\\\\n                                   0 & 0 & 0\n                                  \\end{array}\\right)\\right]{\\mathbf C}.\n\\end{equation}\nThis expression simplifies further if we specialise to the symmetry axis or horizons, using the boundary conditions \\eref{eq:condA}, \\eref{eq:condH1}, \\eref{eq:condH2}, where we consider the co-rotating frames with $\\Omega=\\Omega_1$ or $\\Omega=\\Omega_2$.\n\nNow we can write down the expressions for ${\\mathbf Y}$ and $\\tilde{\\mathbf Y}$ on the three parts of the symmetry axis and on the two horizons. In terms of $K$-dependent $3\\times 3$ matrices ${\\mathbf C}_+$, ${\\mathbf C}_-$, ${\\mathbf C}_0$, ${\\mathbf C}_1$ and ${\\mathbf C}_2$, we have for $\\lambda=+1$,\n\\begin{eqnarray}\n \\label{eq:ApY}\n \\mathcal A_+:\\quad && {\\mathbf Y}={\\mathbf E}{\\mathbf C}_+,\\\\ \n \\label{eq:ApYs}\n           && \\tilde{\\mathbf Y}=\\left[{\\mathbf E}+2\\mathrm{i}\\Omega_{1}(K-\\zeta)\\left(\\begin{array}{ccc}\n                                                      -1 & 0 & 0\\\\\n                                                      1 & 0 & 0\\\\\n                                                      0 & 0 & 0\n                                                     \\end{array}\\right)\\right]{\\mathbf C}_+,\\\\[1.5ex]\n \\mathcal A_0:\\quad && {\\mathbf Y}={\\mathbf E}{\\mathbf C}_0,\\\\\n           && \\tilde{\\mathbf Y}=\\left[{\\mathbf E}+2\\mathrm{i}\\Omega_{1\/2}(K-\\zeta)\\left(\\begin{array}{ccc}\n                                                      -1 & 0 & 0\\\\\n                                                      1 & 0 & 0\\\\\n                                                      0 & 0 & 0\n                                                     \\end{array}\\right)\\right]{\\mathbf C}_0,\\\\[1.5ex]                                                  \n \\label{eq:AmY}\n \\mathcal A_-:\\quad && {\\mathbf Y}={\\mathbf E}{\\mathbf C}_-,\\\\\n           && \\tilde{\\mathbf Y}=\\left[{\\mathbf E}+2\\mathrm{i}\\Omega_{2}(K-\\zeta)\\left(\\begin{array}{ccc}\n                                                      -1 & 0 & 0\\\\\n                                                      1 & 0 & 0\\\\\n                                                      0 & 0 & 0\n                                                     \\end{array}\\right)\\right]{\\mathbf C}_-,\\\\[1.5ex]   \n \\label{eq:H1Y}\n {\\mathcal H_1}:\\quad && {\\mathbf Y}={\\mathbf E}{\\mathbf C}_1,\\\\\n \\label{eq:H1Ys}\n           && \\tilde{\\mathbf Y}=\\left[\\left(\\begin{array}{ccc}\n                               0 & 0 & 0\\\\ 0 & 0 & 0\\\\ 0 & 0 & 1\n                              \\end{array}\\right)\n{\\mathbf E}+2\\mathrm{i}\\Omega_{1}(K-\\zeta)\\left(\\begin{array}{ccc}\n                                                      -1 & 0 & 0\\\\\n                                                      1 & 0 & 0\\\\\n                                                      0 & 0 & 0\n                                                     \\end{array}\\right)\\right]{\\mathbf C}_1,\\\\[1.5ex] \n{\\mathcal H_2}:\\quad && {\\mathbf Y}={\\mathbf E}{\\mathbf C}_2,\\\\\n           && \\tilde{\\mathbf Y}=\\left[\\left(\\begin{array}{ccc}\n                               0 & 0 & 0\\\\ 0 & 0 & 0\\\\ 0 & 0 & 1\n                              \\end{array}\\right)\n{\\mathbf E}+2\\mathrm{i}\\Omega_{2}(K-\\zeta)\\left(\\begin{array}{ccc}\n                                                      -1 & 0 & 0\\\\\n                                                      1 & 0 & 0\\\\\n                                                      0 & 0 & 0\n                                                     \\end{array}\\right)\\right]{\\mathbf C}_2. \n\\end{eqnarray}\nNote that we consider \\emph{both} co-rotating frames with $\\Omega=\\Omega_1$ and $\\Omega=\\Omega_2$ on the axis part $\\mathcal A_0$, but otherwise only that co-rotating frame with the angular velocity of the nearest horizon. Again, the expressions in the Riemannian sheet with $\\lambda=-1$ can be obtained from the above equations using \\eref{eq:sheets}.\n\nSecondly, we consider the LP on the infinitely large semicircle ${\\mathcal C}$. A semicircle with finite radius $R$ can be parametrised by $\\rho=R\\sin\\alpha$, $\\zeta=R\\cos\\alpha$, $0\\le\\alpha\\le\\pi$. On this semicircle, the function $\\lambda$ becomes\n\\begin{equation}\n \\lambda=\\sqrt{\\frac{K-\\mathrm{i} R(\\sin\\alpha-\\mathrm{i}\\cos\\alpha)}{K+\\mathrm{i} R(\\sin\\alpha+\\mathrm{i}\\cos\\alpha)}},\n\\end{equation}\nwhich simplifies to $\\lambda=\\pm\\mathrm{e}^{\\mathrm{i}\\alpha}$ in the limit $R\\to\\infty$. Hence, if we start on $\\mathcal A_+$ with $\\lambda=+1$, then the semicircle ${\\mathcal C}$ leads us to the sheet on $\\mathcal A_-$ with $\\lambda=-1$, and vice versa. This will be important later, when we continuously connect the solutions on the various parts of the boundary. Then we need to compare the $\\lambda=1$ solution on $\\mathcal A_+$ with the $\\lambda=-1$ solution on $\\mathcal A_-$.\n\nIf we consider the LP on $\\mathcal C$, using the asymptotic behaviour \\eref{eq:condC} of the Ernst potentials, we simply obtain ${\\mathbf Y}_{,z}={\\mathbf 0}$ and ${\\mathbf Y}_{,\\bar z}={\\mathbf 0}$. Therefore, ${\\mathbf Y}$ is constant on ${\\mathcal C}$. (More precisely, the Ernst potentials in an asymptotically flat spacetime approach their constant limits at infinity at a rate for which the coefficient matrices on the right-hand side of the LP are of $\\mathcal O (R^{-2})$ on a semicircle with coordinate radius $R$, while the length of the semicircle only increases proportional to $R$ as $R\\to\\infty$.)\n\nNext we have a closer look at the various ${\\mathbf C}$-matrices that appear in the solution to the LP on the different parts of the boundary. These matrices cannot be chosen independently of each other. Instead we have to ensure that the solutions ${\\mathbf Y}$ and $\\tilde{\\mathbf Y}$ are continuous at the points $A$, $B$, $C$, $D$, cf.\\ Fig.~\\ref{fig:IntegrationPath}, and that the solutions on $\\mathcal A_+$ and $\\mathcal A_-$ are correctly connected via ${\\mathcal C}$ as discussed above.\n\nWe start by considering continuity of ${\\mathbf Y}$ at point $A$. For $\\lambda=1$, using \\eref{eq:ApY} and \\eref{eq:H1Y}, we obtain the condition\n\\begin{equation}\\label{eq:Acont}\n {\\mathbf E}{\\mathbf C}_+={\\mathbf E}{\\mathbf C}_1\\quad\\textrm{at}\\quad \\rho=0,\\ \\zeta=K_A.\n\\end{equation}\nThe same condition also ensures continuity of ${\\mathbf Y}$ in the sheet $\\lambda=-1$. Note that the nine components of the matrix condition \\eref{eq:Acont} are not independent. Instead, the second row is the negative of the first row. Hence we will only use the second and third row conditions.\n\nSimilarly, considering continuity of $\\tilde{\\mathbf Y}$ in the frame with $\\Omega=\\Omega_1$, we obtain the condition (for $\\lambda=1$)\n\\begin{eqnarray}\\\n \\fl\n \\left[{\\mathbf E}+2\\mathrm{i}\\Omega_{1}(K-\\zeta)\\left(\\begin{array}{ccc}\n                                                      -1 & 0 & 0\\\\\n                                                      1 & 0 & 0\\\\\n                                                      0 & 0 & 0\n                                                     \\end{array}\\right)\\right]{\\mathbf C}_+\n                                                     \\nonumber\\\\\n \\fl\n \\quad=\\left[\\left(\\begin{array}{ccc}\n                               0 & 0 & 0\\\\ 0 & 0 & 0\\\\ 0 & 0 & 1\n                              \\end{array}\\right)\n{\\mathbf E}+2\\mathrm{i}\\Omega_{1}(K-\\zeta)\\left(\\begin{array}{ccc}\n                                                      -1 & 0 & 0\\\\\n                                                      1 & 0 & 0\\\\\n                                                      0 & 0 & 0\n                                                     \\end{array}\\right)\\right]{\\mathbf C}_1\n                             \\quad\\textrm{at}\\quad \\rho=0,\\ \\zeta=K_A.\n\\end{eqnarray}\nAgain, the nine components are not independent: the second row is the negative of the first row. Moreover, the third row is identical with the third row of \\eref{eq:Acont}. Hence we obtain one new row of conditions.\n\nThe three rows of independent conditions can be combined as follows,\n\\begin{equation}\\fl\n \\left(\\begin{array}{ccc}\n        \\mathcal E_A                     & -1 & -\\Phi_A\\\\\n        \\mathcal E_A+2\\mathrm{i}\\Omega_1(K-K_A) & -1 & -\\Phi_A\\\\\n        2\\bar\\Phi_A              &  0 & 1\n       \\end{array}\\right){\\mathbf C}_+\n =\\left(\\begin{array}{ccc}\n        \\mathcal E_A                     & -1 & -\\Phi_A\\\\\n        2\\mathrm{i}\\Omega_1(K-K_A)      &  0 & 0\\\\\n        2\\bar\\Phi_A              &  0 & 1\n       \\end{array}\\right){\\mathbf C}_1,\n\\end{equation}\nwhere, here and in the following, subscripts $A$, $B$, $C$, $D$ refer to evaluation of function values at the indicated points, i.e.\\ at $\\rho=0$ and $\\zeta=K_A,\\ K_B,\\ K_C$, or $K_D$, respectively. Solving for ${\\mathbf C}_1$, we can also write this condition in the form\n\\begin{equation}\\label{eq:condC1}\n {\\mathbf C}_1=\\left({\\mathbf 1}+\\frac{1}{\\alpha_A}{\\mathbf M}_A\\right){\\mathbf C}_+,\n\\end{equation}\nwhere\n\\begin{equation}\n \\alpha_A:=2\\mathrm{i}\\Omega_1(K-K_A)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:MA}\n {\\mathbf M}_A:=m_A n_A^T,\\quad\n m_A:=\\left(\\begin{array}{c}\n             -1\\\\ \\bar\\mathcal E_A\\\\ 2\\bar\\Phi_A\n            \\end{array}\\right),\\quad\n n_A:=\\left(\\begin{array}{c}\n             -\\mathcal E_A\\\\ 1\\\\ \\Phi_A\n            \\end{array}\\right).\n\\end{equation}\nNote that $n_A\\cdot m_A\\equiv n_A^T m_A=0$ [cf.~\\eref{eq:condf}], which implies ${\\mathbf M}_A^2={\\mathbf 0}$. As a consequence, matrices of the form ${\\mathbf 1}+c{\\mathbf M}_A$ can easily be inverted, and we have\n$({\\mathbf 1}+c{\\mathbf M}_A)^{-1}={\\mathbf 1}-c{\\mathbf M}_A$.\n\nRepeating the above calculations at the points $B$, $C$, and $D$, we obtain the additional conditions\n\\begin{eqnarray}\n  \\label{eq:condC0}\n  {\\mathbf C}_0 &=& \\left({\\mathbf 1}-\\frac{1}{\\alpha_B}{\\mathbf M}_B\\right){\\mathbf C}_1,\n   \\quad \\alpha_B:=2\\mathrm{i}\\Omega_1(K-K_B),\\\\\n  \\label{eq:condC2}\n  {\\mathbf C}_2 &=& \\left({\\mathbf 1}+\\frac{1}{\\alpha_C}{\\mathbf M}_C\\right){\\mathbf C}_0,\n   \\quad \\alpha_C:=2\\mathrm{i}\\Omega_2(K-K_C),\\\\\n  \\label{eq:condCm}\n  {\\mathbf C}_- &=& \\left({\\mathbf 1}-\\frac{1}{\\alpha_D}{\\mathbf M}_D\\right){\\mathbf C}_2,\n   \\quad \\alpha_D:=2\\mathrm{i}\\Omega_2(K-K_D),\n\\end{eqnarray}\nwhere the matrices ${\\mathbf M}_B$, ${\\mathbf M}_C$, ${\\mathbf M}_D$ are defined as in \\eref{eq:MA}, but with the Ernst potentials evaluated at the points $B$, $C$, or $D$, respectively.\n\nIf we would know the exact form of the matrix ${\\mathbf C}_+$, then the above conditions would allow us to compute all of the remaining ${\\mathbf C}$-matrices. This would indeed be possible if we imposed suitable gauge conditions for the LP, see \\cite{Meinel2012}. Here, however, as mentioned before, we intend to demonstrate that the final results can easily be obtained without any particular gauge.\n\nFinally, we consider the transition from the solution on $\\mathcal A_+$ to $\\mathcal A_-$ via ${\\mathcal C}$. Based on our earlier discussion, we arrive at the condition \n\\begin{equation}\n \\lim_{\\zeta\\to\\infty}{\\mathbf Y}(0,\\zeta;K)|_{\\lambda=1}\n =\\lim_{\\zeta\\to-\\infty}{\\mathbf Y}(0,\\zeta;K)|_{\\lambda=-1}.\n\\end{equation}\nWith the explicit solutions \\eref{eq:ApY} and \\eref{eq:AmY}, together with \\eref{eq:sheets} and the asymptotic values \\eref{eq:condC}, the previous equation becomes\n\\begin{equation}\n \\left(\\begin{array}{ccc}\n        1 & 1 & 0\\\\\n        1 &-1 & 0\\\\\n        0 & 0 & 1\n       \\end{array}\\right){\\mathbf C}_+\n ={\\mathbf J}\\left(\\begin{array}{ccc}\n        1 & 1 & 0\\\\\n        1 &-1 & 0\\\\\n        0 & 0 & 1\n       \\end{array}\\right){\\mathbf C}_-{\\mathbf B}.\n\\end{equation}\nThis can be rearranged to\n\\begin{equation}\\label{eq:cond2}\n {\\mathbf C}_-{\\mathbf B}=\\P{\\mathbf C}_+,\n\\end{equation}\nwhere $\\P$ is the following permutation matrix,\n\\begin{equation}\n \\P:=\\left(\\begin{array}{ccc}\n            0 & 1 & 0\\\\\n            1 & 0 & 0\\\\\n            0 & 0 & 1\n           \\end{array}\\right).\n\\end{equation}\n\n\n\\subsection{Parameter conditions}\n\nThe solutions of the LP at the symmetry axis and horizon discussed in the previous subsection depend on the values of the Ernst potentials at the points $A$, $B$, $C$, $D$, the $\\zeta$-coordinates $K_A$, $K_B$, $K_C$, $K_D$ of these points, and the angular velocities $\\Omega_1$, $\\Omega_2$ of the two horizons. These parameters, however, cannot be chosen independently of each other. Instead, we have to impose a number of parameter conditions. In the following, we show how these conditions can be obtained.\n\nCombining \\eref{eq:condC1}, \\eref{eq:condC0}, \\eref{eq:condC2}, \\eref{eq:condCm}, we obtain an equation relating ${\\mathbf C}_-$ and ${\\mathbf C}_+$,\n\\begin{equation}\\fl\n {\\mathbf C}_-=\\left({\\mathbf 1}-\\frac{1}{\\alpha_D}{\\mathbf M}_D\\right)\n         \\left({\\mathbf 1}+\\frac{1}{\\alpha_C}{\\mathbf M}_C\\right)\n         \\left({\\mathbf 1}-\\frac{1}{\\alpha_B}{\\mathbf M}_B\\right)\n         \\left({\\mathbf 1}+\\frac{1}{\\alpha_A}{\\mathbf M}_A\\right){\\mathbf C}_+.\n\\end{equation}\nAnother relation between ${\\mathbf C}_-$ and ${\\mathbf C}_+$ was previously obtained in \\eref{eq:cond2}. Combining those equations, we can eliminate ${\\mathbf C}_-$ and solve for ${\\mathbf B}$,\n\\begin{equation}\\fl\\label{eq:Cpluscond}\n {\\mathbf B}={\\mathbf C}_+^{-1}\n     \\left({\\mathbf 1}-\\frac{1}{\\alpha_A}{\\mathbf M}_A\\right)\n     \\left({\\mathbf 1}+\\frac{1}{\\alpha_B}{\\mathbf M}_B\\right)\n     \\left({\\mathbf 1}-\\frac{1}{\\alpha_C}{\\mathbf M}_C\\right)\n     \\left({\\mathbf 1}+\\frac{1}{\\alpha_D}{\\mathbf M}_D\\right)\\P{\\mathbf C}_+.\n\\end{equation}\nPlugging this into the relation ${\\mathbf B}^2={\\mathbf 1}$ [cf.~\\eref{eq:B2}], we obtain \n\\begin{eqnarray}\n \\fl\n  \\P\n  \\left({\\mathbf 1}-\\frac{1}{\\alpha_D}{\\mathbf M}_D\\right)\n  \\left({\\mathbf 1}+\\frac{1}{\\alpha_C}{\\mathbf M}_C\\right)\n  \\left({\\mathbf 1}-\\frac{1}{\\alpha_B}{\\mathbf M}_B\\right)\n  \\left({\\mathbf 1}+\\frac{1}{\\alpha_A}{\\mathbf M}_A\\right)\\nonumber\\\\\n  \\fl\\quad\n  =\n  \\left({\\mathbf 1}-\\frac{1}{\\alpha_A}{\\mathbf M}_A\\right)\n  \\left({\\mathbf 1}+\\frac{1}{\\alpha_B}{\\mathbf M}_B\\right)\n  \\left({\\mathbf 1}-\\frac{1}{\\alpha_C}{\\mathbf M}_C\\right)\n  \\left({\\mathbf 1}+\\frac{1}{\\alpha_D}{\\mathbf M}_D\\right)\\P.\n\\end{eqnarray}\nAs expected, the gauge dependent matrices ${\\mathbf B}$ and ${\\mathbf C}_+$ have cancelled, and hence the physical restrictions are independent of any gauge choice.\nFinally, simplifying and multiplying both sides by $\\alpha_A\\alpha_B\\alpha_C\\alpha_D$, we obtain the condition that two matrix polynomials of third degree in $K$ must be the same. Equating the coefficients of $K^0$, $K^1$, $K^2$ and $K^3$ leads to a number of constraints for the parameters.\n\nNote that the corresponding conditions in the case of a single black hole in electrovacuum can easily be solved explicitly \\cite{Meinel2012}. In the present case, unfortunately, they are much more involved, and an explicit solution may be difficult to obtain. Fortunately, however, most of the conditions and their solution are not required in the following.\n\nAs an example, we only give the three simplest conditions here, which are\n\\begin{equation}\\label{eq:parcond}\n \\Omega_1(|\\mathcal E_C|^2-|\\mathcal E_D|^2)+\\Omega_2(|\\mathcal E_A|^2-|\\mathcal E_B|^2)=0,\n\\end{equation}\n\\begin{equation}\n \\Omega_1(\\mathcal E_C+\\bar\\mathcal E_C-\\mathcal E_D-\\bar\\mathcal E_D)+\\Omega_2(\\mathcal E_A+\\bar\\mathcal E_A-\\mathcal E_B-\\bar\\mathcal E_B)=0,\n\\end{equation}\n\\begin{equation}\n \\Omega_1[(1-\\bar\\mathcal E_C)\\Phi_C-(1-\\bar\\mathcal E_D)\\Phi_D]\n +\\Omega_2[(1-\\bar\\mathcal E_A)\\Phi_A-(1-\\bar\\mathcal E_B)\\Phi_B]=0.\n\\end{equation}\nFor our derivation of the most general form of the axis potentials on $\\mathcal A_+$ in the next subsection, we will explicitly only require Eq.~\\eref{eq:parcond}.\n\n\n\\subsection{Construction of the axis potentials}\n\nWith the preparations from the previous subsection, we are now in a position to obtain the Ernst potentials $\\mathcal E_+=\\mathcal E(0,\\zeta)$ and $\\Phi_+=\\Phi(0,\\zeta)$ on $\\mathcal A_+$. The key ingredient for this construction is the above-mentioned property that ${\\mathbf Y}$ can generally take on different values on the two Riemannian sheets, but must be unique at the branch points $K_1=\\mathrm{i}\\bar z=\\zeta+\\mathrm{i}\\rho$ and $K_2=-\\mathrm{i} z=\\zeta-\\mathrm{i}\\rho$ where the two sheets are connected. In the limit $\\rho\\to0$, as we approach the $\\zeta$-axis, both branch points converge to $K_1=K_2=\\zeta$, i.e.\\ we have confluent branch points.\n\nWith \\eref{eq:ApY} and \\eref{eq:sheets}, the condition that ${\\mathbf Y}$ for $\\lambda=1$ and ${\\mathbf Y}$ for $\\lambda=-1$ on $\\mathcal A_+$ coincide at $K=\\zeta$ becomes\n\\begin{equation}\n {\\mathbf E}{\\mathbf C}_+={\\mathbf J}{\\mathbf E}{\\mathbf C}_+{\\mathbf B}  \\quad\\textrm{at}\\quad K=\\zeta.\n\\end{equation}\nUsing \\eref{eq:B2}, we can rewrite this equation as\n\\begin{equation}\\label{eq:PCBC}\n {\\mathbf C}_+{\\mathbf B}{\\mathbf C}_+^{-1}\\P={\\mathbf E}^{-1}{\\mathbf J}{\\mathbf E}\\P     \\quad\\textrm{at}\\quad K=\\zeta.\n\\end{equation}\nNow we define\\footnote{Note that ${\\mathbf N}$ generalises the $2\\times2$ matrix $\\mathcal N$ used in the discussion of two-black-hole configurations in vacuum, see Eq.~(22) in \\cite{NeugebauerHennig2009}.}\n\\begin{equation}\\fl\n {\\mathbf N}(\\zeta):={\\mathbf E}^{-1}{\\mathbf J}{\\mathbf E}\\P\n \\equiv\n \\frac{1}{f_+}\\left(\\begin{array}{ccc}\n                    1                  & |\\Phi_+|^2-\\mathrm{i} b_+  & \\Phi_+\\\\\n                    |\\Phi_+|^2+\\mathrm{i} b_+ & |\\mathcal E_+|^2            & -\\Phi_+\\bar\\mathcal E_+\\\\\n                    -2\\bar\\Phi_+       & 2\\bar\\Phi_+\\mathcal E_+     & f_+-2|\\Phi_+|^2\n                   \\end{array}\\right),\n\\end{equation}\nwhere $b=\\Im(\\mathcal E)$, cf.~\\eref{eq:b}. If we reformulate the parameter condition \\eref{eq:Cpluscond} in the form\n\\begin{equation}\\fl\n {\\mathbf C}_+{\\mathbf B}{\\mathbf C}_+^{-1}\\P=\n     \\left({\\mathbf 1}-\\frac{1}{\\alpha_A}{\\mathbf M}_A\\right)\n     \\left({\\mathbf 1}+\\frac{1}{\\alpha_B}{\\mathbf M}_B\\right)\n     \\left({\\mathbf 1}-\\frac{1}{\\alpha_C}{\\mathbf M}_C\\right)\n     \\left({\\mathbf 1}+\\frac{1}{\\alpha_D}{\\mathbf M}_D\\right)\n\\end{equation}\nand specialise to $K=\\zeta$, then, using \\eref{eq:PCBC}, we see that the matrix ${\\mathbf N}$, which contains various combinations of the Ernst potentials, can be obtained from\n\\begin{equation}\\fl\\label{eq:N}\n {\\mathbf N}= \\left.\\left({\\mathbf 1}-\\frac{1}{\\alpha_A}{\\mathbf M}_A\\right)\n     \\left({\\mathbf 1}+\\frac{1}{\\alpha_B}{\\mathbf M}_B\\right)\n     \\left({\\mathbf 1}-\\frac{1}{\\alpha_C}{\\mathbf M}_C\\right)\n     \\left({\\mathbf 1}+\\frac{1}{\\alpha_D}{\\mathbf M}_D\\right)\\right|_{K=\\zeta}.\n\\end{equation}\nNote that the components of the matrix on the right-hand side simplify to rational functions in $\\zeta$ with polynomials of at most fourth degree. \nMoreover, similarly to the parameter conditions, the gauge dependent matrices ${\\mathbf B}$ and ${\\mathbf C}_+$ do not appear in this formula, so the axis potentials are certainly independent of any gauge choice for the LP.\n\nNow we consider the following combinations of components of ${\\mathbf N}$, where we evaluate the right-hand side of \\eref{eq:N} in each case, in order to determine the polynomial structure of the relevant components,\n\\begin{eqnarray}\n \\fl\\label{eq:f+}\n f_+ &=& \\frac{1}{N_{11}} = \\frac{\\pi_4(\\zeta)}{p_4(\\zeta)},\n \\hspace{2cm}  \\pi_4:=(\\zeta-K_A)(\\zeta-K_B)(\\zeta-K_C)(\\zeta-K_D)\\\\\n \\fl\n     &&\\hspace{4.52cm}p_4:\\ \\mbox{real monic polynomial of 4th degree}\\nonumber\\\\\n \\fl\\label{eq:b+}\n b_+ &=&\\frac{N_{21}-N_{12}}{2N_{11}}=\\frac{p_2(\\zeta)}{p_4(\\zeta)},\n \\hspace{0.895cm} p_2:\\ \\mbox{real polynomial of 2nd degree}\\\\\n \\fl\\label{eq:E+2}\n |\\mathcal E_+|^2 &=& \\frac{N_{22}}{N_{11}}=\\frac{q_4(\\zeta)}{p_4(\\zeta)},\n \\hspace{2.08cm} q_4:\\ \\mbox{real monic polynomial of 4th degree}\\\\\n \\fl\\label{eq:Phi+}\n \\Phi_+ &=& \\frac{N_{13}}{N_{11}}=\\frac{p_3(\\zeta)}{p_4(\\zeta)},\n \\hspace{2.06cm} p_3:\\ \\mbox{complex polynomial of 3rd degree}\\\\\n \\fl\\label{eq:Phi+2}\n |\\Phi_+|^2 &=& \\frac{N_{12}+N_{21}}{2N_{11}}=\\frac{q_2(\\zeta)}{p_4(\\zeta)},\\\n \\hspace{0.79cm} q_2:\\ \\mbox{real polynomial of 2nd degree}\\\\\n \\fl\\label{eq:Phi+E+}\n \\Phi_+\\bar\\mathcal E_+ &=& -\\frac{N_{23}}{N_{11}}=\\frac{q_3(\\zeta)}{p_4(\\zeta)}.\n \\hspace{1.64cm} q_3:\\ \\mbox{complex polynomial of 3rd degree}\n\\end{eqnarray}\nAll polynomials in the above formulae can be given explicitly, but the coefficients are rather lengthy expressions depending on the parameters $\\mathcal E_A,\\dots,\\mathcal E_D$, $\\Phi_A,\\dots,\\Phi_D$, $K_A,\\dots,K_D$, $\\Omega_1,\\ \\Omega_2$. Also note that the two polynomials $p_2$ and $q_2$ initially appear to be of third degree, but in both cases the leading coefficients are proportional to the left-hand side of \\eref{eq:parcond} and hence vanish as a consequence of the parameter conditions.\n\nFirstly, we construct $\\mathcal E_+$ using \\eref{eq:f+}, \\eref{eq:b+}, \\eref{eq:Phi+2}, \n\\begin{equation}\\label{eq:E+formula}\n  \\mathcal E_+=f_+-|\\Phi_+|^2+\\mathrm{i} b_+=\\frac{\\pi_4-q_2+\\mathrm{i} p_2}{p_4},\n\\end{equation}\nand compare $|\\mathcal E_+|^2$ as obtained from this expression with \\eref{eq:E+2}. This leads to the condition\n\\begin{equation}\n (\\pi_4-q_2+\\mathrm{i} p_2)(\\pi_4-q_2-\\mathrm{i} p_2)=q_4p_4.\n\\end{equation}\nComparing zeros of both sides and using that the two terms on the left-hand side are complex conjugate polynomials, and the factors on the right-hand side are real polynomials, we observe that each bracket on the left-hand side has two zeros of $q_4$ and two zeros of $p_4$. Hence, in the expression \\eref{eq:E+formula} for $\\mathcal E_+$, two linear factors cancel and we actually have\n\\begin{equation}\\label{eq:E+}\n \\mathcal E_+=\\frac{\\pi_2(\\zeta)}{r_2(\\zeta)},\n\\end{equation}\nwhere $\\pi_2$ and $r_2$ are complex monic polynomials of 2nd degree. \n\nSecondly, we use the expression \\eref{eq:Phi+} for $\\Phi_+$ to calculate $|\\Phi_+|^2$ and compare the result with \\eref{eq:Phi+2}. In this way, we obtain the condition\n\\begin{equation}\n p_3\\bar p_3=q_2p_4.\n\\end{equation}\nSimilarly to the above discussion, we conclude that $p_3$ has two zeros of $p_4$ and one of $q_2$. Hence \\eref{eq:Phi+} simplifies to\n\\begin{equation}\\label{eq:Phi+1}\n \\Phi_+=\\frac{\\pi_1(\\zeta)}{R_2(\\zeta)},\n\\end{equation}\nwhere $\\pi_1$ and $R_2$ are complex polynomials of first and second degrees, respectively, which we choose such that $R_2$ is a monic polynomial.\n\nFinally, we show that, in fact, $R_2=r_2$, i.e.\\ $\\Phi_+$ has the same denominator as $\\mathcal E_+$. For that purpose, we use \\eref{eq:E+} and \\eref{eq:Phi+1} to construct $\\Phi_+\\bar\\mathcal E_+=\\frac{\\pi_1\\bar\\pi_2}{R_2 \\bar r_2}$ and compare with \\eref{eq:Phi+E+}, which shows that\n\\begin{equation}\\label{eq:p4a}\n p_4=R_2\\bar r_2.\n\\end{equation}\n(Since both sides in the previous equation are monic polynomials, we indeed have equality and not just proportionality.)\nWe also use \\eref{eq:Phi+1} to obtain $|\\Phi_+|^2=\\frac{\\pi_1\\bar\\pi_1}{R_2\\bar R_2}$, which, together with \\eref{eq:Phi+2}, implies that \n\\begin{equation}\\label{eq:p4b}\n p_4=R_2\\bar R_2\n\\end{equation}\nis another representation of $p_4$. Combining \\eref{eq:p4a} and \\eref{eq:p4b}, we immediately confirm that $R_2=r_2$ must hold. \n\nHence the previous formulae for the axis potentials finally simplify to\n\\begin{equation}\n \\mathcal E_+=\\frac{\\pi_2(\\zeta)}{r_2(\\zeta)},\\quad\n \\Phi_+=\\frac{\\pi_1(\\zeta)}{r_2(\\zeta)},\n\\end{equation}\ni.e.\\ the axis values are given in terms of complex polynomials $\\pi_2$, $r_2$ and $\\pi_1$ of the indicated degrees, where $\\pi_2$ and $r_2$ are monic polynomials. \n\n\n\\section{Discussion\\label{sec:discussion}}\n\nWe have derived the most general axis data for candidate solutions that could describe axisymmetric and stationary two-black-hole configurations with non-extremal rotating and charged black holes. Necessarily, the axis values of the Ernst potentials must be of the form\n\\begin{equation}\n \\mathcal E_+(\\zeta)=\\frac{(\\zeta-c_1)(\\zeta-c_2)}{(\\zeta-d_1)(\\zeta-d_2)},\\quad\n \\Phi_+(\\zeta)=\\frac{e_1\\zeta+e_2}{(\\zeta-d_1)(\\zeta-d_2)}\n\\end{equation}\nwith complex constants $c_i$, $d_i$, $e_i$, $i=1,2$, which corresponds to 12 real degrees of freedom. Note that we can immediately reduce the available degrees of freedom by comparing the asymptotic expansions of these data,\n\\begin{equation}\n \\mathcal E_+(\\zeta)=1-\\frac{c_1+c_2-d_1-d_2}{\\zeta}+\\mathcal O(\\zeta^{-2}),\\quad\n \\Phi_+(\\zeta)= \\frac{e_1}{\\zeta}+\\mathcal O(\\zeta^{-2}),\n\\end{equation}\nwith the general behaviour of the axis potentials in an asymptotically flat spacetime with total (ADM) mass $M$ and charge $Q$, and without NUT parameter and without magnetic charge,\n\\begin{equation} \n \\mathcal E_+(\\zeta)=1-\\frac{2M}{\\zeta}+\\mathcal O(\\zeta^{-2}),\\quad\n \\Phi_+(\\zeta)=\\frac{Q}{\\zeta}+\\mathcal O(\\zeta^{-2}).\n\\end{equation}\nObviously, we require the constraints\n\\begin{equation}\n \\Im(c_1+c_2-d_1-d_2)=0,\\quad \\Im(e_1)=0,\n\\end{equation}\nwhich leaves us with 10 degrees of freedom.\n\nNote that the boundary data \\eref{eq:bdata1} and \\eref{eq:bdata2a}, \\eref{eq:bdata2b} for the existing 8-parametric exact candidate solutions discussed in Sec.~\\ref{sec:intro} are all of the above form. Hence it remains to decide in future investigations whether these solutions do indeed describe physically acceptable equilibrium configurations, i.e.\\ solutions for which the regularity requirements (i)--(v) from Sec.~\\ref{sec:intro} are all satisfied for particular choices of the parameters. Moreover, it should be studied whether slightly larger solution classes (with the above-mentioned 10 degrees of freedom) need to be considered as well.\n\n\n\n\n\n\n\\section*{Acknowledgments}\nI would like to thank Dominic Searles for commenting on the manuscript. \n \n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section_introduction}}\n\\else\n\\section{Introduction}\n\\label{section_introduction}\n\\fi\n\n\\IEEEPARstart{C}{onvolutional} Neural Networks (CNNs), as the dominant technique of deep learning~\\cite{lecun2015deep}, have shown remarkable superiority in various real-world applications over most machine learning approaches~\\cite{krizhevsky2012imagenet,sainath2013deep,sutskever2014sequence,alphago}. Since the year of 1998 when the first version of the CNN (i.e., LeNet5~\\cite{lecun1998gradient}) was proposed, diverse CNN variants have been developed, such as AlexNet~\\cite{krizhevsky2012imagenet}, VGG~\\cite{simonyan2014very}, GoogleNet~\\cite{szegedy2015going}, ResNet~\\cite{he2016deep} and DenseNet~\\cite{huang2017densely}, to name a few. There is a trend among the CNN variants that their architectures become increasingly deeper. For example, the depth of LeNet5 is six, VGG is $16$, while ResNet~\\cite{he2016identity} achieves at a depth of $1,202$. The principle behind such designs is that a deeper CNN typically has a more powerful capability to address much complex and large-scale data. \n\nThe state-of-the-art CNNs are typically designed by experts who have rich domain knowledge from both investigated data and CNNs. Because the performance of the CNNs strongly relies on the investigated data, it is expected that there is a major limitation to this design manner. For example, researchers who are familiar with the data at hand do not necessarily have the experience in designing the architectures of CNNs, and vice versa. To this end, there is a great demand for developing algorithms which allow researchers without any expertise to automatically derive the best performing CNN for the given data. Indeed, multiple algorithms for this purpose have been proposed in recent years.\n\n\nIn general, the algorithms of designing architectures for CNNs can be divided into two different categories according to their base techniques. The first covers methods using evolutionary algorithms~\\cite{back1996evolutionary}, such as the genetic CNN method (Genetic CNN)~\\cite{xie2017genetic}, the large-scale evolution method (Large-scale Evolution)~\\cite{real2017large}, the hierarchical representation method (Hierarchical Evolution)~\\cite{liu2017hierarchical} and the Cartesian genetic programming method (CGP-CNN)~\\cite{suganuma2017genetic}. These algorithms follow the standard flow of an evolutionary algorithm to heuristically discover the optimal solution. The second refers to algorithms based on reinforcement learning~\\cite{sutton1998reinforcement}, such as the neural architecture search method (NAS)~\\cite{zoph2016neural}, the meta-modelling method (MetaQNN)~\\cite{baker2016designing}, the efficient architecture search method (EAS)~\\cite{cai2018efficient} and the block design method (Block-QNN-S)~\\cite{zhong2017practical}. The algorithms in the second category resemble those in the first category, in addition to the employed heuristic nature that algorithms in the second category utilize the reward-penalty principle of reinforcement learning.\n\nExperimental results of these algorithms have demonstrated promising classification accuracy in the challenging benchmark datasets, such as CIFAR10 and CIFAR100~\\cite{krizhevsky2009learning}, but limitations also exist. Firstly, the algorithms in the first category do not make full use of the advantages of evolutionary algorithms, which results in consuming extensive computational resource and not promising classification accuracy. Secondly, the algorithms in the second category require even more computational resources than the algorithms in the first category due to the nature of reinforcement learning. Thirdly, manual assistances based on domain expertise are required for most algorithms in both categories. To this end, the development of algorithms automatically 1) discovering the best CNN architectures for given data, 2) relying on the limited computational resources and 3) being directly used without any manual refinement or re-composition regarding their discovered CNNs, is still in its infancy. Note that, depending on whether expertise in CNNs is required or not in using these algorithms, they can also be classified into the automatic and the semi-automatic categories. The first includes Large-scale Evolution, CGP-CNN, NAS and Meta-CNN, while the second is composed of Genetic CNN, Hierarchical Evolution, EAS and Block-QNN-S.\n\n\nEvolutionary algorithm~\\cite{back1996evolutionary} is a class of population-based meta-heuristic optimization paradigm inspired by the biological evolution. Typical evolutionary algorithms include genetic algorithms (GAs)~\\cite{davis1991handbook}, genetic programming~\\cite{banzhaf1998genetic}, evolutionary strategy~\\cite{janis1976evolutionary}, etc., among which GAs are the most popular one mainly because of their theoretical evidences~\\cite{schmitt2001theory} and promising performance in solving different optimization problems~\\cite{deb2002fast,sun2018igd,sun2017reference,sun2018improve,transferjiang2018}. It has also been recognized that GAs are capable of generating high-quality optimal solutions by using bio-inspired operators, i.e., mutation, crossover and selection~\\cite{mitchell1998introduction}. Our goal in this paper is to develop an effective and efficient algorithm by using GA, in short, termed as CNN-GA, to automatically discover the best architectures of CNNs for given image classification tasks, so that the discovered CNN can be directly used without any manual refinement or re-composition. The contributions of the proposed CNN-GA method are summarized as follows:\n\n\\begin{enumerate}\n\t\\item The depth of CNNs is not limited to a predefined number in the proposed algorithm, but instead the best depth is discovered during the evolutionary process, by finding the CNN with the best classification accuracy for the given data. Although this design could produce the optimal CNN, the crossover operation, which plays the role of exploitation search (i.e., the local search), cannot work in such a situation due to individuals with unequal (variable) lengths. To address this need, we also design a crossover operator to adapt for these individuals during the evolutionary progress.\n\t\n\t\\item The skip connection, of which the superiority has been theoretically and experimentally proven in effectively training deep architectures, is directly incorporated into the proposed algorithm. In such a way, the evolved CNNs are capable of dealing with complex data by using deep architectures, by avoiding the Gradient Vanishing (GV) problems~\\cite{hochreiter1997long}. Furthermore, this design can also reduce the search space so that the best performance can be achieved within the limited time. In addition, compared to other models with similar performance, the architectures evolved by the proposed algorithm have a much smaller number of parameters.\n\t  \n\t\n\n\t\\item The proposed algorithm is completely automated in discovering the best CNN architectures, and does not require any manual intervention during the evolutionary search. When evolution is finished, the obtained CNNs can be directly used to process the data, and do not need further refinement, such as adding more convolutional or pooling layers. Furthermore, the proposed algorithm can be directly used by other researchers who do not need to do any preparations such as providing a manually tuned network in advance.\n\t\n\t\n\t\\item An asynchronous computational component is designed to make full use of the given computational resources to accelerate the evaluation of fitness for the individuals in the same generation of the proposed algorithm. In addition, a cache component is also developed which is expected to further reduce the fitness evaluation time for the whole population.\n\n\\end{enumerate}\n\nThe remainder of this paper is organized as follows. Firstly, the background is presented in Section~\\ref{section_literature}. Then, the details of the proposed algorithm are documented in Section~\\ref{section_algorithm}. Next, the experimental designs and experimental results are shown in Sections~\\ref{section_exp_settings} and~\\ref{section_exp_results}, respectively. Finally, conclusions and future works are outlined in Section~\\ref{section_conclusion}.\n\n\n\n\\section{Background}\n\\label{section_literature}\n\nIn this section, CNNs, skip connections and GAs, which are the background of the proposed algorithm, are introduced to help readers better understand the related works and the proposed algorithm.\n\n\\subsection{CNNs}\n\\label{sub_sub_section_cnn} \n\nIn this subsection, we mainly introduce the building blocks of CNNs, i.e., the convolutional and pooling layers, which are the basic objects encoded by GAs to represent CNNs.\n\nSpecifically, the convolutional layer employs filters to perform convolutional operations on the input data. One filter can be viewed as a matrix. In this paper, we focus on 2-dimension convolution operators (with a 2-dimension filter) because the proposed algorithm aims at processing image data. During the convolutional operation, the filter horizontally slides (with a given step size), then vertically moves (with another step size) for the next horizontal slide, until the whole image has been scanned. At each position, the filter is applied to the image by multiplying each filter value with the corresponding pixel value, and summing the results to give the output of the filter. The set of filter outputs form a new matrix called the feature map. The horizontal and vertical step sizes are called the width and height of a stride. In a convolutional layer, multiple filters (typically with the same sizes and using the same stride) are allowed to coexist, producing a set of feature maps. The exact number of feature maps used is a parameter in the architecture of the corresponding CNN. The number of filters is derived from the number of resulting feature maps and the spatial size of the input data. In addition, two convolutional operations are applied: the \\textit{same} convolutional operation which pads zeros to the input data when there is no enough area for the\nfilter to overlap, and the \\textit{valid} convolutional operation which does not pad anything. Hence, the parameters of a convolutional layer are the number of feature maps, the filter size, the stride size and the convolutional operation type.\n\nA pooling layer has common components of a convolutional layer except that 1) the filter is called the kernel which has no value, 2) the output of a kernel is the maximal or mean value of the area it stops, and 3) the spatial size of the input data is not changed through a pooling layer. When the maximal value is returned, it is a pooling layer with the type of \\textit{max}, otherwise of \\textit{mean}. Hence, the parameters of a pooling layer are the kernel size, the stride size and the pooling type used.\n\nIn addition, the fully-connected layers are usually incorporated into the tail of a CNN. Because it is very common and also not the building blocks of CNNs, we will not detail it here. But note that, the number of fully-connected layers and the number of neurons in each fully-connected layer are also the parameters of a CNN's architecture, if the fully-connected layers are used in the CNN. In the proposed algorithm, the fully-connected layers are discarded, and the justifications are given in Subsection~\\ref{sub_section_pop_init}.\n\n\n\\subsection{Skip Connections}\n\\label{sub_sub_section_skip_connections}\nCommonly, the connections in CNNs exist between the neurons of two adjacent layers. Analogously, the skip connections refer to those connecting the neurons of the layers that are not adjacent. The skip connection was firstly introduced in~\\cite{hochreiter1997long} as a gate mechanism, effectively training a recurrent neural network with long and short-term memory~\\cite{gers1999learning} and avoiding the GV problems~\\cite{hochreiter1997long}. Specifically, the GV problems refer to the gradient becoming very small or explosion during back propagation training in a deep neural network. Because gradient-based algorithms are the dominant learning algorithms of neural networks, the GV problems are the main obstacle to effectively training \\textit{deep} neural networks. The skip connections were experimentally proven to be able to train very deep neural networks~\\cite{srivastava2015training}. Indeed, the promising performance of ResNet~\\cite{he2016deep}, which was proposed very recently, also benefits from the skip connections. A simple example of using a skip connection is shown in Fig.~\\ref{fig_skip_connection}, where the dashed line denotes the skip connection from the input $X$ to the output of the $N$-th building block, and the symbol ``$\\oplus$'' refers to the element-wise addition.\n\n\\begin{figure}[!htp]\n\t\\centering\n\t\\includegraphics[width=0.3\\columnwidth]{skip_connection}\\\\\n\t\\caption{An example of using the skip connection.}\\label{fig_skip_connection}\n\\end{figure}\n\nOver the past few years, an increasing number of researchers has attempted to theoretically reveal the mechanisms behind the skip connections. For example, the skip connections have also been claimed to be able to eliminate singularities~\\cite{orhan2017skip}. However, a completely satisfactory explanation is still elusive. Among the existing theoretical evidence, skip connections defying the GV problems receives the most recognition~\\cite{gers1999learning,hochreiter1997long}. The GV problems frequently occur when the error has been back-propagated over many layers of a given neural network. Because the skip connections shorten the number of layers of back-propagation, the GV problems should be alleviated. As discussed above, the deeper the CNN is, the more powerful capability it would have to process complex data. Combined with the connections that are not skipped, a CNN with the skip connections can have the capability that a deep architecture has and can also be effectively trained.\n\nOwning to the promising performance of the skip connections, it is naturally incorporated into the proposed algorithm to discover the architectures of an optimal CNN. In addition, the search space can also be reduced if these skip connections are directly used. Consequently, the required computational resources can be minimized, and the promising architectures of CNNs can be discovered within a limited time.\n\n\\begin{figure*}[!htp]\n\t\\centering\n\t\\includegraphics[width=1.8\\columnwidth]{ga_flow_chart}\\\\\n\t\\caption{The flowchart of genetic algorithm.}\\label{fig_ga_flow}\n\\end{figure*}\n\n\\subsection{Genetic Algorithms}\n\\label{sub_sub_section_gas}\nThe flowchart of a GA is shown in Fig.~\\ref{fig_ga_flow}. Specifically, a population of individuals (i.e., CNNs with architecture variants) is randomly initialized first, and then the fitness of each individual is evaluated. The fitness is measured by a deterministic function which is known as the fitness function, based on the context of the problem to be optimized (i.e., performance of CNNs on specific image classification tasks). The input of the function is the decision variables encoded in the individuals. After that, the individuals who have better fitness will be chosen by the selection operation, to hopefully generate offspring with better fitness. Specifically, new offspring are generated by exchanging or varying the encoded information of the selected parent individuals, i.e. by the crossover and mutation operations. These generated offspring are evaluated for the fitness, and then the population surviving into the next generation are selected from the current population, which is composed of the parent population and generated offspring, by the environmental selection. Through repeating a series of these operations, it is expected to find the optimal solution from the population when the GA terminates. Note that, the typical criterion to terminate a GA is a predefined maximal generation number, say $20$ in our experiments.\n\n\\section{The Proposed Algorithm}\n\\label{section_algorithm}\n\nIn this section, we firstly present the framework of the proposed algorithm in Subsection~\\ref{sub_section_alg_overview}, and then detail the main steps in Subsections~\\ref{sub_section_pop_init} to \\ref{sub_section_environmental_selection}. To help readers better understand the proposed algorithm, we will not only document the details of each main step, but also provide the justifications for such designs. \n\\subsection{Algorithm Overview}\n\\label{sub_section_alg_overview}\n\\begin{algorithm}\n\t\\label{alg_framework}\n\t\\caption{Framework of The Proposed Algorithm}\n\t\\KwIn{A set of predefined building blocks, the population size, the maximal generation number, the image dataset for classification.}\n\t\\KwOut{The discovered best architecture of CNN.}\n\t$P_0\\leftarrow$ Initialize a population with the given population size\\;\n\t\\label{alg_framework_line1}\n\t$t\\leftarrow 0$\\;\n\t\\label{alg_framework_line2}\n\t\\While{$t<$ the maximal generation number}\n\t{\n\t\tEvaluate the fitness of each individual in $P_t$\\;\n\t\t\\label{alg_framework_evo_line1}\n\t\t$Q_t\\leftarrow$ Generate offspring by genetic operators from the selected parent individuals\\;\n\t\t\\label{alg_framework_evo_line2}\n\t\t$P_{t+1}\\leftarrow$ Environmental selection from $P_t\\cup Q_t$\\;\n\t\t\\label{alg_framework_evo_line3}\n\t\t$t\\leftarrow t+1$\\;\n\t}\n\t\\textbf{Return} the individual which has the best fitness in $P_{t}$.\n\t\n\\end{algorithm}\nAlgorithm~\\ref{alg_framework} shows the framework of the proposed algorithm. Specifically, by giving a set of predefined building blocks of CNNs, the population size as well as the maximal generation number for the GA and the image classification dataset, the proposed algorithm begins to work, through a series of evolutionary processes, and finally discovers the best architecture of the CNN to classify the given image dataset. During evolution, a population is randomly initialized with the predefined population size, using the proposed encoding strategy to encode the predefined building blocks (line~\\ref{alg_framework_line1}). Then, a counter for the current generation is initialized to zero (line~\\ref{alg_framework_line2}). During evolution, the fitness of each individual, which encodes a particular architecture of the CNN, is evaluated on the given dataset (line~\\ref{alg_framework_evo_line1}). After that, the parent individuals are selected based on the fitness, and then generate new offspring by the genetic operators (line~\\ref{alg_framework_evo_line2}). Then, a population of individuals surviving into the next generation are selected by the environmental selection from the current population (line~\\ref{alg_framework_evo_line3}). Specifically, the current population is composed of the parent population and the generated offspring population. Finally, the counter is increased by one, and the evolution continues until the counter exceeds the predefined maximal generation. As shown in Fig.~\\ref{fig_ga_flow}, the proposed algorithm follows the standard pipeline of a GA (the phases of selection, crossover and mutation, and the offspring population shown in Fig.~\\ref{fig_ga_flow} are collectively described in line~\\ref{alg_framework_evo_line2} of Algorithm~\\ref{alg_framework}). Note that, the genetic operator is composed of the crossover and mutation operations.\n\n\\subsection{Population Initialization}\n\\label{sub_section_pop_init}\n\nAs introduced in Section~\\ref{section_literature}, a CNN is composed of the convolutional layers, pooling layers and occasionally fully-connected layers. The performance of a CNN highly relies on its depth, and the skip connections could turn the \\textit{deep depth} to be a reality. In the proposed encoding strategy, we design a new building block by directly using the skip connections, named the skip layer, to replace the convolutional layer when forming a CNN. In addition, the fully-connected layers are discarded in the proposed encoding strategy (the reason will be given later in this subsection). In summary, only the skip layers and the pooling layers are used to construct a CNN in the proposed encoding strategy.\n\n\\begin{figure}[!htp]\n\t\\centering\n\t\\includegraphics[width=0.3\\columnwidth]{skip_layer}\\\\\n\t\\caption{An example of the skip layer in the proposed encoding strategy.}\\label{fig_skip_layer}\n\\end{figure}\n\nSpecifically, a skip layer is composed of two convolutional layers and one skip connection. The skip connection connects from the input of the first convolutional layer to the output of the second convolutional layer. An example of the skip layer is shown in Fig.~\\ref{fig_skip_layer}. As introduced above, the parameters of a convolutional layer are the number of feature maps, the filter size, the stride size and the convolutional operation type. In the proposed encoding strategy, we use the same settings for the filter sizes, stride sizes and convolutional operations. Particularly, the filter and stride sizes are set to $3\\times 3 $ and $1\\times 1$, respectively, and only the \\textit{same} convolutional operation type is used. To this end, the parameters encoded for a skip layer are the numbers of the feature maps for the two convolutional layers (denoted as $F1$ and $F2$, respectively). In addition, the pooling layers used in the proposed encoding strategy are set to be $2\\times 2$ for both the kernel sizes and the stride sizes. To this end, the parameter encoded for a pooling layer is only the pooling type (denoted as $P1$). Note that, the reasons for this design and adopting the settings for such a design will be explained later in this subsection.\n\n\\begin{algorithm}\n\t\\label{alg_pop_init}\n\t\\caption{Population Initialization}\n\t\\KwIn{The population size $T$.}\n\t\\KwOut{The initialized population $P_0$.}\n\t$P_0\\leftarrow \\emptyset$ \\;\n\t\\While{$|P_0|<T$}\n\t{\n\t\t$L\\leftarrow$ Randomly generate an integer greater than zero\\;\n\t\t\\label{alg_pop_init_line1}\n\t\t$list\\leftarrow$ Create a linked list contains $L$ nodes\\;\n\t\t\\label{alg_pop_init_line2}\n\t\t\\ForEach{node in the linked list}\n\t\t{\\label{alg_pop_init_node_begin}\n\t\t\t$r\\leftarrow$ Uniformly generate a number from $(0,1)$\\;\n\t\t\t\\label{alg_pop_init_node_type}\n\t\t\t\\uIf{$r<0.5$}\n\t\t\t{\\label{alg_pop_init_skip_layer_begin}\n\t\t\t\t$node.type\\leftarrow 1$\\;\n\t\t\t\t$node.F1\\leftarrow Randomly$ generate an integer greater than zero\\;\n\t\t\t\t$node.F2\\leftarrow Randomly$ generate an integer greater than zero\\;\n\t\t\t}\\label{alg_pop_init_skip_layer_end}\n\t\t\t\\Else{\\label{alg_pop_init_pool_begin}\n\t\t\t\t$node.type\\leftarrow 2$\\;\n\t\t\t\t$q\\leftarrow$ Uniformly generate a number from $(0,1)$\\;\n\t\t\t\t\\uIf{$q<0.5$}\n\t\t\t\t{\n\t\t\t\t\t$node.P1\\leftarrow$ \\textit{max}\\;\n\t\t\t\t}\n\t\t\t\t\\Else{\n\t\t\t\t\t$node.P1\\leftarrow$ \\textit{mean}\\;\t\n\t\t\t\t}\n\t\t\t}\\label{alg_pop_init_pool_end}\n\t\t}\t\\label{alg_pop_init_node_end}\n\t\t$P_0\\leftarrow P_0\\cup list$\\;\n\t\t\\label{alg_pop_init_end}\n\t}\n\t\\textbf{Return} $P_{0}$.\n\t\n\\end{algorithm}\nAlgorithm~\\ref{alg_pop_init} shows the details of the population initialization. Briefly, $T$ individuals are initialized in the same manners, and then they are stored into $P_0$. During the individual initialization process, the length (denoted as $L$) of an individual, representing the depth of the corresponding CNN, is randomly initialized at first (line~\\ref{alg_pop_init_line1}). Then, a linked list containing $L$ nodes is created (line~\\ref{alg_pop_init_line2}). After that, each node is configured (lines~\\ref{alg_pop_init_node_begin}-\\ref{alg_pop_init_node_end}), and then the linked list is stored into $P_0$ (line~\\ref{alg_pop_init_end}). During the configuration of each node, a number, $r$, is randomly generated from $(0,1)$ (line~\\ref{alg_pop_init_node_type}). If $r<0.5$, the type of this node is marked as a skip layer by setting its $type$ property to $1$. Otherwise, this node represents a pooling layer by setting its $type$ to $2$. In the case of a skip connection layer, the numbers of the feature maps are randomly generated and then assigned to $node.F1$ and $node.F2$, respectively (lines~\\ref{alg_pop_init_skip_layer_begin}-\\ref{alg_pop_init_skip_layer_end}). Otherwise, the pooling type is determined by the probability of flipping a coin. Particularly, the pooling type, $node.P1$, is set to \\textit{max} when the probability is below $0.5$, and \\textit{mean} otherwise (lines~\\ref{alg_pop_init_pool_begin}-\\ref{alg_pop_init_pool_end}). \n\nNext, we will detail the reasons why discarding the fully-connected layers, using two convolutional layers in a skip layer and the settings for the skip and pooling layers in the proposed algorithm. Typically, multiple fully-connected layers are added to the tail of a CNN. However, the fully-connected layer easily results into the over-fitting phenomenon~\\cite{hawkins2004problem} due to its dense connection~\\cite{srivastava2014dropout}. To reduce the over-fitting, the dropout~\\cite{srivastava2014dropout} randomly removing a part of the connections is commonly used. However, each dropout will introduce one additional parameter. Only a properly specified parameter can lead to the promising performance of the corresponding CNN. Meanwhile, the number of fully-connected layers and the number of neurons in each fully-connected layer are extra parameters hard to tune. If the fully-connected layers are incorporated into the proposed encoding strategy, the search space will be substantially enlarged, increasing the difficulty of searching for the best CNN architecture. The use of two convolutional layers in a skip layer is inspired by the design of ResNet, and the effectiveness of such skip layers have been experimentally proven in literature~\\cite{huang2017densely,he2016identity,real2017large,liu2017hierarchical}. However, the sizes of feature maps in each skip layer of ResNet are set to be equal. In our proposed encoding strategy, the sizes of feature maps can be different, which is believed to be more flexible. Furthermore, setting the convolutional operation type to \\textit{same} and using the $1\\times 1$ stride are to make the dimension of the input data remain the same, which is more flexible for such an automatic design. Please note $1\\times 1$ stride does not change the image size. As far as the settings of filter and kernel sizes as well as the stride size in the pooling layers are concerned, they are all based on the designs of existing hand-crafted CNNs~\\cite{huang2017densely,he2016identity}. Moreover, another important reason for specifying such settings is based on our expertise in manually tuning the architectures of CNNs. The effectiveness of such settings will be shown in Section~\\ref{section_exp_results}.\n\\subsection{Fitness Evaluation}\n\\label{sub_section_fitness_val}\n\\begin{algorithm}\n\t\\label{alg_fitness_eval}\n\t\\caption{Fitness Evaluation}\n\t\\KwIn{The population $P_t$ of the individuals to be evauated, the image dataset for clasification.}\n\t\\KwOut{The population $P_t$ of the individuals with fitness values.}\n\t\\If{t == 0}\n\t{\\label{alg_cache_begin}\n\t\t$Cache \\leftarrow \\emptyset$\\;\n\t\tSet $Cache$ to a global variable\\; \n\t\t\n\t}\\label{alg_cache_end}\n\t\\ForEach{individual in $P_t$}\n\t{\t\n\t\t\\eIf{the identifier of individual in Cache}\n\t\t{\\label{alg_cache_get_from}\n\t\t\t$v\\leftarrow$ Query the fitness by $identifier$ from $Cache$\\;\n\t\t\tSet $v$ to $individual$\\;\n\t\t\t\\label{alg_cache_get_end}\n\t\t}\n\t\t{\\label{alg_cache_not_from}\n\t\t\t\\While{there is available GPU}\n\t\t\t{\n\t\t\t\tasynchronously evaluate $individual$ in an availabel GPU (details shown in Algorithm~\\ref{alg_fitness_individual_eval})\\;\n\t\t\t}\n\t\t}\\label{alg_cache_not_end}\n\t\t\n\t}\n\t\\textbf{Return} $P_{t}$.\n\\end{algorithm}\n\nAlgorithm~\\ref{alg_fitness_eval} details the fitness evaluation of the individuals in the population $P_t$. Briefly, given the population, $P_t$, containing all the individuals for evaluating the fitness, and the image classification dataset on which the best architecture of a CNN is to be discovered, Algorithm~\\ref{alg_fitness_eval} evaluates each individual of $P_t$ in the same manner, and finally returns $P_t$ containing the individuals whose fitness have been evaluated. Specifically, if the fitness evaluation is for the initialized population, i.e., $P_0$, a global cache system (denoted as $Cache$) is created, storing the fitness of the individuals with unseen architectures (lines~\\ref{alg_cache_begin}-\\ref{alg_cache_end}). For each individual (denoted by $individual$) in $P_t$, if $individual$ is found in $Cache$, its fitness is directly retrieved from $Cache$ (lines~\\ref{alg_cache_get_from}-\\ref{alg_cache_get_end}). Otherwise, $individual$ is asynchronously placed on an available GPU for its fitness evaluation (lines~\\ref{alg_cache_not_from}-\\ref{alg_cache_not_end}). Note that, querying an individual from $Cache$ is based on the individual's identifier. Theoretically, arbitrary identifiers can be used, as long as they can distinguish individuals encoding different architectures. In the proposed algorithm, the 224-hash code~\\cite{housley2004224}, which has been implemented by most programming languages, in terms of the encoded architecture is used as the corresponding identifier. Furthermore, the individual is asynchronously placed on an available GPU, which implies that we don't need to wait for the fitness evaluation for the next individual until the fitness evaluation of the current one finishes, but simply place the next individual on an available GPU immediately.\n\n\\begin{algorithm}\n\t\\label{alg_fitness_individual_eval}\n\t\\caption{Individual Fitness Evaluation}\n\t\\KwIn{The individual $individual$, the available GPU, the number of training epochs, the global cache $Cache$, the training data $D_{train}$ and the validation data $D_{valid}$ from the given image classification dataset.}\n\t\\KwOut{The individual $individual$ with its fitness.}\n\tConstruct a CNN with a classifier based on the information encoded in $individual$ and the given image classification dataset\\; \\label{alg_ini_eval_line1}\n\t$v_{best}\\leftarrow 0$\\;\n\t\\ForEach{epoch in the given training epochs}\n\t{\n\t\tTrain the CNN on $D_{train}$ by using the given GPU\\;\n\t\t\\label{alg_ini_eval_train_line1}\n\t\t$v\\leftarrow$ Calculate the classification accuracy on $D_{valid}$\\;\n\t\t\\label{alg_ini_eval_train_line2}\n\t\t\\If{$v>v_{best}$}\n\t\t{\n\t\t\t$v_{best}\\leftarrow v$\\;\n\t\t}\n\t}\n\tSet $v_{best}$ as the fitness of $individual$\\;\\label{alg_ini_eval_train_line3}\n\tPut the identifier of $individual$ and $v_{best}$ into $Cache$\\;\n\t\\label{alg_ini_eval_train_line4}\n\t\\textbf{Return} $individual$.\n\\end{algorithm}\n\nThe details of evaluating the fitness of one individual are shown in Algorithm~\\ref{alg_fitness_individual_eval}. Firstly, a CNN is decoded from $individual$, and a classifier is added to this CNN (line~\\ref{alg_ini_eval_line1}) based on the given image classification dataset. In the proposed algorithm, a softmax classifier~\\cite{nasrabadi2007pattern} is used, and the particular number of classes is determined by the given image dataset. When decoding a CNN, a rectifier activation function~\\cite{glorot2011deep} followed by a batch normalization~\\cite{NIPS2017_6790} operation is added to the output of the convolutional layer, which is based on the conventions of modern CNNs~\\cite{he2016deep,huang2017densely}. In addition, when the spatial number of the skip layer differs from that of the input data, a convolutional layer, which is with the unit filter and the unit stride but the corresponding number of the feature maps, is added to the input data~\\cite{he2016deep,huang2017densely}. After that, the CNN is trained by the Stochastic Gradient Descent (SGD) algorithm~\\cite{bottou2012stochastic} on the training data by using the given GPU (line~\\ref{alg_ini_eval_train_line1}), and the classification accuracy is calculated on the validation data (line~\\ref{alg_ini_eval_train_line2}). Note that, the use of the softmax classifier and SGD training method are based on the conventions of the deep learning community. When the training phase is finished, the best classification accuracy on the validation data is set as the fitness of $individual$ (line~\\ref{alg_ini_eval_train_line3}). Finally, the identifier and fitness of $individual$ are associated and put into $Cache$ (line~\\ref{alg_ini_eval_train_line4}).\n\nNext, the reasons for designing such an asynchronous and a cache components are given. In summary, because the training on CNNs is very time-consuming, ranging from several hours to even several months depending on the particular architecture, they are designed to speed up the fitness evaluation in the proposed algorithm. Specifically, the asynchronous component is a parallel computation platform based on GPUs. Due to the computational nature of calculating the gradients, deep learning algorithms are typically placed on GPUs to speed up the training~\\cite{helfenstein2012parallel}. Indeed, existing deep learning libraries, such as Tensorflow~\\cite{abadi2016tensorflow} and PyTorch~\\cite{paszke2017automatic}, support the calculation on multiple GPUs. However, their parallel calculations are based on the data-parallel and model-parallel pipelines. In the data-parallel pipeline, the input data is divided into several smaller groups, and each group is placed on one GPU for calculation. The reason is that the limited memory of one GPU cannot effectively handle the whole data at the same time. In the model-parallel pipeline, a model is divided into several smaller models, and each GPU carries one smaller model. The obvious reason is the limited computational capability on one GPU cannot run a whole model. However, the designed parallel pipeline obviously does not fall into either of the pipelines, but at a higher level. Hence, such an asynchronous component is designed to make full use of the GPU computational resource, especially for the population-based algorithms. Furthermore, the asynchronous component is widely used in solving a large problem, if the problem can be divided into several independent sub-problems. By parallelly performing these sub-problems in different computational platforms, the total processing time of the whole problem is consequently shortened. In the past, evolutionary algorithms are typically used to solve the problems of which the fitness evaluation is not time-consuming\\footnote{Although there is a type of computationally expensive problems, their fitness is commonly calculated by a surrogate model to bypass the direct fitness evaluation.}, and there is no critical need in developing such asynchronous components. Occasionally, they just use the built-in components based on the adopted programming languages. However, almost all such built-in components are based on CPUs, and they cannot effectively train deep neural networks, mainly because the acceleration platform for neural networks are based on GPUs. Furthermore, the fitness evaluation of each individual is independent, which just satisfies the scenario of using this technique. Motivated by the reasons described above, such an asynchronous component is designed in the proposed algorithm. The cache component is also used to speed up the fitness evaluation, which is based on the considerations of: 1) the individuals surviving into the next generation do not need to evaluate the fitness again if its architecture is not changed, and 2) the architecture, which has been evaluated, could be regenerated with the mutation and crossover operations in another generation.\n\n\\subsection{Offspring Generating}\n\\label{sub_section_offspring_gen}\n\\begin{algorithm}\n\t\\label{alg_offspring_gene}\n\t\\caption{Offspring Generating}\n\t\\KwIn{The population $P_t$ containing individuals with fitness, the probabity for crossover operation $p_c$, the probability for mutation operation$p_m$, the mutation operation list $l_m$, the probabilities of selecting different mutation operations $p_l$.}\n\t\\KwOut{The offspring population $Q_t$.}\n\t\n\t$Q_t\\leftarrow \\emptyset$\\;\n\t\\label{alg_crossover_begin}\n\t\\While{$|Q_t|<|P_t|$}\n\t{\n\t\t$p_1\\leftarrow$ Randomly select two individuals from $P_t$, and from the two then select the one with the better fitness\\;\n\t\t\\label{alg_off_gene_p_selc}\n\t\t$p_2\\leftarrow$ Repeat Line~\\ref{alg_off_gene_p_selc}\\;\n\t\t\\label{alg_off_gene_pp_selc}\n\t\t\\While{$p_2 == p_1$}\n\t\t{\n\t\t\tRepeat Line~\\ref{alg_off_gene_pp_selc}\\;\n\t\t}\\label{alg_off_gene_p_selc_end}\n\t\t\n\t\t$r\\leftarrow$ Randomly generate a number from $(0,1)$\\;\n\t\t\\label{alg_off_gene_r1}\n\t\t\\eIf{$r<p_c$}\n\t\t{\n\t\t\tRandomly choose a point in $p_1$ and divide it into two parts\\;\n\t\t\t\\label{alg_off_cross_r1}\n\t\t\tRandomly choose a point in $p_2$ and divide it into two parts\\;\n\t\t\t$o_1\\leftarrow$ Join the first part of $p_1$ and the second part of $p_2$\\;\n\t\t\t$o_2\\leftarrow$ Join the first part of $p_2$ and the second part of $p_1$\\;\n\t\t\t$Q_t\\leftarrow Q_t\\cup o_1 \\cup o_2$\\;\n\t\t\t\\label{alg_off_cross_r2}\n\t\t}\n\t\t{\n\t\t\t$Q_t\\leftarrow Q_t\\cup p_1 \\cup p_2$\\;\n\t\t\t\\label{alg_off_gene_no_cross}\n\t\t}\n\t}\\label{alg_crossover_end}\n\t\\ForEach{individual p in $Q_t$}\n\t{\\label{alg_mutation_begin}\n\t\t$r\\leftarrow$ Randomly generate a number from $(0,1)$\\;\n\t\t\\label{alg_off_mutation_r1}\n\t\t\\If{$r<p_m$}\n\t\t{\\label{alg_off_mutation_begin}\n\t\t\t$i\\leftarrow$ Randomly choose a point in $p$\\;\n\t\t\t$m\\leftarrow$ Select one operation from $l_m$ based on the probabilities in $p_l$\\;\n\t\t\tDo the mutation $m$ at the point $i$ of $p$\\;\n\t\t}\\label{alg_off_mutation_end}\n\t}\\label{alg_mutation_end}\n\t\\textbf{Return} $Q_t$.\n\\end{algorithm}\nThe details of generating the offspring are shown in Algorithm~\\ref{alg_offspring_gene} which is composed of two parts. The first is the crossover (lines~\\ref{alg_crossover_begin}-\\ref{alg_crossover_end}) and the second is the mutation (lines~\\ref{alg_mutation_begin}-\\ref{alg_mutation_end}). During the crossover operation, there will be totally $|P_t|$ offspring generated, where $|\\cdot|$ measures the size of the collection. Specifically, two parents are selected first, and each is selected from two randomly selected individuals based on the better fitness (lines~\\ref{alg_off_gene_p_selc}-\\ref{alg_off_gene_p_selc_end}). This selection is known as the binary tournament selection~\\cite{miller1995genetic}, which is popularly used in GAs for single-objective optimization. Once the parents are selected, a random number is generated~(line~\\ref{alg_off_gene_r1}), determining whether the crossover will be done or not. If the generated number is not below the predefined crossover probability, these two parent individuals are put into $Q_t$ as the offspring (line~\\ref{alg_off_gene_no_cross}). Otherwise, each parent individual is randomly split into two parts, and the two parts from the two parent individuals are swapped to create two offspring (lines~\\ref{alg_off_cross_r1}-\\ref{alg_off_cross_r2}). During the mutation operation, a random number is generated first (line~\\ref{alg_off_mutation_r1}), and the mutation is performed on the current individual if the generated number is below $p_m$ (lines~\\ref{alg_off_mutation_begin}-\\ref{alg_off_mutation_end}). When mutating an individual, a position (denoted as $i$) is randomly selected from the current individual, and one particular mutation operation (denoted as $m$) is selected from the provided mutation list based on the probabilities defined in $p_l$. Then, $m$ is performed on the position $i$. In the proposed algorithm, the available mutation operations defined in the mutation list are:\n\\begin{itemize}\n\t\\item Adding a skip layer with random settings;\n\t\\item Adding a pooling layer with random settings;\n\t\\item Removing the layer at the selected position;\n\t\\item Randomly changing the parameter values of the building block at the selected position.\n\\end{itemize}\n\nNext, the motivations of designing such a crossover operator and selecting a mutation operation based on a provided probability list are given. Firstly, the designed crossover operator is inspired by the one-point crossover~\\cite{srinivas1994genetic} in traditional GAs. However, the one-point crossover was designed only for the individuals with the equal lengths. The designed crossover operator is used for the individuals with variable lengths. Although the designed crossover operator is simple, it dose improve the performance in discovering better architectures of CNNs, which will be experimentally proven in Section~\\ref{section_exp_results}. Secondly, existing algorithms use an equal probability for choosing the particular mutation operation. In the proposed algorithm, the provided mutation operations are selected with different probabilities. Specifically, we provide a higher probability for the ``adding a skip layer\" mutation, which will encourage a higher probability to increase the depths of CNNs. For other mutation operations, we still employ the equal probabilities. The motivation behind this design is that a deeper CNN tends to have a more powerful capability as mentioned previously. Although the ``adding a pooling layer'' is also capable of increasing the depth of CNNs, the dimension of input data will be reduced to half by using one pooling layer, which results in the unavailability of the discovered CNN. To this end, we do not put a higher probability on it. \n\n\\subsection{Environmental Selection}\n\\label{sub_section_environmental_selection}\n\\begin{algorithm}\n\t\\label{alg_environ_sele}\n\t\\caption{Environmental Selection}\n\t\\KwIn{The parent population $P_t$, the offspring population $Q_t$}\n\t\\KwOut{The population for the next generation $P_{t+1}$.}\n\t$P_{t+1}\\leftarrow\\emptyset$\\;\n\t\\While{$|P_{t+1}|<|P_t|$}\n\t{\\label{alg_env_sele_l1}\n\t\t$p_1, p_2\\leftarrow$ Randomly select two individuals from $Q_t\\cup P_t$\\;\n\t\t$p\\leftarrow$ Select the one who has a better fitness from $\\{p_1,p_2\\}$\\;\n\t\t$P_{t+1}\\leftarrow P_{t+1}\\cup p$\\;\n\t}\\label{alg_env_sele_l2}\n\t$p_{best}\\leftarrow$ Find the individual with the best fitness from $Q_t\\cup P_t$\\;\n\t\\label{alg_env_sele_l3}\n\t\\If{$p_{best}$ is not in $P_{t+1}$}\n\t{\n\t\tReplace the one who has the worst fitness in $P_{t+1}$ by $p_{best}$; \n\t}\\label{alg_env_sele_l4}\n\t\\textbf{Return} $P_{t+1}$.\n\\end{algorithm}\nAlgorithm~\\ref{alg_environ_sele} shows the details of the environmental selection. Firstly, $|P_{t}|$ individuals are selected from the current population ($Q_t\\cup P_t$) by using the binary tournament selection, and then these selected individuals are placed into the next population (denoted $P_{t+1}$) (lines~\\ref{alg_env_sele_l1}-\\ref{alg_env_sele_l2}). Secondly, the best individual is selected to check whether it has been placed into $P_{t+1}$. If not, it will replace the worst individual in $P_{t+1}$ (lines~\\ref{alg_env_sele_l3}-\\ref{alg_env_sele_l4}).\n\nIn principle, only selecting the top $|P_t|$ best individuals for the next generation could cause the premature convergence phenomenon~\\cite{michalewicz1996genetic}, which will lead the algorithm trapped into a local optimum~\\cite{goldberg1988genetic,davis1991handbook}. If we do not explicitly select the best individuals for the next generation, the algorithm will not converge. In principle, a desirable population should contain not only good but also the relatively bad ones for enhancing the diversity~\\cite{anderson2005practical,malik2014preventing}. To this end, the binary tournament selection is typically used for such a purpose~\\cite{zhang2003novel,miller1995genetic}. However, only using the binary tournament selection may miss the best individual, resulting in the algorithm not moving towards a better direction of the evolution. Hence, we explicitly add the best individual into the next population, which is regarded as an elitism strategy in evolutionary algorithms~\\cite{bhandari1996genetic}. \n\n\n\\section{Experimental Designs}\n\\label{section_exp_settings}\n\nIn order to evaluate the performance of the proposed algorithm, a series of the experiments has been conducted on image classification tasks. Specifically, the peer competitors chosen to compare with the proposed algorithm are introduced in Subsection~\\ref{sub_section_peer_competitor}. Then, the used benchmark datasets are detailed in Subsection~\\ref{sub_section_benchmark}. Finally, the parameter settings of the proposed algorithm are shown in Subsection~\\ref{sub_section_parameter}. \n\n\n\\subsection{Peer Competitors}\n\\label{sub_section_peer_competitor}\nIn order to show the effectiveness and efficiency of the proposed algorithm, the state-of-the-art algorithms are selected as the peer competitors to the proposed algorithm. Particularly, the peer competitors are chosen from three different categories. \n\nThe first refers to the state-of-the-art CNNs that are manually designed, including ResNet~\\cite{he2016deep}, DenseNet~\\cite{huang2017densely}, VGG~\\cite{simonyan2014very}, Maxout~\\cite{goodfellow2013maxout}, Network in Network~\\cite{lin2013network}, Highway Network~\\cite{srivastava2015highway} and All-CNN~\\cite{springenberg2014striving}. Specifically, two versions of ResNet, i.e., the ResNet models with a depth of $101$ and $1,202$, are used since both of them have shown promising performance. For the convenience, they are named ResNet (depth=$101$) and ResNet (depth=$1,202$), respectively. Note that, most algorithms from this category are the champions in the large-scale visual recognition challenges in recent years~\\cite{ILSVRC15}.\n\nThe second and the third contain the algorithms discovering the best architectures of CNNs using a semi-automatic way and an automatic manner, respectively. Specifically, Genetic CNN~\\cite{xie2017genetic}, Hierarchical Evolution~\\cite{liu2017hierarchical}, EAS~\\cite{zoph2016neural} and Block-QNN-S~\\cite{zhong2017practical} belong to the second category, while Large-scale Evolution~\\cite{real2017large}, CGP-CNN~\\cite{suganuma2017genetic}, NAS~\\cite{zoph2016neural} and MetaQNN~\\cite{baker2016designing} fall into the third category. In addition, algorithms from these two categories are all proposed in the last two years.\n\nNote that, the term ``semi-automatic'' means that expertise in CNNs must be provided before we use the corresponding architecture discovering algorithms. For example, EAS works on a base CNN which has already shown a good classification accuracy for the given data and QNN presumes a predefined big CNN where its discovered CNNs are embedded. This is opposite to the term ``automatic'' that no expertise in CNNs is needed when using the algorithms from this category. Obviously, the proposed algorithm in this paper is automatic. \n\n\n\\subsection{Benchmark Datasets}\n\\label{sub_section_benchmark}\nThe CIFAR10 and CIFAR100 benchmark datasets~\\cite{krizhevsky2009learning} are chosen as the image classification tasks  in the experiments. The reasons for choosing them are that 1) both datasets are challenging in terms of the image sizes, categories of classification and the noise as well as rotations in each image; and 2) they are widely used to measure the performance of deep learning algorithms, and most of the chosen compared algorithms have publicly reported their classification accuracy on them. \n\n\nSpecifically, the CIFAR10 dataset is an image classification problem for recognizing $10$ classes of natural objects, such as air planes and birds. It consists of $60,000$ RGB images in the dimension of $32\\times 32$. In addition, there are $50,000$ images and $10,000$ images in the training dataset and the testing dataset, respectively. Each category has the equal number of images. The CIFAR100 dataset is similar to CIFAR10, except it has $100$ classes. The objects to be classified in these benchmark datasets typically occupy different areas of the whole image, and their positions are not the same in different images. Their variations are also challenging for the classification algorithms.   \n\nIn the experiments, the training images are split into two parts. The first part accounts for $90\\%$ to serve as the training dataset for training the individuals, while the remaining images serve as the validation dataset for the fitness evaluation. In addition, the images are augmented during the training phases. In order to do a fair comparison, we employ the same augmentation routine as those often used in peer competitors, i.e., each direction of one image is padded by four zeros pixels, and then an image with the dimension of $32\\times32$ is randomly cropped, finally, a horizontal flip is randomly performed on the cropped image with a probability of 0.5~\\cite{he2016deep,huang2017densely}. \n\nTo date, most architecture discovering algorithms do not perform experiments on the CIFAR100 dataset due to its large number of classes. In order to show the superiority of our proposed algorithm over the peer competitors, we perform experiments on CIFAR100 and report the results.\n\n\\begin{table*}\n\t\\renewcommand{\\arraystretch}{1.1}\n\t\\caption{The comparisons between the proposed algorithm and the state-of-the-art peer competitors in terms of the classification accuracy (\\%), number of parameters and the taken GPU days on the CIFAR10 and CIFAR100 benchmark datasets.}\n\t\\label{tab_overview}\n\t\\center\n\t\\begin{tabular}{c|l|c|c|c|c|r}\n\t\t\\hline\n\t\t& &\\textbf{CIFAR10} & \\textbf{CIFAR100} & \\textbf{\\# Parameters} & \\textbf{GPU days} & \\textbf{Manual assistance?} \\\\\n\t\t\\hline\n\t\t\\multirow{8}{*}{\\shortstack[l]{state-of-the-art CNNs}}& ResNet (depth=101) & 93.57 & 74.84 & 1.7M & -- & completely need\\\\\n\t\n\t\t& ResNet (depth=1,202) & 92.07 & 72.18 & 10.2M & -- & completely need\\\\\n\t\n\t\t& DenseNet & 94.17 & 76.58 & 27.2M & -- & completely need\\\\\n\t\n\t\t& VGG & 93.34 & 71.95 & 20.04M & -- & completely need\\\\\n\t\n\t\t& Maxout & 90.70 & 61.40 & -- & --& completely need\\\\\n\t\n\t\t& Network in Network & 91.19 & 64.32 & -- & -- & completely need\\\\\n\t\n\t\t& Highway Network & 92.40 & 67.66 & -- & -- & completely need\\\\\n\t\n\t\t& All-CNN & 92.75 & 66.29 & 1.3M & -- & completely need\\\\\n\t\t\\hline\n\t\t\\multirow{4}{*}{\\shortstack[l]{semi-automatic algorithms}} & Genetic CNN & 92.90 &70.97 & -- & 17 & partially need \\\\\n\t\t& Hierarchical Evolution & 96.37 & -- & -- & 300 & partially need \\\\\n\t\t& EAS & 95.77 & -- & 23.4M & 10 & partially need \\\\\n\t\t& Block-QNN-S & 95.62 & 79.35 & 6.1M & 90 & partially need \\\\\n\t\t\\hline\n\t\t\\multirow{6}{*}{\\shortstack[l]{automatic algorithms}} & Large-scale Evolution & 94.60 & -- & 5.4M & 2,750 & completely not need \\\\\n\t\t& Large-scale Evolution & -- & 77.00 & 40.4M & 2,750 & completely not need \\\\\n\t\t& CGP-CNN & 94.02 & -- & 1.68M & 27 & completely not need \\\\\n\t\t& NAS & 93.99 & -- & 2.5 M & 22,400 & completely not need \\\\\n\t\t& Meta-QNN & 93.08 & 72.86 & -- & 100 & completely not need \\\\\n\t\t\\cline{2-7}\n\t\t& CNN-GA & 95.22 & -- & 2.9M & 35 & completely not need \\\\\n\t\t& CNN-GA & -- & 77.97 & 4.1M  & 40 & completely not need\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\\subsection{Parameter Settings}\n\\label{sub_section_parameter}\nAs have been discussed, the main objective in this paper is to design an automatic architecture discovering algorithm for researchers without domain expertise in CNNs. To further improve applicability of the proposed algorithm, we design it in a way that potential users are not required to have expertise in evolutionary algorithms either. Hence, we simply set the parameters of the proposed algorithm based on the conventions. Specifically, the probabilities of crossover and mutation are set to $0.9$ and $0.2$, respectively, as suggested in~\\cite{back1996evolutionary}. During the training of each individual, the routine in~\\cite{he2016deep} is employed, i.e., the SGD with the learning rate of $0.1$ and the momentum of $0.9$ are used to train $350$ epochs and the learning rate is decayed by a factor of $0.1$ at the $1$-st, $149$-th and $249$-th epochs. Indeed, most peer competitors are also based on this training routine. When the proposed algorithm terminates, we choose the one with the best fitness value, and then train it up to $350$ epochs on the original training dataset. At last, the classification accuracy on the test dataset is tabulated to compare with peer competitors. In addition, the available numbers of feature maps are set to $\\{64, 128, 256\\}$ based on the settings employed by the state-of-the-art CNNs. As for the probabilities of the four mutation operations shown in Subsection~\\ref{sub_section_offspring_gen}, we set the normalized probability of increasing the depth to $0.7$, while others share the equal probabilities. Theoretically, any probability can be set for the adding mutation by keeping it higher than that of others. In addition, the population size and the number of generations are all set to $20$, and the similar settings are also employed by the peer competitors. In theory, the larger population size and larger maximal generation number should result in a better performance, but also cost more computational resources. However, larger settings are not investigated in this paper since the current settings have easily outperformed most of the peer competitors based on the results shown in Table~\\ref{tab_overview}. \n\n\n\n\n\\section{Experimental Results}\n\\label{section_exp_results}\n\nBecause the state-of-the-art CNNs in the first category of peer competitors are hand-crafted, we mainly compare the classification accuracy and the number of parameters. For the algorithms in other two categories, we compare the ``GPU days'' used to discover the corresponding CNN, in addition to the classification accuracy and the number of parameters. Particularly, the unit GPU day means the algorithm has performed one day on one GPU when the algorithm terminates, which reflects the computational resource consumed by these algorithms. For the convenience of summarizing the comparison results, we use the name of the architecture discovering algorithm as the name of the discovered CNN when comparing the classification accuracy and the number of parameters between peer competitors. For example, the proposed algorithm is named CNN-GA which is an architecture discovering algorithm, so its discovered CNN is titled as CNN-GA when comparing it to the chosen state-of-the-art CNNs.\n\nThe comparison results between the proposed algorithm and the peer competitors are shown in Table~\\ref{tab_overview}. In Table~\\ref{tab_overview}, the peer competitors are grouped into three different blocks based on the categories defined in Subsection~\\ref{sub_section_peer_competitor}, and the first column shows the category names. As a supplement, the last column also provides the information regarding how much manual assistance the corresponding CNN requires during discovering the architectures of CNNs. Additionally, the second column denotes the names of the peer competitors. The third and fourth columns refer to the classification accuracy on the CIFAR10 and CIFAR100 datasets, while the fifth column displays the numbers of parameters in the corresponding CNNs. The sixth column shows the used GPU days which are only applicable to the semi-automatic and automatic algorithms. The symbol ``--'' implies there is no result publicly reported by the corresponding algorithm. For ResNet (depth=101), ResNet (depth=1,202), DenseNet, VGG, All-CNN, EAS and Block-QNN-S, they have the same number of parameters on both the CIFAR10 and CIFAR100 datasets, respectively, which is caused by the similar CNNs achieving the best classification accuracy on both datasets. Note that, the results of the peer competitors shown in Table~\\ref{tab_overview} are all from their respective seminal papers\\footnote{It is exceptional for VGG because it does not perform experiments on CIFAR10 and CIFAR100 in its seminal paper. Its results displayed in Table~\\ref{tab_overview} is derived from~\\cite{suganuma2017genetic}.}. For the proposed algorithm, the best CNN discovered by CNN-GA is selected from the population in the last generation, and then trained independently for five runs. The best classification accuracy is selected from the five results to show in Table~\\ref{tab_overview}, which follows the conventions of its peer competitors~\\cite{xie2017genetic,liu2017hierarchical,zoph2016neural,zhong2017practical,real2017large,suganuma2017genetic,baker2016designing}.\n\nFor the peer competitors in the first category, CNN-GA obtains $3.15\\%$, $1.05\\%$ and $1.88\\%$ improvements in terms of the classification accuracy on the CIFAR10 dataset over ResNet (depth=1,202), DenseNet and VGG, respectively, while using merely $28\\%$, $11\\%$ and $14\\%$ of their respective parameters. The number of parameters in CNN-GA is more than that of ResNet (depth=101) and All-CNN on the CIFAR10 dataset, but CNN-GA shows the best classification accuracy among them. In addition, CNN-GA achieves the highest classification accuracy among Maxout, Network in Network and Highway Network on the CIFAR10 dataset. On the CIFAR100 dataset, CNN-GA employs $52\\%$, $85\\%$ and $40\\%$ fewer parameters compared to ResNet (depth=1202), DenseNet and VGG, respectively, but even achieves $5.79\\%$, $1.39\\%$ and $6.02\\%$ improvements over the respective classification accuracy. CNN-GA also outperforms ResNet (depth=101), Maxout, Network in Network, Highway Network and All-CNN in terms of the classification accuracy, although it uses a larger network than that of RestNet (depth=101) and All-CNN. In summary, CNN-GA achieves the best classification accuracy on both the CIFAR10 and CIFAR100 datasets among the state-of-the-art CNNs manually designed. Additionally, it also employs a much fewer number of parameters than most of the state-of-the-art CNNs in the first category.\n\nFor the peer competitors in the second category, the classification accuracy of CNN-GA is better than that of Genetic CNN. CNN-GA shows $1.15\\%$ lower classification accuracy than that of Hierarchical Evolution; however, CNN-GA takes only one-tenth of the GPU days that that of Hierarchical Evolution. Compared to Block-QNN-S, CNN-GA shows slightly worse classification accuracy on the CIFAR10 and CIFAR100 datasets, while CNN-GA consumes about half of the number of the parameters and also half of the CPU days compared to those of Block-QNN-S. Furthermore, the classification accuracy of CNN-GA is competitive to that of EAS, and CNN-GA employs a significantly smaller number of parameters than that of EAS ($87\\%$ fewer parameters used by CNN-GA than that of EAS). Although CNN-GA does not offer the best classification accuracy among the algorithms in this category, the algorithms in this category typically require extended expertise when they are used to solve real-world tasks. For example, EAS requires a manually-tuned CNN on the given dataset, and then EAS is used to refine the tuned CNN. If the tuned CNN is not with promising performance, the developed EAS would perform not well either at the end. In addition, the CNNs discovered by Hierarchical Evolution and Block-QNN-S cannot be directly used. They must be inserted into a larger CNN manually-designed in advance. If the larger network is not designed properly, the final performance of Hierarchical Evolution and Block-QNN-S would also perform poorly. The major advantage of CNN-GA, among the algorithms in this category, is its completely automatic nature.\n\nFor the peer competitors in the third category, CNN-GA performs better than Large-scale Evolution and CGP-CNN, and much better than NAS and Meta-QNN on the CIFAR10 dataset regarding the classification accuracy. Meanwhile, CNN-GA also shows a superiority of the classification accuracy over Large-scale Evolution and Meta-QNN on the CIFAR100 dataset. Furthermore, CNN-GA only has $2.9$M parameters on the CIFAR10 dataset, which is almost half of those of Large-scale Evolution. On the CIFAR100 dataset, CNN-GA has $4.1$M parameters, which saves $90\\%$ parameters compared to those of Large-scale evolution which requires $40.4$M parameters~\\cite{real2017large}. Furthermore, CNN-GA takes only $35$ GPU days on the CIFAR10 dataset and $40$ GPU days on the CIFAR100 dataset, while Large-scale Evolution consumes $2,750$ GPU days on the CIFAR10 dataset and another $2,750$ GPU days on the CIFAR100 dataset. Even more, NAS employs $22,400$ GPU days on the CIFAR10 dataset. Based on the number of parameters and GPU days taken, it can be summarized that CNN-GA shows a promising performance by using $90\\%$ simpler architectures on $99\\%$ less computational resource than the average numbers of competitors in this category\\footnote{The number is calculated by summing up the corresponding number of each peer competitor in this category, and then normalized by the numbers of available classification results. For example, the total GPU days of the peer competitors are $28,027$, and there are only six available classification results. Hence, the average GPU days are $4,671$. In the same way, the average GPU days took by CNN-GA are found to be $37.5$.}. \n\nIn summary, CNN-GA outperforms the state-of-art CNNs manually designed and automatic architecture discovering algorithms in terms of not only the classification accuracy, but also the number of parameters and the employed computational resource. Although the state-of-the-art semi-automatic architecture discovering algorithms show similar (to slightly better) classification accuracy to that of CNN-GA, CNN-GA is competitively automatic and does not require users to have any expertise in CNNs when solving real-world tasks.\n\n\n\n\\section{Conclusions and Future Work}\n\\label{section_conclusion}\nThe objective of this paper is to propose an automatic architecture design algorithm for CNNs by using the GA (in short named CNN-GA), which is capable of discovering the best CNN architecture in addressing image classification problems. This goal has been successfully achieved by designing a new encoding strategy for the GA to encode arbitrary depths of CNNs, incorporating the skip connections to promote deeper CNNs to be produced during the evolution and developing a parallel as well as a cache component to significantly accelerate the fitness evaluation given a limited computational resource. The proposed algorithm is examined on two challenging benchmark datasets, and compared with 16 state-of-the-art peer competitors, including eight manually designed CNNs, four semi-automatic and four automatic algorithms discovering the architectures of CNNs. The experimental results show that CNN-GA outperforms all the manually-designed CNNs as well as the automatic peer competitors, and shows competitive performance with respect to semi-automatic peer competitors in terms of the best classification accuracy. The CNN discovered by CNN-GA has a much smaller number of parameters than those of most peer competitors. Furthermore, CNN-GA also employs significantly less computational resource than most automatic and semi-automatic peer competitors. Moreover, CNN-GA is completely automatic, and researchers can directly use it to address their own image classification problems whether or not they have expertise in CNNs or GAs.\n\nIn CNN-GA, two components have been designed to speed up the fitness evaluation, and much computational resource has been saved. However, the computational resource employed is still fairly large than those of GAs in solving traditional problems. In the field of solving expensive optimization problems, several algorithms based on evolutionary computation techniques have been developed. In future, we will place efforts on developing effective evolutionary computation methods to significantly speed up the fitness evaluation of CNNs.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn this paper we consider a semi-supervised learning approach  which deals with the classification of a large number of unlabeled data when very few labels just are  available. In some  applications  such as  medical images, we have few\ntraining examples which are   labeled.  The aim is  to find  efficient  algorithms with  good performance\n  with these few  labeled examples. In this situation, geometric or topological properties of the unlabeled data has been used to propose and to   improve several algorithms.\n\nA common way to use the unlabeled data in semi-supervised learning is to build a graph over the data e.g., in image classification.\nTo start, we  requires to construct  an adjacency matrix, or weight matrix  $W$, for the data set, which encodes the similarities between pairs of date nodes.\nIf our data set consists of $n$ points  ${\\mathcal X}=\\{x_1,  x_2, \\cdots ,x_n \\}  \\subset \\mathbb{R}^d$, then the  weight matrix $W$   is an  $n \\times n $  symmetric matrix, where the element  $w_{ij}$  represents the similarity between two data points  $x_i$  and $x_j$.\nThe similarity is always nonnegative and should be large when  $x_i$  and $x_j$ are close together spatially,\nand small (or zero), when $x_i$  and $x_j$  are far apart.\nAs a  choice,  the weight matrix can be the  Gaussian weights\n\\[\n w_{ij}  =\\textrm{exp}(-\\frac{| x_{i}-x_{j}|}{2 \\sigma^2}),\n   \\]\nwhere $|\\cdot|$ is the Euclidean norm and  $\\sigma $ is a free parameter that controls the scale at which points are\nconnected by strong similarities  $w_{ij}$  in the weight matrix. In fact, the\nweight matrix $W$  provides  the data set with a graph structure, where each pair of\npoints $(x_i,  x_j)$ is connected by an edge with edge weight $w_{ij}.$ Other choices\nof weight matrix are possible,   such as the $k$-nearest neighbor graph, see \\cite{Lux}.\n\n\nRecently, many  works   aim to transpose and adapt\nPartial Differential Equations (PDEs) on graphs. This  reformulation of continuous problems onto a graph\nis such  that the solution behaves analogously to the continuous formulation see \\cite{A6, DEL}.\n\nIn graph-based semi-supervised learning, we are given a few labeled data on the graph and  we aim  to extend these  labels from a given  set  to the rest of the nodes in graph in  a decisive manner.\nTo model propagating labels in  semi-supervised  learning,  it is   assumed   that  the learned labels   vary   smoothly  and not change fast  within high density regions of the graph (smoothness  assumptions).\nBased on this assumption different approaches have  been proposed.\nOne of the pioneer methods  is {\\it Laplace learning}, \\cite{A19}.\nLater it has been observed that the Laplace learning can give poor results in classification \\cite{A18}.\nThe results are often poor because the solutions have localized spikes near the labeled points, while being almost constant far from them.\nTo overcome this problem several versions of Laplace learning algorithm  have been  proposed, for instance Laplacian regularization,  \\cite{A1}, weighted Laplacian, \\cite{Shi-Weighted, A14} and $p$-Laplace learning, \\cite {A10, slepcev2019analysis}.\nAlso, the limiting  case in $p$-Laplacian when $p$ tends to infinity  is so  called Lipschitz learning is studied in \\cite{K17} and similar to continuum PDEs  is related to finding  the absolutely minimal lipschitz extension of the training data.\nRecently,   in \\cite{A8}  another approach to increase   accuracy of Laplace learning  is   given  and called {\\it Poisson learning}.\n\n\n\n  To explain these methods,  let as before  ${\\mathcal X}={\\{ x_1, \\cdots, x_n}\\}$ denotes the data points or equivalently vertices in a graph.\n  We assume there is a subset of the nodes $ \\Gamma={\\{ x_1, \\cdots, x_m}\\}  \\subset  {\\mathcal X} $  that their labels are given with  a label function $g:\\Gamma\\rightarrow{\\mathbb R}^k$.\nIt is further assumed that $y_i=g(x_i)\\in \\{e_1, \\cdots , e_k\\}$ where $e_i$ is the standard basis in $\\mathbb{R}^k$ and represents the $i\\textsuperscript{th}$ class.\nIn graph-based semi-supervised learning, we aim to  extend labels  to the rest of the vertices $\\{x_{m+1}, \\cdots ,x_n\\}$.\n\n In Laplace learning algorithm the labels are extended by finding the minimizer   $ \\mathbf{ u}: {\\mathcal X}\\rightarrow \\mathbb{R}^k$ for the following problem\n \\begin{equation}\\label{objective}\n \\begin{cases}\n&\\min J_n(\\mathbf{ u}):=\\frac{1}{2}\\sum\\limits_{i=1}^{n}\\sum\\limits_{j=1}^{n} w_{ij} |\\mathbf{ u}(x_{i}) - \\mathbf{ u}(x_{j})|^{2}\\\\\n&\\text{subject to } \\mathbf{ u}(x_i) =y_i,\\quad  \\text{ for } i=1, 2, \\cdots ,m.\n\\end{cases}\n\\end{equation}\nThe minimizer  will be a harmonic function satisfying\n\\begin{equation*\n\\left \\{\n\\begin{array}{ll}\n{\\mathcal L} \\mathbf{ u}(x)=0,  &  x\\in {\\mathcal X}\\setminus\\Gamma ,\\\\\n  \\mathbf{ u}=g,   &   \\text{ on }\\Gamma,\\\\\n  \\end{array}\n\\right.\n\\end{equation*}\n where ${\\mathcal L}$ is the unnormalized graph Laplacian given by\n  \\begin{equation*}\n \\mathcal{L}\\mathbf{ u}(x_i)=\\sum\\limits_{j=1}^{n} w_{ij}\\, (\\mathbf{ u}(x_{i}) - \\mathbf{ u}(x_j)).\n \\end{equation*}\nLet $\\mathbf{ u}=(u_1,\\cdots,u_k)$ be a solution of \\eqref{objective}, the label of node $x_i\\in {\\mathcal X}\\setminus\\Gamma$ is dictated by\n \\begin{equation*\n\\underset{j\\in {\\{1,\\cdots, k}\\}}{ \\textrm{arg max}} {u_{j}(x_i)}.\n  \\end{equation*}\n\nIn $p$-Laplacian algorithm, the object function is replaced with\n\\begin{align*}\n\\min_{\\mathbf{ u}\\in{\\mathcal K}}J_p(\\mathbf{ u}):=\\sum_{i=1}^{n}\\sum\\limits_{j=1}^{n} w_{ij} |\\mathbf{ u}(x_{i}) - \\mathbf{ u}(x_{j})|^{p},\n\\end{align*}\nor for weighted Laplacian the following object is considered\n\\begin{align*}\n\\min_{\\mathbf{ u}\\in{\\mathcal K}}J_\\gamma(\\mathbf{ u}):=\\sum_{i=1}^{n}\\sum\\limits_{j=1}^{n} \\gamma(x_i)w_{ij} |\\mathbf{ u}(x_{i}) - \\mathbf{ u}(x_{j})|^{2},\n\\end{align*}\nwhere $\\gamma(x)\\approx \\text{dist}(x,\\Gamma)^{-\\alpha}$ increases the weights of edges adjacent to labels much more than other edges.\nUsing this method encourages the label functions to be flat (more regular) near labels, thus preventing the appearance of spikes (discontinuous solutions).  (see \\cite{Shi-Weighted, A14} for more details).\n\n\n\nThe authors in \\cite{A8}  have proposed a  scheme, called Poisson learning that  replaces the  label values at training points  as sources and sinks, and solves the Poisson equation on the graph as follows:\n\\[\n \\mathcal{L}\\mathbf{ u}(x_i) =\\sum\\limits_{j=1}^{m}  (y_{j}- \\overline{y})\\delta_{ij},\\qquad   i=1,\\cdots, n,\n\\]\nwith further condition    $\\sum_{i=1}^{n} d(x_i) \\mathbf{ u}(x_i)  =0$, where\n$\n\\overline{y}=\\frac{1}{m}\\sum_{i=1}^{m} y_{i}\n$\nis the average label vector,    $ \\delta_{ij}  $    is  Kronecker delta and $d(x_i)=\\sum_{j=1}^nw_{ij}$ is the degree of vertex $x_i$.\n\n\n\n\n\n\n\n A  major topic in this strand  concerns the continuum limits of these PDEs or functional on  graphs, linking between  the discrete and continuum perspectives and  the study of the consistency of the above methods in the large data limit, we  refer the reader to e.g. \\cite{A14,Abde1,GGHS}.\n\n\n\nLet  $x_1, x_2,\\cdots , x_n $  be a sequence of independent and identically distributed random variables on  $\\Omega$ with  smooth  distribution   $ \\rho(x)$.\n Define the weight matrix\n\\[\nw_{ij}  =\\eta_{\\varepsilon} (| x_{i}-x_{j}|)  = \\frac{1}{ \\varepsilon ^d}  \\eta (\\frac{ | x_{i}-x_{j}|}\\varepsilon)\n\\]\nwhere $\\eta$ is  a radial kernel $\\eta:  [ 0, \\infty )  \\rightarrow [ 0, \\infty )$   which is nonincreasing, continuous at $0$ and given by\n\\begin{equation*\n\t\\eta(t)= \\left \\{\n\t\\begin{array}{ll}\n\t1  &  0 \\le t\\le 1 \\\\\n\t0     & t>2.\n\t\\end{array}\n\t\\right.\n\t\\end{equation*}\nIn \\cite{ACRWJ}  it has been shown that for $u$ sufficiently smooth, with probability one\n\\[\n\\frac{1}{\\varepsilon^{2} n^2} J_{n}(\\mathbf{ u})\\rightarrow   \\sum_{i=1}^k\\int_\\Omega|\\nabla u_i(x)|^{2} \\rho(x) \\, dx:=  J_{\\infty}(\\mathbf{ u}),\n\\]\nwhere  $\\rho$   is density function of a probability measure  that data points are generated.\n\nIn this paper, we propose a novel classification scheme based on the segregation model. Our motivation for the current work is based on\nproperties of a class of reaction diffusion system with highly comparative rate which yields segregation of different components  which means at each point in the domain different components can not coexist. In this model, we solve problem \\eqref{objective} with additional constraint\n\\begin{equation*}\nu_i(x)\\cdot u_j(x)=0, \\qquad\\text{ for all }x\\in {\\mathcal X}, \\;1\\leq i\\neq j\\leq k.\n\\end{equation*}\n\nThe continuous form corresponding to the segregation model has been studied   extensively, for instance  \\cite{CL, F1, Av3}.\nWe state the results related to limiting configuration of  the following coupled system as parameter $ \\varepsilon$ tends to zero.\n\n\n\n \\begin{align}\\label{f20}\n\\begin{cases}\n\\Delta  u_{i}^{\\varepsilon}=   \\frac{ u_{i}^{\\varepsilon}}{\\varepsilon}  \\sum\\limits_{j \\neq i}   u_{j}^{\\varepsilon} (x)\\qquad\\qquad & \\text{ in  } \\Omega,\\\\\nu_{i}^{\\varepsilon} \\ge 0\\;  & \\text{ in  } \\Omega,\\\\\nu^{\\varepsilon}_{i}(x) =\\phi_{i}(x)    &   \\text{ on} \\,  \\partial \\Omega,\\\\\n  \\end{cases}\n\\end{align}\nfor  $i=1,\\cdots, m.$   The boundary values  satisfy\n\\[\n\\phi_{i}(x) \\cdot \\phi_{j}(x)=0,  \\quad i \\neq j \\textrm{ on the boundary}.\n\\]\nFirst, for each fixed  $\\varepsilon $ the exist   unique positive solution  $ (u_{1}^{\\varepsilon},\\cdots   ,u_{m}^{\\varepsilon})$. Next, to explain the asymptotic behaviour of \\ref{f20} by construction barrier functions, one can show that the normal derivative of $ u_{i}^{\\varepsilon}$  is bounded independent of  $ \\varepsilon$, this consequently proves that the  $H^1$-norm of  $ u_{i}^{\\varepsilon}$\nis bounded.    Next integrating  the equation  in \\ref{f20}  over $\\Omega$ indicates\n\\[\n (u_{i}^{\\varepsilon}  \\sum\\limits_{j \\neq i}   u_{j}^{\\varepsilon} (x))   \\rightarrow 0 \\,\\,  \\text{as $\\varepsilon$ tends to zero}.\n\\]\nLet $ (u_1, \\cdots  ,u_m)$ be the limiting configuration, then the  results in \\cite{CL} shows that $u_i$ are pairwise segregated, i.e., $u_{i}(x)\\cdot u_{j}(x)=0,$   harmonic in their supports and satisfy the following differential inequalities\n\n \\begin{itemize}\n\\item  $ -\\Delta u_{i} \\ge 0$,\\\\\n\\item  $ -\\Delta (u_{i}- \\sum\\limits_{j\\neq i}u_j )\\le 0$,\n \\item  Let $x$  belongs to interface    then\n        \\[\n   \\underset{ y\\rightarrow x} {\\text{ lim}} \\ \\nabla u_{i}(y)=-\n\\underset{ y\\rightarrow x}{ \\text{ lim}} \\  \\nabla u_{j}(y)\\quad \\text{Free } \\, \\text{boundary}\\, \\text{condition}.\n  \\]\n \\end{itemize}\n\n\n\n In \\cite{Conti-Terracini}, it has been shown that the limiting solution of (\\ref{f20})   minimizes the following functional\n\n\\begin{equation}\\label{continuous-model}\n\\left\\{\n\\begin{split}\n&\\min J(\\mathbf{ u}):=\\frac{1}{2}\\int_\\Omega\\sum\\limits_{i=1}^k|\\nabla u_i|^2\\, dx\\\\\n&\\text{subject to } u_i=\\phi_i,\\quad  \\text{ on } \\partial\\Omega, \\\\[10pt]\n&\\qquad u_i\\geq0, \\quad\\text{ and }\\quad u_i\\cdot u_j=0\\quad\\text{ in }\\Omega,\n\\end{split}\\right.\n\\end{equation}\n\n In  \\cite{F1, BA, Av1, Av2, Av3}, the authors have proposed and analyzed  the following numerical scheme to solving limiting configuration of system  \\eqref{f20} and  \\eqref{continuous-model}\n \\begin{equation*}\nu_{i}^{t+1} (x) =\\max\\Big(\\overline{u}^{t}_{i}(x)-\\sum\\limits_{j\\neq i}\\overline{u}^{t}_{j}(x),\\,  0\\Big)\n\\end{equation*}\nwhere $\\bar v(x)$   denotes the average of values of  $v$  of neighbors  of    mesh point $x,$ and $t$ refers to iterations.\n\n\n\n\n\n\n\n\\section{Calculus on graphs and setting the  problem }\nThis section is devoted to review some facts about the calculus on graphs and setting our problem.\nLet ${\\mathcal X}=\\{x_1,\\cdots,x_n\\}$ denote the vertices of a  graph with the symmetric edge weight $w_{xy}$ between $x,y\\in {\\mathcal X}$.\nThe degree of a vertex $x$ is given by $d(x)=\\sum_{y\\in {\\mathcal X}} w_{xy}$.  Let $\\ell^{2}({\\mathcal X})$ denote the set of functions $u: {\\mathcal X} \\rightarrow  \\mathbb{R}$ equipped with the inner product\n\\[\n(u,v)=\\sum\\limits_{x\\in {\\mathcal X}}  u(x)v(x),\n\\]\nfor functions $ u,v : {\\mathcal X}\\rightarrow \\mathbb{R}$.\nWe also define a vector field on the graph to be an antisymmetric function $V:{\\mathcal X}\\times {\\mathcal X}\\rightarrow \\mathbb{R}^2$, i.e. $V(x,y)=-V(y,x)$ and denote the space of all vector fields by $\\ell^2({\\mathcal X}^2)$.\nThe gradient of a function $ u\\in \\ell^{2}({\\mathcal X}) $ is the vector field\n\\[\n\\nabla u(x,y)=u(y)-u(x).\n\\]\nFor two vector fields $V_1, V_2$ the inner product is\n\\[\n(V_1, V_2)_{\\ell^{2}({\\mathcal X}^2)}= \\frac{1}{2} \\sum\\limits_{x,y\\in {\\mathcal X}}w_{xy} V_{1} (x, y) V_{2} (x, y),\n\\]\nso the norm of vector field $V$ is $\\|V\\|_{\\ell^{2}({\\mathcal X}^2)}= \\sqrt{(V, V)_{\\ell^{2}({\\mathcal X}^2)}}$.\nThe graph divergence of a vector field $V$ is defined by\n\\[\n{\\rm div\\,} V(x)=\\sum_{y\\in X}w_{xy}V(x,y),\n\\]\nwhich satisfies the divergence formula\n\\[\n(\\nabla u, V)_{\\ell^{2}({\\mathcal X}^2)}=-(u,{\\rm div\\,} V).\n\\]\nThe unnormalized graph laplacian ${\\mathcal L} $ of a function $u\\in \\ell^2({\\mathcal X})$ is defined as\n\\[\n{\\mathcal L} u(x):=-{\\rm div\\,}(\\nabla u)(x)=\\sum_{y\\in {\\mathcal X}}w_{xy}(u(x)-u(y)).\n\\]\nThe operator ${\\mathcal L}$ satisfies\n\\begin{equation}\\label{green-formula}\n({\\mathcal L} u,v)=(\\nabla u,\\nabla v)_{\\ell^2({\\mathcal X}^2)}.\n\\end{equation}\nIn Appendix, we revisit some important tools for PDE on graphs, such as Poincar\\'e inequality and maximum principle.\n\n\nWe consider a subset of the nodes $ \\Gamma  \\subset  {\\mathcal X} $  as the boundary of the graph and define the admissible set\n\\[\n{\\mathcal K}:=\\left\\{\\mathbf{ u}=(u_1,\\cdots, u_k)\\in \\left(\\ell^2({\\mathcal X})\\right)^k: u_i=\\phi_i\\text{ on } \\Gamma\\text{ for }i=1,\\dots,k\\right\\},\n\\]\nwhere the boundary data $\\phi_i$ are known and satisfy the following assumption\n\\begin{equation}\\label{phi-assumption}\n \\phi_i\\in \\{ 0,1\\}.\n\\end{equation}\nWe are going  to solve the optimization problem\n\\begin{align}\\label{L2-Problem}\n&\\min_{\\mathbf{ u}\\in{\\mathcal K}}J(\\mathbf{ u}):=\\|\\nabla\\mathbf{ u}\\|^2=\\sum_{i=1}^k\\|\\nabla u_i\\|_{\\ell^2({\\mathcal X}^2)}^2\\notag\\\\\n & \\textrm{subject to: }\\\\\n & u_i(x)\\geq0, \\ \\forall x\\in {\\mathcal X},  \\\\\n&u_i(x)\\cdot u_j(x)=0\\ \\forall x\\in {\\mathcal X}\\text{ and }i\\neq j.\\notag\n\\end{align}\n\n\nThe following theorem states the existence of solution to problem \\eqref{L2-Problem} and describes some properties of the solution.\n\n\\begin{theorem} \\label{property-minimizers}\nProblem \\eqref{L2-Problem} has a solution. Moreover, the solution satisfies\n\\begin{enumerate}[(i)]\n\\item\n${\\mathcal L} u_i(x)\\leq 0, \\text{ if } u_i(x)=0,\\text { and }x\\in {\\mathcal X}.$\n\n\\item\n${\\mathcal L} u_i(x)=0, \\text{ if } u_i(x)>0,\\text { and }x\\in {\\mathcal X}\\setminus \\Gamma.$\n\n\\item\nFor every $x\\in {\\mathcal X}\\setminus \\Gamma$, there is one component $u_i$ such that $u_i(x)>0$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nConsider a minimizing sequence $\\mathbf{ u}^n\\in {\\mathcal K}$ for problem \\eqref{L2-Problem}. By Poincar\\'e inequality, Proposition \\ref{prop:Poincare inequaity}, we obtain that\n\\[\n\\|u_i^n\\|\\leq \\frac1{\\lambda_1}\\|\\nabla(u_i^n-\\phi_i)\\|+\\|\\phi_i\\|\\leq \\frac1{\\lambda_1}(\\|\\nabla u_i^n\\|+\\|\\nabla \\phi_i\\|)+\\|\\phi_i\\|.\n\\]\nThus for every $i=1,\\cdots, k$ the sequence $\\{u^n_i\\}$ is bounded. Hence, there exists a subsequence such that for every components $i$\n\\[\nu^{n_j}_i\\rightarrow u_i.\n\\]\nIt is obvious that $\\mathbf{ u}=(u_1,\\cdots, u_k)$ satisfies the constraints in \\eqref{L2-Problem} and is a minimizer.\n\n$(i)$ To prove this part of theorem, notice that if $u_i(x)=0$ for some $x\\in {\\mathcal X}$, then\n$${\\mathcal L} u_i(x)=\\sum_{y\\in {\\mathcal X}}w_{xy}(u_i(x)-u_i(y))=-\\sum_{y\\in {\\mathcal X}}w_{xy}u_i(y)\\leq 0.$$\n\n$(ii)$ Now consider the case $u_i(x)>0$ for some fixed node $x\\in {\\mathcal X}\\setminus\\Gamma$. Let us define\n$$ v_i=u_i+t\\delta_x, \\quad v_j=u_j \\text{ when }j\\neq i,$$\nwhere $\\delta_x$ is delta function which is $\\delta_x(y)=0$ for every $y\\neq x$ and $\\delta_x(x)=1$. We also consider some values of $t$ such that $v_i(x)\\geq0$, ($t$ can be negative).\nObviously, $\\mathbf{ v}\\in{\\mathcal K}$ and satisfies the constraints in \\eqref{L2-Problem}.\nTherefore,\n\\begin{align*}\n0\\leq &\\|\\nabla\\mathbf{ v}\\|^2-\\|\\nabla\\mathbf{ u}\\|^2=\\|\\nabla v_i\\|^2-\\|\\nabla u_i\\|^2\\\\\n=&\\frac12\\sum_{y,z\\in {\\mathcal X}}w_{yz}\\left((v_i(z)-v_i(y))^2-(u_i(z)-u_i(y))^2\\right)\\\\\n=&\\sum_{y\\in{\\mathcal X}}w_{xy}\\left(t^2+2t(u_i(x)-u_i(y))\\right)\\\\\n=&t^2d(x)+2t{\\mathcal L} u_i(x).\n\\end{align*}\nSince $u_i(x)>0$, so parameter $t$ can be take some negative values and when $t\\rightarrow 0^\\pm$ we conclude that ${\\mathcal L} u_i(x)=0$.\n\n$(iii)$ Let $A:=\\{x\\in  {\\mathcal X}\\setminus \\Gamma: u_1(x)=\\cdots=u_k(x)=0\\}$. We claim that $A=\\emptyset$.\nOtherwise, there is some $x\\in A$ such that $w_{xy}\\neq0$ and $u_i(y)>0$ for some $i\\in\\{1,\\cdots,k\\}$.\nThus\n\\[\n{\\mathcal L} u_i(x)= - \\sum_{z\\in{\\mathcal X}}w_{xz} u_i(z) \\leq -w_{xy}u_i(y)<0.\n\\]\nNow choose the competitor $\\mathbf{ v}$ with\n\\[\n v_i=u_i+t\\delta_x, \\quad v_j=u_j \\text{ when }j\\neq i,\n\\]\nfor some $t\\geq0$ and repeat the calculation in the previous part to get\n\\[\n0\\leq \\|\\nabla\\mathbf{ v}\\|^2-\\|\\nabla\\mathbf{ u}\\|^2 = t^2d(x)+2t{\\mathcal L} u_i(x).\n\\]\nSince ${\\mathcal L} u_i(x)<0$, we can choose small value for $t$ to arrive at a contradiction.\n\\end{proof}\n\n\n\\begin{remark}\nProblem \\eqref{L2-Problem} has not necessary a unique solution.\nFor example, in a symmetry model, there are different choices for classification.\nIn a toy example, consider a graph with four vertices $A$, $B$, $C$ and $D$.\nLet $w_{AB}=w_{BC}=w_{CD}=w_{AD}=1$ and $w_{AC}=w_{BD}=0$.\nAlso, $A$ and $C$ are boundary points with boundary data $u_1(A)=u_2(C)=1$ and $u_1(C)=u_2(A)=0$.\nThis problem has four solutions\n\n\\begin{enumerate}[(i)]\n\\item\n$u_1(B)=u_1(D)=\\frac12,  \\quad u_2(B)=u_2(D)=0,$\n\\item\n$u_2(B)=u_2(D)=\\frac12,  \\quad u_1(B)=u_1(D)=0,$\n\\item\n$u_1(B)=u_2(D)=\\frac12,  \\quad u_2(B)=u_1(D)=0,$\n\\item\n$u_1(D)=u_2(B)=\\frac12,  \\quad u_1(B)=u_2(D)=0.$\n\\end{enumerate}\n\\end{remark}\n\n\n\n\\section{Gradient projection method}\\label{section:Gradient projection method}\nGradient projection is one method that we use to solve the problem \\eqref{L2-Problem}.\nIn the sequel, we use the following notation for averaging of a function\n\\[\n{\\mathcal A} u(x):= \\frac 1{d(x)}\\sum_{y\\in {\\mathcal X}} w_{xy} u(y),\n\\]\nwhere \n\\[\nd(x) = \\sum_{y\\in {\\mathcal X}} w_{xy},\n\\]\nand the admissible set\n\\[\n{\\mathcal S}:=\\left\\{\\mathbf{ u}=(u_1,\\cdots, u_k)\\in \\left(\\ell^2({\\mathcal X})\\right)^k: u_i=\\phi_i\\text{ on } \\Gamma, u_i\\geq0, u_i\\cdot u_j=0 \\text{ for }i\\neq j\\right\\}.\n\\]\nWe also use the projection ${\\mathcal P}:\\left(\\ell^2({\\mathcal X})\\right)^k\\rightarrow {\\mathcal S}$ which is defined as follows.\nFor every $\\mathbf{ v}\\in \\left(\\ell^2({\\mathcal X})\\right)^k$ and every $x\\in {\\mathcal X}\\setminus \\Gamma$, first we find\n\\[\ni_x:=\\arg\\max_{1\\leq j\\leq k} (v_j(x))^+\n\\]\nand if it has more than one solution we choose the smallest index. ($v^+:=\\max(v,0)$.)\nThen we define $\\mathbf{ u}:={\\mathcal P}\\mathbf{ v}$ with $u_{i_x}(x)=(v_{i_x}(x))^+$ and $u_j(x)=0$ for $j\\neq i_x$.\nFor any $x\\in \\Gamma$, we obviously define $u_i(x)=\\phi_i(x)$.\n\n\nThe following lemma shows why ${\\mathcal P}$ is a projection.\n\n\\begin{lemma}\\label{lem:projection}\nConsider $\\mathbf{ v}\\in \\left(\\ell^2({\\mathcal X})\\right)^k$, then\n\\[\n\\|\\mathbf{ v}-{\\mathcal P}\\mathbf{ v}\\| \\leq \\|\\mathbf{ v}-\\mathbf{ w}\\|,\\quad \\text{ for all }\\mathbf{ w}\\in{\\mathcal S}.\n\\]\n\\end{lemma}\n\\begin{proof}\nConsider $\\mathbf{ w}\\in{\\mathcal S}$ and define the index function $\\sigma:{\\mathcal X}\\rightarrow \\{1,\\cdots, k\\}$ such that  $w_j(x)=0$ for $j\\ne \\sigma(x)$.\nSo,\n\\[\\begin{split}\n\\|\\mathbf{ v}-\\mathbf{ w}\\|^2=&\\sum_{x\\in{\\mathcal X}}\\left((v_{\\sigma(x)}(x)-w_{\\sigma(x)}(x))^2+\\sum_{i\\ne \\sigma(x)}(v_i(x))^2\\right)\\\\\n=&\\sum_{x\\in{\\mathcal X}}\\sum_{i=1}^k (v_i(x))^2 + \\sum_{x\\in{\\mathcal X}}\\left((v_{\\sigma(x)}(x)-w_{\\sigma(x)}(x))^2 - (v_{\\sigma(x)}(x))^2\\right)\n\\end{split}\\]\nSimilarly we have,\n\\[\n\\|\\mathbf{ v}-{\\mathcal P}\\mathbf{ v}\\|^2=\\sum_{x\\in{\\mathcal X}}\\sum_{i=1}^k (v_i(x))^2  + \\sum_{x\\in{\\mathcal X}}\\left((v_{i_x}(x)-(v_{i_x}(x))^+)^2 - (v_{i_x}(x))^2\\right).\n\\]\nIt is enough to show\n\\begin{equation}\\label{grad-proj:eq1}\n(v_{i_x}(x)-(v_{i_x}(x))^+)^2 - (v_{i_x}(x))^2 \\le (v_{\\sigma(x)}(x)-w_{\\sigma(x)}(x))^2 - (v_{\\sigma(x)}(x))^2,\n\\end{equation}\nfor every $x\\in {\\mathcal X}$.\nIf $v_{i_x}(x)\\le 0$, then $v_{\\sigma(x)}(x)\\le 0$ by the definition of $i_x$.\nThus the left hand side of \\eqref{grad-proj:eq1} is zero as well as the right hand side is positive (recall that $w_{\\sigma(x)}(x)\\ge0$).\nIf $v_{i_x}(x)\\ge 0\\ge v_{\\sigma(x)}(x)$, the left hand side of \\eqref{grad-proj:eq1} is negative and the right hand side will be positive.\nIf $v_{i_x}(x)\\ge v_{\\sigma(x)}(x) \\ge 0$, then \\eqref{grad-proj:eq1} will hold trivially.\n\\end{proof}\n\n\nOur algorithm according to the gradient projection method is as follows:\n\\begin{enumerate}[(1)]\n\\item\nChoose an initial guess in ${\\mathcal S}$. It might be an extension of boundary data as $u_{i,0}=\\phi_i$ on $\\Gamma$ and $u_{i,0}=0$ in ${\\mathcal X}\\setminus\\Gamma$.\n\n\\item\nFor $t\\geq 0$, calculate the gradient of the cost function $J$ at $\\mathbf{ u}^t=(u_{1,t},\\cdots,u_{k,t})$. It is equal to\n\\[\n\\delta J(\\mathbf{ u}^t):=({\\mathcal L} u_{1,t},\\cdots,{\\mathcal L} u_{k,t}).\n\\]\n\n\\item\nUpdate the value of each components by\n\\[\n\\mathbf{ u}^{t+1}:={\\mathcal P}(\\mathbf{ u}^{t}-\\frac 1d{\\mathcal L}\\mathbf{ u}^t)={\\mathcal P}({\\mathcal A}\\mathbf{ u}^t).\n\\]\n\n\\item\nIf $\\| \\mathbf{ u}^{t+1} - \\mathbf{ u}^t\\|$ is small then stop the algorithm, otherwise set $t=t+1$ and iterate the previous steps.\n\\end{enumerate}\n\nThe following proposition proves why the algorithm works.\n\n\\begin{proposition}\nAssume $\\mathbf{ u}$ is a solution of problem \\eqref{L2-Problem}.\nConsider an arbitrary point $x\\in {\\mathcal X}\\setminus\\Gamma$ such that $u_i(x)>0$, then\n\\[u_i(x)={\\mathcal A} u_i(x) \\geq {\\mathcal A} u_j(x),\\qquad\\text{ for all }j\\neq i.\\]\n\\end{proposition}\n\\begin{proof}\nFor a fixed index $j\\neq i$,  define a competitor $\\mathbf{ v}\\in{\\mathcal S}$\n\\[\nv_i:=u_i- u_i(x)\\delta_x,\\quad v_j:=u_j+ u_i(x)\\delta_x, \\quad v_\\ell:=u_\\ell,\\text{ for }\\ell\\neq i,j.\n\\]\nTherefore,\n\\[\\begin{split}\n0\\leq & \\|\\nabla\\mathbf{ v}\\|^2-\\|\\nabla\\mathbf{ u}\\|^2 = \\|\\nabla v_i\\|^2+\\|\\nabla v_j\\|^2-\\|\\nabla u_i\\|^2-\\|\\nabla u_j\\|^2 \\\\\n= &\\sum_{y}w_{xy}\\left((v_i(x)-v_i(y))^2+(v_j(x)-v_j(y))^2-(u_i(x)-u_i(y))^2-(u_j(x)-u_j(y))^2\\right)\\\\\n=&\\sum_{y\\neq x}w_{xy}\\left(u_i(y)^2+(u_i(x)-u_j(y))^2-(u_i(x)-u_i(y))^2-u_j(y)^2\\right)\\\\\n=&\\sum_{y\\neq x}2w_{xy}u_i(x)\\left(u_i(y)-u_j(y)\\right).\n\\end{split}\\]\nSince $u_i(x)>0$, we get\n\\[\n\\frac 1{d(x)}\\sum_{y}w_{xy}u_i(y)\\geq  \\frac 1{d(x)}\\sum_{y}w_{xy}u_j(y)={\\mathcal A} u_j(x).\n\\]\nNow apply result $(ii)$ of Theorem \\ref{property-minimizers}, we obtain that $u_i(x)=\\frac 1{d(x)}\\sum\\limits_{y}w_{xy}u_i(y)$.\n\\end{proof}\n\n\nDefine the map ${\\mathcal G}:(\\ell_+^2({\\mathcal X}))^k\\longrightarrow {\\mathcal S}$ with rule ${\\mathcal G}\\mathbf{ u}=\\mathbf{ v}$, where\n\\[\nv_i =\\max \\left( u_i-\\sum_{j\\ne i} u_j , 0\\right),\n\\]\nand $\\ell_+^2({\\mathcal X})$ is the set of nonnegative functions.\nIf we replace this map instead of projection ${\\mathcal P}$ in the gradient projection algorithm, we will obtain  the segregation method.\nWe will study this method in Section \\ref{cluste_seg}.\n\n\n\\begin{proposition}\nSuppose ${\\mathcal P}$ is the projection on ${\\mathcal S}$ defined in Section \\ref{section:Gradient projection method}.\nThen there is a positive constant  $C_0$ such that\n\\[\n\\norm{{\\mathcal P} \\mathbf{ u} -\\mathbf{ u}} \\le \\norm{{\\mathcal G} \\mathbf{ u}-\\mathbf{ u}}\\le C_0\\norm{{\\mathcal P} \\mathbf{ u} -\\mathbf{ u}},\n\\]\nfor every $\\mathbf{ u}\\in (\\ell_+^2({\\mathcal X}))^k$.\n\\end{proposition}\n\\begin{proof}\nThe left inequality will be hold according to Lemma \\ref{lem:projection}.\nFor a fixed node $x\\in{\\mathcal X}$, we need to show\n\\begin{equation}\\label{prop:segr-proj-eq1}\n\\sum_{j=1}^k|v_j(x)-u_j(x)|^2 \\le C_0 \\sum_{j=1}^k |w_j(x)-u_j(x)|^2,\n\\end{equation}\nwhere ${\\mathcal G}\\mathbf{ u}=\\mathbf{ v}$ and ${\\mathcal P}\\mathbf{ u}=\\mathbf{ w}$.\nLet $i:=\\arg\\max_{1\\leq j\\leq k} u_j(x)$.\nIf there is more than one index for $i$, then $\\mathbf{ v}(x)={\\mathcal G}(\\mathbf{ u})(x)=0$.\nThus \\eqref{prop:segr-proj-eq1} holds for $C_0\\ge 2$.\n\nNow assume that $i$ is the unique solution of $i:=\\arg\\max_{1\\leq j\\leq k} u_j(x)$. Hence,\n\\[\nw_i(x)=u_i(x),\\text{ and } w_j(x)=v_j(x)=0\\text{ for }j\\ne i.\n\\]\nTherefore, using the definition of $v_i$ and applying Cauchy-Schwartz inequality we obtain\n\\[\\begin{split}\n\\sum_{j=1}^k|v_j(x)-u_j(x)|^2 &= \\left(\\sum_{j\\ne i} u_j(x)\\right)^2 + \\sum_{j\\ne i}|u_j(x)|^2 \\\\\n&\\le ((k-1)+1)\\sum_{j\\ne i}|u_j(x)|^2 =k\\sum_{j=1}^k|w_j(x)-u_j(x)|^2,\n\\end{split}\\]\nwhich implies that \\eqref{prop:segr-proj-eq1} holds for $C_0\\ge k$.\n\\end{proof}\n\n\n\n\\section{Penalization method}\n\nIn this section, we apply the penalization method to solve problem \\eqref{L2-Problem}.\nSince finding the solution directly is not efficient (the optimization problem \\eqref{L2-Problem} is a problem with $(n-m)^k$ parameters), we would prefer to solve a PDE instead.\nIn this case, we can just find a PDE for points that  $u_i>0$ and this subdomain is not known.\nIn fact, we have a free boundary problem and if we know the domain $\\{u_i>0\\}$, we are able to find the solution.\nIn order to overcome this difficulty, we relax the constraint with a penalty term and try to estimate the solution for the original problem \\eqref{L2-Problem}.\n\n\nIndeed, we consider the following problem\n\\begin{equation}\\label{L2-Problem*}\n\\min_{\\mathbf{ u}\\in{\\mathcal K}}J_\\varepsilon(\\mathbf{ u}):=\\sum_{i=1}^k\\|\\nabla u_i\\|_{\\ell^2(X^2)}^2+\\frac1\\varepsilon\\sum_{i\\neq j}(u_i^2,u_j^2).\n\\end{equation}\nSince the energy function is convex, it is straightforward that the problem has a unique solution which satisfies\n\\begin{equation}\\label{PDE-penalized}\n{\\mathcal L} u_i+\\frac{u_i}\\varepsilon\\sum_{j\\neq i}u_j^2=0,\\quad\\text{ in }X\\setminus\\Gamma.\n\\end{equation}\nFurthermore, we know that the solution is nonnegative due to the maximum principle, Proposition \\ref{Maximum principle}.\n\n\\begin{theorem}\nLet $\\mathbf{ u}^\\varepsilon$ be the solution of \\eqref{L2-Problem*} for every $\\varepsilon>0$.\nFor any sequence $\\varepsilon_n\\rightarrow0$, there is a  subsequence of $\\mathbf{ u}^{\\varepsilon_n}$ which converges to  a minimizer of \\eqref{L2-Problem}.\n\\end{theorem}\n\\begin{proof}\nLet $\\mathbf{ v}$ be an arbitrary minimizer of \\eqref{L2-Problem}, then we have $J_\\varepsilon(\\mathbf{ v})=J(\\mathbf{ v})$ thanks to the constraint in \\eqref{L2-Problem}. So,\n$$J_\\varepsilon(\\mathbf{ u}^\\varepsilon)\\leq J_\\varepsilon(\\mathbf{ v})=J(\\mathbf{ v})=:\\Lambda.$$\nTherefore $\\|\\nabla \\mathbf{ u}^\\varepsilon\\|\\leq \\sqrt\\Lambda$ is uniformly bounded and then by Poincar\\'e inequality we get $\\|\\mathbf{ u}^\\varepsilon\\|$ is bounded, since\n$$\\lambda_1\\|u_i^\\varepsilon-\\phi_i\\|\\leq \\|\\nabla(u_i^\\varepsilon-\\phi_i)\\|\\leq\\sqrt\\Lambda+\\|\\nabla\\phi_i\\|.$$\nHence, toward a subsequence we can assume that $\\mathbf{ u}^{\\varepsilon_n}\\rightarrow\\mathbf{ u}\\in{\\mathcal K}$.\nWe need to show that $\\mathbf{ u}$ is a minimizer of  \\eqref{L2-Problem} and  satisfies its constraints.\nFirst, we have\n$$\n\\frac1{\\varepsilon_n}\\sum_{i\\neq j}\\left((u_i^{\\varepsilon_n})^2,(u_j^{\\varepsilon_n})^2\\right)\\leq J_{\\varepsilon_n}(\\mathbf{ u}^{\\varepsilon_n})\\leq \\Lambda,\n$$\nso,\n$$\n\\sum_{i\\neq j}\\left((u_i^{\\varepsilon_n})^2,(u_j^{\\varepsilon_n})^2\\right)\\longrightarrow0.\n$$\nThus, $(u_i^2,u_j^2)=0$ and taking into account that $u^\\varepsilon_i$ is nonnegative we obtain that the constraint in \\eqref{L2-Problem} holds for $\\mathbf{ u}$.\nTo close the argument, note that\n$$\\|\\nabla\\mathbf{ u}\\|^2=\\lim_{\\varepsilon_n\\rightarrow0}\\|\\nabla\\mathbf{ u}^{\\varepsilon_n}\\|^2\\leq J_{\\varepsilon_n}(\\mathbf{ u}^{\\varepsilon_n})\\leq \\Lambda.$$\nSo, $J(\\mathbf{ u})=\\Lambda$ and $\\mathbf{ u}$ is a minimizer.\n\\end{proof}\n\nIn the sequel we introduce an algorithm to solve problem \\eqref{L2-Problem*} or its equivalent version  \\eqref{PDE-penalized}.\nThe later is a nonlinear system of PDEs and is not easy to solve directly.\nFor an explanation of our algorithm, we define the following sequence which converges to the solution \\eqref{PDE-penalized}.\nFirst, consider the harmonic extension $u_{i,0}$ of boundary data given by\n$$\n\\left\\{\\begin{array}{ll}\n{\\mathcal L} u_{i,0}=0, & \\text{ in }X\\setminus\\Gamma,\\\\[8pt]\nu_{i,0}=\\phi_i& \\text{ on }\\Gamma,\n\\end{array}\\right.\n$$\nwhich is a nonnegative function according to the maximum principle.\nNext, given nonnegative functions $\\mathbf{ u}_m:=(u_{1,m},\\cdots,u_{k,m})$, let $u_{i,m+1}$ be the solution of the following system\n$$\n\\left\\{\\begin{array}{ll}\n{\\mathcal L} u_{i,m+1}+\\frac{u_{i,m+1}}{\\varepsilon}\\sum\\limits_{j\\neq i}u_{j,m}^2=0, & \\text{ in }X\\setminus\\Gamma,\\\\[10pt]\nu_{i,m+1}=\\phi_i,& \\text{ on }\\Gamma,\n\\end{array}\\right.\n$$\nThe following theorem shows that why our algorithm  works for solving problem \\eqref{PDE-penalized}.\n\n\\begin{theorem}\\label{thm-order-seq}\nSuppose that the boundary data $\\phi_i$ satisfy \\eqref{phi-assumption}.\nThen the  sequence $\\mathbf{ u}_m$ makes the following order\n\\begin{equation}\\label{order-sequence-u}\n1\\geq u_{i,0}\\geq u_{i,2}\\geq\\cdots\\geq u_{i,2m}\\geq\\cdots\\geq u_{i,2m+1}\\geq\\cdots\\geq u_{i,3}\\geq u_{i,1}\\geq0.\n\\end{equation}\nMoreover, the limit of this sequence is the solution of \\eqref{PDE-penalized}.\n\\end{theorem}\n\n\n\\begin{proof}\n{\\bf Step 1: } We show that $u_{i,m}$ is nonnegative. \\\\\nThis is a matter of maximum principle, Proposition  \\ref{Maximum principle}.\nFor $m=0$, it is obvious due to maximum principle and taking account that $u_{i,0}$ is a harmonic function. In fact, we consider $p(x)\\equiv0$ in Proposition \\ref{Maximum principle}.\nTo show $u_{i,m+1}\\geq0$, again apply Proposition  \\ref{Maximum principle} for nonnegative function $p(x)=\\frac1{\\varepsilon}\\sum_{j\\neq i}u_{j,m}^2$.\n\n\\medskip\n{\\bf Step 2: } $u_{i,0}\\leq 1$. \\\\\nApply the maximum principle for harmonic function $1 - u_{i,0}$ and recall the assumption \\eqref{phi-assumption}.\n\n\n\\medskip\n{\\bf Step 3: } In this step we show that $u_{i,m}\\leq u_{i,0}$.\\\\\nWe just need to note that ${\\mathcal L} u_{i,m+1}\\leq 0={\\mathcal L} u_{i,0}$. Then maximum principle yields that $u_{i,m+1}\\leq u_{i,0}$.\n\n\\medskip\n{\\bf Step 4:} Here, we claim that $u_{i,2}\\geq u_{i,1}$.\\\\\nBy the result of Step 2, we can write\n\\begin{align*}\n0&={\\mathcal L} u_{i,2}+\\frac{u_{i,2}}{\\varepsilon}\\sum_{j\\neq i}u_{j,1}^2\\\\\n&\\leq {\\mathcal L} u_{i,2}+\\frac{u_{i,2}}{\\varepsilon}\\sum_{j\\neq i}u_{j,0}^2.\n\\end{align*}\nThis together with the equation of $u_{i,1}$,  the maximum principle yields that $u_{i,2}\\geq u_{i,1}$.\n\n\n\\medskip\n{\\bf Step 5:} Now we close the argument with the induction. Assume that\n\\begin{equation}\\label{order-u-induction}\nu_{i,0}\\geq u_{i,2}\\geq\\cdots\\geq u_{i,2m}\\geq u_{i,2m-1}\\geq\\cdots\\geq u_{i,1}\\geq0.\n\\end{equation}\nfor some $m\\geq1$, we will extend the string for $m+1$.\nBy the following inequality\n$$\n0={\\mathcal L} u_{i,2m+1}+\\frac{u_{i,2m+1}}{\\varepsilon}\\sum_{j\\neq i}u_{j,2m}^2\\geq {\\mathcal L} u_{i,2m+1}+\\frac{u_{i,2m+1}}{\\varepsilon}\\sum_{j\\neq i}u_{j,2m-1}^2\n$$\nwe can apply Proposition  \\ref{Maximum principle} for function $(u_{i,2m}-u_{i,2m+1})$ when $p=\\frac{1}{\\varepsilon}\\sum_{j\\neq i}u_{j,2m-1}^2$\nto deduce that $u_{i,2m}\\geq u_{i,2m+1}$.\nSimilarly, we have\n$$\n0={\\mathcal L} u_{i,2m+1}+\\frac{u_{i,2m+1}}{\\varepsilon}\\sum_{j \\neq i}u_{j,2m}^2\\leq {\\mathcal L} u_{i,2m+1}+\\frac{u_{i,2m+1}}{\\varepsilon}\\sum_{j \\neq i}u_{j,2m-2}^2\n$$\naccording the induction assumption $u_{j,2m-2}\\geq u_{j,2m}$.\nComparing with the equation for $u_{i,2m-1}$ we obtain $u_{i,2m+1}\\geq u_{i, 2m-1}$.\nNow repeat this argument to show that $u_{i,2m}\\geq u_{i,2m+2}\\geq u_{i,2m+1}$.\n\n\n\\medskip\n{\\bf Step 6:}\nFrom \\eqref{order-sequence-u}, we know that  there are $\\overline u_i$ and $\\underline u_i$ as the limit of\n\\begin{align*}\nu_{i,2m}\\longrightarrow \\overline u_i,\\\\\nu_{i,2m+1}\\longrightarrow \\underline u_i.\n\\end{align*}\nThese limits satisfy\n\\begin{align}\\label{limit-equation}\n\\left\\{\\begin{array}{ll}\n{\\mathcal L}\\overline u_i+\\frac{\\overline u_i}{\\varepsilon}\\sum_{j\\neq i}\\underline u_j^2=0, \\quad\\text{ in }{\\mathcal X}\\setminus\\Gamma\\\\[10pt]\n{\\mathcal L}\\underline u_i+\\frac{\\underline u_i}{\\varepsilon}\\sum_{j\\neq i}\\overline u_j^2=0, \\quad\\text{ in }{\\mathcal X}\\setminus\\Gamma\\\\[10pt]\n\\overline u_i=\\underline u_i=\\phi_i\\quad\\text{ on }\\Gamma.\n\\end{array}\\right.\n\\end{align}\nMultiply in inner product $\\ell^2({\\mathcal X})$ both equations by $\\overline u_i - \\underline u_i$ and subtract to get\n\\[\n\\varepsilon\\|\\nabla (\\overline u_i - \\underline u_i)\\|_{\\ell^2({\\mathcal X}^2)}^2  = \\Big( \\underline u_i (\\overline u_i - \\underline u_i), \\sum_{j\\neq i}\\overline u_j^2\\Big)-\\Big( \\overline u_i (\\overline u_i - \\underline u_i), \\sum_{j\\neq i}\\underline u_j^2\\Big).\n\\]\nIt is worthwhile noticing that although the equation holds in ${\\mathcal X}\\setminus\\Gamma$, since $\\overline u_i - \\underline u_i=0$ on $\\Gamma$ we are able to utilize the relation \\eqref{green-formula}.\nNow sum over $i$, we obtain\n\\begin{align*}\n\\varepsilon\\|\\nabla (\\overline \\mathbf{ u} - \\underline \\mathbf{ u})\\|_{\\ell^2({\\mathcal X}^2)}^2\n= & \\sum_i \\Big( \\underline u_i \\overline u_i , \\sum_{j\\neq i}(\\overline u_j^2+\\underline u_j^2)\\Big)\n-\\sum_i \\Big(  \\underline u_i^2, \\sum_{j\\neq i}\\overline u_j^2 \\Big)\n-\\sum_i \\Big(  \\overline u_i^2, \\sum_{j\\neq i}\\underline u_j^2 \\Big) \\\\\n= & \\sum_{i,j} \\big( \\underline u_i \\overline u_i , \\overline u_j^2+\\underline u_j^2\\big) - \\sum_i \\big( \\underline u_i \\overline u_i , \\overline u_i^2+\\underline u_i^2\\big) \\\\\n& - 2 \\sum_{i,j} \\big(  \\underline u_i^2, \\overline u_j^2 \\big) +  2 \\sum_i  \\big(  \\underline u_i^2, \\overline u_i^2 \\big)\\\\\n= & \\sum_{x\\in {\\mathcal X}} \\sum_{i,j}  \\underline u_i(x) \\overline u_i(x) (\\overline u_j^2(x)+\\underline u_j^2(x)) \\\\\n& - \\sum_{x\\in{\\mathcal X}} \\sum_i  \\underline u_i(x) \\overline u_i(x) (\\overline u_i(x)-\\underline u_i(x))^2 \\\\\n& -2 \\sum_{x\\in {\\mathcal X}} \\Big(\\sum_i (\\underline u_i(x))^2 \\Big)\\Big( \\sum_i (\\overline u_i(x))^2 \\Big) \\\\\n\\leq & 2 \\sum_{x\\in {\\mathcal X}}  \\left(\\sum_{i}  \\underline u_i(x) \\overline u_i(x)  -  \\Big(\\sum_i (\\underline u_i(x))^2 \\Big)\\Big( \\sum_i (\\overline u_i(x))^2 \\Big) \\right) \\leq 0,\n\\end{align*}\nwhere we have used the relation $0\\leq \\underline u_i \\leq  \\overline u_i \\leq 1$ and in the last line the Cauchy-Schwartz inequality has been applied.\n\nThen $\\|\\nabla (\\overline \\mathbf{ u} - \\underline \\mathbf{ u})\\|_{\\ell^2({\\mathcal X}^2)}^2\\leq 0$ and so, $\\overline \\mathbf{ u} - \\underline \\mathbf{ u}$ is  constant in ${\\mathcal X}$. Taking  this along to the boundary condition  implies that $\\overline \\mathbf{ u} = \\underline \\mathbf{ u}$ in ${\\mathcal X}$.\nRecall \\eqref{limit-equation}, $\\underline u_i =\\overline u_i$ is a solution of \\eqref{PDE-penalized}.\n\\end{proof}\n\n\n\n\n\n\n\n\n \\section{The main algorithm for clustering}\\label{cluste_seg}\n In the previous sections of the paper we considered a minimization problem \\eqref{L2-Problem}, which unfortunately has no unique solution over connected graphs. In the current section in order to overcome the lack of uniqueness we consider different functional and prove the existence and uniqueness of the minimizer. The definition of a new functional is inspired from the numerical results of the spatial segregation of reaction-diffusion systems (see \\cite{Av1}).\n\n\\subsection{Existence and  uniqueness of a minimizer}\n\nWe introduce the discrete counterpart of the spatial segregation problem defined on connected graphs.\nIn the rest of the paper the following notation\n\\[\n\\hat{z}_q=z_q-\\sum_{j\\neq q}z_j,\n\\]\nfor elements  $(z_1,z_2,\\dots,z_k)$, will play a crucial role.\nLet    $\\overline{u}_{i}(x_{l} )$ for $ i=1,2\\cdots k$  denote  the average value  of $u_{i}$ for all neighbor points of $x_{l}:$\n\\[\n\\overline{u}_{i}(x_{l})= \\frac{1}{\\deg(x_{l})} \\sum_{p  \\sim l} w_{lp} u_{i} (x_p),\n\\]\nwhere\n\\[\n\\deg(x_{l})=  \\sum_{(x_l,y)\\in E} w_{x_ly},\n\\]\nand $V$ and $E$ stand for a set of vertices and edges respectively. We will set a graph ${\\mathcal X}$ to be a tuple $(V, E)$ in the rest of this section.\n\n\n\nWhen ${\\mathcal X}$ is a connected graph and also consist of discrete and finite number of points, it turns out that we have to consider slightly different functional (see \\cite[Section $2$]{Av1}). Since $\\mathcal L$ is a self-adjoint operator, then we set:\n \\begin{equation}\n\\label{sun2}\nJ(u_1,\\dots,u_k)=\\frac{1}{2}  \\sum_{i=1}^{k} \\|\\nabla u_i\\|^2_{\\ell^{2}(X^2)}- \\sum_{i\\neq j} \\left(\\nabla u_i, \\nabla u_j\\right)_{\\ell^{2}({\\mathcal X}^2)}, \\end{equation}\nover the set\n\\begin{equation}\\label{disc_min_set}\n\t\\mathcal K=\\left\\{\\mathbf{ u}=(u_1,\\cdots, u_k)\\in \\left(\\ell^2({\\mathcal X})\\right)^k: u_i=\\phi_i\\text{ on } \\Gamma, u_i\\geq0, u_i\\cdot u_j=0 \\text{ for }i\\neq j\\right\\}.\n\\end{equation}\n\n\n\n\n\n\n\n\\begin{theorem}\n\tThe following minimization problem\n\t\\begin{equation}\\label{disc_min_problem}\n\t\\inf_{\\mathcal K}J(u_1,u_2,\\dots,u_k)\\\n\t\\end{equation}\n\thas a solution.\n\\end{theorem}\n\n\\begin{proof}\nThe proof repeats the same lines as in Theorem $2$ in \\cite{Av1}. In \\cite{Av1} the functional is defined for standard difference scheme, but it can be easily concluded for the connected graphs as well.\n\\end{proof}\n\nNow, by following the proofs of Proposition $1$ and Lemma $2$  for $F_l(x,s)=0$ in \\cite{Av1}, We can observe that the similar results can be obtained for connected graphs instead of finite difference discretization. Although, it is worth to notice that the standard finite difference grid is itself a particular case of connected graphs.\n\nThus, we conclude the following  result:\n\n\\begin{theorem}\\label{limit_pde_discrete}\n\tFor every minimizer $(u_1,\\dots,u_k)\\in \\mathcal K,$ the following properties hold:\n\\begin{itemize}\n\t\\item $\\mathcal{L}\\hat{u}_i(x) =0 \\;\\; whenever \\;\\; u_i(x)>0.$\n\t\\item $\\mathcal{L}\\hat{u}_i(x)\\geq 0 \\;\\; whenever \\;\\; x\\in {\\mathcal X} \\setminus\\Gamma.$\n\\end{itemize}\n\\end{theorem}\n\n\n\nTo prove the uniqueness of the minimizer  $(u_1,\\dots,u_k)\\in \\mathcal K$ one needs some technical lemmas.\n\\begin{lemma}\\label{lemma1}\n\tLet ${\\mathcal X}=(V,E)$ be a connected graph. If any two vectors $(u_{1},u_{2},\\dots,u_{k})$ and $(v_{1},v_{2},\\dots,v_{k})$ are  minimizers to the \\eqref{disc_min_problem}, then the following equation holds:\n\t\\[\n\t\\max_{x\\in {\\mathcal X}}\\left(\\hat{u}_l(x)-\\hat{v}_l(x)\\right)=\\max_{\\{ x\\in {\\mathcal X}\\;:\\; u_l(x)\\leq v_l(x)\\}}\\left(\\hat{u}_l(x)-\\hat{v}_l(x)\\right),\n\t\\]\n\tfor all $l=1,2,\\dots,k$.\n\\end{lemma}\n\n\\begin{proof}\n\tWe argue by contradiction. Suppose for some $l_0$ we have\n\t\\begin{equation}\\label{init_assmp}\n\t\t\\begin{multlined}\n\t\t\t\\hat{u}_{l_0}(x_0)-\\hat{v}_{l_0}(x_0)=\n\t\t\t\\max_{x\\in  {\\mathcal X}}(\\hat{u}_{l_0}(x)-\\hat{v}_{l_0}(x))=\\\\=\n\t\t\t\\max_{\\{ x\\in {\\mathcal X}\\;:\\; u_{l_0}(x)> v_{l_0}(x)\\}}(\\hat{u}_{l_0}(x)-\\hat{v}_{l_0}(x))>\n\t\t\t\\max_{\\{ x\\in {\\mathcal X}\\;:\\; u_{l_0}(x)\\leq v_{l_0}(x)\\}}(\\hat{u}_{l_0}(x)-\\hat{v}_{l_0}(x)).\n\t\t\\end{multlined}\n\t\\end{equation}\t\nIt is easy to observe that  the following simple chain of inclusions hold:\n\t\\begin{equation}\\label{incl_chain}\n\t\t\\{u_l(x)> v_l(x)\\}\\subset\\{\\hat{u}_l(x)> \\hat{v}_l(x)\\}\\subset\\{u_l(x)\\geq v_l(x)\\}.\n\t\\end{equation}\n\tWe obviously see that $ u_{l_0}(x_0)> v_{l_0}(x_0)\\geq 0 $ implies\n\t$\\hat{u}_{l_0}(x_0)>\\hat{v}_{l_0}(x_0)$. On the other hand,  Theorem \\ref{limit_pde_discrete}   gives us\n\t$$\n     \\mathcal{L}\\hat{u}_i(x_0)= 0,\n\t$$\n\tand\t\n\t$$\n     \\mathcal{L}\\hat{v}_i(x_0)\\geq 0.\t \t\n\t$$\n\tTherefore \t\n\t\\begin{equation*}\n\t\\mathcal{L}(\\hat{u}_i-\\hat{v}_i)(x_0)\\leq 0.\n\t\\end{equation*}\n\tThus,\n\t\n\t\\begin{align}\n\t\t0  < \\left(\\hat{u}_{l_0}(x_0)- \\hat{v}_{l_0}(x_0)\\right)&\\leq\n\t \\frac{1}{\\deg(x_0)}\\sum_{y\\in {\\mathcal X}}w_{x_{0}y}\\left(\\hat{u}_{l_0}(y)-\\hat{v}_{l_0}(y)\\right)\\nonumber,\n\t\\end{align}\n\twhich implies that $\\hat{u}_{l_0}(x_0)-\\hat{v}_{l_0}(x_0)=\\hat{u}_{l_0}(y)-\\hat{v}_{l_0}(y)>0,$ when $w_{x_{0}y}\\ne0$. Due to the chain  \\eqref{incl_chain}, we apparently have  ${u}_{l_0}(y)\\geq {v}_{l_0}(y)$. According to our assumption \\eqref{init_assmp}, the only possibility is ${u}_{l_0}(y)>{v}_{l_0}(y)$  for all $y\\in {\\mathcal X}$.\n\tNow we can proceed the previous steps for all $y\\in V$ such that $(x_0,y)\\in E,$ and then for each one we will get corresponding neighbours with the same strict inequality and so on. Since the graph ${\\mathcal X}$ is connected, then  one can always find the shortest path from a given vertex $y$ to the vertex belonging $\\Gamma.$   Continuing above procedure along this path we will finally approach to a vertex on $\\Gamma,$ where as we know ${u}_{l_0}(x)={v}_{l_0}(x)={\\phi}_{l_0}(x)$ for all $x\\in\\Gamma$. Hence, the strict inequality  fails, which implies that our initial assumption \\eqref{init_assmp} is false. Observe that the same arguments can be applied if we interchange the roles of ${u}_{l}(x)$ and ${v}_{l}(x)$. Thus, we also have\n\t\\[\n\t\\max_{V}\\left(\\hat{v}_l(x)-\\hat{u}_l(x)\\right)=\\max_{\\{ v_l(x)\\leq u_l(x)\\}}\\left(\\hat{v}_l(x)-\\hat{u}_l(x)\\right),\n\t\\]\n\tfor every $l=1,2,\\dots,m$.\n\t\n\t\n\n\tParticularly, for every fixed $l=1,2.\\dots,m$ and $x\\in V$ we have\t\\begin{multline}\\label{double_ineq}\n\t\t-\\max_{\\{v_l(x)\\leq u_l(x)\\}}(\\hat{v}_l(x)-\\hat{u}_l(x))=\t-\\max\\limits_{x\\in V}(\\hat{v}_l(x)-\\hat{u}_l(x))\\leq\\\\\\leq \\hat{u}_l(x)-\\hat{v}_l(x) \\leq \t\\max\\limits_{x\\in V}(\\hat{u}_l(x)-\\hat{v}_l(x))=\\max_{\\{u_l(x)\\leq v_l(x)\\}}(\\hat{u}_l(x)-\\hat{v}_l(x)).\n\t\\end{multline}\n\\end{proof}\n\nThanks to Lemma \\ref{lemma1} in the sequel we will use the following notations:\n$$\nA:=\\max_l\\;\\left(\\max\\limits_{x\\in V}(\\hat{u}_l(x)-\\hat{v}_l(x))\\right)=\\max_l\\;\\left(\\max\\limits_{\\{u_l(x)\\leq v_l(x)\\}}(\\hat{u}_l(x)-\\hat{v}_l(x))\\right),\n$$\nand\n$$\nB:=\\max_l\\;\\left(\\max\\limits_{x\\in V}(\\hat{v}_l(x)-\\hat{u}_l(x))\\right)=\\max_l\\;\\left(\\max\\limits_{\\{v_l(x)\\leq u_l(x)\\}}(\\hat{v}_l(x)-\\hat{u}_l(x))\\right).\n$$\nNext lemma we write down without a proof. The proof can be easily adapted from  \\cite{Av1}[Lemma $4$].\n\\begin{lemma}\\label{lemma2}\n\tLet ${\\mathcal X}=(V,E)$ be a connected graph. Assume given two vectors $(u_{1},u_{2},\\dots,u_{k})$ and $(v_{1},v_{2},\\dots,v_{k})$ are  minimizers to the \\eqref{disc_min_problem}. For them  we set $A$ and $B$ as defined above. If $A>0$ and it is attained for some $l_0$, then $A=B>0$  and there exists some $t_0\\neq l_0,$ and $y_0\\in V,$ such that\n\t$$\n\t0<A=\\max_{\\{u_{l_0}(x)\\leq v_{l_0}(x)\\}}(\\hat{u}_{l_0}(x)-\\hat{v}_{l_0}(x))=\n\t\\max_{\\{u_{l_0}(x)= v_{l_0}(x)=0\\}}(\\hat{u}_{l_0}(x)-\\hat{v}_{l_0}(x))\n\t=\\hat{v}_{t_0}(y_0)-\\hat{u}_{t_0}(y_0).\n\t$$\t\n\\end{lemma}\nNow, we are ready to proof the uniqueness of the minimizer. The following Theorem is true.\n\\begin{theorem}[Uniqueness]\n\tLet ${\\mathcal X}=(V,E)$ be a connected graph. Then\n\tthere exists  a unique minimizer  $(u_{1},u_{2},\\dots,u_{k})\\in \\mathcal K,$ to minimization problem  \\eqref{disc_min_problem}.\n\n\\end{theorem}\n\\begin{proof}\n\t\n\tLet two vectors $(u_{1},u_{2},\\dots,u_{k})$ and $(v_{1},v_{2},\\dots,v_{k})$ are minimizers to \\eqref{disc_min_problem}. For these vectors we set the definition of $A$ and $B$. Then, we consider two cases $A\\leq 0$ and $A>0$. If we assume that $A\\leq 0,$ then according to Lemma \\ref{lemma2}, we get  $B\\leq 0$. But if  $A$ and $B$ are non-positive, then the uniqueness follows. Indeed,  due to  \\eqref{double_ineq} we have the following obvious inequalities\n\t$$\n\t0\\leq -B\\leq \\hat{u}_l(x)-\\hat{v}_l(x)\\leq A\\leq 0.\n\t$$\n\tThis provides  for every $l=\\overline{1,k}$ and $x\\in V$ we have $\\hat{u}_l(x)=\\hat{v}_l(x),$ which in turn implies $$\n\t{u}_l(x)={v}_l(x).\n\t$$\n\tNow suppose  $A>0$. Our aim is to prove that this case leads to a contradiction. Let the value $A$ is attained for some $l_0\\in\\overline{1,k},$ then\n\tdue to Lemma \\ref{lemma2} there exist $y_0\\in V$ and $t_0\\neq l_0$ such that:\n\t\\begin{align*}\n\t\t0<A=B=&\\max_{\\{u_{l_0}(x)\\leq v_{l_0}(x)\\}}(\\hat{u}_{l_0}(x)-\\hat{v}_{l_0}(x))\\\\&=\\max_{\\{u_{l_0}(x)=v_{l_0}(x)=0\\}}(\\hat{u}_{l_0}(x)-\\hat{v}_{l_0}(x))=\\hat{v}_{t_0}(y_0)-\\hat{u}_{t_0}(y_0).\n\t\\end{align*}\nThus, along with the fact that $\\hat{v}_{t_0}(y_0)>\\hat{u}_{t_0}(y_0)$ implies ${v}_{t_0}(y_0)\\geq {u}_{t_0}(y_0),$ we can repeat the same steps as in the proof of Lemma \\ref{lemma1} to obtain\n\t$$\n\t\\left(\\hat{v}_{t_0}(y_0)- \\hat{u}_{t_0}(y_0)\\right)\\leq \\frac{1}{\\deg(y_0)}\\sum_{(y_0,z)\\in E}w_{y_{0}z}\\left(\\hat{v}_{t_0}(z)-\\hat{u}_{t_0}(z)\\right).\n\t$$\n\tThis  implies  $A=\\hat{v}_{t_0}(y_0)-\\hat{u}_{t_0}(y_0)=\\hat{v}_{t_0}(z)-\\hat{u}_{t_0}(z)>0$ for all $(y_0,z)\\in E$.  The chain \\eqref{incl_chain} provides that  for all $(y_0,z)\\in E,$ we have ${v}_{t_0}(z)\\geq{u}_{t_0}(z)$. Since a graph ${\\mathcal X}$ is connected, then one can always find a shortest path from $y_0$ to some vertex $w\\in\\Gamma.$ Assume the vertices along this path are $y_0;y_1;\\dots;y_{k-1};y_q=w.$ Hence, for every $0 \\leq j\\leq q-1,$ we have $(y_j,y_{j+1})\\in E,$ i.e. every vertex  $y_{j+1}$ is a closest neighbor for $y_{j}$ and $y_{j+2}.$\n\t\n\t\n\t\n\t\n\tAccording to the above arguments for the neighbor vertex $y_1\\in V$ we proceed as follows: If  ${v}_{t_0}(y_1)>{u}_{t_0}(y_1),$ then obviously\n\t$$\n\t\\left(\\hat{v}_{t_0}(y_1)- \\hat{u}_{t_0}(y_1)\\right)\\leq \\frac{1}{\\deg(y_1)}\\sum_{(y_1,z)\\in E}w_{y_{1}z}\\left(\\hat{v}_{t_0}(z)-\\hat{u}_{t_0}(z)\\right).\n\t$$\n\tThis, as we saw a few lines above, leads to  $A=\\hat{v}_{t_0}(y_1)-\\hat{u}_{t_0}(y_1)=\\hat{v}_{t_0}(z)-\\hat{u}_{t_0}(z)>0$ for all $(y_1,z)\\in E$.  In particular,\n\t$A = \\hat{v}_{t_0}(y_2)-\\hat{u}_{t_0}(y_2)> 0.$\n\t\n\tIf  ${v}_{t_0}(y_1)={u}_{t_0}(y_1),$  then due to $\\hat{v}_{t_0}(y_1)-\\hat{u}_{t_0}(y_1)=A=B>0,$ there exists some $\\lambda_0\\neq t_0,$ such that\n\t$$\n\t0<A=\\hat{v}_{t_0}(y_1)-\\hat{u}_{t_0}(y_1)=\n\t\\sum_{l\\neq t_0}\\left({u}_{l}(y_1)-{v}_{l}(y_1) \\right)={u}_{\\lambda_0}(y_1)-\\sum_{l\\neq t_0}{v}_{l}(y_1).\n\t$$\n\tNote that ${u}_{\\lambda_0}(y_1)>0$ implies ${u}_{l}(y_1)=0$ for all $l\\neq \\lambda_0,$ and particularly  ${v}_{t_0}(y_1)={u}_{t_0}(y_1)=0.$\n\tFollowing the definition of $A,$ we get\n\t$$\n\tA={u}_{\\lambda_0}(y_1)-\\sum_{l\\neq t_0}{v}_{l}(y_1)\\geq \\hat{u}_{\\lambda_0}(y_1)-\\hat{v}_{\\lambda_0}(y_1),\n\t$$\n\twhich in turn gives $2\\sum\\limits_{l\\neq \\lambda_0}{v}_{l}(y_1)\\leq 0,$ and therefore\n\t${v}_{l}(y_1)=0$ for all $l\\neq \\lambda_0$. Hence\n\t$$\n\tA={u}_{\\lambda_0}(y_1)-\\sum_{l\\neq t_0}{v}_{l}(y_1)= \\hat{u}_{\\lambda_0}(y_1)-\\hat{v}_{\\lambda_0}(y_1),\n\t$$\n\tThis suggests us to apply the same approach as above to arrive at\n\t$$\n\t\\left(\\hat{v}_{\\lambda_0}(y_1)- \\hat{u}_{\\lambda_0}(y_1)\\right)\\leq \\frac{1}{\\deg(y_1)}\\sum_{(y_1,z)\\in E}w_{y_{1}z}\\left(\\hat{v}_{\\lambda_0}(z)-\\hat{u}_{\\lambda_0}(z)\\right),\n\t$$\n\twhich leads to\n\t$A=\\hat{u}_{\\lambda_0}(y_1)-\\hat{v}_{\\lambda_0}(y_1)=\\hat{u}_{\\lambda_0}(z)-\\hat{v}_{\\lambda_0}(z)>0,$ for all $(y_1,z)\\in E$. In particular,\n\t$A = \\hat{u}_{\\lambda_0}(y_2)-\\hat{v}_{\\lambda_0}(y_2)> 0.$\n\tThus, combining two cases we observe that for $y_2\\in V$ there exist an index $1\\leq l_{y_2}\\leq m$ (in our case $l_{y_2}=t_0$ or $l_{y_2}=\\lambda_0$) such that\n\t\\begin{equation}\\label{contradiction}\n\t\t\\mbox{either}\\;\\;\t\\hat{u}_{l_{y_2}}(y_2)-\\hat{v}_{l_{y_2}}(y_2)=A, \\;\\;\\mbox{or}\\;\\;\n\t\t\\hat{u}_{l_{y_2}}(y_2)-\\hat{v}_{l_{y_2}}(y_2)=-A.\n\t\\end{equation}\n\tIt is not hard to understand that the same procedure can be repeated for a vertex $y_2$ instead of $y_1$ and come to the same conclusion \\eqref{contradiction} for $y_3\\in V$ and some index $l_{y_3}$ and so on. This allows to claim that for every $y_j\\in V$ along the path $(y_0,\\dots,y_q)$  there exist some $l_{y_j}$ such that\n\t\\[\n\t|\\hat{u}_{l_{y_j}}(y_j)-\\hat{v}_{l_{y_j}}(y_j)|=A>0.\n\t\\]\n\tBut this means that above equality holds also for $y_q=w\\in \\Gamma,$ which will lead to a contradiction, because for every $z\\in\\Gamma,$ and $l=1,\\cdots,k$ one has $\\hat{u}_{l}(z)-\\hat{v}_{l}(z)=0$.  This completes the proof of uniqueness.\n\\end{proof}\n\n\n\n\n\n\\subsection{Semi-supervised learning algorithm}\n\n\nUsing definition of graph Laplacian in  $${\\mathcal L} (u_{i}- \\sum\\limits_{j\\neq i}u_j )\\ge 0,$$ yields\n\\begin{equation}\\label{17}\n\t{\\mathcal L} (u_{i} - \\sum\\limits_{j\\neq i}u_{j})(x_l) =\n\t\\sum\\limits_{s=1}^{n} w_{ls}\\, \\left( u_{i}(x_{l})- u_{i}(x_s) -  \\sum\\limits_{j\\neq i}(u_{j}(x_l) - u_{j}(x_s))  \\right).\n\\end{equation}\nTo  obtain   $u_{i} (x_{l})$    from (\\ref{17})  we impose   the following  conditions\n\\[\nu_{i} (x_{l}) \\cdot u_{j} (x_{l}) =0  \\text{ and }    u_{i}(x_l)\\geq 0.\n\\]\nFrom these\n\\[\n\\deg(x_{l}) u_{i} (x_{l})-\\sum\\limits_{s=1}^{n} w_{ls}\\,u_{i} (x_{s})+ \\sum\\limits_{s=1}^{n}\\sum\\limits_{j\\neq i}w_{ls} \\,u_{j} (x_{s})=0\n\\]\nThen\n\\[\nu_{i} (x_{l})=\\overline{u}_{i}(x_{l})-\\sum\\limits_{j\\neq i}\\overline{u}_{j}(x_{l}).\n\\]\n\n\nAccording to the above ideas and following the Theorem \\ref{limit_pde_discrete} we can easily check that if $(u_1,u_2,\\dots,u_k)\\in \\mathcal K$ is a unique minimizer to \\eqref{disc_min_problem}, then  it satisfies the following system of equations:\n\\begin{equation}\\label{scheme_sys}\n\t\\begin{cases}\n\t\tu_{1}(x) =\\max \\left(\\overline{u}_1(x) - \\sum\\limits_{p \\neq 1}  \\overline{u}_p(x), \\,  0\\right),\\;\\;x\\in {\\mathcal X}\\setminus\\Gamma,\\\\\n\t\tu_{2}(x) =\\max \\left(\\overline{u}_2(x) - \\sum\\limits_{p \\neq 2}  \\overline{u}_p(x), \\,  0\\right),\\;\\;x\\in {\\mathcal X}\\setminus\\Gamma,\\\\\n\t\t\\dots\\dots\\dots\\dots\\\\\n\t\tu_{k}(x) =\\max \\left(\\overline{u}_k(x) - \\sum\\limits_{p \\neq k}  \\overline{u}_p(x), \\,  0\\right),\\;\\;x\\in {\\mathcal X}\\setminus\\Gamma,\\\\\n\t\tu_{i}(x) =\\phi_{i}(x),\\;\\;x\\in \\Gamma,\\; \\mbox{for all}\\; i=1,2,\\dots,k.\n\t\\end{cases}\n\\end{equation}\n\\begin{remark}\\label{remark_1}\n\tWe remark that the system \\eqref{scheme_sys} itself implies the disjointness  property, i.e. it is easy to see that  if a vector $(u_1,u_2,\\dots,u_k)$ satisfies the system \\eqref{scheme_sys}, then  $u_i(x)\\cdot u_j(x)=0,$ for every $x\\in V$ and $i\\neq j.$\n\\end{remark}\n\nIn order to approximate   the solution of   system \\eqref{scheme_sys} we propose the iterative  scheme which is  easy to implement   as follows:\nFor  $i=1,\\cdots k,$ and $x_l\\in{\\mathcal X}\\setminus\\Gamma$ we set\n\\begin{equation*}\nu_{i}^{(t+1)} (x_{l}) =\\max\\left(\\overline{u}^{(t)}_{i}(x_{l})-\\sum\\limits_{j\\neq i}\\overline{u}^{(t)}_{j}(x_{l}),\\,  0\\right).\n\\end{equation*}\n\nIn the lite of Remark \\ref{remark_1} it can be seen that for every iteration the disjointness property is preserved. In other words the following lemma is true.\n\\begin{lemma}\\label{lemma}\n\t\tLet ${\\mathcal X}=(V,E)$ be a connected graph. The above iterative method  satisfies\n\t\\[u_i^{(t)}(x)\\cdot u_j^{(t)}(x) =0,\\; \\forall x\\in V,\\; i\\neq j.\\]\n\\end{lemma}\n\n\nThe label decision for vertex $x_l$ is determined by the strictly positive component\n$u_i(x_l),$ i.e. find an index $i_0$ such that $u_{i_0}(x_l)>0$. Thus, in this case the label corresponding to a vertex $x_l$ will be $i_0.$\n\n\n\\\n\n\n\n\\section{Experimental results}\n\nIn this section we are going to test and compare two well-known semi-supervised learning algorithms to the one we have developed based on segregation theory. We note that our taken dataset for visual implementations will be the random generated half-moons and for the statistic analysis we will use the well-known MNIST. Thus, we will depict the predictions for Laplace learning, Poisson learning and our learning (We call it Segregation learning) algorithms.\n\nWe  run the learning algorithms for different number of initial label of classes and for different number of classes (basically we will run for $3, 4$ and $5$ classes). For each  implementation all the classes have the same number of nodes, i.e, either all classes have   $200$ or $300$ nodes. The reader can also observe the red nodes on every figure. They correspond to the randomly chosen initial known labels.\nIn the figures \\ref{fig1}--\\ref{fig15} below one can observe that, when the initial number of labels per class is small, i.e. $2, 3$ or $5$ labels, then the Laplace learning algorithm is performing poorly, whilst both the Poisson and our Segregation learning algorithms are performing  much better and have more or less  the same accuracy.\n\nWhen the initial number of labels per class is $10$ or $20$ labels, then the performance of the Laplace learning becomes more accurate and is getting close to the results depicted for Poisson and Segregation learning algorithms.\n\n\n\nTables \\ref{table_1} and \\ref{table_2}  show the average accuracy over all $100$ trials for various low and high label rates. The implementations have been done on MNIST dataset only for $3$ classes.  We see that for low label rates Laplace learning performs poor as we noted in the depicted figures. On the other hand Poisson and Segregation learning perform better and predicted  more or less with the same accuracy. For high label rates Laplace learning performs much better  and gets close to Poisson and Segregation learning results.\n\\begin{figure}[!htb]\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_3_classes_and_2_labels}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_3_classes_and_2_labels}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_3_classes_and_2_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 3 classes and initial 2 labels per class.}\t\n\t\\label{fig1}\n\n\n\n\\end{figure}\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_3_classes_and_3_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_3_classes_and_3_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_3_classes_and_3_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 3 classes and initial 3 labels per class.}\n\t\\label{fig2}\t\n\n\n\n\\end{figure}\n\\begin{figure}[!htb]\\label{fig3}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_3_classes_and_5_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_3_classes_and_5_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_3_classes_and_5_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 3 classes and initial 5 labels per class.}\t\n\n\n\n\\end{figure}\n\\begin{figure}[!htb]\\label{fig4}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_3_classes_and_10_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_3_classes_and_10_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_3_classes_and_10_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 3 classes and initial 10 labels per class.}\t\t\t\n\\end{figure}\n\\begin{figure}[!htb]\\label{fig5}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_3_classes_and_20_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_3_classes_and_20_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_3_classes_and_20_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 3 classes and initial 20 labels per class.}\t\t\t\n\\end{figure}\n\n\\begin{figure}[!htb]\\label{fig6}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_4_classes_and_2_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_4_classes_and_2_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_4_classes_and_2_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 4 classes and  initial 2 labels per class.}\t\t\t\n\\end{figure}\n\\begin{figure}[!htb]\\label{fig7}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_4_classes_and_3_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_4_classes_and_3_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_4_classes_and_3_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 4 classes and  initial 3 labels per class.}\t\t\t\n\\end{figure}\n\\begin{figure}[!htb]\\label{fig8}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_4_classes_and_5_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_4_classes_and_5_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_4_classes_and_5_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 4 classes and  initial 5 labels per class.}\t\t\t\n\\end{figure}\n\\begin{figure}[!htb]\\label{fig9}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_4_classes_and_10_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_4_classes_and_10_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_4_classes_and_10_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 4 classes and  initial 10 labels per class.}\t\t\t\n\\end{figure}\n\\begin{figure}[!htb]\\label{fig10}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_4_classes_and_20_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_4_classes_and_20_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_4_classes_and_20_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 4 classes and  initial 20 labels per class.}\t\t\t\n\\end{figure}\n\n\\begin{figure}[!htb]\\label{fig11}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_5_classes_and_2_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_5_classes_and_2_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_5_classes_and_2_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 5 classes and  initial 2 labels per class.}\t\t\t\n\\end{figure}\n\\begin{figure}[!htb]\\label{fig12}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_5_classes_and_3_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_5_classes_and_3_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_5_classes_and_3_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 5 classes and  initial 3 labels per class.}\t\t\t\n\\end{figure}\n\n\\begin{figure}[!htb]\\label{fig13}\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_5_classes_and_5_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_5_classes_and_5_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_5_classes_and_5_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 5 classes and  initial 5 labels per class.}\t\t\t\n\\end{figure}\n\n\\begin{figure}[!htb]\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_5_classes_and_10_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_5_classes_and_10_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_5_classes_and_10_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 5 classes and  initial 10 labels per class.}\t\n\t\\label{fig14}\t\t\n\\end{figure}\n\n\\begin{figure}[!htb]\n\t\\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Seg_learning_with_5_classes_and_20_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Poisson_learning_with_5_classes_and_20_labels}  \\includegraphics[width=.32\\linewidth,valign=m]{paper_figures\/Laplace_learning_with_5_classes_and_20_labels}\n\t\\caption{Comparison of Laplace, Poisson and Segregation learning algorithms for 5 classes and  initial 20 labels per class.}\t\n\t\\label{fig15}\t\t\n\\end{figure}\n\n\\begin{table}[h]\n\t\\caption{Average accuracy scores over 100 trials for 3 classes on MNIST dataset.}\n\t\\centering\n\t\\begin{tabular}{cllllll}\n\t\t\\hline\\hline\n\t\tLabels per class &\\textbf{2} & \\textbf{3} & \\textbf{4} & \\textbf{5}& \\textbf{10}&  \\\\ [0.5ex]\n\t\t\\hline\n\t\tLaplace learning &31.3 & 45.4 & 58.2 & 67.7 & 83.4\\\\\n\t\tPoisson learning & 93.6  &  94.5  & 94.9 & 95.3 & 96.7&\\\\\n\t    Segregation learning  & \\textbf{90.3}  & \\textbf{92.1}  & \\textbf{93.5} & \\textbf{95.4} & \\textbf{96.2} &\\\\[1ex]\n\t\t\\hline\\hline\n\t\\end{tabular}\n\t\\label{table_1}\n\\end{table}\n\n\\begin{table}[h]\n\t\\caption{Average accuracy scores over 100 trials for 3 classes on MNIST dataset.}\n\t\\centering\n\t\\begin{tabular}{c llllll}\n\t\t\\hline\\hline\n\t\tLabels per class &\\textbf{20} & \\textbf{40} & \\textbf{80} & \\textbf{100}& \\textbf{120}&\\\\ [0.5ex]\n\t\t\\hline\n\tLaplace learning &88.6  & 91.3 & 93.7 & 95.2  & 97.6\\\\\n\tPoisson learning & 95.6  &  97.2  & 97.9 & 98.3 & 99.4&\\\\\n\tSegregation learning  & \\textbf{94.3}  & \\textbf{96.7}  & \\textbf{98.2} & \\textbf{98.8} & \\textbf{99.2} &\\\\[1ex]\n\t\t\\hline\\hline\n\t\\end{tabular}\n\t \\label{table_2}\n\\end{table}\n\n\\section{Conclusion}\nIn this work we develop several semi-supervised algorithms on graphs. Our main algorithm is a new approach for graph-based semi-supervised learning based on spatial segregation theory. The method is efficient and simple to implement.\n We presented numerical results showing that Segregation Learning performs more or less as Poisson Learning algorithm not only at high label rates, but also at low label rates on MNIST dataset.\n\n\n\n\n\n\n\\renewcommand{\\refname}{REFERENCES }\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\nAn essential task in experimental quantum information\nprocessing is the characterization of quantum states and\ntheir dynamics, which is typically achieved via quantum\nstate tomography (QST)~\\cite{li-pra-2017} and quantum\nprocess tomography (QPT)\\cite{chuang-jmo-09}.  The \nexperimental resources\nrequired  to implement QST and QPT \ngrow exponentially with the size of\nsystem, which makes these methods infeasible beyond a few\nqubits\\cite{mohseni-pra-2008}.  Hence developing techniques\nto reduce the experimental resources required for quantum\nprocess tomography is of paramount importance in scaling up\nquantum technologies.  Several strategies have been designed\nto address these issues, such as methods based on the\nleast-square (LS) linear inversion\ntechnique\\cite{miranowicz-pra-2014}, linear regression\nestimation\\cite{qi-quantum-inf-2017}, maximum likelihood\nestimation (MLE)\\cite{james-pra-2001},  self-guided\ntomography\\cite{rambach-prl-2021} and numerical\nstrategies\\cite{kaznady-pra-2009}.  Several QST protocols\nhave been extended to perform QPT, which include MLE-based\nQPT\\cite{obrien-prl-04}, LS-based\nQPT\\cite{trystan-arxiv-2021}, simplified\nQPT\\cite{kosut-njp-2009}, convex optimization-based\nQPT\\cite{jin-2019}, selective and efficient QPT\n\\cite{perito-pra-2018}, adaptive\nQPT\\cite{pogorelov-pra-2017}, and ancilla-assisted\nQPT\\cite{altepeter-prl-2003}.  These protocols have been\nsuccessfully demonstrated on various physical systems such\nas\nNMR~\\cite{gaikwad-pra-2018,xin-npj-2019,xin-phys-app-2020,Gaikwad-qip-2021,zhao-pra-2021},\nNV-centers\\cite{zhang-prl-2014}, linear\noptics\\cite{paz-prl-2010}, superconducting\nqubits\\cite{neeley-nature-2008,\nchow-prl-2009,gaikwad-ijqi-2020} and ion trap-based quantum\nprocessors\\cite{riebe-prl-2006}.\n\n\nMethods such as Monte-Carlo process\ncertification\\cite{silva-prl-2011} and randomized\nbenchmarking\\cite{knill-pra-2008} have been developed to\naddress scalability issues in standard QST and QPT methods.\nHowever, they are limited in scope as they do not provide\nthe full process matrix and hence cannot be used to identify\ngate errors or improve gate fidelity.  Other methods such as\nancilla-assisted QPT are able to significantly reduce the\nexperimental complexity, however the issues of scalability\nremain. The CS algorithm borrows ideas from classical\nsignal processing which posits that even heavily\nundersampled sparse signals can be efficiently\nreconstructed. The CS algorithm relies on reformulating QST\nand QPT tasks as a constrained convex optimization problem,\nand is able to perform complete and true characterization of\na given quantum process from a heavily reduced data set\nwithout performing actual projective measurements, and does\nnot need any extra resources such as ancilla qubits.  CS-QST\nand CS-QPT have been successfully used to reconstruct\nunknown quantum states from NMR data~\\cite{yang-pra-2017},\nto characterize quantum gates based on superconducting Xmon\nand phase qubits~\\cite{rodionov-prb-2014}, and to perform\nefficient estimation of process matrices of  photonic\ntwo-qubit gates~\\cite{shabani-prl-2011}.\n\n\n\nIn this work we utilize the CS algorithm to perform QPT of\nvarious two- and three-qubit quantum gates on an NMR quantum\nprocessor.  We also demonstrate the efficacy of the CS-QPT\nprotocol in characterizing two-qubit dynamics in a\nthree-qubit system.  We experimentally estimate the full\nprocess matrix corresponding to a given quantum process with\na high fidelity, from a drastically reduced set of initial\nstates and outcomes.  The CS-QPT algorithm is able to\nefficiently characterize a given quantum process provided\nthe corresponding process matrix is sufficiently sparse (\\textit{i.e.~}\nmost of its matrix elements are zero).  We use two different\noperator basis sets to estimate the process matrix using the\nCS algorithm, namely,  the standard Pauli basis and the\nPauli-error basis (where the process matrix is maximally\nsparse~\\cite{rodionov-prb-2014}, \\textit{i.e.~} it contains only one\nnon-zero element).  We also compare the performance of the\nCS-QPT and the LS-QPT methods using significantly reduced\ndata sets in both the standard Pauli basis and the\nPauli-error basis.  We obtained experimental fidelities of\ngreater than 0.9 from a reduced data set of a size\napproximately 5 to 6 times smaller than the size of a full\ndata set, and our results indicate that the CS-QPT method is\nsignificantly more efficient than standard QPT methods.\n\n\nThis paper is organized as follows: In Section~\\ref{qpt} we\ndetail the implementation of the CS algorithm in the context\nof QPT.  The standard QPT protocol is briefly described\nin Section~\\ref{sec2.1}, while the CS-QPT method is\ngiven in Section~\\ref{sec2.2}.\nSection~\\ref{nmr-expt} describes the experimental\nimplementation of the CS-QPT methods using two and three\nNMR qubits.\nIn Sections~\\ref{csqpt2q} and \\ref{csqpt3q},\nwe present the quantum\ncircuit and the corresponding NMR implementation of the\nCS-QPT method for two- and three-qubit quantum gates, respectively. \nSection~\\ref{sec3.3} contains a description of the CS-QPT\nimplementation to capture\ntwo-qubit quantum dynamics embedded in a three-qubit system.\nSection~\\ref{sec3.4} contains a comparison  of the\nCS-QPT and LS-QPT protocols.\nSection~\\ref{concl} contains a few concluding remarks.  \n\\section{QPT for a reduced data set}  \n\\label{qpt}\n\\subsection{Standard QPT and $\\chi$ matrix representation}\n\\label{sec2.1}\nIn a fixed basis set $\\lbrace E_i \\rbrace$, a quantum map\n(a completely positive map)\n$\\Lambda$ can be written as~\\cite{kraus-book-1983}:\n\\begin{equation}\n\\Lambda(\\rho) = \\sum_{m,n} \\chi_{mn} E_m \\rho E_n^{\\dagger}\n\\label{eq2}\n\\end{equation}\nwhere the Kraus operators are expanded as $A_i =\n\\sum_{k}a_{ik}E_k$ and the quantities\n$\\chi_{mn}=\\sum_{i}a_{im}a_{in}^*$ are the elements of the\nprocess matrix $\\chi$ characterizing the quantum map\n$\\Lambda$.  In a $d$-dimensional Hilbert space, $\\chi$ is a\n$d^2 \\times d^2$ dimensional positive semi-definite matrix\nand $d^4$ real independent parameters are required to\nuniquely represent it. The number of required parameters\nreduces from $d^4$ to ($d^4-d^2$) for trace preserving\nprocesses~\\cite{obrien-prl-04}. \n\n\nThe standard QPT protocol estimates the complete $\\chi$\nmatrix by preparing the system in different quantum states,\nletting it evolve under the given quantum process, and then\nmeasuring a set of observables~\\cite{childs-pra-2001}. The\nfull data set for QPT can be acquired using tomographically\ncomplete sets of input states $\\lbrace \\rho_1,\n\\rho_2,....,\\rho_k \\rbrace $, letting them\nundergo the desired quantum process $\\chi$, and \nmeasuring an observable $M$ from the set of measurement operators\n$\\lbrace M_1, M_2,..., M_l \\rbrace$, leading to:\n\\begin{equation}\nB^i_j=\\text{Tr}(M_j\\Lambda(\\rho_i)) = \n\\sum_{m,n} \\chi_{mn} \\text{Tr}(M_j E_m \\rho_i E_n^{\\dagger})\n\\label{eq3}\n\\end{equation}\nFor all input states $\\lbrace \\rho_i \\rbrace$ \nand measurement operators $\\lbrace M_j \\rbrace$ \nin Eq.~(\\ref{eq3}), the relationship between\nthe vector of outcomes and the true process matrix\n can be rewritten in \na compact form~\\cite{childs-pra-2001}:\n\\begin{equation}\n\\overrightarrow{B}(\\chi) = \\Phi \\overrightarrow{\\chi}\n\\label{eq4}\n\\end{equation} \nwhere $\\overrightarrow{B}(\\chi)$ and $\\overrightarrow{\\chi}$\nare vectorized forms of $B^i_j$ and $\\chi_{mn}$\nrespectively, and $\\Phi$ is the coefficient matrix with the\nentries $\\Phi_{ji,mn} = \\text{Tr}(M_j E_m \\rho_i\nE_n^{\\dagger})$. \n\nWe note here that using the standard QPT method  may not\nalways lead to a positive semi-definite experimentally\nconstructed $\\chi$ matrix, due to experimental\nuncertainties. This problem can be resolved by reformulating\nthe linear inversion problem as a constrained convex\noptimization problem as follows\\cite{Gaikwad-qip-2021}:\n \n\\begin{subequations} \n\\begin{alignat}{2}\n\\min_{\\chi}\\quad & \\Vert\n\\overrightarrow{B}^{exp}-\\overrightarrow{B}(\\chi)\\Vert_{l_2}\\label{eq5}\\\\\n\\text{subject to}\\quad       & \\chi \\geq 0,\\label{eq5:constraint1}\\\\\n        &\n\\sum_{m,n}\\chi_{mn}E_m^{\\dagger}E_n =\nI_d.\\label{eq5:constraint2} \\end{alignat} \n\\end{subequations}\nwhere the vector $\\overrightarrow{B}^{exp}$ is constructed\nusing experimental measurement outcomes.  This method is\nreferred to as the least square (LS) optimization method. In\nthis work, we study the performance of the LS-QPT method for\na reduced data set.\n\\subsection{Compressed sensing QPT} \n\\label{sec2.2}\nCompressed sensing methods work well if the process matrix\nis sparse in some known basis and\nrely on compressing the information \ncontained in a process of large size into \none of much smaller size and use \nefficient convex optimization algorithms to \n``unpack'' this compressed information.\nThe\nCS-QPT method hence provides a way to reconstruct the complete and\ntrue $\\chi$ matrix of a given quantum process from a drastically\nreduced data set, provided that the $\\chi$ matrix is\nsufficiently sparse in some known basis \\textit{i.e.~}, the number of\nnon-zero entries in the $\\chi$ matrix is small.\nIt is to be noted that the sparsity is a property of the\nmap representation and not the map itself.\nSpecifically, for quantum gates which are trace-preserving\nunitary quantum processes, one can always find the proper\nbasis in which the corresponding $\\chi$ matrix is maximally\nsparse~\\cite{korotkov-arxiv-2013,kosut-arxiv-2008}.\n\n\nEstimating a sparse process matrix with an unknown\nsparsity pattern from an underdetermined set of\nlinear equations can be done using numerical optimization\ntechniques.\nFor trace-preserving maps, the complete\nconvex optimization problem for CS-QPT is formulated as\nfollows: \n\\begin{subequations} \n\\begin{alignat}{2}\n\\!\\min_{\\chi}        \\quad&\n\\Vert{\\overrightarrow{\\chi}}\\Vert_{l_1}\\label{eq6}\\\\\n\\text{subject to}\\quad       & \\Vert\n\\overrightarrow{B}^{exp}-\\Phi \\overrightarrow{\\chi}\n\\Vert_{l_2} \\leq \\epsilon,\\label{eq6:constraint1}\\\\ &\n\\chi \\geq 0,\\label{eq6:constraint2}\\\\ &\n \\sum_{m,n}\\chi_{mn}E_m^{\\dagger}E_n =\nI_d.\\label{eq6:constraint3} \n\\end{alignat} \n\\end{subequations}\nwhere Eq.~(\\ref{eq6}) is the main objective function which is\nto be minimized and Eq.~(\\ref{eq6:constraint1}) is the\nstandard constraint involved in the CS algorithm;\nEq.~(\\ref{eq6:constraint2}) and Eq.~(\\ref{eq6:constraint3}) denote\nthe positivity and trace preserving constraints of the\nprocess matrix, respectively. The parameter $\\epsilon$\nquantifies the level of uncertainty in the measurement, \\textit{i.e.~}\nthe quantity $\\overrightarrow{B}^{{\\rm exp}}=\\Phi\n\\overrightarrow{\\chi}_0+\\overrightarrow{z}$ is observed,\nwith $\\Vert \\overrightarrow{z} \\Vert_{l_2}  \\leq \\epsilon$,\nwhere $\\overrightarrow{\\chi}_0$ is the vectorized form of\nthe true process matrix and $\\overrightarrow{z}$ is an unknown\nnoise vector. The general $l_p$- norm of\na given vector $ {\\overrightarrow{x}}$ is defined as:\n$\\|x\\|_{p}=\\left(\\sum_{i}\\left|x_{i}\\right|^{p}\\right)^{1 \/\np}$. If the process matrix is sufficiently sparse and the\ncoefficient matrix $\\Phi$ satisfies the restricted isometry\nproperty (RIP) condition, then by solving the optimization\nproblem delineated in Eq.~(\\ref{eq6}), one can accurately\nestimate the process matrix~\\cite{shabani-prl-2011}.\nThe RIP condition is satisfied if the coefficient\nmatrix $\\Phi$ satisfies the following\nconditions~\\cite{shabani-prl-2011,rodionov-prb-2014}:\n\\begin{itemize} \\item[(i)] \\begin{equation} 1-\\delta_s \\leq\n\\frac{\\Vert \\Phi \\overrightarrow{\\chi}_1 - \\Phi\n\\overrightarrow{\\chi}_2 \\Vert_{l_2}^2}\n{\\Vert\\overrightarrow{\\chi}_1 - \\overrightarrow{\\chi}_2\n\\Vert_{l_2}^2}\\leq 1+\\delta_s \\label{rip1} \\end{equation}\nfor all $s$-sparse vectors $\\overrightarrow{\\chi}_1$ and\n$\\overrightarrow{\\chi}_2$.  \nAn $N \\times 1$ dimensional\nvector $\\overrightarrow{x}$ is $s$-sparse, if only $s < N$\nelements are non-zero.  \n\\item [(ii)] The value of the\nisometry constant $\\delta_s < \\sqrt{2}-1$.\nThe restricted isometry constant (RIC) of a\nmatrix $A$ measures how close to an isometry is the action of\n$A$ on vectors with a few nonzero entries, measured in the\n$l_2$-norm~\\cite{emmanuel-crm-2008}. \nSpecifically, the upper and lower RIC of a\nmatrix $A$ of size $n \\times N$ is the maximum and the minimum\ndeviation from unity (one) of the largest and smallest,\nrespectively, square of singular values of $\\text { all\n}\\left(\\begin{array}{c} N \\\\ k \\end{array}\\right) \\text {\nmatrices }$ formed by taking $k$ columns from $A$. \n\\item[(iii)] The size of the data set is sufficiently large\n\\textit{i.e.~}   $m_{\\text{conf}}\\geq C_0 s  \\text{log}(d^4\/s)$ where\n$C_0$ is a constant,  \n$m_{\\text{conf}}$ is the size of the data set, $s$ is \nthe sparsity of\nthe process matrix and $d$ is the dimension of \nthe Hilbert space. \n\\end{itemize} \n\nOnce the basis operators $\\lbrace E_{\\alpha}\n\\rbrace$ and the configuration space  $\\lbrace \\rho_i,\nM_j\\rbrace $ are chosen, the coefficient matrix\n$\\Phi_{\\text{full}}$ corresponding to the entire data set is\nfully defined and does not depend on the measurement\noutcomes. It has been shown that if $\\Phi_{m}$ is built by\nrandomly selecting $m$ rows (\\textit{i.e.~} $m$ number of random\nconfigurations) from $\\Phi_{\\text{full}}$ then it is most\nlikely to satisfy the RIP\nconditions~\\cite{rodionov-prb-2014}.  \nHence the sub-matrix\n$\\Phi_m \\in \\Phi_{\\text{full}} $ together with the\ncorresponding observation vector $\\overrightarrow{B}^{exp}_m\n\\in \\overrightarrow{B}^{exp}_\\text{full}$ can be used to\nestimate the process matrix by solving the optimization\nproblem (Eq.~(\\ref{eq6})). \n  \nIn this study, we use two different operator basis sets,\nnamely the standard Pauli basis (PB) and the Pauli-error\nbasis (PEB).  For both bases, the orthogonality condition is\ngiven by $ \\langle E_{\\alpha} \\vert E_{\\beta} \\rangle = d\n\\delta_{\\alpha \\beta} $.  \nFor\nan $n$-qubit system, \nthe basis operators $P_i$ in the PB set are \n$P_i = \\lbrace I, \\sigma_x, \\sigma_y, \\sigma_z \\rbrace\n^{\\otimes n}$, while the basis operators $E_i$ in \nthe PEB set are:\n$E_i = UP_i$, where $U$ is the desired unitary matrix for\nwhich the process matrix needs to be estimated.  Furthermore,\nthe process matrix in PEB corresponding to the desired $U$,\nis always maximally sparse, \\textit{i.e.~}   it contains only one\nnon-zero element~\\cite{rodionov-prb-2014}.  The convex\noptimization problems involved in LS-QPT and CS-QPT\n(Eq.~(\\ref{eq5}) and Eq.~(\\ref{eq6}), respectively) can be solved\nefficiently using the YALMIP\\cite{lofberg-2004} MATLAB\npackage, which employs SeDuMi\\cite{sturm-oms-1999} as a\nsolver.\n\\begin{figure}\n\\includegraphics[angle=0,scale=1]{fig1-csqpt.pdf} \n\\caption{(Color online) (a) Quantum circuit\nto implement CS-QPT of a CNOT gate.  Single-qubit unitary\noperations $R_{\\phi}^\\theta$  are achieved via rotations by\nan angle $\\theta$ and phase $\\phi$. The first \nblock represents the preparation of the desired input state, while\nthe second and third blocks represent the quantum\nprocess corresponding to the CNOT gate and the measurement,\nrespectively. (b) NMR implementation of the quantum circuit\ngiven in panel (a). The rectangles filled with red, orange,\nyellow and green color denote $\\frac{\\pi}{2}$, $\\pi$,\n$\\frac{\\pi}{4}$ and $\\frac{\\pi}{3}$ pulses, respectively, \nwith the rf phase\nwritten above each pulse. \nThe unfilled rectangles with\nthe phases $\\phi_1$, $\\phi_2$, $\\phi_3$, and $\\phi_4$\nrepresent pulses with flip angles $\\theta_1$,\n$\\theta_2$, $\\theta_3$, and $\\theta_4$, respectively.\nThe gradient line denotes \n$z$-gradient pulses. The evolution time\nperiod $\\tau=\\frac{1}{2J_{CH}}$ where $J_{CH}$ is the scalar\ncoupling constant. (c) ${}^{13}$C-labeled chloroform molecule\nwith ${}^{1}$H and ${}^{13}$C labeling the\nfirst and second qubits, respectively. (d) and (e) depict\nthe\nNMR spectra of ${}^{13}$C and ${}^{1}$H, respectively,\ncorresponding to the configuration $\\lbrace \\vert ++ \\rangle\n\\langle ++ \\vert, IX \\rbrace$.  }   \n\\label{ckt1}\n\\end{figure}\n\\section{Experimental implementation of CS-QPT}\n\\label{nmr-expt} \n\\subsection{CS-QPT of two-qubit gates}\n\\label{csqpt2q} \nWe implemented the CS-QPT protocol for \ntwo, two-qubit nonlocal quantum gates,\nnamely, the CNOT gate and the\ncontrolled-rotation gate.  The controlled-rotation gate is a\nnonlocal gate which rotates the state of the second qubit\nvia $R_x(\\theta)$, if the first qubit is in the state $\\vert\n1 \\rangle $.\n\nFor two qubits the tomographically complete set of input\nstates is given by: $\\lbrace \\vert 0 \\rangle, \\vert 1\n\\rangle, \\vert + \\rangle, \\vert - \\rangle \\rbrace ^{\\otimes\n2}$ where $\\vert + \\rangle = (\\vert 0 \\rangle + \\vert 1\n\\rangle)\/\\sqrt{2} $ and $\\vert - \\rangle = (\\vert 0 \\rangle\n+ i\\vert 1 \\rangle)\/\\sqrt{2} $. \nIn NMR, tomographic measurements are carried out by applying\na set of unitary rotations followed by signal\nacquisition\\cite{long-job-2001}. The time-domain NMR signal\nis recorded as a free induction decay and then Fourier\ntransformed to obtain the frequency spectrum, which\neffectively measures the net magnetization in the transverse\n($x-y$) plane.  For two NMR qubits, the tomographically\ncomplete set of unitary rotations is given\nby~\\cite{li-pra-2017}: $\\lbrace II, IX, IY, XX\\rbrace$ where\n$II$ denotes the no operation on both the qubits, $IX$\ndenotes no operation on the first qubit and\na $90^{\\circ}$ $x$-rotation on the second qubit, $IY$ denotes\nno operation on the first qubit and a $90^{\\circ}$\n$y$-rotation on the second qubit and $XX$ denotes a\n$90^{\\circ}$ $x$-rotation on both qubits. \n\n\nAs an illustration, the quantum circuit and corresponding\nNMR implementation of the CS-QPT protocol for a two-qubit\nCNOT gate is given in Fig.~\\ref{ckt1}.  Fig.\\ref{ckt1}(a)\ndepicts the general quantum circuit to acquire data for\nCS-QPT and contains all possible settings corresponding to a\ntomographically complete set of input quantum states and\nmeasurements. The first block in Fig.\\ref{ckt1}(a) prepares\nthe desired initial input state from $\\vert 00 \\rangle$. In\nthe second block the quantum process (CNOT gate in this\ncase) which is to be tomographed, is applied to the system\nqubits and in the third block, a set of tomographic\noperations are applied, followed by measurements on each\nqubit.  To implement CS-QPT for any other two-qubit quantum\ngate, the CNOT gate should be replaced with the desired\ngate, while the remaining circuit remains unaltered.  The\nfirst block in the Fig.\\ref{ckt1}(b) represents the NMR\npulse sequence which prepares the spin ensemble in the\npseudo pure state (PPS) $\\vert 00 \\rangle$ and then\ngenerates the desired input state from the $\\vert 00 \\rangle\n$ state.  The pulse sequence corresponding to the CNOT gate\n(the quantum process which is to be tomographed) is given in\nthe second block and finally, in the last block, the desired\nset of tomographic pulses are applied and the NMR signal is\nacquired.\n\n\nWe used $^{13}C$-enriched chloroform molecule\n(Fig.\\ref{ckt1}(c)) dissolved in acetone-D6 to\nphysically realize a two-qubit system, with the\n${}^{1}$H and ${}^{13}$C spins denoting the \nfirst and second qubits, respectively. \nThe NMR\nHamiltonian in the rotating\nframe is given by: \n\\begin{equation}\n\\mathcal{H} = - \\sum_{i=1}^2 \\nu_i I_{iz} +  J_{\\rm CH}\nI_{1z} I_{2z} \\label{eq7} \n\\end{equation}\nwhere $\\nu_1$, $\\nu_2$ are the chemical shifts, $I_{1z}$,\n$I_{2z}$ are the $z$-components of the spin angular momentum\noperators of the ${}^{1}$H and ${}^{13}$C spins\nrespectively, and $J_{{\\rm CH}}$ is the scalar coupling\nconstant.  We used the spatial averaging technique to\ninitialize the system in the PPS corresponding to $\\vert 00\n\\rangle$, with the density matrix $\\rho_{00}$ given\nby~\\cite{oliveira-book-07}: \\begin{equation}\n\\rho_{00}=\\frac{1}{4}(1-\\eta)I_4+\\eta \\vert 00\\rangle\n\\langle 00 \\vert \\label{eq8} \\end{equation} where $\\eta$\ncorresponds to the net spin magnetization at thermal\nequilibrium, and $I_4$ is a $4 \\times 4$ identity operator.\nFigs.~\\ref{ckt1}(d) and \\ref{ckt1}(e) depict the NMR spectra\ncorresponding to carbon and hydrogen respectively, obtained\nfor the configuration $\\lbrace \\vert ++ \\rangle, IX  \\rbrace\n$, where $\\lbrace \\vert ++ \\rangle$ refers to the initial\nstate and $IX$ denotes the tomographic pulse set used.  The\nsystem is prepared in the initial input state $\\vert ++\n\\rangle$, a CNOT gate is applied, and finally the\ntomographic pulse $IX$ is applied to obtain the NMR\nspectrum.  For the first qubit, the area under the spectrum\nis related to the density matrix elements $\\rho_{24}$ and\n$\\rho_{13}$, while for the second qubit, the area under the\nspectrum is related to the density matrix elements\n$\\rho_{34}$ and $\\rho_{12}$.  In general, the four readout\nelements of the density matrix are complex numbers; in NMR\nthe imaginary part of the density matrix can be calculated\nby applying a $90^{\\circ}$ phase shift to the spectrum\n(post-processing) and then measuring the\narea~\\cite{long-job-2001}.  Hence a given configuration\ncomprises four data points (two for each qubit).  Since the\nsize of the full configuration space is 64 (16 states\n$\\times$ 4 tomographic rotations), the size of the full data\nset for two qubits is $64 \\times 4 = 256$. The vector\n$\\overrightarrow{B}^{exp}_\\text{full}$ (256$\\times$1\ndimensional)  can be experimentally constructed by computing the area under\nthe spectrum for the full configuration space. One can hence\nconstruct $\\overrightarrow{B}^{exp}_m $ and the\ncorresponding sub-matrix $\\Phi_{m}$ by randomly selecting\n$m$ rows from $\\overrightarrow{B}^{exp}_\\text{full}$ and\n$\\Phi_\\text{full}$ respectively, solving the optimization\nproblem (Eq.~(\\ref{eq6})) for a reduced data set of size $m$,\nand estimating the process matrix; $m$ here refers to  one\nparticular configuration randomly chosen from the set of all possible\n256 configurations.\n\n\\subsection{CS-QPT of three-qubit gates}\n\\label{csqpt3q}\n\\begin{figure}\n\\includegraphics[angle=0,scale=0.95]{fig2-csqpt.pdf}\n\\caption{(Color online) Quantum circuit to implement CS-QPT\nof a Control-NOT-NOT ($U_{\\rm {CNN}}$) gate.  The first\nblock prepares the desired input state while the second and\nthird blocks represent the quantum process corresponding to\nthe ($U_{\\rm {CNN}}$) gate and the measurement,\nrespectively. (b) NMR implementation of the quantum circuit\ngiven in panel (a). Solid black rectangles are refocusing\npulses with flip angle $180^{\\circ}$, while the gray\nrectangles represent pulses with flip angle $45^{\\circ}$;\nthe corresponding rf phases are written below each pulse.\nThe value of $\\beta$ is set to $60^{\\circ}$, while the\nunfilled rectangles with rf phases $\\phi_1$, $\\phi_2$ and\n$\\phi_3$ correspond to flip angles $\\theta_1$, $\\theta_2$\nand $\\theta_3$, respectively.  The black rectangles\nrepresent pulses with flip angle $90^{\\circ}$.  The unfilled\nrectangles in the last block, of phases $\\phi_1$, $\\phi_2$\nand $\\phi_3$ correspond to flip angles $\\theta_1$,\n$\\theta_2$ and $\\theta_3$, respectively which implement\ntomographic operations followed by measurement on each\nqubit; $\\tau_{ij} = \\frac{ 1}{2J_{ij}}$. (c)\n$^{13}C$-labeled diethyl fluoromalonate with ${}^1$H,\n${}^{19}$F and ${}^{13}$C nuclei labeled as the first,\nsecond and third qubits, respectively. NMR spectra depicted\nin (d), (e) and (f) correspond to ${}^{1}$H, ${}^{19}$F and\n${}^{13}$C nuclei respectively, for the configuration\n$\\lbrace \\vert 11+ \\rangle \\langle 11+ \\vert, XYX \\rbrace$. \n}\n\\label{ckt3}\n\\end{figure}\nWe have implemented the CS-QPT protocol to characterize the\nthree-qubit controlled-NOT-NOT ($U_{{\\rm CNN}}$) gate with\nmultiple targets, with the first qubit being denoted the control\nqubit,\nwhile the other two qubits are the target qubits.  The \ncontrolled-NOT-NOT gate\ncan be decomposed using two CNOT gates as:~$U_{{\\rm CNN}} \\equiv$\nCNOT$_{13}.$CNOT$_{12}$, and is widely used in encoding\ninitial input states in error correction codes, fault\ntolerant operations~\\cite{egan-arxiv-2021,shor-pra-1995} and\nin the preparation of three-qubit maximally entangled\nstates~\\cite{mooney-arxiv-2021,singh-pra-2018,dogra-pra-2015}.  \n\nThe NMR Hamiltonian for three qubits\nin the rotating frame is given by:\n\\begin{equation}\n\\mathcal{H} = - \\sum_{i=1}^3 \\nu_i I_{iz} + \\sum_{i,j=1\n(i \\ne j)}^3 J_{ij} I_{iz} I_{jz} \\label{ham3q} \n\\end{equation}\nwhere the indices $i,j$ label the qubit and\n$\\nu_i$ denotes the respective chemical shift. The \nquantity $J_{ij}$ denotes the scalar coupling strengths\nbetween the $i$th and $j$th qubits, while $I_{iz}$ represents\nthe $z$-component of the spin angular momentum of the $i$th qubit. We have\nused ${}^{13}$C-labeled diethyl fluoromalonate\n(Fig.\\ref{ckt3}(c)) dissolved in acetone-D6 to physically\nrealize a three-qubit system, with the\n${}^{1}$H, ${}^{19}$F and ${}^{13}$C nuclei being labeled as the first,\nsecond and third qubits, respectively. State initialization is\nperformed by preparing the system in the PPS\n$\\vert 000 \\rangle$ via the spatial averaging\ntechnique with the corresponding density matrix being given\nby: \n\\begin{equation} \n\\rho_{000} = (\\frac{1-\\epsilon}{8})I_8\n+ \\epsilon \\vert 000 \\rangle \\langle 000 \\vert \\label{pps3}\n\\end{equation} \nwhere $\\epsilon \\approx 10^{-5}$ represents\nthe net thermal magnetization and $I_8$ is the 8$\\times$8 identity\noperator.\n\n \nFor a three-qubit system, the tomographically complete set\nof input states is given by: $\\lbrace \\vert 0 \\rangle, \\vert\n1 \\rangle, \\vert + \\rangle, \\vert - \\rangle \\rbrace\n^{\\otimes 3}$ where $\\vert + \\rangle = (\\vert 0 \\rangle +\n\\vert 1 \\rangle)\/\\sqrt{2} $ and $\\vert - \\rangle = (\\vert 0\n\\rangle + i\\vert 1 \\rangle)\/\\sqrt{2} $ and the tomographically\ncomplete set of unitary rotations is given\nby:~$\\lbrace III, IIY, IYY, YII, XYX, XXY,\nXXX \\rbrace$~\\cite{li-pra-2017}. \nThe quantum circuit and the corresponding NMR\npulse sequence to perform CS-QPT for the three-qubit gate $U_{\\rm\nCNN}$ is given in Fig.~\\ref{ckt3}. The first block in\nFig.\\ref{ckt3}(a) represents the input state\npreparation while the second block represents the\napplication of quantum gate $U_{\\rm{CNN}}$ (i.e. quantum\nprocess which is to be tomographed),\nand tomographic unitary rotations are\napplied in the last block, followed by measurement on each qubit. \nFig.\\ref{ckt3}(b) represents the corresponding NMR\nimplementation of quantum circuit given in the\nFig.\\ref{ckt3}(a).  The spatial averaging\ntechniques are used in the first block~\\cite{singh-pra-2019} \nto initialize system in the desired PPS,\nfollowed by the application of spin-selective rf\npulses to prepare the desired input state. \nIn the second\nblock the pulse sequence corresponding to $U_{\\rm{CNN}}$ is\napplied on the input state and in the last block after\napplication of tomographic pulses, the signal \nof the desired nucleus is recorded.\nThe NMR spectra corresponding to ${}^1$H, ${}^{19}$F and\n${}^{13}$C are given in\nFigs.~\\ref{ckt3}(d), (e) and (f), respectively,\nfor the configuration $\\lbrace \\vert\n11+ \\rangle \\langle 11+ \\vert , XYX \\rbrace $, \\textit{i.e.~} the\ninput state  $ \\vert 11+ \\rangle \\langle 11+ \\vert $ is\nprepared, evolved under the quantum process corresponding\nto $U_{\\rm{CNN}}$, the tomographic set of pulses $XYX$ is\napplied, and finally the NMR signal is recorded. \nFor the first qubit (${}^{1}$H)\nthe area under the four spectral lines \ncorrespond to the\ndensity matrix elements\n$\\rho_{48}$, $\\rho_{26}$, $\\rho_{37}$ and $\\rho_{15}$,\nfor the second qubit (${}^{19}$F)\nthe area under the four spectral lines \ncorrespond to the\ndensity matrix elements\n$\\rho_{57}$, $\\rho_{13}$, $\\rho_{68}$ and $\\rho_{24}$, while\nfor the third qubit (${}^{13}$C),\nthe area under the four spectral lines \ncorrespond to the\ndensity matrix elements\n$\\rho_{56}$, $\\rho_{12}$, $\\rho_{78}$ and \n$\\rho_{34}$, respectively. \nFor a three-qubit\nsystem there are 12 experimental data points (4 per\nqubit) for a given configuration and the total number\nconfigurations are 448 (64 input states $\\times$ 7\ntomographic unitary operations) which yields the\n$\\overrightarrow{B}^{exp}_\\text{full}$ of size = 5376 (448\nconfigurations $\\times $ 12 data points per configuration).\nOne can construct $\\overrightarrow{B}^{exp}_\\text{m}$ by\nrandomly selecting $m$ number of rows from\n$\\overrightarrow{B}^{exp}_\\text{full}$, and using the\ncorresponding coefficient matrix $\\Phi_{m}$ one can solve\nthe optimization problem (Eq.~(\\ref{eq6})), and construct \nthe process matrix for a reduced data\nset of size $m$.\n\n\\subsection{CS-QPT of two-qubit processes in \na three-qubit system}\n\\label{sec3.3}\nIn order to experimentally implement a two-qubit CNOT gate\nin a multi-qubit system, one needs to allow the two system\nqubits to interact with each other \\textit{i.e.~}, let them evolve under\nthe internal coupling Hamiltonian for a finite time. In\nreality, this is non-trivial to achieve experimentally, as\nduring the evolution time the other qubits are also\ncontinuously interacting with system qubits, and one has to\n``decouple'' the system qubits from the other qubits. In the\nlanguage of NMR, this is  referred to as refocusing of the\nscalar $J$-coupling.  \n\nTo implement a two-qubit CNOT gate we need four single-qubit\nrotation gates and one free evolution under the internal\ncoupling Hamiltonian (Fig.~\\ref{ckt1}).  The single-qubit\nrotation gates are achieved by applying very short duration\nrf pulses of length $\\approx 10^{-6}$ s, while the time\nrequired for free evolution under the coupling Hamiltonian\nis $\\approx 10^{-3}$ s. The quality of the experimentally\nimplemented quantum gate depends on the time required for\ngate implementation, which for the two-qubit CNOT gate, is\nprimarily determined by the free evolution under the\ncoupling Hamiltonian.\nWe use the CS-QPT protocol to\nefficiently characterize three coupling evolutions\ncorresponding to $U^J_{ij}$ of the form: \n\\begin{equation}\nU^J_{ij}(t) = e^{-i 2 \\pi J_{ij} I_{iz} I_{jz} t} \\label{hf}\n\\end{equation} \nwhere the indices $i$ and $j$ label the qubits\nand $J_{ij}$ is the \nstrength of the scalar\ncoupling between the $i$th and the $j$th qubit; \nfor the CNOT gate,\n$t = \\vert \\frac{1}{2 J_{ij}} \\vert $. \nA three-qubit system is continuously evolving\nunder all the three $J_{ij}$ couplings, so in order to let a\nsubsystem of two qubits effectively evolve under one \nof these\ncouplings, we have to refocus all the other $J$-couplings.\nFor example, consider the two-qubit subsystem of the $i$th and  $j$th\nqubit with the effective evolution $U_{ij}^{J}(t)$ given by:\n\\begin{equation}\nU_{ij}^{J}(t) = U_{\\rm{int}}(\\frac{t}{2}) R_{x}^k(\\pi)\nU_{\\rm{int}}(\\frac{t}{2}) R_{x}^k(-\\pi)\n\\end{equation}\nwhere $R_{x}^k(\\pm \\pi)$ is an $x$-rotation on the $k$th\nqubit by an angle $\\pm \\pi$ and $U_{{\\rm int}}(\\frac{t}{2})$\nis the unitary operator corresponding to free evolution for\na duration $\\frac{t}{2}$ under the internal Hamiltonian\n$\\mathcal{H}_{{\\rm int}} = \\sum_{i,j=1, i>j}^3 J_{ij} I_{iz}\nI_{jz}$.  The procedure for tomographic reconstruction of\nthe reduced two-qubit density matrix from the full\nthree-qubit density matrix is given in Table.~\\ref{table1}.\nWe were able to successfully characterize all three\n$U_{ij}^{J}(t)$ via the CS-QPT method and constructed the\ncorresponding process matrices, using a heavily reduced data\nset of size $\\approx 20$, with experimental fidelities $>\n0.94$.  Using the information given in Table~\\ref{table1},\none can efficiently characterize a general quantum state as\nwell as the dynamics of a two-qubit subsystem in a\nthree-qubit system, wherein the experimental data is\nacquired by measuring only the two qubits under\nconsideration; hence the complete set of input states and\ntomographic rotations required are the same as for the\ntwo-qubit protocol described in Section~\\ref{csqpt2q}. \n\n\n\\begin{table}[h!] \\caption{\\label{complexity3} Relation\nbetween the readout positions \nof the reduced density matrix of the subsystem ($\\rho_{ij}^{\\prime}$)\nand the readout positions $\\rho_{mn}$.}\n\\begin{ruledtabular}\n\\begin{tabular}{c | c c c c\n~Subsystem ~& \\multicolumn{4}{c}{Readout positions of the reduced density\nmatrix}~ \\\\\n~~ & ~$\\rho_{24}'$ & ~$\\rho_{13}'$ & ~$\\rho_{34}'$ &\n~$\\rho_{12}'$ ~~~~\\\\ \\colrule\n~${}^{1}$H+${}^{19}$F~ & ~$\\rho_{48}+\\rho_{37}$ &\n~$\\rho_{26}+\\rho_{15}$ & ~$\\rho_{57}+\\rho_{68}$ &~\n$\\rho_{13}+\\rho_{24}$ ~~~ \\\\\n~${}^{1}$H+${}^{13}$C~ & ~$\\rho_{48}+\\rho_{26}$ &\n~$\\rho_{37}+\\rho_{15}$ & ~$\\rho_{56}+\\rho_{78}$ &~\n$\\rho_{12}+\\rho_{34}$ ~~~ \\\\\n~${}^{19}$F+${}^{13}$C~ & ~$\\rho_{68}+\\rho_{24}$ &~\n$\\rho_{57}+\\rho_{13}$ & ~$\\rho_{78}+\\rho_{34}$ &~\n$\\rho_{56}+\\rho_{12}$ ~~\n\\label{table1}\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\\begin{figure*}[t]\n\\includegraphics[angle=0,scale=1]{fig3-csqpt.pdf} \n\\caption{(Color online)\nThe top panel represents the average gate fidelity\n$\\overline{\\mathcal{F}}$ corresponding to (a) a three-qubit\n$U_{\\rm {CNN}}$ gate, (b) a CNOT gate and (c) $U^J_{23}$\nagainst the number of data points $m_{\\text{data}}$ on the\n$x$-axis.  The bottom panel represents the standard\ndeviation in average fidelity $\\sigma$, corresponding to (d)\na three-qubit $U_{\\rm {CNN}}$ gate, (e) a CNOT gate, and (f)\n$U^J_{23}$, plotted against the number of data points\n$m_{\\text{data}}$ on the $x$-axis.  The data points in red,\nblue, green and pink correspond to CS-PEB, CS-PB, LS-PEB and\nLS-PB methods, respectively. The CS-PEB method shows the\nbest performance for all three quantum gates.\n}\n\\label{plots}\n\\end{figure*}\n\n\\begin{table*}[t]\n\\caption{\\label{complexity3}\nThe minimum value of $m_{\\rm{data}}$ at which the experimental average gate\nfidelity $\\overline{\\mathcal{F}}$ turns to be $> 0.9$ is computed (alongwith\nthe standard deviation $\\sigma$) for different quantum processes, via the\nCS-PEB, CS-PB, LS-PEB and LS-PB methods.\n}\n\\begin{ruledtabular}\n\\begin{tabular}{r|c c c|c c c|c c c|c c c \n & \\multicolumn{3}{c|}{CS-PEB} & \\multicolumn{3}{c|}{CS-PB}\n& \\multicolumn{3}{c|}{LS-PEB} & \\multicolumn{3}{c}{LS-PB} \\\\\nGate & $m_{\\rm {data}}$ &$\\overline{\\mathcal{F}}$ & $\\sigma$\n& $m_{\\rm {data}}$ &$\\overline{\\mathcal{F}}$ & $\\sigma$&\n$m_{\\rm {data}}$ &$\\overline{\\mathcal{F}}$ & $\\sigma$&\n$m_{\\rm {data}}$ &$\\overline{\\mathcal{F}}$ & $\\sigma$  ~~~\\\\\n\\colrule\n$U_{\\rm{CNN}}$ & 30 & 0.9920 & ~0.0081~ & - & - & - & 320 &\n0.9109 & ~0.0123~ & 290 & 0.9006 & 0.0147 ~~~\\\\\nCNOT & 44  & 0.9798  &  0.0701 & 62  &  0.9203  & ~0.0905~ &\n52    & 0.9514   &    0.0263 & 48  &  0.9308  &\n0.0475~~~\\\\\nC-$R_x^{\\pi}$ & 48  &  0.9728   &   0.0797 & 58  &   0.9068\n& 0.0746 & 48  &  0.9332  &   0.0805 & 52  &  0.9503  &\n0.0504 ~~~\\\\\n$U^J_{12}$ & 14  &  0.9549  &    0.0963 & 24  &  0.9464\n& 0.0468 & 32 &  0.9075  &  0.0459 & 34  &  0.9071  &\n0.0561 ~~~\\\\\n$U^J_{23}$ & 14  &  0.9641  &   0.0734 & 28  &  0.9145  &\n0.0710 & 66  &  0.9019  &  0.0217 & 68  &  0.9048  &\n0.0198~~~\\\\\n$U^J_{13}$ & 18  &  0.9417 &  0.0980 & 38  &  0.9067 & 0.0695\n & -  & -  & -  & -  &  - & -  \n\\label{table2}\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\\subsection{Comparison of CS-QPT and LS-QPT protocols}\n\\label{sec3.4}\n\\begin{table}[h!] \n\\centering\n\\caption{\\label{fid} Experimental quantum process\nfidelities obtained via CS and LS methods using \nthe full data set $m^{\\rm{full}}_{\\rm{data}}$.}\n\\begin{tabular}{r c c c c\n\\hline\nGate & ~~CS-PEB~~& ~~CS-PB~~ & ~~LS-PEB~~& ~~LS-PB\\\\\n\\hline $U_{\\rm{CNN}}$ & 0.9980 & 0.8877 & 0.9542 &~ 0.9542\\\\ \nCNOT & 0.9984 & 0.9843 & 0.9817 &~ 0.9817\\\\ \nC-$R^{\\pi}_x$ & 0.9980 & 0.9744 & 0.9831 &~ 0.9831\\\\\n$U^J_{12}$ & 0.9967 & 0.9894 & 0.9819 &~ 0.9819 \\\\\n$U^J_{23}$ & 0.9976 &  0.9793 & 0.9273 &~ 0.9273 \\\\\n$U^J_{13}$ & 0.9895 & 0.9710 & 0.8942 &~ 0.8942 \\\\\n\\hline \n\\end{tabular}\n\\end{table}\nThe fidelity of the experimentally estimated $\\chi_{{\\rm exp}}$\nis computed using the\nmeasure\\cite{zhang-prl-2014}:\n\\begin{equation}\n{\\mathcal F}(\\chi^{}_{\\rm exp},\\chi^{}_{\\rm ideal})=\n\\frac{|{\\rm Tr}[\\chi^{}_{\\rm exp}\\chi_{\\rm ideal}^\\dagger]|}\n{\\sqrt{{\\rm Tr}[\\chi_{\\rm exp}^\\dagger\\chi^{}_{\\rm exp}]\n{\\rm Tr}[\\chi_{\\rm ideal}^\\dagger\\chi^{}_{\\rm ideal}]}}\n\\label{eq9}\n\\end{equation} \nwhere\n$\\chi_{\\text{ideal}}$ is the theoretically constructed process\nmatrix,\nand as $\\chi_{\\rm{exp}} \\rightarrow\n\\chi_{\\rm{ideal}}$, ${\\mathcal F}(\\chi^{}_{\\rm\nexp},\\chi^{}_{\\rm ideal}) \\rightarrow 1$. \n\n\nWe performed QPT of several two- and three-qubit quantum gates using both\nCS-QPT and LS-QPT protocols on a reduced data set.  The CS-QPT method was\nimplemented for the PEB and PB basis sets.  For a two-qubit system\n$m_{\\text{data}}^{\\rm full} = 256$,  while for a three-qubit system,\n$m_{\\text{data}}^{\\rm full} = 5376$,  where $m_{\\text{data}}^{\\rm full}$\ndenotes the size of the full data set obtained using the complete set of input\nstates and tomographic rotation operators for two and three qubits as given in\nSections~\\ref{csqpt2q} and \\ref{csqpt3q}, respectively.  In the PEB basis,\n$\\chi_{\\text{ideal}}$ is maximally sparse for all unitary quantum gates, while\nin the PB basis, $\\chi_{\\text{ideal}}$ corresponding to the two-qubit CNOT,\ncontrolled-$R_{x}^{\\pi}$ and $U^J_{ij}$ gates have 16, 16 and 4 non-zero\nelements, respectively (out of a total of 256 elements).  For the three-qubit\ngate $U_{\\rm CNN}$, $\\chi_{\\text{ideal}}$ has 16 non-zero elements (out of a\ntotal of 4096 elements).\n \n\nThe performance of the CS-QPT method was compared  with the\nLS-QPT method for six different quantum processes corresponding to: (i) a\nthree-qubit $U_{\\rm CNN}$ gate, (ii) a two-qubit CNOT gate, (iii) a\ncontrolled-$R_x^{\\pi}$ rotation (iv) $U^J_{23}$, (v) $U^J_{13}$ and (vi)\n$U^J_{12}$, of which the results of the quantum process corresponding to (a) a\nthree-qubit $U_{\\rm CNN}$ gate, (b) a two-qubit CNOT gate and (c) $U^J_{23}$,\nare displayed in Fig.~\\ref{plots}. The top panel in Fig.~\\ref{plots} represents\nthe average gate fidelity $\\overline{\\mathcal{F}}$ \nplotted against $m_{\\text{data}}$, while the\nbottom panel represents the standard deviation $\\sigma$ in average gate fidelity\nplotted against $m_{\\text{data}}$. The average gate fidelity is\nobtained using the average process matrix \nestimated via the LS and CS algorithm in the PEB\nand PB bases. \nThe plots in red and blue color represent the results of \nthe CS-QPT method implemented\nin PEB and PB basis respectively, while the \nplots in green and pink color represent\nthe results of the LS-QPT \nmethod implemented in the PEB and PB basis, respectively. The average fidelity\nand the value of $\\sigma$ is computed by implementing the CS-QPT and LS-QPT\nprotocols 50 times for randomly selected $m_{\\text{data}}$ number of data\npoints, and $\\sigma$ is calculated from: \n\\begin{equation}\n\\sigma = \\sqrt{\\frac{\\sum_{i=1}^{N}\n(\\mathcal{F}_i-\\overline{\\mathcal{F}})^2}{N-1}} \\label{eq10}\n\\end{equation} \nwhere $N=50$ and $\\overline{\\mathcal{F}}$ is\nthe average fidelity.\n\n\n\nThe plots in the first column of Fig.~\\ref{plots} correspond to the\nthree-qubit gate $U_{\\rm{CNN}}$, where Fig.~\\ref{plots}(a) depicts the\naccuracy, while Fig.~\\ref{plots}(d) gives the precision in characterizing\n$U_{\\rm{CNN}}$, for a given value of $m_{\\text{data}}$. Similarly, the second\nand third columns in Fig.~\\ref{plots} represent the experimental results\ncorresponding to the CNOT gate and the $U^J_{23}$ quantum \nprocess, respectively. The plots\ncorresponding to the two-qubit controlled-rotation gate (C-R$_{x}^{\\pi}$) is\nsimilar to the CNOT gate, while the plots corresponding to $U^J_{13}$ and\n$U^J_{12}$ are similar to  $U^J_{23}$ (plots not shown). As seen from\nFig.~\\ref{plots}, the CS-QPT method implemented in the PEB basis, performs\nsignificantly better than the LS-QPT and the CS-QPT methods implemented in the\nPB basis, for all the quantum processes considered.  The performance of the\nLS-QPT method is independent of the choice of basis operators.  On the other\nhand, the CS-QPT method may yield a lower fidelity as compared to the LS-QPT\nmethod, if the basis operators are not properly chosen.  Using a reduced data\nset, the overall performance for the three-qubit gate $U_{\\rm{CNN}}$ is CS-PEB\n$>$ CS-PB $>$ LS-PEB $\\approx$ LS-PB, while for the two-qubit CNOT and\nC-R$_x^{\\pi}$ gates, CS-PEB $>$ LS-PEB$\\approx$ LS-PB $>$ CS-PB.  For the\ntwo-qubit $U_{ij}^J$ processes, CS-PEB $>$ CS-PB $>$ LS-PEB $\\approx$ LS-PB.\n\nFor the two-qubit CNOT and C-R$_x^{\\pi}$ gates, the LS algorithm performs\nbetter than the CS algorithm in the PB basis for all values of $m_{\\rm{data}}$,\nwhile for the three-qubit $U_{\\rm{CNN}}$ gate,  the LS algorithm performs\nbetter than the CS algorithm in the PB basis for $m_{\\rm{data}} \\geq 160$,\nwhich clearly shows the importance of selecting an appropriate operator basis\nset while implementing the CS algorithm. \nThe plots given in Fig.~\\ref{plots} provide information about\nthe experimental complexity of the CS and LS algorithms \\textit{i.e.~} the\nnumber of experiments required in each case to characterize a\ngiven quantum process.\nWe note here in passing that the\nstandard deviation in average fidelity ($\\sigma$) is not monotonic.  For small\nvalues of $m_{\\rm {data}}$, the process of randomly selecting $m_{\\rm {data}}$\ndata points to estimate the process matrix is more likely to lead to a lower\nfidelity and higher values of the standard deviation $\\sigma$, and hence lower\nprecision.  For the two-qubit CNOT and controlled-R$_x^{\\pi}$ gates, $\\sigma$\nhas a maximum  around $m_{\\rm {data}}\\approx 20 $, while for the $U_{CNN}$,\n$U^J_{23}$, $U^J_{13}$ and $U^J_{12}$ quantum processes, $\\sigma$ is maximum\naround $m_{\\rm {data}}\\approx 10 $. For all the cases, the CS-PEB method yields\nbetter precision as compared to the CS-PB, LS-PEB and LS-PB methods. \n\n \nThe experimentally obtained minimum value of  $m_{\\rm {data}}$  at which the\nexperimentally computed average gate fidelity is $> 0.9$ is given in\nTable~\\ref{table2}, for all the quantum processes.  For the three-qubit\n$U_{\\rm{CNN}}$ gate, we experimentally obtained\n$\\overline{\\mathcal{F}}_{\\rm{CS-PEB}} = 0.9920 \\pm 0.0081$ for a reduced data\nset of size $m_{\\rm{data}} = 30$.  For the two-qubit CNOT and\ncontrol-$R_x^{\\pi}$ gates, $\\overline{\\mathcal{F}}_{\\rm{CS-PEB}} \\geq  0.9790\n\\pm 0.0701 $ and $\\overline{\\mathcal{F}}_{\\rm{CS-PEB}} \\geq 0.9729 \\pm 0.0797 $\nfor $m_{\\rm{data}} \\geq 44$ and $m_{\\rm{data}} \\geq 48$, respectively. The\nreduced data set is $\\approx 5$ times smaller than the full data set, which\nimplies that the experimental complexity is reduced by $\\approx 80 \\%$ as\ncompared to the standard QPT method. Furthermore, for all the two-qubit quantum\nprocesses corresponding to $U_{ij}^J$, $\\overline{\\mathcal{F}}_{\\rm{CS-PEB}}\n\\geq  0.9417 \\pm 0.0980 $ for $m_{\\rm{data}} \\geq 18$. This reduced data set is\n$\\approx 12$ times smaller than the full data set which implies that the\nexperimental complexity in these cases is reduced by $\\approx 92 \\%$ as\ncompared to the standard QPT method.  \n\\section{Concluding Remarks}\n\\label{concl} \nWe designed a general quantum circuit to acquire experimental data compatible\nwith the CS-QPT algorithm.  The proposed quantum circuit can also be used for\nother experimental platforms and can be extended to higher-dimensional systems.\nWe successfully demonstrated the efficacy of the CS-QPT protocol for various\nquantum processes corresponding to the three-qubit $U_{\\rm{CNN}}$ gate,\ntwo-qubit CNOT and controlled-rotation gates and several two-qubit $U_{ij}^J$\nunitary operations.  Our experimental comparison of the CS-QPT and LS-QPT\nschemes demonstrate that the CS-QPT protocol is far more efficient, provided\nthat the process matrix is maximally sparse and that an appropriate operator\nbasis is chosen.  \n\nStandard QPT protocols do not have access to prior information about the\nintended target unitary and hence require a large number of parameters to\ncompletely characterize the unknown quantum process.  CS methods can be used to\ndramatically reduce the resources required to reliably estimate the full\nquantum process, in cases where there is substantial prior information\navailable about the quantum process to be characterized.  Since the CS-QPT\nmethod is uses fewer resource and is experimentally viable, it can be used to\ncharacterize higher-dimensional quantum gates and to validate the performance\nof large-scale quantum devices. \n\n\n\\begin{acknowledgments}\nAll the experiments were performed on a Bruker Avance-III\n600 MHz FT-NMR spectrometer at the NMR Research Facility of\nIISER Mohali.  Arvind acknowledges financial support from\nDST\/ICPS\/QuST\/Theme-1\/2019\/Q-68.\nK.~D. acknowledges financial support from\nDST\/ICPS\/QuST\/Theme-2\/2019\/Q-74.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nObservation of neutron star properties such as \nmass, size and temperature\nprovides us with important clues to the understanding of the state of matter\nat extremely high densities. \nIn the 1970's, the maximum mass of the neutron star\nwas calculated with the $NN$ potentials available at that time \n\\cite{pand-npa71,bj-npa74,ps-npa75,fp-npa81}\nand mean field models \\cite{wal-ap74,ps-plb75}. \nMost of the calculations done in the 1970's resulted in \nstiff equations of state, \nand thus the maximum mass of \na neutron star was predicted to be larger than $2 M_\\odot$,\nwhere $M_\\odot$ is the solar mass.\nOnly the Reid soft core potential yielded a soft equation of \nstate and consequently a small maximum mass of a neutron star,\n$1.6 M_\\odot$ \\cite{pand-npa71}.\nRecent observations of the masses of binary pulsars \\cite{tc-apj99},\nwhich are candidates of neutron stars indicate that the\nmaximum mass of neutron stars are roughly around $1.5 M_\\odot$,\nsubstantially smaller than most of the predicted values in the 1970's. \n(However, very recent observations seem to suggest possible\nexistence of more massive pulsars in the range $(1.8 - 2.0) M_\\odot$\n\\cite{apj2004,apj2005}, though further confirmation is needed.)\nOn the other hand, exotic forms of matter, i.e., matter consisting\nof degrees of freedom other than the nucleons have been proposed \nalready many years ago.\nSome of the proposed exotic states of matter include those with\ncreation of hyperons \\cite{hyperon}, \nBose-Einstein condensation (pions \\cite{pion} or kaons \\cite{kaon}), \nstrange matter \\cite{strange}, and \nquark deconfinement \\cite{itoh70,bc-plb76,kk-plb76}.\nThese exotic states seem to reduce the maximum mass of a neutron star \nclose to the observations \n\\cite{kpe-prc95, s-pal, nkg-prd92}, \nimplying that exotic degrees of freedom seem to be needed\nto reproduce the observed masses of neutron stars.\n\nIn this work, we consider the strangeness degrees of freedom by including\nboth hyperon creation and kaon condensation in the neutron star matter.\n(It is the anti-kaon that matters here, but we simply refer to\nboth kaons and anti-kaons as kaons for brevity.)\nThe masses and energies of the hyperons and kaons in medium are sensitive to\ntheir interactions with the surrounding matter. \nIn the meson-exchange picture,\nmeson-hyperon and meson-kaon coupling constants can fix the\nstrength of these interactions. \nThe meson-hyperon coupling constants may be\ndetermined from the binding energies of hyperons in hypernuclei.\nThe meson-kaon coupling constants have been studied by using\nthe kaon-nucleon scattering \\cite{Cieply,Kaiser} \nand kaonic atom data \\cite{Cieply}.\nRecently, the magnitudes of the kaon-nucleus potential \nin matter have attracted much attention.\nSome calculations \\cite{Schaffner,oset20,Cieply}\nshow that the real part of the $K^-$-nucleus optical potential $U_{K^-}$ \nis shallow ($U_{K^-} \\approx$ $-$50 MeV), but some other calculations\nsuggest that $U_{K^-}$ can be as large as about $-120$ MeV \n\\cite{gal94,Kaiser} or even close to $-200$ MeV \\cite{Batty}.\n\nAkaishi and Yamazaki predicted possible existence of\ndeeply bound kaonic nuclei \\cite{akaishi},\nin which $U_{K^-}$ at normal density $\\rho_0$ was\nestimated to be about $-120$ MeV.\nThen, experiments at KEK claimed the observation of \ntribaryon kaonic nuclei, \nS$^0$ \\cite{s0} and S$^+$ \\cite{suzuki1}, which seem to suggest that \n$K^-$ may be even more deeply bound than \nthe theoretical prediction \\cite{akaishi}\n(The former claim \\cite{s0}, however, was withdrawn by the experimental group\n\\cite{iwasaki}). \nFINUDA collaboration at DA$\\Phi$NE \\cite{FINUDA} \nand a BNL experiment with $^{16}{\\rm O}(K^-, n)$ reaction \\cite{kishi-npa05} \nalso reported distinct peaks. \nMore recently there was a theoretical work which considered\nlarge kaonic binding energies and calculated widths of kaonic nuclear\nbound states \\cite{Mares}.\nThe identities of these experimental peaks need to be\nstudied further experimentally and theoretically. However, in this work, \nwe consider the possibility of  \ndeep optical potential of kaons in nuclei and \nexplore the consequences in the \ncomposition profile of neutron star matter.\n\nIn this work, for the description of dense matter\nwe employ the modified quark-meson coupling (MQMC) model \n\\cite{mqmc}.\nNucleons and hyperons in the baryon octet are treated as MIT bags.\nThe bag constant $B_B$ and phenomenological constant $Z_B$ for a baryon \n$B$ are fixed to reproduce the free mass of each baryon $B$.\nCoupling constants between ($u$, $d$)-quarks in the bags and ($\\sigma$, \n$\\omega$, $\\rho$)-mesons are adjusted to give us the \nbinding energy per a nucleon $E_b\/A = 16$ MeV and symmetry energy  \n$a_{\\rm sym} = 32.5$ MeV at the\nsaturation density $\\rho_0 = 0.17\\, {\\rm fm}^{-3}$. \nSince the interaction between the $s$-quark and mesons are not well known,\nwe adopt the standard quark counting rule and assume the $s$-quark is\ndecoupled from ($\\sigma$, $\\omega$, $\\rho$)-mesons.\nTo take into account the interactions between $s$-quarks,\nwe introduce $\\sigma^* (980)$ and $\\phi(1020)$ mesons\nfollowing Ref.~\\cite{s-pal} for the baryon and Ref.~\\cite{bb-prc01} \nfor the kaon.\nWe also assume the kaon as a point particle.\nThis treatment allows us to use $U_{K^-}$\nas an input to fix the coupling constant between the $\\sigma$-meson \nand the kaon, $g_{\\sigma K}$. \nIn our model the real part of the kaon optical potential at $\\rho = \\rho_0$\ncan be written as\n$U_{K^-} = - (g_{\\sigma K} \\sigma(\\rho_0)\n+ g_{\\omega K} \\omega(\\rho_0)) $,\nwhere $\\sigma(\\rho_0)$ and $\\omega(\\rho_0)$ are the values of\nthe meson fields at $\\rho_0$. Using the value of $g_{\\omega K}$ \ngiven by the quark counting rule,\nwe can fix $g_{\\sigma K}$ for each given value of $U_{K^-}$. \nOnce the parameters of the model are fixed,\nthe composition profile of neutron star matter can be obtained from\nthe $\\beta$-equilibrium and charge neutrality.\nWe find that the composition of neutron star matter changes \ndramatically depending on the value of $U_{K^-}$.\n\nTo investigate the model dependence of the results\nwe also employ quantum hadrodynamics (QHD)\nmodel \\cite{sw-qhd} for calculating the composition of matter.\nThe parameters of the QHD model are calibrated to produce exactly \nthe same saturation properties as in the MQMC model.\nOur calculations show that the onset densities of \nthe kaon condensation and the compositions of matter\nat high densities are substantially model dependent.\nIn Sect. II, we introduce model Lagrangians and fix the model parameters.\nThe results are discussed in Sect. III.  \nConclusions and discussions follow in Sect. IV.\n\n\\section{Theory}\n\nIn this section we first briefly sketch the MQMC and QHD models\nby presenting the model Lagrangians.\nThe models are calibrated so as to be consistent with each other\nat the saturation density by fixing the \ncoupling constants of both models to produce exactly\nthe same saturation properties; the saturation density,\nthe binding energy, the symmetry energy, the nucleon effective\nmass, and the compression modulus.\nWe then show how the physical quantities that will determine \nthe composition of the neutron star matter can be obtained self-consistently.\n\\subsection{{\\bf Models}}\n\\label{sect:The MQMC model with kaons}\n\nThe model Lagrangian comprises\nthe terms for the octet baryons, exchange mesons, leptons and kaons;\n$\\mathcal L_{tot} = \\mathcal L_B + \\mathcal L_M + \\mathcal L_l + \\mathcal L_K$.\nOctet baryon, exchange meson and lepton terms \nin the mean field approximation can be written as\n\\begin{eqnarray}\n\\mathcal L_B  & =& \\sum_B \\bar \\psi_B \\left[i\\gamma \\cdot \\partial - m_B^* \n(\\sigma, \\sigma^*)\n-\\gamma^0 \\left(g_{\\omega B}\\omega_0 + g_{\\phi B}\\phi_0 + \\frac 12 g_{\\rho B}\n\\tau_z \\rho_{03} \\right) \\right] \\psi_B \\label{eq:lagb} \\\\ \n\\mathcal L_M &=& -\\frac 12 m_\\sigma^2 \\sigma^2 - \\frac 12 m_{\\sigma^*}^2 {\\sigma^*}^2\n+ \\frac 12 m_\\omega^2 \\omega_0^2 + \\frac 12 m_\\phi^2 \\phi_0^2 + \\frac 12 m_\\rho^2 \\rho_{03}^2, \\label{eq:lagm} \\\\ \n\\mathcal L_l &=& \\sum_l \\bar \\psi_l ( i \\gamma \\cdot \\partial - m_l)\\psi_l ,\n\\label{eq:lagl}\n\\end{eqnarray}\nwhere $B$ denotes the sum over all the octet baryons \n($p, ~n, ~\\Lambda, ~\\Sigma^+, ~\\Sigma^0, ~\\Sigma^-, ~\\Xi^0, ~\\Xi^-$),\nand $l$ stands for the sum over the free electrons and muons ($e^-$, $\\mu^-$).\n$\\sigma$, $\\omega$ and $\\rho$ mesons mediate the interactions\nbetween the non-strange light quarks ($u$ and $d$).\n$\\sigma^*$ and $\\phi$ mesons are introduced to take into\naccount the interactions between $s$ quarks.\n$\\mathcal L_B$ is of the identical form for both \nthe MQMC and QHD models,\nbut differs in the definition of\nthe effective baryon mass $m^*_B$\nas will be shown below. \n\n{\\bf MQMC}\n\nIn the MQMC model, a baryon is a composite system with quarks \nin a spherical bag, and its mass is given in terms of bag\nparameters and quark eigenenergy.\nThe effective mass of a baryon in matter $m^{*}_B(\\sigma, \\sigma^*)$\ncan be written as \\cite{s-pal,mqmc,fleck,st,stt}\n\\begin{eqnarray}\nm^*_B = \\sqrt{E^2_B - \\sum_q  \\left(\\frac{x_q}{R} \\right)^2}.\n\\label{eq:efmass}\n\\end{eqnarray}\nThe bag energy of a baryon is given by \n\\begin{eqnarray}\nE_B &=& \\sum_q  \\frac{\\Omega_q}{R} - \\frac{Z_B}{R}\n+ \\frac{4}{3} \\pi\\, R^3\\, B_B,\n\\label{eq:bagery}\n\\end{eqnarray}\nwhere $B_B$ and $Z_B$ are the bag constant and  \na phenomenological constant for the zero-point motion of a baryon $B$,\nrespectively.\n$\\Omega_q = \\sqrt{x^2_q + (R m^*_q)^2}$, \nwhere $m^*_q (= m_q - g^q_\\sigma \\sigma - g^q_{\\sigma^*} \\sigma^*)$\nis the effective mass of a quark whose free mass is $m_q$.\nWe take $m_q =0$ for $q=u,d$ and $m_q = 150$ MeV for $q=s$.\n(Other choices of $m_{q=s}$ values do not make differences in the \nresults~\\cite{Theta+}.)\n$x_q$ is determined from the boundary condition on the bag surface $r = R$,\n\\begin{equation}\nj_0(x_q) = \\beta_q j_1(x_q),\n\\label{eq:bessel}\n\\end{equation}\nwhere\n$\n\\beta_q = \\sqrt{\\frac{\\Omega_q - R m^*_q}{\\Omega_q + R m^*_q}}.\n$\nIn the MQMC model, the bag constant $B_B$ is assumed to depend on density\n\\cite{mqmc,stt}.\nIn this work, we use the extended form in Ref.~\\cite{s-pal} to include the \ncontribution from $\\sigma^*$ as\n\\begin{eqnarray}\nB_B (\\sigma,\\, \\sigma^*)  =  \nB_{B0} \\exp \\left\\{ -\\frac{4}{m_B} \\left({g'}_\\sigma^B \\sum_{q=u,d} n_q \\sigma\n+ {g'}_{\\sigma^*}^B (3-\\sum_{q=u,d}n_q) \\sigma^*\\right) \\right\\},\n\\label{eq:bag}\n\\end{eqnarray}\nwhere $m_B$ is the free mass of the baryon $B$,\nassuming that the $\\sigma$ meson couples to $u$ and $d$ quarks only\nand that the $\\sigma^*$ meson couples to the $s$ quark only.\n\n{\\bf QHD}\n\nIn the QHD model, a baryon is treated as a point particle,\nand thus its effective mass is simply written as \n\\begin{eqnarray}\nm^*_B = m_B - g_{\\sigma B} \\sigma - g_{\\sigma^* B} \\sigma^*.\n\\end{eqnarray}\nIn order to reproduce the same saturation properties \nas obtained in the MQMC model, \nself-interactions of the $\\sigma$-field \\cite{bb-npa77}\n\\begin{eqnarray}\nU^{\\rm QHD}_{\\sigma} = \\frac{1}{3} g_2\\, \\sigma^3 + \n\\frac{1}{4} g_3\\, \\sigma^4 \n\\label{eq:U_QHD}\n\\end{eqnarray}\nare added to Eq.~(\\ref{eq:lagm}) so that\n\\begin{eqnarray}\n\\mathcal L_M^{\\rm QHD} = \\mathcal L_M - U^{\\rm QHD}_{\\sigma}.\n\\end{eqnarray}\nAs mentioned above, the baryon and the lepton Lagrangians for the QHD model \ntake the form given by \nEqs.~(\\ref{eq:lagb}) and (\\ref{eq:lagl}).\n\n{\\bf Kaon}\n\nThe effective Lagrangian for the kaon may be expressed as \\cite{glend}\n\\begin{eqnarray}\n\\mathcal L_K = D_\\mu^* K^* D^\\mu K - {m_K^*}^2 K^* K,\n\\label{eq:lagk}\n\\end{eqnarray}\nwhere \n$\nD_\\mu = \\partial_\\mu + i g_{\\omega K}\\omega_\\mu \n-i g_{\\phi K} \\phi_\\mu + i \\frac 12 g_{\\rho K} \\vec \\tau \\cdot \\vec \\rho_\\mu.\n$\nIn this work we treat the kaon\nas a point particle in both MQMC and QHD models,\nand its effective mass is given by\n\\begin{equation}\nm_K^* = m_K - g_{\\sigma K} \\sigma - g_{\\sigma^* K} \\sigma^*.\n\\label{eq:keffmass}\n\\end{equation}\nThe equation of motion for a kaon is given by   \n\\begin{eqnarray}\n[D_\\mu D^\\mu + {m_K^*}^2] K(x) = 0.\n\\end{eqnarray}\nIn uniform infinite matter, the kaon field $K(x)$ \ncan be written as a plane wave.\nSubstituting the plane wave solution into the equation \nof motion, we obtain the dispersion relation for the anti-kaon\n\\begin{eqnarray}\n\\omega_K = m_K^* - g_{\\omega K} \\omega_0 \n+ g_{\\phi K} \\phi_0 - g_{\\rho K} \\frac 12 \\rho_{03}.\n\\label{eq:kaon-dispersion}\n\\end{eqnarray}\n\n\\subsection{Model parameters}\n\n{\\bf MQMC}\n\nIn the MQMC model, MIT bag parameters $B_{B0}$ and $Z_B$ are determined\nto reproduce the free mass of a baryon $B$,\n$m^*_B \\left|_{\\rho = 0} = m_B \\right. $ \nwith the minimization condition \n$\\left.  \\frac{\\partial m_B}{\\partial R} \\right|_{R = R_0} =0$\nat a free bag radius $R_0$, which we choose as $R_0 = 0.6$ fm.\nThe bag parameters $B_{B0}$ and $Z_B$ for the octet baryons\nare listed in Table~\\ref{tab:b-z}.\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\n~~ $B$ ~~  &  $m_B$ (MeV) & $B^{1\/4}_{B0}$ (MeV) & $Z_B$ \\\\ \\hline\n   $N$     &   939.0 & ~~188.1 ~~& ~~2.030 ~~\\\\ \\hline\n$\\Lambda$  &  1115.6 &   197.6   &   1.926   \\\\ \\hline\n$\\Sigma^+$ &  1189.4 &   202.7   &   1.829   \\\\ \\hline\n$\\Sigma^0$ &  1192.0 &   202.9   &   1.826   \\\\ \\hline\n$\\Sigma^-$ &  1197.3 &   203.3   &   1.819   \\\\ \\hline\n$\\Xi^0   $ &  1314.7 &   207.6   &   1.775   \\\\ \\hline\n$\\Xi^-   $ &  1321.3 &   208.0   &   1.765   \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Bag constants $B_{B0}$ and phenomenological constants $Z_B$ for\noctet baryons to reproduce the free mass of each baryon.  \nThe bag radius is chosen as $R_0=0.6$ fm for all octet baryons, \nand the bare masses of quarks are fixed as\n$m_{u(d)}=0$ MeV and $m_s = 150$ MeV.}\n\\label{tab:b-z}\n\\end{table}\n\nThree saturation\nconditions $\\rho_0$, $E_b\/A$, and $a_{\\rm sym}$ could determine\nthree quark-meson coupling constants $g^{u,d}_\\sigma$,\n$g^{u,d}_\\omega$ and $g^{u,d}_\\rho$, assuming $u$ and $d$ quarks\nto be identical in the isodoublet.\nThe MQMC model, however, introduces an additional constant\n$g'^B_\\sigma$ in Eq.~(\\ref{eq:bag}). \nThus we fix $g^{u,d}_\\sigma = 1$, and adjust the remaining\nthree constants to meet the three conditions.\nThe resulting coupling constants are given in Table~\\ref{tab:coupling}\ntogether with the ratio of the effective mass of the nucleon $m^*_N\/m_N$\nand the compression modulus $K$.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c||c|c|} \\hline\n~~$g_\\sigma^q$~~  & ~~$g_\\omega^q$~~   & ~~${g'}_\\sigma^B$~~  & \n~~$g_\\rho^q$~~    & $m_N^* \/ m_N$  & $K$ (MeV)  \\\\ \\hline\n         1.0      & 2.71           & 2.27       &\n         7.88     & 0.78           & 285.5      \\\\ \\hline\n\\end{tabular}\n\\end{center} \n\\caption{The coupling constants between $(u,\\, d)$-quarks and \n$(\\sigma,\\, \\omega,\\, \\rho)$-mesons in the MQMC \nmodel to reproduce the binding energy $E_b\/A=16$ MeV \nand symmetry energy $a_{\\rm sym}=32.5$ MeV\nat the saturation density 0.17 ${\\rm fm}^{-3}$. \n$m^*_N\/m_N$ and $K$ are the ratio of the \neffective mass to the free mass of \nthe nucleon and the compression modulus at the saturation density,\nrespectively.}\n\\label{tab:coupling}\n\\end{table}\n$m^*_N$ and $K$ are within reasonable ranges: \n$m^*_N = (0.7 \\sim 0.8) m_N$ and $ K = (200 \\sim 300)$ MeV.\n\nThe coupling constants between $s$-quarks and mesons\ncannot be determined from the\nsaturation properties.\nIn principle, experimental data from hypernuclei \nand kaon-nucleus scattering could be used to determine \nthe coupling constants between $s$-quarks and mesons (for example,\nsee Ref.~\\cite{zakout05}).\nHowever, these coupling constants are not well known yet, \nand for simplicity\nwe assume that the quark counting rule holds and that\nthe $s$ quark does not interact with $u$ and $d$ quarks.\nThen we have\n\\begin{equation}\ng^s_{\\sigma} = g^s_{\\omega} = g^s_{\\rho} = \ng^{u,d}_{\\sigma*} = g^{u,d}_\\phi = 0.\n\\label{eq:smcoupling}\n\\end{equation}\nTo fix the meson-baryon coupling constants in the model Lagrangian,\nwe also use the quark counting rule \n\\begin{eqnarray}\n\\frac 13 g_{\\omega N} & = & \\frac 12 g_{\\omega \\Lambda}\n= \\frac 12 g_{\\omega \\Sigma} =g_{\\omega \\Xi} = g_\\omega ^q , \\nonumber \\\\\ng_{\\rho N} & = & g_{\\rho \\Sigma} = g_{\\rho \\Xi} = g_\\rho^q , \n~~g_{\\rho \\Lambda}=0, \\nonumber \\\\\ng_{\\phi \\Lambda} & = & g_{\\phi \\Sigma} = \\frac 12 g_{\\phi \\Xi} = g_\\phi^s,\n\\label{eq:211}\n\\end{eqnarray}\nand the SU(6) symmetry\n\\begin{eqnarray}\ng^s_{\\sigma*} &=& \\sqrt{2} g^{u,d}_\\sigma = \\sqrt{2}, \\nonumber \\\\\ng_\\phi^s &=& \\sqrt 2 g_\\omega^{u,d} = 3.83, \\nonumber \\\\\n{g'}^B_{\\sigma^*} &=& \\sqrt{2}\\, {g'}^B_\\sigma.\n\\label{eq:su6}\n\\end{eqnarray}\nThe quark-meson coupling constants $g^q _{\\omega}$ and $g^q _{\\rho}$\ngiven in Table~\\ref{tab:coupling} and\nthe relations in Eqs.~(15)-(17)\ndetermine all the meson-baryon coupling of the MQMC model.\n\n{\\bf QHD}\n\nIn the QHD model, $g_{\\sigma N}$ and $g_{\\omega N}$ are adjusted\nto yield $\\rho_0$ and $E_b$, and $g_{\\rho N}$ \nis fitted to produce $a_{\\rm sym}$.\n$g_2$ and $g_3$ in $U^{\\rm QHD}_{\\sigma}$ of Eq.~(\\ref{eq:U_QHD})\nare fixed to \nreproduce the same $m^*_N$ and $K$ values as listed in \nTable~\\ref{tab:coupling} for the MQMC model.\nThe coupling constants determined in this way\nare given in Table~\\ref{tab:coupling-qhd}.\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n~~$g_{\\sigma N}$~~ & ~~$g_{\\omega N}$~~ & ~~$g_{\\rho N}$~~ &\n$g_2$ (fm$^{-1}$) & $g_3$ \\\\ \\hline\n8.06 & 8.19 & 7.88 & 12.139 & 48.414 \\\\ \\hline \n\\end{tabular}\n\\end{center}\n\\caption{The meson-nucleon coupling constants and the coefficients\nof the $\\sigma$-meson self interaction terms used in the QHD model.\nThey reproduce the same saturation properties as in the MQMC model; \n$\\rho_0 = 0.17$ fm$^{-3}$, $E_b = 16 A$ MeV, $a_{\\rm sym} = 32.5$ MeV, \n$m^*_N = 0.78 m_N$ and $K = 285.5$ MeV.} \n\\label{tab:coupling-qhd}\n\\end{table}\nIn the MQMC model, meson-baryon coupling constants are\nobtained from the quark-meson coupling constants.\nOn the other hand, in QHD\nmeson-nucleon coupling constants provide the starting point\nfor the determination of remaining other meson-baryon\ncoupling constants.\nOnce meson-nucleon coupling constants are fixed from the saturation\nproperties, meson-hyperon coupling constants can be obtained \nby the quark counting rule (as in Eq.~(\\ref{eq:211}))\nand the SU(6) symmetry (as in Eq.~(\\ref{eq:su6})).\nThe coupling constants between strange mesons and hyperons can be obtained\nby combining the quark counting rule and the SU(6) symmetry,\n{\\it e.g.}, $g_{\\phi \\Lambda} = \\sqrt{2} g_{\\omega N} \/3$ and\n$g_{\\sigma^* \\Lambda} = \\sqrt{2}g_{\\sigma N}\/3$.\n\n{\\bf Kaon}\n\nThere are 5 kaon-meson coupling constants in our models; $g_{\\sigma K}$,\n$g_{\\omega K}$, $g_{\\rho K}$, $g_{\\sigma^* K}$ and $g_{\\phi K}$.\n$g_{\\omega K}$ and $g_{\\rho K}$ can be fixed\nfrom the quark counting rule: $g_{\\omega K} = g_\\omega^q$ and \n$g_{\\rho K}=g_\\rho^q$\nfor the MQMC model, and $g_{\\omega K} = g_{\\omega N}\/3$ and\n$g_{\\rho K}=g_{\\rho N}$ for QHD. \n(Obviously $g_{\\rho K}$ from the MQMC model is the same\nas that from QHD. $g_{\\omega K} (= 2.71)$ from the MQMC model is\nessentially the same as $g_{\\omega K} (=2.73)$ from QHD.)\n$g_{\\sigma^* K}$ may be fixed from $f_0(980)$ decay \\cite{wa76-91}, \nand $g_{\\phi K}$ from the SU(6) relation \n$\\sqrt{2} g_{\\phi K} = g_{\\pi \\pi \\rho} = 6.04$ \\cite{j-schaf}.  \n$g_{\\sigma^* K}$ and $g_{\\phi K}$ thus fixed are 2.65 and 4.27,\nrespectively.\nThe remaining coupling constant, $g_{\\sigma K}$, \ncan be related to the real part of the optical potential \nof a kaon at the saturation density through\n$U_{K^-} = -(g_{\\sigma K}\\sigma + g_{\\omega K}\\omega_0)$.\n$g_{\\sigma K}$ values corresponding to\nseveral values of $U_{K^-}$ are listed in Table~\\ref{tab:gsigmaK}\nfor both MQMC and QHD models.\n\nThus, out of 5 kaon-meson coupling constants\n$g_{\\rho K}$, $g_{\\sigma^* K}$, and $g_{\\phi K}$ are the same for  \nboth models. $g_{\\omega K}$'s are essentially the same for both models.\n$g_{\\sigma K}$ values are also very similar \nin both models for all $U_{K^-}$ values as seen in Table IV.  \nTherefore, all the 5 kaon-meson coupling constants are practically\nthe same for both MQMC and QHD models.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|} \\hline\n$   U_{K^-}  $ (MeV)  & $-80$ & $-100$ & $-120$ & $-140$ & $-160$ \\\\ \\hline\n$g_{\\sigma K}$ (MQMC) & 1.25  & 2.01   & 2.75   & 3.50   & 4.25   \\\\ \\hline\n$g_{\\sigma K}$ (QHD)  & 1.26  & 2.04   & 2.82   & 3.61   & 4.39   \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{$g_{\\sigma K}$ determined for several $U_{K^-}$ values\nin the MQMC and QHD models.}\n\\label{tab:gsigmaK}\n\\end{table}\n\n\\subsection{Other quantities relevant to neutron star matter}  \n\nTo obtain the composition of neutron star matter, \nwe need to determine 16 unknown variables at \neach matter density, which include 5 meson fields \n($\\sigma , \\omega , \\rho , \\sigma^* , \\phi$ ), \n8 octet baryon densities, 2 lepton densities \nand the kaon density $\\rho_K$.\nFive meson fields can be determined from their equations of motion:\n\\begin{eqnarray}\nm_\\sigma^2 \\sigma + \\frac{\\partial}{\\partial \\sigma} U^{\\rm QHD}_{\\sigma}\n= \\sum_B g_{\\sigma B} C_B(\\sigma)\n\\frac{2J_B+1}{2\\pi^2} \\int_0^{k_B} \\frac{m_B^*}{[k^2+{m_B^*}^2]^{1\/2}}k^2 dk\n+ g_{\\sigma K} \\rho_K,\n\\label{sigma}\n\\end{eqnarray}\n\\begin{eqnarray}\nm_{\\sigma^*}^2 \\sigma^* = \\sum_B g_{\\sigma^* B} C_B(\\sigma^*)\n\\frac{2J_B+1}{2\\pi^2} \\int_0^{k_B} \\frac{m_B^*}{[k^2+{m_B^*}^2]^{1\/2}}k^2 dk\n+ g_{\\sigma^* K}\\rho_K,\n\\label{sigmastar}\n\\end{eqnarray}\n\\begin{eqnarray}\nm_\\omega^2 \\omega_0 = \\sum_B g_{\\omega B} (2J_B + 1) k_B^3 \/ (6\\pi^2)\n- g_{\\omega K}\\rho_K,\n\\label{eq:omega}\n\\end{eqnarray}\n\\begin{eqnarray}\nm_\\phi^2 \\phi_0 = \\sum_B g_{\\phi B} (2J_B + 1) k_B^3 \/ (6\\pi^2)\n+ g_{\\phi K}\\rho_K,\n\\label{eq:phi}\n\\end{eqnarray}\n\\begin{eqnarray}\nm_\\rho^2 \\rho_{03} = \\sum_B g_{\\rho B} I_{3B} (2J_B + 1) k_B^3 \/ (6\\pi^2)\n-g_{\\rho K} \\frac 12 \\rho_K ,\n\\label{eq:rho}\n\\end{eqnarray}\nwhere $J_B$ and $I_{3B}$ are the spin and the isospin projection, respectively, \nand $k_B$ is the Fermi momentum of the baryon $B$. \nIn Eq.~(\\ref{sigma}) $\\frac{\\partial} {\\partial \\sigma} U^{\\rm QHD}_{\\sigma}$ \nterm needs to be there only for QHD \nand is not to be included in the MQMC model.\n$C_B(\\sigma)$ and $C_B(\\sigma^*)$ are determined from the relations\n$g_{\\sigma B} C_B(\\sigma) = - \\frac{\\partial m_B^*}{\\partial \\sigma}$\nand\n$g_{\\sigma^* B} C_B(\\sigma^*) = - \\frac{\\partial m_B^*}{\\partial \\sigma^*}$ .\nFor QHD, $C_B(\\sigma) = C_B(\\sigma^*) = 1$.  \nFor MQMC, the explicit forms of $C_B(\\sigma)$ and $C_B(\\sigma^*)$ \nare given in Ref. \\cite{s-pal}.\n\nCharge neutrality condition of neutron star matter is expressed as \n\\begin{eqnarray}\n\\sum_B q_B \\rho_B - \\rho_K - \\rho_e - \\rho_\\mu = 0,\n\\label{eq:baryonconserv}\n\\end{eqnarray}\nwhere $q_B$ is the charge of baryon $B$ and $\\rho_B$ is the number\ndensity of $B$.\nUsing the charge neutrality and the baryon number conservation conditions,\none can fix two quantities, {\\it e.g.,} the density\nof the neutron and the electron.\nWith these two variables fixed,\n$\\beta$-equilibrium conditions of the baryons give us the following 7 relations\nfor the chemical potentials of \n$p,\\, \\Lambda,\\, \\Sigma^+,\\, \\Sigma^-,\\, \\Sigma^0,\\, \\Xi^-$\nand $\\Xi^0$ as\n\\begin{eqnarray}\n\\mu_n = \\mu_\\Lambda &=& \\mu_{\\Sigma^0} = \\mu_{\\Xi^0} , \\nonumber \\\\\n\\mu_n + \\mu_e &=& \\mu_{\\Sigma^-} = \\mu_{\\Xi^-} , \\nonumber \\\\\n\\mu_n - \\mu_e &=& \\mu_p = \\mu_{\\Sigma^+},\n\\label{eq:chemi-eq}\n\\end{eqnarray}\nwhere the chemical potential of baryon $B$ is given by\n$\n\\mu_B = \\sqrt{k_B^2 + {m_B^*}^2(\\sigma,\\sigma^*)} + g_{\\omega B}\\omega_0\n+ g_{\\phi B} \\phi_0 + g_{\\rho B} I_{3B}\\rho_{03}.\n$\nThe chemical potential of a non-interacting lepton $l$ is simply written as\n$\n\\mu_l = \\sqrt{k^2_l + m^2_l}.\n$\nThe $\\beta$-equilibrium condition for leptons\n\\begin{equation}\n\\mu_e = \\mu_\\mu \n\\end{equation}\ndetermines the density of muons.\nAt a density where the condition\n\\begin{equation}\n\\omega_K = \\mu_n - \\mu_p\n\\label{eq:kaonequil}\n\\end{equation}\nis satisfied, kaons are condensed and\nthe kaon density $\\rho_K$ becomes non-zero.\nSolving the Eqs.~(\\ref{sigma}--\\ref{eq:kaonequil})\nself-consistently and simultaneously,\none can determine the 16 variables uniquely.\n\n\\section{Results}\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\epsfig{file=ppl_mqmc120_new.eps, width=6.5cm}\n\\epsfig{file=ppl_qhd120_new.eps, width=6.5cm} \\\\\n\\epsfig{file=ppl_mqmc140_new.eps, width=6.5cm}\n\\epsfig{file=ppl_qhd140_new.eps, width=6.5cm}\\\\\n\\epsfig{file=ppl_mqmc160_new.eps, width=6.5cm}\n\\epsfig{file=ppl_qhd160_new.eps, width=6.5cm}\n\\end{center}\n\\caption{Compositions of neutron star matter calculated from the\nMQMC (left panels) and the QHD (right panels) models.}\n\\label{fig:population}\n\\end{figure}\nFig.~\\ref{fig:population} shows the relative populations,\nthe ratios of the densities of octet baryons, leptons and $K^-$ to the total\nbaryon density, in the neutron star matter as functions of\n$\\rho \/ \\rho_0$ up to $\\rho=10 \\rho_0$.\nThe left panels show the results from the MQMC model and the right \nfrom the QHD model for $U_{K^-} = -120, -140,$ and $-160$ MeV.\n(Figures for $U_{K^-} = -80$ and $-100$ MeV are not shown here since\nthey are not too much different from the one for $U_{K^-} = -120$ MeV\nparticularly for QHD.)\nFigures from both models show that the onset density of the\nkaon condensation $\\rho_{\\rm crit}$ becomes lower as\n$|U_{K^-}|$ increases.\n\nTo see how $\\rho_{\\rm crit}$ changes depending on $U_{K^-}$,\nlet us consider Eq.~(\\ref{eq:kaonequil}), which determines $\\rho_{\\rm crit}$.\nFig.~\\ref{fig:kaonenergy} displays $\\omega_K$ and \n$\\mu_n - \\mu_p$, which are, respectively,\nthe left and the right\nhand sides of Eq.~(\\ref{eq:kaonequil}) \n(computed without including kaon condensation just for producing this figure).\nThe left panel is from the MQMC model, and the right panel from QHD.\nAt a density where the curve of $\\omega_K$ intersects with\nthat of $\\mu_n - \\mu_p$, kaon condensation sets in.\nAmong the coupling constants and meson fields that determine \n$\\omega_K$, only $g_{\\sigma K}$ depends on $U_{K^-}$.  \nThe $\\sigma$-meson contributes to $\\omega_K$ \nattractively, as can be seen in Eq.~(\\ref{eq:kaon-dispersion}).\nThus $\\omega_K$ becomes smaller \nfor a larger $|U_{K^-}|$, as shown in Fig.~\\ref{fig:kaonenergy}.\n\nFigure~\\ref{fig:population} also shows \nthat as $|U_{K^-}|$ increases from $120$ MeV to $140$ MeV\n$\\rho_{\\rm crit}$ changes drastically in both MQMC and QHD models, \nbut as $|U_{K^-}|$ increases further above $140$ MeV,\n$\\rho_{\\rm crit}$ changes only moderately.\nThis can also be seen from Fig.~\\ref{fig:kaonenergy}.\nAs $|U_{K^-}|$ increases from $120$ MeV to $140$ MeV\nthe intersection between the curves for $\\mu_n - \\mu_p$ \nand $\\omega_K$ moves rapidly (particularly for QHD), \nwhereas when $|U_{K^-}|$ increases further above $140$ MeV\nthe intersection \nshifts only a little to lower densities.\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=energy_kvsnp_mqmc.eps, width=7cm}\n\\epsfig{file=energy_kvsnp_qhd.eps, width=7cm}\n\\end{center}\n\\caption{Kaon energies $\\omega_K$ for $U_{K^-}$ = $-120$ MeV (dashed),\n$-140$ MeV (dotted) and $-160$ MeV (dot-dashed) \nare compared with $\\mu_n - \\mu_p$ (solid).\nAt the densities where $\\omega_K$ and $\\mu_n - \\mu_p$ intersect, \nkaons start to condense.\nThe left panel is for the MQMC model, and the right for QHD.} \n\\label{fig:kaonenergy}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=meson_mqmc140.eps, width=7.2cm}\n\\epsfig{file=meson_qhd140.eps, width=7.2cm}\n\\end{center}\n\\caption{Meson fields calculated from the MQMC (left) and the QHD (right)\nmodels as functions of the matter density for $U_{K^-} = -140$ MeV.}\n\\label{fig:meson}\n\\end{figure}\n\nAnother common feature of the two models is that\nregardless of $\\rho_{\\rm crit}$, once the kaon is created,\nthe density of $K^-$ piles up very quickly and overwhelms\nthe population of the hyperons and even the nucleons.\nThis behavior was also obtained by other authors\n\\cite{glend,kpe-prc95,bb-prc01,mpp-prc05}.\nThe reason can be partly attributed to the $\\omega$-meson. \nThe $\\omega$-meson term in the energy of $K^-$ (in Eq.~(14)) \nhas a negative sign and is thus attractive,\nbut it is repulsive for octet baryons.\nFig.~\\ref{fig:meson} shows the $\\omega$-meson\nis a dominant meson at higher densities\nin both MQMC and QHD models. \nThus the $\\omega$-meson enhances the population of $K^-$ \nbut suppresses baryons, and the kaon density increases rapidly.\nIn addition, due to the competition between\nthe negatively charged hyperons and $K^-$\nin the charge neutrality condition, the negative hyperons are highly \nsuppressed and in some cases not even created at all \nas soon as the kaon condensation sets in.\nPositively charged hyperons, on the other hand, receive the \nopposite effects from the kaon condensation, and  \n$\\Sigma^+$ is created at lower densities as $|U_{K^-}|$ increases more.\nThe proton density is also enhanced by large abundance of $K^-$,   \nwhich facilitates in turn the enhancement of $\\Sigma^+$ population\nthrough the chemical equilibrium condition\nof the positively charged hyperons in Eq.~(\\ref{eq:chemi-eq}).\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=mNKstar140.eps, width=7.2cm}\n\\epsfig{file=mYstar140.eps, width=7.2cm}\n\\end{center}\n\\caption{Effective masses of the nucleons, kaon, \nand hyperons are plotted for $U_{K^-} = - 140$ MeV.}\n\\label{fig:mNstar}\n\\end{figure}\n \nLet us now discuss different aspects of the two model calculations.\nFirst, Fig.~\\ref{fig:population} shows that $\\rho_{\\rm crit}$ from \nthe MQMC model is lower than that from QHD.\nFor $U_{K^-} = -120, -140$, and $-160$ MeV, $\\rho_{\\rm crit}$\nvalues are $5.9 \\rho_0$, $3.8 \\rho_0$ and $3.0 \\rho_0$ \nin the MQMC model, respectively, while they are\n$9.8 \\rho_0$, $4.3 \\rho_0$ and $3.3 \\rho_0$ in QHD.\nSecondly, the MQMC model predicts a larger population of the kaon\nthan QHD for a given $U_{K^-}$ value.\nFigure~\\ref{fig:kaonenergy} shows that\n$\\omega_K$ calculated from the MQMC model decreases more\nrapidly with density than $\\omega_K$ from QHD \nfor each $U_{K^-}$ value.\nThe curves for $\\mu_n - \\mu_p$ are more or less the same for both models\nat $\\rho \\lesssim 4\\rho_0$, but at $\\rho > 4\\rho_0$ \n$\\mu_n - \\mu_p$ decreases faster in QHD.\nThus the intersection and kaon condensation occur at lower densities\nin the MQMC model. \nThis behaviour of the intersection in Fig.~\\ref{fig:kaonenergy} \nis well reflected in the kaon condensation onset density \n$\\rho_{\\rm crit}$ in Fig.~\\ref{fig:population}.\nFig.~\\ref{fig:meson} shows that the \n$\\sigma$-meson field calculated by the MQMC model is larger than that\ncalculated by QHD.\nA larger $\\sigma$-field in the MQMC model makes $m^*_K$ and consequently \n$\\omega_K$ smaller.\nOn the other hand, as seen in Fig.~\\ref{fig:kaonenergy},\n$\\mu_n - \\mu_p$ from QHD decreases faster with density\nat higher densities than that from MQMC.\nThus the intersection of the $\\omega_K$ curve with the curve for $\\mu_n - \\mu_p$\noccurs at lower densities with the MQMC model.\nTherefore, $\\rho_{\\rm crit}$ is smaller in the MQMC model.\n\nAnother model dependency of the results can be seen from   \nthe population of kaons, which is larger in the MQMC model.\nThe effective mass of a kaon as a point particle is \ndetermined by $\\sigma$ and $\\sigma^*$ mesons through the relation \n$m_K^* = m_K - g_{\\sigma K}\\sigma -g_{\\sigma^* K} \\sigma^*$\nand is plotted in Fig.~\\ref{fig:mNstar}. \nSince the $\\sigma$ fields are larger in the MQMC model \n(as shown in Fig.~\\ref{fig:meson}), the effective mass and \nthe energy of a kaon are smaller in the MQMC model than QHD.\nThus, kaon condensation takes place more in the MQMC model.\n\nFigures~\\ref{fig:meson} and \\ref{fig:mNstar} also show that\neven though $\\sigma$-meson field from the MQMC model is larger  \nthan that from QHD as the densities increase, \nthe reduction of the effective mass of baryons  \nis smaller (or similar) with the MQMC model.\nIf one could parametrize the effective mass of baryons from \nthe MQMC model in the form of\n$m_B^* = m_B - g_{\\sigma B}(\\sigma) \\sigma \n-g_{\\sigma^* B}(\\sigma^*) \\sigma^*$\nwhere $g_{\\sigma B}(\\sigma)$ and $g_{\\sigma^* B}(\\sigma^*)$ are \nfunctions of $\\sigma$ and $\\sigma^*$, respectively,\nthe results in Fig.~\\ref{fig:mNstar}\nmight imply that $g_{\\sigma B}(\\sigma)$ and \n$g_{\\sigma^* B}(\\sigma^*)$ are decreasing functions \nwith respect to the density.\nThe rate of decrease is rather high since the\nproduct $(g_{\\sigma B}(\\sigma)\\, \\sigma)_{\\rm MQMC}$\nis smaller than (or similar to) $(g_{\\sigma B}\\, \\sigma)_{\\rm QHD}$ \nwhile the $\\sigma$-field value from MQMC is much greater than \nthat from QHD.\nSuch a decrease of $g_{\\sigma B}(\\sigma)$ in the MQMC model\nmay be regarded as partial restoration of the chiral symmetry\nat high densities.\n\nWe have calibrated both the MQMC and QHD model parameters to the same\nsaturation properties. However, we find that the neutron star matter composition\nprofiles from the two models are quite different and\nthat they show significant model dependence.\nQHD assumes the baryons as point particles, \nwhereas the MQMC model treats the baryons as MIT bags. \nThus, the major difference between the two models is in the \ndefinition of the  \neffective mass of baryons, $m^*_B$. \nThe equation of motion for the $\\sigma$-meson field\nis also different accordingly.\n$m^*_B$ in QHD is a simple linear function of \nthe $\\sigma$-field, and the factor $C_B(\\sigma)$ \nin Eq~(\\ref{sigma}), is a constant.\nIn the MQMC model, $m^*_B$ is a\nnon-linear function of $\\sigma$-field, and thus \n$C_B(\\sigma)$ is highly non-linear.\nWhen these non-linear $m^*_B$ and $C_B(\\sigma)$ are expanded\nin powers of the $\\sigma$-field, an infinite number of $\\sigma$-field\nterms would appear. \n(Cubic and quartic terms are explicitly taken into account\nin the QHD model as in Eq.~(\\ref{eq:U_QHD}).)\nHigher order terms can be interpreted as higher order contributions\nsuch as self interactions of meson fields,\nwhich are believed to be more important at high densities.\nBut at high densities it can be questioned \nwhether the non-linear terms of the\n$\\sigma$-meson in the MQMC model \naccount for the higher order effects properly and consistently.\nFor instance, it is generally known that as the baryons come closer\nto each other the interplay of heavy mesons becomes more important.\nAt high enough densities, their self interaction contributions may need\nto be included on the same ground as for the $\\sigma$-meson,\nbut the present MQMC model truncates the heavy\nmeson terms at the leading order. \n\nIt seems worthwhile to discuss at this point two more aspects \nof our results. The first\none is the EoS and the resulting mass-radius relation of the\nneutron star. The second point is\nthe dependence of our results on the $\\Sigma$ hyperon interaction\nin matter, which is not well known yet. \n\n\\begin{figure}[tbp]\n\\begin{center}\n\\epsfig{file=eos_comp.eps, width=7.5cm}\n\\end{center}\n\\caption{Comparison of the EoS with and without kaons in QHD model.\nThe Gibbs condition is used to treat the mixed phase.}\n\\label{fig:eos_comp}\n\\end{figure}\n\nLet us first consider the EoS and the maximum mass of the neutron star.\nAs the kaon ($K^-$) appears and condensates, \nthe number of negatively-charged hyperons decreases to satisfy the\ncharge neutrality.\nThe decrease in the number of baryons will result in the reduction\nof the pressure and lead to softnening the EoS. \nFig.~\\ref{fig:eos_comp} shows such a softening of the EoS due to the \nkaon condensation. \nIn calculating the EoS of a system consisting of multicomponent \nsubstances, as in the case with the kaon condensation, \nGibbs condition has to be employed for the proper description of the \nmixed phase.\nThe curves in Fig.~\\ref{fig:eos_comp}\nare the results obtained with the QHD and Gibbs condition. \nAs kaons appear, the EoS becomes considerably soft and the effect\nbecomes more pronounced with a stronger attraction, \ni.e., for a larger $|U_{K^-}|$ value.\nFor the MQMC model, however, applying the Gibbs conditions do not\ngive us a converging solution. Solving the 16 highly nonlinear \nequations together with Gibbs conditions double the number of equations\nto be solved, and the convergence could not be reached.\nIt is not clear to us whether the convergence problem is due to\nnumerical problems or due to non-linearity which can cause \nbifurcation or chaos. \nTherefore, we have used Maxwell construction for the MQMC model.\n(Some literatures \\cite{Maxwell} show that Maxwell construction\nis a good approximation to the Gibbs condition, but in some other \nliterature \\cite{glend} it was emphasized that Gibbs condition \nproduces significantly different results from those of Maxwell \nconstruction. Below we show that in our case the neutron star mass\nitself does not change much whether we use Maxwell or Gibbs conditions\nfor QHD. Thus our use of Maxwell construction for the MQMC model\nmay be considered as an acceptable approximation.)\nWe solve Tolman-Oppenheimer-Volkoff equation \nto calculate the maximum mass of a neutron star. \nThe results are shown in Tab.~\\ref{tab:mass-radius},\nwhere the central density, the maximum mass, and the corresponding radius\nare listed for both MQMC and QHD models.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{c|ccc|ccc|ccc} \\hline\n & \\multicolumn{3}{c|}{MQMC (Mx)} & \\multicolumn{3}{c|}{QHD (Mx)} &\n\\multicolumn{3}{c}{QHD (Gb)} \\\\\n\\hline\n$U_{K^-}$ &\n$\\rho_c\/\\rho_0$ & $M\/M_\\odot$ & $R$ &\n$\\rho_c\/\\rho_0$ & $M\/M_\\odot$ & $R$ &\n$\\rho_c\/\\rho_0$ & $M\/M_\\odot$ & $R$ \\\\\n\\hline\n$-120$ & 6.2 & 1.61 & 11.8 & 6.1 & 1.50 & 11.4\n& 6.1 & 1.50 & 11.4 \\\\ \\hline\n$-140$ & 4.6 & 1.53 & 12.8 & 5.0 & 1.46 & 12.1\n& 5.0 & 1.45 & 12.1 \\\\ \\hline\n$-160$ & 4.6 & 1.45 & 13.1 & 4.0 & 1.32 & 12.7\n& 4.3 & 1.19 & 12.3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The maximum mass of a neutron star $M$, the corresponding radius $R$\nand the density at the center of the star $\\rho_c$ are listed for\n$U_{K^-} = -120, -140,$ and $-160$ MeV. $U_{K^-}$ is in units of MeV,\nand $R$ in km. ``Mx\" and ``Gb\" refer to Maxwell and Gibbs\nconditions, respectively.}\n\\label{tab:mass-radius}\n\\end{table}\nThe maximum mass calculated with QHD model is roughly 10 \\% smaller than\nthat with MQMC model. For both models the maximum mass becomes smaller\nfor a larger $|U_K |$. \nThe maximum mass calculated with MQMC and $|U_{K^-}| = $ 160 MeV\nis compatible with observation, while\nthe maximum mass calculated with QHD and $|U_{K^-}| = $ 160 MeV \nbecomes too small to be compatible with the observed values.\nHowever, this fact may not necessarily rule out the possibility of\n$|U_{K^-}| $ becoming as large as 160 MeV \nbecause there are other possible mechanisms\nwhich are not included.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\epsfig{file=ppl_repul_Sigma_uk140_mqmc.eps, width=6.5cm}\n\\epsfig{file=ppl_repul_Sigma_uk140_qhd.eps, width=6.5cm}\n\\end{center}\n\\caption{Comparison of the population from the MQMC (left) with\nthat of the QHD (right). \nThe calculations are done with $U_\\Sigma = +30$ MeV and $U_{K^-} = -140$ MeV.}\n\\label{fig:ppl_repul_Sigma}\n\\end{figure}\n\nNow let us consider the second aspect mentioned above.\nIn the calculations made so far, we have assumed quark-counting rule \nin determining the hyperon-meson coupling constants. \nExperiments on $\\Lambda$-hypernuclei indicate that quark counting is a good\napproximation of the realistic interaction of $\\Lambda$ hyperons in nuclei,\nwhich gives the value of $\\Lambda$ optical potential at saturation \ndensity in the range $-40 \\sim -30$ MeV.\nOn the other hand, there is large ambiguity in the $\\Sigma$ hyperon \ninteraction strength. Ref.~\\cite{npa95} shows that $\\Sigma$ hyperon \nfeels repulsion rather than attraction in nuclear medium. \nThere are also experimental indications that the $\\Sigma$ hyperon\ninteraction is repulsive \\cite{saha}.\nIf the $\\Sigma$ interaction is indeed repulsive, the population profile\nof neutron star matter can change\nsignificantly from what is shown earlier in this work since\nwe have used the quark counting rule.\nThe number of $\\Sigma^-$ is closely related to the onset of\n$K^-$ condensation since they compete with each other for\nthe charge neutrality condition.\nTo see the effect of possible repulsive nature of $\\Sigma$ interaction\non the kaon condensation, we have repeated the calculations\nwith repulsive $\\Sigma$ interaction. \nWe first fit the coupling constants $g'^{\\Sigma}_\\sigma$ in MQMC and \n$g_{\\sigma \\Sigma}$ in QHD so that the $\\Sigma$ optical potential value\nat the saturation density equals to $+30$ MeV, and fix the remaining \nmeson-$\\Sigma$ coupling constants with the quark counting rule.\nThe resulting population profiles with the kaon optical potential \n$U_{K^-} = -140$ MeV are shown in Fig.~\\ref{fig:ppl_repul_Sigma}.\nCompared to the quark-counting results in Fig.~\\ref{fig:population},\nthe onset of kaon condensation occurs at slightly lower densities. \nThis minor change happens regardless of $U_{K^-}$ value. \nHowever, the main features of\nkaon condensation, i.e., its onset density, fast increase of population\nand its dominance at high densities are not much affected by the\nchange of $\\Sigma$ interaction in nuclear medium. \n\n\\section{Conclusions and discussions}\n\nUsing the modified quark-meson coupling model,\nwe have obtained the composition profiles of neutron star matter, \nfocusing on the effects of the strange particles of hyperons and kaons.\nMotivated by recent theoretical predictions of deeply bound kaonic states\n\\cite{akaishi} and the subsequent claims of the observations of \ninteresting peaks found \nin KEK\\cite{s0, suzuki1}, FINUDA\\cite{FINUDA}, and BNL\\cite{kishi-npa05}  \nexperiments, large kaon optical potential $U_{K^-}$ was considered.\nBy varying the value of $U_{K^-}$, we have investigated how the \nonset density of the kaon condensation and\nthe composition of the stellar matter change.\nEmploying the QHD model parameters which satisfy exactly \nthe same saturation conditions\nas the MQMC model, we have investigated \nthe model dependence of the results.\n\nWe observed two common features from the two model calculations.\nFirst, a larger $|U_{K^-}|$ produces a smaller onset density\nof the kaon condensation.\nThis behavior is easily understood from the relation between\n$U_{K^-}$ and $g_{\\sigma K}$ together with the role of $g_{\\sigma K}$\nto the energy of the kaon $\\omega_K$.\nSecondly, the number of kaons rapidly increases, \nand the number of negatively charged hyperons is strongly suppressed.\nThis is due to the fact that the $\\omega$-meson \ngives rise to attraction to $K^-$\nwhereas it couples to baryons repulsively.\n\nModel dependence was also observed.\nThe kaon condensation takes place at lower densities\nin the MQMC model. \nThe number of kaons is always larger with the MQMC model \nfor given $U_{K^-}$ values.\nLarger $\\sigma$-meson fields in the MQMC model can explain \nthese behaviors.\nThe differences in the results from the two models become more prominent\nat larger densities.\nGrowing discrepancies at higher densities have its\norigin partly in the effective mass of baryons in\neach model, which greatly affects the self-consistency\ncondition of the $\\sigma$-meson.\nThe factor $C_B(\\sigma)$ in the self-consistency condition\nof the $\\sigma$-meson is highly non-linear in the MQMC model,\nwhich can be interpreted as \nan infinite number of $\\sigma$-meson self-interaction terms. \nThese higher order terms may require more proper and consistent treatment \nat high densities.\n\nAn important issue in the dense matter physics is the restoration \nof the chiral symmetry.\nAccording to Ref.~\\cite{br-prl91}, not only the\nmass but also the pion decay constant and meson-nucleon coupling\nconstants decrease at a similar ratio at around the\nnuclear saturation density.\nIn Ref.~\\cite{rhhj-epja05} \nthe idea of scaling behaviour is applied to the neutron star matter\nusing MQMC and QHD models\nwith only nucleon degrees of freedom, and it is shown that \nthe equation of state becomes stiffer when \nscaling effects are considered.\nThis implies that if we include a scaling behaviour in our present models,\nit may ignite the onset of exotic states earlier than the present results\nwhich do not include a scaling.\n\nIn the kaon sector, \nthe coupling constants of a kaon and exchange mesons\nare currently an important issue.\nWe took various values of the optical potential of $K^-$ as an input\nto fix $g_{\\sigma K}$.\nOther kaon-meson coupling constants are fixed from naive quark counting.\nIt is known, however, that $K^+$ potential is repulsive \nwith the magnitude $U_{K^+} \\sim 10$ MeV \nat the saturation density \\cite{llb-prl97}.\nIf $U_{K^+}$ as well as $U_{K^-}$ is used as an input, $g_{\\sigma K}$ and\n$g_{\\omega K}$ can be determined uniquely.\nFor instance, if we take $U_{K^-} = -120$ MeV and $U_{K^+} = 20$ MeV,\nthen we get $g_{\\sigma K} = 2.041$ and $g_{\\omega K} = 4.187$.\n$g_{\\sigma K}$ becomes smaller than the value listed in Table IV, \nwhile $g_{\\omega K}$ becomes nearly twice of $g_{\\omega K}$\nfixed from the quark counting.\nBoth $\\sigma$ and $\\omega$ mesons contribute to the $K^-$ energy \nattractively, but since the $\\omega$ meson becomes a dominant \ncomponent at higher densities, taking into account of $U_{K^+}$ can produce\nappreciably different results.\nIt may be interesting to see the effects of $U_{K^+}$\non the kaon condensation.\n\nIn our calculations, we have assumed the kaon as a point particle in both \nquark and hadron models. \nComparison of the two models, however, shows that whether we treat \na hadron as a bag (MQMC) or a point particle (QHD) can produce a significant \ndifference.\nTherefore, it is worthwhile to treat the kaon as a bag and \ncompare the corresponding result with that of a point-like kaon.\nIn Ref.~\\cite{mpp-prc05} a kaon is treated as a bag in the framework\nof the QMC model, but no work has been done yet with the MQMC model.\n\nWe assume $m^*_K$ as a linear function of \n$\\sigma$-field, but \nsome authors employ a non-linear form \\cite{kpe-prc95, j-schaf}:\n\\begin{equation}\nm^{*2}_K = m^2_K - g_{\\sigma K} m_K \\sigma.\n\\end{equation}\nIf we expand this expression in powers of $\\sigma\/m_K$, we obtain\n\\[\nm^*_K \\simeq m_K \\left[\n1 - \\frac{1}{2} g_{\\sigma K} \\frac{\\sigma}{m_K} + O(\\sigma^2\/m^2_K)\n\\right].\n\\]\nThe leading order term of the $\\sigma$-field has a factor 1\/2,\nwhich is not present in Eq.~(12).\nDue to the factor 1\/2, \nthe rate of decrease in $m^*_K$ with density \nwould be reduced by a factor 2,\nand this would shift the kaon condensation onset density to higher densities.\nThis dependence on the kaon Lagrangian may be worthwhile to be studied.\n\n\\section*{Acknowledgments}\nSWH thanks B. K. Jennings and TRIUMF for hospitality during his sabbatical\nleave.\nThis work was supported by Korea Research Foundation Grant\nfunded by Korea Government (MOEHRD, Basic Research Promotion Fund)\n(KRF-2005-206-C00007).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecently, Chubanov \\cite{Chubanov2012, Chubanov2015} proposed a new polynomial-time algorithm for solving the problem (${\\rm P}(A)$), \n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}(A)& \\mbox{find}   &  x \\in \\mathbb{R}^n \\\\\n\t&\\mbox{s.t.} &  Ax = \\bm{0}, \\\\\n\t&       &  x > \\bm{0}, \n\\end{array}\n\\notag\n\\end{equation}\nwhere $A$ is a given integer (or rational) matrix and $\\mbox{rank} (A) = m$ and $\\bm{0}$ is an $n$-dimensional vector of $0$s. The method explores the feasibility of the following problem ${\\rm P}_{S_1}(A)$, which is equivalent to ${\\rm P}(A)$ and given by\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}_{S_1}(A)& \\mbox{find}   &  x \\in \\mathbb{R}^n \\\\\n\t&\\mbox{s.t.} &  Ax = \\bm{0}, \\\\\n\t&       &  \\bm{0} < x \\leq \\bm{1},\n\\end{array}\n\\notag\n\\end{equation}\nwhere $\\bm{1}$ is an $n$-dimensional vector of $1$s. Chubanov's method consists of two ingredients, the ``main algorithm'' and the ``basic procedure.'' The structure of the method is as follows: In the outer iteration, the main algorithm calls the basic procedure, which generates a sequence in $\\mathbb{R}^n$ using projection to the set $\\mbox{Ker}A$. The basic procedure terminates in a finite number of iterations returning one of the following:\n\\begin{enumerate}\n\\item a solution of problem ${\\rm P}(A)$, \n\\item a solution of the alternative problem of problem ${\\rm P}(A)$, or\n\\item a cut of ${\\rm P}(A)$, i.e., an index $j \\in \\{ 1,2,\\dots,n \\}$ for which $0 < x_j \\leq \\frac{1}{2} \\notag $ holds for any feasible solution of problem ${\\rm P}_{S_1}(A)$.\n\\end{enumerate}\nIf result 1 or 2 is returned by the basic procedure, then the feasibility of problem ${\\rm P}(A)$ can be determined and the main procedure stops. If result 3 is returned, then the main procedure generates a diagonal matrix $D \\in \\mathbb {R}^{n \\times n} $ with a $ (j, j) $ element of $2$ and all other diagonal elements of $1$ and rescales the matrix as $AD ^ {-1} $. Then, it calls the basic procedure with the rescaled matrix. Chubanov's method checks the feasibility of ${\\rm P}(A) $ by repeating the above procedures.\n\nFor problem ${\\rm P}(A)$, several variations of Chubanov's method have been proposed and computational experiments have been conducted \\cite{Li2015,Roos2018,Wei2019}. Among these, \\cite{Roos2018} proposed a tighter cut criterion of the basic procedure than the one used in \\cite{Chubanov2015}. \\cite{Chubanov2015} used the fact that \n\n\\begin{equation}\nx_j \\leq \\frac{ \\sqrt{n} \\|z\\|_2}{y_j} \\notag\n\\end{equation}\nholds for any $y \\in \\mathbb{R}^n$ satisfying $\\sum_{i=1}^n y_i = 1 , y \\geq 0$ and $y \\notin \\mbox{range} A^T $, $z \\in \\mathbb{R}^n $ obtained by projecting this $y$ onto $\\mbox{Ker} A$, and any feasible solution $x \\in \\mathbb{R}^n$ of ${\\rm P}_{S_1}(A)$, and the basic procedure is terminated if a $y$ is found for which $\\frac{ \\sqrt{n} \\|z\\|_2}{y_j} \\leq \\frac{1}{2}$ holds for some index $j$.\nOn the other hand, \\cite{Roos2018} showed that for $v = y - z$, \n\\begin{equation}\nx_j \\leq \\mbox{\\rm min} \\left( 1 , \\bm{1}^T \\left[ \\frac{-v}{v_j} \\right]^+\\right) \\leq \\frac{ \\sqrt{n} \\|z\\|_2}{y_j} \\notag\n\\end{equation}\nholds if $v_j \\neq 0$, where $\\left[ \\frac{-v}{v_j} \\right]^+$ is the projection of $\\frac{-v}{v_j} \\in \\mathbb{R}^n$ onto the nonnegative orthant and $\\bm{1}$ is the vector of ones, and the basic procedure is terminated if a $y$ is found for which $\\bm{1}^T \\left[ \\frac{-v}{v_j} \\right]^+ \\leq \\frac{1}{2}$ holds.\n\nChubanov's method has also been extended to include the feasibility problem over the second-order cone \\cite{Kitahara2018} and the symmetric cone \\cite{Pena2017,Bruno2019}. The feasibility problem over the symmetric cone is of the form, \n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}(\\mathcal{A})& \\mbox{find}   &  x  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A} (x) = \\bm{0}, \\\\\n\t&       &  x \\in \\mbox{int} \\mathcal{K}, \\notag\n\\end{array}\n\\end{equation}\nwhere $\\mathcal{A}$ is a linear operator, $\\mathcal{K}$ is a symmetric cone, and $\\mbox{int} \\mathcal{K}$ is the interior of the set $\\mathcal{K}$.\n\nAs proposed in \\cite{Pena2017,Bruno2019}, for problem ${\\rm P}(\\mathcal{A})$, the structure of Chubanov's method remains the same; i.e., the main algorithm calls the basic procedure, and the basic procedure returns one of the following in a finite number of iterations: \n\\begin{enumerate}\n\\item a solution of problem ${\\rm P}(\\mathcal{A})$, or\n\\item a solution of the alternative problem of problem ${\\rm P}(\\mathcal{A})$, or\n\\item a recommendation of scaling problem ${\\rm P}(\\mathcal{A})$.\n\\end{enumerate}\nIf result 1 or 2 is returned by the basic procedure, then the feasibility of the problem ${\\rm P}(\\mathcal{A})$ can be determined and the main procedure stops. If result (3) is returned, the problem is scaled appropriately and the basic procedure is called again. It should be noted that the purpose of rescaling differs between \\cite{Bruno2019} and \\cite{Pena2017}.\n\nIn \\cite{Pena2017}, the authors devised a rescaling method so that the following value becomes larger:\n\\begin{equation}\n\\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mathcal{K}) := \\max_x \\left \\{ {\\rm det}(x) \\mid x \\in \\mbox{Ker} \\mathcal{A} \\cap \\mathcal{K} , \\|x\\|_J^2 = r  \\right\\}, \\notag\n\\end{equation}\nwhere $\\mbox{Ker} \\mathcal{A} :=\\{ x \\mid \\mathcal{A}(x) = \\bm{0} \\}$ and $\\|x\\|_J$ is the norm induced by the inner product $\\langle x, y \\rangle = {\\rm trace}(x \\circ y)$ defined in section \\ref{sec: notation}. After showing that their algorithm terminates in $\\log_{1.5} 1 \/ \\max \\left( \\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mathcal{K}) , \\delta (\\mbox{range} \\mathcal{A}^* \\cap \\mathcal{K}) \\right)$ iterations, they proposed four updating schemes to be employed in the basic procedure (the perceptron scheme, von Neumann scheme, smooth perceptron scheme, and von Neumann with the away-step scheme) and conducted numerical experiments to compare the effect of these schemes when the symmetric cone is the nonnegative orthant \\cite{Pena2019}.\n\nIn \\cite{Bruno2019}, the authors assumed that the symmetric cone $\\mathcal{K}$ is given by the Cartesian product of $p$ simple symmetric cones $\\mathcal{K}_1, \\mathcal{K}_2, \\ldots, \\mathcal{K}_p$, and they investigated the feasibility of the problem (${\\rm P}_{S_{1,\\infty}}(\\mathcal{A})$),\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}_{S_{1,\\infty}}(\\mathcal{A})& \\mbox{find}   &  x  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A} (x) =\\bm{0}, \\\\\n\t&       &  \\|x\\|_{1 , \\infty} \\leq 1, \\\\\n\t&       &  x \\in \\mbox{int} \\mathcal{K}, \\notag\n\\end{array}\n\\end{equation}\nwhere for each $x = (x_1, x_2, \\ldots, x_p)  \\in \\mathcal{K} =\\mathcal{K}_1 \\times \\mathcal{K}_2 \\times \\cdots \\mathcal{K}_p$,  $\\|x\\|_{1,\\infty}$ is defined by \n\\begin{equation}\n\\|x\\|_{1,\\infty} := \\max \\{ \\|x_1\\|_1 , \\dots ,\\|x_p\\|_1 \\}, \\notag\n\\end{equation}\nand $\\|x\\|_1$ is the sum of the absolute values of all eigenvalues of $x$. Note that if $p=1$, then problem ${\\rm P}_{S_{1,\\infty}}(\\mathcal{A})$ turns out to be ${\\rm P}_{S_1}(\\mathcal{A})$, which is equivalent to ${\\rm P}(\\mathcal{A})$:\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}_{S_1}(\\mathcal{A})& \\mbox{find}   &  x  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A} (x) = \\bm{0}, \\\\\n\t&       &  \\|x\\|_1 \\leq 1, \\\\\n\t&       &  x \\in \\mbox{int} \\mathcal{K}. \\notag\n\\end{array}\n\\end{equation}\n\nThe authors focused on the volume of the feasible region of ${\\rm P}_{S_{1,\\infty}}(\\mathcal{A})$ and devised a rescaling method so that the volume becomes smaller.\nTheir method will stop when the feasibility of problem {${\\rm P}_{S_{1,\\infty}}(\\mathcal{A})$ or the fact that the minimum eigenvalue of any feasible solution of problem ${\\rm P}_{S_{1,\\infty}}(\\mathcal{A})$ is less than $\\varepsilon$ is determined.\nIt stops in $\\frac{r}{\\varphi(2)} \\log \\left( \\frac{1}{\\varepsilon} \\right) - \\sum_{l=1}^p \\frac{r_\\ell \\log(r_\\ell)}{\\varphi(2)}$ iterations, where $r$ is the rank of $\\mathcal{K}$, $r_\\ell$ is the rank of  $\\mathcal{K}_{\\ell} (\\ell=1,2,\\ldots,p)$, $\\varphi(\\rho) = 2 - \\frac{1}{\\rho} - \\sqrt{ 3 - \\frac{2}{\\rho}}$, and $\\varepsilon$ is a sufficiently small positive value.\n\nThe aim of this paper is to devise a new variant of Chubanov's method for solving ${\\rm P}(\\mathcal{A})$ by extending Roos's method \\cite{Roos2018} to the following feasibility problem (${\\rm P}_{S_{\\infty}}(\\mathcal{A})$) over the symmetric cone $\\mathcal{K}$:\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}_{S_{\\infty}}(\\mathcal{A})& \\mbox{find}   &  x  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A} (x) = \\bm{0}, \\\\\n\t&       &  \\|x\\|_\\infty \\leq 1, \\\\\n\t&       &  x \\in \\mbox{int} \\mathcal{K}, \\notag\n\\end{array}\n\\end{equation}\nwhere $\\|x\\|_\\infty$ is the maximum absolute eigenvalue of $x$.\nThroughout this paper, we will assume that $\\mathcal{K}$ is the Cartesian product of $p$ simple symmetric cones $\\mathcal{K}_1, \\dots, \\mathcal{K}_p$, i.e., $\\mathcal{K} = \\mathcal{K}_1 \\times \\dots \\times \\mathcal{K}_p$.\n\n\\modifySecond{Here, we should mention an important issue about Lemma 4.2 in \\cite{Roos2018}, which is one of the main results of \\cite{Roos2018}.\nThe proof of Lemma 4.2 given in the paper \\cite{Roos2018} is incorrect and a correct proof is provided in the paper \\cite{Wei2019}, while this study derives theoretical results without referring to the lemma.\n}\n\nOur method has a feature that the main algorithm works while keeping information about the minimum eigenvalue of any feasible solution of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ and, in this sense, it is closely related to Louren\\c{c}o et al.'s method \\cite{Bruno2019}. Using the norm $\\| \\cdot \\|_\\infty$ in problem ${\\rm P}_{S_\\infty} (\\mathcal{A})$ makes it possible to \n\\begin{itemize}\n\\item \ncalculate the upper bound for the minimum eigenvalue of any feasible solution of ${\\rm P}_{S_\\infty} (\\mathcal{A})$, \n\\item\nquantify the feasible region of ${\\rm P} (\\mathcal{A})$, and hence, \n\\item\ndetermine whether there exists a feasible solution of ${\\rm P} (\\mathcal{A})$ whose minimum eigenvalue is greater than $\\varepsilon$ as in \\cite{Bruno2019}.\n\\end{itemize}\n\nNote that the symmetric cone optimization includes several types of problems (linear, second-order cone, and semi-definite optimization problems) with various settings and the computational bound of an algorithm depends on these settings. As we will describe in section \\ref{sec: compare}, the theoretical computational bound of our method is \n\\begin{itemize}\n\\item equivalent to that of Roos's original method \\cite{Roos2018} and superior to that of Louren\\c{c}o et al.'s method \\cite{Bruno2019} when the symmetric cone is the nonnegative orthant,\n\\item superior to that of Louren\\c{c}o et al.'s method when the symmetric cone is a Cartesian product of second-order cones, and\n\\item equivalent to that of Louren\\c{c}o et al.'s method when the symmetric cone is the simple positive semidefinite cone, under the assumption that the costs of computing the spectral decomposition and of the minimum eigenvalue are of the same order for any given symmetric matrix.\n\\item superior to that of Pena and Soheili's method~\\cite{Pena2017} for any simple symmetric cones under the feasibility assumption of the problem imposed in~\\cite{Pena2017}.\n\\end{itemize}\n}\n\nAnother aim of this paper is to give comprehensive numerical comparisons of the existing algorithms and our method. \nAs described in section \\ref{sec: numerical experiments}, we generate the following three types of instance:\n\\begin{itemize}\n\\item \\modifyFirst{strongly feasible ill-conditioned instances, i.e.,   $\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\neq  \\emptyset$ and $x \\in \\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K}$ has positive but small eigenvalues,}\n\\item weakly feasible instances, i.e., $\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} = \\emptyset$, but $\\mbox{Ker} \\mathcal{A} \\cap \\mathcal{K} \\setminus \\{ \\bm{0} \\} \\neq  \\emptyset$, and\n\\item infeasible instances, i.e., $\\mbox{Ker} \\mathcal{A} \\cap \\mathcal{K} = \\{ \\bm{0} \\} $\n\\end{itemize}\nfor the simple positive semidefinite cone $\\mathcal{K}$,  and conduct numerical experiments. \nThe results show that our method is reliable and quite a bit faster than the existing algorithms.\nWe focus on comparing our method with Louren\\c{c}o et al.'s  in section \\ref{sec: comparisons} and show that it can reduce the search region more efficiently than Louren\\c{c}o et al.'s.\n\nThe paper is organized as follows:\nSection \\ref{sec:EJA} contains a brief description of Euclidean Jordan algebras and their basic properties.\nSection \\ref{sec: extension} gives a collection of propositions which are necessary to extend Roos's method to  problem ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ over the symmetric cone.\nIn sections \\ref{sec: basic procedure} and \\ref{sec: main algorithm}, we explain the basic procedure and the main algorithm of our variant of Chubanov's method.\n\\modifySecond{Section \\ref{sec: compare} compares the theoretical computational bounds of  Louren\\c{c}o et al.'s method~\\cite{Bruno2019}, Pena and Soheili's method~\\cite{Pena2017} and our method.}\nIn section \\ref{sec: numerical experiments}, we conduct numerical experiments comparing our variant with the existing methods.\nThen, in section \\ref{sec: comparisons}, we make more detailed comparisons of Louren\\c{c}o et al.'s method and our method in terms of the performance of the cut obtained from the basic procedure and the detection performance of an $\\varepsilon$-feasible solution. \nThe conclusions are summarized in section \\ref{sec: concluding remarks}.\n\n\n\\section{Euclidean Jordan algebras and their basic properties}\n\\label{sec:EJA}\n\\modifyThird{\nIn this section, we briefly introduce Euclidean Jordan algebras and symmetric cones. For more details, see~\\cite{Faraut1994}. In particular, the relation between symmetry cones and Euclidean Jordan algebras is given in Chapter III (Koecher and Vinberg theorem) of~\\cite{Faraut1994}.\n}\n\\subsection{Euclidean Jordan algebras}\n\\modifyThird{Let $\\mathbb{E}$ be a real-valued vector space equipped with an inner product $\\langle \\cdot, \\cdot \\rangle$ and a bilinear operation $\\circ$ : $\\mathbb{E} \\times \\mathbb{E} \\rightarrow \\mathbb{E}$,\nand $e$ be the identity element, i.e.,$x \\circ e = e \\circ x = x$ holds for any $ x \\in \\mathbb{E}$.\n}\n\n\\modifyThird{\n$(\\mathbb{E}, \\circ)$ is called a Euclidean Jordan algebra if it satisfies\n\\begin{align*}\nx \\circ y = y  \\circ x,  \\ \\\nx \\circ (x^2 \\circ y) = x^2 \\circ (x \\circ y), \\ \\\n\\langle x \\circ y , z \\rangle = \\langle y , x \\circ z \\rangle\n\\end{align*}\nfor all $x,y,z \\in \\mathbb{E}$ and $x^2 := x \\circ x$.\n}\n\nWe denote $y \\in \\mathbb{E}$ as $x^{-1}$ if  $y$ satisfies $x \\circ y = e$.\n$c \\in \\mathbb{E}$ is called an {\\em idempotent} if it satisfies $c \\circ c = c$, and an idempotent $c$ is called {\\em primitive} if it can not be written as a sum of two or more nonzero idempotents. \nA set of primitive idempotents $c_1, c_2, \\ldots c_k$ is called a {\\em Jordan frame} if $c_1, \\ldots c_k$ satisfy\n\\begin{equation}\n\\notag\nc_i \\circ c_j = 0 \\ (i \\neq j), \\ \\\nc_i \\circ c_i = c_i \\ (i= 1 , \\dots , k), \\ \\\n\\sum_{i=1}^k c_i = e.\n\\end{equation}\nFor $x \\in  \\mathbb{E}$, the {\\em degree} of $x$ is the smallest integer $d$ such that the set $\\{e,x,x^2,\\ldots,x^d\\}$ is linearly independent.\nThe {\\em rank} of $\\mathbb{E}$ is the maximum integer $r$ of the degree of $x$ over all $x \\in \\mathbb{E}$.\nThe following properties are known.\n\n\\begin{proposition}[Spectral theorem  (cf. Theorem III.1.2 of \\cite{Faraut1994})]\n\\label{prop:spectral}\nLet $(\\mathbb{E}, \\circ)$ be a Euclidean Jordan algebra having rank $r$. \nFor any $x \\in \\mathbb{E} $, there exist real numbers $\\lambda_1 , \\dots , \\lambda_r$ and a Jordan frame $c_1 , \\dots , c_r $ for which the following holds:\n\\begin{equation}\nx = \\sum_{i=1}^r \\lambda_i c_i \\notag.\n\\end{equation}\nThe numbers $\\lambda_1 , \\dots , \\lambda_r$ are uniquely determined {\\em eigenvalues} of $x$ (with their multiplicities).\nFurthermore,\n\\begin{equation}\n{\\rm trace}(x) := \\sum_{i=1}^r \\lambda_i, \\  \\ {\\rm det}(x) := \\prod_{i=1}^r \\lambda_i \\notag.\n \\notag\n\\end{equation}\n\\end{proposition}\n\n\n\n\\subsection{Symmetric cone}\n\\label{subsec:symmetric cone}\n\nA proper cone is symmetric if it is self-dual and homogeneous. \nIt is known that the set of squares\n\\begin{equation}\n\\mathcal{K} = \\{ x^2 : x \\in \\mathbb{E} \\} \\notag\n\\end{equation}\nis the symmetric cone of  $\\mathbb{E}$  (cf. Theorems III.2.1 and III.3.1 of \\cite{Faraut1994}).\n\nThe following properties can be derived from the results in \\cite{Faraut1994}, as in  Corollary 2.3 of \\cite{Yoshise2007}:\n\\begin{proposition}\n\\label{prop:lambda-Jordan}\nLet $x \\in \\mathbb{E}$ and let $\\sum_{j=1}^r \\lambda_j c_j$ be a decomposition of $x$ given by Propositoin \\ref{prop:spectral}. Then\n\\begin{description}\n\\item[(i)]\n$x \\in \\mathcal{K}$ if and only if  $\\lambda_j \\geq 0 \\ (j=1,2,\\ldots,r)$, \n\\item[(ii)]\n$x \\in {\\rm int} \\mathcal{K}$ if and only if $\\lambda_j > 0 \\ (j=1,2,\\ldots,r)$.\n\\end{description}\n\\end{proposition}\n\n\\modifyThird{\nFrom Proposition \\ref{prop:lambda-Jordan} and Proposition \\ref{prop:spectral}, for any $x \\in \\mathbb{E}$, its projection $P_\\mathcal{K} (x)$ onto the symmetry cone $\\mathcal{K}$ can be written as an operation to round all negative eigenvalues of $x$ to $0$, i.e., \n\\begin{equation}\nP_\\mathcal{K} (x) = \\sum_{i=1}^r [\\lambda_i]^+ c_i \\notag\n\\end{equation}\nwhere $[\\cdot]^+$ denotes the projection onto the nonnegative orthant. }\n\\modifyThird{\nUsing $P_\\mathcal{K}$, we can decompose any $x \\in \\mathbb{E}$ as follows.\n\\begin{lemma}\n\\label{lemma:-1}\nLet $x \\in \\mathbb{E}$, and $\\mathcal{K}$ be the symmetric cone corresponding to $\\mathbb{E}$.\nThen, $x$ can be decomposed as follows:\n\\begin{equation}\nx = P_\\mathcal{K} (x) - P_\\mathcal{K} (-x).  \\notag \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFrom Propositoin \\ref{prop:spectral}, let $x$ be given as $x = \\sum_{i=1}^r \\lambda_i c_i$. \nLet $I_1$ be the set of indices such that $\\lambda_i \\geq 0$ and $I_2$ be the set of indices such that $\\lambda_i < 0$. Then, we have\n\\begin{equation}\n\\begin{array}{ll}\nP_\\mathcal{K} (x) = \\sum_{i \\in I_1} \\lambda_i c_i,\t&P_\\mathcal{K} (-x) =  \\sum_{i \\in I_2} - \\lambda_i c_i\n\\end{array}\n\\notag\n\\end{equation}\nwhich implies that $x = \\sum_{i=1}^r \\lambda_i c_i = \\sum_{i \\in I_1} \\lambda_i c_i + \\sum_{i \\in I_2} \\lambda_i c_i = P_\\mathcal{K} (x) - P_\\mathcal{K} (-x)$.\n\\end{proof}\n}\n\nA Euclidean Jordan algebra $(\\mathbb{E} , \\circ)$ is called {\\em simple} if it cannot be written as any Cartesian product of non-zero Euclidean Jordan algebras.\nIf the Euclidean Jordan algebra $(\\mathbb{E} , \\circ)$ associated with a symmetric cone $\\mathcal{K}$ is simple, then we say that $\\mathcal{K}$ is {\\em simple}.\nIn this paper, we will consider that $\\mathcal{K}$ is given by a Cartesian product of $p$ simple symmetric cones $\\mathcal{K}_\\ell$,\n\\begin{equation}\n\\mathcal{K} := \\mathcal{K}_1 \\times \\dots \\times \\mathcal{K}_p, \\notag\n\\end{equation}\nwhose rank and identity element are $r_\\ell$ and $e_\\ell$ $(\\ell=1, 2, \\ldots, p)$.\nThe rank $r$ and the identity element of  $\\mathcal{K}$ are given by\n\\begin{equation}\n\\label{eq:r and e}\nr = \\sum_{\\ell=1}^p r_\\ell, \\ \\ e = (e_1 , \\dots , e_p).\n\\end{equation}\nIn what follows, $x_\\ell$ stands for the $\\ell$-th block element of  $x \\in \\mathcal{K}$, i.e., $x = (x_1 , \\dots . x_p) \\in \\mathcal{K}_1 \\times \\dots \\times \\mathcal{K}_p$.\nFor each $\\ell=1, 2, \\ldots, p$, we define \n\\begin{equation}\n\\notag\n\\lambda_{\\min}(x_\\ell) := \\min\\{ \\lambda_1, \\ \\lambda_2, \\ldots,  \\lambda_{r_\\ell} \\}\n\\end{equation}\nwhere $\\lambda_1, \\ \\lambda_2, \\cdots,  \\lambda_{r_\\ell}$ are eigenvalues of $x_\\ell$.\nThe minimum eigenvalue $\\lambda_{\\min}(x)$ of $x \\in \\mathcal{K}$ is given by  \n\\begin{equation}\n\\lambda_{\\min}(x) = \\mbox{min} \\{ \\lambda_{\\min}(x_1) ,  \\lambda_{\\min}(x_2) , \\dots , \\lambda_{\\min}(x_p) \\} . \\notag\n\\end{equation}\n\n\nNext, we consider the {\\em quadratic representation}  $Q_v(x)$ defined by\n\\begin{equation}\nQ_v(x) := 2 v \\circ ( v \\circ x ) - v^2 \\circ x. \\notag\n\\end{equation}\nFor the cone $\\mathcal{K} = \\mathcal{K}_1 \\times \\dots \\times \\mathcal{K}_p$, the quadratic representation $Q_v(x)$ of $x \\in \\mathcal{K}$ is denoted by $Q_v(x) = \\left(Q_{v_1} (x_1) , \\dots , Q_{v_p}(x_p) \\right)$.\nLetting $I_\\ell$ be the identity operator of the Euclidean Jordan algebra $(\\mathbb{E}_\\ell, \\circ_\\ell)$ associated with the cone $\\mathcal{K}_\\ell$, we have $Q_{e_\\ell} = I_\\ell$ for $\\ell=1, 2, \\ldots, p$.\n\n\nThe following properties can also be retrieved from the results in \\cite{Faraut1994} as in Proposition 3 of \\cite{Bruno2019}:\n\n\\begin{proposition}\n\\label{ptop:quadratic}\nFor any  $v \\in {\\rm int}\\mathcal{K}$, $ Q_v (\\mathcal{K}) = \\mathcal{K}$.\n\\end{proposition}\n\n\\modifyThird{\nIt is also known that the following relations hold for the quadratic representation $Q_v$ and $\\det (\\cdot)$ \\cite{Faraut1994}.\n\\begin{proposition}[cf. Proposition II.3.3 and III.4.2-(i), \\cite{Faraut1994}] \n\\label{q-det-relation}\nFor any $v,x \\in \\mathbb{E}$,\n\\begin{enumerate}\n\\item $\\det Q_v(x) = \\det (v)^2 \\det (x)$,\n\\item $Q_{Q_v(x)} = Q_v Q_x Q_v$ (i.e., if $x =e$ then $Q_{v^2} = Q_v Q_v$) .\n\\end{enumerate}\n\\end{proposition}\n}\n\nMore detailed descriptions, including concrete examples of symmetric cone optimization, can be found in, e.g., \\cite{Faraut1994, Faybusovich1997, Schmieta2003, Alizadeh2012}. \n\n\\modifyThird{\nHere, we will use concrete examples of symmetric cones to explain the biliniear operation $\\circ$, the identity element $e$, the inner product $\\langle \\cdot , \\cdot \\rangle$, the eigenvalues $\\lambda_i$, the primitive idempotents $c_i$, the projection $P_\\mathcal{K} (\\cdot)$ on the symmetric cone and the quadratic representation $Q_{v}(\\cdot)$ on the cone.\n}\n\n\\modifyThird{\n\\begin{example}[$\\mathcal{K}$ is the semidefinite cone $\\mathbb{S}^n_+$]\n{\\rm \nLet $\\mathbb{S}^n$ be the set of symmetric matrices of $n \\times n$.The semidefinite cone $\\mathbb{S}^n_+$ is given by \n\\begin{equation}\n\\mathbb{S}^n_+ = \\{ X \\in \\mathbb{S}^n : X \\succeq O \\} . \\notag\n\\end{equation}\nFor any symmetric matrices $X , Y \\in \\mathbb{S}^n$, define the bilinear operation $\\circ$ and inner product as follows: \n\\begin{equation}\nX \\circ Y = \\frac{ XY + YX } {2}, \\ \\\n\\ \\\n\\langle X , Y \\rangle  = \\mbox{tr}(XY) = \\sum_{i=1}^n \\sum_{j=1}^n X_{ij} Y_{ij} . \\notag\n\\end{equation}\nFor any $X \\in \\mathbb{S}^n$, perform the eigenvalue decomposition and let $u_1 , \\dots , \\lambda_n$ be the corresponding normalized eigenvectors for the eigenvalues $\\lambda_1 , \\dots , u_n$:\n\\begin{equation}\nX = \\sum_{i=1}^n \\lambda_i u_i u_i^T , \\notag\n\\end{equation}\nThe eigenvalues of $X$ in the Jordan algebra are $\\lambda_1 , \\dots , \\lambda_n$ and the primitive idempotents are $c_1 = u_1 u_1^T , \\dots , c_n = u_n u_n^T$, which implies that the rank of the semidefinite cone $\\mathbb{S}^n_+$ is $r=n$.\nThe identity element is the identity matrix $I$, and the projection $P_{\\mathbb{S}^n_+}(X)$ onto $\\mathbb{S}^n_+$ is given by $P_{\\mathbb{S}^n_+}(X) = \\sum_{i=1}^n [\\lambda_i]^+ u_i u_i^T$.\nThe quadratic representation of $V \\in \\mathbb{S}^n$ is given by $Q_V(X) = V X V$.\n}\n\\end{example}\n}\n\n\\modifyThird{\n\\begin{example}[$\\mathcal{K}$ is the second-order cone $\\mathbb{L}_n$]\n{\\rm \nThe second order cone is given by\n\\begin{equation}\n\\mathbb{L}_n = \\left \\{  \\begin{pmatrix} x_1 \\\\ \\bm{\\bar{x}} \\end{pmatrix} \\in \\mathbb{R}^n : x_1 \\geq \\| \\bm{\\bar{x}} \\|_2   \\right \\} .  \\notag\n\\end{equation}\nFor any $x , y \\in \\mathbb{R}^n$, define the bilinear operation $\\circ$ and the inner product as follows: \n\\begin{equation}\nx \\circ y = \\begin{pmatrix} x_1 \\\\ \\bm{\\bar{x}} \\end{pmatrix} \\circ \\begin{pmatrix} y_1 \\\\ \\bm{\\bar{y}} \\end{pmatrix} = \\begin{pmatrix} x^Ty \\\\ x_1\\bm{\\bar{y}}  + y_1\\bm{\\bar{x}} \\end{pmatrix}, \n\\\\ \\ \\\n\\langle x , y \\rangle = 2 \\sum_{i=1}^n x_i y_i . \\notag\n\\end{equation}\nFor any $x  \\in \\mathbb{R}^n$, by the decomposition \n\\begin{equation}\nx = \\left( x_1 + \\| \\bm{\\bar{x}}  \\|_2 \\right) \\begin{pmatrix} 1\/2 \\\\ \\frac{\\bm{\\bar{x}}}{2 \\| \\bm{\\bar{x}} \\|_2} \\end{pmatrix}  + \\left(  x_1 - \\| \\bm{\\bar{x}} \\|_2  \\right) \\begin{pmatrix} 1\/2 \\\\ -\\frac{\\bm{\\bar{x}}}{2 \\| \\bm{\\bar{x}} \\|_2} \\end{pmatrix} , \\notag\n\\end{equation}\nwe obtain the eigenvalues and the primitive idempotents as follows:\\begin{equation}\n\\lambda_1 =  x_1 + \\| \\bm{\\bar{x}}  \\|_2  \\ \\ , \\ \\ \\lambda_2 =x_1 - \\| \\bm{\\bar{x}} \\|_2  , \\notag\n\\end{equation}\n\\begin{equation}\nc_1 =\n\\begin{cases}\n\\begin{pmatrix} 1\/2 \\\\ \\frac{\\bm{\\bar{x}}}{2 \\| \\bm{\\bar{x}} \\|_2} \\end{pmatrix} & \\|\\bar{x}\\|_2 \\neq 0 \\\\\n\\\\\n\\begin{pmatrix} 1\/2 \\\\  \\frac{1}{2}z \\end{pmatrix}  & \\|\\bar{x}\\|_2 = 0 \n\\end{cases}\n \\ \\ ,\\ \\ \nc_2 = \n\\begin{cases}\n\\begin{pmatrix} 1\/2 \\\\ -\\frac{\\bm{\\bar{x}}}{2 \\| \\bm{\\bar{x}} \\|_2} \\end{pmatrix} & \\|\\bar{x}\\|_2 \\neq 0 \\\\\n\\\\\n\\begin{pmatrix} 1\/2 \\\\ -\\frac{1}{2}z \\end{pmatrix} & \\|\\bar{x}\\|_2 = 0 \n\\end{cases}\n \\ \\ .\\ \\ \n\\notag\n\\end{equation}\nwhere $z \\in \\mathbb{R}^{n-1}$ is an arbitrary vector satisfying $\\|z\\|_2=1$.\nThe above implies that the rank of the second-order cone $\\mathbb{L}_n$ is $r=2$.\nThe identity element is given by \n$e = \n\\begin{pmatrix}\n1 \\\\\n\\bm{0}\n\\end{pmatrix}\n\\in \\mathbb{R}^n\n$.\nThe projection $P_{\\mathbb{L}_n}(x)$ onto $\\mathbb{L}_n$ is given by\n\\begin{equation}\nP_{\\mathbb{L}_n}(x) = \\left[ x_1 + \\| \\bm{\\bar{x}}  \\|_2 \\right]^+ \\begin{pmatrix} 1\/2 \\\\ \\frac{\\bm{\\bar{x}}}{2 \\| \\bm{\\bar{x}} \\|_2} \\end{pmatrix}  + \\left[  x_1 - \\| \\bm{\\bar{x}} \\|_2  \\right]^+ \\begin{pmatrix} 1\/2 \\\\ -\\frac{\\bm{\\bar{x}}}{2 \\| \\bm{\\bar{x}} \\|_2} \\end{pmatrix} .\n\\notag\n\\end{equation}\nLetting $I_{n-1}$ be the unit matrix of order $n-1$, the quadratic representation $Q_v(\\cdot)$ of $v \\in \\mathbb{R}^n$ is as follows:\n\\begin{equation}\nQ_v(x) = \n\\begin{pmatrix}\n\\| v \\|_2^2 & 2 v_1 \\bm{\\bar{v}}^T \\\\\n2 v_1 \\bm{\\bar{v}} & \\mbox{det}v I_{n-1} + 2 \\bm{\\bar{v}} \\bm{\\bar{v}}^T\n\\end{pmatrix} x.\n\\notag\n\\end{equation}\n}\n\\end{example}\n}\n\n\\subsection{Notation}\n\\label{sec: notation}\n\nThis subsection summarizes the notation used in this paper.\nFor any $x,y \\in \\mathbb{E}$, we define the inner product  $\\langle \\cdot,\\cdot \\rangle$ and the norm $\\|\\cdot\\|_{J}$ as follows:\n\\begin{equation}\n\\langle x , y \\rangle := {\\rm trace}(x \\circ y) \\notag, \\ \\ \\| x \\|_J := \\sqrt{ \\langle x , x \\rangle }. \\notag\n\\end{equation}\nFor any $x \\in \\mathbb{E}$ having decomposition  $x = \\sum_{i=1}^r \\lambda_i c_i$ as in  Proposition \\ref{prop:spectral}, we also define \n\\begin{equation}\n\\notag\n\\| x \\|_1 := |\\lambda_1| + \\dots + |\\lambda_r|, \\ \\  \n\\|x\\|_\\infty := \\max \\{|\\lambda_1| , \\dots , |\\lambda_r| \\}.\n\\end{equation}\nFor $x \\in \\mathcal{K}$, we obtain the following equivalent representations:\n\\begin{equation}\n\\notag \n\\|x\\|_1 = \\langle e , x \\rangle,   \\ \\ \n\\|x\\|_\\infty = \\lambda_{\\max} (x).\n\\end{equation}\n\nThe following is a list of other definitions and frequently used symbols in the paper.\n\\begin{itemize}\n\\item $d$: the dimension of the Euclidean space $\\mathbb{E}$ corresponding to $\\mathcal{K} = \\mathcal{K}_1 \\times \\dots \\times \\mathcal{K}_p$,\n\\item $F_{{\\rm P}_{S_{\\infty}}(\\mathcal{A})}$: the feasible region of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$,\n\\item $P_{\\mathcal{A}} (\\cdot) $: the projection map onto $\\mbox{Ker} \\mathcal{A}$,\n\\item $\\mathcal{P}_\\mathcal{K} (\\cdot) $: the projection map onto $\\mathcal{K}$,\n\\item $\\lambda(x) \\in \\mathbb{R}^r$: an $r$-dimensional vector composed of the eigenvalues of $ x \\in \\mathcal{K}$,\n\\item $\\lambda(x_\\ell) \\in \\mathbb{R}^{r_\\ell}$: an $r_\\ell$-dimensional vector composed of the eigenvalues of $x_\\ell \\in \\mathcal{K}_\\ell$ ($\\ell =1,2,\\ldots,p$),\n\\item \\modifySecond{$c(x_\\ell)_i \\in \\mathcal{K}_\\ell$: the $i$-th primitive idempotent of $x_\\ell \\in \\mathbb{E}_\\ell$. When $\\mathcal{K}$ is simple, it is abbreviated as $c_i$.}\n\\item $\\left[ \\cdot \\right]^+$: the projection map onto the nonnegative orthant, and\n\\item $\\mathcal{A}^*(\\cdot)$: the adjoint operator of the linear operator $\\mathcal{A}(\\cdot)$, i.e., $\\langle \\mathcal{A}(x) , y \\rangle = \\left \\langle x , \\mathcal{A}^* (y) \\right\\rangle$ for all $x \\in \\mathcal{K}$ and $y \\in \\mathbb{R}^m$.\n\\end{itemize}\n\n\n\\section{Extension of Roos's method to the symmetric cone problem}\n\\label{sec: extension}\n\\subsection{Outline of the extended method}\nWe focus on the feasibility of the following problem ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$, which is equivalent to  ${\\rm P}(\\mathcal{A})$: \n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}_{S_{\\infty}}(\\mathcal{A})& \\mbox{find}   &  x  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A} (x) = \\bm{0}, \\\\\n\t&       &  \\|x\\|_\\infty \\leq 1, \\\\\n\t&       &  x \\in \\mbox{int} \\mathcal{K}. \\notag\n\\end{array}\n\\end{equation}\n\nThe alternative problem ${\\rm D}(\\mathcal{A})$ of ${\\rm P}(\\mathcal{A})$ is \n\\begin{equation}\n\\notag\n\\begin{array}{lll}\n{\\rm D}(\\mathcal{A})& \\mbox{find}   &  y  \\\\\n\t\t\t&\\mbox{s.t.} &  y \\in \\mbox{range} \\mathcal{A}^*, \\\\\n\t\t\t&\t& y \\in \\mathcal{K} , y \\neq \\bm{0},\n\\end{array}\n\\end{equation}\nwhere  $\\mbox{range} \\mathcal{A}^*$ is the orthogonal complement of $\\mbox{Ker} \\mathcal{A}$.\nAs we mentioned in section \\ref{subsec:symmetric cone}, \nwe assume that $\\mathcal{K}$ is given by a Cartesian product of $p$ simple symmetric cones $\\mathcal{K}_\\ell (\\ell=1,2,\\ldots,p)$, i.e., $\\mathcal{K} = \\mathcal{K}_1 \\times \\mathcal{K}_2 \\times \\dots \\times \\mathcal{K}_p $.\n\nIn our method, the upper bound for the sum of eigenvalues of a feasible solution of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ plays a key role, whereas the existing work focuses on the volume of the set of the feasible region \\cite{Bruno2019} or the condition number of a feasible solution~\\cite{Pena2017}.\n\n\\modifySecond{\nBefore describing the theoretical results, let us outline the proposed algorithm when $\\mathcal{K}$ is simple. The algorithm repeats two steps: \n\\begin{description}\n\t\\item  Step 1: find a cut for ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$, \n\t\\item  Step 2: scale the problem to an isomorphic problem equivalent to ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ such that the region narrowed by the cut is efficiently explored.\n\\end{description}\nGiven a feasible solution $x$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ and a constant $0<\\xi<1$, the proposed method first searches for a Jordan frame $\\{ c_1 , \\dots, c_r \\}$ such that the following is satisfied:\n\\begin{equation}\n\\notag\n\\langle c_i , x \\rangle  \\leq  \\xi  \\  ( i \\in H), \\ \\  \\langle c_i , x \\rangle \\leq 1 \\ ( i \\notin H),\n\\end{equation}\nwhere $H \\subseteq \\{1, \\dots,r\\}$ and $|H|>0$.\nIn this case, instead of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$, we may consider ${\\rm P}^{\\rm Cut}_{S_{\\infty}}(\\mathcal{A})$ as follows:\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}^{\\rm Cut}_{S_{\\infty}}(\\mathcal{A})& \\mbox{find}   &  x  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A} (x) = \\bm{0}, \\\\\n\t&\t&\\langle c_i, x \\rangle \\leq \\xi, \\ \\ i \\in H \\\\\n\t&\t&\\langle c_i, x \\rangle \\leq 1, \\ \\ i \\notin H \\\\\n\t&      &\\|x\\|_\\infty \\leq 1, \\\\\n\t&      &x \\in \\mbox{int} \\mathcal{K}. \\notag\n\\end{array}\n\\end{equation}\nHere, we define the set $SR^{\\rm Cut} = \\{x \\in \\mathbb{E} : x \\in \\mbox{int} \\mathcal{K}, \\ \\|x\\|_\\infty \\leq 1, \\ \\langle c_i, x \\rangle \\leq \\xi \\ (i \\in H), \\ \\langle c_i, x \\rangle \\leq 1 \\  i \\notin H) \\}$ as the search range for the solutions of the problem ${\\rm P}^{\\rm Cut}_{S_{\\infty}}(\\mathcal{A})$. }\n\n\\modifySecond{\nThe proposed method then creates a problem equivalent and isomorphic to ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ such that $SR^{\\rm Cut}$, the region narrowed by the cut, can be searched efficiently.\nSuch a problem is obtained as follows:\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}_{S_{\\infty}}(\\mathcal{A}Q_g)& \\mbox{find}   &  \\bar{x}  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A}Q_g (\\bar{x}) = \\bm{0}, \\\\\n\t&      &\\|\\bar{x}\\|_\\infty \\leq 1, \\\\\n\t&      &\\bar{x} \\in \\mbox{int} \\mathcal{K}. \\notag\n\\end{array}\n\\end{equation}\nwhere $g$ is given by $g = \\sqrt{\\xi} \\sum_{i \\in H} c_i + \\sum_{i \\notin H} c_i \\in {\\rm int} \\mathcal{K}$ for which $e = Q_{g^{-1}} (u)$ holds for $u = \\sum_{i \\in H} \\xi c_i + \\sum_{i \\notin H} c_i$.\n}\n\n\\modifySecond{\nIn the succeeding sections, we explain how the cut for ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ is obtained from some $v \\in \\mbox{range} \\mathcal{A}^*$; we also explain the scaling method for the problem in detail.} \\modifyThird{To simplify our discussion, we will assume that $\\mathcal {K}$ is simple, i.e., $p=1$, in section \\ref{subsec:simple}. Then, in section \\ref{subsec:non-simple}, we will generalize our discussion to the case of $p \\geq 2$.\n}\n\n\\subsection{Simple symmetric cone case}\n\\label{subsec:simple}\n\nLet us consider the case where $\\mathcal {K}$ is simple, i.e., $p=1$. It is obvious that, for any feasible solution $x$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$, the constraint $\\|x\\|_\\infty \\leq 1$ implies that the sum of eigenvalues has an upper bound $\\langle e , x\\rangle \\leq r$, since $x \\in \\mathcal{K}$. In Proposition \\ref{prop:upper-bound}, we show that this bound may be improved as $\\langle e , x\\rangle <r$ by using a point $v \\in \\mbox{range} \\mathcal{A}^*\\setminus \\{0\\}$. \\modifyFirst{To prove Proposition \\ref{prop:upper-bound}, we need the following Lemma \\ref{lemma:0} and Proposition \\ref{prop:p-d-relation}. }\n\n\\modifyFirst{\n\\begin{lemma}\n\\label{lemma:0}\nLet $(\\mathbb{E}, \\circ)$ be a Euclidean Jordan algebra with the corresponding symmetric cone $\\mathcal{K}$.\nFor any $y \\in \\mathbb{E}$, the following equation holds:\n\\begin{equation}\n\\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}} \\langle y ,x \\rangle  =  \\left \\langle \\mathcal{P}_\\mathcal{K} \\left( y \\right)  ,e \\right \\rangle . \\notag\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nUsing the decomposition $y = \\sum_{i=1}^r \\lambda_i c_i$ obtained by Proposition \\ref{prop:spectral}, we see that \n\\begin{equation}\n\\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}} \\langle y ,x \\rangle  = \\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}} \\left \\langle \\sum_{i=1}^r \\lambda_i c_i , x \\right \\rangle = \\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}}  \\sum_{i=1}^r \\lambda_i  \\left \\langle c_i , x \\right \\rangle . \\label{lemma:0-eq1}\n\\end{equation}\nNoting that $x \\in \\mathcal{K}, e-x \\in \\mathcal{K}$ from $\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}$, since $c_i \\in \\mathcal{K}$ is primitive idempotent, we find that \n\\begin{equation}\n\\begin{array}{ll}\n\\langle c_i,x \\rangle \\geq 0,\t&\\langle c_i, e-x \\rangle \\geq 0 \\ \\ \\mbox{equivalentlly} \\ \\ 1 \\geq \\langle c_i, x \\rangle, \n\\end{array}\n\\notag\n\\end{equation}\ni.e., $0 \\leq \\langle c_i,x \\rangle \\leq 1$ holds.\nThus, letting $I_1$ be the set of indices for which $\\lambda_i \\leq 0$ and $I_2$ be the set of indices for which $\\lambda_i > 0$, if there exists an $x$ satisfying \n\\begin{equation}\n\\langle c_i,x \\rangle =\n\\begin{cases}\n0 & i \\in I_1 \\\\\n1 & i \\in I_2\n\\end{cases}, \n\\label{lemma:0-eq2}\n\\end{equation}\nthen such an $x$ is an optimal solution of (\\ref{lemma:0-eq1}). In fact, if we define $x^* = \\sum_{i \\in I_2} c_i$, then by the dedfinition of the Jordan frame, $x^*$ satisfies (\\ref{lemma:0-eq2}) and $\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}$ and becomes an optimal solution of (\\ref{lemma:0-eq1}). In this case, the optimal value of (\\ref{lemma:0-eq1}) turns out to be\n\\begin{equation}\n\\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}}  \\sum_{i=1}^r \\lambda_i  \\left \\langle c_i , x \\right \\rangle = \\sum_{i=1}^r \\lambda_i  \\left \\langle c_i , x^* \\right \\rangle \n= \\sum_{i \\in I_2} \\lambda_i \n= \\sum_{i=1}^r [\\lambda_i]^+\n= \\langle \\mathcal{P}_\\mathcal{K} (y) , e \\rangle. \n\\notag\n\\end{equation}\n\\end{proof}\n}\n\n\\begin{proposition}\n\\label{prop:p-d-relation} \nLet $(\\mathbb{E}, \\circ)$ be a Euclidean Jordan Algebra with the corresponding symmetric cone  $\\mathcal{K}$.\nFor a given $c \\in \\mathbb{E}$, consider the problem\n\\begin{equation}\n\\notag\n\\begin{array}{lll}\n\\mbox{\\rm max}\t&  \\langle c , x \\rangle  \\\\\n\\mbox{\\rm s.t}\t&  \\mathcal{A}(x) = \\bm{0}, \\\\\n\t\t\t\t\t&  \\bm{0} \\leq \\lambda (x) \\leq \\bm{1}  \\notag.\n\\end{array}\n\\end{equation}\nThe dual problem of the above is \n\\begin{equation}\n\\notag\n\\begin{array}{lll}\n\\mbox{\\rm min}   &  \\left \\langle \\mathcal{P}_\\mathcal{K} \\left( c - u  \\right) , e \\right \\rangle  \\\\\n            \\mbox{\\rm s.t} &  u \\in \\mbox{\\rm range} \\mathcal{A}^*. \\notag\n\\end{array}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof} \\mbox{}\\\\\n\nDefine the Lagrangian function $L(x,w)$ as\n\\begin{equation}\nL(x,w) :=   \\langle c , x \\rangle -  w^\\top \\mathcal{A}(x) \\notag\n\\end{equation}\nwhere $w \\in \\mathbb{R}^m$ is the Lagrange multiplier. \nWe have \n\\begin{align*}\n\\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}} \\min_w L(x,w) &\\leq \\min_w \\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}} L(x,w)  \\\\\n&=  \\min_w \\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}} \\{  \\langle c , x \\rangle - \\langle \\mathcal{A}^*(w) ,x \\rangle \\}  \\\\\n&=  \\min_w \\max_{\\bm{0} \\leq \\lambda(x) \\leq \\bm{1}} \\{  \\langle c -  \\mathcal{A}^*(w) ,x \\rangle \\}  \\\\\n&=  \\min_w   \\left \\langle \\mathcal{P}_\\mathcal{K} \\left( c -  \\mathcal{A}^*(w) \\right)  ,e \\right \\rangle \\ \\ \\ \\modifyFirst{\\mbox{(by  lemma \\ref{lemma:0})}}  \\\\\n&= \\min_{u \\in \\mbox{\\footnotesize range} \\mathcal{A}^*}  \\langle \\mathcal{P}_\\mathcal{K} \\left( c - u \\right) , e \\rangle,\n\\end{align*}\nand the dual problem is \n\\begin{equation}\n\\notag\n\\begin{array}{lll}\n\\mbox{\\rm min}   &  \\langle \\mathcal{P}_\\mathcal{K} \\left( c - u \\right) , e \\rangle  \\\\\n            \\mbox{s.t.} &  u \\in \\mbox{range} \\mathcal{A}^*. \\notag\n\\end{array}\n\\end{equation}\n\\end{proof}\n\nThe following is a key proposition that relates to the stopping criteria of our method.\n\n\\begin{proposition}\n\\label{prop:upper-bound}\n\nSuppose that $v \\in \\mbox{\\rm range} \\mathcal{A}^*$ is given by \n\\begin{equation}\nv = \\sum_{i=1}^r \\lambda_i c_i \\notag\n\\end{equation}\nas in Proposition \\ref{prop:spectral}. \nFor each $i \\in \\{ 1, \\dots , r \\}$ and $\\alpha \\in \\mathbb{R}$, define \n\\begin{equation}\n\\notag\nq_i(\\alpha) := \\left[    1-\\alpha \\lambda_i  \\right]^+ + \\sum_{j \\neq i }^r \\left[ -  \\alpha \\lambda_j  \\right]^+.\n\\end{equation}\nThen, \n\\begin{equation}\n\\label{eq:bound_q}\n\\langle c_i , x \\rangle \\leq  \n\\min_{\\alpha \\in \\mathbb{R}} q_i(\\alpha) = \n\\begin{cases}\n\\min \\left\\{  1 ,  \\left \\langle  e , \\mathcal{P}_{\\mathcal{K}} \\left( -  \\frac{1}{\\lambda_i} v  \\right) \\right \\rangle \\right\\} & \\mbox{if $\\lambda_i \\neq 0$}, \\\\\n1 & \\mbox{if $\\lambda_i = 0$} \n\\end{cases}\n\\end{equation}\nhold for any $x \\in F_{{\\rm P}_{S_{\\infty}}(\\mathcal{A})}$ and $i \\in \\{ 1, \\dots , r \\}$. \n\\end{proposition}\n\n\n\\begin{proof}\n\nFor each $i \\in \\{ 1, 2, \\dots , r \\}$, we have \n\\begin{equation}\n\\mathcal{P}_\\mathcal{K} \\left( c_i -  \\alpha v \\right) = \\mathcal{P}_\\mathcal{K} \\left( c_i - \\alpha \\sum_{j=1}^r \\lambda_j c_j \\right) = \\mathcal{P}_\\mathcal{K} \\left( (1-\\alpha \\lambda_i ) c_i -   \\sum_{j \\neq i }^r \\alpha \\lambda_j c_j  \\right), \\notag\n\\end{equation}\nand hence, \n\\begin{equation}\n\\label{eq:q_i(alpha)}\n\\left \\langle \\mathcal{P}_\\mathcal{K} \\left( c_i -  \\alpha v \\right) , e \\right \\rangle\n=\n\\left \\langle \\mathcal{P}_\\mathcal{K} \\left(    (1-\\alpha \\lambda_i ) c_i -   \\sum_{j \\neq i }^r \\alpha \\lambda_j c_j   \\right) , \\sum_{i=1}^r c_i \\right \\rangle \n= \\left[    1-\\alpha \\lambda_i  \\right]^+ + \\sum_{j \\neq i }^r \\left[ -  \\alpha \\lambda_j  \\right]^+  = q_i(\\alpha) .\n\\end{equation}\n\nNote that, since $q_i(\\alpha)$ is a piece-wise linear convex function, if $\\lambda_i = 0$, it attains the minimum at $\\alpha = 0$ with $q_i(0) = 1$, and if $\\lambda_i \\neq 0$, it attains the minimum at $\\alpha = 0$ with $q_i(0) = 1$ or at $\\alpha =  \\frac{1}{\\lambda_i}$ with\n\\begin{equation}\n\\notag\nq \\left( \\frac{1}{\\lambda_i} \\right) = \\sum_{j \\neq i}^r \\left[ - \\frac{\\lambda_j}{\\lambda_i}  \\right]^+ \n= \\sum_{j=1}^r \\left[ - \\frac{\\lambda_j}{\\lambda_i}  \\right]^+ \n= \\left \\langle e , \\mathcal{P}_{\\mathcal{K}} \\left( -  \\frac{1}{\\lambda_i} v  \\right) \\right \\rangle.\n\\end{equation}\nThus, we obtain equivalence in (\\ref{eq:bound_q}).\n\nSince $\\alpha v \\in \\mbox{range} \\mathcal{A}^*$ for all $\\alpha \\in \\mathbb{R}$, \nfor each $i \\in \\{ 1, \\dots , r \\}$, Proposition \\ref{prop:p-d-relation} and  (\\ref{eq:q_i(alpha)})  ensure that \n\\begin{equation}\n\\langle c_i , x \\rangle \\leq  \\left \\langle \\mathcal{P}_\\mathcal{K} \\left( c_i -  \\alpha v \\right) , e \\right \\rangle = q_i(\\alpha)\n\\notag\n\\end{equation}\nfor all $\\alpha \\in \\mathbb{R}$, which implies the inequality in (\\ref{eq:bound_q}).\n\\end{proof}\n\n\n\\modifyThird{Since $\\sum_{i=1}^r c_i = e$ holds, Proposition \\ref{prop:upper-bound} allows us to compute upper bounds for the sum of eigenvalues of $x$.} The following proposition gives us information about indices whose upper bound for $\\langle c_i , x \\rangle$ in Proposition \\ref{prop:upper-bound} is less than 1.\n\n\\begin{proposition}\n\\label{prop:cut-prb}\n\nSuppose that $v \\in \\mbox{\\rm range} \\mathcal{A}^*$ is given by \n\\begin{equation}\nv = \\sum_{i=1}^r \\lambda_i c_i \\notag\n\\end{equation}\nas in Proposition \\ref{prop:spectral}.\n\\modifyThird{\nIf $v$ satisfies\n\\begin{equation}\n\\left \\langle  e , \\mathcal{P}_{\\mathcal{K}} \\left( -  \\frac{1}{\\lambda_i} v  \\right) \\right \\rangle = \\xi < 1 \\notag\n\\end{equation}\n}\nfor some $\\xi < 1$ and for some $i \\in \\{ 1, \\dots , r \\}$ for which $\\lambda_i \\neq 0$ holds, then $\\lambda_i$ has the  same sign as $\\langle e ,v \\rangle$.\n\\end{proposition}\n\n\\begin{proof}\nFirst, we consider the case where  $\\lambda_i > 0$.\nSince the assumption implies that $\\langle e , \\mathcal{P}_{\\mathcal{K}} (-v) \\rangle = \\lambda_i \\xi$,\nwe have \n\\begin{equation}\n\\notag\n\\langle e , v \\rangle = \\langle e , \\mathcal{P}_{\\mathcal{K}} (v)  \\rangle - \\langle e , \\mathcal{P}_{\\mathcal{K}} (-v)  \\rangle \\\\\n= \\langle e , \\mathcal{P}_{\\mathcal{K}} (v)  \\rangle -  \\lambda_i \\xi \\\\\n\\geq \\lambda_i (1-\\xi) > 0,\n\\end{equation}\n\\modifyThird{where the first equality comes from Lemma \\ref{lemma:-1}.}\n\nFor the case where  $\\lambda_i < 0$, \nsince the assumption also implies that $- \\langle e , \\mathcal{P}_{\\mathcal{K}} (-v) \\rangle = - \\lambda_i \\xi$, \nwe have \n\\begin{equation}\n\\langle e , v \\rangle = \\langle e , \\mathcal{P}_{\\mathcal{K}} (v)  \\rangle - \\langle e , \\mathcal{P}_{\\mathcal{K}} (-v)  \\rangle \\\\\n= - \\lambda_i \\xi   - \\langle e , \\mathcal{P}_{\\mathcal{K}} (-v)  \\rangle \\\\\n\\leq - \\lambda_i \\xi   - ( -\\lambda_i) \\\\\n= (1-\\xi) \\lambda_i < 0.\n\\notag\n\\end{equation}\nThis completes the proof.\n\\end{proof}\n\nThe above two propositions imply that,\nfor any $v \\in \\mbox{\\rm range} \\mathcal{A}^*$ with $v = \\sum_{i=1}^r \\lambda_i c_i$,\nif we compute $\\langle c_i, x \\rangle$ according to Proposition \\ref{prop:upper-bound} for $i \\in \\{ 1, \\dots , r \\}$ having the same sign as the one of $\\langle e , v \\rangle$, we obtain an upper bound for the sum of eigenvalues of $x$ over the set $F_{{\\rm P}_{S_{\\infty}}(\\mathcal{A})}$.\n\\modifyThird{The following proposition concerns the scaling method of problem ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ when we find such a $v \\in \\mbox{\\rm range} \\mathcal{A}^*$.\n }\n\\begin{proposition}\n\\label{prop:scaling}\nSuppose that a nonempty index set $H \\subseteq \\{1 , \\dots r \\}$, Jordan frame $c_1 , \\dots , c_r $, and $0 < \\xi < 1$ satisfy \n\\begin{equation}\n\\notag\n\\langle c_i , x \\rangle  \\leq  \\xi \\  ( i \\in H), \\ \\  \\langle c_i , x \\rangle \\leq 1 \\ ( i \\notin H)\n\\end{equation}\nfor any $x \\in F_{{\\rm P}_{S_{\\infty}}(\\mathcal{A})}$.\nLet us define $g \\in \\mbox{\\rm int} \\mathcal{K}$ as\n\\begin{equation}\n\\label{eq:d}\ng := \\sqrt{\\xi} \\sum_{h \\in H} c_h + \\sum_{h \\notin H} c_h \\ \\ \\ \\ \\mbox{i.e.,} \\ \\ \\ g^{-1} = \\frac{1}{\\sqrt{\\xi}} \\sum_{h \\in H} c_h + \\sum_{h \\notin H} c_h.\n\\end{equation}\n\n\\modifyThird{\nFor the two sets $SR^{\\rm Cut} = \\{x \\in \\mathbb{E} : x \\in \\mbox{{\\rm int}} \\mathcal{K}, \\ \\|x\\|_\\infty \\leq 1, \\ \\langle c_i, x \\rangle \\leq \\xi \\ (i \\in H), \\ \\langle c_i, x \\rangle \\leq 1 \\  (i \\notin H) \\}$ and, \n$SR^{\\rm Scaled} = \\{ \\bar{x} \\in \\mathbb{E} : \\bar{x} \\in \\mbox{{\\rm int}} \\mathcal{K}, \\  \\| \\bar{x} \\|_\\infty \\leq 1 \\}$, the following inclusion relation holds:\n\\begin{equation}\nQ_g( SR^{\\rm Scaled} ) \\subseteq SR^{\\rm Cut}. \\notag\n\\end{equation}\n}\n\\end{proposition}\n\\begin{proof}\n\\modifyThird{\nLet $\\bar{x}$ be an arbitrary point of $SR^{\\rm Scaled} = \\{ \\bar{x} \\in \\mathbb{E} : \\bar{x} \\in \\mbox{int} \\mathcal{K}, \\ \\| \\bar{x} \\|_\\infty \\leq 1 \\}$. It suffices to show that (i) $Q_g(\\bar{x}) \\in \\mbox{{\\rm int}} \\mathcal{K}$ and (ii) $\\| Q_g(\\bar{x}) \\|_\\infty \\leq 1$ hold and (iii) $\\langle c_i, Q_g(\\bar{x}) \\rangle \\leq \\xi \\ (i \\in H), \\ \\langle c_i, Q_g(\\bar{x}) \\rangle \\leq 1 \\ (i \\notin H)$. \\\\\n\\medskip \\\\\n\\noindent (i):\nLet us show that $Q_g(\\bar{x}) \\in \\mbox{{\\rm int}} \\mathcal{K}$.\nSince $g$ and $\\bar{x}$ lie in the set $\\mbox{{\\rm int}} \\mathcal{K}$, from Propositions \\ref{ptop:quadratic} and \\ref{q-det-relation}, we see that \n\\begin{equation}\nQ_g(\\bar{x}) \\in \\mathcal{K}, \\ \\ \\det Q_g(\\bar{x}) = \\det (g)^2 \\det (\\bar{x}) > 0, \\notag \n\\end{equation}\nwhich implies that $Q_g(\\bar{x}) \\in \\mbox{{\\rm int}} \\mathcal{K}$. \\\\\n\\medskip \\\\\n\\noindent (ii)\nNext let us show that $\\| Q_g(\\bar{x}) \\|_\\infty \\leq 1$.\nSince $\\bar{x} \\in SR^{\\rm Scaled}$, we see that $\\bar{x} \\in \\mbox{{\\rm int}} \\mathcal{K}$, $\\| \\bar{x} \\|_\\infty \\leq 1$ and hence  $e-\\bar{x} \\in \\mathcal{K}$.\nSince $g \\in \\mbox{{\\rm int}} \\mathcal{K}$, Proposition \\ref{ptop:quadratic} guarantees that \n\\begin{equation}\n\\label{eq:Q-g}\nQ_g(e-\\bar{x}) \\in \\mathcal{K}.\n\\end{equation}\nBy the definition (\\ref{eq:d}) of $g$,  the following equations hold for $c_1 , \\dots , c_r$:\n\\begin{align*}\n\\mbox{For any $i \\in H$,} \\ \\ \\ \\\nQ_g (c_i) &= 2 g \\circ ( g \\circ c_i ) - (g \\circ g ) \\circ c_i \\\\\n&= 2 g \\circ \\sqrt{\\xi} c_i - \\left( \\xi \\sum_{h \\in H} c_h + \\sum_{h \\notin H} c_h \\right) \\circ c_i \\\\\n&= 2 \\xi c_i - \\xi c_i  = \\xi c_i.\n\\end{align*}\n\\begin{align*}\n\\mbox{For any $ i \\notin H$,} \\ \\ \\ \\\nQ_g (c_i) &= 2 g \\circ ( g \\circ c_i ) - (g \\circ g ) \\circ c_i \\\\\n&= 2 g \\circ c_i - \\left( \\xi \\sum_{h \\in H} c_h + \\sum_{h \\notin H} c_h \\right) \\circ c_i \\\\\n&= 2c_i - c_i = c_i.\n\\end{align*}\nThus, we obtain $Q_g(e) = \\xi \\sum_{i \\in H} c_i + \\sum_{i \\notin H} c_i$.\nCombining this with the facts $c_i \\in \\mathcal{K}$ and $(1-\\xi)>0$  and (\\ref{eq:Q-g}), we have\n\\begin{align*}\n\\mathcal{K} \\ni  (1-\\xi) \\sum_{i \\in H} c_i + Q_g(e - \\bar{x}) \n&=  (1-\\xi) \\sum_{i \\in H} c_i + Q_g(e) -  Q_g(e)\\bar{x} \\\\\n&= (1-\\xi) \\sum_{i \\in H} c_i + \\left( \\xi \\sum_{i \\in H} c_i + \\sum_{i \\notin H} c_i \\right) -  Q_g(e)\\bar{x} \\\\\n&=  e-Q_g(\\bar{x}).\n\\end{align*}\nSince we have shown that  $Q_g(\\bar{x}) \\in \\mbox{{\\rm int}} \\mathcal{K}$, we can conclude that $\\|Q_g(\\bar{x})\\|_\\infty \\leq 1$. \\\\\n\\medskip \\\\\n\\noindent (iii): \nFinally, we compute an upper bound for the value $\\langle Q_g(\\bar{x}) , c_i \\rangle$ over the set $SR^{\\rm Scaled}$. It follows from $c_i \\in \\mathcal{K}$ and (\\ref{eq:Q-g}) that $\\langle Q_g(e - \\bar{x}) , c_i \\rangle \\geq 0$, i.e., $\\langle Q_g(e) , c_i \\rangle \\geq \\langle Q_g(\\bar{x}) , c_i \\rangle$ holds. Since we have shown that $Q_g(e) = \\xi \\sum_{i \\in H} c_i + \\sum_{i \\notin H} c_i$, this implies $\\langle Q_g(\\bar{x}) , c_i \\rangle \\leq \\xi$ holds if $i \\in H$ and $\\langle Q_g(\\bar{x}) , c_i \\rangle \\leq 1$ holds if $i \\notin H$.\n}\n\\end{proof}\n\n\\modifyThird{\nProposition \\ref{prop:scaling} $g$ was proven by focusing on the point $u$ that gives an upper bound for $\\langle e, x \\rangle$ for any feasible solution $x$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$. Specifically, before cut generation, since $\\|x\\|_\\infty \\leq 1$ holds, the point giving an upper bound for the sum of eigenvalues is $u^{\\rm before}=e$, and after the cut generation, since $\\sum_{i=1}^r c_i = e$ holds from the definition of Jordan frame, the point is $u^{\\rm after} = \\sum_{i \\in H} \\xi c_i + \\sum_{i \\notin H} c_i$. $u^{\\rm before}$ and $u^{\\rm after}$ are related by $Q_g(u^{\\rm before}) = u^{\\rm after}$. That is, Proposition \\ref{prop:scaling} implies that if a cut is obtained for ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ based on Proposition \\ref{prop:upper-bound}, we can expect a more efficient search for solutions to problem ${\\rm P}_{S_{\\infty}}(\\mathcal{A}Q_g)$\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P}_{S_{\\infty}}(\\mathcal{A}Q_g)& \\mbox{find}   &  \\bar{x}  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A}Q_g (\\bar{x}) = \\bm{0}, \\\\\n\t&      &\\|\\bar{x}\\|_\\infty \\leq 1, \\\\\n\t&      &\\bar{x} \\in \\mbox{{\\rm int}} \\mathcal{K}. \\notag\n\\end{array}\n\\end{equation}\nwith scaling of $u^{\\rm after}$ to $e$ in the variable space, rather than trying to solve problem ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$.\n}\n\\subsection{Non-simple symmetric cone case}\n\\label{subsec:non-simple}\n\nIn this section, we consider the case where the symmetric cone is not simple; i.e., it is a Cartesian product of $p$ simple symmetric cones $\\mathcal{K} = \\mathcal{K}_1 \\times \\mathcal{K}_2 \\times \\dots \\times \\mathcal{K}_p$ whose rank is given by (\\ref{eq:r and e}). \nPropositions \\ref{prop:upper-bound-direct} and  \\ref{prop:cut-prb-direct} are extensions of Proposition \\ref{prop:upper-bound} and \\ref{prop:cut-prb}, respectively.\n\n\\begin{proposition}\n\\label{prop:upper-bound-direct}\nSuppose that, for any $v \\in \\mbox{\\rm range} \\mathcal{A}^*$, the $\\ell$-th block element $v_\\ell$ of $v \\in \\mathbb{E}$ is decomposed into\n\\begin{equation}\nv_\\ell = \\sum_{i=1}^{r_\\ell} {\\lambda(v_\\ell)}_i {c(v_\\ell)}_i \\notag\n\\end{equation}\nas in Proposition \\ref{prop:spectral}.\nFor each $\\ell \\in \\{ 1,2,\\ldots,p \\}$ and $i \\in \\{1,2,\\ldots,r_p\\}$, define\n\\begin{equation}\n\\label{eq:q_ell,i(alpha)}\nq_{\\ell,i}(\\alpha) := \\left[ 1 - \\alpha {\\lambda(v_\\ell)}_i \\right]^+ + \\sum_{k \\neq i}^{r_\\ell} \\left[ -\\alpha {\\lambda(v_\\ell)}_k \\right]^+ + \\sum_{j \\neq \\ell}^p \\sum_{k=1}^{r_j} \\left[  -\\alpha {\\lambda(v_j)}_k \\right]^+ .\n\\end{equation}\nThen, \n\\begin{equation}\n\\label{eq:upper-bound-direct1}\n\\langle {c(v_\\ell)}_i , x_\\ell \\rangle \\leq  \\min_{\\alpha \\in \\mathbb{R} } q_{\\ell,i}(\\alpha) =\n\\begin{cases}\n\\min \\left\\{  1 ,  \\left \\langle  e , \\mathcal{P}_{\\mathcal{K}} \\left( -  \\frac{1}{{\\lambda(v_\\ell)}_i} v  \\right) \\right \\rangle \\right\\} & \\mbox{if ${\\lambda(v_\\ell)}_i \\neq 0$}, \\\\\n1 & \\mbox{if ${\\lambda(v_\\ell)}_i = 0$} \n\\end{cases}\n\\end{equation}\nholds for any feasible solution $x$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$, $\\ell \\in \\{ 1,2,\\ldots,p \\}$ and $i \\in \\{1,2,\\ldots,r_p\\}$.\n\\end{proposition}\n\n\\begin{proof}\nLet $c \\in \\mathbb {E}$ be an element whose $\\ell$-th block element is $c_\\ell = c (v_\\ell)_i$ and other block elements take $0$.\nFor any real number $\\alpha \\in \\mathbb{R}$, Proposition \\ref {prop:p-d-relation} ensures that\n\\begin{align}\n\\label{eq:upper-bound-direct2}\n\\langle {c(v_\\ell)}_i , x_\\ell \\rangle = \\langle c,x \\rangle &\\leq \\left \\langle \\mathcal{P}_\\mathcal{K} \\left( c -  \\alpha v \\right) , e \\right \\rangle \\notag \\\\\n&= \\left \\langle \\mathcal{P}_{\\mathcal{K}_\\ell} \\left( {c(v_\\ell)}_i -  \\alpha v_\\ell \\right) , e_\\ell \\right \\rangle + \\sum_{j \\neq \\ell}^p \\left \\langle \\mathcal{P}_{\\mathcal{K}_j} \\left( -  \\alpha v_j \\right) , e_j \\right \\rangle \\notag \\\\\n&= \\left[ 1 - \\alpha {\\lambda(v_\\ell)}_i \\right]^+ + \\sum_{k \\neq i}^{r_\\ell} \\left[ -\\alpha {\\lambda(v_\\ell)}_k \\right]^+ + \\sum_{j \\neq \\ell}^p \\sum_{k=1}^{r_j} \\left[  -\\alpha {\\lambda(v_j)}_k \\right]^+  = q_{\\ell,i}(\\alpha). \n\\end{align}\nWe obtain (\\ref{eq:upper-bound-direct1}) by following a similar argument to the one used in the proof of Proposition \\ref{prop:upper-bound}.\n\\end{proof}\n\nThe next proposition follows similarly to Proposition \\ref{prop:cut-prb}, by noting that  $\\langle e,  \\mathcal{P}_\\mathcal{K}(-v) \\rangle = {\\lambda(v_\\ell)}_i \\xi$ holds if ${\\lambda(v_\\ell)}_i >0$ and that $\\langle e,  \\mathcal{P}_\\mathcal{K}(v) \\rangle = - {\\lambda(v_\\ell)}_i \\xi$ if  ${\\lambda(v_\\ell)}_i <0$.\n\n\\begin{proposition}\n\\label{prop:cut-prb-direct}\nSuppose that, for any $v \\in \\mbox{\\rm range} \\mathcal{A}^*$, each $\\ell$-th block element $v_\\ell$ of $v$ is decomposed into\n\\begin{equation}\nv_\\ell = \\sum_{i=1}^{r_\\ell} {\\lambda(v_\\ell)}_i {c(v_\\ell)}_i \\notag\n\\end{equation}\nas in Proposition \\ref{prop:spectral}.\nIf $v$ satisfies\n\\begin{equation}\n\\begin{array}{lll}\n\\label{eq:if}\n{\\lambda(v_\\ell)}_i \\neq 0 &\\mbox{and}& \\left \\langle  e , \\mathcal{P}_{\\mathcal{K}} \\left( -  \\frac{1}{{\\lambda(v_\\ell)}_i} v  \\right) \\right \\rangle = \\xi_\\ell < 1 \n\\end{array}\n\\end{equation}\nfor some $\\xi < 1$, $\\ell \\in \\{1 , \\dots, p \\} $ and $ i \\in \\{1 , \\dots , r_\\ell \\} $, \nthen ${\\lambda(v_\\ell)}_i$ has the same sign as $\\langle e , v \\rangle$.\n\\end{proposition}\n\nFrom Proposition \\ref{prop:upper-bound-direct}, if we obtain $v \\in \\mbox{range} \\mathcal{A}^*$ satisfying  (\\ref{eq:if})  for a block $\\ell \\in \\{1 , \\dots, p \\} $ with an index $i \\in \\{1 , \\dots r_\\ell \\} $, then the upper bound for the sum of the eigenvalues of any feasible solution $x$ of  ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ is reduced by $\\langle e , x \\rangle \\leq r-1 + \\xi_\\ell < r$.\nIn this case, as described below, we can find a scaling such that the sum of eigenvalues of any feasible solution of  ${\\rm P}_{S_\\infty}(\\mathcal{A})$ is bounded by $r$.\n \nLet $H_\\ell$ be the set of indices $i$ satisfying (\\ref{eq:if}) for each block $\\ell$.\nAccording to Proposition \\ref{prop:scaling}, set $g_\\ell = \\sqrt{\\xi_\\ell} \\sum_{h \\in H_\\ell} {c(v_\\ell)}_h + \\sum_{h \\notin H_\\ell} {c(v_\\ell)}_h$ and define the linear operator $Q$ as follows:\n\\begin{equation}\nQ_\\ell := \n\\begin{cases}\nQ_{g_\\ell} & \\mbox{if } | H_\\ell | \\neq 0, \\\\\nI_\\ell &  \\mbox{otherwise},\n\\end{cases}\n\\notag\n\\end{equation}\n\\begin{equation}\nQ(\\mathbb{E}_1 , \\dots , \\mathbb{E}_p) := \\left( Q_1(\\mathbb{E}_1) , \\dots , Q_p(\\mathbb{E}_p) \\right), \\notag\n\\end{equation}\nwhere $I_\\ell$ is the identity operator of the Euclidean Jordan algebra $\\mathbb{E}_\\ell$ associated with the symmetric cone $\\mathcal{K}_\\ell$.\nFrom Proposition \\ref{prop:scaling} and its proof, we can easily see that\n\\begin{equation}\n\\label{eq:prop3.4-1}\nQ_{g_\\ell^{-1}}(c_i) = \\frac{1}{\\xi}c_i \\ (i\\in H_\\ell), \\ \\ \\ Q_{g_\\ell^{-1}}(c_i) =c_i \\ (i \\not\\in H_\\ell),\n\\end{equation}\nand the sum of eigenvalues of any feasible solution of  the scaled problem ${\\rm P}_{S_\\infty}(\\mathcal{A}Q)$ is bounded by $\\langle e,e \\rangle = r = \\sum_{\\ell=1}^{p} r_\\ell$.\n\n\n\\section{Basic procedure of the extended method}\n\\label{sec: basic procedure}\n\\subsection{Outline of the basic procedure}\n\n\nIn this section, we describe the details of our basic procedure. \nFirst, we introduce our stopping criteria and explain how to update $y^k$ when the the stopping criteria is not satisfied. Next, we show that the stopping criteria is satisfied within a finite number of iterations, i.e., finite termination of the basic procedure. Our stopping criteria is new and different from the ones used in \\cite{Bruno2019,Pena2017}, while the method of updating $y^k$ is similar to the one used in \\cite{Bruno2019} or in the von Neumann scheme of \\cite{Pena2017}.\nAlgorithm \\ref{basic procedure} is a full description of our basic procedure. \n\n\\subsection{Termination conditions of the basic procedure}\n\nFor $z^k = P_\\mathcal{A}(y^k)$, $v^k = y^k - z^k$ and a given $\\xi \\in (0,1)$, \nour basic procedure terminates when any of the following four cases occurs:\n\\begin{enumerate}\n\\item $z^k \\in \\mbox{int} \\mathcal{K} $ meaning that $z^k$ is a solution of ${\\rm P}(\\mathcal{A})$,\n\\item $z^k = \\bm{0}$ meaning that $y^k$ is feasible for ${\\rm D}(\\mathcal{A})$,\n\\item $y^k - z^k \\in \\mathcal{K}$ and $y^k - z^k \\neq \\bm{0}$ meaning that $y^k - z^k$ is feasible for ${\\rm D}(\\mathcal{A})$, or\n\\item there exist $\\ell \\in \\{ 1 , \\dots , p\\}$ and $i \\in \\{1 , \\dots , r_\\ell\\}$ for which \n\\begin{equation}\n\\begin{array}{lll}\n{\\lambda(v^k_\\ell)}_i \\neq 0 &\\mbox{and}& \\left \\langle  e , \\mathcal{P}_{\\mathcal{K}} \\left( -  \\frac{1}{{\\lambda(v^k_\\ell)}_i} v^k  \\right) \\right \\rangle = \\xi_\\ell \\leq \\xi < 1, \\label{eq:terminate}\n\\end{array}\n\\end{equation}\nmeaning that $\\langle e,x \\rangle < r$ holds for any feasible solution $x$ of ${\\rm P}_{S_\\infty}(\\mathcal{A})$ (see Proposition \\ref{prop:upper-bound-direct}).\n\\end{enumerate}\n\nCases 1 and 2 are direct extensions of the cases in \\cite{Chubanov2015}, while case 3 was proposed in \\cite{Kitahara2018,Bruno2019}.\nCase 3 helps us to determine the feasibility of ${\\rm P}(\\mathcal{A})$ efficiently,  while we have to decompose $y^k-z^k$ for checking it.  \n\nIf the basic procedure ends with case 1, 2, or 3, the feasibility of ${\\rm P}(\\mathcal{A})$ can be determined, and the basic procedure returns a solution of ${\\rm P}(\\mathcal{A})$ or ${\\rm D}(\\mathcal{A})$ to the main algorithm. \nIf the basic procedure ends with case 4, the basic procedure returns to the main algorithm $p$ index sets $H_1 , \\dots , H_p$ each of which consists of indices $i$ satisfying (\\ref{eq:terminate}) and the set of primitive idempotents $C_\\ell = \\{ {c(v^k_\\ell)}_1, \\dots ,{c(v^k_\\ell)}_{r_\\ell} \\}$ of $v^k_\\ell$ for each $\\ell$.\n\n\\subsection{Update of the basic procedure}\n\n\nThe basic procedure updates $y^k \\in \\mbox{int} \\mathcal{K}$ with $\\langle y^k  , e \\rangle = 1$ so as to reduce the value of $\\| z^k \\|_J$. \nThe following proposition is essentially the same as Proposition 13 in \\cite{Bruno2019}, so we will omit its proof.\n\n\\begin{proposition}[cf. Proposition 13, \\cite{Bruno2019}] \n\\label{prop:update}\nFor $y^k \\in \\mbox{\\rm int} \\mathcal{K}$ satisfying $\\langle y^k  , e \\rangle = 1$, let $z^k = P_\\mathcal{A}(y^k)$.\nIf  $z^k \\notin \\mbox{\\rm int} \\mathcal{K} $ and $z^k \\neq \\bm{0}$, then the following hold.\n\\begin{enumerate}\n\\item \nThere exists $ c \\in \\mathcal{K}$ such that\n\\begin{equation}\n\\langle c , z^k \\rangle =  \\lambda_{\\min} (z^k) \\leq 0, \\ \\langle e , c \\rangle \n=1 \\ \\mbox{and} \\ c\\in \\mathcal{K} . \\label{eq:update_bp1}\n\\end{equation}\n\\item \nFor the above $c$, suppose that $P_\\mathcal{A}(c) \\neq \\bm{0}$ and define\n\\begin{equation} \n\\label{eq:bp2}\n\\alpha= \\frac{ \\langle P_\\mathcal{A}(c) , P_\\mathcal{A}(c) - z^k \\rangle }{\\|z^k-P_\\mathcal{A}(c) \\|^2_J}. \n\\end{equation}\nThen, $y^{k+1} := \\alpha y^k + (1-\\alpha) c$ satisfies \n\\begin{enumerate}\n\\item $y^{k+1} \\in \\mbox{\\rm int} \\mathcal{K}$, \n\\item $\\|y^{k+1}\\|_{1,\\infty} \\geq \\frac{1}{p}$, \n\\item $\\langle y^{k+1} , e \\rangle = 1$, and\n\\item $z^{k+1} := P_\\mathcal{A}(y^{k+1})$ satisfies \n\\begin{equation}\n\\frac{1}{\\| z^{k+1} \\|^2_J} \\geq \\frac{1}{\\|z^k\\|^2_J} + 1.  \\notag\n\\end{equation}\n\\end{enumerate}\n\\end{enumerate}\n\\end{proposition}\n\nA method of accelerating the update of $y^k$ is provided in \\cite{Roos2018}. \nFor $\\ell \\in \\{1,2,\\ldots,p\\}$, let $I_\\ell := \\{ i \\in \\{1,2,\\ldots,r_\\ell\\} \\mid \\lambda_i (z^k_\\ell) \\leq 0\\} $ and set $N = \\sum_{\\ell=1}^p | I_\\ell | $. \nDefine the $\\ell$-th block element of $c \\in \\mathcal{K}$ as\n\\begin{equation}\nc_\\ell = \\cfrac{1}{N} \\sum_{i \\in I_\\ell} {c(z^k_\\ell)}_i . \\notag\n\\end{equation}\nUsing $P_\\mathcal{A} \\left(  c \\right) $, the acceleration method computes $\\alpha$ by (\\ref{eq:bp2}) so as to minimize the norm of $z^{k+1}$ and update $y$ by\n\\begin{equation}\ny^{k+1} = \\alpha y^k +   (1-\\alpha)  c. \\notag\n\\end{equation}\nWe incorporate this method in the basic procedure of our computational experiment and call it the {\\it modified basic procedure}. \n\n\\modifySecond{\nAs described in~\\cite{Pena2017}, we can also use the smooth perceptron scheme~\\cite{Soheili2012, Soheili2013} to update $y^k$ in the basic procedure.\nAs explained in the next section, using the smooth perceptron scheme significantly reduces the maximum number of iterations of the basic procedure.\n}\n\nA detailed description of our basic procedure (Algorithms \\ref{bp-alg-mvn} and  \\ref{bp-alg-sp}) is given in Appendix \\ref{app:modified basic procedure}.\n\n\\subsection{Finite termination of the basic procedure}\n\\label{sec: finite termination of BP}\n\nIn this section, we show that the basic procedure (Proposition \\ref{prop:bp2}) terminates in a finite number of iterations.\nTo do so, we need to prove Lemma \\ref{lemma:1} and Proposition \\ref{prop:bp1}.\n\n\\begin{lemma}\n\\label{lemma:1}\nLet $(\\mathbb{E}, \\circ)$ be a Euclidean Jordan algebra with the corresponding symmetric cone $\\mathcal{K}$\ngiven by the Cartesian product of $p$ simple symmetric cones, i.e., $\\mathcal{K} = \\mathcal{K}_1 \\times \\dots \\times \\mathcal{K}_p$.\nFor any $x \\in \\mathbb{E}$ and $y \\in \\mathcal{K}$, the following inequality holds:\n\\begin{equation}\n[ \\langle x , y \\rangle ]^+ \\leq \\langle \\mathcal{P}_\\mathcal{K} (x)  , y \\rangle. \\notag\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nLet $x \\in \\mathbb{E}$ and suppose that each $\\ell$-th block element $x_\\ell$ of $x$ is given by \n\\begin{equation}\nx_\\ell = \\sum_{i=1}^{r_\\ell} {\\lambda(x_\\ell)}_i {c(x_\\ell)}_i \\notag\n\\end{equation}\nas in Proposition \\ref{prop:spectral}.\nThen, we can see that \n\\begin{align}\n\\left[  \\langle x , y \\rangle  \\right]^+ &=  \\left[   \\sum_{\\ell=1}^p \\left \\langle  \\sum_{i=1}^{r_\\ell} {\\lambda(x_\\ell)}_i {c(x_\\ell)}_i , y_\\ell \\right \\rangle  \\right]^+ \\notag \\\\\n&=  \\left[  \\sum_{\\ell=1}^p \\left( \\sum_{i=1}^{r_\\ell} {\\lambda(x_\\ell)}_i \\left \\langle {c(x_\\ell)}_i , y_\\ell \\right \\rangle  \\right)  \\right]^+ \\notag \\\\\n&\\leq \\sum_{\\ell=1}^p \\sum_{i=1}^{r_\\ell} \\left[   {\\lambda(x_\\ell)}_i \\left \\langle {c(x_\\ell)}_i , y_\\ell \\right \\rangle  \\right]^+ \\notag \\\\\n&= \\sum_{\\ell=1}^p \\sum_{i=1}^{r_\\ell} \\left[  { \\lambda(x_\\ell)}_i \\right]^+ \\langle {c(x_\\ell)}_i , y_\\ell \\rangle   \\notag \\\\\n&= \\sum_{\\ell=1}^p \\left \\langle \\sum_{i=1}^{r_\\ell} \\left[  { \\lambda(x_\\ell)}_i \\right]^+  {c(x_\\ell)}_i , y_\\ell \\right \\rangle \n= \\left \\langle \\mathcal{P}_\\mathcal{K} (x)  , y \\right \\rangle . \\notag\n\\end{align}\nwhere the inequality follows from the fact that ${c(x_\\ell)}_1 , \\dots , {c(x_\\ell)}_{r_\\ell}$, and $y_\\ell$ lie in the symmetric cone $\\mathcal{K}_\\ell$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop:bp1}\nFor a given $y \\in \\mathcal{K}$, define $z = P_\\mathcal{A} (y)$ and $v = y - z$.\nSuppose that $v \\neq 0$ and each $\\ell$-th element $v_\\ell$ is given by $v_\\ell = \\sum_{i=1}^{r_\\ell} {\\lambda(v_\\ell)}_i {c(v_\\ell)}_i$, as in Proposition \\ref{prop:spectral}.\nThen, for any $x \\in F_{{\\rm P}_{S_{\\infty}}(\\mathcal{A})}$, $\\ell \\in \\{ 1, \\dots , p \\}$ and $ i \\in \\{ 1, \\dots , r_\\ell \\}$, \n\\begin{equation}\n\\label{eq:bp1-1}\n\\langle {c(v_\\ell)}_i , x_\\ell \\rangle \\leq \\min_\\alpha q_{\\ell,i}(\\alpha) \\leq \\frac{1}{ \\left \\langle y_\\ell , {c(v_\\ell)}_i \\right \\rangle} \\|z\\|_J \n\\end{equation}\nhold where $q_{\\ell,i}(\\alpha)$ is defined in (\\ref{eq:q_ell,i(alpha)}).\n\\end{proposition}\n\n\\begin{proof}\nThe first inequality of (\\ref{eq:bp1-1}) follows from (\\ref{eq:upper-bound-direct2}) in the proof of Proposition \\ref{prop:upper-bound-direct}. \nThe second inequality is obtained by evaluating  $q_{\\ell, i}(\\alpha)$ at $\\alpha = \\cfrac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle}$, as follows:\n\\begin{align*}\nq_{\\ell,i} \\left(  \\cfrac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\right) \n&= \\left[ 1 - \\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} {\\lambda(v_\\ell)}_i \\right]^+ + \\sum_{k \\neq i}^{r_\\ell} \\left[ - \\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} {\\lambda(v_\\ell)}_k \\right]^+ + \\sum_{j \\neq \\ell}^p \\sum_{k=1}^{r_j} \\left[  -\\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} {\\lambda(v_j)}_k \\right]^+\\\\\n&= \\left[ 1 - \\frac{\\langle y_\\ell -z_\\ell , {c(v_\\ell)}_i \\rangle}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle}    \\right]^+ + \\sum_{k \\neq i}^{r_\\ell} \\left[ - \\frac{\\langle y_\\ell -z_\\ell , c(v_\\ell)_k \\rangle}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle}   \\right]^+ + \\sum_{j \\neq \\ell}^p \\sum_{k=1}^{r_j} \\left[  -\\frac{\\langle y_j - z_j , {c(v_j)}_k \\rangle}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\right]^+ \\\\\n& \\ \\ \\ \\  \\modifySecond{(\\mbox{since} \\ \\lambda(v_\\ell)_i = \\langle v_\\ell, c(v_\\ell)_i \\rangle \\ \\mbox{and} \\ v_\\ell = y_\\ell -z_\\ell )} \\\\\n&= \\left[ \\frac{\\langle z_\\ell , {c(v_\\ell)}_i \\rangle }{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\right]^+ + \\sum_{k \\neq i}^{r_\\ell}  \\left[  \\frac{ \\langle z_\\ell , {c(v_\\ell)}_k \\rangle - \\langle y_\\ell , {c(v_\\ell)}_k \\rangle}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle}  \\right]^+ + \\sum_{j \\neq \\ell}^p \\sum_{k=1}^{r_j} \\left[  \\frac{\\langle z_j , {c(v_j)}_k \\rangle - \\langle y_j , {c(v_j)}_k \\rangle}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\right]^+ \\\\\n&\\leq \\left[ \\frac{\\langle z_\\ell , {c(v_\\ell)}_i \\rangle }{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\right]^+  + \\sum_{k \\neq i}^{r_\\ell} \\left[  \\frac{\\langle z_\\ell , {c(v_\\ell)}_k \\rangle}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle}  \\right]^+ + \\sum_{j \\neq \\ell}^p \\sum_{k=1}^{r_j} \\left[ \\frac{\\langle z_j , {c(v_j)}_k \\rangle}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\right]^+ \\\\\n& \\ \\ \\ \\  \\modifySecond{(\\mbox{since} \\ y_\\ell, c(v_\\ell)_i \\in \\mathcal{K}_\\ell \\ \\mbox{and then} \\ \\langle y_\\ell, c(v_\\ell)_i \\rangle \\geq 0)} \\\\\n&= \\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\left( \\sum_{k=1}^{r_\\ell} \\left[  \\langle z_\\ell , {c(v_\\ell)}_k \\rangle  \\right]^+ + \\sum_{j \\neq \\ell}^p \\sum_{k=1}^{r_j} \\left[ \\langle z_j , {c(v_j)}_k \\rangle \\right]^+ \\right) \\\\\n&\\leq \\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\left(  \\sum_{k=1}^{r_\\ell}   \\langle  \\mathcal{P}_{{\\mathcal{K}_\\ell}}\\left( z_\\ell \\right), {c(v_\\ell)}_k \\rangle + \\sum_{j \\neq \\ell}^p \\sum_{k=1}^{r_j} \\langle  \\mathcal{P}_{{\\mathcal{K}_j}}\\left( z_j \\right), {c(v_j)}_k \\rangle \\right) \\ \\ \\mbox{(by Lemma \\ref{lemma:1})}  \\\\\n&= \\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle}  \\left(  \\langle  \\mathcal{P}_{\\mathcal{K}_\\ell}\\left( z_\\ell \\right), e_\\ell \\rangle  + \\sum_{j \\neq \\ell}^p \\langle  \\mathcal{P}_{\\mathcal{K}_j}\\left( z_j \\right), e_j \\rangle \\right) \\\\\n&= \\frac{\\langle  \\mathcal{P}_\\mathcal{K}\\left( z \\right), e \\rangle  }{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle}\n= \\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\| \\mathcal{P}_{\\mathcal{K}}\\left( z \\right) \\|_1 \n\\leq \\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\| \\mathcal{P}_{\\mathcal{K}}\\left( z \\right) \\|_J \n\\leq \\frac{1}{ \\langle y_\\ell , {c(v_\\ell)}_i \\rangle} \\|  z  \\|_J .\n\\end{align*}\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop:bp2}\nLet $r_{\\max} = \\max \\{ r_1 , \\dots , r_p\\}$.\nThe basic procedure (Algorithm \\ref{basic procedure}) terminates in at most $\\frac{p^2r_{\\max}^2}{\\xi^2} $ iterations. \n\\end{proposition}\n\\begin{proof}\nSuppose that $y^k$ is obtained at the $k$-th iteration of Algorithm \\ref{basic procedure}. Proposition \\ref{prop:update} implies that $\\|y^k\\|_{1,\\infty} \\geq \\frac{1}{p}$ and an $\\ell$-th block element exists for which $\\langle y_\\ell , e_\\ell \\rangle \\geq \\frac{1}{p}$ holds. \nThus, by letting $v^k = y^k - z^k$ and the $\\ell$-th block element $v_\\ell^k$ of $v^k$ be $v_\\ell^k = \\sum_{i=1}^{r_\\ell} {\\lambda(v^k_\\ell)}_i {c(v^k_\\ell)}_i$ as in Proposition \\ref{prop:spectral}, we have\n\\begin{equation}\n\\max_{i=1,\\dots,r_\\ell} \\left \\langle y_\\ell^k , {c(v^k_\\ell)}_i \\right \\rangle \\geq \\frac{1}{pr_\\ell}. \\label{eq:bp2-1}\n\\end{equation}\n\nSince Proposition \\ref{prop:update} ensures that $\\frac{1}{\\|z^k\\|^2_J}  \\geq k$ holds at the $k$-th iteration,\nby setting $k=\\frac{p^2r_{\\max}^2}{\\xi^2}$, we see that \n\\begin{equation}\n\\xi \\geq p r_{\\max}  \\|z^k\\|_J , \\notag\n\\end{equation}\nand combining this with (\\ref{eq:bp2-1}), we have\n\\begin{equation}\n\\notag\n\\xi \\geq p r_{\\max}  \\|z^k\\|_J \\geq p r_\\ell  \\|z^k\\|_J \\geq \\frac{1}{\\max_{i=1,\\dots,r_\\ell} \\left \\langle y_\\ell^k , {c(v_\\ell)}_i \\right \\rangle} \\| z^k \\|_J . \n\\end{equation}\nThe above inequality and Proposition \\ref{prop:bp1} imply that for any $\\ell \\in \\{ 1,2,\\ldots,p \\}$ and $i \\in \\{1,2,\\ldots,r_p\\}$,\n\\begin{equation}\n\\notag\n\\langle c(v^k_\\ell)_i , x_\\ell \\rangle \\leq  \\min_\\alpha q_{\\ell,i} (\\alpha)\n\\leq \\frac{1}{ \\langle y^k_\\ell , {c(v^k_\\ell)}_i \\rangle} \\|z^k\\|_J \\leq \\xi. \n\\end{equation}\nFrom the equivalence in (\\ref{eq:upper-bound-direct1}) and the setting $\\xi \\in (0,1)$, we conclude that Algorithm \\ref{basic procedure} terminates in at most $\\frac{p^2r_{\\max}^2}{\\xi^2} $ iterations by satisfying  (\\ref{eq:terminate}) in the fourth termination condition at an $\\ell$-th block and an index $i$.\n\\end{proof}\n\n\\modifySecond{\nAn upper bound for the number of iterations of Algorithm \\ref{bp-alg-sp} using smooth perceptoron scheme can be found as follows.\n\\begin{proposition}\nLet $r_{\\max} = \\max \\{ r_1 , \\dots , r_p\\}$.\nThe basic procedure (Algorithm \\ref{bp-alg-sp}) terminates in at most $\\frac{2 \\sqrt{2} p r_{\\max}}{\\xi}$ iterations.\n\\end{proposition}\n\\begin{proof}\nFrom Proposition 6 in~\\cite{Pena2017}, after $k \\geq 1$ iterations,  we obtain the inequality $\\|z^k\\|_J^2 \\leq \\frac{8}{(k+1)^2}$.\nSimilarly to the previous proof of Proposition \\ref{prop:bp2}, if $\\xi \\geq p r_{max} \\|z^k\\|_J$ holds, then Algorithm \\ref{bp-alg-sp} terminates. \nThus, $k \\leq \\frac{2 \\sqrt{2} p r_{\\max}}{\\xi}$ holds for a given $k$ satisfying\n\\begin{equation}\n\\left( \\frac{\\xi}{p r_{max}} \\right)^2 \\leq \\frac{8}{(k+1)^2}.\n\\notag\n\\end{equation}\n\\end{proof}\n}\n\nHere, we discuss the computational cost per iteration of Algorithm  \\ref{basic procedure}.\nAt each iteration, the two most expensive operations are computing the spectral decomposition on line 5 and computing $P_{\\mathcal{A}}(\\cdot)$ on lines  24 and  26.\n\nLet $C_{\\ell}^{\\rm sd}$ be the computational cost of the spectral decomposition of an element of $\\mathcal{K}_\\ell$.\nFor example, $C_{\\ell}^{\\rm sd}=\\mathcal{O}(r_\\ell^3)$ if $\\mathcal{K}_\\ell = \\mathbb{S}^{r_\\ell}_+$ and $C_{\\ell}^{\\rm sd}=\\mathcal{O}(r_\\ell)$ if $\\mathcal{K}_\\ell = \\mathbb{L}_{r_\\ell}$, where $\\mathbb{L}_{r_\\ell}$ denotes the $r_\\ell$-dimensional second-order cone. \nThen, the cost $C^{\\rm sd}$ of computing the spectral decomposition of an element of $\\mathcal{K}$ is $C^{\\rm sd} = \\sum_{\\ell=1}^p C_{\\ell}^{\\rm sd}$. Next, let us consider the computational cost of $P_{\\mathcal{A}}(\\cdot)$. \nRecall that $d$ is the dimension of the Euclidean space $\\mathbb{E}$ corresponding to $\\mathcal{K}$. \nAs discussed in \\cite{Bruno2019}, we can compute $P_{\\mathcal{A}} = I - \\mathcal{A}^*(\\mathcal{A}\\mathcal{A}^*)^{-1}\\mathcal{A}$ by using the Cholesky decomposition of  $(\\mathcal{A}\\mathcal{A}^*)^{-1}$.\nSuppose that $(\\mathcal{A}\\mathcal{A}^*)^{-1} = LL^*$, where $L$ is an $m \\times m$ matrix and we store $L^*\\mathcal{A}$ in the main algorithm.\nThen, we can compute $P_{\\mathcal{A}}(\\cdot)$ on lines 24 and 26, which costs $\\mathcal{O}(md)$.\n\\modifySecond{\nThe operation $u_\\mu (\\cdot) : \\mathbb{E} \\rightarrow \\{ u \\in \\mathcal{K} \\mid \\langle u,e \\rangle = 1\\}$ in Algorithm \\ref{bp-alg-sp} can be performed within the cost $C^{\\rm sd}$~\\cite{Soheili2013, Pena2017}.\n}\nFrom the above discussion and Proposition \\ref{prop:bp2}, the total costs of Algorithm \\ref{basic procedure} \\modifySecond{and Algorithm \\ref{bp-alg-sp}} are given by \n\\modifySecond{\n\\begin{equation}\n\\mathcal{O} \\left( \\frac{p^2 r_{\\max}^2}{\\xi^2}  \\max ( C^{\\rm sd} , md ) \\right), \\label{cost:b-p}\n\\end{equation}\n\\begin{equation}\n\\mathcal{O} \\left( \\frac{p r_{\\max}}{\\xi}  \\max ( C^{\\rm sd} , md ) \\right). \\label{cost:b-p-smooth}\n\\end{equation}\n}\n\n\\begin{algorithm}[H]\n \\caption{Basic procedure (von Neumann scheme)}\n \\label{basic procedure}\n \\begin{algorithmic}[1]\n \\renewcommand{\\algorithmicrequire}{\\textbf{Input: }}\n \\renewcommand{\\algorithmicensure}{\\textbf{Output: }}\n \\renewcommand{\\stop}{\\textbf{stop }}\n \\renewcommand{\\return}{\\textbf{return }}\n\n \\STATE \\algorithmicrequire $P_\\mathcal{A}$, $y^1 \\in \\mbox{int} \\mathcal{K}$ such that $\\langle y^1 , e \\rangle = 1$ and a constant $\\xi$ such that $0 < \\xi < 1$\n \\STATE \\algorithmicensure \\modifyThird{(i) a solution to ${\\rm P}(\\mathcal{A})$ or (ii) ${\\rm D}(\\mathcal{A})$ or (iii) a certificate that, for any feasible solution $x$ to ${\\rm P}_{S_\\infty}(\\mathcal{A})$, $\\langle e,x \\rangle < r$}\n \\STATE initialization: $k \\leftarrow 1, z^1 \\leftarrow P_\\mathcal{A}(y^1), v^1 \\leftarrow y^1 - z^1, H_1 , \\dots , H_p  = \\emptyset$\n \\WHILE{ $k \\leq \\frac{p^2 r_{\\max}^2}{\\xi^2}$}\n \\STATE For every $\\ell \\in \\{1 , \\dots , p\\}$, perform spectral decomposition: $z^k_\\ell = \\sum_{i=1}^{r_\\ell} {\\lambda(z_\\ell^k)}_i {c(z_\\ell^k)}_i$ and $v^k_\\ell = \\sum_{i=1}^{r_\\ell} {\\lambda(v_\\ell^k)}_i {c(v^k_\\ell)}_i$\n \\IF {$z^k \\in \\mbox{int } \\mathcal{K}$} \n \\STATE \\stop basic procedure and \\return $z^k$ \\modifyThird{(Output (i))}\n \\ELSIF { $z^k = 0$ or $v^k \\in \\mathcal{K} \\setminus \\{0\\} $}\n \\STATE \\stop basic procedure and \\return $y^k$ or $v^k$ \\modifyThird{(Output (ii))}\n \\ENDIF\n \\IF {$\\langle v^k , e \\rangle  > 0$}\n \\FOR {$\\ell \\in \\{1 , \\dots , p \\}$}\n \\STATE $I_\\ell \\leftarrow \\left \\{ i  \\mid  {\\lambda(v_\\ell^k)}_i > 0 \\right \\}$ and then $H_\\ell \\leftarrow \\left \\{ i \\in I_\\ell | \\left \\langle e , \\mathcal{P}_\\mathcal{K} \\left( -\\frac{1}{{\\lambda(v_\\ell^k)}_i} v \\right) \\right \\rangle \\leq \\xi \\right \\}$\n \\ENDFOR\n \\ELSE \n \\FOR {$\\ell \\in \\{1 , \\dots , p \\}$}\n \\STATE $I_\\ell \\leftarrow \\left \\{ i  \\mid  {\\lambda(v_\\ell^k)}_i < 0 \\right\\}$ and then $H_\\ell \\leftarrow \\left \\{ i \\in I_\\ell | \\left \\langle e , \\mathcal{P}_\\mathcal{K} \\left( -\\frac{1}{{\\lambda(v_\\ell^k)}_i} v \\right) \\right \\rangle \\leq \\xi \\right \\}$\n \\ENDFOR \n \\ENDIF\n \\IF { $ |H_1| + \\dots + |H_p| > 0 $}\n \\STATE For every $\\ell \\in \\{1 , \\dots , p\\}$, let $ C_\\ell $ be  $\\{ {c(v_\\ell^k)}_1 , \\dots , {c(v_\\ell^k)}_{r_\\ell} \\}$. \n \\STATE \\stop basic procedure and \\return $H_1 , \\dots , H_p$ and $C_1 , \\dots , C_p$  \\modifyThird{(Output (iii))} \n \\ENDIF\n \\STATE Let $u$ be an idempotent such that $\\langle e , u \\rangle = 1$ and $\\langle z^k , u \\rangle = \\lambda_{\\min}(z^k)$\n \\STATE $y^{k+1} \\leftarrow \\alpha y^k + (1-\\alpha) u$, where $\\alpha = \\frac{ \\langle P_\\mathcal{A} (u) , P_\\mathcal{A} (u) - z^k \\rangle }{\\|z^k-P_\\mathcal{A} (u)\\|^2_J} $\n \\STATE $k \\leftarrow k+1$ , $z^k \\leftarrow P_\\mathcal{A}(y^k)$ and $v^k \\leftarrow y^k - z^k$\n \\ENDWHILE\n \\STATE \\return basic procedure error\n \\end{algorithmic} \n \\end{algorithm}\n \n\\section{Main algorithm of the extended method}\n\\label{sec: main algorithm}\n\\subsection{Outline of the main algorithm}\n\\label{sec:outline of main alg}\n\nIn what follows, for a given accuracy $\\varepsilon > 0$, we call a feasible solution of ${{\\rm P}_{S_{\\infty}}}(\\mathcal{A})$ whose minimum eigenvalue is $\\varepsilon$ or more an {\\em  $\\varepsilon$-feasible solution of ${{\\rm P}_{S_{\\infty}}}(\\mathcal{A})$}.\n\n\\modifyFirst{\nThis section describes the two main algorithms, Algorithm \\ref{main algorithm} and Algorithm \\ref{main algorithm 2}. The procedures of the algorithms are almost identical, except for one of the termination criteria (the criterion indicating the non-existence of $\\varepsilon$-feasible solutions). Specifically, to set the upper bound for the minimum eigenvalue of any feasible solution $x$ of ${\\rm P}_{S_{S_{\\infty}}}(\\mathcal{A})$, Algorithm \\ref{main algorithm} focuses on the product $\\det (\\bar{x})$ of the eigenvalues of the arbitrary feasible solution $\\bar{x}$ of the scaled problem ${\\rm P}_{S_{S_{\\infty}}}(\\mathcal{A}^k Q^k)$, while Algorithm \\ref{main algorithm 2} focuses on the sum $\\langle \\bar{x} , e \\rangle$ of the eigenvalues. Algorithm \\ref{main algorithm} and Algorithm \\ref{main algorithm 2} work as follows.\n}\n\nFirst, we calculate the corresponding projection $P_{\\mathcal{A}}$ onto $\\mbox{Ker} \\mathcal{A}$ and generate an initial point as input to the basic procedure. \nNext, we call the basic procedure and determine whether to end the algorithm with an $\\varepsilon$-feasible solution or to perform problem scaling according to the returned result, as follows:\n\n\\begin{enumerate}\n\\item \nIf a feasible solution of ${\\rm P}(\\mathcal{A})$ or ${\\rm D}(\\mathcal{A})$ is returned from the basic procedure, the feasibility of ${\\rm P}(\\mathcal {A})$ or ${\\rm D}(\\mathcal {A})$ can be determined, and we stop the main algorithm.\n\\item \nIf the basic procedure returns the sets of indices $H_1 , \\dots , H_p$ and the sets of primitive idempotents $C_1 , \\dots , C_p$ that construct the corresponding Jordan frames, then\n\\begin{description}\n\\item in Algorithm \\ref{main algorithm}:\n\\begin{enumerate}\n\\item\nif $\\mbox{num}_\\ell  \\geq r_\\ell \\frac{\\log \\varepsilon}{\\log \\xi}$ holds for some $\\ell \\in \\{ 1, \\dots p \\}$, we  determine that ${{\\rm P}_{S_{\\infty}}}(\\mathcal{A})$ has no $\\varepsilon$-feasible solution according to Proposition \\ref{prop:lambda-min-upper} and stop the main algorithm, \n\\item \nif $\\mbox{num}_\\ell  < r_\\ell \\frac{\\log \\varepsilon}{\\log \\xi}$ holds for any $\\ell \\in \\{ 1, \\dots p \\}$, we rescale the problem and call the basic procedure again.\n\\end{enumerate}\n\\item \\modifyFirst{in Algorithm \\ref{main algorithm 2}:\n\\begin{enumerate}\n\\item\nif $\\frac{r_\\ell}{\\left( r_\\ell + \\left( \\frac{1}{\\xi} -1 \\right)m_\\ell \\right)} < \\varepsilon$ holds for some $\\ell \\in \\{ 1, \\dots p \\}$, we  determine that ${{\\rm P}_{S_{\\infty}}}(\\mathcal{A})$ has no $\\varepsilon$-feasible solution according to Proposition \\ref{prop:lambda-min-upper-2} and stop the main algorithm, \n\\item \nif $\\frac{r_\\ell}{\\left( r_\\ell + \\left( \\frac{1}{\\xi} -1 \\right)m_\\ell \\right)} \\geq \\varepsilon$ holds for any $\\ell \\in \\{ 1, \\dots p \\}$, we rescale the problem and call the basic procedure again.\n\\end{enumerate}\n}\n\\end{description}\n\\end{enumerate}\n\nNote that our main algorithm is similar to Louren\\c{c}o et al.'s method in the sense that it keeps information about the possible minimum eigenvalue of any feasible solution of the problem. \nIn contrast, Pena and Soheili's method \\cite{Pena2017} does not keep such information.\n\n\\modifyThird{\nWe should also mention that step 24 in Algorithm \\ref{main algorithm} and Algorithm \\ref{main algorithm 2} is not a reachable output theoretically. We have added this step in order to consider the influence of the numerical error in practice.\n}\n\n\\modifyFirst{\nTable \\ref{compare_max_itr_MA} lists upper bounds on the numbers of iterations required by Algorithms \\ref{main algorithm} and \\ref{main algorithm 2}; we will give their proofs in section \\ref{sec: finite termination of MA}. As shown in the table, Algorithm \\ref{main algorithm} can be said to be a polynomial-time algorithm, but Algorithm \\ref{main algorithm 2} is not. On the other hand, the results of the numerical experiments in section \\ref{sec: numerical results and observations} show that Algorithm \\ref{main algorithm 2} is superior to Algorithm \\ref{main algorithm} at detecting $\\varepsilon$-feasibility for the generated instances.\n}\n\n\\begin{table}[H]\n\\caption{\\modifyFirst{Upper bounds on the number of iterations of the main algorithms (cf. section \\ref{sec: finite termination of MA}) }}\n\\begin{center}\n\\label{compare_max_itr_MA}\n\\modifyFirst{\n\\begin{tabular}{lc} \\toprule\nMain Algorithm\t\t\t\t&Upper bound on \\# of iterations\t\\\\ \\midrule\nAlgorithm \\ref{main algorithm}\t\t&$-\\frac{r}{\\log \\xi} \\log \\left( \\frac{1}{\\varepsilon}\\right) - p + 1$\t\\\\\nAlgorithm \\ref{main algorithm 2}\t&$\\frac{\\xi}{1-\\xi} \\left( \\frac{1}{\\varepsilon} -1  \\right) r - p + 1$\t\\\\\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n \\begin{algorithm}[H]\n \\caption{Main algorithm}\n \\label{main algorithm}\n \\begin{algorithmic}[1]\n \\renewcommand{\\algorithmicrequire}{\\textbf{Input: }}\n \\renewcommand{\\algorithmicensure}{\\textbf{Output: }}\n  \\renewcommand{\\stop}{\\textbf{stop }}\n \\renewcommand{\\return}{\\textbf{return }}\n\n \\STATE \\algorithmicrequire $\\mathcal{A}$, $\\mathcal{K}$, $\\varepsilon$ and a constant $\\xi$ such that $0 < \\xi < 1$\n \\STATE \\algorithmicensure a solution to ${\\rm P}(\\mathcal{A})$ or ${\\rm D}(\\mathcal{A})$ or a certificate that there is no $\\varepsilon$ feasible solution.\n \\STATE $k \\leftarrow 1$ , $\\mathcal{A}^1 \\leftarrow \\mathcal{A}$ , $\\mbox{num}_\\ell \\leftarrow 0 $, $\\bar{Q_\\ell} \\leftarrow I_\\ell $ for all $\\ell \\in \\{1,\\dots,p\\}$\n \\STATE Compute $P_{\\mathcal{A}}$ and call the basic procedure with $P_\\mathcal{A}$, $\\frac{1}{r}e$, $\\xi$\n  \\IF {basic procedure returns $z$ }\n \\STATE \\stop main algorithm and \\return $z$ ($z$ is a feasible solution of ${\\rm P}(\\mathcal{A})$)\n \\ELSIF { basic procedure returns $y$ or $v$ }\n \\STATE \\stop main algorithm and \\return $y$ or $v$ ( $y$ or $v$ is a feasible solution of $D(\\mathcal{A})$)\n \\ELSIF  {basic procedure returns $H^k_1 , \\dots , H^k_p$ and $C^k_1 , \\dots , C^k_p$}\n \\FOR {$\\ell \\in \\{1,\\dots,p\\}$}\n \\IF{$|H^k_\\ell| > 0$}\n \\STATE $g_\\ell \\leftarrow \\sqrt{\\xi} \\sum_{h \\in H^k_\\ell} c^k(v_\\ell)_h + \\sum_{h \\notin H_\\ell^k}^{r_\\ell} c^k(v_\\ell)_h$ \n \\STATE $Q_\\ell \\leftarrow Q_{g_\\ell}$\n \\STATE $\\mbox{num}_\\ell \\leftarrow |H^k_\\ell| + \\mbox{num}_\\ell $\n \\IF { $ \\mbox{num}_\\ell  \\geq r_\\ell \\frac{\\log \\varepsilon}{\\log \\xi}$ }\n \\STATE { \\stop main algorithm. There is no $\\varepsilon$ feasible solution.}\n \\ENDIF\n \\STATE $\\bar{Q_\\ell} \\leftarrow  Q_{g_\\ell^{-1}} \\bar{Q_\\ell} $\n \\ELSE\n \\STATE $ Q_\\ell \\leftarrow I_\\ell$\n \\ENDIF\n \\ENDFOR\n \\ELSE\n \\STATE \\return basic procedure error\n \\ENDIF\n \\STATE Let $Q^k = (Q_1 , \\dots , Q_p)$ \n \\STATE \\modifyThird{$\\mathcal{A}^{k+1} \\leftarrow \\mathcal{A}^k Q^k$ , $k \\leftarrow k+1$. Go back to line 4.}\n \\end{algorithmic} \n \\end{algorithm}\n\n\n \\begin{algorithm}[H]\n \\caption{Main algorithm using another criteria for $\\varepsilon$-feasibility}\n \\label{main algorithm 2}\n \\begin{algorithmic}[1]\n \\renewcommand{\\algorithmicrequire}{\\textbf{Input: }}\n \\renewcommand{\\algorithmicensure}{\\textbf{Output: }}\n  \\renewcommand{\\stop}{\\textbf{stop }}\n \\renewcommand{\\return}{\\textbf{return }}\n\n \\STATE \\algorithmicrequire $\\mathcal{A}$, $\\mathcal{K}$, $\\varepsilon$ and a constant $\\xi$ such that $0 < \\xi < 1$\n \\STATE \\algorithmicensure a solution to ${\\rm P}(\\mathcal{A})$ or ${\\rm D}(\\mathcal{A})$ or a certificate that there is no $\\varepsilon$ feasible solution.\n \\STATE $k \\leftarrow 1$ , $\\mathcal{A}^1 \\leftarrow \\mathcal{A}$ , $m_\\ell \\leftarrow 0 $ , $\\bar{Q_\\ell} \\leftarrow I_\\ell $ for all $\\ell \\in \\{1,\\dots,p\\}$\n  \\STATE Compute $P_{\\mathcal{A}}$ and call the basic procedure with $P_\\mathcal{A}$, $\\frac{1}{r}e$, $\\xi$\n \\IF { basic procedure returns $z$ }\n \\STATE \\stop main algorithm and \\return $z$ ($z$ is a feasible solution of ${\\rm P}(\\mathcal{A})$)\n \\ELSIF { basic procedure returns $y$ or $v$ }\n \\STATE \\stop main algorithm and \\return $y$ or $v$ ( $y$ or $v$ is a feasible solution of $D(\\mathcal{A})$)\n \\ELSIF  {basic procedure returns $H^k_1 , \\dots , H^k_p$ and $C^k_1 , \\dots , C^k_p$}\n \\FOR {$\\ell \\in \\{1,\\dots,p\\}$}\n \\IF{$|H^k_\\ell| > 0$}\n \\STATE $g_\\ell \\leftarrow \\sqrt{\\xi} \\sum_{h \\in H^k_\\ell} c^k(v_\\ell)_h + \\sum_{h \\notin H_\\ell^k}^{r_\\ell} c^k(v_\\ell)_h$ \n \\STATE $Q_\\ell \\leftarrow Q_{g_\\ell}$\n \\STATE $m_\\ell \\leftarrow \\left \\langle \\bar{Q_\\ell} \\left( \\sum_{h \\in H^k_\\ell} c^k(v_\\ell)_h \\right) , e_\\ell \\right \\rangle + m_\\ell$\n \\IF {$ \\frac{r_\\ell}{\\left( r_\\ell + \\left( \\frac{1}{\\xi} - 1 \\right) m_\\ell \\right)} \\leq \\varepsilon$}\n \\STATE { \\stop main algorithm. There is no $\\varepsilon$ feasible solution.}\n \\ENDIF\n \\STATE $\\bar{Q_\\ell} \\leftarrow  \\bar{Q_\\ell} Q_{g_\\ell^{-1}}$\n \\ELSE\n \\STATE $ Q_\\ell \\leftarrow I_\\ell$\n \\ENDIF\n \\ENDFOR\n \\ELSE\n \\STATE \\return basic procedure error\n \\ENDIF\n \\STATE Let $Q^k = (Q_1 , \\dots , Q_p)$\n \\STATE \\modifyThird{$\\mathcal{A}^{k+1} \\leftarrow \\mathcal{A}^k Q^k$ , $k \\leftarrow k+1$. Go back to line 4.}\n \\end{algorithmic} \n \\end{algorithm}\n \n\\subsection{Finite termination of the main algorithm}\n\\label{sec: finite termination of MA}\n\nHere, we discuss how many iterations are required until we can determine that the minimum eigenvalue $\\lambda_{\\rm min} (x)$ is less than $\\varepsilon$ for any $x \\in F_{{\\rm P}_{S_{\\infty}}(\\mathcal{A})}$.\n\n\\modifyThird{\nBefore going into the proof, we explain the difference between Algorithm \\ref{main algorithm} and Algorithm \\ref{main algorithm 2} in more detail than in section \\ref{sec:outline of main alg}.\nThe difference between the two algorithms is the processing after the basic procedure returns $H_1, \\dots , H_p$ and $C_1, \\dots , C_p$.\n}\n\n\\modifyThird{\nAt each iteration of Algorithm \\ref{main algorithm}, it accumulates the number of cuts $|H_\\ell^k|$ obtained in the $\\ell$-th block and stores the value in ${\\rm num}_\\ell$. Using ${\\rm num}_\\ell$, we can compute an upper bound for $\\lambda_{\\rm min} (x)$ (Proposition \\ref{prop:lambda-min-upper}). On line 18, $\\bar{Q}_\\ell$ is updated to $\\bar{Q}_\\ell \\leftarrow Q_{g_\\ell^{-1}} \\bar{Q}_\\ell $, where $\\bar{Q}_\\ell$ plays the role of an operator that gives the relation $\\bar{x}_\\ell = \\bar{Q}_\\ell (x_\\ell)$ for the solution $x$ of the original problem and the solution $\\bar{x}$ of the scaled problem. For example, if $|H^1_\\ell| > 0$ for $k=1$ (suppose that the cut was obtained in the $\\ell$-th block), then the proposed method scales $\\mathcal{A}^1_\\ell Q^1_\\ell$ and the problem to yield $\\bar{x}_\\ell = Q_{g_\\ell^{-1}}(x_\\ell)$ for the feasible solution $x$ of the original problem. And if $|H^2_\\ell| > 0$ even for $k=2$, then the proposed method scales $\\bar{x}$ again, so that $\\bar{\\bar{x}}_\\ell = Q_{g_\\ell^{-1}}(\\bar{x}) = \\bar{Q}_\\ell (x_\\ell)$ holds. Note that $\\bar{Q}_\\ell$ is used only for a concise proof of Proposition \\ref{prop:lambda-min-upper}, so it is not essential.\n}\n\n\\modifyThird{\nIn Algorithm \\ref{main algorithm 2}, when a cut is obtained in the $\\ell$-th block, it computes the value of $\\left \\langle \\bar{Q_\\ell} \\left( \\sum_{h \\in H^k_\\ell} c^k(v_\\ell)_h \\right) , e_\\ell \\right \\rangle$ and stores its cumulative value in $m_\\ell$. In fact, using this $m_\\ell$, we can compute an upper bound for $\\lambda_{\\rm min} (x)$ (Proposition \\ref{prop:lambda-min-upper-2}). On line 18, $\\bar{Q}_\\ell$ is updated as $\\bar{Q}_\\ell \\leftarrow \\bar{Q}_\\ell Q_{g_\\ell^{-1}}$, and $\\bar{Q}_\\ell$ of Algorithm \\ref{main algorithm 2} plays the role of an operator that gives the relation $\\langle \\bar{x}_\\ell , a_\\ell \\rangle = \\langle x_\\ell, \\bar{Q}_\\ell (a_\\ell) \\rangle$ for the solution $x$ of the original problem, the solution $\\bar{x}$ of the scaled problem, and any $a \\in \\mathbb{E}$. For example, as before, if $|H^1_\\ell| > 0$ for $k=1$, then $\\langle \\bar{x}_\\ell , a_\\ell \\rangle = \\langle Q_{g_\\ell^{-1}}(x_\\ell) , a_\\ell \\rangle =\\langle x_\\ell , Q_{g_\\ell^{-1}}(a_\\ell) \\rangle$ is valid. And if $|H^2_\\ell| > 0$ even for $k=2$, then the proposed method scales $\\bar{x}$ again, so that $\\langle \\bar{\\bar{x}}_\\ell ,a_\\ell \\rangle = \\langle \\bar{x}_\\ell , Q_{g_\\ell^{-1}}(a_\\ell) \\rangle = \\langle x_\\ell , \\bar{Q}_\\ell (a_\\ell) \\rangle $ holds.\n}\n\n\nNow, let us derive an upper bound for the minimum eigenvalue $\\lambda_{\\min}(x_\\ell)$ of each $\\ell$-th block of $x$ obtained after the $k$-th iteration of Algorithm \\ref{main algorithm}.\nProposition \\ref{prop:iteration-num-ma} gives an upper bound for the number of iterations of Algorithm \\ref{main algorithm}.\n\n\\begin{proposition}\n\\label{prop:lambda-min-upper}\nAfter $k$ iterations of Algorithm \\ref{main algorithm}, for any feasible  solution $x$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ and $ \\ell \\in \\{1 , \\dots , p\\}$, the $\\ell$-th block element $x_\\ell$ of $x$ satisfies\n\\begin{equation}\nr_\\ell \\log {\\left( \\lambda_{\\min}(x_\\ell) \\right)} \\leq \\mbox{\\rm num}_\\ell \\log {\\xi} . \\label{prop:10-0}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n\\modifyThird{\nAt the end of the $k$-th iteration, any feasible solution $\\bar{x}$ of the scaled problem ${\\rm P}_{S_{\\infty}}(\\mathcal{A}^{k+1}) = {\\rm P}_{S_{\\infty}}(\\mathcal{A}^kQ^k)$ obviously satisfies\n\\begin{equation}\n\\det \\bar{x}_\\ell \\leq \\det e_\\ell \\ \\ (\\ell =1,2,\\ldots,p).  \\label{prop:10-1}\n\\end{equation}\nNote that $\\det \\bar{x}_\\ell$ can be expressed in terms of $\\det x_\\ell$. For example, if $|H_\\ell^1|>0$ when $k=1$, then using Proposition \\ref{q-det-relation}, for any feasible solution $\\bar{x}$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A}^{2})$, we find that\n\\begin{equation}\n\\det \\bar{x}_\\ell = \\det Q_{g_\\ell^{-1}} (x_\\ell) = \\det(g_\\ell^{-1})^2 \\det x_\\ell =   {\\left( \\frac{1}{\\sqrt{\\xi}} \\right)}^{2|H_\\ell^1|} \\det x_\\ell =   {\\left( \\frac{1}{\\xi} \\right)}^{|H_\\ell^1|} \\det x_\\ell  .\\notag \n\\end{equation}\nThis means that $\\det \\bar{x}_\\ell$ can be determined from $\\det x_\\ell$ and the number of cuts obtained so far in the $\\ell$-th block. In Algorithm \\ref{main algorithm}, the value of $\\mbox{num}_\\ell$ is updated only when $|H_\\ell^k|>0$. Since $\\bar{x}$ satisfies $\\bar{x}_\\ell = \\bar{Q}_\\ell (x_\\ell) \\ (\\ell=1,2,\\ldots,p)$ for each feasible solution $x$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$, we can see that \n\\begin{equation}\n\\det \\bar{x}_\\ell = \\det \\bar{Q_\\ell} (x_\\ell) =  {\\left( \\frac{1}{\\xi} \\right)}^{|H_\\ell^k|} \\times  {\\left( \\frac{1}{\\xi} \\right)}^{|H_\\ell^{k-1}|} \\dots \\times  {\\left( \\frac{1}{\\xi} \\right)}^{|H_\\ell^1|} \\times \\det x_\\ell =  {\\left( \\frac{1}{\\xi} \\right)}^{\\mbox{num}_\\ell} \\det x_\\ell . \\notag\n\\end{equation}\nTherefore,  (\\ref{prop:10-1}) implies \n\\begin{equation}\n\\det x_\\ell \\leq \\xi^{\\mbox{num}_\\ell} \\det e_\\ell = \\xi^{\\mbox{num}_\\ell}\\notag\n\\end{equation}\nand the fact ${\\left( \\lambda_{\\min}(x_\\ell) \\right)}^{r_\\ell} \\leq \\det x_\\ell$ implies  ${\\left( \\lambda_{\\min}(x_\\ell) \\right)}^{r_\\ell} \\leq {\\xi}^{\\mbox{num}_\\ell}$.\nBy taking the logarithm of both sides of this inequality, we obtain (\\ref{prop:10-0}). \n}\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop:iteration-num-ma}\nAlgorithm \\ref{main algorithm} terminates after no more than\n\\begin{equation}\n-\\frac{r}{\\log \\xi} \\log \\left( \\frac{1}{\\varepsilon}\\right) - p + 1  \\notag\n\\end{equation}\niterations.\n\\end{proposition}\n\n\\begin{proof}\nLet us call iteration $k$ of Algorithm \\ref{main algorithm} {\\em good} if $|H_\\ell^k| > 0$ for some $\\ell \\in \\{1,2, \\dots , p\\}$ at that iteration. Suppose that at least $-\\frac{r_\\ell}{\\log \\xi} \\log \\left( \\frac{1}{\\varepsilon}\\right)$ {\\em good} iterations are observed for a cone $\\mathcal{K}_\\ell$. Then, by substituting $-\\frac{r_\\ell}{\\log \\xi} \\log \\left( \\frac{1}{\\varepsilon}\\right)$ into $\\mbox{num}_\\ell$ of inequality (\\ref{prop:10-0}) in Proposition \\ref{prop:lambda-min-upper}, we have \n\\begin{equation}\n\\log {\\left( \\lambda_{\\min}(x_\\ell) \\right)} \\leq  - \\log \\left( \\frac{1}{\\varepsilon}\\right) = \\log \\varepsilon \\notag\n\\end{equation}\nand hence, $\\lambda_{\\min}(x_\\ell) \\leq \\varepsilon$.\nThis implies that Algorithm \\ref{main algorithm} terminates after no more than \n\\begin{equation}\n\\sum_{\\ell=1}^p \\left( -\\frac{r_\\ell}{\\log \\xi} \\log \\left( \\frac{1}{\\varepsilon}\\right) - 1 \\right) + 1 = -\\frac{r}{\\log \\xi} \\log \\left( \\frac{1}{\\varepsilon}\\right) - p + 1  \\notag\n\\end{equation}\niterations.\n\\end{proof}\n\n\\modifyFirst{Next, let us derive an upper bound for the number of iterations of Algorithm \\ref{main algorithm 2}.} \n\nProposition \\ref{prop:lambda-min-upper} guarantees that $\\varepsilon$-feasibility of the problem ${\\rm P}(\\mathcal{A})$ can be detected by computing $\\det(\\bar{x})$ of any feasible solution of $ {\\rm P}_{S_{\\infty}}(\\mathcal{A}^kQ^k)$. The following proposition ensures that we may use the value $\\langle \\bar{x} , e \\rangle$ of any feasible solution of $ {\\rm P}_{S_{\\infty}}(\\mathcal{A}^kQ^k)$ to detect the $\\varepsilon$-feasibility of problem ${\\rm P}(\\mathcal{A})$, instead of $\\det(\\bar{x})$. While the analysis of the computational complexity in section \\ref{sec: finite termination of MA} does not hold for it, the new criteria is better able to detect $\\varepsilon$-feasibility in the numerical experiments presented in section\\ref{sec: numerical results and observations}.\n\n\\begin{proposition}\n\\label{prop:lambda-min-upper-2}\nAfter $k$ iterations of Algorithm \\ref{main algorithm 2}, for any feasible  solution $x$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ and $ \\ell \\in \\{1 , \\dots , p\\}$, the $\\ell$-th block element $x_\\ell$ of $x$ satisfies\n\\begin{equation}\n\\lambda_{\\min}(x_\\ell) \\leq \\cfrac{r_\\ell}{\\left( r_\\ell + \\left( \\frac{1}{\\xi} - 1 \\right) m_\\ell \\right)} . \\label{prop:11-0}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nIn Algorithm \\ref{main algorithm 2}, $m_\\ell$ is updated only when $|H_\\ell^k|>0$. \nSuppose that, at the end of  the $k$-th iteration of Algorithm \\ref{main algorithm 2}, the last update of $m_\\ell$ had been at the $k'(\\leq k)$-th iteration.\nThen, the stopping criteria of the basic procedure guarantees that at the beginning of the $k'$-th iteration,  $\\bar{Q_\\ell}$\nsatisfies \n\\modifyFirst{\n\\begin{equation}\n\\langle x, \\bar{Q_\\ell}(c^{k'}(v_\\ell)_i) \\rangle \\leq \n\\begin{cases}\n\\xi \t&i \\in  H_\\ell^{k'} \\\\\n1\t&i \\notin H_\\ell^{k'}\n\\end{cases}\n.\n\\label{prop:11-0.5}\n\\end{equation}\n}\n\nThis gives a lower bound for $|H_\\ell^{k'}|$:\n\\begin{equation}\n\\frac{1}{\\xi} \\left \\langle x, \\bar{Q_\\ell} \\left( \\sum_{i \\in H_\\ell^{k'} }c^{k'}(v_\\ell)_i \\right) \\right \\rangle \\leq |H_\\ell^{k'}|. \\label{prop:11-0.75}\n\\end{equation}\n\nUsing the fact that $x_\\ell - \\lambda_{\\min} (x_\\ell) e_\\ell \\in \\mathcal{K}_\\ell$, we obtain\n\\begin{align*}\n\\lambda_{\\min} (x_\\ell) \\langle e_\\ell , \\bar{Q_\\ell} (e_\\ell) \\rangle  &\\leq \\langle x_\\ell , \\bar{Q_\\ell} (e_\\ell) \\rangle \\\\\n&\\modifyFirst{= \\left \\langle x_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\notin H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle + \\left \\langle x_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle} \\\\\n&\\leq r_\\ell - |H_\\ell^{k'}| + \\left \\langle x_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle    \\ \\ \\ \\modifyFirst{\\mbox{(by  (\\ref{prop:11-0.5}))}}  \\\\\n&\\leq r_\\ell  - \\left( \\cfrac{1}{\\xi} - 1 \\right) \\left \\langle x_\\ell  , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle  \\ \\ \\ \\mbox{(by  (\\ref{prop:11-0.75}))} \\\\\n&\\leq r_\\ell  - \\left( \\cfrac{1}{\\xi} - 1 \\right) \\lambda_{\\min} (x_\\ell) \\left \\langle e_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle,\n\\end{align*}\nand hence,  \n\\begin{equation}\n\\lambda_{\\min} (x_\\ell) \\left( \\left \\langle e_\\ell , \\bar{Q_\\ell} (e_\\ell) \\right \\rangle + \\left( \\cfrac{1}{\\xi} - 1 \\right)  \\left \\langle e_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle \\right) \\leq r_\\ell. \\label{prop:11-1}\n\\end{equation}\n\nNext, suppose that, at the beginning of the $k'$-th iteration of Algorithm \\ref{main algorithm 2}, the last update of $m_\\ell$ had been performed at the $i(<k')$-th iteration.\n\nLet $\\bar{Q_\\ell}^{\\rm pre}$ be $\\bar{Q_\\ell}$ obtained at the beginning of the $i$-th iteration of Algorithm \\ref{main algorithm 2}, and let $Q_{g_\\ell}^{\\rm pre}$ and $m_\\ell^{\\rm pre}$ be $Q_{\\ell}$ and $m_\\ell$ obtained after the update at the $i$-th iteration. \nNote that  $\\bar{Q_\\ell}$ at the beginning of the $k'$-th iteration of Algorithm \\ref{main algorithm 2} can be represented by $\\bar{Q_\\ell} = \\bar{Q_\\ell}^{\\rm pre} Q_{g_\\ell^{-1}}^{\\rm pre}$.\nThus, from (\\ref{eq:prop3.4-1}), we see that \n\\begin{align*}\n Q_{g_\\ell^{-1}}^{\\rm pre}(e_\\ell) \n & =  Q_{g_\\ell^{-1}}^{\\rm pre} \\left( \\sum_{j=1}^{r_\\ell} {c^i(v_\\ell)}_j \\right) \\\\\n & =  Q_{g_\\ell^{-1}}^{\\rm pre} \\left( \\sum_{j \\in H^{k'}_\\ell}  {c^i(v_\\ell)}_j \\right) + Q_{g_\\ell^{-1}}^{\\rm pre} \\left( \\sum_{j \\not\\in H^{k'}_\\ell}  {c^i(v_\\ell)}_j \\right) \\\\\n & = \\frac{1}{\\xi} \\sum_{j \\in H^{k'}_\\ell}  {c^i(v_\\ell)}_j  + \\sum_{j \\not\\in H^{k'}_\\ell}  {c^i(v_\\ell)}_j \\\\\n & = e_\\ell + \\left( \\frac{1}{\\xi} - 1 \\right) \\sum_{j \\in H_\\ell^i} {c^i (v_\\ell)}_j \n\\end{align*}\nand hence,  \n\\begin{align}\n\\bar{Q_\\ell}(e_\\ell) = \\bar{Q_\\ell}^{\\rm pre} Q_{g_\\ell^{-1}}^{\\rm pre}(e_\\ell) \n&= \\bar{Q_\\ell}^{\\rm pre} \\left( e_\\ell + \\left( \\frac{1}{\\xi} - 1 \\right) \\sum_{j \\in H_\\ell^i} {c^i (v_\\ell)}_j \\right) \\notag \\\\\n&= \\bar{Q_\\ell}^{\\rm pre} \\left( e_\\ell \\right) + \\left( \\frac{1}{\\xi} - 1 \\right) \\bar{Q_\\ell}^{\\rm pre} \\left(  \\sum_{j \\in H_\\ell^i} {c^i (v_\\ell)}_j \\right). \\label{prop:11-2}\n\\end{align}\n\nBy recursively applying (\\ref{prop:11-2}) to $\\bar{Q_\\ell}^{\\rm pre} \\left( e_\\ell \\right)$, we finally obtain\n\\begin{equation}\n\\left \\langle e_\\ell , \\bar{Q_\\ell} (e_\\ell) \\right \\rangle = r_\\ell +\\left( \\frac{1}{\\xi} - 1 \\right) m_\\ell^{\\rm pre} . \\notag\n\\end{equation}\n\nLet $m_\\ell^{k'}$ be the value of $m_\\ell$ obtained after the update at the $k'$-th iteration.\nThen, \n\\begin{equation}\n\\label{update-m}\nm_\\ell^{k'} = m_\\ell^{\\rm pre} + \\left \\langle e_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle\n\\end{equation}\nand, by (\\ref{prop:11-1}), we obtain\n\\begin{align}\n\\lambda_{\\min} (x_\\ell)  &\\leq \\cfrac{r_\\ell}{\\left( r_\\ell +\\left( \\frac{1}{\\xi} - 1 \\right) m_\\ell^{\\rm pre}  + \\left( \\frac{1}{\\xi} - 1 \\right)  \\left \\langle e_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle \\right)} \\notag \\\\\n&= \\frac{r_\\ell}{\\left( r_\\ell  + \\left( \\cfrac{1}{\\xi} - 1 \\right) m_\\ell^{k'} \\right)} . \\notag\n\\end{align}\n\nSince, at the end of the $k$-th iteration of  Algorithm \\ref{main algorithm 2}, the last update of $m_\\ell$ was at the $k'$-th iteration, we see that $m_\\ell = m_\\ell^{k'}$, and hence (\\ref{prop:11-0}) holds after  $k$ iterations of Algorithm \\ref{main algorithm 2}.\n\\end{proof}\n\n\\modifyFirst{ Using Proposition \\ref{prop:lambda-min-upper-2}, we find an upper bound on the number of iterations of Algorithm \\ref{main algorithm 2}.\n\\begin{proposition}\n\\label{prop:iteration-num-ma-2}\nAlgorithm \\ref{main algorithm 2} terminates at the following iterations.\n\\begin{equation}\n\\frac{\\xi}{1-\\xi} \\left( \\frac{1}{\\varepsilon} - 1\\right) r - p + 1 . \\notag\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWhen $|H_\\ell^k| > 0$ for $ \\ell \\in \\{1 , \\dots , p\\}$ at the $k$-th iteration of Algorithm \\ref{main algorithm 2}, we say that the iteration is ``good'' for the $\\ell$-th block. From Proposition \\ref{prop:lambda-min-upper-2}, since the (meaningful) upper bound of the minimum eigenvalue $\\lambda_{\\min} (x_\\ell)$ of $x_\\ell$ of the $\\ell$-th block of any feasible solution $x$ of ${\\rm P}_{S_{\\infty}}(\\mathcal{A})$ depends on the value of $m_\\ell$, we first calculate a lower bound for the increment of $m_\\ell$ per good iteration in the $\\ell$-th block. \\\\\nSimilar to the proof of Proposition \\ref{prop:lambda-min-upper-2}, suppose that the $k'$-th iteration is a good iteration for the $\\ell$-th block. As shown in equation (\\ref{update-m}), the value of $m_\\ell$ is increased at this time by $\\left \\langle e_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle$ using $\\bar{Q}_\\ell$ at the beginning of the $k'$-th iteration. Let us express $Q_{g_\\ell^{-1}}$ using $g_\\ell$ obtained at the $k$-th iteration as $Q^k_{g_\\ell^{-1}}$, i.e., $\\bar{Q_\\ell} = Q^1_{g_\\ell^{-1}} Q^2_{g_\\ell^{-1}} \\dots Q^{k'-1}_{g_\\ell^{-1}}$. Then, the increment of $m_\\ell$ at the $k'$-th iteration is as follows:\n\\begin{equation}\n\\label{eq-m-value}\n\\left \\langle e_\\ell , \\bar{Q_\\ell} \\left( \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right) \\right \\rangle = \\left \\langle Q^{k'-1}_{g_\\ell^{-1}} \\dots Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) , \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right \\rangle.\n\\end{equation}\nNote that $Q^{k'-1}_{g_\\ell^{-1}} \\dots Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) - e_\\ell \\in \\mathcal{K}_\\ell$ holds, as we will prove below using induction. \\\\\nFirst, if the first iteration is a good one for the $\\ell$ block, then $Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) = \\frac{1}{\\xi} \\sum_{i \\in H_\\ell^1} c^1(v_\\ell)_i+ \\sum_{j \\notin H_\\ell^1} c^1(v_\\ell)_j = e_\\ell + \\left( \\frac{1}{\\xi} -1 \\right)\\sum_{i \\in H_\\ell^1} c^1(v_\\ell)_i$, and if it is not a good iteration, then $Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) =e_\\ell$.\nThus, $Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) - e_\\ell \\in \\mathcal{K}_\\ell$ holds.\nNext, when  $Q^{i}_{g_\\ell^{-1}} \\dots Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) - e_\\ell \\in \\mathcal{K}$ holds, by Proposition \\ref{ptop:quadratic}, $Q^{i+1}_{g_\\ell^{-1}} \\left( Q^{i}_{g_\\ell^{-1}} \\dots Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) - e_\\ell \\right) \\in \\mathcal{K}_\\ell$ holds.\nFurthermore, the same calculation as in the first iteration yields $Q^{i+1}_{g_\\ell^{-1}} \\left( e_\\ell \\right) - e_\\ell \\in \\mathcal{K}_\\ell$, and we see that \n\\begin{align*}\nQ^{i+1}_{g_\\ell^{-1}} \\left( Q^{i}_{g_\\ell^{-1}} \\dots Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) - e_\\ell \\right) \\in \\mathcal{K}_\\ell\n&\\Leftrightarrow Q^{i+1}_{g_\\ell^{-1}} Q^{i}_{g_\\ell^{-1}} \\dots Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) - Q^{i+1}_{g_\\ell^{-1}} \\left( e_\\ell \\right)  \\in \\mathcal{K}_\\ell \\\\\n&\\Rightarrow Q^{i+1}_{g_\\ell^{-1}} Q^{i}_{g_\\ell^{-1}} \\dots Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) - e_\\ell  \\in \\mathcal{K}_\\ell.\n\\end{align*}\nThus, from (\\ref{eq-m-value}), we obtain a lower bound for the increment of $m_\\ell$ as\n\\begin{align*}\n\\left \\langle Q^{k'-1}_{g_\\ell^{-1}} \\dots Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) , \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right \\rangle\n&\\geq \\left \\langle e_\\ell,  \\sum_{j \\in H_\\ell^{k'}} {c^{k'}(v_\\ell)}_j \\right \\rangle \\\\\n&= |H^{k'}_\\ell| \\geq 1,\n\\end{align*}\nwhich means that the value of $m_\\ell$ increases by at least $1$ per good iteration.\nTherefore, if the number of good iterations for the $\\ell$-th block is $\\left( \\frac{r_\\ell}{\\varepsilon} - r_\\ell \\right) \\left( \\frac{\\xi}{1-\\xi} \\right)$ or more, then from Proposition \\ref{prop:lambda-min-upper-2}, we can conclude that $\\lambda_{min}(x_\\ell) \\leq \\varepsilon$ holds; i.e., we obtain an upper bound for the number of iterations of Algorithm \\ref{main algorithm 2} as follows:\n\\begin{equation}\n\\sum_{\\ell=1}^p \\left(  \\left( \\frac{r_\\ell}{\\varepsilon} - r_\\ell \\right) \\left( \\frac{\\xi}{1-\\xi} \\right) -1 \\right) + 1 = \\frac{\\xi}{1-\\xi} \\left( \\frac{1}{\\varepsilon} - 1\\right) r - p + 1. \\notag\n\\end{equation}\n\\end{proof}\n}\n\\modifyThird{\nIt should be noted that the number of iterations required by Algorithm \\ref{main algorithm 2} to detect the non-existence of $\\varepsilon$-feasible solutions is actually likely to be much smaller than the value given in Proposition \\ref{prop:iteration-num-ma-2}.\nThis is because Proposition \\ref{prop:iteration-num-ma-2} calculates the lower bound for the increment of $m_\\ell$ for one good iteration as $1$.\nThe increment of $m_\\ell$ can be calculated using $\\bar{Q}_\\ell$, but it is difficult to calculate the exact increment of $m_\\ell$ because $\\bar{Q}_\\ell$ depends on the results returned by the previous basic procedure. \\\\\nSuppose that both the first and second iterations are good for the $\\ell$-th block. Then, the increment of $m_\\ell$ at the second iteration is \n\\begin{equation}\n\\notag\n\\left \\langle Q^1_{g_\\ell^{-1}} \\left( e_\\ell \\right) , \\sum_{j \\in H_\\ell^{2}} {c^{2}(v_\\ell)}_j \\right \\rangle = \\left \\langle e_\\ell + \\left( \\frac{1}{\\xi} -1 \\right)\\sum_{i \\in H_\\ell^1} c^1(v_\\ell)_i , \\sum_{j \\in H_\\ell^{2}} {c^{2}(v_\\ell)}_j \\right \\rangle,\n\\end{equation}\nbut it is difficult to find a lower bound greater than 0 for $\\left \\langle \\sum_{i \\in H_\\ell^1} c^1(v_\\ell)_i , \\sum_{j \\in H_\\ell^{2}} {c^{2}(v_\\ell)}_j \\right \\rangle$.\n}\n\n\n\\section{Computational costs of the algorithms}\n\\label{sec: compare}\n\nThis section compares the computational costs of Algorithm \\ref{main algorithm}, Louren\\c{c}o et al.'s method~\\cite{Bruno2019} and Pena and Soheili's method~\\cite{Pena2017}. Algorithm \\ref{main algorithm 2} is omitted from the comparison because it is not guaranteed to be a polynomial-time algorithm.\n\nSection \\ref{sec: complexity of MA vs Lourenco} compares the computational costs of Algorithm \\ref{main algorithm} and Louren\\c{c}o et al.'s method, and\nSection \\ref{sec: complexity of MA vs Pena} compares those of Algorithm \\ref{main algorithm} and Pena and Soheili's method under the assumption that  $\\mbox{Ker} \\mathcal{A} \\cap  \\mbox{int} \\mathcal{K} \\neq  \\emptyset$.\n\nBoth the proposed method and the method of Louren\\c{c}o et al. guarantee finite termination of the main algorithm by termination criteria indicating the nonexistence of an $\\varepsilon$-feasible solution, so that it is possible to compare the computational costs of the methods without making any special assumptions. This is because both methods proceed by making cuts to the feasible region using the results obtained from the basic procedure. On the other hand, Pena and Soheili's method cannot be simply compared because the upper bound of the number of iterations of their main algorithm includes an unknown value of $\\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K}) := \\max_x \\left \\{ {\\rm det}(x) \\mid x \\in \\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} , \\|x\\|_J^2 = r \\right\\}$.\n\nHowever, by making the assumption $\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\neq \\emptyset$ and deriving a lower bound for $\\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K})$, we make it possible to compare Algorithm \\ref{main algorithm} (and Algorithm \\ref{main algorithm 2} ) with Pena and Soheili's method without knowing the specific values of $\\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K})$.\n\n\\subsection{Comparison of Algorithm \\ref{main algorithm} and Louren\\c{c}o et al.'s method}\n\\label{sec: complexity of MA vs Lourenco}\n\nlet us consider the computational cost of Algorithm \\ref{main algorithm}.\nAt each iteration, the most expensive operation is computing $P_\\mathcal{A}$ on line 4.\nRecall that $d$ is the dimension of the Euclidean space $\\mathbb{E}$ corresponding to $\\mathcal{K}$.\nAs discussed in  \\cite{Bruno2019}, by considering $P_\\mathcal{A}$ to be an $m \\times d$ matrix, \nwe find that the computational cost of $P_{\\mathcal{A}}$ is  $\\mathcal{O}(m^3+m^2d)$.\nTherefore,  by taking the computational cost  (\\ref{cost:b-p}) of the basic procedure and Proposition \\ref{prop:iteration-num-ma} into consideration, the cost of Algorithm \\ref{main algorithm} turns out to be\n\\begin{equation}\n\\label{eq: computational cost 1}\n\\mathcal{O} \\left( -\\frac{r}{\\log \\xi} \\log \\left( \\frac{1}{\\varepsilon}\\right) \\left( m^3+m^2d +  \\frac{1}{\\xi^2} p^2 r_{\\max}^2  \\left( \\max \\left( C^{\\rm sd} , md \\right) \\right) \\right) \\right)\n\\end{equation}\nwhere $C^{\\rm sd}$ is the computational cost of the spectral decomposition of $x \\in \\mathbb{E}$.\n\nNote that, in \\cite{Bruno2019}, the authors showed that the cost of their algorithm is\n\\begin{equation}\n\\label{eq: computational cost 2}\n\\mathcal{O} \\left( \\left( \\frac{r}{\\varphi(\\rho)} \\log \\left( \\frac{1}{\\varepsilon} \\right) - \\sum_{i=1}^p \\frac{r_i \\log(r_i)}{\\varphi(\\rho)}\\right) \\left( m^3 + m^2d +  \\rho^2 p^3 r_{max}^2 \\left(\\max \\left(C^{\\rm min} , md \\right)  \\right) \\right) \\right) \n\\end{equation}\nwhere $C^{\\rm min}$ is the cost of computing the minimum eigenvalue of $x \\in \\mathbb{E}$ with the corresponding idempotent.\n\nWhen the symmetric cone is simple, by setting $\\xi = \\frac{1}{2}$ and $\\rho = 2$, the maximum number of iterations of the basic procedure is bounded by the same value in both algorithms. Accordingly, we will compare the two computational costs (\\ref{eq: computational cost 1}) and (\\ref{eq: computational cost 2}) by supposing $\\xi = \\frac{1}{2}$ and $\\rho = 2$ (hence, $- \\log \\xi \\simeq 0.69$ and $\\varphi (\\rho) \\simeq 0.09$). As we can see below, the cost (\\ref{eq: computational cost 1}) of our method is smaller than (\\ref{eq: computational cost 2}) in the cases of linear programming and second-order cone problems and is equivalent to (\\ref{eq: computational cost 2}) in the case of semidefinite problems.\nFirst, let us consider the case where $\\mathcal{K}$ is the $n$-dimensional nonnegative orthant $\\mathbb{R}^n_+$.\nHere, we see that $r=p=d=n$, $r_1 = \\cdots = r_p = r_{\\max}=1$, and $\\max \\left( C^{\\rm sd} , md \\right) = \\max \\left( C^{\\rm min} , md \\right) = md$ hold.\nBy substituting these values, the bounds  (\\ref{eq: computational cost 1}) and  (\\ref{eq: computational cost 2}) turn out to be\n\\begin{equation}\n\\mathcal{O} \\left( \\frac{n}{0.69} \\log \\left( \\frac{1}{\\varepsilon} \\right) \\left( m^3 + m^2n + 4 mn^3 \\right) \\right)  \\notag\n\\end{equation}\nand\n\\begin{equation}\n\\mathcal{O} \\left( \\frac{n}{0.09} \\log \\left( \\frac{1}{\\varepsilon} \\right) \\left(  m^3 + m^2n + 4 mn^4 \\right) \\right).  \\notag\n\\end{equation}\nThis implies that for the linear programming case, our method (which is equivalent to Roos's original method \\cite{Roos2018}) is superior to Louren\\c{c}o et al.'s method \\cite{Bruno2019} in terms of bounds  (\\ref{eq: computational cost 1}) and  (\\ref{eq: computational cost 2}) .\n\nNext, let us consider the case where $\\mathcal{K}$ is composed of $p$ simple second-order cones $\\mathbb{L}^{n_i} \\ (i= 1, \\ldots, p)$, i.e., $\\mathcal{K} = \\mathbb{L}^{n_1} \\times \\mathbb{L}^{n_2} \\times \\cdots \\mathbb{L}^{n_p}$.\nIn this case, we see that $d = \\sum_{i=1}^p n_i$, $r_1 = \\cdots = r_p = r_{\\max}=2$ and $\\max \\left( C^{\\rm sd} , md \\right) = \\max \\left( C^{\\rm min} , md \\right) = md$ hold.\nBy substituting these values, the bounds  (\\ref{eq: computational cost 1}) and  (\\ref{eq: computational cost 2}) turn out to be\n\\begin{equation}\n\\mathcal{O} \\left( \\frac{2p}{0.69} \\log \\left( \\frac{1}{\\varepsilon} \\right) \\left( m^3 + m^2d + 16p^2md \\right) \\right)  \\notag\n\\end{equation}\nand\n\\begin{equation}\n\\mathcal{O} \\left(  \\frac{2p}{0.09} \\left( \\log \\left( \\frac{1}{\\varepsilon} \\right) -  \\log 2 \\right) \\left(  m^3 + m^2d + 16  p^3 md \\right) \\right).  \\notag\n\\end{equation}\nNote that $\\varepsilon$ is expected to be very small ($10^{-6}$ or even $10^{-12}$ in practice) and $\\frac{1}{0.69} \\log \\left( \\frac{1}{\\varepsilon} \\right) \\leq \\frac{1}{0.09} \\left( \\log \\left( \\frac{1}{\\varepsilon} \\right)  -  \\log 2 \\right)$ if $\\varepsilon \\leq 0.451$.\nThus, even in this case, we may conclude that our method is superior to Louren\\c{c}o et al.'s method in terms of the  bounds  (\\ref{eq: computational cost 1}) and  (\\ref{eq: computational cost 2}) .\n\nFinally, let us consider the case where $\\mathcal{K}$ is a simple $n \\times n$ positive semidefinite cone. We see that $p=1$, $r = n$, and $d = \\frac{n(n+1)}{2}$ hold, and upon substituting these values, the bounds (\\ref{eq: computational cost 1}) and (\\ref{eq: computational cost 2}) turn out to be\n\\begin{equation}\n\\mathcal{O} \\left( \\frac{n}{0.69} \\log \\left( \\frac{1}{\\varepsilon} \\right) \\left(m^3 + m^2n^2 + 4n^2  \\max \\left( C^{\\rm sd} , mn^2 \\right)  \\right) \\right)  \\notag\n\\end{equation}\nand \n\\begin{equation}\n\\mathcal{O} \\left( \\frac{n}{0.09} \\log \\left( \\frac{1}{\\varepsilon} \\right) \\left(m^3 + m^2n^2 + 4n^2 \\max ( C^{\\rm min} , mn^2) \\right) \\right).  \\notag\n\\end{equation}\nFrom the discussion in Section \\ref{sec: CSD and Cmin}, we can assume $\\mathcal{O}\\left( C^{\\rm sd} \\right) =  \\mathcal{O}\\left( C^{\\rm min} \\right)$, and  the computational bounds of two methods are equivalent.\n\n\\subsection{Comparison of Algorithm \\ref{main algorithm} and Pena and Soheili's method}\n\\label{sec: complexity of MA vs Pena}\n\n\\modifySecond{\nIn this section, we assume that $\\mathcal{K}$ is simple since Pena and Soheili's method does not support the direct product form. We also assume that $\\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K} \\neq \\emptyset$, because Pena and Soheili's method does not terminate if $\\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K} = \\emptyset$ and $\\mbox{{\\rm range}} \\mathcal{A}^* \\cap \\mbox{{\\rm int}} \\mathcal{K} = \\emptyset$. Furthermore, for the sake of simplicity, we assume that the main algorithm of Pena and Soheili's method applies only to $\\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K}$. (Their original method applies the main algorithm to $\\mbox{{\\rm range}} \\mathcal{A}^* \\cap \\mbox{{\\rm int}} \\mathcal{K}$ as well.)\n}\n\n\\modifySecond{\nFirst, we will briefly explain the idea of deriving an upper bound for the number of iterations required to find $x \\in \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K}$ in Pena and Soheili's method. Pena and Soheili derive it by focusing on the indicator $\\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K}) := \\max_x \\left \\{ {\\rm det}(x) \\mid x \\in \\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} , \\|x\\|_J^2 = r \\right\\}$. If $\\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K} \\neq \\emptyset$, then $\\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K}) \\in (0,1]$ holds, and if $e \\in \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K}$, then $\\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K}) =1$ holds. If $e \\in \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K}$, then the basic procedure terminates immediately and returns $\\frac{1}{r}e$ as a feasible solution. Then, they prove that $\\delta (Q_v \\left( \\mbox{Ker} \\mathcal{A} \\right) \\cap \\mbox{int} \\mathcal{K}) \\geq 1.5 \\cdot \\delta (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K})$ holds if the parameters are appropriately set, and derive an upper bound on the number of scaling steps, i.e., the number of iterations, required to obtain $\\delta (Q_v \\left( \\mbox{Ker} \\mathcal{A} \\right) \\cap \\mbox{int} \\mathcal{K}) = 1$.\n}\n\n\\modifySecond{\nIn the following, we obtain an upper bound for the number of iterations of the proposed method using the index $\\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\right) := \\max_x \\left \\{ {\\rm det}(x) \\mid x \\in \\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} , \\|x\\|_J^2 = 1 \\right \\}$. Note that $\\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\right) = r^{\\frac{r}{2}} \\cdot \\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\right)$. In fact, if $x^*$ is the point giving the maximum value of $\\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\right)$, then the point giving the maximum value of $\\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\right)$ is $\\sqrt{r} x^*$. Also, if $\\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K} \\neq \\emptyset$, then $\\delta^{{\\rm supposed}} (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K}) \\in (0,1\/ r^{\\frac{r}{2}} ]$, and if $\\frac{1}{\\sqrt{r}} e \\in \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K}$, then $\\delta^{{\\rm supposed}} (\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K}) =1\/ r^{\\frac{r}{2}}$.\n}\n\n\\modifySecond{\nThe outline of this section is as follows: First, we show that a lower bound for $\\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\right)$ can be derived using the index value $\\delta^{{\\rm supposed}} \\left( Q_{g^{-1}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\right) \\cap \\mbox{int} \\mathcal{K} \\right)$ for the problem after scaling (Proposition \\ref{prop:compare scaling}). Then, using this result, we derive an upper bound for the number of operations required to obtain $\\delta^{{\\rm supposed}} \\left( Q_{g^{-1}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\right) \\cap \\mbox{int} \\mathcal{K} \\right) = 1\/ r^{\\frac{r}{2}}$ (Proposition \\ref{pro:compare iteration}). Finally, we compare the proposed method with Pena and Soheili's method.\n}\n\n\\modifySecond{\nTo prove Proposition \\ref{prop:compare tr} used in the proof of Proposition \\ref{prop:compare scaling}, we use the following propositions from \\cite{JNW1934}.\n}\n\n\\modifySecond{\n\\begin{proposition}[Theorem 3 of~\\cite{JNW1934}]\n\\label{JNW_Th3}\nLet $c \\in \\mathbb{E}$ be an idempotent and $N_\\lambda (c)$ be the set such that $N_\\lambda (c) = \\{ x \\in \\mathbb{E} \\mid c \\circ x = \\lambda x\\}$.\nThen $N_\\lambda(c)$ is a linear maniforld, but if $\\lambda \\neq 0, \\frac{1}{2}$, and $1$, then $N_\\lambda (c)$ consists of zero alone. \\\\\nEach $x \\in \\mathbb{E}$ can be represented in the form\n\\begin{equation}\n\\notag\n\\begin{array}{lccc}\nx = u + v + w,\t&u \\in N_0(c),\t&v \\in N_{\\frac{1}{2}}(c),\t&w \\in N_1(c),\n\\end{array}\n\\end{equation}\nin one and only one way.\n\\end{proposition}\n}\n\n\\modifySecond{\n\\begin{proposition}[Theorem 11 of~\\cite{JNW1934}]\n\\label{JNW_Th11}\n$c \\in \\mathbb{E}$ is a primitive idempotent if and only if $N_1(c) =\\{ x \\in \\mathbb{E} \\mid c \\circ x = x\\} = \\mathbb{R} c$.\n\\end{proposition}\n}\n\n\\modifySecond{\n\\begin{proposition}\n\\label{prop:compare tr}\nLet $c \\in \\mathbb{E}$ be a primitive idempotent.\nThen, for any $x \\in \\mathbb{E}$, $\\langle x, Q_c(x) \\rangle = \\left( \\langle x,c \\rangle \\right)^2$ holds.\n\\end{proposition}\n\\begin{proof}\nFrom Propositions \\ref{JNW_Th3} and \\ref{JNW_Th11}, for any $x \\in \\mathbb{E}$, there exist a real number $\\lambda \\in \\mathbb{R}$ and elements $u \\in N_0(c)$ and $v \\in N_{\\frac{1}{2}}(c)$ such that $x =u + v + \\lambda c $. \\\\\nFirst, we show that $\\langle x,c \\rangle = \\lambda$. For $v \\in N_\\frac{1}{2} (c)$, we see that $\\langle v,c \\rangle = \\langle v, c \\circ c \\rangle = \\langle v \\circ c, c \\rangle = \\langle c \\circ v, c \\rangle = \\frac{1}{2} \\langle v, c \\rangle$, which implies that $\\langle v,c \\rangle =0$. Thus, since $u \\in N_0(c)$ and $u \\circ c = 0$, $\\langle x,c \\rangle$ is given by\n\\begin{equation}\n\\notag\n\\langle x,c \\rangle = \\langle u + v + \\lambda c , c  \\rangle  = \\langle u,c \\rangle + \\langle v,c \\rangle = \\lambda \\langle c,c \\rangle = 0 + 0 + \\lambda.\n\\end{equation}\nOn the other hand, by using the facts $x = u + v + \\lambda c$, $c^2 = c \\circ c = c$, $c \\circ u =0$ and $c \\circ v =\\frac{1}{2} v$ repeatedly, we have \n\\begin{align*}\n\\langle x, Q_c(x) \\rangle \n&= \\langle x, 2 c \\circ (c \\circ x) - c^2 \\circ x \\rangle \\\\\n&=\\langle x, 2 c \\circ (c \\circ (u + v + \\lambda c)) - c \\circ (u + v + \\lambda c)\\rangle \\\\\n&= \\langle x, 2 c \\circ (\\frac{1}{2} v + \\lambda c) - ( \\frac{1}{2}v + \\lambda c ) \\rangle \\\\\n&= \\langle x, (\\frac{1}{2} v + 2 \\lambda c) - ( \\frac{1}{2}v + \\lambda c )  \\rangle \\\\\n&= \\langle x,  \\lambda c  \\rangle =  \\lambda^2.\n\\end{align*}\nThus, we have shown that $\\langle x, Q_c(x) \\rangle = \\left( \\langle x,c \\rangle \\right)^2$ holds.\n\\end{proof}\n}\n\n\\modifySecond{\n\\begin{remark}\n\\label{remark:pena}\nIt should be noted that the proof of Proposition 3 in \\cite{Pena2017} is not correct since equation (14) does not necessarily hold.\nThe above Proposition \\ref{prop:compare tr} also gives a correct proof of Proposition 3 in \\cite{Pena2017}. See the computation $\\langle y , Q_{g^{-2}} (y) \\rangle$ in the proof of Proposition \\ref{prop:compare scaling}.\n\\end{remark}\n}\n\n\\modifySecond{\n\\begin{proposition}\n\\label{prop:compare scaling}\nSuppose that $\\mbox{{\\rm Ker}} \\mathcal{A} \\cap  \\mbox{{\\rm int}} \\mathcal{K} \\neq  \\emptyset$ and that, for a given nonempty index set $H \\subseteq \\{1 , \\dots r \\}$, Jordan frame $c_1 , \\dots , c_r $, and $0 < \\xi < 1$,\n\\begin{equation}\n\\notag\n\\langle c_i , x \\rangle  \\leq  \\xi \\  ( i \\in H), \\ \\  \\langle c_i , x \\rangle \\leq 1 \\ ( i \\notin H)\n\\end{equation}\nholds for any $x \\in F_{{\\rm P}_{S_{\\infty}}(\\mathcal{A})}$.\nDefine $g \\in \\mbox{\\rm int} \\mathcal{K}$ as\n\\begin{equation}\n\\notag\ng := \\sqrt{\\xi} \\sum_{h \\in H} c_h + \\sum_{h \\notin H} c_h \\ \\ \\ \\ \\mbox{i.e.,} \\ \\ \\ g^{-1} = \\frac{1}{\\sqrt{\\xi}} \\sum_{h \\in H} c_h + \\sum_{h \\notin H} c_h.\n\\end{equation}\nThen, the following inequality holds:\n\\begin{equation}\n\\notag\n\\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K} \\right) > \\xi \\cdot \\delta^{{\\rm supposed}} \\left( Q_{g^{-1}} \\left(\\mbox{{\\rm Ker}} \\mathcal{A} \\right) \\cap  {\\rm int} \\mathcal{K} \\right).\n\\end{equation}\n\\end{proposition}\n}\n\n\\begin{proof}\n\\modifySecond{\nFor simplicity of discussion, let $|H|=1$, i.e., $H =\\{i\\}$.\nLet us  define the points $x^*$, $y^*$, and $\\bar{x}^*$ as follows:\n\\begin{equation}\n\\begin{array}{l}\nx^*= \\argmax \\left\\{ \\det (x) \\mid x \\in \\mbox{{\\rm Ker}} \\mathcal{A}  \\cap {\\rm int} \\mathcal{K}, \\| x \\|^2_J = 1 \\right\\}, \\\\\ny^* = \\argmax \\left\\{ \\det (y) \\mid y \\in \\mbox{{\\rm Ker}} \\mathcal{A} \\cap {\\rm int} \\mathcal{K}, \\| Q_{g^{-1}} (y)\\|^2_J = 1 \\right\\}, \\\\\n\\bar{x} ^*= \\argmax \\left\\{ \\det (\\bar{x}) \\mid \\bar{x} \\in Q_{g^{-1}} \\left(\\mbox{{\\rm Ker}} \\mathcal{A} \\right) \\cap {\\rm int} \\mathcal{K}, \\| \\bar{x} \\|^2_J = 1 \\right\\}.\n\\end{array}\n\\notag\n\\end{equation}\nNote that the feasible region with respect to $y$ is the set of solutions whose norm is $1$ after scaling. \n\\medskip \\\\\nFirst, we show that $\\|y\\|_J^2 < 1$, and then $\\det (x^*) > \\det (y^*)$.\nProposition \\ref{q-det-relation} ensures that $\\|Q_{g^{-1}} (y)\\|_J^2 = \\langle Q_{g^{-1}} (y) , Q_{g^{-1}} (y) \\rangle = \\langle y , Q_{g^{-2}} (y) \\rangle$.\nTo expand $Q_{g^{-2}} (y)$, we expand the following equations by letting $a =  \\frac{1}{\\sqrt{\\xi}} - 1 $:\n\\begin{align*}\ng^{-2} &= e + (2a + a^2) c_i, \\\\\ng^{-4} &=  e + \\left( 2(2a + a^2) + (2a+a^2)^2 \\right) c_i \\\\\ng^{-2} \\circ y &= y + (2a+a^2) c_i \\circ y, \\\\\ng^{-2} \\circ ( g^{-2} \\circ y) &= y + 2(2a+a^2) c_i \\circ y + (2a+a^2)^2 c_i \\circ (c_i \\circ y), \\\\\ng^{-4} \\circ y &= y + \\left( 2(2a + a^2) + (2a+a^2)^2 \\right) c_i \\circ y.\n\\end{align*}\nThus, $Q_{g^{-2}} (y)$ turns out to be\n\\begin{align*}\nQ_{g^{-2}} (y)\n&=  2 g^{-2} \\circ ( g^{-2} \\circ y) - g^{-4} \\circ y \\\\\n&=  y +  2(2a+a^2) c_i \\circ y +  2(2a+a^2)^2 c_i \\circ (c_i \\circ y) - (2a+a^2)^2  c_i \\circ y \\\\\n&=  y +  2(2a+a^2) c_i \\circ y +  (2a+a^2)^2 Q_{c_i} (y), \\\\\n\\end{align*}\nand hence, we obtain $\\|Q_{g^{-1}} (y)\\|_J^2$ as \n\\begin{align*}\n\\langle y , Q_{g^{-2}} (y) \\rangle \n&= \\|y\\|_J^2 +  2(2a+a^2) \\langle y, c_i \\circ y \\rangle +  (2a+a^2)^2  \\langle y, Q_{c_i} (y) \\rangle \\\\\n&= \\|y\\|_J^2 +  2(2a+a^2) \\langle y \\circ y, c_i \\rangle +  (2a+a^2)^2  \\left( \\langle y, c_i \\rangle \\right)^2 \\\\\n\\end{align*}\nwhere the second equality follows from Proposition \\ref{prop:compare tr}. \nHere, $y \\in {\\rm int} \\mathcal{K}$ and  $c_i \\in \\mathcal{K}$imply that $\\langle y,c_i \\rangle >0$, and $y \\circ y = y^2 \\in {\\rm int} \\mathcal{K}$ implies  $\\langle y \\circ y,c_i \\rangle > 0$.\nNoting that $a > 0$ and $\\|Q_{g^{-1}} (y)\\|_J^2 = 1$, $\\|y\\|_J^2 < 1$ should hold and hence, $\\frac{1}{\\|y^*\\|_J} >1$, which implies that $\\det \\left(\\frac{1}{\\|y^*\\|_J} y^* \\right) > \\det(y^*)$.\nSince $\\left\\| \\frac{1}{\\|y^*\\|_J} y^* \\right\\|_J^2 = 1$ holds, we find that  $\\det(x^*) > \\det(y^*)$.\n}\n\n\\modifySecond{\nNext, we describe the lower bound for $\\det (y^*)$ using $\\det (\\bar{x}^*)$.\nSince the largest eigenvalue of $\\bar{x}$ satisfying $\\| \\bar{x} \\|_J^2 =1$ is less than $1$, by Proposition \\ref{prop:scaling}, we have:\n\\begin{equation}\n\\left\\{ Q_g(\\bar{x}) \\in \\mathbb{E} \\mid \\bar{x} \\in Q_{g^{-1}} \\left(\\mbox{{\\rm Ker}} \\mathcal{A} \\right) \\cap \\mathcal{K}, \\| \\bar{x} \\|^2_J = 1 \\right\\} \\subseteq \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mathcal{K}. \\notag\n\\end{equation}\nThis implies $\\det (y^*) \\geq \\det \\left( Q_g (\\bar{x}^*) \\right)$, and by Proposition \\ref{q-det-relation}, we have $\\det (y^*) \\geq \\det (g)^2 \\det (\\bar{x}^*) = \\xi^{|H|} \\det (\\bar{x}^*) = \\xi \\det (\\bar{x}^*)$.\nThus, $\\det(x^*) > \\det (y^*) \\geq \\xi \\det (\\bar{x}^*)$ holds, and we can conclude  that \n\\begin{equation}\n\\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K} \\right) > \\xi \\cdot \\delta^{{\\rm supposed}} \\left( Q_{g^{-1}} \\left(\\mbox{{\\rm Ker}} \\mathcal{A} \\right) \\cap  {\\rm int} \\mathcal{K} \\right). \\notag\n\\end{equation}\n}\n\\end{proof}\n\n\\modifySecond{\nNext, using Proposition \\ref{prop:compare scaling}, we derive the maximum number of iterations until the proposed method finds $x \\in \\mbox{{\\rm Ker}} \\mathcal{A} \\cap {\\rm int} \\mathcal{K}$ by using $\\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap {\\rm int} \\mathcal{K} \\right)$ as in Pena and Soheili's method.\n\\begin{proposition}\n\\label{pro:compare iteration}\nSuppose that  $\\mbox{{\\rm Ker}} \\mathcal{A} \\cap  \\mbox{{\\rm int}} \\mathcal{K} \\neq  \\emptyset$ holds.\nAlgorithm \\ref{main algorithm} returns $x \\in \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K}$ after at most $\\log_\\xi \\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K} \\right)$ iterations.\n\\end{proposition}\n\\begin{proof}\nLet $\\mbox{{\\rm Ker}} \\bar{\\mathcal{A}}$ be the linear subspace at the start of $k$ iterations of Algorithm \\ref{main algorithm} and suppose that  $\\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\bar{\\mathcal{A}} \\cap  {\\rm int} \\mathcal{K} \\right) = 1\/r^{\\frac{r}{2}}$ holds.\nThen, from Proposition \\ref{prop:compare scaling}, we find that \n\\begin{equation}\n\\notag\n\\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K} \\right) > \\frac{\\xi^k}{r^{\\frac{r}{2}}}.\n\\end{equation}\nThis implies that $\\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  \\mbox{int} \\mathcal{K} \\right) > \\xi^k$ since $\\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  \\mbox{int} \\mathcal{K} \\right) = r^{\\frac{r}{2}} \\cdot \\delta^{{\\rm supposed}} \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  \\mbox{int} \\mathcal{K} \\right)$ holds.\nBy taking the logarithm base $\\xi$,  we obtain $\\log_\\xi \\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K} \\right ) > k$.\n\\end{proof}\n}\n\n\\modifySecond{\nFrom here on, using the above results, we will compare the computational complexities of the methods in the case that $\\mathcal{K}$ is simple and $\\mbox{{\\rm Ker}} \\mathcal{A} \\cap \\mbox{{\\rm int}} \\mathcal{K} \\neq \\emptyset$ holds. Table \\ref{compare-PS-MA} summarizes the upper bounds on the number of iterations of the main algorithm (UB\\#iterations) of the two methods and the computational costs required per iteration (CC\/iterarion). As in the previous section, the main algortihm requires $\\mathcal{O} (m^3+m^2d)$ to compute the projection $\\mathcal{P}_\\mathcal{A}$. Here, BP shows the computational cost of the basic procedure in each method.\n\\begin{table}[H]\n\\caption{\\modifySecond{Comparison of the proposed method and Pena and Soheili's method in the main algorithm}}\n\\label{compare-PS-MA}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{c|cc}\\toprule\nMethod\t& UB\\#iterations\t\t& CC\/iteration  \\\\ \\midrule\nProposed method \t&$\\log_\\xi \\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K} \\right)$\t&$m^3+m^2d+$ BP \\\\\nPena and Soheili's method &$- \\log_{1.5} \\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K} \\right)$\t&$m^3+m^2d+$ BP\t \\\\ \\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\nThe upper bound on the number of iterations of the main algorithm of the proposed method is given by\n\\begin{equation}\n\\notag\n\\log_\\xi \\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K} \\right) = \\frac{\\log_{1.5} \\delta \\left( \\mbox{{\\rm Ker}} \\mathcal{A} \\cap  {\\rm int} \\mathcal{K} \\right)}{\\log_{1.5} \\xi},\n\\end{equation}\nwhere we should note that $0 < \\xi < 1$.\nSince $0 < \\frac{1}{-\\log_{1.5} \\xi} \\leq 1$ when $\\xi \\leq \\frac{2}{3}$,  if $\\xi \\leq \\frac{2}{3}$, then the upper bound on the number of iterations of the main algorithm of the proposed method is smaller than that of the main algorithm of Pena and Soheili's method.}\n\n\\modifySecond{\nNext, Table \\ref{compare-PS-BP} summarizes upper bounds on the number of iterations of basic procedures in the proposed method (UB\\#iterations) and Pena and Soheili's method and the computational cost required per iteration (CC\/iteration). In particular, it shows cases of using the von Neumann scheme and the smooth perceptron in each method (corresponding to Algorithm \\ref{basic procedure} and Algorithm \\ref{bp-alg-sp} in the proposed method). As in the previous section, $ C^{\\rm sd}$ denotes the computational cost required for spectral decomposition, and $C^{\\min}$ denotes the computational cost required to compute only the minimum eigenvalue and the corresponding primitive idempotent.\n\\begin{table}[H]\n\\caption{\\modifySecond{Comparison of the proposed method and Pena and Soheili's method in the basic procedure}}\n\\label{compare-PS-BP}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{c|cc|cc}\\toprule\n\t\t&\\multicolumn{2}{c|}{von Neumann scheme}\t&\\multicolumn{2}{c}{smooth perceptron} \\\\\nMethod\t& UB\\#iterations\t\t& CC\/iteration   & UB\\#iterations\t\t& CC\/iteration   \\\\ \\midrule\nProposed method\t&$\\frac{r^2}{\\xi^2}$\t&$\\max ( C^{\\rm sd} , md)$\t&$\\frac{2 \\sqrt{2}r}{\\xi} -1$\t&$\\max ( C^{\\rm sd} , md)$ \\\\\nPena and Soheili's method &$16r^4$\t&$\\max (C^{\\min} , md)$\t&$8 \\sqrt{2} r^2-1$\t\t&$\\max ( C^{\\rm sd} , md)$ \\\\ \\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\nNote that by setting $\\xi = \\frac{1}{4r}$, the upper bounds on the number of iterations of the basic procedure of the two methods are the same. If $\\xi = \\frac{1}{4r}$, then $\\frac{1}{-\\log_{1.5} \\xi} = \\frac{1}{\\log_{1.5} 4r} \\leq \\frac{1}{\\log_{1.5} 4} = 0.292$, and the upper bound of the number of iterations of the main algorithm of the proposed method is less than $0.3$ times the upper bound of the number of iterations of the main algorithm of Pena and Soheili's method, which implies that the larger the value of $r$ is, the smaller the ratio of those bounds becomes. \\\\ \nFrom the discussion in Section \\ref{sec: CSD and Cmin}, we can assume $\\mathcal{O}(C^{\\rm sd} = \\mathcal{C^{\\min}})$, and Table \\ref{compare-PS-BP} shows that the proposed method is superior for finding a point $x \\in \\mbox{{\\rm Ker}} \\mathcal{A} \\cap {\\rm int} \\mathcal{K}$.\n}\n\n\\subsection{Computational costs of $ C^{\\rm sd}$ and $C^{\\min}$}\n\\label{sec: CSD and Cmin}\n\n\\modifyEditor{\nThis section discusses the computational cost required for spectral decomposition $ C^{\\rm sd}$ and the computational cost required to compute only the minimum eigenvalue and the corresponding primitive idempotent $C^{\\min}$.}\n\n\\modifyEditor{\nThere are so-called direct and iterative methods for eigenvalue calculation algorithms, briefly described on pp.139-140 of \\cite{Demmel1997}.\n(Note that it is also written that there is no direct method in the strict sense of an eigenvalue calculation since finding eigenvalues is mathematically equivalent to finding zeros of polynomials).\n}\n\n\\modifyEditor{\nIn general, when using the direct method of $\\mathcal{O}(n^3)$, we see that $ C^{\\rm sd}=\\mathcal{O}(n^3)$ and $C^{\\min}=\\mathcal{O}(n^3)$. The Lanczos algorithm is a typical iterative algorithm used for sparse matrices. Its cost per iteration of computing the product of a matrix and a vector once is $\\mathcal{O}(n^2)$. Suppose the number of iterations at which we obtain a sufficiently accurate solution is constant with respect to the matrix size. In that case, the overall computational cost of the algorithm is $\\mathcal{O}(n^2)$. Corollary 10.1.3 in \\cite{Golub2013} discusses the number of iterations that yields sufficient accuracy. It shows that we can expect fewer iterations if the value of \"the difference between the smallest and second smallest eigenvalues \/ the difference between the second smallest and largest eigenvalue\" is larger. However, it is generally difficult to assume that the above value does not depend on the matrix size and is sufficiently large. Thus, even in this case, we cannot take advantage of the condition that we only need the minimum eigenvalue, and we conclude that it is reasonable to consider that $\\mathcal{O}(C^{\\rm sd})=\\mathcal{O}(C^{\\min})$.\n}\n\n\n\n\\section{Numerical experiments}\n\\label{sec: numerical experiments}\n\\subsection{Outline of numerical implementation}\n\\label{sec: outline of numerical implementation}\n\nNumerical experiments were performed\n\\modifyThird{using the authors' implementations of the algorithms}\non a positive semidefinite optimization problem with one positive semidefinite cone $\\mathcal{K} = \\mathbb{S}^n_{+}$ of the form \n\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm P} (\\mathcal{A})& \\mbox{find}   &  X  \\\\\n\t&\\mbox{s.t.} &  \\mathcal{A} (X) = \\bm{0} \\in \\mathbb{R}^m \\\\\n\t&       &  X \\in \\mathbb{S}^n_{++} \\notag\n\\end{array}\n\\end{equation}\nwhere $\\mathbb{S}^n_{++}$ denotes the interior of $\\mathcal{K} = \\mathbb{S}^n_{+}$.\nWe created instances of the following three types:\n\n\\begin{itemize}\n\\item \\modifyFirst{Strongly feasible ill-conditioned instances, i.e.,   $\\mbox{Ker} \\mathcal{A} \\cap  \\mathbb{S}^n_{++} \\neq  \\emptyset$ and $X \\in \\mbox{Ker} \\mathcal{A} \\cap  \\mathbb{S}^n_{++}$ has positive but small eigenvalues.}\n\\item Weakly feasible instances, i.e., $\\mbox{Ker} \\mathcal{A} \\cap  \\mathbb{S}^n_{++} = \\emptyset$, but $\\mbox{Ker} \\mathcal{A} \\cap  \\mathbb{S}^n_{+} \\setminus \\{O\\} \\neq  \\emptyset$.\n\\item Infeasible instances, i.e., $\\mbox{Ker} \\mathcal{A} \\cap  \\mathbb{S}^n_{+} = O $.\n\\end{itemize}\n\nWe will explain how to make each type of instance in section \\ref{sec: instances}.\n\nIn what follows, we refer to Louren\\c{c}o et al.'s method~\\cite{Bruno2019} as Louren\\c{c}o (2019), and Pena and Soheili's method~\\cite{Pena2017} as Pena (2017).\n\nWe set the termination parameter as $\\xi = \\frac{1}{4}$ in our basic procedure, i.e., Algorithm \\ref{basic procedure}.\n\\modifySecond{The reason for setting $\\xi=1\/4$ is to prevent the square root of $\\xi$ from becoming an infinite decimal, and to prevent the upper bound on the number of iterations of the basic procedure from becoming too large.}\nWe also set the accuracy parameter as $\\varepsilon$ = 1e-12, both in our main algorithm (Algorithm \\ref{main algorithm}, Algorithm \\ref{main algorithm 2}) and in Louren\\c{c}o (2019) and  determined whether ${{\\rm P}_{S_\\infty} (\\mathcal{A})}$ or  ${{\\rm P}_{S_1} (\\mathcal {A})}$ has a solution whose minimum eigenvalue is greater than or equal to $\\varepsilon$.\nHere,  we call a solution whose minimum eigenvalue is $\\varepsilon$ or more an {\\em $\\varepsilon$-feasible solution}.\n\nNote that \\cite{Pena2017} proposed various update methods for the basic procedure.\n\\modifySecond{In our numerical experiments, all methods employed the modified von Neumann scheme (Algorithm \\ref{bp-alg-mvn}) with the identity matrix as the initial point and the smooth perceptron scheme (Algorithm \\ref{bp-alg-sp}).}\nThis implies that the basic procedures used in the three methods differ only in the termination conditions for moving to the main algorithm and that all other steps are the same.\n\nAll executions were performed using MATLAB R2022a on an Intel (R) Core (TM) i7-6700 CPU @ 3.40GHz machine with 16GB of RAM. \\modifySecond{Note that we computed the projection $\\mathcal{P}_\\mathcal{A}$ using the MATLAB function for the singular value decomposition. The projection $\\mathcal{P}_\\mathcal{A}$ was given by $\\mathcal{P}_\\mathcal{A} = I - A^\\top (AA^\\top)^{-1} A$ using the matrix $A \\in \\mathbb{R}^{m \\times d}$ which represents the linear operator $\\mathcal{A}(\\cdot)$ and the identity matrix $I$. Here, suppose that the singular value decomposition of a matrix $A$ is given by \n\\begin{equation}\n\\notag\nA = U \\Sigma V^\\top\n=\nU\n(\\Sigma_m \\ O)\nV^\\top\n\\end{equation}\nwhere $U \\in \\mathbb{R}^{m \\times m}$ and $V \\in \\mathbb{R}^{d \\times d}$ are orthogonal matrices, and $\\Sigma_m \\in \\mathbb{R}^{m \\times m}$ is a diagonal matrix with $m$ singular values on the diagonal. \nSubstituting this decomposition into $A^\\top (AA^\\top)^{-1} A$, we have \n\\begin{align*}\nA^\\top (AA^\\top)^{-1} A\n&= A^\\top (U \\Sigma \\Sigma^\\top U^\\top)^{-1} A \\\\\n&= A^\\top U^{-\\top} (\\Sigma_m^2)^{-1} U^{-1} A \\\\\n&= V \\Sigma^\\top \\Sigma_m^{-2} \\Sigma V^\\top \\\\\n&= V\n\\begin{pmatrix}\nI_m & O \\\\\nO & O\n\\end{pmatrix}\nV^\\top \\\\\n&= V_{:, 1:m} V_{:,1:m}^\\top,\n\\end{align*}\nwhere $V_{:, 1:m}$ represents the submatrix from column $1$ to column $m$ of $V$.\nThus, for any $x \\in \\mathbb{E}$, we can compute $\\mathcal{P}_\\mathcal{A}(x) = x - V_{:, 1:m} V_{:,1:m}^\\top x$.\n}\n\n\nFor each method, we observed the total number of iterations of the basic procedure, the number of iterations of the main algorithm, and the total CPU time.\nWe also examined the violation degrees of the output-result, as defined below, and the residual of the constraints for the output-result.\n\nWe classified the output-results into five types: A: an interior feasible solution is found; B: no interior feasible solution is found (ver.1); C: no $\\varepsilon$-feasible solution is found (only for Loren\\c{c}o (2019) and our method); D: no interior feasible solution is found (ver.2; only for Pena (2017)); E: Out-of-time. In what follows, we briefly explain how output-result type D for Pena (2017) differs from output-result type B.\n\n\\cite{Pena2017} pointed out that if ${\\rm P}(\\mathcal{A})$ has no interior feasible solution, meaning that if the main algorithm of Pena (2017) is applied to only ${\\rm P}(\\mathcal{A})$, it does not stop within a finite number of iterations. To overcome this problem, Pena et al. constructed the main algorithm in a way that it applies not only to ${\\rm P}(\\mathcal{A})$ but also to problem ${\\rm Q}(\\mathcal{A})$:\n\n\\begin{equation}\n\\begin{array}{lll}\n{\\rm Q}(\\mathcal{A})&\t\\mbox{find}\t& X   \\\\\n&\\mbox{s.t.}\t& X \\in \\mbox{range} \\mathcal{A}^*, \\\\\n&\t\t& X \\in \\mathbb{S}^n_{++}. \\notag\n\\end{array}\n\\end{equation}\n\nAccordingly, we defined output-result type B as the case where a feasible solution of ${\\rm D} (\\mathcal{A})$ is obtained by applying the main algorithm to ${\\rm P}(\\mathcal{A})$ and defined output-result type D as the case where a feasible solution of ${\\rm Q}(\\mathcal{A})$ is obtained by applying the main algorithm to ${\\rm Q}(\\mathcal{A})$.\n\n\nIn what follows, $\\bar{X} \\in \\mathbb{S}^n$ denotes the output obtained from the main algorithm and $X^*$ the result scaled as the solution of the original problem ${\\rm P}(\\mathcal{A})$ (or ${\\rm D} (\\mathcal{A})$ or ${\\rm Q} (\\mathcal{A})$) multiplied by a real number such that the maximum eigenvalue is 1. We defined the violation degree of the output-result as follows:\n\n\\begin{itemize}\n\\item For output-result type A, the violation degree was defined as the number of eigenvalues of $X^*$ (i.e., the solution of ${\\rm P}(\\mathcal{A})$), whose value is less than or equal to $\\varepsilon$. \n\\item For output-result type B, the violation degree was the number of eigenvalues of $X^*$ (i.e., the solution of ${\\rm D}(\\mathcal{A})$), whose value is less than or equal to $0$.\n\\item For output-result type D, the violation degree was the number of eigenvalues of $X^*$ (i.e., the solution of ${\\rm Q}(\\mathcal{A})$), whose value is less than or equal to $\\varepsilon$. \n\\end{itemize}\nWhen output-result type A was obtained, we defined the residual of the constraints as the value of $\\| \\mathcal{A}(X^*) \\|_2$.\n\n\\modifySecond{\nWe also solved the following problem with a commercial code, Mosek~\\cite{Mosek}, and compared it with the output of Chubanov's methods:\n\\begin{equation}\n\\begin{array}{cccccc}\n{\\rm (P)}\t&\\min\t\t&0\t&{\\rm s.t}\t&\\mathcal{A} (X) = \\bm{0},\t&X \\in \\mathbb{S}^n_{+}.\t\\\\\n{\\rm (D)}\t&\\max\t&\\bm{0}^\\top y\t&{\\rm s.t}\t&-\\mathcal{A}^* y \\in \\mathbb{S}^n_{+}.\t\n\\end{array}\n\\notag\n\\end{equation}\nHere, Mosek solves the self-dual embedding model by using a path-following interior-point method, so if we obtain a solution $(X^*, y^*)$, then $X^*$ and $-\\mathcal {A}^* y^*$ lie in the (approximate) relative interior of the primal feasible region and the dual feasible region, respectively~\\cite{Handbook}. That is, $X^*$ obtained by solving a strongly feasible problem with Mosek is in $\\mathbb{S}^n_{++}$, $X^*$ obtained by solving a weakly feasible problem is in $\\mathbb{S}^n_+ \\setminus \\mathbb{S}^n_{++}$, and $X^*$ obtained by solving an infeasible problem is $X^*=O$ (i.e., $-\\mathcal{A}^* y^* \\in \\mathbb{S}^n_{++}$). As well as for Chubanov's methods, we computed $\\| \\mathcal{A}(X^*) \\|_2$ for the solution obtained by Mosek after scaling so that the largest eigenvalue of $X^*$ would be 1.\n}\n\n\\modifySecond{\nNote that (P) and (D) do not simultaneously have feasible interior points; i.e., the Slater constraint does not hold for both the primal and the dual problems. In general, it is difficult to solve such problems stably by using interior point methods, but since strong complementarity exists between (P) and (D), they can be expected to be stably solved. By applying Lemma 3.4 of ~\\cite{Lourenco2021}, we can generate a problem in which both the primal and dual problems have feasible interior points in which it can be determined whether (P) has a feasible interior point. However, since there was no big difference between the solution obtained by solving the problem generated by applying Lemma 3.4 of ~\\cite{Lourenco2021} and the solution obtained by solving the above (P) and (D), we showed only the results of solving (P) and (D) above.\n}\n\n\n\\subsection{How to generate instances}\n\\label{sec: instances}\nHere, we describe how the strongly feasible instances, weakly feasible instances, and infeasible instances were generated.\n\n\\modifyFirst{\nNote that, due to the rounding error of the numerical computation, the weakly (ill-conditioned strongly) feasible instances generated in this experiment may not have been weakly (ill-conditioned strongly) feasible but rather \"pseudo weakly (pseudo ill-conditioned strongly) feasible.\"\n }\n\nIn what follows, for any natural numbers $m,n$, $\\mbox{rand}(n)$ is a function that returns $n$-dimensional real vectors whose elements are uniformly distributed in the open segment $(0,1)$, and $\\mbox{rand}(m,n)$ is a function that returns an $m \\times n$ real matrix whose elements are uniformly distributed in the open segment $(0,1)$. Furthermore, for any $x \\in \\mathbb{R}^n$ and $X \\in \\mathbb{R}^{m\\times n}$, $\\mbox{diag}(x) \\in \\mathbb{R}^{n \\times n}$ is a function that returns a diagonal matrix whose diagonal elements are the elements of $x$, and $\\mbox{vec}(X) \\in \\mathbb{R}^{mn}$ is a function that returns a vector obtained by stacking the $n$ column vectors of $X$.\n\n\n\\subsubsection{Strongly feasible instances}\n\\label{sec: strongly feasible  instances}\n\nThe strongly feasible instances were generated by extending the method of generating ill-conditioned strongly feasible instances proposed in~\\cite{Pena2019} to the symmetric cone case.\n\n\\begin{proposition}\n\\label{prop: strongly_feasible}\nSuppose that $\\bar{x} \\in \\rm{int} \\mathcal{K}$, $\\| \\bar{x} \\|_\\infty \\leq 1$ and $\\bar{u} \\in \\mathcal{K} , \\| \\bar{u} \\|_1 = r$ satisfy $\\langle \\bar{x} , \\bar{u} \\rangle = r$.\nDefine the linear operator $\\mathcal{A} : \\mathbb{E} \\rightarrow \\mathbb{R}^m$ as $\\mathcal{A}(x) = ( \\langle a_1, x \\rangle , \\langle a_2, x \\rangle , \\dots ,\\langle a_m, x \\rangle)^T$ for which $a_1 = \\bar{u} - \\bar{x}^{-1}$ and $\\langle a_j , \\bar{x} \\rangle = 0$ hold for any $j = 2,\\dots,m$.\nThen, \n\\begin{equation}\n\\bar{x} = \\argmax_x \\left \\{ \\det(x) : x \\in \\mathcal{K} \\cap \\mbox{\\rm ker}\\mathcal{A}, \\| x \\|_\\infty = 1 \\right \\}. \n\\label{eq:0}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nFirst, note that the assertion (\\ref{eq:0}) is equivalent to\n\\begin{equation}\n\\bar{x} = \\argmax_{x \\in \\mathcal{F}}  \\left \\{ \\log \\det(x) \\right \\}  \n\\ \\mbox{where} \\ \n\\mathcal{F} := \\left \\{ x \\in \\mathcal{K} \\cap \\mbox{ker}\\mathcal{A} : \\| x \\|_\\infty \\leq 1 \\right \\}.\n\\label{eq:1}\n\\end{equation}\nFrom the assumptions, we see that $\\bar{x} \\in \\mathcal{K}$, $\\| \\bar{x} \\|_\\infty \\leq 1$ and $\\langle a_1 , \\bar{x} \\rangle = \\langle \\bar{u} - \\bar{x}^{-1} , \\bar{x} \\rangle = r - r = 0$; thus, $\\mathcal{A}(\\bar{x}) = 0$ and $\\bar{x} \\in \\mathcal{F}$. Since $\\nabla \\log \\det (x) = x^{-1}$, if $\\bar{x}$ satisfies \n\\begin{equation}\n\\langle x - \\bar{x} , \\bar{x}^{-1} \\rangle \\leq 0 \\ \\mbox{for any} \\ x \\in \\mathcal{F} \\label{eq:2}\n\\end{equation}\nwe can conclude that (\\ref{eq:1}) holds. In what follows, we show that (\\ref{eq:2}) holds.\n\nFor any $x \\in \\mathcal{F}$, $x \\in \\mbox{ker}\\mathcal{A}$ and hence, $\\langle a_1 , x \\rangle = \\langle \\bar{u} - \\bar{x}^{-1} , x \\rangle = \\langle \\bar{u} - \\bar{x}^{-1} , x \\rangle = 0$, i.e., $\\langle \\bar{u} , x \\rangle = \\langle \\bar{x}^{-1} , x \\rangle$.\nThus, we obtain\n\\begin{align*}\n\\langle x - \\bar{x} , \\bar{x}^{-1} \\rangle &= \\langle \\bar{u} , x \\rangle - r \\\\\n&\\leq \\langle \\bar{u} , x \\rangle - \\| \\bar{u} \\|_1 \\| x \\|_\\infty \\hspace{1cm}(\\mbox{by} \\ \\| \\bar{u} \\|_1 = r \\ \\mbox{and} \\  \\| x \\|_\\infty \\leq 1) \\\\\n&\\leq 0 \\hspace{1cm}(\\mbox{by} \\ \\langle \\bar{u} , x \\rangle \\leq \\| \\bar{u} \\|_1 \\| x \\|_\\infty)\n\\end{align*}\nwhich completes the proof.\n\\end{proof}\n\nProposition \\ref{prop: strongly_feasible} guarantees that we can generate a linear operator $\\mathcal{A}$ satisfying $\\mbox{Ker} \\mathcal{A} \\cap  \\mathbb{S}^n_{++} \\neq  \\emptyset$ by determining an appropriate value $\\mu = \\displaystyle \\max_{ X \\in \\mathcal{F}} \\det(X)$, where $\\mathcal{F} = \\{ X \\in \\mathbb{S}^n : X \\in \\mathbb{S}^n_{++} \\cap \\mbox{Ker} \\mathcal{A} , \\|X\\|_\\infty = 1\\}$.\n\nThe details on how to generate the strongly feasible instances are in Algorithm \\ref{strongly feasible problem}. The input consists of the rank of the semidefinite cone $n$, the number of constraints $m$, an arbitrary orthogonal matrix $P$, \\modifyFirst{and the parameter $\\tau \\in \\mathbb{R}_{++}$ which determines the value of $\\mu$. We made instances for which the value of $\\mu$ satisfies $1e-\\tau \\leq \\mu \\leq 1e-(\\tau-1)$. In the experiments, we set $\\tau \\in \\{50,100,150, 200, 250 \\}$ so that $\\mu$ would vary around 1e-50, 1e-100, 1e-150, 1e-200, and 1e-250; i.e., strongly feasible, but ill-conditioned instances.}\n\n\\modifyFirst{\nNote that Algorithm \\ref{strongly feasible problem} generates instances using $\\bar{x}$ that has a natural eigenvalue distribution. For example, let $n-1=3$ and consider two $X$s where one has 3 eigenvalues of about 1e-2, and the others have 1 each of 1e-1, 1e-2, and 1e-3. $\\det(X) \\simeq $1e-6 is obtained for both $X$s, but the latter is more natural for the distribution of eigenvalues. In our experiment, we generated ill-conditioned instances by using $X$ having a natural eigenvalue distribution as follows:\n\\begin{enumerate}\n\\item Find an integer $s$ that satisfies 1e-$s$ $\\leq l^{\\frac{1}{n-1}} \\leq u^{\\frac{1}{n-1}} \\leq$ 1e-$(s-1)$.\n\\item Generate $t = 2s-1$ eigenvalue classes.\n\\item Decide how many eigenvalues to generate for each class.\n\\end{enumerate}\n}\n\n\\modifyFirst{\nFor example, when $n=13$ and $\\tau = 30$, Algorithm \\ref{strongly feasible problem} yields $s=3$, $t=5$, $a=2$ and $b=2$, and since $b$ is even, we have $num = (2,3,2,3,2)^\\top$. The classes of $t=5$ eigenvalues are shown in Table \\ref{example1} below. Note that $\\left( l^{\\frac{1}{n-1}} \\cdot 10^{s-i} \\right) \\cdot \\left( l^{\\frac{1}{n-1}} \\cdot 10^{s-(t-i+1)} \\right)= l^{\\frac{2}{n-1}}$, $\\left( u^{\\frac{1}{n-1}} \\cdot 10^{s-i} \\right) \\cdot \\left( u^{\\frac{1}{n-1}} \\cdot 10^{s-(t-i+1)} \\right)= u^{\\frac{2}{n-1}}$ holds for the $i$-th and $t-i+1$-th classes. This implies that we obtain $1e-\\tau \\leq \\mu = \\det(X) \\leq 1e-(\\tau-1)$ both when generating $n-1$ eigenvalues in the $s$th class and when generating $n-1$ eigenvalues of $X$ according to $num$. When $n=14, \\tau = 30$, Algorithm \\ref{strongly feasible problem} gives $s=3$, $t=5$, $a=2$, and $b=3$, and since $b$ is an odd number, we have $num = (2,3,3,3,2)^\\top$. Thus, Algorithm \\ref{strongly feasible problem} generates the instances by controlling the frequency so that the geometric mean of the $n-1$ eigenvalues of $X$ falls within the $s$-th class width. \n\\begin{table}[H]\n\\caption{\\modifyFirst{Frequency distribution table of eigenvalues of $X$ generated by Algorithm \\ref{strongly feasible problem} when $n=13$ or $n=14$, $\\tau=30$}}\n\\label{example1}\n\\begin{center}\n\\modifyFirst{\n\\begin{tabular}{ccccc} \\toprule\nClass\t&\\multicolumn{2}{c}{Class width of eigenvalues of $\\bar{x}$}\t\t\t\t\t&\\multicolumn{2}{c}{Frequency($num$)}\\\\ \\midrule\n\t& Lower bound \t\t& Upper bound\t\t\t\t\t\t&$n=13$\t\t&$n=14$\t\\\\ \\midrule\n1\t&$ l^{\\frac{1}{n-1}} \\cdot 10^{2}$\t&$u^{\\frac{1}{n-1}} \\cdot 10^{2}$\t&2\t\t\t&2\\\\\n2\t&$ l^{\\frac{1}{n-1}} \\cdot 10^{1}$\t&$u^{\\frac{1}{n-1}} \\cdot 10^{1}$\t&3\t\t\t&3\\\\\n3\t&$ l^{\\frac{1}{n-1}}$\t\t\t&$u^{\\frac{1}{n-1}}$\t\t\t&2\t\t\t&3\\\\\n4\t&$ l^{\\frac{1}{n-1}} \\cdot 10^{-1}$&$u^{\\frac{1}{n-1}} \\cdot 10^{-1}$\t&3\t\t\t&3\\\\\n5\t&$ l^{\\frac{1}{n-1}} \\cdot 10^{-2}$&$u^{\\frac{1}{n-1}} \\cdot 10^{-2}$\t&2\t\t\t&2\\\\ \\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n}\n\n\\begin{algorithm}[H]\n\\caption{\\modifyFirst{Strongly feasible instance}}\n\\label{strongly feasible problem}\n\\begin{algorithmic}[1]\n \\renewcommand{\\algorithmicrequire}{\\textbf{Input: }}\n \\renewcommand{\\algorithmicensure}{\\textbf{Output: }}\n\n \\STATE \\algorithmicrequire $n, m, \\tau, P$\n \\STATE \\algorithmicensure $A$ \n \\STATE $l \\leftarrow 1{\\rm e}-\\tau$ and $u \\leftarrow 1{\\rm e}-(\\tau-1)$\n \\STATE $s \\leftarrow \\lceil \\frac{\\tau}{n-1} \\rceil$\n \\STATE $t \\leftarrow 2s-1$\n \\STATE $b \\leftarrow (n-1) \\bmod t$ and $a \\leftarrow \\frac{(n-1) - b}{t}$\n \\STATE $num \\leftarrow a \\cdot \\bm{1} \\in \\mathbb{R}^t$\n \\IF {$b$ is odd}\n \\STATE $\\bar{b} \\leftarrow \\frac{b-1}{2}$\n \\STATE $num_i \\leftarrow num_i + 1$ such that $s - \\bar{b} \\leq i \\leq s + \\bar{b}$\n \\ELSE \n \\STATE $\\bar{b} \\leftarrow \\frac{b}{2}$\n \\STATE $num_i \\leftarrow num_i + 1$ such that $s - \\bar{b} \\leq i < s$ or $s < i \\leq  s + \\bar{b}$\n \\ENDIF\n \\STATE $d_1 \\leftarrow 1$\n \\STATE $k \\leftarrow 2$\n \\FOR {$i = 1$ \\mbox{to} $t$}\n \\FOR {$j=1$ \\mbox{to} $num_i$}\n \\STATE $dl \\leftarrow l^{\\frac{1}{n-1}} \\cdot 10^{s-i}$and $du \\leftarrow u^{\\frac{1}{n-1}} \\cdot 10^{s-i}$\n \\STATE $d_k \\leftarrow  dl + \\left( du-dl \\right) \\mbox{rand} \\left( 1 \\right)$ \n \\STATE $k \\leftarrow k+1$\n \\ENDFOR\n \\ENDFOR\n \\STATE $D' \\leftarrow \\mbox{diag}(d)$ and then compute $C \\leftarrow PD'P^T $ and $c \\leftarrow \\mbox{vec}(C)  $\n \\STATE $u \\leftarrow (n , 0_{n-1}^T )^T$ where $0_{n-1}$ denotes the $n-1$-dimensional vector of zeros   \n \\STATE $U \\leftarrow P(\\mbox{diag}(u) - {D'}^{-1})P^T$\n \\STATE $A' \\leftarrow \\mbox{vec}(U)$\n \\STATE $R \\leftarrow I - \\frac{1}{\\|c\\|_2^2} cc^T $\n \\FOR {$ i = 1 \\mbox{ to  }  m-1 $}\n \\STATE $A_i' \\leftarrow \\mbox{rand}(n,n)$ and $A_i \\leftarrow \\left( A_i' + (A_i')^T \\right)\/2 $\n \\STATE $A' \\leftarrow \n\\begin{pmatrix}\nA' \\\\\n\\mbox{vec}(A_i)^T \\\\\n\\end{pmatrix}$\n \\ENDFOR\n \\STATE $\\bar{A} \\leftarrow A'R$\n \\STATE $A \\leftarrow \\begin{pmatrix}\n\\mbox{vec}(U)^T \\\\\n\\bar{A}\n\\end{pmatrix}$\n\\end{algorithmic} \n\\end{algorithm}\n\n\n\\begin{proposition}\nFor any $A \\in \\mathbb{R}^{m \\times n^2}$ returned from Algorithm \\ref{strongly feasible problem},  there exists $X \\in \\mathbb{S}^n_{++}$ satisfying $A\\left( \\mbox{vec} (X) \\right)=\\bm{0}$.\n\\end{proposition}\n\n\\begin{proof}\nWe can see that the matrix $C\\in \\mathbb{S}^n_{++}$ computed on line 7 of  Algorithm \\ref{strongly feasible problem} satisfies \n\\begin{equation}\nA \\mbox{vec}(C) = Ac = \n\\begin{pmatrix}\n\\mbox{vec}(U)^Tc \\\\\n\\bar{A}c\n\\end{pmatrix}= \n\\begin{pmatrix}\nn-n \\\\\nA'Rc\n\\end{pmatrix}= \\bm{0}.\\notag\n\\end{equation}\n\\end{proof}\n\n\n\\subsubsection{Weakly feasible instances}\n\\label{sec: weakly feasible  instances}\n\nThe weakly feasible instances were generated by Algorithm \\ref{weakly feasible instance}.\n\n\\begin{algorithm}[H]\n\\caption{Weakly feasible instance}\n\\label{weakly feasible instance}\n\\begin{algorithmic}[1]\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input: }}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output: }}\n\n \\STATE \\algorithmicrequire $n , m$, $A' = [ \\ \\ ]$\n \\STATE \\algorithmicensure $A$ \n \\STATE $B \\leftarrow \\mbox{rand} (n,n)$        \\hspace{3.3cm}\/\/ $B$ must not be $O$\n \\STATE $C \\leftarrow \\frac{B+B^T}{2}$                     \\hspace{3cm}\/\/ $C \\neq O$ must not be $C \\succeq O$ or  $C \\preceq O$\n \\STATE $C_+ \\leftarrow \\mathcal{P}_{\\mathbb{S}^n_+} (C)$  \\hspace{3.5cm}\/\/ $C_+ \\neq O$ since $C \\neq O$ is not negative semidefinite.\n \\STATE $C_- \\leftarrow -\\mathcal{P}_{\\mathbb{S}^n_+} (-C)$   \\hspace{3cm}\/\/ $C_- \\neq O$ since $C \\neq O$ is not  positive semidefinite.\n \\STATE $c_+ \\leftarrow \\mbox{vec} (C_+) $ and $R \\leftarrow I - \\frac{1}{\\|c_+\\|_2^2} c_+c_+^T $\n \\FOR {$ i = 1 \\mbox{ to  }  m-1 $}\n \\STATE $A_i' \\leftarrow \\mbox{rand}(n,n)$ and $A_i \\leftarrow \\left( A_i' + (A_i')^T \\right)\/2 $\n \\STATE $A' \\leftarrow  \n\\begin{pmatrix}\nA' \\\\\n\\mbox{vec}(A_i)^T \\\\\n\\end{pmatrix}$ \n \\ENDFOR\n \\STATE $A \\leftarrow \n\\begin{pmatrix}\n\\mbox{vec}(C_-)^T \\\\\nA'R\n\\end{pmatrix}$\n\\end{algorithmic} \n\\end{algorithm}\n\n\\begin{proposition}\nFor any $A \\in \\mathbb{R}^{m \\times n^2}$ returned by Algorithm \\ref{weakly feasible instance}, no $X \\in \\mathbb{S}^n_{++}$ exists that satisfies $A\\left( \\mbox{\\rm vec} (X) \\right)= \\bm{0}$, but an $X \\in \\mathbb{S}^n_{+} \\setminus \\{ O \\}$ exists that satisfies $A\\left( \\mbox{\\rm vec} (X) \\right)= \\bm{0}$.\n\\end{proposition}\n\\begin{proof}\nFirst, we show that an $X \\in \\mathbb{S}^n_{+} \\setminus \\{ O \\}$ exists that satisfies $A\\left( \\mbox{vec} (X) \\right)= \\bm{0}$. For the matrix $C_+ \\in \\mathbb{S}^n_{+}$ computed on line 5 of Algorithm \\ref{weakly feasible instance}, we see that $C_+ \\neq O$ and the following holds:\n\\begin{equation}\nA\\left( \\mbox{vec} (C_+) \\right)=  Ac_+ = \n\\begin{pmatrix}\n\\mbox{vec}(C_-)^T \\\\\nA'R\n\\end{pmatrix} c_+ \n=\n\\begin{pmatrix}\n\\mbox{vec}(C_-)^T c_+ \\\\\nA'R c_+\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n\\bm{0} \\\\\nA'( c_+ - c_+)\n\\end{pmatrix}\n= \\bm{0}. \\notag\n\\end{equation}\nNext, we show by contradiction that no $X \\in \\mathbb{S}^n_{++}$ exists that satisfies $A\\left( \\mbox{vec} (X) \\right)= \\bm{0}$. Suppose that an $X \\in \\mathbb{S}^n_{++}$ satisfies $A\\left( \\mbox{vec} (X) \\right)= \\bm{0}$. Since the first row of $A$ is $\\mbox{vec}(C_-)^T$, if $A\\left( \\mbox{vec} (X) \\right)= \\bm{0}$ holds, then $\\mbox{vec}(C_-)^T \\mbox{vec}(X) = 0$, i.e., \n\\begin{align*}\n\\mbox{vec}(C_-)^T \\mbox{vec}(X) &= \\langle C_- , X \\rangle = \\langle PDP^T, QEQ^T \\rangle  \\\\\n&= \\langle D ,P^TQEQ^TP \\rangle = \\sum_{i=1}^n D_{ii} \\left( P^TQEQ^TP \\right)_{ii} = 0\n\\end{align*}\nwhere $C_-=PDP^T$, $X = QEQ^T$, $P$ are $Q$ orthogonal matrices, and $D$ and $E$ are diagonal matrices. Here, $X \\in \\mathbb{S}^n_{++}$ implies $\\left( P^TQEQ^TP \\right)_{ii} > 0$ for any $ i \\in \\{ 1 , \\dots, n \\}$ and hence, $D$ should be $O$ so that $\\sum_{i=1}^n D_{ii} \\left( P^TQEQ^TP \\right)_{ii} = 0$, but this contradicts $C_- \\neq O$. Thus, no $X \\in \\mathbb{S}^n_{++}$ exists satisfying $A\\left( \\mbox{vec} (X) \\right)= \\bm{0}$.\n\\end{proof}\n\n\\subsubsection{Infeasible instances}\n\\label{sec: infeasible  instances}\n\nThe infeasible instances were generated by Algorithm \\ref{infeasible instance}.\n\nIf we define the linear operator $\\mathcal{A} : \\mathbb{S}^n \\rightarrow \\mathbb{R}^m$ as $\\mathcal{A}(X) = ( \\langle A_1, X \\rangle , \\dots ,\\langle A_m, X \\rangle)^T$, then by choosing $A_1 \\in \\mathbb{S}^n_{++}$, we obtain $\\mathcal{A}$ such that $\\mbox{Ker} \\mathcal{A} \\cap \\mathbb{S}^n_{+} = \\{ O \\} $. On the basis of this observation, by introducing a parameter $\\alpha > 0$, we generated a positive definite matrix $A_1$ whose minimum eigenvalue is a uniformly distributed random number in $(0,\\alpha)$. We chose $\\alpha \\in \\{1e-1, 1e-2, 1e-3, 1e-4, 1e-5 \\}$. The input of Algorithm \\ref{infeasible instance} consisted of the rank of the semidefinite cone $n$, the number of constraints $m$, an arbitrary orthogonal matrix $P$, and the parameter $\\alpha > 0$.\n\n\\begin{algorithm}[H]\n\\caption{Infeasible instance}\n\\label{infeasible instance}\n\\begin{algorithmic}[1]\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input: }}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output: }}\n\n \\STATE \\algorithmicrequire $n, m, \\alpha, P$, $A' = [ \\ \\ ]$\n \\STATE \\algorithmicensure $A$ \n \\STATE $B \\leftarrow \\mbox{rand} (n,n)$\n \\STATE $B' \\leftarrow \\frac{B+B^T}{2}$ and then compute an orthogonal matrix $Q$ and diagonal matrix $E$ such that $B' = QDQ^T$\n \\STATE $E_+ = \\mbox{rand} \\left( 1 \\right) \\times \\alpha I + \\mathcal{P}_{\\mathbb{S}^n_+} (E)$ \n \\STATE $d \\leftarrow \\mbox{rand} (n)$ and $D \\leftarrow \\mbox{diag}(d)$\n \\STATE $B_+ \\leftarrow QE_+Q^T $ and $C \\leftarrow PDP^T $ \n \\STATE $c = \\mbox{vec}(C)  $ and $R \\leftarrow I - \\frac{1}{\\|c\\|_2^2} cc^T $\n \\FOR {$ i = 1 \\mbox{ to  }  m-1 $}\n \\STATE $A_i' \\leftarrow \\mbox{rand} (n,n)$ and $A_i \\leftarrow \\left( A_i' + (A_i')^T \\right)\/2 $\n \\STATE $A' \\leftarrow  \n\\begin{pmatrix}\nA' \\\\\n\\mbox{vec}(A_i)^T \\\\\n\\end{pmatrix}$\n \\ENDFOR\n \\STATE $A \\leftarrow \n\\begin{pmatrix}\n\\mbox{vec} (B_+)^T \\\\\nA'R\n\\end{pmatrix}$\n\\end{algorithmic} \n\\end{algorithm}\n\nNote that the first row of the matrix $A$ returned by Algorithm \\ref{infeasible instance} is ${\\mbox{vec}(B_+)}^T$. Since $B_+ \\in \\mathbb{S}^n_{++}$, we see that ${\\mbox{vec}(B_+)}^T \\mbox{vec}(X) > 0$ for any positive definite matrix $X \\in \\mathbb{S}^n_{++}$. Thus, there is no $X \\in \\mathbb{S}^n_{++}$ satisfying $A \\left( \\mbox{vec}(X) \\right)=\\bm{0}$, which implies that the generated instance is infeasible.\n\n\\subsection{Numerical results and observations}\n\\label{sec: numerical results and observations}\n\nWe set the size of the positive semidefinite matrix to $n = 50$, \\modifySecond{so that the computational experiments could be performed in a reasonable period of time. To eliminate bias in the experimental results, we generated instances in which the number of constraints $m$ was controlled using the parameter $\\nu$ for the number $\\frac{n(n+1)}{2}$ of variables in the symmetric matrix of order $n$. Specifically,} the number of constraints $m$ on an integer was obtained by rounding the value of $\\frac{n(n+1)}{2} \\nu$, where $\\nu \\in \\{0.1, 0.3, 0.5, 0.7, 0.9\\}$.\n\nFor each $\\nu \\in \\{0.1, 0.3, 0.5, 0.7, 0.9\\}$, we generated five instances, i.e., 25 instances for each of five strongly feasible cases (corresponding to five patterns of $\\mu \\simeq \\mbox{1e-50}, \\dots, \\mu \\simeq \\mbox{1e-250}$, see section \\ref{sec: strongly feasible instances} for details), 25 instances for a weakly feasible case, and 25 instances for each of five infeasible cases (corresponding to five patterns of $\\alpha = \\mbox{1e-1}, \\dots, \\alpha = \\mbox{1e-5}$, see section \\ref{sec: infeasible instances} for details). Thus, we generated 125 strongly feasible instances, 25 weakly feasible instances, and 125 infeasible instances, totalling 275 instances. We set the upper limit of the execution time to 2 hours and compared the performance of our method (Algorithm \\ref{main algorithm}, \\ref{main algorithm 2}) with those of Louren\\c{c}o (2019) and Pena(2017).\n\n\\modifySecond{\nWe compared the results of the proposed method, two of the existing Chubanov's methods, and Mosek. When using Mosek, we set the primal feasibility tolerance (MSK\\_DPARA\\_INTPNT\\_CO\\_TOL\\_PFEAS) to 1e-12.\n}\n\n\\modifySecond{Tables \\ref{strongly-feasible-MVN-SP1} and \\ref{strongly-feasible-MVN-SP2} list} the results for the (ill-conditioned) strongly feasible case. The ``CO-ratio'' column shows the ratio of correct outputs, the ``times(s)'' column shows the average CPU time of the method, the ``M-iter'' column shows the average iteration number of each main algorithm, \\modifySecond{and the $\\|\\mathcal{A}(X^*)\\|_2$ column shows the residual of the constraints. The ``BP'' column shows which scheme  (the modified von Neumann (MVN) or the smooth perceptron (SP)) was used in the basic procedure.}\nThe values in parentheses () in row $\\mu \\approx \\mbox{1e-100}$ are the average values excluding instances for which the method ended up running out of time. \n\n\\modifySecond{\nFirst, we compare the results when using MVN or SP as the basic procedure in each method.\nFrom Table \\ref{strongly-feasible-MVN-SP1}, we can see that for strongly-feasible problems, using SP as the basic procedure has a shorter average execution time than using MVN.\nNext, we compare the results of each method.\nFor $\\mu \\simeq \\mbox{1e-50}$, there was no significant difference in performance among the three methods.\nFor $\\mu \\leq \\mbox{1e-100}$, the results in the rows BP=MVN show that our method and Louren\\c{c}o (2019) obtained interior feasible solutions for all problems, while Pena (2017) ended up running out of time for 99 instances.}\nThis is because Pena (2017) needs to call its basic procedure to find a solution of $\\mbox{range} \\mathcal{A}^* \\cap \\mathbb{S}^n_{++}$. Comparing our method  with Louren\\c{c}o (2019), we see that it is superior in terms of CPU time. This is probably because it employs a more efficient scaling at each iteration, which will be described in detail in section \\ref{sec: comparisons}. \n\n\\modifySecond{\nFinally, we compare the results for each value of $\\mu$. As $\\mu$ becomes smaller, i.e., as the problem becomes more ill-conditioned, the number of scaling times (M-iter) and the execution time increase, and the accuracy of the obtained solution ($\\| \\mathcal{A} (X^*) \\|_2$)  gets worse.\n}\n\n\\begin{table}[H]\n\\caption{\\modifySecond{Results for ill-conditioned strongly feasible instances (Correct input (CO-) ratio and CPU time)}}\n\\label{strongly-feasible-MVN-SP1}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{ll|rr|rr|rr} \\toprule\n\t\t\t&\t&\\multicolumn{2}{c|}{Algorithm \\ref{main algorithm}} & \\multicolumn{2}{c|}{Louren\\c{c}o (2019)} & \\multicolumn{2}{c}{Pena (2017)} \\\\\nInstance\t\t&BP\t&CO-ratio&time(s)&CO-ratio&time(s)&CO-ratio&time(s)\\\\ \\midrule\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-50}\t\t&MVN\t&25\/25\t&7.81\t\t&25\/25\t&25.94\t&25\/25\t&3.60 \\\\\n\t\t\t\t\t\t\t&SP\t\t&25\/25\t&0.75\t\t&25\/25\t&10.12\t&25\/25\t&0.80 \\\\ \\hline\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-100}\t&MVN\t&25\/25\t&51.62\t&25\/25\t&448.05\t&1\/1\t\t&(4513.59) \\\\\n\t\t\t\t\t\t\t&SP\t\t&25\/25\t&32.11\t&25\/25\t&256.24\t&25\/25\t&123.65 \\\\ \\hline\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-150}\t&MVN\t&25\/25\t&99.39\t&25\/25\t&888.25\t&-\t&- \\\\\n\t\t\t\t\t\t\t&SP\t\t&25\/25\t&76.98\t&25\/25\t&520.73\t&25\/25\t&781.88 \\\\ \\hline\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-200}\t&MVN\t&25\/25\t&144.48\t&25\/25\t&1328.68\t&-\t&- \\\\\n\t\t\t\t\t\t\t&SP\t\t&25\/25\t&118.06\t&25\/25\t&789.29\t&25\/25\t&1874.20 \\\\ \\hline\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-250}\t&MVN\t&25\/25\t&188.11\t&25\/25\t&1827.24\t&-\t&- \\\\\n\t\t\t\t\t\t\t&SP\t\t&25\/25\t&162.67\t&25\/25\t&1074.07\t&25\/25\t&3308.35 \\\\\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[H]\n\\caption{\\modifySecond{Results for ill-conditioned strongly feasible instances (M-iter and $\\|\\mathcal{A}(X^*)\\|_2$)}}\n\\label{strongly-feasible-MVN-SP2}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{ll|rr|rr|rr} \\toprule\n\t\t\t&\t&\\multicolumn{2}{c|}{Algorithm \\ref{main algorithm}} & \\multicolumn{2}{c|}{Louren\\c{c}o (2019)} & \\multicolumn{2}{c}{Pena (2017)} \\\\\nInstance\t\t&BP\t&M-iter&$\\| \\mathcal{A} (X^*)\\|_2$&M-iter&$\\| \\mathcal{A} (X^*)\\|_2$&M-iter&$\\| \\mathcal{A} (X^*)\\|_2$\\\\ \\midrule\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-50}\t\t&MVN\t&3.28\t\t&1.24e-11\t&14.48\t&7.64e-12\t&1.00\t&1.27e-11 \\\\\n\t\t\t\t\t\t\t&SP\t\t&1.00\t\t&1.23e-11\t&14.08\t&8.22e-12\t&1.00\t&1.23e-11 \\\\ \\hline\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-100}\t&MVN\t&53.12\t&9.98e-12\t&329.04\t&1.26e-11\t&(2.00)\t&(1.07e-8) \\\\\n\t\t\t\t\t\t\t&SP\t\t&36.04\t&4.18e-11\t&365.76\t&1.10e-11\t&23.32\t&5.38e-9 \\\\ \\hline\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-150}\t&MVN\t&118.12\t&1.96e-10\t&728.68\t&4.29e-10\t&-\t&- \\\\\n\t\t\t\t\t\t\t&SP\t\t&91.96\t&2.21e-10\t&756.36\t&5.60e-10\t&117.32\t&6.31e-9 \\\\ \\hline\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-200}\t&MVN\t&185.40\t&1.51e-8\t&1145.40\t&4.76e-8\t&-\t&- \\\\\n\t\t\t\t\t\t\t&SP\t\t&151.44\t&1.09e-8\t&1150.20\t&3.81e-8\t\t&236.44\t&1.72e-8 \\\\ \\hline\n\\multirow{2}{*}{$\\mu \\simeq $ 1e-250}\t&MVN\t&251.44\t&9.51e-7\t&1601.20\t&2.58e-6\t&-\t&- \\\\\n\t\t\t\t\t\t\t&SP\t\t&215.12\t&1.72e-6\t&1564.80\t&3.35e-6\t\t&376.24\t&1.73e-6 \\\\\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\\modifySecond{\nTable \\ref{strongly-feasible-Mosek} summarizes the results of our experiments using Mosek to solve strongly feasible ill-conditioned instances. Mosek sometimes returned the error message `'rescode = 10006'' for the $\\mu \\leq 1e-200$ instances. This error message means that \"the optimizer is terminated due to slow progress.\" In this case, the obtained solution is not guaranteed to be optimal, but it may have sufficient accuracy as a feasible solution. Therefore, we took the CO-ratio when the residual $\\|\\mathcal{A}(X^*)\\|_2$ is less than or equal to1e-5 to be the correct output. The reason why we set the threshold to 1e-5 is that the maximum value of $\\|\\mathcal{A}(X^*)\\|_2$ was less than 1e-5 among the $X^*$ values obtained for the strongly feasible ill-conditioned instances by the three methods, Algorithm \\ref{main algorithm}, Louren\\c{c}o (2019) and Pena (2017). On the other hand, for the $\\mu \\leq 1e-200$ instances, the Chubanov methods had higher CO-ratios. That is, when the problem was quite ill-conditioned, the solution obtained by each of the Chubanov methods had a smaller value of $\\| \\mathcal{A}(X^*)\\|_2$ compared with the solution obtained by Mosek, which implies that the accuracy of the solution obtained by each of the Chubanov methods was higher than that of Mosek.\n}\n\n\\begin{table}[H]\n\\caption{\\modifySecond{Results for ill-conditioned strongly feasible instances with Mosek}}\n\\label{strongly-feasible-Mosek}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{l|rrr} \\toprule\nInstance\t\t\t&CO-ratio\t&time(s)\t&$\\| \\mathcal{A} (X^*)\\|_2$\t\\\\ \\midrule\n$\\mu \\simeq $ 1e-50\t&25\/25 \t&1.96\t\t&8.73e-13 \\\\\n$\\mu \\simeq $ 1e-100\t&25\/25 \t&3.18\t\t&1.87e-12 \\\\\n$\\mu \\simeq $ 1e-150\t&25\/25 \t&3.72\t\t&2.48e-10 \\\\\n$\\mu \\simeq $ 1e-200\t&21\/25 \t&6.56 \t\t&2.58e-7 \\\\\n$\\mu \\simeq $ 1e-250\t&1\/25 \t&6.88 \t\t&2.57e-7 \\\\\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\\modifySecond{Table \\ref{infeasible} summarizes} the results for infeasible instances.\nSimilarly to Table \\ref{strongly-feasible-MVN-SP1}, the ``CO-ratio and ``times(s)'' columns respectively show the ratio of correct outputs and the average CPU time of each method (the values in parentheses () in rows $\\alpha = \\mbox{1e-4}$ and $\\alpha = \\mbox{1e-5}$ are the average CPU times of each method excluding the instances for which the method ended up running out of time). \n\n\\modifySecond{When using MVN as the basic procedure,} whereas our method and Louren\\c{c}o (2019) found an element of $\\mbox{range} \\mathcal{A}^* \\cap \\mathbb{S}^n_+$ for all instances, Pena (2017) ended up running out of time for one instance for $\\alpha=\\mbox{1e-4}$ and $\\alpha=\\mbox{1e-5}$. \n\n\\modifySecond{\nFrom the results for infeasible instances, we can observe the following three points. First, our method obtained correct outputs for every instance in a short execution time. This would be because it employed an efficient scaling and found an element of $\\mbox{range} \\mathcal{A}^* \\cap \\mathbb{S}^n_+$. Second, the method of Pena (2017) obtained better results when SP was used as the basic procedure. As shown in Table \\ref{infeasible}, the method of Pena (2017) using SP as the basic procedure solved all problems and had shorter execution times than the method using MVN. Since Pena's (2017) method calls the basic procedure not only to find points in $\\mbox{Ker} \\mathcal{A} \\cap \\mathbb{S}^n_{++}$ but also to find points in $\\mbox{range} \\mathcal{A}^* \\cap \\mathbb{S}^n_{++}$, using SP, which can update basic procedures efficiently, is better than using MVN in terms of execution time. Third, it is not always possible to detect infeasibility (i.e., to find a point in $\\mbox{range} \\mathcal{A}^* \\cap \\mathbb{S}^n_{+}$) in a shorter execution time when using SP than when using MVN. In fact, according to Louren\\c{c}o (2019), the execution time is shorter when using MVN as the basic procedure than when using SP. SP is a more efficient update method than MVN in terms of satisfying a termination criterion (the criterion for moving to scaling) of the basic procedure. On the other hand, from the point of view of finding points in $\\mbox{range} \\mathcal{A}^* \\cap \\mathbb{S}^n_{+}$, it is not possible to determine whether SP or MVN is more suitable. Pena (2017) used SP to significantly reduce the execution time, which is the result of updating the basic procedure for finding points in $\\mbox{range} \\mathcal{A}^* \\cap \\mathbb{S}^n_{++}$ more efficiently than MVN.\n}  \n\\modifySecond{\nMosek obtained a point in $\\mbox{range} \\mathcal{A}^* \\cap \\mathbb{S}^n_{++}$ as a feasible solution to the dual problem for all instances. From the viewpoint of execution time, Mosek was superior to the other methods.\n}\n\n\\begin{table}[H]\n\\caption{\\modifySecond{Results for infeasible instances}}\n\\label{infeasible}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{ll|rr|rr|rr|rr} \\toprule\n\t\t\t&\t&\\multicolumn{2}{c|}{Algorithm \\ref{main algorithm}} & \\multicolumn{2}{c|}{Louren\\c{c}o (2019)} &\\multicolumn{2}{c|}{Pena (2017)} &\\multicolumn{2}{c}{Mosek} \\\\\nInstance\t\t&BP\t&CO-ratio&time(s)&CO-ratio&time(s)&CO-ratio&time(s)\t&CO-ratio&time(s)\\\\ \\midrule\n\\multirow{2}{*}{$\\alpha= $ 1e-1}\t&MVN\t&25\/25\t&1.23\t&25\/25\t&2.37\t\t&25\/25\t&0.79\t&\\multirow{2}{*}{25\/25}\t&\\multirow{2}{*}{1.22} \\\\\n\t\t\t\t\t\t&SP\t\t&25\/25\t&1.01\t&25\/25\t&21.46\t&25\/25\t&0.61 \t\\\\ \\hline\n\\multirow{2}{*}{$\\alpha= $ 1e-2}\t&MVN\t&25\/25\t&4.39\t&25\/25\t&37.93\t&25\/25\t&25.99 &\\multirow{2}{*}{25\/25}\t&\\multirow{2}{*}{1.25}\\\\\n\t\t\t\t\t\t&SP\t\t&25\/25\t&3.87\t&25\/25\t&62.92\t&25\/25\t&1.05 \\\\ \\hline\n\\multirow{2}{*}{$\\alpha= $ 1e-3}\t&MVN\t&25\/25\t&5.38\t&25\/25\t&61.61\t&25\/25\t&61.55 &\\multirow{2}{*}{25\/25}\t&\\multirow{2}{*}{1.25}\\\\\n\t\t\t\t\t\t&SP\t\t&25\/25\t&5.34\t&25\/25\t&84.08\t&25\/25\t&2.08 \\\\ \\hline\n\\multirow{2}{*}{$\\alpha= $ 1e-4}\t&MVN\t&25\/25\t&7.81\t&25\/25\t&88.32\t&24\/24\t&(20.80) &\\multirow{2}{*}{25\/25}&\\multirow{2}{*}{1.24}\\\\\n\t\t\t\t\t\t&SP\t\t&25\/25\t&7.40\t&25\/25\t&98.79\t&25\/25\t&33.48 \\\\ \\hline\n\\multirow{2}{*}{$\\alpha= $ 1e-5}\t&MVN\t&25\/25\t&9.08\t&25\/25\t&76.17\t&24\/24\t&(9.47) &\\multirow{2}{*}{25\/25}\t&\\multirow{2}{*}{1.24}\\\\\n\t\t\t\t\t\t&SP\t\t&25\/25\t&8.00\t&25\/25\t&91.88\t&25\/25\t&55.42 \\\\ \\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\nFor the weakly feasible instances, we compared our method (Algorithm \\ref{main algorithm}), a modified version with another criteria for $\\varepsilon$-feasibility (Algorithm \\ref{main algorithm 2}), Louren\\c{c}o (2019), and Pena (2017). The results are summarized in Table \\ref{weakly-feasible-output}.\n\nAs described in section \\ref{sec: outline of numerical implementation}, we classified the output-results into type A: an interior feasible solution is found; type B: no interior feasible solution is found  (ver.1); type C: no $\\varepsilon$-feasible solution is found (only for Loren\\c{c}o (2019) and our methods); type D: no interior feasible solution is found (ver.2; only for Pena (2017)); type E: Out-of-time. \n\nNote that $\\mbox{B}^*$ indicates that the output was B, but when we converted the obtained solution to a solution of ${\\rm D} (\\mathcal{A})$, it contained a negative eigenvalue and violated the SDP constraint.\n\nFrom Table \\ref{weakly-feasible-output}, we can observe the following:\n\\begin{itemize}\n\\item \\modifySecond{\nFor all the methods, the average execution time was shorter when SP was used as the basic procedure than when MVN was used.\n}\n\\item \\modifySecond{\nAll methods except Algorithm \\ref{main algorithm 2} \n} sometimes obtained output type A (an interior feasible solution is found), \\modifySecond{ and Pena(2017) returned output-result D}, while the obtained solution had $0 \\sim 5$ negative eigenvalues (about -1e-16) and more than 20 positive eigenvalues (less than 1e-12) when we converted it into a solution of ${\\rm P} (\\mathcal{A})$.\n\\item Louren\\c{c}o (2019) obtained output type $\\mbox{B}^*$ (no interior feasible solution is found) but when we converted the obtained solution into a solution of ${\\rm D} (\\mathcal{A})$, it contained a negative eigenvalue and violated the SDP constraint). The obtained solution had $1 \\sim 3$ negative eigenvalues (about -1e-6) and violated the SDP constraint when we converted it into a solution of ${\\rm D} (\\mathcal{A})$.\n\\item Our modified method (Algorithm \\ref{main algorithm 2}) was able to determine the existence of an $\\varepsilon$-feasible solution for all instances. This implies that, \\modifyThird{at least for this specific set of weakly feasible instances,} the criteria focusing on the total value of the eigenvalues used in Algorithm \\ref{main algorithm 2} is more suitable than the criteria focusing on the product of all the eigenvalues.\n\\end{itemize}\n\n\\begin{table}[H]\n\\caption{\\modifySecond{Output types for weakly feasible instances}}\n\\label{weakly-feasible-output}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{cc|ccccc|r}\\toprule\nMethod\t\t\t\t\t\t\t\t\t&BP\t\t&$\\nu=0.1$&$\\nu=0.3$&$\\nu=0.5$&$\\nu=0.7$&$\\nu=0.9$\t&time(s)\\\\ \\midrule\n\\multirow{2}{*}{Algorithm \\ref{main algorithm}}\t&MVN\t&AAAAA\t&AAAAA\t&AAAAA &AAAAA &BBBBB\t&414.42\\\\\n\t\t\t\t\t\t\t\t\t\t&SP\t\t&AAAAA\t&AAAAA\t&AAAAA &AAAAA &ABABB\t&226.25 \\\\ \n\\multirow{2}{*}{Algorithm \\ref{main algorithm 2}} &MVN\t&CCCCC\t&CCCCC\t&CCCCC &CCCCC  &CCCCC\t&301.97\\\\\n\t\t\t\t\t\t\t\t\t\t&SP\t\t&CCCCC\t&CCCCC\t&CCCCC &CCCCC  &CCCCC\t&179.72\\\\ \n\\multirow{2}{*}{Louren\\c{c}o (2019)}\t\t\t&MVN\t&AAAAA & B$\\mbox{B}^*$$\\mbox{B}^*$$\\mbox{B}^*$B & ABAAA & ABA$\\mbox{B}^*$$\\mbox{B}^*$ &BBBBB\t&3512.78\\\\\n\t\t\t\t\t\t\t\t\t\t&SP\t\t&AAAAA &AAAAA\t&AAAAA\t&AAAAA\t&BBBBB\t&1550.76 \\\\ \n\\multirow{2}{*}{Pena (2017)}\t\t\t\t\t&MVN\t&EEEEE\t&EEEEE\t&EEEEE\t&EEEEE\t&EEEEE\t&\\\\ \n\t\t\t\t\t\t\t\t\t\t&SP\t\t&AAAAA\t&DAAAD\t&AAAAA\t&AAAAA\t&DDDDD\t&3239.12 \\\\ \\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\\modifySecond{\nTable \\ref{weakly-feasible-output-Mosek} summarizes the results obtained by Mosek.\nThe error message ``rescode = 10006'' was obtained for 22 instances, similar to the results for the strongly feasible ill-conditioned instances.\nNote that we assumed that feasible solutions were obtained for all instances since the constraint residual $\\|\\mathcal{A}(X^*)\\|_2$ was as small as 1.1e-7 or less for all obtained solutions.\nThere were three instances for which we obtained a feasible solution with a  minimum eigenvalue larger than 1e-12 (These three instances are all included in the instance set of  $\\nu = 0.9$).\nWe set the CO-ratio to $22\/25$ considering that it is difficult to determine whether such a solution satisfies $X \\in \\mathbb{S}^n_{++}$ or $X \\in \\mathbb{S}^n_+$.\n}\n\n\\modifySecond{\nNote that for all problems with $0.1 \\leq \\nu \\leq 0.7$, Algorithm \\ref{main algorithm} and Louren\\c{c}o (2019) using SP for the basic procedure returned output A, i.e., a feasible solution to the original problem. Table \\ref{weakly feasible compare} summarizes the accuracies of the solutions obtained with Algorithm \\ref{main algorithm}, Louren\\c{c}o (2019), and Mosek for all instances with $0.1 \\leq \\nu \\leq 0.7$. \nChubanov's methods sometimes returned incorrect output (output-result type A) for weakly feasible instances, but Table \\ref{weakly feasible compare} shows that the average accuracy of feasible solutions obtained by Chubanov's methods was better than that of Mosek.\n}\n\n\\begin{table}[H]\n\\caption{\\modifySecond{Output types for weakly feasible instances with Mosek}}\n\\label{weakly-feasible-output-Mosek}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{l|rrr} \\toprule\nInstance\t\t\t&CO-ratio\t&time(s)\t\t&$\\| \\mathcal{A} (X^*)\\|_2$\t\\\\ \\midrule\nweakly feasible\t&22\/25 \t\t&4.86 \t\t&1.28e-9 \\\\\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[H]\n\\caption{\\modifySecond{\nAverage of the constraint residuals $\\|\\mathcal{A}(X^*)\\|_2$ of the solution $X^*$ obtained for the weakly feasible instances\n}}\n\\label{weakly feasible compare}\n\\begin{center}\n\\modifySecond{\n\\begin{tabular}{l|ccc} \\toprule\nValue of $\\nu$\t&Algorithm \\ref{main algorithm}\t&Louren\\c{c}o (2019)\t&Mosek \\\\ \\midrule\n$\\nu=0.1$ &1.28e-13\t&5.51e-14\t&1.45e-12\t\\\\\n$\\nu=0.3$ &1.56e-13\t&7.04e-14\t&2.53e-10\t\\\\\n$\\nu=0.5$ &1.40e-13\t&1.05e-13\t&1.29e-9\t\\\\\n$\\nu=0.7$ &3.44e-13\t&1.09e-13\t&3.75e-9\t\\\\\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\nThe detailed numerical results are in Appendix \\ref{sec: detailed numerical results}.\n\n\\section{More comparisons of the basic procedures}\n\\label{sec: comparisons}\n\nIn section \\ref{sec: complexity of MA vs Lourenco}, we showed that the bound of the computational cost of our method is lower than that of Louren\\c{c}o et al. when $\\mathcal{K}$ is the $n$-dimensional nonnegative orthant $\\mathbb{R}^n_+$ or a Cartesian product of simple second-order cones, and that their bounds on their costs are equivalent when $\\mathcal{K}$ is a simple positive semidefinite cone under the assumption that the costs of computing the spectral decomposition and the minimum eigenvalue are the same for an $n \\times n$ symmetric matrix. In this section, we make more detailed comparisons of these algorithms in terms of \\modifyThird{the performance of the cut obtained from the basic procedure} and the detectability of an $\\varepsilon$-feasible solution. Similarly to section \\ref{sec: numerical experiments}, we will refer to Louren\\c{c}o et al.'s method \\cite{Bruno2019} as Louren\\c{c}o (2019) throughout this section.\n\n\\subsection{\\modifyThird{Performance comparison} of the two basic procedures for the simple case}\n\nHere, for the sake of simplicity, we will focus on the case where the symmetric cone is simple, i.e., $p=1$. Let $\\mathbb{E}$ be the Euclidean space corresponding to the symmetric cone $\\mathcal{K}$. For any given $w, v \\in \\mathbb{E}$, Louren\\c{c}o et al. \\cite{Bruno2019} defined $\\mbox{\\rm vol} (w,v)$ as the volume of the intersection $H(w,v) \\cap \\mathcal{K}$, where $H(w,v)$ is the half space given by\n\\begin{equation}\nH(w,v) = \\{ x \\in \\mathbb{E} \\ \\ | \\ \\ \\langle w , x \\rangle \\leq \\langle w,v \\rangle \\}. \\notag\n\\end{equation}\n\n\\modifyThird{\nIn this section, we first identify the half-space $H(w,v)$ that will be transferred to the half-space $H(e, e\/r)$ after scaling and then find the constant $\\mbox{\\textbf{rate}} \\in \\mathbb{R}$ that satisfies $\\mbox{vol} (w,v) \\leq \\mbox{\\textbf{rate}} \\times \\mbox{vol}(e,e\/r)$, so that we can compare the proposed method and Louren\\c{c}o (2019). The proposed method and Louren\\c{c}o (2019) use the basic procedure results to narrow down the original problem's feasible region. It can be interpreted that the algorithm becomes more efficient as the constant $\\mbox{\\textbf{rate}} \\in \\mathbb{R}$ (indicating how much $\\mbox{vol}(w,v)$ is reduced compared with $\\mbox{vol}(e,e\/r)$) gets smaller. In what follows, we call the constant $\\mbox{\\textbf{rate}} \\in \\mathbb{R}$ the reduction rate.\n}\n\n\\modifyThird{\nSection \\ref{sec:reduction 1} derives the reduction rate of the proposed method and section \\ref{sec:reduction 2} that of Louren\\c{c}o (2019). The results in these sections are summarized in Table \\ref{table:reduction}, where the ``UB\\#iterations'' column shows the upper bound on the number of iterations required in the basic procedure. The ``UB\\#iterations'' of Louren\\c{c}o (2019) comes from Proposition 14 of \\cite{Bruno2019} (where the authors showed their result by substituting $\\rho = 2$), whereas that of Algorithm \\ref{basic procedure} comes from Proposition \\ref{prop:bp2} with $\\ell = 1$. The ``Reduction rate'' of Louren\\c{c}o (2019) comes from Theorem \\ref{theo:reduction 2}, whereas that of Algorithm \\ref{basic procedure} comes from (\\ref{eq:reduction 1}) with $(w,v) = (Q_{g^{-1}}(e) , Q_g (e) \/r ) $.\n}\n\n\\modifyThird{\n\\begin{table}[H]\n\\caption{Comparison of reduction rates of the two algorithms: Theoretical results}\n\\label{table:reduction}\n\\begin{center}\n\\begin{tabular}{c|c|c} \\hline\n Basic procedure  & UB\\#iterations & Reduction rate \\\\ \\hline\nLouren\\c{c}o (2019) & $\\rho^2 r_{\\max}^2$ & $ \\mbox{\\rm vol} (w,v) = \\left( \\frac{r^r}{\\det w} \\right)^{\\frac{d}{r}} \\mbox{\\rm vol} (e,e\/r) \\leq \\left( e^{- \\varphi (\\rho)} \\right)^{\\frac{d}{r}} \\mbox{\\rm vol} (e,e\/r) $\\\\ \nAlgorithm \\ref{basic procedure} & $\\cfrac{r_{\\max}^2}{\\xi^2}$  & $ \\mbox{\\rm vol} (w, v)  = \\left( \\xi^N \\right)^{\\frac{d}{r}} \\mbox{\\rm vol} (e,e\/r)  $  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n}\n\n\\modifyThird{\nBy setting $\\rho=2$ and $\\xi = \\frac{1}{2}$, the two bounds in``UB\\#iterations'' have the same value; in this case, the reduction rates turn out to be\n\\begin{align*}\n\\mbox{Louren\\c{c}o (2019):} \\ \\left( \\frac{r^r}{\\det w} \\right)^{\\frac{d}{r}} \\leq \\left( e^{- \\varphi (2)} \\right)^{\\frac{d}{r}} \\simeq (0.918)^{\\frac{d}{r}},  \\hspace{1cm}\n\\mbox{Algorithm \\ref{basic procedure}: }  \\left( \\xi^N \\right) ^{\\frac{d}{r}} \\leq \\left( \\frac{1}{2} \\right)^{\\frac{d}{r}}.\n\\end{align*}\nThe above comparison indicates that Algorithm \\ref{basic procedure} is superior to the basic procedure in \\cite{Bruno2019} in terms of the reduction rate of the feasible region.\n}\n\n\\subsubsection{\\modifyThird{Theoretical reduction rate} of Algorithm \\ref{basic procedure}}\n\\label{sec:reduction 1}\n\nSuppose that Algorithm \\ref{basic procedure} returns a result such that there exists a nonempty index set $I \\subseteq \\{1 ,\\dots , r\\}$ with $|I| = N $ for which \n\\begin{equation}\n\\langle c_i , x \\rangle \\leq \n\\begin{cases}\n\\xi & i \\in I \\\\\n1 & i \\notin I\n\\end{cases} \\label{eq:I}\n\\end{equation}\nholds for any feasible solution $x$ of ${\\rm P}_{S_\\infty(A)}$, where $\\{c_1 , \\dots , c_r\\}$ are primitive idempotents that make up a Jordan frame.\n\nNote that Algorithm \\ref{basic procedure} employs the scaling $\\bar{x} = Q_{g^{-1}} (x)$ with $g^{-1} = \\frac{1}{\\sqrt{\\xi}} \\sum_{i \\in I}  c_i + \\sum_{i \\notin I} c_i$.\nLet us find  $w , v \\in \\mathbb{E}$ which satisfy\n\\begin{equation}\nH(e , e\/r) = Q_{g^{-1}} \\left( H(w,v) \\right).  \\label{appendix-C-1}\n\\end{equation}\nSince (\\ref{appendix-C-1}) and the scaling $\\bar{x} = Q_{g^{-1}} (x)$ imply that\n\\begin{align*}\nH(w,v) &= Q_g \\left( H(e , e\/r)  \\right) \\\\\n&= \\{ Q_g (\\bar{x}) \\in \\mathbb{E}  \\ \\ | \\ \\  \\langle \\bar{x} , e \\rangle \\leq 1 \\} \\\\\n&= \\{ Q_g (\\bar{x}) \\in \\mathbb{E}  \\ \\ | \\ \\  \\langle Q_g (\\bar{x}) , Q_{g^{-1}}(e) \\rangle \\leq 1 \\} \\\\\n&= \\{ x \\in \\mathbb{E}  \\ \\ | \\ \\  \\langle x , Q_{g^{-1}}(e) \\rangle \\leq \\langle Q_{g^{-1}}(e) , Q_g (e)  \/r \\rangle  = 1\\} ,\n\\end{align*}\nby setting $w = Q_{g^{-1}}(e)$ and $v = Q_g (e)  \/r$, we find that the half space $H(w,v)$ is transformed to $H(e,e\/r)$ after the scaling.\nSince $Q_{g^{-1}}(e)  \\in \\mbox{int} \\mathcal{K}$, we can apply the following proposition to $w = Q_{g^{-1}}(e)$.\n\n\\begin{proposition}[Proposition 6 of~\\cite{Bruno2019}]\nSuppose that $w \\in \\mbox{\\rm int} \\mathcal{K}$. Then, \n\\begin{align*}\nQ_{w^{-1\/2} \\sqrt{\\langle w , v \\rangle}} \\left( H(e , e\/r )\\right) &= H(w,v) , \\\\\n\\mbox{\\rm vol} (w,v) &= \\left( \\frac{\\langle w , v \\rangle}{\\sqrt[r]{\\det w}} \\right)^d \\mbox{\\rm vol} (e,e\/r) .\n\\end{align*}\n\\end{proposition}\n\nUsing the above proposition and the assumption $|I| = N$ for the set $I$ in (\\ref{eq:I}), we can see how the volume $\\mbox{\\rm vol} (Q_{g^{-1}}(e) , Q_g (e) \/r)$ of $H (Q_{g^{-1}}(e) , Q_g (e) \/r) \\cap \\mathcal{K}$ decreases compared with $\\mbox{\\rm vol} (e , e\/r)$:\n\\begin{align}\n\\notag\n\\mbox{\\rm vol} (Q_{g^{-1}}(e)  , Q_g (e)  \/r ) &= \\left( \\frac{1}{\\sqrt[r]{\\det Q_{g^{-1}}(e) }} \\right)^d \\mbox{\\rm vol} (e,e\/r) \\\\\n\\notag\n&= \\left( \\frac{1}{\\sqrt[r]{ \\frac{1}{\\xi^N} }} \\right)^d \\mbox{\\rm vol} (e,e\/r) \\\\\n\\label{eq:reduction 1}\n&= \\left( \\xi^N \\right)^{\\frac{d}{r}} \\mbox{\\rm vol} (e,e\/r) .\n\\end{align}\n\n\\subsubsection{\\modifyThird{Theoretical reduction rate} of the basic procedure of Louren\\c{c}o (2019) }\n\\label{sec:reduction 2}\n\nThe following theorem gives the reduction rate of the basic procedure of Louren\\c{c}o (2019).\n\n\\begin{theorem}[Theorem 10 of~\\cite{Bruno2019}]\n\\label{theo:reduction 2}\nLet $\\rho>1$ and $y \\in \\mathcal{K} \\setminus \\{ 0 \\}$ be such that $F_{{\\rm P}_{S_1}(A)} \\subseteq H(y , e\/\\rho r)$.\nLet \n\\begin{align*}\n\\beta &= r - \\left(  \\frac{1}{\\rho} - \\frac{1}{\\sqrt{\\rho(3\\rho-2)}} \\right) , \\\\\nw &=   \\frac{r-\\beta}{\\langle y,e \\rangle} \\rho r y + \\beta e , \\\\\nv &= w^{-1} .\n\\end{align*}\nThen, the following hold:\n\\begin{enumerate}\n\\item $F_{{\\rm P}_S(A)} \\subseteq H(y ,  e \/ \\rho r) \\cap H(e ,  e  \/ r) \\subseteq H(w,v)$\n\\item $Q_{\\sqrt{r} w^{-1 \/ 2}} \\left( H(e ,  e\/r) \\right) = H(w,v) $\n\\item \n\\begin{equation}\n\\mbox{\\rm vol} (w,v) = \\left( \\frac{r^r}{\\det w} \\right)^{\\frac{d}{r}} \\mbox{\\rm vol} (e ,  e\/r) \n\\leq \\left( \\exp \\left( - \\varphi(\\rho) \\right) \\right)^{\\frac{d}{r}} \\mbox{\\rm vol} (e , e\/r)  \\notag\n\\end{equation}\nwhere $\\varphi(\\rho) = 2-\\frac{1}{\\rho} - \\sqrt{3-\\frac{2}{\\rho}} $.\n\nIn particular,  if  $\\rho \\geq 2$, we have $\\mbox{\\rm vol} (w ,v) < (0.918)^{\\frac{d}{r}}  \\mbox{\\rm vol} (e , e\/r)$.\n\n\\end{enumerate}\n\n\\end{theorem}\n\n\n\\subsubsection{Comparison of reduction rates of the two algorithms in numerical experiments}\n\nTo confirm whether similar reduction rates are observed numerically, we conducted an experiment where we used our method (Algorithms \\ref{bp-alg-mvn} and \\ref{main algorithm 2}) with $\\xi = 1\/2$ and Louren\\c{c}o (2019) with modified von Neumann scheme to solve a weakly feasible instance with $\\nu=0.1$. At each iteration of the main algorithms, we recorded the value of $\\frac{r^r}{\\det w} $ of Louren\\c{c}o (2019) and the value of $\\xi^N$ of our method and computed the reduction rates of the search region. The results are summarized in Table \\ref{tabel:reduction2}.\n\n\\begin{table}[H]\n\\caption{\\modifyFirst{Comparison of reduction rates of the two algorithms: Numerical results}}\n\\label{tabel:reduction2}\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c} \\hline\nAlgorithm & \\#iterations of M-A & Output & Average reduction rate & Final reduction rate  \\\\ \\hline\nLouren\\c{c}o (2019) : BP = MVN & 3060 & A &0.864 &3.86e-195 \\\\ \nAlgorithms \\ref{bp-alg-mvn} and \\ref{main algorithm 2}  &618 & C & 0.357 & 9.11e-305  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nThe ``\\#iterations of M-A'' column shows the number of iterations of the main algorithm.\nThe ``Average reduction rate'' column shows the average value of $\\frac{r^r}{\\det w} $ for Louren\\c{c}o (2019) and the average value of $\\xi^N$ for our method (Algorithms \\ref{bp-alg-sp} and \\ref{main algorithm}). The ``Final reduction rate'' column shows the value\n\\begin{equation}\n\\frac{r^{kr}}{\\det w(1) \\times \\det w(2) \\times \\dots \\times \\det w(k)} \\notag\n\\end{equation}\nfor Louren\\c{c}o (2019), where $w(k)$ denotes $w$ computed from the result of the basic procedure at the $k$-th iteration of the main algorithm, or the value \n\\begin{equation}\n\\xi^{N_1 + \\dots + N_k} . \\notag\n\\end{equation}\nfor our method (Algorithms \\ref{bp-alg-mvn} and \\ref{main algorithm 2}), where $N_k$ denotes the number of cuts obtained from the basic procedure at the $k$-th iteration of the main algorithm.\n\nHere, we observed that our method (Algorithms \\ref{bp-alg-mvn} and \\ref{main algorithm 2}) terminated at the 618-th iteration of the main algorithm with a reduction rate of 9.11e-305, while Louren\\c{c}o (2019) attained a reduction rate of 5.88e-40 at the same iteration of the main algorithm.\n\n\n\\subsection{Detection of an $\\varepsilon$-feasible solution}\n\nHere, we discuss the capabilities of our method and Louren\\c{c}o (2019) at detecting an $\\varepsilon$-feasible solution. Both methods terminate their main algorithms by detecting the existence of an $\\varepsilon$-feasible solution. We compared them by computing the reduction in $\\log \\left( \\lambda_{\\min} (x_\\ell) \\right)$ per iteration for parameter settings in which the maximum numbers of iterations of the basic procedures would be the same (i.e., $\\rho=2$ in Louren\\c{c}o (2019) and $\\xi = \\frac{1}{2}$ in our method).\n\nIn \\cite{Bruno2019}, for each block $\\ell$, Lemma 16 ensures that $\\log \\left( \\lambda_{\\min} (x_\\ell) \\right)$ is bounded from above by $\\epsilon_\\ell$, and Theorem 17 ensures that $\\epsilon_\\ell$ decreases at least $\\frac{\\varphi(\\rho)}{r_\\ell}>0$ if a {\\it good} iteration is obtained for the block $\\ell$.\n\nFor our method, Proposition \\ref{prop:lambda-min-upper} ensures that $\\log \\left( \\lambda_{\\min} (x_\\ell) \\right)$ is bounded from above by $\\frac{\\mbox{num}_\\ell}{r_\\ell} \\log \\xi$ and Proposition \\ref{prop:iteration-num-ma} ensures that $\\frac{\\mbox{num}_\\ell}{r_\\ell} \\log \\xi$ decreases $-\\frac{1}{r_\\ell} \\log \\xi > 0 $ in the same situation.\n\nBy substituting $\\rho=2$ and $\\xi = \\frac{1}{2}$ into $\\varphi(\\rho)$ and $- \\log \\xi$ so that the upper bounds for the numbers of iterations of the basic procedures are the same, we obtain\n\\begin{align*}\n\\varphi(2) &= 2 - \\frac{1}{2} - \\sqrt{2} \\simeq 0.085786 , \\\\\n-\\log \\frac{1}{2} &= \\log 2 \\simeq 0.693147\n\\end{align*}\nwhich implies that the rate of reduction in the upper bound $\\log \\left( \\lambda_{\\min} (x_\\ell) \\right)$ of our method is greater than that of Louren\\c{c}o (2019).\n\n\n\n\n\n\\section{Concluding remarks}\n\\label{sec: concluding remarks}\n\nIn this study, we proposed a new version of Chubanov's method for solving the feasibility problem over the symmetric cone by extending Roos's method \\cite{Roos2018} for the feasible problem over the nonnegative orthant, and we conducted comprehensive numerical experiments on the problem over the positive semidefinite cone to compare the performances of our method and the existing ones \\cite{Bruno2019,Pena2017}.\n\nOur method has the following features:\n\\begin{itemize}\n\\item It considers the feasibility problem ${\\rm P}_{S_\\infty} (\\mathcal{A})$, which is equivalent to ${\\rm P} (\\mathcal{A})$, and uses a rescaling focusing on the upper bound for the sum of eigenvalues of any feasible solution of ${\\rm P}_{S_\\infty} (\\mathcal{A})$. \n\\item Using the norm $\\| \\cdot \\|_\\infty$ in problem ${\\rm P}_{S_\\infty} (\\mathcal{A})$ makes it possible to (i) calculate the upper bound for the minimum eigenvalue of any feasible solution of ${\\rm P}_{S_\\infty} (\\mathcal{A})$, (ii) quantify the feasible region of ${\\rm P} (\\mathcal{A})$, and hence (iii) determine whether there exists a feasible solution of ${\\rm P} (\\mathcal{A})$ whose minimum eigenvalue is greater than $\\epsilon$ as in \\cite{Bruno2019}.\n\\item In terms of the computational bound, our method is (i) equivalent to Roos's original method \\cite{Roos2018} and superior to Louren\\c{c}o et al.'s method \\cite{Bruno2019} when the symmetric cone is the nonnegative orthant, (ii) superior to Louren\\c{c}o et al.'s when the symmetric cone is a Cartesian product of second-order cones, (iii) equivalent to Louren\\c{c}o et al.'s when the symmetric cone is the simple positive semidefinite cone, under the assumption that the costs of computing the spectral decomposition and the minimum eigenvalue are of the same order for any given symmetric matrix, and (iv) superior to Pena and Soheili's method~\\cite{Pena2017} for any simple symmetric cones under the assumption that ${\\rm P} (\\mathcal{A})$ is feasible.\n\\end{itemize}\n\nWe conducted comprehensive numerical experiments comparing our method with the methods of Chubanov and Mosek. We generated instances in three types: (i) strongly (but ill-conditioned) feasible instances by using Algorithm \\ref{strongly feasible problem}, (ii) weakly feasible instances by using Algorithm \\ref{weakly feasible instance}, and (iii) infeasible instances by using Algorithm \\ref{infeasible instance}. Our numerical results showed that\n\\begin{itemize}\n\\item Our method (Algorithms \\ref{main algorithm} and \\ref{main algorithm 2}) is superior to the methods proposed in \\cite{Bruno2019,Pena2017} in terms of accuracy and execution time.\n\\item It is considerably faster than the existing methods on ill-conditioned strongly feasible instances.\n\\item A modified version of our method (Algorithm \\ref{main algorithm 2}) can exactly determine whether the instance has no feasible solution whose minimum eigenvalue is less than $\\varepsilon = \\mbox{1e-12}$ for all weakly feasible instances (i.e., having no interior feasible solution), which is in contrast to Louren\\c{c}o et al.'s method, which sometimes returns a solution that does not satisfy the conic constraint of ${\\rm P} (\\mathcal{A})$ or ${\\rm D}(\\mathcal{A})$ and is affected by the large number of iterations of its main algorithm. \n\\item \\modifySecond{\nOur results showed that Mosek was the better than Chubanov's methods in terms of execution time. On the other hand, in terms of the accuracy of the solution (the value of $\\|\\mathcal{A}(X^*)\\|_2$), we found that all of Chubanov's methods are better than Mosek. In particular, we have seen such results for strongly-feasible (terribly) ill-conditioned ($\\mu \\simeq 1e-250$) and weakly-feasible instances.\n}\n\\end{itemize}\n\nOn the basis of the above numerical results, we further examined the number of iterations of Louren\\c{c}o et al.'s method and our method. As a result, we found that the basic procedure of our method is superior to the one of Louren\\c{c}o et al. in terms of both the constant rate of reduction in the volume of the detecting region and the upper bound for the minimum eigenvalue of any feasible solution.\n\nNote that Chubanov's method can find an $x \\in \\mbox{int} \\mathcal{K}$ satisfying $\\mathcal{A} (x) = \\bm{0}$, but not $x \\in \\mathcal{K}$ close to the boundary and satisfying $\\mathcal{A} (x) = \\bm{0}$, and it can determine the feasibility of ${\\rm P} (\\mathcal{A})$ in a finite number of iterations, but not a feasible solution of ${\\rm D}(\\mathcal{A})$ in such a way. \n\nOn the other hand, once we find that ${ \\rm P}_\\infty (\\mathcal{A})$ and ${ \\rm P}_{1,\\infty} (\\mathcal{A})$ have no feasible solution whose minimum eigenvalue is greater than $\\varepsilon$, the next issue is to find an $x \\in \\mathcal{K}\\setminus \\mbox{int} \\mathcal{K}$ satisfying $\\mathcal{A} (x) = \\bm{0}$, or to find the smallest dimensional symmetric cone $\\mathcal{K}_{\\min}$ satisfying $\\mbox{Ker} \\mathcal{A} \\cap \\mbox{int} \\mathcal{K}_{\\min} \\neq \\emptyset$ and $\\mathcal{K}_{\\min} \\subsetneq \\mathcal{K}$. It has been shown that the smallest dimensional symmetric cone $\\mathcal{K}_{\\min}$ can be detected by using a feasible solution of ${\\rm D}(\\mathcal{A})$~\\cite{Waki2013}, and several algorithms have been proposed to find a feasible solution of ${\\rm D}(\\mathcal{A})$ in a finite number of iterations~\\cite{Chubanov2017,Muramatsu2018}.\n\nIt remains as future work to explore whether it is possible to modify Chubanov's method so it can find $x \\in \\mathcal{K}$ close to the boundary and satisfying $\\mathcal{A} (x) = \\bm{0}$ directly or find a feasible solution of ${\\rm D}(\\mathcal{A})$ in a finite number of iterations.\n\n\n\\section*{Acknowledgments}\nWe would like to express our deep gratitude to the reviewers and editors for their many valuable comments. \nTheir comments significantly enriched the content of this paper, especially sections \\ref{sec: extension}, \\ref{sec: main algorithm}, \\ref{sec: compare}, and  \\ref{sec: numerical experiments}.\nWe also would like to sincerely thank Daisuke Sagaki for essential ideas on the proof of Proposition \\ref{prop:compare tr}, and Yasunori Futamura for helpful information about the computational cost of the eigenvalue calculation in Section \\ref{sec: CSD and Cmin}.\nWe could not complete this paper without their support.\nThis work was supported by JSPS KAKENHI Grant Numbers (B)19H02373 and JP 21J20875.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{INTRO}\n\n\n\\noindent\nThe Dyson-Schwinger (DS) approach \n\\cite{ForExample,Alkofer:2000wg,Roberts:2000hi,Roberts:2000aa,Holl:2006ni,Fischer:2006ub} \nto QCD and its modeling is the chirally well-behaved bound-state approach.  Thus, it is \nthe most suitable bound-state approach to treat the light pseudoscalar mesons \n(those composed of the $u, d$ and $s$ quarks), for which \ndynamical chiral symmetry breaking (DChSB) is essential.\nOne solves the (``gap\") DS equations (DSEs) \n${S}_q^{-1} = (S^{\\scriptstyle {\\rm free}}_{q})^{-1} - \\Sigma_q$ \nfor the dynamically dressed quark propagators \n${S}_q$, where $q$ is the quark flavor ($q=u,d,s, ...$),\n$S^{\\scriptstyle {\\rm free}}_q$ is the {free} quark propagator, and \n$\\Sigma_q$ is the quark self-energy.\nThese dressed quark propagator solutions are then employed in \nBethe-Salpeter equations (BSEs)\nfor the bound-state vertex $\\Gamma_{q{\\bar q}'}$ of\nthe meson composed of the quark of the flavor $q$ and antiquark\nof the flavor $q'$:\n\\begin{equation}\n[\\Gamma_{q{\\bar q}'}]_{ef} = \\int [S_q \\Gamma_{q{\\bar q}'} S_{q'} ]_{gh} \n[K]_{ef}^{hg} \\, \\, ,\n\\label{BSE}\n\\end{equation}\nwhere $e,f,g,h$ {\\it schematically} represent spinor, color\nand flavor indices, integration is meant over loop momenta,\nand $K$ is the interaction kernel. Solving Eq. (\\ref{BSE}) \nfor $\\Gamma_{q{\\bar q}'}$ also yields $M_{q\\bar q'}$, \nthe mass eigenvalue of the $q{\\bar q}'$ meson.\n\nTo obtain the chiral behavior as in QCD, DS and BS equations\nmust be solved in a consistent approximation. The rainbow-ladder \napproximation (RLA), where DChSB is well-understood, is still \nthe most usual approximation in phenomenological applications.\nThis also entails that in both DSE and BSE (\\ref{BSE}) we employ \nthe same effective interaction kernel, \n\\begin{equation}\n[K(k)]_{ef}^{hg} = {\\rm i} \\,\n      g^2 D_{\\mu\\nu}^{ab}(k)_{\\mbox{\\rm\\scriptsize eff}} \\,\n[\\frac{\\lambda^a}{2}\\,\\gamma^{\\mu}]_{eg} \\,\n[\\frac{\\lambda^a}{2}\\,\\gamma^{\\nu}]_{hf} \\, ,\n\\label{RLAkernel}\n\\end{equation}\nso that the quark self energy in the gap DSE is\n\\begin{eqnarray}\n \\label{DS-equation}\n\\Sigma_q(p) =  \n- \\int \\!\\!\\frac{d^4\\ell}{(2\\pi)^4} \\,\n  g^2 D_{\\mu\\nu}^{ab}(k)_{\\mbox{\\rm\\scriptsize eff}} \\, \n\\frac{\\lambda^a}{2}\\,\\gamma^{\\mu}\nS_q(\\ell) \\frac{\\lambda^b}{2}\\,\\gamma^{\\nu}~. \n\\end{eqnarray}\nIn Eqs. (\\ref{RLAkernel}) and (\\ref{DS-equation}), \n$D_{\\mu\\nu}^{ab}(k)_{\\mbox{\\rm\\scriptsize eff}}$ is an effective gluon \npropagator. For example, for renormalization-group improved (RGI) \ninteractions (e.g., in Refs. \n\\cite{jain93b,Klabucar:1997zi,Kekez:2000aw,Kekez:2003ri,Kekez:2005ie}), \nit has the form \n\\begin{equation}\ng^2 D_{\\mu\\nu}^{ab}(k)_{\\mbox{\\rm\\scriptsize eff}}\n= \n4\\pi\\alpha_{\\mbox{\\rm\\scriptsize eff}}(k^2)\nD_{\\mu\\nu}^{ab}(k)_{\\scriptstyle {\\rm free}}\n\\end{equation}\nwhere $D_{\\mu\\nu}^{ab}(p)_{\\scriptstyle {\\rm free}}$ is the free gluon propagator,\nand $\\alpha_{\\mbox{\\rm\\scriptsize eff}}(k^2)$ is an {\\it effective} running \ncoupling. For large spacelike momenta ($k^2 \\gg 1$ GeV$^2$), \n$\\alpha_{\\mbox{\\rm\\scriptsize eff}}(k^2)$ approaches the perturbative \nQCD running coupling $\\alpha_{\\mbox{\\rm\\scriptsize s}}(k^2)$ known from the\nQCD renormalization group analysis, although it must be modeled at\nlow momenta.\n\n\nConcretely, in the present paper we recall and utilize the results obtained \n{\\it i)} in Refs. \\cite{Klabucar:1997zi,Kekez:2000aw} by using the  \nRGI of Jain and Munczek \\cite{jain93b}, {\\it ii)} in Ref. \\cite{Kekez:2005ie} \nby using the RGI gluon condensate-induced interaction \\cite{Kekez:2003ri}, \nand {\\it iii)} in Refs. \\cite{Horvatic:2007wu,Horvatic:2007qs} by using \nthe separable interaction \\cite{Blaschke:2000gd}. In any case,\nsuch effective interactions must be modeled at least in the low-energy, \nnonperturbative regime in order to be phenomenologically successful --\nwhich above all means to be sufficiently strong in the low-momentum \ndomain to yield DChSB. \nIn the chiral limit (and {\\it close} to it), light pseudoscalar ($P$) \nmeson $q\\bar q$ bound states ($P=\\pi^{0,\\pm}, K^{0,\\pm}, \\eta$) \nthen {simultaneously} manifest themselves also as \n\\mbox{({\\it quasi-})}Goldstone bosons of DChSB.\nThis enables one to work with the mesons as \nexplicit $q\\bar q$ bound states, \nwhile reproducing \nthe results of the Abelian axial anomaly for the light pseudoscalars, \ni.e., the amplitudes for $P\\rightarrow\\gamma\\gamma$ and \n$\\gamma^\\star \\rightarrow P^0 P^+ P^-$.\nThis is unique among the bound state approaches -- e.g., see\nRefs. \\cite{Roberts:2000hi,Kekez:1998xr,Alkofer:1995jx,Bistrovic:1999dy} \nand references therein. Nevertheless, one keeps the advantage of bound-state \napproaches that from the $q\\bar q$ substructure one can calculate many \nimportant quantities (such as the pion, kaon and $s\\bar s$ pseudoscalar \ndecay constants: $f_\\pi$, $f_K$ and $f_{s\\bar s}$) which are just parameters \nin most of other chiral approaches to the light-quark sector. The treatment \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2001ph,Kekez:2005ie}\nof the $\\eta$-$\\eta'$ complex is remarkable in that it is \nvery successful in spite of the limitations of RLA.\n(Very recently, during the work on the present paper,\nthe first and still simplified DS treatments of $\\eta$ and \n$\\eta'$ beyond RLA appeared \\cite{Lakhina:2007vm,Bhagwat:2007ha}. \nHowever, RLA treatments \nwill probably long retain their usefulness in applications\nwhere simple modeling is desirable, as in the calculationally\ndemanding finite-temperature calculations \\cite{Horvatic:2007qs}.)\nThe RLA treatments of the $\\eta$-$\\eta'$ complex at first determined \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2001ph} the anomalous $\\eta_0$ mass \nparameter by fitting the empirical $\\eta$ and $\\eta'$ masses. More recently,\nthe treatment was improved by avoiding this fitting while retaining the \nphenomenologically successful description \\cite{Kekez:2005ie,Horvatic:2007qs}. \nNamely, the anomalous $\\eta_0$ mass was no longer a free parameter \nbut determined from the lattice results (on QCD topological susceptibility) \nthrough the Witten-Veneziano (WV) relation\n\\cite{Witten:1979vv,Veneziano:1979ec}. However, Shore achieved\n\\cite{Shore:2006mm,Shore:2007yn} what can be considered as a\ngeneralization of the WV relation, and the purpose of the\npresent paper is exploring the usage of\nthis generalization in the DS context.\n\n\nThe paper is organized as follows: in the next section, we recapitulate \nthe procedures and results of our previous treatments \n\\cite{Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs} relying on \nthe WV relation (\\ref{WittenVenez}), and present in Table I also \ntheir extension to the scheme of the four decay constants (and two \nmixing angles) of $\\eta$ and $\\eta'$. \nIn Section \\ref{ShoresRelations}, we expose the usage of the\npertinent Shore's equations \\cite{Shore:2006mm,Shore:2007yn}\nin the context of DS approach.  The last section concludes \nafter giving the results of solving the pertinent equations.\n\n\n\\section{$\\eta$-$\\eta'$ mass matrix from Witten-Veneziano relation}\n\\label{massMatrixAndWVrelation}\n\n\nAll $q\\bar q'$ model masses $M_{q\\bar q'}$ ($q, q' = u,d,s$)\nused in the present paper, and corresponding $q\\bar q'$ bound-state \namplitudes, were obtained in Refs. \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Blaschke:2007ce,Horvatic:2007wu,Horvatic:2007qs}\nin RLA, i.e., \nwith an interaction kernel which \n(irrespective of how one models the dynamics) cannot possibly \ncapture the effects of the non-Abelian, gluon axial anomaly. \nThus, when we form the $\\eta$-$\\eta'$ mass matrix \n\\begin{equation}\n{\\hat M}^2_\\mathrm{NA} =\n\\left[ \\begin{array}{cl}  M_{88}^2 & M_{80}^2\\\\\n                          M_{08}^2 & M_{00}^2\n        \\end{array} \\right]~,\n\\label{M2NA}\n\\end{equation}\nin this case in the octet-singlet basis $\\eta_8$-$\\eta_0$\nof the (broken) flavor-SU(3) states of isospin zero, \n\\begin{equation}\n  \\eta_8\n        =\n  \\frac{1}{\\sqrt{6}}(u\\bar{u} + d\\bar{d} -2 s\\bar{s}),\n\\quad         \n  \\eta_0\n        =\n  \\frac{1}{\\sqrt{3}}(u\\bar{u} + d\\bar{d} + s\\bar{s}),\n\\label{etasdef}\n        \\end{equation}\nthis matrix (\\ref{M2NA}), consisting of our calculated $q\\bar q$ masses,\n\\begin{equation}\nM_{88}^2 \\equiv \\langle \\eta_8 | {\\hat M}^2_\\mathrm{NA} |\\eta_8 \\rangle\n = \\frac{2}{3}\\, (M_{s\\bar{s}}^2 + \\frac{1}{2}M_{u\\bar{u}}^2) \\, ,\n\\end{equation}\n\\begin{equation}\nM_{80}^2 \\equiv \\langle \\eta_8 | {\\hat M}^2_\\mathrm{NA} |\\eta_0 \\rangle\n= M_{08}^2 = \\frac{\\sqrt{2}}{3} ( M_{u\\bar{u}}^2 - M_{s\\bar{s}}^2 )\n< 0,\n\\end{equation}\n\\begin{equation}\nM_{00}^2 \\equiv \\langle \\eta_0| {\\hat M}^2_\\mathrm{NA} |\\eta_0 \\rangle\n= \\frac{2}{3}\\, (\\frac{1}{2}M_{s\\bar{s}}^2 + M_{u\\bar{u}}^2) \\, ,\n\\end{equation}\nis purely non-anomalous ($\\mathrm{NA}$), vanishing in the chiral limit. In \nthe isospin limit, to which we adhere throughout, the pion is strictly \ndecoupled from the gluon anomaly and $M_{u\\bar{u}} = M_{d\\bar{d}}$ is \nexactly our model pion mass $M_{\\pi}$.\nAlso the unphysical $s\\bar s$ quasi-Goldstone's mass $M_{s\\bar s}$ \nresults from RLA BSE and does not include the contribution from the \ngluon anomaly. This is consistent with the fact that due to the \nDashen-Gell-Mann-Oakes-Renner (DGMOR) relation, it is in a good \napproximation \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs} \ngiven by\n\\begin{equation}\nM_{s\\bar s}^2 = 2 M_K^2 - M_{\\pi}^2 \\, ,\n\\label{MssbarMKMpi}\n\\end{equation}\ni.e., by the kaon and pion masses protected from the anomaly\nby strangeness and\/or isospin. \n\nIn our previous DS studies \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Blaschke:2007ce,Horvatic:2007wu,Horvatic:2007qs},\nto which we refer for all model details, the phenomenology of the\nnon-anomalous sector was successfully reproduced, e.g., $f_\\pi$, \n$f_K$, as well as the empirical masses $M_{\\pi}$ and \n$M_K$ (see the upper part of Table \\ref{3WV+Shore}), \nyielding a strongly non-diagonal ${\\hat M}^2_\\mathrm{NA}$ (\\ref{M2NA}). \nIts diagonalization leads to the eigenstates known as the \nnonstrange-strange ($\\mathrm{NS}$-$\\mathrm{S}$) basis,\n\\begin{equation}\n\\eta_\\mathrm{NS} = \\frac{1}{\\sqrt{2}} (u\\bar{u} + d\\bar{d})~,\n\\qquad\n\\eta_\\mathrm{S} = s\\bar{s} \\, ,\n\\label{NS-Sbasis}\n\\end{equation}\nand to \n${\\hat M}^2_\\mathrm{NA}={\\rm diag}[M_{\\pi}^2,M_{s\\bar s}^2]$. \nIn contrast to these mass-squared eigenvalues, the experimental \nmasses are such that $(M_{\\pi}^2)_{exp} \\ll (M_{\\eta}^2)_{exp}$, \nand $\\eta'$ is too heavy, $(M_{\\eta'})_{exp} = 958$ MeV,\nto be considered even as the $s\\bar s$ quasi-Goldstone boson.\nThis is the well-known $U_A(1)$ problem, resolved by the\nfact that the {\\it complete} $\\eta$-$\\eta'$ mass matrix \n${\\hat M}^2$ must contain the anomalous ($A$) part ${\\hat M}^2_\\mathrm{A}$.\nThat is, ${\\hat M}^2 = {\\hat M}^2_\\mathrm{NA} +  {\\hat M}^2_\\mathrm{A}$. \n\nHowever, ${\\hat M}^2_\\mathrm{A}$ is inaccessible to RLA\nwhich yields our Goldstone pseudoscalars. In Refs. \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007wu,Horvatic:2007qs},\n${\\hat M}^2_\\mathrm{A}$ was extracted from lattice data through the WV relation\n[the second equality in Eq. (\\ref{WittenVenez})]. \nThe purpose of the present paper, instead, is to approach \n$\\eta$ and $\\eta'$ through Shore's \\cite{Shore:2006mm,Shore:2007yn}\nrecent generalization of that relation. \n\nBefore that, however, we review the usage of the WV relation in Refs.\n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007wu,Horvatic:2007qs}.\nThe expansion in the large number of colors, $N_c$, indicates that \nthe leading approximation in that expansion describes the bulk\nof main features of QCD.\nThe gluon anomaly is suppressed as $1\/N_c$ and can be viewed as a \nperturbation in the large $N_c$ expansion. In the SU(3) limit\n[compare Eqs. (\\ref{M2A}) and (\\ref{M2AqqX})], it is coupled {\\it only} \nto the singlet combination $\\eta_0$ (\\ref{etasdef}); only the $\\eta_0$ \nmass receives, from the gluon anomaly, a contribution which, unlike \nquasi-Goldstone masses $M_{q\\bar q'}$'s comprising ${\\hat M}^2_\\mathrm{NA}$, \ndoes {\\it not} vanish in the chiral limit.\nAs discussed in Refs. \\cite{Klabucar:1997zi,Kekez:2005ie}, \nin the present bound-state context it is thus meaningful to \ninclude the effect of the gluon anomaly just on the level of\na mass shift for the $\\eta_0$ as the lowest-order effect, and retain\nthe $q\\bar q$ bound-state amplitudes and the corresponding \nmass eigenvalues $M_{q\\bar q}$ as calculated by solving DSEs and\nBSEs with kernels in RLA. \n\nReferences \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007wu,Horvatic:2007qs}\nthus break the $U_A(1)$\nsymmetry, and avoid the $U_A(1)$ problem, by shifting the $\\eta_0$ \n(squared) mass by an amount denoted by $3\\beta$ (in the notation of \nRefs. \\cite{Kekez:2000aw,Kekez:2005ie}). The complete mass matrix \n${\\hat M}^2 = {\\hat M}^2_\\mathrm{NA} +  {\\hat M}^2_\\mathrm{A}$ then contains\nthe anomalous part\n\\begin{equation}\n{\\hat M}^2_\\mathrm{A} = \\mbox{\\rm diag}[0, 3\\beta]~,\n\\label{M2A}\n\\end{equation}\nwhere the anomalous $\\eta_0$ mass shift $3\\beta$ \nis related to the topological susceptibility of the vacuum,\nbut in the present approach must be treated as a parameter to \nbe determined outside of our RLA model, i.e., fixed by\nphenomenology or taken from the lattice calculations \\cite{Alles:1996nm}.\n(The possibility of employing an additional microscopic model for the \ngluon anomaly contribution, such as the one of Ref. \\cite{Blaschke:1996dp},\nis presently not considered.)\n\n\n\nThe SU(3) flavor symmetry breaking and its interplay with the\ngluon anomaly modifies \\cite{Kekez:2005ie} \n${\\hat M}^2_\\mathrm{A}$ (\\ref{M2A}) to \n\\begin{equation}\n{\\hat M}^2_\\mathrm{A} = \n\\beta \\left[ \\begin{array}{cl}\n      \\,\\,  \\frac{2}{3}(1-X)^2 & \\frac{\\sqrt{2}}{3}(1-X)(2+X) \\\\\n  \\frac{\\sqrt{2}}{3}(1-X)(2+X) & \\,\\,\\,\\,\\, \\quad \\frac{1}{3}(2+X)^2\n        \\end{array} \\right]\n \\, ~,\n\\label{M2AqqX}\n\\end{equation}\nwhere $X$ is the flavor symmetry breaking parameter. It is most often\nestimated as $X = f_\\pi\/f_{s\\bar s} \\sim 0.7-0.8$ (see, e.g., Refs. \n\\cite{FeldmannKrollStech98PRD,Feldmann99IJMPA,Kekez:2000aw,Kekez:2005ie},\nalthough there are some other \\cite{Kekez:2000aw}, of course related, \nestimates of $X$). Presently we also adopt $X = f_\\pi\/f_{s\\bar s}$, \nwhich means that $X$ is a calculated quantity in our approach.\nThe employed models achieved good agreement with phenomenology \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs}, \ne.g., fitted the experimental \nvalue of $M_\\eta^2 + M_{\\eta'}^2$ for $\\beta$ around 0.26 -- 0.28 GeV$^2$.\nThe anomaly contribution ${\\hat M}^2_\\mathrm{A}$ then brings the complete $M^2$ \nrather close to a diagonal form for all considered models\n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs}; \nthat is, to diagonalize $M^2$, only a relatively small rotation \n($|\\theta|\\sim 13^\\circ \\pm 2^\\circ$) \nof the $\\eta_{8}$-$\\eta_{0}$ basis states,\n\\begin{equation}\n\\eta = \\cos\\theta \\, \\eta_{8}\n             - \\sin\\theta \\, \\eta_0~,\n\\,\\,\\,\\,\\,\\,\\,\n\\eta^\\prime = \\sin\\theta \\, \\eta_8\n             + \\cos\\theta \\, \\eta_0~,\n\\label{MIXtheta}\n\\end{equation}\nis needed to align them with the mass eigenstates, i.e.,\nwith the physical $\\eta$ and $\\eta^\\prime$.\nIn contrast to this,\nthe $\\eta$-$\\eta^\\prime$ mass matrix in the $\\mathrm{NS}$-$\\mathrm{S}$\nbasis (\\ref{NS-Sbasis}),\n\\begin{eqnarray}\n\\label{M2_NS-S}\n{\\hat M}^2 &=&\n             \\pipsq{\n                \\begin{array}{ll}\n         M_{\\eta_{\\mathrm{NS}}}^2    &  M_{\\eta_{\\mathrm{S}}\\eta_{\\mathrm{NS}}}^2 \\\\\n            M_{\\eta_{\\mathrm{NS}}\\eta_{\\mathrm{S}}}^2 &    M_{\\eta_{\\mathrm{S}}}^2\n                \\end{array}\n        }  \\\\\n     &=&\n       \\pipsq{\n                \\begin{array}{ll}\n      M_{\\pi}^2 + 2 \\beta  & \\quad \\sqrt{2} \\beta X \\\\\n        \\,  \\sqrt{2} \\beta X    & M_{s\\bar{s}}^2 + \\beta X^2\n                \\end{array}\n        }\n\\begin{array}{c} \\vspace{-2mm} \\longrightarrow \\\\ \\phi \\end{array}\n        \\pipsq{\n                \\begin{array}{ll}\n                        M_\\eta^2        & \\,\\,  0 \\\\\n                     \\,   0               & M_{\\eta'}^2\n                \\end{array}\n        },\n\\nonumber\n\\end{eqnarray}\nis then strongly off-diagonal.\nThe indicated diagonalization, given by\n\\begin{equation}\n\\eta = \\cos\\phi \\, \\eta_{\\mathrm{NS}}\n             - \\sin\\phi \\, \\eta_\\mathrm{S}~,\n\\,\\,\\,\\,\\,\\,\\,\n\\eta^\\prime = \\sin\\phi \\, \\eta_\\mathrm{NS}\n             + \\cos\\phi \\, \\eta_\\mathrm{S}~,\n\\label{MIXphi}\n\\end{equation}\nis thus achieved for a large $\\mathrm{NS}$-$\\mathrm{S}$ state-mixing angle\n$\\phi \\sim 42^\\circ \\pm 2^\\circ$. Of course, this is again in agreement \nwith phenomenological requirements \\cite{Kekez:2000aw,Kekez:2005ie}, \nsince $\\phi$ is fixed to the $\\eta_8$-$\\eta_0$ state-mixing angle \n$\\theta$ by the relation \n$\\phi = \\theta + \\arctan\\sqrt{2}  = \\theta + 54.74\\deg$.\nThe masses are\n\\begin{eqnarray}\n\\label{Meta}\n\\nonumber\nM_{\\eta}^2 &=& \\frac{1}{2} \\left[ M_{\\eta_{\\mathrm{NS}}}^2 + M_{\\eta_{\\mathrm{S}}}^2\n  - \\sqrt{(M_{\\eta_{\\mathrm{NS}}}^2  - M_{\\eta_{\\mathrm{S}}}^2)^2 + 8 \\beta^2 X^2} \\right],\n\\\\\nM_{\\eta'}^2 &=& \\frac{1}{2} \\left[ M_{\\eta_{\\mathrm{NS}}}^2 + M_{\\eta_{\\mathrm{S}}}^2\n  + \\sqrt{(M_{\\eta_{\\mathrm{NS}}}^2  - M_{\\eta_{\\mathrm{S}}}^2)^2 + 8 \\beta^2 X^2} \\right].\n\\nonumber\n\\label{MetaPrime}\n\\end{eqnarray}\n\nThe invariant trace of the mass matrix (\\ref{M2_NS-S}), together\nwith Eq. (\\ref{MssbarMKMpi}), \ngives the first equality in \n\\begin{equation}\n   \\beta \\, (2 + X^2) = M_\\eta^2 + M_{\\eta'}^2 - 2 M_K^2 =\n\\frac{6}{f_\\pi^2} \\, \\chi_{\\rm YM}\n \\, .\n\\label{WittenVenez}\n\\end{equation}\nThe second equality is the Witten-Veneziano (WV) relation \n\\cite{Witten:1979vv,Veneziano:1979ec} between the $\\eta$, \n$\\eta'$ and kaon masses and $\\chi_{\\rm YM}$, the topological \nsusceptibility of the pure gauge, Yang-Mills theory.\nThus, $\\beta$ does not need to be a free parameter, but can be\ndetermined from lattice results on $\\chi_{\\rm YM}$, so that \nno fitting parameters are introduced. For the three models\n\\cite{jain93b,Kekez:2003ri,Blaschke:2000gd} utilized in our treatments  \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs} of\n$\\eta$ and $\\eta'$, the bare quark mass parameters and the \ninteraction parameters were fixed already in the non-anomalous sector, \nby requiring the good pion and kaon phenomenology.\n(See the $\\pi$ and $K$ masses and decay constants in\nthe uppermost part of Table \\ref{3WV+Shore}.)\nThen, following Refs. \\cite{Kekez:2005ie,Horvatic:2007qs} in\nadopting the central value of the weighted average of the recent \nlattice results on Yang-Mills topological susceptibility \n\\cite{Lucini:2004yh,DelDebbio:2004ns,Alles:2004vi}, \n\\begin{equation}\n\\chi_{\\rm YM} = (175.7 \\pm 1.5 \\, \\rm MeV)^4 \\, ,\n\\label{chiYMaverage}\n\\end{equation}\nwe have obtained \nthe good descriptions of the $\\eta$-$\\eta'$ phenomenology\n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs}, \nexemplified by the first three columns (one for each DS models used) \nof the middle part of Table \\ref{3WV+Shore}, giving the predictions for \nthe $\\eta$ and $\\eta'$ masses and for the $\\mathrm{NS}$-$\\mathrm{S}$ mixing angle $\\phi$.\n \nThe lowest part of the table, below the second horizontal dividing line, \ncontains the results on the quantities ($\\theta_0$, $\\theta_8$, etc.) \ndefined in the scheme with four $\\eta$ and $\\eta^\\prime$ decay constants \nand two mixing angles, introduced and explained in the following \nSection~\\ref{ShoresRelations}.\nTable \\ref{3WV+Shore} also compares\nthese results of ours (in the first three columns) with the corresponding \nresults of Shore's approach~\\cite{Shore:2006mm,Shore:2007yn}, in which \nthe {\\it experimental} values of the meson masses $M_\\pi$, $M_K$, $M_\\eta$, \nand $M_{\\eta^\\prime}$, as well as the decay constants $f_\\pi$ and $f_K$ \n(in contrast to our $q\\bar q$ bound-state model predictions for these \nquantities) are used as inputs enabling the calculation of various decay \nconstants in the $\\eta$-$\\eta'$ complex and the two mixing angles $\\theta_0$ \nand $\\theta_8$ (corresponding to $\\phi = 38.24^\\circ$ in our approach).\n\n\\begin{table}[b]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nfrom & \\cite{Kekez:2000aw} & \\cite{Kekez:2005ie}     & \\cite{Horvatic:2007qs}     & Shore                            &           \\\\\nRef. &              \\& WV &                 \\& WV &                    \\& WV & \\cite{Shore:2006mm,Shore:2007yn} & Experiment \\\\\n\\hline\n$M_\\pi$            & 137.3   & 135.0      & 140.0   &    & $(138.0)^{\\mathrm{isospin}}_{\\mathrm{average}}$ \\\\\n$M_K$              & 495.7   & 494.9      & 495.0   &    & $(495.7)^{\\mathrm{isospin}}_{\\mathrm{average}}$\\\\\n$M_{s\\bar s}$      & 700.7   & 722.1      & 684.8   &           &                  \\\\\n$f_\\pi$            & 93.1    & 92.9       & 92.0    &           & $92.4 \\pm 0.3 $ \\\\\n$f_K$              & 113.4   & 111.5      & 110.1   &           & $113.0 \\pm 1.0$  \\\\\n$f_{s\\bar s}$      & 135.0   & 132.9      & 119.1   &           &                   \\\\\n\\hline\n$M_\\eta$           & 568.2      & 577.1      & 542.3      &           & $547.75\\pm 0.12$ \\\\\n$M_{\\eta^\\prime}$  & 920.4      & 932.0      & 932.6      &           & $957.78\\pm 0.14$ \\\\\n$\\phi$           & $41.42^\\circ$  & $39.56^\\circ$  & $40.75^\\circ$  & $(38.24^\\circ)$ &                  \\\\\n\\hline\n$\\theta_0$         & $-2.86^\\circ$     & $-5.12^\\circ$     & $-6.80^\\circ$     & $-12.3^\\circ$     & \\\\\n$\\theta_8$         & $-22.59^\\circ$    & $-24.14^\\circ$    & $-20.58^\\circ$    & $-20.1^\\circ$     & \\\\\n$f_0$              & 108.8         & 107.9         & 101.8         & 106.6         & \\\\\n$f_8$              & 122.6         & 121.1         & 110.7         & 104.8         & \\\\\n$f_\\eta^0$         & 5.4           & 9.6           & 12.1          & 22.8          & \\\\\n$f_{\\eta^\\prime}^0$& 108.7         & 107.5         & 101.1         & 104.2         & \\\\\n$f_\\eta^8$         & 113.2         & 110.5         & 103.7         & 98.4          & \\\\\n$f_{\\eta^\\prime}^8$& -47.1         & -49.5         & -38.9         & -37.6         & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The results of employing the WV relation (\\ref{WittenVenez}) \nin our DS approach for the three dynamical models used in Refs. \n\\cite{Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs},\ncompared with the results of Shore's analysis \\cite{Shore:2006mm,Shore:2007yn}\nand with the experimental results.\nThe first column was obtained by the WV-recalculation of the results of \nRef. \\cite{Kekez:2000aw}, which in turn used the Jain-Munczek \n{\\em Ansatz} for the gluon propagator \\cite{jain93b}.\nColumn 2: the results based on Ref. \\cite{Kekez:2005ie}, which used \nthe OPE-inspired, gluon-condensate-enhanced gluon propagator \\cite{Kekez:2003ri}.\nColumn  3: the results based on Ref. \\cite{Horvatic:2007qs}, which utilized the \nseparable {\\em Ansatz} for the dressed gluon propagator \\cite{Blaschke:2000gd}.\nColumn 4: The results of Shore \\cite{Shore:2006mm,Shore:2007yn}, who used the \nlattice result $\\chi_{\\mbox{\\rm\\scriptsize YM}} = (191\\, \\rm MeV)^4$ of \nRef.~\\cite{DelDebbio:2004ns}, and not the weighted average (\\ref{chiYMaverage}),\nin contrast to us.\nColumn 5: the experimental values. All masses and decay constants are in \nMeV, and angles are in degrees. For more details, see text.\n}\n\\label{3WV+Shore}\n\\end{table}\n\n                                                      \n\n\\section{Usage of Shore's equations in DS approach}\n\\label{ShoresRelations}\n\nThe WV relation was derived in the lowest-order approximation\nin the large $N_c$ expansion. \nHowever, considerations by Shore \\cite{Shore:2006mm,Shore:2007yn}\ncontain what amounts to the generalization of the WV relation, \nwhich is valid to all orders in $1\/N_c$. Among the relations he\nderived through the inclusion of the gluon anomaly in DGMOR relations, \nthe following are pertinent for the present paper:\n\\begin{eqnarray}\n\\!\\! (f^0_{\\eta'})^2 M_{\\eta'}^2 + (f^0_{\\eta})^2 M_\\eta^2 =\n{1\\over3} \\bigl(f_\\pi^2 M_\\pi^2 + 2 f_K^2 M_K^2\\bigr) + 6 A~, \n\\quad\n\\label{eq:bn}\\\\\n\\nonumber\\\\\n\\!\\! f^0_{\\eta'} f^8_{\\eta'} M_{\\eta'}^2 + f^0_{\\eta} f^8_{\\eta} M_{\\eta}^2 =\n{2\\sqrt2\\over3}\\bigl(f_\\pi^2 M_\\pi^2 - f_K^2 M_K^2\\bigr)~,  \\qquad\n\\label{eq:bo}\\\\\n\\nonumber\\\\\n\\!\\! (f^8_{\\eta'})^2 M_{\\eta'}^2 + (f^8_{\\eta})^2 M_{\\eta}^2 =\n-{1\\over3}\\bigl(f_\\pi^2 M_\\pi^2 - 4 f_K^2 M_K^2\\bigr)~,    \\qquad\n\\label{eq:bp\n\\end{eqnarray}\nwhere $A$ is the full QCD topological charge parameter, and\n$f^0_{\\eta'}, f^0_{\\eta}, f^8_{\\eta'}, f^8_{\\eta}$ are the {\\it four}\ndecay constants \\cite{Gasser:1984gg,Leutwyler98,KaiserLeutwyler98}\nassociated with the two isoscalar pseudoscalars $\\eta$ and $\\eta'$.\n\nThe nonperturbative parameter $A$ is related to the\nQCD topological susceptibility, quark condensates and\nquark masses \\cite{Shore:2006mm,Shore:2007yn}.\nAt large $N_c$, it should be  well-approximated by the topological susceptibility,\n$A \\approx \\chi$. More precisely, it reduces to the YM topological susceptibility\nin the large $N_c$ limit: $A = \\chi_{\\rm YM} + {\\cal O}({1}\/{N_c})$,\nbut at present it is not known better than that, as there are still\nno lattice data on this nonperturbative QCD parameter.\nTherefore, in his own phenomenological analysis,\nShore himself had to approximate $A$ by a value of\n$\\chi_{\\rm YM}$ \\cite{Shore:2006mm,Shore:2007yn}. In that sense,\nbecause of this crucial assumption based on the lowest-order ${1}\/{N_c}$\napproximation, even his analysis was not (and, because of the lack of the\ncorresponding lattice data, could not be) carried out {\\it numerically}\nconsistently in the orders of ${N_c}$, even though\nhis {\\it formulas} are valid in all orders in the ${1}\/{N_c}$ expansion.\n\nWhile the present bound-state DS approach clearly cannot improve on the\nconsistency aspect, it offers the possibility of a phenomenological analysis\nentirely different from Shore's.\nNamely, in addition to $A \\approx \\chi_{\\rm YM}$, Shore used\nthe experimentally known quantities (pion, kaon, $\\eta$ and $\\eta'$ masses,\nas well as the pion and kaon decay constants) as inputs in\nEqs. (\\ref{eq:bn})-(\\ref{eq:bp}) to obtain the $\\eta$ and $\\eta'$ decay\nconstants $f^0_{\\eta'}, f^0_{\\eta}, f^8_{\\eta'}, f^8_{\\eta}$.\nOn the other hand, the predicting power of our bound-state DS approach is\nmuch larger: not only are pion and kaon masses and decay constants\ncalculated quantities, predicted from the $q\\bar q$ substructure, but once we\nformulate the incorporation of Shore's generalization within the bound-state\nDS approach, it will become obvious that also these four $\\eta$ and $\\eta'$\ndecay constants {\\it and} their masses $M_\\eta$ and $M_{\\eta'}$ come out\nas pure predictions.\nSuch a phenomenological analysis, complementary to Shores, motivates us\nto formulate and perform the treatment based on Shore's generalization,\ninstead of the original WV relation (or fitting the anomalous $\\eta_0$\nmass shift) as in our earlier references \\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2001ph,Kekez:2005ie,Horvatic:2007qs}.\n\nAdding Eqs. (\\ref{eq:bn}) and (\\ref{eq:bp}), one gets the relation\n\\begin{eqnarray}\n(f^0_{\\eta'} )^2 M_{\\eta'}^2  &+& (f^0_{\\eta} )^2 M_\\eta^2 \n+ (f^8_{\\eta})^2 M_\\eta^2 \\nonumber\n\\\\\n&+& (f^8_{\\eta'})^2 M_{\\eta'}^2 - 2f_K^2 M_K^2 = 6A\n\\end{eqnarray}\nwhich is the analogue of the standard WV formula (\\ref{WittenVenez}), \nto which it reduces in the large $N_c$ limit where $A\\to \\chi_{\\rm YM}$, \nthe $f^0_{\\eta '}, f^8_{\\eta }, f_K \\to f_\\pi$ limit, and the limit of\nvanishing subdominant decay constants (since $\\eta$ and $\\eta'$ are \ndominantly $\\eta_8$ and $\\eta_0$, respectively), i.e.,\n$f^0_{\\eta }, f^8_{\\eta'} \\to 0$. However, we will need to use\nnot just this single equation, but the three equations \n(\\ref{eq:bn})-(\\ref{eq:bp}) from Shore's generalization. \n\n\n\nThese four $\\eta$ and $\\eta'$ decay constants are often \nparameterized in terms of two decay constants, $f_8$ and $f_0$,\nand two mixing angles, $\\theta_8$ and $\\theta_0$: \n\\begin{equation}\n\\label{def:f8etaf0eta}\nf^8_\\eta = \\cos\\theta_8\\, f_8~, \\qquad \\,\\,  f^0_\\eta = -\\sin\\theta_0\\, f_0~,\n\\end{equation}\n\\begin{equation}\nf^8_{\\eta'} = \\sin\\theta_8\\, f_8~, \\qquad   f^0_{\\eta'} = \\cos\\theta_0\\, f_0~.\n\\label{def:f8eta'f0eta'}\n\\end{equation}\nThis is the so-called two-angle mixing scheme, which shows explicitly \nthat it is inconsistent to assume that the mixing of the decay constants\nfollows the pattern (\\ref{MIXtheta}) of the mixing of the states \n$\\eta_8$ and $\\eta_0$\n\\cite{Gasser:1984gg,Schechter:1992iz,Leutwyler98,KaiserLeutwyler98,FeldmannKrollStech98PRD,FeldmannKrollStech99PLB,Feldmann99IJMPA}.\n\nThe advantage of our model is that, as we shall see, we are able to calculate\nthe $f_8$ and $f_0$ parts of the physical decay constants\n(\\ref{def:f8etaf0eta})-(\\ref{def:f8eta'f0eta'}) from the $q\\bar q$ substructure.\nHowever, we cannot keep the full generality of Shore's approach, which \nallows for the mixing with the gluonic pseudoscalar operators, and \ntherefore employs the definition \\cite{Shore:2006mm,Shore:2007yn} \nof the decay constants which, in general, due to the gluonic contribution, \ndiffers from the following standard definition \nthrough the matrix elements of the axial currents $A^{a\\,\\mu}(x)$:\n\\begin{equation}\n\\!\\! \\langle 0|A^{a\\,\\mu}(x)|P(p)\\rangle = if^a_P\\, p^\\mu e^{-ip\\cdot x},\n\\,\\,a=8,0;\\,\\,P=\\eta,\\eta^\\prime~.\n\\label{def2angSch}\n\\end{equation}\nNevertheless, Shore's definition \\cite{Shore:2006mm,Shore:2007yn}\ncoincides with the above standard one in the \nnon-singlet channel, where there cannot be any admixture of the pseudoscalar \ngluonic component.  Similarly, since our BS solutions (from Refs. \n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs}) \nare the pure $q\\bar q$ states, without any gluonic components, using Shore's\ndefinition would not help us calculate the gluon anomaly influence on the \ndecay constants. We thus employ the standard definitions (\\ref{def2angSch}), \nalso used by, e.g., Gasser, Leutwyler, and Kaiser \n\\cite{Gasser:1984gg,Leutwyler98,KaiserLeutwyler98},\nas well as by Feldmann, Kroll, and Stech (FKS)\n\\cite{FeldmannKrollStech98PRD,FeldmannKrollStech99PLB,Feldmann99IJMPA}.\n\nIn Eqs. (\\ref{def:f8etaf0eta})-(\\ref{def:f8eta'f0eta'}), the angles are\nchosen so \\cite{Feldmann99IJMPA} that $\\theta_8 = \\theta_0 = \\theta = 0$ \nin the limit of the exact SU(3) flavor symmetry, since only then there \nare just two decay constants, purely octet $f^8_{\\eta} = f_8$ and purely\nsinglet $f^0_{\\eta'} = f_0$, while the off-diagonal decay constants\nvanish, $f^0_{\\eta} = 0 = f^8_{\\eta'}$, in this limit.\nOtherwise, all four decay constants (\\ref{def2angSch}) are different from \nzero due to the breaking of the SU(3) flavor symmetry, since this leads to \n$\\theta \\neq 0$ and gives both $\\eta$ and $\\eta'$ the both components \n$\\eta_8$ and $\\eta_0$. [In the parameterization \n(\\ref{def:f8etaf0eta})-(\\ref{def:f8eta'f0eta'}), the angles $\\theta_8$\nand $\\theta_0$ differ from $\\theta$ since also \n$\\langle 0|A_\\mu^8|\\eta_0\\rangle\\neq 0 \\neq \\langle 0|A_\\mu^0|\\eta_8\\rangle$.]\nThus, although not $\\eta_8$ but $\\eta_0$ couples to the gluon anomaly,  \nthe octet-chanel constants $f^8_{\\eta}$ and $f^8_{\\eta'}$ are influenced by the \ngluon anomaly through its interplay with the SU(3) flavor symmetry breaking\n[similarly to the anomalous mass matrix (\\ref{M2AqqX}) having nonvanishing\n88, 08 and 80 elements when $X \\neq 1$]. \n\n\nEquivalently to $f^0_{\\eta'}, f^8_{\\eta }, f^0_{\\eta }$, and $f^8_{\\eta'}$, \ndefined by Eq. (\\ref{def2angSch}), one has four related but different constants\n$f^{\\mathrm{NS}}_{\\eta'}, f^{\\mathrm{NS}}_{\\eta }, f^\\mathrm{S}_{\\eta }$, and $f^\\mathrm{S}_{\\eta'}$, if instead \nof octet and singlet axial currents ($a=8,0$) in Eq. (\\ref{def2angSch}) \none uses the nonstrange-strange axial currents ($a=\\mathrm{NS},\\mathrm{S}$)\n\\begin{eqnarray}\nA_\\mathrm{NS}^{\\mu}(x) &=&  \\frac{1}{\\sqrt{3}} A^{8\\,\\mu}(x)\n                    + \\sqrt{\\frac{2}{3}} A^{0\\,\\mu}(x)\n\\nonumber\n\\\\\n                &=& \\frac{1}{2} \\left[\n    \\bar{u}(x)\\gamma^\\mu\\gamma_5 u(x)\n    +\n    \\bar{d}(x)\\gamma^\\mu\\gamma_5 d(x)\n                \\right] ,\n\\end{eqnarray}\n\\begin{equation}\nA_\\mathrm{S}^{\\mu}(x) = - \\sqrt{\\frac{2}{3}} A^{8\\,\\mu}(x)\n                   + \\frac{1}{\\sqrt{3}} A^{0\\,\\mu}(x)\n= \\frac{1}{\\sqrt{2}}\\bar{s}(x)\\gamma^\\mu\\gamma_5 s(x)~.\n\\end{equation}\nThe relation between the two equivalent sets is thus\n\\begin{equation}\n\\left[ \\begin{array}{cc}\nf^\\mathrm{NS}_\\eta &\n        f^S_\\eta\n\\\\\nf^\\mathrm{NS}_{\\eta^\\prime} &\n        f^S_{\\eta^\\prime}\n       \\end{array}\n\\right]\n=\n\\left[ \\begin{array}{cc}\nf^8_\\eta &\n        f^0_\\eta\n\\\\\nf^8_{\\eta^\\prime} &\n        f^0_{\\eta^\\prime}\n       \\end{array}\n\\right]\n\\left[ \\begin{array}{cc}\n\\frac{1}{\\sqrt{3}} & -\\sqrt{\\frac{2}{3}}\n\\\\\n\\sqrt{\\frac{2}{3}} & \\frac{1}{\\sqrt{3}}\n       \\end{array}\n\\right]~.\n\\label{TwoMixingAngles:sns-etatap}\n\\end{equation}\nOf course, this other quartet of $\\eta$ and $\\eta'$ decay constants\ncan also be  parameterized in terms of other two constants and two other\nmixing angles:\n\\begin{equation}\n\\label{def:fNSetafSeta}\nf^\\mathrm{NS}_\\eta = \\cos\\phi_\\mathrm{NS}\\, f_\\mathrm{NS}~, \\qquad \\,\\,  f^\\mathrm{S}_\\eta = -\\sin\\phi_\\mathrm{S}\\, f_\\mathrm{S}~,\n\\end{equation}\n\\begin{equation}\nf^\\mathrm{NS}_{\\eta'} = \\sin\\phi_\\mathrm{NS}\\, f_\\mathrm{NS}~, \\qquad  f^\\mathrm{S}_{\\eta'} = \\cos\\phi_\\mathrm{S}\\, f_\\mathrm{S}~,\n\\label{def:fNSeta'fSeta'}\n\\end{equation}\nwhere $f_\\mathrm{NS}$ and $f_\\mathrm{S}$ are given by the matrix elements\n\\begin{equation}\n\\langle 0| A_\\mathrm{NS}^{\\mu}(x) |\\eta_\\mathrm{NS}(p)\\rangle\n=\ni f_\\mathrm{NS}\\, p^\\mu e^{-ip\\cdot x}~,\n\\end{equation}\n\\begin{equation}\n\\langle 0| A_\\mathrm{S}^{\\mu}(x) |\\eta_\\mathrm{S}(p)\\rangle\n=\ni f_\\mathrm{S}\\, p^\\mu e^{-ip\\cdot x}~,\n\\end{equation}\nwhile $\\langle 0| A_\\mathrm{NS}^{\\mu}(x) |\\eta_\\mathrm{S}(p)\\rangle = 0\n= \\langle 0| A_\\mathrm{S}^{\\mu}(x) |\\eta_\\mathrm{NS}(p)\\rangle$.\n\nIn the $\\mathrm{NS}$-$\\mathrm{S}$ basis, it is possible to recover a scheme with a single mixing\nangle $\\phi$ through the application of the Okubo-Zweig-Iizuka (OZI) rule \n\\cite{FeldmannKrollStech98PRD,FeldmannKrollStech99PLB,Feldmann99IJMPA}. \nFor example, $f_\\mathrm{NS} f_\\mathrm{S} \\sin(\\phi_\\mathrm{NS}-\\phi_\\mathrm{S})$ \ndiffers from zero just by an OZI-suppressed term \\cite{Feldmann99IJMPA}. \nNeglecting this term thus implies $\\phi_\\mathrm{NS} = \\phi_\\mathrm{S}$.\n(Refs. \n\\cite{FeldmannKrollStech98PRD,FeldmannKrollStech99PLB,Feldmann99IJMPA}\ndenote $f_\\mathrm{NS}, f_\\mathrm{S}, \\phi_\\mathrm{NS}, \\phi_\\mathrm{S}$ by, respectively,\n$f_q, f_s, \\phi_q, \\phi_s$.)\nIn general, neglecting the OZI-suppressed terms, i.e., \napplication of the OZI rule, leads to the so-called FKS scheme\n\\cite{FeldmannKrollStech98PRD,FeldmannKrollStech99PLB,Feldmann99IJMPA},\nwhich exploits a big practical difference between the (in principle\nequivalent) parameterizations (\\ref{def:f8etaf0eta})-(\\ref{def:f8eta'f0eta'}) \nand (\\ref{def:fNSetafSeta})-(\\ref{def:fNSeta'fSeta'}):\nwhile $\\theta_8$ and $\\theta_0$ differ a lot from each other and\nfrom the octet-singlet {\\it state} mixing angle \n$\\theta \\approx (\\theta_8 + \\theta_0)\/2$, \nthe $\\mathrm{NS}$-$\\mathrm{S}$ decay-constant mixing angles are very close to each other\nand both can be approximated by the state mixing angle:\n$\\phi_\\mathrm{NS} \\approx \\phi_\\mathrm{S} \\approx \\phi$. Therefore one\ncan deal with only this one angle, $\\phi$, and express\nthe physical $\\eta$-$\\eta'$ decay constants as\n\\begin{equation}\n\\! \\! \\left[ \\begin{array}{cc}\nf^8_\\eta &\n   f^0_\\eta\n\\\\\nf^8_{\\eta^\\prime} &\n   f^0_{\\eta^\\prime}\n       \\end{array}\n\\right]\n\\! = \\!\n\\left[ \\begin{array}{cr}\nf_\\mathrm{NS} \\cos\\phi & -f_\\mathrm{S} \\sin\\phi \n\\\\\nf_\\mathrm{NS} \\sin\\phi &  f_\\mathrm{S} \\cos\\phi\n       \\end{array}\n\\right]\n\\!\n\\left[ \\begin{array}{cc}\n\\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\n\\\\\n-\\sqrt{\\frac{2}{3}} & \\frac{1}{\\sqrt{3}}\n       \\end{array}\n\\right].\n\\label{matrixEqLast}\n\\end{equation}\nThis relation is valid also in our approach, where $\\eta$ and $\\eta'$ are \nthe simple $\\eta_\\mathrm{NS}$-$\\eta_\\mathrm{S}$ mixtures (\\ref{MIXphi}).\nThe FKS relations \n\\cite{FeldmannKrollStech98PRD,FeldmannKrollStech99PLB,Feldmann99IJMPA}\n\\begin{equation}\nf_8 = \\sqrt{\\frac{1}{3} f_\\mathrm{NS}^2\n     + \\frac{2}{3} f_\\mathrm{S}^2 }~,\n\\label{f_8}\n\\quad\n\\theta_8 = \\phi - {\\arctan}\\left(\\frac{\\sqrt{2} f_\\mathrm{S}}\n                                    {f_\\mathrm{NS}} \\right)~,\n\\end{equation}\n\\begin{equation}\n\\! f_0 = \\sqrt{\\frac{2}{3} f_\\mathrm{NS}^2\n     + \\frac{1}{3} f_\\mathrm{S}^2 }~,\n\\,\\,\\,\\,\n\\theta_0 = \\phi - \\mbox{\\rm arctan}\\left(\\frac{\\sqrt{2} f_\\mathrm{NS}}\n                                    {f_\\mathrm{S}} \\right)~,\n\\label{f_0}\n\\end{equation}\nequivalent to Eq. (\\ref{matrixEqLast}), were also shown \\cite{Kekez:2000aw}\nto hold in our DS approach.\n\n\nIn our present DS approach, mesons are pure $q\\bar q$\nBS solutions, without any gluonium admixtures, which are \nprominent possible sources of OZI violations. \nTherefore, our decay constants are calculated quantities,\n$f_\\mathrm{NS}=f_{u\\bar u}=f_{d\\bar d}=f_\\pi$ and $f_\\mathrm{S}=f_{s\\bar s}$,\nin agreement with the OZI rule. \nOur DS approach is thus naturally compatible with the FKS scheme,\nand we can use the $\\eta$ and $\\eta'$ decay constants \n(\\ref{matrixEqLast}) with our calculated $f_\\mathrm{NS}=f_\\pi$ and \n$f_\\mathrm{S}=f_{s\\bar s}$ in Shore's equations (\\ref{eq:bn})-(\\ref{eq:bp}).\n\n\n\\begin{table}[b]\n\\begin{center}\n\\begin{tabular}{|c||c|c||c|c||c|c||}\n\\hline\nInputs:     & \\multicolumn{2}{|c||}{from  Ref. \\cite{Kekez:2000aw}}   & \\multicolumn{2}{|c||}{from  Ref. \\cite{Kekez:2005ie}}  & \\multicolumn{2}{|c||}{from  Ref. \\cite{Horvatic:2007qs}  } \\\\\n\\hline\n$\\chi_{\\mbox{\\rm\\scriptsize YM}}^{1\/4}$         & $175.7$ & $191$    & $175.7$ & $191$    & $175.7$  & $191$    \\\\\n\\hline\n$M_\\eta$                                & 485.7     & 499.8      & 482.8     & 496.7      & 507.0      & 526.2      \\\\\n$M_{\\eta^\\prime}$                       & 815.8     & 931.4      & 818.4     & 934.9      & 868.7      & 983.2      \\\\\n$\\phi$                                  & $46.11^\\circ$ & $52.01^\\circ$  & $46.07^\\circ$ & $51.85^\\circ$  & $40.86^\\circ$  & $47.23^\\circ$  \\\\\n\\hline\n$\\theta_0$                              & $1.84^\\circ$  & $7.74^\\circ$   & $1.39^\\circ$  & $7.17^\\circ$   & $-6.69^\\circ$  & $-0.33^\\circ$  \\\\\n$\\theta_8$                              & $-17.90^\\circ$& $-12.00^\\circ$ & $-17.6^\\circ$ & $-11.85^\\circ$ & $-20.47^\\circ$ & $-14.11^\\circ$ \\\\\n$f_0$                                   & 108.8     & 108.8      & 107.9     & 107.9      & 101.8      & 101.8      \\\\\n$f_8$                                   & 122.6     & 122.6      & 121.1     & 121.1      & 110.7      & 110.7      \\\\\n$f_\\eta^0$                              & -3.5      & -14.7      & -2.6      & -13.5      & 11.9       & 0.6        \\\\\n$f_{\\eta^\\prime}^0$                     & 108.8     & 107.9      & 107.9     & 107.1      & 101.1      & 101.8      \\\\\n$f_\\eta^8$                              & 116.7     & 119.9      & 115.4     & 118.5      & 103.7      & 107.4      \\\\\n$f_{\\eta^\\prime}^8$                     & -37.7     & -25.5      & -37.6     & -24.9      & -38.7      & -27.0      \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The results of the three DS models\nobtained through Shore's equations (\\ref{eq:bn})-(\\ref{eq:bp})\nfor the two values of $\\chi_{\\rm YM}$ approximating $A$:\n$(175.7\\rm MeV)^4$ and $(191\\rm MeV)^4$.\nColumns 1 and 2: The results when the non-anomalous inputs for\nEqs. (\\ref{eq:bn})-(\\ref{eq:bp}), namely $M_\\pi, M_K, f_\\pi=f_\\mathrm{NS},\nf_{s\\bar s}=f_\\mathrm{S}$ and $f_K$, are taken from Ref. \\cite{Kekez:2000aw},\nwhich uses Jain--Munczek {\\em Ansatz} interaction \\cite{jain93b}.\nColumns 3 and 4: The results for the non-anomalous\ninputs from Ref. \\cite{Kekez:2005ie}\nusing OPE-inspired interaction nonperturbatively\ndressed by gluon condensates \\cite{Kekez:2003ri}.\nColumns 5 and 6: The results for the\ninputs from Ref. \\cite{Horvatic:2007qs}\nusing the separable {\\em Ansatz} interaction \\cite{Blaschke:2000gd}.\nAll masses and decay constants, as well as\n$\\chi_{\\mbox{\\rm\\scriptsize YM}}^{1\/4}$, are in MeV,\nand angles are in degrees.\n}\n\\label{DGMOR:tab:eta-etap-mixing-all3m-forArticle}\n\\end{table}\n\n\n\\section{Results and conclusions}\n\\label{ResultsAndConclusions}\n\nAll quantities appearing on the right-hand side of Eqs.  (\\ref{eq:bn})-(\\ref{eq:bp}), \nnamely $M_\\pi$, $M_K$, $f_\\pi$, and $f_K$, are calculated in our DS approach \n\\cite{Kekez:2000aw,Kekez:2005ie,Horvatic:2007qs} (for the three dynamical\nmodels \\cite{jain93b,Kekez:2003ri,Blaschke:2000gd}), {\\it except} the full \nQCD topological charge parameter $A$. Since it is at present\nunfortunately not yet known, we follow Shore and approximate it\nby the Yang-Mills topological susceptibility $\\chi_{\\rm YM}$.\n\n\nOn the left-hand side of Eqs. (\\ref{eq:bn})-(\\ref{eq:bp}), \nthe model results for $f_\\mathrm{NS}=f_\\pi$ and $f_\\mathrm{S}=f_{s\\bar s}$ \nand Eq.~(\\ref{matrixEqLast}) reduce the unknown part of the four \n$\\eta$ and $\\eta'$ decay constants $f_\\eta^0$, $f_{\\eta^\\prime}^0$, \n$f_\\eta^8$, and $f_{\\eta^\\prime}^8$, down to the mixing angle $\\phi$.\nThe three Shore's equations (\\ref{eq:bn})-(\\ref{eq:bp}) can then\nbe solved for $\\phi$, $M_\\eta$ and $M_{\\eta^\\prime}$, providing us\nwith the upper three lines of            \nTable \\ref{DGMOR:tab:eta-etap-mixing-all3m-forArticle}.\nFor each of the three different dynamical models\nwhich we used in our previous DS studies\n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Blaschke:2007ce,Horvatic:2007wu,Horvatic:2007qs},\nthese results are displayed\nfor $\\chi_{\\rm YM} = (175.7\\, \\rm MeV)^4$\nas in Refs. \\cite{Kekez:2005ie,Horvatic:2007qs}\nand for $\\chi_{\\rm YM} = (191\\, \\rm MeV)^4$ \\cite{DelDebbio:2004ns}\n(adopted by Shore \\cite{Shore:2006mm,Shore:2007yn}).\nThe lower part of the table, displaying various additional results,\nis then readily obtained through Eq. (\\ref{matrixEqLast}) and\/or \nthe useful relations (\\ref{f_8})-(\\ref{f_0}) which give $f_8$, $f_0$,\n$\\theta_8-\\phi$ and $\\theta_0-\\phi$ in terms of $f_\\mathrm{NS} =f_\\pi$ \nand $f_\\mathrm{S} = f_{s\\bar s}$.\nThus, unlike the mixing angles, $f_0$ and $f_8$ do not result from solving \nof Eqs. (\\ref{eq:bn})-(\\ref{eq:bp}), but are the calculated predictions of a\nconcrete dynamical DS model, independently of Shore's equations.\n\n\nFor all three quite different (RGI \\cite{jain93b,Kekez:2003ri} and\nnon-RGI \\cite{Blaschke:2000gd}) dynamical models\nwhich we used in our previous DS studies\n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Blaschke:2007ce,Horvatic:2007wu,Horvatic:2007qs},\nthe situation with the results turns out to be rather similar.\nSimilar results from various models mean that the usage of Shore's\ngeneralization in conjunction with the DS approach does not help one\nto discriminate between various dynamical models and so draw conclusions\non the dynamics. This is not surprising, as it has been established \n\\cite{ForExample,Alkofer:2000wg,Roberts:2000hi,Roberts:2000aa,Holl:2006ni,Fischer:2006ub}\nthat while a successful reproduction of static properties and other low-energy\nmeson phenomenology requires interaction modeling at low momenta, it is possible\nto achieve a satisfactory description of low-energy phenomenology for many forms\nof model interactions as long as their integrated strength at low momenta\n($p^2 < 1$ GeV$^2$) is sufficient to achieve a realistic DChSB. On the other hand, \nthis (similarity of our results from the very different models) has the\nadvantage that our conclusions further below are not sensitive to the changes\nof the model dynamics.\n\nThe most conspicuous feature of our results is that $\\eta$ and \n$\\eta'$ masses are both much too low when the weighted\naverage $\\chi_{\\rm YM} = (175.7 \\pm 1.5 \\, \\rm MeV)^4$ of Refs.\n\\cite{Lucini:2004yh,DelDebbio:2004ns,Alles:2004vi} is used, in contrast to\nthe results from the standard WV relation, displayed in Table \\ref{3WV+Shore}.\nIf we single out just the highest of these values ($191 \\, \\rm MeV)^4$\n\\cite{DelDebbio:2004ns}), the masses improve somewhat. However,\nother results are spoiled -- e.g., the mixing angle $\\phi$\nbecomes too high to enable agreement with the experimental results on\n$\\eta, \\eta' \\to \\gamma\\gamma$ decays, which require $\\phi \\sim 40^\\circ$\n\\cite{Kekez:2005ie}.\n\nWhen we turn to the lower parts of Tables \\ref{3WV+Shore} and\n\\ref{DGMOR:tab:eta-etap-mixing-all3m-forArticle}, where the\nresults for the $\\eta$ and $\\eta'$ decay constants, and the\ncorresponding two mixing angles $\\theta_0$ and $\\theta_8$,\nare given, we notice a feature common to all our results,\nas well as Shore's (also given in Table \\ref{3WV+Shore}).\nThe diagonal ones, $f_{\\eta^\\prime}^0$ and $f_\\eta^8$, are\nall of the order of $f_\\pi$, being larger by some 10\\% to\n30\\%. The off-diagonal ones, $f_{\\eta^\\prime}^8$ and $f_\\eta^0$,\nare, on the other hand, in general strongly suppressed.\nThis is expected, as $\\eta^\\prime$ is mostly singlet,\nand $\\eta$ is mostly octet. \n\nTo understand the dependence \nof the decay constants on the topological susceptibility \n$\\chi_{\\rm YM}$ (approximating $A$), it is important to note \nthat our $f_0$, which in a full QCD bound-state calculation\nwould be influenced by the gluon anomaly, presently is not, \nsince it is calculated (same as $f_8$) from the modeled\nmeson $q\\bar q$ substructure relying on RLA. \nIn Tables I and II one therefore sees no $\\chi_{\\rm YM}$-dependence of \nnot only $f_8$, but also of $f_0$, since the difference between $f_0$ \nand $f_8$ is presently generated only by their different $\\mathrm{NS}$\nand $\\mathrm{S}$ quark content. This feature is not only consistent with \nthe FKS scheme, but is in fact a general characteristic of this scheme.\nNamely, due to the neglect of OZI-violating contributions\n\\cite{Feldmann99IJMPA}, in the SU(3) flavor symmetry limit \none would have $f_\\mathrm{NS} = f_\\mathrm{S}$ and $f_0 = f_8$.\n(The DS approach will be able to obtain the \ngluon anomaly dependence of $f_0$ only when it manages to go beyond RLA,\nwhich is presently achieved only with schematic, very simplified\n$\\delta$-function-type interactions \\cite{Bhagwat:2007ha}.)\nThus, not only in the present DS calculation, but in fact in any \napplication of the FKS scheme, the $\\chi_{\\rm YM}$-dependence of the \nfour physical $\\eta$-$\\eta'$ decay constants \n(\\ref{def:f8etaf0eta})-(\\ref{def:f8eta'f0eta'})\nstems exclusively from the\n$\\chi_{\\rm YM}$-dependence of the mixing angles $\\phi, \\theta_8$ and \n$\\theta_0$. Its origin, as explained in the previous section, is in \nthe interplay of the anomaly with the flavor symmetry breaking. \nIn fact, the FKS scheme is based on the assumption that the \nflavor symmetry breaking is significantly more important than \nthe OZI-violating contributions (arising beyond RLA in DS approach).\n\nThe feature that may be surprising is that Shore's results\n(which, to be sure, were obtained \\cite{Shore:2006mm,Shore:2007yn}\nin quite a different way from ours) are more similar to our results\nobtained through the standard WV relation, than to our results\nobtained through Shore's Eqs. (\\ref{eq:bn})-(\\ref{eq:bp}).\n\n\nTo summarize: the present paper has explored a modification of \nthe DS treatments of the $\\eta$-$\\eta'$ complex employed in \nRefs. \\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2001ph,Kekez:2005ie}.\nIn Refs. \\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2001ph}, the\nvalue of the anomalous $\\eta_0$ mass shift was obtained by fitting, \nbut Ref. \\cite{Kekez:2005ie} improved the treatment by obtaining \nit from the lattice through the WV relation. A generalization of this\nrelation was recently proposed \\cite{Shore:2006mm,Shore:2007yn}, \nand the purpose of the present paper is to tests the usage thereof \nin the bound-state, DS context, and compare the results with those\nfrom the standard WV relation.\n\n\nAll in all, inspection and comparison of the results \nin Table \\ref{DGMOR:tab:eta-etap-mixing-all3m-forArticle}\nwith the results (in Table \\ref{3WV+Shore}) from the\nanalogous calculations but using the standard WV relation to construct\nthe complete $\\eta$-$\\eta'$ mass matrix, leads to the conclusion that\nthe DS approach with the standard WV relation (\\ref{WittenVenez}) is \nphenomenologically more successful, yielding the masses closer to \nthe experimental ones.  This may seem surprising,\nbut one must be aware that we do not\nyet have at our disposal the full QCD topological charge parameter\n$A$, and that we (along with Shore) had to use its lowest ${1}\/{N_c}$\napproximation, $\\chi_{\\rm YM}$.\nThis in general precludes a {\\it consistently} improved $1\/N_c$ treatment in\nspite of the usage of Shore's relations. The problems with inconsistencies\nin the $1\/N_c$ counting may well cause spoiling of results, especially in an\napproach such as ours, where the $\\eta$ and $\\eta'$ masses are not inputs,\nbut predicted quantities.\nOur results thus add a new argument to the motivation for undertaking\nlattice calculations proposed by Shore \\cite{Shore:2007yn} and aimed\nat proper finding the quantity $A$.\nAlso, we should recall from Sections \\ref{INTRO} and\n\\ref{massMatrixAndWVrelation} that the very usage of the RLA\nassumed that the anomaly is implemented on the level of\nthe anomalous mass only, as a lowest order $1\/N_c$ correction\n\\cite{Klabucar:1997zi,Kekez:2000aw,Kekez:2005ie,Horvatic:2007wu,Horvatic:2007qs}.\nThus, with respect to the orders in ${1}\/{N_c}$,\nusing Shore's generalization in the present formulation of our DS approach\nmay be less consistent than using the standard WV relation, which\nmay well be the cause of its lesser phenomenological success.\n\nIn spite of the lesser phenomenological success (than the standard WV relation) \nin the present context of bound-state DS calculations at zero temperature, the \npresently exposed usage of Shore's generalization will likely find its application \nat finite-temperature calculations in the DS context. Namely, there it may help \nalleviate the difficulties met due to the usage of the standard WV relation in \nthe DS approach at $T>0$, as discussed in Ref. \\cite{Horvatic:2007qs}.\n\n\n\n{\\bf Acknowledgments} \n\nD.H.~and~D.Kl. acknowledge the support of the project No.~119-0982930-1016 \nof MSES of Croatia. D.Kl. also acknowledges the hospitality and support through\nsenior associateship of International Centre for Theoretical \nPhysics at Trieste, Italy, where the present paper was started.  \nD.Kl. also thanks the LIT of JINR for \nits hospitality in Dubna, Russia, in August 2007. D.Ke. acknowledges \nthe support of the Croatian MSES project No.~098-0982887-2872.\nYu.K. thanks for support from Deutsche Forschungsgemeinschaft (DFG) \nunder grant No. BL 324\/3-1, and the work of D.B. was supported by \nthe Polish Ministry of Science and Higher Education under contract \nNo. N N202 0953 33.\n    \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOrigin of instabilities, patterns, and chaos in Rayleigh-B\\'enard convection (RBC) is an interesting problem in fluid dynamics~\\cite[]{chandra:book,busse:TAP_1981,jkb:book,manneville:book, ahlers:RMP_2009}.   Prandtl number, $P$ (ratio of kinematic viscosity $\\nu$ and thermal diffusivity $\\kappa$) and Rayleigh number, $R$ (ratio of buoyancy and dissipative terms) are the two critical parameters for RBC.   Some of the key and difficult problem in this field are related to the onset of convection for zero-Prandtl number (zero-P) and low-Prandtl number (low-P) fluids.  For zero-P and low-P convection, the nonlinear term of the Navier-Stokes equation plays an important role, and it generates vertical vorticity resulting in three-dimensional patterns and the associated secondary oscillatory patterns~\\cite[]{busse:JFM_1972, clever:JFM_1974,  busse:jfm_1984}.  In the present paper we will study various patterns and chaos for zero-Prandtl number convection using bifurcation analysis.  We use direct numerical simulations and a low-dimensional model for this purpose.\n\nLow-Prandtl number fluids, for example, mercury ($P \\approx 0.02$), liquid sodium ($P \\approx 0.01$), solar plasma in the convective zone ($P \\sim 10^{-8}$),  exhibit interesting convective patterns and chaos \\cite[]{croquette:CP_1989_a, croquette:CP_1989_b,cross:RMP_1993, bodenschatz:ARFM_2000, das:PRE_2000, stein:solar}.  In terrestrial experiments, the Prandtl number cannot go below that of liquid sodium ($\\sim 0.01$).   In addition, the visualization of flow patterns inside low-P fluids like mercury and sodium is quite difficult.  Due to these limitations of experiments, numerical simulations are very significant  in the study of low-P and zero-P convection.  The numerical work of \\cite{thual:zeroP_1992} indicate that the properties of low-P convection as $P \\rightarrow 0$ are quite close to that of zero-P convection.  Hence low-P convection as $P \\rightarrow 0$ appears to approach zero-P convection as a limiting case.  Therefore,  zero-P convection is very useful for understanding the properties of low-P convection.   Even though numerical analysis of zero-P convection is quite tricky due to its inherent instabilities near the onset, it provides certain advantages.   For $P=0$  the thermal modes are slaved to the velocity modes, hence the number of independent variables are less than those required for low-P analysis.  Also, the time steps for numerical simulations of very low-P convection could be very small due to the stiffness of the equations, which is not a limitation for zero-P convection \\cite[]{thual:zeroP_1992}.  These features led us to investigate zero-P convection in detail for understanding the convective patterns and chaos in low-P fluids.\n\n\\cite{thual:zeroP_1992} was one of the first to simulate zero-P convection for free-slip and no-slip boundary conditions.  He reported various supercritical oscillatory instabilities, regular and chaotic patterns, etc.\\ in his simulations.   The patterns observed by Thual are  two-dimensional rolls, periodic and quasiperiodic rolls, squares, travelling waves, etc.   For low-P fluids, \\cite{meneguzzi:jfm_1987} performed numerical simulations for $P = 0.2$ under stress-free boundary conditions, and for  $P = 0.025$ under no-slip boundary conditions, and observed similar patterns. \\cite{clever:pf_1990} found travelling wave solutions for low-P convection under no-slip boundary conditions.  These observations indicate that zero-P convection contains pertinent features of low-P convection.  Also, free-slip and no-slip boundary conditions exhibit similar convective flow patterns.  \n\n\nThe oscillatory instabilities, and regular and chaotic patterns found in numerical simulations have also have been observed in convection experiments on mercury,  air, and liquid sodium  \\cite[]{Rossby:lowP, krishnamurti:JFM_1970, krishnamurti:JFM_1973, maurer:JPL_1980,libchaber:JPL_1982, libchaber:physica_1983}.  Some of the most commonly observed patterns in experiments are stationary, periodic, quasiperiodic, chaotic, and travelling rolls, as well as squares, asymmetric squares, and phase locked states.  Chaotic rolls have been observed to appear through both period doubling and quasiperiodic routes in some of these experiments.  \n\nConvection experiments of \\cite{willis:pf_1967}, \\cite{krishnamurti:JFM_1970, krishnamurti:JFM_1973},  \\cite{busse:JFM_1974}, and numerical simulations of \\cite{lipps:jfm_1976}, \\cite{bolton:JFM_1985}, \\cite{clever:pf_1990}, \\cite{meneguzzi:jfm_1987}, \\cite{thual:zeroP_1992}, \\cite{ozoe:NHT_1995} indicate the presence of oscillatory instability of the two-dimensional convective rolls and the resultant stable wavy rolls.  Busse and coworkers~\\cite[]{busse:JFM_1972, clever:JFM_1974,  busse:jfm_1984} showed using perturbative analysis that the two-dimensional rolls become unstable to oscillatory three-dimensional disturbances when the amplitude of the convective motion exceeds a finite critical value.   According to \\cite{busse:JFM_1972}, the condition for the instability takes the form\n\\begin{equation}\n\\frac{R_t}{R_c} -1 \\approx 0.310 P^2\n\\end{equation}\nwhere $R_c$ is the critical Rayleigh number for the onset of convection, and $R_t$ is the Rayleigh number where oscillatory instability starts.  In addition, the time period of the oscillations measured in the units of $d^2\/\\nu$ (viscous time scale) is independent of $P$ \\cite[Eq.~(5.2) of][]{busse:JFM_1972}.  In a related work, \\cite{fauve:1987} investigated the origin of instabilities in low-P convection using phase dynamical equations and argued that the instability always saturates into travelling waves. In the present paper we will explore the origin of oscillatory instability using numerical simulations and bifurcation theory.\n\nOrigin of plethora of convective patterns observed in convective experiments and numerical simulations can be quite intricate.  Each run of a convective simulation takes significantly long time, so it is not possible to scan the parameter space minutely for deciphering detailed bifurcation scenario.  Experiments too have their complexities and limitations.  The large number of modes present in simulations and experiments tend to obscure the underlying dynamics.  These difficulties are circumvented by a  powerful and complimentary approach in which the system is analyzed using an appropriately constructed low-dimensional model.   Relatively low computational cost for running low-dimensional models, and the ease of construction of the bifurcation diagrams are some of the distinct advantages of these models compared to the experiments and simulations.  However care must be taken to take all the relevant modes of the system for constructing the low-dimensional models.  \n\nNumber of active modes near the onset of convection is not very large, so low-dimensional models consisting of these active modes are very useful for analyzing this regime.    \\cite{kumar:JP_1996}  showed using a 6-mode model of zero-P convection that the growth of the 2D rolls saturate through the generation of vertical vorticity (wavy nature). \\cite{kumar:burst_2006} observed critical bursting in the above model during the saturation. \\cite{Pal:PRE_2002} explained the mechanism of selection of the square patterns using a 15-mode model of the zero-P RBC.   Recently \\cite{Pal:EPL_2009} constructed a 13-mode model using the energetic modes of DNS, and performed a bifurcation analysis using the model and simulation results.  Using the bifurcation diagram, \\cite{Pal:EPL_2009}  could explain  the origin of squares (SQ), asymmetric squares (ASQ), oscillating asymmetric squares (OASQ), relaxation oscillations with an intermediate square regime (SQOR), and three kinds of chaotic attractors.  In addition, the above patterns were observed in both DNS and models.  Earlier \\cite{thual:zeroP_1992} had observed SQ and SQOR patterns in his simulations.  \\cite{jenkins:1984} studied the transition from 2D rolls to square patterns in RBC for different Prandtl numbers using analytical tools. \\cite{knobloch:jp_1992}  studied  the stability of the SQ patterns using a complementary procedure called the amplitude equations. \n\n\nA major limitation of the 13-mode model of \\cite{Pal:EPL_2009} was the absence of wavy rolls near the onset.  This limitation is overcome by extending this model to a 27-mode model by incorporating the corresponding dominant modes.  In the present paper we perform a bifurcation analysis of this model and explore the origin of the various convective patterns of zero-P convection with special emphasis on the oscillatory instability and related wavy roll patterns.  All the features of the 13-mode model are reproduced in the 27-mode model by construction.  We will show in our discussion that properties of the wavy rolls of the  27-mode model matches reasonably well with those observed in experiments and simulations.  \n\nThe organization of the paper is as follows: In section \\ref{sec:hydrosystem}, we describe the basic hydrodynamic system considered for the study. The low dimensional model will be derived in section \\ref{sec:lowdmodel}.  Section \\ref{sec:model_results} contains the results of the numerical simulations and the low-dimensional model.  Various bifurcation diagrams are described in this section. A brief study of the wavy rolls observed in the low-dimensional model is presented in section \\ref{sec:wavy_rolls}. We finally conclude in section \\ref{conclusion}.\n\n\\section{Governing equations and direct numerical simulations}\\label{sec:hydrosystem}\n\nThe RBC system consists of a conducting fluid of kinematic viscosity\n$\\nu$, thermal diffusivity $\\kappa$, and coefficient of volume\nexpansion $\\alpha$ confined between two conducting plates separated\nby a distance $d$ and heated from below. In the zero-P limit the\nequations under Boussinesq approximation take the\nform~\\cite[]{spiegel,thual:zeroP_1992}\n\\begin{eqnarray}\n\\partial_t(\\nabla^2 v_3) &=& \\nabla^4 v_3 + R \\nabla^2_H \\theta - \\hat{\\bf e}_3\\cdot\\nabla\\times\n           \\left[(\\mbox{\\boldmath $\\omega$}{\\cdot}\\nabla){\\bf v}\n           -( {\\bf v}{\\cdot}\\nabla)\\mbox{\\boldmath $\\omega$}  \\right],\n\\label{eq:v3}\\\\\n\\partial_t \\omega_3 &=& \\nabla^2 \\omega_3\n          +\\left[(\\mbox{\\boldmath $\\omega$}{\\cdot}\\nabla) v_3\n          -({\\bf v}{\\cdot}\\nabla)\\omega_3\\right],\n\\label{eq:omega3}\\\\\n  {\\nabla}^2 \\theta&=& - v_3,\\label{eq:theta}\\\\\n\\nabla{\\cdot}{\\bf v} &=& 0, \\label{eq:continuity}\n\\end{eqnarray}\nwhere ${\\bf v}(x,y,z,t) \\equiv (v_1,v_2,v_3)$ is the velocity field,\n$\\theta(x,y,z,t)$ is the  deviation in the temperature field from the\nsteady conduction profile, $\\mbox{\\boldmath $\\omega$} \\equiv\n(\\omega_1, \\omega_2, \\omega_3) \\equiv \\nabla\\times{\\bf v}$ is the\nvorticity field, $\\hat{\\bf e}_3 $ is the vertically\ndirected unit vector, and $ \\nabla_H^2 = \\partial_{xx} +\n\\partial_{yy} $  is the horizontal Laplacian. The equations have\nbeen nondimensionalized using  $d$ as the length scale, $d^2\/\\nu$ as\nthe time scale, and $\\nu \\beta d\/\\kappa$ as the temperature scale,\nwhere $\\beta$ is the uniform temperature gradient.   The two\nnondimensional parameters in the equations are the Rayleigh number $R =\n\\alpha \\beta g d^4\/{\\nu\\kappa}$ and the Prandtl number $P = \\nu\/\\kappa$,\nwhere $g$ is the acceleration due to gravity.    In the following\ndiscussions we will also use the reduced Rayleigh number $r =\nR\/R_{c}$ as a parameter.\n\n\nWe consider perfectly conducting boundary conditions for the top and\nbottom plates along with the free-slip boundary condition for the\nvelocity field.  Consequently\n\\begin{equation}\nv_3 = \\partial_{3}v_1 =  \\partial_{3}v_2  = \\theta = 0 ,\\quad\n\\mbox{at}\\quad z = 0, 1. \\label{eq:bc}\n\\end{equation}\nWe assume periodic boundary conditions along the horizontal directions.\n\nEquations~(\\ref{eq:v3}-\\ref{eq:continuity}) are numerically solved\nusing direct numerical simulations (DNS) under the above boundary conditions (Eqs.~\\ref{eq:bc}).  DNS were performed using a pseudo-spectral code TARANG~\\cite[]{canuto,tarang} in a box with aspect ratio $\\Gamma_x = 2\\sqrt{2}, \\Gamma_y = 2\\sqrt{2}$. Various grid resolutions, $32\\times 32\\times 32$, $64\\times 64\\times 64$, have been used for the simulations.   These grids provide well-resolved simulations near the onset of convection.  We used fourth-order Runge-Kutta scheme (RK4) for time advancement. Each run was carried out till the system reaches a steady state. The DNS runs were performed for $ 0.98 \\le r \\le 1.25$.\n\nWe perform around 200 simulation runs, yet it is insufficient to construct the bifurcation diagram of the system in detail.   For this purpose we construct a low-dimensional model, which will be described in the next section.\n\n\\section{Low-dimensional model}\\label{sec:lowdmodel}\n\n We identify 27 Fourier modes that have  significant energy (approximately 1\\% or more of the total energy) in the DNS runs and  create a low-dimensional model.  The modes of the model are shown in Fig.~\\ref{fig:modes_lowdim} and they account for approximately 98\\% of the total energy.  The triangles represent  the interacting  triads. Note that only some of the interacting triads of the model have been shown in the figure.     Care has been taken to include sufficient number of modes so that the model reproduces many features found in experiments and simulations.  Also, we ensure that the model results are reasonably close to the simulation results.   The vertical velocity field ($v_{3}$) and vertical vorticity field ($\\omega_{3}$) in terms of the chosen modes are\n\\begin{eqnarray}\nv_3 &=& W_{101}(t)\\cos(k x)\\sin(\\pi z) + W_{011}(t)\\cos(k y)\\sin(\\pi z)\\nonumber\\\\\n& &+W_{202}(t)\\cos(2 k x)\\sin(2 \\pi z) + W_{022}(t)\\cos(2 k y)\\sin(2 \\pi z)\\nonumber\\\\\n& &+W_{103}(t)\\cos(k x)\\sin(3 \\pi z) + W_{013}(t)\\cos(k y)\\sin(3 \\pi z)\\nonumber\\\\\n& &+W_{301}(t)\\cos(3 k x)\\sin(\\pi z) + W_{031}(t)\\cos(3 k y)\\sin(\\pi z)\\nonumber\\\\\n& &+W_{121}(t)\\cos(k x)\\cos(2 k y)\\sin(\\pi z)+W_{211}(t)\\cos(2 k x)\\cos(k y)\\sin(\\pi z) \\nonumber\\\\\n& &+W_{112}(t)\\cos(k x)\\cos(k y)\\sin(2\\pi z)+W_{111}(t)\\sin(k x)\\sin(k y)\\sin(\\pi z)\\\\\n\\omega_3 &=& Z_{100}(t)\\cos(k x) + Z_{010}\\cos(k y)\\nonumber\\\\\n& & + Z_{110}(t)\\sin(k x)\\sin(k y) + Z_{112}(t)  \\sin(k x)\\sin(k y) \\cos(2 \\pi z) \\nonumber\\\\\n& & + Z_{310}(t)\\sin(3 k x)\\sin(k y) + Z_{130}(t)\\sin(k x)\\sin(3 k y)\\nonumber\\\\\n& & + Z_{120}(t)\\cos(k x)\\cos(2 k y) + Z_{210}(t)\\cos(2 k x)\\cos(k y)\\nonumber\\\\\n& & + Z_{102}(t)\\cos(k x)\\cos(2 \\pi z) + Z_{012}(t)\\cos(k y)\\cos(2 \\pi z)\\nonumber\\\\\n& & + Z_{201}(t)\\cos(2 k x)\\cos(\\pi z) + Z_{021}(t)\\cos(2 k y)\\cos(\\pi z)\\nonumber\\\\\n& & + Z_{111}(t)\\cos(k x)\\cos(k y)\\sin(\\pi z) + Z_{121}(t)\\sin(k x)\\sin(2 k y)\\cos(\\pi z)\\nonumber\\\\\n& &+Z_{211}(t)\\sin(2 k x)\\sin(k y)\\cos(\\pi z)\n\\end{eqnarray}\nwhere $W_{lmn}$ and $Z_{lmn}$ are the Fourier amplitudes of the vertical velocity and vertical vorticity modes respectively with the three subscripts ($l, m, n$) indicating the wavenumber components along the $x$, $y$, and $z$ directions respectively.   The modes $(1,0,1)$ and $(0,1,1)$ are the most important modes of our model, and they represent the rolls along $y$ and $x$ directions respectively.   For the square pattern, the most important participating triad is $\\{ (1,0,1), (0,1,1), (1,1,2) \\}$.  Note that the wavenumbers of the interacting triad satisfy ${\\bf k}={\\bf p}+{\\bf q}$.\n\n\nThe horizontal components of the velocity field can be computed using the incompressibility condition of the velocity field (Eq.~(\\ref{eq:continuity})), and the temperature field $\\theta$ can be computed using Eq.~(\\ref{eq:theta}).  A Galerkin projection of the RBC equations~(\\ref{eq:v3}-\\ref{eq:omega3}) on the above modes provides a  set of $27$-dimensional coupled first-order nonlinear ordinary differential equations for the amplitudes of the above Fourier modes. On this 27-mode model, we perform a detailed bifurcation analysis that we describe in the subsequent sections. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{modes_lowdim.eps}\n\\end{center}\n\\caption{The interacting modes of the 27-mode model. Interactions corresponding to the 13-mode model are shown in blue, whereas new interactions unique to 27-mode model are shown in red. The modes in green are active in the wavy roll flow patterns.} \n\\label{fig:modes_lowdim}\n\\end{figure}\n\nOur 27-mode model is a superset of the 13-mode model of \\cite{Pal:EPL_2009}.  The nonlinear interactions of the 13-mode model are indicated by blue curves in Fig.~\\ref{fig:modes_lowdim}.  Additional interactions induced by the new modes of the 27-mode model are represented by red curves in the figure.   A primary motivation for the 27-mode model is to be able to generate wavy rolls.  The triads $ \\{ (0,1,0), (1,0,1), (1,1,1) \\}$  and $ \\{ (1,0,0), (0,1,1), (1,1,1) \\}$  play a critical role in  inducing wavy rolls along the $y$ and $x$ axes respectively.    In the present paper we will investigate the dynamics of these wavy rolls in RBC using  numerical simulations and the 27-mode model  along with other features that are generated by the inclusion of these modes.\n\nWe numerically solve the 27-mode model by employing accurate ODE solvers of MATLAB.  As a result we observe a variety of convective patterns: squares (SQ), asymmetric squares (ASQ), oscillating asymmetric squares (OASQ), relaxation oscillations with an intermediate square regime (SQOR), wavy rolls, chaotic squares, etc. We have illustrated three snapshots each of  OASQ in Fig.~\\ref{fig:pattern_OASQ}, SQOR in Fig.~\\ref{fig:pattern_SQOR},  and wavy rolls in Fig.~\\ref{fig:pattern_wavyroll}.  For dynamics of these patterns as well as other patterns mentioned in this paper refer to the accompanying videos. Note that all the above patterns were also found in our DNS.  Earlier  \\cite{thual:zeroP_1992} in his DNS of zero-P convection had shown the existence of SQ,  SQOR, oscillatory quasihexagons (SQOS), chaotic squares (SQCH), and chaotic quasihexagon (HXCH).   Thual observed the oscillatory and chaotic quasi-hexagons for Rayleigh numbers beyond the range investigated in this paper.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{OASQ_flow_r1p1.eps}\n\\end{center}\n\\caption{Oscillating asymmetric square (OASQ) pattern in the mid-plane of the convection box in zero-P RBC. The pattern is obtained from the 27-mode model at $r=1.1$. Snapshots at: (a) t = 0, (b) t = T\/4, and (c) t = T\/2, where T is the time period of oscillation.} \\label{fig:pattern_OASQ}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{SQOR_flow_r1p05.eps}\n\\end{center}\n\\caption{Relaxation oscillations with an intermediate square regime (SQOR) pattern observed in the 27-mode model at $r=1.05$. Snapshots at: (a) t = 0, (b) t = T\/4, and (c) t = T\/2.} \\label{fig:pattern_SQOR}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{wavyroll_flow_r1p15.eps}\n\\end{center}\n\\caption{Wavy roll pattern in the 27-mode model at $r=1.15$. \\break Snapshots at: (a) t = 0, (b) t = T\/4, and (c) t = T\/2.} \\label{fig:pattern_wavyroll}\n\\end{figure}\n\n\nWe investigate the origin of various convective flow patterns from the bifurcation diagrams generated using the low dimensional model. To generate the bifurcation diagram, we start first with the fixed points of the system. We compute the fixed points using the Newton-Raphson method for a given $r$, and these fixed points are subsequently continued using a fixed arc-length based continuation scheme for the neighbouring $r$ values~\\cite[]{nanda,wahi}. The stability of the fixed points are ascertained through an eigenvalue analysis of the Jacobian.  New branches of fixed points and limit cycles are born when the eigenvalue(s) become zero (pitchfork) and purely imaginary (Hopf) respectively.  This process is continued on the new branch.   For aperiodic and chaotic solutions, we resort to numerical integration and report the extremum values of the important modes. We use our own MATLAB code as well as MATCONT~\\cite[]{dhooge:matcont} for the analysis. \n\n\n\\section{Bifurcation analysis using model and simulation results }{\\label{sec:model_results}}\n\nIn the present section we numerically solve Eqs.~(\\ref{eq:v3}-\\ref{eq:continuity}) using DNS and the 27-mode model in the range $0.98 \\leq r \\leq 1.25$.  This range of $r$ values is  near the onset of convection. We will  present the bifurcation diagrams associated with the different attractors using the low-dimensional model followed by a detailed comparison of the model results with those obtained from DNS. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=15cm]{3d_bif1_diagram.eps}\n\\end{center}\n\\caption{Three dimensional view of the bifurcation diagram exhibiting the fixed points only. \nSolid and dashed curves represent the stable and unstable fixed points respectively. Black,\nblue, and cyan curves represent stationary squares (SQ), asymmetric stationary squares\n(ASQ), and conduction state respectively. Solid purple lines on the $W_{101}$ and $W_{011}$ axes represent the neutrally stable 2D roll solutions and chained purple lines represent unstable 2D rolls. CB1 (black diamond), CB2 (red dots) and CB3 (green squares) are the codimension-2 bifurcation points on the $r=1$ plane.  The critical eigenvalues at CB1, CB2 and CB3 are denoted by $\\lambda_{c}$ in the figure.}\n\\label{3d_bif}\n\\end{figure}\n\n\\subsection{Fixed points of the system}{\\label{sec:fixed_points}}\nFixed points of a dynamical system and their bifurcations provide important clues about the\nsystem dynamics. Therefore, we start our analysis by locating all\nthe fixed points of the 27-mode model.  In Fig.~\\ref{3d_bif} we\ndisplay the projection of these fixed points on the\n$W_{101}$-$W_{011}$ plane as a function of $r$.  For $r<1$, the only stable fixed point of the system is the origin, which\ncorresponds to the pure conduction state (cyan curve).  At $r=1$,\nthe conduction state loses stability, and neutrally stable pure roll\nsolutions (purple curves of Fig.~\\ref{3d_bif}) and four unstable branches of symmetric square solutions (dotted black curves of Fig.~\\ref{3d_bif}) satisfying $|W_{101}| = |W_{011}|$ are born.  Note\nthat the stability matrix at the origin (CB1) has double zero eigenvalues,\nand hence CB1 is a codimension-2 bifurcation point. \n\nThe neutrally stable 2D roll solutions of the $r=1$ plane become unstable\nthrough another codimension-2 bifurcation  at CB2 \\{($W_{101} \\simeq \\pm 13.44$, $W_{011}=0$); ($W_{011} \\simeq \\pm 13.44$, $W_{101}=0$)\\}, which is represented by large red dots on the $r=1$ plane.  The stability matrix at CB2 has a zero eigenvalue and a purely imaginary pair ($\\lambda_c = (0, \\pm i \\omega$) with $\\omega \\approx 14.2$).     As a consequence of the complex eigenvalues of CB2,  for $r>1$ periodic solutions are born, and the 2D rolls lose their stability (chained purple lines of Fig.~\\ref{3d_bif}). These\nperiodic solutions are also unstable akin to the  symmetric square fixed points associated with CB1.  We will show later that the wavy rolls are associated with CB2.  Note that CB2 and their associated attractors are absent  in the 13-mode model~\\cite[]{Pal:EPL_2009}. \n\nThe unstable roll solutions which persist at $r=1$ with amplitudes greater than $13.44$ subsequently undergo yet another codimension-2\nbifurcation at CB3  \\{($W_{101}=\\pm 26.94, W_{011}=0$); ($W_{101}=0, W_{011}=\\pm 26.94$)\\}, which are represented as large green\nsquares on the $r=1$ plane.  The stability matrix at CB3 has double zero eigenvalues ($\\lambda_c=0,0$).   Consequently unstable asymmetric square solutions with $|W_{101}| \\neq |W_{011}|$ (dotted blue curves) are born.  This bifurcation is also present in the 13-mode model~\\cite[]{Pal:EPL_2009}, and the attractors associated with this bifurcation in the 13-mode model carry over to the 27-mode model as well.   Note that the ASQ solutions of the 27-mode model have two pairs of unstable eigenvalues  as opposed to one unstable pair in the 13-mode model due to the presence of CB2.   \n\nFor low-P convection \\cite{mishra:EPl_2010} have earlier observed that the 2D rolls undergo a pitchfork bifurcation followed by a Hopf.  In the limiting case of zero-P, the Hopf bifurcation point merges with the pitchfork bifurcation point  at $r=1$,  and the critical eigenvalues at this bifurcation point are $(0,0)$ giving rise to a codimension-2 bifurcation CB3. With an increase in $r$, the double zero eigenvalues split into unstable complex conjugate pair giving rise to a scenario very similar to the Takens-Bogdanov bifurcation \\cite[]{guckenheimer:book,kuznetsov:book}.\n\n\n\n\nAs shown in Fig.~\\ref{3d_bif}, on the projection of $W_{101}$-$W_{011}$ plane, there are $13$ unstable fixed points for $r>1$:  one corresponding to the pure conduction state, four satisfying $|W_{101}| = |W_{011}|$ (SQ),  and the remaining eight satisfying $|W_{101}| \\ne |W_{011}|$ (ASQ).   After two successive inverse Hopf bifurcations, to be described later, the unstable ASQ fixed points become stable.  These\nstable fixed points are shown by the solid blue curves in\nFig.~\\ref{3d_bif}. Subsequently at $r\\simeq 1.1690$, these stable\nASQs merge (via a pitchfork bifurcation) with the symmetric square\nsolutions that originate from CB1 and stabilize them (solid black\ncurves in Fig.~\\ref{3d_bif}). Note that there is a small difference in the values\nof $r$ corresponding to the stabilization of the ASQ solutions and\nSQ solutions  for the 13-mode model  and the 27-mode model. \nIn summary, the 3D figure (Fig.~\\ref{3d_bif}) is\nqualitatively similar to the corresponding figure of the 13 mode\nmodel~\\cite[]{Pal:EPL_2009} except the codimension-2 bifurcation  at CB2 which is responsible for the wavy rolls.  In the following discussions we will describe the bifurcation diagrams including limit cycles, chaotic attractors etc.\n\nBifurcation diagrams of the 27-mode model are quite complex.  They include six different types of chaotic attractors, various types of fixed points and periodic solutions.  The model also has multiple coexisting attractors for a given value of $r$.   To disentangle its complexity, we present the bifurcation diagrams as four separate diagrams, ``Bif-13M'', ``Bif-A'', ``Bif-B'', ``Bif-C'', that highlight different features of the dynamics.   First we draw the bifurcation diagram ``Bif-13M'' associated with the 13-mode model~\\cite[]{Pal:EPL_2009},  which is a subset of the 27-mode model.  Later, we will contrast  the bifurcation diagram of the  27-mode model (denoted by ``Bif-A'') with  the diagram of the 13-mode model.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{bifurcation_13mode.eps}\n\\end{center}\n\\caption{Bifurcation diagram Bif-13M of the 13-mode model for $0.98 \\leq r\n\\leq 1.25$~\\cite[adapted from][]{Pal:EPL_2009}. The stable branches corresponding to SQ and ASQ are represented by  solid black and blue lines respectively. Red, green, and brown points represent the extrema of OASQ, SQOR, and chaotic solutions respectively.  A zoomed view of the bifurcation diagram for the chaotic regime is shown in the inset. Branches corresponding to the unstable fixed points are represented by dashed lines. Cyan line represents the conduction state.}\n\\label{fig:bifur13}\n\\end{figure}\n\n\\subsection{Bifurcation diagram  of the 13-mode model}{\\label{sec:bif13M}}\n\nAs illustrated by Fig.~\\ref{fig:modes_lowdim}, the 13-mode model is a subset of the 27-mode model.  If we force only the modes of the 13-mode model to be nonzero, and others to be zero, naturally the bifurcation diagram corresponding to the 13-mode model is reproduced.  \\cite{Pal:EPL_2009} contains a detailed discussion on this diagram and the associated flow patterns (both from DNS and the model).  Here we only provide a brief description. \n\nThe bifurcation diagram for the 13-mode model is illustrated in Fig.~\\ref{fig:bifur13} in which we plot the positive value of $|W_{101}|_{extremum}$ as a function of $r$.   For zero-P convection, chaos is observed at the onset itself.  Therefore, ~\\cite{Pal:EPL_2009} start  their analysis at $r=1.4$ where symmetric square (SQ) patterns are observed.  These states are represented by the solid black curve of Fig.~\\ref{fig:bifur13} (here the diagram is shown only for $r\\le 1.25$).  At around $r \\approx 1.2201$, SQ branches bifurcate to ASQ solutions through a supercritical pitchfork bifurcation. The solid blue curves of Fig.~\\ref{fig:bifur13} correspond to ASQ patterns.  The ASQ branches bifurcate to OASQ solutions (the solid red curves) through a Hopf bifurcation.   The limit cycles thus generated grow in size and touch the saddles (dashed line of the SQ branch) to create a very narrow window of homoclinic chaos.  After this, the system again becomes periodic (SQOR) with the merger of the limit cycles.  The SQOR patterns transform to a chaotic attractor {\\em Ch1} through a homoclinic bifurcation.  {\\em Ch1} turns to {\\em Ch2} and subsequently to {\\em Ch3} through ``crisis''.  \\cite{Pal:EPL_2009}  observed these patterns in both model and DNS.  Earlier, \\cite{thual:zeroP_1992} had observed SQ, ASQ, and SQOR patterns in his DNS runs.\n\nIn the subsequent subsections we will describe the bifurcation scenario for the 27-mode model.\n\n\\subsection{Bifurcation diagram Bif-A of the 27-mode model}{\\label{sec:Bif-A}}\n\nThe square pattern described above is also observed in the 27-mode model.  However, for SQ in the 27-mode model, twelve modes $W_{111}$, $Z_{110}$, $Z_{112}$, $Z_{100}$, $Z_{010}$, $Z_{111}$, $Z_{210}$, $Z_{120}$, $Z_{201}$, $Z_{021}$, $Z_{102}$, and $Z_{012}$ still remain zero.  When we continue the SQ branch of the 27-mode model, we obtain a new bifurcation-diagram called ``Bif-A\"  shown in Fig.~\\ref{fig:bifur27A}.   The\nbifurcation diagram Bif-A is qualitatively similar to Bif-13M except in a  narrow window of $ 1.116 \\le r \\le 1.128$ where additional bifurcations are observed. The ASQ branch of solutions in Bif-A have seventeen active modes with the modes  $W_{111}$, $Z_{100}$, $Z_{010}$, $Z_{111}$, $Z_{210}$, $Z_{120}$, $Z_{201}$, $Z_{021}$, $Z_{102}$, and $Z_{012}$ as zeros.\n\nThe new features of Bif-A are as follows.  At  $r = 1.1260$, the ASQ branch undergoes a  supercritical Hopf bifurcation (H1, see Fig.~\\ref{fig:r_w111_projection}) resulting in a time-periodic convective flow as illustrated in Fig.~\\ref{fig:H1_NS1}(a,b),  where we show a projection of the limit cycle obtained from the DNS and the model on the $W_{111}$-$Z_{010}$ plane.   All the 27 modes are active for these periodic flow patterns.\n\nAs $r$ is reduced further, at $r=1.1257$ a new frequency incommensurate with the original frequency is born through a Neimark-Sacker bifurcation (NS1) and the limit cycle becomes unstable.  Here, a pair of imaginary Floquet multipliers cross the unit\ncircle outwards as illustrated in Fig.~\\ref{fig:Floquet_NS1_NS2}(a).   The phase space\ntrajectory of the system on the $W_{111}$-$Z_{010}$ plane is therefore\nquasiperiodic as demonstrated in Fig.~\\ref{fig:H1_NS1}(c,d) for\nthe DNS and the model respectively. The unstable limit cycle \ncontinues till $r=1.0651$ where they meet the unstable limit cycles\nof the wavy rolls, which will be discussed in~\\S~\\ref{sec:Bif-C}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{bifurcation1_diagram_2.eps}\n\\end{center}\n\\caption{Bifurcation diagram Bif-A of the 27-mode model with the same color convention as Bif-13M shown in Fig.~\\ref{fig:bifur13}.  This diagram is qualitatively similar to Bif-13M.  New features of Bif-A are the H1, NS1, H2, and NS2 bifurcations shown in the boxed region ($1.116 < r < 1.128$) whose zoomed view is shown in the inset.} \n\\label{fig:bifur27A}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{H1_LC_w111_projection.eps}\n\\end{center}\n\\caption{Plot of $W_{111}$ vs. $r$ near the first Hopf bifurcation (H1) of the\nASQ branch. The solid brown curve represents the limit cycles generated after the first Hopf (H1).}\n\\label{fig:r_w111_projection}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{H1_NS1_DNS_model.eps}\n\\end{center}\n\\caption{Projection of the phase space on the $W_{111}$ - $Z_{010}$\nplane. The limit cycles born through H1 in (a) DNS at r=1.138 and (b)\nmodel at r=1.1258. Quasiperiodic attractor born through NS1 in (c)\nDNS at $r=1.131$ and (d) model at $r=1.1245$.} \n\\label{fig:H1_NS1}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{NS1_NS2.eps}\n\\end{center}\n\\caption{Movement of Floquet multipliers in the complex plane during NS1 and NS2 bifurcations. (a) At NS1, a pair of complex Floquet multipliers move out of the unit circle (red dots changing to blue). (b) At NS2, a pair of complex Floquet multipliers  enter the unit circle (red dots again becoming blue).} \n\\label{fig:Floquet_NS1_NS2}\n\\end{figure}\n\nOn further reduction of $r$, at $r= 1.1226$, another Hopf bifurcation\n(H2) takes place on the unstable ASQ branch. The limit cycle born\nfrom H2 is however unstable. At $r = 1.1181$, this limit cycle becomes stable via\nan inverse Neimark-Sacker bifurcation (NS2) wherein the unstable\nFloquet multiplier pair enters the unit circle as evident in\nFig.~\\ref{fig:Floquet_NS1_NS2}(b). The resulting stable limit cycle\nis the oscillatory asymmetric square (OASQ) solution of the 13-mode\nmodel. Figure \\ref{fig:NS2} shows a projection of the limit cycle\ncorresponding to the OASQ solution obtained from the DNS and\nthe model on the $W_{101}$-$W_{011}$ plane.  The\nquasiperiodic solutions exist only in the range\n$r=1.1181$-$1.1257$, i.e., between NS1 and NS2 and disappears after NS2.  Note that the  attractors  between H1 and NS2  contain all the 27 modes, but  beyond NS2 the ten modes $W_{111}$, $Z_{100}$, $Z_{010}$, $Z_{111}$, $Z_{210}$, $Z_{120}$, $Z_{021}$, $Z_{102}$, and $Z_{012}$ again become zero.  \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{NS2_phase_DNS_model.eps}\n\\end{center}\n\\caption{Projections of the two coexisting OASQ solutions on the $W_{101}$ -$W_{011}$ plane just after the NS2 bifurcation point: (a) in DNS at $r=1.0768$ and (b) in the model at $r=1.095$.}\n\\label{fig:NS2}\n\\end{figure}\n\nBeyond NS2, the patterns and associated bifurcations from OASQ to SQOR\nto the chaotic attractors {\\em Ch1, Ch2} and {\\em Ch3} in the decreasing $r$\ndirection are exactly the same as for Bif-13M.    The range of $r$  corresponding to these patterns are approximately $r=1.086$--$1.1181$  for OASQ, $r=1.0046$--$1.086$ for SQOR, $r=1.0034$--$1.0046$ for Ch1, $r=1.0025$--$1.0034$ for {\\em Ch2} and $r=1$--$1.0025$ for {\\em Ch3}.  Comparison with the 13-mode model indicates that the ranges of $r$ for the above patterns as well as that for SQ and ASQ are different.  This is due to the fact that more than 13 modes are active in the present model for Bif-A.  The 27-mode model reproduces the ranges of the patterns obtained from DNS more accurately.  Note that the modes absent  for SQ, ASQ, OASQ, SQOR, {\\em Ch1} to {\\em Ch3} flow patterns are active for wavy rolls that will be described in~\\S\\ref{sec:Bif-C}.  \n\n\n\\subsection{Bifurcation diagram Bif-B of the 27-mode model}{\\label{sec:Bif-B}}\n\nWhen we start from an arbitrary initial condition for $r>1$ near the onset, most often the system tends to another attractor {\\em Ch4}, which differs from {\\em Ch1, Ch2}, and {\\em Ch3}.    The codimension-2 bifurcation point CB3 of Fig.~\\ref{3d_bif} generates the chaotic attractor {\\em Ch4} as well.  The bifurcation diagram ``Bif-B\" shown in Fig.~\\ref{fig:bifur27B} contains the attractor {\\em Ch4}.   The attractor {\\em Ch4} coexists with the chaotic attractors {\\em Ch1, Ch2}, {\\em Ch3} and SQOR for $r=1$--$1.056$.  This feature is illustrated in Fig.~\\ref{fig:multi_attrac} where a phase space projection (both from the model  and DNS) for two different initial conditions at $r = 1.0342$ yield the SQOR (green curve) and the {\\em Ch4} (grey trajectory) attractors. Clearly the trajectories of {\\em Ch4} explore all the four quadrants of the $W_{101}$-$W_{011}$ plane.   The qualitative behavior of the {\\em Ch4} attractor is similar to {\\em Ch2}, but its size is larger than {\\em Ch2} \\cite[compare with Fig. 5 of][]{Pal:EPL_2009}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{bifurcation1_diagram.eps}\n\\end{center}\n\\caption{Bifurcation diagram Bif-B with the same color convention as Bif-13M (Fig.~\\ref{fig:bifur13}). A large chaotic attractor represented by gray dots {\\em Ch4} is born at CB3. The chaotic attractor {\\em Ch4} coexists with {\\em Ch1}, {\\em Ch2}, {\\em Ch3} (shown in the inset using brown dots), and SQOR.}\n\\label{fig:bifur27B}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{multiple_attractor.eps}\n\\end{center}\n\\caption{Phase space projection of coexisting SQOR (green) and chaotic attractor {\\em Ch4} (gray) on the $W_{101}$-$W_{011}$ plane at $r = 1.0342$: (a) obtained from model and (b) obtained from DNS. }\n\\label{fig:multi_attrac}\n\\end{figure}\n\nNote that there are two complex pairs of unstable eigenvalues associated with ASQ.  The first one carries from CB2, and the second one is due to the splitting of the double zero eigenvalues at CB3 into a complex pair (see~\\S\\ref{sec:fixed_points}).  The {\\em Ch4} attractor is associated with the first pair, while the {\\em Ch3} attractor is related to the second pair.    The disappearance of the chaotic attractor {\\em Ch4} is possibly through a boundary crisis \\cite[]{hilborn:book} wherein an unstable periodic\nsolution hits the basin boundaries of {\\em Ch4}. This unstable periodic\nsolution possibly has connections with the unstable periodic\nsolutions originating from the branch point CB2 of Fig.~\\ref{3d_bif}. \n\nIn the next subsection, we will discuss the bifurcations \nassociated with the solutions arising from the branch point CB2 whose flow patterns resemble the wavy rolls.\n\n\n\\subsection{Bifurcation diagram Bif-C of the 27-mode model}{\\label{sec:Bif-C}}\n\n\nThe 27-mode model has another chaotic attractor near the onset that originates from the bifurcation point CB2 of Fig.~\\ref{3d_bif}.  Recall from \\S\\ref{sec:fixed_points} that CB2 is a codimension-2 bifurcation point whose stability matrix has a simple zero eigenvalue and an imaginary pair $(0,\\pm i \\omega)$.  As a consequence, an unstable limit cycle is generated as $r$ is increased beyond 1. The attractors from this branch yield  periodic, quasiperiodic, and chaotic wavy rolls.   See Fig.~\\ref{fig:pattern_wavyroll} for an illustration of a periodic wavy roll.   The diagram Bif-C (Fig.~\\ref{fig:bifur3}) illustrates the bifurcation scenario of this type of solutions. The limit cycles generated through this bifurcation are unstable and they have an unstable torus associated with them as discussed in Section 7.4 of \\cite{guckenheimer:book}. As a result, four chaotic attractors named {\\em Ch5} are born.   A phase space projection of one of the {\\em Ch5} attractor is shown in Fig.~\\ref{fig:roll_QP_model}(a). Its chaotic nature is ascertained by the broad-band power spectrum exhibited  in Fig.~\\ref{fig:roll_QP_model}(b).  As $r$ is increased further, the size of {\\em Ch5} increases till $r \\simeq 1.009$ after which a single large chaotic attractor {\\em Ch6} is generated through an ``attractor-merging crisis''~\\cite[]{hilborn:book}.   A phase space projection and power spectrum of {\\em Ch6}  are shown in Fig.~\\ref{fig:roll_QP_model}(c,d) respectively.  The chaotic attractors {\\em Ch5} and {\\em Ch6} are illustrated in the bifurcation diagram Bif-C.  \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{bif_wavy_roll_lhs.eps}\n\\end{center}\n\\caption{Bifurcation diagram Bif-C of the wavy roll solutions. The blue, red, and brown points represent periodic, quasiperiodic and chaotic ({\\em Ch5} and {\\em Ch6}) solutions respectively. }\n\\label{fig:bifur3}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{QP_route_wavyroll_model.eps}\n\\end{center}\n\\caption{Phase space projections on the $W_{101}$-$W_{111}$  plane for the wavy rolls and their corresponding power spectra obtained from the model:  (a,b) the chaotic attractor {\\em Ch5}  at $r = 1.005$;  (c,d) the chaotic attractor {\\em Ch6}  at $r = 1.05$;  (e,f) frequency locked state with $f1\/f2 \\approx 5$ at $r = 1.078$;  (g,h) quasiperiodic state with $f1\/f2\\approx 6.33$ at $r = 1.10$;  (i,j) periodic state at $r = 1.15$ ($f1=4.3$). }\n\\label{fig:roll_QP_model}\n\\end{figure}\n\n As $r$ is increased further, we observe a series of phase-locked and stable quasiperiodic solutions.  Phase space projections and power spectra of the phase-locked ($r=1.078$) and quasiperiodic ($r=1.10$) states are shown in Fig.~\\ref{fig:roll_QP_model}(e,f) and Fig.~\\ref{fig:roll_QP_model}(g,h) respectively.  The phase-locked and quasiperiodic states are also evident in the bifurcation diagram Bif-C as banded and filled regions.  Further increase in $r$ transforms the quasiperiodic states to a limit cycle through an inverse Niemark-Sacker bifurcation.  A phase space projection and power spectrum of a limit cycle generated for $r=1.15$ are shown in Fig.~\\ref{fig:roll_QP_model}(i,j).  Note that the transformations of phase-locked, quasiperiodic, and periodic states to one another can be understood by the movement of the Floquet multipliers of the underlying limit cycles (see \\S \\ref{sec:Bif-A}).  The above states have also been observed in DNS.  For example, Fig.~\\ref{fig:roll_QP_DNS} illustrates chaotic, quasiperiodic, and periodic states obtained in DNS for $r=1.05, 1.09$, and $1.10$ respectively.\n \nWhen we examine the active modes of Bif-C, we find that only the green colored modes of Fig.~\\ref{fig:modes_lowdim} are active for these convective patterns.  The bifurcation diagram Bif-C has been generated by setting these modes as nonzero and all other modes as zero.  The most important among the active modes of Bif-C are $(1,0,1)$, $(0,1,0)$, and $(1,1,1)$ which are instrumental for the generation of wavy rolls along  the $y$ axis.  Naturally, the wavy rolls along the $x$ axis will have the complimentary set of modes, e.g., $(1,0,0)$ instead of $(0,1,0)$, etc.  \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=8cm]{QP_route_wavyroll_DNS.eps}\n\\end{center}\n\\caption{Phase space projections on the $W_{101}$-$W_{111}$ plane\ncorresponding to the wavy rolls obtained from DNS: (a)\nthe chaotic attractor {\\em Ch6} at $r=1.05$; (b) quasiperiodic state at $r=1.09$; (c) periodic state at $r=1.10$. } \n\\label{fig:roll_QP_DNS}\n\\end{figure}\n\nNote that the above sets of solutions (Bif-A, Bif-B, Bif-C) are observed for different sets of initial conditions.  For both DNS and the model, a random initial condition generally produces Bif-B whose basin of attraction appears to be rather large.  If we keep only the green colored modes of Fig.~\\ref{fig:modes_lowdim} as nonzero, we obtain Bif-C that correspond to the wavy rolls.  In fact, Bif-C is generated by first constructing a limit cycle for $r = 1.25$, and then continuing the solution for lower $r$ values.  The bifurcation diagram Bif-A is generated by starting from the SQ pattern for $r = 1.25$ for which all the modes except $W_{111}$, $Z_{110}$, $Z_{112}$, $Z_{100}$, $Z_{010}$, $Z_{111}$, $Z_{210}$, $Z_{120}$, $Z_{201}$, $Z_{021}$, $Z_{102}$, and $Z_{012}$ are nonzero.  Continuation of the above solution however generates ASQ solutions followed by H1, NS1, H2, and NS2 bifurcations where all the 27 modes are present.  \n\nWavy rolls are one of most studied convective patterns in experiments and numerical simulations.  The bifurcation diagram Bif-C provides a clear explanation for the origin of this pattern.  In the next section we will provide a quantitative comparison of the bifurcation results related to the wavy rolls with those from the experiments and previous simulations.\n\n\\section{Wavy rolls: a quantitative study}\\label{sec:wavy_rolls}\n\n In this section we will analyze time scales of the wavy rolls of Bif-C quantitatively, and compare these values with some of the experimental and numerical results.   At the bifurcation point CB2, the stability matrix has a pair of complex eigenvalues $(0,\\pm i\\omega)$ with $\\omega \\approx 14.2$.  As a result, the unstable limit cycle originating from CB2 has a time period around $2 \\pi\/\\omega \\approx 0.44$ in units of $d^2\/\\nu$ (viscous time scale).  Subsequent periodic, quasiperiodic, and chaotic time series have time scales comparable to the above value since their origin is closely connected to the bifurcation CB2 at $r=1$ in Bif-C.   Our preliminary calculations indicate that the time period of oscillations for these patterns are within a factor of 10 from this value for $r=1$--$1.25$.\n  \n Earlier \\cite{krishnamurti:JFM_1973} observed time-dependent wavy rolls in her convection experiments on mercury ($P\\approx 0.02$).  She observed multiple peaks with the time period ranging from 0.1 to 1 in the time units of $d^2\/\\nu$ \\cite[see Fig.~3 of][]{krishnamurti:JFM_1973}.  Krishnamurti's experimental value for the time-scale of the chaotic wavy rolls is in the same range as our theoretical time-scale estimated above. \\cite{willis:jfm_1970} and \\cite{croquette:wavyroll_1989}  reported the time period of the oscillatory rolls using their experimental data for air ($P=0.7$) to be around 1 in the units of  $d^2\/\\nu$.   Using numerical simulations, \\cite{lipps:jfm_1976} observed time periods of oscillatory rolls to be around $0.24$--$0.27$ in the units of $d^2\/\\kappa$ for $P=0.7$. \\cite{meneguzzi:jfm_1987} found the period of the wavy oscillations to be around 0.065 viscous time units for $P=0.025$.  These results are in  general agreement (within a factor of 10) with our theoretical finding based on the bifurcation analysis.   Note that \\cite{busse:JFM_1972} reported  that the time period  of the oscillatory instability in the units of $d^2\/\\nu$ is independent of the Prandtl number.  Hence the time-scales are not expected to vary appreciably even when we change the Prandtl number by  an order of magnitude which is consistent with the results for mercury ($P\\approx 0.02$) and air ($P\\approx 0.7$). Therefore, a comparison of our results for $P=0$ with those obtained for finite $P$ is also justified. \n \nOscillatory instabilities and their saturation through critical bursting have been studied by Kumar and coworkers \\cite[]{kumar:JP_1996,kumar:burst_2006} using several low-dimensional models.  They show that the growth of the mode $W_{101}$ is saturated by the vorticity mode $Z_{010}$.   In Fig.~\\ref{fig:wavy_roll_energy}(a,b) we plot the time series of $\\langle v_1^2 + v_3^2 \\rangle$ (sum of kinetic energy along $x$ and $z$ axes) and $\\langle v_2^2 \\rangle$ (kinetic energy along $y$ axis) computed from our 27-mode model for $r=1.05$.  These results are in general agreement with the results of Kumar and coworkers.   The panel (c) of Fig.~\\ref{fig:wavy_roll_energy} shows the time series of the modes $W_{101}$, $W_{111}$, and $Z_{010}$ that illustrates their growth and subsequent breakdowns (critical bursting).  Note that the time-scales of oscillations for the modes $W_{111}$ and $Z_{010}$ are around 0.1, which is in  the same range as the theoretical time-scale derived above using the bifurcation analysis.\n\nThe above arguments strongly suggest that the origin of the wavy rolls or the oscillatory instabilities are intimately related to the purely imaginary pair of eigenvalues at CB2 and the limit cycles that originate from it.  \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=!,width=12cm]{Ch6_wavyroll_r1p05_energy.eps}\n\\end{center}\n\\caption{Time series of $\\langle v_1^2 + v_3^2 \\rangle \/ 2$ (panel (a)) and $\\langle v_2^2 \\rangle$\/2 (panel (b)) obtained from the model at $r=1.05$. Panel (c) shows the time series of the modes $W_{101}$, $W_{111}$, and $Z_{010}$ during critical bursting.} \\label{fig:wavy_roll_energy}\n\\end{figure}\n\n\\section{Conclusion}\\label{conclusion}\n\nIn conclusion, we explored various flow patterns of zero-P convection and performed a detailed bifurcation analysis near the onset using direct numerical simulation and a 27-mode low-dimensional model.   The low-dimensional model was constructed using the most energetic modes of DNS.  The results of the DNS and the low-dimensional model are in good agreement with each other.  Several new chaotic attractors and windows of periodic and quasiperiodic rolls  have been reported for the first time for zero-P convection.   The origin and dynamics of all the observed patterns have been explained successfully using the bifurcation diagrams.\n\nThe RBC system for $P=0$ is chaotic at the onset itself.  The stability analysis of the 27-mode model indicates  three codimension-2 bifurcation points that play critical roles in the dynamics of convection near the onset.   The chaotic attractors {\\em Ch1}, {\\em Ch2}, and {\\em Ch3}, described earlier by \\cite{Pal:EPL_2009}, and {\\em Ch4} are all related to the bifurcation point CB3.   Beyond {\\em Ch1} and {\\em Ch4}, we observe SQOR, OASQ, ASQ, SQ etc., that are common to Pal \\etal's 13-mode model.  The other codimension-2 bifurcation point, CB2, generates chaotic attractors {\\em Ch5} and {\\em Ch6}, and the subsequent periodic, quasiperiodic, and phase-locked convective states, which correspond to the wavy roll patterns of RBC observed earlier in experiments and simulations.      In addition,  we find that the frequency of the wavy rolls are connected to the imaginary eigenvalues of the stability matrix at the CB2 bifurcation point.  Thus, the bifurcation analysis presented in the paper provides useful insights into the origin of the wavy rolls of RBC.  \n\nInterestingly, the bifurcation diagram of 30-mode model of \\cite{mishra:EPl_2010} for $P=0.0002$ matches quite closely with Bif-A of our model.  This reinforces earlier observations that zero-P convection is a valid limit of low-P convection as $P \\rightarrow 0$ \\cite[]{thual:zeroP_1992}.   The extension of the present study to low-P convection in relation to wavy rolls will be very valuable for understanding the experimental and numerical findings near the onset.\n\n\\begin{acknowledgments}\nWe are thankful to Krishna Kumar and Pankaj K. Mishra for useful discussions. This work is supported by the Swarnajayanti fellowship grant to MKV by Department of Science and Technology, India.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nTheoretical developments and novel experiments in the area of\nferroelectrics have rapidly evolved over the last ten years,\nallowing further progress in the understanding of this remarkable\nphenomenon. In particular, ``nanoscale\" ferroelectrics have\nattracted considerable\nattention~\\cite{ahn04_1,lee05,was04,chu04,nau04}. The question of\nthe existence of a critical thickness, in other words whether or\nnot ferroelectricity can be maintained at reduced dimensions, is\namongst the most exciting topics of the field today, with very\nactive experimental~\\cite{tyb99,str02,fon04,lic05} and theoretical\nefforts~\\cite{gho00,mey01,jun03_1}.\n\nProbing ferroelectricity in thin films and nanostructures is a\ndifficult task, which requires advanced techniques. Among these,\nscanning probe characterization based on piezoelectric microscopy\nhas allowed a ferroelectric ground state to be identified down to\n40 \\AA\\ in thin Pb(Zr$_{0.2}$Ti$_{0.8}$)O$_3$ films~\\cite{tyb99},\nand x-ray studies on PbTiO$_3$ films suggested that 28 \\AA\\ films\nare ferroelectric~\\cite{lic05}. Dielectric and pyroelectric\nresponse measurements have allowed ferroelectricity to be\nidentified in polymer films down to 10 \\AA\\ (two unit\ncells)~\\cite{bun98}. More recently, ultrahigh vacuum scanning\nprobe characterization based on electrostatic force microscopy was\nused to study ferroelectricity in barium titanate nanowires with\ndiameters as small as 10 nm~\\cite{yun02}. On insulating\nsubstrates, lateral periodicity was observed via x-ray diffraction\nin thin PbTiO$_3$ films down to 12 \\AA\\ and attributed to\nalternately polarized domains~\\cite{str02,fon04}. In all these\nstudies, however, properties {\\it averaged} over the complete\nferroelectric structure were measured and no {\\it local}\ninformation on the atomic displacements was obtained.\n\nIn contrast, the photoemission based photoelectron diffraction\n(XPD) used in our study presents two interesting characteristics:\nit is naturally surface sensitive, due to an electron escape depth\nof approximately 20 \\AA\\ (at the energy used here) ; and atomic\ndisplacements within the unit cell can be directly probed,\nallowing the non-centro-symmetric and tetragonal nature of\nthe crystal lattice to be directly demonstrated. This turns\nout to be crucial for studying ultrathin films, and for\ndiscriminating the behavior of the surface from that of the body of\nthe film.\n\nThe paper is organized as follows. In Section II, we characterize the ferroelectric distortion of PbTiO$_3$ and contrast it to the natural atomic relaxation appearing at surface and interface. In Sec. III A, we discuss aspects of the measurement methods  while, in Sec. III B we show the non-centro-symmetry of a 20 \\AA\\ thick film using XPD. The Sec. III C is dedicated to the tetragonality measurements as a function of film thickness with XPD using Pb as emitter, down to 4 \\AA\\ (one unit cell). We summarize and conclude in Sec. IV.\n\n\\section{Ferroelectric distortion versus surface relaxation}\n\nThe studies were carried out on c-axis oriented perovskite\nPbTiO$_3$ ultrathin films epitaxially grown on conducting\nNb-SrTiO$_3$ substrates. Above 490$^\\circ$C, bulk PbTiO$_3$ is a\nparaelectric insulator and a simple cubic perovskite structure\nwith a lattice parameter of 3.96 \\AA\\ (``para\"-state, see\nFig.~\\ref{despontfig1}(a)). In this structure, the Ti and O atoms\nare in perfectly centro-symmetric positions with respect to the\nsurrounding Pb cage. At lower temperature, the material becomes\ntetragonal and ferroelectric with a- and c-axis parameters of 3.90\n\\AA\\ and 4.17 \\AA\\\/, respectively ~\\cite{nel85,jon97}, as\nillustrated in Fig.~\\ref{despontfig1}(b). The ferroelectric phase is\ncharacterized by a non-centro-symmetric structure where the O and\nTi atoms are unequally shifted with respect to Pb. In a unit cell\nwith the polar c-axis along the $z$-direction, O and Ti move\neither upwards or downwards (with a larger O displacement)\nresulting respectively in a ``down\"- and ``up\"-polarized state, as\ndrawn in Fig.~\\ref{despontfig1}(b).\n\nIn the surface region (five top unit cells) that is probed by the\nXPD technique, the evidence of such a polar atomic distortion could be\nthe signature of a ferroelectric ``up\"- or ``down\"-state but\nmay also arise from the natural atomic relaxation at the film\nsurface and interface already present in the paraelectric phase. A\nproper interpretation of our data therefore requires independent\nquantification of both effects. To that end, a {\\it reference}\n configuration (``para-unrelaxed'') is defined in Fig.~\\ref{despontfig2}(a):\nit corresponds to the truncated bulk paraelectric structure of\nPbTiO$_3$ with the in-plane lattice constant constrained to that\nof SrTiO$_{3}$ ($a_{STO} = 3.90$ \\AA) and a consequent\ntetragonality $c_0\/a_{STO} = 1.03$~\\cite{lic05}.\nFig.~\\ref{despontfig2}(b) (``ferro-unrelaxed'') shows the atomic\ndistortion of the ``up\"-state as determined for bulk tetragonal\nPbTiO$_3$ by Nelmes and Kuhs in Ref. \\cite{nel85} with $c = 4.17$\n\\AA . In order to quantify the surface relaxation, density\nfunctional theory calculations~\\cite{abinit} were performed within\nthe local density approximation (LDA) using the\nABINIT~\\cite{gon02} package. Two different supercells were\nconsidered~: a thick PbTiO$_{3}$ slab in vacuum\n(Fig.~\\ref{despontfig2}(c)) and a SrTiO$_3$\/(one unit cell)\nPbTiO$_3$\/vacuum stack~\\cite{unitcell}(Fig.~\\ref{despontfig2}(d)).\nInsulating SrTiO$_3$ was considered in our simulations since Nb\ndoping is not presently affordable at the first-principles\nlevel~\\cite{STO}. To reproduce the substrate clamping effect, the\nin-plane lattice constant was fixed to the relaxed a-axis value of\nbulk SrTiO$_{3}$~\\cite{acell}. The atomic positions were then\nrelaxed until the maximum residual atomic force was smaller than\n40 meV\/\\AA\\\/. Our calculations were restricted to $(1 \\times 1)$\nsurface periodicity and did not allow for an eventual\nantiferrodistortive (AFD) $c(2 \\times 2)$\nreconstruction~\\cite{bun05,sep05}. The latter is not excluded but,\nas discussed later, was not evidenced on our films at room\ntemperature. The distortions in the upper half of each supercell\nare reported in Fig.~\\ref{despontfig2}(c), (d) (``para-relaxed\"\nstate). For easy comparison with the experiment, because of the\ntypical LDA underestimate of the lattice constant~\\cite{acell},\nthe values are given as a percentage of $c_0$. The magnitudes of\nthe ferroelectric and surface relaxation effects can now be\ncompared. First, the cation-oxygen displacements due to\nferroelectricity (11.6\\% - 8.3\\% of $c_0$,\nFig.~\\ref{despontfig2}(b)) are significantly larger than the\ndisplacements due to surface relaxation\/rumpling (3.4\\% - 1.4\\% of\n$c_0$, Fig.~\\ref{despontfig2} (c) and 3.3\\% - 1.5\\% of $c_0$,\nFig.~\\ref{despontfig2}(d)). Second, the mean layer displacement for\nthe ``up\"-state (Fig.~\\ref{despontfig2}(b)) and for the surface\nrelaxation (Figs.~\\ref{despontfig2}, (c) and (d)) are opposite. Third,\nthe surface relaxation and rumpling effects are globally\nunaffected by the film thickness (Figs.~\\ref{despontfig2}, (c) and\n(d)) and their amplitude decays very quickly in the interior of the\nfilm : they are already negligible two unit cells below the\nsurface. This implies that the XPD measurements will be probing\nboth the narrow relaxed surface region and a few unit cells below,\nessentially not affected by the surface relaxation.\n\n\\section{Experimental results and discussion}\n\\subsection{Experimental details }\n\nThe samples used in this study are epitaxial, c-axis oriented\nPbTiO$_3$ thin films grown on conducting (001) Nb-SrTiO$_3$\nsubstrates using off-axis radio-frequency magnetron\nsputtering~\\cite{eom90,lic04}. Topographic measurements using\natomic force microscopy (AFM) showed that these films are\nessentially atomically smooth, with a root-mean-square roughness\nbetween 2 and 6 \\AA\\ over a 10 $\\times$10 $\\mu$m$^2$ area. Room\ntemperature x-ray diffraction measurements, for films with\nthickness $\\geq $ 28 \\AA\\, allowed us to precisely determine the\nthickness and the c-axis parameter of the films, and to confirm\ntheir epitaxial ``cube-on-cube\" growth.\n\nAfter growth and characterization, the films were transferred\n\\emph{ex-situ} to a modified Vacuum Generators ESCALAB Mk II\nphotoelectron spectrometer. The XPD measurement system comprises a\nhemispherical electron energy analyzer with a three-channel\ndetector, an x-ray photon source with two possible energies\n($h\\nu$ = 1253.6 eV and 1740 eV for MgK$\\alpha$ and SiK$\\alpha$\nradiation, respectively), and a computer-controlled two-axis\ngoniometer capable of rotating the photoelectron emission angle\nover the full hemisphere above the surface~\\cite{ost91,nau93_1}.\n\nThe local geometry around a selected atom can be probed by performing an intensity versus emission-angle\n scan of a chosen photoemission line.\nBecause of the chemical sensitivity of photoemission, a given atom\ntype is then chosen by selecting one of its core levels. The\noutgoing photoemitted electrons exhibit a strongly anisotropic\nangular intensity distribution. This angular distribution is due\nto the interference of the directly emitted photoelectron wave\nwith the scattered electron waves. The analysis of the\ninterference (or diffraction) patterns is facilitated by the\nso-called ``forward focusing\" effect taking place for photoelectron\nkinetic energies greater than $\\approx$ 0.5 keV. When considering\na row of atoms, scattering at the first few atoms along this row\nfocuses the electron flux in the emitter-scatterer direction (for\na review see Ref. \\cite{ege90,fad90}). This enhancement of the\nintensity in the emitter-scatterer direction is schematically\nillustrated by the green curve in Fig.~\\ref{despontfig1}(a) (right\npart) for the centro-symmetric ``para\"-state (continuous line) and\nthe ``up\"-state (dotted line). The forward focusing effect is\nfurther amplified for electron scattering by heavy atoms. In a\nsemi-classical picture this can be understood as the focusing of\nthe electron wave by the high number of protons in high atomic\nnumber atoms~\\cite{ege90}. Note that, despite the forward focusing\neffect, the experimentally measured angles are sensitive to\nmultiple interferences, refraction and possible anisotropic atom\nvibrations at the surface. In the present case of PbTiO$_3$, Pb\nscattering is highly dominant compared to the scattering by other\nelements~\\cite{des05}. As a first step, in order to probe the\nnon-centro-symmetry, O was chosen as emitter-atom (O 1s core\nlevel, E$_{kin}$ = 724.1 eV), since it has the largest\ndisplacement~\\cite{nel85} and has Pb scatterers as nearest\nneighbors (see Fig.~\\ref{despontfig3}). However, the O\ncontribution from the Nb-SrTiO$_{3}$ substrate becomes\nnon-negligible for films thinner than the photoelectron inelastic\nmean free path, making the study of films thinner than 20 \\AA\\\nmore difficult. As a second step, in order to probe the\ntetragonality of the films, i. e., the $c\/a$ ratio of the Pb\nlattice (related to the polarization via the polarization-strain\ncoupling as discussed below and in details in Ref.~\\cite{lic05}),\nPb was chosen as emitter-atom (Pb 4f$_{7\/2}$ core level, E$_{kin}$\n= 1115.5 eV), and Pb-Pb forward focusing directions were used.\nSince Pb atoms are absent from the substrate, this study can be\ndone down to a\n monolayer of ferroelectric material.\n\n\\subsection{Non-centro-symmetric position of oxygen atoms}\n\nFirst, considering oxygen as the emitter atom, fully automated\ncomputer code for calculating electron diffraction in atomic\nclusters (EDAC) via multiple scattering~\\cite{gar01}, based on the\nmuffin-tin potential approximation~\\cite{pen74} was used to\ncalculate the XPD pattern. Fig.~\\ref{despontfig3}(a) shows four O 1s\ncore level emission (E$_{kin}$ = 724.1 eV) interference patterns.\nOne is the measurement made on a 20 \\AA\\ thin film while the three\nothers are multiple scattering EDAC calculations of the ``up\"-,\n``down\"- and ``para\"-state. Intensities are plotted in a\nstereographic projection with the center corresponding to normal\nemission (polar angle $\\theta = 0^\\circ$) and the outer\n border corresponding to grazing emission ($\\theta = 70^\\circ$).\nThe strongest intensities (surrounded by red ellipses) correspond\nto the scattering of O 1s photoelectrons by Pb nearest neighbours\n(see Fig.~\\ref{despontfig3}(b)). The white circle is a guide to the\neye indicating the polar angle of maximum intensity for the\nmeasured interference pattern. It is evident that the polar angle\nposition of this peak, which is directly linked to\n the O-Pb directions, is perfectly reproduced by the ``up\"-state calculation while the ``para\"- and\nthe ``down\"-state simulations predict a different position.\nThe ``up\"-state (down-shifted O position) corresponds to a\nsmaller polar emission angle ($\\theta_{up}$ in\nFig.~\\ref{despontfig3}(b)) appearing closer to normal emission\n(center of the interference pattern in Fig.~\\ref{despontfig3}(a)).\nSuch measurements have also been performed on films with\nthicknesses of $\\sim$ 500, 200, 100, 60, 44 and 28 \\AA\\\/, and all\nperfectly reflect the characteristics of the\n``up\"-state \\footnote{The presence of alternating 180$^{\\circ}$\ndomains in our films can be ruled out from the XPD measurement\nbecause the two characteristic ``up\" and ``down\"-state ``forward\nfocusing\" peaks are not observed simultaneously in the\nexperimental diffractogram, Fig.~\\ref{despontfig3}(a). This\ncontrasts with the results of Fong et al. \\cite{fon04} on a\ninsulating substrates.}.\n\nThese conclusions, drawn from visual inspection of the\ninterference patterns locally around the intensity maximum\n(Fig.~\\ref{despontfig3}(a)), are confirmed by a global matching\napproach using a reliability (R)-factor to evaluate the quality of\nthe fit between\n the complete experimental interference pattern data and theory (Fig.~\\ref{despontfig3}(b)). The c-axis lattice\nconstant value and the O and Ti shifts are the adjustable\nstructural parameters. A cut in the (100) plane, containing Pb and\nO atoms, is shown to facilitate the discussion. In the\ncalculation, O and Ti atoms\n are moved together and the dipole is continuously changed from the ``down\"-state to the ``up\"-state,\ncrossing over the ``para\"-state. The best fit corresponds to the\nminimal R-factor value, which is reached when O and Ti atoms are\nshifted below the centro-symmetric position (parameters used for\nthe ``up\"-state simulation in Fig.~\\ref{despontfig3}(a)), with an\nR-factor value of $\\approx$ 0.34. In comparison, for the same\nc-axis parameter but the opposite O and Ti atom shifts (parameters\nused for the ``down\"-state simulation in Fig.~\\ref{despontfig3}(a)),\nthe calculation gives a much higher R-factor of $\\approx  0.47$.\nIn between (zero O shift), in the centro-symmetric ``para\"-state,\nthe R-factor is $\\approx 0.45$ (parameters used for the\n``para\"-state simulation in Fig.~\\ref{despontfig3}(a)).\n\nIt is important to note that surface relaxation and rumpling,\nneglected here, cannot weaken our conclusions; indeed they would\ngive a picture resembling a ``down\"-state, the corresponding O-Pb\natoms being shifted in the opposite direction than what is\nobserved (see Fig.~\\ref{despontfig2}(c)). Also, the possibility of a\nsurface AFD reconstruction was explored without finding evidence\nfor it in our room temperature experiments.\n\n\nThis R-factor analysis therefore quantitatively confirms the\nobservations made in Fig.~\\ref{despontfig3}(a), namely that the\nmeasured interference pattern is best simulated with the ``up\"-\nstate. This demonstrates unambiguously that, for a film as thin as\n20 \\AA\\,  the O atoms have a non-centro-symmetric position in the\nPb cage corresponding to a non-vanishing spontaneous polarization.\n\nLet us emphasize that piezoelectric AFM measurements performed\nafter the XPD experiments on the thickest films ($\\sim$ 500 \\AA )\nconfirmed the uniform``up\"-state polarization while a uniform\n``down\"-state polarization had been initially found for the same\nfilms just after growth.  This confirms the {\\it monodomain}\ncharacter of the as grown sample and also indicates that the films\nare uniformly switched from ``down\" to ``up\"-state by exposure to\nour conventional x-ray source, attesting for the switchable\ncharacter of the polarization. The details behind the switching\nare presently not known, but we believe that it occurs at the\ninitial stage of the experiment while the measurement itself is\nessentially done in zero field. In fact, our results do not depend\non the x-ray intensity, proving that the films are in equilibrium\nstate during the measurements \\footnote{The influence of the x-ray\nintensity on the tetragonality was measured by investigating \ndifferent x-ray powers. A modification of the\ntetragonality would have indicated, via the polarization-strain\ncoupling, a variation of the spontaneous polarization. However no\nsuch modification was found.}. As discussed below, the\nagreement between the tetragonality deduced from x-ray diffraction\nand XPD also suggests that the measurements are performed in\nsimilar conditions.\n\n\\subsection{Tetragonality via lead emission}\n\nIn a second step, considering Pb as the emitter atom, XPD was used\nto determine the tetragonality. As demonstrated in\nRef.~\\cite{lic05}, below 200 \\AA\\ the tetragonality decreases as\nthe film thickness decreases. This decrease is a consequence of\nthe strong polarization-strain coupling in PbTiO$_3$ and a\nsignature of a reduced polarization in thin films. In\nRef.~\\cite{lic05}, this polarization reduction was attributed to\nimperfect screening of the depolarizing field~\\cite{jun03_1}. With\nXPD, using Pb as emitter, the tetragonality was measured down to\nthe unit cell level as shown in Fig.~\\ref{despontfig4}. The\nabsolute values of  $c\/a$, deduced from the forward focusing\nangles, are particulary large. This might reflect\n a strong enhancement of the polarization in the\nprobed surface region (of the order of $80\\% $ for $c\/a = 1.15$,\nfrom the PTO polarization-strain coupling), even larger than in\nthe theoretical prediction of Ref.~\\cite{gho00}. However, as\npreviously stated, we are not necessarily measuring the precise\natom-atom directions and the anomalously large forward focusing $c\/a$ might\nalso be related to other effects (anisotropic atom\nvibrations at the surface, refraction and multiple scattering\ninterferences). Therefore a comparison\n with x-ray diffraction (XRD)~\\cite{lic05} must be done at the relative level (Fig.~\\ref{despontfig4},\nleft and right scale).\n\nTo study the evolution of the tetragonality as a function\nof the film thickness, the measured XPD values are compared to the $c\/a$ values obtained by XRD.\n The XPD measurement on Fig.~\\ref{despontfig4}\nconfirms the evolution of c\/a obtained from x-ray measurements in\nRef.~\\cite{lic05} and agrees with the theoretical prediction\n(dashed curve) relying on the suppression of polarization due to\nimperfect screening of the depolarizing field~\\cite{lic05}. The\nsimilar thickness dependence for the XPD (very surface sensitive)\nand the x-ray measurements (average on the whole film) implies\nthat the polarization evolves at the surface in the same way as at\nthe interior of the film and that there is no thick paraelectric\ndead layer at the surface.  In addition, the XPD tetragonality\nmeasurement shows a continuous decrease of tetragonality down to\nthe thickness of one unit cell~\\cite{unitcell}. Two ribbons are\ndrawn  in Fig.~\\ref{despontfig4}, labeled with 1 and 2. They\nindicate the regions within which $c\/a$ values of 1.03 and 1.01\nare crossed with respect to both c\/a scales. For film thicknesses\nabove two unit cells, the $c\/a$ values are larger than 1.03, the\nvalue expected at the bulk level for the paraelectric phase\n(resulting from the mechanical constraint imposed by the\nsubstrate, see also Fig.~\\ref{despontfig2}(a)). This observation\ndirectly implies, via the polarization-strain coupling, that the\nfilms still have a finite -although progressively reduced-\nspontaneous polarization. At thicknesses of one or two unit\ncells, as can be seen on Fig.~\\ref{despontfig4}, c\/a drops even\nmore, reaching a value close to 1.01 for the one unit cell thick\nfilm~\\cite{unitcell}. This further decrease highlights that\nmacroscopic elasticity no longer applies at such thicknesses where\nthe interlayer atomic distances are affected by surface relaxation\nand rumpling as shown by the \\emph{ab-initio} calculations\n(Fig.~\\ref{despontfig2}(d)). The measured tetragonality agrees with\nthe computed value of 1.01 for the one unit-cell thick relaxed\nparaelectric film suggesting the absence of any additional\nferroelectric distortion at this thickness.\n\n\\section{Conclusion}\n\nThis study thus directly demonstrates non-centro-symmetry,\nunambiguously a result of ferroelectricity in PbTiO$_3$ thin films\ndown to 20 \\AA . The measurements of the tetragonality, with a\ncontinuous decrease down to the bare substrate, show that even\nextremely thin films (3 unit cells) have a $c\/a$ value larger than\n1.03, attesting for the presence of a non-vanishing spontaneous\npolarization at this thickness scale. As the film thickness is\nreduced to a single unit cell, the experiments, together with\ncalculations, strongly suggest that both non-centro-symmetry and\ntetragonality are governed by surface effects, giving rise for our\ngeometry to a polar relaxed structure but without switchable\nferroelectric distortion.\n\n\n\\section*{Acknowledgements}\nWe would like to thank M. A. Van Hove and C. Battaglia for helpful discussions, P. Paruch for careful reading of the manuscript, and the whole Neuch\\^atel workshop and electric engineering team for efficient technical support. This project has been supported by the Swiss National Science Foundation through the National Center of Competence in\nResearch ``Materials with Novel Electronic Properties-MaNEP\", the European Network of Excellence FAME and the VolkswagenStiftung.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Acceleration. Proofs of \\cref{thm:acceleration_quasiquasarconvexity} and \\cref{thm:riemannian_acceleration}} \\label{app:acceleration}\n\\cite{diakonikolas2017approximate} developed the \\textit{approximate duality gap technique} which is a technique that provides a structure to design and prove first order methods and their guarantees for the optimization of convex problems. We take inspiration from this ideas to apply them to the non-convex problem we have at hand \\cref{thm:acceleration_quasiquasarconvexity}, as it was sketched in \\cref{subsec:sketch_of_my_axgd_proof}. We start with two basic definitions.\n\\begin{definition}\\label{def:bregman_divergence}\n    Given two points $\\tilde{x}, \\tilde{y}$, we define the Bregman divergence with respect to $\\psi(\\cdot)$ as\n    \\[\n        D_{\\psi} (\\tilde{x}, \\tilde{y}) \\defi \\psi(\\tilde{x})-\\psi(\\tilde{y}) - \\innp{\\nabla \\psi(\\tilde{y}), \\tilde{x}-\\tilde{y}}.\n    \\]\n\\end{definition}\n\n\\begin{definition} \\label{def:fenchel_dual}\n    Given a closed convex set $Q$ and a function $\\psi:Q \\to \\R$, we define the convex conjugate of $\\psi$, also known as its Fenchel dual, as the function\n    \\[\n        \\psi^\\ast(\\tilde{z}) = \\max_{\\tilde{x}\\in Q}\\{ \\innp{\\tilde{z}, \\tilde{x}}-\\psi(\\tilde{x})\\}.\n    \\]\n\\end{definition}\nFor simplicity, we will use $\\psi(\\tilde{x}) = \\frac{1}{2}\\norm{\\tilde{x}}^2$ in \\cref{alg:accelerated_gconvex}, but any strongly convex map works. The gradient of the Fenchel dual of $\\psi(\\cdot)$ is $\\nabla\\psi^\\ast(\\tilde{z}) = \\argmin_{\\tilde{z'}\\in\\X}\\{\\norm{\\tilde{z}'-\\tilde{z}}\\}$, that is, the Euclidean projection $\\Pi_Q(\\tilde{z})$ of the point $\\tilde{z}$ onto $Q$. Note that when we apply \\cref{thm:acceleration_quasiquasarconvexity} to \\cref{thm:riemannian_acceleration} our constraint $\\X$ will be a ball centered at $0$ of radius $\\tilde{R}$, so the projection of a point $\\tilde{z}$ outside of $\\X$ will be the vector normalization $\\tilde{R}\\tilde{z}\/\\norm{\\tilde{z}}$. Any continuously differentiable strongly convex $\\psi$ would work, provided that $\\psi^\\ast(z)$ is easily computable, preferably in closed form. Note that by the Fenchel-Moreau theorem we have for any such map that $\\psi^{\\ast\\ast} =\\psi$.\n\nWe recall we assume that $f$ satisfies\n\\begin{align} \\label{eq:appendix_quasiquasarconvexity}\n \\begin{aligned}\n     f(\\tilde{x}) +  \\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}\\leq f(\\tilde{y})  &\\ & & {\\text{ if } \\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}}\\leq 0, \\\\\n     f(\\tilde{x})+\\gammap\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}} \\leq f(\\tilde{y})  &\\ & & {\\text{ if } \\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}} \\geq 0.\n   \\end{aligned}\n\\end{align}\n\nLet $\\alpha_t$ be an increasing function of time $t$. We want to work with continuous and discrete approaches in a unified way so we use Lebesgue-Stieltjes integration. Thus, when $\\alpha_t$ is a discrete measure, we have that $\\alpha_t = \\sum_{i=1}^\\infty a_i \\delta(t-(t_0+i-1))$ is a weighted sum of Dirac delta functions. We define $A_t = \\int_{t_0}^t d\\alpha_\\tau = \\int_{t_0}^t \\dott{\\alpha}_\\tau d\\tau$. In discrete time, it is $A_t = \\sum_{i=1}^{t-t_0+1} a_{i}$.  In the continuous case note that we have $\\alpha_t -A_t = a_{t_0}$.\n\n\nWe start defining a continuous method that we discretize with an approximate implementation of the implicit Euler method. Let $\\tilde{x}_t$ be the solution obtained by the algorithm at time $t$. We define the duality gap $G_t\\defi U_t-L_t$ as the difference between a differentiable upper bound $U_t$ on the function at the current point and a lower bound on $f(x^\\ast)$. Since in our case $f$ is differentiable we use $U_t \\defi f(\\tilde{x}_t)$. The idea is to enforce the invariant $\\frac{d}{dt}(\\alpha_t G_t) = 0$, so we have at any time $f(\\tilde{x}_t)-f(\\tilde{x}^\\ast)\\leq G_t = G_{t_0}\\alpha_{t_0}\/\\alpha_t$. \n\nNote that for a global minimum $\\tilde{x}^\\ast$ of $f$ and any other point $\\tilde{x}\\in Q$, we have $\\innp{\\nabla f(\\tilde{x}), \\tilde{x}^\\ast-\\tilde{x} }\\leq 0$. Otherwise, we would obtain a contradiction since by \\eqref{eq:appendix_quasiquasarconvexity} we would have \n\\[\nf(\\tilde{x}) <f(\\tilde{x})+ \\gammap\\innp{\\nabla f(\\tilde{x}), \\tilde{x}^\\ast-\\tilde{x}} \\leq f(\\tilde{x}^\\ast).\n\\] \nTherefore, in order to define an appropriate lower bound, we will make use of the inequality $f(\\tilde{x}^\\ast) \\geq f(\\tilde{x}) + \\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}), \\tilde{x}^\\ast -\\tilde{x}}$, for any $\\tilde{x}\\in  Q$, which holds true by \\eqref{eq:appendix_quasiquasarconvexity}, for $\\tilde{y}=\\tilde{x}^\\ast$. Combining this inequality for all the points visited by the continuous method we have\n\\[\n    f(\\tilde{x}^\\ast) \\geq \\frac{\\int_{t_{0}}^{t} f(\\tilde{x}_{\\tau}) d \\alpha_{\\tau}}{A_{t}}+\\frac{\\int_{t_{0}}^{t}\\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}_{\\tau}), \\tilde{x}^\\ast-\\tilde{x}_{\\tau}} d \\alpha_{\\tau}}{A_{t}}.\n\\]\nWe cannot compute this lower bound, since the right hand side depends on the unknown point $\\tilde{x}^\\ast$. We could compute a looser lower bound by taking the minimum over $\\tilde{u}\\in Q$ of this expression, substituting $\\tilde{x}^\\ast$ by $\\tilde{u}$. However, this would make the lower bound non-differentiable and we could have problems at $t_0$. In order to solve the first problem, we first add a regularizer and then take the minimum over $\\tilde{u}\\in  Q$.\n\\begin{align*} \n\\begin{aligned}\nf(\\tilde{x}^{\\ast})+& \\frac{D_{\\psi}(\\tilde{x}^{\\ast}, \\tilde{x}_{t_{0}})}{A_{t}} \\\\\n    & \\geq \\frac{\\int_{t_{0}}^{t} f(\\tilde{x}_{\\tau}) d \\alpha_{\\tau}}{A_{t}}+\\frac{\\min_{\\tilde{u} \\in  Q}\\left\\{\\int_{t_{0}}^{t}\\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}_{\\tau}), \\tilde{u}-\\tilde{x}_{\\tau} } d \\alpha_{\\tau}+D_{\\psi}(\\tilde{u}, \\tilde{x}_{t_{0}})\\right\\}}{A_{t}}\n\\end{aligned}\n\\end{align*}\nIn order to solve the second problem, we mix this lower bound with the optimal lower bound $f(\\tilde{x}^\\ast)$ with weight $\\alpha_t -A_t$ (this is only necessary in continuous time, in discrete time this term is $0$). Not knowing $f(\\tilde{x}^\\ast)$ or $D_{\\psi}(\\tilde{x}^\\ast, \\tilde{x}_{t_0})$ will not be problematic. Indeed, we only need to guarantee $\\frac{d}{dt}(\\alpha_t G_t)=0$, so after taking the derivative these terms will vanish. After rescaling the normalization factor, we finally obtain the lower bound\n\\begin{align} \\label{lower_bound_modified_axgd}\n\\begin{aligned}\n    f(\\tilde{x}^\\ast) \\geq L_t \\defi &\\frac{\\int_{t_0}^t f(\\tilde{x}_\\tau) d\\alpha_\\tau}{\\alpha_t} + \\frac{\\min_{\\tilde{u}\\in Q}\\left\\{\\int_{t_0}^t \\innp{\\frac{1}{\\gamman}\\nabla f(\\tilde{x}_\\tau), \\tilde{u}-\\tilde{x}_\\tau} d\\alpha_\\tau + D_{\\psi}(\\tilde{u},\\tilde{x}_{t_0})\\right\\}}{\\alpha_t} \\\\\n    & \\quad + \\frac{(\\alpha_t-A_t)f(\\tilde{x}^\\ast)-D_{\\psi}(\\tilde{x}^\\ast, \\tilde{x}_{t_0})}{\\alpha_t}.\n\\end{aligned}\n\\end{align}\n\nLet $\\tilde{z}_t = \\nabla \\psi(\\tilde{x}_{t_0}) - \\int_{t_0}^t \\frac{1}{\\gamman} \\nabla f(\\tilde{x}_\\tau)d\\alpha_\\tau$. Then, by \\cref{grad_of_fenchel_dual}, we can compute the optimum $\\tilde{u}$ above as\n\\begin{equation}\\label{eq:fenchel_dual_applied_to_z}\n        \\nabla \\psi^\\ast(\\tilde{z}_t) = \\argmin_{\\tilde{u}\\in Q} \\left\\{\\int_{t_0}^t\\innp{\\frac{1}{\\gamman}\\nabla f(\\tilde{x}_\\tau), \\tilde{u}-\\tilde{x}_\\tau} d\\alpha_\\tau + D_\\psi(\\tilde{u},\\tilde{x}_{t_0})\\right\\}.\n\\end{equation}\nRecalling $U_t = f(\\tilde{x}_t)$ and using \\eqref{lower_bound_modified_axgd} and Danskin's theorem in order to differentiate inside the min we obtain:\n\\begin{align*} \n\\begin{aligned}\n    \\frac{d}{dt}(\\alpha_t G_t) &= \\frac{d}{dt} (\\alpha_t f(\\tilde{x}_t)) - \\dott{\\alpha_t}f(\\tilde{x}_t) - \\dott{\\alpha}_t \\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}_t), \\nabla \\psi^\\ast(\\tilde{z}_t)-\\tilde{x}_t} \\\\\n    &=\\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}_t), \\gamman \\alpha_t \\dott{\\tilde{x}} - \\dott{\\alpha}_t(\\nabla \\psi^\\ast(\\tilde{z}_t)-\\tilde{x}_t)}.\n\\end{aligned}\n\\end{align*}\nThus, to satisfy the invariant $\\frac{d}{dt}(\\alpha_t G_t)=0$, it is enough to set $\\gamman \\alpha_t \\dott{\\tilde{x}}_t = \\dott{\\alpha_t}(\\nabla \\psi^\\ast(\\tilde{z}_t)-\\tilde{x}_t)$, yielding the following continuous accelerated dynamics\n\\begin{align}\\label{appendix_modified_accelerated_continuous_dynamics}\n\\begin{aligned}\n    \\dott{\\tilde{z}}_{t}&=-\\frac{1}{\\gamman}\\dott{\\alpha}_{t} \\nabla f(\\tilde{x}_{t}), \\\\\n    \\dott{\\tilde{x}}_{t}&=\\frac{1}{\\gamman}\\dott{\\alpha}_{t} \\frac{\\nabla \\psi^{\\ast}({\\tilde{z}}_{t})-\\tilde{x}_{t}}{\\alpha_{t}},\\\\\n    \\tilde{z}_{(t_{0})}&=\\nabla \\psi(\\tilde{x}_{t_{0}}),\\\\\n    \\tilde{x}_{t_{0}} &\\in  Q \\text { is an arbitrary initial point. }\n\\end{aligned}\n\\end{align}\n\nNow we proceed to discretize the dynamics. Let $ E_{i+1} \\defi A_{i+1}G_{i+1}-A_iG_i$ be the discretization error. Then we have\n\\[\n    G_k = \\frac{A_1}{A_k}G_1 + \\frac{\\sum_{i=1}^{k-1} E_{i+1}}{A_k}.\n\\]\n\\begin{lemma}\\label{lemma:discretization_error_of_modified_axgd}\n    If we have \n    \\begin{equation}\\label{eq:approximate_multiplied_convexity}\n    f(\\tilde{x}_{i+1})-f(\\tilde{x}_i) \\leq \\hat{\\gamma_i} \\innp{\\nabla f(\\tilde{x}_{i+1}), \\tilde{x}_{i+1}-\\tilde{x}_i} + \\hat{\\epsilon}_i,\n    \\end{equation}\n    for some $\\hat{\\gamma}_i, \\hat{\\epsilon}_i \\geq 0$, then the discretization error satisfies\n    \\[\n        E_{i+1} \\leq \\innp{\\nabla f(\\tilde{x}_{i+1}), (A_{i}\\hat{\\gamma}_i +\\frac{a_{i+1}}{\\gamman})\\tilde{x}_{i+1}-\\hat{\\gamma}_i A_i \\tilde{x}_i-\\frac{a_{i+1}}{\\gamman}\\nabla \\psi^\\ast(\\tilde{z}_{i+1}))} - D_{\\psi^\\ast}(\\tilde{z}_i, \\tilde{z}_{i+1}) + A_i\\hat{\\epsilon}_i.\n    \\]\n\\end{lemma}\n\n\\begin{proof}\n    In a similar way to \\cite{diakonikolas2017accelerated}, we could compute the discretization error as the difference between the gap and the gap computed allowing continuous integration rules in the integrals that it contains. However, we will directly bound $E_{i+1}$ as $A_{i+1}G_{i+1}-A_iG_i$ instead. Using the definition of $G_i, U_i, L_i$ we have\n\\begin{align*}\n\\begin{aligned}\n    A_{i+1}&G_{i+1}-A_iG_i \\\\\n    &\\leq (A_{i+1}f(\\tilde{x}_{i+1})-A_i f(\\tilde{x}_i)) -A_{i+1}L_{i+1}+A_iL_i \\\\\n    & \\circled{1}[\\leq] (A_if(\\tilde{x}_{i+1})- A_if(\\tilde{x}_i) + a_{i+1}f(\\tilde{x}_{i+1}))  \\\\\n    &\\quad -\\sum_{j=1}^{i+1} a_j f(\\tilde{x}_j) - \\sum_{j=1}^{i+1} \\frac{a_j}{\\gamman} \\innp{\\nabla f(\\tilde{x}_{j}), \\nabla \\psi^\\ast(\\tilde{z}_{i+1})-\\tilde{x}_j} - D_\\psi(\\nabla \\psi^\\ast(\\tilde{z}_{i+1}), \\tilde{x}_0) \\\\\n    &\\quad +\\sum_{j=1}^{i} a_j f(\\tilde{x}_j) + \\sum_{j=1}^{i} \\frac{a_j}{\\gamman} \\innp{\\nabla f(\\tilde{x}_{j}), \\nabla \\psi^\\ast(\\tilde{z}_{i})-\\tilde{x}_j} + D_\\psi(\\nabla \\psi^\\ast(\\tilde{z}_{i}), \\tilde{x}_0)  \\\\\n    & \\circled{2}[\\leq] A_i(f(\\tilde{x}_{i+1})-f(\\tilde{x}_i)) -\\innp{\\frac{a_{i+1}}{\\gamman}\\nabla f(\\tilde{x}_{i+1}), \\nabla \\psi^\\ast(\\tilde{z}_{i+1})-\\tilde{x}_{i+1}} \\\\\n    &\\quad+ \\sum_{j=1}^i\\innp{\\frac{a_j}{\\gamman}\\nabla f(\\tilde{x}_j), \\nabla \\psi^\\ast(\\tilde{z}_i)-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})}  \\\\\n     &\\quad [- \\innp{\\nabla \\psi(\\tilde{x}_0), \\nabla \\psi^\\ast(\\tilde{z}_i)-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})} + \\psi(\\nabla\\psi^\\ast(\\tilde{z}_{i})) -\\psi(\\nabla \\psi^\\ast(\\tilde{z}_{i+1}))] \\\\\n     & \\circled{3}[\\leq]  A_i(f(\\tilde{x}_{i+1})-f(\\tilde{x}_i)) -\\innp{\\frac{a_{i+1}}{\\gamman}\\nabla f(\\tilde{x}_{i+1}), \\nabla \\psi^\\ast(\\tilde{z}_{i+1})-\\tilde{x}_{i+1}} -D_{\\psi^\\ast}(\\tilde{z}_i, \\tilde{z}_{i+1}) \\\\\n     & \\circled{4}[\\leq] \\innp{\\nabla f(\\tilde{x}_{i+1}), (A_{i}\\hat{\\gamma}_i +\\frac{a_{i+1}}{\\gamman})\\tilde{x}_{i+1}-\\hat{\\gamma}_i A_i \\tilde{x}_i-\\frac{a_{i+1}}{\\gamman}\\nabla \\psi^\\ast(\\tilde{z}_{i+1}))} - D_{\\psi^\\ast}(\\tilde{z}_i, \\tilde{z}_{i+1}) + A_i\\hat{\\epsilon}_i.\n\\end{aligned}\n\\end{align*}\n    In $\\circled{1}$ we write down the definitions of $L_{i+1}$ and $L_{i}$ and split the first summand so it is clear that in $\\circled{2}$ we cancel all the $a_jf(\\tilde{x}_{j})$. In $\\circled{2}$ we also cancel some terms involved in the inner products, we write the definitions of the Bregman divergences and cancel some terms. We recall $\\tilde{z}_i = \\nabla \\psi(x_0) - \\sum_{j=1}^i \\frac{a_j}{\\gamman}\\nabla f(x_j)$ so we use this fact for the second line of $\\circled{2}$ and the first summand of the third line to obtain, along with the last two summands, the term $D_{\\psi}(\\nabla \\psi^\\ast(\\tilde{z}_{i+1}),\\nabla \\psi^\\ast(\\tilde{z}_{i}))$. We use \\cref{prop:duality_of_bregman_divergences} to finally obtain $\\circled{3}$. Inequality $\\circled{4}$ uses \\eqref{eq:approximate_multiplied_convexity}.\n\\end{proof}\n\nWe show now how to cancel out the discretization error by an approximate implementation of implicit Euler discretization of \\eqref{appendix_modified_accelerated_continuous_dynamics}. Note that we need to take into account the assumptions \\eqref{eq:appendix_quasiquasarconvexity} instead of the usual convexity assumption. According to the previous lemma, we can set $\\tilde{x}_{i+1}$ so that the right hand side of the inner product in the bound of $E_{i+1}$ is $0$. Assume for the moment, that the point $\\tilde{x}_{i+1}$ we are going to compute satisfies the assumption of the previous lemma for some $\\hat{\\gamma}_i \\in [\\gammap, 1\/\\gamman]$.  Thus, the implicit equation that defines the ideal method we would like to have is\n\\begin{align*}\n\\begin{aligned}\n    \\tilde{x}_{i+1} = \\frac{\\hat{\\gamma}_iA_i}{A_i \\hat{\\gamma}_i+a_{i+1}\/\\gamman}\\tilde{x}_i + \\frac{a_{i+1}\/\\gamman}{A_i\\hat{\\gamma}_i + a_{i+1}\/\\gamman} \\nabla \\psi^\\ast(\\tilde{z}_i-\\frac{a_{i+1}}{\\gamman} \\nabla f(\\tilde{x}_{i+1})).\n\\end{aligned}\n\\end{align*}\nNote that $\\tilde{x}_{i+1}$ is a convex combination of the other two points so it stays in $Q$. Indeed, $x_0 \\in Q$ and by \\eqref{eq:fenchel_dual_applied_to_z} we have that $\\nabla \\psi^\\ast(\\tilde{z}_j) \\in Q$ for all $j \\geq 0$. However this method is implicit and possibly computationally expensive to implement. Nonetheless, two steps of a fixed point iteration procedure of this equation will be enough to have discretization error that is bounded by the term $A_i\\hat{\\epsilon}_i$: the last term of our bound. The error in the bound of $E_{i+1}$ that the inner product incurs is compensated by the Bregman divergence term. In such a case, the equations of this method become\n\\begin{equation}\\label{appendix_general_rule_modified_quasar_axgd}\n\\left\\{\\begin{array}{l}\n\\tilde{\\chi}_{i}=\\frac{\\hat{\\gamma}_iA_i}{A_i \\hat{\\gamma}_i+a_{i+1}\/\\gamman} \\tilde{x}_{i}+\\frac{a_{i+1}\/\\gamman}{A_i\\hat{\\gamma}_i + a_{i+1}\/\\gamman} \\nabla \\psi^{\\ast}(\\tilde{z}_{i}) \\\\\n\\tilde{\\zeta}_i = \\tilde{z}_{i}-\\frac{a_{i+1}}{\\gamman} \\nabla f(\\tilde{\\chi}_{i}) \\\\\n\\tilde{x}_{i+1}=\\frac{\\hat{\\gamma}_iA_i}{A_i \\hat{\\gamma}_i+a_{i+1}\/\\gamman} \\tilde{x}_{i}+\\frac{a_{i+1}\/\\gamman}{A_i\\hat{\\gamma}_i + a_{i+1}\/\\gamman} \\nabla \\psi^{\\ast}(\\tilde{\\zeta}_i) \\\\\n\\tilde{z}_{i+1} = \\tilde{z}_{i}-\\frac{a_{i+1}}{\\gamman} \\nabla f(\\tilde{x}_{i+1}) \n\\end{array}\\right.\n\\end{equation}\nWe prove now that this indeed leads to an accelerated algorithm. After this, we will show that we can perform a binary search at each iteration, to ensure that even if we do not know $\\tilde{x}_{i+1}$ a priori, we can compute a $\\hat{\\gamma}_i \\in [\\gammap, 1\/\\gamman]$ satisfying assumption \\eqref{eq:approximate_multiplied_convexity}. This will only add a log factor to the overall complexity.\n\n\\begin{lemma}\\label{lemma:approximate_implicit_method}\n    Consider the method given in \\eqref{appendix_general_rule_modified_quasar_axgd}, starting from and arbitrary point $\\tilde{x}_0 \\in  Q$ with $\\tilde{z}_0 = \\nabla \\psi(\\tilde{x}_0)$ and $A_0=0$. Assume we can compute $\\hat{\\gamma}_i$ such that $\\tilde{x}_{i+1}$ satisfies \\eqref{eq:approximate_multiplied_convexity}. Then, the error from \\cref{lemma:discretization_error_of_modified_axgd} is bounded by\n\\begin{align*}\n\\begin{aligned}\n    E_{i+1} &\\leq  \\frac{a_{i+1}}{\\gamman}\\innp{\\nabla f(\\tilde{x}_{i+1}) - \\nabla f(\\tilde{\\chi}_i), \\nabla \\psi^\\ast(\\tilde{\\zeta}_i)-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})} - D_{\\psi^\\ast}(\\tilde{\\zeta}_i,\\tilde{z}_{i+1}) - D_{\\psi^\\ast}(\\tilde{z}_i,\\tilde{\\zeta}_{i}) + A_i\\hat{\\epsilon}_i.\\\\\n\\end{aligned}\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\n    Using \\cref{lemma:discretization_error_of_modified_axgd} and the third line of \\eqref{appendix_general_rule_modified_quasar_axgd} we have\n    \\begin{align*}\n    \\begin{aligned}\n        E_{i+1} -A_i\\hat{\\epsilon}_i &\\leq  \\frac{a_{i+1}}{\\gamman}\\innp{\\nabla f(\\tilde{x}_{i+1}), \\nabla \\psi^\\ast(\\tilde{\\zeta}_i)-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})} - D_{\\psi^\\ast}(\\tilde{z}_i,\\tilde{z}_{i+1})\\\\\n        &\\leq  \\frac{a_{i+1}}{\\gamman}\\innp{\\nabla f(\\tilde{x}_{i+1}) - \\nabla f(\\tilde{\\chi}_i)+ \\nabla f(\\tilde{\\chi}_i), \\nabla \\psi^\\ast(\\tilde{\\zeta}_i)-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})}  - D_{\\psi^\\ast}(\\tilde{z}_i,\\tilde{z}_{i+1})\\\\\n    \\end{aligned}\n    \\end{align*}\n    By the definition of $\\tilde{\\zeta}_i$ we have $(a_{i+1}\/\\gamman) \\nabla f(\\tilde{\\chi}_i) = \\tilde{z}_i - \\tilde{\\zeta}_{i}$. Using this fact and the triangle inequality of Bregman divergences \\cref{prop:triangle_inequality_of_bregman_div}, we obtain\n    \\begin{align*}\n    \\begin{aligned}\n        \\frac{a_{i+1}}{\\gamman}\\innp{\\nabla f(\\tilde{\\chi}_i), \\nabla \\psi^\\ast(\\tilde{\\zeta}_i)&-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})} = \\innp{\\tilde{z}_i-\\tilde{\\zeta}_i, \\nabla \\psi^\\ast(\\tilde{\\zeta}_i)-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})}\\\\\n        &= D_{\\psi^\\ast}(\\tilde{z}_i, \\tilde{z}_{i+1}) - D_{\\psi^\\ast}(\\tilde{\\zeta}_i, \\tilde{z}_{i+1}) - D_{\\psi^\\ast}(\\tilde{z}_i, \\tilde{\\zeta}_i).\n    \\end{aligned}\n    \\end{align*}\n    The lemma follows after combining these two equations.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:accelerated_axgd_modified}\n    Let $Q$ be a convex set of diameter $D$. Let $f: Q \\to \\R$ be an $\\tilde{L}$-smooth function satisfying \\eqref{eq:appendix_quasiquasarconvexity}. Assume there is a point $\\tilde{x}^\\ast \\in Q$ such that $\\nabla f(x^\\ast) = 0$. Let $\\tilde{x}_i, \\tilde{z}_i, \\tilde{\\chi}_i, \\tilde{\\zeta}_i$ be updated according to \\eqref{appendix_general_rule_modified_quasar_axgd}, for $i\\geq 0$ starting from an arbitrary initial point $\\tilde{x}_0 \\in  Q$ with $\\tilde{z}_0 = \\nabla \\psi(\\tilde{x}_0)$ and $A_0=0$, assuming we can find $\\hat{\\gamma}_i$ satisfying \\eqref{eq:approximate_multiplied_convexity}. Let $\\psi: \\B \\to \\R$ be $\\sigma$-strongly convex. If $\\tilde{L} a_{i+1}^2\/\\gamman\\sigma \\leq a_{i+1} + A_i\\gamman\\gammap$, then for all $t\\geq 1$ we have \n    \\[\n        f(\\tilde{x}_t)-f(\\tilde{x}^\\ast) \\leq \\frac{D_{\\psi}(\\tilde{x}^\\ast, \\tilde{\\chi}_0)}{A_t} + \\sum_{i=1}^{t-1} \\frac{A_{i}\\hat{\\epsilon}_i}{A_t}.\n    \\]\n    In particular, if $a_i = \\frac{i}{2} \\cdot \\frac{\\sigma}{\\tilde{L}} \\cdot \\gamman^2 \\gammap$, $\\psi(\\tilde{x}) =\\frac{\\sigma}{2}\\norm{\\tilde{x}}^2$, $\\hat{\\epsilon}_i=\\frac{A_t\\epsilon}{2(t-1) A_{i}}$ and $t=\\sqrt{\\frac{2\\tilde{L}\\norm{\\tilde{x}_0-\\tilde{x}^\\ast}^2}{\\gamman^2\\gammap\\epsilon}}= O(\\sqrt{\\tilde{L}\/(\\gamman^2\\gammap\\epsilon)})$  then \n    \\[\n        f(\\tilde{x}_t)-f(\\tilde{x}^\\ast) \\leq \\frac{2\\tilde{L}\\norm{\\tilde{x}_0 -x^\\ast}^2}{\\gamman^2\\gammap t(t+1)} + \\frac{\\epsilon}{2} < \\epsilon.\n    \\]\n\\end{theorem}\n\n\\begin{proof}\n    We bound the right hand side of the discretization error given by \\cref{lemma:approximate_implicit_method}. Define $a = \\norm{\\nabla \\psi^\\ast(\\tilde{\\zeta}_i)-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})}$ and $b=\\norm{\\nabla \\psi^\\ast(\\tilde{\\zeta}_i)-\\nabla \\psi^\\ast(\\tilde{z}_i)}$. We have\n    \\begin{align*}\n    \\begin{aligned}\n        E_{i+1} -A_i\\hat{\\epsilon}_i&\\circled{1}[\\leq]  \\frac{a_{i+1}}{\\gamman}\\innp{\\nabla f(\\tilde{x}_{i+1}) - \\nabla f(\\tilde{\\chi}_i), \\nabla \\psi^\\ast(\\tilde{\\zeta}_i)-\\nabla \\psi^\\ast(\\tilde{z}_{i+1})} - D_{\\psi^\\ast}(\\tilde{\\zeta}_i,\\tilde{z}_{i+1}) - D_{\\psi^\\ast}(\\tilde{z}_i,\\tilde{\\zeta}_{i})\\\\\n        &\\circled{2}[\\leq]  \\frac{a_{i+1}}{\\gamman}\\tilde{L}\\norm{\\tilde{x}_{i+1}-\\tilde{\\chi}_i}\\cdot a - D_{\\psi^\\ast}(\\tilde{\\zeta}_i,\\tilde{z}_{i+1}) - D_{\\psi^\\ast}(\\tilde{z}_i,\\tilde{\\zeta}_{i})\\\\\n        &\\circled{3}[\\leq]  \\frac{a_{i+1}}{\\gamman}\\tilde{L}\\norm{\\tilde{x}_{i+1}-\\tilde{\\chi}_i}\\cdot a - \\frac{\\sigma}{2}(a^2+b^2)\\\\\n        &\\circled{4}[\\leq]  \\frac{a_{i+1}^2\/\\gamman^2}{A_i \\hat{\\gamma}_i + a_{i+1}\/\\gamman}\\tilde{L}\\cdot ab - \\frac{\\sigma}{2}(a^2+b^2)\\\\\n        &\\circled{5}[\\leq]  ab \\left(\\frac{a_{i+1}^2\/\\gamman^2}{A_i \\hat{\\gamma}_i + a_{i+1}\/\\gamman}\\tilde{L} - \\sigma\\right).\\\\\n    \\end{aligned}\n    \\end{align*}\n    Here $\\circled{1}$ follows from \\cref{lemma:approximate_implicit_method}, $\\circled{2}$ uses the Cauchy-Schwartz inequality and smoothness, $\\circled{3}$ uses \\cref{prop:bounding_breg_div_by_norm_of_difference}, and $\\circled{4}$ uses the fact that by the definition of the method \\eqref{appendix_general_rule_modified_quasar_axgd} we have $\\tilde{x}_{i+1}-\\tilde{\\chi}_i = \\frac{a_{i+1}\/\\gamman}{A_i \\hat{\\gamma}_i + a_{i+1}\/\\gamman} (\\nabla \\psi^\\ast(\\tilde{\\zeta}_i)-\\nabla \\psi^\\ast(\\tilde{z}_i))$. Finally $\\circled{5}$ uses $-(a^2+b^2) \\leq -2ab$, which comes from $(a-b)^2 \\geq 0$. By the previous inequality, if we want $E_{i+1} \\leq A_i\\hat{\\epsilon}_i$, it is enough to guarantee the right hand side of the last expression is $\\leq 0$ which is implied by\n    \\[\n        \\frac{\\tilde{L}}{\\sigma \\gamman}a_{i+1}^2 \\leq a_{i+1} + A_i \\gamman \\gammap,\n    \\]\n    since $\\gammap \\leq \\hat{\\gamma}_i$. By inspection, if we use the value in the statement of the theorem $a_i = \\frac{i}{2}\\cdot \\frac{\\sigma}{\\tilde{L}}\\cdot \\gamman^2\\gammap$ into the previous inequality and noting that $A_i =\\frac{i(i+1)}{4} \\cdot \\frac{\\sigma}{\\tilde{L}}\\cdot \\gamman^2\\gammap $ we have\n    \\begin{align*}\n    \\begin{aligned}\n        \\frac{\\tilde{L}}{\\sigma\\gamman} a_{i+1}^2 &= \\frac{(i+1)^2}{4}\\cdot \\frac{\\sigma}{\\tilde{L}}\\cdot \\gamman^3\\gammap^2 \\\\\n        & \\leq \\left(\\frac{i+1}{2} + \\frac{i(i+1)}{4}\\right) \\frac{\\sigma}{\\tilde{L}}\\cdot \\gamman^3\\gammap^2  \\\\\n        & \\leq \\frac{i+1}{2} \\frac{\\sigma}{\\tilde{L}}\\cdot \\gamman^2\\gammap + \\frac{i(i+1)}{4} \\frac{\\sigma}{\\tilde{L}}\\cdot \\gamman^3\\gammap^2 \\\\\n        &= a_{i+1} + A_i \\gamman\\gammap\n    \\end{aligned}\n    \\end{align*}\n    which holds true. So this choice guarantees discretization error $E_{i+1} \\leq A_i\\hat{\\epsilon}_i$. By the definition of $G_i$ and $E_i$ we have \n    \\[\n        f(\\tilde{x}_t)-f(\\tilde{x}^\\ast) \\leq \\frac{A_1G_1}{G_t} + \\sum_{i=1}^{t} \\frac{A_{i-1}\\hat{\\epsilon}_i}{A_t} \n    \\]\n    So it only remains to bound the initial gap $G_1$. In order to do this, we note that the initial conditions and the method imply the following computation of the first points, from $\\tilde{x}_0 \\in  Q$, which is an arbitrary initial point:\n\\begin{equation}\n\\left\\{\\begin{array}{l}\n        \\tilde{z}_0 = \\nabla \\psi(\\tilde{x}_0) \\\\\n        \\tilde{\\chi}_0 = \\frac{\\hat{\\gamma}_0A_0}{A_0 \\hat{\\gamma}_0+a_{1}\/\\gamman} \\tilde{x}_0 + \\frac{a_{1}\/\\gamman}{A_0 \\hat{\\gamma}_0+a_{1}\/\\gamman} \\nabla \\psi^\\ast(\\tilde{z}_0) = \\nabla\\psi^\\ast(\\nabla \\psi(\\tilde{x}_0)) = \\tilde{x}_0 \\\\\n        \\tilde{\\zeta}_0 = \\tilde{z}_0 -\\frac{a_1}{\\gamman} \\nabla f(\\tilde{\\chi}_0) = \\tilde{z}_0 -\\frac{a_1}{\\gamman} \\nabla f(\\tilde{x}_0)\\\\\n        \\tilde{x}_1 = \\frac{\\hat{\\gamma}_0A_0}{A_0 \\hat{\\gamma}_0+a_{1}\/\\gamman} \\tilde{x}_0 + \\frac{a_{1}\/\\gamman}{A_0 \\hat{\\gamma}_0+a_{1}\/\\gamman} \\nabla \\psi^\\ast(\\tilde{\\zeta}_0)  =  \\nabla \\psi^\\ast(\\tilde{\\zeta}_0)\n\\end{array}\\right.\n\\end{equation}\nWe have used $A_0=0$. Note this first iteration does not depend on $\\hat{\\gamma}_0$. Recall also that, using \\eqref{lower_bound_modified_axgd}, the first lower bound computed is\n    \\begin{align*}\n    \\begin{aligned}\n        L_1 = f(\\tilde{x}_1) + \\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}_1), \\nabla \\psi^\\ast(\\tilde{z}_1)-\\tilde{x}_1}+\\frac{1}{A_1} D_{\\psi}(\\nabla \\psi^\\ast(\\tilde{z}_1), \\tilde{\\chi}_0) - \\frac{1}{A_1} D_{\\psi}(\\tilde{x}^\\ast, \\tilde{\\chi}_0).\n    \\end{aligned}\n    \\end{align*}\n    Using $a_1=A_1$, $\\tilde{x}_1 = \\nabla \\psi^\\ast (\\tilde{\\zeta}_0)$, $(a_1\/\\gamman) \\nabla f(\\tilde{\\chi}_0)=\\tilde{z}_0-\\tilde{\\zeta}_0$, and the triangle inequality for Bregman divergences \\cref{prop:triangle_inequality_of_bregman_div} we obtain\n    \\begin{align}\\label{bounding_g_1_equation_1}\n    \\begin{aligned}\n        \\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{\\chi}_0), \\nabla \\psi^\\ast(\\tilde{z}_1)-\\tilde{x}_1} &= \\frac{1}{A_1} \\innp{\\tilde{z}_0-\\tilde{\\zeta}_0, \\nabla \\psi^\\ast(\\tilde{z}_1) - \\nabla \\psi^\\ast (\\tilde{\\zeta}_0)} \\\\\n    &=\\frac{1}{A_1}\\left(D_{\\psi^\\ast}(\\tilde{z}_0,\\tilde{\\zeta}_0)- D_{\\psi^\\ast}(\\tilde{z}_0,\\tilde{z}_1) +D_{\\psi^\\ast}(\\tilde{\\zeta}_0,\\tilde{z}_1)\\right).\n    \\end{aligned}\n    \\end{align}\n    On the other hand, by smoothness of $f$ and the initial condition we have\n    \\begin{equation}\\label{bounding_g_1_equation_2}\n        \\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}_1)-\\nabla f(\\tilde{\\chi}_0), \\nabla \\psi^\\ast(\\tilde{z}_1)-\\tilde{x}_1} \\geq -\\frac{\\tilde{L}}{\\gamman} \\norm{\\nabla \\psi^\\ast(\\tilde{\\zeta}_0)-\\tilde{\\chi}_0} \\norm{\\nabla \\psi^\\ast(\\tilde{z}_1)-\\tilde{x}_1}.\n    \\end{equation}\n    We can now finally bound $G_1$:\n    \\begin{align*}\n    \\begin{aligned}\n        G_1 &\\circled{1}[\\leq] \\frac{\\tilde{L}}{\\gamman} \\norm{\\nabla \\psi^\\ast(\\tilde{\\zeta}_0)-\\tilde{\\chi}_0} \\cdot \\norm{\\nabla \\psi^\\ast (\\tilde{z}_1)-\\tilde{x}_1} \\\\\n            & \\quad -\\frac{1}{A_1} \\left(D_{\\psi^\\ast}(\\tilde{z}_0,\\tilde{\\zeta}_0)+D_{\\psi^\\ast}(\\tilde{\\zeta}_0, \\tilde{z}_1)\\right) + \\frac{1}{A_1} D_{\\psi}(\\tilde{x}^\\ast, \\tilde{\\chi}_0) \\\\\n            & \\circled{2}[\\leq] \\frac{\\tilde{L}}{\\gamman} \\norm{\\nabla \\psi^\\ast(\\tilde{\\zeta}_0)-\\tilde{\\chi}_0} \\cdot \\norm{\\nabla \\psi^\\ast(\\tilde{z}_1)-\\tilde{x}_1} \\\\\n            &\\quad -\\frac{\\sigma}{2A_1}\\left(\\norm{\\nabla \\psi^\\ast(\\tilde{\\zeta}_0)-\\tilde{\\chi}_0}^2 +\\norm{\\nabla \\psi^\\ast(\\tilde{z}_1)-\\tilde{x}_1}^2\\right) + \\frac{1}{A_1}D_{\\psi}(\\tilde{x}^\\ast, \\tilde{\\chi}_0) \\\\\n            & \\circled{3}[\\leq] \\norm{\\nabla \\psi^\\ast(\\tilde{\\zeta}_0)-\\tilde{\\chi}_0} \\cdot \\norm{\\nabla \\psi^\\ast(\\tilde{z}_1)-\\tilde{x}_1}\\left(\\frac{\\tilde{L}}{\\gamman}-\\frac{\\sigma}{A_1} \\right) + \\frac{1}{A_1}D_{\\psi}(\\tilde{x}^\\ast, \\tilde{\\chi}_0) \\\\\n            & \\circled{4}[\\leq] \\frac{1}{A_1} D_{\\psi}(\\tilde{x}^\\ast, \\tilde{\\chi}_0).\n    \\end{aligned}\n    \\end{align*}\n    We used in $\\circled{1}$ the definition of $G_1=U_1-L_1 = f(\\tilde{x}_1)-L_1$ and we bound the inner product in $L_1$ using $\\eqref{bounding_g_1_equation_1}$, and $\\eqref{bounding_g_1_equation_2}$. Also, since $\\tilde{z}_0 = \\nabla \\psi(\\tilde{\\chi}_0)$ we have $D_{\\psi^\\ast}(\\tilde{z}_0, \\tilde{z}_1) = D_{\\psi}(\\nabla \\psi^\\ast(\\tilde{z}_1), \\nabla\\psi^\\ast(\\tilde{z}_0))= D_{\\psi}(\\nabla \\psi^\\ast(\\tilde{z}_1),\\tilde{\\chi}_0)$, so we can cancel two of the Bregman divergences. In $\\circled{2}$, we used \\cref{prop:bounding_breg_div_by_norm_of_difference}, $\\nabla \\psi^\\ast(\\tilde{z}_0) = \\tilde{\\chi}_0$, and $\\nabla \\psi^\\ast(\\tilde{\\zeta}_0)=\\tilde{x}_1$. In $\\circled{3}$ we used again the inequality $-(a^2+b^2) \\leq -2ab$. Finally $\\circled{4}$ is deduced from $A_1=a_1 \\leq \\sigma \\gamman\/\\tilde{L}$ which comes from the assumption $\\tilde{L}a_{i+1}^2\/\\gamman\\sigma \\leq a_{i+1}+A_i\\gamman\\gammap$ for $i=0$.\n\n    The first part of the theorem follows. The second one is a straightforward application of the first one as we see below. Indeed, taking into account $A_t = \\frac{t(t+1)\\sigma \\gamman^2 \\gammap}{4\\tilde{L}}$ and the choice of $t$ we derive the second statement.\n    \\[\n        f(\\tilde{x}_t) - f(\\tilde{x}^\\ast) \\leq \\frac{A_1 G_1}{A_t} + \\sum_{i=1}^{t-1} \\frac{A_{i}\\hat{\\epsilon}_i}{A_t} \\leq \\frac{\\frac{\\sigma}{2}\\norm{\\tilde{x}_0-\\tilde{x}^\\ast}^2}{A_t} + \\frac{\\epsilon}{2} < \\epsilon.\n    \\]\n\\end{proof}\n\nWe present now the final lemma, that proves that $\\hat{\\gamma}_i$ can be found efficiently. As we advanced in the sketch of the main paper, we use a binary search. The idea behind it is that due to \\eqref{eq:appendix_quasiquasarconvexity} we satisfy the equation for $\\hat{\\gamma}_i=\\frac{1}{\\gamman}$ or $\\hat{\\gamma}_i=\\gammap$, or there is $\\hat{\\gamma}_i\\in(\\gammap, 1\/\\gamman)$ such that $\\innp{\\nabla f(\\tilde{x}_{i+1}), \\tilde{x}_{i+1}-\\tilde{x}_i}=0$. The existence of $\\tilde{x}^\\ast$ that satisfies $\\nabla f(\\tilde{x}^\\ast)=0$ along with the boundedness of $Q$ and smoothness, imply the Lipschitzness of $f$. Both Lipschitzness and smoothness allow to prove that a binary search finds efficiently a suitable point. \n\n\\begin{lemma} \\label{lemma:binary_search}\n    Let $Q \\subset\\R^d$ be a convex set of diameter $2\\tilde{R}$. Let $f:Q\\to\\R$ be a function that satisfies \\ref{eq:appendix_quasiquasarconvexity}, is $\\tilde{L}$ smooth and such that there is $\\tilde{x}^\\ast\\in Q$ such that $\\nabla f(x^\\ast)=0$. Let the strongly convex parameter of $\\psi(\\cdot)$ be $\\sigma = O(1)$. Let $i\\geq 1$ be an index. Given two points $\\tilde{x}_i, \\tilde{z}_i \\in  Q$ and the method in \\eqref{general_rule_modified_quasar_axgd} using the learning rates $a_i=\\frac{i}{2}\\cdot \\frac{\\sigma}{\\tilde{L}}\\cdot \\gamman^2\\gammap$ prescribed in \\cref{thm:accelerated_axgd_modified}, we can compute $\\hat{\\gamma}_i$ satisfying \\eqref{eq:approximate_multiplied_convexity}, i.e.\n    \\begin{equation}\\label{property_result_of_line_search_approximate}\n        f(\\tilde{x}_{i+1})-f(\\tilde{x}_i) \\leq \\hat{\\gamma}_i \\innp{\\nabla f(\\tilde{x}_{i+1}), \\tilde{x}_{i+1}-\\tilde{x}_i} + \\hat{\\epsilon}_i. \n    \\end{equation}\n    And the computation of $\\hat{\\gamma}_i$ requires no more than\n    \\[\n        O\\left(\\log\\left(\\frac{\\tilde{L}\\tilde{R}}{\\gamman\\hat{\\epsilon}_i}\\cdot i\\right)\\right)\n    \\]\n    queries to the gradient oracle. \n\\end{lemma}\n\n\\begin{proof}\n    Let $\\hat{\\Gamma}_i(\\lambda):[\\frac{a_{i+1}}{A_{i+1}}, \\frac{a_{i+1}\/\\gamman}{A_i\\gammap+a_{i+1}\/\\gamman}] \\to \\R$ be defined as \n    \\begin{equation}\\label{def:Gamma}\n        \\hat{\\Gamma}_i\\left(\\frac{a_{i+1}\/\\gamman}{A_i \\mathtt{\\tilde{x}}+a_{i+1}\/\\gamman}\\right) = \\mathtt{\\tilde{x}},\\text{ for }\\mathtt{\\tilde{x}}\\in [\\gammap, \\frac{1}{\\gamman}].  \n    \\end{equation}\n    By monotonicity, it is well defined.  Let $\\tilde{x}_{i+1}^\\lambda$ be the point computed by one iteration of \\eqref{general_rule_modified_quasar_axgd} using the parameter $\\hat{\\gamma}_i = \\hat{\\Gamma}_i(\\lambda)$. Likewise, we define the rest of the points in the iteration \\eqref{general_rule_modified_quasar_axgd} depending on $\\lambda$. We first try $\\hat{\\gamma}_i = 1\/\\gamman$ and $\\hat{\\gamma}_i = \\gammap$ and use any of them if they satisfy the conditions. If neither of them do, it means that for the first choice we had $\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda_1}), \\tilde{x}_{i+1}^{\\lambda_1} - \\tilde{x}_{i}} < 0$ and for the second one, it is $\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda_2}), \\tilde{x}_{i+1}^{\\lambda_2} - \\tilde{x}_{i}} > 0 $, for $\\lambda_1 = \\hat{\\Gamma}_i^{-1}(1\/\\gamman)$ and $\\lambda_2 = \\hat{\\Gamma}_i^{-1}(\\gammap)$. Therefore, by continuity, there is $\\lambda^\\ast \\in [\\lambda_1, \\lambda_2]$ such that $\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda^\\ast}), \\tilde{x}_{i+1}^{\\lambda^\\ast} - \\tilde{x}_{i}}=0$. The continuity condition is easy to prove. We omit it because it is derived from the Lipschitzness condition that we will prove below. Such a point satisfies \\eqref{eq:appendix_quasiquasarconvexity} for $\\hat{\\epsilon}_i=0$. We will prove that the function $G_i:[\\frac{a_{i+1}}{A_{i+1}}, \\frac{a_{i+1}\/\\gamman}{A_i\\gammap+a_{i+1}\/\\gamman}]\\to\\R$, defined as \n    \\begin{equation}\\label{def:G}\n    G_i(\\lambda) \\defi -\\hat{\\Gamma}_i(\\lambda) \\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda}), \\tilde{x}_{i+1}^{\\lambda}-\\tilde{x}_i}+ (f(\\tilde{x}_{i+1}^{\\lambda})-f(\\tilde{x}_i)),\n    \\end{equation}\n    is Lipschitz so we can guarantee that \\eqref{eq:approximate_multiplied_convexity} holds for an interval around $\\lambda^\\ast$. Finally, we will be able to perform a binary search to efficiently find a point in such interval or another interval around another point that satisfies that the inner product is $0$.\n\n    So \n    \\begin{align}\\label{lipschitzness_of_G}\n    \\begin{aligned}\n        \\abs{G_i(\\lambda) - G_i(\\lambda')} &\\leq \\abs{f(\\tilde{x}_{i+1}^{\\lambda})-f(\\tilde{x}_{i+1}^{\\lambda'})} \\\\\n        &\\quad+ \\abs{\\hat{\\Gamma}_i(\\lambda')}\\cdot\\abs{\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda'}), \\tilde{x}_{i+1}^{\\lambda'}-\\tilde{x}_i}-\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda}), \\tilde{x}_{i+1}^{\\lambda}-\\tilde{x}_i}} \\\\\n        &\\quad+\\abs{\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda}), \\tilde{x}_{i+1}^{\\lambda}-\\tilde{x}_i}}\\cdot\\abs{\\hat{\\Gamma}_i(\\lambda')-\\hat{\\Gamma}_i(\\lambda)} \n    \\end{aligned}\n    \\end{align}\n    We have used used the triangular inequality and the inequality \n    \\begin{equation}\\label{simple_inequality_from_triangular}\n    \\abs{\\alpha_1 \\beta_1 -\\alpha_2\\beta_2} \\leq \\abs{\\alpha_1}\\abs{\\beta_1-\\beta_2}+\\abs{\\beta_2}\\abs{\\alpha_1-\\alpha_2},\n   \\end{equation} \n   which is a direct consequence of the triangular inequality, after adding and subtracting $\\alpha_1\\beta_2$ in the $\\abs{\\cdot}$ on the left hand side. We bound each of the three summands of the previous inequality separately, but first we bound the following which will be useful for our other bounds, \n\\begingroup\n\\allowdisplaybreaks\n    \\begin{align} \\label{eq:bound_on_x_i_plus_1}\n    \\begin{aligned}\n      \\norm{\\tilde{x}_{i+1}^{\\lambda'}-\\tilde{x}_{i+1}^{\\lambda}}   & \\circled{1}[=] \\norm{(\\lambda'\\nabla\\psi^\\ast(\\tilde{\\zeta}_i^{\\lambda'})+(1-\\lambda')\\tilde{x}_i)-(\\lambda\\nabla\\psi^\\ast(\\tilde{\\zeta}_i^{\\lambda})+(1-\\lambda)\\tilde{x}_i)} \\\\\n        & \\circled{2}[\\leq] \\norm{\\nabla \\psi^\\ast(\\tilde{\\zeta}_i^\\lambda)-\\tilde{x}_i}\\abs{\\lambda'-\\lambda} + \\norm{\\lambda'\\nabla\\psi^\\ast(\\tilde{\\zeta}_i^{\\lambda'}) -\\lambda'\\nabla\\psi^\\ast(\\tilde{\\zeta}_i^{\\lambda})} \\\\\n        & \\circled{3}[\\leq] 2\\tilde{R}\\abs{\\lambda-\\lambda'} +\\norm{\\nabla\\psi^\\ast(\\tilde{\\zeta}_i^{\\lambda'})-\\nabla\\psi^\\ast(\\tilde{\\zeta}_i^{\\lambda})}  \\\\\n        & \\circled{4}[\\leq] 2\\tilde{R}\\abs{\\lambda-\\lambda'} + \\frac{1}{\\gamman\\sigma} \\norm{\\nabla f(\\tilde{\\chi}_i^{\\lambda})-\\nabla f(\\tilde{\\chi}_i^{\\lambda'})} \\\\\n        & \\circled{5}[\\leq] 2\\tilde{R}\\abs{\\lambda-\\lambda'} + \\frac{\\tilde{L}}{\\gamman\\sigma}\\norm{\\tilde{\\chi}_i^{\\lambda}-\\tilde{\\chi}_i^{\\lambda'}} \\\\\n        & \\circled{6}[\\leq] \\left(2\\tilde{R}+\\frac{2L\\tilde{R}}{\\gamman\\sigma}\\right)\\abs{\\lambda-\\lambda'}\n    \\end{aligned}\n\\end{align}\n\\endgroup\n    Here, $\\circled{1}$ uses the definition of $\\tilde{x}_{i+1}^\\lambda$ as a convex combination of $\\tilde{x}_i$ and $\\nabla\\psi^\\ast(\\tilde{\\zeta}_i^{\\lambda})$. $\\circled{2}$ adds and substracts $\\lambda'\\nabla \\psi^\\ast(\\tilde{\\zeta}_i^\\lambda)$, groups terms and uses the triangular inequality. In $\\circled{3}$ we use the fact that the diameter of $Q$ is $2\\tilde{R}$ and bound $\\lambda'\\leq 1$, and $\\abs{\\lambda} \\leq 1$. $\\circled{4}$ uses the $\\frac{1}{\\sigma}$ smoothness of $\\nabla \\psi^\\ast(\\cdot)$, which is a consequence of the $\\sigma$-strong convexity of $\\psi(\\cdot)$. $\\circled{5}$ uses the smoothness of $f$. In $\\circled{6}$, from the definition of $\\tilde{\\chi}_{i}^{\\lambda}$ we have that $ \\norm{\\tilde{\\chi}_i^{\\lambda}-\\tilde{\\chi}_i^{\\lambda'}}\\leq \\norm{\\tilde{x}_i-\\tilde{z}_i}\\abs{\\lambda-\\lambda'}$. We bounded this further using the diameter of $ Q$.\n\nNote that $f$ is Lipschitz over $Q$. By the existence of $x^\\ast$, $\\tilde{L}$-smoothness, and the diameter of $Q$ we obtain that the Lipschitz constant $L_\\mathtt{p}$ is $L_\\mathtt{p}\\leq 2R^2L$. Now we can proceed and bound the three summands of \\eqref{lipschitzness_of_G}. The first one reduces to the inequality above after using Lipschitzness of $f(\\cdot)$: \n    \\begin{equation}\\label{eq:first_summand}\n        \\abs{f(\\tilde{x}_{i+1}^{\\lambda})-f(\\tilde{x}_{i+1}^{\\lambda'})}  \\leq L_\\mathtt{p}\\norm{\\tilde{x}_{i+1}^{\\lambda'}-\\tilde{x}_{i+1}^{\\lambda}}.\n    \\end{equation}\n    In order to bound the second summand, we note that \n    \\begin{equation}\\label{eq:lipschitzness_of_Gamma}\n    \\abs{(\\hat{\\Gamma}_i^{-1})'(\\mathtt{\\tilde{x}})} = \\absadj{\\frac{A_ia_{i+1}\/\\gamman}{(A_i\\mathtt{\\tilde{x}}+a_{i+1}\/\\gamman)^2}} \\geq  \\frac{\\gamman A_ia_{i+1}}{A_{i+1}^2},\n    \\end{equation}\n    so $\\hat{\\Gamma}_i(\\lambda')$, appearing in the first factor, is bounded by $A_{i+1}^2\/(\\gamman A_i a_{i+1})$. We used $\\mathtt{\\tilde{x}}\\in[\\gammap, 1\/\\gamman]$ for the bound. For the second factor, we add and subtract $\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda}),\\tilde{x}_{i+1}^{\\lambda'}-\\tilde{x}_{i}}$ and use the triangular inequality and then Cauchy-Schwartz. Thus, we obtain\n    \\begin{align}\\label{eq:second_summand}\n    \\begin{aligned}\n        \\abs{&\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda'}), \\tilde{x}_{i+1}^{\\lambda'}-\\tilde{x}_i}-\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda}), \\tilde{x}_{i+1}^{\\lambda}-\\tilde{x}_i}} \\\\\n       &\\leq \\norm{\\nabla f(\\tilde{x}_{i+1}^\\lambda)}\\cdot \\norm{\\tilde{x}_{i+1}^{\\lambda'}-\\tilde{x}_{i+1}^{\\lambda}} +\\norm{\\nabla f(\\tilde{x}_{i+1}^{\\lambda'})-\\nabla f(\\tilde{x}_{i+1}^{\\lambda})}\\cdot\\norm{\\tilde{x}_{i+1}^{\\lambda'}-\\tilde{x}_{i}} \\\\\n        &\\circled{1}[\\leq] (2L_\\mathtt{p} +2\\tilde{L}\\tilde{R})\\norm{\\tilde{x}_{i+1}^{\\lambda'}-\\tilde{x}_{i+1}^{\\lambda}}.\n    \\end{aligned}\n\\end{align}\nIn $\\circled{1}$, we used Lipschitzness to bound the first factor. We also used the diameter of $ Q$ to bound the last factor and the smoothness of $f(\\cdot)$ to bound the first factor of the second summand.\n\nFor the third summand, we will bound the first factor using Cauchy-Schwartz, smoothness of $f(\\cdot)$ and the diameter of $ Q$. We just proved in \\eqref{eq:lipschitzness_of_Gamma} that $\\hat{\\Gamma}_{i}$ is Lipschitz, so use use this property for the second factor. The result is the following\n    \\begin{align}\\label{eq:third_summand}\n    \\begin{aligned}\n        \\abs{\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda}), \\tilde{x}_{i+1}^{\\lambda}-\\tilde{x}_i}}\\cdot\\abs{\\hat{\\Gamma}_i(\\lambda')-\\hat{\\Gamma}_i(\\lambda)}  \\leq 4\\tilde{L}\\tilde{R}^2 \\frac{A_{i+1}^2}{\\gamman A_i a_{i+1}}\\abs{\\lambda'-\\lambda}.\n    \\end{aligned}\n\\end{align}\n\nApplying the bounds of the three summands \\eqref{eq:first_summand}, \\eqref{eq:lipschitzness_of_Gamma}, \\eqref{eq:second_summand}, \\eqref{eq:third_summand} into \\eqref{lipschitzness_of_G} we obtain the inequality $\\abs{G_i(\\lambda') - G_i(\\lambda)} \\leq \\hat{L}\\abs{\\lambda'-\\lambda}$ for \n\\begin{align*}\n    \\begin{aligned}\n        \\hat{L} = \\left(2\\tilde{R}+\\frac{2\\tilde{L}\\tilde{R}}{\\gamman\\sigma}\\right)\\left(L_\\mathtt{p} + (2L_\\mathtt{p}+2\\tilde{L}\\tilde{R})\\frac{A_{i+1}^2}{\\gamman A_i a_{i+1}}\\right)+ 4\\tilde{L}\\tilde{R}^2 \\frac{A_{i+1}^2}{\\gamman A_i a_{i+1}}.\n    \\end{aligned}\n\\end{align*}\n\nWe will use the following to bound $\\hat{L}$. If we use the learning rates prescribed in \\cref{thm:accelerated_axgd_modified}, namely $a_{i} = \\frac{i\\sigma \\gamman^2\\gammap}{2L}$ and thus $A_i=\\frac{i(i+1)\\sigma\\gamman^2\\gammap}{4L}$ we can bound $A^2_{i+1}\/(A_ia_{i+1}) \\leq 3(i+2)$, using that $i\\geq 1$. In our setting, by smoothness and the existence of $\\tilde{x}^\\ast\\in Q$ such that $\\nabla f(\\tilde{x}^\\ast)=0$, we have that $L_\\mathtt{p}\\leq 2\\tilde{R}\\tilde{L}$. Recall we assume $\\sigma=O(1)$. In \\cref{alg:accelerated_gconvex} we use $\\sigma=1$.\n\nRecall we are denoting by $\\lambda^\\ast$ a value such that $\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda^\\ast}), \\tilde{x}_{i+1}^{\\lambda^\\ast} - \\tilde{x}_{i}}=0$ so $G_i(\\lambda^\\ast) \\leq 0$. Lipschitzness of $G$ implies that if $G_i(\\lambda^\\ast) \\leq 0$ then $G_i(\\lambda) \\leq \\hat{\\epsilon}_i$ for $\\lambda \\in [\\lambda^\\ast-\\frac{\\hat{\\epsilon}_i}{\\hat{L}}, \\lambda^\\ast+\\frac{\\hat{\\epsilon}_i}{\\hat{L}}]\\cap[\\Gamma_i^{-1}(\\gamman), \\Gamma_i^{-1}(\\gammap)]$. If the extremal points, $\\Gamma_i^{-1}(\\gamman),\\Gamma_i^{-1}(\\gammap)$ did not satisfy \\eqref{property_result_of_line_search_approximate}, then this interval is of length $\\frac{2\\hat{\\epsilon}_i}{\\hat{L}}$ and a point in such interval or another interval that is around another point $\\bar{\\lambda}^\\ast$ that satisfies $\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\bar{\\lambda}^\\ast}), \\tilde{x}_{i+1}^{\\bar{\\lambda}^\\ast}-\\tilde{x}_i}=0$ can be found with a binary search in at most\n\\[\n    O\\left(\\log\\left(\\frac{\\hat{L}}{\\hat{\\epsilon}_i}\\right)\\right) \\circled{1}[=] O\\left(\\log\\left(\\frac{\\tilde{L}\\tilde{R}}{\\gamman\\hat{\\epsilon}_i}\\cdot i\\right)\\right)\n\\]\niterations, provided that at each step we can ensure we halve the size of the search interval. The bounds of the previous paragraph are applied in $\\circled{1}$.The binary search can be done easily: we start with $[\\Gamma_i^{-1}(\\gamman), \\Gamma_i^{-1}(\\gammap)]$  and assume the extremes do not satisfy \\eqref{property_result_of_line_search_approximate}, so the sign of $\\innp{\\nabla f(\\tilde{x}_{i+1}^\\lambda), \\tilde{x}_{i+1}^\\lambda-\\tilde{x}_i}$ is different for each extreme. Each iteration of the binary search queries the midpoint of the current working interval and if \\eqref{property_result_of_line_search_approximate} is not satisfied, we keep the half of the interval such that the extremes keep having the sign of $\\innp{\\nabla f(\\tilde{x}_{i+1}^\\lambda), \\tilde{x}_{i+1}^\\lambda-\\tilde{x}_i}$ different from each other, ensuring that there is a point in which this expression evaluates to $0$ and thus keeping the invariant. We include the pseudocode of this binary search in \\cref{alg:bin_search}.\n\\end{proof}\n\nWe proceed to prove \\cref{thm:acceleration_quasiquasarconvexity}, which is an immediate consequence of the previous results.\n\n\\begin{proof}[Proof of \\cref{thm:acceleration_quasiquasarconvexity}]\n    The proof follows from \\cref{thm:accelerated_axgd_modified}, provided that we can find $\\hat{\\gamma}_i$ satisfying \\eqref{eq:approximate_multiplied_convexity}. \\cref{lemma:binary_search} shows that this is possible after performing a logarithmic number of queries to the gradient oracle. Note that given our choice of $\\hat{\\epsilon}_i$, $t$ and $a_i$, the number of queries to the gradient oracle \\cref{lemma:binary_search} requires is no more than $O(\\log(\\tilde{L}R\/\\gamman\\epsilon))$ for any $i \\leq t$. So we find an $\\epsilon$-minimizer of $f$ after $\\bigotilde{\\sqrt{\\tilde{L}\/(\\gamma^2\\gammap\\epsilon)}}$ queries to the gradient oracle.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:riemannian_acceleration}]\n    Given the function to optimize $F:\\M\\to\\R$ and the geodesic map $h$, we define $f=F\\circ h^{-1}$. Using \\cref{lemma:smoothness_of_transformed_function} we know that $f$ is $\\tilde{L}$-smooth, with $\\tilde{L}=O(L)$. \\cref{prop:bounding_hyperplane} proves that $f$ satisfies \\eqref{eq:appendix_quasiquasarconvexity} for constants $\\gamman$ and $\\gammap$ depending on $R$. So \\cref{thm:acceleration_quasiquasarconvexity} applies and the total number of queries to the oracle needed to obtain an $\\epsilon$-minimizer of $f$ is $\\bigotilde{\\sqrt{\\tilde{L}\/\\gamman^2\\gammap\\epsilon}} = \\bigotilde{\\sqrt{L\/\\epsilon}}$. The result follows, since $f(\\tilde{x}_t) - f(\\tilde{x}^\\ast) = F(x_t)-F(x^\\ast)$.\n\\end{proof}\n\nWe recall a few concepts that were assumed during \\cref{sec:algorithm} to better interpret \\cref{thm:riemannian_acceleration}. We work in the hyperbolic space or an open hemisphere. The aim is to minimize a smooth and g-convex function defined on any of these manifolds, or a subset of them. The existence of a point $x^\\ast$ that satisfies $\\nabla F(x^\\ast)=0$ is assumed. Starting from an arbitrary point $x_0$, we let $R$ be a bound of the distance between $x_0$ and $x^\\ast$, that is, $R\\geq d(x_0,x^\\ast)$. We let $\\M=\\expon_x(\\bar{B}(0,R))$ so that $x^\\ast\\in\\M$. We assume $F:\\M'\\to\\R$ is a differentiable function, where $\\M' = \\expon_x{B(0,R')}$ and $R'>R$. We define $F$ on $\\M'$ only for simplicity, to avoid the use of subdifferentials. $\\M$ has constant sectional curvature $K$. If $K$ is positive, we restrict $R<\\pi\/2\\sqrt{K}$ so $\\M$ is contained in an open hemisphere and it is uniquely geodesic. We define a geodesic map $h$ from the hyperbolic plane or a open hemisphere onto a subset of $\\R^d$ and define the function $f:h(\\M)\\to\\R$ as $f = F\\circ h^{-1}$. We optimize this function in an accelerated way up to constants and log factors, where the constants appear as an effect of the deformation of the geometry and depend on $R$ and $K$ only. Note the assumption of the existence of $x^\\ast$ such that $\\nabla F(x^\\ast)=0$ is not necessary, since $\\argmin_{x\\in\\expon_{x_0}(\\bar{B}(0,R))}\\{F(x)\\}$ also satisfies the first inequality in \\eqref{eq:appendix_quasiquasarconvexity} so the lower bounds $L_i$ can be defined in the same way as we did. In that case, if we want to perform constrained optimization, one needs to use the Lipschitz constant of $F$, when restricted to $\\expon_x(\\bar{B}(0,R))$, for the analysis of the binary search. \n\n\\begin{algorithm}\n    \\caption{BinaryLineSearch$(\\tilde{x}_i, \\tilde{z}_i, f, \\X, a_{i+1}, A_{i}, \\epsilon, \\tilde{L},  \\gamman, \\gammap)$}\n    \\label{alg:bin_search}\n    \\begin{algorithmic}[1]\n        \\REQUIRE Points $\\tilde{x}_i$, $\\tilde{z}_i$, function $f$, domain $\\X$, learning rate $a_{i+1}$, accumulated learning rate $A_i$, final target accuracy $\\epsilon$, final number of iterations $t$, smoothness constant $\\tilde{L}$, constants $\\gamman, \\gammap$.\n        Define $\\hat{\\epsilon}_i \\gets (A_t\\epsilon)\/(2(t-1)A_i)$ as in \\cref{thm:accelerated_axgd_modified}, i.e. with $A_t=t(t+1)\\gamma_n^2\\gamma_p\/4\\tilde{L}$. $\\hat{\\Gamma}_i$ defined as in \\eqref{def:Gamma} and $G_i$ defined as in \\eqref{def:G} i.e.\n        \\[\n    G_i(\\lambda) \\defi -\\hat{\\Gamma}_i(\\lambda) \\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda}), \\tilde{x}_{i+1}^{\\lambda}-\\tilde{x}_i}+ (f(\\tilde{x}_{i+1}^{\\lambda})-f(\\tilde{x}_i)),\n        \\]\n        for $x_{i+1}^\\lambda$ being the result of method \\eqref{appendix_general_rule_modified_quasar_axgd} when $\\hat{\\gamma}_i = \\hat{\\Gamma}_i(\\lambda)$.\n        \\ENSURE $\\lambda = \\frac{a_{i+1}\/\\gamman}{A_i\\hat{\\gamma}_i+a_{i+1}\/\\gamman}$ for $\\hat{\\gamma}_i$ such that $G_i(\\hat{\\Gamma}_i^{-1}(\\hat{\\gamma}_i)) \\leq \\hat{\\epsilon}_i$.\n        \\IF{$G_i(\\hat{\\Gamma}_i^{-1}(1\/\\gamman)) \\leq \\hat{\\epsilon}_i$} $\\lambda = \\hat{\\Gamma}_i^{-1}(1\/\\gamman)$\n        \\ELSIF{$G_i(\\hat{\\Gamma}_i^{-1}(\\gammap)) \\leq \\hat{\\epsilon}_i$} $\\lambda = \\hat{\\Gamma}_i^{-1}(\\gammap)$\n        \\ELSE\n            \\State $\\varl \\gets \\hat{\\Gamma}_i^{-1}(1\/\\gamman)$\n            \\State $\\varr \\gets \\hat{\\Gamma}_i^{-1}(\\gammap)$\n            \\State $\\lambda \\gets (\\varl + \\varr)\/2$\n            \\WHILE{$G_i(\\lambda) > \\hat{\\epsilon}_i$}\n                \\IF{$\\innp{\\nabla f(\\tilde{x}_{i+1}^{\\lambda}), \\tilde{x}_{i+1}^{\\lambda}-\\tilde{x}_i} <0$} $\\varr \\gets \\lambda$\n                \\ELSE $\\ \\varl \\gets \\lambda$\n                \\ENDIF\n                \\State $\\lambda \\gets (\\varl + \\varr)\/2$\n            \\ENDWHILE\n        \\ENDIF\n        \\State \\textbf{return} $\\lambda$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Auxiliary lemmas}\nThe following are classical lemmas of convex optimization that we used in this section and that we add for completeness.\n\n\\begin{fact}\\label{grad_of_fenchel_dual}\n    Let $\\psi:Q\\to\\R$ be a differentiable strongly-convex function. Then\n    \\[\n        \\nabla \\psi^\\ast(\\tilde{z}) = \\argmax_{\\tilde{x}\\in Q}\\{\\innp{\\tilde{z}, \\tilde{x}}-\\psi(\\tilde{x})\\}.\n    \\] \n\\end{fact}\n\n\\begin{lemma}[Duality of Bregman Div.] \\label{prop:duality_of_bregman_divergences}\n    $D_{\\psi}(\\nabla \\psi^\\ast(\\tilde{z}), \\tilde{x}) = D_{\\psi^\\ast}(\\nabla \\psi(\\tilde{x}), \\tilde{z})$ for all $\\tilde{z}, \\tilde{x}$.\n\\end{lemma}\n\n\\begin{proof}\n    From the definition of the Fenchel dual \\eqref{def:fenchel_dual} and \\eqref{grad_of_fenchel_dual} we have\n    \\[\n        \\psi^\\ast(\\tilde{z}) = \\innp{\\nabla \\psi^\\ast(\\tilde{z}), \\tilde{z}} - \\psi(\\nabla \\psi^\\ast(\\tilde{z})) \\text{ for all } \\tilde{z}.\n    \\] \n    Since by the Fenchel-Moreau Theorem we have $\\psi^{\\ast\\ast}=\\psi$, it holds\n    \\[\n        \\psi(\\tilde{x}) = \\innp{\\nabla \\psi(\\tilde{x}), \\tilde{x}} - \\psi^\\ast(\\nabla \\psi(\\tilde{x})), \\text{ for all } \\tilde{x}.\n    \\] \n    Using the definition of Bregman divergence \\eqref{def:bregman_divergence} and \\eqref{grad_of_fenchel_dual}:\n    \\begin{align*}\n        \\begin{aligned}\n            D_{\\psi}(\\nabla \\psi^\\ast(\\tilde{z}), \\tilde{x}) &=  \\psi(\\nabla \\psi^\\ast(\\tilde{z})) - \\psi(\\tilde{x})-\\innp{\\nabla \\psi(\\tilde{x}), \\nabla\\psi^\\ast(\\tilde{z})-\\tilde{x}} \\\\\n            &= \\psi(\\nabla \\psi^\\ast(\\tilde{z}))+\\psi^\\ast(\\nabla \\psi(\\tilde{x})) - \\innp{\\nabla\\psi(\\tilde{x}), \\nabla\\psi^\\ast(\\tilde{z})} \\\\\n            & = \\psi^\\ast(\\nabla \\psi(\\tilde{x}))-\\psi^\\ast(\\tilde{z}) - \\innp{\\nabla\\psi^\\ast(\\tilde{z}), \\nabla \\psi(\\tilde{x})-\\tilde{z}} \\\\\n            & = D_{\\psi^\\ast}(\\nabla\\psi(\\tilde{x}), \\tilde{z}).\n        \\end{aligned}\n    \\end{align*}\n\\end{proof}\n\n\\begin{lemma}[Triangle inequality of Bregman Divergences] \\label{prop:triangle_inequality_of_bregman_div}\n    For all $\\tilde{x}, \\tilde{y}, \\tilde{z} \\in Q$ we have \n    \\[\n        D_{\\psi^\\ast} (\\tilde{x}, \\tilde{y}) = D_{\\psi^\\ast} (\\tilde{z}, \\tilde{y})  + D_{\\psi^\\ast} (\\tilde{x}, \\tilde{z}) + \\innp{\\nabla \\psi^\\ast(\\tilde{z})-\\nabla \\psi^\\ast(\\tilde{y}), \\tilde{x}-\\tilde{z}}.\n    \\] \n\\end{lemma}\n\n\\begin{proof}\n\\begin{align*}\n    \\begin{aligned}\n        D_{\\psi^\\ast}& (\\tilde{z}, \\tilde{y}) + D_{\\psi^\\ast} (\\tilde{x}, \\tilde{z}) + \\innp{\\nabla \\psi^\\ast(\\tilde{z})-\\nabla \\psi^\\ast(\\tilde{y}), \\tilde{x}-\\tilde{z}} \\\\\n        &= (\\psi^\\ast(\\tilde{z})-\\psi^\\ast(\\tilde{y}) -\\innp{\\nabla \\psi^\\ast(\\tilde{y}), \\tilde{z}-\\tilde{y}}) \\\\\n        & \\quad + (\\psi^\\ast(\\tilde{x})-\\psi^\\ast(\\tilde{z}) -\\innp{\\nabla \\psi^\\ast(\\tilde{z}), \\tilde{x}-\\tilde{z}}) \\\\\n        &\\quad + \\innp{\\nabla \\psi^\\ast(\\tilde{z})-\\nabla \\psi^\\ast(\\tilde{y}), \\tilde{x}-\\tilde{z}}  \\\\\n        & = \\psi^\\ast(\\tilde{x}) -\\psi^\\ast(\\tilde{y}) -\\innp{\\nabla \\psi^\\ast(\\tilde{y}), \\tilde{z}-\\tilde{y}}+ \\innp{-\\nabla \\psi^\\ast(\\tilde{y}), \\tilde{x}-\\tilde{z}} \\\\\n        & =  D_{\\psi^\\ast} (\\tilde{x}, \\tilde{y}).\n    \\end{aligned}\n    \\end{align*}\n\\end{proof}   \n\n\\begin{lemma}\\label{prop:bounding_breg_div_by_norm_of_difference}\n    Given a $\\sigma$-strongly convex function $\\psi(\\cdot)$ the following holds:\n    \\[\n        D_{\\psi^\\ast}(\\tilde{z}_1, \\tilde{z}_2) \\geq \\frac{\\sigma}{2}\\norm{\\nabla \\psi^\\ast(\\tilde{z}_1)-\\nabla \\psi^\\ast(\\tilde{z}_2)}^2.\n    \\] \n\\end{lemma}\n\n\\begin{proof}\n    Using the first order optimality condition of the Fenchel dual and \\eqref{grad_of_fenchel_dual} we obtain\n    \\[\n        \\innp{\\nabla\\psi(\\nabla\\psi^\\ast(\\tilde{z}_1))-\\tilde{z}_1l, \\nabla \\psi^\\ast(\\tilde{z}_2)-\\nabla \\psi^\\ast(\\tilde{z}_1)} \\geq 0\n    \\] \n    Using $\\sigma$-strong convexity of $\\psi$ and the previous inequality we have\n    \\begin{align*}\n    \\begin{aligned}\n        D_{\\psi^\\ast}(\\tilde{z}_1, \\tilde{z}_2) &= \\psi(\\nabla \\psi^\\ast(\\tilde{z}_2)) -\\psi(\\nabla\\psi^\\ast(\\tilde{z}_1)) - \\innp{\\tilde{z}_1, \\nabla \\psi^\\ast(\\tilde{z}_2)-\\nabla \\psi^\\ast(\\tilde{z}_1)} \\\\\n        &\\geq \\frac{\\sigma}{2} \\norm{\\nabla \\psi^\\ast(\\tilde{z}_1)-\\nabla \\psi^\\ast(\\tilde{z}_2)}^2 + \\innp{\\nabla \\psi(\\nabla\\psi^\\ast(\\tilde{z}_1))-\\tilde{z}_1, \\nabla\\psi^\\ast(\\tilde{z}_2)-\\nabla\\psi^\\ast(\\tilde{z}_1)} \\\\\n        &\\geq \\frac{\\sigma}{2} \\norm{\\nabla \\psi^\\ast(\\tilde{z}_1)-\\nabla \\psi^\\ast(\\tilde{z}_2)}^2.\n    \\end{aligned}\n    \\end{align*}\n\\end{proof}   \n\n\\section{Reductions. Proofs of results in \\cref{sec:reductions}.} \\label{app:reductions}\n\n\\begin{proof}[Proof of \\cref{thm:reduction_to_g_convex}]\n    Let $\\mathcal{A}_{\\operatorname{ns}}$ be the algorithm in the statement of the theorem. By strong g-convexity of $F$ and the assumptions on $\\mathcal{A}_{\\operatorname{ns}}$ we have that $\\hat{x}_T$ satisfies\n    \\[\n        \\frac{\\mu}{2}d(\\hat{x}_T, x^\\ast)^2 \\leq F(\\hat{x}_T)-F(x^\\ast) \\leq \\frac{\\mu}{2}\\frac{d(x_0,x^\\ast)^2}{2},\n    \\]\n    after $T=\\timens(L, \\mu, R)$ queries to the gradient oracle. This implies $d(\\hat{x}_T, x^\\ast)^2 \\leq d(x_0,x^\\ast)^2\/2$. Then, by repeating this process $r\\defi \\lceil\\log (\\mu \\cdot d(x_0,x^\\ast)^2\/\\epsilon)-1\\rceil$ times, using the previous output as input for the next round, we obtain a point $\\hat{x}_T^r$ that satisfies\n    \\[\n        F(\\hat{x}_T^r)-F(x^\\ast) \\leq \\frac{\\mu \\cdot d(\\hat{x}_T^{r-1},x^\\ast)^2}{4} \\leq \\cdots \\leq \\frac{\\mu \\cdot d(x_0,x^\\ast)^2}{4 \\cdot 2^{r-1}} \\leq \\epsilon.\n    \\]\n    And the total running time is $\\timens(L, \\mu, R)\\cdot r = O(\\timens(L, \\mu, R)\\log(\\mu \\cdot d(x_0,x^\\ast)^2\/\\epsilon))=O(\\timens(L, \\mu, R)\\log(\\mu\/\\epsilon))$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{coroll:acceleration_st_g_convex}]\n    Let $R$ be an upper bound on the distance between the initial point $x_0$ and an optimum $x^\\ast$, i.e. $d(x_0, x^\\ast)\\leq R$. Note that $\\norm{\\tilde{x}_0-\\tilde{x}^\\ast}\/R$ is bounded by a constant depending on $R$ by \\cref{lemma:deformations}.a). Note that $\\gamman$ and $\\gammap$ are constants depending on $R$ by \\cref{prop:bounding_hyperplane}. As any g-strongly convex function is g-convex, by using \\cref{thm:accelerated_axgd_modified} and \\cref{lemma:binary_search} with $\\epsilon=\\mu \\frac{R^2}{4}$ we obtain that \\cref{alg:accelerated_gconvex} obtains a $\\mu \\frac{R^2}{4}$-minimizer in at most \n\\[\n    T = O\\left(\\frac{\\norm{\\tilde{x}_0-\\tilde{x}^\\ast}}{R}\\sqrt{\\frac{L}{\\mu\\gamman^2\\gammap}}\\log\\left(\\frac{\\norm{\\tilde{x}_0-\\tilde{x}^\\ast}}{R}\\sqrt{\\frac{L}{\\mu\\gamman^2\\gammap}}\\right)\\right) = O\\left(\\sqrt{L\/\\mu}\\log(L\/\\mu)\\right)\n\\] \n    queries to the gradient oracle. Subsequent stages, i.e. calls to \\cref{alg:accelerated_gconvex}, need a time at most equal to this. The analysis is the same, but we start at the previous output point and take into account that the initial distance to the optimum has decreased. Using the reduction \\cref{thm:reduction_to_g_convex} we conclude that given $\\epsilon>0$ and running \\cref{alg:accelerated_gconvex} in stages, we obtain an $\\epsilon$-minimizer of $F$ in \n    \\[\n        O(\\sqrt{L\/\\mu}\\log(L\/\\mu)\\log(\\mu \\cdot d(x_0, x^\\ast)^2\/\\epsilon)) = \\bigoast{\\sqrt{L\/\\mu}\\log(\\mu\/\\epsilon)},\n    \\] \n    queries to the gradient oracle.\n\n    As advanced in the main paper, each of the stages of the algorithm resulting from combining \\cref{thm:reduction_to_g_convex} and \\cref{coroll:acceleration_st_g_convex} reduces the distance to $x^\\ast$ by a factor of $1\/\\sqrt{2}$. This means that subsequent stages can be run using a geodesic map centered at the new starting point, and with the new parameter $R$ being the previous one reduced by a factor of $1\/\\sqrt{2}$. This reduces the constants incurred by the deformation of the geometry which ultimately reduces the overall constant in the rate. Note that in order to perform the method with the recentering steps, we need the function $F$ to be defined over at least $\\expon_{x_0}(\\bar{B}(0, R\\cdot(1+2^{-1\/2})))$, since subsequent centers are only guaranteed to be $\\leq R\/\\sqrt{2}$ close to $x^\\ast$, and they could get slightly farther from $x_0$.\n\\end{proof}\n\n\\subsection{Proof of \\cref{thm:reduction_to_g_st_convex}}\n    The algorithm is the following. We successively regularize the function with strongly g-convex regularizers in this way $F^{(\\mu_i)}(x) \\defi F(x)+ \\frac{\\mu_i}{2}d(x, x_0)^2$ for $i \\geq 0$. For each $i \\geq 0$, we use the algorithm $\\mathcal{A}$ on the function $F^{(\\mu_i)}$ for the time in the statement of the theorem and obtain a point $\\hat{x}_{i+1}$, starting from point $\\hat{x}_i$, where $\\hat{x}_0=x_0$. The regularizers are decreased exponentially $\\mu_{i+1} = \\mu_i\/2$ until we reach roughly $\\mu_T=\\epsilon\/R^2$, see below for the precise value.  Let's see how this algorithm works. We first state the following fact, that says that indeed $\\frac{\\mu_i}{2}d(x, x_0)^2$ is a strongly g-convex  regularizer. Let $D$ be the diameter of $\\M$. We define the following quantities\n\\[\n    \\distorp \\defi \n        \\begin{cases} \n            {1} & {\\text{if } K_{\\max } \\leq 0} \\\\\n            {\\sqrt{K_{\\max }} D \\cot (\\sqrt{K_{\\max }} D)} & {\\text{if }  K_{\\max }>0}\n        \\end{cases}\n\\]\n\\[\n    \\distorn \\defi \n        \\begin{cases} \n            {\\sqrt{-K_{\\min }} D \\operatorname{coth}(\\sqrt{-K_{\\min }} D)} & {\\text{if } K_{\\min }<0} \\\\\n            {1} & {\\text{if } K_{\\min } \\geq 0}\n        \\end{cases}\n\\]\nHere $K_{\\max}$ and $K_{\\min}$ are the upper and lower bounds on the sectional curvature of the manifold $\\mathcal{M}$. In \\cref{thm:reduction_to_g_st_convex}, it is $D=2R$.\n\n\\begin{fact}\\label{strong_convexity_and_smoothness_of_ell_2}\n    Let $\\M=\\expon_{x_0}(\\bar{B}(0,R))$ be a manifold with sectional curvature bounded below and above by $K_{\\min}$ and $K_{\\max}$, respectively, where $x_0\\in\\M$ is a point. The function $f:\\M\\to\\R$ defined as $f(x) = \\frac{1}{2}d(x, x_0)^2$ is $\\distorp$-g-strongly convex and $\\distorn$-smooth.\n\\end{fact}\n\nThe result regarding strong convexity can be found, for instance, in \\cite{alimisis2019continuous} and it is a direct consequence of the following inequality, which can also  be found in \\cite{alimisis2019continuous}:\n\\[\n    d(y,x_0)^2 \\geq d(x,x_0)^2-2\\innp{\\expon^{-1}_x(x_0), y-x} + \\distorp d(x,y)^2,\n\\]\nalong with the fact that $\\operatorname{grad} f(x) = -\\expon^{-1}_x(x_0)$. The result regarding smoothness is, similarly, obtained from the following inequality:\n\\[\n    d(y,x_0)^2 \\leq d(x,x_0)^2-2\\innp{\\expon^{-1}_x(x_0), y-x} + \\distorn d(x,y)^2,\n\\]\nwhich can be found in \\cite{zhang2016first} (Lemma $6$). Alternatively, one can derive these inequalities from upper and lower bounds on the Hessian of $f(x) = \\frac{1}{2}d(x, x_0)$, cf. \\cite{jost2008riemannian}, Theorem 4.6.1, as it was done in \\cite{lezcano2020curvature}.\n\nWe prove now that the regularization makes the minimum to be closer to $x_0$, so the assumption of the Theorem on $\\hat{F}$ holds for the functions we use. Define $x_{i+1}$ as the minimizer of $F^{(\\mu_i)}$.\n\n\\begin{lemma}\\label{claim:reduction_sc}\nWe have $d(x_{i+1},x_0) \\leq d(x^\\ast, x_0)$.\n\\end{lemma}\n\n\\begin{proof}\n    By the fact that $x_{i+1}$ is the minimizer of $F^{(\\mu_i)}$ we have $F^{(\\mu_i)}(x_{i+1})-F^{(\\mu_i)}(x^\\ast) \\leq 0$. Note that by g-strong convexity, equality only holds if $x_{i+1} = x^\\ast$ which only happens if $x_0=x_{i+1}=x^\\ast$. By using the definition of $F^{(\\mu_i)}(x) = F(x) + \\frac{\\mu_i}{2}d(x,x_0)^2$ we have:\n\\begin{align*} \n \\begin{aligned}\n F(x_{i+1}) & + \\frac{\\mu_i}{2} d(x_{i+1},x_0)^2 - F(x^\\ast)-\\frac{\\mu_i}{2}d(x^\\ast,x_0)^2 \\leq 0 \\\\\n    \\Rightarrow  \\ \\ \\ & d(x_{i+1},x_0) \\leq d(x^\\ast,x_0),\n   \\end{aligned}\n\\end{align*}\n    where in the last step we used the fact $F(x_{i+1}) - F(x^\\ast) \\geq 0$ that holds because $x^\\ast$ is the minimizer of $F$.\n\\end{proof}\n\nWe note that previous techniques proved and used the fact that $d(x_{i+1}, x^\\ast) \\leq d(x_0, x^\\ast)$ instead \\cite{allen2016optimal}. But crucially, we need our former lemma in order to prove the bound for our non-Euclidean case. Our variant could be applied to \\cite{allen2016optimal} to decrease the constants of the Euclidean reduction. Now we are ready to prove the theorem.\n\n\\begin{proof}[Proof of \\cref{thm:reduction_to_g_st_convex}]\n    We recall the definitions above. $F^{(\\mu_i)}(x) = F(x)+\\frac{\\mu_i}{2}d(x,x_0)^2$. We start with $\\hat{x}_0=x_0$ and compute $\\hat{x}_{i+1}$ using algorithm $\\mathcal{A}$ with starting point $\\hat{x}_i$ and function $F^{(\\mu_i)}$ for time $\\time(L^{(i)}, \\mu^{(i)}, \\M, R)$, where $L^{(i)}$ and $\\mu^{(i)}$ are the smoothness and strong g-convexity parameters of $F^{(\\mu_i)}$. We denote by $x_{i+1}$ the minimizer of $F^{(\\mu_i)}$. We pick $\\mu_i=\\mu_{i-1}\/2$ and we will choose later the value of $\\mu_0$ and the total number of stages. By the assumption of the theorem on $\\mathcal{A}$, we have that \n    \\begin{equation} \\label{eq:aux_eq_reduction_to_g_st_convex}\n        F^{(\\mu_i)}(\\hat{x}_{i+1})-\\min_{x\\in\\M} F^{(\\mu_i)}(x) = F^{(\\mu_i)}(\\hat{x}_{i+1})- F^{(\\mu_i)}(x_{i+1}) \\leq \\frac{F^{(\\mu_i)}(\\hat{x}_{i})- F^{(\\mu_i)}(x_{i+1})}{4}.\n    \\end{equation}\n\n    Define $D_{i} \\defi F^{\\left(\\mu_{i}\\right)}\\left(\\hat{x}_{i}\\right)-F^{\\left(\\mu_{i}\\right)}\\left(x_{i+1}\\right)$ to be the initial objective distance to the minimum on function $F^{(\\mu_i)}$ before we call $\\mathcal{A}$ for the $(i+1)$-th time. At the beginning, we have the upper bound $D_{0}=F^{(\\mu_{0})}(\\hat{x}_{0})- \\min_{x} F^{(\\mu_{0})}(x) \\leq F(x_{0})-F(x^{\\ast})$. For each stage $i \\geq 1$, we compute that\n\\begin{align*}\n \\begin{aligned}\n     \\quad& D_{i} = F^{\\left(\\mu_{i}\\right)}\\left(\\hat{x}_{i}\\right)-F^{\\left(\\mu_{i}\\right)}\\left(x_{i+1}\\right) \\\\\n    &\\circled{1}[=] F^{\\left(\\mu_{i-1}\\right)}\\left(\\hat{x}_{i}\\right)-\\frac{\\mu_{i-1}-\\mu_{i}}{2}d(x_{0},\\hat{x}_{i})^{2}-F^{\\left(\\mu_{i-1}\\right)}\\left(x_{i+1}\\right)+\\frac{\\mu_{i-1}-\\mu_{i}}{2}d(x_{0},x_{i+1})^{2} \\\\\n    &\\circled{2}[\\leq] F^{\\left(\\mu_{i-1}\\right)}\\left(\\hat{x}_{i}\\right)-F^{\\left(\\mu_{i-1}\\right)}\\left(x_{i}\\right)+\\frac{\\mu_{i-1}-\\mu_{i}}{2}d(x_{0},x_{i+1})^{2} \\\\\n    &\\circled{3}[\\leq] \\frac{D_{i-1}}{4} + \\frac{\\mu_{i}}{2}d(x_{0},x_{i+1})^{2} \\\\\n    &\\circled{4}[\\leq] \\frac{D_{i-1}}{4}+\\frac{\\mu_{i}}{2}d(x_{0},x^\\ast)^{2}.\n   \\end{aligned}\n\\end{align*}\n    Above, $\\circled{1}$ follows from the definition of $F^{(\\mu_i)}(\\cdot)$ and $F^{\\left(\\mu_{i-1}\\right)}(\\cdot)$; $\\circled{2}$ follows from the fact that $x_{i}$ is the minimizer of $F^{\\left(\\mu_{i-1}\\right)}(\\cdot)$. We also drop the negative term $-(\\mu_{i-1}-\\mu_i)d(x_0, \\hat{x}_i)\/2$. $\\circled{3}$ follows from the definition of $D_{i-1}$, the assumption on $\\mathcal{A}$, and the choice $\\mu_i = \\mu_{i-1}\/2$ for $i\\geq 1$; and $\\circled{4}$ follows from \\cref{claim:reduction_sc}. \nNow applying the above inequality recursively, we have\n    \\begin{equation}\\label{eq:aux_eq_bounding_D_T}\nD_T \\leq \\frac{D_0}{4^T} + d(x_0,x^\\ast)^2 \\cdot (\\frac{\\mu_T}{2} + \\frac{\\mu_{T-1}}{8} + \\cdots) \\leq \\frac{F(x_0)-F(x^\\ast)}{4^T} +  \\mu_T \\cdot d(x_0,x^\\ast)^2.\n\\end{equation}\nWe have used the choice $\\mu_i = \\mu_{i-1}\/2$ for the second inequality. Lastly, we can prove that $\\hat{x}_T$, the last point computed, satisfies \n\\begin{align*}\n \\begin{aligned}\n F(\\hat{x}_{T})-F(x^\\ast) &\\circled{1}[\\leq] F^{(\\mu_{T})}(\\hat{x}_{T})-F^{(\\mu_{T})}(x^{\\ast})+\\frac{\\mu_{T}}{2}d(x_{0},x^{\\ast})^{2} \\\\\n         & \\circled{2}[\\leq] F^{(\\mu_{T})}(\\hat{x}_{T})-F^{(\\mu_{T})}(x_{T+1}) +\\frac{\\mu_{T}}{2}d(x_{0},x^{\\ast})^{2} \\\\ \n         &\\circled{3}[=] D_{T}+\\frac{\\mu_{T}}{2}d(x_{0},x^{\\ast})^{2} \\\\\n         &\\circled{4}[\\leq] \\frac{F(x_{0})-F(x^{\\ast})}{4^{T}}+ \\frac{3\\mu_{T}}{2} d(x_{0},x^{\\ast})^{2}.\n   \\end{aligned}\n\\end{align*}\n\n    We use the definition of $F^{(\\mu_{T})}$ in $\\circled{1}$ and drop $-\\frac{\\mu_T}{2}d(x_0,\\hat{x}_T)^2$. In $\\circled{2}$ we use the fact that $x_{T+1}$ is the minimizer of $F^{(\\mu_{T})}$. The definition of $D_T$ is used in $\\circled{3}$. We use inequality \\eqref{eq:aux_eq_bounding_D_T} for step $\\circled{4}$. Finally, by choosing $T = \\lceil \\log_2(\\Delta\/\\epsilon)\/2\\rceil+1$ and $\\mu_0 = \\Delta\/R^2$ we obtain that the point $\\hat{x}_T$ satisfies \n    \\[\n    F(\\hat{x}_T)-F(x^\\ast) \\leq \\frac{F(x_{0})-F(x^{\\ast})}{4\\Delta\/\\epsilon}+ \\frac{3\\mu_0}{8\\Delta\/\\epsilon} d(x_{0},x^{\\ast})^{2} \\leq \\frac{\\epsilon}{4}+\\frac{3\\epsilon}{8}<\\epsilon,\n    \\] \n    and can be computed in time $\\sum_{t=0}^{T-1} \\time(L+2^{-t}\\mu_0 \\distorn, 2^{-t}\\mu_0 \\distorp, \\M, R)$, since by Fact \\ref{strong_convexity_and_smoothness_of_ell_2} the function $F^{(\\mu_t)}$ is $L+2^{-t}\\mu_0 \\distorn$ smooth and $2^{-t}\\mu_0 \\distorp$ g-strongly convex.\n\\end{proof}\n\n\\subsection{\\cref{example:application_of_reduction_to_st_convex}}\nWe use the algorithm in \\cref{coroll:acceleration_st_g_convex} as the algorithm $\\mathcal{A}$ of the reduction of \\cref{thm:reduction_to_g_st_convex}. Given $\\M=\\H$ or $\\M=\\S$, the assumption on $\\mathcal{A}$ is satisfied for $\\time(L, \\mu, \\M, R)=O(\\sqrt{L\/\\mu}\\log (L\/\\mu))$. Indeed, if $\\Delta$ is a bound on the gap $\\hat{F}(x_0)-\\hat{F}(x^\\ast) = \\hat{F}(x_0)-\\min_{x\\in\\M}\\hat{F}(x) = \\hat{F}(x_0)-\\min_{x\\in\\expon_{x_0}(\\bar{B}(0, R))}\\hat{F}(x)$ for some $\\mu$ strongly g-convex $\\hat{F}$, then we know that $d(x_0, x^\\ast)^2 \\leq \\frac{2\\Delta}{\\mu}$. By calling the algorithm in \\cref{coroll:acceleration_st_g_convex} with $\\epsilon = \\frac{\\Delta}{4}$ we require a time that is \n\\begin{align*}\n \\begin{aligned}\n     O( \\sqrt{L\/\\mu}\\log (L\/\\mu)\\log(\\mu \\cdot d(x_0, x^\\ast)^2\/(\\Delta\/4))) &= O(\\sqrt{L\/\\mu}\\log (L\/\\mu)\\log(\\mu \\cdot (2\\Delta\/\\mu)\/(\\Delta\/4))) \\\\\n     &= O(\\sqrt{L\/\\mu}\\log (L\/\\mu)).\n   \\end{aligned}\n\\end{align*}\n\nLet $T = \\lceil \\log_2(\\Delta\/\\epsilon)\/2\\rceil+1$. The reduction of \\cref{thm:reduction_to_g_st_convex} gives an algorithm with rates\n\\begin{align*}\n \\begin{aligned}\n     \\sum_{t=0}^{T-1}& \\time(L+2^{-t}\\mu_0 \\distorn, 2^{-t}\\mu_0 \\distorp, \\M, R)   \\\\\n     &= O\\left(\\sum_{t=0}^{T-1}\\sqrt{\\frac{L}{2^{-t}\\distorp\\Delta\/R^2}+\\frac{\\distorn}{\\distorp}}\\cdot\\log\\left(\\frac{L}{2^{-t}\\distorp\\Delta\/R^2}+\\frac{\\distorn}{\\distorp}\\right)\\right) \\\\\n     &\\circled{1}[=] O\\left(\\left(\\sqrt{\\frac{L}{\\distorp\\epsilon}}+\\sqrt{\\frac{\\distorn}{\\distorp}}\\log(\\Delta\/\\epsilon)\\right)\\log\\left(\\frac{L}{\\distorp\\epsilon}+\\frac{\\distorn}{\\distorp}\\right)  \\right) \\\\\n     &= \\bigotilde{\\sqrt{L\/\\epsilon}}\n   \\end{aligned}\n\\end{align*}\nIn $\\circled{1}$ we have used Minkowski's inequality $\\sqrt{a+b} \\leq \\sqrt{a} + \\sqrt{b}$. We used the value $\\mu_0=\\Delta\/R^2$. In order to group the sum of the first summands, we used the fact that $\\sqrt{1\/\\epsilon} + \\sqrt{1\/2\\epsilon} + \\cdots = O(\\sqrt{1\/\\epsilon})$, along with the fact $2^{-(T-1)}\\mu_0\\geq\\log(\\epsilon\/\\Delta)$. We added up the group of second summands. For the $\\log$ factor, we upper bounded $L\/(2^{-t}\\distorp\\Delta\/R^2) = O(L\/\\distorp\\epsilon)$, for $t< T$.  Note that by $L$-smoothness and the diameter being $2R$, we have $\\Delta \\leq 2LR^2$ so $\\sqrt{\\distorn\/\\distorp}\\log(\\Delta\/\\epsilon) = \\bigotilde{1}$.\n\n\\section{Geometric results. Proofs of Lemmas \\ref{lemma:deformations}, \\ref{prop:bounding_hyperplane} and \\ref{lemma:smoothness_of_transformed_function}} \\label{app:geometric_results}\nIn this section we prove the lemmas that take into account the deformations of the geometry and the geodesic map $h$ to obtain relationships between $F$ and $f$. Namely \\cref{lemma:deformations}, \\cref{prop:bounding_hyperplane} and \\cref{lemma:smoothness_of_transformed_function}. First, we recall the characterizations of the geodesic map and some consequences. Then in \\cref{app:sec_distance_deformation}, \\cref{app:sec_angle_deformation} and \\cref{sec:app_gradient_deformation_and_smoothness} we prove the results related to distances angles and gradient deformations, respectively. That is, each of the three parts of \\cref{lemma:deformations}. In \\cref{sec:app_gradient_deformation_and_smoothness} we also prove \\cref{lemma:smoothness_of_transformed_function}, which comes naturally after the proof of \\cref{lemma:deformations}.c). Finally, in \\cref{app:sec_proof_of_bounding_hyperplane} we prove \\cref{prop:bounding_hyperplane}. Before this, we note that we can assume without loss of generality that the curvature of our manifolds of interest can be taken to be $K\\in\\{1, -1\\}$. One can see that the final rates depend on $K$ through $R$, $L$ and $\\mu$.\n\n\\begin{remark}\\label{remark:rescaling_of_K}\n    For a function $F:\\M\\to\\R$ where $\\M=\\H$ or $\\M=\\S$ is a manifold of constant sectional curvature $K\\not\\in\\{1, -1, 0\\}$, we can apply a rescaling to the Gnomonic or Beltrami-Klein projection to define a function on a manifold of constant sectional curvature $K\\in\\{1, -1\\}$. Namely, we can map $\\M$ to $\\B$ via $h$, then we can rescale $\\B$ by multiplying each vector in $\\B$ by the factor $\\sqrt{\\abs{K}}$ and then we can apply the inverse geodesic map for the manifold of curvature $K\\in\\{1, -1\\}$. If $R$ is the original bound of the initial distance to an optimum, and $F$ is $L$-smooth and $\\mu$-strongly g-convex (possibly with $\\mu=0$) then the initial distance bound becomes $\\sqrt{\\abs{K}}R$ and the induced function becomes $L\/\\abs{K}$-smooth and $\\mu\/\\abs{K}$-strongly g-convex. This is a consequence of the transformation rescaling distances by a factor of $\\sqrt{\\abs{K}}$, i.e. if $r:\\M_{K} \\to \\M_{K\/\\abs{K}}$ is the rescaling function, then $d_{K}(x,y)\\sqrt{\\abs{K}} = d_{K\/\\abs{K}}(r(x),r(y))$, where $d_c(\\cdot, \\cdot)$ denotes the distance on the manifold of constant sectional curvature $c$.\n\\end{remark}\n\n\\subsection{Preliminaries}\nWe recall our characterization of the geodesic map. Given two points $\\tilde{x}, \\tilde{y}\\in \\B$, we have that $d(x,y)$, the distance between $x$ and $y$ with the metric of $\\M$, satisfies\n\\begin{equation}\\label{eq:appendix_characterization_of_geodesic_map_and_metric}\n    C_K(d(x,y))  = \\frac{1+K\\innp{\\tilde{x}, \\tilde{y}}}{\\sqrt{1+K\\norm{\\tilde{x}}^2}\\cdot\\sqrt{1+K\\norm{ \\tilde{y}}^2}}.\n\\end{equation}\nAnd since the expression is symmetric with respect to rotations, $\\X=h(\\M)$ is a closed ball of radius $\\tilde{R}$, with $C_K(R) = (1+K\\tilde{R}^2)^{-1\/2}$. Equivalently, \n\\begin{align} \\label{eq:R_tilde_vs_R}\n \\begin{aligned}\n     \\tilde{R} = \\tan(R)  &\\ & & \\text{ if } K=1, \\\\\n     \\tilde{R} = \\tanh(R)  &\\ & & \\text{ if } K=-1.\n   \\end{aligned}\n\\end{align}\nSimilarly, we can write the distances as\n\\begin{align} \\label{eq:distances}\n \\begin{aligned}\n     d(x, y) = \\arccos\\left(\\frac{ 1 + \\innp{\\tilde{x}, \\tilde{y}}}{\\sqrt{1+\\norm{\\tilde{x}^2}}\\sqrt{1+\\norm{\\tilde{y}^2}}}\\right)  &\\ & & \\text{ if } K=1, \\\\\n     d(x, y) = \\arccosh\\left(\\frac{ 1 - \\innp{\\tilde{x}, \\tilde{y}}}{\\sqrt{1-\\norm{\\tilde{x}^2}}\\sqrt{1-\\norm{\\tilde{y}^2}}}\\right)  &\\ & & \\text{ if } K=-1, \n   \\end{aligned}\n\\end{align}\nAlternatively, we have the following expression for the distance $d(x, y)$ when $K=-1$. Let $\\tilde{a}, \\tilde{b}$ be the two points of intersection of the ball $\\B=B(0,1)$ with the line joining $\\tilde{x}, \\tilde{y}$, so the order of the points in the line is $\\tilde{a}, \\tilde{x}, \\tilde{y}, \\tilde{b}$. Then\n\\begin{equation} \\label{eq:distances_hyper_alt}\n    d(x, y) = \\frac{1}{2}\\log\\left(\\frac{\\norm{\\tilde{a}-\\tilde{y}}\\norm{\\tilde{x}-\\tilde{b}}}{\\norm{\\tilde{a}-\\tilde{x}}\\norm{\\tilde{b}-\\tilde{y}}}\\right) \\text{ if } K=-1.\n\\end{equation}\nWe will use this expression when working with the hyperbolic space. A simple elementary proof of the equivalence of the expressions in \\eqref{eq:distances} and \\eqref{eq:distances_hyper_alt} is the following. We can assume without loss of generality that we work with the hyperbolic plane, i.e. $d=2$. By rotational symmetry, we can also assume that $\\tilde{x} = (x_1, x_2)$ and $\\tilde{y} = (y_1, y_2)$, for $x_1=y_1$. In fact, it is enough to prove it in the case $x_2=0$ because we can split a general segment into two, each with one endpoint at $(x_1, 0)$, and then add their lengths up. So according to \\eqref{eq:distances} and \\eqref{eq:distances_hyper_alt}, respectively, we have\n\\[\n    \\frac{1}{\\cosh^2(d(x, y))} =\\frac{(1-x_1^2)(1-y_1^2-y_2^2)}{(1-x_1^2)^2} =  \\frac{(1-x_1^2-y_2^2)}{1-x_1^2}.\n\\] \n\\begin{align*} \n \\begin{aligned}\n     d(x, y)&=\\frac{1}{2}\\log \\left(\\frac{(\\sqrt{1-y_1^2}+y_2)(\\sqrt{1-x_1^2})}{(\\sqrt{1-x_1^2})(\\sqrt{1-y_1^2}-y_2)}\\right) = \\frac{1}{2}\\log\\left(\\frac{1+y_2\/\\sqrt{1-x_1^2}}{1-y_2\/\\sqrt{1-x_1^2}}\\right) \\\\\n     &= \\arctanh\\left(\\frac{y_2}{\\sqrt{1-x_1^2}}\\right)\n   \\end{aligned}\n\\end{align*}\nwhere we have used the equality $\\tanh(t) = \\frac{1}{2}\\log(\\frac{1+t}{1-t})$. Now, using the trigonometric identity $\\frac{1}{\\cosh^2(t)} = 1-\\tanh^2(t)$, for $t=d(x, y)$, we obtain that the two expressions above are equal. See Theorem 7.4 in \\citep{greenberg1993euclidean} (p. 268) for more details about the distance formula under this geodesic map.\n\nThe spherical case is of a remarkable simplicity. If we have a $d$-sphere of radius $1$ centered at $0$, we can see the transformation as the projection onto the plane $x_d =1$. Given two points $\\mathbf{x}=(\\tilde{x}, 1)$, $\\mathbf{y}=(\\tilde{y}, 1)$ then the angle between these two vectors is the distance of the projected points on the sphere so we have $\\cos(d(x,y))=\\innp{\\mathbf{x}, \\mathbf{y}}\/\\norm{\\mathbf{x}}\\norm{\\mathbf{y}}$ which is equivalent to the corresponding formula in \\ref{eq:distances}.\n\n\\subsection{Distance deformation} \\label{app:sec_distance_deformation}\n\n\\begin{lemma} \\label{lemma:distances_hyperbolic} \n    Let $\\H=\\expon_{x_0}(\\bar{B}(0, R))$ be a subset of the hyperbolic space with constant sectional curvature $K=-1$. Let $x, y\\in\\H$ be two different points. Then, we have\n    \\[\n        1 \\leq \\frac{d(x, y)}{\\norm{\\tilde{x}-\\tilde{y}}} \\leq \\cosh^2(R).\n    \\]\n\\end{lemma}\n\\begin{proof}\n    We can assume without loss of generality that the dimension is $d=2$. As in \\eqref{eq:R_tilde_vs_R}, let $\\tilde{R}=\\tanh(R)$, so any point $\\tilde{x}\\in\\X$ satisfies $\\norm{\\tilde{x}} \\leq \\tilde{R}$, or equivalently $d(x, x_0) \\leq R$.  Recall $\\tilde{x}_0=h(x_0)=0$. Without loss of generality, we parametrize an arbitrary segment of length $\\ell$ in $\\X$ by two endpoints $\\tilde{x}, \\tilde{y}$ with coordinates $\\tilde{x}=(x_1, x_2)$ and $\\tilde{y}=(x_1-\\ell, x_2)$, for $0\\leq x_2\\leq \\tilde{R}$, $0\\leq x_1 \\leq\\sqrt{\\tilde{R}^2-x_2^2}$ and $0<\\ell \\leq x_1+\\sqrt{\\tilde{R}^2-x_2^2}$. Let $\\mathfrak{d}(x_1, x_2, \\ell) \\defi \\frac{d(x,y)}{\\ell}$, the quantity we aim to bound. We will prove the upper bound on $\\mathfrak{d}(x_1, x_2, \\ell)$ in three steps. \n    \\begin{enumerate}\n       \\item  If $x_1 = \\ell$ then $\\mathfrak{d}(\\cdot)$ is larger the larger $x_1$ is. This allows to prove that it is enough to consider points with the extra constraint $\\ell \\leq x_1$.\n\n       \\item The partial derivative of $\\mathfrak{d}(\\cdot)$ with respect to $x_1$, whenever $\\ell \\leq x_1$, is non-negative. So we can just look at the points for which $x_1=\\sqrt{\\tilde{R}^2-x_2^2}$.\n\n       \\item With the constraints above, $\\mathfrak{d}(\\cdot)$ is larger the smaller $\\ell$ is. So we have $\\mathfrak{d}(x_1, x_2, \\ell) \\leq \\lim_{\\ell\\to 0} \\mathfrak{d}(\\sqrt{\\tilde{R}^2-x_2^2}, x_2, \\ell) = \\sqrt{1-x_2^2}\/(1-\\tilde{R}^2)$. This expression is maximized at $x_2=0$ and evaluates to $1\/(1-\\tanh^2(R)) = \\cosh^2(R)$.\n   \\end{enumerate} \n   We proceed now to prove the steps above. For the first step, we note\n   \\[\n       \\mathfrak{d}(x_1, x_2, x_1) = \\frac{1}{2x_1}\\log\\left(\\frac{\\sqrt{1-x_2^2}(\\sqrt{1-x_2^2}+x_1)}{\\sqrt{1-x_2^2}(\\sqrt{1-x_2^2}-x_1)}\\right) = \\frac{1}{2x_1}\\log\\left(1+\\frac{2x_1}{\\sqrt{1-x_2^2}-x_1}\\right).\n   \\]\n   We prove that the inverse of this expression is not increasing with respect to $x_1$. By taking a partial derivative:\n\\begin{align*} \n \\begin{aligned}\n     \\frac{\\partial (1\/\\mathfrak{d}(x_1, x_2,x_1))}{\\partial x_1} = 2\\frac{\\frac{-2x_1\\sqrt{1-x_2^2}}{1-x_2^2-x_1^2}+\\log(1+2x_1\/(\\sqrt{1-x_2^2}-x_1))}{\\log(1+2x_1\/(\\sqrt{1-x_2^2}-x_1))^2} \\stackrel{?}{\\leq} 0  \\\\\n    \\iff \\frac{2x_1\\sqrt{1-x_2^2}}{1-x_2^2-x_1^2}-\\log(1+(2x_1\\sqrt{1-x_2^2}+2x_1^2)\/(1-x_2^2-x_1^2)) \\stackrel{?}{\\geq} 0.\n   \\end{aligned}\n\\end{align*}\nIn order to see that the last inequality is true, note that the expression on the left hand side is $0$ when $x_1 = x_2 = 0$. And the partial derivatives of this with respect to $x_1$ and $x_2$, respectively, are:\n\\[\n   \\frac{4\\sqrt{1-x_2^2} x_1^2}{(1-x_2^2-x_1^2)^2} \\text{ and } \\frac{4x_2x_1^3}{\\sqrt{1-x_2^2}(1-x_2^2-x_1^2)^2} .\n\\]\nBoth are greater than $0$ in the interior of the domain $0\\leq x_2\\leq \\tilde{R}$, $0\\leq x_1 \\leq\\sqrt{\\tilde{R}^2-x_2^2}$ and at least $0$ in the border.\n    Now we use this monotonicity to prove that we can consider $\\ell \\leq x_1$ only. Suppose $\\ell > x_1$. The segment $\\ell$ is divided into two parts by the $e_2$ axis and we can assume without loss of generality that the negative part is no greater than the other, i.e. $x_1 \\geq \\ell - x_1$. Otherwise, we can perform the computations after a symmetry over the $e_2$ axis. Let $\\tilde{r}$ be the point $(0, x_2)$. We want to see that the segment from $\\tilde{x}$ to $\\tilde{r}$ gives a greater value of $\\mathfrak{d}(\\cdot)$:\n    \\begin{align*} \n     \\begin{aligned}\n         \\frac{d(x, r)}{x_1} \\geq \\frac{d(x, y)}{\\ell} & \\iff d(x, r) (x_1 + (\\ell-x_1)) \\geq x_1( d(x, r) + d(r, y) )   \\\\ &\\iff d(x,r)\/x_1 \\geq d(r, y)\/(\\ell-x_1),\n       \\end{aligned}\n    \\end{align*}\n    and the last inequality holds true by the monotonicity we just proved.\n\n    In order to prove the second step, we take the partial derivative of $\\mathfrak{d}(x_1, x_2, \\ell)$ with respect to $x_1$. We have\n    \\[\n        \\mathfrak{d}(x_1, x_2, \\ell) = \\frac{1}{2\\ell}\\log\\left(\\frac{1+\\ell\/(\\sqrt{1-x_2^2}-x_1)}{1-\\ell\/\\sqrt{1-x_2^2}+x_1}\\right),\n    \\]\n    \\[\n        \\frac{\\partial \\mathfrak{d}(x_1, x_2, \\ell)}{\\partial x_1} = \\frac{\\sqrt{1-d^2}(2x_1-\\ell)}{2(1-x_2^2-x_1^2)(1-x_2^2-(x_1-\\ell)^2)}.\n    \\]\n    And the derivative is positive in the domain we are considering.\n\n    We now prove step $3$. We want to show that $\\mathfrak{d}(\\sqrt{\\tilde{R}^2-x_2^2}, x_2, \\ell\\cdot)$ decreases with $\\ell$, within our constraints $\\ell \\leq x_1 = \\sqrt{\\tilde{R}^2-x_2^2}$, $0\\leq x_2\\leq \\tilde{R}$. If we split the segment joining $\\tilde{x}$ and $\\tilde{y}$ in half with, respect to the metric in $\\B$, we see that due to the monotonicity proved in step $1$, the segment that is farther to the origin is longer in $\\mathcal{M}$ than the other one and so $\\mathfrak{d}(\\cdot)$ is greater for this half of the segment than for the original one. In symbols, let $\\tilde{r}$ be the middle point of the segment joining $\\tilde{x}$ and $\\tilde{y}$. We have by monotonicity that $\\mathfrak{d}(x_1, x_2, \\ell\/2) \\geq \\mathfrak{d}(x_1, x_2-\\ell\/2, \\ell\/2)$. So $\\mathfrak{d}(x_1, x_2, \\ell\/2) = \\frac{d(\\tilde{x},\\tilde{r})}{\\ell\/2} \\geq \\frac{d(\\tilde{x},\\tilde{r}) + d(\\tilde{r}, \\tilde{y})}{\\ell}  = \\mathfrak{d}(x_1, x_2, \\ell)$. Thus,\n    \\begin{align*} \n     \\begin{aligned}\n        \\mathfrak{d}(x_1,x_2, \\ell) &\\leq \\lim_{\\ell\\to 0} \\mathfrak{d}(\\sqrt{\\tilde{R}^2-x_2^2},x_2, \\ell) \\\\\n        &= \\lim_{\\ell\\to 0}    \\frac{1}{2\\ell}\\log\\left(\\frac{1+\\ell\/\\left(\\sqrt{1-x_2^2}-\\sqrt{\\tilde{R}^2-x_2^2}\\right)}{1-\\ell\/\\left(\\sqrt{1-x_2^2}+\\sqrt{\\tilde{R}^2-x_2^2}\\right)}\\right) \\\\\n        & \\circled{1}[=] \\lim_{\\ell\\to 0}  \\frac{\\sqrt{1-x_2^2}}{1-\\tilde{R}^2-2\\ell\\sqrt{\\tilde{R}^2-x_2^2}+\\ell^2} \\\\\n        & = \\frac{\\sqrt{1-x_2^2}}{1-\\tilde{R}^2}.\n       \\end{aligned}\n    \\end{align*}\n        We used L'H\u00f4pital's rule for $\\circled{1}$. We can maximize the last the result of the limit by setting $x_2=0$ and obtain that for any two different $\\tilde{x}, \\tilde{y} \\in \\X$\n    \\[\n        \\frac{d(x, y)}{\\norm{\\tilde{x}-\\tilde{y}}} \\leq \\frac{1}{1-\\tilde{R}^2}= \\frac{1}{1-\\tanh^2(R)} = \\cosh^2(R).\n    \\]\n\n    The lower bound is similar, assume that $\\ell>x_1$ and define $\\tilde{r}$ as above. We assume again without loss of generality that $x_1\\geq \\ell-x_1$. Then\n    \\[\n        \\frac{d(x, r)+d(r, y)}{\\ell} \\geq \\frac{d(x, r)}{\\ell-x_1} \\iff \\frac{d(r, y)}{x_1} \\geq \\frac{d(x,r)}{\\ell-x_1}\n    \\] \n    and the latter is true by the monotonicity proved in step $1$. This means that we can also consider $\\ell \\leq x_1$. But this time, according to step $2$, we want $x_1$ to be the lowest possible, so it is enough to consider $x_1=\\ell$. Using step $1$ again, we obtain that the lowest value of $\\mathfrak{d}(\\cdot)$ can be bounded by the limit $\\lim_{\\ell\\to 0} \\mathfrak{d}(\\ell, x_2, \\ell)$ which using L'H\u00f4pital's rule in $\\circled{1}$ is\n    \\begin{align*} \n         \\begin{aligned}\n            \\mathfrak{d}(x_1,x_2, \\ell) &\\geq \\lim_{\\ell\\to 0} \\mathfrak{d}(\\ell,x_2, \\ell) \\\\\n             &= \\lim_{\\ell\\to 0}    \\frac{1}{2\\ell}\\log\\left(1+\\frac{2\\ell}{\\sqrt{1-x_2^2}-\\ell}\\right) \\\\\n             & \\circled{1}[=] \\lim_{\\ell\\to 0}  \\frac{\\frac{2(\\sqrt{1-x_2^2}-\\ell)+2\\ell}{(\\sqrt{1-x_2^2}-\\ell)^2}}{2(1+2\\ell\/(\\sqrt{1-x_2^2}-\\ell))} \\\\\n            & = \\frac{1}{\\sqrt{1-x_2^2}}.\n           \\end{aligned}\n        \\end{align*}\n       The expression is minimized at $x_2=0$ and evaluates to $1$.\n\\end{proof}\n\nThe proof of the corresponding lemma for the sphere is analogous, we add it for completeness.\n\n\\begin{lemma} \\label{lemma:distances_spherical} \n    Let $\\S=\\expon_{x_0}(\\bar{B}(0, R))$ be a subset of the sphere with constant sectional curvature $K=1$ and $R<\\frac{\\pi}{2}$. Let $x, y\\in\\S$ be two different points. Then, we have\n    \\[\n         \\cos^2(R) \\leq \\frac{d(x, y)}{\\norm{\\tilde{x}-\\tilde{y}}} \\leq 1.\n    \\]\n\\end{lemma}\n\n\n\\begin{proof}\n    We proceed in a similar way than with the hyperbolic case. We can also work with $d=2$ only, since $\\tilde{x}, \\tilde{y}$ and $\\tilde{x}_0$ lie on a plane. We parametrize a general pair of points as $\\tilde{x} = (x_1, x_2) \\in \\X$ and $y=(x_1-\\ell,x_2) \\in \\X$, so $x_1^2+x_2^2 \\leq \\tilde{R}^2$, for $\\tilde{R} = \\tan(R)$ and by definition $\\ell=\\norm{\\tilde{x}-\\tilde{y}}$. \n    \n    Let $\\mathfrak{d}(x_1,x_2,\\ell) \\defi d(x,y)\/\\norm{\\tilde{x}-\\tilde{y}}$. We proceed to prove the result in three steps, similarly to the hyperbolic case.\n\\begin{enumerate}\n    \\item If $x_1=\\ell$ then $\\mathfrak{d}(x_1,x_2,\\ell)$ decreases whenever $x_1$ increases. This allows to prove that it is enough to consider points in which $\\ell\\leq x_1$.\n    \\item $\\frac{\\partial \\mathfrak{d}(\\cdot)}{\\partial x_1} \\leq 0$, whenever $\\ell \\leq x_1$. So we can consider $x_1=\\sqrt{\\tilde{R}^2-x_2^2}$ only.\n    \\item With the constraints above, $\\mathfrak{d}(\\cdot)$ increases with $\\ell$, so in order to lower bound $\\mathfrak{d}(\\cdot)$ we can consider $\\lim_{\\ell\\to 0}\\mathfrak{d}(\\sqrt{\\tilde{R}-x_2},x_2, \\ell) = \\sqrt{1+x_2^2}\/(1+\\tilde{R}^2)$. This is minimized at $x_2=0$ and evaluates to $1\/(1+\\tilde{R}^2)$.\n\n\\end{enumerate}\n\nFor the first step, we compute the partial derivative:\n\\begin{equation}\\label{eq:partial_derivative_spherical_deformation}\n    \\frac{\\partial \\mathfrak{d}(x_1, x_2, x_1)}{\\partial x_1} = \\frac{x_1 \\sqrt{1 + x_2^2}\/(1 + x_1^2 + x_2^2) - \\arccos\\left(\\sqrt{(1 + x_2^2)\/(1 + x_1^2 + x_2^2)}\\right)}{x_1^2}.\n\\end{equation}\nIn order to see that it is non-positive, we compute the partial derivative of the denominator with respect to $x_2$ and obtain\n\\[\n    \\frac{2x_1^3 x_2}{\\sqrt{1+x_2^2}(1+x_1^2+x_2^2)} \\geq 0.\n\\]\nso in order to maximize \\eqref{eq:partial_derivative_spherical_deformation} we set $x_2=\\sqrt{\\tilde{R}-x_1^2}$. In that case, the numerator is\n\\begin{equation}\\label{eq:aux_eq_sph_def}\n\\frac{x_1 \\sqrt{1 + R^2 - x_1^2}}{1 + R^2} - \\arccos\\left(\\sqrt{\\frac{1 + R^2 - x_1^2}{1 + R^2}}\\right),\n\\end{equation}\nand its derivative with respect to $x_1$ is\n\\[\n    -\\frac{2 x_1^2}{(1 + R^2) \\sqrt{1 + R^2 - x_1^2}} \\leq 0.\n\\]\nand given that \\eqref{eq:aux_eq_sph_def} with $x_1=0$ evaluates to $0$ we conclude that \\eqref{eq:partial_derivative_spherical_deformation} is non-positive. Similarly to \\cref{lemma:distances_hyperbolic}, suppose the horizontal segment that joins $\\tilde{x}$ and $\\tilde{y}$ passes through $\\tilde{r} \\defi (0,x_2)$. And suppose without loss of generality that $d(x,r) \\geq d(r,y)$, i.e. $x_1 \\geq \\ell-x_1$. Then by the monotonicity we just proved, we have \n\\begin{equation}\\label{eq:aux_mathfrakd_functions}\n    \\frac{d(x,r)}{\\norm{\\tilde{x}-\\tilde{r}}}  = \\mathfrak{d}(x_1, x_2, x_1) \\leq \\mathfrak{d}(\\ell - x_1, x_2, \\ell-x_1) = \\frac{d(r,y)}{\\norm{\\tilde{r}-\\tilde{y}}}.\n\\end{equation}\nAnd this implies $\\mathfrak{d}(x_1, x_2, x_1) \\leq \\mathfrak{d}(x_1, x_2, \\ell)$. Indeed, that is equivalent to show\n\\[\n    \\frac{d(x,r)}{\\norm{\\tilde{x}-\\tilde{r}}} \\leq \\frac{d(x,y)}{\\norm{\\tilde{x}-\\tilde{y}}} = \\frac{d(x,r)+d(r+y)}{\\norm{\\tilde{x}-\\tilde{r}}+\\norm{\\tilde{r}-\\tilde{y}}}.\n\\]\nWhich is true, since after simplifying we arrive to \\eqref{eq:aux_mathfrakd_functions}. So in order to lower bound $\\mathfrak{d}(\\cdot)$, it is enough to consider $\\ell\\leq x_1$.\n\nWe focus on step $2$ now. We have\n\\[\n    \\frac{\\partial \\mathfrak{d}(x_1,x_2,\\ell)}{\\partial x_1}  = \\frac{\\sqrt{1+x_2^2}(\\ell-2 x_1)}{(1+x_2^2+(\\ell-x_1)^2)(1+x_2^2+x_1^2)}.\n\\]\nwhich is non-positive given the restrictions we imposed after step $1$. So in order to lower bound $\\mathfrak{d}(\\cdot)$ we can consider $x_1=\\sqrt{\\tilde{R}-x_2^2}$ only.\n\nFinally, in order to complete step $3$ we compute\n\\begin{align*}\n \\begin{aligned}\n     \\frac{\\partial \\mathfrak{d}(\\sqrt{\\tilde{R}-x_2^2},x_2,\\ell)}{\\partial \\ell} &= \\frac{\\sqrt{1+x_2^2}}{\\ell(1+\\tilde{R}^2) + \\ell^3-2\\ell^2\\sqrt{\\tilde{R}^2-x_2^2}} \\\\\n      &\\quad - \\frac{1}{\\ell^2}\\arccos\\left(\\frac{1+\\tilde{R}^2-\\ell\\sqrt{\\tilde{R}^2-x_2^2}}{\\sqrt{(1+\\tilde{R}^2)(1+\\tilde{R}^2+\\ell^2-2\\ell\\sqrt{\\tilde{R}^2-x_2^2})}}\\right)\n   \\end{aligned}\n\\end{align*}\n\nAnd in order to prove that this is non-negative, we will prove that the same expression is non-negative, when multiplied by $\\ell^2$. We compute the partial derivative of the aforementioned expression with respect to $\\ell$:\n\\begin{align*}\n \\begin{aligned}\n     \\frac{\\partial}{\\partial \\ell}\\left(\\frac{\\partial \\mathfrak{d}(\\sqrt{\\tilde{R}-x_2^2},x_2,\\ell)}{\\partial \\ell} \\ell^2\\right) = \\frac{2\\ell\\sqrt{1+x_2^2}(\\sqrt{\\tilde{R}^2-x_2^2}-\\ell)}{(1+\\tilde{R}^2+\\ell^2-2\\ell\\sqrt{\\tilde{R}^2-x_2^2})^2} \\geq 0.\n   \\end{aligned}\n\\end{align*}\n\nAnd $\\ell^2(\\partial \\mathfrak{d}(\\sqrt{\\tilde{R}-x_2^2},x_2,\\ell)\/\\partial \\ell )$ evaluated at $0$ is $0$ for all choices of parameters $R$ and $x_2$ in the domain. So we conclude that $\\partial \\mathfrak{d}(\\sqrt{\\tilde{R}-x_2^2},x_2,\\ell)\/\\partial \\ell \\geq 0$.\n\nThus, we can consider the limit when $\\ell\\to 0$ in order to lower bound $\\mathfrak{d}(\\cdot)$. In the defined domain, we have \n\\begin{align*}\n \\begin{aligned}\n     \\lim_{\\ell\\to 0}\\mathfrak{d}(\\sqrt{\\tilde{R}-x_2},x_2, \\ell) &= \\lim_{\\ell\\to 0}\\frac{1}{\\ell}\\arccos\\left(\\frac{1+\\tilde{R}^2-x\\sqrt{\\tilde{R}^2-x_2^2}}{\\sqrt{1+\\tilde{R}^2}\\sqrt{1+x_2^2+(\\ell-\\sqrt{\\tilde{R}^2-x_2^2})^2}}\\right)  \\\\\n     &\\circled{1}[=] \\lim_{\\ell\\to 0} \\frac{\\sqrt{1+x_2^2}}{1+\\tilde{R}^2+\\ell^2-2\\ell\\sqrt{\\tilde{R}^2-x_2^2}}\\\\\n     &=\\frac{\\sqrt{1+x_2^2}}{1+\\tilde{R}^2}.\n   \\end{aligned}\n\\end{align*}\nWe used L'H\u00f4pital's rule for $\\circled{1}$. Now, the right hand side of the previous expression is minimized at $x_2=0$ so we conclude that we have\n\\[\n    \\cos^2(R) = \\frac{1}{1+\\tan^2(R)} = \\frac{1}{1+\\tilde{R}^2} \\leq \\mathfrak{d}(x_1, x_2, \\ell) = \\frac{d(p,q)}{\\norm{\\tilde{p}-\\tilde{q}}}.\n\\]\n\n    The upper bound uses again a similar argument. Assume that $\\ell>x_1$ and define $\\tilde{r}$ as above. We assume again without loss of generality that $x_1\\geq \\ell-x_1$. Then\n    \\[\n        \\frac{d(x, r)+d(r, y)}{\\ell} \\leq \\frac{d(x, r)}{\\ell-x_1} \\iff \\frac{d(r, y)}{x_1} \\leq \\frac{d(x,r)}{\\ell-x_1}\n    \\] \n    and the latter is true by the monotonicity proved in step $1$. Consequently we can just consider the points that satisfy $\\ell \\leq x_1$. By step $2$, $\\mathfrak{d}(\\cdot)$ is maximal whenever $x_1$ is the lowest possible, so it is enough to consider $x_1=\\ell$. Using step $1$ again, we obtain that the greatest value of $\\mathfrak{d}(\\cdot)$ can be bounded by the limit $\\lim_{\\ell\\to 0} \\mathfrak{d}(\\ell, x_2, \\ell)$ which using L'H\u00f4pital's rule in $\\circled{1}$ and simplifying is\n    \\begin{align*} \n         \\begin{aligned}\n            \\mathfrak{d}(x_1,x_2, \\ell) &\\leq \\lim_{\\ell\\to 0} \\mathfrak{d}(\\ell,x_2, \\ell) \\\\\n             &= \\lim_{\\ell\\to 0}    \\frac{1}{\\ell}\\arccos\\left(\\sqrt{\\frac{1+x_2^2}{1+\\ell^2+x_2^2}}\\right) \\\\\n             & \\circled{1}[=] \\frac{1}{\\sqrt{1+x_2^2}}.\n           \\end{aligned}\n        \\end{align*}\n       The expression is maximized at $x_2=0$ and evaluates to $1$.\n\\end{proof}\n\n\\subsection{Angle deformation}\\label{app:sec_angle_deformation}\n\\begin{lemma}\\label{lemma:angle_deformation}\n    Let $\\M=\\H$ or $\\M=\\S$ and $K\\in \\{1, -1\\}$. Let $x,y\\in\\M$ be two different points and different from $x_0$. Let $\\tilde{\\alpha}$ be the angle $\\angle x_0xy$, formed by the vectors $x_0-x$ and $y-x$. Let $\\alpha$ be the corresponding angle between the vectors $\\expon_x^{-1}(x_0)$ and $\\expon_x^{-1}(y)$.  The following holds:\n\\begin{align*}\n\\begin{aligned}\n\\sin(\\alpha) = \\sin(\\tilde{\\alpha}) \\sqrt{\\frac{1+K\\norm{\\tilde{x}}^2}{1+K\\norm{\\tilde{x}}^2\\sin^2(\\tilde{\\alpha})}}, \\quad \\quad \\cos(\\alpha) = \\cos(\\tilde{\\alpha}) \\sqrt{\\frac{1}{1+K\\norm{\\tilde{x}}^2\\sin^2(\\tilde{\\alpha})}}.\n\\end{aligned}\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\n    Note that we can restrict ourselves to $\\alpha \\in [0, \\pi]$ because we have $\\widetilde{(-w)} = -\\tilde{w}$ (recall our \\hyperlink{sec:notation}{notation} about vectors with tilde). This means that the result for the range $\\alpha\\in[-\\pi, 0]$ can be deduced from the result for $-\\alpha$.\n\n    We start with the case $K=-1$. We can assume without loss of generality that the dimension is $d=2$, and that the coordinates of $\\tilde{x}$ are $(0,x_2)$, for $x_2\\leq \\tanh(R)$ that $\\tilde{y}=(y_1, y_2)$, for some $y_1\\leq 0$ and $\\tilde{\\delta} \\defi \\angle \\tilde{y}\\tilde{x}_0\\tilde{x} \\in [0, \\pi\/2]$, since we can make the distance $\\norm{\\tilde{x}-\\tilde{y}}$ as small as we want. Recall $\\tilde{x}_0=\\mathbf{0}$. We recall that $d(x, x_0) = \\arctanh(\\norm{\\tilde{x}})$ and we note that $\\sinh(\\arctanh(t)) = \\frac{t}{1-t^2}$, so that $\\sinh(d(x, x_0)) = \\norm{\\tilde{x}}\/\\sqrt{1-\\norm{\\tilde{x}}^2}$, for any $\\tilde{x}\\in\\B$. We will apply the hyperbolic and Euclidean law of sines \\cref{thm:law_of_sines} in order to compute the value of $\\sin(\\alpha)$ with respect to $\\tilde{\\alpha}$. Let $\\tilde{a}$ and $\\tilde{b}$ be points in the border of $\\B$ such that the segment joining $\\tilde{a}$ and $\\tilde{b}$ is a chord that contains $\\tilde{x}$ and $\\tilde{y}$ and $\\norm{\\tilde{a}-\\tilde{x}}\\leq\\norm{\\tilde{b}-\\tilde{y}}$. So $\\norm{\\tilde{a}-\\tilde{x}}$ and $\\norm{\\tilde{b}-\\tilde{y}}$ are $\\sqrt{1-\\norm{\\tilde{x}}^2\\sin(\\tilde{\\alpha})-d\\cos(\\tilde{\\alpha})}$ and $\\sqrt{1-\\norm{\\tilde{x}}^2\\sin(\\tilde{\\alpha})+d\\cos(\\tilde{\\alpha})}$, respectively. We have\n\\begingroup\n\\allowdisplaybreaks\n    \\begin{align*}\n \\begin{aligned}\n     \\sin(\\alpha) &\\circled{1}[=] \\frac{\\sinh(d(x_0, y))\\sin(\\tilde{\\delta})}{\\sinh(d(x, y))} \\\\\n     &\\circled{2}[=] \\frac{\\norm{\\tilde{x}_0-\\tilde{y}}}{\\sqrt{1-\\norm{\\tilde{x}_0-\\tilde{y}}^2}} \\cdot\\frac{\\norm{\\tilde{x}-\\tilde{y}} \\sin(\\tilde{\\alpha})}{\\norm{\\tilde{x}_0-\\tilde{y}}}\\cdot \\frac{1}{ \\sinh(d(x, y))}\\\\\n     &\\circled{3}[=] \\frac{\\sin(\\tilde{\\alpha})}{\\sqrt{1-\\norm{\\tilde{x}}^2+\\norm{\\tilde{x}-\\tilde{y}}(-2\\norm{\\tilde{x}}\\cos(\\tilde{\\alpha})+\\norm{\\tilde{x}-\\tilde{y}})}} \\cdot\\frac{\\norm{\\tilde{x}-\\tilde{y}}}{\\sinh(d(x, y))} \\\\\n     &\\circled{4}[=] \\frac{\\sin(\\tilde{\\alpha})}{\\sqrt{1-\\norm{\\tilde{x}}^2}} \\lim_{d(x, y)\\to 0} \\norm{\\tilde{x}-\\tilde{y}}\\frac{1}{\\sinh(d(x,y))} \\\\\n     &\\circled{5}[=] \\frac{\\sin(\\tilde{\\alpha})}{\\sqrt{1-\\norm{\\tilde{x}}^2}} \\lim_{d(x, y)\\to 0} \\frac{(e^{2d(x, y)}-1)(\\norm{\\tilde{a}-\\tilde{x}}\\cdot \\norm{\\tilde{b}-\\tilde{x}})}{e^{2d(x, y)}\\norm{\\tilde{a}-\\tilde{x}} + \\norm{\\tilde{b}-\\tilde{x}}}\\cdot \\frac{2e^{d(x, y)}}{e^{2d(x, y)}-1}\\\\\n     &=\\frac{\\sin(\\tilde{\\alpha})}{\\sqrt{1-\\norm{\\tilde{x}}^2}}\\cdot\\frac{2 \\norm{\\tilde{a}-\\tilde{x}} \\cdot \\norm{\\tilde{b}-\\tilde{x}}}{\\norm{\\tilde{a}-\\tilde{x}} + \\norm{\\tilde{b}-\\tilde{x}}} \\\\\n     &\\circled{6}[=]\\frac{\\sin(\\tilde{\\alpha})}{\\sqrt{1-\\norm{\\tilde{x}}^2}}\\cdot \\frac{2(1-\\norm{\\tilde{x}}^2\\sin^2(\\tilde{\\alpha})-\\norm{\\tilde{x}}^2\\cos^2(\\tilde{\\alpha}))}{2\\sqrt{1-\\norm{\\tilde{x}}^2\\sin^2(\\tilde{\\alpha})}} \\\\\n     &= \\sin(\\tilde{\\alpha})\\sqrt{\\frac{1-\\norm{\\tilde{x}}^2}{1-\\norm{\\tilde{x}}^2\\sin^2(\\tilde{\\alpha})}}.\n   \\end{aligned}\n\\end{align*}\n\\endgroup\n    In $\\circled{1}$ we used the hyperbolic sine theorem. In $\\circled{2}$ we used the expression above regarding segments that pass through the origin, and the Euclidean sine theorem. In $\\circled{3}$, we simplify and use that the coordinates of $\\tilde{y}$ are $(-\\sin(\\tilde{\\alpha})\\norm{\\tilde{x}-\\tilde{y}}, \\norm{\\tilde{x}}-\\cos(\\tilde{\\alpha})\\norm{\\tilde{x}-\\tilde{y}})$. Then, in $\\circled{4}$, since $\\sin(\\alpha)$ does not depend on $\\norm{\\tilde{x}-\\tilde{y}}$, we can take the limit when $d(x, y) \\to 0$, by which we mean we take the limit $\\tilde{y}\\to\\tilde{x}$ by keeping the angle $\\tilde{\\alpha}$ constant. Since a posteriori the limit of each fraction exists, we compute them one at a time. $\\circled{5}$ uses \\eqref{eq:distances_hyper_alt} and the definition of $\\sinh(d(x, y))$. In $\\circled{6}$ we substitute $\\norm{\\tilde{a}-\\tilde{x}}$ and $\\norm{\\tilde{b}-\\tilde{x}}$ by their values.\n\n    The spherical case is similar to the hyperbolic case. We also assume without loss of generality that the dimension is $d=2$. Define $\\tilde{y}$ as a point such that $\\angle \\tilde{x_0}\\tilde{x}\\tilde{y} = \\tilde{\\alpha} $. We can assume without loss of generality that the coordinates of $\\tilde{x}$ are $(0,x_2)$, that $\\tilde{y}=(y_1, y_2)$, for $y_1\\leq 0$, and $\\tilde{\\delta} \\defi \\angle \\tilde{y}\\tilde{x_0}\\tilde{x} \\in [0, \\pi\/2]$, since we can make the distance $\\norm{\\tilde{x}-\\tilde{y}}$ as small as we want. We recall that by \\eqref{eq:R_tilde_vs_R} we have $d(x_0,x) = \\arctan(\\norm{\\tilde{x}_0-\\tilde{x}})$ and we note that $\\sin(\\arctan(t)) = \\frac{t}{1+t^2}$, so that $\\sin(d(x_0, x)) = \\norm{\\tilde{x}_0-\\tilde{x}}\/\\sqrt{1+\\norm{\\tilde{x}_0-\\tilde{x}}^2}$, for any $\\tilde{x}\\in\\B$. We will apply the spherical and Euclidean law of sines \\cref{thm:law_of_sines} in order to compute the value of $\\sin(\\alpha)$ with respect to $\\tilde{\\alpha}$. We have\n\\begin{align*}\n \\begin{aligned}\n     \\sin(\\alpha) &\\circled{1}[=] \\frac{\\sin(d(x_0, y))\\sin(\\tilde{\\delta})}{\\sin(d(x, y))} \\\\\n     &\\circled{2}[=] \\frac{\\norm{\\tilde{x}_0-\\tilde{y}}}{\\sqrt{1+\\norm{\\tilde{x}_0-\\tilde{y}}^2}} \\cdot\\frac{\\norm{\\tilde{x}-\\tilde{y}} \\sin(\\tilde{\\alpha})}{\\norm{\\tilde{x}_0-\\tilde{y}}} \\frac{1}{ \\sin(d(x, y))}\\\\\n     &\\circled{3}[=] \\frac{\\sin(\\tilde{\\alpha})\\norm{\\tilde{x}-\\tilde{y}}}{\\sqrt{1+\\norm{\\tilde{x}_0-\\tilde{y}}^2}\\sqrt{1-\\frac{(1-\\norm{x}\\cos(\\tilde{\\alpha})\\norm{\\tilde{x}-\\tilde{y}}+\\norm{\\tilde{x}}^2)^2}{(1+\\norm{\\tilde{x}}^2)(1+\\norm{\\tilde{x}_0-\\tilde{y}}^2)}}}\\\\\n     &\\circled{4}[=] \\frac{\\sin(\\tilde{\\alpha})\\norm{\\tilde{x}-\\tilde{y}}}{\\sqrt{\\norm{\\tilde{x}-\\tilde{y}}^2(1+\\norm{\\tilde{x}}^2-\\norm{\\tilde{x}}^2\\cos(\\tilde{\\alpha}))\/(1+\\norm{\\tilde{x}}^2)}} \\\\\n     &\\circled{5}[=] \\sin(\\tilde{\\alpha})\\sqrt{\\frac{1+\\norm{\\tilde{x}}^2}{1+\\norm{\\tilde{x}}^2\\sin^2(\\tilde{\\alpha})}}.\n   \\end{aligned}\n\\end{align*}\n\nIn $\\circled{1}$ we used the spherical sine theorem. In $\\circled{2}$ we used the expression above regarding segments that pass through the origin, and the Euclidean sine theorem. In $\\circled{3}$, we use the fact that the coordinates of $\\tilde{y}$ are $(-\\sin(\\tilde{\\alpha})\\norm{\\tilde{x}-\\tilde{y}}, d-\\cos(\\tilde{\\alpha})\\norm{\\tilde{x}-\\tilde{y}})$, use the distance formula \\eqref{eq:distances} and the trigonometric inequality $\\sin(\\arccos(x)) = \\sqrt{1-x^2}$. Then, in $\\circled{4}$ and $\\circled{5}$, we multiply and simplify. \n\n    Finally, in both cases, the cosine formula is derived from the identity $\\sin^2(\\alpha)+\\cos^2(\\alpha)=1$ after noticing that the sign of $\\cos(\\alpha)$ and the sign of $\\cos(\\tilde{\\alpha})$ are the same. The latter fact can be seen to hold true by noticing that $\\alpha$ is monotonous with respect to $\\tilde{\\alpha}$ and  the fact that $\\tilde{\\alpha}=\\pi\/2$ implies $\\sin(\\alpha)=0$.\n\\end{proof}\n\n\\begin{fact}[Constant Curvature non-Euclidean Law of Sines]\\label{thm:law_of_sines}\n    Let $S_k(\\cdot)$ denote the special sine, defined as $S_K(t) = \\sin(\\sqrt{K}t)$ if $K>0$, $S_K(t) = \\sinh(\\sqrt{-K}t)$ if $K<0$ and $S_k(t)=t$ if $K=0$. Let $a, b,c$ be the lengths of the sides of a geodesic triangle defined in a manifold of constant sectional curvature. Let $\\alpha, \\beta, \\gamma$ be the angles of the geodesic triangle, that are opposite to the sides $a, b, c$. The following holds:\n    \\[\n        \\frac{\\sin(\\alpha)}{S_K(a)} =  \\frac{\\sin(\\beta)}{S_K(b)} = \\frac{\\sin(\\gamma)}{S_K(c)}.\n    \\] \n\\end{fact}\n    We refer to \\cite{greenberg1993euclidean} for a proof of this classical theorem.\n\n\\subsection{Gradient deformation and smoothness of $f$} \\label{sec:app_gradient_deformation_and_smoothness}\n\n\\cref{lemma:angle_deformation}, with $\\tilde{\\alpha} = \\pi\/2$, shows that $e_1 \\perp e_j$, for $j\\neq 1$. The rotational symmetry implies $e_i\\perp e_j$ for $i\\neq j$ and $i,j>1$. As in \\cref{lemma:deformations}, let $x\\in\\M$ be a point and assume without loss of generality that $\\tilde{x}\\in\\operatorname{span}\\{\\tilde{e}_1\\}$ and $\\nabla f(\\tilde{x})\\in \\operatorname{span}\\{\\tilde{e}_1, \\tilde{e}_2\\}$. It can be assumed without loss of generality because of the symmetries. So we can assume the dimension is $d=2$. Using \\cref{lemma:deformations} we obtain that $\\tilde{\\alpha}=0$ implies $\\alpha=0$. Also $\\tilde{\\alpha} = \\pi\/2$ implies $\\alpha=\\pi\/2$, so the adjoint of the differential of $h^{-1}$ at $x$, $(\\mathrm{d} h^{-1})^\\ast_x$ diagonalizes and has $e_1$ and $e_2$ as eigenvectors. We only need to compute the eigenvalues. The computation of the first one uses that the geodesic passing from $x_0$ and $x$ can be parametrized as $h^{-1}(\\tilde{x}_0+\\arctan(\\tilde{\\lambda}\\tilde{e}_1))$ if $K=1$ and $h^{-1}(\\tilde{x}_0+\\arctanh(\\tilde{\\lambda}\\tilde{e}_1))$ if $K=-1$, by \\eqref{eq:appendix_characterization_of_geodesic_map_and_metric}. The derivative of $\\arctan(\\cdot)$ or $\\arctanh(\\cdot)$ reveals that the first eigenvector, the one corresponding to $e_1$, is $1\/(1+K\\norm{\\tilde{x}^2})$, i.e. $\\nabla f(\\tilde{x})_1 = \\nabla F(x)_1 \/ (1+K\\norm{\\tilde{x}^2})$. For the second one, let $x=(x_1, 0)$ and $y=(y_1, y_2)$, with $y_1=x_1$ the second eigenvector results from the computation, for $K=-1$: \n\\begin{align*} \n \\begin{aligned}\n     \\lim_{y_2 \\to 0} \\frac{d(x,y)}{y_2}  &=\\lim_{y_2 \\to 0} \\frac{1}{2y_2} \\log\\left(1+\\frac{2y_2}{\\sqrt{1-x_1^2}-y_2}\\right) \\\\ \n    &\\circled{1}[=] \\lim_{y_2 \\to 0}  \\frac{\\frac{2}{\\sqrt{1-x_1^2}-y_2} + \\frac{2y_2}{(\\sqrt{1-x_1^2}-y_2)^2}}{2+\\frac{4y_2}{\\sqrt{1-x_1^2}-y_2}} \\\\\n    &= \\frac{1}{\\sqrt{1-x_1^2}},\n   \\end{aligned}\n\\end{align*}\nand for $K=1$: \n\\begin{align*} \n \\begin{aligned}\n     \\lim_{y_2 \\to 0} \\frac{d(x,y)}{y_2}  &=\\lim_{y_2 \\to 0} \\frac{1}{y_2} \\arccos\\left(\\frac{\\sqrt{1+x_1^2}}{\\sqrt{1 + x_1^2 + y_2^2}}\\right) \\\\ \n    &\\circled{2}[=] \\lim_{y_2 \\to 0} \\frac{\\sqrt{1 + x_1^2}}{1 + x_1^2 + y_2^2}\\\\\n    &= \\frac{1}{\\sqrt{1+x_1^2}}.\n   \\end{aligned}\n\\end{align*}\nSo, since $x_1=\\norm{\\tilde{x}}$, we have $\\nabla f(\\tilde{x})_2 = \\nabla F(x)_2\/\\sqrt{1+K\\norm{\\tilde{x}}^2}$ for $K\\in\\{1, -1\\}$. We used L'H\u00f4pital's rule in  $\\circled{1}$ and $\\circled{2}$. \n\nAlso note that if $v \\in T_x\\M$ is a vector normal to $\\nabla F(x)$, then $\\tilde{v}$ is normal to $\\nabla f(x)$. It is easy to see this geometrically: Indeed, no matter how $h$ changes the geometry, since it is a geodesic map, a geodesic in the direction of first-order constant increase of $F$ is mapped via $h$ to a geodesic in the direction of first-order constant increase of $f$. And the respective gradients must be perpendicular to all these directions. Alternatively, this can be seen algebraically. Suppose first $d=2$, then $v$ is proportional to $(\\nabla F(x)_2, -\\nabla F(x)_1) = (\\sqrt{1+K\\norm{\\tilde{x}}^2} \\nabla f(\\tilde{x})_2, -(1+K\\norm{\\tilde{x}}^2)\\nabla f(\\tilde{x}_1))$. And a vector $\\tilde{v}'$ normal to $\\nabla f(x)$ must be proportional to $(-\\nabla f(x)_2, \\nabla f(x)_1)$. Let $\\alpha$ be the angle formed by $v$ and $-e_1$, $\\tilde{\\alpha}$ the corresponding angle formed between $\\tilde{v}$ and $-\\tilde{e}_1$, and $\\tilde{\\alpha}'$ the angle formed by $\\tilde{v}'$ and $-\\tilde{e}_1$. Then we have, using the expression for the vectors proportional to $v$ and $\\tilde{v}'$:\n\\[\n    \\sin(\\alpha) = \\frac{-f(x)_2}{\\sqrt{\\nabla f(x)_2^2+(1+\\norm{x}^2)\\nabla f(x)_1^2}} \\text{ and } \\sin(\\tilde{\\alpha}') = \\frac{-f(x)_2}{\\sqrt{\\nabla f(x)_2^2+\\nabla f(x)_1^2}} \n\\] \nand an easy computation yields $\\sin(\\alpha) = \\sin(\\tilde{\\alpha}')\\sqrt{(1+K\\norm{\\tilde{x}^2})\/(1+K\\norm{\\tilde{x}^2}\\sin^2(\\tilde{\\alpha}'))}$, which after applying \\cref{lemma:angle_deformation} we obtain $\\sin(\\tilde{\\alpha}') = \\sin(\\tilde{\\alpha})$ from which we conclude that $\\tilde{\\alpha}' = \\tilde{\\alpha}$ given that the angles are in the same quadrant. So $\\tilde{v} \\perp \\nabla f(x)$. In order to prove this for $d\\geq 3$ one can apply the reduction \\eqref{claim:same_direction_when_projecting} to the case $d=2$ that we obtain in the next section.  \n\nCombining the results obtained so far in \\cref{app:geometric_results}, we can prove \\cref{lemma:deformations}. We continue by proving \\cref{lemma:smoothness_of_transformed_function}, which will generalize the computations we just performed, in order to analyze the Hessian of $f$ and provide smoothness. Then, in the next section, we combine the results in \\cref{lemma:deformations} to prove \\cref{prop:bounding_hyperplane}.\n\n\\begin{proof}[Proof of \\cref{lemma:deformations}]\n    The lemma follows from Lemmas \\ref{lemma:distances_hyperbolic}, \\ref{lemma:distances_spherical}, \\ref{lemma:angle_deformation} and the previous reasoning in this Section \\ref{sec:app_gradient_deformation_and_smoothness}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of \\cref{lemma:smoothness_of_transformed_function}]\n    We will compute the Hessian of $f=F\\circ h^{-1}$ and we will bound its spectral norm for any point $\\tilde{x}\\in\\B$. We can assume without loss of generality that $d=2$ and $\\tilde{x} = (\\tilde{\\ell}, 0)$, for $\\tilde{\\ell} > 0$ (the case $\\tilde{\\ell}=0$ is trivial), since there is a rotational symmetry with $e_1$ as axis. This means that by rotating we could align the top eigenvector of the Hessian at a point so that it is in $\\operatorname{span}\\{e_1, e_2\\}$. Let $\\tilde{y} = (y_1, y_2) \\in \\B$ be another point, with $y_1=\\tilde{\\ell}$. We can also assume that $y_2>0$ without loss of generality, because of our symmetry. Our approach will be the following. We know by \\cref{lemma:angle_deformation} and by the beginning of this section \\ref{sec:app_gradient_deformation_and_smoothness} that the adjoint of the differential of $h^{-1}$ at $y$, $(\\mathrm{d} h^{-1})^\\ast_y$ has $\\expon_y^{-1}(x_0)$ and a normal vector to it as eigenvectors. Their corresponding eigenvalues are $1\/(1+K\\norm{\\tilde{y}}^2)$ and $1\/\\sqrt{1+K\\norm{\\tilde{y}}^2}$, respectively. Consider the basis of $T_x\\M$ $\\{e_1, e_2\\}$ as defined at the beginning of this section, i.e. where $e_1$ is a unit vector proportional to $-\\expon_x^{-1}(x_0)$ and $e_2$ is the normal vector to $e_1$ that makes the basis orthonormal. Consider this basis being transported to $y$ using parallel transport and denote the result $\\{v_y, v_y^\\top\\}$. Assume we have the gradient $\\nabla F(y)$ written in this basis. Then we can compute the gradient of $f$ at $y$ by applying $(\\mathrm{d}h^{-1})^\\ast_y$. In order to do that, we compose the change of basis from $\\{v_y, v_y^\\top\\}$ to the basis of eigenvectors of $(\\mathrm{d} h^{-1})^\\ast_y$, then we apply a diagonal transformation given by the eigenvalues and finally we change the basis to $\\{\\tilde{e}_1, \\tilde{e}_2\\}$. Once this is done, we can differentiate with respect to $y_2$ in order to compute a column of the Hessian.  Let $\\tilde{\\alpha}$ be the angle formed by the vectors $\\tilde{y}$ and $\\tilde{x}$. Note that $\\tilde{\\alpha} = \\arctan(y_2\/y_1)$. Let $\\tilde{\\gamma}$ be the angle formed by the vectors $(\\tilde{y}-\\tilde{x})$ and $-\\tilde{y}$. That is, the angle $\\tilde{\\gamma}=\\pi-\\angle \\tilde{x} \\tilde{y} \\tilde{x}_0$. Since $v_y^\\top$ is the parallel transport of $e_2^\\top$, the angle between $v_y^\\top$ and the vector $\\expon_y^{-1}(x_0)$ is $\\gamma$. Note we use the same convention as before for the angles, i.e. $\\gamma$ is the corresponding angle to $\\tilde{\\gamma}$, meaning that if $\\gamma$ is the angle between two intersecting geodesics in $\\M$, then $\\tilde{\\gamma}$ is the angle between the respective corresponding geodesics in $\\B$. Note the first change of basis is a rotation and that the angle of rotation is $\\gamma-\\pi\/2$. The last change of basis is a rotation with angle equal to the angle formed by a vector $\\tilde{v}$  normal to $-\\tilde{y}$ ( $\\tilde{v}$ is the one such that $-\\tilde{y}\\times \\tilde{v} >0$) and the vector $\\tilde{e}_2$. It is easy to see that this vector is equal to $\\tilde{\\alpha}$. So we have\n\\begin{equation}\\label{eq:equality_gradients}\n    \\nabla f(y) = \n\\begin{pmatrix}\n\\cos(\\tilde{\\alpha}) & -\\sin(\\tilde{\\alpha}) \\\\ \\sin(\\tilde{\\alpha}) & \\cos(\\tilde{\\alpha})\n\\end{pmatrix}\n\\begin{pmatrix}\n    \\frac{1}{1+K(y_1^2+y_2^2)} & 0   \\\\ 0 & \\frac{1}{\\sqrt{1+K(y_1^2+y_2^2)}}\n\\end{pmatrix}\n\\begin{pmatrix}\n    \\sin(\\gamma) & -\\cos(\\gamma) \\\\ \\cos(\\gamma) & \\sin(\\gamma)\n\\end{pmatrix}\n\\nabla F(y)\n\\end{equation}\n\nWe want to take the derivative of this expression with respect to $y_2$ and we want to evaluate it at $y_2=0$. Let the matrices above be $A$, $B$ and $C$ so that $\\nabla f(y) = ABC\\nabla F(y)$. Using \\cref{lemma:angle_deformation} we have\n\\begin{align} \\label{eq:aux_comp_angle_gamma}\n   \\begin{aligned}\n       \\sin(\\gamma) &= \\sin(\\tilde{\\gamma})\\sqrt{\\frac{1+K(y_1^2+y_2^2)}{1+K(y_1^2+y_2^2)\\sin^2(\\tilde{\\gamma})}} \\circled{1}[=] \\cos(\\tilde{\\alpha})\\sqrt{\\frac{1+K(y_1^2+y_2^2)}{1+K(y_1^2+y_2^2)\\cos^2(\\tilde{\\alpha})}}, \\\\\n       \\cos(\\gamma) &= -\\sin(\\tilde{\\alpha})\\sqrt{\\frac{1}{1+K(y_1^2+y_2^2)\\cos^2(\\tilde{\\alpha})}},\n   \\end{aligned}\n\\end{align}\nwhere $\\circled{1}$ follows from $\\sin(\\tilde{\\gamma}) = \\sin(\\tilde{\\alpha}+\\pi\/2) = \\cos(\\tilde{\\alpha})$. Now we can easily compute some quantities\n\\[\n    \\left.A\\right|_{y_2=0} = I,   \\left.B\\right|_{y_2=0} = \n\\begin{pmatrix}\n    \\frac{1}{1+Ky_1^2} & 0 \\\\ 0 & \\frac{1}{\\sqrt{1+Ky_1^2}}\n\\end{pmatrix},\n\\left.C\\right|_{y_2=0} = I,\n\\]        \n\\[ \n        \\left.\\frac{\\partial A}{\\partial y_2}\\right|_{y_2=0} = \\left.\\frac{\\partial \\tilde{\\alpha}}{\\partial y_2}\\right|_{y_2=0} \\cdot\n\\begin{pmatrix}\n    0 & -1 \\\\ 1 & 0\n\\end{pmatrix}\n\\circled{1}[=]\n\\begin{pmatrix}\n    0 & \\frac{-1}{y_1} \\\\ \\frac{1}{y_1} & 0\n\\end{pmatrix}\n,\n\\]\n\\[\n        \\left.\\frac{\\partial B}{\\partial y_2}\\right|_{y_2=0} =\n\\left.\n\\begin{pmatrix}\n    \\frac{2Ky_2}{(1+K(y_1^2+y_2^2))^2} & 0 \\\\ 0 & \\frac{2Ky_2}{2(1+K(y_1^2+y_2^2))^{3\/2}} \n\\end{pmatrix}\n\\right|_{y_2=0} =\n\\begin{pmatrix}\n    0 & 0 \\\\ 0 & 0\n\\end{pmatrix}\n,\n\\]\n\\[\n    \\left.\\frac{\\partial C}{\\partial y_2}\\right|_{y_2=0} \\circled{2}[=]\n\\begin{pmatrix}\n    0 & \\frac{1}{y_1\\sqrt{1+Ky_1^2}} \\\\ \\frac{-1}{y_1\\sqrt{1+Ky_1^2}} & 0\n\\end{pmatrix}\n.\n\\]\nEqualities $\\circled{1}$ and $\\circled{2}$ follow after using \\eqref{eq:aux_comp_angle_gamma}, $\\tilde{\\alpha} = \\arctan(\\frac{y_2}{y_1})$ and taking derivatives. Now we differentiate \\eqref{eq:equality_gradients} with respect to $y_2$ and evaluate to $y_2=0$ using the chain rule. The result is \n\\begin{align*}\n   \\begin{aligned}\n\\begin{pmatrix}\n     \\nabla^2 f(\\tilde{x})_{12} \\\\ \\nabla^2 f(\\tilde{x})_{22} \n\\end{pmatrix}\n        &= \\left.\\left(\\frac{\\partial A}{\\partial y_2} BC\\nabla F(x) + A\\frac{\\partial B}{\\partial y_2} C\\nabla F(x) +AB\\frac{\\partial C}{\\partial y_2} \\nabla F(x) + ABC\\frac{\\partial \\nabla F(x)}{\\partial y_2}  \\right)\\right|_{y_2=0} \\\\\n        &=\n\\begin{pmatrix}\n    \\frac{-\\nabla f(\\tilde{x})_{2}}{y_1\\sqrt{1+Ky_1^2}}  \\\\ \\frac{\\nabla f(\\tilde{x})_{1}}{y_1(1+Ky_1^2)}\n\\end{pmatrix}\n+\n\\begin{pmatrix}\n    0 \\\\ 0\n\\end{pmatrix}\n+\n\\begin{pmatrix}\n \\frac{\\nabla f(\\tilde{x})_{2}}{y_1(1+Ky_1^2)^{3\/2}}  \\\\ \\frac{-\\nabla f(\\tilde{x})_{1}}{y_1(1+Ky_1^2)} \n\\end{pmatrix}\n+\n\\begin{pmatrix}\n   \\frac{\\nabla^2 F(x)_{12}}{(1+Ky_1^2)^{3\/2}}  \\\\ \\frac{\\nabla^2 F(x)_{22}}{1+Ky_1^2} \n\\end{pmatrix}\n   \\end{aligned}\n\\end{align*}\nComputing the other column of the Hessian is easier. We can just consider \\eqref{eq:equality_gradients} with $\\tilde{\\alpha} = 0$ and $\\gamma=\\pi\/2$ and vary $y_1$. Taking derivatives with respect to $y_1$ gives\n\\[\n\\begin{pmatrix}\n    \\nabla^2 f(\\tilde{x})_{11}   \\\\ \\nabla^2 f(\\tilde{x})_{21}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n   \\frac{-2Ky_1\\nabla f(\\tilde{x})_{1}}{(1+Ky_1^2)^{2}}  \\\\ \\frac{-Ky_1\\nabla f(\\tilde{x})_{2}}{(1+Ky_1^2)^{3\/2}} \n\\end{pmatrix}\n+\n\\begin{pmatrix}\n    \\frac{\\nabla^2 F(x)_{11}}{(1+Ky_1^2)^{2}} \\\\ \\frac{\\nabla^2 F(x)_{21}}{(1+Ky_1^2)^{3\/2}} \n\\end{pmatrix}\n.\n\\]\nNote in the computations of both of the columns of the Hessian we have used \n\\[\n    \\frac{\\partial{\\nabla F(y)_i}}{\\partial y_1} = \\nabla F(x)_{i1} \\cdot \\frac{1}{1+Ky_1^2} \\quad \\text{ and } \\quad\\left.\\frac{\\partial{\\nabla F(y)_i}}{\\partial y_2}\\right|_{y_2=0} = \\nabla F(x)_{i2} \\cdot \\frac{1}{\\sqrt{1+Ky_1^2}},\n\\]\nfor $i=1,2$. The eigenvalues of the adjoint of the differential of $h^{-1}$ appear as a factor because we are differentiating with respect to the geodesic in $\\B$ which moves at a different speed than the corresponding geodesic in $\\M$. Note as well, as a sanity check, that the cross derivatives are equal, since \n    \\[\n    -\\frac{1}{y_1\\sqrt{1+Ky_1^2}} + \\frac{1}{y_1(1+Ky_1^2)^{3\/2}} = \\frac{1}{y_1\\sqrt{1+Ky_1^2}}\\left(-1 + \\frac{1}{1+Ky_1^2}\\right) = \\frac{-Ky_1}{(1-y_1^2)^{3\/2}}.\n    \\] \n    Finally, we bound the new smoothness constant $\\tilde{L}$ by bounding the spectral norm of this Hessian. First note that using $y_1=\\tilde{\\ell}$ we have that $\\frac{1}{\\sqrt{1+K\\tilde{\\ell}^2}} = C_K(\\ell)$ and then for $K=-1$ it is $\\tilde{\\ell} = \\tanh(\\ell)$ and for $K=1$ it is $\\tilde{\\ell} = \\tan(\\ell)$, where $\\ell = d(x, x_0) < R$. We have that since there is a point $x^\\ast \\in \\M$ such that $\\nabla F(x^\\ast) = 0$ and $F$ is $L$-smooth, then it is $\\norm{\\nabla F(x)} \\leq 2LR$. Similarly, by $L$-smoothness $\\abs{\\nabla^2 F(x)_{ij}} \\leq L$. We are now ready prove smoothness:\n\\begin{align*} \n \\begin{aligned}\n     \\tilde{L}^2& \\leq\\norm{\\nabla^2 f(\\tilde{x})}_2^2 \\\\\n     & \\leq \\norm{\\nabla^2 f(\\tilde{x})}_F^2 = \\nabla^2 f(\\tilde{x})_{11}+2\\nabla^2 f(\\tilde{x})_{12}+\\nabla^2 f(\\tilde{x})_{22} \\\\\n     &\\leq L^2([C_K^4(R)+4R S_K(R)C_K^3(R)]^2+2[C_K^3(R) +2RS_K(R)C_K^2(R)]^2+C_K^4(R))\n   \\end{aligned}\n\\end{align*}\nand this can be bounded by $ 44 L^2 \\max\\{1,R^2\\}$ if $K=1$ and $44 L^2 \\max\\{1,R^2\\} C_K^8(R)$ if $K=-1$. In any case, it is $O(L^2)$.\n\\end{proof}\n\n\\subsection{Proof of \\cref{prop:bounding_hyperplane}} \\label{app:sec_proof_of_bounding_hyperplane}\n\\begin{proof}[Proof of \\cref{prop:bounding_hyperplane}]\n\nAssume for the moment the dimension is $d=2$. We can assume without loss of generality that $\\tilde{x} = (\\tilde{\\ell}, 0)$. We are given two vectors, that are the gradients $\\nabla F(x)$, $\\nabla f(\\tilde{x})$ and a vector $w\\in T_x\\M$. Let $\\tilde{\\delta}$ be the angle between $\\tilde{w}$ and $-\\tilde{x}$. Let $\\delta$ be the corresponding angle, i.e. the angle between $w$ and $u \\defi \\expon_x^{-1}(x_0)$. Let $\\alpha$ be the angle in between $\\nabla F(x)$ and $u$. Let $\\tilde{\\beta}$ be the angle in between $\\nabla f(\\tilde{x})$ and $-x$. $\\tilde{\\alpha}$ and $\\beta$ are defined similarly. We claim\n\\begin{equation} \\label{inner_product_comparison}\n    \\frac{\\innp{\\frac{\\nabla F(x)}{\\norm{\\nabla F(x)}}, \\frac{w}{\\norm{w}}}}{\\innp{\\frac{\\nabla f(\\tilde{x})}{\\norm{\\nabla f(\\tilde{x})}}, \\frac{\\tilde{w}}{\\norm{\\tilde{w}}}}} = \\sqrt{\\frac{1+K\\tilde{\\ell}^2}{(1+K\\tilde{\\ell}^2 \\sin^2(\\tilde{\\delta}))(1+K\\tilde{\\ell}^2 \\cos^2(\\tilde{\\beta}))}}.\n\\end{equation}\n\n    Let's see how to arrive to this expression. By \\cref{lemma:deformations}.c) we have\n    \\begin{equation}\\label{eq:relationship_tangents_gradients}\n        \\tan(\\alpha) = \\frac{\\tan(\\tilde{\\beta})}{\\sqrt{1+K\\tilde{\\ell}^2}}.\n    \\end{equation}\n    From this relationship we can deduce\n    \\begin{equation} \\label{eq:relationship_cosines_gradients}\n        \\cos(\\alpha)  = \\cos(\\tilde{\\beta}) \\sqrt{\\frac{1+K\\tilde{\\ell}^2}{1+K\\tilde{\\ell}^2\\cos^2(\\tilde{\\beta})}}.\n    \\end{equation}\n    This comes from squaring \\eqref{eq:relationship_tangents_gradients}, reorganizing terms and noting that $\\operatorname{sign}(\\cos(\\alpha)) =\\operatorname{sign}(\\cos(\\tilde{\\beta}))$ which is implied by \\cref{lemma:deformations}.c). We are now ready to prove the claim \\eqref{inner_product_comparison} (for $d=2$). We have  \n\\begin{align*} \n   \\begin{aligned}\n       \\frac{\\innp{\\frac{\\nabla F(x)}{\\norm{\\nabla F(x)}}, \\frac{w}{\\norm{w}}}}{\\innp{\\frac{\\nabla f(\\tilde{x})}{\\norm{\\nabla f(\\tilde{x})}}, \\frac{\\tilde{w}}{\\norm{\\tilde{w}}}}} &= \\frac{\\cos(\\alpha-\\delta)}{\\cos(\\tilde{\\beta}-\\tilde{\\delta})} \\\\\n       &\\circled{2}[=] \\frac{\\cos(\\delta)+\\tan(\\alpha)\\sin(\\delta)}{\\cos(\\tilde{\\beta})\\cos(\\tilde{\\delta})+\\sin(\\tilde{\\beta})\\sin(\\tilde{\\delta})}\\cos(\\alpha) \\\\\n       &\\circled{3}[=] \\frac{\\frac{\\cos(\\tilde{\\delta})}{\\sqrt{1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta})}}+\\frac{\\tan(\\tilde{\\beta})}{\\sqrt{1+K\\tilde{\\ell}^2}}\\frac{\\sin(\\tilde{\\delta)}\\sqrt{1+K\\tilde{\\ell}^2}}{\\sqrt{1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta})}}}{\\cos(\\tilde{\\beta})\\cos(\\tilde{\\delta})+\\sin(\\tilde{\\beta})\\sin(\\tilde{\\delta})}\\cos(\\tilde{\\beta}) \\sqrt{\\frac{1+K\\tilde{\\ell}^2}{1+K\\tilde{\\ell}^2\\cos^2(\\tilde{\\beta})}} \\\\\n       &\\circled{4}[=] \\sqrt{\\frac{1+K\\tilde{\\ell}^2}{(1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta}))(1+K\\tilde{\\ell}^2\\cos^2(\\tilde{\\beta}))}}. \\\\\n   \\end{aligned}\n\\end{align*}\n    Equality $\\circled{1}$ follows by the definition of $\\alpha, \\delta, \\tilde{\\delta}$ and $\\tilde{\\beta}$. In $\\circled{2}$, we used trigonometric identities. In $\\circled{3}$ we used \\cref{lemma:angle_deformation}, \\eqref{eq:relationship_tangents_gradients} and \\eqref{eq:relationship_cosines_gradients}. By reordering the expression, the denominator cancels out with a factor of the numerator in $\\circled{4}$. \n    \n\nIn order to work with arbitrary dimension, we note it is enough to prove it for $d=3$, since in order to bound\n\\[\n\\frac{\\innp{\\frac{\\nabla F(x)}{\\norm{\\nabla F(x)}}, \\frac{v}{\\norm{v}}}}{\\innp{\\frac{\\nabla f(\\tilde{x})}{\\norm{\\nabla f(\\tilde{x})}}, \\frac{\\tilde{v}}{\\norm{\\tilde{v}}}}},\n\\] \nit is enough to consider the following submanifold\n\\[\n  \\M' \\defi  \\expon_x(\\operatorname{span}\\{v, \\expon_x^{-1}(x_0), \\nabla F(x)\\}).\n\\]\n    for an arbitrary vector $v\\in T_x\\M$ and a point $x$ defined as above. The case $d=3$ can be further reduced to the case $d=2$ in the following way. Suppose $\\M'$ is a three dimensional manifold (if it is one or two dimensional there is nothing to do). Define the following orthonormal basis of $T_x\\M$, $\\{e_1, e_2, e_3\\}$ where $e_1=-\\expon_x^{-1}(x_0)\/\\norm{\\expon_x^{-1}(x_0)}$, $e_2$ is a unit vector, normal to $e_1$ such that $e_2\\in\\operatorname{span}\\{e_1, \\nabla F(x)\\}$. And $e_3$ is a vector that completes the orthonormal basis. In this basis, let $v$ be parametrized by $\\norm{v}(\\sin(\\delta), \\cos(\\nu)\\cos(\\delta), \\sin(\\nu)\\cos(\\delta))$, where $\\delta$ can be thought as the angle between the vector $v$ and its projection onto the plane $\\operatorname{span}\\{e_2, e_3\\}$ and $\\nu$ can be thought as the angle between this projection and its projection onto $e_2$. Similarly we parametrize $\\tilde{v}$ by $\\norm{\\tilde{v}}(\\sin(\\tilde{\\delta}), \\cos(\\tilde{\\nu})\\cos(\\tilde{\\delta}), \\sin(\\tilde{\\nu})\\cos(\\tilde{\\delta}))$, where the base used is the analogous base to the previous one, i.e. The vectors $\\{\\tilde{e}_1, \\tilde{e}_2, \\tilde{e}_3\\}$. Taking into account that $e_2 \\perp e_1$, $e_3\\perp e_1$, $\\tilde{e}_2 \\perp \\tilde{e}_1$, $\\tilde{e}_3\\perp \\tilde{e}_1$, and the fact that $e_1$ is parallel to $-\\expon_x(x_0)$, by the radial symmetry of the geodesic map we have that $\\nu = \\tilde{\\nu}$. Also, by looking at the submanifold $\\expon_x(\\operatorname{span}\\{e_1,v\\})$ and using \\cref{lemma:angle_deformation} we have \n\\[\n    \\sin(\\delta) = \\sin(\\tilde{\\delta})\\sqrt{\\frac{1+K\\tilde{\\ell}^2}{1+K\\tilde{\\ell}^2\\sin(\\tilde{\\delta})}}.\n\\]\n    If we want to compare $\\innp{\\nabla F(x), v}$ with $\\innp{\\nabla f(\\tilde{x}), \\tilde{v}}$ we should be able to just zero out the third components of $v$ and $\\tilde{v}$ and work in $d=2$. But in order to completely obtain a reduction to the two-dimensional case we studied a few paragraphs above, we would need to prove that if we call $w\\defi(\\sin(\\delta), \\cos(\\nu)\\cos(\\delta), 0)$ the vector $v$ with the third component made $0$, then $\\tilde{w}$ is in the same direction of the vector $\\tilde{v}$, when the third component is made $0$. The norm of these two vectors will not be the same, however. Let $\\tilde{w}'=(\\sin(\\tilde{\\delta}), \\cos(\\nu) \\cos(\\tilde{\\delta}), 0)$ be the vector $\\tilde{v}$ when the third component is made $0$. Then \n\\begin{equation} \\label{eq:norms_of_projections_from_3d_vectors}\n    \\norm{w} = \\norm{v} \\sqrt{\\sin^2(\\delta)+\\cos^2(\\delta)\\cos^2(\\nu)} \\text{ and } \\norm{\\tilde{w}'} = \\norm{\\tilde{v}}\\sqrt{\\sin^2(\\tilde{\\delta}) + \\cos^2(\\tilde{\\delta})\\cos^2(\\nu)}.\n\\end{equation}\nBut indeed, we claim \n    \\begin{equation}\\label{claim:same_direction_when_projecting}\n    \\tilde{w} \\text{ and } \\tilde{w}' \\text{ have the same direction.}\n\\end{equation}\n     This is easy to see geometrically: since we are working with a geodesic map, the submanifolds $\\expon_x(\\operatorname{span}\\{v, e_3\\})$ and $\\expon_x(\\operatorname{span}\\{e_1, e_2\\})$ contain $w$. Similarly the submanifolds $x+\\operatorname{span}\\{\\tilde{v}, \\tilde{e}_3\\}$ and $x+\\operatorname{span}\\{\\tilde{e}_1, \\tilde{e}_2\\}$ contain $\\tilde{w}'$. If the intersections of each of these pair of manifolds is a geodesic then the geodesic map must map one intersection to the other one, implying $\\tilde{w}$ is proportional to $\\tilde{w}'$. If the intersections are degenerate the case is trivial. Alternatively, one can prove this fact algebraically after some computations. If we call $\\delta^\\ast$ and $\\tilde{\\delta}'$ the angles formed by, respectively, the vectors $e_2$ and $w$, and the vectors $\\tilde{e}_2$ and $\\tilde{w}'$, then we have $\\tilde{w}'$ is proportional to $\\tilde{w}$ iff $\\tilde{\\delta}'=\\tilde{\\delta}^\\ast$, or equivalently $\\delta'=\\delta^\\ast$. Using the definitions of $w$ and $\\tilde{w}'$ we have\n\\begin{align*} \n   \\begin{aligned}\n       \\sin(\\delta^\\ast) = \\sin\\left(\\arctan\\left(\\frac{\\sin(\\delta)}{\\cos(\\nu)\\cos(\\delta)}\\right)\\right) = \\frac{\\tan(\\delta)\/\\cos(\\nu)}{(\\tan(\\delta)\/\\cos(\\nu))^2+1} \\\\\n       = \\frac{\\sin(\\delta)}{\\sqrt{\\sin^2(\\delta)+\\cos^2(\\nu)\\cos^2(\\delta)}},\n   \\end{aligned}\n\\end{align*}\nand analogously\n\\begin{align} \\label{eq:aux_sine_of_angle_3d}\n   \\begin{aligned}\n       \\sin(\\tilde{\\delta}') = \\sin\\left(\\arctan\\left(\\frac{\\sin(\\tilde{\\delta})}{\\cos(\\nu)\\cos(\\tilde{\\delta})}\\right)\\right) = \\frac{\\tan(\\tilde{\\delta})\/\\cos(\\nu)}{(\\tan(\\tilde{\\delta})\/\\cos(\\nu))^2+1} \\\\\n       = \\frac{\\sin(\\tilde{\\delta})}{\\sqrt{\\sin^2(\\tilde{\\delta})+\\cos^2(\\nu)\\cos^2(\\tilde{\\delta})}}.\n   \\end{aligned}\n\\end{align}\nUsing \\cref{lemma:angle_deformation} for the pairs $\\delta'$, $\\tilde{\\delta}'$ and $\\delta^\\ast$, $\\tilde{\\delta}^\\ast$, and the equations above we obtain\n\\begin{align*} \n   \\begin{aligned}\n       \\sin(\\delta^\\ast) = \\frac{\\sin(\\tilde{\\delta})\\sqrt{\\frac{1+K\\tilde{\\ell}^2}{1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta})}}}{\\sqrt{\\sin^2(\\tilde{\\delta})\\frac{1+K\\tilde{\\ell}^2}{1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta})}+\\cos^2(\\nu)\\frac{\\cos^2(\\tilde{\\delta})}{1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta})}}} = \\frac{\\sin(\\tilde{\\delta})\\sqrt{1+K\\tilde{\\ell}^2}}{\\sqrt{\\sin^2(\\tilde{\\delta})(1+K\\tilde{\\ell}^2)+\\cos^2(\\nu)\\cos^2(\\tilde{\\delta})}},\n   \\end{aligned}\n\\end{align*}\nand \n\\begin{align*} \n   \\begin{aligned}\n       \\sin(\\delta') = \\frac{\\sin(\\tilde{\\delta})}{\\sqrt{\\sin^2(\\tilde{\\delta})+\\cos^2(\\nu)\\cos^2(\\tilde{\\delta})}}\\sqrt{\\frac{1+K\\tilde{\\ell}^2}{1+K\\tilde{\\ell}^2\\left( \\frac{\\sin^2(\\tilde{\\delta})}{\\sin^2(\\tilde{\\delta})+\\cos^2(\\nu)\\cos^2(\\tilde{\\delta})}\\right)}\n},\n   \\end{aligned}\n\\end{align*}\nThe two expressions on the right hand side are equal. This implies $\\sin(\\delta') = \\sin(\\delta^\\ast)$. Since the angles were in the same quadrant we have $\\delta' =\\delta^\\ast$ by checking in which sectors the angles must be.\n\nWe can now come back to the study of $\\frac{\\innp{\\nabla F(x), v}}{\\innp{\\nabla f(\\tilde{x}), \\tilde{v}}}$. By  \\eqref{eq:norms_of_projections_from_3d_vectors} we have\n\\begin{align*} \n   \\begin{aligned}\n   \\frac{\\innp{\\nabla F(x), v}}{\\innp{\\nabla f(\\tilde{x}), \\tilde{v}}} = \\frac{\\norm{\\nabla F(x)}}{\\norm{\\nabla f(\\tilde{x})}}\\frac{\\norm{v}}{\\norm{\\tilde{v}}}\\frac{\\innp{\\frac{\\nabla F(x)}{\\norm{\\nabla F(x)}}, \\frac{w}{\\norm{w}}}}{\\innp{\\frac{\\nabla f(\\tilde{x})}{\\norm{\\nabla f(\\tilde{x})}}, \\frac{\\tilde{w}}{\\norm{\\tilde{w}}}}} \\frac{\\sqrt{\\sin^2(\\delta)+\\cos^2(\\delta)\\cos^2(\\nu)}}{\\sqrt{\\sin^2(\\tilde{\\delta})+\\cos^2(\\tilde{\\delta})\\cos^2(\\nu)}} \\end{aligned}\n\\end{align*}\nThe last two factors can be simplified. Using \\eqref{inner_product_comparison} and \\eqref{eq:norms_of_projections_from_3d_vectors} we get that this product is equal to\n\\begin{align*} \n   \\begin{aligned}\n       \\sqrt{\\frac{1+K\\tilde{\\ell}^2}{(1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta}^\\ast))(1+K\\tilde{\\ell}^2\\cos^2(\\tilde{\\beta}))}} \\frac{\\sqrt{\\sin^2(\\tilde{\\delta})\\frac{1+K\\tilde{\\ell}^2}{(1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta}))} +\\cos^2(\\nu)\\frac{\\cos^2(\\tilde{\\delta})}{1+K\\tilde{\\ell}^2\\sin(\\tilde{\\delta})}}}{\\sin^2(\\tilde{\\delta})+\\cos^2(\\tilde{\\delta})\\cos^2(\\nu)}\n   \\end{aligned}\n\\end{align*}\nwhich after using \\eqref{eq:aux_sine_of_angle_3d} (recall $\\tilde{\\delta}^\\ast = \\tilde{\\delta}'$), and simplifying it yields\n\\[\n\\sqrt{\\frac{1+K\\tilde{\\ell}^2}{(1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta}))(1+K\\tilde{\\ell}^2\\cos^2(\\tilde{\\beta}))}}.\n\\] \nSo finally we have\n\\[\n\\frac{\\innp{\\nabla F(x), v}}{\\innp{\\nabla f(\\tilde{x}), \\tilde{v}}} = \\frac{\\norm{\\nabla F(x)}}{\\norm{\\nabla f(\\tilde{x})}}\\frac{\\norm{v}}{\\norm{\\tilde{v}}}\n\\sqrt{\\frac{1+K\\tilde{\\ell}^2}{(1+K\\tilde{\\ell}^2\\sin^2(\\tilde{\\delta}))(1+K\\tilde{\\ell}^2\\cos^2(\\tilde{\\beta}))}}.\n\\] \nWe use now \\cref{lemma:deformations}.a) and \\cref{lemma:deformations}.c), and bound $\\sin^2(\\tilde{\\delta})$ and $\\cos^2(\\tilde{\\beta})$ in order to bound the previous expression. Recall that, by \\eqref{eq:R_tilde_vs_R} we have $1\/\\sqrt{1+K\\tilde{\\ell}^2} = C_K(\\ell)$, for $\\ell=d(x, x_0) \\leq R$. Let's proceed. We obtain, for $K=-1$\n\\[\n    \\cosh^{-3}(R)\\leq \\frac{1}{\\cosh^2(\\ell)} \\cdot 1 \\cdot \\frac{1}{\\cosh(\\ell)} \\leq \\frac{\\innp{\\nabla F(x), v}}{\\innp{\\nabla f(\\tilde{x}), \\tilde{v}}}  \\leq \\frac{1}{\\cosh(\\ell)} \\cdot \\cosh^2(\\ell) \\cdot \\cosh(\\ell) \\leq \\cosh^2(R).\n\\] \nand for $K=1$ it is \n\\[\n    \\cos^{2}(R)\\leq \\frac{1}{\\cos(\\ell)} \\cdot \\cos^2(\\ell) \\cdot \\cos(\\ell) \\leq \\frac{\\innp{\\nabla F(x), v}}{\\innp{\\nabla f(\\tilde{x}), \\tilde{v}}}  \\leq \\frac{1}{\\cos^2(\\ell)} \\cdot 1 \\cdot \\frac{1}{\\cos(\\ell)} \\leq \\cos^{-3}(R).\n\\] \n\nThe first part of \\cref{prop:bounding_hyperplane} follows, for $\\gammap = \\cosh^{-3}(R)$ and $\\gamman = \\cosh^{-2}(R)$ when $K=-1$, and $\\gammap = \\cos^2(R)$ and $\\gamman = \\cos^3(R)$ when $K=1$.\n\nThe second part of \\cref{prop:bounding_hyperplane} follows readily from the first one and g-convexity of $F$, as in the following. It holds\n\\[\n    f(\\tilde{x}) + \\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}} \\circled{1}[\\leq] F(x) + \\innp{\\nabla F(x), y-x} \\circled{2}[\\leq] F(y) = f(\\tilde{y}),\n\\] \nand\n\\[\n    f(\\tilde{x}) + \\gammap\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}} \\circled{3}[\\leq] F(x) + \\innp{\\nabla F(x), y-x} \\circled{4}[\\leq] F(y) = f(\\tilde{y}),\n\\] \nwhere $\\circled{1}$ and $\\circled{3}$ hold if $\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}} \\leq 0$ and $\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}} \\geq 0$, respectively. Inequalities $\\circled{2}$ and $\\circled{4}$ hold by g-convexity of $F$.\n\n\\end{proof}\n\n\\section{Introduction}\n\nAcceleration in convex optimization is a phenomenon that has drawn lots of attention and has yielded many important results, since the renowned Accelerated Gradient Descent (AGD) method of \\citet{nesterov1983method}. Having been proved successful for deep learning \\cite{DBLP:conf\/icml\/SutskeverMDH13}, among other fields, there have been recent efforts to better understand this phenomenon \\cite{allen2014linear,diakonikolas2017approximate,su2014differential,wibisono2016variational}. These have yielded numerous new results going beyond convexity or the standard oracle model, in a wide variety of settings \\cite{allen2016katyusha, allen2017natasha, allen2018katyusha, DBLP:conf\/stoc\/ZhuO15,allen2016even, allen2017much, carmon2017convex, cohen2018acceleration, cutkosky2019matrix, diakonikolas2019generalized, diakonikolas2017accelerated, DBLP:conf\/colt\/GasnikovDGVSU0W19,wang2015unified}. This surge of research that applies tools of convex optimization to models going beyond convexity has been fruitful.  One of these models is the setting of geodesically convex Riemannian optimization. In this setting, the function to optimize is geodesically convex (g-convex), i.e. convex restricted to any geodesic (cf. \\cref{def:g-convex_smooth}). \n\nRiemannian optimization, g-convex and non-g-convex alike, is an extensive area of research. In recent years there have been numerous efforts towards obtaining Riemannian optimization algorithms that share analogous properties to the more broadly studied Euclidean first-order methods: deterministic \\cite{bento2017iteration,wei2016guarantees,zhang2016first}, stochastic \\cite{hosseini2019alternative,khuzani2017stochastic,tripuraneni2018averaging}, variance-reduced \\cite{sato2017riemannian,kasai2018riemannian,zhang2016fast}, adaptive \\cite{kasai2019riemannian}, saddle-point-escaping \\cite{criscitiello2019efficiently,sun2019escaping,zhang2018r,zhou2019faster, criscitiello2020accelerated}, and projection-free methods \\cite{weber2017frank,weber2019nonconvex}, among others. Unsurprisingly, Riemannian optimization has found many applications in machine learning, including low-rank matrix completion \\cite{DBLP:journals\/siamsc\/CambierA16,heidel2018riemannian,mishra2014r3mc,tan2014riemannian,vandereycken2013low}, dictionary learning \\cite{cherian2016riemannian,sun2016complete}, optimization under orthogonality constraints \\cite{edelman1998geometry}, with applications to Recurrent Neural Networks \\cite{DBLP:conf\/nips\/Casado19, DBLP:conf\/icml\/CasadoM19}, robust covariance estimation in Gaussian distributions \\cite{wiesel2012geodesic}, Gaussian mixture models \\cite{hosseini2015matrix}, operator scaling \\cite{allen2018operator}, and sparse principal component analysis \\cite{genicot2015weakly,huang2019riemannian,jolliffe2003modified}.\n\nHowever, the acceleration phenomenon, largely celebrated in the Euclidean space, is still not understood in Riemannian manifolds, although there has been some progress on this topic recently (cf. \\hyperlink{sec:related_work}{Related work}). This poses the following question, which is the central subject of this paper:\n\n\\begin{center}\n\\textit{Can a Riemannian first-order method enjoy the same rates as AGD in the Euclidean space?}\n\\end{center}\n\nIn this work, we provide an answer in the affirmative for functions defined on hyperbolic and spherical spaces, up to constants depending on the curvature and the initial distance to an optimum, and up to log factors. In particular, the main results of this work are the following.\n\n\\paragraph{Main Results:}\n\\begin{itemize}\n    \\item \\textit{Full acceleration}. We design algorithms that provably achieve the same rates of convergence as AGD in the Euclidean space, up to constants and log factors. More precisely, we obtain the rates $\\bigotilde{L\/\\sqrt{\\epsilon}}$ and $\\bigoast{\\sqrt{L\/\\mu}\\log(\\mu\/\\epsilon)}$ when optimizing $L$-smooth functions that are, respectively, g-convex and $\\mu$-strongly g-convex, defined on the hyperbolic space or a subset of the sphere. The notation $\\bigotilde{\\cdot}$ and $\\bigoast{\\cdot}$ omits $\\log(L\/\\epsilon)$ and $\\log(L\/\\mu)$ factors, respectively, and constants. Previous approaches only showed local results \\cite{zhang2018towards} or obtained results with rates in between the ones obtainable by Riemannian Gradient Descent (RGD) and AGD \\cite{ahn2020nesterov}. Moreover, these previous works only apply to functions that are smooth and strongly g-convex and not to smooth functions that are only g-convex. As a proxy, we design an accelerated algorithm under a condition between convexity and \\textit{quasar-convexity} in the constrained setting, which may be of independent interest.\n\n    \\item \\textit{Reductions}. We present two reductions for any Riemannian manifold of bounded sectional curvature. Given an optimization method for smooth and g-convex functions they provide a method for optimizing smooth and strongly g-convex functions, and vice versa.\n\\end{itemize}\n\nIt is often the case that methods and key geometric inequalities that apply to manifolds with bounded sectional curvatures are obtained from the ones existing for the spaces of constant extremal sectional curvature \\cite{grove1997comparison, zhang2016first, zhang2018towards}. Consequently, our contribution is relevant not only because we establish an algorithm achieving global acceleration on functions defined on a manifold other than the Euclidean space, but also because understanding the constant sectional curvature case is an important step towards understanding the more general case of obtaining algorithms that optimize g-convex functions, strongly or not, defined on manifolds of bounded sectional curvature. \n\nOur main technique for designing the accelerated method consists of mapping the function domain to a subset $\\mathcal{B}$ of the Euclidean space via a geodesic map: a transformation that maps geodesics to geodesics. Given the gradient of a point $x\\in\\M$, which defines a lower bound on the function that is affine over the tangent space of $x$, we find a lower bound of the function that is affine over the region of $\\mathcal{B}$ where the previous lower bound was at most $f(x)$, despite the map being non-conformal, deforming distances, and breaking convexity. This allows to aggregate the lower bounds easily. We believe that effective lower bound aggregation is key to achieving Riemannian acceleration and optimality. Using this strategy, we are able to provide an algorithm along the lines of  the one in \\cite{diakonikolas2017accelerated} to define a continuous method that we discretize using an approximate implementation of the implicit Euler method, obtaining a method achieving the same rates as the Euclidean AGD, up to constants and log factors. Our reductions take into account the deformations produced by the geometry to generalize existing optimal Euclidean reductions \\cite{allen2016optimal, allen2014linear}.\n\n\\paragraph{Basic Geometric Definitions.} We recall basic definitions of Riemannian geometry that we use in this work. For a thorough introduction we refer to \\cite{petersen2006riemannian}. A Riemannian manifold $(\\M,\\mathfrak{g})$ is a real smooth manifold $\\M$ equipped with a metric $\\mathfrak{g}$, which is a smoothly varying inner product. For $x \\in \\M$ and any two vectors $v,w  \\in T_x\\M$ in the tangent space of $\\M$, the inner product $\\innp{v,w}_x$ is $\\mathfrak{g}(v,w)$.  For $v\\in T_x\\M$, the norm is defined as usual $\\norm{v}_x \\defi \\sqrt{\\innp{v,v}_x}$. Typically, $x$ is known given $v$ or $w$, so we will just write $\\innp{v,w}$ or $\\norm{v}$ if $x$ is clear from context. A geodesic is a curve $\\gamma : [0,1] \\to \\M$ of unit speed that is locally distance minimizing. A uniquely geodesic space is a space such that for every two points there is one and only one geodesic that joins them. In such a case the exponential map $\\expon_x : T_x\\M\\to \\M$ and inverse exponential map $\\expon_x^{-1}:\\M\\to T_x\\M$ are well defined for every pair of points, and are as follows. Given $x, y\\in\\M$, $v\\in T_x\\M$, and a geodesic $\\gamma$ of length $\\norm{v}$ such that $\\gamma(0) =x$, $\\gamma(1)=y$, $\\gamma'(0)=v\/\\norm{v}$, we have that $\\operatorname{Exp}_x(v) = y$ and $\\operatorname{Exp}_x^{-1}(y) = v$. Note, however, that $\\expon_x(\\cdot)$ might not be defined for each $v\\in T_x\\M$. We denote by $d(x,y)$ the distance between $x$ and $y$. Its value is the same as $\\norm{\\expon_x^{-1}(y)}$. Given a $2$-dimensional subspace $V \\subset T_x\\M$, the sectional curvature at $x$ with respect to $V$ is defined as the Gauss curvature of the manifold $\\operatorname{Exp}_x(V)$ at $x$. \n\n\\paragraph{Notation.}\\hypertarget{sec:notation}{} Let $\\M$ be a manifold and let $\\B\\subset\\R^d$. We denote by $h:\\M\\to\\B$ a geodesic map \\cite{kreyszig1991differential}, which is a diffeomorphism such that the image and the inverse image of a geodesic is a geodesic. Usually, given an initial point $x_0$ of our algorithm, we will have $h(x_0)=0$. Given a point $x\\in\\M$ we use the notation $\\tilde{x} = h(x)$ and vice versa, any point in $\\B$ will use a tilde. Given two points $x, y\\in \\M$ and a vector $v\\in T_{x}\\M$ in the tangent space of $x$, we use the formal notation $\\innp{v, y-x} \\defi -\\innp{v, x-y} \\defi \\innp{v, \\expon^{-1}_{x}(y)}$. Given a vector $v\\in T_x\\M$, we call $\\tilde{v} \\in \\R^d$ the vector of the same norm such that $\\{\\tilde{x}+\\tilde{\\lambda} \\tilde{v}|\\tilde{\\lambda}\\in\\R^+, \\tilde{x}+\\tilde{\\lambda} \\tilde{v} \\in \\B\\} = \\{h(\\expon_x(\\lambda v))|\\lambda\\in I\\subset\\R^+\\}$, for some interval $I$. Likewise, given $x$ and a vector $\\tilde{v}\\in\\R^d$, we define $v\\in T_x\\M$. Let $x^\\ast$ be any minimizer of $F:\\M\\to\\R$. We denote by $R \\geq d(x_0,x^\\ast)$ a bound on the distance between $x^\\ast$ and the initial point $x_0$. Note that this implies that $x^\\ast \\in \\expon_{x_0}(\\bar{B}(0,R))$, for the closed ball $\\bar{B}(0,R) \\subset T_{x_0}\\M$. Consequently, we will work with the manifold that is a subset of a $d$-dimensional complete and simply connected manifold of constant sectional curvature $K$, namely a subset of the hyperbolic space or sphere \\cite{petersen2006riemannian}, defined as $\\expon_{x_0}(\\bar{B}(0,R))$, with the inherited metric.\nDenote by $\\H$ this manifold in the former case and $\\S$ in the latter, and note that we are not making explicit the dependence on $d$, $R$ and $K$. We want to work with the standard choice of uniquely geodesic manifolds \\cite{ahn2020nesterov, liu2017accelerated, zhang2016first, zhang2018towards}. Therefore, in the case that the manifold is $\\S$, we restrict ourselves to $R < \\pi\/2\\sqrt{K}$, so $\\S$ is contained in an open hemisphere. The big $O$ notations $\\bigotilde{\\cdot}$ and $\\bigoast{\\cdot}$ omit $\\log(L\/\\epsilon)$ and $\\log(L\/\\mu)$ factors, respectively, and constant factors depending on $R$ and $K$.\n\nWe define now the main properties that will be assumed on the function $F$ to be minimized.\n\\begin{definition}[Geodesic Convexity and Smoothness] \\label{def:g-convex_smooth}\n    Let $F:\\M \\to \\R$ be a differentiable function defined on a Riemannian manifold $(\\M,\\mathfrak{g})$. Given $L\\geq \\mu > 0$, we say that $F$ is $L$-smooth, and respectively $\\mu$-strongly g-convex, if for any two points $x, y \\in \\M$, $F$ satisfies\n    \\[\n        F(y) \\leq F(x) + \\innp{\\nabla F(x), y-x} + \\frac{L}{2}d(x,y)^2, \\text{ resp. } F(y) \\geq F(x) + \\innp{\\nabla F(x), y-x} + \\frac{\\mu}{2}d(x,y)^2.\n    \\]\n    We say $F$ is g-convex if the second inequality above, i.e. $\\mu$-strong g-convexity, is satisfied with $\\mu=0$.  Note that we have used the formal notation above for the subtraction of points in the inner product.  \n\\end{definition}\n\n\\hypertarget{sec:related_work}{}\\paragraph{Comparison with Related Work.} There are a number of works that study the problem of first-order acceleration in Riemannian manifolds of bounded sectional curvature. The first study is \\cite{liu2017accelerated}. In this work, the authors develop an accelerated method with the same rates as AGD for both g-convex and strongly g-convex functions, provided that at each step a given nonlinear equation can be solved. No algorithm for solving this equation has been found and, in principle, it could be intractable or infeasible. In \\cite{alimisis2019continuous} a continuous method analogous to the continuous approach to accelerated methods is presented, but it is not known if there exists an accelerated discretization of it. In \\cite{alimisis2020practical}, an algorithm presented is claimed to enjoy an accelerated rate of convergence, but fails to provide convergence when the function value gets below a potentially large constant that depends on the manifold and smoothness constant. In \\cite{huang2019extending} an accelerated algorithm is presented but relying on strong geometric inequalities that are not proved to be satisfied. \\citet{zhang2018towards} obtain a \\textit{local} algorithm that optimizes $L$-smooth and $\\mu$-strongly g-convex functions achieving the same rates as AGD in the Euclidean space, up to constants. That is, the initial point needs to start close to the optimum, $O((\\mu\/L)^{3\/4})$ close, to be precise. Their approach consists of adapting Nesterov's estimate sequence technique by keeping a quadratic on $T_{x_t}\\M$ that induces on $\\M$ a regularized lower bound on $F(x^\\ast)$ via $\\expon_{x_t}(\\cdot)$. They aggregate the information yielded by the gradient to it, and use a geometric lemma to find a quadratic in $T_{x_{t+1}}\\M$ whose induced function lower bounds the other one. \\citet{ahn2020nesterov} generalize the previous algorithm and, by using similar ideas for the lower bound, they adapt it to work globally, obtaining strictly better rates than RGD, recovering the local acceleration of the previous paper, but not achieving global rates comparable to the ones of AGD. In fact, they prove that their algorithm eventually decreases the function value at a rate close to AGD but this can take as many iterations as the ones needed by RGD to reach the neighborhood of the previous local algorithm. In our work, we take a step back and focus on the constant sectional curvature case to provide a global algorithm that achieves the same rates as AGD, up to constants and log factors. It is common to characterize the properties of spaces of bounded sectional curvature by using the ones of the spaces of constant extremal sectional curvature \\cite{grove1997comparison, zhang2016first, zhang2018towards}, which makes the study of the constant sectional curvature case critical to the development of full accelerated algorithms in the general bounded sectional curvature case. Additionally, our work studies g-convexity besides strong g-convexity.\n\nAnother related work is the \\textit{approximate duality gap technique} \\cite{diakonikolas2017approximate}, which presents a unified view of the analysis of first-order methods for the optimization of convex functions defined in the Euclidean space. It defines a continuous duality gap and by enforcing a natural invariant, it obtains accelerated continuous dynamics and their discretizations for most classical first-order methods. A derived work \\cite{diakonikolas2017accelerated} obtains acceleration in a fundamentally different way from previous acceleration approaches, namely using an approximate implicit Euler method for the discretization of the acceleration dynamics. The convergence analysis of \\cref{thm:acceleration_quasiquasarconvexity} is inspired by these two works. We will see in the sequel that, for our manifolds of interest, g-convexity is related to a model known in the literature as quasar-convexity or weak-quasi-convexity \\cite{guminov2017accelerated, hinder2019near,nesterov2018primal}.\n\n\n\\section{Algorithm}\\label{sec:algorithm}\nWe study the minimization problem $\\min_{x\\in\\M}{F(x)}$ with a gradient oracle, for a smooth function $F:\\M\\to\\R$ that is g-convex or strongly g-convex. In this section, $\\M$ refers to a manifold that can be $\\H$ or $\\S$, i.e. the subset of the hyperbolic space or sphere $\\expon_{x_0}(\\bar{B}(0, R))$, for an initial point $x_0$. For simplicity, we do not use subdifferentials so we assume $F:\\M\\to\\R$ is a differentiable function that is defined over the manifold of constant sectional curvature $\\M'\\defi \\expon_{x_0}(B(0,R'))$, for an $R'>R$, and we avoid writing $F:\\M'\\to\\R$. We defer the proofs of the lemmas and theorems in this and following sections to the appendix. We assume without loss of generality that the sectional curvature of $\\M$ is $K \\in \\{1,-1\\}$, since for any other value of $K$ and any function $F:\\M\\to\\R$ defined on such a manifold, we can reparametrize $F$ by a rescaling, so it is defined over a manifold of constant sectional curvature $K \\in \\{1,-1\\}$. The parameters $L$, $\\mu$ and $R$ are rescaled accordingly as a function of $K$, cf. \\cref{remark:rescaling_of_K}. We denote the special cosine by $C_K(\\cdot)$, which is $\\cos(\\cdot)$ if $K=1$ and $\\cosh(\\cdot)$ if $K=-1$. We define $\\X = h(\\M)\\subset \\B \\subset \\R^d$. We use classical geodesic maps for the manifolds that we consider: the Gnomonic projection for $\\S$ and the Beltrami-Klein projection for $\\H$ \\cite{greenberg1993euclidean}. They map an open hemisphere and the hyperbolic space of curvature $K\\in\\{1,-1\\}$ to $\\B=\\R^d$ and $\\B =B(0,1)\\subset\\R^d$, respectively. We will derive our results from the following characterization \\cite{greenberg1993euclidean}. Let $\\tilde{x}, \\tilde{y} \\in \\B$ be two points. Recall that we denote $x = h^{-1}(\\tilde{x}), y=h^{-1}(\\tilde{y}) \\in \\M$. Then we have that $d(x,y)$, the distance between $x$ and $y$ with the metric of $\\M$, satisfies\n\\begin{equation}\\label{eq:characterization_of_geodesic_map_and_metric}\n    C_K(d(x,y))  = \\frac{1+K\\innp{\\tilde{x}, \\tilde{y}}}{\\sqrt{1+K\\norm{\\tilde{x}}^2}\\cdot\\sqrt{1+K\\norm{ \\tilde{y}}^2}}.\n\\end{equation}\nObserve that the expression is symmetric with respect to rotations. In particular, the symmetry implies $\\X$ is a closed ball of radius $\\tilde{R}$, with $C_K(R) = (1+K\\tilde{R}^2)^{-1\/2}$. \n\nConsider a point $x\\in\\M$ and the lower bound provided by the g-convexity assumption when computing $\\nabla F(x)$. Dropping the $\\mu$ term in case of strong g-convexity, this bound is affine over $T_x \\M$. We would like our algorithm to aggregate effectively the lower bounds it computes during the course of the optimization. The deformations of the geometry make it a difficult task, despite the fact that we have a simple description of each individual lower bound. We deal with this problem in the following way: our approach is to obtain a lower bound that is looser by a constant depending on $R$, and that is affine over $\\B$. In this way the aggregation becomes easier. Then, we are able to combine this lower bound with decreasing upper bounds in the fashion some other accelerated methods work in the Euclidean space \\cite{allen2014linear, diakonikolas2017accelerated,  diakonikolas2017approximate, nesterov1983method}. Alternatively, we can see the approach in this work as the constrained non-convex optimization problem of minimizing the function $f:\\X\\to\\R$, $\\tilde{x} \\mapsto F(h^{-1}(\\tilde{x}))$:\n\\[\n    \\text{minimize}\\ \\  f(\\tilde{x}),\\quad \\text{ for } \\tilde{x}\\in\\X.\n\\]\nIn the rest of the section, we will focus on the g-convex case. For simplicity, instead of solving the strongly g-convex case directly in an analogous way by finding a lower bound that is quadratic over $\\B$, we rely on the reductions of \\cref{sec:reductions} to obtain the accelerated algorithm in this case.\n\nThe following two lemmas show that finding the aforementioned affine lower bound is possible, and is defined as a function of $\\nabla f(\\tilde{x})$. We first gauge the deformations caused by the geodesic map $h$. Distances are deformed, the map $h$ is not conformal and, in spite of it being a geodesic map, the image of the geodesic $\\expon_x(\\lambda \\nabla F(x))$ is not mapped into the image of the geodesic $\\tilde{x}+\\tilde{\\lambda}\\nabla f(\\tilde{x})$, i.e. the direction of the gradient changes. We are able to find the affine lower bound after bounding these deformations.\n\\begin{lemma}\\label{lemma:deformations}\n    Let $x,y\\in\\M$ be two different points, and in part $b)$ different from $x_0$. Let $\\tilde{\\alpha}$ be the angle $\\angle \\tilde{x}_0\\tilde{x}\\tilde{y}$, formed by the vectors $\\tilde{x}_0-\\tilde{x}$ and $\\tilde{y}-\\tilde{x}$. Let $\\alpha$ be the corresponding angle between the vectors $\\expon_x^{-1}(x_0)$ and $\\expon_x^{-1}(y)$. Assume without loss of generality that $\\tilde{x} \\in \\operatorname{span}\\{\\tilde{e}_1\\}$ and $\\nabla f(\\tilde{x}) \\in \\operatorname{span}\\{\\tilde{e}_1, \\tilde{e}_2\\}$ for the canonical orthonormal basis $\\{\\tilde{e}_i\\}_{i=1}^d$. Let $e_i\\in T_x\\M$ be the unit vector such that $h$ maps the image of the geodesic $\\expon_x(\\lambda e_i)$ to the image of the geodesic $\\tilde{x}+\\tilde{\\lambda}e_i$, for $i=1,\\dots, d$, and $\\lambda, \\tilde{\\lambda} \\geq 0$. Then, the following holds.\n    \\begin{enumerate}[label=\\alph*), leftmargin=*,]\n        \\item Distance deformation:\n \\begin{align*}\n\\begin{aligned}\n    K C_K^2(R) \\leq K \\frac{d(x,y)}{\\norm{\\tilde{x}-\\tilde{y}}} \\leq K.\n\\end{aligned}\n\\end{align*}       \n        \\item Angle deformation:\n\\begin{align*}\n\\begin{aligned}\n\\sin(\\alpha) = \\sin(\\tilde{\\alpha}) \\sqrt{\\frac{1+K\\norm{\\tilde{x}}^2}{1+K\\norm{\\tilde{x}}^2\\sin^2(\\tilde{\\alpha})}}, \\quad \\quad \\cos(\\alpha) = \\cos(\\tilde{\\alpha}) \\sqrt{\\frac{1}{1+K\\norm{\\tilde{x}}^2\\sin^2(\\tilde{\\alpha})}}.\n\\end{aligned}\n\\end{align*}\n        \\item Gradient deformation:\n\\begin{align*}\n\\begin{aligned}\n    \\nabla F(x) = (1+K\\norm{\\tilde{x}}^2)\\nabla f(\\tilde{x})_1 e_1 + \\sqrt{1+K\\norm{\\tilde{x}}^2}\\nabla f(\\tilde{x})_2 e_2 \\quad \\text{ and } \\quad e_i \\perp e_j \\ \\text{ for } i\\neq j.\n\\end{aligned}\n\\end{align*}\n    And if $v \\in T_x\\M$ is a vector normal to $\\nabla F(x)$, then $\\tilde{v}$ is normal to $\\nabla f(x)$. \n    \\end{enumerate}\n\\end{lemma}\n\nThe following uses the deformations described in the previous lemma to obtain the affine lower bound on the function, given a gradient at a point $\\tilde{x}$. Note that \\cref{lemma:deformations}.c implies that we have $\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}= 0$ if and only if $\\innp{\\nabla F(x), y-x}= 0$. In the proof we lower bound, generally, affine functions defined on $T_x\\M$ by affine functions in the Euclidean space $\\B$. This generality allows to obtain a result with constants that only depend on $R$.\n\n\\begin{lemma}\\label{prop:bounding_hyperplane}\n    Let $F:\\M\\to\\R$ be a differentiable function and let $f=F\\circ h^{-1}$.\n    Then, there are constants $\\gamman, \\gammap \\in (0, 1]$ depending on $R$ such that for all $x, y\\in\\M$ satisfying $\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}\\neq 0$ we have: \n    \\begin{align}\\label{eq:quotient_of_inner_products_is_bounded} \n \\begin{aligned}\n     \\gammap \\leq\\frac{\\innp{\\nabla F(x), y-x}}{\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}} \\leq \\frac{1}{\\gamman}.\n   \\end{aligned}\n\\end{align}\nIn particular, if $F$ is g-convex we have:\n\\begin{align} \\label{eq:quasiquasarconvexity}\n \\begin{aligned}\n     f(\\tilde{x}) +  \\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}\\leq f(\\tilde{y})  &\\ & & {\\text{ if } \\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}}\\leq 0, \\\\\n     f(\\tilde{x})+\\gammap\\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}} \\leq f(\\tilde{y})  &\\ & & {\\text{ if } \\innp{\\nabla f(\\tilde{x}), \\tilde{y}-\\tilde{x}}} \\geq 0.\n   \\end{aligned}\n\\end{align}\n\\end{lemma}\n\nThe two inequalities in \\eqref{eq:quasiquasarconvexity} show the affine lower bound. Only the first one is needed to bound $f(\\tilde{x}^\\ast)=F(x^\\ast)$. The first inequality applied to $\\tilde{y}=\\tilde{x}^\\ast$ defines a model known in the literature as quasar-convexity or weak-quasi-convexity \\cite{guminov2017accelerated, hinder2019near,nesterov2018primal}, for which accelerated algorithms exist in the \\textit{unconstrained case}, provided smoothness is also satisfied. However, to the best of our knowledge, there is no known algorithm for solving the constrained case in an accelerated way. The condition in \\eqref{eq:quasiquasarconvexity} is, trivially, a relaxation of convexity that is stronger than quasar-convexity. We will make use of \\eqref{eq:quasiquasarconvexity} in order to obtain acceleration in the constrained setting. This is of independent interest. Recall that we need the constraint to guarantee bounded deformation due to the geometry. We also require smoothness of $f$. The following lemma shows that $f$ is as smooth as $F$ up to a constant depending on $R$. \n\n\\begin{lemma}\\label{lemma:smoothness_of_transformed_function}\n    Let $F:\\M\\to\\R$ be an $L$-smooth function and $f=F\\circ h^{-1}$. Assume there is a point $x^\\ast \\in \\M$ such that $\\nabla F(x^\\ast)=0$. Then $f$ is $O(L)$-smooth.\n\\end{lemma}\n\n\nUsing the \\textit{approximate duality gap technique} \\cite{diakonikolas2017approximate} we obtain accelerated continuous dynamics, for the optimization of the function $f$. Then we adapt AXGD to obtain an accelerated discretization. AXGD \\cite{diakonikolas2017accelerated} is a method that is based on implicit Euler discretization of continuous accelerated dynamics and is fundamentally different from AGD and techniques as Linear Coupling \\cite{allen2014linear} or Nesterov's estimate sequence \\cite{nesterov1983method}. The latter techniques use a balancing gradient step at each iteration and our use of a looser lower bound complicates guaranteeing keeping the gradient step within the constraints. We state the accelerated theorem and provide a sketch of the proof in \\cref{subsec:sketch_of_my_axgd_proof}.\n\n\\begin{theorem} \\label{thm:acceleration_quasiquasarconvexity}\n    Let $Q\\subset \\R^d$ be a convex set of diameter $2R$. Let $f:Q\\to\\R$ be an $\\tilde{L}$-smooth function satisfying \\eqref{eq:quasiquasarconvexity} with constants $\\gamman,\\gammap \\in (0,1]$. Assume there is a point $\\tilde{x}^\\ast\\in Q$ such that $\\nabla f(\\tilde{x}^\\ast)=0$. Then, we can obtain an $\\epsilon$-minimizer of $f$ using $\\bigotilde{ \\sqrt{\\tilde{L}\/(\\gamman^2\\gammap\\epsilon)} }$ queries to the gradient oracle of $f$.\n\\end{theorem}\n\nFinally, we have Riemannian acceleration as a direct consequence of \\cref{thm:acceleration_quasiquasarconvexity}, \\cref{prop:bounding_hyperplane} and \\cref{lemma:smoothness_of_transformed_function}.\n\\begin{theorem}[g-Convex Acceleration] \\label{thm:riemannian_acceleration}\n    Let $F:\\M\\to\\R$ be an $L$-smooth and g-convex function and assume there is a point $x^\\ast\\in\\M$ satisfying $\\nabla F(x^\\ast)=0$. \\cref{alg:accelerated_gconvex} computes a point $x_t\\in\\M$ satisfying $F(x_t)-F(x^\\ast) \\leq \\epsilon$ using $\\bigotilde{\\sqrt{L\/\\epsilon}}$ queries to the gradient oracle.\n\\end{theorem}\n\nWe observe that if there is a geodesic map mapping a manifold into a convex subset of the Euclidean space then the manifold must necessarily have constant sectional curvature, cf. Beltrami's Theorem \\cite{busemann1984general, kreyszig1991differential}.\nThis precludes a straightforward generalization from our method to the case of non-constant bounded sectional curvature.\n\n\\begin{algorithm}\n    \\caption{Accelerated g-Convex Minimization}\n    \\label{alg:accelerated_gconvex}\n\n\\begin{algorithmic}[1] \n    \\REQUIRE Smooth and g-convex function $F:\\M\\to\\R$, for $\\M=\\H$ or $\\M=\\S$.\n    \\Statex Initial point $x_0$;  Constants $\\tilde{L}$, $\\gammap$, $\\gamman$. Geodesic map $h$ satisfying \\eqref{eq:characterization_of_geodesic_map_and_metric} and $h(x_0)=0$.\n    \\Statex Bound on the distance to a minimum $R \\geq d(x_0,x^\\ast)$. Accuracy $\\epsilon$ and number of iterations $t$.\n    \\hrule\n    \\State $\\X \\defi h(\\expon_{x_0}(B(0, R)))\\subset \\B$; $\\quad f \\defi F \\circ h^{-1}\\quad$ and $\\quad\\psi(\\tilde{x}) \\defi \\frac{1}{2}\\norm{\\tilde{x}}^2$ \n    \\State $\\tilde{z}_0 \\gets \\nabla \\psi(\\tilde{x}_0)$; $\\quad A_0 \\gets 0$\n    \\FOR {$i \\text{ \\textbf{from} } 0 \\text{ to } t-1$}\n        \\State $a_{i+1} \\gets (i+1)\\gamman^2\\gammap\/2\\tilde{L}$\n        \\State $A_{i+1} \\gets A_{i} + a_{i+1}$\n        \\State $\\lambda \\gets \\text{BinaryLineSearch}(\\tilde{x}_i, \\tilde{z}_i, f, \\X, a_{i+1}, A_{i}, \\epsilon, \\tilde{L},  \\gamman, \\gammap)$ \\Comment{(cf. \\cref{alg:bin_search} in \\cref{app:acceleration})}\n        \\State $\\tilde{\\chi}_{i} \\gets (1-\\lambda)\\tilde{x}_i + \\lambda \\nabla \\psi^\\ast(\\tilde{z}_i)$ \n        \\State $\\tilde{\\zeta}_{i} \\gets \\tilde{z}_i-(a_{i+1}\/\\gamman)\\nabla f(\\tilde{\\chi}_i)$\n        \\State $\\tilde{x}_{i+1} \\gets (1-\\lambda)\\tilde{x}_i + \\lambda \\nabla \\psi^\\ast(\\tilde{\\zeta}_i)$  \\Comment{$\\big[\\nabla \\psi^\\ast(\\tilde{p})= \\argmin_{\\tilde{z} \\in \\X} \\{\\norm{\\tilde{z}-\\tilde{p}}\\} = \\Pi_\\X(\\tilde{p})\\big]$}\n        \\State $\\tilde{z}_{i+1} \\gets \\tilde{z}_i-(a_{i+1}\/\\gamman)\\nabla f(\\tilde{x}_{i+1})$\n    \\ENDFOR\n    \\State return $x_t$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Sketch of the proof of \\cref{thm:acceleration_quasiquasarconvexity}.} \\label{subsec:sketch_of_my_axgd_proof}\n\nInspired by the \\textit{approximate duality gap technique} \\cite{diakonikolas2017approximate}, let $\\alpha_t$ be an increasing function of time $t$, and denote $A_t=\\int_{t_0}^t d\\alpha_\\tau= \\int_{t_0}^t \\dott{\\alpha}_\\tau d\\tau$. We define a continuous method that keeps a solution $\\tilde{x}_t$, along with a differentiable upper bound $U_t$ on $f(x_t)$ and a lower bound $L_t$ on $f(\\tilde{x}^\\ast)$. In our case $f$ is differentiable so we can just take $U_t=f(x_t)$. The lower bound comes from \n\\begin{equation}\\label{eq:raw_lower_bound}\n    f(\\tilde{x}^\\ast) \\geq \\frac{\\int_{t_{0}}^{t} f(\\tilde{x}_{\\tau}) d \\alpha_{\\tau}}{A_{t}}+\\frac{\\int_{t_{0}}^{t}\\frac{1}{\\gamman}\\innp{\\nabla f(\\tilde{x}_{\\tau}), \\tilde{x}^\\ast-\\tilde{x}_{\\tau}} d \\alpha_{\\tau}}{A_{t}},\n\\end{equation}\nafter applying some desirable modifications, like regularization with a $1$-strongly convex function $\\psi$ and removing the unknown $\\tilde{x}^\\ast$ by taking a minimum over $\\X$. Note \\eqref{eq:raw_lower_bound} comes from averaging \\eqref{eq:quasiquasarconvexity} for $\\tilde{y} = \\tilde{x}^\\ast$. Then, if we define the gap $G_t=U_t-L_t$ and design a method that forces $\\alpha_tG_t$ to be non-increasing, we can deduce $f(x_t)-f(x^\\ast) \\leq G_t \\leq \\alpha_{t_0}G_{t_0}\/\\alpha_t$. By forcing $\\frac{d}{dt}(\\alpha_tG_t) =0$, we naturally obtain the following continuous dynamics, where $z_t$ is a mirror point and $\\psi^\\ast$ is the Fenchel dual of $\\psi$, cf. \\cref{def:fenchel_dual}.\n\\begin{align}\\label{modified_accelerated_continuous_dynamics}\n\\begin{aligned}\n    \\dott{\\tilde{z}}_{t}&=-\\frac{1}{\\gamman}\\dott{\\alpha}_{t} \\nabla f(\\tilde{x}_{t}); \\ \\ \\\n     \\dott{\\tilde{x}}_{t}&=\\frac{1}{\\gamman}\\dott{\\alpha}_{t} \\frac{\\nabla \\psi^{\\ast}({\\tilde{z}}_{t})-\\tilde{x}_{t}}{\\alpha_{t}}; \\ \\ \\\n    \\tilde{z}_{t_{0}}&=\\nabla \\psi(\\tilde{x}_{t_{0}}), \\tilde{x}_{t_{0}} \\in \\X\n\\end{aligned}\n\\end{align}\nWe note that except for the constant $\\gamman$, these dynamics match the accelerated dynamics used in the optimization of convex functions \\cite{diakonikolas2017approximate, diakonikolas2017accelerated, DBLP:conf\/nips\/KricheneBB15}. The AXGD algorithm \\cite{diakonikolas2017accelerated}, designed for the accelerated optimization of convex functions, discretizes the latter dynamics following an approximate implementation of implicit Euler discretization. This has the advantage of not needing a gradient step per iteration to compensate for some positive discretization error. Note that in our case we must use \\eqref{eq:quasiquasarconvexity} instead of convexity for a discretization. We are able to obtain the following discretization coming from an approximate implicit Euler discretization: \n\\begin{equation}\\label{general_rule_modified_quasar_axgd}\n\\left\\{\\begin{array}{l}\n    \\tilde{\\chi}_{i}=\\frac{\\hat{\\gamma}_iA_i}{A_i \\hat{\\gamma}_i+a_{i+1}\/\\gamman} \\tilde{x}_{i}+\\frac{a_{i+1}\/\\gamman}{A_i\\hat{\\gamma}_i + a_{i+1}\/\\gamman} \\nabla \\psi^{\\ast}(\\tilde{z}_{i}); \\ \\ \\ \\ \\ \\ \n    \\tilde{\\zeta}_i = \\tilde{z}_{i}-\\frac{a_{i+1}}{\\gamman} \\nabla f(\\tilde{\\chi}_{i}) \\\\\n    \\tilde{x}_{i+1}=\\frac{\\hat{\\gamma}_iA_i}{A_i \\hat{\\gamma}_i+a_{i+1}\/\\gamman} \\tilde{x}_{i}+\\frac{a_{i+1}\/\\gamman}{A_i\\hat{\\gamma}_i + a_{i+1}\/\\gamman} \\nabla \\psi^{\\ast}(\\tilde{\\zeta}_i); \\ \\\n    \\tilde{z}_{i+1} = \\tilde{z}_{i}-\\frac{a_{i+1}}{\\gamman} \\nabla f(\\tilde{x}_{i+1}) \n\\end{array}\\right.\n\\end{equation}\nwhere $\\hat{\\gamma}_i\\in[\\gammap, 1\/\\gamman]$ is a parameter, $\\tilde{x}_0\\in\\X$ is an arbitrary point, $\\tilde{z}_0=\\nabla \\psi(\\tilde{x}_0)$ and now $\\alpha_t$ is a discrete measure and $\\dott{\\alpha}_t$ is a weighted sum of Dirac delta functions $\\dott{\\alpha}_t=\\sum_{i=1}^\\infty a_i \\delta(t-(t_0+i-1))$. Compare \\eqref{general_rule_modified_quasar_axgd} with the discretization in AXGD \\cite{diakonikolas2017accelerated} that is equal to our discretization but with no $\\gamman$ or $\\hat{\\gamma}_i$. Or equivalently with $\\hat{\\gamma}_i = 1\/\\gamman$ and with no $\\gamman$ for the mirror descent updates of $\\tilde{\\zeta}_i$ and $\\tilde{z}_{i+1}$. However, not having convexity, in order to have per-iteration discretization error less than $\\hat{\\epsilon}\/A_T$, we require $\\hat{\\gamma}_i$ to be such that $\\tilde{x}_{i+1}$ satisfies\n\\begin{equation}\\label{eq:approximate_binary_search_main_paper}\n    f(\\tilde{x}_{i+1})-f(\\tilde{x}_i) \\leq \\hat{\\gamma}_i \\innp{\\nabla f(\\tilde{x}_{i+1}), \\tilde{x}_{i+1}-\\tilde{x}_i} + \\hat{\\epsilon}, \n\\end{equation}\nwhere $\\hat{\\epsilon}$ is chosen so that the accumulated discretization error is $<\\epsilon\/2$, after having performed the steps necessary to obtain an $\\epsilon\/2$ minimizer. We would like to use \\eqref{eq:quasiquasarconvexity} to find such a $\\hat{\\gamma}_i$ but we need to take into account that we only know $\\tilde{x}_{i+1}$ a posteriori. Indeed, using \\eqref{eq:quasiquasarconvexity} we conclude that setting $\\hat{\\gamma}_i$ to $1\/\\gamman$ or $\\gammap$ then we either satisfy \\eqref{eq:approximate_binary_search_main_paper} or there is a point $\\hat{\\gamma}_i \\in (\\gammap, 1\/\\gamman)$ for which $\\innp{\\nabla f(\\tilde{x}_{i+1}), \\tilde{x}_{i+1}-\\tilde{x}_i}=0$, which satisfies the equation for $\\hat{\\epsilon}=0$. Then, using smoothness of $f$, existence of $x^\\ast$ (that satisfies $\\nabla f(x^\\ast)=0$), and boundedness of $\\X$ we can guarantee that a binary search finds a point satisfying \\eqref{eq:approximate_binary_search_main_paper} in $O(\\log (\\tilde{L}i\/\\gamma_n\\hat{\\epsilon}))$ iterations. Each iteration of the binary search requires to run \\eqref{general_rule_modified_quasar_axgd}, that is, one step of the discretization. Computing the final discretization error, we obtain acceleration after choosing appropriate learning rates $a_{i}$. \\cref{alg:accelerated_gconvex} contains the pseudocode of this algorithm along with the reduction of the problem from minimizing $F$ to minimizing $f$. We chose $\\psi(\\tilde{x}) \\defi \\frac{1}{2}\\norm{\\tilde{x}}^2$ as our strongly convex regularizer.\n\n\n\n\\section{Reductions} \\label{sec:reductions}\nThe construction of reductions proves to be very useful in order to facilitate the design of algorithms in different settings. Moreover, reductions are a helpful tool to infer new lower bounds without extra ad hoc analysis. We present two reductions. We will see in \\cref{coroll:acceleration_st_g_convex} and \\cref{example:application_of_reduction_to_st_convex} that one can obtain full accelerated methods to minimize smooth and strongly g-convex functions from our accelerated methods for smooth and g-convex functions and vice versa. These are generalizations of some reductions designed to work in the Euclidean space \\cite{allen2016optimal, allen2014linear}. The reduction to strongly g-convex functions takes into account the effect of the deformation of the space on the strong convexity of the function $F_y(x) = d(x,y)^2\/2$, for $x, y \\in \\M$. The reduction to g-convexity requires the rate of the algorithm that applies to g-convex functions to be proportional to the distance between the initial point and the optimum $d(x_0,x^\\ast)$. The proofs of the statements in this section can be found in the appendix. We will use $\\timens(\\cdot)$ and $\\time(\\cdot)$ to denote the time algorithms $\\mathcal{A}_{\\operatorname{ns}}$ and $\\mathcal{A}$ below require, respectively, to perform the tasks we define below.\n\n\\begin{theorem}\\label{thm:reduction_to_g_convex}\n    Let $\\M$ be a Riemannian manifold, let $F:\\M\\to\\R$ be an $L$-smooth and $\\mu$-strongly g-convex function, and let $x^\\ast$ be its minimizer. Let $x_0$ be a starting point such that $d(x_0,x^\\ast) \\leq R$. Suppose we have an algorithm $\\mathcal{A_{\\operatorname{ns}}}$ to minimize $F$, such that in time $T =\\timens(L, \\mu, R)$ it produces a point $\\hat{x}_T$ satisfying $F(\\hat{x}_T)-F(x^\\ast) \\leq \\mu \\cdot d(x_0,x^\\ast)^2\/4$. Then we can compute an $\\epsilon$-minimizer of $F$ in time $O(\\timens(L, \\mu, R)\\log(R^2\\mu\/\\epsilon))$.\n\\end{theorem}\n\n\\cref{thm:reduction_to_g_convex} implies that if we forget about the strong g-convexity of a function and we treat it as it is just g-convex we can run in stages an algorithm designed for optimizing g-convex functions.  The fact that the function is strongly g-convex is only used between stages, as the following corollary shows by making use of \\cref{alg:accelerated_gconvex}.\n\n\\begin{corollary}\\label{coroll:acceleration_st_g_convex}\n    We can compute an $\\epsilon$-minimizer of an $L$-smooth and $\\mu$-strongly g-convex function $F:\\M\\to\\R$ in $\\bigoast{\\sqrt{L\/\\mu}\\log(\\mu\/\\epsilon)}$ queries to the gradient oracle, where $\\M=\\S$ or $\\M=\\H$.\n\\end{corollary}\n\nWe note that in the strongly convex case, by decreasing the function value by a factor we can guarantee, we decrease the distance to $x^\\ast$ by another factor, so we can periodically recenter the geodesic map to reduce the constants produced by the deformations of the geometry, see the proof of \\cref{coroll:acceleration_st_g_convex}. Finally, we show the reverse reduction.\n\n\\begin{theorem}\\label{thm:reduction_to_g_st_convex}\n    Let $\\M$ be a Riemannian manifold of bounded sectional curvature, let $F:\\M\\to\\R$ be an $L$-smooth and g-convex function, and assume there is a point $x^\\ast\\in\\M$ such that $\\nabla F(x^\\ast)=0$. Let $x_0$ be a starting point such that $d(x_0,x^\\ast) \\leq R$ and let $\\Delta$ satisfy $F(x_0)-F(x^\\ast) \\leq \\Delta $. Assume we have an algorithm $\\mathcal{A}$ that given an $L$-smooth and $\\mu$-strongly g-convex function $\\hat{F}:\\M\\to\\R$, with minimizer in $\\expon_{x_0}(\\bar{B}(0,R))$, and any initial point $\\hat{x}_0\\in\\M$ produces a point $\\hat{x}\\in\\expon_{x_0}(\\bar{B}(0, R))$ in time $\\hat{T}=\\time(L,\\mu,\\M, R)$ satisfying $\\hat{F}(\\hat{x})-\\min_{x\\in\\M}\\hat{F}(x) \\leq (\\hat{F}(\\hat{x}_0) - \\min_{x\\in\\M}\\hat{F}(x))\/4$. Let $T=\\lceil\\log_2(\\Delta\/\\epsilon)\/2\\rceil+1$. Then, we can compute an $\\epsilon$-minimizer in time $\\sum_{t=0}^{T-1}\\time(L+2^{-t}\\Delta\\distorn\/R^2, 2^{-t}\\Delta\\distorp\/R^2, \\M, R)$, where $\\distorp$ and  $\\distorn$ are constants that depend on $R$ and the bounds on the sectional curvature of $\\M$.\n\\end{theorem}\n\n\\begin{example}\\label{example:application_of_reduction_to_st_convex}\n    Applying reduction \\cref{thm:reduction_to_g_st_convex} to the algorithm in \\cref{coroll:acceleration_st_g_convex} we can optimize $L$-smooth and g-convex functions defined on $\\H$ or $\\S$ with a gradient oracle complexity of $\\bigotilde{L\/\\sqrt{\\epsilon}}$. \n\\end{example}\n\nNote that this reduction cannot be applied to the locally accelerated algorithm in \\citep{zhang2018towards}, that we discussed in the related work section. The reduction runs in stages by adding decreasing $\\mu_i$-strongly convex regularizers until we reach $\\mu_i =O(\\epsilon)$. The local assumption required by the algorithm in \\citep{zhang2018towards} on the closeness to the minimum cannot be guaranteed. In \\citep{ahn2020nesterov}, the authors give an unconstrained global algorithm whose rates are strictly better than RGD. The reduction could be applied to a constrained version of this algorithm to obtain a method for smooth and g-convex functions defined on manifolds of bounded sectional curvature and whose rates are strictly better than RGD. \n\n\\section{Conclusion}\n\nIn this work we proposed a first-order method with the same rates as AGD, for the optimization of smooth and g-convex or strongly g-convex functions defined on a manifold other than the Euclidean space, up to constants and log factors. We focused on the hyperbolic and spherical spaces, that have constant sectional curvature. The study of geometric properties for the constant sectional curvature case can be usually employed to conclude that a space of bounded sectional curvature satisfies a property that is in between the ones for the cases of constant extremal sectional curvature. Several previous algorithms have been developed for the optimization in Riemannian manifolds of bounded sectional curvature by utilizing this philosophy, for instance \\cite{ahn2020nesterov,ferreira2019gradient,DBLP:journals\/jota\/WangLY15,zhang2016first,zhang2018towards}. In future work, we will attempt to use the techniques and insights developed in this work to give an algorithm with the same rates as AGD for manifolds of bounded sectional curvature.\n\nThe key technique of our algorithm is the effective lower bound aggregation. Indeed, lower bound aggregation is the main hurdle to obtain accelerated first-order methods defined on Riemannian manifolds. Whereas the process of obtaining effective decreasing upper bounds on the function works similarly as in the Euclidean space---the same approach of locally minimizing the upper bound given by the smoothness assumption is used---obtaining adequate lower bounds proves to be a difficult task. We usually want a simple lower bound such that it, or a regularized version of it, can be easily optimized globally. We also want that the lower bound combines the knowledge that the g-convexity or g-strong convexity provides for all the queried points, commonly an average. These Riemannian convexity assumptions provide simple lower bounds, namely linear or quadratic, but each with respect to each of the tangent spaces of the queried points only. The deformations of the space complicate the aggregation of the lower bounds. Our work deals with this problem by finding appropriate lower bounds via the use of a geodesic map and takes into account the deformations incurred to derive a fully accelerated algorithm. We also needed to deal with other technical problems. Firstly, we needed a lower bound on the whole function and not only on $F(x^\\ast)$, for which we had to construct two different affine lower bounds, obtaining a relaxation of convexity. Secondly, we had to use an implicit discretization of an accelerated continuous dynamics, since at least the vanilla application of usual approaches like Linear Coupling \\cite{allen2014linear} or Nesterov's estimate sequence \\cite{nesterov1983method}, that can be seen as a forward Euler discretization of the accelerated dynamics combined with a balancing gradient step \\cite{diakonikolas2017approximate}, did not work in our constrained case. We interpret that the difficulty arises from trying to keep the gradient step inside the constraints while being able to compensate for a lower bound that is looser by a constant factor.\n\nWe also provided two reductions, which are generally useful for designing new algorithms and proving new lower bounds. Improving the reduction to smooth and g-strongly convex functions with respect to the curvature constants is another interesting direction for future work.\n\n\n\\ackShort{We thank Mario Lezcano-Casado for helpful discussions on this work. We thank Varun Kanade and Patrick Rebeschini for proofreading of this work. This work was supported by EP\/N509711\/1 from the EPSRC MPLS division, grant No 2053152.}\n\n\n\\nolinks\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nReinforcement learning (RL) has achieved impressive performance on many continuous control tasks~\\cite{schulman2015high,lillicrap2015continuous}, and\npolicy optimization is one of the main workhorses for such applications \\cite{duan2016benchmarking,sutton2000policy,schulman2015trust,schulman2017proximal}. \nRecently, there have been extensive research efforts studying the global convergence properties of policy optimization methods on benchmark control problems including linear quadratic regulator (LQR)~\\cite{pmlr-v80-fazel18a,bu2019lqr,malik2019derivative,yang2019provably,mohammadi2021convergence,furieri2020learning,hambly2021policy}, stabilization \\cite{perdomo2021stabilizing,ozaslan2022computing}, linear robust\/risk-sensitive control~\\cite{zhang2021policy,zhang2020stability,gravell2020learning,zhang2021derivative,zhao2021primal,cui2022mixed}, Markov jump linear quadratic control~\\cite{jansch2020convergence,jansch2020policyMDP,jansch2020policy,rathod2021global}, Lur'e system control~\\cite{qu2021exploiting}, output feedback control~\\cite{fatkhullin2020optimizing,zheng2021analysis,li2021distributed,duan2021optimization,duan2022optimization,mohammadi2021lack,zheng2022escaping}, and dynamic filtering \\cite{umenberger2022globally}.\nFor all these benchmark problems, the objective function in the policy optimization formulation is always differentiable over the entire feasible set, and the existing convergence theory heavily relies on this fact.\nConsequently, an important open question remains whether direct policy search can enjoy similar global convergence properties when applied to the famous $\\mathcal{H}_\\infty$ control problem whose objective function can be non-differentiable over certain points in the policy space \\cite{apkarian2006controller,apkarian2006nonsmooth,arzelier2011h2,gumussoy2009multiobjective,burke2020gradient,curtis2017bfgs,noll2005spectral}. \nDifferent from LQR which considers stochastic disturbance sequences,  $\\mathcal{H}_\\infty$ control directly addresses the worst-case disturbance, and provides arguably the most fundamental robust control paradigm  \\cite{zhou96,Dullerud99,skogestad2007multivariable,basar95,doyle1988state,Gahinet1994}. \nRegarding the connection with RL, it has also been shown that $\\mathcal{H}_\\infty$ control can be applied to stabilize the training of adversarial RL schemes in the linear quadratic setup \\cite[Section 5]{zhang2020stability}.\nGiven the fundamental importance of $\\mathcal{H}_\\infty$ control,  we view it as an important benchmark for understanding the theoretical properties of direct policy search in the context of robust control and adversarial RL.  In this work, we study and prove the global convergence properties of direct policy search on the $\\mathcal{H}_\\infty$ state-feedback synthesis problem. \n\n\n\n\n\n\n\n\n\nThe objective of the $\\mathcal{H}_\\infty$ state-feedback synthesis is to design a linear state-feedback policy that stabilizes the closed-loop system and minimizes the $\\mathcal{H}_\\infty$ norm from the disturbance to a performance signal at the same time.\nThe design goal is also equivalent to synthesizing a state-feedback policy that minimizes a quadratic cost subject to the worst-case disturbance. We will present the problem formulation for the $\\mathcal{H}_\\infty$ state-feedback synthesis and discuss such connections in Section~\\ref{sec:PF}. Essentially, $\\mathcal{H}_\\infty$ state-feedback synthesis can be formulated as a constrained policy optimization\nproblem $\\min_{K\\in\\mathcal{K}} J(K)$, where the decision variable $K$ is a matrix parameterizing the linear state-feedback policy, the objective function $J(K)$ is the closed-loop $\\mathcal{H}_\\infty$-norm for given $K$, and the feasible set $\\mathcal{K}$ consists of all the linear state-feedback policies stabilizing the closed-loop dynamics. Notice that the feasible set for the $\\mathcal{H}_\\infty$ state-feedback control problem is the same as the nonconvex feasible set for the LQR policy search problem~\\cite{pmlr-v80-fazel18a,bu2019lqr}. However, the objective function $J(K)$ for the $\\mathcal{H}_\\infty$ control problem\ncan be non-differential over certain feasible points, introducing new difficulty to direct policy search. There has been a large family of nonsmooth $\\mathcal{H}_\\infty$ policy search algorithms developed based on the concept of Clarke subdifferential \\cite{apkarian2006controller,apkarian2006nonsmooth,arzelier2011h2,gumussoy2009multiobjective,burke2020gradient,curtis2017bfgs}. \nHowever, a satisfying global convergence theory is still missing from the literature. Our paper bridges this gap by making the following two contributions.\n\\begin{enumerate}\n\\item We show that all Clarke stationary points for the $\\mathcal{H}_\\infty$ state-feedback policy search problem are also global minimum. \n\\item We identify the coerciveness of the $\\mathcal{H}_\\infty$ cost function and use this property to show that Goldstein's subgradient method \\cite{goldstein1977optimization} and its implementable variants \\cite{pmlr-v119-zhang20p, davis2021gradient,burke2020gradient,burke2005robust,kiwiel2007convergence,kiwiel2010nonderivative} can be guaranteed to stay in the nonconvex feasible set of stabilizing policies during the optimization process and eventually find the global optimal solution of the $\\mathcal{H}_\\infty$ state-feedback control problem. Finite-time complexity bounds for finding $(\\delta,\\epsilon)$-stationary points are also provided.\n\\end{enumerate}\nOur work sheds new light on the theoretical properties of policy optimization methods on $\\mathcal{H}_\\infty$ control problems, and serves as a meaningful initial step towards a general global convergence theory of direct policy search on nonsmooth robust control synthesis.\n\nFinally, it is worth clarifying the differences between $\\mathcal{H}_\\infty$ control and mixed $\\mathcal{H}_2\/\\mathcal{H}_\\infty$ design. For mixed $\\mathcal{H}_2\/\\mathcal{H}_\\infty$ control, the objective is to design a stabilizing policy that minimizes an $\\mathcal{H}_2$ performance bound and satisfies an $\\mathcal{H}_\\infty$ constraint at the same time \\cite{glover1988state,khargonekar1991mixed,kaminer1993mixed,mustafa1991lqg}. In other words, mixed $\\mathcal{H}_2\/\\mathcal{H}_\\infty$ control aims at improving the average $\\mathcal{H}_2$ performance while ``maintaining\" a certain level of robustness by keeping the closed-loop $\\mathcal{H}_\\infty$ norm to be smaller than a pre-specified number. In contrast, $\\mathcal{H}_\\infty$ control aims at ``improving\" the system robustness and the worst-case performance via achieving the smallest closed-loop $\\mathcal{H}_\\infty$ norm. In \\cite{zhang2021policy}, it has been shown that the natural policy gradient method initialized from a policy satisfying the $\\mathcal{H}_\\infty$ constraint can be guaranteed to maintain the $\\mathcal{H}_\\infty$ requirement during the optimization process and eventually converge to the optimal solution of the mixed design problem. However,  notice that the objective function for the mixed $\\mathcal{H}_2\/\\mathcal{H}_\\infty$ control problem is still differentiable over all the feasible points, and hence the analysis technique in \\cite{zhang2021policy} cannot be applied to our $\\mathcal{H}_\\infty$ control setting. \nMore discussions on the connections and differences between these two problems will be given in the supplementary material. \n\n\n\n\n\n\n\n\\section{Problem Formulation and Preliminaries}\n\\label{sec:PF}\n\\subsection{Notation}\nThe set of $p$-dimensional real vectors is denoted as $\\field{R}^p$.\nFor a matrix $A$, we use the notation \\(A^\\mathsf{T}\\), \\( \\|A\\| \\),  \\(\\tr{A}\\), \\(\\sigma_{\\min}(A)\\), \\(\\norm{A}_2\\), and \\(\\rho(A) \\) to denote its transpose, largest singular value, trace, smallest singular value, Frobenius norm, and spectral radius, respectively. \nWhen a\nmatrix $P$ is negative semidefinite (definite), we will use the notation $P \\preceq (\\prec) 0$. When $P$ is positive\nsemidefinite (definite), we use the notation $P \\succeq (\\succ) 0$.\nConsider a (real) sequence $\\mathbf{u}:=\\{u_0,u_1,\\cdots\\}$ where\n$u_t \\in \\field{R}^{n_u}$ for all $t$. This sequence is said to be in $\\ell_2^{n_u}$\nif $ \\sum_{t=0}^\\infty \\| u_t\\|^2<\\infty$ where $\\|u_t\\|$ denotes\nthe standard (vector) 2-norm of $u_t$.  In addition, the $2$-norm for\n$\\mathbf{u} \\in \\ell_2^{n_u}$ is defined as\n$\\|\\mathbf{u}\\|^2:=\\sum_{t=0}^\\infty \\| u_t\\|^2$.  \n\n\n\n\n\n\n\n\\subsection{Problem statement: $\\mathcal{H}_\\infty$ state-feedback synthesis and a policy optimization formulation}\n\n\nWe consider the following linear time-invariant (LTI)  system\n\\begin{align}\\label{eq:lti1}\nx_{t+1}=Ax_t+Bu_t+w_t, \\,\\,x_0=0\n\\end{align}\nwhere $x_t\\in\\field{R}^{n_x}$ is the state, $u_t\\in\\field{R}^{n_u}$ is the control action, and $w_t\\in\\field{R}^{n_w}$ is the disturbance. We have $A\\in \\field{R}^{n_x\\times n_x}$, $B\\in \\field{R}^{n_x\\times n_u}$, and $n_w=n_x$. \nWe denote $\\mathbf{x}:=\\{x_0,x_1,\\cdots\\}$, $\\mathbf{u}:=\\{u_0,u_1,\\cdots\\}$, and $\\mathbf{w}:=\\{w_0, w_1, \\cdots\\}$.\nThe initial condition is fixed as $x_0=0$.\nThe objective of $\\mathcal{H}_\\infty$ control is to choose $\\{u_t\\}$ to minimize the quadratic cost $\\sum_{t=0}^\\infty (x_t^\\mathsf{T} Q x_t+u_t^\\mathsf{T} R u_t)$ in the presence of the worst-case $\\ell_2$ disturbance satisfying $\\norm{\\mathbf{w}}\\le 1$. In this paper, the following assumption is adopted.\n\n\n\n\\begin{assumption}\\label{assump1}\nThe matrices $Q$ and $R$ are positive definite. The matrix pair $(A,B)$ is stabilizable.\n\\end{assumption}\n\n\n\n\n\n In $\\mathcal{H}_\\infty$ control, $\\{w_t\\}$ is considered to be the worst-case disturbance satisfying the $\\ell_2$ norm bound $\\norm{\\mathbf{w}}\\le 1$, and can be chosen in an adversarial manner. \nThis is different from LQR which makes stochastic assumptions on $\\{w_t\\}$. \nWithout loss of generality, we have chosen the $\\ell_2$ upper bound on $\\mathbf{w}$ to be $1$. In principle, we can formulate the $\\mathcal{H}_\\infty$ control problem with any arbitrary $\\ell_2$ upper bound on $\\mathbf{w}$, and there is no technical difference. We will provide more explanations on this fact in the supplementary material.\nTherefore, $\\mathcal{H}_\\infty$ control can be formulated  as the following minimax problem\n\\begin{align}\\label{eq:minmax}\n\\min_{\\mathbf{u}}\\max_{\\mathbf{w}:\\norm{\\mathbf{w}}\\le 1} \\sum_{t=0}^\\infty (x_t^\\mathsf{T} Q x_t+ u_t^\\mathsf{T} R u_t)\n\\end{align}\nUnder Assumption \\ref{assump1}, it is well known that \nthe optimal solution for \\eqref{eq:minmax} can be achieved using a linear state-feedback policy $u_t=-Kx_t$ (see \\cite{basar95}).\nGiven any $K$, \nthe LTI system \\eqref{eq:lti1} can be rewritten as\n\\begin{align}\\label{eq:lti2}\nx_{t+1}=(A-BK)x_t+w_t, \\,x_0=0.\n\\end{align}\nNow we define $z_t=(Q+K^\\mathsf{T} R K)^{\\frac{1}{2}}x_t$. We have $\\norm{z_t}^2=x_t^\\mathsf{T} (Q+K^\\mathsf{T} R K) x_t=x_t^\\mathsf{T} Q x_t+u_t^\\mathsf{T} R u_t$. \nWe denote $\\mathbf{z}:=\\{z_0,z_1,\\cdots\\}$. If $\\mathbf{x}\\in \\ell_2^{n_x}$, then we have $\\norm{\\mathbf{z}}^2=\\sum_{t=0}^\\infty (x_t^\\mathsf{T} Q x_t+u_t^\\mathsf{T} R u_t)<+\\infty$. \nTherefore, the closed-loop LTI system \\eqref{eq:lti2} can be viewed as a linear operator mapping any disturbance sequence $\\{w_t\\}$ to another sequence $\\{z_t\\}$. We denote this operator as $G_K$, where the subscript highlights the dependence of this operator on $K$.\nIf $K$ is stabilizing, i.e. $\\rho(A-BK)<1$, then $G_K$ is bounded in the sense that it maps any $\\ell_2$ sequence  $\\mathbf{w}$ to \nanother sequence $\\mathbf{z}$ in $\\ell_2^{n_x}$. For any stabilizing $K$, the $\\ell_2\\rightarrow \\ell_2$ induced norm of $G_K$ can be defined as:\n\\begin{align}\n\\norm{G_K}_{2\\rightarrow 2}:=\\sup_{0\\neq \\norm{\\mathbf{w}}\\le 1}\\frac{\\norm{\\mathbf{z}}}{\\norm{\\mathbf{w}}}\n\\end{align}\nSince $G_K$ is a linear operator, it is straightforward to show\n\\begin{align*}\n\\norm{G_K}_{2\\rightarrow 2}^2:=\\max_{\\mathbf{w}:\\norm{\\mathbf{w}}\\le 1} \\sum_{t=0}^\\infty x_t^\\mathsf{T} (Q+K^\\mathsf{T} R K) x_t=\\max_{\\mathbf{w}:\\norm{\\mathbf{w}}\\le 1} \\sum_{t=0}^\\infty (x_t^\\mathsf{T} Q x_t+u_t^\\mathsf{T} R u_t).\n\\end{align*}\n\nTherefore, the minimax optimization problem \\eqref{eq:minmax} can be rewritten as the policy optimization problem:\n$\\min_{K\\in\\mathcal{K}}\\norm{G_K}_{2\\rightarrow 2}^2$, where $\\mathcal{K}$ is the set of all linear state-feedback stabilizing policies, i.e. $\\mathcal{K}=\\{K\\in\\field{R}^{n_x\\times n_u}: \\,\\rho(A-BK)<1\\}$.  In the robust control literature \\cite{apkarian2006controller,apkarian2006nonsmooth,arzelier2011h2,gumussoy2009multiobjective,burke2020gradient,curtis2017bfgs}, it is standard to drop the square in the cost function and just reformulate \\eqref{eq:minmax} as $\\min_{K\\in\\mathcal{K}} \\norm{G_K}_{2\\rightarrow 2}$. This is exactly the policy optimization formulation for $\\mathcal{H}_\\infty$ state-feedback control.\nThe main reason why\nthis problem is termed as $\\mathcal{H}_\\infty$ state-feedback control is that in the frequency domain, $G_K$ can be viewed as a transfer function which lives in the Hardy $\\mathcal{H}_\\infty$ space and has an $\\mathcal{H}_\\infty$ norm being  exactly equal to $\\norm{G_K}_{2\\rightarrow 2}$. \nApplying the frequency-domain formula for the $\\mathcal{H}_\\infty$ norm, we can calculate $\\norm{G_K}_{2\\rightarrow 2}$ as\n\\begin{align}\\label{eq:hinfcost}\n\\norm{G_K}_{2\\rightarrow 2}=\\sup_{\\omega\\in[0, 2\\pi]}\\lambda_{\\max}^{1\/2}\\big((e^{-j\\omega}I-A+BK)^{-\\mathsf{T}}(Q+K^{\\mathsf{T}}RK)(e^{j\\omega}I-A+BK)^{-1}\\big),\n\\end{align}\nwhere $I$ is the identity matrix, and $\\lambda_{\\max}$ denotes the largest eigenvalue of a given symmetric matrix.\nTherefore, eventually the $\\mathcal{H}_\\infty$ state-feedback control problem can be formulated as \n\\begin{align}\\label{eq:hinfopt}\n\\min_{K\\in\\mathcal{K}} J(K),\n\\end{align}\nwhere $J(K)$ is equal to the $\\mathcal{H}_\\infty$ norm specified by \\eqref{eq:hinfcost}. Classical $\\mathcal{H}_\\infty$ control theory typically solves \\eqref{eq:hinfopt} via introducing extra Lyapunov variables and\nreparameterizing the problem into a higher-dimensional convex domain\nover which convex optimization algorithms can be applied~\\cite{zhou96,Dullerud99,befb94}.\nIn this paper, we revisit \\eqref{eq:hinfopt} as a benchmark for direct policy search, and discuss how to search the optimal solution of \\eqref{eq:hinfopt} in the policy space directly.  Applying direct policy search to address \\eqref{eq:hinfopt} leads to a nonconvex nonsmooth optimization problem.\nA main technical challenge is that the objective function \\eqref{eq:hinfcost} can be non-differentiable over some important feasible points \\cite{apkarian2006controller,apkarian2006nonsmooth,arzelier2011h2,gumussoy2009multiobjective,burke2020gradient,curtis2017bfgs}.\n\n\n\\subsection{Direct policy search: A nonsmooth optimization perspective}\nNow we briefly review several key facts known for the $\\mathcal{H}_\\infty$ policy optimization problem \\eqref{eq:hinfopt}. \n\\begin{prop} \\label{prop:1}\nThe set $\\mathcal{K}=\\{K: \\rho(A-BK)<1\\}$ is open. In general, it can be unbounded and nonconvex. The cost function \\eqref{eq:hinfcost} is continuous and nonconvex in $K$.\n\\end{prop}\nSee \\cite{pmlr-v80-fazel18a,bu2019topological} for some related proofs.  We have also included more explanations in the supplementary material. An immediate consequence is that \\eqref{eq:hinfopt} becomes a nonconvex optimization problem.\nAnother important fact is that the objective function \\eqref{eq:hinfcost} is also nonsmooth. As a matter of fact, \\eqref{eq:hinfcost} is subject to two sources of nonsmoothness.  Based on \\eqref{eq:hinfcost}, we can see that the largest eigenvalue for a fixed frequency $\\omega$ is nonsmooth, and the optimization step over $\\omega\\in [0, 2\\pi]$ is also nonsmooth. As a matter of fact, the $\\mathcal{H}_\\infty$ objective function \\eqref{eq:hinfcost} can be non-differentiable over important feasible points, e.g. optimal points. \nFortunately, it is well known\\footnote{We cannot find a formal statement of Proposition \\ref{prop:regular} in the literature. However, based on our discussion with other researchers who have worked on nonsmooth $\\mathcal{H}_\\infty$ synthesis for long time, this fact is well known and hence we do not claim any credits in deriving this result. As a matter of fact, although not explicitly stated, the proof of Proposition \\ref{prop:regular} is hinted in the last paragraph of \\cite[Section III]{apkarian2006nonsmooth} given the facts that the $\\mathcal{H}_\\infty$ norm is a convex function over the Hardy $\\mathcal{H}_\\infty$ space (which is a Banach space) and the mapping from $K\\in\\mathcal{K}$ to the (infinite-dimensional) Hardy $\\mathcal{H}_\\infty$ space is strictly differentiable.\nFor completeness, a simple proof of Proposition~\\ref{prop:regular} based on Clarke's chain rule \\cite{clarke1990optimization} is included in the supplementary material.} that the $\\mathcal{H}_\\infty$ objective function \\eqref{eq:hinfcost} has the following desired property so it is Clarke subdifferentiable. \n\\begin{prop}\\label{prop:regular}\nThe $\\mathcal{H}_\\infty$ objective function \\eqref{eq:hinfcost} is  locally Lipschitz and subdifferentially regular over the stabilizing feasible set $\\mathcal{K}$.\n\\end{prop}\n\nRecall that  $J:\\mathcal{K}\\rightarrow \\field{R}$ is locally Lipschitz if for\nany bounded $S\\subset \\mathcal{K}$, there exists a constant $L > 0$ such that $|J(K)-J(K')|\\le L \\norm{K-K'}_2$ for all $K,K'\\in S$. \nBased on Rademacher's theorem, a locally Lipschitz function is differentiable almost everywhere, and the Clarke subdifferential is well defined for all feasible points. Formally, the Clarke subdifferential is defined as \n\\begin{align}\n\\partial_C J(K):=\\conv\\{\\lim_{i\\rightarrow \\infty}\\nabla J(K_i):K_i\\rightarrow K,\\,K_i\\in\\dom(\\nabla J)\\subset \\mathcal{K}\\}\n\\end{align}\nwhere $\\conv$ denotes the convex hull. Then we know that the Clarke subdifferential for the $\\mathcal{H}_\\infty$ objective function \\eqref{eq:hinfcost} is well defined for all $K\\in \\mathcal{K}$. We say that $K$ is a Clarke stationary point if  $0\\in \\partial_C J(K)$. The following fact is also well known.\n\\begin{prop} \\label{pro3}\nIf $K$ is a local min of $J$, then $0\\in\\partial_C J(K)$ and $K$ is a Clarke stationary point.\n\\end{prop}\nUnder Assumption \\ref{assump1}, it is well known that there exists $K^*\\in \\mathcal{K}$ achieving the minimum of~\\eqref{eq:hinfopt}. Since $\\mathcal{K}$ is an open set, $K^*$ has to be an interior point of $\\mathcal{K}$ and hence $K^*$ has to be a Clarke stationary point.  In Section \\ref{sec:land}, we will prove that any Clarke stationary points for \\eqref{eq:hinfopt} are actually global minimum.\n\nNow we briefly elaborate on the subdifferentially regular property stated in Proposition \\ref{prop:regular}.\nFor any given direction $d$ (which has the same dimension as $K$), the generalized Clarke directional derivative of $J$ is defined as\n\\begin{align}\nJ^{\\circ}(K,d):=\\lim_{K'\\rightarrow K}\\sup_{t\\searrow 0} \\frac{J(K'+td)-J(K')}{t}.\n\\end{align}\nIn contrast, the (ordinary) directional derivative is defined as follows (when existing)\n\\begin{align}\nJ'(K,d):=\\lim_{t\\searrow 0} \\frac{J(K+td)-J(K)}{t}.\n\\end{align}\n\nIn general, the Clarke directional derivative can be different from the (ordinary) directional derivative. Sometimes\n the ordinary directional derivative may not even exist.\nThe objective function $J(K)$ is subdifferentially regular if for every $K\\in \\mathcal{K}$, the ordinary directional\nderivative always exists and coincides with the generalized one for every direction, i.e. $J'(K,d)=J^{\\circ}(K,d)$.\nThe most important consequence of the subdifferentially regular property is given as follows.\n\\begin{cor} \\label{cor1}\nSuppose $K ^\\dag\\in\\mathcal{K}$ is a  Clarke stationary point for $J$. If $J$ is subdifferentially regular, then the directional derivatives $J'(K^\\dag, d)$ are non-negative for all $d$.\n\\end{cor}\nSee \\cite[Theorem 10.1]{rockafellar2009variational} for related proofs and more discussions. Notice that having non-negative directional derivatives does not mean that the point $K^\\dag$ is a local minimum. Nevertheless, the above fact will be used in our main theoretical developments.\nNow we briefly summarize two key difficulties in establishing a global convergence theory for direct policy search on the $\\mathcal{H}_\\infty$ state-feedback control problem \\eqref{eq:hinfopt}. First, it is unclear whether the direct policy search method will get stuck at some local minimum. Second, it is challenging to guarantee the direct policy search method to stay in the nonconvex feasible set $\\mathcal{K}$ during the optimization process. Since $\\mathcal{K}$ is nonconvex, we cannot use a projection step to maintain feasibility. Our main results will address these two issues. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Goldstein subdifferential}\n\nGenerating a good descent direction for nonsmooth optimization is not trivial. Many nonsmooth optimization algorithms are based on the concept of Goldstein subdifferential \\cite{goldstein1977optimization}. Before proceeding to our main result, we briefly review this concept here.\n\n\\begin{defn}[Goldstein subdifferential]\nSuppose $J$ is locally Lipschitz. Given a point $K\\in\\mathcal{K}$ and a parameter $\\delta>0$, the Goldstein subdifferential of $J$ at $K$ is defined to be the following set \n\\begin{align} \\label{Gold_sub}\n\\partial_\\delta J(K):=\\conv \\left\\{\\cup_{K'\\in\\mathbb{B}_\\delta(K)} \\partial_C J(K')\\right\\},\n\\end{align}\nwhere $\\mathbb{B}_\\delta(K)$ denotes the $\\delta$-ball around $K$. The above definition implicitly requires $\\mathbb{B}_\\delta(K)\\subset\\mathcal{K}$.\n\\end{defn}\nBased on the above definition, one can further define the notion of $(\\delta,\\epsilon)$-stationarity.\nA point $K$ is said to be $(\\delta,\\epsilon)$-stationary if $\\dist(0, \\partial_\\delta J(K))\\le \\epsilon$.\n It is well-known that the minimal norm\nelement of the Goldstein subdifferential generates a good descent direction. This fact is stated as follows.\n\\begin{prop}[\\cite{goldstein1977optimization}]\nLet $F$ be the minimal norm element in $\\partial_\\delta J(K)$. Suppose $K-\\alpha F\/\\norm{F}_2\\in \\mathcal{K}$ for any $0\\le \\alpha \\le \\delta$. \nThen we have \n\\begin{align}\\label{eq:descent}\nJ(K-\\delta F\/\\norm{F}_2)\\le J(K)-\\delta \\norm{F}_2.\n\\end{align}\n\\end{prop}\nThe idea of Goldstein subdifferential has been used in designing algorithms for nonsmooth $\\mathcal{H}_\\infty$ control \\cite{arzelier2011h2,gumussoy2009multiobjective,burke2020gradient,curtis2017bfgs}. We will show that such policy search algorithms  can be guaranteed to find the global minimum of \\eqref{eq:hinfopt}. It is worth mentioning that there are other notions of enlarged subdifferential~\\cite{apkarian2006nonsmooth} which can lead to good descent directions for nonsmooth $\\mathcal{H}_\\infty$ synthesis. In this paper, we focus on the notion of Goldstein subdifferential and related policy search algorithms. \n\n\n\n\n\\section{Optimization Landscape for $\\mathcal{H}_\\infty$ State-Feedback Control}\n\\label{sec:land}\n\nIn this section, we investigate the optimization landscape of the $\\mathcal{H}_\\infty$ state-feedback policy search problem, and show that any Clarke stationary points of \\eqref{eq:hinfopt} are also global minimum. We start by showing the coerciveness of the $\\mathcal{H}_\\infty$ objective function \\eqref{eq:hinfcost}.\n\\begin{lem}\\label{lem1}\nThe $\\mathcal{H}_\\infty$ objective function $J(K)$ defined by \\eqref{eq:hinfcost} is coercive over the set $\\mathcal{K}$ in the sense that for any sequence $\\{K^l\\}_{l=1}^\\infty\\subset \\mathcal{K}$ we have\n    $J(K^l) \\rightarrow +\\infty$,  \nif either $\\|K^l\\|_2 \\rightarrow +\\infty$, or  $K^l$ converges to an element in the boundary $\\partial \\mathcal{K}$.\n\\end{lem}\n\\begin{proof}\nWe will only provide a proof sketch here. A detailed proof is presented in the supplementary material. Suppose we have a sequence $\\{K^l\\}$ satisfying $\\norm{K^l}_2\\rightarrow +\\infty$. We can choose $\\mathbf{w}=\\{w_0,0,0,\\cdots\\}$ with $\\norm{w_0}=1$ and show \n$J(K^l)\\ge  w_0^\\mathsf{T} (Q+(K^l)^\\mathsf{T} R K^l) w_0 \\ge \\lambda_{\\min}(R) \\norm{K^l w_0}^2$.\nClearly, we have used the positive definiteness of $R$ in the above derivation. Then by carefully choosing $w_0$, we can ensure $J(K^l)\\rightarrow +\\infty$ as $\\norm{K^l}_2\\rightarrow +\\infty$.  Next, we assume $K^l\\rightarrow K\\in \\partial \\mathcal{K}$.\nWe have $\\rho(A-BK)=1$, and hence there exists some $\\omega_0$ such that $(e^{j\\omega_0}I-A+BK)$ becomes singular.\nThen we can use the positive definiteness of $Q$ to show\n$J(K^l)\\ge \\lambda^{1\/2}_{\\min}(Q)   (\\|  (e^{j\\omega_0}I-A+BK^l)^{-1}   \\|\\cdot  \\|  (e^{-j\\omega_0}I-A+BK^l)^{-1}   \\|)^{\\frac{1}{2}}$.\nNotice $\\sigma_{\\min} (e^{\\pm j\\omega_0}I-A+BK^l) \\to 0$ as $l \\to \\infty$, which implies $  \\|  (e^{\\pm j\\omega_0}I-A+BK^l)^{-1}   \\| \\to +\\infty$ as $l \\to \\infty$. Therefore, we have $J(K^l) \\to +\\infty$ as $K^l\\to K\\in \\partial \\mathcal{K}$.\nMore details for the proof can be found in the supplementary material.\n\\end{proof}\nWe want to emphasize that the positive definiteness of $(Q,R)$ are crucial for proving the coerciveness of the cost function \\eqref{eq:hinfcost}.  Built upon Lemma \\ref{lem1}, we can obtain the following nice properties of the sublevel sets of \\eqref{eq:hinfopt}. \n\n\n\\begin{lem}\\label{lem2}\nConsider the $\\mathcal{H}_\\infty$ state-feedback policy search problem \\eqref{eq:hinfopt} with the objective function $J(K)$ defined in \\eqref{eq:hinfcost}. Under Assumption \\ref{assump1}, the sublevel set defined as $\\mathcal{K}_\\gamma:=\\{K\\in \\mathcal{K}: J(K)\\le \\gamma\\}$ is compact and path-connected for every $\\gamma\\ge J(K^*)$ where $K^*$ is the global minimum of \\eqref{eq:hinfopt}.\n\\end{lem}\n\\begin{proof}\nThe compactness of $\\mathcal{K}_\\gamma$ directly follows from the continuity and coerciveness of $J(K)$, and is actually a consequence of \\cite[Proposition 11.12]{bauschke2011convex}. The path-connectedness of the strict sublevel sets for the continuous-time $\\mathcal{H}_\\infty$ control problem has been proved in \\cite{hu2022connectivity}. We can slightly modify the proof in \\cite{hu2022connectivity} to show that the strict sublevel set $\\{K\\in \\mathcal{K}: J(K)<\\gamma\\}$ is path-connected.  Based on the fact that every non-strict sublevel sets are compact, now we can apply \\cite[Theorem 5.2]{martin1982connected} to show $\\mathcal{K}_\\gamma$ is also path-connected.\nAn independent proof based on the non-strict version of the bounded real lemma is also provided in the supplementary material.\n\\end{proof}\nThe path-connectedness of $\\mathcal{K}_\\gamma$ for every $\\gamma$ actually implies the uniqueness of the minimizing set in a certain strong sense \\cite[Sections 2\\&3]{martin1982connected}. Due to the space limit, we will defer the discussion on the uniqueness of the minimizing set to the supplementary material. Here, we present a stronger result which is one of the main contributions of our paper.\n\\begin{thm}\\label{thm1}\nConsider the $\\mathcal{H}_\\infty$ state-feedback policy search problem \\eqref{eq:hinfopt}. Under Assumption \\ref{assump1}, any Clarke stationary point of $J(K)$ is a global minimum.\n\\end{thm}\n\nA detailed proof is presented in the supplementary material.  Here we provide a proof sketch.  Since $Q$ and $R$ are positive definite, the non-strict version of the bounded real lemma\\footnote{The difference between the strict and non-strict versions of the bounded real lemma is quite subtle \\cite[Section 2.7.3]{befb94}. For completeness, we will provide more explanations for the non-strict version of the bounded real lemma in the supplementary material.} states that $J(K)\\le \\gamma$ if and only if there exists a positive definite matrix $P$ such that the following matrix inequality holds\n\\begin{align}\\label{eq:lmi1}\n\\bmat{(A-BK)^\\mathsf{T} P (A-BK) - P & (A-BK)^\\mathsf{T} P \\\\ P(A-BK) & P}+\\bmat{Q+K^\\mathsf{T} R K & 0 \\\\ 0 & -\\gamma^2 I}\\preceq 0.\n\\end{align}\nThe above matrix inequality is linear in $P$ but not linear in $K$. A standard trick from the control theory can be combined with the Schur complement lemma to convert the above matrix inequality condition to another condition which is linear in all the decision variables \\cite{befb94}. Specifically, there exists a matrix function $\\lmi(Y,L,\\gamma)$ which is linear in $(Y,L,\\gamma)$ such that $\\lmi(Y,L,\\gamma)\\preceq 0$ and $Y\\succ 0$ if and only if \\eqref{eq:lmi1} is feasible with $K=L Y^{-1}$ and $P=\\gamma Y^{-1}\\succ 0$. The matrix function $\\lmi(Y,L,\\gamma)$ involves a larger matrix. Hence we present the analytical formula of $\\lmi(Y,L,\\gamma)$ in the supplementary material and skip it here. Since $\\lmi(Y,L,\\gamma)$ is linear in $(Y,L,\\gamma)$, we know $\\lmi(Y,L,\\gamma)\\preceq 0$ is just a convex semidefinite programming condition. Based on this convex necessary and sufficient condition for $J(K)\\le \\gamma$, we can prove the following important lemma. \n\\begin{lem}\\label{lem3}\nFor any $K\\in\\mathcal{K}$ satisfying $J(K)>J^*$, there exists a matrix direction $d\\neq 0$ such that $J'(K,d)\\le J^*-J(K)<0$, where $J^*=J(K^*)$ and $K^*$ is the global minimum of \\eqref{eq:hinfopt}.\n\\end{lem}\n\\begin{proof}\nSuppose we have $K=LY^{-1}$ where $(Y,L, J(K))$ is a feasible point for the convex regime $\\lmi(Y,L,J(K))\\preceq 0$.\nIn addition, we have $K^*=L^* (Y^*)^{-1}$ where $(Y^*,L^*, J(K^*))$ is a point satisfying $\\lmi(Y^*,L^*,J(K^*))\\preceq 0$.\n Since the LMI condition is convex, the line segment between $(Y,L,J(K))$ and $(Y^*,Q^*,J(K^*))$ is also in this convex set. For any $t>0$, we know $(Y+t \\Delta Y,L+t \\Delta L, J(K)+t(J(K^*)-J(K)))$ also satisfies \n$\\lmi(Y+t\\Delta Y, L+t\\Delta L, J(K)+t(J(K^*)-J(K)))\\preceq 0$, \nwhere $\\Delta L=L^*-L$, and $\\Delta Y=Y^*-Y$. \nTherefore, based on the bounded real lemma, we know $J((L+t\\Delta L) (Y+t \\Delta Y)^{-1})\\le J(K)+t(J(K^*)-J(K))$.\nLet's choose $d=\\Delta L Y^{-1} - LY^{-1}\\Delta Y Y^{-1}$. Then we have\n\\begin{align*}\nJ'(K,d)\\le \\lim_{t\\searrow 0} \\left( \\frac{J((L+t\\Delta L) (Y+t \\Delta Y)^{-1})-J(K)}{t}+o(t)\\right)\\le J^*-J(K)<0.\n\\end{align*}\nA detailed verification of the above inequality is provided in the supplementary material.  Notice $d\\neq 0$. If $\\Delta L Y^{-1} -LY^{-1}\\Delta Y Y^{-1}=0$, the above argument still works and we reach to the conclusion  $J'(K,0)<0$. But this is impossible since we always have  $J'(K,0)=0$.  Hence we have $d\\neq 0$. This completes the proof for this lemma.\n\\end{proof}\n\n\n\nNow we are ready to provide the proof for Theorem \\ref{thm1}. Based on Lemma \\ref{lem3} and the fact that $J(\\cdot)$ is subdifferentially regular, the proof  can be done by contradiction.  Suppose $K^*$ is the global minimum, and $K^\\dag$ is a Clarke stationary point. If $K^\\dag$ is not a global minimum. Then by Lemma \\ref{lem3}, there exists $d \\neq 0$ such that $J'(K^\\dag,d) < 0$, this contradicts the fact that $J'(K^\\dag,d) \\ge 0$ for all $d$ by Corollary \\ref{cor1}. Therefore, $K^\\dag$ has to be the global minimum of \\eqref{eq:hinfopt}.\n\n\n\n\n\n\nThe above proof relies on Lemma \\ref{lem3} and the fact that $J$ is subdifferentially regular.\nWithout using the subdifferentially regular property, Lemma \\ref{lem3} itself is not sufficient for proving Theorem \\ref{thm1}.\nIt is also worth mentioning that Lemma \\ref{lem3} can be viewed as a modification of the convex parameterization\/lifting results in \\cite{sun2021learning,umenberger2022globally} for non-differentiable points.\n\n\n\n\n\n\n\\section{Global Convergence of Direct Policy Search on $\\mathcal{H}_\\infty$ State-Feedback Control}\n\\label{sec:main1}\n\nIn this section, we first show that Goldstein's subgradient method \\cite{goldstein1977optimization} can be guaranteed to stay in the nonconvex feasible regime $\\mathcal{K}$ during the optimization process and eventually converge to the global minimum of \\eqref{eq:hinfopt}. \nThe complexity of finding $(\\delta,\\epsilon)$-stationary points of \\eqref{eq:hinfopt} is also presented.\nThen we further discuss the convergence guarantees for various implementable forms of Goldstein's subgradient method.\n\n\n\n\n\n\n\n\\subsection{Global convergence and complexity of Goldstein's subgradient Method}\nWe will investigate the global convergence of Goldstein's subgradient method for direct policy search of the optimal $\\mathcal{H}_\\infty$ state-feedback policy. Goldstein's subgradient method iterates as follows\n\\begin{align}\\label{eq:gold1}\nK^{n+1}=K^n-\\delta^n F^n\/\\norm{F^n}_2,\n\\end{align}\nwhere $F^n$ is the minimum norm element of the Goldstein subdifferential $\\partial_{\\delta^n} J(K^n)$.\nWe assume that an initial stabilizing policy is available, i.e. $K^0\\in\\mathcal{K}$. The same initial policy assumption has also been made in the global convergence theory for direct policy search on LQR \\cite{pmlr-v80-fazel18a}. More recently, some provable guarantees have been obtained for finding such stabilizing policies via direct policy search~\\cite{perdomo2021stabilizing,ozaslan2022computing}. Hence such an assumption on the initial policy $K^0$ is reasonable.  \nOur global convergence result relies on the fact that there is a strict separation between any sublevel set of \\eqref{eq:hinfopt} and the boundary of $\\mathcal{K}$. This fact is formalized as follows.\n\\begin{lem}\\label{lem4}\nConsider the $\\mathcal{H}_\\infty$ state-feedback policy search problem \\eqref{eq:hinfopt} with the cost function $J(K)$ defined in \\eqref{eq:hinfcost}.  Denote the complement of the feasible set $\\mathcal{K}$ as $\\mathcal{K}^c$. \nSuppose Assumption \\ref{assump1} holds and $\\gamma\\ge J^*$. Then there is a strict separation between the sublevel set $\\mathcal{K}_\\gamma$ and $\\mathcal{K}^c$. In other words, we have $dist(\\mathcal{K}_\\gamma, \\mathcal{K}^c)>0$.\n\\end{lem}\n\\begin{proof}\nObviously, the set $\\mathcal{K}_\\gamma \\cap \\mathcal{K}^c$ is empty (since we know $\\mathcal{K}_\\gamma\\subset \\mathcal{K}$). Based on Lemma \\ref{lem2}, we know $\\mathcal{K}_\\gamma$ is compact. Since $\\mathcal{K}$ is open, we know $\\mathcal{K}^c$ is closed. Therefore, there is a strict separation between $\\mathcal{K}_\\gamma$  and $\\mathcal{K}^c$, and we have $dist(\\mathcal{K}_\\gamma, \\mathcal{K}^c)>0$.\n\\end{proof}\n\nNow we are ready to present our main convergence result. \n\\begin{thm} \\label{thm2}\nConsider the $\\mathcal{H}_\\infty$ state-feedback policy search problem \\eqref{eq:hinfopt} with the cost function $J(K)$ defined in \\eqref{eq:hinfcost}. Suppose Assumption \\ref{assump1} holds, and an initial stabilizing policy is given, i.e. $K^0\\in\\mathcal{K}$. Denote $\\Delta_0:=\\dist(\\mathcal{K}_{J(K^0)}, \\mathcal{K}^c)>0$. Choose $\\delta^n=\\frac{c\\Delta_0}{n+1}$ for all $n$ with $c$ being a fixed number in $(0,1)$.\nThen Goldstein's subgradient method \\eqref{eq:gold1} is guaranteed to stay in $\\mathcal{K}$ for all $n$. In addition,  we have $J(K^n)\\rightarrow J^*$ as $n\\rightarrow \\infty$.\n\\end{thm}\n\\begin{proof}\nWe have $\\delta^n\\le c\\Delta_0< \\Delta_0$ for all $n$. Now we use an induction proof to show $K^n\\in \\mathcal{K}_{J(K^0)}$ for all $n$. For $n= 0$, we know $K^0-c\\Delta_0 F^0\/\\norm{F^0}_2$ has to be within the $\\Delta_0$ ball around $K^0$ since we know the norm of $F^0\/\\norm{F^0}_2$ is exactly equal to $1$. Since $\\Delta_0:=\\dist(\\mathcal{K}_{J(K^0)}, \\mathcal{K}^c)>0$, we know $K^0-\\delta^0 F^0\/\\norm{F^0}_2\\in \\mathcal{K}$. As a matter of fact, we know $\\mathbb{B}_{\\delta^0}(K^0)$ has to be a subset of $\\mathcal{K}$. Hence we can apply \\eqref{eq:descent} to show that $K^1$ exists and is also in $\\mathcal{K}_{J(K^0)}$. Similarly, we can repeat this argument to show $K^n\\in \\mathcal{K}_{J(K^0)}$ for all $n$. \nNext, we can apply \\eqref{eq:descent} to every step and then sum the inequalities over all $n$. Then the following inequality holds for all $N$:\n\\begin{align}\\label{eq:iterative}\n\\sum_{n=0}^N \\delta_n \\norm{F^n}_2 \\le J(K^0)-J^*\n\\end{align}\nSince we have $\\sum_{n=0}^\\infty \\delta^n=+\\infty$, we know $\\liminf_{n\\rightarrow\\infty} \\norm{F^n}_2= 0$. There exists one subsequence $\\{i_n\\}$ such that $\\norm{F^{i_n}}_2\\rightarrow 0$. For this subsequence, the resultant policy sequence $\\{K^{i_n}\\}$ is also bounded (notice that the policy parameter sequence stays in the compact set $\\mathcal{K}_{J(K^0)}$ for all $n$) and has a convergent subsequence. We can show that the limit of this subsequence is a Clarke stationary point.\nHence the function value associated with this subsequence converges to $J^*$. Notice that $J(K^n)$ is monotonically decreasing for the entire sequence $\\{n\\}$. Hence we have $J(K^n)\\rightarrow J^*$. \n\\end{proof}\n\n\n\nWe have tried to be brief in giving the above proof. We will present a more detailed proof in the supplementary material. We believe that \nthis is the first result showing that direct policy search can be guaranteed to converge to the global optimal solution of the $\\mathcal{H}_\\infty$ state-feedback control problem. The above result only provides an asymptotic convergence guarantee to ensure $J(K^n)\\rightarrow J^*$. One can use a similar argument to establish a finite-time complexity bound for finding the $(\\delta,\\epsilon)$-stationary points of \\eqref{eq:hinfopt}. Such a result is given as follows.\n\n\\begin{thm} \\label{thm3}\nConsider the $\\mathcal{H}_\\infty$ problem \\eqref{eq:hinfopt} with the cost function \\eqref{eq:hinfcost}. Suppose Assumption \\ref{assump1} holds, and $K^0\\in\\mathcal{K}$. Denote $\\Delta_0:=\\dist(\\mathcal{K}_{J(K^0)}, \\mathcal{K}^c)>0$. For any $\\delta<\\Delta_0$, we can \nchoose $\\delta^n=\\delta$ for all $n$ to ensure that\nGoldstein's subgradient method \\eqref{eq:gold1} stays in $\\mathcal{K}$ and satisfies the following finite-time complexity bound:\n\\begin{align}\n    \\min_{n:0\\le n\\le N} \\norm{F^n}_2\\le \\frac{J(K^0)-J^*}{(N+1)\\delta}\n\\end{align}\nIn other words, we have $\\min_{0\\le n\\le N} \\norm{F^n}_2\\le \\epsilon$ after $N=\\mathcal{O}\\left(\\frac{\\Delta}{\\delta\\epsilon}\\right)$ where $\\Delta:=J(K^0)-J^*$. For any $\\delta<\\Delta_0$ and $\\epsilon>0$, the complexity of finding a $(\\delta,\\epsilon)$-stationary point is $\\mathcal{O}\\left(\\frac{\\Delta}{\\delta\\epsilon}\\right)$.\n\\end{thm}\n\\begin{proof}\nThe above result can be proved using a similar argument from Theorem \\ref{thm2}. \nWe can use the same induction argument to show $K^n\\in\\mathcal{K}_{J(K^0)}$ for all $n$, and \\eqref{eq:iterative} holds with $\\delta^n=\\delta$. Then the desired conclusion directly follows.\n\\end{proof}\nThe complexity for nonsmooth optimization of Lipschitz functions is quite subtle. While the above result gives a reasonable characterization of the finite-time performance of Goldstein's subgradient method on the $\\mathcal{H}_\\infty$ state-feedback control problem, it does not quantify how fast $J(K^n)$ converges to $J^*$. Recall that $(\\delta,\\epsilon)$-stationarity means  $\\dist(0, \\partial_\\delta J(K))\\le \\epsilon$, while $\\epsilon$-stationarity means $\\dist(0, \\partial_C J(K))\\le \\epsilon$.\nAs commented in \\cite{shamir2020can,pmlr-v119-zhang20p,davis2021gradient},    $(\\delta,\\epsilon)$-stationarity does not imply being $\\delta$-close to an\n$\\epsilon$-stationary point of $J$. Importantly, the function value of a $(\\delta,\\epsilon)$-stationary point can be far from $J^*$ even for small $\\delta$ and $\\epsilon$. Theorem 5 in \\cite{pmlr-v119-zhang20p} shows that\nthere is no finite time algorithm that can find $\\epsilon$-stationary points provably for all Lipschitz functions. It is still possible that one can develop some finite time  bounds for $(J(K^n)-J^*)$ via exploiting other advanced properties of the $\\mathcal{H}_\\infty$ cost function \\eqref{eq:hinfcost}. This is an important future task.\n\n\n\n\n\n\n\n\n\n\\subsection{Implementable variants and related convergence results}\n\\label{sec:imple}\n\nIn practice, it can be difficult to evaluate the minimum norm element of the Goldstein subdifferential.\n  Now we discuss implementable variants of Goldstein's subgradient method and  related guarantees. \n\n\\textbf{Gradient sampling  \\cite{burke2020gradient,burke2005robust,kiwiel2007convergence}.} \nThe gradient sampling (GS) method \nis the main optimization algorithm used in  the robust control package HIFOO \\cite{arzelier2011h2,gumussoy2009multiobjective}.\nSuppose we can access a first-order oracle which can evaluate $\\nabla J$ for any differentiable points in the feasible set\\footnote{When $(A,B)$ is known, one can calculate the $\\mathcal{H}_\\infty$ gradient at differential points using the chain rule in \\cite{apkarian2006nonsmooth}. More explanations can be found in the supplementary material.}. Based on Rademacher's theorem, a locally Lipschitz function is differentiable almost everywhere. Therefore, for any $K^n\\in\\mathcal{K}$, we can randomly sample policy parameters over $\\mathbb{B}_{\\delta^n}(K^n)$ and obtain differentiable points with probability one. For all these sampled differentiable points, the Clarke subdifferential at each point is just the gradient. Then the convex hull of these sampled gradients can be used as an approximation for the Goldstein subdifferential $\\partial_{\\delta^n} K^n$. The minimum norm element from the convex hull of the sampled gradients can be solved via a simple convex quadratic program, and is sufficient for generating a reasonably good descent direction for updating $K^{n+1}$ as long as we sample at least $(n_x n_u+1)$ differentiable points for each $n$ \\cite{burke2020gradient}.  In the unconstrained setup, the cluster points of the GS algorithm can be guaranteed to be Clarke stationary \\cite{kiwiel2007convergence,burke2020gradient}. Such a result can be combined with Theorem \\ref{thm1} and Lemma \\ref{lem4} to show the global convergence of the GS method on the $\\mathcal{H}_\\infty$ state-feedback synthesis problem. The following theorem will be treated formally in the supplementary material.\n\\begin{thm}[Informal statement]\\label{thm4}\nConsider the policy optimization problem \\eqref{eq:hinfopt} with the $\\mathcal{H}_\\infty$ cost function defined in \\eqref{eq:hinfcost}. Suppose Assumption \\ref{assump1} holds, and $K^0\\in \\mathcal{K}$. The iterations generated from the trust-region version of the GS method (described in \\cite[Section 4.2]{kiwiel2007convergence} and restated in the supplementary material) can be guaranteed to stay in $\\mathcal{K}$ for all iterations and achieve $J(K^n)\\rightarrow J^*$ with probability~one. \n\\end{thm}\n\n\n\\textbf{Non-derivative sampling (NS) \\cite{kiwiel2010nonderivative}.} The NS method can be viewed as the derivative-free version of the GS algorithm. Suppose we only have the zeroth-order oracle which can evaluate the function value $J(K)$ for $K\\in \\mathcal{K}$. The main difference between NS and GS is that the NS algorithm relies on estimating the gradient from function values via Gupal's estimation method. In the unconstrained setting, the cluster points of the NS method can be guaranteed to be Clarke stationary with probability one \\cite[Theorem 3.8]{kiwiel2010nonderivative}. We can combine \\cite[Theorem 3.8]{kiwiel2010nonderivative} with our results (Theorem \\ref{thm1} and Lemma~\\ref{lem4}) to prove the global convergence of NS in our setting. A detailed discussion is given in the supplementary material.  \n\n\n\\textbf{Model-free implementation of NS.}  When the system model is unknown,  there are various methods available for estimating the $\\mathcal{H}_\\infty$-norm from data \\cite{muller2019gain,muller2017stochastic,rojas2012analyzing,rallo2017data,wahlberg2010non,oomen2014iterative,tu2019minimax,tu2018approximation}.  \nBased on our own experiences\/tests, the multi-input multi-output (MIMO) power iteration method~\\cite{oomen2013iteratively} works quite well as a stochastic zeroth-order oracle for the purpose of implementing NS in the model-free setting.  While the sample complexity for model-free NS is unknown, we will provide some numerical justifications to show that such a model-free implementation closely tracks the convergence behaviors of its model-based counterpart.  \n\n\n\\textbf{Interpolated normalized gradient descent (INGD) with finite-time complexity.} No finite-time guarantees for finding $(\\delta,\\epsilon)$-stationary points have been reported for the GS\/NS methods. In \\cite{pmlr-v119-zhang20p,davis2021gradient}, the INGD method has been developed as another implementable variant of Goldstein's subgradient method, and is proved to satisfy high-probability finite-time complexity bounds for finding $(\\delta,\\epsilon)$-stationary points of Lipschitz functions. INGD uses an iterative sampling strategy to generate a descent direction which serves a role similar to the minimal norm element of the Goldstein subdifferential. A first-order oracle for differentiable points is needed for implementing the version of INGD in \\cite{davis2021gradient}. \nIt has been show \\cite{pmlr-v119-zhang20p,davis2021gradient} that for unconstrained nonsmooth optimization of $L$-Lipschitz functions\\footnote{We slightly abuse our notation by denoting the Lipschitz constant as $L$. Previously, we have used $L$ to denote a particular matrix used in the LMI formulation for $\\mathcal{H}_\\infty$ state-feedback synthesis.}, the INGD algorithm can be guaranteed to find the $(\\delta,\\epsilon)$-stationary point with the high-probability iteration complexity $\\mathcal{O}\\left(\\frac{\\Delta L^2}{\\epsilon^3 \\delta}\\log(\\frac{\\Delta}{p\\delta\\epsilon})\\right)$, where $\\Delta:=J(K^0)-J^*$ is the initial function value gap, and $p$ is the failure probability (i.e. the optimization succeeds with the probability $(1-p)$).\nWe can combine the proofs for\n\\cite[Theorem 2.6]{davis2021gradient} and  Theorem \\ref{thm3} to obtain the following complexity result for our $\\mathcal{H}_\\infty$ setting. A formal treatment is given in the supplementary material.\n\n\\begin{thm}[Informal statement]\\label{thm5}\nConsider the policy optimization problem \\eqref{eq:hinfopt} with the $\\mathcal{H}_\\infty$ cost function defined in \\eqref{eq:hinfcost}. Suppose Assumption \\ref{assump1} holds, and the initial policy is stabilizing, i.e. $K^0\\in \\mathcal{K}$.  Denote $\\Delta_0:=\\dist(\\mathcal{K}_{J(K^0)}, \\mathcal{K}^c)>0$, and let $L_0$ be the Lipschitz constant of $J(K)$ over the set $\\mathcal{K}_{J(K^0)}$. For any $\\delta<\\Delta_0$, we can \nchoose $\\delta^n=\\delta$ for all $n$ to ensure that the iterations of\nthe INGD algorithm stay in $\\mathcal{K}$ almost surely, and find a $(\\delta,\\epsilon)$-stationary point with the high-probability iteration complexity $\\mathcal{O}\\left(\\frac{\\Delta L_0^2}{\\epsilon^3 \\delta}\\log(\\frac{\\Delta}{p\\delta\\epsilon})\\right)$, where $p$ is the failure probability.\n\\end{thm}\n\n\\section{Numerical Simulations}\nTo support our theory, we provide some numerical simulations in this section. The left plot in Figure~\\ref{Simulationrelts} shows that GS, NS, INGD, and model-free NS work well for the following example:\n\\begin{equation} \\label{set_matric}\n    A = \\bmat{1 &0 &-5 \\\\ -1 &1 &0\\\\ 0 &0 &1},\\,\\, B= \\bmat{1 \\\\ 0 \\\\ -1},\\,\\, Q = \\bmat{2 &-1 &0 \\\\ -1 &2 &-1 \\\\ 0 &-1 &2}, \\,\\, R = 1.\n\\end{equation}\nFor this example, we have $J^* = 7.3475$.  We initialize from $K^0 = \\bmat{0.4931 &-0.1368 &-2.2654}$, which satisfies $\\rho(A-BK^0) = 0.5756 < 1$. The hyperparameter choices are detailed in the supplementary material. We can see that model-free NS closely tracks the trajectory of NS and works well.\nIn the middle plot of Figure \\ref{Simulationrelts}, we  test the NS method on randomly generated cases. We set $A\\in \\mathbb{R}^{3\\times 3}$ to be $I + \\xi$,  where each element of $\\xi \\in \\mathbb{R}^{3 \\times 3}$ is sampled uniformly from $[0,1]$. For $B\\in \\mathbb{R}^{3\\times 1}$, each element is uniformly sampled from $[0,1]$. We have $Q = I + \\zeta I \\in \\mathbb{R}^{3\\times 3}$ with $\\zeta$ uniformly sampled from $[0,0.1]$, and $R \\in \\mathbb{R}$ uniformly sampled from $[1,1.5]$. For each experiment, the initial condition $K^0 \\in \\mathbb{R}^{1\\times 3}$ is also randomly sampled such that $\\rho(A-BK^0) < 1$. The NS method converges globally for all the cases. In the right plot, we focus on the model-free setting for \\eqref{set_matric}. We decrease the number of samples used in the $\\mathcal{H}_\\infty$ estimation and show how this increases the noise in the zeroth-order $\\mathcal{H}_\\infty$ oracle and worsens the convergence behaviors of the model-free NS method. \nNevertheless, the model-free NS method tracks its model-based counterpart with enough samples.  More numerical results can be found in the supplementary material. \n\n\\begin{figure}\n\\minipage{0.33\\textwidth}\n  \\includegraphics[width=\\linewidth]{plot\/all_sime.pdf}\n  \\label{fig:awesome_image2}\n\\endminipage\\hfill\n\\minipage{0.33\\textwidth}%\n  \\includegraphics[width=\\linewidth]{plot\/random_exp2.pdf}\n \\label{fig:awesome_image3}\n\\endminipage\\hfill\n\\minipage{0.33\\textwidth}%\n  \\includegraphics[width=\\linewidth]{plot\/vary_N.pdf}\n \\label{fig:awesome_image3}\n\\endminipage\n\\caption{Simulation results. Left: The trajectory of relative error of GS, NS, INGD, and Model-free NS methods on \\eqref{set_matric}. Middle: The trajectory of relative optimality gap of 8 randomly generated cases for NS method. Right: The trajectory of Model-free NS method with more noisy oracle on \\eqref{set_matric}.} \\label{Simulationrelts}\n\\end{figure}\n\n\n\\section{Conclusions and Future Work}\n\nIn this paper, we developed the global convergence theory for direct policy search on the $\\mathcal{H}_\\infty$ state-feedback synthesis problem. Although the resultant policy optimization formulation is nonconvex and nonsmooth, we managed to show that any Clarke stationary points for this problem are actually global minimum, and\nthe concept of Golstein subdifferential can be used to build direct policy search algorithms which are guaranteed to converge to the global optimal solutions. The finite-time guarantees in this paper are developed only for finding $(\\delta,\\epsilon)$-stationary points. \nAn important future task is to investigate the finite-time bounds for the optimality gap (i.e. $J(K^n)-J^*$) as well as\nthe sample complexity of direct policy search on model-free $\\mathcal{H}_\\infty$ control. It is  also of great interests to investigate the convergence properties of direct policy search in nonlinear\/output-feedback settings\\footnote{Some discussions on possible extensions along this direction have been given in the supplementary material.}.\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{ack}\nThis work is generously supported by the NSF award\nCAREER-2048168 and the 2020 Amazon research award.  The authors would like to thank Michael L. Overton, Maryam Fazel, Yang Zheng, Peter Seiler, Geir\nDullerud, Aaron Havens, Darioush Kevian, Kaiqing Zhang, Na Li, Mehran Mesbahi, Tamer Ba\\c{s}ar, Mihailo Jovanovic, and Javad Lavaei for the valuable discussions, as well as the helpful suggestions from the anonymous reviewers of NeurIPS.\n\\end{ack}\n\n\n\n\n\\bibliographystyle{abbrv}\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt's well known that the discontinuity of a Feynman amplitude comes from the singularity of the Feynman propagator \\cite{c0,c1}. In an 1-Particle-Irreducible (1PI) Feynman diagram the discontinuity will contribute imaginary part to the Feynman amplitude. One of the way to calculate the discontinuity is the causal perturbative theory \\cite{c1}, another is called as cutting rules proposed by Cutkosky \\cite{c0}. Here we mainly discuss the cutting rules. We note that although the cutting rules is well-defined, it hasn't given an integrative algorithm to practically calculate the discontinuity. So we want to investigate it further.\n\nOn the other hand, the optical theorem has given a strong constraint on the imaginary part of physical amplitude. Whether they agree with each other hasn't been clearly investigated. So we will discuss their relationship and investigate the origin of the optical theorem. The arrangement of this paper is: firstly we discuss how to ameliorate the cutting rules and investigate its physical meaning; then we calculate the imaginary parts of some Feynman diagrams to see if the ameliorated cutting rules coincides with the conventional integral algorithm; in section IV we investigate the breaking of the optical theorem; in Sect.V we calculate a physical amplitude to see if the physical result keeps gauge independent under the ameliorated cutting rules; lastly we give the conclusion.\n\n\\section{Ameliorate the conventional cutting rules}\n\nWe usually encounter branch cut in Feynman amplitude calculation. Such branch cut will contribute imaginary part to Feynman amplitude in an 1PI Feynman diagram. Consider the Feynman propagator\n\\begin{equation}\n  D_F(x-y)\\,=\\,\\int\\frac{d^4 p}{(2\\pi)^4}\\frac{i}{p^2-m^2+i\\varepsilon}\n  e^{-i p\\cdot(x-y)}\\,,\n\\end{equation}\nthere exist two singularities $p_0=\\pm(({\\bf p}^2+m^2)^{1\/2}-i\\varepsilon)$ in the denominator. They separately represent the processes of the on-shell particle propagating from position $y$ to position $x$ and from position $x$ to position $y$. The cutting rules is just the algorithm to calculate the contribution of the processes to Feynman amplitude. But it isn't suitable for actual calculation since it assumes that only one of the two singularities of Eq.(1) has contribution to Feynman amplitude, however, it doesn't tell which singularity is the case \\cite{c0}. In the follows we investigate this problem from the imaginary part of the unstable particle's self energy. We firstly calculate the imaginary part of the unstable particle's self energy when keeping all of the contributions of the singularities of the Feynman propagators to Feynman amplitude. At one-loop level the result is\n\\begin{eqnarray}\n  Im(-i)\\int\\frac{d^4 k}{(2\\pi)^4}\\frac{1}{k^2-m_1^2+i\\varepsilon}\n  \\frac{1}{(k-p)^2-m_2^2+i\\varepsilon}&\\rightarrow&\n  \\frac{(m^4+m_1^4+m_2^4-2 m^2 m_1^2-2 m^2 m_2^2-2 m_1^2 m_2^2)^{1\/2}}{16\\pi m^2}\n  \\nonumber \\\\\n  &\\times&\\bigl{(} \\theta[m_1-m-m_2]+\\theta[m-m_1-m_2]+\\theta[m_2-m-m_1] \\bigr{)}\\,,\n\\end{eqnarray}\nwhere the external-line momentum $p$ is on shell $p_0=({\\bf p}^2+m^2)^{1\/2}$ and $\\theta$ is the Heaviside function. According to the Breit-Wigner formula the imaginary part of the unstable particle's one-loop self energy is proportional to its decay width: $Im {\\cal M}(p\\rightarrow p)=m\\,\\Gamma$ \\cite{c2}. So one can easily see that the result of Eq.(2) is wrong since only the second term of the right-hand side of Eq.(2) satisfies the Breit-Wigner formula. We can draw a sketch to illustrate the origin of the three terms of the right-hand side of Eq.(2),\n\\vspace{2mm}\n\\begin{center} \\begin{picture}(278,25)\n  \\SetScale{1.1} \\SetWidth{0.45}\n  \\ArrowLine(0,0)(25,0)\n  \\ArrowArcn(38,0)(13,180,0)\n  \\ArrowArcn(38,0)(13,0,180)\n  \\ArrowLine(50,0)(75,0)\n  \\Text(10,8)[]{$m$}\n  \\Text(42,22)[]{$m_1$}\n  \\Text(42,-23)[]{$m_2$}\n  \\ArrowLine(85,0)(110,0)\n  \\ArrowArc(123,0)(13,180,0)\n  \\ArrowArcn(123,0)(13,180,0)\n  \\ArrowLine(135,0)(160,0)\n  \\Text(104,8)[]{$m$}\n  \\Text(134,22)[]{$m_1$}\n  \\Text(134,-23)[]{$m_2$}\n  \\ArrowLine(170,0)(195,0)\n  \\ArrowArc(208,0)(13,0,180)\n  \\ArrowArc(208,0)(13,180,0)\n  \\ArrowLine(220,0)(245,0)\n  \\Text(198,8)[]{$m$}\n  \\Text(230,22)[]{$m_1$}\n  \\Text(230,-23)[]{$m_2$}\n\\end{picture} \\end{center} \\vspace{11mm}\nwhere the arrow denotes the propagator with it is cut and the\nmomentum $q$ along it satisfies the {\\em positive} on-shell\ncondition $q_0=({\\bf q}^2+M^2)^{1\/2}$ ($M$ is the mass of the\npropagator). We call such momentum propagating direction denoted\nby the arrow as {\\em positive-on-shell} momentum propagating\ndirection. The three cut self-energy diagrams separately in turn\nrepresent the origins of the three terms of the right-hand side of\nEq.(2). Obviously only the second cut is acceptable by the\nBreit-Wigner formula, the others must be eliminated. Comparing the\nthree cuts we find that the only difference between them is that\nthe two {\\em positive-on-shell} momentum propagating directions of\nthe cut propagators of the second cut are inverse in the momentum\nloop, but that of the others are equi-directional in the momentum\nloop. We can use this point to constrain which cut is acceptable,\ni.e. the {\\em positive-on-shell} momentum propagating directions\nof the cut propagators must be inverse in every cut momentum loop.\nIn an 1PI Feynman diagram such constraint makes all of the {\\em\npositive-on-shell} momentum propagating directions of the cut\npropagators are along a same direction, i.e. the momentum-energy\npropagating direction of the external-line particles (see Fig.1-5\nfor a rough knowledge). We note that such picture can be used to\ndescribe the on-shell virtual particles' propagating processes\nhappening in the quantum field vacuum, if such processes exist.\nObviously in this picture the on-shell virtual particles satisfy\nthe energy conservation law. Contrarily, if the above constraint\ndoesn't exist, these on-shell virtual particles will not keep the\nenergy conserved because they will provide with energy\nreciprocally in the cut momentum loops thus the total energy of\nthem can be greater than that of the incoming particles.\n\nObviously there are at most two cuts in each momentum loop based on this constraint. On the other hand, according to the knowledge that cutting once in a momentum loop is equivalent to performing the conventional loop momentum integral, each cut momentum loop can only be cut twice. For the uncut propagator, since its singularity has no contribution to Feynman amplitude, it should be replaced by its Cauchy principle value in the loop momentum integral. Summing up all the discussions we obtain the following ameliorated cutting rules:\n\\begin{enumerate}\n\\item Cut through the Feynman diagram in all possible ways such that the cut propagators can simultaneously be put on mass shell and the cut propagators' {\\em positive-on-shell} momentum propagating directions are reverse in every cut momentum loop; keep only two cuts in each cut momentum loop.\n\\item For each cut propagator with definite {\\em positive-on-shell} momentum propagating direction, replace $1\/(p^2-m^2+i\\varepsilon)\\rightarrow -2\\pi\\,i\\,\\theta[p_0]\\,\\delta(p^2-m^2)$ (where the momentum $p$ is along the {\\em positive-on-shell} momentum propagating direction), for each uncut propagator, apply Cauchy principle value to it, then perform the loop integrals.\n\\item Sum the contributions of all possible cuts.\n\\end{enumerate}\n\nOne can easily find that the above ameliorated cutting rules is\ndifferent from the conventional one in some aspects. Contrarily to\nthe conventional cutting rules, the ameliorated one constrains\nthat only two cuts can exist in one momentum loop and implies that\nboth of the two singularities of Eq.(1) can have contributions to\nFeynman amplitude. Besides, it definitely points out that the\nCauchy principle value should be applied to the uncut propagator\nin the loop momentum integrals. We note that to the best of our\nknowledge there exist no similar discussions before. But then\nthere exist some calculations of the imaginary parts of the Green\nfunctions by the causal perturbative theory which coincide with\nour results \\cite{c3} (see section III).\n\nThe above ameliorated cutting rules is very abstract. So we give some illustrations in Fig.1-5. The arrows in Fig.1-5 represent the {\\em positive-on-shell} momentum propagating directions of the cut propagators. In order to explain the cutting conditions we give two examples in Fig.2.\n\\vspace{4mm}\n\\begin{center} \\begin{picture}(218,25)\n  \\SetScale{1.1} \\SetWidth{0.45}\n  \\ArrowLine(0,0)(25,0)\n  \\ArrowLine(25,0)(46,13)\n  \\ArrowLine(46,13)(60,20)\n  \\ArrowLine(25,0)(46,-13)\n  \\ArrowLine(46,-13)(60,-20)\n  \\Line(46,13)(46,-13)\n  \\ArrowLine(70,0)(95,0)\n  \\ArrowLine(95,0)(116,13)\n  \\ArrowLine(116,13)(130,20)\n  \\Line(95,0)(116,-13)\n  \\ArrowLine(116,-13)(130,-20)\n  \\ArrowLine(116,-13)(116,13)\n  \\ArrowLine(140,0)(165,0)\n  \\Line(165,0)(186,13)\n  \\ArrowLine(186,13)(200,20)\n  \\ArrowLine(165,0)(186,-13)\n  \\ArrowLine(186,-13)(200,-20)\n  \\ArrowLine(186,13)(186,-13)\n\\end{picture} \\vspace{10mm} \\\\\n{\\small FIG. 1: Cuts of the one-loop three-point irreducible Feynman diagram.}\n\\end{center}\n\\vspace{2mm}\n\\begin{center} \\begin{picture}(335,90)\n  \\ArrowLine(0,0)(25,0)\n  \\Line(25,0)(50,0)\n  \\ArrowLine(50,0)(75,0)\n  \\ArrowLine(25,25)(25,0)\n  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\\ArrowLine(75,80)(100,80)\n  \\ArrowLine(125,55)(150,55)\n  \\Line(150,55)(175,55)\n  \\ArrowLine(175,55)(200,55)\n  \\ArrowLine(150,80)(150,55)\n  \\ArrowLine(175,80)(175,55)\n  \\ArrowLine(125,80)(150,80)\n  \\Line(150,80)(175,80)\n  \\ArrowLine(175,80)(200,80)\n  \\Text(133,88)[]{$p_1$}\n  \\Text(192,88)[]{$p_2$}\n  \\Text(162,45)[]{$(p_{10}>p_{20})$}\n  \\ArrowLine(225,55)(250,55)\n  \\Line(250,55)(275,55)\n  \\ArrowLine(275,55)(300,55)\n  \\ArrowLine(250,55)(250,80)\n  \\ArrowLine(275,55)(275,80)\n  \\ArrowLine(225,80)(250,80)\n  \\Line(250,80)(275,80)\n  \\ArrowLine(275,80)(300,80)\n  \\Text(233,88)[]{$p_1$}\n  \\Text(292,88)[]{$p_2$}\n  \\Text(262,45)[]{$(p_{10}<p_{20})$}\n\\end{picture} \\vspace{6mm} \\\\\n{\\small FIG. 2: Cuts of the one-loop four-point irreducible Feynman diagram.}\n\\end{center}\n\\vspace{1mm}\n\\begin{center} \\begin{picture}(396,20)\n  \\SetScale{1.1} \\SetWidth{0.45}\n  \\ArrowLine(0,0)(25,0)\n  \\ArrowArcn(40,0)(15,180,90)\n  \\ArrowArc(40,0)(15,180,270)\n  \\Line(40,15)(40,-15)\n  \\CArc(40,0)(15,0,90)\n  \\CArc(40,0)(15,270,360)\n  \\ArrowLine(55,0)(80,0)\n  \\SetOffset(-10,0)\n  \\ArrowLine(95,0)(120,0)\n  \\ArrowArcn(135,0)(15,90,0)\n  \\ArrowArc(135,0)(15,270,360)\n  \\Line(135,15)(135,-15)\n  \\CArc(135,0)(15,90,180)\n  \\CArc(135,0)(15,180,270)\n  \\ArrowLine(150,0)(175,0)\n  \\SetOffset(-20,0)\n  \\ArrowLine(190,10)(215,10)\n  \\Line(215,10)(235,10)\n  \\ArrowArc(235,10)(20,180,270)\n  \\ArrowLine(235,10)(235,-10)\n  \\ArrowArcn(235,-10)(20,90,0)\n  \\Line(235,-10)(255,-10)\n  \\ArrowLine(255,-10)(280,-10)\n  \\SetOffset(-30,0)\n  \\ArrowLine(295,-10)(320,-10)\n  \\Line(320,-10)(340,-10)\n  \\ArrowArcn(340,-10)(20,180,90)\n  \\ArrowLine(340,-10)(340,10)\n  \\ArrowArc(340,10)(20,270,360)\n  \\Line(340,10)(360,10)\n  \\ArrowLine(360,10)(385,10)\n\\end{picture} \\vspace{10mm} \\\\\n{\\small FIG. 3: Cuts of the two-loop two-point irreducible Feynman diagram.}\n\\end{center}\n\\vspace{2mm}\n\\begin{center} \\begin{picture}(376,70)\n  \\SetScale{0.80} \\SetWidth{0.625}\n  \\ArrowLine(0,15)(25,15)\n  \\ArrowLine(25,15)(55,15)\n  \\Line(55,15)(80,15)\n  \\ArrowLine(80,15)(105,15)\n  \\CArc(55,15)(30,180,270)\n  \\ArrowLine(55,-15)(55,15)\n  \\ArrowLine(55,-15)(80,-15)\n  \\Line(80,-15)(80,15)\n  \\ArrowLine(80,-15)(105,-15)\n  \\ArrowLine(120,-15)(145,-15)\n  \\ArrowLine(145,-15)(175,-15)\n  \\Line(175,-15)(200,-15)\n  \\ArrowLine(200,-15)(225,-15)\n  \\CArc(175,-15)(30,90,180)\n  \\ArrowLine(175,15)(175,-15)\n  \\ArrowLine(175,15)(200,15)\n  \\Line(200,-15)(200,15)\n  \\ArrowLine(200,15)(225,15)\n  \\ArrowLine(240,-15)(265,-15)\n  \\Line(265,-15)(320,-15)\n  \\ArrowLine(320,-15)(345,-15)\n  \\ArrowArcn(295,-15)(30,180,90)\n  \\ArrowLine(295,-15)(295,15)\n  \\ArrowLine(320,-15)(320,15)\n  \\Line(295,15)(320,15)\n  \\ArrowLine(320,15)(345,15)\n  \\ArrowLine(360,15)(385,15)\n  \\Line(385,15)(440,15)\n  \\ArrowLine(440,15)(465,15)\n  \\ArrowArc(415,15)(30,180,270)\n  \\ArrowLine(415,15)(415,-15)\n  \\ArrowLine(440,15)(440,-15)\n  \\Line(415,-15)(440,-15)\n  \\ArrowLine(440,-15)(465,-15)\n  \\ArrowLine(30,65)(55,65)\n  \\ArrowLine(55,65)(76,78)\n  \\Line(76,78)(90,85)\n  \\ArrowLine(90,85)(104,92)\n  \\ArrowLine(55,65)(76,52)\n  \\Line(76,52)(90,45)\n  \\ArrowLine(90,45)(104,38)\n  \\Line(76,78)(76,52)\n  \\Line(90,85)(90,45)\n  \\ArrowLine(140,65)(165,65)\n  \\Line(165,65)(186,78)\n  \\ArrowLine(186,78)(200,85)\n  \\ArrowLine(200,85)(214,92)\n  \\Line(165,65)(186,52)\n  \\ArrowLine(186,52)(200,45)\n  \\ArrowLine(200,45)(214,38)\n  \\Line(186,78)(186,52)\n  \\Line(200,85)(200,45)\n  \\ArrowLine(250,65)(275,65)\n  \\Line(275,65)(296,78)\n  \\ArrowLine(296,78)(310,85)\n  \\ArrowLine(310,85)(324,92)\n  \\Line(275,65)(296,52)\n  \\Line(296,52)(310,45)\n  \\ArrowLine(310,45)(324,38)\n  \\Line(296,78)(296,52)\n  \\ArrowLine(310,45)(310,85)\n  \\ArrowLine(360,65)(385,65)\n  \\Line(385,65)(406,78)\n  \\Line(406,78)(420,85)\n  \\ArrowLine(420,85)(434,92)\n  \\Line(385,65)(406,52)\n  \\ArrowLine(406,52)(420,45)\n  \\ArrowLine(420,45)(434,38)\n  \\Line(406,78)(406,52)\n  \\ArrowLine(420,85)(420,45)\n  \\SetScale{1.}\n\\end{picture} \\vspace{10mm} \\\\\n{\\small FIG. 4: Cuts of the two-loop three-point irreducible Feynman diagram.}\n\\end{center}\n\\begin{center} \\begin{picture}(326,90)\n  \\SetScale{0.9} \\SetWidth{0.55}\n  \\ArrowLine(0,0)(30,0)\n  \\ArrowLine(30,0)(50,0)\n  \\Line(50,0)(70,0)\n  \\ArrowLine(70,0)(100,0)\n  \\ArrowLine(50,20)(50,0)\n  \\Line(50,0)(50,-20)\n  \\CArc(50,0)(20,90,180)\n  \\ArrowArcn(50,0)(20,90,0)\n  \\ArrowArc(50,0)(20,180,270)\n  \\CArc(50,0)(20,270,360)\n  \\ArrowLine(130,0)(160,0)\n  \\Line(160,0)(180,0)\n  \\ArrowLine(180,0)(200,0)\n  \\ArrowLine(200,0)(230,0)\n  \\Line(180,20)(180,0)\n  \\ArrowLine(180,0)(180,-20)\n  \\CArc(180,0)(20,90,180)\n  \\ArrowArcn(180,0)(20,90,0)\n  \\ArrowArc(180,0)(20,180,270)\n  \\CArc(180,0)(20,270,360)\n  \\ArrowLine(260,0)(290,0)\n  \\Line(290,0)(310,0)\n  \\ArrowLine(310,0)(330,0)\n  \\ArrowLine(330,0)(360,0)\n  \\ArrowLine(310,0)(310,20)\n  \\Line(310,0)(310,-20)\n  \\ArrowArcn(310,0)(20,180,90)\n  \\CArc(310,0)(20,0,90)\n  \\CArc(310,0)(20,180,270)\n  \\ArrowArc(310,0)(20,270,360)\n  \\ArrowLine(0,60)(30,60)\n  \\ArrowLine(30,60)(50,60)\n  \\Line(50,60)(70,60)\n  \\ArrowLine(70,60)(100,60)\n  \\Line(50,80)(50,40)\n  \\ArrowArcn(50,60)(20,180,90)\n  \\CArc(50,60)(20,0,90)\n  \\ArrowArc(50,60)(20,180,270)\n  \\CArc(50,60)(20,270,360)\n  \\ArrowLine(130,60)(160,60)\n  \\Line(160,60)(180,60)\n  \\ArrowLine(180,60)(200,60)\n  \\ArrowLine(200,60)(230,60)\n  \\Line(180,80)(180,40)\n  \\CArc(180,60)(20,90,180)\n  \\ArrowArcn(180,60)(20,90,0)\n  \\CArc(180,60)(20,180,270)\n  \\ArrowArc(180,60)(20,270,360)\n  \\ArrowLine(260,60)(290,60)\n  \\ArrowLine(290,60)(310,60)\n  \\Line(310,60)(330,60)\n  \\ArrowLine(330,60)(360,60)\n  \\Line(310,60)(310,80)\n  \\ArrowLine(310,40)(310,60)\n  \\ArrowArcn(310,60)(20,180,90)\n  \\CArc(310,60)(20,0,90)\n  \\CArc(310,60)(20,180,270)\n  \\ArrowArc(310,60)(20,270,360)\n\\end{picture} \\\\ \\vspace{11mm}\n{\\small FIG. 5: Cuts of the four-loop two-point irreducible Feynman diagram.}\n\\end{center}\nSince each cut contributes an imaginary factor $i$ to Feynman\namplitude, all of the cut diagrams in Fig.1-5 contribute imaginary\nparts to the Feynman amplitudes. This agrees with the conventional\nknowledge. On the other hand, one can easily find that many\npossible cuts in the conventional cutting rules have been\neliminated by the ameliorated one. For example all of the\npropagators in Fig.1 and 2 can be simultaneously cut in the\nconventional cutting rules \\cite{c0}, but this is forbidden in the\nameliorated one. One can also see that both of the two\nsingularities of many cut propagators have contributions to the\nFeynman amplitudes (e.g. the vertical propagator in Fig.1).\nBesides, some of the cuts in Fig.1-5 have been used to calculate\nthe imaginary parts of the Feynman amplitudes \\cite{c3,c4}.\n\n\\section{Comparison of cutting rules and conventional integral algorithm}\n\nIt's well known that the imaginary part of Feynman amplitude can be directly calculated by the conventional integral algorithm, i.e. by the Feynman parametrization, wick rotation and dimensional regularization \\cite{c5}. In this section we will give four examples to see whether the results obtained by the ameliorated cutting rules agree with the ones obtained by the conventional integral algorithm.\n\nFirstly we calculate the imaginary part of the two-loop two-point irreducible Feynman diagram\n\\vspace{4mm}\n\\begin{center} \\begin{picture}(80,20)\n\\SetScale{1.2} \\SetWidth{0.40}\n  \\ArrowLine(0,2)(25,2)\n  \\Line(25,2)(50,2)\n  \\ArrowLine(50,2)(75,2)\n  \\CArc(38,2)(13,0,180)\n  \\CArc(38,2)(13,180,0)\n  \\Text(14,8)[]{\\small $p$}\n  \\Text(45,24)[]{\\small $m_1$}\n  \\Text(45,8)[]{\\small $m_1$}\n  \\Text(45,-8)[]{\\small $m_1$}\n\\end{picture} \\end{center}\n\\vspace{5mm}\nwhere $p^2=m^2$. Using the conventional integral algorithm we have\n\\begin{eqnarray}\n  &&Im\\int\\frac{d^D k_1}{(2\\pi)^D}\n  \\frac{d^D k_2}{(2\\pi)^D}\\frac{1}{k_1^2-m_1^2+i\\varepsilon}\n  \\frac{1}{k_2^2-m_1^2+i\\varepsilon}\\frac{1}{(k_1+k_2-p)^2-m_1^2+i\\varepsilon}\n  \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{256\\pi^4}Im\\int_0^1 d\\,x\\int_0^1 d\\,y\\,\n  (A-2 m^2 y)(\\frac{2}{\\epsilon}-2\\gamma+\\ln 16\\pi^2+\\ln y-\\ln x(1-x)-\\ln B)\\ln B\n  \\nonumber \\\\\n  =\\hspace{-3mm}&&-\\frac{1}{256\\pi^3}\\int_0^1 d\\,x\\int_0^1 d\\,y\\,\n  (A-2 m^2 y)(\\frac{2}{\\epsilon}-2\\gamma+\\ln 16\\pi^2+\\ln y-\\ln x(1-x)-2\\ln|B|)\n  \\theta[-B] \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{128\\pi^3}\\int_{x_1}^{x_2}d\\,x\\Bigl{(}\n  \\frac{A(A^2-4 m^2 m_1^2)^{1\/2}}{4 m^2}-m_1^2\\ln\\frac{A+(A^2-4 m^2 m_1^2)^{1\/2}}{2 m m_1}\n  \\Bigr{)}\\theta[m-3 m_1]\\,,\n\\end{eqnarray}\nwhere $\\epsilon=4-D$, $\\gamma$ is the Euler constant, and\n\\begin{eqnarray}\n  A\\,=\\hspace{-3mm}&&m^2+m_1^2-\\frac{m_1^2}{x(1-x)}\\,, \\nonumber \\\\\n  B\\,=\\hspace{-3mm}&&m^2 y^2-A\\,y+m_1^2\\,, \\nonumber \\\\\n  x_{1,2}\\,=\\hspace{-3mm}&&\\frac{m-m_1\\mp(m^2-2 m m_1-3 m_1^2)^{1\/2}}{2(m-m_1)}\\,.\n\\end{eqnarray}\nIn Eq.(3) we have used the formula $\\ln a=\\ln|a|-i\\pi\\,\\theta[-a]$. On the other hand, using the ameliorated cutting rules we also have\n\\begin{eqnarray}\n  &&Im\\int\\frac{d^4 k_1}{(2\\pi)^4}\\frac{d^4 k_2}\n  {(2\\pi)^4}\\frac{1}{k_1^2-m_1^2+i\\varepsilon}\\frac{1}{k_2^2-m_1^2+i\\varepsilon}\\frac{1}\n  {(k_1+k_2-p)^2-m_1^2+i\\varepsilon} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{64\\pi^5}\\int d^4 k_1 d^4 k_2\\,\\theta[k_{10}]\\,\\delta(k_1^2-m_1^2)\n  \\,\\theta[k_{20}]\\,\\delta(k_2^2-m_1^2)\\,\\theta[p_0-k_{10}-k_{20}]\\,\\delta((p-k_1-k_2)^2-m_1^2) \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{64 m\\pi^3}\\int_{m_1}^{\\frac{m^2-3 m_1^2}{2 m}}d\\,x\n  \\Bigl{[}(m x+m_1^2)^2-\\frac{m_1^2(m^2-m_1^2)^2}{m^2+m_1^2-2 m x}\\Bigr{]}^{1\/2}\\theta[m-3m_1]\\,.\n\\end{eqnarray}\nThrough numerical calculations we find that Eq.(5) is equal to Eq.(3). The concrete results are shown in Fig.6.\n\\setcounter{figure}{5}\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{file=1.eps,width=7cm} \\\\\n  \\caption{Results of Eq.(3) and Eq.(5).}\n\\end{center}\n\\end{figure}\n\nSecondly we calculate the imaginary part of the one-loop three-point irreducible Feynman diagram\n\\vspace{4mm}\n\\begin{center} \\begin{picture}(70,25)\n  \\SetScale{1.1} \\SetWidth{0.45}\n  \\ArrowLine(0,0)(25,0)\n  \\Line(25,0)(46,13)\n  \\Line(25,0)(46,-13)\n  \\ArrowLine(46,13)(60,20)\n  \\ArrowLine(46,-13)(60,-20)\n  \\Line(46,13)(46,-13)\n  \\Text(10,6)[]{$p$}\n  \\Text(31,12)[]{$m_1$}\n  \\Text(31,-12)[]{$m_1$}\n  \\Text(60,0)[]{$m_1$}\n  \\Text(74,20)[]{$p_1$}\n  \\Text(74,-20)[]{$p_2$}\n\\end{picture} \\end{center}\n\\vspace{8mm}\nwhere $p^2=(p_1+p_2)^2=m_a^2$ and $p_1^2=p_2^2=m_b^2$. Using the conventional integral algorithm we have\n\\begin{eqnarray}\n  &&Im(-i)\\int\\frac{d^4 k}{(2\\pi)^4}\\frac{1}{k^2-m_1^2+i\\varepsilon}\n  \\frac{1}{(k-p_1)^2-m_1^2+i\\varepsilon}\\frac{1}{(k-p_1-p_2)^2-m_1^2+i\\varepsilon}\n  \\nonumber \\\\\n  =\\hspace{-3mm}&&-\\frac{1}{16\\pi^2}\\,Im\\int_0^1 d\\,x\\int_0^{1-x} \\frac{d\\,y}\n  {m_a^2\\,x^2+m_b^2\\,y^2+m_a^2\\,x\\,y-m_a^2\\,x-m_b^2\\,y+m_1^2-i\\varepsilon} \\nonumber \\\\\n  =\\hspace{-3mm}&&-\\frac{1}{16\\pi}\\int_0^1 d\\,x\\int_0^{1-x}d\\,y\\,\n  \\delta(m_a^2\\,x^2+m_b^2\\,y^2+m_a^2\\,x\\,y-m_a^2\\,x-m_b^2\\,y+m_1^2) \\nonumber \\\\\n  =\\hspace{-3mm}&&\\left\\{ \\begin{array}{l}\n  \\frac{1}{16\\pi m_a(m_a^2-4 m_b^2)^{1\/2}}\\ln \\biggl{[} \\frac{m_a^2-2 m_b^2-\n  \\sqrt{(m_a^2-4 m_b^2)(m_a^2-4 m_1^2)}}{m_a^2-2 m_b^2+\n  \\sqrt{(m_a^2-4 m_b^2)(m_a^2-4 m_1^2)}}\\left( \\frac{m_a m_b+\n  \\sqrt{(m_a^2-4 m_b^2)(m_b^2-4 m_1^2)}}{m_a m_b-\n  \\sqrt{(m_a^2-4 m_b^2)(m_b^2-4 m_1^2)}} \\right)^2 \\biggr{]}\n  \\hspace{4mm} \\forall\\, m_1 \\in (0, \\frac{m_b}{2} ) \\,,\\\\\n  \\frac{1}{16\\pi m_a\\sqrt{m_a^2-4 m_b^2}}\\ln\\frac{m_a^2-2 m_b^2-\n  \\sqrt{(m_a^2-4 m_b^2)(m_a^2-4 m_1^2)}}{m_a^2-2 m_b^2+\n  \\sqrt{(m_a^2-4 m_b^2)(m_a^2-4 m_1^2)}}\n  \\hspace{56mm} \\forall\\, m_1\\in  (\\frac{m_b}{2},\\frac{m_a}{2})\n  \\,, \\\\ \\vspace{1mm} \\\\   0\n  \\hspace{127mm} \\forall\\, m_1\\in  (\\frac{m_a}{2},\\infty)  \\,,\n  \\end{array} \\right.\n\\end{eqnarray}\nwhere the mathematical formula ($P$ denotes the Cauchy principle value)\n\\begin{equation}\n  \\frac{1}{a\\pm i\\varepsilon}\\,=\\,P\\frac{1}{a}\\mp i\\pi\\delta(a)\n\\end{equation}\nhas been used in the Feynman parameter integrals. On the other hand, using the ameliorated cutting rules we also have (see Fig.1)\n\\begin{eqnarray}\n  &&Im(-i)\\int\\frac{d^4 k}{(2\\pi)^4}\\frac{1}{k^2-m_1^2+i\\varepsilon}\n  \\frac{1}{(k-p_1)^2-m_1^2+i\\varepsilon}\\frac{1}{(k-p_1-p_2)^2-m_1^2+i\\varepsilon}\n  \\nonumber \\\\\n  =\\hspace{-3mm}&&P\\int\\frac{d^4 k}{8\\pi^2}\\biggl{[} \\frac{\\theta[k_0]\\delta(k^2-m_1^2)\n  \\theta[p_{10}+p_{20}-k_0]\\delta((p_1+p_2-k)^2-m_1^2)}{(k-p_1)^2-m_1^2+i\\varepsilon} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[k_0]\\delta(k^2-m_1^2)\\theta[p_{10}-k_0]\\delta((p_1-k)^2-m_1^2)}\n  {(k-p_1-p_2)^2-m_1^2+i\\varepsilon} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[k_0-p_{10}]\\delta((k-p_1)^2-m_1^2)\\theta[p_{10}+p_{20}-k_0]\n  \\delta((p_1+p_2-k)^2-m_1^2)}{k^2-m_1^2+i\\varepsilon} \\biggr{]} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{\\theta[m_a-2 m_1]}{16\\pi m_a\\sqrt{m_a^2-4 m_b^2}}\n  \\ln\\left|\\frac{m_a^2-2m_b^2-\\sqrt{(m_a^2-4 m_b^2)(m_a^2-4 m_1^2)}}{m_a^2-2m_b^2+\n  \\sqrt{(m_a^2-4 m_b^2)(m_a^2-4 m_1^2)}}\\right| \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[m_b-2m_1]}{8\\pi m_a\\sqrt{m_a^2-4 m_b^2}}\n  \\ln\\left|\\frac{m_a m_b+\\sqrt{(m_a^2-4 m_b^2)(m_b^2-4 m_1^2)}}\n  {m_a m_b-\\sqrt{(m_a^2-4 m_b^2)(m_b^2-4 m_1^2)}}\\right|\\,.\n\\end{eqnarray}\nObviously Eq.(8) is equal to Eq.(6). Eq.(8) also coincides with Eq.(3.24) of Ref.\\cite{c3}.\n\nThirdly we calculate the imaginary part of the one-loop four-point irreducible Feynman diagram\n\\vspace{2mm}\n\\begin{center} \\begin{picture}(100,35)\n  \\ArrowLine(0,0)(25,0)\n  \\Line(25,0)(50,0)\n  \\ArrowLine(50,0)(75,0)\n  \\Line(25,0)(25,25)\n  \\Line(50,0)(50,25)\n  \\ArrowLine(0,25)(25,25)\n  \\Line(25,25)(50,25)\n  \\ArrowLine(50,25)(75,25)\n  \\Text(38,31)[]{$m$}\n  \\Text(38,-6)[]{$m$}\n  \\Text(18,13)[]{$m$}\n  \\Text(57,13)[]{$0$}\n  \\Text(-6,25)[]{$r$}\n  \\Text(-6,0)[]{$q$}\n  \\Text(81,25)[l]{$r=(r_0,p\\,{\\bf e_z})$}\n  \\Text(81,0)[l]{$q=(q_0,-p\\,{\\bf e_z})$}\n\\end{picture} \\end{center} \\vspace{3mm}\nwhere $r_0=q_0=2 p$. Using the conventional integral algorithm and Eq.(7) we have\n\\begin{eqnarray}\n  &&Im(-i)\\int\\frac{d^4 k}{(2\\pi)^4}\\frac{1}{k^2+i\\varepsilon}\n  \\frac{1}{(k-q)^2-m^2+i\\varepsilon}\\frac{1}{k^2-m^2+i\\varepsilon}\n  \\frac{1}{(k+r)^2-m^2+i\\varepsilon} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{16\\pi^2}\\,Im\\int_0^1 d\\,x\\int_0^{1-x}d\\,y\\int_0^{1-x-y}\n  \\frac{d\\,z}{(3 p^2\\,x^2+3 p^2\\,y^2-10 p^2\\,x\\,y-3 p^2\\,x-3 p^2\\,y-m^2\\,z+m^2\n  -i\\varepsilon)^2} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{16\\pi^2 m^2}\\,Im\\int_0^1 d\\,x\\int_0^{1-x}d\\,y\\,\\left[\n  \\frac{1}{3 p^2\\,x^2+3 p^2\\,y^2-10 p^2\\,x\\,y+(m^2-3 p^2)x+(m^2-3 p^2)y-i\\varepsilon}\n  \\right. \\nonumber \\\\\n  -\\hspace{-3mm}&&\\frac{1}{3 p^2\\,x^2+3 p^2\\,y^2-10 p^2\\,x\\,y-3 p^2\\,x-3 p^2\\,y+m^2\n  -i\\varepsilon} \\biggr{]} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{16\\pi m^2}\\int_0^1 d\\,x\\int_0^{1-x}d\\,y\\,\\left[\n  \\delta(3 p^2\\,x^2+3 p^2\\,y^2-10 p^2\\,x\\,y+(m^2-3 p^2)x+(m^2-3 p^2)y)\n  \\right. \\nonumber \\\\\n  -\\hspace{-3mm}&&\\delta(3 p^2\\,x^2+3 p^2\\,y^2-10 p^2\\,x\\,y-3 p^2\\,x-3 p^2\\,y+m^2) \\bigr{]}\n  \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{\\theta[p-\\frac{m}{2}]\\theta[\\frac{m}{\\sqrt{3}}-p]}{128\\pi m^2 p^2}\n  \\ln\\frac{9p^3-m^2 p+2m^2\\sqrt{4p^2-m^2}}{9p^3-m^2 p-2m^2\\sqrt{4p^2-m^2}} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[p-\\frac{m}{\\sqrt{3}}]\\theta[\\frac{2m}{\\sqrt{3}}-p]}{128\\pi m^2 p^2}\n  \\ln\\frac{9p^3-m^2 p+2m^2\\sqrt{4p^2-m^2}}{9(9p^3-m^2 p-2m^2\\sqrt{4p^2-m^2})} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[p-\\frac{2m}{\\sqrt{3}}]}{128\\pi m^2 p^2}\n \\ln\\frac{(9p^3-m^2 p+2m^2\\sqrt{4p^2-m^2})(15p^2+4p\\sqrt{9p^2-12m^2}-4m^2)}\n  {9(9p^3-m^2 p-2m^2\\sqrt{4p^2-m^2})(15p^2-4p\\sqrt{9p^2-12m^2}-4m^2)} \\,.\n\\end{eqnarray}\nOn the other hand, using the ameliorated cutting rules we also have (see Fig.2)\n\\begin{eqnarray}\n  &&Im(-i)\\int\\frac{d^4 k}{(2\\pi)^4}\\frac{1}{k^2+i\\varepsilon}\n  \\frac{1}{(k-q)^2-m^2+i\\varepsilon}\\frac{1}{k^2-m^2+i\\varepsilon}\n  \\frac{1}{(k+r)^2-m^2+i\\varepsilon} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{8\\pi^2}P\\int d^4 k \\Bigl{[} \\frac{\\theta[k_0]\\delta(k^2-m^2)\n  \\theta[q_0+r_0-k_0]\\delta((q+r-k)^2-m^2)}{(k-r)^2((k-r)^2-m^2)} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[k_0]\\delta(k^2-m^2)\\theta[r_0-k_0]\\delta((r-k)^2-m^2)}\n  {k^2((k+q)^2-m^2)}+\\frac{\\theta[k_0]\\delta(k^2-m^2)\\theta[q_0-k_0]\\delta((q-k)^2-m^2)}\n  {k^2((k+r)^2-m^2)} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[k_0]\\delta(k^2)\\theta[r_0-k_0]\\delta((r-k)^2-m^2)}\n  {(k^2-m^2)((k+q)^2-m^2)}+\\frac{\\theta[k_0]\\delta(k^2)\\theta[q_0-k_0]\\delta((q-k)^2-m^2)}\n  {(k^2-m^2)((k+r)^2-m^2)} \\Bigr{]} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{\\theta[p-m\/2]}{128\\pi m^2 p^2}\\ln\\frac{9p^3-m^2 p+2m^2\n  \\sqrt{4p^2-m^2}}{9p^3-m^2 p-2m^2\\sqrt{4p^2-m^2}} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[p-2m\/\\sqrt{3}]}{128\\pi m^2 p^2}\\ln\n  \\frac{15p^2-4m^2+4p\\sqrt{9p^2-12m^2}}{15p^2-4m^2-4p\\sqrt{9p^2-12m^2}}\n  -\\frac{\\theta[p-m\/\\sqrt{3}]}{64\\pi m^2 p^2}\\ln3\\,.\n\\end{eqnarray}\nObviously Eq.(10) is equal to Eq.(9).\n\nLastly we calculate the imaginary part of a little more complex Feynman diagram\n\\begin{center} \\begin{picture}(380,36)\n  \\SetScale{0.9} \\SetWidth{0.55}\n  \\ArrowLine(0,0)(30,0)\n  \\ArrowLine(70,0)(100,0)\n  \\CArc(50,0)(20,0,65)\n  \\BCirc(50,20){9}\n  \\CArc(50,0)(20,115,180)\n  \\CArc(50,0)(20,180,0)\n  \\Text(10,8)[]{$p$}\n  \\Text(26,14)[]{0}\n  \\Text(46,32)[]{$m$}\n  \\Text(46,5)[]{$m$}\n  \\Text(46,-25)[]{$m$}\n  \\Text(64,14)[]{$m$}\n  \\Text(100,0)[]{$\\sim$}\n  \\SetOffset(110,0)\n  \\ArrowLine(0,0)(30,0)\n  \\ArrowLine(70,0)(100,0)\n  \\CArc(50,0)(20,0,65)\n  \\ArrowArcn(50,20)(9,180,0)\n  \\ArrowArc(50,20)(9,180,0)\n  \\CArc(50,0)(20,115,180)\n  \\ArrowArc(50,0)(20,180,0)\n  \\Text(100,0)[]{$+$}\n  \\SetOffset(100,0)\n  \\ArrowLine(130,0)(160,0)\n  \\ArrowLine(200,0)(230,0)\n  \\ArrowArcn(180,0)(20,180,115)\n  \\BCirc(180,20){9}\n  \\CArc(180,0)(20,0,65)\n  \\ArrowArc(180,0)(20,180,0)\n  \\Text(217,0)[]{$+$}\n  \\SetOffset(90,0)\n  \\ArrowLine(260,0)(290,0)\n  \\ArrowLine(330,0)(360,0)\n  \\CArc(310,0)(20,115,180)\n  \\BCirc(310,20){9}\n  \\ArrowArcn(310,0)(20,65,0)\n  \\ArrowArc(310,0)(20,180,0)\n\\end{picture} \\end{center}\n\\vspace{9mm}\nwhere $p^2=M^2$ and the cut diagrams have been shown. Using the conventional integral algorithm and Eq.(7) we have\n\\begin{eqnarray}\n  &&Im\\,\\mu^{2\\epsilon}\\int\\frac{d^D k_1}{(2\\pi)^D}\\frac{d^D k_2}{(2\\pi)^D}\n  \\frac{1}{k_1^2+i\\varepsilon}\\frac{1}{k_2^2-m^2+i\\varepsilon}\n  \\frac{1}{(k_2-k_1)^2-m^2+i\\varepsilon}\\frac{1}{k_1^2-m^2+i\\varepsilon}\n  \\frac{1}{(k_1-p)^2-m^2+i\\varepsilon} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{256\\pi^4}Im\\int_0^1 d\\,x\\int_0^1 d\\,y\\int_0^{1-y}\n  \\frac{d\\,z}{M^2 y^2+(m^2-M^2)y+m^2 z-i\\varepsilon}\\Bigl{[} \\Delta+\n  \\ln\\frac{1-y-z}{x-x^2} \\nonumber \\\\\n  -\\hspace{-3mm}&&\\ln(\\frac{M^2 y^2}{m^2}-\\frac{M^2 y}{m^2}+y+z\n  +\\frac{1-y-z}{x-x^2}-i\\varepsilon)-\\ln(\\frac{M^2 y^2}{m^2}-\\frac{M^2 y}{m^2}+y+z\n  -i\\varepsilon) \\Bigr{]} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{256\\pi^3}\\int_0^1 d\\,x\\int_0^1 d\\,y\\int_0^{1-y}d\\,z\\,\n  \\delta(M^2 y^2+(m^2-M^2)y+m^2 z)\\Bigl{[} \\Delta+\\ln\\frac{1-y-z}{x-x^2} \\nonumber \\\\\n  -\\hspace{-3mm}&&\\ln|\\frac{M^2 y^2}{m^2}-\\frac{M^2 y}{m^2}+y+z\n  +\\frac{1-y-z}{x-x^2}| \\Bigr{]} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{1}{256\\pi^3}P\\int_0^1 d\\,x\\int_0^1 d\\,y\\int_0^{1-y}d\\,z\\,\\frac{\n  \\theta[\\frac{M^2 y}{m^2}-\\frac{M^2 y^2}{m^2}-y-z-\\frac{1-y-z}{x-x^2}]}\n  {M^2 y^2+(m^2-M^2)y+m^2 z} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{1}{512\\pi^3 m^2}Im\\int_0^1 d\\,x\\int_0^1 d\\,y \\Bigl{[}\n  \\ln^2(\\frac{M^2 y^2}{m^2}-\\frac{M^2 y}{m^2}+y-i\\varepsilon)-\n  \\ln^2(\\frac{M^2 y^2}{m^2}-\\frac{M^2 y}{m^2}+1-i\\varepsilon) \\Bigr{]} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{\\theta[M-m]\\,\\theta[2m-M](M^2-m^2)}{256\\pi^3 m^2 M^2}\n  (\\Delta-2\\ln\\frac{M^2-m^2}{m\\,M}+2) \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[M-2m]\\,\\theta[3m-M]}{256\\pi^3 m^2 M^2}\\Bigl{[}\n  (M^2-m^2-M\\sqrt{M^2-4m^2})\\Delta+2(M^2-m^2)(1-\\ln\\frac{M^2-m^2}{m\\,M}) \\nonumber \\\\\n  +\\hspace{-3mm}&&M\\sqrt{M^2-4m^2}(\\ln\\frac{M^2-4m^2}{m^2}+\\frac{\\pi}{\\sqrt{3}}-4) \\Bigr{]}\n  \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[M-3m]}{256\\pi^3 m^2 M^2}\\Bigl{[} (M^2-m^2-M\\sqrt{M^2-4m^2})\n  \\Delta+M\\sqrt{M^2-4m^2}(\\ln\\frac{M^2-4m^2}{m^2}+\\frac{\\pi}{\\sqrt{3}}-4)\n  \\nonumber \\\\\n  +\\hspace{-3mm}&&2(M^2-m^2)(1-\\ln\\frac{M^2-m^2}{m\\,M})\n  +2M^2\\int_{y_1}^{y_2}\\left( F-\\sqrt{3}\\,tg^{-1}\\frac{F}{\\sqrt{3}}\\right)d\\,y \\Bigr{]}\\,,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n  \\Delta&=&\\frac{2}{\\epsilon}+2\\ln4\\pi-2\\gamma-2\\ln\\frac{m^2}{\\mu^2}\\,, \\nonumber \\\\\n  F&=&\\biggl{(}\\frac{M^2 y^2-M^2 y-3m^2 y+4m^2}{y(M^2 y-M^2+m^2)}\\biggr{)}^{1\/2}\\,, \\nonumber \\\\\n  y_{1,2}&=&\\frac{M^2+3m^2\\mp(M^4-10m^2 M^2+9m^4)^{1\/2}}{2M^2}\\,.\n\\end{eqnarray}\nOn the other hand, using the ameliorated cutting rules we also have\n\\begin{eqnarray}\n  &&Im\\,\\mu^{2\\epsilon}\\int\\frac{d^D k_1}{(2\\pi)^D}\\frac{d^D k_2}{(2\\pi)^D}\n  \\frac{1}{k_1^2+i\\varepsilon}\\frac{1}{k_2^2-m^2+i\\varepsilon}\n  \\frac{1}{(k_2-k_1)^2-m^2+i\\varepsilon}\\frac{1}{k_1^2-m^2+i\\varepsilon}\n  \\frac{1}{(k_1-p)^2-m^2+i\\varepsilon} \\nonumber \\\\\n  =\\hspace{-3mm}&&Im\\,\\mu^{2\\epsilon}P\\int\\frac{d^D k_1}{(2\\pi)^D}\\frac{d^D k_2}{(2\\pi)^D}\n  \\Bigl{[} \\frac{4i\\,\\pi^3\\,\\theta[k_{10}]\\delta(k_1^2-m^2)\\theta[k_{20}]\\delta(k_2^2-m^2)\n  \\theta[p_0-k_{10}-k_{20}]\\delta((p-k_1-k_2)^2-m^2)}{(p-k_1)^2((p-k_1)^2-m^2)} \\nonumber \\\\\n  -\\hspace{-3mm}&&\\frac{2\\pi^2\\theta[k_{10}]\\delta(k_1^2)\\theta[p_0-k_{10}]\\delta((p-k_1)^2-m^2)}\n  {(k_2^2-m^2)((k_2-k_1)^2-m^2)(k_1^2-m^2)}-\n  \\frac{2\\pi^2\\theta[k_{10}]\\delta(k_1^2-m^2)\\theta[p_0-k_{10}]\\delta((p-k_1)^2-m^2)}\n  {k_1^2(k_2^2-m^2)((k_2-k_1)^2-m^2)} \\Bigr{]} \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{1}{64\\pi^3}P\\int_0^{\\infty}k_1^2\\,d\\,k_1\n  \\int_0^{\\infty}k_2^2\\,d\\,k_2\\int_{-1}^1 d\\,x\\frac{\\delta(M-\\sqrt{m^2+k_1^2}-\n  \\sqrt{m^2+k_2^2}-\\sqrt{m^2+k_1^2+k_2^2+2 k_1 k_2\\,x})}{\\sqrt{m^2+k_1^2}\n  \\sqrt{m^2+k_2^2}\\sqrt{m^2+k_1^2+k_2^2+2 k_1 k_2\\,x}} \\nonumber \\\\\n  \\times\\hspace{-3mm}&&\\frac{1}{(M^2+m^2-2M\\sqrt{m^2+k_1^2})(M^2-2M\\sqrt{m^2+k_1^2})} \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{2\\pi^2\\mu^{2\\epsilon}\\Gamma(\\epsilon\/2)}{(4\\pi)^{D\/2}m^2}\\int\n  \\frac{d^D k_1}{(2\\pi)^D}\\theta[k_{10}]\\delta(k_1^2)\\theta[p_0-k_{10}]\\delta((p-k_1)^2-m^2) \\nonumber \\\\\n  -\\hspace{-3mm}&&\\frac{2\\pi^2\\mu^{2\\epsilon}\\Gamma(\\epsilon\/2)}{(4\\pi)^{D\/2}m^2}\\int_0^1\n  \\frac{d\\,x}{(x^2-x+1)^{\\epsilon\/2}}\\int\\frac{d^D k_1}{(2\\pi)^D}\n  \\theta[k_{10}]\\delta(k_1^2-m^2)\\theta[p_0-k_{10}]\\delta((p-k_1)^2-m^2) \\nonumber \\\\\n  =\\hspace{-3mm}&&\\frac{\\theta[M-3m]m^2}{64\\pi^3 M}\\int_1^{\\frac{M^3-3m^2}{2m\\,M}}\n  \\frac{\\sqrt{(x^2-1)(M^2-2m\\,M\\,x-3m^2)}}{(M^2+m^2-2m\\,M\\,x)^{3\/2}(M-2m\\,x)}\n  d\\,x \\nonumber \\\\\n  +\\hspace{-3mm}&&\\frac{\\theta[M-m](M^2-m^2)}{256\\pi^3 m^2 M^2}(\\Delta-2\\ln\\frac{M^2-m^2}{m\\,M}+2)\n  \\nonumber \\\\\n  -\\hspace{-3mm}&&\\frac{\\theta[M-2m]\\sqrt{M^2-4m^2}}{256\\pi^3 m^2 M}\n  (\\Delta-\\ln\\frac{M^2-4m^2}{m^2}+4-\\frac{\\pi}{\\sqrt{3}})\\,.\n\\end{eqnarray}\nThrough numerical calculations we find that Eq.(13) is equal to Eq.(11), as shown in Fig.7.\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{file=4.eps,width=7.5cm} \\\\\n  \\caption{Results of Eq.(13) (plotted as dot) and Eq.(11) (plotted as line).}\n\\end{center}\n\\end{figure}\nWe note that a similar discussion has been present in Ref.\\cite{c3}.\n\nAll of the four examples show that the ameliorated cutting rules agrees with the conventional integral algorithm very well. There are also many similar examples which we don't list here.\n\n\\section{Breaking of the optical theorem}\n\nIt's well-known that the optical theorem has put a strong constraint on the imaginary part of physical amplitude. Does it really right? In this section we will discuss this problem.\n\nWe firstly simply review the origin of the optical theorem. It's well known that the S-matrix is unitary: $S^{\\dagger}S=1$. Inserting $S=1+i T$ one has:\n\\begin{equation}\n  -i(T-T^{\\dagger})\\,=\\,T^{\\dagger}T\\,.\n\\end{equation}\nTake the matrix element of this equation between the two-particle states $|{\\bf p}_1{\\bf p}_2\\hspace{-1mm}>$ and $|{\\bf k}_1{\\bf k}_2\\hspace{-1mm}>$. To evaluate the right-hand side, insert a complete set of intermediate states,\n\\begin{equation}\n  <\\hspace{-1mm}{\\bf p}_1{\\bf p}_2|T^{\\dagger}T|{\\bf k}_1{\\bf k}_2\\hspace{-1mm}>\\,=\\,\n  \\sum_f\\int d\\prod_f <\\hspace{-1mm}{\\bf p}_1{\\bf p}_2|T^{\\dagger}|f\\hspace{-1mm}>\n  <\\hspace{-1mm}f|T|{\\bf k}_1{\\bf k}_2\\hspace{-1mm}>\\,,\n\\end{equation}\nwhere the sum runs over all possible sets $f$ of intermediate states and $d\\prod_f$ represents the phase space integral of $f$. Now express the T-matrix elements as invariant matrix elements $\\cal M$ times 4-momentum-conserving delta functions. Eq.(14) then becomes\n\\begin{equation}\n  {\\cal M}(k_1 k_2\\rightarrow p_1 p_2)-{\\cal M}^{\\ast}(p_1 p_2\\rightarrow k_1 k_2)\\,=\\,\n  i\\sum_f \\int d\\prod_f{\\cal M}^{\\ast}(p_1 p_2\\rightarrow f)\n  {\\cal M}(k_1 k_2\\rightarrow f)\\,.\n\\end{equation}\nSetting $k_i=p_i$ and applying the kinematic factors required by Eq.(16) to build a cross section, one obtains the standard form of the optical theorem,\n\\begin{equation}\n  Im{\\cal M}(k_1 k_2\\rightarrow k_1 k_2)\\,=\\,2 E_{cm} p_{cm}\n  \\sigma_{tot}(k_1 k_2\\rightarrow anything)\\,,\n\\end{equation}\nwhere $E_{cm}$ is the total center-of-mass energy and $p_{cm}$ is the momentum of either particle in the center-of-mass frame. Considering the $1\\rightarrow 1$ process one obtains the well-known relationship,\n\\begin{equation}\n  Im{\\cal M}(a\\rightarrow a)\\,=\\,m_a\\Gamma_a\\,,\n\\end{equation}\nwhere $m_a$ and $\\Gamma_a$ is the mass and decay width of particle $a$.\n\nIt seems that the optical theorem is a straightforward consequence of the unitarity of the S-matrix. But we find that there are some severe contradictions in the optical theorem. We firstly discuss a concrete physical amplitude. It is known that the imaginary part of physical amplitude comes from two cases: one is the coupling constant has imaginary part, the other is the Feynman amplitude has branch cut. At one-loop level a physical amplitude can be expressed as\n\\begin{equation}\n  {\\cal M}(i\\rightarrow f)\\,=\\,\\sum_k g_k (a_k+i\\,b_k)\\,,\n\\end{equation} where the sum runs over different interactions, $a_k$ is a\nreal number, $b_k$ is the imaginary part coming from the branch\ncut of Feynman amplitude, and all of the imaginary parts of the\ncoupling constants are included in the coefficient $g_k$. For\nconvenience we define two concepts: the {\\em quasi-imaginary part}\nand the {\\em quasi-real part} of Feynman amplitude, which\nseparately represent the parts of Feynman amplitude coming from\nthe branch cut and normal integral. For example in Eq.(19) the\n{\\em quasi-imaginary part} of the physical amplitude is $\\sum_k\ng_k b_k$ and the {\\em quasi-real part} of the physical amplitude\nis $\\sum_k g_k a_k$. Under the charge conjugation transformation\nonly the imaginary part of the coupling constant changes into its\ncontra-value, all of the other quantities in Eq.(19) keep\nunchanged. So we have from the CPT conservation law \\begin{equation}\n  {\\cal M}^{\\ast}(f\\rightarrow i)\\,=\\,{\\cal M}^{\\ast}(\\bar{i}\\rightarrow \\bar{f})\n  \\,=\\,\\sum_k g_k(a_k-i\\,b_k)\\,.\n\\end{equation}\nFrom Eqs.(19) and (20) we have\n\\begin{equation}\n  {\\cal M}(i\\rightarrow f)-{\\cal M}^{\\ast}(f\\rightarrow i)\\,=\\,2\\,i\\,\\sum_k g_k b_k\\,\\equiv\\,\n  2\\,i\\,\\tilde{Im}{\\cal M}(i\\rightarrow f)\\,,\n\\end{equation}\nwhere $\\tilde{Im}$ takes the {\\em quasi-imaginary part} of the physical amplitude. Obviously this result clearly demonstrates the relationship between the optical theorem and the cutting rules. If the optical theorem is right, one will have for the physical amplitude of top quark decaying into charm quark and gauge boson Z\n\\begin{equation}\n  \\tilde{Im}{\\cal M}(t\\rightarrow c\\,Z)\\,=\\,\\frac{1}{2}\\sum_f \\int d\\prod_f {\\cal M}^{\\ast}(c\\,Z\\rightarrow f)\n  {\\cal M}(t\\rightarrow f)\\,.\n\\end{equation}\nOn the other hand, one can calculate the {\\em quasi-imaginary part} of ${\\cal M}(t\\rightarrow c\\,Z)$ from the branch cuts of the Feynman amplitude. The one-loop Feynman diagrams of $t\\rightarrow c\\,Z$ have been shown in Fig.8.\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{file=tcZ_all.ps,width=12cm} \\\\\n  \\caption{One-loop diagrams of the physical amplitude of $t\\rightarrow c\\,Z$.}\n\\end{center}\n\\end{figure}\nWe note that there is no need to introduce counterterm into the one-loop physical amplitude of $t\\rightarrow c\\,Z$ after including the contribution of the amplitude of Fig.8. From the branch cuts of Fig.8 we find a contradiction in the optical theorem: Eq.(22) forbids the cuts representing the contribution of $\\sum_f\\int d\\prod_f {\\cal M}^{\\ast}(c\\rightarrow f){\\cal M}(t\\rightarrow Z\\,f)$ in which the cuts of $t\\rightarrow c$ two-point diagrams in the seven and eight diagrams of Fig.8 are included, but that cuts of $t\\rightarrow c$ two-point diagrams in the seven and eight diagrams of Fig.8 are permitted by Eq.(18). Obviously the contribution of the cuts of $t\\rightarrow c$ two-point diagrams in the seven and eight diagrams of Fig.8 to the physical amplitude isn't equal to zero.\n\nIn fact there exists a more severe contradiction. From the\nunitarity of S-matrix we also have $T^{\\dagger}T=T T^{\\dagger}$.\nThus Eq.(15) can be changed into \\begin{equation}\n  <\\hspace{-1mm}{\\bf p}_1{\\bf p}_2|T^{\\dagger}T|{\\bf k}_1{\\bf k}_2\\hspace{-1mm}>\\,=\\,\n  \\sum_f\\int d\\prod_f <\\hspace{-1mm}{\\bf p}_1{\\bf p}_2|T|f\\hspace{-1mm}>\n  <\\hspace{-1mm}f|T^{\\dagger}|{\\bf k}_1{\\bf k}_2\\hspace{-1mm}>\\,.\n\\end{equation}\nSince each intermediate state $f$ has its own unique parameters (e.g. mass), Eq.(23) being equal to Eq.(15) means all of the terms about $f$ are separately equal, i.e. for arbitrary $f$\n\\begin{equation}\n  <\\hspace{-1mm}{\\bf p}_1{\\bf p}_2|T^{\\dagger}|f\\hspace{-1mm}>\\,=\\,\n  <\\hspace{-1mm}{\\bf p}_1{\\bf p}_2|T|f\\hspace{-1mm}>\\,, \\hspace{10mm}\n  <\\hspace{-1mm}f|T|{\\bf k}_1{\\bf k}_2\\hspace{-1mm}>\\,=\\,\n  <\\hspace{-1mm}f|T^{\\dagger}|{\\bf k}_1{\\bf k}_2\\hspace{-1mm}>\\,.\n\\end{equation}\nSince $f$ are arbitrary, we ultimately obtain\n\\begin{equation}\n  {\\cal M}(k_1 k_2\\rightarrow p_1 p_2)-{\\cal M}^{\\ast}(p_1 p_2\\rightarrow k_1 k_2)\\,=\\,\n  0\\,.\n\\end{equation}\nObviously Eq.(25) contradicts Eq.(16).\n\nAnother problem about the optical theorem is: if the charge\nconjugation is conserved, from the CPT conservation law the\nleft-hand side of Eq.(16) is a pure imaginary, but the right-hand\nside of Eq.(16) may contain real part which comes from the branch\ncut of the physical amplitudes. Why are there such contradictions\nand problem? In fact the S-matrix is only a symbolistic operator,\nit doesn't have the algebraic algorithm that a true operator\nshould have, e.g. the multiplication and addition. This is because\nthe S-matrix represents the summation of the interactions of the\nphysical world, so it is no possible to express it by a concrete,\nhaving certain algebraic algorithm's, operator. Therefore the\nderivation of Eqs.(14-16) isn't right.\n\n\\section{Gauge independence of physical result under ameliorated cutting rules}\n\nIn order to investigate whether the ameliorated cutting rules is reasonable, we calculate the physical amplitude $t\\rightarrow c\\,Z$ to see if the ameliorated cutting rules keeps the physical result gauge independent.\n\nIn Fig.8 we have shown the one-loop diagrams of the physical process $t\\rightarrow c\\,Z$. Since there is no tree-level diagram, it is no need to introduce counterterm to one-loop level. We note that in the following calculations we have used the program packages {\\em FeynArts, FeynCalc, FormCalc and LoopTools} \\cite{c6}. Our calculations have shown that the {\\em quasi-real part} of ${\\cal M}(t\\rightarrow c\\,Z)$ is gauge-parameter independent. So we only need to consider the {\\em quasi-imaginary part} of ${\\cal M}(t\\rightarrow c\\,Z)$. Using the ameliorated cutting rules we obtain (see also Fig.1)\n\\begin{equation}\n  \\tilde{Im}{\\cal M}(t\\rightarrow c\\,Z)|_{\\xi}\\,=\\,0\\,,\n\\end{equation}\nwhere subscript $\\xi$ takes the gauge-dependent part of the quantity. From Eq.(26) one can easily see that all of the unphysical cuts (e.g. $t\\rightarrow G^{+}d_i$) in Fig.8 have been cancelled out in the physical amplitude by the ameliorated cutting rules. This coincides with the conventional knowledge \\cite{c7}. Obviously such result makes the decay width of $t\\rightarrow c\\,Z$ gauge independent to two-loop level.\n\nBy the way, from the above discussions we know that the optical theorem gives a contradictory result about the {\\em quasi-imaginary part} of ${\\cal M}(t\\rightarrow c\\,Z)$. If keeping the cuts of $t\\rightarrow c$ two-point diagrams in the seven and eight diagrams of Fig.8 and for the first six diagrams of Fig.8 only keeping the first kind cut of Fig.1 required by Eq.(22), we obtain\n\\begin{eqnarray}\n  \\tilde{Im}{\\cal M}(t\\rightarrow c\\,Z)|_{\\xi}&=&\\bar{c}\\,\n  {\\xslash \\epsilon^{\\ast}}\\gamma_L\\,t\\,\\sum_i\\frac{V_{2i}V^{\\ast}_{3i}\\,e^3\n  (3-4 s_W^2)(x_c-\\xi_W-x_{d,i})}{384\\pi\\,c_W\\,s_W^3 x_c} \\nonumber \\\\\n  &\\times&\\left( x_c^2-2(\\xi_W+x_{d,i})x_c+(\\xi_W-x_{d,i})^2 \\right)^{1\/2}\\,\n  \\theta[m_c-m_{d,i}-M_W\\sqrt{\\xi_W}]\\,,\n\\end{eqnarray}\nwhere $\\gamma_L$ is the left-handed helicity operator, $V_{2i}$ and $V_{3i}$ are the CKM matrix elements \\cite{c8}, $e$ is electron charge, $s_W$ and $c_W$ is the sine and cosine of the weak mixing angle, $M_W$ and $\\xi_W$ is the mass and gauge parameter of gauge boson $W$, $m_c$ and $m_{d,i}$ are the masses of charm quark and down-type $i$ quark, $x_c=m_c^2\/M_W^2$ and $x_{d,i}=m_{d,i}^2\/M_W^2$. This result is gauge dependent and will lead to the decay width of $t\\rightarrow c\\,Z$ gauge dependent. One can easily see that the gauge-dependent terms of Eq.(27) come from the unphysical cuts of $t\\rightarrow c$ two-point diagrams in the seven and eight diagrams of Fig.8.\n\n\\section{Conclusion}\n\nIn order to calculate the contribution of the singularity of Feynman propagator to Feynman amplitude we investigate the cutting rules and the optical theorem. We ameliorate the cutting rules in order to make it suitable for actual calculation and give the right result of the imaginary part of physical amplitude. The calculations of the imaginary parts of several Feynman diagrams show that the ameliorated cutting rules agrees with the conventional integral algorithm very well (see Fig.6,7 and Eq.(6,8,9,10)). On the other hand, through careful investigation we find that the optical theorem has severe contradictions and problem thus isn't right. Besides, the calculation of the physical amplitude $t\\rightarrow c\\,Z$ shows that the ameliorated cutting rules keeps the decay width of the physical process gauge independent to two-loop level (see Eq.(26)).\n\nIn the viewpoint of the conventional cutting rules the branch cut of Feynman diagram contributes the imaginary part to physical amplitude. Our investigation further finds that although the Feynman diagram doesn't represent the real physical process, it does provide a physical-like picture to describe the on-shell virtual particles' propagating processes happening in the quantum field vacuum if such processes exist. The ameliorated cutting rules not only gives a feasible algorithm to calculate the contributions of the virtual processes to physical amplitude, but also can help us to discover the deep-seated physical meaning of the quantum field theory.\n\n\\vspace{5mm} {\\bf \\Large Acknowledgments} \\vspace{2mm}\n\nThe author thanks Prof. Cai-dian Lu for the fruitful discussions and the corrections of the words. The author also thanks Yue-long Shen, doctor Jian-feng Cheng, Xian-qiao Yu, Ge-liang Song and Ying Li for the fruitful discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\\subsection{Historical motivation}\n\\label{history}\nEinstein made a number of very pertinent remarks about the foundations of quantum mechanics, often summarised in the famous\n phrase ``God\ndoes not play dice\". For example, in 1935 he wrote \\cite{Einstein1935}:\n\\begin{quote}\n``\nIn any case one does not have the right today to maintain that the foundation must consist in a field theory in the sense of Maxwell.\nThe other possibility, however, leads in my opinion to a renunciation of the time-space continuum and a purely algebraic physics.\nLogically this is quite possible [...] Such a theory doesn't have to be based on the probability concept.\"\n\\end{quote}\nIn 1954, he went further, and opined \\cite{Einstein1954}:\n\\begin{quote}\n``I consider it quite possible that physics cannot be based on the field concept, i.e., continuous structure.\nIn that case, nothing remains of my entire castle in the air, gravitation theory included.\"\n\\end{quote}\nNevertheless, modern physics has stuck with the field concept, and his `castle in the air' remains in general use.\n\nIn 1935, there simply was not enough experimental evidence to provide much of a clue as to the possible structure\nof a `purely algebraic physics', beyond the obvious fact that it must contain a copy of the quaternion group. But today, there is\nan enormous amount of experimental evidence, that essentially shows there is a unique possibility for a purely\nalgebraic physics \\cite{perspective}. Whether it actually works or not, is a separate question. But we should at least try it.\nI begin by considering how much of physics, including continuous approximations  interpreted as\nspacetime and fields \nin spacetime, \ncan be obtained from the quaternion group alone.\n\n\n\n\\subsection{The spin group and the quaternion group}\n\\label{quaternion}\nGroup theory \\cite{Zee}\nfirst entered into quantum mechanics with the discovery of the spin of an electron, which has a direction relative\nto the ambient space, and therefore requires a group $SU(2)$ for its description. On a macroscopic scale, the sum of the spins of a large number of electrons, protons and neutrons gives rise to\nthe phenomenon of magnetism, when there is a sufficiently large bias in the directions of spin. \n\nBut it is experimentally impossible to measure the\ndirection of spin of an individual elementary particle. \nAll one can do is choose a direction, and measure whether the spin is `up' or `down' in that direction. \nIndeed, even for larger particles, such as silver atoms, the Stern--Gerlach experiment \\cite{SternGerlach}\ndemonstrates that the direction of spin is\nquantised.\nTherefore we must assume that the elementary particles have,\nintrinsically, only finite symmetries. The smallest finite group that is available to describe spin symmetries is the\nquaternion group $Q_8$, with $8$ elements $\\pm 1$, $\\pm i$, $\\pm j$ and $\\pm k$ satisfying the rules \n\\begin{eqnarray}\n&ij=k, jk=i, ki=j, \\cr\n&i^2=j^2=k^2=-1.\n\\end{eqnarray}\n\nNow if we take a large number of copies of $Q_8$, and add them together, we obtain \\cite{grouprings} \nthe integral group ring $\\mathbb Z Q_8$, in which all the copies of $Q_8$ are identical, meaning that all electrons are identical, and so on. In macroscopic physics, the individual quanta disappear from view, and it\nbecomes reasonable to approximate the scaled copy of the integers $\\mathbb \\varepsilon \\mathbb Z$ by the real numbers $\\mathbb R$, and study the\ngroup algebra $\\mathbb RQ_8$ instead. There is, up to equivalence, only one faithful representation of $Q_8$, in the quaternion algebra $\\mathbb H$,\nand there are four\n$1$-dimensional real representations, of which one is trivial and three are representations of the three quotients $Q_8\/Z_4\\cong Z_2$.\nTherefore the group algebra has a canonical (Wedderburn) decomposition as \n\\begin{eqnarray}\n\\mathbb R Q_8 &\\cong & 4\\mathbb R + \\mathbb H.\n\\end{eqnarray}\nTaking out the five real scale factors from this algebra, we are left with the group $SU(2)$, which is exactly what we require in order to describe\nmagnetic spin in ordinary macroscopic space.\n\n\\subsection{Analysis of the group algebra}\n\\label{analysis}\nIn the representation theory of finite groups, the usual convention is for the finite group to act on the algebra by right multiplication, and the continuous groups to act by left-multiplication.  This allows us to treat both the discrete and continuous symmetries simultaneously. \nThe macroscopic behaviour, on the other hand, requires the continuous group to act on both sides,\nby conjugation, so that $SU(2)$ acts on $\\mathbb H$ as $SO(3)$, that is, fixing the real part, and rotating the imaginary part as a Euclidean\n$3$-space. \nIt is standard to interpret this $3$-space as real physical space, in the case when $SU(2)$ represents magnetic spin.\nIt would be reasonable, then, to interpret the real part of $\\mathbb H$ as a (non-relativistic) time. This allows the `spin' to be modelled \nas something\nthat takes place \nin space and time, rather than in isolation.\n\nBut the finite group has given us something in addition, namely four copies of the real numbers. These \npresumably represent four physical \nparticles, at least three of which we can\ndetect magnetically. \nThese\nfour particles \nare acted on by the finite group on the right, in three cases to change the sign, so that these three \nappear in a pair of spin states. They are invariant under conjugation by $SU(2)$, so under symmetries of\nmacroscopic space. But they are not invariant under the left action of the finite group. Since we started out modelling spin of electrons\nand protons, these had better be two of the particles, and the most plausible\ninterpretation of the other two is as the neutron\nand (electron) neutrino. The first three appear in two spin states, the last in only one, as we know also from the Wu experiment \\cite{Wu}.\nThus this action describes the weak interaction, in the form of beta decay:\n\\begin{eqnarray}\ne+p &\\leftrightarrow& n+\\nu.\n\\end{eqnarray}\n\nNote that this is an action of the finite group, not of the spin group.\nIn the standard model \\cite{Griffiths}, the weak interaction is described instead by a second copy of $SU(2)$, commuting with the spin group.\nIt is of course perfectly possible to define an action of $SU(2)$ on this $4$-dimensional real space, and have this action\ncommute with the spin $SU(2)$, and try to build a model of physics on top.\nBut this action is not compatible with \nthe group algebra.\nThe standard model therefore puts the weak $SU(2)$ elsewhere, and doubles up the Weyl spinor into a Dirac spinor\nso that there will be no conflict between the finite action and the continuous action. By doing so, however, it breaks the connection\nbetween the weak interaction and the spin group, which then has to be put back in by hand, in the form of a mixing\nbetween the weak force and electromagnetism. In the toy model discussed here, the mixing between magnetism and\nthe weak force happens automatically.\n\nIt is worth analysing the modelling of these particles in a little more detail. In the group algebra, the four $1$-dimensional representations \ncontain the following elements:\n\\begin{eqnarray}\n1a:&& 1+(-1)+i+(-i)+j+(-j)+k+(-k);\\cr\n1b: && 1+(-1)+i+(-i)-j-(-j)-k-(-k);\\cr\n1c:&& 1+(-1)-i-(-i)+j+(-j)-k-(-k);\\cr\n1d:&& 1+(-1)-i-(-i)-j-(-j)+k+(-k).\n\\end{eqnarray}\nWe then see that the elements of the group can be interpreted, not as particles, but as quantum numbers of the particles. Two quantum numbers are sufficient to distinguish these four particles, so that for example, if $\\lambda_i$ and $\\lambda_j$ denote the coefficients of $i$ and $j$ respectively, then we can take the charge to be $(\\lambda_j-\\lambda_i)\/2$, and the weak isospin to be $\\lambda_j\/2$. We do not need to specify the coefficient of $k$, that is $\\lambda_k=\\lambda_i\\lambda_j$.\n\nIn a similar way, the other half of the group algebra has a basis consisting of\n\\begin{eqnarray}\nt&:=& 1-(-1),\\cr\nx&:=& i-(-i),\\cr\ny&:=& j-(-j),\\cr\nz&:=& k-(-k).\n\\end{eqnarray}\nThe names are chosen with the intention of suggesting a potential quantisation of spacetime, with $z$ representing the direction of spin, and the $x,y$ plane relating to the charge and isospin. \nThe (speculative) idea, which I shall explore in more detail below, is that the combination of spin, charge and weak isospin for a (large) collection of interacting elementary particles \nmay be sufficient to define the ambient physical spacetime.\n\nThis \nexample illustrates the general principle of how a macroscopic force, in this case magnetism, emerges from individual quanta. In particular,\nit illustrates how a macroscopic continuous symmetry arises from a microscopic discrete symmetry. It also illustrates how a breaking of the symmetry\nat the microscopic level arises from an interaction between elementary particles, in this case the weak interaction,\nand gives rise to a breaking of symmetry also at the\nmacroscopic level. The quaternion group $Q_8$ on its own is therefore sufficient to model a united magneto-weak force. It is not sufficient for\na complete electro-magneto-weak unification, however, for which we need a larger group.\nThis larger group must be capable of dealing with the fact that the electron comes in three different mass states, that is in three different\ngenerations. \n\n\\subsection{The automorphism group}\n\\label{auto}\nThe symmetries of this finite model of spin are described by the automorphism group of $Q_8$, that is isomorphic to\n$Sym(4)$, the symmetric group on $4$ letters.\nThe automorphisms are of five types, where I denote the letters $W,X,Y,Z$: \n\\begin{itemize}\n\\item the identity element, of order $1$.\n\\item $3$ even permutations of order $2$, such as $(W,X)(Y,Z)$; these are inner automorphisms, that is conjugation by $i$, $j$ and $k$. \n\\item $8$ even permutations of order $3$, such as $(X,Y,Z)$; these can be represented as conjugation by unit quaternions $(-1\\pm i\\pm j\\pm k)\/2$.\n\\item $6$ odd permutations of order $2$, such as $(W,X)$; represented as conjugation by $i\\pm j$, $j\\pm k$ and $k\\pm i$.\n\\item $6$ odd permutations of order $4$, such as $(W,X,Y,Z)$; represented as conjugation by $1\\pm i$, $1\\pm j$ and $1\\pm k$.\n\\end{itemize} \n\nThese symmetries have many different interpretations, depending on what the original copy of $Q_8$ represents. The kind of symmetry that\nit might be useful to look for is an extension of the neutrino\/electron\/proton\/neutron symmetry discussed in the previous section, to\nthree generations of electron, plus the baryon octet \\cite{GellMann}. \nIn this way the baryons would exhibit a $3$-fold symmetry represented by the $3$-cycles,\nthat could perhaps be interpreted as a colour symmetry of the three constituent quarks \\cite{QCD}. Similarly,\nthe electrons would divide into two transpositions, possibly representing the left-handed and right-handed spins. \n\nAlternatively, or in addition,\none might want to consider the odd permutations as representing leptons and quarks. The transpositions could then represent either\nthe left-handed and right-handed electrons, \nor the left-handed parts of both neutrinos and electrons. Similarly,\nthe $4$-cycles could represent quarks, either in up\/down pairs \nor in left\/right pairs.\nAdditional interpretations might arise from consideration of particle interactions, so that the identity element represents electromagnetic\ninteractions, \n the $3$ elements of order $2$ represent the weak interaction,\nand the $8$ elements of order $3$ the strong interaction. The strong interaction manifests itself both in massless form (as gluons) and\nin massive form (as the meson octet, including pions and kaons).\n\nIt may be worth mentioning in passing that if one extends the $S_4$ action by conjugation of the quaternions listed above,\nto include both left and right multiplications, then one obtains the Weyl group of type $F_4$. This may be the fundamental reason\nwhy the Lie group of type $F_4$, and various related types such as $D_4$ and $E_6$, are so attractive as potential ways\nto extend the standard model \\cite{E6,E8,DG,Furey,DMW}. \n\n\\section{The binary tetrahedral group}\n\\label{tetrahedron}\n\\subsection{Overview}\n\\label{overview}\nThere is only one non-trivial way to adjoin a triplet symmetry to $Q_8$,\nand that is to extend \nto the  binary tetrahedral group, $G$ say, of order $24$, by adjoining the quaternion\n\\begin{eqnarray}\nw&:=& (-1+i+j+k)\/2,\n\\end{eqnarray}\nand therefore also its quaternion conjugate\n\\begin{eqnarray}\nv&:=& (-1-i-j-k)\/2.\n\\end{eqnarray}\nThis group and its representation theory are described in some detail in \\cite{perspective}, but we shall require \nyet more detail here. \nI begin with a summary. \n\nThe most important thing to note is the structure of the real group algebra:\n\\begin{eqnarray} \n\\mathbb R G &\\cong & \\mathbb R + \\mathbb C + \\mathbb H + M_2(\\mathbb C) + M_3(\\mathbb R).\n\\end{eqnarray}\nTaking out four real scalars and one complex scalar leaves us with the group\n\\begin{eqnarray}\nU(1) \\times SU(2) \\times SL(2,\\mathbb C)\\times SL(3,\\mathbb R),\n\\end{eqnarray}\nwhich contains all the groups we need for the standard model of particle physics, apart from the fact that we have the\nsplit real form $SL(3,\\mathbb R)$ rather than the compact real form $SU(3)$ of the Lie group of type $A_2$.\n\nThis difference may reflect the fact that we are looking at finite (generation) \nsymmetries that can be observed in\nexperiments, rather than continuous\n(colour) symmetries that are not observable. It may be possible to resolve this\nissue by extending to the complex group algebra, but this process obscures a number of important features of\nthe real group algebra, so I will only do this if absolutely necessary. \nIn any case, we must bear in mind that this difference must be resolved at some point,\nsince it may be the difference between a viable model and an unviable model.\n\n\\subsection{Irreducible representations}\n\\label{irreps}\nRepresentation theory is usually presented first over the complex numbers, since this is the simplest theory, but we shall\nrequire the extra subtlety of the representation theory over real numbers.\nFirst note that the conjugation action of the group on itself divides the group into $7$\nconjugacy classes, of sizes $1,1,6,4,4,4,4$, as listed in this table: \n\\begin{eqnarray}\n&\\begin{array}{cccc}\n\\mbox{Size} & \\mbox{Elements} & \\mbox{Order}\\cr\\hline\n1 & 1 = e& 1\\cr\n1 & -1 = i^2& 2\\cr\n6 & \\pm i, \\pm j, \\pm k & 4\\cr\n4 & w, wi, wj, wk & 3\\cr\n4 & v, -vi,-vj, -vk & 3\\cr\n4 &- w, -wi, -wj, -wk & 6\\cr\n4 & -v, vi,vj, vk & 6\\cr\\hline\n\\end{array}&\n\\end{eqnarray}\n\nThere are therefore $7$ complex irreducible representations, whose characters are listed as rows in the following table. The top row gives a representative of the conjugacy class, and the entries are the traces of the representing matrices, where $\\omega$ and $\\bar\\omega$ are the complex cube roots of unity.\n\\begin{eqnarray}\n\\begin{array}{ccccccc}\n1&-1&i&w&-w&v&-v\\cr\\hline\n1&1&1&1&1&1&1\\cr\n1&1&1&\\omega&\\omega&\\bar\\omega&\\bar\\omega\\cr\n1&1&1&\\bar\\omega&\\bar\\omega&\\omega&\\omega\\cr\n3&3&-1&0&0&0&0\\cr\n2&-2&0&-1&1&-1&1\\cr\n2&-2&0&-\\omega&\\omega&-\\bar\\omega&\\bar\\omega\\cr\n2&-2&0&-\\bar\\omega&\\bar\\omega&-\\omega&\\omega\\cr\\hline\n\\end{array}\n\\end{eqnarray}\nThe three $2$-dimensional representations will eventually play the role of Weyl spinors, but there are three of them, \nnot just the familiar left-handed and right-handed Weyl spinors. \n Of course, the standard model for electro-weak interactions really has three Weyl spinors, two left-handed and one right-handed, so the question is whether it is possible, and if so, how, to match up the three spinors in the two models.\n\nThe structure of the complex group algebra \ncan be read off from the dimensions of the representations, and is\n\\begin{eqnarray}\n3\\mathbb C + M_3(\\mathbb C) + 3M_2(\\mathbb C).\n\\end{eqnarray}\nIn addition to the problem of mixing and matching the three copies of $SU(2)$ or $SL(2,\\mathbb C)$ inside the three copies of $M_2(\\mathbb C)$,\n there is the problem of mixing and matching the three copies of $U(1)$ inside the three copies of $\\mathbb C$. There is too much choice at this stage,\n and the problem becomes rather more tractable if we restrict to the real group algebra. Of course, this restriction means that we lose the group $SU(3)$\nfrom $M_3(\\mathbb C)$, but we have $SL_3(\\mathbb R)$ instead. This different real form may or may not be a satisfactory replacement for the \ngauge group of the strong force. \n\nThe irreducible real representations can be described by taking the sum of a complex representation with its complex conjugate, and at the same time\ntaking the union of each conjugacy class with its inverse class. This reduces the table to five rows and five columns, as follows:\n\\begin{eqnarray}\n\\begin{array}{ccccc}\n1&-1&i&w&-w\\cr\\hline\n1&1&1&1&1\\cr\n2&2&2&-1&-1\\cr\n3&3&-1&0&0\\cr\n4&-4&0&-2&2\\cr\n4&-4&0&1&-1\\cr\\hline\n\\end{array}\n\\end{eqnarray} \nThe first of the two $4$-dimensional representations is quaternionic, so has a continuous $3$-parameter family of possible complexifications.\nThe second of the two $4$-dimensional representations, on the other hand, has a discrete set of exactly two possible complexifications. This interplay between the discrete and the continuous \nmay possibly play an important role in the structure of the Dirac spinor and the Dirac algebra \\cite{Dirac}.\n\nIn order to distinguish these two representations, let us write the first one $4_H$, to denote a Hamiltonian quaternionic representation, \nand the second one $4_C$, to denote a classical complex representation. Then the group algebra, \nas a representation of the finite group $G$, has the structure\n\\begin{eqnarray}\n1+2+3+3+3+4_H+4_C+4_C.\n\\end{eqnarray}\n\n\\subsection{Tensor products}\n\\label{tensors}\nSo far, I have associated the various Lie groups (gauge groups and spin groups) with a single representation of the finite group.\nThe finite group links them together in various ways, described in part by the decompositions of tensor products of representations.\nThese decompositions can be calculated easily from the character table, and are as follows, with $+$ signs omitted to save space:\n\\begin{eqnarray}\n\\begin{array}{c|cccc}\n1&2 & 3 &4_H & 4_C\\cr\\hline\n2 & 13 & 123 & 4_C4_C & 4_H4_C\\cr\n3 & 123\n& 1233 & 4_H4_C4_C & 4_H4_C4_C\\cr\n4_H & 4_C4_C&4_H4_C4_C & 11113333 & 223333\\cr\n4_C & 4_H4_C&4_H4_C4_C & 223333\n& 1123333 \n\\end{array}\n\\end{eqnarray}\n\nThere is clearly plenty of structure in here that might relate to the structure of the standard model. For example the fact that $(1+2)\\otimes 4_H = 4_H+4_C+4_C$ means that the whole of the fermionic part of the algebra can be obtained from a real plus a complex `version' of the spin representation $4_H$.\n\n If one distinguishes the two copies of the real numbers by labelling one of them with $\\gamma_5$, then there is \n some prospect of being able to match up the finite model with the standard model. Similarly, all of the $4\\otimes 4$ representations look like different real slices of the complex Clifford algebra in the standard model, which for a finite (therefore compact) group must split as $1+(1+3)+(3+3)+(1+3)+1$. Again we see a mixing of the top and bottom degrees of the Clifford algebra ($1$ and $i\\gamma_5$) into a real $2$-space, and a similar mixing in the odd part of the Clifford algebra. All this suggests that a careful distinction between $4_H$ and $4_C$ may be able to throw some light on the mixing of quantum electrodynamics and the weak interaction in the standard model. \n\nA similar picture emerges from the bosonic part of the algebra, in which one obtains the whole algebra from the tensor product $(1+3)\\otimes 3 = 1+2+3+3+3$. For the purpose of matching with the standard model, one might wish to go further and observe the isomorphism\n\\begin{eqnarray}\n(1+3)\\otimes (1+3)&\\cong&1+1+2+3+3+3+3\\cr\n&\\cong& 4_C\\otimes 4_C. \n\\end{eqnarray} \nHowever one tries to interpret this equivalence, it seems to imply a mixing between the strong force, with a gauge group acting on $1+3$, and the\nelectroweak forces, with a gauge group acting on $4_C$.\n\nIndeed, there are a number of other suggestive isomorphisms between different tensor product representations. For example, for any\nrepresentation $R$ that does not involve the trivial representation $1$, we have\n\\begin{eqnarray}\n(1+2)\\otimes R &\\cong & 3\\otimes R,\n\\end{eqnarray}\nwhich gives ample scope for mixing a broken $1+2$ symmetry with an unbroken $3$ symmetry.\nAnother example is that\n\\begin{eqnarray}\n3\\otimes 4_H &\\cong& \n3\\otimes 4_C,\n\\end{eqnarray}\nwhich gives plenty of scope for mixing a spinor of type $4_H$ with a spinor of type $4_C$.\n\n\\subsection{Explicit matrices}\n\\label{matrices}\nFor the purposes of explicit calculation, it will be useful to have explicit matrix copies of all the irreducible representations. It is sufficient to specify matrices representing the generators $i$ and $w$. In the $1$-dimensional representation, they both act trivially. In the $2$-dimensional representation, $w$ is a rotation of order $3$, and $i$ acts trivially, so we may take the matrices\n\\begin{eqnarray}\ni\\mapsto \\begin{pmatrix}1&0\\cr 0&1\\end{pmatrix}, && w\\mapsto \\frac12\\begin{pmatrix}-1& \\sqrt 3\\cr -\\sqrt 3 & -1\\end{pmatrix}\n\\end{eqnarray}\nThe $3$-dimensional representation is the representation as symmetries of a regular tetrahedron, which can be embedded as alternate vertices of a cube, so that the matrices can be taken as\n\\begin{eqnarray}\ni\\mapsto\\begin{pmatrix}1&0&0\\cr 0&-1&0\\cr 0&0&-1\\end{pmatrix},&&\nw\\mapsto\\begin{pmatrix}0&1&0\\cr 0&0&1\\cr 1&0&0\\end{pmatrix}.\n\\end{eqnarray}\nThe representation $4_H$  is the representation by right-multiplication on the quaternions, so that the matrices can be taken as\n\\begin{eqnarray}\ni\\mapsto \\begin{pmatrix}0&1&0&0\\cr -1&0&0&0\\cr 0&0&0&-1\\cr 0&0&1&0\\end{pmatrix},&&\nw\\mapsto\\frac12\\begin{pmatrix}-1&1&1&1\\cr -1&-1&-1&1\\cr -1&1&-1&-1\\cr -1&-1&1&-1\\end{pmatrix}\n\\end{eqnarray}\nIn the standard model the quaternionic symmetry is broken, and a particular complex basis is chosen. The following matrices give an example,\nwhich may or may not be similar to what is done in the standard model:\n\\begin{eqnarray}\ni\\mapsto \\begin{pmatrix} i & 0\\cr 0 & -i\\end{pmatrix},&& w\\mapsto \\frac12\\begin{pmatrix}-1+i&1+i\\cr -1+i & -1-i\\end{pmatrix}\n\\end{eqnarray}\nFinally, the representation $4_C$ can be written as \n\\begin{eqnarray}\ni\\mapsto \\begin{pmatrix}0&1&0&0\\cr -1&0&0&0\\cr 0&0&0&-1\\cr 0&0&1&0\\end{pmatrix},&&\nw\\mapsto\\frac14\\begin{pmatrix}\n\\alpha&-\\beta&-\\beta&-\\alpha\\cr \\beta&\\alpha&\\alpha&-\\beta\\cr\n \\alpha&-\\beta&\\beta&\\alpha\\cr \\beta&\\alpha&-\\alpha&\\beta\n \\end{pmatrix}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\alpha&=&1+\\sqrt 3\\cr\n\\beta&=& 1-\\sqrt 3.\n\\end{eqnarray}\nIt is unfortunately not possible to get rid of the annoying square roots of $3$ in this real representation.\nOn the positive side, they are what permits the \nextension of the standard model from one generation of fermions to three. \n\n\\subsection{Projections}\n\\label{projections}\nThe theory of finite group representations contains a canonical set of\nprojections from the group algebra onto the\nvarious matrix subalgebras. \n\nFirst there is the basic division into bosons and fermions, obtained via\nprojection with the idempotents $(e\\pm i^2)\/2$. \nWithin the fermions, there are two projections obtained from the\nsum of all the elements of order $3$, that is\n\\begin{eqnarray}\ns&:=& w+ iw+ jw+ kw + v-iv-jv- kv.\n\\end{eqnarray}\nThese projections are defined by the elements $(2e-s)\/6$ and $(4e+s)\/6$ of the group algebra. Notice that the coefficients\nof the identity element here are $1\/3$ and $2\/3$, which suggests some connection with the charges on the quarks.\n\nWithin the bosons, \nthere are three projections, so that the full list is as follows:\n\\begin{eqnarray}\n&\\begin{array}{cc}\n\\mbox{Subalgebra} & \\mbox{Idempotent}\\cr\\hline\n\\mathbb R & (e+i^2)(e+i)(e+j)(e+w+v)\/24\\cr\n\\mathbb C & (e+i^2)(e+i)(e+j)(2e-w-v)\/24\\cr\nM_3(\\mathbb R) & (e+i^2)(3e-i-j-k)\/8\\cr\\hline\n\\mathbb H & (e-i^2)(2e-s)\/12\\cr\nM_2(\\mathbb C) & (e-i^2)(4e+s)\/12 \\cr\\hline\n\\end{array}&\n\\end{eqnarray}\nIf we omit the subalgebra $M_3(\\mathbb R)$ from this description \nwe obtain the $15$-dimensional subalgebra\n\\begin{eqnarray}\n&\\mathbb R+\\mathbb C+\\mathbb H + M_2(\\mathbb C)&\n\\end{eqnarray}\n in which there is a close parallel between the splitting \n $\\mathbb R+\\mathbb C$ using\n  $(e+w+v)\/3$ and $(2e-v-w)\/3$, and the splitting  $\\mathbb H + M_2(\\mathbb C)$ using $(2e-s)\/6$ and $(4e+s)\/6$.\n \nThese two  \npairs of projections appear to be the finite analogues of the \npair of projections $(1\\pm \\gamma_5)\/2$\nthat split the Dirac spinor into left-handed and right-handed Weyl spinors in the standard model,\nand that distinguish the weak force from electromagnetism. \nHowever, the fact that in the finite model the projections on the scalars are different from the projections on the spinors\nallows the finite model to be more subtle than the standard model, and perhaps incorporate a generation structure\ninto this distinction.  It may be possible, for example, to add in the scalars from $M_3(\\mathbb R)$, to obtain a $16$-dimensional\nalgebra that fulfils the function of the Dirac algebra, but with a more subtle structure derived from the action of the finite group.\n\nIt is worth remarking here that because these projections involve dividing by $2$ and $3$, they can be implemented in the real group algebra\n$\\mathbb RG$ or the rational group algebra $\\mathbb QG$, but not in the integral group ring $\\mathbb ZG$. Indeed, the structure of\n$\\mathbb ZG$ is much more subtle than that of $\\mathbb RG$. Ultimately, to implement the finite model in full we will need to grapple\nwith this structure in detail. For the purposes of the present paper, however, it is enough consider only the real group algebra. \n\nJust to give a flavour of the implications of using the integral group ring, I'll describe a toy model of the proton using the integral group\nring of the group $Z_3$ of order $3$. If we take $e,v,w$ as the elements of $Z_3$, then the idempotents in $\\mathbb RZ_3$ are\n$(e+v+w)\/3$ and $(2e-v-w)\/3$ as above. The former projects onto a real $1$-space containing a down quark, and the latter projects onto a real\n$2$-space containing two up quarks. These are perfectly good real representations of $Z_3$, and give a perfectly good description of the\ninternal structure of a proton.\n\nBut they are not integral representations of $Z_3$. The integral \ngroup algebra $\\mathbb ZZ_3$ does not support any projections, and is\nindecomposable. This I interpret as saying that the proton itself cannot be decomposed into smaller particles, so that, in the real\nuniverse, protons never decay. Thus the quarks are `real' particles, but they are not `whole' (integer) particles.\n\nIt is also possible to use the action of $Z_3$ by conjugation to describe the corresponding bosons. These bosons consist of one `left-handed'\nand one `right-handed' quark, that correspond to `quark' and `anti-quark' in the standard model. So I get three pions this way, consisting of two \ncharged pions, $\\pi^+=u\\bar d$ and $\\pi^-=d\\bar u$, and a neutral pion $\\pi^0=u\\bar u$. While this description of the neutral pion is not the same\nas in the standard model, it is the only possibility in a discrete model, in which quantum superposition cannot be implemented as an intrinsic property\nof an elementary particle, but only as a property of the experiment or the environment.\n\n\n\\section{Implementing the standard model}\n\\subsection{The Dirac equation}\nI have made various tentative suggestions for parts of the group algebra that could be used for various parts of the standard model.\nThe time has come to make these suggestions more definite, and to begin to implement the standard model in this new mathematical framework.\nThe first and most fundamental requirement is to implement the Dirac equation. The most suitable place to do this is surely the matrix\nalgebra $M_2(\\mathbb C)$. Thus we must use these $2\\times 2$ matrices to implement the Dirac spinor, divided into the left-hand\ncolumn and the right-hand column, that should be related in some way to the left-handed and right-handed Weyl spinors in the standard model.\n\nActions on the Dirac spinors can then be obtained from both left-multiplications and right-multiplications, which together generate a $16$-dimensional\ncomplex algebra that we must identify with the complex Clifford algebra in the standard model. For this purpose, we need a choice of\nDirac gamma matrices. There are many possible choices, but since each of the four Dirac matrices swaps the left-handed and right-handed parts of the spinor, the following looks to be a good choice. Each Dirac matrix is written as a pair\nof a left-multiplication and a right-multiplication by Pauli matrices.\n\\begin{eqnarray}\n\\gamma_0 =(1,\\sigma_1)&=& \\left( \\begin{pmatrix}1&0\\cr 0&1\\end{pmatrix}_L, \\begin{pmatrix}0&1\\cr1&0\\end{pmatrix}_R\\right),\\cr\n\\gamma_1 =(\\sigma_1,i\\sigma_2)&=& \\left( \\begin{pmatrix}0&1\\cr 1&0\\end{pmatrix}_L, \\begin{pmatrix}0&1\\cr-1&0\\end{pmatrix}_R\\right),\\cr\n\\gamma_2 =(\\sigma_2,i\\sigma_2)&=& \\left( \\begin{pmatrix}0&-i\\cr i&0\\end{pmatrix}_L, \\begin{pmatrix}0&1\\cr-1&0\\end{pmatrix}_R\\right),\\cr\n\\gamma_3=(\\sigma_3,i\\sigma_2)&=& \\left( \\begin{pmatrix}1&0\\cr 0&-1\\end{pmatrix}_L, \\begin{pmatrix}0&1\\cr-1&0\\end{pmatrix}_R\\right).\n\\end{eqnarray}\n\nWith these definitions, we can write down the Dirac equation in the usual way. The group algebra model, however, contains more structure than\nthe standard model, and in particular contains an important distinction between the Lie group acting on the left, and the finite group acting on the right.\nSeparating out the left-multiplications and the right-multiplications in the above, we have the following right-multiplications:\n\\begin{eqnarray}\ni\\gamma_0&=&(1,i\\sigma_1),\\cr\ni\\gamma_1\\gamma_2\\gamma_3&=&(1,i\\sigma_2),\n\\end{eqnarray}\nwhich together generate the quaternion group $Q_8$.\nHence this Dirac equation can be used to study four particles, as in the toy model discussed earlier.\nThe same equation applies to a variety of different particles, but only to four at a time.\nIt is possible to use\nthis equation to study one generation of elementary fermions, but not to study all three simultaneously.\nIn particular, the standard model has to implement the generation structure outside the Dirac algebra.\n\nOn the left, we have \n\\begin{eqnarray}\ni&=& (i,1)\\cr\n\\gamma_1\\gamma_2&=&(i\\sigma_3,1)\\cr\n\\gamma_2\\gamma_3&=&(i\\sigma_1,1),\n\\end{eqnarray}\nwhich on exponentiation generate the Lie group $U(2)$, that is, a group isomorphic to the electro-weak gauge group in the standard model. \nAlternatively, we can obtain $SL(2,\\mathbb C)$ from $i\\gamma_1\\gamma_2$ and $i\\gamma_2\\gamma_3$.\nNote, however, that this copy of $SL(2,\\mathbb C)$ is distinct from Dirac's relativistic spin group generated by\n$\\gamma_0\\gamma_1$, $\\gamma_1\\gamma_2$ and $\\gamma_2\\gamma_3$. In particular, these two copies of $SL(2,\\mathbb C)$\nhave different physical meanings. At some point it will be worth thinking carefully about the desired interpretations of these groups.\n\n\\subsection{Electro-weak mixing}\nIn the standard model, electro-weak mixing \\cite{Weinberg} is described as a mixing of the gauge groups $U(1)$ and $SU(2)$, but it is fairly clear that\nit is fundamentally a discrete phenomenon rather than a continuous one. In practice, there appears to be a small experimental variation\nof the Weinberg angle from a theoretical maximum of $30^\\circ$.\n\n If we assume for the moment that the underlying discrete property is described\nby an angle of exactly $30^\\circ$, then this angle appears naturally as the angle between the complex numbers $i$ and $\\omega$.\nSince the group algebra model contains both $i$ and $\\omega$, where the standard model only contains $i$, there is every prospect\nthat the group algebra model can explain electro-weak mixing, at least to first order. The small deviation of the Weinberg angle\nfrom $30^\\circ$ will of course require more detailed investigation.\n\nMore specifically, the finite versions of $U(1)$ and $SU(2)$ in the standard model are the scalar group of order $4$ generated by $i$, and the\nquaternion group $Q_8$ generated by $i\\sigma_1$ and $i\\sigma_2$. But the scalars that appear in the finite model at this point are the scalars\n$\\omega$ and $\\bar\\omega$ that are used to convert from the representation $4_H$ to $4_C$, and from one complex version of $4_C$ to the other, by multiplying with the elements of order $3$ in the group $G$. Thus the pair of Weyl spinors defined by $i$ and $-i$ in the standard model becomes a\ntriplet of Weyl spinors defined by $1$, $\\omega$ and $\\bar\\omega$.\n\n\\subsection{The three generations}\nI have shown how the fermionic part of the algebra, $\\mathbb H+M_2(\\mathbb C)$, contains all the structure of the Dirac spinor, the Dirac equation,\nthe Dirac algebra and the basic principles of electro-weak mixing. These parts of the standard model use essentially all of the structure of this\npart of the group algebra. The rest of the standard model must therefore lie in the bosonic part of the algebra $\\mathbb R + \\mathbb C + M_3(\\mathbb R)$.\nI have suggested ways of constructing this part of the algebra via a finite analogue of the construction of the Dirac algebra.\n\nAs far as the Lie group $SL(2,\\mathbb C)$ is concerned, the Dirac algebra is the complex tensor product of two copies of the Dirac spinor representation.\nBut if this structure arises from the structure of the finite group, we must also be able to construct the Dirac algebra from the (complex) tensor product of\ntwo copies of the $4_C$ representation of $G$. Moreover, the complex structure adds nothing of physical significance to the algebra, so that we might\nas well work with the real tensor product, which has structure 1+1+2+3+3+3+3. This contains the whole of the bosonic part of the algebra, together\nwith a spare copy of $1+3$.\n\nThere are two ways of looking at this tensor product. One way is to look at the finite group acting on the spinors on both sides. This gives a\ndescription of the internal symmetries of the elementary particles, without any Lie groups and therefore without any measurements or observations.\nThe other way is to look at the finite group acting on the vectors on the right. This gives a description of things that can be measured\nby experiments that act on the vectors on the left. The representation theory of the finite group links the two viewpoints.\n\nIn particular, the spare copy of $1+3$ must represent things that cannot be measured, for example colours,\nor direction of spin. The corresponding part of the Dirac algebra\nconsists of two $4$-vectors, one of which is usually interpreted as $4$-momentum, so that we might interpret the other as spacetime position,\nand allocate the non-measurability to the Heisenberg uncertainty principle. That leaves us with $1+2+3+3+3$ things that can be\nmeasured, with a macroscopic group $U(1)$ acting on the $2$, and $SL_3(\\mathbb R)$ acting on $3+3+3$.\n\nNow in removing the fourth copy of $3$ we may have removed either momentum or position, but not both. Since position is irrelevant in the\nstandard model, we have presumably kept the momentum. Moreover, we have a real scalar in the remaining copy of $1$, which must similarly\nbe the energy. We then have a macroscopic group $GL(3,\\mathbb R)$ acting on the momentum. This is a little disconcerting, since our everyday\nexperience is that momentum has only an $SO(3)$ symmetry. But we know from the theory of special relativity \\cite{SR} that momentum mixes\nwith mass, and we know from the weak interaction that mass can be converted into momentum, so we should not be too surprised. \n\nIn any case, $3+3+3$ contains a discrete set of $9$ `things' to measure, and a continuous group of dimension $9$ to do the measuring with.\nHence we can measure $9$ masses, for example of $3$ generations of electrons and up and down quarks. What we observe is that\nthree of these masses, the electron masses, are masses of particles whose momentum we can also measure, and they are therefore well-defined\nand precisely measured masses. The other six, the quark masses, are masses of particles whose momentum we cannot measure, so that the\nmasses are quite ill-defined, and vary from one experiment to another. Most of all, they vary from one type of experiment to another, for\nexample between baryon experiments and meson experiments.\n\nThese $8$ mass ratios are closely related in the group algebra to $8$ parameters in the subalgebra $M_2(\\mathbb C)$, but the standard model\ndoes not include a finite group relationship between these two parts of the algebra, so has an entirely different set of $8$ parameters here.\nThese are conventionally written in the symmetric square representation as $3\\times 3$ complex matrices, and are the $8$ mixing angles in\nthe Cabibbo--Kobayashi--Maskawa (CKM) matrix \\cite{Cabibbo,KM} and the Pontercorvo--Maki--Nakagawa--Sakata (PMNS) matrix\n\\cite{Pontecorvo,MNS}.\n\nIn this section I have given only a very brief sketch of how the differing masses of the three generations of fermions can be distinguished,\nbut without offering any physical principles by which the mass is generated. For this, I rely for now on the standard model, and defer the\nquestion of how the experiments give the particular mass values that they do. Answering this question\nrequires a much deeper analysis of\nthe quantum structure of the experimental apparatus itself, and \nis beyond the scope of this paper.\n\n\\section{Field theories}\n\\label{physics}\n\\subsection{Principles}\n\\label{principles}\nThere are three (or four) ways of looking at the groups acting on the group algebra. One can look at the finite groups acting on both left and right, or the Lie groups acting on both left and right, or one of each. The general principle must surely \nbe that if the Lie groups act on both sides, then we see no quantisation,\nand we must recover a reasonable approximation to a description of classical physics. \n\nSimilarly, if the finite group acts on both sides, then all we see are\nquantum numbers, and no macroscopic variables like position, momentum, energy, mass and so on. In between, we see a right-multiplication by a finite group, which flips the quantum numbers, and a left-multiplication by some Lie groups, that permits us to measure the properties of mass, momentum, energy, and so on, that are associated to particular sets of quantum numbers in particular types of experiment \nthat we might contemplate performing.\nThis, at least, must be the general set-up in any hypothetical discrete theory of the type envisaged by Einstein. \n\nThe standard model\ninterprets things differently, and regards the Lie groups as being intrinsic to the elementary particles, despite being unable to provide any\nplausible physical process by which this could happen. (Although in practice, \nthe de Broglie--Bohm  pilot wave scenario is rather close to the approach I am taking.) \n\nWhat is clear, of course, is that the calculations that are done in the standard model are essentially correct. It is only the interpretation\nthat must be somewhat different in a discrete model.\nThe discrete model must \nin fact be capable of explaining not only the standard model, but also classical physics. This is a major challenge for\nany model, and success is hardly to be expected. Nevertheless, let us see how the proposed finite model deals with this challenge.\nThe first and most basic issue is to determine how macroscopic spacetime emerges from the discrete properties of elementary particles\nand their interactions. This must be some kind of generalisation of the toy example already given in Section~\\ref{analysis}.\n\n\\subsection{Classical forces and relativity}\n\\label{classical}\nLooking at the real group algebra first from a macroscopic perspective, with the \nmatrix groups acting on themselves by conjugation,\nwe see three real and two complex scalars, that act trivially, plus three non-trivial symmetry groups: \n\\begin{eqnarray} \nSO(3) \\times SO(3,1) \\times SL(3,\\mathbb R).\n\\end{eqnarray}\nTherefore, in addition to the magneto-weak `spacetime' $\\mathbb H$ with symmetry group $SO(3)$, discussed in Section~\\ref{analysis},\nwe see an electromagnetic (special\nrelativistic) `spacetime' with symmetry group $SO(3,1)$, and a further space (with or without time) with symmetry group\n$SL(3,\\mathbb R)$. \nThe issue then is to decide how the macroscopic spacetime that underlies classical physics relates to these three different\nversions of \nspacetime \nthat seem to emerge in some way from different parts of particle physics. \n\nThere is nothing in classical physics directly corresponding to the group $SL(3,\\mathbb R)$, which allows\nstretching and shearing of space. But \ncombining it with the Lorentz group $SO(3,1)$ acting on the same spacetime gives the group $SL(4,\\mathbb R)$\nof all unimodular coordinate changes, that describes the general covariance of general relativity \\cite{GR1,GR2,GR3}\nas a\ntheory of gravity. So $SL(3,\\mathbb R)$ must in some sense describe a `fluid' gravitational `space'. \n\nClassically, of course, there is only one spacetime, with symmetry group $SO(3)$ \ndefined by the observer. The groups $SO(3,1)$ and $SL(3,\\mathbb R)$ are then interpreted, not as different types of spacetime, but as forces. \nClearly $SO(3,1)$ is the symmetry group of electromagnetism, as most elegantly expressed by Einstein's interpretation of Maxwell's equations\nin the theory of Special Relativity.\nTo be more precise, the electromagnetic field is an association of a (trace $0$) element of $M_2(\\mathbb C)$ with each point in spacetime $\\mathbb H$.\nIt is therefore expressed mathematically as a function \n\\begin{eqnarray}\n\\label{field}\nf&:& \\mathbb H \\rightarrow M_2(\\mathbb C).\n\\end{eqnarray}\n\nMaxwell's equations \nthen describe how the values of the electromagnetic field, in the adjoint representation of $SO(3,1)$, \ntogether with the inertial mass and charge, \nin the scalar part of $M_2(\\mathbb C)$, relate to the ambient physical space, in the adjoint representation of $SO(3)$,\nand to time, in the scalar part of $\\mathbb H$. \nThe theory of special relativity explains how the equations are invariant under coordinate changes on spacetime described by the Lorentz group $SO(3,1)$. That is, any change in spacetime coordinates defined by an element of $SO(3,1)$ can be compensated for by the\ncorresponding change in coordinates for the values of the electromagnetic field, by the corresponding element of $SL(2,\\mathbb C)$ acting\nby conjugation on the trace $0$ matrices.\n\nThe Lorentz group can in turn be interpreted as a change of spacetime coordinates between different observers. But the model under discussion is a model appropriate to a single observer, with a fixed notion of space and time. Therefore the appropriate interpretation of the Lorentz group in this context is as a gauge group, that can be used to choose coordinates for the (electromagnetic) gauge field. The content of the theory of special relativity is then that the theory of electromagnetism does not depend on the choice of coordinates. \n\nAnalogously, we must seek a gravitational field in the adjoint representation of $SL(3,\\mathbb R)$. From the point of view of an individual observer,\nwith $SO(3)$ symmetry, this representation splits up as the sum of a spin $1$ field and a spin $2$ field. Newton's universal theory of gravitation \nalready includes both: the spin $1$ field is the gravitational field, and the spin $2$ field describes the tidal forces that arise from rotations of matter within a gravitational field (or, equivalently, rotations of the gravitational field around the matter). Newton's theory is very good indeed in most circumstances, but has two main drawbacks that have become apparent in the ensuing centuries. One is that it does not distinguish between the gravitational force due to a rotating body, and that due to a stationary body. The other is that it has a static gravitational field, rather than a dynamic field that propagates at a finite speed (presumably the speed of light).\n\nThe former drawback is (at least partly) addressed by Einstein's theory of general relativity, in the sense that\nrotations of the observer (or test particle) are taken into account. But it is not clear that rotations of the gravitating body are fully\naccounted for, \nnor \nthe finite speed of propagation.\nIt is therefore not required for a new model to reproduce general relativity exactly, but merely to reproduce general relativity in the limit where the effects of the finite speed of light can be ignored. There are indeed \ntwo circumstances in which such effects might already have been observed in practice. One is in the effects on fast-moving satellites as the rotate very fast around the Earth. The other is at the edges of galaxies, in which the gravitational attraction of the galactic centre can take hundreds of thousands of years to reach the outer edges.\n \n The former is known as the flyby anomaly \\cite{flyby}, and it is claimed by Hafele \\cite{Hafele} that this anomaly can be entirely explained\n by the finite speed of propagation of the gravitational field of a massive rotating object. The latter goes by many names, and many proposals\n have been made for theoretical explanations for the observed effect, most notably the hypotheses of (a) dark matter, or (b) modified Newtonian \n dynamics, or MOND \\cite{Milgrom1,Milgrom2,Milgrom3,MOND,MOND2}. As far as I am aware, however, there is no proposed explanation that takes into account either the finite speed of propagation\n of the gravitational field, or the very fast rotation of the super-massive black hole (or other objects) at the centre of the galaxy. \n\nThe proposed finite model permits a separation of the $SL(3,\\mathbb R)$ symmetries of gravitational space from the $SO(3,1)$ symmetries of\nelectromagnetic spacetime, and therefore permits a macroscopic theory of gravity that is independent of time, and therefore independent\nof the finite speed of light. But to do that it requires the symmetry group of the gravitating body to be extended from $SO(3)$ to $SL(3,\\mathbb R)$.\nIn other words, the centre of the galaxy must be regarded as a rapidly rotating fluid rather than  a solid object. Given that a typical galaxy has many billions of stars,\nthis is surely a reasonable hypothesis to make. \n\nThe general principle of relativity says that it makes no difference\nto the theory whether the group $SL(3,\\mathbb R)$ is attached to the galactic centre or to the periphery, so that we can interpret the theory\neither way. \nThe latter interpretation seems to be closely related to the dark matter hypothesis: that is the galactic centre is regarded as a Newtonian\npoint mass, and all the\notherwise unexplained behaviour of the outer stars must then be due to some `dark matter halo' surrounding the galaxy. \nBut perhaps the assumption of a Newtonian point mass at the centre of the galaxy is unrealistic?\n\nThe former interpretation is more closely related to the MOND hypothesis: that there is some new unexplained gravitational force,\nthat \ncan only be observed when the Newtonian gravitational field is extremely weak. \nThe proposed finite model\nfavours the MOND interpretation, and suggests (very speculatively) that the new force might be explained in terms of the very fast rotation of the extremely massive\ngalactic centre.\nMost importantly, the group $SL(3,\\mathbb R)$ is now interpreted as the gauge group of the theory of gravity, and is completely separate\nfrom the gauge group $SO(3,1)$ of classical electromagnetism. In the theory of general relativity, these two gauge groups are combined into\na group $SL(4,\\mathbb R)$, which has no meaning in the proposed model. This might explain why attempts to quantise general relativity with\na gauge group $SL(4,\\mathbb R)$ or $GL(4,\\mathbb R)$ have failed \\cite{GL4R1,GL4R2}.\n\n\\subsection{Quantum field theory}\nThe above discussion suggests interpreting the $24$ dimensions of the group algebra as classical fields, divided into $7$ dimensions of scalar fields,\n$3$ dimensions of space, $6$ dimensions of electromagnetic field, and $8$ dimensions of gravitational field. These fields are mathematical\nconstructs, rather than physically real objects. For example, we do not want to interpret the $3$ dimensions of the `space' field as an `aether'\nin the 19th century sense. What is physically real, however, is the propagation of the fields through space, or to be more precise, the propagation\nin spacetime.\n\nI have not precisely identified all of the scalar fields, though four of them appear to be time, inertial mass, charge and gravitational mass.\nEnergy would seem to be separate from both types of mass, and there appears to be a second type of (neutral) charge. This leaves one more\nscalar, which might be identified with the Higgs field in the standard model. Whatever identification is eventually decided on, these scalar\nfields propagate by being attached to matter particles. In other words, it is the movement of matter that defines the propagation of the\nscalar fields. \n\nJust as in the classical case, the fields must be modelled as functions from spacetime to the appropriate subspace of the group algebra.\nIn the quantum case, this function decomposes as a sum of `infinitesimal' quanta, that individually are linear functions. These linear functions\nthemselves lie in representations of the finite group $G$, so that we can use the representation theory to analyse the structure of the\nquantum fields.\nSince the spacetime representation $\\mathbb H$ is self-dual, the quantum fields can be regarded as lying in the tensor product of\n$\\mathbb H$ with the appropriate representation. \n\nThese tensor products then describe the particles that mediate the corresponding forces.   Since the spacetime \n representation is fermionic in this model, the bosonic fields are carried by fermionic\nparticles, and the fermionic fields are carried by bosonic particles. In particular, four of the scalars are bosonic, so carried by fermions, and\nthree of the scalars are fermionic, so carried by bosons.\n\nThe electromagnetic field values are fermionic in this model, and the field \nis therefore propagated by bosons. This agrees with the standard model, in which the propagation is effected by photons, which are parametrised by $3$ dimensions of momentum in $2$ distinct polarisations.\nThe three fermionic scalars, which might be interpreted as inertial mass (dual to Euclidean time, as opposed to energy,\nwhich is dual to Lorentzian time), electric and neutral charge, are \ncarried by the weak bosons, that is the $Z$ and $W$ bosons.\nConversely, in this model the gravitational field values are bosonic, and the field is therefore propagated by fermions. \n\nNo such process exists in the standard model,\nin which all fields are assumed to be carried by bosons.\nBut it would appear to be consistent with experiment \nto suppose that these propagators are neutrinos,\nparametrised by $3$ dimensions of momentum in $3$ distinct generations. This parametrisation would result in a $9$-dimensional\ngravitational field, however, and the model (as well as observation) supports only an $8$-dimensional field. \nIn other words, the $3$ generations of neutrinos cannot be linearly independent. This means that macroscopic rotations act not only on\nthe momentum coordinates, but also on the generation coordinates. The model therefore predicts that the generation of a neutrino\nis not an invariant. Indeed,\nthe non-invariance of neutrino generation is well-attested experimentally, and goes by the name of neutrino oscillation\n\\cite{oscillation,neutrinos,SNO}. \n\n\nAt this point, the neutrinos appear to have taken over the group $SL(3,\\mathbb R)$, that was originally supposed to be allocated to the strong force,\nand in particular to the $8$ gluons. Indeed, since the group $SL(3,\\mathbb R)$ in the proposed model acts on a bosonic field, the corresponding\nmediators must be fermions. We can reconcile the two viewpoints by interpreting the gluons as representing the \\emph{values} of the quantum field,\nrather than the field itself, which is a \\emph{function}. Then the mediators can be interpreted as virtual neutrinos, and the gluons as pairs of virtual neutrinos. However, the model suggests that it may be better not to interpret the gluons as particles at all, but only as symmetries.\n\nAt the same time, we need to address the distinction between the group $SL(3,\\mathbb R)$ used here and the group $SU(3)$ used in the\nstandard model. The latter is a compact group, and fixes a complex inner product, so describes rigid symmetries of a complex $3$-space.\nThe phenomenon of asymptotic freedom \\cite{freedom1,freedom2} suggests that such rigidity does not in fact characterise the strong force. The use of\n$SL(3,\\mathbb R)$, on the other hand, suggests complete freedom to change scale in one direction, providing this is compensated for in\nanother direction. In other words, replacing the (confined) gluons by the (free) neutrinos seems to require a split group, just as the (free)\nphotons are described by the split group\n$SL(2,\\mathbb C)$.\n\n\\subsection\n{Elementary particles and the standard model}\nLet us now turn our attention to the mixed case, with the finite group acting on the right and the Lie groups acting on the left. This is the domain\nof the standard model, where measurements of macroscopic variables such as mass, momentum, energy, angular momentum,\nmagnetic moments and so on are made on individual quanta or `elementary particles'. We then have 24 discrete objects on the right\nthat we can measure, and 24 degrees of freedom for the operators on the left that define what we can measure. The standard model has a kind of duality\nbetween things that can be measured and things that can't, which is sometimes an absolute distinction, and sometimes an uncertainty principle.\n\n\nFor example, the Heisenberg uncertainty principle says you can measure position or momentum, but not both at the same time. Quantum chromodynamics, on the other hand, assigns (unobservable) colours to quarks, and puts the corresponding observable property into the notion of particle `flavour' or generation. In some cases, if the associated representation is a real or complex scalar, the appropriate concept is its own dual, and\nthere is no duality or uncertainty involved. This appears to be the case, for example, with the electric charge.\nThe details are not so important. What is important is, how many distinct measurements can we make?\n\nThe finite model says that we can measure exactly $24$ independent things. \nIf we make a specific choice of the $24$ independent things we want to measure, we obtain $24$ (dimensionless) real numbers that describe everything\nthere is to know about how the elementary particles behave. \nThe standard model has made a particular choice, and has measured precisely $24$ independent things. These $24$ things are usually described as\n$12$ fermion masses ($3$ generations each of neutrino, electron, up and down quark), $3$ boson masses (the $Z$, $W$ and Higgs bosons), (hence $14$ mass ratios), $2$ coupling constants (the finite-structure constant and the strong coupling constant) and $8$ mixing angles ($4$ each in the\nCKM and PMNS matrices)\n\\cite{Cabibbo,KM,Pontecorvo,MNS}, and are regarded as the fundamental parameters.\n\nThe standard model therefore has exactly the right number of parameters to contain a complete description of quantum reality. It produces the right answers, because it contains enough variables and enough equations to calculate everything that can be calculated. There is nothing wrong\nwith the standard model. It is a mathematically correct, and complete, model of everything that can possibly happen. So \nwhy are people\nstill looking for a `theory of everything', if the standard model already \\emph{is} a theory of everything?\n\nThe main reason is that we don't understand where the $24$ parameters come from. We understand perfectly well that the Lorentz group\ncan be interpreted either as a change of coordinates on spacetime between two observers, or as a gauge group for electromagnetism, and that\nthese two interpretations are equivalent, \nso that the theory of electromagnetism is the same for all observers. We understand perhaps a little less well that the same applies to $SL(3,\\mathbb R)$ and the theory of gravity. \n\nSo why don't we understand that the same applies to the\ngauge groups of the standard model? The groups describe the relationship between the observer and the observed, and we have a choice\nbetween regarding the groups as acting on the observed (as in the standard model) or the observer (as in relativity). But we have to make a choice:\nwe cannot both have our cake and eat it. The incompatibility of the standard model with general relativity may be nothing more than the fact that the two\ndisciplines have made incompatible choices of interpretation.\n\nSo to resolve the issue we \nneed to make a consistent choice of interpretation. For practical purposes, it makes no difference which choice we make.\nEither way we do the same calculations, get the same answers, and reconcile them with experiment. But philosophically, there is no contest. The general principle of relativity is such a powerful and obvious philosophical principle that we should not under any circumstances contemplate abandoning it. \nThe consequence of this philosophical viewpoint is that the gauge groups of the standard model have to be transferred to act on the observer, so that\n(most of) the $24$ parameters become parameters associated with the experiment, the environment and the observer, rather than with the\nelementary particles themselves. \n\nI re-iterate that both points of view are mathematically valid, but only one of them is philosophically valid.\nMoreover, since many of the $24$ parameters are known to vary with the experiment, and in particular to `run' with the energy scale, \nit is also the case that only one of the two viewpoints is\nphysically valid. It is not physically reasonable to treat a parameter as a universal constant,\nif experiment shows that it is not.\nIn the model I am proposing, $7$ of the $24$ parameters are scalars, and can therefore be taken as universal constants.\nThe other $17$ must be regarded as properties of the experiment. \n\nIf we look carefully at the experimental evidence, $6$ or $7$ of the $14$ mass\nratios  \nshow evidence of not being constant, namely the $6$ quark masses and the $Z\/W$ mass ratio. All $8$ of the mixing angles are similarly suspect.\nFinally, there is no positive, model-independent, evidence of different masses for the three generations of neutrinos, or indeed any mass different from zero. Moreover, the phenomenon of\nneutrino oscillation suggests that there is no intrinsic difference between the three generations of neutrino anyway, which add another $2$ or $3$\nparameters to make up a total of between $16$ and $18$.  \nIn other words, the proposed finite model is consistent with experiment on this point.\n\n\\subsection{The wave-function}\nThe one thing that one might really hope for from a finite model of elementary particles is an insight into the measurement problem.\nThis was really the focus of Einstein's objections to quantum mechanics throughout his life, and although his attacks, most notably\nthe EPR paradox \\cite{EPR}, were never enough to sink the ship, the fundamental problem has not gone away.\nTo simplify the problem almost to the point of caricature, we may ask, what \\emph{is} the wave-function, and how does it `collapse'?\n\nIn the group algebra model, the wave-function is implemented at the middle level, with the finite group acting on the right\nand the Lie groups on the left. \nThe typical example is equation (\\ref{field}), which describes a\nfunction from spacetime to the Dirac spinors. The `collapse' is some kind of operation that moves down to the discrete level,\nwith the finite group on both sides. In practical terms, the wave-function describes the quantum field in the experiment, and the collapse describes\nthe result of the experiment.\n\nFrom this point of view, the measurement problem arises from the assumption that the wave-function is intrinsic to the elementary\nparticle under investigation. The finite group model, however, does not allow any continuous variables to be associated with an elementary\nparticle. The continuous variables are always associated with the macroscopic measuring apparatus. The model, in other words, does not\n\\emph{solve} the measurement problem. It \\emph{does not find} the measurement problem.\nPhilosophically, this is always the best way to solve problems, namely, to realise that, if looked at in the right way, the problem does not exist.\n\n\\subsection{The real universe}\nThe structure of the model indicates that of the $24$ unexplained dimensionless parameters of the standard model, exactly $7$\nare universal constants, and the rest are dependent in some way on the experiment, the environment and\/or the observer. There is nothing\nlike enough detail in the model as so far developed to indicate which parameters should be regarded as universal constants, nor how the\nother parameters vary. Indeed, there may be \na certain amount of choice as to which parameters can be defined as constant, so that the other\nparameters can be calibrated against them. \n\nWhat is clear, however, is that we must be prepared for the possibility that\ncertain parameters that we are certain\nmust be universal constants, may not be so.\nAt this stage, we can do little more than speculate on these matters, and apply some educated guesswork. There is no real need to use\nthe same fundamental parameters as the standard model, but other mass ratios such as those between electron, proton and neutron might\nalso be worth looking at. The model suggests that approximately $8$ of the fundamental parameters are dependent in some way on the\ngravitational field, including tidal forces. \n\nAt first glance, the very idea is preposterous. But on closer inspection, one realises that while the\nstandard model is in theory based in an inertial frame, the experiments that measure the fundamental parameters are not done in an\ninertial frame, but in a frame which moves with the Earth. It is therefore very hard to rule out, on experimental grounds, the possibility\nthat some of the measurements depend crucially on some dimensionless property of the tides that is constant over all experiments done\non the Earth.\n\nThere are four basic dimensionless parameters of the tides on the Earth, of which two are obvious spatial angles: the angle of tilt of the Earth's axis,\nand the inclination of the Moon's orbit to the ecliptic. The other two are the ratios of the periods of rotation\/revolution, so are angles in Euclidean spacetime. Of course, these\nparameters are not precisely constant, and there are other effects that might be expected to contaminate the results, such as the\ngravitational pull of Jupiter. It will therefore be somewhat tricky to distinguish a real correlation from an unwanted coincidence.\n\nSome $8$ such correlations\/coincidences were presented in \\cite{INI13}, without any very solid justification, but with the suggestion\nthat, while some of them may indeed be pure coincidences, it is very unlikely that they are all coincidences. The parameters discussed\nin that paper include the electron\/proton\/neutron mass ratios, the pion mass ratio, the kaon mass ratio, the kaon\/eta mass ratio,\nthe Cabibbo angle, the Weinberg angle and the CP-violating phase in the CKM matrix. In addition, the paper discusses a number of other\nequations that seem to suggest that some of the fundamental parameters may not be independent of each other. This may be a\nsomewhat less unpalatable suggestion than the suggestion, made originally by Einstein \\cite{gravimass} more than a century ago, \nthat they might depend on the gravitational field! On the other hand, the paper \\cite{INI13} also shows that the observed CP-violating\nbehaviour of neutral kaons \\cite{CP} is quantitatively consistent with the hypothesis that the effect is caused by the small difference in the\ndirection of the gravitational field between the two ends of the experiment.\n\n\\section{Conclusion}\nIn this paper, I have examined what I believe to be the unique possible mathematical model of a discrete algebraic universe,\nin the hope that it will have something useful to say about the seemingly intractable problems in the foundations of physics.\nI have shown how all the essential ingredients of the standard theories of both classical and quantum physics arise from this\nfinite model, and discussed at a general level the relationships between them. \nI have traced the conflict between quantum mechanics and\nrelativity to a conflict in interpretations, that does not affect the mathematics of either theory, and sketched a possible way to\nresolve this conflict.\n\nI have not done all the necessary detailed calculations to show that the proposed model reproduces the standard models exactly,\nso there is still room for doubt as to whether the model I propose is viable. Nevertheless,\n I have shown that the proposed model has enough complexity to incorporate all of the\n subtleties of the standard model of particle physics, including the $24$ dimensionless parameters. \nThe proposed algebraic model seems to be consistent with the experimental fact that the standard model is essentially a complete and correct\ntheory of everything. As for physics `beyond the standard model', the model \nexplains what experiment has \ndemonstrated, namely that, essentially, there is none.\n\n\nSimilarly, it appears to be consistent with general relativity as a theory of gravity, but suggests that the simplifying assumptions\nmade in practical calculations break down in extreme circumstances.\nThe model suggests that incorporating a contribution to gravity from the rotation of\na (not spherically symmetric) gravitating body, \nand taking into account the finite speed of propagation of gravitational waves, may be sufficient to account for the\nanomalous rotation of stars in the outer regions of galaxies, that was the original reason for the hypothesis of\ndark matter. If this is not enough, then the model has room to distinguish gravitational mass from inertial mass,\nand to incorporate a scale factor between the two that is dependent on \nproperties of the motion of the observer.\n\nSo what remains of Einstein's `castle in the air'? The finite model has, if anything, given it slightly firmer foundations. \nSpecial relativity has always been built on solid rock, but the foundations of general relativity are more fluid. \nI see general relativity, therefore, not as a castle in the air, but as an aeroplane,\nthat stays up despite having no visible means of support. My proposed model provides some support,\nin the form of a neutrino wind, to keep it flying. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nOne of interesting effects produced by flare energy release in the solar atmosphere is excitation of helioseismic waves, so-called sunquakes \\citep{Kosovichev1998}. Such waves usually propagate as expanding ripples from local impact sources occupying several pixels in photospheric Dopplergrams. The cause of these events is a subject of intensive debates \\citep[e.g.][]{Donea2011,Kosovichev2014}. Generally, the necessary condition for producing sunquakes is a sudden momentum enhancement in the lower solar atmosphere. One of the possible agents of such disturbance can be chromospheric heating due to injection of accelerated charged particles postulated by the standard model of solar flares \\citep[for a recent review see][]{Fletcher2011}. Models of the gas dynamics processes induced by nonthermal electron beams \\citep[e.g.][]{Kostiuk1975, Fisher1985, Kosovichev1986} predict formation of a shock wave or a chromospheric condensation moving towards the solar photosphere and, thus, transferring momentum to the dense plasma. \\cite{Kosovichev1995} discussed such beam-driven mechanism of sunquakes. However, the plasma momentum transfer is also possible by other mechanisms, such as sharp enhancement of pressure gradient due to flux-rope eruption \\citep[e.g.][]{Zharkov2013} or by an impulse of the Lorenz force which can be stimulated by electric currents in the lower solar atmosphere \\citep{Fisher2012}. Also, it is possible that different sunquake events are caused by different mechanisms. Usually, sunquakes are associated with M and X class flares. However, many X-class flares did not produce sunquakes \\cite[e.g.][]{Donea2011}, whereas these events had been noticed during relatively weak M-class flares \\citep{Martinez-Oliveros2008, Kosovichev2014}.\n\nIn this Letter, we discuss observations of the C7.0 flare of February 17, 2013, which produced a rather strong sunquake initiated during the HXR burst. We use data from four space instruments: EUV observations from SDO\/AIA \\citep{Lemen2012}, vector magnetic field measurements from SDO\/HMI \\citep{Scherrer2012}, integrated soft X-ray emission from GOES, and X-ray spectroscopic imaging data from RHESSI \\citep{Lin2002}. We investigate potential mechanisms of the sunquake initiation, and find that despite the precise temporal coincidence between the HXR impulse and the photospheric impact this event is not consistent with the standard flare model, because the HXR source and the sunquake impact were at different spatial locations, at two different footpoints of the flare loop. Our analysis leads to a suggestion that a significant role in the sunquake initiation may be played by electric currents in the low atmosphere.\n\n\\section{General description of the event and sunquake}\n\nThe flare event of Febrary 17, 2013, was observed in active region NOAA 11675. It consists of two  subflares clearly separated in time and space: the first subflare has the C7.0 GOES X-ray class, and the second subflare reached M1.9 peak intensity. The duration of the whole double flare is about 8 min, starting at 15:46:00 UT and ending at 15:54:00 UT (Fig. 1). The highest energy of the hard X-ray (HXR) emission (maximum at 15:47:20 UT), detected by RHESSI during the first subflare is $\\sim 1$ MeV. The second subflare is characterized by weaker intensity and energy ($<$300 keV) of HXR emission which reached maximum at 15:50:30 UT.\n\nTop panels of Fig. 2 present the AIA images in the 94 $\\rm\\AA$ channel. The temporal and spatial resolutions are  12 seconds and $1.2^{\\prime\\prime}$ (with the angular pixel size of $0.6^{\\prime\\prime}$). The preflare state reveals a compact loop-like structure where the flare process occurs.\n\nThe sunquake is observed as an expanding circular wave in the HMI Dopplergrams filtered in the frequency high range, 5-6 mHz, to isolate the sunquake signal from the convective noise. The propagation of the sunquake wave is shown on the time-distance diagram \\citep{Kosovichev1998,Zharkova2007} presented in Fig.2 (middle right panel) comparing the observed signal with the theoretical ray-path theory prediction (dotted yellow line). The time-distance analysis shows that the sunquake is initiated by the first subflare with the initiation point at the flare impulse signal. The inclined wave pattern above the theoretical curve is associated with the frequency dispersion of the helioseismic wave packet.\n\nThe initiation of the sunquake is observed as a strong localized impulse in the HMI Dopplergrams and line-of-sight magnetogram at $\\approx$15:47:54 UT. The location of the impulse on the HMI magnetogram is illustrated in Fig. 2 (middle left panel). Because of the rapid variations during the flare impulse the Doppler velocity and magnetic field measurements in the impact pixels can be inaccurate. Therefore, we use the original level-1 HMI filtergram data from the two HMI cameras to locate the exact time and place of the flare impact. The bottom panels of Fig. 2 shows the time difference between the HMI filtergram images (from HMI Camera-1). We see that compared to the preflare time during the flare there is enhancement of emission in the pixels associated with the AIA brightnings and the place of the sunquake initiation. The timing of the photospheric impact is illustrated in Fig. 1 (bottom) which shows the signals from both HMI cameras as a function of time. The periodic variations of these curves are due to the line scanning. The plot shows that the photospheric impact coincides with the HXR impulse within 3 sec (the HMI camera resolution).\n\n\\section{Spatial structure of the flare region}\n\nHere we present a description of the spatial structure of the flare region according to the RHESSI and AIA\/SDO observations. RHESSI uses a Fourier technique to reconstruct X-ray emission sources \\citep{Hurford2002}. We apply the CLEAN algorithm to synthesize the X-ray images using detectors 1,3-6,8 (integration times are shown in Fig.3). In Fig. 3, the RHESSI HXR and SXR contour images are compared with the corresponding AIA 94 \\AA ~images for the time interval covering the HXR peaks of both flares. To compare positions of the EUV and X-ray sources with the structure of magnetic field we plot the polarity inversion line from the HMI magnetogram. The structure of the EUV emission sources is rather complicated. There are ribbon-like structures\nlocated on both sides of the magnetic field inversion line. During the HXR burst we observe a loop-like structure with one footpoint associated with very strong HXR emission (25-200 keV), and the other footpoint located in the place of the photospheric impact (sunquake initiation), which also coincides with a weak HXR emission source. The total emission intensity of the weaker X-ray source is approximately five times less than the emission intensity of the stronger HXR source. If the sunquake were initiated by an impact of high-energy electrons, then their impact would be in the place of the intensive energy loss of the accelerated particles, and coincide with the strongest HXR emission source. However, we observe the opposite situation when the sunquake impact correlates with a weaker HXR emission source. This indicates that the sunquake is unlikely be generated by the impact of high-energy electrons.\n\nThe second subflare has SXR source (6-12 keV) coinciding with the HXR source (25-50 keV) and saturated UV emission above the magnetic field inversion line. However, this subflare is located $\\sim$3 Mm away from the place of energy release in the first subflare and according to our analysis is not associated with the sunquake.\n\n\\section{Analysis of RHESSI spectra: accelerated particles and heating}\n\nTo determine properties of the accelerated particles, the plasma and their energetics we use the RHESSI data in the range 5-250 keV. We investigate two spectra taken during the HXR peaks of two subflares. The power-law approximation $f(E)=AE^{-\\gamma}$ ($A$ is normalization coefficient) is considered for the hard X-ray (HXR) nonthermal emission $\\gtrsim 20$ keV. To simulate the presence of the low-energy cutoff we use the broken power law \\citep{Holman2003} with fixed photon spectral index $\\gamma_0=1.5$  below the break energy ($E_{low}$). For the first subflare, an additional break energy ($E_{br}$) is considered, and, thus, for this case we have two spectral indices $\\gamma_1$($E_{low}<E<E_{br}$) and $\\gamma_2$($E>E_{br}$). For the second subflare we consider only one spectral index $\\gamma_1$($E>E_{low}$) and also make a pileup correction as the count rate is sufficient to observe such effect.\n\nThe thermal soft X-ray (SXR) spectrum $\\lesssim 20$ keV is approximated by one-temperature thermal bremsstrahlung emission with two parameters: temperature ($T$) and emission measure ($EM$). The RHESSI spectra are fitted by means of the least squares technique implemented in the OSPEX package with 7 free parameters ($EM$, $T$, $A$, $E_{low}$, $\\gamma_1$, $E_{br}$ and $\\gamma_2$) for the first subflare and 5 parameters ($EM$, $T$, $A$, $E_{low}$ and $\\gamma_1$) for the second subflare. Fig. 4 displays results of the fitting.\n\nFrom the thermodynamics point of view the second subflare is hotter than the first one, but the emission measure is smaller. Volume $V$ of the UV loop estimated in the previous section is $10^{26}$ cm$^3$, so that plasma density $n_1 = \\sqrt{(EM_1\/V)}\\approx 6\\times 10^{10}$ cm$^{-3}$ for the first subflare. The plasma density for the second subflare, assuming the same flare region volume, is $n_2\\approx 2\\times 10^{11}$ cm$^{-3}$. Due to compactness of the flare region the plasma within the magnetic loops is rather dense.\n\nThe HXR photon spectrum is harder for the first HXR burst than for the second one. Normalization coefficient $A$ of the HXR spectrum is also one order of magnitude larger in the case of the first subflare. This means that the acceleration process is more efficient during the first subflare. The total flux $Fl$ [electrons s$^{-1}$] of accelerated electrons can be estimated following the work of \\cite{Syrovatskii1972}:\n\n$$\nFl(E_{low}<E<E_{high}) = 1.02\\times 10^{34}\\frac{\\delta_1^2}{E_{low}\\beta(\\delta_1,1\/2)}\\frac{I_{ph}(E_{low}<E<E_{high})}{[1-(E_{low}\/E_{high})^{\\delta_1}]}\n$$\n\nwhere $E_{high}$ is the upper energy cutoff, $\\delta_1 =\\gamma_1 - 1 $ is the spectral index of accelerated electrons in the HXR emission region, $\\beta(x,y)$ is beta function, and $I_{ph}(E_{low}<E<E_{high})$ photons s$^{-1}$ cm$^{-2}$ is the energy integrated photon spectrum in the range shown in the brackets. From the fitting results using this formula, we obtain electron fluxes $Fl_1\\approx (2.0\\pm 1.2)\\times 10^{35}$ and $Fl_2\\approx (0.10\\pm 0.06)\\times 10^{35} $ electrons s$^{-1}$ for the first and second subflares. Despite the lower GOES class of the first subflare we observe more energetic electrons involved in the acceleration process of this subflare than in the second one. Theoretically, these electrons could contribute to the sunquake initiation. However, the discrepancy between the locations of the sunquake impact and the strongest HXR emission source indicates that the beam-driven origin of the sunquake is unlikely.\n\n\\section{Electric currents in the flare region}\nLocal heating by electric currents or impulsive Lorentz force can also be a source of the sunquake. In this section we consider the evolution of electric currents at the photosphere level. To estimate the horizontal electric currents we use the Faraday law applied to the 45-second line-of-sight HMI magnetograms with the spatial resolution $1^{\\prime\\prime}$ and pixel size $0.5^{\\prime\\prime}$:\n\n$$\n\\oint_C\\vec{E}\\cdot\\vec{dl} = -\\frac{1}{c}\\frac{d}{dt}\\left(\\int_{S_C}\\vec{B}\\cdot \\vec{dS}\\right)\n$$\n\nWe can estimate the average transversal component of the electric field $<E_{\\perp}>=[dF_z\/dt]\/cL$, where $F_z$ is total magnetic flux inside a contour with length $L$, which covers the flare region. The evolution of $<E_{\\perp}>$ presented in Fig.1 (gray histograms in top panels) shows that both subflares correlate with the peaks of $<E_{\\perp}>$.\n\nTo calculate the vertical currents we use the disambiguated HMI vector magnetic field data \\citep{Centeno2014} with time cadence 720 seconds, and the same spatial resolution as of the line-of-sight magnetograms. The vertical electric current density is calculated from the HMI vector magnetograms using the Ampere's law \\citep[e.g.][]{Guo2013}:\n\n$$\nj_z=\\frac{c}{4\\pi}(\\nabla\\times\\vec{B})_z = \\frac{c}{4\\pi}\\left(\\frac{\\partial B_x}{\\partial y} - \\frac{\\partial B_y}{\\partial x}\\right)\n$$\n\nThe resulted $j_z$ map during the flare, effectively averaged over 12 min due to the HMI temporal resolution, is presented in Fig.~3. Figure 5 displays the evolution of $<j_z>$ averaged through the flare region with area $\\approx 1.5\\times 10^{18}$ cm$^2$, and reveals a maximum corresponding to the flare. We estimated errors for $<j_z>$ as the standard deviation of the $j_z$ distribution in the quite Sun regions.\n\nIn Fig.~3 we see that the place of sunquake generation correlates with the strong electric currents, and that there is no significant HXR emission in this place. The HXR is mostly emitted from the source located on the other side of the magnetic field polarity inversion line at the opposite footpoint of the flare loop. Such observation, and the time evolution of $<j_z>$ and $<E_{\\perp}>$ can be an evidence of a non beam-driven origin of sunquake. The correlation with the location of the strongest electric currents suggests that the sunquake event could be initiated due to a local heating or impulsive Lorentz force in the flare region.\n\n\n\\section{Discussion}\n\nIn this section we will discuss the contributions of the electric currents and nonthermal electrons in flare energy release and the generation of the sunquake.\n\nFor the estimated fluxes of accelerated electrons for the first subflare, the total kinetic power $P_{nonth}\\approx 1.5\\times 10^{27}$ erg s$^{-1}$ in the HXR peak. To estimate the Joule heating in the sunquake generation region we need to estimate effective electric conductivity $\\sigma_{eff}$. In the regime of electric currents dissipation the magnetic Reynolds number $Re_m=4\\pi\\sigma_{eff}L^2\/(c^2\\tau)\\sim 1$, where $\\tau$ is a characteristic time of electric current dissipation ($\\sim 100$ s, the duration of the HXR burst), and $L$ is a characteristic length scale ($\\sim 1^{\\prime\\prime}$, the size of the impulsive region on the Dopplergrams). For these characteristic values we get $\\sigma_{eff}\\sim 10^6$ CGS units. This value is substantially higher than the theoretical Spitzer conductivity \\citep{Kopecky1966}. However, recent studies of the partially ionized plasma of the solar chromosphere show that the electric conductivity can be substantially reduced due to Pedersen resistivity \\citep[e.g.][]{Leake2012} or due to small scale MHD turbulence \\citep{Vishniac1999}. The volumetric energy release is $Q_j = j^2\/\\sigma_{eff}\\approx 8\\times 10^3$ erg s$^{-1}$ cm$^{-3}$ for $j\\approx 0.3$ A\/m$^2$. The total energy release due to dissipation of electric currents in the sunquake region is $Q_j^{tot}\\approx 3\\times10^{27}$ erg s$^{-1}$, estimating the volume for a box with length scale $L\\sim 1^{\\prime\\prime}$. So, we see that $P_{nonth}\\sim Q_j^{tot}$, and both types of energy release have the energy budget sufficient to explain heating in the flare according to the GOES data: the change of the plasma internal energy, $d(3nk_BTV)\/dt\\sim 10^{27}$ erg s$^{-1}$, and the radiation losses, $L_{rad}\\sim 5\\times 10^{26}$ erg s$^{-1}$.\n\nTo produce the sunquake we need a strong impulsive force in the lower solar atmosphere. The sunquake momentum can be estimated from the initial impact as $p_{sq}\\sim \\rho L^3v\\sim 10^{22}$ g$\\cdot$cm s$^{-1}$ for $\\rho\\sim 10^{-8}$ g$\\cdot$cm$^{-3}$ (photospheric value) and $v\\sim c_s\\sim 10$ km\/s, where $c_s$ is photospheric sound speed. In principle, the force generating sunquakes can be produced directly by energetic electron beams. The total momentum of injected nonthermal electrons is\n\n$$\np_e=\\tau\\sqrt{2m_e}\\int_{E_{low}}^\\infty f(E)\\sqrt{E}dE\n$$\n\nwhere $m_e$ is the electron mass, $f(E)$ is the distribution function of nonthermal electrons, and $\\tau$ is the characteristic time of the injection. For the first HXR burst, $p_e\\sim 10^{20}$ g$\\cdot$cm s$^{-1}$. As the emission intensity of the weaker HXR source associated with sunquake impact is five times less comparing with strong HXR source, than nonthermal electrons momentum in the footpoint associated with sunquake impact $\\sim 0.2\\times 10^{20}$ g$\\cdot$cm s$^{-1}$.\n\nMomentum of the accelerated protons can be much larger than in the case of electrons and lead to stronger disturbances in the solar atmosphere. Assuming that the protons roughly (not accounting collisions) have energy $E_p\\lesssim E_e$, the momentum contained in the proton beam $p_p\\lesssim p_e\\sqrt{(m_p\/m_e)}\\sim 45p_e\\sim 0.5\\times 10^{22}$ g$\\cdot$cm s$^{-1}$. We see that the momentum of accelerated protons represents a more probable agent of the sunquake initiation than the momentum of electrons.\n\nOur observations show that while the sunquake impact and the HXR impulse are simultaneous in time they are clearly separated in space, and located at the different footpoints of the flare magnetic loop. In addition, we find that the impact location correlates with the strongest electric currents. This suggests that, perhaps, energetic particles are accelerated by electric field in the place of sunquake initiation, and then the particles travel along the flare magnetic loop to the other footpoint and caused the HXR emission.\n\nThe impulsive plasma motion in the lower solar atmosphere may be caused by fast heating due to Joule dissipation or sharp increasing of the Lorentz force. In the first case we can estimate the plasma momentum as $p_J\\sim \\tau V\\nabla P\\sim P\\tau L^2$, where $\\nabla P$ is the pressure gradient on the length scale, $L$. The pressure can be estimated from the energy equation\n\n$$\n\\frac{dP}{dt}=\\frac{j^2}{\\sigma_{eff}}-L_{rad}\n$$\n\nwhere $L_{rad}$ is the radiation heat loss which is the main source of cooling in the lower solar atmosphere. From this equation $P\\lesssim j^2\\tau\/\\sigma_{eff}$ and, hence, $p_J \\lesssim (j\\tau L)^2\/\\sigma_{eff}\\sim 10^{23}$ g$\\cdot$cm s$^{-1}$.\n\nThe plasma momentum, associated with the Lorentz force, is $p_L\\sim jB\\tau L^3\/c\\sim 10^{22}$ g$\\cdot$cm s$^{-1}$, where $B\\sim 100$ G is the magnetic field in the sunquake source, and $c$ is the speed of light.\n\nFrom the estimated values of $Q_j$, $p_J$ and $p_L$ one can conclude that the appearance of strong electric currents in the lower solar atmosphere is sufficient to explain the flare energy release and generation of the sunquake. Moreover, these estimations show that the electric current driven disturbances are sufficiently strong, and also that the electric currents are concentrated in the place of the sunquake initiation while the strongest HXR impulse is $\\sim 3$ Mm away. Therefore, it is likely that not only high-energy particles play significant role in the flares, as assumed by the standard flare model, but also electric currents in the lower solar atmosphere can be also a significant part of flare energy release. In our recent paper, we discuss the relationship between electric currents and the fine structure of flare ribbons \\citep{Sharykin2014}.\n\n\\section{Summary and conclusion}\n\nThe main results of the work are as the following:\n\n\\begin{enumerate}\n\\item We observed a strong sunquake event in a weak C-class flare.\n\\item The sunquake is initiated exactly, within 3 sec observational accuracy, during the burst of the HXR emission.\n\\item The place of the photospheric impact associated with the sunquake generation corresponds to the weaker HXR emission source, while there is no significant photospheric impact in the stronger HXR emission source, which is located at the opposite footpoint of a flare loop observed in the EUV AIA images.\n\\item The place of the photospheric impact associated with sunquake initiation corresponds to the most intense electric currents.\n\\item The total (C7.0-M1.9) flare event temporarily correlates with the maxima of vertical and transversal electric currents estimated in the energy release site.\n\\end{enumerate}\n\nThe main conclusion of the presented observational results is that the helioseismic response (sunquake) and flare energy release in the lower solar atmosphere may have strong connection to photospheric electric currents. The sunquake impact may be initiated by a pressure gradient caused due a rapid current dissipation or impulsive Lorentz force. The discovery of the strong photospheric impact produced by a weak C7 flare, which initiated the helioseismic response, opens new perspectives for studying the flare energy release and transport because such flares usually have relatively simple magnetic topology and do not saturate detectors of space and ground-based telescopes. However, our results show that high spatial and temporal resolutions are needed for these studies.\n\nThe work was partially supported by RFBR grant 13-02-91165, President's grant MK-3931.2013.2, NASA grant NNX14AB70G, and NJIT grant.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecent cosmological observations point to an strong evidence for an spatially flat and accelerated expanding universe\n\\cite{Riess2011, Komatsu2011, Reid2010}. Despite the great agreement of observations with the concordance model\n\\cite{Li2011} \\footnote{The Cosmological Constant Model.}, it is a fact that quintom model, whose Equation of State\n(EoS) can cross the cosmological constant barrier $w=-1$, is not exclude by observations \\cite{Alam2004a,Feng2005,Huterer2005,Nesseris2007,Jassal2010,Novosyadlyj2012,Novosyadlyj2012a,Hinshaw2012}. A popular way\nto realize a viable quintom model and, at the same time, avoid the restrictions imposed by the \\textit{No-Go Theorem} \\cite{Vikman2005,Hu2005, Caldwell2005, Xia2008, Cai2010}\nis the introduction of extra degrees of freedom\\footnote{The only way to realize the crossing\nwithout any ghosts and gradient instabilities in standard gravity and with one single scalar degree of freedom was obtained in \\cite{Deffayet2010}. \n}. Following\nthis recipe, the simple quintom paradigm requires a canonical quintessence scalar\nfield $\\sigma$ and simultaneously a phantom scalar field $\\phi$ where the effective potential can be of arbitrary form,\nwhile the two components can be either coupled \\cite{Zhang2006} or decoupled \\cite{Feng2005, Guo2005}. \n\nThe properties of the quintom models have been studied from different points of view. Among them, the phase space studies,\nusing the dynamical systems tools, are very useful in order to analyze the asymptotic behavior of the model. In quintom models\n this program have been carried out in \\cite{Guo2005, Zhang2006, Lazkoz2006, Lazkoz2007, Setare2009, Setare2008a, Setare2009c, Cai2010}.\n In \\cite{Guo2005} the decoupled case between the canonical and phantom field with an exponential potential is studied shown that\n the phantom-dominated scaling solution is the unique late-time attractor. In \\cite{Zhang2006} the potential considers the interaction\n  between the fields and shows that in the absence of interactions, the solution dominated by the phantom field should be the attractor\n  of the system  and the interaction does not affect its attractor behavior. This result is correct only in the case  in which the\n  existence of the phantom phase excludes the existence scaling attractors \\cite{Lazkoz2006}. Some of these\nresults were extended in \\cite{Lazkoz2007} for arbitrary potentials. In \\cite{Setare2009c} the authors showed that all quintom models\n with nearly flat potentials converge to a single expression for EoS of dark energy, in addition, the necessary conditions for the\n  determination of the direction of the $w=-1$ crossing was found.\n\n\nThe aim of this paper is to extend the study of Refs. \\cite{Guo2005,\nLazkoz2006, Lazkoz2007, Setare2009, Cai2010} -investigation of the\ndynamics of quintom cosmology- to include a wide variety of\npotential beyond the exponential potential without interaction\nbetween the fields, all of them can be constructed using the Bohm\nformalism \\cite{Guzman2007, Socorro2010, Bohm1952} of the quantum\nmechanics under the integral systems premise, which is known as\nquantum potential approach.  This approach makes it possible to\nidentify trajectories associated with the wave function of the\nuniverse \\cite{Guzman2007} when we choose the superpotential\nfunction as the momenta associated to the coordinate field $q$. This\ninvestigation was undertaken within the framework of the\nminisuperspace approximation to quantum theory when we investigate\nthe dynamics of only a finite number of models. Here we make use of\nthe dynamical systems tools to obtain useful information about the\nasymptotic properties of the model. In order to be able to analyze\nself-interaction potentials beyond the exponential one, we rely on\nthe method introduced in Ref. \\cite{Fang2009} in the context of\nquintessence models and that have been generalized to several cosmological contexts\nlike: Randall-Sundrum II and DGP branes \\cite{Leyva2009, Escobar2012a,\nEscobar2012e}, Scalar Field Dark Matter models \\cite{Matos2009},\ntachyon and phantom fields \\cite{Quiros2010, Fang2010a,\nFarajollahi2011} and loop quantum gravity \\cite{Xiao2011j}.\n\nThe plan of the paper is as follow: in section \\ref{ss1} we introduce the quintom model for arbitrary potentials and in section \\ref{ss2}\n we build the corresponding autonomous system. The results of the study of the corresponding critical points,\n their stability properties and the physical discussion are shown in section \\ref{s2}. The section \\ref{ss4} is devoted to conclusions. Finally,  we include in the two appendices \\ref{apen1} and \\ref{apen2} the center manifold calculation of the solutions dominated by either the phantom or quintessence potential. \n\n\\section{The model}\\label{ss1}\nThe starting action of our model, containing the canonical field $\\sigma$ and the phantom field $\\phi$, is \\cite{Feng2005, Guo2005, Zhang2006}:\n\n\\begin{eqnarray}\\label{action}\nS=\\int d^{4}x\\sqrt{-g}\\left(\\frac{1}{2}R-\\frac{1}{2}g^{\\mu\\nu}\\partial_{\\mu}\\sigma\\partial_{\\nu}\\sigma+V_{\\sigma}(\\sigma) + \\right. \\nonumber\\\\\n    \\left. + \\frac{1}{2}g^{\\mu\\nu}\\partial_{\\mu}\\phi\\partial_{\\nu}\\phi+V_{\\phi}(\\phi)\\right),\n\\end{eqnarray}\n\nwhere we used natural units ($8 \\pi G=1$) and $V_{\\sigma}(\\sigma)$ and $V_{\\phi}(\\phi)$ are respectively the self interactions\npotential of the quintessence and phantom fields. \n\nFrom this action the Friedmann equations for a flat geometry reads \\cite{Guo2005, Zhang2006}:\n\\begin{equation}\\label{F1}\n    H^{2}=\\frac{1}{3}\\left( \\frac{\\dot{\\sigma}^{2}}{2}+V_{\\sigma}(\\sigma)-\\frac{\\dot{\\phi}^{2}}{2}+V_{\\phi}(\\phi)\\right)\n\\end{equation}\n\\begin{equation}\\label{F2}\n    \\dot{H}=-\\frac{1}{2}\\left(\\dot{\\sigma}^{2}-\\dot{\\phi}^{2} \\right)\n\\end{equation} where \nwhere $H=\\frac{\\dot{a}}{a}$ is the Hubble parameter and the dot denotes derivative with respect the time. \n\nThe evolution of the quintessence and phantom field are:\n\\begin{equation}\\label{KG1}\n    \\ddot{\\sigma}+3H\\dot{\\sigma}+V_{\\sigma}'(\\sigma)=0\n\\end{equation}\n\\begin{equation}\\label{KG2}\n    \\ddot{\\phi}+3H\\dot{\\phi}- V_{\\phi}'(\\phi)=0,\n\\end{equation} where the coma denotes the derivative of a function with respect to their argument. \n\nAdditionally we can introduce the total energy density and pressure as:\n\\begin{equation}\n    \\rho_{DE}=\\rho_{\\sigma}+\\rho_{\\phi},\\;\\;\\;p_{DE}=p_{\\sigma}+p_{\\phi}\n\\end{equation}\nwhere\n\\begin{equation}\n    \\rho_{\\sigma}=\\frac{\\dot{\\sigma}^2}{2}+V_{\\sigma}(\\sigma),\\;\\;\\;\\rho_{\\phi}=-\\frac{\\dot{\\phi}^2}{2}+V_{\\phi}(\\phi)\n\\end{equation}\n\\begin{equation}\n    p_{\\sigma}=\\frac{\\dot{\\sigma}^2}{2}-V_{\\sigma}(\\sigma),\\;\\;\\;p_{\\phi}=-\\frac{\\dot{\\phi}^2}{2}-V_{\\phi}(\\phi)\n\\end{equation}\nand its equation of state parameter is given by\n\\begin{equation}\\label{weff}\n    w_{eff}=\\frac{p_{\\sigma}+p{\\phi}}{\\rho_{\\sigma}+\\rho_{\\phi}}=\\frac{\\dot{\\sigma}^{2}-\\dot{\\phi}^{2}-2V_{\\sigma}(\\sigma)-2V_{\\phi}(\\phi)}{\\dot{\\sigma}^{2}-\\dot{\\phi}^{2}+2V_{\\sigma}(\\sigma)+2V_{\\phi}(\\phi)}\n\\end{equation}\nand \n\\begin{equation}\\label{omegaaa}\n    \\Omega_{\\sigma}=\\frac{\\rho_{\\sigma}}{\\rho_{DE}},\\;\\;\\Omega_{\\phi}=\\frac{\\rho_{\\phi}}{\\rho_{DE}}\n\\end{equation}\n\\begin{equation}\n    \\Omega_{\\sigma}+ \\Omega_{\\phi}=1\n\\end{equation}\nare the the individual and total dimensionless densities parameters. \n\n\\section{The autonomous system}\\label{ss2}\nIn order to study the dynamical properties of the system\n(\\ref{F1}-\\ref{KG2}) we introduce the following dimensionless phase\nspace variables to build an autonomous system \\cite{Copeland1998, Chen2009}:\n\\begin{equation}\\label{nv1}\nx_{\\sigma}=\\frac{\\dot{\\sigma}}{\\sqrt{6}H}, \\\\\nx_{\\phi}=\\frac{\\dot{\\phi}}{\\sqrt{6}H}, \\\\\ny_{\\sigma}=\\frac{\\sqrt{V_{\\sigma}(\\sigma)}}{\\sqrt{3}H}, \\\\\n\\end{equation}\n\\begin{equation}\n\\lambda_{\\sigma}=-\\frac{V_{\\sigma}'(\\sigma)}{V_{\\sigma}(\\sigma)}, \\\\\n\\lambda_{\\phi}=-\\frac{V_{\\phi}'(\\phi)}{V_{\\phi}(\\phi)}, \\\\\n\\end{equation}\n\nNotice that the phase space variables $\\lambda_{\\sigma}$ and\n$\\lambda_{\\phi}$ are sensitive of the kind of self interactions\npotential chosen for quintessence and phantom component,\nrespectively and are introduced in order to be able to study\narbitrary potentials. Applying the above dimensionless variables to\nthe system (\\ref{F1}-\\ref{KG2}) we obtain the following autonomous\nsystem:\n\n\\begin{eqnarray}\n \\frac{d x_{\\sigma}}{dN}&=& -3 x_{\\sigma} \\left(1+x_\\phi^2-x_\\sigma^2 \\right)+\\sqrt{\\frac{3}{2}} y_{\\sigma}^2 \\lambda_\\sigma \\label{auto1}\\\\\n \\frac{d x_{\\phi}}{dN}&=& -3 x_{\\phi} \\left(1+x_\\phi^2-x_\\sigma^2 \\right)+\\nonumber\\\\&&-\\sqrt{\\frac{3}{2}}\\left(1+x_\\phi^2-x_\\sigma^2-y_\\sigma^2\\right) \\lambda_\\phi \\label{auto2}\\\\\n\\frac{d y_{\\sigma}}{dN} &=&\\frac{1}{2}y_\\sigma \\left(6x_\\sigma^2-\\sqrt{6} x_\\sigma\n   \\lambda_\\sigma -6 x_\\phi^2\\right) \\label{auto3}\\\\\n \\frac{d \\lambda_{\\sigma}}{dN}&=& -\\sqrt{6} x_{\\sigma} f(\\lambda_\\sigma) \\label{auto4}\\\\\n\\frac{d \\lambda_{\\phi}}{dN} &=& -\\sqrt{6}x_\\phi g(\\lambda_\\phi)\\label{auto5}\n\\end{eqnarray}\nwhere $N=\\ln a$ is the number of e-foldings and  $f(\\lambda_\\sigma)= \\lambda_\\sigma\n^2(\\Gamma_{\\sigma}-1)$  and $ g(\\lambda_\\phi)= \\lambda_\\phi\n^2(\\Gamma_{\\phi}-1) $ where:\n\\begin{equation}\n\\Gamma_{\\sigma}=\\frac{V_{\\sigma}(\\sigma)V_{\\sigma}''(\\sigma)}{(V_{\\sigma}'(\\sigma))^{2}},\n\\;\\;\\; \\Gamma_{\\phi}=\\frac{V_{\\phi}(\\phi)V_{\\phi}''(\\phi)}{(V_{\\phi}'(\\phi))^{2}}\n\\end{equation}\nIn order to get from the autonomous equation (\\ref{auto1}-\\ref{auto5})  a closed system of ordinary differential equation\nwe have assumed that the funtions $\\Gamma_{\\sigma}$ and $\\Gamma_{\\phi}$ can be written as a function of the\nvariables $\\lambda_{\\sigma}\\in\\mathbb{R}$ and $\\lambda_{\\phi}\\in\\mathbb{R}$ respectively \\cite{Fang2009}.\n\nThe phase space for the autonomous dynamical system driven by de evolutions of Eqs. (\\ref{auto1}-\\ref{auto5}) can be defined as follows:\n\\begin{eqnarray}\n    \\Psi=\\{(x_{\\sigma},x_{\\phi},y_{\\sigma}):y_{\\sigma}\\geq 0, x_\\sigma^2-x_\\phi^2+y_\\sigma^2\\leq 1 \\}\\times\\nonumber\\\\\n                \\times\\{(\\lambda_{\\sigma},\\lambda_{\\phi})\\in\\mathbb{R}^{2} \\}\n\\end{eqnarray}\n\nWith the aim of explain the physical significance of the critical points of the autonomous system (\\ref{auto1}-\\ref{auto5}) we need to obtain the relevant cosmological parameters in terms of the dimensionless phase space variables (\\ref{nv1}). Following this, the cosmological parameter (\\ref{weff}) and (\\ref{omegaaa}) can be expressed as\n\\begin{equation}\n w_{eff}=-1+2x_{\\sigma}^2-2x_{\\phi}^2\n\\end{equation}\n\\begin{equation}\n \\Omega_{\\sigma}=x_{\\sigma}^2+y_{\\sigma}^2,\\;\\;\\ \\Omega_{\\phi}=1-x_{\\sigma}^2-y_{\\sigma}^2,\n\\end{equation}\nwhile the deceleration parameter becomes\n\\begin{equation}\n q=-\\left[1+\\frac{\\dot{H}}{H^2}\\right]=-1+3x_{\\sigma}^2-3x_{\\phi}^2.\n\\end{equation}\n\n\n\n\\section{Critical points and stability}\\label{s2}\nThe critical points of the system (\\ref{auto1}-\\ref{auto2}) are\nsummarized in Table \\ref{tab1}. The eigenvalues of the corresponding\nJacobian matrices are show in Table \\ref{tab2}. In both cases\n$\\lambda_{\\sigma}^{\\ast}$ and $\\lambda_{\\phi}^{\\ast}$ are the values\nwhich makes the functions\n$f(\\lambda_{\\sigma})=\\lambda_\\sigma^2\\left(\\Gamma_{\\sigma}-1\\right)$\nand $g(\\lambda_{\\phi})=\\lambda_\\phi^2\\left(\\Gamma_{\\phi}-1\\right)$\nvanish respectively.\n\n\n\\begin{table*}\\caption[crit]{Properties of the critical points for the\nautonomous system (\\ref{auto1}-\\ref{auto5})}\n\\begin{tabular}\n{l c c c c c c c c c c}\n\\hline\\hline\\\\[-0.3cm]\n$Label$&$x_{\\sigma}$&$y_{\\sigma}$&$x_{\\phi}$&$\\lambda_{\\sigma}\n$&$\\lambda_{\\phi}$&Existence&$\\Omega_{\\sigma}$&$\\Omega_{\\phi}\n$&$q$&$w_{eff}$\\\\\n\\hline\\\\[-0.2cm]\n\n$P_1^\\pm$ &$0$  &$0$  &$\\pm i$  &$\\lambda_{\\sigma}$  &\n$\\lambda_{\\phi}^{\\ast}$ &  Non real   &$0$  &$1$& $2$ & $1$\\\\[0.2cm]\n\n$P_2^\\pm$ &$\\pm1$  & $0$  &$0$  &$\\lambda_{\\sigma}^{\\ast}$\n&$\\lambda_{\\phi}$&   Always      &$1$ &$0$ &$2$ &$1$ \\\\[0.2cm]\n\n$P_3^\\pm$ &$\\pm\\sqrt{1+x_\\phi^2}$  &$0$  & $x_\\phi$ &$\\lambda_{\\sigma}^{\\ast}$ &\n$\\lambda_{\\phi}^{\\ast}$&   \\textquotedblright      &$1+x_{\\phi}^{2}$ &$-x_{\\phi}^{2}$\n&$2$ &$1$ \\\\[0.2cm]\n\n$P_{4}$ &$\\frac{\\lambda_{\\sigma}^{\\ast}}{\\sqrt{6}}$\n&$\\sqrt{1-\\frac{(\\lambda_{\\sigma}^{\\ast})^{2}}{6}}$&$0$\n&$\\lambda_{\\sigma}^{\\ast}$ &$\\lambda_{\\phi}$&  $-\\sqrt{6}\\leq \\lambda_{\\sigma}^{\\ast} \\leq \\sqrt{6}$       &$1$ &$0$\n&$-1+\\frac{(\\lambda_{\\sigma}^{\\ast})^{2}}{2}$ &\n$-1+\\frac{(\\lambda_{\\sigma}^{\\ast})^{2}}{3}$  \\\\[0.2cm]\n\n$P_5$ & $0$ & $0$ & $0$ & $\\lambda_{\\sigma}$ &\n$0$& Always & $0$  & $1$ &$-1$ & $-1$ \\\\[0.2cm]\n\n$P_6$ &$0$  &$1$  &$0$  &$0$  &$\\lambda_{\\phi}$  & \\textquotedblright & $1$ &\n$0$ & $-1$ &$-1$ \\\\[0.2cm]\n\n$P_{7}$ &$0$  &$y_{\\sigma}$  &$0$  & $0$\n&$0$ &  $0< y_{\\sigma}< 1$     &$y_{\\sigma}^{2}$  &$1-y_{\\sigma}^{2}$ &\n$-1$ &$-1$\\\\ [0.2cm]\n\n$P_{8}$&$0$  &$0$ &$-\\frac{\\lambda_{\\phi}^{\\ast}}{\\sqrt{6}}$\n&$\\lambda_{\\sigma}$ & $\\lambda_{\\phi}^{\\ast}$ &   $\\lambda_{\\sigma}\n\\in \\mathbb{R}$ &$0$ &$1$\n&$-1-\\frac{(\\lambda_{\\phi}^{\\ast})^{2}}{2}$\n&$-1-\\frac{(\\lambda_{\\phi}^{\\ast})^{2}}{3}$ \\\\ [0.2cm] \\hline \\hline\n\\end{tabular}\\label{tab1}\n\\\\ [0.2cm]\n\\end{table*}\n\n\\begin{table*}\\caption[crit2]{Eigenvalues of the linear perturbation matrix associated to each of the critical points displayed in Table \\ref{tab1}}\n\\begin{tabular}\n{l c c c c c c c c c c}\n\\hline\\hline\\\\[-0.3cm]\n$Label$&$m_1$&$m_2$&$m_3$&$m_4$&$m_5$\\\\\n\\hline\\\\[-0.2cm]\n\n$P_1^\\pm$ & $3$ & $0$ & $0$ & $\\mp {i}\\sqrt{6}g'(\\lambda_{\\phi}^{\\ast})$  & $6\\mp{ i}\\sqrt{6}\\lambda_{\\phi}^{\\ast}$\\\\[0.2cm]\n\n$P_2^\\pm$ & $6$ & $0$ & $0$ & $\\mp\\sqrt{6}f'(\\lambda_{\\sigma}^{\\ast})$ & $3\\mp\\sqrt{\\frac{3}{2}}\\lambda_\\sigma^{\\ast}$\\\\[0.2cm]\n\n$P_3^\\pm$ & $0$ & $-\\sqrt{6} g'(\\lambda_{\\phi}^{\\ast}) {x_\\phi}$&$\\mp\\sqrt{6} {f'(\\lambda_{\\sigma}^{\\ast})}\n   \\sqrt{x_\\phi^2+1}$&$3\\mp\\sqrt{\\frac{3}{2}} \\sqrt{x_\\phi^2+1}\n   \\lambda_\\sigma ^{\\ast}$& $6-\\sqrt{6} x_\\phi \\lambda_\\phi ^{\\ast}$ \\\\[0.2cm]\n\n$P_4$ & $0$ &$-{f'(\\lambda_{\\sigma}^{\\ast})} \\lambda \\sigma ^{\\ast}$ & $\\left(\\lambda \\sigma\n   ^{\\ast}\\right)^2$ &$\\frac{1}{2} \\left(\\left(\\lambda \\sigma\n   ^{\\ast}\\right)^2-6\\right)$ &$\\frac{1}{2} \\left(\\left(\\lambda \\sigma\n   ^{\\ast}\\right)^2-6\\right)$\\\\ [0.2cm]\n\n$P_5$ & $-3$ & $0$ & $0$ & $-\\frac{3}{2}\\left(1+\\sqrt{1+\\frac{4}{3}g(0)}\\right)$ & $-\\frac{3}{2}\\left(1-\\sqrt{1+\\frac{4}{3}g(0)}\\right)$\\\\[0.2cm]\n\n$P_6$ & $-3$ & $0$ & $0$ & $-\\frac{3}{2}\\left(1+\\sqrt{1-\\frac{4}{3}f(0)}\\right)$ & $-\\frac{3}{2}\\left(1-\\sqrt{1-\\frac{4}{3}f(0)}\\right)$\\\\[0.2cm]\n\n$P_7$ & $0$ & $\\frac{1}{2} \\left(-\\sqrt{9-12 f(0) y_\\sigma^2}-3\\right)$ & $\\frac{1}{2} \\left(\\sqrt{9-12 f(0) y_\\sigma^2}-3\\right)$ & $\\frac{1}{2} \\left(-\\sqrt{9-12 g(0) \\left(y_\\sigma^2-1\\right)}-3\\right)$ & $\\frac{1}{2} \\left(\\sqrt{9-12 g(0)\n   \\left(y_\\sigma^2-1\\right)}-3\\right)$\\\\[0.2cm]\n\n$P_8$ & $0$ &${g'(\\lambda_{\\phi}^{\\ast})} \\lambda_\\phi ^{\\ast}$& $-\\frac{1}{2} \\left(\\lambda_\\phi\n   ^{\\ast}\\right)^2$&$-\\frac{1}{2} \\left(\\left(\\lambda_\\phi\n   ^{\\ast}\\right)^2+6\\right)$&$-\\frac{1}{2} \\left(\\left(\\lambda_\\phi\n   ^{\\ast}\\right)^2+6\\right)$\\\\ [0.2cm]\n\n\n\n\\hline \\hline\\\\[-0.3cm]\n\\end{tabular}\\label{tab2}\n\\end{table*}\nAs we see from Table \\ref{tab1}, the points $P_1^\\pm$ do not exist in the strict sense ($x_\\phi$ is purely imaginary at the fixed points).\nPoint $P_5$ is associated with a combination of a phantom potential whose first $\\phi$-derivative vanishes at some\/several point\/points, i.e.,\n$\\lambda_\\phi=0$ (this case include the exponential potential whose $\\phi$-derivative at any order vanish everywhere) and an arbitrary self\ninteraction potential for the quintessence component (arbitrary value of $\\lambda_{\\sigma}$). Point $P_6$ is associated with a combination\n of a quintessence potential whose first $\\sigma$-derivative vanishes at some\/several point\/points, i.e., $\\lambda_\\sigma=0$\n (this case include the exponential potential whose $\\sigma$-derivative at any order vanish everywhere) and an arbitrary self\n interaction potential for the phantom component (arbitrary value of $\\lambda_{\\phi}$). Point $P_7$  is associated with a\n combination of a phantom potential whose first $\\phi$-derivative vanishes at some\/several point\/points, i.e., $\\lambda_\\phi=0$\n (this case include the exponential potential whose $\\phi$-derivative at any order vanish everywhere) and a self interaction\n potential for the quintessence component whose first $\\sigma$-derivative vanishes also at some\/several point\/points, i.e., $\\lambda_\\phi=0.$ \nIt is worth noticing that the existence of points  $P_2^\\pm$, $P_3^\\pm$, $P_4$,  $P_8$ and $P_9$  depends of the concrete form\nof the potential. From the table of the eigenvalues, notice, besides, that all the points belongs to nonhyperbolic sets of\ncritical point with a least one null  eigenvalue.\n\n\\subsection{Stability of the critical points}\n\nAlthough  all these critical points are shown in the Tables \\ref{tab1} here we have summarized their basic properties:\n\\begin{itemize}\n    \\item $P_{1}^\\pm$, $P_{2}^\\pm$ and $P_{3}^\\pm$ correspond to a solution dominated by the kinetic energy of the scalar fields\n    (stiff fluid solution: $q=2$ and $\\omega=1$). The exact dynamical behavior differs for each points.  $P_1^\\pm$ corresponds to\n    a phantom kinetic energy dominated  ($\\Omega_{\\sigma}=0$ and $\\Omega_{\\phi}=1$). However, these points have a purely imaginary\n    value of $x_\\phi,$ thus, they do not exists in the strict sense. They have a three$-$dimensional center subspace and a two-dimensional\n    unstable manifold ($m_1=3>0,\\; \\Re(m_5)=6>0$). Thus they cannot be late-time attractors. $P_3^\\pm$ represents an scaling regimen\n    between the kinetics energies of the quintessence and phantom fields ($\\Omega_{\\sigma}=1+x_{\\phi}^{2}$ and $\\Omega_{\\phi}=-x_{\\phi}^{2}$).\n    These points depend of the form of the potentials and under certain conditions they have a four dimensional unstable subspace which could\n    correspond to the past attractor. However, this point is unphysical since $\\Omega_{\\phi}<0$ . $P_{2}^\\pm$ is dominated by the\n    quintessence kinetic term ($\\Omega_{\\sigma}=1$ and $\\Omega_{\\phi}=0$). Since they are non-hyperbolic  due to the existence of\n    two null eigenvalues,  we are not able to extract information about their stability by  using the standard tools of the linear\n    dynamical analysis. However, since these points seems to be  particular cases of $P_3^\\pm,$ they should share the same dynamical\n    behavior. Because all of these points are nonhyperbolic, as we notice before, we cannot rely on the standard  linear\n    dynamical systems analysis for deducing their stability. Thus, we need to rely our analysis on numerical inspection of\n     the phase portrait for specific potentials or use more sophisticated techniques like Center Manifold theory.\n    \\item $P_4$ is an scaling solution between the kinetic and the potential energy of the quintessence component of dark energy. This solution in sensitive to the explicit form of the potential. This is always a saddle equilibrium point in the phase space since $m_2=(\\lambda_{\\sigma}^{\\ast})^{2}$ and $m_4=\\frac{1}{2}((\\lambda_{\\sigma}^{\\ast})^{2}-6)$ are of opposite sign in the existence region of  this point. It represents an accelerated solution for a potential $V_\\sigma(\\sigma)$ whose function $f(\\lambda_{\\sigma})$ vanish for $\\lambda_{\\sigma}=\\lambda_{\\sigma}^{\\ast}$ in the interval $-\\sqrt{2}<\\lambda_{\\sigma}^{\\ast}<\\sqrt{2}$,  leading to a $-1\\leq w_{eff}<-1\/3$. When $\\lambda_{\\sigma}^{\\ast}=0$ the critical point $P_4$ becomes in $P_6$. In the regions $-\\sqrt{6}\\leq\\lambda_{\\sigma}^{\\ast}\\leq-\\sqrt{2}$ or $\\sqrt{2}\\leq\\lambda_{\\sigma}^{\\ast}\\leq\\sqrt{6}$, the critical point $P_4$ represents a non-accelerated phase. A very interesting issue of this critical point appears when, for an specific form of\nthe quintessence potential, $\\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3}$, driving to $w_{eff}=0$. This means that the quintessence field is able to mimic the dark matter behavior.\n    \\item $P_5$, $P_6$ and $P_7$ represents solutions dominated by the potential energies of the potentials (all of them represent de Sitter solutions: $q=-1$ and $w_{eff}=-1$). Once again the exact dynamical nature differs from one point to the other: $P_5$ is dominated by the potential energy of the phantom component ($\\Omega_{\\sigma}=0$ and $\\Omega_{\\phi}=1$). Because of the existence of two null eigenvalues  is not possible to conclude about its dynamics. However it has a three-dimensional stable manifold for $g(0)<0$ (in the interval $g(0)<-\\frac{3}{4}$ it has to  complex conjugated eigenvalues with negative real parts). In this cases it is worthy to analyze its stability using the center manifold theory. $P_6$ is a critical point dominated by the quintessence potential energy term ($\\Omega_{\\sigma}=1$ and $\\Omega_{\\phi}=0$), despite its nonhyperbolicity, it has  three-dimensional stable manifold for $f(0)>0$ (in the case $f(0)>\\frac{3}{4}$ it has to  complex conjugated eigenvalues with\nnegative real parts), thus, it is worthy to analyze its stability using the center manifold theory.  $P_7$ denotes a segment (curve) of non-isolated fixed points, representing a scaling regimen between the quintessence and phantom potential ($\\Omega_{\\sigma}=y_{\\sigma}^{2}$ and $\\Omega_{\\phi}=1- y_{\\sigma}^{2}$).  The existence of one non-zero eigenvalue is due to the fact that it is a curve of fixed points. As an invariant set of non-isolated singular points it is normally-hyperbolic, since the eigenvector associated to the zero eigenvalue, $(0,0,1,0,0)^T,$ is tangent to the curve. Thus its stability is determined by the sign of the remaining non-null eigenvalues. Hence, it is stable for $0<y_\\sigma<1,\\;  f(0)>0, \\; g(0)<0$ or a saddle otherwise.\n\\item $P_8$ is a line of fixed points parameterized by $\\lambda_\\sigma\\in\\mathbb{R}$. The existence of one non-zero eigenvalue is due to the fact that it is a curve of fixed points. As an invariant set of non-isolated singular points it is normally-hyperbolic, since the eigenvector associated to the zero eigenvalue, $(0,0,0,1,0)^T,$ is tangent to the curve. Thus its stability is determined by the sign of the remaining non-null eigenvalues.  From table \\ref{tab2} follows that $P_8$ admits a four dimensional stable subspace provided $g'(\\lambda_{\\phi}^{\\ast}) \\lambda_{\\phi}^{\\ast}<0$, thus, the invariant curve is stable. It represents  accelerated solutions dominated by the phantom potential  providing a crossing through the phantom divide ($\\Omega_{\\sigma}=0$ and $\\Omega_{\\phi}=1$). For every value of $\\lambda_{\\phi}^{\\ast}$ this point provide the typical superaccelerated expansion of quintom paradigm ($w=-1-\\frac{(\\lambda_{\\phi}^{\\ast})^{2}}{3}$) the only exception occurs when $\\lambda_{\\phi}^{\\ast}=0$\nrecovering the behavior of the de Sitter solution  $P_5$  ($\\omega=-1$). This line of critical point corresponds to the stable point $P$ in \\cite{Guo2005} and B in \\cite{Cai2010} (phantom dominated solution). Summarizing, the line $P_8$ is the late time stable attractor provided   $g'(\\lambda_{\\phi}^{\\ast}) \\lambda_{\\phi}^{\\ast}<0$, otherwise, it is a saddle point.\n\\end{itemize}\n\n \\subsection{Cosmological consequences}\n\nAs was shown in the previous subsection the autonomous systems only admits\nseven classes of critical points (some of them are actually curves) \\footnote{$P_{1}^\\pm$  and $P_{3}^\\pm$ are ruled out. The first one because of they lead to imaginary values of dimensionless variable $x_\\phi$. And the last one because of is outside of the physical phase space, representing a critical point with a negative energy density $\\Omega_{\\phi}<0$.}. The curves $P_{2}^\\pm$ correspond to decelerated solutions, with $q=2$, where the Friedmann constraint (\\ref{F1}) is dominated by the kinetic energy of the quintessence field with an equation of state of stiff type, $w_{eff}=1$. These solutions are only relevant a early times and should be unstable \\cite{Copeland1998}. Unfortunately these critical points are nonhyperbolic (it has two zero eigenvalues) meaning that is not possible to obtain conclusions about its stability with the previous linear analysis. However the numerical analysis performed in the next subsection with a particular potentials confirm the previous results in \nliterature.\n\nAn important result come from the stability of critical point $P_4$. This points exists if $-\\sqrt{6}\\leq\\lambda_{\\sigma}^{\\ast}\\leq\\sqrt{6}$ and always behave as a saddle fixed point. The latter means that under certain initial conditions the orbits in the phase space will approach to this point spending some time in its vicinity before being repelled toward the attractor solution of the system. In the case of this point, as we mentioned before, if the quintessence potential fulfill the condition:\n\\begin{equation}\\label{condiDEDM}\n \\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3}\n\\end{equation}\nthen the effective equation of state of this dark energy component would mimic pressureless fluid ($w_{eff}=0$), in other words: it will dynamically behave exactly as cold dark matter. The possibility of this dynamical characteristic impose a fine tunning over the shape of quintessence potentials and a priori there is no guarantee that all possible quintessence potentials may satisfy the above condition (\\ref{condiDEDM}). Let's note that in order to obtain the lower possible dimensionality of the phase space and to studying in a relatively simple way the effects of include arbitrary quintom potentials, we have neglected  the contribution of the usual matter fields: radiation and baryonic matter in our model \\footnote{See Eqs. (\\ref{action}-\\ref{F1}).}. As a result, a full study of important aspects, derived from realization of condition (\\ref{condiDEDM}), such as: transition redshift between the decelerated and accelerated expansion phase and the clustering properties of this \\textit{effective dark matter} \nare beyond the present study and will be left for a future paper.\n\nAnother important characteristics of the model is the presence of three accelerated solutions, described by critical points $P_5$, $P_6$ and $P_7$. All of them are  de Sitter solutions ($w_{eff}=-1$) dominated by the potentials of the scalar fields. As in the case of $P_4$, they behave as saddle points and, depending on the initial conditions, the orbits can evolve from the unstable fixed point ($P_{2}^\\pm$ in our case) towards one or the other of the saddle points. A favorable scenario would be one in which the initial condition lead to an evolution from $P_{2}^\\pm$ to the saddle point $P_4$ \\footnote{we are assuming that if (\\ref{condiDEDM}) is fulfilled, then quintessence field behave as the Dark Matter.} and then, the orbits tend to one of the de Sitter solutions $P_5$, $P_6$ or $P_7$ or to the late time phantom attractor ($P_8$). In terms of the cosmological evolution of the Universe, the above favorable scenario implies that the Universe started at early times from an stage dominated by the kinetic \nterm of \nthe quintessence, then evolve into an epoch dominated by the \\textit{effective dark matter} and finally enter in the final phase of accelerated expansion. This accelerated phase can be the de Sitter solutions or a phantom dominated solution ($w_{eff}<-1$) \\footnote{In fact, these models admits the possibility of having two stable solutions: a de Sitter solution ($P_7$) and a phantom solution ($P_8$), each one within their basin of attraction as was shown in previous subsection.}. This final stage of evolutions towards critical point $P_8$ is consistent  with the recent joint results from \\textit{WMAP}+\\textit{eCMB}+\\textit{BAO}+$H_0$+\\textit{SNe} \\cite{Hinshaw2012} which suggest a mild preference for a dark energy equation-of-state parameter in the phantom region ($w_{eff}<-1$). \n\nFinally, in order to examine the stability of the nonhyperbolic  points that cannot consistently be studied via the present linear analysis, we present  a concrete example. We provide a numerical elaboration of the phase space orbits of the corresponding quintom model.\n\n\\subsection{$V(\\sigma,\\phi)=V_{0}\\sinh^{2}(\\alpha\\sigma)+V_{1}\\cosh^{2}(\\beta\\phi)$}\\label{cosh}\nThis potential is derived, in a Friedmann-Robertson-Walker\ncosmological model, from canonical quantum cosmology under determined\nconditions in the evolution of our universe\\footnote{This is part of a forthcoming paper.}, using the bohmian\nformalism \\cite{Guzman2007}. For this potential:\n\\begin{equation}\\label{f1}\n    f(\\lambda_{\\sigma})=-\\frac{\\lambda_{\\sigma}^{2}}{2}+ 2 \\alpha^{2}, \\;\\;\\lambda_{\\sigma}^{\\ast}=\\pm 2 \\alpha, \\;\\;f'(\\lambda_{\\sigma}^{\\ast})=-{\\lambda_{\\sigma}^{\\ast}}\n\\end{equation}\nand\n\\begin{equation}\\label{f2}\n    g(\\lambda_{\\phi})=-\\frac{\\lambda_{\\phi}^{2}}{2}+ 2 \\beta^{2}, \\;\\;\\lambda_{\\phi}^{\\ast}=\\pm 2 \\beta, \\;\\;g'(\\lambda_{\\phi}^{\\ast})=-{\\lambda_{\\phi}^{\\ast}}.\n\\end{equation}\nFrom the Table \\ref{tab2} and the equation (\\ref{f2}) we see that the condition\nto ensure that Point $P_8$ has a four dimensional stable subspace is\nalways satisfied due to the opposite signs between\n$\\lambda_{\\phi}^{\\ast}$ and $g'(\\lambda_{\\phi}^{\\ast})$. In order to\nhaving achieved success scalar field dark matter domination era we\nneed that $\\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3},$ since this is the only way to have a standard transient matter dominated solution ($P_4$). Recall that for the choice $\\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3},$ the standard quintessence dominated solution mimics dark matter ($w_{eff}=0$).\nImposing the condition $\\lambda_{\\sigma}^{\\ast}=\\pm\\sqrt{3},$  we have as a degree of freedom the potential\nparameter $\\alpha$ that can be adjusted using (\\ref{f1}). Furthermore, we impose\none of the following conditions:\n\\begin{equation}\n    \\lambda_{\\sigma}^{\\ast}=\\sqrt{3},\\;\\lambda_\\phi ^{\\ast}\\leq -\\sqrt{6}, \\;  1<x_\\sigma<\\sqrt{1+\\frac{6}{\\left(\\lambda_\\phi\n   ^{\\ast}\\right)^2}}\\nonumber\n\\end{equation} or\n\\begin{equation}\n    \\lambda_{\\sigma}^{\\ast}=\\sqrt{3},\\;-\\sqrt{6}<\\lambda_\\phi ^{\\ast}<0, \\;  1<x_\\sigma<\\sqrt{2}\\nonumber\n\\end{equation} or\n\\begin{equation}\n    \\lambda_{\\sigma}^{\\ast}=-\\sqrt{3},\\;\\lambda_\\phi ^{\\ast}\\leq -\\sqrt{6}, \\;-\\sqrt{1+\\frac{6}{\\left(\\lambda_\\phi\n   ^{\\ast}\\right)^2}}<x_\\sigma<-1\\nonumber\n\\end{equation} or\n\\begin{equation}\n    \\lambda_{\\sigma}^{\\ast}=-\\sqrt{3},\\;-\\sqrt{6}<\\lambda_\\phi ^{\\ast}<0, \\;-\\sqrt{2}<x_\\sigma<-1\\nonumber.\n\\end{equation}\nto  guarantee that the stiff matter type solution of the quintom cosmology be the past-attractor.\n\nTo finish this section let's discuss some numerical elaborations. In the figure \\ref{fig1} are presented some trajectories in phase space ($x_{\\sigma}$, $y_{\\sigma}$, $y_{\\phi}$) for different sets of initial conditions for potential $V(\\sigma,\\phi)=V_{0}\\sinh^{2}(\\alpha\\sigma)+V_{1}\\cosh^{2}(\\beta\\phi)$. The free parameter have been chosen to be ($\\alpha$, $\\beta$): ($-\\sqrt{3}\/2$, $0.35$). This parameter selection guarantee that point $P_4$, black point in this graphic, represents an scalar field matter (i.e., the scalar field mimicking dark matter) dominated era with a typical saddle dynamics. The late-time attractor is the phantom field dominated solution ($P_8$ in Table \\ref{tab2}). In the figure \\ref{fig2} are displayed some trajectories in phase space ($x_{\\sigma}$, $y_{\\sigma}$) with the same parameter selection as in Fig. \\ref{fig1}. \nFinally, in the figure \\ref{fig3} are prresented trajectories in phase space ($y_{\\sigma}$, $y_{\\phi}$) with the same parameter selection of Fig. \\ref{fig1}. The accelerated de Sitter solution $P_7$, dashed line, is a transient era in the evolution of the Universe being the late time attractor the phantom field dominated solution, black point in this figure, allowing the crossing through the phantom divide.\n\n\\begin{figure}[th!]\n\\begin{center}\n\\includegraphics[scale=0.7]{g1}\\caption{\\label{fig1}Trajectories in phase space ($x_{\\sigma}$, $y_{\\sigma}$, $y_{\\phi}$) for the potential $V(\\sigma,\\phi)=V_{0}\\sinh^{2}(\\alpha\\sigma)+V_{1}\\cosh^{2}(\\beta\\phi)$ with ($\\alpha$, $\\beta$): ($-\\sqrt{3}\/2$, $0.35$). With this parameter selection the scalar field matter dominated era ($P_4$, black point in this graphic) is a saddle, whereas, the phantom field dominated solution ($P_8$ in Table \\ref{tab2}) is the late time attractor.}\n\\end{center}\\end{figure}\n\\begin{figure}[th!]\n\\begin{center}\n\\includegraphics[scale=0.7]{g2}\\caption{\\label{fig2}Trajectories in phase space ($x_{\\sigma}$, $y_{\\sigma}$) with the same parameter selection of Fig. \\ref{fig1}. The critical point $P_4$ represented by the black point is a scalar field matter dominated transient solution.}\n\\end{center}\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=6.5cm,height=5cm]{g3}\\caption{\\label{fig3}Trajectories in phase space ($y_{\\sigma}$, $y_{\\phi}$) with the same parameter selection of Fig. \\ref{fig1}. The accelerated de Sitter solution $P_7$, dashed line, is a transient era in the evolution of the Universe and the late time attractor the phantom field dominated solution, black point in this figure, allowing a crossing through the phantom divide.}\n\\end{center}\\end{figure}\n\n\\section{Conclusions}\\label{ss4}\nIn the present paper a thorough study of the phase space of quintom\nmodel has been undertaken. The results are valid for those\npotential, without interaction between the fields, for which the\nquantities\n$\\Gamma_{\\sigma}=\\frac{V_{\\sigma}(\\sigma)V_{\\sigma}''(\\sigma)}{(V_{\\sigma}'(\\sigma))^{2}}$\nand\n$\\Gamma_{\\phi}=\\frac{V_{\\phi}(\\phi)V_{\\phi}''(\\phi)}{(V_{\\sigma}'(\\phi))^{2}}$\ncan be written as a function of the variables $\\lambda_{\\sigma}$ and\n$\\lambda_{\\phi}$.\n\nIt has been found that for  $g'(\\lambda_{\\phi}^{\\ast}) \\lambda_{\\phi}^{\\ast}<0$, the late time attractor are always the\nphantom dominated solution ($P_8$) generalizing the result\n shown in \\cite{Guo2005, Cai2010} for exponential potentials (but more generally, for potential satisfying\n $\\lambda_{\\sigma}\\approx const$ and $\\lambda_{\\phi}\\approx const$). Otherwise, it is a saddle point. The Universe evolves from a quintessence dominated\n phase to a phantom dominated phase crossing the $w_{eff}=-1$ divide line as a transient stage \\cite{Guo2006}.\n \n \n Center Manifold Theory have been  employed to analyze the stability of de Sitter solution either with phantom ($P_5$) or quintessence potential dominance ($P_6$). After deriving the evolution equation on the center manifolds and making several numerical integrations we have concluded that in both cases the corresponding de Sitter solution is unstable (saddle-like). For $P_5$ we have used an analytical argument, whereas for $P_6$ our conclusion was supported partially on analytical arguments and complemented by  numerical experimentation. \n \n Another important issue is concerning the existence of a point, $P_4,$ corresponding to the standard quintessence dominated solution, which under certain condition on the  potential, can mimick the dark matter behavior. This feature has important cosmological consequences to address de unified description of dark matter and dark energy in a single field. The saddle type character of $P_4$ have been clearly illustrated by resorting to phase plane diagrams for the potential obtained from a canonical quantum cosmology.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the development of General Relativity (GR) Einstein (1907\\cite{A.E.1907}) started  by\nusing the equivalence principle (EP) to analyse the ``apparent'' gravitational field in an\nuniformly accelerated system.  In this analysis he found that the synchronized time\n(coordinate time) could not agree with the local proper time, a fact from which it followed\nimmediately the gravitational redshift, the bending of light trajectories and the dependence\nof light velocity on position when measured using coordinate time. In 1911\\cite{A.E.1911} he\nreviewed and expanded these ideas and conjectured that the effects found in the ``apparent''\ngravitational field of an accelerated system, should also be present in a genuine\ngravitational field as that of the earth or the sun. In particular, he predicted a\ndeflection of light rays grazing the sun of roughly one arc second. However, after a lengthy\nand contorted path he will come to know that, unlike the gravitational redshift and the\ndependence of the velocity of light on position, which hold exactly in a general static\nfield, the deflection of light ray had a hidden assumption on the flatness of space.\nDepending on the curvature of space this deflection could be zero (conformally flat\nspace-time, corresponding to a negative spatial curvature) or twice the quoted value (full\nGR, corresponding to positive spatial curvature around the sun). In 1912 \\cite{A.E.1912I} he\nassumed that a general static gravitational field could be described by a single scalar\nfield, namely, the position dependent velocity of light, $c(\\vec{x})$, that he has found to\nbe the case for the field in an accelerated system. Again, the implicit assumption here was\nthat space remained flat and, as we shall discuss latter, this was bound to give rise to\ndifficulties. Einstein obtained the equation satisfied by the field $c(\\vec{x})$ in an\naccelerated system and through some considerations extended it to the case of a general\nstatic field. The equation was:\n\n\\begin{equation} \\nabla^{2}c(\\vec{x})=4 \\pi G c(\\vec{x}) \\rho(\\vec{x}), \\label{1Juan}\n\\end{equation} where $\\rho(\t\\vec{x})$ is the matter density distribution.  The problem\nwas that one can not arrange (through suitable definition of mass and force within a\ngravitational field) for the simultaneous conservation of energy and momentum. This\ndifficulties led Einstein to advance a modified equation fully consistent with\nenergy-momentum conservation, but not exactly consistent with the EP, which would be\nsatisfied only by sufficiently weak fields. Einstein considered the EP to be the happiest\nthought of his live, thus having to forsake this principle was rather unpalatable to him. In\nthis state of inner dissatisfaction with his latest field equation, he started to muse on\nthe plausibility of his assumption concerning the Euclidean character of  space. Analysing\nthe intrinsic geometry of a rotating disk\\cite{A.E.libro} he realized that, even in the\nabsence of a genuine gravitational field (i.e.  flat space-time), in the co-rotating system,\nwhere an ``apparent'' gravitational field exist, the geometry of space can not be Euclidean.\nThe EP then led  him to believe that the same would be true for an actual gravitational\nfield. It seems that it was at this point that he fully realized that the EP implied the\ntensorial character of the gravitational field, which is essential even in the case of\nstatic fields. From this moment his research took a brisk turn. Once he realized that the\nappropriate formalism for his problem was that of ``absolute calculus'' (the name given in\nthose day to de differential geometry) he set himself to command that calculus and initiated\na long an extremely arduous route culminating three and a half year latter ( in November\n1915) with the fully-fledged theory of GR.  The story of these development is described in\nsome detail in books like Abraham Pais's\\cite{A.Pais} Einsteins's scientific biography or\nHoward $\\&$ Stachel's\\cite{Stachel} history of the development of GR  and it represented for\nEinstein a novel style of research, not only with respect to GR, but with respect to all his\nprevious scientific career. Up to this time physical intuition played the guiding role; in\nthe future that role will be played mostly by  formal considerations. It was after GR was\nobtained that the intuitions for the new physics was gained by extracting its\nimplications.However, the question of which portion of GR could have been anticipated by\nsimple consideration based on EP have, to our knowledge, not been fully clarified. More\nprecisely, the questions of the validity of Eq. (\\ref{1Juan}) and whether the  derivation\nEinstein gave for it was correct,  or which relevant results may be inferred from it, are\nnot found in the most widely used text on the conceptual development  of GR.\n\nIn this work we want to point out  that the Eq. (\\ref{1Juan}) is correct (for negligible\npressure) and that the arguments given by Einstein to prove it from the EP are, with some\nqualifications , correct, being some other assumption that Einstein took that led  to\ncontradictions. The goals that we pursue are three-fold: from the\nconceptual point of view we want to explore how much can one learn about a static\ngravitational field from simple (pre- Riemannian) considerations concerning the EP, as it\nwas the aim of the interrupted Einstein GR research program. From a technical point of view\nwe want to stress that Eq. (\\ref{1Juan}) is a very useful one for static field, and that in\nthe case of spherical symmetry it leads immediately to rather interesting results. From the\nhistoric point of view we want to remark that by a slight twist of events a theory of\ngravitation accounting for most relevant facts (including the three classical test) would\nhave been available in 1912.\n\nThe work is organized as follows: In section 1 we review Einstein derivation of Eq.\n(\\ref{1Juan}) weighing and strengthening the arguments he gave and show that, in the case of\nnegligible pressure,  it follows from Einstein equations of GR. In section 2 we use Eq.\n(\\ref{1Juan})  together with a simple argument relating the $c$ field to the spatial metric\nto derive de Schwarzschild metric. We also consider some other interesting results following\nfrom Eq. (\\ref{1Juan}). In section 3 we discuss Einstein's pre-Riemannian theories of\ngravitation reviewing the arguments and the difficulties that he  found. We also consider\nwhich could have been the course of development of GR if Einstein had been able to continue\nhis initial program, which involved considering particularly meaningful physical situation\nof increasing complexity.\n\n\n\n\n\\section{Einstein's first gravitational field equation}\n\nIn a work published in February 1912\\cite{A.E.1912I} Einstein obtained the gravitational\nfield which according to de EP must exist in an accelerated system with sufficient accuracy\nto show that the Laplacian of the corresponding $c$-field must vanish. He did not elaborate\non the properties of that field beyond this result because he felt some uneasiness about the\nfact that that gravitational field was not homogeneous. In a work by Born\\cite{M.B.1909}\nconcerning the motion of rigid bodies in special relativity it have been shown that  in\norder for a body to be moved rigidly, so that  every portion of it retain its size in the\ncomoving systems, different parts of the body  must experience different accelerations.  An\nuniformly accelerated system must have a position dependent acceleration (although at a\ngiven point it is constant in time) so that the gravitational field implied by de EP is not\nhomogeneous. Einstein knew this result in 1912 and its implication that the EP can only be\napplied locally, but he was so  confused by it that he preferred to avoid mentioning it\nexplicitly. Instead, he just said that the field under consideration is of ``some specific\nkind'', implicitly acknowledging that it is not an uniform field, as he had assumed\npreviously.\n\nThe structure of that field may be investigated using the simple methods used by Einstein,\nbut for a modern reader it may be easier to use Rindler \\cite{Rindler}  metric for an\nuniformly accelerated system. Using  Cartesian spatial coordinates and synchronized time\ncoincident with proper time at the origin we have:\n\n\n\\begin{eqnarray}\n&d^{2}s& =g_{00}d^{2}t-d^{2}x-d^{2}y-d^{2}z;\\nonumber \\\\ \n &g_{00}&=c_{o}^{2}\\left(  1+ \t\\dfrac{a_{0}x}{c_{o}^{2}} \\right)^{2}  \\nonumber \\\\\n &c(x)& \\equiv g_{00}^{1\/2} ; \\nonumber \\\\\n&\\vec{g}&=-\\vec{\\nabla}\\phi= c_{0}^{2}\\dfrac{\\partial}{\\partial \\vec{x}} \\ln g_{00}^{1\/2}= - \\dfrac{a_{0}}{1+a_{0}c_{0}^{-2}x}\\hat{i}; \\nonumber \\\\\n&\\phi &=c_{0}^{2}\\ln g_{00}^{1\/2}\\nonumber \\\\\n&c(x)&= c_{0}e^{\\phi\/c_{0}^{2}}.\n\\label{larga}\n\\end{eqnarray} \n\nwhere $\\vec{g}$ is the gravitational field strength (i.e. the force on the unit mass at\nrest) and $\\phi$ the corresponding potential. We have considered an acceleration in the $x$\ndirection ( $\\vec{i}$  being the unit vector in this direction). The relationship between\n$\\phi$  and  $g_{00}$ is a well-known one for static field (Landau-Lifshtz\\cite{Landau}),\nthat could be derived, anyway, from simple considerations. $a_{0}$ is the acceleration at\nthe origin ($x=0$) and $c_{0}$ is the speed of light in vacuum measured with local proper\ntime (i.e. it is a constant). At $x$ the absolute value of the gravitational field strength\nis $a_{0}(1+a_{0}c_{0}^{-2}x)^{-1}$, in particular at $x=-a_{0}^{-1}c_{0}^{2}$ the \ngravitational field goes to infinity and  is obvious that a solidly moving body can not\nextend beyond that point. According to the EP, this result implies that the proper\nacceleration (i.e. that measured in the comoving system) is  $a_{0}(1+a_{0}x)^{-1}$, that\ndepends on $x$. This is the result that stunned Einstein and that is confusing in at least\ntwo respects: first it is the question of how is it possible that the different portions  of\nthe system accelerate at different rates. At any one time an inertial system (with\ncoordinate time given by local proper time synchronized as usual in special relativity) may\nbe found where all portions of the accelerating system are instantaneously at rest. This\nhave been assumed by Einstein in his treatment of the accelerating system, and it seems to\nfollow necessarily from the very definition of a bodily accelerated system. It is actually\ntrue, but it is in apparent  contradiction with different part of the system having\ndifferent accelerations. Secondly, it is  puzzling that different parts of a system can accelerate at different\nrates relatively to the instantaneously comoving inertial system without deformations. The\nanswer to the last question is as follows: assume that two portions of the accelerated\nsystem, the one at $x=0$ and the one at $x=l$, accelerate at the same rate. If at some time\nboth point are at rest with respect  to an inertial system, at time $t$ each of these\nportion would have displaced with respect to this system by exactly the same amount , since\nboth portions experience the same hyperbolic motion (constant proper acceleration).So the\ndistance between both portions as judged from the inertial system would always be  $l$. But\nas $t$ increases the velocity of those portions with respect to the inertial system\nincreases and if in the inertial system the distance between the two portions remains equal\nto $l$, this means, due to Lorentz-Fitzgerald contraction, that in the accelerating system,\nor in the inertial system where the accelerating system is instantaneously at rest, the\ndistance between those portion must increase. We then see that in order for the system to be\n``rigidly '' accelerated, so that the distance between both portion remains constant in it,\nthe trailing portion must experience a somewhat larger acceleration than the leading one, so\nthat as both portion gain speed with respect to an inertial system the distance between them\nas measured in this system show the contraction corresponding to an intrinsically constant\nsize.The first  questions may be answered as follows:if the system is initially at rest with\nrespect to certain inertial one and at some time start to accelerate and after some time has\ngained certain speed, it is possible to find another inertial system in which the\naccelerating system is instantaneously at rest because, although the trailing portion of the\naccelerating system accelerate at a  larger rate, this portion started its motion latter\nthan the leading one as judged from this latter inertial system, due to the difference in\nsimultaneity with respect to the former inertial system.\n\n\nIt  is convenient to discuss these questions because they are interesting in themself and\nbecause it help understanding Einstein reluctance to discuss the global properties of the\ngravitational field under consideration.\n\nWhat he did to avoid these difficulties was to considerer the local transformation from the\ncomoving inertial to the accelerating system around $t=0$, $x=0$ at second order in $t$. He\nfound for small values of $x$ (i.e. $x<<c_{0}^{2}a^{-1}$)\n\n\\begin{equation*}\nc \\simeq c_{0}+\\dfrac{a}{c_{0}}x\n\\end{equation*}\n\n$a$ is, in fact, the proper acceleration at $x=0$ (what we called $a_{0}$ in Eq.\n(\\ref{larga})), but Einstein avoided the question of the dependence of $a$ on $x$. From \nthis result he concluded that  for this field the Laplacian of $c$ vanishes. This conclusion\nseems rigorous, because the same arguments can be given for any value of x. In fact, the\nexact expression for $c$, given in Eq. (\\ref{larga}), has vanishing Laplacian.\n\nHaving studied the gravitational field in an accelerated system he went on to consider the\ngeneral static case. To this end he made the crucial assumption that  the geometry of space\nwas Euclidean so that the situation could be described with just one function, namely, the\nc-field. It is interesting to note that although this preliminary theory is sometime\nreferred to as ``scalar'', $c(x)$ it is in fact a component of a second rank tensor in four\ndimension ($c(x)=g_{00}^{1\/2}(x)$) and, although Einstein did not possess this language at\nthe time, he was well aware of the fact that $c(x)$ it is an scalar only with respect to\npurely spatial transformation. For a general gravitational field Einstein assumed that, as\nfor the accelerated system, the Laplacian of $c(x)$ vanished outside the matter\ndistribution. This may seem a big jump, and Einstein did not comment on its plausibility,\nbut it may be supported by the following considerations: in an accelerated system the\nLaplacian of $c$ is equal to zero even for arbitrarily large fields (as $x$ goes to\n$-a_{o}\/c^{2}_{0}$ the strength of the field goes to infinity). For a genuine gravitational\nfield, we know that in the weak field limit ($\\phi\/c^{2}_{0}\\ll 1$) Newtonian gravitation\nhold, so that we have:\n\n\\begin{equation*}\n\\nabla ^{2} \\phi \\simeq 0;\\quad  \\nabla ^{2}c \\simeq 0\n\\end{equation*}\n\nBy transforming the weak field generated by a distribution of masses to an accelerated\nsystem it could be shown that the resulting field, which is non trivial and non-weak also\nsatisfies that $\\nabla^{2}c=0$.\n\n From a formal point of view one could say that apart from $c$ itself the only other scalar\n(spatial) that can be formed with $c$ is the modulus of its gradient, but the presence of\nany of this two terms on the field equation in vacuum would be in contradiction with the EP,\nsince we would not have he correct field equation in the case of an accelerating system. We\ncan therefore say that, if not rigorously provable, the equation $\\nabla^{2}c=0$ for vacuum \nis at least very highly plausible.\n\nIn the presence of matter it is clear that in the weak field limit we must have:\n\n\\begin{equation}\nc \\simeq c_{0}(1+\\dfrac{\\phi_{0}}{c^{2}_{0}});\\quad \\nabla^{2}c\\simeq\\dfrac{4 \\pi G \\rho}{c_{0}}.\n\\label{5Juan}\n\\end{equation}\n\nwhere $\\rho$ stands for energy density (all through the rest of this work). For the general\nstatic case Einstein proposed the equation:\n\n \\begin{equation}\n\\nabla^{2}c=k c \\rho, \\quad \\textrm{where} \\ k=\\dfrac{4 \\pi G}{c_{0}^{2}}.\n\\label{6Juan}\n\\end{equation}\n\n The  derivation he provided for it is , such as it stands (at least in the English\n version\\cite{A.E.1912I}) incorrect. Einstein asserted that  both sides of Eq. (\\ref{6Juan})\n should be homogeneous in $c$ because ``it follows immediately from the meaning of $c$ that\n $c$ is determined only up to a constant factor that depends on the constitution of the\n clock with which one measures $t$ at the origin of $K $ (the reference system)''. What\n strictly follows from this observation is merely the trivial fact that both sides of Eq.\n (\\ref{6Juan}) must have the same dimensions so that the equation hold for any choice of the\n time unit. However,  Eq. (\\ref{6Juan}) may be inferred correctly from the fact that any\n point can be taken as origin of the system so that at that point coordinate time agrees\n with proper time and, therefore, $c=c_{0}$. The function $c$ is defined up to a constant\n factor not because time units can be chosen arbitrarily (which it is obviously true, but\n lack physical implications) as the quotation say, but because the origin of the system can\n be chosen arbitrarily (which lastly depend on the E P), as the text most probably was\n intended to mean. In fact, interpreting ``the constitution of clock\" as concerning not the\n physical clock at the origin, but the procedure used to stablish the time coordinate (that\n involves choosing an origin), this would just be the meaning of the quotation. Here we see\n a simple instance of the ubiquous and confusing statement that general covariance is an\n important component of GR. In fact, the physical content of this theory lays entirely on\n the EP in its strong version, that concerns also the gravity field equations and not merely\n the matter motion (weak version). General covariance is a  property of the formalism and\n lack physical meaning. What is physically meaningful is the fact that that general\n covariance is displayed by expressions containing only the metric tensor and quantities\n describing matter. The meaningful fact is that  GR equations do not contain elements (i.e.\n fields) not associated with matter or gravity that  could imply that the equations for\n matter and gravity did not have the same form (locally) in all freely falling systems. In\n other words, the content of ``general covariance'' is simply the absence of foreign\n elements in the equations for matter and gravity, which is a fact implied by EP (strong).\n\n In the presents case, as we have said, the relevant fact to prove that Eq. (\\ref{6Juan})\nmust have the stated form (for negligible pressure) it is not the conventional one that this\nequation must hold for any choice of units, but the fact, implied by the EP (strong), that\n$c$  field can be obtain using a coordinate time that may be chosen to agree with the proper\ntime at any point. \n\n In summary, we have seen that the arguments given by Einstein to obtain Eq. (\\ref{6Juan})\nare  rather cogent, which is not the case for the arguments that led him to drop this\nequation, as we shall latter see.\n\n The arguments leading to Eq. (\\ref{6Juan}) are strong enough  to leave almost no doubt\nabout its validity. Now we shall see that this equation followes from the equations of GR\n(Einstein`s equations) when pressure is negligible. These equations can be written in the\nform\\cite{Landau}\n\n\n\\begin{equation}\nR_{ij}=\\dfrac{8 \\pi G}{c_{0}^{4}} \\left(  T_{ij} - \\dfrac{1}{2} g_{ij} T \\right) \n\\label{7Juan}\n\\end{equation}\n\nwhere $R_{ij}$ is the Ricci Tensor (i, j run from 0 to 3 ), $T_{ij}$ is the energy-momentum\ntensor, and T is its trace ($ T\\equiv T^{i}_{i} $). Using now the following expression for\nthe Riemann tensor\\cite{Landau}:\n\n\\begin{equation}\nR_{iklm}= \\dfrac{1}{2} \\left(  \\dfrac{\\partial ^{2}g_{lm}}{\\partial x^{k}x^{l}}+\n\\dfrac{\\partial ^{2}g_{kl}}{\\partial x^{i}x^{m}} -\n\\dfrac{\\partial ^{2}g_{il}}{\\partial x^{k}x^{m}} -\n\\dfrac{\\partial ^{2}g_{km}}{\\partial x^{i}x^{l}} \\right) + g_{np} \\left( \\Gamma_{kl}^{n}\\Gamma_{im}^{p}-\\Gamma_{km}^{n}\\Gamma_{il}^{p}\\right)\n\\label{8Juan} \n\\end{equation}\n\nand choosing locally Cartesian coordinates at any given point  ($g_{\\mu\\nu}=-\\delta_{\\mu \\nu}$;\n$\\dfrac{\\partial g_{\\mu \\nu}}{\\partial x^{\\lambda}}=0$, with $\\mu, \\nu , \\lambda$ from 1 to\n3) we have for $R_{00}$\n\n\\begin{eqnarray}\nR_{00}=g^{il}R_{i0l0}=-\\dfrac{1}{2} g^{il} \\left( \\dfrac{\\partial ^{2}g_{00}}{\\partial x_{i}x_{l}}\\right)+g_{np}g^{il} \\left(\\Gamma^{n}_{0l}\\Gamma^{p}_{i0}-\\Gamma^{n}_{00}\\Gamma^{p}_{il}\\right)=\\nonumber \\\\ \n=-\\dfrac{1}{2}g^{ii}\\dfrac{\\partial^{2}g_{00}}{\\partial x^{2}_{i}}+g_{nn}g^{ii} \\left(\\Gamma^{n}_{0i}\\Gamma^{p}_{i0}-\\Gamma^{n}_{00}\\Gamma^{p}_{ii}\\right)\n\\label{9Juan} \n\\end{eqnarray}\n\nwhere we have used the fact that $g_{ij}=0$  for  $i\\neq j$. From the definition of\n$\\Gamma$'s we have:\n\n\\begin{equation*}\n\\Gamma^{\\mu}_{0 \\nu}=\\Gamma^{0}_{\\mu \\nu}=0;\\quad \\Gamma^{0}_{\\mu 0}=\\Gamma^{0}_{0 \\mu}= \\dfrac{1}{2}g^{00}\\dfrac{\\partial g_{00}}{\\partial x^{\\mu}} \n\\end{equation*}\n\nwhere use has been made of the static character of the metric (i.e. $\\dfrac{\\partial\ng_{ij}}{\\partial x^{0}}=0$). We then have for $R_{00}$:\n\n\\begin{equation}\nR_{00}=-\\dfrac{1}{2}g^{\\mu \\mu }\\dfrac{\\partial^{2}g_{00}}{\\partial x^{2}_{\\mu}}+g^{\\mu \\mu}g_{00} \\left(\\Gamma^{0}_{\\mu 0}\\right)^{2}.\n\\label{Juan10}\n\\end{equation}\n\nBut in the locally Cartesian spatial coordinates that we are using the first term is simply\none half of the Laplacian of $g_{00}$ and the second:\n\n\n\n\\begin{equation*}\ng^{\\mu \\mu}g_{00} \\left(\\Gamma^{0}_{\\mu 0}\\right)^{2}=-\\dfrac{1}{4g_{00}}\\left(\\dfrac{\\partial g_{00}}{\\mid g_{\\mu \\mu } \\mid^{1\/2}\\partial x^{\\mu}}\\right) ^{2}.\n\\end{equation*}\n\nwhere we have use the fact that $g^{\\mu \\mu}=\\left(g_{\\mu \\mu} \\right)^{-1}$ (note that this\nexpression is valid for any orthogonal system), is simply $-1\/(4g_{00})$ times the square of\nthe gradient (this involves a definition of gradient which is not the most widely used, but\nit has the most direct meaning and is the one used through this work). In the usual\nthree-dimensional notation:\n\n\n\\begin{equation}\nR_{00}=\\nabla ^{2}g_{00}- \\dfrac{1}{4g_{00}}\\vert \\vec{\\nabla}g_{00}\\vert ^{2}\n\\label{11Juan}\n\\end{equation}\n\nUsing the relationship:\n\n\\begin{equation*}\n\\nabla ^{2}f(\\phi)=\\dfrac{df}{d\\phi}\\nabla ^{2}\\phi +\\dfrac{d^{2}f}{d\\phi^{2}}\\vert\\vec{\\nabla \t\\phi}\\vert ^ {2}.\n\\end{equation*}\n\n\nWe finally obtain for $R_{00}$\n\n\\begin{equation}\nR_{00}=g_{00}^{1\/2}\\nabla ^{2}g^{1\/2}_{00}.\n\\label{12Juan}\n\\end{equation}\n\nWe have carried out this derivation in a locally Cartesian system, but, since $R_{00}$  is a\nscalar for spatial transformations, it is clear that Eq. (\\ref{12Juan}) must hold in any\nsystem of spatial coordinates.\n\nAssuming that the source is a perfect fluid with energy density $\\rho$ and pressure $P$, we\nhave for $T$:\n\n\\begin{equation*}\nT=\\left(\\rho - 3P\\right).\n\\end{equation*}\n\nIf the fluid was not perfect we would only need to change $P$ by its average value in the\nthree principal direction. We then have:\n\n\\begin{equation*}\nT_{00}-\\dfrac{1}{2}g_{00}T=g_{00} \\rho -\\dfrac{1}{2}g_{00} \\left(\\rho -3P \\right)=\\dfrac{g_{00}}{2}\\left(\\rho +3P \\right).\n\\end{equation*}\n\nInserting this and de Eq. (\\ref{12Juan}) into Eq. (\\ref{7Juan}), we find:\n\n\\begin{equation}\n\\nabla ^{2} g_{00}^{1\/2}=\\dfrac{4\\pi G}{c_{0}^{4}} g^{1\/2}_{00}\\left(\\rho +3P \\right)\n\\label{13Juan}\n\\end{equation}\n\nNoting that Einstein $c$ field is just $g_{00}^{1\/2}$, we see that in the case of negligible\npressure this equation reduces to Eq. (\\ref{6Juan}), that had been obtained by Einstein from\nsimple considerations. The pressure term is related to the delicate issue of the\ngravitational effect of pressure. We shall latter comment on its origin in GR and on the\nfact that it doesn't seem possible to derive it from simple consideration concerning static\nfields.\n\nIt is interesting to note that if a $\\lambda$ term was considered, that is, if we added to\nthe left hand side of Eq. (\\ref{7Juan}) a term of the form $\\lambda g_{ij}$ ($\\lambda$ being\na constant), in Eq. (\\ref{13Juan}) we should have $\\rho - \\dfrac{\\lambda c_{0}^{2}}{4 \\pi\nG}$ in place of $\\rho$.\n\nWe may summarize this section by saying that the arguments given by Einstein in\n1912\\cite{A.E.1912I} to derive  Eq. (\\ref{6Juan}) were quite solid and that, as a matter of\nfact, this equation is valid in GR when pressure is negligible, as it is apparent by\ncomparing Eq. (\\ref{6Juan}) with the exact equation for $g_{00}$ in an static field in GR,\ngiven by Eq. (\\ref{13Juan}). This equation is quite meaningfull both because it is\ninstrumental in the explanation of several basics facts, as we shall show, and because of\nits close relationship with an equation that played a role in the development of GR (Eq.\n(\\ref{6Juan})).The fact that this equation appear neither on chapters on statics fields in\nthe best known textbook, nor in the books known to us on the historic and conceptual\ndevelopment of GR is quite surprising. This is still more difficult to understand noting\nthat it is a well-known result\\cite{S.Ron}  that Eq.(\\ref{12Juan}) holds for a metric where\n$g_{00}$ it is the only non trivial coefficients. As we have seen, this could have been\neasily generalized to any static metric and with it obtain Eq. (\\ref{13Juan}).\n\n\n\n\n\\section{Spherically symmetric field: derivation of Schwarszchild metric}\n\nShortly after obtaining Eq. (\\ref{6Juan}) and then rejecting it, Einstein started\nquestioning the assumption that in a gravitational field space remains flat. After some \ngroping in the dark he came to the conclusion that the spatial metric must be integrated\ninto a common structure with the ``proper'' gravitational field, $g_{00}$. This structure is\nthe four dimensional tensor $g_{ij}$, that in a sense represent the gravitational field in\nGR. Therefore, in this theory there are ten different ``gravitational potentials'', namely,\nthe ten algebraically independent component of a symmetric second rank tensor in four\ndimension, and to determine them ten equations are needed. This can not be achieved if the\nfield source was only the matter density; a tensorial source is needed. It is then obvious\nthat this source must be the energy momentum tensor, $T_{ij}$ and the ten needed equation\nare those given in Eq. (\\ref{7Juan}).\n\nThe ``proper'' gravitational potential is still given by $g_{00}$ (more precisely\n$c_{0}^{2}\/2 \\ln g_{00}$), in the sense that the gravitational field strength,$\\vec{g}$, is\ngiven by minus its gradient. But this quantity only give us the force that must be exerted\nupon the unit of mass to keep it at a fixed position within the field, while all $g_{ij}$\nare needed to obtain particles trajectories. Furthermore, even if we were only interested in\n$g_{00}$, to obtain it by integrating Eq. (\\ref{6Juan})(or rather, its exact counterpart Eq.\n(\\ref{13Juan})) the spatial metric coefficients, $g_{\\mu \\nu}$, are needed, since they enter\nin the explicit expression for the Laplacian. Thus, Eq.(\\ref{6Juan}), although it is exact\n(for negligible pressure) in any static field it is not of much use without integrating it\nsimultaneously with all the other equation entering in Eq. (\\ref{7Juan}).\n\nFor spherically symmetric fields, however, we shall show that Eq. (\\ref{6Juan}) determines\nuniquely the whole metric (with a simple additional assumption in the non-vacuum case).\nChoosing appropriate polar coordinates, a metric with spherical symmetry can be reduced to\nthe form:\n\n\\begin{equation}\nd^{2}s=g_{00} d^{2}t-g_{rr}d^{2}r-r^{2} \\left( d^{2}\\theta + \\sin^{2} d^{2}\\phi \\right)\n\\label{14Juan}\n\\end{equation}\n\nwhere $\\theta , \\phi$ are the usual polar angles and $r$ is a radial coordinate defined so\nthat the  area of a sphere of radius $r$ is $4 \\pi r^{2}$. There are only two undetermined\nmetric coefficients, $g_{00}$ and $g_{rr}$, the others being predetermined by symmetry\nand the choice of coordinate. \n\nIf we could find a relationship between this two coefficients, we could use Eq.\n(\\ref{13Juan}) to completely determine the metric. We shall now see how to obtain a\nrelationship between $g_{00}$ and $g_{rr}$ in the vacuum case (Schwarzschild metric) by\nmeans of the following argument:\n\nConsider the flat time-space (i.e. Minkowskian) and the usual Galilean reference system,\n$K$, within it. Now consider another system, $K^{\\prime}$, that can be described as a\ncontinuous set of locally Galilean systems,  $k^{\\prime}(\\vec{x})$, so that at each point ,\n$\\vec{x}$, (Cartesian coordinates with respect to $K$), the local system\n$k^{\\prime}(\\vec{x})$ at time $t_{0}$  is moving  with respect to $K$ radially from the\norigin with speed $V(r)$ ($r\\equiv \\vert\\vec{x} \\vert$, $\\vec{V}(\\vec{x})$ is parallel to\n$\\vec{x}$).Let us study the spatial geometry in system $K^{\\prime}$. In system $K$ space is\nflat, so the length of a circumference with radius $r$ in, let us say, plane $z=0$ and\ncentred at the origin is simply $2 \\pi r$. In system $K^{\\prime}$ the length measurements\nare carried out with rods comoving with the local Galilean system $k^{\\prime}(\\vec{x})$. To\nmeasure the length of the circumference this rods must be set at point $\\vec{x}$\nperpendicularly to the velocity of $k^{\\prime}(\\vec{x})$ with respect to $K$, therefore,\nrods measuring  the circumference have the same length in $K$ and in $K^{\\prime}$ and,\nconsequently, the length of the circumference must also be $2\\pi r$ in this last system.\nHowever the radius of the circumference in $K^{\\prime}$ is now longer than $r$, because in\nmeasuring it the rods must be parallel to the velocity of $k^{\\prime}(\\vec{x})$ with respect\nto $K$ so that in this system those rods are affected by the Lorenz-Fitzgerald contraction.\nTherefore, more rods are necessary to cover the radius in $K^{\\prime}$ than in $K$.More\nprecisely ,the proper radius, $\\chi$, in $K^{\\prime}$, due to the contraction is given by:\n\n\\begin{equation}\n\\chi (r)= \\int^{r}_{0} \\dfrac{dr^\\prime}{\\sqrt{1-\\left(\\dfrac{V(r^{\\prime})}{c_{0}}\\right)^{2}}}.\n\\label{15Juan}\n\\end{equation}\n\nFrom the differential relationship between $\\chi$ and $r$ we have immediately:\n\n\n\n\\begin{equation*}\ng_{rr}(r)= \\left[1-\\left (\\dfrac{V(r)}{c_{0}}\\right)^{2} \\right]^{-1\/2}.\n\\end{equation*}\n\nOn the other hand, the relationship between proper time, $\\tau$, and the time coordinate in\nsystem $K$, $t$, is given by:\n\n\\begin{equation*}\nd\\tau \\equiv \\dfrac{g_{00}^{1\/2}}{c_{0}} dt= \\left[1-\\left (\\dfrac{V(r)}{c_{0}}\\right)^{2} \\right]^{1\/2}dt.\n\\end{equation*}\n\nSo we have:\n\n\\begin{equation}\ng_{rr}=c_{0}^{2}g_{00}^{-1}.\n\\label{16Juan}\n\\end{equation}\n\n\n\n\nIt must be noted that the time coordinate in system $K^\\prime$ is also $t$, the Galilean\ntime in $K$. So, by the geometry  of the space in system $K^\\prime$ we mean the metric of\nhypersurface $t=t_{0}$. This metric is not constant, because as the local systems move away\nfrom the origin the relationship between  $V$ and $r$ changes. However, to our purposes it\nwill suffice to obtain the metric at $t_{0}$; a metric that we will compare with the static\nmetric that we are going to discuss next.\n\n In the initial formulation of the EP the effect of a constant gravitational field are\nconsider equivalent to the non-inertial effects in an uniformly accelerated system. In this\nsystem, although different parts of it are in some sense at rest with respect to each other,\nwhen a particle, for example  a photon, move from A to B, by the time the photon arrives at\nB the inertial system  in which B is instantaneously at rest is moving with respect to that\nin which A was at rest at the emission time. This fact determines the metric in the\naccelerated system ($g_{00}$ being the only non-trivial coefficient). In a similar manner,\none way  interpret the spherically symmetric metric in vacuum as the metric in $K^{\\prime}$\nat $t_{0}$, which is not constant, forced to become constant by the presence of the\ngravitational field, which in the process turns space-time from Minkowskian  to a curved\none. It must be noted that for this interpretation to work $V(r)$ must be equal to the\nvelocity attained by a particle that start falling at infinity with zero velocity by the\ntime it reach radii $r$, although this is immaterial with respect to relationship\n(\\ref{16Juan}).\n\nIn  system $K^{\\prime}$, although the metric is instantaneously equal at time $t_{0}$ to\nthat in the gravitational field, the redshift experienced by light when moving within the\nsystem it is not equal to that for the gravitational field (not even for infinitesimal time\nof flight), because this redshift depends not just on the metric in $K^{\\prime}$ at $t_{0}$,\nbut  also on its evolution. For example, for the redshift experienced between $r$ (at\n$t_{0}$) and infinity in $K^{\\prime}$ we have (with $c_{0}=1$):\n\n\\begin{equation*}\n1+\\dfrac{\\Delta \\lambda}{\\lambda}= \\dfrac{1+V(r)}{\\sqrt{1-V(r)^{2}}} \\equiv \\gamma(r) \\left(1+V(r)\\right)\n\\end{equation*}\n\nwhere the factor $1+V(r)$ is the classical Doppler effect associated with the fact that in\n$K^{\\prime}$ the local systems are actually moving. The $\\gamma$ factor correspond to the\ntransversal Doppler effect, associated with the difference between the time at the local\nsystem and that at infinity, which corresponds to the redshift in the gravitational field.\n\nIt is possible to select a ``reference system'' , $K^{\\prime \\prime}$, in Minkowski space\nwhose metric is in closer relationship with that in a gravitational field. The system\n$K^{\\prime \\prime}$ can be conceived as a discrete set of local systems of $K^{\\prime}$ with\nsmall but finite size that for a short period of time, $\\Delta t$, move with velocity $V(r)$\nor $-V(r)$ and then instantaneously change its velocity to  $-V(r)$ or  $V(r)$ . The metric\nin this system and the redshift of light propagating between any two points averaged over\ntimes larger than $\\Delta t$ (the classical Doppler averages out) is the same as in the\ngravitational field.\n\nWhat follows from this interpretation is that in  vacuum the effect of the gravitational\nfield on space and time are both implied by a single quantity (interpreted here as $V(r)$)\nand, therefore, $g_{rr}$ and $g_{00}$ are related, being Eq. (\\ref{16Juan}) the necessary\nform of this relationship. We shall see that this depends on the fact that in vacuum both\n$g_{rr}$ and $g_{00}$ at $r$ depends on the enclosed mass, which is a constant. On the other\nhand, in the general spherically symmetric case, $g_{rr}$ depends on the enclosed mass but\n$g_{00}$ (normalized to the center of the cloud) depends on the  enclosed mass and\npressure-volume  (furthermore, even without pressure $g_{rr}$ depends just on $M(r)$(see Eq.\n(\\ref{Juan21})), while $g_{00}$ depends on $M(r^{\\prime})$ for $r^{\\prime} \\leq r$) and no\ngeneral relationship between $g_{rr}$ and $g_{00}$ can exist, as we shall see by showing\nthat the contrary assumption leads to inconsistencies. In the vacuum case, however, Eq.\n(\\ref{16Juan})must  hold for any value of the cosmological constant and in the case of a\ncharged mass ``point'' (REF Reissner-Nordstron metric).\n\n\nNote that for this relationship to be valid it is implicit that coordinate time have been\nchosen so that $g_{00}$ goes to one at infinity.\n\n\nUsing Eq. (\\ref{16Juan}) in Eq. (\\ref{6Juan}), for vacuum (with $\\lambda =0$), we may\nimmediately derive the whole metric Eq. (\\ref{14Juan}), which has now been reduced to just\none independent quantity, $g_{00}$:\n\n\\begin{equation}\n\\nabla ^{2} c \\equiv \\nabla ^{2}g_{00}^{1\/2}=\\dfrac{1}{r^{2}g_{rr}^{1\/2}} \\dfrac{\\partial}{\\partial r} \\left( \\dfrac{r^{2}}{g_{rr}^{1\/2}} \\dfrac{\\partial g_{00}^{1\/2}}{\\partial r}\\right)=\\dfrac{1}{r^{2}} \\dfrac{g_{00}^{1\/2}}{c_{0}^{2}} \\dfrac{\\partial}{\\partial r}\\left(r^{2}g_{00}^{1\/2} \\dfrac{\\partial g_{00}^{1\/2}}{\\partial r}\\right)=0.\n\\label{17Juan}\n\\end{equation}\n\nwhere the first equality come from the  expression for Laplacian in spherical coordinates\nfor a quantity, $g_{00}^{1\/2}$, that  is spherically symmetric, and the last comes from\nusing Eq. (4).\n\nA first integration of Eq. (\\ref{17Juan}) give:\n\n\\begin{equation*}\nr^{2}g_{00}^{1\/2}\\dfrac{\\partial g_{00}^{1\/2}}{\\partial r}= r^{2}\\dfrac{1}{2}\\dfrac{\\partial g_{00}}{\\partial r}=A\n\\end{equation*}\nwhere $A$ is a constant. Integrating again:\n\\begin{equation*}\ng_{00}=-\\dfrac{2A}{r} +B.\n\\end{equation*}\n\nWith the usual convention of taking the coordinate time to agree with proper time at\ninfinity (which is necessary for Eq. (\\ref{16Juan}) to hold), we must have $B$ equal to\n$c_{0}^{2}$.\n\nAt large values of $r$ the Newtonian limit is valid, so:\n\n\\begin{equation*}\ng_{00}(r)\\simeq c_{0}^{2} \\left( 1-\\dfrac{2 \\phi}{c_{0}^{2}} \\right) =c_{0}^{2}\\left( 1-\\dfrac{2GM}{c_{0}^{2}r}\\right).\n\\end{equation*}\n\nWe must then have $A=GM$, so that we have for $g_{00}$, $g_{rr}$:\n\n\\begin{equation}\ng_{00} =c_{0}^{2}\\left( 1-\\dfrac{2GM}{c_{0}^{2}r}\\right); \\quad g_{rr} =\\dfrac{c_{0}^{2}}{g_{00}}\n\\label{18Juan}\n\\end{equation}\nwhich are the well-known values of these coefficients for Schwarzschild metric.\n\nWe have said before that the quantity that can be called the ``gravitational potential'' in\nGR is  $\\dfrac{c_{0}^{2}}{2} \\ln g_{00}$, in the sense that minus its gradient is equal to\nthe force on a static unit of mass. To study the difference of this potential with respect\nto the Newtonian counterpart, it is covenient to express $\\vec{g}$ in the following form:\n\n\n\\begin{equation}\n\\vec{g}=-\\dfrac{\\partial \\ln g_{00}^{1\/2}}{\\partial \\vec{x}}=-\\dfrac{1}{g_{rr}^{1\/2}}\\dfrac{\\partial}{\\partial r}\\ln g_{00}^{1\/2}\\vec{e}_{r}=c_{0}\\dfrac{\\partial g_{00}^{1\/2}}{\\partial r}\\vec{e}_{r}\n\\label{19Juan}\n\\end{equation}\n\nwhere $\\vec{e}_{r}$ is the unit vector in the $r$ direction and where Eq. (\\ref{16Juan}) has\nbeen used. We see from Eq. (\\ref{17Juan}) that if $g_{rr}$ was not affected by the\ngravitational field, remaining equal to 1, we would obtain:\n\n\\begin{equation*}\ng_{00}^{1\/2}=c_{0}\\left( 1 - \\dfrac{GM}{c_{0}^{2}r}\\right)\n\\end{equation*}\n\nand $\\vec{g}$ would be given by the Newtonian expression. It is then clear that what causes\n$\\vec{g}$ to differ from the Newtonian case is the fact that the gravitational field affect\nthe value of $g_{rr}$, causing it to be , in the present case, equal to\n$g_{00}^{-1}c_{0}^{2}$.\n\nFor the vacuum case with non-zero cosmological constant, we need using Eq. (\\ref{13Juan}) , that, noting that a $\\lambda$ term can formally be treated like a homogeneous fluid with equation of state $p=- \\rho$, takes the form:\n\\begin{equation*}\n\\nabla^{2}g_{00}^{1\/2}=-g_{00}^{1\/2}\\lambda\n\\end{equation*}\n\nUsing again relationship (\\ref{16Juan}), we find:\n\\begin{equation*}\n\\dfrac{\\partial}{\\partial r} \\left(  r^{2} \\dfrac{\\partial g_{00}}{\\partial r} \\right)= -2 \\lambda c_{0}^{2}r^{2}.\n\\end{equation*}\n\nIntegrating this equation with the same conditions as in $\\lambda =0$ case, one obtains immediately:\n\\begin{equation*}\ng_{00}=c_{0}^{2}\\left(  1- \\dfrac{2MG}{c_{0}^{2}r} - \\dfrac{\\lambda r^{2}}{3}\\right), \\quad g_{rr}=c_{0}^{2}g_{00}^{-1}\n\\end{equation*}\n\nwhere  $M$ is a constant. This is the solution for the metric of a ``point'' mass in a\nnon-zero $\\lambda$ vacuum, that was first obtained by Eddington in 1923\\cite{Eddington}.\n\nThe Reissner-Nordstrom metric can be derived in an equally simple manner as we shall show in\na future work.\n\nAssuming in the general spherical case that  relationship (\\ref{16Juan}) holds, leads, together with Eq. (\\ref{13Juan}), amongst other inconsistencies, to the fact that the mass of the distribution (for bounded distributions) depends on the enclosed mass and pressure-volume, that as we shall show in next section it is in contradiction with basics principles. It is therefore clear that this relationship it is not valid, a result that could have been anticipated by more direct (but more involved) argument.\n\n\n\nTo obtain the metric in the general case we shall use the results obtained in the vacuum\ncase together with the assumption that to obtain the field at $r$ only the matter\ndistribution up to $r$ is needed. This is a consequence of Birkhoff Theorem (REFERENCIA),\nderived from G.R. This fact can be proved using Eq. (\\ref{13Juan}) and some general\nconsiderations, but we will simply take it here as a plausible assumption .This assumption,\nthat is simply the supposition that the full gravitational theory will also satisfy Newton's\niron sphere theorem, it is so natural that a derivation containing it can still be\nconsidered to be based on basic principles. With the mentioned assumption and using Eq.\n(\\ref{18Juan}) it is clear that we must have for $g_{rr}$:\n\n\\begin{equation}\ng_{rr}(r)=\\left(1-\\dfrac{2GM(r)}{c_{0}^{2}r}\\right)^{-1}\n\\label{Juan20}\n\\end{equation}\n\nwhere $M(r)$ is the mass enclosed within $r$.$M(r_{0})$ is defined so that if there were no\nmatter beyond $r_{0}$, the asymptotic Newtonian potential would be:\n\n\\begin{equation*}\n\\dfrac{GM(r_{0})}{r}.\n\\end{equation*}\n\nIn next section we shall show from basics considerations that $M(r)$ is given by:\n\n\\begin{equation}\nM(r)=\\dfrac{4 \\pi}{c_{0}^{2}}\\int^{r}_{0} \\rho(r^{\\prime})r^{\\prime 2}dr^{\\prime}\n\\label{Juan21}\n\\end{equation}\nwhere $\\rho$ is the energy density.Using Eq. (\\ref{Juan21}) in Eq. (\\ref{Juan20}) the known result for $g_{rr}$ is recovered\\cite{Landau2}. Inserting this expression for $g_{rr}$ in Eq. (\\ref{13Juan}) we obtain an equation for $g_{00}$\n\\begin{equation}\n\\dfrac{1}{g_{rr}^{1\/2}r^{2}}\\dfrac{\\partial}{\\partial r} \\left( \\dfrac{r^{2}}{g_{rr}^{1\/2}} \\dfrac{\\partial g_{00}^{1\/2}}{\\partial r} \\right)=\\dfrac{4 \\pi G}{c_{0}^{4}}\\left( \\rho + 3P \\right)g_{00}^{1\/2}\n\\label{Juan22}\n\\end{equation}\nwith the conditions:\n\n\\begin{equation*}\n\\dfrac{\\partial g_{00}^{1\/2}}{\\partial r}\\mid_{r=0}=0; \\quad \\quad g_{00}(\\infty)=c_{0}^{2}\n\\end{equation*}\n\nHowever, rather than integrating this equation, we shall obtain the solution at the end of\nnext section by a more physicaly meaningful procedure. \n\nOnce we have the metric, the trajectories of free falling particles can be obtained using\nthe geodesic equation in space-time for this metric, which is the standard procedure in\ntextbook on GR. However, to end this section, we shall indicate how this can be done in a\nsimple manner and in much closer keeping with classical mechanics. \n\nGR implies very small corrections for planetary orbits and we think it is instructive to\ndescribe the effects by mean of small correcting terms added to the Newtonian equations and\nexplain their origin. To say that in GR gravity is not a force, that  planets merely follow\nspace-time geodesics and, in consequence, use a qualitatively different treatment to deal\nwith quantitatively very small effects, does not seem to us the most convenient\npresentation.\n\nIn fact, the very description of gravity as the space-time geometry is an adventitious\ninterpretation and not at the essence of Einstein's theory, as pointed out by\nWeinberg\\cite{Weinberg}. This interpretation may be interesting and compelling for those who\nalready know the theory, but to use it to convey to the layman the meaning of GR does not\nseems the most expedient way.\n\nThe Lagragian for a particle in a static gravitational field may be written in the form:\n\n\\begin{equation}\nL=-m \\dfrac{ds}{dt}=-mc_{o}^{2}\\sqrt{g_{00}-\\left(\\dfrac{dl}{dt}\\right)^{2}};\n\\label{Juan23}\n\\end{equation}\n\\begin{equation*}\nd^{2}l\\equiv g_{\\mu \\nu}dx^{\\mu}dx^{\\mu}\n\\end{equation*}\n\nwhere $s$ is the invariant relativistic interval and $l$ is the arc length ($\\mu , \\nu$ run\nfrom 1 to 3).\n\nIf space was Euclidean, one can easily show that the equation of motion derived from Eq.\n(\\ref{Juan23}) would be:\n\n\n\\begin{eqnarray}\n&\\dfrac{d\\vec{v}}{dt}&=  \\left( -\\dfrac{1}{2}\\vec{\\nabla}g_{00} \\right)_{\\perp}  +\\left( -\\dfrac{1}{2}\\vec{\\nabla}g_{00} \\right)_{\\parallel} \\left( 1-2\\left( \\dfrac{v}{c}\\right)^{2} \\right)=-\\dfrac{1}{2}\\vec{\\nabla}g_{00}+\\dfrac{1}{c^{2}}\\left( \\vec{v} \\cdot \\vec{\\nabla}g_{00}  \\right) \\vec{v}; \\\\\n &\\vec{v}& \\equiv \\dfrac{d \\vec{x}}{dt} \\nonumber\n\\label{Juan24}\n\\end{eqnarray}\n\nwhere the suffixes $\\perp$, $\\parallel$  denotes respectively the components perpendicular\nand parallel to $\\vec{v}$. In this expression both $\\vec{v}$ and its derivative are with\nrespect to coordinate time $t$. Using the local  proper time,  $\\tau$ (not to be confused\nwith particle proper time), we have:\n\n\\begin{equation*}\n\\dfrac{d\\vec{v}}{d\\tau}= \\vec{g}_{\\perp} +\\left(1-\\left( \\dfrac{v}{c_{0}}\\right)^{2}\\right)\\vec{g}_{\\parallel}\n \\end{equation*}\n\n\\begin{equation}\n\\vec{g}= -\\dfrac{c_{0}^{2}}{2g_{00}}\\vec{\\nabla}g_{00};\\qquad \\vec{v}\\equiv\\dfrac{d\\vec{x}}{d\\tau}\n\\label{Juan25}\n \\end{equation}\n \nwhere $\\vec{g}$ is the ``gravitational field'' (i.e. the force on  a unit mass at rest).\nThis equations can be obtained immediately using the fact that in a locally inertial system\n(i.e. free falling) the acceleration vanishes . Therefore, both Eq. (22) and Eq.\n(\\ref{Juan25}) merely express the simple (non-Riemannian) EP, which implies a velocity\ndependent gravitational force. Note that from Eq. (23) it is obvious that $v$ will always\nremain smaller than $c_{0}$, as it must be when time $\\tau$ is used. However Eq. (22) do not\nimply  such restriction, because in time $t$ $v$ can be larger than $c_{0}$, although it\nhave to be smaller than $c(\\vec{x})$. \n\nIn general, space it is not Euclidean, and an extra term must be included in Eq.\n(\\ref{Juan24}). In the spherically symmetric case we have: \n\n\\begin{equation}\n\\left(\\dfrac{dl}{dt}\\right)^{2}=\\left(\\dfrac{d\\chi}{dt}\\right)^{2}+\\left(r\\dfrac{d\\theta}{dt}\\right)^{2},\n\\label{Juan26}\n\\end{equation}\n\\begin{equation*}\nd\\chi \\equiv g_{rr}^{1\/2}dr\n\\end{equation*}\n\nwhere, since it is obvious that orbits must remain in a plane, we are considering only two\ncoordinates $\\chi$ (or $r$) and $\\theta$, that determine the position within that plane.\n\nUsing Eq. (\\ref{Juan26}) in Eq. (\\ref{Juan23}) one obtains the corresponding Lagrangian from\nwhich the equations of evolution for $\\chi$, $\\theta$ can be obtained in the standard\nmanner. But  it is simpler and more enlightening to recall the classical derivation and\nremark the differences. Classically we may write: \n\n\\begin{equation}\n\\dfrac{d\\vec{v}}{dt}=\\dfrac{d}{dt}\\left( \\dot{r}\\vec{e_{r}}+r\\dot{\\theta}\\vec{e_{\\theta}}\\right)=\\vec{F}\n\\label{Juan27}\n\\end{equation}\n\nwhere $\\vec{e_{r}}$, $\\vec{e_{\\theta}}$, are the unit vector along the radial and tangential\ndirections, and $\\vec{F}$ is the total force per unit mass, that in the Newtonian case is\ngiven simply by $\\vec{g}$. Using the following relationship:\n\n \n \\begin{equation}\n \\dot{\\vec{e_{r}}}=\\dot{\\theta}\\vec{e_{\\theta}}; \\quad \\dot{\\vec{e_{\\theta}}}=-\\dot{\\theta}\\vec{e_{r}},\n\\label{Juan28}\n\\end{equation}\nwhere the dot denotes derivation with respect to $t$, we have:\n\\begin{equation*}\n\\left( \\ddot{r} - r \\dot{\\theta}^{2}\\right)\\vec{e_{r}}+\\dfrac{ \\dfrac{d}{dt}\\left( r^{2} \\dot{\\theta}\\right)}{r}\\vec{e_{\\theta}}=\\vec{g}.\n\\end{equation*} \n\nTo obtain the evolution equations in the present case we only need to note that the\nunderlying reasons for Eq. (\\ref{Juan28}) are the relationships satisfied in the Euclidean\nplane. The relationships satisfied in a non-Euclidean plane with radial symmetry around\n$r=0$ can be obtained by equally simple geometrical considerations:\n\n\n\\begin{equation*}\n \\dot{\\vec{e_{r}}}=\\left(\\sin \\rho \\right)\\dot{\\theta}\\vec{e_{\\theta}}; \\quad \\dot{\\vec{e_{\\theta}}}=- \\left( \\sin \\rho\\right) \\dot{\\theta}\\vec{e_{r}}.\n\\end{equation*}\n\n\\begin{equation}\n\\sin \\rho \\equiv \\dfrac{dr}{d\\chi}=g_{rr}^{-1\/2}.\n\\label{Juan29}\n\\end{equation}\n\n\n Representing the actual non-Euclidean plane as an axialy symmetric curved plane embedded in\nEuclidean space and being tangent to the plane $z=0$ at the origin, $\\rho$ is the angle\nbetween the radial direction within the curved plane and the $z$ axis.\n\nIn the present case it also holds that the derivative of $\\vec{v}$ with respect to\ncoordinate time $t$ is equal to the total force per unit mass, $\\vec{F}$, but now $\\vec{v}$\nis given by:\n\n\\begin{equation*}\n\\vec{v}= \\dot{\\chi}\\vec{e_{r}}+ r \\dot{\\theta}\\vec{e_{\\theta}},\n\\end{equation*}\n\nand $\\vec{F}$ is given by the left hand side of Eq. (22). Using \t Eq. (\\ref{Juan29})\nfor the derivative of the base vector, we finally find\n\n\n\\begin{equation}\n\\left( \\ddot{\\chi} - g_{rr}^{-1\/2} r \\dot{\\theta}^{2}\\right)\\vec{e_{r}}+\\left[\\dot{r}\\dot{\\theta}+ \\dfrac{d}{dt}\\left( r \\dot{\\theta} \\right) \\right]\\vec{e_{\\theta}}=-\\dfrac{1}{2}\\vec{\\nabla}g_{00}+\\dfrac{1}{g_{00}}\\left(\\vec{\\nabla}g_{00} \\cdot \\vec{v}\\right)\\vec{v},\n\\label{Juan30}\n\\end{equation}\n\\begin{equation*}\n\\vec{v}\\equiv \\dot{\\chi}\\vec{e_{r}}+r\\dot{\\theta}\\vec{e_{\\theta}}.\n\\end{equation*}\n\n\nFrom the radial and the tangential part of these equation we obtain respectively:\n\\begin{equation*}\ng_{rr}^{-1\/2} \\dfrac{d}{dt} \\left( g_{rr}^{3\/2}\\dot{r}\\right)-r^{2}\\dot{\\theta^{2}}=-\\dfrac{1}{2}\\dfrac{\\partial g_{00}}{\\partial r}; \\quad \\quad \\dfrac{d}{dt} \\left( \\dfrac{r^{2}\\dot{\\theta}}{g_{00}}\\right)=0\n\\end{equation*}\n\nwhere Eq. (\\ref{16Juan}) has been used, so, at variance with Eq. (28), that is valid for any\nspherically symmetric metric, the first of these equations is only valid for vacuum metrics.\nThese equation can be integrated exactly, but we will only discuss simple cases that can\neasily be treated approximately in a more transparent manner.\n\n\n\nFor almost circular orbits, neglecting terms proportional to $\\dot{r}$, we have:\n\n\\begin{equation}\ng_{rr}^{-1\/2}\\ddot{r}-g_{rr}^{-1\/2} r\\dot{\\theta}^{2}=-\\dfrac{1}{2}\\vec{\\nabla}g_{00}\n\\label{Juan31}\n\\end{equation}\n\n\n\\begin{equation*}\n\\dfrac{d\\left(\\dfrac{r^{2}\\dot{\\theta}}{g_{00}}\\right)}{dt}=0  \n\\end{equation*}\n\nThe differences with respect to the  classical case are three fold: First there is the fact\nthat the force per unit mass at rest, $\\vec{g}$,  is not exactly equal to the Newtonian one.\nThen we have the velocity dependence of the gravitational force and, finally we have the\nnon-Euclidean character of space, which is fully described by the function $r(\\chi)$.\nHowever, even when $\\vec{g}$ (related to the acceleration in time $\\tau$) differs from the \nNewtonian one, what appears in the right hand side of Eq. (\\ref{Juan30}) (which correspond\nto time $t$) is exactly equal to the Newtonian case, on the other hand, as we have said, for\nalmost circular orbits the radial velocity dependence is negligible,but not the tangential\none, that renders the precensece of $g_{00}$ in the last of Eq. (29).So, the difference with\nrespect to the Newtonian case are due to the velocity dependence of $\\vec{F}$ and to the\nnon-Euclideanity of space.\n\n The precesion of the perihelion of almost circular orbit can easely be obtained.\nIntegrating the last of Eq. (29) and inserting it in the first, we find:\n\n\\begin{equation*}\n\\ddot{r}=g_{rr}^{-1} \\left( r \\dot{\\theta}^{2}- \\dfrac{1}{2}\\dfrac{\\partial}{\\partial r}g_{00}\\right)=g_{rr}^{-1}\\left( \\dfrac{J^{2}g_{00}^{2}}{r^{3}}-\\dfrac{GM}{r^{2}}\\right)\n\\end{equation*}\n\nwhere $J$ is an integration constant. For a small displacement, $\\Delta r$, from  the value\nof $r$ at  which the last parentheses vanishes, $r_{0}$, we have:\n\n\\begin{equation}\n\\ddot{r}\\simeq -g_{rr}^{-1} \\left(\\dfrac{GM}{r_{0}^{3}}\\left( 1-\\dfrac{4GM}{r_{0}c_{0}^{2}}\\right) \\right)\\Delta r = -\\omega_{r}^{2}\\Delta r\n\\label{primera}\n\\end{equation}\n\nwhere $\\omega_{r}$ is the angular frequency of small radial oscillations. For the angular\nmotion we have:\n\n\\begin{equation*}\n\\omega_{\\theta} \\equiv \\dot{\\theta}= \\left( \\dfrac{GM}{r^{3}_{0}}\\right)^{1\/2}.\n\\end{equation*}\n\nTherefore, the angular frequency of the perihelion, $\\Omega$, is given by:\n\\begin{equation}\n\\Omega= \\omega_{r}-\\omega_{\\theta} \\simeq \\dfrac{3GM}{r_{0}c_{0}^{2}}\\omega_{\\theta}.\n\\label{segunda}\n\\end{equation}\n\nIt must be noticed that when the orbit has finite eccentricity, $r_{0}$ is not the mean\nradius. From its definition:\n\n\\begin{equation*}\nr_{0}=\\dfrac{J^{2}}{GM}=a\\left(1-e^{2} \\right)\n\\end{equation*}\n\nwhere $a$ is the semi-major axis and $e$ is the eccentricity. For finite radial oscilations\nhigher order terms on $\\Delta r$ must be included in Eq. (31), but we know that without the\nrelativistic corrections $\\omega_{r}$ and $\\omega_{\\theta}$ (that for general orbits is\ngiven by the above expression with $r_{0}=a$) remain equall. Therefore, with this value of\n$r_{0}$, the above expression for $\\Omega$ is exact to first order in the potential. Notice\nthat a negative value of $\\Omega$ means that the perihelion moves in the direction of the\nrevolution. From the above computation it is clear that $2\/3$ of $\\Omega$ are due to the\nvelocity dependence of $\\vec{F}$, and this is the value of $\\Omega$ corresponding to\nEinsteins's 1912 theory. The remaining third comes from the presence of $g_{rr}$ in Eq.\n(\\ref{primera}), obviously related to the non-Euclidianity of space. It is interesting to\nnote that in Nordstrom theory, where space-time is comformally flat (therefore, $g_{rr}$ is\nthe inverse of that for GR), the spatial effect has the opposite sign to that for GR, while\nthe effect of the velocity dependence of $\\vec{F}$ is the same. However, in that theory, the\ngravitational potential do not enter Eq. (28) as in the Newtonian theory. The sum of all\neffects render a value o $\\Omega$ that is in magnitude $7\/3$ of that for GR, bu with the\nopposite sign.\n\n\n\nThe deflection of light rays can also be easily computed. From the derivation of Eq.\n(\\ref{Juan30}) it is clear that in quasi-Cartesian coordinates and to first order  on $G$ we\nhave:\n\n\\begin{equation}\n\\dfrac{d \\vec{V}}{dt}= -\\dfrac{1}{2}\\vec{\\nabla}g_{00}+ \\dfrac{1}{2}\\left(\\vec{\\nabla}g_{00}\\right)_{\\parallel} - \\left[  \\dot{\\theta} \\dot{\\chi}\\left( g_{00}^{-1\/2}-1 \\right)\\vec{e_{\\theta}}-r \\dot{\\theta^{2}} \\left(  g_{00}^{-1\/2}-1\\right)\\vec {e_{r}} \\right]\n\\label{Juan?}\n\\end{equation}\n\nwhere we have used the fact that for light $v=c$ and where the last parentheses comes from\nthe fact that  replacing Eq. (\\ref{Juan28}) by  (\\ref{Juan29}) is equivalent  to adding the\napparent force given by the parentheses while retaining the Euclidean affinity structure. We\nhave expressed the parentheses in polar coordinates because it is simpler to derive, but now\nmust be transformed to Cartesians. Considering a ray with impact parameter $r_{0}$ and\ntaking the coordinate system so that the ray moves in the $x$, $y$ plane along the straight\nline (almost) $y=r_{0}$, we then have:\n\n\\begin{equation*}\n\\dfrac{d V_{y}}{dt}=-\\dfrac{GM}{r^2}\\dfrac{r_{0}}{r} -\\dfrac{GM}{r^2}\\dfrac{r_{0}}{r}\n\\end{equation*}\n\nwhere the first term on the right hand side corresponds to the homologous term in Eq.\n(\\ref{Juan?}) and the last one corresponds to the parenthesis. The second term in Eq.\n(\\ref{Juan?}) does not contribute because it goes in the x direction. Taking $t=0$ at the\nmoment of maximum approach and assuming that light traces a straight line at constant speed,\n$c_{0}$ (zeroth order on $G$), it is obvious that the deflection,  $\\Delta \\phi$, is\ngiven by:\n\n\\begin{equation*}\n\\Delta \\phi= \\dfrac{2}{c_{0}} \\int^{\\infty}_{must be0} \\dfrac{dV_{y}}{d t}dt=\\dfrac{2}{c_{0}}\\int^{\\infty}_{0} \\dfrac{2GM}{(r_{0}^{2}+ c_{0}^{2}t^{2})^{3\/2}}dt=\\dfrac{4GM}{r_{0}c_{0}^{2}}\n\\end{equation*}\n\nWhich is the well-known result, that is usually  derived in a rather different manner. We\nhave seen that one half of the effect comes from the non-Riemannian EP (first term of the\nright in Eq. (\\ref{Juan?})) while the other half comes from the non-Euclideanity of space,\n(last parentheses in Eq. (\\ref{Juan?}).\n\n\n\\section{Historic overtones and other topics}\n \n \nWe have said before that the argument that led Einstein to reject Eq. (\\ref{6Juan}) was\nincorrect. The argument\\cite{A.Enstein1} is based in the demonstration of the \nnon-conservation the momentum of a distribution of particle interacting through a\ngravitational field satisfying Eq. (\\ref{6Juan}) and following the dynamic that he had\npreviously derived\\cite{A.Enstein2}. Met with this contradiction Einstein chose to reject\nEq. (\\ref{6Juan}), but the problem lied not with this equation, that we have seen that is\ncorrect (for negligible pressure), but with the assumption that space can remain flat in a\ngravitational field. We have mentioned this fact before; here we shall explicitly show it in\na simple manner. To this end, instead of the general case, we shall consider the case of two\ninteracting point masses. Furthermore, we assume that both masses are instantaneously at\nrest. To obtain the acceleration of the masses we may use Eq. (\\ref{Juan24}). This\nexpression does not fully incorporate non-Euclidianity, since Eq. (\\ref{Juan28}) rather than\nEq. (\\ref{Juan29}) are  implicit, but this is irrelevant for motion in a radial direction.\n\n\\begin{equation}\n\\dfrac{d\\vec{v}}{dt}=-\\dfrac{1}{2}\\vec{\\nabla}g_{00} \\quad \\quad for \\quad \\vec{v}=0\n\\label{32Juan}\n\\end{equation}\n\nNow from Lagrangian Eq. (\\ref{Juan23}) it is clear that the linear momentum is given by:\n\n\n\\begin{equation*}\n\\vec{p}=\\frac{mc_{0}\\vec{v}}{c \\sqrt{1-( \\dfrac{v}{c})^{2}}}.\n\\end{equation*}\n\nThis agrees with the expression derived by Einstein\\cite{A.Enstein2}. It is then clear that\nthe change of momentum of mass 1 and 2 are respectively:\n\n\\begin{equation*}\n\\dfrac{d\\vec{p}_{1}}{dt}=\\dfrac{m_{1}c_{0}}{c_{1}}\\dfrac{d\\vec{v}_{1}}{dt}=-\\dfrac{m_{1}c_{0}}{2c_{1}}\\vec{\\nabla}c_{1}^{2}; \t\\quad \\dfrac{\\vec{dp}_{2}}{dt}=-\\dfrac{m_{2}c_{0}}{2c_{2}}\\vec{\\nabla}c_{2}^{2}\n\\end{equation*}\n\nwhere $c_{1}$, $c_{2}$ are the values of the $c$ field ($\\equiv g_{00}^{1\/2}$) at mass 1 and\nmass 2 respectively.  For the gradient we have:\n\n\\begin{equation*}\n\\vec{\\nabla}c^{2}_{1}=\\dfrac{1}{g^{1\/2}_{rr_{1}}}\\dfrac{dc_{1}^{2}}{dr}\\vec{e}_{r}; \\quad \\vec{\\nabla}c_{2}^{2}=-\\dfrac{1}{g_{rr_{2}}^{1\/2}}\\dfrac{dc^{2}_{2}}{dr} \\vec{e}_{r},\n\\end{equation*}\n\\begin{equation*}\nr \\equiv \\vert \\vec{r}_{2}-\\vec{r}_{1} \\vert\n\\end{equation*}\n\nwhere $\\vec{e}_{r}$ is a unitary vector in the direction of $\\vec{r}$ ($\\equiv \n\\vec{r}_{2}-\\vec{r}_{1}$). Using Eq. (\\ref{18Juan}) for $c$ ($\\equiv g_{00}^{1\/2}$) and\n$g_{rr}$ ($= c^{2}_{0}\/g_{00}$), we find:\n\n\\begin{equation*}\n\\dfrac{d\\vec{p}_{1}}{dt}=m_{1}\\dfrac{dc_{1}^{2}}{dr} \\vec{e}_{r}=\\dfrac{m_{1}m_{2}G}{r^{2}}\\vec{e}_{r}; \t\\quad \\dfrac{d\\vec{p}_{2}}{dt}=-\\dfrac{m_{1}m_{2}G}{r^{2}}\\vec{e}_{r}.\n\\end{equation*}\n\nThe change of the total momentum, therefore vanishes. The analysis of this simple example\nmakes it clear what was wrong with Einstein`s 1912 theory. In the above expression $\\vec{r}$\nis the actual ``metric'' velocity (i.e. $v=\\dfrac{dl}{dt}$, $l$ being the length of arc). In\npolar coordinates in the plane of motion:\n\n\\begin{equation*}\nv=\\sqrt{g_{rr}\\dot{r}^{2}+ \\left(r \\dot{\\theta} \\right)^{2}}\n\\end{equation*}  \n\n\nIn  Einstein's theory space was flat, so in the present case ($\\dot{\\theta}=0$) $v$ is\nidentical to $\\dot{r}$. But in this case instead of Eq. (\\ref{32Juan}) we would have\n\n\\begin{equation*}\ng_{rr}^{1\/2} \\dfrac{d \\vec{v}_{E}}{dt}=-\\dfrac{1}{2}\\vec{\\nabla}g_{00}\n\\end{equation*}\n\nwhere $\\vec{v_{E}}$ is $\\vec{v}$ in Einstein theory (i.e. $\\vert \\vec{v}\\vert =\n\\dot{r}$).Then we would have:\n\n\\begin{equation*}\n\\dfrac{d\\vec{p}_{1}}{dt} \\mid_{E}=\\dfrac{1}{g_{rr_{1}}^{1\/2}}\\dfrac{c_{0}m_{1}m_{2}G}{r^{2}}\\vec{e}_{r}\n\\end{equation*}\n\nand a similar expression for mass 2. This is clearly incompatible with momentum\nconservation.\n\nAs we have seen, confronted with this contradiction Einstein chose to retain the\nEuclidianity of space and to modify Eq. (\\ref{6Juan}) so as to make it compatible with\nmomentum conservation (and Euclidianity) although it was then incompatible with the strict  \n(i.e. for any field strength) EP.\n\nWhen sometime latter he realized that space could not remain Euclidean in a gravitational\nfield, he retook the EP, that  he had rejected very reluctantly, and started a new path\nthrough differential geometry. No discussion and revision of the previous work at the new\nlight can be found in the literature. This may seem quite reasonable: since the\nnon-Euclidianity of space in a gravitational field turns the spatial metric coefficients,\n$g_{\\mu \\nu}$, into some sort of gravitational potentials, obtaining an equation for just\n$g_{00}$ does not look very interesting. But before going after the full theory Einstein\ncould as well have clarified the situation with respect to Eq. (\\ref{6Juan})  and see how\nfar he could  have gone with it in some simple cases. In fact, as we have shown here, if he\nhad followed that course and realized of the argument relating $g_{rr}$ and $g_{00}$ in the\nvacuum case, Schwarzschild's metric and its consequences would have been around since 1912.\nThe mentioned argument is akind to  the one given by Einstein for showing the\nnon-Euclidianity of space in a rotating system, only that somewhat more sophisticated\nbecause it involves comparing spatial geometries in flat space-time and curved space-time.\nThus, we think that this could have been a realistic course of events. Had this been the\ncase, not only would have been available several of the most relevant results of GR for\nastrophysics since that early date, but substantial insight would have been added to the\nfull theory when it became available,  by showing its direct connection with basic principle\nthrough the analysis of  simple cases. Furthermore, we think that, after realizing that\nspace could not remain flat in a gravitational field, Einstein initial project of dealing\nwith the static and stationary case could have been fulfilled. It is true that the task then\nwould has exceeded that of the original Einstein intentions, since equations  must be obtain\nto determine all metric coefficients, not just $g_{00}$. However, that task would have been\nstill much smaller than obtaining the full G.R. equations and, again, would have contributed\nsubstantial insight into it. We have not so far pursued that course, but we now briefly\ndescribe here how we think it could be done.\n\n\n We have said that Einstein found that Eq. (\\ref{6Juan}) can not be derived from a\nvariational principle (so that energy-momentum conservation is not guaranteed), but we know\nthat that equation is correct (for negligible pressure) and, of course, momentum is\nconserved. How are this two fact made compatible by a curved space?. Once it is recognized\nthat the gravitational field affect the space metric, the corresponding metric coefficients\nbecome dynamical quantities and, therefore, must be included in the  Lagrangian for matter\nand the gravitational field. The additional term in the Lagrangian must be invariant under\nspatial coordinate transformation, so it must  be formed out of $c$ (that is constant under\nspatial transformations) and the Ricci scalar of the spatial metric, $R$. Simple\nconsiderations lead to a term of the form $ARc$, with $A$ a constant. This constant must be\nchosen so that Eq. (\\ref{6Juan}) is obtained throug variation of the Lagrangian with respect\nto $c$. Variation with respect to spatial metric coefficients  should provide the other\nequations needed. On the other hand, from the consideration of the flat space-time metric in\na rotating system\\cite{Landau3} it may readily be shown that the term :\n\n\\begin{equation*}\n-2\\dfrac{\\Omega^{2}}{c^{3}}\n\\end{equation*}\n\nwhere $\\Omega$ stand for the rotation speed, must be added to the right hand side of Eq.\n(\\ref{6Juan}) in the presence of rotation (i.e., in a non globally synchronizable system).\nNow, the rotation velocity $\\vec{\\Omega}$ is related to the coefficients $g_{0 \\mu}$ ($\\mu$\nfrom 1 to 3) by\\cite{Landau4}\n\n\\begin{equation}\n\\vec{\\Omega}= \\dfrac{c_{0}}{2} curl \\vec{g}; \\quad \\quad \\vec{g}\\vert_{\\mu}\\equiv - \\dfrac{g_{0\\mu}}{c^{2}_{0}}.\n\\label{33Juan}\n\\end{equation}\n\nInferring the term that must be added to the Lagrangian to obtain the modified Eq.\n(\\ref{6Juan}) and variating the Lagrangian  with respect to all metric coefficients, the ten\nneeded equations could be obtained.\n\nThis line of development might have difficulties, in particular, it must be noted that the\nspatial metric coefficients enter the Lagrangian through derivatives of up to the second\norder ( this is true in GR for all metric coefficients). However, it seem, in principle, a\nreasonable path to follow towards GR and a clarifying analysis once the theory is available.\nIf this approach worked, space-time geometrical formalism would only be needed to deal with\ntime dependent gravitational fields.\n\nWe have seen that Einstein obtained Eq. (\\ref{6Juan}) from simple considerations and that GR\nleads to Eq. (\\ref{13Juan}) which agrees with Eq. (\\ref{6Juan}) only when pressure  is\nnegligible. The question now is what was missing in the arguments leading to Eq.\n(\\ref{6Juan}) and whether Eq. (\\ref{13Juan}) could be derived through some simple\nconsideration that provides that missing element. This question is related to the\ninteresting and very delicate issue of the gravitational effect of pressure and given its\nrelevance for this work we consider it convenient to pay some attention to it.\n\nThe source of the gravitational field in GR is the energy-momentum tensor, $T_{ij}$ (with\n$i,j$ from 0 to 3), whose purely spatial components corresponds to pressure (the stress\ntensor).It is  then no surprise that pressure affect the metric tensor, $g_{ij}$, which are\nin some sense a set of ten gravitational potentials (they are all needed to determine\nparticle trajectories). However, as we stated before,  the coefficient $g_{00}$ is the one\nthat provides the force on a unit mass at rest, so it is in close relationship with the\nNewtonian potential and the fact that pressure contribute to its sources, as seen in Eq.\n(\\ref{13Juan}), seems rather buffling. In one of his 1912 papers\\cite{A.Enstein1}, Einstein\ndiscussed the possibility  of the gravitational action of stresses (i.e. its ability to\nweight in a given gravitational field) and rejected it with an argument that involved the EP\nand the equivalence of mass and energy. He seems to have assumed that what he has proved\ncorrectly for the passive coupling of matter to gravity was also true for the active\ncoupling (i.e. the ability of matter to generate a gravitational field). In the Newtonian\ntheory the superposition principle holds and this implies that momentum conservation can\nonly be satisfied if it holds for any couple of interacting elements (i.e. mass points or\nvolume elements, for extended systems), that implies the action and reaction law. In this\ncase it is clear that the active and passive coupling must be equal. In the present case,\nhowever, the superposition principle does not hold and, therefore, the equality of action\nand reactions is not satisfied. This explains that we may have a passive coupling\nproportional to $\\rho$ while the active coupling is proportional to $\\rho +3P$. But for\nstable and bounded objects interacting at large distances, so that the superposition\nprinciple holds, Einstein's argument applies also to the active coupling, implying a net\ngravitational effect proportional to the total energy of those objects.\n\n\n Einstein considered a box filled with radiation, which is an stable bounded system and,\ntherefore, as he had shown, its gravitational effect at distances much larger than its size\ndepends solely on the sum of the masses in the box (for a small box i.e. negligible binding\nenergy). But, having concluded correctly that the net gravitational effect of stresses has\nto vanish for this system, he arrived at the wrong conclusion (al least implicity) that the\ngravitational effect of stresses must vanish (which is exactly true only for its passive\nrole), because he only considered  the stresses in the containing walls and not the negative\nstresses within it, associated with the radiation pressure. In fact, Einstein argument, that\nwe shall see later in more detail, is compatible with stresses contributing to $g_{00}$,\nbecause in any stable bound system positive an negative stresses cancel each other. In self\ngravitating systems with positive pressure the stabilizing stresses are provided by the\ngravitational field (the energy-momentum tensor of gravity is another delicates issue, but\nwe can not address it here).\n\nKnowing  that there is not a clear argument to exclude pressure from  contributing to the\ngravitational field strength, the question is whether a simple and direct argument can be\ngiven to derive Eq. (\\ref{13Juan}). The answer to this question could be that, after all,\nthe simplest version of GR (with only linear term on the Ricci scalar on the  Lagrangian and\nwithout cosmological term) follows entirely from the strong EP and Eq. (\\ref{13Juan})\nfollowes from it. Thus, the train of thought leading to the presence of the term $3P$ in the\nEq. (\\ref{13Juan}) could be considered to be the argument asked for. Because of its\ninterest, we shall review that train of thought. However this scarcely can be considered a\ndirect argument like  that leading to Eq. (\\ref{6Juan}), because it goes all the way through\nGR. We know of a simple argument showing the gravitational effect of pressure, but it\ninvolves a non-static field and we prefer to present it in a future work. Einstein, on the\nother hand, had no reason in 1912 to search for that argument, and, to our knowledge, it was\nonly on completing GR that he realized of the gravitational effect of pressure in the sense\ndiscussed here.\n\nTo understand the origin of the $3P$ term in Eq. (\\ref{13Juan}) it is interesting to note\nthat  in a preliminary theory, where the field equations were:\n\n\\begin{equation}\nR_{ij}=\\dfrac{8 \\pi G}{c_{0}^{2}}T_{ij}\n\\label{34Juan}\n\\end{equation}\n\nthe equation for $g_{00}$ in the static field would be given by Eq. (\\ref{6Juan}) rather\nthan by Eq. (\\ref{13Juan}). Therefore, it is clear that the key to the questions that we are\nanalysing lies with the origin of the extra term in the right hand side of Eq.\n(\\ref{7Juan}). This equation is equivalent to:\n\n\n\n\\begin{equation}\nR_{ij}^{*}\\equiv R_{ij}-\\dfrac{1}{2}g_{ij}R=\\dfrac{8 \\pi G}{c_{0}^{2}}T_{ij}.\n\\label{35Juan}\n\\end{equation}\n\nOur question has now been reduced to explaining why it is $R_{ij}^{*}$ rather than $R_{ij}$\nthat must appear in the left hand side of the field equations. This equations follows\noutright by variating the simplest GR Lagrangian, but in order to see the connection of our\nquestion with physical principles we shall follow the usual explanation, although with a\nsomewhat different presentation.\n\n Both the metric coefficients and the components of the energy-momentum tensor are sets of\nten algebraically independent quantities. However, since the choice of coordinates is\narbitrary and fixing a coordinate system involves four independent functions (the\ncoordinates conditions), only six of the $g_{ij}$ and $T_{ij}$ are functionally independent.\nThat is, once the coordinates have been fixed, all possible energy-momentum distributions\ncan be described by six independent functions. Any sets of ten functions can describe an a\npriori possible $T_{ij}$ distribution, but it would correspond to certain intrinsic\ndistribution as represented in certain coordinate system. When the latter has been fixed,\nnot all set of ten functions are possible $T_{ij}$ distribution. In other words, the\nintrinsic structural possibilities of both $g_{ij}$ and $T_{ij}$ are span by sets of six\nindependent functions. The same applies to $R_{ij}$, which can be expressed in term of the\n$g_{ij}$. In consequence, we should have just six quantities intrinsically characterizing\n$R_{ij}$ and $T_{ij}$, but since on any given system of coordinates there are ten $R_{ij}$\nand $T_{ij}$, the field equations must be ten equations, although only six must be\nfunctionally independent. However, since the $g_{ij}$ and $T_{ij}$ are different structures\n(in fact $T_{ij}$ is not even specified in terms of the constituting field), the only way in\nwhich the number of independent equation can be reduced to six is by having the side of the\nequation corresponding to gravity, which is geometrical in character and explicitly\nspecified in terms of its constituting fields ($g_{ij}$), satisfying four identities. But\nfrom Bianchi's identities it is known that:\n\n\\begin{equation*}\nR_{\\quad j;i}^{*i}=0.\n\\end{equation*}\n\n$R^{*}_{ij}$ must then be the ``geometrical'' side of the field equations, as shown in Eq.\n(\\ref{35Juan}). Taking the four divergence on both sides of Eq. (\\ref{35Juan}) we\nimmediately have:\n\n\\begin{equation}\nT_{\\quad j;i}^{i}=0\n\\label{36Juan}\n\\end{equation}\n\nthat expresses energy-momentum conservation, that appears now as a direct consequence of\ngeneral covariance (i.e. the laws of physics has the same form in all coordinated\nsystems).We remind, however, that general covariance has no physical content, which is\nprovided by the fact that general covariance,  a conventional requirement that should be met\nby any theory, can be achieved with just matter and gravity, without foreign elements, a\nfact  that follows from the strong EP. If this principle did not hold and certain additional\nfields caused the laws of physics (matter and gravity) not to be the same in all free\nfalling systems, the argument given above would also be valid, but instead of Eq.\n(\\ref{36Juan}) we would have the divergence of $T_{ij}$ plus a tensor build up  from the\nforeign fields set equal to zero. In this case, the conservation that followed from general\ncovariance (that  holds by construction) would not imply the local  conservation of\nenergy-momentum (ordinary) alone.\n\nIn some works one can see the above argument shortened  to saying that, given the empirical\nfact described by Eq. (\\ref{36Juan}), the left hand side of the field equations must have\nidentically vanishing divergence, so it must be $R^{*}_{ij}$. This is a complete reversal \nof the logic of the argument given here, in which the fact of having $R^{*}_{ij}$ on the\nleft hand side of Eq. (\\ref{35Juan}) followed from general covariance, while energy-momentum\nconservation follows from Eq. (\\ref{35Juan}), and it is not a valid reasoning. If energy\n-momentum conservation were merely an empirical  fact or consequence of another symmetry\n(i.e. other than general covariance), Eq. (\\ref{34Juan}) could be valid. We should then \nhave:\n\n\\begin{equation*}\nR^{i}_{\\quad j;i}=0\n\\end{equation*}\n\nwhich is not an identity, but there would be no logical impediment to it. Eq. (35),\nalthouggh it satisfies general covariance, it does not follow from a variational principle\nand, therefore, no conservation law is implied in general by that property. There is a local\nconservation of matter energy-momentuml, that was already there, but not an ordinary\nconservation of ``total'' energy-momentum (including gravity). In fact, it was demanding\nthat this be the case (which follows automaticaly from a variatiional principle) that\nEinstein obtained Eq. (34)\n\nWe have seen that the $3P$ term in Eq. (\\ref{13Juan}) is a  consequence of general\ncovariance, or rather, of the fact, implied by the strong EP, that it can be achieved with\nonly matter and gravity. Now we shall use Einstein argument showing that only mass\ngravitates (valid for stable bounded objects) to obtain an expression for the mass of an\nstatic and bounded spherical distribution of energy-momentum. \n\nConsiderer an static and bounded distribution of mass and pressure. It is clear that at\nlarge enough distances the gravitational potential takes the form:\n\n\\begin{equation}\n\\phi \\sim \\dfrac{GM}{r}.\n\\label{37Juan}\n\\end{equation}\n\nThe constant $M$ we call the mass of the distribution. By considering the acceleration\nexperienced by the whole system within a uniform gravitational field and using the EP\nEinstein showed that  the gravitational mass of the system,$M$, must be equal to the\ninertial mass, that is, it is the latter that couples passively to the gravitational field\n($g_{00}$) and the same is true for the active coupling when computing weak fields. Note\nthat this argument does not apply to a portion of the system. A volume element, $dV$,\ncouples actively to $g_{00}$ not through its mass, but through its mass plus $3P$ times\n$dV$, as we discussed earlier. Now, as pointed out by Einstein, it is clear that the total\nenergy and momentum of the system must depend on the velocity of its center of mass like a\nmass point in special relativity. This is true even when the latter theory does no hold in a\nregion around the system where the field is sufficiently strong and it  can be proved\nrigorously by considering energy-momentum conservation for a set of systems that interact \nweakly (the field on any system due to all other systems being weak) without changing their\ninner structure. From this fact, the proportionality between mass (inertial) and total\nenergy follows immediately. In consequence, $M$ in Eq. (\\ref{37Juan}) must be given by the\ntotal energy of the system. It would be tempting to write:\n\n\n\\begin{equation}\nM=\\int \\rho dV +\\dfrac{1}{2 c_{0}^{2}}\\int \\left(\\dfrac{g_{00}^{1\/2}}{c_{0}}-1 \\right) \\rho dV\n\\label{38Juan}\n\\end{equation}\n\nwhere the second term is the gravitational binding energy.However, writing the binding\nenergy as one half of the total gravitational energy (sum of the gravitational energy of all\nmass elements) is only an approximation, since the presence of $c$ on the right hand side of\nEq. (\\ref{13Juan})(or Eq. (\\ref{6Juan})) makes it clear that the superposition principle does\nnot hold (in consequence, the energy of mass $i$ due to mass $j$ is not symmetric in $i$,\n$j$). But we shall see now that in the case of a spherically symmetric cloud $M(r)$ can be\nobtained exactly using the vacuum case solution. This expression for $M(r)$ can be used to\nobtain the full solution ($g_{00}$ and $g_{rr}$) in the non.homogeneous spherical case, as\nwe have shown in the previous section. If matter and pressure distributions beyond radii $r$\ndo not affect the solution within $r$, the value of $g_{rr}$ at $r$ must be given by Eq.\n(18)\n\n\\begin{equation*}\ng_{rr}(r)= \\left( 1- \\dfrac{2GM(r)}{c_{0}^{2}r}\\right)^{-1}\n\\end{equation*}\n\nwith $M(r)$ the mass enclosed within $r$. The value of $g_{00}(r)$ when normalized so that\n$g_{00}(0)=c_{0}^{2}$ is also unafected by matter and pressure outside $r$, but when, as\nusual, is normalized so that $g_{00}(\\infty)=c_{0}^{2}$, there is a dependence on outside\nmatter.To obtain $M(r)$, we shall assume that there is no matter beyond $r$ (which would be\nirrelevant under our assumption) and compute the total work, $W$, that must be exerted on\nthe  cloud to disperse it  by taking layer after layer  to infinity (obviously, $W$ is\nnumerically equal to the binding energy). $W$ is clearly given by:\n\n\\begin{equation}\nW(r)= - \\int^{r}_{0} \\left(\\dfrac{g_{00}(r^{\\prime })^{1\/2}}{c_{0}}-1\\right)\\rho(r^{\\prime})4 \\pi r^{\\prime 2}g_{rr}^{1\/2}(r^{\\prime})dr^{\\prime}\n\\label{39Juan}\t \n\\end{equation}\n\nwhere $g_{00}^{1\/2}(r)\/c_{0}$ is the total energy of a unit mass at $r$, including the rest\nmass, while $g_{00}^{1\/2}(r)\/c_{0}-1$ is its gravitational energy. We have seen that $M(r)$\nmust be equal to the total energy (divided by $c_{0}^{2}$), which, in turn, must be equal to\nthe energy of the infinitly dispersed cloud minus $W$. But the former is simply the rest\nenergy of the cloud:\n\n\\begin{equation}\n\\int^{r}_{0} \\rho(r^{\\prime})4 \\pi r^{\\prime 2} g_{rr}^{1\/2}(r^{\\prime})dr.\n\\label{40Juan}\n\\end{equation}\n\n\nSubtrating Eq. (\\ref{39Juan}) from this quatity (and divigding by $c_{0}^{2})$, we have for\n$M(r)$:\n\n\\begin{equation}\nM(r)= \\dfrac{1}{c_{0}^{2}}\\int^{r}_{0}g_{00}^{1\/2}(r^{\\prime })\\rho(r^{\\prime})\n4 \\pi r^{\\prime 2}g_{rr}^{1\/2}(r^{\\prime})dr^{\\prime}.\n\\label{41Juan}\n\\end{equation}\n\nIt must be noted that $g_{00}(r^{\\prime})$,$g_{rr}(r^{\\prime})$ in this expression are not\nthe actual ones within the cloud but the ones that will exist after all layers above\n$r^{\\prime}$ had been removed, that is, the ones corresponding to the vacuum case solution\nwith mass $M(r^{\\prime})$. This is so because layer $r^{\\prime}$ is carried to infinity not\nthrough the actual field but through the field generated by $M(r^{\\prime})$. But for the\nvacuum case solution the product: $g_{00}^{1\/2} g_{rr}^{1\/2}\/c_{0}$, is equal to one. Thus,\nwe must have:\n\n\\begin{equation}\nM(r)=\\dfrac{1}{c_{0}^{2}}\\int^{r}_{0} \\rho 4 \\pi r^{\\prime 2} dr^{\\prime}\n\\label{42Juan}\n\\end{equation}\n\nwhich agrees which the result obtained within the full GR\\cite{Landau5}. \n\nIt must be noted that Eq. (\\ref{42Juan}) it is not the sum of the masses, which is given by\nEq. (\\ref{40Juan}). It is the sum of the mass elements multiplied by their corresponding\nvalues of $g_{00}^{1\/2}(r^{\\prime})$. Note also that it is not the sum of the total energies\nof the mass elements, because $g_{00}$ is not the actual one within the cloud extending up\nto $r$, but the vacuum solution with mass $M(r^{\\prime})$.\n\nIt must be remarked that the meaning of $M(r)$ is the mass that must enter the vacuum\nsolution that would exist if all matter beyond $r$ was removed and that it must be equal to\nthe total energy within $r$. But for the latter to be true, the remaining system (the matter\nwithin $r$) must be stable. This can only be achieved by enclosing the system in an\nspherical  container with radius $r$ to withstand the pressure, $P(r)$, after the outside\nmaterial, has been removed. It is only in this circumstances that there will be a vacuum\nsolution beyond $r$ with $M$ given by Eq. (\\ref{41Juan}). The question now is: does the\npresence  of the  container affect the solution at $r$? If the presence of the container,\nthat is massless and arbitrarily thin, was immaterial, $g_{00}(r)$ (normalized to the center\nof the cloud), $g_{rr}(r)$, must be equal to the corresponding value in the actual solution\n(before removing the outer matter), because the outer matter does no affect them. But we\nknow that this can not be, because otherwise $g_{00}$ would not depend on pressure, in\ncontradiction with  Eq. (11).\n\n\nOne may discard  the possibility of  $g_{rr}$ depending  on the enclosed pressure-volume by\nthe following consideration: take an arbitrarily rigid spherically symmetric solid. It is\npossible to redistribute the tension (remaining in equilibrium) increasing it in the outer\nparts and diminishing it in the inner ones spending an arbitrarily small amount of energy.\nThis can be done, for example, by heating the outer layers, since for given thermal\nproperties the energy needed diminishes as the elastic modulus increases. But if  $g_{rr}$\ndepended on the enclosed pressure-volume, its value in the inner parts would change, because\nof the outwardly transferred pressure-volume. This would imply an arbitrarily large increase\nof the elastic energy not compensated by a diminishing gravitational energy (which increases\nslightly due to the small expansion of the body).This is in contradiction with energy\nconservation, although the reader may also note another inconsistency hidden in this\nargument that is independent of energy conservation.\n\nConsequently, $g_{rr}$ must be equal at both sides of the arbitrarily thin container and,\ntherefore, must be given by Eq. (\\ref{Juan20}), that , consistently, tell us that $g_{rr}$ \ndepends only on the enclosed mass. On the other hand, for $g_{00}$   Eq. (\\ref{13Juan})\nshows the relevance of the enclosed pressure-volume.\n\nFrom  Eq. (\\ref{13Juan}) we have:\n\\begin{equation}\n\\vec{\\nabla}g_{00}^{1\/2}(2)-\\vec{\\nabla}g_{00}^{1\/2}(1)=\\dfrac{4 \\pi G}{c_{0}^{4}}\\dfrac{g_{00}^{1\/2}(r) \\int 3 \\bar{P}dV}{r^{2}} \\vec{e}_{r}\n\\label{42}\n\\end{equation}\n\nhere 2, 1 distinguish respectively the solution just  outside and just inside the container\nand where the volume integral is over the walls of the container. $\\bar{P}$ stand for  the\npressure in this wall (in fact a tension) no to be confused with the pressure, $P(r)$,\npositive for usual fluids, which is exerted by the fluid upon the wall. Simple equilibrium \nconsiderations lead to the following  relationship  between the tension, $T$ on surface\nelements whose vectors are within the plane tangent to the container walls and $P$.\n\n\\begin{equation*}\nT= \\dfrac{r P(r)}{2\\Delta r }; \\quad 3\\bar{P}=\\dfrac{rP(r)}{\\Delta r}\n\\end{equation*}\n\nwhere $\\Delta r$  is the physical thickness of the container walls and where the last\nequality follows from the fact  that $\\bar{P}$  is the average of  the pressure in 3\northogonal directions, and in the direction perpendicular to the walls the tension is zero.\nInserting this in  Eq. (\\ref{42}) we find:\n\n\\begin{equation*}\n\\dfrac{1}{g_{rr}^{1\/2}} \\dfrac{\\partial g_{00}^{1\/2}(1)}{\\partial r}=\\dfrac{1}{g_{rr}^{1\/2}} \\dfrac{\\partial g_{00}^{1\/2}(2)}{\\partial r}+ \\dfrac{4 \\pi G}{c_{0}^{4}}g_{00}^{1\/2} r P(r).\n\\end{equation*}\n\nFor $g_{00}$ itself it is no necessary to distinguish between the inner and the outer value,\nbecause it is continuous. It is  convenient to write this equation in the form:\n\n\\begin{equation}\n\\dfrac{1}{2g_{rr}^{1\/2}}\\dfrac{\\partial \\ln g_{00}}{\\partial r}=\\dfrac{1}{2g_{rr}^{1\/2}}\\dfrac{\\partial \\ln \\bar{g}_{00}}{\\partial r}+ \\dfrac{4 \\pi G }{c^{4}_{0}}g_{00}^{1\/2}r P(r)\n\\label{43}\n\\end{equation}\n\nwhich gives the relationship between the inner (actual solution) field strength and the\nouter one given by the vacuum solution, $\\bar{g}_{00}$\n\n\\begin{equation*}\n\\bar{{g}}_{00}(r)=c_{0}^{2} \\left( 1 - \\dfrac{2 GM(r)}{c_{0}^{2}r} \\right).\n\\end{equation*}\n\nIt must be noticed that in derivating $\\bar{g}_{00}$ in Eq. (\\ref{43} ) $M(r)$ mus be hold\nfixed, because there is no matter beyond $r$ in the hypothetical situation that  we are\nconsidering. However, on integrating this expression to obtain the solution for the actual\ndistribution, $M(r)$ is not a constant. We then have for $g_{00}$:\n\n\\begin{equation}\n\\ln g_{00}=- \\dfrac{1}{c_{0}^{2}}\\int_{r}^{\\infty} \\left(\\dfrac{2GM(r^{\\prime})}{r^{\\prime 2}}  + \\dfrac{8 \\pi G r^{\\prime}}{c_{0}^{2}}P(r^{\\prime}) \\right) \\left( 1- \\dfrac{2GM(r^{\\prime})}{c_{0}^{2}r^{\\prime}} \\right)^{-1}dr^{\\prime}\n\\end{equation}\n\nwith the normalization $g_{00}(\\infty)=c_{0}^{2}$. This is equivalent  to the expression\nderived by Weinberg (REF: Weinberg pag 302) with the standard  procedure. In this procedure\nthe equation of hydrostatic equilibrium  is hidden  amongs the three independent equations.\nHere we can derive that equation from direct considerations, obtaining:\n\n\\begin{equation}\n\\dfrac{1}{\\sqrt{g_{00}}}\\dfrac{\\partial P\\sqrt{g_{00}}}{\\partial r}= - \\rho \\dfrac{\\partial \\ln g_{00}^{1\/2}}{\\partial r}\n\\end{equation}\n\nwhere the presence of the factor $\\rho$ is due to the fact that the weight of a volume\nelement in a gravitational field is proportional to  $\\rho$. The presence of $g_{00}$ in the\nleft hand side comes from the fact that the total energy (including gravitational energy)\ntransferred by a unit force  when the body upon which is exerted is displaced by a unit of\nlength it is not one unit of energy, but $g_{00}^{1\/2}\/c_0$.\n\nUsing Eq. (43) in Eq. (45) and if $P$ is a unique function of a second order differential\nequation for $M(r)$ is obtain. Also, for a politropic model ($\\rho=AP^{\\alpha}$, $\\alpha < 1\n$), we can integrate Eq. (45) to obtain a relationship between $P$ or $\\rho$ and $g_{00}$:\n\n\\begin{equation}\nP= \\left(A (g_{00}^{\\frac{\\alpha-1}{2}}-1)\\right)^{\\dfrac{1}{1-\\alpha}}.\n\\end{equation}\n\n\nIt is clear, however, that these models can not correspond exactly to finite mass object:\nfor $\\alpha < 3\/4$ as well as for the case $\\alpha=1$, wich must be treated separately.\n\n\\section{Summary and Conclusions}\n\n\nWe have reviewed the arguments given by Einstein in the derivation of his first , non\nRiemmanian, theory of gravitation (1912). We have weighed carefully these argument and\nstrengthened some points that could look fragile in the original presentation. We concluded\nthat, except for the argument proving that only matter gravitates, these arguments are\ncogent enough as to leave no serious doubt about the validity of Einstein first field\nequation, at least for negligible pressure, and that its ultimate rejection by Einstein was\ndue to a miss-identification of the culprit of the contradiction he was finding. Once we\nconcluded that Einstein first gravity field equation has to be correct (almost), we\nproceeded to demonstrate that this equation follows  from the field equations of GR when\npressure is negligible. In general we find that the source of ``the gravitational field''\n($g_{00}$) is mass density plus three times the pressure, rather than mass density alone. We\nhave shown that with Einstein first field equation (Eq. (\\ref{6Juan})) and a basic argument\nrelating $g_{rr}$ and $g_{00}$ the spherically symmetric vacuum case (called Schwarzschild\nsolution when treated within the full GR) can be obtained immediately both with and without\ncosmological constant, and have pointed out that the metric for a charged mass ``point'' can\neasely be obtained in the same manner. We noted that in the static case the field  equation\nremains formally equal to that in the Newtonian case (Laplacian of the field equal to zero),\nonly that replacing the usual gravitational potential, $\\phi$, by $c$:\n\n\\begin{equation*}\nc \\equiv c_{0} e^{\\phi\/c_{0}^{2}}.\n\\end{equation*}\n\nBut we also noted that there is an important difference between both equations hidden in the\nLaplacian, because while in the Newtonian case, being space flat, $g_{rr}$ is equal to one,\nin the relativistic one $g_{rr}$ depends on $g_{00}$. It have been shown that, although in\nthe case of a general spherically symmetric distribution no algebraic relationship between\n$g_{00}$ and $g_{rr}$ exist, still it may be solved on basic principles. To this end we\nused results obtained in the vacuum case, an expression for the mass enclosed within radii\n$r$ obtained from first principles and the assumption that $g_{rr}(r)$ only depends on\nmatter within $r$. We have written particle trajectories in a form that can easily be\ncompared with its Newtonian counterpart, in order to be able to see in a conceptual manner\n(i.e. not merely studying the differences at various orders in $v$ and $\\phi$) the origins\nof the differences of particles trajectories equations around a ``point'' mass in Newtonian\nand Einstenian theories. We showed that the differences are three fold. First it is the fact\nthat the gravitational field strength  is different from the Newtonian one, a difference\nwhose origin has been explained before. However, it enter in the motion equation in such a\nmanner that the corresponding term agrees exactly with it Newtonian counterpart. Then there\nis a dependence of the gravitational ``force'' on velocity that followes immediately from\nthe EP (non-Riemmanian), with a term proportional to $v^{2}$. This dependence is obviously\ncontained in Einstein's 1912 theory. Finally it is the fact that the geometry of a plane\ncontaining the central point it is not Euclidean. We have shown that this effect can be\nreduced to replacing some simple relationships valid in the Euclidean plane by the\ncorresponding ones in the actual plane, that can be derived from elementary geometrical\nconsiderations. Within this analysis we have computed the angular velocity of the perihelion\nof quasi-Keplerian orbits and found that $2\/3$ of it are due to the velocity dependence\n(i.e. to EP), the other third being due to non-Euclideanity. We have also used the same\napproach for computing the deflection of light rays, showing in a simple manner the well\nknown result that $1\/2$ of the effect is ``Newtonian'' and the other half comes from\nnon-Euclidianity, with the velocity dependent term playing no role.\n\nBy considering the gravitational interaction of two ``point'' masses, we have made patent\nhow the assumption of flat space causes the non-conservation of momentum that plagued\nEinstein first gravitational theory. We have then discussed a possible path of  development\nof GR that could have happened if, after realizing that the flatness of space was untenable,\nEinstein had  removed the error from his first gravitational theory and continued his\ninitial research program.\n\nWe have discussed Einstein arguments showing that pressure can not gravitate and why it\nfails. Then we reviewed the argumental line leading to the presence of pressure in Eq.\n(\\ref{13Juan}), showing that it is a direct consequence of ``general covariance'' and,\ntherefore, closely related to energy-momentum conservation. We have concluded that of all\nquestion treated in this work this is, arguably, the most difficult to have been anticipated\nin 1912 and, although we have thought of an argument proving this fact, we have not\npresented it here because it involves non-static fields. Finally, we have used Einstein\nargument relating gravitational mass and total energy to obtain an expression for the mass\nenclosed within radii $r$ in a general  spherically symmetric distribution, which is\ninstrumental in deriving the metric in this case.This we have done in two manner, first\nusing the expression for $g_{rr}$ obtained with the mentioned argument in Eq. (11) and then\nby extending this argument to obtain $g_{00}$ through direct physical considerations. We\nalso give a direct derivation of the equation of hydrostatic equilibrium and discuss how to\nuse it together with the expression for $g_{00}$ and $P(\\rho)$ to obtain the equilibrium\nconfiguration of a self-gravitating non-rotating cloud. In the standard procedure, using\nEinstein equations (GR), this is obtained in a non very transparent manner.\n\nThe general aim of this work has been to try to understand the most elementary result of GR\nin terms of basics principles. To this end Einstein's 1912 gravity equation has been\ninstrumental along with the EP, energy-momentum conservation and the mass energy\nrelationship. The results that we have derived are not only of heuristic value, which has\nbeen our main goal, but are amongst the most relevant result of GR for astrophysics.\n\nIn all other fields of physics we are used to dealing with trivial cases in a simple manner,\nidentifying the relevant basic principles in a problem to anticipate the aspect of it that\ncan quickly be learned and combining elements of different solutions (not necessarily\nsuperposing them) to gain some insight into the actual problem. In GR, however, this usual\nheuristic is almost entirely lacking. Confronted with a particular problem it is the\ngeometrical symmetry that determine its treatment and then we switch on our elegant\nmathematical formalism to obtain the solution, gaining little insight in the process. Then,\nwhen extracting the physical meaning of the solution, it is not done, in our opinion, in the\nmost enlightening manner, even in the simple cases in which more clarifying procedures seems\npossible.\n\nTake the case of the spherically symmetric vacuum solution. In the homologous case in\nelectromagnetism we know in advance that there is just one quantity, the electrostatic\npotential, to be determined. How could a point endowed with just one type of charge\ndetermine two different (not a function of each other) radial dependences?. In the present\ncase, however, we have two equations for $g_{rr}$ and $g_{00}$ and only after integrating\nthem it is found, in an nontransparent manner, that they are simply related, a relationship\nthat could has been advanced based on the EP.\n\nIn the case of a spherical cloud in equilibrium we have, in GR, three independent equations.\nCertain combination of them must give the hydrostatic equilibrium equation, but this is not\nclear at all in most expositions.\n\nFor the spherically symmetric vacuum solution particle trajectories can be determined\nexactly in the most elegant manner using Hamilton-Jacobi formalism. But then the origin of\nthe relativistic effects is completely masked. Approximate treatments as that given by\nEinstein\\cite{A.E.libro} are somewhat more transparent, but still, in our opinion,\ninsatisfactory. General three-dimensional treatments\\cite{Landau4} are much more helpful,\nbut still, having to compute explicitly the three-dimensional affine connections seems\nwithout proportions with the intrinsic simplicity of the problem. Here we have shown how\nsimply can this problem be treated exactly, and not for the reason of the economy of work\nbut to gain in transparency and insight.\n\nThe questions we have treated here are not specialized ones, but very basic and of interest\nfor any physicist with an eye on GR. The fact that at least some of them has not been\ntreated, or rather, that they are not treated where obviously they should have been, seems\nto us a sort of anomaly, since there is no question here that could not has been treated and\nbe widely known at least 80 years ago.\n\nIt is true that in GR we are deprived of our most valued source of intuition, the\n``solidity'' of space, that time is not what it used to be and then we have the\nnon-linearities, tensorial field and source, etc. But this should be an stimulus to work\nharder and not something to make us stagger back and stick to an infallible but obscure\nformalism renouncing to develop a detailed physical intuition of GR for fears of being\nbetrayed by it too often.\n\nThe terrain is treacherous, and we may atest to it. The simple and, we hope, very easy to\nfollow derivations given here were (some of them) not easy to obtain. When trying to reduce\nthe result searched for to a combination of simple ideas, the above mentioned complexities\nof GR made progress difficult, but now that we have managed to treat the questions presented\nhere in a consistent manner, we can only hope that this questions enhance the understanding\nof GR of the reader as much as it did with ours.\n\nIn the presentations of GR the dependence of the relative rate of two clocks on a\ngravitational field on the diffference of potential between them and the related\ngravitational redshift are directly derived in a simple manner from the the EP . Some of\nthem even discuss the non- Euclidianity of space in a rotating system in flat time-space.\nBut beyond this point direct derivations from basic principles are no longer given , not\neven heuristics ones, being replaced by mathematical considerations. This mimic the\ndevelopment of GR : when Einstein met with overwhelming difficulties in 1912 , he switched\nto a mathematically guided line of research. In consequence, elementary facts and\nconsiderations, as how do genuine gravitational static gravitational fields affect the space\ngeometry,  how and why the passive and active couplings of matter to gravity (the field\n$g_{00}$) do differ and why the latter include pressure, or how the implications of the EP\nin the case of flat space combine with the effect of the spatial curvature in determining\nparticle trajectories, are not discussed. It is the aim of this work to contribute to ease\nthis situation, that would  have been accepted in no other field of physics.\n\nWe are aware of the fact that , although some of the simple derivations given here could be\nused as a base for an elementary exposition of GR , such as it stands it is rather an\nadvanced one , being addressed to those who know well the standard presentations of the\nfacts treated here. Our goal has been to make patent to those readers the concepts and\nprinciples involved in these facts and how simply could they be derived. We also want to\nremark that, even when we make some comments on the history of GR , this is not even\npartially a work on the history of physics. This is a work on physics, and those comments\nhave been made to stress underlying conceptual issues. Nevertheless, we think that the\nhistorical facts that we comment are basically correct, although when we refer to Einsteins\nconceptions with respect to some question at a given date it may be only our best\ninterpretation of what we actually know.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDataflow matrix machines seem to be an attractive platform for general-purpose programming\nand at the same time dataflow matrix machines are representable by matrices of numbers\nin a manner similar to recurrent neural networks (RNNs).\n\nThis opens new possibilities for various interesting applications, such as learning general-purpose software\nvia methods proven effective for learning recurrent neural networks, \nand programming generalized recurrent neural networks within generalized recurrent neural networks themselves.\n\n\\subsection{Architecture of dataflow matrix machines}\n\nA dataflow matrix machine (DMM) can work with various kinds of linear streams of data (the streams of data for\nwhich a notion of linear combinations of several streams is well-defined).\n\nWe currently require to specify a finite number of {\\em kinds} of linear streams, $D_1, \\dots, D_N$. We also currently\nrequire to specify a finite number of {\\em types} of neurons, $C_1, \\dots, C_M$. An input of a neuron\ncomputes linear combinations of linear streams of a particular kind before passing the resulting stream to the neuron.\nA type of neuron specifies how many inputs a neuron of this type has (zero or more),\nwhat kind of streams each input accepts, how many outputs a neuron of this type has (one or more),\nwhat kind of streams each output emits, and the built-in transformation of the collection of input streams into\nthe collection of output streams.\n\nIt is convenient to consider a countable-sized infinite network, where only a finite number of connections have non-zero\nweights at any given moment of time, and only neurons which have some non-zero weights on some connections associated\nwith them are active. Hence only a finite subnetwork is active at any given time, while other neurons are considered to silently produce\nstreams of zero vectors (they don't need to be present in memory and don't spend processing resources).\n\nSo we consider a countable set of neurons of any type $C_i$, and hence we have a countable number of\ninputs and a countable number of outputs of any given kind $D_j$, and a countable-sized matrix of\nconnection weights associated with each kind of streams $D_j$, with only a finite number of those weights being non-zero\nat any given moment of time.\n\nThe topology of the network in question is determined by the sparsity structure of the matrices in question and can also\nevolve with time. \n\nIn this paper, one cannot combine streams of different kinds directly, the only way to perform transformations between streams of different kinds is via neurons\nof appropriate types. We combine countable-sized matrices for all $D_j$ into the single countable-sized matrix $A$ by enforcing the\nsparsity condition that the weights of connections between streams of two different kinds are always zero.\n\nIn this paper, we consider synchronous discrete time which is the most standard choice for RNNs.\n\n\\subsection{Diverse nature of linear streams}\n\nIn this paper, we consider dataflow matrix machines over real numbers.\n\nThere is a vector space $V$ associated with a given kind of linear streams, and a stream associates a representation of\na vector $v_t \\in V$ with each moment of time $t$. We are interested in different kinds of vector spaces $V$: numbers, \nfinite-dimensional vectors of various kinds (including spaces of finite matrices and tensors of fixed shape), and\ninfinite-dimensional vector spaces such the space of signed measures over some $X$.\n\nBecause of our interest in infinite-dimensional vector spaces we have to allow the situations where the representation of\n$v_t$ in the stream is less than perfect. E.g. we might decide to represent signed measures over $X$ by probabilistic samplings\nfrom those measures (with samples marked by the $+$ and $-$ flags to indicate whether they come from  the positive or\nthe negative part of the distribution in question). The linear combination of the streams will be represented by a stochastic sum of streams\nin this case; namely, to pick the next sample from $\\alpha_1*\\mu_1 + \\dots + \\alpha_n*\\mu_n$ we pick the sample associated with\n$\\mu_i$ with the probability $|\\alpha_i |\/ \\sum_j |\\alpha_j|$, and if $\\alpha_i <0$ we toggle its $+\/-$ flag.\n\n\\subsection{Programming the dataflow matrix machines}\n\nIt turns out that the many conventional programming constructions can now be expressed in this formalism.\n\nThe composition (``compound statement\") is expressed via layers similar to ``deep networks\".\n\nThe \\texttt{if} and other conditional constructions correspond to redirecting the flow of information in the network\nand can be achieved by using multiplicative masks on matrix columns. \n\nMultiplicative masks on the matrix itself serve as a convenient mechanism \nto express subnetworks.\n\nAccumulators of values for a particular kind of streams are achieved by allowing the type of neurons expressing\nidentity transformation for this kind of streams, and by making sure that the weight of the connection between\nthe output and the input of the accumulator in question is 1.\n\n\\subsection{Self-referential facilities}\n\nConsider a situation where a vector associated with a stream of the kind $D_1$\nis a  matrix, $A$, with the same structure of indices for rows and columns as matrices describing the dataflow\nmatrix machines in question. Then we can think of\nsuch vectors as descriptions of dataflow matrix machines, and we can think of neurons which take and produce streams of this kind as\ntransformers of dataflow matrix machines.\n\nMore specifically, let's consider the type $C_1$ of neurons to be the identity transformers of the streams of the kind $D_1$.\nAnd let's fix a particular neuron \\texttt{Self} of the type $C_1$, and fix the weight of the connection of its output to its\ninput to be 1 to make it an accumulator, and associate our dataflow matrix machine with the matrix accumulated on the output of\nthe neuron \\texttt{Self}.\n\nThen as long as we make sure to introduce types of neurons doing interesting transformations with streams of the kind $D_1$, we have facilities\nfor self-referential dynamic modification of a dataflow matrix machine while it runs. Like any accumulator, \\texttt{Self}\ntreats contributions from other streams of the kind $D_1$ to which \\texttt{Self}'s input is connected by non-zero weights\nas additive modifications.\n\n\\subsection{Programming language situation}\n\nWe envision a situation where one has an interactive high-level language to describe and modify the dataflow matrix machine.\nIn this paper we start sketching fragments of one possible language of this kind.\n\nThe statements written in that language edit the matrix $A$ defining the dataflow matrix machine in question,\nand neurons whose $D_1$-typed outputs have non-zero connections into \\texttt{Self} are also capable of editing\nthat matrix.\n\n\\subsection{Powerful neurons}\\label{powerful_neurons}\n\nThis approach envisions having a sizable collection of diverse neuron types.  The collection of kinds of streams and\ntypes of neurons is called a {\\em signature} of our dataflow matrix machine,\nand our expectations for dataflow matrix machines to be a convenient\nplatform for general-purpose programming is based on the ability to add sufficiently expressive\ntypes of neurons to the signature.\n\nIn particular, to participate in conditional constructions controlled by multiplicative masks over columns\n(that is, over neuron outputs), a neuron should have a special input for the value of that\nmultiplicative mask. So a neuron would typically have a separate input computing the number associated with\nthe mask, and its associated transform would multiply the values of output streams of the neuron in question by that number.\nAn optimized implementation would be able to omit the associated computations when the mask is zero,\nand to omit the multiplications when the mask is 1.\n\nWe envision having a sufficient collection of types of neurons capable of editing the matrix $A$ to easily\nexpress the edits doable via the interactive high-level language mentioned above. We think about this\ninformally as ``self-referential completeness of the DMM signature relative to the language available to\ndescribe and edit the DMMs\". Note that it is usually easy to trigger a neuron which has a multiplicative\nmask input for exactly one particular moment of time if necessary, e.g. by connecting an external input control \ninto the multiplicative mask input of our neuron. An example of such a type of neurons is given in\nSection~\\ref{higher_order_neuron}.\n\n\\subsection{Bipartite graphs}\n\nFor the case when all neurons have one output, it is sometimes convenient to think about these networks as bipartite graphs.\nOne kind of nodes are neurons with finite fixed input arities. \n\nAnother kind of nodes are {\\em neuron inputs} which have variable\ninput arity. They work as countable linear combinations of the neuron outputs of the appropriate kind,\nso formally speaking they have countable input arity,\nbut only finite number of coefficients are non-zero at any given moment of time. These linear combinations\nare represented by rows of the matrix $A$.\n\nA neuron input has one outgoing link, sending the linear combination of streams to the neuron itself. \nThe situations where one neuron input serves multiple neurons correspond to constraints requiring different rows\nof the matrix $A$ to be equal and are beyond the scope of this paper.\n\n\n\n\n\\section{Related work}\n\n\\subsection{Recurrent neural networks as a computational platform}\n\nThe ideas that recurrent neural networks can be used as a programming platform, and that a recurrent\nneural network can modify other recurrent neural networks including itself are certainly not new.\n\nTuring universality of some classes of recurrent neural networks  became known at least 20 years ago\nand first schemas to compile conventional programming languages into recurrent neural networks\nemerged in the same time frame (see~\\cite{JNetoSiegelmannCosta} and references therein). \n\nThat time period had also seen the first thought experiment in recurrent neural networks involving\n``self-referential\" weight matrix where the network had used its own weights as additional input data and could,\nat least in principle, learn new algorithms for modifying its own weights~\\cite{JSchmidhuber}.\n\nThe first successful instance of using a recurrent neural network to learn a well working learning\nalgorithm for training some recurrent networks appeared in~\\cite{SHochreiterYoungerConwell}.\n\nHowever, none of these became a part of the daily routine of the field. One would be hard-pressed to find recent\ninstances of using RNNs to either manually create general-purpose software or to learn the algorithms of RNN synthesis.\nIn fact, a recent influential post~\\cite{AKarpathy} emphasizes that training recurrent networks should be considered\noptimization over a space of programs, but at the same time cautions  not to ``read too much into\" the theoretical\nresults establishing Turing-completeness of RNNs.\n\nOn the positive side, there is quite a bit of interest recently in using recurrent neural networks and\nrelated machines to learn general-purpose algorithms automatically (see e.g.~\\cite{SReeddeFreitas} and references\ntherein). The question seems to be, how much would one need to modify the traditional RNN architecture\nand learning methods to make general-purpose program learning feasible for daily use.\n\nWe think that the difficulties are to a large extent related to the focus on one-dimensional streams\n(streams of numbers), to configurations with fixed number of neurons, and to per-element\nmatrix modification schemas. Combination of these factors usually leads to awkward encodings,\nwhich make the network behavior overly sensitive to very small changes.\n\nFor example, one might note that Turing universality requires potentially infinite memory. In a typical approach\nto Turing universality of recurrent neural nets one expands the precision of the involved real numbers\nin an unbounded fashion, so that one could  in effect use the binary expansion of a real number as a tape\nfor a Turing machine~\\cite{HSiegelmann}.\n\nAnother way would be to expand the size of the neural net in question in an unbounded fashion.\nSince we already have a countable-sized matrix, this approach seems more natural for our\narchitecture.\nThis approach seems also to be more consistent with the traditional uses of neural nets,\nwhich involve the standard built-in machine representation of real numbers (or, sometimes,\nthe compressed versions of even that, for better performance), and this is the approach\nwe use in the present paper.\n\nWe also tend to modify the network matrix not on per-element basis, but by functional blocks\nin the present paper, and we expect this to also help us to avoid awkward encodings and delicate\ncoordination problems, where a desired modification goes through only for some of the elements\nexpected to be involved in this modification,\ncreating a configuration with uncertain properties.\n\n\n\n\n\\subsection{Dataflow programming with linear streams}\n\nDataflow programming languages such as LabVIEW and Pure Data (which are oriented mostly towards work with streams of continuous data)\nfound some degree of general programming use within their application domains~\\cite{WJohnstonHannaMillar,AFarnell}.\nHowever, those platforms are not purely continuous, and it's not clear how to achieve matrix parametrization of large spaces\nof programs for those platforms.\n\nThe discipline of dataflow programming purely in terms of diverse variety of linear streams was considered in~\\cite{BukatinMatthewsLinear},\nwhere it was also noted that adopting a discipline of bipartite graphs with one class of nodes being associated with general transformations\nand another class of nodes being associated with linear transformations allows to represent large classes of dataflow programs\nover linear streams by matrices of real numbers and to modify programs by continuous change of those numbers, thus achieving\ncontinuous program transformations.\n\nHowever, it was not demonstrated in~\\cite{BukatinMatthewsLinear} that its approach would lead to\na viable general-purpose programming platform, nor was it mentioned in~\\cite{BukatinMatthewsLinear} that the architecture developed there\nwas a generalization of recurrent neural networks.\n\nIn this paper, we observe that the architecture in the spirit of~\\cite{BukatinMatthewsLinear} generalizes recurrent neural networks\nand make progress towards developing this architecture into a viable general-purpose programming platform.\nA particularly novel contribution\nis self-referential modification of matrices controlling the network behavior not on per-element basis, but by additive functional\nblocks. The ability to do so is enabled by going beyond one-dimensional streams of numbers and allowing single neurons to\nwork with streams of countably-sized matrices containing finite number of non-zero elements.\n\n\n\\section{A more detailed description and programming of DMMs}\n\n\\subsection{Neuron types}\n\nConsider kinds of streams $D_1, \\dots, D_N$ equipped with the ability to take linear combinations\nof streams of the same kind with finite number of non-zero coefficients.\n\nDefine a type of neurons $C_i$ as follows. Associate with $C_i$ the non-negative integer number of inputs $\\#I_i$ (neurons which\nare to unconditionally function as inputs for the network have zero inputs)\nand the positive integer number of outputs $\\#O_i$. With every input or output $k$ of $C_i$ associate a kind of streams $D_{j_k}$.\nAlso associate with $C_i$ a transform $F_i$ taking as inputs $\\#I_i$ streams  of appropriate kinds of length $t-1$ and producing as outputs $\\#O_i$ streams of appropriate kinds\nof length $t$ for integer time $t>0$. Require\nthe obvious prefix condition that when $F_i$ is applied to streams of length $t$, the first $t$ elements of the output streams of length $t+1$ are the elements\nwhich $F_i$ produces when applied to the prefixes of the input streams of length $t-1$.\nThe most typical situation is when for $t>1$ the $t$'s elements of the output streams are produced solely\non the basis of elements number $t-1$ of the input streams, but our definition also allows neurons to\naccumulate unlimited history, if necessary.\n\nThe resulting $((D_1, \\dots, D_N), (C_1, \\dots C_M))$ is called the {\\em signature} of our DMM.\n\nWe are going to define some countable collections and we find it convenient to index them by strings rather than\nby numbers. Formally speaking, strings are equivalent to numbers as a countable set\nof indices, since the alphabet for strings can be thought of as digits in the appropriately\nselected system of numbers. But the approach based on strings does\nde-emphasize the order on indices coming from those numbers and encourages\nto use indices for meaningful description of functionality without excessively\ncumbersome G\\\"{o}del-style numbering.\n\nWe use a base alphabet $\\Sigma$ for those strings and additional special characters we always assume to be\noutside of $\\Sigma$. In this paper those special characters are the space and \\texttt{\\#;:,=} characters.\n\n\\subsection{Describing the DMM elements in a programming language}\n\nKeywords in our DMM-oriented language start with the special character \\texttt{\\#} and are followed by the space.\n\nThe kinds of streams are named \\texttt{\\#kind <name1>, ..., \\#kind <name N>}. \nThe user must supply the implementation of the respective streams\nand their linear combinations in the underlying conventional language.\n\nThe types of neurons are named \\texttt{\\#celltype <typename 1>, ..., \\#celltype <typename M>}.\n\nThe types of neurons are specified by stating stream kinds and giving the names to their inputs (if any) and their outputs:\n\\begin{verbatim}\n    #newcelltype <typename> \n        #input  <kindname>:<fieldname> ... #input  <kindname>:<fieldname> \n        #output <kindname>:<fieldname> ... #output <kindname>:<fieldname>;\n\\end{verbatim}\n\nThe user must supply the implementation of the stream transform associated with\neach neuron type in the underlying conventional language.\n\nThe neurons are named using special character \\texttt{:} as follows: \\texttt{\\#neuron <typename>:<cellname>}, where\n\\texttt{<cellname>} is any string in the alphabet $\\Sigma$. We assume that all such neurons silently exist,\nand can be activated by making some of their connections non-zero.\n\nThe matrix element $a_{ij}$ in question describing a connection from an output of a neuron (cell 1) to an input of a neuron (cell 2) can be written as\n\\begin{verbatim}\n    #weight <typename 2>:<cellname 2>:<inputfieldname>\n            <typename 1>:<cellname 1>:<outputfieldname>\n\\end{verbatim}\n\nThis weight can only be non-zero when the kinds of streams associated with the fields\n $i$ = \\texttt{<typename 2>:<cellname 2>:<inputfieldname>} and $j$ = \\texttt{<typename 1>:<cellname 1>:<outputfieldname>}\ncoincide. Otherwise, the weight is forced to be zero.\n\n\n\\subsection{Dynamic modifications of a working DMM}\n\nTo describe a work cycle of a dataflow matrix machine, it is convenient to have the metaphor of\n``two-stroke engine\" in mind. In terms of graphical depiction of the DMM, think about a horizontal layer of all neurons\nwith outputs being above that layer and inputs being below that layer.\n Then on the ``up movement\", neurons compute their outputs based on their\ninputs using transforms associated with their types. On the ``down movement\", {\\em neuron inputs}\ncompute themselves as linear combinations of all neuron outputs using the rows of matrix $A$.\n\nConsider a neuron with the identity transform for some kind of streams.\nIt is important to note that even though the transform is the identity, one can't just\nuse the single shared variable to store the latest values of the input and output\nstreams of that neuron. On the ``down movement\", all neuron inputs including the input for the\nneuron in question are recomputed as linear combinations of all outputs; it is important for\nthe corresponding output of the neuron in question not to be overwritten\nduring the ``down movement\" because it might be used in computing a variety of linear combinations.\nOnly during the ``up movement\" does the identity transform get executed changing the neuron output.\n\nIn particular, this is applicable to the neuron \\texttt{Self} controlling the matrix $A$ itself. The output of\n\\texttt{Self} holds the current value of the matrix $A$. The rows of $A$ are used to compute all the inputs of the\nnetwork on the ``down movement\", including the input of \\texttt{Self} which accumulates the value\nof the matrix $A' = A +$ contributions from other neurons. \nThen on the ``up movement\", the identity transform updates the $A$ to become $A'$. Making this process\nas efficient as possible is a separate topic.\n\n\\subsection{Initialization and expandable memory}\n\nNeurons with all zero incoming and outgoing weights (that is, with all rows of $A$ corresponding to inputs\nof the neuron in question and with all columns of $A$ corresponding to outputs of the\nneuron in question being zero) are only present in the countable address space, but not in memory.\nTheir outputs are considered to be zero vectors, and they don't participate in computations.\n\nA neuron gets into memory and becomes active when one of its associated weights becomes\nnon-zero. Since this can only happen by a matrix update, the first computation which happens\nafter that is ``down movement\", so first the inputs of the neuron in question are updated,\nonly then the neuron itself works.\n\nThere is an option to ``garbage collect\" a neuron from memory, if all its associated rows\nand columns are set to zero.\n\n\\subsection{Programming}\n\nThere is a variety of approaches to neuron names within the DMM, ranging from completely semantically meaningful names to\nautogenerated tokens, and there are various trade-offs associated with this choice. In this paper\nwe explore how far we can go using autogenerated tokens and avoiding the reliance on semantically\nmeaningful names. \n\nAt the same time, in our DMM-oriented language we need identifiers to denote various parts of\nthe DMM in question, e.g. a neuron, a particular input or output of a neuron, a subgraph, etc.\n\nTo avoid ambiguity we always use the word {\\em name} for indices of columns and rows in our matrix\nand similar purposes and we always use the word {\\em identifier} for variables in our DMM-oriented\nlanguage.\n\nAll neurons and links (countably infinite sets) are presumed to exist, and the burden of not spending\nspace and processors on all but the finite number of those is on the implementation.\n\nTherefore a statement \n\\begin{verbatim}\n    #neuron <typename>:<IdNeuron> \n        <fieldname>:<IdOutput1>, ..., <fieldname>:<IdOutputO> =\n        #transformof <fieldname>:<IdInput1>, ..., <fieldname>:<IdInputI>; \n\\end{verbatim}\nsimply introduces an identifier \\texttt{<IdNeuron>} to denote a neuron with a new autogenerated \\texttt{<cellname>} and zero associated weights,  \nand it also introduces identifiers \\texttt{<IdOutput1>, ..., <IdOutputO>, <IdInput1>, ..., <IdInputI>} for the input and output streams \nof this neuron in our language, but it does not change the underlying network.\n\nSimilarly,\n\\begin{verbatim}\n    #stream <kindname>:<IdInputStream> = #neuroninput <IdNeuron>.<fieldname>;\n\\end{verbatim}\nintroduces an identifier for a particular\ninput stream of the neuron denoted by \\texttt{<IdNeuron>}.\n\nBut no functional changes can be obtained in this fashion. The only way to make the network to change its behavior is\nto update the matrix weights. We always do it respecting our accumulator metaphor, for example to update\nthe way a particular input to a neuron is computed from a variety of neuron outputs with the same kind of streams we can write\n\\begin{alltt}\n    #updateweights <IdInputStream> +=\n        \\( \\alpha\\sb{1} \\)* <IdOutputStream\\(\\sb{1}\\)> + \\dots +\\( \\alpha\\sb{k} \\)* <IdOutputStream\\(\\sb{k}\\)>;        (1)\n\\end{alltt}\n\nOr one can add a linear combination of rows corresponding to other input streams of the same kind:\n\n\\begin{alltt}\n    #updateweights <IdInputStream> +=\n        \\( \\beta\\sb{1} \\)* <IdInputStream\\(\\sb{1}\\)> + \\dots +\\( \\beta\\sb{k} \\)* <IdInputStream\\(\\sb{k}\\)>;          (2)\n\\end{alltt}\n\nThis provides one of the mechanisms to set a row to zero:\n\n\\begin{alltt}\n    #updateweights <IdInputStream> += (-1) * <IdInputStream>;          (3)\n\\end{alltt}\n\n\\subsection{Streams of rows and streams of columns}\n\nIn addition to streams of matrices capable of controlling the DMM in question it is convenient to have streams of rows and streams of columns,\nso these are two more kinds of streams to have. The values are used directly as rows and columns of the matrices controlling the DMM and\nalso as multiplicative masks (that is, coefficients of linear combinations; rows are used as ``column masks\", and columns are used as ``row masks\").\n\nThe typical values have only finite number of non-zero elements, but some finitely describable\ninfinite vectors are quite helpful in this context, e.g. infinite rows with all fields corresponding\nto outputs having a particular kind of streams equal to 1 are useful both as trivial column masks to be\napplied to finite row vectors and\nas fake input rows (in the example \\texttt{(1)} of the \\texttt{\\#updateweights} operation\na finite column mask (vector $\\alpha$) is applied to a fake input row of this kind). \n\nThe discipline of only\nallowing non-zero elements for one particular kind of streams in each of our rows and columns might be\npreferable (otherwise one needs to worry about not breaking the prohibition of creating\nnon-zero weight elements between streams of different kinds).\n\nThe general form of the \\texttt{\\#updateweights} statement covering the examples above and more is\n\n\\begin{alltt}\n    #updateweights <FiniteRowMask 1> += <ColumnMask> * <FiniteRowMask 2>;\n\\end{alltt}\n\nIn all the \\texttt{\\#updateweights}  examples above, \\texttt{<FiniteRowMask 1>} has 1 in one place and zeros in the other places, \nbut any finite row mask is allowed and\ncoefficients can be different from 1 and 0 (they are used as multipliers for the right-hand side before the\n\\texttt{+=} is applied for the given row).\n\nIn the example \\texttt{(2)} of \\texttt{\\#updateweights} the vector of $\\beta$'s at the \\texttt{<IdInputStream$_i$>} positions works as a \\texttt{<FiniteRowMask 2>}, and the\nexample \\texttt{(3)} is a particular instance of the example \\texttt{(2)}. The \\texttt{<ColumnMask>} is a trivial mask with infinite number of 1 fields described above in both cases.\n\nThe \\texttt{\\#updateweights} operation is oriented towards working with rows, but a similar operation oriented towards working with\ncolumns can be introduced if necessary.\n\nTake \\texttt{<ColumnMask>} to be vector $\\alpha$, \\texttt{<FiniteRowMask 2>} to be vector $\\beta$, and the left-hand-side \n\\texttt{<FiniteRowMask 1>} to be vector $\\gamma$. Then on the element-wise level the \\texttt{\\#updateweights} operation acts as\n$a_{ij} := a_{ij} + \\gamma_i * \\alpha_j * \\sum_k \\beta_k a_{kj}$.\n\n\\subsection{Editing the DMM matrix with higher-order neurons}\\label{higher_order_neuron}\n\nThe paradigm formulated in Section~\\ref{powerful_neurons} requires us to have a type of higher-order neurons to perform\nthe \\texttt{\\#updateweights} operation. The neurons of this type have to be fairly complex, they need to have at least 5 different inputs.\nOne input is to be of the kind corresponding to the stream of matrices controlling the DMM. Most typically, the output of \\texttt{Self}\nwill be connected to this input with the weight of 1. Two more inputs are needed for\n\\texttt{<FiniteRowMask 1>} and \\texttt{<FiniteRowMask 2>}, and one more input for \\texttt{<ColumnMask>}.\nFinally, at least one more input is needed to control the firing of these neurons, for example a scalar\nmultiplicative mask as discussed in Section~\\ref{powerful_neurons}.\n\nThese neurons would have one output of the kind corresponding to the stream of matrices controlling the DMM.\nThis output would normally emit the result of the application of the \\texttt{<FiniteRowMask 1>}\nto the result of the right-hand side of the \\texttt{\\#updateweights} operation (the result of the application of the \\texttt{<ColumnMask>} to the result of the application of\nthe  \\texttt{<FiniteRowMask 2>} to the matrix itself).\nMost typically, this output would be connected to the input of \\texttt{Self} with the weight of 1.\n\n\n\n\\section{Conclusion and future work}\n\nDataflow matrix machines and recurrent neural networks represent a continuous programming architecture, that is,\nan architecture oriented towards working with the streams of continuous data, with the programs being of\nthe continuous nature themselves. Discrete programming architectures\nsupport a variety of programming styles: imperative, objective-oriented, functional, logical, dataflow, etc.\nSimilarly, a variety of programming styles should be possible for DMMs and RNNs.\n\nFor example, it is relatively easy to support an imperative programming style based on representing\nlinked data structures in the body of a DMM and having neurons traversing those structures, based\non the machinery introduced in the present paper.\n\nIt is also relatively easy to support a programming style based on creating deep copies of subgraphs\nwhile preserving their incoming (and, optionally, outgoing) connections.\nOne can easily create fairly intricate patterns by using this operation\nin a nested way.\n\nWe hope that more styles of programming in this architecture will emerge.\n\nOne of the hopes of this approach is that it eventually results in better ways for\nprogram learning. Currently, a typical result of program learning is a program which\nfunctions, but is almost impossible for a human to read and understand.\nAt the same time, recurrent neural networks have good track record of learning patterns,\nand of generating visually readable patterns back (see e.g.~\\cite{AKarpathy}).\n\nOne would hope that if one starts with a corpus of programs in a DMM-oriented\nlanguage, one could then find a way to train a DMM to reproduce both syntactic patterns and function,\nresulting in programs which not only work, but can also be read and understood by people.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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@addtoreset\\endcsname{equation}{section}\n\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\eeq}[1]{\\label{#1}\\end{equation}}\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\eer}[1]{\\label{#1}\\end{eqnarray}}\n\\newcommand{\\eqn}[1]{(\\ref{#1})}\n\\begin{titlepage}\n\n\n\\begin{center}\n\n\n~\n\\vskip 1 cm\n\n\n\n{\\Large\n\\bf Velocity Laws for Bound States in Asymptotically AdS  Geometries\n}\n\n\n\\vskip 0.7in\n\n {\\bf Dimitrios Giataganas}\n \\vskip 0.1in\n {\\em\n \t${}^1$  Department of Physics,\\\\ National Sun Yat-sen University,  \\\\\n \tKaohsiung 80424, Taiwan\n \t\\vskip .1in\n \t${}^2$  Center for Theoretical and Computational Physics, \\\\\n \tNational Sun Yat-sen University\t\t\t\t\t\n \t\\\\\n \tKaohsiung 80424, Taiwan\\vskip .1in\n \t${}^3$ Physics Division, National Center for Theoretical\n \t\tSciences, \\\\\n \tTaipei 10617, Taiwan\n \\\\\\vskip .15in\n {\\tt ~  dimitrios.giataganas@mail.nsysu.edu.tw\n }\\\\\n }\n\n\\vskip .2in\n\\end{center}\n\n\\vskip .8in\n\n\\centerline{\\bf Abstract}\n\nWe study the behavior of heavy quark bound states in moving plasmas that are dual to theories with generic non-trivial renormalization group flows interpolating between an AdS geometry in the ultraviolet and infrared fixed points with broken symmetries. We investigate analytically the observables associated with the bound state and find their scaling exponents with respect to the Lorentz factor for ultrarelativistic motion. Despite having asymptotically  an AdS geometry, the scaling is not universal and depends on geometric conditions of the Fefferman-Graham expansion in the near boundary regime, or equivalently on the order of the asymptotic background expansion that provides the leading contributions to the Wilson loops. \n\\noindent\n\\end{titlepage}\n\\vfill\n\\eject\n\n\n\n\n\n\\noindent\n\n\n\n\\def1.2{1.2}\n\\baselineskip 19 pt\n\\noindent\n\n\n\\setcounter{equation}{0}\n\n\n\\section{Introduction}\n\nThe Wilson loop operators contain important gauge invariant information about non-perturbative physics of gauge field theories. Their expectation value is an order parameter for the confining-deconfining phase transitions, where confinement indicates an area law. Additionally the Wilson loops above the crossover point of the hadronic state, in the finite temperature strongly coupled deconfined plasma state, are related to a number of different observables which have theoretical interest and some of them are in principle accessible in heavy ion experiments.\n\nThe expectation values of Wilson loops contain ultraviolet divergences which can be realised in the holographic gravity dual theories from the infinite bulk to boundary proper distance where the Wilson loop surface extends, while in the field theory can be realized for example, as coming from propagators connecting coincident points at loop computations. In holography there is a rigorous renormalization formalism  \\cite{Maldacena:1998im,Drukker:1999zq,Chu:2008xg}, established for static Wilson loops, which can be realised with the Legendre transform method, or equivalently with the holographic renormalization, or with the infinite mass subtraction. The latter scheme not only cancels the ultraviolet divergences but contributes certain infrared terms, which can be thought as part of the thermal masses of the infinite heavily bound state when the loops are orthogonal. Therefore the application of the mass renormalization gives the interaction energy of an extremely massive bound state in a strongly coupled environment. Non-static Wilson loops, or boosted static,  have also been studied and can be thought as corresponding to moving bound states in a thermal strongly couple environment. Some of the renormalization schemes have been extended for this case successfully, for example \\cite{Liu:2006he,Chu:2016pea}, despite the complications in the application of the mass renormalization due to the interactions of the color charged singlets in strongly coupled environments  \\cite{Giataganas:2022qgq}.\n\nIt is known that the application of the mass renormalization scheme, leads to a critical value of the renormalized  Wilson loop expectation value, beyond which the state does not exist. For instance, by fixing the scales of the theory, there exists a critical value for the size of the heavy quark  bound state $L$, beyond which the bound state seizes to exist and  becomes energetically non-favourable compared to the free state of its ingredients, the two infinite massive heavy quarks. $L$ naturally depends  on the scales of the theory and the velocity of motion. It has been found holographically that for ultrarelativistic velocities there is a characteristic power that the screening length depends on the Lorentz factor $\\gamma}  \\def\\Gamma{\\Gamma}  \\def\\dot{\\gamma}{{\\dot\\gamma}=1\/\\sqrt{1-v^2}$, which for AdS spacetimes is $1\/4$ when expressed in terms of the velocity $L_{max}\\sim \\prt{1-v^2}^{1\/4} $ \\cite{Liu:2006nn}, while for other type of spacetimes takes different values as expected, especially when other dimensionful scales of the theory are present \\cite{Liu:2006he,Sadofyev:2015hxa,Chernicoff:2012bu,Chernicoff:2006hi,Caceres:2006ta}.\n\nIn this work we investigate further the properties of the non-local observables dependence on the Lorentz factor and how much the degree of symmetry affect the studies. Our motivation is primarily of theoretical interest and our study includes a new type of approach based on the properties of the background's asymptotic expansion, while working on the rest frame of the bound state. The holographic theories we consider have a non-trivial renormalization group flow. Their ultraviolet conformal fixed point is conformal and a Fefferman-Graham asymptotic expansion is applicable in the near boundary regime. Moreover, along the flow we allow rotational symmetry breaking,  as a result the  infrared fixed point is allowed in principle to have reduced symmetries. We point out that our approach is generic, we do not specify the holographic theory and our formalism and results are applicable for  any asymptotically AdS theory with the mentioned symmetries. Example of such specific  holographic theories with non-trivial renormalization group flows include \\cite{Mateos:2011ix,Giataganas:2017koz,Hoyos:2020zeg}.\n\n\nOur work can also be though to be motivated from recent developments on known strongly coupled systems and in particular of the  quark-gluon plasma.  It has been  known for a long time that the suppression in the number of $J\/\\Psi$ mesons \\cite{NA50:2004sgj} may be related to the high transverse momenta. Moreover, recent experimental results show that certain hadrons, for example the  $\\Lambda$-hyperons, which are produced in noncentral collisions exhibit nonzero spin polarization. This  implies that the strongly coupled plasma is rotating and precise measurements suggest that it is the most vortical fluid observed to date \\cite{STAR:2017ckg}. Holographic dual models that describe such fast rotating plasmas, will exhibit breaking of the rotational symmetry due to the rotation at least on a plane. Candidates of such theories are ones that have been deformed by relevant and marginal operators, which can be still asymptotically AdS, while their infrared fixed point has broken symmetry due to a rotation and they can be included in our generic formalism as special cases. The Wilson loop scaling analysis in those theories could have extra minor computational complication but there is no obvious reason that the velocity dependence on the screening length changes for a rotating motion compared to that of a linear motion. In fact in the special case of AdS, the holographic analysis for rotating plasmas supports qualitative the quarkonium suppression observed in heavy ion collisions for linear and rotating motion in the same way. For the spinning bound states in thermal backgrounds their qualitative \\cite{Giataganas:2022qgq} and quantitative \\cite{Peeters:2006iu} dependence of the screening length on the velocity is consistent with the one reported for linear boosts. So we expect that our classification on the scaling powers we have done in this work applies in the case of rotation as well, with potentially modified classification conditions. Recently, there is an extensive effort in holography to understand bound states and other aspects of the fast rotating strongly coupled plasmas, for example \\cite{Giataganas:2022qgq,Garbiso:2020puw,Chen:2022obe,Chen:2020ath, Golubtsova:2021agl,Golubtsova:2022ldm}. It is worthy to note that the behavior of the critical temperature in presence of rotation has been investigated with different type of approaches, see for example \\cite{Jiang:2016wvv,Fujimoto:2021xix,Braguta:2021jgn,Chernodub:2020qah}, where the vast majority of these studies suggests lower critical temperature for the phase transitions and easier melting of the bound states, in agreement with the holographic findings. Notice that these observations are also in agreement with computations in strongly coupled theories where the rotational symmetry is broken by other mechanisms, for example via the backreaction of a strong external field, \\cite{Giataganas:2012zy,Giataganas:2017koz,Gursoy:2021efc,Giataganas:2013lga,\nGiataganasimv}. This alternatively suggests the breaking of the symmetry as one of the dominant causes for the suppression on the heavy quarks and the lowering of the critical temperature of the phase transitions. \n\n\nIn our work we analyze analytically the quantities associated with the heavy bound state in the plasma wind for asymptotically AdS theories, and we find their scaling exponents with respect to the Lorentz factor for ultrarelativistic motion. Despite the fact that our theories are asymptotically AdS, we show that different velocity scaling powers are allowed for the Wilson loop observables. The different scalings are associated with different asymptotic background conditions we specify. As a result there is no universality in the scalings although the set of the scaling values  is finite and discrete. Once the background satisfies an asymptotic condition, a unique scaling power is determined.  The reason that a variety in scalings is possible, is that different background conditions correspond to higher order geometric contributions from the asymptotic expansion on the observables. In section 2, we rotate and boost the plasma that our dipole bound state is placed in order to set up our system in full generality. In section 3, we examine Wilson lines in general holographic boosted backgrounds,  then we apply the formalism in the asymptomatically expanded background in the ultrarelativistic limit. The observables we study for ultrarelativistic motion receive their leading contribution from the near boundary regime.    In section 4, we compute the expectation value of the Wilson loop in generic holographic boosted spacetimes. Then we take the ultrarelativistic limit and assume asymptotically AdS geometries to find how the various quantities scale with the Lorentz factor. For instance, for the screening length  we find $L_{max}\\sim \\prt{1-v^2}^{\\frac{1}{2k}}$, where $k=1,2$ depending on the asymptotic behavior of the metric. We find the conditions of the asymptotic expansion that are associated with each scaling power $k$ and discuss their implications.\n\n\n\n\\section{Gravitational Background: Rotation and Boost}\n\nLet us assume a space  that along the RG flow is allowed to have a potential breaking of rotational symmetry in $d$ spatial dimensions, where the $SO(d)$ boundary symmetry is broken along the flow to infrared to $SO(d-d_1)$  in the $(\\vec{x_i})$ plane and $SO(d_1)$  in the $(\\vec{x_j})$ plane.  To set up our system we consider  a dipole state of random orientation in the space. We examine backgrounds of only radial dependence and therefore our analysis can be reduced without any loss of generality to a three-spatial dimensional configuration with an isotropic plane $x_1 x_2$ preserved along the whole renormalization group flow  and a special direction $x_3$ ($d_1=1$) that breakes the rotational invariance in the bulk. The dipole experiences a wind of velocity $v$ that lies say  in a $x_1 x_3$-plane, and forms an angle $\\theta$ with the $x_3$-direction. We align the new coordinate system at the orientation of the wind with an application of a rotation coordinate transformation of angle $\\theta$  on $x_1 x_3$ plane. Then we can perform a boost along the new aligned direction of the wind that coincides with one axis to eventually obtain the rotated boosted metric\n\\begin{eqnarray}\\nonumber\nds^2= &&\\tilde{g}_{00}(u) dt^2+\\tilde{g}_{11}(u) d\\tilde{x}_1^2+\\tilde{g}_{22}(u) d\\tilde{x}_2^2+\\tilde{g}_{33}(u) d\\tilde{x}_3^2+\\tilde{g}_{uu}(u) du^2\\\\&&+2 \\tilde{g}_{01}(u) d\\tilde{x}_0 d\\tilde{x}_1\n+ 2  \\tilde{g}_{03}(u) d\\tilde{x}_0 d\\tilde{x}_3+ 2\\tilde{g}_{13}(u) d\\tilde{x}_1 d\\tilde{x}_3 ~,\\la{metric1}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\nonumber\n&&\\tilde{g}_{00}(u)= g_{00}(u) \\cosh^2 \\eta +\\prt{g_{33}(u) c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 +g_{11}(u)s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2} \\sinh^2\\eta~,\\quad \\tilde{g}_{11}(u)=  g_{11}(u) c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 +g_{33}(u)s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2~, \\\\ \\nonumber\n&&\\tilde{g}_{33}(u)= g_{00}(u)\\sinh^2 \\eta+\\prt{g_{33}(u) c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 +g_{11}(u)s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2} \\cosh^2\\eta~,\\\\\\nonumber\n&&\\tilde{g}_{01}(u)=  \\frac{1}{2} \\prt{g_{33}(u)   -g_{11}(u) }\\sinh\\eta s_{2\\theta}~, \\quad  \\tilde{g}_{13}(u)=  \\frac{1}{2} \\prt{g_{11}(u)   -g_{33}(u) }\\cosh\\eta s_{2\\theta}~,  \\\\  \\nonumber\n&&\\tilde{g}_{03}(u)= - \\prt{g_{00}(u)   +g_{11}(u)\\sin^2 \\theta+g_{33}(u)c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 }\\sinh\\eta\\cosh\\eta ~.  \n\\end{eqnarray}\nThe tildes refer to the boosted rotated background metric and the metric elements $\\tilde{g}$ that do not appear in the above expressions coincide with the initial holographic background $\\tilde{g}_{ij}=g_{ij}$. The rapidity $\\eta$ is given by the $\\cosh^2 \\eta=\\gamma}  \\def\\Gamma{\\Gamma}  \\def\\dot{\\gamma}{{\\dot\\gamma}^2=1\/\\prt{1-v^2}$, where $\\gamma}  \\def\\Gamma{\\Gamma}  \\def\\dot{\\gamma}{{\\dot\\gamma}$ is the Lorentz factor and $v$ is the velocity of the boost. The setup is particularly convenient to describe the direction of the hot wind and the dipole orientation in a generic space-time. For example, the angle $\\theta=0$ corresponds to having a wind blowing along the $x_3$ direction, boosting therefore the system along the special anisotropic direction, while the angle $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta=\\pi\/2$ corresponds to having the wind within the transverse $x_1 x_2$ plane and the boost of the systems happens  there. When we refer to the dipole orientation is with respect to initial system before any boost and rotation. Notice that analysis is in the probe's rest frame.\n\n\\section{Velocity Laws for Wilson Lines}\n\nLet us start by studying Wilson lines which correspond to free heavy colour charged singlets in the wind. To parametrize the string we fix it to the radial gauge while we embed the string of single boundary endpoint  $\\prt{x_1(\\sigma}  \\def\\Sigma{\\Sigma), x_3(\\sigma}  \\def\\Sigma{\\Sigma)}$, taking into account the symmetry of the theory. The action reads\n\\begin{equation}\\la{act1}\nS=-\\frac{1}{2\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}{}'} \\int d\\sigma}  \\def\\Sigma{\\Sigma \\sqrt{ A_0+ A_1 \\tilde{x}_1{}'(\\sigma}  \\def\\Sigma{\\Sigma)^2 +A_3 \\tilde{x}_3{}'(\\sigma}  \\def\\Sigma{\\Sigma)^2+A_{13} \\tilde{x}_1{}'(\\sigma}  \\def\\Sigma{\\Sigma) \\tilde{x}_3{}'(\\sigma}  \\def\\Sigma{\\Sigma)}~.\n\\end{equation}\nThe functions $A_i \\prt{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta,\\eta,\\sigma}  \\def\\Sigma{\\Sigma}$ are given by\n\\begin{eqnarray}\\la{als}\nA_0=-\\tilde{g}_{tt} \\tilde{g}_{uu}~,\\quad\nA_1=\\tilde{g}_{01}^2 -\\tilde{g}_{00}\\tilde{g}_{11}~,\\quad A_3=\\tilde{g}_{03}^2-\\tilde{g}_{00}\\tilde{g}_{33}~,\\quad A_{13}=2\\prt{\\tilde{g}_{01}\\tilde{g}_{03}-\\tilde{g}_{00}\\tilde{g}_{13}}~,\n\\end{eqnarray}\nwhere the metric elements are of the rotated boosted of the metric \\eq{metric1}.  It is straightforward to obtain the conjugate momenta $\\Pi_1=\\partial {\\cal L}\/\\partial \\tilde{x}_{1}'$ and $\\Pi_3 =\\partial {\\cal L}\/\\partial{\\tilde{x}_{3}'}$,  then we can solve the system of the two equations for $\\tilde{x}_{1}'$ and $\\tilde{x}_{3}'$ to obtain\n\\begin{eqnarray}\\label{p1sol}\n&& \\tilde{x}_{1}'^2=\\frac{4 A_0 \\prt{\\Pi_3 A_{13}-2\\Pi_1 A_3}^2}{\\prt{A_{13}^2-4 A_1 A_3}\\prt{4 \\Pi_1^2 A_3-4 \\Pi_1 \\Pi_3 A_{13}+A_{13}^2+4 A_1\\prt{\\Pi_3^2-A_3}}}~,\\\\\\la{p3sol}\n&& \\tilde{x}_{3}'^2=\\frac{4 A_0 \\prt{\\Pi_1 A_{13}-2\\Pi_3A_1}^2}{\\prt{A_{13}^2-4 A_1 A_3}\\prt{4 \\Pi_1^2 A_3-4 \\Pi_1 \\Pi_3 A_{13}+A_{13}^2+4 A_1\\prt{\\Pi_3^2-A_3}}}~.\n\\end{eqnarray}\nThe requirement of real solutions for the worldsheet, suggests that the momenta remain real along the full string. This imposes the existence of a horizon on the worldsheet, at a critical value $u_c$ that is given by solving the algebraic equations requiring the numerators and the denominators of \\eq{p1sol}, \\eq{p3sol}  to have coinciding zeros\n\\begin{equation}\\la{wshorizon}\n4 A_1 A_3=A_{13} ^2 \\big|_{u=u_c}~, \\qquad \\Pi_1=\\frac{A_{13} \\Pi_3}{2 A_3} \\bigg|_{u=u_c}~,\\qquad  A_{13}= 2 \\Pi_1 \\Pi_3 \\big|_{u=u_c}~.\n\\end{equation}\nNotice the simplicity of the above equations which also leads to a further simplification of the expressions of the momenta evaluated at $u_c$\n\\begin{equation}\\la{momsim}\n\\Pi_3^2=A_{3}\\big|_{u=u_c}~,\\qquad \\Pi_1^2=\\frac{A_{13}^2}{4 A_3}\\bigg|_{u=u_c}~. \n\\end{equation}\nIt is useful to express the above conditions in terms of the initial holographic background elements, since they take a compact form\n\\begin{equation}\n \\Pi_3^2=-g_{00}\\prt{g_{33}c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 +g_{11}s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2}\\big|_{u=u_c}~,\\quad\n \\Pi_1^2=\\frac{-g_{00}\\prt{g_{11}-g_{33}}^2 s_{2\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2  \\cosh^2\\eta}{4 \\prt{g_{33}c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 +g_{11} s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2}}\\bigg|_{u=u_c}~.\n\\end{equation}\nFinally, the on-shell action comes by the substitution of \\eq{p1sol} and \\eq{p3sol} into the action \\eq{act1} and can be written as\n \\begin{equation}\\la{action3}\nS=-\\frac{1}{2\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'}\\int d \\sigma}  \\def\\Sigma{\\Sigma\\sqrt{\\frac{ A_0 \\prt{A_{13}^2-4A_1 A_3}}{A_{13}^2+4 A_1\\prt{\\Pi_3^2-A_3}+4 \\Pi_1^2 A_3-4 \\Pi_1 \\Pi_3 A_{13}}} ~,\n\\end{equation}\nwhich is related to the energy of the system. The system of our equations \\eq{p1sol} and \\eq{p3sol} is solvable analytically only in certain limits, otherwise a numerical treatment is required.\nOf particular interest  is the  ultra-relativistic limit, since the regime that plays a leading role is the near the boundary where we are allowed to investigate the system perturbatively.\n\n \n\n\\subsection{Worldsheet Horizon Equation and Conjugate Momenta}\n\n\nThe existence of the string worldsheet horizon is determined by \\eq{wshorizon}, the equation can be written in the following form\n\\begin{equation}\\la{ws2}\ng_{11} g_{33}\\prt{g_{00}+v^2 \\prt{g_{33}c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2+g_{11}s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2}}\\cdot\\prt{\\prt{g_{11} s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 +g_{33} c_\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta^2}\\prt{1-v^2}}^{-1}=0\\big|_{u=u_c}~.\n\\end{equation}\nwhere we include the denominator in order to comment on the asymptotic behavior of the solutions. \nThis is an algebraic equation that determines the $u_c$ dependence on the velocity in the background under study.\nEven without knowing the exact background the equation the behavior of the criticality in the string worldsheet can be extracted. For zero or low velocity $v\\simeq0$, we have static quarks and the worldsheet critical value $u_c\\simeq u_h$ approaches the black hole horizon resulting to straight string solutions. For these velocities we essentially solve \\eq{ws2} for the blackening factor in $g_{00}$. The opposite ultra-relativistic limit $v\\rightarrow 1$, leads the solution of eq. \\eq{ws2} to the boundary regime in order to compensate the infinity generated by the denominator.\n\nThe equation \\eq{ws2} can be solved for $g_{00}$\n\\begin{equation} \\la{ucsol2}\ng_{00}=-v^2\\prt{g_{33} c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 +g_{11} s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2}\\big|_{u=u_c}~,\n\\end{equation}\nwhich directly simplifies the momenta \\eq{momsim}  \n\\begin{equation}\\la{mom3}\n\\Pi_1^2=\\frac{\\prt{g_{11}-g_{33}}^2s_{2 \\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 ~v^2}{4 \\prt{1-v^2}}\\bigg|_{u=u_c}~,\\qquad \\Pi_3^2= \\prt{g_{33} c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2 + g_{11} s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2}^2 v^2\\big|_{u=u_c}~,\n\\end{equation}\nwhere the metric elements in the right hand side have still implicit dependence on the velocity via the $u_c$ solution of the algebraic equation \\eq{ucsol2}.\n\nNevertheless, from the equations \\eq{mom3} suffices to observe that for an isotropic holographic  renormalization flow, $\\Pi_1=0$. To understand this notice that there is a rotation to align the direction of the wind along the $\\tilde{x}_3$ direction. For isotropic theories, all three directions are equivalent, which means that the non-diagonal $\\tilde{g}_{xi}$ metric elements vanish and therefore only the momentum along the plasma wind is non-zero.  At the same time $\\Pi_3$ becomes independent of $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta$ and it is never zero for the same reason, after the rotation the wind blows always along the direction $\\tilde{x}_3$.   Moreover, even for a non-trivial anisotropic background when the motion happens exclusively along one of the axes $x_1$  or $x_3$ of the initial coordinate system, there is a vanishing $\\Pi_1$  momentum along the orthogonal axis, since the direction of the wind is along  $\\tilde{x}_3$. \n\n\\subsection{Ultra-Relativistic Analysis}\n\nAs we have elaborated in the previous section  $v\\rightarrow 1$ implies that $u_c \\rightarrow (boundary)$. Let us assume a space which is asymptotically AdS, and along the RG flow is allowed to have a potential breaking of rotational symmetry.\n\n\\subsubsection{Asymptotically AdS Renormalization Group  Flows}\n\nThe near boundary expansion of the original theory takes the form\n\\begin{equation}\\la{expans}\nds^2=\\frac{du}{u^2}+\\gamma}  \\def\\Gamma{\\Gamma}  \\def\\dot{\\gamma}{{\\dot\\gamma}_{ij} dx^i dx^j~,\n\\end{equation}\nwith\n\\begin{equation}\n\\gamma}  \\def\\Gamma{\\Gamma}  \\def\\dot{\\gamma}{{\\dot\\gamma}_{ij}=\\frac{1}{u^2}\\prt{\\eta_{ij}+u^2 g^{(2)}_{ij}+u^4 \\prt{ g_{ij}^{(4)}+ h_{ij}^{(4)} \\log u}+\\ldots}~.\n\\end{equation}\nIn order to investigate the expansion in detail and to investigate all the cases let us distinguish the two type of motions with respect to the $x_3$ direction.\n\\newline\n\\textbf{Motion for $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta\\neq \\pi\/2$ }:\\newline\nFor a motion outside the transverse plane, $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta\\neq \\pi\/2$,  the momentum along $\\tilde{x}_1$ reads directly from \\eq{mom3} without the need of solving the algebraic equation for $u_c$ since it is independent of it\n\\begin{equation}\\la{finalmoma1}\n\\Pi_1^2\\simeq\\frac{\\prt{g_{11}^{(2)}-g_{33}^{(2)}}^2\\sin^2 2\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}{4(1-v^2)}~.\n\\end{equation}\nIn contrast, the momentum along $x_3$ depends on $u_c$, so that \\eq{ucsol2} needs to be solved explicitly  to give\n\\begin{equation}\\la{soluca1}\nu_c^2\\simeq \\frac{2\\prt{1-v^2}}{A_5^{(2)}}~,\n\\end{equation}\nwhere $A_5^{(2)}$ is defined via\n\\begin{equation}\\la{a5def}\nA_5^{(2)}:= 2 g_{00}^{(2)}+g_{11}^{(2)}+g_{33}^{(2)}+\\prt{g_{33}^{(2)}-g_{11}^{(2)}} c_{2\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}~.\n\\end{equation}\nNotice the $u_c\\sim \\sqrt{1-v^2}$ dependence which is crucial in what follows and that  the \\eq{soluca1} is assumed for now to be regular. For example, if we had a fully isotropic geometry \\eq{soluca1} would have been  singular. The substitution of $u_c$ in equation \\eq{mom3} leads to the following  expression for the remaining momentum\n\\begin{equation}\\la{finalmoma3}\n \\Pi_3^2=\\frac{ A_5^{(2)}{}^2}{4\\prt{1-v^2}^2}~.\n\\end{equation}\nThe validity of the derived expressions rely on the condition that the expressions are regular otherwise higher order terms need to be considered, a case we examine later.   To obtain eventually the scaling of the action with the velocity, we substitute  the momenta \\eq{finalmoma1}  and \\eq{finalmoma3} to \\eq{action3} to obtain\n\\begin{equation}  \\la{ssb1}\n2\\pi \\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}' S_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta\\neq \\frac{\\pi}{2}}\\simeq -2  \\int_0^{u_c} d u \\frac{1}{u^2}\\prt{1-\\frac{u^2 A_5^{(2)}}{4\\prt{1-v^2}}} ~.\n\\epsilon}        \\def\\vare{\\varepsilon\nIn order to extract the Lorentz factor dependence, we  rescale $u= u_s \\prt{1-v^2}^{1\/2}$ so that we extract the $v$ dependence of the critical point $u_c$. Then the ultrarelativistic expansion scales with the Lorentz factor as\n\\begin{equation}  \\la{ss1}\n2\\pi \\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}' S_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta\\neq \\frac{\\pi}{2}}\\simeq -2  \\prt{1-v^2}^{-\\frac{1}{2}}\\int_0^{u_{sc}} d u_s \\frac{1}{u_s^2}\\prt{1-\\frac{1}{4}u_s^2 A_5^{(2)}\n\\epsilon}        \\def\\vare{\\varepsilon\nThe action is integrated twice to count the energies of both free moving particles and the upper limit $u_{cs}$ is the rescaled worldsheet horizon. It serves as an upper bound in the integration since after this point the string is causally disconnected from the boundary. All the $v$ dependence has been extracted outside the action and we conclude that $S_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta\\neq \\frac{\\pi}{2}} \\sim  \\prt{1-v^2}^{-\\frac{1}{2}}$, when $u_c$ given by the equation \\eq{soluca1} is regular.\n\nLet us examine backgrounds where the expression  \\eq{soluca1} is singular to show that the leading contributions comes from higher orders in the asymptotic expansion. For this to happen the holographic background has to satisfy\n \\begin{equation} \\la{cona1}\n A_5^{(2)}=0~,\n\\end{equation}\nwhich can occur only for a fixed angle $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta$ or for an isotropic background. Let us again work with the generic case and for simplicity we  assume that the leading leading logarithmic contributions are absent, $h_{ij}^{\\prt{4}}=0$, or cancel each other in the expressions. Then the solution of the critical point of the worldsheet becomes\n\\begin{equation}\nu_c^2=\\sqrt{2} \\frac{\\sqrt{1-v^2}}{\\sqrt{A_5^{(4)}}}~,\n\\end{equation}\nwhere we have used the condition \\eq{cona1} and we require the above solution to be real.  $A_5^{(4)}$ is defined in the same way as  $A_5^{(2)}$ in \\eq{a5def} but with the next order terms. The leading contributions in the momenta now are found as\n\\begin{equation}\n\\Pi_1^2=\\frac{\\prt{g_{33}^{(2)}-g_{11}^{(2)}}^2 s_{2\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}^2}{4\\prt{1-v^2}}~, \\qquad\n \\Pi_3^2=\\frac{ A_5^{(4)}}{2\\prt{1-v^2}}~,\n\\end{equation}\nwhile the action takes the form\n\\begin{equation}  \\la{ss2}\nS_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta\\neq \\frac{\\pi}{2}}\\simeq   -\\frac{\\prt{1-v^2}^{-\\frac{1}{4}}}{ 2 \\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'}\\int_0^{u_s} d u_s\\frac{1}{{u_s}^2}\\prtt{1-\\frac{1}{16}\\prt{4 A_5^{(4)}-\\prt{g_{00}^{(4)}+g_{33}^{(4)}}+\\prt{1-c_{4\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta}}\\prt{g_{33}^{(2)}-g_{11}^{(2)}}^2}}   ~,\n\\epsilon}        \\def\\vare{\\varepsilon\nwhere we have rescaled the radial coordinate as $u_s=u \\prt{1-v^2}^{1\/4}$. We point out, that only the leading terms four our study are enough to determine the scaling.\n\nTo summarize the findings so far, for motion with $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta\\neq \\pi\/2$, the free energy scales with the Lorentz factor as $S\\sim \\prt{1-v^2}^{-\\frac{1}{2}}$ as long as the asymptotic condition $A_5^{(2)} \\neq 0$ holds, where $A_5^{(2)}$  is given by \\eq{a5def}. Otherwise, if the background satisfies the above condition for $A_5^{(2)}$  asymptotically,  we obtain  $S\\sim \\prt{1-v^2}^{-\\frac{1}{4}}$.\\newline\n\\textbf{Motion for $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta=\\pi\/2$ }:\\newline\nFor the motion in the transverse plane, $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta=\\pi\/2$, we deal with more compact expressions. Working again in the leading order we obtain from the solution of \\eq{ucsol2} for the $u_c$:\n\\begin{equation}\\la{soluca2}\nu_c^2=\\frac{1-v^2}{g_{00}^{(2)} +g_{11}^{(2)}}~,\n\\end{equation}\nwhich results for the momenta\n\\begin{equation}\\la{finalmoma2}\n\\Pi_1=0~,\\qquad \\Pi_3^2=\\frac{ \\prt{g_{00}^{(2)} +g_{11}^{(2)}}^2 }{\\prt{1-v^2}^2}~.\n\\end{equation}\nThe action \\eq{action3} in the near boundary regime comes by the use of the momenta \\eq{finalmoma2} and the rescaling $u= u_s \\sqrt{1-v^2}$ to isolate the velocity dependence and we obtain\n\\begin{equation}  \\la{ss3}\n \\pi \\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'S_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta= \\frac{\\pi}{2}}\\simeq -\\prt{1-v^2}^{-\\frac{1}{2}}\\int_0^{u_{cs}} d u_s \\frac{1}{u_s^2}\\prt{1-u_s^2 \\frac{g_{00}^{(2)}+g_{11}^{(2)}}{2}} ~.\n\\end{equation}\n\nThe leading contribution changes when the equation \\eq{soluca2} is singular for the holographic background. Then the leading order to the Wilson lines will be the next one in the asymptotic expansion.  In this case the background satisfies \n\\begin{equation}\\la{cona2}\ng_{00}^{(2)} +g_{11}^{(2)}=0~,\n\\end{equation}\nand the worldsheet horizon is found by solving  \\eq{ucsol2} and is equal to\n\\begin{equation}\\la{u2}\nu_c^2\\simeq  \\frac{\\sqrt{1-v^2}}{\\sqrt{g_{00}^{(4)} +g_{11}^{(4)}}}~.\n\\end{equation}\nThe above expression is valid even with logarithmic terms present, as long as $h_{00}^{(4)}=h_{11}^{(4)}$. However, to simplify the presentation let us consider vanishing logarithmic contributions.\nThe momentum $\\Pi_3$ takes the simple form\n\\begin{equation} \\la{finalmom3}\n\\Pi_3^2\\simeq \\frac{g_{00}^{(4)} +g_{11}^{(4)}}{1-v^2}~.\n\\end{equation}\nThe Lorentz scaling in the action  can be found by rescaling $u= u_s \\prt{1-v^2}^{1\/4}$ where we obtain in the near boundary regime the divergent action as\n\\begin{equation} \\la{ss4}\n \\pi \\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}' S_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta= \\frac{\\pi}{2}}\\simeq -\\prt{1-v^2}^{-\\frac{1}{4}}\\int_0^{u_{cs}} d u_s \\frac{1-u_s^4\\prt{g_{00}^{(4)}+g_{11}^{(4)}}}{u_s^2\\prt{1-2 u_s^2 \\prt{g_{00}^{(4)}+g_{11}^{(4)}}} }~.\n\\end{equation}\nIn summary we find from equations \\eq{ss1}, \\eq{ss2}, \\eq{ss3} and \\eq{ss4}, that $S$ scales as $S\\sim \\prt{1-v^2}^{-\\frac{1}{2k}}$, where $k=1,2$ and depends on the angle of motion and the asymptotic behavior. When the leading terms of the asymptotic expansion of the background satisfy \\eq{cona1} and \\eq{cona2} then $k=2$ and the scalings read off the equation \\eq{ss2} and \\eq{ss4}, otherwise $k=1$ and the actions is given asymptotically by  \\eq{ss1} and \\eq{ss3}.\n\n\n\n\\section{Velocity laws for Wilson loops and  the Screening Length}\n\nIn the previous section we have computed the leading contributions of the velocity for the singlets' motion. In this section we compute the expectation value of the Wilson loop from where we read off the velocity dependence for the heavy bound state potential and for the screening length. We will apply the mass renormalization scheme, subtracting the free energy divergences of the bound state from the infinite masses of the singlets we derived in the previous section in order to extract the interaction energy of the bound state. We take the orientation of the dipole to be arbitrary with respect to the velocity of the plasma and the $\\tilde{x}_i$ coordinates. For the Wilson loop computation we fix the radial gauge and have the string embedded along the spatial directions $\\prt{x_1(u),x_2(u),x_3(u)}$. The string has a turning point at $u_0$, and the action reads\n\\begin{equation}\\la{act2}\nS=-\\frac{1}{2\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}{}'} \\int d\\sigma}  \\def\\Sigma{\\Sigma \\sqrt{ A_0+ A_1 \\tilde{x}_1{}'(\\sigma}  \\def\\Sigma{\\Sigma)^2+ A_2 \\tilde{x}_2{}'(\\sigma}  \\def\\Sigma{\\Sigma)^2 +A_3 \\tilde{x}_3{}'(\\sigma}  \\def\\Sigma{\\Sigma)^2+A_{13} \\tilde{x}_1{}'(\\sigma}  \\def\\Sigma{\\Sigma) \\tilde{x}_3{}'(\\sigma}  \\def\\Sigma{\\Sigma)}~,\n\\end{equation}\nwhere the functions $A_i \\prt{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta,\\eta,\\sigma}  \\def\\Sigma{\\Sigma}$  are given by \\eq{als} and additionally\n\\begin{eqnarray}\nA_2=-\\tilde{g}_{00}\\tilde{g}_{22}~.\n\\end{eqnarray}\nWe can solve the system of the momenta  $\\Pi_1=\\partial {\\cal L}\/\\partial \\tilde{x}_{1}'$,  $\\Pi_2=\\partial {\\cal L}\/\\partial \\tilde{x}_{2}'$ and $\\Pi_3 =\\partial {\\cal L}\/\\partial{\\tilde{x}_{3}'}$ with respect to the derivatives of the coordinates to get\n\\begin{eqnarray}\\nonumber\n&&\\tilde{x}_1' =\\frac{2 \\sqrt{A_0 A_2}\\prt{2\\Pi_1 A_3-\\Pi_3 A_{13} }}{\\sqrt{-\\prt{A_{13} ^2- 4 A_1 A_3}{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}}}~,\\\\ \\la{x1}\n&&\\tilde{x}_2'=\\frac{\\Pi_2 \\sqrt{-A_0\\prt{A_{13}^2-4 A_1 A_3}}}{\\sqrt{A_2 {\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}}}~,\\\\ \\nonumber\n&&\\tilde{x}_3'=\\frac{ 2\\sqrt{A_0 A_2}\\prt{2\\Pi_3 A_{1}-\\Pi_1 A_{13}}}{\\sqrt{-\\prt{A_{13} ^2- 4 A_1 A_3}{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}}}~,\n\\end{eqnarray}\nwith a common denominator ${\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}$ which reads\n\\begin{equation}\\la{dd1}\n{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}=A_{13}^2\\prt{\\Pi_2^2-A_2} +4 \\Pi_1 \\Pi_3 A_{13} A_2-4\\prt{\\Pi_1^2 A_2 A_3 +A_1\\prt{A_2\\prt{\\Pi_3^2-A_3}+\\Pi_2^2 A_3}}~.\n\\end{equation}\nThe boundary length of the state is\n\\begin{equation} \\prt{ L_1, L_2, L_3}=\\prt{L s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta_d} s_{\\phi_d}, L s_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta_d} c_{\\phi_d}, L c_{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta_d}}~,\n\\end{equation}\nwhere $\\prt{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta_d, \\phi_d}$ are the bound state orientation angles. To determine it we need to integrate the equations \\eq{x1} as\n\\begin{equation}\\la{lls}\nL_i =2\\int_0^{u_0}  \\tilde{x}_i ~d\\sigma}  \\def\\Sigma{\\Sigma~. \n\\end{equation}\nBy substituting the equations of motion \\eq{x1} to the action \\eq{act2} we can eliminate the derivatives and eventually  express it in terms of the momenta\n \\begin{equation}\\la{actionbs3}\nS=-\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'}\\int_0^{u_0} d \\sigma}  \\def\\Sigma{\\Sigma\\sqrt{\\frac{ - A_0  A_2 \\prt{A_{13}^2-4A_1 A_3}}{{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}}} ~.\n\\end{equation}\nOur aim is to extract the dependence behavior of the interaction energy \\eq{actionbs3} on the Lorentz factor for non-trivial holographic renormalization group flows. The turning point of the string worldsheet is specified by solving the algebraic equation ${\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}=0$ given by the \\eq{dd1}. The algebraic equation  simplifies considerably for high and low velocities. At the limit of high velocities where our primary interest is, it reads\n\\begin{equation}\n\\frac{g_{22}\\prt{g_{00}+g_{33} c_\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta^2+g_{11} s_\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta^2 }\\prtt{g_{11}\\prt{\\Pi_3^2 g_{33}+g_{00}^2\n\t\tg_{33}+g_{00}\\prt{\\Pi_3^2+g_{33}^2}c_\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta^2}+g_{00} g_{33}\\prt{\\Pi_3^2 +g_{11}^2}s_\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta^2}}{\\prt{1-v}^2}=0,\n\\end{equation}\nwith a solution for the turning point $u_0$ that approaches the near boundary regime. So far, we have not made any approximations or assumptions and all the expression derived hold for  any holographic background. To progress further so as to find the explicit velocity dependence, we apply the formalism on non-trivial RG holographic flows that are asymptotically AdS. \nLet us split the analysis again with respect to the direction of the wind  and the values of angles $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta$.\n\n\\subsection{Asymptotically AdS Renormalization Group Flows}\n\\textbf{Motion for $\\prt{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta\\neq \\pi\/2}$}\\newline\nThe state's boundary length integrals \\eq{lls} in the asymptotic regime of the metric \\eq{expans}, after some algebra take the compact form\n\\begin{eqnarray} \\la{ll2}\nL_i\\simeq 2\\int_0^{u_0} du ~\\frac{\\Pi_i u^2}{\\sqrt{D_0}}\\prt{1+ u^2 \\prt{s_i \\frac{A_5^{(2)}}{4\\prt{1-v^2} } +\\tilde{s}_i\\frac{\\prt{g_{33}^{(2)}-g_{11}^{(2)}}s_{2\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta} }{2\\sqrt{1-v^2}}}}~,\n\\end{eqnarray}\nwhere $s_1=s_2=s_3\/3=-1$ and $\\tilde{s}_1=\\tilde{s}_3^{-1}=\\Pi_3\/\\Pi_1~, ~\\tilde{s}_2=0~$ and the $u-$independent $A_5^{(2)}$ has been defined in \\eq{a5def}. Notice that in this section we omit terms in the expansion that will be proven to be subleading with the rescalings below.  ${\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}_0$ is defined through the  asymptotic expansion of ${\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}$ as\n\\begin{equation}\\la{d1exp}\n{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}\\simeq \\frac{4}{u^{12}}\\prtt{1-\\frac{ A_5^{(2)}}{1-v^2} u^2+\\prt{\\frac{ A_5^{(2)}{}^2}{4\\prt{1-v^2}^2}- \\sum_{i=1}^{3}\\Pi_i^2+ \\frac{C_1}{1-v^2}+C_2}u^4 }:=\\frac{4}{u^{12} }{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}_0~,\n\\end{equation}\nwhere $C_i$ are u-independent terms that will be proven subleading in the subsequent analysis.\nThe action for the bound state can be found from \\eq{actionbs3} by putting all the expansions together and reads\n\\begin{equation} \\la{actionb3}\nS\\simeq-\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\int_0^{u_0} d u\\frac{1}{u^2 \\sqrt{D_0}}\\prt{1 - \\frac{3A_5^{(2)} }{4\\prt{1-v^2}} u^2}~.\n\\end{equation}\nAt this stage we have to impose the natural requirement, \\cite{Maldacena:1998im,Drukker:1999zq,Chu:2008xg,Giataganas:2022qgq}, to have an action with leading divergences that are canceled by the ultraviolet divergence of the infinite massive singlets \\eq{ss1}, and that the momenta increase with the increase of the velocity.  To extract the correct scaling dependence, we rescale $u$ and $\\Pi$ at the ultrarelativistic limit as $u= u_s \\sqrt{1-v^2}$ and $\\Pi_i=\\Pi_{s,i} \\prt{1-v^2}^{-1}$.  The energy now becomes in the rescaled $u_s$ coordinates\n\\begin{equation} \\la{actionbsa3}\nS\\simeq -\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\frac{1}{\\sqrt{1-v^2}}\\int_0^{u_{s0}} d u_s \\frac{1}{u_s^2\\sqrt{D_{s0}}}\\prt{1-  \\frac{3A_5^{(2)} }{4}u_s^2}\\simeq -\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\frac{1}{\\sqrt{1-v^2}}\\int_0^{u_{s0}} d u_s \\prt{\\frac{1}{u_s^2} + \\frac{A_5^{(2)} }{4}}~\n\\end{equation}\nand boundary lengths read from equation \\eq{ll2}\n\\begin{equation} \\la{llf}\nL_i\\simeq 2 \\prt{1-v^2}^\\frac{1}{2} \\Pi_{s,i}\\int_0^{u_{s0}} d u_s~ \\frac{u_s^2}{\\sqrt{D_{s0}}}\\prt{1+ s_i \\frac{u_s^2 A_5^{(2)}}{4}}~,\n\\end{equation}\nwhere notice that $D_{s0}$ is obtained from the rescaling in $D_0$ of \\eq{d1exp} and is $v$-independent\n\\begin{equation}\nD_{s0}=1- A_5^{(2)} u_s^2+\\prt{\\frac{ A_5^{(2)}{}^2}{4}- \\sum_{i=1}^{3}\\Pi_{s,i}^2 }u_s^4~.\n\\end{equation}\nThe action \\eq{actionbsa3} indeed has the same ultraviolet asymptotics  with the Wilson line action \\eq{ss1} and the divergences cancel each other as expected.  \n\nNotice that the integrands of $L_i$ in equations \\eq{llf} are independent of the Lorentz factor and therefore $L_i\\sim (1-v^2)^{1\/2}$. The strings under study have a maximum length that depends on the scales of the theory and in the ultrarelativistic limit this is obtained by finding the maximum possible distance that the bound state exist, merely  by the maximization of \\eq{llf}. The velocity dependence has extracted outside the integrands in   \\eq{llf} and the integrals are bounded. Therefore  the maximum separation between a bound quark-antiquark state, which is the screening length for the bound state, scales as $L_{max}\\sim \\prt{1-v^2}^{\\frac{1}{2}}~.$\n\nLet us assume that there exist a scenario for a certain angle that the condition $A_5^{(2)}=0$, like the prior analysis of the singlet \\eq{cona1}.  We expect that the leading action divergences  scales with the Lorentz factor as in \\eq{ss2}.  The asymptotic expansion of ${\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}$ becomes\n\\begin{equation}\n{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}\\simeq \\frac{4}{u^{12}}\\prtt{1+u^4\\prt{-\\frac{  A_5^{(4)}}{\\prt{1-v^2}}- \\sum_{i=1}^{3}\\Pi_i^2  +\\frac{C_1}{1-v^2}+C_2}}=\\frac{4}{u^{12} }{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}_0~,\n\\end{equation}\nwhere the $C_i$ terms will be proved subleading and are not the same as in previous expressions where we have used the same notation. The boundary length now reads\n\\begin{eqnarray} \\la{ll3}\nL_i\\simeq 2\\int_0^{u_0} du ~\\frac{\\Pi_i u^2}{\\sqrt{D_0}}\\prt{1 +\\tilde{s}_i u^2 \\frac{\\prt{g_{33}^{(2)}-g_{11}^{(2)}}s_{2\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta} }{2\\sqrt{1-v^2}}+u^4 \\prt{s_i\\frac{ A_5^{(4)}}{4\\prt{1-v^2}}+\\hat{s}_i \\frac{\\prt{g_{33}^{(2)}-g_{11}^{(2)}}^2\\prt{c_{4\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta} -1}}{8\\prt{1-v^2}}}}~,\n\\end{eqnarray}\nwhere $\\hat{s}_1=\\hat{s}_2=0~,~ \\hat{s}_3=1$ and the subleading terms will be proven that do not contribute. The action takes the form\n\\begin{equation} \\la{actionb4}\nS\\simeq-\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\int_0^{u_0} d u\\frac{1}{u^2 \\sqrt{D_0}}\\prt{1 - \\frac{3 A_5^{(4)}}{4\\prt{1-v^2}} u^4}~.\n\\end{equation}\nActing as in previous cases, we rescale $u= u_s \\prt{1-v^2}^{1\/4}$ and then $\\Pi_i$ needs to be rescaled as  $\\Pi_i=\\Pi_{s,i} \\prt{1-v^2}^{-1\/2}$ to give the energy in the new redial coordinates\n\\begin{equation} \\la{actionb33}\nS\\simeq -\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\frac{1}{\\prt{1-v^2}^{\\frac{1}{4}}}\\int_0^{u_{s0}} d u_s \\frac{1}{u_s^2\\sqrt{D_{s0}}}\\prt{1-  \\frac{3  A_5^{(4)}}{4}u_s^2}~.\n\\end{equation}\nwhere\n\\begin{equation}\nD_{s0}=1- A_5^{(4)}u_s^4- \\sum_{i=1}^{3}\\Pi_{s,i}^2 u_s^4 ~.\n\\end{equation}\nThe leading contributions to the expression of $L_i$ close to the boundary become\n\\begin{equation}\nL_i\\simeq 2\\int_0^{u_{s0}} du_s ~\\frac{\\Pi_{s,i} u_s^2}{\\sqrt{D_{s0}}}\\prt{1 +u_s^4 \\prt{s_i\\frac{A_6}{4}+\\hat{s}_i \\frac{\\prt{g_{33}^{(2)}-g_{11}^{(2)}}^2\\prt{c_{4\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta} -1}}{8}}}~,\n\\end{equation}\nand therefore follows that $L_{max}\\sim  \\prt{1-v^2}^{\\frac{1}{4}}~$ provided that the background satisfies $A_5^{(2)}=0$ for a motion along a certain angle and that the contributions of the logarithmic terms in the asymptotic expansion cancel each other or vanish.\\newline\n\\textbf{Motion for $\\prt{\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta=\\pi\/2}$}\\newline\nFor motion within the transverse plane, $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta=\\pi\/2$, the computation is along the same lines. The asymptotic expansion of ${\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}$ is\n\\begin{equation}\\la{d1exp11}\n{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}\\simeq \\frac{4}{u^{12}}\\prtt{1-\\frac{ 2\\prt{g_{00}^{(2)}+g_{11}^{(2)}}}{1-v^2} u^2+\\prt{\\frac{ \\prt{g_{00}^{(2)}+g_{11}^{(2)}}^2}{\\prt{1-v^2}^2}- \\sum_{i=1}^{3}\\Pi_i^2+\\frac{C_1}{1-v^2}+C_2} u^4 }=\\frac{4}{u^{12} }{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}_0~.\n\\end{equation}\nThe state's boundary length integrals \\eq{lls} in the asymptotic regime of \\eq{expans}, after some algebra take the compact form\n\\begin{eqnarray} \\la{llb}\nL_i\\simeq 2\\int_0^{u_0} du ~\\frac{\\Pi_i u^2}{\\sqrt{D_0}}\\prt{1- s_i u^2 \\frac{\\prt{g_{00}^{(2)}+g_{11}^{(2)}}}{4\\prt{1-v^2} } }~,\n\\end{eqnarray}\nwhere $s_1=s_2=s_3\/3=1$ as defined earlier.  The action for the bound state reads\n\\begin{equation} \\la{actionbb3}\nS\\simeq-\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\int_0^{u_0} d u\\frac{1}{u^2 \\sqrt{D_0}}\\prt{1 - \\frac{3\\prt{g_{00}^{(2)}+g_{11}^{(2)}} }{2\\prt{1-v^2}} u^2+\\frac{3\\prt{g_{00}^{(2)}+g_{11}^{(2)}}^2 }{2\\prt{1-v^2}^2} u^4}~.\n\\end{equation}\nWe require for the action's ultraviolet-divergences to get cancelled with the divergences  of the infinite massive singlets \\eq{ss3}. In the ultrarelativistic limit we rescale $u$ and $\\Pi_i$,  $u= u_s \\sqrt{1-v^2}$ and $\\Pi_i=\\Pi_{s,i} \\prt{1-v^2}^{-1}$ to get the  action for the bound state in new coordinates\n\\begin{equation}\nS\\simeq-\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\frac{1}{\\sqrt{1-v^2}}\\int_0^{u_{s0}} d u_s\\frac{1}{u_s^2 \\sqrt{D_{s0}}}\\prtt{1 - \\frac{3}{2}\\prt{\\prt{g_{00}^{(2)}+g_{11}^{(2)}}  u_s^2+\\prt{g_{00}^{(2)}+g_{11}^{(2)}}^2  u_s^4}}~,\n\\end{equation}\nextracting the right velocity dependence, where $D_{s0}$ in the rescaled coordinates reads\n\\begin{equation}\nD_{s0}= 1- 2\\prt{g_{00}^{(2)}+g_{11}^{(2)}} u_s^2+\\prt{g_{00}^{(2)}+g_{11}^{(2)}}^2u_s^4- \\sum_{i=1}^{3}\\Pi_{s,i}^2 u_s^4 ~.\n\\end{equation}\nThe boundary lengths in the rescaled radial coordinate are equal to\n\\begin{eqnarray} \\la{llbf}\nL_i\\simeq 2\\prt{1-v^2}^{\\frac{1}{2}}\\int_0^{u_{s0}} du_s ~\\frac{\\Pi_{i,s} u_s^2}{\\sqrt{D_{s0}}}\\prt{1- \\frac{1}{4}s_i u_s^2 \\prt{g_{00}^{(2)}+g_{11}^{(2)}}} ~.\n\\end{eqnarray}\nTherefore the screening length of the bound state scales as $L_{max}\\sim \\prt{1-v^2}^{\\frac{1}{2}}$\nwhere we again observe the half-integer power.\n\nWhen the leading contributions come from the next term of the FG expansion we get a different power scaling.  This happens when the asymptotic expansion of the metric satisfies $g_{00}^{(2)}=-g_{11}^{(2)}$. In this case ${\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}$ takes the form\n\\begin{equation}\\la{d1exp22}\n{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}\\simeq \\frac{4}{u^{12}}\\prtt{1+\\prt{-\\frac{ 2\\prt{g_{00}^{(4)}+g_{11}^{(4)}}}{\\prt{1-v^2}}- \\sum_{i=1}^{3}\\Pi_i^2 +C_2}u^4}=\\frac{4}{u^{12} }{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}_0~,\n\\end{equation}\nand $L_i$ reads\n\\begin{eqnarray} \\la{llbk}\nL_i\\simeq 2\\int_0^{u_0} du ~\\frac{\\Pi_i u^2}{\\sqrt{D_0}}\\prt{1- u^2 c_i+\\frac{k_i}{\\prt{1-v^2}}} ~,\n\\end{eqnarray}\nwhere $k_i$ and $c_i$ are u-independent terms. As has been mentioned earlier we omit terms that will be proven that do not affect the scaling. The action computation gives\n\\begin{equation} \\la{actionbb4}\nS\\simeq-\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\int_0^{u_0} d u\\frac{1}{u^2 \\sqrt{{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}_0}}\\prt{1 - \\frac{3\\prt{g_{00}^{(4)}+g_{11}^{(4)}} }{2\\prt{1-v^2}} u^2}~.\n\\end{equation}\nTo extract the right asymptotic of the divergences matching the Wilson line action \\eq{ss4} we rescale the radial coordinate and the momenta as $u= u_s \\prt{1-v^2}^{1\/4}$ and $\\Pi_i=\\Pi_{s,i} \\prt{1-v^2}^{-1\/2}$.\n The action for the bound state becomes\n \\begin{equation} \\la{actionbb4b}\n S\\simeq-\\frac{1}{\\pi\\alpha}      \\def\\dot{\\alpha}{{\\dot\\alpha}'} \\frac{1}{\\prt{1-v^2}^{1\/4}}\\int_0^{u_{s0} }d u_s\\frac{1}{u_s^2 \\sqrt{{\\cal D}} \\def{\\cal E}{{\\cal E}} \\def\\cF{{\\cal F}_{s0}}}\\prt{1 - \\frac{3\\prt{g_{00}^{(4)}+g_{11}^{(4)}} }{2} u_s^2}~, \\end{equation}\n and the boundary length of the bound state scales as\n\\begin{eqnarray} \\la{llbbbb}\nL_i\\simeq 2\\prt{1-v^2}^\\frac{1}{4}\\int_0^{u_{s0}} du_s ~\\frac{\\Pi_{s,i} u_s^2}{\\sqrt{D_{s0}}} ~,\n\\end{eqnarray}\nwhere  the leading contributions of $D_{s0}$ are $v$-independent. Therefore with the same arguing we extract the dependence of the screening length to the bound state $L_{max}\\sim \\prt{1-v^2}^{\\frac{1}{4}}$  for $\\theta} \\def\\Theta{\\Theta} \\def\\vth{\\vartheta=\\pi\/2$ and  $g_{00}^{(2)}=-g_{11}^{(2)}$, assuming that the logarithmic terms of the expansion vanish.\n\nIn summary, we find\n\\begin{eqnarray} \\la{llfff}\nL_{max}\\sim \\prt{1-v^2}^{\\frac{1}{2k}}\n\\end{eqnarray}\nwhere $k=1,2$ depending on the direction of motion and the asymptotic behavior of the metric. $k=2$ when the background conditions \\eq{cona1} and \\eq{cona2} are asymptotically satisfied for the corresponding direction of motion. Our analysis shows that theories dual to non-trivial RG flows which are asymptotically AdS and have a non-trivial renormalization group flow with broken rotational symmetry along the flow, are allowed to have bound states with a non-universal scaling dependence on the velocity, which we explicitly specify.\n\n\n\\textbf{Acknowledgments:}\nThe research work of D.G. is supported by the National Science and Technology Council (NSTC) of Taiwan with the Young Scholar Columbus Fellowship grant 110-2636-M-110-008.\n\n\\bibliographystyle{bibhep}\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:1}\n\n\n\n\nThe exponential decay of unstable systems \nhas been a well-known\nlaw since the early days of quantum theory.\nThe quantum description of those systems, however, allows \ndeviation from exponential decay both at shorter and longer times \\cite{Khalfin(1957)}\nthan those times over which the exponential decay law dominates. \\cite{Fonda(1978),Nakazato(1996)}\nThe short time deviation was actually found in a quantum tunneling experiment,\n\\cite{Wilkinson(1997)}\nwhile the long time deviation seems still not to have been detected in any quantum system. \n\\cite{Greenland(1988)}\nThe main cause that hinders the detection \nis considered as the smallness of the deviation at such long times. \n\\cite{Delgado(2006)}\n\n\n\n\n\nIn a recent study, a method enhancing the long time deviation was proposed.\n\\cite{Jittoh(2005)}\nThe decay of the unstable systems is theoretically modeled in \nthe time evolution of the survival probability of unstable initial state. \nThe survival probability is just the probability of finding the initial state \nin the state at a later time $t$. \nSince it is rewritten in a Fourier integral of the spectral function, \nits behavior at long times \nis determined by that of the spectral function near the threshold of the energy continuum. \n\\cite{Fonda(1978),Nakazato(1996)} \nThe essential aspect of the method is then \ndistorting the spectral function from the Breit-Wigner form and dislocating its peak\ntoward the threshold energy. \nMathematically, this causes a divergence of the spectral function, i.e., \nthe resolvent at the threshold. \nThen, it is expected that the exponential decay period disappears and \nthe survival probability at long times is increased. \nA similar idea was also considered in a related context. \\cite{Rzazewski(1982),Greenland(1988)}\nIn addition, in the analysis of the Friedrichs model \\cite{Friedrichs(1948),Exner(1985)} \nthat is often used for the study on the decays of the unstable systems, \nthe survival probability at long times sometimes exhibits a power decay law \nslower than that in cases of no divergence. \\cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)}\n\n\n\n\nThese facts remind the author \nof the zero energy resonance proposed by Jensen and Kato. \\cite{Jensen(1979)}\nAccording to them,\nsuch zero energy singularities are classified by \nthe zero energy eigenstates of the total Hamiltonian\nthat either belong to or do not belong to the Hilbert space.\nThe cases where such eigenstates exist are called the exceptional cases; \notherwise they are referred to as the regular case.\nThe result in Ref. \\makebox(9,1){\\Large \\cite{Jensen(1979)}}\\hspace{1mm}\nis concerned with the three-dimensional system of the one particle in short-range potentials, \nand they proved that the time evolution operator asymptotically decreases as \n$O(t^{-1\/2})$ for the exceptional cases, \nthat is slower than $O(t^{-3\/2})$ for the regular case. \nHowever, to the author's knowledge, \nthe zero energy resonance for the Friedrichs model seems not to have been examined \nin the previous studies \nincluding Refs. \\makebox(10,1){\\Large \n\\cite{Kofman(1994),Lewenstein(2000),Nakazato(2003),Jittoh(2005)}\n}\\hspace{10mm}, \nin spite of the wide applicability of the model to the various physical systems. \n\\cite{Facchi(1998),Antoniou(2001),Kofman(1994),Rzazewski(1982),Nakazato(2003)}\n\n\n\n\n\n\n\n\n\nIn the present paper, \nwe examine the zero energy singularities of the resolvent at the threshold energy \nfor the Friedrichs model from the viewpoint of the zero energy resonance, \\cite{Jensen(1979)} \nand clarify how the asymptotic behavior of the survival probability at long times is affected. \nThe Friedrichs model\\cite{Friedrichs(1948),Exner(1985)} \ndescribes the system of the finite discrete levels coupled with the continuous spectrum, \nin which the former can be interpreted as the unstable excited levels of atoms and the latter as \nthe environmental electromagnetic fields. \\cite{Kofman(1994),Facchi(1998),Antoniou(2001)}\nWe emphasize that the model is not restricted to the single level case\n\\cite{Friedrichs(1948),Exner(1985),Kofman(1994),Lewenstein(2000),Nakazato(2003),Rzazewski(1982),\nJittoh(2005),Facchi(1998),Antoniou(2001)}\nbut, rather, the $N$-level case, \n\\cite{Exner(1985),Davies(1974),Antoniou(2003),Antoniou(2004),Miyamoto(2004),Miyamoto(2005)}\nIn addition, we assume that the square modulus of the form factors \nvanishes at zero energy with an integer power, \\cite{Facchi(1998),Antoniou(2001),Seke(1994)}\nhowever it is treated without restriction to a specific form to some extent. \nFurthermore, since we only consider the initial state spanned by the discrete states, \nit is sufficient for us to see the reduced resolvent $\\tilde{R}(z)$ \nthat is just the restriction of the resolvent to the subspace spanned by the discrete states. \nThen, \nthe Fourier integral of $\\tilde{R}(z)$ that we call the reduced time evolution operator \n$\\tilde{U}(t)$ \nenables us to calculate the survival probability. \nIn fact \nit is expressed by the square modulus of the expectation value of $\\tilde{U}(t)$ \nin a given initial state. \nWe first study the zero energy eigenstates of the model \nwhich either belong to or do not belong to the Hilbert space.\nIt is then possible to estimate correctly \nthe asymptotic behavior of $\\tilde{R}(z)$ at small energies\nboth in the regular case and the exceptional cases. \nThe latter cases are examined in detail only for the first kind,\nwhere only the zero energy eigenstate not belonging to the Hilbert space exists. \nOn the basis of this analysis, \nwe can derive the long-time asymptotic formula for $\\tilde{U}(t)$ in those cases. \nIn particular,\nthe logarithmic decay proportional to $(\\log t)^{-1}$ of $\\tilde{U}(t)$\nis shown to occur in the exceptional case of the first kind for our form factors,\nwhich is extremely slower than the power decays in the regular case\nand in the exceptional case for another type of form factor. \n\\cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)}\nThese results are shown in Theorems \\ref{thm:rffimLong} and \\ref{thm:rffLong1st}.\n\n\n\n\nThe organization of the paper is as follows.\nWe first explain in Sec. \\ref{sec:3} the $N$-level Friedrichs model \nwith an appropriate Hilbert space, and then \nin Sec. \\ref{sec:4} we introduce the reduced resolvent $\\tilde{R}(z)$. \nSection \\ref{sec:5} is devoted to the identification of\nzero energy eigenstates in this model.\nIt is then possible to obtain\nthe asymptotic expansion of $\\tilde{R}(z)$ at small energies\nin Sec. \\ref{sec:5.5}, where we examine the regular and the exceptional case of the first kind.\nBy making sure of the relation between $\\tilde{R}(z)$ and $\\tilde{U}(t)$ in Sec. \\ref{sec:6},\nthe asymptotic formula for $\\tilde{U}(t)$ in the regular and the exceptional case\nof the first kind\nare derived in Sec. \\ref{sec:7} . Concluding remarks are given in Sec. \\ref{sec:8}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Hilbert space and the $N$-level Friedrichs model}\n\\label{sec:3}\n\nWe shall use bracket notation; however it can be understood in a standard treatment\nbased on functional analysis as in Refs.\n\\makebox(6,1){\\Large \\cite{Exner(1985),Davies(1974)}}\\hspace{7.5mm}.\nThe Hilbert space describing the unstable multilevel systems is here defined by\n\\begin{equation}\n{\\cal H}:=\\mathbb{C}^N \\oplus L^2 ((0,\\infty )).\n\\label{eqn:5.10}\n\\end{equation}\nA vector $\\ket{c} \\in \\mathbb{C}^N$ is expressed by $\\ket{c}=\\sum_{n=1}^{N} c_n \\ket{n}$,\nwhere $\\ket{n}$'s are the orthonormal basis of $\\mathbb{C}^N$,\nso that $\\braket{n}{n'}=\\delta_{nn'}$, where $\\delta_{nn'}$ is Kronecker's delta.\n$L^2 ((0,\\infty ))$ is the Hilbert space of\nthe square-integrable complex function $\\ket{f}$ of the variable $\\omega$ defined on $(0,\\infty )$, i.e.,\n\\begin{equation}\n\\ket{f} \\in L^2 ((0,\\infty )) \\Leftrightarrow \\int_0^{\\infty} |f(\\omega)|^2 d\\omega <\\infty.\n\\label{eqn:5.10b}\n\\end{equation}\nIn a standard notation using\nthe (generalized) eigenstate $\\ket{\\omega}$ of the multiplication operator by $\\omega$,\n$\\ket{f}$ is nothing more than\n\\begin{equation}\n\\ket{f}=\\int_{0}^{\\infty} f(\\omega ) \\ket{\\omega} d\\omega,\n\\label{eqn:5.15}\n\\end{equation}\nwhere $\\braket{\\omega}{\\omega'}=\\delta (\\omega -\\omega')$ and\n$\\delta (\\omega -\\omega')$ is Dirac's delta.\nThen, an arbitrary vector $\\ket{\\Psi} \\in {\\cal H}$\ncomposed of $\\ket{c} \\in \\mathbb{C}^N$ and $\\ket{f} \\in\nL^2 ((0,\\infty ))$\nis denoted by\n\\begin{equation}\n\\ket{\\Psi}:=\n| c \\rangle +\\ket{f} ,\n\\label{eqn:5.20}\n\\end{equation}\nand the inner product between any two vectors $\\ket{\\Psi}$ and\n$\\ket{\\Phi} \\in {\\cal H}$ is defined by \\cite{innerproduct}\n\\begin{equation}\n\\braket{\\Phi}{\\Psi}\n:=\\braket{d}{c}+ \\braket{g}{f}\n=\\sum_{n=1}^{N} d_n^* c_n + \\int_0^{\\infty} g^* (\\omega ) f(\\omega) d\\omega ,\n\\label{eqn:5.30}\n\\end{equation}\nwhere ($^*$) denotes the complex conjugate and\n$\\ket{\\Phi}=\\ket{d}+\\ket{g}$\nwith $\\ket{d} \\in \\mathbb{C}^N$ and $\\ket{g} \\in L^2 ((0,\\infty ))$.\nIn particular, the associated norm of $\\ket{\\Psi}$ is\n$\\| \\Psi \\| := \\sqrt{\\braket{\\Psi}{\\Psi}}$, \nwhich is ensured to be finite for all $\\ket{\\Psi} \\in \\cal{H}$.\n\n\n\n\n\nLet us now introduce the $N$-level Friedrichs model\nfor a description of the decay of the unstable multilevel systems.\nThe Hamiltonian $H$ of this model is defined by\n\\begin{equation}\nH:=H_0 + \\lambda V,\n\\label{eqn:5.60}\n\\end{equation}\nwhere\n$H_0$ is the free part and $V$\nthe interaction part of $H$, respectively,\nand $\\lambda \\in \\mathbb{R}$ is the coupling constant.\n$H_0$ is defined by\n\\begin{equation}\nH_0 :=\\sum_{n=1}^N \\omega_n \\ketbra{n}{n}+\\int_{0}^{\\infty} \\omega \\ketbra{\\omega}{\\omega}d\\omega,\n\\label{eqn:5.40}\n\\end{equation}\nwhere $\\omega_n \\in \\mathbb{R}$ with $\\omega_1 \\leq \\omega_2 \\leq \\cdots \\leq \\omega_N$,\nand its action is prescribed by\n$H_0 \\ket{\\Psi}=\\sum_{n=1}^N \\omega_n c_n \\ket{n}+ \\omega \\ket{f}$\nfor any $\\ket{\\Psi}=\\ket{c}+\\ket{f} \\in D(H_0)$.\n$D(H_0)$ is the domain of $H_0$ defined by\n$\nD(H_0 ):=\\left\\{ \\ket{\\Psi} \\in {\\cal H} ~\\left|~\n\\int_{0}^{\\infty} |\\omega f(\\omega )|^2 d\\omega < \\infty \\right. \\right\\}\n$,\nand then the self-adjointness of $H_0$ is guaranteed.\nThe interaction part $V$ is defined by\n\\begin{equation}\nV:=\\sum_{n=1}^{N} \\int_{0}^{\\infty}\n\\bigl[ v_n^*(\\omega) \\ketbra{n}{\\omega} +v_n(\\omega) \\ketbra{\\omega}{n} \\bigr] d\\omega,\n\\label{eqn:5.70}\n\\end{equation}\nwhere we assumed that $\\ket{v_n} \\in L^2 ((0, \\infty ))$. \n\\cite{threshold}\nWe call the $L^2$-functions $v_n(\\omega)$ the form factors of the system under consideration. \nThe action of $V$ is then given by\n$V \\ket{\\Psi}=\\sum_{n=1}^N \\braket{v_n}{f} \\ket{n} + \\sum_{n=1}^{N} c_n \\ket{v_n}$\nfor any $\\ket{\\Psi}\\in {\\cal H}$.\nNote that since $D(V)=\\cal{H}$ and $V$ is a bounded self-adjoint operator,\n$H$ is self-adjoint with the domain $D(H)=D(H_0)\\cap D(V)=D(H_0)$.\n\n\n\n\n\n\nIn the whole of the paper,\nwe will restrict ourselves to the special kind of the form factor:\nSuppose that the product $v_m^* (\\omega) v_n (\\omega)$\nbetween an arbitrary pair of $v_m^* (\\omega)$ and $v_n (\\omega)$\nis written in a rational function, i.e.,\nit is expressed by\n\\begin{equation}\nv_m^* (\\omega) v_n (\\omega) =\\frac{\\pi_{mn} (\\omega) }{ \\rho_{mn} (\\omega)},\n\\label{eqn:formfactor1}\n\\end{equation}\nwhere $\\pi_{mn} (\\omega)$ and $\\rho_{mn} (\\omega)$ are the polynomials \nof the degree $M_{mn}$ and $N_{nm}$, respectively, \nand we assume that $\\rho_{mn} (\\omega)$ has no zeros in $[0, \\infty )$. \nIt is also assumed that $M_{mn}+2 \\leq N_{mn}$ and $\\pi_{mn} (0)=0$. \nThe former condition ensures that $v_m^* (\\omega) v_n (\\omega)$ is integrable in $[0, \\infty )$ \nand $\\lim_{\\omega\\to\\infty} v_m^* (\\omega) v_n (\\omega)=0$, \nwhile the latter condition implies that \nthe rational function $v_m^* (\\omega) v_n (\\omega)=O(\\omega)$ as $\\omega\\to +0$. \nThe form factors with such properties are often found in actual systems\ninvolving the process of the spontaneous emission of photons from the hydrogen atom,\n\\cite{Facchi(1998),Seke(1994)}\nand quantum dots. \\cite{Antoniou(2001)}\nWe do not treat the algebraic form factor \nthat behaves as $O(\\omega^{1\/2})$ as $\\omega\\to +0$ instead, \nassociated with \nthe photodetachment of electrons from the negative ion\n\\cite{Rzazewski(1982),Haan(1984),Lewenstein(2000),Nakazato(2003)}\nand the spontaneous emission from the atoms in the photonic crystals;\\cite{Kofman(1994)} \nhowever, the discussion developed in the following could be easily extended to such a case.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Reduced resolvent for the $N$-level Friedrichs model}\n\\label{sec:4}\n\n\n\nIn the following, we introduce the reduced resolvent that is simply the restriction\nof the resolvent of $H$ to the $N$ dimensional subspace $\\mathbb{C}^N \\oplus \\{ 0 \\}$.\nSince we only consider the initial state belonging to this subspace,\nthis restriction is sufficient for our study.\nIn a technical sense, this treatment corresponds to the appropriate choice\nof a weighted Sobolev space. \\cite{Jensen(1979),Murata(1982)}\nIn the later sections, we do not distinguish the vector in $\\mathbb{C}^N$ from\nthat in $\\mathbb{C}^N \\oplus \\{ 0 \\}$.\nAfter introducing the reduced resolvent,\nwe see the existence of the boundary values of\nthe reduced resolvent on the positive real line.\nThe large-energy behavior of the reduced resolvent is also examined,\nwhich is necessary for a rigorous estimation of the long time behavior of\nthe reduced time evolution operator.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Reduced resolvent}\n\n\n\nThe resolvent of $H_0$ and that of $H$ are defined by \n$R_0(z ) =(H_0 -z )^{-1}$ and $R(z ) =(H -z )^{-1}$, respectively, \nwhere we assume that $z \\in \\mathbb{C}\\backslash (\\sigma (H_0) \\cup \\sigma (H))$.\n$\\sigma (H_0)$ (or $\\sigma (H)$) is the spectrum of $H_0$ (or $H$),\ni.e., the set of the singular points of $R_0(z)$ (or R(z)).\nThen, we have\n\\begin{eqnarray}\nR(z ) - R_0(z )\n&=& -R_0(z )  V R(z )\n\\label{eqn:2.70} \\\\\n&=& -R_0(z ) V R_0(z ) +R_0(z ) V R_0(z )  V R(z ) .\n\\label{eqn:2.40}\n\\end{eqnarray}\nFrom Eq. (\\ref{eqn:2.70}), one obtains the equation $R(z)= (1+R_0(z ) V )^{-1} R_0(z )$,\nwhich is the starting point of the asymptotic expansion of $R(z)$\nfor the short-range potential systems. \\cite{Jensen(1979)}\nOn the other hand, we instead start from Eq. (\\ref{eqn:2.40}) to obtain\n\\begin{equation}\n[H_0 -z - V R_0(z )  V ] R(z ) = 1 -V R_0(z ) .\n\\label{eqn:2.110}\n\\end{equation}\nThis equation can be solved for our model if \nwe confine ourselves to the state subspace $\\mathbb{C}^N \\oplus \\{ 0 \\}$.\n\\cite{Exner(1985)}\nIn fact, from the fact that $\\bra{n} V R_0(z ) \\ket{n'} =0$\nfor any $\\ket{n}$ and $\\ket{n'} \\in \\mathbb{C}^N \\oplus \\{ 0 \\}$,\nEq. (\\ref{eqn:2.110}) reads\n\\begin{equation}\n\\sum_{m=1}^N\n[(\\omega_{n} -z ) \\delta_{n m} -\\lambda^2 S_{n m}(z) ]\n\\tilde{R}_{m n'}(z )\n=\\delta_{nn'} ,\n\\label{eqn:.20}\n\\end{equation}\nwhere $S (z ) $ and $\\tilde{R} (z ) $ are\nthe $N \\times N$ matrix defined\nwith the matrix components\n\\begin{equation}\nS_{mn}  (z )\n:=\n\\bra{m} V R_0(z ) V \\ket{n}\n=\n\\int_{0}^{\\infty}\n\\frac{v_m^* (\\omega) v_n (\\omega)}{\\omega -z}d \\omega,\n\\mbox{ and }\n\\tilde{R}_{mn}  (z ):=\\bra{m} R(z ) \\ket{n}.\n\\label{eqn:.40}\n\\end{equation}\nWe call $S(z) $ and $\\tilde{R}(z)$ the {\\it self energy} and the {\\it reduced resolvent}, \nrespectively.\nNote that $S(z)$ can be analytically defined for all $z \\in \\mathbb{C}\\backslash [0, \\infty)$.\nFor a later convenience, we also introduce the matrix ${K_0}$ and $K(z)$ by\n\\begin{equation}\n{K_0}_{mn}:=\\bra{m} H_0 \\ket{n}=\\omega_n  \\delta_{m n},\n\\quad \\mbox{and} \\quad\nK_{mn}(z):=[K_0 -\\lambda^2 S(z)]_{mn},\n\\label{eqn:.55}\n\\end{equation}\nrespectively. Then, Eq. (\\ref{eqn:.20}) is equivalent to\n\\begin{equation}\n[K(z)-z]\n\\tilde{R} (z ) =1, ~~~\n\\forall z \\in \\mathbb{C}\\backslash (\\sigma (H_0) \\cup \\sigma (H)),\n\\label{eqn:.60}\n\\end{equation}\nwhich implies that\n${\\rm det}[K(z)-z]\n{\\rm det} [\\tilde{R} (z ) ] =1 $,\nso that\n${\\rm det}[K(z)-z] \\neq 0$ and\n${\\rm det} [\\tilde{R} (z ) ] \\neq 0 $\nfor all $z \\in \\mathbb{C}\\backslash (\\sigma (H_0) \\cup \\sigma (H))$.\nThus, the inverse of $K(z)-z$\nexists, and we have\n\\begin{equation}\n\\tilde{R} (z )\n=\n[K(z)-z]^{-1}, ~~~\n\\forall z \\in \\mathbb{C}\\backslash (\\sigma (H_0) \\cup \\sigma (H)).\n\\label{eqn:.70}\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The boundary values of $\\tilde{R} (z )$ and its large energy behavior}\n\\label{subsec:4.2}\n\nFrom the assumption on the form factors, every $v_m^* (\\omega )v_n (\\omega )$ is \ncontinued to the whole complex plane as a meromorphic function which we merely denote \nas $v_m^* (z)v_n (z)$. \nIt may have a finite number of poles.\nThen, it follows from Lemma \\ref{lm:formfactor1} that\n${S} (z)$ can be reduced to the form\n\\begin{equation}\n{S} (z)\n=\nS(0) + A(z) -(\\log (-z)) {\\mit \\Gamma}(z) ,\n\\label{eqn:.76}\n\\end{equation}\nwhere we choose\n${\\rm arg} (-z) ={\\rm arg} (z)-\\pi$ and $0<{\\rm arg} (z)<2\\pi$.\nThe matrix ${\\mit \\Gamma}(z)$ is defined with the components\n\\begin{equation}\n{\\mit \\Gamma}_{mn} (z):=v_m^* (z)v_n (z),\n\\label{eqn:.100}\n\\end{equation}\nand satisfies ${\\mit \\Gamma}(z)\\to 0$ as $z\\to 0$ in $\\mathbb{C}$.\n$S(0)$ is the limit of $S(z)$ as $z\\to 0$ in $\\mathbb{C}\\backslash [0, \\infty)$,\nwhich turns out to be unique. Indeed, as we see \nfrom the Appendix in Ref. \\makebox(0,1){\\Large \\cite{Miyamoto(2005)}}\\hspace{4mm},\n$S_{mn}(0)=\\int_{0}^{\\infty}v_m^* (\\omega) v_n (\\omega)\/\\omega  \\ d \\omega$.\n$A(z)$ is then defined through Eq. (\\ref{eqn:.76}) and becomes\na Hermitian matrix for real $\\omega$,\nwhose components are the rational functions of $z$\nwithout any singularity on $[0, \\infty )$.\nBy definition, $A(z)$ satisfies $A(z)\\to 0$ as $z\\to 0$.\nOne sees that the boundary values of $S (z )$\nat the half line $(0,\\infty )$ exist and satisfy \\cite{Exner(1985)}\n\\begin{equation}\n\\lim_{\\epsilon \\to +0}\nS (\\omega \\pm i\\epsilon)\n=\n{D} (\\omega ) \\pm \\pi i {\\mit \\Gamma}(\\omega ),\n\\label{eqn:.80a}\n\\end{equation}\nwhere\n\\begin{equation}\n{D} (\\omega )\n:=\nS(0) + A(\\omega) -(\\log \\omega) {\\mit \\Gamma}(\\omega) .\n\\label{eqn:.80b}\n\\end{equation}\nThe matrix ${D} (\\omega )$ is just\nof the components\n\\begin{equation}\n{D}_{mn} (\\omega ):=P \\int_{0}^{\\infty}\n\\frac{v_m^* (\\omega' )v_n (\\omega' )}{\\omega'-\\omega} d\\omega' ,\n\\label{eqn:.90}\n\\end{equation}\nwhere $P$ denotes the principal value of the integral.\nNote that both ${D} (\\omega )$ and ${\\mit \\Gamma}(\\omega )$ are Hermitian\nmatrices and ${\\mit \\Gamma}(\\omega ) \\geq 0$.\n\n\n\n\nIn all the discussion developed in the following, we assume that\n\\begin{equation}\n{\\rm det}[K^\\pm (\\omega)-\\omega]\\neq 0, ~~~ \\forall \\omega >0,\n\\label{eqn:.110}\n\\end{equation}\nwhere we introduced\n\\begin{equation}\nK^\\pm (\\omega):= \\lim_{\\epsilon \\to +0} K(\\omega \\pm i\\epsilon )\n={K_0} -\\lambda^2 {D} (\\omega )\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega ),\n ~~~ \\forall \\omega >0.\n\\label{eqn:.112}\n\\end{equation}\nIt is worth noting that condition (\\ref{eqn:.110}) is\nequivalent to the requirement of no positive eigenvalues of $H_0$,\nwhose eigenstates are normalizable.\nIndeed, if\n${\\rm det}[K^\\pm (\\omega)-\\omega]=0\n$ for some $\\omega >0$, there is a non-zero vector\n$\\ket{\\eta}=\\sum_{n=1}^N \\eta_n \\ket{n} \\in \\mathbb{C}^N$ such that\n$[K^\\pm (\\omega)-\\omega] \\ket{\\eta}=0 $.\nSince both ${D} (\\omega )$ and ${\\mit \\Gamma}(\\omega )$ are Hermitian matrices,\nthe latter equation implies that\n\\begin{equation}\n\\bra{\\eta} [{K_0}-\\omega -\\lambda^2 {D} (\\omega )]\n\\ket{\\eta} =0 ~~\n\\mbox{and} ~~\n\\bra{\\eta} {\\mit \\Gamma}(\\omega ) \\ket{\\eta}\n=\\left|\\sum_{n=1}^{N} v_n(\\omega)\\eta_n \\right|^2=0 .\n\\label{eqn:.167}\n\\end{equation}\nNote that the latter relation means that\n${\\mit \\Gamma}(\\omega ) \\ket{\\eta}=0$ because ${\\mit \\Gamma}(\\omega ) \\geq 0$.\nThus, Eq. (\\ref{eqn:.167}) implies that\n${\\mit \\Gamma}(\\omega ) \\ket{\\eta} =0$ and\n$[{K_0} -\\lambda^2 {D} (\\omega )]\\ket{\\eta}=\\omega \\ket{\\eta}$, i.e.,\n\\begin{equation}\n\\sum_{n=1}^{N} v_n(\\omega)\\eta_n =0,\n~~ \\mbox{and} ~~\n\\sum_{n=1}^{N}\n[\\omega_{m}\\delta_{mn} -\\lambda^2 {D}_{mn} (\\omega )]\n\\eta_n =\\omega \\eta_m ,\n\\label{eqn:.169}\n\\end{equation}\nfor all $m=1, \\ldots, N$.\nThis is merely the condition for the existence of a positive\neigenvalue $\\omega$ of $H$. \\cite{Miyamoto(2005)}\n\n\n\n\n\n\n\\begin{lm\n\\label{lm:100}\nUnder the assumption (\\ref{eqn:.110}), it holds that\n$\\tilde{R}^{\\pm}(\\omega)\n:=\\lim_{\\epsilon \\to +0} \\tilde{R}(\\omega \\pm i \\epsilon )\n$\nexists for all $\\omega >0$ and\n$\\tilde{R}^{\\pm}(\\omega) =[K^\\pm (\\omega)-\\omega]^{-1}$.\n\\end{lm}\n\n\n{\\sl Proof} :\nUnder the assumption (\\ref{eqn:.110}),\n$[K^\\pm (\\omega)-\\omega]^{-1} $ exists. Then\n\\begin{eqnarray}\n&&\n\\|\n[K^\\pm (\\omega)-\\omega]^{-1} -\\tilde{R}(\\omega \\pm i \\epsilon )\n\\|\n\\nonumber \\\\\n&\\leq&\n\\|\n[K^\\pm (\\omega)-\\omega]^{-1}\n\\|\n\\|\n\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\|\n\\|\n\\tilde{R}(\\omega \\pm i \\epsilon )\n\\| .\n\\label{eqn:.120}\n\\end{eqnarray}\nNote that for any nonzero $\\ket{y} \\in \\mathbb{C}^N$ ($\\neq 0$)\nthere is a nonzero $\\ket{x} \\in \\mathbb{C}^N$ such that\n$\\ket{y}=[K(\\omega \\pm i \\epsilon )-\\omega \\mp i \\epsilon]\\ket{x}$.\nWe then obtain\n\\begin{eqnarray}\n\\frac{\n\\|\n\\tilde{R}(\\omega \\pm i \\epsilon )\\ket{y}\n\\|\n}\n{\\| y\\|}\n&\\leq&\n\\frac{\\| x \\|}\n{\n\\bigl|\n\\|\n[K^\\pm (\\omega)-\\omega] \\ket{x}\n\\|\n-\n\\|\n[\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )]\\ket{x}\n\\|\n\\bigr|\n}\n\\nonumber \\\\\n\\!&\\leq&\\!\n\\biggl[\n\\inf_{\\ket{x} \\neq 0, \\ket{x}\\in \\mathbb{C}^N}\\! \\!\n\\frac{\n\\|[K^\\pm (\\omega)-\\omega]\\|\n}{\\| x \\|}\n\\nonumber\\\\\n&&\n-\n\\|\n\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\|\n\\biggr]^{-1} \\hspace{-2mm},\n\\label{eqn:.130}\n\\end{eqnarray}\nwhich implies that\n\\begin{equation}\n\\mathop{\\overline{\\lim}}_{\\epsilon \\to +0}\n\\|\n\\tilde{R}(\\omega \\pm i \\epsilon )\n\\|\n\\leq\n\\left[\n\\inf_{\\ket{x} \\neq 0, \\ket{x}\\in \\mathbb{C}^N}\n\\frac{\n\\|K^\\pm (\\omega)-\\omega\\|\n}{\\| x \\|}\n\\right]^{-1} < \\infty ,\n\\label{eqn:.140}\n\\end{equation}\nwhere\nthe norm of an $N\\times N$ matrix $A$ is defined by\n$\\| A\\|=\n\\sup_{\\ket{x} \\neq 0, \\ket{x}\\in \\mathbb{C}^N} \\| A\\ket{x}\\| \/\\| x\\|$.\nIn Eq. (\\ref{eqn:.130}), we used the fact that\nthere is some $\\epsilon_0 >0$ such that\nfor any positive $\\epsilon < \\epsilon_0$\nand for any non zero $\\ket{x} \\in \\mathbb{C}^N$\n\\begin{eqnarray}\n\\frac{\n\\|\n[K^\\pm (\\omega)-\\omega] \\ket{x}\\|\n}{\\| x \\|}\n&\\geq&\n\\inf_{\\ket{x} \\neq 0, \\ket{x}\\in \\mathbb{C}^N}\n\\frac{\n\\|[K^\\pm (\\omega)-\\omega] \\ket{x}\\|\n}{\\| x \\|}\n\\nonumber \\\\\n&>&\n\\|\n\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\|\n\\nonumber \\\\\n&\\geq&\n\\frac{\n\\|\n[\\pm i\\epsilon +\\lambda^2 S (\\omega \\pm i \\epsilon)\n-\\lambda^2 {D} (\\omega )\n\\mp \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n]\\ket{x}\n\\|\n}{\\| x \\|},\n\\label{eqn:.150}\n\\end{eqnarray}\nwhere the assumption (\\ref{eqn:.110}) is taken into account.\nThus, by using Eq. (\\ref{eqn:.140}), Eq. (\\ref{eqn:.120}) leads us to\n\\begin{equation}\n\\lim_{\\epsilon \\to +0}\n\\|\n[K^\\pm (\\omega)-\\omega]^{-1}\n-\\tilde{R}(\\omega \\pm i \\epsilon )\n\\|\n=0,\n\\label{eqn:.160}\n\\end{equation}\nwhich completes the proof of the lemma.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lm\n\\label{lm:Large-omega}\n:\nUnder the assumption (\\ref{eqn:.110}),\n$\\tilde{R}^\\pm (\\omega)$ is $r$-times differentiable in\n$\\omega \\in (0, \\infty)$, and it behaves as\n\\begin{equation}\n\\frac{d^r \\tilde{R}^\\pm (\\omega)}{d\\omega^r}\n=O(\\omega^{-r-1}) \\mbox{ as } \\omega \\to \\infty .\n\\label{eqn:Large10}\n\\end{equation}\n\\end{lm}\n\n{\\sl Proof} :\nWe first show the statement for $r=0$.\nFrom the assumption on the form factors and Lemma \\ref{lm:formfactor1}, one sees that\n\\begin{equation}\n\\lim_{\\omega \\to \\infty} {D} (\\omega) =0 \\mbox{ and }\n\\lim_{\\omega \\to \\infty} {\\mit \\Gamma} (\\omega) =0.\n\\label{eqn:Large40}\n\\end{equation}\nSince from the assumption (\\ref{eqn:.110})\n$K^\\pm(\\omega)-\\omega$ is invertible\nfor all $\\omega >0$,\nit holds that there is some positive $\\bar{\\omega}  > \\omega_N$\nsuch that for any $\\omega > \\bar{\\omega}$,\n\\begin{eqnarray}\n\\frac{\n\\|\n\\tilde{R}^\\pm (\\omega)\\ket{y}\n\\|\n}\n{\\| y\\|}\n&\\leq&\n\\frac{\\| x \\|}\n{\\displaystyle\n\\|\n({K_0}-\\omega)\\ket{x}\n\\|\n-\\lambda^2\n\\|\n[{D} (\\omega) \\pm \\pi i {\\mit \\Gamma}(\\omega)]\\ket{x}\n\\|\n}\n\\label{eqn:Large30a}\\\\\n&\\leq&\n\\frac{1}\n{\n\\omega -\\omega_N -\\lambda^2\n\\|\n{D} (\\omega) \\pm \\pi i {\\mit \\Gamma}(\\omega)\n\\|\n}\n=\nO(\\omega^{-1}),\n\\label{eqn:Large30}\n\\end{eqnarray}\nwhere\nthe last inequality is obtained as follows:\nwe can choose some positive $\\bar{\\omega}  > \\omega_N$\nsuch that for any $\\omega > \\bar{\\omega}\n$\\begin{equation}\n\\frac{\n\\|\n[{K_0}-\\omega]\\ket{x}\n\\|\n}{\\| x \\|}\n\\geq\n\\min_n \\{ \\omega-\\omega_n \\} =\\omega-\\omega_N\n>\n\\lambda^2 \\|{D} (\\omega) \\pm \\pi i {\\mit \\Gamma}(\\omega) \\|\n\\geq\n\\lambda^2\n\\frac{\n\\|\n[{D} (\\omega)\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega)\n]\\ket{x}\n\\|\n}{\\| x \\|} ,\n\\label{eqn:Large50}\n\\end{equation}\nwhere Eq. (\\ref{eqn:Large40}) was used.\nThus Eq. (\\ref{eqn:Large30}) reads just as Eq. (\\ref{eqn:Large10}) does for $r=0$.\nIn the case of $r\\geq 1$, we first note that from our assumptions on the form factors and\nLemma \\ref{lm:formfactor1} again, $A(\\omega ) $ and ${\\mit \\Gamma}(\\omega ) $,\nwhich are connected through ${D} (\\omega )=S_0 + A(\\omega ) -\\log \\omega {\\mit \\Gamma}(\\omega )$,\nalso satisfy\n\\begin{equation}\n\\frac{d^r A(\\omega )}{d\\omega^r }= O(\\omega^{-1-r}), ~~\n\\frac{d^r \\log \\omega {\\mit \\Gamma}(\\omega )}{d\\omega^r }= O(\\omega^{-1-r} \\log \\omega),\n\\label{eqn:Large80}\n\\end{equation}\nas $\\omega \\to \\infty$, where we used the estimation that\n$\\frac{d^r {\\mit \\Gamma}(\\omega )}{d\\omega^r }= O(\\omega^{-1-r})$.\nThus, for $r=1$, we have\n\\begin{equation}\n\\frac{d \\tilde{R}^{\\pm}(\\omega )}{d\\omega}\n=\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d}{d\\omega}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n= O(\\omega^{-2}),\n\\label{eqn:Large90b}\n\\end{equation}\nas $\\omega \\to \\infty$, where Eq. (\\ref{eqn:Large10}) for $r=0$ was used.\nFor $r\\geq1$, we obtain\n\\begin{equation}\n\\frac{d^r \\tilde{R}^{\\pm}(\\omega )}{d\\omega^r}\n=\n\\sum_{j=1}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\{\n\\prod_{i=1}^{j}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\}\n\\tilde{R}^{\\pm}(\\omega ),\n\\label{eqn:Large100}\n\\end{equation}\nwhere $a^{(r)} (\\{s_i \\}_{i=1}^{j} ) $ is an appropriate positive integer.\nNote that the symbol ($\\ '$) means that\nthe summation over $\\{s_i \\}_{i=1}^{j}$ is taken under the condition that\n$s_i \\geq 1$ for all $i$\nand $\\sum_{i=1}^{j} s_i =r$.\nIf $r=1$, Eq. (\\ref{eqn:Large100}) reproduces Eq. (\\ref{eqn:Large90b})\nwith $a^{(1)} (\\{s_i \\}_{i=1}^{1} ) =1$.\nIn the general case, if Eq. (\\ref{eqn:Large100}) holds\nfor $r=k$, then its derivative is made up of a linear combination of\n\\begin{equation}\n\\left\\{\n\\prod_{i=1}^{j+1}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\} \\tilde{R}^{\\pm}(\\omega ),\n\\label{eqn:Large110}\n\\end{equation}\nwhere $\\sum_{i=1}^{j+1} s_i =k+1$ for $1 \\leq j \\leq k$, and\n\\begin{equation}\n\\left\\{\n\\prod_{i=1}^{j}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\} \\tilde{R}^{\\pm}(\\omega ),\n\\label{eqn:Large120}\n\\end{equation}\nwhere $\\sum_{i=1}^{j} s_i =k+1$ for $1 \\leq j \\leq k$.\nOn the other hand, they are actually included in the right-hand side (rhs) of\nEq. (\\ref{eqn:Large100}) for $r=k+1$. Thus Eq. (\\ref{eqn:Large100}) is\nvalid for all integer $r \\geq 1$.\nLet us now evaluate the asymptotic behavior of\n$d^r \\tilde{R}^{\\pm}(\\omega )\/d\\omega^r$ for large $\\omega$.\nOne can see that\nthe summand for $j=r$ in Eq. (\\ref{eqn:Large100}),\nwhere all $s_i =1$, contributes\n$O(\\omega^{-r-1})$ to $d^r \\tilde{R}^{\\pm}(\\omega )\/d\\omega^r$,\nwhile the other summands for $j<r$ specified by $\\{s_i \\}_{i=1}^{j}$\ncontribute $O(\\omega^{-r-1-2s_0 }(\\log \\omega )^{s_0}))$ at most,\nwhere $s_0$ is a number of $s_i$ satisfying $s_i \\geq 2$ and never vanishes for $j<r$.\nTherefore, the summand dominating for large $\\omega$ is\nthat for $j=r$. Since we recursively show\n$a^{(r)} (\\{s_1 \\}_{i=1}^{r} ) =r!$, which never vanishes, the statement is proved. \n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Classification of the zero-energy singularity of $\\tilde{R}^\\pm (\\omega ) $}\n\\label{sec:5}\n\n\n\nIn order to prescribe the zero energy resonance in the $N$-level Friedrichs model,\nwe should identify the zero energy eigenstates in this model\nwhich either belong to or do not belong to ${\\cal H}$.\nIn the case of the short-range potential systems, \\cite{Jensen(1979)}\nthis task needs some elaborate examination with an appropriately extended Hilbert space.\nOn the other hand, in our case,\nit is rather easily performed, as is seen in the following.\n\n\n\n\n\nLet us first\nsee whether the eigenvector $\\ket{\\psi} \\in \\mathbb{C}^N$ of ${K_0} -\\lambda^2 S(0)$,\nbelonging to the zero eigenvalue,\ncan be actually extended to the eigenvector of $H$ belonging to the zero eigenvalue of $H$.\nIf $\\ket{\\Psi}=\\ket{\\psi}+\\ket{f} \\in D(H) \\subset {\\cal H}$ is a zero eigenvector of $H$,\nit should satisfy $H\\ket{\\Psi}=0$, or equivalently \\cite{Miyamoto(2005)}\n\\begin{equation}\n\\omega_n \\psi_n +\\lambda \\braket{v_n}{f} =0 ~\\mbox{for}~ n=1, \\ldots, N,~~~\\mbox{and}~~~\n\\omega f(\\omega)+ \\lambda \\sum_{n=1}^{N}\\psi_n v_n (\\omega)=0.\n\\label{eqn:.333}\n\\end{equation}\nThe latter equation of Eq. (\\ref{eqn:.333}) is immediately solved as\n\\begin{equation}\nf(\\omega)=-\\lambda \\frac{\\sum_{n=1}^{N}\\psi_n v_n (\\omega)}{\\omega},\n\\label{eqn:.335}\n\\end{equation}\nwhich should be square integrable\nbecause we intend to find $\\ket{\\Psi}$ in ${\\cal H}$.\nIf this is the case, $\\omega f(\\omega) \\in L^2 ((0, \\infty))$, i.e.,\n$\\ket{\\Psi} \\in D(H)$ is ensured,\nand the substitution of Eq. (\\ref{eqn:.335}) into $\\braket{v_n}{f}$ is safely done.\nThen,\nwe find that the former equation of Eq. (\\ref{eqn:.333}) is nothing more than\n\\begin{equation}\n({K_0} -\\lambda^2 S(0) )\\ket{\\psi}=K(0)\\ket{\\psi}=0,\n\\label{eqn:.336}\n\\end{equation}\nwhere $K(0):=K^\\pm (0)={K_0} -\\lambda^2 S(0)$.\nHowever, it is noted that such an $f(\\omega)$ associated with $\\ket{\\psi}$ is not necessarily\nsquare integrable.\nHence, we shall decompose the zero eigenspace of $K(0)$, denoted by\n$M=\\{ \\ket{\\psi} \\in \\mathbb{C}^N |~ K(0)\\ket{\\psi}=0 \\}$,\ninto two kinds of subspaces:\n$M_1=(M_0 \\oplus M_2)^{\\perp}$ \nand $M_2 =\\{ \\ket{\\psi} \\in M | f(\\omega) \\in L^2 ((0, \\infty)) \\}$.\nHere $M_0=M^{\\perp}$, and\n$D^{\\perp}$ denotes the orthogonal complement of the subspace $D$.\nIn short we have $\\mathbb{C}^N = M_0 \\oplus M_1 \\oplus M_2$.\nThen, as is expected from the definition, we have\n\\begin{equation}\nM_1 \\subset \\{ \\ket{\\psi} \\in M | f(\\omega) \\notin L^2 ((0, \\infty)) \\} .\n\\label{eqn:.325}\n\\end{equation}\nNote that in general the subset on the rhs of the above is not a subspace.\nWe call $0$ the {\\it zero energy resonance} (or merely {\\it zero resonance}) \nof $H$ if $M_1$ is not empty.\nWe also introduce the projection operators $Q_0$, $Q_1$, and $Q_2$, associated with\n$M_0$, $M_1$, and $M_2$, respectively.\nWhat we next do is to introduce the terminology \nfollowing the study of Jensen and Kato. \\cite{Jensen(1979)}\n\n\n\n\\begin{df\n\\label{df:regular}\n{\\rm\nWe call the system a {\\it regular} case if it holds that\n$0 \\notin \\sigma (K(0))$, i.e.,\n\\begin{equation}\n{\\rm det}[K(0)]\n\\neq 0.\n\\label{eqn:.300}\n\\end{equation}\nIn this case, $0$ is said to be a {\\it regular} point for $H$.\n}\n\\end{df}\n\n\n\n\n\\begin{df\n\\label{df:exceptional}\n{\\rm \nWe call the system the {\\sl exceptional} case if, instead of Eq. (\\ref{eqn:.300}), it holds that\n$0 \\in \\sigma (K(0))$, i.e.,\n\\begin{equation}\n{\\rm det}[K(0)]\n= 0.\n\\label{eqn:.330}\n\\end{equation}\nIn particular, if $0$ is a resonance but not an eigenvalue\n($Q_1 \\neq 0$, $Q_2 =0$), $0$ is said to be an {\\it exceptional} point for $H$\nof the {\\sl first kind}.\nIf $0$ is not a resonance, but an eigenvalue ($Q_1 = 0$, $Q_2 \\neq 0$),\n$0$ is said to be an exceptional point\nof the {\\sl second kind}.\nIf $0$ is both a resonance and an eigenvalue ($Q_1 \\neq 0$, $Q_2 \\neq 0$),\n$0$ is said to be an exceptional point\nof the {\\it third kind}.\n}\n\\end{df}\n\n\n\n\n\n\nWe here remark that in general a non-trivial solution of Eq. (\\ref{eqn:.336}) does not exist, \nhowever we can find a special case where such a solution surely exists. \nSuppose that $N_+$ eigenvalues $\\omega_n$ of $H_0$ are positive, and \nall form factors $v_n (\\omega)$ satisfying the assumption (\\ref{eqn:formfactor1}) \nare linearly independent. \nThen increasing $\\lambda$ gradually form $0$ to $\\infty$, \nwe can find some critical values of $\\lambda$ \nfor which $K(0)$ has the zero eigenvalue. \nLet us denote the $n$-th eigenvalue of $K(0)$ by $\\kappa_n (0)$ where \n$\\kappa_1 (0)\\leq \\kappa_2 (0) \\leq \\cdots \\leq \\kappa_N (0)$. \nThen, $\\kappa_n (0)$ turns out to satisfy the inequality\n\\begin{equation}\n\\omega_n -\\lambda^2 \\sigma_N (0) \\leq \\kappa_n (0) \\leq \\omega_n -\\lambda^2 \\sigma_1 (0),\n\\label{eqn:zeroeigenvalue}\n\\end{equation}\nwhere both of $\\sigma_1 (0)$ and $\\sigma_N (0)$ \nare positive constants and ensured not to vanish.\\cite{Miyamoto(2005)} \nThus, for a sufficiently small $|\\lambda|$ \n$\\kappa_n (0)$ for each $n\\geq N-N_+ +1$ should be positive, \nwhile for a sufficiently large $|\\lambda|$ they should be negative. \nFurthermore, one easily sees that all $\\kappa_n (0)$ are continuous functions of $\\lambda^2$. \nTherefore, we conclude from the intermediate value theorem \nthat there is at least one critical value of $\\lambda$ to make \n$\\kappa_n (0)=0$ for each $n\\geq N-N_+ +1$. \nWe can actually find such special values of $\\lambda$ in Fig. 1 \ndepicted in Ref. \\makebox(8,1){\\Large \\cite{Miyamoto(2005)}}\\hspace{1mm}. \nThe example mentioned here could be treated in a more general way \nwith resort to the analytic Fredholm theorem \\cite{the analytic Fredholm theorem,Rauch(1978)}\nwhich tells us at most a finite number of the critical values exists. \n\n\n\n\n\nIt is also worth remarking that \nthe existence of the zero energy eigenstates that either belong to or not to \nthe Hilbert space necessarily prescribes the small energy behavior of \nthe form factors in the following way. \nRemember that under the assumption on the form factors, \n${\\mit \\Gamma}(\\omega )$ defined by Eq. (\\ref{eqn:.100}) has an asymptotic form like\n\\begin{equation}\n{\\mit \\Gamma}(\\omega ) = \\sum_{n=1}^{N} \\omega^n {\\mit \\Gamma}_n +O(\\omega^{N+1}),\n\\label{eqn:formfactor2}\n\\end{equation}\nas $\\omega\\to +0$. \nThen, if $\\ket{\\psi} \\in M_1$ exists, it should satisfy \n\\begin{equation}\n\\bra{\\psi} {\\mit \\Gamma}_1 \\ket{\\psi} \\neq 0. \n\\label{eqn:.340}\n\\end{equation}\nIn fact, if $\\bra{\\psi} {\\mit \\Gamma}_1 \\ket{\\psi} = 0$, we see that \n$f(\\omega)$ in Eq. (\\ref{eqn:.335}) has to satisfy \n$|f(\\omega)|^2=\\lambda^2\\bra{\\psi}{\\mit \\Gamma}(\\omega)\\ket{\\psi}\/\\omega^2=O(1)$ \nas $\\omega \\to +0$, however \nwhich concludes that $f(\\omega)$ is square integrable. \nThis contradicts the assumption that $\\ket{\\psi} \\in M_1$. \nIn order to make the condition (\\ref{eqn:.340}) be satisfied, \nat least ${\\mit \\Gamma}_1$ should not vanish identically. \nWe can find such form factors in the physical systems for \nthe spontaneous emission process of photons from the Hydrogen atom \n\\cite{Facchi(1998),Seke(1994)} and the quantum dot. \\cite{Antoniou(2001)} \nOn the other hand, the discussion mentioned here immediately implies the fact that \nif $\\ket{\\psi} \\in M_2$ exists, this time it should satisfy\n\\begin{equation}\n\\bra{\\psi} {\\mit \\Gamma}_1 \\ket{\\psi} = 0, \n\\label{eqn:.350}\n\\end{equation}\nwhich just ensures the requirement that $f(\\omega) \\in L^2 ((0, \\infty))$.\nHowever, note that Eq. (\\ref{eqn:.350}) does not imply ${\\mit \\Gamma}_1=0$ identically \nand only requires ${\\mit \\Gamma}_1=0$ on the subspace $M_2$.\n\n\n\n\n\n\n\n\n\\subsection{The small-energy behavior of $\\tilde{R}^\\pm (\\omega) $ in the regular case}\n\nIn this case, the same as in Eq. (\\ref{eqn:.160}), we can show that\n$\\tilde{R}^{\\pm}(0)\n=(K(0))^{-1}$.\nFurthermore, we can choose some positive $\\omega_0 >0$\nsuch that\n\\begin{equation}\n\\| (K(0))^{-1}\\|\n\\| [\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n] \\|\n< 1,\n\\label{eqn:.170}\n\\end{equation}\nfor all positive $\\omega < \\omega_0$. Then,\n$\\tilde{R}^{\\pm}(\\omega )$ is expanded as a Neumann series,\n\\begin{equation}\n\\tilde{R}^{\\pm}(\\omega )\n=\n\\{\nK(0)\n[\n1- (K(0))^{-1}\n[\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]]\n\\}^{-1}\n=\n\\lim_{N\\to \\infty}S_N(\\omega),\n\\label{eqn:.180}\n\\end{equation}\nwhere\n\\begin{equation}\nS_N(\\omega)\n=\n\\sum_{j=0}^N\n\\{\n(K(0))^{-1}\n[\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]\\}^j\n(K(0))^{-1}\n,\n\\label{eqn:.185}\n\\end{equation}\nfor all positive $\\omega < \\omega_0$ with\n$\\{\n(K(0))^{-1}\n[\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]\\}^0 =1\n$.\nUnder our assumptions on the form factors,\n$A(\\omega ) $ defined in Eq. (\\ref{eqn:.80b}) is asymptotically expanded as\n\\begin{equation}\nA(\\omega) = \\sum_{n=1}^{N} \\omega^n A_n +O(\\omega^{N+1}),\n\\label{eqn:.220}\n\\end{equation}\nas $\\omega \\to 0$. By using Eqs. (\\ref{eqn:formfactor2}) and (\\ref{eqn:.220}), \nit also follows that \n\\begin{equation}\n{D} (\\omega )\n=\nS(0)\n-\\omega \\log \\omega {\\mit \\Gamma}_1\n+\\omega A_1 + O(\\omega^2 \\log \\omega ) ,\n\\label{eqn:.230}\n\\end{equation}\nas $\\omega \\to +0$. \nThen, Eq. (\\ref{eqn:.180}) tells us the dominant asymptotic behavior of $\\tilde{R}^{\\pm}(\\omega )$\nbecomes\n\\begin{equation}\n\\tilde{R}^{\\pm}(\\omega ) =(K(0))^{-1} +O(\\omega \\log \\omega),\n\\label{eqn:.240}\n\\end{equation}\nas $\\omega \\to +0$, where $(K(0))^{-1}$ never vanishes in the regular case.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The small-energy behavior of $\\tilde{R}(z) $\nin the exceptional case of the first kind}\n\\label{subsec:5.C}\n\n\n\nIn the exceptional case of the first kind,\nfrom the definition, $Q_1 \\neq 0$ while $Q_2 = 0$, so that $Q_0 +Q_1 =1$.\nThen, $\\tilde{R} (z)$ is divided into the following four terms,\n\\begin{equation}\n\\tilde{R}(z)\n=\nQ_0 \\tilde{R}(z) Q_1\n+Q_0 \\tilde{R}(z) Q_0\n+Q_1 \\tilde{R}(z) Q_0\n+Q_1 \\tilde{R}(z) Q_1 .\n\\label{eqn:5.C.1.30}\n\\end{equation}\nWe now introduce the four matrices,\n\\begin{equation}\nE_{kl}(z)\n=Q_k [K(z) -z]Q_l\n=\nQ_k [K_0 -z -\\lambda^2 [S(0)+A(z)-(\\log z) {\\mit \\Gamma}(z)+i\\pi {\\mit \\Gamma}(z)]]Q_l,\n\\label{eqn:5.C.1.40}\n\\end{equation}\nwhere $k, l =0, 1$, and $\\log(-z) -\\log z =-i \\pi$ is used.\nFrom the relation that\n$[K(z) -z]\\tilde{R}(z) =1$,\nthey satisfy\n\\begin{equation}\nE_{k0}Q_0 \\tilde{R}Q_l + E_{k1}Q_1 \\tilde{R}Q_l =Q_k \\delta_{kl},\n\\label{eqn:5.C.1.50}\n\\end{equation}\nfor $k,l=0,1$.\nTo solve the above equations we need to check whether\n$E_{11}$ and $E_{00}$ are invertible in the subspaces $M_1$ and $M_0$, respectively.\nBy using Eq. (\\ref{eqn:formfactor2}), $E_{11}(z)$ is rewritten as\n\\begin{equation}\nE_{11}(z)\n=\n\\lambda^2z  (\\log z) Q_1 {\\mit \\Gamma}_1 Q_1\n-z Q_1\n-\\lambda^2 Q_1 \\{ A(z) -(\\log z) [{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n+i\\pi {\\mit \\Gamma}(z)\\} Q_1 ,\n\\label{eqn:5.C.1.60b}\n\\end{equation}\nwhere $Q_1K(0)Q_1 =0$ is used.\nNote that $A(z)=O(z)$ and ${\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 =O(z^2 )$\nfor our form factors,\nso that all terms excepting the first one of the rhs of Eq. (\\ref{eqn:5.C.1.60b}) are\nof the order of $O(z)$.\nFurthermore, the exceptional case of the first kind imposes the fact that\n$Q_1 {\\mit \\Gamma}_1Q_1 \\neq 0$ [see Eq. (\\ref{eqn:.340})], and\n$Q_1 {\\mit \\Gamma}_1Q_1$ is positive definite in $M_1$ and thus invertible in $M_1$.\nHence, $E_{11}(\\omega) $ is invertible for sufficiently small\n$|z|>0$, and the inverse can be expanded by the Neumann series as,\n\\begin{eqnarray}\n\\hspace*{-5mm}\nE_{11}^{-1}(z)\n&=&\n\\sum_{j=0}^{\\infty} (\\tilde{E}_{11}(z))^j\n\\frac{1}{\\lambda^2z  \\log z} (Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\label{eqn:5.C.1.90a}\\\\\n&=&\n\\frac{1}{\\lambda^2z  \\log z} (Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n +O(z^{-1} (\\log z)^{-2})=O(z^{-1} (\\log z)^{-1}) ,\n\\label{eqn:5.C.1.90b}\n\\end{eqnarray}\nfor small $|z|$, where we define\n\\begin{equation}\n\\tilde{E}_{11}(z)\n:=\n\\frac{1}{\\lambda^2z  \\log z} (Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\bigl\\{ z Q_1\n+\\lambda^2 Q_1 \\{ A(z) -(\\log z) [{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n+ \\pi i {\\mit \\Gamma}(z) \\} Q_1\n\\bigr\\} ,\n\\label{eqn:5.C.1.90c}\n\\end{equation}\nwhich behaves as $1\/\\log z$ as $z\\to0$. For $E_{00}$ we have,\n\\begin{equation}\nE_{00}(z)\n=\nQ_0 K(0)Q_0\n-Q_0 [z +\\lambda^2 [A(z)-(\\log z) {\\mit \\Gamma}(z)\n+ i\\pi  {\\mit \\Gamma}(z)]]Q_0 ,\n\\label{eqn:5.C.1.100}\n\\end{equation}\nwhere the first term of the above is invertible in $M_0$,\nand the last term vanishes as $z \\to 0$.\nHence, $E_{00}(z) $ is invertible in $M_0$\nfor sufficiently small $|z|>0$ and is expanded as\n\\begin{equation}\nE_{00}^{-1}(z) \\!\n=\\!\n-\\sum_{j=0}^{\\infty}\n\\{[Q_0 K(0)Q_0 ]^{-1}\n[z +\\lambda^2 [A(z)-(\\log z) {\\mit \\Gamma}(z)\n+ i\\pi {\\mit \\Gamma}(z)]]Q_0 \\}^j\n[Q_0 K(0)Q_0 ]^{-1}\n=\nO(1) ,\n\\label{eqn:5.C.1.110b}\n\\end{equation}\nfor small $|z|$.\nFurthermore, we obtain\n\\begin{equation}\nE_{kl}(z) =\n-\\lambda^2 Q_k [A(z)-(\\log z) {\\mit \\Gamma}(z) +\\pi i {\\mit \\Gamma}(z) ]Q_l =O(z \\log z),\n\\label{eqn:5.C.1.130}\n\\end{equation}\nfor $k\\neq l$ as $z \\to 0$, because\n$Q_0 K(0)Q_1=Q_1 K(0)Q_0=0$.\nSolving Eq. (\\ref{eqn:5.C.1.50}), we obtain \\cite{MatrixAnalysis}\n\\begin{eqnarray}\nQ_0\\tilde{R}Q_0\n&=&\n(E_{00}-E_{01}E_{11}^{-1}E_{10})^{-1} =O(1),\n\\label{eqn:5.C.1.120a}\\\\\nQ_0\\tilde{R}Q_1\n&=&\n-E_{00}^{-1}E_{01} Q_1\\tilde{R}Q_1\n=\n-E_{00}^{-1}E_{01} (E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1}\n=O(1),\n\\label{eqn:5.C.1.120b}\\\\\nQ_1\\tilde{R}Q_0\n&=&\n-Q_1\\tilde{R}Q_1 E_{10} E_{00}^{-1}\n=\n-(E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1} E_{10} E_{00}^{-1}\n=O(1),\n\\label{eqn:5.C.1.120c}\\\\\nQ_1\\tilde{R}Q_1\n&=&\n(E_{11}-E_{10}E_{00}^{-1}E_{01})^{-1} =O(z^{-1} (\\log z)^{-1}).\n\\label{eqn:5.C.1.120d}\n\\end{eqnarray}\nIt is worth noting that \nsince the relation $[K(z) -z]\\tilde{R}(z) =1$\nis analytically continued to the second Riemann sheet\nthrough the cut $[0, \\infty)$, \nthe above-mentioned results are also valid for such a continued region and\nthe estimations obtained here can be applied without any corrections.\n\n\nWhen we consider the small energy behavior of $\\tilde{R}^- (\\omega)$,\nit is convenient to expand $E_{11}$, differently from Eq. (\\ref{eqn:5.C.1.60b}),\nas\n\\begin{eqnarray}\nE_{11}(z)\n&=&\n\\lambda^2z (\\log z -2\\pi i) Q_1 {\\mit \\Gamma}_1 Q_1\n\\nonumber\\\\\n&&\n-z Q_1\n-\\lambda^2 Q_1 \\{ A(z) -(\\log z -2\\pi i)[{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n-i\\pi {\\mit \\Gamma}(z)\\} Q_1 .\n\\label{eqn:5.C.1.125b}\n\\end{eqnarray}\nAll the above-obtained results are only changed by replacing the term\n$\\log z$ with $\\log z -2\\pi i$.\nThen, we can obtain from Eq. (\\ref{eqn:5.C.1.60b})\n\\begin{equation}\nE_{11}^+ (\\omega)\n=\n\\lim_{\\epsilon \\to +0}E_{11}(\\omega+i\\epsilon)\n=\n\\lambda^2 (\\log \\omega) Q_1 {\\mit \\Gamma}(\\omega) Q_1\n-\\omega Q_1 -\\lambda^2 Q_1 [A(\\omega)-i\\pi {\\mit \\Gamma}(\\omega)]Q_1 ,\n\\label{eqn:5.C.1.127a}\n\\end{equation}\nwhile from Eq. (\\ref{eqn:5.C.1.125b})\n\\begin{equation}\nE_{11}^- (\\omega)\n=\n\\lim_{\\epsilon \\to +0}E_{11}(\\omega-i\\epsilon)\n=\n\\lambda^2 (\\log \\omega) Q_1 {\\mit \\Gamma}(\\omega) Q_1\n-\\omega Q_1 -\\lambda^2 Q_1 [A(\\omega)+i\\pi {\\mit \\Gamma}(\\omega)]Q_1 .\n\\label{eqn:5.C.1.127b}\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The small-energy behavior of $\\tilde{R}(z)$\nin the exceptional case of the second kind}\n\\label{subsec:5.D}\n\nIn the exceptional case of the second kind,\nit follows that $Q_1 = 0$, $Q_2 \\neq 0$, and $Q_0 +Q_2 =1$.\nLet us consider the asymptotic behavior of the reduced resolvent at small energies,\nwhich\nis written in the following form,\n$\\tilde{R} (z)=\\sum_{k,l=0,2}Q_k \\tilde{R} (z) Q_l $.\nWe now introduce the four matrices again,\n\\begin{equation}\nE_{kl}(z)\n=Q_k [K(z) -z]Q_l ,\n\\label{eqn:5.C.2.40}\n\\end{equation}\nwhere $k, l =0, 2$.\nFrom the relation that\n$[K(z) -z]\\tilde{R}(z) =1$,\nthey satisfy that\n\\begin{equation}\nE_{k0} Q_0 \\tilde{R} Q_l + E_{k2} Q_2 \\tilde{R} Q_l =Q_k \\delta_{kl},\n\\label{eqn:5.C.2.50}\n\\end{equation}\nfor $k, l=0, 2$.\nThis time,\n$E_{22}$ and $E_{00}$ are invertible in $M_2$ and $M_0$, respectively.\nIn fact, from Eqs. (\\ref{eqn:formfactor2}) and (\\ref{eqn:.220}) we have\n\\begin{equation}\nE_{22}(z) =-z Q_2(1+\\lambda^2 A_1) Q_2 -\\lambda^2 Q_2[A(z)-z A_1 -(\\log z) {\\mit \\Gamma}(z)\n+ i\\pi {\\mit \\Gamma}(z)]Q_2 ,\n\\label{eqn:5.C.2.60b}\n\\end{equation}\nwhere $Q_2K(0)Q_2 =0$ was used.\nNote that since $A(z)-z A_1=O(z^2)$ and $Q_2 {\\mit \\Gamma}_1 Q_2=0$\n[see Eq. (\\ref{eqn:.350})],\nthe second term of the rhs of Eq. (\\ref{eqn:5.C.2.60b}) is of the order of\n$O(z^2 \\log z)$.\nFurthermore, since $Q_2 A_1 Q_2 \\geq 0$ from $Q_2 {\\mit \\Gamma}_1 Q_2=0$\nand Lemma \\ref{lm:1st_order},\n$Q_2(1+\\lambda^2 A_1)Q_2 >0$ and invertible in $M_2$.\nThese facts bring us the fact that $E_{22}(z) $ is invertible in $M_2$ for sufficiently small\n$z>0$, that is\n\\begin{eqnarray}\nE_{22}^{-1}(z)\n&=&\n-\\sum_{j=0}^{\\infty}\n(-\\tilde{E}_{22}(z) )^j \\frac{1}{z}[Q_2 (1+\\lambda^2 A_1 )Q_2 ]^{-1}\n\\label{eqn:5.C.2.90a}\\\\\n&=&\n\\frac{1}{z}[Q_2 (1+\\lambda^2 A_1 )Q_2 ]^{-1}+O(\\log z)=O(z^{-1}),\n\\label{eqn:5.C.2.90b}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\tilde{E}_{22}(z)\n:=\n\\frac{1}{z}[Q_2 (1+\\lambda^2 A_1 )Q_2  ]^{-1}\n\\lambda^2 Q_2[A(z)-z A_1 -(\\log z) {\\mit \\Gamma}(z)+ i\\pi {\\mit \\Gamma}(z)]Q_2.\n\\label{eqn:5.C.2.90c}\n\\end{equation}\nFor $E_{00}$, we next have\n\\begin{equation}\nE_{00}(z)\n=\nQ_0 K(0)Q_0\n-Q_0 [z +\\lambda^2 [A(z)-(\\log z) {\\mit \\Gamma}(z)\n+ i\\pi {\\mit \\Gamma}(z)]]Q_0,\n\\label{eqn:5.C.2.100}\n\\end{equation}\nwhere the first term of the above is invertible in $M_0$,\nand the last term vanishes as $|z| \\to 0$.\nHence, $E_{00}(z) $ is invertible in $M_0$\nfor sufficiently small $|z|>0$, and the inverse is obtained as a Neumann series.\nOn the other hand, $E_{20}$ and $E_{02}$ behave as\n\\begin{equation}\nE_{kl}(z) =\nQ_k [-z(1+\\lambda^2 A_1 )\n-\\lambda^2 [A(z)-z A_1-(\\log z) {\\mit \\Gamma}(z)\n+ \\pi i {\\mit \\Gamma}(z)] ]Q_l =O(z),\n\\label{eqn:5.C.2.130}\n\\end{equation}\nfor small $|z|$ where $k\\neq l$.\nSolving Eqs. (\\ref{eqn:5.C.2.50}) as in Eqs. (\\ref{eqn:5.C.1.120a}) to (\\ref{eqn:5.C.1.120d}),\none sees that\n\nQ_0\\tilde{R}Q_0=O(1),\n$\nfor $k,l=0,2$, except\n\\begin{equation}\nQ_2\\tilde{R}Q_2\n=\n(E_{22}-E_{20}E_{00}^{-1}E_{02})^{-1} =O(z^{-1}),\n\\label{eqn:5.C.2.120d}\n\\end{equation}\nas $z\\to 0$.\nIn particular, the last equation is expanded as,\n\\begin{equation}\nQ_2 \\tilde{R} (z) Q_2\n=\n\\sum_{j=0}^{\\infty}\n\\Bigl[ E_{22}^{-1}E_{20}E_{00}^{-1}E_{02} \\Bigr]^j\nE_{22}^{-1}\n=\n-\\frac{1}{z}[Q_2 (1+\\lambda^2 A_1 )Q_2 ]^{-1} +O(\\log z),\n\\label{eqn:5.C.2.140d}\n\\end{equation}\nfor small $|z|$, where we used Eq. (\\ref{eqn:5.C.2.90b}).\n\n\n\n\n\nWe now remark that the zero energy eigenspace of $H$\ndenoted by ${\\cal N}_0$ is completely characterized by $M_2$.\nThat is, there is a bijection from $M_2\\oplus \\{0\\}$ to ${\\cal N}_0$.\nFrom the discussion concerning Eqs. (\\ref{eqn:.333}), (\\ref{eqn:.335}), and (\\ref{eqn:.336}),\nfor any $\\ket{\\Psi} \\in {\\cal M}_0$, there is a vector $\\ket{\\psi}\\in M_2\\oplus \\{0\\}$ such that\n\\begin{equation}\n\\ket{\\Psi}=\\ket{\\psi}-\\lambda \\int_0^\\infty \\frac{\\sum_{n=1}^N v_n (\\omega)\\psi_n}{\\omega}\n\\ket{\\omega}d\\omega\n=\n[1-\\lambda R_0 (0)V]\\ket{\\psi},\n\\label{eqn:5.C.2.150}\n\\end{equation}\nwhere\n$V$ is restricted to $\\mathbb{C}^N \\oplus \\{0\\}$ and\n$R_0 (0)$ is the (unbounded) multiplication operator of $1\/\\omega$ in $L^2((0,\\infty))$.\nThen we see $V\\ket{\\psi} \\in D(R_0 (0))$ because $\\ket{\\psi}\\in M_2\\oplus \\{0\\}$.\nThus $1-\\lambda R_0 (0)V$ is well defined as an operator from $M_2\\oplus \\{0\\}$ to ${\\cal H}$.\nNow, Eq. (\\ref{eqn:5.C.2.150}) tells us that\n$1-\\lambda R_0 (0)V$ is a surjection from $M_2\\oplus \\{0\\}$ to ${\\cal N}_0$.\nOn the other hand, for any $\\ket{\\Psi} \\in {\\cal N}_0$, if $\\ket{\\Psi}=0$, i.e.,\n$0=\\braket{\\Psi}{\\Psi}$, Eq. (\\ref{eqn:5.C.2.150}) implies that\n$0=\\braket{\\Psi}{\\Psi}\\geq \\braket{\\psi}{\\psi}$. Therefore,\n$1-\\lambda R_0 (0)V$ is also an injection from $M_2\\oplus \\{0\\}$ to ${\\cal N}_0$,\nand the proof is completed.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The small-energy behavior of $\\tilde{R}^\\pm (\\omega) $\nin the exceptional case of the third kind}\n\\label{subsec:5.E}\n\n\n\nIn the exceptional case of the third kind,\nfrom the definition, $Q_1 \\neq 0$, $Q_2 \\neq 0$, and $Q_0 +Q_1 +Q_2 =1$.\nThe reduced resolvent is written in the form,\n$\\tilde{R}^\\pm (\\omega)\n=\n\\sum_{k,l=0}^{2}\nQ_k \\tilde{R}^\\pm (\\omega) Q_l\n$.\nThis time, we need nine matrices,\n\\begin{equation}\nE_{kl}^{\\pm}(\\omega) =Q_k [K^\\pm (\\omega) -\\omega]Q_l ,\n\\label{eqn:5.C.3.40}\n\\end{equation}\nfor $k, l = 0, 1, 2$.\nFrom the relation that\n$[K^\\pm (\\omega) -\\omega]\\tilde{R}^\\pm (\\omega) =1$,\nthey satisfy that\n\\begin{equation}\nE_{k0}^{\\pm} Q_0 \\tilde{R}^\\pm Q_l + E_{k1}^{\\pm} Q_1 \\tilde{R}^\\pm Q_l\n+ E_{k2}^{\\pm} Q_2 \\tilde{R}^\\pm Q_l = Q_k \\delta_{kl},\n\\label{eqn:5.C.3.50}\n\\end{equation}\nfor $k, l = 0, 1, 2$.\nThe asymptotic behaviors of $E_{kl}^{\\pm}(\\omega) $\nare essentially examined\nin the preceding subsections, except for $E_{12}^{\\pm}(\\omega)$ and $E_{21}^{\\pm}(\\omega)$.\nThen, $E_{12}^{\\pm}(\\omega)$ becomes\n\\begin{eqnarray}\nE_{12}^{\\pm}(\\omega)\n&=&-\\omega \\lambda^2 Q_1 A_1 Q_2\n\\nonumber \\\\\n&&\n-\\lambda^2 Q_1 \\bigl[A(\\omega)-\\omega A_1 -(\\log \\omega) [{\\mit \\Gamma}(\\omega)-\\omega {\\mit \\Gamma}_1 ]\n\\pm \\pi i [{\\mit \\Gamma}(\\omega)-\\omega {\\mit \\Gamma}_1 ] \\bigr]Q_2\n\\label{eqn:5.C.3.60b}\\\\\n&=&-\\omega \\lambda^2 Q_1 A_1 Q_2+O(\\omega^2 \\log \\omega ),\n\\label{eqn:5.C.3.60c}\n\\end{eqnarray}\nwhere $Q_1K(0)Q_2 =0$, $Q_1 Q_2 =0$,\nand ${\\mit \\Gamma}_1 Q_2 =0$ are used. The last relation follows from the fact that $Q_2 {\\mit \\Gamma}_1 Q_2 =0$ and\n${\\mit \\Gamma}_1 \\geq 0$.\nIn addition, since $Q_1 {\\mit \\Gamma} (\\omega)Q_2=O(\\omega^2)$,\nwe see that $Q_1 A_1 Q_2=\\int_0^\\infty Q_1 {\\mit \\Gamma} (\\omega) Q_2 \\omega^{-2} d\\omega$.\nBy the same way, we also see that\n\\begin{equation}\nE_{21}^{\\pm}(\\omega)\n=-\\omega \\lambda^2 Q_1 A_1 Q_2+O(\\omega^2 \\log \\omega ).\n\\label{eqn:5.C.3.60d}\n\\end{equation}\nTo solve Eqs. (\\ref{eqn:5.C.3.50}), let us now put the $N\\times N$ matrix $\\bf E$ as\n\\begin{equation}\n{\\bf E}=\n\\left[\n\\begin{array}{ccc}\nE_{00}&E_{01}&E_{02} \\\\\nE_{10}&E_{11}&E_{12} \\\\\nE_{20}&E_{21}&E_{22}\n\\end{array}\n\\right] ,\n\\label{eqn:5.C.3.65a}\n\\end{equation}\nand partition it into\n${\\bf A}=\n\\left[\n\\begin{array}{cc}\nE_{00}&E_{01} \\\\\nE_{10}&E_{11}\n\\end{array}\n\\right]\n$,\n${\\bf B}=\n\\left[\n\\begin{array}{c}\nE_{02} \\\\\nE_{12}\n\\end{array}\n\\right]\n$,\n${\\bf C}=\n\\left[\n\\begin{array}{cc}\nE_{20}&E_{21}\n\\end{array}\n\\right]\n$,\n${\\bf D}=\n\\left[\n\\begin{array}{c}\nE_{22}\n\\end{array}\n\\right]$. %\nThen, from the inverse matrix formula again,\n${\\bf E}^{-1}$ ($=\\tilde{R}$) is expressed as \\cite{MatrixAnalysis}\n\\begin{equation}\n{\\bf E}^{-1}=\n\\left[\n\\begin{array}{cc}\n[{\\bf A}-{\\bf B}{\\bf D}^{-1}{\\bf C}]^{-1}\n& -{\\bf A}^{-1}{\\bf B}[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1}\\\\\n-[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1} {\\bf C} {\\bf A}^{-1}\n&[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1}\n\\end{array}\n\\right] .\n\\label{eqn:5.C.3.67}\n\\end{equation}\nThe validities of ${\\bf A}^{-1}$ and ${\\bf D}^{-1}$ are already ensured in\nthe exceptional cases of the first and second kinds, respectively.\nThen, one sees that\nsince\n${\\bf A}^{-1}=O(\\omega^{-1} (\\log \\omega)^{-1})$,\n${\\bf B}=O(\\omega)$,\n${\\bf C}=O(\\omega)$, and\n${\\bf D}^{-1}=O(\\omega^{-1})$,\nit holds that ${\\bf A}^{-1}{\\bf B}{\\bf D}^{-1}{\\bf C}=O((\\log \\omega)^{-1})$.\nThus, $[{\\bf A}-{\\bf B}{\\bf D}^{-1}{\\bf C}]^{-1}$ exists for small $\\omega$\nand $[{\\bf A}-{\\bf B}{\\bf D}^{-1}{\\bf C}]^{-1}=O(\\omega^{-1}(\\log \\omega)^{-1})$.\nWe also show that $[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1}$ exists for small $\\omega$\nand $[{\\bf D}-{\\bf C}{\\bf A}^{-1}{\\bf B}]^{-1}=O(\\omega^{-1})$.\nTo obtain the asymptotic forms of the matrix components of ${\\bf E}^{-1}$ explicitly,\nsome redundant calculation is required; however, it could be achieved by \na manner as similar to that used in the preceding subsections.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Asymptotic expansion of the reduced resolvent at small $z$ }\n\\label{sec:5.5}\n\n\n\nWe examine the small-energy behavior of the reduced resolvent\nonly for the regular case and the exceptional case of the first kind.\nThis analysis is crucial for determining the asymptotic behavior of\nthe reduced time evolution operator at long times.\n\n\n\n\\subsection{The regular case}\n\nHere, we introduce $\\tilde{A}(\\omega ) := \\omega\/\\lambda^2 +A(\\omega ) $\nand suppose that $\\tilde{A}(\\omega )$ and ${\\mit \\Gamma}(\\omega ) $ behave as\n\\begin{equation}\n\\tilde{A}(\\omega ) := \\frac{1}{\\lambda^2} \\omega +\nA(\\omega ) = \\sum_{n=n_a}^{n_a+N} \\omega^n \\tilde{A}_n +O(\\omega^{n_a+N+1}), ~~~\n{\\mit \\Gamma}(\\omega )\n=\\sum_{n=n_b}^{n_b+N} \\omega^n {\\mit \\Gamma}_n +O(\\omega^{n_b+N+1}),\n\\label{eqn:rff.220}\n\\end{equation}\nas $\\omega \\to 0$, respectively,\nthat is, $\\tilde{A}_{n} =0$ for all $n< n_a$ and ${\\mit \\Gamma}_{n_b}=0$ for all $n< n_b$, \nwhile $\\tilde{A}_{n_a}\\neq0$ and ${\\mit \\Gamma}_{n_b}\\neq0$.\nThen, we obtain\n\\begin{equation}\n\\frac{1}{\\lambda^2} \\omega +\n{D} (\\omega )\n=\nS(0) +\\tilde{A}(\\omega )-(\\log \\omega){\\mit \\Gamma}(\\omega)\n=\nS(0)\n-\\omega^{n_b} \\log \\omega {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\tilde{A}_{n_a} + O(\\omega h(\\omega) ) ,\n\\label{eqn:rff.230}\n\\end{equation}\nas $\\omega \\to +0$, where\n\\begin{equation}\nh(\\omega)\n=\n\\left\\{\\begin{array}{cc}\n\\omega^{n_b} \\log \\omega & (n_b \\leq n_a) \\\\\n\\omega^{n_a}  & (n_b > n_a)\n\\end{array}\\right. .\n\\label{eqn:rff.245}\n\\end{equation}\n\n\nIt is important to note that the values of two parameters $n_a$ and $n_b$ \nare not determined independently. We shall here consider $n_b$ as a controllable one. \nWe first note that if $n_b \\geq 2$ then $n_a =1$ should be concluded, \nbecause from Lemma \\ref{lm:1st_order} we have $A_1 >0$, so that \n$\\tilde{A}_1 =1\/\\lambda^2  +A_1 >0$ holds. \nTherefore, the conditions $n_b \\leq n_a$ and $n_b > n_a$ \ncan be realized only in the situations \n\\begin{equation}\nn_b =1 ~\\mbox{and}~ n_a \\geq 1,\\quad\\mbox{and}\\quad n_b \\geq 2~ \\mbox{and}~ n_a = 1,\n\\label{eqn:rff.246}\n\\end{equation}\nrespectively. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lm} \\label{lm:rffremainder}\n: Assume that $0$ is a regular point for $H$.\nThen the $r$-th derivative of $\\tilde{R}^\\pm (\\omega)$ asymptotically\nbehaves as\n\\begin{equation}\n\\frac{d^r \\tilde{R}^\\pm (\\omega)}{d\\omega^r}\n=\n\\left\\{\n\\begin{array}{cc}\nO(1) & (r=0)\\\\\nO(\\omega^{1-r}(\\log \\omega)^{\\theta(1-r)}) & (r\\geq1)\n\\end{array}\n\\right. ,\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(1) & (r=0)\\\\\nO(\\omega^{[1-r]^+}) &(1\\leq r < n_b) \\\\\nO(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) &(n_b\\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall3}\n\\end{equation}\nfor $n_b =1$, or $n_b \\geq 2$, \nrespectively, as $\\omega \\to 0$,\nwhere $[x]^+ =\\max \\{x, 0 \\}$\nand $\\theta(x)=1$ for $x\\geq 0$ or $0$ for $x<0$.\nIn addition, the $r$-th derivative of $\\tilde{R}^\\pm (\\omega)$\nis approximated by that of a finite series\n\\begin{equation}\n(K(0))^{-1}\n+\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1} ,\n\\label{eqn:rffSmall5}\n\\end{equation}\nthat is, it is shown that\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggl\\{\n\\tilde{R}^\\pm (\\omega)\n\\nonumber \\\\\n&&\n-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\biggr\\}\n\\Biggr\\|\n\\nonumber \\\\\n&=&\n\\begin{array}{cc}\nO(\\omega^{2-r} (\\log \\omega)^{1+\\theta(2-r)})\n& (r \\geq 0)\n\\end{array}\n\\nonumber \\\\\n&&\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[2-r]^+}) & (0\\leq r\\leq n_b) \\\\\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)}) & (n_b+1 \\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall7}\n\\end{eqnarray}\nfor $n_b =1$, or $n_b \\geq 2$, \nrespectively, as $\\omega \\to 0$. Here, $n_a$ is restricted to the condition (\\ref{eqn:rff.246}). \n\\end{lm}\n\n\n\n\n\n\n\n\n\n\n\n{\\sl Proof} :\nThe left-hand side (lhs) of Eq. (\\ref{eqn:rffSmall7}) is written as follows:\n\\begin{eqnarray}\n&&\\hspace*{-10mm}\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggl\\{\n\\tilde{R}^\\pm (\\omega)\n\\nonumber \\\\\n&&\\hspace*{-10mm}\n-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\biggr\\}\n\\Biggr\\|\n\\nonumber \\\\\n&&\\hspace*{-10mm}\n\\leq\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigl\\{\n\\tilde{R}^\\pm (\\omega)-S_1 (\\omega)\n\\bigr\\}\n\\right\\|\n\\nonumber \\\\\n&&\\hspace*{-10mm}\n~~~+\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggl\\{\nS_1 (\\omega)\n\\nonumber \\\\\n&&\\hspace*{-10mm}\n~~~-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\bigg\\}\n\\Biggr\\| ,\n\\label{eqn:rffSmall50}\n\\end{eqnarray}\nwhere $S_N(\\omega)$ is defined by Eq. (\\ref{eqn:.185}).\nWhen $r=0$, the first term on the rhs of the above is estimated from the special case of the below\nfor $N=1$,\n\\begin{equation}\n\\left\\|\n\\tilde{R}^\\pm (\\omega)-S_N (\\omega)\n\\right\\|\n\\! \\leq \\!\n\\frac{\n\\|\n\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n\\|^{N+1}\n\\|\n(K(0))^{-1}\n\\|^{N+2}\n}\n{\n1-\\|(K(0))^{-1}\\|\n\\| [\\omega +\\lambda^2 {D} (\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]\\|\n}\n=O(h(\\omega)^{N+1} ),\n\\label{eqn:rffSmall60c}\n\\end{equation}\nas $\\omega \\to 0$. When $r\\geq 1$, instead we have\n\\begin{eqnarray}\n&&\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigl\\{\n\\tilde{R}^\\pm (\\omega)\n-\nS_1 (\\omega)\n\\bigr\\}\n\\right\\|\n\\nonumber \\\\\n&\\leq&\n\\left\\|\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\{\n\\prod_{i=1}^{j}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\}\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|\n\\nonumber \\\\\n&&\n+\n\\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^r }{d\\omega^r}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n-\n\\frac{d^r}{d\\omega^r}\nS_1 (\\omega)\n\\right\\| ,\n\\label{eqn:rffSmall70b}\n\\end{eqnarray}\nwhere Eq. (\\ref{eqn:Large100}) is used, and here $s_i \\geq 1$ and $\\sum_{i=1}^{j} s_i =r$\nshould be satisfied.\nNote that the first term on the rhs of Eq. (\\ref{eqn:rffSmall70b})\nappears only for $r\\geq 2$, which is estimated in the following.\nIn the following estimations, we temporarily forget the restriction (\\ref{eqn:rff.246}) \nand consider the two general cases: $n_b \\leq n_a$ and $n_b > n_a$.\nIn the case of $n_b \\leq n_a$, we can obtain for $r\\geq 2$\n\\begin{eqnarray}\n&&\n\\left\\|\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\{\n\\prod_{i=1}^{j}\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\}\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|\n\\nonumber \\\\\n&&\\leq\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|^{j+1}\nO(\\omega^{jn_b-r})\n\\prod_{i=1}^{j}\nO((\\log \\omega)^{\\theta(n_b-s_i)})\n\\nonumber \\\\\n&&=\nO(\\omega^{2n_b-r} (\\log \\omega)^{\\theta(n_b+1-r)+\\theta(2n_b+1-r)}),\n\\label{eqn:rffSmall80a}\n\\end{eqnarray}\nas $\\omega \\to 0$.\nFor $n_a < n_b$,\n\\begin{eqnarray}\n&\n\\left\\|\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\tilde{R}^{\\pm}(\\omega )\n\\prod_{i=1}^{j}\n\\left\\{\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\}\n\\right\\|\n\\nonumber \\\\\n&&=\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[2n_a-r]^+}) &(2\\leq r\\leq n_a+n_b-1) \\\\\nO(\\omega^{n_a+n_b-r} (\\log \\omega)^{\\theta(n_a+n_b-r)})\n&(n_a+n_b\\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall80b}\n\\end{eqnarray}\nas $\\omega \\to 0$.\nWe here used that\n\\begin{eqnarray}\n&&\n\\frac{d^r}{d\\omega^r }\n[\n\\omega+\\lambda^2 {D}(\\omega )\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n-\\lambda^2 S(0)\n]\n\\nonumber\\\\\n&&\n=\nO(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)})\n, \\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[n_a-r]^+}) & (0\\leq r< n_b) \\\\\nO(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) & (r\\geq n_b)\n\\end{array}\n\\right.\n,\n\\label{eqn:rffSmall98b}\n\\end{eqnarray}\nfor $n_b\\leq n_a$, or $n_a< n_b$, respectively, as $\\omega \\to 0$.\nEq. (\\ref{eqn:rffSmall98b}) follows from\n\\begin{equation}\n\\frac{d^r \\tilde{A}(\\omega )}{d\\omega^r }\n=O(\\omega^{[n_a-r]^+}) ,\n~~~\n\\frac{d^r {\\mit \\Gamma}(\\omega )}{d\\omega^r }\n=O(\\omega^{[n_b-r]^+}) ,~~~\n\\frac{d^r (\\log \\omega) {\\mit \\Gamma}(\\omega )}{d\\omega^r }\n=O(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) ,\n\\label{eqn:rffSmall90b}\n\\end{equation}\nas $\\omega \\to 0$.\nIncorporating Eqs. (\\ref{eqn:rffSmall80a}), (\\ref{eqn:rffSmall80b}),\nand Eq. (\\ref{eqn:rffSmall98b}), with\n\\begin{eqnarray}\n&&\n\\left\\|\n\\frac{d^r \\tilde{R}^\\pm (\\omega)}{d\\omega^r}\n\\right\\|\n\\nonumber\\\\\n&\\leq&\n\\left\\|\n\\sum_{j=2}^{r}\n{\\sum_{\\{s_i \\}_{i=1}^{j} }}'\na^{(r)} (\\{s_i \\}_{i=1}^{j} )\n\\left\\{\n\\tilde{R}^{\\pm}(\\omega )\n\\prod_{i=1}^{j}\n\\frac{d^{s_i}}{d\\omega^{s_i}}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\}\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|\n\\nonumber \\\\\n&&\n+\n\\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^r }{d\\omega^r}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\| ,\n\\label{eqn:rffSmall99b}\n\\end{eqnarray}\nwe have\n\\begin{eqnarray}\n&&\\hspace*{-10mm}\n\\left\\|\n\\frac{d^r \\tilde{R}^\\pm (\\omega)}{d\\omega^r}\n\\right\\|\n\\nonumber\\\\\n&&\\hspace*{-10mm}\n=\n\\left\\{\n\\begin{array}{cc}\nO(1) & (r=0) \\\\\nO(\\omega^{n_b-r}(\\log \\omega)^{\\theta(n_b-r)})) & (r\\geq 1 )\n\\end{array}\n\\right.\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(1) & (r=0) \\\\\nO(\\omega^{[n_a-r]^+}) & (1\\leq r< n_b) \\\\\nO(\\omega^{n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) & (n_b \\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall99c}\n\\end{eqnarray}\nfor $n_b\\leq n_a$, or $n_a< n_b$, respectively, as $\\omega \\to 0$.\nThen, the first part of the statement can be shown under the restriction (\\ref{eqn:rff.246}).\nLet us next examine the second term on the rhs of Eq. (\\ref{eqn:rffSmall70b}),\nwhich reads for $r\\geq 1$,\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\frac{d^r }{d\\omega^r}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\tilde{R}^{\\pm}(\\omega )\n-\n\\frac{d^r }{d\\omega^r}\nS_1(\\omega)\n\\Biggr\\|\n\\nonumber \\\\\n&&\\leq\n2 \\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n-(K(0))^{-1}\n\\right\\|\n\\left\\|\n\\frac{d^r }{d\\omega^r}\n\\left[\n\\omega + \\lambda^2 {D}(\\omega )\n\\pm \\lambda^2 \\pi i {\\mit \\Gamma}(\\omega )\n\\right]\n\\right\\|\n\\left\\|\n\\tilde{R}^{\\pm}(\\omega )\n\\right\\|\n\\nonumber \\\\\n&&=\nO(\\omega^{2n_b-r} (\\log \\omega)^{1+\\theta(n_b-r)}),\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{n_a+[n_a-r]^+}) & (r< n_b) \\\\\nO(\\omega^{n_a+n_b-r} (\\log \\omega)^{\\theta(n_b-r)}) & (r\\geq n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall100}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$ respectively,\nas $\\omega \\to 0$.\nWe here used Eq. (\\ref{eqn:rffSmall60c}) with $N=0$.\nTherefore, substituting Eqs. (\\ref{eqn:rffSmall80a}),\n(\\ref{eqn:rffSmall80b}), and (\\ref{eqn:rffSmall100}) into\nEq. (\\ref{eqn:rffSmall70b}), one has  for $r\\geq 1$,\n\\begin{eqnarray}\n&&\\hspace*{-12mm}\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigr\\{ \\tilde{R}^\\pm (\\omega)- S_1 (\\omega) \\bigr\\}\n\\right\\|\n\\nonumber\\\\\n&&\\hspace*{-12mm}\n=O(\\omega^{2n_b-r} (\\log \\omega)^{1+\\theta(n_b+1-r)}) ,\n\\mbox{or}\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[2n_a-r]^+}) &(r\\leq n_a+n_b-1) \\\\\nO(\\omega^{n_a+n_b-r} (\\log \\omega)^{\\theta(n_a+n_b-r)})\n&(n_a+n_b\\leq r \\leq 2n_b)\n\\end{array}\n\\right. \\label{eqn:rffSmall110}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$, respectively, as $\\omega \\to 0$.\nNote that this estimation is also valid for $r=0$ because it reproduces\nEq. (\\ref{eqn:rffSmall60c}) for $N=1$.\n\n\n\n\n\nLet us now evaluate the last term in Eq. (\\ref{eqn:rffSmall50}).\nFor $r\\geq 0$, we have\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggr\\{\nS_1 (\\omega)\n\\nonumber\\\\\n&&\n-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\bigg\\}\n\\Biggr\\|\n\\nonumber \\\\\n&&\\leq\n\\left\\|\n(K(0))^{-1}\n\\right\\|^2\n\\nonumber \\\\\n&&\n~~~\\times\n\\left\\|\n\\frac{d^r }{d\\omega^r}\n\\left[\n-(\\log \\omega) \\lambda^2 ({\\mit \\Gamma}(\\omega)- \\omega^{n_b} {\\mit \\Gamma}_{n_b} )\n+\\lambda^2 (\\tilde{A}(\\omega)- \\omega^{n_a} \\tilde{A}_{n_a} ))\n\\pm \\lambda^2  \\pi i\n\\left(\n{\\mit \\Gamma}(\\omega)- \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right)\n\\right]\n\\right\\|\n\\nonumber \\\\\n&&=\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)}),\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[n_a+1-r]^+}) & (0\\leq r\\leq n_b) \\\\\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)}) & (r\\geq n_b+1)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall130b}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$ respectively,\nas $\\omega \\to 0$ for any $r\\geq 0$.\nWe here used that for $r\\geq 0$,\n\\begin{equation}\n\\frac{d^r ({\\mit \\Gamma}(\\omega)- \\omega^{n_b} {\\mit \\Gamma}_{n_b})}{d\\omega^r}\n=O(\\omega^{[n_b+1-r]^+}) ,~~\n\\frac{d^r (\\log \\omega) ({\\mit \\Gamma}(\\omega)- \\omega^{n_b} {\\mit \\Gamma}_{n_b})}{d\\omega^r }\n=\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)})\n\\label{eqn:rffSmall140}\n\\end{equation}\nas $\\omega \\to 0$, and so forth.\nThus,\nsetting Eqs. (\\ref{eqn:rffSmall110}) and (\\ref{eqn:rffSmall130b})\ninto Eq. (\\ref{eqn:rffSmall50}), we conclude that\n\\begin{eqnarray}\n&\n\\Biggl\\|\n\\frac{d^r}{d\\omega^r}\n\\biggl\\{\n\\tilde{R}^\\pm (\\omega)\n\\nonumber\\\\\n&&\n-\n(K(0))^{-1}\n-\n(K(0))^{-1}\n\\left[\n-\\omega^{n_b} (\\log \\omega) \\lambda^2 {\\mit \\Gamma}_{n_b}\n+\\omega^{n_a} \\lambda^2 \\tilde{A}_{n_a} \\pm \\lambda^2  \\pi i \\omega^{n_b} {\\mit \\Gamma}_{n_b}\n\\right]\n(K(0))^{-1}\n\\biggr\\}\n\\Biggr\\|\n\\nonumber \\\\\n&\n=\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{2n_b-r} (\\log \\omega)^{1+\\theta(n_b+1-r)})\n& (r \\geq 0, n_b=1) \\\\\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)})\n& (r \\geq 0, n_b\\geq2)\n\\end{array}\n\\right. ,\n\\nonumber\\\\\n&&\n~~~\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[n_a+1-r]^+}) & (0\\leq r\\leq n_b) \\\\\nO(\\omega^{n_b+1-r} (\\log \\omega)^{\\theta(n_b+1-r)}) & (n_b+1 \\leq r \\leq 2n_b)\n\\end{array}\n\\right. ,\n\\label{eqn:rffSmall60}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$, respectively,\nas $\\omega \\to 0$.\nBy taking into account the restriction (\\ref{eqn:rff.246}), we can show the last part of the lemma.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\nTo estimate the long time behavior of the reduced time evolution operator,\nthe above-mentioned lemma seems not precisely appropriate \nbecause the reduced time evolution operator is obtained from\nthe Fourier transform of the imaginary part of $\\tilde{R}^+ (\\omega)$,\nnot from $\\tilde{R}^+ (\\omega)$ itself, which is explained in the next section.\nHence, the following lemma is more appropriate for our purpose. \n\n\n\\begin{lm} \\label{lm:rffimremainder}\n: Assume that $0$ is a regular point for $H$.\nThen the $r$-th derivative of\n${\\rm Im}\\tilde{R}^+ (\\omega):=(\\tilde{R}^+ (\\omega) -\\tilde{R}^- (\\omega))\/2i$\nis approximated by that of\n\\begin{eqnarray}\n(K(0))^{-1}\n\\lambda^2 \\pi\n\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n(K(0))^{-1} ,\n\\label{eqn:rffimSmall5}\n\\end{eqnarray}\nin the sense that for $0\\leq r \\leq n_b+1$ the remainder is estimated as,\n\\begin{eqnarray}\n&&\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigl\\{\n{\\rm Im}\\tilde{R}^+ (\\omega)-\n(K(0))^{-1}\n\\lambda^2 \\pi\n\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n(K(0))^{-1}\n\\bigr\\}\n\\right\\|\n\\nonumber\\\\\n&&\n=\nO(\\omega^{2-r}\\log \\omega)\n, ~\\mbox{or} ~\nO(\\omega^{1+n_b-r})\n,\n\\label{eqn:rffimSmall60}\n\\end{eqnarray}\nfor $n_b=1$, or $n_b \\geq 2$, respectively, as $\\omega \\to 0$. \nFor $r=n_b +2$, the estimation is replaced by\n$O(\\omega^{-1})$ for $n_b=1$, $O(\\log \\omega)$ for $n_b = 2$, or $O(1)$ for $n_b \\geq 3$, \nrespectively, as $\\omega \\to 0$.\n\\end{lm}\n\n\n\n\n\n\n\n\n{\\sl Proof} :\nSince ${\\rm Im}\\tilde{R}^+ (\\omega)=\\lambda^2 \\pi \\tilde{R}^+ (\\omega) {\\mit \\Gamma}(\\omega)\n\\tilde{R}^- (\\omega)$, one has\n\\begin{eqnarray}\n&&\n\\left\\|\n\\frac{d^r}{d\\omega^r}\n\\bigl\\{\n{\\rm Im}\\tilde{R}^+ (\\omega)\n-\n(K(0))^{-1}\n\\lambda^2 \\pi\n\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n(K(0))^{-1}\n\\bigr\\}\n\\right\\|\n\\label{eqn:rffimSmall40a} \\\\\n&\\leq&\n\\lambda^2 \\pi\n\\sum_{\\scriptsize\n\\begin{array}{c}\ns\\geq0, t\\geq0, u\\geq0, \\\\\n(s+t+u=r)\n\\end{array}}^{r}\n[F_1(s,t,u)+F_2(s,t,u)+F_3(s,t,u)] ,\n\\label{eqn:rffimSmall40c}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&\\displaystyle F_1(s,t,u)=C_{stu}\n\\left\\|\n\\frac{d^s \\tilde{R}^+ (\\omega)}{d\\omega^s}\n\\right\\|\n\\left\\|\n\\frac{d^t {\\mit \\Gamma}(\\omega)}{d\\omega^t}\n\\right\\|\n\\left\\|\n\\frac{d^u}{d\\omega^u}\n\\bigl\\{ \\tilde{R}^- (\\omega)-(K(0))^{-1} \\bigr\\}\n\\right\\| ,&\n\\label{eqn:rffimSmall41a}\\\\\n&\\displaystyle F_2(s,t,u)=C_{stu}\n\\left\\|\n\\frac{d^s \\tilde{R}^+ (\\omega)}{d\\omega^s}\n\\right\\|\n\\left\\|\n\\frac{d^t}{d\\omega^t}\n\\bigl\\{\n{\\mit \\Gamma}(\\omega)\n-\n\\omega^{n_b}({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n\\bigr\\}\n\\right\\|\n&\n\\nonumber\\\\\n&\\hspace*{-57mm}\n\\displaystyle\n\\times\n\\left\\|\n\\frac{d^u(K(0))^{-1}}{d\\omega^u}\n\\right\\|,&\n\\label{eqn:rffimSmall41b}\\\\\n&\\displaystyle\nF_3(s,t,u)=C_{stu}\n\\left\\|\n\\frac{d^s}{d\\omega^s}\n\\bigl\\{\n\\tilde{R}^+ (\\omega) -(K(0))^{-1}\n\\bigr\\}\n\\right\\|\n\\left\\|\n\\frac{d^t}{d\\omega^t}\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} +\\omega^2 {\\mit \\Gamma}_{n_b +2})\n\\right\\|\n&\n\\nonumber\\\\\n&\\hspace*{-67mm}\n\\displaystyle\n\\times\n\\left\\|\n\\frac{d^u(K(0))^{-1}}{d\\omega^u}\n\\right\\| ,\n&\n\\label{eqn:rffimSmall41c}\n\\end{eqnarray}\nwhere $C_{stu}$'s are appropriate constants.\nFor $1\\leq r\\leq n_b +1$,\nthe summation of the first summand in Eq. (\\ref{eqn:rffimSmall40c}) can be estimated as\n\\begin{eqnarray}\n&&\n\\sum_{\\scriptsize\n\\begin{array}{c}\ns\\geq0, t\\geq0, u\\geq0, \\\\\n(s+t+u=r)\n\\end{array}}^{r}\nF_1(s,t,u)\n\\nonumber\\\\\n&&\n=\nF_1(0,r,0)\n+\n\\sum_{t\\geq 0,u\\geq 1}^r\nF_1(0,t,u)\n+\n\\sum_{s\\geq 1,t\\geq 0}^r\nF_1(s,t,0)\n+\n\\sum_{\\min\\{ s, u\\}\\geq 1}^r F_1(s,t,u)\n\\nonumber \\\\\n&&\n=\nO(F_1(0, r-1, 1))\n=\nO(\\omega^{2n_b -r} \\log \\omega ) ,~~\n\\mbox{or}~~\nO(\\omega^{n_a+n_b -r}) ,\n\\label{eqn:rffimSmall47a}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_b > n_a$, respectively, as $\\omega \\to 0$.\nNote that from Eq. (\\ref{eqn:rffSmall60c}) for $N=0$ this estimation is valid for $r=0$ too.\nFor $r=n_b+2$, it is estimated as\n\\begin{equation}\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{n_b-1}\\log \\omega) & (n_b\\geq 2)\\\\\nO((\\log \\omega)^2 ) & (n_b=1)\n\\end{array}\n\\right. ,~~\n\\mbox{or}~~\nO(\\omega^{n_a-1}) ,\n\\label{eqn:rffimSmall47b}\n\\end{equation}\nfor $n_b \\leq n_a$ or $n_b > n_a$, respectively, as $\\omega \\to 0$.\nThe summation of the second summand in Eq. (\\ref{eqn:rffimSmall40c}) for $0\\leq r\\leq n_b +3$\nis also estimated as\n\\begin{equation}\n\\lambda^2 \\pi\n\\sum_{\\scriptsize\n\\begin{array}{c}\ns\\geq0, t\\geq0, u\\geq0, \\\\\n(s+t+u=r)\n\\end{array}}^{r}\nF_2 (s,t,u)\n=\n\\sum_{(s+t=r)}^r\nF_2(s, t, 0)\n=\nO(F_2(0, r, 0))\n=\nO(\\omega^{n_b +3 -r}),\n\\label{eqn:rffimSmall47c}\n\\end{equation}\nboth for $n_b \\leq n_a$ and for $n_b > n_a$, as $\\omega \\to 0$.\nThe summation of the last summand in Eq. (\\ref{eqn:rffimSmall40c}) for $0\\leq r\\leq n_b +1$\nis estimated as\n\\begin{eqnarray}\n\\lambda^2 \\pi\n\\sum_{\\scriptsize\n\\begin{array}{c}\ns\\geq0, t\\geq0, u\\geq0, \\\\\n(s+t+u=r)\n\\end{array}}^{r}\nF_3 (s,t,0)\n&=&\n\\sum_{(s+t=r)}^r\nF_3(s, t, 0)\n=\nO(F_3(1, r-1, 0))\n\\nonumber\\\\\n&=&\nO(\\omega^{2n_b -r } \\log \\omega), ~\n\\mbox{or} ~\nO(\\omega^{n_a +n_b -r}),\n\\label{eqn:rffimSmall47d}\n\\end{eqnarray}\nfor $n_b \\leq n_a$ or $n_a < n_b $,\nrespectively, as $\\omega \\to 0$.\nFor $r=n_b +2$, the estimation is replaced by\n\\begin{equation}\nO(\\omega^{n_b -2}(\\log \\omega)^{\\theta(n_b-2)}) ,\n\\mbox{ or } ~\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[n_a -2]^+}) & (n_b \\geq 3)\\\\\nO(\\log \\omega) & (n_b = 2)\n\\end{array}\n\\right. ,\n\\label{eqn:rffimSmall50a}\n\\end{equation}\nfor $n_b \\leq n_a$ or $n_a < n_b $, respectively,\nas $\\omega \\to 0$.\nThen, by summarizing the above-noted estimations from Eqs.\n(\\ref{eqn:rffimSmall47a}) to (\\ref{eqn:rffimSmall50a}),\nand by taking into account the restriction (\\ref{eqn:rff.246}) again,\nthe proof of the lemma is completed.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The exceptional case of the first kind}\n\nIn this case, we first remember that from the discussion around Eq. (\\ref{eqn:.340}) \nit necessarily holds that ${\\mit \\Gamma}_1\\neq 0$, i.e., $n_b=1$ in Eqs. (\\ref{eqn:rff.220}).\n\n\n\n\\begin{lm} \\label{lm:1stremainder+}\n: Assume that $0$ is an exceptional point of the first kind for $H$.\nThen the $0$-th and the first derivative of $\\tilde{R} (z)$\nare approximated by those of a finite series\n\\begin{equation}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n+\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} ,\n\\label{eqn:1stSmall5}\n\\end{equation}\nthat is, it is shown that\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d^r}{dz^r}\n\\Biggl[\n\\tilde{R} (z)\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\Biggr]\n\\Biggr\\|\n\\nonumber \\\\\n&&=\nO(z^{-1} (\\log z)^{-3}) ~~\\mbox{for}~~r=0,~~~\\mbox{or}~~~\nO(z^{-2} (\\log z)^{-3}) ~~\\mbox{for}~~r=1,\n\\label{eqn:1stSmall7}\n\\end{eqnarray}\nas $z \\to 0$.\n\\end{lm}\n\n\n\n\n{\\sl Proof} :\nLet us first consider the quantity that\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d^r}{dz^r}\n\\Biggl[\n\\tilde{R} (z)\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\Biggr]\n\\Biggr\\|\n\\nonumber \\\\\n&&\\leq\n\\left\\|\n\\frac{d^r }{dz^r}\n\\left[\n\\tilde{R} (z) - Q_1 \\tilde{R} (z) Q_1\n\\right]\n\\right\\|\n+\n\\left\\|\n\\frac{d^r}{dz^r}\n\\Biggl[\nQ_1 \\tilde{R} (z) Q_1\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right.\n\\nonumber \\\\\n&&~~~\n\\left.\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\Biggr]\n\\right\\| .\n\\label{eqn:1stSmall50}\n\\end{eqnarray}\nFor $r= 0$, the first term on the rhs of the above is estimated as follows:\n\\begin{equation}\n\\left\\|\n\\tilde{R} (z)-Q_1 \\tilde{R} (z) Q_1\n\\right\\|\n\\leq\n\\left\\|\nQ_0 \\tilde{R} (z) Q_0\n\\right\\|\n+\n\\left\\|\nQ_0 \\tilde{R} (z) Q_1\n\\right\\|\n+\n\\left\\|\nQ_1 \\tilde{R} (z) Q_0\n\\right\\|\n=O(1),\n\\label{eqn:1stSmall60}\n\\end{equation}\nas $z \\to 0$,\nwhere Eqs. (\\ref{eqn:5.C.1.120a})--(\\ref{eqn:5.C.1.120c}) are used.\nFor $r=1$, one obtains\n\\begin{equation}\n\\left\\|\n\\frac{d\\tilde{R} (z)}{dz}\n-\n\\frac{dQ_1 \\tilde{R} (z) Q_1}{dz}\n\\right\\|\n\\leq\n\\left\\|\n\\frac{dQ_0 \\tilde{R} (z) Q_0}{dz}\n\\right\\|\n+\n\\left\\|\n\\frac{dQ_0 \\tilde{R} (z) Q_1}{dz}\n\\right\\|\n+\n\\left\\|\n\\frac{dQ_1 \\tilde{R} (z) Q_0}{dz}\n\\right\\| =O(z^{-1}).\n\\label{eqn:1stSmall70}\n\\end{equation}\nIn fact, by using expression (\\ref{eqn:5.C.1.120a})\nthe first term on the rhs of the above is estimated as follows:\n\\begin{eqnarray}\n\\frac{d Q_0 \\tilde{R} (z) Q_0}{dz}\n&=&\n-\nQ_0 \\tilde{R} (z) Q_0\n\\left( \\frac{d E_{00}}{dz} -\\frac{d E_{01}}{dz} E_{11}^{-1} E_{10}\n-E_{01} \\frac{dE_{11}^{-1} }{dz} E_{10}\n-E_{01} E_{11}^{-1} \\frac{dE_{10}}{dz}\n\\right)\n\\nonumber \\\\\n&&\n\\times\nQ_0 \\tilde{R} (z) Q_0\n\\label{eqn:1stSmall71b}\\\\\n&=&\nO(\\log z).\n\\label{eqn:1stSmall71c}\n\\end{eqnarray}\nFour derivatives in Eq. (\\ref{eqn:1stSmall71b})\nhave the same order, which can be shown from the use of\nEqs. (\\ref{eqn:5.C.1.60b}), (\\ref{eqn:5.C.1.90b}),\n(\\ref{eqn:5.C.1.100}), and (\\ref{eqn:5.C.1.130}): We here note that\n\\begin{equation}\n\\frac{dE_{00}}{dz}\n=O(\\log z),~~~\n\\frac{dE_{01}}{dz}\n=O(\\log z),~~~\n\\frac{dE_{10}}{dz}\n=O(\\log z),~~~\n\\label{eqn:1stSmall72}\n\\end{equation}\nand from Eqs. (\\ref{eqn:5.C.1.60b}) and (\\ref{eqn:5.C.1.90b})\n\\begin{eqnarray}\n\\frac{dE_{11}^{-1} }{dz}\n&=&\nE_{11}^{-1}\n\\left\\{\n\\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) -\\log z {\\mit \\Gamma}(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1\n\\right\\}\nE_{11}^{-1}\n\\label{eqn:1stSmall73a}\\\\\n&=&\nE_{11}^{-1}\nQ_1 \\left\\{\n1 +\\lambda^2 \\left[ \\frac{dA(z)}{dz}-{\\mit \\Gamma}(z)\/z\n-\\log z \\frac{d{\\mit \\Gamma}(z)}{dz}\n+ \\pi i \\frac{d{\\mit \\Gamma}(z)}{dz} \\right]\n\\right\\} Q_1\nE_{11}^{-1}\n\\label{eqn:1stSmall73b}\\\\\n&=&\nO(z^{-2} (\\log z)^{-1} ).\n\\label{eqn:1stSmall73c}\n\\end{eqnarray}\nIn the same way, the second term on the rhs of Eq. (\\ref{eqn:1stSmall70})\nis also estimated as follows:\n\\begin{eqnarray}\n\\frac{d Q_0 \\tilde{R} (z) Q_1}{dz}\n&=&\n-\\left(\n\\frac{dE_{00}^{-1}}{dz} E_{01}\n+\nE_{00}^{-1} \\frac{dE_{01}}{dz}\n\\right)\nQ_1\\tilde{R}Q_1\n+\nE_{00}^{-1} E_{01}\nQ_1\\tilde{R}Q_1\n\\nonumber \\\\\n&&\n\\times\n\\left( \\frac{d E_{11}}{dz} -\\frac{d E_{10}}{dz} E_{00}^{-1} E_{01}\n-E_{10} \\frac{dE_{00}^{-1} }{dz} E_{01}\n-E_{10} E_{00}^{-1} \\frac{dE_{01}}{dz}\n\\right)\nQ_1\\tilde{R}Q_1\n\\label{eqn:1stSmall74b}\\\\\n&=&\nO(z^{-1}),\n\\label{eqn:1stSmall74c}\n\\end{eqnarray}\nwhere we used Eq. (\\ref{eqn:5.C.1.120b}) and the fact that\n\\begin{equation}\n\\frac{dE_{00}^{-1} }{dz}=O(\\log z),~~~\n\\frac{dE_{11}}{dz}=O(\\log z).\n\\label{eqn:1stSmall75}\n\\end{equation}\nIn a similar manner, we can also show\n\\begin{equation}\n\\frac{d Q_1 \\tilde{R} (z) Q_0}{dz}=O(z^{-1}).\n\\label{eqn:1stSmall76}\n\\end{equation}\n\n\n\nLet us next consider the last term in Eq. (\\ref{eqn:1stSmall50}). For $r=0$, it reads\n\\begin{eqnarray}\n&&\n\\left\\|\nQ_1 \\tilde{R} (z) Q_1\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right.\n\\nonumber \\\\\n&&\n\\left.\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right\\|\n\\label{eqn:1stSmall77a}\\\\\n&\\leq&\n\\left\\|\nE_{11}^{-1}\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right.\n\\nonumber \\\\\n&&\n\\left.\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\right\\|\n\\nonumber \\\\\n&&\n+\\| (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1} -E_{11}^{-1} \\|\n\\label{eqn:1stSmall77c}\\\\\n&\\leq&\n\\biggl\\|\n\\tilde{E}_{11}(z)\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\Biggl\\|\n\\sum_{j=2}^{\\infty}\n(\\tilde{E}_{11}(z))^j\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\Biggr\\|\n+\nO(1)\n\\label{eqn:1stSmall77d}\\\\\n&=&\nO( z^{-1}  (\\log z )^{-3} ),\n\\label{eqn:1stSmall77f}\n\\end{eqnarray}\nwhere in the second inequality we used Eq. (\\ref{eqn:5.C.1.90a}) and that\n\\begin{equation}\n\\| (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1} -E_{11}^{-1} \\|\n\\leq\n\\frac{\\| E_{11}^{-1} E_{10} E_{00}^{-1} E_{01} \\|\n\\| E_{11}^{-1} \\|}{1-\\| E_{11}^{-1} E_{10} E_{00}^{-1} E_{01} \\| }\n=O(1),\n\\label{eqn:1stSmall78}\n\\end{equation}\nas $z \\to 0$.\nSubstituting Eqs. (\\ref{eqn:1stSmall60}) and (\\ref{eqn:1stSmall77f})\ninto Eq. (\\ref{eqn:1stSmall50}), we can obtain the estimation (\\ref{eqn:1stSmall7}) for $r=0$.\n\n\n\n\nFor $r=1$, we can obtain\n\\begin{eqnarray}\n&&\n\\Biggl\\|\n\\frac{d}{dz}\n\\biggl[\nQ_1 \\tilde{R} (z) Q_1\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr]\n\\Biggr\\|\n\\label{eqn:1stSmall80a}\\\\\n&\\leq&\n\\Biggl\\|\n\\frac{d}{dz}\n\\biggl[\nE_{11}^{-1}\n-\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr]\n\\Biggr\\|\n\\nonumber \\\\\n&&\n+\\biggl\\| \\frac{d}{dz}\n\\bigl\\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}E_{11}^{-1}\n-E_{11}^{-1}\n\\bigr\\}\n\\biggr\\|\n\\label{eqn:1stSmall80b}\\\\\n&\\leq&\n\\biggl\\|\nE_{11}^{-1}\n\\left\\{ \\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1 \\right\\}\nE_{11}^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z^2 (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n-E_{11}^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\nE_{11}^{-1}\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\nz\n\\biggl\\{\n\\frac{d}{dz}\n\\frac{1}{\\lambda^4 z^2 (\\log z)^2}\n\\biggr\\}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\\biggl\\|\n\\bigl\\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1} -Q_{11} \\bigr\\}\n\\frac{dE_{11}^{-1}}{dz}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n\\left\\{\n\\frac{d}{dz}\n(Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}\n\\right\\}\nE_{11}^{-1}\n\\biggr\\|\n\\label{eqn:1stSmall80d}\\\\\n&=&O(z^{-2}(\\log z)^{-3}),\n\\label{eqn:1stSmall80e}\n\\end{eqnarray}\nwhere we used the expression for $dE_{11}^{-1}\/dz$\nin Eq. (\\ref{eqn:1stSmall73a}).\nActually, the first term in Eq. (\\ref{eqn:1stSmall80d}) is estimated as\n\\begin{eqnarray}\n&&\\hspace*{-13mm}\n\\biggl\\|\nE_{11}^{-1}\n\\left\\{ \\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1 \\right\\}\nE_{11}^{-1}\n\\nonumber \\\\\n&&\\hspace*{-13mm}\n-\n\\frac{1}{\\lambda^4 z^2 (\\log z)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\\hspace*{-13mm}\n\\leq\n\\biggl\\|\nE_{11}^{-1}-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\biggl\\|\n\\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1\n\\biggr\\|\n\\| E_{11}^{-1} \\|\n\\nonumber \\\\\n&&\\hspace*{-13mm}\n~~~+\n\\frac{\\| ( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\| }{\\lambda^2 |z \\log z|}\n\\biggl\\|\n\\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1\n\\biggr\\|\n\\biggl\\|\nE_{11}^{-1}-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\\hspace*{-13mm}\n~~~+\n\\frac{\\|( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|^2}{\\lambda^4 z^2 (\\log z)^2 }\n\\biggl\\|\n\\frac{d}{dz}\nQ_1 \\left\\{\nz +\\lambda^2 [ A(z) + \\pi i {\\mit \\Gamma}(z) ]\n\\right\\} Q_1\n-\nQ_1 (1+ \\lambda^2 A_1 + \\lambda^2  \\pi i {\\mit \\Gamma}_1 )Q_1\n\\biggr\\|\n\\label{eqn:1stSmall90a}\\\\\n&&\\hspace*{-13mm}\n=\nO(z^{-2}(\\log z)^{-3}),\n\\label{eqn:1stSmall90b}\n\\end{eqnarray}\nas $z \\to 0$.\nOn the other hand, the second term in Eq. (\\ref{eqn:1stSmall80d}) is\nslightly complicated to evaluate:\n\\begin{eqnarray}\n&&\n\\biggl\\|\n-E_{11}^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\nE_{11}^{-1}\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\nz \\biggl\\{\n\\frac{d}{dz}\n\\frac{1}{\\lambda^4 z^2 (\\log z)^2}\n\\biggr\\}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\label{eqn:1stSmall95a}\\\\\n&\\leq&\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\tilde{E}_{11}\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\\tilde{E}_{11}\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n+\nz\n\\frac{2}{(\\lambda^2 z \\log z )^3}\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\sum_{j=2}^{\\infty}\\tilde{E}_{11}^j \\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n\\tilde{E}_{11} \\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\sum_{j=1}^{\\infty}\\tilde{E}_{11}^j \\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n\\sum_{j=2}^{\\infty}\\tilde{E}_{11}^j \\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\nE_{11}^{-1}\n\\biggr\\|\n\\label{eqn:1stSmall95b}\\\\\n&\\leq&\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 \\log z Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\tilde{E}_{11}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n+\nz \\frac{1}{(\\lambda^2 z \\log z )^3}\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\nz (Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\n\\biggl\\|\n-\\tilde{E}_{11}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n+\nz\n\\frac{1}{(\\lambda^2 z \\log z )^3}\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr) ( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+O(z^{-2}(\\log z)^{-3} )\n\\label{eqn:1stSmall95c} \\\\\n&=&O(z^{-2}(\\log z)^{-3} ).\n\\label{eqn:1stSmall95d}\n\\end{eqnarray}\nIn fact, the first term in Eq. (\\ref{eqn:1stSmall95c}) reads\n\\begin{eqnarray}\n&&\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 \\log z Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber\\\\\n&&\n-\\frac{d}{dz}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber\\\\\n&\\leq&\n\\biggl\\|\n-(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n+\\frac{d}{dz}\n\\lambda^2 z (\\log z) Q_1\n\\biggr\\|\n\\frac{\\| ( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|}{(\\lambda^2z  \\log z)^{2}}\n\\label{eqn:1stSmall96b}\\\\\n&\\leq&\n\\lambda^2\n\\left\\{\n\\biggl\\|\n-(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\nQ_1 \\frac{{\\mit \\Gamma}(z)}{z} Q_1\n+\nQ_1\n\\biggr\\|\n+\n\\biggl\\|\n-(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\nQ_1 \\frac{d{\\mit \\Gamma}(z)}{dz} Q_1\n+\nQ_1\n\\biggr\\|\n(\\log z)\n\\right\\}\n\\nonumber \\\\\n&&\n\\times\n\\| ( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|\n\\frac{1}{\\lambda^2 z \\log z}\n\\label{eqn:1stSmall96c}\\\\\n&=&\nO((z \\log z)^{-1}).\n\\label{eqn:1stSmall96d}\n\\end{eqnarray}\nThe second term in Eq. (\\ref{eqn:1stSmall95c}) also reads\n\\begin{eqnarray}\n&&\n\\biggl\\|\n-\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\tilde{E}_{11}\n\\frac{1}{\\lambda^2 z \\log z}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n+\nz \\frac{1}{(\\lambda^2 z \\log z )^3}\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber\\\\\n&\\leq&\n\\|(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\\|\n\\biggl\\|\n-\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n\\nonumber \\\\\n&&\n\\times\n(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\bigl\\{ z Q_1\n+\\lambda^2 Q_1 \\{ A(z) -\\log z [{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n+ \\pi i {\\mit \\Gamma}(z) \\} Q_1\n\\bigr\\}\n\\nonumber \\\\\n&&\n+\nz \\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n\\biggr\\|\n\\frac{\\|( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\| }{(\\lambda^2 z \\log z )^{3}}\n\\label{eqn:1stSmall97b}\\\\\n&\\leq&\n\\|(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\\|\n\\biggl\\|\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr\\|\n\\nonumber \\\\\n&&\n\\times\n\\bigl\\|\n-\n\\bigl\\{ z Q_1\n+\\lambda^2 Q_1 \\{ A(z) -\\log z [{\\mit \\Gamma}(z) -z {\\mit \\Gamma}_1 ]\n+ \\pi i {\\mit \\Gamma}(z) \\} Q_1\n\\bigr\\}\n\\nonumber \\\\\n&&\n+\nz (Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n\\bigr\\|\n\\frac{\\|( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|}{ (\\lambda^2 z \\log z )^{3}}\n\\nonumber \\\\\n&&\n+\n\\|(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\\|\n\\biggl\\|\n-\n\\biggl\\{\n\\frac{d}{dz}\n\\lambda^2 (\\log z) Q_1 {\\mit \\Gamma}(z) Q_1\n\\biggr\\}\n(Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n+\n\\biggl(\n\\frac{d}{dz}\n\\lambda^2 z \\log z\n\\biggr)\nQ_1\n\\biggr\\|\n\\nonumber \\\\\n&&\n\\times\n\\bigl\\|\nz (Q_1+ \\lambda^2 Q_1 A_1 Q_1 + \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n\\bigr\\|\n\\frac{\\|( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} \\|}{ (\\lambda^2 z \\log z )^{3}}\n\\label{eqn:1stSmall97c}\\\\\n&=&\nO((z \\log z)^{-1}),\n\\label{eqn:1stSmall97d}\n\\end{eqnarray}\nas $z \\to 0$.\nThe third term gives the same contribution to the order as the second one does.\nFurthermore, the last term in Eq. (\\ref{eqn:1stSmall95c}) comes from the estimations of\nthe second, third, and last terms in Eq. (\\ref{eqn:1stSmall95b}),\nwhere each contributes the same order as $O(z^{-2}(\\log z)^{-3} )$.\nTherefore, Eq. (\\ref{eqn:1stSmall95d}) is proved.\n\n\nOn the other hand, the third term in Eq. (\\ref{eqn:1stSmall80d}) reads\n\\begin{equation}\n\\biggl\\|\n\\bigl\\{ (Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1} -Q_{11} \\bigr\\}\n\\frac{dE_{11}^{-1}}{dz}\n\\biggr\\|\n\\leq\n\\frac\n{\n\\bigl\\|\nE_{11}^{-1}E_{10} E_{00}^{-1} E_{01}\n\\bigr\\|}\n{\n1-\\bigl\\|\nE_{11}^{-1}E_{10} E_{00}^{-1} E_{01}\n\\bigr\\|\n}\n\\biggl\\|\n\\frac{dE_{11}^{-1}}{dz}\n\\biggr\\|\n=O(z^{-1}),\n\\label{eqn:1stSmall100b}\n\\end{equation}\nas $z \\to 0$,\nwhere Eqs. (\\ref{eqn:5.C.1.90b}), (\\ref{eqn:5.C.1.100}), (\\ref{eqn:5.C.1.130}),\nand (\\ref{eqn:1stSmall73c}) are used.\nIn the same way, the last term in Eq. (\\ref{eqn:1stSmall80d}) reads\n\\begin{eqnarray}\n&&\n\\hspace*{-5mm}\n\\biggl\\| E_{11}^{-1}\n\\frac{d}{dz}\n(Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}\n\\biggr\\|\n\\nonumber\\\\\n&\\leq&\n\\| E_{11}^{-1} \\|\n\\biggl\\|\n(Q_{11} -E_{11}^{-1}E_{10} E_{00}^{-1} E_{01})^{-1}\n\\biggr\\|^2\n\\biggl[\n\\biggl\\|\n\\frac{dE_{11}^{-1}}{dz}\nE_{10} E_{00}^{-1} E_{01}\n\\biggr\\|\n\\nonumber \\\\\n&&\n+\\biggl\\|\nE_{11}^{-1}\\frac{dE_{10} }{dz}\nE_{00}^{-1} E_{01}\n\\biggr\\|\n+\\biggl\\|\nE_{11}^{-1}E_{10} E_{00}^{-1}\\frac{d E_{01}}{dz}\n\\biggr\\|\n+\\biggl\\|\nE_{11}^{-1}E_{10} \\frac{d E_{00}^{-1}}{dz} E_{01}\n\\biggr\\|\n\\biggr]\n\\label{eqn:1stSmall110a}\\\\\n&=&\nO(z^{-1} \\log z ),\n\\label{eqn:1stSmall110b}\n\\end{eqnarray}\nas $z \\to 0$.\nBy substituting Eqs. (\\ref{eqn:1stSmall90b}), (\\ref{eqn:1stSmall95d}), (\\ref{eqn:1stSmall100b}),\nand (\\ref{eqn:1stSmall110b}) into Eq. (\\ref{eqn:1stSmall80d}),\none finally obtains Eq. (\\ref{eqn:1stSmall80e}).\nWe can now show Eq. (\\ref{eqn:1stSmall7}) for $r=1$ by setting Eqs. (\\ref{eqn:1stSmall70})\nand (\\ref{eqn:1stSmall80e}) into Eq. (\\ref{eqn:1stSmall50}).\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\nIf we start with expression (\\ref{eqn:5.C.1.125b}),\nwe obtain the following lemma instead of Lemma \\ref{lm:1stremainder+}.\n\n\n\n\\begin{lm} \\label{lm:1stremainder-}\n: Assume that $0$ is an exceptional point of the first kind for $H$.\nThen the $0$-th and the first derivative of $\\tilde{R} (z)$\nare approximated by those of a finite series\n\\begin{eqnarray}\n&&\n\\frac{1}{\\lambda^2 z (\\log z -2\\pi i)}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber\\\\\n&&\n+\n\\frac{1}{\\lambda^4 z (\\log z -2\\pi i)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 - \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1} ,\n\\label{eqn:1stSmall5-}\n\\end{eqnarray}\nthat is, it is shown that\n\\begin{eqnarray}\n&&\\hspace{-10mm}\n\\biggl\\|\n\\frac{d^r}{dz^r}\n\\biggl[\n\\tilde{R}(z)\n-\n\\frac{1}{\\lambda^2 z (\\log z -2\\pi i)}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\nonumber \\\\\n&&\n-\n\\frac{1}{\\lambda^4 z (\\log z -2\\pi i)^2}( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n(Q_1+ \\lambda^2 Q_1 A_1 Q_1 - \\lambda^2  \\pi i Q_1 {\\mit \\Gamma}_1 Q_1 )\n( Q_1 {\\mit \\Gamma}_1 Q_1 )^{-1}\n\\biggr]\n\\biggr\\|\n\\nonumber \\\\\n&=&\nO(z^{-1} (\\log z -2\\pi i)^{-3}) ~~\\mbox{for}~~r=0,~~~\\mbox{or}~~~\nO(z^{-2} (\\log z -2\\pi i)^{-3}) ~~\\mbox{for}~~r=1,\n\\label{eqn:1stSmall7-}\n\\end{eqnarray}\nas $z \\to 0$.\n\\end{lm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The reduced time evolution operator}\n\\label{sec:6}\n\n\n\nIn this section, we show that the reduced time evolution operator\nis expressed by the Fourier transform of the imaginary part of the reduced resolvent\nboth in the regular case and the exceptional case of the first kind.\nWe here define the reduced time evolution operator by the $N\\times N$ matrix $\\tilde{U}(t)$\nof the components\n$\\tilde{U}_{mn}(t):=\\langle m |Pe^{-itH}P| n \\rangle$,\nwhere $P=E((0, \\infty ))$ and $\\{ E(B) | B \\in \\mathbb{B} \\}$ is\nthe spectral measure of $H$, which is a family of the projection operator.\n$\\mathbb{B}$ is the Borel field of $\\mathbb{R}$.\n\n\n\n\\begin{lm}\n\\label{lm:reduced time evolution operator }\n:\nWe assume that Eq. (\\ref{eqn:.110}) holds so that there is no positive eigenvalue.\nThen, for the system with the rational form-factor (\\ref{eqn:formfactor1}),\nit holds that\n\\begin{equation}\n\\tilde{U}(t)\n=\n\\frac{1}{\\pi}\n\\int_{(0, \\infty )}\ne^{-it\\omega} {\\rm Im} \\tilde{R}^+ (\\omega) d\\omega\n=\n\\lim_{r \\to +0}\n\\frac{1}{\\pi}\n\\int_r^\\infty\ne^{-it\\omega} {\\rm Im} \\tilde{R}^+ (\\omega) d\\omega ,\n\\label{eqn:6.120}\n\\end{equation}\nboth in the regular case and the exceptional case of the first kind,\nwhere\n\\begin{eqnarray}\n{\\rm Im} \\tilde{R}^+ (\\omega)\n:=\n\\frac{1}{2 i} [\\tilde{R}^+(\\omega) -\\tilde{R}^-(\\omega)] ,\n\\label{eqn:6.65}\n\\end{eqnarray}\nwhich is sometimes called the spectral density.\n\\end{lm}\n\n\n\n\n\n\n{\\sl Proof} :\nLet us remember that the matrix $\\tilde{U}(t)$ is expressed by the spectral measure as\n\\begin{equation}\n\\tilde{U}_{mn}(t)\n=\n\\int_{(0, \\infty )}\ne^{-it\\lambda }\nd \\langle m |E(\\lambda )| n \\rangle\n=\n\\int_{(0, \\infty )}\ne^{-it\\lambda }\nd \\tilde{E}_{mn}(\\lambda ) ,\n\\label{eqn:6.110}\n\\end{equation}\nwhere $\\tilde{E}(B)$ is the matrix of the components $\\langle m |E(B)|n \\rangle $.\nTherefore, what we first should do is to clarify the relation between\n${\\rm Im} \\tilde{R}^+ (\\omega)$ and $\\tilde{E}(\\lambda )$.\nResorting to Stone's formula between $E(B)$ and $R(z)$,\nwe clearly see\n\\begin{equation}\n\\frac{1}{2}\n[\\tilde{E}([a,b])+\\tilde{E}((a,b))]\n=\\lim_{\\epsilon \\to +0}\n\\frac{1}{2\\pi i} \\int_a^b [\\tilde{R}(\\omega +i\\epsilon)\n-\\tilde{R}(\\omega -i\\epsilon)] d\\omega,\n\\label{eqn:6.30}\n\\end{equation}\nfor $a, b \\in \\mathbb{R}$ with $a<b$.\nUnder the assumption (\\ref{eqn:.110}),\nLebesgue's dominated convergence theorem and the proof of Lemma \\ref{lm:100} tell us that\nthe exchange between the limit and the integration in Eq. (\\ref{eqn:6.30}) is allowed\nfor $[a,b]\\subset (0,\\infty)$.\nIf $[a,b] \\subset (-\\infty, 0)\\backslash \\sigma(H)$,\nthen $\\tilde{R}^{\\pm} (\\omega) =\\tilde{R} (\\omega) $, and thus\n$\\tilde{E}([a,b])=\\tilde{E}((a,b))=0$.\nIn addition, by the continuity of $\\tilde{R}^{\\pm}(\\omega)$, Eq. (\\ref{eqn:6.30}) tells us\nthat $\\tilde{E}(\\{ a\\} )=0$ for all $a>0$, which leads to\n\\begin{equation}\n\\tilde{E}((a,b))\n=\\tilde{E}([a,b])=\n\\frac{1}{\\pi} \\int_a^b {\\rm Im} \\tilde{R}^+ (\\omega) d\\omega ,\n\\label{eqn:6.60}\n\\end{equation}\nfor all $a, b$ with $b>a>0$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLet us now consider the regular case\nand in particular the validity of the expression (\\ref{eqn:6.60})\nfor the interval including the origin.\nIn this case, \n$\\tilde{R}^\\pm (0):=\\lim_{\\omega \\to +0}\\tilde{R}^\\pm (\\omega)$ exists to be finite.\nFurthermore, $\\lim_{\\omega \\to \\infty}\\tilde{R}^\\pm (\\omega)=0$\nfrom Lemma \\ref{lm:Large-omega}.\nThus, $\\tilde{R}^\\pm (\\omega)$ is uniformly continuous on $(0, \\infty)$.\nTherefore, we can take the limit of Eq. (\\ref{eqn:6.60}) as $a\\to +0$ to obtain \n$\\lim_{a\\to +0}E([a,b])=E((0,b])$.\nWe next see that all components of\n${\\rm Im} \\tilde{R}^+ (\\omega)$\nare integrable, i.e., belong to $L^1 ((0, \\infty ))$.\nSuppose that $| \\psi \\rangle \\in \\mathbb{C}^N$, then\n$\\langle \\psi | \\tilde{E}((0, \\lambda )) | \\psi \\rangle$\nis positive and a monotonically increasing function of $\\lambda$,\nand it is also differentiable in this case. Thus Eq. (\\ref{eqn:6.60}) tells us that\n$\\bra{\\psi} {\\rm Im} \\tilde{R}^+ (\\omega) \\ket{\\psi} \\geq 0$. In addition,\n\\begin{equation}\n\\| \\psi \\|^2\n\\geq \\lim_{\\lambda \\to \\infty }\n\\langle \\psi | \\tilde{E}((0, \\lambda )) | \\psi \\rangle\n= \\lim_{\\lambda \\to \\infty }\n\\frac{1}{\\pi} \\int_0^\\lambda\n\\bra{\\psi} {\\rm Im} \\tilde{R}^+ (\\omega) \\ket{\\psi} d\\omega\n=\n\\frac{1}{\\pi} \\int_0^\\infty\n\\bra{\\psi} {\\rm Im} \\tilde{R}^+ (\\omega) \\ket{\\psi} d\\omega .\n\\label{eqn:6.90}\n\\end{equation}\nHence, from the monotonic convergence theorem,\nwe see that $\\bra{\\psi} {\\rm Im} \\tilde{R}^+ (\\omega) \\ket{\\psi} \\in L^1 ((0, \\infty ))$.\nFrom this fact and the use of the polarization identity,\nwe can prove that all components of the matrix\n${\\rm Im} \\tilde{R}^+ (\\omega)$ are integrable.\nThus, extending the rhs of Eq. (\\ref{eqn:6.60}) to arbitrary\n$B \\in \\{ B \\in \\mathbb{B}| B \\subset (0,\\infty) \\}$,\n$\\int_B {\\rm Im} \\tilde{R}^+ (\\omega) d\\omega$ defines a measure.\nWe can now see from Eq. (\\ref{eqn:6.60}) and from E. Hopf's extension theorem that\n\\begin{equation}\n\\tilde{E}(B)\n=\n\\frac{1}{\\pi} \\int_B\n{\\rm Im} \\tilde{R}^+ (\\omega) d\\omega\n\\label{eqn:6.100}\n\\end{equation}\nholds for all $B \\in \\{ B \\in \\mathbb{B}| B \\subset (0,\\infty) \\}$.\nNote that this expression means that\nthe restriction of $\\tilde{E}_{mn}(B) $ to\n$\\{ B \\in \\mathbb{B}| B \\subset (0,\\infty) \\}$\nis absolutely continuous.\nTherefore, rewriting of $\\tilde{U}(t)$ in Eq. (\\ref{eqn:6.110}) into (\\ref{eqn:6.120})\nis straightforward.\n\n\nIn the exceptional case of the first kind,\nfrom the assumption (\\ref{eqn:.110}), $\\tilde{R}^\\pm (\\omega)$ is continuous on $(0, \\infty)$,\nwhile\n$\\tilde{R}^\\pm (\\omega)=O((\\omega \\log \\omega)^{-1})$ as $\\omega \\to +0$,\nso that it is not integrable around $0$.\nSee, Eq. (\\ref{eqn:5.C.1.120d}).\nHowever, ${\\rm Im} \\tilde{R}^+ (\\omega)$\nis of the order $O(\\omega^{-1} (\\log \\omega)^{-2})$ from Lemmas \\ref{lm:1stremainder+} and\n\\ref{lm:1stremainder-},\nand thus it is integrable around $0$.\nHence, Eq. (\\ref{eqn:6.100}) holds again, and\nEq. (\\ref{eqn:6.120}) is valid for this exceptional case.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\nWe remark that in the case of no negative eigenvalues (point spectrum) of $H$, \n\\cite{Miyamoto(2005)}\nthe restriction of $\\tilde{U}(t)$\nto the continuous energy spectrum is removed because in such a case $P=I$ the identity.\nFurthermore, the connection between $\\tilde{U}(t)$ and the observables is easily found, e.g.,\n$|\\bra{\\psi}\\tilde{U}(t)\\ket{\\psi}|^2\/\\| P\\ket{\\psi} \\|^4$\nfor $| \\psi \\rangle \\in \\mathbb{C}^N$ (or $\\mathbb{C}^N \\oplus \\{ 0\\}$)\nis the survival probability of $P\\ket{\\psi}$\nwhich is the probability of finding the system in the state $P\\ket{\\psi}$\nat the later time $t$,\nwhere $P\\ket{\\psi}$ is just the decaying component of the initial state $\\ket{\\psi}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The asymptotic expansion of the reduced time evolution operator}\n\\label{sec:7}\n\n\n\n\n\nWe can finally show the asymptotic formula for $\\tilde{U}(t)$ at long times\nfor the rational form factors satisfying our assumptions.\nIn the following, we assume that Eq. (\\ref{eqn:.110}) holds, i.e.,\nthere is no positive eigenvalue.\nHowever, this is not explicitly mentioned in the statements of the theorems.\nLet us first consider the regular case. For this purpose,\naccording to Lemma \\ref{lm:rffimremainder},\nwe introduce the remainder $F(\\omega)$ in the following way:\n\\begin{equation}\n\\frac{1}{\\pi}{\\rm Im} \\tilde{R}^+ (\\omega)\n=\n\\lambda^2\n(K(0))^{-1}\n\\omega^{n_b} ({\\mit \\Gamma}_{n_b} +\\omega {\\mit \\Gamma}_{n_b +1} + \\omega^2 {\\mit \\Gamma}_{n_b +2} )\n(K(0))^{-1} +F(\\omega),\n\\label{eqn:rffimbothLong110}\n\\end{equation}\nfor $\\omega >0$.\n\n\n\n\\begin{thm\n\\label{thm:rffimLong}\n: Assume that $0$ is a regular point for $H$.\nFor a system with the rational form factor (\\ref{eqn:formfactor1}) characterized by\nthe positive integers $n_a$ and $n_b$ that satisfy\nthat $n_b \\geq 2$ and $n_a =1$,\nthe reduced time evolution operator $\\tilde{U}(t)$ behaves asymptotically as\n\\begin{equation}\n\\tilde{U}(t)\n=\n\\lambda^2\n\\frac{\\Gamma(1+n_b)}{(it)^{n_b+1}}\n(K(0))^{-1}\n{\\mit \\Gamma}_{n_b} (K(0))^{-1}\n+O(t^{-n_b-2}) ,\n\\label{eqn:rffimbothLong50}\n\\end{equation}\nas $t\\to \\infty$.\nWhen $n_b=1$ and $n_a \\geq 1$, the error term is replaced by\n$O(t^{-3}\\log t)$.\n\\end{thm}\n\n\n\n\n{\\sl Proof} :\nWe first summarize the several properties of ${\\rm Im} \\tilde{R}^+ (\\omega) $.\nBy Lemma \\ref{lm:rffimremainder}, we see that the remainder $F(\\omega)$ in\nEq. (\\ref{eqn:rffimbothLong110}) is arbitrary-times differentiable.\nParticularly it holds that\n$\\lim_{\\omega \\to 0} d^r F(\\omega) \/ d \\omega^r = 0$\nfor all $r \\leq n_b$, and\n\\begin{eqnarray}\n\\left\\| \\frac{d^{n_b+1} F(\\omega)}{d\\omega^{n_b+1}} \\right\\|\n&&=\nO(\\log \\omega), ~\n\\mbox{or} ~\nO(1)\n,\n\\label{eqn:rffimbothLong120}\n\\end{eqnarray}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively, and\n\\begin{equation}\n\\left\\| \\frac{d^{n_b+2} F(\\omega)}{d\\omega^{n_b+2}} \\right\\|\n=\nO(\\omega^{-1}),\n\\mbox{ or }\n\\left\\{\n\\begin{array}{cc}\nO(1) & (n_b \\geq 3)\\\\\nO(\\log \\omega) & (n_b = 2)\n\\end{array}\n\\right. ,\n\\label{eqn:rffimbothLong130}\n\\end{equation}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively,\nas $\\omega \\to 0$.\nOn the other hand, we see from Lemma \\ref{lm:Large-omega} that\n$ (d \/ d \\omega )^r {\\rm Im} \\tilde{R}^+ (\\omega) = O(\\omega^{-r-1})$\nas $\\omega \\rightarrow \\infty $.\nIn particular, if $m \\geq 1$,\n$(d \/d \\omega )^m {\\rm Im} \\tilde{R}^+ (\\omega)$ is integrable\non $[\\delta , \\infty )$ for an arbitrary $\\delta >0$.\n\n\nLet us now split the integral in Eq. (\\ref{eqn:6.120})\ninto two parts by writing\n\\begin{equation}\n{\\rm Im} \\tilde{R}^+ (\\omega ) = \\phi (\\omega ) {\\rm Im} \\tilde{R}^+ (\\omega )\n+ (1- \\phi (\\omega ) ) {\\rm Im} \\tilde{R}^+ (\\omega ) ,\n\\label{eqn:rffimbothLong135}\n\\end{equation}\nwhere $\\phi \\in C_0 ^{\\infty} ([0,\\infty ) )$ and satisfies\n$\\phi (\\omega ) =1$ in a neighborhood of $\\omega = 0$.\nSuch a function is realized by $f (\\omega ) = 1- \\int_0 ^\\omega\ng(x) d x$, where $g(x) = h(x)\/\\int_{\\bf R} h(x) d x$ and\n$h(x) = \\exp(-1\/[a^2-(x-d)^2 ])$ ~($|x-d|<a$) or $0$ ($|x-d| \\geq a$)\nwith $d>a>0$.\n\n\nFrom Lemma 10.1 in Ref. \\makebox(0,1){\\Large \\cite{Jensen(1979)}}\\hspace{2mm}\nand the above-mentioned discussion, we see that\n$(1- \\phi (\\omega ) ) {\\rm Im} \\tilde{R}^+ (\\omega ) $ has a contribution of\n$O(t^{-m})$ to $\\tilde{U}(t)$ for an arbitrary $m\\geq 1$,\ni.e., this term decays faster than any negative power of $t$.\n\n\nOn the other hand, the contribution of\n$\\phi (\\omega ) {\\rm Im} \\tilde{R}^+ (\\omega ) $ to $\\tilde{U}(t)$\ngives\nthe main part of the asymptotic expansion.\nThen, the coefficient of ${\\mit \\Gamma}_{n_b}$, ${\\mit \\Gamma}_{n_b +1}$, and ${\\mit \\Gamma}_{n_b +2}$\nis given by the form \\cite{Copson}\n\\begin{eqnarray}\n\\int_0 ^{\\infty }\n\\phi (\\omega ) \\omega^q e^{-i t \\omega } ~ d \\omega\n& = &\n\\sum_{k=0}^{N-1}\n\\frac{1}{(it)^{k+1}}\n\\left. \\frac{d^k \\omega^q \\phi(\\omega)}{d\\omega^k} \\right|_{\\omega=0}\n+R_N (t)\n=\n\\frac{\\Gamma(1+q)}{(it)^{1+q}}\n+R_N (t) ,\n\\label{eqn:rffimbothLong140}\n\\end{eqnarray}\nfor all $N\\geq 1+q$, where $q$ takes the value $n_b$, $n_b+1$, or $n_b+2$.\nWe here used that\n\\begin{equation}\n\\frac{d^k \\omega^q \\phi(\\omega)}{d\\omega^k}\\biggr|_{\\omega=0}\n=\n\\sum_{j=0}^{\\min \\{ k,q \\}} {{k}\\choose{j}}\n\\frac{d^j \\omega^q }{d\\omega^j }\\biggr|_{\\omega=0}\n\\frac{d^{k-j} \\phi(\\omega)}{d\\omega^{k-j}}\\biggr|_{\\omega=0}\n=\\Gamma(1+q) \\delta_{kq},\n\\label{eqn:rffimbothLong145}\n\\end{equation}\nwhere $\\Gamma(1+n)=\\int_0^\\infty x^n e^{-x} dx$ is the gamma function.\nIn addition, the remainder $R_N (t)$ is bounded above by\n\\begin{equation}\n|R_N (t)|\n\\leq\n\\frac{1}{t^N} \\left| \\int_{0}^{\\infty}\n\\frac{d^N \\omega^q \\phi(\\omega)}{d\\omega^N} e^{-i\\omega t} d\\omega \\right|\n=\no(t^{-N}).\n\\label{eqn:rffimbothLong160}\n\\end{equation}\nNote that\nsince all\nderivatives of $\\phi(\\omega)$ vanish in the neighborhood of $\\omega =0$,\nEq. (\\ref{eqn:rffimbothLong140}) is valid for all $N\\geq q+1$ and thus\n$R_N (t)$ decays faster than any\nnegative power of $t$.\nFurthermore, we understand, by applying Eq. (\\ref{eqn:A.50a}) in\nLemma\\ \\ref{lm:Jensen and Kato, Lemma 10.2} directly to $\\phi (\\omega ) F(\\omega) $\nwith the discussion in the first part of this section, that\nthe contribution of\nthe Fourier transform of the remainder $\\phi (\\omega ) F(\\omega) $\nto $\\tilde{U}(t)$ is\n\\begin{equation}\nO(t^{-3}\\log t)~~\n\\mbox{or} ~~\nO(t^{-n_b-2}) ,\n\\label{eqn:rffimbothLong180}\n\\end{equation}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively, as $\\omega \\to 0$,\nwhere we used the formula of the indefinite integral that\n$\\int (\\log \\omega)^2 d\\omega =\\omega [(\\log \\omega)^2 -2\\log \\omega +2]$.\nSummarizing the above-noted results, we finally obtain that\n\\begin{eqnarray}\n&&\\hspace{-10mm}\n\\left\\| \\tilde{U}(t) -\n\\lambda^2\n(K(0))^{-1}\n\\left[\n\\frac{\\Gamma(1+n_b)}{(it)^{n_b+1}} {\\mit \\Gamma}_{n_b} + \\frac{\\Gamma(2+n_b)}{(it)^{n_b+2}} {\\mit \\Gamma}_{n_b +1} +\\frac{\\Gamma(3+n_b)}{(it)^{n_b+3}} {\\mit \\Gamma}_{n_b +2}\n\\right]\n(K(0))^{-1}\n\\right\\|\n\\nonumber  \\\\\n&&\\hspace{-10mm}\\leq\n\\left\\|\n\\frac{1}{\\pi}\\int_0^{\\infty}\n(1-\\phi(\\omega)){\\rm Im} \\tilde{R}^+ (\\omega ) e^{-i t \\omega } d \\omega\n\\right\\|\n+\n\\left\\|\n\\frac{1}{\\pi}\\int_0^{\\infty}\n\\phi(\\omega) F(\\omega) e^{-i t \\omega } d\\omega\n\\right\\| +O(t^{-N})\n\\nonumber  \\\\\n&&\\hspace{-10mm}=\nO(t^{-3}\\log t)~~\n\\mbox{or} ~~\nO(t^{-n_b-2}) ,\n\\label{eqn:rffimbothLong150}\n\\end{eqnarray}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively,\nas $t\\to \\infty$, where $O(t^{-N})$ is due to the contribution from $R_N (t)$.\nThis is just the asymptotic expansion of $\\tilde{U}(t)$\nin the statement.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\nIt is worth noting that if we resort to Lemma \\ref{lm:rffremainder},\ninstead of Eqs. (\\ref{eqn:rffimbothLong120}) and (\\ref{eqn:rffimbothLong130}),\nwe have\n\\begin{eqnarray}\n\\hspace*{-5mm}\n\\frac{d^r F(\\omega)}{d\\omega^r}\n&&=\nO(\\omega^{2-r} (\\log \\omega)^{1+\\theta (2-r)}), ~\n\\mbox{or} ~\n\\left\\{\n\\begin{array}{cc}\nO(\\omega^{[2-r]^+}) & (0 \\leq r \\leq n_b)\\\\\nO(\\omega^{n_b +1-r} (\\log \\omega)^{\\theta (n_b +1 -r)}) & (r \\geq n_b +1)\n\\end{array}\n\\right.\n,\n\\label{eqn:rffimbothLong170}\n\\end{eqnarray}\nfor $n_b=1$ and $n_a \\geq 1$, or $n_b \\geq 2$ and $n_a =1$, respectively.\nHowever, in the latter case, we see that the Fourier transform of $\\phi(\\omega)F(\\omega)$\ngives the contribution of the order $O(t^{-n_b -1})$,\nwhich is just the same order as that coming from the dominant one.\nHence, we can only obtain an useless estimation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe next show the asymptotic formula for $\\tilde{U}(t)$ at long times \nfor a system with an exceptional point of the first kind. To this end,\nwe write ${\\rm Im} \\tilde{R}^+(\\omega)$ with the remainder $F(\\omega)$\nagain as follows:\n\\begin{equation}\n\\frac{1}{\\pi}{\\rm Im} \\tilde{R}^+ (\\omega)\n=\n\\frac{1}{\\lambda^2 \\omega (\\log \\omega)^2} (Q_1 {\\mit \\Gamma}_1 Q_1)^{-1} +F(\\omega) .\n\\label{eqn:rffimLong1st110}\n\\end{equation}\n\n\n\n\\begin{thm\n\\label{thm:rffLong1st}\n: Assume that $0$ is an exceptional point of the first kind for $H$,\nwhich necessarily imposes that $n_b =1$.\nThen, the reduced time evolution operator $\\tilde{U}(t)$\nfor the rational form factor (\\ref{eqn:formfactor1}) behaves asymptotically as\n\\begin{equation}\n\\tilde{U}(t)\n=\n\\frac{1}{\\lambda^2 \\log t}\n(Q_1 {\\mit \\Gamma}_1 Q_1)^{-1} +O((\\log t)^{-2}),\n\\label{eqn:rffimLong1st50}\n\\end{equation}\nas $t \\to \\infty$.\n\\end{thm}\n\n\n\n\n{\\sl Proof} :\nLet us first look over the some properties of ${\\rm Im} \\tilde{R}^+ (\\omega) $ again.\nBy Lemmas \\ref{lm:1stremainder+} and \\ref{lm:1stremainder-},\nwe see that the remainder $F(\\omega)$ in\nEq. (\\ref{eqn:rffimLong1st110}) is arbitrary-times differentiable,\nsatisfies that $F(\\omega) = O(\\omega^{-1} (\\log \\omega)^{-3})$,\nand $d F(\\omega) \/ d \\omega = O(\\omega^{-2} (\\log \\omega)^{-3}) $,\nas $\\omega \\to +0$.\nOn the other hand, we see from Lemma \\ref{lm:Large-omega} that\n$ (d \/ d \\omega )^r {\\rm Im} \\tilde{R}^+ (\\omega) = O(\\omega^{-r-1})$\nas $\\omega \\rightarrow \\infty $.\nIn particular, if $m \\geq 1$,\n$(d \/d \\omega )^m {\\rm Im} \\tilde{R}^+ (\\omega)$ is integrable\non $[\\delta , \\infty )$ for an arbitrary $\\delta >0$.\n\n\nWe now split the integral in Eq. (\\ref{eqn:6.120})\ninto two parts as in Eq. (\\ref{eqn:rffimbothLong135}) again using\nthe $C_0 ^{\\infty}$-function $\\phi(\\omega )$.\nFrom Lemma 10.1 in Ref. \\makebox(0,1){\\Large \\cite{Jensen(1979)}}\\hspace{2mm}\nand the discussion mentioned above,\n$(1- \\phi (\\omega ) ) {\\rm Im} \\tilde{R}^+ (\\omega ) $ has a contribution of\n$O(t^{-m})$ to $\\tilde{U}(t)$ for an arbitrary $m\\geq 1$.\nOn the other hand, the contribution of\n$\\phi (\\omega ) {\\rm Im} \\tilde{R}^+ (\\omega ) $ to $\\tilde{U}(t)$\ngives\nthe dominant part of the asymptotic expansion.\nThen, the dominant time dependence of the asymptote of $\\tilde{U}(t)$ follows from\nLemma \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral}, that is,\n\\begin{eqnarray}\n\\int_0 ^{\\infty }\n\\phi (\\omega ) (\\omega (\\log \\omega)^2)^{-1} e^{-i t \\omega } ~ d \\omega\n=\n(\\log t)^{-1} +O((\\log t)^{-2} ).\n\\label{eqn:rffimLong1st120}\n\\end{eqnarray}\nFurthermore,\nthe contribution of the Fourier transform of the remainder $\\phi (\\omega ) F(\\omega) $\nto $\\tilde{U}(t)$ can be estimated\nby the similar manner to Lemma\\ \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral},\nrather than Lemma\\ \\ref{lm:Jensen and Kato, Lemma 10.2}.\nBy setting $\\sigma (\\omega)=F(\\omega) e^{-it\\omega}$\ninstead in the proof of Lemma \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral},\nwe can apply it to this case,\nand we have\n\\begin{equation}\n\\int_0^\\infty F(\\omega) \\phi(\\omega) e^{-it\\omega} d\\omega\n=\n-\\lim_{\\omega \\to +0}(\\hat{I} \\sigma)(\\omega)\n+(-1)^N\n\\int_0^\\infty (\\hat{I}^N \\sigma)(\\omega) \\frac{d^N \\phi(\\omega)}{d\\omega^N} d\\omega .\n\\label{eqn:rffimLong1st155}\n\\end{equation}\nThen, corresponding to Eq. (\\ref{eqn:B.530}), we have\n\\begin{equation}\n\\|(\\hat{I}\\sigma)(\\omega)\\|\n=\n\\biggl\\|\ni e^{-it\\omega} \\int_0^\\infty F(\\omega-i\\eta) e^{-t\\eta} d\\eta\n\\biggr\\|\n+\nE(t)\n\\leq\nC\n\\int_0^\\infty \\bigr| (\\omega-i\\eta)^{-1} [\\log (\\omega-i\\eta)]^{-3} \\bigr| e^{-t\\eta} d\\eta\n+\nE(t),\n\\label{eqn:rffimLong1st160}\n\\end{equation}\nwith an appropriate constant $C$.\nNote that in this procedure, $\\tilde{R}^+ (\\omega)$ is analytically continued to\nthe lower plane of the second Riemann sheet, while $\\tilde{R}^- (\\omega)$ still remains\nin the lower plane of the first Riemann sheet.\nThen, both are ensured to contribute the remainders\nof the same order to $F(\\omega-i\\eta)$ in the above integral\nfrom Lemmas \\ref{lm:1stremainder+} and \\ref{lm:1stremainder-}.\nThe remainder term $E(t)$ in Eq. (\\ref{eqn:rffimLong1st160}) \nthat gives the order of $O(e^{-\\gamma t})$ for some $\\gamma >0$ \nis responsible for the possible poles\nof $\\tilde{R}^+ (\\omega)$ continued to the second Riemann sheet,\nthe number of which are guaranteed to be finite from\nthe analytic Fredholm theorem \\cite{the analytic Fredholm theorem}\nand Lemma \\ref{lm:Large-omega} for\nthe continued  $\\tilde{R}^+ (\\omega)$.\nThus, it follows from Lemma \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral}\nthat $\\lim_{\\omega \\to +0} \\|(\\hat{I}\\sigma)(\\omega)\\| =O((\\log t)^{-2})$.\nBy the same argument as in Lemma \\ref{lm:asymptote of inverse-lagarithmic-fourier-integral},\none also sees that the remainder term in Eq. (\\ref{eqn:rffimLong1st155})\nis of the order of $O(t^{-N+1})$.\nSummarizing these arguments, we can finish the proof of the theorem.\n\\raisebox{.6ex}{\\fbox{\\rule[0.0mm]{0mm}{0.8mm}}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding remarks}\n\\label{sec:8}\n\n\n\n\nWe have rigorously derived the asymptotic formula of the reduced time evolution\noperator for the $N$-level Friedrichs model in the context of the zero energy resonance \n\\cite{Jensen(1979)}\nboth for the regular case and the exceptional case of the first kind.\nThen, in the latter case, the logarithmically slow decay proportional to $(\\log t)^{-1}$\nhas been found,\nand the expansion coefficient has been explicitly presented\nby the projection operator associated with the zero energy eigenstates\nof the total Hamiltonian, which is an extended state not belonging to the Hilbert space.\nWe note that the decay involving the logarithmic function \nexpressed by $t^{-j}(\\log t)^{k}$ ($j=1, 2, \\ldots$ and $k=0, \\pm 1, \\ldots$)\ncan occur in the short range potential systems in the even dimensional space. \\cite{Murata(1982)}\nIt should be noted that a realization of the exceptional cases require \nthe parameters, e.g., the coupling constant $\\lambda$, \nto take such special values \nthat the matrix $K(0)$ in Eq. (\\ref{eqn:.336}) has a zero eigenvalue. \nIn addition, some of the form factors $v_n(\\omega)$ have to behave as $|v_n(\\omega)|^2 \\sim c_n \\omega$ \naround $\\omega=0$. \nIn other words, if all of them behave as \n$|v_n(\\omega)|^2 \\sim c_n \\omega^{q_n}$ with $q_n \\geq 2$, \nthe exceptional case of the first kind never occurs \nthough that of the second kind could happen. \nThese circumstances explain how the exceptional cases are surely exceptional. \nThe presented results also enable us to calculate the asymptotic formula for the survival\nprobability of an arbitrary initial state $\\ket{\\psi}$ localized over the $N$ discrete levels.\nIf we choose the special initial state to satisfy\n${\\mit \\Gamma}_{n_b} (K(0))^{-1}\\ket{\\psi}=0$ in Eq. (\\ref{eqn:rffimbothLong50})\nor $(Q_1 {\\mit \\Gamma}_1 Q_1)^{-1}\\ket{\\psi}=0$ in Eq. (\\ref{eqn:rffimLong1st50}),\nour estimations are useless and other decay laws could appear. \\cite{Miyamoto(2004)}\nThe long time behavior of the reduced time evolution operator\nfor the exceptional case of the second and the third kind are not examined.\nAs is expected, in the former case, the non decaying component\nassociated with the localized zero energy eigenstate will appear\ndue to the divergent behavior of $Q_2\\tilde{R}(z)Q_2=O(z^{-1})$ in Eq. (\\ref{eqn:5.C.2.120d}).\nThe latter case can occur in the $N$-level cases of the model only for $N\\geq 3$,\nwhich yields a more complicated situation.\nIn the whole of the paper, we assumed that there is no bound eigenstate\nwith a positive eigenenergy. This situation is actually realized in the weak coupling cases.\n\\cite{Davies(1974),Miyamoto(2005)} However,\nits compatibility with the existence of the extended zero energy eigenstate\nis still not clear in the multilevel cases (except the single level case).\nThe emergence of the logarithmic decay $(\\log t)^{-1}$ is just due to\nthe logarithmic energy dependence of the self energy $S(\\omega)$\nand it comes from the assumption (\\ref{eqn:formfactor1}) \nwhere $|v_n(\\omega)|^2 \\sim c_n \\omega^{q_n}$ with a positive integer $q_n$ is required. \nTherefore, if we choose another type of form factor,\nit is not necessary for such a slow decay to occur even in the exceptional case.\n\\cite{Kofman(1994),Lewenstein(2000),Nakazato(2003)}\nHowever, we stress that our assumption is often satisfied by\nactual systems. \\cite{Facchi(1998),Antoniou(2001)}\nThe experimental realization of the exceptional case\nrequires\nthe setup of parameters like $\\omega_1 \\simeq \\lambda^2 \\Lambda $, \nwhere $\\Lambda$ is a typical cutoff constant. \nThis seems in a strong coupling region to be naturally satisfied\n\\cite{Jittoh(2005),Garcia-Calderon(2001)}, \nand hence it could be suggested to invoke the artificial quantum structures for a realization.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nThe author would like to thank Professor I.\\ Ohba\nand Professor H.\\ Nakazato for useful comments.\nHe also would like to express his gratitude to the organizers\nof the International Workshop TQMFA2005, Palermo, Italy,\nNovember 11-13, 2005,\n{\\sl New Trends in Quantum Mechanics: Fundamental Aspects and Applications}.\nDiscussions during the YITP workshop YITP-W-05-21 on ``Fundamental Problems and Applications of\nQuantum Field Theory'' were useful in completing this work.\nThis research is partly supported by a Grant for The 21st Century COE Program\nat Waseda University from the Ministry of Education, Culture,\nSports, Science and Technology, Japan.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA $k$-{\\it system} $\\mathcal{A}$ of arcs on a punctured surface $S$ is a collection of essential simple arcs on $S$ such that no two arcs of $\\mathcal{A}$ are homotopic or intersect more than $k$ times. We begin with the following observation.\n\n\\begin{remark}\\label{kequalszero}\nIf $k=0$, $S$ is an $n$-punctured sphere with $n \\geq 3$, and the arcs of $\\mathcal{A}$ all join a fixed pair of distinct punctures $p,q$ of $S$, then $|\\mathcal{A}| \\leq n-2$. To see this, fix a complete hyperbolic metric on $S$ of area $2\\pi(n-2)$ and realize the arcs of $\\mathcal{A}$ as geodesics on $S$. Cutting $S$ along $\\mathcal{A}$, we obtain a collection of hyperbolic punctured strips. Since the boundary of each strip consists of two arcs of $\\mathcal{A}$, and since each arc of $\\mathcal{A}$ appears twice as a boundary component of some strip, we count precisely $|\\mathcal{A}|$ strips. The bound on $|\\mathcal{A}|$ now follows from the fact that each of these strips has area at least $2\\pi$. Moreover, this bound is tight since we can easily devise a $0$-system on $S$ whose complement consists entirely of once-punctured strips (see Figure~\\ref{fig:zerosystem}). An area argument also shows that the maximum size of $\\mathcal{A}$ is $2n-5$ if we assume instead that $p=q$.\n\\end{remark}\n\n\\begin{figure}\n\\scalebox{.45}{\n\\includegraphics[trim=0 520 0 0, clip]{0system.eps}\n}\n\\captionof{figure}{A maximum-size $0$-system joining distinct punctures $p,q$ of the $7$-punctured sphere.}\n\\label{fig:zerosystem}\n\\end{figure}\n\nProblems involving bounding the size of a $k$-system of arcs, of which Remark~\\ref{kequalszero} serves as a trivial example, originated in similar problems for curves. Juvan, Malni\\v{c}, and Mohar introduced the term ``$k$-system\" and showed that the maximum size $N(k, \\Sigma)$ of a $k$-system of essential simple closed curves on a fixed compact surface $\\Sigma$ is finite \\cite{juvan1996systems}. Independently, Farb and Leininger inquired about $N(k, g) := N(k, \\Sigma)$ for $\\Sigma$ closed and oriented of genus $g$ and $k=1$. In response, Malestein, Rivin, and Theran provided an upper bound exponential in $g$, and showed that $N(1,2) = 12$ \\cite{malestein2014topological}. Also for $k=1$, Przytycki produced an upper bound on the order of $g^3$ \\cite[Theorem~1.4]{przytycki2015arcs}; since then, tighter bounds on $N(1,g)$ have been found by Aougab, Biringer, and Gaster \\cite{aougab2017packing}, and more recently, Greene \\cite{greene2018curves}. Moreover, Przytycki provided an upper bound on $N(k,g)$ for arbitrary $k$ that grows like $g^{k^2+k+1}$ \\cite[Cor.~1.6]{przytycki2015arcs}. This bound was subsequently improved by Greene to one that grows like $g^{k+1}\\log g$ \\cite{greene2018curvesii}. \n\nIn \\cite{przytycki2015arcs}, so as to prove the aforementioned results about $k$-systems of curves, Przytycki first proved stronger results about $k$-systems of arcs -- for example, that the maximum size of a $k$-system of arcs on a punctured surface $S$ of Euler characteristic $\\chi < 0$ (where distinct arcs are not required to have the same endpoints) grows like $|\\chi|^{k+1}$. In the same article, Przytycki proved the following:\n\n\\begin{theorem}\\cite[Theorem~1.7]{przytycki2015arcs}\\label{przytycki}\nLet $p,q$ be punctures of an $n$-punctured sphere $S$, where $n \\geq 3$. The maximum size of a $1$-system $\\mathcal{A}$ of arcs on $S$ joining $p$ and $q$ is $\\binom{n-1}{2}$. \n\\end{theorem}\n\nNote that $p$ and $q$ are not assumed to be distinct in the statement of Theorem~\\ref{przytycki}. More recently, Bar-Natan showed that for $S,p,q$ as in Theorem~\\ref{przytycki}, if $p=q$ then the maximum size of a $2$-system of arcs on $S$ joining $p$ and $q$ is $\\binom{n}{3}$ \\cite{bar2017arcs}. The main result of this article is that Bar-Natan's maximum holds for $p,q$ distinct:\n\n\\begin{theorem}\\label{maintheorem}\nLet $p,q$ be distinct punctures of an $n$-punctured sphere $S$, where $n \\geq 3$. The maximum size of a $2$-system $\\mathcal{A}$ of arcs on $S$ joining $p$ and $q$ is $\\binom{n}{3}$. \n\\end{theorem}\n\n\\subsection*{Organization} In Section \\ref{example}, we provide an example of a $2$-system of size $\\binom{n}{3}$ joining a fixed pair of distinct punctures of an $n$-punctured sphere for $n \\geq 3$. The remaining sections are concerned with proving that $\\binom{n}{3}$ is an upper bound on the size of such a $2$-system $\\mathcal{A}$. This is proved by induction on $n$; we prove that the number of arcs of $\\mathcal{A}$ that become homotopic after forgetting a puncture of $S$ is not too large. This, in turn, is proved by induction, via the following lemma:\n\n\\begin{lemma}\\label{relation}\nLet $S$ be an $n$-punctured sphere, and let $p,q,r$ be distinct punctures of $S$. Let $\\mathcal{P}, \\mathcal{Q}$ be $1$-systems of arcs starting at $r$ and ending at $p,q$, respectively, so that no arc of $\\mathcal{P}$ intersects an arc of $\\mathcal{Q}$. Suppose $\\mathcal{R} \\subset \\mathcal{P} \\times \\mathcal{Q}$ satisfies the following:\n\\begin{enumerate}[label=(\\roman*)]\n\\item there is at most one point of intersection between any two pairs of arcs in $\\mathcal{R}$;\n\\item the cyclic order around $r$ of any two intersecting pairs $(\\alpha, \\beta), (\\alpha', \\beta') \\in \\mathcal{R}$ with $\\alpha \\neq \\alpha', \\beta \\neq \\beta'$ is alternating.\n\\end{enumerate}\nThen $|\\mathcal{R}| \\leq \\binom{n-1}{2}$. \n\\end{lemma}\n\nSection \\ref{proofoflemma} is devoted to the proof of Lemma~\\ref{relation}. The inductive step again involves forgetting a puncture $s$ of $S$, but this time, we choose $s$ with care in order to control the subsequent behavior of the arcs of $\\mathcal{P}$ and $\\mathcal{Q}$. More precisely, we require that $s$ be $p$-{\\it isolated} (see Section \\ref{definitions} for definitions). \n\nTo guarantee that a puncture with this property exists, we take a detour into annular square diagrams. A $k$-{\\it system annular diagram} $A$ is an annular diagram whose dual curves constitute a $k$-system of arcs joining the boundary paths of $A$. Such a diagram arises as the dual square complex to a $k$-system $\\mathcal{A}$ on a punctured sphere with distinct prescribed endpoints. In Section \\ref{diagrams}, we prove the following:\n\n\n\\begin{theorem}\\label{corner}\nLet $A$ be a $1$-system annular diagram. Then either $A$ is a cycle, or $A$ has a corner on each of its boundary paths.\n\\end{theorem}\n\nRoughly speaking, a corner corresponds to an isolated puncture. Note that Theorem~\\ref{corner} does not hold for $k=2$; see Figure~\\ref{fig:diagrams} (bottom left) for a counterexample, suggested by Przytycki. \n\n\\subsection*{Acknowledgements} I thank my supervisor Piotr Przytycki for his crucial guidance and remarkable patience throughout the course of this article's preparation. I also thank Daniel Wise for providing helpful resources.\n\n\\section{Definitions}\\label{definitions}\n\n\\subsection{Arc systems} A {\\it puncture} is a topological end of a space $S$ obtained from a connected, oriented, compact surface $\\Sigma$ by removing finitely many points $p_1, \\ldots, p_n$ from $\\Sigma$. Note that the punctures of $S$ are in bijection with $p_1, \\ldots, p_n$, and that we allow punctures on the boundary of $\\Sigma$. If $p_1, \\ldots p_n$ are taken from the interior of $\\Sigma$, we call $S$ an $n$-{\\it punctured} $\\Sigma$. \n\nAn {\\it arc} on $S$ is a proper map $\\alpha: (0,1) \\rightarrow S$. A proper map induces a map between ends of topological spaces; in this sense, $\\alpha$ ``maps\" each endpoint of $(0,1)$ to a puncture $p$ of $S$. We call $p$ an {\\it end} of $\\alpha$. If $p,q$ are ends of $\\alpha$, we say $\\alpha$ {\\it starts} at $p$ and {\\it ends} at $q$, or that $\\alpha$ {\\it joins} $p$ and $q$. A {\\it segment} of $\\alpha$ is the restriction of $\\alpha$ to some positive-length subinterval of $(0,1)$. \n\nAn arc $\\alpha$ is {\\it simple} if it is an embedding, in which case we identify $\\alpha$ and its segments with their images in $S$. If $J$ is a subinterval of $(0,1)$ with endpoints $t_1, t_2$ and $\\alpha$ is a simple arc mapping $t_i$ to $x_i$ for $i=1,2$, we denote the segment $\\alpha \\bigr|_J$ by $(x_1x_2)_{\\alpha}$. If $R \\subset S$ is a subset and $p$ is an end of $\\alpha$ corresponding to an endpoint $t_0=0,1$ of $(0,1)$, we say the {\\it $p$-end of $\\alpha$ lies in $R$} if $\\alpha^{-1}(R)$ is a neighbourhood of $t_0$ in $(0,1)$.\n\nA {\\it homotopy} between arcs $\\alpha_1$ and $\\alpha_2$ is a proper map $(0,1) \\times [0,1] \\rightarrow S$ whose restrictions to $(0,1) \\times \\{0\\}$ and $(0,1) \\times \\{1\\}$ are $\\alpha_1$ and $\\alpha_2$, respectively. In particular, a homotopy preserves ends. If $r$ is a puncture of $S$, we say that two arcs on $S$ are {\\it $r$-homotopic} if they are homotopic on the surface $\\bar{S}$ obtained from $S$ by forgetting $r$. Two arcs are in {\\it minimal position} if the number of their intersection points cannot be decreased by a homotopy. Note that if a pair of arcs have a point of intersection that is not transversal, then they are not in minimal position. An arc $\\alpha$ is {\\it essential} if it cannot be homotoped into a puncture, in the sense that there is no proper map $(0,1) \\times [0,1) \\rightarrow S$ whose restriction to $(0,1) \\times \\{0\\}$ is $\\alpha$. Unless otherwise stated, all arcs in the article are simple and essential, and all intersections between arcs are transversal. Note that an arc joining distinct punctures of a punctured surface is automatically essential.\n\nLet $R$ be a closed disc with at most 2 punctures on its boundary and possibly with punctures in its interior. A {\\it region} between arcs $\\alpha_1, \\alpha_2$ on $S$ is a properly embedded $R \\subset S$ such that $\\partial R$ is a union of exactly two segments $\\sigma_1, \\sigma_2$, where $\\sigma_i$ is a segment of $\\alpha_i$ for $i=1,2$ (see Figure~\\ref{fig:region}). We say that $\\alpha_1, \\alpha_2$ (or, more specifically, $\\sigma_1, \\sigma_2$) {\\it form} or {\\it bound} $R$. If $R$ has no punctures in its interior, we say $R$ is {\\it empty}. If $R$ has exactly 0 (respectively, 1, 2) punctures on its boundary and $R \\cap (\\alpha_1 \\cup \\alpha_2) = \\partial R$, we call $R$ a {\\it bigon} (respectively, {\\it half-bigon}, {\\it strip}). We say $R$ is {\\it adjacent} to a puncture $p$ of $S$ if $p$ lies on the boundary of $R$. If $p,s$ are distinct punctures of $S$ and $\\mathcal{A}$ is a collection of arcs on $S$ with $s$  contained in a half-bigon or strip $H$ adjacent to $p$ formed by a pair of arcs of $\\mathcal{A}$ such that $H$ is a component of $S - \\bigcup \\mathcal{A}$, we say that $s$ is $p$-{\\it isolated by} $\\mathcal{A}$. \n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 550 0 0, clip]{region.eps}\n}\n\\captionof{figure}{Regions formed by arcs. The yellow region is a half-bigon. The pink and orange regions are bigons. The orange bigon is empty.}\n\\label{fig:region}\n\\end{figure}\n\nWe will make frequent use of the following lemma.\n\n\\begin{lemma}[The bigon criterion]\\label{bigoncriterion} \\cite[Proposition~1.7]{farb_margalit_2012}\nTwo intersecting arcs on a punctured surface are in minimal position if and only if they form no empty regions.\n\\end{lemma}\n\nSince an empty region bounded by intersecting arcs must contain an empty bigon or half-bigon, we immediately obtain the following corollary.\n\n\\begin{corollary}\\label{corollarybigon}\nTwo intersecting arcs on a punctured surface are in minimal position if and only if they form no empty bigons or half-bigons.  \n\\end{corollary}\n\nA {\\it $k$-system} of arcs on a punctured surface $S$ is a collection $\\mathcal{A}$ of essential simple arcs on $S$ such that no two arcs of $\\mathcal{A}$ are homotopic or have more than $k$ points of intersection. We will mainly consider the case where $S$ is a sphere punctured at least thrice and $\\mathcal{A}$ is a 2-system of arcs joining a fixed pair of distinct punctures $p,q$ of $S$. Note that for any two arcs $\\alpha_1, \\alpha_2$ of such a collection $\\mathcal{A}$, a region bounded by $\\alpha_1, \\alpha_2$ that contains neither $p$ nor $q$ must be a bigon, a half-bigon, or a strip.\n\nIf a punctured surface $S$ has Euler characteristic $\\chi < 0$, then $S$ admits a complete hyperbolic metric of area $2\\pi |\\chi|$. Under such a metric, the homotopy class of any arc contains a unique geodesic representative, and any two distinct geodesic arcs are in minimal position. Thus, for the purposes of determining the size of a $k$-system $\\mathcal{A}$ of arcs on $S$, we may assume that $\\mathcal{A}$ consists of geodesics. \n\n\n\n\n\\subsection{Combinatorial complexes} A map $X \\rightarrow Y$ between CW complexes $X,Y$ is {\\it combinatorial} if its restriction to each open cell of $X$ is a homeomorphism onto an open cell of $Y$. A CW complex $X$ is {\\it combinatorial} if the attaching map of each  cell in $X$ is combinatorial for some subdivision of the sphere. A cell of dimension 0 is a {\\it vertex} and a cell of dimension 1 is an {\\it edge}. The {\\it degree} of a vertex $v$ of $X$ is the number of edges in $X$ incident to $v$, with loops counted twice. \n\n \\subsection{Square complexes} An {$n$-cube} is a copy of $[-1,1]^n$. A {\\it square complex} $X$ is a combinatorial complex whose cells are $n$-cubes with $n \\leq 2$; that is, $X$ is a combinatorial 2-complex each of whose 2-cells is attached via a combinatorial map from a 4-cycle into the 1-skeleton of $X$. The cells of $X$ are called {\\it cubes}, and its 2-cells are called {\\it squares}. \n \n A {\\it midcube} is a subspace of a cube $[-1,1]^n$ obtained by restricting one coordinate to $0$. Let $U$ be a new square complex whose cells are midcubes of $X$ and whose attaching maps are restrictions of attaching maps in $X$ to midcubes. A {\\it dual curve} $\\alpha$ of a cube $c$ in $X$ is a connected component of $U$ containing a midcube of $c$. Note that if $c$ is a square, then it has exactly two dual curves. If $c$ is an edge, we say $\\alpha$ is {\\it dual} to $c$. We call $\\alpha$ an {\\it arc} if it is homeomorphic to an interval (possibly of length 0). There is a natural immersion $\\alpha \\rightarrow X$; if this map is an embedding, we say $\\alpha$ is {\\it simple}. In this case, we identify $\\alpha$ with its image in $X$.\n\n\\subsection{Annular diagrams} An {\\it annular diagram} $A$ is a finite combinatorial cell decomposition of $S^2$ minus two disjoint open 2-cells (see Figure~\\ref{fig:annulardiagram}). The attaching map of each of these 2-cells is a {\\it boundary path} of $A$. \n\\begin{figure}\n\\scalebox{.25}{\n\\includegraphics[trim=0 350 0 10, clip]{boundarypaths.eps}\n}\n\\captionof{figure}{An annular diagram. The boundary paths are indicated in red.}\n\\label{fig:annulardiagram}\n\\end{figure}\n\n\nWe call $A$ a {\\it square annular diagram}, or simply a {\\it diagram}, if it is also a square complex (see Figure~\\ref{fig:diagrams}). A {\\it corner} on a boundary path $P$ of a diagram $A$ is a vertex $v$ on $P$ of degree 2 that is contained in some square of $A$. \n\nLet $c$ be a square of a diagram $A$ with boundary path $P$, and let $x$ be the center of $c$. Suppose the dual curves $\\alpha, \\beta$ of $c$ are dual to consecutive edges $a,b$ on $P$ with shared vertex $v$. Let $\\gamma$ be the loop obtained from the subarcs of $\\alpha, \\beta$ joining $x$ and $P$ and the half-edges of $a,b$ containing $v$. If $\\gamma$ is homotopic in $A$ to a constant path, then we call $c$ a {\\it cornsquare} with {\\it outerpath} $ab$. \n\n\\begin{figure}\n\\scalebox{0.55}{\n\\includegraphics[trim=0 330 0 0, clip]{annulardiagrams2.eps}\n}\n\\captionof{figure}{Square annular diagrams. Dual curves are dashed and colored. Corners are colored red. The bottom left (respectively, bottom right) diagram is a $2$-system (respectively, $1$-system) annular diagram. Note that the square at which the blue and orange dual curves meet in the bottom right diagram is a cornsquare with an outerpath on each boundary path, while the square at which the green and purple dual curves meet is not a cornsquare, even though the latter two dual curves are dual to consecutive edges on each boundary path.} \n\\label{fig:diagrams}\n\\end{figure}\n\nA {\\it hexagon move} on a diagram $A$ is the replacement of three squares forming a subdivided hexagon by an alternate three squares forming a subdivided hexagon (see Figure \\ref{fig:hexagon}). A hexagon move can be visualized as a benign ``sliding\" operation on one of the dual curves of $A$, so that if $A'$ is obtained from $A$ by a hexagon move, there is a natural correspondence between the dual curves of $A$ and those of $A'$. Note that the number of squares of $A$ is preserved under hexagon moves.\n\n\\begin{figure}\n\\scalebox{.3}{\n\\includegraphics[trim=0 570 0 0, clip]{hexagon.eps}\n}\n\\captionof{figure}{A hexagon move and its effect on dual curves.}\n\\label{fig:hexagon}\n\\end{figure}\n\nA square annular diagram $A$ is a {\\it $k$-system annular diagram} if its dual curves are simple arcs joining the boundary paths of $A$ and pairwise intersecting at most $k$ times in $A$. Note that the number of intersections between any pair of dual curves of $A$ is preserved under a hexagon move. Thus, if $A'$ is obtained from a $k$-system annular diagram $A$ by a hexagon move, then $A'$ is also a $k$-system annular diagram. \n\n\n\n\n\n\n\\section{A 2-system of maximum size}\\label{example}\n\nWe provide an example of a 2-system of arcs of size $\\binom{n}{3}$ joining a fixed pair of distinct punctures of an $n$-punctured sphere $S$. This collection was independently discovered by Assaf Bar-Natan.\n\nWe think of $S$ as $\\mathbb{R}^2$ punctured at $p = (-1,0)$ and at the points $r_i = (i - \\frac{1}{2}, 0)$ for $i = 1, \\ldots, n-2$. We construct a 2-system $\\mathcal{A}$ joining $p$ and the puncture $q$ at infinity. \n\nLet $\\alpha_{< -1}$ be the arc given by the ray $\\{(x,0) : \\> x < -1\\}$. For $a, b, c \\in \\{0, 1, \\ldots, n-2\\}$ with $a < b <c$ or $0 < a < b = c = n-2$, let $\\alpha_{abc}$ be the graph of the polynomial function $f_{abc}: (-1, \\infty) \\rightarrow \\mathbb{R}$ given by $x \\mapsto (x+1)(x-a)(x-b)(x-c)$. \n\nNote that for distinct triples $(a,b,c), (a', b',c')$, the difference $f_{abc}-f_{a'b'c'}$ is a cubic polynomial, one of whose roots is $-1$. Thus, the $\\alpha_{abc}$ pairwise intersect at most twice. Furthermore, the $\\alpha_{abc}$ are pairwise non-homotopic \\cite[proof~of~Lemma~4.2]{bar2017arcs}, and $\\alpha_{<-1}$ is not homotopic to any of the $\\alpha_{abc}$ since the complement of $\\alpha_{<-1} \\cup \\alpha_{abc}$ is a pair of punctured strips. \n\nNow fix $M > 0$ such that $M > |f_{abc}(x)|$ for all $x \\in (-1, n-2]$. For each $i,j \\in \\{1, \\ldots, n-2\\}$ with $i < j$, let $\\alpha_{ij}$ be the union of the following horizontal and vertical segments: the segment joining $(-1,0)$ and $(-1, M)$, the segment joining $(-1,M)$ and $(i - \\frac{1}{2} + \\frac{1}{4}, M)$, the segment joining $(i - \\frac{1}{2} + \\frac{1}{4}, M)$ and $(i - \\frac{1}{2} + \\frac{1}{4}, -M)$, the segment joining $(i - \\frac{1}{2} + \\frac{1}{4}, -M)$ and $(-2, -M)$, the segment joining $(-2, -M)$ and $(-2, M+1)$, the segment joining $(-2, M+1)$ and $(j - \\frac{1}{2} - \\frac{1}{4}, M+1)$, and the vertical ray travelling down from $(j - \\frac{1}{2} - \\frac{1}{4}, M+1)$. Note that each $\\alpha_{ij}$ intersects $\\alpha_{<-1}$ exactly once and each $\\alpha_{abc}$ exactly twice (see Figure \\ref{fig:maxfamily}). Furthermore, each $\\alpha_{ij}$ is in minimal position with $\\alpha_{<-1}$ by Corollary~\\ref{corollarybigon}; since $\\alpha_{<-1}$ is disjoint from the $\\alpha_{abc}$, this shows that none of the $\\alpha_{ij}$ is homotopic to any of the $\\alpha_{abc}$. \n\nWe claim that the $\\alpha_{ij}$ are pairwise non-homotopic. Indeed, for $k \\in \\{1, \\ldots, n-3\\}$, let $\\gamma_k$ be the horizontal arc joining the punctures at $x = k- \\frac{1}{2}$ and $x = k + \\frac{1}{2}$, and note that $\\alpha_{ij}$ and $\\gamma_k$ are in minimal position by Corollary~\\ref{corollarybigon}. Since no two of the $\\alpha_{ij}$ share the same number of intersection points with each of the $\\gamma_k$, the $\\alpha_{ij}$ must be pairwise non-homotopic.\n\nWe claim further that the $\\alpha_{ij}$ pairwise intersect at most twice. Indeed, if $i,j,i',j' \\in \\{1, \\ldots, n-2\\}$ with $i \\leq i'$, then the number of intersection points between $\\alpha_{ij}, \\alpha_{i'j'}$ is determined by the order of $j, i', j'$. If $i' < j \\leq j'$, then $\\alpha_{ij}$ and $\\alpha_{i'j'}$ are disjoint (see Figure \\ref{fig:twistingcurves}, top). If $i' < j' < j$, then $\\alpha_{ij}$ and $\\alpha_{i'j'}$ intersect once (see Figure \\ref{fig:twistingcurves}, middle). Otherwise, $j \\leq i'$, and there are two points of intersection between $\\alpha_{ij}$ and $\\alpha_{i',j'}$ (see Figure \\ref{fig:twistingcurves}, bottom). Thus, the family $\\mathcal{A}$ consisting of $\\alpha_{<-1}$, the $\\alpha_{abc}$, and the $\\alpha_{ij}$ is a 2-system of size $1 + (n-3) + \\binom{n-1}{3} + \\binom{n-2}{2} = \\binom{n}{3}$. \n\n\\begin{figure}\n \\begin{tikzpicture}[scale=1]\n\n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n[->] (-2,0)--(4,0); \n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n    [->] (0,-3)--(0,5);\n   \n    \n\\begin{scope}\n\\clip (-2,-3) rectangle (4,5);\n\n\n\n\\draw[thick, green](0.5,0)--(1.5,0);\n\\draw [thick, red, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*\\x*(\\x-1)*(\\x-2)});\n\\draw [thick, pink, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*\\x*(\\x-1)*(\\x-3)});\n\\draw [thick, orange, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*\\x*(\\x-2)*(\\x-3)});\n\\draw [thick, brown, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*(\\x-1)*(\\x-2)*(\\x-3)});\n\\draw [thick, magenta, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*(\\x-1)*(\\x-3)^2});\n\\draw [thick, gray, domain=-1:4 , samples=500] plot (\\x , {0.1*(\\x+1)*(\\x-2)*(\\x-3)^2});\n\\draw[thick, blue](-1,0)--(-2,0);\n\\draw[thick, violet](-1,0)--(-1,2.1);\n\\draw[thick, violet](-1,2.1)--(1.75,2.1);\n\\draw[thick, violet](1.75,2.1)--(1.75,-2.2);\n\\draw[thick, violet](1.75,-2.2)--(-1.5,-2.2);\n\\draw[thick, violet](-1.5,-2.2)--(-1.5,2.6);\n\\draw[thick, violet](-1.5,2.6)--(2.25,2.6);\n\\draw[thick, violet](2.25,2.6)--(2.25,-3);\n\n\\draw [fill=white] (-1,0) circle [radius=0.05];\n\\draw [fill= black] (0,0) circle [radius=0.03];\n\\draw [fill= white] (0.5,0) circle [radius=0.05];\n\\draw [fill= black] (1,0) circle [radius=0.03];\n\\draw [fill= white] (1.5,0) circle [radius=0.05];\n\\draw [fill= black] (2,0) circle [radius=0.03];\n\\draw [fill= white] (2.5,0) circle [radius=0.05];\n\\draw [fill= black] (3,0) circle [radius=0.03];\n\\node at(-1.1,-0.6)[label=$p$]{};\n\\end{scope} \n\\end{tikzpicture}\n\\captionof{figure}{The arcs $\\alpha_{abc}$ on the 5-punctured sphere, together with arc $\\alpha_{<-1}$, drawn in blue, arc $\\alpha_{23}$, drawn in violet, and arc $\\gamma_1$, drawn in green.}\n\\label{fig:maxfamily}\n\\end{figure}\n\n\\begin{figure}\n \\begin{tikzpicture}[scale=1]\n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n[->] (-2,0)--(4.25,0); \n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n    [->] (0,-1)--(0,1.5);\n   \n    \n\\begin{scope}\n\n\n\\draw[thick, violet](-1,0)--(-1,0.5);\n\\draw[thick, violet](-1,0.5)--(0.75,0.5);\n\\draw[thick, violet](0.75,0.5)--(0.75,-0.5);\n\\draw[thick, violet](0.75,-0.5)--(-1.5,-0.5);\n\\draw[thick, violet](-1.5,-0.5)--(-1.5,0.75);\n\\draw[thick, violet](-1.5,0.75)--(2.25,0.75);\n\\draw[thick, violet](2.25,0.75)--(2.25,-1);\n\\draw[thick, orange](-1.05,0)--(-1.05,0.55);\n\\draw[thick, orange](-1.05,0.55)--(1.75,0.55);\n\\draw[thick, orange](1.75,0.55)--(1.75,-0.55);\n\\draw[thick, orange](1.75,-0.55)--(-1.55,-0.55);\n\\draw[thick, orange](-1.55,-0.55)--(-1.55,0.8);\n\\draw[thick, orange](-1.55,0.8)--(3.25,0.8);\n\\draw[thick, orange](3.25,0.8)--(3.25,-1);\n\n\n\n\\draw [fill=white] (-1,0) circle [radius=0.07];\n\\draw [fill= black] (0,0) circle [radius=0.03];\n\\draw [fill= white] (0.5,0) circle [radius=0.07];\n\\draw [fill= black] (1,0) circle [radius=0.03];\n\\draw [fill= white] (1.5,0) circle [radius=0.07];\n\\draw [fill= black] (2,0) circle [radius=0.03];\n\\draw [fill= white] (2.5,0) circle [radius=0.07];\n\\draw [fill= black] (3,0) circle [radius=0.03];\n\\draw [fill= white] (3.5,0) circle [radius=0.07];\n\\node at(-1.1,-0.6)[label=$p$]{};\n\\end{scope} \n\\end{tikzpicture}\n\n \\begin{tikzpicture}[scale=1]\n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n[->] (-2,0)--(4.25,0); \n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n    [->] (0,-1)--(0,1.5);\n   \n    \n\\begin{scope}\n\n\n\\draw[thick, violet](-1,0)--(-1,0.5);\n\\draw[thick, violet](-1,0.5)--(0.75,0.5);\n\\draw[thick, violet](0.75,0.5)--(0.75,-0.5);\n\\draw[thick, violet](0.75,-0.5)--(-1.5,-0.5);\n\\draw[thick, violet](-1.5,-0.5)--(-1.5,0.75);\n\\draw[thick, violet](-1.5,0.75)--(3.25,0.75);\n\\draw[thick, violet](3.25,0.75)--(3.25,-1);\n\\draw[thick, orange](-1.05,0)--(-1.05,0.55);\n\\draw[thick, orange](-1.05,0.55)--(1.75,0.55);\n\\draw[thick, orange](1.75,0.55)--(1.75,-0.55);\n\\draw[thick, orange](1.75,-0.55)--(-1.55,-0.55);\n\\draw[thick, orange](-1.55,-0.55)--(-1.55,0.8);\n\\draw[thick, orange](-1.55,0.8)--(2.25,0.8);\n\\draw[thick, orange](2.25,0.8)--(2.25,-1);\n\n\n\n\\draw [fill=white] (-1,0) circle [radius=0.07];\n\\draw [fill= black] (0,0) circle [radius=0.03];\n\\draw [fill= white] (0.5,0) circle [radius=0.07];\n\\draw [fill= black] (1,0) circle [radius=0.03];\n\\draw [fill= white] (1.5,0) circle [radius=0.07];\n\\draw [fill= black] (2,0) circle [radius=0.03];\n\\draw [fill= white] (2.5,0) circle [radius=0.07];\n\\draw [fill= black] (3,0) circle [radius=0.03];\n\\draw [fill= white] (3.5,0) circle [radius=0.07];\n\\node at(-1.1,-0.6)[label=$p$]{};\n\\end{scope} \n\\end{tikzpicture}\n\n \\begin{tikzpicture}[scale=1]\n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n[->] (-2,0)--(4.25,0); \n\\draw [\n    decoration={markings,mark=at position 1 with {\\arrow[scale=2]{>}}},\n    postaction={decorate},\n    shorten >=0.4pt\n    ]\n    [->] (0,-1)--(0,1.5);\n   \n    \n\\begin{scope}\n\n\n\\draw[thick, violet](-1,0)--(-1,0.5);\n\\draw[thick, violet](-1,0.5)--(0.75,0.5);\n\\draw[thick, violet](0.75,0.5)--(0.75,-0.5);\n\\draw[thick, violet](0.75,-0.5)--(-1.5,-0.5);\n\\draw[thick, violet](-1.5,-0.5)--(-1.5,0.75);\n\\draw[thick, violet](-1.5,0.75)--(1.25,0.75);\n\\draw[thick, violet](1.25,0.75)--(1.25,-1);\n\\draw[thick, orange](-1.05,0)--(-1.05,0.55);\n\\draw[thick, orange](-1.05,0.55)--(1.75,0.55);\n\\draw[thick, orange](1.75,0.55)--(1.75,-0.55);\n\\draw[thick, orange](1.75,-0.55)--(-1.55,-0.55);\n\\draw[thick, orange](-1.55,-0.55)--(-1.55,0.8);\n\\draw[thick, orange](-1.55,0.8)--(3.25,0.8);\n\\draw[thick, orange](3.25,0.8)--(3.25,-1);\n\n\n\n\\draw [fill=white] (-1,0) circle [radius=0.07];\n\\draw [fill= black] (0,0) circle [radius=0.03];\n\\draw [fill= white] (0.5,0) circle [radius=0.07];\n\\draw [fill= black] (1,0) circle [radius=0.03];\n\\draw [fill= white] (1.5,0) circle [radius=0.07];\n\\draw [fill= black] (2,0) circle [radius=0.03];\n\\draw [fill= white] (2.5,0) circle [radius=0.07];\n\\draw [fill= black] (3,0) circle [radius=0.03];\n\\draw [fill= white] (3.5,0) circle [radius=0.07];\n\\node at(-1.1,-0.6)[label=$p$]{};\n\\end{scope} \n\\end{tikzpicture}\n\n\n\\captionof{figure}{The $\\alpha_{ij}$ pairwise intersect at most twice.}\n\\label{fig:twistingcurves}\n\\end{figure}\n\n\\section{Properties of $r$-homotopic arcs intersecting at most twice}\n\nLet $p,q,r$ be distinct punctures of a punctured sphere $S$, and let $\\mathcal{A}$ be a $2$-system of arcs on $S$ joining $p$ and $q$. Let $\\bar{S}$ be the surface obtained from $S$ by forgetting $r$, and for each arc $\\alpha \\in \\mathcal{A}$, let $\\bar{\\alpha}$ be the homotopy class of the corresponding arc on $\\bar{S}$. In order to bound the size of $\\mathcal{A}$ from above, we will need to examine to what extent the map $\\alpha \\mapsto \\bar{\\alpha}$ is injective. In this section, we collect some facts about the fibers of this map. Together, the results of this section show that we can extend $\\mathcal{A}$ so that the size of each fiber is $1$ larger than the number of pairs of disjoint arcs in that fiber. \n\nThe main results of this section are Lemmas~\\ref{rhomotopic}, \\ref{rhomotopicbigon}, and \\ref{extend}. The proofs are rather technical and may be skipped on an initial reading.\n\n\n\n\\begin{lemma}\\label{configurations}\nLet $p,q,r$ be distinct punctures of a punctured sphere $S$, and let $\\alpha_1, \\alpha_2$ be a pair of $r$-homotopic arcs joining $p$ and $q$ and intersecting at most twice. If the $\\alpha_i$ are in minimal position, then they are in one of the configurations shown in Figure \\ref{fig:rhomotopicpairs}, up to relabeling $p$ and $q$. \n\\end{lemma}\n\n\n\\begin{proof}\nIf $\\alpha_1, \\alpha_2$ are disjoint, then they bound a strip whose only puncture is $r$ (see Figure \\ref{fig:rhomotopicpairs}, top left). Otherwise, by Corollary~\\ref{corollarybigon}, $\\alpha_1, \\alpha_2$ bound a half-bigon or bigon $R$ whose only puncture is $r$. If $\\alpha_1, \\alpha_2$ intersect exactly once, then $R$ is a half-bigon and, since the $\\alpha_i$ are $r$-homotopic, all punctures of $S$ distinct from $p,q,r$ lie in the other half-bigon formed by $\\alpha_1, \\alpha_2$ (see Figure \\ref{fig:rhomotopicpairs}, top right). \n\nIf the $\\alpha_i$ intersect twice and $R$ is a half-bigon, then the $\\alpha_i$ do not form a bigon, since otherwise they would not be $r$-homotopic (see Figure~\\ref{fig:nobigonallowed}). Thus, in this case, the $\\alpha_i$ must be as in the bottom right diagram of Figure \\ref{fig:rhomotopicpairs}, and since the $\\alpha_i$ are $r$-homotopic, all punctures of $S$ distinct from $p,q,r$ must lie in the other half-bigon formed by the $\\alpha_i$.\n\nOtherwise, $R$ is a bigon, and the $\\alpha_i$ also bound a pair of punctured half-bigons. These half-bigons must contain all the remaining punctures of $S$ since the $\\alpha_i$ are $r$-homotopic (see Figure \\ref{fig:rhomotopicpairs}, bottom left).\n\\end{proof}\n\n\nThe corollary of the following lemma will be useful in the proofs of Lemmas~\\ref{rhomotopic} and \\ref{rhomotopicbigon}. The former tells us that, in a particular context, if we have a portion of an arc then we can trace out the remainder of that arc.\n\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 250 0 0, clip]{rhomotopicpairs2.eps}\n}\n\\captionof{figure}{The possible configurations of a pair of $r$-homotopic arcs in minimal position and intersecting at most twice, up to relabeling $p$ and $q$.}\n\\label{fig:rhomotopicpairs}\n\\end{figure}\n\n\\begin{figure}\n\\scalebox{.45}{\n\\includegraphics[trim=0 520 0 0, clip]{nobigonallowed.eps}\n}\n\\captionof{figure}{If $\\alpha_1, \\alpha_2$ are in minimal position, intersect exactly twice, and form a bigon that does not contain $r$, then they cannot be $r$-homotopic. The fact that the bigon and half-bigons bounded by the $\\alpha_i$ prior to forgetting $r$ are punctured, and the fact that the arcs on the right are in minimal position, are consequences of Corollary~\\ref{corollarybigon}.}\n\\label{fig:nobigonallowed}\n\\end{figure}\n\n\n\\begin{lemma}\\label{punctureddisc}\nLet $D$ be a disc with at least $2$ punctures in its interior and at least $1$ puncture on its boundary, and let $\\alpha$ be an arc joining an interior puncture $p$ of $D$ to a puncture $x$ on $\\partial D$. If $\\beta$ is another arc joining $p$ and $x$ such that $\\alpha$ and $\\beta$ bound a strip containing all interior punctures of $D$ distinct from $p$, then $\\beta$ is homotopic to exactly one of the arcs $\\alpha_1, \\alpha_2$ shown in Figure \\ref{fig:punctureddisc} (left).\n\\end{lemma}\n\n\\begin{proof}\nSuppose that the $x$-end of $\\beta$ lies to the right of $\\alpha$. Then we may homotope $\\alpha_1$ so that it bounds an empty strip with $\\beta$, as in Figure \\ref{fig:punctureddisc} (right). Thus, in this case, $\\beta$ is homotopic to $\\alpha_1$. Similarly, if the $x$-end of $\\beta$ lies to the left of $\\alpha$, then $\\beta$ is homotopic to $\\alpha_2$.\n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{punctureddisc.eps}\n}\n\\captionof{figure}{}\n\\label{fig:punctureddisc}\n\\end{figure}\n\n\\begin{corollary}\\label{uniquecontinuation}\nLet $p,q,r$ be distinct punctures of an $n$-punctured sphere $S$, with $n \\geq 4$, and let $\\alpha, \\beta$ be $r$-homotopic arcs in minimal position joining $p,q$ and intersecting once or twice. Let $x_1, \\ldots, x_m$ be the points of intersection of $\\alpha, \\beta$ in the order that $\\beta$ traverses them as $\\beta$ travels from $p$ to $q$, and set $x_0=p$, $x_{m+1}=q$. For $i=0, \\ldots, m$, let $\\beta_i$ be the segment of $\\beta$ joining $x_i$ and $x_{i+1}$. If $m=2$, then the homotopy types of $\\beta_0$ and $\\beta_1$ determine that of $\\beta$. If $\\alpha$ and $\\beta$ do not bound a bigon, then the homotopy type of $\\beta_0$ determines that of $\\beta$ for $m=1,2$.\n\\end{corollary}\n\n\\begin{proof} For $i = 0, \\ldots, m$, let $\\alpha_i$ be the segment of $\\alpha$ joining $x_i$ and $x_{i+1}$. We puncture $S$ at $x_1, \\ldots, x_m$. \\\\\n\n\\paragraph{\\textbf{Case 1: $\\alpha$ and $\\beta$ intersect exactly once.}} Cutting $S$ along $\\alpha_0, \\beta_0$ yields two punctured strips. Let $D$ be the strip containing $q$. Note that $x_1$ is now a puncture on $\\partial D$, and that $\\alpha_1$ and $\\beta_1$ are arcs joining $q$ and $x_1$ and bounding a strip containing all the interior punctures of $D$ distinct from $q$. Thus, by Lemma~\\ref{punctureddisc}, the homotopy type of $\\beta_1$ is uniquely determined, since only one of the arcs described in Lemma~\\ref{punctureddisc} produces a $\\beta$ that intersects $\\alpha$ transversally at $x_1$ (in fact, the only other candidate homotopy class of $\\beta_1$ produces a $\\beta$ that is homotopic to $\\alpha$). \\\\\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{bigoncontinuation.eps}\n}\n\\captionof{figure}{By Lemma~\\ref{punctureddisc}, given segments $\\beta_0$ and $\\beta_1$ of $\\beta$, there are at most $2$ homotopy classes of arcs joining $q$ and $x_2$ to which $\\beta_2$ can belong. One such homotopy class produces a $\\beta$ that intersects $\\alpha$ non-transversally at $x_2$, as shown above.}\n\\label{fig:bigoncontinuation}\n\\end{figure}\n\n\\paragraph{\\textbf{Case 2: $\\alpha$ and $\\beta$ form a bigon.}} Let $D$ be the square containing the puncture $q$ obtained by cutting $S$ along $\\alpha_0, \\beta_0, \\alpha_1, \\beta_1$. Now $x_2$ is a puncture on $\\partial D$, and $\\alpha_2, \\beta_2$ are arcs joining $q$ and $x_2$ and bounding a strip containing all the interior punctures of $D$ distinct from $q$, so we may apply Lemma~\\ref{punctureddisc} as in Case 1. Again, only one of the two homotopy classes to which $\\beta_2$ must belong by Lemma~\\ref{punctureddisc} produces a $\\beta$ that intersects $\\alpha$ transversally at $x_2$ (see Figure~\\ref{fig:bigoncontinuation}).  \\\\\n\n\\paragraph{\\textbf{Case 3: $\\alpha$ and $\\beta$ intersect exactly twice but do not form a bigon.}} Let $D$ be the strip containing $q$ obtained by cutting $S$ along $\\alpha_0, \\beta_0$. Since $x_1$ is a puncture on $\\partial D$, and $\\alpha_1, \\beta_1$ are arcs joining $x_1, x_2$ and bounding a strip containing all the interior punctures of $D$ distinct from $x_2$, the homotopy type of $\\beta_1$ is uniquely determined by Lemma~\\ref{punctureddisc} as in the previous cases. Now let $D'$ be the strip containing $q$ obtained by cutting $D$ along $\\alpha_1, \\beta_1$. Since $x_2$ is a puncture on $\\partial D'$, and $\\alpha_2, \\beta_2$ are arcs joining $q, x_2$ and bounding a strip containing all interior punctures of $D'$ distinct from $q$, the homotopy type of $\\beta_2$ is uniquely determined by Lemma~\\ref{punctureddisc}. \n\\end{proof} \n\n\n\\begin{lemma}\\label{rhomotopic}\nLet $p,q,r$ be distinct punctures of an $n$-punctured sphere $S$, with $n \\geq 4$. Let $\\mathcal{A}_r$ be a maximal $2$-system of $r$-homotopic arcs on $S$ joining $p$ and $q$. If $\\mathcal{A}_r$ contains intersecting arcs $\\alpha_1, \\alpha_2 \\in \\mathcal{A}_r$ in minimal position that do not form a bigon, then $\\mathcal{A}_r$ is as in Figure \\ref{fig:halfbigonmaximal}, up to homotopy and relabeling $p$ and $q$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nLet $H$ be the half-bigon formed by the $\\alpha_i$ containing $r$, and assume that $H$ is adjacent to $p$ (see Figures \\ref{fig:case1nobigon}, \\ref{fig:case2nobigon}, left). Let $\\beta \\in \\mathcal{A}_r$. \\\\ \n\n\n\\item \\paragraph{\\textbf{Case 1: The $\\alpha_i$ intersect exactly once.}} Let $x$ be their unique point of intersection. If $\\beta$ is disjoint from the $\\alpha_i$, then $\\beta$ is homotopic to arc $\\beta_1$ in Figure \\ref{fig:case1nobigon} (right). Now suppose $\\beta$ is not disjoint from the $\\alpha_i$, and let $z$ be the first point of intersection of $\\beta$ and the $\\alpha_i$ as $\\beta$ travels from $p$ to $q$. Suppose that $z$ lies on $\\alpha_1$, and that $\\alpha_1, \\beta$ are in minimal position. If the $p$-end of $\\beta$ lies in $H$, then $\\beta$ forms a half-bigon with $\\alpha_1$ whose only puncture is $r$, since otherwise $\\alpha_1$ and $\\beta$ would form an empty half-bigon, contradicting our assumption that $\\alpha_1, \\beta$ are in minimal position (Corollary~\\ref{corollarybigon}). Thus, by Lemma~\\ref{configurations} and Corollary~\\ref{uniquecontinuation}, $\\beta$ is either homotopic to $\\alpha_2$ or to arc $\\beta_2$ in Figure \\ref{fig:case1nobigon} (right). If the $p$-end of $\\beta$ lies outside $H$, then $z$ cannot lie on the segment $(px)_{\\alpha_1}$, since otherwise $\\beta$ and $\\alpha_1$ would form an empty half-bigon. Thus, $z$ lies on $(xq)_{\\alpha_1}$, and so $\\beta$ again forms a half-bigon with $\\alpha_1$ whose only puncture is $r$. Thus, by Corollary~\\ref{uniquecontinuation}, $\\beta$ is again homotopic to one of $\\alpha_2, \\beta_2$. Note that, by the above, $\\mathcal{A}_r$ cannot contain an additional arc $\\beta'$ in minimal position with $\\alpha_2$ and intersecting $\\alpha_2$ first as $\\beta'$ travels from $p$ to $q$. This is because the reflection of $\\beta_2$ across the vertical diameter in Figure \\ref{fig:case1nobigon} (right) intersects $\\beta_2$ thrice. \n\n\\item \\paragraph{\\textbf{Case 2: The $\\alpha_i$ intersect exactly twice.}} Let $x,y$ be the points of intersection of the $\\alpha_i$ in the order that $\\alpha_1$ traverses them as it travels from $p$ to $q$. In this case, $\\beta$ intersects at least one of the $\\alpha_i$ since $p,q$ are in distinct components of the complement of $\\alpha_1 \\cup \\alpha_2$. Let $z$ be the first point of intersection of $\\beta$ and the $\\alpha_i$ as $\\beta$ travels from $p$ to $q$. We suppose again that $z$ lies on $\\alpha_1$, and that $\\alpha_1, \\beta$ are in minimal position. If the $p$-end of $\\beta$ lies in $H$, then, as in Case 1, $\\beta$ forms a half-bigon with $\\alpha_1$ whose only puncture is $r$. Thus, by Corollary \\ref{uniquecontinuation}, $\\beta$ is either homotopic to $\\alpha_2$ or to arc $\\beta_1$ in Figure \\ref{fig:case2nobigon} (right). If the $p$-end of $\\beta$ lies outside $H$, then, as in Case 1, $z$ cannot lie on the segment $(px)_{\\alpha_1}$. Thus, $z$ lies on $(xy)_{\\alpha_1}$. But then $\\beta$ again forms a half-bigon with $\\alpha_1$ whose only puncture is $r$, and so $\\beta$ is either homotopic to $\\alpha_2$ or to $\\beta_1$ as before. Similarly, if $\\beta$ is in minimal position with $\\alpha_2$ and intersects $\\alpha_2$ first as it travels from $p$ to $q$, then $\\beta$ is either homotopic to $\\alpha_1$ or to arc $\\beta_2$ in Figure \\ref{fig:case2nobigon} (right). \n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 520 0 0, clip]{halfbigonmaximal.eps}\n}\n\\captionof{figure}{}\n\\label{fig:halfbigonmaximal}\n\\end{figure}\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{case1nobigon.eps}\n}\n\\captionof{figure}{}\n\\label{fig:case1nobigon}\n\\end{figure}\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{case2nobigon.eps}\n}\n\\captionof{figure}{}\n\\label{fig:case2nobigon}\n\\end{figure}\n\n\n\\begin{lemma}\\label{rhomotopicbigon}\nLet $p,q,r$ be distinct punctures of an $n$-punctured sphere $S$, with $n \\geq 4$. Let $\\mathcal{A}_r$ be a maximal $2$-system of $r$-homotopic arcs on $S$ joining $p$ and $q$. If $\\mathcal{A}_r$ contains arcs $\\alpha_1, \\alpha_2 \\in \\mathcal{A}_r$ in minimal position that form a bigon, then $\\mathcal{A}_r$ is as in Figure \\ref{fig:bigon}, up to homotopy and relabeling $p$ and $q$.\n\\end{lemma}\n\n\\begin{proof}\nLet $H$ be the half-bigon adjacent to $p$ formed by the $\\alpha_i$. Let $x,y$ be the points of intersection of the $\\alpha_i$ in the order that $\\alpha_1$ traverses them as it travels from $p$ to $q$. Let $\\beta \\in \\mathcal{A}_r$, and assume $\\alpha_1, \\alpha_2, \\beta$ are in minimal position. \n\nIf $\\beta$ is disjoint from the $\\alpha_i$, then $\\beta$ is homotopic to the blue arc in Figure \\ref{fig:bigon}. Now suppose $\\beta$ is not disjoint from the $\\alpha_i$, and let $z_1, z_2, \\ldots$ be the points of intersection of $\\beta$ and the $\\alpha_i$ in the order that $\\beta$ traverses them as it travels from $p$ to $q$. We assume that $z_1$ lies on $\\alpha_1$. Note that, by Lemma \\ref{rhomotopic}, $\\beta$ forms a bigon (containing only the puncture $r$) with each of the $\\alpha_i$ that it intersects. \\\\\n\n\\item \\paragraph{\\textbf{Case 1: $z_1$ lies on the segment $(yq)_{\\alpha_1}$.}} In this case, $\\alpha_1$ and $\\beta$ form a half-bigon whose only puncture is $r$. As remarked above, this is impossible.\n\n\\item \\paragraph{\\textbf{Case 2: $z_1$ lies on the segment $(xy)_{\\alpha_1}$.}} In this case, $z_2$ does not lie on $(xy)_{\\alpha_2}$. Otherwise, since $\\alpha_2$ and $\\beta$ are in minimal position, they would form a half-bigon whose only puncture is $r$ (as in Case 1 of Lemma~\\ref{rhomotopic}), but this is impossible. Thus, $z_2$ lies on $(xy)_{\\alpha_1}$, and so $\\beta$ is homotopic to $\\alpha_2$ by Corollary~\\ref{uniquecontinuation}. \n\n\n\n\\item \\paragraph{\\textbf{Case 3: $z_1$ lies on the segment $(px)_{\\alpha_1}$.}} In this case, since $\\alpha_1, \\beta$ are in minimal position, $\\beta$ forms a half-bigon $H'$ with $\\alpha_1$ adjacent to $p$ and containing at least one of the punctures of $H$. \n\nObserve that $z_2$ cannot lie on the segment $(yq)_{\\alpha_2}$, since otherwise $\\alpha_2$ and $\\beta$ would form a half-bigon containing $r$. We also have that $z_2$ cannot lie on $(xy)_{\\alpha_1}$, since otherwise $\\alpha_1$ and $\\beta$ would not form a bigon. Furthermore, if $z_2$ lies on $(px)_{\\alpha_2}$, then so must $z_3$, since otherwise $\\alpha_1$ and $\\beta$ would not form a bigon. But if $z_2$ and $z_3$ both lie on $(px)_{\\alpha_2}$, then $\\alpha_2$ and $\\beta$ form a bigon that does not contain $r$, which is impossible. \n\nNow, if $H'$ contains all the punctures of $H$, then $z_2$ cannot lie on $(xy)_{\\alpha_2}$ since $\\alpha_2, \\beta$ are in minimal position, and if $z_2$ lies on $(yq)_{\\alpha_1}$ then $\\beta$ is homotopic to $\\alpha_2$ by Corollary~\\ref{uniquecontinuation}. Thus, we may assume that $H'$ contains some but not all of the punctures of $H$. \n\nUnder this assumption, $z_2$ cannot lie on $(yq)_{\\alpha_1}$, since otherwise $\\beta$ would be homotopic to the purple arc Figure~\\ref{fig:bigoncases} (left) by Corollary~\\ref{uniquecontinuation}, and so $\\beta$ would intersect $\\alpha_2$ thrice. For the same reason, $z_3$ cannot lie on $(xy)_{\\alpha_1}$ if $z_2$ lies on $(xy)_{\\alpha_2}$. The only case left to consider is that $z_2$ and $z_3$ both lie on $(xy)_{\\alpha_2}$. But then $\\beta$ is homotopic to the orange arc in Figure~\\ref{fig:bigoncases} (right) by Corollary~\\ref{uniquecontinuation}, and so $\\beta$ intersects $\\alpha_1$ thrice. \n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 520 0 0, clip]{bigon.eps}\n}\n\\captionof{figure}{}\n\\label{fig:bigon}\n\\end{figure}\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{bigoncases.eps}\n}\n\\captionof{figure}{}\n\\label{fig:bigoncases}\n\\end{figure}\n\n\n\\begin{lemma}\\label{extend}\nLet $p,q$ be distinct punctures of an $n$-punctured sphere $S$, with $n \\geq 4$. Let $\\alpha_1, \\alpha_2, \\beta$ be arcs on $S$ joining $p,q$ in one of the configurations shown in Figure~\\ref{fig:extend}. Then an arc $\\gamma$ joining $p,q$ that is in minimal position with $\\beta$ and intersects $\\beta$ at least thrice must intersect $\\alpha_1$ or $\\alpha_2$ at least thrice.\n\\end{lemma}\n\n\\begin{proof}\nSet $x_0= p$, and let $x_1, x_2, x_3$ be the first 3 points of intersection of $\\beta$ and $\\gamma$ in the order that $\\gamma$ traverses them as $\\gamma$ travels from $p$ to $q$. For $i=0,1,2$, let $\\beta_i = (x_ix_{i+1})_\\beta, \\gamma_i = (x_ix_{i+1})_\\gamma$, and let $R_i$ be the region not containing $p$ bounded by $\\beta_i, \\gamma_i$. If $\\gamma_0$ does not intersect the $\\alpha_i$, then $R_0$ contains no punctures, and so $\\beta, \\gamma$ are not in minimal position by Lemma~\\ref{bigoncriterion}, contradicting our assumption. Thus, $\\gamma_0$ has at least one point of intersection with the $\\alpha_i$. Similarly, $R_1, R_2$ must each contain at least one puncture of $S$, and so each of $\\gamma_1, \\gamma_2$ has at least 2 points of intersection with the $\\alpha_i$. Thus, $\\gamma$ has at least 3 points of intersection with $\\alpha_1$ or $\\alpha_2$. \n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 600 0 0, clip]{extend.eps}\n}\n\\captionof{figure}{}\n\\label{fig:extend}\n\\end{figure}\n\n\n\n\n\n\n\n \n\n\n\n\\section{1-System annular diagrams}\\label{diagrams}\n\nIn this section, we prove Theorem~\\ref{corner}, which will be useful in the inductive step of the proof of Lemma~\\ref{relation}.\n\n\\begin{lemma}\nLet $S$ be a twice-punctured sphere and $p,q$ its punctures. Let $\\mathcal{A}$ be a finite collection of simple arcs joining $p$ and $q$ and pairwise intersecting at most once. If there is a pair of intersecting arcs of $\\mathcal{A}$, then there is a pair of arcs of $\\mathcal{A}$ forming a half-bigon $H$ adjacent to $p$ such that no other arc of $\\mathcal{A}$ has its $p$-end in $H$. \n\\end{lemma}\n\n\\begin{proof}\nPick $\\alpha, \\beta \\in \\mathcal{A}$ such that $\\alpha$ and $\\beta$ intersect, and let $H$ be the half-bigon adjacent to $p$ formed by $\\alpha, \\beta$. If $H$ is as in the statement of the lemma, then we are done. Otherwise, there is an arc $\\beta' \\in \\mathcal{A}$ whose $p$-end lies in $H$. Since the $q$-end of $\\beta'$ is outside $H$, the arc $\\beta'$ must intersect one of $\\alpha, \\beta$, say $\\alpha$. We now repeat the above steps with arcs $\\alpha, \\beta'$. Since there are finitely many arcs in $\\mathcal{A}$, this process must terminate. \n\\end{proof}\n\nThe following corollary follows immediately.\n\n\\begin{corollary}\\label{cornsquare}\nLet $A$ be a $1$-system annular diagram, and $P$ a boundary path of $A$. If $A$ has at least one square, then $A$ has a cornsquare with outerpath on $P$.\n\\end{corollary}\n\nWe now proceed to the proof of Theorem~\\ref{corner}. \n\n\\begin{proof}[Proof of Theorem~\\ref{corner}]\nWe proceed by induction on the number of squares of $A$. If $A$ has no squares, then $A$ is a cycle. Now suppose $A$ has at least one square, that there is a boundary path $P$ of $A$ without a corner, and that the theorem holds for any annular square complex with fewer squares than $A$. Since $A$ contains a square, $A$ contains a cornsquare with outerpath on $P$ by Corollary~\\ref{cornsquare}. Thus, we may produce a corner on $P$ via a series of hexagon moves \\cite[Figure~3.17]{wise2012riches}. Note that a single hexagon move cannot produce two corners on $P$; otherwise there would be a dual curve beginning and terminating at $P$ (see Figure \\ref{fig:singlecorner}). \n\n\\begin{figure}\n\\scalebox{.3}{\n\\includegraphics[trim=0 580 0 0, clip]{singlecorner.eps}\n}\n\\captionof{figure}{If a single hexagon move produces two corners on $P$, then there is a dual curve beginning and ending at $P$.}\n\\label{fig:singlecorner}\n\\end{figure}\n\nWe perform hexagon moves until the first corner $v$ on $P$ is produced. Note that each neighbor of $v$ has degree at least 4. Indeed, since $P$ had no corners, we had to have performed at least one hexagon move to obtain $v$, but a neighbor of $v$ of degree 3 would correspond to a corner prior to performing that move, contradicting our assumption that $v$ is the first corner produced on $P$ (see Figure~\\ref{fig:vertexdegree}). Thus, by deleting $v$ as well as the two edges and the square incident to $v$, we obtain a 1-system annular diagram with one fewer square than $A$ and without any corners on one of its boundary paths, contradicting the induction hypothesis. \n\\end{proof}\n\n\\begin{figure}\n\\scalebox{.3}{\n\\includegraphics[trim=0 580 0 0, clip]{vertexdegree.eps}\n}\n\\captionof{figure}{If a corner $v$ produced by a hexagon move has a neighbor of degree 3, then that neighbor had to have been a corner prior to performing that move.}\n\\label{fig:vertexdegree}\n\\end{figure}\n\n\n\\begin{corollary}\\label{isolatedpuncture}\nLet $p,q$ be punctures of a punctured sphere $S$, and let $\\mathcal{A}$ be a 1-system of arcs joining $p$ and $q$ such that $|\\mathcal{A}| \\geq 2$ and the arcs of $\\mathcal{A}$ are pairwise in minimal position. There is a puncture $s$ of $S$ distinct from $p,q$ that is $p$-isolated by $\\mathcal{A}$. If $\\mathcal{A}$ contains a pair of intersecting arcs, then $s$ can be chosen so that the component $H$ of $S - \\bigcup \\mathcal{A}$ containing $s$ is a half-bigon.\n\\end{corollary}\n\n\\begin{proof}\nThe dual square complex $A$ to $\\mathcal{A}$ is an annular square complex as in Theorem~\\ref{corner}. Let $P$ be the boundary path corresponding to $p$. Note that $A$ has at least 2 vertices since $|\\mathcal{A}| \\geq 2$, and that $A$ has at least one square if and only if $\\mathcal{A}$ contains at least one pair of intersecting arcs. If $A$ is a cycle, then we may take $H$ to be the strip corresponding to any vertex of $A$. Otherwise, $A$ has a corner $v$ on $P$, and we may take $H$ to be the half-bigon corresponding to $v$. In either case, $H$ is punctured since the arcs of $\\mathcal{A}$ are pairwise non-homotopic and in minimal position.\n\\end{proof}\n\n\n\n\n\\section{Proof of Lemma~\\ref{relation}}\\label{proofoflemma}\n\nIn this section, we prove Lemma~\\ref{relation}, which essentially constitutes the inductive step in the proof of Theorem~\\ref{maintheorem}. We will need the following:\n\n\\begin{lemma}\\label{erdos}\\cite{erdos1946sets}\nA set of pairwise intersecting straight line segments between $\\ell$ points on a circle in $\\mathbb{R}^2$ has size at most $\\ell$. \n\\end{lemma}\n\n\n\n\\begin{proof}[Proof of Lemma~\\ref{relation}] We fix a complete hyperbolic metric on $S$ of area $2\\pi(n-2)$. We may assume that $\\mathcal{P}, \\mathcal{Q}$ are nonempty, and that the arcs of $\\mathcal{P} \\cup \\mathcal{Q}$ are pairwise in minimal position. We divide the proof into steps:\\\\\n\n\n\\item \\paragraph{\\textbf{Step 0. The arcs of $\\mathcal{P}$ (and hence the arcs of $\\mathcal{Q}$) are consecutive at $r$.}}\nIndeed, suppose $\\alpha, \\alpha' \\in \\mathcal{P}$ are distinct, and suppose there is an arc $\\beta \\in \\mathcal{Q}$ whose $r$-end lies in the strip or half-bigon $H$ bounded by $\\alpha, \\alpha'$ and adjacent to $r$. Since $\\beta$ does not intersect $\\alpha, \\alpha'$, the puncture $q$ must lie in $H$. Since no arc of $\\mathcal{Q}$ intersects $\\alpha, \\alpha'$, it follows that the $r$-end of every arc of $\\mathcal{Q}$ must also lie in $H$. \n\n\n\n\\item \\paragraph{We fix an orientation on $S$.} This induces a cyclic order $C$ of the arcs of $\\mathcal{P} \\cup \\mathcal{Q}$ around $r$. By Step 0, this order in turn induces a linear order $<$ on $\\mathcal{P}$, where the minimum and maximum arcs of $\\mathcal{P}$ are those with a successor or predecessor in $\\mathcal{Q}$ under $C$. \n\nWe proceed by induction. If $n=3$, then, up to homotopy, there is a unique arc joining $r$ to each of $p,q$, and the statement of the lemma holds. Now let $n \\geq 4$, and assume the lemma holds if $S$ has fewer punctures. If $\\mathcal{P}$ consists of a single arc then the lemma is trivially satisfied since $|\\mathcal{Q}| \\leq \\binom{n-1}{2}$ by Theorem~\\ref{przytycki}.  Thus, we may assume that $|\\mathcal{P}| \\geq 2$.\n\n\\item \\paragraph{\\textbf{Step 1. There is a puncture $s$ of $S$ distinct from $p,q,r$ that is $p$-isolated by $\\mathcal{P} \\cup \\mathcal{Q}$.}} Indeed, if the arcs of $\\mathcal{P}$ are pairwise disjoint, then since $|\\mathcal{P}| \\geq 2$, we have that $S - \\mathcal{P}$ consists of at least two punctured strips adjacent to $p$ and $r$, and so we may take $s$ to be a puncture of any such strip that does not contain $q$ (see Figure~\\ref{fig:step1}, left). Otherwise, by Corollary~\\ref{isolatedpuncture}, there is a puncture $s$ distinct from $p,r$ that is $p$-isolated by $\\mathcal{P}$ such that the component of $S - \\bigcup\\mathcal{P}$ containing $s$ is a half-bigon (see Figure~\\ref{fig:step1}, right). In this case, $s$ is necessarily distinct from $q$ since we are assuming $\\mathcal{Q}$ to be nonempty, and so there is at least one arc disjoint from the arcs of $\\mathcal{P}$ joining $q$ and $r$.\n\n\\begin{figure}\n\\scalebox{.4}{\n\\includegraphics[trim=0 520 0 0, clip]{step1.eps}\n}\n\\captionof{figure}{An illustration of Step 1.}\n\\label{fig:step1}\n\\end{figure}\n\n\\item \\paragraph{Let $\\bar{S}$ be the surface obtained from $S$ by forgetting the puncture $s$ (endowed with a complete, finite-area hyperbolic metric), and for each arc $\\alpha \\in \\mathcal{P} \\cup \\mathcal{Q}$, let $\\bar{\\alpha}$ be the corresponding arc on $\\bar{S}$.} Let $\\bar{\\mathcal{P}}, \\bar{\\mathcal{Q}}$ be the collection of all $\\bar{\\alpha}$ for $\\alpha \\in \\mathcal{P}, \\mathcal{Q}$, respectively. We tighten the arcs of $\\bar{\\mathcal{P}}\\cup \\bar{\\mathcal{Q}}$ to geodesics, thereby identifying arcs that correspond to $s$-homotopic arcs on $S$.\n\nThe orientation on $S$ induces an orientation on $\\bar{S}$. As above, this gives us a linear order $\\prec$ on $\\bar{\\mathcal{P}}$.\n\n\\item \\paragraph{\\textbf{Step 2. Two distinct arcs in $\\mathcal{Q}$ cannot be $s$-homotopic.}} Otherwise, they would form a strip adjacent to $q,r$ or a half-bigon adjacent to one of $q,r$ whose only puncture is $s$, which cannot happen since the component of $S - \\bigcup(\\mathcal{P}\\cup\\mathcal{Q})$ containing $s$ is adjacent to $p$. Thus, we may identify $\\bar{\\mathcal{Q}}$ with $\\mathcal{Q}$. \n\n\\item \\paragraph{\\textbf{Step 3. Arcs in $\\mathcal{P}$ that are $s$-homotopic must be consecutive at $r$.}} Indeed, suppose that $\\alpha, \\alpha' \\in \\mathcal{P}$ are $s$-homotopic. If $\\alpha, \\alpha'$ bound a strip whose only puncture is $s$, then the $r$-end of any arc $\\beta \\in \\mathcal{P}$ distinct from $\\alpha, \\alpha'$ cannot lie inside this strip, since otherwise $\\beta$ and one of $\\alpha, \\alpha'$ would necessarily form a half-bigon adjacent to $r$ whose only puncture is $s$, contradicting the fact that $s$ is $p$-isolated. \n\nOtherwise, $\\alpha$ and $\\alpha'$ form a half-bigon adjacent to $p$ whose only puncture is $s$, and a half-bigon adjacent to $r$ containing all punctures of $S$ except $p,r,s$. Thus, any arc $\\alpha''$ in $\\mathcal{P}$ distinct from $\\alpha, \\alpha'$ with $r$-end outside the latter half-bigon must be disjoint from $\\alpha,\\alpha'$, in which case $\\alpha, \\alpha', \\alpha''$ are in fact $s$-homotopic. \n\n\\item \\paragraph{\\textbf{Step 4. For any $\\alpha, \\alpha' \\in \\mathcal{P}$, if $\\alpha < \\alpha'$ then $\\bar{\\alpha} \\preceq \\bar{\\alpha'}$.}} Indeed, if $\\bar{\\alpha} \\succ \\bar{\\alpha'}$, then $\\alpha, \\alpha'$ form a half-bigon adjacent to $r$ whose only puncture is $s$. This cannot happen since $s$ is $p$-isolated.\n\n\n\\item \\paragraph{We now extend $\\mathcal{P}$ and $\\mathcal{R}$ as follows.} Suppose $\\mathcal{P}$ contains intersecting $s$-homotopic arcs $\\alpha, \\alpha'$. Then, as discussed in Step 3, an arc $\\alpha''$ disjoint from $\\alpha, \\alpha'$ joining $p$ and $r$ is $s$-homotopic to $\\alpha, \\alpha'$ and is between $\\alpha, \\alpha'$ with respect to the linear order on $\\mathcal{P}$. If $\\alpha''$ is not homotopic to an arc in $\\mathcal{P}$, then we add $\\alpha''$ to $\\mathcal{P}$ (otherwise, name the latter arc $\\alpha''$). If there is an arc $\\beta \\in \\mathcal{Q}$ such that $(\\alpha, \\beta), (\\alpha', \\beta) \\in \\mathcal{R}$ but $(\\alpha'', \\beta) \\notin \\mathcal{R}$, then we add the pair $(\\alpha'', \\beta)$ to $\\mathcal{R}$. Since $\\alpha''$ is disjoint from all arcs in $\\mathcal{P}\\cup \\mathcal{Q}$, we have not violated any of the conditions of the lemma. We do this for each pair of intersecting $s$-homotopic arcs $\\alpha, \\alpha'$.\n\nLet $\\bar{\\mathcal{R}}$ be the image of $\\mathcal{R}$ under the map $\\mathcal{P} \\times \\mathcal{Q} \\rightarrow \\bar{\\mathcal{P}} \\times \\mathcal{Q}, (\\alpha, \\beta) \\mapsto (\\bar{\\alpha}, \\beta)$.\nIt is clear that $\\bar{\\mathcal{R}}$ satisfies condition $(i)$, and it follows from Step 4 that $\\bar{\\mathcal{R}}$ satisfies $(ii)$ as well, so that $|\\bar{\\mathcal{R}}| \\leq \\binom{n-2}{2}$. It remains to show that $|\\mathcal{R}| - |\\bar{\\mathcal{R}}| \\leq n-2$. \n\nLet $\\mathcal{I}$ be the subset of $\\bar{\\mathcal{R}}$ consisting of elements with more than one pre-image under the map $\\mathcal{R} \\rightarrow \\bar{\\mathcal{R}}$. \n\n\\item \\paragraph{\\textbf{Step 5. If $(\\bar{\\alpha}, \\beta), (\\bar{\\alpha'}, \\beta') \\in \\mathcal{I}$ with $\\beta \\neq \\beta'$, then $\\beta, \\beta'$ are disjoint.}} Indeed, let $(\\alpha_1, \\beta), (\\alpha_2, \\beta)$ be pre-images of $(\\bar{\\alpha}, \\beta)$, and $(\\alpha_1', \\beta'), (\\alpha_2', \\beta')$ pre-images of $(\\bar{\\alpha'}, \\beta')$ with $\\alpha_1 \\neq \\alpha_2, \\alpha_1' \\neq \\alpha_2'$. Suppose that $\\beta$ and $\\beta'$ intersect. Note that we cannot have $\\{\\alpha_1, \\alpha_2\\} = \\{\\alpha_1', \\alpha_2'\\}$, since otherwise either $(\\alpha_1, \\beta), (\\alpha_2, \\beta')$ or $(\\alpha_1, \\beta'), (\\alpha_2, \\beta)$ would be two intersecting pairs of arcs in $\\mathcal{R}$ whose cyclic order around $r$ is not alternating, contradicting assumption $(ii)$. \n\nNow suppose $\\{\\alpha_1, \\alpha_2\\} \\cap \\{\\alpha_1', \\alpha_2'\\} \\neq \\emptyset$. Then there are 3 distinct arcs among $\\alpha_1, \\alpha_2, \\alpha_1', \\alpha_2'$ that are $s$-homotopic. Assume without loss of generality that these arcs are $\\alpha_1, \\alpha_2, \\alpha_1'$. Observe that $\\alpha_1'$ cannot intersect $\\alpha_i$ for $i=1,2$, since otherwise $(\\alpha_i, \\beta), (\\alpha_1', \\beta')$ would intersect more than once, contradicting assumption $(i)$. Thus, $\\alpha_1$ intersects $\\alpha_2$, but then $\\alpha_1'$ lies between $\\alpha_1$ and $\\alpha_2$ at $r$, and so there is an $i\\in \\{1,2\\}$ such that $(\\alpha_i, \\beta), (\\alpha_1', \\beta')$ contradict assumption $(ii)$.\n\nWe conclude that $\\alpha_1, \\alpha_2, \\alpha_1', \\alpha_2'$ are distinct; in particular, since at most 3 distinct arcs in $\\mathcal{P}$ can be $s$-homotopic, we must have $\\bar{\\alpha} \\neq \\bar{\\alpha'}$. Thus, by Step 3, the order of $\\alpha_1, \\alpha_2, \\alpha_1', \\alpha_2'$ at $r$ is neither alternating nor nested. By the latter, we mean that $\\alpha_1, \\alpha_2$ do not both lie between $\\alpha_1', \\alpha_2'$ in the linear order on $\\mathcal{P}$, and vice versa. Since each of the pairs of arcs $\\alpha_1, \\alpha_2$ and $\\alpha_1', \\alpha_2'$ bound a strip or half-bigon containing $s$, we must have that $\\alpha_i, \\alpha_j'$ intersect for some $i,j \\in \\{1,2\\}$, but then $(\\alpha_i, \\beta), (\\alpha_j', \\beta')$ intersect more than once, contradicting assumption $(i)$.\n\n\n\n\\item \\paragraph{\\textbf{Step 6. If $(\\bar{\\alpha}, \\beta), (\\bar{\\alpha'}, \\beta') \\in \\mathcal{I}$ with $\\bar{\\alpha} \\neq \\bar{\\alpha'}, \\beta \\neq \\beta'$, then the cyclic order of $(\\bar{\\alpha}, \\beta), (\\bar{\\alpha'}, \\beta')$ at $r$ is alternating.}} Indeed, suppose otherwise, and let $(\\alpha_1, \\beta), (\\alpha_2, \\beta)$ be pre-images of $(\\bar{\\alpha}, \\beta)$, and $(\\alpha_1', \\beta'), (\\alpha_2', \\beta')$ pre-images of $(\\bar{\\alpha'}, \\beta')$ with $\\alpha_1 \\neq \\alpha_2, \\alpha_1' \\neq \\alpha_2'$. Then, by Step 3, the order of $\\alpha_1, \\alpha_2, \\alpha_1', \\alpha_2'$ at $r$ is neither alternating nor nested. Thus, as in Step 5, $\\alpha_i, \\alpha_j'$ must intersect for some $i,j \\in \\{1,2\\}$, but then $(\\alpha_i, \\beta), (\\alpha_j', \\beta')$ are two intersecting pairs of arcs in $\\mathcal{R}$ whose cyclic order around $r$ is not alternating, contradicting assumption $(ii)$ (see Figure~\\ref{fig:nestedalternating}).\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 520 0 0, clip]{nestedalternating.eps}\n}\n\\captionof{figure}{An illustration of Step 6. Here, the pairs $(\\alpha_1, \\beta)$ and $(\\alpha_1', \\beta')$ contradict assumption $(ii)$.}\n\\label{fig:nestedalternating}\n\\end{figure}\n\n\\item \\paragraph{Let $\\mathcal{H}_q$ be the image of $\\mathcal{I}$ under the projection map $\\bar{\\mathcal{P}} \\times \\mathcal{Q} \\rightarrow \\mathcal{Q}$.} Let $\\mathcal{H}_s$ be the collection of all geodesic arcs $a$ joining $r$ and $s$ such that $a$ is contained in a strip bounded by a pair of distinct, disjoint $s$-homotopic arcs in $\\mathcal{P}$. \n\n Let $\\mathcal{H} = \\mathcal{H}_s \\cup \\mathcal{H}_q$, and let $\\mathcal{I}' \\subset \\mathcal{H}_s \\times \\mathcal{H}_q$ be the set of all $(a, \\beta) \\in \\mathcal{H}_s \\times \\mathcal{H}_q$ such that $(\\alpha, \\beta) \\in \\mathcal{I}$ for an arc $\\alpha$ bounding a strip corresponding to $a$. We extended $\\mathcal{R}$ so that the map $\\mathcal{R} \\rightarrow \\bar{\\mathcal{R}}$ is injective outside a set of cardinality $|\\mathcal{I}'|$. Thus, to complete the proof, it suffices to show that $|\\mathcal{I}'| \\leq n-2$. \n\n\\item \\paragraph{\\textbf{Step 7. The complement of $\\mathcal{H}$ consists of punctured strips and a single square, possibly with no punctures.}} Indeed, the arcs of $\\mathcal{H}_s$ are disjoint by construction \\cite[proof~of~Theorem~1.7]{przytycki2015arcs}, and the arcs of $\\mathcal{H}_q$ are disjoint by Step 5, so that the complement of each of $\\mathcal{H}_s, \\mathcal{H}_q$ consists of punctured strips. By Step 0, the arcs in $\\mathcal{H}_q$ (and hence the arcs in $\\mathcal{H}_s$) are consecutive at $r$. Thus, $\\mathcal{H}_s$ is contained in a single strip of $S - \\mathcal{H}_q$ and vice versa. Let $\\beta, \\beta'$ be the arcs in $\\mathcal{H}_q$ bounding the unique strip of $S - \\mathcal{H}_q$ containing $\\mathcal{H}_s$, and let $\\gamma, \\gamma'$ be the arcs in $\\mathcal{H}_s$ bounding the unique strip of $S - \\mathcal{H}_s$ containing $\\mathcal{H}_q$ (note that we do not exclude the possibility that $\\beta = \\beta'$ or $\\gamma = \\gamma'$). Then the complement of $\\mathcal{H}$ consists of the remaining strips of $S - \\mathcal{H}_q$, $S - \\mathcal{H}_s$ and a square bounded by $\\beta, \\beta', \\gamma, \\gamma'$. \n\n\\item \\paragraph{\\textbf{Step 8. $|\\mathcal{H}| \\leq n-1$.}} Indeed, $|\\mathcal{H}|$ is $2$ larger than the number of strips of $S - \\mathcal{H}$, so it suffices to show that there are at most $n-3$ of these strips. This is true by Step 7 since $S$ has area $2\\pi (n-2)$, and a punctured strip and a square each have area at least $2\\pi$. \n\n\\item \\paragraph{\\textbf{Step 9. $|\\mathcal{I}'| \\leq n-2$.}} To show this, we intersect $\\mathcal{H}$ with a small circle $C$ centered at $r$. Each element of $\\mathcal{I}'$ is determined by a pair of points of this intersection, and we connect them by a straight line segment. We also draw a line segment between the outermost points on $C$ corresponding to elements of $\\mathcal{H}_q$. By Step 6, these line segments are pairwise intersecting, so by Lemma~\\ref{erdos}, $|\\mathcal{I}'|+1 \\leq |\\mathcal{H}| \\leq n-1$.\n\\end{proof}\n\n\n\n\n\\section{Proof of Theorem~\\ref{maintheorem}}\n\n\\begin{figure}\n\\scalebox{.5}{\n\\includegraphics[trim=0 520 0 0, clip]{alternating.eps}\n}\n\\captionof{figure}{}\n\\label{fig:alternating}\n\\end{figure}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{maintheorem}]\nWe proceed by induction on $n$. The case $n=3$ is trivial. Now let $n \\geq 4$, and assume the theorem holds if $S$ has fewer punctures. Let $r$ be a puncture of $S$ distinct from $p,q$. Let $\\bar{S}$ be the $(n-1)$-punctured sphere obtained from $S$ by forgetting $r$. For each arc $\\alpha \\in \\mathcal{A}$, let $\\bar{\\alpha}$ be the corresponding arc on $\\bar{S}$, and let $\\bar{\\mathcal{A}} = \\{\\bar{\\alpha} \\> : \\> \\alpha \\in \\mathcal{A}\\}$. We tighten the arcs of $\\bar{\\mathcal{A}}$ to geodesics. Note that $\\bar{\\mathcal{A}}$ is a $2$-system on $\\bar{S}$, and so $|\\bar{\\mathcal{A}}| \\leq \\binom{n-1}{3}$ by the induction hypothesis. Thus, it is enough to show that $|\\mathcal{A}| - |\\bar{\\mathcal{A}}| \\leq \\binom{n-1}{2}$. To that end, we examine the extent to which the map $\\pi: \\mathcal{A} \\rightarrow \\bar{\\mathcal{A}}$, $\\alpha \\mapsto \\bar{\\alpha}$ is injective. \n\nBy Lemmas~\\ref{rhomotopic}, \\ref{rhomotopicbigon}, and \\ref{extend}, we may add arcs to $\\mathcal{A}$ so that for each $\\alpha \\in \\mathcal{A}$,\n\\[\n|\\pi^{-1}(\\bar{\\alpha})| -1 = |\\{ \\{\\alpha_1, \\alpha_2\\} \\in \\pi^{-1}(\\bar{\\alpha}) \\> : \\> \\alpha_1, \\alpha_2 \\text{ distinct  and disjoint} \\}|\n\\]\nLet $\\mathcal{P}$ (resp., $\\mathcal{Q}$) be the collection of all geodesic arcs $\\alpha$ on $S$ starting at $r$ and ending at $p$ (resp., ending at $q$) such that $\\alpha$ is contained entirely in a strip bounded by a pair of distinct, disjoint $r$-homotopic arcs in $\\mathcal{A}$. Let $\\mathcal{R} \\subset \\mathcal{P} \\times \\mathcal{Q}$ be the relation consisting of all pairs $(\\alpha, \\beta)$ such that both $\\alpha$ and $\\beta$ lie in a single such strip. We claim that $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ satisfy the conditions of Lemma~\\ref{relation}, so that $|\\mathcal{R}| \\leq \\binom{n-1}{2}$. Since $|\\mathcal{R}| = |\\mathcal{A}| - |\\bar{\\mathcal{A}}|$, this completes the proof.\n\nWe first note that for $(\\alpha, \\beta), (\\alpha', \\beta') \\in \\mathcal{R}$ corresponding to pairs of disjoint $r$-homotopic arcs $\\gamma_1, \\gamma_2 \\in \\mathcal{A}$ and $\\gamma_1', \\gamma_2' \\in \\mathcal{A}$, respectively, we have for some $i, j \\in \\{1,2\\}$ that $r$ produces at least one point of intersection between $\\gamma_i, \\gamma_j'$. Each point of intersection between the arcs $\\alpha, \\beta, \\alpha', \\beta'$ produces an additional point of intersection between $\\gamma_i, \\gamma_j'$. It follows that there is at most one point of intersection between any two pairs of arcs in $\\mathcal{R}$. \n\nWe now show that no arc in $\\mathcal{P}$ intersects an arc in $\\mathcal{Q}$. Indeed, suppose we have $(\\alpha, \\beta), (\\alpha', \\beta') \\in \\mathcal{R}$ such that $\\alpha$ intersects $\\beta'$. By the above, $\\alpha$ intersects $\\beta'$ exactly once and that is the only point of intersection between the pairs of arcs $(\\alpha, \\beta), (\\alpha', \\beta')$. But then we can find two arcs in $\\mathcal{A}$ that intersect thrice, as shown in Figure~\\ref{fig:alternating} (left).\n\nFinally, if there are two intersecting pairs of arcs in $\\mathcal{R}$ whose cyclic order around $r$ is not alternating, then we can also find two arcs in $\\mathcal{A}$ that intersect thrice, as shown in Figure~\\ref{fig:alternating} (right).\n\\end{proof}\n\n\n\n\n\\bibliographystyle{amsalpha}\n\n\\section*{This is an unnumbered first-level section head}\nThis is an example of an unnumbered first-level heading.\n\n\\specialsection*{THIS IS A SPECIAL SECTION HEAD}\nThis is an example of a special section head%\n\\footnote{Here is an example of a footnote. Notice that this footnote\ntext is running on so that it can stand as an example of how a footnote\nwith separate paragraphs should be written.\n\\par\nAnd here is the beginning of the second paragraph.}%\n.\n\n\\section{This is a numbered first-level section head}\nThis is an example of a numbered first-level heading.\n\n\\subsection{This is a numbered second-level section head}\nThis is an example of a numbered second-level heading.\n\n\\subsection*{This is an unnumbered second-level section head}\nThis is an example of an unnumbered second-level heading.\n\n\\subsubsection{This is a numbered third-level section head}\nThis is an example of a numbered third-level heading.\n\n\\subsubsection*{This is an unnumbered third-level section head}\nThis is an example of an unnumbered third-level heading.\n\n\\begin{lemma}\nLet $f, g\\in  A(X)$ and let $E$, $F$ be cozero\nsets in $X$.\n\\begin{enumerate}\n\\item If $f$ is $E$-regular and $F\\subseteq E$, then $f$ is $F$-regular.\n\n\\item If $f$ is $E$-regular and $F$-regular, then $f$ is $E\\cup\nF$-regular.\n\n\\item If $f(x)\\ge c>0$ for all $x\\in E$, then $f$ is $E$-regular.\n\n\\end{enumerate}\n\\end{lemma}\n\nThe following is an example of a proof.\n\n\\begin{proof} Set $j(\\nu)=\\max(I\\backslash a(\\nu))-1$. Then we have\n\\[\n\\sum_{i\\notin a(\\nu)}t_i\\sim t_{j(\\nu)+1}\n  =\\prod^{j(\\nu)}_{j=0}(t_{j+1}\/t_j).\n\\]\nHence we have\n\\begin{equation}\n\\begin{split}\n\\prod_\\nu\\biggl(\\sum_{i\\notin\n  a(\\nu)}t_i\\biggr)^{\\abs{a(\\nu-1)}-\\abs{a(\\nu)}}\n&\\sim\\prod_\\nu\\prod^{j(\\nu)}_{j=0}\n  (t_{j+1}\/t_j)^{\\abs{a(\\nu-1)}-\\abs{a(\\nu)}}\\\\\n&=\\prod_{j\\ge 0}(t_{j+1}\/t_j)^{\n  \\sum_{j(\\nu)\\ge j}(\\abs{a(\\nu-1)}-\\abs{a(\\nu)})}.\n\\end{split}\n\\end{equation}\nBy definition, we have $a(\\nu(j))\\supset c(j)$. Hence, $\\abs{c(j)}=n-j$\nimplies (5.4). If $c(j)\\notin a$, $a(\\nu(j))c(j)$ and hence\nwe have (5.5).\n\\end{proof}\n\n\\begin{quotation}\nThis is an example of an `extract'. The magnetization $M_0$ of the Ising\nmodel is related to the local state probability $P(a):M_0=P(1)-P(-1)$.\nThe equivalences are shown in Table~\\ref{eqtable}.\n\\end{quotation}\n\n\\begin{table}[ht]\n\\caption{}\\label{eqtable}\n\\renewcommand\\arraystretch{1.5}\n\\noindent\\[\n\\begin{array}{|c|c|c|}\n\\hline\n&{-\\infty}&{+\\infty}\\\\\n\\hline\n{f_+(x,k)}&e^{\\sqrt{-1}kx}+s_{12}(k)e^{-\\sqrt{-1}kx}&s_{11}(k)e^\n{\\sqrt{-1}kx}\\\\\n\\hline\n{f_-(x,k)}&s_{22}(k)e^{-\\sqrt{-1}kx}&e^{-\\sqrt{-1}kx}+s_{21}(k)e^{\\sqrt\n{-1}kx}\\\\\n\\hline\n\\end{array}\n\\]\n\\end{table}\n\n\\begin{definition}\nThis is an example of a `definition' element.\nFor $f\\in A(X)$, we define\n\\begin{equation}\n\\mathcal{Z} (f)=\\{E\\in Z[X]: \\text{$f$ is $E^c$-regular}\\}.\n\\end{equation}\n\\end{definition}\n\n\\begin{remark}\nThis is an example of a `remark' element.\nFor $f\\in A(X)$, we define\n\\begin{equation}\n\\mathcal{Z} (f)=\\{E\\in Z[X]: \\text{$f$ is $E^c$-regular}\\}.\n\\end{equation}\n\\end{remark}\n\n\\begin{example}\nThis is an example of an `example' element.\nFor $f\\in A(X)$, we define\n\\begin{equation}\n\\mathcal{Z} (f)=\\{E\\in Z[X]: \\text{$f$ is $E^c$-regular}\\}.\n\\end{equation}\n\\end{example}\n\n\\begin{xca}\nThis is an example of the \\texttt{xca} environment. This environment is\nused for exercises which occur within a section.\n\\end{xca}\n\nThe following is an example of a numbered list.\n\n\\begin{enumerate}\n\\item First item.\nIn the case where in $G$ there is a sequence of subgroups\n\\[\nG = G_0, G_1, G_2, \\dots, G_k = e\n\\]\nsuch that each is an invariant subgroup of $G_i$.\n\n\\item Second item.\nIts action on an arbitrary element $X = \\lambda^\\alpha X_\\alpha$ has the\nform\n\\begin{equation}\\label{eq:action}\n[e^\\alpha X_\\alpha, X] = e^\\alpha \\lambda^\\beta\n[X_\\alpha X_\\beta] = e^\\alpha c^\\gamma_{\\alpha \\beta}\n \\lambda^\\beta X_\\gamma,\n\\end{equation}\n\n\\begin{enumerate}\n\\item First subitem.\n\\[\n- 2\\psi_2(e) =  c_{\\alpha \\gamma}^\\delta c_{\\beta \\delta}^\\gamma\ne^\\alpha e^\\beta.\n\\]\n\n\\item Second subitem.\n\\begin{enumerate}\n\\item First subsubitem.\nIn the case where in $G$ there is a sequence of subgroups\n\\[\nG = G_0, G_1, G_2, \\ldots, G_k = e\n\\]\nsuch that each subgroup $G_{i+1}$ is an invariant subgroup of $G_i$ and\neach quotient group $G_{i+1}\/G_{i}$ is abelian, the group $G$ is called\n\\textit{solvable}.\n\n\\item Second subsubitem.\n\\end{enumerate}\n\\item Third subitem.\n\\end{enumerate}\n\\item Third item.\n\\end{enumerate}\n\nHere is an example of a cite. See \\cite{A}.\n\n\\begin{theorem}\nThis is an example of a theorem.\n\\end{theorem}\n\n\\begin{theorem}[Marcus Theorem]\nThis is an example of a theorem with a parenthetical note in the\nheading.\n\\end{theorem}\n\n\\begin{figure}[tb]\n\\includegraphics{lion.png}\n\\caption{This is an example of a figure caption with text.}\n\\label{firstfig}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\includegraphics{lion.png}\n\\caption{}\\label{otherfig}\n\\end{figure}\n\n\\section{Some more list types}\nThis is an example of a bulleted list.\n\n\\begin{itemize}\n\\item $\\mathcal{J}_g$ of dimension $3g-3$;\n\\item $\\mathcal{E}^2_g=\\{$Pryms of double covers of $C=\\openbox$ with\nnormalization of $C$ hyperelliptic of genus $g-1\\}$ of dimension $2g$;\n\\item $\\mathcal{E}^2_{1,g-1}=\\{$Pryms of double covers of\n$C=\\openbox^H_{P^1}$ with $H$ hyperelliptic of genus $g-2\\}$ of\ndimension $2g-1$;\n\\item $\\mathcal{P}^2_{t,g-t}$ for $2\\le t\\le g\/2=\\{$Pryms of double\ncovers of $C=\\openbox^{C'}_{C''}$ with $g(C')=t-1$ and $g(C'')=g-t-1\\}$\nof dimension $3g-4$.\n\\end{itemize}\n\nThis is an example of a `description' list.\n\n\\begin{description}\n\\item[Zero case] $\\rho(\\Phi) = \\{0\\}$.\n\n\\item[Rational case] $\\rho(\\Phi) \\ne \\{0\\}$ and $\\rho(\\Phi)$ is\ncontained in a line through $0$ with rational slope.\n\n\\item[Irrational case] $\\rho(\\Phi) \\ne \\{0\\}$ and $\\rho(\\Phi)$ is\ncontained in a line through $0$ with irrational slope.\n\\end{description}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\n\\myparagraph{Motivation} Question answering (QA)\naims to compute\ndirect answers to information needs posed as natural language (NL)\nutterances~\\cite{lu2019answering,sun2019pullnet,clark2018simple,diefenbach2019qanswer,kaiser2021reinforcement,jia21complex,saxena2020improving,he2021improving,saharoy2021question}.\nWe focus on the class of \\textit{factoid questions}\nthat are objective in\nnature and have one or more named entities as answers~\\cite{abujabal2019comqa,dubey2019lc,vakulenko2019message,qiu2020stepwise,christmann2019look,plepi2021context}. \nEarly research~\\cite{ravichandran2002learning,voorhees1999trec} used patterns to \nextract text passages with candidate answers, or had sophisticated\npipelines like the\nproprietary \\textsc{IBM Watson} system~\\cite{ferrucci2010building,ferrucci2012introduction}.\nthat won the Jeopardy! quiz show. \nWith the rise of large knowledge graphs (KGs)\nor knowledge bases (KBs)\nlike YAGO~\\cite{suchanek2007yago}, DBpedia~\\cite{auer2007dbpedia}, \nFreebase~\\cite{bollacker2008freebase}, and\nWikidata~\\cite{vrandevcic2014wikidata}, \nthe focus shifted from text corpora as inputs to these \nstructured RDF data sources,\nrepresented as subject-predicate-object (SPO) triples.\nWe refer to such answering of questions over knowledge graphs as \\textit{KG-QA}.\n\nWhile KGs capture a large\npart of the world's encyclopedic knowledge, they are inherently\nincomplete. This is because they cannot stay up-to-date with\nthe latest information, so that emerging and ephemeral facts \n(e.g., teams losing in semi-finals of sports leagues, or celebrities \ndating each other) are not included. \nAlso, user interests go way beyond the predicates that are modeled in KGs\nlike Wikidata.\nAs a result, answering over text from the open Web, like websites of newspapers or magazines,\nis an absolute necessity. \nWe refer to this paradigm as question answering over text,\n{\\em Text-QA} for short.\n\n\\myparagraph{Limitations of the state-of-the-art} This fragmented landscape\nof QA research has led to a state where methods from one paradigm are\ncompletely incompatible with the other.\nSystems operating over knowledge graphs\nare not equipped at all\nto compute answers from text passages. \nThe reason behind this is two-fold. \nFirst, QA over KGs assumes one of two major paradigms:\n(i) an explicit structured query (in a SPARQL-like language) is constructed \nthat, when executed over the KG, retrieves the answer~\\cite{tanon2018demoing,diefenbach2019qanswer,bhutani2019learning,shen2019multi,abujabal2018never}.\nSuch structured queries cannot be executed over a textual source;\n(ii) they perform approximate graph search without an explicit query,\nusing graph algorithms~\\cite{vakulenko2019message,christmann2019look},\ngraph embeddings~\\cite{saxena2020improving,huang2019knowledge},\nor graph convolutions~\\cite{sun2018open,sun2019pullnet}.\nThe basic idea here is to learn to traverse the graph in symbolic or latent space,\nand these cannot work over text inputs as there is no natural underlying graph representation.\n\nOn the other hand, powerful\ndeep learning methods for text-QA~\\cite{asai2020learning,zhao2021multi,clark2018simple,izacard2021leveraging,chen2017reading}\nare not geared at all\nfor tapping into KGs and other RDF data.\nThis is because these algorithms are basically\nclassifiers trained to predict the start and end position of the most\nlikely answer span in a given piece of text, a sequence-based setup that does not agree\nwith RDF triples.\n\nWorks on {\\em heterogeneous QA}~\\cite{savenkov2016knowledge,xu2016question,sun2018open,sun2019pullnet,oguz2021unikqa,sun2015open,das2017question}, \nincorporate text (typically articles in Wikipedia\nor search results) as a supplement to RDF triples, or\nthe other way around.\nHowever, usually KG and text are combined merely to\nimprove the rankings of candidate answers through additional evidence,\nbut the answers themselves come either from the KG alone or\nfrom text alone.\nMoreover, none of these methods is geared to handle the extreme, but\nrealistic situations, where the input is either only a KG or only a text corpus.\nState-of-the-art methods in heterogeneous QA,\n\\textsc{GRAFT-Net}\\xspace~\\cite{sun2018open} and \\textsc{PullNet}\\xspace~\\cite{sun2019pullnet},\nthat we compare against, suffer from shortcomings that involve\nan indispensable reliance on the KG source.\nSpecifically, they\nrequire trained KG entity embeddings,\nKG entity-linked text,\nshortest paths in KGs for creating training data,\nand can only provide such KG entities as answers. Thus, they cannot operate\nover arbitrary text corpora with an open vocabulary\nwhere ad-hoc phrases could be answers.\nA very recent method, \\textsc{UniK-QA}\\xspace~\\cite{oguz2021unikqa}, takes an approach of unifying sources via verbalization of every knowledge repository~\\cite{thorne2020neural}.\nWe show that our proposal outperforms \\textsc{UniK-QA}\\xspace, that, while being a commendable effort towards an unified view, still cannot deal with more complex information needs.\nSuch cases of complexity are a primary focus in this work, as discussed below.\n\nProblems are particularly exacerbated\nfor complex questions with multiple entities and predicates.\nA running example in this paper is:\n\\begin{Snugshade}\n\t\\textit{\\textbf{Question:} director of the western for which Leo won an Oscar? [\\textbf{Answer:} Alejandro I\\~{n}\\'{a}rritu]}\n\\end{Snugshade}\nHere, SPARQL queries become particularly brittle\nfor KG-QA systems,\ngraph search becomes more complicated,\nor text evidence for Text-QA systems has to \nbe joined from more than one document. \nFor example, the correct query for our running example\nwould be\nsomething like:\n\\begin{Snugshade}\n\\struct{SELECT ?x WHERE\n\\{?y director ?x .\n?y castMember LeonardoDiCaprio .\\\\\n?y genre western .\nLeonardoDiCaprio awarded AcademyAward [forWork ?y] .\\}}\n\\end{Snugshade}\n\nHardly any KG-QA system would get this mapping onto\nKG predicates and entities perfectly right\\footnote{Here, the triple pattern involving \\texttt{[forWork ?y]} shows a simplified representation for matching a fact with qualifiers~\\cite{christmann2022beyond}. In practice this would require matching over a reified fact~\\cite{hernandez2015reifying}.}.\nEven though graph-search methods do not formulate explicit queries, \nthey still need to get the mapping onto KB entities and predicates right.\nFor Text-QA, computing answers requires stitching together\ninformation from several texts, as it is not easy\nto find a single Web page that\ncontains all relevant cues in\nthe same zone.\n\n\\begin{figure} [t]\n\t\\centering\n\t\\begin{subfigure}[h]{0.49\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/xg-kg.pdf}\n\t\t\\caption{$XG(q)$ example for KG as input.}\n\t\t\\label{fig:xg-kg}\n\t\\end{subfigure}\n\\hfill\t\n\n\n\t\\begin{subfigure}[h]{0.49\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/xg-text.pdf}\n\t\t\\caption{$XG(q)$ example for text as input.}\n\t\t\\label{fig:xg-text}\n\t\\end{subfigure}\n\t\\\\\n\t\\begin{subfigure}[h]{\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/xg-kg-text.pdf}\n\t\t\\caption{$XG(q)$ example for KG and text as input.}\n\t\t\\label{fig:xg-hetero}\n\t\\end{subfigure}\n\t\\caption{Context graphs (XG) built by \\textsc{Uniqorn}\\xspace in each of the answering setups for\n\t\tthe question $q=$\n\t\t\\utterance{director of the western for which Leo won an Oscar?}\n\tAnchors are nodes with (partly) underlined labels; answers are in\nbold. Orange subgraphs are Group Steiner Trees.}\n\\label{fig:xg}\n\\vspace*{-0.5cm}\n\\end{figure}\n\n\n\\myparagraph{Approach} To overcome these limitations, we propose \\textsc{Uniqorn}\\xspace, \na \\underline{Uni}fied framework\nfor \\underline{q}uestion answering \\underline{o}ver\n\\underline{R}DF knowledge graphs and \\underline{N}atural language text,\nthat addresses these limitations.\nOur proposal hinges on two key ideas: \n\\squishlist\n\\item Instead of attempting to compute perfect translations of questions\ninto crisp SPARQL queries (which seems elusive for complex questions), \nwe relax the search strategy over KGs by devising graph algorithms\nfor Group Steiner Trees~\\cite{ding2007finding,sun2021finding}, this way connecting question-relevant cues\nfrom which candidate answers can be extracted and ranked.\n\\item Handling KG-QA, Text-QA and heterogeneous setups with the same unified method,\nby building a noisy KG-like context graph from KG or text inputs on-the-fly\nfor each question,\nusing named entity disambiguation (NED), open information extraction (Open IE) and BERT fine-tuning.\n\\squishend\n\nIn a nutshell, \\textsc{Uniqorn}\\xspace works as follows. \nGiven an input question, we first retrieve question-relevant \\textit{evidences} from one or more knowledge sources\nusing a fine-tuned BERT model for sequence pair classification.\nFrom these evidences, that are either KG facts or text fragments, \\textsc{Uniqorn}\\xspace\nconstructs a {\\em context graph (XG)} that contains\nquestion-specific\nentities, predicates, types and candidate answers.\nDepending upon the input source, this XG thus either consists of:\n(i) KG facts\ndefining the neighborhood of the question entities,\nor, (ii) a quasi-KG\ndynamically built by joining Open IE triples extracted from\ntext\nsnippets,\nor, (iii) both, in case of retrieval from heterogeneous sources.\nTriples in the XG originate from evidences that are deemed question-relevant by a fine-tuned BERT model.\nWe identify {\\em anchor nodes} in the XG \nthat match\nphrases in \nthe question.\nTreating the anchors as terminals, Group Steiner Trees \n(GST) are computed \nthat contain candidate answers. \nThese GSTs establish a joint context for \ndisambiguating mentions of\nentities, predicates and types\nin the question.\nCandidate answers are ranked by simple statistical measures rewarding redundancy within the top-$k$\nGSTs. Fig.~\\ref{fig:xg}\nillustrates this unified approach for the settings of KG-QA, Text-QA and heterogeneous-QA.\n\\textsc{Uniqorn}\\xspace thus belongs to the family of methods that locate an answer using graph search and traversal,\nand does not build an explicit structured query~\\cite{saharoy2021question}.\n\n\\myparagraph{Contributions} This work makes the following salient \ncontributions:\n(i) proposing the first unified method\nfor answering\ncomplex\nfactoid questions over knowledge graphs, or text corpora, or both; \n(ii) applying Group Steiner Trees as\na strategy for locating answers to complex questions involving\nmultiple entities and relations;\n(iii) experimentally comparing \\textsc{Uniqorn}\\xspace on\nseveral benchmarks of complex questions against state-of-the-art baselines on KGs, text\nand heterogeneous sources.\nAll code, data, and results for this project are available at:\n\\url{https:\/\/uniqorn.mpi-inf.mpg.de}.\n\n\\myparagraph{Improvements over \\textsc{Quest}\\xspace} \\textsc{Uniqorn}\\xspace is an extension of \\textsc{Quest}\\xspace~\\cite{lu2019answering}, that used GSTs for question answering over quasi KGs induced from text, and was published in SIGIR 2019 as a long paper. \\textsc{Uniqorn}\\xspace contains substantial improvements over \\textsc{Quest}\\xspace, the key factors being the following:\n(i) Unlike \\textsc{Quest}\\xspace that was limited to text inputs, \\textsc{Uniqorn}\\xspace works over KGs, text, and heterogeneous sources;\n(ii) \\textsc{Uniqorn}\\xspace incorporates a novel BERT fine-tuning model for assessing relevance of heterogeneous evidences with respect to the question;\n(iii) \\textsc{Quest}\\xspace was evaluated on a small dataset of $300$ questions. \\textsc{Uniqorn}\\xspace, on the other hand, uses LC-QuAD 2.0~\\cite{dubey2019lc}, a large and recent benchmark for complex KG-QA with about $5000$ test questions (the text corpus counterpart was completely curated by us), along with five other smaller datasets;\nand\n(iv) \\textsc{Uniqorn}\\xspace, over the three setups, includes ten baselines, whereas \\textsc{Quest}\\xspace used just three of them.\n\n\\section{Experimental Results}\n\\label{sec:exp-res}\n\n\\subsection{Key Findings}\n\\label{subsec:keyres}\n\n\\begin{table}[t]\n\t\\centering\n\n\t\\setlength{\\tabcolsep}{0.5em}\n\t\\begin{tabular}{l c c c}\t\\toprule\n\t\n       \n        \\textbf{Method}        \t\t\t\t\t& \\textbf{KG}           & \\textbf{Text}         & \\textbf{KG+Text}      \\\\ \\midrule\n\t\t\\textsc{Uniqorn}\\xspace                                & $\\boldsymbol{0.315}$* & $0.110$               & $\\boldsymbol{0.331}$* \\\\ \\midrule\n\t\t\\textsc{QAnswer}\\xspace~\\cite{diefenbach2020towards}   & $0.308$               & -                     & -                     \\\\\n\t\t\\textsc{Platypus}\\xspace~\\cite{tanon2018demoing}       & $0.036$               & -                     & -                     \\\\ \\midrule\n\t\t\\textsc{PathRetriever}\\xspace~\\cite{asai2020learning}  & -                     & $\\boldsymbol{0.230}$  & -                     \\\\\n\t\t\\textsc{DocumentQA}\\xspace~\\cite{clark2018simple}           & -                     & $0.162$               & -                     \\\\\n\t\t\\textsc{DrQA}\\xspace~\\cite{chen2017reading}            & -                     & $0.112$               & -                     \\\\ \\midrule\n\t\t\\textsc{UniK-QA}\\xspace~\\cite{oguz2021unikqa}           & $0.005$               & $\\boldsymbol{0.240}$  & $0.107$               \\\\ \n\t\t\\textsc{PullNet}\\xspace~\\cite{sun2019pullnet}          & $0.011$               & -                     & $0.010$               \\\\\n\t\t\\textsc{GRAFT-Net}\\xspace~\\cite{sun2018open}            & $0.129$               & -                     & $0.175$               \\\\ \\midrule\n\t\t\\textsc{BFS}\\xspace~\\cite{kasneci2009star}             & $0.179$               & $0.046$               & $0.116$               \\\\\n\t\t\\textsc{ShortestPaths}\\xspace                          \t    & $0.131$               & $0.074$               & $0.110$               \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Comparison of \\textsc{Uniqorn}\\xspace and baselines over the LC-QuAD $2.0$ test set, as measured by P@1 performance. \n\tThe best value per column is in \\textbf{bold}. An asterisk (*) indicates \n\tstatistical significance of \\textsc{Uniqorn}\\xspace over the best baseline in that column.\n\tA hyphen (`-') indicates that the corresponding baseline cannot be applied to that particular setting. An `x' denotes the particular\n\tmetric cannot be computed.}\n\t\\label{tab:main-res}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\begin{table}[t]\n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l ccccc c ccccc c ccccc}\t\\toprule\n\t\t \\multirow{2}{*}{\\textbf{Method}} \n\t\t & \\multicolumn{5}{c}{\\textbf{KG (P@1)}}   \n\t\t && \\multicolumn{5}{c}{\\textbf{Text (P@1)}} \n\t\t && \\multicolumn{5}{c}{\\textbf{KG+Text (P@1)}} \\\\ \n\t\t \\cmidrule{2-6}\\cmidrule{8-12}\\cmidrule{14-18}\n\t\t\t& \\textbf{\\begin{tabular}[c]{@{}c@{}}LCQ1\\end{tabular}}     & \\textbf{ComQA}     & \\textbf{CQ-W}     & \\textbf{CQ-T}     & \\textbf{QALD}       \n\t\t\t&& \\textbf{\\begin{tabular}[c]{@{}c@{}}LCQ1\\end{tabular}}     & \\textbf{ComQA}     & \\textbf{CQ-W}     & \\textbf{CQ-T}     & \\textbf{QALD}      \n\t\t\t&& \\textbf{\\begin{tabular}[c]{@{}c@{}}LCQ1\\end{tabular}}     & \\textbf{ComQA}     & \\textbf{CQ-W}     & \\textbf{CQ-T}     & \\textbf{QALD} \\\\ \\bottomrule\n\t\t\t\\textsc{Uniqorn}\\xspace       & $0.171$    &   $0.099$\t    &    $0.187$    &  $0.107$     &\t$0.114$              && $0.213$ &\t$0.183$ &\t$0.213$ &\t$0.360$ &\t$0.271$  && $0.267$      & $0.208$    & $0.313$        & $0.380$  & $0.171$      \\\\ \\midrule\n\t\t\tQAnswer       & $0.164$\t& $0.238$\t& $0.293$ & $0.147$ & \t$0.114$ && -                      & -                       & -                      & -                      & -                      && -                         & -              & -                  & -            & -               \\\\ \n\t\t\tPlatypus      & $0.021$ & $0.010$ &\t$0.007$\t & $0.007$\t& $0.100$           && -                      & -                       & -                      & -                      & -                      && -                         & -              & -                  & -            & -               \\\\ \\midrule\n\t\t\tPath Retriever         & -                 & -                       & -                      & -                     & -                    && $0.373$ &\t$0.594$ &\t$0.560$ &\t$0.367$ &\t$0.357$             && -                         & -              & -                  & -            & -               \\\\\n\t\t\tDocQA         & -                 & -                       & -                      & -                     & -                    && $0.314$ & $0.609$\t& $0.667$\t& $0.573$\t& $0.229$          && -                         & -              & -                  & -            & -               \\\\ \n\t\t\tDrQA          & -                 & -                       & -                      & -                     & -                    && $0.182$ &\t$0.193$ & $0.300$\t& $0.430$ & $0.057$           && -                         & -              & -                  & -            & -               \\\\ \\midrule\n\t\t\tPullNet       &$0.016$ & $0.040$               & $0.000$                & $0.000$               & $0.000$              && -                      & -                       & -                      & -                      & -                      && $0.040$                   & $0.101$        & $0.014$            & $0.020$      & $0.006$              \\\\ \n\t\t\tGRAFT-Net     & $0.053$           & $0.045$                 & $0.033$                & $0.007$               & $0.000$              && -                      & -                       & -                      & -                      & -                      && $0.116$                   & $0.104$        & $0.087$            & $0.031$      & $0.022$              \\\\\n\t\t\tUniK-QA     & $0.042$ & $0.045$ & $0.013$ & $0.000$ & $0.029$           && $0.207$ & $0.267$ & $0.140$ & $0.087$ & $0.171$       && $0.145$ & $0.149$ & $0.047$ & $0.047$ & $0.086$              \\\\\n\t\t\t\\midrule\n\t\tBFS           &     $0.117$     &       $0.055$     &\t$0.107$     &\t$0.080$     &\t$0.014$              && $0.096$ &\t$0.129$ &\t$0.147$ &\t$0.187$ &\t$0.100$                && $0.118$                   & $0.134$        & $0.083$            & $0.079$      & $0.000$              \\\\ \n\t\t\tSP            &     $0.077$      &   $0.074$  &     $0.133$  &     $0.047$   &   $0.057$   && $0.149$ & $0.243$ &\t$0.193$ &\t$0.273$ & $0.129$                && $0.208$                   & $0.266$        & $0.173$            & $0.291$      & $0.104$              \\\\ \\bottomrule\n\n\t\t\n\t\t\\end{tabular}\n\t}\n\t\\caption{Comparison of \\textsc{Uniqorn}\\xspace and baselines for LC-QuAD 1.0, ComQA, CQ-W, CQ-T, and QALD datasets.\n\t\tThe best value per column is in \\textbf{bold}. An asterisk (*) indicates \n\tstatistical significance of \\textsc{Uniqorn}\\xspace over the best baseline in that column.\n\t\t`-' indicates that the corresponding baseline cannot be applied to this setting.}\n\t\\label{tab:other-res}\n\t\\vspace*{-0.9cm}\n\\end{table}\n\nOur main results are presented in Tables~\\ref{tab:main-res} and~\\ref{tab:other-res}.\nThe following discusses our key findings.\nAll tests of statistical significance hereafter correspond to the two-tailed paired $t$-test,\nwith the $p$-value $\\leq 0.05$.\n\n\\textbf{(i) \\textsc{Uniqorn}\\xspace yields competitive results for most of the settings\nconsidered:} This performance is observed across-the-board,\nby knowledge source, benchmark, baseline (including neural models\nlike PullNet with large-scale training), or metric.\nAchieving this for both\nsettings is a success, given that the competitors\nfor most paradigms do not have any entries in the others\n(as observed by the numerous hyphens in the table).\nNote that PullNet and GRAFT-Net cannot deal with unseen KG entities at test time,\nwhich is very often the case for benchmarks.\nBut to be fair to these strong baselines, we allowed them to have\naccess to questions in the test set during training, so that appropriate\nentity embeddings to be learnt. Additionally, we have only considered questions \nwhere their final classifier receives correct input (i.e. with ground truth in the entity set) \nfrom the previous classification steps.\nIf this is not allowed, their\nperformance would drop drastically: MRRs of $0.062$\nand $0.049$ for PullNet, the better method of the two, for KG+Text and KG, respectively.\nTable~\\ref{tab:other-res} test open-domain performance: ability\nto answer questions on benchmarks (here, the multiple smaller ones) that are different from\nthe one trained on (the largest one).\n\nUnlike many other systems, GRAFT-Net and PullNet can indeed take into\naccount information in qualifiers (these were originally designed for\nFreebase CVTs, we used the analogous ideas when reimplementing for\nWikidata qualifiers).\nWe attribute the relatively poor performance\nof the complex QA system PullNet\nin some scenarios to its tackling of certain classes of complex questions\nonly: namely, the multi-hop ones\n(\\utterance{brother of director of Inception?}).\nHowever, there are also other common classes\n(\\utterance{coach who managed both liverpool and barcelona?}).\n\\textsc{Uniqorn}\\xspace's ability to handle arbitrary types of questions with respect\nto multiple entities and relations contributes this improved numbers.\nThis is a reason why GRAFT-Net systematically shows a better \nperformance than PullNet, even though the latter is an extension of\nthe former.\n\n\\textbf{(ii) Computing GSTs on a question-specific context graph is\na preferable alternative to SPARQL-based systems over RDF graphs:}\nThis is seen with the Hit@5 of $0.414$ for \\textsc{Uniqorn}\\xspace versus $0.318$\nfor QAnswer and $0.109$ for Platypus, the two SPARQL-based systems).\nLet us examine this in more detail.\nFor KGs, QAnswer is clearly the stronger baseline.\nIt operates by mapping names and phrases in the question to KG concepts,\nenumerating all possible SPARQL queries connecting these concepts\nthat return a non-empty answer, followed by a query ranking phase.\nThis approach works really well for simple questions with\nrelatively few keywords. However, the query ranking phase becomes\na bottleneck\nfor complex questions with multiple entities and relations, resulting\nin too many possibilities. This is the reason why \nreliance on the best SPARQL query may be a bad choice.\nAdditionally, unlike \\textsc{Uniqorn}\\xspace, QAnswer cannot leverage\nany qualifier information in Wikidata.\nThis is both due to the complexity of the SPARQL query necessary\nto tap into such records,\nas well as an explosion in the number of query candidates\nif qualifier attributes are allowed into the picture.\nOur GST establishes this common question context \nby a joint disambiguation of all question phrases, and smart answer\nranking helps to cope with the noise and irrelevant information in the XG.\nWhile QAnswer is completely syntax-agnostic, Platypus is the other\nextreme: it relies on accurate dependency parses of the NL question\nand hand-crafted question templates to come up with the best logical\nform. This performs impressively when questions fit the supported\nset of syntactic patterns, but is brittle when exposed to\na multitude of open formulations from an ensemble of QA benchmarks.\nGRAFT-Net performing better than QAnswer also attests to the superiority\nof graph-based search over SPARQL for complex questions.\n\n\\textbf{(iii) \\textsc{Uniqorn}\\xspace's results are better than the MRC systems:}\nIt is \nremarkable that \\textsc{Uniqorn}\\xspace \nsignificantly outperforms the reading comprehension systems DocQA and DrQA.\nBoth systems use sophisticated neural networks with extensive\ntraining and leverage large resources like Wikipedia, SQuAD or\nTriviaQA, whereas \\textsc{Uniqorn}\\xspace is completely unsupervised and could be\nreadily applied to other benchmarks with no further tuning. \n\\textsc{Uniqorn}\\xspace's success comes from its unique ability to stitch\nevidence from multiple documents to faithfully answer a question.\nThe Text-QA baselines can also handle multiple passages, but the\nassumption is still that a single passage or sentence contains\nthe correct answer and\n\\textit{all} question conditions. This is flawed: as an aftermath,\nthe accuracy of these systems is often gained by only satisfying part\nof the question (for \\utterance{Who directed Braveheart and Paparazzi?},\nonly spotting the director of Braveheart suffices).\nAlso, DocQA and DrQA are not able to address such levels of complexity\nas demanded by our challenging benchmarks.\n\n\\textbf{(iv) Graph-based methods open up the avenue for this unified\nanswering process:}\nWhat is especially noteworthy is that\nusing the the joint optimization of interconnections between\nanchors via GSTs is essential and powerful.\nUse of GSTs is indeed required for best performance, and cannot be easily approximated\nwith simpler graph methods like BFS and SP.\nIt is worthwhile to note that in the open-domain experiments (Table~\\ref{tab:other-res}),\nat least for the text setting, BFS performs notably well (ComQA,\nCQ-W, CQ-T, QALD).\nBFS tree search is actually a building block for many Steiner Tree\napproximation algorithms~\\cite{kasneci2009star}.\nShortestPaths (SP) also performs respectably in these settings, \nas notions of shortest paths are also implicit in GSTs.\nThe brittleness of these ad hoc techiques are exposed in\nthe larger and more challenging settings of KG and KG+Text.\n\nWe show representative examples from the various benchmarks in Table~\\ref{tab:anecdotes},\nto give readers a feel of the complexity of information needs that \\textsc{Uniqorn}\\xspace\ncan handle.\n\n\\begin{table} [t]\n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{p{7cm} p{7cm} p{7cm}} \n\t\t\t\\toprule\n\t\t\t\\textbf{KG}\t\t\t\t\t\t\t\t\t\t\t\t  & \\textbf{Text}\t& \\textbf{KG+Text}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\ \\toprule\n\t\t\t\\utterance{Which charitable organization runs Wikipedia ?} (LC-QuAD 2.0) \t                                                                                        & \\utterance{Apollo 13 was nominated for and won which awards?} (LC-QuAD 2.0)  & \\utterance{Who is the master of Leonardo da Vinci, who was employed as a sculptor?}  (LC-QuAD 2.0)   \\\\\t\\midrule\n\t\t\t\n\t\t\t\\utterance{Which research library archives the works of James Joyce?} (LC-QuAD 2.0)\t                                                                        & \\utterance{Who is the singer in the series of Bone Palace Ballet?} (LC-QuAD 2.0)\t                  & \\utterance{What is the name of the reservoir the Grand Coulee Dam made?}\t(LC-QuAD 2.0)\t\\\\\t\\midrule\n\t\t\t\n\t\t\t\\utterance{What church building was constructed by Dutch East India Company?} (LC-QuAD 2.0)\t& \\utterance{In what mountain range is Mt. Anhui located?} (LC-QuAD 2.0)      & \\utterance{What Empire used to have Istanbul as its capital?} (LC-QuAD 2.0) \\\\\t\\midrule\n\t\t\t\n\t\t\t\\utterance{At what university did Jacques Barzun get the doctorate of philosophy?} (LC-QuAD 2.0)  \t                                                                                & \\utterance{What is the location of the formation that is named after Lynyrd Skynyrd?}  (LC-QuAD 2.0)                                     & \\utterance{What is the jurisdiction of the area of East Nusa Tenggara Province?} \t(LC-QuAD 2.0)\t\t\t\t\t\t\t\\\\ \\midrule\n\t\t\t\n\t\t\t\\utterance{What award did David Geffen receive in the year 1990?}  (LC-QuAD 2.0)\t\t                                                                        & \\utterance{What are the video game console model which letter starts with x?} (LC-QuAD 2.0)               & \\utterance{Tell me which logographic writing system is used by the Japanese.}\t(LC-QuAD 2.0)\t\t\t\t\t\t\\\\ \n\t\t\t\\bottomrule\n\t\t\\end{tabular} }\n\t\\caption{Anecdotal examples from all the datasets where \\textsc{Uniqorn}\\xspace was able to compute a correct answer at the top rank (P@1 $= 1$), but none of the baselines could.}\n\t\\label{tab:anecdotes}\n\t\\vspace*{-1cm} \n\\end{table}\n\n\\subsection{Insights and Analysis} \n\\label{subsec:analysis}\n\n\\begin{table} [t]\n\t\\centering\n\t\\setlength{\\tabcolsep}{0.5em}\n\t\\begin{tabular}{l c c c} \\toprule\n\t\t\t\\textbf{Configuration}\t& \\textbf{KG}\t\t\t& \\textbf{Text}\t\t\t& \\textbf{KG+Text}\t\\\\ \\toprule\n\t\t\tFull\t\t        & $0.266$\t                        &  $0.108$\t        &   $\\boldsymbol{0.292}$\t\\\\ \\midrule\n\t\t\tNo types \t\t& $\\boldsymbol{0.279}$                       &  $\\boldsymbol{0.114}$                      &   $0.289$\t\t        \\\\\n\t\t\tNo entity alignment \t&     -                         &  $0.101$                     &   $0.289$ \t\t        \\\\\t\n\t\t\tNo predicate alignment  &     -                         &  $0.097$                      &   $0.240$ \t\t        \\\\\n\t\t\tNo type alignment  &     -                         &  $0.110$                      &   $0.286$ \t\t        \\\\  \\bottomrule \n\t\\end{tabular}\n\t\\caption{Pipeline ablation results on the full LC-QuAD 2.0 dev set with P@1. \n\tBest values per column are in \\textbf{bold}.\n\tStatistically significant differences from the full configuration are marked with *.}\n\t\\label{tab:ablation}\n \t\\vspace*{-0.7cm}\n\\end{table}\n\n\\textbf{Ablation experiments.} To get a proper\nunderstanding of \\textsc{Uniqorn}\\xspace's successes and failures, it is important to\nsystematically ablate its configurational pipeline\n(Table~\\ref{tab:ablation}). We note the following:\n\\squishlist\n\\item \\textbf{[Row 1]} The full pipeline of \\textsc{Uniqorn}\\xspace achieves the best performance, showing that all cogs\nin its wheel play a vital role, as detailed below.\n\\item \\textbf{[Row 2]} ``No NED expansion'' implies directly using entities from NED for KG-lookups.\nWe find that\nexpanding lookups for XG construction with NER + lexicons\ncompensates for NED errors substantially;\n\\item \\textbf{[Row 3]} Using entity types is useful, as seen by the significant\ndrops of $0.327$ to $0.288$\nfor KG+Text and $0.279$ to $0.234$ for text;\n\\item \\textbf{[Rows 4 and 5]} On-the-fly entity and predicate alignments are crucial for addressing\nvariability in surface forms for text-QA (also, as a result, for KG+Text).\n\\squishend\n\n\\begin{table} [t]\n\t\\centering\n\n\t\\setlength{\\tabcolsep}{0.5em}\n\t\\begin{tabular}{l c c c}\n\t\t\\toprule\n\t\t\\textbf{Answer ranking}\t\t& \\textbf{KG}\t\t\t& \\textbf{Text}\t\t\t& \\textbf{KG+Text}\t\\\\ \\toprule\n \t\tGST counts\t\t\t\t\t&   $\\boldsymbol{0.266}$        &   $\\boldsymbol{0.108}$        &   $\\boldsymbol{0.292}$        \\\\ \n\t\tGST costs \t\t\t\t\t&   $0.266$                     &   $0.094$        &   $0.282$        \\\\\n\t\tGST node weights \t\t\t&   $0.264$                       &   $0.098$        &   $0.284$        \\\\\t\t\t\n\t\tAnchor distances        \t&   $0.250$                       &   $0.082$                     &   $0.225$\t \\\\    \n\t\tWeighted anchor distances\t&   $0.260$                     &   $0.082$                     &   $0.227$        \\\\ \\bottomrule\n\t\\end{tabular}\n\n\t\\caption{Different answer ranking results on the LC-QuAD 2.0 dev set on P@1. \n\t\tBest values per column are in \\textbf{bold}.\n\t\tStatistically significant differences from the first row are marked with *.}\n\t\\label{tab:ranking}\n\t\\vspace*{-0.9cm}\n\\end{table}\n\n\\textbf{Answer ranking.} \\textsc{Uniqorn}\\xspace ranks answers by the number of different\nGSTs that they occur in Row 1. However, there arise some natural variants\nthat leverage a weighted sum rather than a simple count\n(Table~\\ref{tab:ranking}). This weight\ncan come from the following sources: (i) the total cost of the GST,\nthe less the better [Row 2]; (ii) the sum of node weights in the GST,\nreflecting question relevance, the more the better [Row 3];\n(iii) total distances of answers to the anchors in the GST, the less\nthe better [Row 4]; and, (iv) a weighted version of anchor distances,\nwhen edge weights are available, the less the better [Row 5].\nWe find that using plain\nGST counts works surprisingly well. Zooming in to the GST and\nexamining anchor proximities is not necessary\n(actually hurts significantly for KG+Text).\n\n\\begin{table} [t]\n\t\\centering\n\t\\setlength{\\tabcolsep}{0.5em}\n\t\\begin{tabular}{l c c c}\n\t\t\\toprule\n\t\t\\textbf{Error scenario}\t                             & \\textbf{KG}\t \t     & \\textbf{Text}      & \\textbf{KG+Text} \\\\ \\toprule\n\t\tAnswer not in corpus\t                             &  $57.99\\%$          \t     \t\t\t& $24.92\\%$\t    & $34.32\\%$\t   \\\\\n\t\tAnswer in corpus but not in XG                       &  $22.59\\%$                     & $24.49\\%$            & $29.63\\%$          \\\\\n\t\tAnswer in XG but not in top-$k$ GSTs \t     \t     &  $2.67\\%$                     & $23.44\\%$             & $7.64\\%$          \\\\\t\t\t\n\t\tAnswer in top-$k$ GSTs but not in top-$5$          &  $6.37\\%$                     & $23.92\\%$             & $28.41\\%$           \\\\    \n\t\tAnswer in top-$5$ but not in top-$1$                 &  $10.38\\%$                     & $3.24\\%$             & $0.00\\%$          \\\\ \\bottomrule\n\t\t\\end{tabular}\n\t\\caption{Percentages of different error scenarios where the answer is not in top-$1$, averaged over the full LC-QuAD 2.0 test set.}\n\t\\label{tab:error}\n\t\\vspace*{-0.9cm}\n\\end{table}\n\n\\textbf{Error analysis.} In Table~\\ref{tab:error}, we extract all questions for which \\textsc{Uniqorn}\\xspace\nproduces an imperfect ranking ($P@1 = 0$), and discuss cases in\na cascaded style. Each column adds up to $100\\%$, and reports the distribution of errors\nfor the specific setting. Note that values across rows are not comparable.\nWe make the following observations:\n\n(i) \\textbf{[Row 1]} indicates \nthe sub-optimal ranking of Google's retrieval from the Web\nwith respect to complex questions.\nStrictly speaking, this is out-of-scope for \\textsc{Uniqorn}\\xspace; nevertheless,\nan ensemble of search engines (Bing+Google)\nmay help improve answer coverage. This cell is null for KGs as all\nthe sampled LC-QuAD $2.0$ questions were guaranteed to\nhave answers in Wikidata.\n\n(ii) \\textbf{[Row 2]} indicates answer presence\nin $KG$ or $KG^D$, but not in the XG, and indicates NED\/NER errors for KG,\nand incorrect phrase matches for text. The largest connected component (LCC)\nis also a likely culprit at this stage, pruning away\nthe question-relevant entities in favor of more popular ones.\nReducing reliance on NERD systems and computing larger neighborhood joins\nas opposed to LCCs could be useful here.\n\n(iii) \\textbf{[Row 3]} Presence of an answer in the XG but not in top-$500$ GSTs\nusually indicates an incorrect anchor matching. This could be due to \nproblems in the similarity functions for entities (currently based\non Jaccard overlaps of mention-entity lexicons, that are inherently\ncomplete) and predicates.\nFor example, incorrect predicate matching due to shortcomings\nof embedding-based similarity (\\phrase{married} and \\phrase{divorced}\nhaving a very high similarity of $0.82$, making the latter an anchor too\nwhen the former is in the question) can happen.\nWe had conducted extensive experiments on hyperparameter-based\nlinear combinations of various similarity measures for both entities\nand predicates (word Jaccard, mention-entity lexical Jaccard, word2vec,\nlexicon priors based on anchor links)\nto see what worked best: the single choices mentioned above performed\nbest and were used. But clearly, there is a big room for improvement\nwith respect to these similarity functions, as both rows 2 and 3\nare affected by this.\n\n(iv) \\textbf{[Rows 4 and 5]} represent cases\nwhen the answer is in the top-$k$ GSTs but languish at lower ranks\nin the candidates.\nExploring weighted rank aggregation by tuning on the development set\nwith variants in Table~\\ref{tab:ranking} is\na likely remedy. A high volume of errors in this bucket is actually one of\npositive outlook: the core GST algorithm generally works well,\nand significant performance gains can be obtained\nby fine-tuning the ranking function with additional parameters.\n\n\\begin{table} [t]\n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{c cc c cc c cc}\n\t\t\\toprule\n\t\t& \\multicolumn{2}{c}{\\textbf{\\textbf{KG}}} &&  \\multicolumn{2}{c}{\\textbf{\\textbf{Text}}}  &&  \\multicolumn{2}{c}{\\textbf{\\textbf{KG+Text}}} \\\\ \n                \\cmidrule{2-3}\\cmidrule{5-6}\\cmidrule{8-9}\n\t\t\t\n \t\t\tGST Ranks            & \\begin{tabular}[c]{@{}c@{}}Avg. \\#Docs\\\\ in GST\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} && \\begin{tabular}[c]{@{}c@{}}Avg. \\#Docs\\\\ in GST\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} && \\begin{tabular}[c]{@{}c@{}}Avg. \\#Docs\\\\ in GST\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} \\\\ \\bottomrule\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n  \t\t\t  1              &       -                                                       & $37.80\\%$                                                                    && $1.557$                                                      & $31.97\\%$                                                                     && $1.715$                                                  & $46.92\\%$                                                                  \\\\\n\t\t\t2              &       -                                                       & $39.42\\%$                                                                    && $1.602$                                                      & $32.37\\%$                                                                     && $1.724$                                                  & $47.41\\%$                                                                  \\\\\n\t\t\t3               &       -                                                       & $40.28\\%$                                                                    && $1.640$                                                      & $32.78\\%$                                                                     && $1.732$                                                  & $47.82\\%$                                                                  \\\\\n\t\t\t4            &       -                                                       & $41.13\\%$                                                                    && $1.679$                                                      & $33.25\\%$                                                                     && $1.742$                                                  & $48.12\\%$                                                                   \\\\\n\t\t\t5            &       -                                                       & $41.90\\%$                                                                    && $1.712$                                                      & $33.63\\%$                                                                     && $1.754$                                                  & $48.53\\%$                                                                  \\\\\\midrule\n\t\t\t\n\t\t\t\\#Evidences in GST        & \\begin{tabular}[c]{@{}c@{}}Avg. Rank \\\\ of GST\\end{tabular}  & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} && \\begin{tabular}[c]{@{}c@{}}Avg. Rank \\\\ of GST\\end{tabular}  & \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} && \\begin{tabular}[c]{@{}c@{}}Avg. Rank \\\\ of GST\\end{tabular}& \\begin{tabular}[c]{@{}c@{}}\\% of Questions with\\\\ Answers in GST\\end{tabular} \\\\ \\midrule\n\t\t\t1                    & -                                                            & -                                                                             && $1.875$                                                    & $7.62\\%$                                                                      && $4.422$                                                  & $18.63\\%$                                                                 \\\\\n\t\t\t2                    & -                                                            & -                                                                             && $1.964$                                                    & $13.21\\%$                                                                     && $2.895$                                                  & $13.03\\%$                                                                  \\\\\n\t\t\t3                    & -                                                            & -                                                                             && $2.030$                                                    & $10.38\\%$                                                                     && $2.679$                                                  & $10.95\\%$                                                                  \\\\\n\t\t\t4                    & -                                                            & -                                                                             && $2.109$                                                    & $4.37\\%$                                                                     && $1.814$                                                  & $5.51\\%$                                                                   \\\\ \n\t\t\t5                    & -                                                            & -                                                                             && $1.992$                                                    & $0.73\\%$                                                                      && $2.112$                                                  & $1.54\\%$                                                                     \\\\ \\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\caption{Effect of multi-document evidence shown via edge contributions by distinct documents to GSTs for the LC-QuAD $2.0$ test set.}\n\t\\label{tab:gst-rank}\n\t\\vspace*{-0.9cm} \n\\end{table}\n\n\\begin{figure} [t]\n\t\\centering\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/gst-k-kg2.pdf}\n\t\t\\caption{KG - GSTs.}\n\t\t\\label{fig:gst-kg}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/TEXT_align_tune2.pdf}\n\t\t\\caption{Text - Alignments.}\n\t\t\\label{fig:align-text}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t~\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/gst-k-text2.pdf}\n\t\t\\caption{Text - GSTs.}\n\t\t\\label{fig:gst-text}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/KG_TEXT_align_tune2.pdf}\n\t\t\\caption{KG+Text - Alignments.}\n\t\t\\label{fig:align-kg-text}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t~\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/gst-k-kg-text2.pdf}\n\t\t\\caption{KG+Text - GSTs.}\n\t\t\\label{fig:gst-kg-text}\n\t\\end{subfigure}\n\t\\caption{Results of parameter tuning for different settings on the LC-QuAD 2.0 development set of \\textsc{Uniqorn}\\xspace.}\n\t\\label{fig:parameters}\n\t\\vspace*{-0.5cm}\n\\end{figure}\n\n\\textbf{Usage of multi-document evidence.} For text-QA, \\textsc{Uniqorn}\\xspace improved over\nDrQA and DocQA on all benchmarks, even\nthough the latter are supervised deep learning methods\ntrained on the large SQuAD and TriviaQA datasets.\nThis is because\nreading comprehension (RC) QA systems\nsearch for the best answer span within a passage, and will not\nwork well unless the passage\nmatches the question tokens and contains the answer.\nWhile DrQA and DocQA\ncan additionally select a good set of documents from a collection,\nit still relies\non the best document to extract the answer from. \\textsc{Uniqorn}\\xspace,\nby joining fragments of evidence across documents via GSTs, thus improves\nover\nthese methods without any training.\n\\textsc{Uniqorn}\\xspace benefits two-fold from multi-document evidence:\n\\squishlist\n\\item Confidence in an answer increases when all conditions for correctness\nare indeed satisfiable (and found) only when looking at multiple documents.\nThis increases\nthe answer's likelihood of appearing in \\textit{some GST}.\n\\item Confidence in an answer increases when it is spotted\nin multiple documents. This increases its likelihood of appearing in\nthe \\textit{top-k} GSTs, as presence in multiple documents increases\nweights\nand lowers costs of the corresponding edges.\n\\squishend\nA detailed investigation of the use of multi-document information is\npresented in Table~\\ref{tab:gst-rank}. We make the following observations:\n(i) Looking at the ``Avg. \\#Docs in GST'' columns in the upper half,\nwe see that \nconsidering the top-$500$ GSTs is worthwhile as all the bins combine\nevidence from multiple ($2+$ on average) documents.\nThis is measured by labeling edges in GSTs with\ndocuments (identifiers) that contribute the corresponding edges.\n(ii) Moreover, they also contain\nthe correct answer uniformly often\n(corresponding\n``\\#Questions with Answers in GST'' columns).\n(iii) The bottom half of the table inspects\nthe inverse phenomenon, and finds that\nconsidering only the top few GSTs is not sufficient\nfor aggregating multi-document evidence.\n(iv) Finally,\nthere is a sweet spot for GSTs aggregating nuggets from multiple documents\nto contain correct answers, and this turns out to be around three\ndocuments (see corresponding ``\\#Questions with Answers in GST'' columns).\nThis, however, is an effect of our questions\nin our benchmarks,\nthat are not complex enough to require stitching evidence\nacross more than three documents.\nDeep-learning-based RC methods over text can handle syntactic complexity\nvery well, but are typically restricted to identifying answer spans from\na single text passage. DrQA and DocQA could not properly tap the answer evidence that\ncomes from combining cues spread across multiple documents.\nAnalogous results for KG and KG+Text are provided for completeness.\n\n\\textbf{Parameter tuning.}\nThere are only five parameters in \n\\textsc{Uniqorn}\\xspace: the alignment edge insertion thresholds on\nnode-node similarity (one for entities $\\tau_{align}^E$ and\none for predicates $\\tau_{align}^P$),\nanalogous anchor selection thresholds on node weight\n($\\tau_{anchor}^E$ and $\\tau_{anchor}^P$),\nand the number of GSTs $k$.\nVariation of\nthese parameters are shown with MRR \nin Fig.~\\ref{fig:parameters} on the LC-QuAD $2.0$ development set.\nWe observe that:\n(i) having several alignment edges in the graph\n(valid only for text- and KG+text-setups)\nfor entities (corresponding to low thresholds)\nactually helps improve\nperformance though apparently inviting noise\n(dark zones indicate good performance).\nFor predicates, relatively high thresholds work\nbest to avoid spurious matches. Note that \nthe exact values of the $x$- and $y$-axes in the heat maps\nare not comparable due to the different similarity functions\nused for the two cases.\nThe predicate threshold exploration is confined to\nthe zone for which the graph algorithm was tractable\n(corresponding to a high cut-off time of thirty minutes\nto produce the answer);\n(ii) There is a similar situation for anchor thresholds:\nlow thresholds for entities work best, while \n\\textsc{Uniqorn}\\xspace is not really sensitive to the exact choice\nfor predicates.\nThe specifics vary across the setups, but as a general\nguideline, most non-extreme choices should work out fine.\nThe white zones in the top right corners\ncorrespond to \nsetting both thresholds to very high values\n(almost no anchors qualify to be in top-$5$, resulting in zero performance);\n(iii) going beyond\nthe chosen value of $k=500$ GSTs\n(for all setups) gives only diminishing returns.\n\n\n\\textbf{Effect of Open IE.} \\textsc{Uniqorn}\\xspace's noisy triple extractor\nresults in quasi KGs that contain the correct answer $85.2\\%$ of\nthe times for CQ-W ($82.3\\%$ for CQ-T). If we use triples\nfrom Stanford OpenIE~\\cite{angeli2015leveraging} to build the\nquasi KG instead, the answer is found only $46.7\\%$ and $35.3\\%$\ntimes for CQ-W and CQ-T, respectively. Thus, in the context\nof QA, losing information with precision-oriented, fewer triples,\ndefinitely hurts more than the adding potentially many noisy ones.\n\n\\begin{figure} [t]\n\t\\centering\n\t\\begin{subfigure}[b]{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.3\\textwidth]{images\/gst-1.pdf}\n\t\n\t\t\\caption{Example GST for \\utterance{Which aspiring model split with Chloe Moretz and is dating Lexi Wood?}}\n\t\t\\label{fig:gst-1}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}[b]{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.7\\textwidth]{images\/gst-2.pdf}\n\t\t\\caption{Example GST for \\utterance{Who played for FC Munich and was born in Karlsruhe?}}\n\t\t\\label{fig:gst-2}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}[b]{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.7\\textwidth]{images\/gst-2.pdf}\n\t\t\\caption{Example GST for \\utterance{Which actor is married to Kaaren Verne and played in Casablanca?}}\n\t\t\\label{fig:gst-3}\n\t\\end{subfigure}\n\t\\caption{GSTs construct interpretable evidence for complex questions.}\n\t\\label{fig:gst-eg}\n\n\\end{figure}\n\n\\textbf{GSTs contribute to explainability.} Finally, we posit that\nGroup Steiner Trees over question-focused context graphs help\nin understanding the process of answer derivation for an end-user.\nWe illustrate\nthis using three anecdotal examples of GSTs for text-QA, in\nFigs.~\\ref{fig:gst-1} through~\\ref{fig:gst-3}. The corresponding\nquestion is in the subfigure caption. Anchor nodes are in orange,\nand answers in light blue. Entities, predicates, and class nodes\nare in rectangles, ellipses, and diamonds, respectively.\nClass edges are dashed, while triple and alignments are in solid lines.\n\n\\begin{figure} [t]\n\t\\centering\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/KG_TEXT_XG_Time.pdf}\n\t\t\\caption{XG construction time.}\n\t\t\\label{fig:xg-time}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t~\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/KG_TEXT_GST_Time.pdf}\n\t\t\\caption{GST computation time.}\n\t\t\\label{fig:gst-time}\n\t\\end{subfigure}\n\t~\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/KG_TEXT_TOTAL_Time.pdf}\n\t\t\\caption{Total time.}\n\t\t\\label{fig:tot-time}\n\t\\end{subfigure}\n\t\\hspace{0.1cm}\n\t\\caption{(a) XG contruction time, (b) GST computation time, and (c) Total time,\n\t\ttaken by \\textsc{Uniqorn}\\xspace on the LC-QuAD 2.0 test set.}\n\t\\label{fig:time} \n\t\\vspace*{-0.5cm}\n\\end{figure}\n\n\\textbf{Runtime analysis.} To conclude our detailed introspection into\nthe inner workings of \\textsc{Uniqorn}\\xspace, we provide a distribution of runtimes.\nThere are two main components that contribute to the total runtime:\nthe construction of the context graph XG, and the computation of the GST\nover the XG. We find that the first step is usually the more expensive\nof the two, as it involves a large number of similarity computations.\nThe use of the fixed-parameter tractable exact algorithm for GSTs\nhelps achieve interactive response times for a large number of questions.\nAs expected, the efficiency for the KG-only setups is much better than\nthe noisier text and heterogeneous setups, as the graphs are much\nbigger in the latter cases. Nevertheless, we find that our runtimes\ncan get quite large for several questions, and this represents the most\npromising direction for future effort for us.\n\n\\myparagraph{Implementation details} All code is in \\texttt{Python}, making use of the popular PyTorch library\\footnote{\\url{https:\/\/pytorch.org}}.\nWhenever a neural model was used, code was run on a GPU (single GPU, NVIDIA Quadro RTX 8000, 48 GB GDDR6).\n\n\n\\section{Conclusions and Future Work}\n\\label{sec:confut}\n\nWe show that \\textsc{Uniqorn}\\xspace\ncan reach performance comparable to state-of-the-art baselines\non benchmarks of complex questions.\n\\textsc{Uniqorn}\\xspace is geared to work over RDF KGs,\ntext corpora, or combinations.\nThrough our proposal of computing Group Steiner Trees on dynamically\nconstructed context graphs, we aim to unify\nthe fragmented paradigms of KG-QA and text-QA.\nEven for canonicalized KGs as input, sole reliance\non accurate SPARQL queries is inferior when it \ncomes to answering complex questions with multiple entities or predicates.\nWe show that relaxing crisp\nSPARQL-style querying to an approximate graph pattern search is vital\nfor question answering over KGs,\nand in the process identify the bridge to text-QA.\nFinally, \\textsc{Uniqorn}\\xspace has the unique design rationale of deliberately allowing noise\nfrom heuristic choices \n(like coreference resolution, triple extraction, and alignment insertion)\ninto early stages of\nthe answering pipeline and coping with it later.\nThis makes it a practical choice for \nproviding direct answers in Web-based QA, with Steiner Trees\nadding the bonus of interpretable insights into answer derivation.\n\n\\section{Concepts and Notation}\n\\label{sec:concepts}\n\nWe now introduce salient concepts necessary for an accurate understanding of \\textsc{Uniqorn}\\xspace.\nA glossary of concepts and notation is provided in Table~\\ref{tab:notation}.\n\n\\subsection{General Concepts}\n\\label{subsec:gen-con}\n\n\\myparagraph{Knowledge graph} An RDF knowledge graph $K$, like Wikidata,\nconsists of \nentities $E$ (like \\phrase{Leonardo DiCaprio}), predicates $P$\n(like \\phrase{award received}), types $\\mathds{T}$ (like \\phrase{film}), and \nliterals $L$ (like \\phrase{07 January 2016}), organized as a set of\nsubject-predicate-object (SPO) triples $\\{T^K\\}$ where $S \\in E$ and\n$O \\; \\in \\; E \\; \\cup \\; \\mathds{T} \\; \\cup \\; L$. Optionally, a triple $T_i^K$ may be accompanied\nby one or more qualifiers as <qualifier-predicate, qualifier-object> tuples that provide additional context.\nThe following is an example of a triple with\n\\textit{qualifiers}\\footnote{\\url{https:\/\/www.wikidata.org\/wiki\/Help:Qualifiers}}:\n\\struct{<LeonardoDiCaprio, awardReceived, AcademyAward;\nforWork, TheRevenant; pointInTime, 2016>}. Each $T_i^K$ represents a fact in $K$.\nUnder the graph model used in this work, each entity, predicate, type and literal becomes a node in the graph.\nEdges connect different components of a fact.\nThey run from the subject to the predicate and on to the object of a fact.\nFor facts with qualifiers, there are additional edges from the main predicate to the qualifier predicate and on to the qualifier object.\nA common alternative model for KGs represents predicates as edge labels, but this makes it difficult to incorporate qualifiers (a more realistic setting).\n\n\\myparagraph{Text corpus} A text corpus $D$ is a collection of\ndocuments, where each document $D_i$ contains a set of natural \nlanguage snippets $\\mathcal{S}$. Each snippet is defined as a span of text in $D_i$ that contains at least two tokens from the question, within a specified context window.\nSuch documents could come from a static collection\nlike ClueWeb12, Common Crawl,\nWikipedia articles, or the open Web. To induce structure on $D$,\nOpen IE~\\cite{del2013clausie,angeli2015leveraging,mausam2016open}\nis performed on each\n$\\mathcal{S}_j \\in D_i$, for every $D_i \\in D$, to return a set of triples\n$\\{T_k^{D_i}\\}$\nin SPO format, where each such triple represents a fact (evidence) mined from some $D_i$.\nThese triples are augmented with those built from Hearst\npatterns~\\cite{hearst1992automatic} run over $D$ that indicate entity-type\nmemberships.\nThis non-canonicalized (open vocabulary) triple store is referred to as\na quasi knowledge graph $K^D$ built from the document corpus, where\n$K^D = \\{T^D\\} = \\bigcup_i T^{D_i}$.\nThus, each $T^D_i$ is a fact in $K^D$.\n\n\\myparagraph{Evidence} We refer to a fact in a KB or a snippet from text by the unifying term ``evidence''. \n\n\\myparagraph{Heterogeneous source} We refer to the mixture of the knowledge\ngraph $K$ and the quasi-KG $K^D$ as a heterogeneous source $K^H = K \\cup K^D$,\nwhere each triple in this heterogeneous KG can come from either \nthe RDF store or the text corpus.\n\n\\myparagraph{Question} A question $q  = \\; \\langle q_1 q_2 \\ldots q_{|q|} \\rangle$ is posed \neither as a full-fledged interrogative sentence\n(\\utterance{Who is the director of the western film for which Leonardo\nDiCaprio won an Oscar?}) or in \ntelegraphic~\\cite{joshi2014knowledge,sawant2019neural}\n\/ utterance~\\cite{abujabal2018never,yih2015semantic} form\n(\\utterance{director of western with Oscar for Leo}), where\nthe $q_i$'s are the tokens in the question,\nand $|q|$ is the total number of tokens in $q$.\nQuestion tokens are either words, or phrases like entity mentions detected by a named entity recognition (NER) system~\\cite{qi2020stanza}.\nStopwords are not considered as question tokens.\n\n\\myparagraph{Complex question} \\textsc{Uniqorn}\\xspace is motivated\nby the need for a unified approach to\ncomplex questions~\\cite{sun2019pullnet,lu2019answering,vakulenko2019message,dubey2019lc,talmor2018web}, as\nsimple questions are already well-addressed\nby prior works~\\cite{petrochuk2018simplequestions,abujabal2018never,bast2015more,yih2015semantic}.\nWe call a question ``simple'' if\nit translates into a query or a logical form\nwith a single entity and a single predicate\n(like \\utterance{capital of Greece?}).\nQuestions where the simplest proper query requires\nmultiple\n$E$ or $P$, are considered ``complex''\n(like \\utterance{director of the western for which Leo won an Oscar?}).\nThere are other notions of complex\nquestions~\\cite{hoffner2017survey}, \nlike those requiring\ngrouping and aggregation,\n(e.g., \\utterance{which director made the highest \nnumber of movies that won an Oscar in any category?}), \nor when questions involve negations\n(e.g., \\utterance{which director has won multiple Oscars but never a Golden Globe?}). \nThese are not considered in this paper.\n\n\\myparagraph{Answer} An answer $a \\in A$ to question $q$ is an\nentity $e \\in E$ or a literal $l \\in L$ in the KG $K$ (like\nthe entity \\struct{The Revenant}\nin Wikidata), or a span of text (a sequence of words)\nfrom some sentence $S_j \\in D_i$ in the corpus $D$\n(like \\phrase{The Revenant film}). $A$ is the set of all correct answers\nto $q$ ($|A| \\geq 1$).\n\n\\begin{table} [t]\n\t\\centering\n\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l l l l}\n\t\t\t\\toprule\n\t\t\t\\textbf{Notation}\t\t\t& \\textbf{Concept}\t\t\t\t& \\textbf{Notation}\t\t\t\t\t\t\t\t\t& \t\\textbf{Concept}\t\t\t\t\t\t\t            \\\\ \\toprule\n\t\t\t$K$\t\t\t\t\t\t\t& Knowledge graph (KG)\t\t\t\t& $\\mathcal{S}$\t\t\t\t\t\t\t\t\t\t&  \tSnippet\t\t\t\t\t\t\t\t\t            \\\\ \n\t\t\t$D$\t\t\t\t\t\t\t& Document corpus\t\t\t\t\t& $q = \\langle q_i\\rangle$\t\t\t\t\t\t\t& \tQuestion, question words\t\t\t\t\t        \\\\ \n\t\t\t$K^D$\t\t\t\t\t\t& Quasi KG from $D$\t\t\t\t\t& $A = \\{a\\}$ \t\t\t\t\t\t\t\t\t\t& \tAnswer to $q$\t\t\t\t\t\t\t\t        \\\\ \n\t\t\t$K^H$\t\t\t\t\t\t& Heterogeneous $KG$\t\t\t\t& $XG^{(\\cdot)}(q)$ \t\t\t\t\t\t\t\t& \tContext graph for $q$ built from $K$\/$K^D$\/$K^H$    \\\\ \n\t\t\t$E = \\{e\\}$\t\t\t\t\t& Entity\t\t\t\t\t\t\t& $N = \\{n\\}$\t\t\t\t\t\t\t\t\t\t& \tNode \t\t\t\t\t\t\t\t\t\t        \\\\\n\t\t\t$P = \\{p\\}$ \t\t\t\t& Predicate \t\t\t\t\t\t& $\\mathcal{E} = \\{\\epsilon\\}$\t\t\t\t\t\t& \tEdge\t\t\t\t\t\t\t\t\t\t        \\\\ \n\t\t\t$\\mathds{T} = \\{t\\}$\t\t& Type\t\t\t\t                & $N^{\\mathcal{T}}, \\mathcal{E}^\\mathcal{T}$\t\t& \tNode types, edge types\t\t\t\t\t\t        \\\\\n\t\t\t$L = \\{l\\}$\t\t\t\t\t& Literal\t\t\t\t\t\t\t& $N^{W}, \\mathcal{E}^W$\t\t\t\t\t\t\t& \tEdge weights, edge weights\t\t\t\t\t        \\\\\n\t\t\t$S$\t\t\t\t\t\t\t& Subject\t\t\t\t\t\t\t& $M_{\\mathcal{T}}^N(\\cdot), M^\\mathcal{E}(\\cdot)$\t& \tMapping function from node\/edge to type\t\t\t    \\\\ \n\t\t\t$O$\t\t\t\t\t\t\t& Object \t\t\t\t\t\t\t& $M_W^N(\\cdot), M_W^\\mathcal{E}(\\cdot)$ \t\t\t& \tMapping function from node\/edge to weight\t\t\t\\\\ \n\t\t\t$T^{(\\cdot)}$\t\t\t\t& Triple in $K$\/$K^D$\/$K^H$\t\t\t& $\\tau_{align}^{(\\cdot)}$\t\t\t\t\t\t\t& \tAlignment threshold for $E\/P$\t\t\t\t        \\\\\n\t\t\t$m$\t\t\t\t\t\t\t& Entity mention\t\t\t\t\t& $\\mathbb{T}_i$\t\t\t\t\t\t            & \tAny tree in $XG^{(\\cdot)}(q)$ connecting anchors\t\\\\ \n\t\t\t$\\mathcal{A}$\t\t\t\t& Anchor node\t\t\t\t\t\t& $\\mathbb{T}$*\t\t\t\t\t\t\t\t\t\t& \tGST\t\t\t\t\t\t\t\t\t\t            \\\\ \\bottomrule\n\t\\end{tabular}\n\t\\caption{Concepts and notation.}\n\t\\label{tab:notation}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\subsection{Graph Concepts}\n\\label{subsec:graph-con}\n\n\\myparagraph{Context graph} A context graph $XG$ for a question $q$ is defined\nas a subgraph of the full \/ quasi \/ heterogeneous knowledge graph, i.e.,\n$XG^K(q) \\subset K$ (KG)\nor $XG^D(q) \\subset K^D$ (text)\nor $XG^H(q) \\subset K^H$ (mixture), such that it contains all\ntriples $T^K_i(q)$ or $T^D_i(q)$ potentially relevant for answering $q$.\nThus, $XG(q)$ is expected to contain every answer entity $a \\in A$.\nAn $XG$ has nodes $N$ and edges $\\mathcal{E}$ with types\n($N^\\mathcal{T}, \\mathcal{E}^\\mathcal{T}$) and weights ($N^W, \\mathcal{E}^W$) as \ndiscussed below. Thus, an $XG$ is always \\textit{question-specific}, and to\nsimplify notation, we often write only $XG$ instead of $XG(q)$.\nFig.~\\ref{fig:xg} shows possible context graphs for our running example question in each of the three setups.\n\n\\myparagraph{Node types} A node $n \\in N$ in an $XG$ is mapped to one of\nfour categories $N^\\mathcal{T}$:\n(i) entity, (ii) predicate, (iii) type, or (iv) literal, via a \nmapping function $M_{\\mathcal{T}}^N(\\cdot)$, where\n$n \\in E \\cup P \\cup \\mathds{T} \\cup L$. Each $n_j$ is produced from\nan $S$, $P$, or $O$, from the triples in $K$ or $K^D$.\nFor\nKG facts,\nwe make no distinction between predicates and qualifier\npredicates, or objects and qualifier objects (no qualifiers\nin\ntext).\n\nLet us use Fig.~\\ref{fig:xg} for reference.\nEntities and literals are shown in rectangles with sharp corners,\nwhile predicates and types are in rectangles with rounded corners.\nEven though it is standard practice to treat predicates as\nedge labels in a KG, we model them as nodes, because this design \nsimplifies the \napplication of our graph algorithms.\n\nNote that predicates originating from different triples are assigned\nunique identifiers in the graph. For instance, for triples\n$T_1 = $ \\struct{<BarackObama, married, MichelleObama>},\nand $T_2 = $ \\struct{<BillGates, married, MelindaGates>}, we will obtain\ntwo \\struct{married} nodes,\nthat will be marked as \\struct{married-1} and \\struct{married-2} in\nthe context graph.\nSuch distinctions prevent false inferences when answering over some XG.\nFor simplicity, we do not show such predicate indexing in Fig.~\\ref{fig:xg}.\n\nFor text-based $XG^D$, we make no\ndistinction between $E$ and $L$, as Open IE markups that produce these\n$T^D$ often lack such ``literal'' annotations. Type nodes $\\mathds{T}$\nin $XG^K$ come from the objects of \\struct{instanceOf} and \\struct{occupation} predicates in $K$\n(e.g., for Wikidata),\nwhile those in $XG^D$ originate from the outputs of Hearst \npatterns~\\cite{hearst1992automatic} in $D$. \nIn Fig.~\\ref{fig:xg-kg}, nodes \\struct{TheRevenant},\n\\struct{director}, \\struct{AcademyAwards}, and \\struct{2016} are\nof type $E$, $P$, $\\mathds{T}$, and $L$, respectively.\nIn Fig.~\\ref{fig:xg-text}, nodes \\phrase{The Revenant},\n\\phrase{directed by}, and \\phrase{2015 American western film}, are\nof type $E$, $P$, and $\\mathds{T}$, respectively.\n\n\\myparagraph{Edge types} An edge $\\epsilon \\in \\mathcal{E}$ in an $XG$ is\nmapped to\none of three categories $\\mathcal{E}^\\mathcal{T}$:\n(i) triple, (ii) type, or (iii) alignment edges, via\na mapping function $M_{\\mathcal{T}}^\\mathcal{E}(\\cdot)$. \nTriples in $\\{T^K\\}$ or $\\{T^D\\}$ or $\\{T^H\\}$, where $M^N(O) = \\{E \\cup L\\}$,\ni.e., the object is of node type entity or literal, contribute triple edges\nto the $XG$. For example, in Fig.~\\ref{fig:xg-kg},\nthe two edges between \\phrase{The Revenant}\nand \\phrase{genre} and between \\phrase{genre} and \\phrase{Western film} are\ntriple edges.\nIn Fig.~\\ref{fig:xg-text},\nthe two edges between \\phrase{Alejandro}\nand \\phrase{director of}, and between \\phrase{director of} and \\phrase{Survival drama Revenant} are\ntriple edges.\n\nTriples where $M_{\\mathcal{T}}^N(O) = \\{\\mathds{T}\\}$ \n(object is a type) are\nreferred to as type triples, and each type triple contributes two\ntype edges to the $XG$. Examples are: edges between \\struct{AcademyAwardForBestDirector} and \\struct{instanceOf}, and \\struct{instanceOf} and \n\\struct{AcademyAwards} in Fig.~\\ref{fig:xg-kg}; \nand \nedges between \\phrase{The Revenant} and \\phrase{type}, and \\phrase{type} and \n\\phrase{2015 American western film} in Fig.~\\ref{fig:xg-text}.\n\nAlignment edges \nrepresent potential synonymy between nodes, and run \\textit{only} between nodes of the same type.\nAlignments are inserted in $XG^D$ or $XG^H$ via\nexternal sources of similarity like aliases or word embeddings.\nThere are no alignment edges in $XG^K$ (or more generally, in $K$) as all items in a KG are canonicalized.\nExamples of alignment edges are\nthe bidirectional dotted edges\nbetween\n\\phrase{The Revenant} and \\phrase{Survival drama Revenant} in \nFig.~\\ref{fig:xg-text},\nand \\struct{director} and \\phrase{directed by} in Fig.~\\ref{fig:xg-hetero}.\nInsertion of alignment edges as opposed to\na naive merging of synonymous nodes is a deliberate choice. This enables\nmore matches with question tokens and a subsequent disambiguation of question concepts. It also precludes\nthe problem of choosing a representative label for merged clusters, and avoids\ntopic drifts arising from the transitive effects of merging nodes at this stage.\n\n\\myparagraph{Node weights} A node $n \\in N$ in an $XG$ is weighted by\na function $M_W^N(\\cdot) \\in [0, 1]$ according to its similarity\nto the question.\nThis is obtained averaging BERT scores of the evidences of which $n$ is a part of.\nNode weights are used as a criterion for answer ranking.\n\n\\myparagraph{Edge weights} Each edge $\\epsilon = (n_i, n_j)$ in an $XG$ is assigned a weight by a function $M_W^\\mathcal{E}(\\cdot) \\in [0, 1]$.\nEdge weights are later converted to edge costs that are vital for computing Group Steiner Trees on the XG.\nIn \\textsc{Uniqorn}\\xspace, edge weights are also the BERT scores of evidences (KB facts or text snippets) from where the edge originates.\nEdge weights are averaged if the same edge is found in multiple evidences. This holds for triple and type edges.\n\nWeights of alignment edges come from similarities between nodes in the XG. For entities, this is computed as the Jaccard overlap of character-level trigrams~\\cite{yih2015semantic} between the node labels that are the endpoints of the edge (for KBs, entity aliases available as part of the KB are appended to entity names before computing the similarity). Character-level $n$-grams make the matching robust to spelling errors. Lexical matching is preferred for entities and literals as we are more interested in \\textit{hard equivalence} rather than a soft relatedness. For predicates and types, lexical matches are not enough and semantic similarity computations are necessary.\nSo alignment scores are computed as pairwise embedding similarities (cosine values) over words in the two node labels (one word from each node), followed by a maximum over these pairs. This is then min-max normalized to $[0, 1]$.\nWikipedia2vec~\\cite{yamada2020wikipedia2vec}, that taps into both corpus statistics and link structure in Wikipedia, was used for computing embeddings.\n\nAn alignment\nedge is inserted into an $XG^D$ or $XG^H$ if the similarity exceeds or equals\nsome threshold $\\tau_{align}$, i.e.,\n$\\textrm{sim}(\\textrm{label}(n_i), \\textrm{label}(n_j)) \\geq \\tau_{align} \\in \n(0,1]$. Zero is not an acceptable value for $\\tau_{align}$ as that would mean\ninserting an edge\nbetween every pair of possibly unrelated nodes. This alignment insertion threshold\n$\\tau_{align}$ could be \npotentially different for entities ($\\tau_{align}^E$) and\npredicates ($\\tau_{align}^P$), due to the use of different similarity functions.\nThese (and other) hyper-parameters are tuned on a \ndevelopment set of question-answer pairs.\n\n\\myparagraph{Anchor nodes} A node $n$ in an $XG$ is an anchor node\n$\\mathcal{A}$ if\nit matches one of the question tokens. Such matches may either be lexical (\\phrase{western} $\\mapsto$ \\struct{2015 American western film} in Fig.~\\ref{fig:xg-text}), or more sophisticated mapping of entity mentions $m$ in questions to KG entities $\\{e\\}$ via named entity recognition and disambiguation (NERD) systems~\\cite{li2020efficient,hoffart2011robust,ferragina2010tagme}\n(\\phrase{Leo} $\\mapsto$ \\struct{Leonardo DiCaprio} in Fig.~\\ref{fig:xg-kg}).\nAnchors are grouped into sets,\nwhere a set\n$\\mathcal{A}^k = \\{\\mathcal{A}^k_1, \\mathcal{A}^k_2, \\ldots \\}$,\ndepending upon which question token $q_k$ the elements of the set match.\nIn other words, more than one XG node can match the same question token, and hence the need for the superscript $k$: the anchor nodes corresponding to $q_2$ would be denoted by $\\{\\mathcal{A}^2_1, \\mathcal{A}^2_2, \\ldots\\}$.\nFor example, \\phrase{director} in the question matches nodes \\phrase{director of}, \\phrase{directed by}, \\phrase{Best Director}, $\\ldots$ in Fig.~\\ref{fig:xg-text}.\nAnchors thus identify the question-relevant nodes of the $XG$,\nin the vicinity of which answers can be expected to be found.\nAny node of category entity, predicate, type, or literal, can qualify\nas an anchor.\n\n\\section{Experimental Setup}\n\\label{sec:exp-setup}\n\n\\subsection{Benchmarks}\n\\label{subsec:data}\n\n\\subsubsection{Knowledge sources}\n\nAs our \\textbf{knowledge graph} we use the NTriples dump of the full Wikidata\\footnote{\\url{https:\/\/dumps.wikimedia.org\/wikidatawiki\/entities\/}} as of 20 April 2021,\nincluding all qualifiers.\nThe original dump consumed about $2$ TB of disk space, and contained about $12$B triples.\nWe use a cleaned version\\footnote{\\url{https:\/\/github.com\/PhilippChr\/wikidata-core-for-QA}}\nthat prunes language tags, external identifiers, additional schema labels and so on (KB cleaning steps in~\\cite{christmann2022beyond}).\nThis left us with about $2$B triples with $40$ GB disk space.\nThe cleaned KB is accessed via the recently proposed \\textsc{Clocq} API\\footnote{https:\/\/clocq.mpi-inf.mpg.de\/}.\nNote that there is nothing in our method specific to Wikidata, and can be easily\nextended to other KGs like YAGO~\\cite{suchanek2007yago} or DBpedia~\\cite{auer2007dbpedia}. Wikidata was chosen as it is one of the popular choices\nin the community today, and\nhas an active community that contributes to growth of the KG, akin to Wikipedia\n(both are supported by the Wikimedia foundation).\nFurther, in 2016, Google ported a large volume of the erstwhile Freebase into\nWikidata~\\cite{pellissier2016freebase}. \n\nFor \\textbf{text}, we create a pool of\n$10$ Web pages per question, obtained from Google Web search\nin June 2021.\nSpecifically, we issue the full question as a query to Google Web search,\nand create a question-specific corpus from these top-$10$ results obtained.\nThis was done using the Web search option inside\nthe Google Custom Search API\\footnote{\\url{https:\/\/developers.google.com\/custom-search\/v1\/overview}}.\nThis design choice of fetching pages from the Web was made to be close\nto the direct-answering-over-the-Web-setting, and not be restricted to specific\nchoices of benchmarks that have associated corpora. This also enables comparing\nbaselines from different QA families on a fair testbed. This was done despite the fact\nthat it entails significant resources, as this has to be done for thousands of questions,\nand \\textit{in addition}, these Web documents have to be entity-linked to Wikidata\nto enable training some of the supervised baselines.\n\nThe \\textbf{heterogeneous} answering setup was created by considering the above two sources together.\nTo be specific, each question from our benchmark is answered over the entire KG and\nthe corresponding text corpus.\n\n\\textit{All baselines were exposed to the same knowledge sources as \\textsc{Uniqorn}\\xspace},\nin all the three setups -- KG, text, or heterogeneous inputs.\n\n\\begin{table} [t] \n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l r l}\n\t\t\t\\toprule\n\t\t\t\\textbf{Benchmark}\t\t\t\t\t\t\t\t\t\t\t\t\t\t& \\textbf{\\#Questions}\t& \\textbf{Answers}\t                                    \\\\ \\toprule\n\t\t\tComplex questions from LC-QuAD $2.0$~\\cite{dubey2019lc}\t\t\t\t\t& $4921$\t\t\t\t& Wikidata entities\t\t\t\t\t\t\t\t        \\\\ \\midrule\n\t\t\tComplex questions from LC-QuAD $1.0$~\\cite{trivedi2017lc}\t\t\t\t& $1459$\t\t\t\t& DBpedia entities, mapped to Wikidata via Wikipedia\t\\\\\n\t\t\tComplex questions from WikiAnswers (CQ-W)~\\cite{abujabal2017automated}\t& $150$\t\t\t\t\t& Freebase entities, mapped to Wikidata via Wikipedia \t\\\\\n\t\t\tComplex questions from Google Trends (CQ-T)~\\cite{lu2019answering}\t\t& $150$\t\t\t\t\t& Entity mention text, manually mapped to Wikidata      \\\\\n\t\t\tComplex questions from QALD $1-9$~\\cite{ngomo20189th}\t\t\t\t\t& $70$\t\t\t\t\t& DBpedia entities, mapped to Wikidata via Wikipedia \t\\\\\n\t\t\tComplex questions from ComQA~\\cite{abujabal2019comqa}\t\t\t\t\t& $202$ \t\t\t\t& Wikipedia URLs, mapped to Wikidata entities\t\t\t\\\\ \\midrule\n\t\t\tTotal number of complex questions\t\t\t\t\t\t\t\t\t\t& $6952$\t\t\t\t& Wikidata entities\t\t\t\t\t\t\t            \\\\ \\bottomrule\n\t\\end{tabular}}\n\t\\caption{Test sets with sampled questions from each benchmark, totaling about $7k$ complex questions.}\n\t\\label{tab:data}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\begin{table} [t] \n\t\\centering\n\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l p{10.5cm}}\n\t\t\t\\toprule\n\t\t\t\\textbf{Benchmark}\t\t\t\t\t\t\t\t&   \\textbf{Complex Questions}\t                                                \\\\ \\toprule\n\t\t\tLC-QuAD $2.0$~\\cite{dubey2019lc}\t\t\t\t&   \\utterance{Which of Danny Elfman's works was nominated for an Academy Award for Best Original Score?}       \\\\\t                                                \\cmidrule{2-2} \n    \t\t\t\t\t\t\t                            &   \\utterance{When did Glen Campbell receive a Grammy Hall of Fame award?}                                        \\\\ \\midrule\n\t\t\tLC-QuAD $1.0$~\\cite{trivedi2017lc}\t\t    \t&   \\utterance{Which award that has been given to James F Obrien, had used Laemmle Theatres for some service?}                                                     \\\\ \\cmidrule{2-2}\n\t\t\t                                                &   \\utterance{Which home stadium of 2011-12 FC Spartak Moscow season is also the location of birth of the Svetlana Gounkina?}\n\t\t\t                                                    \\\\ \\midrule\n\t\t\tCQ-W~\\cite{abujabal2017automated}\t            & \\utterance{Which actor is married to Kaaren Verne and played in Casablanca?} \t        \\\\ \\cmidrule{2-2}\n        \t\t\t\t                                    & \\utterance{Who graduated from Duke University and was the president of US?} \t        \\\\ \\midrule\n\t\t\tCQ-T~\\cite{lu2019answering}\t\t\t\t\t\t& \\utterance{In which event did Taylor Swift and Joe Alwyn appear together?}            \\\\ \\cmidrule{2-2}\n\t\t\t\t\t\t\t\t\t                        & \\utterance{Who played for Barcelona and managed Real Madrid?}                         \\\\ \\midrule\n\t\t\tQALD $1-9$~\\cite{ngomo20189th}\t\t\t\t\t& \\utterance{Which street basketball player was diagnosed with Sarcoidosis?} \t        \\\\ \\cmidrule{2-2}\n\t\t\t\t\t\t\t\t                            & \\utterance{Which recipients of the Victoria Cross fought in the Battle of Arnhem?}    \\\\ \\midrule\n\t\t\tComQA~\\cite{abujabal2019comqa}\t\t\t\t\t& \\utterance{What is the river that borders Mexico and Texas?}\t\t\t                \\\\ \\cmidrule{2-2}\n\t\t\t\t\t\t\t\t                            & \\utterance{Who Killed Dr Martin Luther King Jr's mother?}\t\t\t                    \\\\ \\midrule\n\t\\end{tabular\n\t\\caption{Examples of complex questions from each benchmark.}\n\t\\label{tab:data-eg}\n\t\\vspace*{-1cm}\n\\end{table}\n\n\\subsubsection{Question-answer pairs} \n\nThere is no QA benchmark that is directly suitable for answering complex questions on heterogeneous sources.\nAs a result, we choose six relatively recent benchmarks (Table~\\ref{tab:data}) with realistic questions, proposed for KG-QA, and curate text corpora for questions in these benchmarks\\footnote{Note that the opposite strategy of adapting text-QA benchmarks for KG and heterogeneous QA, is not a practical option. This is because answers in mainstream text-QA may be arbitrary spans of text from given passages, and may not map to crisp entities -- and hence would be out of scope of factoid QA, the focus of this work.}\nOut of these, the most recent one, LC-QuAD $2.0$, served as the main benchmark (owing to its relatively larger size), and the rest were used to test the generalizability of QA systems in open-domain experiments (running pre-trained models on unseen questions).\nBaselines that require supervision were trained on the LC-QuAD 2.0 train set ($18$k questions), and hyperparameter tuning for all methods were done on a random sample of $1000$ questions from the LC-QuAD 2.0 development set ($6$k questions).\n\\textsc{Uniqorn}\\xspace only requires tuning of the hyperparameters inside the BERT model and the alignment thresholds, and for this, used only the $1000$-question sample above.\n\nTo ensure that the benchmark questions pose \nsufficient difficulty to the systems under test, all questions from individual sources\nwere manually examined to ensure that they have at least entity and more than two relations.\nQuestions that do not have a ground truth answer in our our KG Wikidata, and aggregations and existential \nquestions (i.e., questions with counts and yes\/no answers, respectively) were also removed.\nThe number of questions finally contributed by each source is shown in Table~\\ref{tab:data}.\nThese are factoid questions: each question usually has one or a few entities\nas correct answers ($72\\%$ of the total number of questions across all benchmarks have exactly one ground truth answer, and most of the rest have $2-3$ gold answers).\nDetails of specific benchmarks are provided below. Some examples of complex questions from these benchmarks are shown in Table~\\ref{tab:data-eg}.\nAll datasets and code for this project are available at \\url{qa.mpi-inf.mpg.de\/uniqorn}.\n\n(i) \\textbf{LC-QuAD 2.0}~\\cite{dubey2019lc}: This very recent and large QA benchmark from October 2019\nwas compiled using crowdsourcing\nwith human annotators verbalizing KG templates that were shown to them. Answers are Wikidata\/DBpedia SPARQL queries that can be\nexecuted over the corresponding KGs to obtain entities\/literals as answers. \n\n(ii) \\textbf{LC-QuAD 1.0}~\\cite{trivedi2017lc}: This dataset was curated by the same team\nas LC-QuAD 2.0, by a similar process, but where the crowdworkers directly corrected\nan automatically generated natural language question. Answers are DBpedia entities --\nwe linked them to Wikidata for our use via their Wikipedia URLs, that act as\nbridges between popular curated KGs like Wikidata, DBpedia, Freebase, and YAGO.\n\n(iii) \\textbf{CQ-W}~\\cite{abujabal2017automated}: The questions (complex questions from \nWikiAnswers, henceforth CQ-W) were sampled from\nWikiAnswers with multiple entities or relations. Answers are Freebase entities.\nWe mapped them to Wikidata using Wikipedia URLs.\n\n(iv) \\textbf{CQ-T}~\\cite{lu2019answering}: This contains complex questions about emerging\nentities\n(e.g., people dated by celebrities) created from queries in Google Trends\n(complex questions from Google Trends, henceforth CQ-T).\nAnswers are text mentions of entities, that we manually map to Wikidata.\n\n(v) \\textbf{QALD}~\\cite{ngomo20189th}: We collated these questions by going through\nnine years of QA datasets\nfrom the QALD\\footnote{\\url{http:\/\/qald.aksw.org\/}} benchmarking campaigns\n(editions $1-9$).\nAnswers are DBpedia entities, that we map to Wikidata using Wikipedia links.\n\n(vi) \\textbf{ComQA}~\\cite{abujabal2019comqa}: These factoid questions, like~\\cite{abujabal2017automated},\nwere also sampled from the WikiAnswers community QA corpus. Answers are Wikipedia URLs\nor free text, which we map to Wikidata.\n\n\\subsection{Systems under test}\n\\label{subsec:base}\n\n\\begin{table} [t] \n\t\\centering\n\t\\setlength{\\tabcolsep}{0.25em}\n\n\t\\begin{tabular}{l ccc c ccc}\n\t\t\\toprule\n\t\t\\multirow{2}{*}{\\textbf{Setup}} & \\multicolumn{3}{c}{\\textbf{\\#Nodes}}\t&&  \\multicolumn{3}{c}{\\textbf{\\#Edges}} \t\t\\\\\n\t\t\\cmidrule{2-4} \t\t\\cmidrule{6-8}\t\t\t\n\t\t                    & \\textbf{Entity} \t& \\textbf{Predicate}\t& \\textbf{Type} \t&& \\textbf{Triple} \t& \\textbf{Alignment} \t& \\textbf{Type} \\\\ \\midrule\n\t\t\\textbf{KG+Text} \t& $215.2$\t\t\t& $255.6$\t\t\t\t& $12.4$\t\t    && $508.6$\t        & $6151.9$\t            & $2.5$        \\\\ \n\t\t\n\t\t\\textbf{KG} \t\t& $10.3$\t\t\t& $27.6$\t\t\t\t& $11.5$\t\t\t&& $38.7$\t        & N. A.\t\t            & $16.4$\t    \\\\\n\t\t\\textbf{Text} \t\t& $147.9$\t\t\t& $182.3$\t\t\t\t& $3.6$\t\t        && $362.6$\t        & $3.9$\t            & $4.3$        \\\\ \\bottomrule\n\t\\end{tabular\n\t\\caption{Basic properties of \\textsc{Uniqorn}\\xspace's XGs, averaged over LC-QuAD $2.0$ test questions.\n\t\t\tParameters affect these sizes: best values in each setting were used in these measurements.\n\t\tThe graphs typically get bigger and denser as the alignment edge insertion thresholds are lowered.\n\t\tThe KG+Text may not be exact additions of the two settings due the computation of the largest connected component, and slight parameter variations.\n\t\tExpectedly, the heterogeneous setting has the largest number of nodes\n\t\tand edges.\n\t}\n\t\\label{tab:graph-size}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\subsubsection{\\textsc{Uniqorn}\\xspace implementation}\n\\label{subsubsec:init}\n\nAs mentioned in Sec.~\\ref{subsec:evi-retr}, we need NERD systems for the KG-QA pipeline.\nTo improve answer recall, we used two complementary NERD systems on questions -- one biased towards precision (\\textsc{Elq}\\xspace~\\cite{li2020efficient}, with default configuration), and one towards recall (\\textsc{TagMe}\\xspace~\\cite{ferragina2010tagme}, with zero cut-off threshold $\\rho$).\nBoth systems disambiguate to Wikipedia, and we map them to Wikidata using links available in the KG.\nAll baselines were also given the advantage of these same disambiguations.\nFor Text-QA, POS tagging and NER on questions and documents were done using spaCy\\footnote{\\url{https:\/\/spacy.io\/}}.\nFor simplicity, POS tags were mapped to the Google Universal Tagset~\\cite{petrov2012universal}.\nThese POS tags were used for running our open IE pattern extractors, and identifying noun phrases necessary for matching Hearst patterns.\nEntity alignment edges were added using Jaccard coefficients on KG items labels augmented with aliases, and predicate and type alignments using $100$-dimensional Wikipedia2vec~\\cite{yamada2020wikipedia2vec} embedding similarities (Sec~\\ref{subsec:graph-con}).\nBERT fine-tuning needs a train-dev split (the best model is then applied on the test set): the 1000-question LC-QuAD 2.0 development set was split in an $80:20$ ratio for this purpose.\nThe following hyperparameters were found to work best on the development set: a batch size of $50$, a learning rate of $3 \\times 10^{-5}$, and a gradient accumulation of $4$.\nThe maximum token length was $512$.\nHugging Face libraries for the BERT-base-cased model\\footnote{\\url{https:\/\/huggingface.co\/bert-base-cased}} was used, for the sequence pair classification task\\footnote{\\url{https:\/\/huggingface.co\/docs\/transformers\/model_doc\/bert\\#tfbertforsequenceclassification}}.\nThe top-$5$ evidences were returned from the BERT fine-tuning, i.e. $\\varepsilon = 5$.\nThe three hyperparameters for the GST step ($\\tau_{align}^E, \\tau_{align}^P$, and top-$g$ GSTs to consider)\nwere tuned on the $1000$-question development test using grid search in each of the three settings.\nWe obtained:\n(i) for KG+text: $\\tau_{align}^E = 0.8, \\tau_{align}^P = 0.7, g = 10$.\n(ii) for KG: $g = 10$ (no alignment necessary for KGs);\n(iii) for text: $\\tau_{align}^E = 0.5, \\tau_{align}^P = 0.9, g = 10$;\nEffects of these parameters are examined later in Sec.~\\ref{subsec:analysis}.\nUsing these configurations, context graphs in the KG, text, and heterogeneous settings, $XG^K$, $XG^D$ and $XG^H$,\nare constructed in the manner described in text Sec.~\\ref{subsec:xg-build}.\n\\textsc{Uniqorn}\\xspace's XGs in the different setups are characterized in Table~\\ref{tab:graph-size}.\n\n\\subsubsection{Baselines for the heterogeneous setup}\n\nWe use the state-of-the-art systems \\textsc{UniK-QA}\\xspace~\\cite{oguz2021unikqa}, \\textsc{PullNet}\\xspace~\\cite{sun2019pullnet}, and its predecessor \\textsc{GRAFT-Net}\\xspace~\\cite{sun2018open}, as baselines for the KG+text heterogeneous setup.\n\\textsc{UniK-QA}\\xspace is one of the first systems to explore answering over heterogeneous sources by verbalizing all inputs to text sequences (almost opposite to our approach of inducing structure on all sources), and is state-of-the-art in the prominent retrieve-and-read paradigm (to which all our text-QA methods belong, as well) for QA over heterogeneous sources.\n\n(i) \\myparagraph{\\textsc{UniK-QA}\\xspace~\\cite{oguz2021unikqa}} \\textsc{UniK-QA}\\xspace verbalizes all structured inputs (KGs, tables, lists) to sentence form using simple rules, and adds them to the text corputs. This is then queried through a neural retriever (dense passage retrieval, DPR~\\cite{karpukhin2020dense}), and the top-$100$ evidences are fed into a generative reader model (fusion-in-decoder, \\textsc{FiD}\\xspace~\\cite{izacard2021leveraging}). DPR was designed to work over Wikipedia: so we fine-tuned DPR for LC-QuAD 2.0 by using the subset of Wikipedia pages retrieved among the Google results containing a gold answer as positive instances, and randomly choosing a negative instance from the positive instance of a different question (see the original DPR paper~\\cite{karpukhin2020dense} for details). We also fine-tune \\textsc{FiD}\\xspace, originally pre-trained using T5, on LC-QuAD 2.0.\nSome parts of the \\textsc{UniK-QA}\\xspace code is available publicly (\\textsc{DPR}\\xspace, \\textsc{FiD}\\xspace): the rest was reimplemented.\n\n(i) \\myparagraph{\\textsc{PullNet}\\xspace~\\cite{sun2019pullnet}} \\textsc{PullNet}\\xspace uses an iterative process to construct a question-specific subgraph for complex questions, where in each step, a relational graph convolutional neural network (R-GCN) is used to find subgraph nodes that should be expanded (to support multihop conditions) using pull operations on the KG and corpus. After the expansion, another R-GCN is used to predict the answer from the expanded subgraph.\n\\textsc{PullNet}\\xspace code is not public: it was completely reimplemented.\n\n(ii) \\myparagraph{\\textsc{GRAFT-Net}\\xspace~\\cite{sun2018open}} \\textsc{GRAFT-Net}\\xspace (Graphs of Relations Among Facts and Text Networks) also uses R-GCNs specifically designed to operate over heterogeneous graphs of KG facts and entity-linked text sentences. \\textsc{GRAFT-Net}\\xspace uses LSTM-based updates for text nodes and directed propagation using Personalized PageRank (PPR) and tackles the heterogeneous setting with an early fusion philosophy.\n\\textsc{GRAFT-Net}\\xspace code is available for the deprecated KG Freebase, necessary (non-trivial) adaptations were made for answering over Wikidata.\n\\textsc{PullNet}\\xspace and \\textsc{GRAFT-Net}\\xspace need entity-linked text for learning their models, for which we used the same text corpora described earlier ($10$ documents per question from Google Web search) that were tagged with \\textsc{TagMe}\\xspace and \\textsc{Elq}\\xspace\nwith the same $\\rho$ threshold of zero (see Sec.~\\ref{subsubsec:init}).\n\n\\subsubsection{KG-QA baselines}\n\n\\textsc{UniK-QA}\\xspace, \\textsc{PullNet}\\xspace, and \\textsc{GRAFT-Net}\\xspace\ncan be \nrun in KG-only modes as well. So these naturally add to KG-QA baselines\nfor us.\nIn addition, we use the systems\n\\textsc{QAnswer}\\xspace~\\cite{diefenbach2019qanswer,diefenbach2020towards}\\footnote{\\url{https:\/\/qanswer-frontend.univ-st-etienne.fr\/}}\nand\n\\textsc{Platypus}\\xspace~\\cite{tanon2018demoing}\\footnote{\\url{https:\/\/askplatyp.us\/}}\nas baselines for KG-QA.\nAs of March 2021, to the best of our knowledge,\nthese are the only systems \nfor Wikidata with sustained online services and APIs,\nwith \\textsc{QAnswer}\\xspace having state-of-the-art performance.\nWe use public APIs for both systems.\n\n(i) \\myparagraph{\\textsc{QAnswer}\\xspace~\\cite{diefenbach2020towards}} This is an extremely efficient method that relies on an over-generation of SPARQL queries based on simple templates, that are subsequently ranked, and the best query is executed to fetch the answer. The queries are generated by a fast graph breadth-first search (BFS) algorithm relying on HDT indexing\\footnote{\\url{https:\/\/www.rdfhdt.org\/}}.\n\n(ii) \\myparagraph{\\textsc{Platypus}\\xspace~\\cite{tanon2018demoing}} This was designed as a QA system driven by natural language understanding, targeting \\textit{complex questions} using grammar rules and template-based techniques. Questions are translated not exactly to SPARQL, but to a custom logical representation inspired by dependency-based compositional semantics.\n\n\\subsubsection{Text-QA baselines}\n\nThe mainstream manifestation of text-QA in the literature today is in the form machine reading comprehension (MRC), where given a question and a collection of passages, the system derives an answer. There are many, many MRC systems today, along with open retrieval variants where the question-relevant passages are not given but need to be retrieved from a large repository. As for text-QA baselines in this work against which we compare \\textsc{Uniqorn}\\xspace, we focus on distantly supervised methods. These include neural QA models which are pre-trained on large question-answer collections, with additional input from word embeddings. These methods are well-trained for QA in general, but not biased towards specific benchmark collections. We are interested in robust behavior for ad hoc questions, to reflect the rapid evolution and unpredictability of topics in questions on the open Web. Hence this focus on distantly supervised methods. As specific choices, we adopt the well-known \\textsc{PathRetriever}\\xspace~\\cite{asai2020learning}, \\textsc{DocumentQA}\\xspace~\\cite{clark2018simple}, and \\textsc{DrQA}\\xspace~\\cite{chen2017reading} systems as representatives of robust open-source implementations, that can work seamlessly on unseen questions and passage collections. All methods can deal with multi-paragraph documents, and are deep learning-based systems with large-scale training via Wikipedia and reading-comprehension resources like SQuAD~\\cite{rajpurkar2016squad} and TriviaQA~\\cite{joshi2017triviaqa}, and we run pre-trained QA models on our test sets.\n\n(i) \\myparagraph{\\textsc{PathRetriever}\\xspace~\\cite{asai2020learning}} This method is specifically developed for \\textit{complex questions}. The approach used is supervised iterative graph traversal (akin to \\textsc{PullNet}\\xspace~\\cite{sun2019pullnet}) and uses a novel recurrent neural network (RNN) method to learn to sequentially retrieve relevant passages in reasoning paths for complex questions, by conditioning on the previously retrieved documents. It is a very robust method that makes effective use of data augmentation and smart negative sampling to boost its learning, and has a minimal set of hyperparameters. The standard BERT QA model is used as the reader. Code is available at \\url{https:\/\/github.com\/AkariAsai\/learning_to_retrieve_reasoning_paths}. The default configuration was used.\n\n(ii) \\myparagraph{\\textsc{DocumentQA}\\xspace~\\cite{clark2018simple}} The \\textsc{DocumentQA}\\xspace system adapted passage-level reading comprehension to the multi-document setting. It samples multiple paragraphs from documents during training, and uses a shared normalization training objective that encourages the model to produce globally correct output. The \\textsc{DocumentQA}\\xspace model uses a sophisticated neural pipeline consisting of multiple CNN and bidirectional GRU layers, coupled with self-attention. For \\textsc{DocumentQA}\\xspace, we had a number of pre-trained models to choose from, and we use the one trained on TriviaQA (\\texttt{TriviaQA-web-shared-norm}) as it produced the best results on our dev set. Default configuration was used for the remaining parameters\\footnote{\\url{https:\/\/github.com\/allenai\/document-qa}}.\n\n(iii) \\myparagraph{\\textsc{DrQA}\\xspace~\\cite{chen2017reading}} This system combines a search component based on bigram hashing and TF-IDF matching with a multi-layer RNN  model trained to detect answers in paragraphs. Since we do not have passages manually annotated with answer spans, we run the \\textsc{DrQA}\\xspace model pre-trained on SQuAD~\\cite{rajpurkar2016squad} on our test questions with passages from the ten documents as the source for answer extraction. Default configuration settings were used\\footnote{\\url{https:\/\/github.com\/facebookresearch\/DrQA}}.\n\n\\subsubsection{Graph baselines}\n\nWe compare our GST-based method to simpler graph algorithms based on bread-first search (BFS) and \\textsc{ShortestPaths}\\xspace as answering mechanisms:\n\n(i) \\myparagraph{\\textsc{BFS}\\xspace~\\cite{kasneci2009star}} In \\textsc{BFS}\\xspace for graph-based QA, iterators start from each anchor node in a round-robin manner, and whenever at least one iterator from each anchor group meets at some node, it is marked as a candidate answer. At the end of $1000$ iterations, answers are ranked by the number of iterators that met at the nodes concerned.\n\n(ii) \\myparagraph{\\textsc{ShortestPaths}\\xspace} As another intuitive graph baseline, shortest paths are computed between every pair of anchors, and answers are ranked by the number of shortest paths they lie on.\n\nWe perform fine-tuning for the \\textsc{BFS}\\xspace and \\textsc{ShortestPaths}\\xspace baselines by finding the best thresholds for alignment insertion ($\\tau_{align}^E$ and $\\tau_{align}^P$) for these methods, using the development set.\n\n\\subsection{Metrics}\n\\label{subsec:metrics}\n\nMost systems considered return exactly one answer (\\textsc{UniK-QA}\\xspace, \\textsc{PathRetriever}\\xspace, \\textsc{DocumentQA}\\xspace, \\textsc{DrQA}\\xspace, \\textsc{QAnswer}\\xspace, \\textsc{Platypus}\\xspace) and so we use Precision@1 (P@1) as our metric. P@1 is a standard one for factoid QA~\\cite{saharoy2021question}: in several practical use cases like voice-based personal assistants, only one best answer can be returned to the user. The gold answer sets for all our benchmarks are entities and literals grounded (or canonicalized) via the Wikidata KG. Thus, for KG-QA, evaluation is standard: answers are already KG entities and literals, and when the top-$1$ matches any of the ground truth exactly, P@1 = 1, else 0. This is trickier for text, or KG+Text (when the answer source is text) settings, where the system response can be an arbitrary span of text (\\phrase{The Rhine river}, \\phrase{lotr}, \\phrase{two million dollars}, etc.). \nThis also applies for \\textsc{UniK-QA}\\xspace, that uses a generative model for formulating a textual answer, that may not exactly be found in any source evidence.\nWe tackle such cases in a \\textit{KG-grounded evaluation setup} as follows: when the returned answer span matches a Wikidata entity\/literal in the ground truth set exactly, P@1 = 1. If no matches are found, we search for the string in KG entity aliases of the ground truth answers. If an exact match is found, then P@1 = 1, else we proceed to a human evaluation as discussed below.\n\n\\myparagraph{User evaluation} For the remaining cases, we perform a crowdsourced human evaluation via Amazon Mechanical Turk (AMT)\\footnote{\\url{https:\/\/www.mturk.com\/}}. The task is straightforward: given a factoid question and an answer string, verify its correctness by matching against the gold answer set and optionally searching the Web in unsure cases.\nSystem response strings were pooled for every question to reduce annotation overhead (all string responses to a question like \\utterance{Which river flows through Austria and Germany?} were grouped within an AMT HIT, a unit task on the platform).\nResponses from the different systems were shuffled with respect to their ordering in the AMT interface to avoid \\textit{position bias} in annotations.\nTo prevent spam, only Master Turkers\\footnote{\\url{https:\/\/www.mturk.com\/help}} were allowed to participate. Workers were paid at the standard rate of $15$ USD per hour.\nFor being robust to noisy annotations, three labels were collected per QA pair, and the majority vote was taken as the answer.\nFor all questions where a system-generated answer is deemed as correct by human annotation majority vote, P@1 = 1. For all remaining cases, P@1 = 0.\nOn examining anecdotal cases, we found that this human evaluation to be particularly worthwhile: the matching task is non-trivial -- matching abbreviations (St. with Street or Saint), noun phrases (Rhine with The Rhine River), varying date formats, and incomplete KB aliases are some of the particularly tricky cases that would have made a machine evaluation noisy.\n\n\\begin{table} [t] \\small\n\t\\centering\n\t\\caption{Basic statistics for the AMT study.}\n\t \\vspace*{-0.3cm}\n\n\n\t\\begin{tabular}{p{3.5cm} p{9.5cm}}\n\t\t\\toprule\n\t\tTitle\t\t\t            &   Verify correctness of machine-generated answers to fact-based questions\t\\\\\n\t\tDescription\t\t            &   Match machine-generated answers to a fact-based question and its set of correct answers.\n\t\t                                If you consider the pair to match, mark ``yes'', otherwise ``no''.\n\t\t                                When you are unsure, please consult Web Search.\n\t\t                                Cases to look out for are abbreviations, partial matches, and alternative formulations,\n\t\t                                i.e., the machine answer is correct but the exact surface forms vary.       \\\\ \\midrule\n\t\tParticipants\t\t\t    &   $870$ unique Master Turkers                                                  \\\\\n\t\tTime allotted per HIT\t    &   $45$ minutes maximum                                                        \\\\ \n\t\tTime taken per HIT \t        &   $60$ minutes on average                                                     \\\\ \n\t\tAnnotations per QA pair     &   $3$                                                                         \\\\\n\t\tQuestions in one HIT        &   $20$ (About $200$ QA pairs)                                                 \\\\\n\t\tPayment per HIT\t\t\t    &   $5$ USD                                                                     \\\\  \\midrule\n\t\tTotal no. of questions      &   $6,952$ (over all benchmarks, c.f. Table~\\ref{tab:data})                     \\\\\n\t    Total no. of QA pairs       &   $87,387$  (over all benchmarks, systems, and sources)                       \\\\   \n        Central tendency of \\#pairs &   $12.57$ (mean), $11$ (median), $11$ (mode)                                  \\\\  \\bottomrule\n\t\\end{tabular}\n\t\\label{tab:amt}\n\\end{table}\n\n\\section{\\textsc{Uniqorn}\\xspace: Graph Construction}\n\\label{sec:method-1}\n\n\\begin{figure} [t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{images\/uniqorn-block-diagram.pdf}\n\t\\caption{System architecture of \\textsc{Uniqorn}\\xspace.}\n\t\\label{fig:block}\n\t\\vspace*{-0.3cm}\n\\end{figure}\n\nFig.~\\ref{fig:block} gives an overview of the \\textsc{Uniqorn}\\xspace\narchitecture. The two main stages -- construction of the\nquestion-specific context graph (Sec.~\\ref{sec:method-1}), and \ncomputing answers by Group Steiner Trees (Sec.~\\ref{sec:method-2})\non this context graph --\nare described in this section and the next.\n\nWe describe the XG construction, using our example question\nfor illustration:\n\\utterance{director of the western for which Leo won an Oscar?}\nInstantiations of key factors in the two settings, KG-QA and Text-QA, are \nshown\nin Table~\\ref{tab:kg-vs-text}. The heterogeneous setting can be understood\nas simply the union of the two individual cases.\nThe context graph is built in two stages: (i) identifying question-relevant evidences from the knowledge sources, and (iii) creating a graph from these top evidences.\n\n\\begin{table} [t]\n\t\\centering\n\t\\resizebox*{\\textwidth}{!}{\n\t\t\\setlength{\\tabcolsep}{0.5em}\n\t\t\\begin{tabular}{l l l}\n\t\t\t\\toprule\n\t\t\t\\textbf{Scenario}\t\t        & \\textbf{KG}\t\t\t\t\t\t\t    & \\textbf{Text}\t\t\t\t\t\t\\\\ \\toprule\n\t\t\tTriples in XG\t\t\t        & NERD + KG-lookups + BERT\t\t\t\t    & IR system + BERT + Open IE \\\\ \\midrule\n\t\t\tNode types\t\t\t\t        & Entity, predicate, type, literal\t\t    & Entity, predicate, type\t\t\t\\\\\n\t\t\tType nodes \t\t\t            & \\struct{instance of, occupation} triples  & Hearst patterns \t\t\t\t\t\\\\ \\midrule\n\t\t\tEdge types\t\t\t\t        & Triple, alignment, type\t\t\t\t    & Triple, alignment, type \t\t\t\\\\\n\t\t\tEntity alignments\t\t        & None \t\t\t\t\t\t                & With character trigrams   \t\t\\\\\n\t\t\tPredicate and type alignments\t& None\t\t\t\t\t                    & With word embeddings\t\t\t\t\\\\ \\midrule\n\t\t\tNode weights\t\t\t        & With BERT \t\t\t\t                & With BERT \t\t\t            \\\\\n\t\t\tEdge weights\t\t\t        & With BERT\t\t\t\t\t\t\t        & With BERT \t                    \\\\ \\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\caption{Instantiations of different factors of XGs from KG and text corpus.}\n\t\\label{tab:kg-vs-text}\n\t\\vspace*{-0.7cm}\n\\end{table}\n\n\\subsection{Retrieving question-relevant evidences}\n\\label{subsec:evi-retr}\n\nOur underlying goal is to reduce the huge knowledge repositories to a reasonably-sized question-relevant subgraphs, over which graph algorithms can be run with interactive response times.\n\n\\myparagraph{From knowledge graph} \nA typical curated KG $K$ contains billions of facts, with millions of entities and thousands of predicates, occupying multiple terabytes of disk space.\nTo reduce this enormous search space in $K$, \nwe first identify entities $E(q)$\nin the question $q$\nby using methods for named entity recognition and disambiguation (NERD)~\\cite{ferragina2010tagme,hoffart2011robust,van2020rel,li2020efficient}.\nThis produces KG entities\n(\\struct{LeonardoDiCaprio}, \\struct{AcademyAwards})\nas output.\nNext, all KG triples are fetched that are in the $1$-hop\nneighborhood of an entity in $E(q)$\\footnote{One entity to another entity is considered one ``hop''. The $2$-hop\nentity neighborhood in a KG can be enormously large,\nespecially for popular entities\nlike countries, with thousands of 1-hop neighbors.\nThis is further exacerbated by proliferations via type nodes\n(all humans are within two hops of each other on Wikidata). See~\\cite{christmann2022beyond} for more on KG neighborhoods.}.\nTo reduce this large and noisy set of facts (equivalently, KG-evidences) to a few question-relevant ones (to obtain $T^K(q)$), we fine-tune BERT~\\cite{devlin2019bert} (Sec.~\\ref{subsec:bert-tune}).\n\n\\myparagraph{From text corpus} The Web is the analogue of large KGs in text-QA.\nSimilar to the case for the KG, we \ncollect a small set of potentially relevant documents for $q$ from $D$\nusing a commercial search engine (for $D$ being the Web) or\nan IR system like ElasticSearch (for $D$ being a fixed corpus\nsuch as Wikipedia full-text).\n\nKG-style entity-centric retrieval is not a practical approach for open-domain text: it requires entity linking on a potentially large set of sentences, greatly limiting the efficiency of an on-the-fly procedure.\nRather, we take a noisy and recall-oriented process:\nwe locate question tokens (stopwords not considered) within the relevant documents, and consider a window of words (window length = $50$ in all our experiments) to each side of a match as a question-relevant snippet $\\mathcal{S}$. In case two snippets overlap, they are merged to form a longer snippet.\nThese snippets form our candidate text-evidences for locating an answer, and analogous to the KG facts obtained via NERD, are passed on to a BERT-based pruning model.\n\n\\subsection{Finding question-relevant evidences with BERT}\n\\label{subsec:bert-tune}\n\n\\myparagraph{Training a classifier for question-relevance} The goal our fine-tuned BERT model is to classify an evidence as being relevant to the question or not. In other words, we want a model that can score each evidence based on its likelihood of containing the answer. To build such a model, we prepare training data as follows. The (KG fact\/text snippet) retrieval step above feeds a set of evidences to the classifier. Out of these evidences, the ones that contain the gold answer(s) are labeled as \\textit{positive instances}. This is a \\textit{distant or noisy supervision signal} as the mere \\textit{presence} of an answer is not necessarily conclusive evidence of relevance to the question: the fact contained in the evidence may not be the one intended in the question. As an example, consider the question \\utterance{Who played Hugh in The Revenant?}: the KG fact \\struct{The Revenant, award received, Academy Award for Best Actor; winner, Leonardo DiCaprio} would be deemed relevant due to the presence of the gold answer \\struct{Leonardo DiCaprio}. Nevertheless, the method works: such false positives are rare in reality. For each positive instance, we randomly sample five \\textit{negative instances} from the evidences that do not contain the answer. Sampling question-specific negative instances helps learn a discriminative classifier, as all negative instances are guaranteed to contain at least one entity\/keyword from the question. Using \\textit{all facts} that do not contain an answer as negative examples would result in severe class imbalance, as this is much higher than the number of positive instances.\n\nWe then pool together the $\\langle$question, evidence$\\rangle$ paired positive and negative instances for all training questions. Text snippets are already in natural language and amenable for use in a BERT model.\nTo bring KG facts close to an NL form, we \\textit{verbalize} them by concatenating their constituents; qualifier statements are joined using \\phrase{and}~\\cite{oguz2021unikqa}. \nFor example, the verbalizaton of: \\struct{<The Revenant, nominated for, Academy Award for Best Director; nominee, Alejandro Gonz\\'{a}lez I\\~{n}\\'{a}rritu>}. This fact has one qualifier, and would be verbalized simply as \\phrase{The Revenant nominated for Academy Award for Best Director and nominee Alejandro Gonz\\'{a}lez I\\~{n}\\'{a}rritu.}.\nThe questions and the verbalized evidences, along with the binary ground-truth labels, are fed as training input to a \\textit{sequence pair classification model} for BERT: one sequence is the question, the other being the evidence.\n\n\\myparagraph{Applying the classifier}\nFollowing~\\cite{devlin2019bert}, the question and the evidence texts are concatenated with the special separator token \\struct{[SEP]} in between, and the special classification token \\struct{[CLS]} is prepended to this sequence. The final hidden vector corresponding to \\struct{[CLS]}, denoted by $C \\in \\mathbb{R}^H$ ($H$ is the size of the hidden state), is considered to be the accumulated representation. Weights $W$ of a final classification layer are the introduced during fine-tuning, where $W \\in \\mathbb{R}^{\\mathcal{L} \\times H}$, where $\\mathcal{L}$ is the number of class labels ($\\mathcal{L} = 2$ here, an evidence is either question-relevant or it is not). The value $\\log(\\text{softmax}(CW^T))$ is used as the classifier loss. Once this classifier is trained, given a new $\\langle$question, evidence$\\rangle$ pair, it outputs the probability (and the label) of the evidence containing an answer to the question. We make this prediction for all candidate evidences retrieved for a question, and sort them in descending order of this question relevance likelihood. We pick the top-$\\varepsilon$ evidences from here as our question-relevant set for constructing the XG. \n\n\\subsection{Materializing the context graph}\n\\label{subsec:xg-build}\n\n\\myparagraph{For KG evidences} Evidences from the KG are facts, and can be trivially cast into a graph that is a much smaller subset of the entire KG. Entities, predicates, types and literals constitute nodes, while edges represent connections between pieces of the same fact (recall Fig.~\\ref{fig:xg-kg}). Each distinct entity, type, and literal form one node, which (possibly repeated) predicate gets its own node (recall Sec.~\\ref{subsec:graph-con}). \nWe then add type information to this graph, useful for QA in several ways~\\cite{saharoy2021question,abujabal2017automated,yavuz2016improving,ziegler2017efficiency}. For this, we look up the most frequent KG types for each entity (an entity can belong to multiple KG types, but usually one is the most notable) in the qualifying set of evidences, and additionally look up occupations for humans (recall Sec.~\\ref{subsec:graph-con}), and add these triples to the graph.\nTo ensure connectivity in this graph, as far as possible, we add shortest paths from the KG between NERD entities detected in this question to this graph (complex questions often have more than one entity). This often helps reintroduce question-relevant connections between entities in the question. Such shortest paths can be obtained via the \\textsc{Clocq}\\footnote{\\url{https:\/\/clocq.mpi-inf.mpg.de\/}} API.\nThe largest connected component (LCC) is extracted from this graph, and the final structure thus obtained constitute $XG^K(q)$, being made up of individual triples $T^K(q)$. The LCC is expected to cover the most pertinent information and concepts relevant to the question.\n\n\\myparagraph{For text evidences} There is no natural graph structure in the NL snippets, so we induce it using a simple version of open information extraction (open IE). The goal of open IE is to extract informative riples from raw text sources. As off-the-shelf tools like Stanford Open IE~\\cite{angeli2015leveraging},\nOpenIE 5.1~\\cite{saha2018open},\nClausIE~\\cite{del2013clausie}\nor MinIE~\\cite{gashteovski2017minie},\nall have limitations regarding either precision or recall or efficiency, we developed our own custom-made open IE extractor. We start by part-of-speech (POS) tagging and named entity recognition (NER) on the original sentences from $D$ (and not on the snippets $\\mathcal{S}$, to preserve necessary sequence information vital for such taggers).\nThis is followed by light-weight coreference resolution by replacing each third person personal and possessive pronoun (\\phrase{he, him, his, she, her, hers})\nby its nearest preceding entity of type \\struct{PERSON}. We then define\na set of POS patterns that may indicate an entity or a predicate. \nEntities are marked by an unbroken sequence of\n\\struct{nouns, adjectives, cardinal numbers},\nor mention phrases from the NER system\n(e.g., \\phrase{Leonardo DiCaprio}, \\phrase{2016 American western film}, \\phrase{Wolf of Wall Street}). \nTo capture both verb- and noun-mediated relations, predicates correspond to\nPOS patterns \\struct{verb}, \\struct{verb+preposition} or \\struct{noun+preposition}\n(e.g.,\n\\phrase{produced}, \\phrase{collaborated with}, \\phrase{director of}).\nSee node labels in Fig.~\\ref{fig:xg-text} for examples. \nThe patterns are \napplied to\neach snippet $\\mathcal{S}_j \\in D$, to produce a markup like\n$\\mathcal{S}_j \\equiv E_1 \\ldots E_2 \\ldots P_1 \\ldots E_3 \\ldots P_2 \\ldots E_4.$\nThe ellipses ($\\ldots$) denote intervening words in $\\mathcal{S}_j$.\nFrom this markup, \\textsc{Uniqorn}\\xspace finds all $(E_{i_1}, E_{i_2})$ pairs that have\nexactly one predicate $P_k$ between them, this way creating triples.\n$<\\!E_1, P_1, E_3\\!>$ and $<\\!E_2, P_1, E_3\\!>$\n(but excluding\n$<\\!E_1, P_2, E_4\\!>$). \nPatterns from~\\cite{yahya2014renoun}, specially\ndesigned for noun phrases as relations (e.g., \n\\phrase{Oscar winner}), are applied as well.\nSnippets that contain two (or more) entities but no predicate contribute triples with a special \\struct{cooccurs} predicate.\nFor example if a snippet contains three entities $E_1, E_2, E_3$ but zero predicates, we would add triples:\n\\struct{<$E_1$, cooccurs, $E_2$>, <$E_1$, cooccurs, $E_3$>, <$E_2$, cooccurs, $E_3$>} to the open IE triples.\nSuch rules help tap into information in snippets like \\phrase{Leonardo was in Inception}, where the intended relation is implicit (\\phrase{starred}).\n The rationale for this heuristic extractor is to \nachieve high \\textit{recall} towards answers, at the cost of \nintroducing noise. The noise is taken care of in the answering\nstage later.\n\nAs for KG evidences, we would like to to extract entity type information here as well.\nTo this end, we leverage Hearst patterns~\\cite{hearst1992automatic}, like\n\\phrase{$NP_2$ such as $NP_1 \\ldots$}\n(matched by, say, \\phrase{\\underline{western films} such as\n\t\\underline{The Revenant}})\n\\phrase{$NP_1$ is a(n) $NP_2 \\ldots$}\n(matched by, say, \\phrase{\\underline{The Revenant} is a\n\t\\underline{2015 American western film}}),\nor \\phrase{$NP_1$ and other $NP_2 \\ldots$}\n(matched by, e.g., \\phrase{\\underline{Alejandro I\\~{n}\\`{a}rritu} and other\n\\underline{Mexican film directors} $\\ldots$}).\nHere $NP$ denotes a noun phrase, detected by a constituency parser.\nThe resulting triples about type membership\n(of the form: \\struct{<$NP_1$, type, $NP_2$>}\nfor e.g., \\struct{<The Revenant, type, 2015 American western film>})\nare added to the triple collection.\nFinally, as for the KG-QA case,\nall triples are joined by $S$ or $O$ with exact string matches,\nand the LCC is computed, to produce the final $XG^D(q)$. \nTo compensate for \nthe diversity of surface forms where different strings may denote\nthe same entity or predicate, alignment edges\nare inserted into $XG^D(q)$ for node pairs\nas discussed in Sec.~\\ref{subsec:graph-con}.\n$XG^D(q)$ is made up of individual triples $T^D(q)$.\nThus, a large quasi KG $K^D$ is never materialized, and we directly construct $XG^D$, on which our graph algorthims may run.\n\n\\myparagraph{For heterogeneous evidences} The union of $XG^K$ and $XG^D$ form $XG^H$, the heterogeneous context graph for question $q$. Fine-tuned BERT models are run individually on the KG and text repositories for retrieving the source-specific evidences, which are processed as described above.\n\n\n\\section{\\textsc{Uniqorn}\\xspace: Graph Algorithm}\n\\label{sec:method-2}\n\nFor the given context graph $XG(q)$, we find candidate answers $a \\in A$ as follows. First, nodes in the $XG$ are identified\nthat best capture the question; these nodes become anchors.\nFor $XG^K$, question entities detected by NERD systems become entity anchors.\nFor predicates, types and literals, any node with a token in its label that matches any of the question tokens, becomes an anchor (\\struct{cast member} becomes an anchor if the question has \\utterance{Who was \\underline{cast} as ...}). To ensure better semantic coverage, node labels are augmented with aliases from the KB, that are a rich source of paraphrases.\nFor example, Wikidata contains the following aliases for \\struct{cast member}: \n\\phrase{starring}, \\phrase{actor}, \\phrase{actress}, \\phrase{featuring}, and so on.\nThus, \\struct{cast member} will become an anchor node if the question has any of the synonyms above.\nFor $XG^D$, any node with a token in its label that matches any of the question tokens becomes an anchor.\nNode and edge weights in $XG^K$ or $XG^D$ are obtained by using the BERT-based scores that the original evidence (source of the node or edge) received from the fine-tuned model.\nIf a node or edge originates from multiple evidences, their scores are averaged.\nThis helps us harness redundancy of information across evidences.\n\nAnchors are grouped into equivalence classes $\\{\\mathcal{A}^k\\}$ based on\nthe question token that they correspond to.\nAt this stage, we have the directed and weighted\ncontext graph $XG$ as a $6$-tuple:\n$XG^{(\\cdot)}(q) = (N, \\mathcal{E}, N^T, \\mathcal{E}^T, N^W, \n\\mathcal{E}^W)$.\nFor simplification, \nwe disregard the direction of edges, turning $XG$ into an\nundirected graph.\n\n\\myparagraph{Group Steiner Tree} We postulate that the criteria for identifying good answer candidates in the XG are as follows:\n(i) answers lie on \\textit{paths} connecting anchors;\n(ii) \\textit{shorter} paths with \\textit{higher} weights are more likely to contain correct answers;\nand (iii) including at least one instance of an anchor from each \\textit{group} is necessary\n(to satisfy all conditions in a complex question $q$). \nFormalizing these desiderata leads us to the notion of\n\\textit{Steiner Trees}~\\cite{feldmann2016equivalence,bhalotia2002keyword,kacholia2005bidirectional,kasneci2009star}:\nfor an undirected weighted graph, and a subset of nodes \ncalled \\phrase{terminals}, find the tree of least cost that connects \nall terminals.\nIf the number of terminals is two, then this is the weighted\nshortest path problem, and if all nodes of the graph are terminals, this \nbecomes the minimum spanning tree (MST) problem.\nIn our setting, the graph\nis the $XG$, and terminals are the anchor nodes. \nAs anchors are arranged into groups, we pursue the\ngeneralized notion of\n\\textit{Group Steiner Trees (GST)}~\\cite{garg2000polylogarithmic,li2016efficient,ding2007finding,shi2020keyword,chanial2018connectionlens,sun2021finding}:\ncompute a minimum-cost Steiner Tree \nthat connects at least one terminal from each group,\nwhere weights of edges $\\mathcal{E}_k$ are\nconverted into costs\nas $\\textrm{cost}(\\mathcal{E}_k)  = 1 - M_W^\\mathcal{E}(\\mathcal{E}_k)$. \nAt this point, the reader is referred to Fig.~\\ref{fig:xg} again,\nfor illustrations of what GSTs look like (shown\nin orange).\nTo tackle questions with a chain-join~\\cite{saharoy2021question} component (like \\utterance{profession of father of DiCaprio?}), the complete evidence of predicate anchors (resembling predicates dangling from the tree) is admitted into the GST.\n\nFormally, the GST problem in our setting can \nbe defined as follows. Given\na question $q$ with $|q|$ tokens,\nan undirected and weighted graph $XG(q) = (N, \\mathcal{E})$,\nand groups of anchor nodes\n$\\{\\mathcal{A}^1, \\ldots \\mathcal{A}^{|q|}\\}$ with\neach $\\mathcal{A}^k \\subset N$, find the\nminimum-cost tree $\\mathbb{T}^* = (N^*, \\mathcal{E}^*) =\n\\arg \\min_i \\textrm{cost}(\\mathbb{T}_i, XG)$,\nwhere $\\mathbb{T}_i$ is any tree\nthat connects at least one node from each of\n$\\{\\mathcal{A}^1, \\ldots \\mathcal{A}^{|q|}\\}$, such that\n$\\mathcal{A}^k \\cap N^* \\neq \\phi$ for each $\\mathcal{A}_k$,\nand cost of $\\mathbb{T}_i$,\n$\\textrm{cost}(\\mathbb{T}_i) = \\sum_{k}\\textrm{cost}(\\mathcal{E}_k)$,\nwhere $\\mathcal{E}_k \\in \\mathbb{T}_i$.\n\nWhile finding the GST is an NP-hard problem, there are\napproximation algorithms~\\cite{garg2000polylogarithmic}, and also exact methods that are\nfixed-parameter tractable with respect to the number of terminal nodes.\nWe adapted the method\nof Ding et al.~\\cite{ding2007finding} from the latter family, which is exponential in the\nnumber of terminals \nbut $O(n \\; log \\; n)$ in the graph size.\nLuckily for us, the number of anchors\nis indeed typically much less of a bottleneck than the sizes of\nthe $XG$\nin terms of nodes and edges.\nSpecifically, the terminals are the anchor nodes derived\nfrom the question tokens -- so their numbers are not\nprohibitive with respect to computational costs.\n\nThe algorithm is based on \\textit{dynamic programming} and \nworks as follows. It starts from a set of singleton trees, one for\neach terminal group, rooted at one of the corresponding anchor nodes. These\n\\textit{trees are grown} iteratively by exploring immediate \nneighborhoods for\nleast-cost nodes as expansion points. {Trees are merged} \nwhen common nodes are encountered while growing. The trees are stored\nin an efficient implementation of priority queues with \\textit{Fibonacci heaps}.\nThe process terminates when a GST is found that {covers} all terminals \n(i.e., one per group).\nBottom-up dynamic programming guarantees that the\nresult is \\textit{optimal}.\n\n\\myparagraph{Relaxation to top-$k$ GSTs} It is possible that the GST simply\nconnects a terminal from each anchor group, without having any internal \nnodes at all, or with predicates and\/or types as internal nodes. Since we\nneed entities or literals as answers, such\npossibilities necessitate a relaxation of our solution to \ncompute a number of top-$k$\nleast-cost GSTs.\nGST-$k$ ensures that we always get a non-empty\nanswer set, albeit at the cost of making some detours in the graph. \nMoreover, using GST-$k$ provides a natural answer ranking strategy, where\nthe score for an answer can be appropriately reinforced if it appears in\nmultiple such low-cost GSTs. \nThis postulate, and the effect of $k$, is later quantified in our experiments.\nNote that since the tree with the least cost is always kept\nat the top of the priority\nqueue, the $k$ trees can be found in the increasing order of cost, and\nno additional sorting is needed.\nIn other words, the priority-queue-based implementation of the GST algorithm~\\cite{ding2007finding} \neasily supports this top-$k$ computation.\nThe time\nand space complexities for obtaining GST-$k$ is the same as that\n for GST-$1$.\nFig.~\\ref{fig:gst-k} gives an example of GST-$k$ ($k = 3$).\n\n\\begin{figure} [t]\n  \\centering\n    \\includegraphics[width=0.5\\columnwidth]{images\/gst-k.pdf}\n      \\caption{Illustrating GST-$k$, showing edge costs and node weights. Anchors (terminals) and answer candidates (non-terminals A1, A2, A3) are shown in black and white circles respectively. \\{(C11, C12), (C21, C22, C23), (C31)\\} represent anchor groups. Edge costs are used in finding GST-$k$, while node weights are used in answer ranking. A1 is likely to be a better answer due to its presence in two GSTs in the top-3.}\n      \\label{fig:gst-k}\n      \\vspace*{-0.5cm}\n\\end{figure}\n\n\\myparagraph{Answer ranking} Non-terminal entities in GSTs are candidates for final answers. However, this mandates ranking. \nWe use the number of top-$k$ GSTs that an answer candidate lies on, as the ranking criterion. Alternative choices, like weighting these trees by their total node weight, tree cost, or the answers' proximity to anchor nodes, are investigated in Sec.~\\ref{subsec:analysis}. The top-ranking answer is presented to the end user.\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\subsection{QA over Knowledge Graphs}\n\\label{subsec:kg-qa}\n\nThe inception of large KGs like Freebase~\\cite{bollacker2008freebase}, YAGO~\\cite{suchanek2007yago}, DBpedia~\\cite{auer2007dbpedia}, and Wikidata~\\cite{vrandevcic2014wikidata}, gave rise to question answering over knowledge graphs (KG-QA) that typically provides answers as \\textit{single entities} or \\textit{entity lists} from the KG. KG-QA is now an increasingly active research avenue, where the goal is to translate an NL question into a structured query, usually in SPARQL syntax or an equivalent logical form, that is directly executable over the KG triple store containing entities, predicates, classes and literals~\\cite{wu2020perq,qiu2020stepwise,vakulenko2019message,bhutani2019learning,christmann2019look}.\nChallenges in KG-QA include entity disambiguation, bridging the vocabulary gap between phrases in questions and the terminology of the KG, and finally, constructing the SPARQL(-like) query. Early work on KG-QA built on paraphrase-based mappings and question-query templates that typically had a single entity or a single predicate as slots~\\cite{berant2013semantic,unger2012template,yahya2013robust}. This direction was advanced  by~\\cite{bast2015more,bao2016constraint,abujabal2017automated,hu2017answering}, including templates that were automatically learnt from graph patterns in the KG. Unfortunately, this dependence on templates prevents such approaches from coping with arbitrary syntactic formulations in a robust manner. This has led to the development of deep learning-based methods with sequence models, and especially\nkey-value memory networks~\\cite{xu2019enhancing,xu2016question,tan2017context,huang2019knowledge,chen2019bidirectional,jain2016question}.\nThese have been most successful on\nbenchmarks like WebQuestions~\\cite{berant2013semantic} and QALD~\\cite{ngomo20189th}. However, all\nthese methods critically build on sufficient amounts of training data\nin the form of question-answer pairs. In contrast, \\textsc{Uniqorn}\\xspace is fully\nunsupervised and needs neither templates nor training data.\n\n\\textit{Complex question answering} is an area of intense focus in KG-QA \nnow~\\cite{lu2019answering,bhutani2019learning,qiu2020stepwise,ding2019leveraging,vakulenko2019message,hu2018state,jia18tequila}, where the general approach is often \nguided by the existence and detection of substructures for the executable query. \\textsc{Uniqorn}\\xspace treats this as a potential drawback and adopts a joint disambiguation of \nquestion concepts using algorithms for Group Steiner Trees, instead of looking for nested question units that can be mapped to simpler queries. \nApproaches based on question decomposition (explicit or implicit) are brittle due to the huge variety of question formulation patterns (especially for complex \nquestions), and are particularly vulnerable when questions are posed in telegraphic form (\\utterance{oscar-winnning nolan films?}, has to be interpreted as \n\\utterance{Which films were directed by Nolan and won an Oscar?}: this is highly non-trivial).\nAnother branch of complex KG-QA rose from the task of\nknowledge graph reasoning (KGR)~\\cite{das2018go,qiu2020stepwise,dhingra2020differentiable,cohen2020scalable,zhang2018variational}, where the key idea is given a KG entity (\\struct{Albert Einstein}) and a textual relation (\\phrase{nephew}), the best KG path from the input entity to the target entity (answer) is sought. This can be generalized into a so-called QA task where topic (question) entity is known and the question is assumed to be a paraphrase of the multi-hop KG relation (there is an assumption that \\phrase{nephew} is not directly a KG predicate). Nevertheless, this is a restricted view of complex KG-QA, and only deals with such indirection or chain questions (\\phrase{nephew} has to be matched with \\struct{sibling} followed by \\struct{child} in the KG),\nevaluated on truncated subsets of the full KG that typically lack the complexity of qualifier triples.\n\n\n\\subsection{QA over Text}\n\\label{subsec:text-qa}\n\nOriginally, in the late 1990s and early 2000s, question answering considered textual document collections as its underlying source. Classical approaches based on statistical scoring~\\cite{ravichandran2002learning,voorhees1999trec} extracted answers as short text units from passages or sentences that matched most cue words from the question. Such models made intensive use of IR techniques for scoring of sentences or passages and aggregation of evidence for answer candidates. IBM Watson~\\cite{ferrucci2010building}, a thoroughly engineered system that won the Jeopardy! quiz show, extended this paradigm by combining it with learned models for special question types. TREC ran a QA benchmarking series from 1999 to 2007, and more recently revived it as the LiveQA~\\cite{agichtein2015overview} and Complex Answer Retrieval (CAR) tracks~\\cite{dietz2017trec}. \n\n\\textit{Machine reading comprehension (MRC)} was originally motivated by the goal of whether algorithms actually \\textit{understood textual content}. This eventually became a QA variation where a question needs to be answered as a short \\textit{span of words} from a given text paragraph~\\cite{rajpurkar2016squad,yang2018hotpotqa}, and is different from the typical fact-centric answer-finding task in IR. Exemplary approaches in MRC that extended the original single-passage setting to a multi-document one can be found in DrQA~\\cite{chen2017reading} and DocumentQA~\\cite{clark2018simple} (among many, many others). \nIn the scope of such text-QA, we compared with, and outperformed both the aforementioned models, which can both select relevant documents and extract\nanswers from them. Traditional fact-centric QA over text, and multi-document MRC are recently emerging as a joint topic referred to as\n\\textit{open-domain question answering}~\\cite{lin2018denoising,dehghani2019learning,wang2019document}. Open-domain QA tries to combine an IR-based retrieval pipeline and NLP-style reading comprehension, to produce crisp answers extracted from passages retrieved on-the-fly from large corpora.\n\n\\subsection{QA over Heterogeneous Sources}\n\\label{subsec:hybrid-qa}\n\nLimitations of QA over KGs has led to a revival of considering textual sources, in combination with KGs~\\cite{savenkov2016knowledge,xu2016question,sun2018open,sun2019pullnet}. Early methods like PARALEX~\\cite{fader2013paraphrase} and OQA~\\cite{fader2014open} supported noisy KGs in the form of triple spaces compiled via Open IE~\\cite{mausam2016open} on Wikipedia articles or Web corpora. TupleInf~\\cite{khot2017answering} extended and generalized the Open-IE-based PARALEX approach to complex questions, and is geared specifically for \\textit{multiple-choice answer options},\nand is thus inapplicable for our task. TAQA~\\cite{yin2015answering} is another generalization of Open-IE-based QA, by constructing a KG of $n$-tuples from Wikipedia full-text and question-specific search results. \nUnfortunately this method is restricted to questions with prepositional and adverbial constraints only.\nWebAsKB~\\cite{talmor2018web} is an MRC-inspired method that addressed complex questions by decomposing them into a sequence of simple questions, but relies on crowdsourced large-scale training data. Some methods for such hybrid QA start with KGs as a source for candidate answers and use text corpora like Wikipedia or ClueWeb as additional evidence~\\cite{xu2016question,das2017question,sun2018open,sun2019pullnet,sydorova2019interpretable,xiong2019improving}, or start with answer sentences from text corpora and combine these with KGs for giving crisp entity answers~\\cite{sun2015open,savenkov2016knowledge}. Most of these are based on neural networks, and\nare only designed for simple questions like those in the WebQuestions~\\cite{berant2013semantic}, SimpleQuestions~\\cite{bordes2015large},\nor WikiMovies~\\cite{miller2016key} benchmarks. In contrast,\n\\textsc{Uniqorn}\\xspace, through its anchor graphs and GSTs, can handle arbitrary kinds of complex questions and can construct explanatory evidence for its answers -- an unsolved concern for neural methods.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{Intro} Introduction}\n\nThe successful development of atomic laser-cooling techniques has facilitated the study of atomic collisions in the ultracold regime and the macroscopic quantum degeneracy of the Bose--Einstein condensate \\cite{Cornell2002,Ketterle2002}. This progress has inspired interest in the generation of ultracold molecules in prescribed low-energy internal states, opening prospects from ultrahigh precision spectroscopy to the coherent control of chemical reactions and new techniques for quantum computing \\cite{DeMille2002,Tesch2002}. Atomic cooling techniques, however, cannot be generalized even to simple diatomic molecules due to the lack of availability of a closed-loop cooling cycle within the rich molecular internal energy level structure; instead, alternative approaches must be pursued.\n\nVarious alternative direct cooling techniques have been developed that stabilize molecules efficiently into the lowest vibrational levels, but are unable to cool their translational motion significantly below the millikelvin regime \\cite{Weinstein1998,Bethlem2000}. In order to reach microkelvin or nanokelvin temperatures, it is necessary instead to associate ultracold atoms via approaches such as photoassociation \\cite{Thorsheim1987,Fioretti1998}, the manipulation of Feshbach resonances \\cite{Greiner2003,Zwierlein2003}, or three-body collisions \\cite{Jochim2003}. These processes, however, favour the formation of ultracold but vibrationally excited molecules.\n\nRecently, an incoherent optical cycling scheme was demonstrated that accumulates population in the Cs$_2$ singlet ground vibrational state via spontaneous decay \\cite{Viteau2008}. Precise knowledge of the system spectroscopy is not required; however it relies upon the serendipitous availability within Cs$_2$ of a specific continuous-wave (c.w.) pump-decay photoassociative pathway that pre-populates deeply bound vibrational states. Such a pathway is not, in general, available in other dimers, and an optical repumping scheme would require many more spontaneous decay cycles (and hence greater heating of the centre-of-mass motion) before significant enhancement of the ground vibrational state population is obtained. In certain heteronuclear systems, high-power, c.w.\\ photoassociation has been used to produce an incoherent mixture that includes $v=0$ ground-state singlet dimers \\cite{Sage2005,Deiglmayr2008}. In another approach, stimulated Raman adiabatic passage (STIRAP) has been used to transfer molecules to the lowest vibrational ground state in Rb$_2$ triplets \\cite{Lang2008} and Cs$_2$ singlets \\cite{Danzl2009,Mark2009}, as well as heteronuclear KRb molecules to both triplet and singlet $v=0$ states \\cite{Ni2008}. The transfer is coherent but requires full spectroscopic knowledge of the system.\n\nA complementary quantum control strategy to those outlined above is the coherent control \\cite{Warren1993,Rabitz2000} of the photoassociation process using broadband tailored optical fields. The broad coherent bandwidth of a Ti:sapphire ultrafast pulse is well matched to typical diatomic molecule potential binding energies, while the carrier frequency is appropriate for ground- to excited-state transitions. An ultracold scattering event may therefore be steered towards a target state through a variety of vibronic pathways, with phase and amplitude shaping of the excitation pulse offering control over the system Hamiltonian via the dipole interaction. Furthermore, closed-loop control may be applied to the choice of pulse shape in order to identify an empirical optimum. The generality of this approach is thus limited merely by the frequency and bandwidth of the laser system rather than any requirement for precise spectroscopic knowledge.\n\nThe use of ultrafast `pump' and `dump' pulses would permit the fully coherent transfer of population from free atom pairs to deeply bound vibrational states \\cite{Koch2006a}. An excited-state molecular wavepacket is formed by photoassociation, and evolves freely before being optically de-excited at a time that yields optimal Franck--Condon overlap with the target state.\nThrough the exploitation of a non-stationary coherent superposition in the excited state, this wavepacket approach thus offers the advantage of favourable Franck--Condon factors for both `pump' and `dump' stages without relying upon the overlap between individual wavefunctions of specific vibrational states. Previous ultrafast `pump-decay' photoassociation experiments have revealed a coherent quenching effect on pre-formed triplet molecules \\cite{Brown2006,Staanum2006}; a pump-dump approach may enable the coherent enhancement of bound molecular population before the onset of spontaneous decay.\n\nLearning about the dynamics present in the excited state after photoassociation is therefore an important step towards efficient stabilization into the ground state. A useful tool for resolving ultrafast molecular dynamics is the pump-probe experiment \\cite{Machholm1994,Fatemi2001}. A pump pulse generates an excitation which evolves during a controllable time delay before being detected by a probe pulse through a reaction-coordinate--selective mechanism such as ionization or bond fragmentation. Recently, a pump-probe study of the short-delay dynamics of rubidium atoms and molecules after photoassociation revealed coherent transient oscillations during the photoassociation pulse \\cite{Salzmann2008}. In this paper we report on pump-probe experiments investigating the coherent control of bound excited-state dimers over longer timescales as the first step towards the development of a pump-dump scheme to produce deeply bound ground-state Rb$_2$ singlet molecules. We demonstrate the production of bound excited-state molecules and compare the experimental observations to numerical simulations. We characterize the initial state of a background molecular population within the cold atom cloud and identify their association mechanism.\nWe show evidence that the pump pulse addresses free atom pairs with internuclear separations of \\unit[30-60]{a$_0$}, rather than the closer range background molecules. This experimental determination of the initial conditions permits improved comparison between theory and experiment, offering further insight into the reaction process.\n\n\n\\section{Methods and apparatus}\n\\label{sec:method}\n\n\\subsection{Overview}\n\\label{sec:overview}\n\n  \\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figure1.pdf}\n    \\caption{(Colour online). A schematic of the pump-probe experiment. A femtosecond pump pulse photoassociates atom pairs from the ground state, forming a wavepacket spanning a range of vibrational states within the \\mbox{5S + 5P} potential manifold. A typical simulated population distribution is shown (shaded). After a variable time delay $\\tau$, a probe pulse ionizes the excited-state population in preparation for detection via TOF mass spectrometry.} \n    \\label{fig:schematic}\n  \\end{figure}\n\nA schematic of the pump-probe experiment is given in Fig.\\ \\ref{fig:schematic}. A broadband pump pulse photoassociates ultracold ground-state 5S atom pairs within a rubidium magneto-optical trap (MOT), forming long-range molecular wavepackets within the \\mbox{5S + 5P} potential manifold. The excited dimer is allowed to evolve before being ionized by a broadband probe pulse. In order to study the excited-state dynamics over the oscillation timescales of long-range molecules, this pump-probe delay is varied over up to \\unit[250]{ps}. Atomic ions are also produced via nonresonant multi-photon processes. The nascent atomic and molecular ions are detected and distinguished via time-of-flight (TOF) mass spectrometry. In order to ensure that the atom pairs commence the experiment in their ground state, the MOT lasers (which, like the pump pulse, operate on the 5S $\\rightarrow$ 5P transition) are shuttered for a \\unit[2]{$\\mu$s} window spanning the arrival of the pump and probe pulses (see Fig.\\ \\ref{fig:timing}(a)).\n \n\\subsection{Magneto-optical trap}\n\\label{sec:MOT}\n\n  \\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{Figure2.pdf}\n    \\caption{(Colour online). (a) A schematic of the experimental timing. The MOT lasers are shut off for a \\unit[2]{$\\mu$s} window spanning the arrival of the ultrafast pulses. The TOF pushing electrode is switched to a positive voltage \\unit[3.3]{$\\mu$s} after ionization. (b) Experimental layout. Pulses from a CPA laser pass through a beamsplitter, with the majority of the pulse energy pumping the NOPA to form the probe pulses whilst the remainder is spectrally filtered in a 4f line and acts as the pump. The pulses are combined with a variable delay $\\tau$ and focussed onto the MOT. Ionized rubidium atoms and molecules are accelerated towards an MCP detector and distinguished via TOF mass spectrometry.}\n    \\label{fig:timing}\n  \\end{figure}\n\nThe choice of rubidium for this study was motivated by the suitability of its atomic and molecular spectroscopy to Ti:sapphire wavelengths, as well as the extent to which its collisional properties and photoassociation spectroscopy have been studied and catalogued. It therefore provides a convenient test bed for the control scenarios, though the concepts are expected to be of much broader utility.\n\n$^{85}$Rb atoms were cooled and trapped in a MOT generated in an ultrahigh vacuum stainless steel chamber with multiple windows for optical access. Three orthogonal pairs of counter-propagating beams of opposite circular polarizations intersected at the minimum of a quadrupole magnetic field generated by a pair of anti-Helmholtz current coils. Stray magnetic fields in the chamber were nulled using three orthogonal pairs of compensation coils. Rb was loaded into the MOT from a background vapour maintained by the application of a constant current to a series of standard commercial getters sources. \n\nThe master trapping light was generated by a home-built external-cavity diode laser (ECDL). A pick-off of the trapping light was passed through a rubidium vapour cell, and a feedback loop was used to lock its frequency to a feature in the resultant Doppler-free saturated absorption spectrum. The master laser was in turn injected into a higher power, free-running slave diode laser. The output of the slave laser was detuned \\unit[15]{MHz} to the red of the $\\textrm{F}=3 \\rightarrow \\textrm{F}'=4$ trapping transition using an acousto-optic modulator (AOM) before passing through a single-mode polarization-maintaining fibre to the MOT. A total of \\unit[30]{mW} trapping light was split equally over three retro-reflected beams of \\unit[4.0]{mm} half-maximum diameter in intensity. A second ECDL was similarly locked to the $\\textrm{F}=2 \\rightarrow \\textrm{F}'=3$ transition to repump population that accumulated in the dark $\\textrm{F}=2$ lower hyperfine state, with a power of \\unit[3]{mW} at the MOT. The AOMs were additionally employed as fast optical shutters for the trapping and repumping light.\n\nThe total fluorescence and density profile of the MOT cloud were monitored by a photodiode and pair of CCD cameras respectively. $4\\times10^6$ atoms were trapped in the MOT with a maximum density of \\unit[$1.1\\times10^9$]{cm$^{-3}$} and a half-maximum diameter of \\unit[0.8]{mm}. The MOT temperature was measured by the release-and-recapture method \\cite{Lett1988} to be \\unit[110]{$\\mu$}K. \n\n\\subsection{Ultrafast system}\n\\label{sec:ultrafast}\n\n  \\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{Figure3.pdf}\n    \\caption{(Colour online). (a) Spectral intensity and phase of the CPA pulse used to generate the pump and probe (\\unit[53]{fs} FWHM). (b) Typical pump spectrum with cut at Rb D1 line.(\\unit[85]{fs} FWHM). (c) Typical NOPA probe spectrum, filtered at \\unit[480]{nm} (\\unit[390]{fs} FWHM).}\n    \\label{fig:pulses}\n  \\end{figure}\n\nThe ultrafast pulses for this experiment were derived from a regenerative chirped-pulse amplified Ti:sapphire laser (CPA) \\cite{Backus1998} producing \\unit[250]{$\\mu$J}, \\unit[800]{nm} pulses with a \\unit[20]{nm} FWHM at a repetition rate of \\unit[2]{kHz}. Pulse characterization using spectral phase interferometry for direct electric field reconstruction (SPIDER) \\cite{Iaconis1998} revealed near--transform-limited pulses of \\unit[53]{fs} FWHM (see Fig.\\ \\ref{fig:pulses}(a)). The CPA pulses were split into two components: a few microjoules were used to generate the pump pulse while the remainder pumped a non-collinear optical parametric amplifier (NOPA) which acted as the probe \\cite{Cerullo2003}.\n\nThe pump pulse was spectrally filtered by a translatable razor blade in the Fourier plane of a zero-dispersion 4f line (\\unit[1200]{lines\/mm} diffraction gratings separated by \\unit[75]{cm}) with a resolution of \\unit[0.1]{nm} (as determined by the beam waist in the Fourier plane and the 4f line dispersion). The position of the spectral cut was set to eliminate excitation of unbound excited-state population above the \\mbox{5S + 5P$_{1\/2}$} `D1 line' atomic asymptote (see Fig.\\ \\ref{fig:pulses}(b)). Interferometric autocorrelation indicated pump pulses of \\unit[85]{fs} FWHM duration. The NOPA spectrum was tuned to \\unit[500]{nm} (\\unit[20]{nm} FWHM) to correspond to the transitions between the \\mbox{5S + 5P$_{1\/2}$} and molecular ion states (see Fig.\\ \\ref{fig:pulses}(c)).  A long-pass filter with a \\unit[480]{nm} cut-off prevented the resonant ionization of atoms. Intensity autocorrelation indicated probe pulses of \\unit[390]{fs} FWHM duration. A typical pulse energy was \\unit[4]{$\\mu$J} with a fluctuation of less than \\unit[5]{\\%}.\n\nThe pump pulse was passed through a variable delay line, combined with the probe using a dichroic mirror (see Fig.\\ \\ref{fig:timing}(b)) and focussed onto the MOT.\n\n\\subsection{Detection scheme}\n\\label{sec:detection}\n\nThe efficiency of the photoassociation process is limited by the density of the MOT; hence the Rb$_2^+$ ionization rate is exceeded by Rb$^+$ ionization despite the off-resonant nature of the latter process. The challenge is therefore to resolve the molecular ions from the atomic ions with a sensitive detection mechanism. This was achieved using TOF mass spectrometry with a multi-channel plate (MCP) detector. The ions were accelerated towards the MCP by a gated pushing electrode situated opposite the MCP inside the vacuum chamber. The electrode was switched from \\unit[0]{V} to a positive voltage \\unit[3.3]{$\\mu$s} after the arrival of the pump pulse. This provided for an expansion of the ion cloud that reduced the impact of saturation within the central MCP ion channels by distributing the ions over a greater detector surface area \\cite{Kraft2007}.\n\nThe molecular ions arrived at the MCP within a time window narrower than \\unit[100]{ns}. The sensitivity of the detection was changed by varying the pushing electrode voltage from \\unit[100]{V} to \\unit[1100]{V}: a high voltage ensured sensitive detection of  Rb$_2^+$ whilst a low voltage ensured a linear response to  Rb$^+$ without MCP saturation. For each pump-probe delay measurement the Rb$^+$ or Rb$_2^+$ TOF signal was averaged 1000 times by a digital oscilloscope and integrated numerically. This ensured that the duration of the single-point averaging was greater than that of the characteristic NOPA power fluctuations. The signal-to-noise ratio of the pump-probe signal was further improved without compromising immunity to long-term drifts by averaging several successive pump-probe scans.\n\n\\subsection{Bound initial state characterization}\n\\label{sec:REMPI}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Figure4.pdf}\n  \\caption{(Colour online). A schematic of the initial-state characterization experiment. The MOT trapping light photoassociates free atom pairs to loosely bound states within the \\mbox{5S + 5P}$_{3\/2}$ $0_{\\textrm{g}}^-$ potential, from where they spontaneously decay to the $a^3\\Sigma_u^+$ ground triplet potential. These `background' ground triplet molecules may be detected using a two-photon resonant ionization pathway via deeply bound states within the $(2)^3\\Sigma_g^+$ potential. The high-lying populated states within the $a^3\\Sigma_u^+$ ground state are more closely spaced than the deeply bound intermediates within the $(2)^3\\Sigma_g^+$ potential, allowing the ground-state population distribution to be deduced from the resultant spectrum.}\n  \\label{fig:init_schematic}\n\\end{figure}\n\nThe intuitive picture of this pump-probe experiment concerns the photoassociation of a scattering pair of unbound atoms. There are, however, two mechanisms that produce a `background' population of loosely bound ground-state molecules in an $^{85}$Rb MOT in the absence of dedicated photoassociation light: three-body recombination \\cite{Takekoshi1999}, and photoassociation by the MOT trapping light \\cite{Fioretti1998}. Experimental studies in $^{85}$Rb have concluded that the former mechanism dominates at low trapping light intensities \\cite{Gabbanini2000} but that the latter plays a more prominent role at higher intensities \\cite{Caires2005}. \n\nIt is possible that these preformed bound ground-state molecules play an important role in the pump-probe dynamics despite their paucity in comparison to unbound atom pairs, since the low atomic density within the MOT implies a low probability for two neighbouring atoms to interact. A precise characterization of the initial state of the interacting atom pairs is therefore essential for meaningful subsequent numerical simulation of the pump-probe process. The formation mechanism effected by the trapping light involves photoassociation predominantly to the \\mbox{5S + 5P$_{3\/2}$ $0_{\\textrm{g}}^-$} potential followed by spontaneous decay to high-lying states within the $a^3\\Sigma_u^+$ ground triplet potential (see Fig.\\ \\ref{fig:init_schematic}). This photoassociative mechanism is expected to dominate three-body recombination within our setup, as our experimental parameters more closely match those of Ref.\\ \\cite{Caires2005}. A comparison of the measured ground-state occupancy to the calculated population distribution resulting from this decay channel enables this hypothesis to be confirmed. \n\nThe initial distribution of bound ground-state triplet molecules was studied via resonant multi-photon ionization (REMPI). A two-photon process preferentially ionized the Rb$_2$ via a resonant intermediate potential, while atomic ionization followed an off-resonant pathway and required an extra photon. A common REMPI pathway for detection of ground triplet Rb$_2$ employs loosely bound vibrational states within the $(2)^3\\Pi_g$ potential as an intermediate \\cite{Gabbanini2000}. Since the vibrational level spacings of these initial and intermediate states are comparable, it is not easy to unravel the initial distribution from a spectroscopic measurement via this pathway. Instead, this study uses an alternative REMPI route that passes through more deeply bound levels within the $(2)^3\\Sigma_g^+$ potential (see Fig.\\ \\ref{fig:init_schematic}) \\cite{Lozeille2006}. The resultant spectroscopic scan enables distinction between the large energy spacing of the intermediate states and the close spacing of the ground-state triplet Rb$_2$ whose occupancy it is desired to study.\n\nAn estimation of the population distribution of molecules photoassociated by the MOT lasers was made by considering the Franck--Condon overlap of high-lying $0_{\\textrm{g}}^-$ states with each $a^3\\Sigma_u^+$ vibrational state. The expected resultant spectrum for REMPI ionization via the $(2)^3\\Sigma_g^+$ was calculated using the excited-state vibrational level energies given in Ref.\\ \\cite{Lozeille2006}, together with the corresponding transition Franck--Condon factors. A realistic bandwidth for the REMPI laser was assumed.\n\nThe light for the REMPI pathway outlined above was generated with a dye laser (using Pyridine 1 dye) that was pumped by the second harmonic of a Q-switched Nd:YAG laser. Pulses of \\unit[10]{ns} duration and up to \\unit[400]{$\\mu$J} energy were produced at a repetition rate of \\unit[50]{Hz}. The pulse wavelength was tunable between \\unit[685]{nm} and \\unit[705]{nm}, and was calibrated with reference to two-photon atomic transitions to the 4p$^6$6d configuration.\n\n\n\\section{Experimental results}\n\\label{sec:results}\n\n\\subsection{Bound initial state study}\n\\label{sec:initial-state}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Figure5.pdf}\n  \\caption{(Colour online). Spectroscopy of the initial state. High-lying vibrational states within the $a^3\\Sigma_u^+$ ground triplet formed through MOT laser photoassociation are ionized using a REMPI pathway through the $(2)^3\\Sigma_g^+$ potential. The Rb$_2^+$ signal is shown as a function of ionization energy. (a) The full experimental spectrum reveals the large vibrational level spacing within the $(2)^3\\Sigma_g^+$ potential. The atomic resonance used for calibration is indicated. The overall spectrum is modulated by the tuning range of the ionization laser. (b) A close-up plot of the spectral feature at \\unit[14500]{cm$^{-1}$} reveals the initial population distribution within the $a^3\\Sigma_u^+$. A close agreement is found with simulations of the population distribution obtained by MOT photoassociation to the \\mbox{5S + 5P$_{3\/2}$ $0_{\\textrm{g}}^-$} potential. The locations of the highest-lying $a^3\\Sigma_u^+$ vibrational levels are shown and enumerated according to increasing binding energy.}\n  \\label{fig:init-state}\n\\end{figure}\n\nFigure \\ref{fig:init-state}(a) shows the result of a spectroscopic scan of the REMPI ionization signal from the high-lying ground vibrational states populated via background mechanisms. The full spectrum reveals a repeating structure at intervals of \\unit[50]{cm$^{-1}$} that corresponds to the spacing of the vibrational levels within the $(2)^3\\Sigma_g^+$ excited state (as recorded in Fig.\\ 6 of Lozeille et al.\\ \\cite{Lozeille2006}). The detected Rb$_2^+$ signal strength is modulated by the available ionization laser pulse energy as the wavelength is tuned.\n\nAs discussed in Section \\ref{sec:REMPI}, the repeated structure within the spectrum reveals the initial population distribution within the $a^3\\Sigma_u^+$ ground-state occupancy. A close-up of the spectral feature at \\unit[14500]{cm$^{-1}$} is shown in Fig.\\ \\ref{fig:init-state}(b), together with the calculated population distribution formed via MOT laser photoassociation to the $0_{\\textrm{g}}^-$ potential followed by spontaneous decay. The greatest population is found to be contained within the ninth-highest vibrational level, which corresponds to the level with the greatest Franck--Condon overlap from the excited state populated by the MOT light. Around ten bound states are found to be populated in total.\n\nThe experimental spectrum is in close agreement with the calculated population distribution formed by MOT laser photoassociation via the \\mbox{5S + 5P$_{3\/2}$ $0_{\\textrm{g}}^-$} potential. This agreement confirms that this, rather than three-body recombination, is the dominant mechanism for background molecule formation within this experimental setup. The detection of a strong background triplet molecule signal implies that this bound initial population, additionally to unbound scattering atom pairs, is a candidate for contribution to the photoassociation dynamics. This is analyzed further in Section \\ref{sec:bound-dimers}.\n\n\\subsection{Numerical pump-probe simulations}\n\\label{sec:simulations}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics [width=\\columnwidth]{Figure6.pdf}\n  \\caption{(Colour online). Simulations of the time- and position-dependent excited-state population density induced by the pump pulse. Coherent dynamics occur at a range of internuclear separations (represented by the dark shaded areas) despite the incoherence of the initial states. Results are shown for transitions from an unbound initial population to both gerade (a) and ungerade (b) excited states, as well as from a bound initial population to gerade excited states only (c) (single photon transitions to ungerade states are prohibited from the  $a^3\\Sigma_u^+$ potential). }\n  \\label{fig:dynamics}\n\\end{figure}\n\nNumerical simulations were conducted of the excited-state dynamics induced by the pump pulse, with consideration given to both the initial-state scenarios discussed above. The time-dependent Schr\\\"{o}dinger equation was solved for two different initial density matrices: a thermal distribution of scattering states, and the incoherent mixture of bound \\mbox{5S + 5P$_{3\/2}$ $0_{\\textrm{g}}^-$} decay products. The details of the coupled-channel model and the numerical implementation are as discussed in Ref.\\ \\cite{Martay2009}, but with the pump pulse selected to match Fig.\\ \\ref{fig:pulses}(b). The time- and position-dependent population density for both initial density matrices is shown in Fig.\\ \\ref{fig:dynamics}. The dynamics due to transitions from the unbound initial state to excited states of both gerade and ungerade symmetries are shown separately; however dipole selection rules prohibit the population of ungerade excited states from the bound $a^3\\Sigma_u^+$ initial state. The excited-state population exhibits a time-varying internuclear separation distribution, and so a position-sensitive measurement should observe a time-varying signal. Close-range, few-picosecond timescale dynamics are predicted for the bound initial state, compared to slower and longer-range dynamics from the unbound initial state.\n\nThe time-variation is caused by two effects. Firstly, the population density matrix that describes a time-independent distribution in the ground potential, no longer describes a time-independent distribution in the excited states since different potential energy curves govern its motion. The thermal distribution has a nodal structure that, when transferred to the excited state, forms distinct pockets of population that fall inwards. The anharmonicity of the excited potential at large internuclear separation precludes harmonic motion, but coherent interference effects still cause time-varying oscillations in the population at any internuclear separation. The second effect is that the pump-pulse bandwidth and transition Franck--Condon factors only transfer population in a narrow window in internuclear separation. This further improves the coherence of the excited-state population.\n\nThe simulations show that the incoherence of the initial state, whether from thermally distributed scattering states or from preassociated molecules, is not a barrier to the formation of coherently oscillating excited-state population by the pump pulse. The subsequent action of the ionizing probe pulse is less well theoretically understood due to the scope for the ejected electron to carry surplus energy and momentum. Due to the absence of a suitable experimentally verified model for molecular ionization, the probe pulse has been omitted from these calculations. The experimental probe pulse described in Section \\ref{sec:ultrafast} is sufficiently short to resolve these dynamics provided that it permits the requisite position-selective measurement.\n\n\n\n\n\\subsection{Rubidium pump-probe dynamics}\n\\label{sec:pump-probe}\n\n\\begin{figure}\n  \\centering\n   \\includegraphics[width=0.9\\columnwidth]{Figure7.pdf}  \n  \\caption{(Colour online). Typical atomic and molecular ion signals as a function of pump-probe delay (main figure), together with a typical molecular pump-probe signal at larger delays (inset). The atomic ion signal has been scaled for comparison. Both atomic and molecular signals exhibit a step at zero delay and a peak at small positive delays due to transient effects induced by the pump pulse. Fourier analysis of the subsequent signal reveals no dominant oscillatory timescales.}\n  \\label{fig:pu-pr}\n\\end{figure}\n\nFigure \\ref{fig:pu-pr} shows typical atomic and molecular pump-probe signals as a function of pump-probe delay. A low pump-pulse energy of \\unit[85]{nJ} was used in order to allow atomic and molecular dynamics to be monitored simultaneously without excessive atomic ionization. The inset also shows a typical molecular pump-probe signal at longer delays for the higher pulse energy of \\unit[400]{nJ}. The absolute Rb$^+$ and Rb$_2^+$ signal scalings employed in the plots are arbitrary and have been chosen to take a similar magnitude. Both the Rb$^+$ and Rb$_2^+$ experimental signals show a distinct step at zero delay: the signals are greater for a pump-probe sequential pulse order, though significant off-resonant atomic ionization is still evident with probe-pump timings (Fig.\\ \\ref{fig:pu-pr}(a)). In the Rb$_2$ pump-probe experiment of Salzmann et al.\\ \\cite{Salzmann2008}, the sharp spectral cut applied to the pump pulse introduced a long tail into the pump-pulse envelope in the corresponding temporal domain. In exciting an atom pair, the pump pulse induces an electronic dipole which interacts coherently with this long temporal tail, causing oscillations in the excited-state population and hence the Rb$_2^+$ signal. This behaviour is similar to the observations of Zamith et al.\\ in a Rb vapour cell \\cite{Zamith2001}. These dipole transient effects are not apparent in the long-timescale pump-probe signals of this work due to the employment of a less sharp spectral cut of the pump pulse.\n\n\nFourier analysis of the positive-delay signal that follows the coherent transient peak does not reveal any dominant characteristic oscillatory periods. This is in contrast with the picosecond-timescale coherent excited-state dynamics predicted by the simulations (which the \\unit[400]{fs} probe pulse was short enough to resolve), particularly for the case of the bound initial state. The calculations of Section \\ref{sec:simulations} revealed a wide range of conditions under which coherent oscillations may be formed. It therefore seems likely that the absence of such oscillations in the pump-probe signal can be attributed to an insufficiently position-dependent measurement being effected by the probe, for which there are two likely causes. Firstly, the ionization mechanism of the probe pulse is not as well understood as for neutral-neutral transitions (as outlined in Section \\ref{sec:simulations}). In order to attain a sufficiently short pulse to resolve the dynamics, a large bandwidth is required which may therefore compromise selectivity with respect to internuclear separation. Other indirect mechanisms, such as probe-pulse excitation to neutral Rydberg states that couple to the Rb$_2^+$ potential, may also complicate the process. Secondly, the detection mechanism may not be sensitive to very long-range excited-state population. Further information is thus required concerning the nature of the initial population addressed by the pump pulse. These issues are further discussed below in Section \\ref{sec:bound-dimers}.\n\n\\subsection{Bound excited-state dimers}\n\\label{sec:bound-dimers}\n\n\\begin{figure}\n  \\centering\n   \\includegraphics[width=0.9\\columnwidth]{Figure8.pdf}\n  \\caption{(Colour online). (a) Normalized atomic and molecular signals measured at a fixed pump-probe delay for different detunings of the pump-pulse spectral cut from the D1 line. The shaded area indicates evidence for the formation of bound excited-state dimers. (b) Normalized theoretical calculations of the excited-state population transfer induced by a pump pulse as a function of spectral cut detuning (red dashed lines). The different series represent the behaviour of pairs of atoms of different initial separations within the $a^3\\Sigma_u^+$ ground state (labelled in atomic units). The experimental molecular ion signal from (a) is plotted together with an optimized theoretical fit for a distribution of initial separations around \\unit[50]{a$_0$}.}\n  \\label{fig:bound_mols}\n\\end{figure}\n\nThe Rb$_2^+$ ion signal did not appreciably decay over the delays of hundreds of picoseconds that these experiments considered. In order to demonstrate that this large-delay signal is attributable to bound molecular population within the excited-state manifold (rather than further coherent transient effects or photoassociation of an unbound 5S and 5P atom to bound molecular ions), a further study was performed. A fixed, positive pump-probe delay of \\unit[40]{ps} was selected and the atomic and molecular ion signals were recorded as a function of the position of the pump-pulse spectral cut.\n\nAs the pump-pulse spectral cut was further detuned from the \\mbox{5S + 5P$_{1\/2}$} atomic asymptote (increasing the binding energy of the energetically accessible excited states), the Rb$^+$ signal was observed to fall off more quickly than the Rb$_2^+$ signal (\\mbox{Fig.\\ \\ref{fig:bound_mols}(a)}). Indeed, atomic ion formation became negligible at detunings beyond \\unit[10]{cm$^{-1}$}, while a substantial number of molecular ions continued to be formed. The fast decline in Rb$^+$ signal is caused by the loss of spectral intensity resonant with the atomic transition. Hence less atomic population is excited and subsequently ionized. The Rb$_2^+$ signal persists, however, due to excitation into bound vibrational levels below the atomic asymptote. The difference between the Rb$^+$ and Rb$_2^+$ data series (shaded) therefore provides strong evidence for the formation of bound molecular wavepackets by the pump pulse.\n\nIn order to study the initial state that contributed to this pump-probe signal, theoretical calculations were performed of the excited-state population transfer induced by a realistic experimental pump pulse as a function of spectral cut detuning (\\mbox{Fig.\\ \\ref{fig:bound_mols}(b)}), and repeated for pairs of atoms with a range of initial separations. The asymptotic long-range form of the excited-state potentials scales as $R^{-3}$ due to dipole-dipole interactions, whereas the ground-state potentials scale as $R^{-6}$ due to van der Waals interactions. The pump-pulse transition energy therefore becomes more red-detuned from the atomic asymptote at closer range. As a consequence, these simulated population transfers fall off more slowly with spectral cut detuning for closer range initial populations.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.9\\columnwidth]{Figure9.pdf}\n  \\caption{(Colour online). The population distributions of the bound background molecules characterized in Section \\ref{sec:initial-state} and a \\unit[100]{$\\mu$K} scattering state, together with an analytical best fit of the population that contributes to the pump-probe signal. The contributing population is of longer range than the background molecules, but is consistent with the association of unbound scattering atoms by the pump pulse. Dissociation of molecular ions by the TOF electric field is calculated to account for the lack of contribution of very long-range scattering atom pairs to the Rb$_2^+$ signal.}\n  \\label{Pu-pr_init}\n\\end{figure}\n\nThe atom pairs contributing to this pump-probe process will realistically be distributed over a range of internuclear separations rather than located at a single point; indeed, the fact that the experimental data span a number of these theoretical series corroborates this picture. Excellent agreement was found when the initial population distribution was fitted to the experimental data (\\mbox{Fig.\\ \\ref{fig:bound_mols}(b)}). Several analytical forms of the initial distributions were found to fit the data closely; all shared the common characteristic that population was concentrated at internuclear separations of \\unit[30-60]{a$_0$} (see \\mbox{Fig.\\ \\ref{Pu-pr_init}}). By contrast, the outer turning point of the most populated triplet vibrational state according to the study of Section \\ref{sec:initial-state} is located at \\unit[25]{a$_0$}. This provides evidence that the population addressed by the pump and probe pulses is different from the background triplet distribution photoassociated by the MOT light. Instead, the inferred distribution is consistent with the association of unbound atoms, coinciding with several antinodes of the scattering wavefunction. In this case, the detuning of the probe pulse from the Rb$^+$ asymptote might account for the lack of a contribution from the very long-range portion of the scattering wavefunction. As an alternative explanation, our calculations indicate that the TOF electric field is sufficient to dissociate molecular ions (causing \\mbox{$\\textrm{Rb}_2^+ \\rightarrow \\textrm{Rb} ^+ + \\textrm{Rb}$}) of longer range than around \\unit[90]{a$_0$}.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe have performed a spectroscopic measurement of the initial state of the preformed molecules within our ultracold $^{85}$Rb$_2$ photoassociation experiments. The twin mechanisms of three-body recombination and trapping light photoassociation are known to contribute to a background population of loosely bound ground-state triplets within the MOT with relative impacts that are determined by the particular experimental conditions. Our measurement of the initial ground triplet-state occupancy reveals a distribution that closely matches the photoassociative mechanism, confirming this to be the dominant contribution in this case.\n\nWe have carried out a pump-probe study of the excited-state dynamics of Rb$_2$ dimers as the first step towards a fully coherent pump-dump transfer to the ground state. An ultrafast pump pulse was spectrally filtered to populate bound excited states only, and the resultant wavepacket was allowed to evolve for a controllable delay before being ionized with a probe pulse. By observing the large-delay behaviour of the pump-probe signal after the influence of initial transient effects, we have shown the formation of excited-state population that is demonstrably bound in character.\n\nWe have characterized the population that contributes to the pump-probe signal and have found that its behaviour with respect to spectral cut location is consistent with the association of unbound scattering atom pairs, rather than the excitation of background triplet molecules. Calculations indicated that the bulk of the excited-state dynamics induced from such an unbound initial state would occur at larger internuclear separations than are resonantly addressed by the probe pulse. This may therefore be a factor in the lack of experimental observation of pump-probe dynamics.\n\nThe ultimate goal, however, remains the coherent control of this population so as to maximize the Franck-Condon overlap to the dump-pulse target state after a suitably chosen delay; a signal presenting evidence of wavepacket oscillations is therefore required. Simulations reveal that pre-associated molecules in an incoherent mixture of states can be made to oscillate coherently with \\unit[5]{ps} timescales at relatively close range using a single ultrafast pump pulse \\cite{Martay2009}. By contrast, the classical oscillation period of molecules photoassociated from scattering atom pairs is greater by an order of magnitude or more, and occur predominantly at long range where detection is particularly difficult.\nIncreasing the signal-to-noise size at the larger pump-pulse detunings that address the relatively close-range pre-formed molecules should therefore allow this coherent behaviour to be revealed and controlled.\n\nSince the photoassociation rate scales in proportion to the product of the density and the number of trapped atoms, we identify these two parameters as targets for an improvement to our experimental apparatus.  \\mbox{Figure \\ref{fig:bound_mols}(b)} indicates that an order-of-magnitude improvement in photoassociation rate would provide a sufficient signal-to-noise ratio to probe the background molecular population located around \\unit[30]{a$_0$}. To this end we have trialled a dark spontaneous-force optical trap (SPOT) MOT \\cite{Ketterle1993} in order to improve density. This protocol is difficult to implement effectively in our existing Rb MOT, however, as the low branching ratio in Rb to the dark state requires high extinction of the repumper, and we consequently saw no improvement in signal. Instead, we are constructing a new MOT apparatus with which we expect to obtain this necessary improvement in atom number and density.\n\n\\section{Conclusion}\n\\label{sec:conclusions}\n\nThere is much current research interest devoted to the preparation of translationally ultracold molecules in their absolute ground electronic and rovibrational states. A broad range of strategies is at the disposal of the experimentalist in the pursuit of this goal. Some significant recent advances have used the approach of associating ultracold atoms, with the experimental realization of $v=0$ dimers in their absolute ground electronic states via techniques such as STIRAP or incoherent optical cycling. These breakthroughs for certain specific species are complemented by coherent control strategies, which are expected to be broadly applicable to a range of systems without the need for fortuitous initial bound-state preparation pathways, a detailed knowledge of the system spectroscopy or incoherent transfer processes.\n\nIn this work, we have demonstrated the formation of bound excited-state dimers by an ultrafast photoassociation pulse. This represents an important milestone in the design of a coherent pump-dump transfer strategy. We have determined the initial state addressed by the pump pulse and demonstrated that it predominantly comprises longer-range initial population than those background molecules pre-associated by the trapping light. We have identified improvements in MOT number and density as being critical to allow the detection and control of the coherent excited-state dynamics of closer-range population.\n\n\\begin{acknowledgments}\nThe authors are grateful to Jordi Mur-Petit for valuable discussions and suggestions. \\mbox{H.\\ E.\\ L.\\ M.\\ }acknowledges help from Thorsten K\\\"{o}hler. This work was supported by EPSRC grant number EP\/D002842\/1, Marie Curie Initial Training Network CA-ITN-214962-FASTQUAST, and Bulgarian NSF grant number WU-301\/07.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe {\\em Olrov spectrum} of a triangulated category is introduced by Orlov \\cite{O} based on the works of Bondal and Van den Bergh \\cite{BV} and Rouquier \\cite{R}.\nThis categorical invariant is the set of finite generation times of objects of the triangulated category.\nThe generation time of an object is the number of exact triangles necessary to build the category out of the object, up to finite direct sums, direct summands and shifts.\nIn \\cite{R}, Rouquier studies the infimum of the Orlov spectrum, which is called the {\\em Rouquier dimension} of the triangulated category.\nFor more information on Orlov spectra and Rouquier dimensions, we refer the reader to \\cite{BFK, O, R} for instance.\n\nThe Orlov spectrum and Rouquier dimension measure how big a triangulated category is, but it is basically rather hard to calculate them.\nThus, the notions of a level \\cite{ABIM} and a relative Rouquier dimension \\cite{AAITY} are introduced to measure how far one given object\/subcategory from another given object\/subcategory.\nThe main purpose of this paper is to report on these two invariants for the singularity category of a hypersurface of countable (Cohen--Macaulay) representation type.\n\nLet $k$ be an uncountable algebraically closed field of characteristic not two, and let $R$ be a complete local hypersurface over $k$ with countable representation type.\nDenote by $\\operatorname{\\mathsf{D_{sg}}}(R)$ the singularity category of $R$, that is, the Verdier quotient of the bounded derived category of finitely generated $R$-modules by the perfect complexes.\nLet $\\operatorname{\\mathsf{D_{sg}^o}}(R)$ be the full subcategory of $\\operatorname{\\mathsf{D_{sg}}}(R)$ consisting of objects that are zero on the punctured spectrum of $R$.\nThe main results of this paper are the following two theorems; we should mention that (1) and (2a) of Theorem \\ref{b} are essentially shown in the previous papers \\cite{AIT,T3} of some of the authors of the present paper.\n\n\\begin{thm}\\label{a}\nFor all nonzero objects $M$ of $\\operatorname{\\mathsf{D_{sg}}}(R)$ one has\n$$\n\\operatorname{level}_{\\operatorname{\\mathsf{D_{sg}}}(R)}^M(k)\\le1,\n$$\nthat is, $k$ belongs to ${\\langle M\\rangle}_2^{\\operatorname{\\mathsf{D_{sg}}}(R)}$.\n\\end{thm}\n\n\n\\begin{thm}\\label{b}\nLet $\\mathcal{T}$ be a nonzero thick subcategory of $\\operatorname{\\mathsf{D_{sg}}}(R)$, and let $\\mathcal{X}$ be a full subcategory of $\\mathcal{T}$ closed under finite direct sums, direct summands and shifts.\nThen the following statements hold.\n\\begin{enumerate}[\\rm(1)]\n\\item\n$\\mathcal{T}$ coincides with either $\\operatorname{\\mathsf{D_{sg}}}(R)$ or $\\operatorname{\\mathsf{D_{sg}^o}}(R)$.\n\\item\n\\begin{enumerate}[\\rm(a)]\n\\item\nIf $\\mathcal{T}=\\operatorname{\\mathsf{D_{sg}}}(R)$, then\n$$\n\\dim_\\mathcal{X}\\mathcal{T}=\n\\begin{cases}\n0 & (\\text{if } \\mathcal{X} =\\mathcal{T}),\\\\\n1 & (\\text{if } \\mathcal{X} \\neq \\mathcal{T},\\, \\mathcal{X} \\nsubseteq \\operatorname{\\mathsf{D_{sg}^o}}(R)),\\\\\n\\infty & (\\text{if }\\mathcal{X} \\subseteq \\operatorname{\\mathsf{D_{sg}^o}}(R)).\n\\end{cases}\n$$\n\\item\nIf $\\mathcal{T}=\\operatorname{\\mathsf{D_{sg}^o}}(R)$, then\n$$\n\\dim_\\mathcal{X}\\mathcal{T}=\n\\begin{cases}\n0 & (\\text{if } \\mathcal{X}  =\\mathcal{T}),\\\\\n1 & (\\text{if } \\mathcal{X} \\neq \\mathcal{T},\\,\\#\\operatorname{\\mathsf{ind}} \\mathcal{X}=\\infty),\\\\\n\\infty & (\\text{if }\\#\\operatorname{\\mathsf{ind}} \\mathcal{X} <\\infty).\n\\end{cases}\n$$\n\\end{enumerate}\n\\end{enumerate}\n\\end{thm}\n\nAs a consequence of Theorem \\ref{b}, we get the Orlov spectra and Rouquier dimensions of $\\operatorname{\\mathsf{D_{sg}}}(R)$ and $\\operatorname{\\mathsf{D_{sg}^o}}(R)$.\nNote that the equalities of Rouquier dimensions are already known to hold \\cite{AIT,DT}.\n\n\\begin{cor}\\label{c}\nIt holds that\n$$\n\\operatorname{OSpec}\\operatorname{\\mathsf{D_{sg}}}(R)=\\{1\\},\\qquad\n\\operatorname{OSpec}\\operatorname{\\mathsf{D_{sg}^o}}(R)=\\emptyset.\n$$\nIn particular, $\\dim \\operatorname{\\mathsf{D_{sg}}}(R)=1$ and $\\dim\\operatorname{\\mathsf{D_{sg}^o}}(R)=\\infty$.\n\\end{cor}\n\nThe organization of this paper is as follows.\nIn Section 2, we give the definitions of the Orlov spectrum and the (relative) Rouquier dimension of a triangulated category.\nWe also recall the theorem of Buchweitz providing a triangle equivalence for a Gorenstein ring $R$ between the singularity category of $R$ and the stable category of maximal Cohen--Macaulay $R$-modules.\nIn Section 3, we investigate the Orlov spectra and Rouquier dimensions of the singularity categories of hypersurfaces and their double branched covers to reduce to the case of Krull dimension one.\nIn Section 4, using the results obtained in the previous sections together with the classification theorem of maximal Cohen--Macaulay modules over a hypersurface of countable representation type, we prove our main theorems stated above.\n\n\\begin{conv}\nThroughout this paper, all subcategories are assumed to be full.\nWe often omit subscripts and superscripts if there is no risk of confusion.\n\\end{conv}\n\\section{Preliminaries}\n\nWe recall the definitions of several basic notions which are used in the later sections.\n\n\\begin{nota}\nLet $\\mathcal{T}$ be a triangulated category.\n\\begin{enumerate}[(1)]\n\\item\nFor a subcategory $\\mathcal{X}$ of $\\mathcal{T}$ we denote by $\\langle \\mathcal{X} \\rangle$ the smallest subcategory of $\\mathcal{T}$ containing $\\mathcal{X}$ which is closed under isomorphisms, shifts, finite direct sums and direct summands.\n\\item\nFor subcategories $\\mathcal{X}, \\mathcal{Y}$ of $\\mathcal{T}$ we denote by $\\mathcal{X} * \\mathcal{Y}$ the subcategory consisting of objects $M \\in \\mathcal{T}$ such that there is an exact triangle $X\\to M\\to Y \\to X[1]$ with $X\\in \\mathcal{X}$ and $Y\\in \\mathcal{Y}$.\nSet $\\mathcal{X} \\diamond \\mathcal{Y} := \\langle\\langle\\mathcal{X}\\rangle * \\langle\\mathcal{Y}\\rangle\\rangle$.\n\\item\nFor a subcategory $\\mathcal{X}$ of $\\mathcal{T}$ we put ${\\langle \\mathcal{X}\\rangle}_0:=0$, ${\\langle \\mathcal{X}\\rangle}_1:=\\langle \\mathcal{X}\\rangle$, and inductively define ${\\langle \\mathcal{X}\\rangle}_n:= \\mathcal{X} \\diamond {\\langle \\mathcal{X}\\rangle}_{n-1}$ for $n\\geq 2$.\nWe set ${\\langle M \\rangle}_n := {\\langle \\{ M \\}\\rangle}_n$ for an object $M\\in \\mathcal{T}$.\n\\end{enumerate}\n\\end{nota}\n\n\\begin{dfn}\nLet $\\mathcal{T}$ be a triangulated category.\n\\begin{enumerate}[(1)]\n\\item\nThe {\\em generation time} of an object $M\\in\\mathcal{T}$ is defined by\n$$\n\\operatorname{gt}_{\\mathcal{T}}(M):=\\inf\\{ n\\geq 0\\mid\\mathcal{T}={\\langle M \\rangle}_{n+1} \\}.\n$$\nIf $\\operatorname{gt}_{\\mathcal{T}}(M)$ is finite, $M$ is called a {\\it strong generator} of $\\mathcal{T}$.\n\\item\nThe {\\it Orlov spectrum} and {\\em (Rouquier) dimension} of $\\mathcal{T}$ are defined as follows.\n\\begin{align*}\n\\operatorname{OSpec}(\\mathcal{T})&:=\\{ \\operatorname{gt}_{\\mathcal{T}}(M)\\mid M\\text{ is a strong generator of }\\mathcal{T}\\},\\\\\n\\dim\\mathcal{T}&:=\\inf\\operatorname{OSpec}(\\mathcal{T})=\\inf\\{n\\ge0\\mid\\mathcal{T}={\\langle M\\rangle}_{n+1}\\text{ for some }M\\in\\mathcal{T}\\}.\n\\end{align*}\n\\item\nLet $\\mathcal{X}$ be a subcategory of $\\mathcal{T}$.\nThe {\\em dimension} of $\\mathcal{T}$ {\\em with respect to} $\\mathcal{X}$ is defined by\n$$\n\\dim_\\mathcal{X}\\mathcal{T}:=\\inf\\{n\\ge0\\mid \\mathcal{T}={\\langle\\mathcal{X}\\rangle}_{n+1}\\}.\n$$\n\\item\nLet $M, N$ be objects of $\\mathcal{T}$.\nThen the {\\em level} of $N$ with respect to $M$ is defined by\n$$\n\\operatorname{level}_{\\mathcal{T}}^M(N):=\\inf\\{ n\\geq 0\\mid N \\in {\\langle M \\rangle}_{n+1} \\}.\n$$\n\\end{enumerate}\n\\end{dfn}\n\n\\begin{dfn}\nLet $R$ be a Noetherian ring.\n\\begin{enumerate}[(1)]\n\\item\nWe denote by $\\operatorname{\\mathsf{D^b}}(R)$ the bounded derived category of finitely generated $R$-modules.\n\\item\nA {\\em perfect} complex is by definition a bounded complex of finitely generated projective modules.\n\\item\nWe denote by $\\operatorname{\\mathsf{perf}}(R)$ the subcategory of $\\operatorname{\\mathsf{D^b}}(R)$ consisting of complexes quasi-isomorphic to perfect complexes.\n\\item\nThe {\\em singularity category} of $R$ is defined by\n$$\n\\operatorname{\\mathsf{D_{sg}}}(R):=\\operatorname{\\mathsf{D^b}}(R)\/\\operatorname{\\mathsf{perf}}(R),\n$$\nthat is, the Verdier quotient of $\\operatorname{\\mathsf{D^b}}(R)$ by $\\operatorname{\\mathsf{perf}}(R)$.\n\\end{enumerate}\n\\end{dfn}\n\nNote that every object of the singularity category $\\operatorname{\\mathsf{D_{sg}}}(R)$ is isomorphic to a shift of some $R$-module; see \\cite[Lemma 2.4]{sing}.\n\nLet $R$ be a Cohen--Macaulay local ring.\nLet $\\operatorname{\\mathsf{CM}}(R)$ be the category of maximal Cohen-Macaulay $R$-modules, and $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ the stable category of $\\operatorname{\\mathsf{CM}}(R)$.\nThe following theorem is celebrated and fundamental; see \\cite[Theorem 4.4.1]{B}.\n\n\\begin{thm}[Buchweitz]\\label{ragnar}\nLet $R$ be a Gorenstein local ring of Krull dimension $d$.\nThen $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ has the structure of a triangulated category with shift functor $\\Omega^{-1}$, and there exist mutually inverse triangle equivalence functors\n$$\nF:\\operatorname{\\mathsf{D_{sg}}}(R)\\rightleftarrows\\operatorname{\\mathsf{\\underline{CM}}}(R):G,\n$$\nsuch that $GM=M$ for each maximal Cohen--Macaulay $R$-module $M$ and $FN=\\Omega^dN[d]$ for each finitely generated $R$-module $N$.\n\\end{thm}\n\nBy virtue of Theorem \\ref{ragnar}, for a Gorenstein local ring, the study of generation in the singularity category reduces to the stable category of maximal Cohen--Macaulay modules.\n\n\\section{The relationship between the singularity categories of $R$ and ${R^\\sharp}$}\n\nLet $(R,\\mathfrak{m},k)$ be a complete equicharacteristic local hypersurface of (Krull) dimension $d$.\nThen thanks to Cohen's structure theorem we can identify $R$ with a quotient of a formal power series ring over $k$:\n$$\nR=k[\\![x_0,x_1,\\dots,x_d]\\!]\/(f)\n$$\nwith $0\\neq f \\in (x_0,x_1,\\dots,x_d)^2$. \nWe define a hypersurface of dimension $d+1$:\n$$\n{R^\\sharp}=k[\\![x_0,x_1,\\ldots,x_d, y]\\!]\/(f+y^2).\n$$\nNote that the element $y$ is ${R^\\sharp}$-regular and there is an isomorphism ${R^\\sharp}\/y{R^\\sharp}\\cong R$.\nThe main purpose of this section is to compare generation in the singularity categories $\\operatorname{\\mathsf{D_{sg}}}(R)$ and $\\operatorname{\\mathsf{D_{sg}}}({R^\\sharp})$.\nAs both $R$ and ${R^\\sharp}$ are Gorenstein, in view of Theorem \\ref{ragnar} and the remark following the theorem, it suffices to investigate the stable categories of maximal Cohen--Macaulay modules $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\nThe following result is a consequence of \\cite[Proposition 12.4]{Y}, which plays a key role to compare generation in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\n\\begin{lem}\\label{dense}\nThe assignments $M\\mapsto\\Omega_{{R^\\sharp}}M$ and $N\\mapsto N\/yN$ define triangle functors $\\Phi :\\operatorname{\\mathsf{\\underline{CM}}}(R) \\to \\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$ and $\\Psi :\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp}) \\to \\operatorname{\\mathsf{\\underline{CM}}}(R)$ satisfying\n$$\n\\Psi\\Phi(M)\\cong M\\oplus M[1],\\qquad\n\\Phi\\Psi(N) \\cong N\\oplus N[1].\n$$\nIn particular, $\\Phi$ and $\\Psi$ are both equivalences up to direct summands.\n\\end{lem}\n\nApplying this lemma, we deduce relationships of levels in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\n\\begin{prop}\\label{lev}\nOne has the following equalities.\n\\begin{enumerate}[\\rm(1)]\n\\item\n$\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Omega_{{R^\\sharp}} M}(\\Omega_{{R^\\sharp}}^{d+1}k)$ for each $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\\item\n$\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^N(\\Omega_{{R^\\sharp}}^{d+1}k)=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{N\/yN}(\\Omega_{R}^{d}k)$ for each $N\\in\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nWe use Lemma \\ref{dense} and adopt its notation.\nThere are (in)equalities\n\\begin{align*}\n\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)\n\\ge\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Phi(\\Omega_R^dk))\n&\\ge\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi\\Phi M}(\\Psi\\Phi(\\Omega_R^dk))\\\\\n&=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{M\\oplus M[1]}(\\Omega_R^dk\\oplus\\Omega_R^dk[1])\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk),\n\\end{align*}\nwhich show $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Phi(\\Omega_R^dk))$.\nA similar argument gives rise to $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^N(\\Omega_{{R^\\sharp}}^{d+1}k)=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Psi(\\Omega_{R^\\sharp}^{d+1}k))$.\nThere are isomorphisms in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$:\n\\begin{align}\\label{1e}\n\\Psi(\\Omega_{R^\\sharp}^{d+1}k)\n=\\Omega_{R^\\sharp}^{d+1}k\/y\\Omega_{R^\\sharp}^{d+1}k\n&\\cong\\Omega_{{R^\\sharp}\/y{R^\\sharp}}^{d+1}k\\oplus\\Omega_{{R^\\sharp}\/y{R^\\sharp}}^dk\\\\\n\\notag&\\cong\\Omega_R^{d+1}k\\oplus\\Omega_R^dk\n\\cong\\Omega_R^dk[1]\\oplus\\Omega_R^dk,\n\\end{align}\nwhere the first isomorphism follows from \\cite[Corollary 5.3]{T0}.\nApplying $\\Phi$, we obtain\n\\begin{equation}\\label{2e}\n\\Omega_{R^\\sharp}^{d+1}k\\oplus\\Omega_{R^\\sharp}^{d+1}k[1]\\cong\\Phi(\\Omega_R^dk)[1]\\oplus\\Phi(\\Omega_R^dk).\n\\end{equation}\nIt is observed from \\eqref{1e} and \\eqref{2e} that $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Psi(\\Omega_{R^\\sharp}^{d+1}k))=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Omega_R^dk)$ and $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Phi(\\Omega_R^dk))=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Omega_{R^\\sharp}^{d+1}k)$, respectively.\nConsequently we obtain\n\\begin{align*}\n\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)\n&=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Phi(\\Omega_R^dk))\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Phi M}(\\Omega_{R^\\sharp}^{d+1}k)\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^{\\Omega_{R^\\sharp} M}(\\Omega_{R^\\sharp}^{d+1}k),\\\\\n\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}^N(\\Omega_{{R^\\sharp}}^{d+1}k)\n&=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Psi(\\Omega_{R^\\sharp}^{d+1}k))\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{\\Psi N}(\\Omega_R^dk)\n=\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^{N\/yN}(\\Omega_R^dk),\n\\end{align*}\nwhich completes the proof of the proposition.\n\\end{proof}\n\nUsing Lemma \\ref{dense} again, we get relationships of generation times in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\n\\begin{prop}\\label{prop1}\nThe following statements hold true.\n\\begin{enumerate}[\\rm(1)]\n\\item\nIf $M \\in \\operatorname{\\mathsf{\\underline{CM}}}(R)$ is a strong generator, then so is $\\Omega_{R^\\sharp} M\\in\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$, and $\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}(M) = \\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}(\\Omega_{R^\\sharp} M)$.\n\\item\nIf $N \\in \\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$ is a strong generator, then so is $N\/yN\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$, and $\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}(N) = \\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}(N\/yN)$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\n(1) We use Lemma \\ref{dense} and adopt its notation.\nPut $n=\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}(M)$.\nBy definition, it holds that ${\\langle M\\rangle}_{n+1}=\\operatorname{\\mathsf{\\underline{CM}}}(R)\\ne{\\langle M\\rangle}_n$.\nWhat we need to prove is that ${\\langle\\Phi M\\rangle}_{n+1}=\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})\\ne{\\langle\\Phi M\\rangle}_n$.\nFor each $X\\in\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$ we have $\\Psi X\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)={\\langle M\\rangle}_{n+1}$, and $\\Phi\\Psi X\\in{\\langle\\Phi M\\rangle}_{n+1}$.\nSince $X$ is a direct summand of $\\Phi\\Psi X$, it belongs to ${\\langle\\Phi M\\rangle}_{n+1}$.\nTherefore, we get $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})={\\langle\\Phi M\\rangle}_{n+1}$.\nSuppose that the equality $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})={\\langle\\Phi M\\rangle}_n$ holds.\nTaking any $Y\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$, we have $\\Phi Y\\in\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})={\\langle\\Phi M\\rangle}_n$, and $\\Psi\\Phi Y\\in{\\langle\\Psi\\Phi M\\rangle}_n$.\nAs $Y$ is a direct summand of $\\Psi\\Phi Y$, it is in ${\\langle\\Psi\\Phi M\\rangle}_n$.\nHence $\\operatorname{\\mathsf{\\underline{CM}}}(R)={\\langle\\Psi\\Phi M\\rangle}_n={\\langle M\\oplus M[1]\\rangle}_n={\\langle M\\rangle}_n$, which is a contradiction.\nThus $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})\\ne{\\langle\\Phi M\\rangle}_n$.\n\n(2) An analogous argument to the proof of (1) applies.\n\\end{proof}\n\nThe Orlov spectra and Rouquier dimensions of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$ coincide:\n\n\\begin{cor}\\label{cor}\nOne has the following equalities.\n$$\n\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)=\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp}),\\qquad\n\\dim\\operatorname{\\mathsf{\\underline{CM}}}(R)=\\dim\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp}).\n$$\n\\end{cor}\n\n\\begin{proof}\nIt suffices to show the first equality, since the second equality follows by taking the infimums of the both sides of the first equality.\nUsing Proposition \\ref{prop1}(1), we obtain\n\\begin{align*}\n\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)\n&=\\{\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}(M)\\mid\\text{$M$ is a strong generator of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$}\\}\\\\\n&=\\{\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}(\\Omega_{{R^\\sharp}}M)\\mid\\text{$\\Omega_{{R^\\sharp}}M$ is a strong generator of $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$}\\}\\\\\n&\\subseteq\\{\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})}(N)\\mid\\text{$N$ is a strong generator of $\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$}\\}\n=\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp}).\n\\end{align*}\nA similar argument using Proposition \\ref{prop1}(2) shows the opposite inclusion $\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})\\subseteq\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\nWe thus conclude that $\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)=\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}({R^\\sharp})$.\n\\end{proof}\n\n\\section{The singularity category of a hypersurface of countable representation type}\n\nIn this section, we prove our main results, that is, Theorems \\ref{B} and \\ref{A} from the Introduction.\nWe start by the following lemma on exact triangles in a triangulated category (an exact triangle $A\\to B\\to C\\to A[1]$ is simply denoted by $A\\to B\\to C\\rightsquigarrow$).\n\n\\begin{lem}\\label{ladder}\nLet $\\mathcal{T}$ be a triangulated category.\nLet\n$$\nX\\xrightarrow{{\\left(\\begin{smallmatrix}f_1\\\\f_2\\end{smallmatrix}\\right)}}Y_1\\oplus M\\xrightarrow{{\\left(\\begin{smallmatrix}g_1&\\alpha\\end{smallmatrix}\\right)}}N\\overset{p}\\rightsquigarrow,\\qquad\nM\\xrightarrow{{\\left(\\begin{smallmatrix}\\alpha\\\\g_2\\end{smallmatrix}\\right)}}N\\oplus Y_2\\xrightarrow{\\left(\\begin{smallmatrix}h_1&h_2\\end{smallmatrix}\\right)}Z\\overset{q}\\rightsquigarrow\n$$\nbe exact triangles in $\\mathcal{T}$.\nThen there exists an exact triangle in $\\mathcal{T}$ of the form\n$$\nX\\xrightarrow{\\left(\\begin{smallmatrix}f_1\\\\g_2f_2\\end{smallmatrix}\\right)}Y_1\\oplus Y_2\\xrightarrow{\\left(\\begin{smallmatrix}h_1g_1&-h_2\\end{smallmatrix}\\right)}Z\\rightsquigarrow.\n$$\n\\end{lem}\n\n\\begin{proof}\nThere is an isomorphism\n$$\n\\xymatrix@R=3pc@C=5pc{\nM\\ar[r]^-{\\left(\\begin{smallmatrix}\n0\\\\\n1\\\\\n0\n\\end{smallmatrix}\\right)}\\ar@{=}[d] & Y_1\\oplus M\\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\n1 & 0 & 0\\\\\n0 & 0 & 1\n\\end{smallmatrix}\\right)}\\ar[d]_{\\cong}^-{\\left(\\begin{smallmatrix}1&0&0\\\\0&1&0\\\\0&g_2&1\\end{smallmatrix}\\right)} & Y_1\\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\n0 & 0\n\\end{smallmatrix}\\right)}\\ar[d]_{\\cong}^-{\\left(\\begin{smallmatrix}1&0\\\\0&-1\\end{smallmatrix}\\right)} & M[1]\\ar@{=}[d] \\\\\nM\\ar[r]^-{\\left(\\begin{smallmatrix}\n0\\\\\n1\\\\\ng_2\n\\end{smallmatrix}\\right)} & Y_1\\oplus M\\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\n1 & 0 & 0\\\\\n0 & g_2 & -1\n\\end{smallmatrix}\\right)} & Y_1\\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}0&0\\end{smallmatrix}\\right)} & M[1] \n}\n$$\nof sequences.\nThe first row is an exact triangle in $\\mathcal{T}$ since it is the direct sum of exact triangles arising from the identity maps of $M$ and $Y_1\\oplus Y_2$ (see \\cite[Proof of Corollary 1.2.7]{N}).\nHence the second row is an exact triangle in $\\mathcal{T}$ as well.\nWe have a commutative diagram\n$$\n\\xymatrix@R=3pc@C=5pc{\n& X\\ar@{=}[r]\\ar[d]^-{\\left(\\begin{smallmatrix}\nf_1\\\\\nf_2\\\\\n0\n\\end{smallmatrix}\\right)} & X\\ar@{..>}[d]^-{\\left(\\begin{smallmatrix}\ns\\\\\nt\n\\end{smallmatrix}\\right)}\\\\\nM\\ar[r]^-{\\left(\\begin{smallmatrix}\n0\\\\\n1\\\\\ng_2\n\\end{smallmatrix}\\right)}\\ar@{=}[d] & Y_1\\oplus M \\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\n1 & 0 & 0\\\\\n0 & g_2 & -1\n\\end{smallmatrix}\\right)}\\ar[d]^-{\\left(\\begin{smallmatrix}g_1&\\alpha&0\\\\0&0&1\\end{smallmatrix}\\right)} & Y_1\\oplus Y_2\\ar[r]^{\\left(\\begin{smallmatrix}0&0\\end{smallmatrix}\\right)}\\ar@{..>}[d]^-{\\left(\\begin{smallmatrix}u&v\\end{smallmatrix}\\right)} & M[1]\\ar@{=}[d]\\\\\nM\\ar[r]^-{\\left(\\begin{smallmatrix}\n\\alpha\\\\\ng_2\n\\end{smallmatrix}\\right)} & N \\oplus Y_2\\ar[r]^-{\\left(\\begin{smallmatrix}\nh_1 & h_2 \n\\end{smallmatrix}\\right)}\\ar[d]^-{\\left(\\begin{smallmatrix}p&0\\end{smallmatrix}\\right)} & Z\\ar[r]^-q\\ar@{..>}[d] & M[1]\\\\\n& X[1]\\ar@{=}[r] & X[1]\n}\n$$\nwhere the rows are exact triangles in $\\mathcal{T}$, and so is the left column since it is a direct sum of exact triangles (see \\cite[Proposition 1.2.1]{N}).\nUsing the octahedral axiom, we obtain the right column which is an exact triangle in $\\mathcal{T}$.\nThe diagram chasing shows that $\\left(\\begin{smallmatrix}s\\\\t\\end{smallmatrix}\\right)=\\left(\\begin{smallmatrix}f_1\\\\g_2f_2\\end{smallmatrix}\\right)$ and $\\left(\\begin{smallmatrix}u&v\\end{smallmatrix}\\right)=\\left(\\begin{smallmatrix}h_1g_1&-h_2\\end{smallmatrix}\\right)$.\nThus it is an exact triangle we want.\n\\end{proof}\n\nLet $(R,\\mathfrak{m},k)$ be a complete equicharacteristic local hypersurface of dimension $d$.\nAssume that $k$ is uncountable and has characteristic different from two, and that $R$ has {\\em countable (Cohen--Macaulay) representation type}, namely, there exist infinitely but only countably many isomorphism classes of indecomposable maximal Cohen--Macaulay $R$-modules. \nThen $f$ is either of the following; see \\cite[Theorem 14.16]{LW}.\n\\begin{align*}\n(\\mathrm{A}_\\infty) &:\\ x_0^2+x_2^2+\\cdots +x_d^2,\\\\\n(\\mathrm{D}_\\infty) &:\\ x_0^2x_1+x_2^2+\\cdots +x_d^2.\n\\end{align*}\nIn this case, all objects in $\\operatorname{\\mathsf{CM}}(R)$ are completely classified \\cite{BGS,BD,K}.\n\nNow we can state and prove the following result regarding levels in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\n\\begin{thm}\\label{B}\nLet $k$ be an uncountable algebraically closed field of characteristic not two.\nLet $R$ be a $d$-dimensional complete local hypersurface over $k$ of countable representation type.\nThen\n$$\n\\Omega_R^dk\\in{\\langle M\\rangle}_2^{\\operatorname{\\mathsf{\\underline{CM}}}(R)}\n$$\nfor all nonzero objects $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\nIn other words, $\\operatorname{level}_{\\operatorname{\\mathsf{\\underline{CM}}}(R)}^M(\\Omega_R^dk)\\le1$.\n\\end{thm}\n\n\\begin{proof}\nProposition \\ref{lev}(2) reduces to the case $d=1$.\nThus we have the two cases:\n$$\n\\text{(1) }R=k[\\![x,y]\\!]\/(x^2),\\qquad\n\\text{(2) }R=k[\\![x,y]\\!]\/(x^2y).\n$$\n\n(1):\nThanks to \\cite[4.1]{BGS}, the indecomposable objects of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ are the ideals $I_n=(x,y^n)$ with $n\\in\\mathbb{Z}_{>0}\\cup\\{\\infty\\}$, where $I_{\\infty}:=(x)$.\nBy \\cite[6.1]{Sc} there exist exact triangles\n$$\nI_{n}\\to I_{n-1}\\oplus I_{n+1}\\to I_{n}\\rightsquigarrow\\quad(n\\in\\mathbb{Z}_{>0}),\n$$\nwhere $I_0:=0$.\nApplying Lemma \\ref{ladder}, we obtain exact triangles\n$$\nI_{n}\\to I_{1}\\oplus I_{2n-1}\\to I_{n}\\rightsquigarrow\\quad(n\\in\\mathbb{Z}_{>0}),\n$$\nand from \\cite[Proposition 2.1]{AIT} we obtain an exact triangle $I_{\\infty}\\to I_{1}\\to I_{\\infty}\\rightsquigarrow$.\nIt is observed from these triangles that $\\Omega k=I_1$ is in ${\\langle M\\rangle}_2$ for each nonzero object $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\n(2):\nUsing \\cite[4.2]{BGS}, we get a complete list of the indecomposable objects of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$:\n\\begin{align*}\n&X=R\/(x),\n\\ \\Omega_RX,\n\\ M_0^+=R\/(x^2),\n\\ M_0^-=\\Omega M_0^+,\\\\\n&M_n^+=\\mathrm{Coker}\\left(\\begin{smallmatrix}\nx&y^n\\\\\n0&-x\n\\end{smallmatrix}\\right),\n\\ M_n^-=\\Omega M_n^+,\n\\ N_n^+=\\mathrm{Coker}\\left(\\begin{smallmatrix}\nx&y^n\\\\\n0&-xy\n\\end{smallmatrix}\\right),\n\\ N_n^{-}=\\Omega N_n^{+} \\quad(n\\in\\mathbb{Z}_{>0}).\n\\end{align*}\nAccording to \\cite[(6.1)]{Sc}, for each $n\\in\\mathbb{Z}_{>0}$ there are exact triangles\n$$\nM_n^{\\pm}\\to N_{n+1}^{\\pm}\\oplus N_{n}^{\\mp}\\to M_n^{\\mp}\\rightsquigarrow,\\qquad\nN_n^{\\pm}\\to M_{n}^{\\pm}\\oplus M_{n-1}^{\\mp}\\to N_n^{\\mp}\\rightsquigarrow,\n$$\nwhere $N_0^{\\pm}:=0$.\nLemma \\ref{ladder} gives rise to exact triangles\n$$\nM_n\\to N_1 \\oplus N_{2n} \\to M_n\\rightsquigarrow,\\qquad\nN_n\\to N_1 \\oplus N_{2n-1} \\to N_n\\rightsquigarrow,\n$$\nwhere $M_n$ stands for either $M_n^+$ or $M_n^-$, and so on.\nAlso, by \\cite[Proposition 2.1]{AIT} we get an exact triangle $\\Omega X\\to N_{1}^-\\to X\\rightsquigarrow$.\nThus $\\Omega k=N_1^-$ is in ${\\langle C\\rangle}_2$ for any $0\\ne C\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{a}]\nThe assertion is immediate from Theorems \\ref{B} and \\ref{ragnar}.\n\\end{proof}\n\nThe following example shows that Theorem \\ref{a} does not necessarily hold if one replaces $k$ with another nonzero object of the singularity category.\n\n\\begin{ex}\\label{3}\nLet $R=k[\\![x,y]\\!]\/(x^2)$ be a hypersurface over a field $k$.\nThen\n$$\n(x,y^a)R\\notin{\\left\\langle(x,y^b)R\\right\\rangle}_2^{\\operatorname{\\mathsf{D_{sg}}}(R)}\n$$\nfor all positive integers $a,b$ with $a>2b$.\nThus $\\operatorname{level}_{\\operatorname{\\mathsf{D_{sg}}}(R)}^{(x,y^b)R}((x,y^a)R)\\ge2$.\n\\end{ex}\n\n\\begin{proof}\nIn view of Theorem \\ref{ragnar} we replace $\\operatorname{\\mathsf{D_{sg}}}(R)$ with $\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\nLet $I=(x,y^a)R$ and $J=(x,y^b)R$ be ideals of $R$.\nSuppose that $I$ belongs to ${\\langle J\\rangle}_2^{\\operatorname{\\mathsf{\\underline{CM}}}(R)}$.\nSince $\\Omega J\\cong J$, we see that there exists an exact sequence $0 \\to J^{\\oplus m} \\to I\\oplus M \\to J^{\\oplus n} \\to 0$ of $R$-modules with $m,n\\ge0$.\nThis induces an exact sequence\n$$\n\\operatorname{Tor}_1^R(R\/I,J^{\\oplus m}) \\to \\operatorname{Tor}_1^R(R\/I,I)\\oplus\\operatorname{Tor}_1^R(R\/I,M) \\to \\operatorname{Tor}_1^R(R\/I,J^{\\oplus n}).\n$$\nSince $\\operatorname{Tor}_1^R(R\/I,J)\\cong\\operatorname{Tor}_2^R(R\/I,R\/J)$, the first and third Tor modules in the above exact sequence are annihilated by $J$, and hence\n\\begin{equation}\\label{1}\nJ^2\\operatorname{Tor}_1^R(R\/I,I)=0.\n\\end{equation}\nThe minimal free resolution of $R\/I$ is\n$$\nF=(\\cdots\\xrightarrow{\\left(\\begin{smallmatrix}\ny^a&x\\\\\n-x&0\n\\end{smallmatrix}\\right)}R^{\\oplus2}\\xrightarrow{\\left(\\begin{smallmatrix}\n0&-x\\\\\nx&y^a\n\\end{smallmatrix}\\right)}R^{\\oplus2}\\xrightarrow{\\left(\\begin{smallmatrix}\ny^a&x\\\\\n-x&0\n\\end{smallmatrix}\\right)}R^{\\oplus2}\\xrightarrow{(x,y^a)}R\\to0),\n$$\nwhich induces a complex\n$$\nF\\otimes_RR\/I=(\\cdots\\xrightarrow{0}(R\/I)^{\\oplus2}\\xrightarrow{0}(R\/I)^{\\oplus2}\\xrightarrow{0}(R\/I)^{\\oplus2}\\xrightarrow{0}R\/I\\to0).\n$$\nHence $\\operatorname{Tor}_1^R(R\/I,I)\\cong\\operatorname{Tor}_2^R(R\/I,R\/I)=(R\/I)^{\\oplus2}$.\nBy \\eqref{1} we have $J^2(R\/I)^{\\oplus2}=0$, which implies that $J^2$ is contained in $I$.\nTherefore the element $y^{2b}$ is in the ideal $(x,y^a)R$, but this cannot happen since $a>2b$.\n\\end{proof}\n\nRecall that a subcategory of a triangulated category is called {\\em thick} if it is a triangulated subcategory closed under direct summands.\nWe denote by $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ the subcategory of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ consisting of maximal Cohen--Macaulay $R$-modules that are free on the punctured spectrum of $R$.\nThe category $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ is a thick subcategory of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$, and in particular it is a triangulated category.\nFor a subcategory $\\mathcal{X}$ of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ we denote by $\\operatorname{\\mathsf{ind}}\\mathcal{X}$ the set of nonisomorphic indecomposable objects of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ that belong to $\\mathcal{X}$.\nWe can now state and prove the following result concerning relative Rouquier dimensions in $\\operatorname{\\mathsf{\\underline{CM}}}(R)$.\n\n\n\\begin{thm}\\label{A}\nLet $k$ be an uncountable algebraically closed field of characteristic not two.\nLet $R$ be a $d$-dimensional complete local hypersurface over $k$ of countable representation type.\nLet $\\mathcal{T}\\ne0$ be a thick subcategory of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$, and let $\\mathcal{X}$ be a subcategory of $\\mathcal{T}$ closed under finite direct sums, direct summands and shifts.\nThen:\n\\begin{enumerate}[\\rm(1)]\n\\item\n$\\mathcal{T}$ coincides with either $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ or $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$.\n\\item\n\\begin{enumerate}[\\rm(a)]\n\\item\nWhen $\\mathcal{T}=\\operatorname{\\mathsf{\\underline{CM}}}(R)$, one has\n$$\n\\dim_\\mathcal{X}\\mathcal{T}=\n\\begin{cases}\n0 & (\\text{if } \\mathcal{X}=\\mathcal{T}),\\\\\n1 & (\\text{if } \\mathcal{X} \\neq \\mathcal{T}\\text{ and } \\mathcal{X} \\nsubseteq \\operatorname{\\mathsf{\\underline{CM}^o}}(R)),\\\\\n\\infty & (\\text{if } \\mathcal{X} \\subseteq \\operatorname{\\mathsf{\\underline{CM}^o}}(R)).\n\\end{cases}\n$$\n\\item\nWhen $\\mathcal{T}=\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$, one has\n$$\n\\dim_\\mathcal{X}\\mathcal{T}=\n\\begin{cases}\n0 & (\\text{if } \\mathcal{X}=\\mathcal{T}),\\\\\n1 & (\\text{if } \\mathcal{X}\\neq \\mathcal{T}\\text{ and }\\#\\operatorname{\\mathsf{ind}} \\mathcal{X} =\\infty),\\\\\n\\infty & (\\text{if }\\#\\operatorname{\\mathsf{ind}} \\mathcal{X}<\\infty).\n\\end{cases}\n$$\n\\end{enumerate}\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n(1) We combine \\cite[Theorem 6.8]{T3} and \\cite[Theorem 1.1]{AIT}.\nThe singular locus of $R$ consists of two points $\\mathfrak{p}$ and $\\mathfrak{m}$, and its specialization-closed subsets are $\\operatorname{V}(\\mathfrak{p})$, $\\operatorname{V}(\\mathfrak{m})$ and $\\emptyset$.\nThese correspond to the thick subcategories $\\operatorname{\\mathsf{\\underline{CM}}}(R)$, $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ and $0$.\n\n(2) Part (a) follows from \\cite[Theorem 1.1]{AIT}.\nLet us show part (b).\nWhen $\\#\\operatorname{\\mathsf{ind}} \\mathcal{X}<\\infty$, let $X_1,\\dots,X_n$ be all the indecomposable objects in $\\mathcal{X}$.\nSuppose that $\\dim_\\mathcal{X}\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ is finite, say $m$.\nThen it follows that $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)={\\langle\\mathcal{X}\\rangle}_{m+1}={\\langle X\\rangle}_{m+1}$, where $X:=X_1\\oplus\\cdots\\oplus X_n\\in\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$.\nHence $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ has finite Rouquier dimension.\nBy \\cite[Theorem 1.1(2)]{DT}, the local ring $R$ has to have at most an isolated singularity.\nHowever, in either case of the types $(\\mathrm{A}_\\infty)$ and $(\\mathrm{D}_\\infty)$ we see that the nonmaximal prime ideal $(x_0,x_2,\\dots,x_d)R$ belongs to the singular locus of $R$, which is a contradiction.\nConsequently, we obtain $\\dim_\\mathcal{X}\\operatorname{\\mathsf{\\underline{CM}^o}}(R)=\\infty$.\n\nFrom now on we consider the case where $\\mathcal{X}\\ne\\mathcal{T}=\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ and $\\#\\operatorname{\\mathsf{ind}}\\mathcal{X}=\\infty$.\nWe adopt the same notation as in the proof of Theorem \\ref{B}.\n\nAssume that $R$ has type $(A_\\infty)$.\nAs $\\mathcal{X}$ is a proper subcategory, we can find a positive integer $n$ such that $I_n\\notin\\mathcal{X}$.\nSince there are infinitely many indecomposable objects in $\\mathcal{X}$, we can also find an integer $m>n$ such that $I_m\\in\\mathcal{X}$.\nThere exists an exact triangle\n$$\nI_{m}\\to I_{n}\\oplus I_{2m-n}\\to I_{m}\\rightsquigarrow\n$$\nin $\\mathcal{T}=\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$, which shows that $I_n$ belongs to ${\\langle \\mathcal{X}\\rangle}_2$.\nTherefore, we get $\\dim_\\mathcal{X}\\mathcal{T}\\le1$.\nSince $\\langle\\mathcal{X}\\rangle=\\mathcal{X}\\neq \\mathcal{T}$, we have $\\dim_\\mathcal{X}\\mathcal{T}\\ne 0$.\nConsequently, we obtain $\\dim_\\mathcal{X}\\mathcal{T}=1$.\n\nSuppose that $R$ is of type $(D_\\infty)$.\nSimilarly as above, we find two integers $m>n>0$ such that neither $M_n$ nor $N_n$ belongs to $\\mathcal{X}$ and either $M_m$ or $N_m$ is in $\\mathcal{X}$.\nWhen $M_m$ belongs to $\\mathcal{X}$, there are exact triangles\n$$\nM_{m}\\to M_{n}\\oplus M_{2m-n}\\to M_{m}\\rightsquigarrow,\\qquad\nM_{m}\\to N_{n}\\oplus N_{2m-n+1}\\to M_{m}\\rightsquigarrow.\n$$\nWhen $N_m$ is in $\\mathcal{X}$, we have exact triangles\n$$\nN_{m}\\to M_{n}\\oplus M_{2m-n-1}\\to N_{m}\\rightsquigarrow,\\qquad\nN_{m}\\to N_{n}\\oplus N_{2m-n}\\to N_{m}\\rightsquigarrow.\n$$\nIn either case, both $M_n$ and $N_n$ belong to ${\\langle \\mathcal{X}\\rangle}_2$.\nIt follows that $\\dim_\\mathcal{X}\\mathcal{T}\\le1$.\nAs $\\langle\\mathcal{X}\\rangle=\\mathcal{X}\\neq \\mathcal{T}$, we have $\\dim_\\mathcal{X}\\mathcal{T}\\ne 0$.\nNow we conclude $\\dim_\\mathcal{X}\\mathcal{T}=1$.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{b}]\nTheorems \\ref{A} and \\ref{ragnar} immediately deduce the assertion.\n\\end{proof}\n\nAs a corollary of Theorem \\ref{A}, we calculate the Orlov spectra and Rouquier dimensions of $\\operatorname{\\mathsf{\\underline{CM}}}(R)$ and $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ for a hypersurface $R$ of countable representation type.\n\n\\begin{cor}\\label{x}\nLet $k$ be an algebraically closed uncountable field of characteristic not two. \nLet $R$ be a $d$-dimensional complete local hypersurface over $k$ having countable representation type.\nThen one has the following equalities.\n\\begin{enumerate}[\\rm(1)]\n\\item\n$\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}}}(R)=\\{1\\}$.\n\\item\n$\\operatorname{OSpec}\\operatorname{\\mathsf{\\underline{CM}^o}}(R)=\\emptyset$.\n\\end{enumerate}\nIn particular, $\\dim\\operatorname{\\mathsf{\\underline{CM}}}(R)=1$ and $\\dim\\operatorname{\\mathsf{\\underline{CM}^o}}(R)=\\infty$.\n\\end{cor}\n\n\\begin{proof}\n(1) Applying Theorem \\ref{A}(2a) for $\\mathcal{X}=\\langle M\\rangle$, we see that it suffices to verify that\n\\begin{enumerate}[(a)]\n\\item\n$\\langle M\\rangle\\ne\\operatorname{\\mathsf{\\underline{CM}}}(R)$ for all $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)$, and\n\\item\n$\\langle M\\rangle\\ne\\operatorname{\\mathsf{\\underline{CM}}}(R)$ for some $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)\\setminus\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$.\n\\end{enumerate}\nStatement (a) is equivalent to saying that $\\dim\\operatorname{\\mathsf{\\underline{CM}}}(R)>0$, which follows from \\cite[Propositions 2.4 and 2.5]{radius}.\nThere exists an object $M\\in\\operatorname{\\mathsf{\\underline{CM}}}(R)\\setminus\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ since $R$ does not have an isolated singularity, and hence (b) follows from (a).\n\n(2) Let $M\\in\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$ be an object, and let $\\mathcal{X}=\\langle M\\rangle$ be a subcategory of $\\operatorname{\\mathsf{\\underline{CM}^o}}(R)$.\nThen it is observed that $\\#\\operatorname{\\mathsf{ind}}\\mathcal{X}<\\infty$, so applying Theorem \\ref{A}(2b), we obtain $\\dim_\\mathcal{X}\\operatorname{\\mathsf{\\underline{CM}^o}}(R)=\\infty$, which means $\\operatorname{gt}_{\\operatorname{\\mathsf{\\underline{CM}^o}}(R)}(M)=\\infty$.\nThus the assertion follows.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Corollary \\ref{c}]\nCombining Corollary \\ref{x} with Theorem \\ref{ragnar} yields the assertion.\n\\end{proof}\n\n\\begin{ac}\nThe authors thank the referee for reading the paper carefully and giving useful suggestions.\n\\end{ac}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{\\abstractname}%\n       \\else\n         \\small\n         \\begin{center}%\n           {\\bfseries \\ackname\\vspace{-.5em}\\vspace{\\z@}}%\n         \\end{center}%\n         \\quotation\n       \\fi}\n       {\\if@twocolumn\\else\\endquotation\\fi}\n\\fi\n\\makeatother\n\n\\title{On the convergence of stochastic transport equations to a deterministic parabolic one}\n\\author{Lucio Galeati}\n\\date{\\small{Institute of Applied Mathematics\\\\ University of Bonn, Germany}\\\\[1.2ex] lucio.galeati@iam.uni-bonn.de\\\\[1.5ex] \\today}\n\n\\begin{document}\n\\maketitle\n\\thispagestyle{empty}\n\\begin{abstract}\n\\noindent A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed. Our method works in dimension $d\\geq 2$; the case $d=1$ is also investigated but no conclusive answer is obtained. \n\\end{abstract}\n\\section{Introduction}\\label{section 1 - introduction}\n\n\nThroughout this paper, we consider a stochastic transport linear equation of the form\n\\begin{equation}\\label{sec 1 - STLE in compact Stratonovich form}\\tag{STLE}\n\\diff u = b\\cdot\\nabla u \\diff t + \\circ \\diff W\\cdot \\nabla u,\n\\end{equation}\nwhere $b=b(t,x)$ is a given deterministic function and $W=W(t,x)$ is a space-time dependent noise of the form\n\\begin{equation}\\label{sec 1 - explicit description of the noise}\nW(t,x)=\\sum_k \\sigma_k(x)W_k(t).\n\\end{equation}\nHere $\\sigma_k$ are smooth, divergence free, mean zero vector fields, $\\{W_k\\}_k$ are independent standard Brownian motions and the index $k$ might range on an infinite (countable) set; by \\eqref{sec 1 - STLE in compact Stratonovich form} we mean more explicitly the identity\n\\begin{equation}\\label{sec 1 - STLE in non compact Stratonovich form}\n\\diff u = b\\cdot \\nabla u \\diff t + \\sum_k \\sigma_k\\cdot\\nabla u \\circ \\diff W_k,\n\\end{equation}\nwhere $\\circ$ denotes Stratonovich integral. Let us explain the reasons for studying such equation.\n\n\nIn the case of space-independent noise, it has been shown in recent years, starting with \\cite{FlaGub}, that equation \\eqref{sec 1 - STLE in compact Stratonovich form} is well posed under much weaker assumptions on $b$ than its deterministic counterpart (i.e. with $W=0$), for which essentially sharp condition are given by \\cite{DiPLio}, \\cite{Amb}. There is now an extensive literature on the topic of regularization by noise for transport equations, see the review \\cite{Fla2} and the references in \\cite{BecFla}. However, from the modelling point of view, space-independent noise is too simple, since formally the characteristics associated to \\eqref{sec 1 - STLE in compact Stratonovich form} are given by\n\\begin{equation}\\nonumber\n\\diff X_t = -b(t,X_t) \\diff t - \\diff W_t.\n\\end{equation}\nNamely, if we interpret $u$ as an ensemble of ideal particles, the addition of such a multiplicative Stratonovich noise corresponds at the Lagrangian level to non interacting particles being transported by a drift $b$ as well as a random, space independent noise $W$. There are several models, especially those arising in turbulence (see \\cite{Cha} and the discussion in the introduction of \\cite{CogFla}), in which it seems more reasonable to consider all the particles to be subject to the same space-dependent, environmental noise $W$, which is randomly evolving over time and is not influenced by the particles; $W$ may be interpreted as an incompressible fluid in which the particles are immersed. The formal Lagrangian description of \\eqref{sec 1 - STLE in compact Stratonovich form} is\n\\begin{equation}\\label{sec 1 - stochastic characteristics associated, compact Stratonovich form}\n\\diff X_t = -b(t,X_t)\\diff t - \\circ \\diff W(t,X_t),\n\\end{equation}\nwhere the above equation is meaningful once we consider $W$ given by \\eqref{sec 1 - explicit description of the noise} and we explicit the series.\n\nAnother reason to consider a more structured noise is given by the fact that, in the case of nonlinear transport equations, explicit examples in which a space-independent noise doesn't regularize are known, see for instance Section 4.1 of \\cite{Fla}; instead a sufficiently structured, space-dependent noise can provide a partial regularization by avoiding coalescence of particles, as in \\cite{DFV}, \\cite{FlaGub2}.\n\nFinally, if we expect the paradigm \\textquotedblleft the rougher the noise, the better the regularization\\textquotedblright\\ to hold, as it has been observed frequently in regularization by noise phenomena, it is worth to investigate the effect on equation \\eqref{sec 1 - STLE in compact Stratonovich form} of a noise $W$ which has poor regularity in space.\n\nSpecifically, the main goal of this work is not to investigate well posedness of \\eqref{sec 1 - STLE in compact Stratonovich form}, but rather to understand what happens when the space regularity of $W$ is so weak that it's not clear how to give meaning to \\eqref{sec 1 - STLE in compact Stratonovich form} anymore. Indeed, when one writes the corresponding It\\^o formulation of \\eqref{sec 1 - STLE in compact Stratonovich form}, the It\\^o-Stratonovich corrector appearing is finite only if $W$ satisfies a condition of the form\n\\begin{equation}\\label{sec 1 - condition on the regularity of the noise}\n\\mathbb{E}\\Big[\\vert W(1,\\cdot)\\vert_{L^2}^2\\Big]<\\infty.\n\\end{equation}\nIn particular, if the above condition doesn't hold, typically the corrector will be of the form \\textquotedblleft $+\\infty\\,\\Delta u$\\textquotedblright\\ and therefore heuristically one would expect the solution to istantaneously dissipate and become constant, independently of the initial data. A rigorous proof of this assertion, by means of a Galerkin approximation, has been given in a specific case in \\cite[Theorem 1.3]{FlaLuo}, but the technique applied there seems sufficiently robust to be generalized to this setting as well. It turns out that, in order to obtain a non trivial limit when we consider solutions of \\eqref{sec 1 - STLE in compact Stratonovich form} for a sequence of noises $W^N$ whose $L^2$-norm is exploding as $N\\to\\infty$, a suitable sequence of multiplicative coefficients $\\varepsilon^N$ must be introduced. In order to explain better what we mean and to give a rough statement of the main result, we give a  brief description of the setting in which we study \\eqref{sec 1 - STLE in compact Stratonovich form}. More details will be given in the next section.\n\n\nWe consider everything to be defined on the $d$-dimensional torus $\\mathbb{T}^d = \\mathbb{R}^d\/(2\\pi\\mathbb{Z}^d)$ with periodic boundary condition, $d\\geq 2$, with suitable assumptions on $b$.\nWe denote by $\\mathcal{H}$ the closed subspace of $L^2(\\mathbb{T}^d;\\mathbb{R}^d)$ given by divergence free, mean-zero functions (see Section \\ref{subsection 2.1 - notation and functional setting} for the exact definition).\n\nWe fix an a priori given filtered probability space $(\\Omega,\\mathcal{F},\\mathcal{F}_t,\\mathbb{P})$ on which an $\\mathcal{H}$-cylindrical $\\mathcal{F}_t$-Wiener process $\\widetilde{W}$ is defined, see \\cite{DaP}. We apply to $\\widetilde{W}$ a Fourier multiplier $\\Theta$ such that $W:= \\Theta\\widetilde{W}$ satisfies \\eqref{sec 1 - condition on the regularity of the noise}. We consider, for this choice of $W$, the Cauchy problem given by \\eqref{sec 1 - STLE in compact Stratonovich form} together with a deterministic initial condition $u_0\\in L^2(\\mathbb{T}^d)$; we are interested in energy solutions $u$, namely $\\mathcal{F}_t$-progressively measurable processes, with weakly continuous paths, for which equation \\eqref{sec 1 - STLE in compact Stratonovich form} is satisfied when interpreted in an analitically weak sense, i.e. testing against smooth functions, and a suitable energy inequality holds.\nWe stress that we consider $u$ to be a strong solution in the probabilistic sense; we can vary $W$ by considering different choices of $\\Theta$, but the probability space and $\\widetilde{W}$ are fixed and a priori given. The main result can then be loosely stated as follows.\n\n\\begin{res}\\label{result sec 1 - introductory formulation of main theorem}\nAssume that $b$ satisfies suitable conditions together with the following assumption:\n\\begin{itemize}\n\\item[(UN)] (Uniqueness for the parabolic limit equation) $b$  is such that, for any $\\nu>0$, uniqueness holds in the class of weak $L^\\infty(0,T;L^2(\\mathbb{T}^d))$ solutions of the Cauchy problem\n\\begin{equation}\\label{sec 1 - parabolic limit problem}\n\\begin{cases} \\partial_t u = \\nu\\Delta u + b\\cdot\\nabla u\\\\ u(0)=u_0 \\end{cases}.\n\\end{equation}\n\\end{itemize}\nThen for any $\\nu>0$, there exists a class of sequences of Fourier multipliers $\\Theta^N$ and of constants $\\varepsilon^N$, with $\\varepsilon^N$ depending only on $\\nu$ and $\\Theta^N$ for each $N$, such that, denoting $W^N=\\Theta^N\\widetilde{W}$, for any $u_0\\in L^2(\\mathbb{T}^d)$, any sequence of energy solutions $u^N$ of the STLEs\n\\begin{equation}\\label{sec 1 - equations for the approximating u^N}\\begin{cases}\n\\diff u^N = b\\cdot\\nabla u^N\\, \\diff t + \\sqrt{\\varepsilon^N}\\circ \\diff W^N\\cdot \\nabla u^N\\\\\nu^N(0)=u_0\n\\end{cases}\\end{equation}\nconverges in probability (in a suitable topology) as $N\\to\\infty$ to the unique deterministic solution $u$ of the parabolic equation \\eqref{sec 1 - parabolic limit problem}.\n\\end{res}\n\nA more precise statement and the proof will be given in Section \\ref{section 3 - proof of the main result}; let us comment some of the features of the result.\n\n\\begin{itemize}\n\\item[i)] The statement is formulated in the spirit of a multiplicative renormalization: the sequence $\\varepsilon^N$ depends on the chosen $\\Theta^N$, but the limit does not, up to the arbitrary choice of a one dimensional parameter $\\nu>0$. However, as will be discussed in Section \\ref{section 3 - proof of the main result}, this is not a real renormalization due to the presence of some degeneracy: while we need to impose some conditions on $\\Theta^N$, these do not  imply uniqueness of the limit of $\\Theta^N \\widetilde{W}$ and explicit examples of choices leading to different limits, for which the above statement holds, can be given. In a sense, the result is more similar to a weak law of large numbers, as will be discussed in Section \\ref{section 3 - proof of the main result}.\n\n\\item[ii)] The statement provides a sequence of solutions of stochastic transport equations converging to a deterministic parabolic equation. This is rather surprising, not only for the transition from a stochastic problem to a deterministic one, but also for the change in the nature of the equation. The original STLEs are hyperbolic: whenever $W$ is regular enough, they can be solved explicitly by means of the stochastic flow associated to the characteristics \\eqref{sec 1 - stochastic characteristics associated, compact Stratonovich form}; in particular the solutions don't have in general better regularity than the initial data, at least at the level of trajectories. However, when considering the corresponding It\\^o formulation, the It\\^o-Stratonovich corrector gives rise to a Laplacian. It was intuited in \\cite{FlaGub} that equation \\eqref{sec 1 - STLE in compact Stratonovich form} has some parabolic features at the mean level; this has become clear in \\cite{BecFla}.\n\n\\item[iii)] The statements holds for \\textit{any} sequence of energy solutions of \\eqref{sec 1 - equations for the approximating u^N}, even when uniqueness is not known; existence of energy solutions can be shown under suitable assumptions on $b$. We only need well posedness for the limit problem \\eqref{sec 1 - parabolic limit problem} and that's why we require (UN) to hold. In general (UN) is satisfied under very mild assumptions on $b$, much weaker than those required for the associated deterministic transport equation to be well posed. This suggests the possibility to obtain uniqueness for the STLE under the same assumption (UN); in this direction, see the results given in \\cite{Mau}, \\cite{BecFla} and the references therein.\n\n\\item[iv)] From the modelling, perturbative viewpoint, the result could be interpreted in this way: when a system of particles transported by a drift $b$ is subject to an environmental background noise which is very irregular but of very small intensity, in the ideal limit such a disturbance is correctly modelled by a diffusive term $\\nu\\Delta$. This also gives an interesting link between different selection principles for ill posed transport equations, since it hints to the fact that a vanishing viscosity limit and certain types of zero noise limits should behave similarly; observe however that this is not true in general, since in the setting of space-independent noise, examples of transport equations for which the zero noise limit and the vanishing viscosity one do not coincide are provided in \\cite{AttFla}.\n\\end{itemize}\n\nWe believe our main result holds on a wider class of domains and not only on the torus, but there are several technical issues which prevent a straightforward generalization and solving them is currently an open problem. Indeed, if the domain is a bounded open subset of $\\mathbb{R}^d$, then a boundary condition must also be imposed and handled in the limit; in this regard, let us mention the recent work \\cite{Hai}, in which it is shown that in certain scaling regimes (however different from our case) also the boundary condition must undergo a renormalization. If the setting is instead a compact manifold without boundary, the main challenge becomes finding examples of vector fields $\\sigma_k$ for which the It\\^o-Stratonovich correctors, as well as their limit once properly renormalized, can be computed explicitly. On the torus this task is greatly simplified by the presence of many simmetries, as it is shown in Section \\ref{subsection 2.3 - SPDE in Ito form and definition of energy solutions}.\n\nLet us highlight that even if in the discussion we have adopted a perturbative approach, motivating \\eqref{sec 1 - STLE in compact Stratonovich form} as a stochastic variation of an originally deterministic problem, the equation is of interested by itself even when $b=0$ as it is related to the theory of passive scalars and the celebrated Kraichnan model of turbulence, see \\cite{Cha}, \\cite{Fal}. From the mathematical point of view, it has been treated in a very complete but rather technical way in \\cite{LeJ}, \\cite{LeJ2}; in Section \\ref{subsection 4.2 - proof of strong uniqueness in the case b=0} we present a simple proof in the case $b=0$ of pathwise uniqueness of $L^2(\\mathbb{T}^d)$-valued solutions under very mild assumptions on the noise (basically all isotropic divergence free noises for which the equation is well defined are included). To the best of our knowledge this result is new, since even in \\cite{LeJ2} is suggested but not explicitly stated whether pathwise uniqueness can be proved, see the beginning of Section \\ref{subsection 4.2 - proof of strong uniqueness in the case b=0} for more details.\n\nFinally, let us mention the strong similarity between our technique and the one considered in \\cite{FlaLuo2}.\n\n\n\\begin{plan}\nThe paper is structured as follows: in Section \\ref{section 2 - preliminaries}, we introduce our notations and basic definitions; in Section \\ref{section 3 - proof of the main result} we give a more precise statement and the proof of the main result. In Section \\ref{section 4 - discussion of existence and uniqueness}, in order for the main result to be non vacuous, we give a proof of existence of energy solutions and we discuss the problem of their uniqueness. Finally in Section \\ref{section 5 - the case d=1} we treat the case $d=1$, in which we show that we are not able to obtain an equivalent of the main result; still, from the modelling viewpoint, some interesting conclusions can be drawn.\n\\end{plan}\n\n\\section{Preliminaries}\\label{section 2 - preliminaries}\n\n\nIn this section we provide all the notions necessary to give a meaning to \\eqref{sec 1 - STLE in compact Stratonovich form} and its solutions; with this set up we will be able to prove the main result in the next section.\n\\subsection{Notations and functional setting}\\label{subsection 2.1 - notation and functional setting}\n\n\nWe work on the $d$-dimensional torus, $\\mathbb{T}^d=\\mathbb{R}^d\/(2\\pi\\mathbb{Z}^d)$, with periodic boundary condition. We denote by $L^2(\\mathbb{T}^d;\\mathbb{C})$ the set of complex-valued, square integrable function defined on $\\mathbb{T}^d$, which is a Hilbert space endowed with the (normalized) inner product\n\\begin{equation}\\nonumber\n\\langle f, g\\rangle_{L^2} = \\frac{1}{(2\\pi)^d}\\int_{\\mathbb{T}^d} f(x)\\overline{g}(x)\\diff x,\n\\end{equation}\nwhere $\\overline{z}$ denotes the complex conjugate of $z$, $\\vert z\\vert^2 = z\\,\\overline{z}$; we denote by $\\vert\\cdot\\vert_{L^2}$ the norm induced by $\\langle\\cdot,\\cdot\\rangle_{L^2}$. Under this inner product, $\\{e_k\\}_{k\\in\\mathbb{Z}^d}$ given by $e_k(x)= e^{i\\, k\\cdot x}$ is a complete orthonormal system ($k\\cdot x= \\sum_{i=1}^d k_i x_i$ denoting the standard inner product in $\\mathbb{R}^d$). Any element $f\\in L^2(\\mathbb{T}^d;\\mathbb{C})$ can be written uniquely in Fourier series as\n\\begin{equation}\\nonumber\nf = \\sum_{k\\in\\mathbb{Z}^d} f_k\\, e_k, \\quad f_k = \\langle f, e_k\\rangle_{L^2} ,\n\\end{equation}\nwhere the series is convergent in $L^2(\\mathbb{T}^d;\\mathbb{C})$ and it satisfies\n\\begin{equation}\\nonumber\n\\vert f\\vert_{L^2}^2 = \\sum_{k\\in\\mathbb{Z}^d} \\vert f_k\\vert^2.\n\\end{equation}\nAn element $f$ is real-valued if and only if $f_{-k}=\\overline{f}_k$ for every $k\\in\\mathbb{Z}^d$. We denote the set of square integrable, real-valued functions by $L^2(\\mathbb{T}^d)=L^2$. The formulas above hold more generally for $f\\in L^2(\\mathbb{T}^d;\\mathbb{R}^d)$ if we interpret $f_k$ as the $\\mathbb{C}^d$-valued vector with components $f_k^{(j)} = \\langle f^{(j)}, e_k\\rangle_{L^2}$.\n\nWe will always deal with real-valued functions, but for the sake of calculations it is more convenient to use complex Fourier series; for the same reason we work on $\\mathbb{T}^d$ defined as above rather than $\\mathbb{R}^d\/\\mathbb{Z}^d$. We stress however that the results are independent of this choice and can be obtained in the same way by using real Fourier series or $\\mathbb{R}^d\/\\mathbb{Z}^d$.\n\n\nWe consider the Sobolev spaces $H^\\alpha(\\mathbb{T}^d)$, $\\alpha\\in\\mathbb{R}$, given by\n\\begin{equation}\\nonumber\nH^\\alpha(\\mathbb{T}^d) = \\Big\\{f=\\sum_k f_k\\,e_k\\, \\Big\\vert\\, f_{-k}=\\overline{f}_k,\\, \\sum_k \\big(1+\\vert k\\vert^2\\big)^\\alpha \\vert f_k\\vert^2 <\\infty\\Big\\},\n\\end{equation}\nsee \\cite{Tem} for more details. Then the space of test functions $C^\\infty(\\mathbb{T}^d)$ corresponds to $\\cap_\\alpha H^\\alpha(\\mathbb{T}^d)$ and its dual $C^\\infty(\\mathbb{T}^d)'$, the space of distributions, to $\\cup_\\alpha H^\\alpha(\\mathbb{T}^d)$. We denote by $\\langle\\cdot,\\cdot\\rangle$ also the duality pairing between them.\n\nGiven $f\\in L^2(\\mathbb{T}^d;\\mathbb{R}^d)$, we say that $f$ is divergence free in the sense of distributions if\n\\begin{equation}\\nonumber\n\\langle f, \\nabla\\varphi\\rangle = 0 \\qquad \\forall\\, \\varphi\\in C^\\infty(\\mathbb{T}^d).\n\\end{equation}\nIt's easy to check that $f$ is divergence free if and only if $f_k\\cdot k = 0$ for all $k\\in\\mathbb{Z}^d$. Consider the subspace\n\\begin{equation}\\nonumber\n\\mathcal{H}= \\bigg\\{f\\in L^2(\\mathbb{T}^d;\\mathbb{R}^d) \\text{ such that } \\int_{\\mathbb{T}^d} f =0 \\text{ and } f \\text{ is divergence free}\\bigg\\}.\n\\end{equation}\n$\\mathcal{H}$ is a closed linear subspace of $L^2(\\mathbb{T}^d;\\mathbb{R}^d)$ and so the orthogonal projection $\\Pi:L^2(\\mathbb{T}^d;\\mathbb{R}^d)\\to \\mathcal{H}$ is a linear continuous operator. $\\Pi$ can be represented in Fourier series by\n\\begin{equation}\\nonumber\n\\Pi: f=\\sum_{k\\in\\mathbb{Z}^d} f_k\\, e_k \\mapsto \\Pi f = \\sum_{k\\in\\mathbb{Z}^d} P_k f_k\\, e_k,\n\\end{equation}\nwhere $P_k\\in \\mathbb{R}^{d\\times d}$ is the d-dimensional projection on $k^\\perp$, $P_k = I - \\frac{k}{\\vert k\\vert}\\otimes \\frac{k}{\\vert k\\vert}$, whenever $k\\neq 0$ and we set $P_0\\equiv 0$. $\\Pi$ can be extended to a continuous linear operator from $H^\\alpha(\\mathbb{T}^d;\\mathbb{R}^d)$ to itself for any $\\alpha\\in\\mathbb{R}$. We also define the projectors $\\Pi_N$ on the space of Fourier polynomials of degree at most $N$ by\n\\begin{equation}\\nonumber\nf=\\sum_k f_k\\,e_k\\mapsto \\Pi_Nf = \\sum_{k: \\vert k\\vert\\leq N} f_k\\, e_k,\n\\end{equation}\nwhere $\\Pi_N:C^\\infty(\\mathbb{T}^d)'\\to C^\\infty(\\mathbb{T}^d)$.\n\n\\subsection{Construction of the noise $W(t,x)$}\\label{subsection 2.2 - contruction of the noise}\n\n\nWe have introduced the space $\\mathcal{H}$ and the projector $\\Pi$ because we want to deal with an $\\mathcal{H}$-valued noise $W$; the reason for this choice will become clear in Section \\ref{subsection 2.3 - SPDE in Ito form and definition of energy solutions}. We are first going to construct $W$ by giving an explicit Fourier representation, but then we will also provide a more elegant, abstract construction.\n\nSet $\\mathbb{Z}^d_0=\\mathbb{Z}^d\\setminus \\{0\\}$ and consider $\\Lambda\\subset\\mathbb{Z}^d$ such that $\\Lambda$ and $-\\Lambda$ form a partition of $\\mathbb{Z}^d_0$. Consider a collection\n\\begin{equation}\\nonumber\n\\Big\\{ B^{(j)}_k, k\\in\\mathbb{Z}^d_0, 1\\leq j\\leq d-1\\Big\\}\n\\end{equation}\nof standard, real valued, independent $\\mathcal{F}_t$-Brownian motions, defined on a filtered probability space $(\\Omega,\\mathcal{F},\\mathcal{F}_t,\\mathbb{P})$, $\\{\\mathcal{F}_t\\}_{t\\geq 0}$ being a normal filtration (see \\cite{Rev}). Define\n\\begin{equation}\\nonumber\nW^{(j)}_k := \\begin{cases} B^{(j)}_k + iB^{(j)}_{-k}\\qquad & \\text{if }\\ k\\in\\Lambda\\\\\nB^{(j)}_k - iB^{(j)}_{-k} & \\text{if }\\ k\\in-\\Lambda \\end{cases}.\n\\end{equation}\nIn this way, $\\{W^{(j)}_k, k\\in\\mathbb{Z}^d_0, 1\\leq j\\leq d-1\\}$ is a collection of standard complex valued Brownian motions (namely complex processes with real and complex part given by independent real BM) such that $W^{(j)}_{-k}=\\overline{W}^{(j)}_k$ and $W^{(j)}_k, W^{(m)}_l$ are independent whenever $k\\neq \\pm l$, $j\\neq m$. We denote by $[M,N]$ the quadratic covariation process, which is defined for any couple $M$, $N$ of square integrable real semimartingales (see for instance \\cite{Rev}) and we extend it by bilinearity to the analogue complex valued processes.\nObserve that by result of bilinearity it holds\n\\begin{equation}\\nonumber\n\\Big[W^{(j)}_k,W^{(j)}_k\\Big]_t = 0,\\quad \\Big[W^{(j)}_k, W^{(j)}_{-k}\\Big]_t = 2t,\n\\end{equation}\nand therefore\n\\begin{equation}\\nonumber\n\\Big[W^{(j)}_k,W^{(m)}_l\\Big]_t = 2t\\,\\delta_{j,m}\\,\\delta_{k,-l} = 2t\\,\\delta_{j-m}\\,\\delta_{k+l}.\n\\end{equation}\nWe omit the details, but it's easy to check all the stochastic calculus rules, in particular It\\^o formula and It\\^o isometry, can be extended by bilinearity to the case of complex valued semimartingales.\n\nFor any $k\\in\\Lambda$, let $\\{a_k^{(1)},\\ldots, a_k^{(d-1)}\\}$ be an orthonormal basis of $k^\\perp$. Then $\\{k\/\\vert k\\vert, a_k^{(1)},\\ldots, a_k^{(d-1)}\\}$ form an orthonormal basis of $\\mathbb{R}^d$ and it holds\n\\begin{equation}\\nonumber\nP_k = a_k^{(1)}\\otimes a_k^{(1)} + \\ldots +a_k^{(d-1)}\\otimes a_k^{(d-1)} \\quad \\forall\\, k\\in\\Lambda;\n\\end{equation}\nfor $k\\in -\\Lambda$ we can set $a_k^{(j)} = a_{-k}^{(j)}$ and the above identity still holds.\n\nLet $\\{\\theta_k, k\\in\\mathbb{Z}^d_0\\}$ be a collection of real constants such that $\\theta_k=\\theta_{-k}$ and satisfying suitable conditions, which will be specified later. We set\n\\begin{equation}\\label{sec 2.2 - definition of the noise W(t,x)}\nW(t,x):= \\sum_{k\\in\\mathbb{Z}^d_0} \\theta_k \\Bigg(\\sum_{j=1}^{d-1} a_k^{(j)}\\,W_k^{(j)}(t)\\Bigg) e_k(x) .\n\\end{equation}\nFrom now on, whenever it doesn't create confusion, we will only write the indices $k,j$ without specifying their index sets, in order for the notation not to become too burdensome. Observe that, for fixed $t$, $W(t,\\cdot)$ is already written in its Fourier decomposition and by the definitions of $a_k^{(j)}$ and $W_k(j)$ it's a real, mean zero, divergence free random distribution. It only remains to show that, for fixed $t$, $W(t,\\cdot)$ belongs $\\mathbb{P}$-a.s. to $L^2(\\mathbb{T}^d;\\mathbb{R}^d)$. Indeed, denoting by $\\mathbb{E}$ the expectation with respect to $\\mathbb{P}$, we have\n\\begin{equation}\\nonumber\n\\mathbb{E}\\Big[\\,\\vert W(t,\\cdot)\\vert_{L^2}^2\\Big]\n= \\mathbb{E}\\Bigg[ \\sum_k  \\theta_k^2\\, \\Big\\vert \\sum_j a_k^{(j)}W_k^{(j)}(t)\\Big\\vert^2\\Bigg]\n= 2t(d-1)\\sum_k \\theta_k^2\n\\end{equation}\nand therefore, under the conditions\n\\begin{equation}\\label{sec 2.2 - condition on the coefficients}\\tag{H1}\n\\theta_{-k}=\\theta_k\\ \\forall\\, k,\\quad \\sum_k \\theta_k^2<\\infty,\n\\end{equation}\n$W(t,\\cdot)$ is a well defined random variable belonging to $L^2(\\Omega,\\mathcal{F},\\mathbb{P};\\mathcal{H})$. From the point of view of mathematical rigour, we should have first done the above calculation when summing over finite $k$ and then shown that, under condition \\eqref{sec 2.2 - condition on the coefficients}, the sequence of finite sums is Cauchy; this can be easily checked and we omit it for the sake of simplicity. With similar calculations, exploiting Gaussianity and Kolmogorov continuity criterion, it can be shown that, as an $\\mathcal{H}$-valued process, up to modification $W$ has paths in $C^{\\alpha}([0,T];\\mathcal{H})$ for any $\\alpha<1\/2$, see for instance \\cite{DaP}.\n\n\nWe now show an alternative, more abstract construction of $W$. Let $\\theta_k$ be some real coefficients satisfying \\eqref{sec 2.2 - condition on the coefficients} as before and define the Fourier multiplier\n\\begin{equation}\\nonumber\n\\Theta: f=\\sum_k f_k\\, e_k\\mapsto \\Theta f= \\sum_k \\theta_k f_k\\, e_k.\n\\end{equation}\nThen $\\Theta$ is a continuous, self-adjoint operator from $H^\\alpha(\\mathbb{T}^d;\\mathbb{R}^d)$ to itself which commutes with $\\Pi$; condition \\eqref{sec 2.2 - condition on the coefficients} implies that $\\Theta$ is an Hilbert-Schmidt operator, namely $\\Theta^\\ast \\Theta=\\Theta^2$ is a trace class operator:\n\\begin{equation}\\nonumber\n\\Theta^2: f=\\sum_k f_k\\, e_k\\mapsto \\Theta^2 f= \\sum_k \\theta_k^2\\, f_k\\, e_k.\n\\end{equation}\nNow let $\\widetilde{W}$ be a cylindrical Wiener process on $L^2(\\mathbb{T}^d;\\mathbb{R}^d)$ (in the sense of \\cite{DaP}): a Gaussian distribution valued process with covariance\n\\begin{equation}\\nonumber\n\\mathbb{E}\\big[\\langle \\widetilde{W}_t,\\varphi\\rangle\\,\\langle \\widetilde{W}_s,\\psi\\rangle \\big]\n= (t\\wedge s)\\langle \\varphi,\\psi\\rangle\n\\quad \\forall\\, \\varphi,\\psi\\in C^\\infty(\\mathbb{T}).\n\\end{equation}\nThen it can be shown that, up to modifications, $\\widetilde{W}$ has paths in $C^\\alpha([0,T],H^\\beta)$ for any $\\alpha<1\/2$ and for any $\\beta<-d\/2$. If we define\n\\begin{equation}\\nonumber\nW := \\Theta\\Pi \\widetilde{W},\n\\end{equation}\nthen $W$ is a Wiener process on $\\mathcal{H}$. This construction is useful as it shows that we can consider, on a given filtered probability space with a given noise $\\widetilde{W}$, several different $W$ just by varying the deterministic operator $\\Theta$. By construction, $W$ has covariance given by\n\\begin{equation}\\nonumber\\begin{split}\n\\mathbb{E}\\big[\\langle W_t,\\varphi\\rangle\\,\\langle W_s,\\psi\\rangle \\big]\n& = (t\\wedge s)\\langle (\\Theta\\Pi)^\\ast\\varphi,(\\Theta\\Pi)^\\ast\\psi\\rangle\\\\\n& = (t\\wedge s)\\langle \\Theta^2\\Pi\\varphi,\\psi\\rangle\n\\qquad \\quad \\quad \\forall\\, \\varphi,\\psi\\in C^\\infty(\\mathbb{T}).\n\\end{split}\\end{equation}\nIt can be checked that $W$ defined as above is space homogeneous, namely its distribution is invariant under space translations $W(t.\\cdot)\\mapsto W(t,x+\\cdot)$ for any $x\\in\\mathbb{T}^d$. This is a consequence of the fact that $\\widetilde{W}$ is space homogeneous and $W$ is defined by a Fourier multiplier. For our purposes, we want it to be isotropic as well.\n\nLet $E_\\mathbb{Z}(d)$ denote the group of linear isometries of $\\mathbb{R}^d$ into itself which leave $\\mathbb{Z}^d$ invariant; it is the group generated by swaps\n\\begin{equation}\\nonumber\n(x_1,\\ldots,x_i,\\ldots,x_j,\\ldots, x_d)\\mapsto (x_1,\\ldots,x_j,\\ldots,x_i,\\ldots, x_d)\n\\end{equation}\nand reflections\n\\begin{equation}\\nonumber\n(x_1\\ldots x_{i-1},x_i,x_{i+1}\\ldots x_d)\\mapsto (x_1\\ldots x_{i-1},-x_i,x_{i+1}\\ldots x_d).\n\\end{equation}\nTo see this, observe that if $O\\in E_\\mathbb{Z}(d)$, then for any element $e_i$ of the canonical basis it holds $Oe_i\\in\\mathbb{Z}^d$ and $\\vert Oe_i\\vert=1$, which necessarily implies that $Oe_i=\\pm e_j$ for another index $j$.\n$W$ is isotropic if its law is invariant under transformations $W(t,\\cdot)\\mapsto W(t,O\\cdot)$ for all $O\\in E_\\mathbb{Z}(d)$ . In order to have an isotropic noise, we impose the following condition on the coefficients $\\theta_k$:\n\\begin{equation}\\label{sec 2.2 - isotropy condition}\\tag{H2}\n\\theta_k = \\theta_{Ok}\\quad\\, \\forall\\, O\\in E_{\\mathbb{Z}}(d).\n\\end{equation}\nTipical choices of $\\theta_k$ will be of the form $\\theta_k = F(\\vert k\\vert)$, where $F:\\mathbb{R}_{\\geq 0}\\to\\mathbb{R}_{\\geq 0}$ is a function with sufficient decay at infinity; for instance we can take $F$ with finite support, or $F(r)=r^{-\\alpha}$ for some $\\alpha>d\/2$. However all statements in the next section hold in general as long as \\eqref{sec 2.2 - condition on the coefficients} and \\eqref{sec 2.2 - isotropy condition} are satisfied.\n\n\\subsection{STLE in It\\^o form and definition of energy solutions}\\label{subsection 2.3 - SPDE in Ito form and definition of energy solutions}\n\nWe can now write explicitly \\eqref{sec 1 - STLE in compact Stratonovich form} and find the corresponding It\\^o formulation.\nIn order to simplify the exposition, we will do all the computations as if we were summing over a finite number of $k$ and find the right conditions under which every sum is well defined. Rigorously speaking, we should use an approximation argument and check that the finite series form a Cauchy sequence, but we skip this technical part, which can be easily verified.\n\n\nLet $W$ be given as in \\eqref{sec 2.2 - definition of the noise W(t,x)}, then \\eqref{sec 1 - STLE in compact Stratonovich form} can be formulated as\n\\begin{equation}\\label{sec 2.3 - equation in Stratonovich form}\n\\diff u = b\\cdot\\nabla u\\diff t + \\sum_{j,k} \\theta_k\\, e_k\\, a_k^{(j)}\\cdot \\nabla u\\circ \\diff W^{(j)}_k.\n\\end{equation}\nEquation \\eqref{sec 2.3 - equation in Stratonovich form} must be interpreted in integral form: a process $u$ is a strong (from the analytical point of view) solution if $\\mathbb{P}$-a.s. the following identity is satisfied for every $t,x$ (and $u$ is pogressively measurable and sufficiently regular for it to be meaningful):\n\\begin{equation}\\nonumber\\begin{split}\nu(t,x)-u(0,x) = & \\int_0^t b(s,x)\\cdot\\nabla u(s,x)\\diff s\\\\ &+ \\sum_{j,k}\\theta_k \\int_0^t e_k(x)\\,a^{(j)}_k\\cdot\\nabla u(s,x)\\circ \\diff W_k^{(j)}(s).\n\\end{split}\\end{equation}\nSince in general Stratonovich integral is not so easy to control, we prefer to pass to the equivalent formulation in It\\^o form:\n\\begin{equation}\\nonumber\\begin{split}\n\\diff u\n& = b\\cdot\\nabla u\\diff t + \\sum_{j,k} \\theta_k\\, e_k\\, a_k^{(j)}\\cdot \\nabla u \\diff W^{(j)}_k + \\frac{1}{2}\\sum_{j,k} \\theta_k\\, e_k \\diff \\big[a_k^{(j)}\\cdot\\nabla u, W^{(j)}_k\\big]\\\\\n& = b\\cdot\\nabla u\\diff t + \\sum_{j,k} \\theta_k\\, e_k\\, a_k^{(j)}\\cdot \\nabla u\\diff W^{(j)}_k + \\sum_{j,k}  \\theta_k^2\\, e_k\\,a_k^{(j)}\\cdot\\nabla\\Big( e_{-k}\\,a_{-k}^{(j)}\\cdot\\nabla u\\Big) \\diff t\\\\\n& = b\\cdot\\nabla u\\diff t + \\sum_{j,k} \\theta_k\\, e_k\\, a_k^{(j)}\\cdot \\nabla u\\diff W^{(j)}_k + \\sum_{k,j} \\theta_k^2\\, \\text{Tr}\\Big(a_k^{(j)}\\otimes a_k^{(j)}\\, D^2u\\Big)\\diff t\\\\\n& = b\\cdot\\nabla u\\diff t + \\sum_{j,k} \\theta_k\\, e_k\\, a_k^{(j)}\\cdot \\nabla u\\diff W^{(j)}_k + \\text{Tr}\\left( \\Big( \\sum_{k} \\theta_k^2 P_k\\Big) \\, D^2u\\right)\\diff t.\n\\end{split}\\end{equation}\nIn the above computation we exploited many of the properties of $a^{(j)}_k$ and $W^{(j)}_k$ highlighted in the previous section: $d[W^{(j)}_k,W^{(l)}_m] = 2\\delta_{j,l}\\delta_{k,-m}\\,dt$, $a_k^{(j)}\\cdot k=0$, $a_k^{(j)}=a^{(j)}_{-k}$. It remains to compute more explicitly the matrix appearing in the last line on the right hand side:\n\\begin{equation}\\nonumber\n\\sum_k \\theta_k^2 P_k\n= \\sum_k \\theta_k^2 \\left(I - \\frac{k}{\\vert k\\vert}\\otimes \\frac{k}{\\vert k\\vert}\\right)\n= \\left( \\sum_k \\theta_k^2\\right) I - \\sum_k \\theta_k^2 \\frac{k}{\\vert k\\vert}\\otimes \\frac{k}{\\vert k\\vert}.\n\\end{equation}\nBy the isotropy condition \\eqref{sec 2.2 - isotropy condition} of $\\theta_k$, whenever $i\\neq j$, using the change of variables $k\\mapsto \\tilde{k}$ that switches the sign of the $i$-th component, we have\n\\begin{equation}\\nonumber\n\\left( \\sum_k \\theta_k^2 \\frac{k}{\\vert k\\vert}\\otimes \\frac{k}{\\vert k\\vert}\\right)_{ij}\n= \\sum_k \\theta_k^2\\, \\frac{k^{(i)}\\,k^{(j)}}{\\vert k\\vert^2}\n= \\sum_{\\tilde{k}} \\theta_{\\tilde{k}}^2\\, \\frac{\\tilde{k}^{(i)}\\,\\tilde{k}^{(j)}}{\\vert \\tilde{k}\\vert^2}\n= \\sum_k \\theta_k^2\\, \\frac{-k^{(i)}\\,k^{(j)}}{\\vert k\\vert^2} =0.\n\\end{equation}\nInstead, when $i=j$, using a change of variables $k\\mapsto \\tilde{k}$ that swaps the $i$-th component with the $l$-th one, we obtain\n\\begin{equation}\\nonumber\n\\left( \\sum_k \\theta_k^2 \\frac{k}{\\vert k\\vert}\\otimes \\frac{k}{\\vert k\\vert}\\right)_{ii}\n= \\sum_k \\theta_k^2\\, \\frac{{k^{(i)}}^2}{\\vert k\\vert^2}\n= \\sum_k \\theta_k^2\\, \\frac{{k^{(l)}}^2}{\\vert k\\vert^2}\n= \\left( \\sum_k \\theta_k^2 \\frac{k}{\\vert k\\vert}\\otimes \\frac{k}{\\vert k\\vert}\\right)_{ll}\n\\end{equation}\nand therefore, summing over $i$,\n\\begin{equation}\\nonumber\n\\left( \\sum_k \\theta_k^2 \\frac{k}{\\vert k\\vert}\\otimes \\frac{k}{\\vert k\\vert}\\right)_{ii}\n= \\frac{1}{d} \\sum_k \\theta_k^2 \\frac{\\vert k\\vert^2}{\\vert k\\vert^2}\n= \\frac{1}{d} \\sum_k \\theta_k^2.\n\\end{equation}\nIn conclusion, we have obtained\n\\begin{equation}\\label{sec 2.3 - definition of the constant c}\n\\sum_k \\theta_k^2\\, P_k = \\frac{d-1}{d} \\Bigg(\\sum_k \\theta_k^2\\Bigg) I=: c\\,I,\n\\end{equation}\nso that equation \\eqref{sec 2.3 - equation in Stratonovich form} has corresponding It\\^o formulation\n\\begin{equation}\\label{sec 2.3 - equation in Ito form}\n\\diff u = b\\cdot\\nabla u\\diff t + c\\,\\Delta u\\diff t + \\sum_{j,k} \\theta_k\\, e_k\\, a_k^{(j)}\\cdot \\nabla u\\diff W^{(j)}_k.\n\\end{equation}\nWe have actually only shown that formally \\eqref{sec 2.3 - equation in Stratonovich form} implies \\eqref{sec 2.3 - equation in Ito form}, but the same calculations done backward show that the two formulations are equivalent, whenever $u$ is a smooth solution. Observe that condition \\eqref{sec 2.2 - condition on the coefficients} on the coefficients $\\theta_k$ is necessary in order to give a meaning to equation \\eqref{sec 2.3 - equation in Stratonovich form}: otherwise, passing to the It\\^o formulation we would find a term of the form \\textquotedblleft $+\\infty\\,\\Delta u$\\textquotedblright\\  which is ill-defined even in the case $u$ had very good regularity.\nLet us stress that, even if writing the It\\^o formulation we find a diffusion term, this is actually a \\textquotedblleft fake Laplacian\\textquotedblright : the nature of the equation is still hyperbolic and it can be solved by characteristics; moreover, in the case $\\text{div}\\, b=0$, it can be checked that the energy $\\vert u\\vert_{L^2}$ is (formally) invariant, while in a real diffusion it would be dissipated.\nWe have done the computations leading to \\eqref{sec 2.3 - definition of the constant c} explicitly but we could have also derived it by the following reasoning: the matrix\n\\begin{equation}\\nonumber\nA= \\sum_k \\theta_k^2\\, P_k\n\\end{equation}\nis a symmetric and semipositive definite; by the isotropy condition it follows that $O^T A O = A$ for all $O\\in E_{\\mathbb{Z}}(d)$ and therefore necessarily $A=c I$ for some constant $c$. Indeed, if $v$ is an eigenvector for $A$, by the isotropy condition so is $Ov$, with respect to the same eigenvalue, for all $O\\in E_{\\mathbb{Z}}(d)$; this immediately implies that the associated eigenspace is the whole $\\mathbb{R}^d$. But then taking the trace on both sides we find\n\\begin{equation}\\nonumber\n(d-1)\\sum_k\\theta_k^2 = dc,\n\\end{equation}\nwhich gives \\eqref{sec 2.3 - definition of the constant c}. This shows that the presence of $\\Delta$ is strictly related to isotropy of the noise.\n\nSince we are interested in studying weak (in the analytical sense) solutions of equation \\eqref{sec 2.3 - equation in Ito form}, we need to rewrite it in a suitable way by testing against test functions in $C^\\infty(\\mathbb{T}^d)$. Recalling that, for any $k$ and $j$, $x\\mapsto a_k^{(j)}\\,e_k(x)$ is divergence free by construction, the weak formulation then corresponds to:\n\\begin{equation}\\label{sec 2.3 - equation in Ito form, weak formulation in differential form}\\begin{split}\n\\diff\\langle u,\\varphi\\rangle = & -\\langle u, \\text{div}(b\\varphi)\\rangle\\diff t + c\\,\\langle u,\\Delta \\varphi\\rangle\\diff t\\\\ & - \\sum_{j,k} \\theta_k\\, \\langle u, e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k\\quad \\forall\\,\\varphi\\in C(\\mathbb{T}^d),\n\\end{split}\\end{equation}\nwhere as usual the above equation must be interpreted in the integral sense. In order for the term $\\langle u, \\text{div}(b\\varphi)\\rangle = \\langle u, \\text{div}b\\,\\varphi + b\\nabla\\varphi)\\rangle$ to be well defined, we need at least to require the following assumption on $b$:\n\\begin{equation}\\tag{A1}\\label{sec 2.3 - assumption 1 on b}\nb\\in L^2(0,T;L^2),\\quad \\text{div}\\,b\\in L^2(0,T;L^2).\n\\end{equation}\nIt is natural in the definition of weak solution to require weak continuity in time of the solution. We denote by $C([0,T];L^2_w)$ the space of functions $f:[0,T]\\to L^2$ which are continuous w.r.t. the weak topology of $L^2$, namely $f(s)\\rightharpoonup f(t)$ as $s\\to t$. For more details on the weak topology, we refer to \\cite{Bre}. We are now ready to give with the following definition.\n\\begin{definition}\\label{definition sec 2.3 - weak solution}\nLet $(\\Omega,\\mathcal{F},\\mathcal{F}_t,\\mathbb{P})$ be a filtered probability space, with normal filtration $\\{\\mathcal{F}_t\\}$, on which a collection $\\{B_k^{(j)}, k\\in\\mathbb{Z}^d_0, 1\\leq j\\leq d-1\\}$ of independent, standard $\\mathcal{F}_t$-Brownian motions is defined. Let $W$ be defined as in \\eqref{sec 2.2 - definition of the noise W(t,x)}, for given coefficients $\\{\\theta_k\\}_k$ satisfying \\eqref{sec 2.2 - condition on the coefficients}, \\eqref{sec 2.2 - isotropy condition}. We say that an $\\mathcal{F}_t$-progressively measurable, $L^2$-valued process $u$, with paths in $C([0,T];L^2_w)$, satisfying\n\\begin{equation}\\label{sec 2.3 - condition of bounded energy in the definition of weak solution}\n\\int_0^T \\mathbb{E}\\big[\\vert u(t)\\vert^2_{L^2}\\big]\\diff t <\\infty,\n\\end{equation}\nis a \\textbf{weak solution} (in the analytical sense) on the interval $[0,T]$ of the equation\n\\begin{equation}\\label{sec 2.3 - equation in Stratonovich compact form, useful repetition}\n\\diff u = b\\cdot\\nabla u\\diff t + \\circ \\diff W\\cdot\\nabla u\n\\end{equation}\nif, for every $\\varphi\\in C^\\infty(\\mathbb{T}^d)$, $\\mathbb{P}$-a.s. the following identity holds for all $t\\in [0,T]$:\n\\begin{equation}\\label{sec 2.3 - equation in Ito form, weak formulation in integral form}\\begin{split}\n\\langle u(t),\\varphi\\rangle - \\langle u(0),\\varphi\\rangle\n=& -\\int_0^t \\langle u(s), \\text{div}(b\\varphi)\\rangle\\diff s + c\\int_0^t \\langle u(s),\\Delta \\varphi\\rangle\\diff s\\\\\n& - \\sum_{j,k} \\theta_k\\, \\int_0^t \\langle u(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k(s).\n\\end{split}\\end{equation}\n\\end{definition}\n\nIn order to show that it is a good definition, we need to prove that equation \\eqref{sec 2.3 - equation in Ito form, weak formulation in integral form} is meaningful. By assumption \\eqref{sec 2.3 - assumption 1 on b} and condition \\eqref{sec 2.3 - condition of bounded energy in the definition of weak solution}, it holds\n\\begin{equation}\\nonumber\\begin{split}\n\\mathbb{E}&\\bigg[\\bigg\\vert \\int_0^t \\langle u(s), \\text{div}(b\\varphi)\\rangle\\diff s \\bigg\\vert\\bigg]\\\\\n&\\leq \\Vert \\varphi\\Vert_{W^{1,\\infty}} \n\\big(\\Vert b\\Vert_{L^2(0,T;L^2)} + \\Vert \\text{div}b\\Vert_{L^2(0,T;L^2)}\\big)\n\\, \\sqrt{\\int_0^T \\mathbb{E}\\big[\\vert u(t)\\vert^2_{L^2}\\big]\\diff t} <\\infty.\n\\end{split}\\end{equation}\nSince $u$ is $\\mathcal{F}_t$-progressively measurable, the real-valued process $t\\mapsto \\langle u(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle$ is also $\\mathcal{F}_t$-progressively measurable and can be integrated with respect to $W_k^{(j)}$, for any $k$ and $j$. Therefore we only need to check that the infinite series is convergent, in a suitable sense. By It\\^o isometry we have\n\\begin{equation}\\nonumber\\begin{split}\n\\mathbb{E}\\Bigg[\\Big\\vert \\sum_{j,k} \\theta_k & \\int_0^t \\langle u(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k(s) \\Big\\vert^2 \\Bigg]\\\\\n& = 2 \\sum_{j,k}\\theta_k^2\\, \\mathbb{E}\\Bigg[\\int_0^t\\vert \\langle u(s),e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\vert ^2\\diff s\\Bigg]\\\\\n& \\leq 2\\, \\sup_k \\theta_k^2\\, \\sum_{j,k} \\mathbb{E}\\Bigg[\\int_0^T \\vert\\langle u(s)\\,\\nabla\\varphi, a_k^{(j)}\\,e_k\\rangle\\vert^2\\diff s \\Bigg]\\\\\n& \\leq 2\\, \\sup_k \\theta_k^2\\, \\int_0^T\\mathbb{E}\\big[\\vert u(s)\\,\\nabla\\varphi\\vert_{L^2}^2\\big]\\diff s\\\\\n& \\leq 2\\, \\sup_k \\theta_k^2\\, \\Vert\\nabla\\varphi\\Vert_{\\infty}^2  \\int_0^T\\mathbb{E}\\big[\\vert u(s)\\vert_{L^2}^2\\big]\\diff s\n\\end{split}\\end{equation}\nand the last term is finite since $u$ satisfies \\eqref{sec 2.3 - condition of bounded energy in the definition of weak solution} and $\\theta_k$ satisfy \\eqref{sec 2.2 - condition on the coefficients}. In the above calculations we have exploited the fact that $\\{a_k^{(j)}\\, e_k, k\\in\\mathbb{Z}^d_0, 1\\leq j\\leq d-1\\}$ is an (incomplete) orthonormal system in $L^2(\\mathbb{T}^d;\\mathbb{C}^d)$.\n\nLet us now briefly discuss the energy balance for equation \\eqref{sec 2.3 - equation in Stratonovich compact form, useful repetition}. If $u$ were a classical smooth solution of the deterministic linear transport equation\n\\begin{equation}\\nonumber\n\\partial_t u = b\\cdot \\nabla u + v\\cdot\\nabla u,\n\\end{equation}\nwith $b$ as before and $v=v(t,x)$ being a divergence free vector field (both with periodic boundary condition), then we would have\n\\begin{equation}\\nonumber\\begin{split}\n\\frac{\\diff}{\\diff t}\\int_{\\mathbb{T}^d} u^2(t,x)\\diff x\n& = \\int_{\\mathbb{T}^d} 2\\,u(t,x)(b(t,x)+v(t,x))\\cdot \\nabla u(t,x)\\diff x\\\\\n& = \\int_{\\mathbb{T}^d} (b(t,x)+v(t,x))\\cdot \\nabla (u^2)(t,x)\\diff x\\\\\n& = -\\int_{\\mathbb{T}^d} (\\text{div} b)(t,x)\\,u^2(t,x)\\diff x\\\\\n& \\leq \\Vert \\text{div} b(t)\\Vert_\\infty \\int_{\\mathbb{T}^d} u^2(t,x)\\diff x,\n\\end{split}\\end{equation}\nand therefore by Gronwall's lemma we would obtain\n\\begin{equation}\\label{sec 2.3 - energy inequality, preliminary version}\n\\vert u(t)\\vert_{L^2}^2 \\leq \\vert u(0)\\vert_{L^2}^2\\,\\exp\\Big\\{\\int_0^t \\Vert \\text{div} b(s)\\Vert_\\infty\\diff s \\Big\\}.\n\\end{equation}\nIt is therefore natural to impose the following condition on $b$:\n\\begin{equation}\\tag{A2}\\label{sec 2.3 - assumption 2 on b}\n\\text{div}\\,b \\in L^1(0,T;L^\\infty).\n\\end{equation}\nUsing the properties of Stratonovich integral (or if one prefers using a Wong-Zakai approximation technique), it can be shown that, whenever $u$ is a smooth solution of (STLE), the above calculation still holds, since by construction $W(t,\\cdot)$ is divergence free. However, since equation \\eqref{sec 2.3 - equation in Stratonovich compact form, useful repetition} is hyperbolic in nature, we don't expect solutions with initial data only in $L^2$ to become more regular and in this case the above reasoning doesn't hold. By approximation with smooth solutions, we can still at least expect the final inequality \\eqref{sec 2.3 - energy inequality, preliminary version} to hold also for weak solutions.\n\n\nThe above observations lead to the following notion of energy solutions for the Cauchy problem given by \\eqref{sec 2.3 - equation in Stratonovich compact form, useful repetition} and an initial condition $u_0$:\n\\begin{definition}\\label{definition sec 2.3 - energy solution} Given a deterministic initial condition $u_0\\in L^2$, we say that $u$ is an \\textbf{energy solution} of the Cauchy problem\n\\begin{equation}\\nonumber\n\\begin{cases} \\diff u = b\\cdot\\nabla u\\diff t + \\circ \\diff W\\cdot\\nabla u\\\\ u(0)=u_0\n\\end{cases}\n\\end{equation}\nif $u$ is a weak solution of \\eqref{sec 2.3 - equation in Stratonovich compact form, useful repetition}, equation \\eqref{sec 2.3 - equation in Ito form, weak formulation in integral form} is satisfied with $u(0)=u_0$ and the following \\textbf{energy inequality} holds:\n\\begin{equation}\\label{sec 2.3 - energy inequality}\n\\sup_{t\\in [0,T]} \\Big\\{e^{-1\/2 \\int_0^t \\Vert \\text{div} b(s,\\cdot)\\Vert_\\infty\\diff s}\\,\\vert u(t)\\vert_{L^2} \\Big\\} \\leq \\vert u_0\\vert_{L^2} \\qquad \\mathbb{P}\\text{-a.s.}\n\\end{equation}\n\\end{definition}\n\nLet us finally define what we mean by convergence in probability in abstract topological spaces. If $X_n$ is a sequence of random variables defined on the same probability space, with values in $(E,\\tau,\\mathcal{B}(\\tau))$, where $\\tau$ is a topology and $\\mathcal{B}(\\tau)$ is the associated Borel-$\\sigma$ algebra, we say that $X_n\\to X$ in probability if any subsequence of $\\{X_n\\}_n$ contains a subsequence which converges to $X$ $\\mathbb{P}$-a.s.. We need this definition because we will work with convergence in probability in a non metrizable topology.\n\n\\section{Rigorous statement and proof of the main result}\\label{section 3 - proof of the main result}\nIn this section we provide a rigorous statement of the main result and its proof. Throughout the section we consider a fixed, a priori given filtered probability space $(\\Omega,\\mathcal{F},\\mathcal{F}_t,\\mathcal{P})$ together with a collection $\\{B_k^{(j)}, k\\in\\mathbb{Z}^d_0, 1\\leq j\\leq d-1\\}$ of independent, standard $\\mathcal{F}_t$-Brownian motions. However we consider different choices of the parameters $\\{\\theta_k\\}_k$, so that we can obtain different space-time dependent noises $W(t,x)$ constructed from $\\{B_k^{(j)}\\}_{k,j}$ by \\eqref{sec 2.2 - definition of the noise W(t,x)}. All these noises are still defined on the same probability space with respect to the same filtration; the drift $b$ is fixed. Whenever referring to energy solutions of \\eqref{sec 2.3 - equation in Stratonovich compact form, useful repetition} we will therefore consider strong in the probabilistic sense solutions (i.e. progressively measurable w.r.t. $\\mathcal{F}_t$) all defined on the same probability space. The main result can then be stated as follows.\n\\begin{theorem}\\label{theorem sec 3 - main result}\nLet $\\{\\theta_k^N, k\\in\\mathbb{Z}^d_0, N\\in\\mathbb{N} \\}$ be a collection of real coefficients such that:\n\\begin{itemize}\n\\item[i)] For each $N$, $\\{\\theta_k^N\\}_k$ satisfies \\eqref{sec 2.2 - condition on the coefficients} and \\eqref{sec 2.2 - isotropy condition}.\n\\item[ii)] It holds\n\\begin{equation}\\label{sec 3 - condition on the coefficients in the statement of the main theorem}\\tag{H3}\n\\lim_{N\\to\\infty} \\frac{\\sup_k (\\theta_k^N)^2}{\\sum_k (\\theta_k^N)^2} = 0.\n\\end{equation}\n\\end{itemize}\nAssume that $b$ satisfies \\eqref{sec 2.3 - assumption 1 on b}, \\eqref{sec 2.3 - assumption 2 on b} and the following:\n\\begin{itemize}\n\\item[(A3)] $b$  is such that, for any $\\nu>0$, uniqueness holds in the class of weak $L^\\infty(0,T;L^2)$ solutions of the parabolic Cauchy problem\n\\begin{equation}\\label{sec 3 - parabolic cauchy limit problem in the statement of the main theorem}\n\\begin{cases} \\partial_t u = \\nu\\Delta u + b\\cdot\\nabla u\\\\ u(0)=u_0 \\end{cases}.\n\\end{equation}\n\\end{itemize}\nLet $W^N$ denote the divergence free noises constructed from the coefficients $\\{\\theta^N_k\\}_k$ as in \\eqref{sec 2.2 - definition of the noise W(t,x)}. Then for any $\\nu>0$ there exists a sequence of constants $\\varepsilon^N$, which depend on the coefficients $\\{\\theta_k^N\\}$, such that, for any $u_0\\in L^2$, any sequence of energy solutions $u^N$ of the Cauchy problems\n\\begin{equation}\\label{sec 3 - cauchy probelms for uN in the statement of the main theorem}\n\\begin{cases} \\diff u^N= b\\cdot\\nabla u^N\\diff t + \\sqrt{\\varepsilon^N} \\circ \\diff W^N\\cdot\\nabla u^N\\\\\nu(0)=u_0\n\\end{cases}\n\\end{equation}\nconverge in probability, in $L^\\infty(0,T;L^2)$ endowed with the weak-$\\star$ topology, to the unique weak solution of the deterministic Cauchy problem \\eqref{sec 3 - parabolic cauchy limit problem in the statement of the main theorem}.\nIn particular, the constants $\\varepsilon^N$ can be taken as\n\\begin{equation}\\label{sec 3 - choice of the varepsilonN in the statement of the main theorem}\n\\varepsilon^N = \\nu\\,\\frac{d}{d-1}\\Big(\\sum_k(\\theta_k^N)^2 \\Big)^{-1}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe basic idea of the proof is the following: when we rewrite the transport SPDE in It\\^o form, we can see that in general the It\\^o-Stratonovich corrector term is well defined under more restrictive conditions than the It\\^o integral. We can exploit this to our advantage by introducing a multiplicative renormalization $\\sqrt{\\varepsilon^N}$ under which the corrector term is uniformly bounded with respect to $N$, but then under condition \\eqref{sec 3 - condition on the coefficients in the statement of the main theorem} the It\\^o integrals become infinitesimal. To this aim, it is fundamental to have a uniform control on the energy of the solutions $u^N$ and that's why we work with energy solutions. We now formalize this reasoning properly.\n\n\nBy definition of energy solutions, we know that for each $N$ inequality \\eqref{sec 2.3 - energy inequality} holds. In particular it follows that there exists a constant $K$, which only depends on $b$, such that\n\\begin{equation}\\label{sec 3 - uniform energy estimate inside proof of the main theorem}\n\\sup_N \\Vert u^N(\\omega)\\Vert_{L^\\infty(0,T,L^2)}\n= \\sup_N \\sup_{t\\in [0,T]} \\vert u^N(\\omega,t)\\vert_{L^2} \\leq K\\vert u_0\\vert_{L^2} \\quad \\text{for }\\mathbb{P}\\text{-a.e. }\\omega\n\\end{equation}\nand\n\\begin{equation}\\nonumber\n\\sup_N \\int_0^T\\mathbb{E}[\\vert u^N(t)\\vert_{L^2}^2]\\diff t\\leq K\\vert u_0\\vert_{L^2}.\n\\end{equation}\nRewriting the Cauchy problem in It\\^o form, by the definition of energy solution we obtain that, for any $N$ and for any $\\varphi\\in C^{\\infty}(\\mathbb{T}^d)$, it holds\n\\begin{equation}\\nonumber\\begin{split}\n\\langle u^N(t),\\varphi\\rangle - \\langle u_0,\\varphi\\rangle\n=& -\\int_0^t \\langle u^N(s), \\text{div}(b\\varphi)\\rangle\\diff s + \\varepsilon^N c^N \\int_0^t \\langle u^N(s),\\Delta \\varphi\\rangle\\diff s\\\\\n& - \\sqrt{\\varepsilon^N}\\,\\sum_{j,k} \\theta^N_k\\, \\int_0^t \\langle u^N(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k(s),\n\\end{split}\\end{equation}\nwhere $c^N$ is defined as in \\eqref{sec 2.3 - definition of the constant c}. With the choice \\eqref{sec 3 - choice of the varepsilonN in the statement of the main theorem}, the equation becomes\n\\begin{equation}\\label{sec 3 - identity for energy solutions uN inside proof main theorem}\\begin{split}\n\\langle u^N(t),\\varphi\\rangle - \\langle u_0,\\varphi\\rangle\n=& -\\int_0^t \\langle u^N(s), \\text{div}(b\\varphi)\\rangle\\diff s + \\nu \\int_0^t \\langle u^N(s),\\Delta \\varphi\\rangle \\diff s\\\\\n& - \\sqrt{\\varepsilon^N}\\,\\sum_{j,k} \\theta^N_k\\, \\int_0^t \\langle u^N(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k(s).\n\\end{split}\\end{equation}\nUsing estimates similar the ones of Section \\ref{subsection 2.3 - SPDE in Ito form and definition of energy solutions}, it holds\n\\begin{equation}\\nonumber\\begin{split}\n\\varepsilon^N\\, & \\mathbb{E}\\Bigg[ \\Big(\\sum_{j,k}\\theta^N_k \\int_0^T \\langle u^N(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k(s)\\Big)^2 \\Bigg]\\\\\n& \\leq 2\\varepsilon^N\\, (\\sup_k\\theta_k^N)^2\\,\\Vert \\nabla\\varphi\\Vert_{\\infty}^2 \\int_0^T\\mathbb{E}[\\vert u^N(t)\\vert_{L^2}^2] \\diff t\\\\\n& \\leq \\widetilde{K} \\Vert\\nabla\\varphi\\Vert_{\\infty}^2 \\frac{(\\sup_k\\theta_k^N)^2}{\\sum_k (\\theta^N_k)^2}\\to 0 \\text{ as } N\\to\\infty\n\\end{split}\\end{equation}\nby assumption \\eqref{sec 3 - condition on the coefficients in the statement of the main theorem}. Using the properties of It\\^o integral and Doob's inequality, we deduce that for any fixed $\\varphi$\n\\begin{equation}\\nonumber\n\\sup_{t\\in [0,T]} \\bigg\\vert\\sum_{j,k}\\theta^N_k\\int_0^t \\langle u^N(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k(s) \\bigg\\vert\\to 0 \\text{ in probability w.r.t. }\\mathbb{P}.\n\\end{equation}\nLet $\\{\\varphi_n\\}_n$ be a countable dense subset of $C^\\infty(\\mathbb{T}^d)$; by a diagonal extraction argument, we can find a subsequence (which will still be denoted by $N$ for simplicity) and a set $\\Gamma\\subset \\Omega$ with $\\mathbb{P}(\\Gamma)=1$ such that: the above process converges uniformly to 0 for every $\\varphi_n$ and for every $\\omega\\in \\Gamma$; inequality \\eqref{sec 3 - uniform energy estimate inside proof of the main theorem} holds for every $\\omega\\in \\Gamma$. Now let us consider a fixed $\\omega\\in \\Gamma$ and the realizations $\\{u^N(\\omega)\\}_N$. Since they are a bounded sequence $L^\\infty(0,T;L^2)$, we can extract a subsequence (which depends on $\\omega$) which is weak-$\\star$ convergent to some $u\\in L^\\infty(0,T;L^2)$. Taking the limits on both sides of \\eqref{sec 3 - identity for energy solutions uN inside proof main theorem}, since $\\omega\\in\\Gamma$, we find that for every $n$\n\\begin{equation}\\nonumber\n\\langle u(t),\\varphi_n\\rangle - \\langle u_0,\\varphi_n\\rangle = -\\int_0^t \\langle u(s), \\text{div}(b \\varphi_n)\\rangle\\diff s + \\nu \\int_0^t \\langle u(s),\\Delta \\varphi_n\\rangle\\diff s.\n\\end{equation}\nBy density we can extend the above equation for all $\\varphi\\in C^\\infty(\\mathbb{T}^d)$, so that $u$ is a weak solution of the Cauchy problem\n\\begin{equation}\\nonumber\\begin{cases}\n\\partial_t u = \\nu\\Delta u + b\\cdot\\nabla u\\\\ u(0)=u_0\n\\end{cases}.\\end{equation}\nBy assumption (A3), the candidate limit is therefore unique; since the argument applies for any subsequence of $\\{u^N(\\omega)\\}_N$, we conclude that the entire sequence is converging weakly-$\\star$ to the unique solution of the above problem, without the need of selecting an $\\omega$-dependent subsequence. Moreover the reasoning holds for any $\\omega\\in \\Gamma$. Summarising, we have shown the existence of a subsequence of $\\{u^N\\}_N$ such that, for any $\\omega\\in\\Gamma$, this subsequence converges in the weak-$\\star$ topology of $L^\\infty(0,T;L^2)$ to the unique solution of the above deterministic parabolic equation. Since the reasoning holds also for any subsequence of $\\{u^N\\}$, we conclude that convergence in probability in $L^\\infty(0,T;L^2)$ endowed with weak-$\\star$ topology holds.\n\\end{proof}\n\n\n\\begin{remark} Let us make some comments on the above result.\n\n\\begin{itemize}\n\n\\item[i)] For any $u^N$ solving \\eqref{sec 3 - cauchy probelms for uN in the statement of the main theorem}, its expectation $\\tilde{u}(t)=\\mathbb{E}[u^N(t)]$ solves \\eqref{sec 3 - parabolic cauchy limit problem in the statement of the main theorem}. Therefore the result can be expressed as the convergence in probability of $u^N$ to their mean value, which is a weak law of large numbers.\n\n\\item[ii)] Observe that whenever we consider coefficients $\\{\\theta^N_k\\}$ satisfying \\eqref{sec 3 - condition on the coefficients in the statement of the main theorem} such that $\\sup_k \\vert \\theta^N_k\\vert =1$ for all $N$ (some examples will be given shortly), the sequence of noises $W^N(t,x)$ has bounded norm in some distribution spaces, like $H^\\alpha$ for $\\alpha<-d\/2$. Recalling that\n\\begin{equation}\\nonumber\n\\mathbb{E}\\big[\\vert W^N(1,x)\\vert_{L^2}^2\\big]= 2(d-1)\\sum_k(\\theta_k^N)^2,\n\\end{equation}\nwe find that the constants $\\varepsilon^N$ and $\\nu$ must satisfy the relation\n\\begin{equation}\\label{sec 3 - nu measures the irregularity\/magnitude relation}\n\\nu = C(d)\\lim_{N\\to\\infty}\\varepsilon^N\\,\\mathbb{E}\\big[\\vert W^N(1,x)\\vert_{L^2}^2\\big],\n\\end{equation}\nwhere $C(d)=2\/d$ is a dimensional constant, independent of the probability space, the coefficients $\\{\\theta_k^N\\}_{k,N}$ and the noises $W^N$ considered. Therefore the parameter $\\nu$ appearing in the limit equation in front of the dissipation term $\\Delta$ is measuring product of the spatial irregularity of the noise (in terms of its $L^2$ norm) and its magnitude.\n\n\\item[iii)] We illustrate some typical examples of coefficients $\\theta_k^N$, widely used in other contexts, which satisfy \\eqref{sec 2.2 - isotropy condition} and \\eqref{sec 3 - condition on the coefficients in the statement of the main theorem}. Let $F:\\mathbb{R}_{\\geq 0}\\to\\mathbb{R}_{\\geq 0}$ be a smooth, decreasing function of compact support with $F(0)=1$ and consider a sequence of positive real numbers $\\alpha_N\\to 0$; then we can take $\\theta_k^N:= F(\\alpha_N\\vert k\\vert)$. Other choices, for $\\alpha_N$ infinitesimal, are\n\\begin{equation}\\nonumber\n\\theta_k^N = (1+\\alpha_N\\vert k\\vert^2)^\\beta \\text{ for some }\\beta<-d\/2,\\quad \\theta_k^N = (1+\\vert k\\vert^2)^{-d\/2-\\alpha_N}.\n\\end{equation}\nWe can also take $\\theta^N_k=\\mathbbm{1}_{B(0,1)}(\\alpha_N\\vert k\\vert)$, where $\\mathbbm{1}_A$ denotes the characteristic function of $A$. These examples can also be combined together to produce new ones. In terms of Fourier multipliers, some of the above examples are standard rescaled volume cutoffs in Fourier space, others correspond to operators like $(1-\\alpha_N\\Delta)^{-\\beta}$ or $(1-\\Delta)^{-d\/2-\\alpha_N}$.\n\n\\item[iv)] The theorem resembles a renormalization statement: different choices of the coefficients $\\theta_k^N$, which can be spatial regularizations of a space-time white noise, require different multiplicative constants $\\varepsilon^N$, but the final limit solves an equation which is independent of $\\theta_k^N$, up to the choice of a 1-dimensional parameter $\\nu$. We have however already pointed out in the introduction the presence of some degeneracy in our result. Indeed, different choices of the parameters $\\theta_k^N$ can lead to very different limits for $W^N$ in terms of regularity: for instance, taking $(1+\\alpha_N\\vert k\\vert)^{-\\beta}$, the sequence will converge to white noise, while taking $(1+\\vert k\\vert^2)^{-1}\\mathbbm{1}_{B(0,1)}(\\alpha_N\\vert k\\vert)$ it will converge to a Gaussian free field (properly speaking, since we want divergence-free distributions, it will converge to the image under $\\Pi$ of the aforementioned objects). However, in both cases the multiplicative constants $\\varepsilon_N$ will still give convergence to the same limit. It is therefore unclear if the choice of such a renormalization is too strong, in the sense that it is ignoring too much information on the dynamics, and there is some more refined way to recover it, like an \"higher order expansion\" which not only measures the $L^2$-regularity of $W^N$ but also other norms.\n\n\\item[v)] We have required $b$ to satisfy \\eqref{sec 2.3 - assumption 2 on b} in order to deal with energy solutions, but in principle the structure of the proof holds for any sequence $u^N$ of weak solutions satisfying a uniform bound of the form\n\\begin{equation}\\nonumber\n\\sup_N \\int_0^T \\mathbb{E}[\\vert u^N(s)\\vert_{L^2}^2]\\diff s \\leq K\n\\end{equation}\nup to paying the price of restricting ourselves to a weaker notion of convergence, namely weak convergence in $L^2(\\diff \\mathbb{P}\\otimes \\diff t; L^2)$. It's possible that more refined a priori estimates on the solutions $u^N$ provide this kind of bound under milder conditions on $b$ than \\eqref{sec 2.3 - assumption 2 on b}.\n\\end{itemize}\n\\end{remark}\n\nWe now provide explicit sufficient conditions on $b$ under which assumption (A3) is satisfied. We give the statement in full generality, even when assumption \\eqref{sec 2.3 - assumption 2 on b} does not hold.\n\\begin{lemma}\\label{lemma sec 3 - sufficient condition for uniqueness of parabolic problem} Consider $b$ such that \\eqref{sec 2.3 - assumption 1 on b} holds, as well as the following condition:\n\\begin{equation}\\tag{A4}\\label{sec 3 - KR type condition on b}\n\\begin{cases}\nb\\in L^{p_1}(0,T;L^{q_1}(\\mathbb{T}^d))\\quad\n&\\text{with } q_1\\in (d,+\\infty],\\ p_1\\in \\big( \\frac{2 q_1}{q_1 - d},+\\infty\\big]\\\\\n\\textnormal{div}\\, b\\in L^{p_2}(0,T;L^{q_2}(\\mathbb{T}^d)) &\\text{with } q_2\\in \\big(\\frac{d}{2},+\\infty\\big],\\ p_2\\in \\big( \\frac{2 q_2}{2 q_2 - d},+\\infty\\big]\n\\end{cases}.\\end{equation}\nThen (A3) holds, i.e. we have uniqueness in the class of weak $L^\\infty(0,T;L^2(\\mathbb{T}^d))$ solutions of the Cauchy problem\n\\begin{equation}\\label{sec 3 - parabolic Cauchy problem, useful repetition}\\begin{cases}\n\\partial_t u = \\nu\\Delta u + b\\cdot \\nabla u\\\\\nu(0)=u_0 \\in L^2(\\mathbb{T}^d)\n\\end{cases}.\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWithout loss of generality we can assume $\\nu = 1$. By linearity, it suffices to show uniqueness for $u_0=0$. We first show that $u$ is also a mild solution of \\eqref{sec 3 - parabolic Cauchy problem, useful repetition}. Indeed if $u$ is a weak solution of \\eqref{sec 3 - parabolic Cauchy problem, useful repetition}, then for any interval $[s,t]\\subset [0,T]$ and for any $\\varphi\\in C^\\infty([s,t]\\times\\mathbb{T}^d)$ it holds\n\\begin{equation}\\nonumber\\begin{split}\n\\langle u(t),\\varphi(t)\\rangle - \\langle u(s),\\varphi (s)\\rangle & = \\int_s^t \\langle u(r), (\\partial_t+\\Delta)\\varphi(r)\\rangle\\diff r\\\\& - \\int_s^t \\langle u(r), \\text{div}(b(r)\\varphi(r))\\rangle\\diff r.\n\\end{split}\\end{equation}\nThis can be accomplished by taking a partition $s=t_0<t_1<\\ldots<t_n=t$ and applying the standard weak formulation on every interval $[t_i,t_{i+1}]$ with $\\varphi(t_i)$, then summing over $i$ and then letting the mesh of the partition tend to 0. Recall that the heat kernel on the torus is given by\n\\begin{equation}\\nonumber\nP_t = \\sum_k e^{-t\\vert k\\vert^2}\\, e_k.\n\\end{equation}\nThen testing $u$ on any interval $[\\delta, t]$ with $\\delta>0$ against the convolution with the heat kernel $P_{t-s}$ and letting $\\delta\\to 0$, using $u_0=0$ we obtain the mild formulation\n\\begin{equation}\\nonumber\nu(t) = \\int_0^t P_{t-s} (\\nabla (bu)-\\text{div} b\\, u)\\diff s \\quad \\forall\\, t\\in [0,T].\n\\end{equation}\nIn order to conclude it suffices to show that the map\n\\begin{equation}\\nonumber\nu\\mapsto \\int_0^\\cdot P_{\\cdot-s} (\\nabla (bu)-\\text{div} b\\, u)\\diff s\n\\end{equation}\nis a contraction of $L^\\infty([0,T^\\ast],L^2(\\mathbb{T}^d))$ into itself, for $T^\\ast>0$ sufficiently small. If that's the case, then necessarily $u\\equiv 0$ on $[0,T^\\ast]$ and then we can iterate the argument to cover the whole $[0,T]$. We treat separately the two terms\n\\begin{equation}\\nonumber\nu(t)=\\int_0^t P_{t-s}(\\nabla(bu))\\diff s - \\int_0^t P_{t-s}(\\text{div}b\\, u)\\diff s = (I)(t) + (II)(t).\n\\end{equation}\nFor the first term, using regularity of the heat kernel and the fractional Sobolev embeddings, we have\n\\begin{equation}\\nonumber\\begin{split}\n\\vert I(t)\\vert_{L^2}\n& \\leq \\int_0^t \\vert P_{t-s}(\\nabla(bu))\\vert_{L^2}\\diff s\\\\\n& \\leq C \\int_0^t \\Vert P_{t-s}(\\nabla(bu))\\Vert_{W^{\\alpha,r}}\\diff s\\\\\n& \\leq C \\int_0^t (t-s)^{-(1+\\alpha)\/2} \\Vert bu\\Vert_{L^r}\\diff s\\\\\n& \\leq C \\Vert u\\Vert_{L^\\infty(0,t;L^2)} \\int_0^t (t-s)^{-(1+\\alpha)\/2} \\Vert b\\Vert_{L^{q_1}}\\diff s\n\\end{split}\\end{equation}\nwhere $\\frac{1}{r}=\\frac{1}{q_1}+\\frac{1}{2}$ and $W^{\\alpha,r}\\hookrightarrow L^{\\tilde{r}}$, $\\frac{1}{\\tilde{r}} = \\frac{1}{r}-\\frac{\\alpha}{d}$, $\\tilde{r}\\geq 2$ for some $\\alpha<1$ thanks to \\eqref{sec 3 - KR type condition on b}. Young's convolution inequality then gives\n\\begin{equation}\\nonumber\n\\Vert I\\Vert_{L^\\infty(0,T^\\ast;L^2)}\\leq C_1 \\Vert u\\Vert_{L^\\infty(0,T^\\ast;L^2)}\\, \\Vert b\\Vert_{L^{p_1}(0,T;L^{q_1})}\\, \\bigg( \\int_0^{T^\\ast} s^{-p_1^\\ast(1+\\alpha)\/2}\\diff s\\bigg)^{1\/p_1^\\ast},\n\\end{equation}\nwhere $p_1^\\ast$ denotes the conjugate exponent of $p_1$. The last quantity is finite if we can take $\\alpha$ such that $p_1^\\ast(1+\\alpha)<2$, which is guaranteed by \\eqref{sec 3 - KR type condition on b}. In the case of $(II)(t)$ the calculations are similar, with only a slight difference in the initial part; they lead to\n\\begin{equation}\\nonumber\\begin{split}\n\\Vert & II\\Vert_{L^\\infty(0,T^\\ast;L^2)}\\\\ \n& \\leq C_2 \\Vert u\\Vert_{L^\\infty(0,T^\\ast;L^2)}\\, \\Vert \\text{div}b\\Vert_{L^{p_2}(0,T;L^{q_2})}\\, \\bigg( \\int_0^{T^\\ast} s^{-p_2^\\ast(1+\\alpha_2)\/2}\\diff s\\bigg)^{1\/p_2^\\ast}\n\\end{split}\\end{equation}\nfor a suitable $\\alpha_2$ such that the integral is finite. In particular, this shows that for some $T^\\ast$ small enough, the map is a contraction and this concludes the proof.\n\\end{proof}\n\\begin{remark}\nUp to slight modifications, it can be shown with the same type of proof that under \\eqref{sec 3 - KR type condition on b}, uniqueness holds also in the class of weak solutions $u\\in L^r(0,T;L^2(\\mathbb{T}^d))$, with the additional condition $p_1, p_2\\geq r^\\ast$. Observe that condition $p_1> 2q_1\/(q_1-d)$ may be rewritten as\n\\begin{equation}\\nonumber\n\\frac{2}{p_1}+\\frac{d}{q_1}<1,\n\\end{equation}\nwhich is known in literature as Krylov-R\\\"ockner condition, see \\cite{Kry},\\cite{BecFla}.\n\\end{remark}\n\nThe proof of Lemma \\ref{lemma sec 3 - sufficient condition for uniqueness of parabolic problem} is standard (it is a slight improvement of the one contained in \\cite[Lemma 3.2]{Mau}, which is restricted to the case of time independent $b$) but we had to provide it mainly for two reasons. The first one is that a major part of the results in the literature are set in $\\mathbb{R}^d$ and not in $\\mathbb{T}^d$; the second and most important one is that usually uniqueness for \\eqref{sec 3 - parabolic Cauchy problem, useful repetition} is proved among solutions in a more regular class, tipically $H^p_{2,q}:=L^p(0,T;W^{2,q})\\cap W^{1,p}(0,T;L^q)$, see \\cite{Kry2} and the appendix of \\cite{Kry}. If $u$ belongs in this class, then $\\nabla u\\in L^\\infty([0,T];L^\\infty)$ and so there is no need to impose conditions on $\\text{div}\\,b$. Here however, since our solution $u$ is obtained as the limit of solutions of transport equations, we cannot infer that it belongs to $H^p_{2,q}$, which is why we need to impose the stronger condition \\eqref{sec 3 - KR type condition on b}. Maybe further improvements can be done (for instance if both conditions \\eqref{sec 2.3 - assumption 2 on b} and\\eqref{sec 3 - KR type condition on b} are imposed, then \\eqref{sec 2.3 - assumption 1 on b} can be dropped) but we believe the result to be fairly optimal; indeed the Krylov-R\\\"ockner (KR) condition arises naturally as the subcritical regime of a scaling argument and reaching the critical case (usually referred to as Ladyzhenskaya-Prodi-Serrin condition, (LPS) for short)\n\\begin{equation}\\nonumber\n\\frac{2}{p_1}+\\frac{d}{q_1}=1\n\\end{equation}\nis in general very difficult and seems out of reach in a class of functions like $L^\\infty(0,T;L^2)$. For more details on the topic (both the scaling argument and the critical regime) we refer to \\cite{BecFla} and the references therein.\n\\section{Discussion of existence and uniqueness}\\label{section 4 - discussion of existence and uniqueness}\n\nIn order for the statement of Theorem \\ref{theorem sec 3 - main result} to be non vacuous, we discuss in this section existence and uniqueness of energy solutions, even if it is not the main aim of this paper. Existence is accomplished by a standard Galerkin scheme; regarding uniqueness, several references are given, as well as a proof in the special case $b=0$, but a full answer is missing. We stress however that the statement of the main result holds for \\textit{any} sequence of energy solutions, regardless of their uniqueness; indeed the strength of the result also relies on the fact that the limit satisfies an a priori much better posed equation than the approximating sequence.\\\\\nAs in the previous section, we consider an a priori given filtered probability space $(\\Omega,\\mathcal{F},\\mathcal{F}_t,\\mathbb{P})$ with an $\\mathcal{F}_t$-adapted noise $W$, namely we work in the framework of strong solutions in the probabilistic sense.\n\\subsection{Existence of energy solutions}\\label{subsection 4.1 - existence of energy solutions}\nIn this subsection, the existence of energy solutions for any initial data $u_0\\in L^2(\\mathbb{T}^d)$ is shown. The proof is standard and based on a Galerkin approximation scheme.\n\nFirst we need some preparations. Throughout the proof we will adopt the following notation: by $L^2(\\diff \\mathbb{P}\\otimes \\diff t; L^2)$ denotes the space of all $L^2$-valued, square integrable (in the Bochner sense) functions defined on $\\Omega\\times [0,T]$, endowed with the product $\\sigma$-algebra $\\mathcal{F}\\otimes \\mathcal{B}([0,T])$ and the product measure $\\diff\\mathbb{P}\\otimes \\diff t$. $L^2(\\diff \\mathbb{P}\\otimes \\diff t; L^2)$ is a separable Hilbert space with the scalar product\n\\begin{equation}\\nonumber\n\\langle f, g\\rangle = \\int_0^T\\mathbb{E}[\\langle f(t), g(t)\\rangle_{L^2}]\\diff t.\n\\end{equation}\nMorover it's reflexive and closed balls are weakly compact, due to its Hilbert space structure, see \\cite[Proposition 5.1]{Bre}. Also recall that under weak continuity assumptions (which are satisfied by energy solutions by definition), $\\mathcal{F}_t$-adapted processes are actually predictable, namely measurable with respect to the sub-$\\sigma$-algebra $\\mathcal{P}$ of predictable sets, see \\cite[Porposition 3.7]{DaP}. In particular, predictable processes form a closed subspace of $L^2(\\diff \\mathbb{P}\\otimes \\diff t; L^2)$ and therefore they are also closed with respect to weak convergence.\n\\begin{theorem}\\label{theorem sec 4.1 - existence of energy solutions}\nLet $b$ satisfy \\eqref{sec 2.3 - assumption 1 on b} and \\eqref{sec 2.3 - assumption 2 on b}, $\\{\\theta_k\\}_k$ satisfy \\eqref{sec 2.2 - condition on the coefficients} and \\eqref{sec 2.2 - isotropy condition} and $W$ be the associated divergence free noise. Then for any $u_0\\in L^2$ there exists an energy solution $u$ of \\eqref{sec 2.3 - equation in Stratonovich compact form, useful repetition}.\n\\end{theorem}\n\\begin{proof}\nFor any $N>0$, let $\\Pi_N$ denote the the Fourier projector on the modes with magnitude $\\vert k\\vert\\leq N$; define $u_0^N = \\Pi_N u_0$. For any $N$, consider the following Cauchy problem:\n\\begin{equation}\\label{sec 4.1 - Galerkin approximation system inside existence theorem}\n\\begin{cases}\n\\diff u^N = \\Pi_N (b\\cdot \\nabla u^N)\\diff t +c\\Delta u^N\\diff t + \\Pi_N\\left(\\sum_{j,k} \\theta_k\\, e_k\\, a_k^{(j)}\\cdot \\nabla u^N\\diff W^{(j)}_k \\right)\\\\\nu^N(0)=u_0^N\n\\end{cases}\n\\end{equation}\nIt can be checked, by writing explicitly the Fourier decomposition, that the above system only involves the noises $W_k^{(j)}$ belonging to a finite set of indices. It is therefore a linear SDE defined on a finite dimensional space (the space of Fourier polynomials of degree at most $N$) and as such it admits a unique local solution $u^N$ with continuous paths. Moreover, the heuristic calculation regarding the energy balance done in Section \\ref{subsection 2.3 - SPDE in Ito form and definition of energy solutions} in this setting is actually rigorous, since we are summing over a finite series, and therefore $u^N$ is defined on the whole $[0,T]$ and satisfies\n\\begin{equation}\\nonumber\n\\sup_{t\\in [0,T]}\\Big\\{ \\vert u^N(t)\\vert^2_{L^2}\\, e^{-\\int_0^t \\Vert \\text{div}b(s,\\cdot)\\Vert_\\infty\\diff s} \\Big\\}\\leq \\vert u^N_0\\vert^2_{L^2} \\quad\\mathbb{P}\\text{-a.s.};\n\\end{equation}\nin particular, for any $N$,\n\\begin{equation}\\label{sec 4.1 - energy inequality inside existence theorem}\n\\vert u^N(\\omega,t)\\vert^2_{L^2}\\leq e^{\\int_0^t \\Vert \\text{div}b(s,\\cdot)\\Vert_\\infty\\diff s} \\vert u_0\\vert^2_{L^2} \\quad\\text{ for }(\\diff\\mathbb{P}\\otimes \\diff t)\\text{-a.e. }(\\omega,t).\n\\end{equation}\nThis implies that for any $N$, equation \\eqref{sec 4.1 - Galerkin approximation system inside existence theorem} has a unique solution, globally defined on $[0,T]$, and\n\\begin{equation}\\nonumber\n\\int_0^T \\mathbb{E}\\big[\\vert u^N(t)\\vert_{L^2}^2\\big] \\diff t\n\\leq K \\vert u_0\\vert^2_{L^2},\n\\end{equation}\nfor a suitable constant $K$ which only depends on $b$. Therefore the sequence $\\{u^N\\}_N$ is uniformly bounded in $L^2(\\diff\\mathbb{P}\\otimes \\diff t; L^2)$ and we can assume, up to extracting a (not relabelled) subsequence, that it weakly converges to a process $u$. We now proceed to show that there exists a version of $u$ which is a weak solution of \\eqref{sec 2.3 - equation in Stratonovich compact form, useful repetition} with initial data $u_0$. As recalled earlier, $u$ is a predictable process since $u^N$ are so. Fix $\\varphi\\in C^\\infty(\\mathbb{T}^d)$, then by testing $u^N$ against $\\varphi$ we find\n\\begin{equation}\\label{sec 4.1 - weak formulation for u^N inside existence theorem}\\begin{split}\n\\langle u^N(\\cdot),\\varphi\\rangle - \\langle u^N_0,\\varphi\\rangle = & -\\int_0^\\cdot \\langle u^N(s), \\text{div}(b\\Pi_N\\varphi)\\rangle\\diff s + c\\int_0^\\cdot \\langle u^N(s),\\Delta \\varphi\\rangle\\diff s\\\\\n& - \\sum_{j,k} \\theta_k\\, \\int_0^\\cdot \\langle u^N(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\Pi_N\\varphi\\rangle\\diff W^{(j)}_k(s).\n\\end{split}\\end{equation}\nIt's clear that $u^N_0\\to u_0$ in $L^2$; the map $u(\\cdot)\\mapsto \\langle u(\\cdot),\\varphi\\rangle$, from $L^2(\\diff\\mathbb{P}\\otimes \\diff t; L^2)$ to $L^2(\\diff\\mathbb{P}\\otimes \\diff t)$ is linear and continuous and thus also weakly continuous (this is an immediate consequence of the definition of weak convergence). Similarly, the map from $L^2(\\diff\\mathbb{P}\\otimes \\diff t; L^2)$ to $L^2(\\diff\\mathbb{P}\\otimes \\diff t)$ given by\n\\begin{equation}\\nonumber\nu(\\cdot)\\mapsto \\int_0^\\cdot \\langle u(s), \\text{div}(b\\varphi)\\rangle\\diff s\n\\end{equation}\nis linear and continuous since\n\\begin{equation}\\nonumber\\begin{split}\n\\int_0^T & \\mathbb{E}\\Bigg[\\bigg\\vert \\int_0^t \\langle u^N(s), \\text{div}(b\\varphi)\\rangle\\diff s\\bigg\\vert^2\\Bigg]\\diff t\\\\\n& \\leq T\\int_0^T \\vert \\text{div}(b(t)\\varphi)\\vert_{L^2}^2\\diff t\\, \\int_0^T \\mathbb{E}[\\vert u(s)\\vert_{L^2}^2]\\diff t\n\\end{split}\\end{equation}\nand therefore also weakly continuous; since we also have $\\text{div}(b\\Pi^N\\varphi)\\to \\text{div}(b\\varphi)$ strongly, the above estimate shows that overall\n\\begin{equation}\\nonumber\n\\int_0^\\cdot \\langle u^N(s), \\text{div}(b\\Pi_N\\varphi)\\rangle\\diff s \\rightharpoonup \\int_0^\\cdot \\langle u(s), \\text{div}(b\\varphi)\\rangle\\diff s\\quad \\text{weakly}.\n\\end{equation}\nA similar reasoning applies to the processes $\\int_0^\\cdot \\langle u^N(s),\\Delta\\varphi\\rangle\\diff s$. Regarding the stochastic integrals, again we have that for fixed $\\varphi$ the map from $L^2(\\diff\\mathbb{P}\\otimes \\diff t; L^2)$ to $L^2(\\diff\\mathbb{P}\\otimes \\diff t)$ given by\n\\begin{equation}\\nonumber\nu\\mapsto \\sum_{j,k} \\theta_k\\, \\int_0^\\cdot \\langle u^N(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\Pi_N\\varphi\\rangle\\diff W^{(j)}_k(s)\n\\end{equation}\nis linear and continuous, since by the same calculations of Section \\ref{subsection 2.3 - SPDE in Ito form and definition of energy solutions} it holds\n\\begin{equation}\\nonumber\\begin{split}\n\\int_0^T & \\mathbb{E}\\Bigg[\\Big\\vert \\sum_{j,k} \\theta_k\\, \\int_0^t \\langle u^N(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\Pi_N\\varphi\\rangle\\diff W^{(j)}_k(s)  \\Big\\vert^2\\Bigg]\\diff t\\\\\n& \\leq 2T\\,\\sup_k \\theta_k^2\\, \\Vert\\nabla\\varphi\\Vert_{\\infty}^2\\,\\int_0^T \\mathbb{E}[\\vert u(s)\\vert_{L^2}^2]\\diff t\n\\end{split}\\end{equation}\nand as before using the fact that $\\nabla\\Pi_N\\varphi\\to\\nabla\\varphi$ uniformly, we also obtain weak convergence. Taking the weak limit as $N\\to\\infty$ on both sides of \\eqref{sec 4.1 - weak formulation for u^N inside existence theorem} we conclude that $u$ satisfies \n\\begin{equation}\\label{sec 4.1 - weak formulation, useful repetition inside existence theorem}\n\\begin{split}\n\\langle u(\\cdot),\\varphi\\rangle - \\langle u_0,\\varphi\\rangle = & -\\int_0^\\cdot \\langle u(s), \\text{div}(b\\varphi)\\rangle\\diff s + c\\int_0^\\cdot \\langle u(s),\\Delta \\varphi\\rangle\\diff s\\\\\n& - \\sum_{j,k} \\theta_k\\, \\int_0^\\cdot \\langle u(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k(s),\n\\end{split}\n\\end{equation}\nso that $u$ is a candidate weak solution of \\eqref{sec 2.3 - equation in Stratonovich compact form, useful repetition}.\nWe now want to show that there exists a version of $u$ which has paths in $C([0,T];L^2_w)$ and satisfies the energy inequality. Observe that the collection of processes in $L^2(\\diff\\mathbb{P}\\otimes \\diff t; L^2)$ satisfying inequality \\eqref{sec 4.1 - energy inequality inside existence theorem} is a convex, closed subset. Therefore it is also weakly closed (see \\cite{Bre}), which implies that inequality \\eqref{sec 4.1 - energy inequality inside existence theorem} holds also for $u$.\n\n\nFor fixed $\\varphi\\in C^\\infty(\\mathbb{T}^d)$, by standard properties of Lebesgue integral we know that the processes $\\int_0^\\cdot \\langle u(s), \\text{div}(b\\varphi)\\rangle \\diff s$, $\\int_0^\\cdot \\langle u(s),\\Delta \\varphi\\rangle \\diff s$ are $\\mathbb{P}$-a.s. continuous. \nRecall that by construction of the It\\^o integral, for any $k$ and $j$ the process $\\int_0^\\cdot \\langle u(s), e_k\\, a_k^{(j)}\\cdot \\nabla \\varphi\\rangle\\diff W^{(j)}_k(s)$ is a continuous square integrable martingale; moreover, continuous square integrable martingales are closed under $L^2(\\diff \\mathbb{P}\\otimes \\diff t)$-convergence (see \\cite{Rev}) and we have already shown that the infinite series on the r.h.s. of \\eqref{sec 4.1 - weak formulation, useful repetition inside existence theorem} is convergent in this norm. \nTherefore we can conclude that, for a fixed $\\varphi\\in C^\\infty(\\mathbb{T}^d)$, the process appearing on the r.h.s. of \\eqref{sec 4.1 - weak formulation, useful repetition inside existence theorem} is $\\mathbb{P}$-a.s. continuous in time and it coincides up to $(\\diff \\mathbb{P}\\otimes \\diff t)$-negligible sets with $\\langle u,\\varphi\\rangle-\\langle u_0,\\varphi\\rangle$; in particular, $\\mathbb{P}$-a.s. $\\langle u,\\varphi\\rangle$ admits a continuous version.\n\nBut then we can find, also thanks to \\eqref{sec 4.1 - energy inequality inside existence theorem}, a countable dense collection $\\varphi_n$ and a subset $\\Gamma$ of $\\Omega$ with $\\mathbb{P}(\\Gamma)=1$ such that for all $\\omega\\in\\Gamma$ the following holds: $t\\mapsto\\langle u(\\omega,t),\\varphi_n\\rangle$ admits a continuous version for all $n$ and there exists a set $E_\\omega\\subset [0,T]$ of full measure on which $\\vert u(\\omega,\\cdot)\\vert_{L^2}$ is uniformly bounded by some constant $K$. In particular, for any $t\\notin E_\\omega$ and any sequence $\\{t_n\\}_n\\subset E_\\omega$ such that $t_n\\to t$, we can extract a subsequence such that $u(\\omega, t_n)$ admits a weak limit in $L^2$, denoted by $v(\\omega,t)$, whose norm is still bounded by the constant $K$. But since $\\langle u(\\omega,\\cdot),\\varphi_n\\rangle$ all admit continuous versions, the limit $v(\\omega,t)$ is uniquely determined and does not depend on the extracted subsequence, nor on the original sequence $\\{t_n\\}_n$. Reasoning in this way, for fixed $\\omega\\in\\Gamma$, we can define $v(\\omega,t)$ for all $t\\in [0,T]$ and it satisfies the following: $v(\\omega,t)=u(\\omega,t)$ for all $t$ in a set of full Lebesgue measure; $\\vert v(\\omega,t)\\vert_{L^2}\\leq K$ for all $t\\in [0,T]$; $\\langle v(\\omega,t),\\varphi_n\\rangle$ coincides with the continuous version of $\\langle u(\\omega,t),\\varphi_n\\rangle$. But then by the uniform bound and density of $\\varphi_n$ it follows that the map $t\\mapsto \\langle v(\\omega,t)\\varphi\\rangle$ is continuous for every $\\varphi\\in C^\\infty(\\mathbb{T}^d)$ and for every $\\omega\\in \\Gamma$, namely $v$ is a version of $u$ with $\\mathbb{P}$-a.s. weakly continuous paths. Since $v$ also satisfies \\eqref{sec 4.1 - weak formulation, useful repetition inside existence theorem}, we conclude that it is a weak solution.\n\nIt only remains to show that the energy inequality holds, but this is achieved similarly by using the fact that, for all $\\omega\\in \\Gamma$ and all $t\\in \\tilde{E}_\\omega$, $\\tilde{E}_\\omega$ being a full Lebesgue measure set, inequality \\eqref{sec 4.1 - energy inequality inside existence theorem}  indeed holds, and therefore by using lower semicontinuity of $\\vert\\cdot\\vert_{L^2}$ and $v(\\omega,\\cdot)\\in C([0,T];L^2_w)$ it can be extended to all $(\\omega,t)\\in \\Gamma\\times[0,T]$.\n\\end{proof}\n\n\\subsection{Proof of pathwise uniqueness in the case $b=0$}\\label{subsection 4.2 - proof of strong uniqueness in the case b=0}\n\n\nWe prove in this section pathwise uniqueness of solutions in the case $b=0$; before proceeding further, let us mention the already existing results in the literature. Many of them are proved in $\\mathbb{R}^d$ but can be easily generalized to $\\mathbb{T}^d$.\n\nA main result in the topic is the already mentioned work \\cite{FlaGub}, where it is shown that for $b\\in L^\\infty(0,T;C^\\alpha(\\mathbb{R}^d;\\mathbb{R}^d))$ with $\\text{div}\\, b\\in L^p([0,T]\\times\\mathbb{R}^d)$, $\\alpha>0$ and $p\\geq 2$, in the case of a space-independent standard $d$-dim. Brownian motion, pathwise uniqueness holds for \\eqref{sec 1 - STLE in compact Stratonovich form} for any $u_0\\in L^\\infty(\\mathbb{R}^d)$; the proof is based on the existence of a sufficiently regular flow for the associated SDE. Many other results are now available, see the references in \\cite{BecFla}, but tipically the noise considered is space independent or has sufficiently good space regularity. In the stochastic fluid dynamics literature, the use of divergence free transport noise of the form\n\\begin{equation}\\nonumber\nW(t,x)=\\sum_{n\\in\\mathbb{N}} \\sigma_n(x)W_n(t)\n\\end{equation}\nappears fairly often; typical assumptions on this kind of noise are like those contained in \\cite{Brz}, specifically it is required that\\begin{equation}\\label{sec 4.2 brz condition}\n\\bigg\\Vert \\sum_{n\\in\\mathbb{N}} \\vert D\\sigma_k(\\cdot)\\vert^2\\bigg\\Vert_{L^\\infty(\\mathbb{T}^d)}<\\infty.\n\\end{equation}\nIn \\cite{BecFla}, \\eqref{sec 1 - STLE in compact Stratonovich form} is studied mainly in the case of space-independent noise, sufficiently regular initial data and $b\\in L^p(0,T;L^q(\\mathbb{R}^d))$, both in the subcritical (KR) and the critical (LPS) regime, using PDE arguments which do not rely on the existence of a regular flow for the associated SDE. It is stated however in Section 1.9 that all the results generalize to the case of \\textquotedblleft $\\sigma_n$ of class $C^4_b$ with proper summability in $n$ \\textquotedblright\\ such that the SDE\n\\begin{equation}\\nonumber\n\\diff Y=\\sum_{n\\in\\mathbb{N}} \\sigma_n(Y)\\circ \\diff W_n\n\\end{equation}\nhas a sufficiently regular stochastic flow of diffeomorphisms. In particular we expect that at least an analogue requirement to \\eqref{sec 4.2 brz condition} is needed; in our setting, this condition is equivalent to\n\\begin{equation}\\label{sec 4.2 brz condition 2}\n\\sum_{k\\in\\mathbb{Z}^d}\\vert k\\vert^2 \\theta_k^2<\\infty.\n\\end{equation}\nHowever, if instead of pathwise uniqueness one only requires \\textit{Wiener uniqueness} of weak solutions of \\eqref{sec 1 - STLE in compact Stratonovich form}, then the problem greatly simplifies. Wiener uniqueness means uniqueness in the class of processes adapted to the Brownian filtration $\\mathcal{F}^W_t$ and can be established by Wiener chaos expansion techniques, see \\cite{LeJ}, \\cite{LeJ2} and \\cite{Mau}. In particular, even if in \\cite{Mau} only space-independent noise and time-independent drift are considered, the technique seems to easily adapt to our setting, for any $\\{\\theta_k\\}$ satisfying \\eqref{sec 2.2 - condition on the coefficients} and \\eqref{sec 2.2 - isotropy condition} and any $b$ satisfying \\eqref{sec 2.3 - assumption 2 on b}, \\eqref{sec 3 - KR type condition on b}, as it fundamentally only requires wellposedness in a suitable class for the Kolmogorov equation \\eqref{sec 3 - parabolic Cauchy problem, useful repetition}, which holds under the conditions of Lemma \\ref{lemma sec 3 - sufficient condition for uniqueness of parabolic problem}. Wiener uniqueness is however unsatisfactory, for several reasons: if the only information on a solution $u$ is that it is adapted to $\\mathcal{F}_t$, then this strategy only gives pathwise uniqueness of the process $\\tilde{u}(t)=\\mathbb{E}[u(t)\\vert \\mathcal{F}^W_t]$; Wiener uniqueness is also too weak to apply the Yamada-Watanabe theorem (which holds also in infinite dimensions, see \\cite{Roc}) and ill-suited to exploit tools like Girsanov transform, see the discussion in Section 4.7 of \\cite{Fla}.\n\nThe problem of passing from Wiener uniqueness to pathwise uniqueness is not only technical, because if condition \\eqref{sec 4.2 brz condition} is not satisfied, the equation cannot be in general solved by means of characteristics, since phenomena like splitting and coalescence can occurr, as shown in \\cite{LeJ}. In this work, Wiener uniqueness is exploited to construct Markovian statistical solutions $S_t$ which are then studied and classified; in our setting, for $b=0$ and $W$ divergence free, according to the terminology introduced in \\cite{LeJ}, the statistical solution is \\textit{diffusive without hitting} and is not a flow of maps (i.e. does not admit a representation by characteristics), see Theorem 10.1 (by the divergence free condition, in our setting the parameter $\\eta$ is always $1$). In particular splitting can occurr, since the $2$-point motion $(X_t,Y_t)$ associated to $S_t$ starting from $(x,x)$ satisfies $X_t\\neq Y_t$ for all positive $t$, see Definition 6.3. In the work \\cite{LeJ2} statistical solutions are studied more in depth and it is hinted that non uniqueness can happen only in the \\textit{turbulent with hitting} regime, but no explicit proof of pathwise uniqueness in the other regimes is given.\n\nHere instead we adapt the strategy developed in \\cite{Barb}, which yields a relatively simple and short proof of pathwise uniqueness in the special case $b=0$, for any $\\{\\theta_k\\}_k$ satisfying \\eqref{sec 2.2 - condition on the coefficients} and \\eqref{sec 2.2 - isotropy condition}, which is a much weaker condition compared to \\eqref{sec 4.2 brz condition 2}.\n\n\nWe focus on the SPDE\n\\begin{equation}\\label{sec 4.2 - STLE with b=0, Ito form}\n\\diff u = c\\Delta u\\diff t + \\diff W\\cdot\\nabla u,\n\\end{equation}\nwith $W$ as usual given by \\eqref{sec 2.2 - definition of the noise W(t,x)}, $c$ defined in function of $\\{\\theta_k\\}_k$ by \\eqref{sec 2.3 - definition of the constant c}. Given $u_0\\in L^2$, we consider a weak solution of \\eqref{sec 4.2 - STLE with b=0, Ito form} in the sense of Definition \\ref{definition sec 2.3 - weak solution}.  Following \\cite{Barb}, we rewrite the equation in Fourier components. Let $u$ be given by the Fourier series\n\\begin{equation}\\nonumber\nu(t,x)=\\sum_l u_l(t)\\,e_l(x),\n\\end{equation}\nso that\n\\begin{equation}\\nonumber\n\\nabla u = i\\sum_l l\\, u_l\\,e_l,\n\\quad\n\\Delta u = -\\sum_l \\vert l\\vert^2 u_l\\, e_l.\n\\end{equation}\nExplicit calculations give the Fourier expansion for $\\diff W\\cdot\\nabla u$:\n\\begin{equation}\\nonumber\\begin{split}\n\\diff W\\cdot\\nabla u\n& = \\Bigg(\\sum_{j,k} \\theta_k\\,e_k\\,a_k^{(j)}\\diff W_k^{(j)} \\Bigg) \\cdot\\left(i\\sum_l l\\, u_l\\,e_l\\right)\\\\\n& = \\sum_{j,k,l} i\\,\\theta_k\\, a_k^{(j)}\\cdot l\\, u_l\\, e_{k+l}\\diff W_k^{(j)}\\\\\n& = \\sum_k i\\Bigg(\\sum_{j,l}\\theta_{k-l}\\, a_{k-l}^{(j)}\\cdot l\\, u_l\\diff W^{(j)}_{k-l} \\Bigg)\\, e_k\\\\\n& = \\sum_k i\\Bigg(\\sum_{j,l}\\theta_{k-l}\\, a_{k-l}^{(j)}\\cdot k\\, u_l\\diff W^{(j)}_{k-l} \\Bigg)\\, e_k,\n\\end{split}\\end{equation}\nwhere in the last passage we used the fact that $a^{(j)}_{k-l}\\perp k-l$. Uniqueness of the Fourier expansion then gives the following infinite linear system of coupled SDEs for the coefficients $u_k$:\n\\begin{equation}\\label{sec 4.2 - system of SDEs for u_k}\n\\diff u_k = -c\\vert k\\vert^2\\,u_k\\diff t + i \\sum_{j,l}\\theta_{k-l}\\, a_{k-l}^{(j)}\\cdot k\\, u_l\\diff W^{(j)}_{k-l},\n\\end{equation}\nwhere as usual the identity must be interpreted in integral sense:\n\\begin{equation}\\nonumber\nu_k(t) - u_k(0) = -c\\vert k\\vert^2\\int_0^t u_k(s)\\diff s + i \\sum_{j,l}\\theta_{k-l}\\,a_{k-l}^{(j)}\\cdot k \\int_0^t u_l(s)\\diff W^{(j)}_{k-l}(s).\n\\end{equation}\nThe derivation of \\eqref{sec 4.2 - system of SDEs for u_k} was very heuristical, but it can be checked that we would have found the same exact expression in integral form by taking $\\varphi=e_k$ as test functions in \\eqref{sec 2.3 - equation in Ito form, weak formulation in integral form}. Calculations similar to those of Section \\ref{subsection 2.3 - SPDE in Ito form and definition of energy solutions} give\n\\begin{equation}\\nonumber\\begin{split}\n\\mathbb{E}\\Bigg[\\Big\\vert \\sum_{j,l}\\theta_{k-l}\\,a_{k-l}^{(j)}\\cdot k \\int_0^t u_l(s)\\diff W^{(j)}_{k-l}(s) \\Big\\vert^2\\Bigg]\n& \\leq K \\vert k\\vert^2 \\int_0^T \\sum_l \\mathbb{E}\\big[\\vert u_l(s)\\vert^2\\big] \\diff s\\\\\n& = K \\vert k\\vert^2 \\int_0^T \\mathbb{E}\\big[\\vert u(s)\\vert_{L^2}^2\\big] \\diff s.\n\\end{split}\\end{equation}\nfor a suitable constant $K$, so that the infinite series in \\eqref{sec 4.2 - system of SDEs for u_k} is well defined, since $u$ satisfies \\eqref{sec 2.3 - condition of bounded energy in the definition of weak solution}.\nTo prove uniqueness, we need the following result.\n\\begin{lemma}\\label{lemma sec 4.2 - system for averaged energies}\nLet $u$ be a weak solution of \\eqref{sec 4.2 - STLE with b=0, Ito form}, $u_k$ defined as above. Then the real functions $x_k$ defined by $x_k(t)=\\mathbb{E}[\\vert u_k(t)\\vert^2]$ satisfy the following linear infinite system of coupled ODEs:\n\\begin{equation}\\label{sec 4.2 - equation for average energies}\n\\dot{x}_k = -2c\\vert k\\vert^2 x_k + 2\\sum_l \\theta_{k-l}^2\\, \\vert P_{k-l}k\\vert^2\\,x_l.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSince $u$ is a weak solution, we know that $\\{u_k\\}_k$ satisfy system \\eqref{sec 4.2 - system of SDEs for u_k}. Then applying It\\^o formula, using the properties of $W^{(j)}_k$, we have\n\\begin{equation}\\nonumber\\begin{split}\n\\diff(\\vert u_k\\vert^2)\n& = \\diff(u_k\\overline{u}_k)\n= u_k\\diff \\overline{u}_k + \\overline{u}_k\\diff u_k + \\diff [u_k,\\overline{u}_k]\\\\\n& = -c\\vert k\\vert^2 \\vert u_k\\vert^2\\diff t + \\diff M_t -c\\vert k\\vert^2 \\vert u_k\\vert^2\\diff t + \\diff N_t \\\\ & \\quad + 2\\sum_{j,l} \\theta_{k-l}^2 \\vert a_{k-l}^{(j)}\\cdot k\\vert^2 \\vert u_l\\vert^2\\diff t\n\\end{split}\\end{equation}\nwhere $M$ and $N$ are suitable square integrable martingale starting at 0. Taking expectation their contribution disappears and we obtain\n\\begin{equation}\\nonumber\n\\dot{x}_k = -2c\\vert k\\vert^2 x_k + 2\\sum_{j,l} \\theta_{k-l}^2\\, \\vert a_{k-l}^{(j)}\\cdot k\\vert^2\\,x_l.\n\\end{equation}\nObserve that, for any fixed $k$,\n\\begin{equation}\\nonumber\n\\sum_j \\vert a_{k-l}^{(j)}\\cdot k\\vert^2\n= \\Bigg\\vert \\sum_j \\Big(a_{k-l}^{(j)}\\otimes a_{k-l}^{(j)}\\Big)k \\Bigg\\vert^2\n= \\vert P_{k-l} k\\vert^2,\n\\end{equation}\nwhich implies the conclusion.\n\\end{proof}\n\\begin{remark}\\label{remark sec 4.2 - forward equation of markov chain} System \\eqref{sec 4.2 - equation for average energies} can be written as\n\\begin{equation}\\nonumber\n\\dot{x}_k = q_{kk}\\,x_k + \\sum_l q_{kl}\\, x_l,\n\\end{equation}\nwhere $q_{kk}=-2c\\vert k\\vert^2\\in (-\\infty, 0)$, $q_{kl} = 2 \\theta_{k-l}^2\\, \\vert P_{k-l}k\\vert^2\\geq 0$ for $k\\neq l$. Moreover, for any $k$ it holds\n\\begin{equation}\\nonumber\n\\sum_l q_{kl}\n= 2 \\sum_l \\theta_{k-l}^2\\, \\vert P_{k-l}k\\vert^2\n= 2 \\sum_{\\tilde{l}} \\theta_{\\tilde{l}}^2\\, \\vert P_{\\tilde{l}}k\\vert^2\n= 2 c\\vert k\\vert^2 = -q_{kk},\n\\end{equation}\nwhere we used the change of variables $k-l=\\tilde{l}$ and the computation \\eqref{sec 2.3 - definition of the constant c} from Section \\ref{subsection 2.3 - SPDE in Ito form and definition of energy solutions}. Namely, system \\eqref{sec 4.2 - equation for average energies} can be interpreted as the forward equation associated to a $Q$-matrix, which is the generator of a continuous time Markov process on $\\mathbb{Z}^d\\setminus\\{0\\}$; see the similarity with \\cite{BarFla}. This formulation can be useful to study long-time behaviour of solutions: for instance we can deduce immediately that if $u$ is a stationary solution, then it must hold $x_k=c$ for all $k$ and so the only invariant measure with support in $L^2$ is $\\delta_0$. If we expect convergence to equilibrium as $t\\to\\infty$, then all solutions should converge to $0$, even if energy is a formal invariant for equation \\eqref{sec 4.2 - STLE with b=0, Ito form}; indeed in \\cite{BarFla} anomalous dissipation of energy for a similar model is shown. Understanding whether anomalous dissipation takes place in this model will be the subject of future research.\n\\end{remark}\n\\begin{theorem}\\label{theorem sec 4.2 - uniqueness for b=0} Let $\\{\\theta_k\\}_k$ satisfy \\eqref{sec 2.2 - condition on the coefficients} and \\eqref{sec 2.2 - isotropy condition} and $W$ be the associated divergence free noise. If $u$ and $v$ are two weak solutions of \\eqref{sec 4.2 - STLE with b=0, Ito form} with the same initial data $u_0$, then\n\\begin{equation}\\nonumber\n\\mathbb{P}\\big(u(t)=v(t)\\big)=1\\quad \\forall\\,t\\in [0,T].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBy linearity of equation \\eqref{sec 4.2 - STLE with b=0, Ito form}, $w:= u-v$ is a weak solution with initial data $w_0=0$. In order to conclude it suffices to show that $\\mathbb{P}(w(t)=0)=1$ for every $t$. Recall that by the definition of weak solution, $u$ and $v$ satisfy \\eqref{sec 2.3 - condition of bounded energy in the definition of weak solution} and therefore also $w$ does. By Lemma \\ref{lemma sec 4.2 - system for averaged energies}, we know that $x_k=\\mathbb{E}[\\vert w_k\\vert^2]$ satisfy \\eqref{sec 4.2 - equation for average energies} with initial condition $x_k(0)=0$ for all $k$, namely\n\\begin{equation}\\nonumber\nx_k(t) = -2c \\vert k\\vert^2 \\int_0^t x_k(s)\\diff s + 2\\sum_l \\theta_{k-l}^2 \\vert P_{k-l} k\\vert^2 \\int_0^t  x_l(s) \\diff s.\n\\end{equation}\nFix $t\\in [0,T]$ and define $A_k=\\int_0^t x_k(s)\\diff s$, then the above equation becomes\n\\begin{equation}\\label{sec 4.2 - proof of main theorem, internal equation}\nx_k(t) + 2c\\vert k\\vert^2 A_k= 2\\sum_l \\theta_{k-l}^2 \\vert P_{k-l} k\\vert^2 A_l.\n\\end{equation}\nCondition \\eqref{sec 2.3 - condition of bounded energy in the definition of weak solution} implies that $A_k$ is summable:\n\\begin{equation}\\nonumber\n\\sum_k A_k\n= \\sum_k \\int_0^t \\mathbb{E}[\\vert w_k(s)\\vert^2]\\diff s\n\\leq \\int_0^T \\mathbb{E}[\\vert w(s)\\vert_{L^2}^2]\\diff s<\\infty,\n\\end{equation}\nso that $A_k\\to 0$ as $k\\to\\infty$; in particular $\\{A_k\\}_k$ admits a maximum, say at $A_{k_1}$. Since $x_{k_1}(t)\\geq 0$ by construction, we find\n\\begin{equation}\\nonumber\\begin{split}\n2c\\vert k_1\\vert^2 A_{k_1}\n& \\leq x_{k_1}(t) + 2c\\vert k_1\\vert^2 A_{k_1}\n= 2\\sum_l \\theta_{k_1-l}^2 \\vert P_{k_1-l} k_1\\vert^2 A_l\\\\\n& \\leq 2\\max_l A_l \\sum_l \\theta_{k_1-l}^2 \\vert P_{k_1-l} k_1\\vert^2\n= 2c\\vert k_1\\vert^2 A_{k_1},\n\\end{split}\\end{equation}\nwhich implies that all the inequalities are equalities and therefore $x_{k_1}(t)=0$, $A_l=A_{k_1}$ for all $l$ such that $\\theta_{k_1-l}\\neq 0$ and $P_{k_1-l}k_1\\neq 0$ (i.e. $l\\notin \\langle k\\rangle$). We are now going to show that we can construct inductively a sequence $k_n$ such that $k_n\\to\\infty$ and $A_{k_n}=\\max_l A_l$. Recall that $\\{\\theta_k\\}_k$ satisfy the isotropy condition \\eqref{sec 2.2 - isotropy condition}, so if $\\theta_{\\overline{j}}\\neq 0$ for some $\\overline{j}$, then $\\theta_{O\\overline{j}}\\neq 0$ as well. Let $\\Gamma=\\{O\\overline{j}, O\\in E_{\\mathbb{Z}}(d)\\}$. Then we can find $j\\in\\Gamma$ such that $k_2:=k_1-j$ satisfies $k_2\\notin\\langle k_1\\rangle$ and $\\vert k_2\\vert> \\vert k_1\\vert$; since $\\theta_{k_1-k_2}=\\theta_j\\neq 0$ we conclude that $A_{k_2}=\\max_l A_l$. But then we can iterate the reasoning, this time starting from $A_{k_2}$, to find $k_3$ such that $\\vert k_3\\vert>\\vert k_2\\vert$ and $A_{k_3}=\\max_l A_l$, and so on. In this way we find the desidered sequence $\\{k_n\\}_n$; but $A_l\\to 0$ as $l\\to\\infty$, which implies that $\\max_l A_l=0$ and so $A_l=0$ for all $l$.\nSince\n\\begin{equation}\\nonumber\n0=\\sum_l A_l = \\int_0^t \\mathbb{E}[\\vert w(s)\\vert^2_{L^2}]\\diff s\n\\end{equation}\nand the reasoning holds for any $t\\in [0,T]$, we obtain the conclusion.\n\\end{proof}\n\\begin{remark}\\label{remark sec 4.2 - advantages and disadvantages of this line of proof}\nLet us underline both the advantages and the disadvantages of the approach we used. On one side,  the proof could be further generalised: if we had a weaker concept of solution for which the derivation of system \\eqref{lemma sec 4.2 - system for averaged energies} is still rigorous (in principle system \\eqref{sec 4.2 - equation for average energies} is well defined under assumption $\\{x_k\\}_k\\in l^\\infty$), then in order for the proof to work we only need to guarantee that $\\{A_k\\}\\in c_0$ (i.e. $\\{A_k\\}\\in l^\\infty$ and $A_k$ infinitesimal), which could be deduced under milder conditions than \\eqref{sec 2.3 - condition of bounded energy in the definition of weak solution}. The proof also holds for the inhomogeneous equation with an external forcing $f$, since the difference of two solutions of the inhomogeneous system is a solution of the homogenous one.\\\\\nOn the other side, the proof is not easily generalizable on domains different than $\\mathbb{T}^d$ and completely breaks down when treating the case $b\\neq 0$. In fact, we are not able to obtain a closed equation for $\\mathbb{E}[\\vert u_k\\vert^2]$ as in Lemma \\ref{lemma sec 4.2 - system for averaged energies} anymore; it's still possible to find a closed system of ODEs for the terms $x_{k,l}=\\mathbb{E}[u_k\\overline{u}_l]$, but it's not as nice as \\eqref{sec 4.2 - equation for average energies}. The simplification obtained by finding a closed equation for the \"diagonal\" terms $x_{k,k}=\\mathbb{E}[\\vert u_k\\vert^2]$ is the key in our method of proof.\n\\end{remark}\n\nWe immediately obtain the following corollary.\n\n\\begin{corollary}\\label{corollary sec 4.2 - pathwise uniqueness} The following hold.\n\\begin{itemize}\n\n\\item[i)] (Pathwise uniqueness) Let $u$ be a weak solution of \\eqref{sec 4.2 - STLE with b=0, Ito form} with initial data $u_0$. Then $u$ is the unique energy solution of \\eqref{sec 4.2 - STLE with b=0, Ito form} with initial data $u_0$ (up to indistinguishability).\n\n\\item[ii)] (Stability) Let $u$ and $v$ be two weak solutions with respect to initial data $u_0$ and $v_0$. Then\n\\begin{equation}\\nonumber\n\\Vert u-v\\Vert_{L^\\infty(0,T;L^2)}  \\leq \\vert u_0-v_0\\vert_{L^2}\\quad \\mathbb{P}\\text{-a.s.}\n\\end{equation}\n\n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\ni) Let $u$ be as in the hypothesis and $\\tilde{u}$ be an energy solution of \\eqref{sec 4.2 - STLE with b=0, Ito form} with initial data $u_0$. Then by Theorem \\ref{theorem sec 4.2 - uniqueness for b=0}\n\\begin{equation}\\nonumber\n\\mathbb{P}(u(t)=\\tilde{u}(t)\\ \\ \\forall\\, t\\in [0,T]\\cap\\mathbb{Q})=1\n\\end{equation}\nand we conclude that $u$ and $\\tilde{u}$ are indistinguishable, since $u$ and $\\tilde{u}$ both have $\\mathbb{P}$-a.s. continuous paths in the weak topology.\n\nii) By linearity, $u-v$ is a weak solution with initial data $u_0-v_0$. But then it is the unique energy solution and satisfies the energy inequality \\eqref{sec 2.3 - energy inequality} with $b=0$, which gives the conclusion.\n\\end{proof}\n\nAlso observe that we are in the conditions to apply Yamada-Watanabe theorem and therefore not only pathwise but also uniqueness of strong solutions in the probabilistic sense holds; this also implies uniqueness in law.\n\n\\section{The case $d=1$}\\label{section 5 - the case d=1}\nThe proof of our main result required two fundamental features: the use of an incompressible noise, namely $W=W(t,x)$ such that at any fixed time $\\text{div}_x W(t,\\cdot)=0$ in the sense of distributions, and the existence of weak solutions satisfying a uniform energy bound; the existence of such solutions was a consequence of suitable conditions on $b$ and incompressibility of $W$. However, in the case of spatial dimension $d=1$, the divergence-free condition is equivalent to $W$ being space-independent, which doesn't allow to take a sequence of noises which are increasingly rougher in space; it is therefore unclear whether it's possible to obtain an analogue of Theorem \\ref{theorem sec 3 - main result}. One might still look for a suitable sequence of not divergence-free noises, for which existence of energy solutions can be shown, and such that in the limit they converge in a suitable sense to a deterministic PDE. In this section we show that, while this program can still partially be carried out, in the limit we do not expect to find a PDE with a diffusion term.\n\n\nLet us briefly introduce the notation and setting of this section. We consider the one dimensional torus $\\mathbb{T}=\\mathbb{R}\/(0,2\\pi)$, with periodic boundary condition. However, since in one dimension there is no real advantage in working with complex series, we prefer in this case to restrict ourselves to the space of real valued, $2\\pi$-periodic, $L^2$ functions, with the normalised inner product\n\\begin{equation}\\nonumber\n\\langle f,g\\rangle = \\frac{1}{2\\pi}\\int_\\mathbb{T} f(x)g(x)\\diff x.\n\\end{equation}\nA complete orthonormal system for $L^2(\\mathbb{T})$ is given by the real Fourier basis $\\{e_k\\}_{k\\in\\mathbb{Z}}$:\n\\begin{equation}\\nonumber\ne_k(x)=\\begin{cases}\n\\sqrt{2}\\cos(kx)\\qquad & \\text{if } k>0\\\\\n1 & \\text{ if } k=0\\\\\n\\sqrt{2}\\sin(kx) & \\text{ if } k<0\n\\end{cases}.\n\\end{equation}\nThroughout this section, we will assume $\\{W_k\\}_{k\\in\\mathbb{Z}}$ to be a sequence of \\textit{real} independent Brownian motions. In order to understand which kind of noise to use, let us start by considering the deterministic transport equation\n\\begin{equation}\\label{sec 5 - starting deterministic pde}\n\\partial_t u = b\\, \\partial_x u,\n\\end{equation}\nwhere we need to assume at least $b=b(t,x)$ to be in $L^1(0,T;L^2)$ with $\\text{div}\\, b = \\partial_x b \\in L^1(0,T;L^\\infty)$, namely $b\\in L^1(0,T; W^{1,\\infty})$. In the 1-d framework, under such conditions on $b$, equation \\eqref{sec 5 - starting deterministic pde} is already well posed by the DiPerna-Lions theory, see \\cite{DiPLio}. We highlight this aspect as it provides a partial explanation of the fact that in 1-d there doesn't seem to be much space for regularization by space-time dependent noise (even if some results have been obtained, see \\cite{Ges} and the references therein): the deterministic theory is already well posed under sufficiently mild conditions and therefore too \"competitive\" for the noise to perform better. Anyway, observing that for a given smooth deterministic $v=v(t,x)$, the linear PDE\n\\begin{equation}\\nonumber\n\\partial_t u = 2v\\,\\partial_x u + \\partial_x v\\, u\n\\end{equation}\nis formally energy preserving, since\n\\begin{equation}\\nonumber\n\\frac{\\diff}{\\diff t}\\int_\\mathbb{T} u^2\\diff x\n= \\int_\\mathbb{T} \\big(4 v\\,u\\,\\partial_x u + 2\\partial_xv\\, u^2\\big)\\diff x\n= 2\\int_\\mathbb{T} \\partial_x (v\\, u^2)\\diff x = 0,\n\\end{equation}\nit seems reasonable to consider a perturbation of \\eqref{sec 5 - starting deterministic pde} of the form\n\\begin{equation}\\label{sec 5 - linear spde, compact stratonovich formulation}\n\\diff u = b\\, \\partial_x u\\diff t + 2\\circ \\diff W\\, \\partial_x u + \\circ \\diff(\\partial_x W)\\, u,\n\\end{equation}\nwhere $W=W(t,x)$ is a suitable noise which will be described later. As before, $\\circ \\diff W$ denotes Stratonovich integration with respect to the time parameter; observe that, if $W$ is a distribution valued Wiener process, then $\\partial_x W$ is again a  a distribution valued Wiener process and therefore, under proper conditions, Stratonovich integration also with respect to it can be defined. This will become more transparent once we describe $W$ explicitly. By the properties of Stratonovich integral (in particular the chain rule) and the above computation, we expect formally to obtain the same energy balance as for equation \\eqref{sec 5 - starting deterministic pde} and therefore the existence of weak solutions satisfying the energy inequality\n\\begin{equation}\\nonumber\n\\sup_{t\\in [0,T]} \\Big\\{ e^{-1\/2 \\int_0^t \\Vert \\partial_x b(s)\\Vert_\\infty \\diff s}\\, \\vert u(t)\\vert_{L^2} \\Big\\} \\leq \\vert u_0\\vert_{L^2} \\quad \\mathbb{P}\\text{-a.s.}\n\\end{equation}\nObserve however that we have already found a criticality with respect to the approach of the previous sections: in equation \\eqref{sec 5 - linear spde, compact stratonovich formulation}, not only Stratonovich multiplicative noise $\\circ \\diff W$ appears, but also $\\circ \\diff(\\partial_x W)$; if the former, in order to be defined, required $W$ to be an $L^2$-valued random variable, then the latter will require $\\partial_x W$ to be $L^2$-valued as well and therefore $W$ to belong to $H^1$. In particular, the class of noises we can use has one additional degree of regularity with respect to the one we could use in higher dimension. If we expect the paradigm \\textquotedblleft the rougher the noise, the better the regularization\\textquotedblright\\ to hold, then this kind of noise shouldn't be able to regularize very much.\n\n\nLet us give a more precise description of $W$, so that we can give proper meaning to equation \\eqref{sec 5 - linear spde, compact stratonovich formulation} and pass to the It\\^o formulation. Similarly to the previous sections, we consider $W$ given by\n\\begin{equation}\\nonumber\nW(t,x)=\\sum_k \\sigma_k(x) W_k(t),\n\\end{equation}\nwhere $\\sigma_k\\in C^\\infty(\\mathbb{T})$ for each $k$ and the index $k$ is ranging over $\\mathbb{Z}$. In particular, we might consider both a finite or infinite series (in the latter case, convergence is interpreted in the sense of distributions as before). Then, $\\partial_x W$ in the sense of distributions is given by\n\\begin{equation}\\nonumber\n\\partial_x W(t,x)=\\sum_k \\sigma'_k(x) W_k(t),\n\\end{equation}\nso that we can write equation \\eqref{sec 5 - linear spde, compact stratonovich formulation} as\n\\begin{equation}\\label{sec 5 - linear spde, explicit stratonovich formulation}\n\\diff u\n= b\\, \\partial_x u\\diff t + \\sum_k \\big( 2\\sigma_k \\partial_x u + \\sigma_k'\\, u\\big)\\circ \\diff W_k\n= b\\, \\partial_x u\\diff t + \\sum_k \\mathcal{M}_k u\\circ \\diff W_k,\n\\end{equation}\nwhere we consider $\\mathcal{M}_k = 2\\sigma_k \\partial_x + \\sigma_k'$ as a linear (unbounded) operator on $L^2(\\mathbb{T})$. From \\eqref{sec 5 - linear spde, compact stratonovich formulation} we obtain the corresponding It\\^o formulation:\n\\begin{equation}\\nonumber\n\\diff u = b\\, \\partial_x u\\diff t +\\frac{1}{2} \\sum_k \\mathcal{M}_k^2 u\\diff t + \\sum_k \\mathcal{M}_k u\\diff W_k.\n\\end{equation}\nAlgebraic computations yield\n\\begin{equation}\\label{sec 5 - linear spde, general ito formulation}\\begin{split}\n\\diff u = b\\, \\partial_x u\\diff t\n& + \\Bigg(\\sum_k \\sigma_k \\sigma''_k + \\frac{1}{2} (\\sigma_k')^2\\Bigg) u\\diff t\n+ 4 \\Bigg(\\sum_k \\sigma_k \\sigma'_k \\Bigg)\\partial_x u\\diff t\\\\\n&+ 2\\Bigg(\\sum_k \\sigma_k^2 \\Bigg)\\partial_{xx} u\\diff t\n+ \\sum_k \\mathcal{M}_k u\\diff W_k.\n\\end{split}\\end{equation}\nIn particular, in order for the It\\^o-Stratonovich corrector to make sense, we need all the above series to be convergent at any fixed $x$. Moreover, observe that now the term in front of $\\partial_{xx}u$ is in \\textquotedblleft competition\\textquotedblright\\ with those in front of $\\partial_x u$ and $u$; in particular, when we renormalise the noise $W$ by dividing by the term which explodes faster, if the other terms don't grow with the same speed they will disappear in the limit. This is indeed what happens and what determines the failure of Theorem \\ref{theorem sec 3 - main result} in $d=1$, at least when the perturbation is performed by a linear multiplicative noise of the form \\eqref{sec 5 - linear spde, compact stratonovich formulation}.\n\n\nWe illustrate what described above for specific choices of $\\sigma_k$ which allow to perform explicit calculations. Take $\\sigma_k(x) = \\lambda_k\\, e_k(x)$, where $\\lambda_k$ are some real constants (on which we need to impose conditions, see below) such that $\\lambda_k = \\lambda_{-k}$ for all $k$, $\\lambda_0=0$ and $\\{e_k\\}_k$ is the real Fourier basis introduced at the beginning of the section. Then equation \\eqref{sec 5 - linear spde, general ito formulation} becomes\n\\begin{equation}\\label{sec 5 - linear spde, specific ito formulation}\\begin{split}\n\\diff u\n& = b\\, \\partial_x u \\diff t -\\frac{1}{2}\\Bigg( \\sum_k k^2 \\lambda_k^2\\Bigg) u\\diff t + 2\\Bigg( \\sum_k \\lambda_k^2\\Bigg)\\partial_{xx}u\\diff t\\\\\n&\\quad+ \\sum_k \\lambda_k \\big( 2 e_k \\partial_x u + e_k' u\\big) \\diff W_k .\n\\end{split}\\end{equation}\nEquation \\eqref{sec 5 - linear spde, specific ito formulation} confirms the discussion above: the It\\^o-Stratonovich corrector in order to be defined requires the condition\n\\begin{equation}\\nonumber\n\\sum_k k^2 \\lambda_k^2 <\\infty,\n\\end{equation}\nnamely $W$ taking values in $H^1$, and the coefficient in front of $u$ is strictly bigger than the one in front of $\\partial_{xx}u$, at least whenever $\\lambda_k\\neq 0$ for some $k\\notin \\{-1,1\\}$. Let us write the weak formulation of equation \\eqref{sec 5 - linear spde, specific ito formulation} in order to understand if the infinite series of It\\^o integrals is well defined as well and how fast it grows as a function of the parameters $\\lambda_k$: $u$ is a weak solution if, for any $\\varphi\\in C^\\infty(\\mathbb{T})$,\n\\begin{equation}\\label{sec 5 - specific ito formulation, weak formulation}\\begin{split}\n\\diff \\langle u,\\varphi \\rangle =\n& - \\langle u,\\partial_x (b \\varphi) \\rangle\\diff t\n- \\frac{1}{2}\\Bigg( \\sum_k k^2 \\lambda_k^2\\Bigg) \\langle u,\\varphi\\rangle \\diff t\n+ 2\\Bigg( \\sum_k \\lambda_k^2\\Bigg)\\langle u,\\partial_{xx}\\varphi\\rangle\\diff t\\\\\n& - 2 \\sum_k \\lambda_k \\langle u, e_k\\partial_x \\varphi\\rangle\\diff W_k\n- \\sum_k \\lambda_k \\langle u, e'_k\\varphi\\rangle\\diff W_k\n\\end{split}\\end{equation}\nLet us consider the last two series separately (in principle we have already committed an abuse by splitting the series, since we haven't yet proved its convergence, but once convergence is proven the passage is rigorous; otherwise just consider the finite approximations first, for which the splitting is legit, and push to the limit after the convergence of both series is proven). By It\\^o isometry and independence of $\\{W_k\\}_k$, for any $t\\in [0,T]$ we have\n\\begin{equation}\\nonumber\\begin{split}\n\\mathbb{E}\\Bigg[ \\bigg\\vert \\sum_k \\lambda_k \\int_0^t \\langle u(s), e_k\\partial_x\\varphi\\rangle\\diff W_k(s)\\bigg\\vert^2 \\bigg]\n& = \\sum_k \\lambda_k^2 \\int_0^t \\mathbb{E}[\\langle u(s)\\partial_x\\varphi, e_k\\rangle^2]\\diff s\\\\\n& \\leq \\sup_k \\lambda_k^2 \\int_0^T \\mathbb{E}\\bigg[\\sum_k \\langle u(s)\\partial_x\\varphi,e_k\\rangle^2\\bigg]\\diff s\\\\\n& = \\sup_k \\lambda_k^2 \\int_0^T \\mathbb{E}\\big[\\,\\vert u(s)\\partial_x\\varphi\\vert_{L^2}^2\\big]\\diff s\\\\\n& \\leq \\sup_k \\lambda_k^2\\, \\Vert \\partial_x\\varphi\\Vert_\\infty^2\\, \\int_0^T \\mathbb{E}\\big[\\,\\vert u(s)\\vert_{L^2}^2\\big]\\diff s\n\\end{split}\\end{equation}\nTherefore the first series grows as $\\sup_k \\vert \\lambda_k\\vert$ as in the case $d\\geq 2$. For the second series with analogous calculations we find\n\\begin{equation}\\nonumber\n\\mathbb{E}\\Bigg[ \\bigg\\vert \\sum_k k\\lambda_k \\int_0^t \\langle u(s), e_k\\varphi\\rangle\\diff W_k(s)\\bigg\\vert^2 \\Bigg]\n\\leq \\sup_k\\big\\{ k^2 \\lambda_k^2\\big\\}\\, \\Vert \\varphi\\Vert_\\infty^2 \\int_0^T \\mathbb{E}\\big[\\,\\vert u(s)\\vert_{L^2}^2\\big]\\diff s,\n\\end{equation}\nwhich shows that the second series grows in norm as $\\sup_k \\{\\vert k\\lambda_k\\vert\\}$, which is therefore also the leading term of the overall martingale term appearing in equation \\eqref{sec 5 - specific ito formulation, weak formulation}. In particular observe that\n\\begin{equation}\\nonumber\n\\sum_k \\lambda_k^2 \\leq \\sup_k \\{k\\lambda_k\\}^2\\, \\sum_{k\\neq 0} \\frac{1}{k^2} = C\\,\\sup_k \\{k\\lambda_k\\}^2,\n\\end{equation}\nwhich shows that it's not possible to renormalize $W$ in such a way that in the limit the coefficient in front of $\\partial_{xx}u$ survives while the martingale term disappears. The above inequality only holds in dimension $d=1$ and fails for $d$ higher, proving once again that dimension is playing a fundamental role and we can't infer for $d=1$ the same results as for $d\\geq 2$. However, it's still easy to find a collection $\\{\\lambda_k^N, k\\in\\mathbb{Z}, N\\in\\mathbb{N}\\}$ such that $\\lambda^N_k = \\lambda^N_{-k}$, $\\lambda_0^N=0$ for all $N$ and\n\\begin{equation}\\nonumber\n\\lim_{N\\to\\infty}\\frac{\\sup_k \\{k^2(\\lambda^N_k)^2\\}}{\\sum_k k^2(\\lambda^N_k)^2}=0,\n\\qquad \\lim_{N\\to\\infty}\\frac{\\sum_k (\\lambda_k^N)^2}{\\sum_k k^2(\\lambda^N_k)^2}=0,\n\\end{equation}\nwhere we are of course assuming that for fixed $N$ all the quantities appearing are finite. For a given such sequence, if we define the noises $W^N=W^N(t,x)$ as\n\\begin{equation}\\nonumber\nW^N(t,x)= \\sum_k \\lambda^N_k e_k(x) W_k(t)\n\\end{equation}\nand for a fixed $\\nu>0$ we define\n\\begin{equation}\\nonumber\n\\varepsilon_N = 2\\nu \\bigg( \\sum_k k^2(\\lambda^N_k)^2\\bigg)^{-1},\n\\end{equation}\nthen $\\varepsilon_N\\to 0$ as $N\\to\\infty$ and going through the same proof as in Theorem \\ref{theorem sec 3 - main result} we obtain that any weak energy solution of equation\n\\begin{equation}\\nonumber\n\\begin{cases}\n\\diff u^N = b\\, \\partial_x u^N\\diff t + 2\\sqrt{\\varepsilon_N}\\circ \\diff W^N\\, \\partial_x u^N + \\sqrt{\\varepsilon_N}\\circ \\diff (\\partial_x W^N)\\, u^N\\\\\nu^N(0)=u_0\n\\end{cases}\n\\end{equation}\nwill converge in probability as $N$ goes to infinity to $u$ deterministic solution of\n\\begin{equation}\\label{sec 5 - expected deterministic pde limit}\n\\begin{cases}\n\\partial_t u = b\\, \\partial_x u - \\nu\\, u\\\\\nu(0)=u_0\n\\end{cases},\n\\end{equation}\nas long as the weak solution of \\eqref{sec 5 - expected deterministic pde limit} is unique on the interval $[0,T]$. If uniqueness of \\eqref{sec 5 - expected deterministic pde limit} fails, then the proof of Theorem \\ref{theorem sec 3 - main result} breaks down as well; we may still however extract a (not relabelled) subsequence $u^N$ which converges weakly in $L^2(\\Omega\\times[0,T]\\times\\mathbb{T},\\diff\\mathbb{P}\\otimes \\diff t\\otimes \\diff x)$ to a stochastic solution of \\eqref{sec 5 - expected deterministic pde limit}, usually referred to as a \\textit{superposition solution}, see \\cite{Amb}, \\cite{Fla3} ; moreover by properties of weak convergence, this solution still satisfies the energy inequality. We have however lost the main advantage of Theorem \\ref{theorem sec 3 - main result}, where the sequence $u^N$ converges to a deterministic PDE which is in principle much better posed than the approximating sequence, due to the presence of the Laplacian.\n\n\nSuch a result may still be seen, from the modelling point of view, as a mathematical justification of the presence of a friction term $-\\nu u$ in one dimensional PDEs, as the ideal limit of the action of a suitable noise $W$ which is very irregular but of very small intensity; the coefficient $\\nu$ is proportional to the product between the magnitude and the spatial irregularity of the noise (measured by its $H^1$ norm). We underline however that a noise of the form \\eqref{sec 5 - linear spde, compact stratonovich formulation} has been introduced only for mathematical convenience (it allows to obtain an energy inequality for the solutions) and has not been justified from the physical point of view, nor equation \\eqref{sec 5 - linear spde, compact stratonovich formulation} has been derived from first principles (namely from a Lagrangian formulation). Therefore there is still the possibility that the addition of a different multiplicative noise, with a more robust modelling justification, allows to obtain an analogue of Theorem \\ref{theorem sec 3 - main result} also in dimension $d=1$.\n\n\n\\begin{acknowledgements}\nThis work stems from my master thesis (see \\cite{Gal}, Section 4.4) which was developed under the supervision of Prof. David Barbato, to whom I'm deeply indebted and I want to express my gratitude. I also want to thank Prof. Franco Flandoli for the very useful discussions and for encouraging me into writing this work, as well as Prof. Dejun Luo for pointing out a mistake in early calculations and showing me a very simple and elegant way to find the explicit expression for the It\\^o-Stratonovich term. I'm also grateful to Prof. Massimiliano Gubinelli for reviewing the early draft of this work and to Immanuel Zachhuber and Lorenzo Dello Schiavo for the useful suggestions for the proof of Lemma \\ref{lemma sec 3 - sufficient condition for uniqueness of parabolic problem}.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\t\nVortices play significant parts in many fields of theoretical physics including superconductivity theory, cosmology, condensed-matter physics,   electroweak theory, and quantum Hall effect. A tide of research related to vortex equations has been accomplished; see, for example, \\cite{CI, T, TY, WY, Y} and the references therein. Recently, Han, Lin and Yang \\cite{HLY2} investigated a system of relativistic non-Abelian Chern-Simons-Higgs vortex equations whose Cartan matrix $K$ is that of arbitrary simple Lie algebra, they established a general existence result for the doubly periodic solutions of the Chern-Simons-Higgs vortex equations.\n \nIn recent years, equations on graphs have attracted extensive attention; see, for example, \\cite{ ALY, Hu, Hub, HWY, HLY, LP} and the references therein. \n\n Recently, Huang, Lin and Yau \\cite{HLY} proved the existence of solutions to mean field equations \n$$\n\\Delta u+e^{u}=\\rho \\delta_{0}\n$$\nand\n$$\n\\Delta u=\\lambda e^{u}\\left(e^{u}-1\\right)+4 \\pi \\sum_{j=1}^{M} \\delta_{p_{j}}\n$$\non graphs.\n\nLet $G=(V,E)$ is a connected finite graph, and $V$ denotes the vetex set and $E$ denotes the edge set.\n\nInspired by the work of Huang-Lin-Yau \\cite{HLY}, we study the relativistic Chern-Simons-Higgs equations  \n\\begin{equation}\\label{1}\n\t\\Delta u_{i}=\\lambda\\left(\\sum_{j=1}^{n} \\sum_{k=1}^{n} K_{k j} K_{j i} \\mathrm{e}^{u_{j}} \\mathrm{e}^{u_{k}}-\\sum_{j=1}^{n} K_{j i} \\mathrm{e}^{u_{j}}\\right)+4 \\pi \\sum_{j=1}^{N_{i}} \\delta_{p_{i j}}(x), \\quad i=1, \\ldots, n,\n\\end{equation}\non $G$, where $K=(K_{ij})$ is the Cartan matrix of a finite dimensional semisimple Lie algebra $L$, $n\\ge1$ is the rank of $L$ which is the dimension of the Cartan subalgebra of $L$, $p_{ij}$, $j=1,...,N_{i}$, $i=1,...,n$, are arbitrarily chosen distinct vertices on the graph, and $\\delta_{p_{ij}}$ is the Dirac mass at $p_{ij}$. With a view to handling the system in a unified framework, we need some suitable assumption on the matrix K. We suppose that\n\\begin{equation}\\label{41}\n\tK^{T}=PS,\n\\end{equation}\n$P$ is a diagonal matrix satisfying \n\\begin{equation}\\label{33}\n\tP:=diag\\{P_1,...,P_n\\},~P_{i}>0,~i=1,...,n,\n\\end{equation}\n$S$ is a positive definite matrix of the form\n\\begin{equation}\\label{43}\n\tS \\equiv\\left(\\begin{array}{cccccc}\n\t\t\\alpha_{11} & -\\alpha_{12} & \\cdots & \\cdots & \\cdots & -\\alpha_{1 n} \\\\\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t-\\alpha_{i 1} & -\\alpha_{i 2} & \\cdots & \\alpha_{i i} & \\cdots & -\\alpha_{i n} \\\\\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t-\\alpha_{n 1} & -\\alpha_{n 2} & \\cdots & \\cdots & -\\alpha_{n n-1} & \\alpha_{n n}\n\t\\end{array}\\right),\n\\end{equation}\n\\begin{equation}\\label{44}\n\t\\alpha_{i i}>0, \\quad i=1, \\ldots, n, \\quad \\alpha_{i j}=\\alpha_{j i} \\geq 0, i \\neq j=1, \\ldots, n,\n\\end{equation}\nand\n\\begin{equation}\\label{45}\n\t\\text{~all~the~entries~of~}S^{-1} \\text{are positive}.\n\\end{equation}\nBy \\eqref{45}, we conclude that \n\\begin{equation}\\label{46}\n\tR_{i} := \\sum_{j=1}^{n}\\left(\\left(K^{\\tau}\\right)^{-1}\\right)_{i j}>0\n\t\\end{equation}\nfor $i=1,...,n$. \n\n We now state our main results as follows.\n \n \\begin{Theorem}\\label{31}\n \tAssume that the matrix $K$ satisfies \\eqref{41}-\\eqref{45}, then we have the following conclusions: \n \t\t\n \t$(\\mathrm{i})$~Suppose that \\eqref{1} has a solution. Then we have \n \t\\begin{equation}\\label{47}\n \t\t\\lambda>\\lambda_{0} \\equiv \\frac{16 \\pi}{|V|} \\frac{\\sum \\limits_{i=1}^{n} \\sum\\limits_{j=1}^{n} P_{i}^{-1}\\left(K^{-1}\\right)_{j i} N_{j}}{\\sum\\limits_{i=1}^{n} \\sum\\limits_{j=1}^{n} P_{i}^{-1}\\left(K^{-1}\\right)_{j i}}.\n \t\\end{equation}\n \n \t$(\\mathrm{ii})$~There exists a constant $\\lambda_{1}>\\lambda_{0}$ so that if $\\lambda>\\lambda_{1}$,  then \\eqref{1} admits a solution $(u^{\\lambda}_{1},...,u^{\\lambda}_{n})$. \n \\end{Theorem}\n\n\nThe rest of the paper is arranged as below. In Section 2, We present some results that we will use frequently in the following pages. Section 3 and Section 4 are devoted to the proof of Theorem \\ref{31}.\n\n\\section{Preliminary results}\n\nFor each edge $xy \\in E$, We suppose that its weight $w_{xy}>0$ and that $w_{xy}=w_{yx}$. Set $\\mu: V \\to (0,+\\infty)$ be a finite measure. For any function $u: V \\to \\mathbb{R}$, the Laplacian of $u$ is defined by \n\\begin{equation}\\label{l1}\n\t\\Delta u(x)=\\frac{1}{\\mu(x)} \\sum_{y \\sim x} w_{y x}(u(y)-u(x)),\n\\end{equation}\nwhere $y \\sim x$ means $xy \\in E$. The gradient  $\\nabla$ of function $f$ is defined by a vector \n$$\\nabla f (x):=\\left(  \\left[ f(y)-f(x)\\right]  \\sqrt{\\frac{w_{xy}}{2\\mu (x)}} \\right)_{y\\sim x} .$$ The gradient form of $u$ reads \n\\begin{equation}\n\t\\Gamma(u, v)(x)=\\frac{1}{2 \\mu(x)} \\sum_{y \\sim x} w_{x y}(u(y)-u(x))(v(y)-v(x)).\n\\end{equation}\nWe denote the length of the gradient of $u$ by\n\\begin{equation*}\n\t|\\nabla u|(x)=\\sqrt{\\Gamma(u)(x)}=\\left(\\frac{1}{2 \\mu(x)} \\sum_{y \\sim x} w_{x y}(u(y)-u(x))^{2}\\right)^{1 \/ 2}.\n\\end{equation*}\nDenote, for any function $\nu: V \\rightarrow \\mathbb{R}\n$, an integral of $u$ on $V$ by $\\int \\limits_{V} u d \\mu=\\sum\\limits_{x \\in V} \\mu(x) u(x)$. Denote  $|V|$=$ \\text{Vol}(V)=\\sum \\limits_{x \\in V} \\mu(x)$ the volume of $V$. For $p > 0$, denote $|| u ||_{p}:=||u||_{L^{p}(V)}=(\\int \\limits_{V} |u|^{p} d \\mu)^{\\frac{1}{p}}$. Define a sobolev space and a norm on it by \n\\begin{equation*}\n\tW^{1,2}(V)=\\left\\{u: V \\rightarrow \\mathbb{R}: \\int \\limits_{V} \\left(|\\nabla u|^{2}+u^{2}\\right) d \\mu<+\\infty\\right\\},\n\\end{equation*}\nand \\begin{equation*}\n\t\\|u\\|_{H^{1}(V)}=\t\\|u\\|_{W^{1,2}(V)}=\\left(\\int \\limits_{V}\\left(|\\nabla u|^{2}+u^{2}\\right) d \\mu\\right)^{1 \/ 2}.\n\\end{equation*}\n\nTo apply the variational method, we need the following Sobolev embedding, Truding-Moser inequlity and interpolation inequality on graphs.\n\n\\begin{Lemma}\\label{21}\n\t{\\rm (\\cite[Lemma 5]{ALY})} Let $G=(V,E)$ be a finite graph. The sobolev space $W^{1,2}(V)$ is precompact. Namely, if ${u_j}$ is bounded in $W^{1,2}(V)$, then there exists some $u \\in W^{1,2}(V)$ such that up to a subsequence, $u_j \\to u$ in $W^{1,2}(V)$.\n\\end{Lemma}\n\n\\begin{Lemma}\\label{2.2}\n\t{\\rm (\\cite[Lemma 6]{ALY})}\tLet $G = (V, E)$ be a finite graph. For all functions $u : V \\to \\mathbb{R}$ with $\\int \\limits_{V} u d\\mu = 0$, there \n\texists some constant $C$ depending only on $G$ such that $\\int \\limits_{V} u^2 d\\mu \\le C \\int \\limits_{V} |\\nabla u|^2 d\\mu$.\n\\end{Lemma} \n\n\\begin{Lemma}\\label{mt}\n\t{\\rm (\\cite[Lemma 7]{ALY})}\n\tLet $G=(V,E)$ be a finite graph. For any $\\beta\\in \\mathbb{R}$, there exists a constant $C$ depending only on $\\beta$ and $G$ such that for all functions $v$ with $\\int\\limits_{V} |\\nabla v|^{2} d \\mu\\le 1$  and $\\int\\limits_{V} v d \\mu =0$, there holds \n\t\\begin{equation}\n\t\t\\int\\limits_{V} e^{\\beta v^{2}} d\\mu \\le C.\n\t\\end{equation}\n\\end{Lemma} \n\n\\begin{Lemma}\\label{i}\n\t{\\rm (Interpolation inequality for $L^{r}$-norms on graphs) }. Suppose that $\\theta \\in (0,1) $, $0<\\theta r\\le s$, $0<(1-\\theta)r \\le t$ and $\\frac{1}{r}=\\frac{\\theta}{s}+\\frac{1-\\theta}{t}$. Then we have \n\t\\begin{equation}\n\t\t||u||_{L^{r}(V)}\\le ||u||^{\\theta}_{L^{s}(U)} ||u||^{1-\\theta}_{L^{t}(U)} .\n\t\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nBy H$\\ddot{\\text{o}}$lder inequality, we see that  \n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t\\int\\limits_{V}|u|^{r} d \\mu &=\\int\\limits_{V}|u|^{\\theta r}|u|^{(1-\\theta) r} d \\mu \\\\\n\t\t\t& \\leq\\left(\\int\\limits_{V}|u|^{\\theta r \\frac{s}{\\theta r}} d \\mu \\right)^{\\frac{\\theta r}{s}}\\left(\\int\\limits_{V}|u|^{(1-\\theta) r \\frac{t}{(1-\\theta) r}} d \\mu \\right)^{\\frac{(1-\\theta) r}{t}} .\n\t\t\\end{aligned}\n\t\\end{equation}\n\\end{proof}\n\nIn order to establish Theorem \\ref{31}, we need the following result due to Huang-Lin-Yau.\n\\begin{Theorem}\\label{ly}\n\t{\\rm (\\cite[Theorem 2.2]{HLY})}\n\tThere is a critical value $\\lambda_{c}$ depending on $G$ satisfying\t\n\t$$\\lambda_{c}\\ge \\frac{16\\pi M}{|V|},$$\n\tsuch that when $\\lambda>\\lambda_{c}$, the equation\n\t\\begin{equation}\\label{13a}\n\t\t\\Delta u=\\lambda e^{u}\\left(e^{u}-1\\right)+4 \\pi \\sum_{j=1}^{M} \\delta_{p_{j}},~x\\in G,\n\t\\end{equation}\n\thas a solution $u_{\\lambda}$ on $G$, and when $\\lambda<\\lambda_{c}$, the equation \\eqref{13a} has no solution, where $M>0$ is an integer.\n\\end{Theorem}\n\nIn fact, we may establish the following more accurate result.  \n\\begin{Proposition}\n\n\tThe solution $u_{\\lambda}$ obtained in Theorem \\ref{ly} is maximal in the sense that if $u$ is any other solution of \\eqref{1}, then \n\t\\begin{equation}\n\t\tu\\le u_{\\lambda}.\n\t\\end{equation} \n\\end{Proposition}\n\n\\begin{proof}\n\t Suppose that $u$ is any other solution of \\eqref{13a}. Then it is clear that $u$ is a subsolution and hence \n\t\\begin{equation}\\label{b}\n\t\tu\\le v_{n}\n\t\\end{equation} \n\tas in the proof of Lemma 4.2 in \\cite{HLY} for every $n\\in \\mathbb{N}$, where $v_{n}$ is defined by iterative scheme (4.5) in \\cite{HLY}. Letting $n\\to +\\infty$ in \\eqref{b}, we deduce that $u\\le u_{\\lambda}$ and hence that $u_{\\lambda}$ is a maximal solution of \\eqref{13a}. \n\t\n\tWe now complete the proof.\n\\end{proof}\n\nFurthermore, we have the following propositions.\n\n\\begin{Proposition}\n\tIf $\\lambda_{1}>\\lambda_{2}>\\lambda_{c}$, then $u_{\\lambda_{1}}\\ge u_{\\lambda_{2}}$.\n\\end{Proposition}\n\\begin{proof}\n\tBy Lemma 4.4 of \\cite{HLY}, we deduce that $u_{\\lambda_{2}}<0$, and hence that \n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t\\Delta u_{\\lambda_{2}}&=\\lambda_{2} e^{u_{\\lambda_{2}}}(e^{u_{\\lambda_{2}}}-1)+4\\pi\\sum_{j=1}^{M} \\delta_{p_{j}} \\\\\n\t\t\t& >\\lambda_{1} e^{u_{\\lambda_{2}}}(e^{u_{\\lambda_{2}}}-1) +4\\pi\\sum_{j=1}^{M} \\delta_{p_{j}}.\n\t\t\\end{aligned}\n\t\\end{equation}\nThus $u_{\\lambda_{2}}$ is a subsolution of \\eqref{13a} with $\\lambda=\\lambda_{1}$. By the sub-supersolution argument as in the proof of Lemma 4.2 in \\cite{HLY}, and the maximality of $u_{\\lambda_{1}}$, this implies that $u_{\\lambda_{2}} \\le u_{\\lambda_{1}}$.\n\\end{proof}\n\n\\begin{Proposition}\\label{f2}\n\n\t Let $u_{\\lambda}$ be the maximal solution of \\eqref{13a} for $\\lambda>\\lambda_{c}$. We have \n\t\\begin{equation}\n\t\tu_{\\lambda} \\to 0\\text{~as~}\\lambda\\to +\\infty\\text{~uniformly~on~}V.\n\t\\end{equation}\n\\end{Proposition}\n\\begin{proof}\nFix $\\lambda_{0}>\\lambda_{c}$, since $u_{\\lambda}$ is monotone increasing in $\\lambda$, by Lemma 4.4 of \\cite{HLY}, we deduce that $u_{\\lambda_{0}} \\le u_{\\lambda}\\le 0$ and\n\t\\begin{equation}\n\t\t\\int\\limits_{V} e^{u_{\\lambda}} (1-e^{ u_{\\lambda}}) d \\mu= \\frac{4\\pi M}{\\lambda},\n\t\\end{equation} \nfor $\\lambda\\ge \\lambda_{0}$.\n\tLet $\\bar{v}(x):=\\sup\\limits_{\\lambda>\\lambda_{0}} u_{\\lambda} (x) $, $x\\in V$. We deduce that $u_{\\lambda_{0}} \\le \\bar{v} \\le 0$ in $V$ and \n\t\\begin{equation}\n\t\t\\int\\limits_{V} e^{ \\bar{v}} (1-e^{ \\bar{v}}) d \\mu= 0.\n\t\\end{equation} \nIt follows that $\\bar{v}\\equiv 0$ on $V$.\n\nWe now complete the proof.\n\\end{proof}\n\n\n\\section{The constraints}\n\nFor the sake of convience, by applying the translation\n\\begin{equation}\\label{51}\n\tu_{i} \\to u_{i}+ \\text{ln} R_{i},~i=1,...,n,\n\\end{equation}\nin equations \\eqref{1}, we can conclude that\n\\begin{equation}\n\t\\Delta u_{i}=\\lambda\\left(\\sum_{j=1}^{n} \\sum_{k=1}^{n} \\tilde{K}_{j k} \\tilde{K}_{i j} \\mathrm{e}^{u_{j}} \\mathrm{e}^{u_{k}}-\\sum_{j=1}^{n} \\tilde{K}_{i j} \\mathrm{e}^{u_{j}}\\right)+4 \\pi \\sum_{j=1}^{N_{i}} \\delta_{p_{i j}}(x)\n\\end{equation}\nfor $i=1,...,n$.\nSet $u_{i}^{0}$ be the unique solution to \n\\begin{equation}\n\t\\Delta u_{i}^{0}=4 \\pi \\sum_{s=1}^{N_{i}} \\delta_{p_{i s}}-\\frac{4 \\pi N_{i}}{|V|}, \\quad \\int\\limits_{V} u_{i}^{0} \\mathrm{~d} x=0.\n\\end{equation}\nDenote $u_{i}=u_{i}^{0}+v_{i}$, $i=1,...,n$; then $v_{i} (i=1,...,n)$  satisfy\n\\begin{equation}\\label{59}\n\t\\Delta v_{i}=\\lambda\\left(\\sum_{j=1}^{n} \\sum_{k=1}^{n} \\tilde{K}_{j k} \\tilde{K}_{i j} \\mathrm{e}^{u_{j}^{0}+v_{j}} \\mathrm{e}^{u_{k}^{0}+v_{k}}-\\sum_{j=1}^{n} \\tilde{K}_{i j} \\mathrm{e}^{u_{j}^{0}+v_{j}}\\right)+\\frac{4 \\pi N_{i}}{|V|},\n\\end{equation}\nor\n\\begin{equation}\\label{510}\n\t\\Delta \\mathbf{v}=\\lambda \\tilde{K} U \\tilde{K}(\\mathbf{U}-\\mathbf{1})+\\frac{4 \\pi \\mathbf{N}}{|V|},\n\\end{equation}\nwhere  $\\mathbf{v}=(v_1,...,v_n)^{T},$ $\\mathbf{N}=(N_1,...,N_n)^{T}$, $U=diag\\{e^{u_{1}^{0}+v_{1}},...,e^{u_{n}^{0}+v_{n}}\\},$ $\\mathbf{U}=(e^{u_{1}^{0}+v_{1}},...,e^{u_{n}^{0}+v_{n}})^{T},$\n\\begin{equation}\n\t\\tilde{K}:=K^{T}R=PSR,~R:=diag\\{R_1,...,R_n \\}.\n\\end{equation}\n\nWe next establish a necessary condition for the existence of solutions of \\eqref{1}, and then the conclusion (i) of Theorem \\ref{31} follows.\n\n\\begin{Lemma}\n\tSuppose that \\eqref{1} admits a solution. Then \n\t\\begin{equation}\\label{4.7}\n\t\t\\lambda>\\lambda_{0} := \\frac{16 \\pi}{|V|} \\frac{\\sum \\limits_{i=1}^{n} \\sum\\limits_{j=1}^{n} P_{i}^{-1}\\left(K^{-1}\\right)_{j i} N_{j}}{\\sum\\limits_{i=1}^{n} \\sum\\limits_{j=1}^{n} P_{i}^{-1}\\left(K^{-1}\\right)_{j i}}.\n\t\\end{equation} \n\\end{Lemma}\n\\begin{proof}\nDenote \\begin{equation}\\label{57}\n\tA = P^{-1} S^{-1} P^{-1} \\quad \\text { and } \\quad Q = R S R.\n\\end{equation}\t\nIn view of $S$ is positive definite, we see that $A$ ang $Q$ are positive definite. By \\eqref{45}, we deduce that \\begin{equation}\n\t\\mathbf{b} \\equiv\\left(b_{1}, \\ldots, b_{n}\\right)^{T} := 4 \\pi A \\mathbf{N}=4 \\pi P^{-1} S^{-1} P^{-1} \\mathbf{N}>0.\n\\end{equation} \nFrom \\eqref{510}, we deduce that \n\\begin{equation}\n\t\\Delta A \\mathbf{v}=\\lambda U Q(\\mathbf{U}-\\mathbf{1})+\\frac{\\mathbf{b}}{|V|}.\n\\end{equation} \nIt follows that \\begin{equation}\\label{515}\n\t\\int\\limits_{V} \\mathrm{UQ}(\\mathbf{U}-\\mathbf{1}) \\mathrm{d} \\mu +\\frac{\\mathbf{b}}{\\lambda}=\\mathbf{0},\n\\end{equation}\nwhere $\\mathbf{1}:=(1,...,1)^{T}$.\nMultiplying both sides of \\eqref{515} by $\\mathbf{1}^{T}$, we deduce that \n\\begin{equation}\\label{516}\n\t\\int_{\\Omega} \\mathbf{U}^{T} Q(\\mathbf{U}-\\mathbf{1}) \\mathrm{d} \\mu+\\frac{\\mathbf{1}^{T} \\mathbf{b}}{\\lambda}=0.\n\\end{equation}\nFrom \\eqref{41} and \\eqref{46}, we conclude that \n\\begin{equation}\\label{518}\n\t\\left(K^{T}\\right)^{-1} \\mathbf{1}=S^{-1} P^{-1} \\mathbf{1}=R \\mathbf{1}.\n\\end{equation}\nCombining \\eqref{516} and \\eqref{518}, we deduce that\n\\begin{equation}\\label{517}\t\n\t\t\\int\\limits_{V}\\left(\\mathbf{U}-\\frac{\\mathbf{1}}{2}\\right)^{\\tau} Q\\left(\\mathbf{U}-\\frac{\\mathbf{1}}{2}\\right) \\mathrm{d} \\mu \n\t\t=\\frac{|V|}{4} \\mathbf{1}^{\\tau} P^{-1}\\left(K^{\\tau}\\right)^{-1} \\mathbf{1}-\\frac{4 \\pi \\mathbf{1}^{\\tau} P^{-1}\\left(K^{\\tau}\\right)^{-1} \\mathbf{N}}{\\lambda}.\n\\end{equation}\nRecall that $Q$ is positive define, it follows from \\eqref{517} that\n\\begin{equation}\n\t\\frac{|V|}{4} \\mathbf{1}^{\\tau} P^{-1}\\left(K^{\\tau}\\right)^{-1} \\mathbf{1}-\\frac{4 \\pi \\mathbf{1}^{\\tau} P^{-1}\\left(K^{\\tau}\\right)^{-1} \\mathbf{N}}{\\lambda}>0,\n\\end{equation}\nwhich implies that \\eqref{4.7} holds. \n\\end{proof}\n\n\\section{The proof of Theorem \\ref{31}}\nIn this section, we formulate a variational  solution of equations \\eqref{1} by using an equality type constraint. \nDefine the energy functional by \n\\begin{equation}\\label{19}\n\tI(\\mathbf{v})=\\frac{1}{2} \\sum_{j,k=1}^{n} \\int\\limits_{V}   b_{kj} \\Gamma(v_{k},v_{j})  d \\mu+\\frac{\\lambda}{2} \\int\\limits_{V}(\\mathbf{U}-\\mathbf{1})^{T} Q(\\mathbf{U}-\\mathbf{1}) \\mathrm{d} \\mu+\\int\\limits_{V} \\frac{\\mathbf{b}^{T} \\mathbf{v}}{|V|} \\mathrm{d} \\mu,\n\\end{equation}\nwhere $A=(b_{ij})_{n\\times n}$. Due to the fact that $Q$ and $A$ are symmetric, we know that if $\\mathbf{v}$ is a critical point to $I$, then it is a solution to \\eqref{1}.\n\n We could work on the standard space $H^{1}(V):=W^{1,2}(V)$. Denote \n\\begin{equation}\n\tH^{0}:=\\{v\\in W^{1,2}(V) | \\int\\limits_{V} v d \\mu=0 \\}.\n\\end{equation}\nClearly, for any $f\\in H$, there exists a unique $c\\in \\mathbb{R}$ and $f^{'}\\in H^{0}$ such that \n\\begin{equation}\\label{29}\n\tf=c+f^{'}\n\\end{equation}\nIn the sequal, we use $H^{1}(V)$ to denote the spaces of both scalar and vector-valued functions. \n\nSuppose that $\\mathbf{v}=\\mathbf{w}+\\mathbf{c} \\in H^{1}(V)$ given in \\eqref{29} satisfies \\eqref{515}, we deduce that \\begin{equation}\\label{61}\n\t\\operatorname{diag}\\left\\{\\mathrm{e}^{c_{1}}, \\ldots, \\mathrm{e}^{c_{n}}\\right\\} \\tilde{Q}\\left(\\begin{array}{c}\n\t\t\\mathrm{e}^{c_{1}} \\\\\n\t\t\\vdots \\\\\n\t\t\\mathrm{e}^{c_{n}}\n\t\\end{array}\\right)-P^{-1} R \\operatorname{diag}\\left\\{a_{1}, \\ldots, a_{n}\\right\\}\\left(\\begin{array}{c}\n\t\t\\mathrm{e}^{c_{1}} \\\\\n\t\t\\vdots \\\\\n\t\t\\mathrm{e}^{c_{n}}\n\t\\end{array}\\right)+\\frac{\\mathbf{b}}{\\lambda}=\\mathbf{0},\n\\end{equation} \nwhere \\begin{equation}\\label{64}\n\t\ta_{i}  := a_{i}\\left(w_{i}\\right) = \\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+w_{i}} \\mathrm{~d} \\mu ,\n\t\\end{equation}\n\t\t \\begin{equation}\\label{65}\n\t\ta_{i j}  := a_{i j}\\left(w_{i}, w_{j}\\right) = \\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+u_{j}^{0}+w_{i}+w_{j}} \\mathrm{~d} \\mu, \\quad i, j=1, \\ldots, n,\n\\end{equation}\n\\begin{equation}\\label{62}\n\t\\tilde{Q} := \\tilde{Q}(\\mathbf{w}) = R \\tilde{S} R,\n\\end{equation}\nand\n\\begin{equation}\\label{63}\n\t\\tilde{S} \\equiv\\left(\\begin{array}{cccccc}\n\t\ta_{11} \\alpha_{11}  & -\\alpha_{12} a_{12} & \\cdots & \\cdots & \\cdots & -\\alpha_{1 n} a_{1 n} \\\\\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t-\\alpha_{i 1} a_{i 1} & -\\alpha_{i 2} a_{i 2} & \\cdots & \\alpha_{i i} a_{i i} & \\cdots & -\\alpha_{i n} a_{i n} \\\\\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t-a_{n 1} \\alpha_{n 1}  & -\\alpha_{n 2} a_{n 2} & \\cdots & \\cdots & \\cdots & a_{n n} \\alpha_{n n} \n\t\\end{array}\\right).\n\\end{equation}\nBy \\eqref{62} and \\eqref{63}, we deduce that \n\\begin{equation}\n\t\\tilde{Q}~\\text{ is~positive~ positive~definite}. \n\\end{equation}\nWe now write \\eqref{61} as the component form:\n\\begin{equation}\\label{68}\n\t\\mathrm{e}^{2 c_{i}} R_{i}^{2} \\alpha_{i i} a_{i i}-\\mathrm{e}^{c_{i}}\\left(\\frac{R_{i} a_{i}}{P_{i}}+\\sum_{j \\neq i} \\mathrm{e}^{c_{j}} R_{i} R_{j} \\alpha_{i j} a_{i j}\\right)+\\frac{b_{i}}{\\lambda}=0, \\quad i=1, \\ldots, n.\n\\end{equation}\nOf course, \\eqref{59} is a quadratic equation in $t=e^{c}$ which admits a solution if and only if \n\\begin{equation}\\label{67}\n\t\\left(\\frac{R_{i} a_{i}}{P_{i}}+\\sum_{j \\neq i} \\mathrm{e}^{c_{j}} R_{i} R_{j} \\alpha_{i j} a_{i j}\\right)^{2} \\geq \\frac{4 R_{i}^{2} b_{i} \\alpha_{i i} a_{i i}}{\\lambda}, \\quad i=1, \\ldots, n.\n\\end{equation}\nIt is clear that \\eqref{68} follows from the following inequalities\n\\begin{equation}\\label{69}\n\t\\frac{a_{i}^{2}}{a_{i i}} \\geq \\frac{4 \\alpha_{i i} P_{i}^{2} b_{i}}{\\lambda}, \\quad i=1, \\ldots, n.\n\\end{equation}\nDenote \n\\begin{equation}\\label{A}\n\t\\mathscr{A} \\equiv\\left\\{\\mathbf{w} \\mid \\mathbf{w} \\in {H}^{0}(V) \\text { such that } \\eqref{69} \\text { holds }\\right\\}.\n\\end{equation}\nIn this case, we may select $\\mathbf{c}=\\mathbf{c}(\\mathbf{w}):=(c_1,...,c_n)$ in \\eqref{68} to satisfy \n\\begin{equation}\\label{611}\n\t\\begin{aligned}\n\t\t\\mathrm{e}^{c_{i}}=& \\frac{1}{2 R_{i}^{2} \\alpha_{i i} a_{i i}}\\left\\{\\left(\\frac{R_{i} a_{i}}{P_{i}}+\\sum_{j \\neq i} \\mathrm{e}^{c_{j}} R_{i} R_{j} \\alpha_{i j} a_{i j}\\right)\\right.\\\\\n\t\t&+\\sqrt{\\left.\\left(\\frac{R_{i} a_{i}}{P_{i}}+\\sum_{j \\neq i} \\mathrm{e}^{c_{j}} R_{i} R_{j} \\alpha_{i j} a_{i j}\\right)^{2}-\\frac{4 b_{i} R_{i}^{2} \\alpha_{i i} a_{i i}}{\\lambda}\\right\\}} \\\\\n\t\t& =: f_{i}\\left(\\mathrm{e}^{c_{1}}, \\ldots, \\mathrm{e}^{c_{n}}\\right), \\quad i=1, \\ldots, n .\n\t\\end{aligned}\n\\end{equation}\n\n\nTo proof Lemma \\ref{u8} and Lemma \\ref{f1}, we give a priori estimates.\n\n\\begin{Lemma}\n\tFor any $\\mathbf{w} \\in \\mathscr{A}$ and $\\epsilon \\in [0,1]$, if $\\mathbf{t}$ satisfies the following equations\n\t\\begin{equation}\\label{72}\n\t\t\\mathbf{F}(\\epsilon, \\mathbf{t}) \\equiv \\mathbf{t}-\\mathbf{f}(\\epsilon, \\mathbf{t})=\\mathbf{0}, \\quad \\mathbf{t} \\in \\mathbb{R}_{+}^{n}, \\quad \\epsilon \\in[0,1],\n\t\\end{equation}\nwhere \n\\begin{equation}\\label{73}\n\t\\mathbf{f}(\\epsilon, \\mathbf{t}) \\equiv\\left(f_{1}(\\epsilon, \\mathbf{t}), \\ldots, f_{n}(\\epsilon, \\mathbf{t})\\right)^{T},\n\\end{equation}\nand\n\\begin{equation}\\label{74}\n\t\\begin{aligned}\n\t\tf_{i}(\\epsilon, \\mathbf{t}) \\equiv & \\frac{1}{2 R_{i}^{2} \\alpha_{i i} a_{i i}}\\left\\{\\left(\\frac{R_{i} a_{i}}{P_{i}}+\\sum_{j \\neq i} t_{j} R_{i} R_{j} \\alpha_{i j} a_{i j}\\right)\\right.\\\\\n\t\t+&\\left.\\sqrt{\\left(\\frac{R_{i} a_{i}}{P_{i}}+\\sum_{j \\neq i} t_{j} R_{i} R_{j} \\alpha_{i j} a_{i j}\\right)-\\frac{4 \\epsilon b_{i} R_{i}^{2} \\alpha_{i i} a_{i i}}{\\lambda}}\\right\\} \\\\\n\t\ti &=1, \\ldots, n,\n\t\\end{aligned}\n\\end{equation}\nthen \n\\begin{equation}\\label{76}\n\t0<a_{i} t_{i} \\leq|V|, \\quad i=1, \\ldots, n\n\\end{equation}\nand\n\\begin{equation}\\label{77}\n\t0< t_{i} \\leq 1, \\quad i=1, \\ldots, n.\n\\end{equation}\n\\end{Lemma}\n\n\\begin{proof}\n\tFrom \\eqref{72}, \\eqref{73} and \\eqref{74}, we deduce that the left hand side of \\eqref{76} and \\eqref{77} holds.\n\t\n\tBy \\eqref{72}, we see that \n\t\\begin{equation}\n\t\tt_{i}=f_{i}(\\epsilon, \\mathbf{t}) \\leq \\frac{\\frac{R_{i} a_{i}}{P_{i}}+\\sum_{j \\neq i} t_{j} R_{i} R_{j} \\alpha_{i j} a_{i j}}{R_{i}^{2} \\alpha_{i i} a_{i i}}, \\quad i=1, \\ldots, n,\n\t\\end{equation}\nwhich implies that \n\\begin{equation}\\label{k1}\n\t\\tilde{Q} \\mathbf{t} \\leq P^{-1} R \\mathbf{a},\n\\end{equation}\nwhere \\begin{equation}\n\t\\mathbf{a} :=\\left(a_{1}, \\ldots, a_{n}\\right)^{T}.\n\\end{equation}\nFrom \\eqref{64}, \\eqref{65} and\n H$\\ddot{\\text{o}}$lder inequality, we conclude that \n \\begin{equation}\\label{713}\n \ta_{i j}^{2} \\leq a_{i i} a_{j j}, \\quad a_{i} \\leq|V|^{\\frac{1}{2}} a_{i i}^{\\frac{1}{2}}, \\quad i, j=1, \\ldots, n.\n \\end{equation}\n By $\\eqref{713}$ and the fact that $t_{i}>0,$ for $i=1,...,n$, we conclude that \n \\begin{equation}\\label{714}\n \t\\operatorname{diag}\\left\\{a_{11}^{\\frac{1}{2}}, \\ldots, a_{n n}^{\\frac{1}{2}}\\right\\} Q \\operatorname{diag}\\left\\{a_{11}^{\\frac{1}{2}}, \\ldots, a_{n n}^{\\frac{1}{2}}\\right\\} \\mathbf{t} \\leq \\tilde{Q} \\mathbf{t}.\n \\end{equation}\nFrom \\eqref{45} and \\eqref{57}, we conclude that \\begin{equation}\\label{715}\n\t(Q^{-1})_{ij}>0,~i,j=1,...,n.\n\\end{equation}\n Thus, combining \\eqref{713}, \\eqref{714} and \\eqref{715}, we conclude that \n \\begin{equation}\n \t\\begin{aligned}\\label{716}\n \t\t\\mathbf{t} & \\leq \\operatorname{diag}\\left\\{a_{11}^{-\\frac{1}{2}}, \\ldots, a_{n n}^{-\\frac{1}{2}}\\right\\} Q^{-1} \\operatorname{diag}\\left\\{a_{11}^{-\\frac{1}{2}}, \\ldots, a_{n n}^{-\\frac{1}{2}}\\right\\} P^{-1} R \\mathbf{a} \\\\\n \t\t&=\\operatorname{diag}\\left\\{a_{11}^{-\\frac{1}{2}}, \\ldots, a_{n n}^{-\\frac{1}{2}}\\right\\} Q^{-1} \\operatorname{diag}\\left\\{a_{1} a_{11}^{-\\frac{1}{2}}, \\ldots, a_{n} a_{n n}^{-\\frac{1}{2}}\\right\\} P^{-1} R \\mathbf{1} .\n \t\\end{aligned}\n \\end{equation}\n Therefore, by \\eqref{518}, \\eqref{713}, \\eqref{715} and \\eqref{716}, we deduce that \n \\begin{equation}\\label{717}\n \t\\begin{aligned}\n \t\t&\\operatorname{diag}\\left\\{a_{1}, \\ldots, a_{n}\\right\\} \\mathbf{t} \\\\\n \t\t&\\leq \\operatorname{diag}\\left\\{a_{1} a_{11}^{-\\frac{1}{2}}, \\ldots, a_{n} a_{n n}^{-\\frac{1}{2}}\\right\\} Q^{-1} \\operatorname{diag}\\left\\{a_{1} a_{11}^{-\\frac{1}{2}}, \\ldots, a_{n} a_{n n}^{-\\frac{1}{2}}\\right\\} P^{-1} R \\mathbf{1} \\\\\n \t\t&\\leq|V| Q^{-1} P^{-1} R \\mathbf{1} \\\\\n \t\t&=|V| \\mathbf{1}.\n \t\\end{aligned}\n \\end{equation}\n Thus, we get the right hand side of \\eqref{76} holds. It follows from Jensen's inequality that $$\\frac{a_i}{|V|}\\ge e^ { \\frac{\\int\\limits_{V} u_{1}^{0}+w_{i} d \\mu }{|V|} },~i=1,...,n.$$ Then the right hand sides of \\eqref{77} follows from this and \\eqref{717}.\n\t\n\tWe now complete the proof.\n\\end{proof}\n\nThe following result implies that we can solve constraints \\eqref{611}, so constraints \\eqref{68} could be solved.\n\\begin{Lemma}\\label{434}\n\tFor any $\\mathbf{w} \\in$ $ \\mathscr{A}$, the equations \n\t\\begin{equation}\\label{82}\n\t\t\\mathbf{F}(\\mathbf{t}) \\equiv \\mathbf{t}-\\mathbf{f}(\\mathbf{t})=\\mathbf{0}, \\quad \\mathbf{t} \\equiv\\left(t_{1}, \\ldots, t_{n}\\right)^{T} \\in \\mathbb{R}_{+}^{n},\n\t\\end{equation}\n\t admits a solution $\\mathbf{t} \\in (0,\\infty)^{n}$, where $\\mathbb{R}_{+}^{n} \\equiv\\left(\\mathbb{R}_{+}\\right)^{n}, \\mathbf{f}(\\mathbf{t}) \\equiv\\left(f_{1}(\\mathbf{t}), \\ldots, f_{n}(\\mathbf{t})\\right)^{T}$.\t\n\\end{Lemma}\n\\begin{proof}\n\tFor the sake of convience, we write \n\t\\begin{equation}\n\t\t\\left(\\alpha_{1}, \\ldots, \\alpha_{n}\\right)^{\\tau}<(\\leq)\\left(\\beta_{1}, \\ldots, \\beta_{n}\\right)^{\\tau} \\text { if } \\alpha_{i}<(\\leq) \\beta_{i}, i=1, \\ldots, n,\n\t\\end{equation}\nand we use the same notation for matrices. We next find a solution to \\eqref{82} with $\\epsilon=1$.\n\n By \\eqref{77}, we conclude that $\\mathbf{F}(\\epsilon, \\mathbf{t})$ has no zero on the boundary of $\\Omega$ for all $\\mathbf{w} \\in \\mathcal{A}$ and $\\epsilon \\in[0,1]$, where $\\Omega:=(0,r_0)^{n}$ and $r_0>1$ is a constant. Thus, we can define the Brouwer degree $\\operatorname{deg}(\\mathbf{F}(\\epsilon, \\mathbf{t}),\\Omega, \\mathbf{0}).$ Clearly, \n\\begin{equation}\\label{429}\n\t\\mathbf{F}(0, \\mathbf{t})=\\mathbf{0}\n\\end{equation}\nis equivalent to \n\\begin{equation}\\label{237}\n\tt_{i}-\\frac{\\frac{R_{i} a_{i}}{P_{i}}+\\sum\\limits_{j \\neq i} t_{j} R_{i} R_{j} \\alpha_{i j} a_{i j}}{R_{i}^{2} \\alpha_{i i} a_{i i}}=0, \\quad i=1, \\ldots, n.\n\\end{equation}\nWe write \\eqref{237} in its vector form\n\\begin{equation}\n\t\\tilde{Q} \\mathbf{t}=P^{-1} R \\mathbf{a}.\n\\end{equation}\nSince $\\tilde{Q}$ is invertible, we know that \\eqref{429} admits a unique solution \\begin{equation}\n\t\\mathbf{t}=\\tilde{Q}^{-1} P^{-1} R \\mathbf{a},\n\\end{equation}\nBy \\eqref{77},we see that it belong to the interior of $\\Omega$. By the fact that $\\tilde{Q}$ is positive definite, we deduce that the Jacobian of $\\mathbf{F}(0, \\mathbf{t})$ is positive everywhere, and hence that $\\operatorname{deg}(\\mathbf{F}(0, \\mathbf{t}),\\Omega, \\mathbf{0})=1$. It is easy to check that $\\mathbf{F}(\\epsilon, \\mathbf{t})$ is a smooth function for any $\\epsilon\\in [0,1]$. Thus by homotopy invariance, we deduce that \n\\begin{equation}\n\t\\operatorname{deg}(\\mathbf{F}(1, \\mathbf{t}), \\Omega, \\mathbf{0})=\\operatorname{deg}(\\mathbf{F}(0, \\mathbf{t}), \\Omega, \\mathbf{0}).\n\\end{equation}\n \n Now we complete the proof.\n\\end{proof}\n The following Lemma follows from Lemma \\ref{434} immediately. \n \\begin{Lemma}\\label{4}\n \tFor any $\\mathbf{w} \\in \\mathscr{A}$, \\eqref{68} admits a solution   $\\mathbf{c}(\\mathbf{w})=(c_{1}(\\mathbf{w}),...,c_{n}(\\mathbf{w}))^{T}$ which satisfies \\eqref{611}, so that $\\mathbf{v}=\\mathbf{w}+\\mathbf{c}(\\mathbf{w})=(w_1+c_{1}(\\mathbf{w}),...,w_n+c_{n}(\\mathbf{w}))^{T}$ satisfies \\eqref{515}.\n \\end{Lemma} \n\nDefine the constrained functional \n\\begin{equation}\\label{J}\n\tJ(\\mathbf{w}) := I(\\mathbf{w}+\\mathbf{c}(\\mathbf{w})), \\quad \\mathbf{w} \\in \\mathscr{A}.\n\\end{equation}\n  For all $\\mathbf{w} \\in \\mathscr{A}$, since $\\mathbf{v}=\\mathbf{w}+\\mathbf{c}(\\mathbf{w})$ satisfies \\eqref{515}, we conclude that \n  \\begin{equation}\n  \t\\begin{aligned}\n  \t\t\\int\\limits_{V}(\\mathbf{U}-\\mathbf{1})^{\\tau} Q(\\mathbf{U}-\\mathbf{1}) \\mathrm{d} \\mu &=\\int\\limits_{V} \\mathbf{1}^{T} Q(\\mathbf{1}-\\mathbf{U}) \\mathrm{d} \\mu-\\frac{\\mathbf{1}^{T} \\mathbf{b}}{\\lambda} \\\\\n  \t\t&=\\int\\limits_{V} \\mathbf{1}^{T} P^{-1} R(\\mathbf{1}-\\mathbf{U}) \\mathrm{d} \\mu-\\frac{\\mathbf{1}^{\\tau} \\mathbf{b}}{\\lambda}.\n  \t\\end{aligned}\n  \\end{equation}\nBy \\eqref{19}, we deduce that \n\\begin{equation}\n\t\\begin{aligned}\n\t\tJ(\\mathbf{w})=& \\frac{1}{2} \\sum_{j,k=1}^{n} \\int\\limits_{V}   b_{kj} \\Gamma(v_{k},v_{j})  d \\mu+\\frac{\\lambda}{2} \\mathbf{1}^{T} P^{-1} R \\int\\limits_{V}(\\mathbf{1}-\\mathbf{U}) \\mathrm{d} \\mu+\\mathbf{b}^{T} \\mathbf{c}-\\frac{\\mathbf{1}^{T} \\mathbf{b}}{2} \\\\\n\t\t=& \\frac{1}{2} \\sum_{j,k=1}^{n} \\int\\limits_{V}   b_{kj} \\Gamma(v_{k},v_{j})  d \\mu+\\frac{\\lambda}{2} \\sum_{i=1}^{n} \\frac{R_{i}}{P_{i}} \\int\\limits_{V}\\left(1-\\mathrm{e}^{c_{i}} \\mathrm{e}^{u_{i}^{0}+w_{i}}\\right) \\mathrm{d} \\mu \\\\\n\t\t&+\\sum_{i=1}^{n} b_{i} c_{i}-\\frac{1}{2} \\sum_{i=1}^{n} b_{i}.\n\t\\end{aligned}\n\\end{equation}\n\n\nTo prove Lemma \\ref{x4}, we need the following result.\n\n\\begin{Lemma}\\label{jian}\n\tSuppose that $\\mathbf{w} \\in \\mathscr{A}$ and $\\tau \\in(0,1)$. Then  \n\t\\begin{equation}\n\t\t\\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+w_{i}} \\mathrm{~d} \\mu \\leq\\left(\\frac{\\lambda}{4 P_{i}^{2} b_{i} \\alpha_{i i}}\\right)^{\\frac{1-\\tau}{\\tau}}\\left(\\int\\limits_{V} \\mathrm{e}^{\\tau u_{i}^{0}+\\tau w_{i}} \\mathrm{~d} \\mu \\right)^{\\frac{1}{\\tau}}, \\quad i=1, \\ldots, n\n\t\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nLet $a=\\frac{1}{2-\\tau}$. By \\eqref{69} and Lemma \\ref{i}, we conclude that \n\t\\begin{equation}\n\t\t\\begin{aligned}[]\n\t\t\t\t\\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+w_{i}} \\mathrm{~d} \\mu &\\le \\left[ \\int\\limits_{V} \\left( \\mathrm{e}^{u_{i}^{0}+w_{i}} \\right) ^{\\tau}\\mathrm{~d} \\mu\\right] ^{a} \\left[ \\int\\limits_{V} \\mathrm{e}^{2(u_{i}^{0}+w_{i})} \\mathrm{~d} \\mu\\right] ^{1-a} \\\\\n\t\t\t\t&\\le\\left[ \\int\\limits_{V} \\left( \\mathrm{e}^{u_{i}^{0}+w_{i}} \\right) ^{\\tau}\\mathrm{~d} \\mu\\right] ^{a}\n\t\t\t\t\\left[ \\frac{\\lambda}{4 P_{i}^{2} b_{i} \\alpha_{i i}}   \\left( \\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+w_{i}} d \\mu \\right)^{2}      \\right] ^{1-a},~i=1,...,n,\n\t\t\\end{aligned}\n\t\\end{equation}\nand hence that \n\n\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+w_{i}} \\mathrm{~d} \\mu &\\le \\left( \\frac{\\lambda}{4 P_{i}^{2} b_{i} \\alpha_{i i}} \\right) ^{\\frac{1-a}{2a-1}} \\left( \\int\\limits_{V} \\mathrm{e}^{\\tau(u_{i}^{0}+w_{i})} \\mathrm{~d} \\mu \\right) ^{\\frac{a}{2a-1}} \\\\\n\t&\t\\le \\left( \\frac{\\lambda}{4 P_{i}^{2} b_{i} \\alpha_{i i}} \\right) ^{\\frac{1-\\tau}{\\tau}} \\left( \\int\\limits_{V} \\mathrm{e}^{\\tau(u_{i}^{0}+w_{i})} \\mathrm{~d} \\mu \\right) ^{\\frac{1}{\\tau}} ,~i=1,...,n.\t\t\n\t\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\n\\begin{Lemma}\\label{x4}\n\tSuppose that $\\gamma$ is the smallest eigenvalues of $A$. Then \n\t\\begin{equation}\\label{x}\n\t\tJ(\\mathbf{w}) \\geq \\frac{\\gamma}{4} \\sum_{i=1}^{n} \\int\\limits_{V} \\Gamma(w_{i},w_{i}) d \\mu-C(\\ln \\lambda+1),\n\t\\end{equation}\nfor all $\\mathbf{w} \\in \\mathscr{A}$, where $C>0$ is a constant independent of $\\lambda$. \n\\end{Lemma}\n\\begin{proof}\n\tFrom \\eqref{57} and \\eqref{J}, we deduce that \n\t\\begin{equation}\\label{b7}\n\t\tJ(\\mathbf{w}) \\geq \\frac{\\gamma}{2} \\sum_{i=1}^{n}\\left\\|\\nabla w_{i}\\right\\|_{2}^{2}+\\sum_{i=1}^{n} b_{i} c_{i}.\n\t\\end{equation}\nFrom \\eqref{69} and  \\eqref{611}, we deduce that \n\\begin{equation}\n\t\\mathrm{e}^{c_{i}} \\geq \\frac{a_{i}}{2 R_{i} P_{i}  \\alpha_{i i} a_{i i}}\\ge\n\t\t\\frac{2 b_{i} P_{i} }{\\lambda R_{i} a_{i}}=\\frac{2 P_{i} b_{i}}{\\lambda R_{i} \\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+w_{i}} \\mathrm{~d} x}, \\quad i=1, \\ldots, n.\n\\end{equation}\nIt follows from that \n\\begin{equation}\\label{b11}\n\tc_{i} \\geq \\ln \\frac{2 P_{i} b_{i}}{R_{i}}- \\ln \\lambda \\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+w_{i}} \\mathrm{~d} x, \\quad i=1, \\ldots, n.\n\\end{equation}\nBy Cauchy inequlity with $\\epsilon$($\\epsilon$>0) and \\eqref{mt},\nwe deduce that \n\\begin{equation}\\label{g}\n\t\\begin{aligned}\n\t\t\\int\\limits_{V} e^{w} d\\mu &\\le \\int\\limits_{V} e^{\\frac{w}{||\\nabla w||^{2}_{2}} ||\\nabla w||^{2}_{2} } d\\mu\\\\\n\t\t&\\le \\int\\limits_{V} e^{\\frac{w}{ 4\\epsilon||\\nabla w||^{2}_{2}}} d\\mu e^{ {\\epsilon}||\\nabla w||_{2}^{2} }\\\\\n\t\t&=:C(\\epsilon,G) e^{ \\epsilon||\\nabla w||^{2}_{2} }.\n\t\\end{aligned}\n\\end{equation}\nIt is easy to check that \n\\begin{equation}\\label{446}\n\t\\begin{aligned}\n\t\t||\\nabla u^{0}_{i}||^{2}_{2} &= -\\int\\limits_{V} u_{i}^{0} \\Delta u_{i}^{0} d\\mu \\\\\n\t\t&= -\\int\\limits_{V} u^{0}_{i} 4\\pi \\sum_{s=1}^{N_{i}} \\delta_{p_{i s}} d \\mu\\\\\n\t\t&=- 4\\pi \\sum_{s=1}^{N_{i}} u_{i}^{0} (p_{i s }) \\\\\n\t\t&\\le 4\\pi N_{i} \\max_{V} |u_{i}^{0}| \\quad i=1, \\ldots, n .\n\t\\end{aligned}\n\\end{equation}\nBy Lemma \\ref{jian}, \\eqref{446} and \\eqref{g}, we conclude that  \n\\begin{equation}\\label{b1}\n\t\\begin{aligned}\n\t\t\\ln \\int\\limits_{V} \\mathrm{e}^{u_{i}^{0}+w_{i}} \\mathrm{~d} \\mu \\leq & \\frac{1-\\tau}{\\tau}\\left\\{\\ln \\lambda-\\ln \\left(4 P_{i}^{2} b_{i} \\alpha_{i i}\\right)\\right\\}+\\frac{1}{\\tau} \\ln \\int\\limits_{V} \\mathrm{e}^{\\tau u_{i}^{0}+s w_{i}} \\mathrm{~d} \\mu \\\\\n\t\t\\leq & 2\\epsilon \\tau\\left\\|\\nabla w_{i}\\right\\|_{2}^{2}+ 2\\epsilon\\tau 4\\pi N_{i} \\max_{V} |u_{i}^{0}|+ \\frac{1-\\tau}{\\tau}\\left\\{\\ln \\lambda-\\ln \\left(4 P_{i}^{2} b_{i} \\alpha_{i i}\\right)\\right\\} \\\\\n\t\t&+\\frac{\\ln C}{\\tau}, \\quad i=1, \\ldots, n .\n\t\\end{aligned}\n\\end{equation}\nIt follows from \\eqref{b1}, \\eqref{b11} and \\eqref{b7} that \n\\begin{equation}\\label{448}\n\t\\begin{aligned}\n\t\tJ(\\mathbf{w}) \\geq &\\left(\\frac{\\gamma}{2}- 2\\epsilon \\tau \\max _{1 \\leq i \\leq n}\\left\\{b_{i}\\right\\} \\right) \\sum_{i=1}^{n}\\left\\|\\nabla w_{i}\\right\\|_{2}^{2}-\\frac{1}{s} \\sum_{i=1}^{n} b_{i}\\left\\{\\ln \\lambda-\\ln \\left(4 P_{i}^{2} b_{i} \\alpha_{i i}\\right)+\\ln C\\right\\} \\\\\n\t\t&-\\sum_{i=1}^{n} b_{i}\\left\\{\\ln \\left(2 R_{i} P_{i} b_{i} \\alpha_{i i}\\right)+ 2\\epsilon\\tau 4\\pi N_{i} \\max_{V} |u_{i}^{0}| \\right\\}\n\t\\end{aligned}.\n\\end{equation}\nTaking $\\tau$ sufficiently small in \\eqref{448}, we get the desired conclusion \\eqref{x}.\n\\end{proof}\n\nBy Lemma \\ref{x4}, we can select a minimizing sequence $\\{ \\mathbf{w}^{k} \\}:=\\{(w_{1}^{(k)},...,w_{n}^{(k)}) \\}$ of the constrained minimization problem \n\\begin{equation}\n\t\\eta=\\inf\\{J(\\mathbf{w})| w\\in \\mathcal{A} \\}.\n\\end{equation}\nBy Lemma \\ref{x4} and Lemma \\ref{21}, we see that, there exists $$\\mathbf{w}^{0}:=\\{(w_{1}^{(0)},...,w_{n}^{(0)}) \\} \\in H^{0}(V)$$ such that, by passing to a subsequence, denoted still by $\\{\\mathbf{w}^{k} \\}$, \n$${w}_{i}^{(k)} \\to {w}_{i}^{(0)} $$ uniformly for $x\\in V$ as $k\\to +\\infty$ for $i=1,...,n$. By the fact that $$\\lim\\limits_{k\\to +\\infty} J(\\mathbf{w}^{k})=J(\\mathbf{w}^{0}),$$ we deduce that $J(\\mathbf{w}^{0})=\\eta$. Thus $\\mathbf{w}^{0}$ is a minimizer of $J$. Next, we prove that the\nminimizer belongs to the interior of $\\mathscr{A}$.\n\n\\begin{Lemma}\\label{u8}\n\tThere holds the following inequlities\n\t\\begin{equation}\\label{4d}\n\t\t\\inf _{\\mathbf{w} \\in \\partial \\mathscr{A}} J(\\mathbf{w}) \\geq \\frac{|V| \\lambda}{2} \\min _{1 \\leq i \\leq n}\\left\\{\\frac{R_{i}}{P_{i}}\\right\\}-C(1+\\ln \\lambda+\\sqrt{\\lambda}),\n\t\\end{equation}\nwher $C$ is constant independent of $\\lambda$.\n\\end{Lemma}\n\\begin{proof}\nFor any $w\\in \\partial \\mathscr{A}$,\tfrom \\eqref{A}, we know that at least one of the following equalities \n\\begin{equation}\\label{450}\n\t\\frac{a_{i}^{2}}{a_{i i}}=\\frac{4 \\alpha_{i i} P_{i}^{2} b_{i}}{\\lambda}, \\quad i=1, \\ldots, n\n\\end{equation}\nholds.\n\nSuppose the case $i=1$ happens, from \\eqref{713} and \\eqref{717}, we deduce that \n\\begin{equation}\\label{q}\n\t\\begin{aligned}\n\t\ta_{1} \\mathrm{e}^{c_{1}} & \\leq \\frac{R_{1}}{P_{1}}\\left(Q^{-1}\\right)_{11} a_{1}^{2} a_{11}^{-1}+\\sum_{j=2}^{n} \\frac{R_{j}}{P_{j}}\\left(Q^{-1}\\right)_{1 j} a_{1} a_{j} a_{11}^{-\\frac{1}{2}} a_{j j}^{-\\frac{1}{2}} \\\\\n\t\t& \\leq \\frac{R_{1}}{P_{1}}\\left(Q^{-1}\\right)_{11} a_{1}^{2} a_{11}^{-1}+|V|^{\\frac{1}{2}} \\sum_{j=2}^{n} \\frac{R_{j}}{P_{j}}\\left(Q^{-1}\\right)_{1 j} a_{1} a_{11}^{-\\frac{1}{2}}.\n\t\\end{aligned}\n\\end{equation}\nBy \\eqref{q} and \\eqref{450}, we deduce that \n\\begin{equation}\\label{115}\n\ta_{1} \\mathrm{e}^{c_{1}} \\le  \\left(Q^{-1}\\right)_{11} \\frac{4 P_{1} R_{1} b_{1} \\alpha_{11}}{\\lambda}+\\frac{2 P_{1} \\sum_{j=2}^{n} \\frac{R_{j}}{P_{j}}\\left(Q^{-1}\\right)_{1 j}}{\\sqrt{\\lambda}} \\sqrt{b_{1}|\\textsc{V}| \\alpha_{11}}.\n\\end{equation}\nIf other cases occur, we can establish the similar estimate.\nFrom \\eqref{76} and \\eqref{115}, we deduce that \n\\begin{equation}\\label{b15}\n\t\\begin{aligned}\n\t\t&\\frac{\\lambda}{2} \\sum_{i=1}^{n} \\frac{R_{i}}{P_{i}} \\int\\limits_{V}\\left(1-\\mathrm{e}^{c_{i}} \\mathrm{e}^{u_{i}^{0}+w_{i}}\\right) \\mathrm{d} \\mu \\\\\n\t\t&\\geq \\frac{|V| \\lambda R_{1}}{2 P_{1}}-2\\left(Q^{-1}\\right)_{11} P_{1} R_{1} b_{1} \\alpha_{11}-P_{1} \\sum_{j=2}^{n} \\frac{R_{j}}{P_{j}}\\left(Q^{-1}\\right)_{1 j} \\sqrt{b_{1} \\lambda|V| \\alpha_{11}}.\n\t\\end{aligned}\n\\end{equation}\nFrom \\eqref{b1}, \\eqref{b11} and \\eqref{b15}, we obtain the desired conclusion \\eqref{4d}.\n\\end{proof}\n\n\\begin{Lemma}\\label{f1}\n\tThere exists $\\mathbf{w}_{r_{\\varepsilon}}\\in int \\mathscr{A}$ such that\n\t\\begin{equation}\n\t\tJ\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)-\\inf _{\\mathbf{w} \\in \\partial \\mathscr{A}} J(\\mathbf{w})<-1.\n\t\\end{equation}\n\\end{Lemma}\n\\begin{proof}\n \tBy Proposition \\ref{f2}, we see that, for all sufficiently large $r>0$, the problem \n \t\\begin{equation}\n \t\t\\Delta v=r \\mathrm{e}^{u_{i}^{0}+v}\\left(\\mathrm{e}^{u_{i}^{0}+v}-1\\right)+\\frac{4 \\pi N_{i}}{|V|}, \\quad i=1, \\ldots, n,\n \t\\end{equation}\n has solutions $v_{i,r}(i=1,...,n)$ \tso that $v_{i,r} \\to -u_{i}^{0}$ as $r\\to +\\infty$ uniformly for $x\\in V$. Let $c_{i,r}:=\\frac{1}{|V|}\\int\\limits_{V} v_{i,r} d\\mu$. It follows that $w_{i,r}:=v_{i,r}-c_{i,r} \\to -u_{i}^{0}$ as $r\\to +\\infty$, $i=1,...,n.$ Thus we have \n \t\\begin{equation}\\label{456}\n \t\t\\lim _{\\mu \\rightarrow \\infty} a_{i}(w_{i,r})=|V|, \\quad \\lim _{\\mu \\rightarrow \\infty}a_{ij}( w_{i,r},w_{j,r} )=|V|, \\quad i, j=1, \\ldots, n.\n \t\\end{equation}\n \tBy \\eqref{62}, we obtain \n \t\\begin{equation}\\label{457}\n \t\t\\lim _{r \\rightarrow \\infty} \\tilde{Q}\\left(\\mathbf{w}_{r}\\right)=|V| Q .\n \t\\end{equation}\n By \\eqref{457} and \\eqref{456}, we can find $\\sigma>0$ such that for any $\\epsilon\\in (0,1)$, there exists $r_{\\epsilon}>0$ so that \n \\begin{equation}\n \t\\mathbf{w}_{r_{\\varepsilon}}=\\left(w_{1,r_{\\varepsilon}}, \\ldots, w_{n,r_{\\varepsilon}}\\right)^{T} \\in \\text { int } \\mathscr{A}\n \\end{equation}\t\n \tfor all $\\lambda>\\sigma$, and \\begin{equation}\\label{459}\n \t\t\\begin{aligned}\n \t\t\t&a_{i j}\\left(w_{i,r_{\\varepsilon}}, w_{j,r_{\\varepsilon}}\\right)<(1+\\varepsilon)|V|<2|V|, \\quad i, j=1, \\ldots, n \\\\\n \t\t\t&\\frac{(1-\\varepsilon)}{|V|} Q^{-1}<\\tilde{Q}^{-1}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)<\\frac{(1+\\varepsilon)}{|V|} Q^{-1}<\\frac{2}{|V|} Q^{-1}.\n \t\t\\end{aligned}\n \t\\end{equation}                Since $\\mathbf{w}_{r_{\\varepsilon}}\\in \\text { int } \\mathscr{A}$, by \\eqref{67} and the fact that\n $$\\sqrt{1-2x}\\ge 1-2x\\text{~for~}x\\in[0,\\frac{1}{2}] ,$$   we deduce that \\begin{equation}\n \t\\begin{aligned}\n \t\t&\\mathrm{e}^{c_{i}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)}=\\frac{\\frac{R_{i} a_{i}}{P_{i}}+\\sum\\limits_{j \\neq i} \\mathrm{e}^{c_{j}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)} R_{i} R_{j} \\alpha_{i j} a_{i j}}{2 R_{i}^{2} \\alpha_{i i} a_{i i}}\\\\\n \t\t&\\times\\left(1+\\sqrt{1-\\frac{4 b_{i} R_{i}^{2} \\alpha_{i i} a_{i i}}{\\lambda\\left(\\frac{R_{i} a_{i}}{P_{i}}+\\sum\\limits_{j \\neq i} \\mathrm{e}^{\\left.c_{j} R_{i} R_{j} \\alpha_{i j} a_{i j}\\right)^{2}}\\right.}}\\right)\\\\\n \t\t&\\geq \\frac{\\frac{R_{i} a_{i}}{P_{i}}+\\sum_{j \\neq i} \\mathrm{e}^{c_{j}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)} R_{i} R_{j} \\alpha_{i j} a_{i j}}{R_{i}^{2} \\alpha_{i i} a_{i i}}-\\frac{2 b_{i}}{\n \t\t\t\\lambda\\left(\\frac{R_{i} a_{i}}{P_{i}}+\\sum\\limits_{j \\neq i} \\mathrm{e}^{c_{j} R_{i} R_{j} \\alpha_{i j} a_{i j}}\n \t\t\t\\right)\n \t\t}.\n \t\\end{aligned}\n \\end{equation}  \n By \\eqref{k1} and Jensen inequlity, we conclude that \t\n \\begin{equation}\\label{w}\n \t\\mathrm{e}^{c_{i}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)}\n \t\\geq \\frac{\\frac{R_{i}|V|}{P_{i}}+\\sum\\limits_{j \\neq i} \\mathrm{e}^{c_{j}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)} R_{i} R_{j} \\alpha_{i j} a_{i j}}{R_{i}^{2} \\alpha_{i i} a_{i i}}-\\frac{2 P_{i} b_{i}}{\\lambda|V| R_{i}}, \\quad i=1, \\ldots, n.\n \\end{equation}\t\n From now on, we understand\n \\begin{equation}\n \ta_{i}=a_{i}\\left(w_{i,r_{\\varepsilon}} \\right), \\quad a_{i j}=a_{i j}\\left(w_{i,r_{\\varepsilon}}, w_{j,r_{\\varepsilon}}\\right), \\quad i, j=1, \\ldots, n.\n \\end{equation}\t\n\nThen by \\eqref{w} and \\eqref{459}, we deduce that \n\\begin{equation}\\label{eb}\n\t\\begin{aligned}\n\t\tR_{i}^{2} \\alpha_{i i} a_{i i} \\mathrm{e}^{c_{i}\\left(\\mathrm{w}_{r_{\\varepsilon}}\\right)}-\\sum_{j \\neq i} \\mathrm{e}^{c_{j}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)} R_{i} R_{j} \\alpha_{i j} a_{i j} & \\geq \\frac{R_{i}|V|}{P_{i}}-\\frac{2 P_{i} R_{i} b_{i}}{\\lambda|V|} \\alpha_{i i} a_{i i} \\\\\n\t\t& \\geq \\frac{R_{i}|V|}{P_{i}}-\\frac{4 \\alpha_{i i} P_{i} R_{i} b_{i}}{\\lambda}, \\quad i=1, \\ldots, n.\n\t\\end{aligned}\n\\end{equation}\nBy \\eqref{459} and \\eqref{eb}, we conclude that \n\\begin{equation}\\label{l}\n\t\\begin{aligned}\n\t\t&\\left(\\mathrm{e}^{c_{1}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)}, \\ldots, \\mathrm{e}^{c_{n}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)}\\right)^{T} \\\\\n\t\t&\\geq|\\Omega| \\tilde{Q}^{-1}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right) P^{-1} R \\mathbf{1}-\\frac{4 \\tilde{Q}^{-1}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right) P R}{\\lambda} \\operatorname{diag}\\left\\{\\alpha_{11}, \\ldots, \\alpha_{n n}\\right\\} \\mathbf{b} \\\\\n\t\t&\\geq(1-\\varepsilon) Q^{-1} P^{-1} R \\mathbf{1}-\\frac{8 Q^{-1} P R}{\\lambda|\\Omega|} \\operatorname{diag}\\left\\{\\alpha_{11}, \\ldots, \\alpha_{n n}\\right\\} \\mathbf{b} \\\\\n\t\t&=(1-\\varepsilon) \\mathbf{1}-\\frac{8 Q^{-1} P R}{\\lambda|\\Omega|} \\operatorname{diag}\\left\\{\\alpha_{11}, \\ldots, \\alpha_{n n}\\right\\} \\mathbf{b} .\n\t\\end{aligned}\n\\end{equation}\nIt follows that \n\\begin{equation}\n\t\\int\\limits_{V}\\left(1-\\mathrm{e}^{c_{i}\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)} \\mathrm{e}^{u_{i}^{0}+w_{i}^{\\mu_{\\varepsilon}}}\\right) \\mathrm{d} \\mu \\leq|V| \\varepsilon+\\frac{8}{\\lambda} \\sum_{i=1}^{n}\\left(Q^{-1}\\right)_{i j} P_{j} R_{j} b_{j} \\alpha_{j j}, \\quad i=1, \\ldots, n.\n\\end{equation}\nBy \\eqref{77}, there exists a constant $C_{\\epsilon}$ such that\n\t\\begin{equation}\n\t\tJ\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right) \\leq \\frac{|V| \\lambda \\varepsilon}{2} \\sum_{j=1}^{n} \\frac{R_{i}}{P_{i}}+C_{\\varepsilon}.\n\t\\end{equation}\nBy Lemma \\ref{u8}, this implies that there exists $C$ independent of $\\lambda$ so that\n\\begin{equation}\\label{f}\n\tJ\\left(\\mathbf{w}_{r_{\\varepsilon}}\\right)-\\inf _{\\mathbf{w} \\in \\partial \\mathscr{A}} J(\\mathbf{w}) \\leq \\frac{|V| \\lambda}{2}\\left(\\sum_{j=1}^{n} \\frac{R_{i}}{P_{i}} \\varepsilon-\\min _{1 \\leq i \\leq n}\\left\\{\\frac{R_{i}}{P_{i}}\\right\\}\\right)+C(\\sqrt{\\lambda}+\\ln \\lambda+1).\n\\end{equation}\nWe can get \\eqref{f1} by taking $\\epsilon$ suitably small and $\\lambda$ sufficiently large in \\eqref{f}.\n\nWe now prove the lemma.\n\\end{proof}\nFrom Lemmas \\ref{x4} and \\ref{f1}, there exists $\\lambda_{2}:= \\max\\{ \\sigma,\\lambda_{0} \\}$ such that for all $\\lambda>\\lambda_{2}$, we can find  $\\mathbf{w}_{0} \\in int \\mathscr{A}$ such that $\\mathbf{w}_{0}$ is a minimizer of $J$.   It is easy to check that $\\mathbf{v}_{0}:= \\mathbf{w}_{0}+ \\mathbf{c}({\\mathbf{w}_{0}})$ is a critical point of $I$, which implies that $\\mathbf{v}_{0}$ is a solution of equations \\eqref{510}. Thus, we could establish the conclusion (ii) of Theorem \\ref{31}.\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nSpontaneous chiral symmetry breaking (S$\\chi$SB) is fundamental to our\nunderstanding of low energy hadronic phenomena and it is thus\nimportant to demonstrate quantitatively that it is a consequence of\nQCD.  A natural candidate for such investigations is the numerical\nsimulation of QCD on a spacetime lattice. S$\\chi$SB, however, presents\nthe lattice approach with a twofold challenge.\n\nThe first is that spontaneous symmetry breaking does not occur in a\nfinite volume. In QCD, a possible signal of S$\\chi$SB is the presence\nof a non-vanishing quark condensate defined as:\n\\begin{equation}\n-\\Sigma\\equiv\\langle\\bar qq\\rangle =\\lim_{\\mbox{\\tiny $m\\to 0$}}\n\\lim_{\\mbox{\\tiny $V\\to\\infty$}} \\langle\\bar\nqq\\rangle_{m,V}\n\\ ,\n\\label{eq:conddef}\n\\end{equation}\nwhere $\\langle\\bar qq\\rangle_{m,V}$ is the condensate for finite volume $V$\nand mass $m$.  The double limit in \\eq{eq:conddef} is rather\nchallenging numerically!  To get around this problem, we resort to a\nfinite-size scaling analysis.  This involves studying the scaling of\nthe condensate with $V$ and $m$ as the limit of restoration of $\\chi$S\nis approached ($m\\to 0$, $V$ finite).\n\nSuch a study requires good control over the chiral properties of the\ntheory, which is the second challenge. Indeed, at finite lattice\nspacing, ``reasonable'' discretizations of fermions either break\ncontinuum $\\chi$S explicitly or lead to extraneous fermion species\n\\cite{Nielsen:1981hk}. To minimize this problem, we resort to recently\nrediscovered \\cite{Hasenfratz:1997ft,Neuberger:1998wv} Ginsparg-Wilson\n(GW) fermions \\cite{Ginsparg:1982bj} which break continuum $\\chi$S in\na very mild and controlled fashion and actually have a slightly\ngeneralized $\\chi$S even at finite lattice spacing\n\\cite{Luscher:1998pq}.\n\n\\vskip -9.5cm\n\\rightline{CERN-TH\/99-273}\n\\rightline{CPT-99\/PE.3886}\n\\vskip +9.1cm\n\n\n\n\\section{Light quarks on a torus}\n\nIn a large periodic box of volume $V=L^4$ such that $F_\\pi L\\gg 1$,\nfor small quark masses and assuming the standard pattern of S$\\chi$SB\nwith $N_f\\ge 2$, the QCD partition function is dominated by the\nnearly massless pions; the system can be described with the first\nfew terms of a chiral lagrangian \\cite{Gasser:1988zq}. If, in addition,\n$m\\to 0$~\\footnote{We assume here for simplicity that the $N_f$\nflavors all have mass $m$.} so that $M_\\pi\nL\\simeq\\frac{\\sqrt{2m\\Sigma}}{F_\\pi}L\\ll 1$, the global mode of\nthe chiral lagrangian field $U\\in SU(N_f)$ dominates the partition\nfunction, leading to a regime of restoration of\n$\\chi$S \\cite{Gasser:1987ah}.\n\nIn the quenched approximation to which we restrict here, topological\nzero modes of the Dirac operator induce $1\/m$ singularities in\n$\\langle\\bar qq\\rangle_{m,V}$ as $m\\to 0$. To subtract these contributions, we\nwork in sectors of fixed topological charge.  Generalizing the line of\nargument given above, the partition function\n$Z_\\nu$, in a sector of topological charge $\\nu$, was recently\nevaluated \\cite{Osborn:1998qb} for the quenched case\\footnote{The\n  original unquenched treatment is given in \\cite{Leutwyler:1992yt}.}.\nThe quark condensate in sector $\\nu$, proportional to the derivative\nof $\\ln {Z_\\nu}$ w.r.t.\\ $m$, is then $-\\Sigma_\\nu\\equiv\\langle\\bar\nqq\\rangle_{m,V,\\nu}$, such that \\cite{Osborn:1998qb}\n\\begin{equation}\n\\frac{\\Sigma_\\nu}{\\Sigma} = z \\; [ I_\\nu(z) K_\\nu(z) + I_{\\nu+1}(z)\nK_{\\nu-1}(z)]+\\frac{\\nu}{z},\n\\label{qpt}\n\\end{equation}\nwhere $z \\equiv m \\Sigma V$ and $I_\\nu(z)$, $K_\\nu(z)$ are the\nmodified Bessel functions. As advertised, there is a divergence $\\sim\n1\/m$ in sectors with topology. These terms, however, are independent\nof $\\Sigma$.\n\n\\eq{qpt} summarizes the scaling of the quark condensate with the\nvolume and quark mass in the global mode regime, as a function\nof only one non-perturbative parameter: $\\Sigma$. Thus, by fitting the\ndependence of the finite-volume condensate in quark mass and volume to\nMonte Carlo data, we can extract $\\Sigma$ in a perfectly controlled\nmanner.\n\n\n\\section{Ginsparg-Wilson fermions}\n\nTo perform the finite-volume scaling analysis outlined above, we need\nto be able to reach the chiral restoration regime without excessive\nfine-tuning and we need an index theorem to control the contribution\nof topological zero modes. Both these requirements are satisfied by\nGW fermions \\cite{Hasenfratz:1998jp}. In particular, \nthe leading cubic UV divergence of the condensate is known\nanalytically for GW fermions and can thus be subtracted {\\rm\nexactly}. The resulting subtracted condensate,\n$\\Sigma^{sub}_{\\nu}$, however, is still divergent:\n\\begin{eqnarray}\n\\Sigma^{sub}_{\\nu}(a) = C_2 \\frac{m}{a^2} + \\cdots +\n\\Sigma_{\\nu}, \\label{eq:sigsub}\n\\end{eqnarray}\nwhere $a$ is the lattice spacing.  The coefficients of the divergences\nare not known a priori and have to be determined, preferably\nnon-perturbatively. For the values of $m$ and $a$ considered below,\nhowever, only the quadratic divergence is important numerically,\nweaker divergences being suppressed by higher powers of $m$. A final\nmultiplicative renormalization is still required to eliminate a\nresidual logarithmic UV divergence in $\\Sigma_{\\nu}$.\n\nIn the present work, we use Neuberger's implementation of\nGW fermions encoded in the Dirac\noperator \\cite{Neuberger:1997fp,Neuberger:1998wv}:\n\\begin{equation}\naD_N=(1+s)\\left[\n1-A\/\\sqrt{A^{\\dagger}A}\\right]\n\\ ,\\label{eq:Dndef}\n\\end{equation}\nwith $A=1+s - aD_W$ where $D_W$ is the standard Wilson-Dirac operator. The\nparameter $s$ must satisfy $|s|<1$.\n\n\\section{Numerical results}\n\nWe work in the quenched approximation on hypercubic lattices with\nperiodic boundary conditions for gauge and fermion\nfields. We choose $\\beta=5.85$, which corresponds to $a^{-1}\\simeq\n1.5\\mbox{ GeV}$ \\cite{Edwards:1997xf}, and use standard methods to obtain\ndecorrelated gauge-field configurations.\n\nTo evaluate $1\/\\sqrt{A^\\dagger A}$ in \\eq{eq:Dndef}, we use a\nChebyshev approximation, $P_{n,\\epsilon}(A^\\dagger A)$, where\n$P_{n,\\epsilon}$ is a polynomial of degree $n$, which gives an\nexponentially converging approximation to $1\/\\sqrt{x}$ for $x\\in\n[\\epsilon,1]$ \\cite{Hernandez:1998et}. The cost of a multiplication by\n$D_N$ is linear in $n$ and becomes rapidly high. To reduce $n$\nsubstantially, we perform the improvements described in\n\\cite{Hernandez:1999cu}. We take $s=0.6$, a value at which\nNeuberger's operator is nearly optimally local for $\\beta=5.85$\n\\cite{Hernandez:1998et}.\n\nTo determine whether a gauge configuration belongs to the $\\nu=0$ or\n$\\pm 1$ sectors, we compute the few lowest eigenvalues of $D_N^\\dagger\nD_N$ by minimizing the relevant Ritz functional \\cite{ritz}. As pointed\nout in \\cite{Edwards:1998wx}, it is advantageous for this computation,\nas well as for the inversion of $(D_N^\\dagger+m)(D_N+m)$, to stay in a\ngiven chiral subspace. Having determined the topological charge of a\nconfiguration, we then obtain the condensate of \\eq{eq:sigsub} in\nthree volumes ($8^4$, $10^4$ and $12^4$) by computing\n\\begin{equation}\n\\Sigma^{sub}_\\nu= \\frac{1}{V}\\left\\langle\\mathrm{Tr}'\\left\\{\n\\frac{1}{D_N+m}+{\\mathrm h.c.}-\\frac{a}{1+s}\\right\\}\\right\\rangle_\\nu, \n\\label{eq:condtrace}\n\\end{equation}\nwhere the trace is taken in the chiral sector opposite to that\nwith the zero modes \\cite{Edwards:1998wx} and the gauge average is\nperformed in a sector of fixed topology $\\nu$. With this definition, terms\n$\\sim 1\/m$ in \\eq{qpt} are absent\\footnote{Though not shown\n  explicitly in \\protect\\eq{eq:condtrace}, we correctly account for\n  the real eigenvalues of $D_N$ at $2(1+s)\/a$.}. Three gaussian\nsources and a multimass solver \\cite{Jegerlehner:1997rn} were used\nto compute the trace in \\eq{eq:condtrace} for seven values of $m$.\n\nWe show in \\fig{fig:fss} our results for $a^3\\Sigma^{sub}_{\\nu=\\pm\n  1}\/am$ as a function of bare quark mass. We have 15, 10 and 7 gauge\nconfigurations on our $8^4$, $10^4$ and $12^4$ lattices,\nrespectively\\footnote{For the larger volumes, $\\nu=0$ configurations\n  are rare while the calculation of $\\Sigma^{sub}_{\\nu=0}$ requires\n  large statistics \\cite{Hernandez:1999cu}. $|\\nu|>1$ configurations,\n  on the other hand, are rare in the smaller volumes.}. The solid\nlines are a fit of the data to \\eqs{eq:sigsub}{qpt} for all volumes\nand masses.  This fit has only two parameters, namely $\\Sigma$ and the\ncoefficient of the quadratic divergence.  We find\n$a^3\\Sigma=0.0032(4)$ and $C_2=-0.914(8)$.\n\n\n\\begin{figure}[htb]\n\\epsfxsize=7cm\\epsffile[18 270 480 580]{fss_fig.ps} \\caption{\\small\\it\nMass dependence of the condensate for the $8^4$ (circles),\n$10^4$ (squares) and $12^4$ (triangles) lattices. The\ncurves result from a fit to \\protect\\eqs{eq:sigsub}{qpt}.\n\\vspace{-0.5cm}}\n\\label{fig:fss}\n\\end{figure}\n\n\nClearly, the formulae derived in quenched $\\chi$PT give a very good\ndescription of the numerical data. The value of $\\Sigma$ that we\nextract is, in physical units, $\\Sigma(\\mu\\sim\n1.5\\mbox{ GeV})=(221^{+8}_{-9}\\mbox{ MeV})^3$, up to a multiplicative\nrenormalization constant, which has not been computed yet for\nNeuberger's operator. The quoted error on $\\Sigma$ is purely\nstatistical and the statistics are rather small. Quenching and\ndiscretization errors, for instance, as well as possible contributions\nfrom higher orders in $\\chi$PT are not accounted for. Nevertheless,\nthe value obtained and the agreement with q$\\chi$PT support the\nstandard scenario of S$\\chi$SB.\n\nWe further consider the mean value of the lowest non-zero eigenvalue\nof $\\sqrt{D^\\dagger_N D_N}$ in different topological sectors.  In\nRandom Matrix Theory the distributions of these eigenvalues are given\nsolely in terms of $\\Sigma$ \\cite{lmin}. Our determination of $\\Sigma$\ntherefore yields predictions for the mean values. These can then be\ncompared to the average values obtained in simulation. With our $8^4$\nresults, we find agreement within roughly one standard deviation for\n$|\\nu|=1$ (29 configurations) and two for $\\nu=0$ (41 configurations)\n\\cite{Hernandez:1999cu}.\n\n\n{\\small\\it Note: Related work with Neuberger's operator can be found\n  in \\cite{Edwards:1998wx,Edwards:1999ra,Damgaard:1999tk}.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Network Architectures}\n\\label{app:arch}\n\n\\subsection{Autoencoders}\nThe policy and dynamics autoencoders are parameterised by Transformers using stacked self-attention and point-wise, fully connected layers for the encoder, and a fully connected feed-forward network for the decoder. \n\n\\textbf{Encoders:} The encoders consist of one layer composed of two sublayers, followed by another fully connected layer. The first sublayer, is a single-head self-attention mechanism, and the second is a simple fully connected feed-forward network. We employ a residual connection around each of the two sub-layers, followed by layer normalization and dropout. We use a dropout of 0.1 for all experiments. To facilitate these residual connections, all sublayers in the model, as well as the embedding layers, produce outputs of dimension $d_{model} = 64$. The second layer of the encoder projects the output of the first layer into the embedding space (from $d_{model}$ to $d_{emb}$). \n\nThe \\textbf{policy encoder} takes as input a set of state-action pairs $(s_t, a_t)$ from an full trajectory and outputs an embedding for the policy.  \n\nThe \\textbf{dynamics encoder} takes as input a set of state-action-next-state tuples $(s_t, a_t, s_{t+1})$ from a full trajectory and outputs an embedding for the dynamics.\n\nThe dimension of both the policy and dynamics embedding is $d_{emb} = 8$ for all environments, with the exception of swimmer which uses a dynamics embedding of dimension 2.\n\n\\textbf{Decoders:} The decoder is a simple fully connected feed-forward network with three layers and ReLU activations after the first two layers.\n\nThe \\textbf{policy decoder} takes as input the state of the environment and the policy embedding (outputted by the policy encoder) and outputs an action (i.e. the predicted action taken by the agent). \n\nThe \\textbf{environment decoder} takes as input the state of the environment, an action, the dynamics embedding (outputted by the dynamics encoder) and outputs a state (i.e. the predicted next state in the environment). \n\nThe dimensions of the states and actions depend on the given environment. \n\n\\subsection{The Policy-Dynamics Value Function}\n\\label{app:pdvf}\nThe Policy-Dynamics Value Function (PD-VF) takes as inputs the initial state of the environment, as well as a policy embedding and a dynamics embedding, and outputs a scalar representing the predicted expected return.\n\nPD-VF is parameterised by a fully connected feed-forward network. First, the environment state and dynamics embedding are concatanated and passed through a linear layer with output dimension 64 followed by a ReLU nonlinearity. The second layer also has output dimension 64 but is followed by a hyperbolic tangent nonlinearity. The output of the second layer is then passed through another linear layer with output dimension equal to the square of the policy embedding dimension which is 64 in this case. Then, the output of this is rearranged in the form of a lower triangular matrix $L$. This matrix is used to construct a Hermitian positive-definite matrix A using the Cholesky decomposition, $A = L L^T$. Finally, the value outputted by the network is obtained by computing $z_{\\pi}^T A z_{\\pi}$, where $z_{pi}$ is the policy embedding. \n\n\\subsection{Baselines}\n\\label{app:baselines}\nAll the pretrained PPO policies as well as all the baselines (except for CondPolicy as explained below) and the ablations use the same actor-critic network architecture. Both the actor and the critic are parameterised two-layer fully connected networks with hidden size 64 and hyperbolic tangent nonlinearities after each layer. Note that the weights are not shared by the two networks. The critic layer has another linear layer on top that outputs the estimated value. The actor network also has a linear layer on top that outputs a vector with the same number of dimensions as the action space. The actions are sampled from a Gaussian distribution with diagonal covariance matrix and means defined by the vector outputted by the actor network. The CondPolicy baseline has a similar architecture. The only difference is the first layer of both the actor and the critic, which has a larger input dimension due to the fact that these networks also take as input the policy embedding (along with the environment state). \n\n\n\\section{Training Details}\n\\label{app:training}\nFor experiments on Spaceship and Swimmer, we use only $N_d = 1$ steps to infer the dynamics embedding, while for Ant-wind we use $N_d = 2$ and for Ant-legs we use $N_d = 4$. Note that in all four domains, we only need a few steps to infer the environment dynamics, which allows us to quickly find a good policy for acting during the rest of the episode. Consequently, this results in good performance when evaluated on a single episode.  \n\nFirst, we have the \\textbf{reinforcement learning phase}, in which we pretrain 5 different initializations of PPO policies in each of the 20 environments in our distribution (both those used for training and those used for evaluation). We train all policies for $3e6$ environment interactions, which we have found to be enough for all of them to converge to a stable expected return. \n\nThen, in the \\textbf{self-supervised learning phase}, we use the policies pretrained on the training environments (75 policies for each domain) to generate trajectories through the training environments. For each policy-environment pair, we generate 200 trajectories, half of which are used for training the policy and dynamics autoencodesr and the rest are used for evaluation. We train the autoencoders on this data for a maximum of 200 epochs and we save the models with the lowest evaluation loss. Note that the autoencoders are never trained on trajectories generated in the evaluation environments or by policies pretrained on those environments, but only on data produced by interactions with the training environments. \n\nOnce we have the pretrained policy and dynamics autoencoders, we use them for learning the policy-dynamics value function in the \\textbf{supervised training phase}. To do this, we again generate 40 trajectories in the training environments (using only the policies pretrained on those environments). Half of these trajectories are used for training the PD-VF, while the rest are used for evaluation. In our experiments, we have found 20 trajectories from each policy-environment pair to be enough for training the model. For each trajectory, a policy embedding is obtained by passing the full trajectory through the policy encoder. Similarly, a corresponding dynamics embedding is obtained for each trajectory by passing the first few $N_d$ transitions of that trajectory through the dynamics encoder. The initial state and the return of that trajectory are also recorded. Now we have all the data needed for training the PD-VF with supervision. The PD-VF takes as inputs the initial state, the policy and dynamics embeddings and outputs a prediction for the expected return (corresponding to acting with that policy in the given environment). It is trained with $\\ell_2$ loss using the observed return. For the initial training stage of the PD-VF, we use 200 epochs, while for the second stage that include data aggregation for the value function and policy decoder, we use 100 epochs. The second stage is repeated a maximum of 20 times (each training for 100 epochs). We select the model that obtains the lowest loss on the evaluation data (out of all the models after each stage). We use this model for probing performace on the evaluation environments. \n\n\\section{Hyperparameters}\n\\label{app:hyper}\nFor training the PPO policies, as well as the baselines and ablations, we searched for the learning rate in [0.0001, 0.0003, 0.0005, 0.001] \nand found 0.0003 to work best across the boad. The entropy coefficient was set to 0.0, value loss coefficient 0.5, number of PPO epochs 10, number of PPO steps 2048, number of mini batches 32, gamma 0.99, and generealized advantage estimator coefficient 0.95. We also linearly decay the learning rate.\nThese values were not searched over since they have been previously optimized for MuJoCo domains and have been shown to be robust across these environments.\n\nFor MAML, we used the best hyperparameters found in the original paper for MuJoCo, so meta batch size of 20, 10 batches, and 8 workers.\n\nFor the dynamics autoencoder, we did a grid search over the learning rate in [0.0001, 0.001, 0.01] and found 0.001 to be best for the dynamics and 0.01 to be best for the policy. We also searched for the right batch size in [8, 32, 256, 2048] and found 8 to work best for the dynamics and 2048 for the policy. We also did grid searches over $d_{emb} \\in [2, 8, 32]$ and found $d_{emb} = 8$ for the policy autoencoders and $d_{emb} = 2$ for the dynamics autoencoders (except for Ant, in which we use $d_{emb} = 8$. We also searched for the hidden dimension of the transformers $d_{model} \\in [32, 64, 128]$ and found $d_{model} = 64$ to work best for both the policy and the dynamics embeddings.\n\nFor the value function, we tried different values for the number of epochs for the initial training phase $N_{ep, 1} \\in $ [1000, 500, 200, 100] and for the second training phase $N_{ep, 2} \\in $ [500, 200, 100] and we found 200 and 100 (i.e. each of the 20 data aggregation stages has 100 epochs) to work best, respectively. We also tried different learning rates from [0.0005, 0.001, 0.005, 0.01] and found 0.005 to be the best. Similarly, we tried batch sizes in [64, 128, 256] and found 128 to be the best. \n\nAll the results shown in this paper are obtained using the best hyperparameters found in our grid searches.\n\n\\section{Environments}\n\\label{app:envs}\n\n\\subsection{Spaceship Environment}\nThe source code contains the Spaceship domain that we designed, which is wrapped in a \\textit{gym} environment, so it can be easily integrated with any RL algorithm and used to evaluate agents.    \n\nThe task consists of moving a spaceship with a unit point charge from one end of a 2D room through a door at the other end. The action space consists of a fixed-magnitude force vector that is applied at each timestep. The room contains two fixed electric charges that deflect\/attract the ship as it moves through the environment. \n\nAt the beginning of each episode, the agent\\'s location is initialized at the center-bottom of the room with coordinates (2.5, 0.2). The target door is always located at the center-top of the room with coordinates (2.5, 5.0). The size of the room is 5, the size of the door is 1, and the temporal resolution is 0.3 (i.e. the time interval used to compute the next position of the spaceship given the current location and the applied force). The observation consists of the spaceship\\'s 2D location in the room (whose coordinates can be any real number between 0 and 5, the size of the room) and the action consists of the 2D force applied by the agent. The episode ends either when the agent hits a wall, exits the room through the door, or the agent has taken more than 50 steps in the environment. At the end of an episode, the agent receives reward that decreases exponentially with its distance to the target door. The decay factor is set to 3.0. For all other steps, the agent receives no reward. \nThe distribution of dynamics is a centered circle with radius 1.5.\n\n\\subsection{MuJoCo Environments}\nFor Swimmer we use a circle with radius 0.1 to sample the environment dynamics, while Ant-wind uses a radius of 4.0. For all three domains with continuous distribution of dynamics (i.e. Spaceship, Swimmer, and Ant-wind), we sample 15 environments for training and we hold out 5 for evaluation. The evaluation environments have dynamics covering a closed interval from the distribution thus testing the ability of the model to extrapolate (rather than intrapolate) to different dynamics. The Ant-wind domain has a total of 16 environmets, 4 of which are used for evaluation.\n\n\n\\section{Evaluation}\n\\label{app:eval}\nIn this section, we describe in detail the evaluation method and how the results reported here are obtained. For each trained model (i.e. PD-VF, a baseline or an ablation) and for each (unseen) test environment, we use that model to obtain a full trajectory through the given environment. This is repeated 10 times and the average return of the 10 runs is recorded. Then, we compute the mean and standard deviation (of this average return) across 5 different seeds for each model. These are the statistics shown in Figures 4 and 5.\n\nTo generate the t-SNE plots, we generated 10 trajectories for each policy-environment pair, including both the training and the evaluation ones. The encoders are used to obtain policy and dynamics embeddings corresponding to each trajectory. Then, t-Distributed Stochastic Neighbor Embedding (t-SNE) with perplexity 30 is applied to produce Figures 6 and 7. Figure 6 shows the t-SNE for the dynamics embedding, where each point is colored by the environment in which the corresponding trajectory (used to obtain that dynamics embedding) was collected. Conversely, Figure 7 shows the t-SNE for the policy embedding, where each point is colored by the policy which generated the corresponding trajectory (used to obtain that policy embedding). \n\n\n\\begin{figure*}[h!]\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/ablations_train_swim_8.png}}\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/ablations_train_space_8.png}}\n       \n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/ablations_train_ant_8.png}}\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/ablations_train_ant_legs_8.png}}\n    \\caption{\\textbf{Train Performance.} Average return on train environments in Swimmer (top-left), Spaceship (top-right), Ant-wind (bottom-left), and Ant-legs (bottom-right) obtained by PD-VF{} and a few ablations, namely NoDaggerPolicy, NoDaggerValue, Kmeans, and NN. PD-VF{} is comparable with or outperforms the ablations on the train environments. While some of these ablations perform reasonably well on the environments they are trained on, they generalize poorly to unseen dynamics.}\n    \\label{fig:train_ablations}\n\\end{figure*}\n\n\n\\section{Analysis of Learned Embeddings}\n\\label{app:analysis}\n\nFigure~\\ref{fig:policy_embeddings_all} shows a t-SNE plot of the learned policy embeddings for Spaceship, Swimmer, and Ant (from left to right). The top and bottom rows color the embeddings by the policy and environment that generated the corresponding trajectory, respectively. Trajectories produced by the same policy have similar embeddings, while those generated in the same environment are not necessarily close in this embedding space. This shows that the policy embedding preserves information about the policy while disregarding elements of the environment (that generated the corresponding embedded trajectory).\n\n\\begin{figure}[h!]\n        \\subfigure{\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_space_env_env.pdf}}\n        \\subfigure{\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_swim_env_env.pdf}}\n        \\subfigure{\\includegraphics[width=0.37\\columnwidth]{fig\/tsne_ant_env_env.pdf}}\n        \n        \\subfigure{\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_space_env_pi.pdf}}\n        \\subfigure{\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_swim_env_pi.pdf}}\n        \\subfigure{\\includegraphics[width=0.34\\columnwidth]{fig\/tsne_ant_env_pi.pdf}}\n    \\caption{t-SNE plots of the learned environment embeddings $z_{d}$ for Spaceship, Swimmer, and Ant-wind (from left to right). The points are colored by the \\textit{environment} (top) and \\textit{policy} (bottom) used to generate the trajectory of the corresponding dynamics embedding.}\n    \\label{fig:env_embeddings_all}\n\\end{figure}\n\n\nSimilarly, Figure~\\ref{fig:env_embeddings_all} shows a t-SNE plot of the learned dynamics embeddings on the three continuous control domains used for evaluating our method. The top row colors each point by the corresponding environment used to generate the trajectory (from which the embedding is inferred), while the bottom row colors each point by the corresponding policy. One can see that the embedding space retrieves the true dynamics distribution and preserves the smoothness of the 1D manifold.\n\n\\begin{figure}[ht!]\n        \\subfigure{\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_space_pi_pi.pdf}}\n        \\subfigure{\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_swim_pi_pi.pdf}}\n        \\subfigure{\\includegraphics[width=0.34\\columnwidth]{fig\/tsne_ant_pi_pi.pdf}}\n        \n        \\subfigure{\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_space_pi_env.pdf}}\n        \\subfigure{\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_swim_pi_env.pdf}}\n        \\subfigure{\\includegraphics[width=0.37\\columnwidth]{fig\/tsne_ant_pi_env.pdf}}\n    \\caption{t-SNE plots of the learned policy embeddings $z_{\\pi}$ for Spaceship, Swimmer, and Ant-wind (from left to right). The points are colored by the \\textit{policy} (top) and \\textit{environment} (bottom) used to generate the trajectory of the corresponding policy embedding.}\n    \\label{fig:policy_embeddings_all}\n\\end{figure}\n\n\n\nImportantly, this analysis shows that the learned policy and dynamics embeddings are generally disentangled (i.e. information about the dynamics is not contained in the policy space and vice versa). This is important as we want the dynamics space to mostly capture information about the transition function and similarly, we want the policy space to capture variation in the agent behavior. The only exception is the dynamics space of Ant-wind, which contains information about both the environment and the policy. This is because in this environment, the policy is dominated by the force applied to the body of the ant, whose goal is to move forward (while incurring a penalty proportional to the applied force). Thus, depending on the wind direction in the training environment, the agent learns to apply a force of a certain magnitude, a characteristic captured in the embedding space. When evaluated on environments with different dynamics, that policy will still apply a similar force. Our experiments indicate that even if the dynamics space is not fully disentangled (yet it contains information about the environment), the PD-VF is still able to make effective use of the embeddings to find good policies for unseen environments and even outperform other state-of-the-art RL methods. \n\n\\section{The Challenge of Transfer}\n\\label{app:transfer}\nIn this section, we emphasize the fact that the family of environments we designed pose a significant challenge to current state-of-the-art RL methods. To do this, we train PPO policies on each of the environments in our set (until convergence) and evaluate them on all other environments. The results show that any of the policies trained in this way can drastically fail in other environments (with different dynamics) from our training and test sets. This demonstrates that our set of environments provides a wide range of dynamics and that a single policy trained in any of these environments does not generalize well to the other ones. Moreover, when evaluated on a single environments, the performance across the pretrained policies varies greatly, illustrating the diversity of collected behaviors (both optimal and suboptimal). This analysis further supports the need for learning about multiple policies (and their performance in various environments) in order to generalize across widely different scenarios (or dynamics in this case). \n\n\n\\begin{table*}[t!]\n    \\centering\n    \\small\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\n    \\toprule\n     & E1 & E2 & E3 & E4 & E5 & E6 & E7 & E8 & E9 & E10 \\\\ [0.5ex]\n    \\midrule\n    P1  & 598 & 354 & 128 & 372 & 291 & 95.9 & -51.8 & -50.9 & -246 & -299 \\\\ \n    P2  & 512 & 461 & 503 & 349 & 228 & 135 & -1.44 & -46.8 & -28.1 & -271  \\\\ \n    P3  & 689 & 620 & 593 & 500 & 334 & 80 & 0.4 & -152 & -51 & -258 \\\\ \n    P4  & 654 & 665 & 557 & 519 & 29.9 & 180 & 177 & -9.41 & -90.5 & -218.5  \\\\ \n    P5  & 935 & 962 & 947 & 930 & 853 & 648 & 429 & 287 & 155 & -52.9 \\\\ \n    P6  & 811 & 838 & 794 & 778 & 710 & 600 & 386 & 247 & 123 & -5.14 \\\\ \n    P7  & 624 & 659 & 408 & 451 & 351 & 474 & 394 & 82.9 & 151 & 55.9 \\\\ \n    P8  & 500 & 470 & 442 & 393 & 321 & 468 & 410 & 315 & 209 & 124 \\\\ \n    P9  & 303 & 326 & 297 & 295 & 265 & 254 & 243 & 238 & 11.1 & 223 \\\\ \n  P10  & 293 & 54.8 & 294 & 293 & 250 & 226 & 200 & 200 & 180 & -1.28 \\\\ \n  P11  & 473 & 236 & 212 & 218 & 243 & 144 & 83.9 & 132 & 107 & 136 \\\\ \n  P12  & 266 & 264 & 268 & 242 & 214 & 181 & 55.6 & 72.7 & 239.4 & 240 \\\\ \n  P13  & 422 & 669 & 612 & 527 & 401 & 270 & 205 & 128 & 68.5 & 103 \\\\ \n  P14  & 436 & 362 & 424 & 366 & 259 & 296 & 97.3 & 55.7 & 24.9 & -2.44 \\\\ \n  P15  & 420 & 484 & 264 & 125 & 131 & 66.7 & 44.5 & 13.1 & 5.43 & 35.1 \\\\ \n  P16  & 671 & 769 & 573 & 270 & 189 & 212 & 153 & 96.4 & 19.7 & 0.290 \\\\ \n  P17  & 784 & 793 & 683 & 600 & 56.9 & 200 & 56.8 & 12.4 & 4.4 & 19.8  \\\\ \n  P18  & 755 & 703 & 564 & 213 & 170 & 129 & 58.1 & 2.17 & -103 & -43.7  \\\\ \n  P19  & 182 & 593 & 415 & 65.5 & 250 & 112 & 25.8 & 37.1 & -94.9 & -19.2 \\\\ \n  P20  & 297 & 589 & 518 & 350 & 134 & 76.3 & 6.55 & -51.5 & -185.4 & -11.2 \\\\ \n    \\bottomrule\n    \\end{tabular}\n    \\caption{\\textbf{Performance of PPO policies on the Ant-wind domain.} A row shows the mean episode return of a single policy on all environments, while a column shows the mean episode return of all policies on a single environment. This table contains performance on the first 10 environments.}\n     \\label{tab:ppo}\n\\end{table*}\n\n\\begin{table*}[t!]\n    \\centering\n    \\small\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\n    \\toprule\n     & E11 & E12 & E13 & E14 & E15 & E16 & E17 & E18 & E19 & E20 \\\\ [0.5ex]\n    \\midrule\n     P1  & -336 & -136 & -55.2 & -7.47 & -8.11 & 168 & 262 & 473 & 498 & 603 \\\\ \n     P2  & -42.2 & -101 & -28.3 & -112 & 14.2 & 132 & 109 & 345 & 510 & 545 \\\\ \n     P3  & -54.1 & -10.1 & -232 & -6.25 & 59.9 & 120 & 207 & 136 & 563 & 372 \\\\ \n     P4  & -279 & -278 & -135 & -4.81 & 5.49 & 113 & 276 & 150 & 484 & 196 \\\\ \n     P5  & -264 & -236 & -69.8 & 14.0 & 77.6 & 248 & 457 & 602 & 813 & 634 \\\\ \n     P6  & 20.9 & -112 & 98.7 & 147 & 14.9 & 194 & 321 & 524 & 611 & 639 \\\\ \n     P7  & 0.700 & -5.74 & -61.6 & 9.51 & 89.0 & 267 & 212 & 412 & 579 & 536 \\\\ \n     P8  & 39.6 & -33.5 & 11.7 & 53.9 & 21.5 & 206 & 218 & 29.9 & 469 & 509 \\\\ \n     P9  & 7.05 & 216 & 225 & 224 & 248 & 253 & 284 & 266 & 314 & 281 \\\\ \n    P10  & 176 & -159 & 5.98 & 5.03 & 57.8 & 20.6 & 77.0 & 287 & 249 & 539 \\\\ \n    P11  & 92.8 & 148 & 206 & 244 & 277 & 351 & 280 & 485 & 556 & 582 \\\\ \n    P12  & 235 & 235 & 232 & 122 & 139 & 151 & 165 & 299 & 272 & 265 \\\\ \n    P13  & 84.6 & 171 & 230 & 285 & 312 & 300 & 328 & 523 & 640 & 504 \\\\ \n    P14  & 45.2 & 159 & 126 & 302 & 369 & 363 & 537 & 457 & 512 & 396 \\\\ \n    P15  & -4.98 & -70.8 & 41.3 & 360 & 571 & 654 & 687 & 546 & 393 & 537  \\\\ \n    P16  & -7.77 & 19.7 & 63.4 & 248 & 457 & 600 & 740 & 493 & 731 & 804  \\\\ \n    P17  & -20.0 & -102 & -8.87 & 69.8 & 254 & 444 & 515 & 658 & 749 & 703 \\\\ \n    P18  & -93.7 & -59.9 & -11.6 & 125 & 275 & 434 & 577 & 679 & 781 & 781 \\\\ \n    P19  & -10.1 & -16.9 & 1.38 & 18.6 & 86.5 & 126 & 372 & 508 & 489 & 652 \\\\ \n    P20  & -118 & -62.4 & -125 & -81.7 & 20.1 & 82.9 & 131 & 195 & 341 & 392 \\\\ \n    \\bottomrule\n    \\end{tabular}\n    \\caption{\\textbf{Performance of PPO policies on the Ant-wind domain.} A row shows the mean episode return of a single policy on all environments, while a column shows the mean episode return of all policies on a single environment. This table contains performance on the last 10 environments.}\n     \\label{tab:ppo}\n\\end{table*}\n\n\n\n\\begin{table*}[t!]\n    \\centering\n    \\small\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\n    \\toprule\n     & E1 & E2 & E3 & E4 & E5 & E6 & E7 & E8 & E9 & E10 \\\\ [0.5ex]\n    \\midrule\n    P1 & 139.17 & 133.61 & 119.29 & 96.78 & 69.8 & 38.3 & 7.88 & -19.59 & -41.18 & -54.88 \\\\\n    P2 & 197.01 & 191.23 & 176.68 & 154.75 & 125.97 & 95.2 & 64.06 & 36.33 & 13.79 & -1.1 \\\\\n    P3 & 140.38 & 135.81 & 122.1 & 100.15 & 72.56 & 41.97 & 11.69 & -15.47 & -37.79 & -51.59 \\\\ \n    P4 & 139.01 & 134.82 & 121.15 & 98.86 & 71.07 & 41.13 & 10.26 & -17.24 & -39.25 & -54.68 \\\\\n    P5 & 215.38 & 209.66 & 196.33 & 174.77 & 147.62 & 115.54 & 84.68 & 59.61 & 36.2 & 22.31 \\\\\n    P6 & 207.08 & 201.55 & 186.96 & 165.49 & 138.86 & 107.64 & 77.12 & 49.94 & 28.6 & 14.55 \\\\\n    P7 & 210.29 & 205.73 & 191.61 & 169.42 & 141.29 & 112.15 & 81.69 & 53.65 & 32.87 & 18.08 \\\\\n    P8 & 213.98 & 209.7 & 198.61 & 179.11 & 152.56 & 124.49 & 95.18 & 67.78 & 45.47 & 32.82 \\\\\n    P9 & 206.51 & 201.71 & 190.56 & 170.55 & 142.16 & 114.04 & 84.64 & 56.6 & 33.8 & 19.84 \\\\\n    P10 & 204.85 & 201.04 & 186.64 & 168.29 & 141.0 & 113.45 & 80.69 & 53.39 & 33.46 & 19.9 \\\\\n    P11 & 197.34 & 193.67 & 182.94 & 164.05 & 137.93 & 109.44 & 80.39 & 52.76 & 32.46 & 18.77 \\\\\n    P12 & 204.92 & 201.11 & 186.85 & 166.49 & 138.03 & 108.22 & 79.32 & 50.95 & 29.97 & 16.23 \\\\\n    P13 & 202.86 & 204.16 & 174.7 & 174.07 & 139.94 & 130.12 & 88.65 & 45.72 & 24.26 & 15.94 \\\\\n    P14 & 205.89 & 201.64 & 187.07 & 166.63 & 138.54 & 108.87 & 77.48 & 51.66 & 29.76 & 15.68 \\\\\n    P15 & 209.19 & 186.31 & 190.71 & 168.87 & 142.23 & 114.88 & 82.87 & 56.37 & 35.82 & 20.97 \\\\\n    P16 & 214.29 & 204.26 & 188.0 & 165.51 & 140.3 & 109.66 & 80.35 & 56.25 & 37.05 & 23.41 \\\\\n    P17 & 202.78 & 197.79 & 183.82 & 160.93 & 133.64 & 103.02 & 71.89 & 44.94 & 22.01 & 9.32 \\\\\n    P18 & 202.66 & 204.15 & 190.16 & 167.73 & 139.81 & 109.03 & 77.82 & 51.18 & 29.4 & 14.83 \\\\\n    P19 & 208.89 & 204.35 & 191.05 & 168.31 & 139.9 & 109.16 & 79.06 & 51.99 & 28.93 & 14.33 \\\\\n    P20 & 141.37 & 136.28 & 122.22 & 100.69 & 73.13 & 42.51 & 12.4 & -15.19 & -37.6 & -51.74 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{\\textbf{Performance of PPO policies on the Swimmer domain.} A row shows the mean episode return of a single policy on all environments, while a column shows the mean episode return of all policies on a single environment. This table contains performance on the first 10 environments.}\n     \\label{tab:ppo}\n\\end{table*}\n\n\n\\begin{table*}[t!]\n    \\centering\n    \\small\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\n    \\toprule\n     & E11 & E12 & E13 & E14 & E15 & E16 & E17 & E18 & E19 & E20 \\\\ [0.5ex]\n    \\midrule\n     P1 & -59.39 & -54.24 & -40.67 & -19.18 & 9.2 & 39.2 & 69.81 & 97.3 & 119.73 & 134.49 \\\\\n    P2 & -6.11 & -0.47 & 13.55 & 36.02 & 65.06 & 96.16 & 126.34 & 154.61 & 177.55 & 191.34 \\\\\n    P3 & -57.27 & -52.51 & -39.14 & -16.39 & 10.56 & 40.38 & 71.77 & 98.78 & 121.09 & 135.31 \\\\\n    P4 & -58.16 & -53.38 & -39.35 & -18.19 & 9.62 & 40.21 & 70.66 & 98.47 & 120.61 & 134.47 \\\\\n    P5 & 16.13 & 20.32 & 32.89 & 56.59 & 85.07 & 114.87 & 144.05 & 173.91 & 196.76 & 210.27 \\\\\n    P6 & 9.8 & 13.8 & 27.56 & 49.58 & 76.67 & 107.47 & 137.03 & 164.94 & 188.26 & 201.71 \\\\\n    P7  & 14.63 & 19.66 & 34.54 & 56.04 & 82.39 & 114.16 & 143.69 & 171.16 & 190.65 & 205.65 \\\\\n    P8 & 25.48 & 33.05 & 45.05 & 68.32 & 94.65 & 124.24 & 153.37 & 179.63 & 199.71 & 210.72 \\\\\n    P9 & 15.39 & 19.33 & 31.88 & 54.41 & 79.43 & 110.51 & 139.44 & 165.05 & 186.49 & 200.29 \\\\\n    P10 & 12.91 & 16.72 & 32.59 & 52.4 & 79.08 & 107.42 & 138.15 & 161.44 & 184.87 & 199.23 \\\\\n    P11 & 12.84 & 17.28 & 30.41 & 50.77 & 76.78 & 104.17 & 132.33 & 156.83 & 178.78 & 191.66 \\\\\n    P12 & 11.56 & 16.06 & 29.37 & 51.48 & 78.29 & 108.94 & 136.79 & 164.79 & 187.09 & 201.86 \\\\\n    P13 & 3.07 & -8.63 & 30.02 & 59.98 & 72.13 & 110.56 & 128.87 & 148.73 & 173.19 & 192.25 \\\\\n    P14 & 10.65 & 16.23 & 30.17 & 52.24 & 79.07 & 109.76 & 139.88 & 166.59 & 187.43 & 202.78 \\\\\n    P15 & 14.82 & 21.48 & 35.28 & 57.38 & 82.5 & 114.19 & 144.17 & 169.13 & 190.07 & 203.95 \\\\\n    P16 & 18.92 & 23.92 & 36.31 & 57.05 & 88.57 & 116.78 & 147.6 & 172.97 & 196.18 & 209.08 \\\\\n    P17 & 4.5 & 9.35 & 23.38 & 45.9 & 72.68 & 103.65 & 133.52 & 161.88 & 183.18 & 197.69 \\\\\n    P18 & 10.91 & 14.36 & 29.04 & 49.94 & 78.39 & 109.27 & 140.76 & 167.58 & 189.9 & 202.89 \\\\\n    P19 & 9.76 & 14.41 & 27.79 & 49.65 & 76.61 & 109.48 & 140.12 & 167.97 & 188.71 & 203.91 \\\\\n    P20 & -56.61 & -52.49 & -38.36 & -16.25 & 11.95 & 42.14 & 72.88 & 100.43 & 122.74 & 136.87 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{\\textbf{Performance of PPO policies on the Swimmer domain.} A row shows the mean episode return of a single policy on all environments, while a column shows the mean episode return of all policies on a single environment. This table contains performance on the last 10 environments.}\n     \\label{tab:ppo}\n\\end{table*}\n\n\n\\begin{table*}[t!]\n    \\centering\n    \\small\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\n    \\toprule\n     & E1 & E2 & E3 & E4 & E5 & E6 & E7 & E8 & E9 & E10 \\\\ [0.5ex]\n    \\midrule\n    P1  & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\\\\n    P2  & 0.97 & 0.97 & 0.92 & 0.92 & 0.90 & 0.91 & 0.93 & 0.95 & 0.99 & 0.95 \\\\\n    P3  & 0.97 & 0.92 & 0.85 & 0.85 & 0.86 & 0.86 & 0.86 & 0.91 & 0.94 & 0.94 \\\\\n    P4  & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\\\\n    P5  & 0.61 & 0.61 & 0.60 & 0.58 & 0.57 & 0.55 & 0.53 & 0.51 & 0.50 & 0.49 \\\\\n    P6  & 0.96 & 0.90 & 0.87 & 0.87 & 0.88 & 0.90 & 0.91 & 0.92 & 0.96 & 0.95 \\\\\n    P7  & 0.83 & 0.89 & 0.93 & 0.96 & 0.97 & 0.96 & 0.94 & 0.89 & 0.84 & 0.80 \\\\\n    P8  & 0.84 & 0.84 & 0.84 & 0.82 & 0.81 & 0.79 & 0.78 & 0.77 & 0.75 & 0.74 \\\\\n    P9  & 0.97 & 0.93 & 0.90 & 0.87 & 0.86 & 0.86 & 0.87 & 0.87 & 0.87 & 0.86 \\\\\n    P10 & 0.91 & 0.90 & 0.88 & 0.87 & 0.87 & 0.88 & 0.89 & 0.92 & 0.96 & 0.98 \\\\\n    P11 & 0.77 & 0.78 & 0.79 & 0.80 & 0.82 & 0.78 & 0.72 & 0.76 & 0.79 & 0.80 \\\\\n    P12 & 0.67 & 0.56 & 0.39 & 0.46 & 0.58 & 0.41 & 0.39 & 0.46 & 0.49 & 0.47 \\\\\n    P13 & 0.38 & 0.36 & 0.32 & 0.26 & 0.30 & 0.25 & 0.32 & 0.32 & 0.33 & 0.40 \\\\\n    P14 & 0.82 & 0.79 & 0.76 & 0.74 & 0.73 & 0.72 & 0.72 & 0.73 & 0.75 & 0.76 \\\\\n    P15 & 0.67 & 0.66 & 0.65 & 0.63 & 0.62 & 0.62 & 0.61 & 0.61 & 0.61 & 0.61 \\\\\n    P16 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\\\\n    P17 & 0.86 & 0.82 & 0.80 & 0.78 & 0.77 & 0.78 & 0.79 & 0.82 & 0.85 & 0.87 \\\\\n    P18 & 0.69 & 0.68 & 0.66 & 0.65 & 0.64 & 0.63 & 0.63 & 0.63 & 0.63 & 0.63 \\\\\n    P19 & 0.80 & 0.81 & 0.81 & 0.73 & 0.75 & 0.78 & 0.81 & 0.79 & 0.78 & 0.90 \\\\\n    P20 & 0.96 & 0.94 & 0.90 & 0.88 & 0.87 & 0.86 & 0.86 & 0.86 & 0.85 & 0.83 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{\\textbf{Performance of PPO policies on the Spaceship domain.} A row shows the mean episode return of a single policy on all environments, while a column shows the mean episode return of all policies on a single environment. This table contains performance on the first 10 environments.}\n     \\label{tab:ppo}\n\\end{table*}\n\n\\begin{table*}[t!]\n    \\centering\n    \\small\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\n    \\toprule\n     & E11 & E12 & E13 & E14 & E15 & E16 & E17 & E18 & E19 & E20 \\\\ [0.5ex]\n    \\midrule\n    P1  & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\\\\n    P2  & 0.89 & 0.85 & 0.82 & 0.79 & 0.79 & 0.79 & 0.81 & 0.84 & 0.87 & 0.92 \\\\\n    P3  & 0.93 & 0.91 & 0.86 & 0.83 & 0.82 & 0.82 & 0.84 & 0.86 & 0.91 & 0.95 \\\\\n    P4  & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\\\\n    P5  & 0.49 & 0.49 & 0.49 & 0.50 & 0.51 & 0.53 & 0.55 & 0.57 & 0.59 & 0.61 \\\\\n    P6  & 0.91 & 0.85 & 0.82 & 0.79 & 0.78 & 0.78 & 0.80 & 0.82 & 0.86 & 0.92 \\\\\n    P7  & 0.74 & 0.70 & 0.67 & 0.65 & 0.64 & 0.65 & 0.66 & 0.70 & 0.75 & 0.79 \\\\\n    P8  & 0.73 & 0.71 & 0.70 & 0.70 & 0.70 & 0.71 & 0.73 & 0.76 & 0.79 & 0.82 \\\\\n    P9  & 0.84 & 0.82 & 0.80 & 0.79 & 0.80 & 0.81 & 0.84 & 0.87 & 0.93 & 0.97 \\\\\n    P10 & 0.94 & 0.90 & 0.86 & 0.84 & 0.83 & 0.83 & 0.85 & 0.88 & 0.90 & 0.92 \\\\\n    P11 & 0.81 & 0.93 & 0.90 & 0.87 & 0.84 & 0.82 & 0.80 & 0.78 & 0.77 & 0.77 \\\\\n    P12 & 0.46 & 0.83 & 0.61 & 0.44 & 0.77 & 0.74 & 0.64 & 0.59 & 0.59 & 0.63 \\\\\n    P13 & 0.43 & 0.73 & 0.64 & 0.81 & 0.67 & 0.69 & 0.80 & 0.63 & 0.51 & 0.41 \\\\\n    P14 & 0.79 & 0.80 & 0.82 & 0.85 & 0.86 & 0.89 & 0.91 & 0.91 & 0.89 & 0.85 \\\\\n    P15 & 0.61 & 0.62 & 0.62 & 0.63 & 0.64 & 0.65 & 0.66 & 0.67 & 0.68 & 0.68 \\\\\n    P16 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\\\\n    P17 & 0.90 & 0.91 & 0.92 & 0.93 & 0.94 & 0.96 & 0.98 & 0.98 & 0.94 & 0.90 \\\\\n    P18 & 0.63 & 0.63 & 0.64 & 0.64 & 0.65 & 0.67 & 0.68 & 0.69 & 0.70 & 0.70 \\\\\n    P19 & 0.91 & 0.89 & 0.87 & 0.93 & 0.89 & 0.86 & 0.83 & 0.81 & 0.80 & 0.80 \\\\\n    P20 & 0.80 & 0.78 & 0.77 & 0.76 & 0.76 & 0.77 & 0.80 & 0.85 & 0.90 & 0.93 \\\\\n\n    \\bottomrule\n    \\end{tabular}\n    \\caption{\\textbf{Performance of PPO policies on the Spaceship domain.} A row shows the mean episode return of a single policy on all environments, while a column shows the mean episode return of all policies on a single environment. This table contains performance on the last 10 environments.}\n     \\label{tab:ppo}\n\\end{table*}\n\n\n\n\n\\section{Discussion and Future Work}\nIn this work, we propose policy-dynamics value functions (PD-VF{}), a novel framework for fast adaptation to new environment dynamics. The key idea is to learn a value function conditioned on both a policy and a dynamics embedding which are learned in a self-supervised way. At test time, the environment embedding can be inferred from only a few interactions, which allows the selection of a policy that maximizes the learned value function. PD-VF{} has a number of desirable properties: it leverages the structure in both the policy and the dynamics space to estimate the expected return, it only needs a small number of steps to adapt to unseen dynamics, it does not update any parameters at test time, and it does not require dense reward or long rollouts to find an effective policy in a new environment. Empirical results on a set of continuous control domains show that PD-VF{} outperforms other methods on unseen dynamics, while being competitive on training environments. \n\nPD-VF{} opens up many promising directions for future research. First of all, the formulation can be extended to estimate the value function not only for a family of policies and environment dynamics, but also for a family of reward functions. Another avenue for future research is to use a more general class of function approximators (such as neural networks) to parameterise the value estimator instead of a quadratic form. The PD-VF{} framework can, in principle, also be used to evaluate a family of policies and environments on other metrics of interest besides the expected return, such as, for example, reward variance, agent prosociality, deviation from expert behavior, and so on. Another interesting direction is to integrate additional constraints (or prior knowledge) to the optimization problem (\\textit{e.g.}{} maximize expected return while only using policies in a certain region of the policy space). As noted by \\citet{precup2001off}, \\citet{Sutton2011HordeAS}, and \\citet{white2012scaling}, learning about multiple policies in parallel via general value functions can be useful for lifelong learning. Similarly, PD-VF{} can be a useful tool for an agent to continually gather knowledge about various policies and dynamics in the world. Finally, PD-VF{} can also be applied to multi-agent settings for adapting to different opponents or teammates whose behaviors determine the environment dynamics. \n\n\n\\label{conclusion}\n\n\\section{Experiments}\n\\label{experiments}\n\n\\begin{figure}[ht!]\n        \\subfigure[Spaceship]{\\label{fig:space}\\includegraphics[width=0.32\\columnwidth]{fig\/spaceship_diagram.pdf}}\n        \\subfigure[Swimmer]{\\label{fig:swim}\\includegraphics[width=0.32\\columnwidth]{fig\/swimmer.pdf}}\n        \\subfigure[Ant-wind]{\\label{fig:antwind}\\includegraphics[width=0.32\\columnwidth]{fig\/ant_wind.pdf}}\n\n        \\subfigure[Dynamics]{\\label{fig:dynamics}\\includegraphics[width=0.32\\columnwidth]{fig\/dynamics_diagram.pdf}}\n        \\subfigure[Ant-legs-v1]{\\label{fig:antleg1}\\includegraphics[width=0.32\\columnwidth]{fig\/ant_legs_v1.pdf}}\n        \\subfigure[Ant-legs-v2]{\\label{fig:antleg2}\\includegraphics[width=0.32\\columnwidth]{fig\/ant_legs_v2.pdf}}\n    \\caption{(a) - (c) illustrate the continuous control domains used for testing adaptation to unseen environment dynamics. In Spaceship, Swimmer, and Ant-wind, the train and test distribution of the dynamics is continuous as illustrated in (d). (e) and (f) show two instances of the Ant-legs task in which limb lengths sampled from a discrete distribution determine the dynamics.}\n    \\label{fig:diagrams}\n\\end{figure}\n\n\n\\begin{figure*}[ht!]\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/results_eval_swim_8.pdf}}\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/results_eval_space_8.pdf}}\n       \n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/results_eval_ant_8.pdf}}\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/results_eval_ant_legs_8.pdf}}\n    \\caption{\\textbf{Test Performance.} Average return on test environments with unseen  dynamics in Swimmer (top-left), Spaceship (top-right), Ant-wind (bottom-left), and Ant-legs (bottom-right) obtained by PD-VF{}, the upper bound PPOenv, and baselines $RL^2$, MAML, PPOdyn, and PPOall. PD-VF{} outperforms these baselines on most test environments and, in some cases, it is comparable with PPOenv (which was trained directly on the test environments).}\n    \\label{fig:eval_results}\n\\end{figure*}\n\n\\subsection{Experimental Setup}\nWe evaluate PD-VF{} on four continuous control domains, and compare it with an upper bound, four baselines, and four ablations. For each domain, we create a number of environments with different dynamics. Then, we split the set of environments into training and test subsets, so that at test time, the agent has to find a policy that behaves well on unseen dynamics. For all our experiments, we show the mean and standard deviation of the average return (over 100 episodes) across 5 different seeds of each model. The dynamics embeddings are inferred using at most $N_d = 4$ interactions with the environment. \n\n\\subsection{Environments}\n\n\\textbf{Spaceship} is a new continuous control domain designed by us. The task consists of moving a spaceship with a unit point charge from one end of a 2D room through a door at the other end. The action space consists of a fixed-magnitude force vector that is applied at each timestep. The room contains two fixed electric charges that deflect \/ attract the ship as it moves through the environment (see Figure~\\ref{fig:space}). The polarity and magnitude of these charges are parameterised by $d$ and determine the environment dynamics. The distribution of dynamics $\\mathcal{D}$ is chosen to be circular and centered (see Figure~\\ref{fig:dynamics}). Samples $d$ are drawn at intervals of $\\pi\/10$, each forming a different environment instance with charge  configuration $(\\cos(d), \\sin(d))$. The 5 samples in the range $[\\frac{3}{4} 2\\pi, \\ldots, 2\\pi]$ are held out as evaluation environments, the rest being used for training. \n\n\\begin{figure*}[ht!]\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/ablations_eval_swim_8.png}}\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/ablations_eval_space_8.png}}\n       \n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/ablations_eval_ant_8.png}}\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/ablations_eval_ant_legs_8.png}}\n    \\caption{\\textbf{Test Performance.} Average return in Swimmer (top-left), Spaceship (top-right), Ant-wind (bottom-left), and Ant-legs (bottom-right) obtained by PD-VF{}, NoDaggerPolicy, NoDaggerValue, Kmeans, and NN. PD-VF{} is better than these ablations overall.}\n    \\label{fig:eval_ablations}\n\\end{figure*}\n\n\\textbf{Swimmer} is a family of environments with varying dynamics based on MuJoCo's Swimmer-v3 domain \\citep{Todorov2012MuJoCoAP}. The goal is to control a three-link robot in a viscuous fluid to swim forward as fast as possible (Figure~\\ref{fig:swim}). The dynamics are determined by a 2D current within the fluid, whose direction changes between environments (but has fixed magnitude). The current direction is determined by an angle $d$, which is sampled in the same manner as for Spaceship above, \\textit{i.e.}{} train on $3\/4$ of all possible directions and hold out the other $1\/4$ for evaluation.     \n\n\n\\textbf{Ant-wind} is a family of environments based on MuJoCo's Ant-v3 domain in which the goal is to make a four-legged creature walk forward as fast as possible (Figure~\\ref{fig:antwind}). The environment dynamics are determined by the direction of a wind $d$, which is sampled from a continuous distribution in the same way as for Swimmer. \n\n\n\\textbf{Ant-legs} is a second task based on MuJoCo's Ant-v3 domain, in which the dynamics are sampled from a discrete distribution. The training environments are generated by fixing three ankle lengths (short, medium, and long) and generating all possible permutations for the four legs. The length of the ant leg is fixed to medium across all training environments. Symmetries in the training environments are removed by considering ants with the same number of short, medium, or long legs to be the same and choosing one ant from each equivalency class. There are four test environments with both the leg and ankle lengths being either short or long. Note that the test environments are significantly different from all the training ones, thus making Ant-legs a challenging setting for our method. Figures~\\ref{fig:antleg1} and~\\ref{fig:antleg2} show two instances of this environment. \n\n\n\n\\subsection{Baselines}\nWe use PPO \\citep{schulman2017proximal} as the base RL algorithm for all the baselines and for the reinforcement learning phase of training the PD-VF{} (Sec.~\\ref{sec:rltrain}). We use Adam \\citep{kingma2014adam} for optimization. All models use the same network architecture for the policy and value functions. For a given environment, all methods use the same number of steps $N_d$ (at the beginning of each episode) to infer the embedding of the environment dynamics. Then, they each use a single policy network to act in the environment until the end of the episode. We report the cumulative reward obtained by each method throughout an episodes (in which they first infer the environment dynamics which determines the policy used for acting until the end of the episode). We compare with the following baselines: \n\n\n\n\n\\begin{figure*}[h!]\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/results_train_swim_8.png}}\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/results_train_space_8.png}}\n       \n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/results_train_ant_8.png}}\n        \\subfigure{\\includegraphics[width=0.50\\textwidth]{fig\/results_train_ant_legs_8.png}}\n     \\caption{\\textbf{Train Performance.} Average return on train environments in Swimmer (top-left), Spaceship (top-right), Ant-wind (bottom-left), and Ant-legs (bottom-right) obtained by PD-VF{}, the upper bound PPOenv, and baselines $RL^2$, MAML, PPOdyn, and PPOall. PD-VF{} outperforms the baselines and ablations on most test environments and, in some cases, it is comparable with PPOenv (which was trained directly on the test environments). While other methods also perform reasonably well on the training environments, they generalize poorly to new environments with unseen dynamics.}\n    \\label{fig:train_results}\n\\end{figure*}\n\n\n\n\\textbf{PPOenv} trains a PPO policy for each environment in our set. This is used as an upper bound for the other models. \n\n\\textbf{MAML} is the meta-learning algorithm from \\citet{finn2017model}. MAML{} generally requires some amount of training on the test environments, so to make it more comparable to our method and the other baselines, we allow one gradient step using a trajectory of length $N_d$ (i.e. the same length as the one used by PD-VF{} to infer the embedding of the environment dynamics). Thus, MAML{} has an advantage over PD-VF{} which does not make any parameter updates at test time.\n\n$\\textbf{RL}^2$ is the meta-learning algorithm from \\citet{Wang2016LearningTR} and \\citet{Duan2016RL2FR}, which uses a recurrent policy that takes as input the previous action and reward.\n\n\\textbf{PPOdyn} trains (using PPO) a single policy network conditioned on the dynamics embedding. At test time, it first infers the dynamics embedding and then conditions the pretrained policy network on that vector. This is a close implementation of the approach in \\citet{yang2019single}\\footnote{An exact match was not feasible as code for \\citet{yang2019single} was not available.}. \n\n\\textbf{PPOall} trains a single PPO policy on all the training environments and uses it on the test environments without any additional fine-tuning. \n\nWe also compare PD-VF{} with four ablations:\n\n\\textbf{NN} finds the environment that is closest (in Euclidean metric) to the test environment's embedding and uses the PPOenv{} policy trained on that environment to act. This ablation aims to tease out the effect of using both the learned space of policies and that of dynamics to adapt to new environments, from that of only using the learned dynamics space.\n\n\n\\textbf{Kmeans} clusters the environment embeddings (using trajectories collected in Section~\\ref{sec:rltrain}) into $K$ clusters. Then, for each cluster, we train a new PPO policy on all the environments assigned to that cluster. At test time, we find the closest cluster for the given environment embedding and use the policy corresponding to that cluster to act in the environment. \n\n\n\\textbf{NoAggValue} trains a PD-VF{} without using dataset aggregation for the value function (see Section~\\ref{sec:suptrain}).\n\n\\textbf{NoAggPolicy} uses PD-VF{} without using dataset aggregation for the policy decoder (see Section~\\ref{sec:suptrain}).\n\n\n\\section{Introduction}\n\nDeep reinforcement learning (RL) has achieved impressive results on a wide range of complex tasks \\citep{mnih2015human, silver2016mastering,  silver2017mastering, silver2018general, jaderberg2019human, Berner2019Dota2W, vinyals2019grandmaster}. However, recent studies have pointed out that RL agents trained and tested on the same environment tend to overfit to that environment's idiosyncracies and are unable to generalize to even small perturbations \\citep{Whiteson2011ProtectingAE, rajeswaran2017towards, zhang2018study, zhang2018dissection, henderson2018deep, Cobbe2019QuantifyingGI, Raileanu2020RIDERI, Song2020ObservationalOI}.\nIt is often the case that besides the test environments being different from the train environments, they will also have\ncostly interactions, scarce or unavailable feedback, and irreversible consequences. For example, a self-driving car might have to adjust its behavior depending on weather conditions, or a prosthetic control system might have to adapt to a new human. In these cases it is crucial for RL agents to find and execute appropriate policies as quickly as possible.\n\n\n\n\n\n\n\n\nOur approach is inspired by \\citet{Sutton2011HordeAS} who introduced the notion of general value functions (GVFs), which can be used to gather knowledge about the world in the form of predictions. A GVF estimates the expected return of an arbitrary policy on a certain task (as defined by a reward function, a termination function and a terminal-reward function). Similarly, in this work, we aim to learn a value function conditioned on elements of a space of policies and tasks, but here, a ``task'' is specified by the transition function of the MDP instead of the reward function. \n\nMore specifically, we propose PD-VF{}, a novel framework for rapid adaptation to new environment dynamics. PD-VF{} consists of four phases: \n\\begin{enumerate*}[label=(\\roman*)] \n    \\item a \\textit{reinforcement learning phase} in which individual policies are learned for each environment in our training set using standard RL algorithms,\n    \\item a \\textit{self-supervised phase} in which trajectories generated by these policies are used to learn embeddings for both policies and environments,\n    \\item a \\textit{supervised training phase} in which a neural network is used to learn the value function of a certain policy acting in some environment. The network takes as inputs the initial state of the environment, as well as the corresponding policy and environment embeddings (as learned in the previous phase) and is trained with supervision of the cumulative reward obtained during an episode, and finally \n    \\item an \\textit{evaluation phase} in which, given a new environment, its dynamics embedding is inferred using the first few steps of an episode. Then, a policy is selected by finding the policy embedding that maximizes the learned value function. The selected policy is used to act in the environment until the episode ends.\n\\end{enumerate*}\n\nOur framework uses self-supervised interactions with the environment to learn an embedding space of both dynamics and policies. By learning a value function in the policy-dynamics space, PD-VF{} can discover useful patterns in the complex relation between a family of environment dynamics, various behaviors, and the expected return.  The value function is designed to model non-optimal policies along with optimal policies in given environments so that it can understand how changes in dynamics relate to changes in the return of different policies. PD-VF{} uses the learned space of dynamics to rapidly embed a new environment in that space using only a few interactions.  At test time,  PD-VF{} can evaluate or rank policies (from a certain family) on unseen environments without the need of full rollouts (\\textit{i.e.}{} it does not require full trajectories or rewards to update the policy). We evaluate our method on a set of continuous control tasks (with varying dynamics) in MuJoCo \\citep{Todorov2012MuJoCoAP}. The dynamics of each task instance are determined by physical parameters such as wind direction or limb length and can be sampled from a continuous or discrete distribution. Performance is evaluated on a single episode at test time to emphasize rapid adaptation. We show that PD-VF{} outperforms other meta-learning and transfer learning approaches on new environments with unseen dynamics. \n\n\\label{introduction}\n\\section{Policy-Dynamics Value Functions}\n\\label{method}\n\nIn this work, we aim to design an approach that can quickly find a good policy in an environment with new and unknown dynamics, after being trained on a family of environments with related dynamics.\nThe problem can be formalized as a family of Markov decision processes (MDPs) defined by $(\\mathcal{S}, \\mathcal{A}, \\mathcal{T}, \\mathcal{R}, \\gamma)$, where $(\\mathcal{S}, \\mathcal{A}, \\mathcal{R}, \\gamma)$ are the corresponding state space, action space, reward function, and discount factor. Each instance of the family is a stationary MDP with transition function $\\mathcal{T}_d(s'|s, a) \\in \\mathcal{T}$. Each $\\mathcal{T}_d$ has a hidden parameter $d$ that is sampled once from a distribution $\\mathcal{D}$ and held constant for that instance (\\textit{i.e.}{} episode). $\\mathcal{T}_d$ can be continuous or discrete in $d$. By design, the latent variable $d$ that defines the MDP's dynamics cannot be observed from individual states. \n\nWe present Policy-Dynamics Value Functions (PD-VF{}), a novel framework for rapid adaptation across \nsuch MDPs with different dynamics. PD-VF{} is an extension of a value function that not only conditions on a state, but also on a policy and a transition function. \n\nA conventional value function $V: \\mathcal{S} \\rightarrow \\mathcal{R}$ is defined as the expected future return from state $s$ of policy $\\pi$: \n\\begin{equation*}\n    V(s) = \\mathbb{E_{}} \\left[ G_{t} | S_t = s \\right] = \\mathbb{E_{}} \\left[ \\sum_{k = t+1}^{T} \\gamma^k r_{k} | S_t = s \\right]. \n\\end{equation*}\n\nFormally, we define a \\textit{policy-dynamics value function} or PD-VF{} as a function $W: \\mathcal{S} \\times \\Pi \\times \\mathcal{T} \\rightarrow \\mathcal{R}$ with two auxiliary inputs representing the policy $\\pi$ and the dynamics $d$:\n\\begin{equation*}\n    W(s, \\pi, d) = \\mathbb{E} \\left[ G_{t} | S_t = s, A_t \\sim \\pi, S_{t+1} \\sim \\mathcal{T}_d \\right].\n\\end{equation*}\n\n\n\\subsection{Problem setup}\nThe dynamics distribution $\\mathcal{D}$ is partitioned into two disjoint sets $\\mathcal{D}_{train}$ and $\\mathcal{D}_{test}$. These are used to generate a set of training and test environments, each having different transition functions, drawn from their respective distributions. \n\nOur model is learned on the training environments in three stages:\n\\begin{enumerate*}[label=(\\roman*)] \n    \\item a reinforcement learning phase,\n    \\item a self-supervised phase and \n    \\item a supervised phase.\n\\end{enumerate*}\nThe resulting PD-VF model is evaluated on test environments, where it only experiences a single episode in each. This evaluation setting probes PD-VF's ability to very quickly adapt to previously unseen dynamics. \n\n\\subsection{Reinforcement Learning Phase}\n\\label{sec:rltrain}\nThe first phase of training uses standard model-free RL algorithms to acquire experience in the training environments. An ensemble of $N$ policies are trained, each with a different random seed on one of the training environments. For each policy, we save a number of checkpoints at different stages throughout training. Then, we collect trajectories using each of these checkpoints in each of our training environments. This results in experience from a diverse set of policies (some good, some bad) across environments with different dynamics. Importantly, this dataset contains the behaviors of policies in environments they haven't been trained on. In the next section, we describe how the collected trajectories are used to learn policy and dynamics embeddings. \n\n\\subsection{Self-Supervised Learning Phase}\nThe goal of this phase is to learn an embedding space of the dynamics that captures variations in the transition function, as well as an embedding space of the policies that captures variations in the agent behavior. \n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{fig\/emb_diagram.pdf}\n    \\caption{In the \\textbf{self-supervised learning phase}, a pair of autoencoders is trained using transitions generated by a diverse set of policies in a set of environments with different dynamics. By exploiting the Markov property of the environment, distinct latent embeddings of the dynamics $z_d$ and policy $z_\\pi$ are produced.}   \n    \\label{fig:emb_diagram}\n\\end{figure}\nThe space of dynamics is learned using an encoder $E_{d}$ parameterised as a Transformer \\cite{vaswani2017attention}, and a decoder $D_{d}$ parameterised as a feed-forward network. The encoder takes as input a {\\it set} of transitions $\\{(s_t, a_t, s_{t+1})\\}$ from the first $N_d$ steps in each episode and outputs a vector embedding for the dynamics $z_{d}$. The decoder takes as inputs the state $s_t$, action $a_t$ and dynamics embedding $z_{d}$, and predicts the next state $\\hat{s}_{t+1}$. The parameters $\\theta_d$ and $\\phi_d$ of the encoder and decoder are trained to minimize the $\\ell_2$ error of $\\hat{s}_{t+1}$ and ${s}_{t+1}$. Formally,\n\\begin{equation*}\n    z_{d} = E_{d}(\\{(s_t, a_t, s_{t+1})\\} ; \\; \\theta_{d})\n\\end{equation*}\n\\begin{equation*}\n    \\hat{s}_{t+1} = D_{d}(s_t, a_t, z_{d} ;  \\; \\phi_{d}).\n\\end{equation*}\nThis arrangement exploits the inductive bias that, conditioned on $d$, the environment is Markovian. By using no positional encoding in the Transformer, the input transitions lack any temporal ordering, thus preserving the Markov property. The decoder receives no historical information (since it is unnecessary in a Markovian setting), so it is forced to embed information about the dynamics into $z_d$ to make good predictions. Because the input set contains the actions in each triple, the encoder has no incentive to encode policy information into $z_d$. This modeling choice encourages $z_d$ to only contain information about the dynamics, rather than the policy used to generate the transitions.\n\nSimilarly, the space of policies is learned using an encoder $E_{\\pi}$ parameterised as a Transformer and a decoder $D_{\\pi}$ parameterised as a feed-forward network. The encoder takes as input a set (again using the Markov property as an inductive bias) of state-action pairs $\\{(s_t, a_t)\\}$ from a full episode and outputs a vector embedding for the policy $z_{d}$. The decoder takes as inputs the state $s_t$ and the policy embedding $z_{\\pi}$ to predict the action taken by the policy $\\hat{a}_{t}$. Since the policy encoder does not have direct access to full environment transitions, $z_{\\pi}$ is constrained to capture information about the policy without elements of the dynamics. The parameters $\\theta_{\\pi}$ and $\\phi_{\\pi}$ of the encoder and decoder are trained to minimize the $\\ell_2$ error of $\\hat{a}_{t}$ and ${a}_{t}$. Formally,   \n\\begin{equation*}\n    z_{\\pi} = E_{\\pi}(\\{(s_t, a_t)\\} ;  \\; \\theta_{\\pi})\n\\end{equation*}\n\\begin{equation*}\n    \\hat{a}_t = D_{\\pi}(s_t, z_{\\pi} ;  \\; \\phi_{\\pi}).\n\\end{equation*}\n\nBoth the policy and the dynamics embeddings are normalized to have unit $\\ell_2$-norm. \n\nSee Figure~\\ref{fig:emb_diagram} for an overview of the self-supervised learning phase.\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{fig\/pdvf_diagram.pdf}\n    \\caption{In the \\textbf{supervised learning phase}, a parametric value function $W$ is trained to predict the expected return $G$ for an entire space of policies and dynamics. W takes as inputs the initial state $s_0$, policy embedding $z_\\pi$, and dynamics embedding $z_d$ (estimated from a small set of transitions). We train $W$ in a supervised fashion, using Monte-Carlo estimates of the expected return $G$ for policy $\\pi$ in environment with dynamics $\\mathcal{T}_d$. At test time, $z_\\pi$ is optimized to maximize $\\hat{G}$ (red dashed arrow), resulting in $z^*_\\pi$ which is then decoded to an actual policy via $D_\\pi$.}\n\\label{fig:pdvf_diagram}\n\\end{figure}\n\n\\subsection{Supervised Learning Phase}\n\\label{sec:suptrain}\n\nIn this phase, the goal is to train an estimator $W$ of the expected return $\\hat{G}$ for a space of policies and dynamics. More specifically, $W$ is a function approximator conditioned on the learned policy and dynamics embeddings, $z_\\pi$ and $z_d$.\n\nA central idea of our PD-VF framework is that $W$ provides a scoring function over the policy embedding space. It thus provides a mechanism to allow on-the-fly optimization of $z_\\pi$ with respect to the estimated return $\\hat{G}$, without the need for any environment interaction, given an estimate (or embedding) of the environment's dynamics. This is key to PD-VF's ability to rapidly find an effective policy in a new environment, only requiring enough environment interaction to give a reliable estimate of the dynamics embedding $z_d$ (just a few steps in practice). We choose $W$ to have a quadratic form to permit easy optimization with respect to $z_\\pi$:\n\\begin{equation*}\n    \\hat{G} = W(s_0, z_{\\pi}, z_{d}) = z_{\\pi}^T \\, A(s_0, z_{d}; \\psi) \\, z_{\\pi}.\n\\end{equation*}\nThe matrix $A(s_0, z_d; \\psi)$ is a function of the initial environment state $s_0$ as well as the dynamics embedding $z_d$. Note that $A$ only needs to model the initial state $s_0$ rather than an arbitrary state $s$ since the optimization w.r.t $z_\\pi$ occurs only once, at the start of an episode. Since $A$ must be Hermitian positive-definite, a feed-forward network with parameters $\\psi$ is first used to obtain a lower triangular matrix $L(s_0, z_d; \\psi)$. Then $A$ is constructed from $L L^T$. \n\n{\\noindent \\bf Optimizing the policy embedding $z_\\pi$}: The optimization of the policy embedding $z_{\\pi}$ has a closed-form solution which is achieved by performing a singular value decomposition, $A = U S V^T$, and taking the top singular vector of this decomposition $z_{\\pi}^{*}$. Unit $\\ell_2$ normalization is then applied to $z_{\\pi}^{*}$. We refer to this vector $z_{\\pi}^{*}$ as the {\\em optimal policy embedding} (OPE) of the PD-VF{}. \n\n\n{\\noindent \\bf Learning $\\psi$ -- Initial stage}: We collect training data for the PD-VF{} in the following manner. First, we randomly select a policy and an environment from our training set (described in Section~\\ref{sec:rltrain}). Second, we generate full trajectories of that policy in the selected environment and cache the average return obtained across all episodes. This gives us a Monte-Carlo estimate for the expected return of the corresponding policy in that particular environment. Then, we use the first $N_{d}$ steps of that trajectory to infer the dynamics embedding. Similarly, we use the full trajectory to infer the policy embedding (via $E_\\pi$, not the above optimization procedure). After collecting this data into a buffer, we train the estimator $W$ in a supervised fashion by predicting the expected return $G$ given an initial state $s_0$, a policy embedding $z_{\\pi}$ and a dynamics embedding $z_{d}$. \n\n{\\noindent \\bf Learning $\\psi$ -- Data Aggregation for the Value Function}:\nFor the method to work well, it is important that the learned value function $W$ makes accurate predictions for the entire policy space, and especially for the OPE $z_{\\pi}^{*}$ (which correspond to the policies selected to act in the environment). One way to ensure that these estimates are accurate is by adding the OPEs to the training data. After initial training of the PD-VF{} on the original dataset of policy and dynamics embeddings, we use an iterative algorithm that alternates between collecting a new dataset of OPEs and training the PD-VF{} on the aggregated data (including the original data as well as data added from all previous iterations). We use early stopping to select the best value function (\\textit{i.e.}{} the one with the lowest loss) to be used at test time.\n\n{\\noindent \\bf Learning $\\psi$ -- Data Aggregation for the Policy Decoder}:\nSimilarly, the policy decoder may poorly estimate an agent's actions in states not seen during training. Thus, we iteratively train the policy decoder using a combination of the original set of states as well as new states generated by the policy embeddings that maximize the current value function. More specifically, we use the current OPEs (corresponding to the policies that PD-VF{} thinks are best) as inputs to the policy decoder to generate actions and interact with the environment. Then, we add the states visited by this policy to the data. The policy decoder is trained using the aggregated collection of states which includes both the states visited by the original collection of policies as well as the states visited by the current OPEs selected by the PD-VF{}.\n\nSee Figure~\\ref{fig:pdvf_diagram} for an overview of the supervised learning phase.\n\n\\subsection{Evaluation Phase}\n\nAt test time, we want to find a policy that performs well on a single episode of an environment with unseen dynamics. This proceeds as follows: (i) the agent uses one of the pretrained RL policies to act for $N_{d}$ steps; (ii) the generated transitions are then used to infer the dynamics embedding $z_d$; (iii) once an estimate of the dynamics is obtained, the matrix $A(s_0, z_{d};\\; \\psi)$ can be computed; (iv) we employ the closed-form optimization described above to compute the optimal policy embedding $z_{\\pi}^{*}$; (v) the policy decoder, conditioned on the $z_{\\pi}^{*}$ embedding, is then used to take actions in the environment until the end of the episode. Note that only a small number of interactions with a new environment is needed in order to adapt, the policy selection being performed internally within the PD-VF model. Performance is evaluated on a single trajectory of each environment instance. \n\n\n\\section{Related Work}\nOur work draws inspiration from multiple research areas such as transfer learning \\citep{taylor2009transfer, Higgins2017DARLAIZ}, skill and task embedding \\citep{Devin2016LearningMN, Zhang2018DecouplingDA, Hausman2018LearningAE, Petangoda2019DisentangledSE}, and general value functions \\citep{precup2001off, Sutton2011HordeAS, white2012scaling}.\n\n\\textbf{Multi-Task and Transfer Learning.}\n\\citet{taylor2009transfer} presents an overview of transfer learning methods in RL. A popular approach for transfer in RL is multi-task learning \\citep{taylor2009transfer, teh2017distral}, a paradigm in which an agent is trained on a family of related tasks. By simultaneously learning about different tasks, the agent can exploit their common structure, which can lead to faster learning and better generalization to unseen tasks from the same family \\citep{taylor2009transfer, lazaric2012transfer, ammar2012reinforcement, ammar2014automated, parisotto2015actor, borsa2016learning, gupta2017learning, andreas2017modular, oh2017zero, hessel2019multi}. A large body of work has been inspired by the Horde architecture \\citep{Sutton2011HordeAS}, which consists of a number of RL agents with different policies and goals. Each agent is tasked with estimating the value function of a particular policy on a given task, thus collectively representing knowledge about the world. Building on these ideas, other methods leverage the shared dynamics of the tasks \\citep{barreto2017successor, zhang2017deep, madjiheurem2019state2vec} or the similarity among value functions and the associated optimal policies \\citep{schaul2015universal, borsa2018universal, hansen2019fast, siriwardhana2019vusfa}. These approaches assume the same underlying transition function for all tasks. In contrast, we focus on transferring knowledge across tasks with different dynamics.\n\n\\textbf{Meta-Learning and Robust Transfer.}\nA popular approach for fast adaptation to new environments is meta reinforcement learning (meta RL) \\citep{Cully2015RobotsTC, finn2017model, Wang2017RobustIO, Duan2016RL2FR, Xu2018MetaGradientRL, Houthooft2018EvolvedPG, saemundsson2018meta, nagabandi2018learning, humplik2019meta, rakelly2019efficient}. Meta RL methods have been designed to work well with dense reward and recent work has shown that they struggle to learn from a limited number of interactions and optimization steps at test time \\citep{yang2019single}. In contrast, our framework is capable of rapid adaptation to new environment dynamics and does not require dense reward or a large number of interactions to find a good policy. Moreover, PD-VF{} does not update the model parameters at test time, which makes it less computationally expensive than meta RL. Another common approach for transfer across dynamics is model-based RL, which uses  Gaussian processes (GPs) or Bayesian neural networks (BNNs) to estimate the transition function \\cite{DoshiVelez2013HiddenPM, Killian2017RobustAE}. However, such methods require fictional rollouts to train a policy from scratch at test time, which makes them computationally expensive and limits their applicability for real-world tasks. \\citet{Yao2018DirectPT} uses a fully-trained BNN to further optimize latent variables during a single test episode, but requires an optimal policy for each training instance, which makes it harder to scale. Robust transfer methods either require a large number of interactions at test time \\citep{rajeswaran2017towards} or assume that the distribution over hidden variables is known or controllable \\citep{Paul2018FingerprintPO}. An alternative approach was proposed by \\citet{Pinto2017RobustAR} who use an adversary to perturb the system, achieving robust transfer across physical parameters such as friction or mass.  \n\n\n\\textbf{Skill and Task Embeddings.}\nA large body of work proposes the use of learned skill and task embeddings for transfer in RL~\\cite{da2012learning, sahni2017learning, oh2017zero, gupta2017learning, Hausman2018LearningAE, he2018zero}. For example, \\citet{Hausman2018LearningAE} use approximate variational inference to learn a latent space of skills. Similarly, \\citet{Arnekvist2018VPEVP} learn a stochastic embedding of optimal Q-functions for various skills and train a universal policy conditioned on this embedding. In both \\citet{Hausman2018LearningAE} and \\citet{Arnekvist2018VPEVP}, adaptation to a new task is done in the latent space with no further updates to the policy network. \\citet{CoReyes2018SelfConsistentTA} learn a latent space of low-level skills that can be controlled by a higher-level policy, in the context of hierarchical reinforcement learning. This embedding is learned using a variational autoencoder \\citep{Kingma2013AutoEncodingVB}\nto encode state trajectories and decode states and actions.  \\citet{Zintgraf2018FastCA} use a meta-learning approach to learn a deterministic task embedding. \\citet{Wang2017RobustIO} and \\citet{Duan2017OneShotIL} learn embeddings of expert demonstrations to aid imitation learning using variational and deterministic methods, respectively. More recently, \\citet{Perez2018EfficientTL} learn dynamic models with auxiliary latent variables and use them for model-predictive control. \\citet{Zhang2018DecouplingDA} use separate dynamics and reward modules to learn a task embedding. They show that conditioning a policy on this embedding helps transfer to changes in transition or reward function. While the above approaches might learn embeddings of skills or tasks, none of them leverage \\textit{both} the latent space of policies and that of the environments for estimating the expected return and using it to select an effective policy at test time.\n\nMore similar to our work is that of \\citet{yang2019single}, who also focus on fast adaptation to new environment dynamics and evaluate performance on a single episode at test time. \\citet{yang2019single} train an inference model and a probe to estimate the underlying latent variables of the dynamics, which are then used as input to a universal control policy. While similar in scope, our approach is significantly different from that of \\citet{yang2019single}. Importantly, \\citet{yang2019single} does not learn a latent space of policies and instead trains a universal policy on all the environments. Learning a value function in a space of policies and dynamics allows the function approximator to capture relations among dynamics, behaviors (both optimal as well as non-optimal), and rewards that a universal policy cannot learn. Moreover, the learned structure can aid transfer to new dynamics.\n\n\\label{related}\n\\section{Results}\n\\label{results}\n\n\\subsection{Adaptation to New Environment Dynamics}\nAs seen in Figures~\\ref{fig:eval_results} and~\\ref{fig:eval_ablations}, PD-VF{} outperforms all other methods on test environments with new dynamics. In some cases (particularly on Spaceship and Swimmer), our approach is comparable to the PPOenv upper bound which was directly trained on the respective test environment (in contrast, PD-VF{} has never interacted with that environment before). While the strength of PD-VF{} lies in quickly adapting to new dynamics, its performance on training environments is still comparable to that of the other baselines, as shown in Figure~\\ref{fig:train_results}. This result is not surprising since current state-of-the-art RL algorithms such as PPO can generally learn good policies for the environments they are trained on, given enough interactions, updates, and the right hyperparamters. However, as predicted, standard model-free RL methods such as the baseline PPOall do not generalize well to environments with dynamics different from the ones experiences during training. Even meta-learning approaches like MAML or $RL^2$ struggle to adapt when they are allowed to use only a short trajectory for updating the policy at test time, as is the case here. \n\nBut most importantly, PD-VF{} also outperforms the approaches that use the dynamics embedding such as NN, Kmeans, and PPOdyn. This supports our claim that learning a value function for an entire space of policies (rather than for a single optimal policy as standard RL methods do) can be beneficial for adapting to unseen dynamics. By simultaneously estimating the return of a collection of policies in a family of environments with different but related dynamics, PD-VF{} can learn how variations in dynamics relate to differences in the performance of various policies. This allows the model to rank different policies and understand that sub-optimal behaviors in certain environments might be optimal in others. Thus, at least in theory, PD-VF{} has the ability to find policies that are better than the ones seen during training. Our empirical results indicate that this might also hold true in practice. Overall, PD-VF{} proves to be more robust to changes in dynamics relative to the other methods, especially in completely new environments.\n\n\n\\subsection{Analysis of Learned Embeddings}\n\nThe performance of PD-VF{} relies on learning useful policy and dynamics embeddings that capture variations in agent behaviors and transition functions, respectively. In this section, we analyze the learned embeddings. \n\n\\begin{figure}[ht!]\n        \\subfigure[]{\\label{fig:env_space}\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_space_env_env.pdf}}\n        \\subfigure[]{\\label{fig:env_swim}\\includegraphics[width=0.31\\columnwidth]{fig\/\/tsne_swim_env_env.pdf}}\n        \\subfigure[]{\\label{fig:env_ant}\\includegraphics[width=0.36\\columnwidth]{fig\/tsne_ant_env_env.pdf}}\n    \\caption{t-SNE plots of the learned \\textbf{environment embeddings} $z_{d}$ for Spaceship (a), Swimmer (b), and Ant-wind (c). The color corresponds to the \\textit{environment} that generated the transitions used to encode the corresponding dynamics embeddings. The plot contains embeddings of both train and test environments.}\n    \\label{fig:env_embeddings}\n\\end{figure}\n\n\\begin{figure}[ht!]\n        \\subfigure[]{\\label{fig:pi_space}\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_space_pi_pi.pdf}}\n        \\subfigure[]{\\label{fig:pi_swim}\\includegraphics[width=0.31\\columnwidth]{fig\/tsne_swim_pi_pi.pdf}}\n        \\subfigure[]{\\label{fig:pi_ant}\\includegraphics[width=0.34\\columnwidth]{fig\/tsne_ant_pi_pi.pdf}}\n    \\caption{t-SNE plots of the learned \\textbf{policy embeddings} $z_{\\pi}$ for Spaceship (a), Swimmer (b), and Ant-wind (c). The color corresponds to the \\textit{policy} that generated the transitions used to encode the corresponding policy embeddings. The plot contains embeddings of policies trained on both train and test environments.}\n    \\label{fig:policy_embeddings}\n\\end{figure}\n\nFigure~\\ref{fig:env_embeddings} shows a t-SNE plot \\citep{Maaten2008VisualizingDU} of the learned dynamics embeddings on the three continuous control domains used for evaluating our method. Environment $i$ corresponds to dynamics defined by $d = i \\times \\pi \/ 10$ (\\textit{i.e.}{} the direction of the wind in Swimmer's environment 1 is at $\\pi\/10$ degrees). Environments 1 - 15  are used for training, while 16 - 20 are used for evaluation. The latent space captures the continuous nature of the distribution used to generate the environment dynamics. For example, in Figure~\\ref{fig:env_ant}, one can see the wind direction corresponding to a particular environment, indicating that the learned embedding space uncovers the manifold structure of the true dynamics distribution. Even if, during training, the dynamics model never sees trajectories through the test environments, it is still able to embed them within the 1D manifold, thus preserving smoothness in the latent space. \n\nSimilarly, Figure~\\ref{fig:policy_embeddings} shows the corresponding t-SNE \\citep{Maaten2008VisualizingDU} of the learned policy embeddings for Spaceship, Swimmer, and Ant. The embeddings are clustered according to the policy that generated them. \n\n\n\\label{results}\n\n\\section*{Acknowledgements}\nRoberta and Max were supported by the DARPA L2M grant. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nLet $X$ be a (reduced, irreducible) smooth projective curve over an algebraically closed field $k$ of characteristic $p \\ge 0$ and let $k(X)$ be its function field. \nWe consider a morphism $\\varphi: X \\rightarrow \\mathbb{P}^2$, which is birational onto its image.\nIn this situation, Hisao Yoshihara introduced the notion of a Galois point.  \nA point $P \\in \\mathbb{P}^2$ is called a {\\it Galois point}, if the field extension $k(\\varphi(X))\/\\pi_P^*k(\\mathbb{P}^1)$ of function fields induced by the projection $\\pi_P$ from $P$ is a Galois extension (\\cite{miura-yoshihara, yoshihara}).  \nFurthermore, a Galois point $P$ is said to be inner (resp. outer), if $P \\in \\varphi(X) \\setminus {\\rm Sing}(\\varphi(X))$ (resp. if $P \\in \\mathbb{P}^2 \\setminus \\varphi(X)$). \n\nA criterion for the existence of a birational embedding with two Galois points was described by the present author (\\cite{fukasawa2}). \nIt is a natural problem to find a condition for the existence of {\\it three} Galois points (see also \\cite{open}). \nNon-collinear Galois points were considered in \\cite{fukasawa3}. \nIn this article, (three) collinear Galois points are studied. \nThe associated Galois group is denoted by $G_P$, when $P$ is a Galois point.\nThe following criterion is presented. \n\n\\begin{theorem} \\label{main} \nLet $G_1$, $G_2$ and $G_3 \\subset {\\rm Aut}(X)$ be finite subgroups of order at least three, and let $P_1$, $P_2$ and $P_3$ be different points of $X$. \nThen, four conditions\n\\begin{itemize}\n\\item[(a)] $X\/{G_i} \\cong \\Bbb P^1$ for $i=1, 2, 3$,    \n\\item[(b)] $G_i \\cap G_j=\\{1\\}$ for any $i, j$ with $i \\ne j$,  \n\\item[(c)] there exists a divisor $D$ such that $D=P_i+\\sum_{\\sigma \\in G_i}\\sigma(P_j)$ for any $i, j$ with $i \\ne j$, and\n\\item[(d)] $\\dim \\Lambda \\le 2$, for the smallest sublinear system $\\Lambda$ of $|D|$ such that $D, P_i+\\sum_{\\sigma \\in G_i}\\sigma(P_i) \\in \\Lambda$ for $i=1, 2, 3$\n\\end{itemize}\nare satisfied, if and only if there exists a birational embedding $\\varphi: X \\rightarrow \\mathbb P^2$ of degree $|G_1|+1$ such that $\\varphi(P_1)$, $\\varphi(P_2)$ and $\\varphi(P_3)$ are three collinear inner Galois points for $\\varphi(X)$ and $G_{\\varphi(P_i)}=G_i$ for $i=1, 2, 3$. \n\\end{theorem}\n\n\n\\begin{theorem} \\label{main-outer}\nLet $G_1$, $G_2$ and $G_3 \\subset {\\rm Aut}(X)$ be finite subgroups, and let $Q$ be a point of $X$. \nThen, four conditions\n\\begin{itemize}\n\\item[(a)] $X\/{G_i} \\cong \\Bbb P^1$ for $i=1, 2, 3$,    \n\\item[(b)] $G_i \\cap G_j=\\{1\\}$ for any $i, j$ with $i \\ne j$,  \n\\item[(c')] there exists a divisor $D$ such that $D=\\sum_{\\sigma \\in G_i}\\sigma(Q)$ for $i=1, 2, 3$, and\n\\item[(d')] $\\dim \\Lambda \\le 2$, for the smallest sublinear system $\\Lambda$ of $|D|$ such that $\\Lambda_1 \\cup \\Lambda_2 \\cup \\Lambda_3 \\subset \\Lambda$, where $\\Lambda_i$ is the base-point-free linear system induced by the covering map $X \\rightarrow X\/G_i \\cong \\mathbb{P}^1$ for $i=1, 2, 3$\n\\end{itemize}\nare satisfied, if and only if there exists a birational embedding $\\varphi: X \\rightarrow \\mathbb P^2$ of degree $|G_1|$ and three collinear outer Galois points $P_1, P_2$ and $P_3$ exist for $\\varphi(X)$ such that $G_{P_i}=G_i$ for $i=1, 2, 3$, and $\\overline{P_1P_2} \\ni \\varphi(Q)$, where $\\overline{P_1P_2}$ is the line passing through $P_1$ and $P_2$. \n\\end{theorem} \n\n\n\nThe uniqueness of the birational embedding constructed in \\cite{fukasawa2} is also proved. \n\n\\begin{proposition} \\label{uniqueness} \nAssume that the orders of groups $G_1$ and $G_2$ in Facts \\ref{criterion1} and \\ref{criterion2} are at least three. \nThen, a morphism $\\varphi$ described in Fact \\ref{criterion1} (resp. in Fact \\ref{criterion2}) is uniquely determined by a $4$-tuple $(G_1, G_2, P_1, P_2)$ (resp. by a $3$-tupble $(G_1, G_2, Q)$), up to a projective equivalence. \n\\end{proposition} \n\nUsing (the proof of) this Proposition, the following criterion for the extendability of an automorphism $\\sigma \\in G_P$ for an inner Galois point $P$ is presented. \n\n\\begin{proposition} \\label{extendable} \nLet $\\deg \\varphi(X) \\ge 4$, let $\\varphi(P_1)$ and $\\varphi(P_2) \\in \\varphi(X) \\subset \\mathbb{P}^2$ be different inner Galois points, and let $\\sigma \\in G_{\\varphi(P_1)}$ satisfy $P_3=\\sigma(P_2)$. \nThen, there exists a linear transformation $\\tilde{\\sigma}$ of $\\mathbb{P}^2$ such that $\\varphi^{-1}\\circ\\tilde{\\sigma}\\circ\\varphi=\\sigma$, if and only if three conditions\n\\begin{itemize}\n\\item[(a)] $\\sigma(P_1)=P_1$, \n\\item[(b)] $\\varphi(P_3)$ is an inner Galois point, and \n\\item[(c)] $\\sigma^*(P_3+\\sum_{\\gamma \\in G_{\\varphi(P_3)}}\\gamma(P_3))=P_2+\\sum_{\\tau \\in G_{\\varphi(P_2)}}\\tau(P_2)$\n\\end{itemize}\nare satisfied. \n\\end{proposition}\n\n\\begin{corollary} \\label{total flexes}\nLet $\\varphi(P_1), \\varphi(P_2)$ and $\\varphi(P_3)$ be different inner Galois points, and let $\\sigma \\in G_{\\varphi(P_1)}$ satisfy $\\sigma(P_2)=P_3$. \nIf $\\varphi(P_1)$, $\\varphi(P_2)$ and $\\varphi(P_3)$ are total inflection points, then there exists a linear transformation $\\tilde{\\sigma}$ of $\\mathbb{P}^2$ such that $\\varphi^{-1}\\circ\\tilde{\\sigma}\\circ\\varphi=\\sigma$. \n\\end{corollary}\n\n\\section{Preliminaries} \n\nWe recall the criterion presented in \\cite{fukasawa2} for two Galois points. \n\n\\begin{fact} \\label{criterion1} \nLet $G_1$ and $G_2$ be finite subgroups of ${\\rm Aut}(X)$ and let $P_1$ and $P_2$ be different points of $X$.\nThen, three conditions\n\\begin{itemize}\n\\item[(a)] $X\/{G_i} \\cong \\Bbb P^1$ for $i=1, 2$,    \n\\item[(b)] $G_1 \\cap G_2=\\{1\\}$, and\n\\item[(c)] $P_1+\\sum_{\\sigma \\in G_1} \\sigma (P_2)=P_2+\\sum_{\\tau \\in G_2} \\tau (P_1)  $\n\\end{itemize}\nare satisfied, if and only if there exists a birational embedding $\\varphi: X \\rightarrow \\mathbb P^2$ of degree $|G_1|+1$ such that $\\varphi(P_1)$ and $\\varphi(P_2)$ are different inner Galois points for $\\varphi(X)$ and $G_{\\varphi(P_i)}=G_i$ for $i=1, 2$. \n\\end{fact}\n\n\\begin{fact} \\label{criterion2} \nLet $G_1$ and $G_2$ be finite subgroups of ${\\rm Aut}(X)$ and let $Q$ be a point of $X$.\nThen, three conditions\n\\begin{itemize}\n\\item[(a)] $X\/{G_i} \\cong \\Bbb P^1$ for $i=1, 2$,    \n\\item[(b)] $G_1 \\cap G_2=\\{1\\}$, and\n\\item[(c')] $\\sum_{\\sigma \\in G_1} \\sigma (Q)=\\sum_{\\tau \\in G_2} \\tau (Q)  $\n\\end{itemize}\nare satisfied, if and only if there exists a birational embedding $\\varphi: X \\rightarrow \\mathbb P^2$ of degree $|G_1|$ and two outer Galois points $P_1$ and $P_2$ exist for $\\varphi(X)$ such that $G_{\\varphi(P_i)}=G_i$ for $i=1, 2$, and $\\overline{P_1P_2} \\ni Q$. \n\\end{fact}\n\nAccording to \\cite[Lemma 2.5]{fukasawa1}, the following holds. \n\n\\begin{fact} \\label{inner-lemma}\nAssume that $\\deg \\varphi(X) \\ge 4$, and points $\\varphi(P_1)$ and $\\varphi(P_2)$ are distinct inner Galois points for $\\varphi(X)$. \nThen, the line $\\overline{\\varphi(P_1)\\varphi(P_2)}$ is different from the tangent line at $\\varphi(P_1)$. \nIn particular, $\\sigma(P_1) \\ne P_2$ for each automorphism $\\sigma \\in G_{\\varphi(P_1)}$. \n\\end{fact}\n\\section{Proof of Theorems \\ref{main} and \\ref{main-outer}}\n\n\n\\begin{proof}[Proof of Theorem \\ref{main}]\nWe consider the if-part. \nIt follows from conditions (a) and (b) in Fact \\ref{criterion1} that conditions (a) and (b) are satisfied. \nBy Fact \\ref{criterion1}(c), since $\\varphi(P_1), \\varphi(P_2)$ and $\\varphi(P_3)$ are collinear Galois points, condition (c) is satisfied. \nLet $\\Lambda' \\subset |D|$ be the (base-point-free) linear system induced by $\\varphi$. \nSince $\\varphi(P_i)$ is inner Galois, $P_i+\\sum_{\\sigma \\in G_i}\\sigma(P_i) \\in \\Lambda'$, for $i=1, 2, 3$. \nTherefore, $\\dim \\Lambda \\le 2$. \nCondition (d) is satisfied. \n\nWe consider the only-if part. \nBy conditions (a), (b) and (c) and Fact \\ref{criterion1}, for each $i, j$ with $i \\ne j$, there exists a birational embedding $\\varphi_{ij}: X \\rightarrow \\mathbb{P}^2$ such that $\\varphi_{ij}(P_i)$ and $\\varphi_{ij}(P_j)$ are inner Galois points for $\\varphi_{ij}(X)$, $G_{\\varphi_{ij}(P_i)}=G_i$ and $G_{\\varphi_{ij}(P_j)}=G_j$. \nIt follows from Fact \\ref{inner-lemma} that \n$$ G_1P_1 \\ne G_1 P_2, \\mbox{ and } \\sum_{\\sigma \\in G_1}\\sigma(P_1) \\ne \\sum_{\\sigma \\in G_1} \\sigma(P_2). $$\nThen, by condition (a), there exists a function $f \\in k(X) \\setminus k$ such that \n$$ k(X)^{G_1}=k(f), \\ (f)=\\sum_{\\sigma \\in G_1}\\sigma(P_1)-\\sum_{\\sigma \\in G_1}\\sigma(P_2)$$\n(see also \\cite[III.7.1, III.7.2, III.8.2]{stichtenoth}). \nNote that, by condition (c), $(f)_{\\infty}=D-P_1$. \nSimilarly, there exist $g, h \\in k(X) \\setminus k$ such that \n$$ k(X)^{G_2}=k(g), \\ (g)=\\sum_{\\tau \\in G_2}\\tau(P_2)-(D-P_2) $$\nand \n$$ k(X)^{G_3}=k(h), \\ (h)=\\sum_{\\gamma \\in G_3}\\gamma(P_3)-(D-P_3).$$ \nThen, $\\varphi_{12}$ is represented by $(f:g:1)$ (see \\cite[Proofs of Proposition 1 and of Theorem 1]{fukasawa2}). \nLet $\\Lambda \\subset |D|$ be as in condition (d), and let $\\Lambda' \\subset |D|$ be the sublinear system corresponding to $\\langle f, g, 1 \\rangle$.\nSince $D, (f)+D, (g)+D \\in \\Lambda$, it follows that $\\Lambda' \\subset \\Lambda$.  \nBy condition (d), $\\Lambda'=\\Lambda$. \nThis implies that $P_3+\\sum_{\\gamma \\in G_3}\\gamma(P_3) \\in \\Lambda'$. \nTherefore, $h \\in \\langle f, g, 1\\rangle$. \nSince the covering map $X \\rightarrow X\/G_3$ is represented by $\\langle h, 1 \\rangle$, this covering map coincides with the projection from some smooth point of $\\varphi_{12}(X)$. \nSuch a center of projection coincides with $\\varphi_{12}(P_3)$, since the center is determined by ${\\rm supp}(D) \\cap {\\rm supp}((h)+D)$. \nThis implies that $\\varphi_{12}(P_3)$ is an inner Galois point. \nBy condition (c), points $\\varphi_{12}(P_1)$, $\\varphi_{12}(P_2)$ and $\\varphi_{12}(P_3)$ are collinear. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{main-outer}]\nWe consider the if-part. \nIt follows from conditions (a) and (b) in Fact \\ref{criterion2} that conditions (a) and (b) are satisfied. \nBy Fact \\ref{criterion2}(c'), since $P_1, P_2$ and $P_3$ are collinear outer Galois points, condition (c') is satisfied. \nLet $\\Lambda' \\subset |D|$ be the (base-point-free) linear system induced by $\\varphi$. \nSince $P_i$ is outer Galois, the linear system corresponding to $X \\rightarrow X\/G_i \\cong \\mathbb{P}^1$ is contained in $\\Lambda'$, for $i=1, 2, 3$. \nTherefore, $\\dim \\Lambda \\le 2$. \nCondition (d') is satisfied. \n\nWe consider the only-if part. \nBy condition (a), there exists a function $f \\in k(X) \\setminus k$ such that \n$$ k(X)^{G_1}=k(f), \\ (f)_{\\infty}=\\sum_{\\sigma \\in G_1}\\sigma(Q)$$\n(see also \\cite[III.7.1, III.7.2, III.8.2]{stichtenoth}). \nNote that, by condition (c'), $(f)_{\\infty}=D$. \nThe sublinear system corresponding to $\\langle 1, f \\rangle \\subset \\mathcal{L}(D)$ coincides with $\\Lambda_1 \\subset |D|$ as in condition (d'). \nSimilarly, there exist $g, h \\in k(X) \\setminus k$ such that \n$$ k(X)^{G_2}=k(g), \\ k(X)^{G_3}=k(h), \\mbox{ and } \\ (g)_{\\infty}=(h)_{\\infty}=D. $$\nFurthermore, the subspaces $\\langle 1, g \\rangle$ and $\\langle 1, h \\rangle$ correspond to the linear systems $\\Lambda_2$ and $\\Lambda_3$ as in condition (d'), respectively. Then, by conditions (b) and (c'), the morphism $\\varphi$ represented by $(f:g:1)$ is birational onto its image and outer Galois points $P_1$ and $P_2$ exist for $\\varphi(X)$ such that $G_{\\varphi(P_i)}=G_i$ for $i=1, 2$ (see \\cite[Proofs of Proposition 1 and of Theorem 1]{fukasawa2}). \nLet $\\Lambda \\subset |D|$ be as in condition (d'), and let $\\Lambda' \\subset |D|$ be the sublinear system corresponding to $\\langle f, g, 1 \\rangle$. \nSince $\\Lambda_1, \\Lambda_2 \\subset \\Lambda$, it follows that $\\Lambda' \\subset \\Lambda$.  \nBy condition (d'), $\\Lambda'=\\Lambda$. \nThis implies that $\\Lambda_3 \\subset \\Lambda'$. \nTherefore, $h \\in \\langle f, g, 1\\rangle$. \nSince the covering map $X \\rightarrow X\/G_3$ is represented by $\\langle h, 1 \\rangle$, this covering map coincides with the projection from some outer point $P_3 \\in \\mathbb{P}^2 \\setminus \\varphi(X)$. \nThen, $P_3$ is an outer Galois point. \nBy condition (c'), points $P_1, P_2$ and $P_3$ are collinear. \n\\end{proof}\n\n\n\n\\section{Proof of Propositions \\ref{uniqueness} and \\ref{extendable}}\n\n\\begin{proof}[Proof of Proposition \\ref{uniqueness}]\nWe consider inner Galois points. \nAssume that condition (c) in Fact \\ref{criterion1} is satisfied. \nLet $D:=P_1+\\sum_{\\sigma \\in G_1}\\sigma(P_2)=P_2+\\sum_{\\tau \\in G_2}\\tau(P_1)$. \nNote that, by Fact \\ref{inner-lemma}, $P_1+\\sum_{\\sigma \\in G_1}\\sigma(P_1) \\ne D$ and $P_1 \\not\\in {\\rm supp}(P_2+\\sum_{\\tau \\in G_2}\\tau(P_2))$. \nThe uniqueness of the linear system corresponding to a birational embedding follows, since a (base-point-free) linear system $\\Lambda \\subset |D|$ of dimension two such that \n$$D, \\ P_1+\\sum_{\\sigma \\in G_1}\\sigma(P_1), \\ P_2+\\sum_{\\tau \\in G_2}\\tau(P_2) \\in \\Lambda$$\nis uniquely determined. \n\nWe consider outer Galois points. \nAssume that condition (c') in Fact \\ref{criterion2} is satisfied. \nLet $D:=\\sum_{\\sigma \\in G_1}\\sigma(Q)=\\sum_{\\tau \\in G_2}\\tau(Q)$, and let $\\Lambda_i$ be the (base-point-free) linear system corresponding to the covering map $\\pi_i: X \\rightarrow X\/G_i \\cong \\mathbb{P}^1$ for $i=1, 2$. \nThen, $D \\in \\Lambda_i$ and $\\Lambda_i \\subset |D|$. \nIf $\\pi_1$ and $\\pi_2$ are realized as the projections from different outer Galois points for a birational embedding $\\varphi: X \\rightarrow \\mathbb{P}^2$, then $\\varphi$ is determined by a sublinear system $\\Lambda \\subset |D|$ such that $\\dim \\Lambda=2$ and $\\Lambda_1 \\cup \\Lambda_2 \\subset \\Lambda$, up to a projective equivalence.  \nTherefore, the uniqueness follows. \n\\end{proof} \n\n\n\\begin{proof}[Proof of Proposition \\ref{extendable}] \nLet $C:=\\varphi(X)$. \nWe consider the only-if part. \nAssume that there exists a linear transformation $\\tilde{\\sigma}$ of $\\mathbb{P}^2$ such that $\\varphi^{-1}\\circ\\tilde{\\sigma}\\circ\\varphi=\\sigma$.  \nFor a general line $\\ell \\ni \\varphi(P_1)$, $C \\cap \\ell$ contains at least two points (since $\\deg C \\ge 3$), and $\\tilde{\\sigma}((C \\cap \\ell)\\setminus \\{\\varphi(P_1)\\}) \\subset \\ell$. \nSince $\\tilde{\\sigma}$ is a linear transformation, $\\tilde{\\sigma}(\\ell)=\\ell$ follows. \nThis implies that $\\tilde{\\sigma}(\\varphi(P_1))=\\varphi(P_1)$. \nCondition (a) is satisfied. \nSince $\\varphi(P_3)=\\varphi(\\sigma(P_2))=\\tilde{\\sigma}(\\varphi(P_2))$, condition (b) is satisfied. \nSince the divisor $P_2+\\sum_{\\tau \\in G_{\\varphi(P_2)}}\\tau(P_2)$ corresponds to the tangent line of $\\varphi(X)$ at $\\varphi(P_2)$, conditions (c) is also satisfied. \n\nWe consider the if part. \nLet $\\Lambda$ be the linear system corresponding to the birational embedding $\\varphi: X \\rightarrow \\mathbb{P}^2$. \nAs in the proof of Proposition \\ref{uniqueness}, it follows from condition (b) that $\\Lambda$ is the smallest linear system containing the divisors \n$$D, \\ P_1+\\sum_{\\sigma \\in G_{\\varphi(P_1)}}\\sigma(P_1), \\ P_3+\\sum_{\\gamma \\in G_{\\varphi(P_3)}}\\gamma(P_3), $$ \nwhere $D:=P_1+\\sum_{\\sigma \\in G_{\\varphi(P_1)}}\\sigma(P_3)=P_3+\\sum_{\\gamma \\in G_{\\varphi(P_3)}}\\gamma(P_1)$. \nBy condition (a), divisors $D$ and $P_1+\\sum_{\\sigma \\in G_{\\varphi(P_1)}}\\sigma(P_1)$ are invariant under the action of $\\sigma^*$. \nSince $P_2+\\sum_{\\tau \\in G_{\\varphi(P_2)}}\\tau(P_2) \\in \\Lambda$, by condition (c), it follows that $\\sigma^*\\Lambda=\\Lambda$. \n\\end{proof} \n\n\\begin{proof}[Proof of Corollary \\ref{total flexes}] \nWe prove that conditions (a), (b) and (c) in Proposition \\ref{extendable} are satisfied. \nSince $\\varphi(P_1)$ is a total inflection point, by \\cite[III.8.2]{stichtenoth}, condition (a) is satisfied.  \nCondition (b) is satisfied by the assumption. \nSince $\\varphi(P_3)$ is a total inflection point, it follows from \\cite[III.8.2]{stichtenoth} that  \n$$ P_3+\\sum_{\\gamma \\in G_{\\varphi(P_3)}}\\gamma(P_3)=(|G_3|+1) P_3. $$\nTherefore, \n$$ \\sigma^*\\left(P_3+\\sum_{\\gamma \\in G_{\\varphi(P_3)}}\\gamma(P_3)\\right)=(|G_2|+1)P_2= P_2+\\sum_{\\tau \\in G_{\\varphi(P_2)}}\\tau(P_2). $$\nCondition (c) is satisfied.  \n\\end{proof}\n\n\\begin{center} {\\bf Acknowledgements} \\end{center} \nThe author is grateful to Doctor Kazuki Higashine for helpful discussions. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:assumpt-main-results}\nIn this paper we consider  the system of\n$k\\geq 2$ elliptic equations\n\\begin{equation}\\label{modelLK}\n-\\Delta u_i+u_i= f_i(u_i)-\\varkappa \\;u_i\\sum_{j\\neq i}u_j^2,\\quad\\text{ in } \\Omega,\n\\end{equation}\nfor $i=1,\\dots,k$.  It models the steady states of $k$ organisms, each\nof density $u_i$, which coexist in a smooth, connected, bounded domain\n$\\Omega\\subset{\\mathbb R}^N$; their dynamics is ruled out by internal growth $f_i$'s and \nmutual competition of Lotka-Volterra type with parameter $\\varkappa>0$.\nSystems of this form have attracted considerable attention both in\necology and social science since they furnish a relatively simple\nmodel to study the behavior of $k$ populations competing for the same\nresource $\\Omega$.  One of the main question is to investigate whether\n\\emph{coexistence} may occur, namely the existence of equilibrium\nconfigurations where all the densities $u_i$ are strictly positive on\nsets of positive measure, or the internal dynamic leads to\n\\emph{extinction}, that is steady states where one or more densities\nare null.  Many results are nowadays available, dealing mainly with\n$k=2$ populations.  We quote among others\n\\cite{eflg,gl,kl,lm,mhmm,skt}, where for logistic internal growth\n$f_i(u)=u(a_i-u)$, both the situation are proved to be possible\ndepending on the relations between the diffusion rates and the\ncoefficients of intra--specific and\nof inter--specific competitions, see also \\cite{ddh,dd}.\n\nA different perspective is proposed in\n\\cite{cf,ctv2,ctv-asymp,dd3,dg1,dhmp}, where the authors study the\neffect of very strong competition, letting the parameter $\\varkappa$\ngrowing indefinitely.  It is observed (see\nSection~\\ref{sec:asymptotic-analysis}) that the presence of large\ninteractions of competitive type produces, in the limit configuration\nas $\\varkappa\\to \\infty$, the \\emph{spatial segregation} of the\ndensities, meaning that if $(u_i^{\\varkappa})_{i=1,\\dots,k}$\nsolves~(\\ref{modelLK}), then $u_i^{\\varkappa}$ converges (in a suitable\nsense) to some $u_i$ which satisfies\n\\begin{equation}\\label{segrego}\n  u_i(x)\\cdot u_j(x)=0\\text{ a.e. in }\\Omega,\\qquad\\text{for all } i\\neq j.\n\\end{equation}\nA number of qualitative properties of the possible coexistence states\n$u_i$ and their supports is proved in\n\\cite{ctv-var,ctv-asymp,ctv-uniq}, with the aim of describing the way\nthe territory is partitioned by the segregated populations. We refer\nthe interested reader to the above quoted papers for  details on the\nregularity theory so far developed and to \\cite{ctv2,ctv-fucik} for\nsome applications.\n\n\nA further point of interest is to establish if coexistence of the\nspecies is possible in a segregated configuration: do all the species\nsurvive when the intra specific competition becomes larger and larger?\nThe answer cannot be positive in general: \\cite{kw} shows that in any\n\\emph{convex domain} the only stable configurations are those where\nonly one specie is alive.  It is worth pointing out that in\n\\cite{ctv-var,ctv-asymp}, the strict positivity of each component in\nthe limiting configuration is guaranteed by simply forcing\nnon-homogeneous Dirichlet boundary conditions\n\\begin{equation}\\label{dirnonomo}\nu_i=\\phi_i\\quad \\text{on }\\partial\\Omega,\n\\end{equation}\nwith $\\phi_i>0$ on a set of positive $(N-1)$--measure.  Coexistence\nresults for competing systems under more natural homogeneous boundary\nconditions are obtained in \\cite{cf} for the Dirichlet case\n\\begin{equation}\\label{diromo}\nu_i=0\\quad\\text{on }\\partial\\Omega,\n\\end{equation}\nwith interactions of the form $\\varkappa u_i\\sum_{j\\neq i}u_j$.  To avoid the extinction\npredicted by \\cite{kw}, a special class of\nnon-convex domains close to a union of $k$ disjoint balls is considered. Suitable\nnon-degeneracy assumptions on the $f_i$'s allow the application of a\ndomain perturbation technique envisaged in \\cite{d} which strongly relies on the continuity of\nthe eigenvalues of the Laplace operator with respect to the domain.\nIt is well known that such a property does not hold in the case of Neumann boundary conditions,\nsee for instance \\cite{arrieta}.  Hence, in order to treat Neumann\nno-flux boundary conditions, a different approach is needed.\n\nThis is precisely the aim of the present paper: we deal with system\n\\eqref{modelLK} coupled with\n\\begin{equation}\\label{neuomo}\n\\frac{\\partial u_i}{\\partial \\nu}=0\\quad\\text{on }\\partial\\Omega,\n\\end{equation}\nin\na class of non-convex domains $\\Omega=\\Omega_\\e$ suitably approximating a given domain $\\Omega_0$ \ncomposed by $k$ disjoint open sets, see Figure\n\\ref{fig:dd}.\n\\begin{figure}[h]\n  \\centering \\subfigure [example of $\\Omega_{0}$ with $k=3$]{\n    \\begin{psfrags}\n      \\psfrag{B1}{$\\Omega^1$} \\psfrag{B2}{$\\Omega^2$}\n      \\psfrag{B3}{$\\Omega^3$}\n      \\includegraphics[width=5.5cm]{triplo1.eps}\n    \\end{psfrags}\n  } \\quad\\quad\\quad \\subfigure[example of $\\Omega_\\e$ approximating $\\Omega_0$ ] {\n    \\includegraphics[width=5.5cm]{triplo2.eps}\n  }\n  \\caption{}\\label{fig:dd}\n\\end{figure}\n\nDue to the variational structure of  problem \\eqref{modelLK},\nthe following  \\emph{free energy} functional\n\\begin{equation}\\label{freeenergy}\n  J_{\\Omega}(U)=\\sum_{i=1,\\dots, k}\\left\\{\\frac12\n    \\int_\\Omega\\big(|\\nabla u_i(x)|^2+|u_i(x)|^2\\big)dx-\n    \\int_\\Omega F_i(u_i(x))dx\\right\\},\n\\end{equation}\ngiven by the sum of the internal energies of the $k$ densities $u_i$,\neach having internal potential $F_i(x,s)=\\int_0^s f_i(x,u)du$,\nis naturally associated to the system.\n\nOur analysis will highlight how the coexistence of all the densities\nis connected to the following minimization problem: \\emph{finding\n  local minimizers of $J_\\Omega(U)$ in the class of of segregated\n  states}\n\\[{\\mathcal U}=\\left\\{U=(u_1,u_2,\\dots,u_k) \\in\n  \\big(H^1(\\Omega)\\big)^k:\\, u_i\\geq 0,\\ u_i\\cdot u_j=0\\text{ if\n  }i\\neq j,\\text{ a.e. in }\\Omega\\right\\}.\\] The problem of the\nexistence of the \\emph{global minimum} of $J_\\Omega(U)$ in $\\mathcal\nU$ was investigated in \\cite{ctv-var} under the non-homogeneous\nconditions \\eqref{dirnonomo}.  As we shall see in Theorem\n\\ref{p:trivialglobmin}, the global minimizer under homogeneous\nboundary conditions is in general trivial, namely a $k$-tuple with all\nbut one component identically null.  Hence, the only possibility for\nfinding a \\emph{stable} coexistence solution where all the $k$\ndensities survive, consists in looking for \\emph{local} minimizers of\n$J_\\Omega$, see problem ${\\boldsymbol{(P_\\e)}}$ below.\n\nExploiting the variational character of the interaction term in\n\\eqref{modelLK} and developing a suitable domain perturbation\ntechnique, in this paper we give positive answer to both\nquestions of minimization of $J_\\Omega$ and occurrence of coexistence\nstates for the system.  Our main result can be summarized as follows:\nunder suitable assumptions on $f_i$'s ensuring the existence of a\nnon-degenerate solution to the system on the unperturbed domain\n$\\Omega_0$ (see \\eqref{ND} below), \\emph{for all $\\Omega_\\eps$ close\n  enough to $\\Omega_0$ and large parameter $\\varkappa$, there exists\n  $(u_1^\\varkappa,\\dots,u_k^\\varkappa)$ solution to \\eqref{modelLK} in\n  $\\Omega_\\eps$, whose limit configuration as $\\varkappa\\to\\infty$ is a\n  segregated coexistence state $(u_1,\\dots,u_k)$ with $k$ positive\n  components (i.e. each component $u_i\\geq 0$ and $u_i$ is strictly\n  positive on a set of positive measure), characterized as a local\n  minimizer of the free energy~$J_{\\Omega_\\e}$.}\n\nBefore stating rigorously our assumptions and main results, a further\nremark is in order.  As observed in \\cite{ctv-var} (see also Theorem\n\\ref{t:extremality}) any $(u_1,\\dots,u_k)$ which is a local minimizer\nof the free energy $J_\\Omega$ on $\\mathcal U$, is also a solution of\nthe following system of distributional inequalities:\n\\begin{equation}\\label{2kvar}\n    \\begin{cases}\n      \\displaystyle\\int_{\\Omega} \\big(\\nabla u\n      _i(x)\\nabla\\phi(x)+u_i(x) \\phi(x)-\n      f_i(u _i(x))\\phi(x)\\big)dx\\leq 0,\\\\[10pt]\n      \\displaystyle\\int_{\\Omega}\\big(\\nabla \\widehat u\n      _i(x)\\nabla\\phi(x)+\\widehat u_i(x) \\phi(x)-\\widehat f(\\widehat\n      u _i(x))\\phi(x))dx\\geq 0,\n    \\end{cases}\n\\end{equation}\n$i=1,\\dots,k$, for any non-negative $\\phi\\in H^1(\\Omega)$, where we\nhave denoted $\\widehat u_i = u_i-\\sum_{h\\neq i}u_h$ and $\\widehat\nf(\\widehat u_i) =f(u_i)-\\sum_{j\\neq i} f_j(u_j)$.  The link between\nsystems of this form and population dynamics has been pointed out in\n\\cite{cf,ctv-var,ctv-asymp}: as a matter of fact \\emph{all} the\nlimiting configurations as $\\varkappa\\to\\infty$ of the solutions to\n\\eqref{modelLK} are solutions to \\eqref{2kvar}. In other words, the possibility of coexistence\nof many species ruled out by strong competition is governed by the\nsystem of distributional inequalities \\eqref{2kvar}: its independent\nstudy is thus crucial in population dynamics.  In this perspective our\nmain result can be reformulated in the following way: \\emph{ the system of differential\n  inequalities \\eqref{2kvar} has a solution $(u_1,\\dots,u_k)\\in\n  \\mathcal U$ with $k$ positive components.}\n\n\\section{Assumptions and main results}\n\n\\medskip\\noindent {\\bf Description of the domain.} We shall work in a\nclass of smooth non-convex domains $\\Omega_\\eps$ which generalizes the\ndumbbell form with many components as in \\cite{d}. \nLet $N\\geq 2$ and for\n$k\\in{\\mathbb N}$, let\n$$\\Omega_0=\\Omega^1\\cup\\Omega^2\\cup\\dots\\cup\\Omega^k,$$\nwhere  $\\Omega^i\\subset {\\mathbb R}^N$\nare open bounded smooth domains with mutually disjoint closures, i.e.\n\\begin{equation}\\label{unity}\n\\overline{\\Omega^i}\\cap\\overline{\\Omega^j}=\\emptyset\\qquad\\text{ if }i\\neq j.\n\\end{equation}\nFor any $\\e>0$, let\n$R_\\e\\subset({\\mathbb R}^N\\setminus \\Omega_0)$ be a bounded  measurable set  satisfying\nthe properties:\n\\begin{itemize}\n\\item[(i)]$ |R_\\e|\\to 0$ as $\\e\\to 0$\n\\item[(ii)] $\\Omega_0\\cup R_\\e$ is open and connected\n\\item[(iii)]  $\\partial(\\Omega_0\\cup R_\\e)$ is smooth.\n\\end{itemize}\nHere we denote with $|B|$  the Lebesgue measure of any set $B\\subset{\\mathbb R}^N$.\nFinally  we set\n$$\\Omega_\\eps:=\\Omega_0\\cup R_\\e.$$\n\n\\medskip\\noindent {\\bf Assumptions on the nonlinearity.}\nFor every $i=1,\\dots,k$, let $F_i\\in C^2({\\mathbb R})$ with $f_i=F'_i$ satisfying\n\\begin{itemize}\n\\item[(F1)] $F_i(0)=0$ and $f_i(0)=0$;\n\\item[(F2)] there exists $A_i>0$ such that\n$\\mu_i:=F_i(A_i)-\\frac{A_i^2}{2}=\n\\max_{t\\in[0,+\\infty)}\\big(F_i(t)-\\frac{t^2}{2}\\big)$;\n\n\\smallskip\n\\item[(F3)] $f'_i(A_i)<1$.\n\\end{itemize}\nNoticeable examples of nonlinearities satisfying (F1)--(F3) are\nlogistic type functions of the form\n$f_i(u)=\\lambda u-|u|^{p-1}u$ with $p>1$ and $\\lambda>1$.\n\nAssumption (F2) implies that $A_i =f_i(A_i)$ and hence\nthe constant function $u\\equiv A_i$ is a solution to problem\n\\begin{equation*}\n\\begin{cases}\n-\\Delta u+u= f_i(u),&\\text{ in } \\Omega,\\\\[5pt]\n\\;\\;\\;\\dfrac{\\partial u}{\\partial \\nu}=0,&\\text{ on } \\partial\\Omega,\n\\end{cases}\n\\end{equation*}\nin any open smooth domain $\\Omega$; moreover  $u\\equiv A_i$ minimizes\nthe internal energy\n$$\n\\int_\\Omega\\left(\\frac12\n|\\nabla u_i(x)|^2+\\frac12|u_i(x)|^2- F_i(u_i(x))\\right)dx.\n$$\n\n\\medskip\\noindent\nLet us denote $w_i=A_i\\raisebox{1.7pt}{$\\chi$}_{\\Omega^i}$,\n$i=1,\\dots,k$, $W=(w_1,w_2,\\dots,w_k)\\in \\big(H^1(\\Omega_0)\\big)^k$,\nand set\n\\begin{equation}\\label{e:mu}\n\\mu=\\sum_{i=1}^k\\left\\{ \\frac12\\int_{\\Omega_0}\\big(|\\nabla\n  w_i|^2+|w_i|^2\\big)\\,dx-\\int_{\\Omega_0} F_i(w_i)\\,dx\\right\\}\n=-\\sum_{i=1}^k\\mu_i|\\Omega^i|.\n\\end{equation}\nAssumption (F3) implies the following non-degeneracy property:\nfor all $i$ and $u\\in H^1(\\Omega^i)$\n\\begin{equation}\\label{ND}\n  \\int_{\\Omega^i} (|\\nabla u|^2+|u|^2-f_i'(w_i)u^2)dx\n  \\geq\\nu\\int_{\\Omega^i}(|\\nabla u|^2+|u|^2) dx,\n\\end{equation}\nwhere $\\nu:=\\min_{i=1,\\dots,k}\\{1,1-f_i'(A_i)\\}>0$.\n\n\\medskip\nWe are now going to describe the  main results of the present paper,\n starting from the following optimal partition problem.\n\n\\medskip\\noindent\n{\\bf Problem ${\\boldsymbol{(P_\\e)}}$}.\nFind \\emph{nontrivial} local minimizers of the functional\n\\begin{align*}\n&J_{\\Omega_\\e}:\\big(H^1(\\Omega_\\e)\\big)^k\\to(-\\infty,+\\infty],\\\\\n&  J_{\\Omega_\\e}(U)=\\sum_{i=1,\\dots, k}\\left\\{\\frac12\n    \\int_{\\Omega_\\e}\\big(|\\nabla u_i(x)|^2+|u_i(x)|^2\\big)dx-\n    \\int_{\\Omega_\\e} F_i(u_i(x))dx\\right\\},\n\\end{align*}\namong $k$-tuples $U=(u_1,u_2,\\dots,u_k)$ belonging to the class\n\\[\n{\\mathcal U}_\\e=\\left\\{U=(u_1,u_2,\\dots,u_k) \\in\n  \\big(H^1(\\Omega_\\e)\\big)^k:\\, u_i\\geq 0,\\ u_i\\cdot u_j=0\\text{ if }i\\neq\n  j,\\text{ a.e. in }\\Omega_\\e\\right\\}.\n\\]\nBy \\emph{nontrivial} we mean that no component $u_i$\nof the solution $U$ can be null, i.e. $u_i\\not\\equiv 0$ for all\n$i=1,\\dots,k$.\nAs stated in the introduction, we shall prove that in any connected domain and\nfor a wide class of $F_i$'s\nincluding logistic-type nonlinearities, any\n\\emph{global minimizer} of the free energy \\eqref{freeenergy} is indeed trivial.\n\n\\begin{Proposition}\\label{p:trivialglobmin}\n  Let $\\Omega\\subset{\\mathbb R}^N$ be a connected open domain and $F_i\\in\n  C^2({\\mathbb R})$ satisfy  $F_i(0)=0$ and {\\em{(F2)}}.\nThen the infimum\n\\[\n\\lambda:=\\inf_{U\\in \\mathcal U}J_\\Omega(U)\n\\]\nis achieved by $U_0=(u_1^0,\\dots,u_k^0)$\nwith $u_{i_0}^0\\equiv A_{i_0}$ and $u_i^0\\equiv 0$ for $i\\neq i_0$, and\n\\[\\lambda=\n-\\mu_{i_0}|\\Omega|,\n\\]\nwhere $\\mu_{i_0}=\\max_{i\\in \\{1,\\dots,k\\}}\\mu_i$.  Furthermore, any\n$k$-tuple achieving $\\lambda$ has all but one component identically\nnull.\n\\end{Proposition}\nIn view of the above proposition, there is no hope to find\nnontrivial solutions to ${\\boldsymbol{(P_\\e)}}$ by global\nminimization.  On the contrary, by studying $J_{\\Omega_\\e}$ \\emph{near $W$},\nwe can find positive answer to the problem. To this aim let us\ndenote by\n\\[\nB^\\delta_\\e(W):=\\left\\{U\\in \\big(H^1(\\Omega_\\e)\\big)^k:\n  \\,\\|U-W\\|_{(H^1(\\Omega_0))^k}<\\delta\\right\\}\n\\]\nthe set of $k$-tuples $U$ whose restriction to $\\Omega_0$ is close\nwithin $\\delta>0$ to $W$, with respect to the $H^1$ norm\n$\\|V\\|^2_{(H^1(\\Omega_0))^k}= \\sum_{i=1}^k\\|v_i\\|^2_{H^1(\\Omega_0)}$.\nNotice that, if $U=(u_1,\\dots,u_k)\\in B^\\delta_\\e(W)$, then each  $u_i$\nsatisfies  $\\int_{\\Omega_i}|u_i-A_i|^2\\leq \\delta^2$.\nHence, if\n\\begin{equation}\\label{eq:nontrivi}\n\\delta^2<A_i^2 |\\Omega_i|,\\qquad i=1,\\dots,k\n\\end{equation}\nthen $u_i\\not\\equiv0$. Henceforward, $\\delta$ will be supposed\nto satisfy \\eqref{eq:nontrivi}, thus ensuring that any $U\\in\nB^\\delta_\\e(W)$ is nontrivial.\n\\begin{Theorem}\\label{t:partizione}\nAssume that (F1)--(F3) hold and let\n\\[\n\\lambda_{\\e}^{\\delta}:=\\inf_{U\\in{\\mathcal\n      U}_\\e\\cap {B^\\delta_\\e(W)}} J_{\\Omega_\\e}(U).\n\\]\nThen, there exists $\\delta>0$ such that, for every $\\e$ sufficiently\nsmall, $\\lambda_{\\e}^{\\delta}$ is achieved by a $k$-tuple\n$U_{\\e}=(u_1^{\\e},\\dots,u_k^{\\e})$ with $0\\leq u_i^\\e\\leq A_i$ a.e.\nin $\\Omega_\\e$ and $u_i^\\e\\not\\equiv 0$ for all $i=1,\\dots,k$.\n\\end{Theorem}\nThe proof of Theorem \\ref{t:partizione} will be obtained through a\ncareful analysis of the solutions to the original competitive system\n\\eqref{modelLK}, as the parameter $\\varkappa$ of the interspecific\ncompetition grows. Our main result reads as follows:\n\\begin{Theorem}\\label{t:sistemi}\n  Assume that (F1)--(F3) hold. Then, there exists $\\delta>0$ such\n  that, for every $\\e$ sufficiently small and and $\\varkappa>0$ sufficiently\n  large, system \\eqref{modelLK} coupled with \\eqref{neuomo} in\n  $\\Omega_\\e$ admits a solution\n  $U^{\\e,\\varkappa}=(u_1^{\\e,\\varkappa},\\dots,u_k^{\\e,\\varkappa})\\in B^\\delta_\\e(W)$\n  with the following properties:\n  \\begin{enumerate}\n  \\item $u_i^{\\e,\\varkappa}\\not\\equiv 0$ for all $i=1,\\dots,k$.\n  \\item $0\\leq u_i^{\\e,\\varkappa}\\leq A_i$ a.e. in $\\Omega_\\e$ for all\n    $i=1,\\dots,k$.\n  \\item There exists $V^\\e=(v_1^{\\e},...,v_k^{\\e})\\in \\mathcal\n    U_\\e\\cap B^\\delta_\\e(W)$ such that $v_i^\\e\\not\\equiv 0$ for every\n    $i$ and, up to subsequences, $U^{\\e,\\varkappa}\\to V^\\e$ strongly in\n    $(H^1(\\Omega_\\e))^k$ as $\\varkappa\\to+\\infty$.  Furthermore, $V^\\e$ is a\n    local minimizer of $J_{\\Omega_\\e}$, namely\n    $J_{\\Omega_\\e}(V^\\e)=\\lambda_\\e^\\delta$.\n\\end{enumerate}\n\\end{Theorem}\n\\noindent The proof of our results relies on the minimization on\nthe whole $B^\\delta_\\e(W)$ of an auxiliary functional obtained by\npenalizing the internal energy $J_{\\Omega_\\e}$ with a positive competition\nterm. More precisely we shall consider a suitable modification of the\nfunctional\n\\begin{multline*}\n  \\sum_{i=1}^k\\left\\{\n    \\frac12\\int_{\\Omega_\\e}\\big(|\\nabla\n    u_i(x)|^2+|u_i(x)|^2\\big)\\,dx-\\int_{\\Omega_\\e}\n   F_i(u_i(x))\\,dx\\right\\}\n  +\\varkappa \\sum_{\\substack{i,j=1\\\\i\\neq j}}^k \\int_{\\Omega_\\e}\n  u_i(x)^2u_j(x)^2\\,dx\n\\end{multline*}\ndefined on $(H^1(\\Omega_\\e))^k$, see $I_{\\eps,\\varkappa}$ in \\eqref{eq:Iek}\nbelow. Due to the variational character of the competition term in\n\\eqref{modelLK}, by standard Critical Point Theory, any local\nminimizer of the above function is a (weak) solution to the original\nsystem. Section \\ref{s:Iekappa} is devoted to the search for a local\nminimizer of $I_{\\eps,\\varkappa}$ in $B^\\delta_\\e(W)$ and requires the main\ntechnical effort of the paper. By developing a domain perturbation\nargument based on the nondegeneracy condition \\eqref{ND}, we shall\nsucceed in proving the existence of a minimizer in small perturbations\nof the domain $\\Omega_0$, for large values of the competition\nparameter $\\varkappa$.  In this way, we directly obtain the existence of a\npositive solution to the competitive system, at any fixed $\\varkappa$, see\nSection~\\ref{sec:competitive-systems}. In the subsequent Section\n\\ref{sec:asymptotic-analysis} we perform the asymptotic analysis of\nthese solutions as the competition parameter $\\varkappa\\to\\infty$, showing\nthat the steady states segregate in a nontrivial limit configuration\n$V^\\e$. The comparison between the minimal energy levels of\n$I_{\\eps,\\varkappa}$ and $J_{\\Omega_e}$ will allow proving that $V^\\e$\nindeed solves problem ${\\boldsymbol{(P _\\e)}}$ on $B^\\delta_\\e(W)$.\nThis concludes the proof of Theorem \\ref{t:sistemi} and, in turn, that\nof Theorem \\ref{t:partizione}. In the last part of Section\n\\ref{sec:asymptotic-analysis}, we show that any solution to the\noptimal partition problem ${\\boldsymbol{(P _\\e)}}$ satisfies some\nextremality conditions in the form of differential inequalities\n\\eqref{2kvar}.  Finally, in the last section we derive some\nconsequences of this fact, and outline further developments of the\nsubject.\n\n\n\n\\section{A variational problem}\\label{s:Iekappa}\nAim of this section is to study the minimization of a suitable\nfunctional on $(H^1(\\Omega_\\e))^k$, which will reveal to be strongly\nrelated both to problem ${\\boldsymbol{(P _\\e)}}$ and to the original\ncompetitive system. The functional is defined as follows:\n\\begin{multline}\\label{eq:Iek}\n  I_{\\e,\\varkappa}(U)=\\sum_{i=1}^k\\left\\{\n    \\frac12\\int_{\\Omega_\\e}\\big(|\\nabla\n    u_i(x)|^2+|u_i(x)|^2\\big)\\,dx-\\int_{\\Omega_\\e}\n    \\widetilde F_i(u_i(x))\\,dx\\right\\}\\\\\n  +\\varkappa \\sum_{\\substack{i,j=1\\\\i\\neq j}}^k \\int_{\\Omega_\\e}\n  G_i(u_i(x))G_j(u_j(x))\\,dx\n\\end{multline}\nwhere\n\\[\n\\widetilde F_i(t)=\\begin{cases}\n0,\\quad &\\text{if }t\\leq 0,\\\\\nF_i(t),\\quad &\\text{if }0\\leq t\\leq  A_i,\\\\\nA_it+F_i(A_i)-A_i^2,\\quad &\\text{if }t\\geq A_i,\n\\end{cases}\n\\]\nand\n\\[\nG_i(t)=\\begin{cases}\nt^2,\\quad &\\text{if }|t|\\leq A_i,\\\\\n2A_i|t|-A_i^2,\\quad &\\text{if }|t|> A_i.\n\\end{cases}\n\\]\nNotice that  $I_{\\e,\\varkappa}\\in C^1\\big((H^1(\\Omega_\\e))^k,{\\mathbb R}\\big)$.\nAim of this section is to prove\n\\begin{Theorem}\\label{t:auxiliary}\nAssume that (F1)--(F3) hold and let\n\\[\nc_{\\e,\\varkappa}:=\\inf_{U\\in{\\mathcal\n      U}_\\e\\cap {B^\\delta_\\e(W)}} I_{\\e,\\varkappa}(U).\n\\]\nThen, there exists $\\delta>0$ such that, for every for $\\e>0$\nsufficiently small and $\\varkappa>0$ sufficiently large, $c_{\\e,\\varkappa}$ is\nachieved by a $k$-tuple $U^{\\e,\\varkappa}=(u_1^{\\e,\\varkappa},\\dots,u_k^{\\e,\\varkappa})$\nwith $0\\leq u_i^{\\e,\\varkappa}\\leq A_i$ a.e.  in $\\Omega_\\e$ and\n$u_i^{\\e,\\varkappa}\\not\\equiv 0$ for all $i=1,\\dots,k$.\n\\end{Theorem}\n\nThe first step in this direction consists in proving that the minimum\nis achieved on the closure of $B^\\delta_\\e(W)$, namely the set\n\\[\n\\overline{B^\\delta_\\e(W)}:=\\left\\{U\\in \\big(H^1(\\Omega_\\e)\\big)^k:\n  \\,\\|U-W\\|_{(H^1(\\Omega_0))^k}\\leq \\delta\\right\\}.\n\\]\n\\begin{Lemma}\\label{l:Lambdaeps}\n  For every $\\delta$ satisfying \\eqref{eq:nontrivi}, $\\e\n  \\in(0,1)$, and $\\varkappa>0$, the infimum\n  \\[\n  \\Lambda_{\\e,\\varkappa}=\\inf_{U\\in \\overline{B^\\delta_\\e(W)}} I_{\\e,\\varkappa}(U)\n  \\]\n  is achieved by a $k$-tuple\n  $U^{\\e,\\varkappa}=(u_1^{\\e,\\varkappa},\\dots,u_k^{\\e,\\varkappa})$ where\n  $u_i^{\\e,\\varkappa}\\not\\equiv 0$ and\n\\begin{equation}\\label{eq:apriori1sis}\n  0\\leq u_i^{\\e,\\varkappa}(x)\\leq A_i\\quad \\text{for a.e. }x\\in\\Omega_\\e.\n\\end{equation}\n\\end{Lemma}\n\\begin{pf}\n  We first observe that $\\frac12 t^2-\\widetilde F_i(t)\\geq \\frac12\n  A_i^2-F_i(A_i)$ for all $t\\in{\\mathbb R}$, hence, being the coupling term\n  nonnegative, for all $U=(u_1,\\dots,u_k)\\in\n  \\big(H^1(\\Omega_\\e)\\big)^k$\n\\[\nI_{\\e,\\varkappa}(U)\\geq \\sum_{i=1}^k\\int_{\\Omega_\\e}\\bigg(\\frac12\n|u_i|^2-\\widetilde F_i(u_i)\\bigg)dx\\geq\n\\sum_{i=1}^k\\bigg(\\frac{A_{i}^2}2-F_{i}(A_{i})\\bigg)|\\Omega_\\e|,\n\\]\nand hence $\\Lambda_{\\e,\\varkappa}>-\\infty$.  Let\n$\\big\\{U_n=(u_1^n,\\dots,u_k^n)\\big\\}_{n\\in{\\mathbb N}}$ be a minimizing\nsequence, i.e. $ U_n\\in{\\overline{B^\\delta_\\e(W)}}$ and\n$\\lim_{n\\to+\\infty}I_{\\e,\\varkappa}(U_n)=\\Lambda_{\\e,\\varkappa}$.  We notice\nthat, by definition of $\\widetilde F_i$ and the fact that $w_i\\geq\n0$ a.e., we can choose $U_n$ such that $u_i^n\\geq 0$ a.e. in\n$\\Omega_\\e$ for all $i=1,\\dots,k$ (otherwise we take\n$((u_1^n)^+,\\dots,(u_k^n)^+)$ with $(u_i^n)^+:=\\max\\{u_i^n,0\\}$ as\na new minimizing sequence). Letting $V_n=(v_1^n,\\dots,v_k^n)$ with\n$v_i^n=\\min\\{u_i^n,A_i\\}$, it is easy to verify that $V_n\\in\n\\overline{B^\\delta_\\e(W)}$ and $I_{\\e,\\varkappa}(V_n)\\leq\nI_{\\e,\\varkappa}(U_n)$.  Then also $\\big\\{V_n\\big\\}_{n\\in{\\mathbb N}}$ is a\nminimizing sequence.\n\nSince $\\big\\{V_n\\big\\}_{n\\in{\\mathbb N}}$ is a minimizing sequence and it is\nuniformly bounded , it is easy to realize that\n$\\big\\{V_n\\big\\}_{n\\in{\\mathbb N}}$ is bounded in $\\big(H^1(\\Omega_\\e)\\big)^k$,\nhence there exists a subsequence, still denoted as\n$\\big\\{V_n\\big\\}_{n\\in{\\mathbb N}}$, which converges to some\n$V=(v_1,\\dots,v_k)\\in \\big(H^1(\\Omega_\\e)\\big)^k$ weakly in\n$\\big(H^1(\\Omega_\\e)\\big)^k$, strongly in $\\big(L^2(\\Omega_\\e)\\big)^k$\nand a.e. in $\\Omega_\\e$.  A.e.  convergence implies that $0\\leq\nv_i\\leq A_i$ a.e. in $\\Omega_\\e$, while weakly lower semi-continuity\nimplies that $V\\in \\overline{B^\\delta_\\e(W)}$.  From $0\\leq v_i\\leq\nA_i$ and the Dominated Convergence Theorem, it follows that\n\\begin{align*}\n  &\\lim_{n\\to+\\infty}\\int_{\\Omega_\\e}\\widetilde\n  F_i(v_i^n(x))\\,dx=\\int_{\\Omega_\\e}\\widetilde F_i(v_i(x))\\,dx,\\\\\n  &\\lim_{n\\to+\\infty}\\int_{\\Omega_\\e} G_i(v_i^n(x))G_j(v_j^n(x))\\,dx\n  =\\int_{\\Omega_\\e} G_i(v_i(x))G_j(v_j(x))\\,dx,\n\\end{align*}\nfor every $i,j=1,\\dots,k$, which, together with  lower\nsemi-continuity, yields\n\\[\n\\Lambda_{\\e,\\varkappa}\\leq I_{\\e,\\varkappa}(V)\\leq\n\\liminf_{n\\to+\\infty}I_{\\e,\\varkappa}(V_n)=\n\\lim_{n\\to+\\infty}I_{\\e,\\varkappa}(V_n)=\\Lambda_{\\e,\\varkappa},\n\\]\nthus proving that $V$ attains $\\Lambda_{\\e,\\varkappa}$.\n\n Finally, if\n$v_i\\equiv 0$ in $\\Omega_i$ then\n$\\|v_i-A_i\\|_{H^1(\\Omega^i)}^2=\\int_{\\Omega^i} A_i^2 dx\\leq\n\\delta^2$, in contradiction with the choice of $\\delta$ as in \\eqref{eq:nontrivi}.\n\\end{pf}\n\nA major effort is now needed to show that the minimum provided by\nLemma \\ref{l:Lambdaeps} indeed belongs to the open set\n$B_\\delta^\\e(W)$.  The crucial ingredient in this direction\nconsists in providing suitable estimates of the minimal level\n$\\Lambda_{\\e,\\varkappa}$, which require the following technical lemma.\n\\begin{Lemma}\\label{l:continuity}\n  For every $\\eta>0$ there exists $\\delta_{\\eta}>0$ such that if\n  $U\\!=\\!(u_1,\\dots,u_k)\\in \\overline{B^{\\delta_\\eta}_\\e\\!(W)}$ and $|u_i(x)|\\leq\n  A_i$ for a.e. $x\\in\\Omega_0$ and for all $i=1,\\dots,k$, then\n  \\begin{equation}\\label{eq:continuity}\n  \\sum_{i=1}^k\\int_{\\Omega_0}\\bigg[F_i(u_i)-F_i(w_i)-\n  f_i(w_i)(u_i-w_i)-\\frac12 f_i'(w_i)(u_i-w_i)^2\\bigg]\\,dx\\leq\\eta\n  \\|U-W\\|^2_{(H^1(\\Omega_0))^k}.\n\\end{equation}\n\\end{Lemma}\n\\begin{pf}\nWe have\n\\begin{align*}\n  &\\int_{\\Omega_0}\\bigg[F_i(u_i)-F_i(w_i)- F_i'(w_i)(u_i-w_i)-\n  \\frac12 F_i''(w_i)(u_i-w_i)^2\\bigg]\\,dx\\\\\n  &=\\int_{\\Omega_0}\\!\\bigg[\\!\n  \\int_0^1\\!\\bigg[\\!\\bigg(\\!\\frac{d}{dt}F_i(t\\,u_i+(1-t)w_i)\\!\\bigg)\\!-\n  F_i'(w_i)(u_i-w_i)- t\\, F_i''(w_i)(u_i-w_i)^2\\bigg]dt\\bigg]dx\\\\\n&=\\int_{\\Omega_0}\\!\\bigg[\\!\n  \\int_0^1\\!\\big[F_i'(t\\,u_i+(1-t)w_i)-\n  F_i'(w_i)- t\\, F_i''(w_i)(u_i-w_i)\\big](u_i-w_i)\\,dt\\bigg]dx\n\\\\\n&=\\int_{\\Omega_0}\\!\\bigg[\\!\n  \\int_0^1\\!\\!\n  \\bigg(\\int_0^1\\!\\!\\bigg(\\!\\frac{d}{ds}\nF_i'\\big(s(t\\,u_i+(1-t)w_i)+(1-s)w_i\\big)\\bigg)ds\\\\\n&\\hskip7cm-\n  t\\, F_i''(w_i)(u_i-w_i)\\!\\bigg)(u_i-w_i)\\,dt\\bigg]dx.\n\\end{align*}\nHence, by H\\\"older's inequality,\n\\begin{align*}\n  \\int_{\\Omega_0}&\\Big[F_i(u_i)-F_i(w_i)-\n  F_i'(w_i)(u_i-w_i)-\n  \\frac12 F_i''(w_i)(u_i-w_i)^2\\Big]\\,dx\\\\\n  &\\leq \\int_{\\Omega_0}\\!\\bigg[\\!  \\int_0^1\\!  \\bigg(\\int_0^1\\!\\big(\n  F_i''\\big(st(u_i-w_i)+w_i\\big)-\n  F_i''(w_i)\\big)t(u_i-w_i)^2ds\\bigg)\\,dt\\bigg]\\,dx\\\\\n  &\\leq\n  \\|u_i-w_i\\|_{L^{p}(\\Omega_0)}^2\\iint_{(0,1)\\times(0,1)}t\\|F_i''\\big(st(u_i-w_i)\n  +w_i\\big)-F_i''(w_i)\\|_{L^{\\frac{p}{p-2}}(\\Omega_0)}\\,ds\\,dt,\n\\end{align*}\nwhere $p=2^*$ for $N\\geq3$ and $p\\in(2,+\\infty)$ for $N=2$.  The\nconclusion follows now from Sobolev's embeddings and the\ncontinuity of the operator\n\\begin{align*}\n   F_i'':\\big\\{v\\in H^1(\\Omega_0):\\,|v(x)|\\leq 3A_i\\big\\}&\\to\n  L^{\\frac{p}{p-2}}(\\Omega_0),\\\\\n   v&\\mapsto F_i''(v),\n\\end{align*}\nwhich can be easily proved using  the Dominated Convergence\nTheorem.\n\\end{pf}\n\n\\begin{remark}\\label{rem:ipotesi}\n According to Lemma \\ref{l:continuity},\n besides (\\ref{eq:nontrivi}) from now on we assume\n\\begin{equation*}\n0<\\delta\\leq \\delta_0\n\\end{equation*}\nwith $\\delta_0$ small enough in such a way that inequality\n(\\ref{eq:continuity}) with $\\eta=\\min\\{\\frac{\\nu}{4},\\frac18\\}$\nholds for all functions $U=(u_1,\\dots,u_k)\\in\n\\overline{B^{\\delta_0}_\\e(W)}$ satisfying $|u_i(x)|\\leq A_i$ a.e.\nin $\\Omega_0$. We also require that $\\delta_0\\leq A_i^2\/4$ and finally\nthat condition \\eqref{eq:sobo} in Lemma \\ref{l:stima2} is\nsatisfied.\n\\end{remark}\n\n\n\n\\noindent By exploiting the separation of the $\\Omega^i$'s as in \\eqref{unity}, for every $i=1,\\dots,k$, \nwe can construct test functions $\\varphi^i\\in H^1({\\mathbb R}^N)$\nsatisfying\n\\begin{equation}\\label{eq:apriori2}\n0\\leq\\varphi^i(x)\\leq A_i \\quad \\text{a.e. in }{\\mathbb R}^N,\n\\end{equation}\n$\\varphi_i(x)=0$ for all $x\\in \\Omega_0\\setminus \\Omega_i$,\n$\\varphi_i(x)=A_i$ if $x\\in\\Omega_i$, and $\\varphi_i\\cdot\\varphi_j=0$\na.e. in ${\\mathbb R}^N$ if $i\\neq j$.  \nThis allows us to provide an estimate from above of the value $\\Lambda_{\\e,\\varkappa}$\nin terms of the total free-energy of $W$.\n\\begin{Lemma}\\label{l:stima1}\n  For every $\\e\\in(0,1)$, there exists $\\tau_\\e$ such that $\\tau_\\e\\to\n  0$ as $\\e\\to 0$ and, for all $\\varkappa>0$,\n  \\[\n  \\Lambda_{\\e,\\varkappa}\\leq \\mu+\\tau_\\e,\n  \\]\nwith $\\mu$ given by \\eqref{e:mu}.\n\\end{Lemma}\n\\begin{pf}\nLet $\\varphi_\\e^i\\in\nH^1(\\Omega_\\e)$ be the restriction of $\\varphi_i$ to $\\Omega_\\e$.\nNotice that $(\\varphi_\\e^1,\\varphi_\\e^2,\\dots,\\varphi_\\e^k)\\in\n  \\overline{B^\\delta_\\e(W)}$ and that $\\varphi_\\e^i\\cdot\\varphi_\\e^j\\equiv\n  0$ if $i\\neq j$. Hence we have \n\\begin{align*}\n  \\Lambda_{\\e,\\varkappa}&\\leq  I_{\\e,\\varkappa}(\\varphi_\\e^1,\\varphi_\\e^2,\\dots,\\varphi_\\e^k)\\\\\n  &=\\mu+ \\sum_{i=1}^k\\left\\{ \\frac12\\int_{R_\\e}\\big(|\\nabla\n    \\varphi_\\e^i(x)|^2+|\\varphi_\\e^i(x)|^2\\big)\\,dx-\\int_{R_\\e}\n    F_i(\\varphi_\\e^i)\\,dx\\right\\}\\\\\n    &= \\mu+\\tau_\\e,\n\\end{align*}\nwhere  \\begin{equation*}\n\\tau_\\e=\\sum_{i=1}^k\\left\\{ \\frac12\\int_{R_\\e}\\big(|\\nabla\n    \\varphi^i(x)|^2+|\\varphi^i(x)|^2\\big)\\,dx-\\int_{R_\\e}\n    F_i(\\varphi^i)\\,dx\\right\\}.\n\\end{equation*}\nSince $|R_\\e|\\to 0$ as $\\e\\to 0$, then $\\tau_\\e\\to 0$,\nproving the stated estimate.\n\\end{pf}\n\\begin{Lemma}\\label{l:stima2}\n  For every $\\e\\in(0,1)$, there exists $\\sigma_\\e$ such that\n  $\\sigma_\\e\\to 0$ as $\\e\\to0$ and\n  \\[\n  \\|U^{\\e,\\varkappa}-W\\|^2_{(H^1(\\Omega_0))^k}\\leq \\sigma_\\e\n  \\]\nfor every $\\varkappa>\\max\\limits_{i\\neq j }\\frac{2f_i'(0 )}{A_j^2}$.\n\\end{Lemma}\n\\begin{pf}\n  From (\\ref{eq:apriori1sis}), we can write\n  $\\Lambda_{\\e,\\varkappa}=I^1_{\\e,\\varkappa}+I^2_{\\e,\\varkappa}$ where\n\\begin{align*}\n  &I^1_{\\e,\\varkappa}=\\sum_{i=1}^k\\bigg\\{\n  \\frac12\\int_{\\Omega_0}\\!\\!\\big(|\\nabla\n  u_i^{\\e,\\varkappa}|^2+|u_i^{\\e,\\varkappa}|^2\\big)\\,dx-\\!\\!\\int_{\\Omega_0}\\!\\!\n  F_i(u_i^{\\e,\\varkappa})\\,dx+ \\varkappa \\sum_{j\\neq i} \\int_{\\Omega_0}\\!\\!\n  (u_i^{\\e,\\varkappa})^2(u_j^{\\e,\\varkappa})^2\\,dx\\bigg\\}\\\\\n  &I^2_{\\e,\\varkappa}=\\sum_{i=1}^k\\bigg\\{ \\frac12\\int_{R_\\e}\\!\\!\\big(|\\nabla\n  u_i^{\\e,\\varkappa}|^2+|u_i^{\\e,\\varkappa}|^2\\big)\\,dx-\\!\\!\\int_{R_\\e}\\!\\!\n  F_i(u_i^{\\e,\\varkappa})\\,dx + \\varkappa \\sum_{j\\neq i} \\int_{R_\\e}\\!\\!\n(u_i^{\\e,\\varkappa})^2(u_j^{\\e,\\varkappa})^2\\,dx\\bigg\\}.\n\\end{align*}\nSince by assumption $-\\Delta w_i+w_i=f_i(w_i)$ in $\\Omega_0$, we\ncan write each term in $I^1_{\\e,\\varkappa}$ as follows\n\\begin{align*}\n  \\frac12\\int_{\\Omega_0}&\\big(|\\nabla u_i^{\\e,\\varkappa} |^2+|u_i^{\\e,\\varkappa}\n  |^2\\big)\\,dx-\\int_{\\Omega_0} F_i(u_i^{\\e,\\varkappa} )\\,dx+ \\varkappa \\sum_{j\\neq\n    i} \\int_{\\Omega_0}\n  (u_i^{\\e,\\varkappa})^2(u_j^{\\e,\\varkappa})^2\\,dx\\\\\n  &=\\frac12\\int_{\\Omega_0}\\big(|\\nabla w_i |^2+|w_i\n  |^2\\big)\\,dx-\\int_{\\Omega_0}\n  F_i(w_i )\\,dx\\\\\n  &\\quad+ \\frac12\\int_{\\Omega_0}\\big(|\\nabla (u_i^{\\e,\\varkappa}-w_i)\n  |^2+|(u_i^{\\e,\\varkappa}-w_i) |^2\\big)\\,dx\n  -\\int_{\\Omega_0} \\big( F_i(u_i^{\\e,\\varkappa} )-F_i(w_i )\\big)\\,dx\\\\\n  &\\quad+ \\int_{\\Omega_0}\\big(\\nabla w_i \\cdot\\nabla(u_i^{\\e,\\varkappa}-w_i)\n  +w_i (u_i^{\\e,\\varkappa}-w_i) \\big)\\,dx+ \\varkappa \\sum_{j\\neq i} \\int_{\\Omega_0}\n  (u_i^{\\e,\\varkappa})^2(u_j^{\\e,\\varkappa})^2\\,dx\\\\\n  &=-\\mu_i|\\Omega^i| +\\alpha^1_{\\e,\\varkappa,i}+\\alpha^2_{\\e,\\varkappa,i}\\\\\n  &\\quad -\\int_{\\Omega_0} \\big( F_i(u_i^{\\e,\\varkappa} )-F_i(w_i )- f_i(w_i\n  )(u_i^{\\e,\\varkappa}-w_i)-\\frac12f_i'(w_i )(u_i^{\\e,\\varkappa}-w_i)^2 \\big)\\,dx.\n\\end{align*}\nwhere\n\\[\n\\alpha^1_{\\e,\\varkappa,i}=\\frac12\\|u_i^{\\e,\\varkappa}-w_i\\|_{H^1(\\Omega^i)}^2-\n  \\frac12 \\int_{\\Omega^i}f_i'(A_i )(u_i^{\\e,\\varkappa}-w_i)^2 + \\varkappa\n  \\sum_{j\\neq i} \\int_{\\Omega^i}\n  (u_i^{\\e,\\varkappa})^2(u_j^{\\e,\\varkappa})^2\\,dx\n\\]\nand\n\\[\n\\alpha^2_{\\e,\\varkappa,i}=\\frac12 \\sum_{j\\neq i}\\int_{\\Omega^j}\\bigg(\n|\\nabla u_i^{\\e,\\varkappa}|^2+|u_i^{\\e,\\varkappa}|^2- \\Big[f_i'(0 )- 2\\varkappa\n  \\sum_{h\\neq i}(u_h^{\\e,\\varkappa})^2\\Big]|u_i^{\\e,\\varkappa}|^2\\bigg)\\,dx.\n\\]\nFrom (\\ref{ND}) it follows that\n\\begin{equation}\\label{eq:6}\n\\alpha^1_{\\e,\\varkappa,i}\\geq\n\\frac\\nu2\\|u_i^{\\e,\\varkappa}-w_i\\|_{H^1(\\Omega^i)}^2.\n\\end{equation}\nOn the other hand, from H\\\"older's and Sobolev's inequalities it\nfollows that\n\\[\n\\alpha^2_{\\e,\\varkappa,i}\\geq\\frac12 \\sum_{j\\neq i}\\int_{\\Omega^j}\\bigg(\n\\Big(1-\\Big\\|\\Big[f_i'(0 )- 2\\varkappa\n  \\sum_{h\\neq i}(u_h^{\\e,\\varkappa})^2\\Big]^+\\Big\\|_{L^{\\frac{p}{p-2}}(\\Omega^j)}\nS_{p,j}^{-1}\\Big) |\\nabla\nu_i^{\\e,\\varkappa}|^2+|u_i^{\\e,\\varkappa}|^2\\bigg)\\,dx.\n\\]\nwhere $p=2^*$ for $N\\geq3$ and $p\\in(2,+\\infty)$ for $N=2$, and\n$S_{p,j}$ is the best constant in the Sobolev embedding\n$H^1(\\Omega^j)\\hookrightarrow L^p(\\Omega^j)$. Let us denote\n\\[\nA^\\delta_{\\varkappa,j}=\\{x\\in\\Omega^j:|u_j^{\\e,\\varkappa}-A_j|^2>\\delta\\}.\n\\]\nHence\n\\[\n\\delta^2\\geq\\int_{\\Omega^j}|u_j^{\\e,\\varkappa}-A_j|^2\\,dx\\geq \\delta\n|A^\\delta_{\\varkappa,j}|\n\\]\nand then $|A^\\delta_{\\varkappa,j}|<\\delta$. In particular, if $\\delta$\nis such that\n\\begin{equation}\\label{eq:sobo}\n\\delta^{\\frac{p-2}{p}}|(f_i'(0 ))^+|<\\frac{S_{p,j}}2,\n\\end{equation}\nthere holds\n\\begin{equation}\\label{eq:4}\n\\Big\\|\\Big[f_i'(0 )- 2\\varkappa \\sum_{h\\neq\n  i}(u_h^{\\e,\\varkappa})^2\\Big]^+\\Big\\|_{L^{\\frac{p}{p-2}}(A^\\delta_{\\varkappa,j})}\n<\\frac{S_{p,j}}2.\n\\end{equation}\nIn $\\Omega^j\\setminus A^\\delta_{\\varkappa,j}$, there holds\n$u_j^{\\e,\\varkappa}>A_j-\\sqrt \\delta>\\frac{A_j}{2}$ for $\\delta$ small\nas in Remark \\ref{rem:ipotesi}. Then, if $\\varkappa>{2f_i'(0\n)}\/{A_j^2}$,\n\\begin{equation}\\label{eq:5}\nf_i'(0 )- 2\\varkappa \\sum_{h\\neq i}(u_h^{\\e,\\varkappa})^2<0 \\quad \\text{in\n}\\Omega^j\\setminus A^\\delta_{\\varkappa,j}.\n\\end{equation}\nCollecting (\\ref{eq:4}) and (\\ref{eq:5}), we deduce that, for $\\varkappa\n>\\frac{2f_i'(0 )}{A_j^2}$,\n\\[\n\\Big\\|\\Big[f_i'(0 )- 2\\varkappa\n  \\sum_{h\\neq i}(u_h^{\\e,\\varkappa})^2\\Big]^+\\Big\\|_{L^{\\frac{p}{p-2}}(\\Omega^j)}\nS_{p,j}^{-1}<\\frac12,\n\\]\nand therefore\n\\begin{equation}\\label{eq:7}\n\\alpha^2_{\\e,\\varkappa,i}\\geq \\frac14 \\sum_{j\\neq\ni}\\|u_i^{\\e,\\varkappa}-w_i\\|_{H^1(\\Omega^j)}^2.\n\\end{equation}\nFrom (\\ref{eq:6}) and (\\ref{eq:7}), we obtain that\n\\[\n\\alpha^1_{\\e,\\varkappa,i}+\\alpha^2_{\\e,\\varkappa,i}\\geq\n\\min\\Big\\{\\frac\\nu2,\\frac14\\Big\\}\n\\|u_i^{\\e,\\varkappa}-w_i\\|_{H^1(\\Omega_0)}^2.\n\\]\nBy Lemma \\ref{l:continuity} and Remark \\ref{rem:ipotesi} we have\nthat\n\\begin{multline*}\n    \\sum_{i=1}^k\\int_{\\Omega_0}\\big[F_i(u_i^{\\e,\\varkappa} )-F_i(w_i )- f_i(w_i\n  )(u_i^{\\e,\\varkappa}-w_i)-\\frac12f_i'(w_i )(u_i^{\\e,\\varkappa}-w_i)^2 \\big]\\,dx\\\\\n    \\leq\n\\min\\Big\\{\\frac\\nu4,\\frac18\\Big\\}\n\\|U^{\\e,\\varkappa}-W\\|^2_{(H^1(\\Omega_0))^k}.\n  \\end{multline*}\nHence\n\\begin{equation}\\label{eq:i1}\n  I^1_{\\e,\\varkappa}\\geq \\mu+\\min\\Big\\{\\frac\\nu4,\\frac18\\Big\\}\n\\|U_{\\e}-W\\|^2_{(H^1(\\Omega_0))^k}.\n\\end{equation}\nOn the other hand, $I_{\\e,\\varkappa}^2$ can be promptly estimated by\n\\begin{align}\\label{eq:i2}\nI^2_{\\e,\\varkappa}\\geq -|R_\\e|\\sum_{i=1}^k\\mu_i\n\\end{align}\nwith $\\mu_i$ as in (F2).\nCombining inequalities \\eqref{eq:i1} and \\eqref{eq:i2}, it follows\nthat\n\\begin{equation}\\label{eq:sti2}\n\\Lambda_{\\e,\\varkappa}\\geq\n\\mu+\\eta\\|U_{\\e}-W\\|^2_{(H^1(\\Omega_0))^k}-|R_\\e|\\sum_{i=1}^k\\mu_i,\n\\end{equation}\nwhere $\\eta=\\min\\{\\frac{\\nu}{4},\\frac18\\}>0$. From Lemma\n\\ref{l:stima1} and (\\ref{eq:sti2}), we infer that\n$\\|U_{\\e}-W\\|^2_{(H^1(\\Omega_0))^k}\\leq \\sigma_\\e$ with\n$\\sigma_\\e=\\frac{1}{\\eta}(\\tau_\\e+|R_\\e|\\sum_{i=1}^k\\mu_i)$, concluding the proof.\n\\end{pf}\n\n\\medskip\\noindent\n\\begin{pfn}{Theorem \\ref{t:auxiliary}}\n  In order to conclude the proof of the theorem, it is sufficient to\n  consider $U^{\\e,\\varkappa}$ provided by Lemma \\ref{l:Lambdaeps}. If\n  $\\varkappa$ is large enough, we can apply Lemma \\ref{l:stima2} and we\n  infer that $U^{\\e,\\varkappa}\\in B^\\delta_\\e(W)$ provided $\\e$ is\n  sufficiently small, and hence $U^{\\e,\\varkappa}$ attains\n  $c_{\\e,\\varkappa}=\\Lambda_{\\e,\\varkappa}$, i.e. it is a local minimizer of\n  $I_{\\e,\\varkappa}$ on the open set $B^\\delta_\\e(W)$ with all the required\n  properties.\n\\end{pfn}\n\n\n\\section{Competitive systems}\\label{sec:competitive-systems}\nIn this section we prove the existence of solutions to the competitive system\n\\begin{equation}\n\\label{e:nostro}\n\\begin{cases}\n  -\\Delta u_i+u_i=f_i(u_i)- 2\\varkappa u_i\\sum_{j\\neq i}u_j^2,\n  &\\text{ in }\\Omega_\\e,\\\\\n  \\displaystyle\\frac{\\partial u_i}{\\partial \\nu}=0,&\\text{ on\n  }\\partial\\Omega_\\e,\n \\end{cases}\n\\end{equation}\nfor $i=1,\\dots,k$.\n\\begin{Theorem}\\label{t:sistemi2}\n  There exists $\\delta>0$ such that for $\\e>0$ sufficiently small and\n  $\\varkappa>0$ sufficiently large, system \\eqref{e:nostro} admits a\n  solution $U^{\\e,\\varkappa}=(u_1^{\\e,\\varkappa},\\dots,u_k^{\\e,\\varkappa})\\in\n  B^\\delta_\\e(W)$ such that, for all $i=1,\\dots,k$,\n  $u_i^{\\e,\\varkappa}\\not\\equiv 0$ and\n\\begin{equation}\\label{e:apr}\n0\\leq   u_i^{\\e,\\varkappa}\\leq A_i\\quad\\text{a.e. in }\\Omega_\\e.\n\\end{equation}\n\\end{Theorem}\n\\begin{proof}\nBy standard Critical\nPoint Theory, see e.g. \\cite{am}, the critical points of\n$I_{\\e,\\varkappa}$ on $(H^1(\\Omega_\\e))^k$ give rise to weak (and by\nregularity classical) solutions to\n\\begin{equation}\n\\label{e:truncated}\n\\begin{cases}\n  -\\Delta u_i+u_i=\\widetilde f_i(u_i)- \\varkappa g_i(u_i)\\sum_{j\\neq i}G_j(u_j),\n  &\\text{ in }\\Omega_\\e,\\\\\n  \\displaystyle\\frac{\\partial u_i}{\\partial \\nu}=0,&\\text{ on\n  }\\partial\\Omega_\\e,\n \\end{cases}\n\\end{equation}\nwhere\n\\[\n\\widetilde f_i(t)=\\begin{cases}\n0,\\quad &\\text{if }t\\leq 0,\\\\\nf_i(t),\\quad &\\text{if }0\\leq t\\leq  A_i,\\\\\nA_i,\\quad &\\text{if }t\\geq A_i,\n\\end{cases}\n\\]\nand\n\\[\ng_i(t)=\\begin{cases}\n2t,\\quad &\\text{if }|t|\\leq A_i,\\\\\n2A_i\\mathop{\\rm sgn}(t),\\quad &\\text{if }|t|> A_i.\n\\end{cases}\n\\]\nNotice that a solution to \\eqref{e:truncated} satisfying \\eqref{e:apr}\nis also a solution of \\eqref{e:nostro}.  Now the proof of the theorem\nimmediately follows by considering $U^{\\e,\\varkappa}\\in B^\\delta_\\e(W)$ as\nin Theorem \\ref{t:auxiliary}; since it is a local minimizer of\n$I_{\\e,\\varkappa}$, it is a free critical point of $I_{\\e,\\varkappa}$ and hence\nsolves \\eqref{e:truncated}. By the validity of \\eqref{eq:apriori1sis}\nwe finally deduce that $U^{\\e,\\varkappa}$ is actually a solution to\n\\eqref{e:nostro}, thus completing the proof.\n\n\\end{proof}\n\n\n\n\\section{The optimal partition problem}\\label{sec:asymptotic-analysis}\n\nIn this section we deal with problem ${\\boldsymbol{(P_\\e)}}$,\nnamely we look for local minimizers of the free energy on\nsegregated states. The localization of the problem is essentially\nmotivated by the fact that\n any global minimizer of the free energy in a connected\ndomain is trivial, as stated in Proposition\n\\ref{p:trivialglobmin}, the proof of which is given below.\n\n\n\\medskip\\noindent\n\\begin{pfn}{Proposition \\ref{p:trivialglobmin}}\nBy a direct computation, for any $U=(u_1,\\dots,u_k) \\in{\\mathcal\nU}$\n\\begin{align}\\label{eq:1}\n  J_{\\Omega}(U)&\\geq \\sum_{i=1}^k\\int_{\\Omega}\\left[ \\frac{|u_i|^2}{2}-\n    F_i(u_i)\\right]\\,dx\\geq \\sum_{i=1}^k\\left( \\frac{|A_i|^2}2-\n    F_i(A_i)\\right)|\\{x\\in\\Omega:u_i(x)> 0\\}|\\\\\n  \\notag& \\geq -\\mu_{i_0}\\sum_{i=1}^k|\\{x\\in\\Omega:u_i(x)> 0\\}|\\geq\n  -\\mu_{i_0}|\\Omega|=J_{\\Omega}(U_0).\n\\end{align}\nOn the other hand for any nontrivial $k$-uple\n$U=(U_1,\\dots,U_k)\\in{\\mathcal U}$ there exists $j$ such that $|\\nabla\nu_j|\\not\\equiv 0$ and hence the inequality in the first line of\n\\eqref{eq:1} is strict. Therefore $J_{\\Omega}(U)>\nJ_{\\Omega}(U_0)$ and $U$ cannot be a global minimizer.\n\\end{pfn}\n\nA nontrivial solution to the local minimization problem will be\nprovided by a limit configuration of solutions to the competitive\nsystem. To this aim we shall perform the asymptotic analysis of the\nsolutions to (\\ref{e:nostro}) found in Theorem \\ref{t:sistemi2} as\n$\\varkappa\\to+\\infty$.\n\n\\medskip\\noindent\n\\begin{pfn}{Theorem \\ref{t:partizione} and\n\\ref{t:sistemi}}\nLet  $U^{\\e,\\varkappa}=(u_1^{\\e,\\varkappa},...,u_k^{\\e,\\varkappa})$ be the solution\nof   system \\eqref{e:nostro} obtained in  Theorem \\ref{t:sistemi2} by minimizing\n$I_{\\e,\\varkappa}$ on $B^\\e_{\\delta}(W)$,  hence $I_{\\e,\\varkappa}(U^{\\e,\\varkappa})=c_{\\e,\\varkappa}$\nas in Theorem \\ref{t:auxiliary}.\nIn particular\n\\begin{equation}\\label{eq:8}\nc_{\\e,\\varkappa}\\geq \\frac12\\|U^{\\e,\\varkappa}\\|_{(H^1(\\Omega_\\e))^k}^2-\n|\\Omega_\\e|\\sum_i \\max_{t\\in[0,A_i]}|F_i(t)|.\n\\end{equation}\nFor every $U\\in{\\mathcal U}_\\e\\cap B^\\delta_\\e(W)$, define\n$\\widetilde U$ by setting $\\tilde u_i(x)=\\min\\{u_i(x), A_i\\}$.\nThen the following inequalities hold\n$$\nJ_{\\Omega_\\e}(U)\\geq J_{\\Omega_\\e}({\\widetilde U})=I_{\\e,\\varkappa}(U)\\geq c_{\\e,\\varkappa},\n$$\nimplying\n\\begin{equation}\\label{eq:9}\n\\lambda_{\\e}^{\\delta}\\geq c_{\\e,\\varkappa}.\n\\end{equation}\nFrom (\\ref{eq:8}) and (\\ref{eq:9}) we obtain that\n\\[\n\\|U^{\\e,\\varkappa}\\|_{(H^1(\\Omega_\\e))^k}^2\\leq 2\nc_{\\e,\\varkappa}+2|\\Omega_\\e|\\sum_i \\max_{t\\in[0,A_i]}|F_i(t)| \\leq\n2\\lambda_{\\e}^{\\delta}+2|\\Omega_\\e|\\sum_i \\max_{t\\in[0,A_i]}|F_i(t)|.\n\\]\nHence $u_i^{\\e,\\varkappa}$ is bounded in $H^1(\\Omega_\\e)$ uniformly with\nrespect to $\\varkappa$, then there exists a weak limit $v_i^\\e$ such that,\nup to subsequences, $u_i^{\\e,\\varkappa}\\rightharpoonup v_i^{\\e}$ in\n$H^1(\\Omega_\\e)$ as $\\varkappa\\to+\\infty$.  Also, by lower semicontinuity of\nthe norm, we learn that $V^\\e\\in B_\\delta^\\e(W)$, hence, by\n(\\ref{eq:nontrivi}), $v_i^\\e\\not\\equiv 0$ for all $i$.  Let us now\nmultiply the equation of $u_i^{\\e,\\varkappa}$ times $u_i^{\\e,\\varkappa}$ on\naccount of the boundary conditions: then\n\\begin{eqnarray*}\n  \\varkappa\\int_{\\Omega}(u^{\\e,\\varkappa}_i)^2\\sum_{j\\neq\n    i}(u^{\\e,\\varkappa}_j)^2\\quad\\text{is bounded uniformly in }\\varkappa,\n\\end{eqnarray*}\nhence\n\\[\n\\int_{\\Omega}(u^{\\e,\\varkappa}_i)^2\\sum_{j\\neq i}(u^{\\e,\\varkappa}_j)^2\\to\n0,\\qquad \\text{as }\\varkappa\\to\\infty.\n\\]\nBy the pointwise convergence $u_i^{\\e,\\varkappa}(x)\\to v_i^{\\e}(x)$ a.e.\n$x\\in\\Omega_\\e$, we infer that $v_i^\\e(x)\\geq 0$ and\n$v_i^\\e(x)\\cdot v_j^\\e(x)=0$ for almost every $x$, hence\n$V^\\e\\in\\mathcal U_\\e$.\n\nAlso, by the positivity of the interaction term, we know that\n$c_{\\e,\\varkappa}\\leq c_{\\e,\\varkappa'}$ when $\\varkappa\\leq \\varkappa'$: hence the\nsequence of critical levels $c_{\\e,\\varkappa}$ converges to some\n$\\lambda\\leq \\lambda_{\\e}^{\\delta}$ as $\\varkappa\\to +\\infty$.  Since by\nthe Dominated Convergence Theorem (recall that $0\\leq\nu_i^{\\e,\\varkappa}\\leq A_i$)\n$$\n\\int_{\\Omega_\\e} F_i(u_i^{\\e,\\varkappa})\\,dx =\\int_{\\Omega_\\e}\n\\widetilde F_i(u_i^{\\e,\\varkappa})\\,dx\\to \\int_{\\Omega_\\e}\n F_i(v_i^{\\e})\\,dx,\\qquad \\varkappa\\to\\infty,\n$$\nthe following chain of inequalities holds:\n\\begin{align*}\n  \\lambda_{\\e}^{\\delta}&\\geq\\lim_{\\varkappa\\to\\infty}c_{\\e,\\varkappa}\n  =\\lim_{\\varkappa\\to\\infty}I_{\\e,\\varkappa}(U^{\\e,\\varkappa})\\\\\n  & = \\limsup_{\\varkappa\\to\\infty}\\bigg[\n  \\sum_{i=1}^k\\bigg\\{\\frac12\\|u_i^{\\e,\\varkappa}\\|^2_{H^1(\\Omega_\\e)}\n  -\\int_{\\Omega_\\e} \\widetilde F_i(u_i^{\\e,\\varkappa})\\,dx\\bigg\\}+\n  \\varkappa\\sum_{\\substack{i,j=1\\\\i\\neq j}}^k\\int_{\\Omega}\n  (u^{\\e,\\varkappa}_i)^2(u^{\\e,\\varkappa}_j)^2\\bigg]\\\\\n  &\\geq \\limsup_{\\varkappa\\to\\infty}\n  \\sum_{i=1}^k\\bigg\\{\\frac12\\|u_i^{\\e,\\varkappa}\\|^2_{H^1(\\Omega_\\e)}\n  -\\int_{\\Omega_\\e} \\widetilde F_i(u_i^{\\e,\\varkappa})\\,dx\\bigg\\}\\\\\n  &\\geq \\liminf_{\\varkappa\\to\\infty}\n  \\sum_{i=1}^k\\bigg\\{\\frac12\\|u_i^{\\e,\\varkappa}\\|^2_{H^1(\\Omega_\\e)}\n  -\\int_{\\Omega_\\e} \\widetilde F_i(u_i^{\\e,\\varkappa})\\,dx\\bigg\\}\\\\\n  &\\geq \\sum_{i=1}^k\\bigg\\{\\frac12\\|v_i^{\\e}\\|^2_{H^1(\\Omega_\\e)}\n  -\\int_{\\Omega_\\e} F_i(v_i^{\\e})\\,dx\\bigg\\}= J_{\\Omega_\\e}(V^\\e)\\geq\n  \\lambda_{\\e}^{\\delta}.\n\\end{align*}\nTherefore all the above inequalities are indeed equalities. In\nparticular $J_{\\Omega_\\e}(V^\\e)=\\lambda_{\\e}^{\\delta}$, meaning that $V^\\e$\nsolves ${\\boldsymbol{(P_\\e)}}$ on $B_\\delta^\\e(W)$,\ngiving the proof of Theorem \\ref{t:partizione}.\n\n Moreover\n$\\lim_{\\varkappa\\to+\\infty}\\|U^{\\e,\\varkappa}\\|_{(H^1(\\Omega_\\e))^k}=\n\\|V^{\\e}\\|_{(H^1(\\Omega_\\e))^k}$ which, together with weak\nconvergence, implies that the convergence $U^{\\e,\\varkappa}\\to V^\\e$ is\nactually strong in $(H^1(\\Omega_\\e))^k$. We also deduce that\n\\[\n\\lim_{\\varkappa\\to\\infty}\\varkappa\\int_{\\Omega}(u^{\\e,\\varkappa}_i)^2\\sum_{j\\neq\n  i}(u^{\\e,\\varkappa}_j)^2=0.\n\\]\nThe proof of the Theorem \\ref{t:sistemi} is thereby complete.\n\\end{pfn}\n\n\\subsection{Extremality conditions}\\label{sec:extr-cond}\nOnce the existence of a solution for the optimal partition problem\n${\\boldsymbol{(P_\\e)}}$ is known, we can appeal to \\cite{ctv-var} to\nderive some interesting properties of $U_\\e$.  In particular, since\n$U^\\e$ is a local minimizer of the free energy $J_{\\Omega_\\e}$ we can\nprove that its components are solution of a remarkable system of\ndifferential inequalities.\n\\begin{Theorem}\\label{t:extremality} Let\n  $U^\\e\\in B^\\delta_\\e(W)$ be a solution to problem\n  ${\\boldsymbol{(P_\\e)}}$. Then $U^\\e$ is a solution of the $2k$\n  distributional inequalities \\eqref{2kvar}, namely, for every $i$ and\n  every $\\phi \\in H^1(\\Omega_\\e)$ such that $\\phi\\geq 0$ a.e. in\n  $\\Omega_\\e$, there holds\n\\begin{align*}\n \\begin{cases}\n{\\displaystyle   \\int_{\\Omega_\\e}}\\big(\\nabla u^\\e_i\\nabla\\phi+u_i^\\e\\phi-f_i(u^\\e_i)\n\\phi\\big)\\,dx\\leq 0,\\\\[10pt]\n{\\displaystyle \\int_{\\Omega_\\e}}\\big(\\nabla \\widehat\nu^\\e_i\\nabla\\phi+\\widehat u_i^\\e\\phi-\\widehat f_i(\\widehat\nu^\\e_i)\\phi\\big)\\,dx\\geq 0,\n\\end{cases}\n\\end{align*}\nwhere $\\widehat u_i = u_i-\\sum_{h\\neq i}u_h$ and $\\widehat\nf(\\widehat u_i) =f_i(u_i)-\\sum_{j\\neq i} f_j(u_j)$.\n\\end{Theorem}\nThe proof can be obtained as in \\cite[Theorem 5.1]{ctv-var}, the only\ndifference being that here we are dealing with local (and not\nglobal) minima of the free energy.\n  For the reader's convenience, we\n  sketch the main steps.\n\n \\medskip\\noindent\n \\begin{pf}   We argue by contradiction, hence, to prove the\n  first inequality, we assume that there exists one index $j$ and\n  $\\phi\\in H^1(\\Omega_\\e)$ such that $\\phi\\geq 0$ and\n\\begin{equation}\\label{e:ass1}\n\\int_{\\Omega_\\e}\\big(\\nabla u^\\e_j\\nabla\n\\phi+u_j^\\e\\phi-f_j(u_j^\\e)\\phi\\big)>0.\n\\end{equation}\nFor $t\\in (0,1)$ we consider $V=(v_1,\\dots,v_k)$ defined as\n$$\nv_{i}=\\left\\{\\begin{array}{ll}u^\\e_{i} &\\mbox{ if $i\\neq j$}\n    \\\\(u_i^\\e-t\\phi)^+&\\mbox{ if $i=j$.}\\end{array}\\right.\n$$\nWe notice that $V\\in{\\mathcal U}_\\e$. Moreover, since that map\n$z\\mapsto [z]^+$ is continuous from $H^1(\\Omega_\\e)$ to\n$H^1(\\Omega_\\e)$ and $U^\\e\\in B_\\delta^\\e(W)$, we learn that $V\\in\nB_\\delta^\\e(W)$ for all $t$ small enough.  In light of \\eqref{e:ass1}\nit is immediate to check that\n$J_{\\Omega_\\e}(V)<J_{\\Omega_\\e}(U^\\e)=\\min\\{J_{\\Omega_\\e}(U),\\,U\\in\nB_\\delta^\\e(W)\\cap{\\mathcal U}_\\e\\}$ for $t$ small enough, a\ncontradiction.  Let now $j$ and $\\phi\\in H^1(\\Omega_\\e)$, $\\phi\\geq\n0$, such that\n$$\n\\int_\\Omega\\big(\\nabla \\widehat u^\\e_j\\nabla\\phi+u^\\e_i\\phi -\\widehat\nf(\\widehat u^\\e_j)\\phi\\big)<0.\n$$\nAgain, we show that the value of the functional can be lessen by\nreplacing $U$ with an appropriate new function $V$ close to $W$.\nThis is defined as  $V=(v_1,\\dots,v_k)$ with\n$$\nv_i=\\left\\{\n\\begin{array}{ll}\n\\left(\\widehat u_j+t\\phi\\right)^+, &\\mbox{if $i=j$}\\\\\n\\left(\\widehat u_j+t\\phi\\right)^-\\chi_{\\supp(u_i)}, &\\mbox{if\n$i\\neq j$}.\n\\end{array}\\right.\n$$\nSimple computations lead to\n$$\nJ_{\\Omega_\\e}(V)-J_{\\Omega_\\e}(U^\\e)=t\\int_{\\Omega_\\e}\\big(\\nabla \\widehat\nu^\\e_j\\nabla\\phi+\\widehat u^\\e_i\\phi-\\widehat f_j(\\widehat\nu^\\e_j)\\phi\\big)+o(t),\n$$\nwhich leads to a contradiction if $t$ is small enough.\n\\end{pf}\n\n\n\n\\section{Conclusions and final remarks}\nAs a final step of our study, we have proved the existence of an\nelement $(u_1,\\dots,u_k)$ with $k$ non-trivial components in the\nfunctional class\n$${\\mathcal S}(\\Omega)=\\left\\{\n\\begin{array}{c}\n(u_1,\\cdots,u_k)\\in (H^1(\\Omega))^k:\\\n u_i\\geq0,\\, u_i\\not\\equiv 0,\\, u_i\\cdot u_j=0\\mbox{ if }\\,i\\neq j,\\\\[5pt]\n  \\int_{\\Omega}\\big(\\nabla u_i\\nabla\\phi+u_i\\phi-f_i(u_i)\\phi\\big)\n\\leq 0\\text{ and }\n  \\int_{\\Omega}\\big(\\nabla \\widehat u_i\\nabla\\phi+\\widehat\n  u_i\\phi-\\widehat f(\\widehat u_i)\\phi\\big)\\geq 0\\\\[5pt]\n\\text{for every }i=1,\\dots,k \\text{ and }\\phi\\in H^1(\\Omega)\n \\text{ such that }\\phi\\geq0\\text{ a.e. in }\\Omega\n \\end{array}\\right\\}\n$$\nwhen $\\Omega=\\Omega_\\e$ with small $\\e$.\n\nIn particular, by choosing test functions $\\phi$ with compact support\nin $\\Omega_\\e$, we learn that any element of ${\\mathcal S}(\\Omega_\\e)$\nis a solution (in distributional sense) of the following $2k$\ndifferential inequalities:\n\\begin{equation}\\label{e:diffineq}\n\\begin{cases}\n  -\\Delta u_i\\leq f_i(u_i),\\quad &\\text{in }\\Omega_\\e,\\\\\n  -\\Delta \\widehat u_i\\geq \\widehat f(\\widehat u_i),\n  &\\text{in }\\Omega_\\e.\n\\end{cases}\n\\end{equation}\nBy appealing to the interior regularity theory developed in\n\\cite[Section 8]{ctv-var}, we know that any $u_i$ is locally Lipschitz\ncontinuous and, in particular, the set $\\omega_i=\\{x\\in\\Omega_\\e:\\;\nu_i(x)>0\\}$ is an open (nonempty) set. Hence by \\eqref{2kvar} we\nobtain that $u_i\\big|_{\\omega_i}$ is solution of\n$$-\\Delta u_i+u_i=f_i(u_i),\\qquad \\text{in }\\omega_i,$$\nsubject to the boundary condition $$\\frac{\\partial\n  u_i}{\\partial\\nu}=0,\\qquad\\text{on }\\partial\\Omega_\\e\\cap \\omega_i.$$\nThis suggest that the validity of \\eqref{2kvar} not only implies the\n differential inequalities \\eqref{e:diffineq} in\n$\\Omega_\\e$, but it also contains boundary conditions on\n$\\partial\\Omega_\\e$ in some Neumann form, the major difficulty being\nto give functional sense to ``$\\frac{\\partial u_i}{\\partial\\nu}$'' on the\nwhole of $\\partial\\Omega_\\e$.  A rigorous analysis of this point\nrequires the development of a regularity theory for the class\n$\\mathcal S(\\Omega)$ up to the boundary, that will be object of future\nstudies.\\\\\n\n\\noindent\\textbf{Acknowledgements.} We are indebted to \nProf. Susanna Terracini for her interesting comments and\nsuggestions. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: The $\\Delta N\\protect\\gamma $ Transition Form Factors\nand Multipoles}\n\n\\label{i}\n\nThe $\\Delta N\\gamma $ transition form factors\\cite{devenish77} $G_{M}^{\\ast\n}(q^{2})$, $G_{E}^{\\ast }(q^{2})$, and $G_{C}^{\\ast }(q^{2})$ and their\nmultipole counterparts $M_{1^{+}}$, $E_{1^{+}}$, and $S_{1^{+}}$ in nuclear\nand elementary particle physics are very important in nuclear and elementary\nparticle physics because they provide a basis for testing theories of\neffective quark forces or production models.\n\n\\begin{itemize}\n\\item  They are especially important in the understanding of perturbative\nQCD (PQCD) models\\cite{stoler93} involving gluon exchange mechanisms, tensor\ninteractions, or possible hybrid baryonic states;\n\n\\item  They are important in enhanced quark models in which the transition\nform factors may be calculated as a function of $q^{2}$; in\nelectroproduction and photoproduction processes; in symmetry schemes such as\nSU(6) and U(6,6), and Melosh transformations; in bag models; in dispersion\nrelation and Bethe-Salpeter approaches; in current algebra baryon sum rules;\nand in nonperturbative methods such as lattice QCD, QCD sum rules, and\nalgebraic formulations.\n\n\\item  The fundamental reason that the transition form factors are such good\nQCD probes lies in the fact that in many quark, symmetry, and potential\nmodels, $G_{E}^{*}(q^{2})$ and\/or $G_{C}^{*}(q^{2})$ are identically zero\nthus giving rise to pure magnetic dipole $M_{1^{+}}$ transitions\n\n\\item  In the na\\\"{i}ve quark model, it can be shown that the quantity $%\nE_{1^{+}}\/M_{1^{+}}=-G_{E}^{*}\/G_{M}^{*}\\equiv $ ratio of the electric\nquadrupole moment to the magnetic dipole moment $\\;\\equiv \\;R_{EM}=\\;0$ .\nExperimentally, however, the $R_{EM}$ appears to be non-zero but small in\nmagnitude and of the order of a few percent. Most analyses predict the $%\nR_{EM}$ to be small and negative at small momentum transfer, however a\nrecent analysis\\cite{burkert} extracted the value $R_{EM}(q^{2}=-3.2$ $%\n(GeV\/c)^{2})\\approx (+6\\pm 2\\pm 3)\\%$. Subsequently, another even more\nrecent analysis\\cite{beck}predicts that $R_{EM}=-(2.5\\pm 0.2\\pm 0.2)\\%$ at\nthe maximum of the $\\Delta (1232)$\\ resonance and $R_{EM}=-3.5\\%$ when\nbackground scattering amplitude contributions are taken into consideration.\n\n\\item  Clearly, the capability of any particular theoretical model\n(including ours) to predict accurately and precisely non-zero $R_{EM}$\nvalues of the right sign and magnitude for particular values of $q^{2}$ in\nagreement with experiment is critical. As has been noted, the $R_{EM}$ ratio\nis especially effective for testing effective quark forces such as occur in\nQCD one-gluon exchange tensor interactions, various types of enhanced quark\nmodels, symmetry schemes such as $SU(6)$, $U(6,6)$, melosh transformations,\ndispersion relations and sum rules (where the $\\Delta $ always plays an\nimportant role).\n\\end{itemize}\n\n\n\\subsection{Importance of the ${\\bf {\\Delta \\rightarrow N+\\protect\\gamma }}$\nTransition Form Factors}\n\n\\begin{itemize}\n\\item  Provide a basis for testing theories of effective quark forces and\nproduction models\n\n\\item  QCD: One gluon exchange mechanisms, tensor interactions, and possible\nhybrid baryonic states.\n\n\\item  Enhanced Quark Models: They should be capable of predicting\naccurately the $\\Delta $--$N$ transition form factors as a function of $%\nq^{2} $.\n\n\\item  Bag models of hadrons\n\n\\item  Current Algebra approaches to hadron physics\n\n\\item  Non-perturbative approaches to hadron physics such as lattice QCD\n\n\\item  Electroproduction and Photoproduction: Important for correct\ntheoretical description.\n\n\\item  Symmetry Schemes: The $\\Delta $ always plays an important role in\nmodels involving $SU(6)$, $U(6,6)$, etc., and melosh transformations.\n\n\\item  Dispersion relations: The $\\Delta $ always plays an important role.\n\n\\item  Baryon Sum Rules: The $\\Delta $ always plays an important role.\n\\end{itemize}\n\n\n\n\n\\subsection{\\ The ${\\bf {\\Delta \\rightarrow N+\\protect\\gamma }}$ Transition\nForm Factors and Transition Amplitudes}\n\nIn general one may write for the $\\Delta \\rightarrow N+\\gamma $ transition\namplitude the following expression\\cite{devenish77}:\n\n\\begin{equation}\n\\left\\langle p(\\vec{p},\\lambda _{p})\\right| j_{\\mu }(0)\\left| \\Delta ^{+}(%\n\\vec{p}^{*},\\lambda _{\\Delta })\\right\\rangle =\\frac{1}{{(}2\\pi {)^{3}}}\\sqrt{%\n\\frac{mm^{*}}{E_{p}E_{\\Delta }}}\\bar{u}_{P}(\\vec{p},\\lambda _{P})\\left[\n\\Gamma _{\\mu \\beta }\\right] u_{\\Delta }^{\\beta }(\\vec{p}^{\\,*},\\lambda\n_{\\Delta })  \\label{eq1}\n\\end{equation}\n\nwhere\n\n\\begin{equation}\n\\begin{array}{ccl}\n\\Gamma _{\\mu \\beta } & = & i%\n{\\displaystyle{3(m^{*}+m) \\over 2m}}%\n(G_{M}^{*}-3G_{E}^{*})\\Theta ^{-1}m^{*}q_{\\beta }\\epsilon _{\\mu }(qp\\gamma )\n\\\\ \n& - & \n{\\displaystyle{3(m^{*}+m) \\over 2m}}%\n(G_{M}^{*}+G_{E}^{*})\\Theta ^{-1}[2\\epsilon _{\\beta \\sigma }(p^{*}p)\\epsilon\n_{\\mu }^{\\;\\sigma }(p^{*}p)\\gamma _{5}-im^{*}q_{\\beta }\\epsilon _{\\mu\n}(qp\\gamma )] \\\\ \n& + & \n{\\displaystyle{3(m^{*}+m) \\over m}}%\nG_{C}^{*}\\Theta ^{-1}q_{\\beta }[p\\cdot qq_{\\mu }-q^{2}p_{\\mu }]\\gamma _{5}.\n\\end{array}\n\\label{eq2}\n\\end{equation}\n\nIn Eqs. (\\ref{eq1}) and (\\ref{eq2}), the electromagnetic current is denoted\nby $j_{\\mu }$, $q\\equiv p^{\\ast }-p$, $p^{\\ast }$ and $p$ are the\nfour-momenta of the $\\Delta ^{+}$ and nucleon respectively. $\\Theta\n^{-1}\\equiv \\lbrack ((m^{\\ast }+m)^{2}-q^{2})((m^{\\ast }-m)^{2}-q^{2})]^{-1}$\nis a kinematic factor which depends on $q^{2}$, $m^{\\ast }$ (the $\\Delta ^{+}\n$ mass), and $m$ (the proton mass); $\\lambda _{P}$ and $\\lambda _{\\Delta }$\nare the helicities of the proton and $\\Delta ^{+}$ respectively. We note\nthat the first, second, and third terms in Eq.(\\ref{eq2}) induce transverse $%\n\\frac{1}{2}$ ($h_{3}$), transverse $\\frac{3}{2}$ ($h_{2}$), and longitudinal\nhelicity transitions ($h_{1}$) respectively in the rest frame of the $\\Delta\n^{+}$ isobar\\cite{devenish77}. $G_{M}^{\\ast }$, $G_{E}^{\\ast }$, and $%\nG_{C}^{\\ast }$ are related to the helicity form factors $h_{1}$, $h_{2}$,\nand $h_{3}$ by the relations:\n\n\\begin{eqnarray}\nh_{3} &=&-\\frac{3(m^{*}+m)}{2m}(G_{M}^{*}-3G_{E}^{*})  \\label{eq3} \\\\\nh_{2} &=&-\\frac{3(m^{*}+m)}{2m}(G_{M}^{*}+G_{E}^{*})  \\nonumber \\\\\nh_{1} &=&\\frac{3(m^{*}+m)}{m}G_{C}^{*}.  \\nonumber\n\\end{eqnarray}\n\n\nFor the transition amplitude governing the virtual process $p\\rightarrow\np+\\gamma $, we have similarly\n\n\\begin{equation}\n\\left\\langle p(\\vec{s},\\lambda )\\right| j_{\\mu }(0)\\left| p(\\vec{t},\\lambda\n^{*})\\right\\rangle =\\frac{1}{{(2\\pi )^{3}}}\\sqrt{\\frac{m{^{2}}}{E{_{\\vec{p}}}%\nE_{\\vec{p}^{*}}}}\\bar{u}_{p}({\\vec{s}},\\lambda )\\left[ {\\Gamma _{\\mu }}%\n\\right] u_{p}(\\vec{t},\\lambda ^{*})  \\label{eq4}\n\\end{equation}\n\nwhere\n\n\\begin{equation}\n\\Gamma _{\\mu }=[1-\\widetilde{q}^{2}\/(4m{^{2})}]^{-1}[(i\/(4m^{2}))G_{M}(%\n\\tilde{q}^{2})\\epsilon _{\\mu }\\left( \\tilde{P}\\tilde{q}\\gamma \\right) \\gamma\n_{5}+(1\/(2m))G_{E}(\\tilde{q}^{2})\\tilde{P}_{\\mu }],  \\label{eq4.1}\n\\end{equation}\n\n$\\tilde{P}{_{\\mu }}\\equiv \\widetilde{p}+\\widetilde{p}^{*}$ , $\\widetilde{q}=%\n\\widetilde{p}^{*}-\\widetilde{p}$ with $\\widetilde{p}^{*}=(\\widetilde{p}^{*0},%\n\\vec{t})$ and $\\widetilde{p}=(\\widetilde{p}^{0},\\vec{s})$. $G_{M}(\\widetilde{%\nq}^{2})$ and $G_{E}(\\widetilde{q}^{2})$ in Eq.~(\\ref{eq4.1}) are the\nfamiliar nucleon Sachs form factors.\n\n\\subsection{Relationship Between the ${\\bf \\Delta N\\protect\\gamma }$\\ Form\nFactors and Multipoles}\n\nThe magnetic, electric, and coulombic multipole transition moments given by $%\nM_{1^{+}}(q^{2})$, $E_{1^{+}}(q^{2})$, and $S_{1^{+}}(q^{2})$ can be written\n\\cite{cott} in terms of $G_{M}^{\\ast }(q^{2})$, $G_{E}^{\\ast }(q^{2})$, and $%\nG_{C}^{\\ast }(q^{2})$ . Indeed one has\n\n\\begin{eqnarray}\nM_{1^{+}} &=&\\alpha _{1}\\;\\sqrt{Q^{-}}\\;G_{M}^{*}  \\label{eq4.2} \\\\\nE_{1^{+}} &=&\\alpha _{2}\\;\\sqrt{Q^{-}}\\;G_{E}^{*}  \\nonumber \\\\\nS_{1^{+}} &=&\\alpha _{3}\\;Q^{-}\\sqrt{Q^{+}}\\;G_{C}^{*}  \\nonumber\n\\end{eqnarray}\n\nwhere $\\alpha _{1}$, $\\alpha _{2}=-\\alpha _{1}$, $\\alpha _{3}$\\ are\nfunctions of parameters governing the process $\\Gamma(\\Delta \\rightarrow \\pi\nN)$ (and in particular are dependent on the $\\Delta $\\ mass $m^{*}$) and\nwhere $Q^{\\pm }\\equiv \\sqrt{(m^{*}\\pm m)^{2}-q^{2}}$.\n\n\n\\subsection{${\\bf {A_{\\frac{1}{2}}}}$ and ${\\bf {A_{\\frac{3}{2}}}}$ Photon\nDecay Helicity Amplitudes}\n\nThe total $\\Delta $ radiative width $\\equiv \\Gamma _{\\gamma }^{T}$, for\ndecay into $p+\\gamma $ is given by:\n\n\\begin{equation}\n\\Gamma _{\\gamma }^{T}=\\frac{mq_{c}^{2}}{2m^{\\ast }\\pi }\\sum_{\\lambda =\\frac{1%\n}{2},\\frac{3}{2}}A_{\\lambda }^{2}  \\label{eq5}\n\\end{equation}\nwhere \n\\begin{equation}\nA_{\\frac{1}{2}}=-e\\left( \\frac{\\sqrt{3}}{8}\\right) {\\left[ \\frac{{m^{\\ast }}%\n^{2}-m^{2}}{m^{3}}\\right] }^{\\frac{1}{2}}\\left[ G_{M}^{\\ast\n}(0)-3G_{E}^{\\ast }(0)\\right] ,  \\label{eq6}\n\\end{equation}\n\\vspace*{0.3in} $q_{c}^{2}=$ CM momentum, and \n\\begin{equation}\nA_{\\frac{3}{2}}=-e\\left( \\frac{3}{8}\\right) {\\left[ \\frac{{m^{\\ast }}%\n^{2}-m^{2}}{m^{3}}\\right] }^{\\frac{1}{2}}\\left[ G_{M}^{\\ast }(0)+G_{E}^{\\ast\n}(0)\\right]   \\label{eq7}\n\\end{equation}\n\\vspace*{0.5in} Experimentally\\cite{pdg98},\n\n\\begin{equation}\nA_{\\frac{1}{2}}\\cong (-140\\pm 5)\\times 10^{-3}GeV^{-1\/2}\\text{\\qquad and\n\\qquad }A_{\\frac{3}{2}}\\cong (-258\\pm 6)\\times 10^{-3}GeV^{-1\/2}\n\\label{eq7.1}\n\\end{equation}\n\n\n\\section{Calculation}\n\n\\begin{itemize}\n\\item  Our treatment is non-perturbative and performed in a broken symmetry\nhadronic world\\cite{milt}. We do not require the use of ``mean'' mass\napproximations;\n\n\\item  Physical masses are used at all times. Thus, $G_{E}^{*}$ is not\nforced to equal zero as in the na\\\"{i}ve quark model;\n\n\\item  Our treatment is completely relativistic. Current conservation is\nguaranteed. Additionally, the correct electromagnetic transition operator is\nused in all calculations;\n\n\\item  We use the infinite-momentum frame for calculations or\nequivalently--- ``infinite'' Lorentz boosts that are not always in the\nz-direction, thus implicitly (and often explicitly) bringing into play\nWigner rotations resulting in mixed helicity particle states.\n\n\\item  In order to proceed with the calculation of $G_{M}^{*}(q^{2})$, and $%\nG_{E}^{*}(q^{2})$, we consider helicity states with $\\lambda =\\pm {1\/2}$ and \n$\\lambda _{\\Delta }=1\/2$ (i.e. spin flip and non-flip sum rules) and the\nnon-strange ($S=0$) $L=0$ ground state baryons ($J^{PC}=\\frac{1}{2}^{+},%\n\\frac{3}{2}^{+}$). We will ultimately find two independent constraint\nequations which will then allow one to calculate $G_{M}^{*}(q^{2})$, and $%\nG_{E}^{*}(q^{2})$, and their multipole counterparts.\n\\end{itemize}\n\n\n\\begin{itemize}\n\\item  It is well-known\\cite{milt} that if one defines the axial-vector\nmatrix elements: $\\,\\left\\langle p,1\/2\\right| A_{\\pi ^{+}}\\left|\nn,1\/2\\right\\rangle \\equiv f=g_{A}(0)$, $\\left\\langle {\\Delta ^{++},1\/2}%\n\\right| A_{\\pi ^{+}}\\left| {\\Delta ^{+},1\/2}\\right\\rangle \\equiv -\\sqrt{{%\n\\frac{3}{2}}}\\,g$, and $\\left\\langle {\\Delta ^{++},1\/2}\\right| A_{\\pi\n^{+}}\\left| {p,1\/2}\\right\\rangle \\equiv -\\sqrt{6}\\,h$, and applies\nasymptotic level realization to the chiral $SU(2)\\otimes SU(2)$ charge\nalgebra $\\left[ {A_{\\pi ^{+}},A_{\\pi ^{-}}}\\right] =2V_{3}$, then $h^{2}={%\n(4\/25)}f^{\\;2}$ (the sign of $h=+(2\/5)f$ , can be fixed by requiring that $%\nG_{M}^{*}(0)>0$)and $g={(-}\\sqrt{2}\/5)f$.\n\n\\item  If one inserts the algebra $\\left[ j_{3}^{\\mu }(0){,}A_{\\pi ^{+}}%\n\\right] =A_{\\pi ^{+}}^{\\mu }{(0)}$ ($j^{\\mu }\\equiv j_{3}^{\\mu }+j_{S}^{\\mu }\n$ , where $j_{3}^{\\mu }\\equiv $ isovector part of $j^{\\mu }$ and $j_{S}^{\\mu\n}$ is isoscalar) between the ground states $\\left\\langle B(\\alpha ,\\lambda\n=\\pm 1\/2,\\overrightarrow{s})\\right| $and $\\left| B^{\\prime }(\\alpha ,\\lambda\n=1\/2,\\overrightarrow{t})\\right\\rangle $ with $\\left| \\overrightarrow{s}%\n\\right| \\rightarrow \\infty ,\\left| \\overrightarrow{t}\\right| \\rightarrow\n\\infty $, where $\\left\\langle B(\\alpha )\\right| $ and $\\left| B^{\\prime\n}(\\beta )\\right\\rangle $ are the following $SU_{F}(2)$ related combinations: \n$\\left\\langle p,n\\right\\rangle $, $\\left\\langle p,\\Delta ^{0}\\right\\rangle $%\n, $\\left\\langle \\Delta ^{++},p\\right\\rangle $, $\\left\\langle n,\\Delta\n^{-}\\right\\rangle $, $\\left\\langle \\Delta ^{++},\\Delta ^{+}\\right\\rangle $, $%\n\\left\\langle {\\Delta ^{+}},{\\Delta ^{0}}\\right\\rangle $, $\\left\\langle\n\\Delta ^{0},\\Delta ^{-}\\right\\rangle $ and $\\left\\langle \\Delta\n^{+},n\\right\\rangle $, then one obtains (we use $<N|\\,j_{S}^{\\mu }|\\,\\Delta\n>=0$ ) the following two independent sum rule constraints:\n\\end{itemize}\n\n\\begin{center}\nSpin Non-Flip Sum Rule\n\\end{center}\n\n\\begin{eqnarray}\n<p,1\/2,\\vec{s}\\,|j^{\\mu }(0)|\\,\\Delta ^{+},1\/2,\\vec{t}> &=&\\frac{5\\sqrt{2}}{4%\n}\\left\\{ -\\frac{\\left\\langle p,1\/2,\\vec{s}\\right| A_{\\pi ^{+}}^{\\mu\n}(0)\\left| n,1\/2,\\vec{t}\\right\\rangle }{2f}\\right.  \\label{eq8} \\\\\n&&+\\left. \\left\\langle p,1\/2,\\vec{s}\\,|j_{3}^{\\mu }(0)|\\,p,1\/2,\\vec{t}%\n\\right\\rangle \\right\\} .  \\nonumber\n\\end{eqnarray}\n\nand\n\n\\begin{center}\nSpin Flip Sum Rule\n\\end{center}\n\n\\begin{equation}\n<p,-1\/2,\\vec{s}\\,|j^{\\mu }(0)|\\,\\Delta ^{+},1\/2,\\vec{t}>=\\frac{5}{8}\\sqrt{2}%\n\\left\\langle p,-1\/2,\\vec{s}\\right| j_{3}^{\\mu }(0)\\left| p,1\/2,\\vec{t}%\n\\right\\rangle  \\label{eq9}\n\\end{equation}\n\n\nNow take the limit $\\left| \\overrightarrow{t}\\right| \\rightarrow \\infty $\nand $\\left| \\overrightarrow{s}\\right| =\\rightarrow \\infty $ and evaluate\ndirectly each of the matrix elements in Eq.(\\ref{eq8}) and Eq. (\\ref{eq9}).\nWe find respectively that:\n\n\\begin{equation}\nG_{M}^{*}(q^{2})+\\left[ \\frac{2m^{*}(4m-m^{*})+m^{2}-2q^{2}}{%\n2(m^{*2}+m^{2}-q^{2})}\\right] G_{E}^{*}(q^{2})=  \\label{eq10}\n\\end{equation}\n\n\\[\n\\frac{5\\sqrt{3}}{3}\\left[ \\frac{2m^{*}m^{2}}{(m^{*}+m)(m^{*2}+m^{2}-q^{2})}%\n\\sqrt{\\frac{(m^{*}+m)^{2}-q^{2}}{4m^{2}-q^{2}}}\\right] G_{M}^{V}(q^{2}) \n\\]\n\nand\n\n\\begin{equation}\nG_{M}^{*}(q^{2})-3G_{E}^{*}(q^{2})=\\left[ \\frac{5\\sqrt{3}m\\sqrt{-{\\tilde{q}}%\n^{2}}}{3(m^{*}+m)\\left[ (m^{*}-m)^{2}-q^{2}\\right] ^{1\/2}}\\right] G_{M}^{V}({%\n\\tilde{q}}^{2})\\text{,}  \\label{eq11}\n\\end{equation}\n\nwhere in Eq. (\\ref{eq11}), a collinear limit of $\\left| \\overrightarrow{t}%\n\\right| $ and $\\left| \\overrightarrow{s}\\right| $ was taken in such a\nfashion that $\\left| \\overrightarrow{s}\\right| =r\\left| \\overrightarrow{t}%\n\\right| $ ($\\overrightarrow{s}$ and $\\overrightarrow{t}$ are taken along the\nz-axis, $0<r\\leq m^{2}\/m^{*2}$) and ${\\tilde{q}}^{2}$ and $q^{2}$ are\nrelated by the equations ${\\tilde{q}}^{2}=(1-r)\\left[ q^{2}-(1-r)m^{*2}%\n\\right] $ and $q^{2}=\\frac{(1-r)}{r}(m^{*^{2}}r-m^{2})$.\n\nEq. (\\ref{eq10}) and Eq. (\\ref{eq11}) may be solved for $G_{M}^{*}(q^{2})$\nand $G_{E}^{*}(q^{2})$. The numerical results (Figure 1.) are as follows:\n\n\n\\section{Results and Conclusions}\n\n\\begin{itemize}\n\\item  $G_{M}^{*}(q^{2})$, $G_{E}^{*}(q^{2})$, $R_{EM}(q^{2})$, $A_{\\frac{1}{%\n2}}$, and $A_{\\frac{3}{2}}$ are computed in good agreement with experiment (Figure 1).\n\n\\begin{enumerate}\n\\item  At the $\\Delta $ pole mass, we find that $G_{M}^{*}(0)=3.18$, $%\nG_{E}^{*}(0)=+0.07$, and $R_{EM}(0)=-2.19\\%$. We also determine that $A_{%\n\\frac{1}{2}}\\cong -134\\times 10^{-3}GeV^{-1\/2}$, and $A_{\\frac{3}{2}}\\cong\n-254\\times 10^{-3}GeV^{-1\/2}$.\n\n\\item  When the $\\Delta $ mass is taken to be $1.232$ $GeV\/c$, $%\nG_{M}^{*}(0)=3.09$, $G_{E}^{*}(0)=+0.12$, and $R_{EM}(0)=-3.94\\%$. We also\ndetermine that $A_{\\frac{1}{2}}\\cong -128\\times 10^{-3}GeV^{-1\/2}$, and $A_{%\n\\frac{3}{2}}\\cong -262\\times 10^{-3}GeV^{-1\/2}$;\n\\end{enumerate}\n\n\\item  We find that $R_{EM}(q^{2})$ is negative for $0\\leq -q^{2}\\lesssim\n5\\;GeV^{2}\/c^{2}$, changes sign in the region $-q^{2}\\approx\n6-7\\;GeV^{2}\/c^{2}$, and very slowly approaches $1$ as $-q^{2}\\rightarrow\n\\infty $;\n\n\\item  The PQCD prediction that $R_{EM}(q^{2})\\rightarrow 1$ as $%\n-q^{2}\\rightarrow \\infty $ is verified but only as an asymptotic condition\napplicable only at very high momentum transfer\\cite{stuart98};\n\n\\item  The $R_{EM}$ is particularly sensitive to the $\\Delta $ mass (i.e.\nmass parametrization used) when $0\\leq -q^{2}\\lesssim 1\\;GeV^{2}\/c^{2}$\n(i.e. photoproduction).\n\\end{itemize}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMicrofluidics is an indispensable tool of modern bio-physical\nresearch. It allows to perform complex single-cell experiments\nwith an immense throughput and high level of control. A flexible\ndesign allows for custom geometries and control of flows and\nchemical reactions. Recently, to probe the dynamics of actin\npolymerization, as well as \nto use the geometry of microfluidic device having the main supply channel with\nnumerous identical side channels or chambers of different shapes\nThe main channel serves to deliver and fill the\nside chambers with reagents where the corresponding reacting can\nbe and observed. The flow in the main channel and diffusion in the\nside-channels are dominating means of transport in such devices.\nRemarkably, the process of diffusion in this particular geometry\nwas extensively studied in the context of anomalous diffusion. It\nis known as a comb model and it was demonstrated that the\ntransport of particles along the main channel (called backbone in\nthe model) can become subdiffusive when the particles get trapped\nby diffusing into the side channels. Until recently it was mostly\nan abstract model, which was, however, extremely useful in\nunderstanding the principles of anomalous subdiffusive transport\n(many references here). The experimental setup on actin\npolymerization \\cite{pfohl1,pfohl2} is the direct implementation of the\ncomb model \\cite{white,weiss,havlin}, where the effects of complex diffusion should have a\nsubstantial effect on the observed phenomena. Interestingly, the\ncomb structure not only leads to an anomaly in transport but also\nto a very remarkable effects on the propagation of chemical\nreactions \\cite{iomend}.\n\nThe goal of this paper is to combine the consideration of\nanomalous transport and reaction dynamics to provide the\ntheoretical grounds for the corresponding experimental efforts.\nOur analytical results on reaction propagation can help to guide\nthe design of microfluidic devices but also can lead to real\nexperimental tests of anomalous diffusion and reaction dynamics.\nFor the reaction of polymerization, depending on the concentration\nof reagents we can recover such remarkable phenomena as the\nfailure of reaction front propagation \\cite{sokolov,iomsok} or a finite speed or a\nFisher-KPP long time asymptotic regime \\cite{kpp,fisher}.\n\n\n\\section{Mapping of the Laplace operator on a Comb equation}\n\nMapping of the Laplace operators in the $3D$ complicated continuous-discrete geometry on a continuous comb model equation, relates to averaging over $xyz$-space.\nIn reality, we have two Laplace operators acting on distribution functions in a bulk $P_b(x,y,z)$ and in fingers (where reactions take place) $P_f(x,y,z)$.\n\n(\\textit{i}) In the bulk of infinite length along the $x$ coordinate and  $yz$ surface cross-section $a\\times a$ we have\nfor the Laplace operator\n\\[D\\Delta P_b(x,y,z)=D(\\partial_x^2+\\partial_y^2+\\partial_z^2)P_b(x,y,z)\\]\nwith the diffusivity of magnesium $D$ and the boundary conditions\n\\[\\partial_xP|_{x=0}=\\partial_xP|_{x=a}=\\partial_yP|_{y=0}=\\partial_yP|_{y=a}=0\\, ,\\]\nthe following algorithm of mapping can be suggested.\n\nIntegration over $z$ leads to the disappearance of the $z$ component due to the boundary condition.\nIntegration over $y$ in the bulk yields zero except those $y$ regions where the bulk is connected with the fingers. Plunging the fingers inside the bulk, one obtains that this region is at $y=0$.\nTherefore, we have\n\\[D\\Delta  P_b(x,y,z)\\Rightarrow aD\\partial_x^2P(x,y=0)\\, .\\]\n\n(\\textit{ii}) The fingers have height $h$ (along the $y$) and the $xz$ cross-section is of size $b\\times b$. Therefore,\nwe have the following Laplace operator with diffusivity $d$ inside the fingers\n\\[d\\Delta P_f(x,y,z)=d(\\partial_x^2+\\partial_y^2+\\partial_z^2)P_f(x,y,z)\\, .\\]\n\nIntegration\/averaging over $x$ and $z$ leads to zero for everywhere, except for $\\partial_y^2$ in periodic regions\n\\[\\partial_y^2P(x,y)\\sum_n\\delta(x-nL)\\Rightarrow \\frac{b}{L}\\partial_y^2P(x,y)\\, ,\\]\nwhere $L$ is a distance between the fingers.\nCombining both equations, one obtains a kinetic equation for the magnesium transport along the comb structure\nwith parameters related to experimental geometry $a,b,L,h$ and diffusion coefficients $D$ and $d$\n\\begin{equation}\\label{comb}\n\\partial_tP=D\\delta(y\/h)\\partial_x^2P+\\rho d\\partial_y^2P\\, .\n\\end{equation}\nHere $\\rho=bh\/La$ is a dimensionless density of the fingers. In what follows we will use the notation for the coefficient\n\\[d_{\\rho}=d\\cdot\\rho\\, .\\]\nHowever, the scaling of the $\\delta$ function by $h$ can be inconvenient in the limit $h\\rightarrow\\infty$. Therefore, in what necessary, we scale the $\\delta$ function by\n\\begin{equation}\\label{hb}\nh_b=\\sqrt{hb}\\, ,~~~~~~~~~~ \\delta(y\/h)\\rightarrow \\delta(y\/h_b)\\, .\n\\end{equation}\nIn this case,\nwe have $h\\rightarrow\\infty$ and $b\\rightarrow 0$, and in these limits $hb={\\rm const}$. This scaling supports the density $\\rho$. Say $hb=aL$ and in this subdiffusive limit $\\rho=1$.\n\nThe boundary conditions are\n\\[P(x=\\pm\\infty,y,t)=\\partial_xP(x=\\pm\\infty,y,t)=0~\\mbox{and}~ \\partial_yP(x,y=\\pm h,t)=0\\, .\\]\nThe initial condition is\n\\[P(x,y,t=0)=P_0(x)\\delta(y)\\, .\\]\n\nAmending comb equation (\\ref{comb}) with reaction term inside fingers $\\rho\\cdot C(P)$, we have\n\\begin{equation}\\label{comb_r}\n\\partial_tP=D\\delta(y\/h_b)\\partial_x^2P+ d_{\\rho}\\partial_y^2P-\\rho C(P)\\, ,\n\\end{equation}\n\nProbably, another mapping on the comb model can be made in the framework of a Dykhne model\n\\cite{dkm2004,dvork2009,dvorkm2010}. In this case, dimensionless density is $\\rho=\\frac{b^2h}{a^2L}$.\nIn some extend, this density sounds more physically. But this leads to additional argumentation of limits, for example\n$h\\rightarrow \\infty$ and $b\\rightarrow 0$. Namely $hb^2\\rightarrow a^2L$ and\n$h_b\\rightarrow h_{bb}=[hb^2]^{\\frac{1}{3}}$.\n\n\n\\section{Few points on reaction for definition of the reaction term}\n\nFor the reaction-polymerization, we use the following \\textit{stoichiometry} expression\n\\[ A+B \\Rightarrow C\\, ,\\]\nwhere\n\\begin{eqnarray}\n& [A] & ~ \\mbox{is concentration of actin} \\nonumber \\\\\n& [B] &  ~ \\mbox{is concentration of magnesium}  \\nonumber \\\\\n& [C]  &~ \\mbox{is concentration of polymer}  \\nonumber\n\\end{eqnarray}\n\nIn general case, we consider the second order reaction, and also take into account that $[B]=P(x,y,t)$.\nTherefore, we have\n\\[\\frac{d\\,P}{d\\,t}=-kP\\cdot [A]\\]\nwith the the reaction rate $k$ and initial condition $[B(t=0)]\\equiv P(t=0)=P_0,~[A(t=0)]=[A]_0=N,~[C]_0=0$.\n\nLet us express $[A]$ by $P$. We have\n\\begin{eqnarray*}\n& P\\equiv [B]=[B]_0-[C]=P_0-[C]\\, , \\\\\n& [A]=[A]_0-[C]= [A]_0-P_0+P\\, .\n\\end{eqnarray*}\nTherefore the reaction equation reads\n\\begin{equation}\\label{e_react2}\n\\frac{d\\,P}{d\\,t}=-k(N-P_0+P)P\\, .\n\\end{equation}\nIf during the reaction time, $N-P_0\\gg P$ and $N\\gg P_0$, the second order reaction equation (\\ref{e_react2}) becomes of the first order\n\\begin{equation}\\label{e_react1}\n\\frac{d\\,P}{d\\,t}=-kNP\\, .\n\\end{equation}\nSubstituting Eq. (\\ref{e_react1}) in Eq. (\\ref{comb_r}) and denoting $C=\\rho\\cdot k\\cdot N$ yields\n\\begin{equation}\\label{comb_react1_a}\n\\partial_tP=Dh_b\\delta(y)\\partial_x^2P+ d_{\\rho}\\partial_y^2P- CP\\, .\n\\end{equation}\n\n\\section{Solution of linear reaction-transport comb equation: extinction dynamics}\n\nThe reaction term disappears from Eq. (\\ref{comb_react1_a}) by substitution\n\\[P=e^{-Ct}\\bar{P}\\, .\\]\nThe next step is the Laplace transform $\\hat{\\cal{L}}[\\bar{P}](t)=\\tilde{P}(s)$.\nThis yields\n\\begin{equation}\\label{comb_react1_b}\ns\\tilde{P}=Dh_b\\delta(y)\\partial_x^2\\tilde{P}+d_{\\rho}\\partial_y^2\\tilde{P}+P_0\\delta(y)\\, ,\n\\end{equation}\nand the solution can be considered as a product $\\tilde{P}(x,y,s)=\\tilde{n}(y,s)f(x,s)$\nThe solution inside fingers is found from the equation in the Laplace domain\n\\[s\\tilde{n}(y)=d_{\\rho}\\partial_y^2\\tilde{n}(y)\\]\nand the boundary condition $\\partial_y\\tilde{n}(y)|_{y=\\pm h}=0$, which yields\n\\[\\tilde{n}(y,s)=\n\\frac{\\cosh\\Big[(h-|y|)\\sqrt{s\/d_{\\rho}}\\Big]}{\\cosh\\Big[(h\\sqrt{s\/d_{\\rho}}\\Big]}\\]\nand $n(y=0)=1$.\n\nThe solution is looked for in the form\n\\begin{equation}\\label{solution}\n\\tilde{P}(x,y,s)=\\tilde{n}(y,s)f(x,s)\\, ,\n\\end{equation}\nthat yields for the second derivative over $y$\n\\[\\partial_y^2\\tilde{P}=\\Big\\{-2\\delta(y)\\sqrt{s\/d_{\\rho}}\n\\frac{\\sinh\\Big[(h-|y|)\\sqrt{s\/d_{\\rho}}\\Big]}{\\cosh\\Big[(h\\sqrt{s\/d_{\\rho}}\\Big]}\n+[s\/d_{\\rho}]\\tilde{n}(y)\\Big\\}f(x,s)\\, .\\]\nTherefore extinct diffusion of magnesium in the balk, determined by $f(x,s)$, is described by equation\n\\begin{equation}\\label{bulk_dif1}\nDh_b\\partial_x^2f-2\\sqrt{sd_{\\rho}}\\tanh\\Big[h\\sqrt{s\/d_{\\rho}}\\Big]f+P_0=0.\n\\end{equation}\nThe presence of the hyperbolic tanh distinguishes two time scales\/regioons.  At a short time scale, when\n$h\\sqrt{s\/d_{\\rho}}\\gg 1$,\nwe have $\\tanh[h\\sqrt{s\/d_{\\rho}}]\\approx 1$ that yields\n\\begin{equation}\\label{bulk_dif2}\nD_{\\frac{1}{2}}\\partial_x^2f-\\sqrt{s}f+f_0=0\\, ,\n\\end{equation}\nwhere $$D_{\\frac{1}{2}}=\\frac{Dh_b}{2\\sqrt{d_{\\rho}}}=\\frac{D\\sqrt{aL}}{2\\sqrt{d}}$$\nis a generalized diffusion coefficient and $f_0(x)=P_0(x)=\nP_0\\delta(x)$, for simplicity. \\textit{It should be admitted that the generalized diffusion coefficient\nin the $x$ direction is independent of the finger's geometry parameters $h$ and $b$.}\n\nPerforming the Fourier transform $\\bar{f}(k)=\\hat{\\cal F}[f(x)]$, one obtains\n\\begin{equation}\\label{bulk_dif3}\n\\bar{f}(k,s)=\\frac{P_0}{\\sqrt{s}+D_{\\frac{1}{2}}k^2}\\, .\n\\end{equation}\nPerforming the Laplace inversion of the solution (\\ref{bulk_dif3}) and taking into account a definition of the Mittag-Leffler function \\cite{bateman}\n\\[ E_{\\alpha,\\beta}(z)=\\frac{1}{2\\pi i}\\int_{G}\\frac{r^{\\alpha-\\beta}e^r}{r^{\\alpha}-z}dr\\, ,~~~~\\alpha,\\beta>0\\, , \\]\none obtains\n\\begin{equation}\\label{bulk_dif4}\n\\bar{f}(k,t)=\\frac{P_0}{2\\pi i}\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}\\frac{e^{st}ds}{\\sqrt{s}+D_{\\frac{1}{2}}k^2}=\n\\frac{P_0}{\\sqrt{t}}E_{\\frac{1}{2}\\, ,\\frac{1}{2}}\\left(-D_{\\frac{1}{2}}k^2t^{\\frac{1}{2}}\\right)\\, .\n\\end{equation}\nThis solution is valid for the ``short'' time-scale $ t \\ll \\frac{hLa}{bd}=\\frac{h^2}{d_{\\rho}} $. Therefore,\nthe argument of the Mittag-Leffler function is small, and the latter reads \\cite{bateman,klafter}\n\\[ E_{\\frac{1}{2}\\, ,\\frac{1}{2}}\\left(-D_{\\frac{1}{2}}k^2t^{\\frac{1}{2}}\\right)\\approx\n\\exp\\left[-\\frac{D_{\\frac{1}{2}}k^2t^{\\frac{1}{2}}}{\\Gamma(3\/2)}\\right]\\, ,\\]\nwhere $\\Gamma(3\/2)=(1\/2)\\Gamma(1\/2)=\\sqrt{\\pi}\/2$ is the gamma function\\footnote{\nNote, that this formula is true only for the Mittag-Leffler function $E_{1\/2,1}(z)=E_{1\/2}(z)$. However, we can use it approximately here, admitting that the denominator in the exponential $\\Gamma(3\/2)$ is not correct. It should be stressed that this approximation does not affect the following analysis.}.\nNow the Fourier inversion can be easily performed that yields\n\\[\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{-ikx}e^{-ak^2}dk=\n\\left[\\frac{\\Gamma(3\/2)}{4\\pi D_{\\frac{1}{2}}t}\\right]^{\\frac{1}{2}}\n\\exp\\left[-\\frac{x^2\\Gamma(3\/2)}{D_{\\frac{1}{2}}\\sqrt{t}}\\right]\\, .\\]\nTaking into account the reaction term, one obtains the solution for the short-time scale exstinc diffusion in\nthe bulk\n\\begin{equation}\\label{bulk_dif5}\nP(x,y=0,t<t_0)=\\frac{P_0e^{-Ct}}{\\sqrt{8D_{\\frac{1}{2}}\\pi t^3}}\n\\exp\\left(-\\frac{\\sqrt{\\pi}x^2}{2D_{\\frac{1}{2}}\\sqrt{t}}\\right)\\, .\n\\end{equation}\nNote, that this result is independent of $h$ and $b$ and coincides with the subdiffusive comb limit\n$h\\rightarrow\\infty$.\n\nOn a long time-scale the hyperbolic tanh in Eq. (\\ref{bulk_dif1}) has a small argument, which yields\n\\[\\tanh[h\\sqrt{s\/d_{\\rho}}]\\approx h\\sqrt{s\/d_{\\rho}}\\, .\\]\nThis corresponds to an equation for normal diffusion\n\\begin{equation}\\label{bulk_dif6}\nD\\sqrt{b\/h}\\partial_x^2f-sf+P_0\/h=0\\, ~~~~~~~~D\\sqrt{b\/h}\\equiv D_{bh}\n\\end{equation}\nwith a well-known solution for normal diffusion. Finally this yields the long-time scale solution in the bulk\n\\begin{equation}\\label{bulk_dif7}\nP(x,y=0,t>t_0)=e^{-Ct}\\int dx'P_0(x')\n\\frac{1}{\\sqrt{\\pi h^2D_{bh} t}}\\exp\\left(-\\frac{(x-x')^2}{4D_{bh}t}\\right)\\, .\n\\end{equation}\nWhen $P_0(x)=\\delta(x)$, the Green function coincides with the distribution.\nIn Eqs. (\\ref{bulk_dif5}) and (\\ref{bulk_dif7}), the transient time parameter $t_0=\\frac{hLa}{bd}$ is determined from the geometry of the experiment.\n\n\n\\section{Reaction front propagation in a case of high concentration of magnesium}\n\nIn a case when the concentration of magnesium is high enough, the approximation (\\ref{e_react1}) is not valid anymore,\nand one has to take into account the second order reaction (\\ref{e_react2}). Thus the reaction term reads\n\\begin{equation}\\label{nlinCP}\nC(P)=CP+C_1P^2\\, ~~~~C=\\rho\\cdot k\\cdot (N-P_0)\\, , ~~~~C_1=\\rho\\cdot k\\, .\n\\end{equation}\nIn this nonlinear case, an exact analytical treatment\nof Eq. (\\ref{comb_r}) is impossible, and we apply an analytical approximation to find the overall velocity of the reaction-polymerization front propagation without resolving an exact shape of the front, namely without knowledge of the exact distribution function $P(x,y,t)$.\n\nSince we are seeking for the front propagation in the $x$ direction, the exact shape in the $y$ direction is not important, and we consider the distribution as a function of the $x$ coordinate only $P_1(x,t)$.\nIn other words, the detail dynamics inside\nfingers is not important, and we take into account its overall contribution to diffusion in the $x$ axis .\nTo this end, the $y$ coordinate is integrated out\n\\begin{equation}\\label{P1}\nP_1(x,t)=\\int_{-h}^hP(x,y,t)dy\\, .\n\\end{equation}\nTherefore, this integration in Eq. (\\ref{comb_r}) with reaction due to Eq. (\\ref{nlinCP}) yields an equation in the unclosed form. First of all, we take into account integration\n\\[\\int_{-h}^h\\partial_y^2P(x,y,t)dy=\\partial_yP(x,y,t)\\Big|_{y=-h}^{y=h}=0\\, .\\]\nThis yields the equation in task\n \\begin{equation}\\label{combP1}\n\\partial_tP_1(x,t)=h_bD\\partial_x^2P(x,y=0,t)-CP_1-C_1\\int_{-h}^hP^2(x,y,t)dy\\, .\n\\end{equation}\nWe have here two problematic terms. The first one is $P(x,y=0,t)$, which will be expressed by $P_1(x,t)$.\nTo this end we use the relation (\\ref{solution}), where we pay attention on $\\tilde{P}(x,y=0,s)=f(x,s)$.\nIntegrating Eq. (\\ref{solution}) over $y$ yields\n\\[ \\tilde{P}_1(x,s)=\\tilde{P}(x,y=0,s)\\int_{-h}^{h}\\tilde{n}(y,s)dy=\n2\\tilde{P}(x,y=0,s)\\int_0^h\\tilde{n}(y,s)dy\\, .\\]\nTherefore, one obtains\n\\begin{equation}\\label{PtoP1_a}\n\\tilde{P}(x,y=0,s)=\\sqrt{s\/4d_{\\rho}}\\tanh^{-1}\\Big[h\\sqrt{s\/d_{\\rho}}\\Big]\\cdot  \\tilde{P}_1\\, .\n\\end{equation}\nPerforming the inverse Laplace transform, one obtains from Eq. (\\ref{PtoP1_a})\n\\begin{eqnarray}\\label{PtoP1_b}\nP(x,y=0,t)&=&\\frac{1}{2\\pi i}\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}ds\\int_{0}^{\\infty}dt_1dt_2P_1(x,t_1)\n\\mathcal{R}_1(t_2)\\cdot e^{-st_1}\\cdot e^{-st_2}\\cdot e^{st} \\nonumber \\\\\n&=&\\int_0^{\\infty}dt_1\\int_{0}^{\\infty}dt_2P_1(t_1)\\mathcal{R}_1(t_2)\\delta(t-t_1-t_2)=\n\\int_{0}^{t}dt'P_1(t-t')\\mathcal{R}_1(t')\\, .\n\\end{eqnarray}\nHere we take into account the causality principle, saying that $P(t-t_1)=0$ for $t_1>t$ and\nintroduce the kernel $\\mathcal{R}_1(t)$ through the Laplace inversion\n\\begin{equation}\\label{kernelR1}\n\\mathcal{R}_1(t)=\\hat{\\mathcal{L}}^{-1}\\left[\\sqrt{s\/4d_{\\rho}}\n\\tanh^{-1}\\Big(h\\sqrt{s\/d_{\\rho}}\\Big)\\right]\\, .\n\\end{equation}\n\nThe second problematic term is integration of the nonlinear reaction. We present it in a form convenient for\nthe further analytical treatment. Therefore, we have the following chain of transformations:\n\\begin{eqnarray*}\nN.L.R.T.&\\equiv &\\int_{-h}^{h}P^2(x,y,t)dy=\\frac{1}{(2\\pi i)^2}\n\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}e^{s_1t}ds_1e^{s_2t}ds_2\n\\int_{-h}^h\\tilde{P}(x,y,s_1)\\tilde{P}(x,y,s_2)dy=  \\\\\n& =&\\frac{1}{(2\\pi i)^2}\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}e^{s_1t}ds_1e^{s_2t}ds_2\\tilde{P}(x,y=0,s_1)\n\\tilde{P}(x,y=0,s_2)\\int_{-h}^h\\tilde{n}(y,s_1)\\tilde{n}(s_2,y)dy = \\\\\n&=& \\frac{1}{(2\\pi i)^2}\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}\\tilde{\\mathcal{R}}(s_1,s_2)\\tilde{P}_1(x,s_1)\n\\tilde{P}_1(x,s_2)\\frac{\\sqrt{s_1s_2}}{4d_{\\rho}}\\cdot\n\\tanh^{-1}\\Big[h\\sqrt{s_1\/d_{\\rho}}\\Big]\\cdot\n\\tanh^{-1}\\Big[h\\sqrt{s_2\/d_{\\rho}}\\Big] \\\\\n&\\times&\ne^{s_1t}\\,e^{s_2t}ds_1\\,ds_2=  \\int_{0}^{\\infty}d\\tau_1d\\tau_2P_1(x,\\tau_1)P_1(x,\\tau_2) \\\\\n&\\times&\\frac{1}{(2\\pi i)^2}\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}\\tilde{\\mathcal{R}}(s_1,s_2)\\cdot\n\\frac{\\sqrt{s_1s_2}}{4d_{\\rho}}\\cdot\n\\tanh^{-1}\\Big[h\\sqrt{s_1\/d_{\\rho}}\\Big]\\times\\tanh^{-1}\\Big[h\\sqrt{s_2\/d_{\\rho}}\\Big]\\cdot\ne^{s_1(t-\\tau_1)}\\,e^{s_2(t-\\tau_2)}ds_1\\,ds_2 \\, ,\n\\end{eqnarray*}\nwhere\n\\[\\mathcal{R}(s_1,s_2)=\\int_{-h}^h\\tilde{n}(y,s_1)\\tilde{n}(s_2,y)dy\\, .\\]\nIntroducing new variables $t-\\tau_1=t_1$ and $t-\\tau_2=t_2$ yields\n\\[\\int_0^{\\infty}d\\tau_1=-\\int_t^{-\\infty}dt_1=\\int_{-\\infty}^tdt_1\\]\nand the same for $\\tau_2$. Therefore, we have\n\\begin{equation}\\label{nlrt}\nN.L.R.T.=\\int_{0}^{\\infty}dt_1dt_2P_1(x,t-t_1)P_1(x,t-t_2)\\cdot\n\\frac{1}{(2\\pi i)^2}\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}\\tilde{\\mathcal{R}}(s_1,s_2)\ne^{s_1t_1}\\,e^{s_2t_2}ds_1\\,ds_2 \\, .\n\\end{equation}\nIn what follow we will keep this form as the N.L.R.T.\n\nEventually, equation for the $F_1(x,t)$ reads\n\\begin{equation}\\label{EqFront}\n\\partial_tP_1=hD\\partial_x^2\\int_0^t\\mathcal{R}_1(t')P_1(t-t')dt'-CP_1-\nC_1\\int_{-\\infty}^t\\mathcal{R}(t',t^{\\prime\\prime})P_1(t-t')\nP_1(t-t^{\\prime\\prime})dt'dt^{\\prime\\prime}\\, ,\n\\end{equation}\nwhere two-time-point kernel $\\mathcal{R}(t',t^{\\prime\\prime})$ is defined in the Laplace space in Eq. (\\ref{nlrt}).\n\n\\section{Hyperbolic scaling for the overall velocity of the reaction front propagation}\n\nTo evaluate the overall velocity of the asymptotic front,\nwe follow the hyperbolic scaling consideration, developed in \\cite{Fr,F} and adopted in \\cite{iomend}\nfor consideration of the reaction transport front propagation in comb structures. Let us\nintroduce a small parameter, say $\\varepsilon$, at the derivatives\nwith respect to time and space \\cite{Fr,F}. To this end we\nre-scale\n$$x\\rightarrow x\/\\varepsilon~~ \\mbox{and} ~~~ t\\rightarrow\nt\/\\varepsilon\\, , \\mbox{and}~~~P_1(x,t)\\rightarrow\nP_1^{\\varepsilon}(x,t)=P_1\\left(\\frac{x}{\\varepsilon},\n\\frac{t}{\\varepsilon}\\right)$$. Therefore, one looks for the\nasymptotic solution in the form of the Green's approximation\n\\begin{equation}\\label{WKB}\nP_1^{\\varepsilon}(x,t)=\n\\exp\\left[-\\frac{G^{\\varepsilon}(x,t)}{\\varepsilon}\\right]\\, .\n\\end{equation}\nThe main strategy of implication of this construction is the limit\n$\\varepsilon\\rightarrow 0$ that yields an asymptotic behavior at finite $x$ and $t$, where\nwe have\n\\[\\exp\\left[-\\frac{G^{\\varepsilon}(x,t)}{\\varepsilon}\\right]=0\\, ,\\]\nexcept for the condition when\n\\[G^{\\varepsilon}(x,t)=0\\, .\\]\n\nThis equation determines the position of the reaction spreading front, and\nin this limit,\n$G(x,t)=\\lim_{\\varepsilon\\to 0}G^{\\varepsilon}(x,t)$ is accounted as the principal Hamiltonian function\n\\cite{Fr,F}. Therefore, the Hamiltonian\napproach can be applied to calculate the propagation front velocity. In\nthis case partial derivatives of $G(x,t)$ with respect to time and\ncoordinate have the physical senses of the Hamiltonian and the momentum:\n\\begin{equation}\\label{Action}\n\\frac{\\partial G(x,t)}{\\partial t}=-H\\, ,~~~~~~~~~\n\\frac{\\partial G(x,t)}{\\partial x}=p\\, .\n\\end{equation}\n\nNow, ansatz (\\ref{WKB}) for the\nprobability distribution function inside the bulk is inserted\nin Eq. (\\ref{EqFront}), where we also make the scaling change\n$x\\rightarrow\\frac{x}{\\varepsilon}$ and $t\\rightarrow\\frac{t}{\\varepsilon}$.\n\nLet us start from the last term\nin Eq. (\\ref{EqFront}), which is the reaction term.\nFirst of all we take into account the following change\nin the upper limit of integrations\n\\[\\int_{-\\infty}^t dt'\\Rightarrow\\lim_{\\varepsilon\\to 0}\\int_{-\\infty}^{\\frac{t}{\\varepsilon}}dt'\n=\\int_{-\\infty}^{\\infty}dt'\\, .\\]\nThen, we make the following expansion for\n$P_1^{\\varepsilon}(x,\\frac{t}{\\varepsilon}-t')\\equiv P_1^{\\varepsilon}(t-\\varepsilon t')$,\nwhich reads\n\\[P_1^{\\varepsilon}(t-\\varepsilon t')=\\exp\\left[-\\frac{1}{\\varepsilon}G^{\\varepsilon}(t-\\varepsilon t')\\right]\\approx\n\\exp\\left[-\\frac{1}{\\varepsilon}G^{\\varepsilon}(t)+t'\\partial_tG^{\\varepsilon}(t)\\right]\\, .\\]\nTherefore, the nonlinear reaction term reads\n\\begin{equation}\\label{hyperbolic1}\nN.L.R.T. =e^{-2\\frac{G^{\\varepsilon}}{\\varepsilon}}\\cdot\\int_{-\\infty}^{\\infty}dt^{\\prime}dt^{\\prime\\prime}\n\\frac{1}{(2\\pi i)^2}\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}\\tilde{\\mathcal{R}}(s^{\\prime},s^{\\prime\\prime})\ne^{-(H-s^{\\prime})t^{\\prime}}\\,\ne^{-(H-s^{\\prime\\prime})t^{\\prime\\prime}}ds^{\\prime}ds^{\\prime\\prime}\\, ,\n\\end{equation}\nwhere we use the first equation in (\\ref{Action}). Integration over $s^{\\prime}$ and $s^{\\prime\\prime}$\nis performed with some care, since $\\tilde{\\mathcal{R}}$\nis singular at $s^{\\prime}=\\pm s^{\\prime\\prime}$. The main result here is that $\\tilde{\\mathcal{R}}$ is finite and\n$N.L.R.T.\\sim  e^{-2\\frac{G^{\\varepsilon}}{\\varepsilon}}$, that in the limit $\\varepsilon\\rightarrow 0$ is\nof the order of $o\\left(e^{-\\frac{G^{\\varepsilon}}{\\varepsilon}}\\right)$ and, as will be seen in this limit,\ndoes not contribute to the finite result, since the rest of the equation is of the order of\n$O\\Big(e^{-\\frac{G^{\\varepsilon}}{\\varepsilon}}\\Big)$.\n\nNow we consider the the kinetic term taking the time integration with $\\mathcal{R}_1$ kernel. This reads\n\\begin{eqnarray*}\n\\int_0^{\\frac{t}{\\varepsilon}}\\mathcal{R}_1(t^{\\prime})\n&{}& e^{-\\frac{G^{\\varepsilon}(t-\\varepsilon t^{\\prime},x)}{\\varepsilon}}dt^{\\prime}=\ne^{-\\frac{G^{\\varepsilon}(t)}{\\varepsilon}}\n\\int_0^{\\infty}\\mathcal{R}_1(t^{\\prime})e^{-Ht^{\\prime}}dt^{\\prime}\\\\\n&=&\ne^{-\\frac{G^{\\varepsilon}(t)}{\\varepsilon}}\\tilde{\\mathcal{R}}_1(H)\\, ,\n\\end{eqnarray*}\nwhere $\\tilde{\\mathcal{R}}_1(H)$ is defined in Eq. (\\ref{kernelR1}).\n\nFinally, differentiating in the limit $\\varepsilon\\rightarrow 0$ where  ($N.L.R.T.=0$) and taking\ninto account that the Hamiltonian $H$ and the momentum $p$ in Eq.\n(\\ref{Action}) are independent of $x$ and $t$ explicitly (which\nleads to the absence of mixed derivatives), one obtains that the kinetic equation\n(\\ref{EqFront}) becomes a  Hamilton-Jacobi equation:\n\\begin{equation}\\label{HJE}\n-\\partial_tG=h_bD\\Big(\\partial_xG\\Big)^2\\tilde{\\mathcal{R}}_1(H)-C\\, ,\n\\end{equation}\nwhere $G\\equiv G(x,t)=\\lim_{\\varepsilon\\to 0}G^{\\varepsilon}$ is the action (principal Hamiltonian function)\n\\[G(x,t)=\\int_0^t[p(\\tau)\\dot{x}(\\tau)-H(p(\\tau),x(\\tau))]d\\tau\\, .\\]\nThe rate $v$ at\nwhich the front moves is determined at the condition $G(x,t)=0$.\nTogether with the Hamilton equations, this yields\n\\begin{equation}\\label{frontV}\nv=\\dot{x}=\\frac{\\partial H}{\\partial p}\\, ,~~~v=\\frac{H}{p}\\, . %\n\\end{equation}\nThe first equation in (\\ref{frontV}) reflects the\ndispersion condition, while the second one is a result of the\nasymptotically free particle dynamics, when the action is\n$G(x,t)=px-Ht$. Taking into account $x=vt$, one obtains Eq.\n(\\ref{frontV}). The combination of these two equations can be\nreplaced by\n\\begin{equation}\\label{minmin}\nv=\\min _{H>0}\\frac{H}{p(H)}=\\min _{p>0}\\frac{H(p)}{p}.\n\\end{equation}\nTo proceed, we consider two time limits of the kernel\/dispersion term $\\tilde{\\mathcal{R}}_1(H)$ in Eq. (\\ref{HJE}).\n\n\\subsection{short time limit $h\\sqrt{H\/d_{\\rho}}\\gg 1$}\n\nIn the short time limit, which corresponds to subdiffsion on the intermediate asymptotic times,\nwe have $\\tanh\\Big[h\\sqrt{H\/d_{\\rho}}\\Big]=1$ that yields $ \\tilde{\\mathcal{R}}_1(H)\\approx \\sqrt{H\/4d_{\\rho}}$.\nTaking this into account, one obtains from Eq. (\\ref{HJE})\n\\begin{equation}\\label{Vshort1}\n\\frac{(H+C)^2}{H}=\\frac{(h_bD)^2}{4d_{\\rho}}p^4\\, ,\n\\end{equation}\nand $p(H)=\\frac{(4d_{\\rho})^{\\frac{1}{4}}}{\\sqrt{hD}}\\cdot\\frac{\\sqrt{H+C}}{H^{\\frac{1}{4}}}$.\nTaking logarithm from Eq. (\\ref{Vshort1}) and differentiation over $p$, one obtains\n\\[\\frac{2}{H+C}=\\frac{5}{H}\\, .\\]\nThis equation has no solution for $C>0$. Therefore, for $C>0$, $v=\\min _{H>0}\\frac{H}{p(H)}=0$.\nThis means the failure of the front propagation, as expected for the initial concentration of magnesium less than actin.\n\nFor $C<0$ that corresponds $P_0>N$, one obtains the solution $H=5|C|\/3$ that yields a nonzero velocity of the reaction\nfront propagation at subdiffusion at the initially short times\n  \\[v=\\Big(\\frac{5}{3}\\Big)^{\\frac{5}{4}}\\cdot\\Big(\\frac{2}{3}\\Big)^{\\frac{1}{2}}\n\\Big(\\frac{D^2h_b^2|C|^3}{4d_{\\rho}}\\Big)^{\\frac{1}{4}}=\n\\Big(\\frac{5}{3}\\Big)^{\\frac{5}{4}}\\cdot\\Big(\\frac{2}{3}\\Big)^{\\frac{1}{2}}\n\\Big(\\frac{DaL|C|^3}{4d}\\Big)^{\\frac{1}{4}} \\, .\\]\nAgain, we admit that this result is independent of the finger geometry and correspondingly, has the comb limit\n$h\\rightarrow\\infty$.\n\n\\subsection{large time limit $h\\sqrt{H\/d_{\\rho}}\\ll 1$}\nIn the opposite case of the large time asymptotics, $\\tanh\\Big[h\\sqrt{H\/d_{\\rho}}\\Big]\\approx h\\sqrt{H\/d_{\\rho}}$,\nand $\\tilde{\\mathcal{R}}_1(H)\\approx\\frac{1}{2h}$. Therefore, as seen from  Eqs. (\\ref{bulk_dif6}) and (\\ref{bulk_dif7}), we have from Eq. (\\ref{HJE})\n\\begin{equation}\\label{Vlong1}\nH=\\frac{D_{bh}}{2}p^2-C\\, ,\n\\end{equation}\nand $p=\\sqrt{2(H+C)\/D_{bh}}$. This immediately yields\n\\[v=\\min_{H>0}\\Big[\\frac{HD_{bh}^{\\frac{1}{2}}}{\\sqrt{2(H+C)}}\\Big]=0 \\]\nfor $H=0$, as expected for $C>0$.\n\nThe situation changes dramatically for $C<0$ ($P_0>N)$. In this case, the equation\n\\[\\frac{\\partial H}{\\partial p}=\\frac{H}{p}\\]\nyields\n\\[2\\bar{D}p=\\bar{D}p+\\frac{|C|}{p}\\\\, .\\]\nHere we scale the diffusion coefficient $\\bar{D}=\\frac{D_{bh}}{2}$.\nThis yields the solution for $p=\\sqrt{|C|\/\\bar{D}}$. Therefore, substitution in the velocity equation yields\n\\[v=\\min_{p>0}\\frac{H(p)}{p}=\\frac{\\bar{D}p^2+|C|}{p}=2\\sqrt{\\bar{D}|C|}\\, .\\]\nThis is a well known Fisher-KPP result for the reaction front velocity with the scaled diffusivity $\\bar{D}$.\n\n\\section{Conclusion}\n\nIn the paper we considered anomalous transport and reaction dynamics by providing\nthe theoretical grounds for the possible experimental realization of actin polymerization in \ncomb-like geometry. Different regimes for the reaction of polymerization have been \nconsidered, and depending on the concentration\nof reagents (magnesium and actin), we recovered both the failure of reaction front propagation and a finite speed corresponding to the Fisher-KPP long time asymptotic regime.\nFinally, we repeat that our analytical results on reaction propagation can help to guide\nthe design of microfluidic devices but also can lead to real\nexperimental tests of anomalous diffusion and reaction dynamics.\n\nThis research was supported by the Israel Science Foundation (ISF). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn a gravitational system, the scattering of radiation off\nabsorbing rotating objects produce waves with amplitude larger\nthan incident one under certain conditions which is known as\nrotational superradiance \\cite{ZEL0, ZEL1}. In 1971, Zel'dovich\nshowed that scattering of radiation off rotating absorbing\nsurfaces result in waves with a larger amplitude as $\\omega <\nm\\Omega$ where $\\omega $ is the frequency of the incident\nmonochromatic radiation with m, the azimuthal number with respect\nto the rotation axis and $\\Omega$ is the angular velocity of the\nrotating gravitational system. For review we would like to mention\nthe lecture notes \\cite{REVIEW}, and the references therein.\nRotational superradiance belongs to a wider class of classical\nproblems displaying stimulated or spontaneous energy emission,\nsuch as the Vavilov-Cherenkov effect, the anomalous Doppler\neffect. When quantum effects were incorporated, it was argued that\nrotational superradiance would become a spontaneous process and\nthat rotating bodies including black holes would slow down by\nspontaneous emission of photons. From the historic perspective,\nthe discovery of black-hole evaporation \\cite{HAW} was well\nunderstood from the studies of black-hole superradiance.\n\nInterest in the study of black-hole superradiance has recently\nbeen revived in different areas, including astrophysics,\nhigh-energy physics via the gauge\/gravity duality along with\nfundamental issues in General Relativity. Superradiant\ninstabilities can be used to constrain the mass of ultralight\ndegrees of freedom \\cite{INST00, INST0, INST1, INST2}, with\nimportant applications to dark-matter searches. The black hole\nsuperradiance is also associated with the existence of new\nasymptotically flat hairy black-hole solutions \\cite{HAIR} and\nwith phase transitions between spinning or charged black objects\nand asymptotically anti-de Sitter (AdS) spacetime \\cite{ADS0,\nADS1, ADS2} or in higher dimensions \\cite{MSHO}. Finally, the\nknowledge of superradiance is instrumental in describing the\nstability of black holes and in determining the fate of the\ngravitational collapse in confining geometries \\cite{ADS1}.\n\nDuring the last few decades, the standard theories of general\nrelativity have been continuing to explain many important\nexperimental results. However, there is still some room to the use\nof alternative theories of the general theory of relativity. From\na theoretical viewpoint, having an ultraviolet complete theory of\ngeneral relativity is complimentary as well as supportive.\nMoreover from the observational point of view, general relativity\nhas shortcomings to describe some gravitational phenomena at a\nlarge scale such as the dark side of the universe. These\nshortcomings automatically demand modified theories of general\nrelativity. The modifications may render some imprints in\nastrophysical phenomena where it is expected that the strong\ngravity triggers the events in the vicinity of celestial bodies\nlike astrophysical black holes and neutron stars. Black holes can\nbe used as a potential probe to investigate the possible\nhigh-energy modifications to general relativity in the regime\nwhere gravity is sufficiently strong. In this respect, the use of\nalternative theories of gravity would be of cardinal importance to\nstudy the astrophysical aspects of black hole. The important\nastrophysical phenomena namely black hole superradiance is\nextremely sensitive to the spacetime geometries linked with it.\nRecently, several investigations have been carried out on\nsuperradiance phenomena and on the issues closely linked to it\nwith the extended framework of modified theories of gravity\n\\cite{MODG0, MODG1, MODG2, MODG3, MODG4, MODG5, MODG6, MODG7,\nMODG8, MODG9, MODG10, MODG11, MODG12, MODG13}. As an extension in\nthis direction, an attempt has been made here to study the\nsuperradiance of the spinning black holes within the framework of\nLorentz-violating gravity. It is commonly known as the\n'Einstein-bumblebee model \\cite{KOSTEL0} which involves the\ninnovative 'spontaneous Lorentz symmetry breaking' principle. From\nthe theoretical point of view, it arrived from one of the standard\nissues of quantization of gravity through string theory. Although\nthe Lorentz symmetry is the fundamental underlying symmetry of two\nsuccessful field theories describing the universe, i.e. GR and the\nstandard model of particle, however, it is more or less accepted\nfrom all corners that it may break at quantum gravity scales. The\nLSB has been introduced through the formulation of an effective\nfield theory, known as 'standard model extension(SME), where\nparticle standard model along with GR has been attempted to bring\ntogether in one framework, and every operator is expected to break\nthe Lorentz symmetry \\cite{KOSTEL1, COLL0, COLL1, COLE}. Standard\nmodel extension provides essential inputs to probe LSB both in\nhigh energy particle physics and astrophysics. The SME can be used\nin analysis of most modern experimental results indeed. Einstein-\nbumblebee model is essentially a simple model that contains\nLorentz symmetry breaking scenario in a significant manner in\nwhich the physical Lorentz symmetry breaks down through an axial\nvector field known as the bumblebee field. The breaking of the\nLorentz symmetry in a local Lorentz frame takes place when at\nleast one quantity carrying local Lorentz indices receives a\nnon-vanishing vacuum expectation value. In the Einstein bumblebee\nmodel, it is the bumblebee field that receives it. Over the last\nfew years, a remarkable enthusiasm has been noticed among the\nphysicist to study the different interesting physical phenomena in\nthe framework of Einstein Bumblebee model \\cite{SEIF, JPAR, DCAP,\nCASANA, OLIV, OAV, OAV1, OLIVE, DING}. Recently, the superradiance\nphenomenon corresponding to Kerr black hole is studied in\n\\cite{MK} in this framework. The black hole solution considered\nthere was Kerr-like. The study of superradiance phenomena of black\nholes through the Einstein bumblebee model using Kerr-sen-like\nblack hole solution is a natural extension. This new investigation\nis likely to be useful in the study of black holes in the quantum\ngravity realm since it would be possible to compare the\ncontribution of LV to the superradiance phenomenon.\n\nThe article is organized as follows. In Sec.2 a brief discussion\nof Einstein-bumblebee gravity with Kerr-Sen-like black hole\nsolutionis given. A subsection of Sec.2 contains the discussion of\nHorizon, Ergosphere, and static limit surface. Sec.3 is devoted\nwith the superradience scattering of scalar field off\nKerr-Sen-like black hole. Amplification factor  for superradiance\nsacttering off Kerr-Sen-like black hole has been calculated in\nSec.4 and a subsection of which is devoted with the superradiant\ninstability issue for Kerr-Sen-like black hole. In Sec.5\nConstraining of the parameter of this black hole is made from the\nobserved data for $\\mathrm{M}87^{*}$. Final Sec. 6 contains a\nbrief summary and discussions.\n\n\\section{EXACT KERR-SEN LIKE BLACK HOLE SOLUTION IN EINSTEIN-BUMBLEBEE MODEL}\nEinstein-bumblebee theory is an extension of Einstein's theory\nwhere a vector boson is involved that plays a pivotal role in the\nexisting symmetry of Einstein's theory \\cite{CASANA, ARS, ARS1,\nOLIVE, DING, ARS2}. It is an effective classical field theory\nwhere the vector field involved in the theory receives vacuum\nexpectation when spontaneous braking of an existing symmetry of\nthe action takes place and a Lorentz violation enters into the\ntheory as an outcome. Einstein-bumblebee theory is described by\nthe action\n\\begin{eqnarray}\n\\mathcal{S}=\\int d^{4} x \\sqrt{-g}\\left[\\frac{1}{16 \\pi\nG_{N}}\\left(\\mathcal{R}+\\varrho B^{\\mu} B^{\\nu} \\mathcal{R}_{\\mu\n\\nu}\\right)-\\frac{1}{4} B^{\\mu \\nu} B_{\\mu\n\\nu}-V\\left(B^{\\mu}\\right)\\right]. \\label{ACT}\n\\end{eqnarray}\nHere $\\varrho^{2}$ stands for the real coupling constant. It\ncontrols the non-minimal gravity interaction to the bumblebee field\n$B_{\\mu}$ (with the mass dimension 1). The coupling constant\n$\\varrho^{2}$ has mass dimension $-1$.\n\nThe action (\\ref{ACT}) leads to the following gravitational field\nequation in vacuum\n\\begin{eqnarray}\n\\mathcal{R}_{\\mu \\nu}-\\frac{1}{2} g_{\\mu \\nu} \\mathcal{R}=\\kappa\nT_{\\mu \\nu}^{B},\n\\end{eqnarray}\nwhere $\\kappa=8 \\pi G_{N}$ is the gravitational coupling. The\nbumblebee energy momentum tensor $T_{\\mu \\nu}^{B}$ reads\n\\begin{eqnarray}\\nonumber\nT_{\\mu \\nu}^{B}&=&B_{\\mu \\alpha} B_{\\nu}^{\\alpha}-\\frac{1}{4}\ng_{\\mu \\nu} B^{\\alpha \\beta} B_{\\alpha \\beta}-g_{\\mu \\nu} V+2\nB_{\\mu} B_{\\nu} V^{\\prime} \\\\\\nonumber\n&+&\\frac{\\varrho}{\\kappa}[\\frac{1}{2} g_{\\mu \\nu} B^{\\alpha}\nB^{\\beta} R_{\\alpha \\beta}\n-B_{\\mu} B^{\\alpha} R_{\\alpha \\nu}-B_{\\nu} B^{\\alpha} R_{\\alpha \\mu} \\\\\n&+&\\frac{1}{2} \\nabla_{\\alpha} \\nabla_{\\mu}\\left(B^{\\alpha}\nB_{\\nu}\\right)+\\frac{1}{2} \\nabla_{\\alpha}\n\\nabla_{\\nu}\\left(B^{\\alpha} B_{\\mu}\\right)-\\frac{1}{2}\n\\nabla^{2}\\left(B^{\\mu} B_{\\nu}\\right)-\\frac{1}{2} g_{\\mu \\nu}\n\\nabla_{\\alpha} \\nabla_{\\beta} \\left(B^{\\alpha} B^{\\beta}\\right)].\n\\end{eqnarray}\nHere prime(') denotes the differentiation with respect to the\nargument.\n\nEinstein's equation in the present  situation is  generalized to\n\\begin{eqnarray}\\nonumber\n&&\\mathcal{R}_{\\mu \\nu}-\\kappa b_{\\mu \\alpha}\nb_{\\nu}^{\\alpha}+\\frac{\\kappa}{4} g_{\\mu \\nu} b^{\\alpha \\beta}\nb_{\\alpha \\beta}+\\varrho b_{\\mu} b^{\\alpha} \\mathcal{R}_{\\alpha\n\\nu}+\\varrho b_{\\nu} b^{\\alpha} \\mathcal{R}_{\\alpha\n\\mu}\\\\\\nonumber &&-\\frac{\\varrho}{2} g_{\\mu \\nu} b^{\\alpha}\nb^{\\beta} \\mathcal{R}_{\\alpha \\beta}\n-\\frac{\\varrho}{2}\\left[\\nabla_{\\alpha}\n\\nabla_{\\mu}\\left(b^{\\alpha} b_{\\nu}\\right) +\\nabla_{\\alpha}\n\\nabla_{\\nu}\\left(b^{\\alpha} b_{\\mu}\\right)\n-\\nabla^{2}\\left(b_{\\mu} b_{\\nu}\\right)\\right]=0\\\\\n&&\\Rightarrow\\bar{R}_{\\mu \\nu}=0, \\label{MEQ}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\\nonumber\n\\bar{R}_{\\mu \\nu}&=&\\mathcal{R}_{\\mu \\nu}-\\kappa b_{\\mu \\alpha}\nb_{\\nu}^{\\alpha} +\\frac{\\kappa}{4} g_{\\mu \\nu} b^{\\alpha \\beta}\nb_{\\alpha \\beta} +\\varrho b_{\\mu} b^{\\alpha} \\mathcal{R}_{\\alpha\n\\nu} +\\varrho b_{\\nu} b^{\\alpha} \\mathcal{R}_{\\alpha \\mu}\n-\\frac{\\varrho}{2} g_{\\mu \\nu} b^{\\alpha} b^{\\beta}\n\\mathcal{R}_{\\alpha \\beta}\n+\\bar{B}_{\\mu \\nu}, \\\\\n\\bar{B}_{\\mu \\nu}&=&-\\frac{\\varrho}{2}\\left[\\nabla_{\\alpha}\n\\nabla_{\\mu}\\left(b^{\\alpha} b_{\\nu}\\right)+\\nabla_{\\alpha}\n\\nabla_{\\nu}\\left(b^{\\alpha}\nb_{\\mu}\\right)-\\nabla^{2}\\left(b_{\\mu} b_{\\nu}\\right)\\right].\n\\end{eqnarray}\nIf we now adopt the standard Boyer-Lindquist coordinates we\nfind that the underlying generalized gravity model admits a\nKerr-Sen-like black hole solution:\n\\begin{eqnarray}\nd s^{2}=-\\left(1-\\frac{2 M r}{\\rho^{2}}\\right) d t^{2}-\\frac{4 M r\na \\sqrt{1+\\ell} \\sin ^{2} \\theta}{\\rho^{2}} d t d\n\\varphi+\\frac{\\rho^{2}}{\\Delta} d r^{2}+\\rho^{2} d\n\\theta^{2}+\\frac{A \\sin ^{2} \\theta}{\\rho^{2}} d \\varphi^{2},\n\\label{FINAL}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n \\rho^{2}=r(r+b)+(1+\\ell) a^{2} \\cos ^{2} \\theta,\\Delta=\\frac{r(r+b)\n -2 M r}{1+\\ell}+a^{2}, A=\\left[r(r+b)+(1+\\ell)\na^{2}\\right]^{2}-\\Delta(1+\\ell)^{2} a^{2} \\sin ^{2} \\theta.\n\\end{eqnarray}\nHere  $a=\\frac{J}{M}$ represents the rotation (Kerr) parameter  $b=\\frac{Q^2}{M}$, the Sen parameter related to the electric charge, and  $\\ell$ the Lorentz-violating parameter. M, J, and  Q are representing respectively the mass, angular momentum, and charge of the black hole. Note that when  $\\ell \\rightarrow 0 $  it recovers the usual\nKerr-Sen metric \\cite{ASEN}.\n\\subsection{Horizon, Ergosphere, and static limit surface}\nThe metric is singular when $\\Delta=0$. The roots of the equation\ndepend on the parameter $M$,$a$, $b$, and $\\ell$. Having a maximum\nof two real roots, or two equal roots, and no real roots are the\npossibilities to occur from the condition$\\Delta=0$ \\cite{SGG,\nSGG1, SGG2, TJDP}. The horizons correspond to the two real roots\nwhich are given by\n\\begin{eqnarray}\nr_{\\pm}=M-\\frac{b}{2} \\pm\\frac{\n\\sqrt{(b-2M)^{2}-4a^{2}(1+\\ell)}}{2},\n\\end{eqnarray}\nwhere $\\pm$ signs correspond to the outer and inner horizon\nrespectively. The event horizon and Cauchy horizon are labelled\nwith $r_{eh}$ and $r_{ch}$ respectively. We will have a black hole\nonly when\n\\begin{equation}\n|b-2M|\\geq 2a\\sqrt{1+l}.\n\\end{equation}\nWhen $r_{-} = r_{+}$ we have extremal black hole. The parameter\nspace $(a,b)$ for three different values values of Lorentz\nviolating parameter $\\ell= -0.5, 0$, and $+0.5$ are shown here.\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.52\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{parameter_space_Black_hole_l_-.5.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.52\\textwidth}\n\\centering\n  \\includegraphics[width=.8\\linewidth]{parameter_space_Black_hole_l_0.eps}\n\\end{subfigure}\n\\begin{subfigure}{.52\\textwidth}\n\\centering\n  \\includegraphics[width=.8\\linewidth]{parameter_space_Black_hole_l_.5.eps}\n\\end{subfigure}%\n\\caption{Parameter space($a$,$b$) for various values of $l$.}\n\\label{fig:test}\n\\end{figure}\n\nFig.1 shows that parameter space for which we have a black hole is\nshrinking with an increase in the LV parameter $\\ell$. So increase\nin $\\ell$ makes the system less probable for having a black hole\nand the reverse is the case when LV parameter $\\ell$ decreases. We\nnow plot the $\\Delta$ versus $r\/M$ for various different variation\nof $a$, $b$, and $\\ell$. In the plots when the variation of one\nparameter is considered the other two are held fixed.\n\nFrom Fig. 2 and Fig. 3, we readily observe that there exist a\ncritical value  $a_{c}$ for the parameter $a$ for fixed values of\n$bl$ and $l$. Similarly, there is a critical value $b_{c}$ for the\nparameter $b$ when the parameter $a$ and $l$ care kept fixed. For\nfixed values of the parameter $b$ and $a$, $l_{c}$ comes out as\ncritical value for the parameter $\\ell$. At these critical values\ntwo roots of the  $\\Delta=0$ becomes identical which indicate\nextremal black holes. For instance, when $b=0.7, l=0.4$ we have\n$a_{c}=0.54935$, when  $a=0.6, l=0.4$ we have $b_{c}=0.580141$,\nand when $a=0.6, b=0.7$ we have $l_{c}=0.173611$. Therefore, for\n$a < a_{c}$ we have black hole and for $a\n> a_{c}$ we have naked singularity. Similarly, $b < b_{c}$\nindicates existence of a black hole and for $b > b_{c}$ it is a\nnaked singularity. For $l < l_{c}$, in a similar way, we have\nblack hole and $l > l_{c}$ signifies naked singularity.\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{delta_various_b_a_.6_l_.4.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{delta_various_a_l_.4_b_.7.eps}\n\\end{subfigure}\n\\caption{The left one gives variation of $\\Delta$ for various\nvalues of $b$ with $a=0.6$ and $l=0.4$ and the right one gives\nvariation for various values of $a$ with $b=0.7$ and $l=0.4$.}\n\\label{fig:test}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{delta_various_l_a_.6_b_.7.eps}\n\\end{subfigure}\n\\caption{It gives variation of $\\Delta$for various values of $l$\nwith $b=0.7$ and $a=0.6$.} \\label{fig:test}\n\\end{figure}\n\n\nLet us now turn towards the static limit surface (SLS) where the\nasymptotic time-translational Killing vector becomes null which\ngives\n\\begin{equation}\ng_{tt}=\\rho^{2}-2Mr=0.\n\\end{equation}\nThe real positive solutions of the above equation give radial\ncoordinates of the ergosphere given by\n\\begin{equation}\nr_{\\pm}^{e r g o}=M\n-\\frac{b}{2}\\pm \\frac{\\sqrt{(b-2M)^{2}-4a^{2}(1+\\ell) \\cos ^{2}\n\\theta}}{2}.\n\\end{equation}\nInside the SLS no observer can stay static and\nthey are bound to co-rotate around the black hole. The region\nbetween the SLS and the event\nhorizon is called the ergosphere shown below in Fig. 4. According to Penrose\n \\cite{PENR, PENR1} energy can be extracted from black hole's ergosphere.\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.3\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\linewidth]{cross_section_a_.5_l_.3_b_.7.eps}\n\\end{subfigure}%\n\\hspace{1.5em}%\n\\begin{subfigure}{.31\\textwidth}\n  \\centering\n  \\includegraphics[width=.95\\linewidth]{cross_section_a_.5_l_.6_b_.7.eps}\n\\end{subfigure}%\n\\hspace{1.5em}%\n\\begin{subfigure}{.32\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{cross_section_a_.5_l_.9_b_.7.eps}\n\\end{subfigure}\n\\par\\bigskip\n\\begin{subfigure}{.3\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\linewidth]{cross_section_a_.6_l_.3_b_.4.eps}\n\\end{subfigure}%\n\\hspace{1.5em}%\n\\begin{subfigure}{.31\\textwidth}\n  \\centering\n  \\includegraphics[width=.95\\linewidth]{cross_section_a_.6_l_.3_b_.6.eps}\n\\end{subfigure}%\n\\hspace{1.5em}\n\\begin{subfigure}{.33\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{cross_section_a_.6_l_.3_b_.8.eps}\n\\end{subfigure}\n\\caption{The cross-section of event horizon (outer red line),\nSLS (outer blue dotted line) and ergoregion of Kerr-Sen like black holes.}\n \\label{fig:test}\n\\end{figure}\nFig. 4 shows that the shape of the ergosphere closely depends on\nparameters $a, b,$ and $l$. A careful look reveals that the size\nof the ergosphere is enhancing with the increase in the LV\nparameter $\\ell$ when $a$ and $b$ remain fixed. The value of the\nparameter $b$ also has an influence on the size of the ergosphere.\nThe size of the ergosphere increases with the increase in the\nvalue of the parameter $b$ as well when $a$ and $l$ remain\nunchanged.\n\nHorizon angular velocity is found out to be\n\\begin{equation}\n\\hat{\\Omega}=\\frac{a\\sqrt{1+l}}{r_{eh}\\left(r_{eh}+b\\right)+a^{2}\\left(1+\\ell\\right)}.\n\\end{equation}\n\n\\section{Superradience scattering of scalar field off Kerr-Sen-like black hole}\nTo study the superradiance scattering of a scalar field $\\Phi$ with\nmass $\\mu$ we consider the Klein-Gordon equation in curved\nspacetime\n\\begin{eqnarray}\n\\left(\\bigtriangledown_{\\alpha}\\bigtriangledown^{\\alpha}+\\mu^{2}\\right)\\Phi(t,r,\\theta,\\phi)\n= \\left[\\frac{-1}{\\sqrt{-g}}\\partial_{\\sigma}\\left(g^{\\sigma\n\\tau}\\sqrt{-g}\\partial_{\\tau}\\right)+\\mu^{2}\\right]\\Phi(t,r,\\theta,\\phi)&=&0.\n\\label{KG}\n\\end{eqnarray}\nAdopting the separation of variables method on the equation\n(\\ref{KG})  it  is possible to separate it into radial and\nangular part using the following ansatz in the standard\nBoyer-Lindquist coordinates $(t, r, \\theta, \\phi)$\n\\begin{eqnarray}\n\\Phi(t, r, \\theta, \\phi)=R_{\\omega j m}(r) \\Theta(\\theta) e^{-i\n\\omega t} e^{i m \\phi}, \\quad j \\geq 0, \\quad-j \\leq m \\leq j,\n\\quad \\omega>0, \\label{PHI}\n\\end{eqnarray}\nwhere $R_{\\omega j m}(r)$ is the radial function and\n$\\Theta(\\theta)$ is the oblate spheroidal wave function. The\nsymbols $j$, $m$, and $\\omega$ respectively stand for the angular\neigenfunction, angular quantum number, and the positive frequency\nof the field, which is under investigation, as measured by a far\naway observer. Using the the ansatz ({\\ref{PHI}) the differential\nequation (\\ref{KG}), is found to  get separated into the following\ntwo ordinary differential equations. For radial part the equation\nreads\n\\begin{eqnarray}\n&&\\frac{d}{d r}(\\Delta \\frac{d R_{\\omega j m}(r)}{d\nr})+(\\frac{((r(r+b)+a^{2}(1+\\ell)) \\omega-am\\sqrt{1+\\ell}\n)^{2}}{\\Delta(1+\\ell)})R_{\\omega l m}(r)\n \\nonumber \\\\\n&&-(\\mu^{2} r(r+b)+j(j+1)+a^{2}(1+\\ell) \\omega^{2}-2 m \\omega\na\\sqrt{1+\\ell}) R_{\\omega l m}(r)=0, \\label{RE}\n\\end{eqnarray}\nand for the angular part it reads\n\\begin{eqnarray}\n&&\\sin \\theta \\frac{d}{d \\theta}\\left(\\sin \\theta \\frac{d\n\\Theta_{\\omega j m}(\\theta)}{d \\theta}\\right)+\\left(j(j+1) \\sin\n^{2} \\theta-\\left(\\left(a\\sqrt{1+\\ell} \\omega \\sin ^{2}\n\\theta-m\\right)^{2}\\right)\\right)\\Theta_{\\omega j m}(\\theta)\\nonumber \\\\\n&& + a^{2}(1+\\ell) \\mu^{2} \\sin ^{2} \\theta \\cos ^{2} \\theta~\n\\Theta_{\\omega j m}(\\theta)=0.\n\\end{eqnarray}\nWe can have a general solution of the radial equation (\\ref{RE})\nusing the earlier investigation \\cite{BEZERRA, KRANIOTIS}. We have\ngiven it as an appendix-A. However we are intended to study the\nscattering of the field $\\Phi$  following the articles\n\\cite{STRO1, STRO2, TEUK, PAGE}, and in this situation we have\nused the asymptotic matching procedure which is explicitely used\nin \\cite{RAN}. This article however is an extension of the\nimportant works \\cite{STRO1, STRO2, TEUK, PAGE}. Use of asymptotic\nmatching also has been found in \\cite{MK}. This led us to land\nonto the required result without using the general solution. Let\nus first focus on the radial equation. To deal with the radial\nequation according to our need we apply a Regge-Wheeler-like\ncoordinate $r_{*}$ which is defined by\n\\begin{eqnarray}\nr_{*} \\equiv \\int d r \\frac{r(r+b)+a^{2}(1+\\ell)}{\\Delta},\n\\quad\\left(r_{*} \\rightarrow-\\infty \\quad \\text{at event horizon},\n\\quad r_{*} \\rightarrow \\infty \\quad \\text{at infinity} \\right)\n\\end{eqnarray}\nTo transform the equation into the desired shape, we introduce a\nnew radial function $\\mathcal{R}_{\\omega j\nm}\\left(r_{*}\\right)=\\sqrt{r(r+b)+a^{2}(1+\\ell)} R_{\\omega j\nm}(r)$. After a few steps of algebra, we obtain the radial\nequation with our desired form where an effective potential took\nits entry into the picture.\n\\begin{equation}\n\\frac{d^{2} \\mathcal{R}_{\\omega l m}\\left(r_{*}\\right)}{d\nr_{*}^{2}}+V_{\\omega j m}(r) \\mathcal{R}_{\\omega j\nm}\\left(r_{*}\\right)=0. \\label{RE1}\n\\end{equation}\nThe effective potential that has the crucial role on the\nscattering reads\n\\begin{eqnarray}\nV_{\\omega j m}(r)&=&\\frac{1}{1+\\ell}\\left(\\omega-\\frac{m\n\\hat{a}}{r(r+b)+\\hat{a}^{2}}\\right)^{2}-\\frac{\\Delta}{\\left(r(r+b)\n+\\hat{a}^{2}\\right)^{2}}\\left[\\right. j(j+1)+\\hat{a}^{2}\n\\omega^{2}-2 m \\hat{a} \\omega+\\mu^{2} r(r+b)\n\\\\\\nonumber\n&&\\left.+\\sqrt{r(r+b)+\\hat{a}^{2}}\\frac{d}{dr}\\left(\\frac{\\Delta\n(2r+b)}{2\\left(r(r+b)+\\hat{a}^{2}\\right)^{\\frac{3}{2}}}\\right)\\right],\n\\label{POT}\n\\end{eqnarray}\nwhere $\\tilde{a}=a(1+\\ell)^{\\frac{1}{2}}$. We are intended to\nstudy the scattering of the scalar field $\\Phi$ under this\neffective potential. In this context, it is beneficial to study\nthe asymptotic behavior of the scattering potential at the event\nhorizon and at spatial infinity. In the  asymptotic limit the\npotential at the event horizon looks\n\\begin{eqnarray}\n\\lim _{r \\rightarrow r_{eh}} V_{\\omega j\nm}(r)=\\frac{1}{1+\\ell}\\left(\\omega-m \\tilde{\\Omega}_{h}\\right)^{2}\n\\equiv k_{e h}^{2},\n\\end{eqnarray}\n\\begin{equation}\n\\lim _{r \\rightarrow \\infty} V_{\\omega j m}(r)=\\omega^{2}-\\lim _{r\n\\rightarrow \\infty} \\frac{\\mu^{2} r(r+b)\n\\Delta}{\\left(r(r+b)+\\tilde{a}^{2}\\right)^{2}}=\\frac{\\omega^{2}}{1+\\ell}\n-\\hat{\\mu}^{2}\\equiv k_{\\infty}^{2}, ~~~\n\\hat{\\mu}=\\frac{\\mu}{\\sqrt{\\ell+1}}.\n\\end{equation}\nNote that at the two extremal points, event horizon and spatial\ninfinity, the potential asymptotically shows constant behavior.\nHowever, the values of the constants are different indeed.\n\nWe are now in a position to see  the asymptotic  behavior of  the\nradial equation.  It is found that the radial equation (\\ref{RE1})\nhas the following asymptotic solutions\n\\begin{equation}\\label{AS}\nR_{\\omega j m}(r) \\rightarrow\\left\\{\\begin{array}{cl}\n\\frac{A_{i n}^{eh} e^{-i k_{eh} r_{*}}}{\\sqrt{r_{e h}(r_{eh}+b)\n+\\hat{a}^{2}}} & \\text { for } r \\rightarrow r_{e h} \\\\\n\\mathcal{A}_{i n}^{\\infty} \\frac{e^{-i k_{\\infty}\nr_{*}}}{r}+\\mathcal{A}_{r e f}^{\\infty} \\frac{e^{i k_{\\infty}\nr_{*}}}{r} & \\text { for } r \\rightarrow \\infty\n\\end{array}\\right.\n\\end{equation}\nHere $\\mathcal{A}_{in}^{eh}$ represents the amplitude of the\nincoming scalar wave at event horizon(\"eh\"), and\n$\\mathcal{A}_{in}^{\\infty}$ is the corresponding quantity of the\nincoming scalar wave at infinity$(\"\\infty\")$. Along with these,\nthe amplitude of the reflected part of  scalar wave at infinity\n$(\"\\infty\")$ is $\\mathcal{A}_{ref}^{\\infty}$.\n\nLet us now compute the Wronskian for the region adjacent to the\nevent horizon and at infinity. It is found that  Wronskian for\nthis region is\n\\begin{equation}\nW_{eh}=\\left(R_{\\omega j m}^{e h} \\frac{d R_{\\omega j m}^{* e\nh}}{d r_{*}}-R_{\\omega j m}^{* eh} \\frac{d R_{\\omega j m}^{eh}}{d\nr_{*}}\\right),\n\\end{equation}\nand the Wronskian at infinity  reads\n\\begin{equation}\nW_{\\infty}=\\left(R_{\\omega j m}^{\\infty} \\frac{d R_{\\omega j m}^{*\n\\infty}}{d r_{*}}-R_{\\omega j m}^{* \\infty} \\frac{d R_{\\omega j\nm}^{\\infty}}{d r_{*}}\\right).\n\\end{equation}\nThe solutions  are linearly independent. From the knowledge of\nstandard theory of ordinary differential equation it can be\nunderstandable that the Wronskian corresponding to the solutions\nwill be independent of $r^*$. Thus, the Wronskian evaluated at\nhorizon does amenable  to equate with the Wronskian evaluated at\ninfinity. In physical sense, it is reflecting the flux\nconservation \\cite{REVIEW}. It results an important relation\nbetween the amplitudes of incoming and reflected waves at\ndifferent regions of interest.\n\\begin{equation}\n\\left|\\mathcal{A}_{r e f}^{\\infty}\\right|^{2}=\\left|\\mathcal{A}_{i\nn}^{\\infty}\\right|^{2}-\\frac{k_{e\nh}}{k_{\\infty}}\\left|\\mathcal{A}_{i n}^{e h}\\right|^{2}.\n\\label{AMP}\n\\end{equation}\nThe above equation transpires that if $\\frac{k_{e\nh}}{k_{\\infty}}<0$ i.e.,  $\\omega<m \\hat{\\Omega}_{e h}$, the\nscalar wave will be superradiantly amplified, because in this\nsituation, the relation  $\\left|\\mathcal{A}_{r e\nf}^{\\infty}\\right|^{2}>\\left|\\mathcal{A}_{i\nn}^{\\infty}\\right|^{2}$  holds explicitly.\n\\section{Amplification factor $Z_{jm}$ for superradiance}\nWe now rewrite the radial equation (\\ref{RE}) as\n\\begin{eqnarray}\\nonumber\n&&\\Delta^{2} \\frac{d^{2} R_{\\omega j m}(r)}{d r^{2}}+\\Delta\n\\frac{d \\Delta}{d r} \\cdot \\frac{d R_{\\omega j m}(r)}{d r}\\\\\n &&+\\left(\\frac{\\left(\\left(r(r+b)+\\hat{a}^{2}\\right)\n\\omega-\\hat{a} m\\right)^{2}}{1+\\ell}-\\Delta\\left(\\mu^{2}\nr(r+b)+j(j+1)+\\hat{a}^{2} \\omega^{2}-2 m \\hat{a}\n\\omega\\right)\\right) R_{\\omega j m}(r)=0. \\label{RE2}\n\\end{eqnarray}\nWe now turn to  derive the near-region as well as  the far-region\nsolution and try to find out a single solution matching the\nnear-region solution at infinitely with the far-region solution at\nits initial point such that this single solution works in the\nvicinity of the cardinal region. We apply the change of variable\n$x=\\frac{r-r_{eh}}{r_{eh}-r_{ch}}$. Using this change of variable\nequation (\\ref{RE2}) under the approximation $\\hat{a} \\omega \\ll\n1$ turns into\n\\begin{eqnarray}\n&&\\frac{x^{2}(x+1)^{2}}{(\\ell+1)^{2}} \\frac{\\mathrm{d}^{2}\nR_{\\omega j m}(x)}{\\mathrm{d} x^{2}}+\\frac{x(x+1)(2\nx+1)}{(\\ell+1)^{2}} \\frac{\\mathrm{d} R_{\\omega j m}(x)}{\\mathrm{d}\nx} \\\\\\nonumber &&+\\left(\\frac{P^2\nx^{4}}{1+\\ell}+\\frac{B^{2}}{1+\\ell}-\\frac{j(j+1)}{\\ell+1}\nx(x+1)-\\frac{\\hat{\\mu}^{2} P^{2}}{\\omega^{2}}\nx^{3}(x+1)-\\hat{\\mu}^{2} r_{e h}^{2} x(x+1)-\\frac{2 \\hat{\\mu}^{2}\nr_{e h} P}{\\omega} x^{2}(x+1) \\right. \\\\\\nonumber\n&&\\left.-\\frac{\\hat{\\mu}^{2}Pb}{\\omega}x^{2}(1+x)-\\hat{\\mu}^{2}br_{eh}x(1+x)\\right)\nR_{\\omega j m}(x)=0,\n\\end{eqnarray}\nwhere $P=\\left(r_{e h}-r_{c h}\\right) \\omega$ and\n$B=\\frac{(\\omega-m \\hat{\\Omega})}{r_{e h}-r_{c h}} r_{e h}^{2}$.\nFor near-region we have  $P x \\ll 1$ and $\\hat{\\mu}^{2} r_{e\nh}^{2} \\ll 1$ and hence the above equation reduces to\n\\begin{eqnarray}\nx^{2}(x+1)^{2} \\frac{\\mathrm{d}^{2} R_{\\omega j m}(r)}{\\mathrm{d}\nx^{2}}+x(x+1)(2 x+1) \\frac{\\mathrm{d} R_{\\omega j\nm}(r)}{\\mathrm{d} x}+\\left((\\ell+1)B^{2}-j(j+1)(\\ell+1)\nx(x+1)\\right) R_{\\omega j m}(r)=0.\n\\end{eqnarray}\nThe approximation $\\left(\\hat{\\mu}^{2} r_{e h}^{2} \\ll 1\\right)$\nis originated from the consideration that the Compton wavelength\nof the boson participating in the scattering process is much\nsmaller than the size of the black hole. The general solution of\nthe above equation in terms of associated Legendre function of the first kind\n$P_{\\lambda}^{\\nu}(y)$ can be written down as\n\\begin{eqnarray}\nR_{\\omega j m}(x)=c\nP^{2i\\sqrt{1+\\ell}B}_{\\frac{\\sqrt{1+4j(j+1)\n(l+1)}-1}{2}}(1+2x).\n\\end{eqnarray}\nWe now use the relation\n\\begin{equation}\nP_{\\lambda}^{\\nu}(z)=\\frac{1}{\\Gamma(1-\\nu)}\\left(\\frac{1+z}{1-z}\\right)^{\\nu\n\/ 2}{ }_{2} F_{1}\\left(-\\lambda, \\lambda+1 ; 1-\\nu ;\n\\frac{1-z}{2}\\right).\n\\end{equation}\nIt enables us to  express $R_{\\omega j m}(x)$ in terms of the\nordinary hypergeometric functions ${ }_{2} F_{1}(a, b ; c ; z)$ :\n\\begin{equation}\nR_{\\omega j m}(x)=c\\left(\\frac{x}{x+1}\\right)^{-i\\sqrt{\\ell+1} B}{\n}_{2} F_{1}\\left(\\frac{1-\\sqrt{1+4(\\ell+1) j(j+1)}}{2},\n\\frac{1+\\sqrt{1+4(\\ell+1) j(j+1)}}{2} ; 1-2 i\\sqrt{\\ell+1} B\n;-x\\right).\\label{NEAR}\n\\end{equation}\nAs we have mentioned, we require a single solution using the\nmatching condition at the desired position where the two solutions\nmingle with each other. In this respect, we need to observe the\nlarge $x$ behavior of the above expression. The Eqn. (\\ref{NEAR})\nfor large x $(x \\to\\infty$) turns into\n\\begin{eqnarray}\nR_{\\text {near-large } x} \\sim c &&\\left(\\frac{\\Gamma(\\sqrt{1+4(\\ell+1) j(j+1)})\n \\Gamma(1-2 i\\sqrt{\\ell+1} B)}{\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1) j(j+1)}}{2}\n -2 i\\sqrt{\\ell+1} B\\right) \\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1) j(j+1)}}{2}\\right)}\n x^{\\frac{\\sqrt{1+4(\\ell+1) j(j+1)}-1}{2}}+\\right.\\\\\n&& \\frac{\\Gamma(-\\sqrt{1+4(\\ell+1) j(j+1)}) \\Gamma(1-2\ni\\sqrt{\\ell+1} B)}{\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)\nj(j+1)}}{2}\\right) \\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)\nj(j+1)}}{2}-2 i\\sqrt{\\ell+1} B\\right)}\nx^{\\left.-\\frac{\\sqrt{1+4(\\ell+1) j(j+1)}+1}{2}\\right)} .\n\\label{NF}\n\\end{eqnarray}\nFor the far-region, we can use the  approximations $x+1 \\approx x$\nand $\\hat{\\mu}^{2} r_{e h}^{2} \\ll 1$. We may drop all the terms\nexcept those which describe the free motion with momentum $j$ and\nthat reduces equation (\\ref{RE2}) to\n\\begin{equation}\n\\frac{\\mathrm{d}^{2} R_{\\omega j m}(x)}{\\mathrm{d}\nx^{2}}+\\frac{2}{x} \\frac{\\mathrm{d} R_{\\omega j m}(x)}{\\mathrm{d}\nx}+\\left(k_{l}^{2}-\\frac{j(j+1)(\\ell+1)}{x^{2}}\\right) R_{\\omega j\nm}(x)=0, \\label{FAR}\n\\end{equation}\nwhere $k_{l} \\equiv \\frac{P\\sqrt{1+\\ell}}{\\omega}\n\\sqrt{\\omega^{2}-\\mu^{2}}$. Equation (\\ref{FAR}) has the\ngeneral solution\n\\begin{eqnarray}\\nonumber\nR_{\\omega j m, \\text { far }}=e^{-i k x}\\left(d_{1} x^{\\frac{\\sqrt{1+4(\\ell+1)\n j(j+1)}-1}{2}} M\\left(\\frac{1+\\sqrt{1+4(\\ell+1) j(j+1)}}{2},\n 1+\\sqrt{1+4(\\ell+1) l(l+1)}, 2 i k_{l} x\\right)+\\right. \\\\\n\\left.d_{2} x^{-\\frac{\\sqrt{1+4(\\alpha+1) j(j+1)}+1}{2}}\nM\\left(\\frac{1-\\sqrt{1+4(\\ell+1) j(j+1)}}{2}, 1-\\sqrt{1+4(\\ell+1)\nj(j+1)}, 2 i k_{l} x\\right)\\right), \\label{FARR}\n\\end{eqnarray}\nwhere $M(a, b, y)$ refers to the confluent hypergeometric Kummer\nfunction of first kind. In order to match the solution with\n(\\ref{NF}), we look for  the  small $x$ behavior of the solution\n(\\ref{FARR}). For small $x (x \\to 0)$, the equation (\\ref{FARR})\ntakes the form\n\\begin{equation} R_{\\omega\nj m, \\text { far-small } \\mathrm{x}} \\sim d_{1}\nx^{\\frac{\\sqrt{1+4(\\ell+1) j(j+1)}-1}{2}}+d_{2}\nx^{-\\frac{1+\\sqrt{1+4(\\ell+1) j(j+1)}}{2}}. \\label{FN}\n\\end{equation}\nThe solution (\\ref{NF}) and (\\ref{FN}) are susceptible for\nmatching, since these two have common region of interest. The\nmatching of the asymptotic solutions (\\ref{NF}) and (\\ref{FN})\nenables us to compute the scalar wave flux at infinity resulting in\n\\begin{eqnarray}\nd_{1}=&\\left.c \\frac{\\Gamma(\\sqrt{1+4(\\ell+1) j(j+1)})\n\\Gamma(1-2 i\\sqrt{\\ell+1} B)}{\\left.\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1) j(j+1)}}{2}\\right)\n-2 i\\sqrt{\\alpha+1} B\\right) \\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1) j(j+1)}}{2}\\right.}\\right), \\\\\\\nd_{2}=& c \\frac{\\Gamma(-\\sqrt{1+4(\\ell+1) j(j+1)}) \\Gamma(1-2\ni\\sqrt{\\ell+1} B)}{\\left.\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)\nj(j+1)}}{2}\\right)-2 i\\sqrt{\\alpha+1} B\\right)\n\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1) j(j+1)}}{2}\\right)}.\n\\label{DD}\n\\end{eqnarray}\nWe expand equation (\\ref{FARR}) around infinity which after\nexpansion results\n\\begin{eqnarray}\nd_{1} \\frac{\\Gamma(1+\\sqrt{1+4(\\ell+1) j(j+1)})}{\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1)\nj(j+1)}}{2}\\right)} k_{l}^{-\\frac{1+\\sqrt{1+4(\\ell+1)\nj(j+1)}}{2}}\\left((-2 i)^{-\\frac{1+\\sqrt{1+4(\\ell+1)\n j(j+1)}}{2}} \\frac{e^{-i k_{l} x}}{x}+(2 i)^{-\\frac{1+\\sqrt{1+4(\\ell+1) j(j+1)}}{2}}\n  \\frac{e^{i k_{l} x}}{x}\\right)+ \\\\\\nonumber\nd_{2} \\frac{\\Gamma(1-\\sqrt{1+4(\\ell+1)\nj(j+1)})}{\\frac{1-\\sqrt{1+4(\\ell+1) j(j+1)}}{2}}\nk_{l}^{\\frac{\\sqrt{1+4(\\ell+1) j(j+1)}-1}{2}}\\left((-2\ni)^{\\frac{\\sqrt{1+4(\\ell+1) j(j+1)}-1}{2}} \\frac{e^{-i k_{l}\nx}}{x}+(2 i)^{\\frac{\\sqrt{1+4(\\ell+1) j(j+1)}-1}{2}} \\frac{e^{i\nk_{l} x}}{x}\\right).\n\\end{eqnarray}\nWith the approximations $\\frac{1}{x} \\sim \\frac{P}{\\omega} \\cdot\n\\frac{1}{r}, \\quad e^{\\pm i k_{l} x} \\sim e^{\\pm i\n\\sqrt{(1+\\ell)(\\omega^{2}-\\mu^{2})} r}$, if we match the above\nsolution with the radial solution \\eqref{AS}\n$$\nR_{\\infty}(r) \\sim \\mathcal{A}_{i n}^{\\infty} \\frac{e^{-i\n\\sqrt{\\frac{\\omega^{2}}{1+\\ell}-\\hat{\\mu}^{2}}\nr^{*}}}{r}+\\mathcal{A}_{r e f}^{\\infty} \\frac{e^{i\n\\sqrt{\\frac{\\omega^{2}}{1+\\ell}-\\hat{\\mu}^{2}} r^{*}}}{r}, \\quad\n\\text { for } \\quad r \\rightarrow \\infty\n$$\nwe get\n$$\n\\begin{array}{c}\n\\mathcal{A}_{i n}^{\\infty}=\\frac{P}{\\omega}\\left(d_{1}(-2 i)^{-\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}}\n \\frac{\\Gamma(1+\\sqrt{1+4(\\ell+1)j(j+1)})}{\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}\\right)}\n k_{l}^{-\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}}+\\right. \\\\\n\\left.d_{2}(-2 i)^{\\frac{\\sqrt{1+4(\\ell+1)j(j+1)}-1}{2}}\n\\frac{\\Gamma(1-\\sqrt{1+4(\\ell+1)j(j+1)})}{\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)j(j+1)}}{2}\\right)}\nk_{l}^{\\frac{\\sqrt{1+4(\\ell+1)j(j+1)}-1}{2}}\\right),\n\\end{array}\n$$\nand\n$$\n\\begin{array}{l}\n\\mathcal{A}_{r e f}^{\\infty}=\\frac{P}{\\omega}\\left(d_{1}(2 i)^{-\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}}\n \\frac{\\Gamma(1+\\sqrt{1+4(\\ell+1)j(j+1)})}{\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}\\right)}\n  k_{l}^{-\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}}+\\right. \\\\\n\\left.d_{2}(2 i) \\frac{\\sqrt{1+4(\\ell+1)j(j+1)}-1}{2}\n\\frac{\\Gamma(1-\\sqrt{1+4(\\ell+1)j(j+1)})}{\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)j(j+1)}}{2}\\right)}\nk_{l}^{\\frac{\\sqrt{1+4(\\ell+1)j(j+1)}-1}{2}}\\right) .\n\\end{array}\n$$\nSubstituting  the expressions of $d_{1}$ and $d_{2}$ from Eqn.\n(\\ref{DD}) into the above expressions we have\n\\begin{eqnarray}\n\\mathcal{A}_{in}^{\\infty}&=&\\frac{c(-2\ni)^{-\\frac{1+\\sqrt{1+4(\\ell+1)\nj(j+1)}}{2}}}{\\sqrt{(1+\\ell)(\\omega^{2}-\\mu^{2})}} \\cdot\n\\frac{\\Gamma(\\sqrt{1+4(\\ell+1) j(j+1)}) \\Gamma(1+\\sqrt{1+4(\\ell+1)\nj(j+1)})}{\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}-2\ni\\sqrt{\\ell+1}\nB\\right)\\left(\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}\\right)\\right)^{2}}\\times\n\\\\\\nonumber &&\\Gamma(1-2 i\\sqrt{\\alpha+1} B)\nk_{l}^{\\frac{1-\\sqrt{1+4(\\ell+1)j(j+1)}}{2}}+\\frac{c(-2\ni)^{\\frac{\\sqrt{1+4(\\ell+1)j(j+1)}-1}{2}}}{\\sqrt{(1+\\ell)(\\omega^{2}-\\hat{\\mu}^{2})}}\n\\times \\\\\\nonumber &&\\frac{\\Gamma(1-\\sqrt{1+4(\\ell+1)j(j+1)})\n\\Gamma(-\\sqrt{1+4(\\ell+1)j(j+1)})}{\\left(\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)j(j+1)}}{2}\\right)\\right)^{2}\n\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)j(j+1)}}{2}-2 i\\sqrt{\\ell+1}\nB\\right)} \\Gamma(1-2 i\\sqrt{\\ell+1} B)\nk_{l}^{\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\mathcal{A}_{ref}^{\\infty}&=&\\frac{c(2\ni)^{-\\frac{1+\\sqrt{1+4(\\ell+1)\nj(j+1)}}{2}}}{\\sqrt{(1+\\ell)(\\omega^{2}-\\mu^{2})}} \\cdot\n\\frac{\\Gamma(\\sqrt{1+4(\\ell+1) j(j+1)}) \\Gamma(1+\\sqrt{1+4(\\ell+1)\nj(j+1)})}{\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}-2\ni\\sqrt{\\ell+1}\nB\\right)\\left(\\Gamma\\left(\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}\\right)\\right)^{2}}\\times\n\\\\\\nonumber &&\\Gamma(1-2 i\\sqrt{\\alpha+1} B)\nk_{l}^{\\frac{1-\\sqrt{1+4(\\ell+1)j(j+1)}}{2}}+\\frac{c(2\ni)^{\\frac{\\sqrt{1+4(\\ell+1)j(j+1)}-1}{2}}}{\\sqrt{(1+\\ell)(\\omega^{2}-\\hat{\\mu}^{2})}}\n\\times \\\\\\nonumber &&\\frac{\\Gamma(1-\\sqrt{1+4(\\ell+1)j(j+1)})\n\\Gamma(-\\sqrt{1+4(\\ell+1)j(j+1)})}{\\left(\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)j(j+1)}}{2}\\right)\\right)^{2}\n\\Gamma\\left(\\frac{1-\\sqrt{1+4(\\ell+1)j(j+1)}}{2}-2 i\\sqrt{\\ell+1}\nB\\right)} \\Gamma(1-2 i\\sqrt{\\ell+1} B)\nk_{l}^{\\frac{1+\\sqrt{1+4(\\ell+1)j(j+1)}}{2}}.\n\\end{eqnarray}\nThe amplification factor ultimately results out to be\n\\begin{equation}\nZ_{j m} \\equiv \\frac{\\left|\\mathcal{A}_{r e\nf}^{\\infty}\\right|^{2}}{\\left|\\mathcal{A}_{i\nn}^{\\infty}\\right|^{2}}-1. \\label{AMPZ}\n\\end{equation}\nEquation (\\ref{AMPZ}) is a general expression of the amplification\nfactor obtained by making use of the asymptotic matching method.\nWhen\n$\\frac{\\left|\\mathcal{A}_{ref}^{\\infty}\\right|^{2}}{\\left|\\mathcal{A}_{i\nn}^{\\infty}\\right|^{2}}$ acquires a value greater than unity there\nwill be a gain in amplification factor that corresponds to\nsuperradiance phenomena. However, a negative value of the\namplification factor indicates a loss that corresponds to the\nnonappearance of superradiance. To study the effect of Lorentz\nviolation on the superradiance phenomena, it will be useful to\nplot $Z_{j m}$ versus $M\\omega$ for different LV parameters. In\nFig. (6), we present the variation $Z_{j m}$ versus $M\\omega$ for the\nleading multipoles $J= 1$, and $2$ taking different values (both\nnegative and positive) of LV Parameter. From the fig. (5) along with fig. (6), it is\nevident that superradiance for a particular $j$ occurs when the\nallowed values of $m$ are restricted to $m > 0$.\n\nFor negative $m$ amplification factor takes negative value which refers to the\nnonoccurrence of superradiance. The plots also show transparently\nthat with the decrease in the value of the LV paraneters the\nsuperradiance process enhances and the reverse is the case when\nthe value of the LV parameter decreases. In Fig. (8) we have also\nstudied the effect of the parameter $b=\\frac{Q^2}{M}$ on the\nsuperradiance scenario. It shows that the superradiance scenario\ngets diminished with the increase in the value of the parameter\n$b$. In \\cite{ARS} we have noticed that the size of the shadow\ndecreases with the increase in the value of both the parameters\n$l$ and $b$. The only difference is that $l$ can take both\npositive values, however, $b$ as per definition can not be\nnegative. Therefore, an indirect relation of superradiance with\nthe size of the shadow is being revealed through this analysis. A\ndecrease in the value of $b$ and $l$ indicate the increase in the\nsize of the shadow.\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z1-1_various_l_a_.4_b_.1.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z10_various_l_a_.4_b_.1.eps}\n\\end{subfigure}\n\\caption{Variation of amplification factors with $\\ell$ for\nnon-superradiant multipoles with $\\hat{\\mu}=0.1,b=0.1$, and\n$\\hat{a}=0.4$.} \\label{fig:test}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z11_b_.1.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n\n  \\includegraphics[width=.8\\linewidth]{z21_b_.1.eps}\n\\end{subfigure}\n\\begin{subfigure}{.58\\textwidth}\n\\centering\n  \\includegraphics[width=.8\\linewidth]{z22_b_.1.eps}\n\\end{subfigure}\n\\caption{Variation of amplification factors with $\\ell$ for\nvarious multipoles with $\\hat{\\mu}=0.1,b=0.1$, and $\\hat{a}=0.9$.}\n\\label{fig:test}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z11_various_a_b_.1_l_-.5.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z11_various_a_b_.1_l_1.eps}\n\\end{subfigure}\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z22_various_a_b_.1_l_-.5.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z22_various_a_b_.1_l_1.eps}\n\\end{subfigure}\n\\caption{Variation of amplification factors with $\\hat{a}$ for\nvarious multipoles with $\\hat{\\mu}=0.1$ and $b=0.1$. For left ones\n$\\ell=-0.5$ and for right ones $\\ell=1$.} \\label{fig:test}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z11_various_b_a_.7_l_-.5.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z11_various_b_a_.7_l_1.eps}\n\\end{subfigure}\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z22_various_b_a_.7_l_-.5.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{z22_various_b_a_.7_l_1.eps}\n\\end{subfigure}\n\\caption{Variation of amplification factors with $b$ for various\nmultipoles with $\\hat{\\mu}=0.1$ and $\\hat{a}=0.7$. For left ones\n$\\ell=-0.5$ and for right ones $\\ell=1$.} \\label{fig:test}\n\\end{figure}\n\\subsection{Superradiant instability for Kerr-Sen-like black hole}\nFrom equation (\\ref{RE}) we have\n\\begin{eqnarray}\n\\Delta \\frac{d}{d r}\\left(\\Delta \\frac{d R_{\\omega j m}}{d\nr}\\right)+\\xi R_{\\omega j m}=0, \\label{MRE}\n\\end{eqnarray}\nwhere for a slowly rotating black hole $(\\hat{a} \\omega \\ll 1)$\n$$\n\\xi \\equiv\\frac{\\left(\\left(r(r+b)+\\hat{a}^{2}\\right) \\omega-m\n\\hat{a}\\right)^{2}}{1+\\ell}+\\Delta\\left(2 m \\hat{a}\n\\omega-l(l+1)-\\mu^{2}r(r+b)\\right).\n$$\nDemanding the black hole bomb mechanism, we should have the\nfollowing solutions for the radial equation (\\ref{MRE})\n$$\nR_{\\omega j m} \\sim\\left\\{\\begin{array}{ll}\ne^{-i(\\omega-m \\hat{\\Omega}) r_{*}} & \\text { as } r \\rightarrow r_{e h}\n\\left(r_{*} \\rightarrow-\\infty\\right) \\\\\n\\frac{e^{-\\sqrt{\\mu^{2}-\\omega^{2} r_{*}}}}{r} & \\text { as } r\n\\rightarrow \\infty  \\left(r_{*} \\rightarrow \\infty\\right)\n\\end{array}\\right.\n$$\nThe above solution represents the physical boundary conditions\nthat the scalar wave at the black hole horizon is purely ingoing\nwhile at spatial infinity it is decaying exponentially (bounded)\nsolution, provided that $\\omega^{2}<\\mu^{2}$. With the new radial\nfunction\n$$\n\\psi_{\\omega j m} \\equiv \\sqrt{\\Delta} R_{\\omega j m},\n$$\nthe radial equation (\\ref{MRE}) becomes\n$$\n\\left(\\frac{d^{2}}{d r^{2}}+\\omega^{2}(1+\\ell)-V\\right)\n\\psi_{\\omega j m}=0.\n$$\nwith\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{parameter_space_l_-.5.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{parameter_space_l_0.eps}\n\\end{subfigure}\n\\begin{subfigure}{.58\\textwidth}\n  \\includegraphics[width=.8\\linewidth]{parameter_space_l_.5.eps}\n\\end{subfigure}%\n\\caption{Parameter space($m\\Omega$-$\\mu$) for massive scalar field\nwhere colored area represents region with stable dynamics and\nnon-colored area represents region with unstable dynamics.}\n\\label{fig:test}\n\\end{figure}\n$$\n\\omega^{2}(1+\\ell)-V=\\frac{\\xi\\left(1+\\ell\\right)+M^{2}-\\hat{a}^{2}+\\frac{b^{2}}{4}-bM}{\\Delta^{2}},\n$$\nwhich is the Regge-Wheel equation. By discarding the terms\n$\\mathcal{O}\\left(1 \/ r^{2}\\right)$ the asymptotic form of the\neffective potential $V(r)$ reads\n$$\nV(r)=\\mu^{2}\\left(1+\\ell\\right)-\\left(1+\\ell\\right)\\frac{4 M\n\\omega^{2}}{r}+(\\ell+1) \\frac{2 M \\mu^{2}}{r}.\n$$\nTo realize the trapping meaningfully by the above effective\npotential it is necessary that its asymptotic derivative be\npositive i.e. $V^{\\prime} \\rightarrow 0^{+}$ as $r \\rightarrow\n\\infty[33]$.  This along with the fact that superradiance\namplification of scattered waves occur when $\\omega<m\n\\hat{\\Omega}$ we get the regime\n$$\n\\frac{\\mu}{\\sqrt{2}}<\\omega^{2}<m \\hat{\\Omega},\n$$\nin which the integrated system of Kerr-Sen bumblebee black hole\nand massive scalar field may experience a superradiant instability,\nknown as the black hole bomb. The dynamics of the massive\nscalar field in Kerr-Sen like black hole will remain stable when\n$\\mu\\geq\\sqrt{2}m\\hat{\\Omega}$.\n\n\\section{Constraining from the observed data for $\\mathrm{M}87^{*}$}\nThis section is devoted to constraining the parameters from the\nobserved data for $\\mathrm{M}87^*$. After the announcement of the\ncapturing of the shadow there have been several attempts to\nconstraint the parameters used in different modified theories of\ngravity \\cite{MISBAM87, MK, SGM87}. However, before going towards\nconstraining parameters let us give a brief description of the\nphoton orbit in this Kerr-Sen-like spacetime background and see\nhow the shadow gets deformed with the\nadditional parameters of this Lorentz violating spacetime.\n\\subsection{Mathematical formulation of the deviation from circularity}\nThe Hamiltonian for a massless particle like the photon is given\nby\n\\begin{equation}\nH\\left(x^{\\tau}, p_{\\tau}\\right)=\\frac{1}{2}\\left[g^{\\tau \\nu}\np_{\\tau} p_{\\nu} -\\left(p_{0} \\sqrt{-g^{00}}\\right)^{2}\\right].\n\\end{equation}\nThe standard definitions $x^{\\tau}=\\partial H \/ \\partial p_{\\tau}$,\nand $\\dot{p_{\\tau}}=\\partial H \/ \\partial x^{\\tau}$ renders  the\nequations of motion for the photon. Then the null\ngeodesics in the bumblebee rotating black hole spacetime in terms\nof $\\xi$ are given by\n\\begin{eqnarray}\\nonumber\n\\rho^{2} \\frac{d r}{d \\lambda}=\\pm \\sqrt{R},\\qquad \\rho^{2} \\frac{d\n\\theta}{d \\lambda}=\\pm \\sqrt{\\Theta}, \\\\\\nonumber\n(1+\\ell) \\Delta \\rho^{2} \\frac{d t}{d \\lambda}=A-2 \\sqrt{1+\\ell} \\operatorname{Mra\\xi}, \\\\\n(1+\\ell) \\Delta \\rho^{2} \\frac{d \\phi}{d \\lambda}=2 \\sqrt{1+\\ell}\nM r a+\\frac{\\xi}{\\sin ^{2} \\theta}\\left(\\rho^{2}-2 M r\\right),\n\\label{GEO1}\n\\end{eqnarray}\nwhere $\\lambda$ is the affine parameter and\n\\begin{eqnarray}\nR(r)=\\left[\\frac{r(r+b)+(1+\\ell) a^{2}}{\\sqrt{1+\\ell}}-a\n\\xi\\right]^{2}-\\Delta\\left[\\eta+(\\xi-\\sqrt{1+\\ell}a)^{2}\\right],\n\\Theta(\\theta)=\\eta+(1+\\ell) a^{2} \\cos ^{2} \\theta-\\xi^{2} \\cot\n^{2} \\theta. \\label{GEO2}\n\\end{eqnarray}\nIn the Eqns. (\\ref{GEO1}) and (\\ref{GEO2}), we introduce two\nconserved parameters $\\xi$ and $\\eta$ as usual which are defined\nby\n\\begin{eqnarray}\n\\xi=\\frac{L_{z}}{E} \\quad \\textrm{and} \\quad\n\\eta=\\frac{\\mathcal{Q}}{E^{2}},\n\\end{eqnarray}\nwhere $E$, $L_{z}$, and $\\mathcal{Q}$ are the energy, the axial\ncomponent of the angular momentum, and the $Carter~constant$\nrespectively. The radial equation of motion can be cast into the\nknown form\n\\begin{eqnarray}\n\\left(\\rho^{2} \\frac{d r}{d \\lambda}\\right)^{2}+V_{e f f}=0.\n\\end{eqnarray}\nThe effective potential $V_{e f f}$ in this situation is written\ndown as\n\\begin{eqnarray}\nV_{e f f}=-[\\frac{r(r+b)+(1+l) a^{2}}{\\sqrt{1+l}}-a\n\\xi]^{2}+\\Delta\\left[\\eta+(\\xi-\\sqrt{1+l}a)^{2}\\right]\n-\\frac{\\left[r(r+b)+a^{2}(1+l)\\right]^{2}}{1+l}.\n\\end{eqnarray}\nNote that it has explicit dependence on the LV factor $\\ell$ and\n$a$. So it is natural that the structure the of photon orbit will\ndepend on the parameters $a$ and $l$. The unstable spherical orbit\non the equatorial plane is given by the following equations\n\\begin{eqnarray}\n\\theta=\\frac{\\pi}{2},\\quad R(r)=0,\\quad \\frac{d R}{d r}=0,\\quad \\frac{d^{2} R}{d\nr^{2}}<0,\\quad \\mathrm{and}\\quad \\eta=0.\\label{CONDITION1}\n\\end{eqnarray}\nFor more generic orbits $\\theta \\neq \\pi \/ 2$ and $\\eta \\neq 0,$\nthe solution of Eqn. (\\ref{CONDITION1}) $ r=r_{s}$, gives the $r-$\nconstant orbit, which is also called spherical orbit and the\nconserved parameters of the spherical orbits can be expressed in\nthe following form\n\\begin{eqnarray}\n\\begin{aligned}\n\\xi_{s} &=\\frac{a^{2}\\left(1+\\mathrm{l}\\right)\\left(2 M+2\nr_{\\mathrm{s}} +b\\right)+r_{\\mathrm{s}}\\left(2 r_{\\mathrm{s}}^{2}\n+3 b r_{\\mathrm{s}}+b^{2}-2 M\\left(3 r_{\\mathrm{s}}\n+b\\right)\\right)}{a\\sqrt{1+\\mathrm{l}}\\left(2 M-2 r_{\\mathrm{s}}-b\\right)}, \\\\\n\\eta_{s} &=-\\frac{r_{\\mathrm{s}}^{2}\\left(-8\na^{2}\\left(1+\\mathrm{l}\\right) M\\left(2 r_{\\mathrm{s}}+b\\right)\n+\\left(2 r_{\\mathrm{s}}^{2}+3b r_{\\mathrm{s}} +b^{2} -2 M\\left(3\nr_{\\mathrm{s}}+b\\right)\\right)^{2}\\right)}{a^{2}\\left(1+\\mathrm{l}\\right)\\left(2\nM-2 r_{\\mathrm{s}}-b\\right)^{2}} .\n\\end{aligned}\n\\end{eqnarray}\nThe two celestial coordinates which are used to describe the\nshape of the shadow that an observer see in the sky,\ncan be given by\n\\begin{eqnarray}\\nonumber\n\\alpha(\\xi, \\eta ; \\theta)&=&\\lim _{r \\rightarrow \\infty} \\frac{-r\np^{(\\varphi)}}{p^{(t)}} = -\\xi_{s} \\csc \\theta,\\\\\\nonumber\n\\beta(\\xi, \\eta ; \\theta)&=&\\lim _{r \\rightarrow \\infty} \\frac{r\np^{(\\theta)}}{p^{(t)}}\n=\\sqrt{\\left(\\eta_{s}+a^{2} \\cos ^{2} \\theta-\\xi_{s}^{2} \\cot ^{2} \\theta\\right)},\\\\\n\\end{eqnarray}\nwhere $(p^{(t)}, p^{(r)}, p^{(\\theta)}$, and $p^{(\\phi)})$ are the\ntetrad components of the photon momentum with respect to locally\nnon-rotating reference frames \\cite{BARDEEN}.\n\\subsection{Constraining with respect to deviation from circularity data: $\\Delta C \\le 0.10$}\nWe now proceed towards constraining the parameters involved in\nthis spacetime metric from the available experimental findings of\nthe $M87^*$ as a new window for testing gravity in the\nstrong-field regime has been opened after the announcement of the\nnews of capturing the image of supermassive black hole $M87^*$ at\n$1.3mm$ wavelength with the angular resolution of $20 \\mu as$. The\nangular diameter of the shadow of $M87^*$ was fund to be $\\theta_d\n= 42 \\pm 3\\mu as$ and the deviation from circularity was $\\Delta C\n\\le 10$ which was consistent with the Kerr black hole's image as\npredicted from the theory of General Relativity \\cite{EHT1, EHT2,\nEHT3, EHT4, EHT5, EHT6}. Let us first proceed to constrain the\nparameter from the observation of $M87^*$ concerning $\\Delta C \\le\n.10$. We have considered Kerr-Sen-like black holes, which have\nadditional parameters $b$, $l$ along with the Kerr black hole\nparameters, and the parameters $b$ and $l$ produce deviation from\nKerr geometry with a considerably good configuration. It is also\nfound that the LV parameter quantitatively influences the\nstructure of the event horizon by reducing its radius\nsignificantly than that of the Kerr black hole \\cite{ARS}, for a\ngiven $b$ and $a$, and the resulting increase in ergosphere area\nis thereby likely to have an impact on energy extraction\n\\cite{ARS}. The  boundary of the shadow is described by the  polar\ncoordinate $(R(\\phi),\\phi)$ with the origin at the center of the\nshadow $(\\alpha_{C}, \\beta_{C})$, where\n$\\alpha_{C}=\\frac{|\\alpha_{max}+\\alpha_{min}|}{2}$ and\n$\\beta_{C}=0$. If a point $(\\alpha, \\beta)$ over the boundary of\nthe image subtends an angle $\\phi$ on the $\\alpha$ axis at the\ngeometric center, $\\left(\\alpha_{C}, 0\\right)$ and  $R(\\phi)$ be\nthe distance between the point $(\\alpha, \\beta)$ and\n$\\left(\\alpha_{C}, 0\\right)$, then the average radius\n$R_{\\text{avg}}$ of the image is given by \\cite{CBK}\n\\begin{equation}\nR_{\\text {avg}}^{2}  \\equiv \\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} d \\phi R^{2}(\\phi), \\\\\n\\end{equation}\nwhere $R(\\phi) \\equiv\n\\sqrt{\\left(\\alpha(\\phi)-\\alpha_{C}\\right)^{2}+\\beta(\\phi)^{2}}$,\nand  $\\phi = tan^{-1}\\frac{\\beta(\\phi)}{\\alpha(\\phi)-\\alpha_{C}}$\n\nWith the above inputs, the  circularity deviation  $\\Delta C$ is\ndefined by \\cite{TJDP},\n\\begin{equation}\n\\Delta C \\equiv 2\\sqrt{\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} d\n\\phi\\left(R(\\phi)-R_{\\text {avg }}\\right)^{2}}.\n\\end{equation}\nIn the figures below, the deviation\nfrom circularity $\\Delta C$ is shown for Kerr-Sen-like black holes\nfor inclination angles $\\theta=90^{o}$ and $17^{o}$ respectively.\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.25\\textwidth}\n\\centering\n \\includegraphics[scale=.65]{deviation_a_b_with_l_.1_theta_90.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.28\\textwidth}\n\\centering\n\\raisebox{.2\\height}{\\includegraphics[scale=.55]{deviation_a_b_with_l_.1_theta_901.eps}}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n\\centering\n \\includegraphics[scale=.65]{deviation_a_l_with_b_.1_theta_902.eps}\n\\end{subfigure}%\n\\begin{subfigure}{.28\\textwidth}\n\\centering\n\\raisebox{.19\\height}{\\includegraphics[scale=.6]{deviation_a_l_with_b_.1_theta_903.eps}}\n\\end{subfigure}\n\\caption{The left one is for $\\ell=0.1$ and the right one is for\n$b=0.1$ where the inclination angle is $90^{o}$. The black solid\nlines correspond to $\\Delta C=0.1$.}\n\\end{figure}\n\\smallskip\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.25\\textwidth}\n\\centering\n \\includegraphics[scale=.65]{deviation_a_b_with_l_.1_theta_17.eps}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.28\\textwidth}\n\\centering\n\\raisebox{.2\\height}{\\includegraphics[scale=.55]{deviation_a_b_with_l_.1_theta_171.eps}}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n\\centering\n \\includegraphics[scale=.65]{deviation_a_l_with_b_.1_theta_172.eps}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.28\\textwidth}\n\\centering\n\\raisebox{.19\\height}{\\includegraphics[scale=.6]{deviation_a_l_with_b_.1_theta_173.eps}}\n\\end{subfigure}\n\\caption{The left one is for $\\ell=0.1$ and the right one is for\n$b=0.1$ where the inclination angle is $17^{o}$.}\n\\end{figure}\nWe compare the shadows produced from the numerical calculation by\nthe Kerr-Sen-like black holes with the observed one for the\n$\\mathrm{M}87^{*}$ black hole. For comparison, we consider the\nexperimentally obtained astronomical data for the circularity\ndeviation $\\Delta \\leq 0.10$ in this subsection. The next section\nis devoted to constrain the parameter from observation of angular\ndiameter $\\theta_{d}=42\\pm 3 \\mu as$ \\cite{EHT1, EHT2, EHT3, EHT4,\nEHT5, EHT6}.\n\n\\subsection{Constraining from the observation of angular diameter $\\theta_{d}=42\\pm 3 \\mu as$}\nWe now consider the shadow angular diameter which is define by\n\\begin{equation}\n\\theta_{d}=\\frac{2}{d}\\sqrt{\\frac{A}{\\pi}},\n\\end{equation}\nWhere $A=2\\int_{r_{-}}^{r_{+}} \\beta d\\alpha $ is the shadow area\nand $d=16.8 Mpc$ is the distance of $M87^{*}$ from the earth.\nThese relations enable us to accomplish a comparison between the\ntheoretical predictions for Kerr-Sen-like black-hole shadows and\nthe experimental findings of the Event Horizon Telescope\ncollaboration. In the figures below, the angular diameter\n$\\theta_{d}$ is shown for Kerr-Sen-like black holes for\ninclination angles $\\theta=90^{o}$ and $17^{o}$ respectively.\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.25\\textwidth}\n\\centering\n \\includegraphics[scale=.65]{theta_a_b_with_l_.1_theta_90a.eps}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.28\\textwidth}\n\\centering\n\\raisebox{.2\\height}{\\includegraphics[scale=.55]{theta_a_b_with_l_.1_theta_90b.eps}}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n\\centering\n \\includegraphics[scale=.65]{theta_a_l_with_b_.1_theta_90c.eps}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.28\\textwidth}\n\\centering\n\\raisebox{.19\\height}{\\includegraphics[scale=.6]{theta_a_l_with_b_.1_theta_90d.eps}}\n\\end{subfigure}\n\\caption{The left one is for $\\ell=0.1$ and the right one is for\n$b=0.1$ where the inclination angle is $90^{o}$.The black solid lines correspond to $\\theta_{d}=39 \\mu as$. }\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.25\\textwidth}\n\\centering\n \\includegraphics[scale=.65]{theta_a_b_with_l_.1_theta_17a.eps}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.28\\textwidth}\n\\centering\n\\raisebox{.2\\height}{\\includegraphics[scale=.55]{theta_a_b_with_l_.1_theta_17b.eps}}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n\\centering\n \\includegraphics[scale=.65]{theta_a_l_with_b_.1_theta_17c.eps}\\hspace{1.5em}%\n\\end{subfigure}%\n\\begin{subfigure}{.28\\textwidth}\n\\centering\n\\raisebox{.19\\height}{\\includegraphics[scale=.6]{theta_a_l_with_b_.1_theta_17d.eps}}\n\\end{subfigure}\n\\caption{The left one is for $\\ell=0.1$ and the right one is for\n$b=0.1$ where the inclination angle is $17^{o}$.The black solid lines correspond to $\\theta_{d}=39 \\mu as$. }\n\\end{figure}\n\\section{Summary and discussions}\nIn this article, we have studied the superradiance phenomena of\nthe scalar field scattered off Kerr-sen-like black holes along\nwith the study of some salient features of the Kerr-sen-like\nLorentz violating spacetime. The LV parameter enters into the\nKerr-Sen-like background via a spontaneous symmetry breaking when\nthe pseudovector field of the bumblebee field receives a vacuum\nexpectation value. The Kerr-Sen-like spacetime is a solution to\nEinstein's bumblebee gravity model. Along with the parameter $M$,\n$a$ (which are contained in the Kerr spacetime metric) the\nKerr-Sen-like metric has two more parameters, $b$ and the LV\nparameter $\\ell$.The presence of these four parameters has a\ncrucial role in the spacetime geometry associated with the matric.\nDepending on the value of the parameter we can have non-extremal\nand extremal cases. We can even have a naked singularity. We\nobserved that the parameter space formed by $a$ and $b$ that\ncorresponds to black hole singularity is getting reduced with an\nincrease in the value of the LV parameter. We have also observed\nthat the size of the ergosphere gets enhanced with the increase in\nthe value LV parameter. The increase in the value of $b$ also\nresults in an increase in the size of the ergosphere like the\nincrease in the value of $\\ell$.\n\nIf we look towards the superradiance phenomena of the scalar field\nscattered off Kerr-sen-like black holes we find that the LV effect\nhas a great influence on the superradiance phenomena. How the\nsuperradiance phenomena and the instability associated with it get\ninfluenced by the Lorentz violation effect is studied in detail.\nWe consider the Klein-Gordon equation in the Kerr-Sen-like\nbackground and employ asymptotic matching of the scalar wave, and\nestablish transparently from the equation (\\ref{AMP}) that in the\nlow-frequency limit, i.e., for $\\omega < m \\hat{\\Omega}_{e h}$,\nthe scalar waves show the superradiant mode, i.e. it becomes\namplified in a superradiant manner. The numerical computation,\nhowever, shows that for $m \\leq 0$ the massive scalar field has a\nnon-superradiant mode. For $m \\ge 0$ it is superradiant. The role\nof the LV parameter in the superradiant phenomena as we have\nobserved is as follows. The superradiant process enhances with the\ndecrease in the value of the LV parameter and reverse is the case\nwhen the LV parameter increases irrespective of the sign of the\nvalue of this parameter please vide Fig.6.\n\nOur observation also transpires that the superradiant process gets\ninfluenced by the parameter $b$ also. From Fig. 8, it is clear\nthat the superradiance enhances with the decrease of the positive\nvalue of the parameter $b$. Note that $b=\\frac{Q^2}{M}$. So it\ncannot be negative.\n\nExtending the issue of the black hole bomb, the analytical study\nof superradiant instability is made. Fig. 9 related to the study\nof superradiant instability reveals that the LV parameter\nremarkably affects the instability regime. In the background with\nnegative LV Parameter, the scalar field has more chances to\nacquire unstable dynamics and for the positive values of the LV\nparameter, these chances are less. Therefore, the LV has a\nsignificant influence on the superradiance scattering phenomena\nand the corresponding instability linked with it.\n\nIn his article we have also tried to put constraints on parameters\ncontained in the Kerr-Sen-like spacetime metric from the\nobservation of $M87^*$, and observed that the circularity\ndeviation $\\delta C \\le 0.1$  \\cite{EHT1, EHT2, EHT3, EHT4, EHT5,\nEHT6} is satisfied exhaustively for the entire $(a-l)$ and $(b-l)$\nparameter space at $\\theta = 17^0$ inclination angle but at\ninclination angle $\\theta = 90^0$ it is satisfied for a finite\n$(a-l)$ and $(b-l)$ space. The angular diameter satisfies\n$\\theta_d = 42 \\pm 3 \\mu as$ within the $1 \\sigma$ region\n\\cite{EHT1, EHT2, EHT3, EHT4, EHT5, EHT6}over a finite $(a-b)$ and\n$(a-l)$ space.} However, when we compare with angular diameter\n$\\theta_d= 42 \\pm 3 \\mu as$ data it is found that agreeable\n$(a-b)$ parameter space is smaller than the $(a-l)$ parameter\nspace and it is more restricted.\n\nBesides we should make some comments in connection with the recent article \\cite{MALUF}.\n The comment of the article although does not target our study of superradiance directly,\n the $b=0$ limit of our metric has to pass through this unfavorable situation. In fact,\n  the comment made in \\cite{MALUF} is all about the inconsistency of the black\n  hole solution obtained in \\cite{DING}\nwhich is a special case: b=0 of the metric used here. Even if that\ninconsistency is taken as granted in the metric developed in\n\\cite{DING},  it has been pointed out in the article \\cite{KANGI},\nthat in the slow rotating limit $a\\rightarrow 0$ the metric has a\nconsistent outcome. With this limit,\n the metric turns into a true slowly rotating black hole solution of\n Einstein-bumblebee gravity \\cite{DINGC}. Moreover, for $a=b=0$,\n one lands onto the flawless Schwarzschild-like solution of the\n Einstein-bumblebee gravity presented in \\cite{CASANA}.\n\n\n\\vspace{.5cm}\n Appendix-A\n\nFollowing the procedure as followed in \\cite{BEZERRA} the general\nsolution of equation (\\ref{RE}) is given by\n\\begin{equation}\n\\begin{aligned}\n&R(z)=\\frac{M}{\\sqrt{\\Delta(1+1)}} e^{\\frac{1}{2} \\alpha z}(z-1)^{\\frac{1}{2}(1+\\gamma)} z^{\\frac{1}{2}(1+\\beta)} \\times\\\\\n&\\left\\{C_{1}\\text {HeunC }(\\alpha, \\beta, \\gamma, \\delta, \\eta, z)+C_{2} z^{-\\beta} \\text { HeunC }(\\alpha,- \\beta, \\gamma, \\delta, \\eta, z)\\right\\}\\\\\n\\end{aligned}\n\\end{equation}\n\\text{with}\\\\\n\\begin{equation}\\nonumber\n\\begin{aligned}\n&2Md=r_{+}-r_{-},z=-\\frac{1}{2Md}(r-r_{+})\\\\\n&\\alpha=4 d M\\left(\\mu^{2}-\\omega^{2}\\right)^{1\/2} \\sqrt{1+l}\\\\\n&\\beta=\\sqrt{1-\\frac{4 A}{M^{2}}},\\\\\n&\\gamma=\\sqrt{1-\\frac{4 C}{M^{2}}}, \\delta=-\\frac{2 d}{M^{2}}(B+D)\\\\\n&\\eta=\\frac{1}{2}+\\frac{2Bd}{M^{2}}\\\\\n\\end{aligned}\n\\end{equation} where\n\\begin{equation}\\nonumber\n\\begin{aligned}\nA &=\\frac{1}{4 d^{2}}\\left[4 M^{4} \\omega^{2}(1+l)(d+1)^{2}+\\left(M b-\\frac{b^{2}}{4}\\right) \\omega^{2}(1+l)+\\hat{a}^{2} m^{2}(1+l)\\right.\\\\\n&+4 \\hat{a} \\omega mMb(1+l)+M^{2} d^{2}+\\left(2 M b \\omega^{2}\\left(Mb-\\frac{b^{2}}{4}\\right)(1+l)\\right.\\\\\n&\\left.-4 M^{2}\\left(M b-\\frac{b^{2}}{4}\\right) \\omega^{2}(1+l)-4 \\hat{a} \\omega m M^{2}(1+l)\\right)(d+1) \\\\\n&\\left.+\\left(M^{2} b^{2}-4 M^{3} b\\right) \\omega^{2}(d+1)^{2}(1+l)\\right] \\\\\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\nonumber\n\\begin{aligned}\nB &=\\frac{1}{4 d^{3}}\\left[4 M^{4} \\omega^{2}(d+1)^{2}(2 d-1)(1+l)+\\left(M^{2} b^{2}-4 M^{3} b\\right) \\omega^{2}\\left(d^{2}-1\\right)(1+l)\\right.\\\\\n&-\\left(M b-\\frac{b^{2}}{4}\\right) \\omega^{2}(1+l)-\\hat{a}^{2} m^{2}(1+l)-4 \\hat{a} \\omega mM b(1+l)-M^{2} d^{2} \\\\\n&-\\left(2 M b \\omega^{2}\\left(M b-\\frac{b^{2}}{4}\\right)(1+l)-4 M^{2}\\left(M b-\\frac{b^{2}}{4}\\right) \\omega^{2}(1+l)\\right.\\\\\n&-\\left.4 \\hat{a} \\omega m M^{2}(1+l)\\right)+2 d^{2}(d+1)\\left(-2 M^{3} b \\omega^{2}(1+l)\\right.\\\\\n&\\left.+\\mu^{2} M^{3} b(1+l)\\right)+2 d^{2}\\left(-2 M^{2} \\omega^{2}(1+l)\\left(M b-\\frac{b^{2}}{4}\\right)\\right.\\\\\n&-\\left.\\left.\\mu^{2} M^{4}(d+1)^{2}(1+l)-\\left(\\hat{a}^{2}\n\\omega^{2}+j(1+j)\\right) M^{2}(1+l)\\right)\\right]\\\\\\nonumber\nC&=\\frac{1}{4 d^{2}}\\left[4 M^{4} \\omega^{2}(d-1)^{2}(1+l)+\\left(M^{2} b^{2}-4 M^{3} b\\right) \\omega^{2}(d-1)^{2}(1+l)\\right. \\\\\n&+\\left(M b-\\frac{b^{2}}{4}\\right) \\omega^{2}(1+l)+\\hat{a}^{2} m^{2}(1+l)+4 \\hat{a} \\omega mMb(1+l)+M^{2} d^{2} \\\\\n&-(d-1)\\left(2 M b \\omega^{2}\\left(M b-\\frac{b^{2}}{4}\\right)(1+l)-4 M^{2}\\left(Mb-\\frac{b^{2}}{4}\\right) \\omega^{2}(1+l)\\right. \\\\\n&\\left.\\left.-4 \\hat{a} \\omega m M^{2}(1+l)\\right)\\right] \\\\\nD&=\\frac{1}{4 d^{3}}\\left[4 M^{4} \\omega^{2}(d-1)^{2}(2 d+1)(1+l)-\\left(d^{2}-1\\right)\\left(M^{2} b^{2}-4 M^{3} b\\right) \\omega^{2}(1+l)\\right. \\\\\n&+\\left(M b-\\frac{b^{2}}{4}\\right) \\omega^{2}(1+l)+\\hat{a}^{2} m^{2}(1+l)+4 \\hat{a} \\omega mMb(1+l)+M^{2} d^{2} \\\\\n&+2 M b \\omega^{2}\\left(M b-\\frac{b^{2}}{4}\\right)(1+l)-4 M^{2}\\left(M b-\\frac{b^{2}}{4}\\right) \\omega^{2}(1+l) \\\\\n&-4 \\hat{a} \\omega m M^{2}(1+l)+2 d^{2}(d-1)\\left(-2 M^{3} b \\omega^{2}(1+l)\\right. \\\\\n&\\left.+\\mu^{2} M^{3} b(1+l)\\right)+2 d^{2}\\left(2 M^{2} \\omega^{2}(1+l)\\left(M b-\\frac{b^{2}}{4}\\right)\\right. \\\\\n&\\left.\\left.+\\mu^{2} M^{4}(d+1)^{2}(1+l)+\\left(\\hat{a}^{2}\n\\omega^{2}+j(1+j)\\right) M^{2}(1+l)\\right)-2 \\mu^{2}\nM^{4}(1+l)\\right]\n\\end{aligned}\n\\end{equation}\nIn the limit $z \\rightarrow 0 \\Rightarrow r \\rightarrow r_{+}$ we\nhave\n$$\nR(r)=\\frac{M}{\\sqrt{\\Delta(1+1)}}\\left[C_{1}\\left\\{-\\frac{1}{2 M\nd}\\left(r-r_{+}\\right)\\right\\}^{\\frac{1}{2}(1+\\beta)}+C_{2}\\left\\{-\\frac{1}{2\nMd}\\left(r-r_{+}\\right)\\right\\}^{\\frac{1}{2}(1-\\beta)}\\right]\n\\text {,}\n$$\nand in the limit $z \\rightarrow \\infty \\Rightarrow r \\rightarrow\n\\infty$ we land onto\n$$\n\\begin{aligned}\nR(r)&=\\frac{M}{\\sqrt{\\Delta(1+l)}}\\left[C_{1} e^{-i \\sqrt{\\omega^{2}-\\mu^{2}} \\sqrt{l+l}\\left(r-r_{+}\\right)} z^{k}\\right. \\\\\n&\\left.+C_{2} e^{i \\sqrt{\\omega^{2}-\\mu^{2}} \\sqrt{1+l}\\left(r-r_{+}\\right)} z^{-k}\\right] \\\\\n\\text{where}\\\\\n&k=-\\delta \/ \\alpha\n\\end{aligned}\n$$\nThe solution of near and far region should agree with the solution\nused hare in our study of superradence. However it is an involved\nmathematical issue and it is beyond the scope of this article.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Nomenclature}\n{\\renewcommand\\arraystretch{1.0}\n\\noindent\\begin{longtable*}{@{}l @{\\quad=\\quad} l@{}}\n    $\\mathcal{B}$ & body-fixed frame \\\\\n    $C_{diff}$ & diffusion coefficient \\\\\n    $\\bar{g}$ & reaction wheel spin axis unit vector resolved in $\\mathcal{B}$ frame \\\\\n    $\\mathcal{I}$ & inertial frame \\\\\n    $J$ & inertia matrix of the cuboid bus in $\\mathcal{B}$ frame \\\\\n    $J_w$ & moment of inertia of reaction wheel about spin axis \\\\\n    $l_i$ & distance between spacecraft's center of mass and geometric center; $i \\in \\{x, y, z\\}$, m \\\\\n    $L_i$ & dimension of cuboid bus; $i \\in \\{x, y, z\\}$, m \\\\\n    $p$ & number of operational reaction wheels \\\\\n    $\\Delta t$ & sampling period \\\\\n    $u$ & control input vector \\\\\n    $\\hat{u}_S$ & unit vector in the direction from the spacecraft to the Sun \\\\\n    $U$ & control admissible set \\\\\n    $W$ & matrix containing reaction wheel spin axes as column vectors \\\\\n    $x$ & state vector \\\\\n    $X$ & prescribed set defined by state constraints \\\\\n    $\\kappa$ & time-before-exit \\\\\n    $\\lambda$ & Lagrange multiplier \\\\\n    $\\phi$ & Euler angle (roll), rad \\\\\n    $\\theta$ & Euler angle (pitch), rad \\\\\n    $\\psi$ & Euler angle (yaw), rad \\\\\n    $\\bar{\\nu}$ & reaction wheel spin rate vector, rad\/s\\\\\n    $\\bar{\\omega}$ & angular velocity vector resolved in $\\mathcal{B}$ frame, rad\/s \\\\\n    $\\vec{\\tau}_{srp}$ & solar radiation pressure torque vector, N$\\cdot$m \\\\\n    $\\Phi_s$ & solar flux, W\/m$^2$ \\\\\n    \\multicolumn{2}{@{}l}{Subscripts} \\\\\n    E & Earth \\\\\n    $k$ & time instant \\\\\n    S & Sun \\\\\n    SC & spacecraft \\\\\n    srp\t& solar radiation pressure\n\\end{longtable*}}\n\n\\section{Introduction}\\label{sec:1}\n\nConstraints are ubiquitous and crucial to the safe and efficient operation of many engineered systems. A variety of control strategies have been developed to manage constraints, such as model predictive control (MPC) \\cite{camacho2013model}, reference governors \\cite{garone2017reference}, and control barrier function \\cite{ames2016control}. In certain scenarios, all trajectories of system state under admissible control inputs will eventually drift out of a desired operating region defined by a set of constraints, which makes constraint violation in finite time inevitable. It is not uncommon to observe such behaviors in systems subject to persistent disturbances and limited actuator capability\/resources. For instance, a geostationary satellite will eventually fall out of its designated position because of orbital perturbations and limited amount of onboard fuel.\n\nFor aforementioned situations where constraint violation in finite time is inevitable, drift counteraction optimal control (DCOC) has been proposed in which one seeks to maximize a functional representing the total time or yield before the first occurrence of constraint violation \\cite{kolmanovsky2018drift}. In this paper, we focus on a specific class of DCOC problems in which one seeks to maximize the time before system state exits a prescribed set defined by constraints. This time is referred to as the \\textit{time-before-exit}. \n\nIn the optimal control literature, DCOC problems are often referred to as ``exit-time'' problems or ``optimal stopping time'' problems \\cite{barles1988exit}. For continuous-time deterministic systems, it has been shown that optimal control policy in such problems can be computed via finding the viscosity solutions of the corresponding Hamilton-Jacobi-Bellman (HJB) equation \\cite{blanc1997deterministic}. However, it is generally difficult to solve an HJB equation either analytically or numerically, especially considering that it may not admit a smooth solution. As a result, recent advances in DCOC have been in a discrete-time setting \\cite{kolmanovsky2008discrete,zidek2015deterministic,zidek2017drift,zidek2016stochastic,zidek2017auto,li2017training,zidek2017lp,zidek2017receding,zidek2018spacecraft}, which is more computationally tractable. Apart from computational benefits, a discrete-time formulation also leads to control solutions that are easily implementable with digital micro-controllers. For discrete-time systems, DCOC approaches based on dynamic programming (DP) \\cite{kolmanovsky2008discrete,zidek2015deterministic,zidek2017drift,zidek2016stochastic,zidek2017auto,li2017training} and mixed-integer programming (MIP) \\cite{zidek2017lp,zidek2017receding,zidek2018spacecraft} have been developed. However, both DP-based and MIP-based approaches are faced with computational challenges: The former can treat DCOC problems with general yield functional \\cite{kolmanovsky2008discrete} but suffer from the \\textit{curse of dimensionality} \\cite{bertsekas1995dynamic}; this makes them computationally prohibitive to treat higher-order systems. An equivalent MIP reformulation of a discrete-time open-loop DCOC problem is proposed in \\cite{zidek2017lp,zidek2017receding,zidek2018spacecraft}. However, because the number of integer variables in this MIP formulation is proportional to the planning horizon and the worst-case computational complexity of MIP grows combinatorially with the number of integer variables \\cite{richards2005mixed}, computing numerical solutions to this MIP problem can be very challenging, especially for systems\/problems that require longer planning horizons or shorter sampling periods. To alleviate the computational burden, an approach based on linear programming (LP) relaxation of the MIP formulation that replaces integer variables with continuous ones has been proposed in \\cite{zidek2017lp} for linear systems. For nonlinear systems, an MPC-based strategy based on linearized prediction models and the LP reformulation is investigated in \\cite{zidek2017receding,zidek2018spacecraft}. However, these continuous relaxation-based approaches only provide approximate solutions to the original DCOC problem and do not guarantee maximum \\textit{time-before-exit}.\n\nIn this paper, we present a novel continuous optimization approach (i.e., based on optimization with only continuous variables) to discrete-time DCOC. In this approach, inspired by a recent result in minimum-time control \\cite{verschueren2017stabilizing}, we employ exponentially weighted penalties in the cost function to encourage later constraint violation and show that this leads to guaranteed optimality of the control solution in terms of maximizing the \\textit{time-before-exit}. This approach avoids the need for integer variables in optimization and is thus computationally more tractable for larger-dimensional problems. Compared to our preliminary conference paper \\cite{tang2021continuous} published in AIAA sponsored American Control Conference, in this journal version we present a more complete treatment including re-worked and more streamlined theoretical results, the treatment of time-varying constraints, and a new case study that highlights the potential for application of these results to spacecraft attitude control.\n\nHigh-precision attitude control is crucial for successfully performing imaging missions in deep space exploration. Because gas thrusters are often unable to achieve required precision and their use expends the onboard fuel, reaction wheels (RWs) are commonly used to control and\/or maintain the spacecraft orientation in a prescribed region. However, RW failures are not uncommon, which can severely impair the spacecraft's mission effectiveness. For instance, two out of four RWs on the Kepler telescope malfunctioned \\cite{cowen2013wheels}. When the number of functioning RWs is less than three, a control moment in an arbitrary direction cannot be generated and spacecraft becomes underactuated \\cite{crouch1984spacecraft}. \nUnder the assumption of zero angular momentum, a spacecraft with two independent RWs is shown to be small-time locally controllable in \\cite{krishnan1995attitude}, and various strategies have been proposed for controlling such a spacecraft, including open-loop reorientaion methods \\cite{rui2000nonlinear} and discontinuous feedback control \\cite{horri2003attitude}. In the case of non-zero angular momentum, a procedure for constructing open-loop controls is introduced in \\cite{boyer2007further}, and a method for recovering linear controllability by exploiting solar radiation pressure is proposed in \\cite{petersen2016recovering}.\n\nThe DCOC can be used to maximize the time duration before spacecraft drifting out of a specified region desired for effectively accomplishing mission objectives (i.e., maximize the \\textit{time-before-exit}). In this paper, we apply our continuous optimization approach to the original nonlinear spacecraft attitude dynamics model and demonstrate the effectiveness of our approach in maximizing the \\textit{time-before-exit} for the cases of two and three functioning RWs in numerical examples. In \\cite{zidek2017receding} and \\cite{zidek2018spacecraft}, a DCOC approach based on LP relaxation of an MIP formulation is implemented based on a linearized spacecraft attitude dynamics model. Our work here improves upon the previous work of \\cite{zidek2017receding} and \\cite{zidek2018spacecraft} in the following two aspects: Firstly, when applied to a same model, our approach produces optimal solutions in terms of maximizing the \\textit{time-before-exit}, while the approach of \\cite{zidek2017receding} and \\cite{zidek2018spacecraft} produces only approximate solutions to the original DCOC problem that may not achieve maximum \\textit{time-before-exit}. Secondly, by applying our approach directly to the original nonlinear model, we avoid the errors due to linearization, which can negatively affect the performance of computed control solutions. Note that the LP-based approach of \\cite{zidek2017receding} and \\cite{zidek2018spacecraft} recomputes the control sequence over a receding horizon in an attempt to compensate for the mismatch between linear model and nonlinear system. Although our approach is directly applicable to more accurate nonlinear models and the principle of optimality \\cite{bertsekas1995dynamic} holds for DCOC, a receding horizon approach could also be beneficial in our case if there is a significant model mismatch.\n\nThis paper is organized as follows: In Section~\\ref{sec:2}, we introduce the general DCOC problem and present our continuous nonlinear programming (NLP)-based approach to DCOC. In Section~\\ref{sec:3}, we establish theoretical results that verify the equivalence between the original DCOC problem and our NLP reformulation, and thereby verify the optimality of control solutions obtained by our approach in terms of maximizing the \\textit{time-before-exit}. In Section~\\ref{sec:4}, we consider the application of our DCOC approach to high-precision spacecraft attitude control, including a nominal case of a fully-actuated spacecraft with three RWs and a case of an underactuated spacecraft with two functioning RWs, where we also demonstrate the computational efficiency of our NLP-based approach. The paper is concluded in Section~\\ref{sec:5}.\n\n\\section{A Continuous Optimization Approach to Drift Counteraction Optimal Control}\\label{sec:2}\n\nThe objective of DCOC is to compute an optimal control that maximizes the time duration before the state vector exits a prescribed set. Consider a dynamic system that can be represented by the following discrete-time model,\n\\begin{equation}\n    x_{k+1} = f_d(x_k, u_k),\n\\end{equation}\nwhere $x_k \\in \\mathbb{R}^{n_x}$ denotes the state vector at the time instant $k$, and $u_k \\in U \\subset \\mathbb{R}^{n_u}$ denotes the control input vector at $k$. We assume $f_d: \\mathbb{R}^{n_x} \\times \\mathbb{R}^{n_u} \\to \\mathbb{R}^{n_x}$ is a twice continuously differentiable function (i.e., $f_d \\in C^2(\\mathbb{R}^{n_x} \\times \\mathbb{R}^{n_u} \\to \\mathbb{R}^{n_x})$).\n\nTime-dependent state constraints are considered defined by the sets\n\\begin{equation}\\label{equ:X_k}\n    X(k) = \\{x_k \\in \\mathbb{R}^{n_x}: H_k(x_k) \\leq h_k\\}, \\quad k = 0, 1, 2, \\dots, \n\\end{equation}\nwhere $H_k \\in C^2(\\mathbb{R}^{n_x} \\to \\mathbb{R}^{n_{h_k}})$ and $h_k \\in \\mathbb{R}^{n_{h_k}}$. When a finite horizon of length $N \\in \\mathbb{Z}$ is considered, we define the \\textit{time-before-exit} as follows:\n\\begin{definition}\\label{def:kappa}\n    Given an initial condition $x_0 \\in X(0)$ and control inputs $\\{u_k\\}_{k=0}^{N-1}$, the time-before-exit is defined as \n    \\begin{equation}\\label{eqn:kappa_1}\n        \\kappa (x_0, \\{u_k\\}_{k=0}^{N-1}) = \\max \\left\\{k \\in \\mathbb{Z}_{[1,N]} : x_i \\in X(i), i = 0, 1, \\dots, k \\right\\}.\n    \\end{equation}\n    Given an initial condition $x_0 \\in X(0)$, the maximum time-before-exit is defined as \n    \\begin{equation}\\label{eqn:kappa_2}\n        \\kappa^*(x_0) = \\max \\left\\{\\kappa(x_0, \\{u_k\\}_{k=0}^{N-1}): u_k \\in U, k = 0, 1, \\dots, N-1 \\right\\}.\n    \\end{equation}\n\\end{definition}\n\n\\begin{remark}\n    An alternative notion, ``first exit-time,'' is introduced in \\cite{zidek2017receding}, which is defined as the first time instant where constraint violation occurs. As a result, the definition of ``first exit-time'' requires that constraint violation must happen during the considered finite horizon of length $N$ (i.e., $\\kappa^*(x_0) + 1 \\leq N$). In contrast, the notion ``time-before-exit'' considered in this paper enables us to treat a more general class of DCOC problems where constraints can be satisfied over the horizon (in this case $\\kappa^*(x_0) = N$).\n\\end{remark}\n \nAssuming the control admissible set $U \\subset \\mathbb{R}^{n_u}$ is compact and convex, the DCOC problem can be formally expressed as the following optimal control problem:\n\\begin{subequations}\\label{equ:ocp}\n    \\begin{align}\n        \\max_{u_0, u_1, \\dots, u_{N-1}} &\\, \\kappa (x_0, \\{u_k\\}_{k=0}^{N-1}) \\\\\n        \\text{subject to } &\\, x_{k+1} = f_d(x_k, u_k), \\\\\n        &\\, u_k \\in U, \\quad k = 0, 1, \\dots, N-1.\n    \\end{align}\n\\end{subequations}\nAs shown in \\cite{zidek2017lp} (see also Remark~2), the above \\eqref{equ:ocp} is a well-defined optimization problem.\n\nWe consider the following NLP problem with only continuous variables as a computationally efficient reformulation of the DCOC problem \\eqref{equ:ocp}:\n\\begin{subequations}\\label{equ:NLP}\n    \\begin{align}\n        \\min_{\\substack{u_0, u_1, \\dots, u_{N-1} \\\\ \\epsilon_0, \\epsilon_1, \\dots, \\epsilon_N}} &\\, J_{\\eqref{equ:NLP}} = \\sum_{k = 0}^{N} \\theta^{N-k} \\epsilon_k \\\\\n        \\text{subject to } \n        &\\, x_{k+1} = f_d(x_k, u_k), \\label{equ:NLP_1} \\\\\n        &\\, u_k \\in U, \\\\\n        &\\, 0 \\leq \\epsilon_k \\leq \\epsilon_{k+1}, \\quad k = 0, 1, \\dots, N-1, \\label{equ:NLP_d} \\\\\n        &\\, H_k(x_k) \\leq h_k + \\textbf{1} M \\epsilon_k, \\quad k = 0, 1, \\dots, N, \\label{equ:NLP_2}\n    \\end{align}\n\\end{subequations}\nwhere $M > 0$ is a sufficiently large positive number, and $\\theta > 1$ is a weighting parameter and chosen to be sufficiently large. We will discuss theoretical properties of the above NLP problem \\eqref{equ:NLP}, in particular, show the equivalence between \\eqref{equ:ocp} and \\eqref{equ:NLP} in the next section.\n\n\n\\begin{remark}\n    An MIP-based reformulation of the DCOC problem \\eqref{equ:ocp} has been proposed in \\cite{zidek2017lp}, which uses binary variables in place of the continuous variables $\\epsilon_k$ in \\eqref{equ:NLP} and does not use exponential weighting in the cost function. Theorem~1 of \\cite{zidek2017lp} establishes the equivalence between this MIP reformulation and the original DCOC problem \\eqref{equ:ocp}. Compared to the MIP-based approach of \\cite{zidek2017lp}, our continuous optimization approach (i.e., based on the continuous NLP problem \\eqref{equ:NLP}) is significantly more computationally efficient while maintaining the equivalence and optimality guarantee in terms of maximizing the \\textit{time-before-exit}.\n\\end{remark}\n\n\\section{Theoretical Results}\\label{sec:3}\n\nIn this section, we discuss the relationship between the DCOC problem \\eqref{equ:ocp} and our continuous NLP problem \\eqref{equ:NLP}. The connection is established through two related optimization problems with only continuous variables. Based on sensitivity analysis and exact penalty method, we first derive intermediate results in Lemmas~\\ref{lemma:dcoc-int}--\\ref{lemma:absolute}, and then use them to prove our main results in Theorems~\\ref{thm:main}--\\ref{thm:convexity}. \n\nThe first related problem is the following parameter-dependent optimization problem with the cost function $\\phi(\\epsilon_{\\kappa^*(x_0)+1}, \\dots, \\epsilon_{N}) \\triangleq \\sum_{k = \\kappa^*(x_0)+1}^{N}\\theta^{N-k} \\epsilon_k$:\n\\begin{subequations}\\label{equ:eqcons-ocp}\n\\begin{align}\n    \\min_{\\substack{u_0, u_1, \\dots, u_{N-1} \\\\ \\epsilon_0, \\epsilon_1, \\dots, \\epsilon_N } } & \\,\\phi(\\epsilon_{\\kappa^*(x_0)+1}, \\dots, \\epsilon_{N}) \\\\\n    \\text{subject to }\n    &\\, x_{k+1} = f_d(x_k, u_k), \\\\\n    &\\, u_k \\in U, \\\\\n    &\\, \\epsilon_k \\leq \\epsilon_{k+1}, \\quad k = \\kappa^*(x_0), \\dots, N-1, \\\\\n    &\\, H_k(x_k) \\leq h_k + \\textbf{1}M\\epsilon_k, \\quad k = 0, 1, \\dots, N, \\label{equ:eqcons-ocp_1} \\\\\n    &\\, \\epsilon_{k} = \\eta_k, \\quad k = 0, 1, \\dots, \\kappa^*(x_0), \\label{equ:eqcons-ocp_2}\n\\end{align}\n\\end{subequations}\nwhere $\\eta_k$, $k = 0, 1, \\dots, \\kappa^*(x_0)$, are parameters, and their nominal values are $\\eta_k = 0$. To condense the notations, we define the following vectors: $\\epsilon = [\\epsilon_{\\kappa^*(x_0)+1}, \\dots, \\epsilon_{N}]^\\top$, $\\eta = [\\eta_0, \\eta_1, \\dots, \\eta_{\\kappa^*(x_0)}]^\\top$, and $\\eta^{(k)} = \\eta_k e_k$, where $e_k$, $k = 0, 1, \\dots, \\kappa^*(x_0)$, are the standard basis vectors of $\\mathbb{R}^{\\kappa^*(x_0)+1}$. Note that when all parameters take their nominal values (i.e., $\\eta_k = 0$ for all $k = 0, 1, \\dots, \\kappa^*(x_0)$), any feasible solution $z = (\\{u_k\\}_{k=0}^{N-1}, \\{\\epsilon_k\\}_{k=0}^{N})$ must satisfy $\\epsilon_k = 0$, for $k = 0, 1, \\dots, \\kappa^*(x_0)$, due to the constraints in \\eqref{equ:eqcons-ocp_2}. Then, the following lemma connects \\eqref{equ:eqcons-ocp} and the original DCOC problem \\eqref{equ:ocp}:\n\n\\begin{lemma}\\label{lemma:dcoc-int}\n    For $\\eta = \\textbf{0}$, let $z = (\\{u_k\\}_{k=0}^{N-1}, \\{\\epsilon_k\\}_{k=0}^{N})$ be a feasible solution to \\eqref{equ:eqcons-ocp}. Then, $\\{u_k\\}_{k=0}^{N-1}$ is a global optimizer of the DCOC problem \\eqref{equ:ocp}.\n\\end{lemma}\n\\begin{proof}\nBecause $\\eta = \\textbf{0}$ and $z$ is a feasible solution, \\eqref{equ:eqcons-ocp_1} and \\eqref{equ:eqcons-ocp_2} yield that $H_k(x_k) \\leq h_k$ (i.e., $x_k \\in X(k)$) holds for $k = 0, 1, \\dots, \\kappa^*(x_0)$. Then, according to \\eqref{eqn:kappa_1} and \\eqref{eqn:kappa_2} in Definition~\\ref{def:kappa}, we have $\\kappa (x_0, \\{u_k\\}_{k=0}^{N-1}) = \\kappa^*(x_0)$, i.e., $\\{u_k\\}_{k=0}^{N-1}$ is a global optimizer of \\eqref{equ:ocp}.\n\\end{proof}\n\nNote that \\eqref{equ:eqcons-ocp} cannot be constructed without the knowledge of the maximum \\textit{time-before-exit} $\\kappa^*(x_0)$, which is typically \\textit{a-priori} unknown. Therefore, we now introduce the second related problem, which replaces the parameter-dependent equality constraints $\\epsilon_k = \\eta_k, k = 0, 1, \\dots, \\kappa^*(x_0)$, in \\eqref{equ:eqcons-ocp_2} with penalty terms $\\theta^{N-k}|\\epsilon_k|$ in the cost function:\n\\begin{subequations}\\label{equ:exact-ocp}\n    \\begin{align}\n        \\min_{\\substack{u_0, u_1, \\dots, u_{N-1} \\\\ \\epsilon_0, \\epsilon_1, \\dots, \\epsilon_{N}}} &\\, J_{\\eqref{equ:exact-ocp}} = \\sum_{k = 0}^{\\kappa^*(x_0)}\\theta^{N-k} |\\epsilon_k| + \\phi(\\epsilon_{\\kappa^*(x_0)+1}, \\dots, \\epsilon_{N}) \\label{equ:exact-ocp_1}\\\\\n        \\text{subject to }\n        &\\, x_{k+1} = f_d(x_k, u_k), \\\\\n        &\\, u_k \\in U, \\\\\n        &\\, \\epsilon_k \\leq \\epsilon_{k+1}, \\quad k = \\kappa^*(x_0), \\dots, N-1, \\label{equ:exact-ocp_d}\\\\\n        &\\, H_k(x_k) \\leq h_k + \\textbf{1}M\\epsilon_k, \\quad k = 0, 1, \\dots, N.\n    \\end{align}\n\\end{subequations}\nWe now make the following two assumptions about \\eqref{equ:eqcons-ocp}, which will facilitate establishing the relationship between \\eqref{equ:eqcons-ocp} and \\eqref{equ:exact-ocp}:\n\n\\begin{assumption}\n    For a given $\\eta \\in \\mathbb{R}^{\\kappa^*(x_0)+1}$, we denote a minimizer of \\eqref{equ:eqcons-ocp} by $z(\\eta) = (\\{u_k(\\eta)\\}_{k=0}^{N-1}, \\{\\epsilon_k(\\eta)\\}_{k=0}^N) \\in \\mathbb{R}^{n_z}$ and its associated Lagrange multiplier vector by $\\lambda(\\eta) \\in \\mathbb{R}^{n_{\\lambda}}$. We assume that for $\\eta = \\textbf{0}$, the pair $\\left(z(\\textbf{0}), \\lambda(\\textbf{0})\\right)$ satisfies the strong second-order sufficient conditions (see Theorem~2 of \\cite{buskens2001sensitivity}).\n    \\label{assum:SSC}\n\\end{assumption}\n\nUnder Assumption~\\ref{assum:SSC} and according to Theorem~3 of \\cite{buskens2001sensitivity}, there exists a neighborhood of $\\textbf{0}$, $V \\subset \\mathbb{R}^{\\kappa^*(x_0)+1}$, and continuously differentiable $z(\\eta): V \\to \\mathbb{R}^{n_z}$ and $\\lambda(\\eta): V \\to \\mathbb{R}^{n_{\\lambda}}$ such that for all $\\eta \\in V$, the pair $\\left(z(\\eta), \\lambda(\\eta)\\right)$ satisfies the strong second-order sufficient conditions. Furthermore, the following sensitivity result holds \\cite{buskens2001sensitivity}: For $k = 0, 1, \\dots, \\kappa^*(x_0)$,\n\\begin{equation}\\label{equ:sensitivity}\n    \\lim_{\\eta_k \\to 0^+} \\frac{\\phi(\\epsilon(\\eta^{(k)})) - \\phi(\\epsilon(\\textbf{0}))}{\\eta_k - 0} = -\\lambda_k(\\textbf{0}),\n\\end{equation}\nwhere $\\lambda_k(\\textbf{0})$ is the Lagrange multiplier associated with the equality constraint $\\epsilon_k = \\eta_k = 0$.\n\n\\begin{assumption}\\label{assum:sens}\n    There exists $L > 0$ such that,\n    \\begin{equation}\\label{equ:Lsens}\n    \\bigg\\rvert \\lim_{\\eta_k \\to 0^+} \\frac{\\epsilon_i(\\eta^{(k)}) - \\epsilon_i(\\textbf{0})}{\\eta_k - 0} \\bigg\\rvert \\leq L,\n    \\end{equation}\n    for all $i = \\kappa^*(x_0)+1, \\dots, N$, all $k = 0, 1, \\dots, \\kappa^*(x_0)$, and all $\\theta > 1$.\n\\end{assumption}\n\nUnder Assumptions~\\ref{assum:SSC} and \\ref{assum:sens}, the relationship between \\eqref{equ:eqcons-ocp} and \\eqref{equ:exact-ocp} is stated in the following lemma:\n\n\n\\begin{lemma}\\label{lemma:equal}\n    Under Assumptions~\\ref{assum:SSC} and \\ref{assum:sens}, there exists $\\theta_0 > 1$ such that if $\\theta \\geq \\theta_0$, then \\eqref{equ:eqcons-ocp} with $\\eta = \\textbf{0}$ and \\eqref{equ:exact-ocp} share the same set of (local) minimizers.\n\\end{lemma}\n\\begin{proof}\nSee Lemma~1 of \\cite{tang2021continuous}.\n\\end{proof}\n\n    \n\nBased on Lemma~\\ref{lemma:equal}, we now state the relationship between \\eqref{equ:NLP} and \\eqref{equ:exact-ocp} in the following lemma:\n\n\\begin{lemma}\\label{lemma:absolute}\n    Under Assumptions~\\ref{assum:SSC} and \\ref{assum:sens}, when $\\theta > 1$ is sufficiently large, \\eqref{equ:NLP} and \\eqref{equ:exact-ocp} share the same set of global minimizers.\n\\end{lemma}\n\\begin{proof}\n    Let $z^* = (\\{u^*_k\\}_{k=0}^{N-1}, \\{\\epsilon^*_k\\}_{k=0}^N)$ be a global minimizer of \\eqref{equ:NLP}. Since the set of constraints of \\eqref{equ:exact-ocp} is a subset of the constraints of \\eqref{equ:NLP}, $z^*$ is a feasible point of \\eqref{equ:exact-ocp}. Let $z^{\\prime} = (\\{u^{\\prime}_k\\}_{k=0}^{N-1}, \\{\\epsilon^{\\prime}_k\\}_{k=0}^N)$ be a global minimizer of \\eqref{equ:exact-ocp}. By Lemma~\\ref{lemma:equal}, $z^{\\prime}$ is at least a local minimizer of \\eqref{equ:eqcons-ocp} with $\\eta = \\textbf{0}$ and hence satisfies $\\epsilon^{\\prime}_{k} = \\eta_k = 0$ for $k = 0, 1, \\dots, \\kappa^*(x_0)$. In turn, $z^{\\prime}$ is a feasible point of \\eqref{equ:NLP}. Since $z^*$ and $z^{\\prime}$ are global minimizers of \\eqref{equ:NLP} and \\eqref{equ:exact-ocp}, respectively, and are feasible points of \\eqref{equ:exact-ocp} and \\eqref{equ:NLP}, respectively, we have $J_{\\eqref{equ:NLP}}(z^*) \\le J_{\\eqref{equ:NLP}}(z^{\\prime})$ and $J_{\\eqref{equ:exact-ocp}}(z^{\\prime}) \\le J_{\\eqref{equ:exact-ocp}}(z^*)$. However, for any point $z$ that is feasible to \\eqref{equ:NLP} (and hence is also feasible to \\eqref{equ:exact-ocp}), we have\n    \\begin{align*}\n        J_{\\eqref{equ:NLP}}(z) &= \\sum_{k = 0}^{\\kappa^*(x_0)}\\theta^{N-k} \\epsilon_k + \\sum_{k = \\kappa^*(x_0)+1}^{N}\\theta^{N-k} \\epsilon_k \\\\\n        &= \\sum_{k = 0}^{\\kappa^*(x_0)}\\theta^{N-k} |\\epsilon_k| + \\sum_{k = \\kappa^*(x_0)+1}^{N}\\theta^{N-k} \\epsilon_k = J_{\\eqref{equ:exact-ocp}}(z).\n    \\end{align*}\n    Note that we have used $\\epsilon_k \\ge 0$ (due to the constraints in \\eqref{equ:NLP_d}) to derive the equality. This implies $J_{\\eqref{equ:NLP}}(z^*) \\le J_{\\eqref{equ:NLP}}(z^{\\prime}) = J_{\\eqref{equ:exact-ocp}}(z^{\\prime}) \\le J_{\\eqref{equ:exact-ocp}}(z^*) = J_{\\eqref{equ:NLP}}(z^*)$. Therefore, we have $J_{\\eqref{equ:NLP}}(z^*) = J_{\\eqref{equ:NLP}}(z^{\\prime})$ and $J_{\\eqref{equ:exact-ocp}}(z^{\\prime}) = J_{\\eqref{equ:exact-ocp}}(z^*)$, which implies that $z'$ and $z^*$ are not only feasible points but indeed also global minimizers of \\eqref{equ:NLP} and \\eqref{equ:exact-ocp}, respectively. This proves Lemma~\\ref{lemma:absolute}.\n\\end{proof}\n\nWe are now ready to establish the connection between the DCOC problem \\eqref{equ:ocp} and our NLP problem \\eqref{equ:NLP} in the following theorem:\n\n\\begin{theorem}\\label{thm:main}\n    Given $x_0 \\in X(0)$ and under Assumptions~\\ref{assum:SSC} and \\ref{assum:sens}, when $\\theta > 1$ is sufficiently large, any global minimizer of \\eqref{equ:NLP}, $(\\{u^*_k\\}_{k=0}^{N-1},\\{\\epsilon^*_k\\}_{k=0}^N)$, must satisfy $\\epsilon^*_k = 0$ for all $k = 0, 1,\\dots,\\kappa^*(x_0)$, where $\\kappa^*(x_0)$ is the maximum \\textit{time-before-exit} defined in \\eqref{eqn:kappa_2}. In turn, the control input sequence $\\{u^*_k\\}_{k=0}^{N-1}$ is a global optimizer of the DCOC problem \\eqref{equ:ocp}.\n\\end{theorem}\n\\begin{proof}\n    Firstly, by Lemma~\\ref{lemma:absolute}, any global minimizer of \\eqref{equ:NLP}, $z^* = (\\{u^*_k\\}_{k=0}^{N-1}, \\{\\epsilon^*_k\\}_{k=0}^N)$, must also be a global minimizer of \\eqref{equ:exact-ocp}. Then, by Lemma~\\ref{lemma:equal}, $z^*$ must be a (local) minimizer of \\eqref{equ:eqcons-ocp} with $\\eta = \\textbf{0}$. In particular, $z^*$ is a feasible point of \\eqref{equ:eqcons-ocp} and thus satisfies $\\epsilon^*_k = \\eta_k = 0$ for $k = 0, 1, \\dots, \\kappa^*(x_0)$. Finally, by Lemma~\\ref{lemma:dcoc-int}, $\\{u^*_k\\}_{k=0}^{N-1}$ is indeed a global optimizer of the DCOC problem \\eqref{equ:ocp}.\n\\end{proof}\n\nTheorem~\\ref{thm:main} says that a globally optimal solution to our NLP problem \\eqref{equ:NLP} provides a control input sequence $\\{u^*_k\\}_{k=0}^{N-1}$ that\nmaximizes the \\textit{time-before-exit} \\eqref{eqn:kappa_1}, i.e., solves the DCOC problem \\eqref{equ:ocp}. In practice, NLP problems are frequently solved using gradient-based algorithms (such as the interior-point method and the sequential quadratic programming method), which converge to only local minimizers. We now make the following additional assumption, which will enable us to extend Theorem~\\ref{thm:main} to local minimizers:\n\n\n\\begin{assumption}\\label{assum:convex}\n    Let $Z \\subset \\mathbb{R}^{n_z}$ denote the feasible region of \\eqref{equ:NLP} defined by the constraints \\eqref{equ:NLP_1}-\\eqref{equ:NLP_2}, and let $z^* = (\\{u^*_k\\}_{k=0}^{N-1}, \\{\\epsilon^*_k\\}_{k=0}^N)$ be a global minimizer of \\eqref{equ:NLP}. For any $z_0 \\in Z$, there exists $r_0(z_0) > 0$ such that $z_0 + r (z^* - z_0) \\in Z$ for all $r \\in [0, r_0(z_0)]$.\n\\end{assumption}\n\nAssumption~\\ref{assum:convex} holds for many cases. For instance, if $f_d$ is linear and all components of $H_k$ are convex functions, then the feasible region $Z$ is convex, and in this case Assumption~\\ref{assum:convex} holds true. More generally, if $Z$ is star-shaped with $z^*$ as a star center, then Assumption~\\ref{assum:convex} also holds true. Under Assumption~\\ref{assum:convex}, we have the following result:\n\n\\begin{theorem}\\label{thm:convexity}\n    Under Assumption~\\ref{assum:convex}, local minimizers of \\eqref{equ:NLP} are all global minimizers.\n\\end{theorem}\n\\begin{proof}\nSee Theorem~2 of \\cite{tang2021continuous}.\n\\end{proof}\n\nTheorem~\\ref{thm:convexity} guarantees that, under Assumption~\\ref{assum:convex}, any local minimizer of our NLP problem \\eqref{equ:NLP} (e.g., obtained by a gradient-based algorithm) solves the DCOC problem \\eqref{equ:ocp}.\n\n\n\\section{Case Study: Spacecraft Attitude Control}\\label{sec:4}\n\nHigh-precision pointing is desirable in many space missions including deep space telescope imaging and optical communication. High-precision pointing is often achieved using internal torque actuators such as reaction wheels (RWs). Failure of some RWs results in inaccessible spacecraft dynamics \\cite{crouch1984spacecraft}, which may lead to failure of certain mission objectives. In this section, we present a case study of high-precision attitude control for a spacecraft with three RWs and an underactuated spacecraft with two functioning RWs. \n\n\\subsection{Spacecraft attitude dynamics model}\n\nWe consider a spacecraft consisting of a cuboid bus and $p$ reaction wheels. An inertial frame $\\mathcal{I}$ and a body-fixed frame $\\mathcal{B}$ are used to represent spacecraft orientation. Throughout this section, a physical vector is denoted as $\\vec{r}$, and its corresponding mathematical vector resolved in a frame $\\mathcal{H}$ is denoted as $\\vec{r}|_{\\mathcal{H}}$. For simplicity, we denote the mathematical vector of $\\vec{r}$ resolved in the body-fixed frame as $\\Bar{r}$, i.e., $\\Bar{r} = \\vec{r}|_{\\mathcal{B}}$. The spacecraft state and control input vectors are defined as\n\\begin{equation}\n    x = \\begin{bmatrix}\n        \\phi & \\theta & \\psi & \\omega_1 & \\omega_2 & \\omega_3 & \\nu_1 & \\dots & \\nu_p\n    \\end{bmatrix}^T, \\quad u = \\begin{bmatrix}\n        \\dot{\\nu}_1 & \\dots & \\dot{\\nu}_p\n    \\end{bmatrix}^T,\n\\end{equation}\nwhere $\\phi, \\theta, \\psi$ are the 3-2-1 Euler angles which characterize the sequence of rotations from the inertial frame $\\mathcal{I}$ to the body-fixed frame $\\mathcal{B}$, $\\Bar{\\omega} = [\\omega_1, \\omega_2, \\omega_3]^T$ is the angular velocity vector of the spacecraft resolved in the body-fixed frame, $\\Bar{\\nu} = [\\nu_1, \\dots, \\nu_p]^T$ are the spin rates of $p$ RWs, and $\\dot{\\nu}_1, \\dots, \\dot{\\nu}_p$ are the angular acceleration of $p$ RWs. \n\nAssuming all $p$ RWs are identical and off-spin-axis moments of inertia of each RW are sufficiently small, let $J_w$ be the moment of inertia of each RW about its spin axis. In the body-fixed frame $\\mathcal{B}$, the unit direction vector from the cuboid bus to the $i$th RW is denoted as $\\Bar{g}_i$, and we define $W = [\\Bar{g}_1, \\dots, \\Bar{g}_p]$. Let $J = diag(J_1, J_2, J_3)$ denote the inertia matrix of the cuboid bus in frame $\\mathcal{B}$, and define the locked inertia as $\\Bar{J} = J + J_w W W^T$. Based on Newton's second law for rotation and kinematic equations, the nonlinear dynamics model in continuous-time takes the form,\n\\begin{subequations}\\label{equ:cont-dyn}\n    \\begin{align}\n    \\begin{bmatrix}\n        \\dot{\\phi} \\\\ \\dot{\\theta} \\\\ \\dot{\\psi}\n    \\end{bmatrix} &= \\frac{1}{\\cos(\\theta)}\\begin{bmatrix}\n        \\cos(\\theta) & \\sin(\\phi)\\sin(\\theta) & \\cos(\\phi)\\sin(\\theta)\\\\\n        0 & \\cos(\\phi)\\cos(\\theta) & -\\sin(\\phi) \\cos(\\theta)\\\\\n        0 & \\sin(\\phi) & \\cos(\\phi)\n    \\end{bmatrix} \\begin{bmatrix}\n        \\omega_1 \\\\ \\omega_2 \\\\ \\omega_3\n    \\end{bmatrix}\\\\\n    \\dot{\\Bar{\\omega}} &= \\Bar{J}^{-1}(\\Bar{\\tau}_{srp} - S[\\Bar{\\omega}](\\Bar{J}\\Bar{\\omega} + J_w W \\Bar{\\nu}) - J_w W u)\\\\\n    \\dot{\\Bar{\\nu}} &= u\n\\end{align}\n\\end{subequations}\nwhere $\\Bar{\\tau}_{srp}$ refers to the torque vector generated by solar radiation pressure (SRP) resolved in the body-fixed frame, which is a nonlinear function of the Euler angles $\\phi, \\theta, \\psi$ and taken from \\cite{petersen2016recovering}. Let $\\dot{x} = f(x,u)$ be the short-hand notation for the continuous-time model \\eqref{equ:cont-dyn}. Using Euler's method with sampling period $\\Delta t$, a discrete-time model, $x_{k+1} = f_d(x_k, u_k) = x_k + f(x_k, u_k) \\Delta t$, is obtained.\n\n\\subsection{Attitude control simulation results}\n\nIn the numerical experiments, we consider two scenarios where two or three RWs are operational, i.e. $p = 2, 3$. When $p = 3$, assuming the direction vectors of RW spinning axes are linearly independent, the reaction wheels are able to generate torque in an arbitrary direction in the body-fixed frame. When $p < 3$, the spacecraft becomes underactuated. In the sequel we consider the case $p = 2$ where only two RWs are operational to demonstrate the capability of our NLP-based approach to DCOC to maintain the attitude in the desired range for maximum time duration. The parameters used in the experiments are reported in Table~\\ref{tab:parameter}.\n\\begin{table}[H]\n\\caption{Parameter values.}\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n    \\hline\n       Quantity & Value & Unit \\\\\n    \\hline\n        $J_1, J_2, J_3$  & 430, 1210, 1300 & $kg\/m^2$\\\\\n    \\hline\n        $J_w$ & 0.043 & $kg\/m^2$\\\\\n    \\hline \n        $L_x, L_y, L_z$ & 2, 2.5, 5 & $m$\\\\\n    \\hline\n        $l_x, l_y, l_z$ & 0, 0.5, 0 & $m$\\\\\n    \\hline\n        $\\Phi_s$ & 1367 & $W\/m^2$\\\\\n    \\hline\n        $C_{diff}$ & 0.2 & [-]\\\\\n    \\hline \n        $\\hat{u}_{S}$ & $[1\/\\sqrt{3}, 1\/\\sqrt{3}, 1\/\\sqrt{3}]^T$ & [-]\\\\\n    \\hline\n    \\hline\n    \\end{tabular}\n    \\label{tab:parameter}\n\\end{table}\n\n\\subsubsection{Three reaction wheels: $p = 3$}\n\nIn the numerical experiments with three operational reaction wheels, the unit vectors of the RW spinning axis directions resolved in the body-fixed frame $\\mathcal{B}$ are defined as\n\\begin{equation*}\n    W = [\\Bar{g}_1, \\Bar{g}_2, \\bar{g}_3] = \\begin{bmatrix}\n        1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\n    \\end{bmatrix}^T.\n\\end{equation*}\n\nThe following constraints for state and control input variables are considered,\n\\begin{align*}\n    -0.003 [rad] &\\leq \\phi \\leq 0.002 [rad], \\quad -0.00065 [rad] \\leq \\theta \\leq 0.00135 [rad], \\quad -0.01 [rad] \\leq \\psi \\leq 0.01 [rad]\\\\\n    20 [rad\/s]&\\leq ||\\Bar{\\nu}||_1 \\leq 80 [rad\/s], \\quad 0 [rad\/s^2]\\leq ||u||_1 \\leq 2 [rad\/s^2]\n\\end{align*}\nwhere $||\\cdot||_1$ is the 1-norm of a vector. Note that the spinning rates of three RWs are all lower bounded by $20\\ [rad\/s]$, which prevents zero speed crossings and additional RW wear. The initial condition for state variables are chosen as \n\\begin{align*}\n    [\\phi_0, \\theta_0, \\psi_0] &= [-1\\times 10^{-3}, 3.5\\times 10^{-4}, -5\\times 10^{-4}] \\quad [rad]\\\\\n    [\\Bar{\\omega}_{1, 0}, \\Bar{\\omega}_{2, 0}, \\Bar{\\omega}_{3, 0}] &= [-5, 2, 5]\\times 10^{-4} \\quad [rad\/s]\\\\\n    [\\Bar{\\nu}_{1, 0}, \\Bar{\\nu}_{2, 0}, \\bar{\\nu}_{3, 0}] &= [50, 50, 50] \\quad [rad\/s]\n\\end{align*}\n\nWe choose a sampling period of $\\Delta t = 2\\ [sec]$, and a prediction horizon of $N = 75$. Using ``ipopt'' solver and $\\theta = 1.1$, the numerical simulation results of the proposed NLP approach to DCOC \\eqref{equ:NLP} are reported in Fig.~\\ref{fig:3RW_1} below.\n\\begin{figure}[H]\n\\begin{center}\n\\begin{picture}(400, 370)\n\\put(  0,  185){\\epsfig{file=plots\/RW3_u_1.eps,height=2.5in}} \n\\put(  200,  185){\\epsfig{file=plots\/RW3_nu_1.eps,height=2.5in}} \n\\put(  0,  0){\\epsfig{file=plots\/RW3_angle_1.eps,height=2.5in}} \n\\put(  200, 0){\\epsfig{file=plots\/RW3_phase_1.eps,height=2.5in}}\n\\small\n\\put(180, 360){(a)}\n\\put(380, 360){(b)}\n\\put(180, 180){(c)}\n\\put(380, 180){(d)}\n\\normalsize\n\\end{picture}\n\\end{center}\n      \\caption{High precision pointing with three operational RWs (No constraint violation).}\n      \\label{fig:3RW_1}\n      \\vspace{-0.2in}\n\\end{figure}\n\nThe time histories of control variables, spinning accelerations of each reaction wheel are plotted in (a). The time histories of the reaction wheel spinning rates, and the Euler angles are presented in (b) and (c), respectively. A trajectory of Euler angles ($\\phi, \\theta, \\psi$) is plotted as the red line in (d), where the blue cubic region represents the desired operating ranges for Euler angles. During the prediction horizon of $150\\ [sec]$, our optimal control policy manages to keep all state variables within given ranges. Such results are expected since three RWs are sufficient to generate torque in an arbitrary direction to counteract the non-zero angular momentum of the spacecraft. \n\nIn the next example, we present a simulation with same condition as the previous one except the initial spinning rate of the second reaction wheel $\\bar{\\nu}_{2, 0}$. The initial spinning rate vector of three reaction wheels are given as,\n\\begin{equation*}\n    [\\Bar{\\nu}_{1, 0}, \\Bar{\\nu}_{2, 0}, \\bar{\\nu}_{3, 0}] = [50, 75.2, 50] \\quad [rad\/s]\n\\end{equation*}\n\nThe simulation results with modified initial condition are shown in Fig.~\\ref{fig:3RW_2} below.\n\\begin{figure}[H]\n\\begin{center}\n\\begin{picture}(400, 370)\n\\put(  0,  185){\\epsfig{file=plots\/RW3_u_2.eps,height=2.5in}} \n\\put(  200,  185){\\epsfig{file=plots\/RW3_nu_2.eps,height=2.5in}} \n\\put(  0,  0){\\epsfig{file=plots\/RW3_angle_2.eps,height=2.5in}} \n\\put(  200, 0){\\epsfig{file=plots\/RW3_phase_2.eps,height=2.5in}}\n\\small\n\\put(180, 360){(a)}\n\\put(380, 360){(b)}\n\\put(180, 180){(c)}\n\\put(380, 180){(d)}\n\\normalsize\n\\end{picture}\n\\end{center}\n      \\caption{High precision pointing with three operational RWs (Constraint violation observed).}\n      \\label{fig:3RW_2}\n      \\vspace{-0.2in}\n\\end{figure}\n\nThe modified second RW initial spinning rate $\\bar{\\nu}_{2, 0}$ is close to its operating limit, 80 [$rad\/s$]. With the modification, Euler angles cannot be kept within the specified range over the prediction horizon. Specifically, $\\phi$ and $\\theta$ first violate their constraints at $t = 48\\ [sec]$, and $\\psi$ exits its desired range shortly afterwards at $t = 52\\ [sec]$ as shown in (c). It is observed in (b) that the second reaction wheel reaches its maximum spinning rate at the beginning of the simulation, which restricts the RWs to generate sufficient torque to counteract the angular momentum. Despite the inevitable constraint violation, the optimal control slows down the Euler angles from drifting out of the prescribed sets as shown in (c), for example, $\\phi$ is maintained at its lower bound without exiting the desired region between $8\\ [sec]$ and $48\\ [sec]$. \n\n\n\\subsubsection{Two reaction wheels: $p = 2$}\n\nNow we consider an underactuated spacecraft which only has two operational reaction wheels. We present two cases with different initial conditions, control admissible sets, reaction wheel spinning axis directions, and state constraints. In the first case, the unit vectors of RW spinning axis directions in the body-fixed frame $\\mathcal{B}$ are \n\\begin{equation*}\n    W = [\\Bar{g}_1, \\Bar{g}_2] = \\begin{bmatrix}\n        1\/\\sqrt{3} & 1 \\\\ 1\/\\sqrt{3} & 0 \\\\ 1\/\\sqrt{3} & 0\n    \\end{bmatrix}^\\top.\n\\end{equation*}\n\nThe state and control input constraints are defined as,\n\\begin{align*}\n    -0.003 [rad] &\\leq \\phi \\leq 0.002 [rad], \\quad -0.0014 [rad] \\leq \\theta \\leq 0.0026 [rad], \\quad -0.02 [rad] \\leq \\psi \\leq 0.02 [rad]\\\\\n    20 [rad\/s]&\\leq ||\\Bar{\\nu}||_1 \\leq 80 [rad\/s], \\quad 0 [rad\/s^2]\\leq ||u||_1 \\leq 4 [rad\/s^2]\n\\end{align*}\n\nThe initial condition for state variables are chosen as,\n\\begin{align*}\n    [\\phi_0, \\theta_0, \\psi_0] &= [-1 \\times 10^{-3}, 6\\times 10^{-4}, -5\\times 10^{-4}] \\quad [rad]\\\\\n    [\\Bar{\\omega}_{1, 0}, \\Bar{\\omega}_{2, 0}, \\Bar{\\omega}_{3, 0}] &= [-5, 2, 3]\\times 10^{-4} \\quad [rad\/s]\\\\\n    [\\Bar{\\nu}_{1, 0}, \\Bar{\\nu}_{2, 0}] &= [50, 50] \\quad [rad\/s]\n\\end{align*}\n\nThe sampling period is $\\Delta t = 2\\ [sec]$, and the prediction horizon length is $N = 75$. Using `ipopt' solver and parameter $\\theta = 1.1$, the numerical simulation results of our NLP approach to DCOC are plotted in Fig.~\\ref{fig:2RW_1} below.\n\\begin{figure}[H]\n\\begin{center}\n\\begin{picture}(400, 360)\n\\put(  0,  180){\\epsfig{file=plots\/RW2_u_1.eps,height=2.4in}} \n\\put(  200,  180){\\epsfig{file=plots\/RW2_nu_1.eps,height=2.4in}} \n\\put(  0,  0){\\epsfig{file=plots\/RW2_angle_1.eps,height=2.4in}} \n\\put(  200, 0){\\epsfig{file=plots\/RW2_phase_1.eps,height=2.4in}}\n\\small\n\\put(180, 340){(a)}\n\\put(380, 340){(b)}\n\\put(180, 165){(c)}\n\\put(380, 165){(d)}\n\\normalsize\n\\end{picture}\n\\end{center}\n      \\caption{High precision pointing with two operational RWs (No constraint violation).}\n      \\label{fig:2RW_1}\n      \\vspace{-0.1in}\n\\end{figure}\n\nDespite the spacecraft is underactuated, in this example, DCOC still manages to maintain all state variables within their desired range over the prediction horizon. Although two reaction wheels can only generate torque on a 2-D plane in the body-fixed frame, it appears that there is some coordination between the two reaction wheels that allows them to counteract angular momentum in all three axes. Specifically, spikes are observed in both RW accelerations (as shown in (a)) around $t = 70\\ [sec]$. At the same time instant, it is observed in (c) that $\\psi$ is maintained at current value for about $15\\ [sec]$ before it keeps growing, which avoids constraint violation over the prediction horizon.\n\nIn the second case, the unit vectors of the RW spinning axis directions resolved in the body-fixed frame $\\mathcal{B}$ are\n\\begin{equation*}\n    W = [\\Bar{g}_1, \\Bar{g}_2] = \\begin{bmatrix}\n        1\/\\sqrt{3} & 0 \\\\ 1\/\\sqrt{3} & 1 \\\\ 1\/\\sqrt{3} & 0\n    \\end{bmatrix}^\\top.\n\\end{equation*}\n\nThe following constraints for state and control input variables are considered,\n\\begin{align*}\n    0.09 [rad] &\\leq \\phi \\leq 1.01 [rad], \\quad -0.02 [rad] \\leq \\theta \\leq 0.02 [rad], \\quad -0.05 [rad] \\leq \\psi \\leq 0.05 [rad]\\\\\n    20 [rad\/s]&\\leq ||\\Bar{\\nu}||_1 \\leq 100 [rad\/s], \\quad 0 [rad\/s^2]\\leq ||u||_1 \\leq 1 [rad\/s^2]\n\\end{align*}\n\nThe initial condition for state variables are chosen as \n\\begin{align*}\n    [\\phi_0, \\theta_0, \\psi_0] &= [1, 3\\times 10^{-4}, -0.01] \\quad [rad]\\\\\n    [\\Bar{\\omega}_{1, 0}, \\Bar{\\omega}_{2, 0}, \\Bar{\\omega}_{3, 0}] &= [4, 4, -50]\\times 10^{-5} \\quad [rad\/s]\\\\\n    [\\Bar{\\nu}_{1, 0}, \\Bar{\\nu}_{2, 0}] &= [80, 20] \\quad [rad\/s]\n\\end{align*}\n\nThe sampling period is $\\Delta t = 2\\ [sec]$, and the prediction horizon length is $N = 75$. Using `ipopt' solver and parameter $\\theta = 1.1$, the numerical simulation results of our NLP approach to DCOC are plotted in Fig.~\\ref{fig:2RW_2} below.\n\\begin{figure}[H]\n\\begin{center}\n\\begin{picture}(400, 370)\n\\put(  0,  185){\\epsfig{file=plots\/RW2_u_2.eps,height=2.4in}} \n\\put(  200,  185){\\epsfig{file=plots\/RW2_nu_2.eps,height=2.4in}} \n\\put(  0,  0){\\epsfig{file=plots\/RW2_angle_2.eps,height=2.4in}} \n\\put(  200, 0){\\epsfig{file=plots\/RW2_phase_2.eps,height=2.4in}}\n\\small\n\\put(180, 360){(a)}\n\\put(380, 360){(b)}\n\\put(180, 180){(c)}\n\\put(380, 180){(d)}\n\\normalsize\n\\end{picture}\n\\end{center}\n      \\caption{High precision pointing with two operational RWs (Constraint violation observed).}\n      \\label{fig:2RW_2}\n      \\vspace{-0.2in}\n\\end{figure}\n\nIn this example, the control authority is further restricted with smaller admissible RW accelerations. As a result, constraint violations occur at $t = 84\\ [sec]$ in the pitch angle $\\theta$. Shortly afterwards, at $t = 88\\ [sec]$, constraints on raw and yaw angles, $\\phi$, $\\psi$ are both violated. Although the optimal control cannot prevent Euler angles from drifting out of their desired regions over the prediction horizon, it is observed in (a) that $\\phi$ is kept constant at its lower bound between $24$ and $88\\ [sec]$, hence postpones the \\textit{time-before-exit}, which is expected from a DCOC solution.\n\n\\section{Conclusions}\\label{sec:5}\n\nThis paper presented a continuous optimization approach to DCOC and its application to spacecraft high-precision attitude control. The approach computes a control input sequence that maximizes the \\textit{time-before-exit} by solving an NLP problem with an exponentially weighted cost function and purely continuous variables. Based on results from sensitivity analysis and exact penalty method, we proved the optimality guarantee of our approach. The practical application of our approach was demonstrated through a spacecraft high-precision attitude control example. A nominal case with  three functional RWs and an underactuated case with only two functional RWs were considered. Simulation results illustrated the effectiveness of our approach as a contingency method for extending spacecraft's effective mission time in the case of RW failures.\n\n\\section*{Funding Sources}\n\nThis research was supported by the National Science Foundation award ECCS 1931738 and the Air Force Office of Scientific Research grant FA9550-20-1-038.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThis paper is concerned with the problem of supervised classification, a topic of interest in both statistics and machine learning.\nHastie {\\it et al} \\citeyear{has:tib:fri00} gives a description of various classification methods. We outline our\nproblem as follows. We have a collection of training data $\\{(x_i,y_i),i=1,\\dots,n\\}$. The values in the\ncollection $\\hbox{\\boldmath$x$} = \\{x_1,\\dots,x_n\\}$ are often called features and can be conveniently\nthought of as covariates.\nWe denote the class labels as $\\hbox{\\boldmath$y$}=\\{y_1,\\dots,y_n\\}$, where each $y_i$ takes one of the\nvalues $1,2,\\dots,G$. Given a collection of incomplete\/unlabelled\ntest data $\\{(x_i,y_i),i=n+1,\\dots,n+m\\}$, the problem amounts to predicting the class labels\nfor $\\hbox{\\boldmath$y$}^*=\\{y_{n+1},\\dots,y_{n+m}\\}$ with corresponding feature vectors\n$\\hbox{\\boldmath$x$}^*=\\{x_{n+1},\\dots,x_{n+m}\\}$.\n\nPerhaps the most common approach to classification is the well-known $k-$nearest neighbours ($k-$nn) algorithm.\nThis algorithm amounts to classifying an unlabelled $y_{n+i}$ as the most common class among the\n$k$ nearest neighbours of $x_{n+i}$ in the training set $\\{(x_i,y_i),i=1,\\dots,n\\}$.\nWhile this algorithm is easy to implement, and often gives good performance, it can be\ncriticised since it does not allow any uncertainty to be associated to the test class labels, and\nto the value to $k$. Indeed the choice of $k$ is crucial to the performance of the algorithm. The value of\n$k$ is often chosen on the basis of leave-one-out cross-validation.\n\n\n\n\n\nThere has been some interest in extending the $k-$nearest neighbours algorithm to allow for\nuncertainty in the test class labelling, most notably by \\cite{hol:ada02}, \\cite{hol:ada03} and\nmore recently \\shortcite{cuc:mar08}. Each of these probabilistic variants of the $k-$nearest neighbour\nalgorithm, is based on defining a neighbourhood of each point $x_i$, consisting of the $k$ nearest neighbours\nof $x_i$. But moreover, each of these neighbouring points has equal influence in determining the missing\nclass label for $y_i$, regardless of distance from $x_i$. In this article we present a class of models,\nthe \\textit{distance nearest neighbour} model, which shares many of the advantages of these probabilistic approaches, but in contrast to these approaches, the relative influence of neighbouring points depends\non the distance from $x_i$. Formally, the distance nearest neighbour model is a discrete-valued Markov random field, and, as is typical with such models, depends on an intractable normalising constant. To overcome this problem we use the exchange algorithm of Murray {\\textit{et al.}} \\citeyear{Murray06} and illustrate that this provides a computationally efficient algorithm with very good mixing properties.\nThis contrasts with the difficulties encountered by Cucala \\textit{et al.} \\citeyear{cuc:mar08} in their implementation of the sampling scheme of M{\\o}ller \\textit{et al} \\citeyear{mol:pet06}.\n\nThis article is organised as follows. Section 2 presents a recent overview of recent probabilistic approaches\nto supervised classification. Section 3 introduces the new distance nearest neighbour model and outlines\nhow it compares and contrasts to previous probabilistic nearest neighbour approaches. We provide a computationally\nefficient framework for carrying out inference for the distance nearest neighbour model in Section 4. The performance\nof the algorithm is illustrated in Section 5 for a variety of benchmark datasets, as well as challenging\nhigh-dimensional datasets. Finally, we present some closing remarks in Section 6.\n\n\n\n\\section{Probabilistic nearest neighbour models}\n\nHolmes and Adams \\citeyear{hol:ada03} attempted to place the $k-$nn algorithm in a probabilistic setting\ntherefore allowing for uncertainty in the test class labelling. In their approach the full-conditional\ndistribution for a training label is written as\n\\[ \\pi(y_i|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,k) \\propto \\exp\\left( \\beta\\sum_{j\\sim^k i} I(y_i=y_j)\/k \\right), \\]\nwhere the summation is over the $k$ nearest neighbours of $x_i$ and where $I(y_i=y_j)$ is an indicator\nfunction taking the value $1$ if $y_i=y_j$ and $0$ otherwise. The notation, $j\\sim^k i$ means that\n$x_j$ is one of the $k$ nearest neighbours of $x_i$.\nHowever, as pointed out in \\shortcite{cuc:mar08}, there is a difficulty with this formulation, namely that\nthere will almost never be a joint probability for $\\hbox{\\boldmath$y$}$ corresponding to this collection of\nfull-conditionals. The reason is simply because the $k-$nn neighbourhood system is usually asymmetric.\nIf $x_i$ is one of the $k$ nearest neighbours of $x_j$, then it does not necessarily follow that\n$x_j$ is one of the $k$ nearest neighbours of $x_i$.\n\nCucala \\textit{et al.} \\citeyear{cuc:mar08} corrected the issue surrounding the asymmetry of the $k$-nn\nneighbourhood system. In their probabilistic $k$-nn ($pk$-nn) model, the full-conditional for class label $y_i$ appears as\n\\begin{equation}\n \\pi(y_i|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,k) \\propto \\exp\\left( \\beta\/k\n    \\left\\{\\sum_{j\\sim^k i} I(y_i=y_j) + \\sum_{i\\sim^k j} I(y_i=y_j) \\right\\} \\right),\n\\label{eqn:cuc}\n\\end{equation}\nand this gives rise to the joint distribution\n\\[ \\pi(\\hbox{\\boldmath$y$}|\\hbox{\\boldmath$x$},\\beta,k) \\propto \\exp\\left( \\beta\/k \\sum_{i=1}^n \\sum_{j\\sim^k i} I(y_i=y_j) \\right). \\]\nTherefore under this model, following~(\\ref{eqn:cuc}), mutual neighbours are given double weight, with respect to non-mutual neighbours and for this reason the model could be seen, perhaps, as an ad-hoc solution to this problem.\n\nIt is important to also note that both Holmes and Adams \\citeyear{hol:ada02} and Cucala \\textit{et al.} \\citeyear{cuc:mar08}\nallow the value of $k$ to be a variable. Therefore the neighbourhood size can vary. Holmes and Adams \\citeyear{hol:ada02}\nargue that allowing $k$ to vary has a certain type of smoothing effect.\n\n\\section{Distance nearest neighbours}\n\\label{sec:dnn}\n\nMotivated by the work of Holmes and Adams \\citeyear{hol:ada02} and Cucala \\textit{et al.} \\citeyear{cuc:mar08} our\ninterest focuses on modelling the distribution of the training data as a Markov random field. Similar to these\napproaches, we consider a Markov random field based approach, but in contrast our approach\nexplicitly models and depends on the distances between points in the training set. Specifically,\nwe define the full-conditional distribution of the class label $y_i$ as\n\\[ \\pi(y_i|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,\\sigma) \\propto \\exp\\left( \\beta\\sum_{j=1, j\\neq i}^n\n  w_j^iI(y_j=y_i)  \\right). \\]\nPositive values of the Markov random field parameter $\\beta$ encourage aggregation of the\nclass label. When $\\beta=0$, the class labels are uncorrelated.\nIn contrast to the $pk$-nn model, here the neighbourhood set of $x_i$ is constructed to be\n\\[ \\hbox{\\boldmath$x$}\\setminus\\{x_i\\} = \\{x_1,\\dots,x_{i-1},x_{i+1},\\dots,x_n\\} \\]\nand is therefore of maximal size. We consider three possible models depending on how\nthe collection of weights $\\{w_i^j\\}$ for $j=1,\\dots,i-1,i+1,\\dots,n$ are defined.\n\n\\begin{enumerate}\n\\item $d-$nn$_1$:\n\\[ w_i^j \\propto \\exp\\left\\{ -\\frac{d(x_i,x_j)^2}{2\\sigma^2}\\right\\},\\; \\mbox{for}\\; j=1,\\dots,i-1,i+1,\\dots,n,  \\]\nwhere $d$ is a distance measure such as Euclidean.\n\\item $d-$nn$_2$:\n\\[ w_i^j \\propto \\epsilon + (1-\\epsilon) I(d(x_i,x_j)<\\sigma),\\; \\mbox{for}\\; j=1,\\dots,i-1,i+1,\\dots,n,  \\]\nagain where, $I$ is an indicator function taking value $1$, if $d(x_i,x_j)<\\sigma$ and $0$, otherwise.\nFurther, $\\epsilon\\in (0,1)$ is defined as a constant, and is set to a value close to $0$. (Throughout\nthis paper we assign the value $\\epsilon=10^{-10}$.)\nA non-zero value of $\\epsilon$ guarantees that if there are no features within a distance $\\sigma$ of $x_i$ then the class of $y_i$ is modelled using the marginal proportions of the class labels.\n\\item $d-$nn$_3$:\n\\[ w_i^j \\propto \\exp\\left\\{ -d(x_i,x_j)\\sigma \\right\\},\\; \\mbox{for}\\; j=1,\\dots,i-1,i+1,\\dots,n.  \\]\n\\end{enumerate}\n\nClearly the neighbour system for both models is symmetric, and so the Hammersley-Clifford theorem guarantees that the joint distribution of the class\nlabels is a Markov random field. This joint distribution is written as\n\\begin{equation}\n \\pi(\\hbox{\\boldmath$y$}|\\hbox{\\boldmath$x$},\\beta,\\sigma) = \\frac{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})}{z(\\beta,\\sigma)} =\n \\frac{\\exp\\left( \\beta\\sum_i \\sum_{j=1, j\\neq i}^n w_j^iI(y_j=y_i) \\right) }{z(\\beta,\\sigma)}.\n \\label{eqn:joint}\n\\end{equation}\n\nAs usual, the normalising constant of such a Markov random field is difficult to evaluate in all but\ntrivial cases. It appears as\n\\begin{equation}\n z(\\beta,\\sigma) = \\sum_{y_1}\\dots\\sum_{y_n} \\exp\\left( \\beta\\sum_i\n  \\sum_{j=1, j\\neq i}^n w_j^iI(y_j=y_i)  \\right).\n\\label{eqn:nc}\n\\end{equation}\nSome comments:\n\\begin{enumerate}\n \\item The $k-$nn algorithm and its probabilistic variants always contain neighbourhoods of size $k$, regardless\nof how far each of neighbouring points are from the center point, $x_i$. Moreover, each neighbouring point $x_j$\nhas equal influence, regardless of distance from $x_i$. It could therefore be argued that these algorithms are\nnot sensitive to outliers. By contrast the distance nearest neighbour models deal with outlying points in a more\nrobust manner, since if a point $x_j$ lies further away from other neighbours of $x_i$, then it will have a relatively\nsmaller weight, and consequently less influence in determining the likely class label of $y_i$.\n\n\\item The formulation of distance nearest neighbour models includes every training point in the neighbourhood set, but the value of $\\sigma$\ndetermines the relative influence of points in the neighbourhood set. For the $d-$nn$_1$ model, small values of $\\sigma$ imply that only those points with small distance from the centre point will be influential, while for large values of $\\sigma$, points in the neighbourhood set are more uniformly weighted. Similarly, for the $d-$nn$_2$ model, points within a $\\sigma$ radius of the center point\nare weighted equally, while those outside a $\\sigma$ radius of the center point will have relatively\nlittle weight, when $\\epsilon$ is very close to $0$.\nBy contrast, for the $d-$nn$_3$ model, large values of the parameter\n$\\sigma$ imply that points close to the centre point will be influential.\n\\item For the $d-$nn$_2$ model, if there are no features in the training set within a distance $\\sigma$\nof $x_i$, then\n\\[ \\pi(y_i=j|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,\\sigma) \\propto \\exp\\left( \\beta p_j^i \\right),\\; \\mbox{for}\\; j=1,\\dots,G,  \\]\nwhere $p_j^i$ denotes the proportion of class labels $j$ in the set $\\hbox{\\boldmath$y$}\\setminus\\{y_i\\}$.\nThe parameter $\\beta$ determines the dependence on the class proportions.  A large value of $\\beta$ typically predicts the class label to be the class with the largest proportion, whereas a small value of $\\beta$ results in a prediction which is almost uniform over all possible classes.\nConversely,\nif there any feature vectors within a radius $\\sigma$ of $x_i$, then the class labels for these features\nwill most influence the class label of $y_i$.\n\\item As $\\beta\\rightarrow\\infty$, the most frequently occurring training label in the neighbourhood\nof a test point will be chosen with increasing large probability. The $\\beta$ parameter can be thought of, in a\nsense, as a tempering parameter. In the limit as $\\beta\\rightarrow\\infty$, the modal class label in the neighbourhood\nset has probability $1$.\n\n\\end{enumerate}\n\nThere has been work on extending the $k-$nearest neighbours algorithm to weight neighbours within the neighbourhood of size $k$.  For example, \\cite{dud75} weighted neighbours using the distance in a linear manner while standardizing weights to lie in $[0,1]$.\n\nA model similar to the $d-$nn$_1$ model appeared in \\cite{zhu:gha02}, but it does not contain\nthe $\\beta$ Markov random field parameter to control the level of aggregation in the spatial\nfield. Moreover, the authors outline some MCMC approaches, but note that inference for this model is\nchallenging. The aim of this paper is to illustrate how this model may be generalised and to\nillustrate an efficient algorithm to sample from this model. We now address the latter issue.\n\n\n\\section{Implementing the distance-nearest neighbours algorithm}\n\nThroughout we consider a Bayesian treatment of this problem. The posterior distribution of test\nlabels and Markov random field parameters can be expressed as\n\\begin{displaymath}\n \\pi(\\hbox{\\boldmath$y$}^*,\\beta,\\sigma|\\hbox{\\boldmath$y$},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$x$}^*) \\propto \\pi(\\hbox{\\boldmath$y$},\\hbox{\\boldmath$y$}^*|\\beta,\\sigma,\\hbox{\\boldmath$x$},\\hbox{\\boldmath$x$}^*)\\pi(\\beta)\\pi(\\sigma),\n\\end{displaymath}\nwhere $\\pi(\\beta)$ and $\\pi(\\sigma)$ are prior distributions for $\\beta$ and $\\sigma$, respectively.\nNote, however that the first term on the right hand side above depends on the intractable normalising\nconstant (\\ref{eqn:nc}). In fact, the number of test labels is often much greater than the number of\ntraining labels, and so the resulting normalising constant for the distribution $\\pi(\\hbox{\\boldmath$y$},\\hbox{\\boldmath$y$}^*|\\beta,\\sigma,\\hbox{\\boldmath$x$},\\hbox{\\boldmath$x$}^*)$\ninvolves a summation over $G^{n+m}$ terms, where as before $n,m$ and $G$ are the number of test data points, training\ndata points and class labels, respectively. A more pragmatic alternative is to consider the posterior distribution of the unknown parameters for\nthe training class labels,\n\\begin{displaymath}\n \\pi( \\beta,\\sigma|\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$}) \\propto \\pi(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$}) \\pi(\\beta) \\pi(\\sigma),\n\\end{displaymath}\nwhere now the normalising constant depends on $G^n$ terms. Test class labels can then be predicted by averaging\nover the posterior distribution of the training data,\n\\begin{displaymath}\n \\pi( y_{n+i}|x_{n+i},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$}) = \\int \\pi(y_{n+i}|x_{n+i},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$},\\beta,\\sigma) \\pi( \\beta,\\sigma|\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$}) d\\beta d\\sigma.\n\\end{displaymath}\nObviously, this assumes that the test class labels, $\\hbox{\\boldmath$y$}^*$ are mutually independent, given the training data, which\nwill typically be an unreasonable assumption.\nThe training class labels are modelled as being mutually independent. Clearly, this is not ideal\nfrom the Bayesian perspective.\nNevertheless, it should reduce the computational complexity of the\nproblem dramatically.\n\nIn practice, we can estimate the predictive probability of $y_{n+i}$ as an ergodic average\n\\begin{displaymath}\n \\pi( y_{n+i}|x_{n+i},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$}) \\approx \\frac{1}{J}\\sum_{j=1}^J \\pi(y_{n+i}|x_{n+i},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$},\\beta^{(j)},\\sigma^{(j)}),\n\\end{displaymath}\nwhere $\\beta^{(j)},\\sigma^{(j)}$ are samples from the posterior distribution $\\pi( \\beta,\\sigma|\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$})$.\n\n\\subsection{Pseudolikelihood estimation}\n\nA standard approach to approximate the distribution of a Markov random field is to use a pseudolikelihood\napproximation, first proposed in \\cite{bes74}. This approximation consists of a product of easily normalised\nfull-conditional distributions. For our model, we can write a pseudolikelihood approximation as\n\\begin{displaymath}\n \\pi(\\hbox{\\boldmath$y$}|\\hbox{\\boldmath$x$},\\beta,\\sigma) \\approx \\prod_{i=1}^n \\pi(y_i|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,\\sigma) =\n\\prod_{i=1}^n \\frac{\\exp\\left( \\beta\\sum_{j=1, j\\neq i}^n w_j^iI(y_j=y_i) \\right) }\n{\\sum_{k=1}^G \\exp\\left( \\beta\\sum_{j=1, j\\neq i}^n w_j^iI(y_j=k) \\right) }.\n\\end{displaymath}\nThis approximation yields a fast approximation to the posterior distribution, however it does ignore dependencies\nbeyond first order.\n\n\\subsection{The exchange algorithm}\n\nThe main computational burden is sampling from the posterior distribution\n\\begin{eqnarray*}\n \\pi( \\beta,\\sigma | \\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$} ) &\\propto& \\pi(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$}) \\pi(\\beta) \\pi(\\sigma) \\\\\n &=& \\frac{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})}{z(\\beta,\\sigma)} \\pi(\\beta) \\pi(\\sigma).\n\\end{eqnarray*}\nA naive implementation of a Metropolis-Hastings algorithm proposing to move from $(\\beta,\\sigma)$ to\n$(\\beta',\\sigma')$ would require calculation of the following ratio at each sweep of the algorithm\n\\begin{equation}\n \\frac{q(\\hbox{\\boldmath$y$}|\\beta',\\sigma',\\hbox{\\boldmath$x$})\\pi(\\beta')\\pi(\\sigma')}{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta)\\pi(\\sigma)}\n \\times \\frac{z(\\beta,\\sigma)}{z(\\beta',\\sigma')}.\n\\label{eqn:MHratio}\n\\end{equation}\nThe intractability of the normalising constants, $z(\\beta,\\sigma)$ and $z(\\beta',\\sigma')$, makes this\nalgorithm unworkable. There has been work which has tackled the problem of sampling from such complicated\ndistributions, for example, \\shortcite{mol:pet06}. The algorithm presented in this paper overcomes the\nproblem of sampling from a distribution with intractable normalising constant, to a large extent. However\nthe algorithm can result in an MCMC chain with poor mixing among the parameters. The algorithm in\n\\shortcite{mol:pet06} has been extended and improved in \\cite{Murray06}.\n\nThe algorithm samples from an augmented distribution\n\\begin{equation}\n \\pi(\\beta',\\sigma',\\hbox{\\boldmath$y$}',\\sigma,\\beta|\\hbox{\\boldmath$y$},\\hbox{\\boldmath$x$}) \\propto \\pi(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta)\\pi(\\sigma)\n h(\\beta',\\sigma'|\\beta,\\sigma) \\pi(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$}),\n\\label{eqn:exchange}\n\\end{equation}\nwhere $\\pi(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})$ is the same distance nearest-neighbour distribution as the\ntraining data $\\hbox{\\boldmath$y$}$. The distribution $h(\\beta',\\sigma'|\\beta,\\sigma)$ is any arbitrary distribution\nfor the augmented variables $(\\beta',\\sigma')$ which might depend on the variables $(\\beta,\\sigma)$, for\nexample, a random walk distribution centred at $(\\beta,\\sigma)$. It is clear that the marginal distribution\nof (\\ref{eqn:exchange}) for variables $\\sigma$ and $\\beta$ is the posterior distribution of interest.\n\nThe algorithm can be written in the following concise way:\n\\begin{algorithm}\n\\step{1.} Gibbs update of $(\\beta',\\sigma',\\hbox{\\boldmath$y$}')$:\\\\\n(i) Draw $(\\beta',\\sigma') \\sim h(\\cdot,\\cdot|\\beta,\\sigma)$.\\\\\n(ii) Draw $\\hbox{\\boldmath$y$}' \\sim \\pi(\\cdot|\\beta',\\sigma',\\hbox{\\boldmath$x$})$.\n\\step{2.} Propose to move from $(\\beta,\\sigma,\\hbox{\\boldmath$y$}),(\\beta',\\sigma',\\hbox{\\boldmath$y$}')$ to $(\\beta',\\sigma',\\hbox{\\boldmath$y$}),(\\beta,\\sigma,\\hbox{\\boldmath$y$}')$.\n(Exchange move) with probability\n\\begin{displaymath}\n \\min\\left( 1, \\frac{q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta')\\pi(\\sigma')h(\\beta,\\sigma|\\beta',\\sigma')q(\\hbox{\\boldmath$y$}|\\beta',\\sigma',\\hbox{\\boldmath$x$})}\n{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta)\\pi(\\sigma)h(\\beta',\\sigma'|\\beta,\\sigma)q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})}\n\\times \\frac{z(\\beta,\\sigma)z(\\beta',\\sigma')}{z(\\beta,\\sigma)z(\\beta',\\sigma')} \\right).\n\\end{displaymath}\n\\end{algorithm}\n\nNotice in Step 2, that all intractable normalising constants cancel above and below the fraction.\nThe difficult step of the algorithm in the context of the $d-$nn model is Step 1 (ii), since this requires a draw from $\\pi(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})$.\nPerfect sampling \\cite{pro:wil96} is often possible for Markov random field models, however a pragmatic alternative\nis to sample from $\\pi(\\cdot|\\beta',\\sigma',\\hbox{\\boldmath$x$})$ by standard MCMC methods, for example, Gibbs sampling, and take\na realisation from a long run of the chain as an approximate draw from the distribution.\nNote that this is the approach that Cucala \\textit{et al.} \\citeyear{cuc:mar08}\ntake. They argue that perfect sampling is possible for the $pk-$nn algorithm\nfor the case where there are two classes, but that the time to coalescence can be prohibitively large. They note that\nperfect sampling for more than two classes is not yet available.\n\nNote that this algorithm has some similarities with Approximate Bayesian Computation (ABC)\nmethods \\cite{sis:fan:tan07} in the sense that ABC algorithms also rely on drawing exact values from\nanalytically intractable distributions. By contrast however, ABC algorithms rely on comparing summary statistics of the auxiliary\ndata to summary statistics of the observed data.\nFinally, note that the Metropolis-Hastings ratio in step $2$ above, after re-arranging some terms, and assuming that\n$h(\\beta,\\sigma|\\beta',\\sigma')$ is symmetric can be written as\n\\[\n \\frac{q(\\hbox{\\boldmath$y$}|\\beta',\\sigma',\\hbox{\\boldmath$x$})\\pi(\\beta')\\pi(\\sigma')q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})}\n{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta)\\pi(\\sigma)q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})}.\n\\]\nComparing this to~(\\ref{eqn:MHratio}), we see that the ratio of normalising constants, $z(\\beta,\\sigma)\/z(\\beta',\\sigma')$,\nis replaced by $q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})\/q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})$, which itself can be interpreted as an importance\nsampling estimate of $z(\\beta,\\sigma)\/z(\\beta',\\sigma')$, since\n\\[ {\\mathbf{E}}_{\\hbox{\\boldmath$y$}'|\\beta',\\sigma'} \\Big[\\frac{q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})}{q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})}\\Big] =\n \\int \\frac{q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})}{q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})} \\frac{q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})}{z(\\beta',\\sigma')}\\;d\\hbox{\\boldmath$y$}'\n = \\frac{z(\\beta,\\sigma)}{z(\\beta',\\sigma')}.\n\\]\n\n\n\\section{Results}\n\nThe performance of our algorithm is illustrated in a variety of settings. We begin by testing\nthe algorithm on a collection of benchmark datasets and follow this by exploring two real\ndatasets with high-dimensional feature vectors. Matlab computer code and all of the datasets\n(test and training) used in this paper can be found at \\texttt{mathsci.ucd.ie\/$\\sim$nial\/dnn\/}.\n\n\\subsection{Benchmark datasets}\n\nIn this section we present results for our model and in each case we compare results with the $k-$nn algorithm\nfor well known benchmark datasets.\nA summary description of each dataset is presented in Table~\\ref{tab:bench}.\n\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{l|ccc}\n       & $G$ & $F$ & $N$ \\\\\n\\hline\nPima   & $2$  & $8$ & $532$ \\\\\nForensic glass & $4$  & $9$ & $214$ \\\\\nIris   & $3$   & $4$  & $150$  \\\\\nCrabs  & $4$  & $5$ & $200$ \\\\\nWine   & $3$   & $13$  & $178$ \\\\\nOlive  & $3$   & $9$  & $572$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Summary of the benchmark datasets: $G,F,N$ correspond to the number of classes, the number\nof features and the overall number of observations, respectively.}\n\\label{tab:bench}\n\\end{table}\n\nIn all situations, the training dataset was approximately $25\\%$ of the size of the overall dataset,\nthereby presenting a challenging scenario for the various algorithms. Note that the sizes of each dataset\nranges from quite small in the case of the iris dataset, to reasonably large in the case of the forensic\ndataset. In all examples, the data was standardised to give transformed features with zero mean and unit variance.\nIn the Bayesian model, non-informative $N(0,50^2)$ and $U(0,100)$ priors were chosen for $\\beta$ and $\\sigma$,\nrespectively. Each $d-$nn algorithm was run for $20,000$ iterations, with the first $10,000$ serving as\nburn-in iterations. The auxiliary chain within the exchange algorithm was run for $1,000$ iterations.\nThe $k-$nn algorithm was computed for values of $k$ from $1$ to half the number of features in the training\nset. In terms of computational run time, the $d-$nn algorithms took, depending on the size of the dataset, between\n$1$ to $12$ hours to run using Matlab code on a $2$GHz desktop machine.\n\nA summary of misclassification error rates is presented in Table~\\ref{tab:bench_res} for various benchmark datasets. In almost all of the situations $d-$nn$_1$ and $d-$nn$_3$ performs at least as well as $k-$nn and often considerably better. In general, $d-$nn$_1$ and $d-$nn$_3$ performed better than $d-$nn$_2$.\nA possible explanation for this may be due to the\ncut-off nature of the weight function in the $d-$nn$_2$ model, since if a point $x_i$ has no neighbours inside a ball of\nradius $\\sigma$, then $w_i^j$ is uniform over the entire test set, and consequently there is no effect of distance. By contrast,\nboth the $d-$nn$_1$ and $d-$nn$_3$ models, have weight functions which depend on distance, and smoothly converge to a uniform\ndistribution as $\\sigma\\rightarrow \\infty$ and $\\sigma\\rightarrow 0$, respectively. \n\n\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{l|cccc}\n  & $k-$nn & $d-$nn$_1$ & $d-$nn$_{2}$ & $d-$nn$_{3}$ \\\\\n\\hline\nPima   & $30\\%$  & $29\\%$ & $32\\%$ & $30\\%$\\\\\nForensic glass & $35\\%$  & $33\\%$ & $39\\%$ & $31\\%$ \\\\\nIris   & $6\\%$   & $5\\%$  & $5\\%$ & $6\\%$ \\\\\nCrabs  & $16\\%$  & $16\\%$ & $23\\%$ & $16\\%$ \\\\\nWine   & $6\\%$   & $4\\%$  & $6\\%$ &  $4\\%$    \\\\\nOlive  & $1\\%$   & $3\\%$  &$4\\%$ & $2\\%$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Misclassification error rates for various benchmark dataset. The value of $k$ in the $k-$nn algorithm\nwas chosen as the value that minimises the leave-one-out cross-validation error rate. (In the case of a tie,\nthe smallest value of $k$ was selected.)}\n\\label{tab:bench_res}\n\\end{table}\n\n\n\\subsection{Classification with large feature sets: food authenticity}\n\nHere we consider two datasets concerned with food authentication. The first example involves\nsamples of Greek olive oil from $3$ different regions, and the second example involves\nsamples of $5$ different types of meat. In both situations each sample was analysed using\nnear infra-red spectroscopy giving rise to $1050$ reflectance values for wavelengths in the\nrange $400-2098$nm. These $1050$ reflectance values serve as the feature vector for each sample.\nThe objective in both examples is to authenticate a test sample based on a training set of\ncomplete data (both reflectance values and class labels). Details of how both datasets were collected\nappear in \\shortcite{mcel:dow:fea99}, and were analysed using a model-based clustering approach in \\shortcite{dea:mur:dow06}.\n\n\\subsubsection{Classifying meat samples}\n\nHere $231$ samples of meat were collected.\nThe aim of this study was to see if these measurements could be used to classify each meat sample according to whether it is chicken, turkey, pork, beef or lamb. The data were randomly split into $60$ training samples and $171$ test samples. The respective number of samples in each class is given in the table below.\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{ccc}\n& Training & Test \\\\\n\\hline\nChicken & 15 & 40 \\\\\nTurkey & 20 & 35 \\\\\nPork & 13 & 42 \\\\\nBeef & 11 & 21 \\\\\nLamb & 11 & 23 \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Number of samples within each class for both the training and test datasets}\n\\end{table}\n\nAs before, non-informative normal, $N(0,50^2)$ and uniform $U(0,10)$ priors were chosen for $\\beta$ and $\\sigma$,\nrespectively. In the exchange\nalgorithm, the auxiliary chain was run for $1000$ iterations, and the overall chain ran for $20,000$ of which\nthe first $10,000$ were discarded as burn-in iterations. The overall acceptance rate for the exchange algorithm was around $25\\%$ for each of the $d-$nn models.\n\nThe misclassification error rate for leave-one-out cross-validation on the training dataset is minimised for\n$k=3$ and $k=4$. See Figure~\\ref{fig:meat_crossval} (a). At both of these values, the $k-$nn algorithm yielded a misclassification error rate of $35\\%$ and $39\\%$, respectively, for the test dataset.\nSee Figure~\\ref{fig:meat_crossval} (b). By comparison, the $d-$nn$_1$, $d-$nn$_2$  and $d-$nn$_3$ models\nachieved misclassification error rates of $29\\%$, $33\\%$ and $27\\%$,\nrespectively, for the test dataset.\nThis example further illustrates the value of the $d-$nn models.\n\n\n\\begin{figure}[htp]\n\\begin{center}\n \\hspace*{-3cm}\\includegraphics[scale=0.95]{meat_knn.pdf}\n \\caption{Meat dataset: (a) Training data: misclassification rates of leave-one-out cross-validation for $k-$nn algorithm for varying values of $k$. (b) Test data: misclassification rates for $k-$nn algorithm for varying values of $k$.}\n\\label{fig:meat_crossval}\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection{Classifying Greek olive oil}\n\nThis example concerns classifying Greek oil samples, again based on infra-red spectroscopy. Here\n$65$ samples of Greek virgin olive-oil were collected.\nThe aim of this study was to see if these measurements could be used to classify each olive-oil sample to\nthe correct geographical region. Here there were $3$ possible classes (Crete (18 locations), Peloponnese\n(28 locations) and other regions (19 locations).\n\nIn our experiment the data were randomly split into a training set of $25$ observations and a test set of $40$ observations. In the training dataset the proportion of class labels was similar to that in the complete dataset.\n\nIn the Bayesian model,  non-informative $N(0,50^2)$ and $U(0,100)$ priors were chosen for\n$\\beta$ and $\\sigma$. In the exchange algorithm, the auxiliary chain was run for $1000$ iterations, and the overall chain ran for $50,000$ of which the first $20,000$ were discarded as burn-in iterations. The overall acceptance rate\nfor the exchange algorithm was around $15\\%$ for each of the Markov chains.\n\nThe $d-$nn$_1$, $d-$nn$_2$ and $d-$nn$_3$ models achieved misclassification rates\nof $20\\%$, $26\\%$ and $20\\%$, respectively.\nIn terms of comparison with the $k-$nn algorithm, leave-one-out cross-validation was minimised for\n$k=3$ for the training dataset. See Figure~\\ref{fig:olive_crossval} (a). The misclassification rates at\nthis value of $k$ was $29\\%$ for the test dataset. See Figure~\\ref{fig:olive_crossval} (b).\n\n\n\\begin{figure}[htp]\n\\begin{center}\n \\hspace*{-3cm}\\includegraphics[scale=0.95]{olive_knn.pdf}\n \\caption{Olive oil dataset: (a) Training data: misclassification rates of leave-one-out cross-validation for $k-$nn algorithm for varying values of $k$. (b) Test data: misclassification rates for $k-$nn algorithm for varying values of $k$.}\n  \\label{fig:olive_crossval}\n\\end{center}\n\\end{figure}\n\n\nIt is again encouraging that the $d-$nn algorithms yielded  improved misclassification rates by\ncomparison.\n\n\n\\section{Concluding remarks}\n\\label{sec:conc}\n\nIn terms of providing a probabilistic approach  to a Bayesian analysis of supervised learning, our work builds on that of Cucala \\textit{et al} \\citeyear{cuc:mar08} and shares many of the advantages of the approach there, providing a sound setting for Bayesian inference.  The most likely allocations for the test dataset can be evaluated and also the uncertainty that goes with them. So this makes it possible to determine regions where allocation to specific classes is uncertain. In addition, the Bayesian framework allows for an automatic approach to choosing weights for neighbours or neighbourhood sizes.\n\nThe present paper also addresses the computational difficulties related to the well-known issue of the intractable normalising constant for discrete exponential family models. While Cucala \\textit{et al} \\citeyear{cuc:mar08}\ndemonstrated that MCMC sampling is a practical  alternative to the perfect sampling scheme of M{\\o}ller \\textit{et al} \\citeyear{mol:pet06}, there remain difficulties with their implementation of the approach of \\shortcite{mol:pet06}, namely the choice of an auxiliary distribution.  To partially overcome the difficulties of a poor choice, Cucala \\textit{et al} \\citeyear{cuc:mar08} use an adaptive algorithm where the auxiliary distribution is defined by using historical values in the Monte Carlo algorithm. We use an alternative approach based on the exchange algorithm which avoids this choice or adaptation and has very good mixing properties and therefore also has\ncomputational efficiency.\n\nAn issue with the neighbourhood model of Cucala \\textit{et al} \\citeyear{cuc:mar08}, which is an Ising or Boltzmann type model, is that it is necessary to define an upper value for the association parameter $\\beta$.  This parameter value arises from the phase change of the model and which is known for a regular neighbourhood structure but has to be investigated empirically for the probabilistic neighbourhood model.  Our distance nearest neighbour models avoid this difficulty.\n\nOur approach is robust to outliers whereas the nearest neighbour approaches will always have an outlying point having neighbours and therefore classified according to assumed independent distant points which are the nearest neighbours.\n\n\\paragraph*{Acknowledgements:} Nial Friel was supported by a Science Foundation Ireland Research Frontiers\nProgram grant, 09\/RFP\/MTH2199. Tony Pettitt's research was supported by an Australian Research Council Discovery Grant.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\nAxisymmetric magnetic activity on the Sun and sun-like stars increases the frequencies of the modes of acoustic oscillation. However, it is unclear how a corotating patch of activity  affects the oscillations, since such a perturbation is unsteady in the frame of the observer. In this paper we qualitatively describe the asteroseismic signature of a large active region in the power spectrum of the dipole ($\\ell=1$) and quadrupole ($\\ell=2$)  p modes.\nFirst we calculate the frequencies and the relative amplitudes of the azimuthal modes of oscillation in a frame that corotates with the active region, using first-order perturbation theory. For the sake of simplicity, the influence of the active region is approximated by a near-surface increase in sound speed.  In the corotating frame the perturbations due to (differential) rotation and the active region completely lift the ($2\\ell+1$)-fold azimuthal  degeneracy of the frequency spectrum of modes with harmonic degree $\\ell$. \nThen we transform to an inertial frame to obtain the observed power spectrum. In the frame of the observer, the unsteady nature of the perturbation leads to the appearance of $(2\\ell+1)^2$ peaks in the power spectrum of a multiplet. These peaks blend into each other to form asymmetric line profiles. In the limit of a small active region (angular diameter less than $30^\\circ$), we approximate the power spectrum of a multiplet in terms of $2\\times(2\\ell+1)$ peaks, whose amplitudes and frequencies depend on the latitude of the active region and the inclination angle of the star's rotation axis.\nIn order to check the results and to explore the nonlinear regime, we perform numerical simulations using the 3D time-domain pseudo-spectral linear pulsation code GLASS. \nFor small sound-speed perturbations, we find a good agreement between the simulations and linear theory. Larger perturbation amplitudes will induce mode mixing and lead to additional complex changes in the predicted power spectrum.\n{  However linear perturbation theory provides useful guidance to search for the observational signature of large individual active regions in stellar oscillation power spectra.}\n\n\\tiny\n \\fontsize{8}{11}\\helveticabold { \\section{Keywords:} asteroseismology, stars: oscillations, stars: activity, stars: rotation, starspots} \n\\end{abstract}\n\n\n\n\\section{Introduction}\n\n\nSurface magnetic activity shifts the frequencies of the global modes of acoustic oscillation during solar and stellar activity cycles  \\citep[e.g.,][]{Palle1989,Garcia2010,Santos2016,Kiefer2017}. Asteroseismology can in turn inform us about the strength and the latitude distribution of a band of magnetic activity on the stellar surface  \\citep{Gizon2002,Chaplin2007a,Gizon2016}. However, the signature of a single large active region in stellar p-mode oscillation power spectra has not been discussed so far in detail.  A complication inherent to this problem comes from the fact that the perturbation associated with a rotating active region is not steady in the observer's frame. \nYet, this problem is relevant, since large starspots are detected in photometric variability \\citep[{{e.g.}},][]{Mosser2009a} and using Doppler or Zeeman-Doppler imaging  \\citep[{{e.g.}},][]{Strassmeier2009}. \n\n\n\n \n\n\nWe build on preliminary work by \\citet{gizon1995, gizon1998} who considered the effect of a single sunspot in corotation on the low-degree modes of solar oscillation. \nThe problem of unsteady perturbations has been considered in the past in different contexts. The interaction with acoustic modes with an inclined magnetic field with respect to the stellar rotation axis was first studied by \\citet{Kurtz1982} to explain the oscillation power spectra of roAp stars \\citep[see also][]{Kurtz2008}.\nIn the oblique pulsator model the effect of the magnetic field dominates over rotation, and the pulsation axis is aligned with the magnetic axis of the star. Only modes of oscillations symmetric with respect to the magnetic field axis are excited.\nThe oblique pulsator model was extended by \\citet{Dziembowski1985} to include the first-order effects of the Coriolis and Lorentz forces, and then  by \\citet{Shibahashi1993}  to account for the distortion in the eigenfunctions.\nIn parallel \\citet{Dziembowski1984} and \\citet{Gough1984}  discussed the combined influence of rotation and of an inclined magnetic field in corotation  on multiplets of solar acoustic oscillations. They explicitly mentioned that each multiplet consist of $(2\\ell+1)^2$ components in the power spectrum \\citep[this was already hinted at by][]{Dicke1982}.\n\nWe focus on stars with a level of activity higher than the Sun, which may have active regions with larger surface coverage, and therefore better chances for detection. Following the same approach of  \\citet{goode1992}, we investigate the linear changes  induced in a $n\\ell$-multiplet by an unsteady perturbation, that mimics an active region (AR) rotating with the star. In particular we study the power spectra of the dipole and quadrupole multiplets. For the active region we consider a simple two-parameter model, where near-surface sound-speed perturbations have a given amplitude and surface coverage.\n\nAs a complement to the linear analysis we also explore the nonlinear regime of the active-region perturbation by means of 3D numerical simulations, by studying the combined effect of rotation and mode mixing on the observed power spectra using the wave propagation code GLASS \\citep[see][]{Hanasoge2007,Papini2015}. \n{ We note that the non-linear regime was studied in the context of strong magnetic fields in roAp stars by, e.g., \\citet{Cunha2000},  \\citet{Bigot2002}, \\citet{Saio2004}, and  \\citet{Cunha2006}.}\n\n\\section{Methods}\n\n\\subsection{Signature of an active region in the oscillation power spectrum: linear theory}\n\\label{cha:rotation:sec:lineartheory}\n\n\\subsubsection{Linear problem in the corotating frame}\n\nThe normal modes of oscillation of a spherically symmetric non-rotating star are identified by three integer numbers: the radial order $n$, the angular degree $\\ell$, and the azimuthal order $m$, with $|m|\\le \\ell$.\nIn the absence of attenuation the degenerate mode frequencies, $\\omega_{n\\ell}^{(0)}$, are real and the displacement eigenvectors ${\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}(\\vet r) \\exp(-{{\\mathrm{i}}} \\omega_{n\\ell}^{(0)} t)$ solve\nthe linearized equation of motion\n\\begin{gather}\n\\label{eq:H0xi0}\n {\\mathcal{L}}^{(0)} [ {\\mbox{\\boldmath${\\xi}$} }_{n\\ell m}^{(0)} ]  = \\omega_{n\\ell}^{(0)2} {\\mbox{\\boldmath${\\xi}$} }_{n\\ell m}^{(0)} ,\n\\end{gather}\nwhere ${\\mathcal{L}}^{(0)}$ is a linear spatial differential operator \\citep[see, {{e.g.}}, ][]{Unno1979}. \n{ Hereafter superscripts ''(0)'' denote quantities associated with the non-rotating stellar model.}\nIn spherical polar coordinates $\\vet r=(r,\\theta,\\phi)$, the displacement eigenfunctions can be written as\n\\begin{gather}\n\\label{eq:xi_0}\n {\\mbox{\\boldmath${\\xi}$} }_{n\\ell m}^{(0)}(\\vet r)= \n      \\left [\\xi_{r,n\\ell}(r) {\\mbox{\\boldmath${e}$} }_r +\n      \\xi_{h,n\\ell}(r) \\left ( {\\mbox{\\boldmath${e}$} }_\\theta \\PD{ }{\\theta} + \n             \\frac{1}{\\sin\\theta}{\\mbox{\\boldmath${e}$} }_\\phi\\PD{ }{\\phi} \\right )\\right] Y_l^m(\\theta,\\phi) ,\n\\end{gather}\nwhere $Y_l^m(\\theta,\\phi)$ are spherical harmonics and $\\xi_{r,n\\ell}$ and $\\xi_{h,n\\ell}$ are functions of radius only that can be calculated numerically for a given stellar model \\citep[see, e.g.,][]{Aerts2010}.\n\nWe now consider two perturbations, the first perturbation due to rotation \\citep[e.g.,][]{Hansen1977} and the second arising from the presence of an active region that rotates with the star \\citep[see, e.g.,][for the effect of a sunspot on high-degree modes]{Schunker2013}.   The latter perturbation is unsteady in any inertial frame of reference. \nHere we aim to study how these two effects may affect the fine structure of the modes within a fixed multiplet $(n,\\ell)$.\nThe two perturbations taken together completely remove the $(2\\ell+1)$-fold degeneracy in $m$.\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{fig1.jpg}\n \\caption[reference frames]{Reference frames and angles of the problem. Arrows show the polar axes of the coordinate systems ${{\\mathcal{R}}}$, ${{\\mathcal{R}}}_\\beta$, and $\\widehat{{\\mathcal{R}}}_\\beta$. The frame ${{\\mathcal{R}}}$ is the inertial frame of the observer.\n The rotation axis of the star is inclined by an angle $i$ with respect to the line of sight. \n Both frames ${{\\mathcal{R}}}_\\beta$ and $\\widehat{{\\mathcal{R}}}_\\beta$ corotate with the active region (shaded area) at a constant angular velocity $\\Omega_\\beta$. The polar axis of ${{\\mathcal{R}}}_\\beta$ is aligned with the rotation axis of the star. In ${{\\mathcal{R}}}_\\beta$ the active region has a colatitude $\\beta$ and in $\\widehat{{\\mathcal{R}}}_\\beta$ it is at the pole. \n }\n \\label{fig:referenceframes}\n\\end{figure}\n\n\nProvided that there is only one active region, it is much more convenient to first tackle the problem in a reference frame that is corotating with the active region, where both perturbations are steady \\citep[e.g.,][]{Dziembowski1984,goode1992}. \n In Fig. \\ref{fig:referenceframes} we define three frames of reference, ${{\\mathcal{R}}}$, ${{\\mathcal{R}}}_\\beta$, and $\\widehat{{{\\mathcal{R}}}}_\\beta$, all three with the same origin at the center of the star.\nFrame ${{\\mathcal{R}}}$ is an inertial frame of reference, with polar axis $\\vet z$ pointing towards the observer. \nWe denote by $\\beta$ the colatitude of the active region in frame ${{\\mathcal{R}}}$. The other two frames are both corotating with the active region at the angular velocity $\\Omega_\\beta$ about the rotation axis of the star. \nThe polar axis of ${{\\mathcal{R}}}_\\beta$ is directed along the stellar rotation axis and is inclined by an angle $i$ with respect to $\\vet z$. The polar axis of the frame $\\widehat{{\\mathcal{R}}}_\\beta$ is inclined by the angle $\\beta$ with respect to the rotation axis. In ${\\mathcal{\\widehat{R}}}_\\beta$ the center of the active region is at the north pole. \nProvided that the starspot has bno proper motion, the angular velocity $\\Omega_\\beta$ is equal to the surface rotational angular velocity of the star at  colatitude $\\beta$.\nWe call $\\vet r=(r,\\theta,\\phi)$, $\\vet r_\\beta=(r,\\theta_\\beta,\\phi_\\beta)$, and $\\widehat\\vet r_\\beta=(r,\\widehat\\theta_\\beta,\\widehat\\phi_\\beta)$ the spherical-polar coordinates associated with ${{\\mathcal{R}}}$, ${{\\mathcal{R}}}_\\beta$, and ${\\mathcal{\\widehat{R}}}_\\beta$ respectively. \n\n\nWe consider the effects of rotation and the active region on the acoustic oscillations as small perturbations. In the frame ${{\\mathcal{R}}}_\\beta$ each mode is identified by the index $M$, $-\\ell \\leq M \\leq \\ell$, and we expand the displacement eigenvectors and eigenfunctions as \n\\begin{equation}\n\\label{eq:xi_pert}\n{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} (\\vet r_\\beta) = \\sum_{m=-\\ell}^{\\ell} A^{M}_{m}  {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m} (\\vet r_\\beta) + \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} (\\vet r_\\beta) +\\cdots \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:freq_pert}\n\\omega_{n\\ell M} = \\omega^{(0)}_{n\\ell} + \\delta \\omega_{n\\ell M} + \\cdots ,\n\\end{equation}\nwhere $\\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}$ is orthogonal to each unperturbed eigenvector ${\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}$ with the same $\\ell$ and $n$ \\citep[{{e.g.}}][]{1990gough}, \nand { the coefficients $A^{M}_{m}$ are (real) amplitudes}.\nWe write the wave operator as\n\\begin{equation}\n{\\mathcal{L}} = {\\mathcal{L}}^{(0)} + \\delta {\\mathcal{L}}  + \\cdots, \n\\end{equation}\nwith\n\\begin{equation}\n\\delta {\\mathcal{L}}  =   {\\mathcal{L}}_\\Omega +  {\\mathcal{L}}_{\\rm AR},\n\\end{equation}\nwhere ${\\mathcal{L}}_{\\Omega}$ accounts for  the effects of rotation and ${\\mathcal{L}}_{\\rm AR}$  for the effects of the active region.\nTo first order, the linearized equation of motion reduces to \n\\begin{equation}\n  \\sum_{m=-\\ell}^\\ell A_{m}^M \\left ({\\mathcal{L}}_\\Omega+{\\mathcal{L}}_\\RM{AR} \\right )[{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}]  + {\\mathcal{L}}^{(0)} [ \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} ] = \n 2 \\omega^{(0)}_{n\\ell}  \\delta \\omega_{n\\ell M}  \\sum_{m=-\\ell}^\\ell  A^{M}_{m}  {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}  +     \\omega_{n\\ell}^{(0)2} \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}.\n\\label{eq:HMxiM}\n\\end{equation}\nWe define the inner product between two vectors ${\\mbox{\\boldmath${\\xi}$} }(\\vet r_\\beta)$ and ${\\mbox{\\boldmath${\\eta}$} }(\\vet r_\\beta)$ on the Hilbert space of displacement vectors as\n\\begin{gather}\n \\inner{{\\mbox{\\boldmath${\\xi}$} }}{{\\mbox{\\boldmath${\\eta}$} }} =\\int_V {\\mbox{\\boldmath${\\xi}$} }^* \\cdot {\\mbox{\\boldmath${\\eta}$} }\\,\\rho\\mathrm{d} V  ,\n\\end{gather}\nwhere $^*$ denotes the complex conjugate and $V$ is the stellar volume.\nThe unpertubed eigenmodes are normalized such that\n$$\\inner{{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'}}{{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}}=\\delta_{m'm}.$$\nWe take the inner product of Eq. (\\ref{eq:HMxiM}) with ${\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'}$ to obtain\n\\begin{equation}\n  \\sum_{m=-\\ell}^\\ell A_{m}^M \\inner{{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'}}{ ({\\mathcal{L}}_\\Omega+{\\mathcal{L}}_{AR})[{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m} ] }  \n+ \\inner{ {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} }{  {\\mathcal{L}}^{(0)} [ \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} ] }  \n=2 \\omega^{(0)}_{n\\ell}  \\delta \\omega_{n\\ell M}   A_{m'}^M .   \n\\label{eq:inner1}\n\\end{equation}\nBecause ${\\mathcal{L}}^{(0)}$ is Hermitian symmetric and $\\langle  {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} ,\\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} \\rangle =0$, the second term on the left-hand side of the above equation vanishes\n\\begin{align*}\n\\inner { {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} } \n\t   {{\\mathcal{L}}^{(0)} [ \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} ] } \n        = \\inner {{\\mathcal{L}}^{(0)} [{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'}] } \n       {   \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}  } =0 .\n\\end{align*}\nIntroducing the perturbation matrix elements\n\\begin{align}\nO_{m' m} = O^{\\Omega}_{m' m} + O^{\\rm AR}_{m' m},\n\\end{align}\nwhere\n\\begin{align}\n& O^{\\Omega}_{m' m} =   \\frac{1}{2\\omega^{(0)}_{n\\ell}} \n\\inner {{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} }{ {\\mathcal{L}}_\\Omega [ {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m} ]} \\\\\n& O^{\\rm AR}_{m' m}  =  \\frac{1}{2\\omega^{(0)}_{n\\ell}} \n\\inner {{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} }{ {\\mathcal{L}}_{\\rm AR} [ {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m} ] } ,\n\\end{align}\nEquation (\\ref{eq:inner1}) becomes\n\\begin{align}\n    \\sum_{m=-\\ell}^\\ell O_{m'm}  A^{M}_{m}      = \n   \\delta \\omega_{M}   A^M_{m'}.   \n\\end{align}\nTo simplify the notation we dropped the indices $n\\ell$ on $\\delta\\omega_M$.\nIn matrix form,\n\\begin{align}\n\\label{eq:eigen_problem}\n    \\vet O \\vec{A}^{M} = \n   \\delta \\omega_{M}  \\;  \\vec{A}^{M},\n\\end{align}\nwhere\n$\n\\vec{A}^M= [A_{-\\ell}^M \\; A_{-\\ell+1}^M \\;\\dots \\; A_{\\ell}^M]^T\n$\nis the vector of amplitudes. \nTo find $\\vec{A}^M$ and $\\delta\\omega_M $  we have to solve the above eigenvalue problem, Eq. (\\ref{eq:eigen_problem}).\n\n\nThe rotation perturbation matrix $\\vet O^{\\Omega}$ is diagonal in the frame ${{\\mathcal{R}}}_\\beta$.\nThe active region perturbation matrix $\\vet O^{\\mathrm{AR}}$ is not diagonal in ${{\\mathcal{R}}}_\\beta$, \n but it can be obtained in  terms of \n the diagonal perturbation matrix ${\\vet{\\widehat{O}}}^{\\mathrm{AR}}$ expressed in the frame ${\\mathcal{\\widehat{R}}}_\\beta$, \n\\begin{gather}\n\\label{eq:OtildeAR_toOAR}\n\\vet O^\\mathrm{AR}={\\vet{R}}^{(\\ell)}{\\vet{\\widehat{O}}}^\\mathrm{AR} \n({{\\vet{R}}^{(\\ell)}})^{-1}. \n\\end{gather}\nwhere the matrix ${\\vet{R}}^{(\\ell)}$ performs \na clockwise rotation of $\\beta$ about the $y$ axis that transforms the frame ${\\mathcal{\\widehat{R}}}_\\beta$ into the frame ${{\\mathcal{R}}}_\\beta$.\nMore explicitly, the elements of the rotation matrix are given by\n\\begin{gather}\n\\label{eq:rotylmabg}\nR_{m\\,m'}^{(\\ell)}= r^{(\\ell)}_{m\\,m'}(-\\beta) = r^{(\\ell)}_{m'm}(\\beta),\n\\end{gather}\nwhere  $r^{(\\ell)}_{m\\,m'}(\\beta)$ is given by \\citet{1959messiah}.\n \n \n \n\\label{sec:corot}\n\n\\subsubsection{Frequency splittings due to rotation}\n\n\nIn the corotating frame ${{\\mathcal{R}}}_\\beta$ the rotation perturbation matrix is diagonal:\n\\begin{align}\n O^\\Omega_{m'm} = \\delta_{m'm}\\delta\\omega_m^\\Omega \n    \\label{eq:rot_splitting}\n\\end{align}\nwith \n\\begin{equation}\n\\delta\\omega_m^\\Omega =m \\int_V K_{n\\ell m} (r,\\theta) \\left[ \\Omega (r,\\theta)-\\Omega_\\beta \\right] \\mathrm{d}{V} \n- m\\Omega_\\beta C_{n\\ell} +  \\eta Q_{2 \\ell m}\\,  \\omega_{n\\ell}^{(0)},\n    \\label{eq:rot_splitting2}\n\\end{equation}\nwhere $\\Omega(r,\\theta)$ is the internal angular velocity in an inertial frame.\nBy construction, the angular velocity $\\Omega_\\beta$ of the frame ${{\\mathcal{R}}}_\\beta$ is $\\Omega_\\beta = \\Omega( R,\\beta)$, where $R$ is the radius of the star.\nThe first and second terms on the right-hand side of Eq. (\\ref{eq:rot_splitting2}) describe the effect of differential rotation, where the functions $K_{n\\ell m} (r,\\theta)$ are the rotational sensitivity kernels \\citep{Hansen1977} and $C_{n\\ell}$ are the Ledoux constants \\citep{1951ledoux} that account for the effect of the Coriolis force. \nThe last term describes the quadrupole distortion of the star due to the centrifugal forces \\citep[e.g.][]{Saio1981,Aerts2010} and is proportional to the ratio of the centrifugal to the gravitational forces at the surface $\\eta=\\Omega^2 R^3\/(G M)$, where $M$ is the mass of the star and $G$ is the universal constant of gravity.\nThe term $Q_{2\\ell m}$ accounts for the quadrupolar component of the centrifugal distorsion \n\\begin{gather}\n Q_{2 \\ell m}\\simeq \\frac{2\/3 \\int_{-1}^{1} P_2(x) \\left[ P_\\ell^{|m|}(x)\\right]^2 \\mathrm{d}{x}}\n {\\int_{-1}^{1}  \\left[ P_\\ell^{|m|}(x)\\right]^2 \\mathrm{d}{x}} \n = \\frac{2\\ell(\\ell+1) - 6 m^2 }{3(2\\ell+3) (2\\ell-1)},\n\\end{gather}\nwhere $P_\\ell^m(x)$ are the associated Legendre functions and $P_2$ is the Legendre polynomial of second order.\nThe centrifugal term is very small in the case of slow rotators like the Sun \\citep{1992dziembowski}, however it increases rapidly with rotation and it is not negligible anymore for stars rotating a few times faster than the Sun \\citep[e.g.][]{2004gizon}.\n\n\n\n\\subsubsection{Frequency splittings due to the active region}\n\nIn this section we parametrize the effects  of the active region on the oscillation frequencies in the corotating frame ${\\mathcal{\\widehat{R}}}_\\beta$, where the active region is at the north pole.\n\nModeling the complex influence of surface magnetic fields on acoustic oscillations is challenging \\citep[{{e.g.}},][]{Gizon2010,Schunker2013}.\nHere we choose to drastically simplify the physics and to focus on the geometrical aspects of the problem. \n\nAssuming that the area of the active region covers a polar cap with $0\\le \\widehat \\theta_\\beta \\le \\alpha$ (see Fig.\n\\ref{fig:referenceframes}) and that the structure of the active region is separable in $r$ and $\\widehat\\theta_\\beta$, we can parametrize the perturbation matrix as follows:\n\\begin{gather}\n\\label{eq:omat_mag_epsilon}\n \\widehat O^{\\mathrm{AR}}_{m''m'} = \\delta_{m''m'} \\delta\\omega_{m'}^{\\rm AR} = \\delta_{m''m'} \\omega_{n\\ell}^{(0)} \\,\\varepsilon_{n\\ell}\\, G_\\ell^{m''}(\\alpha) \\, ,\n\\end{gather}\nwhere\n\\begin{gather}\n\\label{eq:glm}\n  G_\\ell^{m''}(\\alpha) = \\frac{(2\\ell+1)(\\ell-|m''|)!}{2(\\ell+|m''|)!} \n      \\int_{\\cos\\alpha}^1 \\left [P_\\ell^{|m''|}(x)\\right ]^2 \\mathrm{d} x\n\\end{gather}\nis a geometrical weight factor that accounts for the surface coverage of the active region.\nFor small values of $\\alpha$, $G_\\ell^{m''}(\\alpha)$ decreases fast as $|m''|$ increases (Fig. \\ref{fig:glmalpha}). \n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{fig2.jpg}\n \\caption{ Left panel: Geometrical weight factor $G_\\ell^{m''}(\\alpha)$ defined by Eq.~(\\ref{eq:glm}) as a function of angular degree $\\ell$ for $m=0$, $\\pm1$, $\\pm2$, at fixed $\\alpha=8^\\circ$. Right panel: $G_\\ell^{m''}(\\alpha)$ as a function of  $\\alpha$ for $\\ell=2$.\n{ The red curves show the parabolic approximations for $G_\\ell^0(\\alpha)$ in the limit of small $\\alpha$ (see Sec. \\ref{sec:small_ar}, Eq. \\ref{eq:Gl0_approx}).} \n }\n \\label{fig:glmalpha}\n\\end{figure}\nSince the value of $G_\\ell^{m''}(\\alpha)$ does not depend on the sign of $m''$, the eigenvalues of $\\vet O^{\\rm AR}$ are degenerate in $|m|$ \\citep[see also, {{e.g.}},][]{Kurtz1986}. \n\nThe parameter $\\varepsilon_{n\\ell}$ is a measure of the relative magnitude of the active region perturbation and contains all the physics. \nFrom studies of local helioseismology \\citep[{{e.g.}}][]{Gizon2009, Moradi2010, Schunker2013}, it is known that the net effect of an active region is to increase the frequencies of acoustic modes, i.e. waves propagate faster in magnetic regions and thus $\\varepsilon_{n\\ell}$ is positive.\nSince the active region introduces a perturbation that is strongly localized near the surface, the value of $\\varepsilon_{n\\ell}$ increases with radial order $n$. \nA proper calculation of the value of $\\varepsilon_{n\\ell}$ goes beyond the scope of the present study.\nInstead, we parametrize the active region perturbation as an increase in sound speed near the surface.\nFollowing \\citet[][]{Papini2015}, we write\n\\begin{align}\n\\label{eq:epsilon_nldef}\n \\varepsilon_{n\\ell}= & \\int_\\mathrm{AR} \\Delta c^2(r) \\left( \\frac{1}{r} \\PD{\\left(r^2\\xi_{r,n\\ell}(r)\\right)}{r} - \\ell(\\ell+1)\\xi_{h,n\\ell} (r)\\right)^2 \\rho_0(r) \\mathrm{d}{r} \\nonumber\\\\\n &\\times \\left ( 2 \\omega_{n\\ell}^{(0)2} \\int_V ||{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}||^2\\rho_0(r)\\mathrm{d}{V}\n \\right)^{-1} ,\n\\end{align}\nwhere $\\rho_0(r)$ is the density of the unperturbed stellar background and $\\Delta c^2(r)$ is the radial change in the squared sound speed.\nHere the integral is over the radial extent of the active region.  In Section~\\ref{sec:AR_parameters} we will further specify the effective sound-speed perturbation.\n\n\nUsing Eq.~(\\ref{eq:OtildeAR_toOAR}), the perturbation matrix elements  in frame ${{\\mathcal{R}}}_\\beta$ are\n\\begin{gather}\nO^{\\mathrm{AR}}_{m'm} = \\omega_{n\\ell}^{(0)} \\varepsilon_{n\\ell} \n\\sum_{m''=-\\ell}^\\ell G_\\ell^{m''}(\\alpha) r^{(\\ell)}_{m'' m'}(\\beta) r^{(\\ell)}_{m ''\\, m}(\\beta).\n\\end{gather}\nWe note that for $\\ell\\leq 3 $ and $\\alpha \\lesssim 30^\\circ$, we have $G_\\ell^{0}(\\alpha) \\gg  G_\\ell^{m''}(\\alpha)$, where $|m''| \\neq 0$,\nand therefore the dominant eigenvalue is \n$\nO_{00}^\\mathrm{AR}~=~\\omega_{n\\ell}^{(0)}\\varepsilon_{n\\ell} G_\\ell^0(\\alpha)\n$, see Eq. (\\ref{eq:omat_mag_epsilon}).\n\n\n\n\\subsection{Numerical setup for the linear problem}\n\n\nWe now introduce the internal rotation model and the active region parameters, used to illustrate the theory.\nWe consider a star with a rotation period of $8$~days, about one third the rotation period of the Sun. \n{ This choice of rotation period ensures that the azimuthal modes in a multiplet are well separated in frequency space.}\nFor the internal rotation profile, we take\n\\begin{gather}\n  {\\Omega(r,\\theta)}\/{2\\pi} = \n  \\left\\{ \n  \\begin{array}{ll}\n  (1447 - 183 \\cos^2\\theta - 253 \\cos^4\\theta)\n  \\text{  nHz } &  r>0.7R_\\odot , \\\\\n 1447\\text{  nHz } &  r< 0.7 R_\\odot ,\n \\end{array}\n \\right.\n \\label{eq.rotprofile}\n\\end{gather}\nwhich is a scaled model of solar differential rotation as in \\citet{2004gizon}. The centrifugal term has a significant effect. It shifts the $m=0$ mode and introduces an asymmetry in the shifts for positive and negative azimuthal orders $m$, with a maximum frequency shift of more than $100~{\\mbox{$\\mathrm{nHz}$}}$ in the case of a multiplet near $3$~mHz.\nTherefore this term must be included when performing the analysis.\n\n\n\\label{sec:AR_parameters}\n\nFrom observations of $p$-mode frequency changes during the solar cycle, \\citet{Libbrecht1990} showed that the (positive) frequency shifts are almost independent of $\\ell$ and increase with frequency, thus indicating that the effects of magnetic activity on acoustic oscillations are confined to the surface.\nAssuming that the perturbation covers two pressure scale heights below the photosphere and setting $\\Delta c^2\/c^2 \\simeq 10\\%$ there, Eq. (\\ref{eq:epsilon_nldef}) gives $\\varepsilon_{n\\ell}\\simeq 0.003$  at frequencies near $3$~mHz.\nThe surface coverage of a stellar active region, as inferred from Doppler imaging, ranges from a percent up to $10\\%$ \\citep{Strassmeier2009}. Here we consider two different surface coverages of either $4\\%$ or  $7\\%$, corresponding to $\\cos\\alpha=0.92$ or $0.86$ ($\\alpha\\simeq23^\\circ$ and $30^\\circ $).\nFinally, we consider an active region at a colatitude of either $\\beta=20^\\circ$ (near the pole) or $80^\\circ$ (near the equator).\n\nIn the following section we focus on two multiplets with $(\\ell, n)=(1, 18)$ and $(2,18)$. For each of these modes we solve the eigenvalue problem (\\ref{eq:eigen_problem}) by means of Jacobi's method \\citep{numrecipes}. \nFor the calculation of the unperturbed eigenmodes $(\\omega_{n\\ell}^{(0)},{\\mbox{\\boldmath${\\xi}$} }_{n\\ell m}^{(0)})$ we use the ADIPLS software package \\citep{Christensen-Dalsgaard2008} and Solar Model S  as the reference structure model \\citep{JCD1996}.\nThe unperturbed frequencies $\\omega_{n\\ell}^{(0)}\/2\\pi$ of the dipole and quadrupole modes are $2695.40~\\mu{\\mbox{$\\mathrm{Hz}$}}$ and $2756.95~\\mu{\\mbox{$\\mathrm{Hz}$}}$ respectively.\n{ We note that the choice of reference solar model is unimportant for the present study.}\n\n\n\\subsection{Power spectrum in the observer's frame: $(2\\ell+1)^2$ peaks}\n\\label{sec:linear_PS}\n\nGiven particular values for $\\alpha$, $\\beta$, and $\\varepsilon_{n\\ell}$ the eigenvalue problem (\\ref{eq:eigen_problem}) is fully specified and can be solved.\nIn this section we use the solutions (\\ref{eq:xi_pert}) and (\\ref{eq:freq_pert}) to build a synthetic power spectrum in the observer's frame, in order to relate the results to observations.\n\nWe need to find an expression that connects the eigenmodes to the observed intensity fluctuations.\nFor the sake of simplicity, we assume that the variation $I(\\theta_\\beta,\\phi_\\beta,t)$ induced by the acoustic oscillations in the emergent photospheric intensity is proportional to the Eulerian pressure perturbation $p$ of the acoustic wavefield \\citep[{{e.g.}}][]{Toutain1993}, as measured  at the stellar surface $r=R$. \nThe pressure perturbation $p$ is related to the wavefield  displacement ${\\mbox{\\boldmath${\\xi}$} }$ through the linearized adiabatic equation\n\\begin{gather*}\np = -\\rho_0 c_0^2 \\nabla \\cdot {\\mbox{\\boldmath${\\xi}$} } - {\\mbox{\\boldmath${\\xi}$} }\\cdot \\nabla P_0,\n\\end{gather*}\nwhere $c_0$ and $P_0$ are respectively the sound speed and the pressure of the unperturbed stellar model.\nThe pressure perturbation $p_{n\\ell M}(r,\\theta_\\beta,\\phi_\\beta)$ of the mode $M$ is\n\\begin{equation}\np_{n\\ell M} = -\\rho_0 c_0^2 \\nabla \\cdot {\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} - {\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}\\cdot \\nabla P_0,\n\\end{equation}\nwhere ${\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}$ is given by Eq. (\\ref{eq:xi_pert}).\nTo  leading order, we have\n\\begin{gather}\np_{n\\ell M}({\\vet r_\\beta})= \\sum_{m=-\\ell}^\\ell A_m^M p_{n\\ell m}^{(0)}(\\vet r_\\beta)\n\\end{gather}\nwith \n\\begin{gather*}\np_m^{(0)}(R,\\theta_\\beta,\\phi_\\beta) = \n -\\rho_0 c_0^2 \\nabla \\cdot {\\mbox{\\boldmath${\\xi}$} }_m^{(0)} - {\\mbox{\\boldmath${\\xi}$} }_m^{(0)}\\cdot \\nabla P_0 ,\n\\end{gather*}\nwhere we dropped the subscripts $n\\ell$.\nAcoustic oscillations in stars are stochastically excited and damped by turbulent convection, therefore $I(\\theta_\\beta,\\phi_\\beta,t)$ is a realization of a random process. \nSince  the perturbation is steady in the frame ${{\\mathcal{R}}}_\\beta$, this random process is stationary in that frame.\nAn expression for the intensity fluctuations $I(\\theta_\\beta,\\phi_\\beta,\\omega)$ in  Fourier space with the required  statistical properties is\n\\begin{align}\nI(\\theta_\\beta,\\phi_\\beta,\\omega) \n& \\propto \\sum_{M=-\\ell}^\\ell p_M(R,\\theta_\\beta,\\phi_\\beta)\nL_M^{1\/2}(\\omega) {{\\mathcal{N}}}_M(\\omega) \\nonumber \\\\\n & \\propto \\sum_{M=-\\ell}^\\ell \\sum_{m=-\\ell}^\\ell A_m^M {Y_\\ell^m}(\\theta_\\beta,\\phi_\\beta)\nL_M^{1\/2}(\\omega)\n    {{\\mathcal{N}}}_M(\\omega) ,\n\\label{eq:Iomega_corot}\n\\end{align}\nwhere the ${{\\mathcal{N}}}_M(\\omega)$ are independent complex Gaussian random variables, with zero mean and unit variance:\n\\begin{gather}\n\\label{eq:covNMNMp}\n  E \\left [ {{{\\mathcal{N}}}_{M'}}^*(\\omega'){{\\mathcal{N}}}_M(\\omega)\\right ] = \\delta_{M'M} \\delta_{\\omega'\\omega}.\n\\end{gather}\nIn Equation (\\ref{eq:Iomega_corot}) and (\\ref{eq:covNMNMp}) we only consider the positive-frequency part of the spectrum ($\\omega$ and $\\omega'>0$) since $I(\\theta_\\beta,\\phi_\\beta,t)$ is real. The negative-frequency part is related to the positive part by $I(\\theta_\\beta,\\phi_\\beta,-\\omega)= I^*(\\theta_\\beta,\\phi_\\beta,\\omega)$.\nThe function $L_M(\\omega)$ is a Lorentzian  \n\\begin{gather}\nL_M(\\omega)=\\left [ 1+\\left (\\frac{\\omega-\\omega_M}{\\Gamma\/2}  \\right )^2 \\right ]^{-1},\n\\end{gather}\nappropriate for describing the power spectrum of an exponentially damped oscillator  with full width at half maximum (FWHM) $\\Gamma$ \\citep[see ,e.g.,][]{Anderson1990}.\n\nWe transform Eq. (\\ref{eq:Iomega_corot}) back to the time domain by inverse Fourier transformation to obtain $I(\\theta_\\beta,\\phi_\\beta,t)$.\nThe intensity $I(\\theta,\\phi,t)$ as seen by the observer in the inertial frame ${{\\mathcal{R}}}$ is obtained by applying a passive rotation of Euler angles $(0,-i,\\Omega_\\beta t)$ to\nexpress ${Y_\\ell^m}(\\theta_\\beta,\\phi_\\beta)$ in terms of $\\theta$ and $\\phi$ \\citep{1959messiah}:   \n\\begin{gather}\nY_\\ell^{m}(\\theta_\\beta,\\phi_\\beta) = \ne^{-\\mathrm{i} m\\Omega_\\beta t}\n\\sum_{m'=-\\ell}^{\\ell} Y_\\ell^{m'}(\\theta,\\phi) r_{m'm}^{(\\ell)}(-i) .\n\\end{gather}\nIn the frequency domain, the intensity fluctuations become\n\\begin{equation}\nI (\\theta,\\phi,\\omega)  = \\sum_{M=-\\ell}^\\ell \\sum_{m=-\\ell}^\\ell\\sum_{m'=-\\ell}^\\ell \nA_m^M Y_\\ell^{m'}(\\theta,\\phi) r_{m\\,m'}^{(\\ell)}(i) \\;  L_M^{1\/2}(\\omega - m\\Omega_\\beta)\n    {{\\mathcal{N}}}_M(\\omega- m\\Omega_\\beta) ,\n\\label{eq:I_thetaphiomega}\n\\end{equation}\nwhere we used the property $r_{m'm}^{(\\ell)}(-i)=r_{m\\,m'}^{(\\ell)}(i)$.\nSince $L_M(\\omega - m\\Omega_\\beta)$ peaks at frequency $\\omega=\\omega_M + m\\Omega_\\beta$, the intensity spectrum observed in the inertial frame has $(2\\ell+1)^2$ peaks, corresponding to all combinations of $m$ and $M$.\n\nTo obtain the full-disk integrated intensity fluctuations  $I_\\text{obs}(\\omega)$ we perform an integration over the visible disk of the star:\n\\begin{gather*}\n I_\\text{obs}(\\omega) = \n\\int_0^{2\\pi} \\mathrm{d}\\phi \\int_0^{\\pi\/2} \\mathrm{d}\\theta \\;\n I(\\theta,\\phi,\\omega)  W(\\theta)\\cos{\\theta}\\sin\\theta \\,,\n\\end{gather*}\nwhere $W(\\theta)$ is the limb-darkening function.\nThe components with $m'\\ne 0$ vanish upon integration over $\\phi$, thus \n\\begin{gather}\n\\label{eq:Iobs_final}\nI_\\text{obs} (\\omega) \n =  \\sum_{M=-\\ell}^\\ell \\sum_{m=-\\ell}^\\ell\nB_m^M L_M^{1\/2}(\\omega-m\\Omega_\\beta)\n    {{\\mathcal{N}}}_M(\\omega- m\\Omega_\\beta) ,\n\\end{gather}\nwith\n$B_m^M =  A_m^M \\,V_\\ell \\; r_{m0}^{(\\ell)}(i)$ and\n\\begin{gather}\n\\label{eq:V_l}\nV_\\ell=\n\\int_0^{2\\pi} \\mathrm{d}\\phi \\int_0^{\\pi\/2} \\mathrm{d}\\theta\\; \n Y_\\ell^0(\\theta,\\phi)  W(\\theta)\\cos{\\theta}\\sin\\theta .\n \\end{gather}\nThe matrix elements $r_{m0}^{(\\ell)}(i)$ are written explicitly in terms of the associated Legendre polynomials \\citep{1959messiah}:\n\\begin{gather}\n\\label{eq:r0lm}\n\n {r}_{m0}^{(\\ell)}(i) =\n (-1)^m \\sqrt{\\frac{(\\ell-m)!}{(\\ell+m)!}} P_\\ell^m(\\cos i).\n\\end{gather}\n\n\n\n\\begin{figure*}\n\\includegraphics[width=20cm]{fig3a.jpg}\n\\includegraphics[width=20cm]{fig3b.jpg}\n \\caption{ Left side of the diagram: Perturbations to the mode eigenfrequencies in the frame that is corotating with the active region,  for the multiplets with $(\\ell,n)=(1,18)$ (top) and with $(\\ell,n)=(2,18)$ (bottom). The star is Sun-like and rotates with a scaled solar differential rotation profile (rotation period is approximately $8~\\text{days}$). The active region perturbation is specified by  $\\varepsilon_{n\\ell}=0.003$, $\\alpha=23^\\circ$, and $\\beta=80^\\circ$. The rotational frequency of the active region at colatitude $\\beta$ is $\\Omega_\\beta\/2\\pi$ is $1.504~\\mu{\\mbox{$\\mathrm{Hz}$}}$.\nThe frequency of the mode $M=0$  is the most shifted. Right side of the diagram: The  $(2\\ell+1)^2$ peaks of the power spectrum as seen in the observer's frame, for an inclination angle $i=80^\\circ$. For each $m$, the $M$-components are identified with different colors: red for $M=0$, black for $M=1$, blue for $M=-1$, orange for $M=2$, and pink for $M=-2$.  \n The peaks with the same colors are statistically correlated to each other (according to  Eq.~\\ref{eq:PS_correlation}).\n }\n \\label{fig:degenerate2obs}\n\\end{figure*}\n\n\n\nA realization of the power spectrum is given by   \n\\begin{align}\nP(\\omega) &= |I_\\text{obs}(\\omega)|^2 \\nonumber\\\\\n &=  \\left |\\sum_{M=-\\ell}^\\ell \\sum_{m=-\\ell}^\\ell\n     B_m^M L_M^{1\/2}(\\omega-m\\Omega_\\beta)\n    {{\\mathcal{N}}}_M(\\omega- m\\Omega_\\beta) \\right |^2,\n\\label{eq:powerspectrum_realization}\n\\end{align}\nwhich depends on $2\\ell+1$ independent realizations of complex Gaussian random variables. The expectation value of $ P(\\omega)$ is\n\\begin{gather}\n\\label{eq:powerspectrum}\n\\mc{P}(\\omega) = E\\left [P(\\omega)\\right ]=  \\sum_{m=-\\ell}^\\ell \\sum_{M=-\\ell}^\\ell\n P_m^M L_M(\\omega - m\\Omega_\\beta).\n\\end{gather}\nAs mentioned earlier, the power spectrum displays $(2\\ell+1)^2$ Lorentzian peaks, centered at frequencies \n\\begin{gather}\n\\label{eq:observed_frequencies}\n \\omega_m^M := \\omega_M + m \\Omega_\\beta = \\omega_{n\\ell}^{(0)}+\\delta\\omega_M + m \\Omega_\\beta  ,\n\\end{gather}\n with peak power\n\\begin{gather}\n\\label{eq:observed_amplitude}\n P_m^M := ({B_m^M})^2 = \\frac{(\\ell-|m|)!}{(\\ell+|m|)!} \\left [V_\\ell A_m^M P_\\ell^{|m|}(\\cos i) \\right ]^2.\n\\end{gather}\nExample power spectra for a dipole and a quadrupole multiplet are shown in Fig. \\ref{fig:degenerate2obs}.\nThe frequencies and amplitudes of the $(2\\ell+1)^2$ peaks are obtained from Eqs.  (\\ref{eq:observed_frequencies}) and (\\ref{eq:observed_amplitude}),  using the limb-darkening function quoted by \\citet{Pierce2000} to calculate $V_\\ell$.\n\nFigure \\ref{fig:degenerate2obs} also displays the different contributions to the frequency splittings due to rotation and to the active region perturbation in the corotating frame ${{\\mathcal{R}}}_\\beta$.\nFor both multiplets, the $M=0$ peaks are shifted by the largest amount, they are the most affected by the AR perturbation and in the frame ${\\mathcal{\\widehat{R}}}_\\beta$  \\citep[see also][]{Papini2015}.\nThis feature, which arises from geometrical considerations only,  is preserved in the spectrum as seen in the observer's frame, where  the $M=0$ peaks are clearly visible.\nWith increasing AR surface coverage the frequency shifts of the $M\\neq 0$ modes increase and the peaks get less clustered.\n\n\\section{Results}\n\n\\subsection{Dipole and quadrupole power spectra}\nIn this section we describe the changes  imprinted by a large active region in the spectrum of two multiplets with $(\\ell,n)=(1,18)$ and $(2,18)$, \nfor a star rotating with a period of $8~\\text{days}$.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.9\\columnwidth]{fig4.jpg}\n \\end{center}\n \\caption{Oscillation power spectra for the $(\\ell,n)= (2,18)$ multiplet observed at two inclination angles $i=30^\\circ$ (panels a and c) and $80^\\circ$ (panels b and d),\n for a star with a rotation period of $8~\\text{days}$ and for an active region with $\\varepsilon_{n\\ell}=0.003$, $\\beta=20^\\circ$ (panels a and b) or $80^\\circ$ (panels c and d), and for a surface coverage with $\\alpha=23^\\circ$. {The power spectra are normalized with respect to $V_2$ (Eq. \\ref{eq:V_l}).}  The vertical line segments show the theoretical frequencies and amplitudes for $M=\\pm1,\\pm 2$ modes (black) and the $M=0$ mode (red). \n The envelopes of the power spectra (solid black curves) are obtained by summing over Lorentzians with widths of $1~\\mu{\\mbox{$\\mathrm{Hz}$}}$. The dashed black curve shows the envelope of the pure rotational power spectrum, which includes the centrifugal distortion (Eq. \\ref{eq:rot_splitting}).\n }\n \\label{fig:l2n18spectrum}\n\\end{figure}\nFigure \\ref{fig:l2n18spectrum} shows the results for the quadrupole multiplet, for four combinations of the values of $\\alpha$ and $\\beta$ selected in Sect.~\\ref{sec:AR_parameters}: the observed power spectra are plotted for two angles of observation, $i=30^\\circ\\text{ and}~80^\\circ$, and are \nnormalized with respect to $V_2$, {{i.e.}}~with respect to the $m=0$ peak of the pure rotational spectrum seen with $i=0$.\nThe corresponding theoretical Lorentzian envelope (solid line) has been calculated \nfrom Eq. (\\ref{eq:powerspectrum}), by setting a value for the FWHM of $\\Gamma\/2\\pi=1~\\mu{\\mbox{$\\mathrm{Hz}$}}$, typical for this multiplet in the Sun \\citep[see, {{e.g.}},][]{Chaplin2005}.\nDue to the finite lifetime of the modes of oscillation, it is clear that is not possible to resolve all the $(2\\ell+1)^2$ peaks, and an observer would identify not many more than $(2\\ell+1)$ peaks in a multiplet.\nIn the cases shown here, it is possible to identify from 5 to 6 peaks for $i=80^\\circ$, the additional peak coming from the uppermost shifted $m=2,M=0$ peak. \nWe note that the Lorentzian envelope displays an asymmetric profile.\nBecause of their large shifts in frequency, the $M=0$ peaks blend with peaks from different $m$-quintuplets. Blending increases with activity level. Figure \\ref{fig:l2n18spectrum}a shows a case for which the $(M,m)=(0,0)$ and  $(M,m)=(1,1)$ peaks have close frequencies and comparable amplitudes, they contribute equally to a single peak in the power spectrum.\n \nThe  envelope of the power spectrum  is very sensitive to the latitudinal position of the AR and to the inclination angle: in Fig. \\ref{fig:l2n18spectrum}c the power spectrum is near the standard rotationally-split spectrum, while the same configuration observed from a different inclination angle (Fig. \\ref{fig:l2n18spectrum}d) shows a more asymmetric profile with additional peaks. \n{This is better seen in Fig. \\ref{fig:contours}, which shows contours of the acoustic power as a function of inclination angle, for both the $\\ell=1$ and $\\ell=2$ multiplets, for the same active region parameters as in Fig. \\ref{fig:l2n18spectrum}. For an active region at high latitude (middle panels of Fig. \\ref{fig:contours}), the central peak  shows a significant shift and overlaps with the $m=1$ peak. \nFor a low-latitude active region (bottom panels) the envelope of the power spectrum displays more than $2\\ell+1$ peaks. A distinct feature is the presence of  two peaks  instead of one when observing an $\\ell=2$ multiplet at zero inclination angle.}\n{The sensitivity of the spectrum to the colatitude of the AR, shown in Fig \\ref{fig:contours_beta}, is due in part to the variation with $\\beta$ of the non-diagonal elements of the rotation matrix ${{\\vet{R}}}^{(\\ell)}$. }\n\nAn observed power spectrum is, of course, much more difficult to interpret than its expectation value. The power spectrum in Fig.~\\ref{fig:noise_1} includes realization noise due to the stochastic nature of stellar oscillations and to additional shot noise. At each frequency the observed power is a realization of an exponential distribution (a chi-squared with two degrees of freedom) with standard deviation and mean equal to the expectation value of the power spectrum. Realization noise considerably degrades the spectrum, however in some cases it is still possible to distinguish between the pure rotational spectrum and a spectrum with the active region.\n\n\n\\begin{figure}\n\\includegraphics[width=0.97\\columnwidth]{fig5.jpg\n \\caption{\n Expectation value of power spectra of oscillation,  as functions of inclination angle $i$.\n The left panels are for the dipole multiplet $\\ell=1,n=18$, and the right panels for the quadrupole multiplet $\\ell=2,n=18$.\n The top panels are for the pure-rotation case, the middle panels are for an active region at co-latitude $\\beta=20^\\circ$, and the bottom panels for  $\\beta=80^\\circ$.\nThe active region parameters are  $\\varepsilon_{n\\ell}=0.003$, $\\alpha=23^\\circ$, and the stellar rotation period is $8$~days.\n }\n \\label{fig:contours}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth]{fig6.jpg\n \\caption{Expectation value of power spectra of oscillation, as functions of active-region colatitude $\\beta$, at fixed inclination angle $i=80^\\circ$. The other physical parameters are the same as in Fig.~\\ref{fig:contours}.\n }\n \\label{fig:contours_beta}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth]{fig7.jpg\n \\caption{A realization of the power spectrum of an $\\ell=2$ multiplet (gray) and its expectation value (black solid curve, also shown in  Fig.~\\ref{fig:l2n18spectrum}d). The observation duration is 6 months and the signal-to-noise ratio is 50.  The dashed black curve shows the expectation value of the pure rotational spectrum.\n }\n \\label{fig:noise_1}\n\\end{figure}\n\n\n\n\\subsection{Asymptotics}\n\\subsubsection{Limit of small latitudinal differential rotation}\n\\label{sec:small_diff_rot}\nThe examples shown so far suggest that the power spectrum of a multiplet may be approximated by the sum of $2(2\\ell+1)$ peaks, for two reasons.\nFirst, in the corotating frame the splitting due to rotation is small compared to the shift induced by the AR perturbation (see, {{e.g.}}, Fig. \\ref{fig:degenerate2obs}).\nSecond, the AR perturbation induces a shift that is largest for the $M=0$ mode. Therefore, for each $m$, all the $M\\neq0$ peaks are clustered near the pure rotational frequencies and appear as a single peak, while the $M=0$ peaks are well separated. \n\nHere we wish to find an approximation for the power spectrum of a multiplet in terms of $2(2\\ell+1)$ Lorentzians only. \nIn the following we assume the rotation perturbation to be small compared to the active region perturbation, and seek an approximate solution to the eigenvalue problem (\\ref{eq:eigen_problem}). \n\nIt is convenient to solve the eigenvalue problem in the frame where the dominant perturbation is diagonal, {{i.e.}}, the frame ${\\mathcal{\\widehat{R}}}_\\beta$ of the active region. The following results are similar to Sec. 19.5 of \\citet[][]{Unno1979}. \nWe rewrite the elements of the full matrix $\\widehat{\\vet O}$ in ${\\mathcal{\\widehat{R}}}_\\beta$ as\n\\begin{gather}\n\\widehat{O}_{m''m'} = \\delta_{m''m'}\\delta\\omega_{m'}^{\\rm AR} +\n\\sum_{m=-\\ell}^{\\ell}\nr_{m''m}^{(\\ell)} (\\beta) r_{m'm}^{(\\ell)} (\\beta)\n\\delta\\omega_{m}^\\Omega ,\n\\end{gather}\nwhere the term with the sum in the right hand side correspond to $\\widehat O_{m''m'}^\\Omega $.\nWe look for solutions of the form\n\\begin{gather}\n\\delta\\omega_M = \\delta\\omega_{M}^{\\rm AR} + \\delta\\omega_M^{(1)}\n\\end{gather}\nand \n\\begin{gather}\n\\widehat\\boldsymbol{A}{}^M = \\widehat\\boldsymbol{A}{}^{M,\\rm AR} + \\widehat\\boldsymbol{A}{}^{M,(1)},\n\\end{gather}\nwhere $\\left (\\delta\\omega_M^{(1)},\\widehat\\boldsymbol{A}{}^{M,(1)}\\right )$ are the perturbations to the (partially degenerate) eigenvalues $\\delta\\omega_{M}^\\mathrm{AR}$ and eigenvectors $\\widehat\\boldsymbol{A}{}^{M,\\rm AR}$ of $\\widehat\\vet O{}^\\mathrm{AR}$, with $\\widehat{A}_m^{M,\\mathrm{AR}}=\\delta_{mM}$. \nThe eigenvector perturbation is orthogonal to the reference eigenspace  \\citep[see, {{e.g.}},][p. 687]{1959messiah}, \n\\begin{equation}\n\\widehat{\\boldsymbol{A}}^{M,(1)} \\cdot \\widehat{\\boldsymbol{A}}^{\\pm M,\\mathrm{AR}}=0,\n\\label{eq:orthoA'A}\n\\end{equation}\nwhich gives \n$\n\\widehat A_{\\pm M}^{M,(1)} = 0.\n$\nThe eigenvectors  $\\boldsymbol{A}^M$ in the frame ${{\\mathcal{R}}}_\\beta$ are given by\n\\begin{gather}\n\\boldsymbol{A}^M = {\\vet{R}}^{(\\ell)} \\widehat\\boldsymbol{A}{}^M \n\\end{gather}\nvia the rotation matrix ${\\vet{R}}^{(\\ell)}$ (see Eq.~\\ref{eq:rotylmabg}).\n\nTo first order , the eigenvalue problem (Eq.~\\ref{eq:eigen_problem})  becomes\n\\begin{gather}\n \\left ( \\widehat{\\vet O}^\\Omega - \\delta\\omega_M^{(1)} \\vet I \\right ) \\widehat\\boldsymbol{A}{}^{M,\\mathrm{AR}} + \\left ( \\widehat\\vet O^\\text{AR} -\\delta\\omega_{M}^\\mathrm{AR} \\vet I \\right )\\widehat\\boldsymbol{A}{}^{M,(1)} = 0,\n\\label{eq:eigenproblem_small}\n\\end{gather}\nwhere $\\vet I$ is the identity matrix. \nTo calculate $\\widehat\\boldsymbol{A}{}^{M,(1)}$ and $\\delta\\omega_M^{(1)}$ we multiply the above equation on the left by the transpose of $\\widehat\\boldsymbol{A}{}^{m',\\text{AR}}$ to obtain:\n\\begin{gather}\n\\widehat O_{m'M}^\\Omega  - \\delta\\omega_M^{(1)} \\delta_{m'M} + \n\\left (\\delta\\omega_{m'}^\\mathrm{AR} -\\delta\\omega_{M}^\\mathrm{AR} \\right ) \\widehat A_{m'}^{M,(1)}  = 0  ,  \\quad   -\\ell \\le m' \\le \\ell.\n\\label{eq:eigenproblem_small_component}\n\\end{gather}\nWe find the perturbed eigenvalues by setting $m'=M$ in the above equation\n\\begin{equation}\n\\delta\\omega_M^{(1)} =  \\sum_{m=-\\ell}^{\\ell}\n[r_{Mm}^{(\\ell)} (\\beta) ]^2 \n\\delta\\omega_{m}^\\Omega .\n\\end{equation}\nThe non-zero elements of $\\boldsymbol{A}^{M,(1)}$ are obtained from the $m'\\neq \\pm M$ components of Eq. (\\ref{eq:eigenproblem_small_component}):\n\\begin{gather}\n\\widehat A_{m'}^{M,(1)} = \\frac{\\widehat O_{m'M}^\\Omega}{\\delta\\omega_{M}^\\mathrm{AR}-\\delta\\omega_{m'}^\\mathrm{AR}} \\quad \\text{ for } m'\\neq \\pm M .\n\\end{gather}\nThe explicit expressions for the eigenvalues and eigenvectors of the combined perturbations are\n\\begin{align}\n\\label{eq:shifts_small_rotation}\n\\delta\\omega_M = & \\,\\delta\\omega_{M}^\\mathrm{AR} + \\sum_{m=-\\ell}^{\\ell}\n[r_{Mm}^{(\\ell)} (\\beta) ]^2 \n\\delta\\omega_{m}^\\Omega \n\\end{align}\nand\n\\begin{gather}\n\\label{eq:amplitudes_small_rotation}\n\\widehat A_{m'}^M = \\left \\{\n\\begin{aligned}\n&1 & \\text{for } m'&=M ,\\\\\n&0 & \\text{for } m'&=-M ,\\\\\n& \\ffrac{\\sum_{m=-\\ell}^{\\ell}\nr_{m'm}^{(\\ell)} (\\beta) r_{Mm}^{(\\ell)} (\\beta)\n\\delta\\omega_{m}^\\Omega}\n{\\delta\\omega_{M}^\\mathrm{AR}-\\delta\\omega_{m'}^\\mathrm{AM}}\n& \\text{for } m' &\\neq M .\n\\end{aligned}\n\\right .\n\\end{gather}\nThe explicit expression for the amplitudes $A_m^M$ in the frame ${{\\mathcal{R}}}_\\beta$ is then\n\\begin{align}\nA_m^M  = &\\sum_{m'=-\\ell}^\\ell  \\widehat A_{m'}^M r_{m'm}^{(\\ell)}(\\beta) \\nonumber\\\\\n= &r_{Mm}^{(\\ell)}(\\beta) \n + \\sum_{m'=-\\ell,m'\\neq \\pm M}^{\\ell}\n\\frac{r_{m'm}^{(\\ell)}(\\beta)}{\\delta\\omega_{M}^\\mathrm{AR}-\\delta\\omega_{m'}^\\mathrm{AR}}\n\\sum_{m''=-\\ell}^{\\ell}\nr_{m'm''}^{(\\ell)} (\\beta) r_{Mm''}^{(\\ell)} (\\beta)\n\\delta\\omega_{m''}^\\Omega .\n\\label{eq:amplitudes_asymptotics}\n\\end{align}\n\n\\subsubsection{Neglecting latitudinal differential rotation}\n\nIf we neglect differential rotation then Eq. (\\ref{eq:rot_splitting2}) reduces to\n\\begin{equation} \n\\delta\\omega_m^\\Omega = -m\\Omega_\\beta C_{n\\ell} +  \\eta Q_{2\\ell m} \\, \\omega_{n\\ell}^{(0)}.\n\\label{eq:domega_nodiff_rot}\n\\end{equation}\nThen, by using the identities \\citep{Unno1979,1990gough}\n\\begin{equation}\n\t\\sum_{m''=-\\ell}^\\ell \n    \\left [r_{m'm''}^{(\\ell)}(\\beta) \\right ]^2 m'' = m' \\cos\\beta \n\\end{equation}\nand\n\\begin{equation}\n  \\sum_{m''=-\\ell}^\\ell \\left [r_{m'm''}^{(\\ell)}(\\beta) \\right ]^2\n    Q_{2 \\ell m''}=P_2(\\cos\\beta) Q_{2 \\ell m'},\n\\end{equation}\nthe frequency shifts (Eq.~\\ref{eq:shifts_small_rotation}) simplify to\n\\begin{align}\n\\delta\\omega_M = \\delta\\omega_{M}^\\mathrm{AR} \n- M C_{n\\ell} \\Omega_\\beta \\cos\\beta \n+ \\eta P_2(\\cos\\beta) Q_{2 \\ell M} \\, \\omega_{n\\ell}^{(0)} .\n \\label{eq:lin_dom_no_diff}\n\\end{align}\nNote that for moderately fast rotating stars the Coriolis term  in the above equation is much smaller than centrifugal distortion term \\citep[e.g.,][]{2004gizon}.\n\nDue to the clustering of the $M\\neq0$ peaks, the power spectrum in the observer's frame can be approximately modeled by $2(2\\ell+1)$ Lorentzians, some of which may overlap. Half of these correspond to the peaks with $M=0$. The remaining $2\\ell+1$ (approximate) Lorentzians are obtained by summing over the $M\\neq0$ peaks; their mean frequency shifts are given by a power weighted average of the $M\\neq0$ frequency shifts $\\delta\\omega_M^m = \\delta\\omega_M +m\\Omega_\\beta$. We denote by $\\langle \\delta\\omega\\rangle_m $ this average:\n\\begin{align}\n{\\langle \\delta\\omega\\rangle}_m = \n(\\Sigma_m )^{-1}\n\\left( \\sum_{1\\leq |M| \\leq \\ell} ({A_m^M})^2 \\delta\\omega_M  \\right)\n+m\\Omega_\\beta,\n\\label{eq:linear_freq_shift_Mneq0}\n\\end{align}\nwith \n\\begin{align}\n\\Sigma_m = \\sum_{1\\leq |M| \\leq \\ell} {(A_m^M)}^2.\n\\end{align}\nThe corresponding averaged power amplitudes $\\langle P\\rangle_m $ are (see Eq.~\\ref{eq:observed_amplitude})\n\\begin{gather}\n \\langle P \\rangle_m = \\frac{(\\ell-|m|)!}{(\\ell+|m|)!} \\left [V_\\ell P_\\ell^{|m|}(\\cos i) \\right ]^2 \n \\Sigma_m.\n \\label{eq:linear_amplitude_shift_Mneq0}\n\\end{gather}\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig8.jpg}\n\\caption{Acoustic power spectrum for the same case as in Fig. \\ref{fig:l2n18spectrum}d, (black solid line) and the power spectrum resulting from neglecting differential rotation and by assuming a small rotation perturbation with respect to the AR perturbation (black dashed line), as calculated from Eqs. (\\ref{eq:amplitudes_asymptotics}, \\ref{eq:domega_nodiff_rot}, \\ref{eq:lin_dom_no_diff}).\nVertical red lines denote the $M=0$ peaks, and black lines the averaged $M\\neq 0$ peaks from Eqs. (\\ref{eq:linear_freq_shift_Mneq0}, \\ref{eq:linear_amplitude_shift_Mneq0}). }\n\\label{fig:spectrum_small_diff_rot}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{fig9.jpg\n\\caption{\nBlack lines: Averaged frequency shifts $\\langle \\delta\\omega\\rangle_m$ vs. stellar rotation rate, as given by Eq. (\\ref{eq:linear_freq_shift_Mneq0}) and calculated using Eq. (\\ref{eq:eigen_problem}) (solid lines) and using the approximations of Eqs. (\\ref{eq:amplitudes_asymptotics}, \\ref{eq:domega_nodiff_rot}, \\ref{eq:lin_dom_no_diff}) (dashed lines).\nRed lines: Approximate frequency shifts $\\delta\\omega_{M=0}+m\\Omega_\\beta$ of the $M=~0$ peaks as given by Eq. (\\ref{eq:lin_dom_no_diff}) (dashed lines) and first-order exact shifts (solid lines).\n}\n\\label{fig:linear_qnl_AR}\n\\end{figure}\n\nFigure \\ref{fig:spectrum_small_diff_rot} shows how good is the $2(2\\ell+1)$-Lorentzian model  in reproducing the expected power spectrum for the case of Fig.\\ref{fig:noise_1}. The vertical red lines are for the $(2\\ell+1)$  peaks with $M=0$,  while the vertical black lines refer to the $(2\\ell+1)$ peaks with power  $\\langle P\\rangle_m $ and frequency shifts $\\langle \\delta\\omega\\rangle_m $. The envelope of power of the $2(2\\ell+1)$-model compares well with the envelope  obtained by summing over all the $(2\\ell+1)^2$ peaks.\n\nFigure \\ref{fig:linear_qnl_AR} shows the frequency shifts $\\langle \\delta\\omega\\rangle_m $ (black lines) and the shifts of the $M=0$ peaks (red lines) calculated from the  solution of Eq. (\\ref{eq:eigen_problem}) (solid lines) and from the approximate solutions (dashed lines), as a function of stellar rotation rate and for an active region with a surface coverage of $4\\%~(\\alpha=23^\\circ)$ and $\\beta = 80^\\circ$.\nThe linear approximation successfully returns the frequency shifts of the $2(2\\ell+1)$ Lorentzians, even for moderately high rotation rates.\n\n\n\n\n\\subsubsection{Limit of small-size active region ($\\alpha \\lesssim 15^\\circ$)}\n\\label{sec:small_ar}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.5\\columnwidth]{fig10a.jpg}\n\\includegraphics[width=0.5\\columnwidth]{fig10b.jpg\n\\caption{\nLeft: Contour plot of the absolute difference \n$|\\overline{\\delta\\omega}_{M=0} - \\delta\\omega_{M=0} |$ \nbetween the approximate shifts $\\overline{\\delta\\omega}_{M=0}$ (see Eq.~\\ref{eq:small_ar_shiftsM0}) and the first-order shifts ${\\delta\\omega}_{M=0}$, as a function of stellar rotation rate and surface coverage of the active region. \nRight: Contour plot of the maximum of the absolute difference\n$| \\langle \\overline{\\delta\\omega}\\rangle_m - \\langle \\delta\\omega\\rangle_m | $ over all values of $m$, as a function of stellar rotation rate and surface coverage of the active region.\n}\n\\label{fig:alphaomega_contour}\n\\end{figure*}\n\nThe above formulae for the frequency shifts simplify further when the active region has a small surface area. For small values of $\\alpha$, the integral in Eq.~(\\ref{eq:glm}) can be approximated,\n\\begin{align}\n\\int_{\\cos\\alpha}^1 \\left [P_\\ell^{|m''|}(\\mu)\\right ]^2 \\mathrm{d}{\\mu}  & \n\\underset{\\alpha \\rightarrow 0}{\\sim} \n\\left [P_\\ell^{|m''|}(0)\\right ]^2\\frac{\\alpha^2}{2} + {{\\mathcal{O}}}(\\alpha^4)  = \\delta_{m''0} \\frac{\\alpha^2}{2} + {{\\mathcal{O}}}(\\alpha^4),\n\\end{align}\nsuch that $G_\\ell^{m''}(\\alpha)$ becomes\n\\begin{gather}\n\\label{eq:Gl0_approx}\nG_\\ell^{m''}(\\alpha) = \\delta_{m''0}\\frac{\\alpha^2}{4} (2\\ell+1) + {{\\mathcal{O}}}(\\alpha^4).\n\\end{gather}\nAs shown in Fig.~\\ref{fig:glmalpha}, the above approximation is very good for $\\alpha \\lesssim 15^\\circ$ and $\\ell \\le 2$. \nUp to order $\\alpha^2$, the active region induces a shift only in the frequency of the $M=0$ mode\n\\begin{gather}\n\\delta\\omega_M^{\\rm AR} = \\delta_{M0}\\, \\omega_{n\\ell}^{(0)} \\varepsilon_{n\\ell} \\frac{\\alpha^2}{4} (2\\ell+1) + {{\\mathcal{O}}}(\\alpha^4).\n\\end{gather}\nPerturbed frequency shifts and amplitudes are obtained by following the calculation described in Sect.~\\ref{sec:small_diff_rot}, but taking into account the fact that the eigenfrequencies $\\delta\\omega_M^{\\rm AR}$ are now degenerate for $M\\neq0$.\nThe frequencies in the observer's frame can then be approximated by\n\\begin{gather}\n\\omega_m^M = \\omega_{n\\ell}^{(0)} +\\overline{\\delta\\omega}_M +m\\Omega_\\beta ,\n\\end{gather}\nwhere \n\\begin{align}\n\\overline{\\delta\\omega}_M = & \\delta_{M0} \\, \\omega_{n\\ell}^{(0)} \\varepsilon_{n\\ell} \\frac{\\alpha^2}{4} (2\\ell+1) + (1-\\delta_{M0})\\eta P_2(\\cos\\beta) Q_{2\\ell M}\\, \\omega_{n\\ell}^{(0)}\n\\,.\n\\label{eq:small_ar_shiftsM0}\n\\end{align}\nThe peak amplitudes in the power spectrum, $P_m^M$, are given by Eq.~(\\ref{eq:observed_amplitude}), where $A_m^M$ is replaced by \n\\begin{gather}\nA_m^M = \\sum_{m'=-\\ell}^\\ell \\overline{A}_{m'}^M \\; r_{m'm}^{(\\ell)}(\\beta)\n\\label{eq:linear_ampl_shifts_smallAR}\n  \\end{gather}\nwith\n\\begin{align}\n\\overline{A}_{m'}^M = \\delta_{Mm'} + (\\delta_{M0} - \\delta_{m'0})\n\\frac{\\eta\\omega_{n\\ell}^{(0)}}\n{\\delta\\omega_0^{\\rm AR}} \\sum_{m''=-\\ell}^\\ell \nr_{m'm''}^{(\\ell)}(\\beta) r_{Mm''}^{(\\ell)}(\\beta) Q_{2\\ell m''}.\n\\end{align}\n\n Figure~\\ref{fig:alphaomega_contour} shows the frequency error in $\\mu$Hz introduced by the small-$\\alpha$ approximation.\nDark blue shades indicate the regions in parameter space ($\\alpha$ and $\\Omega_\\beta$) where the approximation is very good.\nFor an active region with a  surface coverage below $2\\%$, i.e. $\\alpha<15^\\circ$, the frequency shift $\\overline{\\delta\\omega}_{M=0}$ is within $\\sim 0.1\\; \\mu$Hz of ${\\delta\\omega}_{M=0}$, even for fast rotation rates. \n\n\n\n\n\\subsection{Correlations in frequency space}\n\\label{sec:theory_correlations}\nDue to the fact that  the active region perturbation is unsteady in the observer's frame, the intensity fluctuations (Eq.~\\ref{eq:Iobs_final}) are not statistically independent in frequency space. \nGiven two frequencies $\\omega$ and $\\omega'$, the intensity covariance  is \n\\begin{align}\n \\text{Cov} & \\left [ {I_\\text{obs}}(\\omega), \\, I_\\text{obs}(\\omega')\\right ] \\nonumber\\\\\n&\n= E\\left [ {I^*_\\text{obs}}(\\omega) I_\\text{obs}(\\omega')\\right ] \\nonumber \\\\ \n& =\\sum_{M,m} \\sum_{M',m'}  B_m^M  B_{m'}^{M'} L^{1\/2}_M(\\omega-m\\Omega_\\beta)  L^{1\/2}_{M'}(\\omega'-m'\\Omega_\\beta)\nE[{{\\mathcal{N}}}^*_M(\\omega- m\\Omega_\\beta) {{\\mathcal{N}}}_{M'}(\\omega'- m'\\Omega_\\beta)]\n\\nonumber\\\\\n& =\\sum_{M=-\\ell}^{\\ell} \\sum_{m=-\\ell}^{\\ell} \\sum_{m'=-\\ell}^{\\ell} B_m^M B_{m'}^M L_M(\\omega-m\\Omega_\\beta)\\delta_{m',m+(\\omega'-\\omega)\/\\Omega_\\beta}   \\,  .\n\\label{eq:I_correlation}\n\\end{align}\nTo simplify the analysis we assumed that $k=(\\omega'-\\omega)\/\\Omega_\\beta$ is an integer (this is not a weakness of the theory though). The above expression vanishes unless   $| k | \\le 2 \\ell $.  The quantities $I_\\text{obs} (\\omega)$ and $I_\\text{obs} (\\omega')$ are correlated for frequency separations $\\Delta\\omega = \\omega'-\\omega=(m'-m)\\Omega_\\beta$. \n\n\nThe power spectrum  is also correlated for  \n$\\Delta \\omega =(m-m')\\Omega_\\beta$. Using the formulae given in appendix C of \\citet{fournier2014}, we find\n\\begin{align}\n \\text{Cov}\\left [ P(\\omega),P(\\omega') \\right ] \n &= \n \\text{Cov}\\left [ I^*_\\text{obs}(\\omega)I_\\text{obs}(\\omega) , \\, I^*_\\text{obs}(\\omega')I_\\text{obs}(\\omega') \\right ]  \\nonumber \\\\\n & = E \\left [ I^*_\\text{obs}(\\omega)I_\\text{obs}(\\omega')  \\right ]  E \\left [ I_\\text{obs}(\\omega)I^*_\\text{obs}(\\omega')  \\right ]  \\nonumber \\\\\n&= \\left | \\text{Cov} \\left [ I_\\text{obs}(\\omega), \\, I_\\text{obs}(\\omega') \\right ] \\right |^2  ,\n\\label{eq:PS_correlation}\n\\end{align}\nwhere the intensity covariance is given above.\nWe note that the values of the intensity and power spectrum covariances  (Eqs.~\\ref{eq:I_correlation} and~\\ref{eq:PS_correlation}) depend on the parameters of the model. However the existence of a  correlation is a general feature, which arises from the fact that the active region rotates with the star.   \nRemarkably, the expectation value  of the power spectrum $\\mc{P}(\\omega)$ (see Eq.~\\ref{eq:powerspectrum})  is as if all the terms in Eq.~(\\ref{eq:Iobs_final}) were statistically independent.\n\n\n\n\n\\section{Discussion}\n\\subsection{Nonlinear frequency shifts and amplitudes from numerical simulations}\n\\label{cha:rotation:sec:simulation}\n\nIn order to compare with the perturbation theory, we study the nonlinear regime by means of numerical simulations.\nWe use the GLASS wave propagation code, with the same numerical setup employed by  \\citet{Papini2015}.\n\n\\subsubsection{Rotation in post-processing}\nRunning different numerical simulations for different values of $\\beta$ and different perturbation amplitudes is computationally expensive. \nInstead we performed simulations for a 3D polar perturbation to the sound speed and for a star with no rotation, that is equivalent to solve the numerical problem in the reference frame ${\\mathcal{\\widehat{R}}}_\\beta$.\nWe introduced the effect of rotation later in processing the output.\nThis approach has the advantage that, for a given amplitude of the AR perturbation, we only need to run one simulation in order to calculate the power spectrum for any given value of $\\beta$ and rotation period.\nHowever we can only reproduce solid body rotation, and it is not possible to include the effects of the centrifugal distortion and of the Coriolis force,\ntherefore in each $n\\ell$-multiplet we expect to find only $(2\\ell+1)(\\ell+1)$ peaks.\nNonetheless the results are useful for exploring the nonlinear regime.\nWe note that, as a consequence of neglecting these rotational effects, $M$ can be identified with the azimuthal degree of the spherical harmonics $Y_\\ell^M (\\widehat\\theta_\\beta,\\widehat\\phi_\\beta) $ in the frame ${\\mathcal{\\widehat{R}}}_\\beta$ (see Sect.~\\ref{sec:corot}).\n\\subsubsection{Sound-speed perturbation}\nAs was done earlier, we approximate the perturbation introduced by the starspot by a local increase in sound speed.\nWe consider separable perturbations in the square sound speed of the form\n\\begin{equation}\n\\label{eq:deltacs2spot}\n {\\Delta c^2(r,\\widehat\\theta_\\beta)} = \\mathrm{\\epsilon} \\, c_0^2(r)  \\, f(r) \\, g(\\widehat\\theta_\\beta), \n \\end{equation}\n where $\\epsilon>0$ is a positive amplitude, $f$ is a radial profile, and $g$ a latitudinal profile.\nExplicitly, $g = 1\/2 + \\cos(\\kappa \\theta)\/2$ is a raised cosine for $0\\le \\kappa\\theta<\\pi$ and is zero otherwise, where $\\pi\/(2\\kappa)= 0.65$~rad~$=37.5^\\circ$. The function $f=\\exp(-|r-r_c|^2\/2\\sigma^2) [1\/2 + \\cos(|r-r_c|\/\\sigma)\/2]$  is a Gaussian function centered at radius $r_c=0.9985~R_\\odot$ with dispersion $\\sigma=0.004~R_\\odot$, multiplied by a raised cosine. \nThis functional form is the same as the one described by \\citet{Papini2015}.\nThe perturbation is thus placed along the polar axis at a depth of $4$~Mm, with a surface coverage of $12\\%$.  \n\n\\subsection{Synthetic power spectrum}\n\n\\label{cha:rotation:sec:simulationpowerspectrum}\nAs in Sect.~\\ref{sec:linear_PS}, we assume that the intensity fluctuations are proportional to the Eulerian pressure perturbation measured at $r_0=R+200~\\mathrm{km}$ above the surface \\citep[see ][]{Papini2015}. \nTo calculate the expectation value of the observed power spectrum, we performed a first set of simulations for which all the modes were excited with the same phase at the initial time.\n\n\nThe approximation that the changes in the eigenfunction of a mode $M$ are limited to the same angular degree $\\ell$ (Eq. \\ref{eq:xi_pert}) does not hold in the nonlinear case: the horizontal shape of an eigenfunction is a combination of spherical harmonics of different $\\ell$ values. \nHowever, since the perturbation is axisymmetric (see Eq.~\\ref{eq:deltacs2spot}), \nthe eigenfunction for a mode $M$ is a combination of spherical harmonics  $Y_\\ell^M(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta)$ of different angular degree $\\ell$ and same $M$.\nTherefore the intensity fluctuations  $I_{M}(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta,t)$ due to all the modes with the same $M$ take the form\n\\begin{gather}\n\\label{eq:I_Mglass}\n I_{M}(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta,t) \\propto \\sum_{\\ell=|M|}^{\\ell_\\text{max}}\\Re \\left \\{ \n {p}_{\\ell M}(r_0,t)  \n Y_\\ell^M(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta) \\right \\},\n\\end{gather}\nwhere ${p}_{\\ell M}(r_0,t)$ are the coefficients of the spherical harmonic decomposition of the pressure wavefield $p(r_0,\\widehat\\theta_\\beta,\\widehat\\phi_\\beta,t)$  returned by the GLASS code  in the frame $\\widehat{{\\mathcal{R}}}_\\beta$. The index $\\ell_\\mathrm{max} $ is set by the spectral resolution of the spherical harmonic transform.\n\n\nTo obtain the full-disk integrated intensity in the observer's frame \nwe express each $Y_\\ell^M(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta)$ in terms of the spherical harmonics in the frame ${{\\mathcal{R}}}$ \n\\begin{gather}\nY_\\ell^{M}(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta) = \n\\sum_{m=-\\ell}^{\\ell} \nr_{mM}^{(\\ell)}(\\beta) e^{-\\mathrm{i} m\\Omega_\\beta t} \n\\sum_{m'=-\\ell}^{\\ell}\nY_\\ell^{m'}(\\theta,\\phi) \\; r_{m' m}^{(\\ell)}(-i) \n  .\n\\end{gather}\nby means of two consecutive rotations of the Euler angles $(0,-i,\\Omega_\\beta t)$ and $(0,\\beta,0)$. Combining this equation with  Eq.~(\\ref{eq:I_Mglass}) and integrating over the visible disk, we obtain the full-disk integrated intensity of each $mM$-component:\n\\begin{gather}\n\\label{eq:integrated_intensity_sim}\nI_{mM}(t)= \n\\sum_{\\ell=\\max\\{|m|,|M|\\}}^{\\ell_{\\max}} V_\\ell\\; r^{(\\ell)}_{m0}(i) \\, r^{(\\ell)}_{mM}(\\beta) \\, \n\\Re \\left \\{{p}_{\\ell M}(r_0,t)  e^{-im\\Omega_\\beta t} \\right \\}.\n\\end{gather}\nWe then perform a Fourier transform to calculate the intensity $I_{mM}(\\omega)$ in the frequency domain. Finally, we derive the expression for the expectation value of the  power spectrum\n\\begin{gather}\n\\label{eq:powerspectrum_sim}\n\\mc{P}(\\omega)=  \\sum_{m=-\\ell_{\\max}}^{\\ell_{\\max}}\\sum_{M=-\\ell_{\\max}}^{\\ell_{\\max}} \n\\left |I_{mM}(\\omega)\\right |^2,\n\\end{gather}\nthat is analogous to Eq. (\\ref{eq:powerspectrum}), but for the entire wavefield. \n\n\n\\begin{figure*}\n \\includegraphics[width=0.5\\columnwidth]{fig11a.jpg}\n \\includegraphics[width=0.5\\columnwidth]{fig11b.jpg}\n \\caption{Numerical simulations of oscillation power spectra for two quadrupole multiplets with $n=12$ (left panels) and $n=18$ (right panels), for an inclination angle $i=80^\\circ$ and a \nstellar  rotation period of $8~\\text{days}$ (solid body rotation).\nThe active region has a colatitude  $\\beta=80^\\circ$ and different perturbation amplitudes of $\\epsilon=0.1$ (top), $0.2$ (middle), and $0.3$ (bottom), with $r_c=0.9985~R_\\odot$, $\\sigma = 0.004~R_\\odot$, and $\\kappa=2.4$ (see Eq. \\ref{eq:deltacs2spot}).\n The black curves show the power spectra computed with  the GLASS code, the red dot-dashed curves indicate the power spectra computed using  linear perturbation theory. Vertical lines show the  peaks from linear theory, in red for $M=0$ and in black $M\\neq0$.\n The blue curves in the right panels display the contribution of the $M=0$ peaks to the simulated power spectrum.\n The $n=12$ peaks  have a FWHM of $\\Gamma\/2\\pi\\simeq0.2~\\mu{\\mbox{$\\mathrm{Hz}$}}$, while for $n=18$ the peaks have   $\\Gamma\/2\\pi\\simeq1~\\mu{\\mbox{$\\mathrm{Hz}$}}$.\n }\n \\label{fig:l2spectrum_sim}\n\\end{figure*}\n\n\nFor the nonlinear study we chose three perturbation amplitudes   $\\mathrm{\\epsilon}=0.1$, $0.2$, and $0.3$ which, for the multiplet $(\\ell,n)=(2,18)$, correspond to $\\epsilon_{n\\ell}\\simeq 0.005$, $0.010$, and $0.015$, {{i.e.}}, roughly twice to six times the value used in the linear analysis.\nThe simulations run for $80~\\text{days}$ (stellar time), in order to reach an accuracy of $\\sim 0.14~\\mu{\\mbox{$\\mathrm{Hz}$}}$ in the frequency domain.\nThe wavefield computed by GLASS includes some numerical damping that increases with frequency with an exponential dependence.\nWe took advantage of this damping and  selected two $\\ell=2$ multiplets: one with $n=18$ and a FWHM comparable to the solar value, the other with $n=12$ and a FWHM small enough to resolve all the $mM$ peaks in the multiplet.  \nModel~S is convectively stabilized \\citep{Papini2014}, which implies that the unperturbed frequencies of the quadrupole modes are $1970.50$~$\\mu$Hz for $n=12$ and $2783.62$~$\\mu$Hz for $n=18$.\nFigure \\ref{fig:l2spectrum_sim} shows the observed power spectra of the two selected multiplets in the case   $\\beta=80^\\circ$ and $i=80^\\circ$, normalized with respect to the highest $M\\neq0$ peak.\nIn the case $\\epsilon=0.1$, the simulated power spectrum (black curve) of the $(\\ell,n)=(2,12)$ multiplet (top left panel) is well reproduced by the power spectrum computed with linear theory (red dot-dashed curves), except for the peaks corresponding to $M=0$ (vertical red curves show the $M=0$ peaks from linear theory), which are less shifted in frequency and have smaller amplitudes than predicted. \nFor the $(\\ell,n)=(2,18)$ multiplet (top right panel) the nonlinear effects are less visible, due to the overlapping  Lorentzian profiles. A blue curve, displaying the contribution to the power spectrum of the $M=0$ peaks, shows that also for this multiplet the $M=0$ component of the power spectrum deviates from the linear behaviour, both in frequency and amplitude. \n This plot also shows an example of the combined action of mode mixing and mode visibility, which almost suppresses the $m=0$, $M=0$ peak located at a frequency shift of $\\sim 3.1~\\mu{\\mbox{$\\mathrm{Hz}$}}$.\n\n\nWith increasing $\\epsilon$ (middle and bottom panels) the interaction of the wavefield  with the active region becomes strongly nonlinear, and for a perturbation with $\\epsilon=0.3$ results in a massive distortion of the power spectrum with respect to that  one predicted by linear theory. \nHere two different behaviors are evident: in the $(\\ell,n)=(2,12)$ multiplet the $m=0, M=0$  peaks are almost suppressed, while  \nthe peak with $m=0, M=0$ in the $(\\ell,n)=(2,18)$ multiplet, that was suppressed in the case $\\epsilon=0.1$, increases in amplitude as $\\epsilon$ increases (middle and bottom right panels).\nMoreover the peaks with $M=1$ start to deviate from the linear prediction.\nThis is in agreement with what found in the non-rotating case by \\citet{Papini2015}, who also showed that second-order perturbations would correct most of the differences.\n\n\\subsection{Correlations in synthetic power spectra}\n\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{fig12.jpg}\n \\caption{ Autocorrelation of the intensity spectrum as defined by  Eq. (\\ref{eq:autocorrelation_PS_I}) for a starspot with $\\epsilon=0.1$ (red curve).\n  A stellar rotation period $2\\pi\/\\Omega_\\beta = 8$~days, an inclination angle $i=80^\\circ$, and a starspot colatitude $\\beta=80^\\circ$ were chosen for the post processing. Vertical dotted lines denote frequency separations $\\Delta\\omega=j\\Omega_\\beta$ where $j=1,2,3, 4$.  For comparison, the black curve is the case with no starspot.\n  }\n  \\label{fig:correlation_glass}\n \\end{figure}\n \nIn Sect.~\\ref{sec:theory_correlations} we showed that in presence of an active region rotating with the star, the power spectrum of a multiplet is correlated at frequency separations that are multiple of the rotational frequency of the active region. However, for small perturbations this correlation is too weak to be observed. Here we ask whether such a correlation can be measured in presence of a perturbation of moderate amplitude. \nFor that purpose, we ran a second set of simulations, in which the acoustic waves were excited by applying a random forcing function at  $150~\\text{km}$ below the surface at each time step, as described by \\citet{Hanasoge2007}.\nThe duration of the simulation is 80 days (stellar time).\nIn the frequency domain, the observed intensity is \n\\begin{equation}\nI_\\mathrm{obs} (\\omega) = \\sum_{m=-\\ell_{\\max}}^{\\ell_{\\max}}\\sum_{M=-\\ell_{\\max}}^{\\ell_{\\max}} I_{mM}(\\omega) ,\n\\end{equation}\nwhere $I_{mM}(\\omega)$ is obtained by Fourier transformation of  Eq.~(\\ref{eq:integrated_intensity_sim}) using the numerical realizations of $p_{\\ell M}(r_0, t)$.\n\n\nThe autocorrelation of  the intensity spectrum is\n\\begin{gather}\n C_I(\\Delta\\omega) = \n \\frac{\\sum_{n=15}^{25}  \\int_{[\\omega_{n\\ell}]}  \n I_\\text{obs}^*(\\omega) I_\\text{obs}(\\omega+\\Delta\\omega)\\mathrm{d}{\\omega} }\n {\\sum_{n=15}^{25} \\int_{[\\omega_{n\\ell}]}  \n |I_\\text{obs}(\\omega)|^2 \\mathrm{d}{\\omega} } ,\n \\label{eq:autocorrelation_PS_I} \n\\end{gather}\nwhere $[\\omega_{n\\ell}]$ denotes an appropriate frequency interval of size $\\sim20~\\mu$Hz containing the multiplet $n\\ell$ but excluding the nearby $l=0$ mode. \nThe average is performed over all $\\ell=2$ multiplets with $n$ ranging from 15 to 25.\n\nFigure \\ref{fig:correlation_glass} shows the real part of $C_I$  for the special case of an inclination angle  $i=80^\\circ$ and a starspot at colatitude  $\\beta=80^\\circ$ with $\\epsilon=0.1$.\nA correlation is clearly visible at frequency separations $\\Delta\\omega=2\\Omega_\\beta$ and  $4\\Omega_\\beta$.\nThis suggests that the frequency-domain  autocorrelation function could be used as a diagnostic tool to identify \nunsteady perturbations in the time series of stellar oscillation. \nWe note that the imaginary part of $C_I$ contains no visible signal above the noise level.\n\n\n\n\n\n\\subsection{Towards a physical model for mode interaction with an active region}\n\\label{SectPhys}\n\n{ \nIn this paper we replaced the  active region  by a localized  increase in sound-speed near the stellar surface.\nWe focused on the geometrical aspects of the problem rather than on the physics.\nOne may ask, however,  what would be the difference in the obtained results if we had instead considered a realistic model for the magnetic active region.\nAlthough we will not answer this question here, it is worth listing some of the steps involved.\n\n\n\nA typical solar active region consists of a pair of sunspots surrounded by plage with strong vertical field. \nLocal helioseismology of the visible disk and the far side indicates that p modes are strongly scattered by both sunspots and extended plage \\citep[see, e.g.,][and references therein]{Gizon2009}.\nSome studies of the interaction of high-degree p~modes with magnetostatic sunspots  \\citep[e.g.,][]{Moradi2010,Cameron2011} have been carried out using MHD wave propagation codes \\citep{Cameron2008,Felipe2016}. \nOther studies are based on  numerical magneto-convective simulations \\citep[e.g.,][]{Rempel2009,DeGrave2014}. \nThe main conclusion of these simulations is that the interaction takes place in the top few hundred kilometres below the surface, where the direct effects of the magnetic field and the indirect effects due to changes in thermodynamic structure with respect to the reference atmosphere (e.g. the Wilson depression) are large.\nWave simulations indicate that the outgoing p modes are phase shifted with respect to the incoming p modes in such a way that the effective wave speed is increased, as observed. \nThe physical interaction involves the conversion of p modes  into fast and slow magnetoacoustic modes in the sunspot  \\citet[e.g.][]{Cameron2008,Khomenko2006}. A fraction of the incoming p-mode  energy is tunnelled downward in the form of slow MHD waves, leading to absorption  \\citep{Braun1995,ZhaoH2016}. See also, e.g.,  \\citet{Saio2004} and  \\citet{Cunha2006} for mode conversion calculations in roAp stars.\n\n\n\nUsing 2D ray tracing, \\citet{Liang2013} showed that high-degree helioseismic waveforms can be reproduced by increasing the effective wave speed by 10\\%  in the sunspot.\n This provides some justification for the values that we have used in the present paper, although the extension to low-degree p modes has not been studied.\nClearly, much additional work will be needed to determine the correct active-region perturbation amplitude from first principles. Until then, a simple calibration can be obtained from the observational study by \\citet{Santos2016} who estimated empirically the contribution of sunspots to the low-degree p-mode frequency shifts associated with the solar cycle. By  combining our Eq. (\\ref{eq:omat_mag_epsilon}) with Eq. (3) from \\citet{Santos2016}, we find  $\\varepsilon_{n\\ell}=-\\Delta\\delta_\\mathrm{ch}\/I_\\ell$, where $\\Delta\\delta_\\mathrm{ch}$ is the integral phase difference introduced in the mode eigenfunction by a sunspot and $I_\\ell$ is related to mode inertia. Using  the value $\\Delta\\delta_\\mathrm{ch}=-0.44$  estimated by \\citet{Santos2016}, we have  $\\varepsilon_{n\\ell} \\sim 0.05$ for a quadrupole mode, {{i.e.}}\\  a larger value than proposed by \\citet{Liang2013} and used in the present paper. This may suggest that for large active regions the spectra calculations may have to be carried out in the nonlinear regime. However, only realistic numerical modelling would help settle this question. The full problem would also have to include multiple scattering by collections of flux tubes in plage  \\citep[see, e.g.,][]{Hanson2015}. \n}\n\n\n\\section{Conclusions}\n\n\n\nIn this paper we investigated the changes in global acoustic oscillations caused by a localised sound-speed perturbation on a rotating star mimicking  a large active region, using both linear perturbation theory and 3D numerical simulations. In an inertial frame, the active region perturbation is unsteady.\n\nWe find that the power spectra of low-degree modes have a complex structure. The combined effects of the active region and differential rotation cause each $n\\ell$-multiplet to appear as $(2\\ell+1)^2$ peaks, each with a different amplitude. \nMost of the peaks are clustered near the classical rotationally-split frequencies, and only $2\\ell+1$ peaks (the $M=0$ peaks, which correspond to the axisymmetric mode in the reference frame of the AR) are shifted to higher frequencies. This leads to an apparent asymmetry in the line profiles. However, due to the finite lifetime of acoustic oscillations, most of the peaks cannot be resolved.\n{ For solar-type stars, the results are not very sensitive to the choice of latitudinal differential rotation profile.}\n\nThe structure of the power spectra  is sensitive to the latitudinal position of the active region and to the inclination angle $i$ of the stellar rotation axis. The latter plays a major role in determining the relative visibility of the individual peaks.\nWe find that the envelope of the power spectrum becomes more complex as the latitude of the active region decreases. \nIn practice, it would be very difficult to perform a fit of the $(2\\ell+1)^2$ peaks in a multiplet, due to peak blending and noise. However, by neglecting differential rotation it is possible to derive a simplified formula that approximates the power spectrum of a multiplet to a sum of only $2(2\\ell+1)$ Lorentzian profiles. For small-area  active regions, the formula further simplifies and directly links the frequencies of the peaks in the power spectrum  to the active region parameters. Such formula may find applications in the analysis of real asteroseismic observations.\n\n\nNumerical simulations were performed to explore the nonlinear regime of the perturbation. We find that the $M=0$ peaks deviate from the linear behavior for active-region perturbation amplitudes $\\varepsilon_{n\\ell} \\gtrsim 0.005$. Depending on  each particular case, the amplitude of these peaks is either reduced or enhanced compared to first-order linear theory, due to mixing  with modes with other values of $\\ell$ and $m$ \\citep{Papini2015}. \n\n\nWe found that there are correlations in the power spectrum at frequency separations that are multiples of the active-region rotation rate. In the linear regime the correlation signal is too weak to be observed. However the numerical simulations show that for active-region perturbations of moderate amplitude, such a correlation might be detectable, provided that  the frequency intervals are carefully selected  to increase the signal to noise ratio.\n\nWe note that perturbation theory can easily be extended to compute the effect of multiple active regions provided that latitudinal differential rotation is small. The  treatment of  several active regions rotating at different rotation rates  would require a different setup, since there is no frame in which these perturbations are steady. A numerical approach would be preferable in such a case.\n\n\n{ The work presented in this paper uses simplified physics, but  it should provide useful guidance to identify the seismic signature of a large active region in the power spectrum of stellar oscillations. \nGiven that the values of $\\epsilon_{n\\ell}$ are uncertain, we believe that it is worth searching  for low-degree multiplets consisting of $2(2\\ell+1)$ components in available asteroseismic observations.}\nIdeal targets are stars that are known to have high-quality oscillation power spectra \\citep[high SNR, narrow line profiles, clear rotational splitting, see e.g.][]{Nielsen2014} \nand  show evidence for long-lived starspots \\citep[e.g.][]{Nielsen2013, Nielsen2018}. \nThe catalogue of potential targets, currently limited to CoRoT and {\\it Kepler},  will increase fast with  TESS \\citep{TESS2015} and  PLATO \\citep{plato2014}.\n\n\n\n\\section*{Conflict of Interest Statement}\n\nThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.\n\n\\section*{Author Contributions}\n\nL.G. provided the basic theory and E.P. the numerical applications. Both authors contributed to the analysis of the results and to the writing of the article.\n\n\n\\section*{Acknowledgments}\n\nThis paper is a contribution to PLATO PSM activity `Seismic diagnostics of stellar activity'. We acknowledge financial support from the German Aerospace Center DLR and from the Max Planck Society.\nWe thank Shravan Hanasoge for making the GLASS code available. \nComputational resources were provided by the German Data Center for SDO.\nThis work appeared in part in the PhD thesis of \\citet{Papini-thesis}. \n\n\\bibliographystyle{frontiersinSCNS_ENG_HUMS} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nModern $\\gamma$-ray~  telescopes operating at energies above the MeV window, provide\nevent-resolved observational data. Each event (after the reconstruction process)\nis typically described by a tuple (i.e. an ordered list of elements) storing sky \ncoordinates, arrival time, and energy. \nDetection of discrete sources (either point-like or extended) is\nperformed using various methods. Given the discrete topological nature of $\\gamma$-ray~\nimages, methods based on cluster search, like the {\\it  { Minimum} Spanning Three}\n(MST) \\citep{MSTI,MSTII} have successfully been used. One of the main advantages of topometric\nmethods, { compared to methods using the spatial binning},  is to minimize \nthe impact of the poor energy-dependent Point Spread Function (PSF), typical of\n$\\gamma$-ray~ telescopes, preserving the spatial information of each event. Moreover,\nthese methods are able to detect sources compounded by a small amount of events,\nbut they need to be fine tuned to take into account properly the background. The\nproblem of background rejection is the most penalizing feature of topometric\nmethods, for this reason in this paper, for the first time, we present a \nmethod based on the DBSCAN~ algorithm \\citep{Ester96DBScan}. \nThe DBSCAN~ is a topometric algorithm used to\ncluster spatial data that are affected by background noise. Compared to other\ntopometric methods, it has the advantage to embed inside the algorithm itself the\ndiscrimination between signal (cluster) and background (noise), according to the\nlocal density of events within a typical scanning brush { i.e. within \na given scanning area}.\n\n{ The aim of the present paper is to show the potentialities of the method,\nand  its statistical characterization when applied to astrophysical $\\gamma$-ray~ data.} \nWe apply this method to the detection of point-like sources in the {\\it Fermi}-LAT~ data. \nWe explore a large volume of the $\\gamma$-ray DBSCAN~ parameter space, by means  of simulated \ndata, and we provide a statistical characterization the $\\gamma$-ray DBSCAN, finding signatures \nthat differentiate  purely random fields, from fields with sources. We define a \nsignificance level for the detected clusters, and we successfully test this significance \nwith our simulated data. We apply the method to real {\\it Fermi}-LAT~ $\\gamma$-ray~ data, and we \nfind an excellent agreement with the results  obtained with simulated data.\n\n\nIn a companion paper \\citep{dbscan_inprep}, we will apply the method\nto the\t {\\it Fermi}-LAT~ sky, {investigating specific issues related to the {\\it Fermi}-LAT~ response \nfunctions}, showing the potentiality for the discovery of new sources,\nin particular of small clusters located at high galactic latitude, or   \nclusters on the galactic plane, affected by a strong background.\n\nThe paper is organized as follows. In Sec. \\ref{sec:dbscan} we describe the logic of\nthe DBSCAN~ method, and we present the algorithm implemented to analyse $\\gamma$-ray~\ndata, the $\\gamma$-ray DBSCAN.\nIn Sec. \\ref{sec:caveat} we discuss some caveats regarding the\napplication of the $\\gamma$-ray DBSCAN~ algorithm to $\\gamma$-ray~ data. \nIn Sec. \\ref{sec:stats_sim} we study the statistical properties of the\n$\\gamma$-ray DBSCAN~ detection, using a simulated test field with only noise,\nand five simulated test fields with noise plus point-like sources.\nIn Sec. \\ref{sec:det_sim}  we evaluate the detection  performance of the method \nin terms of positional accuracy, cluster reconstruction,  and rejection of spurious clusters.\nIn Sec. \\ref{sec:signif_sim} we investigate the significance of the clusters,\nand describe our algorithmic implementation.\nIn Section \\ref{sec:realdata} we finally use our method with real {\\it Fermi}-LAT~ data,\ninvestigating the detection performance, and comparing the $\\gamma$-ray DBSCAN~ clusters \nsignificance, to that returned by the Maximum Likelihood method with\nstandard {\\it Fermi}-LAT~ software \\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/overview.html}.\nIn Section \\ref{sec:conclusions}, we present our conclusions, and we discuss\nfuture developments and applications.\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{l} \n\t\\includegraphics[width=18cm]{sim_detection_plot.pdf}\\\\\n\\end{tabular}\n\\caption{ Photon map for the {\\it sky} test field 1, \nwith the result of the $\\gamma$-ray DBSCAN~ detection for $K = 5$ and $\\varepsilon=0.17$ deg.\nThe blue crosses refer to the simulated sources, the green boxes  \nto  51 detected {\\it true} clusters, and the red boxes to the 2 {\\it fake} ones. \nThe black dots represent the background events, the remaining colors indicate\ncluster events.}\n\\label{fig:sim_detection}\n\\end{figure*}\n\n\n\\section{The $\\gamma$-ray~ DBSCAN~ algorithm}\n\\label{sec:dbscan} \nThe DBSCAN~ \\citep{Ester96DBScan} is a topometric algorithm used  to cluster\nspatial data that are affected by background noise.  Some modifications have\nbeen developed to adapt the original DBSCAN~ algorithm to our study. Our\nalgorithm is mainly built upon the following criteria: \n\\begin{enumerate}\n\\item{} given a list of photons $D$, where each element $p_i$ is  a  tuple storing  \npositional sky coordinates, let $\\rho(p_k,p_l)$ be the angular distance between \ntwo photons $p_k$ and $p_l$.\n\\item{} We iterate over the full photon list $D$. A seed cluster $C_m^*$ is built \nwhen a minimum number of photons $K+1$ is enclosed within a circle of radius $\\varepsilon$~\ncentered on $p_i$ \n\\item{} For each photon  $p_l \\in C^*_m$,  we build the photon list $C_m^+$ \nby collecting all the photons $p_k$ respecting the condition:  \n$\\rho(p_l,p_k)<\\varepsilon$, and $p_k \\notin C^*_m$. \n\\item For each photon  $p_j \\in C_m^+$, if the number of photons enclosed within a circle of  \nradius $\\varepsilon$~ centered on $p_j$ is $\\leq K$ and $p_j \\notin C^*_m$, then $p_j$ will be attached to the \n{ final photon list of the cluster   without a recursive search for further neighbours}, \nthese points are defined  {\\it density-reachable}.\n\\item For each photon  $p_j \\in C_m^+$, if the number of photons enclosed within a circle of \nradius $\\varepsilon$~ centered on $p_j$ is $>K$ and $p_j \\notin C^*_m$, $p_j$ is attached to the $C_m^*$, and  \nand step { 3} is repeated recursively.\n\\item{} { When both conditions at step 4 and 5 are false, the cluster $C_m$ is built by joining\nthe {\\it density-reachable} events to those in the $C^*_m$ and in the  $C_m^+$  lists.}\n{ \\item{} The process starts again from step 1 searching for new clusters, skipping all the\nevents already flagged as {\\it noise} or {\\it clusters}, until all the\nevents in $D$ are flagged as {\\it cluster}, or {\\it noise}, or\n{\\it density-reachable}, events. }\n\\item{} At the end of the process the full photon list will be partitioned as \nfollows: \n\\begin{eqnarray}\nD &=& D_{cls} \\cup D_{noise}= (p_i \\in \\cup_m C_m) \\cup (p_i \\notin  \\cup_m C_m) \\\\\n\\emptyset&=&D_{cls} \\cap D_{noise} \\nonumber\n\\end{eqnarray}\n\\end{enumerate}\nIn this way high densely populated areas are classified as clusters (sources),\nconversely  low densely populated areas  are classified as noise (background). \nThe recursive call  of step 3, is not implemented in the original\nDBSCAN~ algorithm, and { represents} a novelty. This new feature, allows to reconstruct\nclusters with a size significantly larger than the $\\varepsilon$~ radius, making rare the  \npossibility to fragment a single clusters in small satellite clusters. Moreover, allows\nthe possibility to reconstruct extended { structures}, in particular extended sources, or\nfilamentary structures in the background.\n\n\n\n\n\n\nAfter the clustering process, each photon in $D$ will be described by a tuple, storing:\n{ \nthe photon position (both in galactic  and celestial  coordinates),\nthe photon class type ({\\it noise} or {\\it cluster}),\nand the ID of the cluster the photon belongs to. \nEach cluster $C_m$,  will be described by a tuple storing \nthe position of the centroid  with his positional error,   the ellipse  \nof the cluster containment, the cluster effective radius ($r_eff$),\nand number of photons in the cluster ($N_p$).\nThe ellipse of the cluster containment, is defined by major and \nminor semi-axis ($\\sigma_x$  and $\\sigma_y$, respectively), and the inclination angle\n($\\sigma_{alpha}$) of the major semi-axis w.r.t. the latitudinal coordinate. ($b$ or $DEC$). \nTo evaluate the ellipse axis we use the Principal Component Analysis method (PCA) \\citep{Jolliffe1986}. This method uses the eigenvalue decomposition of the \ncovariance matrix of the  the two position arrays $\\mathbf x$, and $\\mathbf y$. \nBy definition, the square root of the first eigenvalue will correspond \nto $\\sigma_x$, and the second to $\\sigma_y$.\nThe axes represent the two orthogonal directions of maximum variance of the cluster.\nThe effective radius is defined as $r_{eff}=\\sqrt{\\sigma_x^2 + \\sigma_y^2}$.\nTo find the centroid of the cluster and its uncertainty, we use a weighted average of the \nposition of each photon in $C_m$, as follows:\n\\begin{itemize}\n\\item{} we define the first order centroid ($C_{ave}$) as the average of the \nposition of each cluster photon: $C_{ave}=(<\\mathbf x>,<\\mathbf y>)$.\n\\item{} We define the weight array, according to the distance between\n$p_k \\in C_m$ and $C_{ave}$: $w_k=1\/\\rho(p_k,C_{ave})$.\n\\item {} The cluster centroid $C_{ctr}$ will result from average of the position  of\neach cluster point weighted by $w_k$.\n\\item {} The centroid position uncertainty ($pos_{err}$) is determined by propagating the  \nerror on the weighted average of $C_{ctr}$.  \nWe have numerically verified that  $pos_{err}$ corresponds to a $\\approx 95\\%$ \npositional uncertainty. \n\n\\end{itemize}\n\n\n\\begin{figure}[!h]\n\\includegraphics[width=9cm]{sim_det_zooming.pdf}\n\\caption{\nA close-up of two {\\it true} clusters reported in Fig. \\ref{fig:sim_detection}. \nThe  ellipses correspond to the ellipse of  the cluster containment. \nThe purple and orange points represent the cluster points, the black dots represent the background\nevents, the blue crosses the position of the simulated sources, and green boxes the position of \nthe clusters centroid\n}\n\\label{fig:closeup}\n\\end{figure}\n\n\n\n\n\n\\section{Caveat on the application to $\\gamma$-ray~ data }\n\\label{sec:caveat}\nThe application of clustering methods, such as the $\\gamma$-ray DBSCAN, leads to deal\nwith practical difficulties, related mostly to the instrument PSF, and\nto gradient and\/or structures in the background. In order to deal with these \nissues, without biasing the detection results, it's recommended to apply some\ncriteria that we discuss in the following.\n\n \n As first, we comment on the PSF impact. The PSF imposes a limit on the capability\nof an instrument to resolve sources separated by a distance smaller then the PSF\nsize. Sources with sizes below the PSF { } classified as point-like, { otherwise} are classified as extended. A further complication is that the PSF\noften depends on the energy; in the case of {\\it Fermi}-LAT,  the $68\\%$ containment angle of the \nreconstructed incoming photon direction, for normal incidence photons, has a typical \nsize of a couple of degrees at 100 MeV \\citep{LATCalib}, and scales down to few tenths of \ndegree above the GeV energies \n\\footnote{http:\/\/www.slac.stanford.edu\/exp\/glast\/groups\/canda\/\\\\\tlat\\_Performance.html}. \nThe size of the PSF is strongly connected to the size $\\varepsilon$~ of the \n$\\gamma$-ray DBSCAN~ scanning brush. Indeed, if $\\varepsilon$~ is much smaller than the PSF size, it \nmight occur the risk to loose clusters characterized by small $N_p$, or to fragment a \ncluster with large $N_p$ in smaller fake {\\it satellite} clusters.  \nWe stress that the formation of {\\it satellite} clusters is a very rare event, \nthanks to our recursive  DBSCAN implementation, that is explained in Sec. \\ref{sec:dbscan}. \nOn the contrary, if $\\varepsilon$~ is much larger w.r.t the PSF, it is likely to build extended \n{ clusters} contaminated by the background, or by  close sources. \\\\\n{\nA careful and self-consistent analysis of the effects of the energy dependence\nof the PSF, and in general of  issues related to the {\\it Fermi}-LAT~ response function, \nis beyond the scope of this paper, where we focus mostly on a statistical \ncharacterization  of the method.\nThese subjects will be investigated in the companion paper \\citep{dbscan_inprep}.\n}\n\nA second relevant issue, is the inhomogeneity of the background, that affects\nboth the choice of $\\varepsilon$~ and $K$. If the background is homogeneous over the\nentire field, the optimal choice of a single pair of values of $\\varepsilon$~ and $K$,\nguarantees a safe rejection of the background. Indeed, values of $\\varepsilon$~ and $K$,\nsuch that the average density of photons within $\\varepsilon$~ is significantly larger\nthe the average density of the background photons, make rare the chance to grow\na cluster from a background fluctuation. Unfortunately, the $\\gamma$-ray~ sky shows\nstrong gradients of background, in particular at low galactic latitudes. To\nsolve this issue, one could think to adapt the value of $\\varepsilon$~ and $K$ according\nto a local value of the background photon density. Since $\\varepsilon$~ has a strong\nconstraint imposed by the PSF, one should tune mostly the value of $K$. The\ndrawback is that as we increase the value of $K$ to compensate for the\nbackground, we decrease the capability to detect cluster with small $N_p$. To\n{ overcome} this difficulty, we adopt an alternative solution. We use a unique pair\nof values of $\\varepsilon$~ and $K$, for each field, where $\\varepsilon$~ is mostly constrained by\nthe PSF, and $K$, by the field average background, ad we take into account the\nbackground inhomogeneities by defining a significance level of the cluster,\naccording to the signal to noise ratio \\citep{LiMa1983}, evaluated  from the\nlocal background. This is explained in detail in Section \\ref{sec:signif_sim}. The\ncapability to reject clusters according to a low significance level, allows to\nrelax the constrain on  $\\varepsilon$~ and $K$, increasing the number of clusters \ndetected, hence increasing the detection ratio, and at the same time allows to\nreject spurious sources, due to the significance threshold. \nAnyhow, to avoid that the background is { so high}, that the fluctuations \nin the background events, can lead to densities comparable to those\nof weak sources, it's recommended to apply a cut in energy, to make\nthis possibility rare. \nIn order   to optimize the ratio between background and clusters events, in the \nfollowing we use a threshold energy of 3 GeV, that mitigates the possible bias\ndue to the background fluctuations.\n\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{r68_rand.pdf}&\n\\includegraphics[width=9cm]{r68_campo1.pdf}\\\\\n\\end{tabular}\n\\caption{{\\it Panel a:} distribution of the values of $\\log_{10} r_{eff}$ for \nthe {\\it random} field case, for  the full parameter space (black line) and fit \nby means  of Gaussian distribution (blue line). {\\it Panel b:} \nthe same as in the top panel, for the case of $K=3$ and $\\varepsilon=0.3$ deg. \n{\\it Panel c:}  distribution of $\\log_{10} r_{eff} $ for the case of\nthe {\\it sky} test field 1, for {\\it fake} clusters (red solid line), \nand {\\it true} clusters (blue solid line, hatched histogram). \n{ the  dashed lines represent a Gaussian best fit.}\n}\n\\label{fig:r_68}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{r68_vs_Eps_stat_rand.pdf}&\n\\includegraphics[width=9cm]{r68_vs_Eps_stat.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\\it: Left panel:} the $r_{eff}\/\\varepsilon$ statistical distribution as a function of $\\varepsilon$, \nfor the {\\it random} field case. The blue solid circles represent the median, and the grey shaded \narea represents  the 1-$\\sigma$ confidence level region,  for each value of $\\varepsilon$. \n{\\it Right panel:} the same as in the bottom left panel, for the case of the {\\it sky} test\nfield 1. The red solid circles represent the median of the {\\it true} clusters case,  and\nthe grey area the 1-$\\sigma$ confidence level region. The  dashed line shows the \n1-$\\sigma$ confidence level region, for the case of the {\\it fake} clusters.\n}\n\\label{fig:r_68_vs_Eps}\n\\end{figure*}\n\n\t\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{Np_poiss_vs_lognorm.pdf}&\n\\includegraphics[width=9cm]{Np.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\\it Left panels:} the distribution of $N_p$ for the {\\it random} test field,\nfor the case of $K=2$, $\\varepsilon=0.20$ deg (panel a, red solid boxes). The empty blue bars\nline represent a Poissonian best fit. The panel b shows the case of\n $K=2$ $\\varepsilon=0.30$ deg (purple solid triangles). \n The panel c shows the case for the full $K$-$\\varepsilon$~ parameter space,\n the solid black line represent a  log-normal best fit.\n{\\it Right panels:} panel c shows the distribution of $r_{eff}^2$ (black solid line),\nand it's best fit by means of a log-normal distribution (red dashed line).\nPanel  d shows the $N_p$ distribution for the {\\it fake} clusters in the {\\it sky} test { field} 1 (red \nsolid circles),  and the blue empty bars a Poissonian best fit. Panel  e shows the $N_p$ distribution \nfor the {\\it fake} clusters in the {\\it sky} test { field} 1   (blue hatched histogram),  the log-normal best\nfit (red dashed line), and the Poissonian fit (solid black line.)\n}\n\\label{fig:Np}\n\\end{figure*}\n\n\\section{Statistical properties of the $\\gamma$-ray DBSCAN~ clusters}\n\\label{sec:stats_sim}\n\\subsection{The test fields}\nIn this section we study the statistical properties of the clusters,\nlooking for signatures that characterize random Poissonian fields,\nand fields with point-like sources. To accomplish this task we\ncompare results obtained for a test field with only noise ({\\it random} test\nfield), and the five test  fields with noise plus point-like sources ({\\it sky} test\nfields 1-5).\n\nAs {\\it sky } test fields we use the same fields used in the\n\\cite{MSTII}. Each of these five {\\it sky} fields covers a broad sky region, with\na galactic longitude extension of $80^\\circ<l<170^\\circ$, and a galactic latitude\nextension of $40^\\circ<b<65^\\circ$. The $\\gamma$-ray~ background has been simulated\nusing the standard {\\ttfamily gtobssim}\n\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/help\/gtobssim.\ntxt} tool, developed by the {\\it Fermi}-LAT~ collaboration, simulating both the Galactic\nand isotropic components for a 2-year long period, using a threshold energy of 3\nGeV, for a total amount of 9322 photons.  To this photon list  { we added} 70\nsimulated sources: for each source, the number of photons was chosen from a\nprobability distribution given by a power-law, with exponent 2, from a minimum\nvalue of 4 up to 40 photons, joined to a constant tail up to 240 photons. \n{ The number of the sources is similar to that reported in  the Fermi-LAT \nSecond Source Catalog  \\citep[2FGL hereafter]{2FGL}, in the same region of  the sky.} \nThe source events are spatially distributed with a bivariate Gaussian \nprobability density function (PDF) with $\\sigma^{sim}_x=\\sigma^{sim}_y = 0.2$ \ndeg., centered at the source location. \nFive simulated test fields have been generated, adding the simulated sources to\nthe diffuse background. The only difference in the five realizations is the\nsource location,  randomly chosen to have different brightness contrast between\nsources and the background.\nThe {\\it random} test field covers the same area of the {\\it sky} test fields,\nand a number of events equal to  the {\\it sky} test field-1 (background and\nsources), for a total amount of   11044 events.\n\nIn Fig. \\ref{fig:sim_detection} we show  the photon map\nfor the {\\it sky} test field 1, and the result of the $\\gamma$-ray DBSCAN~ detection\nfor $K=5$ and $\\varepsilon=0.17$ deg. \nWe detect 51 {\\it true} clusters, and only 2 {\\it fake} ones. \nA cluster is defined {\\it true}, { if the position of the simulated source  falls within \na circle centered  on the cluster centroid, with a radius equal to $2pos_{err}$. }\nWe call {\\it fake}, the remaining clusters. \nIn Fig. \\ref{fig:closeup}, we show a close-up of two\n{\\it true} clusters. The black ellipses correspond to the ellipses\nof the positional error, and the purple  and orange thick points represent the cluster\npoints, while the black thick dots represent the background.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{Np_vs_K_stat_rand.pdf}&\n\\includegraphics[width=9cm]{Np_vs_K_stat.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\\it Panel b:}  the $N_p$ statistical distribution as a function of $K$, for the {\\it  \nrandom}  field case. The blue solid circles represent the median, and the grey shaded area represents  \nthe 1-$\\sigma$ confidence level region around the median, for each value of $K$. The dashed black line represents the  \n$N_p=K+1$ law.\n{\\it Panel e:} same as in  panel b, for the {\\it sky} test field 1 case.\n{\\it Panel a:} number of detected clusters for the {\\it random} test field case (blue solid   \npoints), as a function of $K$, and best fit by means of a Poissonian survival  function  \n(red empty bars).\n{\\it Panel c:} number of detected cluster for the {\\it sky} test field 1 case (black solid  \npoints),  for the case of  {\\it fake} clusters, as a function of $K$, and best fit by means of a  \nPoissonian survival function (red empty bars).\n{\\it Panel d:} number of detected cluster for the {\\it sky} test field 1 case (red solid  \nboxes),  for the case of  {\\it true} clusters, as a function of $K$, and best fit by means of a  \nPoissonian survival  function (black empty boxes).\n}\n\\label{fig:Np_vs_K}\n\\end{figure*}\n\n\\subsection{Test strategy}\n\n\nWe want to investigate the statistical properties of the $\\gamma$-ray DBSCAN~\nclusters, in particular signatures that distinguish purely random fields\nfrom fields with point-like sources,  and their dependence on $K$ and $\\varepsilon$. \nTo investigate systematically a broad volume of the parameter space, \nwe use a parametric approach. We set the range of $\\varepsilon$~ in [0.1$\\div$0.50] deg. with a \nstep of 0.01 deg., and the range of $K$ in [2$\\div$15],  with a step of 1. The\ntotal amount of detection trials for each test field is 574. \nWe collect the statistics of the trials, and we investigate the distribution\nof  $r_{eff}$ and $N_p$, and their connection with $\\varepsilon$~ and $K$, respectively.\n\n\n\n\n\\subsection{Statistics of $r_{eff}$ and connection with $\\varepsilon$}\nWe start by investigating the distribution of the $\\log(r_{eff})$ values, in the\ncase of the  {\\it random}, and of the {\\it sky} test field-1. The distribution for the\ndetections collected over the full $K$-$\\varepsilon$~  parameter space (top left panel of\nFig. \\ref{fig:r_68}),  shows a symmetric shape \nwell fitted by a Gaussian distribution (log-normal w.r.t. $r_{eff}$), with\nthe mean value of $<\\log_{10}(r_{eff})>\\simeq -0.45$ (corresponding to\n$<r_{eff}>\\simeq 0.3$ deg) and a dispersion of $\\sigma_{\\log_{10}( r_{eff})}\\simeq$ 0.23.\n{\nThe log-normal distribution  provides a reasonable\ndescription of the empirical distributions also for individual pairs of ($K$,$\\varepsilon$) values.\nAn example is\ngiven in panel c of  Fig. \\ref{fig:r_68}, for the case $K=3$, $\\varepsilon=0.3$ deg.,\nwhere the best fit values are $<\\log_{10}(r_{eff})>\\simeq -0.51$, and $\\sigma_{\\log_{10}( r_{eff})}\\simeq$ 0.16.\nWe now investigate the empirical distribution of  $\\log_{10}(r_{eff})$ for \nfields with point-like sources. In the right panel of Fig. \\ref{fig:r_68},\nwe show the case of  the {\\it  sky}  test field-1. The distributions of $\\log_{10}(r_{eff})$ are \nstill described by a by a normal. In the case of {\\it fake} clusters (red dashed line), the \nbest fit values of the mean ($<\\log_{10}(r_{eff})>\\simeq -0.46$) and of the\ndispersion ($\\sigma_{\\log_{10} r_{eff}}\\simeq$ 0.24), are very similar to those found  in the\ncase of the {\\it random} test field. On the contrary, the {\\it true} cluster distribution (blue hatched histogram) \nis peaking around the value of $\\log_{10}(r_{eff})\\simeq-  0.67$ deg, corresponding \nto $r_{eff}\\simeq 0.21$. deg., very close to  the value of the \ndispersion $\\sigma^{sim}=0.20$ deg., used to simulate the sources. }\nSince the simulation parameter $\\sigma^{sim}$ reproduces the\neffect of the instrumental PSF, we observe that for non-{\\it random} fields, the\ntypical size of the reconstructed clusters is constrained by the PSF, suggesting\nthe empirical rule to set the value of $\\varepsilon$~ of the order of the PSF size.\n\n\nTo investigate more accurately the connection between $\\varepsilon$~ and the PSF, we analyse \nthe statistical properties of the quantity  $r_{eff}\/\\varepsilon$ as a function of $\\varepsilon$. \nFor each value of $\\varepsilon$, we determine the median, and the  two-sided 1-$\\sigma$ \nconfidence level (CL) interval around the median,  of the $r_{eff}\/\\varepsilon$ distributions.\nIn the left  panel of Fig. \\ref{fig:r_68_vs_Eps} we plot the $r_{eff}\/\\varepsilon$ \nmedian   (blue solid circles) and 1-$\\sigma$  CL region, as  a function of $\\varepsilon$,\nfor the {\\it random} field.  \nWe note that  the $r_{eff}\/\\varepsilon$ trend is slightly increasing with $\\varepsilon$, \nand that the 1-$\\sigma$ CL region is consistent with the case $r_{eff}\/\\varepsilon=1$,\nbut  the upper boundary shows a systematic increase, compared to the lower\nboundary, for $\\varepsilon$$\\gtrsim 0.30$ deg.\n{ The trend for the case of the {\\it true} clusters in {\\it sky} test field 1 \n(right panel Fig.\\ref{fig:r_68_vs_Eps}), shows a different behaviour. \nThe median of $r_{eff}\/\\varepsilon$ (red solid circles) is slightly decreasing with\n$\\varepsilon$, showing that, for {\\it true } clusters, $r_{eff}$ is not sensitive to the \nsize of $\\varepsilon$, being mostly constrained by the simulated PSF size. As expected,  for the case of \n{\\it fake} clusters (blue dashed line), the trend is almost identical to that \nof the clusters in the {\\it random} field.\n}\n \n\n \n\n\n\n\n\n\n\n\n\n\\subsection{Statistics of $N_p$ and connection with $K$}\nWe now investigate the statistics of the distribution of the number of \nphotons per cluster. In the case of {\\it random} fields, we expect  that\nthe number of photons in a cluster attends a Poisson distribution. Indeed,\nfor a generic two-dimensional Poisson process,  the probability to observe\na number of events ($N(S)=j$) enclosed by a surface $S$  is given by:\n\\begin{equation}\nP(N(S)=j)=\\frac{(\\lambda |S|)^j \\exp{(-\\lambda |S|)}}{j!},\n\\label{eq:Poisson}\n\\end{equation}\nwhere $\\lambda$ is the average spatial density.\nTranslating $S$ in terms of $\\varepsilon^2$, we can rewrite:\n\\begin{equation}\nP(N(\\varepsilon^2)=j)=\\frac{(\\lambda |\\varepsilon^2|)^j \\exp{(-\\lambda |\\varepsilon^2|)}}{j!},\n\\label{eq:Poisson_Eps}\n\\end{equation}\nfrom which follows that, given the value of $K$ and $\\varepsilon$, the probability\nto find a cluster  as function of $K$ and $\\varepsilon$~ will be given by\n\\begin{equation}\nP_{clus}(\\varepsilon,K)=P(N(\\varepsilon^2)>K)=1- \\sum\\limits_{j=0}^{K} \\frac{(\\lambda |\\varepsilon^2|)^j\\exp{(-\\lambda |\\varepsilon^2|)}}{j!},\n\\label{eq:P_clus}\n\\end{equation}\nnamely the Poissonian  survival function. \nAnyhow, due to  the logic of the  DBSCAN~ \nclustering process, the Poisson statistics can't be extended  from $\\varepsilon$~ \nto $r_{eff}$, for any value of $\\varepsilon$.\nIndeed, a cluster  is not a simple collection of points enclosed\nwithin a surface $S$, this holds only within the $\\varepsilon$-sized circle, namely the\n{\\it seed} of the cluster ($C^*$). If we consider the annulus defined  between $\\varepsilon$~ \nand the cluster radius  $r_{clus}$, not all the points in the annulus will be cluster\nmember, but only those that are at least density reachable. \nThis implies that we expect a deviation from the Poisson \nstatistics, when $r_{eff}$  is significantly larger than $\\varepsilon$, i.e. $\\varepsilon$ $\\gtrsim$ 0.3 deg. \n(according to the analysis presented in the previous section).  This expected deviation from the \nPoissonian statistics,  is confirmed by the plots in the left panels of Fig. \\ref{fig:Np}.\nIn panel a we show the distribution of $N_p$ for the\ncase $K=2$ and $\\varepsilon=0.20$ deg. We note that the Poisson distribution (Eq. \\ref{eq:Poisson_Eps}) \ngives a reasonable  description of the empirical distribution. On the contrary, for the case of\n$\\varepsilon=0.30$ deg. (panel b), we observe that the Poisson distribution shows larger deviations,\nin particular for $K>6$.  When we take into account the $N_p$ distribution for the full \nparameter space (panel c), we note the Possonian distribution is failing in  providing\na reasonable description of the empirical distribution, whilst a log-normal one gives a \ngood fit.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{param_run.png}\n\\caption{\nIsolevel maps for $D_{fake}$ (panel a),  $D_{true}$ (panel b), $D_{eff}$ (panel c), and $Q$ \n(panel d), for the {\\it sky} test field 1. The white lines show the\n$\\mathrm{isolevel}=0$, the black lines show the $\\mathrm{isolevel}=0.68$, and the blue lines \nshow the $\\mathrm{isolevel}=0.95$.\n}\n\\label{fig:param_run}\n\\end{figure*}\n\n\n\nThe log-normal trend of $N_p$ is consistent with the log-normal trend of the  \ndistribution of $r_{eff}$.  Since the number of photons in a cluster will be \napproximatively $N_p \\propto \\lambda r_{eff}^2$,\nwe can write the PDF of $N_p$:\n\n\\begin{equation}\nf(N_p)\\propto f(r_{eff}^2)\\lambda.\n\\label{eq:f_N_p}\n\\end{equation}\n\nTo evaluate the distribution of $ r_{eff}^2$ we can use the\nstandard theory of the transformation of Random Variables (RV) \\citep{papoulis}.\nIt can be easily proved that, given a RV $X$ having a log-normal distribution,\n\n\\begin{equation}\nf(X)=\\frac{1}{X\\sqrt{2\\pi\\sigma^2}}\\exp\\Big({\\frac{-\\ln(X)-\\mu}{2\\sigma^2}}\\Big), \n\\end{equation}\nthe RV $Y=X^2$, will follow a log-normal distribution given by:\n\n\\begin{equation}\nf(Y)=\\frac{1}{2Y\\sqrt{2\\pi\\sigma^2}}\\exp\\Big({\\frac{-\\ln(Y)-2\\mu}{4\\sigma^2}}\\Big). \n\\end{equation}\t\t\t\n\nIndeed, our $r_{eff}^2$ distribution, for the {\\it random} field (panel d, \nFig. \\ref{fig:Np}), is fitted by a log-normal distribution peaking at $\\simeq 0.03$ deg$^2$.\nHence, according to Eq. \\ref{eq:f_N_p} we expect that also $f(N_{p})$ will follow a log-normal\ndistribution, when $N_p$ is not ruled by a Poissonian statistics.\n\nWe verify, that the same statistical trends, describe the {\\it real} sky fields. \nThe  panels e and f in Fig. \\ref{fig:Np}, show \nthe statistical distribution of $N_p$ for the {\\it sky} test field 1 case. \nIn agreement with the analysis concerning the {\\it random} test field, \nwe see that the {\\it fake } clusters  ($\\varepsilon=0.30$ deg., panel e in \nFig. \\ref{fig:Np}), are described  by a Possonian statistic, whilst, the  {\\it true } \nclusters (panel f in Fig.  \\ref{fig:Np}), are  better described by a log-normal \ndistribution (red dashed line), compared to a Poissonian  one (solid black line). \nWe  also observe that the log-normal law,  describes reasonably \nthe empirical distribution, only for values of $N_p\\lesssim 50$, whilst shows \nsignificant deviation  in the tail, consistent with the statistics of our simulated sources    \npopulation.\n \n\n\nTo complete this statistical characterization, we investigate the distribution\nof the number of detected clusters, as a function of the threshold $K$.\nAccording to Eq. \\ref{eq:P_clus}, we expect that the number of detected cluster,\nfor a {\\it random} field, follows a Poisson survival distribution. The  plot a of \nFig. \\ref{fig:Np_vs_K} confirms our hypothesis, indeed\nthe Poisson  survival function provides a reasonable description of the \nempirical distribution. The same holds for the case of {\\it fake} clusters of\nthe {\\it sky} test field 1 (plot c Fig. \\ref{fig:Np_vs_K}). On the\ncontrary, in the case of {\\it true} clusters (panel d Fig.\n\\ref{fig:Np_vs_K}), the Poisson survival distribution is not able to reproduce\nthe observed trend, consistently with the non-poissonian statistic of the\nsimulated clusters . The panels b and e of Fig. \\ref{fig:Np_vs_K}, show the\n$1-\\sigma$ CL region for the $N_p$, as a function of $K$. We note, that both in the\ncase of {\\it random} and {\\it sky} field {\\it true} clusters, the lower boundary\nof the region is constrained by the equation $y=K+1$, that is consistent with\nthe $\\gamma$-ray DBSCAN~ logic.\nOn the contrary, the upper boundary shows a different\nbehaviour. In the case of the {\\it random} field, the upper boundary\ndeviates from the lower boundary compatibly with the fluctuations \nof the events around the $\\varepsilon$~ circle, and ranges from about 8 to about 16.\nOn the contrary, in the case of {\\it sky} field {\\it true} the upper boundary\nis constrained by the statistics of the number of events in the simulated sources,\nand rages from about 60 to 100.\n \n\n\n\n\n\n\n\\section{Testing  the detection performance with simulated $\\gamma$-ray~ data}\n\\label{sec:det_sim}\nIn this section we investigate the detection performance of the\n$\\gamma$-ray DBSCAN.\nAs first point, we study the dependency of the detection efficiency on $K$ and $\\varepsilon$, \nand their impact on the spurious ratio, and on the detection efficiency. \nThen, we investigate the capability of\nthe algorithm to reconstruct the simulated clusters, and the positional accuracy\nof the reconstructed centroids.  \nWe test the detection performance of the $\\gamma$-ray DBSCAN~ using as benchmark the \nfive {\\it sky} test fields used in the previous section, and exploring\nthe same parameter space.\n\n\n\n\\subsection{Detection efficiency and spurious ratio as a function of $K$ and \n$\\varepsilon$}\n\nTo investigate the detection performance of the $\\gamma$-ray DBSCAN,  we   run, \nfor each of the five {\\it sky} test fields, and  for each pair of values $K$,$\\varepsilon$,\na $\\gamma$-ray DBSCAN~ detection. For each detection run, we build a  {\\it cluster catalog}.\nStarting from the {\\it cluster catalog}, we build the corresponding {\\it candidate catalog}. \nThe {\\it candidate catalog} is a list of  sources built by  taking into  \naccount two possible biases, the {\\it confusion}, and the {\\it multiple association}, \nin detail:\n\\begin{itemize}\n\\item{} a cluster is defined {\\it true}, i.e. with a possible counterpart, \n{    if the position of the simulated source \n falls within a circle centered  on the cluster centroid, with a radius equal to \n $2 pos_{err}$. }\n\\item{} Two or more {\\it true} clusters are defined {\\it confused}, if they have the same counterpart\n\\item{} A {\\it true} cluster has a  {\\it multiple association}, if has more than one counterpart.\n\\end{itemize}\nWe stress that, the number of confused clusters is negligible, \n{ indeed the average number of \n{\\it confused} clusters per run is about 0.08, and no  {\\it confused} clusters \nare found for $K>4$, and that  the average number of  {\\it multiple associations}\nper run is about 0.2.\n}\n\n\n\n\nThe final {\\it candidate catalog} will count a number  of candidate sources $N_{src}$,  \neach identified by a unique $SRC_{ID}$.  The  number of spurious sources  will be \n$N_{fake}=N_{src}-N_{true}$. In order to to characterize the performance, we\ndefine the following parameters:\n\n\\begin{itemize}\n\\item{} the detection efficiency:\n\\begin{equation}\nD_{eff}= \n\\left\\{\n\\begin{array}{c l}\n\\frac{N_{true}-N_{fake}}{N_{sim}(N_p sim.>K)}, &\\mathrm{if~}  (N_{true}-N_{fake}) \\leq N_{sim}(N_p sim.>K) \\\\\n1.0                                          , &\\mathrm{if~}  (N_{true}-N_{fake})>N_{sim}(N_p sim.>K) \\end{array} \n\\right.\n\\label{eq:Deff}\n\\end{equation}\nwhere $N_{sim}(N_p sim.>K)$ is the number of simulated sources\nwith a number of simulated events larger than $K$\n\\item{} the true detection ratio  $D_{true}=N_{true}\/N_{src}$\n\\item{} the spurious detection ratio  $D_{fake}=N_{fake}\/N_{src}$\n\\item{} the overall detection quality factor ($Q$), that takes into account \nthe tradeoff between $D_{eff}$  and $D_{fake}$, defined as:\n\\begin{equation}\nQ=D_{eff}\\Big(1-\\frac{N_{fake}}{N_{src}}\\Big)\n\\label{eq:Qeff}\n\\end{equation}\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{position_recon_vs_Np.pdf}&\n\\includegraphics[width=9cm]{position_recon.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\n{\\it Panel a:} red solid boxes show the mean positional error of the centroid, for\n{\\it true} clusters in {\\it sky} test field 1, and the standard deviation (vertical error bar), vs. $N_p$.  \nThe clusters are  binned in $N_p$, with the bin width indicated by the horizontal error bar. \nThe black solid circles represent the corresponding trend for the distance between the cluster \ncentroid and the simulated source position.\n} \n{\\it Panel b:} the distribution of the distance between the simulated source position and the \ncluster centroid, expressed in arcsec, for the case of $\\varepsilon=0.10$ deg. (black line), \n$\\varepsilon=0.15$ deg. (blue line), and $\\varepsilon=0.20$ deg. (red lines). {\\it Panel c:} the cumulative \ndistributions  corresponding to panel b.\n}\n\\label{fig:Pos_prec}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{Np_sym_vs_detected.pdf}\n\\caption{\n{\n{\\it Top panel:} the average number  of photons associated to each clusters \n$N_p$, and their dispersion (vertical bar) vs. the number of photons  \nsimulated  ($N_p$ sim). The red points refer to\nthe sub parameter space $\\varepsilon=0.15$ deg., and the solid blue  circles to the\n$\\varepsilon=0.20$ deg. sub space. The solid green lines represent the\nlaw  $N_p=N_p$ sim. The dashed lines represent the law $N_p=N_p$ sim. $\\pm$ 10.\n{\\it Bottom panel:} The corresponding fractional deviation $(N_p-N_p sim.)\/N_p sim$.\n}\n}\n\\label{fig:Np_rec}\n\\end{figure}\n\n\n\n{ The $D_{eff}$ parameter shows the fraction of simulated clusters, \nabove the threshold $N_p sim=K$, detected by the method,\nnet of the {\\it fake} ones. Hence, does not provide an indication\nof the spurious contamination. For this reason we have introduced\nthe $Q$ parameter, which rescale the $D_{eff}$ according to the ratio \nbetween {\\it fake} clusters, and found clusters $N_{src}$.}\nWe remind that, according to the $D_{eff}$ definition in Eq. \\ref{eq:Deff},\nit's possible to obtain values of $D_{eff}>1.0$. Assume to have a simulated\ncluster such that, for a given $K$ and $\\varepsilon$,  the corresponding seed cluster  \nhas a size $N^*=N_p sim.=K$. In the case of no background events within\nthe circle of radius $\\varepsilon$, this cluster  will be rejected. \nIf we have one or more background events contained within the circle\t\nof radius $\\varepsilon$, i.e.  $N^*>K$, the cluster will be detected.\nFor this reason, in such a case, we report a value of $D_{eff}=1.0$. \nThe same applies to $Q$.\n\nIn Fig.  \\ref{fig:param_run} we summarize the detection runs for the \ncase of {\\it sky} test field 1, for the full parameters space with \n$K>2$. The  panel a shows the isolevel map of the {\\it fake} \nclusters detection ratio. The gradient in the isolevel map is quite \nsharp, and  roughly half of the parameter space shows no {\\it \nfake} clusters (white isolevel line). To have a better understanding of \nthe impact of {\\it fake} clusters, it's interesting to compare the \n$D_{fake}$ isolevel map  to the $D_{true}$ isolevel map (panel b \nFig. \\ref{fig:param_run}). Also in this case the map shows a \nsharp  gradient, and the region with $D_{true}>0.95$ overlaps the \n$D_{fake}=0$ region. These two maps, clearly show the region\nof the parameter space where the algorithm has the best performance,\nbut the $D_{true}$ and $D_{fake}$ ratios do not provide information on the { ratio} \nbetween the number of {\\it true} detected clusters and the number\nof simulated clusters. At this regard more information are provided\nby the  $D_{eff}$ isolevel map (panel c, Fig. \\ref{fig:param_run}). To focus on the \n\"effective\" volume of the parameter space,  we hide by a white area \nthe region where $D_{eff}<0$. \nWe note that the isolevel lines $D_{eff}=0$ \nand the isomap lines in the maximum gradient area show a positive \ncorrelation between  $K$ and $\\varepsilon$, meaning that  an increased  value of $\\varepsilon$~, \nrequires an increased value of $K$, to have better background rejection. \nTo evaluate better the trade-off \nbetween $D_{true}$ and $D_{fake}$, we plot in the panel d  \nof Fig. \\ref{fig:param_run}, the isolevel map of $Q$. This plot \nshows that the area corresponding to $Q>0.95$, is consistent with that found in the \ncase of $D_{eff}$.\nIn  Tab. 1 we report the $D_{eff}$ values obtained for all the five \n{\\it  sky} fields, for  detections with a number  of {\\it fake}  sources \n$\\leq$ 6.  We note that the average values of {\\it true} clusters ranges\nbetween 44 and 51, with the {\\it fake} ones ranging between 1 and 3, \nand an average  $D_{eff}$ between 0.96 and 1.0.\nThis is a very promising result.  \n \n\n\n\n\n\n\n\n\n\n\\subsection{Cluster reconstruction, and positional accuracy}\nThe positional accuracy of the topometric methods, is probably\nthe most important feature of this class of algorithms.\nIn Sec. \\ref{sec:dbscan}, we have described our weighting \nmethod to reconstruct the centroid of the cluster.\n\n\n\n{\nThe panel a of Fig. \\ref{fig:Pos_prec} shows by red solid boxes\nthe mean positional error of the clusters centroid and the standard \ndeviation (vertical error bar) vs. $N_p$, for the {\\it true} clusters \nof the {\\it sky} test field 1 with  $\\varepsilon \\leq 30$ deg.\nThe clusters are binned in $N_p$, with the bin width indicated by the horizontal \nerror bar. \t\nAs expected, the uncertainty on the reconstructed cluster centroid \nis  $pos_{err} \\approx \\sigma_{sim}\/\\sqrt N_p$ (solid red line).\nThe solid black circles represent the corresponding trend \nfor the separation between the simulated cluster position\nand the reconstructed cluster centroid. For $N_p \\gtrsim 30$,\nthe separation is below $2'$. \nIn  the panel b   of Fig. \\ref{fig:Pos_prec} we plot the\nhistogram of the distribution of the angular separation \nbetween the position of the simulated source, and the\nposition of the cluster centroid. \nFor the three cases of $\\varepsilon=0.10$ deg., $\\varepsilon=0.15$ deg.,  \nand  $\\varepsilon=0.20$ deg., the positional error is below the $1.5'$, for \nthe $68\\%$ of the sample. \n}\n\n\\begin{table}\n\\input{spur_Cut_det_stat_table.tex}\n\\label{tab:tab1}\n\\caption{\nSummary of the\tdetections obtained for all the five {\\it sky} fields, \nfor detections with a number of fake sources $\\leq 6$. $N_{sim}^{cut}>K$, is is the number of \nsimulated sources with a number of simulated events larger than $K$, the number separated by the \n$\/$ symbol, indicates the full number of simulated sources.\n}\n\\end{table}\n\nBesides positional accuracy, is also important to understand\nthe capability of the $\\gamma$-ray DBSCAN~ to reconstruct the simulated\ncluster in terms of number of photons. Indeed, this information\ngives an idea of the average number of background photons\ncontaminating the reconstructed cluster. In the top left panel \nof Fig. \\ref{fig:Np_rec}, we show the scatter plot\nof $N_p$ vs. the number of simulated events ($N_p$ sim.). \nThe solid points represent the average value of $N_p$,\nfor a given value of  $N_p$ sim., and the error bar, corresponds \nto the standard deviation.\nThe solid green line represents the case $N_p=N_p$ sim.,\nand the dashed upper and lower lines, represent $N_p=N_p\\pm 10$ sim.,\nrespectively. \nBoth for the cases of $\\varepsilon=0.15$ deg.,  and  $\\varepsilon=0.20$ deg.,\nthe scatter is bounded by the dashed lines, showing that the largest\nexcess in the $N_p$ is about 10 photons, independently of $N_p$ sim.\nWe note, that in the case of $\\varepsilon=0.15$ deg., the number of reconstructed\nphotons, systematically underestimates the simulated number, whilst, \nthe $\\varepsilon=0.20$ deg. case does not shows this bias.\nIt's possible to appreciate better this effect, in the  bottom left panel   \nof Fig. \\ref{fig:Np_rec}, where we show\nthe fractional reconstruction error ($N_p-N_p$ sim.)\/$N_p$ sim., vs. \n$N_p$ sim. The solid green line represent the case with 0 error, \nand the dashed lines represent the $\\pm 20\\%$ boundaries. \nThe bias on $N_p$ in the case of $\\varepsilon=0.15$ deg., shows again\nthe strong correlation between $\\varepsilon$~ and the PSF radius. When $\\varepsilon$~ is\nsmaller then the $\\sigma^{sim}$ (that in our simulations reproduces\nthe PSF effect), the number of reconstructed events $N_p$ is systematically\nsmaller than $N_p$ sim., on the contrary, when the $\\varepsilon$~ radius matches the PSF\nradius size ($\\varepsilon=0.20$ deg.), the bias disappears.\n \n\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9.0cm]{sig_distr_fake.pdf}&\n\\includegraphics[width=8.5cm]{param_run_sigcut4.png}\\\\\n\\end{tabular}\n\\caption{\n{\\it Left panel:} The distribution (blue line) of the square of the significance, for \nthe {\\it fake} clusters in the {\\it sky} test field 1, for the full $K$,$\\varepsilon$~ parameter space, \ncompared to a $\\chi^2$ distribution with one degree of freedom.\n{\\it Right panel:} the  spurious  ratio\n$D_{fake}$ for $S_{cls}>4.0$, the white line shows the isolevel $D_{fake}=0.0$. \n}\n\\label{fig:signif}\n\\end{figure*}\n\n\n\n\\section{Cluster significance, background inhomogeneities, and  rejection of spurious clusters}\n\\label{sec:signif_sim}\n\nEven though, we have identified the region of the $K$-$\\varepsilon$~ parameter space,  \nwhere the detection efficiency is larger,  and the probability to detect {\\it fake} \ncluster is lower, in the application to real data, it's mandatory to \nprovide a significance level, expressing the probability of  a \ncluster being not originated in a background fluctuation.\nWe propose a method derived   from the \\cite{LiMa1983} approach, \nbased on the evaluation of the  signal to noise (S\/N) ratio. \nA significance method based on the S\/N ratio fits well the the  \n$\\gamma$-ray DBSCAN~ implementation, because the algorithm directly provides \na partition of the photon list in {\\it cluster} and {\\it noise} \nevents. Hence, for each  cluster we can evaluate easily the S\/N \nratio, knowing the exact nature of each event. The procedure to \nevaluate the significance is summarized by the following items:\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{MST_region_clean_3GeV_Cmap_crop.pdf}\n\\caption{Aitoff projection of the Fermi sky region. The purple boxes represent the $\\gamma$-ray DBSCAN~\n sources ($K=8,\\varepsilon=0.21$ deg.). The green crosses are the 2FGL sources with $TS>16$, the\n red one those with $TS\\leq16$. There are no {\\it fake} sources, and the $\\gamma$-ray DBSCAN~ finds all the\n sources with $TS>16$, except only one, enclosed by the red circle, and with the center positioned \n at the edge of the field.}\n \\label{fig:Fermi_Cmap}\n\\end{figure*}\n\n\\begin{enumerate}\n\\item{} for each cluster, we define an annular region,  with \nan inner radius  $r_{in}$, and an external radius  $r_{out}$.\n\\item{} $r_{in}$ is set to an initial value of $r_{in}=2 r_{eff}$, and\n is adaptively increased with a step of $r_{in}\/10$, for a\nmaximum of 10 trials, until at least the $95\\%$ of the cluster events are enclosed within $r_{in}$.\n\\item{} $r_{out}$ is set to $3 r_{in}$.\n\\item{} We count all the cluster events $N^{in}_{src}$  and all the background events   \n$N^{in}_{bkg}$, enclosed  within the circle with  radius $r_{in}$ and centered on the cluster  \ncentroid.  \n\\item{} We determine the $N^{out}_{bkg}$ background level, rescaling the number of \nbackground events in  $r_{in}<r<r_{out}$, to a circle with  radius $r_{in}$.  \n\\item{} To evaluate possible gradients in the background, we select a region enough far from the \ncluster to sample properly the background level, and enough close to the cluster, to measure \na \"local\" background level. At this regard, we define the radius $r_{out}^{ave}=(r_{out}+r_{in})\/2$, \nand we evaluate the average background  level ($N^{local}_{bkg}$)  in a circle of radius $r_{in}$, centered on each point in  $r_{out}^{ave}<r<r_{out}$.\n\\item{} If no background points are { found} in  $r_{out}^{ave}<r<r_{out}$, we set \n$N^{local}_{bkg}=N^{out}_{bkg}$.  \n\\item {} By\tcomparing\t$N^{local}_{bkg}$ to $N^{in}_{bkg}$,\twe\tevaluate the\tfraction\nof {\\it noise} already resolved by the $\\gamma$-ray DBSCAN, and we evaluate the effective background level \n$N^{eff}_{bkg}$, by correcting $N^{local}_{bkg}$ for $N^{in}_{bkg}$.\n\\item{}  we evaluate the significance according to the Likelihood Ratio Test (LRT)\nmethod proposed by \\cite{LiMa1983}:\n\n\\begin{equation}\nS_{cls}=\\sqrt{2 \\Big  ( N^{in}_{src} \\ln \\Big[ \\frac{2 N^{in}_{src}}{N^{in}_{src}+N^{eff}_{bkg}} \\Big]+\nN^{eff}_{bkg} \\ln \\Big[ \\frac{2N^{in}_{src}}{ N^{in}_{src}+N^{eff}_{bkg}} \\Big ]  \\Big  )}\n\\label{eq:signif}\n\\end{equation}\n\n\\end{enumerate}\n\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=9cm]{position_err_cluster_vs_pos_err_2FGL.pdf}\n\n\\caption{\n{\nThe scatter plot the positional error of the $\\gamma$-ray DBSCAN~ clusters vs. the \npositional error of the corresponding associated 2FGL$_{TS>16}$  sources. \nFor each  2FGL$_{TS>16}$  source associated to one or more $\\gamma$-ray DBSCAN~ clusters, \nwe plot the error on the position  of the reconstructed cluster centroid  and \nits standard deviation (represented by the error bar). \nThe dashed red line represents a linear best fit with a slope\nof $\\simeq$ 0.99, and an intercept of $\\simeq$ 9.53. \n}\n}\n\\label{fig:Pos_rec_RealSky}\n\\end{figure}\n\n\n\nUnder the hypothesis that a cluster is due to a background fluctuation,\nthe variable $S_{cls}^2$, is expected to follow a chi square distribution,\nwith one degree of freedom ($\\chi(1)^2$).  \nIn the left panel of Fig.  \\ref{fig:signif}, we plot the  distribution of $S_{cls}^2$,\nfor the fake clusters in the {\\it sky} test field 1  (blue histogram), \ncompared to a  $\\chi(1)^2$ distribution. \nThe empirical distribution, is well described by the expected $\\chi(1)^2$ distribution, proofing \nthat the value of $S_{cls}$, can be used  as  the \"significance\" of the detected cluster. \nA very illustrative example of the power of $S_{cls}$ in rejecting {\\it fake} clusters,  is  \ngiven by the plot in the right panel in Fig \\ref{fig:signif}, where we plot the\n$D_{fake}$ ratio isolevel map, applying the selection  $S_{cls}>4.0$. { The {\\it fake} ratio is 0 \nfor the parameter space with $\\varepsilon\\lesssim 0.25$ deg.  For 0.25 deg. $\\lesssim \\varepsilon \\lesssim$ 0.35 deg.,  \nthere are { fluctuations} showing  $D_{fake}\\lesssim 0.05$. Only for $\\varepsilon$$\\gtrsim 0.40$ deg. and \n$K\\lesssim 8$, the  {\\it fake} ratio shows a significant increase, but we stress that in this \nregion of the parameter  space, $\\varepsilon$~ is more then double of the PSF size, hence \nthis is a region of the parameter space that should not be used in the detection with real data.\n}\n\n\n\n\n\\section{Application to real {\\it Fermi}-LAT~ data }\n\\label{sec:realdata}\n\n\n\nThe last step in our investigation of the $\\gamma$-ray DBSCAN, is the application\nto real {\\it Fermi}-LAT~ $\\gamma$-ray~ data. We select the same region of the sky\nused for the simulated test { field} ( $80^\\circ<l<170^\\circ$, and $40^\\circ<b<65^\\circ$),\nand we extract all the photons with energy $E>3$ GeV. The photons\nare collected for the same time span of the 2FGL\ncatalog . We repeat the detection \ntest performed in the case of simulated data (see Sec. \\ref{sec:det_sim} and  \nSec. \\ref{sec:signif_sim}), restricting the parameter space to $2\\leq K \\leq10$,  \nand $0.10\\leq \\varepsilon \\leq0.30$ deg.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{l} \n\\includegraphics[width=17cm]{Fermi_Sky_det_2FGL_SigCut4.png}\\\\\n\\includegraphics[width=17cm]{Fermi_Sky_det_SigCut2_2FGL_SigCut4.png}\\\\\n\\includegraphics[width=17cm]{Fermi_Sky_det_SigCut4_2FGL_SigCut4.png}\n\\end{tabular}\n\\caption{\n$D_{fake}$ (left panels) and $D_{eff}$ (right panels), for the\nreal sky detections, using the 2FGL$_{TS>16}$ catalog.\n{\\it Panels a,b:} no cut on $S_{cls}$ applied.\n{\\it Panels c,d:} $S_{cls}>2.0$\n{\\it Panels e,f:} $S_{cls}>4.0$ \n}\n\\label{fig:Fermi_Det}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{sig_vs_TS.pdf}&\n\\includegraphics[width=9cm]{sig_distr_fake_Fermi.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\\it Left panel: }scatter plot of $S_{cls}$ vs. $\\sqrt{TS}$. For \neach source in our 2FGL$_{TS>16}$ list, associated to one or more $\\gamma$-ray DBSCAN~ cluster, we plot the  $\\sqrt{TS}$ in the 3-10 GeV band, \nv.s. the average values of $S_{cls}$ and its standard deviation\n(represented by the error bar).\n{\\it Right panel: }the distribution (blue line) of the square of the significance, for \nthe {\\it fake} clusters in the {\\it Fermi}-LAT~ real sky, for the full $K$,$\\varepsilon$~ parameter space, \ncompared to a $\\chi^2$ distribution with one degree of freedom.\n}\n\\label{fig:sig_vs_true_rate_RealSky}\n\\end{figure*}\n\n\nTo properly understand the detection performance, we need\nto take into account that the 2FGL catalog has been built using\nphotons with an energy threshold of 100 MeV, whilst we use\na value of 3 GeV. A possibility is to select sources with\na reported flux larger than zero,  \nin  the 3-10 GeV band flux column of the 2FGL.\nThis flux-based selection, is not the best way to \nstudy the detection performance of the $\\gamma$-ray DBSCAN~, indeed\nthe flux does not contain a unambiguous relation with\nthe significance of the detection, for that energy threshold. \nA more reliable  criterion is to select the sources according \nto the significance  reported in the 2FGL. The 2FGL detection \nsignificance is given by the $\\sqrt{TS}$.  The $TS$ is the test \nstatistic  defined as  $TS=2(\\log L(\\mathrm{source}) - \\log L(\\mathrm{no ~ source}) )$,\nwhere $L$ is the likelihood of the data given the model with or without \na source present at a given position on the sky \\citep{2FGL}.\nWe apply a selection according to $\\sqrt{TS}>4$, and we refer\nto the corresponding source list (counting 35 sources) as\n2FGL$_{TS>16}$. \n\n\n\n\nAn example of the application of the $\\gamma$-ray DBSCAN~ to real {\\it Fermi}-LAT~ \ndata is given in Fig. \\ref{fig:Fermi_Cmap}, where we report an Aitoff \nprojection in galactic coordinates of the analysed $\\gamma$-ray~ sky region. The red\ncrosses represent the 2FGL sources with $TS<16$ in the 3-10 GeV band, and the \ngreen crosses represents those with $TS\\geq 16$. The purple boxes\nrepresent the $\\gamma$-ray DBSCAN~ sources found for $K=8,~\\varepsilon=0.21$ deg. \nFor this choice of parameters, we find no {\\it fake} sources,  and we find all \nthe sources with with $TS>16$, except only one, enclosed by the red circle, and \npositioned at the edge of the sky region, with a galactic latitude $l=64.85$ deg.\nIn  Tab. 2 we summarize the detection performance, for  detections with a number  \nof {\\it fake}  sources $\\leq$ 4.  We note that  values of {\\it true} clusters ranges\nbetween 35 and 34, out of the 35 present in the 2FGL$_{TS>16}$. \nThe {\\it fake} ones range between { 1} and 4, and we obtain an average detection\nefficiency of  $D_{eff}= 0.94$. \n\n\nIn  Fig. \\ref{fig:Pos_rec_RealSky} we compare the localization \nperformance of the $\\gamma$-ray DBSCAN~ algorithm  with that returned by the \nlikelihood analysis implemented in the Fermi Science Tools. \nFor each source in our 2FGL$_{TS>16}$ list, \nassociated to one or more $\\gamma$-ray DBSCAN~ clusters, we plot the  the error on the position \nof the reconstructed cluster centroid  and its standard deviation (represented by the error bar) \nvs. the 95$\\%$ positional  uncertainty reported in the 2FGL. We evaluate the\n2FGL  95$\\%$ positional  uncertainty as $\\sqrt{\\sigma_{95,min}~\\sigma_{95,max}}$,\nwhere $\\sigma_{95,min}$ and $\\sigma_{95,max}$, are the semimajor and semiminor axes of the \n95$\\%$ confidence source location region, respectively.\nThe dashed red line represents a linear best fit, with a slope\nof $\\simeq$ 0.99, and an intercept of $\\simeq$ 9.53, showing that \nthe error on the position of the reconstructed cluster centroid, performed with a threshold of 3 GeV, is \nof the same order of the $95\\%$ positional uncertainty reported in the 2FGL catalog, performed \nabove 100 MeV.\n\n\nTo test the reliability of  the significance $S_{cls}$ to reject spurious sources, \n in Fig. \\ref{fig:Fermi_Det}  we plot the $D_{fake}$ and $D_{eff}$, based\non the 2FGL$_{TS>16}$ catalog. The panels a and b,  correspond to the case of\nno selection on $S_{cls}$. Both the $D_{fake}$  and the $D_{eff}$ trends are \nvery similar to the case of the simulated sky.\nIf we apply a significance cut of $S_{ls}>2.0$ (panels c,d), we observe that\nthe number of spurious ratio is $D_{fake}\\leq$ 0.05 for  almost half of the parameter space\n(region to the right of the purple line). \n The more severe cut of $S_{cls}>4.0$\n(panels d,e), removes  all the {\\it fake} clusters except two, { for \n$\\varepsilon \\lesssim$ 0.15 deg. Only for  $\\varepsilon \\gtrsim$ 0.25 deg., the $D_{fake}$ \nratio shows a significant increase, ranging from 0.05  up to $\\simeq$ 0.1.  \nIn agreement with our analysis on simulated data, the region of the parameter space \nwhere $\\varepsilon$~ is comparable to the PSF size, gives the better performance.\n} \n\nTo have a further confirmation about the robustness of our significance,\nwe plot in the right panel of Fig. \\ref{fig:sig_vs_true_rate_RealSky},\n$S_{cls}$ vs. $\\sqrt{TS}$. For each source in our 2FGL$_{TS>16}$ list, \nassociated to one or more $\\gamma$-ray DBSCAN~ cluster, we plot the  $\\sqrt{TS}$ \nin the 3-10 GeV band, \nv.s. the average value of $S_{cls}$ and its standard deviation\n(represented by the error bar). The average value of $S_{cls}$  and its\nstandard deviation  are evaluated from  the list of all the cluster associated\nto  the same  2FGL source. The solid blue boxes represent the full $K$,$\\varepsilon$~\nparameter space case, and the red solid circles represent the $\\varepsilon=0.10$ deg. case.\nThe dashed black line represents a linear best fit. \nThe slope of the linear fit is $\\simeq 0.5$. The strong correlation in\nthe scatter plots ($r\\simeq 0.98$, for both data sets), proves that  our\nsignificance implementation is consistent with the $\\sqrt{TS}$  reported\nin the 2FGL, and the slope of the linear fit suggests that $S_{cls}\\simeq 0.5 \\sqrt{TS}$. \n\n\n\n\n\n\n\n\n\n\n\n\\begin{table}\n\\input{spur_Cut_det_stat_realsky_table.tex}\n\\label{tab:tab2}\n\\caption{Summary of the detection performance for the\nreal {\\it Fermi}-LAT~  field, for detections with a number of fake sources $\\leq 4$.\n}\n\\end{table}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nFor the first time, we have used the DBSCAN~ for the detection of sources \nin $\\gamma$-ray~  astrophysical images. We have implemented a new version of\nthe  DBSCAN, the $\\gamma$-ray DBSCAN, that is optimized for the application to $\\gamma$-ray~   \nastrophysical images, with relevant background noise. Our $\\gamma$-ray DBSCAN, presents\nthe novelty of recursive call of the DBSCAN~ algorithm, that allows an\nexcellent reconstruction of the cluster, with an effective background\nrejection.\nWe have tested the algorithm with a sample of simulated $\\gamma$-ray~{\\it Fermi}-LAT~ fields,\nto give a statistical characterization of the method, and to benchmark\nthe detection performance. The results, with the simulated $\\gamma$-ray~ data,\nare summarized by the following items:\n\\begin{itemize}\n\\item{} The radius of $\\gamma$-ray DBSCAN~ scanning brush $\\varepsilon$~, has a strong \ncorrelation with the instrumental PSF radius. We find that \nthe typical size of the reconstructed {\\it true} cluster is of the\norder of the simulated  PSF size  $\\sigma^{sim}$,\n{ and that the precision of the reconstructed\ncentroid is of the order of  $\\sigma^{sim}\/\\sqrt{N_p}$. \n}\n\\item{} The number of reconstructed events $N_p$ is ruled by the \nPoissonian statistics in the {\\it random } fields, and for\nthe {\\it fake} clusters. On the contrary, for {\\it true} clusters,\nthe statistics of $N_p$, is ruled by that of the simulated sources.\n\\item{} The fractional error  on the reconstructed events number is \nof the order of $20\\%$ for  $N_p sim. \\lesssim 50$, and is negligible\nfor larger values, with best performance obtained when \n$\\varepsilon$$\\simeq \\sigma^{sim}$.\n\\item{} We have investigated the detection performance, for \na wide range of the $K$,$\\varepsilon$~ parameter space, and we have\nidentified the region with the best performance in terms\nof detection efficiency, and spurious ratio.\n\\item{} We have implemented an algorithm for the estimate \nof the Signal to Noise (S\/N) ratio, able to deal with local\nbackground inhomogeneities and nearby sources contamination, \nand we have successfully used the S\/N estimate to determine \nthe significance of the clusters, using the definition in \\cite{LiMa1983}.\n\\item{} Our cluster significance, $S_{cls}$, for random clusters,\nfollows the $\\chi(1)^2$ statistics, and can be used to\nreject spurious sources. The chance to find spurious\nsources for $S_{cls}>4$, is negligible. This means, that\nour $S_{cls}$ is a robust a reliable tool to reject spurious sources,\nand that $\\chi(1)^2$ statistics can be used to evaluate the probability of\na cluster to be spurious.\n\\end{itemize}\n\nWe have successfully applied the $\\gamma$-ray DBSCAN~ to real {\\it Fermi}-LAT~ data.\nWe have found an excellent agreement with  results from the simulated fields.\nWe tested our detection performance using as catalog, the  2FGL\nsourced with a $\\sqrt(TS)>4$ cut.\nThe results, with the real {\\it Fermi}-LAT~ $\\gamma$-ray~ data,\nare summarized by the following items:\n\\begin{itemize}\n{\n\\item{} the error on the position of the reconstructed cluster centroid, \nperformed with a threshold of 3 GeV, is of the same order of the 95$\\%$ \npositional uncertainty reported in the 2FGL,  performed above 100 MeV. \n}\n\\item{} We tested the $\\gamma$-ray DBSCAN~ significance, finding that it is\nstrongly correlated with the $TS$ provided in the 2FGL.\nThe significance cut, allows to remove safely spurious \nclusters.\n\\item{} The detection efficiency with real data is excellent,\nwe are able to find all the 35 sources with $\\sqrt(TS)>4$.\n\\item{}  When working with $\\varepsilon$~ of the order of the instrumental\nPSF size, we obtain the best performance, in terms of spurious \nrejection, and detection efficiency \n\\end{itemize}\n\nIn general, we find that the $\\gamma$-ray DBSCAN~ is a very powerful detection method\nto find clusters in $\\gamma$-ray~ images, corresponding to real sources.\nIt has the great advantage to deal self-consistently  with\ngradient in the background, providing an effective rejection of spurious\nclusters. Our implementation of the detection significance, in addition\nto the algorithm to evaluate local fluctuations in the background,\nallows to apply statistically significant selection, making even\nmore effective the rejection of spurious sources.\n\nIn a companion paper \\citep{dbscan_inprep}, we will a apply the method\nto the {\\it Fermi}-LAT~ sky, showing the potentiality for the discovery of new sources,\nin particular of small clusters located at high galactic latitude, or\ncluster on the galactic plane, affected by a strong background.\n{\nWe will also investigate how to plug the energy dependence\nof the PSF into the $\\gamma$-ray DBSCAN~ algorithm, and how to improve\nthe detection performance taking into account other \n{\\it Fermi}-LAT~ calibration properties.\n}\n\nWe  remark that, since the $\\gamma$-ray DBSCAN~ provides also density maps,\nit can potentially be used in the detection of large  scale structures \nin the galactic $\\gamma$-ray~ background, providing patterns to compare to the\ninterstellar gas distribution.  \nWe also stress, that the application of this method are not limited \nto $\\gamma$-ray~ images, but can be potentially used for any application\nrelated to the detection of spatial, and\/or spatio\/temporal clusters.\n \n\n\n\n\n\n\n\\begin{acknowledgements}\nWe are grateful to Enrico Massaro, Riccardo Camapana, and Enrico Bernieri, \nfor helpful comments, and for providing us the simulated test fields.\nWe are grateful to Gino Tosti, for helpful comments.\nWe thank the anonymous referee for providing us with constructive comments and useful suggestions.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa} %\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe remarkable similarity between the evolution of the star formation\nhistory of galaxies and the emissivity of AGN from $z\\sim1$ to the\npresent \\citep[e.g., ][]{bo98,fr99,me04,sh07,si08b} suggests a common\nmechanism that regulates their growth.  Major mergers of galaxies\n\\citep{mi96} and secular evolution \\citep{ko04,hop06} can both\npotentially funnel gas to the nuclear region that can then power\nconcurrent star formation and accretion onto supermassive black holes.\nFully consistent with this scheme, young stellar populations are known\nto be prevalent within the bulges of nearby Seyfert galaxies\n\\citep[e.g.,][]{te90,go01,wh03} and luminous quasars\n\\citep[]{ja04a,let07}.  Enhanced levels of ongoing star formation are\nalso evident in nearby Seyfert 2s \\citep{gu06} and the hosts of some\nmid-infrared selected quasars \\citep{la07} although are not high\nenough to put them in the same class as the Ultraluminous Infrared\nGalaxies \\citep{st06}.  On the other hand, the lack of star formation\nbased on [OII] emission in the hosts of a significant sample of PG\nquasars \\citep{ho05} and type 1 AGN \\citep{ki06} from the Sloan\nDigital Sky Survey (SDSS) may indicate an elevated role for AGN in\nsuppressing star formation \\citep[][]{gr04,sp05,cr06,cav07,hop08a}.\nHowever, there do appear to be high levels of star formation in the\nhosts of PG quasars when considering their far-infrared emission\n\\citep{sc06,ne07}, and type 2 quasars from the SDSS\n\\citep[$0.3<z<0.8$; ][]{za03,ki06} that may hint at an underlying\nevolutionary sequence (i.e., a modification to the unified model with\na correlation between star formation and nuclear obscuration) or the\nsystematic increase in the global star formation history of galaxies\n\\citep[e.g., ][]{li95,ho06,no07,tr07,zh07}.  These seemingly\ndiscrepant results highlight the complexity in determining if a casual\nrelationship between star formation and AGN activity exists.  This is\nmost likely due to the challenges in measuring the host galaxy\nproperties under the glare of a bright AGN, widely varying selection\nmethods, and inadequate control samples.\n\nThe ability to study the host galaxies of AGN selected from a large\nparent sample of galaxies has dramatically improved in the last few\nyears.  The SDSS has generated an unprecedented sample of galaxies to\nselect large numbers of low redshift ($z<0.3$) narrow-line AGN\n\\citep{ka03b,ke06,sta06,he06,wild07} to cleanly study their host\nproperties.  Powerful AGN have been clearly shown to reside in\nbulge-dominated galaxies with young stellar populations \\citep{ka03b}.\nIn conjunction with GALEX observations, AGN have optical-ultraviolet\ncolors placing them in an intermediate region \\citep {ma07,sa07,sch07}\nbetween the red and blue galaxy populations thus stirring interest in\ntheir relation to transitional galaxies \\citep{hop07}.  Remarkably,\n\\citet{ka07} find a strong correlation between the mass accretion rate\nonto SMBHs and the presence of young stars in the bulge that is\ntypically accompanied by a star-forming outer disk that may be\nsupplying the nuclear fuel.  It is imperative to extend such studies\nto epochs closer to the peak ($z\\sim1$) of the cosmic AGN and star\nformation history since AGNs may have an impact on the gas content of\ngalaxies and their subsequent demise.\n\nX-ray emission is a unique probe of AGN activity due to its ability to\npenetrate a moderately obscuring medium ($N_H \\lesssim 10^{24}$\ncm$^{-2}$), the lack of confusion with stellar sources, and because it\nallows a more direct probe of the mass accretion rate \\citep[see][for\na review]{br05}.  Selection of narrow line AGN (i.e., obscured; type\n2) by X-rays \\citep[e.g., ][]{ba03,sz04,si05,br07} can compensate for\nthe deficiencies of optical selection based on emission line ratios at\n$z>0.3$.  X-ray observations with both $Chandra$ and XMM-$Newton$ are\nbeing exploited to cleanly study the host galaxies of X-ray selected\nAGN \\citep{gr05,na07,pi07,ga08}.  Recently, \\citet{si08a} show that the\nbulge-dominated hosts of X-ray selected AGN in the Extended $Chandra$\nDeep Field - South have rest-frame colors that are bluer with\nincreasing redshift, possibly related to the star formation history of\ngalaxies.  In a related study, the mean star formation rate based on\nbroad-band photometry of 58 X-ray selected AGN ($z\\sim0.5-1.4$) in the\n1 Msec CDF-S has been shown to be similar to IRAC-selected galaxies in\nan equivalent mass regime \\citep{ah07}.  To date, few studies using a\nsignificant sample of AGN selected from a large parent sample of\ngalaxies with optical spectroscopy have been undertaken at these\nhigher redshifts ($z>0.3$).\n\nTo do so, we utilize the rich multiwavelength observations of the\nCOSMOS field \\citep{sc07} to carry out such a study at $0.5 \\lesssim z\n\\lesssim1.0$.  The COSMOS survey is roughly a 2 square degree region\nof the sky selected to be accessible from all major observatories both\nfrom the ground (e.g., Subaru, VLT) and space (e.g., $HST$, $Spitzer$,\nXMM-$Newton$, $Chandra$).  The zCOSMOS survey \\citep{lilly07} targets\nobjects for optical spectroscopy with the VLT in two separate\nobserving programs. A bright sample ($i<22.5$) is observed with a red\ngrism to provide a wavelength coverage of $5500-9500$ {\\rm \\AA}\nideal for identifying galaxies ($L_*$) up to $z \\sim 1.2$.  A deep\nprogram, not utilized in the present study, targets faint galaxies\n($B<25$), selected to be in the redshift range $1.5\\lesssim z\\lesssim\n2.5$, using a blue grism with a wavelength coverage of\n$3600<\\lambda<6700$ {\\rm \\AA}.  Here, we select a sample of galaxies\nbased on their stellar mass with reliable spectroscopic redshifts from\nthe zCOSMOS bright program.  Those that host AGN are identified by\ntheir X-ray emission as detected by XMM-$Newton$ \\citep{ha07,cap07}.\nWe measure the strength of emission lines (i.e., [OII]$\\lambda$3727,\n[OIII]$\\lambda$5007), using our automated pipeline\n\\citep[\"platefit\\_vimos\";][]{lam08} to determine star formation rates\nfor the entire galaxy sample including those hosting AGN.  An\nadditional spectral indicator ($D_n4000$) enables us to discern the\nage of the stellar populations on longer timescales.  X-ray\nluminosities of the AGN give us a handle on their bolometric output,\nless affected by obscuration, to infer mass accretion rates and\ndetermine any trends with star formation rate and redshift.  Finally,\nwe point out that a companion paper \\citep{si08c} based on the zCOSMOS\n10k catalog expands on the current study by investigating the\nenvironmental impact on AGN activity and their host galaxy properties.\n\nThroughout this work, we assume $H_0=70$ km s$^{-1}$ Mpc$^{-1}$,\n$\\Omega_{\\Lambda}=0.75$, and $\\Omega_{\\rm{M}}=0.25$.\n\n\\section{Data and derived broad-band properties}\n\n\\subsection{Parent galaxy sample and AGN identification}\n\nWe use the zCOSMOS 10k spectroscopic 'bright' catalog to construct a\nwell-defined sample of galaxies including those hosting X-ray selected\nAGN up to $z\\sim1$.  Specifically, we identify 7543 galaxies with a\nselection magnitude $i_{ACS} \\leq 22.5$ and high quality spectra hence\nreliable redshifts up to $z=1.02$.  A galaxy is included in our sample\nif the redshift has a quality flag 2.0 or higher that amounts to a\nconfidence of $\\sim99\\%$ for the overall sample.  The redshift success\nrate is not strongly dependent on galaxy color over $0.5\\lesssim z\n\\lesssim1.0$ given the quality of the spectra and presence of strong\nfeatures (i.e., 4000 \\AA~break; Ca H+K, [OII]).  Full details on data\nacquisition, reduction, redshift measurements and quality assurance\ncan be found in \\citet{lilly07,lilly08}.\n\nWe use the catalog of X-ray sources \\citep{cap07} generated from the\nuniform XMM-$Newton$ coverage ($\\sim50$ ks depth) of the COSMOS field\n\\citep{ha07} to identify galaxies within our parent sample that harbor\nAGN.  The X-ray catalog includes 1848 point--like sources detected\nabove a given threshold using a maximum likelihood method in either\nthe soft (0.5--2 keV), hard (2--10 keV) or ultra-hard (5--10 keV)\nbands down to a limiting flux of 5$\\times 10^{-16}$, 2$\\times\n10^{-15}$ and 5$\\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ in the\nrespective band.  The adopted threshold ($Likelihood > 10$)\ncorresponds to a probability $\\sim 4.5\\times10^{-5}$ that a source is\na background fluctuation.  An additional 26 faint XMM sources\ncoincident with diffuse emission (Finoguenov et al. in preparation)\nwere excluded.  Both the soft (0.5-2.0 keV) and hard band (2.0-10.0\nkeV) detections comprise our AGN sample in order to include the\nlow-to-moderate luminosity and\/or absorbed sources.  The higher\nsensitivity of the XMM-$Newton$ observations in the soft band enables\nus to probe lower luminosity AGN ($log~L_X\\sim43$) with $0.8\\lesssim z\n\\lesssim 1.0$ not sufficiently represented in the hard-band catalog.\n\nOptical and near-infrared counterparts to the XMM-$Newton$ X-ray\nsources are found using a maximum-likelihood method \\citep{br07}.\nMost X-ray sources (84\\%) have reliable counterparts while the\nremaining are uncertain due to the presence of multiple objects within\nthe X-ray error box that have similar probababilities of being the\ntrue counterpart.  We use the $Chandra$ observations (Elvis et al. in\npreparation; Civano et al. in preparation) that cover the central 1\nsquare degree to further refine our identifications.  We elect to\ninclude counterparts with lower confidence given that $\\sim50\\%$ of\nthem are likely to be correct based on the overlap between\nXMM-$Newton$ and $Chandra$ identifications (M. Brusa, private\ncommunication).  All results presented here are confirmed based on the\nslightly smaller catalog of highly reliable counterparts.  \n\nThe zCOSMOS 'Bright' program provides optical spectra for 357 of the\n1093 optical counterparts to X-ray sources having $i<22.5$.  Reliable\nspectroscopic redshifts are available for 90\\% of which 164 have\nredshifts between $0<z<1.02$.  Higher redshift AGN up to $z\\sim4$ have\nbeen identified by the zCOSMOS 'Bright' program but are not the focus\nof this study.  Since a high fraction of the zCOSMOS galaxies with\nassociated X-ray emission have detections in the hard 2-10 keV band\n(84\\%), we do not expect the inclusion of soft-selected sources to\ninduce any significant bias.  A subset (54\\%) of the X-ray sources has\nbeen designated as 'Compulsory' targets when designing the VIMOS masks\nwhich essentially lessens their randomness and results in a 71.9\\%\nsampling rate compared to the current rate of 29.8\\% for galaxies.  We\nset priority to these sources since they were not included in the\nspectroscopic followup program using Magellan \\citep{trump07}.  Since\nthe number of AGNs is not overwhelmingly large, we chose to not select\na smaller pseudo-random sample but rather incorporate correction\nfactors applied to derived measurements where applicable.\n\n\\begin{figure}\n\n\\epsscale{1.2}\n\n\\plotone{f1.eps}\n\\caption{X-ray luminosity (0.5-10 keV) versus redshift for 152 AGN\n(large symbols).  Filled circles are AGN primarily characterized by\nnarrow emission lines while open symbols denote those with at least one\nbroad optical emission line falling within the observed spectral\nrange.  The upper limit to the broad-band X-ray luminosity for each\nzCOSMOS galaxy is shown by the small grey dots.  The solid horizontal\nline is the minimum X-ray luminosity that we enforce to measure the\nfraction of galaxies hosting AGN to ensure a significant underlying\ngalaxy population.  The dotted line shows our chosen upper limit to\navoid AGN contamination.}\n\n\\label{lx_z}\n\\end{figure}\n\nModerate-luminosity AGN ($L_X\\sim10^{43}$ erg s$^{-1}$), detected in\nmedium-to-deep X-ray observations, have been shown to be ideal\nlaboratories for the study of their host galaxies due to their low\noptical brightness which in many cases is due to their nuclear\nobscuration \\citep[e.g.,][]{to06,main07}.  Our AGNs predominately have\nluminosities below that of typical QSOs ($L_X>10^{44}$ erg s$^{-1}$)\nbut are the dominant contributor to the Cosmic X-ray Background\n\\citep{gi07}.  In specific cases, we further select AGN based on their\nX-ray luminosity ($42.0 < log~L_{0.5-10.0~{\\rm keV}}<43.7$), as done\nin \\citet{si08a}, to isolate a sample for which we can cleanly\ndetermine their host properties (i.e., stellar masses, rest-frame\n$U-V$, $D_n4000$).  \\citet{ga08} have demonstrated that the optical\nemission from a nearly equivalent AGN sample in COSMOS is primarily\nattributed to their host galaxies; AGN dominated galaxies are\npreferentially at $z>0.8$ due to the fact that their X-ray\nluminosities are typically above our cutoff ($L_X\\sim10^{43.7}$ erg\ns$^{-1}$).  We conclude that the derived properties of the host\ngalaxies of AGN in zCOSMOS are likely to be reliable.  Where feasible,\nwe include higher luminosity AGN (QSOs; 16 with $L_{0.5-10~{\\rm\nkeV}}>10^{44}$ erg s$^{-1}$), since we can measure their stellar\nproperties (i.e., [OII] strength) even under the glare of an\nintrinsically, bright AGN (see Section~\\ref{sfr_measure} for details).\nFor these cases, we refrain from analyses based on their host\nproperties such as mass-weighted SFR or rest-frame color.  One caveat\nof our luminosity-selected sample is a potential bias towards specific\naccretion modes such as the \"Seyfert mode\" \\citep{hop06} or specific\nevolutionary stages in their fueling such as pre- or post-mergers.\nFortunately, we can test the later since merger-driven models\n\\citep{hop08a} predict vastly different properties (i.e., SFRs,\nmorphology) of the hosts of AGNs before and after the coalescence of\nmassive disk galaxies.\n\nWe list in Table~\\ref{sample} the statistics of the parent galaxy\nsample and those hosting X-ray selected AGN.  The redshift and X-ray\nluminosity distribution of the full AGN sample (152) is shown in\nFigure~\\ref{lx_z} up to $z=1.02$.  Symbols denote their optical\nspectral properties with 82\\% lacking broad ($FWHM>1000$ km s$^{-1}$)\nemission lines and most characterized as narrow emission-line\ngalaxies.  As further detailed in subsequent sections, we isolate\nvarious subsamples based on redshift, galaxy mass, and AGN luminosity\nto optimize our analyses.  For instance, as described above, we\nenforce an upper limit to the X-ray luminosity\n($log~L^{max}_{X}=43.7$) of the AGN when measuring a quantity with\nrespect to its host galaxy.  Also, we slightly increase our minimum\nX-ray luminosity ($log~L^{min}_{X}$=42.48) when determining the\nfraction of galaxies hosting AGN to ensure that a significant parent\nsample of zCOSMOS galaxies are capable of detecting each AGN to avoid\neffects based on limited statistics\\footnote{Specifically, an\noverabundance of AGN at $z\\sim0.35$ falling at the flux limit of the\n$XMM$-Newton observations, seen in Figure~\\ref{lx_z}, have an adverse\neffect on our determination of the fraction of galaxies hosting an\nAGN, due to the limited sample of zCOSMOS galaxies capable of\ndetecting AGN with $L_X\\sim10^{42}$ erg s$^{-1}$, that disappears when\nimplementing a slightly higher selection on luminosity.}.\n\n\\subsection{Stellar mass measurements}\n\nStellar masses, including rest-frame absolute magnitudes (AB system;\n$M_U$, $M_V$), are derived from fitting stellar population synthesis\nmodels from the library of \\citet{bc03} to both the broad-band optical\n\\citep[CFHT: $u$, $i$, $K_s$; Subaru: $B$, $V$, $g$, $r$, $i$,\n$z$;][]{capak07} and near-infrared \\citep[$Spitzer$\/IRAC: $3.6 \\mu $,\n$4.5 \\mu $;][]{sand07} photometry using a chi-square minimization for\neach galaxy.  The measurement of stellar mass (M$_{*}$) includes (1)\nthe assumption of a Chabrier initial mass function, (2) a star\nformation history with both a constant rate and an additional\nexponentially declining component covering a range of time scales\n($0.1 < \\tau < 30$ Gyr), (3) extinction ($0<A_V<3$) following\n\\citet{ca00}, and (4) solar metallicities.  Further details on mass\nmeasurements can be found in \\citet{bo08} and \\citet{me08}.  For\ncompatibility with the star formation rate calibration of \\citet{mo06}\nimplemented in subsequent sections, we convert all masses to an\nequivalent based upon a Salpeter IMF by applying a multiplicative\nfactor of 1.7 \\citep{po07}.  We show the mass distribution for the\nparent galaxy population, including those harboring X-ray selected\nAGN\\footnote{It is worth mentioning that there may be a potential\nproblem that galaxies with even moderate-luminosity AGN may have\ninaccurate mass estimates; a bluer continuum will essentially reduce\nthe stellar age and hence lower the mass measurements since the\nderived mass-to-light ratio depends strongly on the spectrum.}  with\n$log~L_{0.5-8.0~{\\rm keV}}>42.0$, as both a function of redshift\n(Figure~\\ref{selection}) and rest-frame color ($U-V$;\nFigure~\\ref{color_mass}).  All masses are expressed in solar units\n(M$_{\\sun}$) throughout this work.\n\n\\begin{figure}\n\\epsscale{1.2}\n\n\\plotone{f2.eps}\n\\caption{Stellar-mass versus redshift for 7543 zCOSMOS galaxies (small\ngrey circles). The box marks the mass-selected subsample of galaxies\n($log~M>10.6$; $0.48<z<1.02$) for which (specific) star formation\nrates are determined using [OII]$\\lambda$3727.  Galaxies hosting X-ray\nselected AGN are further marked by a larger black circle (filled:\n$42.0<log~L_{0.5-10~{\\rm keV}}<43.7$; open: $log~L_{0.5-10~{\\rm\nkeV}}>43.7$).}\n\n\\label{selection}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[angle=90,scale=0.75]{f3.eps}\n\n\\caption{Rest-frame color $U-V$ versus stellar mass split into three\nredshift intervals for galaxies and AGN with symbols described in\nFigure~\\ref{selection}.  The vertical line marks our imposed mass\nlimit.}\n\n\\label{color_mass}\n\\end{figure*}\n\nWe determine a minimum mass threshold that all galaxies must satisfy\nup to $z\\sim 1.0$.  The mass limit is set in order to ensure a fairly\ncomplete representation of both blue and red galaxies at all redshifts\nconsidered.  In Figure~\\ref{color_mass}, it is clearly evident that\nthe mass limit of $log~M=10.6$ is essentially imposed by the red\ngalaxy population at $z\\gtrsim0.8$ (rightmost panel).  The lack of red\ngalaxies below this limit is due to our initial selection on apparent\nmagnitude.  \\citet{me08} estimate based on a series of mock catalogs\nfrom the Millennium simulation that the zCOSMOS 'Bright sample' is\nessentially complete for galaxies with $log~M\\approx10.6$ at $z=0.8$\nwhile the completeness drops to $\\sim50\\%$ at $z=1$.  In total, we\nhave a sample of 2540 galaxies ($0.1<z<1.02$) above this mass limit of\nwhich 105 host X-ray selected AGN (see Table~\\ref{sample} for\nstatistics regarding subsamples employed herein.).\n\n\n\n\\section{[OII] as a star-formation rate indicator and the AGN contribution}\n\\label{sfr_measure}\n\nOur primary aim is to use the emission line [OII]$\\lambda$3727 to\nmeasure the SFRs of a well-defined sample of galaxies including those\nhosting AGN, as commonly used for star-forming galaxies at these\nredshifts \\citep[e.g., ][]{ke04,co07}.  The availability of [OII]\nwithin the spectral window of the zCOSMOS 'bright' program requires\nour sample of galaxies to fall within redshift range $z=0.48-1.02$.\nThe [OII] line is one of a suite of spectral features (e.g.,\nH$\\alpha$$\\lambda$6563, H$\\beta$$\\lambda$4861, [OIII]$\\lambda$5007)\nmeasured by an automated routine \"platefit\\_vimos\" \\citep{lam08} that\nsimultaneously fits all lines with a gaussian function to determine\nline flux and equivalent widths for the entire zCOSMOS 10k sample.\nEmission line fluxes are corrected for slit loss based on a comparison\nof their spectroscopic and photometric ($i_{ACS}$) magnitudes.  Line\nmeasurements with a significance less than 1.15$\\sigma$ are quoted\nhere as upper limits.  \n\nAn assessment of the AGN contribution to the [OII] emission line is\nrequired even though the production of this low ionization line has\nbeen shown to be relatively weak. \\citet{cr02} postulate, based on a\ncomparison of the equivalent width of [OII] in the composite 2dF QSO\nspectrum to that in a composite 'normal' galaxy spectrum, that [OII]\nis mainly produced by star-formation in the host galaxies of AGN over\na wide range of absolute magnitude.  The [OII] strength has been\nobserved to be $\\sim10-30\\%$ of the [OIII]$\\lambda$5007 emission line\nflux \\citep{fe86,ho05,ki06}.  \\citet{ho05} has explored the\nfeasibility of using [OII] as an indicator of ongoing star formation\nfor a sample of PG quasars that in many cases provided only an upper\nlimit.  Upon further investigation using emission line ratio\ndiagnostics, \\citet{ki06} conclude based on a sample of $\\sim3600$\ntype 1 AGN ($z<0.3$) selected from the SDSS that [OII] emission is\nmainly attributed to AGN photoionization and does not have a strong\nHII component.    \n\nThe aforementioned results do not rule out the potential effectiveness\nof using [OII] as a SFR indicator for different samples of AGNs such\nthose having higher luminosities, significant intrinsic obscuration or\nbeing at higher redshifts.  For example, the stellar populations of\n'strong' type 2 AGN from the SDSS have young ages similar to\nlate-type galaxies \\citep{ka03b} based on the age indicator $D_n4000$.\nWe illustrate that this is likely related to the presence of ongoing\nstar formation.  In Figure~\\ref{emlines}, we show the [OII]\/[OIII]\nratio for type 1 (panel $a$) and type 2\\footnote{The luminosity\nselection of type 2 AGNs from the SDSS is equivalent to the\ndefinition of 'strong' AGN ($L>10^7$ L$_{\\sun}$) given in\n\\citet{ka03b}.  This luminosity selection, based on\nextinction-corrected values, guarantees that the sample is dominated\nby Seyfert galaxies comparable to the luminosities of the zCOSMOS\nAGN.}  (panel $b$;$L_{\\rm [OIII]}>10^{40.58}$ erg s$^{-1}$) AGNs from\nthe SDSS having $z<0.3$.  The data for 25000 type 2 AGNs is taken\nfrom the high level products available from MPA based on the SDSS DR4\nrelease; we require a $S\/N>3$ detection for [OII], H$\\beta$, [OIII]\nand H$\\alpha$.  First of all, it is worth highlighting that the\nbest-fit linear relation to the type 1 AGN (panel $a$) illustrates a\nscenario where [OII] emission scales with [OIII] ($log~L_{\\rm\n[OII]}\\propto 0.36\\times L_{\\rm [OIII]}$) but not with a one-to-one\ncorrespondence.  Even without correcting for extinction, we see that\nthere is evidence (Fig.~\\ref{emlines}$b$) for elevated [OII]\/[OIII]\nfor type 2 AGNs as seen by the flatter slope of the best-fit relation\n(slanted dashed line; $log~L_{\\rm [OII]}\\propto 0.58\\times L_{\\rm\n[OIII]}$) compared to that of the type 1 AGNs.  The change in slope\nappears to signify that the strength of an additional component to\n[OII] (i.e., star formation) increases with AGN luminosity possibly\nrelated to the results of \\citet{ka07} where a decline in stellar age\nis seen for AGNs with higher mass accretion rates.  Finally, we see\nconclusively that the AGN host galaxies in zCOSMOS, while having\nhigher [OIII] luminosities, have elevated [OII]\/[OIII] ratios compared\nto type 1 AGNs (panel $a$), and exhibit a similar slope to the type 2\nAGNs in the SDSS and slightly further enhanced [OII] emission.\n\n\\begin{figure*} \n\n\\includegraphics[angle=90,scale=0.75]{f4.eps} \n\n\\caption{Emission-line properties of AGN.  ($a$) Observed (i.e., no\nextinction correction) line ratio [OII]\/[OIII] versus [OIII]\nluminosity for zCOSMOS galaxies ($log~M>10.6$; small green points) and\nthose hosting AGN shown by larger red circles.  For comparison,\nemission line properties of type 1 AGN from the SDSS \\citep[$z<0.3$;\n][]{ki06} are marked by the small black dots that are further\ncharacterized by the best fit linear relation (solid line) and mean\nratio $<[OII\/[OIII]>=0.27$ (open white square).  The dashed horizontal\nline denotes our single assumption for the value of a purely AGN\ndominated [OII]\/[OIII] ratio (see text for further details).  ($b$)\nSame as panel $a$ but with type 1 AGNs replaced by type 2 AGNs from\n\\citet{ka03b}.  The slanted lines are the best fit linear relation to\ntype 1 (solid) and type 2 (dashed) AGNs.  ($c$) Histogram of the\nextinction-corrected [OII]\/[OIII] distribution for the type 1\/2 AGN\n(short dash\/long dash) from the SDSS and our zCOSMOS sample (solid)\nall having $log~L_{\\rm [OIII]}>41.3$.}\n\n\\label{emlines} \n\\end{figure*} \n\nWhen correcting for internal extinction based on the Balmer\ndecrements, we find that the type 2 AGNs have a significantly higher\ndistribution of [OII]\/[OIII] than the type 1s (Fig.~\\ref{emlines}$c$).\nThis further suggests that star formation is more prevalant in the\ntype 2 AGN population compared to those with an observable broad line\nregion; such a hypothesis has been put forth by \\citet{ki06} to\nexplain the [OII] emission ($<[OII]\/[OIII]>=-0.12$) evident in the\nmore luminous, type 2 QSOs \\citep{za03}.  To further justify our use\nof [OII], we demonstrate in Figure~\\ref{type2_sdss} that a correlation\nbetween total [OII] emission (HII+AGN) and the spectral index\n$D_n4000$ exists and is in agreement with significant levels of\nongoing star formation seen in the hosts of type 2 AGN \\citep{gu06,yan06}.\n\n\\begin{figure}\n\\epsscale{1.0}\n\\plotone{f5.eps}\n\n\\caption{Relation between $D_n4000$ and (a) [OII] luminosity and (b)\n[OII]\/[OIII] ratio for 4609 luminous($L_{\\rm [OIII}> 10^7$\nL$_{\\sun}$), type 2 AGN from the SDSS.  In panel $b$, the median\n[OII]\/[OIII] ratio is shown (large red circles) in bins of $D_n4000$\nhaving a width of 0.1.}\n\\label{type2_sdss}\n\\end{figure}\n\nFor our purpose, we use the higher ionization line [OIII]$\\lambda5007$\nwhen present in our spectra, and an empirical relation between [OII]\nand [OIII] to statistically remove the component of the detected [OII]\nline that can be attributed to the AGN.  This entails an assumption\nthat the [OIII] line is purely of AGN origin \\citep{ka03b} that may\nlead us to underestimate SFRs due to a significant stellar\ncontribution but only in very few cases since the [OIII] luminosities\nof Seyfert galaxies in SDSS, and also zCOSMOS AGN hosts (see\nFigure~\\ref{emlines}$a$), are substantially higher than that of\ntypical HII galaxies \\citep{ke06}.  \\citet{ki06} find that the median\nvalue of the observed [OII]\/[OIII] ratio is 0.27 with significant\ndispersion (see Fig.~\\ref{emlines}$a$).  We chose to use a slightly\nlower ratio ([OII]\/[OIII]=0.21) that is the mean value for type 1 AGN\nin the SDSS sample (M. Kim private communication) with\n$log~L_{[OIII]}> 41.5$, a luminosity regime similar to our zCOSMOS AGN\nsample.  This higher luminosity cut guarantees that the ratio best\nreflects that produced by the AGN, while effectively minimizing a\ncontribution from HII regions.  This results in slightly higher SFRs\nfor our AGN sample than if we chose to use the mean of the entire type\n1 SDSS sample but our final results in this study are consistent using\neither value.  We make no attempt to use a luminosity-dependent\ncorrection, as inferred by the SDSS type 1 AGN, since a simple linear\nrelation may not be evident at higher luminosities\n($log~L_{[OIII]}>42$).  Also, there is no need to correct this\nrelation for extinction given the low levels of dust attenuation\n\\citep[$A_V\\approx0.2$ mag;][]{ki06} observed in these type 1 AGN.  We\nstrongly note, given the high dispersion in the [OII]\/[OIII] ratio for\ntype 1 SDSS AGN, that this is only a statistical correction applied to\nthe population as a whole to infer the global or mean properties of\nthe sample; the individual measurements of [OII] strength associated\nwith star formation in a particular galaxy are expected to be\ninaccurate.  We further note that even with this correction the more\nluminous AGN tend to have higher SFRs as evident by the best-fit\nrelation for type 2 AGNs in the SDSS (Fig~\\ref{emlines}$b$;\n$log~L_{\\rm [OII]}\\propto 0.37\\times L_{\\rm [OIII]}$).  In addition,\nwe assess the impact of the dispersion in the [OII]\/[OIII] ratio for\ntype 1 SDSS AGNs by assuming a normal distribution of the line ratio\n($<log [OII]\/[OIII]>=-0.69$; $\\sigma=0.25$) and perform many\niterations in our determination the distribution of SFR.\n\nWe consider extinction due to dust as an important factor in\ndetermining the component of the observed [OII] emission attributed to\nAGN photoionization given that our sample is predominantly composed of\ntype 2 AGN (see Fig.~\\ref{lx_z}).  An extinction-corrected [OIII]\nluminosity is found for each zCOSMOS AGN with detected [OIII] line\nemission using $A_V=0.8$.  Since most of our sample does not have\nH$\\alpha$ and H$\\beta$ within our observed spectral window, we chose\nto implement a level of extinction\\footnote{This amount of optical\nattenuation is similar to that of X-ray selected AGN; we find a median\n$A_V$ of 0.98 based on the Balmer decrements of 20 AGN in our sample\nwith significant $H\\alpha$ and $H\\beta$ line measurements (Mainieri et\nal. in preparation).} based on the mean Balmer decrement of type 2\nSeyferts in SDSS \\citep{ke06}.  This same level of attenuation is then\nreapplied to the inferred AGN component to [OII], based on the\nempirical relation mentioned above, and subtracted from the observed\n(i.e. no dust correction) [OII] emission line luminosity to provide an\nestimate free of any AGN contribution.  We are confident that this\nmethod is applicable to the zCOSMOS sample given that the [OII]\nluminosities of AGN hosts are all systematically higher\n($\\sim3-4\\times$) than that expected from gas photoionized by an AGN\nas indicated by the SDSS type 1 AGN (Fig.~\\ref{emlines}$c$) of similar\nluminosities ($log~L_{\\rm [OIII]}>41.3$).  Also, we highlight that the\nenhancement of [OII] relative to [OIII] for our zCOSMOS AGN is not\npurely induced by our extinction corrections as evident in observed\nrelation (i.e., no extinction correction; Fig.~\\ref{emlines}$a,b$).\n\n\\begin{figure} \n\\epsscale{1.15}\n\\plotone{f6a.eps}\n\\epsscale{1.05}\n\\plotone{f6b.eps} \n\n\n\\caption{($a$) Extinction-corrected [OIII]$\\lambda$5007 luminosity\nversus X-ray luminosity (2-10 keV) for AGN.  A clear correlation is\nevident as shown by the best-fit linear relation to these data (solid\nline) and local Seyferts \\citep[dotted line; ][]{pa06}.  ($b$)\nFraction of the [OII] line luminosity attributed to an AGN and split\nby the method of determining the AGN contribution.  The number\ndistribution of galaxies with detected [OIII] emission, used to\nestimate the amount of [OII] due to AGN activity, is given by the\nsolid histogram.  Galaxies without [OIII] detections are indirectly\nassessed by their hard X-ray luminosity (dashed histogram).  There is\nno evidence for a systematic offset between both methods.}\n\n\\label{o3_lx} \n\\end{figure} \n\n\nFor AGN host galaxies at $z\\gtrsim0.8$, we need to modify our method\nsince [OIII] is no longer within our observed spectral window.\nFortunately, we can utilize the strong correlation between hard X-ray\n(2-10 keV) and [OIII] luminosity \\citep{he05,pa06} to estimate the AGN\ncontribution for these cases.  It is important to highlight that our\nAGN are X-ray selected by the 0.5-10 keV band thus limiting the\ninclusion of more heavily absorbed type 2 AGN such as the\nCompton-thick population \\citep{fi08}, for which this correlation is\nless evident if the X-ray luminosities are not corrected for X-ray\nabsorption \\citep{he05} as is the case here.  In\nFigure~\\ref{o3_lx}$a$, we plot the [OIII] line luminosity, corrected\nfor extinction as done in the previous paragraph, as a function of the\nrest-frame 2-10 keV luminosity for AGN in the zCOSMOS sample with\nredshifts over a larger baseline ($0.2<z<1.0$).  We use these lower\nredshift AGN in order to provide higher statistics to measure the\nbest-fit linear relation between these two quantities.  We determine\nthe best fit $L_X-L_{[OIII]}$ relation (Equation~\\ref{lx-o3}) for our\nsample by implementing a bivariate linear regression (EM) algorithm\nwith ASURV \\citep[Survival Analysis for Astronomy package Rev. 1.2;\n][]{la92} that considers limits as well as detections.\n\n\\begin{equation}\nlog~L_{[OIII]}=(0.729\\pm0.101)~log~L_{2-10~keV}+(10.307\\pm4.381)\n\\label{lx-o3}\n\\end{equation}\n\n\\noindent This relation (Fig.~\\ref{o3_lx}$a$), as shown by the solid\nline, is similar to previous studies based on a sample of low redshift\nAGN \\citep[e.g.,][]{pa06} shown by the dotted line.  We find a nearly\nequivalent slope although a higher normalization (i.e., larger values\nof $L_{{\\rm [OIII]}}$ for a given $L_{2-10~{\\rm keV}}$).  The strong\nlinear relation (Pearson correlation coefficient $r=0.63$) strengthens\nour assumption that most of the [OIII] line luminosity is due to AGN\nphotoionization.  Any significant contribution from stars to the\n[OIII] luminosity would increase our derived star formation rates in\nAGN hosts.  In Figure~\\ref{o3_lx}$b$, we demonstrate that there are no\nsystematic offsets by introducing this indirect probe of the [OIII]\nline luminosity; both methods essentially remove a similar fraction\n($\\lesssim 60\\%$) of the [OII] luminosity attributed to AGN\nphotoionization.\n\nWe determine SFRs using the empirical calibration of \\citet{mo06}.\nThis relation considers interdependent factors (i.e., dust extinction,\nmetallicity, and ionization) to provide a SFR tracer that is\napplicable to samples for which these quantities are inadequately\nknown.  To derive the following empirical relation, we fit a linear\nrelation to the bright end ($log~L(B)>9.5$) of the dependence of SFR\non absolute $B$-band magnitude \\citep[$M_B$; see Figure 19 of\n][]{mo06}:\n\n\\begin{equation}\nlog~SFR({\\rm [OII]})=log~L_{{\\rm [OII]}}-41-0.195 \\times M_{B}-3.434\n\\label{eq:sfr}\n\\end{equation}\n\n\\noindent The AGN component to the emission line luminosity\n$L_{[OII]}$ is removed as described above and no correction for dust\nextinction is applied that is in essence an assumption that star\nformation is external to a dusty NLR and circumvents the potential\nproblem that extinction of the nuclear region may differ from that of\nHII regions \\citep[see][]{gu06}.  We do not attempt to adjust (i.e.,\nlower) the upper limits on nondetections by considering the\ncontribution of an AGN.  It is worth noting that an additional caveat\nof our method is that there is an underlying assumption that the\nmetallicity and extinction of galaxies with or without AGN are\nsimilar.\n\n\\section{Physical properties of AGN host galaxies}\n\n\\subsection{Stellar masses}\n\\label{mass}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{f7.eps}\n\\caption{Stellar mass distribution of galaxies hosting AGN in the\nredshift range $0.1<z<0.5$.  Fifty-two galaxies ($log~M_*>9.5$) with\nX-ray selected AGN ($42<Log~L_{0.5-10~{\\rm keV}}<43.7$) are shown by\nthe solid histogram.  The observed parent distribution of 3356\ngalaxies (dotted histogram) is renormalized to match the AGN\ndistribution.  The data points represent the AGN fraction\n($42.48<Log~L_{0.5-10~{\\rm keV}}<43.7$; scale on the right hand\nvertical axis) in fixed mass intervals (shown by the horizontal bars\ncentered on each data point) with $1\\sigma$ error bars.}\n\n\\label{mass_frac}\n\\end{figure}\n\nWe measure the fraction of galaxies that host X-ray selected AGN as a\nfunction of their stellar mass to determine the characteristic mass\nabove which AGN activity is most prevalent.  First of all, it is\napparent in Figure~\\ref{selection} that galaxies hosting X-ray\nselected AGN are preferentially massive ($log~M\\gtrsim10.5$) as\nevident over the full redshift range and in agreement with most\nstudies to date \\citep[e.g.,][]{ka03b,bu08}.  For this exercise, we\nconsider the redshift interval $0.1<z<0.5$ for which we have a high\ndegree of completeness for galaxies over a wide mass baseline ($log~M\n\\gtrsim 9.5$).  We follow the technique discussed in $\\S$~3.1 of\n\\citet{le07} to determine the AGN fraction of our parent population of\ngalaxies that accounts for the spatially varying sensitivity limits of\nthe {\\it XMM} observations of the COSMOS field \\citep[see Fig.~17\nof][]{cap07}.  The necessity of this approach is demonstrated in\nFigure~\\ref{lx_z}, which shows the limiting X-ray luminosity as a\nfunction of redshift for the entire galaxy sample.  Even though the\nsensitivity of the $XMM$ coverage is remarkably uniform, some\ndispersion is present as shown by the relatively narrow distribution\nof the X-ray upper limits at each redshift.  To properly account for\nthe luminosity-redshift relation, we determine the contribution of\neach AGN separately to the total fraction.  The AGN fraction ($f$; see\nequation{~\\ref{fraction} below) is determined by summing over the full\nsample of AGN ($N$) with $N_{\\rm gal,i}$ representing the number of\ngalaxies in which we could have detected an AGN with X-ray luminosity\n$L^{i}_{\\rm X}$.  The different sampling rates, based soley on slit\nplacement, of the galaxies ($S_{gal}$) and AGN (i.e., X-ray\nsources\\footnote{In many cases, X-ray sources have redshifts based on\na random placement of a slit for which the sampling reflects the\noverall galaxy population ($S_x=S_{gal}$)}, $S_x$) are taken into\naccount in equation~\\ref{fraction}.  We estimate the associated\n1$\\sigma$ error (equation~\\ref{fraction_error}) using binomial\nstatistics where $N^{eff}_{agn}$ is the number of AGN that would be\ndetected if all galaxies have the same limiting X-ray sensitivity and\nthe sample of AGN was randomly selected.  Here, we only consider AGN\nwith $42.48<log~L_{0.5-10 {\\rm keV}}<43.7$.  As previously mentioned,\nthe lower limit ensures that we have a statistically significant\nsample of parent galaxies ($\\gtrsim700$) that could host each AGN\nwhile the upper limit restricts the sample to low-to-moderate\nluminosities thus securing the accuracy of their host-galaxy masses.\nWe refer the reader to \\citet{si08a} for further details and results\nemploying this method.\n\n\\begin{equation}\nf=\\sum_{i=1}^{N}\\frac{1\/S_x}{N_{\\rm gal,i}\/S_{gal}}\n\\label{fraction}\n\\end{equation}\n\n\\begin{equation}\n\\sigma^2=N^{eff}_{agn}\\times(N_{gal}-N^{eff}_{agn})\/N^3_{gal}\n\\label{fraction_error}\n\\end{equation}\n\nIn Figure~\\ref{mass_frac}, we plot the fraction of galaxies that host\nAGN and the number distribution (Table~\\ref{sample}; Sample A) of both\ngalaxies and those with AGN over a slightly wider luminosity interval\n($42.0<log~L_{0.5-10 {\\rm keV}}<43.7$).  The absolute fraction is low\n($\\sim1-4\\%$) due to our restrictions on X-ray luminosity.\n\\citet{ka03b} measure a fraction $\\sim5\\times$ higher although they\ninclude in their sample AGN $\\sim80\\times$ fainter.  The difference in\nthese fractions is mainly due to the steep faint-end slope of the\nluminosity function at low redshifts \\citep{sh06}.  Here, we are\nmainly interested in the relative change of the AGN fraction with\ngalaxy mass.  Clearly, we see that the fraction of galaxies that host\nAGN monotonically increases with stellar mass similar to results based\non obscured \\citep{ka03b} and radio-loud \\citep{be05} AGN in the SDSS.\n\n\\subsection{Star formation rates}\n\nIn Figure~\\ref{sfr1}, we show the SFRs for 1820 zCOSMOS galaxies\n(Table~\\ref{sample}: Sample B\\footnote{The sample is slightly smaller\nthan that given in Table~\\ref{sample}; four AGNs have [OII] falling\noutside the observed spectral window that can vary slightly from\nslit-to-slit.}) including 43 of them that host moderate-luminosity AGN\n($42.0<log~L_{0.5-10.0~{\\rm keV}}<43.7$).  The results are presented\nbefore (panels a and b) and after (panels c and d) the correction for\nthe AGN contribution to the [OII] luminosity based on a single-value\n(0.21) of the observed, median [OII]\/[OIII] ratio of type 1 AGNs in\nthe SDSS.  First of all, we see that 88\\% of AGNs have detectable\n[OII] emission\\footnote{The fraction of AGN with detected [OII]\nemission is 68\\% ($S\/N>2$) and 61\\% ($S\/N>3$) depending on the given\nline significance.  We have confirmed our results based on the lower\nS\/N ratio (1.15) with those based on these smaller, high confidence\nsamples of AGN.} (panel $a$).  It is apparent by comparing the SFR\ndistribution for AGN hosts in panels $b$ and $d$ that a significant\nshift in the distribution occurs when removing the AGN contribution to\nthe [OII] emission line.  Even so, we find that the SFRs of AGN host\ngalaxies are almost exclusively between 1-100 M$_\\sun$ yr$^{-1}$\n(Fig.~\\ref{sfr1}$c$,$d$), a range consistent with analytic models of\nAGN hosts with star-forming disks \\citep{bal08} and supportive of\nconstraints on the cosmic IR background \\citep{bal07}.\n\n\\begin{figure*}\n\\epsscale{0.9}\n\\plotone{f8.eps}\n\n\\caption{Star formation rates ($M_{\\sun}$ yr$^{-1}$) versus stellar\nmass for galaxies ($0.48<z<1.02$; $log~M_*>10.6$) before (a) and after\n(c) the removal of the AGN contribution.  Measurements are shown by\neither a small black (emission-line galaxy) or large red (AGN;\n$42.0<log~L_{0.5-10~{\\rm keV}}<43.7$) circle.  Upper limits are shown\nas an arrow with their color descriptive of the type as given above.\n($b$, $d$) For ease of comparison, the distributions, including those\nwith upper limits, are shown by either a solid (AGN hosts) or dashed\n(all galaxies) histogram with the full galaxy sample scaled down to\nmatch the AGNs.}\n\n\\label{sfr1}\n\\end{figure*}\n\nTo quantitatively test whether the SFR distribution of AGN hosts\ndiffers from that of the underlying galaxy population, we have\ndetermined the SFR distribution, including those with upper limits,\nwhile also considering the dispersion in the AGN contribution to the\nobserved [OII] emission-line luminosity.  We assume a log normal\ndistribution of the [OII]\/[OIII] ratio with parameters given in\nSection~\\ref{sfr_measure}.  Based on one thousand iterations, we\ndetermine the mean SFR distribution and perform statistical tests on\neach individual iteration compared to the parent galaxy population.\nThe results of this exercise are shown in Figure~\\ref{sfr2} for two\nAGN samples of differing luminosities.  First, we consider AGN, as\ndone above, that fall within the luminosity interval $42<log~L_X<43.7$\nfor which we have high confidence in our well-matched, mass-selected\nparent sample of galaxies.  Based on this limited sample of AGNs, we\ncannot significantly discriminate between the two distributions\n(Fig.~\\ref{sfr2}$a$) given the results of the individual K-S tests\n(panel $b$).  Although, we do find that the median SFRs shown in panel\n$c$ are for the most part higher than that of the full galaxy sample\n(4.7 M$_\\sun$ yr$^{-1}$) in each iteration.\n\nWe are able to use a less restrictive sample that includes higher\nluminosity AGN ($log~L_X>43.7$) thus improving our statistics although\nat the expense of maintaining reliable stellar masses across our\nsample.  In Figure~\\ref{sfr2}$d-f$, we present the equivalent analysis\nfor AGNs with $log~L_X>42$.  Based on this larger sample, we see that\nthe SFR distribution of AGN hosts is shifted to higher values than the\noverall population (panel $d$).  The K-S tests on the individual\ndistributions now reject the null hypothesis (i.e., distributions are\nequivalent) at the $\\sim99\\%$ level ($>2.5\\sigma$) in essentially all\niterations (panel $e$).  Moreover, the median SFRs are all\nsystematically higher ($SFR\\sim9.5$ M$_\\sun$ yr$^{-1}$) than that of\nthe full galaxy sample.  We further point out that the SFR\ndistribution of AGN hosts while being elevated in comparison to the\nunderlying massive galaxy population is essentially equivalent to\nthose forming stars (i.e., emission-line galaxies).  Therefore, we\nconclude that a plentiful gas reservoir is a necessary ingredient for\nthe fueling of AGN as indicated by the presence of significant star\nformation with rates reaching up to $\\sim100$ M$_\\sun$ yr$^{-1}$.\n\n\\begin{figure*}\n\n\\includegraphics[angle=90,scale=0.75]{f9.eps}\n\n\\caption{Star formation rates of AGN hosts with dispersion in the\napplied correction for the AGN contribution.  The two rows are given\nfor AGN spanning a different luminosity range: ($a-c$)\n$42<log~L_X<43.7$, ($d-f$) $log~L_X>42$.  The left panels ($a$, $d$),\nequivalent to those in Figure~\\ref{sfr1}, show the mean distribution\nof SFR, including upper limits, based on 1000 iterations.  The middle\npanels ($b$, $e$) give the probability distribution based on a KS\ntests that the two samples are equivalent.  The right panels ($c$,\n$d$) display the distribution of the median SFR for each iteration\nwith the median SFR (4.73 $M_{\\sun}$ yr$^{-1}$) of all galaxies\ndepicted by the vertical dashed line.}\n\n\\label{sfr2}\n\\end{figure*}\n\n\nWe investigate how star formation in galaxies hosting AGN is evolving\ncompared to that of the parent galaxy population.  In\nFigure~\\ref{sfr_evol}, we show SFR as a function of redshift for\ngalaxies with $log~M>10.6$ and those hosting AGN (Table~\\ref{sample};\nSample B).  A systematic shift of the distribution of zCOSMOS galaxies\ntowards higher SFRs with increasing redshift is clearly evident with\nthose hosting AGN exhibiting a similar behavior.  Therefore, we find\nthat the SFRs of AGN hosts are dictated by that of the underlying\ngalaxy population thus solidifying similar evidence based on the color\nevolution of AGN hosts in the E-CDF-S \\citep{si08a}.  As a\nconsequence, ongoing star formation is most likely responsible for the\nsignificant population of AGN hosts over the redshift range\n$0.5\\lesssim z \\lesssim 1.4$ having blue rest-frame colors\n\\citep{sa04,bo07,na07} with evidence for such a trend remaining in\nplace for quasar hosts at higher redshifts $1.8<z<2.8$ \\citep{ja04b}.\n\nIt is illuminating to compare the SFRs from zCOSMOS to AGN at lower\nredshifts.  For consistency with zCOSMOS, we measure SFRs of the type\n2 AGN from SDSS \\citep[$z<0.3$;][]{ka03b} using the [OII]$\\lambda3727$\nand [OIII]$\\lambda5007$ emission line fluxes from \\citet{br04} to\ndetermine an AGN-corrected [OII] luminosity.  We further select only\nthose galaxies having stellar masses above our zCOSMOS limit\n($log~M_*>10.6$) based on the conversion factor (Kroupa to Salpeter\nIMF) given in \\citet{br04}.  Here, we use the calibration of\n\\citet{ke04} to derive SFRs\\footnote{We have confirmed that the\n\\citet{mo06} relation gives the same results.} in order to have\nconsistency with the SFRs measured for the type 1 SDSS sample\n\\citep{ki06}.  Initially, we demonstrate that the ongoing SFRs shown\nin Figure~\\ref{sfr_evol} are in agreement with the findings of\n\\citet{ka03b} based on $D_n(4000)$: weak AGNs ($L_{[OIII]}< 10^7\nL_{\\sun}$; blue points) have low SFRs ($\\sim$0.2 M$_\\sun$ yr$^{-1}$)\nwhile strong AGN ($L_{[OIII]}> 10^7 L_{\\sun}$; small red points) are\nmore actively forming stars ($\\sim$1 M$_\\sun$ yr$^{-1}$).  For ease of\nvisualization, we show the best-fit relation $log~SFR \\propto\nlog(1+z)$ for only the zCOSMOS galaxies with significant [OII]\nemission (upper limits are not considered; black curve) and those\nhosting AGN (red curve).  Remarkably, we further find that the SDSS\nAGN may be the low redshift analogs of the AGNs in the zCOSMOS survey\ngiven their close proximity to an extrapolation of the evolution of\nzCOSMOS galaxies.  In addition, the low-to-moderate levels of star\nformation ($\\approx 0.5-3$ M$_\\sun$ yr$^{-1}$) in type 1 SDSS AGN\n\\citep[shown roughly by the green triangle in\nFig.~\\ref{sfr_evol};][]{ki06} are in agreement with the type 2 AGN\nfrom SDSS and the aforementioned passive evolutionary scenario.  In\nlight of these results from the SDSS that effectively extend our\nredshift baseline, we reiterate our conclusion that the SFRs of AGN\nhost galaxies are reflective of the overall star formation history of\ngalaxies and provide no indication of the suppression or truncation\ndue to a mechanism related to the AGN itself.\n\n\\begin{figure*}\n\\epsscale{0.8}\n\\plotone{f10.eps}\n\n\\caption{Cosmic evolution of star formation.  At $z>0.48$, we show the\nSFR-$z$ distribution for all zCOSMOS galaxies with $log~M>10.6$ (small\nblack circles and crosses) and those hosting AGN with $log ~L_X>42$\n(73; large red circles and arrows).  The best-fit linear relation for\nthe emission-line objects is shown for both zCOSMOS populations\n(black: galaxies; red: AGN hosts) with an extrapolation to lower\nredshifts (dashed lines).  For comparison, we plot SFRs of AGN hosts\nfrom the SDSS with an equivalent selection on stellar mass; obscured\nAGN (type 2) from the sample of \\citet{ka03b} are shown with strong\nAGN ($log~L_{OIII}>40.5$) in red and those of lower luminosity in\nblue.  A large green triangle marks the mean value of the SFR for SDSS\ntype 1 AGN \\citep{ki06}.}\n\n\\label{sfr_evol}\n\\end{figure*}\n\n\\subsection{Stellar ages}\n\nTo complement our study of the ongoing SFRs of AGN hosts, we use the\nspectral index $D_n(4000)$ \\citep{ba99}, determined by our\n\"platefit\\_vimos\" routine for each galaxy in the zCOSMOS sample to\ninfer the age of the overall stellar population on longer timescales\n($>0.1$ Gyr).  This index is the ratio of the average flux density\n$F_{\\lambda}$ in the continuum bands 3850-3950 {\\rm \\AA} and\n4000-4100 {\\rm \\AA}.  This is essentially a measure of the strength\nof the 4000-{\\rm \\AA} break with galaxies having experienced a\nrecent episode of star formation exhibiting a smaller index due to the\npresence of young stars.  In Figure~\\ref{age}, we show the values of\n$D_n(4000)$ for galaxies with respect to their specific SFR. We\nimplement two mass limits ($log~M>10.6$: panels $a$, $b$;\n$log~M>11.1$: panels $c$, $d$) in order to check for consistency when\nthe underlying $D_n(4000)$ distribution of galaxies is significantly\ndifferent: the higher mass cut results in a distribution dominated by\nmore evolved galaxies as evident by its peak at $D_n(4000)\\sim1.8$.\nInitially, it is apparent (Fig.~\\ref{age}$a$) that there is a good\ncorrespondence between $D_n(4000)$ and specific SFR for all galaxies\nincluding those with AGNs.  We then use the relation given in\n\\citet{ka03a} that assumes an instantaneous burst model of\nsolar-metallicity to infer stellar age from the value of $D_N(4000)$\nas given by the scale bar in Figure~\\ref{age}a; this relation provides\nus with a rough assessment of the actual ages since many of the\nzCOSMOS galaxies including those with AGN may have had a more sedative\nexistence in their recent past.  As a result, we find that most\ngalaxies with AGN contain a young stellar component since 70\\% have\n$D_n(4000)<1.6$ ($age \\lesssim 2~ Gyr$; Fig~\\ref{age}a, c).\nFurthermore, we measure the fraction of galaxies hosting an AGN as a\nfunction of $D_n(4000)$ and demonstrate (Fig.~\\ref{age}$b$, $d$) that\na galaxy is more likely to have an accreting SMBH if there is a\nsufficient supply of gas, fully consistent with the mass measurements\nof HI \\citep{ho08} and CO \\citep{sco03} in AGN hosts, given the higher\nrate of occurance in galaxies with younger ages.  These results\neffectively extend such studies based on $D_n(4000)$ at low redshift\n\\citep{ka03b} up to $z\\sim1$.\n\n\\begin{figure*}\n\\epsscale{1.0}\n\n\\plotone{f11.eps} \n\n\\caption{($a$) Specific SFR versus the stellar age indicator\nD$_n$(4000).  Symbols are equivalent to those in Figure~\\ref{sfr1}a.\nAn approximate age scale is given based on Figure 2 of\n\\citet{ka03a}. ($b$) Number distribution and the fraction of galaxies\nhosting an AGN as a function of $D_n(4000)$.  (c, d) The equivalent\nplots are shown for a higher mass cut as given.  AGN activity is\nprimarily associated with young stellar populations over a broad range\nin host galaxy mass.}\n\n\\label{age} \n\\end{figure*}\n\n\nThere is a noteable discrepancy between our claim for a higher level\nof AGN activity for star-forming galaxies and the location of AGN\nhosts on the color-magnitude diagram.  Both X-ray selected\n\\citep{na07,si08a} and optically-selected AGN \\citep{ma07} exhibit a\nlow AGN fraction along the red sequence, enhanced activity in the\nintermediate region (i.e., \"green valley\") between the blue and red\ngalaxy populations, and a dropoff towards very blue galaxies.  On the\ncontrary, the mass-selected AGN sample of \\citet{ka03b} clearly have\nstellar properties very similar to late-type galaxies (see their\nFig. 14) and, thus a low fraction of AGN within the 'blue cloud'\nreported by the aforementioned studies is surprising.  Interestingly,\n\\citet{le08} find that the fraction of the stacked X-ray signal of\nlate-type galaxies attributed to AGN emission continuously rises with\nSFR.\n\nTo investigate this further, we measure the fraction of galaxies\nhosting AGN as a function of their rest-frame optical color ($U-V$)\nfor both a luminosity and mass-selected sample\n(Figure~\\ref{color_mag_mass}$a$, $b$).  Since we are not constrained\nby the use of [OII] for this exercise, we have extended the redshift\nbaseline $0.1<z<1.02$ to improve the statistics.  In the first panel,\nwe find that the fraction of galaxies with X-ray selected AGN is\nstrongly peaked in the \"green valley\" for the luminosity-selected\nsample (panel a) thus agreeing with the results of \\citet{si08a}.  In\ncontrast, the mass-selected sample, behaves similarly for colors\n$U-V>1.4$, but does not have a decline towards bluer colors $U-V<1.4$.\nThis is easily understood since blue galaxies are known to have lower\nmass-to-light ratios and the AGN fraction rises substantially with\nhost galaxy mass as demonstrated in Section~\\ref{mass}.  We conclude\nthat blue, star forming galaxies do have enhanced levels of AGN\nactivity and there is no evidence for a significant delay in the\nemergence of nuclear activity with respect to the onset of star\nformation.  This result is consistent with the recent findings of\n\\citet{si08a}, based the morphology of the host galaxy, that blue\n($U-V<0.7$), bulge-dominated ($n_{sersic}>2.5$) galaxies have the\nhighest incidence (21.3\\%) of AGN activity, and the blue rest-frame\ncolors of low redshift ($z<0.2$) quasars irrespective of their\nmorphology \\citep{ja04a}.  However, there remains the possibility that\nthe lower fraction of red galaxies hosting AGN is due to some form of\nself-regulating feedback that effectively reduces the AGN luminosity\nbelow our flux limit.  Nonetheless, these results lend no support for\na simple model prescription that attributes the truncation of star\nformation or the color evolution of galaxies to AGN feedback and bring\ninto question whether studies based on the optical emission line\ndiagostics of AGN activity miss a significant number of those residing\nin star-forming galaxies \\citep[see][for a discussion on this\ntopic]{sch07}.\n\n\\begin{figure*}\n\\epsscale{1.0}\n\n\\plotone{f12.eps}\n\n\\caption{Rest-frame color distribution $U-V$ of galaxies\n($0.1<z<1.02$) and the fraction hosting AGN.  We show the distribution\nfor 58 galaxies (solid histogram) selected by optical luminosity\n($M_V<-21.5$; panel a) and 65 galaxies above a fixed mass limit\n($log~M>10.6$; panel b) that host AGN.  For comparison, the\ndistribution for the parent galaxy population (dashed histogram) is\nnormalized to match the AGNs in each panel.  The decline in the AGN\nfraction towards the bluest colors (panel a) appears to be due to the\ninclusion of galaxies with low mass-to-light ratios that are not\npresent in the mass-selected sample (panel b).}\n\n\\label{color_mag_mass}\n\\end{figure*}\n\n\\section{AGN-star formation connection and co-evolution}\n\nHaving established an association between AGN activity and star\nformation, we are motivated to determine how closely these phenomena\nare related on a case-by-case basis that can potentially signify an\nunderlying causal connection.  The existence of such a relationship\nmay be realized given the recent findings that AGN accretion power as\nprobed by [OIII] luminosity is higher for galaxies with younger\nstellar populations \\citep{ka07}.  Ideally, we would like to\ninvestigate such a relation using a quantity more closely associated\nwith the accretion process, namely X-ray emission, known to originate\ncloser to the black hole and an indicator of the instantaneous star\nformation rate.\n\nTo do so, we have plot in Figure~\\ref{agn_gal_relations}a the SFRs of\nour AGN sample, as measured by [OII] strength, versus hard X-ray\nluminosity (2-10 keV) and inferred mass accretion rate while\nimplementing a bolometric correction as given in \\citet{ma04} and an\naccretion efficiency of 0.1.  We find that a weak correlation exists\nbased upon a Pearson correlation coefficient of 0.17 and a linear fit\nthat has a shallow slope with significant dispersion in its value\n($0.28\\pm0.22$).  We conclude that underlying complexities such as the\nefficiency of transferring gas to nuclear region over kiloparsec\nscales, and varying duty cycles for star formation and accretion may\ncontribute to the large dispersion in these relations.\n\n\\begin{figure}\n\n\\plotone{f13.eps}\n\n\\caption{AGN-galaxy relations: ($a$) SFR versus hard X-ray luminosity\nand mass accretion rate including the best fit relation.  Absorbed AGN\nwith detected star formation are further highlighted by an open box.\n($b$) Ratio of mass accretion to SFR versus redshift.  The horizontal\ndashed line marks the median ratio.  Measurements are shown by\na solid circle in both panels while limits either upper ($a$) or lower\n($b$) are given by an arrow.}\n\n\\label{agn_gal_relations} \n\n\\end{figure}\n\nThere have been some recent claims that an obscured phase, coupled\nwith enhanced star formation, may represent an early stage in the\nsubsequent evolution of AGN\n\\citep[e.g.,][]{pa04,al05,ki06,hop08a,po08}.  To test this scenario,\nwe have marked those AGN in Figure~\\ref{agn_gal_relations}a that have\nexcessive X-ray absorption ($N_{\\rm H} \\gtrsim 10^{22}$ cm$^{-2}$)\nbased on their hardness ratio [$HR=(H-S)\/H+S)> -0.2$] determined by\nthe X-ray counts in the soft (S: 0.5-2.0 keV) and hard (H: 2-10 keV)\nbands assuming a powerlaw spectrum with a photon index of 1.9 at an\neffective redshift of $z\\sim0.7$.  We find that the absorbed sources\nspan the same range of SFR as the unabsorbed sources.  A KS test give\na probability of 65\\% that both absorbed and unabsorbed AGN could be\ndrawn from the same parent population.  These results are not\ndependent on the chosen division in hardness ratio.  Therefore, we\nconclude that an amendment, such as the aforementioned evolutionary\nscenario, to the unification model \\citep{an93} for these\nmoderate-luminosity AGNs is not supported by our findings.  Although,\nthere remains substantive evidence that enhanced star formation may be\nassociated with nuclear obscuration for the more luminous QSOs\n\\citep[e.g.,][]{pa04,la07,za08} possibly due to different physical\nmechanisms for triggering mass accretion on SMBHs.\n\nIt is of much interest to determine the relative growth rate of a\ngalaxy and its central SMBH as a function of redshift in light of the\nwell-established local relation\n$M_{BH}\/M_{bulge}\\approx1.5\\times10^{-3}$ \\citep[e.g.,][]{mc02,ha04}.\nWe plot in Figure~\\ref{agn_gal_relations}b the ratio of mass accretion\nrate onto a SMBH to the SFR as a function of redshift.  The median\nvalue ($1.9\\times10^{-2}$) is roughly an order-of-magnitude higher\nthan the local ratio $M_{BH}\/M_{bulge}$.  This difference is likely\ndue to the varying timescales between SMBH accretion and star\nformation $\\Delta t_{BH}\/\\Delta t_{gal}\\approx M_{BH}\/M_{bulge} \\times\nSFR\/dM_{accr} dt^{-1}\\approx 0.1$.  We note that our SFR measurements\nmost likely include a significant disk component thus a more rigorous\nassessment is required and beyond the scope of this work.\nNonetheless, by assuming that star formation occurs over rougly a\ndynamical timescale $\\Delta t_{gal}\\approx10^9$ yr, the duty cycle for\nAGN activity ($\\approx10^8$ yr) is consistent with the\nluminosity-dependent model predictions of \\citet{hop08b} for SMBHs\nwith $M_{BH}\\sim10^8$ M$_{\\sun}$ and accreting above an Eddington\nratio of 0.01, both within a physical regime spanned by our sample\nassuming these AGNs have already settled on a SMBH - bulge relation at\ntheir respective redshifts.  Alternatively, if we assume a timescale\nfor galaxy growth to be the inverse of their sSFRs as shown in\nFigure~\\ref{age}, the AGN lifetimes can reach up to $\\sim10^{10}$ yr\nthat fully illustrates that these AGNs and their host galaxies are\nclose to being fully matured although have SMBH growth times at least\nan order-of-magnitude less than those in the SDSS \\citep{he04}.\nFurthermore, we find that there is no dependence of the ratio\n($dM_{accr} dt^{-1}\/SFR$) on redshift thus supporting a\nco-evolutionary scenario where both the average SFR and mass accretion\nrates onto SMBHs are rapidly declining with equivalent rates from\n$z\\sim1$ to the present possibly due to diminishing fuel supplies.\n\nThese comparisons suggest that beyond the local universe the $M_{BH} -\nM_{bulge}$ relation should display higher intrinsic dispersion given\nthe large spread over two orders of magnitude ($10^{-3}-10^{-1}$) in\nthe relation between accretion rate and SFR\n(Fig.~\\ref{agn_gal_relations}$b$).  \\citet{ro06} point out that AGN\nsamples at high redshift may have larger intrinsic scatter in the\nvelocity dispersion of their hosts due to their immature dynamical\nstate. The observational situation is so far unclear: limited AGN\nsamples at higher redshifts \\citep{wo06,wo08} show preferentially\nhigher black hole masses relative to that expected by the local $M -\n\\sigma$ relation, while other studies of low redshift AGNs, undergoing\nsubstantial rates of accretion ($L_{Bol}\/L_{Edd}>0.1$), have black\nhole masses lower than that of inactive galaxies of equivalent host\nmass \\citep{ho08} or luminosity \\citep{ki08}.  Notwithstanding the\nlarge observational biases \\citep{lau07}, it is possible that the\ncoeval growth of galaxies and their SMBHs may be intermittent with\neither stars or SMBHs growing somewhat faster or slower than each\nother, as indicated by our findings exemplified in\nFigure~\\ref{agn_gal_relations}$b$, while still resulting in a tight\nblack hole-bulge mass relation for inactive galaxies at $z=0$ as\nsuggested by \\citet{wo06}.\n\n\\section{Summary and conclusions}\n\nWe have utilized the zCOSMOS 10k catalog of galaxies with reliable\nspectroscopic redshifts to investigate the properties of those that\nhost AGN.  X-ray observations with $XMM$-Newton enable us to identify\n152 AGN, from a parent sample of 7543 zCOSMOS galaxies, that include\nthose with significant obscuration and of low optical luminosity.  The\nderived properties of the full zCOSMOS sample such as stellar mass,\nrest-frame colors, and spectral properties (i.e., emission line\nstrength, $D_n4000$) enable us to determine the prevalence of AGN\nactivity as a function of galaxies with the aforementioned\ncharacteristics.\n\nSpecifically, we measure the SFR of galaxies using the\n[OII]$\\lambda$3727 line luminosity.  We account for the contribution\nfrom the underlying AGN component most likely arising from the narrow\nline region by using the [OIII]$\\lambda$5007 luminosity and the\ntypical [OII]\/[OIII] ratio found from previous studies of AGN and\nquasars \\citep{ho05,ki06}.  The [OIII] line luminosity is measured\ndirectly from our spectra if present.  For the subsample with [OIII]\noutside our observed spectral bandpass, we infer the [OIII] strength\nfrom the hard (2--10 keV) X-ray luminosity and the well known\ncorrelation between these two quantities.\n\nOverall, we find that the two main requirements for a galaxy to host\nan actively, accreting SMBH are (1) its stellar mass and (2) a\nsignificant amount of gas content as inferred by the observed levels\nof star formation and the age of their stellar populations.  These\nfindings essentially extend those of the SDSS \\citep{ka03b} out to\n$z\\sim1$.\n\nWe particularly draw the following conclusions:\n\n\\begin{itemize}\n\n\\item We confirm with many previous studies that the fraction of\ngalaxies hosting AGN rises with increasing mass with most host\ngalaxies having $M_*>4\\times10^{10}$ $M_{\\sun}$.\n\n\\item The host galaxies of AGN have ongoing star formation with a\nbroad range of rates ($\\approx1-100$ $M_{\\sun}$ yr$^{-1}$) higher than\nthat of the overall massive ($log~M>10.6$) galaxy population and\nessentially equivalent to those forming stars (i.e., emission-line\ngalaxies).  The association of AGN activity and young stellar\npopulations is further substantiated by an observed increase in the\nfraction of galaxies harboring AGN at low values of the spectral index\n$D_n(4000)$ and blue rest-frame color $U-V$.\n\n\\item The enhancement of AGN in young, star-forming galaxies with\n$1.0<D_n(4000)<1.4$ and $0.8<U-V<1.5$ lessens the evidence for AGN\nprefering to reside in galaxies of intermediate colors and their role\nin galaxy evolution.  We demonstrate that previous studies based on\nluminosity-selected samples are misleading due to the inclusion of\ngalaxies with low mass-to-light ratios that are less likely to harbor\nAGN given their lower masses.\n\n\\item The SFRs of AGN hosts evolves with redshift in an equivalent\nmanner to the overall star-forming galaxy population.  This\nessentially brings the evidence for a co-evolution scenario between\naccretion onto SMBHs and the star-formation history of galaxies to a\ncloser physical scale (i.e., within the same galaxies).\n\n\\item A direct relationship between the consumption of gas into stars\nand that accreted onto SMBHs is weak, suggesting that additional\nphysical complexities or varying timescales may be inherent.  On\naverage, a co-evolution scenario for the overall population is clearly\nevident given the constancy of the ratio ($\\sim10^{-2}$) between mass\naccretion rate onto SMBHs and SFR with redshift possibly indicative of\ndepleting gas reservoirs from $z\\sim1$ to the present.  The\norder-of-magnitude increase in this ratio compared to the locally\nmeasured value of $M_{BH}\/M_{bulge}$, is consistent with an AGN\nlifetime substantially shorter than that of star formation.\nFurthermore, the significant dispersion in this ratio may be\nindicative of larger scatter in the BH-bulge relations at higher\nredshift.\n\n\\end{itemize}\n\nOur results put important constraints on physical models of AGNs and\ntheir evolution.  The considerable rates of star formation ($\\sim10$\nM$_\\sun$ yr$^{-1}$) in their hosts, coupled with the lack of\nstructural signs of galaxy interactions \\citep{gr05,ga08} and\ntimescales for accretion, indicate that a \"Seyfert mode\" of accretion,\ndriven by secular processes, is more likely than merger-driven models\nfor this class of moderate-luminosity AGN.  In particular, there is no\nindication of the suppression or truncation of star formation at\nlevels expected from models implementing AGN feedback.  Therefore, we\nfind no conclusive evidence that AGNs, undergoing this mode of\naccretion, are a key factor in the evolution of galaxies.\n\n\\acknowledgments\n\nWe are most grateful to the referee for providing a critical review\nthat significantly improved the paper and to Minjin Kim for allowing\nus to use his emission line measurements of SDSS AGNs and fielding\nsubsequent communications.  This work is fully based on observations\nundertaken at the European Southern Observatory (ESO) Very Large\nTelescope (VLT) under the Large Program 175.A-0839 (P.I., Simon\nLilly).\n\n{\\it Facilities:} \\facility{XMM}, \\facility{VLT:Melipal (VIMOS)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nVarious approaches of perturbative QCD including the\nnext-to-leading order calculations (NLO QCD) have been applied \nto analyze the heavy flavour particle production in hadron \nreactions at high energies. \n\nSuch reactions are usually analyzed\nwithin the hard parton scattering model (HPSM) suggested in \n\\cite{Efremov,FF}. This model was applied to\nthe charmed meson production both in proton-proton and meson-proton \ninteractions at high energies, see for example \\cite{Bednyakov:1995}. \nThe HPSM is significantly improved by applying the QCD parton approach \n\\cite{Nasson,Kniehl2}, see details in \\cite{LKSB:09} and references\ntherein.\nUnfortunately the QCD approach including the NLO \n has some uncertainties related to the renormalization\nparameters especially at small transverse momenta $p_t$ \\cite{LKSB:09}. \n\nIn this paper we study the charmed and beauty meson production\nwithin the QGSM \\cite{kaid1} or the dual parton model (DPM) \\cite{Capella:1994}\nin $pp$ and $p{\\bar p}$ collisions at high energies based on the $1\/N$ \nexpansion in QCD \\cite{tHooft:1974,Veneziano:1974}.   \nWe show that this approach can be applied rather successfully at not very large values\nof $p_t$.\nIn addition, we investigate the open charm and beauty baryon production\nin $pp$ collisions at LHC energies and very small $p_t$ within the QGSM to find\nnew information on the sea charmed and beauty quark distributions in the proton.\nAnd at the end of the paper we use this approach to analyze the charmed meson production\nin the $p{\\bar p}$ collision at not large energies because the obtained results would be\nvery interesting for the future experiments at the GSI (Darmstadt) planned by the PANDA\nCollaboration.\n\\section{Charmed and beauty hadron production in \n$pp$ collisions at high energies }\n\\subsection{Heavy flavour meson production }\nFirst, let us analyze the $D$ and $B$ meson production in the $pp$ \ncollisions within the QGSM \nincluding the transverse motion of quarks and diquarks in\ncolliding protons \\cite{LAS}. As is known, the cylinder type\ngraphs for the $pp$ collision presented in Fig.1 make\nthe main contribution to this process \\cite{kaid1}. \nThe left diagram of Fig.1, the so-called\none-cylinder graph, corresponds to the case where two colorless\nstrings are formed between the quark\/diquark ($q\/qq$) and the\ndiquark\/quark ($qq\/q$) in colliding protons; then, after their\nbreakup, $q{\\bar q}$ pairs are created and fragmentated to a hadron,\nfor example, $D$ meson. The right diagram of Fig.1, the\nso-called multicylinder graph, corresponds to creation of the same\ntwo colorless strings and many strings between sea\nquarks\/antiquarks $q\/{\\bar q}$ and sea antiquarks\/quarks ${\\bar\nq}\/q$ in the colliding protons.\n\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{cylgr_2_1.eps}\n\\caption{The one-cylinder graph (left diagram) and the multi-cylinder \ngraph (right diagram) for the inclusive $p p\\rightarrow h X$ process.}\n\\label{Fig.1}\n\\end{figure}\n\nThe general form for the invariant inclusive hadron spectrum\nwithin the QGSM is \\cite{kaid2,LAS}\n\n\\begin{align}\nE\\frac{d\\sigma}{d^3{\\bf p}}\\equiv\n\\frac{2E^*}{\\pi\\sqrt{s}}\\frac{d\\sigma}{d x d p_t^2}=\n\\sum_{n=1}^\\infty \\sigma_n(s)\\phi_n(x,p_t)~, \n\\label{def:invsp}\n\\end{align}\n\nwhere $E,{\\bf p}$ are the energy and the three-momentum of the\nproduced hadron $h$ in the laboratory system (l.s.) of colliding protons, \n$E^*,s$ are the energy of $h$ and the square of the initial energy in the\nc.m.s of $pp$; $x$ and $p_t$ are the Feynman variable and the transverse\nmomentum of $h$; $\\sigma_n$ is the cross section for production of\nthe $n$-Pomeron chain (or $2n$ quark-antiquark strings) decaying\ninto hadrons, calculated within the ``eikonal approximation''\n\\cite{Ter-Mart}, the function $\\phi_n(x,p_t)$ has the following\nform \\cite{LAS}:\n\n\\begin{eqnarray}\n\\phi_n(x,p_t)=\\int_{x^+}^1 d x_1\\int_{x_-}^1 \nd x_2\\psi_n(x,p_t;x_1,x_2)~, \n\\label{def:phin}\n\\end{eqnarray}\nwhere\n\\begin{align}\n&\\psi_n(x,p_t;x_1,x_2) = \\notag \\\\\n&\\quad F_{qq}^{(n)}(x_+,p_t;x_1)F_{q_v}^{(n)}(x_-,p_t;x_2)\/F_{q_v}^{(n)}(0,p_t) \n\\notag \\\\\n&\\quad +F_{q_v}^{(n)}(x_+,p_t;x_1)F_{qq}^{(n)}(x_-,p_t;x_2)\/F_{qq}^{(n)}(0,p_t) \n\\notag \\\\\n&\\quad +2(n-1)F_{q_s}^{(n)}(x_+,p_t;x_1) \\notag \\\\\n&\\qquad\\qquad \\times F_{{\\bar q}_s}^{(n)}(x_-,p_t;x_2)\/F_{q_s}^{(n)}(0,p_t)~,\n\\label{def:psin}\n\\end{align}\nand $x_{\\pm}=(\\sqrt{x^2+x_t^2}\\pm x)\/2$, \\\\\n$x_t=2\\sqrt{(m_h^2+p_t^2)\/s}$,\n\n\\begin{align}\n&F_\\tau^{(n)}(x_\\pm,p_t;x_{1,2}) \\notag \\\\\n&\\quad =\\int d^2k_t{\\tilde f}_\\tau^{(n)}(x_{1,2},k_t){\\tilde G}_{\\tau\\rightarrow h}\n\\left(\\frac{x_\\pm}{x_{1,2}},k_t;p_t\\right)~, \n\\label{def:Ftaux} \\\\\n&F_\\tau^{(n)}(0,p_t) \\notag \\\\\n&\\quad = \\int_0^1 dx^\\prime d^2k_t {\\tilde\nf}_\\tau^{(n)}(x^\\prime,k_t) {\\tilde G}_{\\tau\\rightarrow\nh}(0,p_t)  \\notag \\\\ \n&\\quad ={\\tilde G}_{\\tau\\rightarrow h}(0,p_t)~.\n\\label{def:Ftauzero}\n\\end{align}\n\nHere $\\tau$ means the flavor of the valence (or sea) quark or\ndiquark, ${\\tilde f}_\\tau^{(n)}(x^\\prime,k_t)$ is the quark\ndistribution function depending on the longitudinal momentum\nfraction $x^\\prime$ and the transverse momentum $k_t$ in the\n$n$-Pomeron chain; ${\\tilde G}_{\\tau\\rightarrow h}(z,k_t;p_t)=\nz{\\tilde D}_{\\tau\\rightarrow h}(z,k_t;p_t)$, ${\\tilde\nD}_{\\tau\\rightarrow h}(z,k_t;p_t)$ is the fragmentation function\n(FF) of a quark (antiquark) or diquark of flavour $\\tau$ into a hadron\n$h$ ($D$ meson in our case). \nWe present the quark distributions\n and the FF in the factorized forms ${\\tilde f}_\\tau(x,k_t)=\nf_\\tau(x)g_\\tau(k_t)$, and, according to \\cite{LS:1992},\n${\\tilde G}_{\\tau\\rightarrow h}(z,k_t;p_t)=G_{\\tau\\rightarrow h}(z)\n{\\tilde g}_{\\tau\\rightarrow h}({\\tilde k}_t)$, where ${\\tilde{\\bf k}}_t=\n{\\bf p}_t-z{\\bf k}_t$.\n We take the quark distributions $f_\\tau(x)$ and the FF  $G_{\\tau\\rightarrow h}(z)$\nobtained within the QGSM from \\cite{kaid2,Shabelsky,Piskunova},\nwhereas their $k_t$ distributions are chosen in the form suggested in\n\\cite{Piskunova:1984,LS:1992} (see the details in \\cite{LKSB:09})\n\\begin{align}\n& \\qquad g_\\tau(k_t)=(B_q^2\/2\\pi)\\exp(-B_q A(k_t))~,\n\\label{def:ktdistr} \\\\\n& \\qquad {\\tilde g}_{\\tau\\rightarrow h}({\\tilde k}_t)=(B_c^2\/2\\pi)\n\\exp(-B_c A(k_t))~,\n\\label{def:ktFF}\n\\end{align}\nwhere $A(k_t)=(\\sqrt{k^2_t+m^2_D}-m_D))$ and $m^2_{Dt}=p_t^2+m_D^2$\nAfter the integration of eq.~(\\ref{def:Ftaux}) over $d^2k_t$ \nwe have, according to \\cite{LS:1992}, \n\\begin{align}\n&F_\\tau^{(n)}(x_\\pm,p_t;x_{1,2}) \\notag \\\\\n&\\quad={\\tilde f}_\\tau^{(n)}(x_{1,2})\nG_{\\tau\\rightarrow h}(z)I_n(z,p_t)~, \n\\label{def:Fnew}\n\\end{align}\nwhere $z=x_\\pm\/x_{1,2}$,\n$I_n(z,p_t)=B_z^2\/(2\\pi(1+B_zm_D))\\exp(-B_z(m_{Dt}-m_D))$, \n$B_z=B_c\/(1+n\\rho z^2)$, $\\rho=B_c\/B_q$. \nThe function $B_z$ also can be presented in the equivalent form\n$B_z=B_q\/({\\tilde\\rho}+n z^2)$, where ${\\tilde\\rho}=B_q\/B_c$. \nThe differential cross section $d\\sigma\/d p_t^2$\nfor $D$ mesons produced in $pp$ collisions is written in the  \nform \\cite{kaid2}\n\\begin{align}\n\\qquad \\frac{d\\sigma}{d p_t^2}=\\frac{\\pi}{2}\n\\sqrt{s}\\sum_{n=1}^\\infty\\sigma_n(s) \\int\\phi_n(x,p_t)\\frac{d x}{E^*}~. \n\\label{def:dsigdptsq}\n\\end{align} \nTo describe the experimental \ndata on the $p_t$-spectra in the $p_t$ region where the NLO QCD calculation has a big uncertainty\nwe chose $\\tilde{\\rho}=7$ both for the Tevatron and the LHC energies.\nWhen $B_c~<~1$ (GeV\/c)$^{-1}$, Eq.(\\ref{def:ktFF}) can be approximately presented in the form \n${\\tilde g}_{\\tau\\rightarrow h}({\\tilde k}_t)\\simeq a^2\/(a^2+{\\tilde k}^2_t)$ at\n${\\tilde k}^2_t<2m_D$, where $a=\\sqrt{2m_D\/B_c}$ \\cite{Greco:1994}. \nThis form is similar to the form for the FF of heavy quarks obtained within the perturbative \nQCD. \nNote that the function $I_n(z,p_t)$ in Eq.(\\ref{def:Fnew}) was obtained in \n\\cite{LS:1992,LAS} on the assumption of the consequent sharing of the transverse  momentum\n$p_t$ in the proton (antiproton) between $n$-Pomeron chains. It allowed us to describe rather \nsatisfactorily the experimental data on the inclusive $p_t$ spectra of charmed and beauty mesons\nproduced in $p{\\bar p}$ collisions at moderate values of the transverse momentum $p_t~<~10$ GeV\/c\n\\cite{LKSB:09}.\nAs is shown in  \\cite{LS:1992}, this version of the QGSM describes\nrather satisfactorily the experimental data on inclusive spectra of $D$\nmesons produced in $pp$ collisions at $\\sqrt{s}=27.4$ GeV.\n\nIt allows us to make \nthe predictions for inclusive $p_t$ spectra of $D^0$ and\n$B^+$ mesons produced in the $pp$ collision at LHC energies and\ncompare our results with the NLO QCD calculation for the produced charmed quarks\n\\cite{ALICE}, see  Figs.2, 3.\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{D0lhcB4r7_02_new.eps}\n\\caption{The inclusive spectrum for $D^0$ mesons produced in the\n$pp$ collision at the LHC energy $\\sqrt{s}=14$ TeV obtained within\nthe QGSM for charmed mesons \\cite{LKSB:09}\nand the NLO QCD for $c$ quarks\n \\cite{ALICE}.}\n\\label{Fig.2}\n\\end{figure}\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{B+mes_lhc13_04_09_new.eps}\n\\caption{The inclusive spectrum for $B^+$ mesons produced in the\n$pp$ collision at the LHC energy $\\sqrt{s}=14$ TeV obtained within\nthe QGSM for beauty mesons \\cite{LKSB:09}\nand the NLO QCD for single\n$b$ quarks \\cite{ALICE}.}\n\\label{Fig.3}\n\\end{figure}\nThe solid  lines in Figs.2, 3 correspond to our calculations within the\nQGSM, whereas the hatched regions show the calculations within the\nNLO QCD including uncertainties \\cite{ALICE}. A big difference\nbetween the QGSM and NLO QCD calculations at $p_t>10$\nGeV\/c for $D$ and $B$ mesons can be due to the following.\nFirst, the NLO QCD calculation \\cite{ALICE}\ndoes not include the hadronization of quarks to heavy mesons,\nwhereas the QGSM calculation includes it. Second, we do not\ninclude the contribution of gluons and their hard scatterings off\nquarks and gluons which can be sizable at large values of $p_t$.\n\\subsection{Heavy flavor baryon production}\nNow let us analyze the charmed and beauty baryon production in the $pp$\ncollision at LHC energies and very small $p_t$ within the soft QCD, e.g.,\nthe QGSM. This study can be interesting for it may allow \npredictions for future LHC experiments like TOTEM and ATLAS and \nan opportunity to find new information on the distribution\nof sea charmed ($c$) and beauty ($b$) quarks at very low $Q^2$. \nAccording to the QGSM, the distribution of $c({\\bar c})$ quarks in the $n$th Pomeron \nchain (Fig.1(right)) is, see for example \\cite{LAS} and references therein,\n\\begin{eqnarray}\n\\lefteqn{\nf_{c({\\bar c})}^{(n)}(x) = C_{c({\\bar c})}^{(n)}\\delta_{c({\\bar c})}\nx^{-\\alpha_\\psi(0)}} \\nonumber \\\\\n& {\\times}(1-x)^{\\alpha_\\rho(0)-2\\alpha_B(0)+(\\alpha_\\rho(0)-\\alpha_\\psi(0))\n+n-1}~,\\quad\n\\end{eqnarray}\nwhere $\\delta_{c({\\bar c})}$ is the weight of charmed pairs in the quark sea, \n$C_{c({\\bar c})}^{(n)}$\nis the normalization coefficient \\cite{kaid2},\n $\\alpha_\\psi(0)$ is the intercept of the $\\psi$- Regge trajectory.\nIts value can be $-2.18$ assuming that this trajectory $\\alpha_\\Psi(t)$ \nis linear and the intercept and the slope $\\alpha_\\Psi^\\prime(0)$ can be\ndetermined by drawing the trajectory through the $\\Psi$-meson mass \n$m_\\Psi=3.1$ GeV and the $\\chi$-meson mass $m_\\chi=3.554$ GeV \n\\cite{Boresk-Kaid:1983}. Assuming that the $\\psi$- Regge trajectory is \nnonlinear one can get  $\\alpha_\\psi(0)\\simeq 0$, which follows from perturbative \nQCD, as it was shown in \\cite{Kaid-Pisk:1986}. \nThe  distribution of $b({\\bar b})$ quarks in the $n$th Pomeron \nchain (Fig.1(right)) has the similar form  \n\\begin{eqnarray}\n\\lefteqn{\nf_{b({\\bar b})}^{(n)}(x) = C_{b({\\bar b})}^{(n)}\\delta_{b({\\bar b})}\nx^{-\\alpha_\\Upsilon(0)}}\n\\nonumber \\\\\n &{\\times}(1-x)^{\\alpha_\\rho(0)-2\\alpha_B(0)+(\\alpha_\\rho(0)-\n\\alpha_\\Upsilon(0))\n+n-1}~,\\quad\n\\end{eqnarray}\nwhere $\\alpha_\\rho(0)=1\/2$ is the well known intercept of the $\\rho$-trajectory; $\\alpha_B(0)\\simeq -0.5$\nis the intercept of the baryon trajectory, $\\alpha_\\Upsilon(0))$ is the intercept  of the \n$\\Upsilon$- Regge trajectory, its value also has an uncertainty. Assuming its linearity \none can get $\\alpha_\\Upsilon(0)=-8, -16$, while for nonlinear ($b{\\bar b}$) Regge trajectory\n$\\alpha_\\Upsilon(0)\\simeq 0$, see details in \\cite{Piskunova}.\nInserting these values to the form\nfor $f_{c({\\bar c})}^{n)}(x)$ and $f_{b({\\bar b})}^{n)}(x)$ we get the large sensitivity\nfor the $c$ and $b$ sea quark distributions in the $n$th Pomeron chain.\nNote that the FFs also depend on the parameters of these Regge trajectories. Therefore,\nthe knowledge of the intercepts and slopes of the heavy-meson Regge trajectories is\nvery important for the theoretical analysis of open charm and beauty production in\nhadron processes.\n \n\\begin{figure}[ht]\n   \\begin{center}\n {\\epsfig{file=lambda_c_v2.eps,width=0.85\\linewidth  }}\n \\caption[Fig.10]{The differential cross section $d\\sigma\/dx$ for the\n inclusive process $pp\\rightarrow\\Lambda_c X$ at $\\sqrt{s}=62~\\mathrm{GeV}$.\n } \n   \\end{center}\n \\end{figure}\n\nThe information on the charmonium ($c{\\bar c}$) and botomonium ($b{\\bar b}$) Regge trajectories\ncan be found from the experimental data on the charmed and beauty baryon production in $pp$ \ncollisions at high energies. For example, Fig.4 illustrates the sensitivity of the inclusive\nspectrum $d\\sigma\/dx$ of the produced charmed  baryons $\\Lambda_c$ to different values for \n$\\alpha_\\psi(0)$. \nThe solid line corresponds to $\\alpha_\\psi(0)=0$, whereas the dashed curve corresponds to \n$\\alpha_\\psi(0)=-2.18$.\nUnfortunately the experimental data presented in Fig.4 have big uncertainties; \ntherefore, one can't extract the information on the $\\alpha_\\psi(0)$ values from the \nexisting experimental data.\n\nA high sensitivity of the inclusive spectrum $d\\sigma\/dx$ of the produced beauty baryons \n$\\Lambda_b$ to different values for $\\alpha_\\Upsilon(0)$ is presented in Fig.5 (left).\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\mbox{\\epsfig{file=lambda_b_v2.eps,width=0.45\\linewidth}} &\n\\mbox{\\epsfig{file=lambda_b_pt2.eps,width=0.45\\linewidth}}\n\\end{tabular}\n\\end{center}\n \\caption[Fig.11]{The differential cross section $d\\sigma\/dx$ (left) and\n$d\\sigma\/dP_t^2$ (right) for the\n inclusive process $pp\\rightarrow\\Lambda_b X$ at $\\sqrt{s}=4~\\mathrm{TeV}$.\n}\n\\end{figure}\nThe $p_t$-inclusive spectrum of $\\Lambda_b$ has much lower sensitivity to this quantity,\naccording to the results presented in Fig.5 (right).\nActually, our results presented in Fig.5 could be considered as some predictions for \nfuture experiments at LHC.\n\\section{Charmed and beauty hadron production in \n$p{\\bar p}$ collisions}\n\\subsection{Heavy flavour meson production at high energies }\nThe production of heavy\nmesons like $D$ and $B$ mesons in proton-antiproton collisions at\nhigh energies is usually analyzed within the different schemes of\nQCD. To study these processes within the QGSM we have to include\nat least one additional graph corresponding to the creation of\nthree chains between quarks in the initial proton and antiquarks\nin the colliding antiproton, as is illustrated in Fig.6(c). \n\nThe diagrams in Fig.6(a,b) are similar to the\none-cylinder and multicylinder diagrams for the $pp$ collision in\nFig.1 with a following difference. In the $p{\\bar p}$\ncollision two colorless strings between quark\/diquark ($q\/qq$) in\nthe initial proton and antiquark\/antidiquark (${\\bar q}\/{\\bar\nqq}$) are created. Many quark-antiquark ($q-{\\bar q}$) strings for\n$p{\\bar p}$ collision (Fig.6b) are the\nsame as for the $pp$ collision (Fig.1, right diagram).\nTherefore, the invariant inclusive spectrum of hadrons produced in\nthe $p{\\bar p}$ collision calculated within the QGSM has the\nfollowing form:\n\n\\begin{align}\nE\\frac{d\\sigma^{p{\\bar p}}}{d^3{\\bf p}} & =\n\\sigma_1(s)[\n(1-\\omega)\\phi^{p{\\bar p}}_1(x,p_t)+\\omega{\\tilde\\phi}(x,p_t)]\n\\notag \\\\\n&\\quad +\\sum_{n=2}^\\infty \\sigma_n(s)\\phi^{p{\\bar p}}_n(x,p_t)\n\\label{def:sppbarp}\n\\end{align}\n\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{cyl1_3chains_new.eps}\n\\caption{The one-cylinder graph (a) and\n the multi-cylinder graph (b), and the three-chains graph \n(c) for the $p{\\bar p}\\rightarrow h X$ inclusive process.}\n\\label{Fig.6}\n\\end{figure}\n\nwhere $1-\\omega$ is the probability of contribution of the cut\none-cylinder (one-Pomeron exchange) and cut multicylinder\n(multiPomeron exchanges) graphs (the left and right\ndiagrams in Fig.6), whereas $\\omega$ is the probability\nof the contribution of the three-chain diagram \nto the inclusive spectrum. The value of $\\omega$ can be estimated\nas the ratio of the $p{\\bar p}$ annihilation cross section\n$\\sigma_{p{\\bar p}}^{ann}$ to the total $p{\\bar p}$ cross section\n$\\sigma_{p{\\bar p}}^{tot}$. The cross section $\\sigma_{p{\\bar\np}}^{tot}$ is well known in the wide range of the initial energies\nto the Tevatron energy, whereas experimental data on\n$\\sigma_{p{\\bar p}}^{ann}$ are available only for the antiproton\ninitial energy up to $10$ GeV, see \\cite{Uzh02} and references\ntherein. However, some theoretical predictions, for example\n\\cite{GosNus80,BZK:1988}, show that asymptotically\n$\\sigma_{p{\\bar p}}^{ann}$ goes to about $2-4$ mb. It corresponds\nto $\\omega\\simeq \\sigma_{p{\\bar p}}^{ann}\/\\sigma_{p{\\bar\np}}^{tot}~<~ 0.1$ at the Tevatron energy. Note that in addition\nto the graph of Fig.6c there can be diagrams consisting of these three chains \nand multicilynder chains between sea quarks and antiquarks. However, as our estimations \nshow, their contribution to the inclusive spectrum is much smaller than the contribution\nfrom the three-chain graph (Fig.6c). Therefore, we neglect \nit. \n\nThe form for the function $\\phi^{p{\\bar p}}_n(x,p_t)$ is similar\nto $\\phi_n(x,p_t)$ entering into (3) by replacing\n$F_{q_v}^{(n)}(x_-,p_t;x_2)$, $F_{q_v}^{(n)}(0,p_t)$ to $F_{\\bar\n{qq}}^{(n)}(x_-,p_t;x_2)$, $F_{\\bar {qq}}^{(n)}(0,p_t)$\nrespectively, and replacing $F_{qq}^{(n)}(x_-,p_t;x_2)$,\n$F_{qq}^{(n)}(0,p_t)$ to $F_{{\\bar q}}^{(n)}(x_-,p_t;x_2)$ and\n$F_{{\\bar q}}^{(n)}(0,p_t)$ respectively. The additional term\n${\\tilde\\phi}(x,p_t)$ in (\\ref{def:sppbarp}) has the\nfollowing form\n\n\\begin{eqnarray}\n{\\tilde\\phi}(x,p_t)=3{\\tilde F}_{q_v}(x_+,p_t){\\tilde F}_{{\\bar\nq}_v}(x_-,p_t)\/{\\tilde F}_q(0,p_t)~, \\label{def:tphi}\n\\end{eqnarray}\nwhere ${\\tilde F}_{q_v({\\bar\nq}_v)}(x_\\pm,p_t)=F^{(n=1)}_{q_v({\\bar q}_v)}(x_\\pm,p_t)$ and\n${\\tilde F}_q(0,p_t)=F_q^{(n=1)}(0,p_t)$.\n\nThe inclusive $p_t$ spectra of $D^0$ and $B^+$ mesons produced in\nthe $p{\\bar p}$ collision at the Tevatron energy $\\sqrt{s}=1.96$\nTeV are presented in Figs.(7,8), see \\cite{LKSB:09}.\nThe hatched regions in\nFigs.(7.8) show the calculations within the NLO\nQCD including uncertainties \\cite{CDF_Dmes}. \nNote that the our calculation showed that the contribution\nof the three-chain graph (Fg.6c) is very small at the\nTevatron energy. It is due to small values of the $p{\\bar p}$ annihilation cross \nsection at very high energies \\cite{GosNus80,BZK:1988}.   \n\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.40\\textwidth]{D0Tev13_04_09_new.eps}\n\\caption{ The inclusive $p_t$-spectrum\nfor $D^0$ mesons produced in the $p{\\bar p}$ collision at the\nTevatron energy $\\sqrt{s}=1.96$ TeV obtained within the QGSM \n\\cite{LKSB:09} (the solid line) and within the NLO QCD\n\\cite{CDF_Dmes} (the hatched regions). \nThe experimental data are taken from \\cite{CDF1,CDF2}}\n\\label{Fig.7}\n\\end{figure}\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.40\\textwidth]{BPTev13_04_09_01_new.eps}\n\\caption{The inclusive $p_t$-spectrum\nfor $B^+$ mesons produced in the $p{\\bar p}$ collision at the\nTevatron energy $\\sqrt{s}=1.96$ TeV obtained within the QGSM\n\\cite{LKSB:09} \n(the solid line) and within the NLO QCD\n\\cite{CDF_Dmes} (the hatched regions).\nThe experimental data \nare taken from \\cite{CDF1,CDF2}}\n\\label{Fig.8}\n\\end{figure}\n\\subsection{Charmed meson production at intermediate energies } \nLet us now use the QGSM to analyze the inclusive charmed meson production in\n${\\bar p}p$ collisions at not large energies, less than 15-20 GeV because it\nwould be interesting for future experiments at the GSI (Darmstadt) planned by the PANDA\nCollaboration. At these energies the contribution of the three-chain graph (Fig.6c)\ncan be sizable because the cross section of the ${\\bar p}p$ annihilation\nis not small \\cite{Uzh02}. Note that at energies close to the threshold of the $D$-meson \nproduction the binary process ${\\bar p}p\\rightarrow{\\bar D}D$ is dominant, according\nto \\cite{Kaid-Volkov:94}. Therefore, the total cross section of $D$ mesons produced in\n${\\bar p}p$ collisions at energies starting from the threshold is the sum of cross\nsections for the binary process $\\sigma_{{\\bar p}p\\rightarrow{\\bar D}D}$ and the inclusive reaction \n $\\sigma_{{\\bar p}p\\rightarrow D X}$. The cross section  $\\sigma_{{\\bar p}p\\rightarrow{\\bar D}D}$\nwas calculated in \\cite{Kaid-Volkov:94} within the simple Regge pole model.   \n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{p_pbar_g2.eps}\n\\caption{ The cross section for $D({\\bar D})$ meson production in \n${\\bar p}-p$ collision as a function of the antiproton momentum $p_{lab}$ in the \nl.s. }\n\\label{Fig.9}\n\\end{figure}\nIn Fig.9 the cross sections $\\sigma_{{\\bar p}p\\rightarrow{\\bar D}D}$ (dashed curve)\nand $\\sigma_{{\\bar p}p\\rightarrow D X}$ (dash-dotted line and dotted line)\nare presented as functions of the incident momentum of the antiproton in the l.s. \nof ${\\bar p}p$. \nThe dash-dotted curve corresponds to our calculation of the\ninclusive process ${\\bar p}p\\rightarrow D({\\bar D})X$ with neglect of the graph of Fig.6c\n($\\omega=0$), and the dotted line corresponds to the similar calculation but with\nthe Fig.6c graph included; the probability for the three-chain graph (Fig.6c) \n$\\omega=\\sigma_{{\\bar p}p}^{ann}\/\\sigma_{{\\bar p}p}^{tot}$ was taken from the fit of\nthe experimental data \\cite{Uzh02}.\nThe solid line is the total yield of $D$ mesons produced in ${\\bar p}p$\ncollision.\n\\section{Conclusion}\n We have shown that the modified QGSM including the\nintrinsic longitudinal and transverse motion of quarks\n(antiquarks) and diquarks in colliding protons allowed us to\ndescribe rather satisfactorily the existing experimental data on\ninclusive spectra of heavy flavour mesons produced in $pp$ collisions \nand to make some predictions for similar spectra at LHC energies. \n\nTo verify whether these predictions can be reliable or not we apply\nthe QGSM to the analysis of charmed and beauty meson production in\nproton-antiproton collisions at Tevatron energies including graphs\nlike those in Fig.6c corresponding to annihilation of\nquarks and antiquarks in colliding $p$ and ${\\bar p}$, and\nproduction of heavy flavour mesons. \nWe got a satisfactory QGSM description ($p_t~<~10$ GeV\/c)\nof the experimental data on\n$p_t$ spectra of $D^0$ and $B^+$ mesons produced in the $p{\\bar\np}$ collisions which were obtained by the CDFII Collaboration at\nthe Tevatron \\cite{CDF_Dmes}. \n\nTo describe these spectra and make some predictions for the future LHC experiments\nin a wide region of transverse momenta one can combine the ``soft QCD'' (the QGSM)\nat small values of $p_t$ with the NLO QCD at large $p_t$.\n\nAt the Tevatron energies the contribution of the three-chain graph (Fig.6c) \nto the inclusive spectra of heavy mesons is too small, as was\nshown in \\cite{LKSB:09}; therefore, it can be neglected. However, at the\nantiproton energies about a few GeV the three-chain graph contribution is\nsizable and can  amount to 30-40 percent, as is shown in Fig.9.\nIt has also been shown that at the incident momenta of antiprotons\n$p_{lab}$ above $8-9$ GeV\/c the inclusive production of $D$ mesons in \n${\\bar p}p$ collisions should be included in addition to the binary \n${\\bar p}p\\rightarrow {\\bar D}D$ process to get the total yield of \n$D$ mesons at $p_{lab}\\leq 14-15$ GeV\/c. These results would be interesting\nfor future experiments at the GSI (Darmstadt) with the antiproton beam.\n\nWe also made some predictions for future LHC forward experiments on the beauty \nbaryon production in $pp$ collisions which can give us new information on\nthe beauty quark distribution in the proton and very interesting information on\nthe Regge trajectories of ($b{\\bar b}$) mesons.\n\n\n\\section{Acknowledgments}\n We thank P. Braun-Munzinger, W. Cassing, M. Deile, S. Dubnicka, A. Z. Dubnickova,\nA. V. Efremov, K. Eggert, D. Elia, A.Galoyan, S. B. Gerasimov, A. B. Kaidalov, Z. M. Karpova, \nB. Z. Kopeliovich, A. D. Martin, K. Peters, T. V. Lyubushkina, M. Poghosyan, K. Safarik, \nV. V. Uzhinsky and U. Wiedner for very useful discussions. \nThis work was supported in part by the RFBR grant N 08-02-01003.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\n\nThe discrete Fourier transform (DFT) can be applied in error correcting codes and code-based cryptography. \nThe cyclotomic DFT method \\cite{Fedorenko03,Fedorenko04} is the best one for computing the DFT over finite field. \nAlexey Maevskiy pointed that formula \\hbox{\\cite[(6)]{Fedorenko04}} in the example of paper \\cite{Fedorenko04} \nhad not been proved. \nWe have corrected this mistake and introduced the proof.\n\n\\section{BASIC NOTIONS AND DEFINITIONS}\n\nThe DFT of length ${n \\mid 2^m-1}$ of a vector \n$f = (f_i)$, \\, $i \\in [0,n-1]$, $f_i \\in GF(2^m)$,\nis the vector $F = (F_j)$\n$$F_j = \\sum_{i=0}^{n-1} f_i \\alpha^{ij}, \\quad j \\in [0,n-1],$$ \nwhere $\\alpha$ is an element of order $n$ in $GF(2^m)$. \nLet us write the DFT in matrix form\n\n\\begin{equation}\nF = Wf,\n\\end{equation}\nwhere $W = (\\alpha^{ij})$, \\, $i,j \\in [0,n-1]$, is a Vandermonde matrix.\nWe assume that the length of the $n$-point Fourier transform\nover $GF(2^m)$ is $n = 2^m - 1$. \n\nLet us consider cyclotomic cosets modulo\n$n=2^m-1$ over $GF(2)$ \n\n\\begin{eqnarray*}\n\\{c_0\\} = \\{0\\}\\\\\n \\{c_1,c_1 2,c_1 2^2, \\ldots,c_1 2^{m_1-1}\\},\\\\\n\\ldots,\\\\\n \\{c_l,c_l 2,c_l 2^2, \\ldots,c_l 2^{m_l-1}\\},\n\\end{eqnarray*}\nwhere $c_k \\equiv c_k 2^{m_k} \\bmod n$,\n$l+1$ is the number of cyclotomic cosets modulo $n$ over $GF(2)$.\n\nLet us introduce the set of indices modulo $n$\n\n$$\nZ = (Z_i) = (c_0,c_1,c_12,c_12^2, \\ldots,c_12^{m_1-1}, \\ldots,$$\n$$c_l,c_l2,c_l2^2, \\ldots,c_l2^{m_l-1}), \\quad i \\in [0,n-1].\n$$\n\nThen, we define a permutation matrix $\\Pi=(\\Pi_{i,j})$, \\, $i,j \\in [0,n-1]$,\n\n$$ \n\\Pi_{i,j} = \n\\left\\{\n\\begin{matrix}\n1, & \\hbox{ if } j = Z_i, \\quad i \\in [0,n-1] \\cr\n0, & \\hbox{ otherwise}.                    \\cr\n\\end{matrix}\n\\right.\n$$\n\nLet us denote a basis\n$\\beta_k=\\left(\\beta_{k,0}, \\ldots ,\\beta_{k,m_k-1}\\right)$ of the subfield $GF(2^{m_k}) \\subset GF(2^m)$.\n\nThen we can write the cyclotomic DFT \\cite{Fedorenko03,Fedorenko04}\n\n\\begin{equation*}\nF_j=\\sum_{k=0}^l\\sum_{s=0}^{m_k-1}a_{k,j,s}\\left(\\sum_{p=0}^{m_k-1}\n\\beta_{k,s}^{2^p}f_{c_k 2^p}\\right),\n\\end{equation*}\nwhere $a_{k,j,s}\\in GF(2)$. \n\nThis equation can be represented in matrix form as\n\n\\begin{equation}\nF=AL (\\Pi f),\n\\end{equation} \nwhere $A$ is a matrix with elements $a_{k,j,s}\\in GF(2)$ \nand $L$ is a block diagonal matrix with elements $\\beta_{k,s}^{2^p}$.\nIf one chooses the normal basis $\\beta_k$, then all the blocks of \nthe matrix $L$ are circulant matrices. \n\nThe inverse DFT in the field $GF(2^m)$ is  \n$$\nf=W^{-1}F.\n$$\n\nIt is easily shown that \n\\begin{equation}\nW^{-1} = E W,\n\\end{equation} \nwhere $E$ is a matrix\n\n$$ \nE =\n\\begin{pmatrix}\n1     & 0     & 0    & \\cdot & 0    & \\cdot & 0    & 0     \\cr\n0     & 0     & 0    & \\cdot & 0    & \\cdot & 0    & 1     \\cr\n0     & 0     & 0    & \\cdot & 0    & \\cdot & 1    & 0     \\cr\n\\cdot & \\cdot &\\cdot & \\cdot & \\cdot& \\cdot & \\cdot& \\cdot \\cr\n0     & 0     & 0    & \\cdot & 1    & \\cdot & 0    & 0     \\cr\n\\cdot & \\cdot &\\cdot & \\cdot & \\cdot& \\cdot & \\cdot& \\cdot \\cr\n0     & 0     & 1    & \\cdot & 0    & \\cdot & 0    & 0     \\cr\n0     & 1     & 0    & \\cdot & 0    & \\cdot & 0    & 0     \\cr\n\\end{pmatrix}.\n$$\n\n\\bigskip\n\n\\section{THE INVERSE CYCLOTOMIC DFT}\n\nSince both matrices $A$ and $L$ are invertible, from (2) \nthe following representation of the inverse DFT can be derived\n\\begin{equation*}\n\\Pi f = L^{-1} A^{-1} F.\n\\end{equation*}\n\n\\begin{Lemma}[\\cite{Hong}]\n\\label{lemma1}\nSuppose $\\beta_k$ are the normal bases,\nthen it is possible to show that blocks of $L^{-1}$ consist\nof elements of bases $\\beta_{k}'$ which are dual to $\\beta_{k}$,\nthat is, the blocks of $L^{-1}$ are also circulant matrices. \n\\end{Lemma}\n\n\\begin{Theorem}\n\\label{theorem1}\nThe inverse cyclotomic DFT for $GF(2^m)$ is\n$$\n(\\Pi E) F = L^{-1} A^{-1} f.\n$$\n\\end{Theorem}\n\n\\begin{IEEEproof}\nFrom (1) and (2) we have \n\n$$\nW = AL\\Pi\n$$ \nand \n$$\nW^{-1} = \\Pi^{-1} L^{-1} A^{-1}.\n$$\n\nFrom the last formula and (3) we obtain \n$$\nW^{-1} = E W = \\Pi^{-1} L^{-1} A^{-1} \n$$\nand\n$$\nW = E^{-1} \\Pi^{-1} L^{-1} A^{-1}. \n$$\n\nHence,\n$$\nF = W f = (E^{-1} \\Pi^{-1}) L^{-1} A^{-1} f.\n$$\nand\n$$\n(\\Pi E) F = L^{-1} A^{-1} f.\n$$\n\\end{IEEEproof}\n\n\\section{EXAMPLES}\n\n\\subsection{DFT of length $n=7$}\n\nLet $(\\gamma,\\gamma^2,\\gamma^4)$ be a normal basis of $GF(2^3)$, \nwhere $\\gamma=\\alpha^3$ and $\\alpha$ is a root of the primitive polynomial $x^3+x+1$.  \nThen the cyclotomic DFT can be represented as\n\n$$\nF =\n\\begin{pmatrix}\nF_0\\\\\nF_1\\\\\nF_2\\\\\nF_3\\\\\nF_4\\\\\nF_5\\\\\nF_6\\\\\n\\end{pmatrix}=\n\\begin{pmatrix}\n1&1&1&1&1&1&1\\\\\n1&0&1&1&1&0&0\\\\\n1&1&0&1&0&1&0\\\\\n1&1&0&0&1&0&1\\\\\n1&1&1&0&0&0&1\\\\\n1&0&0&1&0&1&1\\\\\n1&0&1&0&1&1&0\n\\end{pmatrix}\n$$\n$$\n\\times\n\\begin{pmatrix}\n1&0       &0       &0         &0       &0       &0       \\\\\n0&\\gamma^1&\\gamma^2&\\gamma^4  &0       &0       &0       \\\\\n0&\\gamma^2&\\gamma^4&\\gamma^1  &0       &0       &0       \\\\\n0&\\gamma^4&\\gamma^1&\\gamma^2  &0       &0       &0       \\\\\n0&0       &0       &0         &\\gamma^1&\\gamma^2&\\gamma^4\\\\\n0&0       &0       &0         &\\gamma^2&\\gamma^4&\\gamma^1\\\\\n0&0       &0       &0         &\\gamma^4&\\gamma^1&\\gamma^2\n\\end{pmatrix}\n$$\n$$\n\\times\n\\begin{pmatrix}\n 1 &0 &0 &0 &0 &0 &0 \\\\\n 0 &1 &0 &0 &0 &0 &0 \\\\\n 0 &0 &1 &0 &0 &0 &0 \\\\\n 0 &0 &0 &0 &1 &0 &0 \\\\\n 0 &0 &0 &1 &0 &0 &0 \\\\\n 0 &0 &0 &0 &0 &0 &1 \\\\\n 0 &0 &0 &0 &0 &1 &0 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nf_0\\\\\nf_1\\\\\nf_2\\\\\nf_3\\\\\nf_4\\\\\nf_5\\\\\nf_6\\\\\n\\end{pmatrix}\n= AL \\Pi f.\n$$\n\nUsing Theorem~\\ref{theorem1}, we obtain the inverse cyclotomic DFT \n\n$$\n(\\Pi E) F\n$$\n$$ =\n\\begin{pmatrix}\n 1 &0 &0 &0 &0 &0 &0 \\\\\n 0 &1 &0 &0 &0 &0 &0 \\\\\n 0 &0 &1 &0 &0 &0 &0 \\\\\n 0 &0 &0 &0 &1 &0 &0 \\\\\n 0 &0 &0 &1 &0 &0 &0 \\\\\n 0 &0 &0 &0 &0 &0 &1 \\\\\n 0 &0 &0 &0 &0 &1 &0 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n 1 &0 &0 &0 &0 &0 &0 \\\\\n 0 &0 &0 &0 &0 &0 &1 \\\\\n 0 &0 &0 &0 &0 &1 &0 \\\\\n 0 &0 &0 &0 &1 &0 &0 \\\\\n 0 &0 &0 &1 &0 &0 &0 \\\\\n 0 &0 &1 &0 &0 &0 &0 \\\\\n 0 &1 &0 &0 &0 &0 &0 \\\\\n\\end{pmatrix}\n$$\n$$\n\\times\n\\begin{pmatrix}\nF_0\\\\\nF_1\\\\\nF_2\\\\\nF_3\\\\\nF_4\\\\\nF_5\\\\\nF_6\\\\\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1&0       &0       &0         &0       &0       &0       \\\\\n0&\\gamma^1&\\gamma^2&\\gamma^4  &0       &0       &0       \\\\\n0&\\gamma^2&\\gamma^4&\\gamma^1  &0       &0       &0       \\\\\n0&\\gamma^4&\\gamma^1&\\gamma^2  &0       &0       &0       \\\\\n0&0       &0       &0         &\\gamma^1&\\gamma^2&\\gamma^4\\\\\n0&0       &0       &0         &\\gamma^2&\\gamma^4&\\gamma^1\\\\\n0&0       &0       &0         &\\gamma^4&\\gamma^1&\\gamma^2\n\\end{pmatrix}\n$$\n$$\n\\times\n\\begin{pmatrix}\n1&1&1&1&1&1&1\\\\\n1&0&0&1&1&1&0\\\\\n1&1&0&1&0&0&1\\\\\n1&0&1&0&0&1&1\\\\\n1&1&0&0&1&0&1\\\\\n1&1&1&0&0&1&0\\\\\n1&0&1&1&1&0&0\n\\end{pmatrix}\n\\begin{pmatrix}\nf_0\\\\\nf_1\\\\\nf_2\\\\\nf_3\\\\\nf_4\\\\\nf_5\\\\\nf_6\\\\\n\\end{pmatrix}\n= L^{-1} A^{-1} f.\n$$\n\nFinally note that the last formula coincides with formula \\hbox{\\cite[(6)]{Fedorenko04}}.\n\n$$\n\\begin{pmatrix}\nF_0\\\\\nF_6\\\\\nF_5\\\\\nF_3\\\\\nF_4\\\\\nF_1\\\\\nF_2\\\\\n\\end{pmatrix}\n= L^{-1} A^{-1}\n\\begin{pmatrix}\nf_0\\\\\nf_1\\\\\nf_2\\\\\nf_3\\\\\nf_4\\\\\nf_5\\\\\nf_6\n\\end{pmatrix}.\n$$\n\nThis algorithm requires 6 multiplications and \n24 additions and appears to be the best known 7-point DFT for $GF(2^3)$.\n\n\\subsection{DFT of length $n=15$}\n\nLet $(\\alpha^{ 3},\\alpha^{ 6},\\alpha^{12},\\alpha^{ 9})$ and $(\\alpha^{11},\\alpha^{ 7},\\alpha^{14},\\alpha^{13})$\nbe the normal bases of $GF(2^3)$, \nwhere $\\alpha$ is a root of the primitive polynomial $x^4+x+1$.  \nThen the cyclotomic DFT can be represented as formula (4) or $F=AL (\\Pi f)$.\nThe inverse cyclotomic DFT can be written as formula (5) or $(\\Pi E) F = L^{-1} A^{-1} f$.\n\n\\section*{ACKNOWLEDGMENT}\n\nThe author would like to thank Alexey Maevskiy for pointing out a methodological mistake in the paper \\cite{Fedorenko04},\nand Peter Trifonov for helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we shall use a manifestly covariant form of statistical\nmechanics which has more general structure than the standard forms\nof relativistic statistical mechanics, but which reduces to those theories\nin a certain limit, to be described precisely below. These theories,\nwhich are characterized classically by mass-shell constraints, and the\nuse, in quantum field theory, of fields which are constructed on the\nbasis of on-mass-shell free fields, are associated with the\nstatistical treatment of {\\it world lines} and hence, considerable\ncoherence (in terms of the macroscopic structure of whole world lines\nas the elementary objects of the theory) is implied. In\nnonrelativistic statistical mechanics, the elementary objects of the\ntheory are points.\n The relativistic  analog of this essentially structureless foundation for a statistical theory is the set of points in spacetime, i.e., the so-called {\\it events}, not the world lines (Currie, Jordan and Sudarshan \\cite {Sudarshan} have discussed the difficulty of constructing a relativistic mechanics on the basis of world lines).\n\nThe mass of particles in a mechanical theory of events is necessarily a dynamical variable, since the classical phase space of the relativistic set of events consists of the spacetime and energy-momentum coordinates $\\{{\\bf q}_i, t_i;\\  {\\bf p}_i,E_i\\}$, with no {\\it a priori} constraint on the relation between the ${\\bf p}_i$ and the $E_i$, and hence  such theories are ``off-shell''. It is well known from the work of Newton and Wigner \\cite{NW} that on-shell relativistic quantum theories such as those governed by Klein-Gordon or Dirac type equations do not provide local descriptions (the wave functions corresponding to localized particles are  spread out); for such theories the notion of ensembles over local initial conditions is difficult to formulate. The off-shell theory that\nwe shall use here is,  however, precisely local in both its first and\nsecond quantized forms \\cite{AH,Shnerb}.\n\nWe finally remark that the standard formulations of  quantum relativistic\nstatistical mechanics, and quantum field theory at finite temperature, lack\nmanifest covariance on a fundamental level. As for nonrelativistic statistical\nmechanics, the partition function is described by the Hamiltonian, which is not\nan invariant object, and hence thermodynamic mean values do not have tensor\nproperties. [One could consider the invariant $p_\\mu n^\\mu$ in place of the\nHamiltonian \\cite{Synge}, where $n^\\mu$ is a unit four-vector; this construction (supplemented by a spacelike vector othogonal to $n^\\mu$) implies an induced\nrepresentation for spacetime. The quantity that takes the place of the\nparameter $t$ is then $x_\\mu n^\\mu$; in the corresponding quantum mechanics,\nthe space parts of (induced form of) the momentum do not commute with\nthis time variable. Some of the problems associated with this construction are closely related to those pointed out by Currie, Jordan and Sudarshan \\cite{Sudarshan}, for which\ndifferent world lines are predicted dynamically by the change in the form of the effective Hamiltonian in different frames.] Since the form of such a theory is not constrained by covariance requirements, its dynamical structure\nand predictions may be different than for a theory which satisifies these\nrequirements. For example, the canonical distribution of Pauli \\cite{Pauli}\nfor the free Boltzmann gas has a high temperature limit in which the energy is\ngiven by $3k_BT$, which does not correspond to  any known\nequipartition rule, but for the corresponding distribution for the\nmanifestly covariant theory, the limit is $2k_BT$, corresponding to\n${1\\over 2}k_BT$ for each of the four relativistic degrees of freedom\n\\cite{HSP,BH1}. For the quantum field theories at finite temperature,\nthe path integral formulation \\cite {Kap1} replaces the Hamiltonian in\nthe canonical exponent by the Lagrangian due to the infinite product of factors $\\langle\\phi \\vert \\pi\\rangle$ (transition matrix element of the\ncanonical field and its conjugate required to give a Weyl ordered Hamiltonian\nits numerical value). However, it is the $t$ variable which is analytically\ncontinued to construct the finite temperature canonical ensemble, completely\nremoving the covariance of the theoretical framework. One may argue that some\nframe has to be chosen for the statistical theory to be developed, and perhaps\neven for temperature to have a meaning, but as we have remarked above, the\nrequirement of relativistic covariance has dynamical consequences (note that the model Lagrangians used in the non-covariant formulations are established with\nthe criterion of relativistic covariance in mind), and we argue that the choice\nof a frame, if necessary for some physical reason, such as the definition  and\nmeasurement of temperature, should be made in the framework of a manifestly\ncovariant  structure.\n\nIn the framework of a manifestly covariant relativistic statistical\nmechanics, the dynamical evolution of a system of $N$ particles, for the\nclassical case, is governed by equations of motion that are of the form\nof Hamilton equations for the motion of $N$ $events$ which generate the\nspace-time trajectories (particle world lines) as functions of a\ncontinuous Poincar\\'{e}-invariant parameter $\\tau $ \\cite{Stu,HP}, usually \nreferred to as a ``proper time''. These events are characterized by\ntheir positions $q^\\mu =(t,{\\bf q})$ and energy-momenta $p^\\mu =(\nE,{\\bf p})$ in an $8N$-dimensional phase-space. For the quantum case, the\nsystem is characterized by the wave function $\\psi _\\tau (q_1,q_2,\\ldots\n,q_N)\\in L ^2(R^{4N}),$ with the measure $d^4q_1d^4q_2\\cdots d^4q_N\\equiv\nd^{4N}q,$ $(q_i\\equiv q_i^\\mu ;\\;\\;\\mu =0,1,2,3;\\;\\;i=1,2,\\ldots ,N),$\ndescribing the distribution of events, which evolves with a generalized\nSchr\\\"{o}dinger equation \\cite{HP}. The collection of events (called\n``concatenation'' \\cite{AHL}) along each world line corresponds to a\n{\\it particle,} and hence, the evolution of the state of the $N$-event\nsystem describes, {\\it a posteriori,} the history in space and time of\nan $N$-particle system.\n\nFor a system of $N$ interacting events (and hence, particles) one takes\n\\cite{HP}\n\\beq\nK=\\sum _i\\frac{p_i^\\mu p_{i\\mu }}{2M}+V(q_1,q_2,\\ldots ,q_N),\n\\eeq\nwhere $M$ is a given fixed parameter (an intrinsic property of the\nparticles), with the dimension of mass, taken to be the same for all the\nparticles of the system. The Hamilton equations are\n$$\\frac{dq_i^\\mu }{d\\tau }=\\frac{\\partial K}{\\partial p_{i\\mu }}=\\frac{p_\ni^\\mu }{M},$$\n\\beq\n\\frac{dp_i^\\mu }{d\\tau }=-\\frac{\\partial K}{\\partial q_{i\\mu }}=-\\frac{\n\\partial V}{\\partial q_{i\\mu }}.\n\\eeq\nIn the quantum theory, the generalized Schr\\\"{o}dinger equation\n\\beq\ni\\frac{\\partial }{\\partial \\tau }\\psi _\\tau (q_1,q_2,\\ldots ,q_N)=K\n\\psi _\\tau (q_1,q_2,\\ldots ,q_N)\n\\eeq\ndescribes the evolution of the $N$-body wave function\n$\\psi _\\tau (q_1,q_2,\\ldots ,q_N).$ To illustrate the meaning of this\nwave function, consider the case of a single free event. In this case\n(1.3) has the formal solution\n\\beq\n\\psi _\\tau (q)=(e^{-iK_0\\tau }\\psi _0)(q)\n\\eeq\nfor the evolution of the free wave packet. Let us represent $\\psi _\\tau\n(q)$ by its Fourier transform, in energy-momentum space:\n\\beq\n\\psi _\\tau (q)=\\frac{1}{(2\\pi )^2}\\int d^4pe^{-i\\frac{p^2}{2M}\\tau }\ne^{ip\\cdot q}\\psi _0(p),\n\\eeq\nwhere $p^2\\equiv p^\\mu p_\\mu ,$ $p\\cdot q\\equiv p^\\mu q_\\mu ,$ and $\\psi\n_0(p)$ corresponds to the initial state. Applying the Ehrenfest arguments\nof stationary phase to obtain the principal contribution to $\\psi _\\tau\n(q)$ for a wave packet at $p_c^\\mu ,$ one finds ($p_c^\\mu $ is the peak\nvalue in the distribution $\\psi _0(p))$\n\\beq\nq_c^\\mu \\simeq \\frac{p_c^\\mu }{M}\\tau ,\n\\eeq\nconsistent with the classical equations (1.2). Therefore,\nthe central peak of the wave packet moves along the classical\ntrajectory of an event, i.e., the classical world line.\n\nIt is clear from the form of (1.3) that one can construct  relativistic\ntransport theory in a form analogous to that of the nonrelativistic theory; a\nrelativistic Boltzmann equation and its consequences, for example, was studied\nin ref. \\cite{HSS}.\n\nSince, in the Hilbert space $L^2(R^4)$ the operators $x^\\mu ,p^\\nu $ obey the canonical commutation relations $(g^{\\mu \\nu }={\\rm diag}(-,+,+,+))$\n\\beq\n[x^\\mu ,p^\\nu ]=i\\hbar g^{\\mu \\nu },\n\\eeq\nthe uncertainty relations\n\\beq\n\\triangle x\\;\\triangle p\\geq\\frac{\\hbar }{2},\\;\\;\\;\n\\triangle t\\;\\triangle E\\geq\\frac{\\hbar }{2}\n\\eeq\nfollow directly from the mathematical structure of the theory, and are\non the same footing (with the usual statistical interpretation\n\\cite{FW}). The dispertion $\\triangle t$ is a property of the wave\nfunction $\\psi _\\tau (q)$ at a given $\\tau .$\n\nArshansky and Horwitz \\cite{AH} have studied thought experiments analogous to those discussed by Landau and Peierls \\cite{LP}, within the framework of the manifestly covariant relativistic quantum theory which we are using here, and derived the (causal) Landau-Peierls relation\n\\beq\n\\triangle t\\;\\triangle p\\stackrel{>}{\\sim }\\frac{\\hbar }{c},\n\\eeq\nconcerning the time interval $\\triangle t$ during which the momentum of a particle is measured, and the momentum dispersion of the state. \n\nIn this paper we shall discuss another uncertainty relation following\ndirectly from the mathematical structure of the theory which is\nrealized on a statistical mechanical level (for an ensemble of events),\n\\beq\nT_{\\triangle V}\\;\\triangle m\\stackrel{>}{\\sim }\\frac{2\\pi \\hbar}{c^2},\n\\eeq\nwhere $\\triangle m$ is the mass dispersion around the on-shell value\ndue to the off-shellness of the events making up the ensemble, and\n$T_{\\triangle V}$ is the average passage interval in $\\tau $ for the\nevents which pass through the small (typical) four-volume $\\triangle\nV$ in the neighborhood of the $R^4$-point.\\footnote{This result is\nanalogous to the {\\it nonrelativistic} $\\triangle t\\;\\triangle E$\nrelation, which, as for (1.10), does not follow from commutation\nrelations and the Schwartz inequality.  The nonrelativistic time-energy \nuncertainty relation, in fact, follows from (1.10) and (1.11) in the\n  nonrelativistic \nlimit for which $\\tau \\rightarrow t,$ $\\langle E\\rangle \/M \\rightarrow 1,$\nand $\\triangle m\\;c^2 \\rightarrow \\triangle E$. This result implies\nthe existence, in the non-relativistic limit, of a residual ensemble\nover $t$, consistently with the treatment of the non-relativistic\nlimit given in ref. [7].} The four-volume $\\triangle V$ is the smallest that can be considered a macrovolume in representing the ensemble. $T_{\\triangle V}$ is related to the (average) extent of the ensemble along the time axis, through the Hamilton equation (1.2) for $\\mu =0$ (in the sense of a statistical average), \n\\beq\n\\frac{\\triangle t}{T_{\\triangle V}}=\\frac{\\langle E\\rangle }{M},\n\\eeq\nif the ensemble is constructed with the minimum time span to\ncharacterize the physical system.\n\n\\section{Ideal relativistic gas of events}\nTo describe an ideal gas of events in the grand canonical ensemble, we\nuse the expression for the number of events given in \\cite{HSP} (in the following we shall use the system of units in which $\\hbar =c=k_B=1,$ unless otherwise specified),\n\\beq\nN=\\sum _{p^\\mu }n_{p^\\mu }=\\sum _{p^\\mu }\\frac{1}{e^{(E-\\mu -\\mu _K\\frac{m^2}{2M})\/T}\\mp 1},\n\\eeq\nwhere $E\\equiv p^0,$ $m^2 \\equiv  -p^2 = - p^\\mu p_\\mu ,$ and the sign in the denominator of (2.1) is determined by the event statistics in the usual way;\n$\\mu _K$ is an additional mass potential \\cite{HSP}, which arises in the\ngrand canonical ensemble as the derivative of the free energy with respect to\nthe value of the dynamical evolution function $K$, interpreted as the invariant\nmass of the system. In the kinetic theory \\cite{HSP}, $\\mu _K$ enters as a Lagrange multiplier for the equilibrium distribution for $K,$ as $\\mu $ is for $N$ and $1\/T$ for $E.$  In order to simplify subsequent considerations, we\nshall take it to be a fixed parameter. \n\nWe restrict ourselves, in the following, to the case of the events obeying Bose-Einstein statistics and use, therefore, the relation (2.1) with the minus sign in the denominator. \nTo ensure a positive-definite value for $n_{p^\\mu },$ the number density of bosons with four-momentum $p^\\mu ,$ we require that\n\\beq\nm-\\mu -\\mu _K\\frac{m^2}{2M}\\geq 0.\n\\eeq\nThe discriminant for the l.h.s. of the inequality must be nonnegative,\ni.e.,\n\\beq\n\\mu \\leq \\frac{M}{2\\mu _K}.\n\\eeq\nFor such $\\mu ,$ (2.2) has the solution\n\\beq\nm_1\\equiv \\frac{M}{\\mu _K}\\left( 1-\\sqrt{1-\\frac{2\\mu \\mu _K}{M}}\\right) \\leq\nm\\leq\n\\frac{M}{\\mu _K}\\left( 1+\\sqrt{1-\\frac{2\\mu \\mu _K}{M}}\\right) \\equiv m_2.\n\\eeq\nFor small $\\mu \\mu _K\/M,$ the region (2.4) may be approximated by\n\\beq\n\\mu \\leq m\\leq \\frac{2M}{\\mu _K}.\n\\eeq\nOne sees that $\\mu _K$ plays a fundamental role in determining an upper\nbound of the mass spectrum, in addition to the usual lower bound $m\\geq\n\\mu .$ In fact, small $\\mu _K$ admits a very large range of off-shell\nmass, and hence can be associated with the presence of strong\ninteractions \\cite{MS}. For our present purposes it will be sufficient\nto assume that the mass distribution has a finite range  $m_1\\leq\nm\\leq m_2$ around the on-shell value $m_c= M\/\\mu_K$ corresponding to the \nlimiting value for which the inequality (2.3) becomes an equality.\n\nIn order to show that our results hold independent of the dimensionality of spacetime, we shall consider our ensemble in \none temporal and $D$ spatial dimensions, $D\\geq 1.$\n\nReplacing the sum over $p^\\mu $ (2.1) by an integral, $$\\sum _{p^\\mu }\\Longrightarrow \\frac{V^{(1+D)}}{(2\\pi )^{1+D}}\\int d^{1+D}p,$$\nwhere $V^{(1+D)}$ is the system's $(1+D)$-volume, and using the relation $(p^\\mu =(p^0,{\\bf p}))$ $$d^{(1+D)}p=\\frac{d^Dp}{2p^0}dm^2,\\;\\;m^2\\equiv -p^\\mu p_\\mu ,\\;\\;\\mu =0,1,\\ldots ,D,$$ one obtains for the density of events per unit $(1+D)$-volume, $n\\equiv N\/V^{(1+D)},$\n\\beq\nn=\\int _{m_1}^{m_2}\\frac{dm\\;m}{2\\pi }\\int \\frac{d^D{\\bf p}}{(2\\pi )^Dp^0}f(p),\n\\eeq\nwith $f(p)\\equiv n_{p^\\mu },$ as given in Eq. (2.1). \nTypical average values are given by the relations\n\\beq\n\\langle p^\\mu \\rangle =\\frac{1}{n}\\int _{m_1}^{m_2}\\frac{dm\\;m}{2\\pi }\n\\int \\frac{d^D{\\bf p}}{(2\\pi )^Dp^0}\\;p^\\mu f(p),\n\\eeq\n\\beq\n\\langle p^\\mu p^\\nu  \\rangle =\\frac{1}{n}\\int _{m_1}^{m_2}\\frac{dm\\;m}{2\\pi }\n\\int \\frac{d^D{\\bf p}}{(2\\pi )^Dp^0}\\;p^\\mu p^\\nu f(p),\\;\\;\\;{\\rm etc.}\n\\eeq\n\nTo find the expressions for the pressure and energy density in our ensemble, we study the particle energy-momentum tensor defined by the relation\\footnote{The corresponding relation of ref. \\cite{HSS} is given in four-dimensional spacetime.} \\cite{HSS}\n\\beq\nT^{\\mu \\nu }(q)=\\sum _i\\int d\\tau \\frac{p_i^\\mu p_i^\\nu }{m_c}\n\\delta ^{1+D}(q-q_i(\\tau )),\n\\eeq\nin which $m_c$ is the value around which the mass of the events making up the \nensemble is distributed. Upon integrating over a small $(1+D)$-volume \n$\\triangle V$ and taking the ensemble average, (2.9) reduces to \\cite{HSS}\n\\beq\n\\langle T^{\\mu \\nu }\\rangle =\\frac{T_{\\triangle V}}{m_c}n\\langle\np^\\mu p^\\nu \\rangle .\n\\eeq\nIn this formula, $n=N\/V$, and $T_{\\triangle V}$ is the average passage interval in\n$\\tau $ for the events which pass through $\\triangle V,$ which we discussed above. The formula (2.10) reduces, through Eq. (2.8), to\n\\beq\n\\langle T^{\\mu \\nu }\\rangle =\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\int \\frac{d^D{\\bf p}}{(2\\pi )^Dp^0}\\;p^\\mu p^\\nu f(p).\n\\eeq\nUsing the standard expression\n\\beq\n\\langle T^{\\mu \\nu }\\rangle =pg^{\\mu \\nu }-(p+\\rho )u^\\mu u^\\nu ,\n\\eeq\nwhere $p$ and $\\rho $ are the particle pressure and energy density,\nrespectively, we obtain $$\\rho =\\langle T^{00}\\rangle,\\;\\;\\;p=\\frac{1}{D}g^{ii}\\langle T_{ii}\\rangle ,\\;\\;i=1,2,\\ldots, D,$$ and therefore, through (2.11),\n\\beq\np=\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\frac{{\\bf p}^2}{Dp^0}f(p),\n\\eeq\n\\beq\n\\rho =\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\;p^0 f(p).\n\\eeq\nWe now calculate the particle number density per\nunit $D$-volume. The particle $D+1$-current is given by the\nformula \\cite{HSS}\n\\beq\nJ^\\mu (q)=\\sum _i\\int d\\tau \\frac{p^\\mu _i}{m_c}\\delta ^{1+D}(q-q_i(\\tau )),\n\\eeq\nwhich upon integrating over a small $(1+D)$-volume and taking the\naverage reduces to\n\\beq\n\\langle J^\\mu \\rangle =\\frac{T_{\\triangle V}}{m_c}n\\langle p^\\mu \\rangle ;\n\\eeq\nthen the {\\it particle} number density \\cite{Synge,Hakim} is\n\\beq\nN_0\\equiv \\langle J^0\\rangle =\\frac{T_{\\triangle V}}{m_c}n\\langle E\\rangle ,\n\\eeq\nso that, with the help of Eq. (2.7),\n\\beq\nN_0=\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\;f(p).\n\\eeq\nSince\n\\beq \np=\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\frac{{\\bf p}^2}{Dp^0}f(p)\\equiv p(m),\n\\eeq\n\\beq\n\\rho =\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\;p^0 f(p)\\equiv \\rho (m)\n\\eeq\nand\n\\beq\nN_0=\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\;f(p)\\equiv N_0(m)\n\\eeq\nare the standard expressions for the pressure, energy density and particle number density in $1+D$ dimensions, respectively \n\\cite{HW,Act,1+D}, we have the following relations:\n\\beq\np=\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;p(m),\n\\eeq\n\\beq\n\\rho =\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;\\rho (m),\n\\eeq\n\\beq\nN_0=\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;N_0(m).\n\\eeq\nIt is seen in these relations that the manifestly covariant framework\nprovides {\\it a linear mass spectrum, independent of the dimensionality of spacetime}. In order to obtain the expressions for the basic thermodynamic quantities, one has to integrate the standard (on-shell) results over this spectrum within the range of the mass distribution. \n\nUsing the formulas (2.22)-(2.24), one can establish the uncertainty relation (1.10) for a narrow mass width around the on-shell value: as $m\\rightarrow m_c,$ \n\\beq\n\\int _{m_1}^{m_2}dm\\;m\\;f(m)\\rightarrow \\triangle m\\;m_c\\;f(m_c),\n\\eeq\nwhere $f(m)$ stands for each of the $p(m),\\rho (m),N_0(m),$ and $\\triangle m$ is the (infinitesimal) width of the mass distribution around $m_c.$ The requirement that the results for $p,\\rho $ and $N_0$ coincide with those of the usual on-shell theories implies $p=p(m_c),\\rho =\\rho (m_c),N_0=N_0(m_c)$ in Eqs. (2.22)-(2.24), which leads, with Eq. (2.25), to the relation\\footnote{In c.g.s. units, this relation has a factor $\\hbar \/c^2$ on the right hand side.}\n\\beq\nT_{\\triangle V}\\triangle m=2\\pi ,\n\\eeq\nthe case of equality in the relation (1.10),\nin agreement with the results obtained earlier in ref. \\cite{glim}.\nOne can understand this relation, up to a numerical factor, in terms of\nthe uncertainty principle (rigorous in the $L^2(R^4)$ quantum theory)\n$\\triangle E\\cdot \\triangle t \\geq 1\/2.$ Since the time\ninterval for the particle to pass the volume $\\triangle V$ (this smallest\nmacroscopic volume is bounded from below by the size of the wave packets)\n$\\triangle t\\simeq E\/M \\;\\triangle \\tau ,$ and the dispersion of $E$ due\nto the mass distribution is $\\triangle E\\sim m\\triangle m\/E,$ one obtains\na lower bound for $T_{\\triangle V}\\triangle m$ of order unity.\n\n\\section{Mass-proper time uncertainty relation}\nWe now wish to prove the relation (1.10) for the general case, not\nonly for the case of a narrow mass width as done in the previous\nsection. First, we note that we previously considered a relativistic\nensemble without degeneracy; therefore no degeneracy factor appeared\nin the expressions for the basic thermodynamic quantities. Now suppose\nthat we have $\\nu$ internal degrees of freedom in our ensemble which correspond\nto degeneracy. In this case considerations made previously will remain valid and lead to the formulas (2.22)-(2.24) with the extra factor of $\\nu$ on their right hand side:\n\\beq\np=\\frac{T_{\\triangle V}\\nu}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;p(m),\n\\eeq\n\\beq\n\\rho =\\frac{T_{\\triangle V}\\nu}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;\\rho (m),\n\\eeq\n\\beq\nN_0=\\frac{T_{\\triangle V}\\nu}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;N_0(m).\n\\eeq\nOn the other hand, according to our previous arguments,\none can consider the $\\nu$ degrees of freedom as being distributed in the mass interval $m_1\\leq m\\leq m_2$ with a linear mass spectrum, \n\\beq \n\\tau (m)=Cm,\n\\eeq\nwhich leads to the formula\n\\beq\np=\\int _{m_1}^{m_2}dm\\;\\tau (m)\\;p(m),\n\\eeq\nand analogous relations for $\\rho $ and $N_0$ (similar to the\ntreatment of a strongly interacting system by means of a particle\nresonance spectrum \\cite{Shu}). In fact, the linear mass spectrum finds its confirmation in the experimental hadronic resonance spectrum: if one calculates, for example, the pressure in the hadronic resonance gas by summing up the individual contributions of a finite number of the different hadronic species with the corresponding degeneracies, $$p=\\sum _ig_i\\;p(m_i),\\;\\;\\;p(m_i)=\\frac{T^2m_i^2}{2\\pi ^2}K_2(m_i\/T),$$ and by using the formula (3.5), in which $m_1$ and $m_2$ are the masses of the lightest and the heaviest species, respectively, the results coincide \\cite{linear}.\n\nThe normalization constant $C$ is determined by the condition\n\\beq\n\\int _{m_1}^{m_2}\\tau (m)\\;dm=\\nu;\n\\eeq\ntherefore \n\\beq\nC=\\frac{2\\nu}{m_2^2-m_1^2}=\\frac{\\nu}{\\triangle m\\;(m_1+m_2)\/2},\n\\eeq\nwhere $\\triangle m=m_2-m_1$ is the width of the mass distribution in a general case. Since $m_c$ should be associated with one of the averages $\\langle m\\rangle $ or $\\langle m^2\\rangle $ which both are closer to $m_2$ than to $m_1$ for a linear spectrum [$(m_1+m_2)\/2\\stackrel{<}{\\sim } m_c$], \n\\beq\nC\\stackrel{>}{\\sim }\\frac{\\nu}{m_c\\;\\triangle m}.\n\\eeq\n\nDirect comparison of the formulas (3.1)-(3.3) with the relation (3.5) and analogous formulas for $\\rho $ and $N_0,$ with $\\tau (m)$ given by (3.4), \n\\beq\np=C\\int _{m_1}^{m_2}dm\\;m\\;p(m),\n\\eeq\n\\beq\n\\rho =C\\int _{m_1}^{m_2}dm\\;m\\;\\rho (m),\n\\eeq\n\\beq\nN_0=C\\int _{m_1}^{m_2}dm\\;m\\;N_0(m),\n\\eeq\nleads to the relation $$C=\\frac{T_{\\triangle V}\\nu}{2\\pi m_c},$$ which reduces,\nthrough (3.8), to\n\\beq\nT_{\\triangle V}\\triangle m\\stackrel{>}{\\sim }2\\pi ,\n\\eeq \nthe relation (1.10) for the general case of the finite range mass distribution.\n\nIn order that our considerations be valid, the effective degeneracy in\na relativistic ensemble should be large, $\\nu >>1,$ so that one could\nspeak of the distribution of $\\nu$ degrees of freedom in a finite mass\nrange. This is the case for realistic physical systems such as high temperature strongly interacting hadronic matter \\cite{Shu}. \n\n\\section{Concluding remarks}\nIn this paper we have proved the uncertainty relation (1.10) for the\ngeneral case of a finite range of mass distribution in a relativistic\nensemble. This relation allows one to admit the following picture of a\nstrongly interacting system: one can consider a strongly interacting\nsystem as a distribution of free particles which  temporarily go off-shell\nwhile undergoing an interaction. Then $T_{\\triangle V}$ may be\nassociated with the time for the particle to remain very close to its mass shell. So, for weakly interacting systems, $\\triangle m\\simeq 0,$ $T_{\\triangle\nV}\\simeq \\infty ,$ i.e., the particle remains on-shell almost all the\ntime. In contrast, for a strongly interacting system, $\\mu _K\\simeq 0$\n\\cite{MS}; then, in view of (2.5), $\\triangle m\\simeq \\infty ,$ and\n$T_{\\triangle V}\\simeq 0,$ i.e., the particle is off-shell almost\nalways (because it undergoes interaction almost continuously).\n\nWe have remarked that the non-relativistic limit of the mass-``proper\ntime'' uncertainty relation provides a new derivation of the $\\triangle\nE \\triangle t  \\stackrel{<}{\\sim } \\hbar $ relation of the\nnon-relativistic theory.  This result implies, as we pointed out, the\nexistence of a residual ensemble over $t$, even in the\nnon-relativistic limit. Such an ensemble has been introduced recently\n\\cite{lax} in order to achieve an exact semigroup (exponential decay)\nlaw of evolution for the reduced motion of an unstable system.  The\nusual derivation\\cite{jammer} of the non-relativistic energy-time uncertainty\nrelation studies the motion of the system under the action of the full\nHamiltonian relative to the eigenstates of unperturbed energy; this\nprocedure corresponds precisely to that of Weisskopf and\nWigner\\cite{wigner} for the description of the decay of unstable\nsystems. It is argued in refs. \\cite{lax} that the introduction of an\nensemble over $t$ is necessary for achieving the semigroup property as\nwell as for the consistency of the interpretation.\n\n\nThe uncertainty relation (1.10) may be useful in practical calculations concerning realistic strongly interacting systems in which the particles are necessarily off-shell. For example, it may allow one to estimate the relaxation times for the quark-gluon plasma created in ultrarelativistic heavy-ion collisions.\n\n\\bigskip\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDue to the strong field dynamics, critical phenomena arises in gravitational collapse in the threshold of black hole formation. In a similar way to phase transitions in thermodynamics, taking the mass of black hole as an order parameter, critical gravitational collapse can be classified as type I or type II. In type I collapse the final black hole mass has a minimum finite value, whereas in type II collapse the black hole mass can be arbitrary small.\n\t\nHistorically, M. Choptuik discovered a critical phenomena in gravitational collapse while studying numerically the gravitational collapse of a real massless scalar field, a result which was later named critical gravitational collapse of type II~\\cite{PhysRevLett.70.9}. He found that for a family of initial data parametrized by some arbitrary parameter $p$, the scalar field is completely dispersed to infinity for $p<p^*$, with $p^*$ some critical value, while for $p>p^*$ the final black hole mass follows a power-law scaling relation of the form:\n\\begin{equation}\nM \\propto (p-p^*)^\\gamma \\; .\n\\end{equation}\nThe critical solution $p=p^*$ that separates both states exhibits universality, {\\em i.e.} it does not depend on the way in which the family is parameterized.  Additionally, for different types of matter the critical solution can have either continuous self-similarity (CSS), or discrete self-similarity (DSS). In particular, for the case of a real massless scalar field the critical solution wa found to have DSS. This property is best appreciated in a logarithmic time defined as: \n\\begin{equation}\nT = -\\ln ( \\tau^*-\\tau ) \\; ,\n\\end{equation}\nwith $\\tau$ some measure of time which is usually taken as the proper time at the origin, and $\\tau^*$ the so-called accumulation time. In this logarithmic time $T$ the solution is periodic with period $\\Delta$. This property is known as ``scale echoing''. For the real massless scalar field the critical exponents have been found to be $\\gamma \\approx 0.374$ and $\\Delta\\approx3.445$, via both numerical simulations and semi-analytical studies \\cite{PhysRevLett.70.9,Rinne:2020asi,PhysRevD.49.890,PhysRevD.51.5558,Hamad__1996,PhysRevD.92.084037}.\n    \nOn the other hand, critical collapse of type I was later discovered by Choptuik et. al. \\cite{PhysRevLett.77.424} while studying the critical collapse of a Yang-Mills field. In contrast to the critical collapse of type II, in this case the final black hole mass has a minimum finite value, and there is a different scaling law of the form:\n\\begin{equation}\n\\label{eq:tau_scaling}\n\\tau \\propto -\\gamma \\ln|p-p^*| \\; ,\n\\end{equation}\nwhere $\\tau$ now measures the time that a given solution remains near the critical solution. Additionally, the critical solution itself is either stationary or periodic in time.\n    \nOne can expect type I critical collapse when in the field equations there exists either a mass or a length scale that is relevant to the dynamics. On the contrary, when the equations do not contain a length scale, or when such a length scale is not relevant, type II phenomena occur. Both types can coexists in different regions in the parameter space of initial data, as for example in the case of a real massive scalar field~\\cite{PhysRevD.56.R6057}, where the type of critical phenomena depends on the size of the Compton wavelength of the field when compared to the width of initial data (an excellent review about these type of critical phenomena can be found in~\\cite{Gundlach:2007gc}).\n   \nIn this paper we study the critical collapse of a massive complex scalar field. A previous study was done by Hawley and Choptuik in \\cite{PhysRevD.62.104024}, showing that the critical solution for the complex scalar field corresponds to stationary solutions to the Einstein-Klein-Gordon known in the literature as boson stars~\\cite{PhysRev.172.1331,PhysRev.187.1767,Visinelli:2021uve,Liebling:2012fv}. These solutions are determined by assuming spherical symmetry, and by the requirement that the metric coefficients must be static, while the complex scalar field has a harmonic time dependence of the form:\n\\begin{equation}\\label{eq:BS_ansatz}\n\\Phi(t,r) = \\varphi(r)e^{i\\omega t} \\; ,\n\\end{equation}\nwith $\\omega$ a real valued frequency, and $\\varphi(r)$ a real valued function of radius only. Taking the mass parameter of the complex scalar field as $m$, the maximum possible mass of a boson star has been found to be $M_{max}\\approx0.633 \\: M_{Planck}^2\/m$, corresponding to a central value of the scalar field of $\\varphi_{max} = \\varphi(0) \\approx 0.271$, see for example~\\cite{PhysRevD.42.384,GLEISER1989733} (though this value can change depending on the normalization, see below for our normalization choice).  This central value of the field separates the boson star configurations into two branches depending on their stability properties. If $\\varphi(0)<\\varphi_{max}$ the boson star is stable under small perturbations, whereas for $\\varphi(0)>\\varphi_{max}$ the configurations are unstable. The lowest energy solution for a boson star for a given value of $\\varphi(0)$, also known as the ground state, has no nodes in the scalar field. Excited states are classified depending on the number nodes of the field in the radial direction.\n\nThe critical solutions found by Hawley and Choptuik corresponded to unstable boson stars in the ground state, and were obtained by perturbing a stable boson star that interacted gravitationally with a small pulse of a massless scalar field that acts as the perturbation. In our study we take a different approach, and we begin with a simple gaussian pulse in the complex field with a variable width. We then evolve this initial data and vary the amplitude of the gaussian pulse until a critical solution is found. \n    \nSince the mass of the complex scalar field introduces a scale, we expect our system to display both types of critical phenomena depending on the width of the initial gaussian pulse. In a similar way as in the case of a real massive scalar field~\\cite{PhysRevD.56.R6057}, we will explore both types of critical behaviour by changing the width of our initial pulse. Furthermore, if one performs a linear perturbative analysis for critical phenomena of both type I and II, the critical exponent $\\gamma$ can be shown to be the inverse of the so-called Lyapunov exponent $\\chi$ of the system $\\gamma=1\/\\chi$~\\cite{Gundlach:2007gc}.\\footnote{The Lyapunov exponent measures the stability of a system due to changes in its initial conditions. For close trajectories in phase space parametrized by $t$, and initial points separated by an infinitesimal distance $\\delta$, the Lyapunov exponent quantifies their rate of separation as $F(t,x_0+\\delta)-F(t,x_0)\\approx\\delta e^{\\chi t}$, with $\\chi$ the Lyapunov exponent of the system. For $\\chi>0$ the trajectories diverge, whereas for $\\chi<0$ they do not.}\n\nFor the case of critical collapse of type I, we will compare our critical solutions with the known solutions for stationary boson stars. Furthermore, we can also compare our critical exponents with the Lyapunov exponents for the unstable modes of boson stars. On the other hand, if the critical phenomena of type II we will limit our study to finding the critical exponent $\\gamma$ and the echoing period $\\Delta$.\n    \nThis paper is organized as follows.  In Section~\\ref{sec:scalarfield} we discuss the field equations for a complex massive scalar field and our initial data. In Section~\\ref{sec:code} we discuss our numerical code, gauge conditions and diagnostic tools. Section~\\ref{sec:results} show the results of our numerical simulations.\n\n\n\n\\section{Complex scalar field}\n\\label{sec:scalarfield}\n\n\n\n\\subsection{The Einstein Klein-Gordon equations}\n\nOur matter model consists of a massive complex scalar field $\\Phi$ coupled minimally to gravity, which can be described by the action (in units such that $G=c=1$):\n\\begin{equation}\nS = \\int d^4x \\sqrt{-g} \\left[ \\frac{R}{16\\pi}\n- \\frac{1}{2} \\left( \\nabla^\\mu\\Phi\\nabla_\\mu \\Phi^*\n+ m^2\\Phi\\Phi^* \\right) \\right] \\; ,\n\\end{equation}\nwhere $R$ is the Ricci scalar of the spacetime and $m$ is the mass parameter of the field (notice that this fixes our normalization choice). Varying the action with respect to the metric and the scalar field one obtains the Einstein field equations:\n\\begin{equation}\nR_{\\mu \\nu} - \\frac{1}{2} g_{\\mu \\nu} R = 8 \\pi T_{\\mu \\nu} \\; ,\n\\end{equation}\nand the Klein--Gordon equation:\n\\begin{equation}\\label{eq:KG}\n\\nabla^{\\mu} \\nabla_{\\mu} \\Phi-m^{2} \\Phi=0 \\; ,\n\\end{equation}\nwhere the stress-energy tensor $T_{\\mu \\nu}$ for the scalar field is given by:\n\\begin{equation}\nT_{\\mu\\nu} = \\frac{1}{2} \\left[ \\left( \\nabla_{\\mu} \\Phi \\nabla_{\\nu} \\Phi^{*}\n+ \\nabla_{\\nu} \\Phi \\nabla_{\\mu} \\Phi^{*} \\right)\n- g_{\\mu\\nu} \\left( \\nabla^\\alpha \\Phi \\nabla_\\alpha \\Phi^* + m^2 \\Phi \\Phi^* \\right) \\right] \\; .\n\\end{equation}\n\nIn order to study numerically the evolution of the system, we will use the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of general relativity \\cite{PhysRevD.52.5428,PhysRevD.59.024007}, which is known to be strongly hyperbolic \\cite{PhysRevD.66.064002}. Particularly, as we are only interested in the case of spherical symmetry, we will use the BSSN formulation adapted to curvilinear coordinates as described in~\\cite{PhysRevD.79.104029,Alcubierre:2011pkc}. In spherical symmetry, we will adopt the line element given by:\n\\begin{equation}\n\\label{eq:ds4}\nds^2 = -\\alpha^2 dt^2 + \\psi^4 \\left( A dr^2 + r^2 B d \\Omega^2 \\right) \\; ,\n\\end{equation}\nwhere $(\\alpha,\\psi,A,B)$ are functions of $(t,r)$ only, and $d\\Omega^2=d\\theta^2+\\sin^2\\theta\\ d\\varphi^2$ is the standard solid angle element.\n\nIn order to recast the Klein--Gordon equation as a first order system we define the following auxiliary variables\n\\begin{equation}\n\\Pi := \\frac{\\partial_t\\Phi}{\\alpha} \\; , \\qquad \\chi := \\partial_r \\Phi \\; .\n\\end{equation}\nWith these definitions, the Klein--Gordon equation \\eqref{eq:KG} can be rewritten as:\n\\begin{eqnarray}\n\\partial_t\\Phi &=& \\alpha\\Pi \\; , \\\\\n\\partial_t\\chi &=& \\alpha \\partial_r \\Pi + \\Pi \\partial_r \\alpha \\; , \\\\\n\\partial_t\\Pi &=& \\frac{\\alpha}{A\\psi^4} \\left[ \\partial_{r} \\chi\n+ \\chi \\left( \\frac{2}{r} - \\frac{\\partial_{r} A}{2 A} + \\frac{\\partial_{r} B}{B}\n+ 2 \\partial_{r} \\ln \\psi \\right) \\right]\n+ \\frac{\\chi \\partial_{r} \\alpha}{A \\psi^{4}} + \\alpha K \\Pi - \\alpha m^2 \\;L ,\n\\end{eqnarray}\nwith $K:=K^m_m$ the trace of the extrinsic curvature of the spatial hypersurfaces of constant time.\n\nFrom the orthogonal decomposition of the stress-energy tensor:\n\\begin{equation}\nT^{\\mu \\nu} = S^{\\mu \\nu} + J^{\\mu} n^{\\nu} + n^{\\mu} J^{\\nu} + \\rho n^{\\mu} n^{\\nu} \\;,\n\\end{equation}\nwe obtain the energy density $\\rho:=n^{\\mu} n^{\\nu} T_{\\mu \\nu}$, the momentum density $J_\\mu:= -P_{\\ \\mu}^{\\nu} n^{\\lambda} T_{\\nu \\lambda}$, and the stress tensor \\mbox{$S_{\\mu \\nu}:=P_{\\mu}^{\\sigma} P_{\\nu}^{\\lambda} T_{\\sigma \\lambda}$}, where $n^\\mu=(1\/\\alpha,0,0,0)$ is the unit normal vector to the spatial hypersurfaces and $P^\\mu_{\\ \\nu}:=\\delta^\\mu_{\\ \\nu}+n^\\mu n_\\nu$ is the projection operator.  For the complex scalar field we find in particular:\n\\begin{eqnarray}\n\\rho &=& \\frac{1}{2} \\left( |\\Pi|^{2} + \\frac{|\\chi|^{2}}{A \\psi^{4}} + m^2|\\Phi|^2 \\right) \\; , \\\\\nJ_r &=& -\\frac{1}{2} \\left( \\rule{0mm}{5mm} \\chi \\Pi^{*} + \\Pi \\chi^{*} \\right) \\; , \\\\\nS_r^r &=& \\frac{1}{2} \\left( |\\Pi|^{2} + \\frac{|\\chi|^{2}}{A \\psi^{4}} - m^2|\\Phi|^2 \\right) \\; , \\\\\nS_{\\theta}^{\\theta} &=& \\frac{1}{2} \\left( |\\Pi|^{2} - \\frac{|\\chi|^{2}}{A \\psi^{4}}\n- m^{2}|\\Phi|^{2} \\right) \\; ,\n\\end{eqnarray}\n\n\n\n\\subsection{Initial data}\n\nIn \\cite{PhysRevD.62.104024}, Hawley and Choptuik showed that the critical solution for the cas of a massive complex scalar field is an unstable boson star. In their study the critical solution was obtained by perturbing a boson star in the stable branch with a real massless scalar field, and tuning the amplitude of the massless field up to the threshold of black hole formation. Here we will take a different approach, by considering as our initial condition a simple pulse of complex scalar field with the following gaussian profile:\n\\begin{eqnarray}\n\\Phi(t=0,r) &=& \\Phi_0 e^{-r^2\/\\sigma^2} \\label{eq:phi} \\; , \\\\\n\\Pi(t=0,r) &=& i \\kappa \\Phi_0 e^{-r^2\/\\sigma^2} \\label{eq:pi} \\; ,\n\\end{eqnarray}\nwhere $\\Phi_0,\\sigma,\\kappa$ are real parameters, and $\\Phi_0$ is the tuning amplitude of the initial pulse. In order to find the initial data for the geometry we assume a conformally flat spatial metric, so that $A=B=1$ in~\\eqref{eq:ds4}, and proceed to solve the constraint equations. Notice that even though equations \\eqref{eq:phi} and \\eqref{eq:pi} do not formally represent an instant of time symmetry, the momentum density $J_r$ is still zero, so the momentum constraint is trivially satisfied. On the other hand, at $t=0$ the hamiltonian constraint becomes a nonlinear second order differential equation for conformal factor $\\psi$ of the form:\n\\begin{equation}\n\\partial^2_r\\psi + \\frac{2}{r} \\: \\partial_r \\psi + 2 \\pi \\psi^5 \\rho = 0 \\; ,\n\\label{eq:ham_const}\n\\end{equation}\nwith the energy density given by:\n\\begin{equation}\n\\rho = \\frac{1}{2} \\left[ | \\Pi |^2 + \\frac{|\\partial_r \\Phi|^2}{\\psi^4} + m^2 |\\Phi|^2 \\right] \\; .  \n\\end{equation}\nThe above non-linear equation is solved numerically by using an iterative method. Boundary conditions for equation \\eqref{eq:ham_const} are obtained from the asymptotically flatness condition, which implies:\n\\begin{equation}\n\\psi(r)|_{r\\rightarrow\\infty} = 1 \\; .\n\\end{equation} \nIn practice, however, we use a Robin boundary type condition at a finite radius corresponding to the edge of our numerical grid:\n\\begin{equation}\n\\partial_r\\psi = \\frac{1-\\psi}{r} \\; .\n\\end{equation}\nThis condition reflects the fact that as $r\\rightarrow\\infty$ we have $\\psi\\rightarrow1+{\\cal O}(r^{-1})$. On the other hand, regularity at the origin implies that $\\psi$ must be an even function of $r$, so that:\n\\begin{equation}\n\\partial_r \\psi|_{r=0} = 0 \\; .\n\\end{equation}\nThe initial conditions given by equations~\\eqref{eq:phi}-\\eqref{eq:pi} were not chosen randomly. At $t=0$ they are similar to the harmonic ansatz for boson stars, Eq.~\\eqref{eq:BS_ansatz}, with $\\kappa$ taking the role of the oscillation frequency $\\omega$ of the boson star, but instead of using the profile for a stable boson star we use a simple gaussian profile. Notice that $\\kappa$ is a free\nparameter, which we later choose to be equal to the mass $m$ of the scalar field for simplicity.\n\n\n\n\\section{Numerical code and diagnostics}\n\\label{sec:code}\n\n\\subsection{Code and gauge choices}\n\nWe integrate the Einstein--Klein--Gordon system with the OllinSphere code, a numerical relativity finite-difference code suited for spherical symmetry which evolves the BSSN formulation of the Einstein equations. This code has been previously used for example in~\\cite{Alcubierre:2019qnh,Degollado:2020lsa}, but it has now been updated to include the possibility of fixed mesh refinement around the origin, so that if the outer boundary is located at $r_{max}$ with grid resolution $\\Delta r$, the local boundary of the $N$ refinement level is situated at $r_{max}\/2^{N-1}$ with  resolution $\\Delta r\/2^{N-1}$, and the grid structure remains fixed during the evolution.\n\nTo close the system we also need to specify the lapse function $\\alpha$ and the shift vector $\\beta$. In our simulations we choose for the lapse the standard $1+\\log$ slicing condition:\n\\begin{equation}\n\\partial_t\\alpha = -2 \\alpha K \\; ,\n\\end{equation}\nwith $K$ the trace of the extrinsic curvature. We choose for the initial value of the lapse a pre-collapsed profile of the form $\\alpha(t=0)=\\psi^{-2}$, with $\\psi$ the initial conformal factor.\nAlso, for simplicity we choose a vanishing shift vector for all our simulations. Indeed, this choice was already assumed in the line element~\\eqref{eq:ds4}.\n\n\n\n\\subsection{Diagnotics}\n\nAs is usual in the study of critical behavior, the final state of the evolution is classified depending on the strength of the initial data. We report our initial amplitude precision in finding the critical solution via the dimensionless quantity: \n\\begin{equation}\n\\delta\\Phi = \\frac{\\Phi_c - \\Phi_d}{\\Phi_d} \\; ,\n\\end{equation}\nwhere $\\Phi_c$ is the highest amplitude for which the initial data is dispersed and leaves behind Minkowski spacetime, and $\\Phi_d$ is the lowest amplitude for which a black hole is formed. The critical value for the amplitude $\\Phi^*$ is a metastable state which separates the two behaviors described previously. We have found that in order to obtain the critical exponents correctly we need an accuracy of at least $\\delta\\Phi\\sim10^{-6}$, o even higher. This value can be improved by using a finer grid, and results in less uncertainty in the value of the critical exponents, and also in a longer evolution near the critical solution for type I critical collapse.\n\nSince we are mostly interested in the subcritical case, for the supercritical simulations we will not follow the evolution until the black hole settles down to equilibrium, which in any case would require a non-zero shift vector. With our slicing choice, the lapse function at the origin will return to one if the initial scalar field pulse is dispersed to infinity. Otherwise, if a black hole is formed, the lapse will collapse to zero at the origin.\n\nIn order to detect when a black hole is formed we will search at every time step for an apparent horizon. This procedure is done by looking for a location where the expansion of outgoing null geodesics becomes zero (see for example~\\cite{Alcubierre:1138167}):\n\\begin{equation}\n\\frac{1}{\\psi^{2} \\sqrt{A}} \\left( \\frac{2}{r} + \\frac{\\partial_{r} B}{B}\n+ 4 \\frac{\\partial \\psi}{\\psi} \\right) - 2 K_{\\theta}^{\\theta} = 0 \\; .\n\\end{equation}\nHere $K_\\theta^\\theta$ is simply the angular component of the extrinsic curvature with mixed indices.\n\n\n\n\\subsection{Characterizing type I critical solutions}\n\nIn Eq.~\\eqref{eq:tau_scaling} we take $\\tau$ as the proper time measured by an observer located at the origin $r=0$ at a point in the evolution when a first apparent horizon is located. As explained before, the critical solutions of type I for the complex scalar field should correspond to an unstable boson star. We should emphasize, however, that our initial conditions for the complex scalar field given by Eqs.~\\eqref{eq:phi}-\\eqref{eq:pi} with the critical amplitude $\\Phi_0^*$ do not correspond to a boson star at $t=0$, unstable or otherwise. This implies that for our near critical simulations the excess of scalar field will be radiated to infinity and the remaining content should approach a boson star in the unstable branch.\n\nSince boson stars do not have a well defined boundary, one can describe their size by means of the so-called $R_{95}$ or $R_{99}$ radius, which correspond to the areal radius of a sphere containing $95\\%$ or $99\\%$ of the total mass $M_T$, respectively. Furthermore, since the system is not stationary the integrated mass will be a function of $M(t,r)$. To determine if a compact object has formed, we inspect the compactness function defined as:\n\\begin{equation}\\label{eq:compactness}\nC(t,r) = \\frac{M(t,r)}{R(t,r)} \\; ,\n\\end{equation}\nwhere $R(t,r)$ is the areal radius of a sphere at a given time $t$ and coordinate radius $r$. We look for the global maximum as a function of $r$ for every time step. We expect that if a boson star has formed, there will be a maximum mean value of $C(t,r)$, plus some small oscillations around it corresponding to perturbations of this star. This behavior will tell us if a compact object has formed or not, and will also provide us with an approximate lifetime of the critical solution obtained.\n\nIn Eq.~\\eqref{eq:compactness} we estimate the mass of the configuration by using the Kodama mass~\\cite{Kodama:1979vn, PhysRevD.91.084057,PhysRevD.96.064047,Racz:2005pm}, which is a quasi-local conserved energy in a spherically symmetric spacetime. The Kodama vector is defined by:\n\\begin{equation}\nK^A = \\epsilon^{AB} \\partial_{B} R \\; ,\n\\end{equation}\nwhere $R$ is the areal radius of a sphere at constant $t$ and $r$, $\\epsilon^{A B}$ is the totally antisymetric tensor in the two-dimensional manifold with coordinates $(t,r)$, and the indices $(A,B)$ run over $(0,1)$. The vector $K^A$ can be naturally extended to the four dimensional manifold  \nby setting to zero the remaining components. Next, we define the four vector $S^\\mu$ as follows:\n\\begin{equation}\n{\\cal{S}}^\\mu = T^{\\mu \\nu} K_\\nu \\; ,\n\\end{equation}\nwhere $T^{\\mu \\nu}$ is the stress-energy tensor. It is possible to show that $S^\\mu$ is a conserved current, so it satisfies the conservation law:\n\\begin{equation}\n\\partial_\\mu \\left( \\sqrt{-g} {\\cal S}^\\mu \\right) = 0 \\; ,\n\\end{equation}\nIn a sphere of radius $r_0$ at constant $t$, we can then define a conserved mass (the so-called Kodama mass) as:\n\\begin{equation\nM\\left(t, r_{0}\\right) := \\int_{\\text {sphere }} S^{t} \\alpha \\sqrt{\\gamma} \\:d x^{3} \\;,\n\\end{equation}\nwhere $\\gamma$ is the determinant of the 3-metric, and where we used the fact that $-g = \\alpha \\gamma$. Using our expression for the spatial metric this reduces to:\n\\begin{equation}\\label{eq:M_Kodama}\nM(t r_{0}) := 4 \\pi \\int_0^{r_0}  \\alpha {\\cal S}^{t} r^2 \\psi^6 A^{1\/2} B \\: dr \\; ,\n\\end{equation}\nOn the other hand, from the above definitions the quantity $S^t$ can now be expressed as:\n\\begin{equation}\n{\\cal S}^t = \\frac{r}{\\alpha} \\sqrt{\\frac{B}{A}} \\left[ \\rho \\left( 2 r \\chi\n+ \\frac{\\partial_r B}{2 B} + \\frac{1}{r} \\right)\n- J_r K_\\theta^\\theta\\right] \\; .\n\\end{equation}\nIn a similar way, one can also define the integrated energy flux through the sphere of radius $r_0$ as:\n\\begin{equation}\n\\label{eq:P_Kodama}\nP(t_1,t_2;r_0) := 4 \\pi \\int_{t_1}^{t_2} \\left. \\left( \\alpha {\\cal S}^r \\sqrt{\\gamma} \\right) \\right|_{r_0} dt \\; ,\n\\end{equation}\nwhere ${\\cal S}^r$ is now given by: \n\\begin{equation}\n{\\cal S}^r = r \\sqrt{\\frac{B}{A}} \\left[ \\frac{1}{\\psi^4A} \\left( 2 r \\chi\n+ \\frac{\\partial_r B}{2B} + \\frac{1}{r} \\right)\n- S_r^r K_\\theta^\\theta \\right] \\; .\n\\end{equation}\nWith the definitions \\eqref{eq:M_Kodama} and \\eqref{eq:P_Kodama}, the conservation law can be rewritten as:\n\\begin{equation}\n\\partial_t \\left[ M(t,r_0) + P(t_0,t;r_0) \\right] = 0 \\; .\n\\end{equation}\n\nWe still need to estimate the integration radius $r_0$. In order to do this we use the fact that stationary boson stars are well characterized in the literature, and our code is able to find such solutions (see for example~\\cite{Alcubierre:2018ahf}).  For a near critical simulation we then first obtain the mean value of the scalar field amplitude at the origin, $\\left< \\Phi(t,r=0) \\right>$. Having found this mean amplitude, we construct the corresponding boson star stationary solution with that same amplitude, and choose $r_0$ as the $R_{99}$ radius of that stationary solution.\n\n\n\n\\subsection{Characterizing type II critical collapse}\n\nAs the mass of the scalar field introduces a length (or mass) scale, following \\cite{PhysRevD.56.R6057} we expect that if $\\sigma m \\ll 1$ we should observe critical collapse of type II, whereas if $\\sigma m \\gg 1$ we should find collapse of type I.  Finding the value of the critical exponent $\\gamma$ using the black hole mass scaling is somewhat difficult since we would need to follow the black hole until it reaches an equilibrum configuration, something that is not trivial to do numerically. Instead, we will consider subcritical evolutions since in a critical collapse of type II the maximum value of the 4D Ricci scalar will then follow the scaling law:\n\\begin{equation}\nR_{max} \\approx |\\Phi_0^*-\\Phi_0|^{-2\\gamma} \\; ,\n\\end{equation}\nwhere the $-2$ factor in the exponent is there because the Ricci scalar has units of lenght$^{-2}$. Additionally to this behavior, the discrete self-similarity of the phenomena adds a fine structure to the scaling law \\cite{PhysRevD.55.R440}, so the Ricci scalar in fact will behave as:\n\\begin{equation}\n\\label{eq:4DR_1}\n\\ln R_{max} = c - 2 \\gamma \\ln |\\Phi_0^* - \\Phi_0| + f( \\ln |\\Phi_0^*-\\Phi_0| ) \\; ,\n\\end{equation}\nwith $c$ some constant, and where $f$ is a periodic function with angular frequency:\n\\begin{equation}\n\\omega = \\Delta \/2 \\gamma \\; ,\n\\end{equation}\nwith $\\Delta$ is the so-called echoing period. To leading order, $f$ can be approximated by:\n\\begin{equation}\nf(x) = a_0 \\sin \\left( \\omega x + \\varphi \\right) \\; ,\n\\end{equation}\nwith $\\varphi$ some arbitrary phase. The 4D Ricci scalar then behaves as:\n\\begin{equation}\n\\label{eq:4DR_2}\n\\ln R_{max} = c - 2\\gamma \\ln |\\Phi_0^*-\\Phi_0|\n+ a_0 \\sin \\left( \\omega \\ln |\\Phi_0^*-\\Phi_0| + \\varphi \\right) \\; ,\n\\end{equation}\nwhere the constants $c,a_0,\\varphi$ depend on the form of the initial data family.\n\n\n\n\\section{Numerical results}\n\\label{sec:results}\n\nAll our simulations were done with fourth order centered differences in space, and fourth order Runge--Kutta for the evolution in time. For simplicity, we fix the scalar field mass to $m=1$. Also, for the family of initial data~\\eqref{eq:phi}-\\eqref{eq:pi} we set $\\kappa=m=1$ for all cases. The values chosen for the width parameter $\\sigma$ will be reported below. The parameter $\\Phi_0$ is then adjusted until we find the black hole formation threshold with the desired accuracy.\n\nTo reduce the source of errors in our simulations we use constraint preserving boundary conditions. These have already been described and used for example in~\\cite{Alcubierre:2014joa,PhysRevD.86.104044}, and they help to reduce the errors coming in from the boundaries by a factor of about $10^3$ when compared with the standard Sommerfeld (radiative) boundary conditions. The error introduced by the finite difference method can also be diminished by using Kreiss--Oliger numerical dissipation. In all our evolutions we use sixth-order dissipation in order to be compatible with the fourth-order discretization. The artificial dissipation dampens high frequency modes that would otherwise spoil the numerical stability of the near-critical solutions. Resolution also affects the critical behavior, for this reason we will report relevant quantities for our highest resolution simulations.\n\n\n\n\\subsection{Type II critical collapse}\n\nAs already stated before, we expect type II critical collapse for $\\sigma m \\ll 1$. To check this, we choose $\\sigma\\leq0.5$ and proceed to find the critical amplitude $\\Phi_0^*$ using a bisection method. Since studying critical phenomena requires high numerical precision, instead of using many levels of refinement, which introduce reflections at the refinement boundaries, we will use just one grid level with a radial transformation of coordinates from the original coordinate radius $r$ to a new coordinate $\\tilde{r}$ related to $r$ through:\n\\begin{equation}\n\\label{eq:rnew}\n\\frac{dr}{d\\tilde{r}} = \\frac{1}{1+e^{\\beta r^2+\\delta}} \\; .\n\\end{equation}\nWith this transformation a uniform grid in $\\tilde{r}$ becomes a non-uniform grid in $r$. This coordinate transformation was first used in~\\cite{PhysRevD.105.064071} for studying the critical behavior of scalar-tensor theories of gravity in the Jordan frame. In Eq. \\eqref{eq:rnew}, $\\delta$ adjusts the resolution near the origin $\\tilde{r}=0$, while $\\beta$ measures how fast $\\tilde{r}$ approaches $r$ far away. Notice that as the transformed radial coordinate approaches infinity $\\tilde{r}\\rightarrow\\infty$, we have $dr\/d\\tilde{r}\\rightarrow 1$. For our simulations we use $\\delta=5$ and $\\beta=-1$, with a grid spacing $\\Delta \\tilde{r}=0.005$, and $N_r=2500$ points in radial direction. We also use an adaptive time step in order to satisfy the Courant--Friedrichs--Levi (CFL) stability condition. With these settings we were able to find the critical amplitude with a precision of $\\delta\\Phi\\approx10^{-12}$.\n\nFigure~\\ref{fig:lnRmax} shows the maximum value of the 4D Ricci scalar at the origin obtained from subcritical evolutions for $\\sigma=0.5$. From the Figure we can clearly see the expected behaviour for type II critical collapse. All cases $\\sigma\\leq0.5$, fitting the function \\eqref{eq:4DR_2} results in very similar critical exponent $\\gamma \\approx 0.38 \\pm 0.01$ and echoing period $\\Delta \\approx 3.4\\pm 0.1$. These values are very similar to those found in the literature for the case of a real massless scalar field.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/lnRmax.pdf}\n\\caption{Scaling of the maximum value of the 4D Ricci scalar for subcritical simulations of a massive complex scalar field, using the initial data family \\eqref{eq:phi}-\\eqref{eq:pi}, with gaussian width $\\sigma=0.5$.}\n\\label{fig:lnRmax}\n\\end{figure}\n\n\n\n\\subsection{Type I critical collapse}\n\nAs already mentioned, for the case when $\\sigma m\\gg 1$ we expect to find type I critical collapse. To investigate this, we start from $\\sigma=2.5$ and consider higher values of the gaussian width. Once the value of $\\sigma$ has been chosen, for each case we proceed to find the critical amplitude $\\Phi_0^*$ with the bisection method. For these simulations we have used fixed mesh refinement with $N=4$ levels. We have observed that the lifetime of the nearest critical solution obtained initially increases as $\\sigma$ is increased, reaching its maximum at $\\sigma \\approx 4.0$, and then decreases again for higher values of $\\sigma$.\n\nTo set the position of the outer boundary, we first estimate the time of black hole formation $t_{BH}$ using low resolution runs. We then multiply $t_{BH}$ the asymptotic gauge speed $v_g=\\sqrt{2}$ of the $1+\\log$ slicing condition (as it is larger than the coordinate speed of light $v_l=1$). A perturbation that starts at the origin and bounces at the boundary will take at least twice this time to return to the origin (in fact longer since the lapse is smaller than one near the origin), so we set the outer boundary at $r_{max}=t_{BH}\/\\sqrt{2}$. The position of the outer boundary $r_{max}$, and the resolution of the base grid $\\Delta r$, are displayed in Table~\\ref{tab:resolution_bdry}. In all these simulations we use a fixed time step compatible with the CFL condition.\n\n\\begin{table}\n\\begin{tabular}{@{}|c|c|c|@{}}\n\t\\hline\n\t\\multicolumn{1}{|c|}{$\\sigma$} & \\multicolumn{1}{c|}{$\\Delta r$} & \\multicolumn{1}{c|}{$r_{max}$ } \\\\ \\hline\n\t2.5      &            & 150       \\\\\n\t2.75     & 0.1        & 200       \\\\\n\t3        &            & 250       \\\\ \\hline\n\t3.5      &            & 400       \\\\\n\t4        & 0.15       & 550      \\\\\n\t5        &            & 400       \\\\ \\hline\n\t6        &            & 350       \\\\\n\t7        &            & 325       \\\\\n\t8        & 0.1        & 325       \\\\\n\t9        &            & 290       \\\\\n\t10       &            & 225       \\\\ \\hline\n\\end{tabular}\n\\caption{Resolution and position of the outer boundary for each value of the width parameter $\\sigma$.}\n\\label{tab:resolution_bdry}\n\\end{table}\n\nFigure~\\ref{fig:compactness-norm} shows the maximum value of the compactness function (top panel), and the norm of the complex scalar field at the origin (bottom panel), for a near critical solution with initial width $\\sigma=2.5$. Since the initial profile consists of a gaussian pulse, the scalar field has not agglomerated to form a compact object. After $t \\approx 25$ a portion of the scalar field has been radiated away and the remainder starts to oscillate around a mean value, indicating that a compact object has been formed. As this state is unstable, the object eventually disperses after $t \\approx 225$. It is important to mention, however, that for some values of the initial amplitude $\\Phi_0$ the scalar field does not disperse completely, and the remaining bulk oscillates around the origin. This behavior is similar to that observed by Lai and Choptuik in~\\cite{Lai:2007tj}. In their study, the remaining bulk can be described as excitations of the fundamental mode of stable boson stars. However we will leave the study of this phenomenon for a future work.\n\nIn the time interval $25 < t < 225$, we obtain the mean value of the norm of the complex field at the origin, and then look for the $R_{99}$ of the corresponding boson star. Figure~\\ref{fig:KodamaMass} shows the Kodama mass for the same subcritical evolution measured at that radius. Again, we observe an oscillation around a mean value, and the dispersion of the object after some time. We can also obtain the oscillation frequency of the scalar field by applying a Fast Fourier Transform (FFT) to the central value of its real and imaginary parts. Figure~\\ref{fig:freq_sigma_2p5} plots the frequency obtained after applying the FFT. We can clearly see a very narrow peak centered at $\\omega=0.7933$.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/compactness_s=2.5.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/complex_phi_norm_s=2.5.pdf}\n\\caption{Top panel: Maximum value of the compactness function for a near critical evolution with $\\sigma=2.5$. Bottom panel: Norm of the scalar field at the origin.  Initially we see no indication that a compact object has formed. However, from $t \\approx 25$ up to $t \\approx 225$ we can appreciate a clear oscillation around a mean value. Since this is a subcritical case, we see dispersion of the object for $t > 225$.}\n\\label{fig:compactness-norm}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/mass_s=2.5.pdf}\n\\caption{Kodama mass for the same simulation of Figure~\\ref{fig:compactness-norm}. After obtaining the mean value of the norm of the scalar field at the origin we find the $R_{99}$ of the corresponding boson star. We then evaluate the Kodama mass at that radius.}\n\\label{fig:KodamaMass}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/w-sigma_2p5.pdf}\n\\caption{Fourier transform of the central value of the scalar field for $\\sigma=2.5$. Both the real and imaginary parts have a very narrow peak centered at a frequency $\\omega=0.7933$.}\n\\label{fig:freq_sigma_2p5}\n\\end{figure}\n\nWe summarize the results for all our simulations in Table~\\ref{tab:all_data}, where we report the mean value of the norm of the scalar field at the origin, its oscillation frequenc\n, and the Kodama mass of the critical solution. The uncertainty in the norm of the complex field and its mass are calculated from the standard deviation, and the uncertainty in the frequency is reported as half of the peak width in the FFT. The uncertainty in the critical exponent is obtained from the method of least squares applied to equation~\\eqref{eq:tau_scaling}.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|}\n   \t\t\\hline\n   \t\t$\\sigma$ & $|\\bar{\\Phi}(r=0)|$        & $\\bar{M}$               & $\\omega$          & $\\gamma$           \\\\ \\hline\n   \t\t2.5      & 0.127$\\pm$0.008 & 0.590 $\\pm$ 0.003 & 0.793 $\\pm$ 0.008 & 5.075 $\\pm$ 0.024  \\\\\n   \t\t2.75     & 0.115$\\pm$0.007 & 0.606 $\\pm$ 0.002 & 0.804 $\\pm$ 0.006 & 6.884 $\\pm$ 0.022  \\\\\n   \t\t3        & 0.106$\\pm$0.006 & 0.615 $\\pm$ 0.003 & 0.813 $\\pm$ 0.004 & 9.168 $\\pm$ 0.023  \\\\\n   \t\t3.5      & 0.092$\\pm$0.004 & 0.629 $\\pm$ 0.002 & 0.833 $\\pm$ 0.008 & 15.592 $\\pm$ 0.042 \\\\\n   \t\t4        & 0.086$\\pm$0.004 & 0.631 $\\pm$ 0.002 & 0.842 $\\pm$ 0.006 & 20.887 $\\pm$ 0.061 \\\\\n   \t\t5        & 0.092$\\pm$0.003 & 0.628 $\\pm$ 0.002 & 0.832 $\\pm$ 0.008 & 15.494 $\\pm$ 0.023 \\\\\n   \t\t6        & 0.098$\\pm$0.004 & 0.623 $\\pm$ 0.003 & 0.826 $\\pm$ 0.009 & 11.567 $\\pm$ 0.035 \\\\\n   \t\t7        & 0.104$\\pm$0.004 & 0.621 $\\pm$ 0.003 & 0.820 $\\pm$ 0.011 & 9.572 $\\pm$ 0.022  \\\\\n   \t\t8        & 0.108$\\pm$0.005 & 0.613 $\\pm$ 0.004 & 0.813 $\\pm$ 0.012 & 8.423 $\\pm$ 0.023  \\\\\n   \t\t9        & 0.112$\\pm$0.006 & 0.609 $\\pm$ 0.004 & 0.810 $\\pm$ 0.014 & 7.608 $\\pm$ 0.016  \\\\\n   \t\t10       & 0.115$\\pm$0.007 & 0.605 $\\pm$ 0.005 & 0.806 $\\pm$ 0.006 & 7.051 $\\pm$ 0.024  \\\\ \n   \t\t\\hline\n\\end{tabular}\n\\caption{Summary of our numerical results for the critical solutions. Since $\\Phi(r=0)$ and $M$ have an oscillatory behavior, we report the mean value with an uncertainty given by the standard deviation. The frequency is obtained using a FFT applied to the real and imaginary parts of the field at the origin. The critical exponent $\\gamma$ is calculated using a least squares fit to eq.~\\eqref{eq:tau_scaling}. We notice that as $\\sigma$ increases, the mass of the critical solution first approaches the highest possible value for the mass for a boson star $M \\sim 0.633$, reaching the maximum value for $\\sigma=4.0$, while for higher values of $\\sigma$ the mass decreases again.}\n\\label{tab:all_data}\n\\end{table}\n\n\\vspace{5mm}\n\nWe now turn to the question of whether our critical solutions do indeed correspond to unstable boson stars as found Hawley and Choptuik in~\\cite{PhysRevD.62.104024}, even if our initial data is very different. As a first comparison, Figure \\ref{fig:BS-s4-10} shows the norm of the complex scalar field for our critical solution as a function of areal radius for the cases with $\\sigma=4$ (top pannel) and $\\sigma=10$ (bottom panel), superimposed with the norm of the complex scalar field for an unstable boson star with the same amplitude. We notice that the critical solutions obtained have no nodes in the field, so the corresponding boson star is in its ground state. We can clearly see that the profiles of our critical solutions follow very closely the expected profile\nfor the boson stars.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/BS-s4.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/BS-s10.pdf}\n\\caption{Comparison of the norm of the complex scalar field of our critical solutions with that of unstable boson stars with the same amplitude, for the cases $\\sigma=4$ (top pannel) and $\\sigma=10$ (bottom panel). The dots represents the critical solutions, and the solid lines the corresponding boson stars.}\n\\label{fig:BS-s4-10}\n\\end{figure}\n\nNext, in Figure~\\ref{fig:mass_freq} we show a plot of the mass vs. the frequency of oscillation for our critical solutions, corresponding to the data in table~\\ref{tab:all_data}, compared to the same plot for boson star solutions (solid lines).  We separate the data into two plots to make more evident the fact that the mass of the critical solution first increases with $\\sigma$ up to $\\sigma=4$, and then decreases again with higher values of $\\sigma$. Figure~\\ref{fig:mass_phi} shows a similar plot but now of the mass vs. the central value of the norm of the scalar field. As can be seen in the plots, our critical solutions fall directly in the line for stationary boson stars.  Moreover, they are all to the left of the maximum mass in Figure~\\ref{fig:mass_freq} and to the right of the maximum in Figure~~\\ref{fig:mass_phi}, which correspond to the unstable branch for boson stars.  The critical solution for $\\sigma=4$ is almost at the maximum mass. Figure~\\ref{fig:scaling_tau} shows the scaling of $\\tau$, the lifetime of the near critical solutions for the different values of $\\sigma$. We can see that the scaling shows good agreement with Eq.~\\eqref{eq:tau_scaling}, but with different critical exponents for the different values of $\\sigma$.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/M-w_1.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/M-w_2.pdf}\n\\caption{Mass and oscillation frequency for our critical solutions compared with the curve for stationary boson star solutions. Circles corresponds to the specific value of $\\sigma$ and the solid line to the known values for stationary boson stars. Top panel: Critical solutions for $\\sigma \\leq 4$. Bottom panel: Critical solutions for $\\sigma \\geq 4$. We observe that the maximum value for the mass is reached for $\\sigma$ such that $3.5 < \\sigma < 5$.}\n\\label{fig:mass_freq}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/phi-M_1.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/phi-M_2.pdf}\n\\caption{Mass and scalar field norm at the origin for our critical solutions compared with the curve for stationary boson star solutions. Circles corresponds to the specific value of $\\sigma$ and the solid line to the known values for stationary boson stars. Top panel: Critical solutions for $\\sigma \\leq 4$. Bottom panel: Critical solutions for $\\sigma \\geq 4$.}\n\\label{fig:mass_phi}\n\\end{figure}\n\nSince the ADM mass of the initial pulse increases monotonically relative to $\\sigma$, we could have expected that the mass of the critical solution approaches asymptotically the maximum value for boson stars $M_{ADM} \\sim 0.633$, which separates the unstable from the stable regions,  as $\\sigma$ is increased. But as can be seen from the plots and the data ofTable~\\ref{tab:all_data}, instead we find that for $\\sigma \\lesssim 4$ the mass of the critical solution increases, while for higher values of $\\sigma$ the mass decreases and moves away from the maximum mass value.  Our data indicates that the maximum possible mass for a boson star will probably be attained for $\\sigma $ between $3.5< \\sigma <5$. This behavior is also reflected in the values of critical exponent $\\gamma$, which also reaches its maximum value between $3.5 < \\sigma < 5$.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/scaling_tau.pdf}\n\\caption{Scaling of the lifetime of near critical solutions for different values of the gaussian width $\\sigma$. With the values of $\\sigma$ tested, the maximum value of the critical exponent $\\gamma$ is reached at $\\sigma=4$. }\n\\label{fig:scaling_tau}\n\\end{figure}\n\nFinally, for boson stars the imaginary part of Lyapunov exponent $\\lambda$ can be related to the critical exponent $\\gamma$, by \\mbox{$\\text{Im}(\\chi) = 1\/\\gamma$} (details about the procedure to obtain the Lyapunov exponents can be consulted in~\\cite{PhysRevD.62.104024}). Figure~\\ref{fig:lyapunov} compares the square of the Lyapunov exponent for boson stars obtained through a linear perturbation analysis (data provided by A.~Bernal~\\cite{BernalA_2021}), with the critical exponents $1\/\\gamma^2$ measured in our simulations. We can see an excellent agreement between both data sets.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/Lyapunov_1.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/Lyapunov_2.pdf}\n\\caption {Comparison of the square of Lyapunov exponent for unstable modes of boson stars. Circles correspond to the $1\/\\gamma^2$ for the specific values of $\\sigma$, while the solid lines correspond to the Lyapunov exponents obtaned from a linear perturbation analysis of boson star solutions. Top panel: for $\\sigma \\leq 4$. Bottom panel: $\\sigma \\geq 4$.}\n\\label{fig:lyapunov}\n\\end{figure}      \n\n\n\\section{Conclusions}\n\nWe performed numerical simulations of a massive complex scalar field using a numerical code adapted to spherical symmetry in order to study critical gravitational collapse. Our initial conditions for the complex field are somewhat similar to the harmonic boson star ansatz, but crucially they do not correspond to a stationary boson star solution.\n\nWe find that, depending on the width of initial data, the critical collapse behaves in two very different ways. For $\\sigma \\leq 0.5$ we can measure the 4D Ricci scaling, which is indicative of type II critical collapse. We obtain values for the critical exponent $\\gamma = 0.38 \\pm 0.01$ and echoing period $\\Delta = 3.4 \\pm 0.1$, which are very similar to those found in the literaure for the case of a real massless scalar field. On the other hand, for $\\sigma \\geq 2.5$ we obtain the scaling of the lifetime of near critical solutions, which is characteristic of type I critical collapse. For type I collapse we observe that the critical exponent depends on the initial gaussian width $\\sigma$: as this width increases the critical exponent reaches its highest value for $\\sigma \\approx 4$, while for higher values of $\\sigma$ the value of the critical exponent decreases again.\n\nIn a similar way to Hawley and Choptuik~\\cite{PhysRevD.62.104024}, we also find that the critical solutions obtained correspond to boson stars in the ground state in the unstable branch. We validate our results by contrasting the curves of $|\\Phi(0)|$ vs. $M_{ADM}$, and $\\omega$ vs $M_{ADM}$. Up to our uncertainties we find that our critical solutions do fall on the curves for unstable stationary boson stars. Also, for our simulations the maximum mass of the critical solution is obtained for $\\sigma \\simeq 4$, which leads us to conjecture that the maximum mass of a boson star at the boundary between the stable and unstable branches, $M \\sim 0.633$,  will be attained for $\\sigma$ somewhere in the range $3.5<\\sigma<5.0$.  Furthermore, we also confirm that the inverse of the critical exponent $\\gamma$ for our critical solutions does indeed correspond to the imaginary part of Lyapunov exponents for unstable boson stars obtained through a linear perturbation analysis.\n\nOne final comment about the transition from type I critical collapse to type II. Since we have obtained the two different behaviors by varying the value of $\\sigma$, we can in principle study the transition between both types of collapse by concentrating in the region $0.5 < \\sigma < 2.5$. On the other hand, as can be seen from our plots, for type I collapse as $\\sigma$ decreases the critical solution moves further into the unstable branch for boson stars. This raises the questions as to how far down this branch we can go before transitioning to type II critical collapse. We will leave this question for a future study.\n\n\n\n\\acknowledgments\n\nThis work was partially supported by CONACyT Network Projects No. 376127 and No. 304001. EJ was also supported by a CONACyT National Graduate Grant.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDirected relations are essential to explaining pairwise dependencies among multiple interacting units. \nIn gene network analysis, regulatory gene-to-gene relations are a focus of biological investigation \\citep{sachs2005causal}, while in a human brain network, scientists investigate causal influences among regions of interest to understand how the brain functions through the regional effects of neurons \\citep{liu2017effective}. \nIn such a situation, inferring directed effects without other information is generally impossible because of the lack of identifiability in a Gaussian directed acyclic graph (DAG) model, and hence external interventions are introduced to treat a non-identifiable situation \\citep{heinze2018causal}. \nFor instance, the genetic variants such as single-nucleotide polymorphisms (SNPs) can be, and indeed are increasingly, treated as external interventions to infer inter-trait causal relations in a quantitative trait network \\citep{brown2020phenome} and gene interactions in a gene regulatory network \\citep{teumer2018common,molstad2021covariance}. \nIn neuroimaging analysis, scientists use randomized experimental stimuli as interventions \nto identify causal relations in a functional brain network \\citep{grosse2016identification,bergmann2021inferring}.\nHowever, the interventions in these studies often have unknown targets and off-target effects \\citep{jackson2003expression,eaton2007exact}. \nConsequently, inferring directed relations while identifying useful interventions for inference is critical.\nThis paper focuses on the simultaneous inference of directed relations subject to unspecified interventions, that is, without known targets.\n\nIn a DAG model, the research has been centered on the reconstruction of directed\nrelations in observational and interventional studies \\citep{van2013ell,oates2016estimating,heinze2018causal,zheng2018dags,yuan2019constrained}. \nFor inference, Bayesian methods \\citep{friedman2003being,luo2011bayesian,viinikka2020towards} have been popular. \nYet, statistical inference remains under-studied, especially for interventional models in high dimensions \\citep{peters2016causal,rothenhausler2019causal}.\nRecently, for observational data, \\citet{jankova2018inference} propose a debiased test of the strength of \na single directed relation and \\citet{li2019likelihood} derives a constrained likelihood ratio test for multiple directed relations.\n\nDespite progress, challenges remain. First, inferring directed relations requires identifying a certain DAG topological order \\citep{van2013ell}, while the identifiability in a Gaussian DAG with unspecified interventions remains under-explored. Second, the inferential results should agree with the acyclicity requirement for a DAG. As a result, degenerate and intractable situations can occur, making inference greatly different from the classical ones. Third, likelihood-based methods for learning the DAG topological order often use permutation search \\citep{van2013ell} or continuous optimization subject to the acyclicity constraint \\citep{zheng2018dags,yuan2019constrained}, \nwhere a theoretical guarantee of the actual estimate (instead of the global optimum) has not been established for these approaches.\n{Recently, an important line of work \\citep{ghoshal2018learning,rajendran2021structure,rolland2022score} has focused on order-based algorithms with computational and statistical guarantees. However, in Gaussian DAGs, existing methods often rely on some error scale assumptions \\citep{bulhmann2013inde}, which is sensitive to variable scaling like the common practice of standardizing variables. This drawback could limit their applications, especially in causal inference, as causal relations are typically invariant to scaling.}\n\n{\n To address the above issues, we develop structure learning and inference methods for a Gaussian DAG with unspecified additive interventions. Unlike the existing approach where structure discovery and subsequent inference are treated separately, our proposal integrates DAG structure learning and testing directed relations (also known as the natural direct effect in causal inference), accounting for the uncertainty of structure learning for inference. With suitable interventions called instrumental variables (IVs), the proposed approach removes the restrictive error scale assumptions and delivers creditable outcomes with theoretical guarantees in low order polynomial time. This result indicates IVs, a well-known tool in causal inference, can play important roles in structure learning even if some interventions do not meet the IV criterion.\n More specifically, our contributions are summarized as the following aspects.}\n\n\\begin{itemize}\n \\item For modeling, we establish the identifiability conditions for a Gaussian DAG with unspecified interventions. In particular, the conditions allow interventions on more than one target, which is suitable for multivariate causal analysis \\citep{murray2006avoiding}. Moreover, we introduce the concepts of nondegeneracy and regularity for hypothesis testing under a DAG model to characterize the behavior of a test.\n\n \\item For methodology, we develop likelihood ratio tests for directed edges and pathways in a super-graph of the true DAG, where the super-graph is formed by ancestral relations and candidate interventional relations, offering the topological order for inference. \nFurthermore, we reconstruct the super-graph by the peeling algorithm, which automatically meets the acyclicity requirement. {By integrating structure learning with inference, we account for the uncertainty of the super-graph estimation for the proposed tests via a novel data perturbation (DP) scheme, which effectively controls the type-I error while enjoying high statistical power.} \n\n  \\item For theory, we prove that the proposed peeling algorithm based on nodewise regressions yields consistent results in {$O(p \\times \\log\\kappa_{\\max}^\\circ \\times (q^3 + nq^2) )$ operations almost surely, where $p,q$ are the numbers of primary and intervention variables, $n$ is the sample size, and $\\kappa_{\\max}^\\circ$ is the sparsity.} On this basis, we justify the proposed DP inference method by establishing the convergence of the DP likelihood ratio to the null distribution and desired power properties.\n  \n  \\item The numerical studies and real data analysis demonstrate the utility and effectiveness of the proposed methods. The implementation of the proposed tests and structure learning method is available at \\url{https:\/\/github.com\/chunlinli\/intdag}.\n\n\\end{itemize}\n\n{The rest of the article is structured as follows.\nSection \\ref{section:model} establishes model identifiability and introduces the concepts of nondegeneracy and regularity.\nSection \\ref{section:method} develops the proposed methods for structure learning and statistical inference.\nSection \\ref{section:theory} presents statistical theory to justify the proposed methods.\nSection \\ref{section:simulation} performs simulation studies, followed by an application to infer gene pathways with gene expression and SNP data in Section \\ref{section:real-data}. Section \\ref{section:conclusion} concludes the article. \nThe Appendix contains illustrative examples and technical proofs.}\n\n\\section{Gaussian directed acyclic graph with additive interventions}\\label{section:model}\n\nTo infer directed relations among primary variables (variables of primary interest) $\\bm Y =\n(Y_1,\\ldots,Y_p)^\\top$, \nconsider a structural equation model with additive interventions:  \n\\begin{eqnarray}\n    \\label{equation:model}\n    \\bm Y= \\bm U^{\\top} \\bm Y + \\bm W^{\\top} \\bm X + \\bm \\varepsilon, \n    \\quad \\bm \\varepsilon \\sim N(\\bm 0, \\bm\\Sigma), \\quad \\bm \\Sigma = \\text{Diag}(\\sigma^2_1,\\ldots,\\sigma^2_p),\n\\end{eqnarray}\nwhere $\\bm X=(X_1,\\ldots, X_q)^\\top$ is a vector of additive intervention variables,\n$\\bm U\\in \\mathbb R^{p \\times p}$ and $\\bm W \\in \\mathbb R^{q \\times p}$ are unknown coefficient matrices, \nand $\\bm \\varepsilon = (\\varepsilon_1,\\ldots,\\varepsilon_p)^\\top$ is a vector of random errors with $\\sigma_j^2>0$; $j=1,\\ldots,p$. In \\eqref{equation:model}, $\\bm \\varepsilon$ is independent\nof $\\bm X$ but components of $\\bm X$ can be dependent.\nThe matrix $\\bm U$ specifies the directed relations among $\\bm Y$, \nwhere $U_{kj}\\neq 0$ if $Y_k$ is a direct cause of $Y_j$, denoted by $Y_k \\rightarrow Y_j$, {and $Y_k$ is called a {parent} of $Y_j$ or $Y_j$ a {child} \nof $Y_k$.}\nThus, $\\bm U$ represents a directed graph, which is required to be acyclic to ensure the validity of the local Markov property \\citep{spirtes2000causation}.\nThe matrix $\\bm W$ specifies the targets and strengths of interventions, \nwhere $W_{lj}\\neq 0$ indicates $X_l$ intervenes on $Y_j$, denoted by \n$X_l\\to Y_j$. In \\eqref{equation:model}, no directed edge from a primary \nvariable $Y_j$ to an intervention variable $X_l$ is permissible. \n\n{In what follows, we will focus on the DAG $G$ with nodes $\\bm Y=(Y_1,\\ldots,Y_p)$, primary variable edges $\\mathcal E = \\{ (k,j) : U_{kj}\\neq 0 \\}$, and intervention edges $\\mathcal I=\\{ (l,j) : W_{lj}\\neq 0 \\}$.}\n\n\n\\subsection{Identifiability}\\label{section:identifiability}\n\n{\nModel \\eqref{equation:model} is generally non-identifiable without interventions \n($\\bm W = \\bm 0$) when errors do not meet\nsome requirements such as the equal-variance assumption \\citep{bulhmann2013inde} and its variants\n\\citep{ghoshal2018learning,rajendran2021structure}.}\nMoreover, a model can be identified when $\\bm \\varepsilon$ in \\eqref{equation:model}\nis replaced by non-Gaussian errors \\citep{shimizu2006linear}\nor linear relations are replaced by nonlinear ones \n\\citep{hoyer2009nonlinear}. Regardless, suitable interventions can make \n\\eqref{equation:model} identifiable. \nWhen intervention targets are known, \nthe identifiability issue has been studied \n\\citep{oates2016estimating,chen2018two}. \nHowever, it is less so when the exact targets and strengths of interventions are unknown,\nas in many biological applications \\citep{jackson2003expression,kulkarni2006evidence}, \nreferring to the case of \\emph{unspecified} or \\emph{uncertain} interventions \n\\citep{heinze2018causal,eaton2007exact,squires2020permutation}. \n\n\nWe now categorize interventions as instruments and invalid instruments.\n\n\\begin{definition}[Instrument] \\label{definition:instrument}\nAn intervention is an instrument in a DAG if it satisfies:\n\\begin{enumerate}\n  \\item [(A)] (Relevance) it intervenes on at least one primary variable; \n  \\item [(B)] (Exclusion) it does not intervene on more than one primary variable.\n\\end{enumerate}\nOtherwise, it is an invalid instrument.\n\\end{definition}\n\nHere, (A) requires an intervention \nto be active, while (B) prevents simultaneous interventions of a single \nintervention variable on multiple primary variables. This is critical to identifiability\nbecause in this situation an instrument on a cause variable $Y_1$ helps reveal its directed effect on \nan outcome variable $Y_2$, which would otherwise be impossible. \nThus, the instruments are introduced to break the symmetry that results in non-identifiability of directed relations $Y_1 \\rightarrow Y_2$ and $Y_2 \\rightarrow Y_1$.\n\n\nNext, we make some assumptions on intervention variables to yield an identifiable \nmodel, where dependencies among intervention variables are permissible.\n \n\\begin{assumption}\\label{assumption:identifiability}\n  Assume that model \\eqref{equation:model} satisfies the following conditions.\n  \\begin{enumerate}\n    \\item [(1A)] (Positive definiteness) $\\operatorname{\\mathbb E}\\bm X\\bm X^\\top$ is positive definite, where $\\operatorname{\\mathbb E}$ is the expectation.\n      \n    \\item [(1B)] (Local faithfulness) $\\operatorname{Cov}(Y_j, X_l \\mid \\bm X_{\\{1,\\ldots,q\\}\\setminus\\{l\\}})\\neq 0$ when $X_l$ intervenes on $Y_j$, where $\\operatorname{Cov}$ denotes the covariance.\n\n    \\item [(1C)] (Instrumental sufficiency) Each primary variable is intervened by at least one instrument.\n  \\end{enumerate}\n\\end{assumption}\n\nAssumption 1A imposes mild distributional restrictions on $\\bm X$, permitting discrete variables such as SNPs. Assumption 1B requires interventional effects not to cancel out as multiple targets from an invalid instrument are permitted. \nImportantly, if either Assumption 1B or Assumption 1C fails, model \\eqref{equation:model} is generally not identifiable, as shown in Example \\ref{example:identifiability} of Appendix \\ref{section:identifiability-example}. \nIn Section \\ref{section:simulation}, we empirically examine the situation when Assumption 1C is not met.\n\n\n\\begin{proposition}[Identifiability]\\label{proposition:identifiability}\nUnder Assumption \\ref{assumption:identifiability},  {the parameters $(\\bm U, \\bm W, \\bm\\Sigma)$ in model \\eqref{equation:model} are identifiable.}\n\\end{proposition}\n\nProposition \\ref{proposition:identifiability} {(proved in Appendix \\ref{proof:identifiability})} is derived for a DAG model with unspecified interventions. \nThis is in contrast to Proposition 1 of \\citet{chen2018two}, which proves the identifiability of the parameters in a directed graph with target-known instruments on each primary variable.\nMoreover, it is worth noting that the estimated graph in \\citet{chen2018two} may be cyclic and lacks the local Markov property for causal interpretation \\citep{spirtes2000causation}. \n\n\\subsection{Likelihood-based inference for a DAG}\\label{section:testability}\n\nOur primary goal is to perform statistical inference of directed edges and pathways. \nLet $\\mathcal{H}\\subseteq \\{ (k,j):k\\neq j, 1\\leq k,j\\leq p \\}$ be an edge set among primary variables $\\{ Y_1,\\ldots,Y_p \\}$, where $(k,j) \\in \\mathcal{H}$ \nspecifies a (hypothesized) directed edge $Y_k \\to Y_j$ in \\eqref{equation:model}. \nWe shall focus on two types of testing with null and alternative hypotheses $H_0$ and $H_a$.\nFor simultaneous testing of directed edges,\n\\begin{eqnarray}\n\\label{equation:link-test}\nH_0: U_{kj}= 0; \\text{ for all } (k,j) \\in \\mathcal{H} \\quad \\mbox{ versus } \n\\quad H_a: U_{kj}\\neq 0 \\text{ for some } (k,j) \\in \\mathcal{H}; \n\\end{eqnarray}\nfor simultaneous testing of pathways, \n\\begin{eqnarray}\n\\label{equation:pathway-test}\nH_0: U_{kj}= 0; \\text{ for some } (k,j) \\in \\mathcal{H} \\quad \\mbox{ versus }\n\\quad H_a: U_{kj} \\neq  0 \\text{ for all } (k,j) \\in \\mathcal{H},\n\\end{eqnarray}\nwhere $(\\bm U_{\\mathcal {H}^c},\\bm W, \\bm \\Sigma)$ \nare unspecified nuisance parameters and $^c$ is the set complement. \n\n{Note that $H_0$ in \\eqref{equation:pathway-test} is a composite hypothesis that can be decomposed into sub-hypotheses\n  \\begin{equation*}\n  H_{0,\\nu}: U_{k_\\nu,j_\\nu}= 0, \\quad \\mbox{ versus } \n  \\quad {H}_{a,\\nu}: U_{k_\\nu,j_\\nu}\\neq 0; \\quad \\nu=1,\\ldots,|\\mathcal H|,\n  \\end{equation*}\nwhere $\\mathcal{H}=\\{ (k_1,j_1),\\ldots,(k_{|\\mathcal H|},j_{|\\mathcal H|}) \\}$ and each sub-hypothesis is a directed edge test.}\n\nNow consider likelihood inference for \\eqref{equation:link-test}.\nGiven an independent and identically distributed sample $(\\bm Y_i,\\bm X_i)_{i=1}^n$ from \n\\eqref{equation:model}, the log-likelihood is \n\\begin{equation}\n\\label{equation:likelihood}\n    L(\\bm\\theta,\\bm\\Sigma)= -\n   \\frac{1}{2} \\sum_{i=1}^n\n    \\|\\bm\\Sigma^{-1\/2} \n    ((\\bm I - \\bm U^\\top)\\bm Y_i - \\bm W^\\top \\bm X_i ) \\|^2_2 - \n    n\\log\\sqrt{\\det(\\bm\\Sigma)},\n\\end{equation}\nwhere $\\bm\\theta = (\\bm U,\\bm W)$, $\\bm \\Sigma=\\text{Diag}(\\sigma_1^2,\\ldots,\\sigma_p^2)$, and $\\bm U$ is subject to the acyclicity constraint \\citep{zheng2018dags,yuan2019constrained} that no directed cycle is permissible in\na DAG. The likelihood ratio is defined as \n$\\text{Lr} = L(\\widehat{\\bm\\theta}^{(1)},\\widehat{\\bm\\Sigma}) - L(\\widehat{\\bm\\theta}^{(0)},\\widehat{\\bm\\Sigma})$,\nwhere $\\widehat{\\bm\\theta}^{(1)}$ is a maximum likelihood estimate (MLE) in \\eqref{equation:likelihood} subject to the acyclicity constraint,\n$\\widehat{\\bm\\theta}^{(0)}$ is an MLE subject to an additional requirement $\\bm U_{\\mathcal H} = \\bm 0$,\nand $\\widehat{\\bm\\Sigma}$ is an estimator of $\\bm\\Sigma$.\n\nIn many statistical models, {a likelihood ratio has a nondegenerate and tractable distribution}, for instance, a chi-squared distribution with degrees of freedom $|\\mathcal{H}|$.\nHowever, in \\eqref{equation:model}, because of the acyclicity constraint, {some hypothesized edges in $\\mathcal H$ can be absent given the edge subset $\\mathcal E\\cap \\mathcal H^c$ defined by the nonzero elements of nuisance parameter $\\bm U_{\\mathcal H^c}$, where $\\mathcal{E}$ denotes the edge set of the whole DAG.} \nAs a result, $\\text{Lr}$ may converge to a distribution with degrees of freedom less than $|\\mathcal H|$ and the distribution may be even intractable, making inference for a DAG greatly different from the classical ones, as illustrated by Example \\ref{example:testability}.\n\n\\begin{figure}\n  \\centering\n \n    \\includegraphics[width=0.45\\textwidth]{example1a.pdf}\n   \n \n  \\caption{A true DAG structure of five primary variables \n$Y_1,\\ldots,Y_5$ and five intervention variables $X_1,\\ldots,X_5$, where\ndirected edges are represented by solid arrows while \ndependencies among $\\bm X$ are not displayed.} \n\\label{figure:testability}\n\\end{figure}\n\n\\begin{example}\\label{example:testability} Consider the likelihood ratio test under null $H_0$ \nand alternative $H_a$ for the DAG displayed in Figure \\ref{figure:testability}. {Let $\\mathcal E$ be the edge set of the DAG.}\n\\begin{itemize}\n  \\item $H_0: U_{21} = 0$ versus $H_a: U_{21}\\neq 0$, where $\\mathcal H=\\{ (2,1) \\}$. {Here, $(2,1)$ forms a cycle together with the edges in $\\mathcal E \\cap \\mathcal H^c$ (namely the edges not considered by the hypothesis), violating the acyclicity constraint.} Note that the maximum likelihood value $L(\\widehat{\\bm\\theta},\\widehat{\\bm\\Sigma})$ with $U_{21}\\neq 0$ tends to be smaller than that with $U_{21}=0$ when a random sample is obtained under $H_0$, especially so when the asymptotics kicks in as the sample size increases. Consequently, the likelihood ratio $\\text{Lr}$ becomes zero, constituting a degenerate situation.\n\n  \\item $H_0: U_{45} = U_{53} = 0$ versus $H_a: U_{45}\\neq 0$ or $U_{53}\\neq 0$, where $\\mathcal H=\\{ (4,5),(5,3)\\}$.\n  {In this case, $\\{(4,5),(5,3)\\}$ forms a cycle with the edges in $\\mathcal E\\cap\\mathcal H^c$, violating the acyclicity constraint.} \n  Since the likelihood value $L({\\bm\\theta},\\bm\\Sigma)$ tends to be maximized under the true graph when data is sampled under $H_0$, we have $L(\\widehat{\\bm\\theta}^{(1)},\\widehat{\\bm\\Sigma}) = \\max(L(\\widehat{U}_{45},\\widehat{U}_{53}=0,\\widehat{\\bm U}_{\\mathcal H^c},\\widehat{\\bm W},\\widehat{\\bm\\Sigma}), L(\\widehat{U}_{45}=0,\\widehat{U}_{53},\\widehat{\\bm U}_{\\mathcal H^c},\\widehat{\\bm W},\\widehat{\\bm\\Sigma}))$. As a result, the likelihood ratio distribution becomes complicated in this situation due to the dependence between the two components in $L(\\widehat{\\bm\\theta}^{(1)},\\widehat{\\bm\\Sigma})$.\n\\end{itemize}\n\\end{example}\n\n\n\nMotivated by Example \\ref{example:testability}, we introduce the concepts of nondegeneracy and regularity.\n\n\\begin{definition}[Nondegeneracy and regularity with respect to DAG]\\label{definition:testability} ~\n\\begin{enumerate}[(A)]\n  \\item An edge $(k,j)\\in \\mathcal{H}$ is nondegenerate with respect to DAG $G$ if $\\{(k,j)\\}\\cup \\mathcal{E}$ contains no directed cycle,\n  where $\\mathcal{E}$ denotes the edge set of $G$. Otherwise, $(k,j)$ is degenerate.\n  Let $\\mathcal{D} \\subseteq \\mathcal{H}$ be the set of all nondegenerate edges with respect to $G$.\n  A null hypothesis $H_0$ is nondegenerate with respect to DAG $G$ if $\\mathcal{D}\\neq\\emptyset$.\n  Otherwise, $H_0$ is degenerate.\n  \\item A null hypothesis $H_0$ is said to be regular with respect to DAG $G$ if $\\mathcal{D} \\cup \\mathcal{E}$ contains no directed cycle, where $\\mathcal{E}$ denotes the edge set of $G$. Otherwise, $H_0$ is called irregular.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\n  In practice, $\\mathcal{D}$ is unknown and needs to be estimated from data.\n\\end{remark}\n\nNondegeneracy ensures nonnegativity of the likelihood ratio. \n{In testing \\eqref{equation:link-test}, \nregularity excludes intractable situations for the null distribution.\nIn testing \\eqref{equation:pathway-test}, if $H_0$ is irregular, \nthen $\\mathcal D\\cup\\mathcal{E}$ has a directed cycle,\nwhich means the hypothesized directed pathway cannot exist due to the acyclicity constraint. \nThus, regularity excludes the degenerate situations in testing \\eqref{equation:pathway-test}.}\n\nIn what follows, we mainly focus on nondegenerate and regular hypotheses. \nFor the degenerate case, the p-value is defined to be one.\n{For the irregular case of edge test \\eqref{equation:link-test},  \nwe decompose the hypothesis into regular sub-hypotheses and conduct multiple testing.\nFor the irregular case of pathway test \\eqref{equation:pathway-test}, the p-value is defined to be one.\nMore discussions on the implementation in irregular cases are provided in Appendix \\ref{section:appendix-irregular-hypothesis}.}\n\n\n\nFinally, we introduce some notations for a DAG $G$ with primary \nvariables $\\{Y_1,\\ldots,Y_p\\}$, intervention variables $\\{ X_1,\\ldots,X_q \\}$, a directed edge set $\\mathcal E\\subseteq \n\\{(k,j):k\\neq j, 1\\leq k,j\\leq p \\}$, and an intervention edge set $\\mathcal I \\subseteq \\{ (l,j) : 1\\leq l\\leq q, 1\\leq j\\leq p \\}$. \nIf there is a directed path $Y_k\\to\\cdots\\to Y_j$ in $G$, \n$Y_k$ is called an ancestor of $Y_j$ or $Y_j$ a descendant of $Y_k$.\nLet $\\mathcal A = \\{ (k,j) : Y_k\\to\\cdots\\to Y_j\\}$ be the set of all ancestral relations. \nIn a DAG $G$, let $\\textnormal{\\textsc{pa}}_G(j) = \\{ k : (k,j)\\in\\mathcal E \\}$, \n$\\textnormal{\\textsc{an}}_G(j) = \\{ k: (k,j)\\in \\mathcal A \\}$, $\\textnormal{\\textsc{d}}_G(j) = \\{ k: (k,j) \\in\\mathcal D \\}$, and $\\textnormal{\\textsc{in}}_G(j)=\\{ l : (l,j)\\in\\mathcal I \\}$ be the parent, ancestor, nondegenerate hypothesized parent, and intervention sets of $Y_j$, respectively.\n\n\n\n\\section{Methodology}\\label{section:method}\n\nThis section develops the main methodology, \nincluding the peeling algorithm for structure learning and the data perturbation inference for \nsimultaneous testing of directed edges \\eqref{equation:link-test} and pathways \\eqref{equation:pathway-test}.\n\nOne major challenge to the likelihood approach is optimization subject to the acyclicity requirement. This constraint imposes difficulty on not only computation but also asymptotic theory. As a result, there is a gap between the asymptotic distribution based on a global maximum and that on the actual estimate which could be a local maximum \\citep{jankova2018inference,li2019likelihood}. Moreover, the actual estimate may give an imprecise topological order, tending to impact adversely on inference.\n\nTo circumvent the acyclicity requirement, we propose using a super-graph $S$ of the true DAG $G$, where $S$ is constructed by a novel peeling method without explicitly imposing the acyclicity constraint. As a result, our method is scalable with a statistical guarantee of the actual estimate.\n{We will focus on a specific super-graph $S$ consisting of primary variable edges $\\mathcal E_{S} := \\mathcal A\\supseteq \\mathcal E$} and candidate intervention edges $\\mathcal I_{S} := \\mathcal C \\supseteq \\mathcal I$, \nwhere $\\mathcal{E}$ and $\\mathcal I$ are sets of primary variable edges and intervention edges in the original graph $G$, {and set $\\mathcal{C}$ is to be defined shortly after \\eqref{equation:model2}}. Note that $S$, whose primary variable edges consist of all ancestral relations of $G$, is also a DAG. \nGiven the super-graph $S$, the true DAG $G$ can be reconstructed by nodewise constrained regressions.\n\nLet $\\bm \\theta_{S}=\\big(\\bm U_{\\mathcal A}, \\bm W_{\\mathcal C}\n\\big)$ be $\\bm \\theta$ restricted to $S$. Note that \n$\\bm\\theta_{S^c} = (\\bm U_{\\mathcal A^c},\\bm W_{\\mathcal C^c}) = \\bm 0$ in the original graph $G$.\nGiven $S$, the log-likelihood $L(\\bm \\theta,\\bm\\Sigma)$ in \\eqref{equation:likelihood} can be rewritten as\n\\begin{equation}\n\\label{equation:likelihood-supergraph}\n  \\begin{split}\n L_S(\\bm \\theta_S,\\bm \\Sigma)\n    &= -\\sum_{j = 1}^p \\Big(\\text{RSS}_j(\\bm \\theta_{S})\/2\\sigma_j^2 + \\log\\sqrt{2\\pi\\sigma_j^2}\\Big),\\\\\n    \\text{RSS}_j(\\bm\\theta_{S})\n    &= \\sum_{i=1}^n \\Big(Y_{ij} - \\sum_{(k,j)\\in\\mathcal A} U_{kj}Y_{ik} - \\sum_{(l,j)\\in\\mathcal C} W_{lj}X_{il} \\Big)^2,\\\\\n  \\end{split}\n\\end{equation} \nwhich involves $|\\mathcal{A}|+|\\mathcal C|$ parameters for $\\bm \\theta_{S}$.\n\nOur plan is as follows. \nIn Section \\ref{section:causal-discovery}, we hierarchically construct ${S}$ via nodewise regressions of $Y_j$ over $\\bm X$; $j=1,\\ldots,p$ without the acyclicity constraint for $\\bm U$.\nThen in Section \\ref{section:dp-inference}, on the ground of constructed super-graph $S$ and \\eqref{equation:likelihood-supergraph}, we develop likelihood ratio tests for hypotheses \\eqref{equation:link-test} and \\eqref{equation:pathway-test}.  \n\n\n\\subsection{Structure learning via peeling}\\label{section:causal-discovery}\n\nThis section develops a novel structure learning method based on a\npeeling algorithm and nodewise constrained regressions to construct \n$S=(\\mathcal{A},\\mathcal C)$ in a hierarchical manner.\n\nFirst, we observe an important connection between primary variables and \nintervention variables. \nRewrite \\eqref{equation:model} as\n\\begin{equation}\n\\label{equation:model2}\n\\bm Y = \\bm V^\\top \\bm X + \\bm \\varepsilon_V, \\quad \\bm \\varepsilon_V=(\\bm I-\\bm U^\\top)^{-1}\\bm \\varepsilon \\sim N(\\bm 0,\\bm\\Omega^{-1}),\n\\end{equation}\nwhere $\\bm\\Omega= (\\bm I - \\bm U)\\bm\\Sigma^{-1}(\\bm I -\\bm U^\\top)$ is a precision matrix \nand $\\bm V = \\bm W (\\bm I - \\bm U)^{-1}$. \nNow, define the set of candidate interventional relations $\\mathcal C := \\{ (l,j) : V_{lj}\\neq 0 \\}$.\n\n\n\\begin{proposition} \\label{proposition:peeling} \n   Suppose Assumption \\ref{assumption:identifiability} is satisfied. Then $\\mathcal C\\supseteq \\mathcal I$. Moreover,\n    \\begin{enumerate}[(A)]\n        \\item if $V_{lj}\\neq 0$, then $X_{l}$ intervenes on $Y_j$ or an ancestor of $Y_j$; \n        \\item $Y_j$ is a leaf variable (having no child)\n        if and only if there is an instrument $X_l$ such that $V_{lj}\\neq 0$ and $V_{lj'}=0$ for $j'\\neq j$. \n    \\end{enumerate}\n\\end{proposition}\n\n{The proof of Proposition \\ref{proposition:peeling} is deferred to Appendix \\ref{proof:peeling}.}\nIntuitively, $V_{lj}\\neq 0$ implies the dependence of $Y_j$ on $X_{l}$ through a directed path $X_l\\to \\cdots \\to Y_j$, and \nhence that $X_l$ intervenes on $Y_j$ or an ancestor of $Y_j$. \nThus, the instruments on a leaf variable are independent of the other primary variables conditional on the rest of interventions. \nThis observation suggests a method to reconstruct the DAG topological order by sequentially identifying the leaves in the DAG and removing the identified leaf variables. \n\nNext, we discuss the estimation of $\\bm V$ and construction of $S$.\n\n\\subsubsection{Nodewise constrained regressions}\n\nWe propose nodewise $\\ell_0$-constrained regressions to estimate\nnonzero elements of $\\bm V$:\n\\begin{equation}\n    \\label{pseudo-likelihood}\n    \\widehat{\\bm V}_{\\cdot j} = \\mathop{\\rm arg\\, min\\,}_{\\bm V_{\\cdot j}} \\ \n    \\sum_{i=1}^n (Y_{ij} - \\bm V_{\\cdot j}^\\top \\bm X_i)^2\n    \\quad \\text{s.t.} \\quad \\sum_{l=1}^q \\operatorname{I}(V_{lj}\\neq 0) \\leq \\kappa_j; \\quad j=1,\\ldots,p,\n\\end{equation}\nwhere $1 \\leq \\kappa_j\\leq q$ is an integer-valued tuning parameter controlling the sparsity of $\\bm V_{\\cdot j}$ and can be chosen by BIC or cross-validation.\n\nTo solve \\eqref{pseudo-likelihood}, we use $J_{\\tau_j}(z) = \\min(|z|\/\\tau_j,1)$ as a surrogate of the $\\ell_0$-function $\\operatorname{I}(z \\neq 0)$ \\citep{shen2012likelihood} and {develop a difference-of-convex (DC) algorithm with the $\\ell_0$-projection to improve globality of the solution of \\eqref{pseudo-likelihood}.} \nAt $(t+1)$th iteration, \nwe solve a weighted Lasso problem, \n\\begin{eqnarray}\n  \\label{equation:penalized-regression}\n  \\widetilde{\\bm V}_{\\cdot j}^{[t+1]}=\\mathop{\\rm arg\\, min\\,}_{\\bm V_{\\cdot j}} \\\n  \\sum_{i=1}^n (Y_{ij} -\n  \\bm V_{\\cdot j}^\\top \\bm X_i)^2+2n\\gamma_j \\tau_j \\sum_{l=1}^q \\operatorname{I}\\big(|\\widetilde{V}^{[t]}_{lj}|\\leq \\tau_j\\big) \n  |V_{lj}|; \\quad j=1,\\ldots,p, \n\\end{eqnarray}\nwhere $\\gamma_j >0$ is a tuning parameter and $\\widetilde{\\bm V}^{[t]}_{\\cdot j}$ is the solution of \\eqref{equation:penalized-regression} at the $t$-th iteration. \nThe DC algorithm terminates at $\\widetilde{\\bm V}_{\\cdot j} = \\widetilde{\\bm V}_{\\cdot j}^{[t]}$ such that $\\|\\widetilde{\\bm V}_{\\cdot j}^{[t+1]}-\\widetilde{\\bm V}_{\\cdot j}^{[t]}\\|_{\\infty}\\leq \\sqrt{\\text{tol}}$, where $\\text{tol}$ is the machine precision.\nThen, we obtain the solution $\\widehat{\\bm V}_{\\cdot j}$ of \\eqref{pseudo-likelihood} by projecting $\\widetilde{\\bm V}_{\\cdot j}$ onto the \n$\\ell_0$-constrained set $\\{\\|\\bm V_{\\cdot j}\\|_0 \\leq \\kappa_j\\}$. \n\n\n\n\\begin{algorithm}\n\nSpecify $\\gamma_j>0$, $\\tau_j>0$, and $1\\leq \\kappa_j\\leq q$.\nInitialize $\\widetilde{\\bm V}_{\\cdot j}^{[0]}$ with $|\\{l:\\widetilde{V}^{[0]}_{l j} >\\tau_j \\}|\\leq \\kappa_j$.  \n\nCompute $\\widetilde{\\bm V}_{\\cdot j} = \\widetilde{\\bm V}^{[t]}_{\\cdot j}$ such that   \n$\\|\\widetilde{\\bm V}_{\\cdot j}^{[t+1]}-\\widetilde{\\bm V}_{\\cdot j}^{[t]}\\|_{\\infty}\\leq \\sqrt{\\text{tol}}$.\n\n\n{($\\ell_0$-projection)}\nLet $B_j = \\{ l :  \\sum_{l'=1}^q \\operatorname{I}( |\\widetilde{V}_{l'j}| \\geq |\\widetilde{V}_{lj}| ) \\leq \\kappa_j   \\}$. \nSet \n$$\\widehat{\\bm V}_{\\cdot j}= \\mathop{\\rm arg\\, min\\,}_{\\bm V_{\\cdot j}} \\\n    \\|\\bm{\\mathsf{Y}}_{j} - \\bm V_{\\cdot j}^\\top \\bm{\\mathsf{X}}\\|_2^2 \\quad\\text{s.t.}\\quad\nV_{lj}=0 \\text{ for } l \\notin B_j.$$\n\n\\caption{Constrained minimization via DC programming + $\\ell_0$ projection} \\label{algorithm:nodewise-regression}\n\\end{algorithm}\n\n\\begin{algorithm}\n\n \n  Let $\\widehat{\\bm V}^{[0]}=\\widehat{\\bm V}$. \n  Begin iteration with $h=0,\\ldots$: at iteration $h$, \n  \n \n  In the current sub-graph identify all instrument-leaf pairs $X_l\\to Y_j$ \n  satisfying $l = \\mathop{\\rm arg\\, min\\,}_{l':\\|\\widehat{\\bm V}^{[h]}_{l'\\cdot}\\|_0 \\neq 0} \\|\\widehat{\\bm V}^{[h]}_{l' \\cdot}\\|_0$ and\n  $j=\\mathop{\\rm arg\\, max\\,}_{j'} |\\widehat{V}^{[h]}_{l j'}|$. \n  Let $h_{\\widehat G}(j)=h$.\n  \n \n  \n  For a leaf $Y_k$, identify all directed relations $Y_k \\to Y_j$ with $h_{\\widehat G}(j)=h-1$ when $\\widehat{V}_{lj}\\neq 0$ for each instrument $X_l$ of $Y_k$ in the current sub-graph. \n  \n  \n  Peeling-off all the leaves $Y_j$\n  by deleting $j$th columns from $\\widehat{\\bm V}^{[h]}$ to yield\n  $\\widehat{\\bm V}^{[h+1]}$. \n  \n  Repeat Steps 2-4 until all primary variables are removed. \n   \n  Compute $\\widehat{S} = (\\widehat{\\mathcal{A}},\\widehat{\\mathcal{C}})$ and $\\widehat{\\mathcal D}$: \n  \\begin{equation*}\n    \\begin{split}\n      \\widehat{\\mathcal{A}}&=\\Big\\{(k,j): Y_k \\to \\cdots \\to Y_j\\Big\\},  \\quad \n      \\widehat{\\mathcal D}=\\Big\\{(k,j)\\in \\mathcal{H}: (j,k)\\notin \\widehat{\\mathcal A}  \\Big\\},\\\\\n      \\widehat{\\mathcal{C}}&=\\Big\\{(l,j): |\\widehat{V}_{lj}|> \\tau_j \\Big\\}, \\text{ where $\\tau_j$ is specified in Algorithm \\ref{algorithm:nodewise-regression}.}\n    \\end{split}\n  \\end{equation*}\n  \n  \\caption{Reconstruction of topological layers by peeling} \\label{algorithm:peeling}\n  \\end{algorithm}\n\n\n\\subsubsection{Peeling}\n\nNow, we describe a \\emph{peeling} algorithm to estimate $S=(\\mathcal{A},\\mathcal C)$ based on $\\bm V$. \n\nBefore proceeding, we introduce the notion of \\emph{topological height} to characterize the DAG topological order. Given a DAG $G$, the height $h_G(j)$ of $Y_j$ is defined as the maximal length of a directed path from $Y_j$ to a leaf variable, where $0\\leq h_G(j) \\leq p-1$ with a leaf having zero height. \nThen the primary variables of the same height form a topological layer and the layers constitute a partition of primary variables in $G$. \nFor instance, in Figure \\ref{figure:testability}, \n$\\{ Y_4,Y_5 \\},\\{ Y_3 \\},\\{ Y_2 \\},\\{Y_1\\}$ are topological layers with height $0,1,2,3$, respectively.\n  \n  Observe that the topological layer of height $h$ are leaves in the sub-graph with primary variables of height no less than $h$. Based on Proposition \\ref{proposition:peeling}, the peeling \nalgorithm sequentially identifies and peels off the primary variables of height $h$;\n  $h=0,1,2,\\ldots$. Note that instruments in the sub-graph may not be necessarily the \ninstruments in the graph $G$ after some primary variables are removed. However, \nthe next proposition indicates that the instruments in the corresponding sub-graph remain \nuseful to reconstruct ancestral relations.\n\n\n\\begin{proposition}\\label{proposition:ancestral-relation}\n  Let $Y_k$ have height $h_G(k)$. Let $G'$ be the sub-graph with primary variables of height $\\geq h_G(k)$. Under Assumption \\ref{assumption:identifiability},\nfor a pair $(k,j)$ with $h_G(j) = h_G(k)-1$, we have $Y_k \\to Y_j$ if and only if $V_{lj}\\neq 0$ for \neach instrument $X_l$ of $Y_k$ in $G'$.\n\\end{proposition}\n\n\nProposition \\ref{proposition:ancestral-relation} {(proved in Appendix \\ref{proof:ancestral-relation})} permits the construction of\ndirected relations between the topological layers of heights $h$ and $h-1$; $h\\geq 1$ and thus ancestral relations.\n\n\n\nThe peeling algorithm is summarized in Algorithm \\ref{algorithm:peeling} and \na detailed illustration is presented in Example \\ref{example:peeling} of Appendix \\ref{section:peeling-example}. \n\n\n\n\\begin{remark}[Recovery of true graph $G$]\n$\\widehat{\\mathcal A}$ estimates a superset of directed relations of $\\bm U$ and $\\widehat{\\mathcal C}$  \na superset of interventional relations of $\\bm W$. Therefore, under some technical conditions, with suitable tuning parameters,\nthe true graph $G$ can be reconstructed by estimating $(\\bm U,\\bm W)$ \nthrough $\\ell_0$-constrained regressions given the estimated super-graph\n$\\widehat{S} = (\\widehat{\\mathcal A},\\widehat{\\mathcal{C}})$,\n\\begin{equation}\\label{directed-likelihood}\n  \\begin{split}\n    \\min_{(\\bm U_{\\textnormal{\\textsc{an}}_{\\widehat{S}}(j), j}, \\bm W_{\\textnormal{\\textsc{in}}_{\\widehat{S}}(j), j})}   \n      \\sum_{i=1}^n \\left(Y_{ij} -\\bm U_{\\textnormal{\\textsc{an}}_{\\widehat{S}}(j), j}^{\\top} \\bm Y_{i,\\textnormal{\\textsc{an}}_{\\widehat{S}}(j)} - \n\\bm W_{\\textnormal{\\textsc{in}}_{\\widehat S}(j), j}^{\\top} \\bm X_{i,\\textnormal{\\textsc{in}}_{\\widehat S}(j)}\\right)^2\\\\\n    \\text{s.t.} \\quad  \\sum_{k\\in\\textnormal{\\textsc{an}}_{\\widehat S}(j)} \\operatorname{I}(U_{kj}\\neq 0) \n    + \\sum_{l\\in\\textnormal{\\textsc{in}}_{\\widehat S}(j)} \\operatorname{I}(W_{lj}\\neq 0)\n     \\leq \\kappa_j'.\n  \\end{split}\n\\end{equation}\nThe final estimates $\\widehat{\\bm U}_{\\cdot j} = \n\\Big(\\widehat{\\bm U}_{\\textnormal{\\textsc{an}}_{\\widehat S}(j),j},\\bm 0_{\\textnormal{\\textsc{an}}_{\\widehat S}(j)^c}\\Big)$\nand $\\widehat{\\bm W}_{\\cdot j} = \n\\Big(\\widehat{\\bm W}_{{\\textnormal{\\textsc{in}}}_{\\widehat S}(j),j},\\bm 0_{{\\textnormal{\\textsc{in}}}_{\\widehat S}(j)^c}\\Big)$;\n$j=1,\\ldots,p$.\n\\end{remark}\n\n\n\n\n\\subsection{Likelihood ratio test via data perturbation}\\label{section:dp-inference}\n\n{This subsection proposes an inference method for testing \\eqref{equation:link-test} and \\eqref{equation:pathway-test}.}\nWe first construct a likelihood ratio $\\text{Lr}(S,\\bm \\Sigma)$ in given $(S,\\bm\\Sigma)$ and then plug-in an estimate $(\\widehat{S},\\widehat{\\bm \\Sigma})$.\nNext we perform a test via data perturbation, accounting for the uncertainty of constructing $S$.\n\n\\subsubsection{Likelihood ratio given $S$}\n\nConsider the test in \\eqref{equation:link-test}. From \\eqref{equation:likelihood-supergraph}, the likelihood ratio given $(S,\\bm\\Sigma)$ is\n\\begin{equation*}\n  \\text{Lr}(S,\\bm \\Sigma) = L_S(\\widehat{\\bm\\theta}_{S}^{(1)},\\bm \\Sigma) - L_S(\\widehat{\\bm\\theta}_{S}^{(0)}, \\bm \\Sigma)\n= \\sum_{j=1}^p \\frac{\\text{RSS}_j(\\widehat{\\bm\\theta}^{(0)}_{S}) - \\text{RSS}_j(\\widehat{\\bm\\theta}^{(1)}_{S})} {2\\sigma_j^2}\n\\end{equation*}\nwhere $\\widehat{\\bm\\theta}_{S}^{(1)}$ is the MLE given $(S,\\bm\\Sigma)$, and $\\widehat{\\bm\\theta}_{S}^{(0)}$ is \nthe MLE given $(S,\\bm\\Sigma)$ subject to $\\bm U_{\\mathcal H} = 0$. \n\n{Now, recall that $\\textnormal{\\textsc{d}}_G(j) = \\{ k:(k,j)\\in\\mathcal{D} \\}$, where $\\mathcal D$ is the set of nondegenerate edges of $H_0$ with respect to $G$ (Definition \\ref{definition:testability}). \nOf note, $\\textnormal{\\textsc{d}}_S(j) = \\textnormal{\\textsc{d}}_G(j)$ since an edge is nondegenerate with respect to $S$ if and only if it is nondegenerate with respect to $G$.\nTo eliminate the nuisance parameters, observe that if $\\textnormal{\\textsc{d}}_S(j) = \\emptyset$, then \n${\\text{RSS}}_{j}(\\widehat{\\bm\\theta}^{(0)}_{S})-{\\text{RSS}}_{j}(\\widehat{\\bm\\theta}^{(1)}_{S})\n= 0$,\nbecause no $H_0$ constraint is imposed for $Y_{j}$.  \nHence, $\\text{Lr}(S,\\bm \\Sigma)$ only summarizes the contributions\nfrom the primary variables with nondegenerate hypothesized edges,\n\\begin{equation}\\label{equation:likelihood-ratio-edge}\n  \\text{Lr}(S,\\bm\\Sigma)\n = \\sum_{j : \\textnormal{\\textsc{d}}_S(j)\\neq \\emptyset} \n \\frac{{\\text{RSS}}_j(\\widehat{\\bm\\theta}^{(0)}_{S})\n - {\\text{RSS}}_j(\\widehat{\\bm\\theta}^{(1)}_{S})}{2\\sigma_j^2},\n\\end{equation}\nwhere the nuisance parameters for $Y_j$ with $\\textnormal{\\textsc{d}}_S(j)=\\emptyset$ are eliminated.}\n\nThe likelihood ratio test statistic becomes \n$\\text{Lr}=\\text{Lr}(\\widehat{S},\n\\widehat{\\bm \\Sigma})$.\nHere, $\\widehat{S}$ is constructed by Algorithm \\ref{algorithm:peeling} in Section \\ref{section:causal-discovery} \nand $\\widehat{\\bm \\Sigma}=\\text{Diag}(\\widehat{\\sigma}^2_1,\\ldots,\\widehat{\\sigma}^2_p)$,\n\\begin{eqnarray}\n\\label{degree-freedom}\n\\widehat{\\sigma}^2_j=\n\\frac{\\text{RSS}_j(\\widehat{\\bm\\theta}^{(1)}_{\\widehat{S}})}{n-|{\\textnormal{\\textsc{an}}}_{\\widehat S}(j)|-|{\\textnormal{\\textsc{in}}}_{\\widehat S}(j)|},\n\\quad j=1,\\ldots,p, \n\\end{eqnarray}\nwhere ${\\textnormal{\\textsc{an}}}_{\\widehat{S}}(j)$ and ${\\textnormal{\\textsc{in}}}_{\\widehat S}(j)$\nare the estimated ancestor and candidate intervention sets of $Y_j$.\n\n\\subsubsection{Testing directed edges \\eqref{equation:link-test} via data perturbation}\n\nThe likelihood ratio requires an estimation of $S$, \nwhere we must account for the uncertainty of $\\widehat{S}$ for accurate finite-sample inference.\n\nTo proceed, we consider the test statistic $\\text{Lr}$ based on a ``correct'' super-graph $S\\supseteq S^\\circ$, \nwhere $S=(\\mathcal{A}, \\mathcal{C}) \\supseteq S^\\circ=(\\mathcal{A}^\\circ, \\mathcal{C}^\\circ)$ means that \n$\\mathcal{A}\\supseteq\\mathcal A^\\circ$ and $\\mathcal C\\supseteq\\mathcal C^\\circ$ \nand $^\\circ$ denotes the truth.  \nIntuitively, {a ``correct'' super-graph distinguishes descendants and nondescendants and thus can help to infer the true directed relations defined by the local Markov property \\citep{spirtes2000causation} without introducing model errors}, yet may lead to a less powerful test when $S$ is much larger than $S^\\circ$.\nBy comparison, a ``wrong'' graph $S \\not \\supseteq S^\\circ$ provides {an incorrect topological order}, and a test based on a ``wrong'' graph may be biased, \naccompanied by an inflated type-I error. \n\nNow, let $\\bm{\\mathsf{Z}}_{n\\times(p+q)} = (\\bm{\\mathsf X}, \\bm{\\mathsf Y})$ be the data matrix, where \n$\\bm{\\mathsf Y}_{n\\times p}=(\\bm Y_1,\\ldots,\\bm Y_n)^\\top$\nand $\\bm{\\mathsf X}_{n\\times q} = (\\bm X_1,\\ldots,\\bm X_n)^\\top$.\nMoreover, let $\\bm{\\mathsf{e}}_j \\in \\mathbb R^n$ be the $j$th column of error matrix\n$\\bm{\\mathsf{e}}_{n\\times p} = (\\bm\\varepsilon_1,\\ldots,\\bm\\varepsilon_n)^\\top$. \nFrom \\eqref{equation:likelihood-ratio-edge}, assuming $\\widehat{S}\\supseteq S^\\circ$ is a ``correct'' super-graph, \nthe likelihood ratio becomes \n\\begin{equation} \\label{equation:likelihood-ratio}\n  \\begin{split}\n    \\text{Lr} \n    &=  \\sum_{j:{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)\\neq\\emptyset} \n    \\frac{\\bm{\\mathsf Y}_j^\\top \n    (\\bm{\\mathsf{P}}_{ \\widehat A_j } - \\bm{\\mathsf{P}}_{ \\widehat B_j}) \n    \\bm{\\mathsf Y}_j }{ 2\\widehat{\\sigma}_j \n       } \\\\ \n    &= \\sum_{j:{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)\\neq\\emptyset}\n    \\frac{( \\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j),j} +  \\bm{\\mathsf e}_j)^\\top (\\bm{\\mathsf{P}}_{ \\widehat A_j } - \\bm{\\mathsf{P}}_{ \\widehat B_j})\n    (\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j),j} + \\bm{\\mathsf e}_j)}{2\\widehat{\\sigma}_j}, \\\\\n    \\widehat{\\sigma}_j &= \\frac{\\bm{\\mathsf Y}_j^\\top(\\bm I-\\bm{\\mathsf{P}}_{ \\widehat A_j })\\bm{\\mathsf Y}_j}{n-|\\widehat A_j|}=\\frac{\\bm{\\mathsf e}_j^\\top(\\bm I-\\bm{\\mathsf{P}}_{ \\widehat A_j })\\bm{\\mathsf e}_j}{n-|\\widehat A_j|}; \\quad 1\\leq j \\leq p, \n  \\end{split}\n\\end{equation}\nwhere $\\bm{\\mathsf{P}}_A = \\bm{\\mathsf Z}_A(\\bm{\\mathsf Z}_A^\\top\\bm{\\mathsf Z}_A)^{-1}\\bm{\\mathsf Z}_A^\\top$ is the projection matrix onto the span of columns $A$ in $\\bm{\\mathsf Z}$,\n$\\widehat{A}_j = {\\textnormal{\\textsc{an}}}_{\\widehat{S}}(j)\\cup {\\textnormal{\\textsc{in}}}_{\\widehat S}(j)\\cup{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)$, and \n{$\\widehat{B}_j = ({\\textnormal{\\textsc{an}}}_{\\widehat{S}}(j)\\cup {\\textnormal{\\textsc{in}}}_{\\widehat S}(j))\\setminus{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)$}.\nOf note, $\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_S(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_S(j),j} = \\bm 0$ for all $j$ under $H_0$,\nwhile $\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_S(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_S(j),j} \\neq \\bm 0$ for some $j$ under $H_a$. \n\nWe propose a data perturbation (DP) method \\citep{shen2002adaptive,breiman1992little} to approximate the null distribution of \\text{Lr} in \\eqref{equation:likelihood-ratio}. The idea behind DP is to assess the sensitivity of the estimates through perturbed data $\\bm{Y}^*_i = \\bm{Y}_i + \\bm \\varepsilon^*_i$, where $\\bm \\varepsilon_i^* \\sim N(0,\\widehat{\\bm\\Sigma})$; $i=1,\\ldots,n$. \n\nLet $(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*) = (\\bm{\\mathsf X},\\bm{\\mathsf Y}^*,\\bm{\\mathsf e}^*)$ be\nperturbed data, where $\\bm{\\mathsf e}^*_{n\\times p}=(\\bm{\\mathsf e}^*_1,\\ldots,\\bm{\\mathsf e}^*_p)=(\\bm\\varepsilon^*_1,\\ldots,\\bm\\varepsilon^*_n)^\\top$ denotes perturbation errors. \nGiven $(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*)$, we obtain a perturbation estimate $\\widehat{S}^*$. \nIn \\eqref{equation:likelihood-ratio}, under $H_0$, $\\text{Lr} = f(\\bm{\\mathsf X}, \\bm \\mu, \\bm{\\mathsf e})$ is\na function of intervention data $\\bm{\\mathsf X}$, conditional mean  \n$\\bm\\mu = \\operatorname{\\mathbb E}(\\bm{\\mathsf{Y}} \\mid \\bm{\\mathsf{X}}) \n\\in \\mathbb R^{n\\times p}$ and unobserved error $\\bm{\\mathsf e}$.\nHowever, the perturbation error $\\bm{\\mathsf e}^*$ is \\emph{known} given the DP sample \n$(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*)$, suggesting a form $\\text{Lr}^*$ based on\n$\\text{Lr}$:  \n\\begin{equation}\\label{equation:likelihood-ratio-dp}\n  \\begin{split}\n    \\text{Lr}^* &\\equiv \n      f(\\bm{\\mathsf X}, \\bm{\\mathsf Y},\\bm{\\mathsf e}^*)= \n      \\sum_{j:{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)\\neq\\emptyset} \n          \\frac{(\\bm{\\mathsf e}_j^*)^\\top (\\bm{\\mathsf{P}}_{\\widehat A^*_j} - \\bm{\\mathsf{P}}_{\\widehat B^*_j})\n          \\bm{\\mathsf e}^*_j}{2\\widetilde{\\sigma}_j^*}, \\quad \n  \\widetilde{\\sigma}^*_j = \\frac{(\\bm{\\mathsf e}_j^*)^\\top(\\bm I-\\bm{\\mathsf{P}}_{\\widehat A^*_j})\\bm{\\mathsf e}^*_j}{n-|\\widehat A^*_j|}, \\quad 1\\leq j\\leq p,\n  \\end{split}\n\\end{equation}\nwhich mimics \\eqref{equation:likelihood-ratio} when $\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_S(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_S(j)} = \\bm 0$ under $H_0$,\nwith $\\bm{\\mathsf e}_j$ replaced by $\\bm{\\mathsf e}^*_j$. \nAs a result, when $\\widehat{S}^*\\supseteq S^\\circ$, the conditional distribution of \n$\\text{Lr}^*$ given $\\bm{\\mathsf Z}$ well approximates\nthe null distribution of $\\text{Lr}$, \nwhere the model selection effect is accounted for by assessing the variability of $\\widehat A^*_j$ and $\\widehat{B}^*_j$ over different realizations of $(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*)$.\n\nIn practice, we use Monte-Carlo to approximate the distribution\nof $\\text{Lr}^*$ given $\\bm{\\mathsf Z}$ in \\eqref{equation:likelihood-ratio-dp}. In particular, we generate $M$ perturbed \nsamples $(\\bm{\\mathsf{Z}}^*_m,\\bm{\\mathsf e}_m^*)_{m=1}^M$ independently and compute $\\text{Lr}^*_m$; $m=1,\\ldots,M$.\nThen, we examine the condition $\\widehat{S}^*_m\\supseteq S^\\circ$ by checking its empirical counterpart \n$\\widehat{S}^*_m\\supseteq \\widehat{S}$.\nThe DP p-value of edge test in \\eqref{equation:link-test} is defined as \n\\begin{equation}\\label{pval:link}\n  \\text{Pval} = \\left(\\sum_{m=1}^{M} \n  \\operatorname{I}(\\text{Lr}_m^*\\geq \\text{Lr},\\widehat{S}^*_m\\supseteq\\widehat{S})\\right)\/\n  \\left( \\sum_{m=1}^{M} \\operatorname{I}(\\widehat{S}^*_m\\supseteq \\widehat{S})\\right),\n\\end{equation}\nwhere $\\operatorname{I}(\\cdot)$ is the indicator function.\n\n\n\n\\begin{remark}\n  Assume \n$\\widehat{S}^*\\supseteq S^\\circ$ is a ``correct'' super-graph. \nA naive likelihood ratio based on \nperturbed data $(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*)$ is \n\\begin{equation*}\n  \\text{Lr}(\\widehat{S}^*,\\widehat{\\bm\\Sigma}^*)=\\sum_{j:{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)\\neq \\emptyset} \\frac{(\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j),j} +  \\bm{\\mathsf e}_j + \\bm{\\mathsf e}^*_j)^\\top\n  (\\bm{\\mathsf{P}}_{\\widehat{A}^*_j} - \\bm{\\mathsf{P}}_{\\widehat{B}^*_j})(\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j),j} + \\bm{\\mathsf e}_j + \\bm{\\mathsf e}^*_j) }{ 2 \\widehat{\\sigma}_j^*}.\n\\end{equation*} \nNote that the error $\\bm{\\mathsf e}_j$ given $\\bm{\\mathsf Z}$ is deterministic and does not vanish under either $H_0$ or $H_a$; $j=1,\\ldots,p$. \nThus, the conditional distribution of $\\text{Lr}(\\widehat{S}^*,\\widehat{\\bm\\Sigma}^*)$ given $\\bm{\\mathsf Z}$ does not approximate the \nnull distribution of $\\text{Lr}$ in \\eqref{equation:likelihood-ratio},\nin contrast to the DP likelihood ratio $\\text{Lr}^*$ in \\eqref{equation:likelihood-ratio-dp}.\n\\end{remark}\n\n\n\\subsubsection{Extension to hypothesis testing for a pathway}\n\nNext, we extend the DP inference for \\eqref{equation:pathway-test}. \nDenote $\\mathcal{H}=\\{ (k_1,j_1),\\ldots,(k_{|\\mathcal{H}|},j_{|\\mathcal{H}|}) \\}$. \nThen the test of pathways in \\eqref{equation:pathway-test} can be reduced to testing sub-hypotheses\n\\begin{equation*}\n  H_{0,\\nu}: U_{k_\\nu,j_\\nu}= 0, \\quad \\mbox{ versus } \n  \\quad {H}_{a,\\nu}: U_{k_\\nu,j_\\nu}\\neq 0; \\quad \\nu=1,\\ldots,|\\mathcal H|,\n\\end{equation*}\nwhere each sub-hypothesis is a directed edge test. \nGiven $(S,\\bm\\Sigma)$, the likelihood ratio \nfor $\\text{H}_{0,\\nu}$ is \n$ \\text{Lr}_\\nu(S,\\bm \\Sigma)\n= L_S(\\widehat{\\bm\\theta}_{S}^{(1)},\\bm \\Sigma) \n- L_S(\\widehat{\\bm \\theta}_{S}^{(0,\\nu)},\\bm\\Sigma)$,\nwhere $\\widehat{\\bm\\theta}_{S}^{(0,\\nu)}$ is the MLE under \nthe constraint that $U_{k_\\nu,j_{\\nu}}=0$. \nThen for $\\widehat{S}\\supseteq S^\\circ$, \nwe have \n\\begin{equation}\n  \\begin{split}\n    \\text{Lr}_\\nu &= \\frac{1}{2}\n  \\frac{( \\bm{\\mathsf Y}_{k_\\nu} U_{k_\\nu, j_{\\nu}} + \\bm{\\mathsf e}_{j_\\nu})^\\top (\\bm{\\mathsf{P}}_{\\widehat A_{j_\\nu}} - \\bm{\\mathsf{P}}_{\\widehat B_{j_\\nu}})\n  (\\bm{\\mathsf Y}_{k_\\nu} U_{k_\\nu, j_\\nu} + \\bm{\\mathsf e}_{j_\\nu})}{(\\bm{\\mathsf e}_{j_\\nu})^\\top(\\bm I-\\bm{\\mathsf{P}}_{\\widehat A_{j_\\nu}})\\bm{\\mathsf e}_{j_\\nu}\/(n-|\\widehat A_{j_\\nu}|)}, \n  \\\\\n  \\text{Lr}_\\nu^* &=\n  \\frac{1}{2} \n  \\frac{(\\bm{\\mathsf e}^*_{j_\\nu})^\\top (\\bm{\\mathsf{P}}_{\\widehat A^*_{j_\\nu}} - \\bm{\\mathsf{P}}_{\\widehat B^*_{j_\\nu}})\n  \\bm{\\mathsf e}^*_{j_\\nu}}{(\\bm{\\mathsf e}^*_{j_\\nu})^\\top(\\bm I-\\bm{\\mathsf{P}}_{\\widehat A^*_{j_\\nu}})\\bm{\\mathsf e}^*_{j_\\nu}\/\n  (n-|\\widehat A^*_{j_\\nu}|)},\n  \\end{split}\n\\end{equation}\nwhere the distributions of $\\text{Lr}_\\nu^*$ given $\\bm{\\mathsf Z}$\napproximates the null distributions of $\\text{Lr}_\\nu$. \nFinally, define the p-value of pathway test in \\eqref{equation:pathway-test} as\n\\begin{equation}\\label{pval:path}\n  \\text{Pval} = \\max_{1\\leq \\nu\\leq |\\mathcal{H}|} \n  \\left(\\sum_{m=1}^{M} \\operatorname{I}(\\text{Lr}_{\\nu,m}^*\\geq \\text{Lr}_{\\nu},\\widehat{S}^*_{m}\\supseteq\\widehat{S})\\right)\n  \/\\left( \\sum_{i=1}^{M} \\operatorname{I}(\\widehat{S}^*_m\\supseteq \\widehat{S})\\right).\n\\end{equation}\nNote that if any $\\text{H}_{0,\\nu}$ is degenerate, then $\\text{Pval} = 1$.\n\nAlgorithm \\ref{algorithm:data-perturbation} summarizes the DP method for hypothesis testing.\n\n\n\\begin{algorithm}\nSpecify the Monte Carlo size $M$. \n\nCompute Lr and $\\widehat{\\bm \\Sigma}=\\text{Diag}(\\widehat{\\sigma}^2_1,\\ldots,\n\\widehat{\\sigma}^2_p)$\nvia Algorithm \\ref{algorithm:peeling} with original data $\\bm{\\mathsf Z}$.\n\n{(Parallel DP)} \nGenerate perturbed data $(\\bm{\\mathsf Z}^*_m,\\bm{\\mathsf e}_{m}^{*}) = (\\bm{\\mathsf X},\\bm{\\mathsf Y}_{m}^*,\\bm{\\mathsf e}_{m}^{*})$ in parallel, \nwhere $\\bm\\varepsilon_{m,i}^{*}\\sim N(\\bm 0,\\widehat{\\bm\\Sigma})$.\nFor \\eqref{equation:link-test}, compute $\\text{Lr}^*_m$ based on $(\\bm{\\mathsf Z}^*_m,\\bm{\\mathsf e}_{m}^{*})$, while, for \\eqref{equation:pathway-test}, \ncompute $\\text{Lr}^*_{\\nu,m}$; $\\nu=1,\\ldots,|\\mathcal{H}|$ based on $(\\bm{\\mathsf Z}^*_m,\\bm{\\mathsf e}_{m}^{*})$;\n$m=1,\\ldots,M$.\n\nCompute the p-value of a test as \\eqref{pval:link} or \\eqref{pval:path} accordingly.\n \n\\caption{DP likelihood ratio test} \\label{algorithm:data-perturbation}\n\\end{algorithm}\n\n\n\n\\begin{remark}[Computation]\n  To accelerate the computation, we consider a parallelizable procedure in Algorithm \\ref{algorithm:data-perturbation}. In addition, we use the original estimate $\\widehat{\\bm\\theta}$ as a warm-start initialization for the DP estimates, which effectively reduces the computing time. \n\\end{remark}\n\n\\begin{remark}[Connection with bootstrap]\n  One may consider parametric or nonparametric bootstrap for $\\text{Lr}$. The parametric bootstrap requires a good initial estimate of $(\\bm U,\\bm W)$. Yet, it is rather challenging to correct the bias of this estimate because of the acyclicity constraint. By comparison, DP does not rely on such an estimate. On the other hand, nonparametric bootstrap resamples the original data with replacement. In a bootstrap sample, only about 63\\% distinct observations in the original data are used in model selection and fitting, leading to deteriorating performance \\citep{kleiner2012big}, especially in a small sample. As a result, nonparametric bootstrap may not well-approximate the distribution of $\\text{Lr}$, while DP provides a better approximation of $\\text{Lr}$, taking advantage of a full sample. \n\\end{remark}\n\n\n\n\n\n\n\n\\section{Theory}\\label{section:theory}\n\nThis section provides the theoretical justification for the proposed methods.\n\n\\subsection{Convergence and consistency of structure learning}\\label{section:consistency}\n\nFirst, we introduce some technical assumptions to derive statistical \nand computational properties of Algorithms \\ref{algorithm:nodewise-regression} and \\ref{algorithm:peeling}. \nLet $(\\bm{\\mathsf Y}, \\bm{\\mathsf X}) \\in \\mathbb R^{n\\times (p+q)}$ be the data matrix. \nDenote by $\\bm\\zeta$ a generic vector and $\\bm\\zeta_A\\in \\mathbb R^{|A|}$ be the projection \nof $\\bm \\zeta$ onto coordinates in $A$. Let \n$\\kappa^\\circ_j = \\|\\bm V^\\circ_{\\cdot j}\\|_0$ and \n$\\kappa^\\circ_{\\max}=\\max_{1\\leq j\\leq p} \\kappa_j^\\circ$.\n\n\\begin{assumption}[Restricted eigenvalues]\\label{assumption:eigen}\n  For a constant $c_1>0$, \n\\begin{equation*}\n  \\min_{ A : |A|\\leq 2 \\kappa^\\circ_{\\max} }\n\\min_{\\{ \\bm\\zeta : \\|\\bm\\zeta_{A^c}\\|_1\\leq 3\\|\\bm\\zeta_{A}\\|_1 \\}} \n\\frac{\\|\\bm{\\mathsf X} \\bm\\zeta \\|_2^2}{n\\| \\bm\\zeta \\|_2^2} \\geq c_1. \n\\end{equation*}\n\\end{assumption}\n\n\n\\begin{assumption}[Interventions]\\label{assumption:intervention}\n  For a constant $c_2>0$, \n\\begin{equation*}\n  \\max\\Big(\\max_{1\\leq l\\leq q} n^{-1}(\\bm{\\mathsf X}^\\top\\bm{\\mathsf X})_{ll},\n    \\max_{A: |A|\\leq 2 \\kappa^\\circ_{\\max}}\\max_{1\\leq l \\leq |A|} n((\\bm{\\mathsf X}_A^\\top\\bm{\\mathsf X}_A)^{-1})_{ll} \\Big)\n    \\leq c_2^2.\n\\end{equation*}\n\\end{assumption}\n\n\\begin{assumption}[Nuisance signals]\\label{assumption:signal}\n  \\begin{equation}\\label{equation:signal}\n    \\min_{ V^0_{lj} \\neq 0}|V^0_{lj}|\n  \\geq 100 \\frac{c_2}{c_1} \\max_{1 \\leq j \\leq p}\\sqrt{\\Omega_{jj}^{-1}\\Big(\\frac{\\log(q)}{n} + \\frac{\\log(n)}{n}\\Big)}.\n  \\end{equation}\n\\end{assumption}\n    \n\nAssumptions \\ref{assumption:eigen}-\\ref{assumption:intervention}, \nas a replacement of Assumption 1A, are satisfied with a probability tending to one\nfor isotropic subgaussian or bounded $\\bm X$ \n\\citep{rudelson2012reconstruction}. \nAssumption \\ref{assumption:eigen} is a common condition for proving the convergence rate of Lasso regression \n\\citep{bickel2009simultaneous,zhang2014lower}. \nAssumption \\ref{assumption:signal}, as an alternative to Assumption 1B, specifies the minimal signal strength over candidate interventions. \nA signal strength requirement like Assumption \\ref{assumption:signal} is a condition for establishing selection consistency in\nhigh dimensional variable selection \\citep{fan2014strong,loh2017support,zhao2018pathwise}.\nMoreover, Assumption \\ref{assumption:signal} can be further relaxed to a less \nintuitive condition, Assumption \\ref{assumption:relax-signal}; see Appendix \\ref{section:relax-signal} for details.\n\n\n\\begin{theorem} [Finite termination and consistency] \\label{theorem:consistent-structure-learning}\nUnder Assumptions \\ref{assumption:identifiability}-\\ref{assumption:signal} with constants $c_1 < 6c_2$, if the machine precision $\\text{tol}\\ll 1\/n$ is negligible, and tuning parameters of Algorithm \\ref{algorithm:nodewise-regression} satisfy:\n\\begin{enumerate}[(1)]\n  \\item $\\gamma_j \\leq  c_1\/6$,\n  \\item $\\sqrt{32c_2^2\\Omega_{jj}^{-1} (\\log (q) + \\log(n))\/n }\/\\gamma_j \\leq \\tau_j \\leq \n  0.4 \\min_{ V^0_{lj} \\neq 0}|V^0_{lj}|$,\n  \\item $\\kappa_j = \\kappa_j^\\circ$,\n\\end{enumerate}\nfor $j=1,\\ldots,p$, \nthen Algorithm \\ref{algorithm:nodewise-regression} yields the global minimizer $\\widehat{\\bm V}_{\\cdot j}$ of \n\\eqref{pseudo-likelihood} in at most $1 + \\lceil \\log (\\kappa_{\\max}^\\circ) \/ \\log 4\\rceil$ DC iterations almost surely, where $\\lceil \\cdot\\rceil$ is the ceiling function. Moreover, \nAlgorithm \\ref{algorithm:peeling} recovers $\\mathcal{A}^\\circ$,\n$\\mathcal C^\\circ \\equiv \\{ l: V^\\circ_{lj}\\neq 0 \\}\\supseteq \\mathcal I^\\circ$, and $\\mathcal D^\\circ$ almost surely as $n\\to\\infty$. \n\\end{theorem}\n\n\n\nBy Theorem \\ref{theorem:consistent-structure-learning} {(proved in Appendix \\ref{proof:consistent-structure-learning})}, \nthe computational complexity of Algorithm \\ref{algorithm:nodewise-regression} is the number of iterations multiplied\nby that of solving a weighted Lasso regression, $O(\\log \\kappa^\\circ_{\\max}) \\times\nO(q^3+n q^2)$ \\citep{efron2004least}. \n{Also note that in Steps 2-6 of Algorithm \\ref{algorithm:peeling}, no heavy computation is involved. Thus, the overall time complexity of Algorithm \\ref{algorithm:peeling} is bounded by $O(p\\times \\log \\kappa^\\circ_{\\max} \\times (q^3+n q^2) )$ for computing $\\widehat{\\bm V}$ matrix.}\nFinally, the peeling algorithm does not apply to observational data ($\\bm W = \\bm 0$). In a sense, interventions play an essential role.\n\nAlthough Theorem \\ref{theorem:consistent-structure-learning} establishes\nthe consistent reconstruction of the peeling algorithm \nfor the super-graph $S$, it does not provide any uncertainty \nmeasure of the presence of some directed edges of the true DAG. \nIn what follows, we will develop a theory for hypothesis tests to make valid inference concerning directed edges of interest. \n\n\n\n\n\\subsection{Inferential theory}\\label{section:inference-theory}\n\n\\begin{assumption}[Hypothesis-specific dimension restriction]\\label{assumption:dimension}\nFor a constant $\\rho$, \n\\begin{equation*}\n  \\max_{j : \\textnormal{\\textsc{d}}_{G^\\circ}(j)\\neq \\emptyset} \\frac{|\\textnormal{\\textsc{an}}_{G^\\circ}(j)| + |\\textnormal{\\textsc{d}}_{G^\\circ}(j)| +{\\kappa_j^\\circ}}{n} \\leq \\rho < 1, \\quad \\text{ as } n \\rightarrow \\infty.\n\\end{equation*}\n\\end{assumption}\n\n\n Assumption \\ref{assumption:dimension} is a hypothesis-specific condition restricting the underlying dimension of the testing problem. \n Usually, $|\\textnormal{\\textsc{an}}_{G^\\circ}(j)| \\asymp \\kappa_j^\\circ \\ll p$; $j=1,\\ldots,p$, \nwhich dramatically relaxes the condition \n$n\\gg {|\\mathcal D|^{1\/2} p\\log(p)}$ for \nthe constrained likelihood ratio test \n\\citep{li2019likelihood}. \n\n\n\n\\begin{theorem}[Empirical p-values]\n  \\label{theorem:dp-null-distribution}\nSuppose Assumptions \\ref{assumption:identifiability}-\\ref{assumption:dimension} are met and $H_0$ is regular.  \nIf the tuning parameters in Algorithm \\ref{algorithm:nodewise-regression}\nsatisfy (1)-(3) in Theorem \\ref{theorem:consistent-structure-learning}, then, \n  \\begin{enumerate}[(A)]\n    \\item (Test of directed edges \\eqref{equation:link-test})  \n    \\begin{equation*}\n      \\begin{split} \n       \\lim_{\n       \\substack{n\\to\\infty\\\\\n       \\bm \\theta^\\circ \\textnormal{ satisfies H}_0 \\textnormal{ in } \\eqref{equation:link-test}}\n       } \n       \\lim_{M \\rightarrow \\infty} \\operatorname{\\mathbb P}_{\\bm\\theta^\\circ}(\\textnormal{Pval} < \\alpha) = \\alpha, \n       &\\quad \\text{ if } H_0 \\text{ is nondegenerate};\\\\\n       \\lim_{\n       \\substack{n\\to\\infty\\\\\n       \\bm \\theta^\\circ \\textnormal{ satisfies H}_0 \\textnormal{ in } \\eqref{equation:link-test}}\n       } \n      \\lim_{M \\rightarrow \\infty} \\operatorname{\\mathbb P}_{\\bm\\theta^\\circ}(\\textnormal{Pval} = 1) = 1, &\\quad \\text{ if } H_0 \\text{ is degenerate}.\n      \\end{split}\n    \\end{equation*}\n    \\item (Test of directed pathways \\eqref{equation:pathway-test})\n    \\begin{equation*}\n      \\begin{split}\n        \\limsup_{\n       \\substack{n\\to\\infty\\\\\n       \\bm \\theta^\\circ \\textnormal{ satisfies H}_0 \\textnormal{ in } \\eqref{equation:pathway-test}}\n       }  \n     \\lim_{M \\rightarrow \\infty} \\operatorname{\\mathbb P}_{\\bm\\theta^\\circ}(\\textnormal{Pval} < \\alpha) = \\alpha, &\\quad \\text{ if } H_0 \\text{ is nondegenerate with } |\\mathcal D|=|\\mathcal H|;\\\\\n     \\limsup_{\n      \\substack{n\\to\\infty\\\\\n      \\bm \\theta^\\circ \\textnormal{ satisfies H}_0 \\textnormal{ in } \\eqref{equation:pathway-test}}\n      }  \n      \\lim_{M \\rightarrow \\infty} \\operatorname{\\mathbb P}_{\\bm\\theta^\\circ}(\\textnormal{Pval} = 1) = 1, &\\quad \\text{ if } |\\mathcal D|<|\\mathcal H|.\n      \\end{split}\n    \\end{equation*}\n  \\end{enumerate}\n\\end{theorem} \n\nBy Theorem \\ref{theorem:dp-null-distribution}, the DP likelihood ratio test yields a valid p-value for \\eqref{equation:link-test} and \\eqref{equation:pathway-test}.\nNote that $|\\mathcal D|$ is permitted to depend on $n$. \nMoreover, Proposition \\ref{proposition:asymptotic} summarizes the asymptotics for \ndirected edge test \\eqref{equation:link-test}. \n\n\\begin{proposition} [Asymptotic distribution of $\\textnormal{Lr}$ of \\eqref{equation:link-test}] \\label{proposition:asymptotic}\n  Assume that the assumptions of Theorem \\ref{theorem:dp-null-distribution} are met. Under a regular $H_0$, as $n\\to\\infty$, \n  \\begin{equation*}\n    \\begin{split}\n      2 \\textnormal{Lr}  \\overset{d}{\\longrightarrow} \\chi^2_{|\\mathcal D|},  & \\text{ if } H_0 \\text{ is nondegenerate with }  |\\mathcal D| > 0 \\text{ being fixed};  \\\\\n   \\frac{2 \\textnormal{Lr} - |\\mathcal D|}{\\sqrt{2|\\mathcal D|}} \\overset{d}{\\longrightarrow} N(0,1), & \\text{ if } H_0 \\text{ is nondegenerate with } |\\mathcal D| \\rightarrow \\infty, \n   \\ \\frac{|\\mathcal D|\\log |\\mathcal D|}{n} \\rightarrow 0.\n    \\end{split}\n\\end{equation*}\n\\end{proposition}\n\n{The proofs of Theorem \\ref{theorem:dp-null-distribution} and Proposition \\ref{proposition:asymptotic} are given in Appendix \\ref{proof:dp-null-distribution} and \\ref{proof:asymptotic}.}\n\n\\begin{remark}[Local structure for hypothesis testing]\n  Theorem \\ref{theorem:dp-null-distribution} or Proposition \\ref{proposition:asymptotic} only requires correct identification of the local structures of the DAG such as, $\\textnormal{\\textsc{an}}_S(j) = \\textnormal{\\textsc{an}}_G(j)$, $\\textnormal{\\textsc{d}}_S(j) = \\textnormal{\\textsc{d}}_G(j)$, and $\\textnormal{\\textsc{in}}_S(j)$, for variables $Y_j$ with $\\textnormal{\\textsc{d}}_S(j)\\neq\\emptyset$, as opposed to those of entire $S$.\n\\end{remark}\n\n\nNext, we analyze the local limit power of the proposed tests for \\eqref{equation:link-test} and \\eqref{equation:pathway-test}. \nAssume $\\bm \\theta^\\circ = (\\bm U^\\circ,\\bm W^\\circ)$ satisfies $H_0$. \nLet $\\bm \\Delta\\in\\mathbb R^{p\\times p}$ satisfy\n$\\bm \\Delta_{\\mathcal D^c}=\\bm 0$ so that $\\bm U^\\circ+\\bm \\Delta$ represents a DAG. \nFor nondegenerate and regular $H_0$, consider an alternative $H_a$: $\\bm U_{\\mathcal H} = \\bm U_{\\mathcal H}^\\circ + \\bm \\Delta_{\\mathcal H}$,\nand define the power function as \n\\begin{equation}\n  \\beta(\\bm\\theta^\\circ,\\bm \\Delta) = \\operatorname{\\mathbb P}_{H_a}( \\text{Pval} < \\alpha ).\n\\end{equation}\n\n\n\n\\begin{proposition}[Local power of edge test]\n\\label{proposition:power-edge}\nAssume that $H_0$ is nondegenerate and regular. \nLet $\\| \\bm \\Delta\\|_{F} = \n\\|\\bm \\Delta_{\\mathcal H}\\|_{F} = n^{-1\/2} \\delta$ when $|\\mathcal D| > 0$ is fixed \nand $\\|\\bm \\Delta\\|_{F} = |\\mathcal D|^{1\/4} n^{-1\/2}h$ when $|\\mathcal D|\\to\\infty$, \nwhere $\\delta>0$ and $\\|\\cdot\\|_{F}$ is the matrix Frobenius norm. \nIf the assumptions of Theorem \\ref{theorem:dp-null-distribution} are met, then under $H_a$, as $n,M\\to\\infty$,\n\\begin{equation*}\n    \\beta(\\bm\\theta^\\circ,\\bm \\Delta) \\geq\n    \\begin{cases}\n      \\operatorname{\\mathbb P}\\Big(\\|\\bm Z + c_l\\sqrt{n}\\bm \\Delta \\|_2^2 > \\chi^2_{|\\mathcal D|,1-\\alpha} \\Big) & \\text{if } |\\mathcal D|>0 \\text{ is fixed};\\\\\n      \\operatorname{\\mathbb P}\\Big(Z > z_{1-\\alpha} - {c_l \\|\\bm \\Delta\\|^2_2}\/{\\sqrt{2|\\mathcal D|}} \\Big) & \\text{if } |\\mathcal D|\\to\\infty, \\frac{|\\mathcal D|\\log|\\mathcal D|}{n}\\to 0,\n    \\end{cases}\n\\end{equation*}\nwhere $\\bm Z\\sim N(\\bm 0,\\bm I_{|\\mathcal D|})$, $Z\\sim N(0,1)$, and $\\chi^2_{|\\mathcal D|,1-\\alpha}$ and $z_{1-\\alpha}$ are the $(1-\\alpha)$th quantile of distributions $\\chi^2_{|\\mathcal D|}$ and $N(0,1)$, respectively.\nIn particular, $\\lim_{\\delta\\to\\infty} \\lim_{n\\to\\infty} \\beta(\\bm\\theta^\\circ,\\bm \\Delta) = 1$.\n\\end{proposition}\n\n\\begin{proposition} [Local power of pathway test]\n  \\label{proposition:power-path}\nAssume that $H_0$ is nondegenerate and regular with $|\\mathcal D| = |\\mathcal H|$. \nLet $\\min_{(k,j)\\in \\mathcal H}|U^\\circ_{kj} + \\Delta_{kj}| = n^{-1\/2}\\delta$ when $|\\mathcal H|>0$ is fixed\nand $\\min_{(k,j)\\in \\mathcal H}|U^\\circ_{kj} + \\Delta_{kj}| = n^{-1\/2}\\delta\\sqrt{\\log|\\mathcal H|}$ when $|\\mathcal H|\\to\\infty$.\nIf the assumptions of Theorem \\ref{theorem:dp-null-distribution} are met, then under $H_a$, as $n,M\\to\\infty$,\n  \\begin{equation*}\n      \\beta(\\bm\\theta^\\circ,\\bm \\Delta) \\geq 1 - \\frac{|\\mathcal H|}{\\sqrt{2\\pi}} \n      \\exp\\Big( -\\Big(\\delta\\sqrt{\\log|\\mathcal H|}\/\\max_{1\\leq j\\leq p}\\Omega_{jj} - \\sqrt{\\chi^2_{1,1-\\alpha}}\\Big)^2\/2   \\Big),\n  \\end{equation*}\n  where $Z \\sim N\\Big(\\delta^2\/\\max_{1\\leq j\\leq p}\\Omega_{jj},1\\Big)$ and \n  $\\chi^2_{1,1-\\alpha}$ is the $(1-\\alpha)$th quantile of distribution $\\chi^2_{1}$.\n  Then, $\\lim_{\\delta\\to\\infty} \\lim_{n\\to\\infty} \\beta(\\bm\\theta^\\circ,\\bm \\Delta) = 1$.\n\\end{proposition}\n\n{The proofs of Propositions \\ref{proposition:power-edge} and \\ref{proposition:power-path} are deferred to Appendix \\ref{proof:power-edge} and \\ref{proof:power-path}.}\n\n\\section{Simulations}\\label{section:simulation}\n\nThis section investigates the operating characteristics of the proposed tests and the peeling algorithm via simulations.  In simulations, we consider two setups for generating $\\bm U\\in\\mathbb{R}^{p\\times p}$, representing random and hub DAGs, respectively.\n\\begin{itemize}\n  \\item \\textbf{Random graph.} The upper off-diagonal entries $U_{kj}$; $k<j$ are sampled independently from $\\{0,1\\}$ according to Bernoulli$(1\/p)$, while other entries are zero.\n  This generates a random graph with a sparse neighborhood. \n\n  \\item \\textbf{Hub graph.} Set $U_{1,2j+1} = 1$ and $U_{2,2j+2} = 1$ for $j= 1,\\ldots, \\lfloor p\/2\\rfloor -2$, while other entries are zero. This generates a hub graph, where nodes 1 and 2 are hub nodes with a dense neighborhood.\n\\end{itemize}\n\nMoreover, we consider three setups for intervention matrix $\\bm W\\in\\mathbb{R}^{q\\times p}$, representing different scenarios. Setups A and B are designed for inference, whereas\nSetup C in Section \\ref{section:structure-learning-simulation} is designed to compare with the method of \n\\citet{chen2018two} for structure learning.\nLet $\\bm W = (\\bm A^\\top, \\bm B^\\top, \\bm 0^\\top)^\\top$,\nwhere $\\bm A,\\bm B\\in\\mathbb{R}^{p\\times p}$ and $\\bm 0\\in\\mathbb{R}^{(q-2p)\\times p}$.\n\\begin{itemize}\n  \\item \\textbf{Setup A.} Set $A_{jj} = B_{jj} = B_{j,j+1} = 1$; $j = 1,\\ldots,p-1$, $A_{pp}= 1$, while other entries of $\\bm A$, $\\bm B$ are zero. Then, $X_1,\\ldots,X_p$ are instruments for $Y_1,\\ldots,Y_p$, respectively, $X_{p+1},\\ldots,X_{2p-1}$ are invalid instruments with two targets, and $X_{2p},\\ldots,X_q$ represent inactive interventions. \n\n  \\item \\textbf{Setup B.} Set $A_{jj}=A_{j,j+1} = B_{jj} = B_{j,j+1}=1$; $j=1,\\ldots,p-1$, $A_{pp} = 1$,\nwhile other entries of $\\bm A$, $\\bm B$ are zero. Here, the only valid instrument is $X_{p}$ on $Y_p$, and the other intervention variables either have two targets or are inactive. Importantly, the instrumental sufficiency (Assumption 1C) is not met.\n    \n\\end{itemize}\n\nTo generate $(\\bm X,\\bm Y)$ for each setup, we sample $\\bm X\\sim N(\\bm 0,\\bm\\Sigma_X)$ with $(\\Sigma_X)_{ll'} = 0.5^{|l-l'|}$; $1\\leq l,l'\\leq q$ and sample $\\bm Y$ according to \\eqref{equation:model} with $(\\bm U,\\bm W,\\sigma_1^2,\\ldots,\\sigma_p^2)$, where $\\sigma_1^2,\\ldots,\\sigma_p^2$ are set to be equally spaced from $0.5$ to $1$. \n\n\n\n\\subsection{Inference}\\label{section:inference-simulation}\n\nWe compare three tests in empirical type-I errors and powers in simulated examples, \nnamely, the DP likelihood ratio test (DP-LR) in Algorithm \\ref{algorithm:data-perturbation},\nthe asymptotic likelihood ratio test (LR), and the oracle likelihood ratio test (OLR). Here LR uses $\\text{Lr}(\\widehat{S},\\widehat{\\bm\\Sigma})$, while \nOLR uses $\\text{Lr}(S^\\circ,\\widehat{\\bm\\Sigma})$ assuming that \nthe super-graph $S^\\circ$ were known in advance.\nThe p-values of LR and OLR are computed via Proposition \\ref{proposition:asymptotic}. \nThe implementation details of these tests are in Appendix \\ref{section:implementation}.\n\nFor the empirical type-I error of a test, we compute the percentage of times \nrejecting $H_0$ out of 500 simulations when $H_0$ is true. \nFor the empirical power of a test, we report the percentage of times\nrejecting $H_0$ out of 100 simulations when $H_a$ is true under alternative hypotheses $H_a$. \n\\begin{itemize}\n  \\item \\textbf{Test of directed edges.} \n  For \\eqref{equation:link-test}, we examine two different hypotheses: \n  \n  (i) $H_0: U_{1, 20} = 0$ versus $H_a:  U_{1,20} \\neq  0$. In this case, $|\\mathcal D|=1$.\n  \n  (ii) $H_0: \\bm U_{\\mathcal H} = \\bm 0$ versus $H_a: \\bm U_{\\mathcal H} \\neq  \\bm 0$, where $\\mathcal{H} = \\{ (k,20): k=1,\\ldots,15 \\}$. In this case, $|\\mathcal D|=15$.\n\n  Moreover, five alternatives \n  $H_a: U_{1,20} = 0.1 l$ and $\\bm U_{\\mathcal H\\setminus\\{ (1,20)\\}}= \\bm 0$; $l=1,2,3,4,5$,\n  are used for the power analysis in (i) and (ii).\n  The data are generated by modifying $\\bm U$ accordingly.\n\n  \\item \\textbf{Test of directed pathways.} \n  We examine the test of a directed path \n  $Y_1\\to Y_5\\to Y_{10}\\to Y_{15}\\to Y_{20}$, namely\n  $\\mathcal H = \\{ (1,5),(5,10),(10,15),(15,20) \\}$ in \\eqref{equation:pathway-test}.\n  Since \\eqref{equation:pathway-test} is a test of composite null hypothesis, the data are generated under a graph with parameters $(\\bm U,\\bm W,\\bm \\Sigma)$ satisfying $H_0$, \n  where $\\bm U_{\\mathcal H} = \\bm 0$. Five alternatives $H_a: \\bm U_{\\mathcal H} = \\bm{0.1} l $; $l=1,2,3,4,5$ are used for the power analysis.\n\\end{itemize}\n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{inference.pdf}\n  \\caption{Empirical type-I errors and powers of tests of directed edges.\n  The black dotted line marks the nominal level of significance $\\alpha=0.05$.}\n  \\label{figure:edge-inference}\n\\end{figure}\n\n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{path_inference.pdf}\n  \\caption{Empirical type-I errors and powers of tests of a directed pathway.\n  The black dotted line marks the nominal level of significance $\\alpha=0.05$.}\n  \\label{figure:path-inference}\n\\end{figure}\n\nFor testing directed edges, as displayed in Figure \\ref{figure:edge-inference}, DP-LR and LR perform well compared to the ideal test OLR in Setup A with Assumption \\ref{assumption:identifiability} satisfied. In setup B with \nAssumption 1C not fulfilled, DP-LR appears to have control of \ntype-I error, whereas LR has an inflated empirical type-I error compared to\nthe nominal level $\\alpha = 0.05$. This discrepancy is attributed to the data\nperturbation scheme accounting for the uncertainty of identifying \n$S$. However, both suffer a loss of power compared to the oracle test OLR in this \nsetup. This observation suggests that without Assumption 1C, the peeling algorithm \ntends to yield an estimate $\\widehat{S}\\supseteq S^\\circ$, which overestimates $S^\\circ$, \nresulting in a power loss. \n\nFor testing directed pathways, as indicated in Figure \\ref{figure:path-inference},\nwe observe similar phenomena as in the previous directed edge tests.\nOf note, both LR and DP-LR are capable in controlling type-I error of directed path tests.\n\nIn summary, DP-LR has a suitable control of type-I error when there are invalid instruments and Assumption 1C is violated. \nConcerning the power, DP-LR and LR are comparable in all scenarios and \ntheir powers tend to one as the sample size $n$ or the signal strength \nof tested edges increases. Moreover, DP-LR and LR perform nearly as well\nas the oracle test OLR when Assumption 1 is satisfied. These empirical findings\nagree with our theoretical results. \n\n\n\n\\subsection{Structure learning}\\label{section:structure-learning-simulation}\n\nThis subsection compares the peeling algorithm with the Two-Stage Penalized Least Squares (2SPLS, \\citet{chen2018two}) in terms of the structure learning accuracy. For peeling, we consider Algorithm \\ref{algorithm:peeling} with an additional step \\eqref{directed-likelihood} for structure learning of $\\bm U$. \nFor 2SPLS, we use the R package \\texttt{BigSEM}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{structure_learning.pdf}\n  \\caption{Structural Hamming distances (SHDs) for the peeling algorithm \nand 2SPLS, where a smaller value of SHD indicates a better result.}\n  \\label{figure:structure-learning}\n\\end{figure}\n\n2SPLS requires that all the intervention variables to be target-known instruments in addition to Assumption 1C. Thus, we consider \nan additional Setup C.\n\\begin{itemize}\n  \\item \\textbf{Setup C.} Let $\\bm W = (\\bm I_{p\\times p},\\bm 0)^\\top\\in\\mathbb{R}^{q\\times p}$. Then $X_1,\\ldots,X_p$ are valid instruments for $Y_1,\\ldots,Y_p$, respectively, and other intervention variables are inactive.\n\\end{itemize}\nFor 2SPLS, we assign each active intervention variable to its most correlated primary variable in Setups A-C. \nIn Setup C, this assignment yields a correct identification of valid instruments, meeting all the requirements of 2SPLS.\n\n\n\nFor each scenario, we compute the average structural Hamming distance $\\text{SHD}(\\widehat{\\bm U}, \\bm U^\\circ) = \\sum_{k,j}|\\operatorname{I}(\\widehat{U}_{kj}\\neq 0)-\\operatorname{I}(U^\\circ_{kj}\\neq 0)|$ over 100 runs.\nAs shown in Figure \\ref{figure:structure-learning}, the peeling algorithm \noutperforms 2SPLS, especially when there are invalid \ninstruments and Assumption 1C is violated. \n\n\n\\subsection{Comparison of inference and structure learning}\\label{section:comparison-simulation}\n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{additional.pdf}\n  \\caption{{Rejection rates for $H_0:U_{1,20}=0$ versus $H_a:U_{1,20}=1\/\\sqrt{n}$ by DP inference and structure learning. For structure learning, $H_0$ is rejected if $\\widehat{U}_{1,20}\\neq 0$. The red lines indicate the results of DP inference using the significance level $\\alpha=0.05$. The other colored lines display the results of structure learning using different sparsity parameter values $\\kappa = 1, 2, 3, 4, 5$.\n  The simulation is repeated for 500 times and $\\kappa = 2$ is chosen by BIC in over 90\\% cases.}}\n  \\label{figure:comparison-inference-learning}\n\\end{figure}\n\n\n{\nThis subsection compares the proposed DP testing method against the proposed structure learning method in \\eqref{directed-likelihood} in terms of inferring the true graph structure.\nTo this end, we consider Setup A in Section \\ref{section:inference-simulation} with $p = 30, q = 100$, and the hypotheses \n\\begin{center}\n  $H_0: U_{1,20} = 0$ versus $H_a: U_{1,20} = 1\/\\sqrt{n}$.\n\\end{center}\nFor DP inference, we use $\\alpha = 0.05$ and choose the tuning parameters by BIC as in previous experiments; see Appendix \\ref{section:implementation} for details. \nFor structure learning, we reject the null hypothesis when $\\widehat{U}_{1,20}\\neq 0$. \n\nAs displayed in Figure \\ref{figure:comparison-inference-learning},\nwhen the null hypothesis $H_0$ is true, the DP testing method controls type-I error very close to the nominal level of 0.05, whereas the type-I error of the structure learning varies greatly depending on the tuning parameter selection. Under the alternative hypothesis $H_a$, the DP inference enjoys high statistical power than structure learning methods when $n\\geq 200$. Interestingly, the power of structure learning diminishes as $n$ increases. This observation is in agreement with our theoretical results in Theorem \\ref{theorem:consistent-structure-learning}, suggesting that consistent reconstruction requires the smallest size of nonzero coefficients to be of order $\\sqrt{\\log(n)\/n}$ with the tuning parameter $\\tau$ of the same order (fixing $p,q$).\nIn this case, the edge $U_{1,20}$ is of order $1\/\\sqrt{n}$, which is less likely to be reconstructed as $n$ increases. In contrast, Proposition \\ref{proposition:power-edge} indicates that a DP test has a non-vanishing power when the hypothesized edges are of order $1\/\\sqrt{n}$.\n\nFigure \\ref{figure:comparison-inference-learning} demonstrates some important distinctions between inference and structure learning. When different tuning parameters are used, the structure learning results correspond to different points on an ROC curve. Although it is asymptotically consistent when optimal tuning parameters are used, structure learning lacks an uncertainty measure of graph structure identification. As a result, it is nontrivial for structure learning methods to trade-off the false discovery rate and detection power in practice. This makes the interpretation of such results hard, especially when they heavily rely on hyperparameters as in Figure \\ref{figure:comparison-inference-learning}. By comparsion, DP inference aims to maximize {statistical power while controlling} type-I error at a given level, offering a clear interpretation of its result. \nThis observation agrees with the discussions in the literature on variable selection and inference \\citep{wasserman2009high,meinshausen2010stability,lockhart2014significance,candes2018panning} and it justifies the demand for inferential tools for directed graphical models.\n}\n\n\n\n\n\n\n\\section{ADNI data analysis}\\label{section:real-data}\n\n\nThis section applies the proposed tests to analyze an Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset. \nIn particular, we infer gene pathways related to Alzheimer's Disease (AD) to highlight some gene-gene interactions differentiating patients with AD\/cognitive impairments and healthy individuals.\n\nThe raw data are available in the ADNI database (\\url{https:\/\/adni.loni.usc.edu}), including gene expression, whole-genome sequencing, and phenotypic data. After cleaning and merging, we have a sample size of 712 subjects. \nFrom the KEGG database \\citep{kanehisa2000kegg}, we extract the AD reference pathway (hsa05010, \\url{https:\/\/www.genome.jp\/pathway\/hsa05010}), including 146 genes in the ADNI data.\n\nFor data analysis, we first regress the gene expression levels on five covariates \n-- {gender}, {handedness}, {education level}, {age}, and {intracranial volume}, and \nthen use the residuals as gene expressions in the following analysis. \nNext, we extract the genes with at least one SNP at a marginal significance level below $10^{-3}$, yielding $p=63$ genes as primary variables.\nFor these genes, we further extract their marginally most correlated two SNPs, resulting in $q = 63\\times2 = 126$ SNPs as unspecified intervention variables, in subsequent data analysis.\nAll gene expression levels are normalized. \n\nThe data contains individuals in four groups, namely, Alzheimer's Disease (AD), Early Mild Cognitive Impairment (EMCI), Late Mild Cognitive Impairment (LMCI), and Cognitive Normal (CN). \nFor our purpose, we treat 247 CN individuals as controls while the remaining 465 individuals as cases (AD-MCI). {Then, we use the gene expressions and the SNPs to reconstruct the ancestral relations and infer gene pathways for 465 AD-MCI and 247 CN control cases, respectively.}\n\n\n\n\\begin{figure}\n  \\centering\n  \\subfloat[AD-MCI]{\\includegraphics[width=0.35\\textwidth]{AD-MCI-subnet.pdf}\\label{figure:ad-mci-subnet}}\n  \\hfill\n  \\subfloat[CN]{\\includegraphics[width=0.35\\textwidth]{CN-subnet.pdf}\\label{figure:cn-subnet}}\\\\\n  \\subfloat[AD-MCI]{\\includegraphics[width=0.35\\textwidth]{2SPLS-AD-MCI-subnet.pdf}\\label{figure:2sls-ad-mci-subnet}}\n  \\hfill\n  \\subfloat[CN]{\\includegraphics[width=0.35\\textwidth]{2SPLS-CN-subnet.pdf}\\label{figure:2sls-cn-subnet}}\n  \\caption{\n    Display of the subnetworks associated with genes APP and CASP3.\n    (a) and (b): Solid\/dashed arrows indicate significant\/insignificant edges at $\\alpha=0.05$\n    after adjustment for multiplicity by the Bonferroni-Holm correction.\n    (c) and (d): Solid arrows indicate the reconstructed edges using 2SPLS \\citep{chen2018two}.}\n  \\label{figure:network}\n\\end{figure}\n\n\\begin{figure}\t\n  \\centering\n  \\subfloat[AD-MCI]{\\includegraphics[width=0.35\\textwidth]{AD-MCI-path.pdf}\\label{figure:ad-mci-path}}\n  \\hfill\n  \\subfloat[CN]{\\includegraphics[width=0.35\\textwidth]{CN-path.pdf}\\label{figure:cn-path}}\n  \\caption{The p-values of pathway tests \\eqref{equation:pathway-test} by the proposed\ntests for the AD-MCI and CN groups, where p-values are adjusted for multiplicity by \nthe Bonferroni-Holm correction and solid\/dashed arrows indicate significant\/insignificant pathways\nat $\\alpha=0.05$.}\n  \\label{figure:PSEN1}\n\\end{figure}\n\n\n\nIn the literature, genes APP, CASP3, and PSEN1 are well-known to be associated with AD, reported to play different roles in AD patients and healthy subjects \\citep{julia2017genetics,su2001activated,kelleher2017presenilin}.\n{For this dataset, \nwe conduct hypothesis testing on edges and pathways related to genes APP, CASP3, and PSEN1 in the KEGG AD reference (hsa05010) to evaluate the proposed DP inference by checking if DP inference can discover the differences that are reported in the biomedical literature.}\nFirst, we consider testing  \n$H_0: U_{kj}=0$ versus \n$H_a: U_{kj}\\neq0$,\nfor each edge $(k,j)$\nas shown in Figure \\ref{figure:network} (a) and (b).\nMoreover, we consider two hypothesis tests of pathways\n$H_0: U_{kj}=0$ for some $(k,j)\\in\\mathcal{P}_\\ell$ versus \n$H_a: U_{kj}\\neq 0$ for all $(k,j)\\in\\mathcal{P}_\\ell$; $\\ell=1,2$,\nwhere the two pathways are specified by \n$\\mathcal{P}_1 = \\{ \\text{PSEN1}\\to\\text{CAPN1}\\to\\text{CDK5R1} \\}$,\nand $\\mathcal{P}_2 = \\{ \\text{PSEN1}\\to \\text{CAPN2}\\to\\text{CDK5R1} \\}$.\nSee Figure \\ref{figure:PSEN1}.\n{Of note, for clear visualization, Figure \\ref{figure:network} (a), (b) and Figure \\ref{figure:PSEN1} only display the edges related to hypothesis testing, whereas the ancestral relations are reconstructed using $p=63$ genes and $q = 126$ SNPs for AD-MCI and CN groups separately.}\n\n\n{In Figures \\ref{figure:network}-\\ref{figure:PSEN1}, \nthe significant results under the level $\\alpha=0.05$ after the Holm-Bonferroni adjustment for \n$2\\times (9 + 2)= 22$ tests\nare displayed.} In Figures \\ref{figure:network},\nthe edge test in \\eqref{equation:link-test} exhibits a strong evidence for the \npresence of directed connectivity $\\{$APP $\\to$ APBB1, APP $\\to$ GSK3B, FADD $\\to$ CASP3$\\}$ \nin the AD-MCI group, but no evidence in the CN group. \nMeanwhile, this test suggests the presence of connections \n$\\{$TNFRSF1A $\\to$ CASP3, FADD $\\to$ CASP8$\\}$ in the CN group but not so in the AD-MCI group. \nIn both groups, we identify directed connections $\\{$TNFRSF1A $\\to$ FADD, TNFRSF1A $\\to$ CASP8$\\}$. \nIn Figure \\ref{figure:PSEN1},\nthe pathway test \\eqref{equation:pathway-test} supports the presence of a pathway \nPSEN1 $\\to$ CAPN1 $\\to$ CDK5R1 in the AD-MCI group with a p-value of $0.044$ \nbut not in the CN group with a p-value of $0.33$. \nThe pathway PSEN1 $\\to$ CAPN2 $\\to$ CDK5R1 appears insignificant at $\\alpha=0.05$ for\nboth groups. Also noted is that some of our discoveries agree with the literature\naccording to the AlzGene database (\\url{alzgene.org}) and the AlzNet database (\\url{https:\/\/mips.helmholtz-muenchen.de\/AlzNet-DB}).\nSpecifically, GSK3B differentiates AD patients from normal subjects;\nas shown in Figures \\ref{figure:network},\nour result indicates the presence of connection APP $\\to$ GSK3B for the AD-MCI group,\nbut not for the CN group, the former of which is confirmed by \nFigure 1 of \\citet{vanleuven2011gsk3}.\nThe connection APP $\\to$ APBB1 also differs in AD-MCI and CN groups, \nwhich appears consistent with Figure 3 of \\citet{bu2009apolipoprotein}.\nMoreover, the connection CAPN1 $\\to$ CDK5R1, in the pathway PSEN1 $\\to$ CAPN1 $\\to$ CDK5R1 discovered in AD-MCI group, \nis found in the AlzNet database (interaction-ID 24614, \\url{https:\/\/mips.helmholtz-muenchen.de\/AlzNet-DB\/entry\/show\/1870}).\nFinally, as suggested by Figure \\ref{figure:residuals}, the normality assumption in \\eqref{equation:model} is adequate for both groups.\n\nBy comparison, as shown in Figure \\ref{figure:network} (c) and (d), gene APP in the reconstructed networks by 2SPLS \\citep{chen2018two} is not connected with other genes, \n{indicating no regulatory relation of APP with other genes in the AD-MCI and CN groups. However, as a well-known gene associated with AD, APP is reported to play different roles in controlling the expressions of other genes for AD patients and healthy people \\citep{matsui2007expression,julia2017genetics}. Our results in Figure \\ref{figure:network} (a) and (b) are congruous with the studies: the connections of APP with other genes are different in our estimated networks for AD-MCI and CN groups.}\n\n\n\n\n\n\\begin{figure}\t\n  \\centering\n  \\includegraphics[width=0.7\\textwidth]{residuals.pdf}\n  \\caption{Normal quantile-quantile plots of studentized residuals of the AD-MCI and CN groups. }\n  \\label{figure:residuals}\n\\end{figure}\n\nIn summary, our findings seem to agree with those in the literature \\citep{julia2017genetics,su2001activated,kelleher2017presenilin}, {where the subnetworks of genes APP, CASP3 in Figure \\ref{figure:network} and PSEN1 in  Figure \\ref{figure:PSEN1} differentiate the AD-MCI from the CN groups}. Furthermore, the pathway PSEN1 $\\to$ CAPN1 $\\to$ CDK5R1 in Figure \\ref{figure:PSEN1} seems to differentiate these groups, which, however, requires validation in biological\nexperiments.\n\n\n\n\n\\section{Summary}\\label{section:conclusion}\n\nThis article proposes structure learning and inference methods for a Gaussian DAG with interventions, where the targets and strengths of interventions are unknown.\nA likelihood ratio test is derived based on an estimated super-graph formed by ancestral relations and candidate interventional relations. \nThis test accounts for the statistical uncertainty of the construction of the super-graph based on a novel data perturbation scheme. \nMoreover, we develop a peeling algorithm for the super-graph construction. \nThe peeling algorithm allows scalable computing and yields a consistent estimator.\nThe numerical studies justify our theory and demonstrate the utility of our methods.\n\nThe proposed methods can be extended to many practical situations beyond biological applications with independent and identically distributed data. An instance is to infer directed relations between multiple autoregressive time series \\citep{pamfil2020dynotears}, where the lagged variables and covariates can serve as interventions for each time series.\n\n\n\\acks{The authors would like to thank the action editor and anonymous referees for helpful comments and suggestions.\nThe research is supported by NSF grants DMS-1712564, DMS-1952539\nand NIH grants R01GM126002, R01HL116720, R01HL105397, R01AG069895, R01AG065636, R01AG074858, and U01AG073079.\n}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn interplay  between arithmetic and geometry is an established research area,\nsee e.g.  [Weil 1949] \\cite{Wei1}.  \nThe proof of  Weil's Conjectures was a  motivation  for the notion  of a scheme\n[Grothendieck 1960]  \\cite{Gro1}.  Recall that $Spec~R$  \nof  a commutative ring $R$ is  the set of all prime ideals of $R$ endowed \nwith the Zariski topology.   The topology of  $Spec~R$ is not  Hausdorff, yet it  admits a \ncohomology theory and an analog of the Lefschetz Fixed-Point Theorem  [Grothendieck 1960]  \\cite{Gro1}.\nIt appears to be enough to prove Weil's Conjectures. \n\nLet  $\\mathbf{Z}$ be  the ring of integers.  It was observed,  that $Spec~\\mathbf{Z}$\n ``looks  like''   the Riemann sphere $\\mathbf{C}P^1$  [Eisenbud \\& Harris 1999] \\cite[p. 83]{EH}.\nSuch an analogy extends to the inclusions $\\mathbf{Z}\\subset O_K$, where  $O_K$ is the ring of integers \nof a number field $K$.  In this case,  one gets  a  ramified covering $\\mathscr{R}\\to\\mathbf{C}P^1$,  where  \n$\\mathscr{R}$ is a Riemann surface.  The  theory of schemes alone  cannot explain  this analogy,\nsee e.g.  [Manin 2006] \\cite[Section 2.2]{Man1}. \n\n\\medskip\nThe aim of our note  is a proof of the relation  between  $\\mathbf{Z}$ and $\\mathbf{C}P^1$.\nWe prove  that  the inclusion  $\\mathbf{Z}\\subset O_K$ defines  \na  covering $\\mathscr{R}\\to\\mathbf{C}P^1$ ramified over three points $\\{0,1,\\infty\\}$ (theorem \\ref{thm1.3}).\nOur approach  is based on the notion of a Serre $C^*$-algebra \\cite[Section 5.3.1] {N}.   \n To formalize our results, we shall need the following definitions. \n\n\\medskip\nLet $V$ be a complex projective variety. Denote by $B(V, \\mathcal{L}, \\sigma)$ a twisted \nhomogeneous coordinate ring of $V$, where $\\mathcal{L}$ is an invertible sheaf and $\\sigma$\nis an automorphism of $V$ [Stafford \\& van ~den ~Bergh 2001]  \\cite[p. 173]{StaVdb1}. \nRecall that a {\\it Serre $C^*$-algebra} $\\mathscr{A}_V$  is the norm closure of a\nself-adjoint representation of the ring  $B(V, \\mathcal{L}, \\sigma)$ by the bounded linear \noperators  on a Hilbert space $\\mathscr{H}$;  such an algebra depends on $V$ alone,\nsince the values of $\\mathcal{L}$ and $\\sigma$ are fixed by the $\\ast$-involution of algebra\n$B(V, \\mathcal{L}, \\sigma)$  \\cite[Section 5.3.1] {N}. \nThe map $V\\mapsto \\mathscr{A}_V$ is a functor. \n Namely, if $V$ and $V'$ are  defined over a number field $K\\subset\\mathbf{C}$,\nthen $V$ is $K$-isomorphic to $V'$ if and only if the algebra $\\mathscr{A}_V$ is isomorphic to  $\\mathscr{A}_{V'}$. \nIn  contrast, the variety $V$ is $\\mathbf{C}$-isomorphic to $V'$ if and only if  $\\mathscr{A}_V$ is Morita equivalent to  \n$\\mathscr{A}_{V'}$, i.e.  $\\mathscr{A}_V\\otimes\\mathscr{K}\\cong \\mathscr{A}_{V'}\\otimes\\mathscr{K}$,\nwhere  $\\mathscr{K}$  is the $C^*$-algebra of  compact operators \\cite[Corollary 1.2]{Nik1}.\nThus, the tensor  product  $\\mathscr{A}_V\\otimes\\mathscr{K}$ is  an analog of the change of base  from $K$\nto $\\mathbf{C}$. \n\n\nThe above observation can be used to ``geometrize''  the ring $O_K$. \nNamely, recall that  $B(V, \\mathcal{L}, \\sigma)\\cong M_2(R)$, \nwhere $R$ is the  homogeneous coordinate ring of a variety $V$ \n[Stafford \\& van ~den ~Bergh 2001]  \\cite[Section 8]{StaVdb1}. \nIf $R\\cong O_K$,  then the norm closure of a self-adjoint representation of  the ring \n$M_2(O_K)$ is  a $C^*$-algebra denoted by  $\\mathscr{A}_{O_K}$.\n Notice that  the  $\\mathscr{A}_{O_K}$ is no longer  a Serre $C^*$-algebra. \nHowever, changing the base from $K$ to $\\mathbf{C}$,  we conclude that  \n  the  tensor  product $\\mathscr{A}_{O_K}\\otimes\\mathscr{K}$ must  be isomorphic  \nto a Serre  $C^*$-algebra.   Thus,  one gets  the following definition. \n\\begin{definition}\nThe complex projective variety $V$ is called a {\\it karma}\n\\footnote{For the lack of a better word meaning the ``correspondence'' or ``relationship''.} \n of the ring $O_K$, if there exists  a $C^*$-algebra homomorphism \n\n \\begin{equation}\n h: \\mathscr{A}_V\\to\\mathscr{A}_{O_K}\\otimes\\mathscr{K}. \n \\end{equation}\n\n\\end{definition}\n\\begin{example}\nIf $R$ is the homogeneous coordinate ring of a complex projective variety $V$, then \n$V$ is a karma of  $R$.  In this case,   $\\mathscr{A}_{R}\\otimes\\mathscr{K}\\cong\\mathscr{A}_V$,\ni.e. the map $h$ is a $C^*$-algebra isomorphism. \n\\end{example}\nOur main result can be formulated as follows. \n\\begin{theorem}\\label{thm1.3}\nLet $\\mathbf{Z}$ be the ring of rational integers and let $O_K$ be the ring of algebraic \nintegers of a number field $K$.   Then:\n\n\n\\medskip\n(i) the Riemann sphere $\\mathbf{C}P^1$ is a karma of the ring $\\mathbf{Z}$;\n\n\\smallskip\n(ii)  there exists a Riemann surface $\\mathscr{R}=\\mathscr{R}(K)$,  \nsuch that $\\mathscr{R}$ is a karma of the  ring  $O_K$;\n\n\\smallskip\n(iii)  the inclusion $\\mathbf{Z}\\subset O_K$ defines a covering $\\mathscr{R}\\to \\mathbf{C}P^1$\nramified over three  points $\\{0,1, \\infty\\}$. \n\\end{theorem}\nThe article is organized as follows. In Section 2 we briefly review \nnoncommutative algebraic geometry and arithmetic groups.  \nTheorem \\ref{thm1.3} is proved  in Section 3. \nAs an application of theorem \\ref{thm1.3},  we give a new short proof of the Belyi Theorem \n [Belyi 1979] \\cite[Theorem 4]{Bel1}. \n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\nWe  review some facts of noncommutative algebraic geometry and \n arithmetic groups. The reader is referred to [Humphreys 1980] \\cite{H} and \n[Stafford \\& van ~den ~Bergh 2001]  \\cite{StaVdb1} for a detailed account.  \n\n\n\\subsection{Noncommutative algebraic geometry}\nLet $V$ be a projective variety over the field $k$.  Denote by $\\mathcal{L}$ an invertible\nsheaf of the linear forms on $V$.  If $\\sigma$ is an automorphism of $V$,  then\nthe pullback of $\\mathcal{L}$ along $\\sigma$ will be denoted by $\\mathcal{L}^{\\sigma}$,\ni.e. $\\mathcal{L}^{\\sigma}(U):= \\mathcal{L}(\\sigma U)$ for every $U\\subset V$. \nThe graded $k$-algebra\n\n\\begin{equation}\\label{eq2.1}\nB(V, \\mathcal{L}, \\sigma)=\\bigoplus_{i\\ge 0} H^0\\left(V, ~\\mathcal{L}\\otimes \\mathcal{L}^{\\sigma}\\otimes\\dots\n\\otimes  \\mathcal{L}^{\\sigma^{ i-1}}\\right)\n\\end{equation}\nis called a {\\it twisted homogeneous coordinate ring} of $V$.  Such a ring is \nalways non-commutative,  unless the automorphism $\\sigma$ is trivial. \nA multiplication of sections of  $B(V, \\mathcal{L}, \\sigma)=\\oplus_{i=1}^{\\infty} B_i$ is defined by the \nrule  $ab=a\\otimes b$,   where $a\\in B_m$ and $b\\in B_n$.\nAn invertible sheaf $\\mathcal{L}$ on $V$  is called $\\sigma$-ample, if for \nevery coherent sheaf $\\mathcal{F}$ on $V$,\n the cohomology group $H^k(V, ~\\mathcal{L}\\otimes \\mathcal{L}^{\\sigma}\\otimes\\dots\n\\otimes  \\mathcal{L}^{\\sigma^{ n-1}}\\otimes \\mathcal{F})$  vanishes for $k>0$ and\n$n>>0$.   If $\\mathcal{L}$ is a $\\sigma$-ample invertible sheaf on $V$,  then\n\\begin{equation}\\label{eq2.2}\nMod~(B(V, \\mathcal{L}, \\sigma)) \/ ~Tors ~\\cong ~Coh~(V),\n\\end{equation}\nwhere  $Mod$ is the category of graded left modules over the ring $B(V, \\mathcal{L}, \\sigma)$,\n$Tors$ is the full subcategory of $Mod$ of the torsion  modules and  $Coh$ is the category of \nquasi-coherent sheaves on a scheme $V$.  In other words, the $B(V, \\mathcal{L}, \\sigma)$  is  \na coordinate ring of the variety $V$.\n\\begin{example}\\label{ex2.1}\n([Stafford \\& van den Bergh 2001]  \\cite[p.173]{StaVdb1})\nDenote by $P^1(k)$  a projective line over the field $k$.\nConsider an automorphism $\\sigma$ of the $P^1(k)$\ngiven by the formula $\\sigma(u)=qu$, where $u\\in P^1(k)$\nand $q\\in k^{\\times}$.  Then  $B(P^1(k), \\mathcal{L}, \\sigma)\\cong U_q$,\nwhere   $U_q$  is  the $k$-algebra of polynomials in \nvariables $x_1$ and $x_2$ satisfying  a commutation relation:\n\\begin{equation}\\label{eq2.3}\nx_2x_1=qx_1x_2.\n\\end{equation}\n\\end{example}\n\\begin{example}\n([Stafford \\& van den Bergh 2001]  \\cite[p.197]{StaVdb1})\nDenote by \n$\\mathcal{E}(k)=\\{(u,v,w,z)\\in P^3(k) ~|~u^2+v^2+w^2+z^2 =\n{1-\\alpha\\over 1+\\beta}v^2+{1+\\alpha\\over 1-\\gamma}w^2+z^2 = 0\\}$  \nan elliptic curve  over the field $k$,  where $\\alpha,\\beta,\\gamma\\in k$\nare constants, such that  $\\beta\\ne -1$ and  $\\gamma\\ne 1$.\nLet  $\\sigma$  be a shift automorphism of the $\\mathcal{E}(k)$.\nThen  $B(\\mathcal{E}(k), \\mathcal{L}, \\sigma)\\cong S(\\alpha,\\beta,\\gamma)$,\nwhere   $S(\\alpha,\\beta,\\gamma)$  is  the  Sklyanin algebra on four  generators\n$x_i$ satisfying  the commutation relations:\n\n\\begin{equation}\\label{eq2.4}\n\\left\\{\n\\begin{array}{ccc}\nx_1x_2-x_2x_1 &=& \\alpha(x_3x_4+x_4x_3),\\\\\nx_1x_2+x_2x_1 &=& x_3x_4-x_4x_3,\\\\\nx_1x_3-x_3x_1 &=& \\beta(x_4x_2+x_2x_4),\\\\\nx_1x_3+x_3x_1 &=& x_4x_2-x_2x_4,\\\\\nx_1x_4-x_4x_1 &=& \\gamma(x_2x_3+x_3x_2),\\\\ \nx_1x_4+x_4x_1 &=& x_2x_3-x_3x_2,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\alpha+\\beta+\\gamma+\\alpha\\beta\\gamma=0$.  \n\\end{example}\n\\begin{example}\\label{ex2.3}\n(\\cite[Lemma 3.1]{Nik2})\nLet $\\mathscr{R}$ be an arithmetic Riemann surface, i.e. given by the \nAF-algebra  of stationary type \\cite[Section 5.2]{N}. \n(Such Riemann surfaces can be identified with the complex \nalgebraic curves  defined over a number field.).  Then \n\\begin{equation}\nB(\\mathscr{R}, \\mathcal{L}, \\sigma)\\cong R[\\pi_1(S^3 \\backslash \\mathscr{L})],\n\\end{equation}\nwhere  $\\mathscr{L}$ is a link embedded in the three-sphere $S^3$ \nand  $R[\\pi_1(S^3 \\backslash \\mathscr{L})]$\nis the group ring of the fundamental group $\\pi_1(S^3 \\backslash \\mathscr{L})$. \n\\end{example}\n\n\n\\subsection{Arithmetic groups}\nLet $G$ be a linear algebraic group defined over the field $\\mathbf{Q}$.\nDenote by $G_{\\mathbf{Z}}$ the group of integer points of $G$. A subgroup \n$\\Gamma\\subset G$ is called {\\it arithmetic} if $\\Gamma$ is commensurable with \nthe  $G_{\\mathbf{Z}}$, i.e. $\\Gamma\\cap G_{\\mathbf{Z}}$ has a finite index both\nin $\\Gamma$ and $G_{\\mathbf{Z}}$. Informally, the arithmetic group is a discrete \nsubgroup of the group $GL_n(\\mathbf{C})$ defined by some arithmetic properties.\nFor instance, $\\mathbf{Z}\\subset\\mathbf{R}$, $GL_n(\\mathbf{Z})\\subset GL_n(\\mathbf{R})$ and\n$SL_n(\\mathbf{Z})\\subset SL_n(\\mathbf{R})$ are  examples of the arithmetic groups. \n\n\nDenote by $\\mathcal{O}$ the ring of algebraic integers of all finite  \nextensions of the number field $\\mathbf{Q}$. \nLet $\\mathbb{H}^3$ be the hyperbolic 3-dimensional space.\nThe following remarkable result establishes \na deep link between arithmetic groups and  topology.\n\\begin{theorem}\\label{thm2.4}\n{\\bf ([Maclachlan \\& Reid  2003] \\cite[p. 169]{MR})}\nLet $M=\\mathbb{H}^3\/\\Gamma$ be a finite volume hyperbolic  3-manifold.\nThen $\\Gamma$ is conjugate to a subgroup of the group $PSL_2(\\mathcal{O})$.  \n\\end{theorem}\n\\begin{example}\\label{ex2.5}\nLet $\\mathscr{L}$ be a hyperbolic link, i.e.  $S^3\\backslash\\mathscr{L}\\cong \\mathbb{H}^3\/\\Gamma$\nfor an arithmetic group  $\\Gamma$. Then \n\\begin{equation}\\label{eq2.5} \n\\pi_1(S^3 \\backslash\\mathscr{L})\\cong\\Gamma.\n\\end{equation}\n\\end{example}\n\n\n\n\\section{Proof of theorem \\ref{thm1.3}}\n(i)  Let us show that  the  $\\mathbf{C}P^1$ is a karma of  $\\mathbf{Z}$.\nIndeed, in this case $R\\cong\\mathbf{Z}$ and $\\mathscr{A}_{\\mathbf{Z}}$ is the closure of a self-adjoint representation of\nthe ring $M_2(\\mathbf{Z})$. Consider the group $PSL_2(\\mathbf{Z})=SL_2(\\mathbf{Z})\/\\pm I$, where  $SL_2(\\mathbf{Z})$ is the group \nof invertible  elements of $M_2(\\mathbf{Z})$. \nRecall that the group $PSL_2(\\mathbf{Z})$ is  generated by the matrices: \n\\begin{equation}\\label{eq3.1}\nu=\\left(\n\\begin{matrix}\n0 & 1\\cr -1 & 0\n\\end{matrix}\n\\right)\n\\quad\n\\hbox{and}\n   \\quad v=\\left(\n\\begin{matrix}\n0 & -1\\cr 1 & -1\n\\end{matrix}\n\\right)\n\\end{equation}\nwhich satisfy the relations modulo $\\pm I$:\n\\begin{equation}\\label{eq3.2}\nu^2= v^3=1.\n\\end{equation}\n\n\n\\bigskip\nOn the other hand, consider Example \\ref{ex2.1} with $k\\cong\\mathbf{Q}$ and assume  \nthat $q=-1$ in relation (\\ref{eq2.3}).  In other words, one gets a relation:\n\\begin{equation}\\label{eq3.3}\nx_2x_1=-x_1x_2.\n\\end{equation}\n\nConsider a substitution:\n\\begin{equation}\\label{eq3.4}\n\\left\\{\n\\begin{array}{ccl}\nu&=& x_2x_1x_2^{-1}x_1^{-1}\\\\\nv &=& x_2. \n\\end{array}\n\\right.\n\\end{equation}\n\nThe reader can verify, that substitution (\\ref{eq3.4}) and relation (\\ref{eq3.3}) \nreduces relations (\\ref{eq3.2}) to the form: \n\\begin{equation}\\label{eq3.5}\nx_2^3=1.\n\\end{equation}\n\nLet $\\mathscr{I}$ be a two-sided ideal in the algebra $B(P^1(\\mathbf{Q}), \\mathcal{L}, \\sigma)$ of \nExample \\ref{ex2.1} generated by relation (\\ref{eq3.5}).  In view of (\\ref{eq3.2})-(\\ref{eq3.5}), one gets a\nring  isomorphism: \n\\begin{equation}\\label{eq3.6}\nB(P^1(\\mathbf{Q}), \\mathcal{L}, \\sigma)\/\\mathscr{I}\\cong M_2(\\mathbf{Z}). \n\\end{equation}\n\nLet $\\rho$ be a self-adjoint representation of the ring $B(P^1(\\mathbf{Q}), \\mathcal{L}, \\sigma)$\nby the linear operators on a Hilbert space $\\mathscr{H}$. Notice that such a representation \nexists, because relation (\\ref{eq3.3}) is invariant under the involution $x_1^*=x_2$ and \n$x_2^*=x_1$. Since $\\rho(B(P^1(\\mathbf{Q}), \\mathcal{L}, \\sigma))=\\mathscr{A}_{P^1(\\mathbf{Q})}$\nand $\\rho(M_2(\\mathbf{Z}))=\\mathscr{A}_{\\mathbf{Z}}$, it follows from (\\ref{eq3.6}) that there exists a \n$C^*$-algebra homomorphism\n\\begin{equation}\\label{eq3.7}\nh:  \\mathscr{A}_{P^1(\\mathbf{Q})}\\to\\mathscr{A}_{\\mathbf{Z}},\n\\end{equation}\nwhere  $Ker ~h=\\rho(\\mathscr{I})$.  \nThe homomorphism $h$ extends to a homomorphism between the \n products\n\\begin{equation}\\label{eq3.8}\nh:  \\mathscr{A}_{P^1(\\mathbf{Q})}\\otimes\\mathscr{K}\\to\\mathscr{A}_{\\mathbf{Z}}\\otimes\\mathscr{K}, \n\\end{equation}\nwhere $\\mathscr{K}$ is the $C^*$-algebra of compact operators. \nBut $\\mathscr{A}_{P^1(\\mathbf{Q})}\\otimes\\mathscr{K}\\cong \\mathscr{A}_{\\mathbf{C}P^1}$\nand, therefore, one gets a $C^*$-algebra homomorphism\n\\begin{equation}\\label{eq3.9}\nh: \\mathscr{A}_{\\mathbf{C}P^1} \\to\\mathscr{A}_{\\mathbf{Z}}\\otimes\\mathscr{K}. \n\\end{equation}\n\nIn other words, the Riemann sphere $\\mathbf{C}P^1$ is a karma of the ring $\\mathbf{Z}$. \n\n\n\n\n\n\n\\bigskip\n(ii)  Let us show that if $K$ is a number field, then there exists a Riemann surface $\\mathscr{R}$, \nsuch that $\\mathscr{R}$ is a karma of the ring $O_K$. \nIndeed,   we can always assume that $K$ has at least one complex embedding and fix one of such embeddings \n$K\\not\\subset\\mathbf{R}$.  \n (For otherwise, we replace $K$ by a CM-field of $K$, i.e. a totally imaginary quadratic extension\nof the totally real field $K$.  This case corresponds to the double covering $\\mathscr{R}'$of the Riemann surface  $\\mathscr{R}$.)  \nFor simplicity, let $R\\cong O_K$ and $\\Gamma\\cong PSL_2(O_K)$.  (The case of a non-maximal order $\\Lambda\\subseteq O_K$ is treated likewise and corresponds to the covering of \nthe Riemann surface $\\mathscr{R}$.)  In view of  (\\ref{eq2.5}),   there exists a hyperbolic link $\\mathscr{L}$,  \nsuch that: \n\\begin{equation}\\label{eq3.10} \nPSL_2(O_K)\\cong\\pi_1(S^3 \\backslash\\mathscr{L}).\n\\end{equation}\n\n\\bigskip\nOn the other hand,  it is known that\n\\begin{equation}\\label{eq3.11} \nR[\\pi_1(S^3 \\backslash\\mathscr{L})]\\cong B(\\mathscr{R}, \\mathcal{L}, \\sigma), \n\\end{equation}\nwhere $R[\\pi_1(S^3 \\backslash\\mathscr{L})]$ is the group ring of $\\pi_1(S^3 \\backslash\\mathscr{L})$ and \n $\\mathscr{R}$ is a Riemann surface, see example \\ref{ex2.3}. In particular, it follows from (\\ref{eq3.10}) \n that  \n\\begin{equation}\\label{eq3.12} \nB(\\mathscr{R}, \\mathcal{L}, \\sigma)\\cong R[PSL_2(O_K)]. \n\\end{equation}\n\n\nLet $\\rho$ be a self-adjoint representation of the ring $B(\\mathscr{R}, \\mathcal{L}, \\sigma)$\nby the linear operators on a Hilbert space $\\mathscr{H}$. \nThe norm closure of $\\rho(B(\\mathscr{R}, \\mathcal{L}, \\sigma))$\nis the Serre $C^*$-algebra $\\mathscr{A}_{\\mathscr{R}}$.  \n\n\nOn the other hand, it follows from (\\ref{eq3.12}) that  taking the norm closure of \n$\\rho(R[PSL_2(O_K)])$,  one gets a $C^*$-algebra $\\mathscr{A}_{O_K}$, \nsuch that \n\\begin{equation}\\label{eq3.13} \n\\mathscr{A}_{O_K}\\otimes\\mathscr{K}\\cong \\mathscr{A}_{\\mathscr{R}}. \n\\end{equation}\n\nIn other words,  there exists an isomorphism:\n\\begin{equation}\\label{eq3.14} \nh: \\mathscr{A}_{\\mathscr{R}}\\to \\mathscr{A}_{O_K}\\otimes\\mathscr{K}. \n\\end{equation}\n\nIt follows from (\\ref{eq3.14}) that  the Riemann surface $\\mathscr{R}$ is a karma of \nthe ring $O_K$. \n\n\n\\bigskip\n(iii) Finally, let us show that \n the inclusion $\\mathbf{Z}\\subset O_K$ defines a covering $\\mathscr{R}\\to \\mathbf{C}P^1$\nramified over three  points $\\{0,1, \\infty\\}$. \n\n\nIn the lemma below we shall prove a stronger result. \nNamely, let $\\mathfrak{K}$ be a category of the Galois extensions of the field $\\mathbf{Q}$,\nsuch that the  morphisms in $\\mathfrak{K}$ are inclusions $K\\subseteq K'$, where $K,K'\\in\\mathfrak{K}$.  \n Likewise, let $\\mathfrak{R}$ be a category of the Riemann surfaces,\nsuch that the  morphisms in $\\mathfrak{R}$ are holomorphic maps $\\mathscr{R}\\to \\mathscr{R}'$, \nwhere $\\mathscr{R}, \\mathscr{R}'\\in\\mathfrak{R}$.  \nLet $F: \\mathfrak{K}\\to\\mathfrak{R}$ be a map acting by the formula $O_K\\mapsto\\mathscr{R}$,  where\n$\\mathscr{R}$ is the Riemann surface defined by the isomorphism (\\ref{eq3.12}).  \n\\begin{remark}\\label{rmk3.1}\nThe category $\\mathfrak{R}$ consists of the Riemann surfaces, which are algebraic curves defined over a number field.\nIn particular,  the morphisms in  $\\mathfrak{R}$ can be  defined over the number field.  Both facts follow from  \nthe property of the AF-algebra $\\mathscr{A}_{\\mathscr{R}}$ being of a stationary type  \\cite[Section 5.2]{N}. \nWe refer the reader to Example \\ref{ex2.3} and \\cite[Lemma 3.1]{Nik2}. \n\\end{remark}\n\\begin{lemma}\\label{lm3.1}\nThe map $F: \\mathfrak{K}\\to\\mathfrak{R}$ is  a covariant functor, i.e. $F$  transforms  inclusions \nin the category $\\mathfrak{K}$  to  holomorphic maps in the category $\\mathfrak{R}$. \n\\end{lemma}\n\\begin{proof}\nLet $K\\in\\mathfrak{K}$ be a number field and let $\\mathscr{R}=F(K)$ be the corresponding\nRiemann surface $\\mathscr{R}\\in\\mathfrak{R}$. Let $K\\subseteq K'$ be an inclusion, where \n$K'\\in \\mathfrak{K}$.\n\nUsing isomorphism (\\ref{eq3.13}),  one gets  an inclusion of the corresponding Serre $C^*$-algebras:  \n\\begin{equation}\n\\mathscr{A}_{\\mathscr{R}}\\subseteq \\mathscr{A}_{\\mathscr{R}'}. \n\\end{equation}\n\nOn the other hand, it is known the algebra $\\mathscr{A}_{\\mathscr{R}}$\nis a coordinate ring of the Riemann surface $\\mathscr{R}$ \\cite[Theorem 5.2.1]{N}. \n In particular,  if $h: \\mathscr{A}_{\\mathscr{R}'}\\to \\mathscr{A}_{\\mathscr{R}}$ is a homomorphism,\n  one gets a holomorphic map $w: \\mathscr{R}'\\to\\mathscr{R}$ defined by  a commutative  diagram in Figure 1. \n\\begin{figure}[h]\n\\begin{picture}(300,110)(-70,-5)\n\\put(20,70){\\vector(0,-1){35}}\n\\put(130,70){\\vector(0,-1){35}}\n\\put(45,23){\\vector(1,0){60}}\n\\put(45,83){\\vector(1,0){60}}\n\\put(13,20){$ \\mathscr{A}_{\\mathscr{R}'}$}\n\\put(75,30){$h$}\n\\put(75,90){$w$}\n\\put(123,20){$ \\mathscr{A}_{\\mathscr{R}}$}\n\\put(17,80){$\\mathscr{R}'$}\n\\put(122,80){$\\mathscr{R}$}\n\\end{picture}\n\\caption{Holomorphic map $w$.}\n\\end{figure}\n\n\nThus $F$ is a functor, which maps the inclusion   $K\\subseteq K'$\ninto a holomorphic map $w: \\mathscr{R}'\\to\\mathscr{R}$. The reader \ncan verify that $F$ is a covariant functor. Lemma \\ref{lm3.1} is proved. \n\\end{proof}\n\n\\begin{lemma}\\label{lm3.3}\n The inclusion $\\mathbf{Z}\\subset O_K$ defines a covering $\\mathscr{R}\\to \\mathbf{C}P^1$\nramified over three  points $\\{0,1, \\infty\\}$.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathscr{U}$ be the Riemann sphere $\\mathbf{C}P^1$ without three points,\nwhich we always assume to be $\\{0,1,\\infty\\}$ after a proper M\\\"obius transformation. \nIt is easy to see, that the fundamental group $\\pi_1(\\mathscr{U})\\cong \\mathfrak{F}_2$,  where \n $\\mathfrak{F}_2$ is a free group on two generators $u$ and $v$. \n \n \n Since the the Riemann surface $\\mathscr{U}$ corresponds to \n  an unlink $\\mathscr{L}\\cong S^1\\cup S^1$,    one gets an isomorphism:\n\n \\begin{equation}\\label{eq3.16}\n B(P^1(\\mathscr{U}, \\mathcal{L}, \\sigma))\\cong R[\\mathfrak{F}_2].\n \\end{equation}\n\n\n\n\n\\smallskip\nConsider a two-sided ideal $\\mathscr{I}\\subset  B(P^1(\\mathscr{U}, \\mathcal{L}, \\sigma))$\ngenerated by relations (\\ref{eq3.2}). In view of  (\\ref{eq3.16}), we have:\n\n \\begin{equation}\\label{eq3.17}\n B(P^1(\\mathscr{U}, \\mathcal{L}, \\sigma))\/\\mathscr{I}\\cong R[PSL_2(\\mathbf{Z})].\n \\end{equation}\n\n\n\\smallskip\nIn other words, one gets  a homomorphism between the $C^*$-algebras:\n \\begin{equation}\\label{eq3.18}\n \\mathscr{A}_{\\mathscr{U}}\\to \\mathscr{A}_{\\mathbf{C}P^1}. \n \\end{equation}\n\n\nUsing the commutative diagram in Figure 1, we get a holomorphic map between \nthe corresponding Riemann surfaces:\n \\begin{equation}\\label{eq3.19}\n \\mathscr{U}\\to \\mathbf{C}P^1.  \n \\end{equation}\n\n\n\n \n \n\n\n   \n \\bigskip \n Let now $\\mathbf{Z}\\subset O_K$ be an inclusion, where $K$ is a number field. \n By item (ii) of theorem \\ref{thm1.3} there exists a Riemann surface $\\mathscr{R}\\in\\mathfrak{R}$ \n corresponding to $O_K$.  By lemma \\ref{lm3.1},  there exists a holomorphic map:\n \\begin{equation}\\label{eq3.20}\n  \\mathscr{R}\\to \\mathbf{C}P^1. \n \\end{equation}\n\n \nUsing (\\ref{eq3.19}) and (\\ref{eq3.20}), one gets a commutative digram in Figure 2. \n\\begin{figure}[h]\n\\begin{picture}(300,90)(-90,10)\n\\put(20,70){\\vector(0,-1){35}}\n\\put(45,30){\\vector(3,2){60}}\n\\put(45,83){\\vector(1,0){60}}\n\\put(13,20){$\\mathscr{U}$}\n\\put(17,80){$\\mathscr{R}$}\n\\put(122,80){$\\mathbf{C}P^1$}\n\\end{picture}\n\\caption{The map $\\mathscr{R}\\to \\mathscr{U}$.}\n\\end{figure}\n\n\n \n  \\medskip\nWe use the  diagram in Figure 2  to define a holomorphic map:\n\n \\begin{equation}\\label{eq4.4}\n  \\mathscr{R}\\to \\mathscr{U}.\n \\end{equation}\n\n  \n  \\smallskip\n  Since $\\mathscr{U}=\\mathbf{C}P^1\\backslash\\{0,1,\\infty\\}$, one gets the conclusion of \n  lemma \\ref{lm3.3}.   \n \\end{proof}\n\n\n\\bigskip\nItem (iii)  of theorem \\ref{thm1.3}  follows from lemmas \\ref{lm3.1} and \\ref{lm3.3}. \n\n\n\n\\bigskip\nTheorem \\ref{thm1.3} is proved. \n\n \n\n\\section{Belyi's Theorem}\nBelyi's Theorem says that the algebraic curve $\\mathscr{R}$ can be defined over \na number field  if and only if there exist a covering $\\mathscr{R}\\to \\mathbf{C}P^1$\nramified over three points of  the Riemann sphere \n$\\mathbf{C}P^1$.  This remarkable result was proved by [Belyi 1979] \\cite[Theorem 4]{Bel1}. \nIn this section we show that Belyi's Theorem follows from  theorem \\ref{thm1.3} and remark \\ref{rmk3.1}. \n\\begin{theorem}\\label{thm4.1}\n\\textbf{(Belyi's Theorem)}\nA complete non-singular algebraic curve over $\\mathbf{C}$\n can be defined over an algebraic number field if and only if such a curve is a covering\nof the Riemann sphere $\\mathbf{C}P^1$ ramified over three points. \n\\end{theorem}\n\\begin{proof}\nWe identify the Riemann surface $\\mathscr{R}\\in\\mathfrak{R}$ with a complete non-singular algebraic \ncurve over  the field of characteristic zero (Chow's Theorem). \n\n\\smallskip\nIn view of the remark \\ref{rmk3.1},  each $\\mathscr{R}\\in\\mathfrak{R}$  is the algebraic curve\n defined  over a finite extension of the field $\\mathbf{Q}$.  \n On the other hand, item (iii) of theorem \\ref{thm1.3}  says  that each  Riemann surface \n$\\mathscr{R}\\in\\mathfrak{R}$ is a covering of the  $\\mathbf{C}P^1$ ramified over the points\n$\\{0,1,\\infty\\}$.  The ``only if'' part  of Belyi's Theorem follows. \n \n\\smallskip \nLet $\\mathscr{R}$ be a covering of the  $\\mathbf{C}P^1$ ramified over the points\n$\\{0,1,\\infty\\}$. Using lemma \\ref{lm3.1},  one can construct a ring $O_K$ corresponding\nto the Riemann surface $\\mathscr{R}$.  By item (ii) of theorem \\ref{thm1.3} and remark \\ref{rmk3.1} we have\n$\\mathscr{R}\\in\\mathfrak{R}$.   In other words,  $\\mathscr{R}$ is an algebraic curve defined over an algebraic number field.   \nThe ``if'' part of of Belyi's Theorem is proved.\n \\end{proof}\n\\begin{remark}\nIt is an  interesting problem to calculate the  ramification data and  equations of  the Belyi curves $\\mathscr{R}$\nin terms of the  orders $\\Lambda\\subseteq O_K$ and number fields $K$ obtained in  theorem \\ref{thm1.3}. \nHowever this problem is out of the scope of present paper. \n\\end{remark}\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{EXIT Charts and the Matching Condition for BEC}\nTo start, let us review the case of transmission over the $\\ensuremath{\\text{BEC}}(\\ensuremath{{\\tt{h}}})$\nusing a degree distribution pair $(\\ensuremath{{\\lambda}}, \\ensuremath{{\\rho}})$.\nIn this case density evolution is equivalent to the EXIT chart\napproach and the condition for successful decoding under \\BP\\ reads\n\\begin{align*}\nc(x) \\defas 1-\\ensuremath{{\\rho}}(1-x) \\leq \\ensuremath{{\\lambda}}^{-1}(x\/\\ensuremath{{\\tt{h}}}) \\defas v^{-1}_{\\ensuremath{{\\tt{h}}}}(x).\n\\end{align*}\nThis is shown in Fig.~\\ref{fig:becmatching} for the degree distribution pair\n$(\\ensuremath{{\\lambda}}(x)=x^3, \\ensuremath{{\\rho}}(x)=x^4)$. \n\\begin{figure}[hbt]\n\\centering\n\\setlength{\\unitlength}{1.0bp}%\n\\begin{picture}(110,110)\n\\put(0,0){\\includegraphics[scale=1.0]{exitchartbecldpcc}}\n\\put(112, 0){\\makebox(0,0)[l]{\\small $x$}}\n\\put(40, 60){\\makebox(0,0){\\small $c(x)$}}\n\\put(16, 89){\\makebox(0,0){\\small $v^{-1}_{\\ensuremath{{\\tt{h}}}}(x)$}}\n\\put(58, 102){\\makebox(0,0)[b]{{\\small $\\ensuremath{{\\tt{h}}}=0.58$}}}\n\\put(100, -2){\\makebox(0,0)[tr]{$h_{\\text{out-variable}}=h_{\\text{in-check}}$}}\n\\put(-4,100){\\makebox(0,0)[tr]{\\rotatebox{90}{$h_{\\text{out-check}}=h_{\\text{in-variable}}$}}}\n\\end{picture}\n\\caption{\\label{fig:becmatching} The EXIT chart method for\nthe degree distribution $(\\ensuremath{{\\lambda}}(x)=x^3, \\ensuremath{{\\rho}}(x)=x^4)$ and\ntransmission over the $\\ensuremath{\\text{BEC}}(\\ensuremath{{\\tt{h}}} = 0.58)$.}\n\\end{figure}\nThe area under the curve $c(x)$ equals $1-\\int \\!\\ensuremath{{\\rho}}$ and the \narea to the left of the curve $v^{-1}_{\\ensuremath{{\\tt{h}}}}(x)$ is equal to \n$\\ensuremath{{\\tt{h}}} \\int \\!\\ensuremath{{\\lambda}}$. By the previous remarks, a necessary condition\nfor successful \\BP\\ decoding\nis that these two areas do not overlap.\nSince the total area equals $1$ we get the necessary condition\n$\\ensuremath{{\\tt{h}}} \\int \\ensuremath{{\\lambda}}+1-\\int \\ensuremath{{\\rho}}\\leq 1$. Rearranging terms, this \nis equivalent to the condition\n\\begin{align*}\n1-C_{\\Shsmall} = \\ensuremath{{\\tt{h}}} \\leq \\frac{\\int \\ensuremath{{\\rho}}}{\\int \\ensuremath{{\\lambda}}}= 1 - r(\\ensuremath{{\\lambda}}, \\ensuremath{{\\rho}}).\n\\end{align*}\nIn words, the rate $r(\\ensuremath{{\\lambda}}, \\ensuremath{{\\rho}})$ of any LDPC ensemble which, for\nincreasing block lengths, allows successful \ndecoding over the $\\ensuremath{\\text{BEC}}(\\ensuremath{{\\tt{h}}})$, can not surpass the Shannon limit\n$1-\\ensuremath{{\\tt{h}}}$. \nAs pointed out in the introduction, an argument very similar to the above was introduced\nby Shokrollahi and Oswald \\cite{Sho00,OsS01} (albeit not using the language and geometric\ninterpretation of EXIT functions and applying a slightly different range of integration).\nIt was the first bound on the performance of iterative systems in which the Shannon capacity\nappeared explicitly using only quantities of density evolution.\nA substantially more general version of this bound can be found in \\cite{AKtB02a,AKTB02,AKtB04}\n(see also Forney \\cite{For05}).\n\nAlthough the final result (namely that transmission above capacity\nis not possible) is trivial, the method of proof is well worth the effort\nsince it shows how capacity enters in the calculation of the performance \nof iterative coding systems. By turning this bound around, we \ncan find conditions under which iterative systems achieve capacity:\nIn particular it shows that the two component-wise\nEXIT curves have to be matched perfectly. Indeed, all currently known\ncapacity achieving degree-distributions for the BEC can be derived \nby starting with this perfect matching condition and working backwards.\n\n\\section{GEXIT Charts and the Matching Condition for BMS Channels}\nLet us now derive the equivalent result for general BMS channels.\nAs a first ingredient we show how to interpolate the sequence of densities\nwhich we get from density evolution so as to form a complete family of\ndensities.\n\n\\begin{definition}[Interpolating Channel Families]\n\\label{def:interpolation}\nConsider a degree distribution pair $(\\ensuremath{{\\lambda}}, \\ensuremath{{\\rho}})$\nand transmission over the BMS channel characterized by its\n$L$-density $\\Ldens{c}$. Let $\\Ldens{a}_{-1}=\\Delta_0$\nand $\\Ldens{a}_0=\\Ldens{c}$ and set $\\Ldens{a}_{\\alpha}$,\n$\\alpha \\in [-1, 0]$, to \n$\\Ldens{a}_{\\alpha}=-\\alpha \\Ldens{a}_{-1} + (1+\\alpha) \\Ldens{a}_0$.\nThe {\\em interpolating density evolution families} \n$\\{\\Ldens{a}_{\\alpha}\\}_{\\alpha=-1}^{\\infty}$\nand $\\{\\Ldens{b}_{\\alpha}\\}_{\\alpha=0}^{\\infty}$ are then defined as follows:\n\\begin{align*}\n\\Ldens{b}_{\\alpha} & = \\sum_{i} \\ensuremath{{\\rho}}_i \\Ldens{a}_{\\alpha-1}^{\\boxast (i-1)},\n\\;\\;\\;\\;\\; \\alpha \\geq 0,\\\\\n\\Ldens{a}_{\\alpha} & =  \n\\sum_{i} \\ensuremath{{\\lambda}}_i \\Ldens{c} \\star \\Ldens{b}_{\\alpha}^{\\star (i-1)},\n\\;\\;\\;\\;\\;\\alpha \\geq 0,\n\\end{align*}\nwhere $\\star$ denotes the standard convolution of densities and\n$\\Ldens{a} \\boxast \\Ldens{b}$ denotes the density at the output of\na check node, assuming that the input densities are $\\Ldens{a}$ and $\\Ldens{b}$,\nrespectively.\n\\end{definition}\nDiscussion: First note that $\\Ldens{a}_{\\ell}$ ($\\Ldens{b}_{\\ell}$), \n$\\ell \\in \\ensuremath{\\mathbb{N}}$,\nrepresents the sequence of $L$-densities of density evolution\nemitted by the variable (check) nodes in the $\\ell$-th iteration.\nBy starting density evolution not only with $\\Ldens{a}_{0}=\\Ldens{c}$\nbut with all possible convex combinations of $\\Delta_0$ and\n$\\Ldens{c}$, this discrete sequence of densities is completed to\nform a continuous family of densities ordered by physical degradation.\nThe fact that the densities are ordered by physical degradation\ncan be seen as follows: note that the computation tree for $\\Ldens{a}_{\\alpha}$ \ncan be constructed by taking  \nthe standard computation tree of $\\Ldens{a}_{\\lceil \\alpha \\rceil}$ \nand independently erasing the observation associated to each variable leaf node with probability\n$\\lceil \\alpha \\rceil-\\alpha$. It follows that we can convert the computation tree of \n$\\Ldens{a}_{\\alpha}$ to that of $\\Ldens{a}_{\\alpha-1}$ by erasing all\nobservations at the leaf nodes and by independently erasing\neach observation in the second (from the bottom) row of variable nodes\nwith probability $\\lceil \\alpha \\rceil-\\alpha$.\nThe same statement is true for $\\Ldens{b}_{\\alpha}$.\nIf $\\lim_{\\ell \\rightarrow \\infty} \\entropy(\\Ldens{a}_{\\ell})=0$, i.e., \nif \\BP\\\ndecoding is successful in the limit of large blocklengths, then\nthe families are both complete.\n\n\\begin{example}[Density Evolution and Interpolation]\nConsider transmission over the $\\ensuremath{\\text{BSC}}(0.07)$ using a\n$(3, 6)$-regular ensemble. Fig.~\\ref{fig:debsc} depicts\nthe density evolution process for this case.\n\\begin{figure}[htp]\n\\setlength{\\unitlength}{0.5bp}%\n\\begin{center}\n\\begin{picture}(740,170)\n\\put(0,0)\n{\n\\put(0,0){\\includegraphics[scale=0.5]{de1}}\n\\put(120,50){\\includegraphics[scale=0.5]{b1}}\n\\put(0,120){\\includegraphics[scale=0.5]{a0}}\n\\put(50, 110){\\makebox(0,0)[c]{\\tiny $a_{0}$}}\n\\put(170, 40){\\makebox(0,0)[c]{\\tiny $b_{1}$}}\n\\put(50, -2){\\makebox(0,0)[t]{\\tiny $\\entropy(\\Ldens{a})$}}\n\\put(-2, 50){\\makebox(0,0)[r]{\\tiny \\rotatebox{90}{$\\entropy(\\Ldens{b})$}}}\n}\n\\put(260,0)\n{\n\\put(0,0){\\includegraphics[scale=0.5]{de2}}\n\\put(120,50){\\includegraphics[scale=0.5]{b2}}\n\\put(0,120){\\includegraphics[scale=0.5]{a1}}\n\\put(50, 110){\\makebox(0,0)[c]{\\tiny $a_{1}$}}\n\\put(170, 40){\\makebox(0,0)[c]{\\tiny $b_{2}$}}\n\\put(50, -2){\\makebox(0,0)[t]{\\tiny $\\entropy(\\Ldens{a})$}}\n\\put(-2, 50){\\makebox(0,0)[r]{\\tiny \\rotatebox{90}{$\\entropy(\\Ldens{b})$}}}\n}\n\\put(500,0)\n{\n\\put(0,0){\\includegraphics[scale=0.5]{de25}}\n\\put(120,50){\\includegraphics[scale=0.5]{b13}}\n\\put(0,120){\\includegraphics[scale=0.5]{a12}}\n\\put(50, 110){\\makebox(0,0)[c]{\\tiny $a_{12}$}}\n\\put(170, 40){\\makebox(0,0)[c]{\\tiny $b_{13}$}}\n\\put(50, -2){\\makebox(0,0)[t]{\\tiny $\\entropy(\\Ldens{a})$}}\n\\put(-2, 50){\\makebox(0,0)[r]{\\tiny \\rotatebox{90}{$\\entropy(\\Ldens{b})$}}}\n}\n\\end{picture}\n\\end{center}\n\\caption{\\label{fig:debsc} Density evolution for $(3, 6)$-regular ensemble over $\\ensuremath{\\text{BSC}}(0.07)$.}\n\\end{figure}\nThis process gives rise to the sequences of densities $\\{\\Ldens{a}_{\\ell}\\}_{\\ell =0}^{\\infty}$,\nand $\\{ \\Ldens{b}_{\\ell}\\}_{\\ell=1}^{\\infty}$. Fig.~\\ref{fig:interpolation} shows\nthe interpolation of these sequences for the choices $\\alpha=1.0, 0.95, 0.9$ and $0.8$\nand the complete such family.\n\\begin{figure}[htp]\n\\setlength{\\unitlength}{0.6bp}%\n\\begin{center}\n\\begin{picture}(650,110)\n\\put(0,0){\\includegraphics[scale=0.6]{de25}}\n\\put(50, 102){\\makebox(0,0)[b]{\\tiny $\\alpha=1.0$}}\n\\put(50, -2){\\makebox(0,0)[t]{\\tiny $\\entropy(\\Ldens{a})$}}\n\\put(-2, 50){\\makebox(0,0)[r]{\\tiny \\rotatebox{90}{$\\entropy(\\Ldens{b})$}}}\n\\put(130,0){\\includegraphics[scale=0.6]{de52}}\n\\put(180, 102){\\makebox(0,0)[b]{\\tiny $\\alpha=0.95$}}\n\\put(180, -2){\\makebox(0,0)[t]{\\tiny $\\entropy(\\Ldens{a})$}}\n\\put(108, 50){\\makebox(0,0)[r]{\\tiny \\rotatebox{90}{$\\entropy(\\Ldens{b})$}}}\n\\put(260,0){\\includegraphics[scale=0.6]{de53}}\n\\put(310, 102){\\makebox(0,0)[b]{\\tiny $\\alpha=0.9$}}\n\\put(310, -2){\\makebox(0,0)[t]{\\tiny $\\entropy(\\Ldens{a})$}}\n\\put(258, 50){\\makebox(0,0)[r]{\\tiny \\rotatebox{90}{$\\entropy(\\Ldens{b})$}}}\n\\put(390,0){\\includegraphics[scale=0.6]{de54}}\n\\put(440, 102){\\makebox(0,0)[b]{\\tiny $\\alpha=0.8$}}\n\\put(440, -2){\\makebox(0,0)[t]{\\tiny $\\entropy(\\Ldens{a})$}}\n\\put(388, 50){\\makebox(0,0)[r]{\\tiny \\rotatebox{90}{$\\entropy(\\Ldens{b})$}}}\n\\put(520,0){\\includegraphics[scale=0.6]{de55}}\n\\put(570, -2){\\makebox(0,0)[t]{\\tiny $\\entropy(\\Ldens{a})$}}\n\\put(518, 50){\\makebox(0,0)[r]{\\tiny \\rotatebox{90}{$\\entropy(\\Ldens{b})$}}}\n\\end{picture}\n\\end{center}\n\\caption{\\label{fig:interpolation} Interpolation of densities.}\n\\end{figure}\n\\end{example}\n\nAs a second ingredient we recall from \\cite{MMRU04} the definition of GEXIT functions. \nThese GEXIT functions fulfill the Area Theorem for the case of general\nBMS channels.\nUp to date, GEXIT functions have been mainly\nused to derive upper bounds on the \\MAP\\ threshold of iterative\ncoding systems, see e.g., \\cite{MMRU04,MMRU05}. Here we will apply them\nto the components of LDPC ensembles.\n\n\\begin{definition}[The GEXIT Functional]\nGiven two families of $L$-densities\n$\\{\\Ldens{c}_\\cp\\}$ and $\\{\\Ldens{a}_\\cp\\}$ parameterized by $\\epsilon$ define\nthe GEXIT functional $\\gentropy(\\Ldens{c}_{\\cp}, \\Ldens{a}_{\\cp})$ by\n\\begin{align*}\n\\gentropy(\\Ldens{c}_{\\cp}, \\Ldens{a}_{\\cp}) & =\n\\int_{-\\infty}^{\\infty} \\Ldens{a}_{\\cp}(z) \\gexitkl {\\Ldens{c}_{\\cp}} z \\text{d}z,\n\\end{align*}\nwhere\n\\begin{align*}\n\\gexitkl {\\Ldens{c}_{\\cp}} z\n& =\n\\frac{\\int_{-\\infty}^{\\infty} \\frac{\\text{d} \\Ldens{c}_{\\cp}(w)}{\\text{d} \\cp} \n\\log(1+e^{-z-w}) \\text{d}w}{\n\\int_{-\\infty}^{\\infty} \\frac{\\text{d} \\Ldens{c}_{\\cp}(w)}{\\text{d} \\cp}\n\\log(1+e^{-w}) \\text{d}w}.\n\\end{align*}\nNote that the kernel is normalized not with respect to $d \\epsilon$ but\nwith respect to $d \\ensuremath{{\\tt{h}}}$, i.e., with respect to changes in the entropy.\nThe families are required to be smooth in the sense that \n$\\{\\entropy(\\Ldens{c}_{\\cp}), \\gentropy(\\Ldens{c}_{\\cp}, \\Ldens{a}_{\\cp})\\}$\nforms a piecewise continuous curve.\n\\end{definition}\n\\begin{lemma}[GEXIT and Dual GEXIT Function]\n\\label{lem:dualgexit}\nConsider a binary code $C$ and transmission over a complete\nfamily of BMS channels characterized by their family of $L$-densities\n$\\{\\Ldens{c}_\\cp\\}$. Let $\\{\\Ldens{a}_{\\cp}\\}$ denote\nthe corresponding family of (average) extrinsic \\MAP\\ densities.\nThen the standard GEXIT curve is given in parametric form by\n$\\{\\entropy(\\Ldens{c}_{\\cp}), \\gentropy(\\Ldens{c}_{\\cp}, \\Ldens{a}_{\\cp})\\}$.\nThe {\\em dual}\nGEXIT curve is defined by \n$\\{\\gentropy(\\Ldens{a}_{\\cp}, \\Ldens{c}_{\\cp}), \\entropy(\\Ldens{a}_{\\cp})\\}$. \nBoth, standard and dual GEXIT curve have an area equal to \n$r(C)$, the rate of the code.\n\\end{lemma}\nDiscussion:\nNote that both curves are ``comparable'' in that they first component measures\nthe channel $\\Ldens{c}$ and the second argument measure the \\MAP\\ density\n$\\Ldens{a}$. The difference between the two \nlies in the choice of measure which is applied to each component.\n\n\\begin{proof}\nConsider the entropy\n$\\entropy(\\Ldens{c}_{\\cp} \\star \\Ldens{a}_{\\cp})$. We have\n\\begin{align*}\n\\entropy(\\Ldens{c}_{\\cp} \\star \\Ldens{a}_{\\cp}) & =\n\\int_{-\\infty}^{\\infty} \\Bigl(\\int_{-\\infty}^{\\infty}\n\\Ldens{c}_{\\cp}(w) \\Ldens{a}_{\\cp}(v-w) \\text{d}w \\Bigr) \\log(1+e^{-v}) \\text{d}v \\\\\n& = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty}\n\\Ldens{c}_{\\cp}(w) \\Ldens{a}_{\\cp}(z) \\log(1+e^{-w-z}) \\text{d}w \\text{d}z\n\\end{align*}\nConsider now $\\frac{\\text{d} \\entropy(\\Ldens{c}_{\\cp} \\star \\Ldens{a}_{\\cp})}{\\text{d} \\cp}$.\nUsing the previous representation we get\n\\begin{align*}\n\\frac{\\text{d} \\entropy(\\Ldens{c}_{\\cp} \\star \\Ldens{a}_{\\cp})}{\\text{d} \\cp} & =\n\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty}\n\\frac{\\text{d}\\Ldens{c}_{\\cp}(w)}{\\text{d} \\cp} \\Ldens{a}_{\\cp}(z) \\log(1+e^{-w-z}) \\text{d}w \\text{d}z + \\\\\n& \\phantom{=} \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty}\n\\Ldens{c}_{\\cp}(w) \\frac{\\text{d} \\Ldens{a}_{\\cp}(z)}{\\text{d} \\cp} \\log(1+e^{-w-z}) \\text{d}w \\text{d}z.\n\\end{align*}\nThe first expression can be identified with the standard GEXIT curve \nexcept that it is parameterized by a generic parameter $\\cp$.\nThe second expression is essentially the same, but the roles of\nthe two densities are exchanged.\n\nIntegrate now this relationship over the whole range of $\\cp$ and\nassume that this range goes from ``perfect'' (channel) to ``useless''. \nThe integral on the left clearly equals 1. To perform the integrals\nover the right reparameterize the first expression with respect to \n$\\ensuremath{{\\tt{h}}} \\defas \\int_{\\infty}^{\\infty} \\Ldens{c}_{\\cp}(w) \\log(1+e^{-w}) \\text{d} w$\nso that it becomes the standard GEXIT curve given by \n$\\{\\entropy(\\Ldens{c}_{\\cp}), \\gentropy(\\Ldens{c}_{\\cp}, \\Ldens{a}_{\\cp})\\}$.\nIn the same manner reparameterize the second expression by\n$\\ensuremath{{\\tt{h}}} \\defas \\int_{\\infty}^{\\infty} \\Ldens{a}_{\\cp}(w) \\log(1+e^{-w}) \\text{d} w$\nso that it becomes the curve given by\n$\\{\\entropy(\\Ldens{a}_{\\cp}), \\gentropy(\\Ldens{a}_{\\cp}, \\Ldens{c}_{\\cp})\\}$.\nSince the sum of the two areas equals one and the area under the \nstandard GEXIT curve equals $r(C)$, it follows that the area under\nthe second curve equals $1-r(C)$. Finally, note that if we consider the inverse\nof the second curve by exchanging the two coordinates, i.e., if we consider the\ncurve \n$\\{\\gentropy(\\Ldens{a}_{\\cp}, \\Ldens{c}_{\\cp}), \\entropy(\\Ldens{a}_{\\cp})\\}$,\nthen the area under this curve is equal to $1-(1-r(C))=r(C)$, as claimed. \n\\end{proof}\n\\begin{example}[EXIT Versus GEXIT]\nFig.~\\ref{fig:exitversusgexit} compares the EXIT function to the\nGEXIT function for the $[3,1,3]$ repetition code and the $[6,5,2]$ single parity-check\ncode when transmission takes place over the \\ensuremath{\\text{BSC}}. As we can see, the two curves\nare similar but distinct. In particular note that the areas\nunder the GEXIT curves are equal to the rate of the codes but that this is not\ntrue for the EXIT functions.\n\\begin{figure}[htp]\n\\setlength{\\unitlength}{1.0bp}%\n\\begin{center}\n\\begin{picture}(300,120)\n\\put(0,0)\n{\n\\put(0,0){\\includegraphics[scale=1.0]{gexitchartrep}}\n\\put(90,30){\\makebox(0,0)[c]{\\small $\\frac13$}}\n\\put(60, -2){\\makebox(0,0)[t]{\\small $\\entropy(\\Ldens{c}_{\\ensuremath{{\\tt{h}}}})$}}\n\\put(-2, 60){\\makebox(0,0)[r]{\\small \\rotatebox{90}{$\\entropy(\\Ldens{a}_{\\ensuremath{{\\tt{h}}}}), \\gentropy(\\Ldens{c}_{\\ensuremath{{\\tt{h}}}}, \\Ldens{a}_{\\ensuremath{{\\tt{h}}}})$}}}\n\\put(-2, -2){\\makebox(0,0)[rt]{\\small $0$}}\n\\put(120,-2){\\makebox(0,0)[t]{\\small $1$}}\n\\put(-2,120){\\makebox(0,0)[r]{\\small $1$}}\n\\put(60,80){\\makebox(0,0)[b]{\\small $[3, 1, 3]$}}\n}\n\\put(180,0)\n{\n\\put(0,0){\\includegraphics[scale=1.0]{gexitchartspc}}\n\\put(90,30){\\makebox(0,0)[c]{\\small $\\frac56$}}\n\\put(60, -2){\\makebox(0,0)[t]{\\small $\\entropy(\\Ldens{c}_{\\ensuremath{{\\tt{h}}}})$}}\n\\put(-2, 60){\\makebox(0,0)[r]{\\small \\rotatebox{90}{$\\entropy(\\Ldens{a}_{\\ensuremath{{\\tt{h}}}}), \\gentropy(\\Ldens{c}_{\\ensuremath{{\\tt{h}}}}, \\Ldens{a}_{\\ensuremath{{\\tt{h}}}})$}}}\n\\put(-2, -2){\\makebox(0,0)[rt]{\\small $0$}}\n\\put(120,-2){\\makebox(0,0)[t]{\\small $1$}}\n\\put(-2,120){\\makebox(0,0)[r]{\\small $1$}}\n\\put(60,40){\\makebox(0,0)[b]{\\small $[6, 5, 2]$}}\n}\n\\end{picture}\n\\end{center}\n\\caption{\\label{fig:exitversusgexit} A comparison of the EXIT with the GEXIT function for the\n$[3,1,3]$ and the $[6, 5, 2]$ code.} \n\\end{figure}\n\\end{example}\n\\begin{example}[GEXIT Versus Dual GEXIT]\nFig.~\\ref{fig:gexitanddualgexit} shows the standard\nGEXIT function and the dual GEXIT function for the $[5, 4, 2]$ code\nand transmission over the $\\ensuremath{\\text{BSC}}$. Although the two curves have quite\ndistinct shapes, the area under the two curves is the same.\n\\begin{figure}[htp]\n\\setlength{\\unitlength}{1.0bp}%\n\\begin{center}\n\\begin{picture}(400,120)\n\\put(0,0)\n{\n\\put(0,0){\\includegraphics[scale=1.0]{dualgexitbsc1}}\n\\put(60, -2){\\makebox(0,0)[t]{\\small $\\entropy(\\Ldens{c}_{\\ensuremath{{\\tt{h}}}})$}}\n\\put(-2, 60){\\makebox(0,0)[r]{\\small \\rotatebox{90}{$\\gentropy(\\Ldens{c}_{\\ensuremath{{\\tt{h}}}}, \\Ldens{a}_{\\ensuremath{{\\tt{h}}}})$}}}\n\\put(60, 30){\\makebox(0,0)[t]{\\small standard GEXIT}}\n\\put(-2, -2){\\makebox(0,0)[rt]{\\small $0$}}\n\\put(120,-2){\\makebox(0,0)[t]{\\small $1$}}\n\\put(-2,120){\\makebox(0,0)[r]{\\small $1$}}\n}\n\\put(140,0)\n{\n\\put(0,0){\\includegraphics[scale=1.0]{dualgexitbsc2}}\n\\put(60, -2){\\makebox(0,0)[t]{\\small $\\gentropy(\\Ldens{a}_{\\ensuremath{{\\tt{h}}}}, \\Ldens{c}_{\\ensuremath{{\\tt{h}}}})$}}\n\\put(-2, 60){\\makebox(0,0)[r]{\\small \\rotatebox{90}{$\\entropy(\\Ldens{a}_{\\ensuremath{{\\tt{h}}}})$}}}\n\\put(60, 30){\\makebox(0,0)[t]{\\small dual GEXIT}}\n\\put(-2, -2){\\makebox(0,0)[rt]{\\small $0$}}\n\\put(120,-2){\\makebox(0,0)[t]{\\small $1$}}\n\\put(-2,120){\\makebox(0,0)[r]{\\small $1$}}\n}\n\\put(280,0)\n{\n\\put(0,0){\\includegraphics[scale=1.0]{dualgexitbsc3}}\n\\put(60, -2){\\makebox(0,0)[t]{\\small $\\entropy(\\Ldens{c}_{\\ensuremath{{\\tt{h}}}})$, $\\gentropy(\\Ldens{a}_{\\ensuremath{{\\tt{h}}}}, \\Ldens{c}_{\\ensuremath{{\\tt{h}}}})$}}\n\\put(-2, 60){\\makebox(0,0)[r]{\\small \\rotatebox{90}{$\\gentropy(\\Ldens{c}_{\\ensuremath{{\\tt{h}}}}, \\Ldens{a}_{\\ensuremath{{\\tt{h}}}})$,$\\entropy(\\Ldens{a}_{\\ensuremath{{\\tt{h}}}})$}}}\n\\put(60, 30){\\makebox(0,0)[t]{\\small both GEXIT}}\n\\put(-2, -2){\\makebox(0,0)[rt]{\\small $0$}}\n\\put(120,-2){\\makebox(0,0)[t]{\\small $1$}}\n\\put(-2,120){\\makebox(0,0)[r]{\\small $1$}}\n}\n\\end{picture}\n\\end{center}\n\\caption{\\label{fig:gexitanddualgexit} Standard and dual GEXIT function of $[5, 4, 2]$\ncode and transmission over the $\\ensuremath{\\text{BSC}}$.}\n\\end{figure}\n\\end{example}\n\\begin{lemma}\nConsider a degree distribution pair $(\\ensuremath{{\\lambda}}, \\ensuremath{{\\rho}})$\nand transmission over an BMS channel characterized by its\n$L$-density $\\Ldens{c}$ so that density evolution converges to \n$\\Delta_{\\infty}$. \nLet $\\{\\Ldens{a}_{\\alpha}\\}_{\\alpha=-1}^{\\infty}$\nand $\\{\\Ldens{b}_{\\alpha}\\}_{\\alpha=0}^{\\infty}$ denote the interpolated\nfamilies as defined in Definition \\ref{def:interpolation}.\n\nThen the two GEXIT curves parameterized by\n\\begin{align*}\n\\{ \\entropy(\\Ldens{a}_{\\alpha}), \n\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha+1}) \\}, \\tag*{GEXIT of check nodes} \\\\\n\\{ \\entropy(\\Ldens{a}_{\\alpha}),\n\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha}) \\}, \\tag*{inverse of dual GEXIT of variable nodes}\n\\end{align*}\ndo not overlap and faithfully represent density evolution.\nFurther, the area under the ``check-node'' GEXIT function\nis equal to $1-\\int \\!\\ensuremath{{\\rho}}$ and the area to the left of the\n``inverse dual variable node'' GEXIT function is equal to $\\entropy(\\Ldens{c}) \\int \\!\\ensuremath{{\\lambda}}$.\nIt follows that $r(\\ensuremath{{\\lambda}}, \\ensuremath{{\\rho}}) \\leq 1-\\entropy(\\Ldens{c})$, i.e., \nthe transmission rate can not exceed the Shannon limit.\n\nThis implies that transmission approaching capacity requires\na perfect matching of the two curves.\n\\end{lemma}\n\\begin{proof}\nFirst note that $\\{ \\entropy(\\Ldens{a}_{\\alpha}),\n\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha+1}) \\}$\nis the standard GEXIT curve representing the action\nof the check nodes: $\\Ldens{a}_{\\alpha}$ corresponds to\nthe density of the messages {\\em entering} the check nodes and\n$\\Ldens{b}_{\\alpha+1}$ represents the density of the corresponding \noutput messages.\nOn the other hand,\n$\\{ \\entropy(\\Ldens{a}_{\\alpha}),\n\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha}) \\}$\nis the inverse of the dual GEXIT curve \ncorresponding to the action at the variable nodes:\nnow the input density to the check nodes is \n$\\Ldens{b}_{\\alpha}$ and $\\Ldens{a}_{\\alpha}$ denotes the \ncorresponding output density.\n\nThe fact that the two curves do not overlap can be seen as follows.\nFix an entropy value. This entropy value corresponds to a\ndensity $\\Ldens{a}_{\\alpha}$ for a unique value of $\\alpha$.\nThe fact that\n$G(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha}) \\geq\nG(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha+1})$ now follows from\nthe fact that $\\Ldens{b}_{\\alpha+1} \\prec \\Ldens{b}_{\\alpha}$ and\nthat for any symmetric $\\Ldens{a}_{\\alpha}$ this relationship\nstays preserved by applying the GEXIT functional.\n\nThe statements regarding the areas of the two curves \nfollow in a straightforward manner from the GAT and Lemma \\ref{lem:dualgexit}.\nThe bound on the achievable rate follows in the same manner as for\nthe BEC: the total area of the GEXIT box equals one and the two curves do not\noverlap and have areas $1-\\int \\ensuremath{{\\rho}}$ and $\\entropy(\\Ldens{c})$.\nIt follows that\n$1-\\int \\!\\ensuremath{{\\rho}} + \\entropy(\\Ldens{c}) \\int \\!\\ensuremath{{\\lambda}} \\leq 1$,\nwhich is equivalent to the claim $r(\\ensuremath{{\\lambda}}, \\ensuremath{{\\rho}}) \\leq 1-\\entropy(\\Ldens{c})$.\n\\end{proof}\n\nWe see that the matching condition still holds even for general channels.\nThere are a few important differences between the general case and the simple\ncase of transmission over the BEC. For the BEC, the intermediate densities\nare always the BEC densities independent of the degree distribution.\nThis of course enormously simplifies the task. Further, for the BEC, given\nthe two EXIT curves, the progress of density evolution is simply given\nby a staircase function bounded by the two EXIT curves. For the general case,\nthis staircase function still has vertical pieces but the ``horizontal''\npieces are in general at an angle. This is true since the $y$-axis for\nthe ``check node'' step measures \n$\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha+1})$, but\nin the subsequent ``inverse variable node'' step \nit measures \n$\\gentropy(\\Ldens{a}_{\\alpha+1}, \\Ldens{b}_{\\alpha+1})$.\nTherefore, one should think of two sets of labels on the $y$-axis,\none measuring $\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha+1})$,\nand the second one measuring $\\gentropy(\\Ldens{a}_{\\alpha+1}, \\Ldens{b}_{\\alpha+1})$. The ``horizontal'' step then consists of first\nswitching from the first $y$-axis to the second, so that the labels\ncorrespond to the same density $\\Ldens{b}$ and then drawing a horizontal\nline until it crosses the ``inverse variable node'' GEXIT curve.\nThe ``vertical'' step stays as before, i.e., it really corresponds to\ndrawing a vertical line. All this is certainly best clarified by\na simple example.\n\\begin{example}[$(3, 6)$-Regular Ensemble and Transmission over $\\ensuremath{\\text{BSC}}$]\nConsider the $(3, 6)$-regular ensemble and transmission over the $\\ensuremath{\\text{BSC}}(0.07)$.\nThe corresponding illustrations are shown in Fig.~\\ref{fig:componentgexit}.\nThe top-left figure shows the standard GEXIT curve for the check node side.\nThe top-right figure shows the dual GEXIT curve corresponding to the\nvariable node side. In order to use these two curves in the same figure,\nit is convenient to consider the inverse function for the variable\nnode side. This is shown in the bottom-left figure. In the bottom-right\nfigure both curves are shown together with the ``staircase'' like function\nwhich represents density evolution. As we see, the two curves to not overlap\nand have both the correct areas.\n\\begin{figure}[hbt]\n\\centering\n\\setlength{\\unitlength}{1.5bp}\n\\begin{picture}(220,220)\n\\put(0,120){\n\\put(0,0){\\includegraphics[scale=1.5]{componentgexit1}}\n\\put(50, -2){\\makebox(0,0)[t]{\\small $\\entropy(\\Ldens{a}_{\\alpha})$}}\n\\put(102, 50){\\makebox(0,0)[l]{\\small \\rotatebox{90}{$\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha+1})$}}}\n\\put(50, 40){\\makebox(0,0)[t]{\\small GEXIT: check nodes}}\n\\put(50, 30){\\makebox(0,0)[t]{\\small $\\text{area}=\\frac56$}}\n\\put(50, 10){\\makebox(0,0)[c]{$\\Ldens{b}_{\\alpha+1} = \\sum_{i} \\ensuremath{{\\rho}}_i \\Ldens{a}_{\\alpha}^{\\boxast (i-1)} $}}\n}\n\\put(120, 120)\n{\n\\put(0,0){\\includegraphics[scale=1.5]{componentgexit2}}\n\\put(50, -2){\\makebox(0,0)[t]{\\small {$\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha})$}}}\n\\put(-2, 50){\\makebox(0,0)[r]{\\small  \\rotatebox{90}{$\\entropy(\\Ldens{a}_{\\alpha})$}}}\n\\put(50, 70){\\makebox(0,0)[t]{\\small dual GEXIT: variable nodes}}\n\\put(50, 60){\\makebox(0,0)[t]{\\small $\\text{area}=\\frac13 h(0.07)$}}\n\\put(102, 36.6){\\makebox(0,0)[l]{\\small \\rotatebox{90}{$h(0.07) \\approx 0.366$}}}\n\\put(50, 40){\\makebox(0,0)[c]{$\\Ldens{a}_{\\alpha} = \\Ldens{c} \\star \\sum_{i} \\ensuremath{{\\lambda}}_i \\Ldens{b}_{\\alpha}^{\\star (i-1)} $}}\n}\n\\put(0,0)\n{\n\\put(0,0){\\includegraphics[scale=1.5]{componentgexit3}}\n\\put(50, -2){\\makebox(0,0)[t]{\\small $\\entropy(\\Ldens{a}_{\\alpha})$}}\n\\put(-2, 50){\\makebox(0,0)[r]{\\small \\rotatebox{90}{$\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha})$}}}\n\\put(50, 30){\\makebox(0,0)[t]{\\small inverse of dual GEXIT:}}\n\\put(50, 20){\\makebox(0,0)[t]{\\small variable nodes}}\n\\put(36.6, 102){\\makebox(0,0)[b]{\\small $h(0.07) \\approx 0.366$}}\n}\n\\put(120,0)\n{\n\\put(0,0){\\includegraphics[scale=1.5]{componentgexit4}}\n\\put(50, -2){\\makebox(0,0)[t]{\\small $\\entropy(\\Ldens{a}_{\\alpha})$}}\n\\put(-2, 50){\\makebox(0,0)[r]{\\small \\rotatebox{90}{$\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha})$}}}\n\\put(102, 50){\\makebox(0,0)[l]{\\small \\rotatebox{90}{$\\gentropy(\\Ldens{a}_{\\alpha}, \\Ldens{b}_{\\alpha+1})$}}}\n\\put(36.6, 102){\\makebox(0,0)[b]{\\small $h(0.07) \\approx 0.366$}}\n}\n\\end{picture}\n\\caption{\n\\label{fig:componentgexit}\nFaithful representation of density evolution by two non-overlapping component-wise\nGEXIT functions which represent the ``actions'' of the check nodes and variable nodes,\nrespectively. The area between the two curves equals is equal to the additive\ngap to capacity.\n}\n\\end{figure}\n\\end{example}\n\nAs remarked earlier, one potential use of the matching condition\nis to find capacity approaching degree distribution pairs. Let us\nquickly outline a further such potential application. Assuming that\nwe have found a sequence of capacity-achieving degree distributions,\nhow does the number of required iterations scale as we approach capacity.\nIt has been conjectured that the the number of required iterations\nscales like $1\/\\delta$, where $\\delta$ is the gap to capacity.\nThis conjecture is based on the geometric picture which the\nmatching condition implies. To make things simple, imagine\nthe two GEXIT curves as two parallel lines, lets say both\nat a 45 degree angle, a certain distance\napart, and think of density evolution as a staircase function.\nFrom the previous results, the area between the lines is proportional\nto $\\delta$. Therefore, if we half $\\delta$ the distance between\nthe lines has to be halved and one would expect that we need\ntwice as many steps. Obviously, the above discussion was \nbased on a number of simplifying assumptions. It remains to \nbe seen if this conjecture can be proven rigorously.  \n\n\n\n\\section*{Acknowledgments}\n\nThe work of A.~Montanari was partially supported by the European Union under \nthe project EVERGROW.\n\\bibliographystyle{ieeetr} \n\n\\newcommand{\\SortNoop}[1]{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Acknowledgements}\n\nThe first author gratefully acknowledges support for this work by the Swiss National Science Foundation, Project 146356.\n\n\\newpage\n\\input{introduction}\n\n\\input{prelim}\n\n\\input{benchmark}\n\n\\input{approx}\n\n\\input{conclusion}\n\n\\input{int_est\n\n\\pagestyle{plain}\n\n\\printbibliography[heading=bibintoc]\n\n\n\\end{document}\n\n\n\\section{Approximation Properties}\\label{sec:approx}\n\nCompared to the discussion in \\autoref{sec:introduction}, the only addition we are now able to make is to formulate the goals in terms of conditions on the ridgelet coefficient sequence, as well as a finer statement about the relation between the decay of $f$ \\eqref{eq:f_glob_decay} and the deviation $\\delta$ in \\autoref{th:approx_intro}. \\cite[Thm. 4.1]{mutilated} shows that for a function $f\\in H^t(\\bbR^d\\setminus \\{h_i\\})$,  supported (compactly) on $[0,1]^d$, the ridgelet coefficient sequence $\\parens*{\\inpr*{\\phi_{j,\\ell,k},f}}_{j,\\ell,k}$  belongs to the $\\ell^p_w$ space with the best possible $p=p^*:=\\parens*{\\frac td + \\frac 12}^{-1}$, which implies the optimal $N$-term approximation rate $\\frac td$ in the $L^2$-norm.\n\nIn contrast, we will show that the coefficient sequence $\\inpr{\\Phi,f}:=\\parens*{\\inpr{\\varphi_\\lambda,f}}_\\lambda$ with $f\\in H^t(\\bbR^d\\setminus \\{h_i\\})$ satisfying the following polynomial decay condition,\n\\begin{align\n\t\\abs{f(\\vec x)} \\lesssim \\reg{\\vec x}^{-2m},\n\\end{align}\nis in $\\ell^{p^*+\\frac dm}_w$. In particular, if the above decay holds for all $m\\in\\bbN$ (which is satisfied for example if $f$ has exponential decay or compact support), then $\\inpr{\\Phi,f}\\in \\ell^{p^*+\\delta^*}_w$ for arbitrary $\\delta^*>0$.\n\nFinally, in the context of CDD-schemes (see \\autoref{ssec:cdd}) ---  we need to consider $N$-term approximation with regard to the $\\Hs$-norms instead of the $L^2$ norm. Due to \\eqref{eq:RidgeletFrame}, this corresponds to bounding the $\\ell^p_w$-norm of the weighted coefficient sequence ${\\mathbf{W}}\\inpr{\\Phi,f}$, compare also with \\autoref{cor:approx} in this regard. Since the weights $w_{\\lambda}\\sim \\reg*{2^j \\vec s\\cdot \\vec s_\\jl}$ grow with scale $j$, this causes extra work as well.\n\n\\subsection{Main Theorem}\n\nWe begin straight away by formulating the goal of this section, \\autoref{th:approx}. Once we will have proved this result, we will have satisfied ingredient \\ref{itm:ingr:4} from \\autoref{ssec:cdd}.\n\n\\begin{theorem}\\label{th:approx}\n\tLet $f$ be a function in $L^2(\\bbR^d)$ such that $f \\in H^t(\\bbR^d)$ apart from $N$ hyperplanes --- i.e.~of the form \\eqref{eq:Ht_except_hyp} --- such that the following decay condition (recalling $\\reg{y}=\\sqrt{1+|y|^2}$)\n\t%\n\t\\begin{align}\\label{eq:f_glob_decay}\n\t\t\\abs{f(\\vec x)} \\lesssim \\reg{\\vec x}^{-2m}\n\t\\end{align}\n\t%\n\tholds for a certain $m\\in \\bbN$. Then, if $\\kappa\\in H^{\\ceil{t}+\\frac d2}$ (for the terms involving $\\CS[f]$),\n\t%\n\t\\begin{align}\n\t\t\\inpr{\\Phi,f}_{L^2} &:= \\parens*{\\inpr{\\varphi_\\lambda,f}_{L^2}}_{\\lambda\\in\\Lambda}\\in \\ell^{p^* + \\frac dm}_w,\\\\\n\t\t{\\mathbf{W}}\\inpr{\\Phi,\\CS[f]}_{L^2} &:= \\parens*{w_\\lambda\\inpr{\\varphi_\\lambda,\\CS[f]}_{L^2}}_{\\lambda\\in\\Lambda}\\in \\ell^{p^* + \\frac dm}_w,\n\t\\end{align}\n\t%\n\ti.e.~the (weighted) ridgelet coefficient sequences belong to the $\\ell^p_w$-space with $p=p^* + \\frac dm$, where $p^*= (\\frac td + \\frac 12)^{-1}$ would be the best possible $p$ for $f\\in H^t(\\bbR^d)$, \\emph{even without singularities}(!), and the deviation from the optimal value\\footnote{For $\\inpr{\\Phi,f}$, the absence of ${\\mathbf{W}}$ is not completely without effect, in that the deviation from the optimal $p$ can be bounded by $\\frac{d}{2n}+\\varepsilon$ with arbitrarily small $\\varepsilon>0$.} decays linearly with $m$.\n\t\n\tIf the decay condition \\eqref{eq:f_glob_decay} holds for arbitrarily large $m\\in\\bbN$ --- which is satisfied for exponential decay or compact support, for example --- then, for arbitrary $\\delta^*>0$, $\\inpr{\\Phi,f}_{L^2} \\in \\ell^{p^* + \\delta^*}_w$ and  ${\\mathbf{W}} \\inpr{\\Phi,\\CS[f]}_{L^2} \\in \\ell^{p^* + \\delta^*}_w$.\n\\end{theorem}\n\nWe split some essential parts of the proof into separate results, namely the \\emph{localisation in angle} of the ridgelet coefficients in \\autoref{prop:loc_angle} and --- after proving decay rates for (modified) ridgelets in space in \\autoref{lem:decay_ridgelet} --- the \\emph{localisation in space} for the response to the singularity in \\autoref{prop:loc_space}, as well as the general decay of the coefficients for functions satisfying \\eqref{eq:f_glob_decay} in \\autoref{lem:ridge_coeff_tail_decay}.\n\nBefore we continue, let us briefly state the following corollary of \\autoref{th:approx}, recalling the non-linear set $\\Sigma_N$ of functions being linear combinations of less than $N$ terms from $\\Phi$ (from \\autoref{def:nonlin_basics}).\n\n\\begin{corollary}\\label{cor:approx}\n\tFor $f \\in H^t(\\bbR^d\\setminus \\{h_i\\})$ such that \\eqref{eq:f_glob_decay} is satisfied for all $n\\in\\bbN$, we have, for arbitrary $\\delta>0$,\n\t%\n\t\\begin{align}\n\t\t&\\inf_{g\\in \\Sigma_N(\\Phi)} \\norm{f-g}_{L^2} \\lesssim N^{-\\frac{t}{d}+\\delta},\\\\\n\t\t&\\inf_{g\\in \\Sigma_N(\\Phi)} \\norm{\\CS[f]-g}_{\\Hs} \\lesssim N^{-\\frac{t}{d}+\\delta}.\n\t\\end{align}\n\\end{corollary}\n\n\\begin{proof}\n\tThis statement is a direct consequence of \\autoref{prop:wlp_Nterm_frame} and \\eqref{eq:RidgeletFrame}, respectively, that $\\Phi$ is also a frame (without the weight ${\\mathbf{W}}$) for $L^2$.\n\\end{proof}\n\n\\subsection{Localisation in Angle}\\label{sec:loc_angle}\n\nIn \\autoref{lem:fourier_decomp}, we considered how the Fourier transform of a function $f(\\vec x)=H(\\vec x\\cdot \\vec n-v) g(\\vec x)$ splits into two parts, compare \\eqref{eq:sing_fourier_rot}. Based on this decomposition, we will likewise split the coefficient sequence in two, where our main interest is with the \\emph{singular} contribution arising from the cut-off.\n\nThe crucial property of ridgelets we want to explore in this section concerns the fact that large coefficients will only appear for ridgelets that are aligned with the singularity, and we are able to describe the decay of the coefficient in terms of the smoothness of $g$ and the angle between $\\vec s_\\jl$ and $\\vec n$.\n\n\\begin{proposition}\\label{prop:loc_angle}\n\tLet $f(\\vec x)=H(\\vec x\\cdot \\vec n-v) g(\\vec x)$ with $g\\in H^t$, $t\\in\\bbN$. Furthermore, let $\\kappa\\in H^{t+\\frac d2}$ and $\\kappa\\ge \\gamma>0$. Then, the sequences\n\t%\n\t\\begin{align}\n\t\t\\alpha_\\lambda&:=\\inpr*{\\varphi_\\lambda,f}=\\inpr*{\\hat\\varphi_\\lambda,\\hat f}=\\alpha^\\mathrm{r}_\\lambda+\\alpha^\\mathrm{s}_\\lambda,\\\\\n\t\t\\beta_\\lambda&:=\\inpr*{\\varphi_\\lambda,\\CS[f]}=\\inpr*{\\hat\\varphi_\\lambda,\\CF[\\CS[f]]}= \\beta^\\mathrm{r}_\\lambda+\\beta^\\mathrm{s}_\\lambda,\n\t\\end{align}\n\t%\n\tcan be split into a \\emph{regular contribution} $\\alpha^\\mathrm{r}$ resp.~$\\beta^\\mathrm{r}$ (coming from the function) and a \\emph{singular contribution} $\\alpha^\\mathrm{s}$ resp.~$\\beta^\\mathrm{s}$ (coming from the cut-off). The regular part of both sequences shows decay with scale $j$ proportional to $t$,\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\ell,\\vec k} \\abs*{\\alpha^\\mathrm{r}_{\\smash[t]{{j\\!\\?\\?,\\?\\ell\\!\\?\\?,\\?\\vec k}}}}^2 &\\lesssim \\varepsilon_j^2 2^{-2jt}\\norm{g}_{H^t}^2,\\\\\n\t\t\\sum_{\\ell,\\vec k} \\abs*{w_{j\\!\\?\\?,\\?\\ell\\!\\?\\?,\\?\\vec k} \\beta^\\mathrm{r}_{\\smash[t]{{j\\!\\?\\?,\\?\\ell\\!\\?\\?,\\?\\vec k}}}}^2 &\\lesssim \\varepsilon_j^2 2^{-2jt}\\norm{g}_{H^t}^2, \\label{eq:gamma_est_reg}\n\t\\end{align}\n\t%\n\twhere $\\sum_j \\varepsilon_j^2 \\le 1$. Defining the angle $\\theta_{j\\!\\?\\?,\\ell}(\\vec n):=\\arccos (\\vec s_\\jl\\cdot\\vec n)$ that the vectors $\\vec s_\\jl$ enclose with $\\vec n$ and setting\n\t%\n\t\\begin{align}\\label{eq:Lambda_jr}\n\t\t\\Lambda^j_r(\\vec n):=\\begin{cases}\n\t\t\t\\set{\\ell}{2^{-r}\\le \\abs*{\\sin \\theta_{j\\!\\?\\?,\\ell}(\\vec n)} \\le 2^{-r+1}}, & 1\\le r<j, \\\\\n\t\t\t\\set{\\ell}{\\abs*{\\sin \\theta_{j\\!\\?\\?,\\ell}(\\vec n)} \\le 2^{-j+1}}, & r=j,\n\t\t\\end{cases}\n\t\\end{align}\n\t%\n\tthe singular part of both sequences is \\emph{localised in angle} for a given scale $j$ as follows,\n\t%\n\t\\begin{align}\n\t\\begin{split}\\label{eq:gamma_est_sum_Lambda}\n\t\t&\\sum_{\\ell\\in\\Lambda^j_r(\\vec n)} \\!\\! \\sum_{\\vec k} \\abs*{\\alpha^\\mathrm{s}_{\\smash[t]{{j\\!\\?\\?,\\?\\ell\\!\\?\\?,\\?\\vec k}}}}^2 \\lesssim 2^{-j} 2^{-(j-r)(2t-1)}\\norm{g}_{H^t}^2, \\\\\n\t\t&\\sum_{\\ell\\in\\Lambda^j_r(\\vec n)} \\!\\! \\sum_{\\vec k} \\abs*{w_{j\\!\\?\\?,\\?\\ell\\!\\?\\?,\\?\\vec k} \\beta^\\mathrm{s}_{\\smash[t]{{j\\!\\?\\?,\\?\\ell\\!\\?\\?,\\?\\vec k}}}}^2\\lesssim 2^{-j} 2^{-(j-r)(2t-1)}\\norm{g}_{H^t}^2.\n\t\\end{split}\n\t\\end{align}\n\\end{proposition}\n\n\\begin{proof\n\tFrom \\cite{grohs1}, we know that\n\t%\n\t\\begin{align}\n\t\t&\\norm{f}_{L^2} \\sim \\norm{\\inpr{\\Phi,f}}_{\\ell^2}=\\norm{\\alpha_\\Lambda}_{\\ell^2} \\label{eq:ridge_frame}\\\\\n\t\t&\\norm{\\CS[f]}_{\\Hs} \\sim \\norm{{\\mathbf{W}} \\inpr{\\Phi,\\CS[f]}}_{\\ell^2}=\\norm{{\\mathbf{W}} \\beta_\\Lambda}_{\\ell^2}, \\label{eq:ridge_Hs_stable}\n\t\\end{align}\n\t%\n\twhere $u:=\\CS[f]\\in\\Hs$ for $f\\in L^2$ (cf. \\eqref{eq:Hs_elliptic}). We define\n\t%\n\t\\begin{align}\n\t\tP^j_r(\\vec n):=\\bigcup_{\\ell \\in \\Lambda^j_r(\\vec n)} P_{j\\!\\?\\?,\\ell},\n\t\\end{align}\n\t%\n\tand let $\\hat\\chi$ be the indicator function of this set. Obviously, $\\hat \\chi$ has an inverse Fourier transform, which we name $\\chi$. Since $\\curly{j}\\times \\Lambda^j_r\\times \\bbZ^d$ is a subset of all ridgelets intersecting $P^j_r$, we can apply \\eqref{eq:ridge_frame} to $f*\\chi$ (which, by looking at the Fourier side, is obviously in $L^2$) we arrive at\n\t%\n\t\\begin{align}\\label{eq:alpha_loc_fourier}\n\t\t\\sum_{\\ell\\in\\Lambda^j_r}\\sum_{\\vec k} \\abs*{\\alpha_\\lambda}^2\n\t\t\\lesssim \\int \\abs*{\\hat f \\hat \\chi}^2 \\d \\vec \\xi\n\t\t=\\int_{P^j_r(\\vec n)} \\abs*{\\hat f}^2 \\d \\vec \\xi.\n\t\\end{align}\n\t%\n\tTo get the same (localised!) estimate in Fourier space for second sequence, we need to do a bit more. First, we split the absorption into its parts\n\t%\n\t\\begin{align}\n\t\tAu=\\vec s \\cdot \\nabla u + (\\gamma+\\kappa_0)u = f \\quad \\Longleftrightarrow \\quad A_{\\mathrm{c}} u:=\\vec s\\cdot \\nabla u+\\gamma u=f-\\kappa_0 u =:f_{\\mathrm{c}},\n\t\\end{align}\n\t%\n\tand observe that $u$ is also the solution of the differential equation $A_{\\mathrm{c}} u=f_{\\mathrm{c}}$. From \\autoref{prop:sol_smooth_except_hyp} we know that $u(\\vec x)=u_0(\\vec x)+H(\\vec x\\cdot\\vec  n-v)u_1(\\vec x)$ is of the same form as $f$,  with $\\norm{u_i}_{H^t}\\lesssim\\norm{g}_{H^t}$, and again, the multiplication with $\\kappa$ also does not pose a problem (see \\cite[Prop. 3.3.5]{my_thesis}).\n\t\n\tThus, $f_{\\mathrm{c}}(\\vec x)= g_0(\\vec x) + H(\\vec x\\cdot\\vec  n-v)g_1(\\vec x)$ with $\\norm{g_i}_{H^t}\\lesssim\\norm{g}_{H^t}$. The gain of this change of differential equation is that $A_{\\mathrm{c}}$ trivially satisfies the requirements for \\eqref{eq:Hs_elliptic}, and therefore $\\norm{u}_{\\Hs}\\sim\\norm[]{A_{\\mathrm{c}} u}_{L^2}$ holds for $A_{\\mathrm{c}}$ as well. Now, crucially, due to the constant absorption term in $A_{\\mathrm{c}}$, the operator commutes with convolution, i.e.~$A_{\\mathrm{c}}(u*\\chi)= (A_{\\mathrm{c}} u)*\\chi=f_{\\mathrm{c}}*\\chi$ with $\\chi$ as above. Thus, using the same argument as for \\eqref{eq:alpha_loc_fourier} and applying \\eqref{eq:ridge_Hs_stable} for $u*\\chi$,\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\ell\\in\\Lambda^j_r}\\sum_{\\vec k} \\abs*{w_\\lambda\\beta_\\lambda}^2\n\t\t&\\lesssim \\norm*{u*\\chi}_{\\Hs}^2 \\sim \\norm*{A_{\\mathrm{c}} (u*\\chi)}_{L^2}^2 = \\norm*{f_{\\mathrm{c}}*\\chi}_{L^2}^2 \\\\\n\t\t&= \\int_{P^j_r} \\abs*{\\hat{f_{\\mathrm{c}}}}^2 \\d \\vec \\xi\n\t\t\\le \\int_{P^j_r(\\vec n)} \\abs*{\\hat g_0}^2 \\!+ \\abs*{\\CF[H(\\vec x\\cdot\\vec  n-v)g_1(\\vec x)](\\vec \\xi)}^2\\d \\vec \\xi.\n\t\\end{align}\n\t%\n\tThe term involving $\\hat g_0$ will contribute to the regular part ${\\mathbf{W}} \\beta^\\mathrm{r}$, and the required inequality \\eqref{eq:gamma_est_reg} is easily proved since, for $t\\in\\bbR^+$, $\\norm{h}_{H^t}\\sim\\norm*{\\reg*{\\vec \\xi}^{t}\\hat h(\\vec \\xi)}_{L^2}$, thus\n\t%\n\t\\begin{align}\n\t\t\\int_{P^j_r(\\vec n)} \\abs*{\\hat g_0}^2\\d \\vec \\xi =\\int_{P^j_r(\\vec n)} \\frac{\\reg*{\\vec \\xi}^{2t}}{\\reg*{\\vec \\xi}^{2t}} \\abs*{\\hat g_0}^2\\d \\vec \\xi \\lesssim 2^{-2jt}\\int_{P^j_r(\\vec n)} \\reg*{\\vec \\xi}^{2t} \\abs*{\\hat g_0}^2\\d \\vec \\xi  \\lesssim 2^{-2jt} \\norm{g_0}_{H^t}  \\lesssim 2^{-2jt} \\norm{g}_{H^t},\n\t\\end{align}\n\t%\n\tusing the fact that $|\\vec \\xi|\\gtrsim 2^j$ for $\\vec \\xi \\in P^j_r(\\vec n)$.\n\t\n\tAside from the contribution above, both sequences have now been dominated by the same kind of integral, and so we can deal with both cases together from now on. We will work with $g$ (and thus the estimate for $\\alpha$), but by replacing any occurrence of it with $g_1$ (and recalling $\\norm{g_1}_{H^{t}}\\lesssim \\norm{g}_{H^{t}}$) will automatically yield the estimate for ${\\mathbf{W}}\\beta$.\n\t\n\tSetting\n\t%\n\t\\begin{align}\n\t\tf(\\vec x)=\\tilde f(\\Rn \\vec x) \\quad \\text{where} \\quad \\tilde f(\\vec x) := H(x_1-v)\\tilde g(\\vec x) \\quad \\text{and} \\quad \\tilde g(\\vec x):=g(\\Rn^{-1}\\vec x)\n\t\\end{align}\n\t%\n\tlike in the proof of \\autoref{lem:fourier_decomp}, we continue from \\eqref{eq:alpha_loc_fourier} as follows. Transforming with $\\vec \\zeta=\\Rn\\vec \\xi$, we use \\eqref{eq:sing_fourier_rot} and the fact that $(a+b)^2\\le 2(a^2+b^2)$,\n\t%\n\t\\begin{align}\n\t\t\\int_{P^j_r(\\vec n)} \\abs*{\\hat f(\\vec \\xi)}^2 \\d \\vec \\xi\n\t\t&= \\int_{\\Rn P^j_r(\\vec n)} \\abs*{\\hat f(\\Rn^{-1}\\vec \\zeta)}^2 \\d \\vec \\zeta\\\\\n\t\t&\\lesssim \\int_{\\Rn P^j_r(\\vec n)}  \\abs*{\\vec\\zeta}^{-2} \\abs*{\\CF[H(x_1-v)\\nabla \\tilde g(\\vec x)](\\vec \\zeta)}^2 + \\abs*{\\vec\\zeta}^{-2} \\abs*{\\wh{g\\normalr|_{h}}(\\CP_{\\vec e_1}\\vec \\zeta)}^2 \\d \\vec \\zeta.\n\t\\end{align}\n\t%\n\tEach $P_{j\\!\\?\\?,\\ell}$ making up $P^j_r(\\vec n)$ is a polar rectangle (compare \\eqref{eq:supp_Pjl}), and with the help of the transform $\\CR(r,\\vec s):=r\\vec s$, we can easily describe effect of $\\Rn$,\n\t%\n\t\\begin{align}\n\t\t\\Rn P_{j\\!\\?\\?,\\ell} = \\Rn\\CR\\parens*{[2^{j-1},2^{j+1}]\\times B_{{\\mathbb{S}^{d-1}}}(\\vec s_\\jl, 2^{-j+1})} = \\CR\\parens*{[2^{j-1},2^{j+1}]\\times B_{{\\mathbb{S}^{d-1}}}(\\Rn\\vec s_\\jl, 2^{-j+1})}.\n\t\\end{align}\n\t%\n\tThus,\n\t%\n\t\\begin{align}\n\t\t\\sin\\theta_{\\Rn\\vec s_\\jl}(\\vec e_1) = \\arccos (\\Rn\\vec s_\\jl\\cdot \\vec e_1) = \\sin\\theta_{j\\!\\?\\?,\\ell}(\\vec n) \\quad \\text{and therefore} \\quad \\Rn P^j_r(\\vec n) = P^j_r(\\vec e_1).\n\t\\end{align}\n\t\n\tWe begin the induction with $t=0$, where the claim follows trivially with $\\alpha^\\mathrm{s}_\\lambda\\equiv0$ resp.~$\\beta^\\mathrm{s}_\\lambda\\equiv0$. Suppose now that the claim holds for $t-1\\in \\bbN$, i.e.~$g\\in H^{t-1}$ and thus also $\\tilde g\\in H^{t-1}$. Then, by the above, the induction hypothesis implies\n\t%\n\t\\begin{align}\n\t\n\t\t\\int_{P^j_r(\\vec n)} \\abs*{\\hat f}^2 \\d \\vec \\xi\n\t\t&\\lesssim \\int_{P^j_r(\\vec e_1)}  \\abs*{\\vec\\xi}^{-2} \\abs*{\\CF[H(x_1-v)\\nabla \\tilde g(\\vec x)](\\vec \\xi)}^2 \\d \\vec \\xi + \\! \\int_{P^j_r(\\vec e_1)} \\abs*{\\vec\\xi}^{-2} \\abs*{\\wh{g\\normalr|_{h}}(\\CP_{\\vec e_1}\\vec \\xi)}^2 \\d \\vec \\xi \\\\\n\t\t&\\lesssim  \\varepsilon_j^2 (\\delta^j_r)^2 2^{-j(2t-1)-2j}+ 2^{-3j} 2^{-(j-r)(2(t-1)-1)} + 2^{-2j} \\!\\! \\int_{P^j_r(\\vec e_1)} \\abs*{ \\wh{g\\normalr|_{h}} (\\CP_{\\vec e_1} \\vec \\xi)}^2 \\d \\vec \\xi, \\label{eq:gamma_est_ind_hyp}\n\t\\end{align}\n\t%\n\tbecause, again, $\\abs*{\\vec \\xi}\\gtrsim 2^j$ on $P^j_r(\\vec e_1)$. The factors $\\varepsilon_j$ and $\\delta^j_r$ satisfy\n\t%\n\t\\begin{align}\n\t\t\\sum_{r=1}^j (\\delta^j_r)^2 =1 \\qquad \\text{and} \\qquad \\sum_{j=1}^\\infty \\varepsilon^2_j =1.\n\t\\end{align}\n\t%\n\tThe last integral can be treated (for $1\\le r<j$) as follows\n\t%\n\t\\begin{align}\n\t\t\\int_{P^j_r(\\vec e_1)} \\abs*{ \\wh{g\\normalr|_{h}} (\\CP_{\\vec e_1} \\vec \\xi)}^2 \\d \\vec \\xi\n\t\t&= \\int_{P^j_r(\\vec e_1)} \\abs*{ \\wh{g\\normalr|_{h}} (\\CP_{\\vec e_1} T(r \\cos \\theta, r \\sin\\theta, \\phi))}^2 r^{d-1} (\\sin\\theta)^{d-2}\\d r \\d \\phi \\d \\theta,\n\t\t\\intertext{\n\t%\n\twhere, after having transformed into spherical coordinates, we split off the first component $r\\cos \\theta$  ($\\theta$ being the angle to the $\\xi_1$-axis) and treat the rest as $d-1$-dimensional spherical coordinates with radius $r\\sin\\theta$, i.e.~$T(x,s,\\phi)=(x,s\\cos\\phi_1,s\\sin\\phi_1\\cos\\phi_2,\\ldots)^\\top$. Consequently, we transform with $r'=r\\sin\\theta$,\n\t%\n\t\t}\n\t\n\t\t&\\lesssim \\int_{\\CP_\\theta P^j_r(\\vec e_1)} \\frac{2^j}{\\abs{\\sin \\theta}} \\int_{\\bbS^{d-2}} \\int_{2^{j-1}\\abs{\\sin\\theta}}^{2^{j+1}\\abs{\\sin\\theta}} \\abs*{\\wh{g\\normalr|_{h}}(r',\\phi)}^2 (r')^{d-2} \\d r' \\d \\phi \\d \\theta,\n\t\\end{align}\n\t%\n\twhere one power of $r\\lesssim 2^j$ has been estimated, $\\abs{\\sin \\theta}$ is bounded from below due to $r<j$ and $\\CP_\\theta$ denotes the projection onto the relevant interval $[0,2\\pi)$ for $\\theta$ (which results in two intervals symmetric around $\\pi$ due to the way $\\Lambda^j_r$ is defined; we can treat both cases as one since only the absolute value of $\\sin\\theta$ is involved).\n\t\n\tThe integrals over $r'$ and $\\phi$ can again be interpreted as a $d-1$-dimensional integral, allowing us to continue the estimation due to $g\\normalr|_{h}\\in H^{t-\\frac 12}(\\bbR^{d-1})$, for the same reason as above (i.e.~the identification of the $H^t$-norm with a weighted $L^2$-norm in Fourier space),\n\t%\n\t\\begin{align}\n\t\t\\int_{P^j_r(\\vec e_1)} \\abs*{\\wh{g\\normalr|_{h}} (\\CP_{\\vec e_1} \\vec \\xi)}^2 \\d \\vec \\xi\n\t\t&\\lesssim \\int_{\\CP_\\theta P^j_r(\\vec e_1)} \\frac{2^j}{\\abs{\\sin \\theta}} \\int_{\\bbS^{d-2}}\\int_{2^{j-1}\\abs{\\sin\\theta}}^{2^{j+1}\\abs{\\sin\\theta}} \\abs*{\\wh{g\\normalr|_{h}}(r',\\phi)}^2 (r')^{d-2} \\frac{\\reg{r'}^{2t-1}}{\\reg{r'}^{2t-1}} \\d r' \\d \\phi \\d \\theta \\\\\n\t\t&\\lesssim \\int_{\\CP_\\theta P^j_r(\\vec e_1)} \\frac{2^j}{\\abs{\\sin \\theta}} \\frac{1}{\\reg{2^j\\sin\\theta}^{2t-1}} \\int_{\\bbR^{d-1}} \\abs*{\\wh{g\\normalr|_{h}}(\\vec \\xi{}')}^2 \\reg*{\\vec \\xi{}'}^{2t-1} \\d \\vec \\xi{}' \\d \\theta \\\\\n\t\t&\\lesssim \\int_{\\CP_\\theta P^j_r(\\vec e_1)} \\frac{2^j}{\\abs{\\sin \\theta}} \\frac{1}{\\reg{2^j\\sin\\theta}^{2t-1}} \\norm{g\\normalr|_{h}}_{H^{t-\\frac 12}(\\bbR^{d-1})}^2 \\d \\theta\\\\\n\t\t&\\le 2^{-j(2t-2)} \\int_{\\CP_\\theta P^j_r(\\vec e_1)} \\abs{\\sin\\theta}^{-2t} \\d \\theta \\,\\norm{g}_{H^{t}(\\bbR^{d})}^2. \\label{eq:branch_r1}\n\t\\end{align}\n\t%\n\tFor $1<r<j$, we continue by transforming with $y=\\sin\\theta$ (the functional determinant $(\\cos \\theta)^{-1}$ is bounded from above by a constant due to the restriction that $r<j$)\n\t%\n\t\\begin{align}\n\t\n\t\t\\int_{\\CP_\\theta P^j_r} \\abs{\\sin\\theta}^{-2t} \\d \\theta \\lesssim \\bracket*{-y^{-2t+1}}_{2^{-r}-2^{-j}}^{2^{-r+1}+2^{-j}} = 2^{(-2t+1)(-r-1)}-2^{(-2t+1)(-r+2)}\\lesssim 2^{r(2t-1)},\n\t\\end{align}\n\t%\n\tbecause $r>1$. For $r=1$, we have $\\abs{\\sin\\theta}\\gtrsim1$ and thus, the integral in \\eqref{eq:branch_r1} can be estimated by a constant, and is trivially less than (a constant independent of $j$ times) $2^{r(2t-1)}$.\n\t\n\tFinally, for $r=j$, the support $P^j_j$ is contained in a cylinder aligned with the $\\xi_1$-axis (cf. \\cite[Prop. A.6]{compress}), and therefore\n\t%\n\t\\begin{align}\n\t\t\\int_{P^j_j} \\abs*{\\wh{g\\normalr|_{h}} (\\CP_{\\vec e_1} \\vec \\xi)}^2 \\d \\vec \\xi \\le \\int_{[-2^{j+1},2^{j+1}]\\times B_{\\bbR^{d-1}}(0,8)} \\abs*{\\wh{g\\normalr|_{h}} (\\CP_{\\vec e_1} \\vec \\xi)}^2 \\d \\vec \\xi \\lesssim 2^j \\norm*{\\wh{g\\normalr|_{h}}}_{L^2(\\bbR^{d-1})}^2 \\le 2^j \\norm{g}_{H^{t}(\\bbR^{d})}^2.\n\t\\end{align}\n\t%\n\tReturning to \\eqref{eq:gamma_est_ind_hyp} and collecting all the factors, we see that the latter two terms (combined into $\\alpha^\\mathrm{s}_\\lambda$ resp.~$\\beta^\\mathrm{s}_\\lambda$) satisfy \\eqref{eq:gamma_est_sum_Lambda}, as claimed.\n\\end{proof}\n\n\\subsection{Localisation in Space}\\label{sec:loc_space}\n\nSimilarly to the localisation in angle (i.e.~$\\ell$), we require a localisation in space (i.e.~$\\vec k$) for a function cut off across a hyperplane, which we will establish in this section. Before we are able to tackle the proof, we need to investigate the spatial decay of the ridgelets (and a modified variant) in physical space.\n\nFinally, after having dealt with the singular functions in \\autoref{prop:loc_space}, we investigate how spatial decay of a general $L^2$-function transfers to the ridgelet coefficients in \\autoref{lem:ridge_coeff_tail_decay}.\n\n\\begin{lemma}\\label{lem:decay_ridgelet}\n\tFor arbitrary $n\\in\\bbN$ and an arbitrary rotation $R\\in \\mathrm{SO}(d)$, we have that,\n\t%\n\t\\begin{align}\n\t\t\\abs{\\varphi_\\lambda(\\vec x)} &\\lesssim \\frac{2^{\\frac j2}}{\\reg*{U_{j\\!\\?\\?,\\ell}^{-1}\\vec x-\\vec k}^{2m}}, \\label{eq:ridge_est} \\\\\n\t\t\\abs[\\Big]{\\CF^{-1}\\bracket[\\Big]{\\frac{\\xi_1}{\\abs*{\\vec \\xi}^2}\\hat\\varphi_\\lambda (R \\vec \\xi)}(\\vec x)} &\\lesssim \\frac{2^{-\\frac j2}}{\\reg*{U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k}^{2n}},\\label{eq:mod_ridge_est}\n\t\\end{align}\n\t%\n\twhere the implicit constants depends only on $n$ (and the choice of window function used to construct $\\hat\\psi$).\n\\end{lemma}\n\n\\begin{proof}\n\tWe start by proving \\eqref{eq:mod_ridge_est}, inserting the definition and transforming by $\\vec \\xi = R^{-1}U_{j\\!\\?\\?,\\ell}^{-\\top}\\vec\\eta$, which yields (because, for rotations, $R^{-1}=R^{\\top}$)\n\t%\n\t\\begin{align}\n\t\t\\CF^{-1}\\bracket[\\Big]{\\frac{\\xi_1}{\\abs*{\\vec \\xi}^2}\\hat\\varphi_\\lambda (R\\vec \\xi)}(\\vec x)\n\t\t&= 2^{-\\frac{j}{2}}\\int \\frac{\\xi_1}{\\abs*{\\vec \\xi}^2} \\hat \\psi_{j\\!\\?\\?,\\ell}(R\\vec \\xi) \\exp\\parens*{2\\pi\\mathrm{i} (\\vec \\xi\\cdot \\vec x-R\\vec \\xi \\cdot U_{j\\!\\?\\?,\\ell} \\vec k)} \\d \\vec \\xi\\\\\n\t\t&= 2^{-\\frac{j}{2}}\\int \\frac{\\xi_1}{\\abs*{\\vec \\xi}^2} \\hat \\psi_{j\\!\\?\\?,\\ell}(R\\vec \\xi) \\exp\\parens*{2\\pi\\mathrm{i} R\\vec \\xi  \\cdot U_{j\\!\\?\\?,\\ell}( U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x- \\vec k)} \\d \\vec \\xi\\\\\n\t\t&=2^{\\frac{j}{2}}\\int \\frac{\\parens*{R^{-1}U_{j\\!\\?\\?,\\ell}^{-\\top} \\vec \\eta}_1}{\\abs*{U_{j\\!\\?\\?,\\ell}^{-\\top} \\vec \\eta}^2} \\hat \\psi_{({j\\!\\?\\?,\\ell})}(\\vec \\eta) \\exp\\parens*{2\\pi\\mathrm{i}\\vec \\eta \\cdot (U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k)} \\d \\vec \\eta. \\label{eq:mod_ridge_est_trafo}\n\t\\end{align}\n\t%\n\tdue to the representation from \\autoref{assump:psi_smooth}. The idea now is to use integration by parts and\n\t%\n\t\\begin{align}\\label{eq:laplace_exp}\n\t\t\\Delta_{\\vec \\eta} \\exp\\parens*{2\\pi\\mathrm{i} \\vec \\eta \\cdot (U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k)} = - (2\\pi)^2 \\abs*{U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k}^2 \\exp\\parens*{2\\pi\\mathrm{i} \\vec \\eta \\cdot  (U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k)}.\n\t\\end{align}\n\t%\n\tto generate sufficiently high powers of $\\abs*{U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k}$ in the denominator.\n\t\n\tThe support $U_{j\\!\\?\\?,\\ell}^{\\top}P_{j\\!\\?\\?,\\ell} \\subseteq \\bracket*{\\frac 14,2}\\times B_{\\bbR^{d-1}}(0,4)$ is bounded from above and below (compare again \\cite[Prop. A.6]{compress}) and so we just have to bound the derivatives independently of $j$. Due to \\autoref{assump:psi_smooth}, this is not a problem for the $\\psi_{({j\\!\\?\\?,\\ell})}(\\vec \\eta)$. If we can show the same for the fraction in front, we will have the desired estimate due to the product rule.\n\t%\n\t\\begin{align}\n\t\td_{j\\!\\?\\?,\\ell}(\\vec \\eta):=\\frac{\\parens*{R^{-1}U_{j\\!\\?\\?,\\ell}^{-\\top} \\vec \\eta}_1}{\\abs*{U_{j\\!\\?\\?,\\ell}^{-\\top} \\vec \\eta}^2} =\\frac{\\parens*{R^{-1}R_\\jl^{-1} D_{2^{j}} \\vec \\eta}_1}{\\abs*{D_{2^{j}} \\vec \\eta}^2} = \\frac{2^jc_1\\eta_1+c_2\\eta_2+\\ldots +c_d\\eta_d}{2^{2j}\\eta_1^2+\\eta_2^2+\\ldots +\\eta_d^2}.\n\t\\end{align}\n\t%\n\tDue to the fact that the factor $2^j$ only appears in connection with $\\eta_1$, we only need to consider what happens when deriving by $\\eta_1$, as all other derivatives just produce higher powers of the denominator without adding powers of $2^j$ in the numerator. By splitting the term into $\\eta_1$ and the rest, respectively, and adding\n\t%\n\t\\begin{align}\n\t\t0=-\\mathrm{i} c_1\\sqrt{\\eta_2^2+\\ldots+\\eta_d^2}+\\mathrm{i} c_1\\sqrt{\\eta_2^2+\\ldots+\\eta_d^2}\n\t\\end{align}\n\t%\n\tin the numerator of the first term, we see that (since the quantity we're interested is clearly real),\n\t%\n\t\\begin{align}\n\t\t\\Dpn{k}{\\eta_1} d_{j\\!\\?\\?,\\ell} = \\Re\\parens[\\bigg]{\\Dpn{k}{\\eta_1}\\frac{c_1}{2^j\\eta_1+\\mathrm{i}\\sqrt{\\eta_2^2+\\ldots+\\eta_d^2}}} + \\Dpn{k}{\\eta_1}\\frac{c_2\\eta_2+\\ldots +c_d\\eta_d}{2^{2j}\\eta_1^2+\\eta_2^2+\\ldots +\\eta_d^2}.\n\t\\end{align}\n\t%\n\tThe first term resolves to\n\t%\n\t\\begin{align}\n\t\t\\Re\\parens[\\Bigg]{\\frac{c_1 (-1)^k k! 2^{jk}}{\\parens[\\Big]{2^j\\eta_1+\\mathrm{i}\\sqrt{\\eta_2^2+\\ldots+\\eta_d^2}}^{k+1}}},\n\t\\end{align}\n\t%\n\twhere, clearly, the power of $2^j$ is always at least one higher in the denominator than in the numerator. For the second term, we anticipate Fa\\`{a} di Bruno's formula \\eqref{app:eq:faa_di_bruno} --- the generalisation of the chain rule to arbitrary order of differentiation --- and the generating function machinery of \\autoref{ssec:genfunc} to deal with the Bell polynomials that are involved, i.e.~\\eqref{app:eq:genfunc_bell}. Then, with ``inner'' function $g(x):=2^{2j}x^2$,\n\t%\n\t\\begin{align}\n\t\tB_{n,k}(2^{2j+1}x,2^{2j+1},0,\\ldots)= 2^{2jk}\\frac{n!}{k!} \\parens[\\Big]{\\Dn{n}{t} t^k(2x+t)^k}\\Bigr|_{t=0}, \\quad \\text{for} \\quad \\ceil{\\frac n2}\\le k\\le n,\n\t\\end{align}\n\t%\n\tand thus\n\t%\n\t\\begin{align}\n\t\t\\Dpn{k}{\\eta_1} \\frac{c_2\\eta_2+\\ldots +c_d\\eta_d}{2^{2j}\\eta_1^2+\\eta_2^2+\\ldots +\\eta_d^2} = \\sum_{k=\\ceil{\\frac n2}}^n \\frac{2^{2jk} p_k(\\eta_1) (c_2\\eta_2+\\ldots +c_d\\eta_d)}{(2^{2j}\\eta_1^2+\\eta_2^2+\\ldots +\\eta_d^2)^{k+1}},\n\t\\end{align}\n\t%\n\twhere $p_k(\\eta_1)=(-1)^k n!\\parens{\\Dn{n}{t} t^k(2\\eta_1+t)^k}\\bigr|_{t=0}$ is a polynomial in $\\eta_1$ independent of $2^j$. In particular, in terms of powers of $2^j$, the power in the denominator is always at least two larger than in the numerator.\n\t\n\tTaken together, we can see that because  $\\abs{\\vec \\eta}\\sim 1$ for $\\vec \\eta \\in U_{j\\!\\?\\?,\\ell}^{\\top}P_{j\\!\\?\\?,\\ell}$, for any multi-index $\\alpha$ with $\\abs{\\alpha}\\ge 0$,\n\t%\n\t\\begin{align}\\label{eq:mod_ridge_djl}\n\t\t\\Dpi{\\alpha}[d_{j\\!\\?\\?,\\ell}]{\\vec\\eta}(\\vec \\eta) \\lesssim 2^{-j}.\n\t\\end{align}\n\t\n\tThis allows us to apply \\eqref{eq:laplace_exp} to \\eqref{eq:mod_ridge_est_trafo} --- boundary terms vanish because of compact support. Consequently, using \\cite[Cor. C.3]{compress}, we achieve\n\t%\n\t\\begin{align}\n\t\\MoveEqLeft\n\t\t\\abs[\\Big]{\\CF^{-1}\\bracket[\\Big]{\\frac{\\xi_1}{\\abs*{\\vec \\xi}^2}\\hat\\varphi_\\lambda (R\\vec \\xi)}(\\vec x)}\\\\\n\t\t&= \\abs[\\bigg]{2^{\\frac j2} \\frac{(-4\\pi^2)^n}{\\abs*{U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k}^{2n}} \\int_{U_{j\\!\\?\\?,\\ell}^{\\top}P_{j\\!\\?\\?,\\ell}} \\Delta^n\\parens{d_{j\\!\\?\\?,\\ell}(\\vec\\eta) \\hat \\psi_{({j\\!\\?\\?,\\ell})}(\\vec \\eta)} \\exp\\parens*{2\\pi\\mathrm{i}\\vec \\eta \\cdot (U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k)} \\d \\vec \\eta}\\\\\n\t\t&\\lesssim \\frac{2^{\\frac j2}}{\\abs*{U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k}^{2n}} \\int_{U_{j\\!\\?\\?,\\ell}^{\\top}P_{j\\!\\?\\?,\\ell}} \\abs*{d_{j\\!\\?\\?,\\ell}(\\vec \\eta)}_{\\CC^{2n}} \\, \\abs*{\\hat \\psi_{({j\\!\\?\\?,\\ell})}(\\vec \\eta)}_{\\CC^{2n}} \\d \\vec \\eta \\lesssim  \\frac{2^{-\\frac j2}}{\\abs*{U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k}^{2n}}.\n\t\\end{align}\n\t%\n\tFinally, this implies\n\t%\n\t\\begin{align}\n\t\t\\abs[\\Big]{\\CF^{-1}\\bracket[\\Big]{\\frac{\\xi_1}{\\abs*{\\vec \\xi}^2}\\hat\\varphi_\\lambda (R\\vec \\xi)}(\\vec x)}\n\t\t&\\lesssim  \\frac{2^{-\\frac j2}}{\\reg*{U_{j\\!\\?\\?,\\ell}^{-1}R\\vec x-\\vec k}^{2n}},\n\t\\end{align}\n\t%\n\tbecause the compact support of $\\hat\\varphi_\\lambda$ implies that the function on the left-hand side belongs to $\\CC^\\infty(\\bbR^d)$ --- i.e.~has no singularities --- with respect to $\\vec x$ (and is bounded independently of $\\vec k$) and therefore, we may replace the absolute value with its regularised version (up to a constant).\n\t\n\tThe proof of \\eqref{eq:ridge_est} proceeds along the same lines, with the simplification that we do not have to consider $d_{j\\!\\?\\?,\\ell}$ or its derivatives --- which also removes the factor $2^{-j}$ that we had gained from \\eqref{eq:mod_ridge_djl}. This finishes the proof.\n\\end{proof}\n\nWith these tools in hand, we proceed to the promised localisation in space for a function cut off across a hyper plane.\n\n\\begin{proposition}\\label{prop:loc_space}\n\tLet $f(\\vec x)=H(\\vec x \\cdot \\vec n-v)g(\\vec x)$ with $g\\in H^{\\frac 12}(\\bbR^d)$, cut off across the hyperplane $h=\\set{\\vec x}{\\vec x\\cdot \\vec n=v}$, such that the restriction $g\\normalr|_{h}\\in L^2(\\bbR^{d-1})$ satisfies\n\t%\n\t\\begin{align}\n\t\t\\abs*{g(\\vec x)\\delta_{\\{\\vec x\\cdot \\vec n=v\\}}}=\\abs*{g\\normalr|_{h}(\\vec x{}'_h)} &\\lesssim \\reg{\\vec x{}'_h}^{-2m} = \\reg{\\CP_{\\vec n}\\vec x}^{-2m} \\quad \\text{almost everywhere}, \\label{eq:decay_sing_f}\n\t\t\\intertext{\n\t%\n\twhere $\\vec x{}'_h\\in\\bbR^{d-1}$ is identified with $\\CP_{\\vec n}\\vec x=\\vec x - (\\vec x\\cdot \\vec n)\\vec n$.\n\tFor the solution $u=\\CS[f]$, the mutilated part (cf. \\autoref{prop:sol_smooth_except_hyp}) is of the same form, i.e.~$u(\\vec x)=u_0(\\vec x)+H(\\vec x \\cdot \\vec n-v)u_1(\\vec x)$, and we require\n\t%\n\t}\n\t\t\\abs*{u_1(\\vec x)\\delta_{\\{\\vec x\\cdot \\vec n=v\\}}}=\\abs*{u_1\\normalr|_{h}(\\vec x{}'_h)} &\\lesssim \\reg{\\vec x{}'_h}^{-2m} = \\reg{\\CP_{\\vec n}\\vec x}^{-2m}\\quad \\text{almost everywhere}.\\label{eq:decay_sing_u}\n\t\\end{align}\n\t%\n\tCondition \\eqref{eq:decay_sing_f} does not necessarily imply \\eqref{eq:decay_sing_u}, but the latter follows for example from the stronger assumption that $\\abs{g(\\vec x)} \\lesssim \\reg{\\vec x}^{-2m}$, compare \\autoref{lem:decay_sol}.\n\t\n\tIf the ridgelets are constructed with a window function such that \\eqref{eq:mod_ridge_est} holds for $m$ sufficiently large, we have the following \\emph{localisation in space} in terms of a modified translation parameter $\\vec t(\\vec k)$,\n\t%\n\t\\begin{align}\\label{eq:gamma_est_w}\n\t\t\\abs*{\\alpha^\\mathrm{s}_\\lambda}&\\lesssim \\frac{1}{a} \\frac{2^{-\\frac j2}}{(\\abs[]{\\vec t{}''} + \\rho_1^2+\\rho_2^2+1)^m}, &\n\t\t\\abs*{w_\\lambda\\beta^\\mathrm{s}_\\lambda}&\\lesssim \\frac{1}{a} \\frac{2^{\\frac j2}}{(\\abs[]{\\vec t{}''} + \\rho_1^2+\\rho_2^2+1)^m},\n\t\t\\intertext{\n\t%\n\twhere, with $\\theta_{\\vec n}:=\\arccos(\\vec n\\cdot \\vec e_1)$ and $\\phi:=\\theta_{j\\!\\?\\?,\\ell}-\\theta_{\\vec n}$,\n\t%\n\t\t}\n\t\t\\vec t(\\vec k)&=\\vec k-vU_{j\\!\\?\\?,\\ell}^{-1}\\vec n, & \\rho_1(\\vec t)&:= \\lcopywidth{\\frac 1a}{\\frac{1}{a^2}} (\\cos \\phi\\, t_1-2^j\\sin\\phi\\, t_2),\\\\\n\t\ta&:=\\sqrt{2^{2j} \\sin^2 \\phi + \\cos^2\\phi}, &\n\t\t\\rho_2(\\vec t)&:= \\frac{1}{a^2}\\parens*{2^j\\sin\\phi\\, t_1 + \\cos\\phi\\, t_2}.\n\t\\end{align}\n\t%\n\tThe essence \\eqref{eq:gamma_est_w} is, in slightly obscured form, that the magnitude of the coefficients of the modified translation parameter $\\vec t$ appear unchanged --- for $d-1$ dimensions --- in the decay of the ridgelet coefficients, while in one dimension (basically, along $t_1$), the magnitude is attenuated by a factor $\\sim 2^{-j}$, as soon as $\\sin\\phi$ is not negligible. This behaviour corresponds directly to the way the grid of translations in \\autoref{def:phi_jlk} is refined in one direction (resp.~the shape of the ridgelets themselves).\n\\end{proposition}\n\n\\begin{proof}\n\tSince $g$ and $u_1$ satisfy exactly the same condition (i.e.~\\eqref{eq:decay_sing_f} and \\eqref{eq:decay_sing_u}), we can deal with both cases at the same time (here exemplarily for $\\alpha^\\mathrm{s}_\\lambda$).\n\tTransforming with $\\vec\\zeta=\\Rn\\vec \\xi$, using \\eqref{eq:sing_fourier_rot} and applying the Plancherel identity, this results in (because $\\zeta_1$ is real)\n\t%\n\t\\begin{align}\n\t\t\\alpha^\\mathrm{s}_\\lambda\n\t\t&=\\frac{-\\mathrm{i}}{2\\pi}\\int \\overline{\\hat \\varphi_\\lambda (\\vec \\xi)} \\frac{(\\Rn\\vec \\xi)_1}{\\abs*{\\vec\\xi}^2}  \\wh{g\\normalr|_{h}}(\\CP_{\\vec n}\\vec \\xi) \\d \\vec \\xi\n\t\n\t\n\t\t=\\frac{ -\\mathrm{i}}{2\\pi}\\int \\overline{\\hat \\varphi_\\lambda (\\Rn^{-1}\\vec \\zeta) \\frac{\\zeta_1}{\\abs*{\\vec \\zeta}^2}} \\CF\\bracket*{g(\\Rn^{-1}\\vec x)\\delta_{\\{\\vec x\\cdot \\vec e_1=v\\}}}(\\vec \\zeta) \\d \\vec \\zeta \\\\\n\t\n\t\n\t\n\t\t&= \\frac{-\\mathrm{i}}{2\\pi} \\int \\overline{\\CF^{-1}\\bracket[\\Big]{\\frac{\\xi_1}{\\abs*{\\vec \\xi}^2}\\hat\\varphi_\\lambda (\\Rn^{-1}\\vec \\xi)}\\!(\\vec x)} \\, g(\\Rn^{-1}\\vec x)\\delta_{\\{x_1=v\\}} \\d \\vec x\\\\\n\t\t&= \\frac{-\\mathrm{i}}{2\\pi} \\int_{[\\vec e_1]^\\bot} \\overline{\\CF^{-1}\\bracket[\\Big]{\\frac{\\xi_1}{\\abs*{\\vec \\xi}^2}\\hat\\varphi_\\lambda (\\Rn^{-1}\\vec \\xi)}\\!\\binom{v}{\\vec x{}'}} \\, g\\parens{\\Rn^{-1}\\binom{v}{\\vec x{}'}} \\d \\vec x{}',\n\t\\end{align}\n\t%\n\twhere $[\\vec e_1]$ denotes the span of $\\vec e_1$, and $[\\vec e_1]^\\bot$ its orthogonal complement. Furthermore, it is easy to check that $\\Rn^{-1}\\binom{v}{\\vec x{}'}= \\Rn^{-1}\\binom{0}{\\vec x{}'}+v\\vec n\\in h$, as well as $\\CP_{\\vec n} \\Rn^{-1}\\binom{v}{\\vec x{}'} = \\Rn^{-1}\\binom{0}{\\vec x{}'}$. Therefore, using \\eqref{eq:mod_ridge_est} and \\eqref{eq:decay_sing_f},\n\t%\n\t\\begin{align}\n\t\t\\abs{\\alpha^\\mathrm{s}_\\lambda}\n\t\t&\\lesssim 2^{-\\frac j2}\\int_{[\\vec e_1]^\\bot} \\frac{1}{\\reg*{U_{j\\!\\?\\?,\\ell}^{-1}\\Rn^{-1}\\binom{v}{\\vec x{}'}-\\vec k}^{2n}} \\frac{1}{\\reg*{\\Rn^{-1} \\binom{0}{\\vec x{}'}}^{2m}} \\d \\vec x{}'\\\\\n\t\t&= 2^{-\\frac j2}\\int_{[\\vec e_1]^\\bot} \\frac{1}{\\reg*{U_{j\\!\\?\\?,\\ell}^{-1}\\Rn^{-1}\\binom{0}{\\vec x{}'}-\\vec t}^{2n}} \\frac{1}{\\reg*{\\binom{0}{\\vec x{}'}}^{2m}} \\d \\vec x{}',\n\t\\end{align}\n\t%\n\twhere we have split off the component involving $v$ in the first denominator, i.e.\n\t%\n\t\\begin{align}\n\t\tU_{j\\!\\?\\?,\\ell}^{-1}\\binom{v}{\\vec x{}'}-\\vec k=U_{j\\!\\?\\?,\\ell}^{-1}\\binom{0}{\\vec x{}'}-\\vec t, \\quad \\text{using} \\quad \\vec t:=\\vec k-vU_{j\\!\\?\\?,\\ell}^{-1}\\Rn^{-1}\\vec e_1=\\vec k-vU_{j\\!\\?\\?,\\ell}^{-1}\\vec n.\n\t\\end{align}\n\t%\n\tSince $U_{j\\!\\?\\?,\\ell}^{-1}\\Rn^{-1}[\\vec e_1]^\\bot$ is still a hyperplane, the first denominator attains its minimum when the left term corresponds to the orthogonal projection of $\\vec t$ onto that plane. The normal vector transforms with the transpose of the inverse of the transformation, i.e.\n\t%\n\t\\begin{align}\n\t\tU_{j\\!\\?\\?,\\ell}^{-1}\\Rn^{-1}[\\vec e_1]^\\bot \\bot (U_{j\\!\\?\\?,\\ell}^{-1}\\Rn^{-1})^{-\\top}\\vec e_1 = U_{j\\!\\?\\?,\\ell}^\\top \\vec n,\n\t\t\\quad \\text{and we set} \\quad\n\t\t\\vec s{}^*:= \\frac{U_{j\\!\\?\\?,\\ell}^\\top \\vec n}{\\abs*{U_{j\\!\\?\\?,\\ell}^\\top \\vec n}}.\n\t\\end{align}\n\t%\n\tThus, the minimiser of the first denominator is  $\\tau(\\vec t):= \\Rn U_{j\\!\\?\\?,\\ell}(\\vec t-\\vec s{}^*(\\vec s{}^*\\cdot \\vec t)) \\in [\\vec e_1]^\\bot$. By the transformation $\\binom{0}{\\vec y{}'}=\\binom{0}{\\vec x{}'}+\\tau(\\vec t)$, we have\n\t%\n\t\\begin{align}\n\t\t\\abs{\\alpha^\\mathrm{s}_\\lambda}&\\lesssim 2^{-\\frac j2}\\int_{[\\vec e_1]^\\bot} \\frac{1}{\\reg*{U_{j\\!\\?\\?,\\ell}^{-1}\\Rn^{-1}\\binom{0}{\\vec y{}'} - \\vec s{}^* (\\vec s{}^*\\cdot \\vec t)}^{2n}} \\frac{1}{\\reg*{\\binom{0}{\\vec y{}'} - \\tau(\\vec t)}^{2m}} \\d \\vec y{}'\\\\\n\t\t&= 2^{-\\frac j2}\\int_{[\\vec e_1]^\\bot} \\frac{1}{\\parens*{\\abs*{U_{j\\!\\?\\?,\\ell}^{-1}\\Rn^{-1}\\binom{0}{\\vec y{}'}}^2 + (\\vec s{}^*\\cdot \\vec t)^2+1}^{n}} \\frac{1}{\\reg*{\\binom{0}{\\vec y{}'} - \\tau(\\vec t)}^{2m}} \\d \\vec y{}', \\label{eq:gamma_est_y_trafo}\n\t\\end{align}\n\t%\n\twhere the two summands in the first denominator are now orthogonal to each other, and the last equality follow by Pythagoras' theorem.\n\t\n\tWe will now try to construct a rotation, which will transform the terms in the integral in a way that it can be dealt with. For reasons of clarity, this will be a two-stop process, first constructing the main rotation $Q$, and then a small correction $T$. The condition on the rotation (that $R_\\jl \\vec s_\\jl=\\vec e_1$) fixes the first component of $R_\\jl\\vec e_1$ to be $\\cos \\theta_{j\\!\\?\\?,\\ell}$. Otherwise, we have the freedom to choose $R_\\jl$ such that\n\t%\n\t\\begin{align}\n\t\tR_\\jl \\vec e_1=\\begin{pmatrix}\n\t\t\\phantom{-}\\cos \\theta_{j\\!\\?\\?,\\ell} \\\\ -\\sin\\theta_{j\\!\\?\\?,\\ell}\\\\0\\\\ \\vdots\n\t\t\\end{pmatrix},\n\t\\qquad \\text{and in the same way,} \\qquad\n\t\t\\Rn \\vec e_1=\\begin{pmatrix}\n\t\t\\phantom{-}\\cos \\theta_{\\vec n} \\\\ -\\sin\\theta_{\\vec n}\\\\0\\\\ \\vdots\n\t\t\\end{pmatrix}.\n\t\\end{align}\n\t%\n\tWe may choose further images for our rotations, as long as we maintain the property that $R\\vec s\\cdot R\\vec s{}'=\\vec s\\cdot\\vec s{}'$ for any pair of images $R\\vec s,R\\vec s{}'$ we choose for vectors $\\vec s, \\vec s{}'\\in{\\mathbb{S}^{d-1}}$, because rotations maintain the intrinsic (geodesic) distance between points on the sphere. Defining $\\vec s_\\jl':=P_{\\vec e_1}\\vec s_\\jl$ (with norm $\\abs*{\\sin\\theta_{j\\!\\?\\?,\\ell}}$), we may therefore choose\n\t%\n\t\\begin{align}\n\t\t\\Rn \\frac{\\vec s_\\jl'}{\\sin\\theta_{j\\!\\?\\?,\\ell}}=\\begin{pmatrix}\n\t\t\\sin \\theta_{\\vec n} \\\\ \\cos\\theta_{\\vec n}\\\\0\\\\ \\vdots\n\t\t\\end{pmatrix},\n\t\\end{align}\n\t%\n\tsince, by our choice for $\\Rn \\vec e_1$ above, $\\Rn \\vec e_1\\cdot \\Rn \\frac{\\vec s_\\jl'}{\\sin\\theta_{j\\!\\?\\?,\\ell}} = 0 =  \\vec e_1 \\cdot \\frac{\\vec s_\\jl'}{\\sin\\theta_{j\\!\\?\\?,\\ell}}$. In case that $\\vec s_\\jl=\\vec e_1$ (and thus $\\sin\\theta_{j\\!\\?\\?,\\ell}=0$), we take $\\Rn e_2= (\\sin \\theta_{\\vec n}, \\cos\\theta_{\\vec n}, 0,\\ldots)^\\top$, which is a permissible choice for the same reason.\n\t\n\tOnce more in the same fashion, we are able to determine a rotation $Q$ in the following way\n\t%\n\t\\begin{align}\n\tQ^{-1} \\vec e_1=\\Rn \\vec e_1 \\qquad \\text{and} \\qquad Q^{-1}\\vec e_2 = \\Rn\\frac{\\vec s_\\jl'}{\\sin\\theta_{j\\!\\?\\?,\\ell}},\n\t\\end{align}\n\t%\n\twhereby we achieve\n\t%\n\t\\begin{align}\n\t\tQ^{-1} &= \\left(\\begin{array}{cc|c}\n\t\t\t\\phantom{-}\\cos \\theta_{\\vec n} & \\sin \\theta_{\\vec n} & 0 \\\\\n\t\t\t-\\sin\\theta_{\\vec n} & \\cos\\theta_{\\vec n} & 0 \\\\ \\hline\n\t\t\t0 & 0 &  (Q'')^{-1}\n\t\t\\end{array}\\right),\\\\\n\t\tR_\\jl \\Rn^{-1}Q^{-1} &= \\left(\\begin{array}{cc|c}\n\t\t\t\\phantom{-}\\cos \\theta_{j\\!\\?\\?,\\ell} & \\sin \\theta_{j\\!\\?\\?,\\ell} & 0 \\\\\n\t\t\t-\\sin\\theta_{j\\!\\?\\?,\\ell} & \\cos\\theta_{j\\!\\?\\?,\\ell} & 0  \\\\ \\hline\n\t\t\t0 & 0 &  (V'')^{-1}\n\t\t\\end{array}\\right). \\label{eq:RjlQ}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\end{align}\n\t%\n\tThe form of the second matrix can be seen since, due to the choice of $Q^{-1}\\vec e_1$, the first column is just $R_\\jl \\vec e_1$, while the second column is calculated as follows. The first entry evaluates to\n\t%\n\t\\begin{align}\n\t\t\\vec e_1 \\cdot R_\\jl \\Rn^{-1} Q^{-1} \\vec e_2\n\t\t&= R_\\jl^{-1} \\vec e_1 \\cdot \\Rn^{-1} \\Rn \\frac{\\vec s_\\jl'}{\\sin\\theta_{j\\!\\?\\?,\\ell}} = \\vec s_\\jl \\cdot \\frac{\\CP_{\\vec e_1} \\vec s_\\jl}{\\sin\\theta_{j\\!\\?\\?,\\ell}}  = \\frac{\\sum_{k=2}^d(\\vec s_\\jl)_k^2}{\\sin\\theta_{j\\!\\?\\?,\\ell}} = \\frac{1-(\\vec s_\\jl)_1^2}{\\sin\\theta_{j\\!\\?\\?,\\ell}} \\\\\n\t\t&=\\frac{1-\\cos^2 \\theta_{j\\!\\?\\?,\\ell}}{\\sin\\theta_{j\\!\\?\\?,\\ell}} = \\sin \\theta_{j\\!\\?\\?,\\ell},\n\t\\end{align}\n\t%\n\tsince $(\\vec s_\\jl)_1=\\vec s_\\jl \\cdot \\vec e_1= \\cos \\theta_{j\\!\\?\\?,\\ell}$. As the norm of the first row is thus one already, the other entries must be zero. Similarly,\n\t%\n\t\\begin{align}\n\t\t\\vec e_2 \\cdot R_\\jl \\Rn^{-1}Q^{-1} \\vec e_2 = \\vec e_2 \\cdot  R_\\jl \\parens{\\!\\vec s_\\jl - \\begin{pmatrix}\n\t\t(\\vec s_\\jl)_1\\\\0\\\\ \\vdots\n\t\t\\end{pmatrix}\\!} \\frac{1}{\\sin\\theta_{j\\!\\?\\?,\\ell}} = \\frac{\\vec e_2}{\\sin\\theta_{j\\!\\?\\?,\\ell}} \\!\\cdot\\! (\\vec e_1 - \\cos\\theta_{j\\!\\?\\?,\\ell} R_\\jl \\vec e_1) = \\cos\\theta_{j\\!\\?\\?,\\ell}\n\t\\end{align}\n\t%\n\timplies the zeros in the second row and column. The matrices $Q''$ resp.~$V''$ (restricted to have determinant one) represent the degrees of freedom we still have in choosing the rotation $Q$.\n\t\n\tFinally, we define the two matrices\n\t%\n\t\\begin{align}\n\t\tS_{\\theta} = \\begin{pmatrix}\n\t\t\t\\cos \\theta &- \\sin\\theta \\\\ \\sin\\theta & \\phantom{-}\\cos \\theta\n\t\t\\end{pmatrix}, \\qquad T:= \\begin{pmatrix}\n\t\t\tS_{\\theta_{\\vec n}} & 0 \\\\ 0 & \\bbI_{d-2}\n\t\t\\end{pmatrix},\n\t\\end{align}\n\t%\n\twhere $S_{\\theta}$ is the standard rotation matrix in two dimensions satisfying $S_{\\theta}S_{\\phi}=S_{\\theta+\\phi}$. Inverting \\eqref{eq:RjlQ}, we see that\n\t%\n\t\\begin{align}\n\t\tQ&= \\begin{pmatrix}\n\t\t\tS_{\\theta_{\\vec n}} & 0 \\\\ 0 & Q''\n\t\t\\end{pmatrix},\n\t\t& Q\\RnR_\\jl^{-1} &= \\begin{pmatrix}\n\t\t\tS_{\\theta_{j\\!\\?\\?,\\ell}} & 0 \\\\ 0 & V''\n\t\t\\end{pmatrix},\n\t\t& T^{-1}Q &= \\begin{pmatrix}\n\t\t\t\\bbI_{2} & 0 \\\\ 0 & Q''\n\t\t\\end{pmatrix},\n\t\\end{align}\n\t%\n\tand consequently, setting $\\phi:=\\theta_{j\\!\\?\\?,\\ell}-\\theta_{\\vec n}$,\n\t%\n\t\\begin{align}\\label{eq:RjlQ_inv}\n\t\tT^{-1}Q\\RnR_\\jl^{-1} &= \\begin{pmatrix}\n\t\t\tS_{\\phi} & 0 \\\\ 0 & V''\n\t\t\\end{pmatrix},\n\t\t&R_\\jl \\Rn^{-1} &=R_\\jl \\Rn^{-1}Q^{-1}Q\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t=\\begin{pmatrix}\n\t\t\tS_{-\\phi} & 0 \\\\ 0 & (V'')^{-1}Q''\n\t\t\\end{pmatrix}.\n\t\\end{align}\n\t\n\tComing back to \\eqref{eq:gamma_est_y_trafo}, for reasons that will become apparent shortly, we transform with $\\binom{0}{\\vec z{}'} = T^{-1}Q\\binom{0}{\\vec y{}'}$ (since $T^{-1}Q$ leaves the first component invariant!) to arrive at\n\t%\n\t\\begin{align}\n\t\t\\abs{\\alpha^\\mathrm{s}_\\lambda}\n\t\t&\\!\\begin{multlined}[t][\\textwidth-\\mathindent-\\widthof{$\\abs{\\alpha^\\mathrm{s}_\\lambda}$}-\\multlinegap]\n\t\t\t\\lesssim 2^{-\\frac j2}\\int_{[\\vec e_1]^\\bot} \\! \\parens{\\abs*{D_{2^j}R_\\jl \\Rn^{-1}Q^{-1}T\\vec z{}'}^2 + (\\vec s{}^*\\cdot \\vec t)^2+1}^{-n} \\cdot \\ldots \\\\\n\t\t\t\\ldots \\cdot \\reg*{Q^{-1}T\\parens*{(0,\\vec z{}')^\\top - T^{-1}Q\\tau(\\vec t)}}^{-2m} \\d \\vec z{}'\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\textwidth-\\mathindent-\\widthof{$\\abs{\\alpha^\\mathrm{s}_\\lambda}$}-\\multlinegap]\n\t\t\t=2^{-\\frac j2}\\int_{[\\vec e_1]^\\bot} \\! \\parens{\\abs*{\\parens*{-2^j \\sin \\phi\\, z_2, \\cos \\phi\\, z_2, (V'')^{-1}\\vec z{}''}^\\top}^2 + (\\vec s{}^*\\cdot \\vec t)^2+1}^{-n} \\cdot \\ldots \\label{eq:gamma_est_z_trafo} \\\\\n\t\t\t\\ldots \\cdot \\reg*{(0,\\vec z{}')^\\top - T^{-1}Q\\tau(\\vec t)}^{-2m} \\d \\vec z{}',\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\twhere we used the (inverse of the) first equality from \\eqref{eq:RjlQ_inv} to evaluate the first denominator. The next step is to calculate $T^{-1}Q\\tau(\\vec t)$, which we tackle as follows, this time with the second equality in \\eqref{eq:RjlQ_inv},\n\t%\n\t\\begin{align}\n\t\tU_{j\\!\\?\\?,\\ell}^\\top \\vec n &= D_{2^{-j}}R_\\jl\\Rn^{-1}\\vec e_1 = \\parens*{2^{-j}\\cos \\phi,-\\sin \\phi,0,\\ldots}^\\top,\\\\\n\t\n\t\t\\abs*{U_{j\\!\\?\\?,\\ell}^\\top \\vec n}&=2^{-j}\\sqrt{2^{2j} \\sin^2\\phi +\\cos\\phi}=:2^{-j} a.\n\t\\end{align}\n\t%\n\tFrom there on, we compute\n\t%\n\t\\begin{align}\n\t\t\\vec s{}^* = \\frac{1}{a} \\parens*{\\cos \\phi,-2^j\\sin \\phi,0,\\ldots}^\\top, \\qquad \\vec s{}^* \\cdot \\vec t =\\frac{1}{a}(\\cos\\phi \\,t_1 - 2^j\\sin\\phi \\,t_2) = \\rho_1(\\vec t),\n\t\\end{align}\n\t%\n\tand therefore, again with \\eqref{eq:RjlQ_inv},\n\t%\n\t\\begin{align}\n\t\tT^{-1}Q\\tau(\\vec t)\n\t\t&=T^{-1}Q\\Rn R_\\jl^{-1}D_{2^{-j}}\\parens*{\\vec t - (\\vec s{}^* \\cdot \\vec t)\\vec s{}^*}\\\\\n\t\t&=T^{-1}Q\\Rn R_\\jl^{-1} \\parens{\n\t\t\\begin{pmatrix}\n\t\t\t2^{-j}t_1\\\\t_2\\\\ \\vec t{}''\n\t\t\\end{pmatrix}\n\t\t-\\frac{\\rho_1(\\vec t)}{a}\n\t\t\\begin{pmatrix}\n\t\t\t2^{-j}\\cos\\phi\\\\-2^j\\sin\\phi\\\\ 0\n\t\t\\end{pmatrix}} \\\\\n\t\t&= \\begin{pmatrix}\n\t\t\t2^{-j}\\cos \\phi \\, t_1 - \\sin \\phi \\, t_2\\\\2^{-j}\\sin\\phi \\, t_1 + \\cos\\phi \\, t_2\\\\ V''\\vec t{}''\n\t\t\\end{pmatrix}\n\t\t-\\frac{\\rho_1(\\vec t)}{a}\n\t\t\\begin{pmatrix}\n\t\t\t2^{-j}\\cos^2\\phi+2^j\\sin^2\\phi\\\\2^{-j}\\sin\\phi\\cos\\phi -2^j\\sin\\phi\\cos\\phi\\\\ 0\n\t\t\\end{pmatrix} \\\\\n\t\t&= \\begin{pmatrix}\n\t\t\t0 \\\\ \\frac{1}{a^2}\\parens*{2^{j} \\sin \\phi \\, t_1 + \\cos \\phi \\, t_2} \\\\ V''\\vec t{}''\n\t\t\\end{pmatrix}\n\t\t= \\begin{pmatrix}\n\t\t\t0 \\\\ \\rho_2(\\vec t) \\\\ V''\\vec t{}''\n\t\t\\end{pmatrix},\n\t\\end{align}\n\t%\n\twhere the first two components simplify substantially, after inserting $\\rho_1(\\vec t)$, and expanding the first term with $\\frac{a^2}{a^2}$. We also note that $\\rho_1^2+a^2\\rho_2^2=t_1^2+t_2^2$.\n\t\n\tContinuing from \\eqref{eq:gamma_est_z_trafo}, we can now bring this in the following form (for convenience, we will not further denote the dependence of $\\rho_1,\\,\\rho_2$ on $\\vec t$)\n\t%\n\t\\begin{align}\n\t\t\\abs{\\alpha^\\mathrm{s}_\\lambda}\n\t\t&\\lesssim 2^{-\\frac j2} \\int_{-\\infty}^{\\infty} \\int_{[\\vec e_1,\\vec e_2]^\\bot} \\parens*{|\\vec z{}''|^2 + {\\underbrace{a^2z_2^2+\\rho_1^2+1}_{=:p^2}}}^{-n} \\parens*{\\abs[]{\\vec z{}''-V''\\vec t{}''}^2+{\\underbrace{(z_2 - \\rho_2)^2+1}_{=:q^2}}}^{-m}  \\d \\vec z{}'' \\d z_2.\n\t\\end{align}\n\t%\n\tUsing \\autoref{cor:Imn_higher_dim} for $\\vec z{}''$ (basically applying \\autoref{th:Imn_est} in direction $V''\\vec t{}''$, transforming to polar coordinates and then once more \\autoref{th:Imn_est} with respect to the radius), this yields\n\t%\n\t\\begin{align}\n\t\t\\abs{\\alpha^\\mathrm{s}_\\lambda}\n\t\t&\\lesssim 2^{-\\frac j2} \\int_{-\\infty}^{\\infty} p^{-2n} \\parens{\\abs[]{\\vec t{}''} + p^2+q^2}^{-m} \\d z_2\\\\\n\t\n\t\t&\\le 2^{-\\frac j2} \\int_{-\\infty}^{\\infty} \\parens{a^2z_2^2+ \\rho_1^2+1}^{-m} \\parens{(z_2 - \\rho_2)^2+\\abs[]{\\vec t{}''} + \\rho_1^2+1}^{-m}  \\d z_2\n\t\t\\intertext{\n\t%\n\tand again with the help of \\eqref{eq:Imn_est},\n\t%\n\t\t}\n\t\t&\\lesssim  \\frac{2^{-\\frac j2}}{(a^2\\rho_2^2+a^2(\\abs[]{\\vec t{}''} + \\rho_1^2+1)+\\rho_1^2+1)^m} \\parens[\\bigg]{\\frac{a^{2m-1}}{(\\rho_1^2+1)^{\\frac{2m-1}2}} + \\frac{1}{(\\abs[]{\\vec t{}''} + \\rho_1^2+1)^{\\frac{2m-1}2}}}\\\\\n\t\t&\\le \\frac{2^{-\\frac j2}}{(\\abs[]{\\vec t{}''} + \\rho_1^2+\\rho_2^2+1)^m} \\parens{\\frac{1}{a} + \\frac{1}{a^{2m}}} \\le \\frac{1}{a} \\frac{2^{-\\frac j2}}{(\\abs[]{\\vec t{}''} + \\rho_1^2+\\rho_2^2+1)^m},\n\t\\end{align}\n\t%\n\tsince $1\\le a\\le 2^{j}$, which is what we wanted to show for $\\alpha^\\mathrm{s}_\\lambda$.\n\t\n\tRepeating the whole procedure for $u_1$ to estimate $w_\\lambda\\beta_\\lambda^\\mathrm{s}$ increases the right-hand side by $2^j$ at worst, due to the weight (it is not apparent if or how the additional smoothness in the direction of $\\vec s$ can be utilised, since we are only integrating over a hyperplane), and the proof is complete.\n\\end{proof}\n\nAs the last result in this section, we consider how general decay of the function being tested against the ridgelet frame transfers to the coefficients.\n\n\\begin{lemma}\\label{lem:ridge_coeff_tail_decay}\n\tFor a function $f\\in L^2$ satisfying the decay condition\n\t%\n\t\\begin{align}\n\t\t\\abs{f(\\vec x)} \\lesssim \\reg{\\vec x}^{-2m}\n\t\\end{align}\n\t%\n\tfor some $n\\in\\bbN$, the ridgelet coefficients satisfy\n\t%\n\t\\begin{align}\\label{eq:ridge_coeff_tail_decay}\n\t\t\\abs*{\\inpr{\\varphi_\\lambda,f}}\\lesssim \\frac{2^{-\\frac j2}}{\\parens*{\\parens*{\\frac{k_1}{2^j}}^2+\\abs[]{\\vec k{}'}^2+1}^m}.\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tUsing \\eqref{eq:ridge_est} and the required decay of $f$, we estimate and then transform by $\\vec y = R_\\jl \\vec x$, yielding\n\t%\n\t\\begin{align}\n\t\t\\abs*{\\inpr{\\varphi_\\lambda,f}}\n\t\t&=\\abs[\\bigg]{\\int  \\overline{\\varphi_\\lambda(\\vec x)} f(\\vec x) \\d \\vec x}\n\t\t\\le \\int  \\frac{2^{\\frac j2}}{\\reg*{U_{j\\!\\?\\?,\\ell}^{-1}\\vec x-\\vec k}^{2n}} \\frac{1}{\\reg*{\\vec x}^{2m}} \\d \\vec x\n\t\t= \\int \\frac{2^{\\frac j2}}{\\reg*{D_{2^j}\\vec y-\\vec k}^{2n}}  \\frac{1}{\\reg*{\\vec y}^{2m}} \\d \\vec y\\\\\n\t\t&=2^{\\frac j2} \\int_{-\\infty}^{\\infty}\\int_{\\bbR^{d-1}} \\frac{1}{\\parens*{\\abs[]{\\vec y{}'-\\vec k{}'}^2+(2^jy_1-k_1)^2+1}^{n}}  \\frac{1}{\\parens*{\\abs[]{\\vec y{}'}^2+(y_1)^2+1}^{m}} \\d \\vec y{}' \\d y_1.\n\t\\end{align}\n\t%\n\tBy \\autoref{cor:Imn_higher_dim}, this can be estimated in the following way,\n\t%\n\t\\begin{align}\n\t\t\\abs*{\\inpr{\\varphi_\\lambda,f}}\n\t\t&\\lesssim 2^{\\frac j2} \\int_{-\\infty}^{\\infty} \\frac{1}{\\parens*{\\abs[]{\\vec k{}'}^2+(2^jy_1-k_1)^2+y_1^2+1}^{m}}  \\frac{1}{\\parens*{(2^jy_1-k_1)^2+1}^{m}} \\d y_1\\\\\n\t\t&\\le 2^{\\frac j2} \\int_{-\\infty}^{\\infty} \\frac{1}{\\parens*{2^{2j}\\parens*{y_1-\\frac{k_1}{2^j}}^2+1}^{m}} \\frac{1}{\\parens*{y_1^2+\\abs[]{\\vec k{}'}^2+1}^{m}} \\d y_1,\n\t\t\\intertext{\n\t%\n\twhich, by \\eqref{eq:Imn_est}, is less than a multiple of\n\t%\n\t\t}\n\t\t&\\lesssim  \\frac{2^{\\frac j2}}{\\parens*{2^{2j}\\parens*{\\frac{k_1}{2^j}}^2+2^{2j}(\\abs[]{\\vec k{}'}^2+1)+1}^{m}} \\parens[\\bigg]{2^{j(2m-1)}  + \\frac{1}{\\parens*{\\abs[]{\\vec k{}'}^2+1}^{\\frac{2m-1}{2}}}} \\le \\frac{2^{-\\frac j2}}{\\parens*{\\parens*{\\frac{k_1}{2^j}}^2+\\abs[]{\\vec k{}'}^2+1}^m}.\n\t\\end{align}\n\\end{proof}\n\n\\subsection{Proof of \\autoref{th:approx}}\\label{ssec:proof_approx}\n\nWith the results of Sections \\ref{sec:loc_angle} and \\ref{sec:loc_space}, we are now in a position to prove \\autoref{th:approx}.\n\n\\begin{proof}[Proof of \\autoref{th:approx}]\n\tRecall that $\\alpha_\\lambda=\\inpr*{\\varphi_\\lambda,f}$ and $\\beta_\\lambda=w_\\lambda\\inpr*{\\varphi_\\lambda,\\CS[f]}$ from \\autoref{prop:loc_angle}. Thus, the claim of the theorem will follow if we are able to prove\n\t%\n\t\\begin{align}\n\t\t\\norm{\\alpha_\\Lambda}_{\\ell^{p^* + \\frac dm}_w} \\lesssim \\sum_{i=0}^N\\norm{f_i}_{H^t} \\qquad \\text{and} \\qquad\n\t\t\\norm{{\\mathbf{W}} \\beta_\\Lambda}_{\\ell^{p^* + \\frac dm}_w} \\lesssim \\sum_{i=0}^N\\norm{f_i}_{H^t},\n\t\\end{align}\n\t%\n\twhere the deviation $\\frac dn$ from the best possible value $p^*=\\parens*{\\frac td + \\frac 12}^{-1}$ (for $f_i\\in H^t$), decays with $n$.\n\t\n\t\\step{Preparations}\n\tFirst off, we note that due to the linearity of the ridgelet coefficients, resp.~the differential equation \\eqref{eq:LinTrans}, it suffices to treat a function $f(\\vec x)=H(\\vec x\\cdot \\vec n-v)g(\\vec x)$ and the rest follows through superposition. Also, we will be able to prove the results for $\\alpha_\\Lambda$ and $\\beta_\\Lambda$ almost completely simultaneously, and will mostly deal with $\\beta_\\Lambda$ (noting, where appropriate, the mitigating factors in the easier $\\alpha_\\Lambda$-case). The only additional thing we have to check for $u=\\CS[f]$ is that the decay conditions for \\autoref{lem:ridge_coeff_tail_decay} holds, but this follows directly from \\autoref{lem:decay_sol}.\n\t\n\tFurthermore, we recall from \\autoref{prop:sol_smooth_except_hyp} that $u(\\vec x)=u_0(\\vec x)+H(\\vec x\\cdot \\vec n-v)u_1(\\vec x)$ with $\\norm{u_i}_{H^t}\\lesssim \\norm{g}_{H^t}$. Due to \\eqref{eq:f_glob_decay}, we also have the necessary conditions to apply \\autoref{prop:loc_space},\n\t%\n\t\\begin{align}\n\t\t\\abs*{g\\normalr|_{h}(\\CP_{\\vec n}\\vec x)} &\\lesssim \\reg{\\CP_{\\vec n}\\vec x}^{-2m}, \\qquad \\abs*{u_1\\normalr|_{h}(\\CP_{\\vec n}\\vec x)}\\lesssim \\reg{\\CP_{\\vec n}\\vec x}^{-2m}.\n\t\\end{align}\n\t\n\tWe begin by choosing $\\delta>0$, and will show membership of ${\\mathbf{W}}\\beta_\\Lambda$ in $\\ell^{p'}_w$, where $p'=p^* + \\delta^{*}$ and $\\delta^{*}=\\delta\\parens{t+\\frac d2}^{-1}$. The modified $\\delta^{*}$ will have to satisfy a lower bound to ensure that all the arguments hold, and we will then that $\\frac dm$ is enough to achieve this bound.\n\tFollowing the decomposition $\\beta_\\lambda=\\beta^\\mathrm{r}_\\lambda+\\beta^\\mathrm{s}_\\lambda$, we will show this for both subsequences separately.\n\t\n\t\\step{Singular part, large coefficients}\n\tWe begin with ${\\mathbf{W}}\\beta^\\mathrm{s}_\\Lambda$, splitting the sequence into two parts, namely the \\emph{tail}\n\t%\n\t\\begin{align}\\label{eq:approx_main_sing_tail}\n\t\tT_{j\\!\\?\\?,\\ell}&:=\\set[\\Big]{\\vec k\\in\\bbZ^d}{\\abs*{V_{j\\!\\?\\?,\\ell}\\vec t(\\vec k)} > \\parens*{2^{j+1}\\abs*{\\sin\\theta_{j\\!\\?\\?,\\ell}}}^{\\frac \\delta d}+\\frac{\\sqrt{d}}{2}},\n\t\\end{align}\n\t%\n\tand its complement $T_{j\\!\\?\\?,\\ell}^\\mathsf{c}:=\\bbZ^d\\setminus T_{j\\!\\?\\?,\\ell}$. Here, $\\vec t(\\vec k)$ is defined as in \\autoref{prop:loc_space}, and similarly, $a$ and $\\phi$ are reused to define  $V_{j\\!\\?\\?,\\ell}$ as the following block-diagonal matrix (with determinant $\\frac 1a$),\n\t\\begin{align}\n\tV_{j\\!\\?\\?,\\ell}= \\left(\\begin{array}{cc|c}\n\t\t\\frac{\\cos\\phi}{a} & \\frac{2^j\\sin\\phi}{a} & 0 \\\\\n\t\t\\frac{2^j\\sin\\phi}{a^2} & \\frac{\\cos\\phi}{a^2} & 0 \\\\ \\hline\n\t\t0 & 0 &\\bbI_{d-2}\n\t\\end{array}\\right).\n\t\\end{align}\n\t\n\tAs the first step, we will show membership of the sequence ${\\mathbf{W}}\\beta^\\mathrm{s}_\\Lambda\\normalr|_{\\{\\lambda:\\vec k\\not\\in T_{j\\!\\?\\?,\\ell}\\}}$ in $\\ell^{p'}_w$. \n\tTaking the number of elements whose absolute value exceeds $\\varepsilon$, i.e.\n\t%\n\t\\begin{align}\n\t\tN(\\varepsilon):=\\card*{\\lambda}{k \\not\\in T_{j\\!\\?\\?,\\ell} \\land \\abs{w_\\lambda\\beta^\\mathrm{s}_\\lambda}\\ge \\varepsilon},\n\t\\end{align}\n\t%\n\tan equivalent definition of the $\\ell^{p'}_w$-norm is\n\t%\n\t\\begin{align}\n\t\t\\norm*{{\\mathbf{W}}\\beta^\\mathrm{s}_\\Lambda\\normalr|_{\\{\\lambda:\\vec k\\not\\in T_{j\\!\\?\\?,\\ell}\\}}}_{\\ell^{p'}_w}\\sim\\sup_{\\varepsilon>0} \\varepsilon N^{\\frac 1{p'}}(\\varepsilon) \\lesssim \\norm{g}_{H^t},\n\t\\end{align}\n\t%\n\twhich is what we will show in the following.\n\t\n\tTo do so, we define a further subset of $N(\\varepsilon)$,\n\t%\n\t\\begin{align}\n\t\tN_{j,r}(\\varepsilon):=\\card*{(\\ell,\\vec k)}{\\ell\\in\\Lambda^j_r \\land k \\not\\in T_{j\\!\\?\\?,\\ell} \\land \\abs{w_\\lambda\\beta^\\mathrm{s}_\\lambda}\\ge \\varepsilon}.\n\t\\end{align}\n\t%\n\tClearly, \\eqref{eq:gamma_est_sum_Lambda} implies that $\\abs{w_\\lambda\\beta^\\mathrm{s}_\\lambda}^2 \\le C 2^{-j} \\norm{g}^2_{H^t}$, and therefore $N_{j,r}(\\varepsilon)=0$ if $2^j\\ge C\\varepsilon^{-2}\\norm{g}^2_{H^t}$.\n\t\n\tNext, we need to determine the cardinality of $T_{j\\!\\?\\?,\\ell}^\\mathsf{c}$ for $\\ell \\in \\Lambda^j_r$. First we estimate the sum by an integral (where the maximum of each cell of size $[0,1]^d$ can be at most $\\frac{\\sqrt{d}}2$ away from the value of the function at $\\vec k \\in \\bbZ^d$), and then transform with $\\vec x=V_{j\\!\\?\\?,\\ell}\\vec t(\\vec k)=V_{j\\!\\?\\?,\\ell} (\\vec k-vU_{j\\!\\?\\?,\\ell}^{-1}\\vec n)$ --- introducing a factor $a$ from the determinant --- to estimate\n\t%\n\t\\begin{align}\n\t\t\\# T_{j\\!\\?\\?,\\ell}^\\mathsf{c}\n\t\t&\\le \\int \\mathbbm{1}_{\\curly*{\\abs[]{V_{j\\!\\?\\?,\\ell} \\vec t(\\vec k)}\\le \\parens{2^{j+1}\\abs{\\sin\\theta_{j\\!\\?\\?,\\ell}}}^{\\frac \\delta d}+\\sqrt{d}}} \\d \\vec k = \\int a \\,\\mathbbm{1}_{\\curly*{\\abs[]{\\vec x}\\le \\parens{2^{j+1}\\abs{\\sin\\theta_{j\\!\\?\\?,\\ell}}}^{\\frac \\delta d}+\\sqrt{d}}}\\d \\vec x\\\\\n\t\t&= a \\,\\mu\\parens[\\Big]{B_{\\bbR^{d}}\\parens*{0, \\parens*{2^{j+1}\\abs*{\\sin\\theta_{j\\!\\?\\?,\\ell}}}^{\\frac \\delta d}+\\sqrt{d}}} \\lesssim 2^{(j-r)(1+\\delta)}. \\label{eq:card_Sjl}\n\t\\end{align}\n\t%\n\tHere, $\\mu$ is the $d$-dimensional Lebesgue measure.\n\tThe restriction defining $T_{j\\!\\?\\?,\\ell}$ may now seem somewhat arbitrary, but the result is that the number of translations in the singular set is essentially one-dimensional (even though the set of translations is $d$-dimensional!), which gives rise to the following bound for the cardinality $N_{j,r}$ (regardless of the condition that the absolute value be larger than $\\varepsilon$),\n\t%\n\t\\begin{align}\\label{eq:card_Njr}\n\t\tN_{j,r}(\\varepsilon)\\lesssim \\underbrace{2^{(j-r)(d-1)}}_{\\mathllap{\\#\\Lambda^j_r}\\lesssim} \\cdot \\underbrace{2^{(j-r)(1+\\delta)}}_{\\mathllap{\\#T_{j\\!\\?\\?,\\ell}^\\mathsf{c}}\\lesssim} = 2^{(j-r)(d+\\delta)}.\n\t\\end{align}\n\t%\n\tAs we shall see, the factor $d+\\delta$ will directly influence the best $p$ possible for the $\\ell^p_w$-norm of the subsequence involving $\\vec k \\in T_{j\\!\\?\\?,\\ell}$, in the sense that we will achieve\n\t%\n\t\\begin{align}\\label{eq:est_p4sing}\n\t\tp=\\frac{2(d+\\delta)}{2t+(d+\\delta)} \\le \\frac{2(d+\\delta)}{2t+d} = \\parens[\\Big]{\\frac td + \\frac 12}^{-1} + \\frac{2\\delta}{2t+d} = p^*+\\delta^{*} =p',\n\t\\end{align}\n\t%\n\twhere we recall $\\delta^{*}=\\delta\\parens*{t+\\frac d2}^{-1}$.\n\t\n\tOf course, restricting the translations to achieve this bound is only half the battle --- everything we exclude now needs to be bounded afterwards --- but this shall work out in our favour (and make sense of the definition of $T_{j\\!\\?\\?,\\ell}$), because the tail is precisely defined to contain only the small coefficients.\n\t\n\tLet $\\eta:=\\varepsilon\/\\norm{g}_{H^t}$.  Then, by \\eqref{eq:gamma_est_sum_Lambda} and \\eqref{eq:card_Njr},\n\t%\n\t\\begin{align}\n\t\tN_{j,r}(\\varepsilon)\\lesssim \\min\\parens*{2^{(j-r)(d+\\delta)},\\eta^{-2} 2^{-j} 2^{-(j-r)(2t-1)}}.\n\t\\end{align}\n\t%\n\tCalculating the maximal $r$ such that the second term is the minimum, we find\n\t%\n\t\\begin{align}\n\t\t\\eta^{-2} 2^{-j} 2^{-(j-r)(2t-1)} \\le 2^{(j-r)(d+\\delta)} \\qquad \\Longleftrightarrow \\qquad r\\le j- \\log_2(\\eta^{-\\frac 2\\sigma} 2^{-\\frac{j}\\sigma}),\n\t\\end{align}\n\t%\n\twhere $\\sigma:=d+\\delta+2t-1$. Therefore,\n\t%\n\t\\begin{align}\n\t\tN_j(\\varepsilon)=\\sum_{r=1}^j N_{j,r}(\\varepsilon) \\lesssim \\min\\parens*{2^{j(d+\\delta)},\\eta^{-2} 2^{-j} (\\eta^{-\\frac 2\\sigma} 2^{-\\frac{j}\\sigma})^{2t-1}}=\\min\\parens*{2^{j(d+\\delta)},\\eta^{-\\frac{2(d+\\delta)}\\sigma} 2^{-\\frac{j(d+\\delta)}\\sigma}}\n\t\\end{align}\n\t%\n\tAgain, we determine the critical $j$ where the minimum switches from one term to the other, and see that\n\t%\n\t\\begin{align}\n\t\t\\eta^{-\\frac{2(d+\\delta)}\\sigma} 2^{-\\frac{j(d+\\delta)}\\sigma} \\le 2^{j(d+\\delta)} \\qquad \\Longleftrightarrow \\qquad \\eta^{-\\frac{2}{\\sigma+1}} \\le 2^j,\n\t\\end{align}\n\t%\n\twhich implies\n\t%\n\t\\begin{align}\n\t\tN_j(\\varepsilon)\\lesssim\\begin{cases}\n\t\t\t2^{j(d+\\delta)}, & 2^j\\le  \\eta^{-\\frac{2}{\\sigma+1}}, \\\\\n\t\t\t\\eta^{-\\frac{2(d+\\delta)}\\sigma} 2^{-\\frac{j(d+\\delta)}\\sigma}, & \\eta^{-\\frac{2}{\\sigma+1}} \\le 2^j \\le C\\eta^{-2},\\\\ \n\t\t\t0, & 2^j\\ge C\\eta^{-2}.\n\t\t\\end{cases}\n\t\\end{align}\n\t%\n\tFinally, we have\n\t%\n\t\\begin{align}\n\t\tN(\\varepsilon)\n\t\t&=\\sum_{j=0}^\\infty N_j(\\varepsilon)\\lesssim \\sum_{j\\colon 2^j \\le \\eta^{-2\/(\\sigma+1)}} 2^{j(d+\\delta)} + \\sum_{j\\colon\\eta^{-2\/(\\sigma+1)}\\le 2^j} \\eta^{-\\frac{2(d+\\delta)}\\sigma} 2^{-\\frac{j(d+\\delta)}\\sigma} \\\\\n\t\t&\\lesssim \\eta^{-\\frac{2(d+\\delta)}{\\sigma+1}}+ \\eta^{-\\frac{2(d+\\delta)}\\sigma + \\frac{d+\\delta}\\sigma \\frac{2}{\\sigma+1}} \\lesssim \\eta^{-\\frac{2(d+\\delta)}{2t+d+\\delta}} \\le \\eta^{-p'},\n\t\\end{align}\n\t%\n\tby \\eqref{eq:est_p4sing} since $\\eta<1$, which finishes the argument for the first subsequence.\n\t\n\t\\step{Singular part, tail}\n\tNext we will show membership of the sequence ${\\mathbf{W}}\\beta^\\mathrm{s}_\\Lambda\\normalr|_{\\{\\lambda:\\vec k\\in T_{j\\!\\?\\?,\\ell}\\}}$ in $\\ell^{p'}\\subseteq \\ell^{p'}_w$. We begin by taking $q<1$, and --- after applying \\eqref{eq:gamma_est_w} --- again estimate the sum by an integral (recalling that the maximum per cell can be at most $\\frac{\\sqrt{d}}2$ away from the value at $\\vec k$, which cancels with the shift in the right-hand side of the defining inequality of $T_{j\\!\\?\\?,\\ell}$),\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\vec k \\in T_{j\\!\\?\\?,\\ell}} \\abs{w_\\lambda \\beta^\\mathrm{s}_\\lambda}^q\n\t\t&\\lesssim \\sum_{\\vec k \\in T_{j\\!\\?\\?,\\ell}} \\frac{2^{\\frac {jq}2} a^{-q}}{\\parens*{|\\vec t{}''|^2+\\rho_1^2+\\rho_2^2+1}^{mq}} \\\\\n\t\t&\\le \\int \\mathbbm{1}_{\\curly*{\\abs[]{V_{j\\!\\?\\?,\\ell} \\vec t}> \\sqrt[d\/\\delta]{2^{j+1}\\abs{\\sin\\theta_{j\\!\\?\\?,\\ell}}}}} \\frac{2^{\\frac {jq}2} a^{-q}}{\\parens*{|\\vec t{}''|^2+\\rho_1^2+\\rho_2^2+1}^{mq}} \\d \\vec k.\n\t\\end{align}\n\t%\n\tCalculating $V_{j\\!\\?\\?,\\ell} \\parens*{t_1, t_2, t_3,\\ldots}^\\top=\\parens*{\\rho_1, \\rho_2, t_3,\\ldots}^\\top$,\n\n\n\n\n\n\twe transform the first term by $\\vec x=V_{j\\!\\?\\?,\\ell}^{-1} (\\vec k-U_{j\\!\\?\\?,\\ell}^{-1}\\vec n)$ to yield\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\vec k \\in T_{j\\!\\?\\?,\\ell}} \\abs{w_\\lambda \\beta^\\mathrm{s}_\\lambda}^q \n\t\t&\\lesssim \\int \\mathbbm{1}_{\\curly*{\\abs[]{\\vec x} > \\parens[]{2^{j+1}\\abs{\\sin\\theta_{j\\!\\?\\?,\\ell}}}^{\\frac \\delta d}}} \\frac{2^{\\frac {jq}2}}{(|\\vec x|^2+1)^{mq}} a^{1-q}\\d \\vec x \\\\\n\t\t&\\lesssim  \\int_{\\parens[]{2^{j+1}\\abs{\\sin\\theta_{j\\!\\?\\?,\\ell}}}^{\\frac \\delta d}}^\\infty \\frac{2^{\\frac {jq}2}}{(r^2+1)^{mq}} r^{d-1} a^{1-q} \\d r \\\\\n\t\n\t\t&\\lesssim  2^{\\frac {jq}2} 2^{(j-r)(-\\frac{2mq-d}{d}\\delta+1-q)}\n\t\t\\lesssim 2^{\\frac {jq}2}2^{(j-r)(-\\frac{2mq\\delta}{d}+1+\\delta)}. \\label{eq:est_transl_sing}\n\t\\end{align}\n\t%\n\n\tSumming over $\\ell \\in \\Lambda^j_r$, which has cardinality $\\# \\Lambda^j_r\\lesssim 2^{(j-r)(d-1)}$, this implies\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\ell\\in\\Lambda^j_r}\\sum_{\\vec k \\in T_{j\\!\\?\\?,\\ell}} \\abs{w_\\lambda \\beta^\\mathrm{s}_\\lambda}^q \\lesssim 2^{\\frac {jq}2}2^{-(j-r)(\\frac{2mq\\delta}{d}-d-\\delta)}.\n\t\\end{align}\n\t\n\tAt first glance this estimate might seem of questionable benefit, since the first term has a positive power of $j$. However, \\eqref{eq:gamma_est_sum_Lambda} and the following interpolation inequality\\footnote{Log-convexity of $L^p$-norms; follows from applying H\\\"older's inequality to $\\abs{f}=\\abs{f}^\\theta \\abs{f}^{1-\\theta}$.} for a sequence $(c_i)_{i\\in I}$ will save the day,\n\t%\n\t\\begin{align}\n\t\t\\norm{c}_{\\ell^p} \\le \\norm{c}_{\\ell^q}^\\theta \\norm{c}_{\\ell^2}^{1-\\theta} \\qquad \\text{where} \\qquad \\frac 1p = \\frac{\\theta}q + \\frac{1-\\theta}{2}.\n\t\\end{align}\n\t%\n\tIn particular, we have that\n\t%\n\t\\begin{align}\n\t\t\\parens[\\bigg]{\\sum_{\\ell\\in\\Lambda^j_r}\\sum_{\\vec k \\in T_{j\\!\\?\\?,\\ell}} \\abs{w_\\lambda \\beta^\\mathrm{s}_\\lambda}^p}^{\\frac 1p} \n\t\t&\\lesssim \\parens{2^{\\frac {j}2}2^{-(j-r)(\\frac{2m\\delta}{d}-\\frac{d+\\delta}{q})}}^\\theta \\parens{2^{-\\frac j2} 2^{-(j-r)(t-\\frac12)}}^{1-\\theta}\\\\\n\t\t&=2^{-j(\\frac 12 -\\theta)} 2^{-(j-r)\\parens{\\theta(\\frac{2m\\delta}{d}-\\frac{d+\\delta}{q})+(1-\\theta)(t-\\frac12)}}, \\label{eq:sum_gamma_p_theta}\n\t\\end{align}\n\t%\n\twhich implies that for $\\theta<\\frac 12$ (i.e.~$q$ sufficiently small) and $n$ sufficiently large, we have achieved a finite $\\ell^p$-norm for the tail of the singular part, as we can now simply sum over $r$:\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\ell}\\sum_{\\vec k \\in T_{j\\!\\?\\?,\\ell}} \\abs{w_\\lambda \\beta^\\mathrm{s}_\\lambda}^p\n\t\t&\\lesssim 2^{-jp(\\frac 12 -\\theta)}.\n\t\\end{align}\n\t%\n\tThe condition $\\theta<\\frac 12$ prescribes an upper bound for $q$ depending on the desired $p$, namely that $\\theta<\\frac 12 \\Longleftrightarrow \\frac 1q > \\frac 2p-\\frac 12$.\n\t\n\tIn the same way as above, we get that\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\ell\\in\\Lambda^j_r}\\sum_{\\vec k \\not\\in S_{j\\!\\?\\?,\\ell}} \\abs{\\alpha^\\mathrm{s}_\\lambda}^q \\lesssim 2^{-\\frac {jq}2}2^{-(j-r)(\\frac{2nq\\delta}{d}-d-\\delta)},\n\t\\end{align}\n\t%\n\twhich lets us set $q=p'$ directly without having to interpolate (which leads to a slightly improved estimate for $\\delta^*$ than the one below, compare also Step 7).\n\t\n\t\\step{Singular part, estimating $\\delta^{*}$}[\\label{stp:approx:sing_delta}]\n\tTo determine a lower bound for $\\delta^{*}$ in dependence of $n$, we observe that $\\frac{2m\\delta}{d}>\\frac{d+\\delta}{q}$ has to hold for the second exponent in \\eqref{eq:sum_gamma_p_theta} to be negative (actually this is a source for potential improvement for $\\delta$, since, due to $\\theta<\\frac 12$, we could afford to let the first term become slightly negative; we will not deal with this, though). Together with the condition for $\\theta$ and for the $p$ we want to achieve, this implies\n\t%\n\t\\begin{align}\n\t\t\\frac{2m\\delta}{d(d+\\delta)}>\\frac{1}{q}>\\frac{2}{p^*+\\delta^{*}}-\\frac 12 = \\frac{2t+d}{d+\\delta}-\\frac 12,\n\t\\end{align}\n\t%\n\twhere we recalled $p^*=(\\frac td+\\frac 12)^{-1}$ and $\\delta^{*}=\\delta(t+\\frac d2)^{-1}$. From this, we can use this to deduce a worst-case lower bound for the deviation from the optimal $p^*$ to make our argument work, namely\n\t%\n\t\\begin{align}\n\t\t\\delta\\parens[\\Big]{\\frac{2m}{d}+\\frac 12}> 2t+\\frac d2 \\qquad \\Longrightarrow \\qquad \\delta^{*}>2 \\smash{\\underbrace{\\frac{t+\\frac d4}{t+\\frac d2}}_{<1}} \\frac{1}{\\frac {2m}d+\\frac 12},\n\t\\end{align}\n\t%\n\twhich is satisfied in particular if\n\t%\n\t\\begin{align}\n\t\\delta^{*}\\ge \\frac{4d}{4n+d}.\n\t\\end{align}\n\t%\n\tSaid otherwise, everything we showed is true if we choose $\\delta^{*}=\\frac dm$, which, conversely, means that the coefficient sequence is \\emph{at least} in the space $\\ell^{p^*+\\frac dm}_w$, and possibly in an even smaller space $\\ell^{p^*+\\bar{\\delta}}_w$, i.e.~with $\\bar{\\delta}\\le\\frac dm$.\n\t\n\t\\step{Regular part, large coefficients}[\\label{stp:approx:reg_large}]\n\tLike for the singular part, we split the sequence into two parts, again defining a \\emph{tail}\n\t%\n\t\\begin{align}\\label{eq:tail_reg}\n\t\tT&:=\\set[\\Big]{\\vec k\\in\\bbZ^d}{\\abs*{D_{2^{-j}}\\vec k} > 2^{j\\frac{\\delta}{d}}+\\frac{\\sqrt{d}}2},\n\t\\end{align}\n\t%\n\tonly this time, both subsequences will be summable in $\\ell^{p'}\\subseteq \\ell^{p'}_w$. Like before, we estimate the cardinality of the complement $T^\\mathsf{c}:=\\bbZ^d\\setminus T$,\n\t%\n\t\\begin{align}\n\t\t\\#T^\\mathsf{c} \\le \\int \\mathbbm{1}_{\\{D_{2^{-j}}\\vec k<2^{j\\frac{\\delta}{d}}+\\sqrt{d}\\}} \\d \\vec k = 2^j \\int_{0}^{2^{j\\frac{\\delta}{d}}+\\sqrt{d}} r^{d-1} \\d r \\lesssim 2^{j(1+\\delta)}.\n\t\\end{align}\n\t\n\tThe following estimate will be the key to finish this step. For sequences $f,g$ with $g>0$ almost everywhere and some $q>1$, we apply H\\\"older's inequality,\n\t%\n\t\\begin{align}\n\t\t\\norm*{\\abs{f}^{\\frac 1q}}_{\\ell^1} = \\norm*{\\abs{fg}^{\\frac 1q}\\abs{g}^{-\\frac 1q}}_{\\ell^1} \\le \\norm*{\\abs{fg}^{\\frac 1q}}_{\\ell^q} \\norm*{\\abs{g}^{-\\frac 1q}}_{\\ell^{q'}} = \\norm*{fg}_{\\ell^1}^{\\frac 1q} \\norm*{\\abs{g}^{-\\frac 1{q-1}}}_{\\ell^{1}}^{\\frac{q-1}q}\n\t\\end{align}\n\t%\n\twhere $q'=\\frac{q}{q-1}$, which is sometimes called the \\emph{reverse H\\\"older inequality}, because then, by taking $q$\\nth powers and bringing the last factor to the left-hand side, ``$\\norm{fg}_{\\ell^1}\\ge \\norm{f}_{\\ell^{\\frac 1q}} \\norm{g}_{\\ell^{-\\frac{1}{q-1}}}$'', if we allowed negative exponents in the $\\ell^p$-quasi-norms.\n\t\n\tIn our case, we chose $q=\\frac 2{p'}$, $f:=\\abs{{\\mathbf{W}}\\beta^\\mathrm{r}_\\Lambda}^2$ and $g=\\mathbbm{1}_{\\{j,\\vec k\\in T^\\mathsf{c}\\}}$, allowing us to apply the above inequality,\n\t%\n\t\\begin{align}\n\t\t\\norm*{{\\mathbf{W}}\\beta^\\mathrm{r}_\\Lambda\\normalr|_{j,T^\\mathsf{c}}}_{\\ell^{p'}}^2 = \\parens[\\bigg]{\\sum_{\\ell=1}^{L_j} \\sum_{\\vec k\\in T^\\mathsf{c}}\\abs{w_\\lambda \\beta^\\mathrm{s}_\\lambda}^{p'}}^{\\frac 2{p'}} = \\norm*{{\\mathbf{W}}\\beta^\\mathrm{r}_\\Lambda\\normalr|_{j,T^\\mathsf{c}}}_{\\ell^{\\frac 1q}} \\le \\norm*{{\\mathbf{W}}\\beta^\\mathrm{r}_\\Lambda\\normalr|_{j,T^\\mathsf{c}}}_{\\ell^2}^2 \\norm*{\\mathbbm{1}_{\\{j,\\vec k\\in T^\\mathsf{c}\\}}}_{\\ell^{1}}^{q-1}.\n\t\\end{align}\n\t%\n\tSince $g$ is constant, the last norm simply evaluates to $L_j\\cdot\\#T^\\mathsf{c}\\lesssim 2^{j(d+\\delta)}$ and therefore, using \\eqref{eq:gamma_est_reg}, we have\n\t%\n\t\\begin{align}\n\t\t\\norm*{{\\mathbf{W}}\\beta^\\mathrm{r}_\\Lambda\\normalr|_{j,T^\\mathsf{c}}}_{\\ell^{p'}}^2 \\lesssim \\varepsilon_j^2 2^{-2jt} 2^{j(d+\\delta)(\\frac 2{p'}-1)}.\n\t\\end{align}\n\t%\n\tComputing\n\t%\n\t\\begin{align}\n\t\t\\frac{1}{p'}=\\parens[\\Big]{p^*+\\frac{\\delta}{t+\\frac d2}}^{-1} =\\parens[\\Big]{p^*+\\frac{\\delta}{t+\\frac d2}}^{-1}=\\frac{\\frac td +\\frac 12}{1+ \\frac \\delta d}\n\t\\end{align}\n\t%\n\tmakes it clear that the second exponent simplifies to\n\t%\n\t\\begin{align}\n\t\t2jd\\parens[\\Big]{1+\\frac \\delta d}\\parens[\\bigg]{\\frac{\\frac td +\\frac 12}{1+ \\frac \\delta d}-\\frac 12}\n\t\t=2jd\\parens[\\Big]{\\frac td - \\frac{\\delta}{2d}}=2j\\parens[\\Big]{t - \\frac{\\delta}{2}},\n\t\\end{align}\n\t%\n\tand thus\n\t%\n\t\\begin{align}\n\t\t\\norm*{{\\mathbf{W}}\\beta^\\mathrm{r}_\\Lambda\\normalr|_{j,T^\\mathsf{c}}}_{\\ell^{p'}} \\lesssim \\varepsilon_j 2^{-j\\frac{\\delta}{2}},\n\t\\end{align}\n\t%\n\twhich is summable in $j$.\n\t\n\t\\step{Regular part, tail}\n\tFor the tail of ${\\mathbf{W}}\\beta^\\mathrm{r}_\\Lambda$, we need to use the decay of $f$ resp.~$\\CS[f]$ to apply \\autoref{lem:ridge_coeff_tail_decay},\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\vec k\\in T} \\abs{w_\\lambda \\beta^\\mathrm{r}_\\lambda}^{p'}\n\t\t&\\lesssim \\sum_{\\vec k\\in T}  \\frac{2^{-\\frac {jp'}2}}{\\parens*{\\parens*{\\frac{k_1}{2^j}}^2+\\abs[]{\\vec k{}'}^2+1}^{mp'}}\n\t\t\\le 2^{-\\frac {jp'}2} \\int \\mathbbm{1}_{\\{\\abs[]{D_{2^{-j}}\\vec k} > 2^{j\\frac{\\delta}{d}}\\}} \\frac{1}{\\parens*{\\parens*{\\frac{k_1}{2^j}}^2+\\abs[]{\\vec k{}'}^2+1}^{mp'}} \\d \\vec k.\n\t\t\\intertext{\n\t%\n\tTransforming with $\\vec t=D_{2^{-j}}\\vec k$, we are able to continue,\n\t%\n\t\t}\n\t\t&= 2^{-\\frac {jp'}2} \\int_{\\abs[]{\\vec k}>2^{j\\frac{\\delta}{d}}}  \\frac{2^j}{\\parens*{\\abs[]{\\vec t}^2+1}^{mp'}} \\d \\vec t\n\t\t= 2^{j\\parens[]{1-\\frac{p'}2}} \\!\\! \\int_{2^{j\\frac{\\delta}{d}}}^\\infty  r^{d-1-2mp'} \\d r = 2^{-j\\parens[]{(2mp'-d)\\frac{\\delta}{d}+\\frac{p'}2-1}}.\n\t\\end{align}\n\t%\n\tSince this estimate is independent of $\\ell$, we can sum over the $L_j\\sim 2^{j(d-1)}$ directions on scale $j$ to arrive at\n\t%\n\t\\begin{align}\\label{eq:sum_reg_tail}\n\t\t\\sum_{\\ell=1}^{L_j} \\sum_{\\vec k\\in T} \\abs{w_\\lambda \\beta^\\mathrm{r}_\\lambda}^q \\lesssim 2^{-j\\parens{\\frac{2mp'\\delta}{d}+\\frac{p'}2-d-\\delta}}.\n\t\\end{align}\n\t%\n\tFor $m$ sufficiently large, resp.~$\\delta$ not too small (see below), the exponent is negative, and we can thus sum in $j$ and have shown that the tail of the regular part is in $\\ell^{p'}\\subseteq \\ell^{p'}_w$.\n\t\n\t\\step{Regular part, estimating $\\delta^{*}$}\n\tSimilarly to \\autoref{stp:approx:sing_delta}, we check which condition $\\delta$ (resp.~$\\delta^{*}$) needs to satisfy so that the results from above hold (i.e.~that the exponent \\eqref{eq:sum_reg_tail} is negative). Apart from the negligible term $\\frac{p'}2$ in the exponent, this is the same condition we had for the tail of the singular part, only that now, we don't need to interpolate,\n\t%\n\t\\begin{align}\n\t\t\\frac{2m\\delta}{d(d+\\delta)}>\\frac{1}{p'}=\\frac{1}{p^*+\\delta^{*}} = \\frac{t+\\frac d2}{d+\\delta}.\n\t\\end{align}\n\t%\n\tTherefore, the condition for $\\delta^{*}$ becomes\n\t%\n\t\\begin{align}\n\t\t\\delta>\\frac{d}{2m}\\parens[\\Big]{t+\\frac d2} \\qquad \\Longrightarrow \\qquad \\delta^{*}> \\frac{d}{2m},\n\t\\end{align}\n\t%\n\twhich is weaker than the condition from \\autoref{stp:approx:sing_delta}, and satisfied in particular for $\\frac dm$.\n\t\n\tThis allows us to round up the results thus far, and we observe that, taken together, Steps 1--\\thestep\\ prove the claim of the theorem for $t\\in\\bbN$.\n\t\n\t\\step{Interpolating in $t$}\n\tWhat remains to show is to extend this to the half-line $t>0$ via interpolation theory. Taking the operators defined by\n\t%\n\t\\begin{align}\n\t\tT_1 f:=\\inpr{\\Phi,f}, \\qquad T_2 f:={\\mathbf{W}}\\inpr{\\Phi,\\CS[f]},\n\t\\end{align}\n\t%\n\twe know that, by the frame property, resp.~by \\eqref{eq:ridge_Hs_stable},\n\t%\n\t\\begin{align}\n\t\t\\norm{T_i}_{L^2\\to\\ell^2}&<\\infty, \\quad i=1,2.\n\t\\end{align}\n\t%\n\tOn the other hand, as we have just proved above for any $t\\in\\bbN$, the $T_i$ also satisfy\n\t%\n\t\\begin{align}\n\t\t\\norm{T_i}_{H^t\\to\\ell^{p^*+\\delta^*}_w}&<\\infty, \\quad i=1,2,\n\t\\end{align}\n\t%\n\twhere $\\delta^*\\le \\frac dm$ and $\\frac{1}{p^*}=\\frac{t}{d}+\\frac 12$. Using the results from \\autoref{thm:interp}, we see that\n\t%\n\t\\begin{align}\n\t\t\\norm{T_i}_{H^{t\\theta}\\to\\ell^{\\bar{p}}_2}&<\\infty,\n\t\\end{align}\n\t%\n\twhere $\\frac{1}{\\bar{p}}=\\frac{1-\\theta}{2} + \\frac{\\theta}{p^*+\\delta^*}$. Letting $\\bbR^+\\ni \\bar{t}=t\\theta$ (regardless of the choice of $t\\in\\bbN$ and $0<\\theta<1$), we compute\n\t%\n\t\\begin{align}\n\t\t\\frac{1}{\\bar{p}}=\\frac{1-\\theta}{2} + \\frac{\\theta(t+\\frac d2)}{d+\\delta}=\\frac{\\bar{t}}{d+\\delta}+\\frac{\\theta}{2}\\parens[\\Big]{\\frac{d}{d+\\delta}-1} +\\frac 12,\n\t\\end{align}\n\t%\n\tand thus, since $\\theta<\\frac 12$, we can deduce\n\t%\n\t\\begin{align}\n\t\t\\bar{p}=\\frac{d+\\delta}{\\bar{t}+\\frac d2 + \\frac{\\delta}{2}(1-\\theta)} \\le \\frac{d+\\delta}{\\bar{t}+\\frac d2} = \\parens[\\Big]{\\frac {\\bar{t}}d+\\frac 12}^{-1} +\\delta^{*}.\n\t\\end{align}\n\t%\n\tFinally, we remark that $\\ell^{\\bar{p}}_2\\subseteq \\ell^{\\bar{p}}_w$, and thus the proof is finished, since for arbitrary $t\\in\\bbR^+$, we have shown that for $f\\in H^{t}(\\bbR^d)$ with solution $\\CS[f]$, it holds that\n\t%\n\t\\begin{align}\n\t\\inpr{\\Phi,f}\\in\\ell^{p^*+\\frac dm}_w \\quad \\text{as well as} \\quad {\\mathbf{W}}\\inpr{\\Phi,\\CS[f]}\\in\\ell^{p^*+\\frac dm}_w \\quad \\text{where} \\quad p^*=\\parens[\\Big]{\\frac td+\\frac 12}^{-1}.\n\t\\end{align}\n\\end{proof}\n\n\\subsection{Implications}\n\nTo conclude this section, we briefly discuss differences to the proof and results in \\cite{mutilated}.\n\n\\begin{remark}\n\tAn obvious question that presents itself with regard to \\cite{mutilated} is why Cand\\`es was able to achieve $p=p^*$, while we only achieve this value up to an arbitrarily small $\\delta^*$. Aside from the fact that the functions we treat do not have to have compact support, one major source for this loss is the fact that we have so many more translations to deal with ($d$ dimensions vs. one in \\cite{mutilated}).\n\t\n\tTo achieve $p=p^*$, we would have to:\n\t%\n\t\\begin{enumerate}[(A)]\n\t\\item\n\t\tBound \\eqref{eq:card_Sjl} in a way that depends purely linearly from $2^{j-r}$ --- since we can see that the deviation $\\delta$ directly impacts \\eqref{eq:est_p4sing} through \\eqref{eq:card_Njr}.\n\t\\item\n\t\tOn the other hand, the $\\ell^q$ norm of the tail in \\eqref{eq:est_transl_sing} still has to have an exponent of $2^{j-r}$ that is negative for sufficiently large $m$.\n\t\\end{enumerate}\n\t%\n\tThe transformation $V_{j\\!\\?\\?,\\ell}$ in $T_{j\\!\\?\\?,\\ell}$ is already chosen in a way to be able to optimally estimate \\eqref{eq:est_transl_sing} by achieving radial symmetry, and due to its functional determinant $a\\sim 2^{j-r}$, (A) becomes almost impossible. One may squeeze a slight improvement out of \\eqref{eq:gamma_est_w} by pulling a factor $a^{\\frac 1m}$ in the denominator, which would reduce the determinant of transforming with $V_{j\\!\\?\\?,\\ell}$ to $a^{1-\\frac{d}{2m}}$. Still, this improvement is not enough, as it would force $\\delta\\le\\frac{d}{2m}$ in (A), making (B) impossible, since $m$ would disappear from the exponent. Also, the $\\ell^p$-interpolation does not save us, due to the fact that the condition on $\\theta$ would necessitate $q<0$.\n\t\n\tOne solution would be if the fraction $\\frac 1a$ in \\eqref{eq:gamma_est_w} had any power that grew with $m$, even if it were as slow as $\\log(m)$ --- as this would save (B) in the above scenario. But in general, due to the sharpness of \\eqref{eq:Imn_est}, this seems unlikely to be possible. Nevertheless, we do not rule out that $\\delta^*$ could be eliminated through other proof techniques for functions with compact support.\n\\end{remark}\n\n\\begin{remark}\n\tFinally, while we were able to follow the general approach of \\cite{mutilated} --- i.e.~localisation in angle, localisation in space, split off tails etc. --- we had very different issue to deal with. On the one hand, we were able to avoid the sampling estimates in Fourier space (which is essentially due to the fact that Cand\\`es construction is supported on $[\\vec s_\\jl]$ in frequency, while our construction diffuses this to the sets $P_{j\\!\\?\\?,\\ell}$ where we are able to integrate), but on the other hand, issues like the localisation in space became much more intricate and required ``heavy machinery'' (in the form of \\autoref{th:Imn_est}) to deal with. The removal of the condition of compact support --- resp.~being also able to quantify the coefficient decay for functions with only polynomial decay --- is a welcome bonus, as is the fact that we were able to avoid the numerous case distinctions that are often necessary in other proofs of optimality (compare e.g.~\\cite{curvelet_cartoon}).\n\\end{remark}\n\\section{$N$-Term Approximation $\\&$ Benchmark Rates}\\label{sec:benchmark}\n\nWe briefly recall some core concepts of (non-linear) approximation theory, see \\cite{DeVore1998} for a survey.\n\n\\begin{definition}\\label{def:nonlin_basics}\n\tConsider a (relatively) compact class $\\FC\\subseteq \\CH$, where $\\CH$ is a separable Hilbert space, as well as a dictionary $\\Phi=(\\varphi_\\lambda)_{\\lambda\\in\\Lambda}\\in\\CH$. To measure the mentioned quality of approximation in dependence of $N$, we introduce the set\n\t%\n\t\\begin{align}\n\t\t\\Sigma_N(\\Phi):=\\set[\\bigg]{f=\\sum_{\\lambda\\in\\Lambda_N} c_\\lambda \\varphi_\\lambda}{\\#\\Lambda_N\\le N},\n\t\\end{align}\n\t%\n\twhich is non-linear (as $f,g\\in \\Sigma_N(\\Phi)$ generally don't imply $f+g\\in\\Sigma_N(\\Phi)$) --- consequently, analysis of the quantity\n\t%\n\t\\begin{align}\n\t\t\\varsigma_N(f,\\Phi):=\\inf_{g\\in \\Sigma_N(\\Phi)} \\norm{f-g}_\\CH\n\t\\end{align}\n\t%\n\tis referred to as the study of \\emph{non-linear approximation}.\n\\end{definition}\n\nComparing, $\\varsigma_N(f,\\Phi)$ with $N^{-\\alpha}$, we have the following definition of approximation spaces (which can be done in much greater generality, see \\cite[Sec. 4.1]{DeVore1998}).\n\n\\begin{definition}\n\tLet\n\t%\n\t\\begin{align}\n\t\t\\CA^\\alpha_q(\\Phi):=\\set{f\\in \\CH}{\\norm{f}_{\\CA^\\alpha_q}:=\\snorm{f}_{\\CA^\\alpha_q}+\\norm{f}_\\CH <\\infty},\n\t\\end{align}\n\t%\n\twhere\n\t%\n\t\\begin{align}\n\t\t\\snorm{f}_{\\CA^\\alpha_q} :=\n\t\t\\begin{cases}\n\t\t\t\\parens{\\sum_{n\\in\\bbN} \\parens*{n^\\alpha \\varsigma_n(f,\\Phi)}^q \\frac 1n}^{\\frac 1q}, & 0<q<\\infty,\\\\\n\t\t\t\\sup_{n\\in\\bbN} n^\\alpha \\varsigma_n(f,\\Phi), & q=\\infty.\n\t\t\\end{cases}\\tag*{\\qedhere}\n\t\\end{align}\n\t%\n\\end{definition}\n\nThese spaces are fairly well-understood --- especially if $\\Phi$ forms a basis for $\\CH$ --- and we will see that the similarity to the definition of Lorentz spaces is more than incidental.\n\n\\begin{theorem}\n\tFor $\\alpha>0$ and $0<p<q\\le \\infty$, we have $\\CA^\\alpha_p(\\Phi)\\subseteq \\CA^\\alpha_q(\\Phi)$. More importantly, the approximation spaces are interpolation spaces; for $0<\\alpha<r$ and $0<p,q\\le \\infty$, we have (\\cite[Thm. 2]{DeVore1998})\n\t%\n\t\\begin{align}\n\t\t\\CA^\\alpha_q(\\Phi)=(\\CH,\\CA^r_p(\\Phi))_{\\frac{\\alpha}{r},q}.\n\t\\end{align}\n\t%\n\tFinally, if $\\Phi$ is a basis for $\\CH$, we have the following complete characterisation of the approximation spaces $\\CA^\\alpha_q(\\Phi)$, namely that (\\cite[Thm. 4]{DeVore1998})\n\t%\n\t\\begin{align}\\label{eq:approx_charact_basis}\n\t\tf\\in \\CA^\\alpha_q(\\Phi) \\quad \\Longleftrightarrow \\quad \\inpr{\\Phi,f}:=\\parens{\\inpr{\\varphi_\\lambda,f}}_{\\lambda\\in\\Lambda} \\in \\ell^{p(\\alpha)}_q \\quad \\text{where} \\quad p(\\alpha)=\\parens[\\Big]{\\alpha+\\frac 12}^{-1}.\n\t\\end{align}\n\\end{theorem}\n\nFor arbitrary dictionaries, \\eqref{eq:approx_charact_basis} does not hold in general, but at least we can salvage one implication for frames (follows from \\cite[Lem. 3.1]{kutyniok_lemvig}, which is carried out in \\cite[Prop. 2.3.5]{my_thesis}; see also \\cite[Sec. 8]{DeVore1998}).\n\n\\begin{proposition}\\label{prop:wlp_Nterm_frame}\n\tFor a frame $\\Phi$ with canonical dual frame $\\wt \\Phi$, we have the implication that\n\t%\n\t\\begin{align}\n\t\t\\inpr{\\Phi,f} \\in \\ell^{p(\\alpha)}_w \\quad \\text{with} \\quad p(\\alpha)=\\parens[\\Big]{\\alpha+\\frac 12}^{-1} \\quad \\Longrightarrow \\quad f\\in \\CA^\\alpha(\\wt\\Phi).\n\t\\end{align}\n\\end{proposition}\n\n\nConsequently, we can ask how well a given dictionary approximates $\\FC$. This is easily done by defining\n\\begin{align}\n\t\\sigma^*(\\FC,\\Phi):=\\sup\\set{\\sigma>0}{\\FC\\subseteq A^\\sigma(\\wt \\Phi)}.\n\\end{align}\nThe more interesting question is, how well \\emph{any} dictionary could possibly approximate $\\FC$ (in terms of $N$-term approximation). This can be done very abstractly with encoding\/decoding schemes, see \\cite{hypercubes_don,hypercubes}.\n\n\\begin{definition}\\label{def:encoding_rate}\nFor a class  of signals $\\FC\\subseteq \\CH$ with $\\CH$ a separable Hilbert space, we define:\n\t\\begin{itemize}\n\t\\item\n\t\tAn \\emph{encoding\/decoding pair} $(E,D)\\in \\CE\\CD(R)$ consists of two mappings\n\t\t%\n\t\t\\begin{align}\n\t\t\tE:\\FC\\to \\{0,1\\}^R, \\quad D:\\{0,1\\}^R\\to \\CH,\n\t\t\\end{align}\n\t\t%\n\t\twhere we call $R$ the \\emph{runlength} of $(E,D)$ and $\\CE\\CD:=\\bigcup_{R\\in\\bbN} \\CE\\CD(R)$ is the set of all such pairs.\n\t\\item\n\t\tThe \\emph{distortion} of $(E,D)$ is defined as\n\t\t%\n\t\t\\begin{align}\n\t\t\t\\delta(E,D):=\\sup_{f\\in\\FC} \\norm{ f-D(E(f))}_\\CH.\n\t\t\\end{align}\n\t\\item\n\t\tThe \\emph{optimal encoding rate} is defined as\n\t\t%\n\t\t\\begin{align}\n\t\t\t\\sigma^*(\\FC):= \\sup\\set{\\sigma>0}{ \\exists C>0 \\;\\forall N\\in \\bbN \\;\\exists (E_N,D_N)\\in \\CE\\CD(N): \\delta(E_N,D_N)\\le C N^{-\\sigma}}\\tag*{\\qedhere}\n\t\t\\end{align}\n\t\\end{itemize}\n\\end{definition}\n\n\\begin{remark}\n\tThe quantity $\\sigma^*(\\FC)$ limits the best approximation rate of $\\FC$ for any decoding scheme, in particular, for the discretisation with \\emph{any} dictionary (as long as the natural restriction of polynomial depth search\\footnote{This is to exclude pathological behaviour --- namely, choosing arbitrary functions from a countable set $\\Phi$ which is dense in $\\CH$ This would give a perfect $1$-term approximation, but no way to efficiently compute the approximation or even store the dictionary. Polynomial depth search means that the set $\\Lambda_N$ we select for our $N$-term approximation may only search through the first $\\pi(N)$ dictionary elements, where $\\pi$ is an arbitrary polynomial.} is imposed), i.e.\n\t%\n\t\\begin{align}\n\t\t\\sigma^*(\\FC,\\Phi)\\le \\sigma^*(\\FC).\n\t\\end{align}\n\t\n\t\n\tIt is also intimately related to the \\emph{Kolmogorov} (or metric) $\\varepsilon$-entropy (see e.g. \\cite{kolmog_entr}) --- which we will denote by $H_\\varepsilon(\\FC)$ --- in the sense that (\\cite[Rem. 5.10]{hypercubes})\n\t%\n\t\\begin{align}\n\t\t\\sigma^*(\\FC)=\\sup \\set[\\Big]{\\sigma>0}{\\sup_{\\varepsilon>0} \\varepsilon^{\\frac 1\\sigma} H_\\varepsilon(\\FC) < \\infty}.\\tag*{\\qedhere}\n\t\\end{align}\n\\end{remark}\n\nThe interesting thing --- despite its very general definition --- is that $\\sigma^*(\\FC)$ can be estimated with the help of the following definition and \\autoref{thm:hypercubes}.\n\n\\begin{definition}\\label{def:hypercubes}\n\tLet $\\FC\\subseteq \\CH$ be a signal class in a separable Hilbert space $\\CH$.\n\t\\begin{itemize}\n\t\\item\n\t\tWe say $\\FC$ \\emph{contains an embedded orthogonal hypercube of dimension $m$ and sidelength $\\delta$} if there exist $f_0\\in\\FC$ and orthogonal functions $(\\psi_i)_{i=1}^m\\in \\CH$ with $\\norm{\\psi_i}_\\CH=\\delta$, such that the collection of vertices\n\t\t%\n\t\t\\begin{align}\n\t\t\t\\Fh\\parens*{f_0,(\\psi_i)_{i=1}^m}:=\\set[\\bigg]{h=f_0+\\sum_{i=1}^{m}\\varepsilon_i \\psi_i}{\\varepsilon_i\\in\\{0,1\\}}\n\t\t\\end{align}\n\t\t%\n\t\tcan be embedded into $\\FC$.\n\t\\item\n\t\tFor $p>0$, $\\FC$ is said to \\emph{contain a copy of} $\\ell^p_0$ if there exists a sequence of orthogonal\n\t\thypercubes $(\\Fh_k)_{k\\in\\bbN}$ with dimensions $m_k$ and sidelength $\\delta_k$ embedded in $\\FC$, such that $\\delta_k\\to0$ and for some constant $C>0$\n\t\t%\n\t\t\\begin{align}\\label{eq:def_lp0}\n\t\t\t\\delta_k\\ge C m_k^{-\\frac 1p}, \\quad \\forall k\\in\\bbN.\n\t\t\\end{align}\n\t\\end{itemize}\n\\end{definition}\n\n\\begin{remark}\\label{rem:hypercubes}\n\tThe motivation behind \\autoref{def:hypercubes} is, on the one hand, that we know precisely how many bits we need to encode the vertices of a hypercube, and if $\\FC$ contains such hypercubes, then it must be at least as complex.\n\t\n\tOn the other hand, $\\ell^p$ trivially contains orthogonal hypercubes of dimension $m$ and sidelength $m^{-\\frac 1p}$; in fact every $\\ell^p_q$ with $q>0$ contains a copy of $\\ell^p_0$, which partly motivates the choice of notation (since $\\ell^p_{q_1}\\subseteq \\ell^p_{q_2}$ for $q_1\\le q_2$).\n\t\n\tEven more, since $\\ell^{p_1}\\subseteq \\ell^{p_2}$ for $p_1\\le p_2$, it is perhaps not surprising (and easy to check from \\eqref{eq:def_lp0}) that $\\ell^p_q$ with $q>0$ contains a copy of $\\ell^\\tau_0$ for all $\\tau \\le p$.\n\\end{remark}\n\nAs hinted in \\autoref{rem:hypercubes}, being able to precisely understand the complexity of hypercubes allowed \\cite[Thm. 2]{hypercubes_don} to prove the following landmark result. Since the proof is highly non-trivial, we also refer to \\cite[Thm. 5.12]{hypercubes}, where an elementary proof of this result was given.\n\n\\begin{theorem}\\label{thm:hypercubes}\n\tIf the signal class $\\FC\\subseteq \\CH$ contains a copy of $\\ell^p_0$ for $p\\in(0,2]$, the optimal encoding rate satisfies\n\t%\n\t\\begin{align}\n\t\t\\sigma^*(\\FC) \\le \\frac{2-p}{2p}.\n\t\\end{align}\n\\end{theorem}\n\nFinally, since we are interested in functions in the Sobolev space $H^t$, computing the optimal encoding rate of this class with the tools we have just introduced is a simple adaptation of \\cite[Thm. 5.17]{hypercubes}.\n\n\\begin{lemma}[{\\cite[Lem. 2.3.11]{my_thesis}}]\\label{lem:Ht_complexity}\n\tThe Sobolev space $H^t(\\bbR^d)\\subseteq L^2(\\bbR^d)$ contains a copy of $\\ell^{p^*}_0$, where $p^*=\\parens*{\\frac td + \\frac 12}^{-1}$. In particular, $\\sigma^*\\parens*{H^t(\\bbR^d)}\\le \\frac td$.\n\\end{lemma}\n\n\\begin{remark}\\label{rem:benchmark_Ht_ex_hyp}\n\tThe model class of functions $H^t(\\bbR\\setminus\\{h_i\\})$ --- see \\autoref{def:Ht_except_hyp} --- we will deal with regarding \\eqref{eq:LinTrans} consists of functions in $H^t$ that are allowed to have cut-offs across hyperplanes. This is trivially a superset of $H^t$, and so it is clear that\n\t%\n\t\\begin{align}\n\t\t\\sigma^*\\parens*{H^t(\\bbR\\setminus\\{h_i\\})}\\le \\sigma^*\\parens*{H^t(\\bbR^d)} \\le \\frac td.\n\t\\end{align}\n\t%\n\tIn other words, these ``mutilated'' $H^t$-functions have --- potentially --- even worse approximation rates than functions in $H^t$.\n\t\n\tNevertheless, although the possibilities of achieving this benchmark seem slim (considering the very abstract and extremely general definition of $\\sigma^*(\\FC)$), there is some hope: there \\emph{are} constructions for several classes $\\FC$ that \\emph{do} achieve $\\sigma^*(\\FC)$, cf. \\cite[Chap. 5]{hypercubes}, and even in the more complicated case of allowing cut-offs, wavelets (for example) can be shown to achieve the best possible approximation rate for piece-wise $\\CC^k$-functions over an interval, cf. \\cite[Cor. 5.36]{hypercubes}, and point-like singularities in $\\bbR^d$ in general.\n\t\n\tTo summarise, we have now found the definitive benchmark to aim for, since, in view of \\autoref{prop:wlp_Nterm_frame}, we know that $\\inpr{\\Phi,f}\\in \\ell^{p^*}_w$ with $p^*=\\parens*{\\frac td + \\frac 12}^{-1}$ is the \\emph{absolute best} that is (even theoretically) possible for generic functions in $H^t$ (or the superset  $H^t(\\bbR\\setminus\\{h_i\\})$) --- any $p<p^*$ would imply an $N$-term approximation rate that we have proved (see \\autoref{lem:Ht_complexity}) to be impossible.\n\\end{remark}\n\\section{Conclusion}\\label{sec:conclusion}\n\nWrapping up, the fact that ridgelets approximate mutilated Sobolev functions with the benchmark rate --- up to arbitrarily small $\\delta>0$ --- for this class (as set out in \\autoref{sec:benchmark}) allows us to combine \\autoref{th:approx} with \\cite{compress} to achieve \\autoref{th:main_result}. To the best of our knowledge,\nthis is the first construction of an optimally adapted PDE solver with non-standard frames and for non-elliptic\nproblems.\n\nAs indicated in \\autoref{ssec:impact}, this can be utilised for solving more involved transport equations \\eqref{eq:RTE}, based on the fact that the ridgelet frame $\\Phi$ covers all direction $\\vec s$ simultaneously, while the multiscale structure makes it possible to alleviate the curse of dimensionality, for example by the ``sparse discrete ordinates method'' (see e.g. \\cite{grella}).\n\nFinally, as a numerical check of the claimed results, we return to \\autoref{fig:sing_sol}, which is the solution of \\eqref{eq:LinTrans} with a source function that is a box function times a Gaussian, rotated so that the edges of the box are aligned with the transport direction $\\vec s$ (which is chosen to not coincide with any of the $\\vec s_\\jl$ in the frame). Since --- apart from the singularities --- the function is $\\CC^\\infty$, we even observe super-polynomial decay, in the sense that in the doubly logarithmic plot \\autoref{fig:sing_sol:nterm}, the curve has non-zero curvature (as far as we were able to calculate) and overtakes any straight line (which would correspond to a fixed power $N^{-\\sigma}$).\n\nFurthermore, the localisation in angle discussed in \\autoref{prop:loc_angle}, is also confirmed theoretically. Indeed, the results are again very convincing, in that, for at least the first $100'000$ largest coefficients (on scales $1$--$10$), the corresponding rotational parameters $\\ell$ are all either the $\\vec s_\\jl$ closest to the direction $\\vec s$ or its immediate left\/right neighbours --- across all scales! To put this into perspective, there are $2^{j+1}$ different $\\vec s_\\jl$ on each scale, and we can discard \\emph{all but three} of them without discernible loss in accuracy. For further numerical results in this context see \\cite[Sec. 8.2]{my_thesis}.\n\n\\begin{figure}\n\t\\setlength{\\plotsize}{0.5\\linewidth}\n\t\n\t\\subcaptionbox{Solution for singular $f$\\label{fig:sing_sol:rhs}}{%\n\t\t\\tikzsetnextfilename{box_sol_sml}%\n\t\t\\input{figures\/source\/box_sol_sml.tikz}}}%\n\t\\hfill\\setlength{\\plotsize}{0.35\\linewidth}%\n\t\\subcaptionbox{$N$-term approximation rate\\label{fig:sing_sol:nterm}}{%\n\t\t\\tikzsetnextfilename{Nterm_sing}%\n\t\t\\input{figures\/source\/Nterm_sing.tikz}%\n\n\t}\n\t\\caption{Solution to the advection equation for a singular right-hand side, as well the $N$-term approximation rate of the ridgelet frame}\\label{fig:box_sol}\n\\end{figure}\n\n\n\n\n\n\n\\section{An Integral (In)Equality}\\label{sec:int_est}\n\nThe following result turned out to be necessary for the proof of \\autoref{th:approx}, but has proven useful in other contexts as well (e.g.~\\autoref{lem:decay_sol}). It is very well-suited for quantifying interactions between (offset) decaying functions --- particularly for convolutions (see \\autoref{app:cor:Imn_higher_dim}) --- and substantially stronger than other results of this type we are aware of (see \\autoref{rem:grafakos}).\n\n\\subsection{Main Theorem \\texorpdfstring{$\\&$}{and} Consequences}\n\nThe following theorem is formulated not in its most general form, but in a form that any one-dimensional problem of this type can be transformed into\n\n\\begin{theorem}\\label{app:th:Imn_est}\n\tFor $m,n\\in\\bbN$, $a\\in\\bbR^+_0$, $b\\in\\bbR$, $c,d\\in\\bbR^+$, we have\n\t%\n\t\\begin{align}\n\t\tI_{m,n}&:=\\int_{-\\infty}^{\\infty} \\frac{1}{\\parens{a^2(x-b)^2+c^2}^m} \\frac{1}{\\parens{x^2+d^2}^n} \\d x \\notag \\\\\n\t\t&\\phantom{:}= \\frac{\\pi}{\\parens{(c+ad)^2+a^2 b^2}^{m+n-1}} \\frac{1}{c^{2m-1}} \\frac{1}{d^{2n-1}} \\sum_{\\substack{i+j+2k=2(m+n)-3\\\\i\\ge 2m-1 \\, \\lor\\, j\\ge 2n-1}} c^{m,n}_{i,j} c^{i} (ad)^{j} (ab)^{2k}\\notag \\\\\n\t\t&\\phantom{:}\\lesssim  \\frac{a^{2n-1}}{\\parens{(c+ad)^2+a^2 b^2}^n} \\frac{1}{c^{2m-1}} + \\frac{1}{\\parens{(c+ad)^2+a^2 b^2}^m} \\frac{1}{d^{2n-1}} \\label{app:eq:Imn_est_best}\\\\\n\t\t&\\phantom{:}\\le \\frac{a^{2n-1}}{\\parens{a^2b^2+a^2d^2+c^2}^n} \\frac{1}{c^{2m-1}} + \\frac{1}{\\parens{a^2b^2+a^2d^2+c^2}^m} \\frac{1}{d^{2n-1}}. \\label{app:eq:Imn_est}\n\t\\end{align}\n\t%\n\tFurthermore, the generating function for $I_{m,n}$ is\n\t%\n\t\\begin{align}\\label{app:eq:genfunc_Imn_intro}\n\t\t\\sum_{m,n\\ge 1} I_{m,n} y^m z^n = \\frac{\\pi yz}{\\sqrt{c^2-\\smash{y}}\\sqrt{d^2-z}} \\frac{\\sqrt{c^2-\\smash{y}}+a\\sqrt{d^2- z}}{(\\sqrt{c^2-\\smash{y}}+a\\sqrt{d^2- z})^2+a^2b^2},\n\t\\end{align}\n\t%\n\tand with $h(v,w):=(v+w)^2+1$, we also have a generating function for the coefficients,\n\t%\n\t\\begin{multline}\\label{app:eq:genfunc_coeff}\n\t\t\\sum_{\\substack{i,j\\ge 0\\\\m,n\\ge 1}} c^{m,n}_{i,j} v^i w^j y^m z^n \\\\\n\t\n\t\t=\\frac{h(v,w)yz}{\\sqrt{1-h(v,w)y}\\sqrt{1-h(v,w)z}} \\frac{v\\sqrt{1-h(v,w)y}+w\\sqrt{1-h(v,w)z}}{\\parens*{v\\sqrt{1-h(v,w)y}+w\\sqrt{1-h(v,w)z}}^2+1}.\\qquad\n\t\\end{multline}\n\t%\n\tThe coefficients are zero \\emph{unless} the following conditions are satisfied,\n\t%\n\t\\begin{align}\n\ti+j&\\equiv 1\\bmod{2}, & i+j&\\le 2(m+n)-3, & (i&\\ge 2m-1 \\lor j\\ge 2n-1).\n\t\\end{align}\n\t%\n\tIf these are satisfied, the coefficients can be calulated as follows,\n\t%\n\t\\begin{align}\n\t\tc^{m,n}_{i,j}\n\t\t\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$c^{m,n}_{i,j}$}-\\multlinegap]\n\t\t\t=\\sum_{r=0}^{i}\\sum_{s=0}^{j} \\delta_{\\{r+s\\equiv1\\bmod{2}\\}} (-1)^{m+n+\\frac{r+s-1}{2}} \\binom{r+s}{s}  \\cdot\\ldots \\\\\n\t\t\t\\ldots \\cdot  \\binom{\\frac{r-1}{2}}{m-1}\\binom{\\frac{s-1}{2}}{n-1} \\binom{m+n-1}{m+n-1-\\frac{i+j-r-s}{2}} \\binom{i+j-r-s}{i-r}.\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tAnother representation of the coefficients can be found in \\autoref{app:prop:coeff_expl_long}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{theorem}\n\n\\begin{remark}\\label{rem:grafakos}\n\tIn \\cite[App. B.1]{grafakos}, it is shown that for dimension $k\\ge 1$, powers $m,n>k$, factors $p,q>0$ and vectors $\\vec r, \\vec s\\in \\bbR^k$, the following inequality holds\n\t%\n\t\\begin{align}\n\t\t\\int_{\\bbR^k} \\frac{p^k}{\\parens{1+p\\abs{\\vec x-\\vec r}}^m} \\frac{q^k}{\\parens{1+q\\abs{\\vec x-\\vec s}}^n} \\mathrm{d} \\vec x \\lesssim \\frac{\\min(p,q)^k}{\\parens{1+\\min(p,q)\\abs{\\vec r-\\vec s}}^{\\min(m,n)}}.\n\t\\end{align}\n\t%\n\tApplied to our context (using the fact that $1+x^2\\ge \\frac 12 (1+\\abs{x})^2\\ge \\frac 12 (1+x^2)$, resp.~$k=1$), implies\n\t%\n\t\\begin{align}\n\t\tI_{m,n} \\lesssim \\frac{1}{a} \\frac{1}{(b^2+d^2)^{\\min(m,n)}} \\frac{1}{c^{2m-1}} \\frac{1}{d^{2n-2\\min(m,n)}} + \\frac{1}{(a^2b^2+c^2)^{\\min(m,n)}} \\frac{1}{c^{2m-2\\min(m,n)}} \\frac{1}{d^{2n-1}}.\n\t\\end{align}\n\t%\n\tEven though we know that in our case, we can take $m$ to be arbitrarily large (but fixed), this only yields\n\t%\n\t\\begin{align}\\label{app:eq:grafakos}\n\t\tI_{m,n} \\lesssim \\frac{1}{a} \\frac{1}{(b^2+d^2)^{n}} \\frac{1}{c^{2m-1}} + \\frac{1}{\\parens{a^2b^2+c^2}^{n}} \\frac{1}{c^{2(m-n)}} \\frac{1}{d^{2n-1}}.\n\t\\end{align}\n\t%\n\tIn one of the key estimates that we need for \\autoref{th:approx} (see the proof of \\autoref{prop:loc_space}), $c$ and $d$ will contain variables in other dimensions to be integrated over, and the fact that one term now has three factors renders an second application of \\eqref{app:eq:grafakos} impossible in this case. We would need to accept considerable slack in the estimates (by dropping one factor in the second term of the right-hand side of \\eqref{app:eq:grafakos}), which would make achieving sufficiently strong estimates much more difficult (if not impossible).\n\t\n\tCompare also with \\autoref{app:cor:Imn_higher_dim}, where we are able to leverage \\eqref{app:eq:Imn_est} into higher dimensions as well --- in our case obtaining a denominator that contains both $p$ and $q$ (in the notation of this remark), see \\eqref{eq:Imn_higher_dim}.\n\\end{remark}\n\n\\begin{remark}\\label{rem:Imn_est_decay_a}\n\tThe estimate has one deficiency in terms of the behaviour of $a$ --- namely, that $I_{m,n}$ always decreases with increasing $a$ (albeit much slower than might be expected; the decay with $b,c,d$ is much more pronounced), whereas the first term in the estimates increases with $a$ until around\n\t%\n\t\\begin{align}\n\t\ta\\sim \\sqrt{\\frac{n c^2}{b^2+d^2}} \\quad \\text{for \\eqref{app:eq:Imn_est}}\n\t\t\\quad \\text{resp.} \\quad\n\t\ta\\sim \\frac{ncd+\\sqrt{n(b^2+d^2)c^2}}{b^2+d^2} \\quad \\text{for \\eqref{app:eq:Imn_est_best}}\n\t\\end{align}\n\t%\n\tand only then starts to decrease with $a$.\n\t\n\tThis is not avoidable, as the first term of \\eqref{app:eq:Imn_est_best} actually appears as such (modulo a constant) in the explicit representation of $I_{m,n}$, but there, its growth is eliminated by the decay with $a$ of terms like the second one in \\eqref{app:eq:Imn_est_best}, which have a higher weight in practice.\n\t\n\tConsequently, barring a more precise analysis of the constant's dependence on $m$ and $n$, the estimate can be made more efficient in some cases by directly estimating away $a\\ge1$ within $I_{m,n}$ and then applying \\eqref{app:eq:Imn_est_best} --- namely when\n\t%\n\t\\begin{align}\n\t\t1& \\ll a \\lesssim \\frac{(b^2+(c+d)^2)^n}{(b^2+d^2)^n}\n\t\\end{align}\n\t%\n\tif only the first term should be minimised, which is the case if $a\\sim c \\gg b,d$, for example. Considering both terms simultaneously, estimating $a\\ge1$ is still beneficial in the following regime,\n\t%\n\t\\begin{align}\n\t\t1& \\ll a \\lesssim \\frac{(b^2+(c+d)^2)^n}{(b^2+d^2)^n}\\frac{(b^2+(c+d)^2)^{m}d^{2n-1}}{(b^2+(c+d)^2)^m d^{2n-1}+(b^2+(c+d)^2)^n c^{2m-1}}.\\tag*{\\qedhere}\n\t\\end{align}\n\\end{remark}\n\n\\begin{proof}[Proof of \\autoref{app:th:Imn_est}]\n\tThe proof is split into several parts. First, we need to determine the partial fraction decomposition (PFD) of the integrand, which we do in \\autoref{app:prop:pfd}. Since $m$ and $n$ are arbitrary, we will only be able to formulate a recursion at first. However, we can leverage this recursion into explicit generating functions for the terms appearing in the PFD, which we do in \\autoref{app:prop:genfunc}. This machinery is necessary, unfortunately, since mere induction is hopelessly inadequate for the task at hand.\n\t\n\tWith the help of these two tools, we are able to calculate the generating function \\eqref{app:eq:genfunc_Imn_intro}\n\twhich we prove in \\autoref{app:prop:genfunc_Imn}.\n\t\n\tIn the form \\eqref{app:eq:genfunc_Imn_intro}, we have already achieved (essentially) the crucial cancellation (compared to the PFD) that eliminates the ``bad'' factors from the denominator. However, what remains to be shown compared to \\autoref{app:th:Imn_est} is that $c^{m,n}_{i,j}=0$ if $i\\le 2(m-1) \\land j\\le 2(n-1)$. To gain explicit control over these coefficients, we first ``disassemble'' the function \\eqref{app:eq:genfunc_Imn_intro} into its parts (by differentiation) in \\autoref{app:prop:coeff_expl_long}, which yields another formula for $c^{m,n}_{i,j}$ (which is more complex, but without binomial coefficients of non-integers).\n\t\n\tThen, inserting the ``indicators'' we need, we put it back together to arrive at the formula \\eqref{app:eq:genfunc_coeff} in \\autoref{app:prop:genfunc_coeff}. Finally, we take apart \\eqref{app:eq:genfunc_coeff} one last time in a different way that allows us to conclude that the required coefficients are actually zero in \\autoref{app:prop:coeff_zero}. This will finish the proof. Finally, in \\autoref{rem:conject_coeff_pos}, we mention the conjecture that, always, $c^{m,n}_{i,j}\\ge0$, which, however, we have not (seriously) attempted to prove.\n\\end{proof}\n\nBefore we continue, we record an corollary of \\autoref{app:th:Imn_est} for higher dimensions.\n\n\\begin{corollary}\\label{app:cor:Imn_higher_dim}\n\tFor $m,n\\in\\bbN$  and $c,d>0$, as well as vectors $\\vec r,\\vec s \\in\\bbR^k$ and invertible matrices $A,B$ such that $AB^{-1}$ is diagonalisable\\footnote{The restriction that $AB^{-1}$ has to be diagonalisable is obviously artificial and can be removed in principle (although the formula would become much more complicated).}, we assume that two functions satisfy\n\t%\n\t\\begin{align}\n\t\t\\abs{f(\\vec x)}\\lesssim \\parens*{\\abs*{A(\\vec x+\\vec r)}^2+c^2}^{-m}\n\t\t\\quad \\text{and} \\quad\n\t\t\\abs{g(\\vec x)}\\lesssim \\parens*{\\abs*{B(\\vec x+\\vec s)}^2+d^2}^{-n}.\n\t\\end{align}\n\t%\n\tThen, if $m,n> \\ceil*{\\frac{k}{2}}$, we have the following estimate for the convolution of $f$ and $g$,\n\t%\n\t\\begin{multline}\n\t\t\\abs{[f*g](\\vec t)}\n\t\t\\lesssim\\frac{1}{\\abs{\\det B}} \\biggl( \\frac{\\norm[]{BA^{-1}}^{k}}{\\parens*{|B(\\vec r+\\vec s +\\vec t)|^2+d^2+\\|BA^{-1}\\|^2 c^2}^{n}} \\frac{1}{c^{2m-k-1}} + \\ldots \\\\\n\t\t\t\t\t\\ldots+\\frac{\\norm[]{BA^{-1}}^{2m}}{\\parens*{|B(\\vec r+\\vec s +\\vec t)|^2+d^2+\\|BA^{-1}\\|^2 c^2}^{m}} \\frac{1}{d^{2n-k-1}} \\biggr)\n\t\n\t\n\t\\end{multline}\n\t\n\tand --- in a dual way --- the same estimate holds after concurrently switching $\\vec r\\leftrightarrow\\vec s$, $A\\leftrightarrow B$, $c\\leftrightarrow d$ and $m\\leftrightarrow n$, i.e.~we can choose the minimum of the two.\n\t\n\tSpecialising to $A=B=\\bbI$ and $\\vec r=\\vec s =0$, we see that\n\t%\n\t\\begin{align}\\label{eq:Imn_higher_dim}\n\t\t\\abs{[f*g](\\vec t)} \\lesssim \\frac{1}{\\parens*{|\\vec t|^2+c^2+d^2}^n}\\frac{1}{c^{2m-k-1}} + \\frac{1}{\\parens*{|\\vec t|^2+c^2+d^2}^m}\\frac{1}{d^{2n-k-1}}.\n\t\\end{align}\n\\end{corollary}\n\n\\begin{proof}\n\tWe begin by inserting the definition, using the assumed estimates and transforming by $\\vec y=B(\\vec x+\\vec s)$.\n\t%\n\t\\begin{align}\n\t\t\\abs{[f*g](\\vec t)}\n\t\t&= \\abs[\\bigg]{\\int f(\\vec t-\\vec x) g(\\vec x) \\d \\vec x}\n\t\t\\lesssim \\int \\parens*{\\abs*{A(\\vec t-\\vec x+\\vec r)}^2+c^2}^{-m} \\parens*{\\abs*{B(\\vec x+\\vec s)}^2+d^2}^{-n} \\d \\vec x \\label{app:eq:est_conv_def}\\\\\n\t\t&=\\frac{1}{\\abs{\\det B}} \\int \\parens*{\\abs*{A(\\vec r+\\vec t-B^{-1}\\vec y +\\vec s)}^2+c^2}^{-m} \\parens*{\\abs{\\vec y}^2+d^2}^{-n} \\d \\vec y.\n\t\\end{align}\n\t%\n\tDecomposing $AB^{-1}=VDV^{-1}$ with $D$ being a diagonal matrix with eigenvalues sorted by descending absolute value and $V$ being the (unitary) matrix of corresponding eigenvectors, we set $\\vec u:=V^{-1}A(\\vec r+\\vec s+ \\vec t)$ and continue by transforming with $\\vec z= V^{-1}\\vec y$,\n\t%\n\t\\begin{align}\n\t\t\\abs{[f*g](\\vec t)}\n\t\t&\\lesssim \\frac{1}{\\abs{\\det B}} \\int \\parens*{\\abs*{-VDV^{-1}\\vec y+A(\\vec r+\\vec s+ \\vec t)}^2+c^2}^{-m} \\parens*{\\abs{\\vec y}^2+d^2}^{-n} \\d \\vec y\\\\\n\t\t&= \\frac{1}{\\abs{\\det B}} \\int \\parens*{\\abs*{D\\vec z-\\vec u}^2+c^2}^{-m} \\parens*{\\abs{\\vec z}^2+d^2}^{-n} \\d \\vec z.\n\t\\end{align}\n\t%\n\tThe problem we face now is that the above integral behaves ``elliptically'' in some sense --- and in a way we can't remove by suitable stretching --- because each component appears \\emph{both} as $z_i$ and as $\\lambda_i z_i$. With \\eqref{app:eq:Imn_est} in mind, there are two ways out of this. On the one hand, we could shave off dimension after dimension (since the $z_i$ are decoupled, we can apply \\eqref{app:eq:Imn_est} in each dimension sequentially), but this blows up the number of terms to (potentially) $2^k$, and we would ``lose'' at least half a power of either denominator in each step (or every second step, if one is careful).\n\t\n\tThe second way --- which we will choose --- is to (effectively) make the matrix $D$ a multiple of the identity (thus removing the ``elliptic'' influences), by estimating it with its smallest eigenvalue (by magnitude) as follows below. In view of \\autoref{rem:Imn_est_decay_a}, it is not unlikely that this might even be the more efficient estimate in many cases. A combination of the two methods is of course also possible, in fact, if $AB^{-1}$ is not diagonalisable, it is necessary to ``cut apart'' the Jordan blocks in the way described above.\n\t\n\tEach entry of $D\\vec z-\\vec u$ contributes a term $(\\lambda_i z_i- u_i)^2=\\abs{\\lambda_i}^2\\parens*{z_i- \\frac{u_i}{\\lambda_i}}^2$ to the absolute value, and we can estimate this from below (thus estimating the integral from above) by $|\\lambda_d|^2(z_i-\\frac{u_i}{\\lambda_i})^2$. For notational ease, we set $a:=\\abs{\\lambda_d}$ as well as $\\vec v:=D^{-1}\\vec u$, and continue from above,\n\t%\n\t\\begin{align}\n\t\t\\int \\parens*{\\abs*{D(\\vec z-D^{-1}\\vec u)}^2+c^2}^{-m} \\parens*{\\abs{\\vec z}^2+d^2}^{-n} \\d \\vec z\n\t\t&\\le \\int \\parens*{a^2\\abs{\\vec z-\\vec v}^2+c^2}^{-m} \\parens*{\\abs{\\vec z}^2+d^2}^{-n} \\d \\vec z.\n\t\\end{align}\n\t%\n\tNow we set $R:=R_{\\vec v\/ \\abs{\\vec v}}$ (compare \\autoref{def:rot_Rs}) --- satisfying $R\\vec v=\\parens*{|\\vec v|,0,\\ldots}^\\top$ --- and transform with $\\vec w:=R\\vec z$ using the invariance of the Euclidian norm under rotations,\n\t\\begin{multline}\n\t\t\\int \\parens*{a^2\\abs{\\vec z-\\vec v}^2+c^2}^{-m} \\parens*{\\abs{\\vec z}^2+d^2}^{-n} \\d \\vec z\\\\\n\t\t=\\int_{\\bbR^k} \\frac{1}{\\parens*{a^2(w_1 - |\\vec v|)^2 + a^2|\\vec w'|^2 +c^2}^m} \\frac{1}{\\parens*{w_1^2+|\\vec w'|^2+d^2}^n} \\d \\vec w,\n\t\\end{multline}\n\twhere $\\vec w=\\binom{w_1}{\\vec w'}$, i.e.~$\\vec w'$ represents the $k-1$ lower components of $\\vec w$.\n\t\n\tWe split off the integration in $w_1$ and apply \\eqref{app:eq:Imn_est},\n\t%\n\t\\begin{align}\n\t\t\\MoveEqLeft\n\t\t\\int_{\\bbR^{k-1}} \\int_{-\\infty}^{\\infty } \\frac{1}{\\parens*{a^2(w_1 - |\\vec v|)^2 + a^2|\\vec w'|^2 +c^2}^m} \\frac{1}{\\parens*{w_1^2+|\\vec w'|^2+d^2}^n} \\d w_1 \\d \\vec w'\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-2em-\\multlinegap]\n\t\t\t\\lesssim \\int_{\\bbR^{k-1}}  \\frac{a^{2n-1}}{\\parens*{a^2|\\vec v|^2 + 2a^2|\\vec w'|^2+c^2+a^2d^2}^n} \\frac{1}{\\parens*{a^2|\\vec w'|^2+c^2}^{\\frac{2m-1}{2}}} + \\ldots\\\\\n\t\t\t\\ldots + \\frac{1}{\\parens*{a^2|\\vec v|^2 + 2a^2|\\vec w'|^2+c^2+a^2d^2}^m} \\frac{1}{\\parens*{|\\vec w'|^2+d^2}^{\\frac{2n-1}{2}}} \\d \\vec w'\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-2em-\\multlinegap]\n\t\t\t\\le \\int_{0}^\\infty  \\frac{a^{2n-k}}{\\parens*{r^2+a^2|\\vec v|^2+c^2+a^2d^2}^n} \\frac{r^{k-1}}{\\parens*{r^2+c^2}^{\\frac{2m-1}{2}}} \\d r + \\ldots\\\\\n\t\t\t\\ldots + \\int_{0}^\\infty \\frac{1}{\\parens*{a^2r^2+a^2|\\vec v|^2+c^2+a^2d^2}^m} \\frac{r^{k-1}}{\\parens*{r^2+d^2}^{\\frac{2n-1}{2}}} \\d r\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-2em-\\multlinegap]\n\t\t\t\\le \\frac{1}{2} \\int_{-\\infty}^\\infty  \\frac{a^{2n-k}}{\\parens*{r^2+a^2|\\vec v|^2+c^2+a^2d^2}^n} \\frac{1}{\\parens{r^2+c^2}^{m-\\ceil{\\frac{k}{2}}}} \\frac{1}{(c^2)^{\\ceil{\\frac{k}{2}}-\\frac{k}{2}}} \\d r + \\ldots\\\\\n\t\t\t\\ldots + \\frac{1}{2}\\int_{-\\infty}^\\infty \\frac{1}{\\parens*{a^2r^2+a^2|\\vec v|^2+c^2+a^2d^2}^m} \\frac{1}{\\parens{r^2+d^2}^{n-\\ceil{\\frac{k}{2}}}} \\frac{1}{(d^2)^{\\ceil{\\frac{k}{2}}-\\frac{k}{2}}} \\d r,\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\twhere we split the integrals and transformed the first term by $\\vec x'=\\frac{1}{a}\\vec w'$ before changing to polar coordinates. We then extended the integral over $r$ to $-\\infty$ in order to be able to apply \\eqref{app:eq:Imn_est} once more,\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t\\lesssim \\frac{a^{2n-k}}{\\parens*{a^2|\\vec v|^2 +c^2+a^2d^2}^{m-\\ceil{\\frac k2}+n-\\frac 12}}\\frac{1}{(c^2)^{\\ceil{\\frac{k}{2}}-\\frac{k}{2}}}  + \\frac{a^{2n-k}}{\\parens*{a^2|\\vec v|^2 +c^2+a^2d^2}^{n}} \\frac{1}{c^{2m-k-1}} + \\ldots \\\\\n\t\t\t\\ldots + \\frac{a^{2n-2\\ceil{\\frac{k}{2}}-1}}{\\parens*{a^2|\\vec v|^2 +c^2+a^2d^2}^{n-\\ceil{\\frac k2}+m-\\frac 12}} \\frac{1}{(d^2)^{\\ceil{\\frac{k}{2}}-\\frac{k}{2}}}  + \\frac{1}{\\parens*{a^2|\\vec v|^2 +c^2+a^2d^2}^{m}} \\frac{1}{d^{2n-k-1}}\n\t\t\\end{multlined}\\\\\n\t\t&\\lesssim \\frac{a^{2n-k}}{\\parens*{a^2|\\vec v|^2 +c^2+a^2d^2}^{n}} \\frac{1}{c^{2m-k-1}} + \\frac{1}{\\parens*{a^2|\\vec v|^2 +c^2+a^2d^2}^{m}} \\frac{1}{d^{2n-k-1}}, \n\t\\end{align}\n\t%\n\tbecause, obviously, $a^2|\\vec v|^2 +c^2+a^2d^2>c^2,a^2d^2$. Now, $a=\\abs{\\lambda_d}$, the smallest eigenvalue of $AB^{-1}$, corresponds to the inverse of the largest eigenvalue of $BA^{-1}$, which itself is equal to the matrix norm $\\norm{BA^{-1}}$. Furthermore, $|D^{-1}\\vec u|=|VD^{-1}\\vec u|=|B(\\vec r+\\vec s +\\vec t)|$. Putting everything together, we arrive at\n\t%\n\t\\begin{align}\n\t\t\\abs{[f*g](\\vec t)}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$\\abs{[f*g](\\vec t)}$}-\\multlinegap]\n\t\t\t\\lesssim \\frac{1}{\\abs{\\det B}} \\biggl( \\frac{\\norm[]{BA^{-1}}^{k}}{\\parens*{|B(\\vec r+\\vec s +\\vec t)|^2+d^2+\\|BA^{-1}\\|^2 c^2}^{n}} \\frac{1}{c^{2m-k-1}} + \\ldots \\\\\n\t\t\t\\ldots+\\frac{\\norm[]{BA^{-1}}^{2m}}{\\parens*{|B(\\vec r+\\vec s +\\vec t)|^2+d^2+\\|BA^{-1}\\|^2 c^2}^{m}} \\frac{1}{d^{2n-k-1}} \\biggr).\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tDepending on the quantities in question (but certainly in the case that $\\norm{AB^{-1}}\\ll\\norm{BA^{-1}}$), we transform differently from \\eqref{app:eq:est_conv_def},\n\t%\n\t\\begin{align}\n\t\t\\abs{[f*g](\\vec t)}\n\t\t&\\lesssim \\int \\parens*{\\abs*{A(\\vec t-\\vec x+\\vec r)}^2+c^2}^{-m} \\parens*{\\abs*{B(\\vec x+\\vec s)}^2+d^2}^{-n} \\d \\vec x \\\\\n\t\t&=\\frac{1}{\\abs{\\det A}} \\int \\parens*{\\abs{\\vec y}^2+c^2}^{-m} \\parens*{\\abs*{B(\\vec t+\\vec s-A^{-1}\\vec y+\\vec r}^2+d^2}^{-n} \\d \\vec y \\\\\n\t\t&=\\frac{1}{\\abs{\\det A}} \\int \\parens*{\\abs{\\vec y}^2+c^2}^{-m} \\parens*{\\abs*{BA^{-1}\\vec y - A(\\vec r+ \\vec s + \\vec t)}^2+d^2}^{-n} \\d \\vec y.\n\t\\end{align}\n\t%\n\tProceeding like before, this means that the convolution \\emph{also} satisfies\n\t%\n\t\\begin{align}\n\t\t\\abs{[f*g](\\vec t)}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$\\abs{[f*g](\\vec t)}$}-\\multlinegap]\n\t\t\t\\lesssim \\frac{1}{\\abs{\\det A}} \\biggl( \\frac{\\norm[]{AB^{-1}}^{2n}}{\\parens*{|A(\\vec r+ \\vec s + \\vec t)|^2+c^2+\\|AB^{-1}\\|^2 d^2}^{n}} \\frac{1}{c^{2m-k-1}} + \\ldots \\\\\n\t\t\t\\ldots+\\frac{\\norm[]{AB^{-1}}^{k}}{\\parens*{|A(\\vec r+ \\vec s + \\vec t)|^2+c^2+\\|AB^{-1}\\|^2 d^2}^{m}} \\frac{1}{d^{2n-k-1}} \\biggr).\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tand we can choose the one that is smaller.\n\\end{proof}\n\n\\subsection{Some Basic Generating Function Theory}\\label{ssec:genfunc}\n\nThe main idea of the approach of generating functions --- see e.g.~\\cite{wilf} --- can be described as follows: take a sequence $g_n$ (recursively defined, for example) and calculate\n\\begin{align}\n\tG(x):=\\ops{g_n}{x^n}:=\\sum_{n\\ge 0} g_n x^n,\n\\end{align}\nwhere ``ops'' stands for \\emph{ordinary power series} (as opposed to exponential power series, which we will not need).\nIf $G$ can be identified with a known function, $g_n$ can be recovered as\n\\begin{align}\\label{app:eq:def_coeff_op}\n\tg_n=\\frac{1}{n!}\\Dn{n}[G(x)]{x}\\biggr|_{x=0}=: \\coeff{x^n} G(x),\n\\end{align}\nwhich is often possible, even if the recursion for $g_n$ cannot be resolved by induction. The notation $[x^n]$ will henceforth denote the $n$\\nth coefficient of $G$ with respect to $x$. Two properties follow immediately from the definition,\n\\begin{align}\n\t\\coeff{x^n}(x^k G(x)) &= \\coeff*{x^{n-k}} G(x), \\label{app:eq:coeff_shift} \\\\\n\t\\coeff{x^n}G(\\beta x) &=\\beta^n\\coeff{x^n}G(x), \\label{app:eq:coeff_factor}\n\\end{align}\nwhere $\\beta\\in\\bbR$, as a simple consequence of the chain rule.\n\nThe main tool to calculate $G$ are basic identities from the theory of power series, with the advantage that we can do all calculations purely formally at first, while ultimately, if the resulting $G$ turns out to be convergent in a ball of radius $R>0$ then all our formal calculations are actually justified analytically as well. This kind of freedom is especially useful if $g_n$ is itself a partial sum of the sequence $f_{n,k}$, since we can freely interchange the order of summation, i.e.\n\\begin{align}\n\tG(x):=\\sum_{n\\ge 0} \\sum_{k=0}^n f_{n,k} x^n = \\sum_{n\\ge 0} \\sum_{k\\ge 0} \\mathbbm{1}_{\\{k\\le n\\}} f_{n,k} x^n = \\sum_{k\\ge 0} \\sum_{n\\ge k} f_{n,k} x^n = \\sum_{k\\ge 0} x^k \\smash{\\overbrace{\\sum_{n\\ge 0} f_{n-k,k} x^n}^{=F_k(x)}}.\n\\end{align}\nIf --- as will often be the case --- we can calculate $F_k(x)$ and consequently $G(x)$, we may then find $g_n=\\sum_{k=0}^n f_{n,k}$ as $\\coeff{x^n} G$.\n\nWe consider --- again only formally --- a function $F(x)=\\ops{f_n}{x^n}$ generated by $\\{f_n\\}$, as well as another one generated by $\\{g_n\\}$, $G(x)=\\ops{g_n}{x^n}$; then\n\\begin{align}\n\t\\ops{f_{n+1}}{x^n}&=\\frac{F(x)-F(0)}{x}, \\label{app:eq:ops_shift}\\\\\n\tF(x)G(x)&=\\ops[\\bigg]{\\sum_{k=0}^n f_n g_{n-k}}{x^n}. \\label{app:eq:ops_prod}\n\\end{align}\n\nFurthermore, the last essential ingredient is having the identification of as many power series as possible with known functions. But before we define the binomial coefficient for $\\alpha\\in\\bbC$ (we will only need $\\alpha\\in\\bbR$) and $n\\in\\bbN_0$,\n\\begin{align}\\label{eq:binom_alpha}\n\t\\binom{\\alpha}{n}:=\\frac{\\alpha(\\alpha-1)\\cdot\\ldots\\cdot(\\alpha-n+1)}{n!}= \\frac{\\Gamma(\\alpha+1)}{\\Gamma(n+1)\\Gamma(\\alpha-n+1)},\n\\end{align}\nwhere the last equality holds in the limit $\\alpha'\\to\\alpha$ if one of the $\\Gamma$-terms has a singularity at $\\alpha$.\nBy reversing the signs and order of the factors in the numerator, we see that the following identity holds\n\\begin{align}\\label{app:eq:binom_alpha}\n\t\\binom{\\alpha}{n}=(-1)^n\\binom{n-\\alpha-1}{n}.\n\\end{align}\nWe only list the power series identities we will need (\\cite[Sec 2.5]{wilf}):\n\\begin{alignat}{2}\n\t\\frac{1}{1-x} & =\\sum_{n\\ge 0} x^n, && \\label{app:eq:ops_geom} \\\\\n\t\\frac{1}{(1-x)^{k+1}} & =\\sum_{n\\ge 0} \\binom{n+k}{n} x^n, && k\\in\\bbN_0, \\label{app:eq:ops_geom_k} \\\\\n\t(1+x)^\\alpha & =\\sum_{n\\ge 0} \\binom{\\alpha}{n} x^n, &\\quad& \\alpha\\in\\bbR, \\label{app:eq:ops_geom_alpha} \\\\\n\t\\frac{1}{\\sqrt{1-4x}} & =\\sum_{n\\ge 0}\\binom{2n}{n} x^n, && \\label{app:eq:ops_sqrt} \\\\\n\t\\parens[\\bigg]{\\frac{1-\\sqrt{1-4x}}{2x}}^k & =\\sum_{n\\ge 0} \\frac{k}{n+k}\\binom{2n+k-1}{n} x^n, && k\\in\\bbN. \\label{app:eq:ops_sqrt_k}\n\\end{alignat}\nNote that, due to \\eqref{app:eq:binom_alpha}, \\eqref{app:eq:ops_geom_alpha} generalises both \\eqref{app:eq:ops_geom} and \\eqref{app:eq:ops_geom_k}.\n\n\\subsection{Partial Fraction Decomposition}\n\nBefore we can think about the integration in $I_{m,n}$, we first need to figure out the partial fraction decomposition (PFD) of the integrand.\n\n\\begin{proposition}\\label{app:prop:pfd}\n\tUsing\n\t%\n\t\\begin{align}\\label{app:eq:pfd_Delta}\n\t\t\\Delta:=\\delta_+\\delta_-:=((c+ad)^2+a^2 b^2)((c-ad)^2+a^2 b^2)\n\t\\end{align}\n\t%\n\tit holds that\n\t%\n\t\\begin{align}\n\t\\begin{split}\\label{app:eq:pfd}\n\t\tP_{m,n}&:=\\frac{1}{\\parens{a^2(x-b)^2+c^2}^m} \\frac{1}{\\parens{x^2+d^2}^n} = \\\\\n\t\t&\\phantom{:}= \\sum_{k=1}^{m} \\frac{a^{2n}}{\\Delta^{n+k-1}} \\frac{r^n_k + a^2b\\,x\\,s^n_k}{(a^2(x-b)^2+c^2)^{m-k+1}} + \\sum_{\\ell=1}^{n} \\frac{a^{2(\\ell-1)}}{\\Delta^{m+\\ell-1}} \\frac{t^\\ell_m +a^2b\\,x\\,u^\\ell_m}{(x^2+d^2)^{n-\\ell+1}},\n\t\\end{split}\n\t\\end{align}\n\t%\n\twhere, for $k,\\ell\\in\\bbN$, the coefficients can be calculated recursively:\n\t%\n\t\\mathtoolsset{showonlyrefs=false}\n\t\\begin{subequations}\\label{app:eqs:pfd_recursion}\n\t\\begin{align}\n\t\tr^\\ell_k&=\\sum_{k'=1}^k r^1_{k-k'+1}r^{\\ell-1}_{k'} + a^2b^2 (\\Delta u^1_{k-k'}- (a^2b^2+c^2) u^1_{k-k'+1}) u^{\\ell-1}_{k'} \\label{app:eq:pfd_recursion:r} \\\\\n\t\tt^\\ell_k&=\\sum_{k'=1}^k t^1_{k-k'+1}r^{\\ell-1}_{k'} + a^4b^2d^2u^1_{k-k'+1} u^{\\ell-1}_{k'} \\label{app:eq:pfd_recursion:t} \\\\\n\t\tu^\\ell_k&=\\sum_{k'=1}^k  u^{1}_{k-k'+1} r^{\\ell-1}_{k'}-t^1_{k-k'+1}u^{\\ell-1}_{k'}  = -s^\\ell_k \\label{app:eq:pfd_recursion:u}\n\t\\end{align}\n\t\\end{subequations}\n\t%\n\tThe calculation first needs to resolve $\\ell\\to\\ell-1\\to\\ldots\\to 1$ back to initial values\n\t%\n\t\\begin{subequations}\\label{app:eqs:pfd_init}\n\t\\begin{align}\n\t\tr^1_k&=r^1_1 t^{1}_{k-1} + a^2b^2(a^2b^2+c^2) u^1_1 u^1_{k-1}, \\label{app:eq:pfd_init:r}\\\\\n\t\tt^1_k&=t^1_1 t^{1}_{k-1} - a^4b^2d^2 u^1_1 u^1_{k-1}, \\label{app:eq:pfd_init:t}\\\\\n\t\tu^1_k&=u^1_1 t^{1}_{k-1} + t^1_1 u^1_{k-1} = -s^1_k, \\label{app:eq:pfd_init:u}\n\t\\end{align}\n\t\\end{subequations}\n\t%\n\twhich themselves can be resolved by recurring back (in $k$) to\n\t%\n\t\\begin{subequations}\\label{app:eqs:pfd_init_init}\n\t\\begin{align}\n\t\tr^1_1&=3a^2b^2+a^2d^2-c^2, & && r^1_0&=-1, \\label{app:eq:pfd_init_init:r}\\\\\n\t\tt^1_1&=\\phantom{3}a^2b^2-a^2d^2+c^2, &\\text{resp.}&& t^1_0&=1, \\label{app:eq:pfd_init_init:t}\\\\\n\t\tu^1_1&=2 = -s^1_1, & && u^1_0&=0=s^1_0. \\label{app:eq:pfd_init_init:u}\n\t\\end{align}\n\t\\end{subequations}\\mathtoolsset{showonlyrefs=true}%\n\t%\n\tFurthermore, we have the following important relation between the coefficients\n\t%\n\t\\begin{align}\\label{app:eq:r_plus_t_eq_u}\n\t\tr^\\ell_k+t^\\ell_k= 2a^2b^2 u^\\ell_k.\n\t\\end{align}\n\\end{proposition}\n\n\\begin{proof}\n\tFirst off, we calculate the PFD for the case $m,n=1$,\n\t%\n\t\\begin{align\n\t\t\\frac{1}{a^2(x-b)^2+c^2} \\frac{1}{x^2+d^2} = \\frac{a^{2}}{\\Delta} \\frac{3a^2 b^2+a^2 d^2-c^2 -2a^2 b\\,x}{a^2(x-b)^2+c^2} + \\frac{1}{\\Delta} \\frac{a^2 b^2-a^2 d^2+c^2 +2a^2 b\\,x}{x^2+d^2}\n\t\\end{align}\n\t%\n\tfrom which we can read off the left half of \\eqref{app:eqs:pfd_init_init}. Next, assuming \\eqref{app:eq:pfd} for $m$ and $n=1$ we use induction in $m$,\n\t%\n\t\\begin{align}\\label{app:eq:pfd_indm_hyp}\n\t\tP_{m+1,1}\n\t\t&= \\sum_{k=1}^{m} \\frac{a^{2}}{\\Delta^{k}} \\frac{r^1_k +a^2b\\,x\\,s^1_k}{(a^2 (x-b)^2+c^2)^{m-k+2}} +  \\frac{1}{\\Delta^{m}} \\frac{t^1_m +a^2b\\,x\\,u^1_m}{(a^2y^2+c^2)(x^2+d^2)}.\n\t\\end{align}\n\t%\n\tWe continue by splitting the second term with the help of \\eqref{app:eqs:pfd_init_init},\n\t%\n\t\\begin{align}\n\t\\MoveEqLeft\n\t\t\\frac{1}{\\Delta^{m}} \\frac{t^1_m +a^2b\\,x\\,u^1_m}{(a^2(x-b)^2+c^2)(x^2+d^2)} \\\\\n\t\t&= \\frac{a^{2}}{\\Delta^{m+1}} \\frac{(r^1_1 +a^2b\\,x\\,s^1_1)(t^1_m +a^2b\\,x\\,u^1_m)}{a^2 (x-b)^2+c^2} + \\frac{1}{\\Delta^{m+1}} \\frac{(t^1_1 +a^2b\\,x\\,u^1_1)(t^1_m +a^2b\\,x\\,u^1_m)}{x^2+d^2}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-2em-\\multlinegap]\n\t\t\t=\\frac{a^{2}}{\\Delta^{m+1}} \\frac{r^1_1 t^1_m +a^2b\\,x(r^1_1 u^1_m+s^1_1 t^1_m)+a^4 b^2 x^2 s^1_1 u^1_m}{a^2 (x-b)^2+c^2} +\\ldots \\\\\n\t\t\t\\ldots+ \\frac{1}{\\Delta^{m+1}} \\frac{t^1_1 t^1_m +a^2b\\,x(t^1_1 u^1_m+u^1_1 t^1_m)+a^4 b^2 x^2 u^1_1 u^1_m)}{x^2+d^2}\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tConsidering \\eqref{app:eq:pfd_init_init:u}, we collect the $x^2$-terms in the numerators,\n\t%\n\t\\begin{align}\n\t\t\\frac{2a^4 b^2 u^1_m}{\\Delta^{m+1}} \\parens[\\bigg]{\\frac{x^2}{x^2+d^2} - \\frac{a^2 x^2}{a^2 (x-b)^2 + c^2}} =\\frac{2a^4 b^2 u^1_m}{\\Delta^{m+1}} \\parens[\\bigg]{1 - \\frac{d^2}{x^2+d^2} - 1 + \\frac{a^2b^2+c^2-2a^2b\\,x}{a^2 (x-b)^2 + c^2}}.\n\t\\end{align}\n\t%\n\tComing back to \\eqref{app:eq:pfd_indm_hyp}, we set the abbreviation $f_1(x):= a^2 (x-b)^2 + c^2$ for the first denominator, and compute that\n\t%\n\t\\begin{multline}\n\t\tP_{m+1,1} = \\sum_{k=1}^{m} \\frac{a^{2n}}{\\Delta^{k}} \\frac{r^1_k +a^2b\\,x\\,s^1_k}{f_1(x)^{m-k+2}} +   \\frac{\\smash{\\overbrace{t^1_1 t^1_m - a^4 b^2 d^2 u^1_1 u^1_m}^{t^1_{m+1}}}+ a^2 b\\,x (\\smash{\\overbrace{t^1_1 u^1_m + u^1_1 t^1_m}^{u^1_{m+1}}})}{\\Delta^{m+1}(x^2+d^2)}  + \\ldots\\\\\n\t\t\\ldots+\\frac{a^{2}}{\\Delta^{m+1}f_1(x)} \\parens*{{\\underbrace{r^1_1 t^1_m\\! + a^2 b^2 (a^2b^2\\!+c^2) u^1_1 u^1_m}_{r^1_{m+1}}}\\!+ a^2 b\\,x ({\\underbrace{r^1_1 u^1_m\\! + s^1_1 t^1_m-4a^2b^2 u^1_m}_{=-u^1_1 t^1_m-t^1_1 u^1_m=s^1_{m+1}}})},\n\t\\end{multline}\n\t%\n\tbecause $r^1_1-4a^2 b^2 =-t^1_1$ and $s^1_1=-u^1_1$. This coincides with the recurrences in \\eqref{app:eqs:pfd_init}, as claimed.\n\t\n\tAs claimed in \\eqref{app:eq:r_plus_t_eq_u}, the identity $r^1_1+t^1_1= 4a^2 b^2$ can be extended (here first for $\\ell=1$),\n\t%\n\t\\begin{align}\n\t\\begin{split}\\label{app:eq:r_plus_t_eq_u:one}\n\t\tr^1_k+t^1_k\n\t\t&=(r^1_1+t^1_1) t^1_{k-1} +2a^2 b^2 (a^2 b^2-a^2d^2+c^2) u^1_{k-1}\\\\\n\t\t&= 4a^2b^2 t^1_{k-1}+2a^2 b^2 t^1_1 u^1_{k-1}=2a^2b^2u^1_k,\n\t\\end{split}\n\t\\end{align}\n\t%\n\twhich will help cut short some computations below.\n\t\n\tNow we come to the induction in $n$. The case for arbitrary $m$ and $n=1$ has been proved above, which covers the base case. Under the induction hypothesis that \\eqref{app:eq:pfd} holds for $m,n\\in\\bbN$, we have\n\t%\n\t\\begin{align}\\label{app:eq:pfd_indn_hyp}\n\t\tP_{m,n+1}\n\t\t= \\sum_{k=1}^{m} \\frac{1}{\\Delta^{n+k-1}} \\frac{a^{2n}(r^n_k + a^2b\\,x\\,s^n_k)}{f_1(x)^{m-k+1}}\\frac{1}{x^2+d^2} + \\sum_{\\ell=1}^{n} \\frac{1}{\\Delta^{m+\\ell-1}} \\frac{a^{2(\\ell-1)}(t^\\ell_m +a^2b\\,x\\,u^\\ell_m)}{(x^2+d^2)^{n-\\ell+2}}.\n\t\\end{align}\n\t%\n\tClearly, we can now apply our previous knowledge (the case with $n=1$), i.e.\n\t%\n\t\\begin{multline}\n\t\t\\frac{1}{(a^2(x-b)^2+c^2)^{m-k+1}}\\frac{1}{x^2+d^2}\\\\\n\t\t=\\sum_{k'=1}^{m-k+1} \\frac{1}{\\Delta^{k'}} \\frac{a^{2}(r^1_{k'} +a^2b\\,x\\,s^1_{k'})}{(a^2 (x-b)^2+c^2)^{m-k+1-k'+1}} +  \\frac{1}{\\Delta^{m-k+1}} \\frac{t^1_{m-k+1} +a^2b\\,x\\,u^1_{m-k+1}}{x^2+d^2}\n\t\\end{multline}\n\t%\n\tInserting this into \\eqref{app:eq:pfd_indn_hyp} yields\n\t%\n\t\\begin{align}\n\t\\MoveEqLeft\n\t\t\\sum_{k=1}^{m} \\frac{a^{2n}}{\\Delta^{n+k-1}} \\frac{r^n_k + a^2b\\,x\\,s^n_k}{(a^2(x-b)^2+c^2)^{m-k+1}}\\frac{1}{x^2+d^2} \\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-2em-\\multlinegap]\n\t\t\t=\\sum_{k=1}^{m}\\sum_{k'=1}^{m-k+1} \\frac{a^{2(n+1)}}{\\Delta^{n+k+k'-1}} \\frac{r^1_{k'} r^n_k + a^2b\\,x(r^1_{k'} s^n_k+s^1_{k'}r^n_k)+a^4 b^2 x^2 s^1_{k'}s^n_k}{(a^2(x-b)^2+c^2)^{m-k-k'+2}} + \\ldots \\\\\n\t\t\t\\ldots + \\sum_{k=1}^{m} \\frac{a^{2n}}{\\Delta^{m+n}} \\frac{t^1_{m-k+1} r^n_k +a^2b\\,x(t^1_{m-k+1}s^n_k + u^1_{m-k+1} r^n_k)+a^4 b^2 x^2 u^1_{m-k+1}s^n_k}{x^2+d^2}.\n\t\t\n\t\t\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tWe replace the undesired quadratic part $a^4b^2x^4$ depending on the denominator of the term we're dealing with,\n\t%\n\t\\begin{align}\n\t\ta^4 b^2 x^2&=a^2 b^2 \\parens*{a^2 (x-b)^2 + c^2}\\parens[\\Big]{1-\\frac{a^2b^2+c^2-2a^2b\\,x}{a^2 (x-b)^2 +c^2}},\\\\\n\t\ta^4 b^2 x^2&=a^4 b^2 \\parens*{x^2+ d^2}\\parens[\\Big]{1-\\frac{d^2}{x^2+d^2}},\n\t\\end{align}\n\t%\n\tand change the summation indices as follows (demonstrated for only one term):\n\t%\n\t\\begin{align}\n\t\t\\sum_{k=1}^{m}\\sum_{k'=1}^{m-k+1} r^1_{k'} r^n_k\n\t\t&=\\sum_{k=0}^{m-1}\\sum_{k'=1}^{m-k} r^1_{k'} r^n_{k+1}\n\t\t=\\sum_{k=0}^{m-1}\\sum_{k'=0}^{m-k-1}  r^1_{k'+1} r^n_{k+1}\n\t\t=\\sum_{k''=0}^{m-1}\\sum_{k+k'=k''}  r^1_{k+1} r^n_{k'+1}\\\\\n\t\t&=\\sum_{k''=0}^{m-1}\\sum_{k'=0}^{k''} r^1_{k''-k'+1} r^n_{k'+1}\n\t\t=\\sum_{k=1}^{m}\\sum_{k'=0}^{k-1} r^1_{k-k'} r^n_{k'+1}\n\t\t=\\sum_{k=1}^{m}\\sum_{k'=1}^{k}  r^1_{k-k'+1} r^n_{k'}\n\t\\end{align}\n\t%\n\tThis leads to\n\t%\n\t\\begin{align}\n\t\\MoveEqLeft\n\t\t\\sum_{k=1}^{m} \\frac{a^{2n}}{\\Delta^{n+k-1}} \\frac{r^n_k + a^2b\\,x\\,s^n_k}{(a^2(x-b)^2+c^2)^{m-k+1}}\\frac{1}{x^2+d^2} \\\\\n\t\t&= \\sum_{k=1}^{m} \\frac{a^{2(n+1)}}{\\Delta^{n+k}} \\biggl(\\frac{\\sum_{k'=1}^{k} r^1_{k-k'+1} r^n_{k'}-a^2 b^2 (a^2b^2+c^2) s^1_{k-k'+1}s^n_{k'}}{(a^2(x-b)^2+c^2)^{m-k+1}} + \\ldots\\\\\n\t\t&\\mathrel{\\phantom{=}}\\ldots + \\frac{a^2b\\,x\\sum_{k'=1}^{k} s^1_{k-k'+1}r^n_{k'}+(r^1_{k-k'+1}-2a^2b^2 u^1_{k-k'+1})s^n_{k'}}{(a^2(x-b)^2+c^2)^{m-k+1}}  + \\ldots \\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-2em-\\multlinegap]\n\t\t\t\\mathrel{\\phantom{=}}\\ldots + \\frac{\\sum_{k'=1}^{k} a^2 b^2 s^1_{k-k'+1}s^n_{k'}}{(a^2(x-b)^2+c^2)^{m-k}} \\biggr)  + \\sum_{k'=1}^{m} \\frac{a^{2n}}{\\Delta^{m+n}} \\biggl(\\frac{t^1_{m-k'+1} r^n_{k'} + a^4 b^2 d^2 s^1_{m-k'+1}s^n_{k'}}{x^2+d^2}+ \\ldots \\\\\n\t\t\t\\mathrel{\\phantom{=}}\\ldots + \\frac{a^2b\\,x(t^1_{m-k'+1}s^n_{k'} + u^1_{m-k'+1} r^n_{k'})}{x^2+d^2} +  a^4 b^2 u^1_{m-k'+1}s^n_{k'} \\biggr).\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tThe terms with $x^2+d^2$ in the denominator exactly match the claimed $t$- and $u$-terms from \\eqref{app:eqs:pfd_recursion} for $k=m$ and $\\ell=n+1$, and thus we only have to deal with the remaining terms having powers of $a^2(x-b)^2+c^2$ in the denominator (as well as the very last term). To harmonise those powers, we perform an index shift for the third term (except the last summand, which will cancel with the very last term above), to arrive at\n\t%\n\t\\begin{align}\n\t\t\\sum_{k=2}^{m} \\frac{a^{2(n+1)}}{\\Delta^{n+k-1}} \\frac{\\sum_{k'=1}^{k-1} a^2 b^2 s^1_{k-k'}s^n_{k'}}{(a^2(x-b)^2+c^2)^{m-k+1}} + \\frac{a^{2(n+1)}}{\\Delta^{m+n}} \\sum_{k'=1}^{m} a^2 b^2 s^n_{k'} (\\smash{\\overbrace{s^1_{m-k'+1}+u^1_{m-k'+1}}^{=0}}).\n\t\\end{align}\n\t%\n\tHere, we can extend the summation to $\\sum_{k=1}^m\\sum_{k'=1}^k$, because all additional terms are zero.\n\t\n\tFurthermore, due to \\eqref{app:eq:r_plus_t_eq_u:one}, we can simplify the second term (containing all terms with the factor $a^2 b\\,x$) to\n\t%\n\t\\begin{align}\n\t\t\\sum_{k=1}^{m} \\sum_{k'=1}^{k} \\frac{a^{2(n+1)}}{\\Delta^{n+k}} \\frac{a^2b\\,x(t^1_{k-k'+1} u^n_{k'}-u^1_{k-k'+1}r^n_{k'})}{(a^2(x-b)^2+c^2)^{m-k+1}}.\n\t\\end{align}\n\t%\n\tTherefore,\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\tP_{m,n+1}= \\sum_{k=1}^{m} \\frac{a^{2(n+1)}}{\\Delta^{n+k}} \\biggl( \\frac{\\sum_{k'=1}^{k} r^1_{k-k'+1} r^n_{k'}+a^2 b^2 u^n_{k'}(\\Delta u^1_{k-k'} - (a^2b^2+c^2) u^1_{k-k'+1})}{(a^2(x-b)^2+c^2)^{m-k+1}} +\\ldots \\\\\n\t\t\t\\ldots+ \\frac{a^2b\\,x\\sum_{k'=1}^{k} t^1_{k-k'+1} u^n_{k'}-u^1_{k-k'+1}r^n_{k'}}{(a^2(x-b)^2+c^2)^{m-k+1}} \\biggr) + \\sum_{\\ell=1}^{n+1} \\frac{a^{2(\\ell-1)}}{\\Delta^{m+\\ell-1}} \\frac{t^\\ell_m +a^2b\\,x\\,u^\\ell_m}{(x^2+d^2)^{n-\\ell+2}},\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\twhich now also matches the $r$- and $s$-terms in \\eqref{app:eqs:pfd_recursion}, and in particular also $u^{n+1}_m=-s^{n+1}_m$.\n\t\n\tFinally, we want to extend \\eqref{app:eq:r_plus_t_eq_u:one} to \\eqref{app:eq:r_plus_t_eq_u}, which can be done as follows,\n\t%\n\t\\begin{align\n\t\tr^\\ell_k+t^\\ell_k\n\t\t&=\\sum_{k'=1}^k (r^1_{k-k'+1}+t^1_{k-k'+1})r^{\\ell-1}_{k'} + a^2b^2 (\\Delta u^1_{k-k'}- t^1_1 {\\underbrace{u^1_{k-k'+1}}_{=\\mathrlap{u^1_1 t^{1}_{k-k'} + t^1_1 u^1_{k-k'}}}}) u^{\\ell-1}_{k'}\\\\[-0.245cm]\n\t\t&=\\sum_{k'=1}^k 2a^2b^2 u^1_{k-k'+1} r^{\\ell-1}_{k'} + a^2b^2 \\parens*{\\underbrace{(\\overbrace{\\Delta-(t^1_1)^2}^{=4a^4b^2d^2}) u^1_{k-k'}- 2t^1_1 t^1_{k-k'}}_{=-2t^1_{k-k'+1}}} u^{\\ell-1}_{k'}=2a^2b^2 u^\\ell_k.\\tag*{\\qedhere}\n\t\\end{align}\n\\end{proof}\n\n\\begin{remark}\n\tWe note that the recursion \\eqref{app:eqs:pfd_init} is formulated in terms of $t$- and $u$-terms because we chose to induce in $m$ first --- inducting over $n$ first would lead to a different update rule \\eqref{app:eqs:pfd_init}! \n\\end{remark}\n\n\\subsection{Generating Functions for the PFD}\n\nThe following result about coupled recursions answers an obvious question to ask in this context, and as such, is certainly known already. However, since the solution is quite trivial, we have not searched for a reference, but rather proved it ourselves.\n\n\\begin{proposition}\\label{app:prop:genfunc_2d_rec}\n\tFor a recursion\n\t%\n\t\\begin{align}\n\t\t\\binom{g_k}{h_k}=M \\binom{g_{k-1}}{h_{k-1}} = M^{k-i} \\binom{g_i}{h_i} \\qquad \\text{with} \\qquad M=\\begin{pmatrix}\n\t\t\tm_{1,1}& m_{1,2}\\\\ m_{2,1} & m_{2,2}\n\t\t\\end{pmatrix},\n\t\\end{align}\n\t%\n\twith arbitrary $M$ and initial value $\\binom{g_i}{h_i}$, where $i\\ge 0$, the generating functions $G(y)=\\ops*{g_k}{y^k}$ and $H(y)=\\ops*{h_k}{y^k}$ are given by\n\t%\n\t\\begin{align}\\label{app:eq:genfunc_2d_rec}\n\t\tG(y)=y^i \\frac{(m_{1,2}h_i-m_{2,2}g_i)y+g_i}{\\det(M) y^2 - \\mathrm{tr}(M) y +1}, \\qquad H(y)=y^i \\frac{(m_{2,1}g_i- m_{1,1}h_i)y+h_i}{\\det(M) y^2 - \\mathrm{tr}(M) y +1}.\n\t\\end{align}\n\\end{proposition}\n\n\\begin{proof}\n\tLet us begin by assuming that $M$ is diagonalisable. Then there exists an invertible matrix $Q$ such that\n\t%\n\t\\begin{align}\n\t\tM=Q\\begin{pmatrix}\n\t\t\t\\lambda_1 & 0 \\\\ 0 & \\lambda_2\n\t\t\\end{pmatrix} Q^{-1}.\n\t\\end{align}\n\t%\n\tSetting $\\binom{g'_i}{h'_i}:=Q^{-1}\\binom{g_i}{h_i}$, we can calculate --- because $g_k=0$ for $k<i$ --- that\n\t%\n\t\\begin{align}\n\t\t\\binom{G(y)}{H(y)}\n\t\t&=\\sum_{k\\ge 0} \\binom{g_{k}}{h_k} y^k \n\t\t= \\sum_{k\\ge i} Q\\begin{pmatrix}\n\t\t\t\\lambda_1^{k-i} & 0 \\\\ 0 & \\lambda_2^{k-i}\n\t\t\\end{pmatrix} \\binom{g'_i}{h'_i} y^k\n\t\t= y^i Q\\sum_{k\\ge 0} \\binom{g'_i\\lambda_1^k y^k}{h'_i\\lambda_2^k y^k} = y^i Q \\binom{\\frac{g'_i}{1-\\lambda_1 y}}{\\frac{h'_i}{1-\\lambda_2 y}}.\n\t\\end{align}\n\t%\n\tSince $\\lambda_1 \\lambda_2=\\det(M)$ and $\\lambda_1+\\lambda_2=\\mathrm{tr}(M)$, this implies\n\t%\n\t\\begin{align}\n\t\t\\binom{G(y)}{H(y)}&\n\t\n\t\t= \\frac{y^i}{\\det(M) y^2 - \\mathrm{tr}(M) y +1} Q \\binom{g'_i(1-\\lambda_2 y)}{h'_i(1-\\lambda_1 y)}.\n\t\\end{align}\n\t%\n\tBy explicitly calculating the eigenvalues and eigenvectors in terms of the $m_{i,j}$, one can check that (note the switched eigenvalues!)\n\t%\n\t\\begin{align}\n\t\tQ\\begin{pmatrix}\n\t\t\t\\lambda_2 & 0 \\\\ 0 & \\lambda_1\n\t\t\\end{pmatrix} Q^{-1} = \\begin{pmatrix}\n\t\t\t\\phantom{-}m_{2,2} & -m_{1,2} \\\\ -m_{2,1} & \\phantom{-}m_{1,1}\n\t\t\\end{pmatrix},\n\t\\end{align}\n\t%\n\tand therefore\n\t%\n\t\\begin{align}\n\t\tQ \\binom{g'_i(1-\\lambda_2 y)}{h'_i(1-\\lambda_1 y)}=Q Q^{-1}\\binom{g_i}{h_i} -\n\t\ty\\begin{pmatrix}\n\t\t\t\\phantom{-}m_{2,2} & -m_{1,2} \\\\ -m_{2,1} & \\phantom{-}m_{1,1}\n\t\t\\end{pmatrix} \\binom{g_i}{h_i}\n\t\t= \\binom{g_i+y(m_{1,2}h_i-m_{2,2}g_i)}{h_i+y(m_{2,1}g_i-m_{1,1}h_i)},\n\t\\end{align}\n\t%\n\twhich finishes the proof for diagonalisable $M$.\n\t\n\tIf $M$ is not diagonalisable, we can choose a sequence $M_n$ of diagonalisable matrices which converge to $M$ in an appropriate norm (e.g.~the Frobenius-norm), since, in the space $\\bbC^{d\\times d}$, non-diagonalisable matrices are a subset of the hypersurface where the discriminant of the characteristic polynomial is zero.\n\t\n\tFor $G_n(y)$ and $H_n(y)$ corresponding to $M_n$, we see that they depend continuously on the coefficients $(m_{i,j})_n$ of $M_n$ --- taking an appropriate norm with respect to $y$ over a sufficiently small ball around the origin (such that the denominator is always positive) --- \\emph{with no blow-up} for $M_n$ converging to non-diagonalisable $M$, and therefore $G_n(y)$ and $H_n(y)$ converge to the $G(y),\\,H(y)$ formed with $M$. This finishes the proof.\n\\end{proof}\n\n\\begin{proposition}\\label{app:prop:genfunc}\n\tLetting $q:=a^2b^2+a^2d^2-c^2$, we can calculate the generating functions for the coefficients of the PFD of $P_{m,n}$,\n\t%\n\t\\mathtoolsset{showonlyrefs=false}\n\t\\begin{subequations}\\label{app:eqs:pfd_genfunc}\n\t\\begin{alignat}{3}\n\t\tR(y,z)&:=\\sum_{\\ell\\ge 1} R^\\ell(y) z^\\ell&&:= \\sum_{\\ell\\ge 1} \\sum_{k\\ge 1} r^\\ell_k y^k z^\\ell &&=yz\\frac{\\phantom{-}\\Delta(y-z) +r^1_1}{\\Delta (y-z)^2 - 2t^1_1 y -2q z+1},\\hspace{-0.5cm} \\label{app:eq:pfd_genfunc:r} \\\\\n\t\tT(y,z)&:=\\sum_{\\ell\\ge 1} T^\\ell(y) z^\\ell&&:= \\sum_{\\ell\\ge 1} \\sum_{k\\ge 1} t^\\ell_k y^k z^\\ell &&=yz\\frac{-\\Delta(y-z)+t^1_1}{\\Delta (y-z)^2 - 2t^1_1 y -2q z+1},\\hspace{-0.5cm} \\label{app:eq:pfd_genfunc:t} \\\\\n\t\tU(y,z)&:=\\sum_{\\ell\\ge 1} U^\\ell(y) z^\\ell&&:= \\sum_{\\ell\\ge 1} \\sum_{k\\ge 1} u^\\ell_k y^k z^\\ell &&=yz\\frac{2}{\\Delta (y-z)^2 - 2t^1_1 y -2q z+1},\\hspace{-0.5cm} \\label{app:eq:pfd_genfunc:u}\n\t\\end{alignat}\n\t\\end{subequations}\\mathtoolsset{showonlyrefs=true}%\n\t%\n\tand $U(y,z)=-S(y,z)$. Then,\n\t%\n\t\\begin{align}\n\t\tr^\\ell_k = \\coeff*{y^k z^\\ell} R(y,z), \\qquad t^\\ell_k = \\coeff*{y^k z^\\ell} T(y,z), \\qquad u^\\ell_k = \\coeff*{y^k z^\\ell} U(y,z).\n\t\\end{align}\n\\end{proposition}\n\n\\begin{proof}\n\tWe begin by considering $T^1$ and $U^1$ --- if we write \\eqref{app:eq:pfd_init:t} and \\eqref{app:eq:pfd_init:u} in matrix form, we see that\n\t%\n\t\\begin{align}\n\t\t\\binom{t^1_{k}}{u^1_k}=M\\binom{t^1_{k-1}}{u^1_{k-1}} = M^{k-1} \\binom{t^1_1}{u^1_1}\n\t\t\\qquad \\text{with} \\qquad\n\t\tM:= \\begin{pmatrix}\n\t\t\tt^1_1 & -2a^4 b^2d^2 \\\\ 2 & t^1_1\n\t\t\\end{pmatrix}.\n\t\\end{align}\n\t%\n\tEven though going back to initial values $\\binom{r^1_0}{u^1_0}=\\binom{1}{0}$ would be possible, we omit it on purpose, because the case for $m=0$ is not of interest to us (compare the definition of $I_{m,n}$).\n\t\n\tFortunately, it turns out that we know the determinant of $M$ already from somewhere,\n\t%\n\t\\begin{align}\n\t\t\\det(M)=\\Delta, \\qquad \\mathrm{tr}(M)=2t^1_1,\n\t\\end{align}\n\t%\n\tand therefore, by \\eqref{app:eq:genfunc_2d_rec}, we see that\n\t%\n\t\\begin{align\n\t\tT^1(y)=y\\frac{-\\parens*{{\\overbrace{4a^4b^2d^2+(t^1_1)^2}^{=\\Delta}}} y+t^1_1}{\\Delta y^2 -2t^1_1 y+1}, \\qquad U^1(y)=\\frac{2y}{\\Delta y^2 -2t^1_1 y+1} \n\t\\end{align}\n\t%\n\tSince \\eqref{app:eq:r_plus_t_eq_u} implies that $R^1(y)+T^1(y)=2a^2b^2 U^1(y)$, we can easily calculate\n\t%\n\t\\begin{align\n\t\tR^1(y)=y\\frac{\\Delta y+r^1_1}{\\Delta y^2 -2t^1_1 y+1}.\n\t\\end{align}\n\t\n\tNow we come to the full recursion, recalling that $R^{\\ell}(y)=\\sum_{k\\ge 1} r^\\ell_k y^k$ (and correspondingly for $U$, $T$) --- which effectively means that we have set $r^\\ell_0$ to zero for the function $R^\\ell$, resp.~$R$.\n\tStarting from \\eqref{app:eq:pfd_recursion:r} and applying \\eqref{app:eq:ops_shift} --- which is now easier due to the choice regarding $r^\\ell_0$ --- it follows that\n\t%\n\t\\begin{align}\n\t\t\\frac{R^\\ell(y)}{y}\n\t\t&= \\sum_{k\\ge 0} r^\\ell_{k+1} y^{k}\n\t\t=\\sum_{k\\ge 0}  y^{k} \\sum_{k'=1}^{k+1} r^1_{k-k'+2}r^{\\ell-1}_{k'} + a^2b^2 (\\Delta u^1_{k-k'+1}- (a^2b^2+c^2) u^1_{k-k'+2}) u^{\\ell-1}_{k'}.\n\t\\end{align}\n\t%\n\tWe treat the terms separately, using \\eqref{app:eq:ops_prod}, to compute\n\t%\n\t\\begin{align}\n\t\t\\sum_{k\\ge 0} y^k  \\sum_{k'=1}^{k+1} r^1_{k-k'+2}r^{\\ell-1}_{k'}\n\t\t&= \\sum_{k\\ge 0} y^k  \\sum_{k'=0}^{k} r^1_{k-k'+1}r^{\\ell-1}_{k'+1}\n\t\t= \\parens[\\bigg]{\\sum_{k\\ge 0}r^1_{k+1}y^k}\\parens[\\bigg]{\\sum_{k\\ge 0}r^{\\ell-1}_{k+1}y^k}\n\t\t= \\frac{R^1(y)}{y} \\frac{R^{\\ell-1}(y)}{y}.\n\t\\end{align}\n\t%\n\tApplying same procedure to the other terms, we see that, due to the choice to omit $r^\\ell_0$ (and because $u^\\ell_0$ is zero anyway),\n\t%\n\t\\begin{align}\n\t\tR^\\ell(y)=\\frac{\\Delta y+r^1_1}{\\Delta y^2 -2t^1_1 y+1} R^{\\ell-1}(y)+2a^2b^2\\frac{\\Delta y-a^2b^2-c^2}{\\Delta y^2 -2t^1_1 y+1} U^{\\ell-1}(y).\n\t\\end{align}\n\t%\n\tis a well-defined recursion for $R^\\ell$ with initial value $\\ell=1$. For the $U$-recursion, we proceed in the same way to arrive at\n\t%\n\t\\begin{align}\n\t\tU^\\ell(y)&=\\frac{2}{\\Delta y^2 -2t^1_1 y+1} R^{\\ell-1}(y)+\\frac{\\Delta y - t^1_1}{\\Delta y^2 -2t^1_1 y+1}U^{\\ell-1}(y).\n\t\\end{align}\n\t\n\tThis allows us to again use \\autoref{app:prop:genfunc_2d_rec} with $i=1$ and $z$ instead of $y$, by writing\n\t%\n\t\\begin{align}\n\t\t\\binom{R^\\ell(y)}{U^\\ell(y)}&=\\parens[\\Bigg]{{\\underbrace{\\frac{1}{\\Delta y^2 -2t^1_1 y+1}\\begin{pmatrix} \\Delta y+r^1_1 & 2a^2b^2(\\Delta y-a^2b^2-c^2) \\\\ 2 & \\Delta y - t^1_1 \\end{pmatrix}}_{=:N}}}^{\\ell-1} \\binom{R^1(y)}{U^1(y)}.\n\t\\end{align}\n\t%\n\tAgain, some factors in the determinant cancel,\n\t%\n\t\\begin{align}\n\t\t\\det(N)=\\frac{\\Delta}{\\Delta y^2 -2t^1_1 y+1}, \\qquad \\mathrm{tr}(N)=2\\frac{\\Delta y+\\smash{\\overbrace{a^2b^2+a^2d^2-c^2}^{=q}}}{\\Delta y^2 -2t^1_1 y+1},\n\t\\end{align}\n\t%\n\tand we see that\n\t%\n\t\\begin{align}\n\t\tU(y,z)\n\t\t&=z\\frac{\\Delta y^2 -2t^1_1 y+1}{\\Delta z^2-2(\\Delta y+q)z+\\Delta y^2 -2t^1_1 y+1} \\frac{(\\smash{\\overbrace{N_{2,1}(\\Delta y+r^1_1) -2N_{1,1}}^{=0}})yz+2y}{\\Delta y^2 -2t^1_1 y+1}\\\\\n\t\t&=\\frac{2yz}{\\Delta (y-z)^2 - 2t^1_1 y -2q z+1}, \\label{app:eq:pfd_genfunc:U} \\\\\n\t\tR(y,z)\n\t\t&=z\\frac{(2 N_{1,2} -N_{2,2}(\\Delta y+r^1_1)) yz+(\\Delta y+r^1_1)y}{\\Delta z^2-2(\\Delta y+q)z+\\Delta y^2 -2t^1_1 y+1}\n\t\n\t\t=yz\\frac{\\Delta(y-z) +r^1_1}{\\Delta (y-z)^2 - 2t^1_1 y -2q z+1}, \\label{app:eq:pfd_genfunc:R}\n\t\\end{align}\n\t%\n\tbecause $2 N_{1,2} -N_{2,2}(\\Delta y+r^1_1)=-\\Delta$.\n\t\n\tTo conclude, we can use the fact that \\eqref{app:eq:r_plus_t_eq_u} implies $R(y,z)+T(y,z)=2a^2b^2 U(y,z)$, which implies\n\t%\n\t\\begin{align\n\t\tT(y,z)=yz\\frac{-\\Delta(y-z)+t^1_1}{\\Delta (y-z)^2 - 2t^1_1 y -2q z+1}.\\tag*{\\qedhere}\n\t\\end{align}\n\\end{proof}\n\n\\subsection{Generating Function for \\texorpdfstring{$I_{m,n}$}{Imn}}\n\nWe begin by noting that for values at half-integers, the Gamma function can be evaluated as follows ($m\\in\\bbN_0$)\n\\begin{align}\\label{app:eq:gamma_half_int}\n\t\\Gamma\\parens[\\Big]{\\frac 12+m}=\\frac{(2m)!}{4^m m!}\\sqrt{\\pi}, \\qquad \\Gamma\\parens[\\Big]{\\frac 12-m}=\\frac{(-4)^m m!}{(2m)!}\\sqrt{\\pi},\n\\end{align}\nwhich can be shown by induction using the functional equation $\\Gamma(x+1)=x\\Gamma(x)$ and the well-known fact that $\\Gamma\\parens*{\\frac 12}=\\sqrt{\\pi}$.\n\nThis allows us to explicitly calculate the constants of the integral over one of the terms in the PFD, transforming first with $y=\\frac ac (x-b)$, using the fact that the resulting function is even, and finally transforming with $t=\\frac{1}{y^2+1}$,\n\\begin{align}\n\t\\int_{-\\infty}^{\\infty} \\frac{1}{(a^2 (x-b)^2+c^2)^m} \\d x\n\t&= \\frac{2}{c^{2m}}\\int_{0}^{\\infty} \\frac{1}{(y^2+1)^m} \\frac{c}{a} \\d y= \\frac{2}{a\\,c^{2m-1}}\\int_{1}^{0} t^m \\frac{-1}{2\\sqrt{t}^3\\sqrt{1-t}} \\d t \\\\\n\t&=\\frac{1}{a\\,c^{2m-1}}\\int_{0}^{1} t^{m-\\frac 12 -1} (1-t)^{\\frac 12-1} \\d t=\\frac{1}{a\\,c^{2m-1}} B\\parens[\\Big]{m-\\frac 12,\\frac 12} \\\\\n\t&= \\frac{1}{a\\,c^{2m-1}} \\frac{\\Gamma(m-\\frac 12)\\Gamma(\\frac 12)}{\\Gamma(m)} = \\frac{\\pi}{a\\,c^{2m-1}} \\frac{\\binom{2(m-1)}{m-1}}{4^{m-1}}, \\label{app:eq:single_factor_expl_int}\n\\end{align}\nusing the definition of the Beta function as well as \\eqref{app:eq:gamma_half_int}.\n\n\\begin{proposition}\\label{app:prop:genfunc_Imn}\n\tThe generating function for $I_{m,n}$ is\n\t%\n\t\\begin{align}\n\t\t\\sum_{m\\ge 1}\\sum_{n\\ge 1} I_{m,n} y^m z^n\n\t\t&=\\frac{\\pi yz}{\\sqrt{c^2-\\smash{y}}\\sqrt{d^2-z}} \\frac{\\sqrt{c^2-\\smash{y}}+a\\sqrt{d^2- z}}{(\\sqrt{c^2-\\smash{y}}+a\\sqrt{d^2- z})^2+a^2b^2}.\n\t\\end{align}\n\t%\n\tIn particular, for $m,n\\in\\bbN$, we have\n\t%\n\t\\begin{align}\\label{app:eq:Imn_as_coeff}\n\t\tI_{m,n} =\\coeff*{y^{m-1} z^{n-1}} \\frac{\\pi}{\\sqrt{c^2-\\smash{y}}\\sqrt{d^2-z}} \\frac{\\sqrt{c^2-\\smash{y}}+a\\sqrt{d^2- z}}{(\\sqrt{c^2-\\smash{y}}+a\\sqrt{d^2- z})^2+a^2b^2}.\n\t\\end{align}\n\\end{proposition}\n\n\\begin{proof}\n\tWe begin by inserting \\eqref{app:eq:pfd},\n\t%\n\t\\begin{align}\n\t\tI_{m,n}&=\\int_{-\\infty}^\\infty P_{m,n} \\d x\n\t\t=\\int_{-\\infty}^\\infty \\sum_{k=1}^{m} \\frac{a^{2n}}{\\Delta^{n+k-1}} \\frac{r^n_k + a^2b\\,x\\,s^n_k}{(a^2(x-b)^2+c^2)^{m-k+1}} + \\sum_{\\ell=1}^{n} \\frac{a^{2(\\ell-1)}}{\\Delta^{m+\\ell-1}} \\frac{t^\\ell_m +a^2b\\,x\\,u^\\ell_m}{(x^2+d^2)^{n-\\ell+1}} \\d x.\n\t\\end{align}\n\t%\n\tThe $u$-terms vanish because the integral is an odd function, and for the $s$-terms --- after shifting $x$ accordingly for the first sum --- the same happens. However, we need to pay attention to the terms with $k=m$ and $\\ell=n$, because they are not integrable in $x$ by themselves, but we will see that together, they are.\n\t%\n\t\\begin{multline}\\label{app:eq:int_PFD_split}\n\t\tI_{m,n}\n\t\t=\\int_{-\\infty}^\\infty \\sum_{k=1}^{m-1} \\frac{a^{2n}}{\\Delta^{n+k-1}} \\frac{r^n_k + a^2b^2s^n_k}{(a^2x^2+c^2)^{m-k+1}} + \\sum_{\\ell=1}^{n-1} \\frac{a^{2(\\ell-1)}}{\\Delta^{m+\\ell-1}} \\frac{t^\\ell_m}{(x^2+d^2)^{n-\\ell+1}} + \\ldots\\\\\n\t\t\\ldots+ \\frac{a^{2(n-1)}}{\\Delta^{n+m-1}} \\parens[\\bigg]{\\frac{r^n_m}{a^2x^2+c^2}+\\frac{t^n_m}{x^2+d^2}+a^2 s^n_m \\parens[\\bigg]{\\frac{a^2 b(x+b)}{a^2x^2+c^2}-\\frac{b\\,x}{x^2+d^2}} } \\d x.\n\t\\end{multline}\n\t%\n\tWe begin by treating the last term, first calculating the definite integral for arbitrary $0<X_-,X_+<\\infty$, splitting and transforming appropriately,\n\t%\n\t\\begin{align}\n\t\\MoveEqLeft\n\t\t\\int_{-X_-}^{X_+} \\frac{a^2 b(x+b)}{a^2x^2+c^2}-\\frac{b\\,x}{x^2+d^2} \\d x\\\\\n\t\t&=\\frac{b}{2}\\int_{-X_-}^{X_+} \\frac{2a^2 x}{a^2x^2+c^2} \\d x + \\int_{-X_-}^{X_+}\\frac{a^2 b^2}{a^2x^2+c^2} \\d x-\\frac{b}{2}\\int_{-X_-}^{X_+} \\frac{2x}{x^2+d^2} \\d x \\\\\n\t\t&=\\frac{b}{2}\\parens[\\bigg]{\\int_{0}^{a^2X_+^2}-\\int_{0}^{a^2X_-^2}} \\frac{1}{u+c^2} \\d u + \\frac{a\\, b^2}{c}  \\int_{-\\frac{a}{c}X_-}^{\\frac{a}{c}X_+}  \\frac{1}{u^2+1} \\d u-\\frac{b}{2}\\parens[\\bigg]{\\int_{0}^{X_+^2}-\\int_{0}^{X_-^2}} \\frac{1}{u+d^2} \\d u\\\\\n\t\t&=\\frac{b}{2}\\log\\parens[\\Big]{\\frac{a^2X_+^2+c^2}{a^2X_-^2+c^2}} + \\frac{a\\,b^2}{c}\\parens[\\Big]{\\arctan\\parens[\\Big]{\\frac{a}{c}X_+}-\\arctan\\parens[\\Big]{-\\frac{a}{c}X_-}\\!\\?} -\\frac{b}{2}\\log\\parens[\\Big]{\\frac{X_+^2+d^2}{X_-^2+d^2}} \\\\\n\t\t&=\\frac{b}{2}\\parens[\\Big]{\\log\\parens[\\Big]{\\frac{a^2X_+^2+c^2}{X_+^2+d^2}}-\\log\\parens[\\Big]{\\frac{a^2X_- ^2+c^2}{X_-^2+d^2}}\\!\\?} + \\frac{a\\,b^2}{c}\\parens[\\Big]{\\arctan\\parens[\\Big]{\\frac{a}{c}X_+ }+\\arctan\\parens[\\Big]{\\frac{a}{c}X_-}\\!\\?}.\n\t\\end{align}\n\t%\n\tWe can see that the integral does not only exist as a principal value, but converges to a finite number in $X_-$ and $X_+$ separately, and therefore,\n\t%\n\t\\begin{align}\n\t\t\\lim_{X_-\\to\\infty} \\lim_{X_+\\to\\infty}\n\t\t\\int_{-X_-}^{X_+} \\!\\frac{a^2 b(x+b)}{a^2x^2+c^2}-\\frac{b\\,x}{x^2+d^2} \\d x\n\t\t= \\frac{b}{2}\\parens*{\\log(a^2)-\\log(a^2)} +\\frac{a\\,b^2}{c}\\parens[\\Big]{\\frac{\\pi}{2}+\\frac{\\pi}{2}} = \\frac{\\pi a\\,b^2}{c}.\n\t\\end{align}\n\t%\n\tContinuing from \\eqref{app:eq:int_PFD_split}, we apply \\eqref{app:eq:single_factor_expl_int} and see that the terms we have just treated separately actually conform to the pattern exhibited by the other terms --- i.e.~we can extend the summation to $k=m$ and $\\ell=n$ again.\n\t%\n\t\\begin{align}\n\t\tI_{m,n}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$I_{m,n}$}-\\multlinegap]\n\t\t\t=\\sum_{k=1}^{m}  \\frac{1}{\\Delta^{n+k-1}} \\frac{a^{2n-1}}{c^{2(m-k)+1}} \\frac{\\pi}{4^{m-k}} \\binom{2(m-k)}{m-k} \\parens*{r^n_k + a^2b^2s^n_k} + \\ldots\\\\\n\t\t\t\\ldots+ \\sum_{\\ell=1}^{n} \\frac{1}{\\Delta^{m+\\ell-1}} \\frac{a^{2(\\ell-1)}}{d^{2(n-\\ell)+1}} \\frac{\\pi}{4^{n-\\ell}} \\binom{2(n-\\ell)}{n-\\ell} t^\\ell_m\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$I_{m,n}$}-\\multlinegap]\n\t\t\t=\\frac{\\pi}{\\Delta^{m+n-1}c^{2m-1}d^{2n-1}}\\biggl(\\sum_{k=1}^{m}  \\frac{\\Delta^{m-k}  c^{2(k-1)} (ad)^{2n-1}\\binom{2(m-k)}{m-k}}{4^{m-k}} \\parens*{r^n_k + a^2b^2s^n_k} + \\ldots\\\\\n\t\t\t\\ldots+\\sum_{\\ell=1}^{n}  \\frac{\\Delta^{n-\\ell} c^{2m-1}  (ad)^{2(\\ell-1)}}{4^{n-\\ell}} \\binom{2(n-\\ell)}{n-\\ell} t^\\ell_m \\biggr)\n\t\t\\end{multlined}\\\\\n\t\t&=: \\frac{\\pi}{\\Delta^{m+n-1}c^{2m-1}d^{2n-1}}Q_{m,n}.\n\t\\end{align}\n\t%\n\tWe deal with the two sum separately (reversing the order of $k$), calculating the generating function that will have the term we're looking for as the $m\\nth$ coefficient.\n\t%\n\t\\begin{align}\n\t\\MoveEqLeft\n\t\t\\sum_{m\\ge 0} \\sum_{k=1}^{m} \\Delta^{m-k} c^{2(k-1)} (ad)^{2n-1} \\frac{1}{4^{m-k}}\\binom{2(m-k)}{m-k} \\parens*{r^n_k + a^2b^2s^n_k} y^m\\\\\n\t\t&=\\sum_{m\\ge 0} \\sum_{k\\ge 0} \\mathbbm{1}_{\\{k\\le m-1\\}} \\Delta^{k} c^{2(m-k-1)} (ad)^{2n-1} \\frac{1}{4^{k}}\\binom{2k}{k} \\parens*{r^n_{m-k} - a^2b^2u^n_{m-k}} y^m  \\\\\n\t\t&=(ad)^{2n-1} \\sum_{k\\ge 0} \\parens[\\Big]{\\frac{\\Delta}{4c^2}}^k \\binom{2k}{k} c^{-2} \\sum_{m\\ge k+1} \\parens*{r^n_{m-k} - a^2b^2u^n_{m-k}} (c^2y)^m \\\\\n\t\t&=(ad)^{2n-1} \\sum_{k\\ge 0} \\parens[\\Big]{\\frac{\\Delta}{4}}^k \\binom{2k}{k} y^{k+1} \\sum_{m\\ge 0} \\parens*{r^n_{m+1} - a^2b^2u^n_{m+1}} (c^2y)^m\\\\\n\t\t&=(ad)^{2n-1} \\sum_{k\\ge 0} \\parens[\\Big]{\\frac{\\Delta}{4}}^k \\binom{2k}{k} y^{k+1} \\parens[\\Big]{\\frac{R^n(c^2y)-a^2b^2U^n(c^2y)}{c^2y}}\\\\\n\t\t&=\\coeff{z^n}\\frac{(ad)^{2n-1}y}{\\sqrt{1-\\Delta y}}\\parens[\\Big]{\\frac{R(c^2y,z)-a^2b^2U(c^2y,z)}{c^2y}}\n\t\\end{align}\n\t%\n\twhere we used \\eqref{app:eq:ops_shift} --- recalling that $R^n$ and $U^n$ have no constant term --- and the generating functions of \\autoref{app:prop:genfunc}, as well as \\eqref{app:eq:ops_sqrt} for the remaining sum in $k$.\n\t\n\tUsing the same procedure, we see that\n\t%\n\t\\begin{align}\n\t\t\\sum_{\\ell=1}^{n} \\Delta^{n-\\ell} c^{2m-1} (ad)^{2(\\ell-1)} \\frac{1}{4^{n-\\ell}} \\binom{2(n-\\ell)}{n-\\ell} t^\\ell_m =\\coeff{y^m z^n}\\frac{c^{2m-1}z}{\\sqrt{1-\\Delta z}} \\frac{T(y,a^2d^2z)}{a^2d^2z},\n\t\\end{align}\n\t%\n\tsince $t^0_m=0$ for all $m\\in\\bbN$.\n\t\n\tThen, for $m,n\\ge 1$, using \\eqref{app:eq:coeff_factor},\n\t%\n\t\\begin{align}\n\t\tQ_{m,n}\n\t\t&=\\coeff*{y^m z^n} \\frac{y}{ad\\sqrt{1-\\Delta y}} \\frac{R(c^2y,a^2d^2z)-a^2b^2U(c^2y,a^2d^2z)}{c^2y} + \\frac{z}{c\\sqrt{1-\\Delta z}} \\frac{T(c^2y,a^2d^2z)}{a^2d^2z}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$Q_{m,n}$}-\\multlinegap]\n\t\t\t=\\coeff*{y^m z^n} \\frac{adyz}{\\sqrt{1-\\Delta y}} \\frac{\\Delta(c^2y-a^2d^2z) +r^1_1-2a^2b^2}{\\Delta (c^2y-a^2d^2z)^2 - 2t^1_1 c^2y -2q \\,a^2d^2z+1} + \\ldots\\\\\n\t\t\t\\ldots+\\frac{cyz}{\\sqrt{1-\\Delta z}} \\frac{-\\Delta(c^2y-a^2d^2z)+t^1_1}{\\Delta (c^2y-a^2d^2z)^2 - 2t^1_1 c^2y -2q\\, a^2d^2z+1}\n\t\t\\end{multlined}\\\\\n\t\t&=\\coeff*{y^{m-1} z^{n-1}} \\frac{ad\\sqrt{1-\\Delta z}(\\Delta(c^2y-a^2d^2z) +q)-c\\sqrt{1-\\Delta y}(\\Delta(c^2y-a^2d^2z) -t^1_1)}{\\sqrt{1-\\Delta y}\\sqrt{1-\\Delta z}(\\Delta (c^2y-a^2d^2z)^2 - 2c^2 t^1_1 y -2a^2d^2 q z+1)},\n\t\\end{align}\n\t%\n\twhere we have used \\eqref{app:eq:coeff_shift} for the last equality.\n\t\n\tThe next crucial realisation is to (try to) factor everything in terms of $v(y):=\\sqrt{1-\\Delta y}$ and $w(z):=\\sqrt{1-\\Delta z}$, which works out almost miraculously well. First for the last factor of the denominator (multiplied by $\\Delta$),\n\t%\n\t\\begin{align\n\t\\MoveEqLeft\n\t\t\\Delta\\parens*{\\Delta (c^2y-a^2d^2z)^2 - 2c^2 t^1_1 y -2a^2d^2q z+1}\\\\\n\t\t&=\\parens*{c^2(1-v^2)-a^2d^2(1-w^2)}^2-2c^2 t^1_1(1-v^2)-2a^2d^2 q(1-w^2)+\\Delta\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-2em-\\multlinegap]\n\t\t\t=c^4v^4-2a^2c^2d^2v^2w^2+ a^4d^4w^4+ 2c^2v^2(\\underbrace{t^1_1+a^2d^2-c^2}_{a^2b^2}) +\\ldots \\\\[-0.245cm]\n\t\t\t\\ldots+2a^2d^2w^2({\\overbrace{q-a^2d^2+c^2}^{\\cramped{a^2b^2}}})+\\smash{\\overbrace{c^4-2a^2c^2d^2 +a^4d^4 -2c^2t^1_1-2a^2d^2q+\\Delta}^{\\cramped{a^4b^4}}}\n\t\t\\end{multlined}\\\\\n\t\t&=(c^2v^2+a^2d^2w^2+a^2b^2)^2-4a^2c^2d^2v^2w^2\\\\\n\t\t&=(c^2v^2+a^2d^2w^2+a^2b^2+2acdvw)(c^2v^2+a^2d^2w^2+a^2b^2-2acdvw)\\\\\n\t\t&=\\parens*{(cv+adw)^2+a^2b^2}\\parens*{(cv-adw)^2+a^2b^2}\n\t\\end{align}\n\t%\n\tand then, in similar fashion for the numerator\n\t%\n\t\\begin{align}\n\t\tadw(\\Delta(c^2y-a^2d^2z) +q)-cv(\\Delta(c^2y-a^2d^2z) -t^1_1)\n\t\t= (cv+adw)\\parens*{(cv-adw)^2+a^2b^2},\n\t\\end{align}\n\t%\n\twhich finally achieves the cancellation that will turn out to be the key to get rid of the factor $\\delta_-$ in the denominator of the PFD. Taken together, this yields\n\t%\n\t\\begin{align}\n\t\tQ_{m,n}=\\coeff*{y^{m-1} z^{n-1}} \\frac{\\Delta}{v(y)w(z)}\\frac{cv(y)+adw(z)}{(cv(y)+adw(z))^2+a^2b^2},\n\t\\end{align}\n\t\n\tFinally, recalling $I_{m,n}=  \\pi Q_{m,n} \/ (\\Delta^{m+n-1}c^{2m-1}d^{2n-1})$, we apply \\eqref{app:eq:coeff_factor} to arrive at\n\t%\n\t\\begin{align}\n\t\tI_{m,n} &= \\coeff*{y^{m-1} z^{n-1}}  \\frac{\\pi}{c^{2m-1}d^{2n-1}} \\frac{1}{\\sqrt{1-\\smash{y}}\\sqrt{1-z}}\\frac{c\\sqrt{1-\\smash{y}}+ad\\sqrt{1- z}}{(c\\sqrt{1-\\smash{y}}+ad\\sqrt{1- z})^2+a^2b^2},\n\t\\end{align}\n\t%\n\tand once more using \\eqref{app:eq:coeff_factor}, to yield the representation \\eqref{app:eq:Imn_as_coeff}. This proves the claimed generating function for $I_{m,n}$.\n\\end{proof}\n\n\\subsection{Determining the Coefficients}\n\nCompared to \\autoref{app:th:Imn_est}, \\eqref{app:eq:Imn_as_coeff} already has the right structure, but what remains to be proved at this stage is that the coefficients $c^{m,n}_{i,j}$ are zero when $i\\le 2(m-1) \\land j\\le2(n-1)$. Showing this will concern us for the remainder of this section.\n\nThe generalised chain rule is expressed through \\emph{Fa\\`{a} di Bruno's formula}\n\\begin{align}\\label{app:eq:faa_di_bruno}\n\t\\Dn{n}{x} f(g(x))=\\sum_{k=1}^n f^{(k)}(g(x)) B_{n,k}\\parens*{g'(x),g''(x),\\ldots,g^{(n-k+1)}(x)},\n\\end{align}\nwhere\n\\begin{align}\n\tB_{n,k}(y_1,\\ldots,y_{n-k+1})=\\sum_{\\substack{\\sum_i j_i=k\\\\ \\sum_i i \\cdot j_i=n}} \\frac{n!}{j_1!j_2!\\cdot \\ldots \\cdot j_{n-k+1}!} \\parens[\\Big]{\\frac{y_1}{1!}}^{j_1}\\parens[\\Big]{\\frac{y_2}{2!}}^{j_2}\\cdot \\ldots \\cdot\\parens[\\Big]{\\frac{y_{n-k+1}}{(n-k+1)!}}^{j_{n-k+1}}\n\\end{align}\nare the so-called \\emph{Bell polynomials}, which satisfy the following identity for their generating function,\n\\begin{align}\\label{app:eq:genfunc_bell}\n\t\\sum_{n=k}^\\infty B_{n,k}(y_1,y_2,\\ldots,y_{n-k+1}) \\frac{t^n}{n!}=\\frac{1}{k!}\\parens[\\Big]{\\sum_{m=1}^\\infty y_m \\frac{t^m}{m!}}^k.\n\\end{align}\n\n\\begin{lemma}\n\tFor $1\\le k\\le n$ and $c>0$, we have\n\t%\n\t\\begin{align}\\label{app:eq:bell_sqrt}\n\t\tB_{n,k}\\parens[\\Big]{\\D{y}\\sqrt{c^2-y},\\ldots,\\Dn{n-k+1}{y}\\sqrt{c^2-y}}\\Bigr|_{y=0} = (-2c)^{k-2n} \\frac{(2n-k-1)!}{(k-1)!(n-k)!}.\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tUsing \\eqref{app:eq:ops_geom_alpha} we see that\n\t%\n\t\\begin{align}\n\t\t\\coeff{y^m}\\sqrt{c^2-y}=\\coeff{y^m} c\\sqrt{1-\\frac{y}{c^2}}=\\binom{\\frac 12}{m}(-1)^m c^{1-2m}\n\t\\end{align}\n\t%\n\tInserting this into \\eqref{app:eq:genfunc_bell}, we obtain by virtue of \\eqref{app:eq:ops_shift} and \\eqref{app:eq:ops_sqrt_k} that\n\t%\n\t\\begin{align}\n\t\tB_{n,k}\\parens[\\Big]{\\D{y}\\sqrt{c^2-y},\\ldots}\\Bigr|_{y=0}\n\t\t&= \\coeff{t^n} \\frac{n!}{k!} c^k \\parens*{\\sqrt{1-t\/c^2}-1}^k = \\coeff{t^n} \\frac{n!}{k!} c^{k}  \\parens[\\Big]{\\frac{1-\\sqrt{1-t\/c^2}}{t\/(2c^2)}}^k \\parens[\\Big]{\\frac{-t}{2c^2}}^k\\\\\n\t\t&= \\coeff*{t^{n-k}} \\frac{n!}{k!} (-2c)^{-k}  \\parens[\\Big]{\\frac{1-\\sqrt{1-t\/c^2}}{t\/(2c^2)}}^k \\\\\n\t\t&= \\frac{n!}{k!} (-2c)^{-k} \\frac{k}{n}\\binom{2n-k-1}{n-k} (4c^2)^{k-n},\n\t\\end{align}\n\t%\n\twhich yields the desired result after rearranging.\n\\end{proof}\n\n\\begin{proposition}\\label{app:prop:coeff_expl_long}\n\tBy deriving \\eqref{app:eq:Imn_as_coeff} $m-1$ times in $y$ and $n-1$ times in $z$, the coefficients in \\eqref{app:eq:Imn_est_best} can be calculated as follows\n\t%\n\t\\begin{align}\n\t\tc^{m,n}_{i,j}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$c^{m,n}_{i,j}$}-\\multlinegap]\n\t\t\t= \\frac{\\delta_{\\{i+j\\equiv1\\bmod{2}\\}}}{4^{m+n-2}}  \\sum_{k=1}^{m-1}\\sum_{\\ell=1}^{n-1}\\sum_{p=0}^{\\ell}\\sum_{q=0}^{k}\\sum_{r=0}^{k-q}\\sum_{s,t\\ge 0} \\frac{2^k k}{m-1} \\binom{2m-k-3}{\\phantom{2}m-k-1} \\frac{2^\\ell \\ell}{n-1} \\binom{2n-\\ell-3}{\\phantom{2}n-\\ell-1}  \\cdot\\ldots\\\\\n\t\t\t\\shoveright{\\ldots\\cdot \\binom{\\ell+1}{p} \\binom{\\ell-p}{q} \\binom{\\ell+r}{r} \\binom{m+n-2-\\ell-r}{s} \\binom{m+n-2-p-q-r-s}{t} \\cdot\\ldots} \\\\\n\t\t\t\\ldots\\cdot \\binom{m+n-1}{i-p-s} \\binom{m+n-1-i+p+s}{j-q-r-t} (-1)^{\\frac{3(i+j)+1}{2}+p+s+r+t}.\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tFurthermore, $c^{m,n}_{i,j}$ is zero for $i+j>2(m+n)-3$.\n\\end{proposition}\n\n\\begin{proof}\n\tLetting $v(y):=\\sqrt{c^2-y}$ and $w(z):=\\sqrt{d^2-z}$ (which have the same behaviour but different constants to their namesakes above), we begin by rewriting \\eqref{app:eq:Imn_as_coeff},\n\t%\n\t\\begin{align}\n\t\tI_{m,n} &=\\coeff*{y^{m-1} z^{n-1}} \\frac{\\pi}{vw}\\parens{\\frac{v+aw-\\mathrm{i} ab}{(v+aw+\\mathrm{i} ab)(v+aw-\\mathrm{i} ab)} + \\frac{\\mathrm{i} ab}{(v+aw)^2+a^2b^2}}\\\\\n\t\t&=\\Re \\coeff*{y^{m-1} z^{n-1}}  \\frac{\\pi}{v(y)w(z)}\\frac{1}{v(y)+aw(z)+\\mathrm{i} ab}, \\label{app:eq:Imn_simplif_frac}\n\t\\end{align}\n\t%\n\tsince, clearly, $I_{m,n}$ is always real (and the second summand always imaginary).\n\t\n\tFor now, we assume that both $n,m\\ge2$, and apply \\eqref{app:eq:faa_di_bruno} to derive $n-1$ times in $z$, arriving at\n\t%\n\t\\begin{align}\n\t\tI_{m,n}&=\\Re\\coeff*{y^{m-1}} \\frac{\\pi}{v(y)} \\frac{1}{(n-1)!}\\sum_{\\ell=1}^{n-1} B_{n-1,\\ell}\\parens[\\Big]{\\D{z}w(z),\\ldots}\\Bigr|_{z=0} \\Dn{\\ell}{w} \\frac{1}{w}\\frac{1}{v(y)+aw+\\mathrm{i} ab}\\Bigr|_{w=d\n\t\n\t\\end{align}\n\t%\n\tUsing the PFD in $w$, we see that\n\t%\n\t\\begin{align}\n\t\t\\Dn{\\ell}{w} \\frac{1}{w}\\frac{1}{v(y)+aw+\\mathrm{i} ab}\\Bigr|_{w=d} \n\t\t&= \\Dn{\\ell}{w} \\frac{1}{v(y)+\\mathrm{i} ab}\\parens[\\Big]{\\frac{1}{w}-\\frac{a}{v(y)+aw+\\mathrm{i} ab}}\\Bigr|_{w=d} \\\\\n\t\t&=\\frac{\\ell!(-1)^\\ell}{v(y)+\\mathrm{i} ab}\\parens[\\Big]{\\frac{1}{w^{\\ell+1}}-\\frac{a^{\\ell+1}}{(v(y)+aw+\\mathrm{i} ab)^{\\ell+1}}}\\Bigr|_{w=d} \\\\\n\t\t&=\\frac{\\ell!(-1)^\\ell}{v(y)+\\mathrm{i} ab}\\frac{1}{d^{\\ell+1}}\\parens[\\Big]{1-\\frac{(ad)^{\\ell+1}}{(v(y)+ad+\\mathrm{i} ab)^{\\ell+1}}}.\n\t\\end{align}\n\t%\n\tTogether with \\eqref{app:eq:bell_sqrt}, we can continue manipulating $I_{m,n}$\n\t%\n\t\\begin{align}\n\tI_{m,n}&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$I_{m,n}$}-\\multlinegap]\n\t\t= \\Re\\coeff*{y^{m-1}} \\frac{\\pi}{v(y)}\\frac{1}{v(y)+\\mathrm{i} ab} \\frac{1}{(n-1)!}\\cdot\\ldots \\\\\n\t\t\\ldots\\cdot\\sum_{\\ell=1}^{n-1} \\frac{(-2d)^{\\ell-2n+2}(2n-\\ell-3)!}{(\\ell-1)!(n-\\ell-1)!} \\frac{\\ell!(-1)^\\ell}{d^{\\ell+1}}\\parens[\\Big]{1-\\frac{(ad)^{\\ell+1}}{(v(y)+ad+\\mathrm{i} ab)^{\\ell+1}}}\n\t\\end{multlined}\\\\\n\t&= \\Re\\coeff*{y^{m-1}} \\frac{\\pi}{v(y)}\\frac{1}{v(y)+\\mathrm{i} ab} \\sum_{\\ell=1}^{n-1} \\frac{2^{\\ell+1}}{(2d)^{2n-1}} \\frac{\\ell}{n-1} \\binom{2n-\\ell-3}{\\phantom{2}n-\\ell-1} \\parens[\\Big]{1-\\frac{(ad)^{\\ell+1}}{(v(y)+ad+\\mathrm{i} ab)^{\\ell+1}}}.\n\t\\end{align}\n\t%\n\tUsing\n\t%\n\t\\begin{multline}\n\t\t1-\\frac{(ad)^{\\ell+1}}{(v(y)+ad+\\mathrm{i} ab)^{\\ell+1}} \\\\\n\t\t=\\frac{1}{(v(y)+ad+\\mathrm{i} ab)^{\\ell+1}}\\parens[\\bigg]{\\sum_{p=0}^{\\ell+1}\\binom{\\ell+1}{p}(ad)^p (v(y)+\\mathrm{i} ab)^{\\ell+1-p} -(ad)^{\\ell+1}}\n\t\\end{multline}\n\t%\n\tthis collapses further to\n\t%\n\t\\begin{align}\n\t\tI_{m,n}&=\\Re\\coeff*{y^{m-1}} \\frac{\\pi}{v(y)} \\sum_{\\ell=1}^{n-1} \\sum_{p=0}^{\\ell}  \\frac{2^{\\ell+1}}{(2d)^{2n-1}} \\frac{\\ell}{n-1} \\binom{2n-\\ell-3}{\\phantom{2}n-\\ell-1} \\binom{\\ell+1}{p} \\frac{(ad)^p (v(y)+\\mathrm{i} ab)^{\\ell-p}}{(v(y)+ad+\\mathrm{i} ab)^{\\ell+1}}.\n\t\\end{align}\n\t%\n\tApplying the product rule, we observe first that\n\t%\n\t\\begin{align}\n\t\t\\Dn{j}{v}\\frac{1}{v(v+ad+\\mathrm{i} ab)^{\\ell+1}}\n\t\t&=\\sum_{r=0}^{j}\\binom{j}{r}(-1)^j \\frac{r!(\\ell+j-r)!}{\\ell!} \\frac{1}{v^{r+1}(v+ad+\\mathrm{i} ab)^{\\ell+1+j-r}}\\\\\n\t\n\t\t&=j!(-1)^j\\sum_{r=0}^{j}\\binom{\\ell+r}{r}\\frac{1}{v^{j-r+1}(v+ad+\\mathrm{i} ab)^{\\ell+1+r}},\n\t\\end{align}\n\t%\n\twhere we reversed the order of summation for the last equation. Consequently, with $j=k-q$, once more due to the product rule, $I_{m,n}$ is equal to\n\t%\n\t\\begin{align}\n\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\\mathrel{\\phantom{=}} \\Re \\frac{\\pi}{(m-1)!} \\sum_{\\ell=1}^{n-1} \\sum_{p=0}^{\\ell}  \\frac{2^{\\ell+1}}{(2d)^{2n-1}} \\frac{\\ell}{n-1} \\binom{2n-\\ell-3}{\\phantom{2}n-\\ell-1} \\! \\binom{\\ell+1}{p} \\! \\sum_{k=1}^{m-1} \\! B_{m-1,k}\\parens[\\Big]{\\D{y}v(y),\\ldots}\\Bigr|_{y=0} \\! \\cdot\\ldots\\hspace{-2cm}\\\\\n\t\t\\ldots\\cdot\\sum_{q=0}^{\\mathclap{\\min(k,\\ell-p)}} \\hspace{0.3cm}\\binom{k}{q}\\frac{(\\ell-p)!}{(\\ell-p-q)!} (c+\\mathrm{i} ab)^{\\ell-p-q}  \\sum_{r=0}^{k-q} \\binom{\\ell+r}{r}\\frac{(ad)^p (k-q)!(-1)^{k-q}}{c^{k-q-r+1}(c+ad+\\mathrm{i} ab)^{\\ell+1+r}}\n\t\\end{multlined}\\\\\n\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t=\\Re \\pi \\sum_{k=1}^{m-1}\\sum_{\\ell=1}^{n-1} \\frac{2^{k+1}}{(2c)^{2m-1}} \\frac{k}{m-1} \\binom{2m-k-3}{\\phantom{2}m-k-1} \\frac{2^{\\ell+1}}{(2d)^{2n-1}} \\frac{\\ell}{n-1} \\binom{2n-\\ell-3}{\\phantom{2}n-\\ell-1} \\cdot\\ldots\\\\\n\t\t\\ldots\\cdot\\sum_{p=0}^{\\ell}\\sum_{q=0}^{k} \\sum_{r=0}^{k-q} \\binom{\\ell+1}{p} \\binom{\\ell-p}{q} \\binom{\\ell+r}{r} \\frac{(-1)^q  c^{q+r} (ad)^p (c+\\mathrm{i} ab)^{\\ell-p-q}}{(c+ad+\\mathrm{i} ab)^{\\ell+r+1}}.\n\t\\end{multlined}\n\t\\end{align}\n\t%\n\tTo find the respective powers of $c$ and $ad$, we pull out the maximal power of the denominator --- multiplied by its conjugate to make it real; recalling $\\delta_+=(c+ad)^2+a^2b^2=(c+ad+\\mathrm{i} ab)(c+ad-\\mathrm{i} ab)$ --- to find\n\t%\n\t\\begin{multline}\\label{eq:Imn_coeff_unified_denom}\n\t\tI_{m,n}\n\t\t=\\frac{4\\pi}{(4\\delta_+)^{m+n-1}c^{2m-1}d^{2n-1}} \\Re\\!\\!\\sum_{k,\\ell,p,q,r} \\!\\frac{2^k k}{m-1} \\binom{2m-k-3}{\\phantom{2}m-k-1} \\frac{2^\\ell \\ell}{n-1} \\binom{2n-\\ell-3}{\\phantom{2}n-\\ell-1} \\! \\binom{\\ldots}{p,q,r}  \\cdot\\ldots\\\\\n\t\t\\ldots\\cdot(-1)^q c^{q+r} (ad)^p (c+\\mathrm{i} ab)^{\\ell-p-q}(c+ad+\\mathrm{i} ab)^{m+n-2-\\ell-r} (c+ad-\\mathrm{i} ab)^{m+n-1},\n\t\\end{multline}\n\t%\n\twhere we will denote\n\t%\n\t\\begin{align}\n\t\t\\binom{\\ldots}{k,\\ell,m,n,p,q,r}:=\\frac{2^k k}{m-1} \\binom{2m-k-3}{\\phantom{2}m-k-1} \\frac{2^\\ell \\ell}{n-1} \\binom{2n-\\ell-3}{\\phantom{2}n-\\ell-1} \\binom{\\ell+1}{p} \\binom{\\ell-p}{q} \\binom{\\ell+r}{r}\n\t\\end{align}\n\t%\n\tfor brevity, and will often collect consecutive letters with a hyphen, e.g.~$k-n$ for $k,\\ell,m,n$. As the penultimate step, we expand the last line of \\eqref{eq:Imn_coeff_unified_denom} to\n\t%\n\t\\begin{multline}\n\t\n\t\n\t\n\t\t\\sum_{s\\ge 0} \\sum_{t\\ge 0} \\sum_{g\\ge 0} \\sum_{h\\ge 0} \\binom{m+n-2-\\ell-r}{s} \\binom{m+n-2-p-q-r-s}{t}  \\binom{m+n-1}{g} \\cdot\\ldots \\\\\n\t\t\\ldots\\cdot \\binom{m+n-1-g}{h} (-1)^{g+h+m+n+q+1} c^{h+q+r+t} (ad)^{g+p+s} (\\mathrm{i} ab)^{2(m+n)-3-g-h-p-q-r-s-t}.\n\t\\end{multline}\n\t%\n\tFinally, we collect like powers of $c$ and $ad$ as powers of $i=h+q+r+t$ and $j=g+p+s$, respectively,\n\t%\n\t\\begin{align}\n\tI_{m,n}\n\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$I_{m,n}$}-\\multlinegap]\\label{eq:Imn_repr_coeff}\n\t\t=\\phantom{:}\\frac{\\pi}{\\delta_+^{m+n-1}c^{2m-1}d^{2n-1}} \\Re\\!\\! \\sum_{i,j=0}^{2(m+n)-3} \\!\\! \\sum_{k,\\ell,p,q,r,s,t} \\binom{\\ldots}{k,\\ell,m,n,p,q,r,s,t} \\binom{m+n-1}{j-p-s} \\cdot\\ldots\\\\\n\t\t\\ldots\\cdot \\binom{m+n-1-j+p+s}{i-q-r-t} \\frac{(-1)^{i+j+p+r+s+t+m+n+1}}{4^{m+n-2}} c^i (ad)^j (\\mathrm{i} ab)^{2(m+n)-3-i-j}\n\t\\end{multlined}\\\\\n\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$I_{m,n}$}-\\multlinegap]\n\t\t=\\phantom{:}\\frac{\\pi}{\\delta_+^{m+n-1}c^{2m-1}d^{2n-1}} \\Re\\!\\! \\sum_{i,j=0}^{2(m+n)-3} \\!\\! \\sum_{k,\\ell,p,q,r,s,t} \\binom{\\ldots}{k,\\ell,m,n,p,q,r,s,t} \\binom{m+n-1}{j-p-s} \\cdot\\ldots\\\\\n\t\t\\ldots\\cdot \\binom{m+n-1-j+p+s}{i-q-r-t} \\frac{\\mathrm{i}^{-i-j+1}(-1)^{p+r+s+t}}{4^{m+n-2}} c^i (ad)^j (ab)^{2(m+n)-3-i-j}\n\t\\end{multlined}\\\\\n\t&=:\\frac{\\pi}{\\delta_+^{m+n-1}c^{2m-1}d^{2n-1}} \\sum_{i,j=0}^{2(m+n)-3} c^{m,n}_{i,j} c^i (ad)^j (ab)^{2(m+n)-3-i-j}. \n\t\\end{align}\n\t%\n\tObserve that due to the fact that we only consider the real parts of the sum, $i+j$ has to be an odd number, so that the power of $\\mathrm{i}$ is even.\n\t\n\tThe claim that $i+j$ has to be less than $2(m+n)-3$ follows easily from the binomial coefficient in $i$, because it implies that (estimating $t$ by its maximal value, which can be read off from the corresponding binomial coefficient)\n\t%\n\t\\begin{multline}\n\t\ti-q-r-t\\le m+n-1-j+p+s \\\\\n\t\t\\Longrightarrow \\quad\n\t\ti+j\\le m+n-1+p+q+r+s+t\\le 2(m+n)-3.\\tag*{\\qedhere}\n\t\\end{multline}\n\\end{proof}\n\nNow, after having disassembled $I_{m,n}$ into its parts, we are able to insert the indicators we later want to look for --- meaning we multiply by $v^i w^j y^m z^n$ --- and try to reassemble the whole thing once more.\n\n\\begin{proposition}\\label{app:prop:genfunc_coeff}\n\tDenoting $g(v,w):=(cv+adw)^2+a^2b^2$, it can be seen that the generating function of the numerator in $I_{m,n}$ is\n\t%\n\t\\begin{align}\n\t\\MoveEqLeft\n\t\tN(v,w,y,z):=\\sum_{i,j\\ge 0,\\,m,n\\ge 1} c^{m,n}_{i,j} (cv)^i (adw)^j  (ab)^{2(m+n)-3-i-j} y^m z^n\\\\\n\t\t&=\\Re \\frac{yz}{\\sqrt{1-g(v,w)y}\\sqrt{1-g(v,w)z}} \\frac{g(v,w)}{cv\\sqrt{1-g(v,w)y} +adw\\sqrt{1-g(v,w)z}+\\mathrm{i} ab} \\label{app:eq:Imn_ops_numerator}\\\\\n\t\t&=\\frac{g(v,w)yz}{\\sqrt{1-g(v,w)y}\\sqrt{1-g(v,w)z}} \\frac{cv\\sqrt{1-g(v,w)y}+adw\\sqrt{1-g(v,w)z}}{(cv\\sqrt{1-g(v,w)y}+adw\\sqrt{1-g(v,w)z})^2+a^2b^2}, \n\t\\end{align}\n\t%\n\tand for the coefficients alone, with $h(v,w):=(v+w)^2+1$,\n\t%\n\t\\begin{align}\n\t\tC(v,w,y,z)\n\t\t&:=\\sum_{i,j\\ge 0,\\,m,n\\ge 1} c^{m,n}_{i,j} v^i w^j y^m z^n\\\\\n\t\t&\\phantom{:}=\\Re \\frac{yz}{\\sqrt{1-h(v,w)y}\\sqrt{1-h(v,w)z}} \\frac{h(v,w)}{v\\sqrt{1-h(v,w)y}+w\\sqrt{1-h(v,w)z}+\\mathrm{i}}\\\\\n\t\t&\\phantom{:}=\\frac{h(v,w)yz}{\\sqrt{1-h(v,w)y}\\sqrt{1-h(v,w)z}} \\frac{v\\sqrt{1-h(v,w)y}+w\\sqrt{1-h(v,w)z}}{(v\\sqrt{1-h(v,w)y}+w\\sqrt{1-h(v,w)z})^2+1}.\n\t\\end{align}\n\\end{proposition}\n\n\\begin{remark}\n\tUsing \\eqref{app:eq:Imn_as_coeff} and \\eqref{app:eq:coeff_factor}, we can deduce that\n\t%\n\t\\begin{align}\n\t\tI_{m,n}=\\frac{\\pi}{\\delta_+^{m+n-1}c^{2m-1}d^{2n-1}}\\coeff*{y^{m}z^{n}}\n\t\n\t\t\\Re \\frac{yz}{\\sqrt{1-\\delta_+y}\\sqrt{1-\\delta_+z}} \\frac{\\delta_+}{c\\sqrt{1-\\delta_+y}+ad\\sqrt{1-\\delta_+z}+\\mathrm{i} ab}\\qquad  \\label{app:eq:Imn_ops_rescaled}\n\t\n\t\n\t\\end{align}\n\t%\n\tComparing this with \\eqref{app:eq:Imn_ops_numerator}, we immediately see the correspondence (by setting $v,w=1$, because $g\\bigr|_{v=1,w=1}=\\delta_+$) --- the knowledge we gain with $N(v,w,y,z)$ is the ability to determine a single coefficient as opposed to the whole numerator.\n\\end{remark}\n\n\\begin{proof}[Proof of \\autoref{app:prop:genfunc_coeff}]\n\tAs outlined in \\autoref{ssec:genfunc}, we take \\eqref{eq:Imn_repr_coeff} and extend the summation to $\\infty$ for all parameters except $\\ell$ (the binomial coefficients ensure that only finitely many terms are non-zero, except for $r$, where we need change the order of summation and use $k\\ge q+r$ to enforce $r\\le k-q$), and interchange order so that we can sum, first in $j$ and $i$,\n\t%\n\t\\begin{align}\n\t\tN(v,w,y,z)&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$N(v,w,y,z)$}-\\multlinegap]\n\t\t\t=\\Re\\sum_{j,k,\\ell,m,n,p,q,r,s,t}  \\binom{\\ldots}{k,\\ell,m,n,p,q,r,s,t} \\binom{m+n-1}{j-p-s} \\frac{(-1)^{m+n+1+j+p+r+s+t}}{4^{m+n-2}} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot   (adw)^j (\\mathrm{i} ab)^{2(m+n)-3-j} y^m z^n \\sum_{i=q+r+t}^\\infty \\binom{m+n-1-j+p+s}{i-q-r-t} \\parens[\\Big]{\\frac{-cv}{\\mathrm{i} ab}}^i\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$N(v,w,y,z)$}-\\multlinegap]\n\t\t\t=\\Re\\sum_{j,k,\\ell,m,n,p,q,r,s,t}  \\binom{\\ldots}{k-n,p-t} \\binom{m+n-1}{j-p-s} \\frac{(-1)^{m+n+1+j+p+r+s+t}}{4^{m+n-2}} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot (adw)^j  (\\mathrm{i} ab)^{2(m+n)-3-j} y^m z^n \\parens[\\Big]{\\frac{-cv}{\\mathrm{i} ab}}^{q+r+t} \\parens[\\Big]{1-\\frac{cv}{\\mathrm{i} ab}}^{m+n-1-j+p+s}\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$N(v,w,y,z)$}-\\multlinegap]\n\t\t\t=\\Re\\sum_{k,\\ell,m,n,p,q,r,s,t}  \\binom{\\ldots}{k-n,p-t} \\frac{(-1)^{m+n+1+q}}{4^{m+n-2}} (cv)^{q+r+t} (\\mathrm{i} ab)^{2(m+n)-3-q-r-t} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot   y^m z^n \\parens[\\Big]{1-\\frac{cv}{\\mathrm{i} ab}}^{m+n-1} \\parens[\\Big]{\\frac{adw}{\\mathrm{i} ab}}^{p+s} \\sum_{j=0}^\\infty \\binom{m+n-1}{j} \\parens[\\Big]{\\frac{-adw}{\\mathrm{i} ab-cv}}^{j}\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$N(v,w,y,z)$}-\\multlinegap]\n\t\t\t=\\Re\\sum_{k,\\ell,m,n,p,q,r,s,t}  \\binom{\\ldots}{k-n,p-t} \\frac{(-1)^{m+n+1+q}}{4^{m+n-2}} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot  (cv)^{q+r+t} (adw)^{p+s} (\\mathrm{i} ab)^{2(m+n)-3-p-q-r-s-t} y^m z^n \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}^{m+n-1},\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\twhere the last line uses $(1-\\frac{cv}{\\mathrm{i} ab})(1-\\frac{adw}{\\mathrm{i} ab-cv})=1-\\frac{cv+adw}{\\mathrm{i} ab}$. We continue by summing in $t$ and $s$, which are just applications of the binomial theorem,\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re  \\sum_{k-n,p-s}  \\binom{\\ldots}{k-n,p-s} \\frac{(-1)^{m+n+1+q}}{4^{m+n-2}} (cv)^{q+r} (adw)^{p+s}  (\\mathrm{i} ab)^{2(m+n)-3-p-q-r-s} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot  y^m z^n\\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}^{m+n-1}   \\sum_{t=0}^{m+n-2-\\smash{p-q}-r-s} \\binom{m+n-2-p-q-r-s}{t} \\parens[\\Big]{\\frac{cv}{\\mathrm{i} ab}}^t\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re \\sum_{k-n,p-r} \\binom{\\ldots}{k-n,p-r} \\frac{(-1)^{m+n+1+q}}{4^{m+n-2}}    (\\mathrm{i} ab)^{2(m+n)-3-p-q-r}  y^m z^n \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}^{m+n-1} \\!\\!\\!\\! \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot  (cv)^{q+r}  (adw)^{p}  \\parens[\\Big]{1+\\frac{cv}{\\mathrm{i} ab}}^{m+n-2-p-q-r} \\sum_{s=0}^{m+n-2-\\ell-r} \\binom{m+n-2-\\ell-r}{s} \\parens[\\Big]{\\frac{adw}{cv+\\mathrm{i} ab}}^s\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re \\sum_{k-n,p-r} \\binom{\\ldots}{k-n,p-r} \\frac{(-1)^{m+n+1+q}}{4^{m+n-2}} (cv)^{q+r}  (adw)^{p}   (\\mathrm{i} ab)^{2(m+n)-3-p-q-r}  y^m z^n  \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot  \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}^{m+n-1}  \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{m+n-2-p-q-r}  \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell}.\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tNext we sum in $n$, setting $g(v,w):=(cv+adw)^2+a^2b^2=-(\\mathrm{i} ab)^2(1-\\frac{cv+adw}{\\mathrm{i} ab})(1+\\frac{cv+adw}{\\mathrm{i} ab})$,\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re \\sum_{k-m,p-r} \\binom{\\ldots}{k-m,p-r} \\frac{(-1)^{m+1+q}}{4^{m-2}} (cv)^{q+r} (adw)^{p} (\\mathrm{i} ab)^{2m-3-p-q-r}  y^m \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}^{m-1}\\!\\!\\!\\!\\! \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot   \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{m-2-p-q-r}  \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell} \\sum_{n=\\ell+1}^{\\infty}\\frac{2^\\ell \\ell}{n-1} \\binom{2n-\\ell-3}{\\phantom{2}n-\\ell-1} \\parens[\\Big]{\\frac{gz}{4}}^n\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re \\sum_{k-m,p-r} \\binom{\\ldots}{k-m,p-r} \\frac{(-1)^{m+1+q}}{4^{m-2}} (cv)^{q+r} (adw)^{p} (\\mathrm{i} ab)^{2m-3-p-q-r}  y^m \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}^{m-1}\\!\\!\\!\\!\\! \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{m-2-p-q-r}  \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell}\\parens[\\Big]{\\frac{gz}{4}}^{\\ell+1} 2^\\ell \\sum_{n=0}^{\\infty}\\frac{\\ell}{n+\\ell}  \\binom{2n+\\ell-1}{n} \\parens[\\Big]{\\frac{gz}{4}}^n\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re \\sum_{k-m,p-r} \\binom{\\ldots}{k-m,p-r} \\frac{(-1)^{m+q}}{4^{m-1}} (cv)^{q+r} (adw)^{p} (\\mathrm{i} ab)^{2m-1-p-q-r}  y^m z  \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}^{m} \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{m-1-p-q-r}  \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell}  \\parens*{1-\\sqrt{1-gz}}^\\ell,\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\twhere we used \\eqref{app:eq:ops_sqrt_k} for the last equation and distributed one surplus power back to its constituent factors.\n\t\n\tIn the case of $\\ell=0$, \\eqref{app:eq:ops_sqrt_k} is not applicable, so we would have to subtract the entire term with $\\ell$ set to $0$. As is immediately obvious from the last display, this only affects the case that $n=1$ (because the only remaining power of $z$ is $z^1$), and we will not deal with the case $m=1 \\lor n=1$ here, as they can be done in exactly the same way as for the general case (e.g.~starting from \\eqref{app:eq:Imn_simplif_frac}, where either $y$ or $z$ can then be set to zero and the derivatives only need to be considered for the other variable).\n\t\n\tSimilarly, we tackle the sum in $m$ (again ignoring the error for $k=0$), allowing us to also deal with the sum in $k$,\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re  \\sum_{k,\\ell,p-r}  \\binom{\\ldots}{p-r} 4(-1)^{q} (cv)^{q+r} (adw)^{p} (\\mathrm{i} ab)^{-1-p-q-r} z \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{-1-p-q-r} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot   \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell}  \\parens*{1-\\sqrt{1-gz}}^\\ell 2^k \\parens[\\Big]{\\frac{gy}{4}}^{k+1}\\sum_{m=0}^\\infty \\frac{ k}{m+k} \\binom{2m+k-1}{m} \\parens[\\Big]{\\frac{gy}{4}}^{m}\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re  \\sum_{k,\\ell,p-r}  \\binom{\\ldots}{p-r} (-1)^{q+1} (cv)^{q+r} (adw)^{p}  (\\mathrm{i} ab)^{1-p-q-r} yz \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot  \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{-p-q-r} \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell}  \\parens*{1-\\sqrt{1-gz}}^\\ell \\parens*{1-\\sqrt{1-gy}}^k\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re  \\sum_{\\ell,p-r}  \\binom{\\ldots}{p-r} (-1)^{q+1} (cv)^{q+r} (adw)^{p}  (\\mathrm{i} ab)^{1-p-q-r} yz \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}  \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{-p-q-r} \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell}  \\parens*{1-\\sqrt{1-gz}}^\\ell \\sum_{k=q+r}^\\infty \\parens*{1-\\sqrt{1-gy}}^k\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re\\frac{yz}{\\sqrt{1-gy}}\\sum_{\\ell,p-r}  \\binom{\\ldots}{p,q,r}  (-1)^{q+1} (cv)^{q+r} (adw)^{p}  (\\mathrm{i} ab)^{1-p-q-r} \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}  \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot  \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{-p-q-r} \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell}  \\parens*{1-\\sqrt{1-gz}}^\\ell \\parens*{1-\\sqrt{1-gy}}^{q+r}.\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tNext, we sum in $r$, by \\eqref{app:eq:ops_geom_alpha},\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re\\frac{yz}{\\sqrt{1-gy}} \\sum_{\\ell,p,q} \\binom{\\ldots}{p,q\n\t\t\t(-1)^{q+1} (cv)^{q} (adw)^{p}  (\\mathrm{i} ab)^{1-p-q} \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}} \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{-p-q} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p+q-\\ell}  \\parens*{1-\\sqrt{1-gz}}^\\ell \\parens*{1-\\sqrt{1-gy}}^{q} \\;\\smash[b]{\\underbrace{\\sum_{r=0}^\\infty \\binom{\\ell+r}{r} \\parens[\\bigg]{\\frac{cv\\parens*{1-\\sqrt{1-gy}}}{cv+adw+\\mathrm{i} ab}}^r,}_{\\parens*{\\frac{cv+adw+\\mathrm{i} ab}{cv\\sqrt{1-gy}+adw+\\mathrm{i} ab}}^{\\ell+1}}} \\vphantom{\\underbrace{\\sum_{r=0}}_{a}}\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tthen in $q$,\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re\\frac{-yz}{\\sqrt{1-gy}}\\sum_{\\ell,p}  \\binom{\\ell+1}{p} (adw)^{p}  (\\mathrm{i} ab)^{1-p} \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}} \\parens[\\Big]{1+\\frac{cv+adw}{\\mathrm{i} ab}}^{-p} \\parens[\\Big]{1+\\frac{adw}{cv+\\mathrm{i} ab}}^{p-\\ell} \\!\\!\\!\\!\\! \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot \\parens*{1-\\sqrt{1-gz}}^\\ell \\parens[\\Big]{\\frac{cv+adw+\\mathrm{i} ab}{cv\\sqrt{1-gy}+adw+\\mathrm{i} ab}}^{\\ell+1} \\sum_{q=0}^{\\infty} \\binom{\\ell-p}{q} \\parens[\\bigg]{\\frac{-cv\\parens*{1-\\sqrt{1-gy}}}{cv+\\mathrm{i} ab}}^q,\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tas well as in $p$,\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re\\frac{-yz}{\\sqrt{1-gy}}\\sum_{\\ell=0}^\\infty \\mathrm{i} ab \\parens[\\Big]{1-\\frac{cv+adw}{\\mathrm{i} ab}}  \\parens[\\Big]{\\frac{cv+adw+\\mathrm{i} ab}{cv+\\mathrm{i} ab}}^{-\\ell} \\parens*{1-\\sqrt{1-gz}}^\\ell \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot  \\parens[\\Big]{\\frac{cv+adw+\\mathrm{i} ab}{cv\\sqrt{1-gy}+adw+\\mathrm{i} ab}}^{\\ell+1} \\parens[\\bigg]{\\frac{cv\\sqrt{1-gy}+\\mathrm{i} ab}{cv+\\mathrm{i} ab}}^\\ell \\sum_{p=0}^{\\ell} \\binom{\\ell+1}{p} \\parens[\\bigg]{\\frac{adw}{cv\\sqrt{1-gy}+\\mathrm{i} ab}}^p\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re\\frac{yz}{\\sqrt{1-gy}}  \\sum_{\\ell=0}^\\infty \\frac{\\parens{cv+adw-\\mathrm{i} ab}\\parens{cv+adw+\\mathrm{i} ab}}{cv\\sqrt{1-gy}+adw+\\mathrm{i} ab} \\parens*{1-\\sqrt{1-gz}}^\\ell \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot   \\parens[\\bigg]{\\frac{cv\\sqrt{1-gy}+\\mathrm{i} ab}{cv\\sqrt{1-gy}+adw+\\mathrm{i} ab}}^\\ell \\parens{\\frac{\\parens{cv\\sqrt{1-gy}+adw+\\mathrm{i} ab}^{\\ell+1}-(adw)^{\\ell+1}}{(cv\\sqrt{1-gy}+\\mathrm{i} ab)^{\\ell+1}}}.\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tFinally, we wrap things up by summing in $\\ell$,\n\t%\n\t\\begin{align}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\multlinegap]\n\t\t\t=\\Re\\frac{yz}{\\sqrt{1-gy}} \\frac{(cv+adw)^2+a^2b^2}{cv\\sqrt{1-gy}+\\mathrm{i} ab} \\Biggl(\\sum_{\\ell=0}^\\infty  \\parens*{1-\\sqrt{1-gz}}^{\\ell} - \\ldots \\\\\n\t\t\t\\ldots -\\sum_{\\ell=0}^\\infty\\frac{adw}{cv\\sqrt{1-gy}+adw+\\mathrm{i} ab}\\parens[\\bigg]{\\frac{adw\\parens*{1-\\sqrt{1-gz}}}{cv\\sqrt{1-gy}+adw+\\mathrm{i} ab}}^\\ell\\Biggr)\n\t\t\\end{multlined}\\\\\n\t\t&=\\Re\\frac{yz}{\\sqrt{1-gy}} \\frac{g}{cv\\sqrt{1-gy}+\\mathrm{i} ab} \\parens[\\bigg]{\\frac{1}{\\sqrt{1-gz}}-\\frac{adw}{cv\\sqrt{1-gy}+adw\\sqrt{1-gz}+\\mathrm{i} ab}}\\\\\n\t\t&=\\Re\\frac{yz}{\\sqrt{1-gy}\\sqrt{1-gz}} \\frac{g}{cv\\sqrt{1-gy}+adw\\sqrt{1-gz}+\\mathrm{i} ab}.\n\t\\end{align}\n\t%\n\tAs it turns out, this formula also holds for the case $m=1\\lor n=1$ that we have omitted thus far. The proof is left as an exercise to the reader.\n\t\n\tThe form of the function $C(v,w,y,z)$ is easily seen by using \\eqref{app:eq:coeff_factor} --- or more heuristically, setting $a=b=c=d=1$ in $N(v,w,y,z)$. This finishes the proof.\n\\end{proof}\n\n\\begin{proposition}\\label{app:prop:coeff_zero}\n\tIf any of the following conditions is \\emph{not} met, the coefficient $c^{m,n}_{i,j}$ is zero,\n\t%\n\t\\begin{align}\n\ti+j&\\equiv 1\\bmod{2}, & i+j&\\le 2(m+n)-3, & (i&\\ge 2m-1 \\lor j\\ge 2n-1).\n\t\\end{align}\n\t%\n\tIf the first two conditions \\emph{are} met, the coefficients can be calculated as follows\n\t\\begin{align}\n\t\tc^{m,n}_{i,j}\n\t\t\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$c^{m,n}_{i,j}$}-\\multlinegap]\n\t\t\t=\\sum_{r=0}^{i}\\sum_{s=0}^{j} \\delta_{\\{r+s\\equiv1\\bmod{2}\\}} (-1)^{m+n+\\frac{r+s-1}{2}} \\binom{r+s}{s}  \\cdot\\ldots \\\\\n\t\t\t\\ldots \\cdot  \\binom{\\frac{r-1}{2}}{m-1}\\binom{\\frac{s-1}{2}}{n-1} \\binom{m+n-1}{m+n-1-\\frac{i+j-r-s}{2}} \\binom{i+j-r-s}{i-r}.\n\t\t\\end{multlined}\n\t\\end{align}\n\\end{proposition}\n\n\\begin{proof}\n\tInstead of explicit differentiation, we will reverse the process of calculating a generating function, and ``disassemble'' $C(v,w,y,z)$ in a particular way. However, we will still use the information we have already, i.e.~that $i+j$ must be odd and less than $2(m+n)-3$ (e.g.~from the representation in \\autoref{app:prop:coeff_expl_long}). First we calculate\n\t%\n\t\\begin{align}\n\t\tc^{m,n}_{i,j}\n\t\t&=\\Re\\coeff*{v^i w^j y^{m}z^{n}}\\frac{yz}{\\sqrt{1-hy}\\sqrt{1-hz}} \\frac{h}{v\\sqrt{1-hy}+w\\sqrt{1-hz}+\\mathrm{i}} \\\\\n\t\t&=\\Re\\coeff*{v^i w^j y^{m}z^{n}} \\frac{hyz}{\\sqrt{1-hy}\\sqrt{1-hz}} \\sum_{r\\ge 0}\\frac{(-v\\sqrt{1-hy})^r}{(w\\sqrt{1-hz}+\\mathrm{i})^{r+1}}\\\\\n\t\t&=\\Re\\coeff*{v^i w^j y^{m}z^{n}} \\frac{hyz}{\\sqrt{1-hz}} \\sum_{r\\ge 0} (-v)^r (\\sqrt{1-hy})^{r-1} (-\\mathrm{i})^{r+1} \\sum_{s\\ge 0} \\binom{r+s}{s} \\parens*{\\mathrm{i} w\\sqrt{1-hz}}^s\\\\\n\t\t&=\\Re\\coeff*{v^i w^j y^{m}z^{n}} -hyz \\sum_{r,s\\ge 0} \\mathrm{i}^{r+s+1} \\binom{r+s}{s} v^r w^s \\sum_{k\\ge 0} \\binom{\\frac{r-1}{2}}{k} (-hy)^k \\sum_{\\ell\\ge 0} \\binom{\\frac{s-1}{2}}{\\ell} (-hz)^\\ell,\n\t\n\t\n\t\n\t\n\t\\end{align}\n\t%\n\thaving used \\eqref{app:eq:ops_geom}, \\eqref{app:eq:ops_geom_k} and \\eqref{app:eq:ops_geom_alpha}. Extracting the appropriate powers of $y,z$, we continue\n\t%\n\t\\begin{align}\n\t\tc^{m,n}_{i,j}\n\t\t&=\\Re\\coeff*{v^i w^j} \\sum_{r,s\\ge 0} (-1)^{m+n+1} \\mathrm{i}^{r+s+1} \\binom{r+s}{s} \\binom{\\frac{r-1}{2}}{m-1} \\binom{\\frac{s-1}{2}}{n-1} v^r w^s h^{m+n-1}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$c^{m,n}_{i,j}$}-\\multlinegap]\n\t\t\t=\\Re\\coeff*{v^i w^j} \\sum_{r,s,t,u\\ge 0}  (-1)^{m+n+1} \\mathrm{i}^{r+s+1} \\binom{r+s}{s} \\binom{\\frac{r-1}{2}}{m-1} \\binom{\\frac{s-1}{2}}{n-1} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot\\binom{m+n-1}{t} \\binom{2(m+n-1-t)}{u} v^{r+u} w^{s+2(m+n-1-t)-u} 1^t\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$c^{m,n}_{i,j}$}-\\multlinegap]\n\t\t\t=\\Re\\coeff*{v^i w^j} \\sum_{i',j',r,s\\ge 0} (-1)^{m+n+1} \\mathrm{i}^{r+s+1} \\binom{r+s}{s} \\binom{\\frac{r-1}{2}}{m-1} \\binom{\\frac{s-1}{2}}{n-1} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot \\binom{m+n-1}{m+n-1-\\frac{i'+j'-r-s}{2}} \\binom{i'+j'-r-s}{i'-r} v^{i'} w^{j'}\n\t\t\\end{multlined}\\\\\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$c^{m,n}_{i,j}$}-\\multlinegap]\n\t\t\t= \\Re\\sum_{r=0}^{i}\\sum_{s=0}^{j} (-1)^{m+n+1} \\mathrm{i}^{r+s+1} \\binom{r+s}{s} \\binom{\\frac{r-1}{2}}{m-1} \\binom{\\frac{s-1}{2}}{n-1} \\cdot\\ldots \\\\\n\t\t\t\\ldots\\cdot\\binom{m+n-1}{m+n-1-\\frac{i+j-r-s}{2}} \\binom{i+j-r-s}{i-r}.\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tFirstly, the restriction of the range in $r,s$ comes from the last binomial coefficient, because clearly the lower entry needs to be positive, and the upper one needs to be larger than the lower one. Also, like $i+j$, we see that $r+s$ has to be odd for the term to contribute to the (real part of the) sum. From this we can read off the formula for the coefficients claimed in the proposition.\n\t\n\tWe use $r+s\\equiv 1\\bmod{2}$ to split the sum into ($r=2p$ even, $s=2q+1$ odd) and ($r=2p+1$ odd, $s=2q$ even) for $p,q\\in\\bbN$,\n\t%\n\t\\begin{align}\n\t\tc^{m,n}_{i,j}\n\t\t&\\!\\begin{multlined}[t][\\linewidth-\\mathindent-\\widthof{$c^{m,n}_{i,j}$}-\\multlinegap]\n\t\t\t=\\sum_{p=0}^{\\floor{\\frac{i}{2}}}  \\sum_{q=0}^{\\floor{\\frac{j-1}{2}}}  (-1)^{m+n+p+q} \\binom{2p+2q+1}{2q+1}  \\binom{p-\\frac{1}{2}}{m-1}  \\binom{q}{n-1} \\cdot\\ldots \\\\\n\t\t\t\\shoveright{\\ldots\\cdot\\binom{m+n-1}{m+n+p+q-\\frac{i+j+1}{2}}  \\binom{i+j-2p-2q-1}{i-2p}  +\\ldots}\\\\\n\t\t\t\\shoveleft{\\mathrel{\\phantom{=}} \\ldots+ \\sum_{p=0}^{\\floor{\\frac{i-1}{2}}}  \\sum_{q=0}^{\\floor{\\frac{j}{2}}}  (-1)^{m+n+p+q} \\binom{2p+2q+1}{2q}  \\binom{p}{m-1} \\cdot \\ldots}\\\\\n\t\t\t\\ldots\\cdot\\binom{q-\\frac{1}{2}}{n-1}  \\binom{m+n-1}{m+n+p+q-\\frac{i+j+1}{2}}  \\binom{i+j-2p-2q-1}{i-2p-1}.\n\t\t\\end{multlined}\n\t\\end{align}\n\t%\n\tNow, if $i\\le 2(m-1)$, we see that $p<m-1$ in the second sum, and similarly, if $j\\le 2(n-1)$ then $q<n-1$ in the first sum, which means all terms in the sums are zero due to the binomial coefficients $\\binom{p}{m-1}$ and $\\binom{q}{n-1}$. This finishes the proof.\n\\end{proof}\n\n\\begin{remark}\\label{rem:conject_coeff_pos}\n\tAfter having investigated the quantity $I_{m,n}$ intensely while trying to prove \\autoref{app:th:Imn_est}, we are convinced that\n\t%\n\t\\begin{align}\n\t\tc^{m,n}_{i,j}\\ge 0, \\qquad \\forall m,n\\in\\bbN,\\, \\forall i,j\\in\\bbN_0.\n\t\\end{align}\n\t%\n\tHowever, we have not seriously attempted to prove this.\n\\end{remark}\n\n\\section{Introduction}\\label{sec:introduction}\n\nOver the past two decades, the field of Applied Harmonic Analysis has produced a wide range of multiscale systems which are exceptionally well-adapted to different signal classes. Wavelets \\cite{wavelets} are the classical construction in this context of course, which are able to achieve ($N$-term) approximation rates of functions with point singularities as if the singularities were not there (in stark contrast to Fourier series, for example).\n\n\\subsection{Higher-Dimensional Singularities}\n\nThe number of possible construction increases dramatically when going from point-like singularities to functions that are singular on a higher-dimensional manifold, and has spawned an entire ``zoo'' of constructions, among which are ridgelets \\cite{Can98}, curvelets \\cite{Candes2005a,Candes2005b,CDDY06}, shearlets \\cite{KuLaLiWe,shearlets_book} and contourlets \\cite{DV05} --- the mentioned constructions are well-adapted to line singularities (ridgelets, see \\cite{mutilated}), resp. curved singularities (the rest, see e.g. \\cite{curvelet_cartoon,par-mol}). Since many of the proofs (for example of approximation rates) often resemble each other between constructions, some effort has been made recently to unify them by discovering and clarifying the underlying concepts --- curvelets, shearlets and contourlets fall into the framework of so-called ``parabolic molecules'' \\cite{par-mol}, while all of the mentioned systems (including wavelets and ridgelets) are encompassed by the even broader framework of $\\alpha$-molecules \\cite{alpha-mol}.\n\nThe well-adaptedness of such a dictionary (say, $\\Phi$) to its target classes is reflected in the fact that, for a function $f$ from the respective class, the coefficient vector $\\inpr{\\Phi,f}:=(\\inpr{\\varphi_\\lambda,f})_{\\lambda\\in\\Lambda}$ contains few large coefficients and the rest is small or negligible --- allowing tremendously improved performance for data compression, data denoising, data reconstruction etc. compared to other representation systems.\n\n\\subsection{Motivation: CDD-Schemes}\\label{ssec:cdd}\n\nBut aside from image applications (where the dominating features --- mostly edges --- can often be modelled by such singularities), many of these function classes appear in the context of various PDEs as well. The seminal work \\cite{cdd} showed that, in the context of elliptic PDEs exhibiting point singularities, wavelets not only sparsify the solution, but also the Galerkin matrix of the discretised operator. This allows, roughly speaking, to construct a numerically feasible algorithm that recovers $N$ of the largest coefficients of the solution in $\\CO(N)$ floating point operations (flops) --- which is the best that is even theoretically possible.\n\nThis line of thought led to many further papers (e.g. \\cite{cdd2,Stevenson2004,quadr_unit_cost_gant_ste,Dahlke2007,dahlke_steepest_descent}) and these methods are now referred to as CDD-schemes. Considering the very strong results obtained for wavelets with this approach, the question arose whether such a scheme could also be developed for a class of PDEs exhibiting higher-dimensional singularities --- with the well-adapted representation systems mentioned above being the obvious candidates for discretising the PDE.\n\nTo formulate some sufficient conditions for transferring the results to other PDEs, consider a differential operator $A:\\CH\\to\\CH'$, mapping from a Hilbert space $\\CH$ to its dual and inducing a variational formulation\n\\begin{align}\\label{eq:intro_gen_var_prb}\n\ta(v,u)=\\ell(v), \\quad \\forall v\\in \\CH.\n\\end{align}\nBased on \\cite{cdd,Stevenson2004,Dahlke2007},  the following list of ingredients makes it possible to transfer the results achieved by \\cite{cdd} for wavelets to a discretisation for $A$ (see e.g. \\cite[Sec. 1.2]{compress}, \\cite[Sec. 4.3.1]{my_thesis}):\n\n\\begin{enumerate}[(I)]\n\\item\\label{itm:ingr:1}\n\tThe bilinear form $a$ is bounded and coercive with respect to the norm of the Hilbert space $\\CH$, i.e.\n\t%\n\t\\begin{align}\n\t\ta(v,v) \\sim \\norm{v}_{\\CH}^2\n\t\t\\quad \\text{and} \\quad\n\t\ta(v,u)\\lesssim \\norm{v}_{\\CH} \\norm{u}_{\\CH}.\n\t\\end{align}\n\\item\\label{itm:ingr:2}\n\tThe system is discretised with a frame\\footnote{Actually, more specifically, a Gel'fand frame for the Gel'fand triple $(\\CH,L^2,\\CH')$, see \\cite{Dahlke2007}.} $\\Phi$ for $\\CH$, in other words, there is a diagonal weight ${\\mathbf{W}}=\\mathrm{diag}\\parens*{(w_\\lambda)_{\\lambda\\in\\Lambda}}$ such that\n\t%\n\t\\begin{align}\n\t\t\\norm{u}_{\\CH} \\sim \\norm*{\\inpr{\\Phi,u}_\\CH}_{\\ell_{{\\mathbf{W}}}^2}:= \\norm*{\\parens*{w_\\lambda\\inpr{\\varphi_\\lambda,u}_\\CH}_{\\lambda\\in\\Lambda}}_{\\ell^2}.\n\t\\end{align}\n\\item\\label{itm:ingr:3}\n\tThe Galerkin matrix ${\\mathbf{A}}$ --- i.e. the discretisation of \\eqref{eq:intro_gen_var_prb} by $\\Phi$ (and preconditioned by ${\\mathbf{W}}$) --- needs to be ``almost diagonal''\\footnote{This name tries to encapsulate the underlying principle without going into details; the needed concept is called compressibility, see \\cite[Def. 3.6]{cdd}.}.\n\\item\\label{itm:ingr:4}\n\tThe frame $\\Phi$ is optimal for a class $\\FC$ of model solutions to \\eqref{eq:intro_gen_var_prb}, in the sense that it allows $N$-term approximation rates of functions in this class with the benchmark rate $\\sigma^*(\\FC)$; see \\autoref{sec:benchmark}.\n\\end{enumerate}\n\nThese ingredients suffice to formulate the desired algorithm --- under the assumption of unit quadrature cost\\footnote{Removing this assumption is the subject of future work based on ideas in \\cite{quadr_unit_cost_gant_ste}.} for evaluating the inner products $\\inpr{\\Phi,u}_\\CH$ and within ${\\mathbf{A}}$ --- see \\autoref{th:main_result} below.\n\n\\subsection{Linear Advection Equations}\n\n\\begin{figure}\n\t\\setlength{\\plotsize}{0.6\\linewidth}\n\t\\tikzsetnextfilename{box_sol}%\n\t\\input{figures\/source\/box_sol.tikz}\n\t\\caption[Solution for a singular right-hand side]{Solution to the advection equation \\eqref{eq:LinTrans} for a singular right-hand side}\\label{fig:sing_sol}\n\\end{figure}\n\nRegarding differential equations exhibiting solutions with more complicated singularities than points, such features already appear in the context of linear advection equations. More precisely, we will deal with the unidirectional \\emph{advection--reaction equation},\n\\begin{align}\\label{eq:LinTrans}\n\tAu:=\\vec s \\cdot \\nabla u(\\vec x)  + \\kappa(\\vec x) u(\\vec x)= f(\\vec x),\n\\end{align}\nwhich describes the stationary distribution of the unknown quantity $u$ under absorption $\\kappa$, emission $f$ and transport in direction $\\vec s$. The corresponding Hilbert space to \\eqref{eq:LinTrans} is the \\emph{anisotropic Sobolev space}\n\\begin{align}\n\t\\Hs(\\bbR^d)&:=\\set*{f\\in L^2(\\bbR^d)}{(\\vec s\\cdot\\nabla)f\\in L^2(\\bbR^d)},\n\\end{align}\nwhich is equipped with the norm\n\\begin{align}\n\t\\norm{f}_{\\Hs(\\bbR^d)}&:=\\sqrt{\\norm{f}_{L^2(\\bbR^d)}^2+\\norm{(\\vec s\\cdot\\nabla)f}_{L^2(\\bbR^d)}^2}.\n\\end{align}\n\nRoughly speaking, the operator $A$ smoothes only in the direction of $\\vec s$, and singularities orthogonal to this direction may remain, compare for example \\autoref{fig:sing_sol}. Typical methods for the numerical solution of \\eqref{eq:LinTrans} include (\\cite[Chap. 5]{ern}):\n\\begin{itemize}\n\t\\item Galerkin Least Squares: $\\inpr{Av,Au}=\\inpr{Av,f}\\quad \\forall v \\in \\CV_{\\mathrm{test}}$\n\t\\item Discontinuous Galerkin Methods\n\t\\item Streamline Upwinding Petrov Galerkin (SUPG) \\cite{supg}\n\\end{itemize}\nAll of these suffer from the fact that the transport term $s \\cdot \\nabla u$ means that \\eqref{eq:LinTrans} is not $H^1$-elliptic, which leads to ill-conditioned systems of equations (with no rigorous results about the choice or efficiency of preconditioners).\n\nEven wavelets do not perform optimally for \\eqref{eq:LinTrans} --- heuristically, the break-down can also be explained with the fact that it takes many wavelets (which are adapted to point singularities) to resolve a singularity along a line --- picture a ``cliff'' or compare again with \\autoref{fig:sing_sol}.\nTo model this behaviour of the solutions, we will also introduce a class based on ``mutilated'' Sobolev functions (cf. \\cite{mutilated}) in \\autoref{ssec:prop_advec}.\n\nRidgelets turned out to be particularly well-suited for discretising \\eqref{eq:LinTrans} --- \\cite{grohs1} essentially proved \\ref{itm:ingr:2} of the ingredients mentioned above, while \\cite{compress} showed \\ref{itm:ingr:1} $\\&$ \\ref{itm:ingr:3}. In this sense, the present work complements and completes these previous works by aiming to show the last remaining ingredient \\ref{itm:ingr:4}, which will then allow to achieve the desired highly efficient algorithms, see \\autoref{th:main_result}.\n\n\\subsection{Main Result}\n\nThe core of this paper is to show that ridgelets (in the version of \\cite{grohs1}) achieve the best possible $N$-term approximation rate for functions in $H^t(\\bbR^d)$ that are allowed to be singular across hyperplanes, stated in rough strokes in the following ``theorem''.\n\n\\begin{theorem}[Sketch of \\autoref{th:approx}]\\label{th:approx_intro}\n\tLet $f$ be a function in $f \\in H^t(\\bbR^d)$ apart from hyperplanes\\footnote{We will make this statement precise in \\autoref{def:Ht_except_hyp}.}, such that the decay condition\n\t%\n\t\\begin{align}\\label{eq:intro_f_glob_decay}\n\t\t\\abs{f(\\vec x)} \\le \\frac{C_m}{\\reg{\\vec x}^{2m}}\n\t\\end{align}\n\t%\n\tholds for all $m\\in \\bbN$, where $\\reg{\\vec x}=\\sqrt{1+|\\vec x|^2}$. Then, for arbitrary $\\delta>0$ and $N\\in\\bbN$, the function $f_N$ --- reconstructed from only the $N$ largest ridgelet coefficients of $f$ --- satisfies\n\t%\n\t\\begin{align}\\label{eq:intro_approx_f}\n\t\t\\norm{f-f_N}_{L^2} \\le C_\\delta N^{-\\frac{t}{d}+\\delta},\n\t\\end{align}\n\t%\n\twhich (up to $\\delta$) is the best theoretically possible rate.\n\t\n\tAdditionally, if $u$ is the solution to \\eqref{eq:LinTrans} with the $f$ from above, then, for $\\kappa$ smooth enough, the reconstruction $u_N$ from the $N$ largest ridgelet coefficients of $u$ similarly satisfies (note the different norm)\n\t%\n\t\\begin{align}\\label{eq:intro_approx_u}\n\t\t\\norm{u-u_N}_{\\Hs} \\le C'_\\delta N^{-\\frac{t}{d}+\\delta}.\n\t\\end{align}\n\\end{theorem}\n\nAs hinted at in the beginning, the original ridgelet definition by \\cite{Can98} already showed \\eqref{eq:intro_approx_f} in \\cite{mutilated}. However, by its construction, it is not possible to incorporate it into the kind of CDD-schemes we want to achieve. In particular, the frame elements of \\cite{mutilated} have unbounded support (which is made mathematically feasible by restricting the domain to $\\Omega=[0,1]^d$), and therefore, the necessary sparsity of the matrix that \\ref{itm:ingr:3} alludes to is impossible to achieve.\n\nIn contrast, \\cite{grohs1} constructs a frame for the full $L^2(\\bbR^d)$ (resp. $\\Hs(\\bbR^d)$), but this necessitates frame elements $\\varphi_\\lambda$ which are intrinsically in $L^2(\\bbR^d)$ themselves, and thus, have much more localised support. The price for this is that we need the full grid $\\bbZ^d$ (under a certain transformation) of translations to cover all of $\\bbR^d$ with our frame elements, while \\cite{Can98} is able to make do with a one-dimensional grid. Since the proof involves counting large coefficients (in some sense), these additional $d-1$ dimensions make the proof of our result substantially more involved and require some fairly delicate estimates to work out. In this respect, we believe that the auxiliary estimates (\\autoref{th:Imn_est} and \\autoref{cor:Imn_higher_dim}) we have proved for this purpose are of independent interest due to their increased strength compared to previous results (see \\autoref{rem:grafakos}).\n\nFinally, for the CDD-machinery to work, we need approximation estimates in the norm of the Hilbert space $\\CH=\\Hs$ in which the solution lives, compare \\eqref{eq:intro_approx_u}. This norm corresponds to multiplying the coefficient sequence (element-wise) with a growing weight (namely the ${\\mathbf{W}}$ from \\ref{itm:ingr:2}), which further complicates matters.\n\nHowever, once we achieve the proof of \\autoref{th:approx_intro}, we will have showed the following, compare \\cite[Cor. 6.1, Thm. 6.2]{compress}\\footnote{Note that the requirements of \\cite[Thm. 6.2]{compress} are stated erroneously, in the sense that the decay of the $f_i$ needs to be global (compare \\eqref{eq:intro_f_glob_decay}) and not just across the interfaces of the hyperplanes.}. For a more detailed account of combining ingredients \\ref{itm:ingr:1}--\\ref{itm:ingr:4} into the result below, we refer to \\cite[Thm. 7.1.1]{my_thesis}.\n\n\\begin{theorem}\\label{th:main_result}\n\tFor arbitrary $\\vec s\\in{\\mathbb{S}^{d-1}}$, consider \\eqref{eq:LinTrans} with right-hand side $f\\in H^t$ that is allowed to be singular across hyperplanes and satisfies the decay condition\n\t%\n\t\\begin{align}\n\t\t\\abs{f(\\vec x)} \\le \\frac{C_m}{\\reg{\\vec x}^m}\n\t\\end{align}\n\t%\n\tfor all $m\\in\\bbN$ (which is possible for compact support or exponential decay, for example).\n\t\n\tAssuming that the absorption coefficient satisfies $\\kappa\\ge\\gamma>0$ and $\\kappa\\in H^{4(t+d+1)}$, an approximand $u_\\varepsilon$ to the solution of $Au=f$ satisfying\n\t%\n\t\\begin{align}\n\t\t\\norm{u-u_\\varepsilon}_{\\Hs} \\le \\varepsilon\n\t\\end{align}\n\t%\n\tcan be found with the help of a numerically feasible ridgelet-based algorithm, such that, for arbitrary $\\sigma<\\frac td$ (and ignoring quadrature cost),\n\t%\n\t\\begin{align}\n\t\t\\#\\curly*{\\text{arithmetic operations necessary to compute $u_\\varepsilon$}}\\lesssim\\varepsilon^{-\\frac 1\\sigma}.\n\t\\end{align}\n\\end{theorem}\n\n\\begin{remark}\\label{rem:sing_kappa}\n\tAs a matter of fact, in \\cite[Sec. 7.2]{my_thesis}, we show that \\autoref{th:main_result} can be extended to $\\kappa\\in H^{4(t+d+1)}$ that is also allowed to be singular across hyperplanes\\footnote{As long as any potential singularities of $f$ lie in hyperplanes that are \\emph{parallel} to the singularity in $\\kappa$.}, which is somewhat surprising, since \\ref{itm:ingr:3} depends crucially on the smoothness of $\\kappa$. In a nutshell, one shifts the singularity from $\\kappa$ to the right-hand side $f$, where it is harmless (see above), which is possible mainly due to having an explicit formula for the solution of \\eqref{eq:LinTrans}.\n\\end{remark}\n\n\\subsection{Impact}\\label{ssec:impact}\n\nAlthough \\eqref{eq:LinTrans} is quite simple, having a highly efficient solver for such an equation opens the door to efficiently solve more complicated equations like the radiative transport equation (RTE),\n\\begin{equation}\\label{eq:RTE}\n\tB u:=\\vec s \\cdot \\nabla u(\\vec x, \\vec x) + \\beta(\\vec x) u(\\vec x,\\vec s) = f(\\vec x) + \\sigma(\\vec x) \\int_{\\bbS^{d-1}} K(\\vec s, \\vec s{}') u(\\vec x,\\vec s{}') \\d \\vec s{}'.\n\\end{equation}\nwhich couples the different directions $\\vec s, \\vec s{}'$; see e.g. \\cite[Sec. 9.5]{modest2013radiative} for an introduction, resp. \\cite{appl_RTE,nummeth_RTE,RTE_schwab,RTE_schwab2}, for examples of applications of \\eqref{eq:RTE} and existing methods to solve it.\n\nThere are several ways to utilise a solver for \\eqref{eq:LinTrans} to solve \\eqref{eq:RTE}. If we neglect the scattering term for the moment, the ridgelet-based solver we develop could be used to solve \\eqref{eq:RTE} by either tensor product or collocation methods in $\\vec s$ (similar to techniques used in \\cite{grella}, where one can additionally make use of the multiscale structure of $\\Phi$ to alleviate the curse of dimensionality by balancing resolution in angle with resolution in space).\n\nOne possibility to reintroduce the scattering term is via an iterative scheme --- for example by evaluating the integral for the previous iterand and adding the result to the right-hand side. We refer to \\cite{FFT-paper}, where an \\ttt{FFT}-based ridgelet discretisation based on this ``source iteration'' has been implemented.\n\nOne crucial aspect in these procedures is that solutions for different $\\vec s$ can be added together easily, which is satisfied by our construction (since the ridgelets achieve optimal approximation for all directions $\\vec s$ simultaneously!). While we have already mentioned that uniformly refined FE methods (even if adaptive) do not work well in this context, one might have the idea to adapt the FEM mesh anisotropically. The problem with this is that the meshes for different directions $\\vec s \\neq \\vec s{}'$ would have to be combined somehow, making cumbersome interpolation between such meshes necessary.\n\nConsidering the work already carried out in \\cite{grohs1,compress}, this paper completes the picture regarding the necessary ingredients for developing a CDD-scheme for \\eqref{eq:LinTrans}. The result of this --- \\autoref{th:main_result} --- is very strong:  complexity here is measured in terms of arithmetic operations to be carried out by a processor and the solution is even allowed to possess singularities along lines (resp. hyperplanes in  higher dimensions) --- for the right-hand side $f$, as well as the absorption coefficient $\\kappa$ (see \\autoref{rem:sing_kappa}).\n\nTo illustrate the strength of these results, consider a function $f\\in H^t$ apart from a finite number of line singularities (in arbitrary directions) in two dimensions. Then, the approximation error of using just $N$ ridgelets is $\\CO(N^{-\\frac t2})$ and the number of flops to find these coefficients is of order $\\CO(N)$. For functions with medium to high Sobolev regularity (apart from the line singularities), this approximation rate represents an improvement of many orders of magnitude over wavelet or FE Methods, with respective $N$-term approximation rates of $\\CO(N^{-\\frac 12})$ and $\\CO(N^{-\\frac 14})$, irrespective of the magnitude of $t$. In terms of complexity, the advantage is greater still because the linear systems for other methods cannot usually be solved in linear time.\n\nOn the other hand, the convergence results are confined to linear advection equations \\eqref{eq:LinTrans} and our analysis assumes that $\\vec x$ belongs to the full space $\\bbR^d$. The latter fact poses no problem if for instance the source term $f$ is compactly supported but in many applications one needs to restrict $\\vec x$ to a finite domain $D\\subseteq \\bbR^d$ and impose inflow boundary conditions. The efficient incorporation of boundary conditions will require the construction of ridgelet frames on finite domains, which is the subject of future work\\footnote{To be more precise, incorporation of inflow boundary conditions is possible with the code developed in \\cite{FFT-paper} but a rigorous analysis is still lacking.}. With such a construction at hand the theoretical analysis carried out in this paper would essentially go through also for finite domains. In this regard we mention that, very recently, shearlet frames were successfully constructed on domains (\\cite{shearlet-domains}), raising the hope that this approach can be transferred to the closely-related ridgelets.\n\n\\subsection{Outline}\n\nThe outline of the paper is as follows. Below, we wrap up the section by briefly introducing the most important notational conventions we will use throughout this paper. In \\autoref{sec:prelim}, we cover some crucial estimates, the most important properties of the advection equation \\eqref{eq:LinTrans}, as well as the model class of ``mutilated'' Sobolev functions. Furthermore, we recall the ridgelet construction of \\cite{grohs1} and a few classical results we will need later on.\n\n\\autoref{sec:benchmark} deals with a minimalistic introduction to $N$-term approximation and how one can determine the best theoretically possible approximation rate of \\emph{any} discretisation for a given class of functions $\\FC$.\n\nThe core of the thesis is contained in \\autoref{sec:approx}; aside from the differences in the ridgelet constructions (see discussion after \\autoref{th:approx_intro}) and the different techniques necessary to treat them, we are able to follow the structure of \\cite{mutilated} relatively closely --- establishing the localisation of the ridgelet coefficients for a function cut off at a hyper plane first in angle in \\autoref{sec:loc_angle}, and then in space in \\autoref{sec:loc_space}, before we proceed to the proof of the main result, \\autoref{th:approx}, in \\autoref{ssec:proof_approx}.\n\nThe main part of the paper wraps up with the conclusion in \\autoref{sec:conclusion}, while the postponed proof of the crucial estimates in \\autoref{ssec:key_est} follows in \\autoref{sec:int_est}.\n\n\n\\subsection{Notation}\n\nThis subsections lists the most important conventions we will use throughout this paper. As usual, we conclude proofs by $\\Box$; additionally, we mark the end of definitions and remarks by $\\triangle$.\n\nThe letter $\\bbN$ denotes the natural numbers \\emph{without zero}, while $\\bbN_0$ includes it. Similarly, $\\bbR^+:=(0,\\infty)$, while $\\bbR^+_0:=[0,\\infty)$.\n\nWe let $B_X(x,r):=\\{x'\\in X:\\, \\mathrm{dist}_X(x,x')<r\\}$ be the open ball in the metric space $X$. Occasionally we omit the space if it is clear from the context. To distinguish the Euclidian norm from the other norms, we denote it by $|\\vec x|$. The inner product on $\\bbR^d$ is simply denoted by $\\vec x \\cdot \\vec x'$, all other inner products are denoted by $\\inpr[\\CH]{\\cdot,\\cdot}$, where the \\emph{first} argument is antilinear and the \\emph{second} is linear (which is closer to the interpretation as a functional (see e.g. Bra-ket notation) and has several advantages, in our opinion).\n\nThe Fourier transform we use is\n\\begin{align*}\n\t\\hat f(\\vec\\xi) := \\bracket*{\\CF (f)} (\\vec\\xi) := \\int_{\\bbR^d} f(\\vec x)\\mathrm{e}^{-2\\pi\\mathrm{i} \\vec x \\cdot \\vec\\xi} \\d \\vec x,\n\\end{align*}\nwhere we will mostly omit the square brackets for improved legibility if the second term has to be used. In order to limit the amount of constants we have to carry, we define the following relation,\n\\begin{align*}\n\tA(y) \\lesssim B(y) \\, :\\Longleftrightarrow \\, \\exists\\, c>0:\\, A(y)\\le c B(y),\n\\end{align*}\nwhere the constant has to be independent of $y$.\nSimilarly, $A\\sim B$ denotes the case that both $A\\lesssim B$ and $B\\lesssim A$ hold.\n\nFor vector variables (and occasionally multi-indices), $i$ primes ($i=1,\\ldots,3$) will always indicate the last $d-i$ components of that vector, i.e. $\\vec k{}''=(k_3,k_4,\\ldots,k_d)^\\top\\in\\bbR^{d-2}$.\n\nSquare brackets around a vector --- i.e. $[\\vec e_1]$ --- denote the linear span, while $[\\vec e_1]^\\bot$ denotes its orthogonal complement. The orthogonal projection along $\\vec n$ is denoted by $\\CP_{\\vec n}$.\n\nFinally, we let $H(y):=\\mathbbm{1}_{\\bbR^+}(y)$ denote the Heaviside step function, and define the \\emph{regularised absolute value} $\\langle\\vec x\\rangle:=\\sqrt{1+|\\vec x|^2}$ (to avoid problems with division by zero).\n\n\n\\section{Preparations}\\label{sec:prelim}\n\nIn this section, we set up the foundations on which the rest of the paper will be built. In \\autoref{ssec:key_est}, we introduce a crucial estimate that will be necessary later on (but whose proof we postpone to \\autoref{sec:int_est}), while in \\autoref{ssec:prop_advec}, we deal with the properties of the advection equation and the model class of solutions we will consider. In \\autoref{ssec:ridgeframes}, we briefly recall the ridgelet construction of \\cite{grohs1} in the necessary detail to prove our results, and we wrap up this section with some classical results about interpolation in \\autoref{ssec:interp}.\n\n\\subsection{An Integral Estimate}\\label{ssec:key_est}\n\nAs one of the key tools for the main proof, we introduce the following integral inequality, the proof of which we postpone to \\autoref{sec:int_est}.\n\n\\begin{theorem}[{\\autoref{app:th:Imn_est}}]\\label{th:Imn_est}\n\tFor $m,n\\in\\bbN$, $a\\in\\bbR^+_0$, $b\\in\\bbR$, $c,d\\in\\bbR^+$, we have\n\t%\n\t\\begin{align}\n\t\tI_{m,n}&:=\\int_{-\\infty}^{\\infty} \\frac{1}{\\parens{a^2(x-b)^2+c^2}^m} \\frac{1}{\\parens{x^2+d^2}^n} \\d x \\notag \\\\\n\t\t&\\phantom{:}= \\frac{\\pi}{\\parens{a^2b^2+(ad+c)^2}^{m+n-1}} \\frac{1}{c^{2m-1}} \\frac{1}{d^{2n-1}} \\sum_{\\substack{i+j+2k=2(m+n)-3\\\\i\\ge 2m-1 \\, \\lor\\, j\\ge 2n-1}} c^{m,n}_{i,j} c^{i} (ad)^{j} (ab)^{2k}\\notag \\\\\n\t\n\t\t&\\phantom{:}\\lesssim \\frac{a^{2n-1}}{\\parens{a^2b^2+a^2d^2+c^2}^n} \\frac{1}{c^{2m-1}} +  \\frac{1}{\\parens{a^2b^2+a^2d^2+c^2}^m} \\frac{1}{d^{2n-1}}. \\label{eq:Imn_est}\n\t\\end{align}\n\t%\n\tFor an explicit representation of the constants $c^{m,n}_{i,j}$, as well as the generating functions of $I_{m,n}$ and $c^{m,n}_{i,j}$, see \\autoref{sec:int_est}.\n\\end{theorem}\n\nAs a simple corollary for higher dimensions, we also record the following corollary of \\autoref{th:Imn_est}, which is proved in \\autoref{sec:int_est} as well.\n\n\\begin{corollary}[{\\autoref{app:cor:Imn_higher_dim}}]\\label{cor:Imn_higher_dim}\n\tFor $m,n\\ge\\ceil{\\frac k2}$ $c,d>0$, we have the following inequality,\n\t%\n\t\\begin{align}\n\t\t\\int_{\\bbR^d} \\frac{1}{\\parens*{|\\vec x - \\vec t|^2 +c^2}^m} \\frac{1}{\\parens*{|\\vec x|^2+d^2}^n} \\d \\vec x  \\lesssim \\frac{1}{\\parens*{|\\vec t|^2+c^2+d^2}^n}\\frac{1}{c^{2m-k-1}}+\\frac{1}{\\parens*{|\\vec t|^2+c^2+d^2}^m}\\frac{1}{d^{2n-k-1}}.\n\t\\end{align}\n\\end{corollary}\n\n\\subsection{The Advection Equation $\\&$ Mutilated Functions}\\label{ssec:prop_advec}\n\nTo rotate the transport direction $\\vec s$ in \\eqref{eq:LinTrans} --- resp. the normals of the hyperplanes appearing in \\autoref{def:Ht_except_hyp} --- into a canonical unit vector, we need the following rotation matrices.\n\n\\begin{definition}\\label{def:rot_Rs}\n\tFor any vector $\\vec s\\in{\\mathbb{S}^{d-1}}$, let $R_{\\vec s}$ be a matrix that maps $\\vec s$ to $\\vec e_1=(1,0,\\ldots)^\\top$, and let $R_{\\vec s}^{-1}=R_{\\vec s}^\\top$ be its inverse. This rotation is not unique in dimensions $d\\ge3$, however, the ambiguity will be irrelevant. We also define the respective pullbacks for $f\\in L^2(\\bbR^d)$ by\n\t%\n\t\\begin{align}\n\t\t\\rho_{\\vec s} f(\\vec x):= f(R_{\\vec s}^{-1} \\vec x), \\qquad \\rho_{\\vec s}^{-1} f(\\vec x) := f(R_{\\vec s} \\vec x),\n\t\\end{align}\n\t%\n\tthus (for continuous $f$), $\\rho_{\\vec s} f(\\vec e_1) =f (\\vec s), \\; \\rho_{\\vec s}^{-1} f(\\vec s) = f(\\vec e_1)$.\n\\end{definition}\n\n\\begin{remark}\n\tUsing these pullbacks, it's easy to see that\n\t%\n\t\\begin{align}\\label{eq:trans_dir_e1}\n\t\t\\vec s \\cdot \\nabla u + \\kappa u= f\\quad \\Longleftrightarrow \\quad \\vec e_1 \\cdot \\nabla \\rho_{\\vec s} u  + \\rho_{\\vec s} \\kappa \\rho_{\\vec s} u= \\rho_{\\vec s} f.\n\t\\end{align}\n\t%\n\tThis makes an explicit calculation possible (see \\cite[Sec. 1]{compress}), namely that with\n\t%\n\t\\begin{align}\\label{eq:lintrans_expl_sol}\n\t\ty(x_1,\\vec x{}') := \\mathrm{e}^{-K(x_1,\\vec x{}')} \\int_{-\\infty}^{x_1} \\rho_{\\vec s} f (t,\\vec x{}') \\mathrm{e}^{K(t,\\vec x{}')} \\d t , \\quad \\text{where } \\quad K(t,\\vec x{}')= \\int_0^t \\rho_{\\vec s} \\kappa (r,\\vec x{}') \\d r,\n\t\\end{align}\n\t%\n\tsetting $u:=\\rho_{\\vec s}^{-1}y$ yields an explicit solution to \\eqref{eq:LinTrans} for arbitrary $f\\in L^2(\\bbR^d)$, as long as $0<\\kappa_0 \\le \\kappa(\\vec x) < \\infty$ almost everywhere. With the $y$ from \\eqref{eq:lintrans_expl_sol}, we define the solution operator for $A$ as follows,\n\t%\n\t\\begin{align}\n\t\t\\CS[f]:=\\rho_{\\vec s}^{-1}y, \\qquad \\CS\\colon L^2\\to \\Hs.\\tag*{\\qedhere}\n\t\\end{align}\n\\end{remark}\n\nThe following theorem shows that $\\CS$ is bounded from $H^t$ to itself.\n\n\\begin{proposition}[{\\cite[Thm. 2.2]{compress}, \\cite[Thm. 3.3.3]{my_thesis}}]\\label{prop:sol_op_bounded}\n\tFor $\\kappa\\in H^{\\ceil{t}+\\frac d2}(\\bbR^d)$ with $t\\ge 0$, such that $\\kappa\\ge \\gamma>0$, the operator $\\CS$ and is bounded (at least) from $H^t\\to H^t$. Furthermore\n\t%\n\t\\begin{align}\\label{eq:Hs_elliptic}\n\t\t\\norm{\\CS[f]}_{\\Hs}\\sim \\norm{Au}_{L^2}=\\norm{f}_{L^2}.\n\t\\end{align}\n\\end{proposition}\n\nHowever, what we are mainly interested in in this paper are functions of the following form.\n\n\\begin{definition}\\label{def:Ht_except_hyp}\n\tWe say that a function $f$ is in $H^t$ \\emph{except for $N$ hyperplanes} if there are $N$ hyperplanes $h_i$ with corresponding (normalised) orthogonal vectors $\\vec n_i$ and offsets $v_i$, as well as functions $f_0,f_1,\\ldots, f_N\\in H^t(\\bbR^d)$ such that\n\t%\n\t\\begin{align}\\label{eq:Ht_except_hyp}\n\t\tf(\\vec x)=f_0(\\vec x)+\\sum_{i=1}^N f_i(\\vec x) H(\\vec x \\cdot \\vec n_i - v_i),\n\t\\end{align}\n\t%\n\twhere, as mentioned, $H$ is the Heaviside step function. To abbreviate the concept notationally, we will sometimes write $f\\in H^t(\\bbR^d\\setminus\\{(h_i)_{i=1}^N\\})$, or $f\\in H^t(\\bbR^d\\setminus\\{h_i\\})$ for short. This is justified in the sense that --- by factoring with the right choice of equivalence relation --- these are in fact Hilbert spaces again (see \\cite[Rem. 3.2.2]{my_thesis}).\n\\end{definition}\n\nThe analysis of the ridgelet coefficients of such a function will require calculating the Fourier transform of such mutilated Sobolev functions, which --- for each term --- splits into a regular and a singular contribution (from the function and the cut-off, respectively). This follows immediately from \\cite[Eq. (3.3)]{mutilated}, but we formulate it in the version we will use later on.\n\n\\begin{lemma}\\label{lem:fourier_decomp}\n\tThe Fourier transform of a function $f(\\vec x)=H(\\vec x\\cdot \\vec n-v) g(\\vec x)$, where $g\\in H^t$ and $t> \\frac 12$, in terms of a singularity aligned with $\\vec e_1$ is\n\t%\n\t\\begin{align}\\label{eq:sing_fourier_rot}\n\t\t\\hat f(\\Rn^{-1} \\vec \\xi)\n\t\t&= - \\frac{\\mathrm{i}}{2\\pi\\abs*{\\vec\\xi}^2} \\vec \\xi \\cdot \\CF[H(x_1-v)\\nabla g(\\Rn^{-1}\\vec x)](\\vec \\xi) - \\frac{\\mathrm{i}\\, \\xi_1}{2\\pi\\abs*{\\vec\\xi}^2} \\wh{g\\normalr|_{h}}(\\CP_{\\vec e_1}\\vec \\xi),\n\t\\end{align}\n\t%\n\twhere $g\\normalr|_{h}\\in H^{t-\\frac 12}(\\bbR^{d-1})$ is the restriction to the hyperplane $h=\\set{\\vec x\\in\\bbR^d}{\\vec x\\cdot\\vec  n=v}$. Furthermore, $\\CP_{\\vec n}$ is the orthogonal projection along $\\vec n$ and we identify $\\CP_{\\vec n}\\vec \\xi$ with $\\vec \\xi{}'_h\\in \\bbR^{d-1}$, while $\\wh{g\\normalr|_{h}}$ is shorthand for $\\CF_{\\bbR^{d-1}}[g\\normalr|_{h}]$.\n\\end{lemma}\n\nSince we are dealing with half-spaces due to the cut-off with the Heaviside function, the following extension result (see e.g. \\cite[Sec. 4.5]{triebel2}) will be useful.\n\n\\begin{theorem}\\label{th:half_space_ext}\n\tFor a function $f\\in H^t$, the restriction to the half-space $\\set{\\vec x\\in\\bbR^d}{x_1>0}$ can be extended to the full space in a bounded fashion, i.e. there is $g\\in H^t$ such that\n\t%\n\t\\begin{align}\n\t\t\\norm{g}_{H^t(\\bbR^d)}  \\lesssim \\norm{f}_{H^t(\\set{\\vec x\\in\\bbR^d}{x_1>0})} \\qquad \\text{as well as} \\qquad f(\\vec x) = g(\\vec x) \\quad \\forall \\vec x \\in \\set{\\vec x\\in\\bbR^d}{x_1>0}.\n\t\\end{align}\n\t%\n\tObviously, by rotating and translating $f$ (which leaves the norms invariant), this result also holds for half-spaces separated by arbitrary hyperplanes.\n\\end{theorem}\n\n\n\\begin{remark}\\label{rem:hyp_normals_neg}\n\tUsing \\autoref{th:half_space_ext}, we see that, for arbitrary $\\vec s$, we can restrict the representation of \\eqref{eq:Ht_except_hyp} to hyperplanes satisfying $\\vec s \\cdot \\vec n_i \\le 0$, since, if $\\vec s \\cdot \\vec n_i>0$, we can extend the mutilated $f_i$ to the full space --- absorbing it into $f_0$ --- and subtracting its extension on the other side (hence, the vector $n_i$ in the Heaviside function flips signs).\n\\end{remark}\n\nThe following result extends the boundedness of $\\CS$ from \\autoref{prop:sol_op_bounded} to the spaces $H^t(\\bbR^d\\setminus\\{h_i\\})$, which can be seen as confirmation that this class of functions is well-chosen for the behaviour of \\eqref{eq:LinTrans}.\n\n\\begin{proposition}\\label{prop:sol_smooth_except_hyp}\n\tFor $f\\in H^t(\\bbR^d\\setminus\\{h_i\\})$, the solution $u=\\CS[f]\\in H^t(\\bbR^d\\setminus\\{h_i\\})$ is of the same form \\eqref{eq:Ht_except_hyp}, and furthermore, $\\norm{u_i}_{H^t} \\lesssim \\norm{f_i}_{H^t}$ for $i=1,\\ldots,N$, as well as $\\norm{u_0}_{H^t}\\lesssim \\sum_{i=0}^N \\norm{f_i}_{H^t}$. In other words, the solution operator $\\CS$ is bounded on this space,\n\t%\n\t\\begin{align}\n\t\t\\norm{\\CS}_{H^t(\\bbR^d\\setminus\\{h_i\\})\\to H^t(\\bbR^d\\setminus\\{h_i\\})}<\\infty.\n\t\\end{align}\n\\end{proposition}\n\n\\begin{proof}\n\tWe begin by noting that, since the differential equation \\eqref{eq:LinTrans} is linear, it suffices to deal with one term $f(\\vec x) H(\\vec x \\cdot \\vec n-v)$, and the rest follows via superposition. Furthermore, using \\eqref{eq:trans_dir_e1} and \\autoref{rem:hyp_normals_neg}, we can assume without loss of generality that $\\vec s=\\vec e_1$, and that $(\\vec n)_1\\le 0$. \n\t\n\tAssuming $(\\vec n)_1<0$ (i.e.~a strict inequality) for the moment, we use \\eqref{eq:lintrans_expl_sol} with this right-hand side, and observe that, if $x_1$ is before the cut-off, the Heaviside function does not matter, and for everything beyond that, the integral goes just until the interface of the hyperplane,\n\t%\n\t\\begin{align}\n\t\tu(\\vec x)\n\t\t&= \\mathrm{e}^{-K(\\vec x)} \\int_{-\\infty}^{x_1} f (t,\\vec x{}') H\\parens*{(t,\\vec x{}')^\\top \\cdot \\vec n -v} \\mathrm{e}^{K(t,\\vec x{}')} \\d t\\\\\n\t\t&=H(\\vec x\\cdot \\vec n-v) \\mathrm{e}^{-K(\\vec x)} \\int_{-\\infty}^{x_1} f (t,\\vec x{}') \\mathrm{e}^{K(t,\\vec x{}')} \\d t +H(v-\\vec x\\cdot \\vec n) \\mathrm{e}^{-K(\\vec x)} \\int_{-\\infty}^{\\frac{v-\\vec x{}' \\cdot \\vec n'}{(\\vec n)_1}} f (t,\\vec x{}') \\mathrm{e}^{K(t,\\vec x{}')} \\d t\\\\\n\t\t&=H(\\vec x\\cdot \\vec n-v) \\CS[f](\\vec x) +H(v-\\vec x\\cdot \\vec n) \\CS[f]\\binom{\\frac{v-\\vec x{}' \\cdot \\vec n'}{(\\vec n)_1}}{\\vec x{}'} \\mathrm{e}^{K\\parens*{(\\frac{v-\\vec x{}' \\cdot \\vec n'}{(\\vec n)_1},\\vec x{}')^\\top}} \\mathrm{e}^{-K(\\vec x)}.\n\t\\end{align}\n\t%\n\tDue to \\autoref{prop:sol_op_bounded}, $\\CS[f]\\in H^t(\\bbR^d)$, with norm bounded by $\\norm{f}_{H^t}$. In order for the second term to appear, $v-\\vec x\\cdot \\vec n$ has to be greater than zero, and therefore (since $(\\vec n)_1<0$) we have $x_1>\\frac{v-\\vec x{}' \\cdot \\vec n'}{(\\vec n)_1}$. As the absorption satisfies $\\kappa\\ge\\gamma>0$, $K(\\cdot,\\vec x{}')$ is (strictly) monotonically increasing, and this implies that the term $\\mathrm{e}^{K\\parens*{(\\frac{v-\\vec x{}' \\cdot \\vec n'}{(\\vec n)_1},\\vec x{}')^\\top}} \\mathrm{e}^{-K(\\vec x)}$ is bounded. Since the coordinate transformation in the first component is linear, we conclude that\n\t%\n\t\\begin{align}\n\t\t\\CS[f]\\binom{\\frac{v-\\vec x{}' \\cdot \\vec n'}{(\\vec n)_1}}{\\vec x{}'} \\mathrm{e}^{K\\parens*{(\\frac{v-\\vec x{}' \\cdot \\vec n'}{(\\vec n)_1},\\vec x{}')^\\top}-K(\\vec x)} \\in H^t\\parens*{\\set*{\\vec x\\in\\bbR^d}{\\vec x\\cdot \\vec n-v<0}},\n\t\\end{align}\n\t%\n\tby since multiplication with smooth enough $\\kappa$ is bounded from $H^t$ to itself (which can be showing that the derivatives remain in $L^2$ and interpolation), and noting that the exponential function does not decrease the smoothness. Again by \\autoref{th:half_space_ext}, we can extend this term to $y\\in H^t(\\bbR^d)$, and thus\n\t%\n\t\\begin{align}\n\t\tu(\\vec x) &= H(\\vec x\\cdot \\vec n-v) \\parens*{\\CS[f](\\vec x)-y(\\vec x)} + y(\\vec x).\n\t\\end{align}\n\t%\n\tDue to the boundedness of $\\CS$ on $H^t$ and $\\norm{y}_{H^t} \\lesssim \\norm{f}_{H^t}$, this implies $\\norm{u}_{H^t(\\bbR^d\\setminus h)}\\lesssim \\norm{f}_{H^t(\\bbR^d\\setminus h)}$.\n\t\n\tIn the last remaining case --- that $(\\vec n)_1=0$ (i.e.~$\\vec n \\bot \\vec s$) --- the path of the integral never crosses the hyperplane, and thus $u=H(\\vec x\\cdot \\vec n-v) \\CS[f](\\vec x)$.\n\\end{proof}\n\nLastly, we discuss how decay properties transfer to the solution.\n\n\\begin{lemma}\\label{lem:decay_sol}\n\tIf $f\\in L^2$ satisfies\n\t%\n\t\\begin{align}\n\t\t\\abs{f(\\vec x)} &\\lesssim \\reg{\\vec x}^{-2n},\n\t\\end{align}\n\t%\n\tthen the solution $u=\\CS[f]$ to \\eqref{eq:LinTrans} also satisfies\n\t%\n\t\\begin{align}\n\t\t\\abs{u(\\vec x)} &\\lesssim \\reg{\\vec x}^{-2n}.\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tUsing the solution formula from \\eqref{eq:lintrans_expl_sol} (for $\\vec s =\\vec e_1$), we calculate\n\t%\n\t\\begin{align}\n\t\t\\abs{u(\\vec x)} &\\lesssim \\int_{-\\infty}^{x_1} \\reg*{(t,\\vec x{}')^\\top}^{-2n} \\mathrm{e}^{K(t,\\vec x{}')-K(x_1,\\vec x{}')} \\d t\n\t\t\\le \\int_{-\\infty}^{x_1} \\reg*{(t,\\vec x{}')^\\top}^{-2n} \\mathrm{e}^{-\\gamma(x_1-t)} \\d t,\n\t\\end{align}\n\t%\n\tsince the strict monotonicity of $K$ (due to $\\gamma>0$) implies $K(t,\\vec x{}')-K(x_1,\\vec x{}')\\le \\gamma(t-x_1)$ for $t\\le x_1$. At this point, we use the fact that for arbitrary $m$,\n\t%\n\t\\begin{align}\n\t\t\\mathrm{e}^{-\\gamma y} \\lesssim \\reg{y}^{-2m}, \\qquad \\text{for} \\qquad y\\ge 0.\n\t\\end{align}\n\t%\n\tInserting this into the above and resolving the square root from $\\reg{y}=\\sqrt{1+|y|^2}$, we continue\n\t%\n\t\\begin{align}\n\t\t\\abs{u(\\vec x)}\n\t\t&\\lesssim \\int_{-\\infty}^{x_1} \\reg*{(t,\\vec x{}')^\\top}^{-2n} \\reg{x_1-t}^{-2m} \\d t \\le \\int_{-\\infty}^{\\infty} \\frac{1}{(t^2+|\\vec x{}'|^2+1)^{n}} \\frac{1}{((t-x_1)^2+1)^{m}} \\d t \\\\\n\t\t&\\lesssim \\frac{1}{(x_1^2+|\\vec x{}'|^2+2)^n} + \\frac{1}{(x_1^2+|\\vec x{}'|^2+2)^m}\\frac{1}{(|\\vec x{}'|^2+1)^{n-\\frac 12}}\n\t\t\\le \\frac{1}{(|\\vec x|^2+1)^n} = \\reg{\\vec x}^{-2n},\n\t\\end{align}\n\t%\n\twhere the last line uses \\eqref{eq:Imn_est}, and $m\\ge n$ (which we we are free to choose this large).\n\\end{proof}\n\n\\subsection{The Ridgelet Construction \\cite{grohs1}}\\label{ssec:ridgeframes}\n\nWe recall the definition of a ridgelet frame from \\cite{grohs1,compress}, where details on the construction can be found. The starting point is a (sufficiently) smooth partition of unity in frequency (constructed from a given window function),\n\\begin{align}\\label{eq:partititon_unity}\n\t\\parens*{\\psi_{j\\!\\?\\?,\\ell}}_{j\\in\\bbN_0,\\, \\ell\\in\\{0,\\ldots,L_j\\}} \\quad \\text{such that} \\quad \\sum_{j=0}^\\infty \\sum_{\\ell=0}^{L_j} \\hat \\psi_{j\\!\\?\\?,\\ell}^2 = 1,\n\\end{align}\nsupported on (approximate) polar rectangles\n\\begin{align}\\label{eq:supp_Pjl}\n\t\\mathrm{supp}\\mathop{} \\hat \\psi_{j\\!\\?\\?,\\ell} \n\t&\\subseteq P_{j\\!\\?\\?,\\ell}:= \\set[\\Big]{\n\t\\vec \\xi \\in \\bbR^d}{ 2^{j-1}<|\\vec \\xi| < 2^{j+1}, \\, \\smash{\\frac{\\vec \\xi}{|\\vec \\xi|}} \\in B_{\\mathbb{S}^{d-1}}(\\vec s_{j\\!\\?\\?,\\ell}, 2^{-j+1})},\n\\end{align}\nwhere the vectors $\\vec s_{j\\!\\?\\?,\\ell}$ are an approximately uniform sampling of points on the sphere for each scale $j$, with average distance $\\sim 2^{-j}$ and cardinality $\\#\\{\\vec s_{j\\!\\?\\?,\\ell}\\}_{\\ell} \\lesssim 2^{j(d-1)}$, see \\cite{borup} for the construction or \\cite{grohs1,compress} for more details on their relation to ridgelets.\n\n\\begin{definition}\\label{def:phi_jlk}\n\tUsing the functions $\\hat \\psi_{j\\!\\?\\?,\\ell}$, a Parseval frame for $L^2(\\bbR^d)$ is defined by \n\t%\n\t\\begin{align}\n\t\t\\varphi_{j\\!\\?\\?,\\?\\ell\\!\\?\\?,\\?\\vec k} = \\, 2^{-\\frac{j}{2}} T_{U_{j\\!\\?\\?,\\ell} \\vec k}\\,\\psi_{j\\!\\?\\?,\\ell}, \\quad\n\t\tj\\in\\bbN_0, \\, \\ell \\in \\{0,\\ldots,L_j\\}, \\, \\vec k \\in \\bbZ^d, \n\t\\end{align}\n\t%\n\twith $T$ the translation operator, $T_{\\vec y}f(\\cdot) := f(\\cdot-\\vec y)$, and $U_{j\\!\\?\\?,\\ell}:= R_\\jl^{-1} D_{2^{-j}}$, where $R_\\jl:=R_{\\vec s_{j\\!\\?\\?,\\ell}}$ is the transformation introduced at the beginning of the section, and $D_a$ dilates the first component, $D_a \\vec k := (a \\, k_1, k_2,\\ldots,k_d)^\\top$. The rotation $R_\\jl$ is arbitrary (to the extent that it is ambiguous) but fixed. Whenever possible, we will subsume the indices of $\\varphi$ by $\\lambda=({j\\!\\?\\?,\\?\\ell\\!\\?\\?,\\?\\vec k})$.\n\\end{definition}\n\n\\begin{assumption}\\label{assump:psi_smooth}\n\tThe window functions\n\t%\n\t\\begin{align}\\label{eq:psi_trafo}\n\t\t\\hat \\psi_{({j\\!\\?\\?,\\ell})} (\\vec \\eta):= \\hat\\psi_{j\\!\\?\\?,\\ell}(U_{j\\!\\?\\?,\\ell}^{-\\top}\\vec \\eta)\n\t\\end{align}\n\t%\n\thave bounded derivatives \\emph{independently} of $j$ and $\\ell$. Thus, for all $n$ up to an upper bound $N$ dependent on the differentiability of the window functions (or possibly for all $n\\in\\bbN$ if the window functions are $\\CC^\\infty$), we have the estimate\n\t%\n\t\\begin{align\n\t\t\\norm*{\\hat \\psi_{({j\\!\\?\\?,\\ell})}}_{\\CC^n} \\le\\beta_{n}.\n\t\\end{align}\n\tIn \\cite[Lem. B.1]{compress}, it is shown that this assumption can be satisfied with a reasonable (and still quite flexible) choice of window functions.\n\\end{assumption}\n\n\\begin{definition}\n\tWe consider the diagonal matrix\n\t%\n\t\\begin{align}\n\t\t{\\mathbf{W}}=({\\mathbf{W}}_{\\lambda,\\lambda'})_{\\lambda,\\lambda'\\in\\Lambda} \\quad \\text{with} \\quad {\\mathbf{W}}_{\\lambda,\\lambda'}:= \\begin{cases}\n\t\t0, & \\lambda\\neq\\lambda',\\\\\n\t\t1+2^j \\abs{\\vec s\\cdot \\vec s_{j\\!\\?\\?,\\ell}}, & \\lambda=\\lambda',\n\t\\end{cases}\n\t\\end{align}\n\t%\n\twhich is the right choice of weight to make $\\Phi$ a frame for $\\Hs$ (see \\cite[Thm. 10]{grohs1}, \\cite[Thm. 4.3]{compress}), i.e.\n\t%\n\t\\begin{align}\\label{eq:RidgeletFrame}\n\t\t\\norm{f}_{\\Hs} \\sim \\norm{{\\mathbf{W}} \\inpr{\\Phi,f}}_{\\ell^2}\n\t\\end{align}\n\\end{definition}\n\n\\subsection{Interpolation between Spaces}\\label{ssec:interp}\n\nWe recall the scale of sequence spaces necessary for our analysis, as well as some classical interpolation results in the context we will work in (the mentioned sequence spaces, Sobolev spaces, and operators between them).\n\n\\begin{definition}\n\tThe \\emph{Lorentz spaces} of sequences are defined as follows (\\cite[p. 89]{DeVore1998}),\n\t%\n\t\\begin{align}\n\t\t\\ell^p_q:=\\set[\\bigg]{(c_n)_{n\\in\\bbN}}{\\snorm{c_\\bbN}_{\\ell^p_q}:=\\parens[\\bigg]{\\sum_{n\\in\\bbN}(c^*_n)^q n^{\\frac{q}{p}-1}}^{\\frac{1}{q}}<\\infty},\n\t\\end{align}\n\t%\n\twhere $c^*_n$ is the decreasing rearrangement of $(\\abs{c_n})_{n\\in\\bbN}$, $0< p <\\infty$ and $0< q < \\infty$. For $q=\\infty$, we have\n\t%\n\t\\begin{align}\n\t\t\\ell^p_\\infty:=\\set{(c_n)_{n\\in\\bbN}}{\\snorm{c_\\bbN}_{\\ell^p_\\infty}:=\\sup_{n\\in\\bbN} c^*_n n^{\\frac 1p} <\\infty}=:\\ell^p_w,\n\t\\end{align}\n\t%\n\twhich is also called the \\emph{weak $\\ell^p$-space}. It should be noted that, technically, the norms involved are generally only quasinorms, i.e.\n\t%\n\t\\begin{align}\\label{eq:lpw_triangle_ineq}\n\t\t\\snorm{a_\\bbN+b_\\bbN}_{\\ell^p_q}\\le K_{p,q} \\parens*{\\snorm{a_\\bbN}_{\\ell^p_q}+\\snorm{b_\\bbN}_{\\ell^p_q}}.\n\t\\end{align}\n\t%\n\tAdditionally, it can be shown that $\\ell^p_p=\\ell^p$, as well as $\\ell^p_{q_1}\\subseteq \\ell^p_{q_2}$ for $q_1\\le q_2$.\n\\end{definition}\n\n\n\\begin{theorem}\\label{thm:interp}\n\tFor Sobolev spaces with $t>0$, the real interpolation with $L^2$ depending on $0<\\theta<1$ yields (\\cite[Thm. 6.4.5]{interp_sp})\n\t%\n\t\\begin{align}\\label{eq:interp_sob}\n\t\t(L^2,H^t)_{\\theta,2}=H^{t\\theta}.\n\t\\end{align}\n\t%\n\tOn the sequence space side, we have that (\\cite[Thm. 5.3.1]{interp_sp})\n\t%\n\t\\begin{align}\\label{eq:interp_seq_sp}\n\t\t(\\ell^2,\\ell^p_w)_{\\theta,2}=\\ell^{\\bar{p}}_2, \\quad \\text{where} \\quad \\bar{p}=\\parens[\\Big]{\\frac{1-\\theta}{2} + \\frac{\\theta}{p}}^{-1}.\n\t\\end{align}\n\t%\n\tFinally, for compatible couples\\footnote{I.e. both spaces are linear subspaces of a larger Hausdorff topological vector space and the embeddings are continuous, c.f. \\cite[Def. 3.1.1]{interp_op}} $(X_0,X_1)$ and $(Y_0,Y_1)$ and an admissible linear operator $T$, i.e.\n\t%\n\t\\begin{align}\n\t\tT:X_0+X_1 \\to Y_0+Y_1, \\quad \\text{such that} \\quad T\\bigr|_{X_i}:X_i\\to Y_i \\quad \\text{and} \\quad \\norm{T}_{X_i\\to Y_i}<\\infty \\quad \\text{for}\\quad  i=0,1,\n\t\\end{align}\n\t%\n\tit holds that (\\cite[Thm. 3.1.2]{interp_sp}, \\cite[Thm. 5.1.12]{interp_op}), for all $\\theta\\in(0,1)$,\n\t%\n\t\\begin{align}\\label{eq:interp_op}\n\t\t\\norm{T}_{(X_0,X_1)_{\\theta,q}\\to(Y_0,Y_1)_{\\theta,q}}< \\infty.\n\t\\end{align}\n\\end{theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\\input{sections\/intro.tex}\n\n\\section{Related Work}\n\\input{sections\/related_work.tex}\n\n\\section{Challenges in Multi-Object Tracking}\\label{sec:challenges}\n\\input{sections\/mot_challenges}\n\n\\section{The MOTCOM Metrics}\\label{sec:motcom_metrics}\n\\input{sections\/mot_task.tex}\n\\input{sections\/occlusion_method.tex}\n\\input{sections\/motion_method.tex}\n\\input{sections\/appearance_method.tex}\n\\input{sections\/motcom_method.tex}\n\n\\section{Evaluation}\\label{sec:evaluation}\n\\input{sections\/evaluation.tex}\n\n\\section{Results}\\label{sec:results}\n\\input{sections\/results.tex}\n\n\\section{Discussion}\\label{sec:discussion}\n\\input{sections\/discussion.tex}\n\n\\section{Conclusion}\n\\input{sections\/conclusion.tex}\n\n\\subsubsection*{Acknowledgements}\nThis work has been funded by the Independent Research Fund Denmark under case number 9131-00128B.\n\n\\clearpage\n\\bibliographystyle{splncs04}\n\n\\subsubsection{Visual Similarity.}\nThe visual appearance of objects can vary widely depending on the type of object and type of scene.\nAppearance is especially important when tracking is lost, for example, due to occlusion, and re-identification is a common tool for associating broken tracklets. \nThe complexity of this process depends on the visual similarity between objects, but intra-object similarity also plays a role.\nAs an object moves through a scene, its appearance can change from the perspective of the viewer.\nThe object may turn around, increase its distance to the camera, or the illumination conditions may change. \nAside from the visual cues, the object's position is also critical.\nIntuitively, it becomes less likely to confuse objects as the spatial distance between them increases.\n\\subsection{Visual Similarity Metric}\nIn order to define a metric that links an object's visual appearance with tracking complexity, we investigate how similar an object in one frame is compared to itself and other objects in the next frame.\nTwo objects may look similar, but they cannot occupy the same spatial position.\nTherefore, we propose a spatial-aware visual similarity metric called VCOM.\n\n\\begin{figure}[!b]\n    \\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/VCOM_example.pdf}\n        \\caption{}\n        \\label{fig:blurred}\n    \\end{subfigure}\n    \\qquad \\quad\n    \\begin{subfigure}[b]{0.35\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/distance_func_drawing.pdf}\n        \\caption{}\n        \\label{fig:distance}\n    \\end{subfigure}\n    \\caption{a) Example showing three images with the object in focus and a blurred background produced from a frame from the MOT17-05 sequence. b) The distance ratio, $r$, affects the FDR when other objects are in the proximity of the target. The red dot is the nearest neighbor, the green dot is the true positive match, and the remaining dots are other objects.}\n\\end{figure}\n\nVCOM consists of a preprocessing, feature extraction, and distance evaluation step.\nFor every object $k \\in K$ in every frame $t \\in F$ an image $I^k_t$ is produced with the object's bounding box in focus and a heavy blurred background.\nWe blur the image using a discrete Gaussian function, except in the region of the object's bounding box as visualized in \\figref{blurred}.\n\nA feature embedding is then extracted from each of the preprocessed images.\nAs opposed to looking at the bounding box alone, using the entire image allows us to retain and embed spatial information in the feature vector.\nThe object's location is especially valuable in scenes with similarly looking objects and the blurred background contributes with low frequency information of the surroundings.\n\nWe blur the image with a Gaussian kernel with a fixed size of 201 and a sigma of 38 and extract the image features using an ImageNet \\cite{ImageNet_2009_CVPR} pre-trained ResNet-18 \\cite{ResNet_2016_CVPR} model.\nWe measure the similarity between the feature vector of the target object in frame $t$ and the feature vectors of all the objects in frame $t+1$ by computing the Euclidean distance.\nThe uncertainty increases if more objects are located within the proximity of the target. \nTherefore, we do not only look for the nearest neighbor, but rather the number of objects within a given distance, $d(r)$, from the target feature vector\n\\begin{equation}\\label{eq:region}\n    d(r) = d_{\\mathrm{NN}} + d_{\\mathrm{NN}} \\cdot r\n\\end{equation}\nwhere $d_{\\mathrm{NN}}$ is the distance to the nearest neighbor and $r$ is a distance ratio.\nThe ratio is multiplied by the distance to the nearest neighbor in order to account for the variance in scale, e.g., as induced by object resolution or distinctiveness.\n\nAn object within the distance boundary that shares the same identity as the target object is considered a true positive (TP) and all other objects are considered false positives (FP).\nBy measuring the complexity based on the false discovery rate, $\\mathrm{FDR} = \\frac{\\mathrm{FP}}{\\mathrm{FP} + \\mathrm{TP}}$, we get an output in the range $[0,1]$ where a higher number indicates a more complex task.\nAn illustrative example of how the FDR is determined based on the distance ratio $r$ can be seen in \\figref{distance}.\nIt is ambiguous to choose a single optimal distance ratio $r$.\nTherefore, we calculate VCOM based on the average of distance ratios from the set $R = \\{0.01,0.02,...,1.0\\}$\n\\begin{equation}\n\\mathrm{VCOM} = \\frac{1}{|R|}\\sum^{R}_{r}\\frac{1}{|F|}  \\sum_t^F \\frac{1}{|K^t|} \\sum_k^{K^t} FDR_{d(r)}(k) \n\\end{equation}\n\\subsubsection{Computing MOTCOM.}\nWe have evaluated four averaging methods for combining the sub-metrics into MOTCOM.\nThe four methods are the arithmetic, quadratic, geometric, and harmonic means and they are presented in \\equationref{arithmetic}, \\equationref{quadratic},\\equationref{geometric}, and \\equationref{harmonic}, respectively.\n\\begin{table*}[!ht]\n\\begin{tabular}{cc}\n\\begin{minipage}{0.45\\linewidth}\n    \\begin{equation}\\label{eq:arithmetic}\n        \\mathrm{arithmetic} = \\frac{1}{n} \\sum^{n}_{i=1} m_i\n    \\end{equation}\n\\end{minipage}   \n&  \n\\begin{minipage}{0.45\\linewidth}\n    \\begin{equation}\\label{eq:quadratic}\n        \\mathrm{quadratic} = \\sqrt{\\frac{1}{n} \\sum^{n}_{i=1} m_i^2}\n    \\end{equation}\n\\end{minipage}\n\\\\\n\\begin{minipage}{0.45\\linewidth}\n    \\begin{equation}\\label{eq:geometric}\n        \\mathrm{geometric} = \\sqrt[n]{\\prod^{n}_{i=1} m_i}\n    \\end{equation}    \n\\end{minipage}\n    & \n\\begin{minipage}{0.45\\linewidth}\n    \\begin{equation}\\label{eq:harmonic}\n        \\mathrm{harmonic} = \\frac{n}{\\sum^{n}_{i=1} \\frac{1}{m_i}}\n    \\end{equation}   \n\\end{minipage}\n\\end{tabular}\n\\end{table*}\n\n\\begin{figure}[]\n\\centering\n    \\includegraphics[width=0.6\\linewidth]{figs\/corr_spearman_top-30_MOT17_MOT20_means.pdf}\n    \\caption{Spearman's correlation matrix. The entries represent the MOTCOM values when the sub-metrics are combined using the four different averaging methods. The HOTA performance is the average of the top-30 ranked trackers. The scores are based on the combined MOT17 and MOT20 test splits.}\n    \\label{fig:correlation_means}\n\\end{figure}\n\nWe present the four variations of MOTCOM in \\figref{correlation_means} computed on the combined MOT17 and MOT20 test splits. \nWe see that they all correlate negatively with the HOTA score.\nHowever, the arithmetic mean has the strongest negative correlation and it correlates positively with all the sub-metrics.\nTherefore, we suggest to compute MOTCOM as the arithmetic mean of the sub-metrics.\n\n\\subsubsection{Complete Spearman's Correlation Matrix for MOT17 and MOT20.}\nIn Figure 9 in the main paper we presented a partial Spearman's correlation matrix based on the MOT17 and MOT20 sequences.\nWe used the matrix to evaluate the monotonic relationship between the three complexity metrics (MOTCOM, \\textit{density}, and \\textit{tracks}) and HOTA, MOTA, and IDF1.\nIn \\figref{spearman_top-30} we present the complete Spearman's correlation matrix, which shows additional details on the relationship between the entries.\n\n\\begin{figure}[]\n\\centering\n    \\centering\n    \\includegraphics[width=0.75\\linewidth]{figs\/corr_spearman_top-30_MOT17_MOT20.pdf}\n    \\caption{Spearman's correlation matrix. Based on the average performance of the top-30 trackers on MOT17 and MOT20 test split.}\n    \\label{fig:spearman_top-30}\n\\end{figure}\n    \n\\newpage\n\n\\subsubsection{Complete Spearman's Correlation Matrix for MOTSynth.}\nIn the main paper we presented the Spearman's Footrule Distance and the complexity scores for the MOTSynth sequences.\nTo expand upon this, we include the complete Spearman's correlation matrix for the MOTSynth train split in \\figref{spearman_motsynth}.\nThe matrix gives a detailed overview of the monotonic relationship between the entries.\n\n\\begin{figure}[]\n        \\centering\n    \\includegraphics[width=0.75\\linewidth]{figs\/corr_spearman_top-None_MOTSynth.pdf}\n    \\caption{Spearman's correlation matrix. Based on the CenterTrack performance on the MOTSynth train split.}\n    \\label{fig:spearman_motsynth}\n\\end{figure}\n\n\\subsection{Ground Truth} \nIn order to create a strong foundation for the evaluation, we are in need of benchmark datasets with consistent annotation standards and leader boards with a wide range of state-of-the-art trackers.\nTherefore, we evaluate MOTCOM on the popular MOT17 \\cite{milan2016mot16} and MOT20 \\cite{dendorfer2020motchallenge} datasets\\footnote{With permission from the MOTChallenge benchmark authors.}.\nThere are seven sequences in the test split of MOT17 and four sequences in the test split of MOT20, some of which are presented in \\figref{motchl_samples}.\nFurthermore, leader boards are provided for both benchmarks with results from 212 trackers for MOT17 and 80 trackers for MOT20. \nWe use the results from the top-30 ranked trackers\\footnote{Leader board results obtained on March 4, 2022.} based on the average HOTA score, so as to limit unstable and fluctuating performances.\n\nIn order to strengthen and support the evaluation, we include the training split of the fully synthetic MOTSynth dataset \\cite{fabbri2021motsynth} which contains 764 varied sequences of pedestrians.\nA few samples from the dataset can be seen in \\figref{motsynth_samples}.\nIn order to obtain ground truth tracker performance for MOTSynth, we train and test a CenterTrack model \\cite{zhou2020tracking} on the data.\nWe have chosen CenterTrack as it has been shown to perform well when trained on synthetic data \\cite{fabbri2021motsynth}.\n\n\\subsection{Evaluation Metrics}\nWe evaluate and compare the dataset complexity metrics by their ability to rank the MOT sequences according to the HOTA score of the trackers.\nWe rank the sequences from simple to complex by their \\textit{density}, \\textit{number of tracks} (abbr. \\textit{tracks}), MOTCOM score, and HOTA score.\nDepending on the metric, the ranking is in decreasing (HOTA) or increasing order (\\textit{density}, \\textit{tracks}, MOTCOM).\nThe absolute difference between the ranks, known as Spearman's Footrule Distance (FD) \\cite{diaconis1977spearman}, gives the distance between the ground truth and estimated ranks \n\\begin{equation}\\label{eq:md}\n    \\mathrm{FD} = \\sum^{n}_{i=1} |\\mathrm{rank}(x_i) - \\mathrm{rank}(\\mathrm{HOTA}_i)|,\n\\end{equation}\nwhere $n$ is the number of sequences and $x$ is \\textit{density}, \\textit{tracks}, or MOTCOM.\nIn order to directly compare results of sets of different lengths, we normalize the FD by the maximal possible distance $\\mathrm{FD}_{\\mathrm{max}}$ which is computed as \n\\begin{equation}\n    \\mathrm{FD}_{\\mathrm{max}} = \n    \\begin{cases}\n        \\sum^{n}_{i=1}i - \\frac{n}{2} & \\quad \\{n ~ | ~ 2m ~, ~ m \\in ~ \\mathbb{Z}^{+} \\}\\\\\n        \\sum^{n}_{i=1}i - \\frac{n+1}{2} & \\quad \\{n ~ | ~ 2m-1 ~,~ m\\in ~ \\mathbb{Z}^{+} \\}        \n    \\end{cases}.\n\\end{equation}\nFinally, we compute the normalized FD, $\\mathrm{NFD} = \\frac{\\mathrm{FD}}{\\mathrm{FD}_{\\mathrm{max}}}$.\n\n\n\\subsubsection{Preliminaries}\nWe define a MOT sequence as a set of frames $F = \\{1, 2, \\dots\\}$ containing a set of objects $K = \\{k_1, k_2, \\dots\\}$. \nThe objects do not have to be present in every frame, therefore, we define the set of frames where a given object is present by $F^{k} = \\{t_1, t_2, \\dots\\}$.\nThe objects present in a given frame $t$ are defined as the set $K^t = \\{k | k \\in K \\land~t \\in F^k\\}$.\nAt each frame $t$ an object $k$ is represented by its center-position in image coordinates and the height and width of the surrounding bounding box $k_t = (x,y,h,w)$.\n\\subsection{MOTCOM}\nOcclusion alone does not necessarily indicate an overwhelming problem if the object follows a known motion model or if it is visually distinct.\nThe same is true for erratic motion and visual similarity when viewed in isolation.\nHowever, the combination of occlusion, erratic motion, and visual similarity becomes increasingly difficult to handle.\n\nTherefore, we combine the occlusion, erratic motion, and visual similarity metrics into a single MOTCOM metric that describes the overall complexity of a sequence.\nMOTCOM is computed as the weighted arithmetic mean of the three sub-metrics and is given by\n\\begin{equation}\n\\mathrm{MOTCOM} = \\frac{w_{\\mathrm{OCOM}}\\cdot\\mathrm{OCOM}+w_{\\mathrm{MCOM}}\\cdot\\mathrm{MCOM}+w_{\\mathrm{VCOM}}\\cdot\\mathrm{VCOM}}{w_{\\mathrm{OCOM}}+w_{\\mathrm{MCOM}}+w_{\\mathrm{VCOM}}}\n\\end{equation}\nwhere $w_{\\mathrm{OCOM}}$, $w_{\\mathrm{MCOM}}$, and $w_{\\mathrm{VCOM}}$ are the weights for the three sub-metrics.\nEqual weighting can be obtained by setting $w_{\\mathrm{OCOM}}=w_{\\mathrm{MCOM}}=w_{\\mathrm{VCOM}}$, while custom weights may be suitable for specific applications.\nDuring evaluation we weight the sub-metrics equally as we deem each of the sub-problems equally difficult to handle. \n\\subsubsection{Complexity Score Plots for MOT17 and MOT20}\nIn the main paper we evaluate MOTCOM, \\textit{density}, and \\textit{tracks} on the MOT17 and MOT20 test splits.\nWe focus mainly on the ranking capabilities of the metrics as we expect tracker performance to have a monotonic, but not necessarily linear, relationship with complexity.\nThe ranks of MOTCOM, \\textit{density}, and \\textit{tracks} presented in the main paper are based on the scores displayed in \\figref{complexity_metrics_hota}.\n\\begin{figure}[]\n    \\centering\n    \\begin{subfigure}[b]{0.55\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/MOTCOM_HOTA_top30_MOT20_MOT17.pdf}\n    \\caption{MOTCOM vs. HOTA.}\n    \\label{fig:motcom_hota_top30}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.45\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/density_HOTA_top30_MOT20_MOT17.pdf}\n    \\caption{Density vs. HOTA.}\n    \\label{fig:density_hota_top30}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.45\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/tracks_HOTA_top30_MOT20_MOT17.pdf}\n    \\caption{Tracks vs. HOTA.}\n    \\label{fig:tracks_hota_top30}\n    \\end{subfigure}\n    \\caption{Average HOTA performance of the top-30 trackers on MOT17 and MOT20 test split against a) MOTCOM, b) \\textit{density}, and c) \\textit{tracks}. Square markers represent MOT20 sequences and crosses are MOT17 sequences.}\n    \\label{fig:complexity_metrics_hota}\n\\end{figure}\n\nThe position of the marker indicates the average score and the error bar is the standard deviation.\nThe marker of the MOT17 sequences is a cross and the MOT20 sequences are represented by a square.\nWe see that the MOT20 sequences have significantly higher densities and more tracks compared to the MOT17 sequences, but the HOTA performance is not correspondingly low.\nThis illustrates that \\textit{density} and \\textit{tracks} do not suffice to describe the complexity of MOT sequences.\n\\subsubsection{Erratic Motion.}\nWe use motion as a term for an object's spatial displacement between frames.\nThis is typically caused by the locomotive behavior of the object itself, camera motion, or a combination.\nAs the number of factors that influence the observed motion increases, the motion becomes harder to predict.\nAn example of two objects exhibiting different types of motion is presented in \\figref{erratic_motion}.\nThe blue object moves with approximately the same direction and speed between the time steps.\nPredicting the next state of the object seems trivial and the search space is correspondingly small.\nOn the other hand, the red object behaves erratically and unpredictably while the motion model is less confident as illustrated by the larger search space.\n\\subsection{Motion Metric}\nThe proposed motion metric, MCOM, is based on the assumption that objects move linearly when observed at small time steps.\nIf this assumption is not upheld, it is a sign of erratic motion and thereby a more complex MOT sequence.\n\nInitially, the displacement vector, $P^k_t$, between the object's position in the current and past time step is calculated as\n\\begin{equation}\n    P^k_{t} = p^k_{t} - p^k_{t-\\beta},\n\\end{equation}\nwhere $p_t$ is the position of object $k$ at time $t$, defined by its $x$- and $y$-coordinates, and $\\beta$ describes the temporal step size.\nWhen calculating the displacement between two consecutive frames $\\beta = 1$.\nThe displacement vector in the first frame of a trajectory is set to zero and $\\beta$ is capped by the first and last frame of a trajectory when the object is not present at time $t\\pm\\beta$.\n\\begin{figure}[!t]\n    \\centering\n    \\begin{subfigure}[b]{0.25\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/displacement.pdf}\n        \\caption{}\n        \\label{fig:displacement}\n    \\end{subfigure}\n    \\qquad\\quad\n    \\begin{subfigure}[b]{0.25\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/dist_area.pdf}\n        \\caption{}\n        \\label{fig:dist_area}\n    \\end{subfigure}\n    \\qquad\\quad\n    \\begin{subfigure}[b]{0.25\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{figs\/dist_area_change.pdf}\n        \\caption{}\n        \\label{fig:dist_area_change}\n    \\end{subfigure}\n    \\caption{a) Illustrative example of how the positional error $\\Delta_t$ is calculated as the distance between the true position $p_{t+\\beta}$ and estimated position $\\lambda_{t+\\beta}$. b) The three objects have traveled an equal distance. Relative to their size, the two smaller objects are displaced by a larger amount and the bounding box overlap disappears. c) If the size of an object increases between two time steps the displacement is relatively less important, compared to when the size of the object decreases.}\n    \\label{fig:displacement_and_area}\n\\end{figure}\n\nThe position in the next time step is predicted using a linear motion model with constant velocity based on the current position and the calculated displacement vector.\nThe position is predicted by\n\n\\begin{equation}\n    \\lambda^k_{t+\\beta} = P^k_{t} + p^k_t.\n\\end{equation}\nThe error between the predicted and true position of the object is calculated by\n\\begin{equation}\n    \\Delta^k_{t} = \\ell_2(p^k_{t+\\beta}, \\lambda^k_{t+\\beta})\n\\end{equation}\nwhere $\\ell_2$ is the Euclidean distance function and a larger $\\Delta^k_t$ indicates a more complex motion.\nSee \\figref{displacement} for an illustration of how the displacement error is calculated.\nThis approach may seem overly simplified, but it encapsulates changes in both direction and velocity.\nFurthermore, it is deliberately sensitive to low frame rates and camera motion, as both factors add to the complexity of tracking sequences.\n\nInspired by the analysis of decreasing tracking performance with respect to smaller object sizes by Bergmann et al. \\cite{Bergmann_2019_ICCV}, the size is also taken into consideration.\nThe combination of size and movement affects the difficulty of predicting the next state of the object.\nIn \\figref{dist_area}, the rectangles are equally displaced but do not experience the same displacement relative to their size. \nIntuitively, if a set of objects are moving at similar speeds, it is harder to track the smaller objects due to their lower spatio-temporal overlap.\n\n\\begin{wrapfigure}{r}{0.45\\textwidth}\n\\centering\n    \\includegraphics[width=\\linewidth]{figs\/plots\/weighted_func_MCOM.pdf}\n    \\caption{$\\alpha$ controls the growth of the function $g(x,\\alpha)$ and decides when an output value of 0.5 is reached. The dashed line illustrates $g(x,\\alpha)$ when using the average of a set of $\\alpha$ values.}\n    \\label{fig:weighted_func}\n\\end{wrapfigure}\n\nAccordingly, the motion-based complexity measure is based on the displacement relative to the size of the object.\nAs illustrated in \\figref{dist_area_change}, the size of the object may change between two time steps.\nThe direction of the change is critical as the displacement is less distinct if the size of the object is increasing, compared to the opposite situation.\nTherefore, we multiply the current size of the object with the change in object size to get the transformed object size\n\\begin{equation}\n    \\rho^k_t = s^k_t \\cdot \\frac{s^k_{t+\\beta}}{s^k_t}=s^k_{t+\\beta},\n\\end{equation}\nwhere $s_t^k = \\sqrt{w^k_{t} \\cdot h^k_{t}}$ and $h^k_t$ and $w^k_t$ are the height and width of object $k$ at time step $t$, respectively. \nThe motion complexity measure is then calculated as the mean size-compensated displacement across all frames, $F$, and all objects at each frame, $K^t$, and weighted by the log-sigmoid function $g(x,\\alpha)$\n\\begin{equation}    \n\\mathrm{MCOM} = \\frac{1}{|A|} \\sum_{\\alpha}^{A} g\\left( \\frac{1}{\\sum^K_k|F^k|}\\sum_k^K\\sum^{F^k}_{t} \\frac{\\Delta^{k}_{t}}{\\rho^k_t} , \\alpha \\right ),\n\\end{equation}\nwhere the average of $A = \\{0.01,0.02,...,1.0\\}$ is used to avoid manually deciding on a specific value for $\\alpha$.\nThe use of the function $g(x,\\alpha)$ is motivated by the aim of having an output in the range $[0,1]$, where a higher number describes a more complex motion. \nThe function $g(x,\\alpha)$ is given by\n\\begin{equation}\\label{eq:weighted_func}\n    g(x,\\alpha) = \\frac{1}{1+e^{-\\log(x)}\\alpha} = \\frac{1}{1+\\frac{\\alpha}{x}} = \\frac{x}{x+\\alpha},\n\\end{equation}\nwhere $\\alpha$ affects the gradient of the monotonically increasing function and indicates the point where the output of the function will reach 0.5 as illustrated in \\figref{weighted_func}.\nThe function is designed such that displacements in the lower ranges are weighted higher.\nThe argument for this choice is based on the assumption that minor increments to an extraordinarily erratic locomotive behavior have less impact on the complexity.\n\\subsubsection{Occlusion.}\nOcclusion describes situations where the visual information of an object within the camera view is partially or fully hidden.\nThere are three types of occlusion: \\textit{self-occlusion}, \\textit{scene-occlusion}, and \\textit{inter-object-occlusion} \\cite{andriyenko2011analytical}.\nSelf-occlusion can reduce the visibility of parts of an object, e.g., if a hand is placed in front of a face, but defining the level of self-occlusion is non-trivial and depends on the type of object.\nScene-occlusion occurs when a static object is located in the line of sight between the camera and the target object, thereby decreasing the visual information of the target.\nA scene-occlusion is marked by the red box in \\figref{occlusion_types}, where flowers partially occlude a sitting person.\n\n\\begin{figure}[!b]\n    \\centering\n    \\begin{subfigure}[t]{0.4\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{figs\/occlusion_types.pdf}\n        \\caption{}\n        \\label{fig:occlusion_types}\n    \\end{subfigure}\n    \\qquad \\qquad\n    \\begin{subfigure}[t]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{figs\/erratic_motion.png}\n        \\caption{}\n        \\label{fig:erratic_motion}\n    \\end{subfigure}\n    \\caption{a) Sample from MOT17-04 \\cite{milan2016mot16}. The yellow boxes illustrate objects partly occluded by scene-occlusion (red) and inter-object-occlusion (blue). b) The blue object displays nearly linear motion, whereas the red object is behaving erratically. The ellipsoids symbolize the confidence of an artificial underlying motion model.}\n\\end{figure}\nInter-object-occlusion is typically the most difficult to handle, especially if the objects are of the same type, as the trajectories of multiple objects cross.\nAn example can be seen in \\figref{occlusion_types}, where the blue box marks a person that partially occludes another person. \n\n\\subsection{Occlusion Metric}\nAs mentioned in \\sectionref{challenges}, occlusion can be divided into three types: self-, scene- and inter-object occlusion.\nIn order to quantify the occlusion rate in a sequence, one should ideally account for all three types.\nHowever, it is most often non-trivial to determine the level of self-occlusion and it is commonly not taken into account in MOT.\nPedersen et al. \\cite{Pedersen_2020_CVPR} used the ratio of intersecting object bounding boxes to determine the inter-object occlusion rate.\nSimilarly, the MOT16, MOT17, and MOT20 datasets include a visibility score based on the intersection over area (IoA) of both inter- and scene-objects \\cite{dendorfer2020motchallenge}, where IoA is formulated as the area of intersection over the area of the target.\n\nFollowing this trend, we omit self-occlusion and base the occlusion metric, OCOM, on the IoA and compute it as\n\\begin{equation}\n    \\mathrm{OCOM} = \\frac{1}{|K|} \\sum_k^K \\bar{\\nu}^k,\n\\end{equation}\nwhere $\\bar{\\nu}^k$ is the mean level of occlusion of object $k$. $\\nu^k_t$ is in the interval $[0,1]$ where 0 is fully visible and 1 is fully occluded.\nIt is assumed that terrestrial objects move on a ground plane which allows us to interpret their y-values as pseudo-depth and decide on the ordering.\nAnnotations are needed to calculate the occlusion level for objects moving in 3D.\nOCOM is defined in the interval $[0,1]$ where a higher value means more occlusion and a harder problem to solve.\n\n\\subsubsection{Occlusion.}\nOcclusions can be difficult to handle and they are often simply treated as missing data \\cite{andriyenko2011multi}.\nHowever, in scenes were the objects have weak or similar visual features this can be harmful for the tracking performance \\cite{andriyenko2011analytical,milan2013continuous,stadler2021improving}.\n\nMost authors state that a higher occlusion rate makes tracking harder \\cite{Cao2020,liu2019model,luo2014bi}, but they seldom quantify such statements.\nAn exception is the work proposed by Bergmann et al. \\cite{Bergmann_2019_ICCV} where they analyzed the tracking results with respect to object visibility, the size of the objects, and missing detections.\nMoreover, Pedersen et al. \\cite{Pedersen_2020_CVPR} argued that the number of objects is less critical than the amount and level of occlusion when it comes to multi-object tracking of fish.\nThey described the complexity of their sequences based on occlusions alone.\n\n\\subsubsection{Erratic Motion.}\nPrior information can be used to predict the next state of an object which minimizes the search space and hence reduces the impact of noisy or missing detections.\nA linear motion model assuming constant velocity is a simple, but effective method for predicting the movement of non-erratic objects like pedestrians \\cite{luo2020multiple,milan2013continuous}.\nIn scenes that include camera motion or complex movement more advanced models may improve tracker performance.\nPellegrini et al. \\cite{Pellegrini2009} proposed incorporating human social behavior into their motion model and Kratz et al. \\cite{kratz2010tracking} proposed utilizing the movement of a crowd to enhance the tracking of individuals.\nA downside of many advanced motion models is an often poor ability to generalize to other types of objects or environments.\n\n\\subsubsection{Visual Similarity.}\nVisual cues are commonly used in tracklet association and re-identification and are well studied for persons \\cite{ye2021personreid}, vehicles \\cite{khan2019vehiclereid}, and animals \\cite{schneider2019animalreid} such as zebrafish \\cite{haurum2020zebrafishreid} and tigers \\cite{schnedier2020animalreid}. \nModern trackers often solve the association step using CNNs, like Siamese networks, based on a visual affinity model \\cite{Bergmann_2019_ICCV,leal2016learning,xiang2015learning,yin2020unified}.\nSuch methods rely on visual dissimilarity between the objects. \nHowever, tracklet association becomes more difficult when objects are hard to distinguish purely by their appearance.\n\n\\subsubsection{Dataset Complexity.}\nDetermining the complexity of a dataset is a non-trivial task.\nOne may have a ``feeling'' or intuition about which datasets are harder than others, but this is subjective and can differ depending on who you ask, as well as differ depending on the task at hand.\nIn order to objectively determine the complexity of a dataset, one has to develop a task-specific framework.\nAn early attempt at this was the suite of 12 complexity measures (c-measures) by Ho and Basu \\cite{HoBasu2002}, based on concepts such as inter-class overlap and linear separability.\nHowever, these c-measures are not suitable for image datasets due to unrealistic assumptions, such as the data being linearly separable.\nTherefore, Branchaud-Charron et al. \\cite{Branchaud-Charron_2019_CVPR} developed a complexity measure based on spectral clustering, where the inter-class overlap is quantified through the eigenvalues of an approximated adjacency matrix.\nThis approach was shown to correlate well with the CNN performance on several image datasets.\nSimilarly, Cui et al. \\cite{cui2019measuring} presented a framework for evaluating the fine-grainedness of image datasets, by measuring the average distance from data examples to the class centers.\nBoth of these approaches rely on embedding the input images into a feature space by using, e.g., a CNN, and determining the dataset complexity without any indication of what makes the dataset difficult.\n\nIn contrast, dataset complexity in the MOT field has so far been determined through simple statistics such as the number of tracks and density.\nThese quantities are currently displayed for every sequence alongside other stats such as resolution and frame rate for the MOTChallenge benchmark datasets \\cite{dendorfer2020motchallenge}.\nThe preliminary works of Bergmann et al. \\cite{Bergmann_2019_ICCV} and Pedersen et al. \\cite{Pedersen_2020_CVPR} have attempted to further explain what makes a MOT sequence difficult by investigating the effect of occlusions. \nHowever, there is no clear way of describing the complexity of MOT sequences and the current methods have not been verified.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{quote}\n{\\em \nPang Juan was an ancient Chinese military general of the Wei state during the Warring States period. Both he and Sun Bin studied under the tutelage of the hermit Guiguzi, but later led the opposing armies of two countries (resp.) Wei and Qi at war. Ambushed by Qi, the Wei army suffered a crushing defeat and Pang Juan committed suicide. In traditional folklore, Sun Bin carved the words ``Pang Juan dies under this tree'' on a tree at the ambush area. When Pang and his men arrived, he saw that there were carvings on the tree so he lit a torch for a closer look. At that moment, the Qi troops lying in ambush attacked and Pang Juan committed suicide under that very tree. \\quad \\\\ \\quad \\hfill {(Based on \\url{https:\/\/en.wikipedia.org\/wiki\/Pang_Juan}) }\n}\n\\end{quote}\n\n\nThe Pang Juan example demonstrates that the announcement of something false may make it true. (A detailed analysis will be given later.) This is reminiscent of the logic of truthful public announcements, wherein the announcement of something true may make it false. The archetypical example is the announcement of $p \\wedge \\neg \\Box p$, for example when you are being told: ``You do not know that Pang Juan died in 342 BC.'' To get $p \\wedge \\neg \\Box p$ we have to read this sentence as its implicature ``Pang Juan died in 342 BC and you do not know that.''  This phenomenon may even be called the reason that this logic exists, that it is considered interesting, and various puzzles build upon the phenomenon. But in truthful public announcement logic we cannot announce something false thus making it true. The announced formulas are supposed to be true. In an alternative semantics for public announcement logic, that of conscious update \\citep{gerbrandyetal:1997}, and that is also known as {\\em believed (public) announcement logic}, the announcement of a formula is independent of its truth. In that logic, we can call an announcement a lie if the announced formula is false. A true lie is then the announcement of something false that makes it true. \n\nThis analysis of a lie is not entirely according to the traditional view of agents lying to each other: you are lying to me if you believe $\\neg\\phi$ but you say to me that $\\phi$ with the intention to make me believe that $\\phi$.\\footnote{This analysis of lying goes back to \\cite{Augustine:dm}. Lying has been a thriving topic in the philosophical community since then \\citep{siegler:1966,bok:1978,mahon:2006,mahon.stanford:2008}. More recent modal logical analyses that can be seen as a continuation of this philosophical tradition include  \\cite{baltag:2002,steiner:2006,kooietal:2011,hvdetal.lying:2012,sakama:2011,liuetal:2013,hvd.lying:2014}. In modal logics with only belief operators the intentional aspect of lying is not modelled.} If we model the realization of this intention (namely that indeed I believe you), and incorporate all belief aspects, we get $\\Box_a\\neg\\phi \\rightarrow [\\Box_a\\phi]\\Box_b \\Box_a\\phi$ (where agent $a$ lies to agent $b$). Abstracting from all belief aspects gives us $\\neg\\phi \\rightarrow [\\phi]\\phi$. Although the former is more precise, it is not uncommon, if not customary, to abstract from the source and the justification of the lie so that we get $\\neg\\phi \\rightarrow [\\phi]\\Box_b \\phi$, from where it is not a big step to $\\neg\\phi \\rightarrow [\\phi]\\phi$, the main focus of our investigation. (In order to make this identification, we can think of agent $b$ as the special agent who is able to distinguish all states, for whom it holds that $\\Box_b \\phi \\leftrightarrow \\phi$.) This simplied schema $\\neg\\phi \\rightarrow [\\phi]\\phi$ is clearly in opposition to the schema $\\phi \\rightarrow [\\phi]\\phi$ for the so-called {\\em successful formulas}. (In truthful public announcement logic the successful formulas are those for which $[\\phi]\\phi$ is valid. In that logic $[\\phi]\\phi$ is equivalent to $\\phi \\rightarrow [\\phi]\\phi$. But in believed public announcement logic, where announcements are independent of the truth of the announced formula, $[\\phi]\\phi$ is not equivalent to $\\phi \\rightarrow [\\phi]\\phi$.) In this work we mainly investigate true lies, successful formulas, and yet other notions in believed announcement logic, and also the iteration of such announcements.\n\nWe also present results in this work that are not for believed announcement logic, but that are still related to iterated announcements, or to lying. \n\nIn the logic of believed public announcements, interpreted on ${\\mathcal{KD}45}$ models encoding belief, the announced formula can be false and true before the announcement and can also be false and true after the announcement; and even beyond that, when iterating that same announcement, this value can change in arbitrary ways after each next announcement. In the logic of truthful public announcements, interpreted on ${\\mathcal{S}5}$ models encoding knowledge, lying is not possible. But it can still be that formulas other than the announcement keep changing their value when iterating a given announcement, in ways otherwise very similar to the results for belief.\n\nIn the Pang Juan example (as we will explain later) it can be argued that the action involved is not so much an announcement but an assignment. Assignments model that propositional variables change their value (it is also known as ontic change): the proposition ``Pang Juan dies under this tree'' was false, and afterwards it is true: factual change. We present a scenario involving lying wherein private (i.e., non-public) announcements and private assignments both play a role: two friends hesitating to go to a party are both being lied to that the other one is going to the party, subsequent to which they both change their minds (factual change) and go to the party.\n\nWe now give an overview of our results and of the contents of our paper. Section~\\ref{sec.two} recalls the logic of believed announcements, and the subsequent Section \\ref{sec.twoplus} introduces all novel terminology to put our results in perspective, and also gives an overview of our results. These results are then developped in detail in the remaining sections. In Section \\ref{sec.2valid} we present results on successful formulas, true lies, self-refuting formulas, and (so-called) impossible lies (that remain false even after announcement). In Section \\ref{sec.three} we elaborate on the result that there are models and announcements such that by iterating that announcement in the model we can change its value arbitrarily often and in arbitrarily chosen ways. In Section \\ref{sec.four} we syntactically characterize the single-agent true lies, i.e., formulas that satisfy $\\neg\\phi \\rightarrow [\\phi]\\phi$. In Section \\ref{sec.five} we discuss iteration of truthful public announcements on models encoding knowledge, and in Section \\ref{sec.six} the interaction of private announcements and private assignments, by way of the above-mentioned example of two friends going to party. Section \\ref{sec.last} presents an integration of our different results in view of future research.\n\n\\section{Logic of believed announcements} \\label{sec.two}\n\nWe recall the modelling of lying and truthtelling (after \\citep{hvdetal.lying:2012}) in {\\em believed (public) announcement logic}, also known as `arrow elimination' (not necessarily truthful) public announcement logic \\citep{gerbrandyetal:1997}, which is a lesser known alternative for the better known `state elimination' (truthful) public announcement logic \\citep{plaza:1989,baltagetal:1998}. Its language, structures, and semantics are as follows. Given are a finite set of agents $A$ and a countable set of propositional variables $P$ (let $a\\inA$ and $p\\inP$).\n\n\\begin{definition}[Language] \\[ \\ensuremath{\\mathcal{L}} \\ \\ni \\ \\phi ::= p \\ | \\ \\neg \\phi \\ | \\ (\\phi \\wedge \\phi) \\ | \\ \\Box_a \\phi \\ | \\ [\\phi]\\phi \\] \\end{definition}\nOther propositional connectives are defined by abbreviation. For $\\Box_a \\phi$, read `agent $a$ believes formula $\\phi$'. If there is a single agent only, we may omit the index and write $\\Box \\phi$ instead. Agent variables are $a,b,\\agentc,\\dots$. For $[\\phi]\\psi$, read `after public announcement of $\\phi$, $\\psi$'. If $\\Box_a \\neg\\phi$, we say that $\\phi$ is {\\em unbelievable} (for $a$) and, consequently, if $\\neg \\Box_a \\neg\\phi$, for which we write $\\Diamond_a \\phi$, we say that $\\phi$ is {\\em believable} (for $a$). This is also read as `agent $a$ considers it possible that $\\phi$'. Shared belief $\\Box_A \\phi$ (everybody believes that $\\phi$) is defined as $\\bigwedge_{a\\inA} \\Box_a \\phi$. We say that {\\em $\\phi$ is believable} iff all agents consider $\\phi$ believable, i.e., $\\bigwedge_{a\\inA} \\Diamond_a \\phi$, for which we write $\\Diamond_A \\phi$ (however, this is not the dual of $\\Box_A\\phi$: we may have that $\\neg\\Box_A\\neg\\phi$ but not $\\Diamond_A\\phi$). Believability plays an important role in our setting.\n\n\\begin{definition}[Structures]\nAn {\\em epistemic model} $M = ( S, R, V )$ consists of a {\\em domain} $S$ of {\\em states} (or `worlds'), an {\\em accessibility function} $R: A \\rightarrow {\\mathcal P}(S \\times S)$, where each $R(a)$, for which we write $R_a$, is an accessibility relation, and a {\\em valuation} $V: P \\rightarrow {\\mathcal P}(S)$, where each $V(p)$ represents the set of states where $p$ is true. For $s \\in S$, $(M,s)$ is an {\\em epistemic state}. \\end{definition} An epistemic state is also known as a pointed Kripke model. We often omit the parentheses in $(M,s)$. Without any restrictions we call the model class ${\\mathcal K}$. The class of models where all accessibility relations are transitive and euclidean is called ${\\mathcal K45}$, and if they are also serial it is called ${\\mathcal KD45}$ (this class is the main focus of our investigations, because it encodes agents with consistent beliefs). The class of models where all accessibility relations are equivalence relations is ${\\mathcal S5}$. Class ${\\mathcal KD45}$ is said to have the {\\em properties of belief}, and ${\\mathcal S5}$ to have the {\\em properties of knowledge}.\n\\begin{definition}[Semantics] \\label{def.truthlyingpub}\nAssume an epistemic model $M = ( S, R, V )$.  \n\\[ \\begin{array}{lcl}\nM,s \\models p &\\mbox{iff} & s \\in V_p \\\\ \nM,s \\models \\neg \\phi &\\mbox{iff} & M,s \\not \\models \\phi \\\\ \nM,s \\models \\phi \\wedge \\psi &\\mbox{iff} & M,s \\models \\phi  \\text{ and } M,s \\models \\psi \\\\  \nM,s \\models \\Box_a \\phi &\\mbox{iff} & \\mbox{for all \\ }  t \\in S: R_a(s,t) \\text{ implies } M,t  \\models \\phi \\\\  \nM,s \\models [\\phi] \\psi &\\mbox{iff} & M|\\phi,s \\models \\psi \n\\end{array} \\] \nwhere epistemic model $M|\\phi = ( S, R^\\phi, V )$ is as $M$ except that for all $a\\inA$, $R^\\phi_a \\ := \\ R_a \\cap \\ (S \\times \\II{\\phi}_M)$ (and where $\\II{\\phi}_M := \\{ s\\inS \\mid M,s\\models \\phi\\}$). \n\\end{definition} \n\nIn our semantics, $[\\phi]\\Box_a\\psi$ means that, regardless of the truth of $\\phi$, agent $a$ believes $\\psi$ after public announcement of $\\phi$. In particular, $[p]\\Box_\\agentp$ is valid: after the announcement of a propositional variable, the agent believes that it is true; regardless of the value of that variable. This is why it is called {\\em believed (public) announcement} of $\\phi$, in contrast to the {\\em truthful (public) announcement} of $\\phi$ \\citep{plaza:1989}, wherein we restrict the model to the subdomain of all states where $\\phi$ is true (see Appendix A). As said, believed announcement logic originates with \\cite{gerbrandyetal:1997}, where it is called the logic of conscious updates. In believed announcement logic new information is accepted by the agents independent from the truth of that information. In truthful announcement logic new information is only incorporated if it is true. \n\nIt should be noted that in believed announcement logic announcements can be truthful (namely when true) and lying (namely when false), whereas in truthful announcement logic announcements can only be truthful. In \\cite{hvdetal.lying:2012} the believed public announcement of $\\phi$ is modelled as non-deterministic choice between such truthful and lying public announcement of $\\phi$, so that `after truthful announcement of $\\phi$, $\\psi$' corresponds to $\\phi \\rightarrow [\\phi] \\psi$, and `after lying announcement of $\\phi$, $\\psi$' (after the lie that $\\phi$, $\\psi$) corresponds to $\\neg\\phi \\rightarrow [\\phi] \\psi$.\n\nThe announcing agent is not modelled in announcement logics, but only the effect of its announcements on the audience, the set of all agents. The interaction between announcement and belief can be formalized as $[\\phi] \\Box_a \\psi \\leftrightarrow \\Box_a (\\phi \\rightarrow [\\phi]\\psi)$ \\citep{gerbrandyetal:1997}.\n\nBelieved and truthful announcement are therefore closely related. And even in a technical sense: whenever a believed announcement is true, the semantics delivers results that are indistinguishable in the logics. This is because on any epistemic model, the model restriction semantics and the arrow restriction semantics result in bisimilar models, on the part of the model wherein the announcement is true. Investigations of these correspondences are made in \\cite{kooi.jancl:2007} and \\cite{hvdetal.lying:2012}. \n\n\\section{Iteration of lying and truthful announcements} \\label{sec.twoplus}\n\n\\subsection{What is a true lie?}\n\nLet $p$ be the proposition `Pang Juan dies under this tree' (Chinese does not have a future tense).  As a consequence of Pang Juan observing `Pang Juan dies under this tree', he dies under this tree. The logic of public announcements is a logic of public observations, and observing written text is the typical action that has a straightforward formalization in logic: it corresponds to the announcement of the proposition representing that text. We could imagine Pang Juan being uncertain if he would die under this tree ($\\neg \\Box p \\wedge \\neg \\Box \\neg p$), such that observing $p$ removes this uncertainty.\n\nStill, this analysis is unsatisfactory: If the written text had been  `Pang Juan does {\\bf not} die under this tree', i.e., $\\neg p$, he would still have died. If the written text had been `The moon is made of green cheese', i.e., the announcement of some unrelated proposition $q$, he would still have died. Even if there had nothing been written on that tree (it wasn't carvings, but merely a gnarled old tree trunk), i.e., the announcement of the trivial proposition $\\top$, he would still have died. It was giving away his position by lighting the torch that caused his death, irrespective of what was written on the tree. Therefore the scenario cannot be modelled as initial uncertainty that is reduced by an informative action: it does not fit the schema $\\neg \\phi \\rightarrow [\\phi] \\phi$. \n\nAlternatively, with some justification it can be said that $p$ is false before the observation of $p$ but after the observation of $p$, $p$ is true. Changing the value of a variable is factual\/ontic change. It could then be a lying public announcement of $p$ (observing $p$ while $p$ is false) linked to a public assignment changing the value of $p$ into true. However, this analysis has the same shortcomings as the previous one: what is observed does not matter. Also, Pang Juan can hardly be seen as enacting his own death. The actors are the soldiers of his enemy Sun Bin. Of course we can see the visual observation (informative) and the subsequent death (ontic) as two distinct, but related, actions.\n\nIndeed \\cite{peteretal:2016} argue that this action is neither epistemic (informative) nor performative (factual).\n\n\\medskip\n\nIn the True Lies movie Jamie Lee Curtis (unsuccessfully) plays the role of a timid slovenly housewife, who starts enacting the lie that she is a spy in order to make her husband jealous and thus seduce him. As Arnold Schwarzenegger really is a spy, because of her enactment she also really becomes a spy (and incidentally also a sparkling, extrovert, and well-dressed woman, a role that fits her rather well). This story of gradual and (exclusively) factual change also does not fit a logic of exclusively informational change. However, this factual changes interacts with belief (informative) change.\n\n\\medskip\n\nIn Section \\ref{sec.six} we will see that the combination of (private) informational and (private) factual change after all makes for a pretty good but different kind of true lie.  With that background we will, in Section \\ref{sec.last}, review once more the Pang Juan and the Arnold Schwarzenegger examples. In the current section (and the subsequent three sections, altogether the core of our contribution) we focus exclusively on true lies as informational change, i.e., $\\phi$ is a true lie iff $\\neg\\phi \\rightarrow [\\phi]\\phi$ is valid in the logic of believed announcements. Also relevant is the satisfiability of $\\neg\\phi \\wedge [\\phi]\\phi$. These different perspectives are, of course, related, and we will now proceed to explain how. (In Section \\ref{sec.term} they are properly defined as special cases of a more general setup.)\n\n\\medskip\n\nConsider the formula $\\phi = p \\wedge \\Box p$ and the model $M$ consisting of two states $s$ and $t$ wherein a proposition $p$ is false and true, respectively, and that are indistinguishable for the agent. We then have that $M,t \\models \\neg (p \\wedge \\Box p)$  whereas $M|(p \\wedge \\Box p),t \\models p \\wedge \\Box p$ so that \\[ M,t \\models \\neg (p \\wedge \\Box p) \\rightarrow [p \\wedge \\Box p] (p \\wedge \\Box p). \\] Therefore, the formula $p \\wedge \\Box p$ is a true lie in state $t$ of the model. On the other hand, although $M,s \\models \\neg (p \\wedge \\Box p)$, we have that $M|(p\\wedge\\Box p),s\\not\\models p \\wedge \\Box p$ for the simple reason that $p$ is false in $s$. So \\[ M,s \\not\\models \\neg (p \\wedge \\Box p) \\rightarrow [p \\wedge \\Box p] (p \\wedge \\Box p). \\] We illustrate both transitions below:\n\n\\[ \\begin{array}{lll}\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (2,0) {$p$};\n\\node (0s) at (0,-0.5) {$s$};\n\\node (1t) at (2,-0.5) {$t$};\n\\draw[<->] (0) to (1);\n\\draw[->] (0) edge[loop above,looseness=15] (0); \n\\draw[->] (1) edge[loop above,looseness=15] (1); \n\\end{tikzpicture}\n&\n\\quad \\stackrel {p\\wedge\\Box p} \\Rightarrow \\quad\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (2,0) {$p$};\n\\node (0s) at (0,-0.5) {$s$};\n\\node (1t) at (2,-0.5) {$t$};\n\\end{tikzpicture}\n\\end{array} \\]\nFor another example, note that \\[ M|(p\\wedge\\Box p),t \\models \\neg (p \\wedge \\Box p) \\rightarrow [p \\wedge \\Box p] (p \\wedge \\Box p) \\] for the simple reason that $M|(p\\wedge\\Box p),t \\not\\models \\neg(p \\wedge \\Box p)$. This seems somewhat undesirable: you do not want the implication to be true because the antecedent is false. In a given model $(N,u)$ you only want to call $\\phi$ a true lie if $N,u \\models \\neg \\phi$ but $N,u \\models [\\phi] \\phi$, in other words, if $N,u \\models \\neg \\phi \\wedge [\\phi] \\phi$. \n\nThe formula $p \\wedge \\Box p$ is sometimes a true lie and sometimes not. Of additional interest are the formulas that are always true lies, i.e., formulas $\\phi$ such that $\\neg\\phi \\rightarrow [\\phi]\\phi$ is valid. Consider model class ${\\mathcal{K}45}$. We show that $\\Box p$ and $p \\vee \\Box p$ are true lies. However, the latter is more interesting than the former.\n\nLet us first show that $\\Box p$ is a true lie. Consider an epistemic state $(M,s)$, where $M = (S,R,V)$, and such that $M,s \\models \\neg \\Box p$. Then there is a state $t$ with $(s,t) \\in R$ and such that $M,t \\models \\neg p$. Now consider the believed announcement update $\\Box  p$. Let $u$ be any state with $(s,u)\\in R$. In a ${\\mathcal{K}45}$ model $M|\\Box p$ with accessibility relation $R^{\\Box p}$ we have that $(s,u) \\not \\in R^{\\Box p}$, because from $(s,t)\\in R$ and $(s,u)\\in R$ follows (euclidicity) $(u,t)\\in R$ and thus $M,u \\models \\neg \\Box p$, so $(s,u)\\not\\in R^{\\Box p}$. In $M|\\Box p$ no state is accessible from $s$. Thus, $M|\\Box p, s \\models \\Box p$ and therefore $M, s \\models \\neg \\Box p \\rightarrow [\\Box p]\\Box p$. \n\nIf the $\\Box$ modality represents belief, this is not a very interesting form of lying, as in the resulting model the agent's beliefs are inconsistent: $M|\\Box p,s\\models \\Box p$ but also $M|\\Box p,s\\models \\Box \\neg p$. Differently said, as the agent did not believe $\\Box p$, she does not consider it possible that the lie $\\Box p$ is the truth. More interesting are what we will call the {\\em believable} true lies that we associate with the validity of $(\\neg \\phi \\wedge \\Diamond \\phi) \\rightarrow [\\phi]\\phi$ and with the satisfiability of $(\\neg \\phi \\wedge \\Diamond \\phi) \\wedge [\\phi]\\phi$ (and with the model class ${\\mathcal{KD}45}$ --- believable true lies preserve consistent beliefs). Formula $\\Box p$ is not believable in the last sense, as $\\neg \\Box p \\wedge \\Diamond \\Box p$ implies (in ${\\mathcal{KD}45}$) that $\\neg \\Box p$ and $\\Box p$ are both true, which is inconsistent.\n\nWe now show that $p \\vee \\Box p$ is a true lie. Consider a ${\\mathcal{K}45}$ epistemic state $(M,s)$, where $M = (S,R,V)$, and such that $M,s \\models \\neg p \\wedge \\neg \\Box p$. Again, because $M,s \\models \\neg p \\wedge \\neg \\Box p$ and because $M$ is a ${\\mathcal{K}45}$ model, none of the accessible states satisfy $\\Box  p$. But of course, some may satisfy $p$. Therefore, $M|(p \\vee \\Box p), s \\models \\Box p$ and thus $M|(p \\vee \\Box p), s \\models p \\vee \\Box p$. Formula $p \\vee \\Box p$ is therefore also a believable true lie, and even in the non-trivial sense that $\\neg (p \\vee \\Box p) \\wedge \\Diamond (p \\vee \\Box p) \\wedge [p \\vee \\Box p] (p \\vee \\Box p)$ is satisfiable (namely when there are accessible $p$ worlds as well from $s$ in $M$). If $p \\vee \\Box p$ is believable, this lie will preserve seriality in the updated model (as there must be an accessible $p$ state), and therefore preserves ${\\mathcal{KD}45}$.\n\nWe illustrate the difference between these two true lies on the same model as above. (We recall that $p \\wedge \\Box p$ is {\\em not} a true lie in both states of the model, but only in state $t$.)\n\n\n\\[ \\begin{array}{lll}\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (2,0) {$p$};\n\\node (0s) at (0,-0.5) {$s$};\n\\node (1t) at (2,-0.5) {$t$};\n\\draw[<->] (0) to (1);\n\\draw[->] (0) edge[loop above,looseness=15] (0); \n\\draw[->] (1) edge[loop above,looseness=15] (1); \n\\end{tikzpicture}\n&\n\\quad \\stackrel {\\Box p} \\Rightarrow \\quad\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (2,0) {$p$};\n\\node (0s) at (0,-0.5) {$s$};\n\\node (1t) at (2,-0.5) {$t$};\n\\end{tikzpicture} \\\\ \\ \\\\ \n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (2,0) {$p$};\n\\draw[<->] (0) to (1);\n\\draw[->] (0) edge[loop above,looseness=15] (0); \n\\draw[->] (1) edge[loop above,looseness=15] (1); \n\\end{tikzpicture}\n&\n\\quad \\stackrel {p \\vee \\Box p} \\Rightarrow \\quad\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (2,0) {$p$};\n\\draw[->] (0) to (1);\n\\draw[->] (1) edge[loop above,looseness=15] (1); \n\\end{tikzpicture}\n\\end{array} \\]\n\n\nIn this work we wish to investigate in depth which formulas may be true lies on a given model, and which formulas may be true lies on any model, and for which classes of models. In addition to true lies that become true after lying announcements, there are the successful formulas that remain true after truthful announcements, and we can consider two other options: self-refuting formulas that become false after truthful announcements, and what one might call `impossible lies' that remain false after lying announcements. We will therefore take a more general perspective: what formulas may {\\em change their value or keep their value} after their announcement, how does this depend on whether they were true or false before the announcement, and what happens when we iterate these announcements? We now first introduce terminology to address these matters. \n\n\\subsection{Terminology for iterated announcements} \\label{sec.term}\n\nGiven $\\sigma\\in \\{0,1\\}^*$ and $k\\leq |\\sigma|$, $\\sigma_k$ denotes the $k$th digit of $\\sigma$ and $\\sigma|k$  the prefix consisting of the first $k$ elements of $\\sigma$. We abuse notation and also view $\\sigma_k$ as a function $\\sigma_k:\\mathcal{L} \\to \\mathcal{L}$ on formulas such that\n$$\\sigma_k(\\phi)=\\left\\{\\begin{array}{ll}\n\\phi & \\text{if } \\sigma_k=1\\\\ \n\\neg\\phi & \\text{if } \\sigma_k=0\n\\end{array}\\right. $$\n\\begin{definition} \\label{def.above}\nLet ${\\mathcal X}$ be a class of models and $\\sigma\\in \\{0,1\\}^*$ with $n = |\\sigma| \\geq 2$. Formula $\\phi\\in\\ensuremath{\\mathcal{L}}$ is {\\em $\\sigma$-satisfiable} in ${\\mathcal X}$ iff $\\sigma_1(\\phi) \\wedge [\\phi]\\tau_2(\\phi)$ is satisfiable in ${\\mathcal X}$, and $\\phi$ is {\\em $\\sigma$-valid}  in ${\\mathcal X}$ iff $\\sigma_1(\\phi) \\rightarrow [\\phi]\\tau_2(\\phi)$ is valid in ${\\mathcal X}$, where:\n$$\\tau_k(\\phi)=\\left\\{\\begin{array}{ll}\n\\sigma_k(\\phi)\\land [\\phi]\\tau_{k+1}(\\phi) & 1< k<n \\\\\n\\sigma_k(\\phi) & k=n\n\\end{array}\\right.$$\nFormula $\\phi$ is {\\em non-trivially $\\sigma$-valid} iff it is $\\sigma$-valid and $\\sigma$-satisfiable.\n \nFor $\\sigma \\in  \\{0,1\\}^\\omega$ (i.e.\\  $\\sigma: \\Nat^+ \\rightarrow \\{0,1\\}$), $\\phi\\in\\ensuremath{\\mathcal{L}}$ is $\\sigma$-satisfiable in ${\\mathcal X}$ iff there is a ${\\mathcal X}$ model $(M,s)$ such that $\\phi$ is $\\sigma|k$-satisfiable in $(M,s)$ for all $k \\in \\Nat$ (with $k \\geq 2$).  Formula $\\phi\\in\\ensuremath{\\mathcal{L}}$ is $\\sigma$-valid in ${\\mathcal X}$ iff it is $\\sigma|k$-valid for all $k \\in \\Nat$ (with $k \\geq 2$).\n\\end{definition} \nFor example, if $|\\sigma|=3$ then $\\phi$ is a $\\sigma$-valid formula in ${\\mathcal K}$ iff  \n$\\sigma_1(\\phi)\\to [\\phi](\\sigma_2(\\phi)\\land [\\phi]\\sigma_3(\\phi))$ is valid in ${\\mathcal K}$; if $\\sigma = 001$ we  get $\\models \\neg\\phi\\to [\\phi](\\neg\\phi\\land [\\phi]\\phi)$ (in ${\\mathcal K}$). We typically only consider model classes ${\\mathcal K}$ and ${\\mathcal{KD}45}$.\n\nWe can alternatively describe $\\sigma$-satisfiability as follows. For that, given a model $M$, a formula $\\phi\\in\\ensuremath{\\mathcal{L}}$, and $k \\in \\Nat$, we first define $M|\\phi^0 = M$ and $M|\\phi^{k+1} = (M|\\phi^k)|\\phi$. Then,  $\\phi\\in\\ensuremath{\\mathcal{L}}$ is $\\sigma$-satisfiable in ${\\mathcal X}$ iff there is a ${\\mathcal X}$ model $(M,s)$ such that for all $1 \\leq k \\leq |\\sigma|$, $M|\\phi^{k-1}, s \\models \\sigma^k(\\phi)$. \n\n\\begin{definition}\nFormula $\\phi\\in\\ensuremath{\\mathcal{L}}$ is {\\em believable $\\sigma$-satisfiable} in ${\\mathcal X}$ iff $\\sigma_1(\\phi) \\wedge \\Diamond_A\\phi \\wedge [\\phi]\\tau_2(\\phi)$ is satisfiable in ${\\mathcal X}$, and $\\phi$ is {\\em believable $\\sigma$-valid} in ${\\mathcal X}$ iff $(\\sigma_1(\\phi) \\wedge \\Diamond_A \\phi) \\rightarrow [\\phi]\\tau_2(\\phi)$ is valid in ${\\mathcal X}$, where:\n$$\\tau_k(\\phi)=\\left\\{\\begin{array}{ll}\n\\sigma_k(\\phi)\\land \\Diamond_A \\phi\\wedge [\\phi]\\tau_{k+1}(\\phi) & 1< k<n \\\\\n\\sigma_k(\\phi) & k=n\n\\end{array}\\right.$$\n\\end{definition}\n{\\em Non-trivially believable $\\sigma$-valid} is believable $\\sigma$-valid and believable $\\sigma$-satisfiable, analogously to above, in Def.~\\ref{def.above}. Believable $\\sigma$-satisfiable and believable $\\sigma$-valid for infinite strings are also analogously defined as above. For the believable $\\sigma$-satisfiable\/valid formulas we typically consider model class ${\\mathcal{KD}45}$.\n\nIf $\\phi$ is $\\sigma$-satisfiable then it may not be believable $\\sigma$-satisfiable, as the latter implies the former. If $\\phi$ is $\\sigma$-valid then it is also believable $\\sigma$-valid, as the former implies the latter (the condition in the implication is stronger). However, if $\\phi$ is non-trivially $\\sigma$-valid then it may not be non-trivially believable $\\sigma$-valid, and vice versa. \n\n\nFor strings $\\sigma$ of length 2 we use corresponding English terminology that reflects common usage in the literature.\n\\[ \\begin{array}{l|l|l|l}\n\\sigma & \\text{\\bf name} & \\text{\\bf $\\sigma$-valid} & \\text{\\bf $\\sigma$-satisfiable} \\\\ \\hline\n10 & \\text{self-refuting} & \\models \\phi \\rightarrow [\\phi]\\neg\\phi & \\exists (M,s) : (M,s) \\models \\phi \\wedge [\\phi] \\neg\\phi \\\\ \n01 & \\text{true lie} & \\models \\neg\\phi \\rightarrow [\\phi]\\phi & \\exists (M,s) : (M,s) \\models \\neg\\phi \\wedge [\\phi]  \\phi \\\\ \n11 & \\text{successful} &  \\models \\phi \\rightarrow [\\phi]\\phi & \\exists (M,s) : (M,s) \\models \\phi \\wedge [\\phi]  \\phi \\\\ \n00 & \\text{impossible lie} & \\models \\neg\\phi \\rightarrow [\\phi]\\neg\\phi & \\exists (M,s) : (M,s) \\models \\neg\\phi  \\wedge [\\phi]  \\neg\\phi \\\\ \n\\end{array} \\]\nIt is also common to refer to a formula that is $11$-satisfiable in a given model $(M,s)$ as a {\\em successful update}, and to a formula that is not, as an {\\em unsuccessful update}, etc. The successful formulas are the most well-known from the literature, and have been investigated on the class of ${\\mathcal{S}5}$ models \\citep{hvdetal.synthese:2006,hollidayetal:2010}. The archetypical unsuccessful update and even self-refuting formula in that setting is $p \\wedge \\neg \\Box p$ \\citep{moore:1942,hintikka:1962,hollidayetal:2010}.\n\nIt is easy to come up with $\\sigma$-satisfiable formulas for any finite string $\\sigma$. But it is even easier to see finite strings as prefixes of infinite strings, and to show that infinite $\\sigma$ are always satisfiable. For infinite $\\sigma$, in view of iterated relativization \\citep{milleretal:2005,sadzik:2006}, formulas that can be announced arbitrarily often in a given model without stabilizing to the value of the announcement formula or its negation have our special attention. Such a formula we call {\\em unstable}.\n\n\\begin{definition}[Unstable]\nA $\\sigma \\in \\{0,1\\}^\\omega$ is called {\\em unstable} iff for all $n \\in \\mathbb N$ there are $m,k \\geq n$ such that $\\sigma_m = 1$ and $\\sigma_k = 0$. A formula $\\phi\\in \\ensuremath{\\mathcal{L}}$ is an unstable formula iff it is $\\sigma$-satisfiable for an unstable $\\sigma$.\n\\end{definition}\nIn other words, a formula $\\phi$ is unstable iff there is a $(M,s)$ such that for all $n \\in \\mathbb N$ there are $m,k \\geq n$ such that $M|\\phi^m,s \\models \\phi$ and $M|\\phi^k,s \\models \\neg\\phi$. In the continuation we will see that there are $\\sigma$-satisfiable formulas for any $\\sigma \\in \\{0,1\\}^\\omega$ (this includes the unstable $\\sigma$s), whereas for $\\sigma$-validity we only need to consider finite strings (this does not imply stability in a particular model, but recurrence of a finite pattern).\n\n\\subsection{Overview of results and conjectures}\n\nIn this section we present results and conjectures for $\\sigma$-satisfiable and $\\sigma$-valid formulas, with respect to the classes ${\\mathcal K}$ and ${\\mathcal{KD}45}$. We wish to encourage the reader to obtain further results. We start with two rather obvious observations. Let $\\sigma$ be a finite or infinite $\\{0,1\\}$ string:\n\\begin{itemize}\n\\item If $\\phi$ is $\\sigma$-satisfiable and $\\tau$ is a prefix of $\\sigma$, then $\\phi$ is $\\tau$-satisfiable; and if $\\phi$ is $\\sigma$-valid and $\\tau$ is a prefix of $\\sigma$, then $\\phi$ is $\\tau$-valid.\n\\item If $\\phi$ is believable $\\sigma$-satisfiable on class ${\\mathcal{KD}45}$ in $(M,s)$, then for all $m < |\\sigma|$, $M|\\phi^m \\in {\\mathcal{KD}45}$.\n\\end{itemize}\nWe now start with results for $\\sigma$-satisfiability and continue with results for $\\sigma$-validity. Various results are accompanied by lengthy proofs and drawn-out examples which are found in separate sections.\n\n\\begin{proposition} \\label{prop.a}\nLet $\\sigma\\in\\{0,1\\}^\\omega$. Then there are $\\sigma$-satisfiable formulas on class ${\\mathcal K}$.\n\\end{proposition}\n\n\\begin{proof}\nSee Section \\ref{sec.threeone}. We provide a single formula, namely $\\neg \\Box \\bot \\wedge ((\\Diamond \\Box \\bot \\wedge \\Diamond \\neg\\Box \\bot) \\rightarrow \\Diamond (p \\wedge \\Box \\bot))$, and a single frame $F$ (a frame is a model without the valuation, i.e., a pair pair $(S, R)$) with designated state $s \\in S$ such that for any $\\sigma \\in \\{0,1\\}^\\omega$, by properly choosing the valuation $V$ on $F$, we can establish that $\\neg \\Box \\bot \\wedge ((\\Diamond \\Box \\bot \\wedge \\Diamond \\neg\\Box \\bot) \\rightarrow \\Diamond (p \\wedge \\Box \\bot))$ is $\\sigma$-satisfiable in $((F,V),s)$. The root $s$ is infinitely branching. \n\\end{proof}\n\n\\begin{proposition} \\label{prop.b}\nLet $\\sigma\\in\\{0,1\\}^\\omega$. Then there are believable $\\sigma$-satisfiable formulas on class ${\\mathcal{KD}45}$.\n\\end{proposition}\n\n\\begin{proof}\nSee Section \\ref{sec.threetwo}. The result is obtained by adapting the formula and model used in Proposition \\ref{prop.a}. This is according to fairly standard method to transform a model with a directed single-agent accessibility relation into a model with two undirected accessibility relations for different agents. (The method has been used for multi-${\\mathcal{S}5}$ models but not for multi-${\\mathcal{KD}45}$ models, to our knowledge.) \n\\end{proof}\n\nWe continue with results on $\\sigma$-validity. Our special interest have non-trivial $\\sigma$-valid formulas and non-trivial believable $\\sigma$-valid formulas. (We recall that `non-trivial' means that they are also $\\sigma$-satisfiable.)\n\nFirst, a negative and a positive result for single updates.\n\n\\begin{proposition} \\label{prop.nonon}\nThere are no non-trivial $01$-valid formulas on class ${\\mathcal K}$.\n\\end{proposition}\n\n\\begin{proof}\nIf $\\phi$ is a trivial $01$-valid formula then $\\phi$ is equivalent to $\\top$. Let us therefore assume that $\\neg\\phi$ is satisfiable.\n\nSuppose towards contradiction that such a non-trivial true lie exists. Call it $\\phi$. Then $\\neg\\phi$ is satisfiable on a model. Without loss of generality suppose it is a tree, with the root of the tree as the designated state.\n\nNow consider a singleton model with an empty accessibility relation, i.e., a dead end. Call it $(M,s)$. If $(M,s)$ satisfies $\\neg\\phi$ then $\\phi$ cannot be a true lie, since the announcement of $\\phi$ does not change the model (the accessibility relation is already empty). Therefore $(M,s)$ satisfies $\\phi$. \n\nNow consider any model $(N,t)$ such that $t$ only has successors that are dead ends. Using the above result for singletons we know that those successors also satisfy $\\phi$. Now if $(N,t)$ were to satisfy $\\neg\\phi$, then $\\phi$ is not a true lie since the announcement of $\\phi$ would not change the model either: all successors are $\\phi$ states. Therefore $(N,t)$ has to satisfy $\\phi$.\n\nContinuing this reasoning, it will be clear that $\\phi$ must be satisfied in all finite-depth trees. Therefore, if a non-trivial true lie exists, it can only be satisfiable on an infinite-depth tree.\n\nLet $(M,s)$ be an infinite-depth tree such that $(M,s)$ satisfies $\\neg\\phi$ and $[\\phi]\\phi$. The formula $\\neg\\phi\\wedge[\\phi]\\phi$ is equivalent to an announcement-free formula $\\psi$ (the logic of believed announcement is equally expressive as the base modal logic, via a rewriting system). Let $d$ be the modal depth of $\\psi$.\n\nFor $d'> d$, let $(N,t)$ be a finite-depth tree that is $d'$-bisimilar to $(M,s)$ (we refer to \\cite{blackburnetal:2001} for this notion). Then for all formulas $\\eta$ of modal depth at most $d'$, $(M,s) \\models \\eta$ iff $(N,t) \\models \\eta$. Therefore, as $(M,s) \\models \\psi$ and the modal depth of $\\psi$ is $d < d'$, also $(N,t) \\models \\psi$, i.e., $(N,t) \\models \\neg\\phi \\wedge [\\phi]\\phi$. This contradicts that finite-depth trees cannot satisfy true lies as already obtained above.\n\nAs neither finite-depth nor infinite-depth trees may satisfy $\\neg\\phi$, no tree satisfies $\\neg\\phi$ and thus no pointed model. Therefore non-trivial true lies do not exist on class ${\\mathcal K}$.\n\\end{proof}\n\n\\begin{proposition} \nLet $\\sigma\\in\\{0,1\\}^2$. Then there are non-trivial $\\sigma$-valid formulas and non-trivial believable $\\sigma$-valid formulas on class ${\\mathcal{KD}45}$.\n\\end{proposition}\n\n\\begin{proof}\nSee Section \\ref{sec.2valid}.  The section also gives references for the origin of the corresponding English terminology, in the context of validity and satisfiability.\n\\end{proof}\nWe recall that a believable true lie is a believable $01$-valid formula. The following result is a syntactical characterization of this semantic notion, on the ${\\mathcal{KD}45}$ models. The term `disjunctive lying form'  is defined syntactically in Section \\ref{sec.four}.\n\\begin{proposition}[Characterization of believable true lies] \\label{prop.char} \\label{th:char}\nA formula is a believable true lie iff it is not equivalent to a disjunctive lying form.\n\\end{proposition}\n\n\\begin{proof}\nSee Section \\ref{sec.four}. The proof closely follows the proof of the syntactic characterization of successful formulas in \\cite{hollidayetal:2010}.\n\\end{proof}\nThere are successful formulas that are not true lies, such as booleans. Clearly $\\models p \\rightarrow [p]p$,  whereas  $\\not\\models \\neg p \\rightarrow [p]p$. There are also successful formulas that are true lies, such as $\\neg \\Box p$ and (in the universal fragment:) $p \\vee \\Box p$. At some stage we conjectured that {\\em all true lies are successful formulas}. But this is not the case.\n\n\\begin{proposition}[Some true lies are not successful formulas] \\label{prop.conjone}\nOn class ${\\mathcal{K}45}$ (${\\mathcal{KD}45}$), some 01-valid formulas are not 11-valid. \n\\end{proposition}\n\\begin{proof}\nLet $\\phi=\\Box\\bot\\lor (p\\land \\Diamond p\\land \\Diamond \\neg p)\\lor(\\neg p \\land \\Diamond p \\land \\Box p)$. We show that on ${\\mathcal{K}45}$ $\\phi$ is 01-valid but not 11-valid. \n\nDue to transitivity and euclidicity, a pointed ${\\mathcal{K}45}$ model consists of a designated point possibly pointing at a cluster of indistinguishable points. The formula $\\phi$ actually specifies its ${\\mathcal{K}45}$ models modulo bisimilarity. We list all eight ${\\mathcal{K}45}$ pointed models (modulo bisimilarity) for one variable $p$, where we distinguish those where $\\phi$ is true from those where $\\phi$ is false, and we also give the result of the $\\phi$ update. The underlined state is the designated state.\n\n\\bigskip\n\n\\noindent --- $\\phi$ is true at these four models:\n\\vspace{-.5cm}\n\\begin{center}\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$a:$};\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (1.5,0) {\\underline{$p$}};\n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$b:$};\n\\node (0) at (0,0) {\\underline{$\\neg p$}};\n\\node (1) at (1.5,0) {$p$};\n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$c:$};\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (1.5,0) {\\underline{$p$}};\n\\draw[<->] (0) to (1);\n\\draw[->] (0) edge[loop above,looseness=15] (0); \n\\draw[->] (1) edge[loop above,looseness=12] (1); \n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$d:$};\n\\node (0) at (0,0) {\\underline{$\\neg p$}};\n\\node (1) at (1.5,0) {$p$};\n\\draw[->] (0) to (1);\n\\draw[->] (1) edge[loop above,looseness=15] (1); \n\\end{tikzpicture}\n\\end{center}\n--- annoucing $\\phi$ in those models delivers:\n\\vspace{-.5cm}\n\\begin{center}\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$a':$};\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (1.5,0) {\\underline{$p$}};\n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$b':$};\n\\node (0) at (0,0) {\\underline{$\\neg p$}};\n\\node (1) at (1.5,0) {$p$};\n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$c':$};\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (1.5,0) {\\underline{$p$}};\n\\draw[->] (0) to (1);\n\\draw[->] (1) edge[loop above,looseness=12] (1); \n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$d':$};\n\\node (0) at (0,0) {\\underline{$\\neg p$}};\n\\node (1) at (1.5,0) {$p$};\n\\end{tikzpicture}\n\\end{center}\n\\noindent --- $\\phi$ is false at these four models:\n\\begin{center}\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$e:$};\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (1.5,0) {\\underline{$p$}};\n\\draw[->] (1) edge[loop above,looseness=12] (1); \n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$f:$};\n\\node (0) at (0,0) {\\underline{$\\neg p$}};\n\\node (1) at (1.5,0) {$p$};\n\\draw[->] (0) edge[loop above,looseness=12] (0); \n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$g:$};\n\\node (0) at (0,0) {\\underline{$\\neg p$}};\n\\node (1) at (1.5,0) {$p$};\n\\draw[<->] (0) to (1);\n\\draw[->] (0) edge[loop above,looseness=12] (0); \n\\draw[->] (1) edge[loop above,looseness=15] (1); \n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$h:$};\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (1.5,0) {\\underline{$p$}};\n\\draw[<-] (0) to (1);\n\\draw[->] (0) edge[loop above,looseness=12] (0); \n\\end{tikzpicture}\n\\end{center}\n\\noindent --- announcing $\\phi$ in those models delivers:\n\\vspace{-.5cm}\n\\begin{center}\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$e':$};\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (1.5,0) {\\underline{$p$}};\n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$f':$};\n\\node (0) at (0,0) {\\underline{$\\neg p$}};\n\\node (1) at (1.5,0) {$p$};\n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$g':$};\n\\node (0) at (0,0) {\\underline{$\\neg p$}};\n\\node (1) at (1.5,0) {$p$};\n\\draw[->] (0) to (1);\n\\draw[->] (1) edge[loop above,looseness=15] (1); \n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (min) at (-.7,0) {$h':$};\n\\node (0) at (0,0) {$\\neg p$};\n\\node (1) at (1.5,0) {\\underline{$p$}};\n\\end{tikzpicture}\n\\end{center}\nModels $e',f',g',h'$ are bisimilar to $a, b, d, a$ respectively, thus $\\neg \\phi\\to [\\phi]\\phi$ holds on $e, f, g, h$, i.e., $\\phi$ is 01-valid. However, $c'$ is bisimilar to $e$ thus $\\phi\\to [\\phi]\\phi$ is false at model $c$, i.e., $\\phi$ is not 11-valid. \n\nWe not only have that $\\phi$ is 01-valid but not 11-valid on ${\\mathcal{K}45}$ but also that $\\phi$ is 01-valid (obvious) but not 11-valid (not obvious: the transition between $c$ and $c'$ is between ${\\mathcal{KD}45}$ models) on ${\\mathcal{KD}45}$. The reader may further observe that $\\phi$ is also non-trivially believable 01-valid (the transition from $g$ to $g'$ is between ${\\mathcal{KD}45}$ models).\n\\end{proof}\nA natural question to ask is whether for all $\\sigma \\in \\{0,1\\}^* \\cup \\{0,1\\}^\\omega$ there are (non-trivial) $\\sigma$-valid formulas. The answer is (un)fortunately negative. For example, there are no (non-trivial) $001$-valid formulas. We recall that $001$-valid means $\\vDash \\neg \\phi\\to [\\phi](\\neg\\phi \\land[\\phi]\\phi)$. Let $(M,s)$ be such that $M,s\\nvDash \\phi$. Then $M,w\\vDash [\\phi](\\neg \\phi \\land [\\phi]\\phi)$, i.e., $M|\\phi,s\\vDash \\neg \\phi$ and $M|\\phi,s\\models [\\phi]\\phi$. From the latter we get $M|\\phi|\\phi,s\\models \\phi$. However, as $\\neg \\phi\\to [\\phi](\\neg\\phi \\land[\\phi]\\phi)$ is valid it also holds on $(M|\\phi, s)$, so that $M|\\phi,s\\models\\neg \\phi\\to [\\phi](\\neg\\phi \\land[\\phi]\\phi)$, and in particular $M|\\phi,s\\models \\neg \\phi\\to [\\phi]\\neg\\phi$, so that, as $M|\\phi,s\\models \\neg\\phi$, we have $M|\\phi|\\phi,s\\models \\neg\\phi$ . Contradiction.\n\nWe already observed above that if $\\sigma$ is a prefix of $\\tau$ and a formula is $\\tau$-valid then it is also $\\sigma$-valid. In the other direction this is very obviously false, for example there are impossible lies (such as $\\neg\\Box p$, i.e., $00$-valid formulas). But we saw in the previous paragraph that there are no $001$-valid formulas. However, sometimes, if $\\sigma$ is a prefix of $\\tau$ and a formula is $\\sigma$-valid, then it is also $\\tau$-valid. \n\nGiven $\\sigma\\in\\{0,1\\}^\\omega$ and prefix $\\tau$ of $\\sigma$, then $\\sigma$ and $\\tau$ (and $\\tau$ and $\\sigma$) are {\\em completion equivalent} iff for all intermediate strings $\\tau'$ (for all prefixes $\\tau'$ of $\\sigma$ such that $\\tau$ is a prefix of $\\tau'$), a formula is $\\sigma$-valid iff it is $\\tau'$-valid. This relation is clearly an equivalence relation. The shortest sequence of a completion equivalence class is the {\\em representative} and the longest (possibly infinite) sequence of this class is the {\\em completion}.\n\n\\begin{conjecture}\nThe following completion equivalence classes of $\\sigma$-valid formulas are all non-empty and all different, and there are no other classes.\n\\[\\begin{array}{llll}\n 01^k \\ (k>0) \\quad & \\quad 10^k \\ (k>0) \\quad  & \\quad 01^k0 \\ (k\\geq 0) \\quad & \\quad  10^k1 \\ (k\\geq 0)\n\\end{array}\\]\n\\end{conjecture}\n\nIf $\\sigma = 01^k0$ (for $k\\geq 0$), then $\\tau = 0(1^k0)^\\omega$ is its completion; and similarly if $\\sigma = 10^k1$ (for $k\\geq 0$), then $\\tau = 1(0^k1)^\\omega$ is its completion. For example, the $11$-valid formulas are also $111$-valid, or $1111$-valid, or $111\\dots$-valid: they are all completion equivalent; $111\\dots$ is the completion of this class and $11$ is its representative. On the other hand, $\\sigma$s of shape $01^k$ and $10^k$ (for $k > 0$) may have no infinite completions and be already maximal. \n\nThere are $011$-validities on ${\\mathcal{K}45}$. A true lie $p \\vee \\Box p$ is an example. \nWe recall that a true lie is by definition a $01$-validity. But formula $p \\vee \\Box p$ is also $011$-valid, because if announced when false, it results in a ${\\mathcal{K}45}$ model satisfying $\\Box p$ and such that further updates have no informative effect. Therefore, it is also $0111$-valid, and \\dots and $01^\\omega$-valid. Similarly, a {\\em believable} true lie $p \\vee \\Box p$ is an example of a $011$-validity on ${\\mathcal{KD}45}$. As it is believable, this guarantees the existence of an accessible $p$ state. That state will be preserved after the $p \\vee \\Box p$ update, and thus the resulting model is serial. More interesting would be a $011$-validity that is not a $0111$-validity. For another example, from Proposition \\ref{prop.nonon} follows that there are no $01^k$-validities on class ${\\mathcal K}$. But we have not investigated systematically which of the conjectured $\\sigma$-valid types are non-empty and for which classes of models.\n\n\\section{The cases 01, 11, 10, and 00} \\label{sec.2valid}\n\nIn this section we restrict ourselves to validities on class ${\\mathcal{KD}45}$ (or ${\\mathcal{S}5}$, included in ${\\mathcal{KD}45}$).\n\n\\weg{\n@@\nTypical examples of @@check!!!@@ non-trivial believable $\\sigma$-valid formulas on class ${\\mathcal{KD}45}$ are: \n\\[ \\begin{array}{|l|l|}\n\\hline\n00 & \\neg \\Box p \\\\ \n01 & \\Box p \\\\\n10 & p \\wedge \\neg \\Box p \\\\\n11 & p , \\Box p \\\\\n\\hline \n\\end{array} \\]\n@@\n}\n\n\\paragraph{Successful formulas}\n\n\\[ \\models \\phi \\rightarrow [\\phi]\\phi \\]\n\nA successful update in some given model is a $11$-satisfiable formula (in that model). We then call a formula `successful' if it is always a successful update, i.e., if it is not merely $11$-satisfiable but $11$-valid. The typical example of a successful update is `no child steps forward' in the muddy children problem \\citep{mosesetal:1986,hvdetal.puzzle:2015}. Given $k$ muddy children out of $n$ children, this is a succesful update for the first $k-2$ times that father makes his request to step forward if you know whether you are muddy, and only an unsuccessful update the $(k-1)$th time, as the $k$th time that the request is made, the muddy children step forward. The formula in question, for the example of three muddy children, is $\\neg (\\Box_a m_a \\vee \\Box_a \\neg m_a) \\wedge \\neg (\\Box_b m_b \\vee \\Box_b \\neg m_b) \\wedge \\neg (\\Box_c m_c \\vee \\Box_c \\neg m_c)$ (where $m_i$ stands for ``child $i$ is muddy''). However, the archetypical unsuccessful update is $p \\wedge \\neg \\Box p$ \\citep{moore:1942,hintikka:1962}. The original 1940s setting is the incoherence of a person saying ``I went to the pictures [movies] last Tuesday, but I don't believe that I did'' \\cite[p.\\ 543]{moore:1942}. In fact, $p \\wedge \\neg \\Box p$ is a self-refuting formula: after announcing it truthfully, it always becomes false.\n\nThe successful formulas are the most well-known from the literature, and have been investigated on the class of ${\\mathcal{S}5}$ models \\citep{hvdetal.synthese:2006,hollidayetal:2010}. On the class ${\\mathcal{S}5}$, and using the (state eliminating) semantics of truthful public announcement logic, the successful formulas are those for which $\\models [\\phi]\\phi$ (and the self-refuting formulas are then those for which $\\models [\\phi]\\neg\\phi$). On ${\\mathcal{S}5}$ (but of course not in ${\\mathcal{KD}45}$),  $\\phi \\rightarrow [\\phi]\\psi$ is equivalent to $[\\phi]\\psi$. A formula is successful in believed announcement logic if and only if it is successful in truthful announcement logic. This we can easily see:\n\nIn truthful announcement logic, $M,s \\models [\\phi]\\phi$ iff ($M,s \\models \\phi$ implies $M|_\\phi,s \\models \\phi$), where $M|_\\phi$ denotes the submodel consisting of the states satisfying $\\phi$ (see Appendix A); whereas in believed announcement logic, $M,s \\models \\phi \\rightarrow [\\phi]\\phi$ iff  ($M,s \\models \\phi$ implies $M|\\phi,s \\models \\phi$). Using that, whenever the announcement formula $\\phi$ is true in a model $(M,s)$, the model $(M|_\\phi,s)$ resulting from state elimination is bisimilar to the model $(M|\\phi,s)$ resulting from arrow elimination.\n \n\nA characterization of single-agent successful formulas in ${\\mathcal{S}5}$ is known \\citep{hollidayetal:2010} (we will use it below to obtain results for true lies), but only incidental results and no characterization is known for multi-agent successful formulas. Results are that: the positive fragment is successful \\citep{hvdetal.synthese:2006} (the positive or universal fragment of the language of public announcement logic is inductively defined by $\\phi ::= p \\mid \\neg p \\mid \\phi \\vee \\phi \\mid \\phi \\wedge \\phi \\mid \\Box_a \\phi \\mid [\\neg\\phi]\\phi$), publicly known formulas are successful (formulas $\\Box^*_A \\phi$, where $\\Box^*_A$ stands for common knowledge for the set of all agents --- however note that $\\Box^*_B \\phi$ may be unsuccessful if $B \\subset A$), $\\neg \\Box_a p$ is successful (and non-trivially believable) \\citep{qian:2002} --- an older result than \\cite{hollidayetal:2010}, but subsumed by the later characterization of \\cite{hollidayetal:2010}.\n\nA strong intuitive and technical motivation for the successful formulas in truthful announcement logic is that they describe the so-called {\\em substitution free fragment} of the logic. In general we do not have for dynamic epistemic logics that $\\models \\phi$ iff $\\models \\phi[p\/\\psi]$, for example $[p]p$ is valid but $[p \\wedge \\neg \\Box p](p \\wedge \\neg \\Box p)$ is invalid. But this {\\em uniform substitution} holds for the fragment of the language consisting of the successful formulas \\citep{hollidayetal:2013}.\n\n\\paragraph{Self-refuting formulas}\n\n\\[ \\models \\phi \\rightarrow [\\phi]\\neg\\phi \\]\n\nSimilarly to the successful formulas, a formula is self-refuting in believed announcement logic if and only if it is self-refuting in truthful announcement logic. As said, a well-known self-refuting formula (and that is also non-trivially believable) is $p \\wedge \\neg \\Boxp$. A syntactic characterization of single-agent self-refuting formulas in ${\\mathcal{S}5}$ is also presented in \\cite{hollidayetal:2010}, but a more general multi-agent characterization is also not known to us (researchers have been looking for this in vain for some considerable time).\n\n\\paragraph{Impossible lies}\n\n\\[ \\models \\neg\\phi \\rightarrow [\\phi]\\neg\\phi \\]\n\nClearly $\\top$ is an impossible lie, as this makes the antecedent $\\neg\\phi$ of the implication false, but this is the trivial case. Slightly more interesting impossible lies are the booleans. (And all these are non-trivially believable.) For example, $p$ is an impossible lie because $\\models \\neg p \\rightarrow [p] \\neg p$. This is obvious, as announcements do not change the value of propositional variables.\n\nAnother impossible lie is $\\neg \\Box p$. It is trivial: $\\models \\Box p \\rightarrow [\\neg \\Box p] \\Box p$ holds because in any model satisfying $\\Box p$ the accessibility relation of the agent is empty after the $\\neg \\Box p$ update. \n\nThe formula $\\neg \\Box p$ is also a successful formula. No characterization results are known for impossible lies. \n\n\\paragraph{True lies}\n\n\\[ \\models \\neg\\phi \\rightarrow [\\phi]\\phi \\]\n\nAbove we have already seen examples of true lies. The formula $\\top$ (or any other valid formula) is a trivial true lie. Examples of single-agent true lies are:  $\\Box p$, $p \\vee \\Box p$ (which is non-trivially believable). \n\nUnsatisfiable formulas are not true lies, as $[\\bot]\\bot$ is equivalent to $\\bot$ (in believed announcement logic, not in truthful announcement logic), so that $\\neg \\bot \\rightarrow \\bot$ is equivalent to $\\bot$. \n\nSection \\ref{sec.four} proves a syntactical characterization of single-agent believable true lies in class ${\\mathcal{KD}45}$ (Proposition \\ref{prop.char}). \n\n\\section{An unstable formula} \\label{sec.three}\n\nIn this section we investigate some models and formulas where the iteration of the same update never stabilizes and continues to transform the model, forever and ever. First comes an example with a ${\\mathcal K}$ model and a single agent, and an alternating $010101...$-satisfiable update (Section \\ref{sec.threeone}). Then comes another example generalizing this to arbitrary boolean functions $\\sigma$ and $\\sigma$-satisfiability  (Section \\ref{sec.threearb}). Following that comes a multi-agent example for ${\\mathcal{KD}45}$ agents (with consistent beliefs)  (Section \\ref{sec.threetwo}).\n\n\\subsection{Example of an unstable formula}  \\label{sec.threeone}\n\n{\\em We demonstrate that the formula $\\neg \\Box \\bot \\wedge ((\\Diamond \\Box \\bot \\wedge \\Diamond \\neg\\Box \\bot) \\rightarrow \\Diamond (p \\wedge \\Box \\bot))$ is unstable for the model defined below and for the boolean function $\\sigma \\in \\{0,1\\}^\\omega$ (i.e.\\  $\\sigma: \\Nat^+ \\rightarrow \\{0,1\\}$) such that $\\sigma_n = 1$ for $n$ even and  $\\sigma_n = 0$ for $n$ odd.}\n\n\\bigskip\n\nConsider a model $M$ in Figure \\ref{fig.unstable} on the left consisting of a root $s$ with branches of length $n$ for each positive natural number $n$. We also consider a single propositional variable $p$ that is false in the leaves, and from there on towards the root alternatingly true and false. We finally take the value of $p$ in the root to be true, but in fact that does not matter.  (In the figure, nodes where $p$ is true are denoted $\\circ$ and nodes where $p$ is false are denoted $\\bullet$.) \n\nWe now consider the update of $(M,s)$ with the formula $\\phi = \\neg \\Box \\bot \\wedge ((\\Diamond \\Box \\bot \\wedge \\Diamond \\neg\\Box \\bot) \\rightarrow \\Diamond (p \\wedge \\Box \\bot))$. The formula $\\phi$ describes, in other words: ``I am not a leaf node and if I am the root (i.e., if I am the unique node that is branching) then there is a branch of length 1 wherein $p$ is true in the leaf of that branch.'' \n\nWe take the arrow elimination update, that is, arrows pointing to states where the formula is false are deleted, and all other arrows are preserved. The result is the middle structure in Figure \\ref{fig.unstable}. The underlying frames of that model and the left model in the figure are isomorphic. This we can see by rotating the middle structure counterclockwise by 45 degrees and removing the isolated (unreachable) points. That result is pictured on the right in the figure.\n\nIf we now update again with $\\phi$, the original model reappears (in the strong sense that the root-generated submodel is bisimilar --- this is the ${\\raisebox{.3ex}[0mm][0mm]{\\ensuremath{\\medspace \\underline{\\! \\leftrightarrow\\!}\\medspace}}}$ notation --- and even isomorphic): \\[ \\begin{array}{l} M,s \\models\\neg\\phi \\\\ M|\\phi,s \\models\\phi \\\\mathcal{M}|\\phi|\\phi,s \\models\\neg\\phi \\hspace{1cm} \\hfill \\text{such that } (M|\\phi|\\phi,s) {\\raisebox{.3ex}[0mm][0mm]{\\ensuremath{\\medspace \\underline{\\! \\leftrightarrow\\!}\\medspace}}} (M,s) \\end{array} \\]\nand in general we have in this case (we recall that $M|\\phi^0 = M$ and $M|\\phi^{n+1} = (M|\\phi^n)|\\phi$)\n \\[ \\begin{array}{l} M|\\phi^n,s \\models\\phi \\hspace{1cm} \\hfill \\text{ for $n\\in\\mathbb N$ odd} \\\\\nM|\\phi^n,s \\models\\neg\\phi \\hfill \\text{ for $n\\in\\mathbb N$ even or $n=0$} \\\\\n(M|\\phi^n,s) {\\raisebox{.3ex}[0mm][0mm]{\\ensuremath{\\medspace \\underline{\\! \\leftrightarrow\\!}\\medspace}}} (M|\\phi^{n+2},s) \\hspace{5cm} \\hfill \\text{for all } n \\in \\Nat \\\\\n\\end{array} \\]\n\n\\begin{figure}\n\\scalebox{.88}{\n\\begin{tikzpicture}[->,thick]\n\\node (0) at (0,0) {$\\circ$};\n\\node (11) at (-1,0) {$\\bullet$};\n\\node (21) at (-.85,.85) {$\\circ$};\n\\node (22) at (-1.7,1.7) {$\\bullet$};\n\\node (31) at (0,1) {$\\bullet$};\n\\node (32) at (0,2) {$\\circ$};\n\\node (33) at (0,3) {$\\bullet$};\n\\node (41) at (.85,.85) {$\\circ$};\n\\node (42) at (1.7,1.7) {$\\bullet$};\n\\node (43) at (2.55,2.55) {$\\circ$};\n\\node (44) at (3.4,3.4) {$\\bullet$};\n\\node (0a) at (0,-.3) {\\color{white} $\\neg\\phi$};\n\\node (21a) at (-1.15,.85) {$\\phi$};\n\\node (31a) at (0.3,1) {$\\phi$};\n\\node (32a) at (0.3,2) {$\\phi$};\n\\node (41a) at (1.15,.85) {$\\phi$};\n\\node (42a) at (2,1.7) {$\\phi$};\n\\node (43a) at (2.85,2.55) {$\\phi$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (11);\n\\draw[->] (0) to (21);\n\\draw[->] (21) to (22);\n\\draw[->] (0) to (31);\n\\draw[->] (31) to (32);\n\\draw[->] (32) to (33);\n\\draw[->] (0) to (41);\n\\draw[->] (41) to (42);\n\\draw[->] (42) to (43);\n\\draw[->] (43) to (44);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n{$\\stackrel {\\phi!} \\Rightarrow$}\n\\begin{tikzpicture}[->,thick]\n\\node (0) at (0,0) {$\\circ$};\n\\node (11) at (-1,0) {$\\bullet$};\n\\node (21) at (-.85,.85) {$\\circ$};\n\\node (22) at (-1.7,1.7) {$\\bullet$};\n\\node (31) at (0,1) {$\\bullet$};\n\\node (32) at (0,2) {$\\circ$};\n\\node (33) at (0,3) {$\\bullet$};\n\\node (41) at (.85,.85) {$\\circ$};\n\\node (42) at (1.7,1.7) {$\\bullet$};\n\\node (43) at (2.55,2.55) {$\\circ$};\n\\node (44) at (3.4,3.4) {$\\bullet$};\n\\node (0a) at (0,-.3) {$\\phi$};\n\\node (31a) at (0.3,1) {$\\phi$};\n\\node (41a) at (1.15,.85) {$\\phi$};\n\\node (42a) at (2,1.7) {$\\phi$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (21);\n\\draw[->] (0) to (31);\n\\draw[->] (31) to (32);\n\\draw[->] (0) to (41);\n\\draw[->] (41) to (42);\n\\draw[->] (42) to (43);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n{${\\raisebox{.3ex}[0mm][0mm]{\\ensuremath{\\medspace \\underline{\\! \\leftrightarrow\\!}\\medspace}}}$}\n\\begin{tikzpicture}[->,thick]\n\\node (0) at (0,0) {$\\circ$};\n\\node (11) at (-1,0) {$\\circ$};\n\\node (21) at (-.85,.85) {$\\bullet$};\n\\node (22) at (-1.7,1.7) {$\\circ$};\n\\node (31) at (0,1) {$\\circ$};\n\\node (32) at (0,2) {$\\bullet$};\n\\node (33) at (0,3) {$\\circ$};\n\\node (41) at (.85,.85) {$\\bullet$};\n\\node (42) at (1.7,1.7) {$\\circ$};\n\\node (43) at (2.55,2.55) {$\\bullet$};\n\\node (44) at (3.4,3.4) {$\\circ$};\n\\node (0a) at (0,-.3) {$\\phi$};\n\\node (21a) at (-1.15,.85) {$\\phi$};\n\\node (31a) at (0.3,1) {$\\phi$};\n\\node (32a) at (0.3,2) {$\\phi$};\n\\node (41a) at (1.15,.85) {$\\phi$};\n\\node (42a) at (2,1.7) {$\\phi$};\n\\node (43a) at (2.85,2.55) {$\\phi$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (11);\n\\draw[->] (0) to (21);\n\\draw[->] (21) to (22);\n\\draw[->] (0) to (31);\n\\draw[->] (31) to (32);\n\\draw[->] (32) to (33);\n\\draw[->] (0) to (41);\n\\draw[->] (41) to (42);\n\\draw[->] (42) to (43);\n\\draw[->] (43) to (44);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n}\n\\caption{Example unstable formula}\n\\label{fig.unstable}\n\\end{figure}\n\n\\subsection{Arbitrary boolean function} \\label{sec.threearb}\n\nLet now any boolean function $\\sigma \\in \\{0,1\\}^\\omega$ be given. Consider again the frame of the model $M$ of Subsection \\ref{sec.threeone} and the same formula $\\phi := \\neg \\Box \\bot \\wedge ((\\Diamond \\Box \\bot \\wedge \\Diamond \\neg\\Box \\bot) \\rightarrow \\Diamond (p \\wedge \\Box \\bot))$. The value in the root of the model is irrelevant. In that model, $\\circ$ no longer means that $p$ is true. Instead, we decorate the model with values for atom $p$ according to the string $\\sigma$, as below. This ensures that for all $n \\in \\mathbb N$, $M|\\phi^n,s \\models \\sigma_{n+1}(\\phi)$. \n\nFor an example, let be $\\sigma = 011100011\\dots$. Then $\\sigma_1 = 0$ (the first digit), $\\sigma_2 = 1$, etc. Therefore $p$ is false at the leaf of the branch of length 1 (false means value 0), and in the branch of length 2 $p$ is true halfway ($\\sigma_3$) and again false at the leaf, and so on. For $n=0$, we should get $M|\\phi^0,s \\models \\sigma_1(\\phi)$. As $M|\\phi^0 = M$, and $\\sigma_1(\\phi) = \\neg\\phi$, we get $M,s \\models \\neg\\phi$. And this is indeed the case, because the subformula $\\Diamond (p \\wedge \\Box \\bot)$ of $\\phi$ is false in the root, because $p$ is false ($\\sigma_1 = 0$) at the leaf of the branch of length 1.\n\nAnnouncement of $\\phi$ causes all leaves to be eliminated from the model, as before, with the result that all leaves are now decorated with propositions $p$ with value $\\sigma_2 = 1$. So we should have $M|\\phi,s \\models \\phi$, i.e., $M|\\phi^1,s \\models \\sigma_2(\\phi)$. And so it is, because $M|\\phi,s \\models \\Diamond (p \\wedge \\Box \\bot)$. And so on.\n\n\\begin{center}\n\\scalebox{.9}{\n\\begin{tikzpicture}[->,thick]\n\\node (0) at (0,0) {$\\circ$};\n\\node (11) at (-1,0) {$\\circ$};\n\\node (21) at (-.85,.85) {$\\circ$};\n\\node (22) at (-1.7,1.7) {$\\circ$};\n\\node (31) at (0,1) {$\\circ$};\n\\node (32) at (0,2) {$\\circ$};\n\\node (33) at (0,3) {$\\circ$};\n\\node (41) at (.85,.85) {$\\circ$};\n\\node (42) at (1.7,1.7) {$\\circ$};\n\\node (43) at (2.55,2.55) {$\\circ$};\n\\node (44) at (3.4,3.4) {$\\circ$};\n\\node (11a) at (-1,-.3) {$\\sigma_1$};\n\\node (21a) at (-1.15,.85) {$\\sigma_2$};\n\\node (22a) at (-2,1.7) {$\\sigma_1$};\n\\node (31a) at (0.3,1) {$\\sigma_3$};\n\\node (32a) at (0.3,2) {$\\sigma_2$};\n\\node (33a) at (0.3,3) {$\\sigma_1$};\n\\node (41a) at (1.15,.85) {$\\sigma_4$};\n\\node (42a) at (2,1.7) {$\\sigma_3$};\n\\node (43a) at (2.85,2.55) {$\\sigma_2$};\n\\node (44a) at (3.7,3.4) {$\\sigma_1$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (11);\n\\draw[->] (0) to (21);\n\\draw[->] (21) to (22);\n\\draw[->] (0) to (31);\n\\draw[->] (31) to (32);\n\\draw[->] (32) to (33);\n\\draw[->] (0) to (41);\n\\draw[->] (41) to (42);\n\\draw[->] (42) to (43);\n\\draw[->] (43) to (44);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}$\\stackrel {\\phi} \\Rightarrow$\n\\begin{tikzpicture}[->,thick]\n\\node (0) at (0,0) {$\\circ$};\n\\node (11) at (-1,0) {$\\circ$};\n\\node (21) at (-.85,.85) {$\\circ$};\n\\node (22) at (-1.7,1.7) {$\\circ$};\n\\node (31) at (0,1) {$\\circ$};\n\\node (32) at (0,2) {$\\circ$};\n\\node (33) at (0,3) {$\\circ$};\n\\node (41) at (.85,.85) {$\\circ$};\n\\node (42) at (1.7,1.7) {$\\circ$};\n\\node (43) at (2.55,2.55) {$\\circ$};\n\\node (44) at (3.4,3.4) {$\\circ$};\n\\node (11a) at (-1,-.3) {$\\sigma_2$};\n\\node (21a) at (-1.15,.85) {$\\sigma_3$};\n\\node (22a) at (-2,1.7) {$\\sigma_2$};\n\\node (31a) at (0.3,1) {$\\sigma_4$};\n\\node (32a) at (0.3,2) {$\\sigma_3$};\n\\node (33a) at (0.3,3) {$\\sigma_2$};\n\\node (41a) at (1.15,.85) {$\\sigma_5$};\n\\node (42a) at (2,1.7) {$\\sigma_4$};\n\\node (43a) at (2.85,2.55) {$\\sigma_3$};\n\\node (44a) at (3.7,3.4) {$\\sigma_2$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (11);\n\\draw[->] (0) to (21);\n\\draw[->] (21) to (22);\n\\draw[->] (0) to (31);\n\\draw[->] (31) to (32);\n\\draw[->] (32) to (33);\n\\draw[->] (0) to (41);\n\\draw[->] (41) to (42);\n\\draw[->] (42) to (43);\n\\draw[->] (43) to (44);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n$\\stackrel {\\phi} \\Rightarrow \\hspace{1cm} \\dots$\n}\n\\end{center}\n\n\\noindent \nThis proves Proposition \\ref{prop.a} that there are $\\sigma$-satisfiable formulas on class ${\\mathcal K}$ for any string $\\sigma$.\n\n\\subsection{An unstable formula for agents with consistent beliefs} \\label{sec.unstablecons}  \\label{sec.threetwo}\n\nA common way to transform a model for a directed single-agent (unimodal) setting exhibiting a certain desirable property into a model for a symmetric multi-agent setting with the same property, is to replace an arrow by a pair of (undirected) links for different agents, i.e., a pair $(x,y) \\in R$ by two pairs $(x,z) \\in R_a$ and $(z,y) \\in R_b$.  In a multi-agent ${\\mathcal{S}5}$ setting, this means chaining equivalence classes of size two for different agents $a$ and $b$. Instead of arbitrarily long finite chains, we then build arbitrarily long alternating and interlocking $a\/b$ chains. The technique has been used to great effect for undecidability arguments for ${\\mathcal{S}5}$ logics \\citep{frenchetal:2008} and also to great effect for expressivity arguments for such logics \\citep{hvdetal.del:2007,kooi.jancl:2007}. This chaining procedure can be extended in a natural way to ${\\mathcal{KD}45}$ models (for consistent belief), namely when (at least) the root of the model is an unreachable state wherein incorrect belief is possible, but where (almost) all remaining states constitute equivalence classes (wherein belief is correct). We apply this procedure to get a symmetric multi-agent model from an asymmetric single-agent model, to the model $M$ of Subsection \\ref{sec.threeone}, in order to get an example of an unstable formula in a multi-agent ${\\mathcal{KD}45}$ setting. \n\nStarting with the model $M$ of that section, we thus get the following model $N$. The accessibility relation is solid for agent $a$ and it is dashed for agent $b$. Intuitively, to get $N$ from $M$, all nodes are replaced by two nodes with the same valuation and that are indistinguishable for an agent $b$, whereas one might say the previous directed arrows are replaced by indistinguishability for an agent $a$.  Except for the root, wherein we do something special: all pairs in  the accessibility relation for $a$ starting in the root are directed, and we add another node to the model that is accessible from the root for agent $b$ only (in the figure: the node below the root), and from where all nodes accessible from the root for agent $a$ are also $a$-accessible (in the figure we have only drawn one of those arrows, for visual clarity). This is a ${\\mathcal{KD}45}$ model. We assume the usual visual conventions for such models: all directed arrows point to {\\em clusters} of indistinguishable nodes for that agent (see Appendix B). However, merely in order to emphasize the similarity with the model $M$, from the root of the model we have many outgoing arrows (whereas according to the visual convention one would have been enough).\n\n\\bigskip\n\n\\scalebox{0.8}{\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\circ$};\n\\node (11) at (-1,0) {$\\bullet$};\n\\node (12) at (-2,0) {$\\bullet$};\n\\node (21) at (-.85,.85) {$\\circ$};\n\\node (22) at (-1.7,1.7) {$\\circ$};\n\\node (23) at (-2.55,2.55) {$\\bullet$};\n\\node (24) at (-3.4,3.4) {$\\bullet$};\n\\node (31) at (0,1) {$\\bullet$};\n\\node (32) at (0,2) {$\\bullet$};\n\\node (33) at (0,3) {$\\circ$};\n\\node (34) at (0,4) {$\\circ$};\n\\node (35) at (0,5) {$\\bullet$};\n\\node (36) at (0,6) {$\\bullet$};\n\\node (41) at (.85,.85) {$\\circ$};\n\\node (42) at (1.7,1.7) {$\\circ$};\n\\node (43) at (2.55,2.55) {$\\bullet$};\n\\node (44) at (3.4,3.4) {$\\bullet$};\n\\node (45) at (4.25,4.25) {$\\circ$};\n\\node (46) at (5.1,5.1) {$\\circ$};\n\\node (47) at (5.95,5.95) {$\\bullet$};\n\\node (48) at (6.8,6.8) {$\\bullet$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (11);\n\\draw[dashed,-] (11) to (12);\n\\draw[->] (0) to (21);\n\\draw[dashed,-] (21) to (22);\n\\draw[-] (22) to (23);\n\\draw[dashed,-] (23) to (24);\n\\draw[->] (0) to (31);\n\\draw[dashed,-] (31) to (32);\n\\draw[-] (32) to (33);\n\\draw[dashed,-] (33) to (34);\n\\draw[-] (34) to (35);\n\\draw[dashed,-] (35) to (36);\n\\draw[->] (0) to (41);\n\\draw[dashed,-] (41) to (42);\n\\draw[-] (42) to (43);\n\\draw[dashed,-] (43) to (44);\n\\draw[-] (44) to (45);\n\\draw[dashed,-] (45) to (46);\n\\draw[-] (46) to (47);\n\\draw[dashed,-] (47) to (48);\n\\draw[-] (11) to (21);\n\\draw[-] (21) to (31);\n\\draw[-] (31) to (41);\n\\node (x) at (1,0.05) {$ \\ $};\n\\draw[dotted,-] (41) to (x);\n\\node (newroot) at (0,-1) {$\\bullet$};\n\\draw[dashed,->] (0) to (newroot);\n\\draw[->] (newroot) to (11);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n}\n\n\\noindent Instead of \\[ \\phi = \\neg \\Box \\bot \\wedge ((\\Diamond \\Box \\bot \\wedge \\Diamond \\neg\\Box \\bot) \\rightarrow \\Diamond (p \\wedge \\Box \\bot)) \\] we then get a more or less corresponding multi-agent formula \\[ \\psi = \\neg\\Diamond_b(\\Box_a p \\vee \\Box_a \\neg p) \\wedge ((p\\wedge\\Box_b\\negp) \\rightarrow \\Diamond_a\\Diamond_b \\Box_a p)\\] To get the latter from the former, first replace each $\\Diamond$ by $\\Diamond_a\\Diamond_b$. We recall that in the model, we replaced each directed unlabeled arrow by an $a$ step plus a $b$ step. Then, replace $\\Box \\bot$ by $\\Diamond_b(\\Box_a p \\vee \\Box_a \\neg p)$. We recall that $\\Box \\bot$ defined a leaf node in $M$. As ${\\mathcal{KD}45}$ models are serial, we now observe that the leaves of model $N$ are the unique nodes wherein $a$ knows whether $p$: $\\Box_a p \\vee \\Box_a \\neg p$, such that  $\\Diamond_b (\\Box_a p \\vee \\Box_a \\neg p)$ is only true in the final two nodes of each branch. \n\nThe extra node is needed to allow for a formula that distinguishes the root node from other nodes: the root of $N$ is the unique node wherein agent $b$ has incorrect belief: $p \\wedge \\Box_b \\neg p$. In all other nodes, agent $b$ (correctly) knows the value of $p$. (We did not know how to distinguish the root node from other nodes without introducing such an additional node.)\n\nThe nodes $a$-accessible from the root form an infinite $a$-equivalence class. The $a$ arrows leading from the extra node below the root to that class are there so that this node is not a leaf.\n \nWith the `translation' so far, the final part $\\Diamond (p \\wedge \\Box \\bot)$ of the formula $\\phi$ becomes $\\Diamond_a\\Diamond_b(p \\wedge \\Diamond_b(\\Box_a p \\vee \\Box_a \\neg p))$, which is ${\\mathcal{KD}45}$ equivalent to $\\Diamond_a\\Diamond_b\\Box_a p$, as found in the formula $\\psi$. \n\nIn plain English, $\\psi$ describes that: $(i)$ $b$ does not consider it possible that $a$ knows whether $p$, and that $(ii)$ in the node where $b$'s belief is incorrect (the root note), $a$ considers it possible that $b$ considers it possible for $a$ to know {\\em that} (not whether!) $p$. Condition $(i)$ is always false in the leaves and in the nodes just below the leaves, whereas condition $(ii)$ is only true in the root if there is a path of length 2 to a leaf node satisfying $p$. If we now update model $N$ with $\\psi$ we get this model $N|\\psi$ --- wherein $\\psi$ is now true in the root, as the length 2 path is now to a $p$ node:\n\n\\bigskip\n\n\\scalebox{.8}{\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\circ$};\n\\node (11) at (-1,0) {$\\circ$};\n\\node (12) at (-2,0) {$\\circ$};\n\\node (21) at (-.85,.85) {$\\bullet$};\n\\node (22) at (-1.7,1.7) {$\\bullet$};\n\\node (23) at (-2.55,2.55) {$\\circ$};\n\\node (24) at (-3.4,3.4) {$\\circ$};\n\\node (31) at (0,1) {$\\circ$};\n\\node (32) at (0,2) {$\\circ$};\n\\node (33) at (0,3) {$\\bullet$};\n\\node (34) at (0,4) {$\\bullet$};\n\\node (35) at (0,5) {$\\circ$};\n\\node (36) at (0,6) {$\\circ$};\n\\node (41) at (.85,.85) {$\\bullet$};\n\\node (42) at (1.7,1.7) {$\\bullet$};\n\\node (43) at (2.55,2.55) {$\\circ$};\n\\node (44) at (3.4,3.4) {$\\circ$};\n\\node (45) at (4.25,4.25) {$\\bullet$};\n\\node (46) at (5.1,5.1) {$\\bullet$};\n\\node (47) at (5.95,5.95) {$\\circ$};\n\\node (48) at (6.8,6.8) {$\\circ$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (11);\n\\draw[dashed,-] (11) to (12);\n\\draw[->] (0) to (21);\n\\draw[dashed,-] (21) to (22);\n\\draw[-] (22) to (23);\n\\draw[dashed,-] (23) to (24);\n\\draw[->] (0) to (31);\n\\draw[dashed,-] (31) to (32);\n\\draw[-] (32) to (33);\n\\draw[dashed,-] (33) to (34);\n\\draw[-] (34) to (35);\n\\draw[dashed,-] (35) to (36);\n\\draw[->] (0) to (41);\n\\draw[dashed,-] (41) to (42);\n\\draw[-] (42) to (43);\n\\draw[dashed,-] (43) to (44);\n\\draw[-] (44) to (45);\n\\draw[dashed,-] (45) to (46);\n\\draw[-] (46) to (47);\n\\draw[dashed,-] (47) to (48);\n\\draw[dotted,-] (0) to (dots);\n\\draw[-] (11) to (21);\n\\draw[-] (21) to (31);\n\\draw[-] (31) to (41);\n\\node (x) at (1,0.05) {$ \\ $};\n\\draw[dotted,-] (41) to (x);\n\\node (newroot) at (0,-1) {$\\bullet$};\n\\draw[dashed,->] (0) to (newroot);\n\\draw[->] (newroot) to (11);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n}\n\n\\noindent The next iteration restores $N$: $(N|\\psi|\\psi,s) {\\raisebox{.3ex}[0mm][0mm]{\\ensuremath{\\medspace \\underline{\\! \\leftrightarrow\\!}\\medspace}}} (N,s)$. Again, we get:\n\\[ \\begin{array}{l} N|\\psi^n,s \\models\\psi \\hspace{1cm} \\hfill \\text{ for $n\\in\\mathbb N$ odd} \\\\\nN|\\psi^n,s \\models\\neg\\psi \\hfill \\text{ for $n\\in\\mathbb N$ even or $n=0$} \\\\\n(N|\\psi^n,s) {\\raisebox{.3ex}[0mm][0mm]{\\ensuremath{\\medspace \\underline{\\! \\leftrightarrow\\!}\\medspace}}} (N|\\psi^{n+2},s) \\hspace{5cm} \\hfill \\text{for all } n \\in \\Nat \\\\\n\\end{array} \\]\n\\weg{\n\n\\subsection{A communicative scenario for the multi-agent setting}  \\label{sec.threethree}\n\n{\\em The infinitely branching model $N$ of the previous subsection seems to fall out of the sky. In this section we provide a story and a communicative scenario involving truthtelling and lying producing a very similar model with the same iterated update behaviour, as a variation on a well-known epistemic riddle. This merely shows `the power of updates': what we will do is nothing but customize an epistemic model based on the semantic operations of public announcements and agent announcements \\citep{hvd.lying:2014}. It is shows by example that {\\em any} multi-agent ${\\mathcal{KD}45}$ structure can be justified and backed up by communicative scenarios involving truthtelling and lying.}\n\n\n\\bigskip\n\nConsider two agents Anne and Bill, and a quizmaster, Rineke, performing a number of actions and informing Anne and Bill about such actions. Anne and Bill each have a sticker with a natural number on their forehead. These numbers are consecutive. Anne and Bill have to discover what the number is. (So, we have the standard setting of the consecutive number riddle \\citep{littlewood:1953,hvdetal.puzzle:2015}.) \n\nWe further assume that the actual numbers are that Anne has 0 and Bill has 1.\n\nSo far, so good. However, from here on things proceed differently as usual... For a start, Anne's sticker has been attached to her forehead with superglue. Which she does not know yet but will become relevant later in the story.\n\nGiven that Anne has 0 and Bill has 1, Anne and Bill have common knowledge that Anne's number is even and that Bill's number is odd, so without loss of generality we may assume that in our modelling. Some of the announcements below will be lies. We assume that Anne and Bill are {\\em skeptical agents} (in the technical sense of \\cite{hvd.lying:2014}) that believe lies if they consider it possible that the lie is true but that otherwise do not believe the lie; i.e., they do not change their beliefs if they already believe the opposite of the lie.\n\n\\begin{enumerate}\n\\item The quizmaster tells Anne and Bill that she will inform them what the maximum even number is that Anne may have, except when Anne has 2 and Bill has 1, in which case she will only inform Bill, and when Anne has 0 and Bill has 1, in which case she informs neither Anne nor Bill. She then takes two envelopes, puts a sheet of paper into each envelope with the promised information (for example, two sheets with the number 6; or an empty sheet intended for Anne and a sheet with 0 intended for Bill), seals the envelopes, and hands the envelopes to Anne and to Bill.\n\n{\\em But this is not all.}\n\n\\item The quizmaster tells Anne and Bill that if the aforementioned maximum for Anne's number is a multiple of 4 (including 0), then the backside of Anne's sticker has superglue and she will need surgery to have it removed later, whereas otherwise (for example, when Anne has 2) it is as usual easily detachable. (Of course she really has superglue on the backside, as this is more fun.)\n\n\\item If Bill has the number 1, he says to Anne: ``You do not have the number 0.'' (This is a lie, because Anne has the number 0.)\n\n\\item The quizmaster tells Anne and Bill that Anne doesn't have superglue on her sticker. (This is lie.)\n\n\\end{enumerate}\n\nThe model $Q$ (with root $\\mathbf{01}$) resulting from these announcements is as follows, below on the left. (The Appendix ``Customizing Consecutive Numbers'' on page \\pageref{appendix.a} provides details.)\n\nNow consider the proposition $\\eta$: \\begin{quote} {\\em ``Bill considers it possible that (Anne believes that she knows her number and that Anne believes that she knows whether she has superglue on her sticker), and if Bill incorrectly believes that Anne has no superglue on her sticker, then Anne considers it possible that Bill considers it possible that (Anne believes that she knows her number and that Anne believes {\\bf that} she has superglue on her sticker).'' } \\end{quote}\nAs before, this proposition is only true in the final two nodes of each branch and in the root $\\mathbf{01}$ in case the $01\\text{---}21\\text{---}23$ branch is one wherein Anne has superglue. (Initally, not.)\n\nThe updated model $Q|\\eta$ is therefore as follows, below on the right (and the update $Q|\\eta|\\eta$ of that is again isomorphic to the initial model $Q$, etc.).\n\n\\bigskip\n\n\\noindent\n\\scalebox{.7}{\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\mathbf{01}$};\n\\node (11) at (-1,0) {$21$};\n\\node (12) at (-2,0) {$23$};\n\\node (21) at (-.85,.85) {$\\mathbf{21}$};\n\\node (22) at (-1.7,1.7) {$\\mathbf{23}$};\n\\node (23) at (-2.55,2.55) {$\\mathbf{43}$};\n\\node (24) at (-3.4,3.4) {$\\mathbf{45}$};\n\\node (31) at (0,1) {$21$};\n\\node (32) at (0,2) {$23$};\n\\node (33) at (0,3) {$43$};\n\\node (34) at (0,4) {$45$};\n\\node (35) at (0,5) {$65$};\n\\node (36) at (0,6) {$67$};\n\\node (41) at (.85,.85) {$\\mathbf{21}$};\n\\node (42) at (1.7,1.7) {$\\mathbf{23}$};\n\\node (43) at (2.55,2.55) {$\\mathbf{43}$};\n\\node (44) at (3.4,3.4) {$\\mathbf{45}$};\n\\node (45) at (4.25,4.25) {$\\mathbf{65}$};\n\\node (46) at (5.1,5.1) {$\\mathbf{67}$};\n\\node (47) at (5.95,5.95) {$\\mathbf{87}$};\n\\node (48) at (6.8,6.8) {$\\mathbf{89}$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (11);\n\\draw[dashed,-] (11) to (12);\n\\draw[->] (0) to (21);\n\\draw[dashed,-] (21) to (22);\n\\draw[-] (22) to (23);\n\\draw[dashed,-] (23) to (24);\n\\draw[->] (0) to (31);\n\\draw[dashed,-] (31) to (32);\n\\draw[-] (32) to (33);\n\\draw[dashed,-] (33) to (34);\n\\draw[-] (34) to (35);\n\\draw[dashed,-] (35) to (36);\n\\draw[->] (0) to (41);\n\\draw[dashed,-] (41) to (42);\n\\draw[-] (42) to (43);\n\\draw[dashed,-] (43) to (44);\n\\draw[-] (44) to (45);\n\\draw[dashed,-] (45) to (46);\n\\draw[-] (46) to (47);\n\\draw[dashed,-] (47) to (48);\n\\draw[-] (11) to (21);\n\\draw[-] (21) to (31);\n\\draw[-] (31) to (41);\n\\node (x) at (1,0.05) {$ \\ $};\n\\draw[dotted,-] (41) to (x);\n\\node (newroot) at (0,-1) {$01$};\n\\draw[dashed,->] (0) to (newroot);\n\\draw[->] (newroot) to (11);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n}\n\\ \\ \\ \n\\scalebox{.7}{\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\mathbf{01}$};\n\\node (11) at (-1,0) {$\\mathbf{21}$};\n\\node (12) at (-2,0) {$\\mathbf{23}$};\n\\node (21) at (-.85,.85) {$21$};\n\\node (22) at (-1.7,1.7) {$23$};\n\\node (23) at (-2.55,2.55) {$43$};\n\\node (24) at (-3.4,3.4) {$45$};\n\\node (31) at (0,1) {$\\mathbf{21}$};\n\\node (32) at (0,2) {$\\mathbf{23}$};\n\\node (33) at (0,3) {$\\mathbf{43}$};\n\\node (34) at (0,4) {$\\mathbf{45}$};\n\\node (35) at (0,5) {$\\mathbf{65}$};\n\\node (36) at (0,6) {$\\mathbf{67}$};\n\\node (41) at (.85,.85) {$21$};\n\\node (42) at (1.7,1.7) {$23$};\n\\node (43) at (2.55,2.55) {$43$};\n\\node (44) at (3.4,3.4) {$45$};\n\\node (45) at (4.25,4.25) {$65$};\n\\node (46) at (5.1,5.1) {$67$};\n\\node (47) at (5.95,5.95) {$87$};\n\\node (48) at (6.8,6.8) {$89$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[->] (0) to (11);\n\\draw[dashed,-] (11) to (12);\n\\draw[->] (0) to (21);\n\\draw[dashed,-] (21) to (22);\n\\draw[-] (22) to (23);\n\\draw[dashed,-] (23) to (24);\n\\draw[->] (0) to (31);\n\\draw[dashed,-] (31) to (32);\n\\draw[-] (32) to (33);\n\\draw[dashed,-] (33) to (34);\n\\draw[-] (34) to (35);\n\\draw[dashed,-] (35) to (36);\n\\draw[->] (0) to (41);\n\\draw[dashed,-] (41) to (42);\n\\draw[-] (42) to (43);\n\\draw[dashed,-] (43) to (44);\n\\draw[-] (44) to (45);\n\\draw[dashed,-] (45) to (46);\n\\draw[-] (46) to (47);\n\\draw[dashed,-] (47) to (48);\n\\draw[-] (11) to (21);\n\\draw[-] (21) to (31);\n\\draw[-] (31) to (41);\n\\node (x) at (1,0.05) {$ \\ $};\n\\draw[dotted,-] (41) to (x);\n\\node (newroot) at (0,-1) {$01$};\n\\draw[dashed,->] (0) to (newroot);\n\\draw[->] (newroot) to (11);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n}\n\n}\n\n\\noindent This completes our example of a believable $(01)^\\omega$-satisfiable formula for ${\\mathcal{KD}45}$. From this example now directly follows Proposition \\ref{prop.b} that there are believable $\\sigma$-satisfiable formulas on class ${\\mathcal{KD}45}$ for any string $\\sigma$: this is similar to how we generalized the example of $(01)^\\omega$-satisfiability on ${\\mathcal K}$ to a proof of $\\sigma$-satisfiability on ${\\mathcal K}$ for any $\\sigma\\in\\{0,1\\}^\\omega$, using the same formula and the same model except for a different decoration of the variable $p$.\n\n\\section{Characterization of single-agent believable true lies} \\label{sec.four}\n\nExactly which formulas are true lies? Can they been characterized syntactically, and if so, how? The corresponding problem for successful and self-refuting formulas has been studied and solved for the single-agent case \\citep{hollidayetal:2010}. These characterizations are given for ${\\mathcal{S}5}$ (see Section \\ref{sec.2valid}) but are said to hold also for ${\\mathcal{KD}45}$ ``with minor changes'' \\citep{hollidayetal:2010}. The problem is still open for the multi-agent case, for both classes of formulas. Two main reasons that the single-agent case is easier, are that in that case, first, it is well known that any formula is (${\\mathcal{S}5}$ or ${\\mathcal{KD}45}$) equivalent to a formula in disjunctive normal form without any nestings of modalities, and, second, that satisfiability of a formula in disjunctive normal form can be checked syntactically \\citep{hollidayetal:2010} (see below for details). Based on these two properties, the characterizations in \\cite{hollidayetal:2010} check that certain disjuncts (conditionally) exist in the (disjunctive normal form) formula, which is shown to ensure that the formula is of the required type.\n\nIn this section we give a similar characterization of believable true lies. It is heavily inspired by \\cite{hollidayetal:2010}. Indeed, we take the technique directly from them, as well as notation and concepts. Just as \\cite{hollidayetal:2010} syntactically characterize the set of successful formulas, we syntactically characterize the \\emph{complement} of the set of believable true lies. The differences are in the details of the characterization and the corresponding proof.\n\nIn the rest of this section, a ``formula'' always refers (single-agent) formula in the standard modal language. (Note that any formula in $\\ensuremath{\\mathcal{L}}$ is equivalent to such an announcement-free formula.) We now formally define the two properties mentioned above, disjunctive normal forms and syntactic satisfiability checking, and then give the characterization in Definition \\ref{def:syntactic}.\n\n\\begin{definition}[Disjunctive normal form] A formula $\\phi$ is in \\emph{disjunctive normal form} iff it is a disjunction of conjunctions of the form \\[\\delta = \\alpha \\wedge \\Box \\beta_1 \\wedge \\ldots \\wedge \\Box \\beta_n \\wedge \\Diamond \\gamma_1 \\wedge \\ldots \\wedge \\Diamond \\gamma_m\\] where $\\alpha$ and each $\\gamma_i$ are conjunctions of literals and each $\\beta_i$ is a disjunction of literals. \n\\end{definition}\nWe write $\\ourneg{l}$ to denote the negation of a literal $l$ where double negations are removed. Similarly, when $\\beta$ is a disjunction of literals, $\\ourneg{\\beta}$ denotes the conjunction of negated literals with single negation removed. For a given conjuction $\\delta$ of the form above, in the following we will write $\\delta^\\alpha$ for $\\alpha$ and $\\delta^{\\Box\\Diamond}$ for $\\Box \\beta_1 \\wedge \\ldots \\wedge \\Box \\beta_n \\wedge \\Diamond \\gamma_1 \\wedge \\ldots \\wedge \\Diamond \\gamma_m$. Every formula is ${\\mathcal{KD}45}$-equivalent to one in disjunctive normal form.\n\n\\begin{definition}[Clarity \\citep{hollidayetal:2010}] \nGiven a conjunction or disjunction $\\chi$ of literals, $L(\\chi)$ denotes the set of  literals occurring in it. Set $L(\\chi)$ is \\emph{open} iff no literal in $L(\\chi)$ is the negation of any other. A conjunction $\\delta = \\alpha \\wedge \\Box \\beta_1 \\wedge \\ldots \\wedge \\Box \\beta_n \\wedge \\Diamond \\gamma_1 \\wedge \\ldots \\wedge \\Diamond \\gamma_m$ in disjunctive normal form is \\emph{clear} iff (i) $L(\\alpha)$ is open; (ii) there is an open set of literals $\\{l_1,\\ldots,l_n\\}$ with $l_i \\in L(\\beta_i)$; and (iii) for every $\\gamma_k$ there is a set of literals $\\{l_1,\\ldots,l_n\\}$ with $l_i \\in L(\\beta_i)$ such that $\\{l_1,\\ldots,l_n\\} \\cup L(\\gamma_k)$ is open. A disjunction in disjunctive normal form is clear iff at least one of the disjuncts is clear.\n\\end{definition}\nThe key point about clarity is that a formula is ${\\mathcal{KD}45}$-satisfiable iff it is clear \\cite[Lemma 3.6]{hollidayetal:2010}.\n\nWe define a formula that we call a \\emph{disjunctive lying form}. We will then show that, modulo logical equivalence, the class of these formulas corresponds exactly to the class of formulas that are not believable true lies. The characterization will be explained after the definition. \n\n\\begin{definition}[Disjunctive lying form]\n  \\label{def:syntactic}\n A formula $\\phi$ in disjunctive normal form is a \\emph{disjunctive lying form} iff there exist (possibly empty) sets $S$ and $T$ of disjuncts of $\\phi$ and a conjunct $\\Box \\beta_\\theta$ of each $\\theta \\in T$, such that any disjunctive normal form of\n\\[\\chi = \\neg \\phi \\wedge \\Diamond \\phi \\wedge \\chi_1 \\wedge \\chi_2 \\wedge \\chi_3\\]\nis clear\\footnote{The definition allows both $S$ and $T$ to be empty. However, if $S$ is empty then (any disjunctive form of) $\\chi$ is not clear ($\\chi_3$ is a contradiction), so it follows that a disjunctive lying form actually must have a non-empty $S$ as a witness. If $T$ is empty then $\\chi_3$ is a tautology. There exist disjunctive lying forms with an empty $T$ as a witness (one example is $p \\wedge \\Box p$).}\n, where\n\\[\\chi_1 = \\bigwedge_{\\theta \\in T} t(\\theta) \\wedge \\bigwedge_{\\theta \\in \\overline{T}} \\neg t(\\theta)\\quad\\quad\\quad t(\\theta) = \\theta^\\alpha \\wedge \\bigwedge_{\\Diamond \\gamma \\mbox{ in } \\theta} \\bigvee_{\\sigma \\in S} \\Diamond(\\sigma^\\alpha \\wedge \\gamma)\\]\n\\[\\chi_2 = \\bigwedge_{\\sigma \\in S}\\sigma^{\\Box\\Diamond} \\wedge \\bigwedge_{\\sigma \\in \\overline{S}}\\neg \\sigma^{\\Box\\Diamond}\\]\n\\[\\chi_3 = \\bigwedge_{\\theta \\in T}\\bigvee_{\\sigma \\in S} \\Diamond(\\sigma^\\alpha \\wedge \\ourneg{\\beta_\\theta})\\]\nand $\\overline{X}$ denotes the set of disjuncts of $\\phi$ that are not in $X$.\n\\end{definition}\n\nWe now recall (page \\pageref{th:char}):\n\n\\bigskip\n\n\\noindent {\\bf Proposition \\ref{th:char}} \\ \\ \n{\\em A formula is a believable true lie iff it is not equivalent to a disjunctive lying form.}\n\n\\bigskip\n\n\\noindent Before we prove Proposition \\ref{th:char}, let us explain the intuition behind the definition. This will also serve as a guide to the proof of Proposition \\ref{th:char}. If $\\chi$ is true in some pointed model (is clear), then $\\neg \\phi \\wedge \\Diamond\\phi$ is true. The role of $\\chi_1-\\chi_3$ is to ensure that $\\neg\\phi$ remains false in the updated model --- which holds iff $\\phi$ is not a believable true lie --- by ensuring that certain disjuncts (conditionally) exist in $\\phi$.\n\nThe role of $S$ is to syntactically encode which states are still accessible from the current state after the update. These are exactly the states where $\\sigma^\\alpha$ is true for some $\\sigma \\in S$.  Since $\\phi$ is false in the initial pointed model, there is only one way it can become true in the updated pointed model: if, for some disjunct $\\theta$ of $\\phi$ where $\\theta^\\alpha$ is already true and all $\\Diamond\\gamma$ are already true and \\emph{stay} true in the updated model, all $\\Box\\beta$ that were false in the initial pointed model \\emph{become} true in the updated model. We must avoid that. Those disjuncts that can potentially become true are exactly those in $T$. Thus, we must make sure that for every disjunct $\\theta \\in T$, there is at least one $\\Box\\beta$ such that $\\beta$ is false in at least one state that is still accessible after the update. That is ensured by $\\chi_3$.\n\nAs an example, to see that the believable true lie $p \\vee \\Box p$ (already in disjunctive normal form) is not a disjunctive lying form, we need to check for all possible subsets of disjuncts $S$ and $T$ over the set of all disjuncts $\\{p, \\Box p\\}$, whether $\\chi$ is clear. There are only two possibilities for $T$: $T = \\emptyset$ or $T = \\{\\Box p\\}$. In the former case we get that $\\chi_1 = \\neg p \\wedge \\bot$ (not clear). In the latter case, there are four possibilities for $S$: if $S=\\emptyset$ then $\\chi_3 = \\bot$; if $S=\\{p\\}$ then $\\chi_3 = \\Diamond(p \\wedge \\neg p)$; if $S=\\{\\Box p\\}$ then $\\chi_2 = \\Box p$ and $\\chi_3 = \\Diamond \\neg p$; and if $S = \\{p,\\Box p\\}$ then $\\chi_2 = \\Box p$ and $\\chi_3 = \\Diamond \\neg p \\wedge \\Diamond (p \\wedge \\neg p)$. In all cases $\\chi$ is not clear.\n\nAs another example, consider $p \\wedge \\Box p$ which is \\emph{not} a believable true lie. To see that it is a disjunctive lying form, take $S = \\{p \\wedge \\Box p\\}$ and $T = \\emptyset$. We get that $\\chi_1 = \\neg p$, and $\\chi_2 = \\Box p$ and $\\chi_3 = \\top$. It is easy to see that $\\chi$ is clear (it is equivalent to $\\neg p \\wedge \\Box p \\wedge \\Diamond (p \\wedge \\Box p)$).\n\n\\begin{proof}[of Prop.~\\ref{th:char}]  Without loss of generality, let $\\phi$ be a disjunctive normal form.\n \nConsider first the implication towards the left: assume that $\\phi$ is a disjunctive lying form and let $S$, $T$, $\\chi$ and $\\beta_\\theta$ for each $\\theta \\in T$ be as in Def. \\ref{def:syntactic}. We must show that there exists $M,s$ such that $M,s \\models \\neg \\phi \\wedge \\Diamond \\phi$ and $\\upd{M}{\\phi},s \\models \\neg \\phi$.  Since any disjunctive normal form of $\\chi$ is clear iff it is satisfiable, so let $M,s$ be such that $M,s \\models \\chi$. Then $M,s \\models \\neg \\phi \\wedge \\Diamond \\phi$. Assume, towards a contradiction, that $\\upd{M}{\\phi},s \\models \\phi$. Then there exists a disjunct $\\theta$ in $\\phi$ such that $\\upd{M}{\\phi},s\\models \\theta$.  As explained above, we are going to show that $\\chi_3$ ensures that this is impossible. To do that, we must first show that it must be the case that $\\theta \\in T$ (intuitively defined as the set of disjuncts that can potentially become true in the updated model), and then that $\\chi_3$ ensures that the conjunct $\\Box\\beta_\\theta$ is false in $\\upd{M}{\\phi},s$, leading to a contradiction.\n\n\\medskip\n\nThus, we first show that it must be the case that $\\theta \\in T$. From $\\upd{M}{\\phi},s\\models \\theta^\\alpha$ it follows that $M,s\\models \\theta^\\alpha$ ($\\theta^\\alpha$ is propositional). We now show that also $M,s \\models \\bigwedge_{\\Diamond \\gamma \\mbox{ in } \\theta} \\bigvee_{\\sigma \\in S} \\Diamond(\\sigma^\\alpha \\wedge \\gamma)$, and it follows from $M,s \\models \\chi_1$ that $\\theta \\in T$. Let $\\Diamond\\gamma$ be a conjunct in $\\theta$. We have that $\\upd{M}{\\phi},s \\models \\Diamond\\gamma$, i.e., that there exists a state $t$ such that $Rst$ and $\\upd{M}{\\phi},t\\models \\gamma$ and, since $t$ is a state in $\\upd{M}{\\phi}$, $M,t \\models \\phi$. From the latter it follows that $M,t \\models \\sigma$ for some disjunct $\\sigma$ of $\\phi$.  From $M,t \\models \\sigma^{\\Box\\Diamond}$ and $Rst$ it follows that $M,s \\models \\sigma^{\\Box\\Diamond}$ by standard ${\\mathcal{KD}45}$ reasoning, and thus that $\\sigma \\in S$ by $\\chi_2$.  Since $\\gamma$ is propositional, $\\upd{M}{\\phi},t\\models \\gamma$ implies that $M,t \\models \\gamma$. Thus, $M,s \\models \\Diamond(\\sigma^\\alpha \\wedge \\gamma)$ for some $\\sigma \\in S$. Since $\\Diamond\\gamma$ was arbitrary, we get that $M,s \\models \\bigwedge_{\\Diamond \\gamma \\mbox{ in } \\theta} \\bigvee_{\\sigma \\in S} \\Diamond(\\sigma^\\alpha \\wedge \\gamma)$ and thus that $\\theta \\in T$.\n\nHaving shown that $\\theta \\in T$, we now get a contradiction from the fact that $M,s \\models \\chi_3$. It follows that there exists a $\\sigma \\in S$ such that $M,s \\models \\Diamond(\\sigma^\\alpha \\wedge \\ourneg{\\beta_\\theta})$. That means that there exists a state $t$ such that $Rst$ and $M,t \\models \\sigma^\\alpha \\wedge \\ourneg{\\beta_\\theta}$. From the fact that $\\sigma \\in S$ we know that $M,s \\models \\sigma^{\\Box\\Diamond}$ and thus, by standard ${\\mathcal{KD}45}$ reasoning, that $M,t \\models \\sigma^{\\Box\\Diamond}$. That means that $M,t \\models \\sigma$ and thus $M,t \\models \\phi$. The latter means that $t$ is accessible from $s$ in $\\upd{M}{\\phi}$, and thus that $\\upd{M}{\\phi},s \\models \\neg\\Box \\beta_\\theta$. But this contradicts $\\upd{M}{\\phi},s \\models \\Box \\beta_\\theta$ which follows from the assumption that $\\upd{M}{\\phi},s \\models \\theta$. This concludes the implication towards the left.\n\nFor the direction towards the right, assume that $\\phi$ is not a true lie, i.e., that there exist $M,s$ such that $M,s \\models \\neg\\phi \\wedge \\Diamond \\phi$ and $\\upd{M}{\\phi},s \\models \\neg \\phi$. We now define $T$, $S$, and $\\beta_\\theta$ for each $\\theta \\in T$, as follows. We will then show that the resulting $\\chi$ is satisfiable. For any disjunct $\\delta$ in $\\phi$, let\n  \\[\\delta \\in T \\Leftrightarrow M,s \\models t(\\delta) \\quad\\quad\\quad \\delta \\in S \\Leftrightarrow M,s \\models \\delta^{\\Box\\Diamond}\\]\nLet $\\theta \\in T$. In order to define $\\beta_\\theta$, we first show that for every conjunct $\\Diamond \\gamma$ in $\\theta$, we have that $\\upd{M}{\\phi},s \\models \\Diamond \\gamma$.  Let $\\Diamond \\gamma$ be a conjunct in $\\theta$. From the fact that $M,s \\models t(\\theta)$ we get that there is a $\\sigma \\in S$ such that $M,s \\models \\Diamond(\\sigma^\\alpha \\wedge \\gamma)$. In other words, there exists a state $t$ such that $Rst$ and $M,t \\models \\sigma^\\alpha \\wedge \\gamma$. From $\\chi_2$ and the fact that $\\sigma \\in S$ and standard ${\\mathcal{KD}45}$ reasoning we have that $M,t \\models \\sigma^{\\Box\\Diamond}$. Thus, $M,t \\models \\sigma$. That means that $M,t \\models \\phi$ and that $t$ is accessible from $s$ in $\\upd{M}{\\phi}$, so $\\upd{M}{\\phi},s \\models \\Diamond \\gamma$, for all $\\Diamond \\gamma$ in $\\theta$. Since $\\theta^\\alpha$ is propositional, $\\upd{M}{\\phi},s \\models \\theta^\\alpha$ follows from $M,s\\models t(\\theta)$, and thus $\\upd{M}{\\phi},s \\models \\neg \\theta$ (which follows from $\\upd{M}{\\phi},s \\models \\neg \\phi$) implies that there must be a conjunct $\\Box\\beta$ in $\\theta$ such that $\\upd{M}{\\phi},s \\models \\neg \\Box\\beta$. Let $\\beta_\\theta$ be one such $\\beta$.\n\nFinally, we show that $\\chi$ is satisfiable. It follows that any disjunctive normal form of $\\chi$ is clear, and thus that $\\phi$ is a disjunctive lying form. We show that $M,s \\models \\chi$ (where $M,s$ is the pointed model above). We have that $M,s \\models \\neg\\phi \\wedge \\Diamond\\phi$ by assumption, and $M,s \\models \\chi_1 \\wedge \\chi_2$ by definition. It remains to be shown that $M,s \\models \\chi_3$. Let $\\theta \\in T$.  By definition of $\\beta_\\theta$, $\\upd{M}{\\phi},s \\models \\Diamond \\ourneg{\\beta_\\theta}$. That means that there exists a state $t$ such that $t$ is accessible from $s$ in the updated model and $\\upd{M}{\\phi},t \\models \\ourneg{\\beta_\\theta}$. Since $t$ is accessible from $s$ in $\\upd{M}{\\phi}$, $M,t \\models \\phi$. Let $\\sigma$ be a disjunct of $\\phi$ such that $M,t \\models \\sigma$. Thus $M,t \\models \\sigma^{\\Box\\Diamond}$, and from that and standard ${\\mathcal{KD}45}$ reasoning, $M,s \\models \\sigma^{\\Box\\Diamond}$, and thus $\\sigma \\in S$. From $M,t \\models \\sigma$ we know that $M,t \\models \\sigma^\\alpha$, and since $\\ourneg{\\beta_\\theta}$ is propositional, $M,t \\models \\ourneg{\\beta_\\theta} \\wedge \\sigma^\\alpha$ for some $\\sigma \\in S$ and thus $M,s \\models \\Diamond(\\ourneg{\\beta_\\theta} \\wedge \\sigma^\\alpha)$.  Since $\\theta \\in T$ was arbitrary, $M,s \\models \\chi_3$.\n\\end{proof}\n\n\\section{Knowledge and iterated announcement whether} \\label{sec.five}\n\nThe focus of our story is on lying and therefore on the believed announcement logic, not on the truthful announcement logic wherein lying is impossible. However some of our questions are also meaningful in a logic of knowledge change, interpreted on models with equivalence relations. As an example, let us address the matter of arbitrarily often iterated updates and unstable formulas. For ${\\mathcal{KD}45}$ models we could affirmatively answer the following question (Section \\ref{sec.threetwo}):\n\\begin{quote} {\\em \nAre there a model $(M,s)$ and a formula $\\phi$, such that if we continue to announce in this model that $\\phi$, the value of $\\phi$ never stabilizes? \\hfill (i)}\n\\end{quote}\nThis cannot be in ${\\mathcal{S}5}$, as $\\phi$ cannot be truthfully announced when $\\phi$ is false in the actual state. An obvious question in the setting of ${\\mathcal{S}5}$ models and {\\em truthful} public announcements is:\n\\begin{quote} {\\em \nAre there a model $(M,s)$ and a formula $\\phi$, such that if we continue to announce in this model \\pmb{whether} $\\phi$, the value of $\\phi$ never stabilizes? \\hfill (ii)}\n\\end{quote}\nThe (truthful public) {\\em announcement whether} $\\phi$ announces the {\\em value} of $\\phi$. If $\\phi$ is true, it is a truthful announcement of $\\phi$, whereas if $\\phi$ is false, it is a truthful announcement of $\\neg\\phi$. To question (ii) we do not know the answer. We conjecture that the answer is: {\\bf no}. However, a related question is:\n\\begin{quote} {\\em \nAre there a model $(M,s)$ and formulas $\\phi$ \\pmb{and} $\\pmb{\\psi}$, such that if we continue to announce whether $\\phi$, the value \\pmb{of} $\\pmb{\\psi}$ never stabilizes? \\hfill (iii)}\n\\end{quote}\nThis question we can answer positively, and we think that it may be found in various epistemic scenarios of interest. In the scenario we present here, we do slightly better than (iii):\n\\begin{quote} {\\em \nAre there a model $(M,s)$ and formulas $\\phi$ and $\\psi$, such that if we continue to announce \\pmb{that} $\\phi$, the value of $\\psi$ never stabilizes?  \\hfill (iv)}\n\\end{quote}\nAn answer to (iv) is also an answer to (iii).\n\nLack of stability in the sense of (iv) is more formally defined as follows. Let $\\sigma \\in \\{0,1\\}^\\omega$ be {\\em unstable}; we recall that this means that for all $n \\in \\mathbb N$ there are $m,k \\geq n$ such that $\\sigma_m = 1$ and $\\sigma_k = 0$. Then we wish to find $(M,s)$, $\\phi$, and $\\psi$ such that for all $n \\in \\Nat$, $M|_{\\phi^n},s \\models \\sigma_{n+1}(\\psi)$, and where $M$ is an ${\\mathcal{S}5}$ model. (Where we further recall that $\\sigma_n(\\psi) = \\psi$ if $\\sigma_n=1$ and else $\\sigma_n(\\psi) =\\neg\\psi$; and that $M|_\\phi$ is the (${\\mathcal{S}5}$ preserving) state elimination semantics for truthful public announcement---see Section \\ref{sec.two} and Appendix A.)\n\nNow consider $\\sigma = (01)^\\omega$ ($\\sigma = 010101...$), the alternating string. Our solution is a simple adjustment of the model $N$ from Section \\ref{sec.threetwo} to realize an unstable formula for believed announcement logic. The model below is the equivalence closure (reflexive, symmetric, and transitive closure) of that model $N$. To achieve that visually, we only have to replace the directed arrows involving the root and the node below the root by undirected links, and assume transitivity. The alternating values of $\\sigma_i$ have been made explicit in three branches, for an example. As in Section \\ref{sec.threetwo}, we decorate the states of $N$ with the variable $p$ according to $\\sigma$; $\\bullet$ means that $p$ is false ($\\sigma_1 = \\sigma_3 = 0$), and $\\circ$ that $p$ is true ($\\sigma_2 = 1$). \n\n\\bigskip\n\n\\scalebox{.8}{\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\circ$};\n\\node (11) at (-1,0) {$\\bullet$};\n\\node (12) at (-2,0) {$\\bullet$};\n\\node (21) at (-.85,.85) {$\\circ$};\n\\node (22) at (-1.7,1.7) {$\\circ$};\n\\node (23) at (-2.55,2.55) {$\\bullet$};\n\\node (24) at (-3.4,3.4) {$\\bullet$};\n\\node (31) at (0,1) {$\\bullet$};\n\\node (32) at (0,2) {$\\bullet$};\n\\node (33) at (0,3) {$\\circ$};\n\\node (34) at (0,4) {$\\circ$};\n\\node (35) at (0,5) {$\\bullet$};\n\\node (36) at (0,6) {$\\bullet$};\n\\node (41) at (.85,.85) {$\\circ$};\n\\node (42) at (1.7,1.7) {$\\circ$};\n\\node (43) at (2.55,2.55) {$\\bullet$};\n\\node (44) at (3.4,3.4) {$\\bullet$};\n\\node (45) at (4.25,4.25) {$\\circ$};\n\\node (46) at (5.1,5.1) {$\\circ$};\n\\node (47) at (5.95,5.95) {$\\bullet$};\n\\node (48) at (6.8,6.8) {$\\bullet$};\n\\node (11a) at (-1,-.3) {$\\sigma_1$};\n\\node (12a) at (-2,-.3) {$\\sigma_1$};\n\\node (21a) at (-1.15,.85) {$\\sigma_2$};\n\\node (22a) at (-2,1.7) {$\\sigma_2$};\n\\node (23a) at (-2.85,2.55) {$\\sigma_1$};\n\\node (24a) at (-3.7,3.4) {$\\sigma_1$};\n\\node (31a) at (-.3,1.15) {$\\sigma_3$};\n\\node (32a) at (-.3,2) {$\\sigma_3$};\n\\node (33a) at (-.3,3) {$\\sigma_2$};\n\\node (34a) at (-.3,4) {$\\sigma_2$};\n\\node (35a) at (-.3,5) {$\\sigma_1$};\n\\node (36a) at (-.3,6) {$\\sigma_1$};\n\\node (dots) at (3,0) {$ \\ $};\n\\draw[-] (0) to (11);\n\\draw[dashed,-] (11) to (12);\n\\draw[-] (0) to (21);\n\\draw[dashed,-] (21) to (22);\n\\draw[-] (22) to (23);\n\\draw[dashed,-] (23) to (24);\n\\draw[-] (0) to (31);\n\\draw[dashed,-] (31) to (32);\n\\draw[-] (32) to (33);\n\\draw[dashed,-] (33) to (34);\n\\draw[-] (34) to (35);\n\\draw[dashed,-] (35) to (36);\n\\draw[-] (0) to (41);\n\\draw[dashed,-] (41) to (42);\n\\draw[-] (42) to (43);\n\\draw[dashed,-] (43) to (44);\n\\draw[-] (44) to (45);\n\\draw[dashed,-] (45) to (46);\n\\draw[-] (46) to (47);\n\\draw[dashed,-] (47) to (48);\n\\draw[dotted,-] (0) to (dots);\n\\draw[-] (11) to (21);\n\\draw[-] (21) to (31);\n\\draw[-] (31) to (41);\n\\node (x) at (1,0.05) {$ \\ $};\n\\draw[dotted,-] (41) to (x);\n\\node (newroot) at (0,-1) {$\\bullet$};\n\\draw[dashed,-] (0) to (newroot);\n\\draw[-] (newroot) to (11);\n\\draw[dotted,-] (0) to (dots);\n\\end{tikzpicture}\n}\n\n\\bigskip\n\nIn order to get $N|_{\\phi^n},s \\models \\sigma_{n+1}(\\psi)$ we decorate as before the model with values $\\sigma_n$ for the propositional variable $p$ (where the model above demonstrates strict alternation in the first three values) and we take $\\phi := \\neg\\Diamond_b(\\Box_a p \\vee \\Box_a \\neg p)$ and $\\psi := \\neg(\\Box_bp\\vee\\Box_b\\negp) \\rightarrow \\Diamond_a\\Diamond_b \\Box_a p$. In other words, we split the unstable formula $\\neg\\Diamond_b(\\Box_a p \\vee \\Box_a \\neg p) \\wedge ((p\\wedge\\Box_b\\negp) \\rightarrow \\Diamond_a\\Diamond_b \\Box_a p)$ that we employed in Section \\ref{sec.threetwo} into a part doing the job of always removing the leaf and the node before the leaf of every branch, and another part checking whether there is a branch of length two from the root to a $p$ node, where the distinguishing formula for the root has become $b$'s ignorance of $p$ (there are now two nodes that can be the root).\n\n\\bigskip\n\nFor a different example of $M|_{\\phi^n},s \\models \\sigma_{n+1}(\\psi)$, consider an infinite number of (possibly) muddy children, of which an infinite subset are in fact muddy, let $\\phi$ be the usual proposition `nobody steps forward' (nobody knows whether he or she is muddy; see Section \\ref{sec.2valid}), and let $\\psi$ be the proposition `it is common knowledge that at least $n$ children are muddy but it is not common knowledge that at least $n+1$ children are muddy, where $n$ is even', and let $\\sigma = (01)^\\omega$, again. After father's initial announcement that at least one child is muddy (which serves as the initial model $N$ in this case), $\\psi$ is false (because $n=1$ which  is odd). Whereas after nobody steps forward following the first request by father to do so if you know whether you are muddy, $\\psi$ is true (because now it is common knowledge that at least two children must be muddy: $n=2$ which is even). And so on, ad infinitum.\n\n\\bigskip\n\nIf we consider non-public events, there are yet other interesting forms of unstable iteration on ${\\mathcal{S}5}$ models. Running ahead of the next Section \\ref{sec.six} (and see Appendix A): there are {\\em action models} such that their iterated execution does not stabilize on a given model. The following is a vintage example in a two-agent setting. \n\nConsider a two-state epistemic model for two agents Anne (solid access) and Bob (dashed access) such that Anne is uncertain whether $p$ but Bob knows whether $p$. Actually, $p$ is true (the $\\circ$ state, boxed). Iterated execution of the action model with two actions with preconditions $\\neg \\Box_a p$ and $p \\wedge \\Box_b \\neg \\Box_a p$ transforms this two-state epistemic model into a three-state model, and vice versa, ad infinitum. We could think of the formula $\\Box_b \\neg \\Box_a p$ being alternatingly true and false in the $p$-world (the $\\circ$-world) about which agent $a$ is uncertain. The execution can be visualized as follows. \n\n\\[ \\begin{array}{lllll}\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {\\framebox{$\\circ$}};\n\\node (1) at (2,0) {$\\bullet$};\n\\draw[-] (0) to (1);\n\\node (0b) at (0,-2) {\\color{white} $\\circ$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\times$};\n\\node (0b) at (0,-2) {\\color{white} $\\circ$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\ensuremath{\\mathsf{s}}$};\n\\node (1) at (2,0) {$\\ensuremath{\\mathsf{t}}$};\n\\node (0b) at (0,-2) {\\color{white} $\\circ$};\n\\node (0x) at (0.3,-0.5) {\\footnotesize{$\\neg \\Box_a p$}};\n\\node (1x) at (2.5,-0.5) {\\footnotesize{$p \\wedge \\Box_b\\neg \\Box_a p$}};\n\\draw[dashed,-] (0) to (1);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$=$};\n\\node (0b) at (0,-2) {\\color{white} $\\circ$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {\\framebox{$\\circ$}};\n\\node (1) at (2,0) {$\\bullet$};\n\\node (0b) at (0,-2) {$\\circ$};\n\\draw[-] (0) to (1);\n\\draw[-,dashed] (0) to (0b);\n\\end{tikzpicture}\n\\\\ \\ \\\\\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {\\framebox{$\\circ$}};\n\\node (1) at (2,0) {$\\bullet$};\n\\node (0b) at (0,-2) {$\\circ$};\n\\draw[-] (0) to (1);\n\\draw[-,dashed] (0) to (0b);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\times$};\n\\node (0b) at (0,-2) {\\color{white} $\\circ$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$\\ensuremath{\\mathsf{s}}$};\n\\node (1) at (2,0) {$\\ensuremath{\\mathsf{t}}$};\n\\node (0b) at (0,-2) {\\color{white} $\\circ$};\n\\node (0x) at (0.3,-0.5) {\\footnotesize{$\\neg \\Box_a p$}};\n\\node (1x) at (2.5,-0.5) {\\footnotesize{$p \\wedge \\Box_b\\neg \\Box_a p$}};\n\\draw[dashed,-] (0) to (1);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {$=$};\n\\node (0b) at (0,-2) {\\color{white} $\\circ$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[thick]\n\\node (0) at (0,0) {\\framebox{$\\circ$}};\n\\node (1) at (2,0) {$\\bullet$};\n\\draw[-] (0) to (1);\n\\node (0b) at (0,-2) {\\color{white} $\\circ$};\n\\end{tikzpicture}\n\\end{array}\\]\nIn the regrettably unpublished \\citep{sadzik:2006} the matter of stabilization after action model execution is discussed at great length.\n\n\\section{Private lies} \\label{sec.six}\n\n\\begin{quote}\n{\\em {\\bf True Lies and Butterflies } \\\\\nMei wants to invite two friends Zhu Yingtai and Liang Shanbo to a party. She knows that they are dying to get close to each other. Thus one will come if and only if (s)he believes that the other will come. Obviously they do not yet wish to admit this to each other, because as far as they know they are both still very uncertain about each other's feelings. Given this uncertainty, both in fact don't intend to come to the party. Mei now lies to Yingtai that Shanbo will come to the party and she lies to Shanbo that Yingtai will come to the party. As a result, they will both come to the party. \\\\ (This story is a free adaptation of the {\\em Butterfly Lovers}, a famous Chinese folktale, see \\url{https:\/\/en.wikipedia.org\/wiki\/Butterfly_Lovers}.)\n}\n\\end{quote}\n\nWe can consider this an example of a true lie, because when Mei is telling to to Yingtai that Shanbo plans to come, in fact Shanbo is not planning to come, and when she is telling to Shanbo that Yingtai plans to come, in fact Yingtai is not (yet) planning to come. (For modelling convenience we assume that Yingtai is slow in making up her mind after Mei informs her about Shanbo.) After that, they both change their minds, and both lies have become true.\n\nIn order to model this story we will depart in two respects from the previous setting of believed announcement logic. In the first place these announcements are not public but private. In the second place the agents that we model change their minds. As we consider the factual propositions `Yingtai plans to go to the party' and `Shanbo plans to go to the party', changing your mind involves factual change. A logic allowing both formalizations is called action model logic (with factual change) \\citep{baltagetal:1998,hvdetal.del:2007,jfaketal.lcc:2006}. This logic can be seen as a straightforward generalization of believed announcement logic. (Alternatively, we can model the dynamics in more protocol oriented dynamic epistemic logics, such as \\cite{Wang10:phd,hvdetal.aij:2014}.) We only present some examples of action models and assume familiarity with the framework. For technical details, see Appendix A on page \\pageref{sec.appendixa} or the above references.\n\nFor an initial model, we assume that Yingtai and Shanbo know of themselves whether they intend to go to the party but do not know it of the other one (and that this is known, that this is the background knowledge). This comes with the following model, wherein solid access represents the uncertainty of Yingtai and dashed access represents the uncertainty of Shanbo, and where worlds are named with the facts $p_y$ (`Yingtai comes to the party') and $p_s$ (`Shanbo comes to the party') that are true there, where $01$ stands for `$p_y$ is false and $p_s$ is true', etc. The designated point of the model is boxed: initially both do not intend to go to the party.\n\n\\bigskip\n\n\\begin{tikzpicture}[thick]\n\\node (00m) at (4,0) {\\fbox{$00$}};\n\\node (10m) at (6,0) {$10$};\n\\node (01m) at (4,2) {$01$};\n\\node (11m) at (6,2) {$11$};\n\\draw[dashed,-] (00m) to (10m);\n\\draw[dashed,-] (01m) to (11m);\n\\draw[-] (00m) to (01m);\n\\draw[-] (10m) to (11m);\n\\end{tikzpicture}\n\n\\bigskip\n\nMei now lies to Yingtai, in private, that Shanbo goes to the party. For the convenience of the reader informed about action model logic, we can model this as a three-pointed action model as follows, on the left---where for convenience we have put the similar private lie to Shanbo about Yingtai next to it, on the right.\n\n\\bigskip\n\n\\begin{tikzpicture}[thick]\n\\node (x) at (0,0) {\\fbox{$\\neg p_s$}};\n\\node (y) at (2,0) {$p_s$};\n\\node (z) at (1,2) {$\\top$};\n\\draw[->] (x) to (y);\n\\draw[dashed,->] (x) to (z);\n\\draw[dashed,->] (y) to (z);\n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (x) at (0,0) {\\fbox{$\\neg p_y$}};\n\\node (y) at (2,0) {$p_y$};\n\\node (z) at (1,2) {$\\top$};\n\\draw[dashed,->] (x) to (y);\n\\draw[->] (x) to (z);\n\\draw[->] (y) to (z);\n\\end{tikzpicture}\n\n\\bigskip\n\nThe result of the first of these lying actions is as follows.\n\n\\bigskip\n\n\\begin{tikzpicture}[thick]\n\\node (00m) at (4,0) {\\fbox{$00$}};\n\\node (10m) at (6,0) {$10$};\n\\node (01m) at (4,2) {$01$};\n\\node (11m) at (6,2) {$11$};\n\\node (00r) at (8,0) {$00$};\n\\node (10r) at (10,0) {$10$};\n\\node (01r) at (8,2) {$01$};\n\\node (11r) at (10,2) {$11$};\n\\draw[dashed,-] (00r) to (10r);\n\\draw[dashed,-] (01r) to (11r);\n\\draw[-] (00r) to (01r);\n\\draw[-] (10r) to (11r);\n\\draw[dashed,->, bend left=20] (00m) to (00r);\n\\draw[dashed,->, bend left=20] (01m) to (01r);\n\\draw[dashed,->, bend left=20] (10m) to (10r);\n\\draw[dashed,->, bend left=20] (11m) to (11r);\n\\draw[->] (00m) to (01m);\n\\draw[->] (10m) to (11m);\n\\end{tikzpicture}\n\n\\bigskip\n\nNow Mei lies privately to Shanbo that Yingtai goes to the party. The result of that action is as follows. For the convenience of the reader we also depict (on the right) the restriction of this model to the submodel generated by the point $00$:\n\n\\bigskip\n\n\\scalebox{.8}{\n\\noindent\n\\begin{tikzpicture}[thick]\n\\node (00mb) at (4,0) {\\fbox{$00$}};\n\\node (10mb) at (6,0) {$10$};\n\\node (01mb) at (4,2) {$01$};\n\\node (11mb) at (6,2) {$11$};\n\\node (00rb) at (8,0) {$00$};\n\\node (10rb) at (10,0) {$10$};\n\\node (01rb) at (8,2) {$01$};\n\\node (11rb) at (10,2) {$11$};\n\n\\node (00m) at (4,4) {$00$};\n\\node (10m) at (6,4) {$10$};\n\\node (01m) at (4,6) {$01$};\n\\node (11m) at (6,6) {$11$};\n\\node (00r) at (8,4) {$00$};\n\\node (10r) at (10,4) {$10$};\n\\node (01r) at (8,6) {$01$};\n\\node (11r) at (10,6) {$11$};\n\n\\draw[dashed,-] (00r) to (10r);\n\\draw[dashed,-] (01r) to (11r);\n\\draw[-] (00r) to (01r);\n\\draw[-] (10r) to (11r);\n\\draw[dashed,->, bend left=20] (00m) to (00r);\n\\draw[dashed,->, bend left=20] (01m) to (01r);\n\\draw[dashed,->, bend left=20] (10m) to (10r);\n\\draw[dashed,->, bend left=20] (11m) to (11r);\n\\draw[->] (00m) to (01m);\n\\draw[->] (10m) to (11m);\n\n\\draw[dashed,->] (00rb) to (10rb);\n\\draw[dashed,->] (01rb) to (11rb);\n\\draw[->, bend left=20] (00rb) to (00r);\n\\draw[->, bend left=20] (10rb) to (10r);\n\\draw[->, bend left=20] (01rb) to (01r);\n\\draw[->, bend left=20] (11rb) to (11r);\n\\draw[dashed,->, bend left=20] (00mb) to (10rb);\n\\draw[dashed,->, bend left=20] (01mb) to (11rb);\n\\draw[dashed,->, bend left=20] (10mb) to (10rb);\n\\draw[dashed,->, bend left=20] (11mb) to (11rb);\n\\draw[->, bend left=20] (00mb) to (01m);\n\\draw[->, bend left=20] (10mb) to (11m);\n\\draw[->, bend left=20] (01mb) to (01m);\n\\draw[->, bend left=20] (11mb) to (11m);\n\\end{tikzpicture}\n\\quad\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (00mb) at (4,0) {\\fbox{$00$}};\n\\node (10rb) at (10,0) {$10$};\n\\node (01m) at (4,6) {$01$};\n\n\\node (00r) at (8,4) {$00$};\n\\node (10r) at (10,4) {$10$};\n\\node (01r) at (8,6) {$01$};\n\\node (11r) at (10,6) {$11$};\n\n\\draw[dashed,-] (00r) to (10r);\n\\draw[dashed,-] (01r) to (11r);\n\\draw[-] (00r) to (01r);\n\\draw[-] (10r) to (11r);\n\\draw[dashed,->, bend left=20] (00mb) to (10rb);\n\\draw[->, bend left=20] (00mb) to (01m);\n\\draw[->, bend left=20] (10rb) to (10r);\n\\draw[dashed,->, bend left=20] (01m) to (01r);\n\\end{tikzpicture}\n}\n\n\\bigskip\n\nThe above model formalizes that: Yingtai is not going to the party, believes that Shanbo goes to the party, and believes that Shanbo is uncertain whether she goes to the party; and that: Shanbo is not going to the party, believes that Yingtai goes to the party, and believes that Yingtai is uncertain whether he goes to the party.\n\n\\bigskip\n\nWe now first let Yingtai change her mind and then Shanbo change his mind. Yingtai changing her mind can again be formalized as an action model, namely as a private assignment to Yingtai; and similarly, Shanbo changing his mind as a private assignment to Shanbo. It is important here that a {\\em public} assignment is an improper way to formalize this action: a public assignment of Yingtai going to the party if she believes that Shanbo goes to the party would be informative to Shanbo in case he were to believe that she believed that he was going to the party. Because in case he was uncertain if she would go to the party, he would then learn from this public assignment that she would come to the party for his sake. Boring. Because exactly the absence of this sort of knowledge of the other's intentions makes first lovers' meetings so thrilling. That kind of uncertainty should {\\em not} be resolved. Therefore, we formalize it as a private assignment. Interestingly, in the current model the result of a public and of a private assignment (the result of Yingtai privately changing her mind or publicly changing her mind) is the same. But that is because both Yingtai and Shanbo believe that the other is uncertain whether they go to the party.\n\nBelow on the left is the action model for Yingtai changing her mind, and on the right, the one for Shanbo changing his mind. (So, for example, according to our conventions, in the left action model the solid relation, that of Yingtai, has identity access, and the dashed relation, for Shanbo, has only a reflexive arrow in the point that the dashed arrow is pointing to.) \n\n\\bigskip\n\n\\begin{tikzpicture}[thick]\n\\node (x) at (0,0) {\\fbox{$p_y := \\Box_y p_s$}};\n\\node (y) at (4,0) {$\\top$};\n\\draw[dashed,->] (x) to (y);\n\\end{tikzpicture}\n\\quad\\quad\\quad\n\\begin{tikzpicture}[thick]\n\\node (x) at (0,0) {\\fbox{$p_s := \\Box_s p_y$}};\n\\node (y) at (4,0) {$\\top$};\n\\draw[->] (x) to (y);\n\\end{tikzpicture}\n\n\\bigskip\n\nWe now depict, from left to right, once more the model before they change their minds, the model resulting from executing the action of Yingtai changing her mind, and the model resulting from Shanbo changing his mind, where once again we restrict the actually resulting models to the point-generated subframes.\n\n\\bigskip\n\n\\scalebox{.8}{\n\\noindent\n\\begin{tikzpicture}[thick]\n\\node (00mb) at (6.5,2.5) {\\fbox{$00$}};\n\\node (10rb) at (10,2.5) {$10$};\n\\node (01m) at (6.5,6) {$01$};\n\n\\node (00r) at (8,4) {$00$};\n\\node (10r) at (10,4) {$10$};\n\\node (01r) at (8,6) {$01$};\n\\node (11r) at (10,6) {$11$};\n\n\\draw[dashed,-] (00r) to (10r);\n\\draw[dashed,-] (01r) to (11r);\n\\draw[-] (00r) to (01r);\n\\draw[-] (10r) to (11r);\n\\draw[dashed,->] (00mb) to (10rb);\n\\draw[->] (00mb) to (01m);\n\\draw[->] (10rb) to (10r);\n\\draw[dashed,->] (01m) to (01r);\n\\end{tikzpicture}\n$\\stackrel {\\text{Yingtai}} \\Rightarrow$\n\\begin{tikzpicture}[thick]\n\\node (00mb) at (6.5,2.5) {\\fbox{$10$}};\n\\node (10rb) at (10,2.5) {$10$};\n\\node (01m) at (6.5,6) {$11$};\n\n\\node (00r) at (8,4) {$00$};\n\\node (10r) at (10,4) {$10$};\n\\node (01r) at (8,6) {$01$};\n\\node (11r) at (10,6) {$11$};\n\n\\draw[dashed,-] (00r) to (10r);\n\\draw[dashed,-] (01r) to (11r);\n\\draw[-] (00r) to (01r);\n\\draw[-] (10r) to (11r);\n\\draw[dashed,->] (00mb) to (10rb);\n\\draw[->] (00mb) to (01m);\n\\draw[->] (10rb) to (10r);\n\\draw[dashed,->] (01m) to (01r);\n\\end{tikzpicture}\n$\\stackrel {\\text{Shanbo}} \\Rightarrow$\n\\begin{tikzpicture}[thick]\n\\node (00mb) at (6.5,2.5) {\\fbox{$11$}};\n\\node (10rb) at (10,2.5) {$11$};\n\\node (01m) at (6.5,6) {$11$};\n\n\\node (00r) at (8,4) {$00$};\n\\node (10r) at (10,4) {$10$};\n\\node (01r) at (8,6) {$01$};\n\\node (11r) at (10,6) {$11$};\n\n\\draw[dashed,-] (00r) to (10r);\n\\draw[dashed,-] (01r) to (11r);\n\\draw[-] (00r) to (01r);\n\\draw[-] (10r) to (11r);\n\\draw[dashed,->] (00mb) to (10rb);\n\\draw[->] (00mb) to (01m);\n\\draw[->] (10rb) to (10r);\n\\draw[dashed,->] (01m) to (01r);\n\\end{tikzpicture}\n}\n\n\\bigskip\n\nNow that Yingtai and Shanbo have changed their minds, the lies have become the truth! They both go the party, and they will both expect the other to be surprised to find them at the party.\\footnote{This is in accordance with the resulting model, above on the right; for example, in the root \\fbox{11} of that model Yingtai believes ---move to the top 11--- that Shanbo believes ---move to 01 and 11--- that Yingtai is uncertain whether Shanbo comes to the party, both when she comes to the party ---the equivalence class consisting of 00 and 01--- as when she does not come to the party ---the cluster consisting of 10 and 11.} They declare their love to each other and they live happily ever after.\n\n\\weg{\n\n\\medskip\n\nWe conclude with two further technical observations on the model constructions and the analysis.\n\nFirstly, one can imagine Yingtai already changing her mind before Mei informs Shanbo that Yingtai is going to the party---in which case Mei would no longer have been lying. But that is a modelling artifact. To avoid such a scenario we simply assume that Mei simultaneously sends two {\\em letters} to Yingtai and to Shanbo containing the lies. At that moment both are indeed lies. Then again, the information change affected in Yingtai and Shanbo depends on the moment the letter is opened... It does not greatly matter: Yingtai changing her mind can be modelled both before and after Mei talking to Shanbo and the resulting pointed models are bisimilar. But lying twice is far more interesting than lying once only.\n\nSecondly, as mentioned, when executing the private assignments we restricted ourselves to point-generated subframes. Without that restriction, for example, based on the model on the left above, the model in the middle above would look as follows:\n\n\\bigskip\n\n\\scalebox{.8}{\n\\noindent\n\\begin{tikzpicture}[thick]\n\\node (00mb) at (6.5,2.5) {\\fbox{$10$}};\n\\node (10rb) at (10,2.5) {$10$};\n\\node (01m) at (6.5,6) {$11$};\n\n\\node (00r) at (8,4) {$00$};\n\\node (10r) at (10,4) {$10$};\n\\node (01r) at (8,6) {$01$};\n\\node (11r) at (10,6) {$11$};\n\n\\node (x00mb) at (12.5,2.5) {$00$};\n\\node (x10rb) at (16,2.5) {$10$};\n\\node (x01m) at (12.5,6) {$01$};\n\n\\node (x00r) at (14,4) {$00$};\n\\node (x10r) at (16,4) {$10$};\n\\node (x01r) at (14,6) {$01$};\n\\node (x11r) at (16,6) {$11$};\n\n\\draw[dashed,->, bend left=20] (00r) to (x00r);\n\\draw[dashed,->, bend left=20] (01r) to (x01r);\n\\draw[dashed,->, bend left=20] (10r) to (x10r);\n\\draw[dashed,->, bend left=20] (11r) to (x11r);\n\\draw[-] (00r) to (01r);\n\\draw[-] (10r) to (11r);\n\\draw[dashed,->, bend left=20] (00mb) to (x10rb);\n\\draw[->] (00mb) to (01m);\n\\draw[->] (10rb) to (10r);\n\\draw[dashed,->, bend left=20] (01m) to (x01r);\n\n\\draw[dashed,-] (x00r) to (x10r);\n\\draw[dashed,-] (x01r) to (x11r);\n\\draw[-] (x00r) to (x01r);\n\\draw[-] (x10r) to (x11r);\n\\draw[dashed,->] (x00mb) to (x10rb);\n\\draw[->] (x00mb) to (x01m);\n\\draw[->] (x10rb) to (x10r);\n\\draw[dashed,->] (x01m) to (x01r);\n\\end{tikzpicture}\n}\n\n\\bigskip\n\nClearly the simplified visualization is better.\n\n}\n\n\\section{What lies in the future?} \\label{sec.last}\n\nWe have modelled the true lie $\\phi$ in believed announcement logic as the validity \n\\[ \\neg\\phi \\rightarrow [\\phi]\\phi \\]\nor as the satisfiability of \\[ \\neg \\phi \\wedge [\\phi]\\phi, \\] \nwhere the believed announcement models {\\em informative change} and not factual change. Finding satisfiable true lies, of which there are many, seems a first step towards finding valid true lies, of which there are few. The latter seem to give more insight. They have the shape of correctness statements $\\phi \\rightarrow [\\alpha] \\psi$ in Hoare logic; and indeed we have seen that valid true lies come with strong syntactic restrictions. In the private lie example we also modelled actions involving {\\em factual change}. With this tool (and action model logic in its full generality) in hand we can also formalize the Pang Juan and the Arnold Schwarzenegger true lies, and make them fit a similar pattern. However, in this case we stop at satisfiability and the mere description of the examples. We defer a more general treatment and generic patterns involving validity to future research.\n\nThe Pang Juan true lie can be formalized as a two-pointed action model where one action has precondition $p$, the other action has precondition $\\neg p$, where the postcondition is $p := \\top$ in both actions (`no matter what was the case, you will now die'), and where the agent only considers the action with precondition $p$ possible. The designated point is the action where $p$ is false (Pang Juan does not die). Let us call this action $\\alpha(p)$.\n\n\\medskip\n\n\\begin{tikzpicture}[thick]\n\\node (x) at (0,0) {\\fbox{$\\neg p; p := \\top$}};\n\\node (y) at (4,0) {$p; p := \\top$};\n\\draw[->] (x) to (y);\n\\end{tikzpicture}\n\n\\medskip\n\nAlternatively, we can separate the observation part from the ontic change part, and consider the composition of those two actions. The former is then the {\\em lie that $p$}, i.e., the believed announcement that $p$ when $p$ is false. The action model equivalent of a believed announcement is a two-point action model (see Appendix A). The latter a singleton action called public assignment (namely of $\\top$ to $p$). So we get:\n\n\\medskip\n\n\\begin{tikzpicture}[thick]\n\\node (x) at (0,0) {\\fbox{$\\neg p$}};\n\\node (y) at (2,0) {$p$};\n\\draw[->] (x) to (y);\n\\end{tikzpicture}\n\\quad \n\\begin{tikzpicture}[thick]\n\\node (x) at (0,0) {composed with};\n\\end{tikzpicture}\n\\quad \n\\begin{tikzpicture}[thick]\n\\node (x) at (0,0) {\\fbox{$p := \\top$}};\n\\end{tikzpicture}\n\n\\medskip\n\nAs the observing agent is Pang Juan and the killing agent one of his opponents, one may prefer the sequence of two epistemic actions over the single epistemic action. On the other hand, as Pang Juan's observation and his subsequent death seem inextricably linked, one might prefer the single epistemic action. However, there is no notion of agency in dynamic epistemic logics: the composition of two actions is indistinguishable from the single action that combines the informative and ontic part: in the logic we have that $[\\alpha(p)]\\phi$ is equivalent to $[p][p := \\top]\\phi$, for all $\\phi$. So it does not matter. (We also recall that Pang Juan would die no matter which observation he made. We can satisfy this requirement by replacing the announcement that $p$ by the announcement that $q$ in the modelling with two epistemic actions. But that does not give any insight.) We end up with the satisfiability of \n\\[ \\neg p \\wedge [\\alpha(p)]p \\] but without being able to generalize this to a validity of shape $\\neg p \\rightarrow [\\alpha(p)]p$. This would require stronger or different logics, involving agency, or other notions of action or interaction \\citep{peteretal:2016}, or of {\\em self-fulfilling prophesy}.\n\nIn the Arnold Schwarzenegger example there is a lot of belief change and factual change going on. The epistemic goals of both agents, and their respective protocols in order to reach those goals, seem to play an important part. \n\nLet $p_j$ represent `Jamie Lee Curtis is a spy' and $p_a$ represent `Arnold Schwarzenegger is a spy'. At first Arnold is a spy ($p_a$), Arnold believes that he is a spy ($B_a p_a$ ---of course, the reader might say; but hang on!), Jamie believes he is not a spy ($B_j \\neg p_a$), and Arnold's goal is to keep it that way ($B_j \\neg p_a$). Jamie is not a spy ($\\neg p_j$), she believes that she is not a spy ($B_j \\neg p_j$), and Arnold also believes this ($B_a \\neg p_j$). Jamie's goal also is to keep it that way. So they both believe that the other is not a spy, and this is a stable situation, because both want the other to believe that. We now get a number of transitions where beliefs, facts, and goals all change. The initial situation is (i). \\[\\begin{array}{l|llllll|l|l}\n&&&&&&& \\text{goal } a & \\text{goal } j \\\\\n\\hline\n(i) & p_a & \\neg p_j & B_a p_a & B_j \\neg p_a & B_a \\neg p_j & B_j \\neg p_j & B_j \\neg p_a & B_a \\neg p_j \\\\\n(ii) & p_a & \\neg p_j & B_a p_a & B_j \\neg p_a & B_a \\neg p_j & \\pmb{B_j p_j} & B_j \\neg p_a & \\pmb{B_a p_j} \\\\\n(iii) & p_a & \\neg p_j & B_a p_a & \\pmb{B_j p_a} & B_a \\neg p_j & B_j p_j & B_j \\neg p_a & B_a p_j \\\\\n(iv) & p_a & \\pmb{p_j} & B_a p_a & B_j p_a & \\pmb{B_a p_j} & B_j p_j & \\pmb{B_j p_a} & B_a p_j \\\\\n\\end{array} \\]\nThe epistemic goals of the agents are only realized in (i) and (iv); only there they are stable. Indeed, the last stage is the happy ending. (For details, please see the movie.) In (ii), Jamie has started to enact a spy in order to make Arnold believe that she is a spy; she even believes that she has become a spy, without realizing that she has been set up: she is not really a spy. In (iii), she finds out that Arnold is a spy (because unlike Arnold, third parties believe that she is a spy and that she is collaborating with Arnold). In (iv) ---abstracting from intervening details involving helicopter fights and Miami skyscrapers--- they end up collaborating as spies and believing in each other, now truthfully so. Again, a stable situation.\n\nLet us call the composition of all these actions $\\alpha(p_a,p_j)$. Again we have satisfiability of \n\\[ (p_a \\wedge \\neg p_j) \\wedge [\\alpha(p_a,p_j)](p_a \\wedge p_j) \\]\nbut no general pattern involving validity of $(p_a \\wedge \\neg p_j) \\rightarrow [\\alpha(p_a,p_j)](p_a \\wedge p_j)$ or any other additional insight. Lacking in our formal analysis in dynamic epistemic logic are much stronger notions of {\\em epistemic protocol} (how does one realize an epistemic goal given a belief state?), protocol logics, and, again, notions of agency \\citep{bolanderetal:2011,Wang10:phd}. Second-order false belief scenarios are modelled in \\cite{brauneretal:2016,arslanetal:2015}.\n\n\n\\section{Conclusions}\n\nWe have presented a large variety of communicative scenarios involving agents truthtelling and lying while keeping their beliefs consistent. Of particular interest were iterations of announcements where the beliefs of the agents never stabilize: the models keep changing with every next announcement. The investigation can be seen as the obvious next step given existing in-depth investigations of successful and self-refuting updates changing agents' knowledge, i.e., on models where epistemic stances are interpreted with equivalence relations. We have carried this forward not merely to the ${\\mathcal{KD}45}$ models for believing agents, but also to models without structural properties. Two new types of update then come to the fore: true lies (that become true after a lying announcement) and impossible lies (that stay false even after a lying announcement), and also iterations of such announcements. An open question on these iterations is which $\\sigma$-valid formulas are realizable.\n\n\\section*{Acknowledgements}\n\nYanjing Wang thanks C\\'edric D\\'egremont and Andreas Witzel for discussions on the example in Section \\ref{sec.five}, and he acknowledges support from the National Program for Special Support of Eminent Professionals. Hans van Ditmarsch acknowledges support from ERC project EPS 313360. He is also affiliated to IMSc (Institute for Mathematical Sciences), Chennai, India. We thank the reviewers of Synthese for their helpful comments.\n\n\\bibliographystyle{natbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{intro} Introduction}\n\n\\begin{figure}[tp]\n\\scalebox {0.6}{\\includegraphics{scfig1}}\n\\caption{Shown schematically is a magnetic flux tube which exits the solar \nphotosphere (shaded region) and enters the solar corona (clear region) through \na sunspot on the chromosphere boundary. The magnetic flux tube then exits the \nsolar corona and enters back into the photosphere through a second sunspot on \nthe chromosphere boundary. The walls of magnetic flux tube are a vortex \nof circulating electric currents.}\n\\label{fig1}\n\\end{figure}\n\nFor physical reasons which are presently not entirely clear, dark sunspots exist \non the optical solar surface. These have long been observed ever since Galileo saw \nsunspots with his optical telescope. It was later found that magnetic flux tubes \nexit out of some solar sunspots and enter back into \nothers\\cite{Hale:1908,Hale:1919}. Employing modern X-ray telescopes,\nspectacular pictures have been taken of magnetic flux \ntubes\\cite{Priest:1998} which arch into and out of the solar corona,\nwell above the optical solar surface. The situation is schematically shown in \nFIG.\\ref{fig1}. The closed magnetic flux tubes are pictured in both the solar \nphotosphere and the solar corona. The closed flux tube magnetic field lines enter \ninto the solar corona through one sunspot and exit out of the solar corona \nthrough another sunspot. The floating flux tubes in the solar corona are held up by \na magnetic buoyancy\\cite{Parker:1955}. The outer walls of the magnetic flux tube consist \nof large circulating electric currents forming a turbulent vortex with a darker \ncomparatively quiet magnetic core. When the magnetic flux tubes explode\\cite{Parker:1957} \ninto a solar flare, with or without a coronal mass ejection\\cite{Kahler:2007}, \ncharged particles with very high energy are  produced\\cite{Forbush:1946,Dorman:1993,Reams:2004,Belov:2005,Vashenyuka:2005,Bostanjyan:2007}, \nsay up to \\begin{math} \\sim 10^2 \\end{math} GeV.  These relativistic particles can escape \nthe sun and be observed on earth as ground level cosmic ray \nenhancements\\cite{Shea:2001,Cliver:2006,L3:2006} induced by solar flares.\n\nOur purpose is to discuss how such energetic particles arise in the solar corona \nand how these particles induce nuclear reactions well above the solar \nphotosphere. The central feature of our explanation, which centers around \nFaraday's law, is the notion of a solar accelerator closely analogous to the \nbetatron\\cite{Kirst:1941,Serber:1941}. Conceptually, the betatron is a step up \ntransformer whose secondary coil is a toroidal ring of accelerating charged \nparticles circulating about a Faraday law (time varying) magnetic flux tube.\n\n\n\n\n\n\n\nActing as a step up transformer, it is possible for a solar magnetic flux tube \nto transfer circulating charged particle kinetic energy upward from the \nphotosphere to circulating charged particles located in the corona. Circulating \ncurrents located deep in the photosphere can be viewed conceptually as a net \ncurrent \\begin{math} I_{\\cal P} \\end{math} circulating around a primary coil.  \nCirculating currents found high in the corona can be viewed as a net current \n\\begin{math} I_{\\cal S} \\end{math} circulating around a secondary coil. \nIf \\begin{math} K_{\\cal P} \\end{math} and \\begin{math} K_{\\cal S} \\end{math} \nrepresent, respectively, the charged particle kinetic energies in the primary and \nsecondary coils, then one finds the step up transformer power equation \n\\begin{math} \n\\dot{K}_{\\cal P}=V_{\\cal P}I_{\\cal P} =V_{\\cal S}I_{\\cal S}=\\dot{K}_{\\cal S} \n\\end{math}, \nwherein \\begin{math} V_{\\cal P} \\end{math} and \\begin{math} V_{\\cal S} \\end{math} \nrepresent, respectively, the voltages across the primary and secondary coils. \nThe total kinetic energy transfer  \n\\begin{math} \n\\Delta K_{\\cal P}=\\int V_{\\cal P}I_{\\cal P}dt=\\int V_{\\cal S}I_{\\cal S}dt=\\Delta K_{\\cal S} \n\\end{math}. \nThe essence of the step up transformer mechanism is that the kinetic energy distributed \namong a very large number of charged particles in the photosphere can \nbe transferred via the magnetic flux tube to a distributed kinetic energy shared among \na distant much smaller number of charged particles located in the corona, i.e.\na small accelerating voltage in the primary coil produces a large accelerating voltage \nin the secondary coil. The resulting transfer of kinetic energy is \n{\\em collective} from a large group of charged particles to a smaller group of charged \nparticles. The kinetic energy per charged particle of the dilute gas in the corona \nmay then become much higher than the kinetic energy per charged particle of the \nmore dense fluid in the photosphere. In terms of the connection between temperature \nand kinetic energy, the temperature of the dilute gas in corona will be much higher \nthan the temperature of the more dense fluid photosphere. \n\nIf the kinetic energy of the circulating currents in that part of flux tubes floating \nin the corona becomes sufficiently high, then the flux tubes can explode violently into \na solar flare which may be accompanied by a coronal mass ejection. \nThe loss of magnetic energy during the flux tube explosion is rapidly converted into \ncharged particle kinetic energy. The relativistic high energy products of the explosion \nyield both nuclear and elementary particle interactions. These processes are discussed \nin Sec.\\ref{flare}. For magnetic flux tubes of smaller diameter which do not explode \ninto a flare and\/or a coronal mass ejection, one may still have low energy nuclear \nreactions that occur in a roughly steady state by continual conversion of magnetic \nfield energy into charged particle energy. Such processes can account for the fact \nthat the solar corona remains continually much hotter than the photosphere.\nSteady state low energy nuclear processes are discussed in Sec.\\ref{lenr}. In the \nconcluding Sec.\\ref{conc} we further discuss the notion that not all nuclear \nprocesses necessarily take place near within the solar core. \n\n\n\\section{\\label{flare} Solar Flares}\n\n\\begin{figure}[bp]\n\\scalebox {0.6}{\\includegraphics{scfig2}}\n\\caption{Boson exchange diagrams for electron-proton scattering \ninto a lepton plus ``anything'' $\\{\\ e^- +p^+ \\to l+X\\ \\}$ include photon \n$\\gamma $ and $Z$ exchange wherein the final lepton is an electron,  \nas well as charged $W^-$ exchange wherein the final state lepton is a neutrino. \nOn an energy scale of $\\sim 300\\ {\\rm GeV}$, all of these exchange processes have \namplitudes of similar orders of magnitude.}\n\\label{fig2}\n\\end{figure}\n\nThe magnetic flux through a cylindrical tube of inner cross sectional area  \n\\begin{math} \\Delta S \\end{math} and mean magnetic field \\begin{math} B \\end{math}, \nis \n\\begin{equation}\n\\Delta \\Phi =B\\Delta S.\n\\label{flare1}\n\\end{equation}\nIf a tube explodes in a time period \\begin{math} \\Delta t \\end{math}, \nthen the resulting loss of magnetic flux yields a mean Faraday law \naccelerator voltage around the tornado walls as given by  \n\\begin{equation}\n\\overline{\\rm V}=\\frac{\\Delta \\Phi}{\\Delta t},\n\\label{flare2}\n\\end{equation}\ni.e. \n\\begin{equation}\ne\\overline{\\rm V}=ecB\\left(\\frac{\\Delta S}{\\Lambda }\\right) \n\\ \\ \\ {\\rm wherein}\\ \\ \\ \\Lambda=c\\Delta t.\n\\label{flare3}\n\\end{equation}\nA useful identity for numerical estimates is \n\\begin{equation}\necB\\equiv 29.9792458\n\\left[\\frac{\\rm GeV}{\\rm kilometer}\\right]\n\\left[\\frac{B}{\\rm kiloGauss}\\right].\n\\label{flare4}\n\\end{equation}\nFor a coronal mass ejection exploding coil with a Faraday flux loss \ntime \\begin{math} \\Delta t \\sim 10^2\\ {\\rm second}  \\end{math} and \nwith substantial sun spots at the ends of the magnetic flux coil, one \nmay estimate\\cite{Dikpati:2002,Lozitska:1994,Benz:2008}    \n\\begin{eqnarray}\n\\Delta S\\approx \\pi R^2 ,\n\\nonumber \\\\ \nR\\sim 10^4\\  {\\rm kilometer}, \n\\nonumber \\\\ \nB\\sim 1\\ {\\rm kiloGauss},\n\\nonumber \\\\ \n\\Lambda \\sim 3\\times 10^7\\ {\\rm kilometer}, \n\\nonumber \\\\  \ne\\overline{\\rm V}\\sim 300\\ {\\rm GeV}.\n\\label{flare5}\n\\end{eqnarray}\n\nThe uncharged walls of the circulating vortex are represented roughly by  \nan electron beam circulating in one direction and proton beam circulating \nin the other direction. These two colliding beams are hit with a flare or \ncoronal mass ejecting Faraday law voltage pulse, as in Eq.(\\ref{flare5}), \nsetting an electron proton collision energy scale of \n\\begin{math} E\\sim 300\\ {\\rm GeV}  \\end{math}. At such a high energy \nscale, electron-proton scattering\\cite{Roberts:1990} is ruled by electro-weak \nexchange interactions all of the same order of magnitude in probability. \nShown in FIG.\\ref{fig2} is the electro-weak boson exchange Feynman diagram \nfor electron-proton scattering  \n\\begin{equation}\ne^- + p^+ \\to l+X.\n\\label{flare6}\n\\end{equation}\nThe final state lepton is an electron for the case of photon \n\\begin{math} \\gamma \\end{math} or \\begin{math} Z \\end{math} \nexchange and the final state lepton is neutrino for the case  \nof \\begin{math} W^- \\end{math} exchange.\n\n\n\n\n\n\n\nA solar flare or coronal mass ejecting event is thereby accompanied \nby an increased emission of solar neutrinos over a broad energy scale \nas well as relativistic protons\\cite{Bostanjyan:2007}, \nneutrons\\cite{Hurford:2006,Murphy:1999,Hua:2002,Murphy:2007, Watanabe:2003} \nand electrons\\cite{Kahler:2007}. The full \nplethora\\cite {Ryan:2006,Li:2007} of final \\begin{math} X \\end{math} \nstates including electron, muon and pion particle anti-particle pairs \nshould also be present in such events. The conversion of magnetic field \nenergy into relativistic particle kinetic energy via the Faraday law voltage \npulse is collective in that the magnetic flux in the core of the vortex depends \non the rotational currents of {\\em all} of the initial protons and electrons. \n\n\n\\section{\\label{lenr} Low Energy Nuclear Reactions}\n \nEven without spectacular solar flare explosions ejecting mass through the \nsolar corona, there exists (within flux tubes) collective magnetic \nenergy which allows for a significant occurrence of low energy nuclear reactions \nat many different locations in and around the sun. \nIn particular, let us consider the inverse beta decay reaction \n\\begin{equation}\nW_{\\rm magnetic}+e^- + p^+ \\to \\nu_e + n\n\\label{lenr1}\n\\end{equation}  \nwherein the final state lepton is a neutrino, the final state \n``\\begin{math} X \\end{math}'' is a neutron \\begin{math} n \\end{math} \nand \\begin{math} W_{\\rm magnetic} \\end{math} is magnetic field energy \nfed into the reaction.  \n\nIn a steady state flux tube (which does {\\em not} explode) entering \ninto the solar corona from one sunspot and exiting out of the solar \ncorona through another sunspot, there is a substantial amount of \nstored magnetic energy. If there \nis a small change \\begin{math} \\delta I \\end{math} in the current going \naround the vortex circumference, then the small change in the magnetic \nfield energy \n\\begin{math} \\delta {\\cal E}  \\end{math} obeys \n\\begin{equation}\n\\delta {\\cal E}=\\Phi \\delta I. \n\\label{lenr2}\n\\end{equation}\nIf \\begin{math} L  \\end{math} denotes the length of the vortex circumference of the   \nmagnetic flux tube, then the change in current due to the weak interaction reaction \nEq.(\\ref{lenr1}) is given by \n\\begin{equation}\n\\delta I=-\\frac{ev}{L}\\ ,\n\\label{lenr3}\n\\end{equation}\nwherein \\begin{math} v \\end{math} is the relative velocity component \n(tangent to the circumference) between the proton and electron. \nPutting \\begin{math} \\Phi = B\\Delta S  \\end{math} and \n\\begin{math} \\delta {\\cal E}=-W_{\\rm magnetic} \\end{math} yields \n\\begin{equation}\nW_{\\rm magnetic}=ec\\left(\\frac{\\Phi }{L}\\right)\\frac{v}{c}=\necB\\left(\\frac{\\Delta S}{L}\\right)\\frac{v}{c}\\ .\n\\label{lenr4}\n\\end{equation}\nThe product \\begin{math} (ecB)  \\end{math} is given in Eq.(\\ref{flare4}). \nFor the case of a cylindrical flux tube, \n\\begin{equation}\n\\frac{\\Delta S}{L}=\\left(\\frac{\\pi R^2}{2\\pi R}\\right)=\\frac{R}{2}\\ .\n\\label{lenr5}\n\\end{equation}\nyielding \n\\begin{equation}\nW_{\\rm magnetic}\\approx 15{\\rm \\ GeV} \n\\left[\\frac{R}{\\rm kilometer}\\right]\n\\left[\\frac{B}{\\rm kiloGauss}\\right]\n\\frac{v}{c}\\ .\n\\label{lenr6}\n\\end{equation}\nEmploying the estimates \n\\begin{eqnarray} \nR\\sim 10^2\\  {\\rm kilometer}, \n\\nonumber \\\\ \nB\\sim 1\\ {\\rm kiloGauss},\n\\nonumber \\\\ \n\\frac{v}{c}\\sim 10^{-2}, \n\\nonumber \\\\ \nW_{\\rm magnetic}\\sim 15\\ {\\rm GeV}.\n\\label{lenr7}\n\\end{eqnarray}\nOn the energy scale \\begin{math} W_{\\rm magnetic} \\ll 300\\ {\\rm GeV} \\end{math} of \nEq.(\\ref{lenr7}), the weak interaction \\begin{math} p^+ e^-   \\end{math}  \nprocesses Eq.(\\ref{lenr1}) that produce neutrons proceed  more slowly than the purely \nelectromagnetic \\begin{math} p^+ e^- \\end{math} processes. Nevertheless one finds \nappreciable neutron production in the solar corona. The production of neutrons among \nthe protons allows for the creation of nuclei with higher mass numbers via neutron-capture \nnuclear reactions and subsequent beta decays. \n\n \n\n\n\\section{\\label{conc} Conclusion}\n\nMagnetic flux tubes arise\\cite{Dikpati:2002,Lozitska:1994} out of\nthe turbulent magneto-fluid mechanics of a solar fluid plasma with high electrical \nconductivity. Turbulent magneto-fluid flows yield a full spectrum of magnetic \nfield values\\cite{Benz:2008} expected to vary randomly over many different \nlength scales. The estimates of the magnetic field \ndiscussed in this work are merely order of magnitude. They are based on observations \nof the magnetic flux tubes entering into and exiting out of sunspots and also into and \nout of smaller crevices and holes that are commonly observed on the sun's optical surface.\n\nFor many years, the source of relativistic particle \nfluxes\\cite{Yousef:2005,Roussev:2004} often observed to emanate from the solar corona \nhas been theoretically obscure. Our explanation  for these fluxes is simply derived from \nFaraday's law \n\\begin{equation}\n-\\frac{\\partial {\\bf B}}{\\partial t}=curl{\\bf E}\n\\label{conc1} \n\\end{equation} \nas it appears in well understood transformer and inductor circuits. Circulating currents\naround the walls of a flux tube can transfer energy into some parts of the magnetic field \nconfiguration while removing equal amounts of energy from other distant parts of the \nmagnetic field configuration. This transformer action is very well understood. \n\n\n\n\n\nThe resulting energy balance allows a large number of low energy charged particles to \ncollectively transfer  their kinetic energy to a significantly smaller number of charged \nparticles whose energy per particle then becomes very high. When the charged particle \nenergy of low density solar corona particles is made sufficiently high, reactions of \nthe form in Eqs.(\\ref{flare6}) and (\\ref{lenr1}) clearly can take place, \nleading to neutron production. Once neutrons are created and added to the \nelectron-proton plasma, a variety of nuclear synthesis reactions \nbecome possible\\cite{Fowler:1965}.  If that is the case, then Coulomb barrier-penetrating \nfusion reactions in the sun's core are not necessarily the sun's {\\em only} significant \nsource of solar nuclear energy. Relativistic particle fluxes have been clearly observed \nemanating from regions located well above the sun's surface and very far away \nfrom the solar core. Finally, it has not escaped our notice that the energetic particle \nproduction via the collective mechanisms discussed in this work may shed some light on the \norigin of the anomalous short-lived isotopes observed on other astronomical \nobjects\\cite{Cowley:2004}. \n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nExceptionally bright quasars with redshifts up to $z\\sim7$ have been discovered \\citep[e.g.][]{Fan2001,Fan2003,Mortlock2011}. These quasars are thought to be powered by the thin disk accretion of gas onto super-massive black-holes at the centres of galaxies. Their maximum (Eddington) luminosity depends on the mass of the black-hole, and the brighter quasars are inferred to have black-holes with masses of more than a few billion solar masses. Since their discovery \\citep[][]{Fan2001,Fan2003}, the existence of such massive black-holes at $z\\ga6$ has posed a challenge to models for their formation. This is because a $\\sim10^9$M$_\\odot$ black-hole accreting at the Eddington rate with a radiative efficiency of 10\\% requires almost the full age of the Universe at $z\\sim6$ to grow from a stellar mass seed. Many authors have therefore discussed solutions to this apparent mystery by including a significant build-up of mass through mergers \\citep[][]{Haiman2001}, collapse of low-spin systems \\citep[][]{Eisenstein1995}, and suppression of molecular line cooling via a large Lyman-Werner flux \\citep[][]{Dijkstra2008}. Other authors have also attempted to explain the fast growth of black-holes at high redshifts based on super-massive stars \\citep[][]{Loeb1994,Bromm2003} and more recently, quasi-stars \\citep[][]{Volonteri2010,Begelman2010}. \n\n\\citet[][]{Volonteri2005} discussed the possibility of super-Eddington accretion of black-hole seeds in high redshift galaxies. They pointed out that seed black-holes are located at the centres of isothermal disks where the conditions for quasi-spherical Bondi accretion should be prevalent. At high redshift these disk centres are sufficiently dense that the Bondi accretion rate greatly exceeds the Eddington rate. \\citet[][]{Volonteri2005} point out that this super-Eddington accretion provides a route by which a large fraction of the mass $e$-foldings needed to grow a super-massive black-hole by redshift six could be accommodated within a small fraction of the age of the Universe. However the calculations in \\citet[][]{Volonteri2005} ignored feedback effects like gas heating, which may raise the sound-speed, and hence lower the density and therefore the Bondi accretion rate. For example \\citet[][]{Milosavljev2009} show that photoheating and radiation pressure from photoionization significantly reduce the steady-state accretion rate and potentially render a quasi-radial accretion flow unsteady and inefficient. They find that the time-averaged accretion rate is always a small fraction of the Bondi accretion rate. Thus, the very high accretion rates implied by the Bondi accretion in the centre of a high redshift isothermal disk might never be reached.  On the other hand, if the accretion rate is sufficiently high that the emergent photons are trapped within the accretion flow, then these feedback effects cannot operate \\citep[][]{Begelman1979}, and so the accretion rate can reach arbitrarily high levels. \n\nIn this paper we find that at sufficiently high redshift, the central densities of galaxies imply Bondi accretion rates that exceed the rate required to trap radiation and advect it into a black-hole \\citep[][]{Begelman1979}. Thus, we find that there were periods in the growth of black-holes at high redshift where the growth was super-Eddington and feedback mechanisms could not halt the accretion flow.  Our goal is not to make a self-consistent model for both the transport of material from large galactic radii and central black-hole accretion. Rather, we note that there is a significant literature looking at the problem of super-Eddington accretion assuming large mass-delivery rates to the region of the black-hole, and identify the cosmological conditions that would provide sufficient accretion rates to allow this by trapping radiation.  We begin in \\S~\\ref{model} with a description of our simple model, before presenting our results in \\S~\\ref{results}. We finish with a discussion in \\S~\\ref{discussion}, and conclusions in \\S~\\ref{conclusion}.  In our numerical examples, we adopt the standard set of cosmological parameters \\citep[][]{Komatsu2011}, with values of $\\Omega_{\\rm b}=0.04$, $\\Omega_{\\rm m}=0.24$ and $\\Omega_\\Lambda=0.76$ for the matter, baryon, and dark energy fractional density respectively, $h=0.73$, for the dimensionless Hubble constant, and $\\sigma_8=0.82$.\n\n\n\n\\section{Model}\n\n\\label{model}\nThe basis of this paper is a comparison between the Bondi accretion rate, and the accretion rate required to trap photons within the accretion flow. We discuss these in turn.\n\n\\subsection{The Bondi accretion rate}\n\nWe begin with the expression for the Bondi accretion rate \\citep[][]{Bondi1952}  onto a central black-hole of mass $M_{\\rm bh}$ \n\\begin{equation}\n\\label{bondi}\n\\dot{M}_{\\rm Bondi} = 4\\pi\\rho_0 r_{\\rm Bondi}^2 v_{\\rm ff} = 4\\pi n_0 \\mu m_{\\rm p} r_{\\rm Bondi}^2 \\sqrt{\\frac{G M_{\\rm bh}}{r_{\\rm Bondi}}},\n\\end{equation}\nwhere $v_{\\rm ff}$ is the free-fall time at the Bondi radius\n\\begin{equation}\nr_{\\rm Bondi} = \\frac{G M_{\\rm bh}}{c_{\\rm s}^2},\n\\end{equation}\nand $c_{\\rm s}$ is the sound speed, which for an isothermal gas we assume corresponds to a temperature of $10^4$K.\nTo evaluate $\\dot{M}_{\\rm Bondi}$ we specify the central number density of a self gravitating disk \\citep[][]{Schaye2004} \n\\begin{equation}\n\\label{n0}\nn_0 = \\frac{G M_{\\rm disk}^2}{12 \\pi c_{\\rm s}^2 R_{\\rm d}^4 \\mu m_{\\rm p}}.  \n\\end{equation}\nHere $R_{\\rm d}$ is the characteristic radius of an exponential disk of surface density profile\n\\begin{equation}\n\\Sigma(r) = \\Sigma_0 \\exp(-r\/R_{\\rm d}),\n\\label{eq:expdensprof}\n\\end{equation}\nwith $\\Sigma_0 = M_{\\rm disk}\/ 2\\pi R_{\\rm d}^2$, and the disk scale length is given by\n\\begin{equation}\nR_{\\rm d} = {\\lambda \\over \\sqrt{2}} r_{\\rm vir},\n\\end{equation}\nwhere $\\lambda$ is the dimensionless spin parameter of the halo.\nIn equation~(\\ref{n0}), $M_{\\rm disk}=m_{\\rm d}M_{\\rm halo}$ is the disk mass, $m_{\\rm p}$ is the mass of hydrogen, and $\\mu=1.22$ is the molecular weight of primordial neutral gas. At the high redshifts of interest, most of the virialized galactic gas is expected to cool rapidly and assemble into the disk. We therefore assume $m_{\\rm d}=0.17$. The corresponding mass density is $\\rho_{\\rm 0} = m_{\\rm p} n_0$. The virial radius of a halo with mass $M_{\\rm halo}$ is given by the expression\n\\begin{equation}\n\\label{eps}\n\\nonumber r_{\\rm vir}= 0.784 h^{-1}\\,\\mbox{kpc} \\left(\\frac{M_{\\rm halo}}{10^{8}M_{\\odot}h}\\right)^{\\frac{1}{3}}\n[\\zeta(z)]^{-\\frac{1}{3}}\\left(\\frac{1+z}{10}\\right)^{-1},\n\\end{equation}\nwhere $\\zeta(z)$ is close to unity and\ndefined as $\\zeta\\equiv [(\\Omega_{\\rm m}\/\\Omega_{\\rm m}^z)(\\Delta_c\/18\\pi^2)]$,\n$\\Omega_{\\rm m}^z \\equiv [1+(\\Omega_\\Lambda\/\\Omega_{\\rm m})(1+z)^{-3}]^{-1}$,\n$\\Delta_c=18\\pi^2+82d-39d^2$, and $d=\\Omega_{\\rm m}^z-1$ \\citep[see equations~22--25 in][for more details]{Barkana2001}. From equations~(\\ref{n0}) and (\\ref{eps}) we see that the central density $n_0$ and hence the Bondi accretion rate $\\dot{M}_{\\rm Bondi}$ scales as $(1+z)^4$, and that as a result accretion rates are expected to be much larger at high redshift.  \n\nDormant central black-holes are ubiquitous in local galaxies \\citep[][]{Magorrian1998}. The\nmasses of these super-massive black-holes scale with physical properties of their hosts\n\\cite[e.g.][]{Magorrian1998,Merritt2001,Tremaine2002}.  However at high redshift the relations observed in the local Universe may not be in place. We therefore do not impose a model for the relation between black-hole and halo mass in this paper, and instead explore a range of values. Indeed, our results indicate that feedback, which is thought to drive the black-hole -- halo-mass relation relation, would not be effective at early times. The grey curves in Figure~\\ref{fig1} show the Bondi accretion rate as a function of redshift for different values of halo and black-hole mass. Here we assume $\\lambda=0.05$ corresponding to the mean spin parameter for dark-matter halos \\citep[][]{Mo1998}.\n\n\n\n \\begin{figure*}\n\\begin{center}\n\\vspace{3mm}\n\\includegraphics[width=17.5cm]{fig1.pdf}\n\\caption{\\label{fig1} The grey curves show the Bondi accretion rate as a function of redshift. The dark lines show the critical accretion rate for which the photon diffusion speed is smaller than the gravitational free-fall speed. In cases where the Bondi accretion rate is larger than the critical accretion rate, photons are trapped and the AGN is obscured. Three cases are considered for the vertical structure of the disk, which is assumed to be set by the sound speed of the gas ({\\em Left Panels}), the turbulent velocity associated with a $Q=1$ disk ({\\em Central Panels}), and the virial velocity ({\\em Right Panels}) respectively. In each case the {\\em Upper} and {\\em Lower} panels correspond to cases with halo masses of $M=10^{9}$M$_\\odot$ and  $M=10^{10}$M$_\\odot$. Three curves in each case correspond to black-hole masses of $10^3$M$_\\odot$, $10^4$M$_\\odot$ and $10^5$M$_\\odot$. The cross-section was assumed to be $F_{\\rm sig}=100$ times larger than the Thomson cross-section, the minimum radius was $F_{\\rm min}=10^3$ times $r_{\\rm g}$, and the spin parameter $\\lambda=0.05$.}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Photon trapping by the accretion flow}\n\nIf the diffusion velocity of photons at a radius $r$ is smaller than the free-fall velocity of the material at radius $r$, then photons become trapped in the accretion flow \\citep[][]{Begelman1979}. In such cases, the black-hole would be obscured. In this section we estimate the accretion rate required to achieve this photon trapping at radius $r$. \n\nThe free-fall time from radius $r$ to a smaller radius $r_{\\rm min}$ is \n\\begin{equation}\n\\label{tff}\nt_{\\rm ff}\\sim \\frac{(r-r_{\\rm min})}{v_{\\rm ff}},\n\\end{equation}\nwhere $v_{\\rm ff}\\sim \\sqrt{G M_{\\rm bh}\/r}$ within the Bondi radius. This should be compared with the diffusion time of \n\\begin{equation}\n\\label{tdiff}\nt_{\\rm diff}\\sim \\tau \\frac{(r-r_{\\rm min})}{c}.\n\\end{equation}\nHere the optical depth is given by \n\\begin{equation}\n\\tau \\sim \\bar{\\rho} \\frac{F_{\\rm sig}\\sigma_{\\rm T}}{m_{\\rm p}}(r-r_{\\rm min}),\n\\end{equation}\nwhere $\\sigma_{\\rm T}$ is the Thomson cross-section, $F_{\\rm sig}$ is the ratio of the scattering cross-section of the accreting gas to the emergent radiation in units of $\\sigma_{\\rm T}$, and $\\bar{\\rho}$ is the line-of-sight averaged density between radii $r_{\\rm min}$ and $r$. The density within the Bondi radius scales as $\\rho(r)=\\rho_0 (r\/r_{\\rm Bondi})^{-1.5}$, yielding\n\\begin{eqnarray}\n\\nonumber\n\\bar{\\rho}&=&2\\rho_0 \\frac{r_{\\rm Bondi}}{r-r_{\\rm min}}  [(\\frac{r_{\\rm min}}{r_{\\rm Bondi}})^{-0.5}  - (\\frac{r}{r_{\\rm Bondi}})^{-0.5}]\\\\\n\\nonumber\n&=&2\\rho \\frac{r}{r-r_{\\rm min}}  [(\\frac{r}{r_{\\rm min}})^{0.5}  - 1]\\hspace{5mm}\\mbox{for}\\hspace{5mm}r<r_{\\rm Bondi}.\n\\end{eqnarray}\n\n\nThe condition $t_{\\rm diff}>t_{\\rm ff}$ at radius $r$ is satisfied for accretion rates $\\dot{M}>\\dot{M}_{\\rm lim}$ where\n\\begin{equation}\n\\dot{M}_{\\rm lim}=\\frac{\\rho}{\\bar{\\rho}}\\left(\\frac{4\\pi m_{\\rm p} c}{F_{\\rm sig}\\sigma_{\\rm T}}\\right)\\frac{r^2}{r-r_{\\rm min}},\n\\end{equation}\nin which we have used the relation $\\dot{M}_{\\rm Bondi} = 4\\pi\\rho r^2 v_{\\rm ff}$. We find that the accretion rate required for photon trapping has a minimum value (i.e. $d\\dot{M}_{\\rm lim}\/dr=0$) at a radius of $r=4r_{\\rm min}$, yielding\n a minimum required accretion rate for photon trapping of\n\\begin{equation}\n\\label{lim}\n\\dot{M}_{\\rm lim}=\\left(\\frac{8\\pi m_{\\rm p} c}{F_{\\rm sig}\\sigma_{\\rm T}}\\right)r_{\\rm min}.\n\\end{equation}\nSince the Eddington accretion rate at an efficiency $\\epsilon$ is $\\dot{M}_{\\rm Edd}=(4\\pi G M_{\\rm bh} m_{\\rm p})\/(c\\sigma_{\\rm T}\\epsilon)$, we find \n\\begin{equation}\n\\label{lim2}\n\\dot{M}_{\\rm lim}=2\\left(\\frac{\\epsilon}{F_{\\rm sig}}\\right) \\left(\\frac{r_{\\rm min}}{r_{\\rm g}}\\right) \\dot{M}_{\\rm Edd},\n\\end{equation} \nwhere $r_{\\rm g}=GM_{\\rm bh}\/c^2$. \n\nThe limiting accretion rate depends on the emergent radiation spectrum. As described below in \\S~\\ref{xray}, we find that the condition for photon trapping is more readily achieved for X-ray photons than for optical\/UV photons. We therefore frame our discussion around trapping of optical\/UV photons. In the case of optical\/UV photons the cross-section would be dominated by dust at radii larger than a sublimation radius $r_{\\rm min} = r_{\\rm sub} $, beyond which the opacity of the gas to optical\/UV photons could be larger than the Thomson opacity by as much as two or three orders of magnitude due to the presence of dust \\citep[][]{Laor1993}. The existence of dust can be questioned for high-redshift galaxies \\citep[e.g.][]{Bouwens2010d}, however high metallicities are inferred from the broad emission lines of all quasars out to $z=7.1$ and so metal enrichment (due to star formation) is known to precede the growth of the black-hole in the galactic nuclei of interest \\citep[][]{Hamann2010}. On the other hand, the diffusion time may be lessened if the gas is clumpy, corresponding to a lower effective opacity. We therefore take a typical value for the cross-section that is $F_{\\rm sig}=100$ times larger than the Thomson cross-section (additional cases are presented below in \\S~\\ref{structure}). Defining $r_{\\rm min}=F_{\\rm min} r_{\\rm g}$ we assume a typical value of $F_{\\rm min}=10^3$ to describe the sublimation radius \\citep[][]{Netzer1993}. \n\nLimiting accretion rates corresponding to these default values are plotted in the left hand panels of Figure~\\ref{fig1} (dark lines) for the case where the vertical structure of the disk is set by the sound speed of the gas. The {\\em Upper} and {\\em Lower} panels correspond to cases of halo masses of $M_{\\rm halo}=10^{9}$M$_\\odot$ and  $M_{\\rm halo}=10^{10}$M$_\\odot$. The larger value corresponds to the lower end of the inferred halo masses of the Lyman-break population at $z\\ga6$ \\citep[][]{trenti2010}. \n\n\\subsection{Disk structure}\n \\label{structure}\n\nBefore proceeding we first note that we have so far assumed the thickness of the gaseous disk at the centre of the galaxy to be set by the sound speed of gas at $10^4$K. However from equation~(\\ref{ratio}) we see a strong dependence on the value for the effective sound speed. For example, in turbulent disks, the turbulent velocity replaces the isothermal velocity in determining the effective sound speed. Recently, \\citet[][]{Genzel2010} inferred a Toomre's $Q$ value of $Q=1$ in ULIRGS, implying a value for the turbulent velocity of $c_{\\rm T}\\sim G \\Sigma\/\\Omega=\\sqrt{G\\Sigma_0 r\/\\pi}$, where $v^2 = G\\Sigma_0 \\pi r^2 \/ r $,  yielding $\\Omega = v\/r = \\sqrt{G\\Sigma_0\\pi\/r}$. Evaluating at the Bondi radius,  we get\n\\begin{equation}\nc_{\\rm T} = \\left(\\frac{G^2\\Sigma_0 M_{\\rm bh}}{\\pi}\\right)^{0.25}.\n\\end{equation}\nThis value of $c_{\\rm T}$ is the maximum value possible for a disk at large radius (as a higher $c_{\\rm T}$ would imply an unphysical disk with $h>r$). Therefore, for a $Q=1$ disk the turbulent velocity should decrease towards small $r$. At sufficiently small radii, $c_{\\rm T}<c_{\\rm s}$, implying that the minimum thickness of the disk at its centre is set by the sound speed in the gas. Thus, although the turbulent velocity can be significant at large radii in the disk, we find that when evaluated at the Bondi radius, the turbulent velocity is smaller than the sound speed unless the halo is very massive and at high redshift ($\\ga10^{11}$ at $z>4$). \n\nThere may also be sources of energy injection that heat the gas to temperatures that correspond to velocities much larger than $c_{\\rm T}$ (or $c_{\\rm s}$), although in order to remain bound the velocity of the gas must be smaller than the virial velocity of the halo. We therefore show two cases in addition to $c_{\\rm s}\\sim10$km\/s, namely $c_{\\rm T}$ and $c_{\\rm v}\\sim f v_{\\rm vir}$ where $f\\sim0.5$ in order to bracket the range of interest, with corresponding limiting accretion rates plotted in the central and right hand panels of Figure~\\ref{fig1} (dark lines). The results are almost unchanged where $c_{\\rm T}$ is used to set the disk height. This is because we find that for the halo masses, black-hole masses and redshifts of interest, the sound speed is smaller than the turbulent velocity when evaluated at the Bondi-radius. However if the disk height is set by the maximum velocity $v_{\\rm c}$, then larger black-holes and smaller halos are needed in order for the Bondi accretion rate to exceed the limiting rate, and at $z\\sim6$ black-hole accretion would not be obscured in this case. In the remainder of this paper we restrict our attention to the case of an isothermal disk.\n\n\n\\section{Photon Trapping}\n\\label{results}\n\nIn cases where the Bondi accretion rate is larger than the critical accretion rate, rest frame optical\/UV photons are trapped and the active galactic nucleus (AGN) can be obscured. Thus the cross-over of the limiting and Bondi accretion rate curves in Figure~\\ref{fig1} represents the redshift beyond which accretion traps radiation for an AGN powered by the particular black-hole masses shown. If the vertical disk structure is set by the sound speed, we find that a $10^3$M$_\\odot$ black-hole in a $10^{10}$M$_\\odot$ halo will result in trapped radiation at $z>4$. In this section we discuss the range of halo and black-hole masses that result in photon trapping.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=17.5cm]{fig2.pdf}\n\\caption{\\label{fig2}  Contours of the redshift at which the Bondi accretion rate becomes larger than the critical accretion rate as a function of black-hole and halo mass. We show contours for cases where the vertical structure of the disk is assumed to be set by the sound speed of the gas. In the {\\em Left}, {\\em Central} and {\\em Right} panels we show contours for parameter combinations $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=1$, $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=0.3$ and $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=3$. For illustration, the grey regions show the portion of parameter space at $z\\sim6$ that does not result in photon trapping accretion flows. The example assumes $\\lambda=0.05$ and $m_{\\rm d}=0.17$.}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Scaling relations}\n\nThe conditions for the halo mass, black-hole mass, and redshift that conspire to provide accretion rates that trap the optical\/UV radiation can be obtained by combining equations~(\\ref{bondi}) and (\\ref{lim}). Evaluating we find\n\\begin{eqnarray}\n\\label{ratio}\n\\nonumber\n&&\\hspace{-7mm}\\frac{\\dot{M}_{\\rm bondi}}{\\dot{M}_{\\rm lim}} \\sim  37.5\\left(\\frac{M_{\\rm bh}}{10^5\\mbox{M}_\\odot}\\right) \\left(\\frac{M_{\\rm halo}}{10^{10}\\mbox{M}_\\odot}\\right)^{2\/3}    \\left(\\frac{1+z}{7}\\right)^4  \\left(\\frac{m_{\\rm d}}{0.17}\\right)^2  \\\\\n&&\\hspace{2mm} \\times \\left(\\frac{F_{\\rm min}}{10^3}\\right)^{-1} \\left(\\frac{c_{\\rm s}}{10\\,\\mbox{km\/s}}\\right)^{-5}\\left(\\frac{F_{\\rm sig}}{100}\\right)\\left(\\frac{\\lambda}{0.05}\\right)^{-4}.\n\\end{eqnarray}\nThis expression makes explicit the dependencies that lead to photons being more easily trapped (i.e. $\\dot{M}_{\\rm bondi}\/\\dot{M}_{\\rm lim}>1$), namely larger black-holes in larger halos, and at higher redshift. We note that the condition for photon trapping has a much steeper dependence on redshift than obscuration via dust absorption, which scales as $(1+z)^2$.\n\nThis critical redshift at which the Bondi accretion rate exceeds the critical rate for photon trapping is shown graphically in Figure~\\ref{fig2}. Here we  plot contours of the redshift at which the Bondi accretion rate becomes larger than the critical accretion rate, as a function of halo and black-hole mass. This figure shows the mass combinations that give a cross over from AGN in which radiation can escape to those in which the radiation is trapped. Based on equation~(\\ref{ratio}) we plot contours for the parameter combinations $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=1$, $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=0.3$ and $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=3$ in the {\\em Left}, {\\em Central} and {\\em Right} panels of Figure~\\ref{fig2}. Smaller (larger) values of this combination lead to larger (smaller) inflow rates being needed to trap the radiation, and so larger (smaller) black-holes are needed to trap the radiation at fixed halo mass and redshift (see equation~\\ref{ratio}). \n\nFrom equation~(\\ref{lim2}), the relation between the limiting rate for photon trapping and the Eddington rate can be illustrated  via the condition \n\\begin{equation}\n\\label{ratio2}\n\\frac{\\dot{M}_{\\rm lim}}{\\dot{M}_{\\rm Edd}} \\sim 2 \\left(\\frac{\\epsilon}{0.1}\\right)\\left(\\frac{F_{\\rm sig}}{10^2}\\right)^{-1} \\left(\\frac{F_{\\rm min}}{10^3}\\right).\n\\end{equation}\nAn accretion rate that leads to photon trapping is likely to be in excess of the Eddington rate, and hence associated with a very rapid build-up of black-hole mass. \n\n \n\n\\subsection{Trapping of X-rays}\n\\label{xray}\n\nUp until now we have found that the critical rate at which the accretion flow traps optical\/UV photons can be reached in the centres of high redshift galaxies owing to gas with large opacity (i.e. $F_{\\rm sig}\\gg1$) beyond a radius $F_{\\rm min}r_{\\rm g}$. The results of \\citet[][]{Laor1993} indicate that while $F_{\\rm sig}\\gg1$ is expected for UV photons, the X-ray component of the spectrum will see an opacity to the inflowing gas that is set by the Thomson cross-section (i.e. $F_{\\rm sig,X}=1$). Moreover, if the X-ray photon component of the spectrum were not trapped, it could halt the accretion flow via radiation pressure if it exceeded the Eddington rate by itself, thus preventing trapping of the optical\/UV radiation. On the other hand, X-rays see this Thomson opacity at radii much smaller than the sublimation radius, with $F_{\\rm min,X}\\sim10$. \n\nTo illustrate the importance of X-rays in this context we modify equation~(\\ref{ratio2}) describing the relation between the limiting rate for optical\/UV photon trapping and the Eddington rate that is specific to the X-ray portion of the spectrum ($\\dot{M}_{\\rm Edd,X}$). We define $F_{\\rm X}$ to be the fraction of the luminosity in X-rays, which based on the spectrum in \\citet[][]{Elvis1994} takes a value of $F_{\\rm X}\\sim10\\%$. We therefore find \n\\begin{equation}\n\\label{ratio3}\n\\frac{\\dot{M}_{\\rm lim}}{\\dot{M}_{\\rm Edd,X}} \\sim 0.2 \\left(\\frac{\\epsilon}{0.1}\\right)\\left(\\frac{F_{\\rm sig}}{10^2}\\right)^{-1} \\left(\\frac{F_{\\rm min}}{10^3}\\right)\\left(\\frac{F_{\\rm X}}{0.1}\\right).\n\\end{equation}\nHere $F_{\\rm sig}\\gg1$ and $F_{\\rm min}\\gg1$ correspond to the values for UV photon trapping. Equation~(\\ref{ratio3}) shows that in order for X-rays to exceed the Eddington limit, the parameters corresponding to optical\/UV trapping need to be $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}\\ga5$. At these large accretion rates we find that when optical\/UV photons are trapped, so too are the X-rays unless the minimum radius at which the X-rays encounter opacity ($r_{\\rm min,X}=F_{\\rm min,X}\\,r_{\\rm g}$) is described by a value of $F_{\\rm min,X}>50$. To see this we note that the ratio of the Bondi accretion rate to the rate needed to trap X-rays is \n\\begin{equation}\n\\frac{\\dot{M}_{\\rm Bondi}}{\\dot{M}_{\\rm lim,X}}=\\left(\\frac{F_{\\rm min}}{F_{\\rm min,X}}\\right)\\left(\\frac{1}{F_{\\rm sig}}\\right)\\frac{\\dot{M}_{\\rm Bondi}}{\\dot{M}_{\\rm lim}}.\n\\end{equation}\nSince we expect $F_{\\rm min,X}\\sim6$, corresponding to the innermost stable circular orbit, we expect that an accretion flow which traps the optical\/UV radiation will also trap the X-rays. As a result we do not consider the effect of X-rays on the accretion flow for the remainer of this paper.\n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=17.5cm]{fig3.pdf}\n\\caption{\\label{fig3_new} Regions of black-hole -- halo mass parameter space which result in photon trapping. {\\em Left:} Contours of $\\lambda_{\\rm c}$ as a function of black-hole and halo mass. The grey shading shows the region of parameter space which does not result in photon trapping for the mean disk, because the critical spin parameter is smaller than $\\lambda=0.05$. {\\em Center:} Contours of $\\lambda_{\\rm c,B}$ as a function of black-hole and halo mass. The grey shading shows the region of parameter space which does not result in photon trapping for the mean disk, because the Bondi radius is larger than the scale height at the disk centre. {\\em Right:} Contours of the probability that accretion will result in photon trapping, corresponding to the spin parameter $\\lambda$ lying between the two critical values, i.e. $\\lambda_{\\rm c,B}<\\lambda<\\lambda_{\\rm c}$. To calculate the probability we assume the distribution is Gaussian in the natural logarithm $\\ln{\\lambda}$, with variance $\\sigma_\\lambda=0.5$ and a mean at $\\lambda=0.05$. Contours are shown for $P= 1\\%$,  $P= 10\\%$ and  $P= 50\\%$. The examples assume $z=6$, $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=1$, and $m_{\\rm d}=0.17$. \n}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Critical spin parameter}\n\n\nEquation~(\\ref{ratio}) shows that the trapping of photons is very sensitive to the value of the spin parameter $\\lambda$ which governs the density of the galactic disk. In particular, photons may be more easily trapped within disks of low spin parameter. The assembly of dark matter halos leads to a distribution of spin parameters. To illustrate the parameter space we recast equation~(\\ref{ratio}). Photon trapping at all wavelengths occurs for spin parameters $\\lambda<\\lambda_{\\rm c}$ where\n\\begin{eqnarray}\n\\label{ratio4}\n\\nonumber\n&&\\hspace{-7mm}\\lambda_{\\rm c} \\sim 0.12 \\left(\\frac{M_{\\rm bh}}{10^5\\mbox{M}_\\odot}\\right)^{1\/4} \\left(\\frac{M_{\\rm halo}}{10^{10}\\mbox{M}_\\odot}\\right)^{1\/6}    \\left(\\frac{1+z}{7}\\right)  \\left(\\frac{m_{\\rm d}}{0.17}\\right)^{1\/2}  \\\\\n&&\\hspace{7mm} \\times \\left(\\frac{F_{\\rm min}}{10^3}\\right)^{-1\/4} \\left(\\frac{c_{\\rm s}}{10\\,\\mbox{km\/s}}\\right)^{-5\/4} \\left(\\frac{F_{\\rm sig}}{10^2}\\right)^{1\/4} ,\n\\end{eqnarray}\n\nIn the left panel of Figure~\\ref{fig3_new} we plot contours of $\\lambda_{\\rm c}$ as a function of black-hole and halo mass at $z=6$. The example assumes $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=1$, and $m_{\\rm d}=0.17$. The mean spin parameter $\\lambda=0.05$ is shown in black, and the grey shaded region illustrates the region of parameter space where trapping is not possible for a mean disk. Figure~\\ref{fig3_new} shows that emergent photons from black-holes with $M_{\\rm bh}\\ga10^{3-4}$M$_\\odot$ will be trapped at all wavelengths in the centres of high redshift galaxies ($\\sim10^{8-10}$M$_\\odot$) that are at the mean of the spin-parameter distribution . \n\n\n\n\\subsection{Breakout of the Bondi radius}\n\\label{breakout}\n\nIn the previous sections we have illustrated that the large densities expected at the centres of high redshift galaxies lead to conditions where accretion rates may be sufficient to trap photons within the accretion flow. These calculations are based on the density at the centre of a pressure supported self-gravitating disk. In this section, we point out that the calculation applies only to black-hole masses for which the Bondi radius is smaller than the scale-height at the disk centre. Moreover, we note that the trapping of radiation cannot be realised once the black-hole grows sufficiently that its Bondi radius exceeds the scale height. \n\nWe again utilise the self-gravitating disk model. The scale height at the disk centre is \n\\begin{equation}\n\\label{height}\nz_0 = \\frac{c_{\\rm s}^2}{\\pi G \\Sigma_0}.\n\\end{equation}\nThis expression ignores the gravitational contribution from the black-hole which would serve to reduce the scale height, and is valid when $r_{\\rm Bondi}< z_0$. \nPhoton trapping and obscuration are only possible in this regime. \nWe therefore calculate the ratio of Bondi radius to the central disk scale height as\n\\begin{equation}\n\\frac{r_{\\rm Bondi}}{z_0} = \\frac{G^2}{c_{\\rm s}^4}\\frac{M_{\\rm bh}m_{\\rm d}M}{\\lambda^2R_{\\rm vir}^2}.\n\\end{equation}\nPutting in characteristic values we obtain\n\\begin{eqnarray}\n\\label{ratio5}\n\\nonumber\\frac{r_{\\rm Bondi}}{z_0} &=& 2.6 \\left(\\frac{M_{\\rm bh}}{10^5\\mbox{M}_\\odot}\\right) \\left(\\frac{M_{\\rm halo}}{10^{10}\\mbox{M}_\\odot}\\right)^{1\/3}    \\left(\\frac{1+z}{7}\\right)^2  \\\\\n&&\\hspace{9mm}\\times\\left(\\frac{m_{\\rm d}}{0.17}\\right)  \\left(\\frac{c_{\\rm s}}{10\\,\\mbox{km\/s}}\\right)^{-4}\\left(\\frac{\\lambda}{0.05}\\right)^{-2}.\n\\end{eqnarray}\n\nThus, within $M\\sim10^{10}$M$_\\odot$ halos at $z\\sim6$, black-holes in excess of $M_{\\rm bh}\\sim10^5$M$_\\odot$ have Bondi radii which are larger than the disk scale-height and so may not be obscured. We note that the ratio is very sensitive to the value of the sound speed, with a value larger than the fiducial $10\\,$km$\\,$s$^{-1}$ significantly reducing the ratio, allowing larger black-holes to accrete in the photon trapping mode. A small gas fraction also reduces the ratio. However the breakout of the Bondi radius implies that high mass black-holes such as those observed within the SDSS quasars \\citep[e.g.][]{Fan2001,Fan2003} could not have their emergent radiation trapped.\n\nTo better understand the constraints imposed on black-hole masses where photon trapping can occur, we evaluate the critical value of spin parameter $\\lambda_{\\rm c,B}$ at which $r_{\\rm Bondi}=z_0$\n\\begin{eqnarray}\n\\label{lambda_critB}\n\\nonumber\n&&\\hspace{-7mm}\\lambda_{\\rm c,B} \\sim 0.08 \\left(\\frac{M_{\\rm bh}}{10^5\\mbox{M}_\\odot}\\right)^{\\frac{1}{2}} \\left(\\frac{M_{\\rm halo}}{10^{10}\\mbox{M}_\\odot}\\right)^{\\frac{1}{6}}    \\left(\\frac{1+z}{7}\\right)  \\left(\\frac{m_{\\rm d}}{0.17}\\right)^{\\frac{1}{2}}  \\\\\n&&\\hspace{10mm} \\times  \\left(\\frac{c_{\\rm s}}{10\\,\\mbox{km\/s}}\\right)^{1\/2}.\n\\end{eqnarray}\nIn the central panel of Figure~\\ref{fig3_new} we plot contours of $\\lambda_{\\rm c,B}$ as a function of black-hole and halo mass at $z=6$. The example again assumes $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=1$, and $m_{\\rm d}=0.17$. The mean spin parameter $\\lambda=0.05$ is shown in black, and the grey shading illustrates the region of parameter space where photon trapping is not possible for the mean disk because the Bondi radius is larger than the scale-height. Figure~\\ref{fig3_new} shows that black-holes with $M_{\\rm bh}\\ga10^{4.5}$M$_\\odot$ within high redshift galaxies ($\\sim10^{10}$M$_\\odot$) having the mean of the spin-parameter cannot form a photon trapping accretion flow. \n\n\n\\subsection{When and where could photon trapping occur?}\n\nThe right panel of Figure~\\ref{fig3_new} shows the probabilities that the spin parameter $\\lambda$ lies between the two critical values needed for the conditions of $i)$ photon trapping, and $ii)$ a Bondi radius that is contained within the disk scale height (i.e. $\\lambda_{\\rm c,B}<\\lambda<\\lambda_{\\rm c}$). To calculate this probability\n\\begin{equation}\nP = \\frac{1}{\\sqrt{2\\pi}\\sigma_\\lambda}\\int_{\\lambda_{\\rm c,B}}^{\\lambda_{\\rm c}}  \\exp{\\left[-\\frac{(\\ln{\\lambda}-\\ln{\\bar{\\lambda}})^2}{2\\sigma_\\lambda^2}\\right]}\\,d\\lambda\n\\end{equation}\n we assume the distribution of spin parameters is Gaussian in the natural logarithm $\\ln{\\lambda}$, with variance $\\sigma_\\lambda=0.5$ and a mean at $\\bar{\\lambda}=0.05$ \\citep[][]{Mo1998}. The probability is $P=0$ if $\\lambda_{\\rm c}<\\lambda_{\\rm c,B}$, indicating a black-hole -- halo mass combination that cannot produce a photon trapping accretion flow. Contours are shown that represent black-hole -- halo mass combinations for which $P= 1\\%$,  $P= 10\\%$ and  $P= 50\\%$ of disks would have densities that result in photon trapping. As before, the example assumes $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=1$, and $m_{\\rm d}=0.17$. The mean disk at $z\\sim6$ has a central density that leads to photon trapping for black-hole masses up to $M_{\\rm bh}\\sim10^5$M$_\\odot$ within halos up to $M_{\\rm halo}\\sim10^{9}$M$_\\odot$. However $\\sim10\\%$ of disks are dense enough that photon trapping will occur for black-hole masses up to $M_{\\rm bh}\\sim10^5$M$_\\odot$ within larger halos up to $M_{\\rm halo}\\sim10^{11}$M$_\\odot$. Thus, we would expect photon trapping to be common in galaxies hosting $M_{\\rm bh}\\la10^5$M$_\\odot$ black-holes at $z\\sim6$. Moreover, since from equation~(\\ref{ratio}) we see that this growth is super-Eddington, we find that photon trapping provides a mechanism by which rapid black-hole growth could proceed at high redshift, helping to explain how super-massive black-holes grow less than a billion years after the Big-Bang.\n\nIn Figure~\\ref{fig4_new} we explore how the conclusions regarding the black-hole and halo mass-ranges that produce photon trapping accretion flows are effected by redshift and the parameter combination  $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}$. We choose three values of redshift, $z=1$, $z=6$ and $z=10$. For $z=6$ and $z=10$ we choose $m_{\\rm d}=0.17$. However at later times we expect that gas is less plentiful and so assume $m_{\\rm d}=0.025$ at $z\\sim1$, corresponding to a comparison with low redshift disks \\citep[][]{Mo1998}. These cases are shown in the upper central and lower rows respectively. In each case we show examples with $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=0.3$, $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=1$ and $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=3$ ({\\em Left}, {\\em Central} and {\\em Right} panels respectively). In all panels we show contours for the probabilities that the spin parameter $\\lambda$ lies between the two critical values needed for the conditions of $i)$ photon trapping, and $ii)$ a Bondi radius that is contained within the disk scale height (i.e. $\\lambda_{\\rm c,B}<\\lambda<\\lambda_{\\rm c}$). As before contours are shown that represent black-hole -- halo mass combinations for which $P= 1\\%$,  $P= 10\\%$ and  $P= 50\\%$ of disks would have central densities that lead to photon trapping.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=17.5cm]{fig4.pdf}\n\\caption{\\label{fig4_new}  Contours of the probability  in black-hole -- halo mass parameter space that accretion will result in photon trapping, corresponding to the spin parameter $\\lambda$ lying between the two critical values, i.e. $\\lambda_{\\rm c,B}<\\lambda<\\lambda_{\\rm c}$. Contours are shown for $P= 1\\%$,  $P= 10\\%$ and  $P= 50\\%$. Examples are shown for three values of redshift, $z=1$, $z=6$ and $z=10$ ({\\em Top} to {\\em Bottom}). For $z=6$ and $z=10$ we choose $m_{\\rm d}=0.17$. We assume $m_{\\rm d}=0.025$ at $z\\sim1$.  In each case we show examples with $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=0.3$, $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=1$ and $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}=3$ (from {\\em Left} to {\\em Right}). The thick grey curve is the predicted value for black-hole mass halo mass relation (equation~\\ref{relation}).}\n\\end{center}\n\\end{figure*}\n\n\nWe find that larger values of $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}$ lead to more massive black-holes in smaller halos having their emergent radiation trapped. At redshift $z\\sim6-10$ we find that black-holes with masses up to $M_{\\rm bh}\\sim10^5$M$_\\odot$ have their radiation trapped in 10\\%-50\\% of cases. At lower redshift,  black-holes with masses of $M_{\\rm bh}\\sim10^{5.5}$M$_\\odot$ could have radiation trapped in up to 10\\% of cases, but only if the halo mass and  $(F_{\\rm sig}\/10^2)(F_{\\rm min}\/10^3)^{-1}$ are large. Thus, photon trapping is likely to be a phenomenon dominated by low mass ($M_{\\rm bh}\\la10^5$M$_\\odot$) black-holes in high redshift ($z\\ga6$) galaxies ($M_{\\rm halo}\\sim10^{10}$M$_\\odot$).\n\n\\subsection{Comparison with $M_{\\rm bh}-M_{\\rm halo}$ models}\n\nAs noted in the introduction, the relations observed between black-hole mass and galaxy properties in the local Universe may not be in place at high redshift. For this reason we have not imposed a model for the relation between black-hole and halo mass in this paper, and instead have explored a range of values. However it is interesting to compare the range of black-hole masses found to be accreting in the photon trapping mode with expectations of the simple models relating black-hole and halo masses that have been successful in describing some of the properties of high redshift quasars. \n\nMotivated by local observations \\citep[][]{Ferrarese2002}, we consider a model in which the central black-hole mass is correlated with the halo\ncircular velocity. This scenario is supported by the results of \\citet[][]{Shields2003} who studied quasars out to $z\\sim3$ and demonstrated that the\nrelation between black-hole masses and the stellar velocity dispersion does not\nevolve with redshift. This is expected if the mass of the black-hole is\ndetermined by the depth of the gravitational potential well in which it\nresides, as would be the case if growth is regulated by feedback from\nquasar outflows \\cite[e.g.][]{Silk1998,Wyithe2003b}.  Expressing\nthe halo virial velocity, $v_c$, in terms of the halo mass, $M_{\\rm halo}$, and\nredshift, $z$, the redshift dependent relation between the super-massive black-hole and halo\nmasses may be written as\n\\begin{equation}\n\\label{relation}\nM_{\\rm bh} = \\epsilon_{\\rm bh} M_{\\rm halo} \\left(  \\frac{M_{\\rm halo}}{10^{10}\\mbox{M}_\\odot}\\right)^{2\/3}[\\zeta(z)]^{5\/6} \\left(\\frac{1+z}{7}\\right)^{5\/2}.\n\\end{equation}\nThe normalising constant in this relation has an\nobserved\\footnote{ We have used the normalization derived by\n\\citep[][]{Ferrarese2002} under the simplifying assumption\nthat the virial velocity of the halo\nrepresents its circular velocity.} value of $\\epsilon_{\\rm bh}\\approx 10^{-4.3}$ based on calibration of equation~(\\ref{relation}) at $z=0$ \\citep[][]{Ferrarese2002}, where we take the underlying\nassumption that the halo mass profile resembles a Singular Isothermal Sphere. In \\citet[][]{Wyithe2005} this model was shown to be consistent with the clustering and luminosity function data of the 2dF quasar redshift survey \\citep[][]{Croom2002}. \n\nInterestingly, the predicted value for super-massive black-hole masses in galaxy mass halos (as shown by the thick grey curve in Figure~\\ref{fig4_new}) is comparable to the range at which we would expect photon trapping in galaxies with $z\\ga6$. Thus, prior to obtaining masses where the self regulating mechanisms thought to be responsible for the relations between black-hole mass and halo properties take effect we would expect black-holes in high redshift galaxies to reach the level where super-Eddington accretion would become a natural part of their evolution. However at low redshift, Figure~\\ref{fig4_new} shows that photon trapping requires black-hole masses that are greater than the observed black-hole -- halo mass relation. Thus, we do not expect expect photon trapping within low redshift galaxies. \n\n\n\\section{Discussion}\n\\label{discussion}\n\nThe findings in this paper point to some potentially important implications for the growth of high redshift super-massive black-holes. Recent simulations by \\citet[][]{Li2011} show that self gravity overcomes radiative feedback and that accretion onto intermediate mass black-holes reaches the Eddington rate. However, at high enough accretion rates, a spherical inflow is not subject to the Eddington limit (though the emergent radiation is), and the Bondi accretion rate can take arbitrarily large values in high density galactic centres \\citep[][]{Begelman1979}. The conditions for spherical accretion may only rarely be realised at the centres of high redshift galaxies. On the other hand, there is a class of slim disk models \\citep[e.g.][]{Abramowicz1988,Chen1995,Ohsuga2005,Watarai2006,Ohsuga2011}, for which super-Eddington accretion flows are possible.  These models include a large viscosity, and so rapid transport of material through the disk within a small multiple of the free fall time \\citep[e.g.][]{Watarai2006}, and are optically thick and advection dominated \\citep[][]{Chen1995}. Photon trapping plays an important role within these disks, even those with complex flows, leading to very inefficient energy conversion and a luminosity that is independent of accretion rate \\citep[e.g.][]{Ohsuga2005}. Simulations indicate that at sufficient densities, the mass accretion rate can reach hundreds of Eddington, although the photon luminosity does not \\citep[e.g.][]{Ohsuga2011}. These super-Eddington accretion disks will generate outflows \\citep[which cannot be generated in spherical accretion,][]{Begelman1979} that are likely to be channelled along the poles, and so will not suppress the accretion along the equatorial plane. The calculations presented in this paper provide the boundary condition for super-Eddington flows. These rapid accretion events may provide the seeds for super-massive black-hole growth.\n\n\nA plausible scenario therefore includes two episodes of black-hole growth. Initially the accretion may have been spherical or via a slim disk with high viscosity, so that the Eddington rate was greatly exceeded by the accretion rate. We find that radiation would have been trapped  and advected into the central black-hole by the very large Bondi accretion rates at high redshift. This phase of growth would be obscured at both optical\/UV and X-ray wavelengths. Once the Bondi radius became larger than the scale-height of the galaxy disk, the accretion rate dropped, and higher\nangular momentum gas would have settled into a thin disk accretion mode, in which the\naccretion time was much longer than the free-fall time, allowing the radiation to escape. For a pressure supported disk, we have found that the Bondi radius is expected to exceed the scale-height of the disk for black-holes in excess of $\\sim10^{5}$M$_\\odot$.\nThe luminous quasars discovered at $z\\gsim6$ with black-hole masses in excess of $10^8$M$_\\odot$ are therefore thought to be shining in this later mode, un-obscured in the optical and with accretion rates close to, but smaller than Eddington. However a portion of their prior growth, when the black-hole mass was $\\sim10^{4-5}$M$_\\odot$ would have been in the photon trapping mode.\n\nSuper-Eddington accretion rates arising from the large densities in the cores of high redshift galaxies were previously considered by \\citet[][]{Volonteri2005}. As noted in the introduction, the calculations in \\citet[][]{Volonteri2005} neglected feedback effects like gas heating, which may lower the the Bondi accretion rate \\citep[e.g.][]{Milosavljev2009}.  In this paper we note that if the accretion rate is sufficiently high that the emergent photons are trapped within the accretion flow, then feedback effects cannot operate \\citep[][]{Begelman1979}. Our model also makes two different assumptions which modify our conclusions relative to \\citet[][]{Volonteri2005}. Firstly, \\citet[][]{Volonteri2005} assume that once the gas is enriched, metal line cooling allows the gas to cool to temperatures much lower than $10^4$K, so that it fragments to form stars and the super-Eddington accretion episode is ended. This assumption was necessary in order that super-Eddington accretion not lead to black-hole densities in excess of those observed. However, we expect that fragmentation and associated star-formation will reheat the gas via radiative and supernova feedback so that the pressure support at the centre of the disk is maintained and accretion can continue. This scenario is supported by the observations of \\citet[][]{Genzel2010} which imply significant turbulent velocities in high redshift galaxies. Rather than make an arbitrary assumption that growth stops when the size of the accretion disk grows by a factor of five, we instead assume that the super-Eddington photon trapping accretion mode would be regulated by the time when the Bondi radius exceeds the scale height of the pressure supported disk. We find that this condition prevents super-Eddington accretion onto high mass black-holes.   \n\n\n\n\\subsection{Obscured accretion}\n\nOur results have some relevance to the recent discussion surrounding obscured accretion in high redshift galaxies. While luminous optical quasars in the most massive halos ($\\ga10^{12}$M$_\\odot$) dominate the observations of high redshift super-massive black-holes, \\citet[][]{Treister2011} recently presented evidence that most of the black-hole accretion at $z\\ga6$ is actually optically obscured, and in galaxies below halo masses of $\\sim10^{10-11}$M$_\\odot$. Since high redshift Lyman-break galaxies are thought not to be dusty \\cite[e.g.][]{Bouwens2010d}, our results might have provided a mechanism by which the AGN can be obscured even in the absence of a large dusty component. However our results do not support a photon trapping explanation for this result. Firstly, X-ray photons would be more strongly trapped by the accretion flow than the UV photons, indicating that we would not expect to observe X-rays without optical detection. In addition, we also find that photon trapping is only expected in a fraction of galaxies rather than in all galaxies as implied by \\citet[][]{Treister2011}. \n\nWhile other mechanisms may lead to buried accretion, the findings of \\citet[][]{Treister2011} have also been disputed by a number of authors \\citep[][]{Fiore2011,Cowie2011,Willott2011}, who do not observe the same level of X-ray emission. These authors find limits on the average X-ray luminosity in the rest frame 0.5-2keV band of $L_{0.5-2}<4\\times10^{41}\\,$erg\/s for $z\\sim6.5$ dropout galaxies. This luminosity can be related to black-hole mass as\n\\begin{equation}\nL_{0.5-2}\\sim 3\\times10^{41}\\left(\\frac{M_{\\rm bh}}{10^5\\mbox{M}_\\odot}\\right)\\eta\\,\\mbox{erg}\\,\\mbox{s}^{-1},\n\\end{equation}\nwhere $\\eta=1$ is the fraction of the Eddington accretion rate, and we have assumed the spectral energy distribution of \\citet[][]{Elvis1994}. Observed luminosities must have $\\eta<1$ even if the accretion rate is super-Eddington \\citep[][]{Begelman1979}. The observed luminosity limit therefore corresponds to observed black-hole masses of $M_{\\rm bh}\\la1.3\\times 10^5$M$_\\odot$. We do not find that black-holes with masses $M_{\\rm bh}\\ga10^5$M$_\\odot$ produce photon trapping accretion flows, and so photon trapping does not explain the lack of observed X-ray sources among the $z\\sim6.5$ dropouts \\citep[][]{Fiore2011,Cowie2011,Willott2011}. Since the stacked observations in these studies are based on only $\\sim10^2$ galaxies, this lack of detection could follow from the low-duty cycle of AGN \\citep[which is likely a few percent,][]{Wyithe2003b}. However our results do suggest that 90\\%-100\\% of disks with black-holes below this mass would be in the photon trapping mode. Thus, we would expect that deeper and wider field X-ray observations using future X-ray observatories, to display a cut-off in the X-ray luminosity function at about $L_{0.5-2}\\sim 3\\times10^{41}$erg$\\,$s$^{-1}$. Conversely, the discovery of X-ray AGN with luminosities an order of magnitude lower than current limits would therefore rule out photon trapping accretion as a mechanism for rapid growth of early black-holes. Finally, we note that since super-Eddington accretion at high redshift is obscured at both optical and X-ray wavelengths, rapid growth of seed black-holes could not provide a significant source of X-ray for reionization of the IGM \\citep[][]{Volonteri2005}.\n\n\n\\subsection{Seed black-hole growth}\n\nThe photon trapping mode is  likely to be important for the rapid growth of seed super-massive black-holes with masses of $\\sim10^{4-5}$M$_\\odot$. Because the Bondi accretion rates in these high redshift galaxies could be  orders of magnitude larger than the Eddington rate, the photon trapping mechanism helps alleviate the difficulty of growing super-massive black-holes of more than a billion solar masses \\citep[corresponding to the most distant known quasars,][]{Mortlock2011} within the first billion years of the Universe's age. This point was made in detail in \\citet[][]{Volonteri2005}. To illustrate we note that accretion at the Eddington rate (with $\\epsilon=0.1)$ leads to an $e$-folding time of $t=4\\times10^7$ years. Assuming that a black-hole accretes with a duty cycle of unity, the number of e-folding times available by $z\\sim7$ is therefore $\\sim20$. This should be compared with the 20 $e$-folds needed to grow a $1$M$_\\odot$ black-hole seed up to a mass of $10^9$M$_\\odot$.  Thus there is only just enough time during the age of the Universe at $z\\ga6$ for a stellar black-hole seed to grow to a super-massive black-hole. We suggest that the period of obscured growth in some galaxies would provide a path toward growing these super-massive black-holes. \n\n\n\n\n\\section{Conclusion}\n\\label{conclusion}\n\nIn this paper we have determined the cosmological regime in which photons produced through accretion onto a central black-hole are trapped by infalling material, so that $i)$ radiation feedback on the infall of gas outside the dust sublimation radius is suppressed, allowing accretion rates far in excess of the Eddington limit, $ii)$ AGN appear obscured, and $iii)$ the growth time of black-hole is short. Specifically we find that a large fraction of galaxies at $z\\ga6$ with masses up to those of the observed Ly-break population (halo masses of $\\sim10^{9-11}$M$_\\odot$) exhibit Bondi-accretion rates onto $M_{\\rm bh}\\sim10^{3-5}$M$_\\odot$ black-holes that are sufficiently high to trap the resulting rest frame optical\/UV\/X-ray radiation. The obscuration due to photon trapping is found only to occur for black-holes with masses up to $\\sim10^5$M$_\\odot$ because larger black-holes have Bondi radii that exceed the scale height of the disk from which they accrete gas, so that a photon trapping mechanism cannot operate. As a result, we find a natural distinction between obscured, photon trapping accretion onto $\\sim10^5$M$_\\odot$ black-holes in galaxies of halo mass $\\la10^{10}$M$_\\odot$, and the luminous accretion seen in the brightest quasars with black-hole masses of $\\sim10^{8-9}$M$_\\odot$ within halos of mass $\\sim10^{11-12}$M$_\\odot$. At lower redshift photon trapping requires black-holes that are larger than expected from the black-hole -- halo mass relation, and so is not expected to be observed. Our results indicate that super-Eddington accretion of mass to form seed black-holes of $\\sim10^5$M$_\\odot$ provided a mechanism by which super-massive black-holes were able to form prior to $z\\sim6$. \n\n\\vspace{5mm}\n\n{\\bf Acknowledgments} JSBW acknowledges\nthe support of the Australian Research Council.  AL was supported in\npart by NSF grant AST-0907890 and NASA grants NNX08AL43G and\nNNA09DB30A.\n\n\n\n\\newcommand{\\noopsort}[1]{}\n\n\\bibliographystyle{mn2e}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn \\cite{k1937}, Kolmogorov posed the problem of characterizing the subset of the complex plane, denoted by $\\Theta_n$, that consists of the individual eigenvalues of all $n$-by-$n$ stochastic matrices. \n\nOne can easily verify that for each $n \\geq 2$, the region $\\Theta_n$ is closed, inscribed in the unit-disc, star-convex (with star-centers at zero and one), and symmetric with respect to the real-axis. Furthermore, it is clear that $\\Theta_n \\subseteq \\Theta_{n+1}$, $\\forall n \\in \\bb{N}$. In view of these properties, $\\partial \\Theta_n = \\{ \\lambda \\in \\Theta_n : \\alpha \\lambda \\not \\in \\Theta_n,\\forall \\alpha > 1\\}$, and each region is determined by its boundary.   \n\nDmitriev and Dynkin \\cite{dd1946} obtained a partial solution to Kolmogorov's problem, and Karpelevi{\\v{c}}~\\cite[Theorem B]{k1951}, expanding on the work of \\cite{dd1946}, resolved it by showing that the boundary of $\\Theta_n$ consists of curvilinear arcs (herein, \\emph{Karpelevi{\\v{c}}~arcs} or K-arcs), whose points satisfy a polynomial equation that is determined by the endpoints of the arc (which are consecutive roots of unity). {\\DJ}okovi{\\'c} \\cite[Theorem 4.5]{d1990} and Ito \\cite[Theorem 2]{i1997} each provide a simplification of this result. However, noticably absent in the Karpelevi{\\v{c}}~Theorem (and the above-mentioned works) are \\emph{realizing-matrices} (i.e., a matrix whose spectrum contains a given point) for points on these arcs. \n\nThis problem has been addressed previously in the literature. Dmitriev and Dynkin \\cite[Basic Theorem]{dd1946} give a schematic description of such matrices for points on the boundary of $\\Theta_n \\backslash \\Theta_{n-1}$ and Swift \\cite[\\S 2.2.2]{s1972} provides such matrices for $3\\leq n \\leq 5$.\n\nOur main result is providing, for every $n$ and for each arc, a single parametric matrix that realizes the entire $K$-arc as the parameter runs from 0 to 1. Aside from the theoretical importance -- after all, the original problem posed by Kolmogorov is intrinsically matricial -- possession of such matrices is instrumental in the study of nonreal \\emph{Perron similarities} in the longstanding \\emph{nonnegative inverse eigenvalue problem} \\cite{jp2017}, and provides a framework for resolving Conjecture 1 \\cite{lpk2015} vis-\\`{a}-vis the results in \\cite{j1981}. \n\nIn addition, we provide some partial results on the differentiability of the Karpelevi{\\v{c} arcs. We demonstrate that some powers of certain realizing-matrices realize other arcs. Finally, we pose several problems that appeal to a wide variety of mathematical interests.\n\n\\section{Notation \\& Background}\n\nThe algebra of complex (real) $n$-by-$n$ matrices is denoted by $\\mat{n}{\\bb{C}}$ ($\\mat{n}{\\bb{R}}$). A real matrix is called \\emph{nonnegative} (\\emph{positive}) if it is an entrywise nonnegative (positive) matrix. If $A$ is nonnegative (positive), then we write $A \\geq 0$ ($A > 0$). \n\nAn $n$-by-$n$ nonnegative matrix $A$ is called \\emph{(row) stochastic} if every row sums to unity; \\emph{column stochastic} if every column sums to unity; and \\emph{doubly stochastic} if it is row stochastic and column stochastic.\n\nGiven $n \\in \\bb{N}$, the set ${F}_n := \\{ p\/q : 0\\leq p < q \\leq n,~\\gcd(p,q)=1 \\}$ is called the \\emph{set of Farey fractions of order n}. If $p\/q$, $r\/s$ are elements of ${F}_n$ such that $p\/q < r\/s$, then $(p\/q,r\/s)$ is called a \\emph{Farey pair (of order $n$)} if $x \\not\\in {F}_n$ whenever $p\/q < x < r\/s$. The Farey fractions $p\/q$ and $r\/s$ are called \\emph{Farey neighbors} if $(p\/q,r\/s)$ or $(r\/s, p\/q)$ is a Farey pair.\n\nThe following is the celebrated Karpelevi{\\v{c}}~Theorem in a form due to Ito \\cite{i1997}. \n\n\\begin{thm}[Karpelevi{\\v{c}}] \n\\label{thm:karpito}\nThe region $\\Theta_n$ is symmetric with respect to the real axis, is included in the unit-disc $\\{ z \\in \\bb{C} : |z| \\leq 1\\}$, and intersects the unit-circle $\\{ z \\in \\bb{C} : |z| = 1\\}$ at the points $\\{ e^{2\\pi\\ii p\/q} : p\/q \\in {F}_n \\}$. The boundary of $\\Theta_n$ consists of these points and of curvilinear arcs connecting them in circular order. \n\nLet the endpoints of an arc be $e^{2\\pi\\ii p\/q}$ and $e^{2\\pi\\ii r\/s}$ ($q < s$). Each of these arcs is given by the following parametric equation:  \n\\begin{equation}\nt^{s} \\left( t^{q} - \\beta \\right)^{\\floor{n\/q}} = \\alpha^{\\floor{n\/q}} t^{q\\floor{n\/q}},~\\alpha \\in [0,1], ~\\beta:=1-\\alpha \\label{ito_eq}.\n\\end{equation} \n\\end{thm}\n\n\\hyp{Figure}{fig:karpregions} contains the regions $\\Theta_3$, $\\Theta_4$, and $\\Theta_5$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.32\\textwidth}\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\naxis lines=none,\naxis equal image,\nscale=0.32,\nxlabel={$\\Re{(\\lambda)}$},\nylabel={$\\Im{(\\lambda)}$},\nylabel style={rotate=-90,, anchor=north},\nxmin=-1,\nxmax=1,\nymin=-1.0,\nymax=1.0,\nxtick={-1,1},\nytick={-1,1}\n]\n\n\\addplot[thick,black] coordinates{\n(1,0) \n(-.5,.866025403784439) \n(-.5,-.866025403784439)\n(1,0)}; \n\\addplot[thick,black] coordinates{(-1,0) (-.5,0)};                              \t\t\t\n\n\\draw[color=gray] (axis cs:0,0) circle (1);\n\\end{axis}\n\\end{tikzpicture}\n\\caption{$\\Theta_3$}\n\\label{fig:karpregion3}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}{.32\\textwidth}\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\naxis lines=none,\naxis equal image,\nscale=0.32,\nxlabel={$\\Re{(\\lambda)}$},\nylabel={$\\Im{(\\lambda)}$},\nylabel style={rotate=-90, anchor=north},\nxmin=-1,\nxmax=1,\nymin=-1.0,\nymax=1.0,\nxtick={-1,1},\nytick={-1,1}\n]\n\n\\addplot[thick,black] coordinates{(1,0) (0,1)};                              \n\\addplot[thick,black] coordinates{(1,0) (0,-1)};                             \n\\addplot[thick,black] table {aarc1a.dat};                                    \n\\addplot[thick,black] table {aarc1b.dat};                                    \n\\addplot[thick,black] table {aarc2a.dat};                                    \n\\addplot[thick,black] table {aarc2b.dat};                                    \n\n\\draw[color=gray] (axis cs:0,0) circle (1);\n\\end{axis}\n\\end{tikzpicture}\n\\caption{$\\Theta_4$}\n\\label{fig:karpregion4}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}{.32\\textwidth}\\centering\n\\begin{tikzpicture}\n\\begin{axis}[\naxis lines=none,\naxis equal image,\nscale=0.32,\nxlabel={$\\Re{(\\lambda)}$},\nylabel={$\\Im{(\\lambda)}$},\nylabel style={rotate=-90, anchor=north},\nxmin=-1,\nxmax=1,\nymin=-1.0,\nymax=1.0,\nxtick={-1,1},\nytick={-1,1}\n]\n\n\\addplot[thick,black] coordinates{(1,0) (.309016994374947,.951056516295154)};                              \n\\addplot[thick,black] coordinates{(1,0) (.309016994374947,-.951056516295154)};                              \n\\addplot[thick,black] table {fivearcs1.dat};                                     \t\t\t\t\t\n\\addplot[thick,black] table {fivearcs2.dat};                                    \n\\addplot[thick,black] table {fivearcs3.dat};                                    \n\\addplot[thick,black] table {fivearcs4.dat};                                    \n\\addplot[thick,black] table {fivearcs5.dat};                                    \n\\addplot[thick,black] table {fivearcs6.dat};                                    \n\\addplot[thick,black] table {aarc1a.dat};                                    \n\\addplot[thick,black] table {aarc1b.dat};                                    \n\n\\draw[color=gray] (axis cs:0,0) circle (1);\n\\end{axis}\n\\end{tikzpicture}\n\\caption{$\\Theta_5$}\n\\label{fig:karpregion5}\n\\end{subfigure}\n\\caption{$\\Theta_n$, $3 \\leq n \\leq 5$}\n\\label{fig:karpregions}\n\\end{figure}\n\nFor $n \\in \\bb{N}$, we call the collection of such arcs \\emph{the K-arcs (of order $n$)} and we denote by $K(p\/q,r\/s) = K_n(p\/q,r\/s)$ the arc connecting $e^{2 \\pi \\ii p \/q}$ and $e^{2 \\pi \\ii r \/s}$, when $p\/q$ and $r\/s$ are Farey neighbors. Notice that the number of K-arcs equals $| F_n| = 1 + \\sum_{k=1}^n \\phi(k)$, where $\\phi$ denotes \\emph{Euler's totient function}. \n\nFor Farey neighbors $p\/q$ and $r\/s$, $q < s$, we call the collection of equations \\eqref{ito_eq} the \\emph{Ito equations (with respect to $\\{p\/q,r\/s\\}$)} and the collection of polynomials \n\\[ f_\\alpha (t) := t^{s} \\left( t^{q} - \\beta \\right)^{\\floor{n\/q}} - \\alpha^{\\floor{n\/q}} t^{q\\floor{n\/q}},~\\alpha \\in [0,1]\\] \nthe \\emph{Ito polynomials (with respect to $\\{p\/q,r\/s\\}$)}.   \n\nA \\emph{directed graph} (or simply \\emph{digraph}) $\\Gamma = (V,E)$ consists of a finite, nonempty set $V$ of \\emph{vertices}, together with a set $E \\subseteq V \\times V$ of \\emph{arcs}. For $A \\in \\mat{n}{\\bb{C}}$, the \\emph{directed graph} (or simply \\emph{digraph}) of $A$, denoted by $\\Gamma = \\dg{A}$, has vertex set $V = \\{ 1, \\dots, n \\}$ and arc set $E = \\{ (i, j) \\in V \\times V : a_{ij} \\neq 0\\}$. \n\nA digraph $\\Gamma$ is called \\emph{strongly connected} if for any two distinct vertices $i$ and $j$ of $\\Gamma$, there is a path in $\\Gamma$ from $i$ to $j$. Following \\cite{br1991}, we consider every vertex of $V$ as strongly connected to itself. A strong digraph is \\emph{primitive} if the greatest common divisor of all its cycle-lengths is one, otherwise it is \\emph{imprimitive}. \n\nFor $n \\geq 2$, an $n$-by-$n$ matrix $A$  is called \\emph{reducible} if there exists a permutation matrix $P$ such that\n\\begin{align*}\nP^\\top A P =\n\\begin{bmatrix}\nA_{11} & A_{12} \\\\\n0 & A_{22}\n\\end{bmatrix},\n\\end{align*}\nwhere $A_{11}$ and $A_{22}$ are nonempty square matrices. If $A$ is not reducible, then A is called \\emph{irreducible}. It is well-known that a matrix $A$ is irreducible if and only if $\\dg{A}$ is strongly connected (see, e.g., \\cite[Theorem 3.2.1]{br1991} or \\cite[Theorem 6.2.24]{hj2013}). \n\nAn irreducible nonnegative matrix is called \\emph{primitive} if, in its digraph, the set of cycle-lengths is relatively prime; otherwise it is \\emph{imprimitive}. \n\nFor $n \\in \\bb{N}$, denote by $C_n$ the \\emph{basic circulant}, i.e.,  \n\\[ C_n =\n\\left[ \n\\begin{array}{cc}\n0 & I_{n-1} \\\\\n1 & 0\n\\end{array} \\right]. \\]\nNote that the digraph of $C_n$ is a cycle of length $n$.\n\nGiven an $n$-by-$n$ matrix $A$, the \\emph{characteristic polynomial of $A$}, denoted by $\\chi_A$, is defined by $\\chi_A = \\det{(tI - A)}$. The \\emph{companion matrix} $C = C_f$ of a monic polynomial $f(t) = t^n + \\sum_{k=1}^{n} c_{k} t^{n - k}$ is the $n$-by-$n$ matrix defined by\n\\[ C = \n\\left[\\begin{array}{cc}\n0 & I_{n-1} \\\\\n-c_n & -c\n\\end{array} \\right], \\]\nwhere $c = [c_{n-1}~\\cdots~c_1]$. It is well-known that $\\chi_C = f$. Notice that $C$ is irreducible if and only if $c_n \\neq 0$. \n\n\n\\section{Realizing-matrices}\n\n\\begin{lem}\\label{lem:det}\nLet $A \\in \\mat{n}{\\bb{C}}$. If $B = A + \\alpha e_k e_\\ell^\\top$, then $\\det(B) = \\det(A) + (-1)^{k + \\ell} \\alpha \\det(A_{k\\ell})$.\n\\end{lem}\n\n\\begin{proof}\nTake either a Laplace-expansion along the $k$-th row or the $k$-th column of $B$. \n\\end{proof} \n\n\\begin{thm}\\label{thm:main}\nFor each K-arc $K_n (p\/q,r\/s)$, there is a parametric, stochastic matrix $M = M(\\alpha)$, $0\\leq \\alpha \\leq 1$, such that each point $\\lambda = \\lambda(\\alpha)$ of the arc is an eigenvalue of $M$. Furthermore, if $\\alpha \\in (0,1)$, then $M$ is primitive.\n\\end{thm}\n\n\\begin{proof}\nLet $p\/q$ and $r\/s$ be Farey neighbors, where $q < s$. Note that $s \\neq q\\floor{n\/q}$ since $q$ and $s$ are relatively prime. \n\nFirst, we consider the case in which  $p\/q = 0$ and $r\/s = 1\/n$ (which we call the \\emph{Type 0 arc}). Then \\eqref{ito_eq} reduces to $(t  - \\beta)^n - \\alpha^n = 0$. If \n\\[ M = M(\\alpha) := \\alpha  C_n + \\beta I  \n= \n\\begin{bmatrix}\n\\beta & \\alpha & \t \t\\\\\n & \\beta & \\alpha \t\t\\\\\n & & \\ddots & \\ddots \t\\\\\n & & & \\beta & \\alpha \t\\\\\n\\alpha & & & & \\beta\n\\end{bmatrix} \\in \\mat{\\bb{R}}{n}, \\]\nthen \n\\begin{align*} \n\\chi_M (t) \n&= \\det{(tI - (\\alpha  C_n + \\beta I))} \\\\\n&= \\det{((t - \\beta)I - \\alpha  C_n)} \\\\\n& = \\chi_{\\alpha C_n} (t - \\beta) \\\\\n& = (t - \\beta)^n - \\alpha^n.\n\\end{align*} \nIf $\\alpha \\in (0,1)$, then $\\dg{M}$ contains directed-cycles of length one and $n$. Hence, $M$ is irreducible and since the greatest common divisor of all cycle-lengths of $\\dg{M}$ is obviously one, $M$ is primitive.\n\nNext, we consider the case in which $\\floor{n\/q}= 1$ (herein referred to as a \\emph{Type I arc}). Then \\eqref{ito_eq} reduces to $t^{s}  - \\beta  t^{s-q} - \\alpha = 0$. If \n\\begin{align} \nM = M(\\alpha) :=  \n\\begin{bmatrix}\nz & I \\\\\n\\alpha & \\beta e_{s-q}^\\top \n\\end{bmatrix} \\in \\mat{s}{\\bb{R}}, \\label{typeonemats}\n\\end{align} \nthen $M \\geq 0$ and $\\chi_{M} (t) = t^{s}  - \\beta  t^{s-q} - \\alpha$. If $\\alpha \\in (0,1)$, then $\\dg{M}$ contains $\\dg{C_n}$. Hence, $M$ is irreducible, and, since $\\gcd{(s-(s-q),s)} = \\gcd{(q,s)}=1$, it must be primitive.\n\nNext, we consider the case in which $\\floor{n\/q}> 1$ and $s < q\\floor{n\/q}$ (which we call a \\emph{Type II arc}). Then \\eqref{ito_eq} reduces to \n\\begin{align*}\n(t^q  - \\beta)^{\\floor{n\/q}} - \\alpha^{\\floor{n\/q}} t^{q\\floor{n\/q} - s} = 0.\n\\end{align*}\nConsider the nonnegative matrix $M = M(\\alpha) := \\alpha X +  \\beta Y$, where $X$ is the nonnegative companion matrix of the polynomial $t^{q\\floor{n\/q}} - t^{q\\floor{n\/q} - s}$, and\n\\begin{align*} \nY := \\bigoplus_{k=1}^{\\floor{n\/q}} C_q =\n\\begin{bmatrix}\nC_q & \\\\\n & \\ddots & \\\\\n & & C_q\n\\end{bmatrix} \\in \\mat{q\\floor{n\/q}}{\\bb{R}}.\n\\end{align*} \nSince $1 < q\\floor{n\/q} - s + 1 \\leq n - s + 1 < q + 1$, it follows that \n\\begin{align*} \nM =\n\\left[ \n\\begin{array}{*{14}{c}}\n& 1 & & & \\vline & & & & & \\vline \t\t\t\t\t\t\t\t\t\t\t\\\\\n& & \\ddots & & \\vline & & & & & \\vline \t\t\t\t\t\t\t\t\t\t\\\\\n& & & 1 & \\vline & & & & & \\vline \t\t\t\t\t\t\t\t\t\t\t\\\\\n\\beta & & & & \\vline & \\alpha & & & & \\vline \t\t\t\t\t\t\t\t\t\\\\\n\\hline \n& & & & \\vline & \\multicolumn{4}{c}{\\multirow{4}{*}{\\Large $\\ddots$}} & \\vline & & &  \t\\\\\n& & & & \\vline & & & & & \\vline & & &  \t\t\t\t\t\t\t\t\t\t\\\\\n& & & & \\vline & & & & &\\vline & & &  \t\t\t\t\t\t\t\t\t\t\\\\\n& & & & \\vline & & & & & \\vline & \\alpha \t\t\t\t\t\t\t\t\t\t\\\\\n\\hline\n& & & & \\vline & & & & & \\vline & & 1 \t\t\t\t\t\t\t\t\t\t\\\\\n& & & & \\vline & & & & & \\vline & & & \\ddots \t\t\t\t\t\t\t\t\t\\\\\n& & & & \\vline & & & & & \\vline & & & & 1 \t\t\t\t\t\t\t\t\t\\\\\n\\multicolumn{4}{c}{\\alpha e_{q\\floor{n\/q} - s + 1}^\\top} & \\vline & & & & & \\vline & \\beta & &   \n\\end{array} \n\\right],\n\\end{align*}\nwhere $e_{q\\floor{n\/q} - s + 1} \\in \\bb{R}^q$. Because $M - \\alpha e_{q\\floor{n\/q}} e_{q\\floor{n\/q} - s + 1}^\\top$ is block upper-triangular, it follows from \\hyp{Lemma}{lem:det} that \n\\begin{align*}\n\\chi_M (t) \n&= (t^q - \\beta)^{\\floor{n\/q}} + \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\n&\\quad (-1)^{2 q\\floor{n\/q} -s + 1} (-\\alpha) t^{q\\floor{n\/q} - s} (-\\alpha)^{\\floor{n\/q} -1} (-1)^{q\\floor{n\/q} - 1 - (q\\floor{n\/q} - s) - (\\floor{n\/q} -1)}\t\\\\\n&= (t^q - \\beta)^{\\floor{n\/q}} + (-1)^{2 q\\floor{n\/q}+ 1} \\alpha t^{q\\floor{n\/q} - s} \t\t\t\t\t\t\t\t\t\t\t\\\\\n&= (t^q - \\beta)^{\\floor{n\/q}} - \\alpha t^{q\\floor{n\/q} - s}.\n\\end{align*}\nIf $\\alpha \\in (0,1)$, then the directed graph contains $\\floor{n\/q}$ strongly connected components and the graph on these components, determined whether off-diagonal blocks are nonzero, is also strongly connected; hence, the entire graph is strongly connected, i.e., $M$ is irreducible. Furthermoe, since $\\dg{M}$ contains cycles of length $q$ and $q\\floor{n\/q} - (q\\floor{n\/q} - s + 1) + 1 = s$, it follows that $M$ is primitive .\n\nFinally, we consider the case when $\\floor{n\/q}> 1$ and $s > q\\floor{n\/q}$ (herein referred to as a \\emph{Type III arc}). For convenience, let $d = s - q\\floor{n\/q}$. Then \\eqref{ito_eq} reduces to \n\\begin{align*}\nt^{d} (t^q  - \\beta)^{\\floor{n\/q}} - \\alpha^{\\floor{n\/q}}  = 0.\n\\end{align*}\nConsider the nonnegative matrix $M = M(\\alpha) := \\alpha C_s + \\beta Y$, where\n\\begin{align*} \nY =\n\\begin{bmatrix}\n\\jord{d}{0}  & \t\t\t\t\\\\\n\t\t& C_q & & \t\t\t\\\\\n\t\t&        & \\ddots & \t\t\\\\\n\t\t& \t&\t    & C_q\n\\end{bmatrix} + e_d e_{d+1}^\\top \\in \\mat{s}{\\bb{R}}.\n\\end{align*}\nThen \n\\[ \nM =\n\\kbordermatrix{& & & & & d & & \t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\n  & 0 & 1 & & & & \\vline & & & & & \\vline & & & & & \\vline & & & & \t\t\t\t\t\t\t\t\\\\\n  & & 0 & 1 & & & \\vline & & & & & \\vline & & & & & \\vline & & & &\t\t\t\t\t\t\t\t\\\\\n  & & & \\ddots & \\ddots & & \\vline & & & & & \\vline & & & & & \\vline & & & &\t\t\t\t\t\t\\\\\n  & & & & 0 & 1 & \\vline & & & & & \\vline & & & & & \\vline & & & & \t\t\t\t\t\t\t\t\\\\\nd & & & & & 0 & \\vline & 1 & & & & \\vline & & & & & \\vline& & & & \t\t\t\t\t\t\t\t\\\\\n\\cline{2-20}\n& & & & & & \\vline & & 1 & & & \\vline & & & & & \\vline & & & & \t\t\t\t\t\t\t\t\\\\\n& & & & & & \\vline & & & \\ddots  & & \\vline & & & & & \\vline & & & &\t\t\t\t\t\t\t\\\\\n& & & & & & \\vline & & & & 1 & \\vline & & & & & \\vline & & & &\t\t\t\t\t\t\t\t\\\\\n& & & & & & \\vline & \\beta & & & & \\vline & \\alpha & & & & \\vline & & & & \t\t\t\t\t\t\t\\\\\n\\cline{2-20}\n& & & & & & \\vline & & & & & \\vline & \\multicolumn{4}{c}{\\multirow{4}{*}{\\Large $\\ddots$}} & \\vline & & & \t\\\\\n& & & & & & \\vline & & & & & \\vline & & & & & \\vline\t\t\t\t\t\t\t\t\t\t\\\\\n& & & & & & \\vline & & & & & \\vline & & & & & \\vline\t\t\t\t\t\t\t\t\t\t\\\\\n& & & & & & \\vline & & & & & \\vline & & & & & \\vline & \\alpha\t\t\t\t\t\t\t\t\t\\\\\n\\cline{2-20}\n& & & & & & \\vline & & & & & \\vline & & & & & \\vline & & 1 & & \t\t\t\t\t\t\t\t\\\\\n& & & & & & \\vline & & & & & \\vline & & & & & \\vline & & & \\ddots &  \t\t\t\t\t\t\t\\\\\n& & & & & & \\vline & & & & & \\vline & & & & & \\vline & & &  & 1 \t\t\t\t\t\t\t\t\\\\\n& \\alpha & & & & & \\vline & & & & & \\vline & & & & & \\vline & \\beta & & & }.\n\\]\nSince $M - \\alpha e_s e_1^\\top$ is block upper-triangular, following \\hyp{Lemma}{lem:det}, \n\\begin{align*}\n\\chi_M (t) \n&= t^d (t^q - \\beta)^{\\floor{n\/q}} + (-1)^{s+1}(-\\alpha)(-\\alpha)^{\\floor{n\/q}-1}(-1)^{s - 1 -(\\floor{n\/q}-1)} \t\\\\\n&= t^d (t^q - \\beta)^{\\floor{n\/q}} + (-1)^{2s+1}\\alpha^{\\floor{n\/q}}\t\t\t\t\t\t\t \\\\\n&= t^d (t^q - \\beta)^{\\floor{n\/q}} - \\alpha^{\\floor{n\/q}}.\n\\end{align*}\nIf $\\alpha \\in (0,1)$, then $\\dg{M}$ contains $\\dg{C_n}$ as a subgraph. Hence, $M$ is irreducible, and since $\\dg{M}$ clearly contains cycles of length $q$ and $s$, $M$ is primitive.\n\\end{proof}\n\n\\begin{rem}\nNotice that the realizing matrices for arcs of Type I, II, and II all have trace zero. \n\\end{rem}\n\n\\begin{ex}\n\\hyp{Table}{tabone} contains realizing matrices illustrating each type of arc when $n=9$ (the smallest order for which each arc-type appears). \n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{ccc}\n$K\\left(\\frac{p}{q},\\frac{r}{s} \\right)$ & \\emph{Type} & $M(\\alpha)$, $\\beta := 1 - \\alpha$\t\\vspace*{5pt} \\\\\n$K\\left(\\frac{1}{9},\\frac{1}{8} \\right)$ & I & \n$\\begin{bmatrix}\n0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n\\alpha & \\beta & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\end{bmatrix}$ \\vspace*{5pt}\t\t\t\t\t\t\t\t\\\\ \n$K\\left(\\frac{2}{7},\\frac{1}{3} \\right)$ & II & \n$ \n\\left[ \\begin{array}{*{11}{c}}\n0 & 1 & 0 & \\vline &  0 & 0 & 0 & \\vline & 0 & 0 & 0 \\\\\n0 & 0 & 1 &  \\vline &0 & 0 & 0 & \\vline & 0 & 0 & 0 \\\\\n\\beta & 0 & 0 & \\vline & \\alpha & 0 & 0 & \\vline & 0 & 0 & 0 \\\\\n\\cline{1-11}\n0 & 0 & 0 & \\vline & 0 & 1 & 0 & \\vline & 0 & 0 & 0 \\\\\n0 & 0 & 0 &  \\vline &0 & 0 & 1 & \\vline & 0 & 0 & 0 \\\\\n0 & 0 & 0 &  \\vline &\\beta & 0 & 0 & \\vline & \\alpha & 0 & 0 \\\\\n\\cline{1-11}\n0 & 0 & 0 &  \\vline &0 & 0 & 0 & \\vline & 0 & 1 & 0 \\\\\n0 & 0 & 0 &  \\vline &0 & 0 & 0 & \\vline & 0 & 0 & 1 \\\\\n0 & 0 & \\alpha & \\vline & 0 & 0 & 0 & \\vline & \\beta & 0 & 0\n\\end{array} \\right]$ \\vspace*{5pt} \t\t\t\t\t\t\t\\\\ \n$K\\left(\\frac{2}{9},\\frac{1}{4} \\right)$ & III & \n$\n\\left[ \\begin{array}{*{11}{c}}\n0 & \\vline & 1 & 0 & 0 & 0 & \\vline & 0 & 0 & 0 & 0 \\\\\n\\cline{1-11}\n0 & \\vline & 0 & 1 & 0 & 0 & \\vline & 0 & 0 & 0 & 0 \\\\\n0 & \\vline & 0 & 0 & 1 & 0 & \\vline & 0 & 0 & 0 & 0 \\\\\n0 & \\vline & 0 & 0 & 0 & 1 & \\vline & 0 & 0 & 0 & 0 \\\\\n0 & \\vline & \\beta & 0 & 0 & 0 & \\vline & \\alpha & 0 & 0 & 0 \\\\\n\\cline{1-11}\n0 & \\vline & 0 & 0 & 0 & 0 & \\vline & 0 & 1 & 0 & 0 \\\\\n0 & \\vline & 0 & 0 & 0 & 0 & \\vline & 0 & 0 & 1 & 0 \\\\\n0 & \\vline & 0 & 0 & 0 & 0 & \\vline & 0 & 0 & 0 & 1 \\\\\n\\alpha & \\vline & 0 & 0 & 0 & 0 & \\vline & \\beta & 0 & 0 & 0\n\\end{array} \\right]$\n\\end{tabular}\n\\caption{Realizing matrices for arcs of Type I, II, and II when $n=9$.}\n\\label{tabone}\n\\end{table}\n\nLet $\\mathcal{M} := \\{ M(\\alpha) : \\alpha \\in [0,1] \\}$ be the set of realizing matrices for the arc $K(1\/9,1\/8)$. For $d \\in \\bb{N}$, let $\\mathcal{M}^d = \\{ M(\\alpha)^d : M(\\alpha) \\in \\mathcal{M} \\}$. \\hyp{Theorem}{thm:arcpowers} shows that certain powers of the realizing matrices for the arc realize other arcs: in particular, $\\mathcal{M}^2$, $\\mathcal{M}^3$, and $\\mathcal{M}^4$ form a set of realizing matrices for the arcs $K(2\/9,1\/4)$, $K(1\/3,3\/8)$, and $K(4\/9,1\/2)$, respectively. \n\\end{ex}\n\n\\section{Differentiability of the Arcs}\nWe investigate here the smoothness of the K-arcs, a natural question not previously addressed. \n\nTo that end, let $f$ and $g$ be monic polynomials of degree $n$. For $\\alpha \\in [0,1]$, let $c_\\alpha := \\alpha f + (1 - \\alpha) g$. Since the roots of a polynomial vary continuously with respect to its coefficients, it follows that the locus $L(f,g) := \\left\\{ t \\in \\bb{C}: c_\\alpha(t) = 0,~\\alpha \\in [0,1] \\right\\}$ consists of $n$ continuous paths (counting multiplicities), each of which connects a root of $g$ to a root of $f$, whose points depend continuously on the parameter $\\alpha$ (if $f$ and $g$ share a root, then there is a degenerate path at this root). \n\n\nDenote by $P(\\mu, \\lambda)$ the path that starts at the root $\\mu$ of $g$ and terminates at the root $\\lambda$ of $f$ ($\\mu \\neq \\lambda$). If $r = r(\\alpha) \\in P(\\mu, \\lambda)$, $\\alpha \\in (0,1)$, then \n\\begin{align*}\n0 = \\alpha f(r) + (1 - \\alpha) g(r).\n\\end{align*}\nDifferentiating with respect to $\\alpha$ yields\n\\begin{align*}\n0 = f(r) + \\alpha g'(r) r' - g(r) + (1-\\alpha)g'(r)r' = f(r) - g(r) + r' c'_\\alpha (r).\n\\end{align*}\nIf $c_\\alpha'(r) \\neq 0$ (i.e., if $r$ is not a multiple root of $c_\\alpha$), then \n\\begin{equation*}\nr' = \\frac{g(r) - f(r)}{c_\\alpha'(r)}.\n\\end{equation*} \nThus, the path $P(\\mu, \\lambda)$ is differentiable at $r$ if $r$ is not a multiple root of $c_\\alpha$ \\cite{i2011}.\n\n\n\\begin{prop}\\label{distincteigs}\nFor $n \\geq 4$, let \n\\begin{equation}\nf_\\alpha (t) := t^n - \\beta t - \\alpha, ~\\alpha \\in [0,1], ~\\beta := 1 - \\alpha. \\label{polyalpha}\n\\end{equation} \n\\begin{enumerate}\n[label=(\\roman*)]\n\\item If $n$ is even, then $f_\\alpha$ has $n$ distinct roots.\n\\item If $n$ is odd and $\\alpha \\geq \\beta$,  then $f_\\alpha$ has $n$ distinct roots.\n\\item If $n$ is odd and $\\alpha < \\beta$,  then $f_\\alpha$ has a multiple root if and only if\n\\[ n^n \\alpha^{n-1} - (n-1)^{n-1} \\beta^n = n^n \\alpha^{n-1} + (n-1)^{n-1} (\\alpha -1)^n = 0. \\]  \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof} Notice that $f_\\alpha(1) = 0$, and, since $C_f$ is primitive, if $f_\\alpha(\\lambda) = 0$, $\\lambda \\neq 1$, then \n\\begin{equation}\n|\\lambda| < 1. \\label{dominate}\n\\end{equation} \n\nIt is well-known that a polynomial has a multiple root if and only if it shares a root with its formal derivative. Thus, $f_\\alpha$ has a multiple root $\\lambda \\in \\bb{C}$ if and only if $f_\\alpha (\\lambda) = f_\\alpha'(\\lambda) = 0$, i.e., if and only if  \n\\begin{align}\n\\lambda^n - \\beta \\lambda - \\alpha &= 0 \t\\label{poly}\t\\\\\nn\\lambda^{n-1} - \\beta &= 0\t\t\t\\label{deriv}.\n\\end{align}\nSolving for $\\beta$ in \\eqref{deriv} and substituting the result in \\eqref{poly} yields\n\\begin{equation}\n\\lambda^n = -\\frac{\\alpha}{n-1}\t.\t\t\\label{lambdan}\n\\end{equation}\nSubstituting for $\\lambda^n$ in \\eqref{poly} yields\n\\begin{equation}\n\\lambda = -\\frac{\\alpha n}{\\beta(n-1)} < 0. \t\\label{lambdafrac}\n\\end{equation} \n\nWe now consider each part separately:\n\n\\begin{enumerate}\n[label=(\\roman*)]\n\\item For contradiction, if $f_\\alpha$ has a multiple root, then it must be negative \\eqref{lambdafrac}; however, because $f_\\alpha(-t) = t^n + \\beta t - \\alpha$, Descartes' Rule of Signs ensures that $p$ has at most one negative root, a contradiction. \n \n\\item Suppose that $n$ is odd and $\\alpha \\geq \\beta$. For contradiction, if $f_\\alpha$ has a multiple root, then, following \\eqref{lambdafrac},  \n\\begin{equation*}\n|\\lambda| = \\frac{\\alpha n}{\\beta(n-1)} \\geq \\frac{n}{n-1} > 1,\n\\end{equation*}\ncontradicting \\eqref{dominate}.\n\n\\item It is well-known that a polynomial $f$ has a multiple root if and only if its \\emph{resultant} $R(f,f')$ vanishes. If \n\\[ \nS(f_\\alpha,f_\\alpha') =\n\\kbordermatrix{\n   & 1 & \\cdots & n-1 & & n & n+1 & & 2n-1 \t\t\\\\\n1 & 1                         & & & \\vrule & -\\beta & -\\alpha \t\\\\\n\\vdots & & \\ddots & & \\vrule & & \\ddots & \\ddots \t\\\\\nn-1 & & & 1 & \\vrule & & & -\\beta & -\\alpha \t\t\\\\\n\\cline{2-9}\nn & n & & & \\vrule & -\\beta \t\t\t\t \\\\\n\\vdots & & \\ddots & & \\vrule & & \\ddots \t\t\\\\\n2n & & & n & \\vrule & & & -\\beta \\\\\n2n-1 & 0  & \\cdots & 0 & \\vrule & n & & & -\\beta\n}, \\] \nthen $R(f_\\alpha,f_\\alpha') = | S(f_\\alpha,f_\\alpha')| = | D - C B|$, where $B$, $C$, and $D$ denote the upper-right, lower-left, and lower-right blocks of $S(f_\\alpha,f_\\alpha')$. Since \n\\[ D - CB =\n\\begin{bmatrix}\n(n-1) \\beta & n\\alpha & &  \\\\\n& \\ddots & \\ddots & \\\\\n& & (n-1) \\beta & n\\alpha \\\\\nn & & & -\\beta\n\\end{bmatrix}, \\]\nand $n$ is odd, it follows that $R(f_\\alpha,f_\\alpha')  = n^n \\alpha^{n-1} - (n-1)^{n-1} \\beta^n$ and the result is established.\n\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{rem}\\label{rem:negmult}\nIf $n$ is odd and $f_\\alpha$ has a multiple root $\\lambda$ (which, folllowing \\eqref{lambdafrac}, must be negative), then Descartes' Rule of Signs applied to $f_\\alpha (-t) = -t^n +\\beta t - \\alpha$ forces the multiplicity of $\\lambda$ as a root of $f_\\alpha$ to be exactly two. \n\\end{rem}\n\n\\begin{rem}\n\\label{polypi}\nUnder the hypotheses of part (iii) of \\hyp{Proposition}{distincteigs}, the resultant $R(f_\\alpha,f'_\\alpha) = \\pi(\\alpha) = n^n \\alpha^{n-1} + (n-1)^{n-1} (\\alpha -1)^n$ for the polynomial $f_\\alpha$ defined in \\eqref{polyalpha}, is a univariate polynomial in $\\alpha$. Since $\\pi(0) = -(n-1)^{n-1} < 0$ and $\\pi(1) = n^n > 0$, it folllows that $\\pi$ must have a root in $(0,1)$. However, $\\pi'(\\alpha) = n^n (n-1) \\alpha^{n-2} + (n-1)^{n-1} (\\alpha -1)^{n-1}$ and because $n$ is odd, we have $\\pi(\\alpha) \\geq 0$ for all $\\alpha \\geq 0$. Thus, $\\pi$ is strictly increasing on $(0,\\infty)$ and hence has exactly one root in $(0,1)$. \n\\end{rem}\n \n\\begin{cor} \\label{cor:diffarcs}\nLet $n \\geq 4$ be a positive integer.\n\\begin{enumerate}\n[label=(\\roman*)]\n\\item If $n$ is even and $\\floor{n\/2} \\leq m \\leq n$, then the K-arc $K_n \\left( {1}\/{m},{1}\/{m-1} \\right)$ is differentiable. \n\\item If $n$ is odd and $\\floor{n\/2}+1 \\leq m \\leq n$, then the K-arc $K_n \\left( {1}\/{m},{1}\/{m-1} \\right)$ is differentiable. \n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}\nIn view of \\hyp{Proposition}{distincteigs}, it suffices to consider the case when $n$ is odd and $\\alpha < \\beta$, where $f_\\alpha$ is defined as in \\eqref{polyalpha}; however, this case is clear as well since \\hyp{Remark}{rem:negmult} ensures that if $f_\\alpha$ has a multiple multiple root, then $\\lambda$ is real.   \n\\end{proof}\n\n\\section{Powers of Realizing-matrices}\n\nFor each of the arc types listed in the proof of \\hyp{Theorem}{thm:main}, we refer to the collection of polynomials \n\\begin{align}\nf_\\alpha(t) &= (t  - \\beta)^n - \\alpha^n \t\t\t\t\t\t\t\t\\tag{Type 0}\t\t\\\\\nf_\\alpha(t) &= t^{s}  - \\beta  t^{s-q} - \\alpha \t\t\t\t\t\t\t\\tag{Type I}\t\t\\\\\nf_\\alpha(t) &= (t^q  - \\beta)^{\\floor{n\/q}} - \\alpha^{\\floor{n\/q}} t^{q\\floor{n\/q} - s}\t\\tag{Type II} \t\\\\\nf_\\alpha(t) &= t^{s - q\\floor{n\/q}} (t^q  - \\beta)^{\\floor{n\/q}} - \\alpha^{\\floor{n\/q}} \t\t\t\\tag{Type III}\t\n\\end{align}\nas the \\emph{reduced Ito polynomials}. \n\nThe following result is readily deduced  from several well-known theorems concerning Farey pairs (see, e.g., \\cite[pp. 28--29]{hw2008}).\n\n\\begin{lem} \n\\label{lem:farey_pair}\nIf $p\/q$, $r\/s$ are elements of ${F}_n$, then $(p\/q,r\/s)$ is a Farey pair of order $n$  if and only if $qr- ps= 1$ and $q + s > n$. \n\\end{lem}\n\n\\begin{lem}\n\\label{lem:divisor}\nIf $d$ is a positive integer such that $1 < d < n$, then $(d\/n,d\/n-1)$ is a Farey pair of order $n$ if and only if $d$ divides $n$ or $d$ divides $n-1$. \n\\end{lem}\n\n\\begin{proof}\nIf there is a positive integer $k$ such that $n = dk$, then $(d\/n,d\/n-1) = (1\/k,d\/n-1)$. Since $dk - (n-1) = 1 $, it follows that $d\/n-1 \\in {F}_n$.  Because $k > 1$, it follows that $k + n - 1 >  n$. Following \\hyp{Lemma}{lem:farey_pair}, $(1\/k,d\/n-1)$ is a Farey pair. A similar argument demonstrates that $(d\/n,1\/k)$ is a Farey pair if $d$ divides $n-1$.\n\nConversely, if $d$ does not a divisor of either $n$ or $n-1$, then $dn - d(n-1) = d \\neq 1$. The result now follows from \\hyp{Lemma}{lem:farey_pair}.   \n\\end{proof}\n\n\\begin{cor}\n\\label{cor:divisor}\nLet $d$, $m$, and $n$ be positive integers such that $d < m \\leq n$, and suppose that $(1\/m,1\/m-1)$ is a Farey pair of order $n$. \n\\begin{enumerate}[label=(\\roman*)]\n\\item If $d$ divides $m$ and $k:=m\/d$, then $(1\/k, d\/m-1)$ is a Farey pair of order $n$ if and only if $k + m - 1 > n$.   \n\\item If $d$ divides $m-1$ and $k:=(m-1)\/d$, then $(d\/m,1\/k)$ is a Farey pair of order $n$ if and only if $m + k > n$.\n\\end{enumerate}\n\\end{cor}\n\n\n\\begin{thm} \\label{thm:arcpowers}\nLet $d$, $m$, and $n$ be positive integers such that $1< d < m \\leq n$. Suppose that $(1\/m,1\/m-1)$ and $(d\/m,d\/m-1)$ are Farey pairs of order $n$. \nWe distinguish the following cases:\n\\begin{enumerate}[label=(\\roman*)]\n\\item d divides m: For $f_\\alpha(t) = t^m - \\beta t - \\alpha$, let $M(\\alpha)$ be defined as in \\eqref{typeonemats}. If $\\mathcal{M} := \\{ M(\\alpha) : \\alpha \\in [0,1] \\}$, then $\\mathcal{M}^d := \\{ M(\\alpha)^d : \\alpha \\in [0,1]\\}$ forms a set of realizing-matrices for $K_n(1\/k,d\/m-1)$, where $k = m\/d$.  \n\n\\item d divides $m-1$ and $m > k\\floor{n\/k}$, where $k = m\/d$: For $f_\\alpha(t) = t^m - \\beta t - \\alpha$, let $M(\\alpha)$ be defined as in \\eqref{typeonemats}. If $\\mathcal{M} := \\{ M(\\alpha) : \\alpha \\in [0,1] \\}$, then $\\mathcal{M}^d := \\{ M(\\alpha)^d : \\alpha \\in [0,1]\\}$ forms a set of realizing-matrices for $K_n(d\/m,1\/k)$, where $k = m\/d$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof} Part (i): Since $(1\/k,d\/m-1)$ is a Farey pair, following \\hyp{Corollary}{cor:divisor}, $n < m + k - 1$; consequently,  \n\\begin{align*}\nd = \\frac{m}{k} \\leq \\frac{n}{k} < \\frac{m + k - 1}{k} = d + 1 - \\frac{1}{k} < d + 1\n\\end{align*}\nand hence $\\floor{n\/k} = d$. The Ito equations for $(1\/k,d\/m-1)$ are given by  \n\\begin{align*}\nt^{m-1} \\left( t^k - \\beta \\right)^{\\floor{n\/k}} = \\alpha^{\\floor{n\/k}} t^{k\\floor{n\/k}},~\\alpha \\in [0,1],~\\beta:=1-\\alpha,\n\\end{align*}\nand the reduced Ito polynomials for this arc are given by \n\\begin{align}\nq_\\alpha (t) = (t^k - \\beta)^d - \\alpha^d t,~\\alpha \\in [0,1],~\\beta:=1-\\alpha. \n\\end{align}\nNotice that $\\deg{(q_\\alpha)} = m$, for every $\\alpha \\in [0,1]$. \n\nLet $\\lambda = \\lambda(\\alpha) \\in K(1\/k,d\/m-1)$. Consider the reduced Ito polynomial $p_\\beta (t) = t^m -  \\alpha t - \\beta$ and its nonnegative companion matrix $M = M(\\beta)$. The Cayley-Hamilton theorem (see, e.g., \\cite[p.~109]{hj2013}) ensures that $M^m - \\beta I = \\alpha M$; hence  \n\\begin{align*}\nq_\\alpha(M^d) = (M^{dk} - \\beta I)^d - \\alpha^d M^d = (M^m - \\beta I)^d  - (\\alpha M)^d = 0,\t\t\n\\end{align*}\ni.e., $q_\\alpha$ is an \\emph{annihilating polynomial} for $M^d$.  \n\nDenote by $\\psi_M$ the \\emph{minimal polynomial} of $M$, i.e., $\\psi_M$ is the unique monic polynomial of minimum degree that annihilates $M$ (see, e.g., \\cite[p.~192]{hj2013}). Since $M$ is a companion matrix, $\\psi_M = \\chi_M$ (\\cite[Theorem 3.3.14]{hj2013}). Hence, if $J = \\inv{S} M S$ is a Jordan canonical form of $M$, then $J$ is \\emph{nonderogatory} (\\cite[Theorem 3.3.15]{hj2013}), i.e., $J$ contains exactly one \\emph{Jordan block} corresponding to every distinct eigenvalue. Since $M^d = S J^d \\inv{S}$, it follows that any Jordan canonical form of $J^d$ is nonderogatory -- indeed, if $f(x) = x^d$, then $f'(x) = d x^{d-1}$ and $f'(x) = 0$ if and only if $x=0$; since zero is not a repeated root \\eqref{lambdafrac} (and hence not associated with a nontrivial Jordan block), the claim follows from \\cite[p.~424, Theorem 6.2.25]{hj1994}) -- thus, $M^d$ is nonderogatory and, following \\cite[Theorem 3.3.15]{hj2013}, $\\psi_{M^d} = \\chi_{M^d}$ and $\\deg{\\left(\\psi_{M^d}\\right)} = m$. Since $\\psi_{M^d}$ is the unique polynomial of minimum degree that annihilates $M$, and since $\\deg{(q_\\alpha)} = m$, it must be the case that $\\chi_{M^d} = \\psi_{M^d} = q_\\alpha$. Hence, $M^d$ is a realizing-matrix for $\\lambda$.\n\nPart (ii): By hypothesis, \n\\begin{align*}\nd = \\frac{m-1}{k} < \\frac{m}{k} \\leq \\frac{n}{k}, \n\\end{align*}\nhence $d \\leq \\floor{n\/k}$. Since $m > k\\floor{n\/k}$, it follows that $m - k\\floor{n\/k} \\geq 1$ and $\\floor{n\/k} \\leq (m-1)\/k = d$. Hence, $d = \\floor{n\/k}$.  \n\nThe Ito equations for $(d\/m,1\/k)$ are given by \n\\begin{align*}\nt^m \\left( t^k - \\beta \\right)^d = \\alpha^d t^{m-1},~\\alpha \\in [0,1],~\\beta:=1-\\alpha, \n\\end{align*}\nand the reduced Ito polynomials for this arc are given by\n\\begin{align*}\nq_\\alpha (t) = t(t^k - \\beta)^d - \\alpha^d,~\\alpha \\in [0,1],~\\beta:=1-\\alpha.\n\\end{align*}\nNotice that $\\deg{(q_\\alpha)} = m$, for every $\\alpha \\in [0,1]$. \n\nLet $\\lambda = \\lambda(\\alpha) \\in K(d\/m,1\/k)$. Consider the reduced Ito polynomial $f_\\alpha (t) = t^m -  \\beta t - \\alpha$ and its nonnegative companion matrix $M = M(\\alpha)$. The Cayley-Hamilton theorem ensures that $M(M^{m-1} - \\beta I) = M^m - \\beta M = \\alpha I$; hence  \n\\begin{align*}\nq_\\alpha(M^d) = M^d (M^{m-1} - \\beta I)^d - \\alpha^d I = (M^m - \\beta M)^d  - (\\alpha I)^d = 0,\t\t\n\\end{align*}\ni.e., $q_\\alpha$ is an \\emph{annihilating polynomial} for $M^d$. \n\nUsing exactly the same argument as in part (i), it can be shown that $\\chi_{M^d} = \\psi_{M^d} = q_\\alpha$. Hence, $M^d$ is a realizing-matrix for $\\lambda$. \n\\end{proof}\n\n\\section{Additional Questions}\n\nIn this section, we pose several problems and conjectures for further inquiry.\n\n\\subsection{Karpelevi{\\v{c}}~Arcs}\n\n\\hyp{Theorem}{thm:main} establishes the existence of parametric realizing-matrices for the K-arcs. Suppose that $M$ is a realizing-matrix for a given point on a given arc, and let $M_k$ be the irreducible component that realizes the arc. Clearly, $M_k^\\top$ and $P M_k P^\\top$ are also realizing-matrices. With the aforementioned in mind, we offer the following.  \n\n\\begin{prob}\nTo what extent are the realizing-matrices unique? \n\\end{prob}\n\n\\hyp{Corollary}{cor:diffarcs} and \\hyp{Theorem}{thm:arcpowers} show that many, but not all arcs are differentiable. Given the empirical evidence, we pose the following.\n\n\\begin{conj}\nAll K-arcs of order $n$ are differentiable for every $n$.\n\\end{conj} \n\nFor $S\\subseteq\\bb{C}$, let $S^d := \\{ \\lambda^d : \\lambda \\in S \\}$. \\hyp{Theorem}{thm:arcpowers} demonstrates that $\\sig{M}^d = \\sig{M^d}$. Although the evidence is ample, a demonstration that the powered K-arc $K_n^d(1\/m,1\/m-1)$ corresponds to $K_n(1\/k,d\/m-1)$ ($d$ divides $m$) or $K_n(d\/m,1\/k)$ ($d$ divides $m-1$ and $m>k\\floor{n\/k}$) has proven elusive. Thus, we offer the following.\n\n\\begin{conj}\nLet $d$, $m$, and $n$ be positive integers such that $1< d < m \\leq n$. Suppose that $(1\/m,1\/m-1)$ and $(d\/m,d\/m-1)$ are Farey pairs of order $n$. \n\\begin{enumerate}[label=(\\roman*)]\n\\item If d divides m, then $K_n^d (1\/m,1\/m-1) = K_n(1\/k,d\/m-1)$, where $k = m\/d$.  \n\n\\item If d divides $m-1$ and $m > k\\floor{n\/k}$, then $K_n^d (1\/m,1\/m-1) = K_n(d\/m,1\/k)$, where $k = m\/d$. \n\\end{enumerate}\n\\end{conj}\n\nLet $K$ be a K-arc and let $d_K : [0,1] \\longrightarrow \\mathbb{R}_0^+$ be the function defined by $\\alpha \\longmapsto |\\lambda|$, where $\\lambda = \\lambda(\\alpha)$ is the point on $K$ corresponding to $\\alpha \\in [0,1]$. From \\hyp{Figure}{fig:karpregions}, we pose the following. \n\n\\begin{conj}\nIf $K$ is any K-arc, then the function $d_K$ is strictly convex. \n\\end{conj}\n\n\\subsection{The Levick-Pereira-Kribs Conjecture}\n \nFor a natural number $n$, denote by $\\Pi_n$ the convex-hull of the $n$\\textsuperscript{th} roots-of-unity, i.e., \n\\[ \\Pi_n = \\left\\{ \\sum_{k=0}^{n-1} \\alpha_k \\exp{(2\\pi\\ii k\/n)} : \\alpha_k \\geq 0,~\\sum_{k=0}^{n-1} \\alpha_k =1 \\right\\}. \\]\nDenote by $\\Omega_n$ the subset of the complex-plane containing all single eigenvalues of all $n$-by-$n$ doubly stochastic matrices. Perfect and Mirsky \\cite{pm1965} conjectured that $\\Omega_n = \\bigcup_{k=1}^n \\Pi_k$ and proved their conjecture when $1 \\leq n \\leq 3$. Levick et al.~\\cite{lpk2015} proved Perfect-Mirsky when $n=4$ but a counterexample when $n=5$ was given by Mashreghi and Rivard \\cite{mr2007}. Levick et al.~conjectured that $\\Omega_n = \\Theta_{n-1} \\cup \\Pi_n$ (\\cite[Conjecture 1]{lpk2015}). \n\nIn \\cite{j1981}, necessary and sufficient conditions were found for a stochastic matrix to be similar to a doubly stochastic matrix. Thus, it is possible to investigate the Levick-Pereira-Kribs Conjecture via the realizing matrices given in \\hyp{Theorem}{thm:main} vis-\\`{a}-vis the results in \\cite{j1981}. In particular, if $M$ is a realizing matrix for $\\lambda$ on the boundary of $\\Theta_n$ excluding the unit-circle (this case is clear), and $M \\oplus 1$ is similar to a doubly stochastic matrix $D$, then $\\Theta_{n-1} \\cup \\Pi_n \\subseteq \\Omega_n$.\n\n\\section{Acknowledgment}\n\nWe would like to thank University of Washington Bothell undergraduate student Amber R.~Thrall for proving that the polynomial $\\pi$ is \\hyp{Remark}{polypi} has only one root in $(0,1)$.\n\n\n\\bibliographystyle{abbrv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThere has been a long-standing interest in the question of the amenability of Richard Thompson's group $F$, introduced in Thompson's notes of 1965 (see the survey \\cite{CFP} for a general background on the three Thompson groups $F<T<V$), with many failed attempts to prove either the amenability or non-amenability of $F$.  The groups $F<T<V$ arise in many areas of mathematics for reasons which are not entirely understood.  One plausible explanation is that they express in some fundamental way connections through Category Theory with associativity and versions of commutativity (see \\cite{BrinCat, GeoGuz, DehGeo} for some discussion of these connections), which of course are fundamental aspects of any theory involving products. Regardless of the cause, it is still the case that these groups arise naturally in many areas of mathematics including dynamics, logic, topology, and more obviously geometric group theory.  One fetching example of such an appearance is in e.g., the relationship between the group $F$ and the theory of associahedra, and in particular, the theory related to the the proof of the Four Colour Theorem \\cite{BowlinBrin}.  In any case, it is well known now that the Richard Thompson groups are fundamental.   In this paper, we will be investigating some structures related to $T$ which have implications towards the amenability of $F$, by exploring some related questions from the theory of $C^*$ algebras.\n\nSince the foundational paper  \\cite{PowersFreeAlgebraSimplicity} of Powers, there has been a long-standing interest in whether there could exist a group $G$ with unique trace, for which its reduced $C^*$ algebra $\\cstar{G}$ is not simple.  The paper  \\cite{PowersFreeAlgebraSimplicity}  contains various conditions on a group which imply uniqueness of trace and\/or simplicity of the algebra.  (See Section \\ref{tests} for a discussion of an often-used condition of Powers.) The question of the equivalence of these two properties is explicitly stated in the papers \\cite{dlH}, \\cite{dlH2} of Pierre de la Harpe, and his excellent overview of the problem \\cite{dlH3}.  For more on the general problem we refer the reader to the paper \\cite{Robin} of Robin Tucker-Drob.\n\n  Uffe Haagerup and Kristian Knudsen Olesen in  \\cite{HaagerupOlesen} show that the simplicity of $\\cstar{T}$ implies the non-amenability of $F$ via a construction given below (see Theorem \\ref{HO-IFF} below), while it is well known that $T$ is a group with unique trace.  Thus, in this paper we begin to investigate the ideal structure of $\\cstar{T}$.\n \n Haagerup and Olesen's idea showing that that the simplicity of the algebra $\\cstar{T}$ implies the non-amenability of $F$  runs as follows.  Consider $T$ ``acting'' on the interval $[0,1]$.  Assume that the stabiliser of $0$, which is the standard copy of the Thompson group $F$ in $T$, is amenable.  Since the action of $T$ on $\\mathbb{Z}[\\frac{1}{2}]$ is transitive, we have that the representation induced by this action, $\\pi:T\\rightarrow B(l_2(\\mathbb{Z}[\\frac{1}{2}]))$, is weakly contained in the left regular representation. From this one sees that there is a unique $*$-homomorphism from $\\cstar{T}$ into the $C^*$-algebra generated by $\\pi(T)$.  \n \nConsider now a finite subset $H$ of $T$ so that  $H=H_1\\sqcup H_2$ and with\n\n  \\[\n  \\sum_{g\\in H_1}\\pi(g)-\\sum_{h\\in H_2}\\pi(h)=0.\n  \\]\n\nThe simplicity of $\\cstar{T}$ now implies that the ideal generated by  $\\sum_{g\\in H_1}\\lambda(g)-\\sum_{h\\in H_2}\\lambda(g)$ is proper.  However, this is not possible since $\\pi$ is non-trivial, so $F$ must be non-amenable.\n\nOur main result is to show a (partial) converse of the above program (we except the simplicity of $\\cstar{T}$).\n\n\\begin{theorem}\\label{HO-IFF}\nThompson's group F is non-amenable if and only if there exists  a finite set $H$ which can be decomposed as  disjoint union of sets $H_1$ and $H_2$ with $\\sum_{g\\in H_1}\\pi(g)=\\sum_{h\\in H_2}\\pi(h)$ and such that the closed ideal generated by $\\sum_{g\\in H_1}\\lambda(g)-\\sum_{h\\in H_2}\\lambda(h)$ coincides with $C^*_\\lambda(T)$.\n\\end{theorem}\n\nThe rest of this article proceeds as follows. \n\nIn Section \\ref{ideals} we present the proof of the main result, modulo our Lemma \\ref{fixed-point}.  \n\nIn the Section \\ref{tests} we discuss a theorem of Powers and another of Kesten which together produce Condition \\ref{kesten-test}, a main test which is often used to detect the simplicity of $\\cstar{G}$ for a group $G$.  We show that Condition \\ref{kesten-test} fails to apply to Thompson's group $T,$  and we offer a new test (Condition \\ref{weak-kesten}) which appears to be a natural weakening of Condition \\ref{kesten-test}, and which may be of use for various groups $G$ to show that $\\cstar{G}$ is simple. Like Condition \\ref{kesten-test}, the new Condition \\ref{weak-kesten} gives the simplicity of $\\cstar{G}$ for a group $G$ if for all finite subsets $H$ of $G\\,\\backslash\\!\\left\\{e\\right\\}$, we can carry out a certain construction (creating large free subgroups in a certain way from $H$). We also show here that Condition \\ref{weak-kesten} is still not weak enough to show that the algebra $\\cstar{T}$ is simple, while giving some evidence that Condition \\ref{weak-kesten} is properly weaker than the commonly used Condition \\ref{kesten-test}. \n\nThe authors are unaware of any group with unique trace, for which Condition \\ref{kesten-test} fails.  We believe $T$ is a first example of such a group.\n\n  In Section \\ref{p-pong} we provide a short discussion of the historical Ping-Pong Lemma and we prove a version of the Ping-Pong Lemma (Lemma \\ref{free-powering-one}) useful for detecting free subgroups when considering a group generated by two homeomorphisms of $S^1$ with rational rotation numbers.  Lemma \\ref{free-powering-one} is an essential ingredient in the  proof of Lemma \\ref{fixed-point}.  \n  \n  In Section \\ref{trying-conditions}, we give proofs that we can carry out the construction of Condition \\ref{weak-kesten} for many cases of finite $H\\subset T\\,\\backslash\\!\\left\\{e\\right\\}$, including the cases we need to prove Lemma \\ref{fixed-point}.  We also describe some cases of $H$ where we cannot carry out the construction of Condition \\ref{weak-kesten}, but where related constructions do produce large free subgroups.  \n  \n  In our final Section \\ref{conjectures} we state some remaining questions which we find interesting.\n  \n{\\it Acknowledgments:} The authors are grateful to Uffe Haagerup and Kristian Knudsen Olesen for sharing the early drafts of their work \\cite{HaagerupOlesen} with us, and to Martin Kassabov and Justin Tatch Moore for feedback on this article while it was under construction.\n\\section{Non-amenability of $F$ and a condition on ideals of its $C^*$-algebra.}\\label{ideals}\n\n\\begin{theorem}\nThe Thompson group F is not amenable if and only if there exists  a finite set $H$ which can be decomposed as  disjoint union of sets $H_1$ and $H_2$ with $\\sum_{g\\in H_1}\\pi(g)=\\sum_{h\\in H_2}\\pi(h)$\nand such that the closed ideal generated by $\\sum_{g\\in H_1}\\lambda(g)-\\sum_{h\\in H_2}\\lambda(h)$ coincides with $C^*_\\lambda(T)$.\n\\end{theorem}\n\\begin{proof}\n\nOne part of the theorem follows from the draft \\cite{HaagerupOlesen} of Haagerup and Olesen. \\\\\n\nIt is left to show that if $F$ is not amenable then there is a finite set $H$ which satisfies the conditions of the theorem.  Let $x_1$ and $x_2$ be the standard generators of the copy of Thompson's group $F$ in $T$, supported in $[\\frac{1}{2},1]\\subset S^1$, with supports in $[\\frac{1}{2},1]$ and $[\\frac{3}{4},1]$ respectively. Let $g_1=x_1^r$ and $g_2=x_2^r$ be the conjugated copies of these generators, where $r$ is rotation by $1\/2$, so that $g_1$ and $g_2$ act on the interval $[0,\\frac{1}{2}]\\subset S^1$ and generate a copy of $F$ there, with trivial action on $[\\frac{1}{2},1]=[\\frac{1}{2},0]$. Define $H=\\{e, g_1, g_2, x_1, x_2, g_1x_1,g_2x_2 \\}$. Define $H_1=\\{g_1, g_2, x_1,x_2\\}$ and $H_2$ to be the rest of the set $H$. Obviously, $\\sum_{g\\in H_1}\\pi(g)-\\sum_{h\\in H_2}\\pi(h)=0$. \n\nLet us show  that the ideal, $J$, generated by $\\sum_{g\\in H_1}\\lambda(g)-\\sum_{h\\in H_2}\\lambda(h)$ is the whole reduced $C^*$-algebra of $T$. Note that $\\|1+\\lambda(g_1)+\\lambda(g_2)\\|\\leq C<3$ by assumption. Moreover, the point $p=\\frac{7}{8}$ is not  fixed  by any element from  the set $E=\\{ x_1, x_2, g_1x_1,g_2x_2  \\}$. Thus we can apply Lemma \\ref{fixed-point} for the set $E$: let $\\varepsilon>0$ and let $g$ and $c_1,\\ldots, c_n\\in C_T(g)$ be such that \n$$\\| \\sum_{s\\in E} \\sum_{i=1}^n\\lambda((sg)^{c_i}) \\|\\leq \\varepsilon n.$$\n\nNote that the element $b=\\lambda(g-\\frac{1}{3}[e+g_1+g_2]g+\\frac{1}{3} \\sum_{s\\in E} sg)$ is in $J$, thus $\\frac{1}{n}\\sum_{i=1}^n\\lambda(c_i)b\\lambda({c_i}^{-1}) \\in J$. The distance between the element $\\frac{1}{n}\\sum_{i=1}^n\\lambda(c_i)b\\lambda({c_i}^{-1})$ and $\\lambda(g)$ is strictly smaller then $1$ for large $n$. Indeed,\n\n\\begin{align*}\n\\|\\lambda(g)-b\\|\\leq& \\frac{1}{3}\\|1+\\lambda(g_1)+\\lambda(g_2)\\| + \\frac{1}{3n}\\|\\sum_{s\\in E} \\sum_{i=1}^n\\lambda((sg)^{c_i})  \\|\\leq C+\\varepsilon.\n\\end{align*}\nthus we have found an invertible element in $J$, therefore $J=C^*_\\lambda(T)$.\n\\end{proof}\n\n\\section{Powers' test}\\label{tests}\nIn \\cite{PowersFreeAlgebraSimplicity} Powers' gives the following test for the simplicity of the algebra $\\cstar{G}$ over a group $G$.\n\n\\begin{theorem}\\label{sumTest}\n  If for all non-empty $H\\subset G$ with $|H|<\\infty$, $e\\not\\in H$ and for all positive integers $n$ there is a set $\\{c_1,c_2,\\ldots,c_n\\}\\subset G$ so that \n\\[\n\\lim_{n\\to\\infty}\\frac{1}{n}||\\Sigma_{i=1}^n \\lambda(c_ih{c_i}^{-1})||=0,\\,\\forall h\\in H,\n\\]\nthen $\\cstar{G}$ is simple.\n\\end{theorem}\n\nLet $G$ be a group generated by a finite set $S$ with $S=S^{-1}$, then $\\frac{1}{|S|}||\\Sigma_{h\\in S} \\lambda(h)||$ is equal to the spectral radius of the simple random walk on the Cayley graph of $G$ with respect to $S$, denoted by $\\rho(G,S)$. The spectral radius of the simple random walk have been computed for many groups. Kesten, \\cite{kesten}, showed that if $S=\\{g_1,\\ldots,g_n\\}$ is {\\it a free set}, i.e., $g_1,\\ldots,g_n$ are standard generators of the free group of rank $n$, then the spectral radius is\n$$\\rho(G,S)=\\frac{\\sqrt{2n-1}}{n}$$\n\nThus the following condition implies the hypothesis of the Theorem \\ref{sumTest}.\n\\begin{condition}\\label{kesten-test}\nFor all finite subsets $H\\subset G$ with $e\\not\\in H$ and for all positive integers $n$ there is a set $\\{c_1,c_2,\\ldots,c_n\\}\\subset G$ so that \n\\[\n\\langle h^{c_1},h^{c_2}, \\ldots, h^{c_n}\\rangle\n\\]\nis a free subgroup of $G$ of rank $n$ for all $h\\in H$.\n\\end{condition}\n\n\nIf $g$ is a bijection from a set $X$ to itself, denote by $Supp(g):=\\left\\{x\\in X\\mid g\\cdot x\\neq x\\right\\}$ and $Fix(g):=X\\backslash Supp(g)$, the support and the set of points fixed by  $g$, respectively.\n\nThe following remark holds true for groups of permutations of a set $X$.\n\\begin{remark}\\label{disjointSupportsAndConjugation}\nLet $X$ be a set, and $G$ the group of bijections from $X$ to $X$.  \nSuppose $h_1$, $h_2\\in G\\backslash\\left\\{1\\right\\}$ so that $Supp(h_1)\\cap Supp(h_2)=\\emptyset$.  If $c_1$, $c_2\\in G$ so that $Supp(h_1^{c_1})\\cup Supp( h_1^{c_2})=X$ then $Supp(h_2^{c_1})\\cap Supp(h_2^{c_2})=\\emptyset$.\n\\end{remark}\n\n\\begin{proof}\nSuppose \n\\[\nX=Supp(h_1^{c_1})\\cup Supp(h_1^{c_2}) (= c_1\\cdot Supp(h_1)\\cup c_2\\cdot Supp(h_1)).\n\\]\nIf there is $x\\in X$ so that  $x\\in Supp(h_2^{c_1})\\cap Supp(h_2^{c_2})$, then $x=c_1\\cdot y$ and $x=c_2\\cdot z$, where $y$ and $z$ are in $Fix(h_1)$.  In particular, $x\\in c_1\\cdot Fix(h_1)\\cap c_2\\cdot Fix(h_1)$.  This implies that $c_1\\cdot Supp(h_1)\\cup c_2\\cdot Supp(h_1)\\neq X$.\n\\end{proof}\n\nRemark \\ref{disjointSupportsAndConjugation} immediately implies that we cannot  use Condition \\ref{kesten-test} when approaching the question of the simplicity of the algebra $\\cstar{T}$.\n\n\\begin{corollary}\\label{negative}\nSuppose that $H\\subset T$ admits elements $h_1$ and $h_2$ so that $Supp(h_1)\\cap Supp(h_2)=\\emptyset $.  Then for $n\\geq 2$ there is no set of elements $\\left\\{c_1,c_2\\ldots,c_n\\right\\}$ so that $\\langle h^{c_1},h^{c_2}, \\ldots,h^{c_n}\\rangle$ is a free group on $n$ generators for all $h\\in H$.\n\\end{corollary}\n\\begin{proof}\nSuppose $H:=\\left\\{h_1,h_2,\\ldots,h_k\\right\\}$ is a finite set with cardinality at least two, and $h_1$ and $h_2$ are in $H$ so that $Supp(h_1)\\cap Supp(h_2)=\\emptyset.$  Further suppose that $n\\geq 2$ is fixed and $c_1$, $c_2$,$\\ldots$, $c_n$ are chosen so that for all $h\\in H$, we have $\\langle h^{c_1},h^{c_2},\\ldots, h^{c_n}\\rangle$ is free on $n$ generators.  As proven and Brin and Squier's paper \\cite{BSPLR}, the group of piecewise linear homeomorphisms of the unit interval has no non-abelian free subgroups, so we see immediately that $Supp(h_1^{c_1})\\cup Supp(h_1^{c_2})=S^1$.  Now by Remark \\ref{disjointSupportsAndConjugation} we know that  $Supp(h_2^{c_1})\\cap Supp(h_2^{c_2})=\\emptyset$  Therefore $\\langle h_2^{c_1},h_2^{c_2}\\rangle\\cong \\mathbb{Z}\\times\\mathbb{Z}$.\n\\end{proof}\n\nWe now offer an apparently weaker version of Condition \\ref{kesten-test} which will be used throughout the remainder of this article.  First, we need a supporting theorem.\n\nBelow, let $C_G(g)$ be the centralizer of an element $g$ in $G$.\n\n\\begin{theorem}\nLet $H\\subset G$ be a finite set and there is an element $w\\in H$ such that for all positive integers $n$ there is a set $\\{c_1,c_2,\\ldots,c_n\\}\\subset G$ and $r,s\\in G$ such that $c_i \\in C_G(swr)$ for all $i$ and \n\\[\n\\lim_{n\\to\\infty}\\frac{1}{n}||\\Sigma_{i=1}^n \\lambda(c_isgr{c_i}^{-1})||=0,\\,\\text{ for all } g\\in H\\backslash \\{w\\},\n\\]\nthen for all coefficients $\\beta_g$ indexed by $H$ with $\\beta_w\\neq 0$, the ideal generated by $\\sum_{g\\in H}\\beta_g \\lambda(g)$ is equal to  $\\cstar{G}$.\n\n\n\\end{theorem}\n\\begin{proof}\nLet $I$ be an ideal in $C^*_{\\lambda}(G)$ generated by $b:=\\sum_{g\\in H}\\beta_g \\lambda(g)$. Assume that $I$ is proper. The closure of $I$ is proper, thus we can assume $I$ is closed. Note that $\\Sigma_{i=1}^n \\lambda(c_is)b\\lambda(rc_i^{-1})\\in I$. Since $c_i \\in C_G(swr)$ we have \n\\begin{align*}\n\\|\\lambda (swr)-\\frac{1}{\\beta_w n}\\Sigma_{i=1}^n \\lambda(c_is)b\\lambda(rc_i^{-1})\\|=&\\frac{1}{\\beta_w n} \\|\\Sigma_{g\\in H\\backslash \\{w\\}}\\Sigma_{i=1}^n \\beta_g\\lambda(c_isgrc_i^{-1})\\| \\\\\n\\leq& \\frac{1}{\\beta_w} \\Sigma_{g\\in H\\backslash \\{w\\}}|\\beta_g|\\frac{1}{n}\\|\\Sigma_{i=1}^n \\lambda(c_isgrc_i^{-1})\\|\\\\\n=&\\max(|\\beta_g|\/ \\beta_w:g\\in H)\\cdot \\max(\\frac{1}{n}\\|\\Sigma_{i=1}^n \\lambda(c_isgrc_i^{-1})\\|:g\\in H).\n\\end{align*}\nBy our assumptions, the last quantity can be arbitrarily small for large $n$. Thus there is an element in $I$ which is on distance less then $1$ to a unitary operator, this implies that it is invertible and $I=\\cstar{G}$.\n\\end{proof}\n\nApplying the theorem above  to the set $H\\cup \\{e\\}$ shows that the following condition implies simplicity of $C^*_{\\lambda}(G)$ :\n\n\\begin{condition}\\label{weak-kesten}\nFor all finite non-empty subsets $H\\subset G$, $e\\not\\in H$ and for all positive integers $n$ there are $r$, $s\\in G$ and a set $\\{c_1,c_2,\\ldots,c_n\\}\\subset C_G(sr)$  such that the set $\\{c_k(sgr)c_k^{-1}: k=1,\\ldots,n\\}$ is free for all $g\\in H$.\n\\end{condition}\n\n\nCondition \\ref{kesten-test} implies Condition \\ref{weak-kesten} and it seems that the other implication is false.  However, Condition \\ref{weak-kesten} is still inadequate for showing that $\\cstar{T}$ is simple.\n\\begin{lemma}\nThere are $g_1$, $g_2\\in T\\backslash \\{e\\}$ so that for any $r$, $s\\in T$ there are no elements $c_1$, $c_2$, $c_3$, and $c_4\\in C_T(sr)$ with both $G_1=\\langle (sg_1r)^{c_1},(sg_1r)^{c_2}, (sg_1r)^{c_3},(sg_1r)^{c_4}\\rangle$ and $G_2=\\langle (sg_2r)^{c_1},(sg_2r)^{c_2}, (sg_2r)^{c_3},(sg_2r)^{c_4}\\rangle$ free on four generators.\n\\end{lemma}\n\\begin{proof}\nLet $g_1$, $g_2\\in T$ so that $Supp(g_1)=(0,1\/2)$ and $Supp(g_2)=(1\/2,1)$.  Let $r$ and $s\\in T$ and suppose $c_1$, $c_2$, $c_3$ and $c_4\\in C_T(sr)$.  Set $k_{ij}=(sg_ir)^{c_j}$, for $i$, $j\\in \\{1,2,3,4\\}$, and suppose that  $c_1$, $c_2$, $c_3$ and $c_4$ were so chosen so that $G_i=\\langle k_{i1},k_{i2}, k_{i3}, k_{i4}\\rangle$ is free on four generators for $1\\leq i\\leq 2$.\n\nConsider the intervals $X_{i1}=(c_1r^{-1})\\cdot Fix(g_i)$, $X_{i2}=(c_2r^{-1})\\cdot Fix(g_i)$, $X_{i3}=(c_3r^{-1})\\cdot Fix(g_i)$, and $X_{i4}=(c_4r^{-1})\\cdot Fix(g_i)$.  If $x_{ij}\\in X_{ij}$, then $k_{ij}\\cdot x_{ij} = c_jsg_irc_j^{-1}\\cdot x_{ij} =(c_jsg_i)\\cdot y_{ij}=(c_js)\\cdot y_{ij} = (c_jsrc_j^{-1})\\cdot x_{ij} = (sr)\\cdot x_{ij}$, as for all $i$, $j$ we have $y_{ij}\\in Fix(g_i)$.  That is, $k_{ij}$ acts as $sr$ over $X_{ij}$.\n\nFurther, consider the elements $f_{i,ab}= k_{ia}^{-1}k_{ib}$, where $i\\in \\{1,2\\}$ and $a\\neq b \\in \\{1,2,3,4\\}$.  It is immediate that $\\langle f_{i,ab},f_{i,cd}\\rangle$ is free on two generators if either $b\\neq c$ or $d\\neq a$.  Therefore, by Brin and Squier's result (from \\cite{BSPLR}) that $PL_o(I)$ has no non-abelian free subgroups, we know that $Fix(f_{i,ab})\\cap Fix(f_{i,cd})=\\emptyset$ for $i\\in\\{1,2\\}$ and either $b\\neq c$ or $d\\neq a$.  Now, for instance, if there is an index $i$ and some point $p\\in X_{i1}\\cap X_{i2}\\cap X_{i3}$,  then both $f_{i,12}$ and $f_{i,13}$ must fix $p$, which is a contradiction.  Therefore we see that $X_{i1}$, $X_{i2}$, and $X_{i3}$ cannot share a common point for any index $i$.  By the same argument, for any valid indices $i$, $a$, $b$, and $c$ (where $i\\in \\{1,2\\}$ and $a\\neq b\\neq c\\neq a$) we see that $X_{ia}\\cap X_{ib}\\cap X_{ic}=\\emptyset$.\n\nOne now sees immediately that for any valid indices $i$, $a$,$b$, and $c$ (where $i\\in \\{1,2\\}$ and $a\\neq b\\neq c\\neq a$) we must also have that $X_{ia}\\cup X_{ib}\\cup X_{ic}=S^1$.  This follows as otherwise there is some point $p$ in the intersection $X_{ja}\\cap X_{jb}\\cap X_{jc}$ for the index $j\\neq i$ (since $X_{1*} = \\overline{S^1\\backslash X_{2*}}$ for any index $*$).  \n\nSuppose that for some indices $i$, $a\\neq b$ we have that $X_{ia}\\subset X_{ib}$, and let $c$ and $d$ be the two remaining distinct indices of $\\{1,2,3,4\\}\\backslash\\{a,b\\}$.  Let $p$ be an endpoint of $X_{ib}$.  We have that $p$ must be in both $X_{ic}$ and $X_{id}$, otherwise their will be some point $q \\in S^1\\backslash X_{ib}$ which is near to $p$ so that $q$ is not in either of $X_{ia}\\cup X_{ib}\\cup X_{ic}= X_{ib}\\cup X_{ic}$ or $X_{ia}\\cup X_{ib}\\cup X_{id}= X_{ib}\\cup X_{id}$.  But this contradicts the fact that $X_{ib}\\cap X_{ic}\\cap X_{id}= \\emptyset$.\n\nIt now immediately follows that for any index $i$ and two distinct indices $a$ and $b$, we have that $X_{ia}\\cap X_{ib}$ is a non-empty closed interval (possibly a single point) while $X_{ia}\\cup X_{ib}$ is also a closed interval which misses some points in $S^1$.\n\nBut now we are done as follows.  For any index $i$ the intervals $X_{i1}$ $X_{i2}$ and $X_{i3}$ cover the circle, and have the properties that each pair of sets intersects in an interval, and  no pair covers the whole circle.  Now consider $X_{i4}$.  It must likewise intersect both $X_{i1}$ and $X_{i2}$ non-trivially, and the union of $X_{i1}$, $X_{i2}$ and $X_{i4}$ also covers the whole circle.  Therefore the end of $X_{i1}$ which is not in $X_{i2}$ is in both $X_{i3}$ and $X_{i4}$.  Hence $X_{i1}\\cap X_{i3}\\cap X_{i4}\\neq \\emptyset$, which implies that the group $G_i$ cannot be free on four generators, as $f_{i,13}$ and $f_{i,14}$ share a common fixed point and will not generate a free subgroup of $G_i$.\n\\end{proof}\n\n\\begin{remark}We observe that it is still plausible that even with $g_1$ and $g_2$ as in the proof above (supports over $(0,1\/2)$ and $(1\/2,1)$, respectively), one could plausibly find $r$, $s$, and $c_1$, $c_2$, and $c_3\\in C_T(sr)$ so that setting $k_{ij}= c_jsg_irc_j^{-1}$ as above we would have $H_r=\\langle k_{r1}, k_{r2}, k_{r3}\\rangle$ free on three generators for both $r=1$ and $r=2$, where the related claim for even two generator free groups could not be conceived of under Condition \\ref{kesten-test}.\n \\end{remark}\n \n\\section{A Ping-Pong Lemma for orientation preserving homeomorphisms of $S^1$}\\label{p-pong}\n\nIn this section, we prove a version of the Ping-Pong Lemma which we are using in our main argument.  In the notations below we write all actions as left actions, in keeping with the tradition in the $C^*$ literature, although much Thompson groups literature uses right action. In particular, if $x\\in S^1$ and $s$,$t\\in T$, we write $tx$ for the image of $x$ under $t$, and the conjugation $s^t:= sts^{-1}$, which means, apply $s^{-1}$ first, then $t$, and then $s$. We consider finite sets with repetitions. \n\nIn support of that lemma we ask the reader to recall an ordinary statement of Fricke and Klein's Ping-Pong Lemma (first proven in \\cite{FrickeKlein}, but we give a different statement), and two further facts, one quite classical.\n\n\n\n\\begin{lemma}(Ping-Pong Lemma)\\label{ping-pong} Let $G$ be a group of permutations on a set $X$, and let $a$, $b\\in G$, where $b^2\\neq 1$.  If $X_a$ and $X_b$ are two subsets of $X$ so that neither is contained in the other, and for all integers $n$ we have $b^n\\cdot X_a\\subset X_b$ whenever $b^n\\neq 1$, and $a^n\\cdot X_b\\subset X_a$ whenever $a^n\\neq 1$, then $\\langle a,b\\rangle$ factors naturally as the free product of $\\langle a\\rangle$ and $\\langle b\\rangle$.   In particular, $\\langle a,b\\rangle\\cong \\langle a\\rangle *\\langle b\\rangle$.\n\\end{lemma}\n\nSuppose that $f:S^1\\rightarrow S^1$ is an orientation preserving homeomorphism of the circle $S^1=\\mathbb{R}\/\\mathbb{Z}$, then $f$ may be lifted to a homeomorphism of $\\mathbb{R}$ by $F(x+m)=F(x)+m$ for every  $x$ and $m$. The rotation number of $f$ is defined to be $Rot(f)=\\lim_{n\\rightarrow \\infty} (F^n(x)-x)\/{n}$.\nThe following theorem is generally relevant to the arguments in the final section of this paper, and appears first in \\cite{GhysSergiescu}, although there now exist many different proofs, the shortest of which appears to be in \\cite{BKMStructure}.\n\\begin{theorem}\nEvery element of Thompson's group $T$ has rational rotation number.\n\\end{theorem}\nThe last tool we need in order to establish our own version of the Ping Pong lemma is the following classical result of Poincar\\`e.\n\\begin{lemma}(Poincar\\`es Lemma, circa 1905)\nIf $f$ is an orientation preserving homeomorphism of $S^1$ and $f$ has rotation number $p\/q$ in lowest terms, then there is an orbit in $S^1$ of size exactly $q$ under the action of $\\langle f\\rangle$.\n\\end{lemma}\n\n\n\nWe are now in a good position to quote and prove our main technical tool.\n\n\n\\begin{lemma}\\label{free-powering-one}\nSuppose $a$ and $b$ are orientation preserving homeomorphisms of the circle $S^1$ with rational rotation numbers $Rot(a)=p\/q$ and $Rot(b)=r\/s$ in lowest non-negative terms where\n\\begin{enumerate}\n\\item  $b$ is not torsion, and \n\\item  if $x\\in Fix(b^s)$ and $j\\in\\mathbb{Z}$ with $a^j\\neq 1_T$, then we have $a^jx\\not\\in Fix(b^s)$,\n\\end{enumerate}\nthen, there is a positive integer $k$ so that $a$ and $b^k$ are a free basis for the group $\\langle a, b^k\\rangle$.\n\\end{lemma}\n\\begin{proof}\nIn the proof below, let us take $a$, $b\\in T$ and $p$, $q$, $r$, $s\\in \\mathbb{N}$ as in the statement of the lemma. \nSet $b_0:=b$.  We will occasionally update to a new version of $b$, which will be given by a new index.  The new $b$ will always be an integral power of the previous indexed $b$.\n\nSet $b_1:=b_0^s$.  The element $b_1$ will have rotation number $0\/1$ in lowest non-negative terms.  For $b_1$, we have $Fix(b_1)$ is not empty, and also not the whole circle (else $b$ was originally a torsion element in $T$).\n\n\n\\newcommand{under the action of $\\langle a\\rangle$}{under the action of $\\langle a\\rangle$}\n\\newcommand{under the action of $\\langle b\\rangle$}{under the action of $\\langle b\\rangle$}\n\\newcommand{\\mathscr{I}}{\\mathscr{I}}\n\nLet $\\mathscr{I}\\subset S^1$ be such that for each component $C$ of $Supp(b_1)$, we have $|C\\cap \\mathscr{I}|=1$, and associate each such $C$ with its unique point in $\\mathscr{I}$, so that $\\mathscr{I}$ becomes an index set for the components of $Supp(b_1)$.  We observe that $\\mathscr{I}$ comes with an inherent circular order as a subset of $S^1$.  Let $L_b$ represent the set of limit points of $\\mathscr{I}$ which are not in $\\mathscr{I}$, and observe that $L_b\\subset Fix(b_1)$.\n\nFor each positive integer $d$, set $\\Delta_d:=[-d,d]\\cap(\\mathbb{Z}\\backslash\\{0\\})$, the set of non-zero integers a distance $d$ or less from zero.  Now for all positive integers $d$ we can set $\\epsilon_d$ to be one half of the distance from $Fix(b_1)$ to the set $\\cup _{i\\in\\Delta_d} a^i\\cdot Fix(b_1)$.  Noting that these $\\epsilon_d$ are all well defined and non-zero (unless $a$ is torsion) as the sets involved are compact and as $a^m\\cdot Fix(b_1)\\cap a^n\\cdot Fix(b_1) \\neq \\emptyset$ implies that either $m=n$ or that $a$ is torsion and $n-m$ is divisible by the order of $a$.\n\n\n Our analysis now splits, depending on whether or not $a$ is torsion.  In the case that $a$ is torsion, our proof is somewhat easier, so we will execute that proof immediately.\n\n\\vspace{.1 in}\n{\\flushleft {\\it \\underline{Case}: $a$ is torsion with order $q$.}}\n\nIn this case, the value $\\epsilon_{q-1}$ explicitly measures one half of the distance between $Fix(b_1)$ and the union of the images of $Fix(b_1)$ under the action of non-trivial powers of $a$.  Set $\\mathcal{U}$ to be the open $\\epsilon_{q-1}$ neighbourhood of $Fix(b_1)$, and observe that for each integer $i\\in \\{1,2,\\ldots ,q-1\\}$ we have $a^i\\cdot Fix(b_1)\\cap \\mathcal{U}=\\emptyset$.  For each non-zero $i\\in \\left\\{1,2,\\ldots q-1\\right\\}$ set $\\mathcal{U}_i$ to be the $\\epsilon_{q-1}$ neighbourhood of $a^i\\cdot Fix(b)$.  Again, for all such indices $i$, $\\mathcal{U}\\cap\\mathcal{U}_i=\\emptyset$.  Set \n\\[\nX_b:= \\mathcal{U}\\bigcap_{1\\leq i<q}(a^{q-i}\\cdot\\mathcal{U}_i). \n\\]  Now by construction we have that the image set $a^i\\cdot X_b\\subset \\mathcal{U}_i$, but $\\mathcal{U}_i\\cap X_b=\\emptyset$ since $\\mathcal{U}_i\\cap \\mathcal{U}=\\emptyset$ and $X_b\\subset \\mathcal{U}$.  Thus, $X_b$ is an open set containing $Fix(b_1)$ that is disjoint from its image under the action of any nontrivial power of $a$.\n\nAs $X_b$ and the components of support of $b_1$ altogether cover the circle, there is a finite set of open interval components of $Supp(b_1)$ which together with $X_b$ covers the circle.  In turn, this implies there is a minimal positive integer $v$ so that for all $x\\in S^1\\backslash X_b$, we have $b_1^v x\\in X_b$ and $b_1^{-v} x\\in X_b$. \n\n Now we can set\n\\[\nX_a:= \\cup_{1\\leq i<q} (a^i\\cdot X_b).\n\\]\nWith this choice of $X_a$ and $X_b$ we have arranged that we satisfy the hypotheses of Lemma \\ref{ping-pong} for the elements $a$ and $a^k$ where $k=v\\cdot s$.\n\n{\\flushleft {\\it \\underline{Case}: $a$ is not torsion.}}\n\nThroughout this case, given a set $X\\subset S^1$, and $\\epsilon>0$, we shall use the notation $N_\\epsilon(X)$ to denote the open $\\epsilon$-neighbourhood of $X$, that is, all points in $S^1$ a distance less than $\\epsilon$ from some point in $X$.\n\nIn this case with $a$ not torsion, we must specify the set $F_a:= Fix(a^q)$, which is a closed non-empty subset of the circle which is disjoint from $Fix(b_1)$.  Choose a specific $\\epsilon>0$ so that  $N_\\epsilon(Fix(b_1))\\cap N_\\epsilon(F_a)= \\emptyset$, noting that such an epsilon value exists as $Fix(b_1)$ and $F_a$ are disjoint compact subsets of $S^1$.\n\nLet $m$ be a positive integer so that both $a^{mq}\\cdot Fix(b_1)\\subset N_\\epsilon(F_a)$ and $a^{-mq}\\cdot Fix(b_1)\\subset N_\\epsilon(F_a)$.  This $m$ exists as $a^{q}$ acts as a monotone strictly increasing, or as a monotone strictly decreasing function over each component of its support, and as the limit point of any point in a component of support of $a^q$ under increasing powers of $a^q$ must be a fixed point of $a^q$ (and similarly under a negative powers of $a^q$), and as $Fix(b_1)$ is a compact set and hence is contained in a union of finitely many components of support of $a^q$.\n\nWe now observe that for $n$ an integer with $|n|>m$, we have that $a^{nq}\\cdot Fix(b_1)\\subset F_a$ as well.  We would like to argue a stronger result now that there is a positive constant $N$ so that for all $j>N$ we have $a^{j}\\cdot Fix(b_1)\\subset N_\\epsilon(F_a)$ and $a^{-j}\\cdot Fix(b_1)\\subset N_\\epsilon(F_a)$.\n\nTo make this argument, the main point to observe is that there is an induced action of $\\langle a\\rangle$ on the set of components of support of $a^q$ which partitions these components into (possibly infinitely many) orbits of size $q$.  Further, as $a$ commutes with $a^q$ and $a$ is orientation preserving, it is easy to see that each such orbit consists of components of support where the action of $a^q$ is increasing on all components of the orbit, or decreasing on all components of the orbit.\n\nIt is also the case that there are only finitely many components of support of $a^q$ which are not already wholly contained in $N_\\epsilon(F_a)$.  Let $C_1$, $C_2$, $\\ldots$, $C_w$ represent these components, and observe that $Fix(b_1)$ is contained in the union $K$ of these compact intervals. For each component $C_j$, let $I_j$ be the closed interval $C_j\\backslash N_\\epsilon(F_a)$.  Now each of these components $C_j$ are in an orbit of length $q$ amongst the components of support of $a^q$, and in each such orbit the action of $a^q$ on each component is in the same direction.  Hence there is a finite number $N$ so that for all $j>N$ and intervals $I_m$, we have that $a^j\\cdot I_m\\subset N_\\epsilon(F_a)$, and also $a^{-j}\\cdot I_m\\subset N_\\epsilon(F_a)$.\n\nNow define $J$ as below:\n\\[\nJ:=\\cup_{i\\in\\Delta_N} ((a^i\\cdot Fix(b_1))\\cap K).\n\\]\nWhere we recall that $\\Delta_n=[-n,n]\\cap(\\mathbb{Z}\\backslash\\{0\\})$ for any particular $n\\in\\mathbb{N}$.\n\nIt is immediate that $J$ is a compact set which is disjoint from $Fix(b_1)$.  As such, there is a $\\delta>0$ so that $\\delta<\\epsilon$ and the $\\delta$-neighbourhood $N_\\delta(Fix(b_1))$ of $Fix(b_1)$ is disjoint from the set $V_\\delta$ defined as\n\\[\nV_\\delta:=\\cup_{i\\in\\Delta_N} (a^i\\cdot N_\\delta(Fix(b_1)))\n\\]\n and noting that as $\\delta<\\epsilon$ we also have that $N_\\delta(Fix(b_1))$ is disjoint from $N_\\epsilon(F_a)$.\n\nNow set $X_b:=N_\\delta(Fix(b_1))$ and $X_a:=N_\\epsilon(F_a)\\cup V_\\delta$.\n\nBy construction, there is an integer $z>0$ so that $b_1^z$ takes the complement of $X_b$ (and so, $X_a$) into $X_b$, while all non-trivial powers of $a$ take $X_b$ into $X_a$.  Hence the integer $k=s\\cdot z$ has the property that  $a$ and $b^k$ freely generate a free group of rank $2$.\n\\end{proof}\n\n\\section{Applying Condition \\ref{weak-kesten}, and variants, in $T$}\\label{trying-conditions}\nHere we list lemmas, where the Condition \\ref{weak-kesten} can be used.\n\n\\begin{lemma}\\label{fixed-point}\nLet $H$ be a finite set of nontrivial elements in $T$ so that there is some point $p\\in \\cap_{h\\in H} Supp(h).$   Then, for any positive integer $n$ there is an element $g\\in T$  and $\\{c_1,c_2,\\ldots,c_n\\}$ so that $c_i\\in C_T(g)$ for all $i$, and so that for all $h\\in H$ we have the set\n\\[\nG_h:=\\left\\{ (gh)^{c_i}\\mid i\\in \\left\\{1,2,\\ldots, n\\right\\}\\right\\}\n\\]\nis a free basis for a free group of rank $n$.\n\\end{lemma}\n\\begin{proof}\nLet $H$ and $p$ as in the statement of the lemma, and let $n\\in \\mathbb{N}$ be given.  For each $h\\in H$, let $Rot(h):=r_h\/s_h$ written in lowest terms (NB, any finite periodic orbit under the action of $\\langle h\\rangle$ is of length $s_h$).  By the definition of $p$, we see there is a non-empty interval $(a,b)$ with $p\\in (a,b)$ so that  for all $h\\in H$ we have $(a,b)\\cdot h^j\\cap (a,b)=\\emptyset$ for all $1\\leq j<s_h$.\n\nNow let $g\\in T$ be an element with rotation number $Rot(g)=0$ which fixes exactly the point $p$.  We can choose $g$ so that for all $h\\in H$ the product $gh$ has precisely two fixed points which are generated from the contact of the graph of $g$ with the graph of the element $h^{-1}$, by choosing the graph of $g$ to be near to a step function (the nearly vertical component of the graph of $g$ should be steeper than any slope of any element of $H$, and the nearly horizontal component of the graph of $g$ should have slope closer to zero than any slope of any element of $H$, and, in a small epsilon box around $(p,p)$ which misses the graphs of the elements of $H$, the graph of $g$ is unrestrained).  As $g$ fixes the point $p$, it is the case that that for all elements $h\\in H$, we have $k_h\\cdot p=gh\\cdot p=g\\cdot(h\\cdot p) \\neq p$ and further that under the action of $\\langle k_h\\rangle$, $p$ is in an infinite orbit which limits to the two ends of the component of support of $k_h$ which contains $p$.\n\\newcommand{\\mathop{LCM}}{\\mathop{LCM}}\nIn these conditions we can immediately apply Lemma \\ref{free-powering-one} to claim that for each $h\\in H$, there is a power $g^{\\rho_h}$ of $g$ so that $k_h$ and the element $g^{\\rho_h}$ freely generate a free group of rank $2$.  Now take $\\theta:=\\mathop{LCM}\\{\\rho_h\\mid h\\in H\\}$, so that $g^\\theta$ is free for $k_h$ for each $h$.\n\nNow, it is immediate that setting $c_i:= g^{i\\cdot \\theta}$ for indices $i$ in the range $1\\leq i\\leq n$,  we can create sets $X_h:=\\{(gh)^{c_i}\\}_{i=1}^n$ so that $\\langle X_h\\rangle$ is free on $n$-generators for all $h\\in H$.\n\n\\end{proof}\nThe following is some weakening of Condition \\ref{weak-kesten} which we can achieve in Thompson's group $T$.  Unfortunately, by raising to the power $p$, we find a free subgroup of $K:=\\left\\{ ((hg)^{c_i})\\mid i\\in \\left\\{1,2,\\ldots, n\\right\\}\\right\\}$ which is generally not finite index.\n\n\\begin{lemma}\\label{powerfree}  Let $H$ be a finite set of nontrivial elements in $T$ with cardinality $p$.  Then there is an element $g\\in T$ such that for any positive integer $n$ there are elements  $\\{c_1,c_2,\\ldots,c_n\\}$ so that $c_i\\in C_T(g)$ for all $i$, and so that for all $h\\in H$  the set\n\\[\nG_h:=\\left\\{ ((hg)^{c_i})^p\\mid i\\in \\left\\{1,2,\\ldots, n\\right\\}\\right\\}\n\\]\nfreely generates a free group of rank $n$.\n\\end{lemma}\n\\begin{proof}\nIndex the $h\\in H$ so that $H=\\left\\{h_1,h_2,\\ldots,h_p\\right\\}$ for some minimal positive integer $p$.  As each element $t\\in T$ has a rational rotation number $Rot(t)\\in \\mathbb{Q}\/\\mathbb{Z}$, let  $Rot(h_i)=r_i\/s_i$ expressed in lowest positive terms.  For each index $i$, inductively choose a point $x_i\\in Supp(h_i)$ so that the set $$X_i:=\\left\\{ h_j^{-1}\\cdot x_i, x_i, h_j\\cdot x_i\\mid 1\\leq j\\leq i\\right\\}$$ is disjoint from the union $\\cup _{j<i}X_j$, and where each $x_i$ is chosen as a dyadic rational so that no point in $X_i$ is an end of a component of support of $h_j^{s_j}$ for any index $j$.  Note that in all cases we are choosing an $x_i$ so that a finite set of images miss a finite set of points in the circle, and this is easy to do.  Set $P:=\\left\\{x_i\\mid 1\\leq i\\leq p\\right\\}$.  Reindex $P$ so that $0\\leq x_1<x_2<\\cdots<x_p<1$, and reindex the sets $X_i$ correspondingly.\n\nNow choose, for each $i$, an interval $(a_i,b_i)$ centred around $x_i$, and set $$I_i:=\\left\\{h_j^{-1}\\cdot(a_i,b_i), (a_i,b_i), h_j\\cdot (a_i,b_i)\\mid 1\\leq j\\leq i\\right\\}$$ where each $(a_i,b_i)$ is chosen small enough so that each element of $I_i$ is disjoint from the union $\\cup_{j<i, B\\in I_j}B$, and where $(a_i,b_i)$ is disjoint from $h_j^{-1}\\cdot(a_i,b_i)$  and $h_j\\cdot(a_i,b_i)$ whenever $x_i\\in Supp(h_j)$, and where $h_j^{-1}\\cdot(a_i,b_i)$  and $h_j\\cdot(a_i,b_i)$ intersect each other non-trivially only if $h_j$ is torsion of order two.  We can insist these intervals are possibly smaller still, so that the complement of the union $\\tilde{U}:=\\cup_{1\\leq j\\leq p, B\\in I_j}B$  is a union of closed intervals each of which has non-empty interior.  We observe that by our choices of the sets $X_i$, we can so choose our intervals $(a_i,b_i)$.\n\nNow let $\\tilde{g}$ be a non-torsion element with rotation number $1\/p$ which admits exactly one orbit of size $p$ under the action of $\\langle\\tilde{g}\\rangle$, where the orbit is the set $P$, and where $\\tilde{g}$ cyclically permutes the $x_i$ in order, taking $x_1$ to $x_2$ and etc. \n\nThe element $\\gamma:=\\tilde{g}^p$ has $p$ components of support $S_1:=(x_1,x_2)$, $S_2:=(x_2,x_3)$, $\\dots$, $S_p:=(x_p,x_1)$, and $\\gamma$ is either increasing on each component of support, or decreasing on each component of support.  We re-choose $\\tilde{g}$ if necessary so that it satisfies all previous conditions and so that $\\gamma$ is increasing on each component of support, and note that such elements $\\tilde{g}$ exist.\n\nNow set $U:=\\cup_{1\\leq j\\leq p} (a_i,b_i)$, which union also has the property that the complement $C:= S^1\\backslash U$ is a union of $p$ disjoint closed intervals $D_i:=[b_i,a_{i+1}]$ (where we set $a_{p+1}:= a_1$) each of which is not a point.  As such is the case, we can find a positive integer $d$ so that $\\gamma^d\\cdot C \\subset U$ as after $d$ iterations of $\\gamma$, the set $D_i$ will be moved so as to have image just to the left of $x_{i+1}$ (where again, we set $x_{p+1}:= x_1$).\n\nNow set $g:= \\gamma^d\\tilde{g}$.  The element $g$ so constructed has rotation number $1\/p$ as before, and acts on the set $P$ as $\\tilde{g}$ does, but it has the further property that for all integers $s\\neq 0$, $g^s\\cdot C\\subset U.$\n\nNow set, for each index $i$, $k_i:=g h_i$.  We will now show that for all $k_i$, the point $x_j$ is not in any finite periodic orbit of $k_i$, for any index $j$.  \n\nLet $i$ and $j$ be indices so that $k_i$ and $x_j$ are an element and point as defined above, respectively.  Let $q$ be an index and consider the interval $(a_q,x_q]$ under the action of $g$, and then consider the interval $(a_j,x_j]$ under the action of $k_i$.\n\nUnder the action of $g$, we know already that $x_q\\mapsto x_{q+1}$, and that the component $[b_{q-1},a_q]$ of $C$ maps into the interval $(a_{q+1},x_{q+1})$.  In particular, the whole interval $[b_{q-1},x_q]$ is mapped into $(a_{q+1},x_{q+1}]$ by $g$ and so the interval $[a_q,x_q]$ is mapped into $(a_{q+1},x_{q+1}]$ as well, and precisely the point $x_q$ will map to $x_{q+1}$.\n\nNow consider the action of $k_i$ on $(a_j,x_j]$.  If $x_j$ is in the support of $h_i$, then $h_i\\cdot(a_j,x_j]\\subset C$, whereupon that image is moved into some interval $(a_*,x_*)$ by the action of $g$.\n\nIn particular, $k_i^z\\cdot x_j = x_{j+z}$ (in cyclic order) as long as for all $0\\leq y\\leq z$ we have $x_{j+y}\\not\\in Supp(h_i)$.  If instead for some minimal integer $s\\in\\left\\{0,1,\\ldots,p-1\\right\\}$ we have $x_{j+s}\\in Supp (h_i)$ then on $k_i^{s+1}$ will move $x_j$ out of $P$ as we will have $k_i^{s+1}\\cdot x_j\\in (a_{*},x_{*})$ for some index $*$.  Furthermore, if this happens, all further iterates of $x_j$ will fail to re-enter $P$.\n\nHowever, $x_i$ is itself in the support of $h_i$, so this eventuality happens (in $p$ steps or fewer), and so no $x_j$ is on a finite periodic orbit under the action of $\\langle k_i\\rangle$, for any indices $j$ and $i$.\n\nIn particular, for all indices $j$, we see that $ (k_j^{tp}\\cdot P)\\cap P=\\emptyset$ for all $t\\in \\mathbb{Z}\\backslash\\left\\{0\\right\\}$. \n\nWe can now quote Lemma \\ref{free-powering-one}, using $a:= k_j^p$ for each index $j$ and $b=g$ to claim that for each index $j$ there is an integer $z_j$ so that the group $\\langle k_i^p, g^{z_j}\\rangle$ is free on two generators  (in this setup, $P=Fix(b^s)=Fix(g^p)$).  Now set $z:=LCM(\\left\\{z_1,z_2,\\ldots,z_p\\right\\})$ the least common multiple of the values $z_j$, and then we have that for all indices $j$, the elements $k_j^p$ and $g^z$ generate a free group.  In particular, setting $c_i:= g^{iz}$ for $i\\in \\left\\{1,2,\\ldots, n\\right\\}$ we see that $\\Gamma_j:=\\left\\{ (k_j^p)^{c_i}=(k_j^{c_i})^p\\mid 1\\leq i\\leq n, i\\in\\mathbb{Z}\\right\\}$ freely generates a free group of rank $n$.\n\nBut now, recalling that $k_i=g h_i$, and that therefore $[g,c_i]=1$, we have proved our claim. \n\\end{proof}\n\n\\section{Some questions}\\label{conjectures}\nThe last result in particular shows that our Condition \\ref{weak-kesten} is almost (in some sense) enough to show that $\\cstar{T}$ is simple.  But not quite! This, together with our partial converse of the Haagerup-Olesen result,  encourages us to ask the foliowing question.\n\\begin{question}\nIs the non-amenability of Thompson's group $F$ equivalent to the simplicity of the algebra $\\cstar{T}$.\n\\end{question}\n\nAs the setup of Condition \\ref{weak-kesten} is more flexible than that of Condition \\ref{kesten-test}, we find Condition \\ref{weak-kesten} easier to use.  Still, we have not actually proven that the conditions are not equivalent.  Thus it would be quite useful to give a positive answer to the following question.\n\\begin{question}\nIs there a group $G$ which fails to satisfy Condition \\ref{kesten-test} but which does satisfy Condition \\ref{weak-kesten}?\n\\end{question}\n\nIn \\cite{JM}, \\cite{JS} and \\cite{JNS} several approaches to amenability via group actions were developed. It is an interesting question to relate these approaches to properties of a group's $C^*$-algebra.\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\n\nSparse modeling in linear regression has been a topic of fervent interest in recent years \\cite{hastie,bvdg}. This interest has taken several forms, from substantial developments in the theory of the Lasso to advances in algorithms for convex optimization. Throughout there has been a strong emphasis on the increasingly high-dimensional nature of linear regression problems; in such problems, where the number of variables $p$ can vastly exceed the number of observations $n$, sparse modeling techniques are critical for performing inference.\n\n\n\\subsubsection*{Context}\n\nOne of the fundamental approaches to sparse modeling in the usual linear regression model of $\\mb y={\\mb X}{\\bs\\beta}+\\bs\\epsilon$, with $\\mb y\\in\\mathbb{R}^n$ and ${\\mb X}\\in\\mathbb{R}^{n\\times p}$, is the\nbest subset selection \\cite{miller2002subset} problem:\n\\begin{equation}\\label{eqn:a10}\n\\displaystyle\\min_{\\|{\\bs\\beta}\\|_0\\leq k}\\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2,\n\\end{equation}\nwhich seeks to find the best choice of $k$ from among $p$ features that best explain the response in terms of the least squares loss function.\nThe problem \\eqref{eqn:a10} has received extensive attention from a variety of statistical and optimization perspectives---see for example \\cite{bkm} and references therein. One can also consider the Lagrangian, or penalized, form of \\eqref{eqn:a10}, namely,\n\\begin{equation}\\label{eqn:a10p}\n\\displaystyle\\min_{{\\bs\\beta}}\\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\mu \\|{\\bs\\beta}\\|_0,\n\\end{equation}\nfor a regularization parameter $\\mu>0$. One of the advantages of \\eqref{eqn:a10} over \\eqref{eqn:a10p} is that it offers direct control over estimators' sparsity via the discrete parameter $k$, as opposed to the Lagrangian form \\eqref{eqn:a10p} for which the correspondence between the continuous parameter $\\mu$ and the resulting sparsity of estimators obtained is not entirely clear. For further discussion, see \\cite{convspen}.\n\n\nAnother class of problems that have received considerable attention in the statistics and machine learning literature is the following:\n\\begin{equation}\\label{eqn:a1}\n\\min_{\\bs\\beta} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + R({\\bs\\beta}),\n\\end{equation}\nwhere $R({\\bs\\beta})$ is a choice of regularizer which encourages sparsity in ${\\bs\\beta}$.\nFor example, the popularly used Lasso \\cite{tibshirani} takes the form of problem \\eqref{eqn:a1} with $R({\\bs\\beta})=\\mu\\|{\\bs\\beta}\\|_1$, where $\\|\\cdot\\|_1$ is the $\\ell_1$ norm; in doing so, the Lasso simultaneously selects variables and also performs shrinkage. \nThe Lasso has seen widespread success across a variety of applications.\n\nIn contrast to the convex approach of the Lasso, there also has been  been growing interest in considering richer classes of regularizers $R$ which include nonconvex functions. Examples of such penalties include the $\\ell_{q}$-penalty (for $q\\in [0,1]$), minimax concave penalty (MCP) \\cite{mcp}, and the smoothly clipped absolute deviation (SCAD) \\cite{scad}, among others. Many of the nonconvex penalty functions considered are \\emph{coordinate-wise separable}; in other words, $R$ can be decomposed as\n$$R({\\bs\\beta}) = \\sum_{i=1}^p \\rho(|\\beta_i|),$$\nwhere $\\rho(\\cdot)$ is a real-valued function \\cite{zhangzhang}. There has been a variety of evidence suggesting the promise of such nonconvex approaches in overcoming certain shortcomings of Lasso-like approaches.\n\nOne of the central ideas of nonconvex penalty methods used in sparse modeling is that of creating a continuum of estimation problems which bridge the gap between convex methods for sparse estimation (such as Lasso) and subset selection in the form \\eqref{eqn:a10}. However, as noted above, such a connection does not necessarily offer direct control over the desired level of sparsity of estimators.\n\n\\subsection*{The trimmed Lasso}\n\nIn contrast with coordinate-wise separable penalties as considered above, we consider a family of penalties that are not separable across coordinates. One such penalty which forms a principal object of our study herein is \n$$\\tk{{\\bs\\beta}} := \\min_{\\substack{\\|\\bs \\phi\\|_0\\leq k}} \\|\\bs \\phi-{\\bs\\beta}\\|_1.$$\nThe penalty $T_k$ is a measure of the distance from the set of $k$-sparse estimators as measured via the $\\ell_1$ norm. In other words, when used in problem \\eqref{eqn:a1}, the penalty $R=T_k$ controls the amount of shrinkage towards sparse models. \n\nThe penalty $T_k$ can equivalently be written as \n$$\\tk{{\\bs\\beta}} = \\sum_{i=k+1}^p |\\beta_{(i)}|,$$\nwhere $|\\beta_{(1)}|\\geq |\\beta_{(2)}|\\geq \\cdots\\geq |\\beta_{(p)}|$ are the sorted entries of ${\\bs\\beta}$. In words, $\\tk{{\\bs\\beta}}$ is the sum of the absolute values of the $p-k$ smallest magnitude entries of ${\\bs\\beta}$. The penalty was first introduced in \\cite{thiao,hempel,gotoh1,gotoh2}. We refer to this family of penalty functions (over choices of $k$) as the \\emph{trimmed Lasso}.\\footnote{The choice of name is our own and is motivated by the least trimmed squares regression estimator, described below} The case of $k=0$ recovers the usual Lasso, as one would suspect. The distinction, of course, is that for general $k$, $T_k$ no longer shrinks, or biases towards zero, the $k$ largest entries of ${\\bs\\beta}$.\n\nLet us consider the least squares loss regularized via the trimmed lasso penalty---this leads to the following optimization criterion:\n\\begin{equation}\\label{eqn:rmaux1}\n\\displaystyle\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda \\tk{{\\bs\\beta}},\n\\end{equation}\nwhere $\\lambda>0$ is the regularization parameter. The penalty term shrinks the smallest $p-k$ entries of ${\\bs\\beta}$ and does not impose any penalty on the largest $k$ entries of ${\\bs\\beta}$. If $\\lambda$ becomes larger, the smallest $p-k$ entries of ${\\bs\\beta}$ are shrunk further; after a certain threshold---as soon as $\\lambda \\geq \\lambda_0$ for some finite $\\lambda_0$---the smallest $p-k$ entries are set to zero.\nThe existence of a finite $\\lambda_0$ (as stated above) is an attractive feature of the trimmed Lasso and is known as its \\emph{exactness} property, namely, for $\\lambda$ sufficiently large, the problem \\eqref{eqn:rmaux1} exactly solves constrained best subset selection as in problem \\eqref{eqn:a10} (\\emph{c.f.} \\cite{gotoh1}). Note here the contrast with the separable penalty functions which correspond instead with problem \\eqref{eqn:a10p}; as such, the trimmed Lasso is distinctive in that it offers precise control over the desired level of sparsity vis-\\`a-vis the discrete parameter $k$. Further, it is also notable that many algorithms developed for separable-penalty estimation problems can be directly adapted for the trimmed Lasso.\n\nOur objective in studying the trimmed Lasso is distinctive from previous approaches. In particular, while previous work on the penalty $T_k$ has focused primarily on its use as a tool for reformulating sparse optimization problems \\cite{thiao,hempel} and on how such reformulations can be solved computationally \\cite{gotoh1,gotoh2}, we instead aim to explore the trimmed Lasso's structural properties and its relation to existing sparse modeling techniques.\n\nIn particular,  a natural question we seek to explore is, what is the connection of the trimmed Lasso penalty with existing separable penalties commonly used in sparse statistical learning? For example, the trimmed Lasso bears a close resemblance to the clipped (or capped) Lasso penalty \\cite{cl}, namely,\n$$\\sum_{i=1}^p \\mu\\min\\{\\gamma|\\beta_i|,1\\},$$\nwhere $\\mu,\\gamma>0$ are parameters (when $\\gamma$ is large, the clipped Lasso approximates $\\mu\\|{\\bs\\beta}\\|_0$).\n\n\n\n\n\n\\subsection*{Robustness: robust statistics and robust optimization}\n\nA significant thread woven throughout the consideration of penalty methods for sparse modeling is the notion of robustness---in short, the ability of a method to perform in the face of noise. Not surprisingly, the notion of robustness has myriad distinct meanings depending on the context. Indeed, as Huber, a pioneer in the area of robust statistics, aptly noted:\n\\begin{quote}\n``The word `robust' is loaded with many---sometimes inconsistent---connotations.'' \\cite[p. 2]{huber}\n\\end{quote}\nFor this reason, we consider robustness from several perspectives---both the robust statistics~\\cite{huber} and robust optimization~\\cite{RObook} viewpoints.\n\nA common premise of the various approaches is as follows: that a robust model should perform well even under small deviations from its underlying assumptions; and that to achieve such behavior, some efficiency under the assumed model should be sacrificed. Not surprisingly in light of Huber's prescient observation, the exact manifestation of this idea can take many different forms, even if the initial premise is ostensibly the same.\n\n\\subsubsection*{Robust statistics and the ``min-min'' approach}\n\nOne such approach is in the field of robust statistics \\cite{huber,robRegBook,robStatsSurvey}. In this context, the primary assumptions are often probabilistic, i.e. distributional, in nature, and the deviations to be ``protected against'' include possibly gross, or arbitrarily bad, errors. Put simply, robust statistics is primary focused on analyzing and mitigating the influence of outliers on estimation methods. \n\n\n\nThere have been a variety of proposals of different estimators to achieve this. One that is particularly relevant for our purposes is that of \\emph{least trimmed squares} (``LTS'') \\cite{robRegBook}. For fixed $j\\in\\{1,\\ldots,n\\}$, the LTS problem is defined as\n\\begin{equation}\\label{eqn:introlts}\n\\min_{\\bs\\beta} \\sum_{i=j+1}^n |r_{(i)}({\\bs\\beta})|^2,\n\\end{equation}\nwhere $r_i({\\bs\\beta}) = y_i-\\mb x_i'{\\bs\\beta}$ are the residuals and $r_{(i)}({\\bs\\beta})$ are the sorted residuals given ${\\bs\\beta}$ with $|r_{(1)}({\\bs\\beta})|\\geq |r_{(2)}({\\bs\\beta})|\\geq\\cdots\\geq |r_{(n)}({\\bs\\beta})|$. In words, the LTS estimator performs ordinary least squares on the $n-j$ smallest residuals (discarding the $j$ largest or worst residuals). \n\n\n\nFurthermore, it is particularly instructive to express \\eqref{eqn:introlts} in the equivalent form (\\emph{c.f.} \\cite{bmlqs})\n\\begin{equation}\\label{eqn:introltsalt}\n\\min_{\\bs\\beta}\\min_{\\substack{I\\subseteq \\{1,\\ldots,n\\}:\\\\|I|=n-j }} \\sum_{i\\in I} |r_i ({\\bs\\beta})|^2.\n\\end{equation}\nIn light of this representation, we refer to LTS as a form of ``min-min'' robustness. One could also interpret this min-min robustness as \\emph{optimistic} in the sense the estimation problems \\eqref{eqn:introltsalt} and, \\emph{a fortiori}, \\eqref{eqn:introlts} allow the modeler to also choose observations to discard. \n\n\n\n\\subsubsection*{Other min-min models of robustness}\n\nAnother approach to robustness which also takes a min-min form like LTS is the classical technique known as \\emph{total least squares} \\cite{tls,tlsoverview}. For our purposes, we consider total least squares in the form\n\\begin{equation}\\label{eqn:introeiv}\n\\min_{{\\bs\\beta}}\\min_{\\bs\\Delta} \\frac{1}{2}\\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2^2 + \\eta\\|\\bs\\Delta\\|_{2}^2,\n\\end{equation}\nwhere $\\|\\bs\\Delta\\|_2$ is the usual Frobenius norm of the matrix $\\bs\\Delta$ and $\\eta>0$ is a scalar parameter. In this framework, one again has an optimistic view on error: find the best possible ``correction'' of the data matrix ${\\mb X}$ as ${\\mb X}+\\bs\\Delta^*$ and perform least squares using this corrected data (with $\\eta$ controlling the flexibility in choice of $\\bs\\Delta$).\n\nIn contrast with the penalized form of \\eqref{eqn:introeiv}, one could also consider the problem in a constrained form such as\n\\begin{equation}\\label{eqn:introeivcon}\n\\min_{\\bs\\beta} \\min_{\\bs\\Delta\\in\\mathcal{V}} \\frac{1}{2}\\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2^2,\n\\end{equation}\nwhere $\\mathcal{V}\\subseteq\\mathbb{R}^{n\\times p}$ is defined as $\\mathcal{V}= \\{\\bs\\Delta: \\|\\bs\\Delta\\|_2\\leq \\eta'\\}$ for some $\\eta'>0$. \nThis problem again has the min-min form, although now with perturbations $\\bs\\Delta$ as restricted to the set $\\mathcal{V}$.\n\n\n\n\n\n\\subsubsection*{Robust optimization and the ``min-max'' approach}\n\nWe now turn our attention to a different approach to the notion of robustness known as robust optimization \\cite{RObook,ROsurvey}. In contrast with robust statistics, robust optimization typically replaces  distributional assumptions with a new primitive, namely, the deterministic notion of an \\emph{uncertainty set}. Further, in robust optimization one considers a worst-case or pessimistic perspective and the focus is on perturbations from the nominal model (as opposed to possible gross corruptions as in robust statistics).\n\nTo be precise, one possible robust optimization model for linear regression takes form \\cite{xu,RObook,bcrobreg}\n\\begin{equation}\\label{eqn:roprimitive}\n\\min_{\\bs\\beta}\\max_{\\bs\\Delta\\in\\mathcal{U}} \\frac{1}{2}\\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2^2,\n\\end{equation}\nwhere $\\mathcal{U}\\subseteq\\mathbb{R}^{n\\times p}$ is a (deterministic) uncertainty set that captures the possible deviations of the model (from the nominal data ${\\mb X}$). \nNote the immediate contrast with the robust models considered earlier (LTS and total least squares in \\eqref{eqn:introlts} and \\eqref{eqn:introeiv}, respectively) that take the min-min form; instead, robust optimization focuses on ``min-max'' robustness. For a related discussion contrasting the min-min approach with min-max, see \\cite{worstbest,optimisticrobust} and references therein.\n\nOne of the attractive features of the min-max formulation is that it gives a re-interpretation of  several statistical regularization methods. For example, the usual Lasso (problem \\eqref{eqn:a1} with $R=\\mu\\ell_1$) can be expressed in the form \\eqref{eqn:roprimitive} for a specific choice of uncertainty set:\n\\begin{proposition}[e.g. \\cite{xu,RObook}]\\label{prop:lasso}\nProblem \\eqref{eqn:roprimitive} with uncertainty set $\\mathcal{U} = \\{\\bs\\Delta: \\|\\bs\\Delta_i\\|_2\\leq \\mu\\;\\forall i\\}$\nis equivalent to the Lasso, i.e., problem \\eqref{eqn:a1} with $R({\\bs\\beta})=\\mu\\|{\\bs\\beta}\\|_1$, where $\\bs\\Delta_i$ denotes the $i$th column of $\\bs\\Delta$.\n\\end{proposition}\n\\noindent For further discussion of the robust optimization approach as applied to statistical problems, see \\cite{bcrobreg} and references therein.\n\n\n\\subsubsection*{Other min-max models of robustness}\n\nWe close our discussion of robustness by considering another example of min-max robustness that is of particular relevance to the trimmed Lasso. In particular, we consider problem \\eqref{eqn:a1} with the SLOPE (or OWL) penalty \\cite{slope,owl}, namely,\n$$R_{\\textsc{SLOPE}(\\mb w)}({\\bs\\beta}) = \\sum_{i=1}^p w_i |\\beta_{(i)}|, $$\nwhere $\\mb w$ is a (fixed) vector of weights with $w_1\\geq w_2\\geq \\cdots\\geq w_p\\geq 0$ and $w_1>0$. In its simplest form, the SLOPE penalty has weight vector $\\tilde{\\mb w}$, where $\\tilde{w}_1=\\cdots=\\tilde{w}_k=1$, $\\tilde{w}_{k+1}=\\cdots=\\tilde{w}_p=0$, in which case we have the identity\n$$R_{\\textsc{SLOPE}(\\tilde{\\mb w})}({\\bs\\beta})  =\\| {\\bs\\beta} \\|_{1} - T_{k}({{\\bs\\beta}}).$$\n\n\n\nThere are some apparent similarities but also subtle differences between the SLOPE penalty and the trimmed Lasso. From a high level, while the trimmed Lasso focuses on the smallest magnitude entries of ${\\bs\\beta}$, the SLOPE penalty in its simplest form focuses on the \\emph{largest} magnitude entries of ${\\bs\\beta}$. As such, the trimmed Lasso is generally nonconvex, while the SLOPE penalty is always convex; consequently, the techniques for solving the related estimation problems will necessarily be different.\n\nFinally, we note that the SLOPE penalty can be considered as a min-max model of robustness for a particular choice of uncertainty set:\n\n\\begin{proposition}\\label{prop:slope}\nProblem \\eqref{eqn:roprimitive} with uncertainty set\n$$\\mathcal{U} =\\left\\{\\bs\\Delta :\n\\begin{array}{c}\n\\bs\\Delta \\text{ has at most $k$ nonzero}\\\\\n\\text{columns and } \\|\\bs\\Delta_i\\|_2\\leq \\mu\\;\\forall i\n\\end{array}\\right\\}\n$$\nis equivalent to problem \\eqref{eqn:a1} with $R({\\bs\\beta})=\\mu R_{\\textsc{SLOPE}(\\tilde{\\mb w})}({\\bs\\beta}) $, where $\\tilde{w}_1=\\cdots=\\tilde{w}_k=1$ and $\\tilde{w}_{k+1}=\\cdots=\\tilde{w}_p=0$.\n\\end{proposition}\n\\noindent We return to this particular choice of uncertainty set later. (For completeness, we include a more general min-max representation of SLOPE in Appendix \\ref{app:slope}.)\n\n\n\\subsection*{Computation and Algorithms}\n\nBroadly speaking, there are numerous distinct approaches to algorithms for solving problems of the form \\eqref{eqn:a10}--\\eqref{eqn:a1} for various choices of $R$. We do not attempt to provide a comprehensive list of such approaches for general $R$, but we will discuss existing approaches for the trimmed Lasso and closely related problems. Approaches typically take one of two forms: heuristic or exact.\n\n\\subsubsection*{Heuristic techniques}\n\nHeuristic approaches to solving problems \\eqref{eqn:a10}--\\eqref{eqn:a1} often use techniques from convex optimization \\cite{BV2004}, such as proximal gradient descent or coordinate descent (see \\cite{scad,sparsenet}). Typically these techniques are coupled with an analysis of  local or global behavior of the algorithm. For example, global behavior is often considered under additional restrictive assumptions on the underlying data; unfortunately, verifying such assumptions can be as difficult as solving the original nonconvex problem. (For example, consider the analogy with compressed sensing \\cite{crt,donoho1,gitta} and the hardness of verifying whether underlying assumptions hold \\cite{tillman,bandeira}).\n\n\nThere is also extensive work studying the local behavior (e.g. stationarity) of heuristic approaches to these problems. For the specific problems \\eqref{eqn:a10} and \\eqref{eqn:a10p}, the behavior of augmented Lagrangian methods \\cite{admmsilva,admmteng} and complementarity constraint techniques \\cite{mpccportfolio,burdakov,compconl0,asc} have been considered. For other local approaches, see \\cite{folded}.\n\n\n\\subsubsection*{Exact techniques}\n\nOne of the primary drawbacks of heuristic techniques is that it can often be difficult to verify the degree of suboptimality of the estimators obtained. For this reason, there has been an increasing interest in studying the behavior of exact algorithms for providing certifiably optimal solutions to problems of the form \\eqref{eqn:a10}--\\eqref{eqn:a1} \\cite{bkm,bmlqs,mipgo,discretedantzig}. Often these approaches make use of techniques from \\emph{mixed integer optimization} (``MIO'')\\cite{bonami} which are implemented in a variety of software, e.g. Gurobi \\cite{gurobi}. The tradeoff with such approaches is that they typically carry a heavier computational burden than convex approaches. For a discussion of the application of MIO  in statistics, see \\cite{bkm,bmlqs,mipgo,discretedantzig}.\n\n\n\n\n\\subsection*{What this paper is about}\n\nIn this paper, we focus on a detailed analysis of the trimmed Lasso, especially with regard to its properties and its relation to existing methods. In particular, we explore the trimmed Lasso from two perspectives: that of sparsity as well as that of robustness. We summarize our contributions as follows:\n\n\\begin{enumerate}\n\n\n\\item We study the robustness of the trimmed Lasso penalty. In particular, we provide several min-min robustness representations of it. We first show that the same choice of uncertainty set that leads to the SLOPE penalty in the min-max robust model \\eqref{eqn:roprimitive} gives rise to the trimmed Lasso in the corresponding min-min robust problem \\eqref{eqn:introeivcon} (with an additional regularization term). This gives an interpretation of the SLOPE and trimmed Lasso as a complementary pair of penalties, one under a pessimistic (min-max) model and the other under an optimistic (min-min) model.\n\nMoreover, we show another min-min robustness interpretation of the trimmed Lasso by comparison with the ordinary Lasso. In doing so, we further highlight the nature of the trimmed Lasso and its relation to the LTS problem \\eqref{eqn:introlts}. \n\n\n\\item We provide a detailed analysis on the connection between estimation approaches using the trimmed Lasso and separable penalty functions. In doing so, we show directly how penalties such as the trimmed Lasso can be viewed as a generalization of such existing approaches in certain cases. In particular, a trimmed-Lasso-like approach always subsumes its separable analogue, and the containment is strict in general. We also focus on the specific case of the clipped (or capped) Lasso \\cite{cl}; for this we precisely characterize the relationship and provide a necessary and sufficient condition for the two approaches to be equivalent. In doing so, we highlight some of the limitations of an approach using a separable penalty function.\n\n\\item Finally, we describe a variety of algorithms, both existing and new, for trimmed Lasso estimation problems. We contrast two heuristic approaches for finding locally optimal solutions with exact techniques from mixed integer optimization that can be used to produce certificates of optimality for solutions found via the convex approaches. We also show that the convex envelope \\cite{rockafeller} of the trimmed Lasso takes the form\n$$\\left(\\|{\\bs\\beta}\\|_1 - k\\right)_+,$$\nwhere $(a)_+:=\\max\\{0,a\\}$, a ``soft-thresholded'' variant of the ordinary Lasso. Throughout this section, we emphasize how techniques from convex optimization can be used to find high-quality solutions to the trimmed Lasso estimation problem. An implementation of the various algorithms presented herein can be found at\n\\begin{center}\n\\url{https:\/\/github.com\/copenhaver\/trimmedlasso}.\n\\end{center}\n\n\n\\end{enumerate}\n\n\n\n\\subsubsection*{Paper structure}\n \n\nThe structure of the paper is as follows. In Section \\ref{sec:basic}, we study several properties of the trimmed Lasso, provide a few distinct interpretations, and highlight possible generalizations. In Section \\ref{sec:rob}, we explore the trimmed Lasso in the context of robustness. Then, in Section \\ref{sec:ncpm}, we study the relationship between the trimmed Lasso and other nonconvex penalties. In Section \\ref{sec:algs}, we study the algorithmic implications of the trimmed Lasso. Finally, in Section \\ref{sec:conc} we share our concluding thoughts and highlight future directions.\n\n\n\n\n\n\\section{Structural properties and interpretations}\\label{sec:basic}\n\nIn this section, we provide further background on the trimmed Lasso: its motivations, interpretations, and generalizations. Our remarks in this section are broadly grouped as follows: in Section \\ref{ssec:defn} we summarize the trimmed Lasso's basic properties as detailed in \\cite{thiao,hempel,gotoh1,gotoh2}; we then turn our attention to an interpretation of the trimmed Lasso as a relaxation of complementarity constraints problems from optimization (Section \\ref{ssec:compcon}) and as a variable decomposition method (Section \\ref{ssec:vardecomp}); finally, in Sections \\ref{ssec:gens} and \\ref{ssec:dantzig} we highlight the key structural features of the trimmed Lasso by identifying possible generalizations of its definition and its application. These results augment the existing literature by giving a deeper understanding of the trimmed Lasso and provide a basis for further results in Sections \\ref{sec:rob} and \\ref{sec:ncpm}.\n\n\n\\subsection{Basic observations}\\label{ssec:defn}\n\nWe begin with a summary of some of the basic properties of the trimmed Lasso as studied in \\cite{thiao,hempel,gotoh1}.\nFirst of all, let us also include another representation of $T_k$:\n\\begin{lemma}\\label{lemma:miprep}\nFor any ${\\bs\\beta}$,\n$$\\begin{array}{lll}\n\\tk{{\\bs\\beta}} =  \\smash{\\displaystyle\\min_{\\substack{I\\subseteq\\{1,\\ldots,p\\}:\\\\|I| = p-k}} \\sum_{i\\in I} |\\beta_i| } = &\\displaystyle\\min_{\\mb z} & \\langle\\mb z,|{\\bs\\beta}|\\rangle\\\\\n & \\operatorname{s.t.}& \\displaystyle\\sum_{i} z_i =p- k\\\\ & &\\displaystyle \\mb z\\in\\{0,1\\}^p,\n\\end{array}$$\nwhere $|{\\bs\\beta}|$ denotes the vector whose entries are the absolute values of the entries of ${\\bs\\beta}$.\n\\end{lemma}\n\\noindent In other words, the trimmed Lasso can be represented using auxiliary binary variables.\n\n\nNow let us consider the problem\n\\begin{equation*}\n\\min_{{\\bs\\beta}} \\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2+\\lambda \\tk{{\\bs\\beta}},\\tag{$\\textsc{TL}_{\\lambda,k}$}\n\\end{equation*}\nwhere $\\lambda>0$ and $k\\in\\{0,1,\\ldots,p\\}$ are parameters. Based on the definition of $T_k$, we have the following:\n\\begin{lemma}\\label{lemma:vdrep}\nThe problem $\\tla{\\lambda,k}$ can be rewritten exactly in several equivalent forms:\n\\begin{align*}\n\\tla{\\lambda,k} &=\\min_{\\substack{{\\bs\\beta},\\bs \\phi:\\\\\\|\\bs \\phi\\|_0\\leq k}}\\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|^2 + \\lambda \\|{\\bs\\beta}-\\bs \\phi\\|_1\\nonumber\\\\\n&= \\min_{\\substack{{\\bs\\beta},\\bs \\phi,\\bs\\epsilon:\\\\{\\bs\\beta}=\\bs \\phi+\\bs\\epsilon\\\\\\|\\bs \\phi\\|_0\\leq k}}\\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|^2 + \\lambda \\|\\bs\\epsilon\\|_1\\nonumber\\\\\n& =\\min_{\\substack{\\bs \\phi,\\bs\\epsilon:\\\\\\|\\bs \\phi\\|_0\\leq k}}\\frac{1}{2}\\|\\mb y-{\\mb X}(\\bs \\phi+\\bs\\epsilon)\\|^2 + \\lambda \\|\\bs\\epsilon\\|_\n\\end{align*}\n\\end{lemma}\n\n\n\n\n\\subsubsection*{Exact penalization}\n\nBased on the definition of $T_k$, it follows that $T_k({\\bs\\beta})=0$ if and only if $\\|{\\bs\\beta}\\|_0\\leq k$. Therefore, one can rewrite problem \\eqref{eqn:a10} as\n\\begin{equation*}\n\\min_{T_k({\\bs\\beta})=0} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2.\n\\end{equation*}\nIn Lagrangian form, this would suggest an approximation for \\eqref{eqn:a10} of the form\n\\begin{equation*}\n\\min_{{\\bs\\beta}} \\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda T_k({\\bs\\beta}),\n\\end{equation*}\nwhere $\\lambda>0$. As noted in the introduction, this approximation is in fact exact (in the sense of \\cite{bert76,bertexact}), summarized in the following theorem; for completeness, we include in Appendix \\ref{app:proof} a full proof that is distinct from that in \\cite{gotoh1}.\\footnote{The presence of the additional regularizer $\\eta\\|{\\bs\\beta}\\|_1$ can be interpreted in many ways. For our purposes, it serves to make the problems well-posed.}\n\n\\begin{theorem}[\\emph{c.f.} \\cite{gotoh1}]\\label{thm:exactEquiv}\nFor any fixed $k\\in\\{0,1,2,\\ldots,p\\}$, $\\eta>0$, and problem data $\\mb y$ and ${\\mb X}$, there exists some $\\overline{\\lambda}=\\overline{\\lambda}(\\mb y,{\\mb X})>0$ so that for all $\\lambda>\\overline{\\lambda}$, the problems \n$$\\displaystyle\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda \\tk{{\\bs\\beta}} + \\eta \\|{\\bs\\beta}\\|_1 $$\nand\n$$\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta}}& \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\eta\\|{\\bs\\beta}\\|_1\\\\\n\\operatorname{s.t.}& \\|{\\bs\\beta}\\|_0\\leq k\n\\end{array}$$\nhave the same optimal objective value and the same set of optimal solutions.\n\\end{theorem}\n\n\n\n\n\nThe direct implication is that trimmed Lasso leads to a continuum (over $\\lambda$) of relaxations to the best subset selection problem starting from ordinary least squares estimation; further, best subset selection lies on this continuum for $\\lambda$ sufficiently large. \n\n\n\n\n\n\n\\subsection{A complementary constraints viewpoint}\\label{ssec:compcon}\n\n\nWe now turn our attention to a new perspective on the trimmed Lasso as considered via mathematical programming with complementarity constraints (``MPCCs'') \\cite{scholtes,mpcclin,kanzow0,kanzow1,kanzow2,burdakov}, sometimes also referred to as mathematical programs with equilibrium constraints \\cite{bilevel}. By studying this connection, we will show that a penalized form of a common relaxation scheme for MPCCs leads directly to the trimmed Lasso penalty. This gives a distinctly different optimization perspective on the trimmed Lasso penalty.\n\n\nAs detailed in \\cite{mpccportfolio,burdakov,compconl0}, the problem \\eqref{eqn:a10} can be exactly rewritten as\n\\begin{equation}\\label{eqn:BSSr}\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta},\\mb z}&\\displaystyle \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 \\\\%+ R({\\bs\\beta})\\\\\n\\operatorname{s.t.}& \\sum_iz_i=p-k\\\\\n& \\mb z\\in[0,1]^p\\\\\n& z_i\\beta_i = 0.\n\\end{array}\n\\end{equation}\nby the inclusion of auxiliary variables $\\mb z\\in[0,1]^p$. In essence, the auxiliary variables replace the combinatorial constraint $\\|{\\bs\\beta}\\|_0\\leq k$ with \\emph{complementarity} constraints of the form $z_i\\beta_i=0$. Of course, the problem as represented in \\eqref{eqn:BSSr} is still not directly amenable to convex optimization techniques.\n\nAs such, relaxation schemes can be applied to \\eqref{eqn:BSSr}. One popular  method from the MPCC literature is the Scholtes-type relaxation \\cite{kanzow1}; applied to \\eqref{eqn:BSSr} as in \\cite{burdakov,compconl0}, this takes the form\n\\begin{equation}\\label{eqn:BSSr2}\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta},\\mb z}& \\displaystyle\\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2\\\\% + R({\\bs\\beta})\\\\\n\\operatorname{s.t.}& \\sum_iz_i=p-k\\\\\n& \\mb z\\in[0,1]^p\\\\\n& |z_i\\beta_i|\\leq t,\n\\end{array}\n\\end{equation}\nwhere $t>0$ is some fixed numerical parameter which controls the strength of the relaxation, with $t=0$ exactly recovering \\eqref{eqn:BSSr}. In the traditional MPCC context, it is standard to study local optimality and stationarity behavior of solutions to \\eqref{eqn:BSSr2} as they relate to the original problem \\eqref{eqn:a10}, \\emph{c.f.} \\cite{compconl0}.\n\nInstead, let us consider a different approach. In particular, consider a penalized, or Lagrangian, form of the Scholtes relaxation \\eqref{eqn:BSSr2}, namely,\n\\begin{equation}\\label{eqn:BSSr3}\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta},\\mb z}& \\displaystyle\\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2   + \\lambda\\sum_i (|z_i\\beta_i|-t)\\\\% + R({\\bs\\beta})\\\\\n\\operatorname{s.t.}& \\sum_iz_i=p-k\\\\\n& \\mb z\\in[0,1]^p\n\\end{array}\n\\end{equation}\nfor some fixed $\\lambda\\geq0$.\\footnote{To be precise, this is a \\emph{weaker} relaxation than if we had separate dual variables $\\lambda_i$ for each constraint $|z_i\\beta_i|\\leq t$, at least in theory.} Observe that we can minimize \\eqref{eqn:BSSr3} with respect to $\\mb z$ to obtain the equivalent problem\n$$\\min_{{\\bs\\beta}} \\displaystyle \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2  + \\lambda T_k({\\bs\\beta})  - p\\lambda t,$$\nwhich is precisely problem $(\\textsc{TL}_{\\lambda,\\ell})$ (up to the fixed additive constant). In other words, the trimmed Lasso can also be viewed as arising directly from a  penalized form of the MPCC relaxation, with auxiliary variables eliminated. This gives another view on Lemma \\ref{lemma:miprep} which gave a representation of $T_k$ using auxiliary binary variables.\n\n\n\n\n\n\\subsection{Variable decomposition}\\label{ssec:vardecomp}\n\n\nTo better understand the relation of the trimmed Lasso to existing methods, it is also useful to consider alternative representations. Here we focus on representations which connect it to variable decomposition methods. Our discussion here is an extended form of related discussions in \\cite{hempel,gotoh1,gotoh2}.\n\nTo begin, we return to the final representation of the trimmed Lasso problem as shown in Lemma \\ref{lemma:vdrep}, viz.,\n\\begin{equation}\\label{eqn:dtl}\n\\tla{\\lambda,k}=\\min_{\\substack{\\bs \\phi,\\bs\\epsilon:\\\\\\|\\bs \\phi\\|_0\\leq k}}\\frac{1}{2}\\|\\mb y-{\\mb X}(\\bs \\phi+\\bs\\epsilon)\\|^2 + \\lambda \\|\\bs\\epsilon\\|_1\n\\end{equation}\nWe will refer to $\\tla{\\lambda,k}$ in the form \\eqref{eqn:dtl}\nas the \\emph{split} or \\emph{decomposed} representation of the problem. This is because in this form it is clear that we can think about estimators ${\\bs\\beta}$ found via $\\tla{\\lambda,k}$ as being decomposed into two different estimators: a sparse component $\\bs \\phi$ and another component $\\bs\\epsilon$ with small $\\ell_1$ norm (as controlled via $\\lambda$).\n\nSeveral remarks are in order. First, the decomposition of ${\\bs\\beta}$ into ${\\bs\\beta}=\\bs \\phi+\\bs\\epsilon$ is truly a decomposition in that if ${\\bs\\beta}^*$ is an optimal solution to $\\tla{\\lambda,k}$ with $(\\bs \\phi^*,\\bs\\epsilon^*)$ a corresponding optimal solution to the split representation of the problem \\eqref{eqn:dtl}, then one must have that $\\phi_i^*\\epsilon_i^*=0$ for all $i\\in\\{1,\\ldots,p\\}$. In other words, the supports of $\\bs \\phi$ and $\\bs\\epsilon$ do not overlap; therefore, ${\\bs\\beta}^*=\\bs \\phi^*+\\bs\\epsilon^*$ is a genuine decomposition.\n\nSecondly, the variable decomposition \\eqref{eqn:dtl} suggests that the problem of finding the $k$ largest entries of ${\\bs\\beta}$ (i.e., finding $\\bs \\phi$) can be solved as a best subset selection problem with a (possibly different) convex loss function (without $\\bs\\epsilon$). To see this, observe that the problem of finding $\\bs \\phi$ in \\eqref{eqn:dtl} can be written as the problem\n$$\\min_{\\|\\bs \\phi\\|_0\\leq k} \\widetilde{L}(\\bs \\phi),$$\nwhere\n$$\\widetilde{L}(\\bs \\phi) = \\min_{\\bs\\epsilon} \\frac{1}{2}\\|\\mb y-{\\mb X}(\\bs \\phi+\\bs\\epsilon)\\|_2^2 + \\lambda \\|\\bs\\epsilon\\|_1.$$\nUsing theory on duality for the Lasso problem \\cite{lassodual}, one can argue that $\\widetilde{L}$ is itself a convex loss function. Hence, the variable decomposition gives some insight into how the largest $k$ loadings for the trimmed Lasso relates to solving a related sparse estimation problem.\n\n\\subsubsection*{A view towards matrix estimation}\n\nFinally, we contend that the variable decomposition of ${\\bs\\beta}$ as a sparse component $\\bs \\phi$ plus a ``noise'' component $\\bs\\epsilon$ with small norm is a natural and useful analogue of corresponding decompositions in the matrix estimation literature, such as in factor analysis \\cite{mardia,anderson2003,barth} and robust Principal Component Analysis \\cite{candesrpca}. For the purposes of this paper, we will focus on the analogy with factor analysis.\n\nFactor analysis is a classical multivariate statistical method for decomposing the covariance structure of random variables; see \\cite{fabcm} for an overview of modern approaches to factor analysis. Given a covariance matrix $\\bs\\Sigma\\in\\mathbb{R}^{p\\times p}$, one is interested in describing it as the sum of two distinct components: a low-rank component $\\bs\\Theta$ (corresponding to a low-dimensional covariance structure common across the variables) and a diagonal component $\\bs\\Phi$ (corresponding to individual variances unique to each variable)---in symbols, $\\bs\\Sigma=\\bs\\Theta+\\bs\\Phi$.\n\nIn reality, this \\emph{noiseless} decomposition is often too restrictive (see e.g.\\cite{guttman1958extent,shapirorankred,ten1998some}), and therefore it is often better to focus on finding a decomposition $\\bs\\Sigma=\\bs\\Theta+\\bs\\Phi+\\mathcal{N}$, where $\\mathcal{N}$ is a noise component with small norm. As in \\cite{fabcm}, a corresponding estimation procedure can take the form\n\\begin{equation}\\label{eqn:fa}\n\\begin{array}{ll}\n\\displaystyle\\min_{\\bs\\Theta,\\bs\\Phi}& \\|\\bs\\Sigma-(\\bs\\Theta+\\bs\\Phi)\\|\\\\\n\\operatorname{s.t.}& \\operatorname{rank}(\\bs\\Theta)\\leq k\\\\\n& \\bs\\Phi = \\operatorname{diag}(\\Phi_{11},\\ldots,\\Phi_{pp}) \\cg\\mb0\\\\\n& \\bs\\Theta\\cg\\mb0,\n\\end{array}\n\\end{equation}\nwhere the constraint $\\mb A\\cg\\mb0$ denotes that $\\mb A$ is symmetric, positive semidefinite, and $\\|\\cdot\\|$ is some norm. One of the attractive features of the estimation procedure \\eqref{eqn:fa} is that for common choices of $\\|\\cdot\\|$, it is possible to completely eliminate the combinatorial rank constraint and the variable $\\bs\\Theta$ to yield a smooth (nonconvex) optimization problem with compact, convex constraints (see \\cite{fabcm} for details).\n\nThis exact same argument can be used to motivate the appearance of the trimmed Lasso penalty. Indeed, instead of considering estimators ${\\bs\\beta}$ which are exactly $k$-sparse (i.e., $\\|{\\bs\\beta}\\|_0\\leq k$), we instead consider estimators which are approximately $k$-sparse, i.e., ${\\bs\\beta}=\\bs \\phi+\\bs\\epsilon$, where $\\|\\bs \\phi\\|_0\\leq k$ and $\\bs\\epsilon$ has small norm. Given fixed ${\\bs\\beta}$, such a procedure is precisely\n$$\\min_{\\|\\bs \\phi\\|_0\\leq k} \\|{\\bs\\beta}-\\bs \\phi\\|.$$\nJust as the rank constraint is eliminated from \\eqref{eqn:fa}, the sparsity constraint can be eliminated from this to yield a continuous penalty which precisely captures the quality of the approximation ${\\bs\\beta}\\approx\\bs \\phi$. The trimmed Lasso uses the choice $\\|\\cdot\\|=\\ell_1$, although other choices are possible; see Section \\ref{ssec:gens}.\n\nThis analogy with factor analysis is also useful in highlighting additional benefits of the trimmed Lasso. One of particular note is that it enables the direct application of existing convex optimization techniques to find high-quality solutions to $\\tla{\\lambda,k}$.\n\n\n\n\\subsection{Generalizations}\\label{ssec:gens}\n\nWe close this section by considering some generalizations of the trimmed Lasso. These are particularly useful for connecting the trimmed Lasso to other penalties, as we will see later in Section \\ref{sec:ncpm}.\n\nAs noted earlier, the trimmed Lasso measures the distance (in $\\ell_1$ norm) from the set of $k$-sparse vectors; therefore, it is natural to inquire what properties other measures of distance might carry. In light of this, we begin with a definition:\n\\begin{definition}\nLet $k\\in\\{0,1,\\ldots,p\\}$ and $g:\\mathbb{R}_+\\to\\mathbb{R}_+$ be any unbounded, continuous, and strictly increasing function with $g(0)=0$. Define the corresponding $k$th projected penalty function, denoted $\\pi_k^g$, as\n$$\\pi_k^g({\\bs\\beta}) = \\min_{\\|\\bs \\phi\\|_0\\leq k}  \\sum_i g(|\\phi_i-\\beta_i|).$$\n\\end{definition}\n\n\\noindent It is not difficult to argue that $\\pi_k^g$ has as an equivalent definition\n$$\\pi_k^g({\\bs\\beta}) = \\sum_{i>k} g(|\\beta_{(i)}|).$$\nAs an example, $\\pi_k^g$ is the trimmed Lasso penalty when $g$ is the absolute value, viz. $g(x)=|x|$, and so it is a special case of the projected penalties. Alternatively, suppose $g(x) = x^2\/2$. In this case, we get a trimmed version of the ridge regression penalty: $\\sum_{i>k} |\\beta_{(i)}|^2\/2$.\n\nThis class of penalty functions has one notable feature, summarized in the following result:\\footnote{An extended statement of the convergence claim is included in Appendix \\ref{app:proof}.}\n\n\\begin{proposition}\\label{prop:asymp}\nIf $g:\\mathbb{R}_+\\to\\mathbb{R}_+$ is an unbounded, continuous, and strictly increasing function with $g(0)=0$, then for any ${\\bs\\beta}$, \n$\\pi_k^g({\\bs\\beta}) = 0$ if and only if $\\|{\\bs\\beta}\\|_0\\leq k$. \nHence, the problem $\\displaystyle\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda\\pi_k^g({\\bs\\beta})$ converges in objective value to $\\displaystyle\\min_{\\|{\\bs\\beta}\\|_0\\leq k} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2$ as $\\lambda\\to\\infty$.\n\\end{proposition}\n\n\n\n\nTherefore, any projected penalty $\\pi_k^g$ results in the best subset selection problem \\eqref{eqn:a10} asymptotically. While the choice of $g$ as the absolute value gives the trimmed Lasso penalty and leads to exact sparsity in the non-asymptotic regime (\\emph{c.f.} Theorem \\ref{thm:exactEquiv}) , Proposition \\ref{prop:asymp} suggests that the projected penalty functions have potential utility in attaining approximately sparse estimators. We will return to the penalties $\\pi_k^g$ again in Section \\ref{sec:ncpm} to connect the trimmed Lasso to nonconvex penalty methods.\n\nBefore concluding this section, we briefly consider a projected penalty function that is different than the trimmed Lasso. As noted above, if $g(x) = x^2\/2$, then the corresponding penalty function is the trimmed ridge penalty $\\sum_{i>k} |\\beta_{(i)}|^2\/2$.\nThe estimation procedure is then\n$$\\min_{{\\bs\\beta}} \\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 +\\frac{\\lambda}{2} \\sum_{i>k}|\\beta_{(i)}|^2,$$\nor equivalently in decomposed form (\\emph{c.f.} Section \\ref{ssec:vardecomp}),\\footnote{Interestingly, if one considers this trimmed ridge regression problem and uses convex envelope techniques \\cite{rockafeller,BV2004} to relax the constraint $\\|\\bs \\phi\\|_0\\leq k$, the resulting problem takes the form $\\min_{{ \\bs \\phi ,\\bs\\epsilon }} \\|\\mb y-{\\mb X}(\\bs \\phi+\\bs\\epsilon)\\|_2^2\/2 + \\lambda \\|\\bs\\epsilon\\|_2^2 + \\tau\\|\\bs \\phi\\|_1$, a sort of ``split'' variant of the usual elastic net \\cite{zhel}, another popular convex method for sparse modeling.}\n$$\\min_{\\substack{\\bs \\phi,\\bs\\epsilon:\\\\\\|\\bs \\phi\\|_0\\leq k}}\\frac{1}{2}\\|\\mb y-{\\mb X}(\\bs \\phi+\\bs\\epsilon)\\|_2^2 +\\frac{ \\lambda}{2} \\|\\bs\\epsilon\\|_2^2.$$\nIt is not difficult to see that the variable $\\bs\\epsilon$ can be eliminated to yield\n\\begin{equation}\\label{eqn:ridge}\n\\min_{\\|\\bs \\phi\\|_0\\leq k} \\frac{1}{2}\\left\\|\\mb A(\\mb y-{\\mb X}\\bs \\phi)\\right\\|_2^2,\n\\end{equation}\nwhere $\\mb A = (\\mb I-{\\mb X}({\\mb X}'{\\mb X}+\\lambda\\mb I)^{-1}{\\mb X}')^{1\/2}$. It follows that the largest $k$ loadings are found via a modified best subset selection problem under a different loss function---precisely a variant of the $\\ell_2$ norm. This is in the same spirit of observations made in Section \\ref{ssec:vardecomp}.\n\n\n\\begin{obs}\\label{obs:ridge}\nAn obvious question is whether the norm in \\eqref{eqn:ridge} is genuinely different. Observe that this loss function is the same as the usual $\\ell_2^2$ loss if and only if $\\mb A'\\mb A$ is a non-negative multiple of the identity matrix. It is not difficult to see that this is true iff ${\\mb X}'{\\mb X}$ is a non-negative multiple of the identity. In other words, the loss function in \\eqref{eqn:ridge} is the same as the usual ridge regression loss if and only if ${\\mb X}$ is (a scalar multiple of) an orthogonal design matrix.\n\\end{obs}\n\n\n\n\\subsection{Other applications of the trimmed Lasso: the (Discrete) Dantzig Selector}\\label{ssec:dantzig}\n\n\n\nThe above discussion which pertains to the least squares loss data-fidelity term can be generalized to other loss functions as well.\nFor example, let us consider a data-fidelity term given by the \nmaximal absolute inner product between the features and residuals, given by $\\|{\\mb X}'(\\mb y-{\\mb X}{\\bs\\beta})\\|_\\infty$. An $\\ell_{1}$-penalized version of this data-fidelity term, \npopularly known as the Dantzig Selector~\\cite{dantzig2,dasso}, is given by the following linear optimization problem:\n\\begin{equation}\\label{eqn-DS}\n\\min_{\\bs\\beta}  \\|{\\mb X}'(\\mb y-{\\mb X}{\\bs\\beta})\\|_\\infty  + \\mu\\|{\\bs\\beta}\\|_1.\n\\end{equation}\nEstimators found via \\eqref{eqn-DS} have statistical properties similar to the Lasso.\nFurther, problem \\eqref{eqn-DS} may be interpreted as an $\\ell_{1}$-approximation to the cardinality constrained version: \n\\begin{equation}\\label{eqn-DS-L0}\n\\min_{\\|{\\bs\\beta}\\|_0\\leq k}  \\|{\\mb X}'(\\mb y-{\\mb X}{\\bs\\beta})\\|_\\infty,\n\\end{equation}\nthat is, the Discrete Dantzig Selector, recently proposed and studied in~\\cite{discretedantzig}. The statistical properties of~\\eqref{eqn-DS-L0} are similar to the best-subset selection problem \\eqref{eqn:a10}, but  may be more attractive from a computational viewpoint as it relies on mixed integer \\emph{linear} optimization as opposed to mixed integer \\emph{conic} optimization (see \\cite{discretedantzig}).\n\n\nThe trimmed Lasso penalty can also be applied to the data-fidelity term $\\|{\\mb X}'(\\mb y-{\\mb X}{\\bs\\beta})\\|_\\infty$, leading to the following estimator:\n$$\\min_{\\bs\\beta}  \\|{\\mb X}'(\\mb y-{\\mb X}{\\bs\\beta})\\|_\\infty+\\lambda\\tk{{\\bs\\beta}}+\\mu\\|{\\bs\\beta}\\|_1.$$\nSimilar to the case of the least squares loss function, the above estimator yields $k$-sparse solutions for any $\\mu>0$ and for $\\lambda>0$ sufficiently large.\\footnote{For the same reason, but instead with the usual Lasso objective, the proof of Theorem \\ref{thm:exactEquiv} (see Appendix \\ref{app:proof}) could be entirely omitted; yet, it is instructive to see in the proof there that the trimmed Lasso truly does set the \\emph{smallest} entries to zero, and not simply all entries (when $\\lambda$ is large) like the Lasso.} While this claim follows \\emph{a fortiori} by appealing to properties of the Dantzig selector, it nevertheless highlights how any exact penalty method with a separable penalty function can be turned into a trimmed-style problem which offers direct control over the sparsity level.\n\n\n\n\\section{A perspective on robustness}\\label{sec:rob}\n\n\nWe now turn our attention to a deeper exploration of the robustness properties of the trimmed Lasso. We begin by studying the min-min robust analogue of the min-max robust SLOPE penalty; in doing so, we show under which circumstances this analogue is the trimmed Lasso problem. Indeed, in such a regime, the trimmed Lasso can be viewed as an optimistic counterpart to the robust optimization view of the SLOPE penalty. Finally, we turn our attention to an additional min-min robust interpretation of the trimmed Lasso in direct correspondence with the least trimmed squares estimator shown in \\eqref{eqn:introlts}, using the ordinary Lasso as our starting point.\n\n\n\n\\subsection{The trimmed Lasso as a min-min robust analogue of SLOPE}\n\nWe begin by reconsidering the uncertainty set that gave rise to the SLOPE penalty via the min-max view of robustness as considered in robust optimization:\n$$\\mathcal{U}_k^\\lambda:=\\left\\{\\bs\\Delta :\n\\begin{array}{c}\n\\bs\\Delta \\text{ has at most $k$ nonzero}\\\\\n\\text{columns and } \\|\\bs\\Delta_i\\|_2\\leq \\lambda\\;\\forall i\n\\end{array}\\right\\}.\n$$\nAs per Proposition \\ref{prop:slope}, the min-max problem \\eqref{eqn:roprimitive}, viz., \n\\begin{equation*\n\\min_{\\bs\\beta}\\max_{\\bs\\Delta\\in\\mathcal{U}_k^\\lambda} \\frac{1}{2}\\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2^\n\\end{equation*}\nis equivalent to the SLOPE-penalized problem\n\\begin{equation}\\label{eqn:slopepen}\n\\min_{\\bs\\beta} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda R_{\\textsc{SLOPE}(\\tilde{\\mb w})} ({\\bs\\beta}).\n\\end{equation}\nfor the specific choice of $\\tilde{\\mb w}$ with $\\tilde w_1=\\cdots=\\tilde w_k=1$ and $\\tilde w_{k+1}=\\cdots=\\tilde w_{p}=0$.\n\nLet us now consider the form of the min-min robust analogue of the the problem \\eqref{eqn:roprimitive} for this specific choice of uncertainty set. As per the discussion in Section \\ref{sec:intro}, the min-min analogue takes the form of problem \\eqref{eqn:introeivcon}, i.e., a variant of total least squares:\n\\begin{equation*\n\\min_{\\bs\\beta} \\min_{\\bs\\Delta\\in\\mathcal{U}_k^\\lambda} \\frac{1}{2}\\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2^2,\n\\end{equation*}\nor equivalently as the linearly homogenous problem\\footnote{In what follows, the linear homogeneity is useful primarily for simplicity of analysis, \\emph{c.f.} \\cite[ch. 12]{RObook}. Indeed, the conversion to linear homogeneous functions is often hidden in equivalence results like Proposition \\ref{prop:slope}.}\n\\begin{equation}\\label{eqn:eivconhom}\n\\min_{\\bs\\beta} \\min_{\\bs\\Delta\\in\\mathcal{U}_k^\\lambda} \\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2.\n\\end{equation}\n\n\n\n\\noindent It is useful to consider problem \\eqref{eqn:eivconhom} with an explicit penalization (or regularization) on ${\\bs\\beta}$\n\\begin{equation}\\label{eqn:eivconhompen}\n\\min_{\\bs\\beta} \\min_{\\bs\\Delta\\in\\mathcal{U}_k^\\lambda} \\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2 + r({\\bs\\beta}),\n\\end{equation}\nwhere $r(\\cdot)$ is, say, a norm (the use of lowercase is to distinguish from the function $R$ in Section \\ref{sec:intro}).\n\nAs described in the following theorem, this min-min robustness problem \\eqref{eqn:eivconhompen} is equivalent to the trimmed Lasso problem for specific choices of $r$. The proof is contained in Appendix \\ref{app:proof}.\n\n\\begin{theorem}\\label{thm:robeivInterp}\n\nFor any $k$, $\\lambda>0$, and norm $r$, the problem \\eqref{eqn:eivconhompen} can be rewritten exactly as\n\\begin{equation*\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta} } &\\displaystyle\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + r({\\bs\\beta}) - \\lambda \\sum_{i=1}^k|\\beta_{(i)}| \\\\\n\\operatorname{s.t.} & \\displaystyle\\lambda\\sum_{i=1}^k|\\beta_{(i)}| \\leq \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2.\n\\end{array}\n\\end{equation*}\n\\end{theorem}\n\nWe have the following as an immediate corollary:\n\n\\begin{corollary}\\label{cor:slope}\nFor the choice of $r({\\bs\\beta}) = \\tau \\|{\\bs\\beta}\\|_1$, where $\\tau > \\lambda$, the problem \\eqref{eqn:eivconhompen} is precisely\n\\begin{equation}\\label{eqn:cor1}\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta} } &\\displaystyle\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + (\\tau-\\lambda)\\|{\\bs\\beta}\\|_1+ \\lambda \\tk{{\\bs\\beta}} \\\\\n\\operatorname{s.t.} & \\displaystyle\\lambda\\sum_{i=1}^k|\\beta_{(i)}| \\leq \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2.\n\\end{array}\n\\end{equation}\nIn particular, when $\\lambda>0$ is small, it is approximately equal (in a precise sense)\\footnote{For a precise characterization and extended discussion, see Appendix \\ref{app:proof} and Theorem \\ref{thm:corprecise}. The informal statement here is sufficient for the purposes of our present discussion.} to the trimmed Lasso problem\n$$\\min_{{\\bs\\beta} } \\displaystyle\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + (\\tau-\\lambda)\\|{\\bs\\beta}\\|_1+ \\lambda \\tk{{\\bs\\beta}}.$$\n\\end{corollary}\n\nIn words, the min-min problem \\eqref{eqn:eivconhompen} (with an $\\ell_1$ regularization on ${\\bs\\beta}$) can be written as a variant of a trimmed Lasso problem, subject to an additional constraint. It is instructive to consider both the objective and the constraint of problem \\eqref{eqn:cor1}. To begin, the objective has a combined penalty on ${\\bs\\beta}$ of $(\\tau-\\lambda)\\|{\\bs\\beta}\\|_1 + \\lambda \\tk{{\\bs\\beta}}$. This can be thought of as the more general form of the penalty $T_k$. Namely, one can consider the penalty $T_{\\mb x}$ (with $0\\leq x_1\\leq x_2\\leq\\cdots\\leq x_p$ fixed) defined as\n$$T_{\\mb x} ({\\bs\\beta}) := \\sum_{i=1}^p x_i |\\beta_{(i)}|.$$\nIn this notation, the objective of \\eqref{eqn:cor1} can be rewritten as $\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + T_{\\mb x}({\\bs\\beta})$, with\n$$\\mb x=(\\underbrace{\\tau-\\lambda,\\ldots,\\tau-\\lambda}_{k \\text{ times}},\\underbrace{\\tau,\\ldots,\\tau}_{p-k \\text{ times}}).$$\nIn terms of the constraint of problem \\eqref{eqn:cor1}, note that it takes the form of a model-fitting constraint: namely, $\\lambda$ controls a trade-off between model fit $\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2$ and model complexity measured via the SLOPE norm $\\sum_{i=1}^k |\\beta_{(i)}|$.\n\nHaving described the structure of problem \\eqref{eqn:cor1}, a few remarks are in order. First of all, the trimmed Lasso problem (with an additional $\\ell_1$ penalty on ${\\bs\\beta}$) can be interpreted as (a close approximation to) a min-min robust problem, at least in the regime when $\\lambda$ is small; this provides an interesting contrast to the sparse-modeling regime when $\\lambda$ is large (\\emph{c.f.} Theorem \\ref{thm:exactEquiv}). Moreover, the trimmed Lasso is a min-min robust problem in a way that is the \\emph{optimistic} analogue of its min-max counterpart, namely, the SLOPE-penalized problem \\eqref{eqn:slopepen}. Finally, Theorem \\ref{thm:robeivInterp} gives a natural representation of the trimmed Lasso problem in a way that directly suggests why methods from difference-of-convex optimization \\cite{dcSummary} are relevant (see Section \\ref{sec:algs}).\n\n\n\n\\subsubsection*{The general SLOPE penalty}\n\n\n\nLet us briefly remark upon SLOPE in its most general form (with general $\\mb w$); again we will see that this leads to a more general trimmed Lasso as its (approximate) min-min counterpart. In its most general form, the SLOPE-penalized problem \\eqref{eqn:slopepen} can be written as the min-max robust problem \\eqref{eqn:roprimitive} with choice of uncertainty set\n$$\\mathcal{U}_\\mb w^\\lambda =\\left\\{\\bs\\Delta : \\|\\bs\\Delta\\bs \\phi\\|_2 \\leq  \\lambda \\sum_i w_i|\\phi_{(i)}|\\;\\forall\\bs \\phi\\right\\}\n$$\n(see Appendix \\ref{app:slope}). In this case, the penalized, homogenized min-min robust counterpart, analogous to problem \\eqref{eqn:eivconhompen}, can be written as follows:\n\n\\begin{proposition}\\label{prop:robeivslope}\nFor any $k$, $\\lambda>0$, and norm $r$, the problem\n\\begin{equation}\\label{eqn:auxslopepf1}\n\\min_{\\bs\\beta} \\min_{\\bs\\Delta\\in\\mathcal{U}_\\mb w^\\lambda} \\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2 + r({\\bs\\beta})\n\\end{equation}\ncan be rewritten exactly as\n\\begin{equation*\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta} } &\\displaystyle\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + r({\\bs\\beta}) - \\lambda R_{\\textsc{SLOPE}(\\mb w)}({\\bs\\beta}) \\\\\n\\operatorname{s.t.} & \\displaystyle\\lambda R_{\\textsc{SLOPE}(\\mb w)}({\\bs\\beta}) \\leq \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2.\n\\end{array}\n\\end{equation*}\nFor the choice of $r({\\bs\\beta}) = \\tau \\|{\\bs\\beta}\\|_1$, where $\\tau > \\lambda w_1$, the problem \\eqref{eqn:auxslopepf1} is\n\\begin{equation*\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta} } &\\displaystyle\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + T_{\\tau\\mb 1 - \\lambda \\mb w}({\\bs\\beta}) \\\\\n\\operatorname{s.t.} & \\displaystyle\\lambda R_{\\textsc{SLOPE}(\\mb w)}({\\bs\\beta}) \\leq \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2.\n\\end{array}\n\\end{equation*}\nIn particular, when $\\lambda>0$ is sufficiently small, problem \\eqref{eqn:auxslopepf1} is approximately equal to the generalized trimmed Lasso problem\n$$\\min_{{\\bs\\beta} } \\displaystyle\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + T_{\\tau\\mb 1 - \\lambda \\mb w}({\\bs\\beta}).$$\n\\end{proposition}\n\nPut plainly, the general form of the SLOPE penalty leads to a generalized form of the trimmed Lasso, precisely as was true for the simplified version considered in Theorem \\ref{thm:robeivInterp}.\n\n\\subsection{Another min-min interpretation}\n\nWe close our discussion of robustness by considering another min-min representation of the trimmed Lasso. We use the ordinary Lasso problem as our starting point and show how a modification in the same spirit as the min-min robust least trimmed squares estimator in \\eqref{eqn:introlts} leads directly to the trimmed Lasso.\n\nTo proceed, we begin with the usual Lasso problem\n\\begin{equation}\\label{eqn:lasso}\n\\min_{\\bs\\beta} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda\\|{\\bs\\beta}\\|_1.\n\\end{equation}\nAs per Proposition \\ref{prop:lasso}, this problem is equivalent to the min-max robust problem \\eqref{eqn:roprimitive} with uncertainty set\n$\\mathcal{U} = \\mathcal{L}^\\lambda = \\{\\bs\\Delta:\\|\\bs\\Delta_i\\|_2 \\leq \\lambda\\;\\forall i\\}$:\n\\begin{equation}\\label{eqn:lassorob}\n\\min_{\\bs\\beta}\\max_{\\bs\\Delta\\in\\mathcal{L}^\\lambda} \\frac{1}{2}\\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2^2.\n\\end{equation}\nIn this view, the usual Lasso \\eqref{eqn:lasso} can be thought of as a least squares method which takes into account certain feature-wise adversarial perturbations of the matrix ${\\mb X}$. The net result is that the adversarial approach penalizes all loadings equally (with coefficient $\\lambda$).\n\n\nUsing this setup and Theorem \\ref{thm:exactEquiv},\nwe can re-express the trimmed Lasso problem $\\tla{\\lambda,k}$ in the equivalent min-min form\n\\begin{equation}\\label{eqn:a2}\n\\min_{\\bs\\beta}\\min_{\\substack{\\\\mb I\\subseteq\\{1,\\ldots,p\\}:\\\\|I|=p-k}} \\max_{\\bs\\Delta\\in\\mathcal{L}^\\lambda_I} \\frac{1}{2}\\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2^2,\n\\end{equation}\nwhere $\\mathcal{L}^\\lambda_I\\subseteq \\mathcal{L}^\\lambda$ requires that the columns of $\\bs\\Delta\\in\\mathcal{L}^\\lambda_I$ are supported on $I$:\n$$\\mathcal{L}^\\lambda_I = \\{\\bs\\Delta: \\|\\bs\\Delta_i\\|_2\\leq \\lambda \\;\\forall i, \\; \\bs\\Delta_i = \\mb 0\\;\\forall i\\notin I\\}.$$\nWhile the adversarial min-max approach in problem \\eqref{eqn:lassorob} would attempt to ``corrupt'' all $p$ columns of ${\\mb X}$, in estimating ${\\bs\\beta}$ we have the power to optimally discard $k$ out of the $p$ corruptions to the columns (corresponding to $I^c$). In this sense, the trimmed Lasso in the min-min robust form \\eqref{eqn:a2} acts in a similar spirit to the min-min, robust-statistical least trimmed squares estimator shown in problem \\eqref{eqn:introltsalt}.\n\n\n\n\n\n\n\\section{Connection to nonconvex penalty methods}\\label{sec:ncpm}\n\n\nIn this section, we explore the connection between the trimmed Lasso and existing, popular nonconvex (component-wise separable) penalty functions used for sparse modeling.  We begin in Section \\ref{ssec:ncpoverview} with a brief overview of existing approaches. In Section \\ref{ssec:ncpreform} we then highlight how these relate to the trimmed Lasso, making the connection more concrete with examples in Section \\ref{ssec:ncpeg}. Then in Section \\ref{ssec:ncpgenerality} we exactly characterize the connection between the trimmed Lasso and the clipped Lasso \\cite{cl}. In doing so, we show that the trimmed Lasso subsumes the clipped Lasso; further, we provide a necessary and sufficient condition for when the containment is strict. Finally, in Section \\ref{ssec:ncpunbounded} we comment on the special case of unbounded penalty functions.\n\n\n\n\n\\subsection{Setup and Overview}\\label{ssec:ncpoverview}\n\nOur focus throughout will be the penalized $M$-estimation problem of the form\n\\begin{equation}\\label{eqn:ncpm\n\\min_{{\\bs\\beta}} L({\\bs\\beta}) + \\sum_{i=1}^p \\rho(|\\beta_i|;\\mu,\\gamma),\n\\end{equation}\nwhere $\\mu$ represents a (continuous) parameter controlling the desired level of sparsity of ${\\bs\\beta}$ and $\\gamma$ is a parameter controlling the quality of the approximation of the indicator function $I\\{|\\beta|>0\\}$. A variety of nonconvex penalty functions and their description in this format is shown in Table \\ref{tab:ncp} (for a general discussion, see \\cite{zhangzhang}). In particular, for each of these functions we observe that \n$$\\lim_{\\gamma\\to\\infty} \\rho(|\\beta|;\\mu,\\gamma) = \\mu \\cdot I\\{|\\beta|>0\\}.$$\nIt is particularly important to note the \\emph{separable} nature of the penalty functions appearing in \\eqref{eqn:ncpm}---namely, each coordinate $\\beta_i$ is penalized (via $\\rho$) independently of the other coordinates.\n\nOur primary focus will be on the bounded penalty functions (clipped Lasso, MCP, and SCAD), all of which take the form\n\\begin{equation}\\label{eqn:pff}\n\\rho(|\\beta|;\\mu,\\gamma) = \\mu \\min\\{g(|\\beta|;\\mu,\\gamma),1\\}\n\\end{equation}\nwhere $g$ is an increasing function of $|\\beta|$. We will show that in this case, the problem \\eqref{eqn:ncpm} can be rewritten exactly as an estimation problem with a (non-separable) trimmed penalty function:\n\\begin{equation}\\label{eqn:ncpt\n\\min_{{\\bs\\beta}} L({\\bs\\beta}) + \\mu\\sum_{i=\\ell+1}^p g(|\\beta_{(i)} |)\n\\end{equation}\nfor some $\\ell\\in\\{0,1,\\ldots,p\\}$ (note the appearance of the projected penalties $\\pi_k^g$ as considered in Section \\ref{ssec:gens}). In the process of doing so, we will also show that, in general, \\eqref{eqn:ncpt} cannot be solved via the separable-penalty estimation approach of \\eqref{eqn:ncpm}, and so the trimmed estimation problem leads to a richer class of models. Throughout we will often refer to \\eqref{eqn:ncpt} (taken generically over all choices of $\\ell$) as the \\emph{trimmed counterpart} of the separable estimation problem \\eqref{eqn:ncpm}.\n\n\n\\begin{table*}\n\\centering\n  \\begin{tabular}{| c | c | c | c|}\n\\hline \nName  & Definition & Auxiliary Functions\\\\\\hline\\hline\nClipped Lasso &\\multirow{2}{*}{$ \\mu\\min\\{\\gamma|\\beta|,1\\} $ } &\\multirow{4}{*}{ \\small $g_1(|\\beta|) =  \\left\\{\\begin{array}{rc} 2\\gamma|\\beta|-\\gamma^2\\beta^2, & |\\beta|\\leq 1\/\\gamma,\\\\1 ,& |\\beta|>1\/\\gamma.\\end{array}\\right.$  }\\\\\n\\cite{cl} &  & \\\\\\cline{1-2}\nMCP & \\multirow{2}{*}{$\\mu\\min\\{g_1(|\\beta|),1\\}$ } &   \\\\\n\\cite{mcp} & & \\\\\\cline{1-2}\nSCAD & \\multirow{2}{*}{$\\mu\\min\\{g_2(|\\beta|),1\\}$} &  \\multirow{6}{*}{ \\small $ g_2(|\\beta|) = \\left\\{\n\\begin{array}{rc}\n|\\beta|\/(\\gamma\\mu),&  |\\beta| \\leq 1\/\\gamma,\\\\\n \\frac{\\beta^2 +(2\/\\gamma-4\\mu\\gamma)|\\beta| +1\/\\gamma^2}{ 4\\mu - 4\\mu^2\\gamma^2} , &  1\/\\gamma < |\\beta| \\leq 2\\mu\\gamma-1\/\\gamma,\\\\1, &  |\\beta| > 2\\mu\\gamma-1\/\\gamma.\n\\end{array}\\right.   $  }   \\\\\n\\cite{scad} & & \\\\\\cline{1-2}\n$\\ell_q$ ($0<q<1$) & \\multirow{2}{*}{$\\mu|\\beta|^{1\/\\gamma}$ }& \\\\\n\\cite{lq,log} & & \\\\\\cline{1-2}\nLog & \\multirow{2}{*}{$\\displaystyle\\mu {\\log(\\gamma|\\beta| + 1)}\/{\\log(\\gamma+1)}$ } & \\\\\n\\cite{log} & & \\\\\\hline\n  \\end{tabular}\n  \\caption{Nonconvex penalty functions $\\rho(|\\beta|;\\mu,\\gamma)$ represented as in \\eqref{eqn:ncpm}. The precise parametric representation is different than their original presentation but they are equivalent. We have taken care to normalize the different penalty functions so that $\\mu$ is the sparsity parameter and $\\gamma$ corresponds to the approximation of the indicator $I\\{|\\beta|>0\\}$. For  SCAD, it is usually recommended to set $2\\mu>3\/\\gamma^2$.\n}\n  \\label{tab:ncp\n\\end{table*}\n\n\n\n\n\n\n\\subsection{Reformulating the problem \\eqref{eqn:ncpm}}\\label{ssec:ncpreform}\n\nLet us begin by considering penalty functions $\\rho$ of the form \\eqref{eqn:pff} with $g$ a non-negative, increasing function of $|\\beta|$. Observe that for any ${\\bs\\beta}$ we can rewrite $\\sum_{i=1}^p \\min\\{g(|\\beta_i|),1\\}$ as\n\\begin{align*}\n&\\min\\left\\{\\sum_{i=1}^p g(|\\beta_{(i)}|),1 + \\sum_{i=2}^p g(|\\beta_{(i)}|),\\ldots, p-1 + g(|\\beta_{(p)}|),p  \\right\\}\\\\\n&=\\min_{\\ell\\in\\{0,\\ldots,p\\}} \\left\\{ \\ell + \\sum_{i>\\ell}  g(|\\beta_{(i)}|)  \\right\\}.\n\\end{align*}\nIt follows that \\eqref{eqn:ncpm}\n can be rewritten \\emph{exactly} as \n\\begin{equation}\\label{eqn:ncptna}\n\\min_{\\substack{{\\bs\\beta},\\\\\\ell\\in\\{0,\\ldots,p\\}}} \\left(L({\\bs\\beta}) + \\mu\\sum_{i>\\ell} g(|\\beta_{(i)}|) + \\mu\\ell\\right)\n\\end{equation}\nAn immediate consequence is the following theorem:\n\n\\begin{theorem}\\label{thm:MasT}\nIf ${\\bs\\beta}^*$ is an optimal solution to \\eqref{eqn:ncpm}, where $\\rho(|\\beta|;\\mu,\\gamma) = \\mu\\min\\{g(|\\beta|;\\mu,\\gamma),1\\}$, then there exists some $\\ell^*\\in\\{0,\\ldots,p\\}$ so that ${\\bs\\beta}^*$ is optimal to its trimmed counterpart\n\\begin{equation*\n\\min_{\\bs\\beta} L({\\bs\\beta}) + \\mu\\sum_{i>\\ell^*} g(|\\beta_{(i)}|).\n\\end{equation*}\nIn particular, the choice of $\\ell^* = |\\{i: g(|\\beta_i^*|) \\geq1 \\}|$ suffices.\n Conversely, if ${\\bs\\beta}^*$ is an optimal solution to \\eqref{eqn:ncptna}, then ${\\bs\\beta}^*$ in an optimal solution to \\eqref{eqn:ncpm}.\n\\end{theorem}\n\n\nIt follows that the estimation problem \\eqref{eqn:ncpm}, which decouples each loading $\\beta_i$ in the penalty function, can be solved using ``trimmed'' estimation problems of the form \\eqref{eqn:ncpt} with a trimmed penalty function that couples the loadings and only penalizes the $p-\\ell^*$ smallest. Because the trimmed penalty function is generally nonconvex by nature, we will focus on comparing it with other nonconvex penalties for the remainder of the section.\n\n\\subsection{Trimmed reformulation examples}\\label{ssec:ncpeg}\n\nWe now consider the structure of the estimation problem \\eqref{eqn:ncpm} and the corresponding trimmed estimation problem for the clipped Lasso and MCP penalties. We use the $\\ell_2^2$ loss throughout.\n\n\\subsubsection*{Clipped Lasso}\n\nThe clipped (or capped, or truncated) Lasso penalty \\cite{cl,shen12} takes the component-wise form\n$$\\rho(|\\beta|;\\mu,\\gamma) = \\mu \\min\\{\\gamma|\\beta|,1\\}.$$\nTherefore, in our notation, $g$ is a multiple of the absolute value function. A plot of $\\rho$ is shown in Figure \\ref{fig:ncpa}. In this case, the estimation problem with $\\ell_2^2$ loss is\n\\begin{equation}\\label{eqn:clm}\n\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\mu\\sum_i \\min\\{\\gamma|\\beta_i|,1\\}.\n\\end{equation}\nIt follows that the corresponding trimmed estimation problem (\\emph{c.f.} Theorem \\ref{thm:MasT}) is exactly the trimmed Lasso problem studied earlier, namely,\n\\begin{equation}\\label{eqn:clt}\n\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\mu\\gamma \\tk{{\\bs\\beta}}.\n\\end{equation}\nA distinct advantage of the trimmed Lasso formulation \\eqref{eqn:clt} over the traditional clipped Lasso formulation \\eqref{eqn:clm} is that it offers direct control over the desired level of sparsity vis-\\`a-vis the discrete parameter $k$. We perform a deeper analysis of the two problems in Section \\ref{ssec:ncpgenerality}.\n\n\n\\subsubsection*{MCP}\n\nThe MCP penalty takes the component-wise form\n$$\\rho(|\\beta|;\\mu,\\gamma) = \\mu \\min\\{g(|\\beta|),1\\}$$\nwhere $g$ is any function with $g(|\\beta|) = 2\\gamma|\\beta|-\\gamma^2\\beta^2$ whenever $|\\beta| \\leq1\/\\gamma$ and $g(|\\beta|)\\geq1$ whenever $|\\beta| > 1\/\\gamma$. An example of one such $g$ is shown in Table \\ref{tab:ncp}.\nA plot of $\\rho$ is shown in Figure \\ref{fig:ncpa}. Another valid choice of $g$ is $g(|\\beta|) = \\max\\{2\\gamma|\\beta|-\\gamma^2\\beta^2,\\gamma|\\beta|\\}$. In this case, the trimmed counterpart is\n\\begin{equation*\n\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|^2 + \\mu\\gamma \\sum_{i>\\ell}\\max\\left\\{ 2|\\beta_{(i)}| -\\gamma\\beta_{(i)}^2,|\\beta_{(i)}|\\right\\}.\n\\end{equation*}\n\nNote that this problem is amenable to the same class of techniques as applied to the trimmed Lasso problem in the form \\eqref{eqn:clt} because of the increasing nature of $g$, although the subproblems with respect to ${\\bs\\beta}$ are no longer convex (although it is a usual MCP estimation problem which is well-suited to convex optimization approaches; see \\cite{sparsenet}).\nAlso observe that we can separate the penalty function into a trimmed Lasso component and another component:\n$$\\sum_{i>\\ell} |\\beta_{(i)}|\\text{\\quad and \\quad} \\sum_{i>\\ell} \\left(|\\beta_{(i)}|-\\gamma\\beta_{(i)}^2\\right)_+.$$\nObserve that the second component is uniformly bounded above by $(p-\\ell)\/(4\\gamma)$, and so as $\\gamma\\to\\infty$,\nthe trimmed Lasso penalty dominates.\n\n\n\\begin{figure*\n\\centering\n\\begin{subfigure}{0.49\\linewidth}\n        \\centering\n\\begin{tikzpicture}\n  \\begin{axis}[ \n    xlabel=$|\\beta|$,\n   \n    xmin = 0,\n    xmax = 2,\n    ymin = 0,\n    ymax = 2,\n    scale = .8,\n    minor y tick num=0,\n          yticklabels={,,,,$\\mu$},\n    minor x tick num=0,\n          xticklabels={,0,,,$1\/\\gamma$},\n          legend entries = {$\\rho_\\text{CL}$,$\\rho_\\text{MCP}$},\n          legend style = {at={(.93,.07)}, anchor = south east},\n  ] \n    \\addplot[samples=\\ns, color=blue, thick, dashed] { (x<1.5)*(x) + (x>1.5)*1.5};\n    \\addplot[samples=\\ns, color=red, thick] { (x<1.5) * (2*x-x^2\/1.5)  + (x>1.5)*(1.5) };\n  \\end{axis}\n\\end{tikzpicture}\n        \\caption{Clipped Lasso and MCP}\n        \\label{fig:ncpa}\n    \\end{subfigure}\n\\begin{subfigure}{0.49\\linewidth}\n        \\centering\n \\begin{tikzpicture}\n  \\begin{axis}[ \n    xlabel=$|\\beta|$,\n   \n    xmin = 0,\n    xmax = 4,\n    ymin = 0,\n    ymax = 4,\n    scale = .8,\n    minor y tick num=0,\n          yticklabels={,,,,$\\mu$},\n    minor x tick num=0,\n          xticklabels={,0,,,$1$},\n    legend entries = {$\\rho_\\text{log}$,$\\rho_\\text{$\\ell_q$}$},\n              legend style = {at={(.93,.07)}, anchor = south east},\n  ] \n    \\addplot[samples=\\ns, color=blue, thick, dashed] { 3*ln(2*x\/3+1)\/ln(3) };\n    \\addplot[samples=\\ns, color=red, thick] { 3*(x\/3)^(1\/2) };\n\\end{axis}\n\\end{tikzpicture}\n\\caption{Log and $\\ell_q$}\n\\label{fig:ncpb}\n\\end{subfigure}\\\\[1ex]\n\\caption{Plots of $\\rho(|\\beta|;\\mu,\\gamma)$ for some of the penalty functions in Table \\ref{tab:ncp}.}\n\\label{fig:ncp}\n\\end{figure*}\n\n\n\n\\subsection{The generality of trimmed estimation}\\label{ssec:ncpgenerality}\n\nWe now turn our focus to more closely studying the relationship between the separable-penalty estimation problem \\eqref{eqn:ncpm} and its trimmed estimation counterpart. The central problems of interest are the clipped Lasso and its trimmed counterpart, viz., the trimmed Lasso:\\footnote{One may be concerned about the well-definedness of such problems (e.g. as guaranteed vis-\\`a-vis coercivity of the objective, \\emph{c.f.}\\cite{rockafeller}). In all the results of Section \\ref{ssec:ncpgenerality}, it is possible to add a regularizer $\\eta\\|{\\bs\\beta}\\|_1$ for some fixed $\\eta>0$ to both $(\\textsc{CL}_{\\mu,\\gamma})$ and $(\\textsc{TL}_{\\lambda,\\ell})$ and the results remain valid, \\emph{mutatis mutandis}. The addition of this regularizer implies coercivity of the objective functions and, consequently, that the minimum is indeed well-defined. For completeness, we note a technical reason for a choice of $\\eta\\|{\\bs\\beta}\\|_1$ is its positive homogeneity; thus, the proof technique of Lemma \\ref{lem:key} easily adapts to this modification.}\n\\begin{align*}\n&\\displaystyle \\min_{\\bs\\beta} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|^2_2 + \\mu \\sum_i \\min\\{\\gamma|\\beta_i|,1\\}\\tag{$\\textsc{CL}_{\\mu,\\gamma}$}\\\\\n&\\displaystyle\\min_{\\bs\\beta} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|^2_2 + \\lambda \\tka{\\ell}{{\\bs\\beta}}.\\tag{$\\textsc{TL}_{\\lambda,\\ell}$}\n\\end{align*}\nAs per Theorem \\ref{thm:MasT}, if ${\\bs\\beta}^*$ is an optimal solution to $(\\textsc{CL}_{\\mu,\\gamma})$, then ${\\bs\\beta}^*$ is an optimal solution to $(\\textsc{TL}_{\\lambda,\\ell})$, where $\\lambda=\\mu\\gamma$ and $\\ell=|\\{i:|\\beta_i^*|\\geq1\/\\gamma\\}|$. We now consider the converse: given some $\\lambda>0$ and $\\ell\\in\\{0,1,\\ldots,p\\}$ and a solution ${\\bs\\beta}^*$ to $(\\textsc{TL}_{\\lambda,\\ell})$, when does there exist some $\\mu,\\gamma>0$ so that ${\\bs\\beta}^*$ is an optimal solution to $(\\textsc{CL}_{\\mu,\\gamma})$? As the following theorem suggests, the existence of such a $\\gamma$ is closely connected to an underlying discrete form of ``convexity'' of the sequence of problems $\\tla{\\lambda,k}$ for $k\\in\\{0,1,\\ldots,p\\}$. We will focus on the case when $\\lambda=\\mu\\gamma$, as this is the natural correspondence of parameters in light of Theorem \\ref{thm:MasT}.\n\n\n\\begin{theorem}\\label{thm:clconv}\nIf $\\lambda>0$, $\\ell\\in\\{0,\\ldots,p\\}$, and ${\\bs\\beta}^*$ is an optimal solution to $(\\textsc{TL}_{\\lambda,\\ell})$, then there exist $\\mu,\\gamma>0$ with $\\mu\\gamma=\\lambda$ and so that ${\\bs\\beta}^*$ is an optimal solution to $(\\textsc{CL}_{\\mu,\\gamma})$ if and only if\n\\begin{equation}\\label{eqn:clconv}\nZ\\tla{\\lambda,\\ell_e} < \\frac{j-\\ell_e}{j-i} Z\\tla{\\lambda,i} + \\frac{\\ell_e-i}{j-i} Z\\tla{\\lambda,j}\n\\end{equation}\nfor all $0\\leq i< \\ell_e < j \\leq p$, where $Z(\\textsc{P})$ denotes the optimal objective value to optimization problem $\\textsc{(P)}$ and $\\ell_e = \\min\\{\\ell,\\|{\\bs\\beta}^*\\|_0\\}$.\n\\end{theorem}\n\nLet us note why we refer to the condition in \\eqref{eqn:clconv} as a discrete analogue of convexity of the sequence $\\{z_k :=Z\\tla{\\lambda,k},\\; k=0,\\ldots,p\\}$. In particular, observe that this sequence satisfies the condition of Theorem \\ref{thm:clconv} if and only if the function defined as the linear interpolation between the points $(0,z_0)$, $(1,z_1)$, \\ldots, and $(p,z_p)$ is strictly convex about the point $(\\ell,z_\\ell)$.\\footnote{To be precise, we mean that the real-valued function that is a linear interpolation of the points has a subdifferential at the point $(\\ell,z_\\ell)$ which is an interval of strictly positive width.}\n\nBefore proceeding with the proof of the theorem, we state and prove a technical lemma about the structure of $(\\textsc{TL}_{\\lambda,\\ell})$.\n\n\\begin{lemma}\\label{lem:key}\nFix $\\lambda>0$ and suppose that ${\\bs\\beta}^*$ is optimal to $(\\textsc{TL}_{\\lambda,\\ell})$.\n\\begin{enumerate}[(a)]\n\\item The optimal objective value of $(\\textsc{TL}_{\\lambda,\\ell})$ is $Z(\\textsc{TL}_{\\lambda,\\ell}) = (\\|\\mb y\\|_2^2-\\|{\\mb X}{\\bs\\beta}^*\\|_2^2)\/2$.\n\\item If ${\\bs\\beta}^*$ is also optimal to $\\tla{\\lambda,\\ell'}$, where $\\ell<\\ell'$, then $\\|{\\bs\\beta}^*\\|_0\\leq\\ell$ and ${\\bs\\beta}^*$ is optimal to $\\tla{\\lambda,j}$ for all integral $j$ with $\\ell<j<\\ell'$.\n\\item If $\\kappa:=\\|{\\bs\\beta}^*\\|_0 <\\ell$, then  ${\\bs\\beta}^*$ is also optimal to $\\tla{\\lambda,\\kappa}$, $\\tla{\\lambda,\\kappa+1}$, \\ldots, and $\\tla{\\lambda,\\ell-1}$. Further, ${\\bs\\beta}^*$ is \\emph{not} optimal to $\\tla{\\lambda,0}$, $\\tla{\\lambda,1}$, \\ldots, nor $\\tla{\\lambda,\\kappa-1}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSuppose ${\\bs\\beta}^*$ is optimal to $(\\textsc{TL}_{\\lambda,\\ell})$. Define\n$$a(\\epsilon):= \\|\\mb y-\\epsilon{\\mb X}{\\bs\\beta}^*\\|_2^2\/2 + \\epsilon\\lambda \\tka{\\ell}{{\\bs\\beta}^*}.$$\nBy the optimality of ${\\bs\\beta}^*$, $a(\\epsilon)\\geq a(1)$ for all $\\epsilon\\geq0$. As $a$ is a polynomial with degree at most two, one must have that $a'(1) = 0$. This implies that\n$$a'(1) = -\\langle\\mb y,{\\mb X}{\\bs\\beta}^*\\rangle + \\|{\\mb X}{\\bs\\beta}^*\\|_2^2 + \\lambda\\tka{\\ell}{{\\bs\\beta}^*} = 0.$$\nAdding $(\\|\\mb y\\|_2^2-\\|{\\mb X}{\\bs\\beta}^*\\|_2^2)\/2$ to both sides, the desired result of part (a) follows.\n\nNow suppose that ${\\bs\\beta}^*$ is also optimal to $\\tla{\\lambda,\\ell'}$, where $\\ell'>\\ell$. By part (a), one must necessarily have that $Z(\\textsc{TL}_{\\lambda,\\ell})=Z\\tla{\\lambda,\\ell'} = (\\|\\mb y\\|_2^2-\\|{\\mb X}{\\bs\\beta}^*\\|_2^2)\/2$. Inspecting $Z(\\textsc{TL}_{\\lambda,\\ell})-Z\\tla{\\lambda,\\ell'}$, we see that\n$$0=Z(\\textsc{TL}_{\\lambda,\\ell})-Z\\tla{\\lambda,\\ell'} = \\lambda \\sum_{i=\\ell+1}^{\\ell'} |\\beta_{(i)}^*|.$$\nHence, $|\\beta_{(\\ell+1)}^*|=0$ and therefore $\\|{\\bs\\beta}^*\\|_0\\leq \\ell$.\n\nFinally, for any integral $j$ with $\\ell\\leq j\\leq \\ell'$, one always has that $Z(\\textsc{TL}_{\\lambda,\\ell}) \\geq Z\\tla{\\lambda,j} \\geq Z\\tla{\\lambda,\\ell'}$. As per the preceding argument, $Z(\\textsc{TL}_{\\lambda,\\ell}) = Z\\tla{\\lambda,\\ell}$ and so $Z(\\textsc{TL}_{\\lambda,\\ell}) = Z\\tla{\\lambda,j}$, and therefore ${\\bs\\beta}^*$ must also be optimal to $\\tla{\\lambda,j}$ by applying part (a). This completes part (b).\n\nPart (c) follows from a straightforward inspection of objective functions and using the fact that $Z\\tla{\\lambda,j}\\geq Z\\tla{\\lambda,\\ell}$ whenever $j\\leq\\ell$.\n\\end{proof}\n\nUsing this lemma, we can now proceed with the proof of the theorem.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:clconv}]\nLet $z_k = Z\\tla{\\lambda,k}$ for $k\\in\\{0,1,\\ldots,p\\}$. Suppose that $\\mu,\\gamma>0$ is so that $\\lambda = \\mu\\gamma$ and ${\\bs\\beta}^*$ is an optimal solution to $(\\textsc{CL}_{\\mu,\\gamma})$. Let $\\ell_e = \\min\\{\\ell,\\|{\\bs\\beta}^*\\|_0\\}$. Per equation \\eqref{eqn:ncptna}, ${\\bs\\beta}^*$ must be optimal to \n\\begin{equation}\\label{eqn:pf1}\n\\min_{\\bs\\beta} \\min_{k\\in\\{0,\\ldots,p\\}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\mu k + \\mu\\gamma \\tk{{\\bs\\beta}}.\n\\end{equation}\nObserve that this implies that if $k$ is such that $k$ is a minimizer of ${\\min}_k \\mu k + \\mu\\gamma \\tk{{\\bs\\beta}^*}$, then ${\\bs\\beta}^*$ must be optimal to $\\tla{\\lambda,k}$.\n\nWe claim that this observation, combined with Lemma \\ref{lem:key}, implies that\n$$\\ell_e=\\underset{ {k\\in\\{0,\\ldots,p\\}} }{\\argmin} \\mu k + \\mu\\gamma \\tk{{\\bs\\beta}^*}.$$\nThis can be shown as follows:\n\\begin{enumerate}[(a)]\n\\item Suppose   $\\ell\\leq \\|{\\bs\\beta}^*\\|_0$ and so $\\ell_e = \\min\\{\\ell,\\|{\\bs\\beta}^*\\|_0\\} = \\ell$. Therefore, by Lemma \\ref{lem:key}(b), ${\\bs\\beta}^*$ is not optimal to $\\tla{\\lambda,j}$ for any $j< \\ell$, and thus\n$$\\underset{ {k\\in\\{0,\\ldots,p\\}} }{\\min} \\mu k + \\mu\\gamma \\tk{{\\bs\\beta}^*} = \\underset{ {k\\in\\{\\ell,\\ldots,p\\}} }{\\min} \\mu k + \\mu\\gamma \\tk{{\\bs\\beta}^*}.$$\n\n\n\nIf $k>\\ell$ is such that $k$ is a minimizer of ${\\min}_k \\mu k + \\mu\\gamma \\tk{{\\bs\\beta}^*}$, then ${\\bs\\beta}^*$ must be optimal to $\\tla{\\lambda,k}$ (using the observation), and hence by Lemma \\ref{lem:key}(b), $\\|{\\bs\\beta}^*\\|_0\\leq \\ell$. Combined with $\\ell\\leq \\|{\\bs\\beta}^*\\|_0$, this implies that $\\|{\\bs\\beta}^*\\|_0=\\ell$. Yet then,\n$\\mu\\ell = \\mu\\ell + \\mu\\gamma \\tka{\\ell}{{\\bs\\beta}^*} <  \\mu k + \\mu\\gamma\\tk{{\\bs\\beta}^*}$, contradicting the optimality of $k$. Therefore, we conclude that $\\ell_e=\\ell$ is the \\emph{only} minimizer of $\\min_k \\mu k + \\mu\\gamma\\tk{{\\bs\\beta}^*}$. \n\n\\item Now instead suppose that $\\ell_e = \\|{\\bs\\beta}^*\\|_0 < \\ell$. Lemma \\ref{lem:key}(c) implies that any optimal solution $k$ to $\\min_k \\mu k + \\mu\\gamma \\tk{{\\bs\\beta}^*}$ must satisfy $k\\geq\\|{\\bs\\beta}^*\\|_0$ (by the second part combined with the observation). As before, if $k>\\|{\\bs\\beta}^*\\|_0=\\ell_e$, then $\\mu k > \\mu \\ell_e$, and so $k$ cannot be optimal. As a result, $k=\\ell_e=\\|{\\bs\\beta}^*\\|_0$ is the unique minimum.\n\n\n\\end{enumerate}\nIn either case, we have that $\\ell_e$ is the unique minimizer to $\\min_k \\mu k + \\mu\\gamma \\tk{{\\bs\\beta}^*}$.\n\nIt then follows that $Z(\\text{problem }\\eqref{eqn:pf1}) = z_{\\ell_e} + \\mu \\ell_e$. Further, by optimality of ${\\bs\\beta}^*$, $z_{\\ell_e} + \\mu  \\ell_e < z_i + \\mu  i$ for all $0\\leq i\\leq p$ with $i\\neq\\ell_e$. For $0\\leq i < \\ell_e$, this implies $\\mu < (z_i-z_{\\ell_e})\/(\\ell_e-i) $ and for $j>\\ell_e$, $\\mu > (z_{\\ell_e}-z_j)\/(j-\\ell_e)$. In other words, for $0\\leq i < \\ell_e < j\\leq p$,\n$$\\frac{z_{\\ell_e}-z_j}{j-\\ell_e} < \\frac{z_i - z_{\\ell_e}}{\\ell_e-i}, \\quad \\text{i.e., }\\;   z_{\\ell_e} <\\frac{j-\\ell_e}{j-i}z_i + \\frac{\\ell_e-i}{j-i} z_j.$$\nThis completes the forward direction. The reverse follows in the same way by taking any $\\mu$ with\n\\begin{equation*}\n\\mu\\in \\left( \\max_{j> \\ell_e} \\frac{z_{\\ell_e} -z_j}{j-\\ell_e},  \\min_{i<\\ell_e} \\frac{z_i-z_{\\ell_e}}{\\ell_e-i} \\right).\n\\end{equation*}\n\\end{proof}\n\n\nWe briefly remark upon one implication of the proof of Theorem \\ref{thm:clconv}. In particular, if ${\\bs\\beta}^*$ is a solution to $(\\textsc{TL}_{\\lambda,\\ell})$ and $\\ell <\\|{\\bs\\beta}^*\\|_0$, then ${\\bs\\beta}^*$ is not the solution to $\\tla{\\lambda,k}$ for any $k\\neq \\ell$.\n\nAn immediate question is whether the convexity condition \\eqref{eqn:clconv} of Theorem \\ref{thm:clconv} always holds. While the sequence $\\{Z\\tla{\\lambda,k} : k=0,1,\\ldots,p\\}$ is always non-increasing, the following example shows that the convexity condition need not hold in general; as a result, there exist instances of the trimmed Lasso problem whose solutions \\emph{cannot} be found by solving a clipped Lasso problem.\n\n\\begin{example}\\label{eg:cl}\nConsider the case when $p=n=2$ with\n$$\\mb y = \\begin{pmatrix}1\\\\1\\end{pmatrix} \\text{ \\quad and \\quad } \\mb X = \\begin{pmatrix} 1 & -1\\\\-1&2\\end{pmatrix}.$$\nLet $\\lambda =1\/2$ and $ \\ell = 1$, and consider $\\min_{{\\bs\\beta}} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2\/2 + |\\beta_{(2)}|\/2 = \\min_{\\beta_1,\\beta_2} (1-\\beta_1+\\beta_2)^2\/2 + (1+\\beta_1-2\\beta_2)^2\/2 + |\\beta_{(2)}|\/2.$\nThis has unique optimal solution ${\\bs\\beta}^* = (3\/2,1)$ with corresponding objective value $ z_1 = 3\/4$. One can also compute $z_0 = Z\\tla{1\/2,0} = 39\/40$ and $z_2 = Z\\tla{1\/2,2} = 0$. Note that $z_1 = 3\/4 > (39\/40)\/2 + (0)\/2 = z_0\/2+z_2\/2$, and so there do not exist any $\\mu,\\gamma>0$ with $\\mu\\gamma=1\/2$ so that ${\\bs\\beta}^*$ is an optimal solution to $\\cla{\\mu,\\gamma}$ by Theorem \\ref{thm:clconv}. Further, it is possible to show that ${\\bs\\beta}^*$ is not an optimal solution to $(\\textsc{CL}_{\\mu,\\gamma})$ for \\emph{any} choice of $\\mu,\\gamma\\geq0$. (See Appendix \\ref{app:proof}.)\n\\end{example}\n\nAn immediate corollary of this example, combined with Theorem \\ref{thm:MasT}, is that the class of trimmed Lasso models contains the class of clipped Lasso models as a \\emph{proper} subset, regardless of whether we restrict our attention to $\\lambda=\\mu\\gamma$. In this sense, the trimmed Lasso models comprise a richer set of models. The relationship is depicted in stylized form in Figure \\ref{fig:mc}.\n\n\\subsubsection*{Limit analysis}\n\nIt is important to contextualize the results of this section as $\\lambda\\to\\infty$. This corresponds to $\\gamma\\to\\infty$ for the clipped Lasso problem, in which case $(\\textsc{CL}_{\\mu,\\gamma})$ converges to the penalized form of subset selection:\n\\begin{equation*}\n\\min_{\\bs\\beta} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\mu\\|{\\bs\\beta}\\|_0.\\tag{$\\textsc{CL}_{\\mu,\\infty}$}\n\\end{equation*}\nNote that penalized problems for all of the penalties listed in Table \\ref{tab:ncp} have this as their limit as $\\gamma\\to\\infty$.  On the other hand, $(\\textsc{TL}_{\\lambda,\\ell})$ converges to constrained best subset selection:\n\\begin{equation*}\n\\min_{\\|{\\bs\\beta}\\|_0\\leq \\ell} \\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2.\\tag{$\\textsc{TL}_{\\infty,k}$}\n\\end{equation*}\nIndeed, from this comparison it now becomes clear why a convexity condition of the form in Theorem \\ref{thm:clconv} appears in describing when the clipped Lasso solves the trimmed Lasso problem. In particular, the conditions under $\\cla{\\mu,\\infty}$ solves the constrained best subset selection problem $\\tla{\\infty,k}$ are precisely those in Theorem \\ref{thm:clconv}.\n\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n  \\coordinate (c1) at (0,0);\n      \\pgfmathsetmacro{\\colmixer}{mod(10*1,100)}%\n      \\path [draw, fill=blue!\\colmixer, postaction=decorate] (c1) ellipse (2 and 1.5);\n      \\pgfmathsetmacro{\\colmixer}{mod(10*2,100)}%\n      \\path [dashed, draw, fill=blue!\\colmixer, postaction=decorate] (c1) ellipse (1.5 and .9) ;\n \\path [ decoration={text along path, text={Clipped Lasso}, reverse path, text align={align=center}}, postaction=decorate] (2.5,-2.1) arc (0:180:2.5);\n \\path [ decoration={text along path, text={Trimmed Lasso}, reverse path, text align={align=center}}, postaction=decorate] (2.6,-1.5) arc (0:180:2.6);\n\\end{tikzpicture}\n\\caption{Stylized relation of clipped Lasso and trimmed Lasso models. Every clipped Lasso model can be written as a trimmed Lasso model, but the reverse does not hold in general.}\n\\label{fig:mc}\n\\end{figure}\n\n\n\n\n\\subsection{Unbounded penalty functions}\\label{ssec:ncpunbounded}\n\nWe close this section by now considering nonconvex penalty functions which are unbounded and therefore do not take the form $\\mu\\min\\{g(|\\beta|),1\\}$. Two such examples are the $\\ell_q$ penalty ($0<q<1$) and the log family of penalties as shown in Table \\ref{tab:ncp} and depicted in Figure \\ref{fig:ncpb}. Estimation problems with these penalties can be cast in the form\n\\begin{equation}\\label{eqn:unbpm}\n\\min_{\\bs \\phi} \\frac{1}{2} \\|\\mb y-{\\mb X}\\bs \\phi\\|_2^2 + \\mu \\sum_{i=1}^p g(|\\phi_i|;\\gamma)\n\\end{equation}\nwhere $\\mu,\\gamma>0$ are parameters, $g$ is an unbounded and strictly increasing function, and $ g(|\\phi_i|;\\gamma) \\xrightarrow{\\gamma\\to\\infty} I\\{|\\phi_i|>0\\}$. The change of variables in \\eqref{eqn:unbpm} is intentional and its purpose will become clear shortly.\n\nObserve that because $g$ is now unbounded, there exists some $\\overline{\\lambda} = \\overline{\\lambda}(\\mb y,{\\mb X},\\mu,\\gamma)>0$ so that for all $\\lambda>\\overline{\\lambda}$ any optimal solution $(\\bs \\phi^*,\\bs\\epsilon^*)$ to the problem\n\\begin{equation}\\label{eqn:unbpm-aux}\n\\min_{\\bs \\phi,\\bs\\epsilon} \\frac{1}{2} \\|\\mb y-{\\mb X}(\\bs \\phi+\\bs\\epsilon)\\|_2^2 + \\lambda\\|\\bs\\epsilon\\|_1+\\mu \\sum_{i=1}^p g(|\\phi_i|;\\gamma)\n\\end{equation}\nhas $\\bs\\epsilon^*=\\mb0$.\\footnote{The proof involves a straightforward modification of an argument along the lines of that given in Theorem \\ref{thm:exactEquiv}. Also note that we can choose $\\overline{\\lambda}$ so that it is decreasing in $\\gamma$, \\emph{ceteris paribus}.} Therefore, \\eqref{eqn:unbpm} is a special case of \\eqref{eqn:unbpm-aux}. We claim that in the limit as $\\gamma\\to\\infty$ (all else fixed), that \\eqref{eqn:unbpm-aux} can be written exactly as a trimmed Lasso problem $\\tla{\\lambda,k}$ for some choice of $k$ and with the identification of variables ${\\bs\\beta} = \\bs \\phi+\\bs\\epsilon$.\n\nWe summarize this as follows:\n\n\\begin{proposition}\nAs $\\gamma\\to\\infty$, the penalized estimation problem \\eqref{eqn:unbpm} is a special case of the trimmed Lasso problem.\n\\end{proposition}\n\\begin{proof}\nThis can be shown in a straightforward manner: namely, as $\\gamma\\to\\infty$, \\eqref{eqn:unbpm-aux} becomes\n\\begin{equation*\n\\min_{\\bs \\phi,\\bs\\epsilon} \\frac{1}{2} \\|\\mb y-{\\mb X}(\\bs \\phi+\\bs\\epsilon)\\|_2^2 + \\lambda\\|\\bs\\epsilon\\|_1+\\mu \\|\\bs \\phi\\|_0\n\\end{equation*}\nwhich can be in turn written as \n\\begin{equation*\n\\min_{\\substack{\\bs \\phi,\\bs\\epsilon:\\\\\\|\\bs \\phi\\|_0\\leq k}} \\frac{1}{2} \\|\\mb y-{\\mb X}(\\bs \\phi+\\bs\\epsilon)\\|_2^2 + \\lambda\\|\\bs\\epsilon\\|_1\n\\end{equation*}\nfor some $k\\in\\{0,1,\\ldots,p\\}$. But as per the observations of Section \\ref{ssec:vardecomp}, this is exactly $\\tla{\\lambda,k}$ using a change of variables ${\\bs\\beta}=\\bs \\phi+\\bs\\epsilon$. In the case when $\\lambda$ is sufficiently large, we necessarily have ${\\bs\\beta}=\\bs \\phi$ at optimality.\n\\end{proof}\n\nWhile this result is not surprising (given that as $\\gamma\\to\\infty$ the problem is \\eqref{eqn:unbpm} is precisely penalized best subset selection), it is useful for illustrating the connection between \\eqref{eqn:unbpm} and the trimmed Lasso problem even when the trimmed Lasso parameter $\\lambda$ is not necessarily large: in particular, $\\tla{\\lambda,k}$ can be viewed as estimating ${\\bs\\beta}$ as the sum of two components---a sparse component $\\bs \\phi$ and small-norm (``noise'') component $\\bs\\epsilon$. Indeed, in this setup, $\\lambda$ precisely controls the desirable level of allowed ``noise'' in ${\\bs\\beta}$. From this intuitive perspective, it becomes clearer why the trimmed Lasso type approach represents a continuous connection between best subset selection ($\\lambda$ large) and ordinary least squares ($\\lambda$ small).\n\nWe close this section by making the following observation regarding problem \\eqref{eqn:unbpm-aux}. In particular, observe that regardless of $\\lambda$, we can rewrite this as\n\\begin{equation*\n\\min_{{\\bs\\beta}} \\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\sum_{i=1}^p \\widetilde{\\rho}(|\\beta_i|)\n\\end{equation*}\nwhere $\\widetilde{\\rho}(|\\beta_i|)$ is the new penalty function defined as\n$$\\widetilde{\\rho}(|\\beta_i|) = \\min_{\\phi+\\epsilon = \\beta_i} \\lambda|\\epsilon| + \\mu g(|\\phi|;\\gamma).$$\nFor the unbounded and concave penalty functions shown in Table \\ref{tab:ncp}, this new penalty function is quasi-concave and can be rewritten easily in closed form. For example, for the $\\ell_q$ penalty $\\rho(|\\beta_i|) = \\mu|\\beta_i|^{1\/\\gamma}$ (where $\\gamma>1$), the new penalty function is\n$$\\widetilde{\\rho}(|\\beta_i|) = \\min\\{\\mu|\\beta_i|^{1\/\\gamma},\\lambda|\\beta_i|\\}.$$\n\n\n\n\n\n\\section{Algorithmic Approaches}\\label{sec:algs}\n\n\n\nWe now turn our attention to algorithms for estimation with the trimmed Lasso penalty. Our principle focus throughout will be the same problem considered in Theorem \\ref{thm:exactEquiv}, namely\n\\begin{equation}\\label{eqn:alg}\n\\displaystyle\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda \\tk{{\\bs\\beta}} + \\eta \\|{\\bs\\beta}\\|_1\n\\end{equation}\nWe present three possible approaches to finding potential solutions to \\eqref{eqn:alg}: a first-order-based alternating minimization scheme that has accompanying local optimality guarantees and was first studied in \\cite{gotoh1,gotoh2}; an augmented Lagrangian approach that appears to perform noticeably better, despite lacking optimality guarantees; and a convex envelope approach. We contrast these methods with approaches for certifying global optimality of solutions to \\eqref{eqn:alg} (described in \\cite{thiao}) and include an illustrative computational example.\nImplementations of the various algorithms presented can be found at\n\\begin{center}\n\\url{https:\/\/github.com\/copenhaver\/trimmedlasso}.\n\\end{center}\n\n\n\n\n\n\\subsection{Upper bounds via convex methods}\\label{ssec:ub}\n\nWe start by focusing on the application of convex optimization methods to finding to finding potential solutions to \\eqref{eqn:alg}. Technical details are contained in Appendix \\ref{app:algsupp}.\n\n\n\\subsubsection*{Alternating minimization scheme}\n\n\nWe begin with a first-order-based approach for obtaining a locally optimal solution of \\eqref{eqn:alg} as described in \\cite{gotoh1,gotoh2}. The key tool in this approach is the theory of difference of convex optimization (``DCO'') \\cite{anThesis,taoan97,dcSummary}. Set the following notation:\n$$\\begin{array}{lll}\nf({\\bs\\beta}) &=& \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2\/2 + \\lambda \\tk{{\\bs\\beta}}+ \\eta \\|{\\bs\\beta}\\|_1,\\\\\nf_1({\\bs\\beta}) &=& \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2\/2 + (\\eta+\\lambda) \\|{\\bs\\beta}\\|_1,\\\\\nf_2({\\bs\\beta}) &=& \\lambda \\sum_{i=1}^k |\\beta_{(i)}|.\n\\end{array}\n$$\nLet us make a few simple observations:\n\\begin{enumerate}[(a)]\n\\item Problem \\eqref{eqn:alg} can be written as $\\displaystyle\\min_{\\bs\\beta} f({\\bs\\beta})$.\n\n\\item For all ${\\bs\\beta}$, $f({\\bs\\beta}) = f_1({\\bs\\beta})-f_2({\\bs\\beta})$.\n\n\\item The functions $f_1$ and $f_2$ are convex.\n\\end{enumerate}\n\nWhile simple, these observations enable one to apply the theory of DCO, which focuses precisely on problems of the form\n$$\\min_{{\\bs\\beta}} f_1({\\bs\\beta})-f_2({\\bs\\beta}),$$\nwhere $f_1$ and $f_2$ are convex. In particular, the optimality conditions for such a problem have been studied extensively \\cite{dcSummary}. Let us note that while it may appear that the representation of the objective $f$ as $f_1-f_2$ might otherwise seem like an artificial algebraic manipulation, the min-min representation in Theorem \\ref{thm:robeivInterp} shows how such a difference-of-convex representation can arise naturally.\n\nWe now discuss an associated alternating minimization scheme (or equivalently, a sequential linearization scheme), shown in Algorithm \\ref{alg:1}, for finding local optima of \\eqref{eqn:alg}.\nThe convergence properties of Algorithm \\ref{alg:1} can be summarized as follows:\\footnote{To be entirely correct, this result holds for Algorithm \\ref{alg:1} with a minor technical modification---see details in Appendix \\ref{app:algsupp}.}\n\n\\begin{theorem}[\\cite{gotoh1}, Convergence of Algorithm \\ref{alg:1}]\\label{thm:altConvProp}\n\\begin{enumerate}[(a)]\n\\item The sequence $\\{f({\\bs\\beta}^\\ell):\\ell=0,1,\\ldots\\}$, where ${\\bs\\beta}^\\ell$ are as found in Algorithm \\ref{alg:1}, is non-increasing.\n\n\\item The set $\\{\\bs\\gamma^\\ell: \\ell=0,1,\\ldots\\}$ is finite and eventually periodic.\n\n\\item Algorithm \\ref{alg:1} converges in a finite number of iterations to local minimum of \\eqref{eqn:alg}.\n\n\\item The rate of convergence of $f({\\bs\\beta}^\\ell)$ is linear.\n\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{algorithm}[ht]\n\\begin{enumerate}\n\\item Initialize with any ${\\bs\\beta}^{0}\\in\\mathbb{R}^p$ ($\\ell = 0$); for $\\ell \\geq 0$, repeat Steps 2-3 until $f({\\bs\\beta}^\\ell) = f({\\bs\\beta}^{\\ell+1})$.\n\n\\item Compute $\\bs\\gamma^\\ell$ as\n\\begin{equation}\\label{eqn:wrtbg}\n\\bs\\gamma^\\ell \\in\\begin{array}{ll}\n\\underset{\\bs\\gamma}{\\operatorname{argmax}} & \\langle \\bs\\gamma,{\\bs\\beta}^\\ell\\rangle\\\\\n\\operatorname{s.t.}& \\displaystyle\\sum_i |\\gamma_i| \\leq \\lambda k\\\\\n& \\displaystyle|\\gamma_i|\\leq \\lambda\\;\\forall i.\n\\end{array}\n\\end{equation}\n\n\\item Compute ${\\bs\\beta}^{\\ell+1}$ as \n\\begin{equation}\n{\\bs\\beta}^{\\ell+1} \\in\\underset{{\\bs\\beta}}{\\operatorname{argmin }} \\;  \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 +(\\eta+\\lambda)\\|{\\bs\\beta}\\|_1 - \\langle{\\bs\\beta},\\bs\\gamma^\\ell\\rangle.\\label{eqn:wrtbb}\n\\end{equation}\n\n\n\n\n\\end{enumerate}\n\\caption{An alternating scheme for computing a local optimum to \\eqref{eqn:alg}}\\label{alg:1}\n\\end{algorithm}\n\n\n\n\n\n\\begin{obs}\nLet us return to a remark that preceded Algorithm \\ref{alg:1}. In particular, we noted that Algorithm \\ref{alg:1} can also be viewed as a sequential linearization approach to solving \\eqref{eqn:alg}. Namely, this corresponds to sequentially performing a linearization of $f_2$ (and leaving $f_1$ as is), and then solving the new convex linearized problem.\n\nFurther, let us note why we refer to Algorithm \\ref{alg:1} as an alternating minimization scheme. In particular, in light of the reformulation \\eqref{eqn:mainReform} of \\eqref{eqn:alg}, we can rewrite \\eqref{eqn:alg} exactly as \n$$\\eqref{eqn:alg} = \\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta},\\bs\\gamma} & f_1({\\bs\\beta}) - \\langle\\bs\\gamma,{\\bs\\beta}\\rangle\\\\\n\\operatorname{s.t.}& \\displaystyle\\sum_i |\\gamma_i| \\leq \\lambda k\\\\\n& \\displaystyle|\\gamma_i|\\leq \\lambda\\;\\forall i.\n\\end{array}$$\nIn this sense, if one takes care in performing alternating minimization in ${\\bs\\beta}$ (with $\\bs\\gamma$ fixed) and in $\\bs\\gamma$ (with ${\\bs\\beta}$ fixed) (as in Algorithm \\ref{alg:1}), then a locally optimal solution is guaranteed.\n\\end{obs}\n\nWe now turn to how to actually apply Algorithm \\ref{alg:1}. Observe that the algorithm is quite simple; in particular, it only requires solving two types of well-structured convex optimization problems. The first such problem, for a fixed ${\\bs\\beta}$, is shown in \\eqref{eqn:wrtbg}. This can be solved in closed form by simply sorting the entries of $|{\\bs\\beta}|$, i.e., by finding $|\\beta_{(1)}|,\\ldots,|\\beta_{(p)}|$. \nThe second subproblem, shown in \\eqref{eqn:wrtbb} for a fixed $\\bs\\gamma$, is precisely the usual Lasso problem and is amenable to any of the possible algorithms for the Lasso \\cite{tibshirani,lars,hastie}. \n\n\n\n\n\\subsubsection*{Augmented Lagrangian approach}\n\nWe briefly mention another technique for finding potential solutions to \\eqref{eqn:alg} using an Alternating Directions Method of Multiplers (ADMM) \\cite{admm} approach. To our knowledge, the application of ADMM to the trimmed Lasso problem is novel, although it appears closely related to \\cite{admmteng}. \nWe begin by observing that \\eqref{eqn:alg} can be written exactly as\n$$\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta},\\bs\\gamma}& \\frac{1}{2}\\left\\|\\mb y-{\\mb X}{\\bs\\beta}\\right\\|_2^2 + \\eta\\left\\|{\\bs\\beta}\\right\\|_1 + \\lambda \\tk{\\bs\\gamma} \\\\\n\\operatorname{s.t.} & {\\bs\\beta}=\\bs\\gamma,\n\\end{array}$$\nwhich makes use of the canonical variable splitting. Introducing dual variable $\\mb q\\in\\mathbb{R}^p$ and parameter $\\sigma>0$, this becomes in augmented Lagrangian form\n\\begin{align}\n\\displaystyle\\min_{{\\bs\\beta},\\bs\\gamma} \\max_{\\mb q}\\;&\\frac{1}{2}\\left\\|\\mb y-{\\mb X}{\\bs\\beta}\\right\\|_2^2 + \\eta\\left\\|{\\bs\\beta}\\right\\|_1 +\\lambda \\tk{\\bs\\gamma} + \\nonumber\\\\\n& \\langle \\mb q, {\\bs\\beta}-\\bs\\gamma\\rangle + \\frac{\\sigma}{2}\\left\\|{\\bs\\beta}-\\bs\\gamma\\right\\|_2^2.\\label{eqn:admm}\n\\end{align}\n\nThe utility of such a reformulation is that it is directly amenable to ADMM, as detailed in Algorithm \\ref{alg:admm}. While the problem is nonconvex and therefore the ADMM is not guaranteed to converge, numerical experiments suggest that this approach has superior performance to the DCO-inspired method considered in Algorithm \\ref{alg:1}\n\nWe close by commenting on the subproblems that must be solved in Algorithm \\ref{alg:admm}. Step 2 can be carried out using ``hot'' starts. Step 3 is the solution of the trimmed Lasso in the orthogonal design case and can be solved by performed by sorting $p$ numbers; see Appendix \\ref{app:algsupp}.\n\n\\begin{algorithm}[ht]\n\\begin{enumerate}\n\\item Initialize with any ${\\bs\\beta}^0,\\bs\\gamma^0,\\mb q^0 \\in\\mathbb{R}^p$ and $\\sigma>0$. Repeat, for $\\ell\\geq 0$,  Steps 2, 3, and 4 until a desired numerical\nconvergence tolerance is satisfied.\n\n\\item Set\n\\begin{align*}\n{\\bs\\beta}^{\\ell+1} \\in  \\displaystyle\\underset{{\\bs\\beta}}{\\operatorname{argmin}}\\; &\\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2+\\eta\\|{\\bs\\beta}\\|_1\\;+\\\\\n&\\langle \\mb q^\\ell, {\\bs\\beta}\\rangle + \\frac{\\sigma}{2}\\|{\\bs\\beta}-\\bs\\gamma^\\ell\\|_2^2.\n\\end{align*}\n\n\\item Set\n$$\\bs\\gamma^{\\ell+1} \\in\\underset{\\bs\\gamma}{\\operatorname{argmin} } \\;\\lambda \\tk{\\bs\\gamma} + \\frac{\\sigma}{2}\\|{\\bs\\beta}^{\\ell+1}-\\bs\\gamma\\|_2^2  - \\langle \\mb q^\\ell,\\bs\\gamma\\rangle.$$\n\n\\item Set $\\mb q^{\\ell+1}  = \\mb q^{\\ell} + \\sigma\\left({\\bs\\beta}^{\\ell+1} - \\bs\\gamma^{\\ell+1}\\right)$. \n\n\\end{enumerate}\n\\caption{ADMM algorithm for \\eqref{eqn:admm}}\\label{alg:admm}\n\\end{algorithm}\n\n\n\n\n\n\n\n\\subsubsection*{Convexification approach}\\label{ssec:convenv}\n\n\nWe briefly consider the convex relaxation of the problem \\eqref{eqn:alg}. We begin by computing the convex envelope \\cite{rockafeller,BV2004} of $T_k$ on $[-1,1]^p$ (here the choice of $[-1,1]^p$ is standard, such as in the convexification of $\\ell_0$ over this set which leads to $\\ell_1$). The proof follows standard techniques (e.g. computing the biconjugate\\cite{rockafeller}) and is omitted.\n\n\\begin{lemma}\\label{lem:convenv}\nThe convex envelope of $T_k$ on $[-1,1]^p$ is the function $\\overline{T_k}$ defined as\n$$\\overline{T_k}({\\bs\\beta}) = \\left(\\|{\\bs\\beta}\\|_1-k\\right)_+.$$\n\\end{lemma}\n\nIn words, the convex envelope of $T_k$ is a ``soft thresholded'' version of the Lasso penalty (thresholded at level $k$). This can be thought of as an alternative way of interpreting the name ``trimmed Lasso.''\n\nAs a result of Lemma \\ref{lem:convenv}, it follows that the convex analogue of \\eqref{eqn:alg}, as taken over $[-1,1]^p$, is precisely\n\\begin{equation}\\label{eqn:convenv}\n\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\eta\\|{\\bs\\beta}\\|_1 + \\lambda\\left(\\|{\\bs\\beta}\\|_1-k\\right)_+.\n\\end{equation}\nProblem \\eqref{eqn:convenv} is amenable to a variety of convex optimization techniques such as subgradient descent \\cite{BV2004}.\n\n\n\n\\subsection{Certificates of optimality for \\eqref{eqn:alg}}\\label{ssec:mio}\n\nWe close our discussion of the algorithmic implications of the trimmed Lasso by discussing techniques for finding certifiably optimal solutions to \\eqref{eqn:alg}. All approaches presented in the preceding section find potential candidates for solutions to \\eqref{eqn:alg}, but none is necessarily globally optimal. Let us return to a representation of \\eqref{eqn:alg} that makes use Lemma \\ref{lemma:miprep}:\n\\begin{equation*\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta},\\mb z} & \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\eta\\|{\\bs\\beta}\\|_1 + \\lambda\\langle\\mb z,|{\\bs\\beta}|\\rangle\\\\\n\\operatorname{s.t.}& \\displaystyle\\sum_i z_i=p-k\\\\\n&\\mb z\\in\\{0,1\\}^p.\n\\end{array}\n\\end{equation*}\nAs noted in \\cite{gotoh1}, this representation of \\eqref{eqn:alg} is amenable to mixed integer optimization (``MIO'') methods \\cite{bonami} for finding globally optimal solutions to \\eqref{eqn:alg}, in the same spirit as other MIO-based approaches to statistical problems \\cite{bmlqs,bkm}.\n\n\n\nOne approach, as described in \\cite{thiao}, uses the notion of ``big $M$.'' In particular, for $M>0$ sufficiently large, problem \\eqref{eqn:alg} can be written exactly as the following linear MIO problem:\n\n\\begin{equation}\\label{eqn:pf7}\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta},\\mb z,\\mb a} &\\displaystyle \\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\eta\\|{\\bs\\beta}\\|_1 + \\lambda\\sum_i a_i\\\\\n\\operatorname{s.t.}& \\displaystyle\\sum_i z_i=p-k\\\\\n&\\mb z\\in\\{0,1\\}^p\\\\\n& \\mb a \\geq {\\bs\\beta} + M\\mb z- M\\mb 1\\\\\n& \\mb a \\geq -{\\bs\\beta} + M\\mb z- M\\mb 1\\\\\n& \\mb a\\geq \\mb0.\n\\end{array}\n\\end{equation}\nThis representation as a linear MIO problem enables the direct application of numerous existing MIO algorithms (such as \\cite{gurobi}).\\footnote{There are certainly other possible representations of \\eqref{eqn:mainReform}, such as using special ordered set (SOS) constraints, see e.g. \\cite{bkm}. Without more sophisticated tuning of $M$ as in \\cite{bkm}, the SOS formulations appear to be vastly superior in terms of time required to prove optimality. The precise formulation essentially takes the form of problem \\eqref{eqn:BSSr}. An SOS-based implementation is provided in the supplementary code as the default method of certifying optimality.} Also, let us note that the linear relaxation of \\eqref{eqn:pf7}, i.e., problem \\eqref{eqn:pf7} with the constraint $\\mb z\\in\\{0,1\\}^p$ replaced with $\\mb z\\in[0,1]^p$, is the problem\n\\begin{equation*\n\\displaystyle\\min_{{\\bs\\beta}}  \\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\eta\\|{\\bs\\beta}\\|_1 + \\lambda \\left(\\|{\\bs\\beta}\\|_1-Mk\\right)_+,\n\\end{equation*}\nwhere we see the convex envelope penalty appear directly. As such, when $M$ is large, the linear relaxation of \\eqref{eqn:pf7} is the ordinary Lasso problem $\\min_{{\\bs\\beta}}  \\frac{1}{2} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\eta\\|{\\bs\\beta}\\|_1$.\n\n\n\n\\subsection{Computational example}\n\nBecause a rigorous computational comparison is not the primary focus of this paper, we provide a limited demonstration that describes the behavior of solutions to \\eqref{eqn:alg} as computed via the different approaches. Precise computational details are contained in Appendix \\ref{app:compdetail}. We will focus on two different aspects: sparsity and approximation quality.\n\n\n\\subsubsection*{Sparsity properties}\n\nAs the motivation for the trimmed Lasso is ostensibly sparse modeling, its sparsity properties  are  of particular interest. We consider a problem instance with $p=20$, $n=100$, $k=2$, and signal-to-noise ratio 10 (the sparsity of the ground truth model ${\\bs\\beta}_\\text{true}$ is $10$). The relevant coefficient profiles as a function of $\\lambda$ are shown in Figure \\ref{fig:coeffpath}. In this example none of the convex approaches finds the optimal two variable solution computed using mixed integer optimization. Further, as one would expect \\emph{a priori}, the optimal coefficient profiles (as well as the ADMM profiles) are not continuous in $\\lambda$. Finally, note that by design of the algorithms, the alternating minimization and ADMM approaches yield solutions with sparsity at most $k$ for $\\lambda$ sufficiently large.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.8\\textwidth]{coeffpath.pdf}\n\\caption{}\n\\label{fig:coeffpath}\n\\end{figure}\n\n\n\\subsubsection*{Optimality gap}\n\nAnother critical question is the degree of suboptimality of solutions found via the convex approaches. We average optimality gaps across 100 problem instances with $p=20$, $n=100$, and $k=2$; the relevant results are shown in Figure \\ref{fig:optgap}. The results are entirely as one might expect. When $\\lambda$ is small and the problem is convex or nearly convex, the heuristics perform well. However, this breaks down as $\\lambda$ increases and the sparsity-inducing nature of the trimmed Lasso penalty comes into play. Further, we see that the convex envelope approach tends to perform the worst, with the ADMM performing the best of the three heuristics. This is perhaps not surprising, as any solution found via the ADMM can be guaranteed to be locally optimal by subsequently applying the alternating minimization scheme of Algorithm \\ref{alg:1} to any solution found via Algorithm \\ref{alg:admm}.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.8\\textwidth]{optgap.pdf}\n\\caption{}\n\\label{fig:optgap}\n\\end{figure}\n\n\n\\subsubsection*{Computational burden}\n\nLoosely speaking, the heuristic approaches all carry a similar computational cost per iteration, namely, solving a Lasso-like problem. In contrast, the MIO approach can take significantly more computational resources. However, by design, the MIO approach maintains a suboptimality gap throughout computation and can therefore be terminated, before optimality is certified, with a certificate of suboptimality. We do not consider any empirical analysis of runtime here.\n\n\n\\subsubsection*{Other considerations}\n\nThere are other additional computational considerations that are potentially of interest as well, but they are primarily beyond the scope of the present work. For example, instead of considering optimality purely in terms of objective values in \\eqref{eqn:alg}, there are other critical notions from a statistical perspective (e.g. ability to recover true sparse models and performance on out-of-sample data) that would also be necessary to consider across the multiple approaches.\n\n\n\n\n\n\n\\section{Conclusions}\\label{sec:conc}\n\nIn this work, we have studied the trimmed Lasso, a nonconvex adaptation of Lasso that acts as an exact penalty method for best subset selection. Unlike some other approaches to exact penalization which use coordinate-wise separable functions, the trimmed Lasso offers direct control of the desired sparsity $k$. Further, we emphasized the interpretation of the trimmed Lasso from the perspective of robustness. In doing so, we provided contrasts with the SLOPE penalty as well as comparisons with estimators from the robust statistics and total least squares literature.\n\nWe have also taken care to contextualize the trimmed Lasso within the literature on nonconvex penalized estimation approaches to sparse modeling, showing that penalties like the trimmed Lasso can be viewed as a generalization of such approaches in the case when the penalty function is bounded. In doing so, we also highlighted how precisely the problems were related, with a complete characterization given in the case of the clipped Lasso.\n\nFinally, we have shown how modern developments in optimization can be brought to bear for the trimmed Lasso to create convex optimization optimization algorithms that can take advantage of the significant developments in algorithms for Lasso-like problems in recent years.\n\nOur work here raises many interesting questions about further properties of the trimmed Lasso and the application of similar ideas in other settings. We see two particularly noteworthy directions of focus: algorithms and statistical properties. For the former, we anticipate that an approach like trimmed Lasso, which leads to relatively straightforward algorithms that use close analogues from convex optimization, is  simple to interpret and to implement. At the same time, the heuristic approaches to the trimmed Lasso presented herein carry no more of a computational burden than solving convex, Lasso-like problems. On the latter front, we anticipate that a deeper analysis of the statistical properties of estimators attained using the trimmed Lasso would help to illuminate it in its own right while also further connecting it to existing approaches in the statistical estimation literature.\n\n\n\\begin{appendices}\n\n\\section{General min-max representation of SLOPE}\\label{app:slope}\n\nFor completeness, in this appendix we include the more general representation of the SLOPE penalty $R_{\\textsc{SLOPE}(\\mb w)}$ in the same spirit of Proposition \\ref{prop:slope}. Here we work with SLOPE in its most general form, namely,\n$$R_{\\textsc{SLOPE}(\\mb w)}({\\bs\\beta}) = \\sum_{i=1}^p w_i |\\beta_{(i)}|, $$\nwhere $\\mb w$ is a (fixed) vector of weights with $w_1\\geq w_2\\geq \\cdots\\geq w_p\\geq 0$ and $w_1>0$.\n\nTo describe the general min-max representation, we first set some notation. For a matrix $\\bs\\Delta\\in\\mathbb{R}^{n\\times p}$, we let $\\boldsymbol\\nu(\\bs\\Delta)\\in\\mathbb{R}^p$ be the vector $(\\|\\bs\\Delta_1\\|_2,\\ldots,\\|\\bs\\Delta_p\\|_2)$ with entries sorted so that $\\nu_1\\geq \\nu_2\\geq \\cdots \\geq \\nu_p$. As usual, for two vectors $\\mb x$ and $\\mb y$, we use $\\mb x\\leq\\mb y$ to denote that coordinate-wise inequality holds. With this notation, we have the following:\n\\begin{proposition}\\label{prop:slopefullgenerality}\nProblem \\eqref{eqn:roprimitive} with uncertainty set\n$$\\mathcal{U}_\\mb w =\\left\\{\\bs\\Delta : \\boldsymbol\\nu(\\bs\\Delta) \\leq \\mb w \\right\\}\n$$\nis equivalent to problem \\eqref{eqn:a1} with $R({\\bs\\beta})=R_{\\textsc{SLOPE}(\\mb w)}({\\bs\\beta}) $. Further,\nproblem \\eqref{eqn:roprimitive} with uncertainty set\n$$\\mathcal{U}_{\\mb w} =\\left\\{\\bs\\Delta : \\|\\bs\\Delta\\bs \\phi\\|_2 \n\\leq R_{\\textsc{SLOPE}(\\mb w)}(\\bs \\phi) \\;\\forall \\bs \\phi \\right\\}\n$$\nis equivalent to problem \\eqref{eqn:a1} with $R({\\bs\\beta})=R_{\\textsc{SLOPE}(\\mb w)}({\\bs\\beta}) $.\n\\end{proposition}\n\nThe proof, like the proof of Proposition \\ref{prop:slope}, follows basic techniques described in \\cite{RObook} and is therefore omitted.\n\n\n\n\n\\section{Additional proofs}\\label{app:proof}\n\nThis appendix section contains supplemental proofs not contained in the main text.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:exactEquiv}]\nLet $\\overline{\\lambda} = \\|\\mb y\\|_2\\cdot\\left(\\max_j\\|\\mb x_j\\|_2\\right)$, where $\\mb x_j$ denotes the $j$th row of ${\\mb X}$. We fix $\\lambda>\\overline{\\lambda}$, $k$, and $\\eta>0$ throughout the entire proof. We begin by observing that it suffices to show that any solution ${\\bs\\beta}$ to\n\\begin{equation}\\label{eqn:thmmain}\n\\displaystyle\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda \\tk{{\\bs\\beta}} + \\eta \\|{\\bs\\beta}\\|_1\n\\end{equation}\nsatisfies $\\tk{{\\bs\\beta}} = 0$, or equivalently, $\\|{\\bs\\beta}\\|_0\\leq k$. As per Lemma \\ref{lemma:miprep}, problem \\eqref{eqn:thmmain} can be rewritten exactly as\n\\begin{equation}\\label{eqn:mainReform}\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta},\\mb z} & \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda\\langle\\mb z,|{\\bs\\beta}|\\rangle+ \\eta\\|{\\bs\\beta}\\|_1 \\\\\n\\operatorname{s.t.}& \\displaystyle\\sum_i z_i=p-k\\\\\n&\\mb z\\in\\{0,1\\}^p.\n\\end{array}\n\\end{equation}\nLet $({\\bs\\beta}^*,\\mb z^*)$ be any solution to \\eqref{eqn:mainReform}. Observe that necessarily ${\\bs\\beta}^*$ is also a solution to the problem\n\\begin{equation}\\label{eqn:pfsupp1}\n\\min_{{\\bs\\beta}}\\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2  + \\lambda\\langle\\mb z^*,|{\\bs\\beta}|\\rangle+ \\eta\\|{\\bs\\beta}\\|_1.\n\\end{equation}\nNote that, unlike \\eqref{eqn:thmmain}, the problem in \\eqref{eqn:pfsupp1} is readily amenable to an analysis using the theory of proximal gradient methods \\cite{combetteswasj,bauschke}. In particular, we must have for any $\\gamma>0$ that \n\\begin{equation}\\label{eqn:pfsupp2}\n{\\bs\\beta}^* = \\operatorname{prox}_{\\gamma R} \\left({\\bs\\beta}^* - \\gamma({\\mb X}'{\\mb X}{\\bs\\beta}^* - {\\mb X}'\\mb y)\\right),\n\\end{equation}\nwhere $\\displaystyle R({\\bs\\beta}) = \\eta\\|{\\bs\\beta}\\|_1 + \\lambda \\sum_{i\\;:\\:z_i^*=1} |\\beta_i|$. Suppose that $\\tk{{\\bs\\beta}^*}>0$. In particular, for some $j\\in\\{1,\\ldots,p\\}$, we have $\\beta_j^* \\neq 0$ and $z_j^*=1$. Yet, as per \\eqref{eqn:pfsupp2},\\footnote{This is valid for the following reason: since $\\beta_j^*\\neq 0$ and $\\beta_j^*$ satisfies \\eqref{eqn:pfsupp2}, it must be the case that $\\left|\\beta_j^* - \\gamma \\mb x_{j}'({\\mb X}{\\bs\\beta}^* - \\mb y)\\right| > \\gamma(\\eta+\\lambda)$, for otherwise the soft-thresholding operator at level $\\gamma(\\eta+\\lambda)$ would set this quantity to zero.}\n$$\\left|\\beta_j^* - \\gamma \\langle\\mb x_{j},{\\mb X}{\\bs\\beta}^* - \\mb y\\rangle\\right| > \\gamma(\\eta+\\lambda)\\;\\quad \\text{ for all } \\gamma>0,$$\nwhere $\\mb x_j$ denotes the $j$th row of ${\\mb X}$. This implies that\n$$\\left|\\langle\\mb x_j,{\\mb X}{\\bs\\beta}^*-\\mb y\\rangle\\right| \\geq \\eta+\\lambda.$$\nNow, using the definition of $\\overline{\\lambda}$, observe that\n\\begin{align*}\n\\eta+\\lambda\\leq \\left|\\langle\\mb x_j,{\\mb X}{\\bs\\beta}^*-\\mb y\\rangle\\right|& \\leq \\|\\mb x_j\\|_2 \\|{\\mb X}{\\bs\\beta}^*-\\mb y\\|_2\\\\\n&\\leq \\|\\mb x_j\\|_2 \\|\\mb y\\| \\leq \\overline{\\lambda}<\\lambda,\n\\end{align*}\nwhich is a contradiction since $\\eta>0$. Hence, $\\tk{{\\bs\\beta}^*}=0$, completing the proof.\n\\end{proof}\n\n\n\n\n\\subsubsection*{Extended statement of Proposition \\ref{prop:asymp}}\n\nWe now include a precise version of the convergence claim in Proposition \\ref{prop:asymp}. Let us set a standard notion: we say that ${\\bs\\beta}$ is $\\epsilon$-optimal (for $\\epsilon>0$)  to an optimization problem $(\\textrm{P})$ if the optimal objective value of $(\\textrm{P})$ is within $\\epsilon$ of the objective value of ${\\bs\\beta}$. We add an additional regularizer $\\eta\\|{\\bs\\beta}\\|_1$, for $\\eta>0$ fixed, to the objective in order to ensure coercivity of the objective functions.\n\n\\begin{proposition}[Extended form of Proposition \\ref{prop:asymp}]\nLet $g:\\mathbb{R}_+\\to\\mathbb{R}_+$ be an unbounded, continuous, and strictly increasing function with $g(0)=0$. \nConsider the problems\n\\[\\label{eqn:cvg1}\n\\displaystyle\\min_{{\\bs\\beta}} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\lambda\\pi_k^g({\\bs\\beta}) + \\eta\\|{\\bs\\beta}\\|_1\n\\]\nand\n\\[\\label{eqn:cvg2}\n\\displaystyle\\min_{\\|{\\bs\\beta}\\|_0\\leq k} \\frac{1}{2}\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + \\eta\\|{\\bs\\beta}\\|_1.\n\\]\nFor every $\\epsilon>0$, there exists some $\\underline{\\lambda}=\\underline{\\lambda}(\\epsilon)>0$ so that for all $\\lambda>\\underline{\\lambda}$,\n\\begin{enumerate}\n\\item For every optimal $\\bb^*$ to \\eqref{eqn:cvg1}, there is some $\\widehat{\\bb}$ so that $\\|\\bb^*-\\widehat{\\bb}\\|_2\\leq \\epsilon$, $\\widehat{\\bb}$ is feasible to \\eqref{eqn:cvg2}, and $\\widehat{\\bb}$ is $\\epsilon$-optimal to \\eqref{eqn:cvg2}.\n\\item Every optimal ${\\bs\\beta}^*$ to \\eqref{eqn:cvg2} is $\\epsilon$-optimal to \\eqref{eqn:cvg1}.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThe proof follows a basic continuity argument that is simpler than the one presented below in Theorem \\ref{thm:corprecise}. For that reason, we do not include a full proof. Observe that the assumptions on $g$ imply that $g^{-1}$ is well-defined on, say, $g([0,1])$. If we let $\\epsilon>0$ and suppose that $\\bb^*$ is optimal to \\eqref{eqn:cvg1}, where $\\lambda > \\underline{\\lambda} := \\|\\mb y\\|_2^2\/(2g(\\epsilon\/p))$, and if we define $\\widehat{\\bb}$ to be $\\bb^*$ with all but the $k$ largest magnitude entries truncated to zero (ties broken arbitrarily), then $\\pi_k^g(\\bb^*)\\leq \\|\\mb y\\|_2^2\/(2\\lambda)$ and $\\pi_k^g (\\bb^*) = \\sum_{i=1}^p g(|\\beta_i^*-\\widehat{\\beta}_i|)$ so that $|\\beta_i^*-\\widehat{\\beta}_i| \\leq g^{-1}(\\|\\mb y\\|_2^2\/(2\\lambda)) \\leq \\epsilon\/p$ by definition of $\\underline{\\lambda}$. Hence, $\\|\\bb^*-\\widehat{\\bb}\\|_1\\leq \\epsilon$, and all the other claims essentially follow from this.\n\\end{proof}\n\n\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:robeivInterp}]\nWe begin by showing that for any ${\\bs\\beta}$,\n\\begin{equation*\n\\min_{\\bs\\Delta\\in\\mathcal{U}_k^\\lambda}\\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2 =  \\left(\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 - \\lambda \\sum_{i=1}^k |\\beta_{(i)}|   \\right)_+\n\\end{equation*}\nwhere $(a)_+:=\\max\\{0,a\\}$. Fix ${\\bs\\beta}$ and set $\\mb r = \\mb y-{\\mb X}{\\bs\\beta}$. We assume without loss of generality that $\\mb r\\neq \\mb0$ and that ${\\bs\\beta}\\neq\\mb0$. For any $\\bs\\Delta$, note that $\\|\\mb r - \\bs\\Delta{\\bs\\beta}\\|_2 \\geq 0$ and $\\|\\mb r - \\bs\\Delta{\\bs\\beta}\\|_2 \\geq \\|\\mb r\\|_2 - \\|\\bs\\Delta{\\bs\\beta}\\|_2$ by the reverse triangle inequality. Now observe that for $\\bs\\Delta\\in\\mathcal{U}_k^\\lambda$,\n$$\\|\\bs\\Delta{\\bs\\beta}\\|_2  \\leq \\sum_i |\\beta_i| \\|\\bs\\Delta_i\\|_2 \\leq \\sum_{i=1}^k \\lambda |\\beta_{(i)}|.$$\nTherefore, $\\|\\mb r-\\bs\\Delta{\\bs\\beta}\\|_2 \\geq \\left(\\|\\mb r\\|_2 - \\lambda\\sum_{i=1}^k |\\beta_{(i)}| \\right)_+$. Let $I\\subseteq\\{1,\\ldots,p\\}$ be a set of $k$ indices which correspond to the $k$ largest entries of ${\\bs\\beta}$ (if $|\\beta_{(k)}|=|\\beta_{(k+1)}|$, break ties arbitrarily). Define $\\bs\\Delta\\in\\mathcal{U}_k^\\lambda$ as the matrix whose $i$th column is\n$$\\left\\{\\begin{array}{rl}\n\\underline{\\lambda}\\operatorname{sgn}(\\beta_i)\\mb r \/ \\|\\mb r\\|_2,&i\\in I\\\\\n0,&i\\notin I,\n\\end{array}\\right.$$\nwhere $\\underline{\\lambda} = \\min\\left\\{\\lambda, \\|\\mb r\\|_2\/\\left(\\sum_{i=1}^k|\\beta_{(i)}|\\right)\\right\\}$. It is easy to verify that $\\bs\\Delta\\in\\mathcal{U}_k^\\lambda$ and  $\\|\\mb r-\\bs\\Delta{\\bs\\beta}\\|_2 = \\left(\\|\\mb r\\|_2 - \\lambda\\sum_{i=1}^k |\\beta_{(i)}| \\right)_+$. \nCombined with the lower bound, we have\n$$\\min_{\\bs\\Delta\\in\\mathcal{U}_k^\\lambda} \\|\\mb y-({\\mb X}+\\bs\\Delta){\\bs\\beta}\\|_2 = \\left(\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 -\\lambda\\sum_{i=1}^k |\\beta_{(i)}|\\right)_+$$\nwhich completes the first claim.\n\nIt follows that the problem \\eqref{eqn:eivconhompen} can be rewritten exactly as\n\\begin{equation}\\label{eqn:pfsupp3}\n\\min_{{\\bs\\beta}} \\left(\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 - \\lambda \\sum_{i=1}^k |\\beta_{(i)}|   \\right)_+ + r({\\bs\\beta}).\n\\end{equation}\n\nTo finish the proof of the theorem, it suffices to show that if ${\\bs\\beta}^*$ is a solution to \\eqref{eqn:pfsupp3}, then\n$$\\|\\mb y-{\\mb X}{\\bs\\beta}^*\\|_2 - \\lambda\\sum_{i=1}^k |\\beta_{(i)}^*| \\geq 0.$$\nIf this is not true, then $\\|\\mb y-{\\mb X}{\\bs\\beta}^*\\|_2 - \\lambda\\sum_{i=1}^k |\\beta_{(i)}^*| <0$ and so ${\\bs\\beta}^*\\neq \\mb 0$. However, this implies that for $1>\\epsilon>0$ sufficiently small, ${\\bs\\beta}_\\epsilon:=(1-\\epsilon){\\bs\\beta}^*$ satisfies $\\|\\mb y-{\\mb X}{\\bs\\beta}_\\epsilon\\|_2 - \\lambda\\sum_{i=1}^k |(\\beta_\\epsilon)_{(i)}| <0$. This in turn implies that\n$$\\begin{array}{l}\n\\left(\\|\\mb y-{\\mb X}{\\bs\\beta}_\\epsilon\\|_2 - \\lambda\\sum_{i=1}^k |(\\beta_\\epsilon)_{(i)}| \\right)_+ + r({\\bs\\beta}_\\epsilon)\\\\\n<  \\left(\\|\\mb y-{\\mb X}{\\bs\\beta}^*\\|_2 - \\lambda\\sum_{i=1}^k |\\beta_{(i)}^*| \\right)_+ +r({\\bs\\beta}^*),\n\\end{array}$$\nwhich contradicts the optimality of ${\\bs\\beta}^*$. (We have used the absolute homogeneity of the norm $r$ and that ${\\bs\\beta}^*\\neq\\mb0$.) Hence, any optimal ${\\bs\\beta}^*$ to \\eqref{eqn:pfsupp3} necessarily satisfies $\\|\\mb y-{\\mb X}{\\bs\\beta}^*\\|_2 - \\lambda\\sum_{i=1}^k |\\beta_{(i)}^*| \\geq 0$ and so the desired results follows.\n\\end{proof}\n\n\\emph{N.B.} The assumption that $r$ is a norm can be relaxed somewhat (as is clear in the proof), although the full generality is not necessary for our purposes.\n\n\n\\subsection*{Corollary \\ref{cor:slope} and related discussions}\n\nHere we include a precise statement of the ``approximate'' claim in Corollary \\ref{cor:slope}. After the proof, we include a discussion of related technical issues\n\n\\begin{theorem}[Precise statement of Corollary \\ref{cor:slope}]\\label{thm:corprecise}\nFor $\\tau>\\lambda>0$, consider the problems\n\\begin{equation}\\label{eqn:ced1\n\\begin{array}{ll}\n\\displaystyle\\min_{{\\bs\\beta} } &\\displaystyle\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + (\\tau-\\lambda)\\|{\\bs\\beta}\\|_1+ \\lambda \\tk{{\\bs\\beta}} \\\\\n\\operatorname{s.t.} & \\displaystyle\\lambda\\sum_{i=1}^k|\\beta_{(i)}| \\leq \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2.\n\\end{array}\n\\end{equation}\nand\n\\begin{equation}\\label{eqn:ced2\n\\min_{{\\bs\\beta} } \\displaystyle\\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2 + (\\tau-\\lambda)\\|{\\bs\\beta}\\|_1+ \\lambda \\tk{{\\bs\\beta}}.\n\\end{equation}\nFor all $\\epsilon>0$, there exists $\\overline{\\lambda}=\\overline{\\lambda}(\\epsilon)>0$ so that whenever $\\lambda\\in(0,\\overline{\\lambda})$,\n\\begin{enumerate}\n\\item Every optimal ${\\bs\\beta}^*$ to \\eqref{eqn:ced1} is $\\epsilon$-optimal to \\eqref{eqn:ced2}.\n\\item For every optimal ${\\bs\\beta}^*$ to \\eqref{eqn:ced2}, there is some $\\widehat{\\bb}$ so that $\\|{\\bs\\beta}^*-\\widehat{\\bb}\\|_2\\leq \\epsilon$, $\\widehat{\\bb}$ is feasible to \\eqref{eqn:ced1}, and $\\widehat{\\bb}$ is $\\epsilon$-optimal to \\eqref{eqn:ced1}. \n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nFix $\\tau>0$ throughout. We assume without loss of generality that $\\mb y\\neq\\mb0$, as otherwise the claim is obvious. We will prove the second claim first, as it essentially implies the first.\n\nLet us consider two situations. In particular, we consider whether there exists a nonzero optimal solution to\n\\[\\label{eqn:ced3}\n\\min_{\\bs\\beta} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2+\\tau\\|{\\bs\\beta}\\|_1.\n\\]\n\n\\subsubsection*{Case 1---existence of nonzero optimal solution to \\eqref{eqn:ced3}}\nWe first consider the case when there exists a nonzero solution to problem \\eqref{eqn:ced3}. We show a few lemmata:\n\n\\begin{enumerate}\n\\item We first show that the norm of solutions to \\eqref{eqn:ced2} are uniformly bounded away from zero, independent of $\\lambda$. To proceed,\nlet $\\widehat{\\bb}$ be any nonzero optimal solution to \\eqref{eqn:ced3}. Observe that if $\\bb^*$ is optimal to \\eqref{eqn:ced2}, then\n\\begin{align*}\n\\|\\mb y-{\\mb X}\\bb^*\\|_2 + (\\tau-\\lambda)\\|\\bb^*\\|_1 + \\lambda T_k({\\bb^*}) & \n\\leq \\|\\mb y-{\\mb X}\\widehat{\\bb}\\|_2 + (\\tau-\\lambda)\\|\\widehat{\\bb}\\|_1 + \\lambda T_k(\\widehat{\\bb})\\\\\n&\\leq \\|\\mb y-{\\mb X}\\bb^*\\|_2 + \\tau\\|\\bb^*\\|_1 - \\lambda \\|\\widehat{\\bb}\\|_1 + \\lambda T_k(\\widehat{\\bb}),\n\\end{align*}\nimplying that $\\|\\widehat{\\bb}\\|_1 - T_k(\\widehat{\\bb}) \\leq \\|\\bb^*\\|_1 - T_k(\\bb^*)$. In other words, $\\sum_{i=1}^k|\\widehat{\\beta}_{(i)}| \\leq \\sum_{i=1}^k |{\\beta}_{(i)}^*|\\leq \\|\\bb^*\\|_1$. Using the fact that $\\widehat{\\bb}\\neq\\mb0$, we have that any solution $\\bb^*$ to \\eqref{eqn:ced2} has strictly positive norm:\n$$\\|\\bb^*\\|_1 \\geq C>0,$$\nwhere $C:=\\sum_{i=1}^k|\\widehat{\\beta}_{(i)}|$ is a universal constant depending only on $\\tau$ (and not $\\lambda$).\n\n\\item We now upper bound the norm of solutions to \\eqref{eqn:ced2}.  In particular, if $\\bb^*$ is optimal to \\eqref{eqn:ced2}, then\n$$\\|\\mb y-{\\mb X}\\bb^*\\|_2 + (\\tau-\\lambda)\\|\\bb^*\\|_1+ \\lambda T_k({\\bb^*}) \\leq \\|\\mb y\\|_2 + 0 + 0 = \\|\\mb y\\|_2,$$\nand so $\\|\\bb^*\\|_1\\leq \\|\\mb y\\|_2\/(\\tau-\\lambda)$. (This bound is not uniform in $\\lambda$, but if we restrict our attention to, say $\\lambda\\leq \\tau\/2$, it is.)\n\n\\item We now lower bound the loss for scaled version of optimal solutions. In particular, if $\\sigma\\in[0,1]$ and $\\bb^*$ is optimal to \\eqref{eqn:ced2}, then by optimality we have that\n$$\\|\\mb y-{\\mb X}\\bb^*\\|_2 + (\\tau-\\lambda)\\|\\bb^*\\|_1 + \\lambda T_k(\\bb^*) \\leq \\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2 + (\\tau-\\lambda)\\sigma\\|\\bb^*\\|_1 + \\lambda \\sigma T_k(\\bb^*),$$\nwhich in turn implies that\n\\begin{align*}\n\\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2 &\\geq \\|\\mb y-{\\mb X}\\bb^*\\|_2 + (\\tau-\\lambda)(1-\\sigma) \\|\\bb^*\\|_1 + \\lambda (1-\\sigma) T_k(\\bb^*)\\\\\n& \\geq \\|\\mb y-{\\mb X}\\bb^*\\|_2 + (\\tau-\\lambda)(1-\\sigma) C\\geq (\\tau-\\lambda)(1-\\sigma)C\n\\end{align*}\nby combining with the first observation.\n\n\\end{enumerate}\n\n\nUsing these, we are now ready to proceed. Let $\\epsilon>0$; we assume without loss of generality that $\\epsilon<2\\|\\mb y\\|_2\/\\tau$. Let\n$$\\overline{\\lambda}:= \\min\\left\\{\\frac{\\epsilon\\tau^3 C}{4\\|\\mb y\\|_2(2\\|\\mb y\\|_2 - \\epsilon\\tau)},\\frac{\\tau}{2}\\right\\}.$$\nFix $\\lambda\\in(0,\\overline{\\lambda})$ and let $\\bb^*$ be any optimal solution to \\eqref{eqn:ced2}.   Define \n$$\\sigma:= \\left(1-\\frac{\\epsilon\\tau}{2\\|\\mb y\\|_2} \\right) \\text{\\quad and \\quad}\\widehat{\\bb} := \\sigma\\bb^*.$$\nWe claim that $\\widehat{\\bb}$ satisfies the desired requirements of the theorem:\n\n\\begin{enumerate}\n\\item We first argue that $\\|\\bb^*-\\widehat{\\bb}\\|_2 \\leq \\epsilon$. Observe that\n$$\\|\\bb^*-\\widehat{\\bb}\\|_2 = \\epsilon\\tau\\|\\bb^*\\|_2\/({2\\|\\mb y\\|_2} ) \\leq {\\epsilon\\tau}\\|\\bb^*\\|_1\/({2\\|\\mb y\\|_2} ) \\leq {\\epsilon\\tau} \\|\\mb y\\|_2\/({2\\|\\mb y\\|_2}( \\tau-\\lambda)) \\leq\\epsilon.\n$$\n\n\\item We now show that $\\widehat{\\bb}$ is feasible to \\eqref{eqn:ced1}. This requires us to argue that\n$\\lambda \\sum_{i=1}^k |\\widehat{\\beta}_{(i)}| \\leq \\|\\mb y-{\\mb X}\\widehat{\\bb}\\|_2$. Yet,\n\\begin{align*}\n\\lambda\\sum_{i=1}^k |\\widehat{\\beta}_{(i)}| &\\leq \\lambda \\|\\widehat{\\bb}\\|_1 = \\lambda \\sigma\\|\\bb^*\\|_1\\leq  2\\lambda\\sigma\\|\\mb y\\|_2\/\\tau\\leq  \\frac{\\tau}{2} (1-\\sigma) C\\\\\n& \\leq (\\tau-\\lambda) (1-\\sigma)C \\leq \\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2 = \\|\\mb y-{\\mb X}\\widehat{\\bb}\\|_2,\n\\end{align*}\nas desired. The only non-obvious step is the inequality $2\\lambda\\sigma\\|\\mb y\\|_2\/\\tau\\leq \\tau(1-\\sigma)C\/2$, which follows from algebraic manipulations using the definitions of $\\sigma$ and $\\overline{\\lambda}$.\n\n\\item Finally, we show that $\\widehat{\\bb}$ is $\\left(\\epsilon\\|{\\mb X}\\|_2\\right)$-optimal to \\eqref{eqn:ced1}. Indeed, because $\\bb^*$ is optimal to \\eqref{eqn:ced2} which necessarily lowers bound problem \\eqref{eqn:ced1}, we have that the objective value gap between $\\widehat{\\bb}$ and an optimal solution to \\eqref{eqn:ced1} is at most\n\\begin{align*}\n&\\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2 -\\|\\mb y-{\\mb X}\\bb^*\\|_2 + (\\tau-\\lambda)(\\sigma-1)\\|\\bb^*\\|_1 + \\lambda(\\sigma-1)T_k(\\bb^*)\\\\\n&\\leq (1-\\sigma)\\|{\\mb X}\\bb^*\\|_2 + 0 + 0 \\leq (1-\\sigma)\\|{\\mb X}\\|_2\\|\\bb^*\\|_2 \\leq 2(1-\\sigma) \\|{\\mb X}\\|_2\\|\\mb y\\|_2\/\\tau   \\\\\n&=2\\epsilon\\tau\/(2\\|\\mb y\\|_2)\\|{\\mb X}\\|_2\\|\\mb y\\|_2\/\\tau = \\epsilon\\|{\\mb X}\\|_2.\n\\end{align*}\n\\end{enumerate}\nAs the choice of $\\epsilon>0$ was arbitrary, this completes the proof of claim 2 in the theorem in the case when $\\mb0$ is not a solution to \\eqref{eqn:ced3}.\n\n\\subsubsection*{Case 2---no nonzero optimal solution to \\eqref{eqn:ced3}}\n\nIn the case when there is no nonzero optimal solution to \\eqref{eqn:ced3}, $\\mb 0$ is optimal and it is the only optimal point. Our analysis will be similar to the previous approach, with the key difference being in how we lower bound the quantity $\\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2$ where $\\bb^*$ is optimal to \\eqref{eqn:ced2}. Again, we have several lemmata:\n\n\\begin{enumerate}\n\\item As before, if $\\bb^*$ is optimal to \\eqref{eqn:ced2}, then $\\|\\bb^*\\|_1\\leq \\|\\mb y\\|_2\/(\\tau-\\lambda)$.\n\n\\item We now lower bound the quantity $\\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2$, where $\\bb^*$ is optimal to \\eqref{eqn:ced2} and $\\sigma\\in[0,1]$. As such, consider the function\n$$f(\\sigma):= \\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2 + \\sigma\\tau\\|\\bb^*\\|_1.$$\nBecause $f$ is convex in $\\sigma$ and the unique optimal solution to \\eqref{eqn:ced3} is $\\mb0$, we have that\n$$f(\\sigma)\\geq f(0) + \\sigma f'(0)\\;\\;\\;\\forall\\sigma\\in[0,1]\\text{\\quad and \\quad} f'(0)\\geq0$$\n(It is not difficult to argue that $f$ is differentiable at $0$.) An elementary computation shows that $f'(0) = \\tau\\|\\bb^*\\|_1 -\\langle\\mb y,{\\mb X}\\bb^*\\rangle\/\\|\\mb y\\|_2$. Therefore, we have that\n$$\\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2 + \\sigma\\tau\\|\\bb^*\\|_1 \\geq \\|\\mb y\\|_2 + \\sigma \\left(\\tau\\|\\bb^*\\|_1 - \\langle \\mb y,{\\mb X}\\bb^*\\rangle\/\\|\\mb y\\|_2\\right),$$\nimplying that\n$$\\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2 \\geq \\|\\mb y\\|_2 - \\sigma \\langle \\mb y,{\\mb X}\\bb^*\\rangle\/\\|\\mb y\\|_2 \\geq \\|\\mb y\\|_2 - \\sigma \\tau\\|\\bb^*\\|_1\\geq \\|\\mb y\\|_2 - \\sigma\\tau\\|\\mb y\\|_2\/(\\tau-\\lambda),$$\nwith the final step following by an application of the previous lemma.\n\n\\end{enumerate}\n\n\n\nWe are now ready to proceed. Let $\\epsilon>0$; we assume without loss of generality that $\\epsilon<2\\|\\mb y\\|_2\/\\tau$. Let\n$$\\overline{\\lambda}:= \\min\\left\\{\\frac{\\epsilon\\tau^2}{4\\|\\mb y\\|_2 - \\epsilon\\tau},\\frac{\\tau}{2}\\right\\}.$$\nFix $\\lambda\\in(0,\\overline{\\lambda})$ and let $\\bb^*$ be any optimal solution to \\eqref{eqn:ced2}.   Define \n$$\\sigma:= \\left(1-\\frac{\\epsilon\\tau}{2\\|\\mb y\\|_2} \\right) \\text{\\quad and \\quad}\\widehat{\\bb} := \\sigma\\bb^*.$$\nWe claim that $\\widehat{\\bb}$ satisfies the desired requirements:\n\n\\begin{enumerate}\n\\item The proof of the claim that $\\|\\bb^*-\\widehat{\\bb}\\|_2 \\leq \\epsilon$ is exactly as before.\n\n\\item We now show that $\\widehat{\\bb}$ is feasible to \\eqref{eqn:ced1}, which requires a different proof. Again this requires us to argue that\n$\\lambda \\sum_{i=1}^k |\\widehat{\\beta}_{(i)}| \\leq \\|\\mb y-{\\mb X}\\widehat{\\bb}\\|_2$. Yet,\n\\begin{align*}\n\\lambda\\sum_{i=1}^k |\\widehat{\\beta}_{(i)}| &\\leq \\lambda \\|\\widehat{\\bb}\\|_1 = \\lambda \\sigma\\|\\bb^*\\|_1\\leq  \\lambda\\sigma\\|\\mb y\\|_2\/(\\tau-\\lambda)\\leq \\|\\mb y\\|_2 - \\sigma\\tau\\|\\mb y\\|_2\/(\\tau-\\lambda)\\\\\n&\\leq \\|\\mb y-\\sigma{\\mb X}\\bb^*\\|_2 = \\|\\mb y-{\\mb X}\\widehat{\\bb}\\|_2,\n\\end{align*}\nas desired. The only non-obvious step is the inequality $\\lambda \\sigma\\|\\mb y\\|_2\/(\\tau-\\lambda) \\leq \\|\\mb y\\|_2 - \\sigma\\tau\\|\\mb y\\|_2\/(\\tau-\\lambda)$, which follows from algebraic manipulations using the definitions of $\\sigma$ and $\\overline{\\lambda}$.\n\n\\item Finally, the proof that $\\widehat{\\bb}$ is $\\left(\\epsilon\\|{\\mb X}\\|_2\\right)$-optimal to \\eqref{eqn:ced1} follows in the same way as before.\n\\end{enumerate}\n\nTherefore, we conclude that in the case when $\\mb 0$ is the unique optimal solution to \\eqref{eqn:ced3}, then again we have that the claim 2 of the theorem holds.\n\n\n\nFinally, we show that claim 1 holds: any solution $\\bb^*$ to \\eqref{eqn:ced1} is $\\epsilon$-optimal to \\eqref{eqn:ced2}. This follows by letting $\\overline{{\\bs\\beta}}$ be any optimal solution to \\eqref{eqn:ced2}. By applying the entire argument above, we know that the objective value of some $\\widehat{\\bb}$, feasible to \\eqref{eqn:ced1} and close to $\\overline{{\\bs\\beta}}$, is within $\\epsilon$ of the optimal objective value of \\eqref{eqn:ced1}, i.e., the objective value of $\\bb^*$, and within $\\epsilon$ of the objective value of \\eqref{eqn:ced2}, i.e., the objective value of $\\overline{{\\bs\\beta}}$. This completes the proof.\n\\end{proof}\n\n\nIn short, the key complication is that the quantity $\\|\\mb y-{\\mb X}{\\bs\\beta}^*\\|_2$ does not need to be uniformly bounded away from zero for solutions ${\\bs\\beta}^*$ to problem \\eqref{eqn:ced2}. This is part of the complication of working with the homogeneous form of the trimmed Lasso problem. For a concrete example, if one considers the homogeneous Lasso problem with $p=n=1$, $\\mb y=(1)$, and ${\\mb X}=(1)$, then the homogeneous Lasso problem $\\min_{\\bs\\beta} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2+\\eta\\|{\\bs\\beta}\\|_1$ is\n$$\\min_{\\beta} |1-\\beta| + \\eta|\\beta|.$$\nFor $\\eta\\in[0,1]$, $\\beta^*=1$ is an optimal solution to this problem with corresponding error $\\|\\mb y-{\\mb X}{\\bs\\beta}^*\\|=0$. If we make an assumption about the behavior of $\\|\\mb y-{\\mb X}{\\bs\\beta}^*\\|$, then we do not need the setup as shown above. \n\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:robeivslope}]\nThe proof is entirely analogous to that of Theorems \\ref{thm:robeivInterp} and \\ref{thm:corprecise} and is therefore omitted.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of validity of Example \\ref{eg:cl}]\n\nLet us consider the problem instance where $p=n=2$ with\n$$\\mb y = \\begin{pmatrix}1\\\\1\\end{pmatrix} \\text{ \\quad and \\quad } \\mb X = \\begin{pmatrix} 1 & -1\\\\-1&2\\end{pmatrix}.$$\nLet $\\lambda =1\/2$ and $ \\ell = 1$, and consider the problem\n\\begin{equation}\\label{eqn:as1}\n\\min_{{\\bs\\beta}} \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2 + |\\beta_{(2)}| = \\min_{\\beta_1,\\beta_2} (1-\\beta_1+\\beta_2)^2 + (1+\\beta_1-2\\beta_2)^2 + |\\beta_{(2)}|.\n\\end{equation}\nWe have omitted the factor of $1\/2$ as shown in the actual example in the main text in order to avoid unnecessary complications.\n\nSolving problem \\eqref{eqn:as1} and its related counterparts (for $\\ell\\in\\{0,2\\}$) can rely on convex analysis because we can simply enumerate all possible scenarios. In particular, the solution to \\eqref{eqn:as1} is ${\\bs\\beta}^*=(3\/2,1)$ based on an analysis of two related problems:\n\\begin{align*}\n\\min_{\\beta_1,\\beta_2} (1-\\beta_1+\\beta_2)^2 + (1+\\beta_1-2\\beta_2)^2 + |\\beta_1|.\\\\\n\\min_{\\beta_1,\\beta_2} (1-\\beta_1+\\beta_2)^2 + (1+\\beta_1-2\\beta_2)^2 + |\\beta_2|.\n\\end{align*}\n(We should be careful to impose the additional constraints $|\\beta_1|\\leq |\\beta_2|$ and $|\\beta_1|\\geq|\\beta_2|$, respectively, although a simple argument shows that these constraints are not required in this example.) A standard convex analysis using the Lasso (e.g. by directly using subdifferentials) shows that the problems have respective solutions $(1\/2,1\/2)$ and $(3\/2,1)$, with the latter having the better objective value in \\eqref{eqn:as1}. As such, ${\\bs\\beta}^*$ is indeed optimal. The solution in the cases of $\\ell\\in\\{0,2\\}$ follows a similarly standard analysis.\n\nIt is perhaps more interesting to study the general case where $\\mu,\\gamma\\geq0$. In particular, we will show that ${\\bs\\beta}^*=(3\/2,1)$ is not an optimal solution to the clipped Lasso problem\n\\begin{equation}\\label{eqn:as2}\n\\min_{\\beta_1,\\beta_2} (1-\\beta_1+\\beta_2)^2 + (1+\\beta_1-2\\beta_2)^2 + \\mu\\min\\{\\gamma|\\beta_1|,1\\}+\\mu\\min\\{\\gamma|\\beta_2|,1\\}\n\\end{equation}\nfor any choices of $\\mu$ and $\\gamma$. While in general such a problem may be difficult to fully analyze, we can again rely on localized analysis using convex analysis. To proceed, let\n$$f(\\beta_1,\\beta_2) = (1-\\beta_1+\\beta_2)^2 + (1+\\beta_1-2\\beta_2)^2 + \\mu\\min\\{\\gamma|\\beta_1|,1\\}+\\mu\\min\\{\\gamma|\\beta_2|,1\\},$$\nwith the parameters $\\mu$ and $\\gamma$ implicit. We consider the following exhaustive cases:\n\n\\begin{enumerate}\n\\item $\\boxed{\\gamma>1}$ : In this case, $f$ is convex and differentiable in a neighborhood of ${\\bs\\beta}^*$. Its gradient at ${\\bs\\beta}^*$ is $\\nabla f({\\bs\\beta}^*)=(0,-1)$, and therefore ${\\bs\\beta}^*$ is neither locally optimal nor globally optimal to problem \\eqref{eqn:as2}.\n\n\\item $\\boxed{\\gamma<2\/3}$ : In this case, $f$ is again convex and differentiable in a neighborhood of ${\\bs\\beta}^*$. Its gradient at ${\\bs\\beta}^*$ is $\\nabla f({\\bs\\beta}^*)=(\\mu\\gamma,\\mu\\gamma-1)$. Again, this cannot equal $(0,0)$ and therefore ${\\bs\\beta}^*$ is neither locally nor globally optimal to problem \\eqref{eqn:as2}.\n\n\\item $\\boxed{2\/3<\\gamma<1}$ : In this case, $f$ is again convex and differentiable in a neighborhood of ${\\bs\\beta}^*$. Its gradient at ${\\bs\\beta}^*$ is $\\nabla f({\\bs\\beta}^*)=(0,\\mu\\gamma-1)$. As a necessary condition for local optimality, we must have that $\\mu\\gamma=1$, implying that $\\mu>1$. Further, if ${\\bs\\beta}^*$ is optimal to \\eqref{eqn:as2}, then $f({\\bs\\beta}^*)\\leq f(0,0)$. Yet,\n\\begin{align*}\nf({\\bs\\beta}^*) &= 1\/2 + \\mu + \\mu\\gamma = 3\/2 + \\mu\\\\\nf(0,0) & = 2 ,\n\\end{align*}\nimplying that $\\mu\\leq 1\/2 $, in contradiction of $\\mu>1$. Hence, ${\\bs\\beta}^*$ cannot be optimal to \\eqref{eqn:as2}.\n\n\n\\item $\\boxed{\\gamma=2\/3}$ : In this case, we make two comparisons, using the points ${\\bs\\beta}^*$, $(0,0)$, and $(3,2)$:\n\\begin{align*}\nf({\\bs\\beta}^*) &= 1\/2 + \\mu + 2\\mu\/3 = 1\/2 + 5\\mu\/3\\\\\nf(0,0) & = 2\\\\\nf(3,2)&= 2\\mu.\n\\end{align*}\nAssuming optimality of ${\\bs\\beta}^*$, we have that $f({\\bs\\beta}^*)\\leq f(0,0)$, i.e., $\\mu\\leq 9\/10$; similarly, $f({\\bs\\beta}^*)\\leq f(3,2)$, i.e., $\\mu\\geq3\/2$. Clearly both cannot hold, and so therefore ${\\bs\\beta}^*$ cannot be optimal.\n\n\\item $\\boxed{\\gamma=1}$ : Finally, we see that $f({\\bs\\beta}^*)\\leq f(3,2)$ would imply that $1\/2+2\\mu\\leq 2\\mu$, which is impossible; hence, ${\\bs\\beta}^*$ is not optimal to \\eqref{eqn:as2}. (This argument can clearly also be used in the case when $\\gamma>1$, although it is instructive to see the argument given above in that case.)\n\n\\end{enumerate}\n\n\\noindent In any case, we have that ${\\bs\\beta}^*$ cannot be a solution to the clipped Lasso problem \\eqref{eqn:as2}. This completes the proof of validity of Example \\ref{eg:cl}.\n\\end{proof}\n\n\n\n\n\\section{Supplementary details for Algorithms}\\label{app:algsupp}\n\nThis appendix contains further details on algorithms as discussed in Section \\ref{sec:algs}. The presentation here is primarily self-contained. Note that the alternating minimization scheme based on difference-of-convex optimization can be found in \\cite{gotoh1}.\n\n\\subsection{Alternating minimization scheme}\n \nLet us set the following notation\n$$\\begin{array}{lll}\nf({\\bs\\beta}) &=& \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2\/2 + \\lambda \\tk{{\\bs\\beta}}+ \\eta \\|{\\bs\\beta}\\|_1,\\\\\nf_1({\\bs\\beta}) &=& \\|\\mb y-{\\mb X}{\\bs\\beta}\\|_2^2\/2 + (\\eta+\\lambda) \\|{\\bs\\beta}\\|_1,\\\\\nf_2({\\bs\\beta}) &=& \\lambda \\sum_{i=1}^k |\\beta_{(i)}|.\n\\end{array}\n$$\n\n\\begin{definition}\nFor any function $F:\\mathbb{R}^p\\to\\mathbb{R}$ and $\\epsilon\\geq0$, we define the $\\epsilon$-subdifferential of $F$ at ${\\bs\\beta}_0\\in\\mathbb{R}^p$ to be the set $\\partial_\\epsilon F({\\bs\\beta}_0)$ defined as\n$$\\left\\{ \\bs\\gamma\\in\\mathbb{R}^p\\; : \\; F({\\bs\\beta}) - F({\\bs\\beta}_0) \\geq \\langle\\bs\\gamma,{\\bs\\beta}-{\\bs\\beta}_0\\rangle - \\epsilon \\;\\forall \\;{\\bs\\beta}\\in\\mathbb{R}^p\\right\\}.$$\nIn particular, when $\\epsilon=0$, we refer to $\\partial_0 F({\\bs\\beta}_0)$ as the subdifferential of $F$ at ${\\bs\\beta}_0$, and we will denote this as $\\partial F({\\bs\\beta}_0)$.\n\\end{definition}\n\n\n\nUsing this definition, we have the following result precisely characterizing local and global optima of \\eqref{eqn:alg}.\n\n\\begin{theorem}\\label{thm:optCharac}\n\\begin{enumerate}[(a)]\n\\item A point ${\\bs\\beta}^*$ is a local minimum of $f$ if and only if $\\partial f_2({\\bs\\beta}^*) \\subseteq \\partial f_1({\\bs\\beta}^*)$.\n\n\\item A point ${\\bs\\beta}^*$ is a global minimum of $f$ if and only if $\\partial_\\epsilon f_2({\\bs\\beta}^*) \\subseteq \\partial_\\epsilon f_1({\\bs\\beta}^*)$ for all $\\epsilon\\geq0$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nThis is a direct application of results in \\cite[Thm. 1]{taoan97}. Part (b) is immediate. The forward implication of part (a) is immediate as well; the converse implication follows by observing that $f_2$ is a \\emph{polyhedral} convex function \\cite[Thm. 1(ii)]{dcSummary} (see definition therein).\n\\end{proof}\n\n\nLet us note that $\\partial f_1$ and $\\partial f_2$ are both easily computable, and hence,  local optimality can be verified given some candidate ${\\bs\\beta}^*$ per Theorem \\ref{thm:optCharac}.\\footnote{For the specific functions of interest, verifying local optimality of a candidate ${\\bs\\beta}^*$ can be performed in $O(p\\min\\{n,p\\}+p\\log p)$ operations; the first component relates to the computation of ${\\mb X}'{\\mb X}{\\bs\\beta}^*$, while the second captures the sorting of the entries of ${\\bs\\beta}^*$. See Appendix \\ref{app:alg1supp} for details.\n}\nWe now discuss the associated alternating minimization scheme (or equivalently, as a sequential linearization scheme), shown in Algorithm \\ref{alg:1} for finding local optima of \\eqref{eqn:alg} by making use of Theorem \\ref{thm:optCharac}. Through what follows, we make use of the standard notion of a conjugate function, defined as follows:\n\n\\begin{definition}\nFor any function $F:\\mathbb{R}^p\\to\\mathbb{R}$, we define its conjugate function $F^*:\\mathbb{R}^p\\to\\mathbb{R}$ to be the function\n$$F^*(\\bs\\gamma) = \\sup_{{\\bs\\beta}} \\langle\\bs\\gamma,{\\bs\\beta} - F({\\bs\\beta})\\rangle.$$\n\\end{definition}\n\n\nWe will make the following minor technical assumption: in step 2) of Algorithm \\ref{alg:1}, we assume without loss of generality that the $\\bs\\gamma^\\ell$ so computed satisfies the additional criteria:\n\\begin{enumerate}\n\\item it is an extreme point of the relevant feasible region, \n\\item and that if $\\partial f_2({\\bs\\beta}^\\ell) \\not\\subseteq \\partial f_1({\\bs\\beta}^\\ell)$, then $\\bs\\gamma^\\ell$ is chosen such that $\\bs\\gamma^\\ell \\in \\partial f_2({\\bs\\beta}^\\ell) \\setminus\\partial f_1({\\bs\\beta}^\\ell)$.\n\\end{enumerate}\nSolving \\eqref{eqn:wrtbg} with these additional assumptions can nearly be solved in closed form by simply sorting the entries of $|{\\bs\\beta}|$, i.e., by finding $|\\beta_{(1)}|,\\ldots,|\\beta_{(p)}|$\nWe must take some care to ensure that the second without loss of generality condition on $\\bs\\gamma$ is satisfied. This is straightforward but tedious; the details are shown in Appendix \\ref{app:alg1supp}.\n\n\nUsing this modification, the convergence properties of Algorithm \\ref{alg:1} can be proven as follows:\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:altConvProp}]\nThis is an application of \\cite[Thms. 3-5]{taoan97}. The only modification is in requiring that $\\bs\\gamma^\\ell$ is chosen so that $\\bs\\gamma^\\ell \\in \\partial f_2({\\bs\\beta}^*) \\setminus\\partial f_1({\\bs\\beta}^*)$ if ${\\bs\\beta}^\\ell$ is not a local minimum of $f$---see \\cite[\\textsection3.3]{taoan97} for a motivation and justification for such a modification. Finally, the correspondence between $\\bs\\gamma^\\ell\\in \\partial f_2({\\bs\\beta}^\\ell)$ and \\eqref{eqn:wrtbg}, and between ${\\bs\\beta}^{\\ell+1}\\in \\partial f_1^*(\\bs\\gamma^\\ell)$ and \\eqref{eqn:wrtbb}, is clear from an elementary argument applied to subdifferentials of variational formulations of functions.\n\\end{proof}\n\n\\subsection{Algorithm \\ref{alg:1}, Step 2}\\label{app:alg1supp}\n\nHere we present the details of solving \\eqref{eqn:wrtbg} in Algorithm \\ref{alg:1} in a way that ensures that the associated without loss of generality claims hold. In doing so, we also implicitly study how to verify the conditions for local optimality (\\emph{c.f.} Theorem \\ref{thm:optCharac}). \nThroughout, we use the $\\operatorname{sgn}$ function defined as \n$$\\operatorname{sgn}(x) = \\left\\{\\begin{array}{rl}\n1,& x>0\\\\\n-1,&x<0\\\\\n0,&x=0.\n\\end{array}\\right.$$\n\nFor fixed ${\\bs\\beta}$, the problem of interest is\n\\begin{equation*\n\\begin{array}{ll}\n\\displaystyle\\max_{\\bs\\gamma} & \\langle{\\bs\\beta},\\bs\\gamma\\rangle\\\\\n\\operatorname{s.t.}& \\displaystyle\\sum_i |\\gamma_i| \\leq \\lambda k\\\\\n& \\displaystyle|\\gamma_i|\\leq \\lambda\\;\\forall i.\n\\end{array}\n\\end{equation*}\nWe wish to find a maximizer $\\bs\\gamma$ for which the following hold:\n\\begin{enumerate}\n\\item $\\bs\\gamma$ is an extreme point of the relevant feasible region, \n\\item and that if $\\partial f_2({\\bs\\beta}) \\not\\subseteq \\partial f_1({\\bs\\beta})$, then $\\bs\\gamma$ is such that $\\bs\\gamma \\in \\partial f_2({\\bs\\beta}) \\setminus\\partial f_1({\\bs\\beta})$.\n\\end{enumerate}\nAs the problem on its own can be solved by sorting the entries of ${\\bs\\beta}$, the crux of the problem is ensuring that 2) holds.\n\nGiven the highly structured nature of $f_1$ and $f_2$ in our setup, it is simple, albeit tedious, to ensure that such a condition is satisfied. Let $I = \\{i: |\\beta_i|=|\\beta_{(k)}|\\}$. If $|I|=1$, the optimal solution is unique, and there is nothing to show. Therefore, we will assume that $|I|\\geq2$. We will construct an optimal solution $\\bs\\gamma$ which satisfies the desired conditions. First observe that we necessarily must have that 1) $\\gamma_i = \\lambda\\operatorname{sgn}(\\beta_i)$ if $|\\beta_i|> |\\beta_{(k)}|$ and 2) $\\gamma_i=0$ if $|\\beta_i|<|\\beta_{(k)}|$. We now proceed to define the rest of the entries of $\\bs\\gamma$. We consider two cases:\n\n\\begin{enumerate}\n\\item First consider the case when $|\\beta_{(k)}|>0$. We claim that $\\partial f_2({\\bs\\beta}) \\not\\subseteq \\partial f_1({\\bs\\beta})$. To do so, we will inspect the $i$th entries of $\\partial f_1({\\bs\\beta})$ for $i\\in I$; as such, let $P_i^j =\\{\\delta_i: \\bs\\delta\\in \\partial f_j({\\bs\\beta})\\}$ for $j\\in\\{1,2\\}$ and $i\\in I$ (a projection). For each $i\\in I$, we have using basic convex analysis that $P_i^1$ is a singelton: $P_i^1 = \\{\\langle{\\mb X}_i,{\\mb X}{\\bs\\beta}-\\mb y\\rangle + (\\eta+\\lambda)\\operatorname{sgn}(\\beta_i)\\}$, where ${\\mb X}_i$ is the $i$th column of ${\\mb X}$. In contrast, because $|I|\\geq2$, the set $P_i^2$ is an interval with strictly positive length for each $i\\in I$ (it is either $[-\\lambda,0]$ or $[0,\\lambda]$, depending on whether $\\beta_i<0$ or $\\beta_i>0$, respectively). Therefore, $\\partial f_2({\\bs\\beta}) \\not\\subseteq \\partial f_1({\\bs\\beta})$, as claimed.\n\nFix an arbitrary $j\\in I$. Per the above argument, we must have that $\\langle{\\mb X}_j,{\\mb X}{\\bs\\beta}-\\mb y\\rangle+ (\\eta+\\lambda)\\operatorname{sgn}(\\beta_j)\\neq 0$ or $\\langle{\\mb X}_j,{\\mb X}{\\bs\\beta}-\\mb y\\rangle + (\\eta+\\lambda)\\operatorname{sgn}(\\beta_j)\\neq\\lambda\\operatorname{sgn}(\\beta_j)$. In the former case, set $\\gamma_i=0$, while in the latter case we define $\\gamma_i=\\lambda\\operatorname{sgn}(\\beta_i)$ (if both are true, either choice suffices). It is clear that it is possible to fill in the remaining entries of $\\gamma_i$ for $i\\in I\\setminus\\{j\\}$ in a straightforward manner so that $\\bs\\gamma\\in \\partial f_2({\\bs\\beta})$.  Further, by construction, $\\bs\\gamma\\notin \\partial f_1({\\bs\\beta})$, as desired.\n\n\n\\item Now consider the case when $|\\beta_{(k)}|=0$. Using the preceding argument, we see that $P_i^1$ is the interval $[\\langle{\\mb X}_i,{\\mb X}{\\bs\\beta}-\\mb y\\rangle-(\\eta+\\lambda),\\langle{\\mb X}_i,{\\mb X}{\\bs\\beta}-\\mb y\\rangle+\\eta+\\lambda] $ for $i\\in I$. In contrast, $P_i^2$ is the interval $[-\\lambda,\\lambda]$ for $i\\in I$. If for all $i\\in I$ one has that $P_i^2\\subseteq P_i^1$, then the choice of $\\gamma_i$ for $i\\in I$ is obvious: any optimal extreme point $\\bs\\gamma$ of the problem will suffice. (Note here that it may or may not be that $\\partial f_2({\\bs\\beta}) \\subseteq \\partial f_1({\\bs\\beta})$. This entirely depends on $\\beta_i$ for $i\\notin I$.)\n\nTherefore, we may assume that there exists some $j\\in I$ so that $P_j^2\\not\\subseteq P_j^1$. (It follows immediately that $\\partial f_2({\\bs\\beta}) \\not\\subseteq \\partial f_1({\\bs\\beta})$.) We must have that $\\langle{\\mb X}_j,{\\mb X}{\\bs\\beta}-\\mb y\\rangle -(\\eta+\\lambda)>-\\lambda$ or $\\langle{\\mb X}_j,{\\mb X}{\\bs\\beta}-\\mb y\\rangle + (\\eta+\\lambda)<\\lambda$. In the former case, set $\\gamma_i=-\\lambda$, while in the latter case we define $\\gamma_i=\\lambda$ (if both are true, either choice suffices). It is clear that it is possible to fill in the remaining entries of $\\gamma_i$ for $i\\in I\\setminus\\{j\\}$ in a straightforward manner so that $\\bs\\gamma\\in \\partial f_2({\\bs\\beta})$. By construction, $\\bs\\gamma\\notin \\partial f_1({\\bs\\beta})$, as desired.\n\n\n\\end{enumerate}\n\n\nIn either case, we have that one can choose $\\bs\\gamma\\in\\partial f_2({\\bs\\beta})$ so that 1) $\\bs\\gamma$ is an extreme point of the feasible region $\\{\\bs\\gamma:\\sum_i|\\gamma_i|\\leq\\lambda k,\\; |\\gamma_i|\\leq \\lambda\\;\\forall i\\}$ and that 2) $\\bs\\gamma \\in \\partial f_2({\\bs\\beta}) \\setminus\\partial f_1({\\bs\\beta})$ whenever $\\partial f_2({\\bs\\beta}) \\not\\subseteq \\partial f_1({\\bs\\beta})$. This concludes the analysis; thus, we have shown the validity (and computational feasibility) of the without loss of generality claim present in Algorithm \\ref{alg:1}. Indeed, per our analysis, Step 2 in Algorithm \\ref{alg:1} can be solved in $O(p\\min\\{n,p\\}+p\\log p)$ operations (sorting of ${\\bs\\beta}$ in $O(p\\log p)$ followed by $O(p)$ conditionals and gradient evaluation in $O(np)$). In reality, if we keep track of gradients in Step 3, there is no need to recompute gradients in Step 2, and therefore in practice Step 2 is of the same complexity of sorting a list of $p$ numbers. (We assume that ${\\mb X}'\\mb y$ has been computed offline and store throughout for simplicity.)\n\n\n\n\\subsection{Algorithm \\ref{alg:admm}, Step 3}\\label{app:admmsupp}\n\nHere we show how to solve Step 3 in Algorithm \\ref{alg:admm}, namely, solving the orthogonal design trimmed Lasso problem\n\\begin{equation}\\label{eqn:pf6}\n\\min_{\\bs\\gamma} \\lambda \\tk{\\bs\\gamma} + \\frac{\\sigma}{2}\\|{\\bs\\beta}-\\bs\\gamma\\|_2^2  - \\langle \\mb q,\\bs\\gamma\\rangle,\n\\end{equation}\nwhere ${\\bs\\beta}$ and $\\mb q$ are fixed. This is solvable in closed form. Let $\\bs\\alpha={\\bs\\beta}-\\mb q\/\\sigma$. First observe that we can rewrite \\eqref{eqn:pf6} as\n\\begin{align*}\n\\eqref{eqn:pf6} &= \\min_{\\bs\\gamma} \\lambda \\tk{\\bs\\gamma} + \\sigma\\|\\bs\\gamma-\\bs\\alpha\\|_2^2\/2\\\\\n&= \\min_{\\substack{\\bs\\gamma,\\mb z:\\\\\\sum_i z_i=p-k\\\\\n\\mb z\\in\\{0,1\\}^p}}\\lambda\\langle \\mb z,|\\bs\\gamma|\\rangle + \\sigma\\|\\bs\\gamma-\\bs\\alpha\\|_2^2\/2\\\\\n&= \\min_{\\substack{\\bs\\gamma,\\mb z:\\\\\\sum_i z_i=p-k\\\\\n\\mb z\\in\\{0,1\\}^p}}\\sum_i \\left(\\lambda z_i|\\gamma| +   \\sigma(\\gamma_i-\\alpha_i)^2\/2\\right).\n\\end{align*}\nThe penultimate step follows via Lemma \\ref{lemma:miprep}. Per this final representation, the solution becomes clear. In particular, let $I$ be a set of $k$ indices of $\\bs\\alpha$ corresponding to $\\alpha_{(1)}$, $\\alpha_{(2)}$, \\ldots, $\\alpha_{(k)}$. (If $|\\alpha_{(k)}| = |\\alpha_{(k+1)}|$, we break ties arbitrarily.) Then a solution $\\bs\\gamma^*$ to \\eqref{eqn:pf6} is\n$$\\gamma_i^* = \\left\\{\\begin{array}{rl}\n\\alpha_i, & i\\in I\\\\\n\\operatorname{soft}_{\\lambda\/\\sigma}(\\alpha_i),&i\\notin I,\n\\end{array}\\right.$$\nwhere $\\operatorname{soft}_{\\lambda\/\\sigma}(\\alpha_i) = \\operatorname{sgn}(\\alpha_i) \\left|\\alpha_i-\\lambda\/\\sigma\\right|$.\n\n\n\n\\subsection{Computational details}\\label{app:compdetail}\n\nFor completeness and reproducibility, we also include all computational details. For Figure \\ref{fig:coeffpath}, the following parameters were used to generate the test instance: $n = 100$, $p = 20$, $\\text{SNR} = 10$, \\texttt{julia} seed = 1, $\\eta=0.01$, $k=2$. The example was generated from the following true model:\n\\begin{enumerate}\n\\item ${\\bs\\beta}_\\text{true}$ is a vector with ten entries equal to 1 and all others equal to zero. (So $\\|{\\bs\\beta}_\\text{true} \\|_0=10$.)\n\\item covariance matrix $\\bs\\Sigma$ is generated with $\\Sigma_{ij} = .8^{|i-j|}$.\n\\item ${\\mb X}\\sim N(\\mb 0,\\bs\\Sigma)$.\n\\item $\\epsilon_i\\stackrel{\\text{i.i.d.}}{\\sim} N(0, {\\bs\\beta}_0'\\bs\\Sigma{\\bs\\beta}_0\/\\text{SNR})$\n\\item $\\mb y$ is then defined as ${\\mb X}{\\bs\\beta}_0+\\bs\\epsilon$\n\n\\end{enumerate}\n\nThe 100 examples generated for Figure \\ref{fig:optgap} were using the following parameters: $n = 100$, $p = 20$, $\\text{SNR} = 10$, \\texttt{julia} seed $\\in\\{ 1,\\ldots,100\\}$, $\\eta=0.01$, $k=2$, $\\text{bigM} = 20$. MIO using Gurobi solver. Max iterations: alternating minimization---1000; ADMM (inner)---2000; ADMM (outer)---10000. ADMM parameters: $\\sigma=1$, $\\tau=0.9$. The examples themselves had the same structure as the previous example. The optimal gaps shown are relative to the objective in \\eqref{eqn:alg}. The averages are computed as geometric means (relative to optimal 100\\%) across the 100 instances, and then displayed relative to the optimal 100\\%.\n\n\n\\end{appendices}\n\n\\section*{Acknowledgments}\n\n\nCopenhaver was partially supported by the Department of Defense, Office of Naval Research, through the National Defense Science and Engineering Graduate (NDSEG) Fellowship. Mazumder was partially supported by ONR Grant N000141512342.\n\n\n\n\\bibliographystyle{IEEEtranS}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Reanalysis of CADIS data}\n\nIn two previous papers (Phleps et al. 2000, 2005) we have studied the\ndistribution of stars in the Milky Way using deep star counts based on Calar \nAlto Deep Imaging Survey data. For this purpose we analyzed five fields which \ncontained in total 1627 faint stars (15.5 $ < R < $ 23). We do not repeat here \nany details of our analysis, but give again in Table 1 the coordinates \nof the fields. At the time of writing the papers we were mainly interested in \ndetermining descriptive parameters of the density distributions\nof the various components of the Milky Way \nsuch as the vertical scale heights of the thin and thick disks, respectively, \ntheir relative normalization and the parameters of the halo density law. We\nrealized, however, already then that, while the 1h, 16h, and 32h fields gave \nvery consistent results, the 9h and to a lesser degree the 13h field lead \nto discrepant results in the sense that the vertical density distribution \nof the stars in these fields had a few kpc above the midplane a shallower slope \nthan in the three other fields. With the advent of the Data Release 5 of the\nSloan Digital Sky Survey and its subsequent analyses by Juri\\'c et al.~(2006) \nand Belokurov et al.~(2006a, b) overdensely populated parts of the density \ndistribution of the stars in the Milky Way have been revealed in such rich \ndetail that we can interpret now in retrospect also the results of the 13h\nfield in a consistent way, whereas the nature of the overdensity in 9h \nremains at present less clear. By pure chance two out of the five \nline--of--sights of the CADIS fields have crossed two separate of such \noverdense regions! Of course such few line--of--sights did not allow the \nidentification of the excess densities as distinct isolated overdensely \npopulated regions in the Milky Way. \n\n\\begin{table}\n\\caption[]{Pointings of the CADIS fields}\n         \\label{poi}\n\\[\n\\begin{tabular}{rrrrr}\n\\hline\n \\noalign{\\smallskip}\n \\hline\n \\noalign{\\smallskip}\n  field & R.A. & Dec & l & b \\\\ \n       & \\multicolumn{4}{c}{deg}\\\\\n \\noalign{\\smallskip}\n \\hline\n \\noalign{\\smallskip}\n 1\\,h &\t 27 &\t  2 &   150 &\t-59 \\\\\n 9\\,h & 138 &\t 46 &\t175 &\t 45 \\\\\n13\\,h &\t207 &\t  6 &\t335 &\t 60 \\\\\n16\\,h & 246 &\t 56 &\t 85 &\t 45 \\\\\n23\\,h & 349 &\t 12 &\t 90 &\t-43 \\\\\n \\noalign{\\smallskip}\n \\hline\n \\end{tabular}\n \\]\n\\end{table}\n\nSuch features in the density distribution of stars have attracted recently\ngreat interest in the literature, because they almost certainly represent debris\nof satellite galaxies which have fallen into the Milky Way and were then\ndisrupted. Particularly striking are the large filaments like the Sgr stream or \nthe newly discovered `Orphan' stream (Ibata et al.~1997, Majewski et al.~2003, \nBelokurov et al.~2006a, b, Grillmair 2006) which are interpreted as tidal \ntails of dwarf galaxies presently in the process of being cannibalized. \nObviously such accretion events played an important role in the formation \nhistory of the Milky Way.\n\n\nIn this {\\em note} we present a reanalysis of the CADIS data, because in our \nview these data still contribute valuable information on the overdensities. \nIn particular we can trace the overdensities right into the disk of the Milky\nWay, which is not possible with the SDSS data. Moreover, we can provide \nestimates of the masses of the overdensities. \n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8.7cm]{5698f1.ps}\n  \\caption[]{Number density distribution of stars [**\/kpc$^3$]  \n  perpendicular to the  Galactic plane derived from star counts in CADIS \n  fields. The five fields are coded as: 1h (squares), 9h (asterisks),\n  13h (circles), 16h (triangles), 23h (pentagons). Only data above $z$ = 2 kpc \n  are shown (colour coded in the electronic version). The solid line is the \n  vertical density profile at the position of the Sun of a smooth Galaxy model \n  fitted to the data of the 1h, 16h and 23h fields.}\n  \\label{cadidat}\n  \\end{figure}\n\n The CADIS star counts were carried out along the lines--of--sight\nwhose coordinates are given in Table 1. In Fig.~1 we show the inferred number \ndensities of stars not as function of heliocentric distance, but as function of\nthe distance from the midplane, i.e.~as vertical density profiles perpendicular \nto the Galactic plane. In this way the five fields are projected onto each \nother which allows a direct comparison of their density distributions. The \ndistributions of the 1h and\n23 h fields have been flipped up. As can be seen from Fig.~1 the distribution \nof stars is traced from the outer halo into the disk of the Milky Way. Data \nbelow $z$ = 2 kpc are not shown. The density distributions derived from the \nstar counts in the 1h, 16h, and 23h fields, respectively, are within statistical\nuncertainties consistent with each other. The inclinations of the\nlines--of--sight of the CADIS fields relative to the vertical axis hardly affect\nthe shape of the density profiles at heights of more then 5 kpc above the\nmidplane. At lower heights part of the scatter among the data shown in Fig.~1\ncan be ascribed to the varying viewing directions of the CADIS fields. For the \npresent purposes the 1h, 16h and 23h fields define a vertical reference profile.\nHaving realized this we have repeated the fit of the smooth Galaxy model of\nPhleps et al.~(2005, Eqns.~3 and 5) using only these three fields. As in the\nprevious paper we adopt a vertical scale height of the thin disk of 283 pc. The\nfit to the density distribution in the three reference fields leads to a \nslightly reduced vertical scale height of the thick disk of 900 pc, but to the \nsame normalization of the local density of the thick disk at 4 percent of the \ntotal density at the midplane. In the case of a spherical halo model we find an \nindex of the halo density law of $\\alpha = 3.25\\,\\pm\\,0.10$ and in the case of \na halo flattened as $(c\/a)=0.6$ an index of $\\alpha= 2.69\\,\\pm\\,0.09$. The \ndensity profile perpendicular to the Galactic plane ($b=90^\\circ$) of the\nsmooth Galaxy model, which is shown as a solid line in Fig.~1, is in excellent \nagreement with the Galaxy model of Juri\\'c et al.~(2006).\n\nThe 13h field shows a statistically significant excess density relative  \nto the reference profile which can be traced from $z$ = 2 to 14 kpc with the\nmaximal deviation at about $z$ = 4 kpc. The heliocentric distances range \nfrom 2.3 to 16 kpc. The coordinates of this field point towards the fringe \nof the Virgo overdensity which is discussed in detail by Juri\\'c et al.~(2006, \ncf.~their Fig.~24). Indeed, Fig.~1 can be directly compared with Fig.~22 \nof Juri\\'c et al.~(2006), where they delineate the overdensity by subtracting \nfrom the density distribution of stars observed in the meridional section of \nthe Milky Way which contains the Virgo overdensity their smooth Galaxy model.\nAbove $z$ = 5 kpc, which corresponds to a galactocentric radius of $R$ = 5.5\nkpc in the 13h field, the vertical profile of the overdensity found in the \nSDSS DR5 and the CADIS data, respectively, are absolutely consistent with each \nother. However, in the CADIS data it can be traced right down into the disk of \nthe Milky Way, confirming the supposition of Juri\\'c et al.~(2006) that this might\nbe the case. \n\nMoreover, both data sets have been analyzed as Hess diagrams. In \ntheir Fig.~24 Juri\\'c et al.~(2006) show by subtracting the Hess diagram of a \ncontrol field from the Hess diagram of the Virgo field that the excess \npopulation of stars in the Virgo field is primarily found in the blue branch of\nthe Hess diagram at $(g-r) \\approx 0.4$, where halo stars are located. Precisely\nthe same is found in the Hess diagram of the 13h field shown in Fig.~2. There \nis a clear excess in the blue branch of the halo stars at $(b-r) \\approx 0.1$ \nwith respect to the smooth Galaxy model, which has been recalculated using the\nparameters given above. We conclude from this discussion that the excess \ndensity in the 13h CADIS field can be identified as part of  the Virgo\noverdensity.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8.7cm]{5698f2.ps}\n  \\caption[]{Hess diagram of the 13h field. The observed distribution of stars\n  in the $R - (b-r)$ plane is colour coded according to the number density of\n  stars per bin of size 0.1 $\\times$ 0.1 in magnitude and colour, respectively. \n  The colour table is shown at the top. A smooth model of the Galaxy, which has\n  been fitted to the reference fields, is shown as isodensity contours. The \n  equally spaced contour levels span the same range as the colour table.}\n  \\label{hess1}\n  \\end{figure}\n\nSubtracting the reference profile from the observed density profile of the 13h \nfield, both now reckoned along the line--of--sight, allows us to\ndetermine the mass of the excess population of stars. We find that 276 out of\nthe 517 stars in the 13h field seem to belong to the density excess. Phleps et\nal.~(2005) have identified in the CNS4 catalogue (Jahrei{\\ss} \\& Wielen 1997)\nthe analogues of the blue halo stars in the CADIS fields in the local volume\n(Fuchs \\& Jahrei{\\ss} 1998).  They have shown that the extrapolation of the \nouter\nhalo density law towards the Galactic midplane agrees ideally with the number\ndensity of stars with the same colours and absolute magnitudes in the local\nsample. The average mass of these stars is 0.66 ${\\mathcal{M}}_\\odot$\nand the average mass--to--light ratio is ${\\mathcal{M}}\/{\\mathcal{L}_{\\rm V}}\n= 2.7 {\\mathcal{M}}_\\odot\/{\\mathcal{L}}_{{\\rm V}\\odot}$. \nThus a number density of 10$^5$ stars per kpc$^3$ corresponds to a mass \ndensity of $6.6\\,\\cdot\\,10^{-5}$ ${\\mathcal{M}}_\\odot\\,pc^{-3}$. We estimate \nfrom the area preserving Lambert projection in Fig.~24 of Juri\\'c et al.~(2006) \nthat the Virgo overdensity subtends an area of 846 square degrees. One CADIS \nfield has a size of 121 square arcminutes. If the mass of the excess population \nof stars in the 13h field is representative for the rest of the Virgo \noverdensity the latter contains a mass of $4.6\\,\\cdot\\,10^6$ \n${\\mathcal{M}}_\\odot$. This is in our view a clear indication that the Virgo\noverdensity is the relic of a shredded dwarf galaxy. Adopting the \nmass--to--light ratio of  ${2.7\\,{\\mathcal{M}}_\\odot\/\n{\\mathcal{L}}_{{\\rm V}\\odot}}$ found above the stars in the Virgo overdensity\nwould have a total luminosity of $1.7\\,\\cdot\\,10^6$ \n${\\mathcal{L}}_{{\\rm V}\\odot}$ or an absolute magnitude of M$_{\\rm V}$ =\n -- 11 mag, which is quite typical for Local Group dwarf spheroidal galaxies. \nAlso the inferred mass--to--light ratio is typical for the stellar populations \nof dwarf spheroidals (Mateo 1998).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8.7cm]{5698f3.ps}\n  \\caption[]{The same as Fig.~2 but for the 9h field.}\n  \\label{hess2}\n  \\end{figure}\n\nThe 9h field shows at distances between $z$ = 2 and 15 kpc above the midplane\nan overdensity relative to the reference profile, which is even more \npronounced than that in the 13h field. The maximal deviation is found around \n$z$ = 9 kpc. The corresponding heliocentric distances range from 3 to 21 kpc.\nHowever, this overdensity is more difficult to associate confidently with one of\nthe overdensities seen in the SDSS data (Juri\\'c et al.~2006, Belokurov et \nal.~2006a, b). We note that most of the density excess in the Hess diagram of\nthe 9h field shown in Fig.~3 is found in the colour range\n$(b-r) \\approx 0.5 - 1$ and falls right into the gap\nbetween the two branches of the disk and halo stars, respectively. Judging from\nthe Hess diagrams presented by Juri\\'c et al.~(2006) for the Virgo field and its\ncontrol field stars of such intermediate colour have been eliminated by the\ncolour cut $(g - r) < 0.4$ from the sample of Belokurov et al.~(2006a).  \nIn Fig.~1 of Belokurov (2006b) there is also the distribution of stars with\ncolours $0.4 < (g - r) \\leq 0.6$ shown and we find the 9h field at the eastern \nfringe of the northern part of the Monoceros stream (Newberg et al.~2002) which\nseems to be in this colour range much more extended than in the bluer \n$(g - r) < 0.4$ colours (Belokurov et al.~2006a). Similarly the Monoceros \nstream is seen in the data of Juri\\'c et al.~(2006, their Fig.~9) mainly at \nslightly lower galactic latitudes than that of the 9h field. Its \nline--of--sight passes though through the \ninner fringe of the overdensity in anticenter direction in the panels of\nFig.~9 which show the density distribution of stars with colours $0.1 < \n(r - i) \\leq 0.15$ at heights of 4 and 5 kpc above the midplane. Juri\\'c et\nal.~(2006) do study the distribution of redder stars, but cannot trace it beyond\na few kpc from the Sun.  Pe\\~narrubia et al.~(2005) have modelled the Monoceros\nstream by numerical simulations as a tidal stream. Their simulations  \nshow that part of the stream might be very well seen in the direction of the 9h \nfield and at the heliocentric distances of the overdensity in the 9h field.\nThus this overdensity may be tentatively associated with the Monoceros \nstream. However, there is a further aspect of the interpretation of the\noverdensity in the 9h field. The 9h field is positioned exactly on the \ngalactocentric great circle on which the Orphan stream lies and which passes \nalso through the high velocity cloud complex A (Belokurov et al.~2006a, b,\nWakker 2001). The position is roughly in the middle between the northern tip of\nthe Orphan stream and Complex A. Belokurov et al.~(2006b) have determined \na heliocentric distance of the northern tip of the Orphan stream of 35$\\pm$10\nkpc, whereas the distance to Complex A is 10.1$\\pm$0.9 kpc. This discrepancy \ncan be resolved in a natural way if Complex A is on a different wrap around the\nGalactic center as the Orphan stream. The density profile of the 9h field\ngives the impression that there is an extra `hump' in the density excess of\nstars relative to the reference fields which is centered around z\n$\\approx$ 9 kpc or a heliocentric distance of 13 kpc. It is tempting to\nspeculate that the `hump stars' are associated with the second wrap of\nthe Orphan stream. The width of the Orphan stream is estimated to be only a few\n100 pc (Belokurov et al.~2006b). That would be consistent with a narrow feature\nin the density profile of the overdensity. The fairly long elongated \noverdensity in the density profile of the 9h field must be then still ascribed\nto the Monoceros stream.\n\nIn summary, we conclude that the Virgo overdensity has almost certainly been \nseen in the 13h CADIS field, although it was not recognized previously by us as\nsuch. We have found in the 9h field another overdensity which is as significant\nas the Virgo overdensity in the 13h field which we tentatively attribute to the\nMonoceros and Orphan streams. Interestingly both features could be traced to \ndistances less than 3 kpc from the Sun. If they are related to any of the star \nstreams identified as fine structure in the phase space distribution function \nof stars in the solar neighbourhood (Helmi et al.~1999, Chiba \\& Beers 2001, \nNavarro et al.~2004, Helmi et al.~2006, Arifyanto \\& Fuchs 2006) is at present \nunclear.\n\n\\acknowledgements{We thank Vasily Belokurov, Wyn Evans and \nHans--Walter Rix for very helpful discussions.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe third observing run of the global gravitational wave network has not only produced a plethora of varied and unique astrophysics events \\cite{Abbott-AAP20-PopulationProperties, Abbott-AAP20-GWTC2Compact}, it has defined a milestone in quantum metrology: that the LIGO, VIRGO and GEO600 observatories are now all reliably improving their scientific output by incorporating squeezed quantum states \\cite{Tse-PRL19-QuantumEnhancedAdvanced, VirgoCollaboration-PRL19-IncreasingAstrophysical, Lough-PRL21-FirstDemonstration}. This marks the transition where optical squeezing, a widely researched, emerging quantum technology, has become an essential component producing new observational capability.\n\nFor advanced LIGO, observing run three provides the first peek into the future of quantum enhanced interferometry, revealing challenges and puzzles to be solved in the pursuit of ever more squeezing for ever greater observational range. Studying quantum noise in the LIGO interferometers is not simple. The audio-band data from the detectors contains background noise from many optical, mechanical and thermal sources, which must be isolated from the purely quantum contribution that responds to squeezing. All the while, the interferometers incorporate optical cavities, auxiliary optical fields, kg-scale suspended optics, and radiation pressure forces. The background noise and operational stability of the LIGO detectors is profoundly improved in observing run three \\cite{Buikema-PRD20-SensitivityPerformance}, enabling new precision observations of the interactions between squeezed states and the complex optomechanical detectors.\n\nQuantum radiation pressure noise (QRPN) is the most prominent new observation from squeezing \\cite{Yu-N20-QuantumCorrelations, Acernese-PRL20-QuantumBackaction}. QRPN results from the coupling of photon momentum from the amplitude quadrature of the light into the phase quadrature, as radiation force fluctuation integrates into mirror displacement uncertainty. When vacuum states enter the interferometer, rather than squeezed states, QRPN imposes the so-called standard quantum limit\n\\cite{Braginsky-S80-QuantumNondemolition, Braginsky-92-QuantumMeasurement, Braginsky-RMP96-QuantumNondemolition},\nbounding the performance of GW interferometers. Because the QRPN coupling between quadratures is coherent, squeezed states allow the SQL to be surpassed\n\\cite{Kimble-PRD01-ConversionConventional, Yu-N20-QuantumCorrelations}.\nBoth surpassing the SQL and increasing the observing range is possible by using a frequency-dependent squeezing (FDS) source implemented with a quantum filter cavity\n\\cite{Kimble-PRD01-ConversionConventional, McCuller-PRL20-FrequencyDependentSqueezing, Zhao-PRL20-FrequencyDependentSqueezed, Oelker-PRL16-AudioBandFrequencyDependent, Chelkowski-PRA05-ExperimentalCharacterization, Whittle-PRD20-OptimalDetuning, Khalili-PRD10-OptimalConfigurations, Evans-PRD13-RealisticFilter, Kwee-PRD14-DecoherenceDegradation}. LIGO is including such a source in the next observing run as part of its ``A+'' upgrade\\cite{Whittle-PRD20-OptimalDetuning, McCuller-PRL20-FrequencyDependentSqueezing}. To best utilize its filter cavity squeezing source, the frequency-dependence of LIGO's quantum response must be precisely understood.\n\nDegradations to squeezing from optical loss and ``phase noise'' fluctuations of the squeezing angle are also prominently observed in LIGO. Whereas QRPN's correlations cause frequency dependent effects, loss and phase noise are typically described as causing frequency independent, broadband changes to the quantum noise spectrum. This work analyzes the quantum response of both LIGO interferometers to injected squeezed states, indicating that QRPN and broadband degradations, taken independently, are insufficient to fully describe the observed quantum response to squeezing.\n\nThe first sections of this work expand the response and degradation model of squeezing to examine and explain the LIGO quantum noise data by decomposing it into independent, frequency-dependent parameters. The latter sections relate the parameter decomposition back to interferometer models, to navigate how squeezing interacts with cavities that have internal losses, transverse-mode selectivity, and radiation pressure interactions. The spectra at LIGO are explained using a set of broadly applicable analytical expressions, without the need for elaborate and specific computer simulations. The analytical models elucidate the physical basis of LIGO's squeezed state degradations, prioritizing transverse-mode quality using wavefront control of external relay optics\n\\cite{Cao-OEO20-EnhancingDynamic, Cao-AOA20-HighDynamic, Perreca-PRD20-AnalysisVisualization} to further improve quantum noise. This analysis also demonstrates the use of squeezing as a diagnostic tool\\cite{Mikhailov-PRA06-NoninvasiveMeasurements}, examining not only the cavities but also the radiation pressure interaction. These diagnostics show further evidence of the benefit of balanced homodyne detection \\cite{Fritschel-OEO14-BalancedHomodyne}, another planned component of the ``A+'' upgrade. The description of squeezing in this work expands the modeling of degradations in filter cavities \\cite{Kwee-PRD14-DecoherenceDegradation}, explicitly defining an intrinsic, non-statistical, form of dephasing. Finally, the derivations of the quantum response metrics in sec. \\ref{sec:derivations} show how to better utilize internal information inside interferometer simulations, simplifying the analysis of squeezing degradations for current and future gravitational wave detectors.\n\n\\section{Squeezing Response Metrics}\n\\label{sec:metrics}\n\nTo introduce the frequency-dependent squeezing metrics, it is worthwhile to first describe the metrics used for standard optical squeezing generated from an optical parametric amplifier (OPA), omitting any interferometer. For optical parametric amplifiers, the squeezing level is determined by three parameters. The first is the normalized nonlinear gain, $y$, which sets the squeezing level and scales from 0 for no squeezing to 1 for maximal squeezing at the threshold of amplifier oscillation. For LIGO, $y$ is determined from a calibration measurement of the parametric amplification \\cite{Xiao-PRL87-PrecisionMeasurement, Aoki-OEO06-Squeezing946nm, Takeno-OEO07-ObservationDB, Khalaidovski-CQG12-LongtermStable, Schnabel-OC04-SqueezedLight, Dwyer-OEO13-SqueezedQuadrature}. The second parameter is the optical efficiency $\\eta$ of states from their generation in the cavity all the way to their observation at readout. Losses that degrade squeezed states are indicated by $\\eta < 1$. Finally, there is the squeezing phase angle, $\\phi$, which determines the optical field quadrature with reduced noise and the quadrature with the noise increase mandated by Heisenberg uncertainty, anti-squeezing. By correlating the optical quadratures, variations in $\\phi$ continuously rotate between squeezing and anti-squeezing. These parameters relate to the observable noise as:\n\\begin{align}\n  N(\\phi) &= \\left( 1 - \\frac{4 \\eta y}{(1 + y)^2} \\right)\\cos^2(\\phi) + \\left( 1 + \\frac{4 \\eta y}{(1 - y)^2} \\right)\\sin^2(\\phi)\n\\end{align}\nThe noise, $N(\\phi)$, can be interpreted as the variance of a single homodyne observation of a single squeezed state, but for a continuous timeseries of measurements, $N$ can be considered as a power spectral density, relative to the density of shot-noise. Using relative noise units, $N=1$ corresponds to observing vacuum states rather than squeezing. While the nonlinear gain parameter $y$ may be physically measured and is common in experimental squeezing literature, theoretical work more commonly builds states from the squeezing operator, parameterized by $r$, which constructs an ideal, ``pure'' squeezed state that adjusts the noise power by $e^{\\pm 2r}$. State decoherence due to optical efficiency is then incorporated as a separate, secondary process. This is formally related to the previous expression using:\n\\begin{align}\n  N(\\phi)\n  &= \\eta\\left(\n    e^{-2r}\\cos^2(\\phi) + e^{+2r}\\sin^2(\\phi)\n    \\right) + \\left(1 - \\eta\\right)\n    \\label{eq:etaSQZ_basic1}\n  \\\\\n  e^{-2r}\n  &=\n    1 - \\frac{4y}{(1 + y)^2} \\label{eq:OPA_y_SQZ}\n    , \\hspace{2.5em} e^{+2r} = 1 + \\frac{4y}{(1 - y)^2}\n \n\\end{align}\nIn experiments, the squeezing angle drifts due to path length fluctuations and pump noise in the amplifier, but is monitored using additional coherent fields at shifted frequencies and stabilized by feedback control. This stabilization is imperfect, resulting in a root-mean-square (RMS) phase noise,  $\\ensuremath{\\phi^2_{\\text{rms}}}$, that mixes squeezing and antisqueezing. Using $\\hat{\\phi}$ to represent the statistical distribution of the squeezing angle, and $E[\\cdot]$ the expectation operation, phase noise can be incorporated as a tertiary process given the expectation values:\n\\begin{align}\n \\ensuremath{\\phi^2_{\\text{rms}}} &= E\\left[\\sin^2(\\delta \\hat{\\phi})\\right]\n             &\n \\ensuremath{\\phi} &= E\\left[\\hat{\\phi}\\right]\n  &\n\\delta \\hat{\\phi} &= \\hat{\\phi} - \\ensuremath{\\phi}\n\\end{align}\nresulting in the ensemble average noise $\\overline{N}$, relative shot noise.\n\\begin{align}\n  \\overline{N}(\\phi) &= E\\left[N(\\ensuremath{\\phi} + \\delta \\hat{ \\phi})\\right]\n              \\\\\n \n   &= \\eta\\left(1 - \\ensuremath{\\phi^2_{\\text{rms}}}\\right)\\left(e^{-2r}\\cos^2(\\ensuremath{\\phi}) + e^{+2r}\\sin^2(\\ensuremath{\\phi}) \\right)\n     \\nonumber\\\\ &\\hspace{1em}\n                   + \\eta\\ensuremath{\\phi^2_{\\text{rms}}} \\left(e^{+2r}\\cos^2(\\ensuremath{\\phi}) + e^{-2r}\\sin^2(\\ensuremath{\\phi}) \\right) + (1 - \\eta)\n    \\label{eq:etaSQZ_basic2}\n\\end{align}\nAgain, the relative noise $\\overline{N}$ is computed as a single value here, but represents a power spectral density that is experimentally measured at many frequencies. These equations, as they are typically used, represent a change to the quantum noise that is constant across all measured frequencies. Notably, the $\\ensuremath{\\phi^2_{\\text{rms}}}$ phase noise term, which caps at $1\/2$, enters as a weighting factor that averages the anti-squeezing noise increase with squeezing noise reduction, while $\\eta$ mixes squeezing with standard vacuum.\n\nIncorporating an interferometer such as LIGO requires extending these equations to handle frequency-dependent effects. The equations must include terms to represent multiple sources of loss entering before, during, and after the interferometer, as well as terms for the frequency-dependent scaling of the quantum noise due to QRPN and the interferometer's suspended mechanics. The extension of the metrics is described by the following equations and parameters:\n\\begin{align}\n  N(\\Omega)\n  &\\equiv \\Gamma(\\Omega) \\cdot \\Big( \\eta(\\Omega) S(\\Omega) + \\Lambda_\\ensuremath{\\text{IRO}}(\\Omega) \\Big) \n    \\label{eq:metric_N}\n  \\\\\n \n \n \n \n \n \n \n \n  S(\\Omega)\n  &\\equiv S_{\\!{-}}\\cos^2\\Big(\\ensuremath{\\phi} - \\theta(\\Omega)\\Big) + S_{\\!{+}} \\sin^2\\Big(\\ensuremath{\\phi} - \\theta(\\Omega)\\Big)\n    \\\\\n  S_{\\!\\pm} &\\equiv \\big(1 - \\Xi'(\\Omega)\\big)e^{\\pm2r} + \\Xi'(\\Omega) e^{\\mp2r}\n              \\label{eq:metric_S}\n  \\\\\n  \\Lambda_\\ensuremath{\\text{IRO}}(\\Omega) &\\equiv (1 - \\eta_\\ensuremath{\\text{I}})\\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}} + \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}}) + (1 - \\eta_\\ensuremath{\\text{O}}) \/ \\Gamma\n    \\label{eq:metric_Lambda}\n\\end{align}\nThese metrics are composed of the following variables:\n\\renewcommand{\\descriptionlabel}[1]{\\hspace{\\labelsep}{#1}:}\n\\begin{description}[noitemsep, nolistsep]\n \\item[$N(\\Omega)$] the power spectrum of quantum noise in the readout, relative to the vacuum power spectral density, $\\hbar \\omega \/ 2$, of broadband shot noise.\n \\item[$\\Gamma(\\Omega)$] The quantum noise gain of the interferometer optomechanics. While $N(\\Omega)$ is relative shot-noise, QRPN causes interferometers without injected squeezing to exceed shot noise at low frequencies, resulting in $\\Gamma > 1$. For optical systems with $\\Gamma\\ne1$, the system cannot be passive, and must apply internal squeezing\/antisqueezing to the optical fields.\n \\item[$e^{2r}, e^{-2r}$] The ``pure'' injected squeezing and anti-squeezing level, before including any degradations. This level is computed for optical parametric amplifier squeezers using \\cref{eq:OPA_y_SQZ}.\n \\item[$S_{\\!-}, S_{\\!+}$] The minimum and maximum relative noise change from squeezing at any squeezing angle, ignoring losses.\n \\item[$S(\\Omega)$] The potentially observable injected squeezing level, before applying losses or noise gain.\n  \\item[$\\ensuremath{\\phi}$] The frequency independent squeezing angle chosen between the source and readout. This is usually stabilized with a co-propagating coherent control field and feedback system.\n  \\item[$\\theta(\\Omega)$] the squeezing angle rotation due to the propagation through intervening optical system. In a GW interferometer, this can be due to a combination of cavity dispersion and optomechanical effects. Quantum filter cavities target this term to create frequency dependent squeeze rotation.\n  \\item[$\\eta_\\ensuremath{\\text{I}}(\\Omega)$, $\\eta_\\ensuremath{\\text{O}}(\\Omega)$, $\\eta_\\ensuremath{\\text{R}}(\\Omega)$] The individually budgeted transmission efficiencies of the squeezed field at input, reflection and output paths of the interferometer. $1 - \\eta_{\\text{I,R,O}}$ indicates optical power lost in that component.\n  \\item[$\\eta(\\Omega)$] The collective transmission efficiency of the squeezed field. This is usually the product of the efficiencies in each path, $\\eta = \\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}$, but can deviate from this when $\\Gamma \\ne 1$ and interferometer losses affect both $\\Gamma$ and $\\eta_\\ensuremath{\\text{R}}$.\n  \\item[$\\Lambda_\\ensuremath{\\text{IRO}}(\\Omega)$] The total transmission loss over the squeezing path that contaminates injected squeezed states with standard vacuum. When $\\Gamma \\approx 1$, then $\\Lambda_\\ensuremath{\\text{IRO}} \\approx 1 - \\eta$.\n  \\item[$\\Xi'(\\Omega)$] This is a squeezing-level dependent decoherence mechanism called dephasing. It\n    incorporates both statistical $\\ensuremath{\\phi^2_{\\text{rms}}}$ phase fluctuations and the fundamental degradation arises from optical losses with unbalanced cavities, denoted $\\Xi(\\Omega)$. It can also arise from QRPN with structural or viscous mechanical damping. \\Cref{sec:effective_dephasing} shows how to incorporate fundamental dephasing  $\\Xi(\\Omega)$, standard phase uncertainty, $\\ensuremath{\\phi^2_{\\text{rms}}}$, and cavity tuning fluctuations, $\\ensuremath{\\theta^2_{\\text{rms}}}(\\Omega)$, into $\\Xi'(\\Omega)$ to make a total effective dephasing factor. When small, these factors sum to approximate the effective total $\\Xi'$\n\\end{description}\n\nAfter the data analysis of the next section, these quantum response metrics are derived in \\cref{sec:derivations}. These squeezing metrics indicate three principle degradation mechanisms, all frequency-dependent. These are losses, where $\\Lambda_\\ensuremath{\\text{IRO}}(\\Omega)\\approx 1 {-} \\eta(\\Omega) > 0$; Mis-phasing, from $\\phi{-}\\theta(\\Omega)\\ne 0$; and de-phasing, $\\Xi(\\Omega) > 0$.\n\nThe interaction of squeezing with quantum radiation pressure noise is described within these terms. Broadband Squeezing naively forces a trade-off between increased measurement precision and increased quantum back-action. When squeezing is applied in the phase quadrature, it results in anti-squeezing of the amplitude quadrature. The amplitude quadrature then pushes the mirrors and increases QRPN; thus, the process of reducing imprecision seemingly increases back-action. In other terms, QRPN causes the interferometer's ``effective'' observed quadrature\\footnote{The observed quadrature in this context is with respect to the injected quantum states, be they squeezed or vacuum. It is dependent on the quadrature of the interferometer's homodyne readout, but does not refer specifically to it. The effective observed quadrature also does not refer to the specific quadrature that the interferometer signal is modulated into.} to transition from the phase quadrature at high frequencies to the amplitude quadtrature at low frequencies. In the context of these metrics, the observation quadrature is captured in the derivation of $\\theta(\\Omega)$. The associated back-action trade-off can be considered a mis-phasing degradation, allowing the SQL to be surpassed using the quantum quadrature correlations introduced by varying the squeezing angle\\cite{Yu-N20-QuantumCorrelations}. Frequency dependent squeezing, viewed as a modification of the squeezing source, can be considered as making $\\phi(\\Omega)$ frequency-dependent, tracking $\\theta(\\Omega)$. Alternatively, it can be viewed as a modification of the interferometer, to maintain $\\theta(\\Omega) \\approx 0$. While a quantum filter cavity is not explicitly treated in this work, the derivations of \\cref{sec:derivations} are setup to be able to include a filter cavity as a modification to the input path of the interferometer.\n\nWhile mis-phasing can be compensated using quantum filter cavities, the other two degradations are fundamental. For squeezed states, they establish the noise limit:\n\\begin{align}\n  N(\\Omega) &\\ge \\Gamma{\\cdot}\\left(2\\eta\\sqrt{\\Xi'(1-\\Xi')} + \\Lambda_\\ensuremath{\\text{IRO}}\\right),\n  &\n  e^{-2r} &= \\sqrt{\\Xi'(\\Omega)}\n  \\label{eq:N_limit}\n\\end{align}\nSetting the squeezing level as $\\sqrt{\\Xi'}$ solves for the optimal noise given the dephasing. Squeezing is then further degraded from losses, producing the noise limit. Notably, the optimal squeezing is generally frequency-dependent due to $\\Xi(\\Omega)$, indicating that for typical broadband squeezing sources, this bound cannot always be saturated at all frequencies.\n\n\\subsection{Ideal Interferometer Response}\n\\label{sec:ideal_IFO_metrics}\nBefore analyzing quantum noise data to utilize the squeezing metrics of \\crefrange{eq:metric_N}{eq:metric_Lambda}, it is worthwhile to first review the quantum noise features expected in the LIGO detector noise spectra\\cite{Kimble-PRD01-ConversionConventional, Aasi-CQG15-AdvancedLIGO}, under ideal conditions and without accounting for realistic effects present in the interferometer. The derivations later will then extend how the well-established equations below generalize to incorporate increasingly complex interferometer effects, both by extracting features from matrix-valued simulation models, as well as by extracting features from scalar boundary-value equations for cavities.\n\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_IFO, \n  label={SQZ_mm_IFO},\n  caption={\n    This simplified diagram of the interferometer layout shows the propagation of the source laser (solid red) and squeezed beam (dashed burgundy). At (a), the squeezed beam is sourced from a parametric amplifier cavity and circulated to the interferometer with a Faraday isolator. At (b), the squeezing field reflects from the interferometer. Depending on the frequency and transverse beam profile, the states partially transit the interferometer cavities, but also partially reflect promptly. The squeezing that enters the interferometer symmetrically is beam split inside the signal recycling cavity, coherently resonates in both arms, and recombines again at the beamsplitter, effectively experiencing the two branches as a single linear coupled cavity. Injected at a different port, the red laser field carries substantial laser power and is symmetrically split to pump the arm cavities. Differential length signals are sourced by modulating the circulating pump field, creating a phase-quadrature field that resonates in the same effective linear cavity as the squeezing. The signal is emitted at (b) follows with the reflected squeezing. The transverse beam profile (mode) of the signal and squeezing is then selected using the output mode cleaning cavity at (c). Ultimately, the signal and noise are read as timeseries in photodetectors at (d). This effect of coherent interference between prompt and cavity-circulated squeezing from this sequence is formulated, measured, and analyzed in the following sections.\n},\n}\n\n\nOther than shot noise imprecision, the dominant quantum effect in gravitational wave interferometers arises from radiation pressure noise. In an ideal, on-resonance interferometer, this noise is characterized by the interaction strength $\\ensuremath{\\mathcal{K}}(\\Omega)$ that correlates amplitude fluctuations entering the interferometer to phase fluctuations that are detected along with the signal. $\\ensuremath{\\mathcal{K}}$ is generated from the circulating arm power $P_\\ensuremath{\\text{A}}$ creating force noise that drives the mechanical susceptibility $\\chi(\\Omega)$. The susceptibility relates force to displacement on each of the four identical mirrors of mass $m$ in the GW arm cavities. The QRPN effect is enhanced by optical cavity gain $g(\\Omega)$ which resonantly enhances quantum fields entering the arm cavities and signal fields leaving them.\n\\begin{align}\n  \\mathcal{K}(\\Omega)\n  &= 16k\\frac{P_\\ensuremath{\\text{A}}}{c} g^2(\\Omega)\\chi(\\Omega),\n    &\n  g(\\Omega) &= \\frac{\\sqrt{{\\gamma_\\ensuremath{\\text{A}} c}\/{L_{\\ensuremath{\\text{a}}}}}}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n              \\label{eq:optical_gain_g}\n\\end{align}\nHere, $k$ is the wavenumber of the interferometer laser and $c$ the speed of light.\nThe arm cavity gain $g(\\Omega)$ is a function of the signal bandwidth $\\gamma_\\ensuremath{\\text{A}}$, derived later, and the interferometer arm length $L_\\ensuremath{\\text{a}}$. Unlike in past works, this expression of $\\ensuremath{\\mathcal{K}}(\\Omega)$ here is kept complex, holding the phase shift that arises from the interferometer cavity transfer function. The phase of $\\ensuremath{\\mathcal{K}}(\\Omega)$ is useful for later generalizations. $\\ensuremath{\\mathcal{K}}(\\Omega)$ adds the amplitude quadtrature noise power to the phase quadrature fluctuations directly reflected from the interferometer, setting the noise gain $\\Gamma(\\Omega)$\n\\begin{align}\n  \\Gamma(\\Omega) &= 1 + |\\mathcal{K}(\\Omega)|^2\n  &\n  \\theta(\\Omega) &= \\arctan(|\\ensuremath{\\mathcal{K}}(\\Omega)|)\n                   \\label{eq:GammaTheta_standard}\n\\end{align}\nThe relationship between $\\Gamma(\\Omega)$ and $\\theta(\\Omega)$ from $\\ensuremath{\\mathcal{K}}(\\Omega)$ is stated above as reference, but it will more appropriately handle the complex $\\ensuremath{\\mathcal{K}}(\\Omega)$ when it is derived later. The value $|\\ensuremath{\\mathcal{K}}(\\Omega_\\ensuremath{\\text{sql}})| \\equiv 1$ defines the crossover frequency $\\Omega_\\ensuremath{\\text{sql}}$ between noise contributions from shotnoise imprecision and QRPN, corresponding to $\\Gamma(\\Omega_\\ensuremath{\\text{sql}})= 2$ and $\\theta(\\Omega_\\ensuremath{\\text{sql}})=-45^\\circ$. For the $\\chi(\\Omega)$ susceptibility of a free test mass, the factor $\\ensuremath{\\mathcal{K}}(\\Omega)$ can be expressed using only frequency scales.\n\\begin{align}\n  \\mathcal{K}(\\Omega)\n  &= -\\frac{\\Omega^2_\\ensuremath{\\text{sql}}}{\\Omega^2}\\left( \\frac{\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\\right)^2,\n  &&\\text{given}&\n  \\chi(\\Omega) &\\equiv \\frac{-1}{m \\Omega^2}\n    \\label{eq:Kchi_standard}\n\\end{align}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/dual_d.pdf}\n\t\\caption{\n    This figure plots the total quantum and classical noise measured in the LIGO detectors in displacement amplitude spectral density units. The black trace plots a reference measurement of the total noise without injected squeezing at 0.25Hz resolution over 1.5Hr integration for LLO and 1.1Hr for LHO. The orange shotnoise measurement shows the displacement calibration, $\\sqrt{G(\\Omega)}$, in amplitude density units. Subtracting the shotnoise level from the reference yields the gray datapoints, which have been rebinned using a median statistic applied after the subtraction and with a logararithmic bin spacing. The subtraction primarily shows the classical noise but also contains QRPN. Multiple measurements are taken at varied squeezing angles, with 5 of 12 plotted for Livingston (LLO) and 5 of 34 plotted for Hanford (LHO), using the same median rebinning method as the gray subtraction. The variation in the data errorbars results from the binning span of each datapoint, $\\Delta F$, and the measurement integration time, $\\Delta T$. The measured spectra error relative to the total noise and proportional to $1\/\\sqrt{\\Delta F \\Delta T}$. The squeezing angle of $-3.9^\\circ$ and $1.4^\\circ$ datasets at LLO used ${\\sim}1$Hr integration, and the remainder used 15 min each. The squeezing angle $4.5^\\circ$ dataset at LHO used ${\\sim}1$Hr integration, while all others use 2 minutes each. The squeezing level $e^{\\pm2r}$ is constant over all angles, but different between the two sites. This accounts for the difference in the yellow, ${\\sim}30^{\\circ}$, dataset at each site.\n  }\n\t\\label{fig:data_h}\n\\end{figure*}\n\nFrequency independent losses are applied to squeezing before and after the interferometer using $\\eta=\\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{R}}\\eta_\\ensuremath{\\text{O}}$ where $\\eta_\\ensuremath{\\text{I}} < 1$, $\\eta_\\ensuremath{\\text{O}} < 1$. The ideal interferometer assumption of the formulas above enforce $\\eta_\\ensuremath{\\text{R}}=1$. Phase noise in squeezing is included in this ideal interferometer case using $\\Xi' = \\ensuremath{\\phi^2_{\\text{rms}}}$.\n\nThe above expressions relate the optical noise $N(\\Omega)$ of \\cref{eq:metric_N} to past models of the quantum strain sensitivity of GW interferometers\\cite{Kimble-PRD01-ConversionConventional, Buonanno-PRD01-QuantumNoise, Buonanno-PRD03-ScalingLaw}. Since $N(\\Omega)$ is relative to shot-noise, it must then be converted to strain or displacement using the optical cavity gain $g(\\Omega)$, by how it affects the GW signal through the calibration factor $G(\\Omega)$. This factor $G(\\Omega)$ relates strain modulations to optical field phase modulations in units of optical power.\n\\begin{align}\n  \\text{PSD}_{\\text{strain}}(\\Omega) &= G(\\Omega)N(\\Omega),\n                                       &\n  G(\\Omega) &= \\frac{\\hbar c}{\\eta_\\ensuremath{\\text{O}} L_{\\ensuremath{\\text{a}}}^2|g(\\Omega)|^{2} k P_\\ensuremath{\\text{A}} }\n              \\label{eq:optical_calibration_G}\n\\end{align}\nTogether, these relations allow one to succinctly calculate the effect of squeezing on the strain power spectrum in the case of an ideal interferometer. These factors and the calculations behind them will be revisited as non-idealities are introduced.\n\n\\section{Experimental Analysis and Results}\n\\label{sec:experiment}\n\nA goal of this paper is to use the squeezing response metrics of \\crefrange{eq:metric_N}{eq:metric_Lambda} to relate measurements of the instrument's noise spectrum to the parameters of the squeezer system, namely its degradations due to loss $1-\\eta$, radiation pressure from mis-phasing $\\phi{-}\\theta(\\Omega)$, and dephasings $\\Xi'(\\Omega)$. This section presents measurements from the LIGO interferometers that are best described using the established frequency-dependent metrics. The measurements then motivate the remaining discussion of the paper that construct simple interferometer models to describe this data in the context of the metrics. This section refers to and relates to the later sections to provide early experimental motivation for the discussions that follow. The reader may prefer instead to skip this section and first understand the models before returning to see their application to experimental data.\n\nThe main complexity in analyzing the LIGO data is that the detectors have additional classical noises, preventing a direct measurement of $N(\\Omega)$. The many frequency-dependent squeezing parameters must also be appropriately disentangled. To address both of these issues, the unknown squeezing parameters are fit simultaneously across multiple squeezing measurements. The classical noise contribution is determined by taking a reference dataset where the squeezer is disabled, such that $S(\\Omega)=1$, and then subtracting it from the datasets where squeezing is injected.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/dual_Q.pdf}\n  \\caption{\n    This figure shows the data of \\cref{fig:data_h} processed as per \\cref{sec:experiment} for each LIGO site. The processing subtracts away the classical noise determined from the unsqueezed reference dataset. The top panels show the relative noise change $Q_k(\\Omega_i)$ of \\cref{eq:Q_processing} computed using $\\Gamma_k(\\Omega_i)$ from the exact interferometer model of \\cref{sec:matrix_coupled_cavity} using the parameters of \\cref{tab:LIGO_params}. The top panel includes dots with errorbars for the processed data and lines for the best-fit $Q_k(\\Omega_i)$. The middle panel shows the best-fit frequency dependent loss as data points, with errorbars propagated through the fit. For LLO, two sets of loss datapoints are shown, corresponding to interferometer models with different readout angles $\\zeta$.\n    The loss plots also show $1{-}\\eta(\\Omega)$ as computed from the exact matrix model, along with a phenomenological fit against the model of \\cref{eq:Lambda_MM} of \\cref{sec:modeling_TMM}. The phenomenological fit assumes frequency independent losses from the input and output squeezing path with a frequency-dependent addition attributed to transverse mismatch. The bottom panels show the frequency-dependent fit to the observed squeezing angle $\\theta_k(\\Omega_i)$, using the convention of $\\theta(2\\pi{\\cdot}3\\text{kHz}) = 0$. It also plots $\\theta(\\Omega)$ as computed using the exact matrix model. For the LLO data, the $\\zeta=0^\\circ$ model is typically assumed for Michelson-like interferometers such as LIGO; However, the model at that readout angle implies losses at low frequencies that are not favored by the $\\eta(\\Omega)$ models explored in this paper. Alternatively, the $\\zeta\\approx -13^\\circ$ model is consistent with both the fitted losses and the fitted squeezing angles.\n  }\n  \\label{fig:data_Q}\n\\end{figure*}\n\nRepresentative strain spectra from the LIGO Livingston (LLO) and LIGO Hanford (LHO) observatory datasets are plotted in \\cref{fig:data_h}. The Livingston dataset is also reported in \\cite{Yu-N20-QuantumCorrelations}, which details the assumptions and error propagation for the classical noise components and calibration. Only statistical uncertainty is considered in this analysis, in order to propagate error to the parameter fits. The strain spectra of \\cref{fig:data_h} include a reference dataset where the squeezer is disabled, shown in black and at the highest frequency resolution. Additionally, the shotnoise ($N=1$) is plotted in orange, indicating the calibration $\\sqrt{G(\\Omega)}$ of \\cref{eq:optical_calibration_G}. The gray subtraction curve depicts the total classical noise contribution summed with the radiation pressure noise $G(\\Omega)\\ensuremath{\\mathcal{K}}^2(\\Omega)$. The gray dataset can equivalently be computed using a cross correlation of the two physical photodetectors at the interferometer readout\\cite{martynov-pra17-quantumcorrelation}. The equivalence of subtraction and cross correlation is used to precisely experimentally determine the shot-noise scale $G(\\Omega)$ from the displacement-calibrated data.\n\n\\subsection{Analysis}\n\nEach squeezing measurement, indexed by $k$, is indicated by $M_{\\text{ref}, k}(\\Omega_i)$, with a value at each frequency indexed by $i$. The reference dataset is denoted $M_{\\text{ref}}(\\Omega_i)$. The two are subtracted to cancel the stationary classical noise component. The calibration $G(\\Omega)$ is removed to result in the differential quantum noise measurement $D_k(\\Omega_i)$.\n\\begin{align}\n  D_k(\\Omega_i)\n  &\\equiv\n    \\frac{M_{\\text{sqz}, k}(\\Omega_i) - M_{\\text{ref}}(\\Omega_i)}{G(\\Omega_i)}\n    \\label{eq:D_processing}\n\\end{align}\nFor these datasets, the squeezing level $e^{\\pm 2r}$, is held constant and independently measured using the nonlinear gain technique\\cite{Dwyer-OEO13-SqueezedQuadrature} to derive $y$ of \\cref{eq:OPA_y_SQZ}. Each differential data $D_k(\\Omega_i)$ is taken at some squeezing angle $\\phi_k$, which is either fit (LLO) or derived from independent measurements (LHO). The parameters $\\eta_i$ and squeezing rotation $\\theta_i$ are independent at every frequency $\\Omega_i$ but fit simultaneously. All $\\phi_k$ are also fit simultaneously across all datasets. Nonlinear least squares fitting was performed using the Nelder-Mead simplex algorithm \\cite{Gao-COA12-ImplementingNelderMead} implemented in SciPy \\cite{Virtanen-NM20-SciPyFundamental}. The residual minimized by least squares fitting is\n\\begin{align}\n  \\mathcal{R} &= \\sum_{\\substack{i=0\\\\k=0}}^{\\mathcal{N}} \\left( \\frac{D_k(\\Omega_i) - \\overline{D}_k(\\Omega_i)}{\\Delta D_k(\\Omega_i)}\\right)^2\n\\end{align}\nThe measurement statistical uncertainty $\\Delta D$, dominated by the statistical uncertainty in power-spectrum estimation, was propagated through the datasets per \\cite{Yu-N20-QuantumCorrelations}. $\\overline{D}_k(\\Omega_i)$ is the model of the data that is a function of the fit parameters, $\\eta_i, \\theta_i, \\phi_k$ as well as independently measured parameters such as $e^{-2r}$. $\\Xi'(\\Omega_i)$ is not fit using this data since the squeezing level $e^{2r}$ is not varied across the datasets. This is discussed below. These given fit parameters affect are propagated through the squeezing metric functions create a model of this particular differential quantum noise measurement.\n\\begin{align}\n  \\overline{S}_k(\\Omega_i) &\\equiv\n  e^{-2r}\\cos^2\\Big(\\ensuremath{\\phi}_k - \\theta_i\\Big) + e^{+2r}\\sin^2\\Big(\\ensuremath{\\phi}_k - \\theta_i\\Big)\n  \\\\\n  \\overline{D}_k(\\Omega_i)\n  &\\equiv \\left.N(\\Omega_i)\\right|_{S = \\overline{S}_k(\\Omega_i)} - \\left.N(\\Omega_i)\\right|_{S = 1}\n\\end{align}\nWhich simplifies to\n\\begin{align}\n  \\overline{D}_k(\\Omega_i)\n  &= \\left(\\overline{S}_k(\\Omega_i) - 1\\right) \\eta_i \\Gamma(\\Omega_i)\n\\end{align}\nNotably, the individual efficiencies $\\eta_\\ensuremath{\\text{I}}, \\eta_\\ensuremath{\\text{R}}, \\eta_\\ensuremath{\\text{O}}$ cannot be individually measured and only the ``total'' efficiency $\\eta(\\Omega)$ is measurable using this differential method, where the classical noise is subtracted using a reference dataset with squeezing disabled. Additionally, the optical efficiency $\\eta$ can only be inferred given some knowledge or assumption of $\\Gamma(\\Omega)$. In effect, the product $\\eta \\Gamma$ is the primary measurable quantity, rather than its decomposition into separate $\\eta$ and $\\Gamma$ terms; However, for the purposes of modeling, decomposing the two is conceptually useful. Furthermore, to characterize physical losses, the efficiency $\\eta$ or loss $\\Lambda_\\ensuremath{\\text{IRO}}\\approx 1 {-} \\eta$ is easier to plot and interpret than the product $\\eta\\Gamma$.\n\nFor these reasons, the differential data $D_k(\\Omega_i)$ is further processed, creating the measurement $Q_k(\\Omega_i)$ with a form similar to \\cref{eq:etaSQZ_basic1}\n\\begin{align}\n  Q_k(\\Omega_i)\n  &\\equiv \n    \\frac{D_k}{\\Gamma(\\Omega_i)} + 1 \\approx S_k(\\Omega_i)\\eta_i + (1 - \\eta_i) + \\Delta Q\n    \\label{eq:Q_processing}\n\\end{align}\nThe LIGO squeezing data expressed in dB's of $Q_k(\\Omega_i)$ are plotted in the upper panels of \\cref{fig:data_Q}. The data and error bars are in discrete points, while the parameter fits to $Q_k$ using $\\eta_i$, $\\theta_i$ and $\\phi_k$ are the solid lines between the data points. The spectra in each set are calculated using the Welch method a median statistic at each frequency to average all of the frames through the integration time. This prevents biases due to instrumental glitches adding non-stationary classical noise. This technique is detailed in \\cite{Yu-N20-QuantumCorrelations}.\n\nAfter computing $Q_k(\\Omega_i)$ at full frequency resolution, the data is further rebinned to have logararithmic spacing by taking a median of the data points within the frequency range of each bin. This rebinning greatly improves the statistical uncertainty at high frequencies, where many points are collected. At lower frequencies, the relative error benefits less from binning; however, both the LLO and the LHO datasets use a long integration time for their reference measurement and at least one of the squeezing angle measurements. Using the median removes narrow-band lines visible in the strain spectra of \\cref{fig:data_h}. Fitting combines the few long-integration, low-error datasets with many short-integration, high-error sets at many variations of the operating parameters. The few low-error datasets reduce the absolute uncertainty in the resulting fit parameters, whereas the many variations reduce co-varying error that would otherwise result from modeling parameter degeneracies.\n\nThe relative statistical error in each bin of the original PSD $M_k(\\Omega_i)$ is approximately $(\\Delta F \\Delta T)^{-{1}\/{2}}$ given the integration time $\\Delta T$ of 2 minutes to 1 hour and bin-width $\\Delta F$ of 0.25Hz. This relative error is converted to absolute error and propagated through the processing steps of \\crefrange{eq:D_processing}{eq:Q_processing}. At low frequencies, the classical noise contribution to each $M_k$ is larger than the quantum noise. Although it is subtracted away to create $D_k(\\Omega_i)$, the classical noise increases the absolute error, and, along with less rebinning, results in the larger relative errors at low-frequency in \\cref{fig:data_Q}. After fitting the squeezing parameters, the Hessian of the reduced chi-square is computed from the Jacobian of the fit residuals with respect to the parameters. This Hessian represents the Fisher information, and the diagonals of its inverse provides the variances indicated by the plotted loss and angle parameter error bars.\n\nFor the LHO data, the fit parameters $\\phi_k$ are determined by mapping the demodulation angles of its coherent control feedback system\\cite{Tse-PRL19-QuantumEnhancedAdvanced, Dooley-OEO15-PhaseControl, Vahlbruch-PRL06-CoherentControl, Chelkowski-PRA07-CoherentControl} back to the squeezing angle. That mapping has 3 unknown parameters, an offset in demodulation angle, an offset in squeezing angle, and a nonlinear compression parameter, all of which are fit simultaneously in all datasets. This $\\phi_k$ mapping was not performed on the LLO data, as some systematic errors in the demodulation angle records bias the results. Despite fitting more independent parameters, the longer integration time of the LLO data gives it sufficiently low statistical uncertainty at frequencies below $\\Omega_\\ensuremath{\\text{sql}}$ that the model and parameter degeneracy between $\\phi_k$, $\\theta_i$ and $\\eta_i$ is not an issue.\n\n\\subsection{Results}\n\nThe middle panels of \\cref{fig:data_Q} show the fits to $\\eta_i$, though plotted as loss $1{-}\\eta_i$ to represent $\\Lambda_\\ensuremath{\\text{IRO}}$. Both datasets additionally include a red loss model curve fit, assembled using the equations in \\cref{sec:modeling}. The orange exact model curves use \\cref{sec:matrix_coupled_cavity}. The data and model curve fit shows a variation in the efficiency, where losses increase from low to high frequencies. This increase in loss can be attributed either to losses within the signal recyling cavity of the interferometer, or to a coherent effect resulting from transverse Gaussian beam parameter mismatch between the squeezer and interferometer cavities. At low frequencies, the optical efficiency is similar between the two LIGO sites, indicating that frequency independent component to the loss are consistent between the implementations at both LIGO sites. The differing high-frequency losses can reasonably be ascribed to variations in the optical beam telescopes of the squeezing system and are analyzed in \\cref{sec:modeling_TMM}.\n\nThe LLO middle panel of \\cref{fig:data_Q} shows two separate inferred loss $1 - \\eta_i$ datasets. These differ in their underlying model of $\\Gamma(\\Omega_i)$. The following section \\ref{sec:derivations} discusses how variations in $\\Gamma$ arise and describes the local oscillator angle $\\zeta$. The $\\zeta = 0$ data reflects the standard, ideal radiation pressure noise model of \\crefrange{eq:Kchi_standard}{eq:GammaTheta_standard}. This model is disfavored given the frequency dependency of $\\eta(\\Omega_i)$ derived using optical cavity models later in this paper. The $\\zeta = -13^\\circ$ model presents an alternative that is compatible with models of the optical efficiency. The need for this alternative indicates that squeezing metrics must account for variations in interferometer noise gain $\\Gamma$. Physically, these variations arise from the readout angle adjusting the prevalence of radiation pressure versus pondermotive squeezing. The $\\zeta = -13^\\circ$ model results in a smaller noise gain $\\Gamma$ at $40 \\text{Hz}$ than does the $\\zeta = 0^\\circ$ model. Since the lower $\\Gamma$ model is favored, this dataset provides some, moderate, evidence that LLO currently benefits from the quantum correlations introduced by the mirrors near $\\Omega_{\\text{SQL}}$, while experiencing lessened sensitivity elsewhere.\n\nThis data demonstrates that the readout angle has an effect on the interferometer sensitivity and the optimal local oscillator is not necessarily $\\xi = 0$ due to radiation pressure.  The quantum benefit of decreased $\\Gamma$ from the readout angle $\\xi$ is a method to achieve sub-SQL performance that is an alternative to injecting squeezing. Like squeezing, it has a frequency dependent enhancement known as the ``variational readout'' technique \\cite{Kimble-PRD01-ConversionConventional, Khalili-PRD07-QuantumVariational}, that a sensitivity increase from lowering $\\Gamma$ while minimizing the sensitivity decrease of the frequency-independent form. For LLO, the reduced sensitivity from $\\xi \\ne 0$ masquerades as a $5\\%$ loss of signal power, but does not actually affect the $\\eta$ or $\\Lambda_\\ensuremath{\\text{IRO}}$ contributions to the squeezing level.\n\nThe bottom panels of \\cref{fig:data_Q} show the fits of $\\theta_i$ of each dataset. The magnitude of $e^{\\pm 2r}$ provides a ``lever arm'' in the variation of $S_k(\\Omega_i)$ that strongly constrains the $\\phi_k{-}\\theta_i$ effective squeezing angle. These leveraged constraints result in small errorbars to the fitted $\\theta_i$. The LLO data are plotted with two models of the $\\theta_i$ based on the assumed local oscillator angle $\\zeta$. The $\\zeta=0^\\circ$ model follows the standard radiation pressure model of \\cref{eq:GammaTheta_standard} at low frequencies and includes a filter-cavity type rotation around the interferometer cavity bandwidth $\\gamma \\approx 2\\pi \\cdot 450\\text{ Hz}$. This rotation is modeled in \\cref{sec:modeling_cavities}. The $\\zeta=-13^{\\circ}$ model is computed using the coupled cavity model of \\cref{sec:matrix_coupled_cavity} and internally includes a weak optical spring effect along with the shifted readout angle $\\zeta$. Together, these effects modify the effective squeezing angle $\\theta$ away from \\cref{eq:GammaTheta_standard} at low frequencies, and agree well with the dataset. This agreement provides further evidence of the reduced radiation pressure noise gain $\\Gamma(\\Omega)$ in LLO that results from the effective LO readout angle $\\zeta$. A nonzero readout angle $\\zeta$ is reasonable to expect due to unequal optical losses in the LIGO interferometer arm cavities. The arm mismatch results in imperfect subtraction of the fringe-light amplitude quadrature at the beamsplitter, creating a static field that adds to the phase-quadrature light created from the Michelson offset and results in $\\zeta \\ne 0$. Past diagnostic measurements\nconclude that some power in the readout diodes must be in the amplitude quadrature, but until now could not determine the sign.\n\nAlthough the squeezing angle parameters $\\phi_k$ and $\\theta_i$ are fit, the frequency-dependent dephasing parameter $\\Xi_i$ cannot be reliably determined from these datasets given the accuracy to which $e^{\\pm 2r}$ is measured. Additionally, the squeezing level $e^{\\pm 2r}$ is not varied in this data, nor is it sufficiently large to resolve an influence from $\\Xi(\\Omega) < 10^{-3}$. This $\\sqrt{\\Xi} \\approx \\ensuremath{\\phi_{\\text{rms}}}$ is expected from independent measurements of phase jitter that propagate through the coherent control scheme of the squeezer system\\cite{Tse-PRL19-QuantumEnhancedAdvanced}. A large source of optically induced $\\Xi$ is not expected has the interferometer cavities are not sufficiently detuned. Measurements of the squeezing system indicate $\\ensuremath{\\phi_{\\text{rms}}} \\lesssim 30\\text{ mRad}$. Future LIGO measurements should include additional datasets that vary $r$ along a third indexing axis $j$ and should increase the injected squeezing level $e^{2r} > 30$ to measure, or at least bound, $\\Xi$ and its frequency-independent contribution $\\ensuremath{\\phi^2_{\\text{rms}}}$. The model fits described above are consistent with the data while assuming $\\Xi = \\ensuremath{\\phi^2_{\\text{rms}}} \\equiv 0$.\n\n\\section{Decomposition Derivation}\n\\label{sec:derivations}\n\nThe factors $\\eta(\\Omega), \\theta(\\Omega)$ and $\\Xi(\\Omega)$ from \\cref{sec:metrics} each describe an independent way for squeezing to degrade. $\\Gamma(\\Omega)$ indicates how the quantum noise scales above or below the shot noise level from squeezing and from quantum radiation pressure within the interferometer. They represent a natural extension of standard squeezing metrics that incorporates frequency dependence, and, as scalar functions, they are simple to plot and to relate with experimental measurements. This section delves into their derivation by employing matrices in the two photon formalism \\cite{Caves-PRA85-NewFormalism, Schumaker-PRA85-NewFormalism} to represent the operations of squeezing, adding loss, shifting the squeezing phase, reflecting from the interferometer, and final projection of the quantum state into the interferometer readout. The derived formulas can be used in frequency-domain simulation tools that compute noise spectra using matrix methods, so that the quantum response metrics can be provided in addition to opaquely propagating squeezing to an simulation result of $N(\\Omega)$. \n\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_cavities, \n  label=SQZ_mm_cavities, \n  caption={\n    The two-photon transformation matrices experienced by squeezing through the sequence of \\cref{fig:SQZ_mm_IFO}. The effective linear coupled cavity, including the optomechanical effect of radiation pressure, is collected and computed into the transformation $\\Tmat{H}_R$. The middle cavity is the signal recycling cavity and the rightmost cavity represents the coherent combination of both arms. Each cavity adds losses from each mirror. For simplicity, these are collected into round-trip cavity loss contributions, $\\Lambda_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{s}}}$, and $\\Lambda_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{s}}}$ that inject standard optical vacuum into the cavities, circulating and transforming into the loss terms $\\Tmat{T}_{\\ensuremath{\\text{R}},\\ensuremath{{\\,\\mu}}}$ while lowering the efficiency $\\eta_\\ensuremath{\\text{R}}$. Transformations of the squeezing at the input and output are included with the terms, $\\Tmat{H}_{\\ensuremath{\\text{I}}}$, $\\Tmat{H}_{\\ensuremath{\\text{O}}}$ and any additive vacuum contributions,  $\\Tmat{T}_{\\ensuremath{\\text{I}},\\ensuremath{{\\,\\mu}}}$, $\\Tmat{T}_{\\ensuremath{\\text{O}},\\ensuremath{{\\,\\mu}}}$.\n  },\n}\n\nTwo-photon matrices are an established method to represent transformations of the optical phase space of Guassian states in an input-output Heisenberg representation of the instrument\\cite{Danilishin-LRR12-QuantumMeasurement}. They are concise yet rigorous when measuring noise spectra from squeezed states using the quantum measurement process of Homodyne readout. Section II of \\cite{Danilishin-LRR19-AdvancedQuantum} provides a review of their usage in the context of gravitational-wave interferometers. Here, two-photon matrices are indicated by doublestruck-bold lettering, and are given strictly in the amplitude\/phase quadrature basis.\n\nEach matrix represents the transformation of the optical phase space of a single optical ``mode'' as it propagates through each physical element towards the readout. The term ``mode'' refers to a basis vector in a linear decomposition of optical field the transverse optical plane of many physical ports\\footnote{Simulations also typically span multiple optical frequencies, but this is not treated here.}. Each plane is further decomposed into transverse spatial modes using a Hermite or Laguerre-Gaussian basis. In this decomposition, each optical mode is indexed by the placeholder $\\mu$ and acts as a continuous transmission channel for optical quantum states. The phase space transformations of these continuous optical states is indexed by time or, more conveniently, frequency. Optical losses and mixing from transverse mismatch behave like beamsplitter operations, serving to couple multiple input modes, generally carrying vacuum states, to the mode of the readout where states are measured.\n\nThe mode of the injected squeezed states, and their specific transformations during beam propagation, must be distinguished from all of the lossy elements that couple in vacuum states. The squeezed states experience a sequence of transformations by the input elements, interferometer, and output elements, denoted $\\Tmat{H}_\\ensuremath{\\text{I}}(\\Omega)$,  $\\Tmat{H}_\\ensuremath{\\text{R}}(\\Omega)$,  $\\Tmat{H}_\\ensuremath{\\text{O}}(\\Omega)$. This sequence multiplies to formulate the total squeeze path propagation $\\Tmat{H}$. \n\\begin{align}\n  \\Tmat{H}(\\Omega) &= \n\\Tmat{H}_{\\text{O}}\\Tmat{H}_{\\text{R}}\\Tmat{H}_{\\text{I}}\n\\end{align}\nLossy optical paths mix the squeezed states with additional standard vacuum states. These are collected into sets of transformation matrices corresponding to each individual loss source, $\\{\\Tmat{T}_\\mu\\}$. See \\cref{fig:SQZ_mm_cavities}. The sets are grouped by their location along the squeezing path where the lossy element is incorporated. The beamsplitter-like operation that couples each loss is given by a $\\Tmat{\\Lambda}_\\mu$, indexed by its location and source along the squeezing path. Loss transformations $\\Tmat{\\Lambda}_\\mu$ are generally frequency-independent. $\\Tmat{\\Lambda}_{\\text{R}, i}$ are an exception, as they occur within the cavities of the interferometer and include some cavity response. The vacuum states associated with each loss then propagate along with the squeezed states and experience the remaining transformations that act on squeezing.\n\\begin{align}\n  \\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{H}_{\\text{O}}\\Tmat{H}_{\\text{R}}\\Tmat{\\Lambda}_{\\text{I},\\ensuremath{{\\,\\mu}}}\n    \\\\\n  \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{H}_{\\text{O}}\\Tmat{\\Lambda}_{\\text{R},\\ensuremath{{\\,\\mu}}}\n    \\\\\n  \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{\\Lambda}_{\\text{O},\\ensuremath{{\\,\\mu}}}\n  \\\\\n  \\left\\{\\Tmat{T}\\right\\} &= \\left\\{\\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}; \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}; \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}  \\right\\}\n\\end{align}\nTogether, all of the transformations of $\\Tmat{H}$ and  $\\{\\Tmat{T}\\}$ define the output states at the readout of the interferometer in terms of the input states entering through the squeezer and loss elements. The two quadrature observables of the optical states are given with the convention $\\hat{q}$ being the amplitude quadrature and $\\hat{p}$ being the phase, and they are indexed to distinguish their input port and transverse mode.\n\\begin{align}\n  \\begin{bmatrix}\\hat{q}_{\\text{out}}(\\Omega) \\\\ \\hat{p}_{\\text{out}}(\\Omega)\\end{bmatrix}\n  &=\n    \\Tmat{H} \\begin{bmatrix}\\hat{q}_{\\text{in}}(\\Omega) \\\\ \\hat{p}_{\\text{in}}(\\Omega)\\end{bmatrix} + \n  \\sum_{\\Tmat{T}_\\ensuremath{{\\,\\mu}}\\in \\left\\{\\Tmat{T}\\right\\}} \\Tmat{T}_\\ensuremath{{\\,\\mu}} \\begin{bmatrix}\\hat{q}_\\ensuremath{{\\,\\mu}}(\\Omega) \\\\ \\hat{p}_\\ensuremath{{\\,\\mu}}(\\Omega)\\end{bmatrix}\n  \\label{eq:qp_out}\n\\end{align}\nThe two-photon matrices $\\Tmat{H}$ and $\\Tmat{T}_\\ensuremath{{\\,\\mu}}$ must preserve commutation relations, namely $[\\hat{q}_\\text{out}, \\hat{p}_\\text{out}] = [\\hat{q}_\\ensuremath{{\\,\\mu}}, \\hat{p}_\\ensuremath{{\\,\\mu}}] = i\\hbar$. In doing so, the matrices ensure that losses within $\\Tmat{H}$ couple ancillary vacuum states that degrade squeezing.\n\nThe readout carries a continuous coherent optical field known as the ``local oscillator'' and the output states are read using homodyne readout. The phase of the local oscillator, $\\zeta$, defines the observed quadrature, $\\hat{m}$, for the homodyne measurement. Gravitational Wave interferometers typically use a ``Michelson offset''\\cite{Fricke-CQG12-DCReadout, Hild-CQG09-DCreadoutSignalrecycled, Ward-CQG08-DcReadout} in the paths adjacent their beamsplitter to operate slightly off of dark fringe. This offset couples a small portion of their pump carrier light to their output as the local oscillator field. This is a form of homodyne readout that fixes $\\zeta$ to measure in the phase quadrature, defined here to be when $\\zeta=0$. Imperfect interference at the beamsplitter can couple some amplitude quadrature and shift $\\zeta$ away from $0$. Balanced homodyne readout is an alternative implementation proposed for LIGO's ``A+'' upgrade and will allow $\\zeta$ to be freely chosen\\cite{Fritschel-OEO14-BalancedHomodyne}. Regardless of the implementation, the homodyne observable is $\\hat{m}$,\n\\begin{align}\n  \\hat{m} &=\n  \\ensuremath{\\Tvec{v}^{\\dagger}}\\begin{bmatrix}\\hat{q}_{\\text{out}}(\\Omega) \\\\ \\hat{p}_{\\text{out}}(\\Omega)\\end{bmatrix}\n           \n  &\n  \\ensuremath{\\Tvec{v}^{\\dagger}}(\\zeta)\n  &=\n    \\begin{bmatrix}\n      \\sin(\\zeta) & \\cos(\\zeta)\n    \\end{bmatrix}\n \\label{eq:homodyne_observable}\n\\end{align}\nHomodyne readout enforces a symmetrized expectation operator, denoted here with the subscript HR, for all measurements of the optical quantum states.\nFurther details of the measurement process are beyond the scope of this work, but the following quadratic expectations arise when computing the noise spectrum and are sufficient to simplify the homodyne expectation values of $\\hat{m}$.\n\\begin{align}\n  1 &= \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}^2}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}^2}_{\\text{HR}},\n      \\label{eq:vacuum_expectations}\n  \\hspace{1em}\n  0 = \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}\\hat{p}_\\ensuremath{{\\,\\mu}}}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}\\hat{q}_\\ensuremath{{\\,\\mu}}}_{\\text{HR}}\n      \\\\\n  0 &= \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}\\hat{q}_\\ensuremath{{\\,\\nu}}}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}\\hat{p}_\\ensuremath{{\\,\\nu}}}_{\\text{HR}} \\text{  for } \\nu \\ne \\mu\n      \\label{eq:vacuum_expectations2}\n\\end{align}\nAs a result of these expectations, the vector norm suffices to evaluate noise power using this matrix formalism.\nThe addition of squeezing can be seen either as a modification of the input states $\\hat{q}_{\\text{in}}$, $\\hat{q}_{\\text{in}}$, which violate \\cref{eq:vacuum_expectations}. This work uses the alternative picture, where an additional squeezing transformation occurring at the very start of the squeezing path $\\Tmat{H}$ that acts on $\\hat{q}_{\\text{in}}$, $\\hat{q}_{\\text{in}}$ that are also vacuum states. The squeezing transformation is defined by the squeezing level $r$ and the squeezing angle $\\phi$, which act via the matrices:\n\\begin{align}\n  \\Tmat{R}(\\phi)\n  &\\equiv\n  \\begin{bmatrix}\n    \\cos(\\phi) & {-}\\sin(\\phi)\\\\\n    \\sin(\\phi) & \\phantom{-}\\cos(\\phi)\n  \\end{bmatrix}\n &\n  \\Tmat{S}(r) &\\equiv\n  \\begin{bmatrix}\n    e^r & 0\\\\\n    0 & e^{-r}\n  \\end{bmatrix}\n\\end{align}\nWhen added to the squeezing path, the resulting quantum noise is calculated from the observable $\\hat{m}$.\n\\begin{align}\n  N(\\Omega) &= \\Braket{\\hat{m}^\\dagger\\hat{m}}_{\\text{HR}} = \n              \\left| \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}\\Tmat{R}(\\phi) \\Tmat{S}(r) \\right|^2 + \n\\sum_{\\Tmat{T}_\\ensuremath{{\\,\\mu}}\\in \\left\\{\\Tmat{T}\\right\\}} \\left| \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_\\ensuremath{{\\,\\mu}} \\right|^2\n                             \\label{eq:N_no_sqz_derivation}\n\\end{align}\nThe first term of which is one of the factors in \\cref{eq:metric_N}\n\\begin{align}\n  \\eta(\\Omega)\\cdot S(\\Omega, \\phi) \\cdot \\Gamma(\\Omega, \\zeta)\n \n \n \n \n \n   &= \\left|\\ensuremath{\\Tvec{v}^{\\dagger}}\n     \\Tmat{H}\\Tmat{R}(\\phi)\n     \\Tmat{S}(r)\n     \\right|^2\n    \\label{eq:decomposition_relation}\n\\end{align}\nAt this point, the factors can be separated because: $\\Tmat{R}\\Tmat{S}$ determines the factor $S(\\Omega, \\phi)$; $\\Tmat{H}$ has been ``reduced'' by loss, indicating when $\\eta(\\Omega) < 1$; and the benchmark noise level is defined by $\\Gamma(\\Omega)$, contained in the interferometer's optomechanical element $\\Tmat{H}_{\\text{R}}$.\n\nTo distinguish these terms, further manipulations are necessary. The first is to examine just the vector $\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}$ to determine how the later term $\\Tmat{R}\\Tmat{S}$ results in $S(\\Omega)$. Basis vectors for the two quadrature observables are defined, and the local oscillator is represented using them.\n\\begin{align}\n  \\ensuremath{\\Tvec{v}^{\\dagger}}(\\zeta)\n  &=\n  \\Tvec{e}_p^\\dagger\\Tmat{R}(\\zeta)\n  &\n    \\Tvec{e}_q &= \n    \\begin{bmatrix}\n      1 \\\\ 0\n    \\end{bmatrix}\n  &\n    \\Tvec{e}_p &= \n    \\begin{bmatrix}\n      0 \\\\ 1\n    \\end{bmatrix}\n\\end{align}\nThe basis vectors then allow the vector norm to be split into its two components $\\ensuremath{m_q}$ and $\\ensuremath{m_p}$, defining the \\textit{observed noise quadrature}.\n\\begin{align}\n  \\ensuremath{m_q}(\\Omega) &= \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H} \\Tvec{e}_q & \\ensuremath{m_p}(\\Omega) &= \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H} \\Tvec{e}_p\n\\end{align}\nThe vector $\\vec{m}$ contains the magnitude and angle of a projection of the quantum state $\\hat{q}_{\\text{in}}$, $\\hat{p}_{\\text{in}}$ at each frequency, but it also contains the complex phase shift from propagation delay in the interferometer and squeezing path. This later phase contribution does not affect noise calculations, but must be properly handled. Projecting it away requires maintaining phase information, and this is why the optomechanical factor $\\ensuremath{\\mathcal{K}}$ is complex in this work.\n\nThe squeezing angle rotation $\\Tmat{R}(\\phi)$ can be viewed through its left-multiplication, applying a rotation to the observed noise quadtrature rather than to the squeezing. In this picture, the angle $\\phi$ can align the observed quadtrature with either the squeezing or anti-squeezing quadrature. The rotation needed to do so determines $\\theta(\\Omega)$, again with the caveat that both $m_q$ and $m_p$ are complex. Their common phase carries the delay information, but their differential phase causes dephasing. In short, differential phase forces $\\vec{m}$ to project into both quadratures at any rotation $\\Tmat{R}(\\phi)$. This has the effect of always adding anti-squeezing to squeezing and vice-versa, resulting in the factor $\\Xi(\\Omega)$. The relations are fully derived in \\cref{sec:phase_noise_composition} using a singular value decomposition to identify the principle noise axes. It leads to the expressions\n\\begin{align}\n  \\theta(\\Omega) &\\approx -\\arctan\\left(\\Re\\!\\left\\{{\\frac{\\ensuremath{m_q}}{\\ensuremath{m_p}}}\\right\\}\\right)\n                   \\label{eq:theta_calculation}\n  \\\\\n  \\Xi(\\Omega) &= \\frac{1}{2} -  \\sqrt{\n        \\frac{\\left(|\\ensuremath{m_q}|^2 - |\\ensuremath{m_p}|^2\\right)^2 + 4 \\Re\\left\\{ \\ensuremath{m_q}\\ensuremath{\\conj{m}_p} \\right\\}}\n        {4\\left(|\\ensuremath{m_q}|^2 + |\\ensuremath{m_p}|^2\\right)^2}\n        } \n                   \\label{eq:Xi_calculation}\n\\end{align}\nThe observation vector $\\vec{m}$, and \\cref{eq:theta_calculation} generalizes the observed noise quadrature description of quantum radiation pressure noise. With it, the observed quadrature angle $\\theta(\\Omega)$ may be computed for any readout angle $\\zeta$ and for more complex interferometers $\\Tmat{H}_\\ensuremath{\\text{R}}$. The ideal interferometer example is demonstrated in \\cref{sec:ideal_IFO_example}\n\nThe phase and magnitude of of the previous argument allows one to determine $S(\\Omega)$ from the form of $\\Tmat{S}$ applied to $\\vec{m}\\Tmat{R}(\\phi)$. Factoring $S$ away, the magnitude of $\\vec{m}$ carries the efficiency of transmitting the squeezed state, along with the noise gain applied to it.\n\\begin{align}\n  \\eta(\\Omega) \\cdot \\Gamma(\\Omega, \\zeta)\n  &=\n    |\\ensuremath{m_q}|^2 + |\\ensuremath{m_p}|^2\n    \\label{eq:eta_gamma}\n\\end{align}\n$\\Gamma(\\Omega)$ expresses the total noise from the interferometer when squeezing is not applied, applying radiation pressure or optomechanical squeezing to both the squeezing path vacuum and internally loss-sourced vacuum. $\\eta \\Gamma$ is affected by all losses, but some of them affect $\\Gamma(\\Omega)$ as well. Using squeezing or a coherent field to probe $\\Tmat{H}$ always measures the product $\\eta \\Gamma$, so the noise gain factor $\\Gamma$ serves primarily as a benchmark. As a benchmark, it relates the dependence of $N(\\Omega)$ to $S(\\Omega)$ and separates the scaling by the efficiency $\\eta$ so that the physical losses may be determined. For this reason, there is freedom to define $\\Gamma$ to make it as independent from the losses as possible, so that it best serves as a benchmark. Here, it is defined using the simulated knowledge of the total noise from the interferometer elements alone:\n\\begin{align}\n \\Gamma(\\Omega) \n  &= \n\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}_{\\text{R}}\\right|^2 + \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{\\Lambda}_{\\text{R},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\label{eq:gamma_HR}\n\\end{align}\n$\\eta$ is then determined by dividing \\cref{eq:eta_gamma} by \\cref{eq:gamma_HR}. Under this definition of $\\Gamma$, $\\eta \\propto \\eta_\\ensuremath{\\text{I}}$ and $\\eta \\propto \\eta_\\ensuremath{\\text{O}}$. Losses within the interferometer affect $\\Gamma(\\Omega)$ slightly, and  $\\eta \\propto \\eta_\\ensuremath{\\text{R}}$ is only approximate. \\Cref{sec:radiation_pressure_calc} gives an example of how losses affect $\\eta$ and $\\Gamma$. The primary alternative definition is to use $\\Gamma = N\\big|_{S=1}$, but this definition makes $\\eta_\\ensuremath{\\text{O}}$ both less physically intuitive and also sensitive to interferometer parameters.\n\nSubtracting $\\eta\\Gamma$ from \\cref{eq:N_no_sqz_derivation} and factorizing by the optical paths provides the definition of the remaining efficiency terms.\n\\begin{align}\n  (1 - \\eta_\\ensuremath{\\text{O}})\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\\\\n  \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}}) \\Gamma\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\\\\n  \\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}(1 - \\eta_\\ensuremath{\\text{I}}) \\Gamma\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}\\right|^2\n\\end{align}\nWhich add together to create the loss term in \\cref{eq:metric_N}.\n\\begin{align}\n    \\Lambda_\\ensuremath{\\text{IRO}}\\Gamma &= \\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}(1 - \\eta_\\ensuremath{\\text{I}})\\Gamma\n    + \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}})\\Gamma\n    + (1 - \\eta_\\ensuremath{\\text{O}})\n \n \n \n \n \n \n \n\\end{align}\n\n\\subsection{Ideal Interferometer Example}\n\\label{sec:ideal_IFO_example}\n\nThe derivations are now extended to recreate and generalize the ideal noise model of \\cref{sec:ideal_IFO_metrics}, using \\cref{eq:Kchi_standard} for $\\ensuremath{\\mathcal{K}}$. The two-photon matrix corresponding to the interferometer in \\cref{fig:SQZ_mm_cavities} is given below for the lossless interferometer that is perfectly on resonance.\n\\begin{align}\n  \\Tmat{H}_\\ensuremath{\\text{R}}(\\Omega) &\\simeq\n  \\begin{bmatrix}\n    \\ensuremath{\\mathfrak{r}}(\\Omega) & 0\\\\\n    \\ensuremath{\\mathcal{K}}(\\Omega) & \\ensuremath{\\mathfrak{r}}(\\Omega)\n  \\end{bmatrix},\n  &\n  \\ensuremath{\\mathfrak{r}}(\\Omega) &\\simeq\n\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega},\n                 &\\Tmat{\\Lambda}_\\ensuremath{\\text{R}} &= \\Tmat{0}\n \n \n \n \n \n \n \n \n \n \n \n \n \n\\end{align}\nIn the ideal lossless case, the input and output paths also have perfect efficiency $\\eta_\\ensuremath{\\text{I}} \\simeq 1$ with $\\Tmat{H}_\\ensuremath{\\text{I}}(\\Omega) = \\eta_\\ensuremath{\\text{I}}\\Tmat{1},\\; \\Tmat{\\Lambda}_\\ensuremath{\\text{I}}(\\Omega) = \\sqrt{1-\\eta_\\ensuremath{\\text{I}}}\\Tmat{1}$ and similarly for the output. These can be used to compute $\\Tmat{H}$ and $\\vec{m}$.\n\\begin{align}\n  m_q &= \\cos(\\zeta)\\ensuremath{\\mathcal{K}}(\\Omega) + \\sin(\\zeta)\\ensuremath{\\mathfrak{r}}(\\Omega)\n  ,&\n  m_p &= \\cos(\\zeta)\\ensuremath{\\mathfrak{r}}(\\Omega)\n\\end{align}\nThe equations above maintain the correct phase information for this ideal case analysis. Interestingly, $\\ensuremath{\\mathcal{K}}$ and $\\ensuremath{\\mathfrak{r}}(\\Omega)$ have different magnitude responses resulting from different factors of $\\gamma_\\ensuremath{\\text{A}} \\pm i\\Omega$, yet their phase response is the same. This Kramers-Kronig coincedence ensures $\\Xi(\\Omega)=0$ as long as the $\\chi(\\Omega)$ contribution to $\\ensuremath{\\mathcal{K}}(\\Omega)$ is purely real. Thus, lossy mechanics will cause QRPN to dephase injected squeezing. This will not happen to any meaningful level for LIGO, but is noteworthy for optomechanics experiments operating on mechanical resonance.\n\nThe $\\vec{m}$ above also includes the effect of the readout angle. For $\\zeta=0$, it recovers \\crefrange{eq:optical_gain_g}{eq:Kchi_standard}. More generally, it gives\n\\begin{align}\n  \\Gamma(\\Omega) &= |\\cos(\\zeta)\\ensuremath{\\mathcal{K}}(\\Omega)|^2 + \\sin(2\\zeta)|\\ensuremath{\\mathcal{K}}(\\Omega)| + 1\n  \\\\\n  \\theta(\\Omega) &= \\arctan\\big(|\\ensuremath{\\mathcal{K}}(\\Omega)| - \\tan(\\zeta)\\big)\n\\end{align}\nThe exact expressions above can be simplified to better relate them to the LIGO data. Firstly, the squeezing angle is modified to be 0 at high frequencies, to match the conventions of the data. This modified angle is $\\theta'(\\Omega) = \\theta(\\Omega) - \\theta(\\Omega{\\gg}\\gamma_\\ensuremath{\\text{A}})$. Secondly, small shifts of the homodyne angle are linearized.\n\\begin{align}\n  \\Gamma'(\\Omega) &\\approx\n                  \n                  \n                   \\big(1 - |K(\\Omega)|\\big)^2 + 2\\big(1 + \\zeta\\big)|K(\\Omega)|\n \n  \\\\\n  \\theta'(\\Omega) &\\approx \\arctan\\big(|\\ensuremath{\\mathcal{K}}(\\Omega)|) - \\zeta \\frac{|\\ensuremath{\\mathcal{K}}(\\Omega)|^2}{1 + |\\ensuremath{\\mathcal{K}}(\\Omega)|^2}\n\\end{align}\nThe linearized $\\Gamma'$ shows that , when $\\zeta=-13^\\circ=-0.23$, for frequencies near $\\Omega_\\ensuremath{\\text{sql}}$, $\\ensuremath{\\mathcal{K}}(\\Omega_\\ensuremath{\\text{sql}}) \\simeq 1$, the interferometer quantum noise is reduced by about 23\\% with respect to a nominal $\\xi=0$ readout. This change is shown in the blue vs. grey plotted data for the Livingston loss plot in \\cref{fig:data_Q} of $1-\\eta$. There, $\\eta$ changes as the $\\Gamma$ model changes since only $\\eta\\Gamma$ can be measured due to subtracting an unsqueezed reference dataset. The $23\\%$ noise reduction corresponds to approximately 1dB improvement from pondermotive quantum correlations. The angle formula above indicates that for frequencies $\\Omega\\lesssim\\Omega_\\ensuremath{\\text{sql}}$, the local oscillator also adds some additional shift to $\\theta$ at low frequency, which is also observed in the LLO angle fits.\n\nThis analysis gives an example of how the derivations of this section are applied to extend the existing ideal interferometer models towards the real instruments. Exact models including more optical physics are yet more analytically opaque, but give a more complete complete picture if implemented numerically. \\Cref{sec:matrix_coupled_cavity} shows the full matrix solution, including the cavities, to recover these equations while also handling cavity length offset detunings. It also includes transverse modal mismatch in its description. \\Cref{sec:radiation_pressure_calc} gives the minimal extension of this ideal lossless interferometer to incorporate transverse mismatch, showing how the noise gain, $\\Gamma$,  and rotation angle $\\theta$ change specifically from mismatch. In particular, it shows that relating a measurement of $\\Omega_\\ensuremath{\\text{sql}}$ using squeezing back to the arm power $P_\\ensuremath{\\text{A}}$ using \\cref{eq:optical_gain_g} and  \\cref{eq:Kchi_standard} is biased by transverse mismatch.\n\n\\section{Cavity Modeling and Metrics}\n\\label{sec:modeling}\n\nThe previous section derives the general form of the squeezing metrics using matrices of the two photon formalism.\nFor passive systems, the optical transfer function, $\\ensuremath{\\mathfrak{h}}(\\Omega)$, given at every sideband frequency, is sufficient to characterize the response to externally-supplied squeezing. The conceptual simplification and restriction to using only transfer functions is useful for interferometer modeling. Transfer functions, being complex scalar functions, are suitable for analytic calculations of cavity response and can be decomposed into rational function forms to inspect the rational roots, zeros and poles, and the overall gain of the response.\n\nThis section analyzes the coupled cavity system of the interferometer, depicted in \\cref{fig:SQZ_mm_cavities}, through its decomposition into roots. More complicated transverse modal simulations analyze the frequency response of the interferometer cavities for each optical mode to every other mode. Modal simulations thus output a matrix of transfer functions, $\\mat{H}(\\Omega)$, which is difficult to analytically manipulate, but \\cref{sec:modeling_TMM} shows how it can be projected back to a single scalar transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$ and further simplified into the squeezing metrics.\n\nThe transfer function techniques of this section elucidate new squeezing results by avoiding the combined complexity of both two-photon and modal vector spaces. The full generality of two-photon matrices is only required for active systems that introduce internal squeezing, parametric gain or radiation pressure. Passive systems have the property that $\\hat{q}_\\text{out}$, $\\hat{p}_\\text{out}$ also obey the expectations of \\cref{eq:vacuum_expectations,eq:vacuum_expectations2}. Following the notation of \\cref{sec:derivations}, this results in the following condition.\n\\begin{align}\n  \\Tmat{1} &= \\Tmat{H}\\Tmat{H}^\\dagger\n  +\\sum_{\\ensuremath{{\\,\\mu}}}\\Tmat{T}_{\\ensuremath{{\\,\\mu}}}\\Tmat{T}^\\dagger_{\\ensuremath{{\\,\\mu}}}\n             \\label{eq:passivity_condition}\n\\end{align}\nAdditionally, $\\Gamma = 1$ is implied by that condition.\nWithout parametric gain, photons at upper and lower sideband frequencies are never correlated by a passive system. By the passivity condition and manipulations between sideband and quadrature basis, \\cref{sec:passive_derivations} derives the squeezing metrics purely in terms of the transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$.\n\\begin{align}\n  \\theta(\\Omega)\n  &=\n    \\big(\\arg\\big(\\ensuremath{\\mathfrak{h}}(+\\Omega)\\big) + \\arg\\big(\\ensuremath{\\mathfrak{h}}(-\\Omega)\\big)\\big)\/2\n    \\label{eq:theta_passive}\n  \\\\\n  \\eta(\\Omega)\n  &=\n    \\big( |\\ensuremath{\\mathfrak{h}}(+\\Omega)|^2 + |\\ensuremath{\\mathfrak{h}}(-\\Omega)|^2\\big)\/2\n    \\label{eq:eta_passive}\n  \\\\\n  \\Xi(\\Omega)\n  &=\n    \\big( |\\ensuremath{\\mathfrak{h}}(+\\Omega)| - |\\ensuremath{\\mathfrak{h}}(-\\Omega)| \\big)^2\/4\\eta\n    \\label{eq:Xi_passive}\n\\end{align}\nQuantum filter cavities are a method to use an entirely optical system to reduce the radiation pressure associated with squeezed light\\cite{McCuller-PRL20-FrequencyDependentSqueezing, Zhao-PRL20-FrequencyDependentSqueezed}. They are passive cavities, and provide a useful example to study these squeezing metric formulas.\nThe first of these, \\cref{eq:theta_passive}, is a well-established formula for the filter cavity design. It indicates that for cavities with an asymmetric phase response, usually due to being off-resonance or ``detuned'', that the squeezing field picks up a frequency-dependent quadrature rotation. Such a rotation applied in $\\Tmat{H}_\\ensuremath{\\text{I}}$ can be generated by a cavity with transfer function $\\ensuremath{\\mathfrak{h}}_\\ensuremath{\\text{I}}(\\Omega)$ before the interferometer. This cavity rotation compensates the $\\theta(\\Omega)$ due to $\\Tmat{H}_\\ensuremath{\\text{R}}$. Together, the product $\\Tmat{H}_\\ensuremath{\\text{R}}\\Tmat{H}_\\ensuremath{\\text{I}}$ has $\\theta(\\Omega) = 0$, allowing a single choice of squeezing angle $\\phi$ to optimize $N(\\Omega)$ at all frequencies.\n\nThe formulas \\cref{eq:eta_passive} and \\cref{eq:Xi_passive} indicate how losses represented in a transfer function translate to loss-like and dephasing degradations from cavity reflections. For filter cavities, these degradations are investigated in \\cite{Kwee-PRD14-DecoherenceDegradation}, but this new factorization into scalar functions clarifies the discussion. The efficiency $\\eta(\\Omega)$ behaves as expected, an average of the loss in each sideband. The form of $\\Xi(\\Omega)$ is less expected, showing how the combination of loss and detuning in filter cavities creates noise that scales with the squeezing level. A simple picture for the dephasing effect is that when optical quadratures are squeezed, the noise power in both upper and lower sidebands is strictly increased. The sideband correlations allow the increased noise to subtract away for squeezed quadrature measurements but to add for measurements in the anti-squeezed quadrature. The asymmetric losses of detuned cavities preserve the noise increase on one sideband, while degrading the correlations. This ruins the subtraction for the squeezed quadrature and introduces $\\Xi(\\Omega) > 0$. This source of noise is $e^{\\pm r}$ squeezing level dependent but entirely unrelated to fluctuations of the squeezing phase $\\ensuremath{\\phi_{\\text{rms}}}$.\n\n\\subsection{Single Cavity Model for Interferometers}\n\\label{sec:modeling_cavities}\n\nThis section analyzes the effect of the interferometer cavities on squeezing. It starts by considering an interferometer with only one cavity - either in the Michelson arms or from a mirror at the output port, but not both. It represents the first generation of GW detectors. This single cavity scenario is also similar to a quantum filter cavity, in the regime of small detuning\\cite{Komori-PRD20-DemonstrationAmplitude, Corbitt-PRD04-OpticalCavities, Khalili-PRD08-IncreasingFuture}. Advanced LIGO uses a coupled cavity system, depicted in \\cref{fig:SQZ_mm_cavities}, and the transfer function equations for the reflection from the resonant sideband extraction cavity is extended in the next subsection to include the loss and detuning of the additional cavity.\n\nA single cavity operated near resonance may be described using the scale parameters of the cavity bandwidth $\\gamma_\\ensuremath{\\text{A}}$, loss rate $\\lambda_\\ensuremath{\\text{A}}$ and detuning frequency $\\delta_\\ensuremath{\\text{A}}$, which are computed from the physical parameters of the mirror transmissivity $T_\\ensuremath{\\text{a}}$, round-trip loss $\\Lambda_\\ensuremath{\\text{a}}$, cavity length $L_\\ensuremath{\\text{a}}$, and microscopic length detuning $\\Delta L_\\ensuremath{\\text{a}}$.\n\\begin{align}\n  \\gamma_\\ensuremath{\\text{A}}  &= \\frac{c T_\\ensuremath{\\text{a}}}{4 L_\\ensuremath{\\text{a}}}\n  & \n  \\lambda_\\ensuremath{\\text{A}} &= \\frac{c \\Lambda_\\ensuremath{\\text{a}}}{4 L_\\ensuremath{\\text{a}}}\n  &\n  \\delta_\\ensuremath{\\text{A}}  &= -ck\\frac{\\Delta L_\\ensuremath{\\text{a}}}{L_\\ensuremath{\\text{a}}}\n                 \\label{eq:cavity_relations}\n\\end{align}\nThese relations are accurate in the high-finesse limit $T_\\ensuremath{\\text{a}} \\ll 1$, and combine to give the transfer function of the frequency-dependent cavity reflection.\n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_1(\\Omega)\n \n \n  &\\approx\n    -\\frac{(\\gamma_\\ensuremath{\\text{A}} - \\lambda_\\ensuremath{\\text{A}}) - i(\\Omega - \\delta_\\ensuremath{\\text{A}})}{(\\gamma_\\ensuremath{\\text{A}} + \\lambda_\\ensuremath{\\text{A}}) + i(\\Omega - \\delta_\\ensuremath{\\text{A}})}\n    \\label{eq:single_cavity}\n\\end{align}\nNotably, the sign of the reflectivity for a high-finesse cavity on resonance $\\ensuremath{\\mathfrak{r}}_1(\\Omega \\ll \\gamma_\\ensuremath{\\text{A}})=-1$,\nbut outside of resonance $\\ensuremath{\\mathfrak{r}}_1(\\Omega \\gg \\gamma_\\ensuremath{\\text{A}})=1$. This sign determines constructive or destructive interference in transverse mismatch loss analyzed in the next section. The internal losses of the cavity $\\Lambda_\\ensuremath{\\text{a}}$ become cavity-enhanced in the reflection, causing squeezing to experience losses of $\\Lambda_\\ensuremath{\\text{A}}$.\n\\begin{align}\n  \\Lambda_\\ensuremath{\\text{A}}\n  &\\equiv 1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1\\\\ |\\Omega| \\ll \\gamma_\\ensuremath{\\text{A}}}}\n \n \n \n  \\approx \\frac{4\\lambda_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}}\n                            \\approx \\frac{4\\Lambda_\\ensuremath{\\text{a}}}{T_\\ensuremath{\\text{a}}}\n\\end{align}\nFurthermore, detuning the cavity off of resonance causes a rotation of reflected squeezing. For small detunings, the rotation can be approximated.\n\\begin{align}\n  \\theta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1\\\\ k\\Delta L_\\ensuremath{\\text{a}} \\ll T_\\ensuremath{\\text{a}}}}\n  &\\approx\n  \\frac{2\\delta_\\ensuremath{\\text{A}}\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\approx\n   -k \\Delta L_\\ensuremath{\\text{a}}\\frac{8}{T_\\ensuremath{\\text{a}}}\\frac{\\gamma_\\ensuremath{\\text{A}}^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n\\end{align}\nFluctuations in $\\Delta L_\\ensuremath{\\text{a}}$ or $\\delta_\\ensuremath{\\text{A}}$ lead to a phase noise analogous to $\\ensuremath{\\phi_{\\text{rms}}}$, but with the frequency dependence from the above equation\\cite{Kwee-PRD14-DecoherenceDegradation}. Additionally, losses in the cavity lead to intrinsic dephasing $\\Xi(\\Omega)$, calculated below. This calculation is valid at any detuning $\\delta_\\ensuremath{\\text{A}}$, even those larger than the cavity width $\\gamma_\\ensuremath{\\text{A}}$. Its validity only requires being in the overcoupled cavity regime, where losses $\\lambda_\\ensuremath{\\text{A}} \\lesssim \\gamma_\\ensuremath{\\text{A}}\/2$.\n\\begin{align}\n  \\Xi(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1}}\n  &\\approx\n    \\left(\n  \\frac{4\\gamma_\\ensuremath{\\text{A}}\\lambda_\\ensuremath{\\text{A}}\\delta_\\ensuremath{\\text{A}}\\Omega}{\\left( \\gamma_\\ensuremath{\\text{A}}^2 + (\\Omega - \\delta_\\ensuremath{\\text{A}})^2 \\right)\\left( \\gamma_\\ensuremath{\\text{A}}^2 +  (\\Omega + \\delta_\\ensuremath{\\text{A}})^2 \\right)}\\right)^2\n\\end{align}\nWhen plotted, this expression for $\\Xi(\\Omega)$ has a Lorentzian-like profile, with a peak at $\\Omega_{\\Xi\\text{max}}$. Above $|\\delta_\\ensuremath{\\text{A}}| \\gtrsim \\gamma_\\ensuremath{\\text{A}}$, where the cavity resonance acts entirely either on upper or lower sidebands, the peak dephasing reaches a maximum. At small detunings, $|\\delta_\\ensuremath{\\text{A}}| \\lesssim \\gamma_\\ensuremath{\\text{A}}$, the sideband loss asymmetry scales with the detuning.\n\\begin{align}\n  \\Omega_{\\Xi{\\text{max}}} &\\approx \\sqrt{\\gamma^2_\\ensuremath{\\text{A}}\/4 + \\delta^2 },\n  & \\Xi_{\\text{max}} \n  &\\approx \n    \\frac{\\lambda^2_\\ensuremath{\\text{A}}}{\\gamma^2_\\ensuremath{\\text{A}}}{\\cdot}\\frac{8\\delta^2_\\ensuremath{\\text{A}}}{5\\gamma^2_\\ensuremath{\\text{A}} + 8\\delta^2_\\ensuremath{\\text{A}}}\n    \\label{eq:detuning_dephasing}\n\\end{align}\nThis single cavity model is also useful for analyzing quantum filter cavities and, like the $\\Xi$ metric itself, these peak values have not been calculated in past frequency-dependent squeezing work. Conventional squeezing phase uncertainty, $\\ensuremath{\\phi_{\\text{rms}}}$, can be cast into the units RMS radians of phase deviation, leading to the noise suppression limit for squeezing $S(\\Omega)\\ge 2\\ensuremath{\\phi_{\\text{rms}}}$, by \\cref{eq:N_limit}. For highly detuned cavities such as quantum filter cavities, $\\sqrt{\\Xi(\\Omega)} \\approx \\Lambda_\\ensuremath{\\text{fc}} \/ T_\\ensuremath{\\text{fc}}$. Using the parameters of the A+ filter cavity \\cite{Whittle-PRD20-OptimalDetuning}, $\\Lambda_{\\ensuremath{\\text{fc}}}{\\approx}60\\text{ppm}$ and $T_{\\ensuremath{\\text{fc}}} {=} 1000 \\text{ppm}$ indicates that optical dephasing is of order $60\\text{mRad}$. For an optimal filter cavity with low losses\\cite{Whittle-PRD20-OptimalDetuning}, this dephasing maximum occurs at $\\Omega_{\\Xi{\\text{max}}} = \\sqrt{5\/8}\\Omega_{\\text{SQL}}$. This level of dephasing is commensurate with or even exceeds the expected residual phase uncertainty $\\ensuremath{\\phi_{\\text{rms}}} < 30\\text{mRad}$.\n\nOptical dephasing from the LIGO interferometer cavities is not expected to be large for as they are stably operated on-resonance; however, detuned configurations of LIGO\\cite{Ganapathy-PRD21-TuningAdvanced} are limited by dephasing from the unbalanced response and optical losses in the signal recycling cavity.\n\n\\subsection{Double Cavity Model for Interferometers}\n\nFor interferometers using resonant sideband extraction, like LIGO, the arm cavities have a length $L_\\ensuremath{\\text{a}}$, an input transmissivity of $T_\\ensuremath{\\text{a}}$, and are each on resonance to store circulating laser power. The signal recycling cavity (SRC) has a length $L_\\ensuremath{\\text{s}}$ and a signal recycling mirror (SRM) of transmissivity $T_\\ensuremath{\\text{s}}$. The SRM forms a cavity with respect to the arm input mirror that resonantly increases the effective transmissivity experienced by the arm cavities to be larger than $T_\\ensuremath{\\text{a}}$, broadening the signal bandwidth. While the SRC is resonant with respect to the arm input mirror, it is anti-resonant with respect to the arm cavity, due to the negative sign of \\cref{eq:single_cavity}. The anti-resonance leads to the opposite sign in the reflection transfer function below, \\cref{eq:double_cavity}. The discrepancy in resonance vs. anti-resonance viewpoints is why the signal recycling cavity is also called the signal extraction cavity in GW literature.\n\nThe coupled cavity forms two bandwidth scales for the system, $\\gamma_\\ensuremath{\\text{A}}$, the modified effective arm bandwidth, and $\\gamma_\\ensuremath{\\text{S}}$, the bandwidth of the signal recycling cavity. The arm and signal cavities have their respective round-trip losses $\\Lambda_\\ensuremath{\\text{a}}$ and $\\Lambda_\\ensuremath{\\text{s}}$, as well as length detunings $\\Delta L_\\ensuremath{\\text{a}}$, $\\Delta L_\\ensuremath{\\text{s}}$. In practice, the arm length detuning is expected to be negligible to maximize the power storage, but the signal recycling cavity detuning can be varied by modifying a bias in the control system that stabilizes $\\Delta L_\\ensuremath{\\text{s}}$.\n\nThe scale parameters for the cavity transfer function are approximated from the physical parameters:\n\\begin{align}\n  u_\\ensuremath{\\text{a}} &= 1 - \\sqrt{1 - T_\\ensuremath{\\text{a}}}\n  &\n  u_\\ensuremath{\\text{s}} &= 1 - \\sqrt{1 - T_\\ensuremath{\\text{s}}}\n           \\label{eq:u_factors}\n  \\\\\n  \\gamma_\\ensuremath{\\text{A}}  &= \\frac{c u_\\ensuremath{\\text{a}}}{2 L_\\ensuremath{\\text{a}}} \\cdot\\frac{2 - u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{s}}}\n  & \n    \\gamma_\\ensuremath{\\text{S}}   &= \\frac{c u_\\ensuremath{\\text{s}}}{2 L_\\ensuremath{\\text{s}}}\n                    \\label{eq:double_cavity_gamma}\n  \\\\\n  \\lambda_\\ensuremath{\\text{A}} &= \\frac{c}{L_\\ensuremath{\\text{a}}} \\left(\\frac{\\Lambda_\\ensuremath{\\text{a}}}{4} - \\frac{u_\\ensuremath{\\text{a}} \\Lambda_\\ensuremath{\\text{s}}}{2 u_\\ensuremath{\\text{s}}^2} + \\frac{u_\\ensuremath{\\text{a}} \\Lambda_\\ensuremath{\\text{s}}}{4 u_\\ensuremath{\\text{s}}}\\right)\n  &\n    \\lambda_\\ensuremath{\\text{S}}  &= \\frac{c\\Lambda_\\ensuremath{\\text{s}}}{4L_\\ensuremath{\\text{s}}}\\left(1 - \\frac{u_\\ensuremath{\\text{s}}}{2} \\right)\n  \\\\\n  \\delta_\\ensuremath{\\text{A}} &= -ck\\frac{\\Delta L_\\ensuremath{\\text{a}}}{L_\\ensuremath{\\text{a}}}  - \\frac{\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{S}}}\\delta_\\ensuremath{\\text{S}}\n  &\n  \\delta_\\ensuremath{\\text{S}} &= -ck\\frac{\\Delta L_\\ensuremath{\\text{s}}}{L_\\ensuremath{\\text{s}}} \n           \\label{eq:delta_factors}\n\\end{align}\nThese approximations are valid for the LIGO mirror parameters, see \\cref{tab:LIGO_params}, and model the loss and detuning to {\\color{red} $5\\%$} accuracy. They are derived in \\cref{sec:scalar_coupled_cavity} from Taylor expansions, solving roots, and selectively removing terms. Expanding in the $u$ factors of \\cref{eq:u_factors} gives lower error than expanding in transmissivity or reflectivity factors directly, due to the low effective finesse of the coupled cavity system and the high transmissivity of the SRM.\n\\begin{table}\n\\centering\n\\caption{Parameters of LIGO for data fitting and modeling}\n\\begin{ruledtabular}\n\\begin{tabular}{l|c|cc}\nParameter & Symbol & LLO Value & LHO Value \\\\\n\\hline\narm input transmissivity & $T_\\ensuremath{\\text{a}}$&0.0148 & 0.0142 \\\\\narm length & $L_\\ensuremath{\\text{a}}$ & \\multicolumn{2}{c}{3995 m}\\\\\narm round-trip loss & $\\Lambda_{\\ensuremath{\\text{a}}}$ & \\multicolumn{2}{c}{${\\sim}80$ppm}\\\\\nSRM transmission& $T_\\ensuremath{\\text{s}}$&\\multicolumn{2}{c}{0.325}  \\\\\nSRC length & $L_\\ensuremath{\\text{s}}$ &\\multicolumn{2}{c}{55 m}\\\\\nSRC round-trip loss & $\\Lambda_\\ensuremath{\\text{s}}$ & \\multicolumn{2}{c}{$\\lesssim 3000$ppm}\\\\\n  \\hline\nMirror mass& $m$ &  \\multicolumn{2}{c}{39.9kg}\\\\\nArm power& $P_\\ensuremath{\\text{A}}$ & $200{\\pm} 10\\text{ kW}$  &  $190{\\pm}10\\text{ kW}$ \\\\\nQRPN crossover& $\\Omega_{\\ensuremath{\\text{sql}}}$ & $2\\pi\\cdot 33$ Hz & $2\\pi\\cdot 30$ Hz \\\\\n  \\hline\narm signal band& $\\gamma_\\ensuremath{\\text{A}}$& $2\\pi\\cdot450$ Hz &  $2\\pi\\cdot410$ Hz \\\\\nSRC band& $\\gamma_\\ensuremath{\\text{S}}$& \\multicolumn{2}{c}{$2\\pi\\cdot80$kHz} \\\\\nArm length detuning & $\\Delta L_{\\ensuremath{\\text{a}}}$ & \\multicolumn{2}{c}{0nm}\\\\\nSRC length detuning & $\\Delta L_{\\ensuremath{\\text{s}}}$ & 1.02nm & 1.23nm\\\\\n  \\hline\narm resonant loss& $\\Lambda_\\ensuremath{\\text{A}}$ & \\multicolumn{2}{c}{$\\lesssim 2000$PPM}\\\\\nSRC resonant loss& $\\Lambda_\\ensuremath{\\text{S}}$ & \\multicolumn{2}{c}{${\\sim} 1\\%$ to $3\\%$}\\\\\narm\/SRC detuning& $\\delta_{\\ensuremath{\\text{A}}}$ & $2\\pi\\cdot 10.1$Hz & $2\\pi\\cdot 11.2$Hz \\\\\n  \\hline\nInjected squeezing & $e^{\\pm 2r}$ & {${\\pm}9.7\\text{ dB}$}  &  {${\\pm}8.7\\text{ dB}$} \\\\\nSQZ-OMC mismatch& $\\Upsilon_{\\ensuremath{\\text{O}}}$ & $2\\%$ & $4\\%$ \\\\\nReflection mismatch (fit)& $\\Upsilon_{\\ensuremath{\\text{R}}}$ & $12\\%$ & $35\\%$ \\\\\nAdditional SQZ loss (fit)& $\\Lambda_{\\text{IO}}=1{-}\\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{O}}$ & $31\\%$ & $34\\%$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tab:LIGO_params}\n\\end{table}\nThe scale factors result in the following reflectivity transfer function.\n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_2(\\Omega)\n  &=\n    \\frac{(\\gamma_\\ensuremath{\\text{A}} - \\lambda_\\ensuremath{\\text{A}}) - i(\\Omega - \\delta_\\ensuremath{\\text{A}})}{(\\gamma_\\ensuremath{\\text{A}} + \\lambda_\\ensuremath{\\text{A}}) + i(\\Omega - \\delta_\\ensuremath{\\text{A}})}\n   \n    {\\cdot}\\frac{(\\gamma_\\ensuremath{\\text{S}} - \\lambda_\\ensuremath{\\text{S}}) - i(\\Omega - \\delta_\\ensuremath{\\text{S}})}{(\\gamma_\\ensuremath{\\text{S}} + \\lambda_\\ensuremath{\\text{S}}) + i(\\Omega - \\delta_\\ensuremath{\\text{S}})}\n    \\label{eq:double_cavity}\n\\end{align}\nNotably, this reflectivity is $\\ensuremath{\\mathfrak{r}}_2(\\pm\\Omega \\ll \\gamma)=1$ and $\\ensuremath{\\mathfrak{r}}_2(\\gamma_\\ensuremath{\\text{A}} \\ll \\pm\\Omega \\ll \\gamma_\\ensuremath{\\text{S}})=-1$ which has an opposite overall sign to that of single cavity interferometers. On reflection, the squeezing field experiences different cavity enhanced losses depending on the frequency.\n\\begin{align}\n  \\Lambda_{\\ensuremath{\\text{S}}} &\\equiv \n1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ \\gamma_\\ensuremath{\\text{A}} \\ll |\\Omega| \\ll \\gamma_\\ensuremath{\\text{S}}}}\n&&\\hspace{-3em}\\approx \\frac{2 - u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{s}}}\\Lambda_\\ensuremath{\\text{s}}\n  \\\\\n  \\Lambda_{\\ensuremath{\\text{A}}} &\\equiv \n1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ |\\Omega| \\ll \\gamma_\\ensuremath{\\text{A}}}}\n&&\\hspace{-3em}\\approx \\frac{4\\lambda_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}}+\\Lambda_\\ensuremath{\\text{s}} \\approx \\frac{u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{a}}}\\Lambda_\\ensuremath{\\text{a}}\n\\end{align}\nThe dataset of \\cref{sec:experiment} shows frequency dependent losses, where the loss increases $12\\%$ for LLO and $33\\%$ for LHO. Assuming the losses result from the equations above, this corresponds to round-trip losses in the LIGO signal recycling cavities, $\\Lambda_\\ensuremath{\\text{s}}$, of $1.1\\%$ to $3.2\\%$, which is not realistic. Most mechanisms that introduce loss in the SRC would also introduce it into the power recycling cavity in an obvious manner. The current power recycling factors exclude this possibility, and independent measurements of $\\gamma_\\ensuremath{\\text{A}}$ bound $\\Lambda_\\ensuremath{\\text{s}}$ losses to ${\\le} 3000$ppm. The next section investigates how transverse mismatch can result in this level of observed losses.\n\nIn addition to the losses, \\cref{eq:double_cavity} can be used to determine the cavity-induced squeeze state rotation from the detuning of the signal recycling cavity.\n\\begin{align}\n  \\theta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ \\delta_\\ensuremath{\\text{S}} \\ll \\gamma_\\ensuremath{\\text{S}} \\\\ \\Delta L_\\ensuremath{\\text{a}} = 0}}\n  &\\approx\n  \\frac{2\\delta_\\ensuremath{\\text{A}}\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    +\n  \\frac{2\\delta_\\ensuremath{\\text{S}}\\gamma_\\ensuremath{\\text{S}}}{\\gamma_\\ensuremath{\\text{S}}^2 + \\Omega^2}\n \n \n \n \n \n \n \n \n    \\\\\n  &\\approx\n    k\\Delta L_\\ensuremath{\\text{s}}\\frac{4}{u_\\ensuremath{\\text{s}}}\n    \\left(\n    \\frac{\\gamma^2_\\ensuremath{\\text{S}}}{\\gamma_\\ensuremath{\\text{S}}^2 + \\Omega^2}\n    -\n    \\frac{\\gamma^2_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n \\right)\n    \\label{eq:double_cav_theta}\n \n\\end{align}\nThis indicates the surpising result that detuning the SRC length does not affect the squeezing within the effective\narm bandwidth to first order. Instead, it adds the squeezing rotation in the middle band above the arm bandwidth but below the SRC bandwidth. In the data analysis of \\cref{sec:experiment} and \\cref{fig:data_Q}, the convention for $\\theta(\\Omega)$ is set to be 0 at ``high'' frequencies in this intermediate cavity band, in which case it appears to cause a rotation around $\\gamma_\\ensuremath{\\text{A}}$. This convention used for the data corresponds to omitting the first, $\\gamma_\\ensuremath{\\text{S}}$-scaled term of \\cref{eq:double_cav_theta}.\n\n\\section{Transverse Mismatch Model}\n\\label{sec:modeling_TMM}\n\nSqueezing, as it is typically implemented for GW interferometers, modifies the quantum states in a single optical mode. For LIGO, this mode is the fundamental Gaussian beam resonating in the parametric amplifier cavity serving as the squeezed state source. The cavity geometry establishes a specific complex Gaussian beam parameter that defines a modal basis decomposition into Hermite Gaussian (HG) or Laguerre Guassian (LG) modes. That basis is transformed and redefined during the beam propagation through free space and through telescope lenses on its way to and from the interferometer. The cavities of the interferometer each define their own resonating beam parameters and respective HG or LG basis of optical modes.\n\nIn practice, the telescopes propagating the squeezed beam to and from the interferometer imperfectly match the complex beam parameters, so basis transformations must occur that mix the optical modes. The mismatch of complex beam parameters is called here ``transverse mismatch''. Non-fundamental HG or LG transverse modes do not enter the OPA cavity, and so carry standard vacuum rather than squeezing. Basis mixing from transverse mismatch thus leads to losses; however, unlike typical losses such basis transformations are coherent and unitary, which leads to the constructive and destructive interference effects studied in this section.\n\nThe interferometer transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$ is a single scalar function representing the frequency dependence of the squeezing channel from source to readout, but the optical fields physically have many more channels. The cavities visited by the squeezed states each have a transfer function matrix in their local basis, given by $\\mat{H}_{\\ensuremath{\\text{I}}}$, $\\mat{H}_{\\ensuremath{\\text{R}}}$, $\\mat{H}_{\\ensuremath{\\text{O}}}$ for the squeezing input, interferometer reflection, and system output respectively. The diagonals of these matrices indicate the frequency response during traversal for every transverse optical mode. The off-diagonals represent the coupling response between modes that result from scattering and optical wavefront errors.\n\nBetween the cavities, $\\mat{U}$ matrices represent the basis transformations due to transverse mismatch. Here, $\\vec{e}_{\\text{sqz}}$, $\\vec{e}_{\\text{read}}$ are basis vectors for projecting from the single optical mode of the emitted squeezed states and to the single mode of the optical homodyne readout defined by its local oscillator field.\n\\begin{align}\n\\ensuremath{\\mathfrak{h}}(\\Omega) &= {\\vec{e}_{\\text{read}}}^{\\,\\dagger} \\mat{H}(\\Omega)\\vec{e}_{\\text{sqz}}\n                  \\label{eq:TF_projection}\n  \\\\\n\\mat{H}(\\Omega) &=\\mat{H}_\\ensuremath{\\text{O}}\\mat{U}_{\\ensuremath{\\text{O}}, \\ensuremath{\\text{R}}}\\mat{H}_\\ensuremath{\\text{R}}\\mat{U}_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{I}}}\\mat{H}_\\ensuremath{\\text{I}}\n                  \\label{eq:TF_matrix}\n\\end{align}\n\\Cref{eq:TF_projection} and \\cref{eq:TF_matrix} give the general, basis independent, form to compose the effective transfer function for the squeezed field using a multi-modal simulation of a passive interferometer. This is complicated in the general case, but the following analysis develops a simpler, though general, model for how transverse mismatch manifests as squeezing losses.\n\nTransverse mismatch is often physically measured as a loss of coupling efficiency, $\\ensuremath{\\Upsilon}$, of an external Gaussian beam to a cavity measured as a change in optical power. Realistically, more than two transverse modes are necessary to maintain realistic and unitary basis transformations, but, for small mismatches of complex beam parameters, $\\Upsilon < 10\\%$. In this case, only the two lowest modes in the Laguerre-Gauss basis have significant cross-coupling. For low losses, the fundamental Gaussian mode, LG0, loses most of its power to the radially symmetric LG1 mode, assuming low astigmatism and omitting azimuthal indices. This motivates the following simplistic two-mode model to analyze the effect of losses on $\\ensuremath{\\mathfrak{h}}(\\Omega)$. In this model $\\mat{U}$ gives the unitary, though not perfectly physical, basis transformation:\n\\begin{align}\n  \\mat{U}(\\ensuremath{\\Upsilon}, \\psi, \\phi)\n  &\\equiv\n    e^{i\\phi}\\begin{bmatrix}\n    \\sqrt{\\ensuremath{1{-}\\MM}} & -e^{i\\psi}\\sqrt{\\ensuremath{\\Upsilon}}\n    \\\\\n    e^{-i\\psi}\\sqrt{\\ensuremath{\\Upsilon}} & \\sqrt{\\ensuremath{1{-}\\MM}}\n  \\end{bmatrix}\n                           \\label{eq:U_def}\n\\end{align}\nThis unitary transformation includes two unknown phase parameters. The first, $\\psi$, is the phase of the mismatch, which characterizes whether beam size error or wavefront phasing error dominates the overlap integral of the external LG0 and cavity LG1 modes. The second, $\\phi$, is the mismatch phase error from the external LG0 to the cavity LG0. The $\\phi$ term is included above to fully express the unitary freedom of $\\mat{U}$, but is indistinguishable from path length offsets, physically controlled to be $0$, and ignored in further expressions.\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_chain, \n  caption={\n    Propagation of the squeezed beam and unsqueezed higher order transverse beam modes from source to readout. The stages (a)-(d) correspond to the components in \\cref{fig:SQZ_mm_IFO}, depicting the matrix math of \\crefrange{eq:H_refl}{eq:h_chain}. $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ represents the transverse mismatch loss of the squeezing to interferometer, and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$ is the mismatch of the squeezing to readout via the output mode cleaner. These mismatches cause beamsplitter-like mixing between the LG0 and LG1+ modes through \\cref{eq:U_def}. $\\psi_\\ensuremath{\\text{I}}$,  $\\psi_\\ensuremath{\\text{O}}$,  $\\psi_{\\text{G}}$ are unmeasured phasing terms of the interferometer and output mismatch and of the Gouy-phase advance from the beam propagating to the output mode cleaner.\n  },\n  label=SQZ_mm_chain,\n}\n\nIn the case of a GW interferometer with an output mode cleaner, there are two mode matching efficiencies expressed as individually measurable parameters. The first is the coupling efficiency (in power) and phasing associated between the squeezer and interferometer $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}, \\ensuremath{\\psi_\\text{I}}$. The second are parameters for efficiency and phasing between the squeezer and output mode cleaner, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}, \\ensuremath{\\psi_\\text{O}}$, which defines the mode of the interferometer's homodyne readout. Both cases represent a basis change from the Laguerre-Gauss modes of the squeezer OPA cavity into the basis of each respective cavity. In constructing $\\mat{H}(\\Omega)$, however, the squeezing is transformed to the interferometer basis, reflects, and then transforms back to the squeezing basis. This corresponds to the operations of \\cref{fig:SQZ_mm_chain}. There are also parameters to express the coupling efficiency and phase, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{F}}, \\ensuremath{\\psi_\\text{F}}$, between the interferometer cavity and the OMC cavity. The $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{F}}$ parameter is less natural to analyze squeezing is not independent from  $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$. It is considered at the end of this section, as it can also be independently measured.\n\n\\cref{fig:SQZ_mm_chain} is implemented into \\cref{eq:TF_matrix} through this simplistic two-mode representation by assuming that the interferometer reflection transfer function $\\ensuremath{\\mathfrak{r}}(\\Omega)$ applies to the LG0 mode in the interferometer basis. The LG1 mode picks up the reflection transfer function $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}$, which is approximately ${\\sim}1$ due to high order modes being non-resonant in the interferometer cavities and thus directly reflecting. %\n\\begin{align}\n  \\mat{H}_\\ensuremath{\\text{R}} &= \n  \\begin{bmatrix}\n    \\ensuremath{\\mathfrak{r}}(\\Omega) & 0\n    \\\\\n    0 & \\ensuremath{\\mathfrak{r}_{\\text{hom}}}(\\Omega)\n  \\end{bmatrix},\n  &\n  \\mat{G} &= \n    \\begin{bmatrix}\n    1 & 0\n    \\\\\n    0 & e^{i\\psi_G}\n  \\end{bmatrix}\n        \\label{eq:H_refl}\n        \\\\\n  \\ensuremath{\\mathfrak{r}}(\\Omega) &= \\ensuremath{\\mathfrak{r}}_2(\\Omega) \\text{ or } \\ensuremath{\\mathfrak{r}}_1(\\Omega)\n  &\n  \\ensuremath{\\mathfrak{r}_{\\text{hom}}}(\\Omega) &= e^{i\\theta_{\\text{hom}}} \\approx 1\n\\end{align}\nThe reflection term $\\ensuremath{\\mathfrak{r}}(\\Omega)$ can use either the single, \\cref{eq:single_cavity}, or double,  \\cref{eq:double_cavity}, cavity forms. LIGO, using resonant sideband extraction, uses $\\ensuremath{\\mathfrak{r}}_2(\\Omega)$. Frequencies where the reflection takes a negative sign will be shown to experience destructive interference from modal basis changes, increasing squeezing losses. The $\\mat{G}$ matrix includes a phasing factor due to additional Gouy phase of higher-order-modes. This factor is degenerate with the mismatch phasings $\\psi_\\ensuremath{\\text{I}}$ and $\\psi_\\ensuremath{\\text{O}}$ in observable effects. These matrices are composed per \\cref{fig:SQZ_mm_chain} to formulate the overall transfer function of the squeezed field.\n\\begin{align}\n  \\mat{H}(\\Omega) &= \n\\underbrace{\\mat{U}(\\ensuremath{\\Upsilon_{\\text{O}}}, \\ensuremath{\\psi_\\text{O}})\n\\mat{G}\n\\mat{U}^\\dagger(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})}_{\\mat{U}_{\\text{O,R}}}\n\\mat{H}_\\ensuremath{\\text{R}}\n                    \\underbrace{\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})\n                    }_{\\mat{U}_{\\text{R,I}}}\n                    \\label{eq:H_chain}\n                    \\\\\n  \\ensuremath{\\mathfrak{h}}(\\Omega) &= \n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}^T\n  \\mat{H}(\\Omega)\n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}\n                    \\label{eq:h_chain}\n  ,\n  \\text{   and using }\n\\mat{H}_\\ensuremath{\\text{O}} = \\mat{H}_\\ensuremath{\\text{I}} = \\mat{1}\n\\end{align}\nIgnoring intra-cavity losses and detunings, the two reflection forms $\\ensuremath{\\mathfrak{r}}_1$, $\\ensuremath{\\mathfrak{r}}_2$ can be simplified to give their respective transfer functions $\\ensuremath{\\mathfrak{h}}_1$, $\\ensuremath{\\mathfrak{h}}_2$.\n\nFor quantum noise below $\\Omega < \\gamma_\\ensuremath{\\text{S}}$, the double cavity reflectivity $\\ensuremath{\\mathfrak{r}}_2(\\Omega)$ behaves like a single cavity, using the $\\gamma_\\ensuremath{\\text{A}}$ of \\cref{eq:double_cavity_gamma} and with the opposite reflection sign as \\cref{eq:single_cavity}. \n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_2(\\Omega) &\\approx +\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n  &\\Rightarrow&&\n  \\ensuremath{\\mathfrak{h}}_2(\\Omega) &= \\sqrt{\\textstyle \\ensuremath{1{-}\\MMO}}\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\ensuremath{\\alpha}\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n                   \\label{eq:h2_TMM}\n                  \\\\\n  \\ensuremath{\\mathfrak{r}}_1(\\Omega) &= -\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n  &\\Rightarrow&&\n  \\ensuremath{\\mathfrak{h}}_1(\\Omega) &= \\sqrt{\\textstyle \\ensuremath{1{-}\\MMO}}\\frac{i\\Omega - \\ensuremath{\\alpha}\\gamma_\\ensuremath{\\text{A}}}{i\\Omega + \\gamma_\\ensuremath{\\text{A}}}\n\\end{align}\nUsing the factor\n\\begin{align}\n  \\ensuremath{\\alpha} &\\equiv 1 - 2\\ensuremath{\\Upsilon_{\\text{I}}} + 2\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}e^{i\\ensuremath{\\psi_\\text{R}}}\n             \\intertext{where:}\n  \\ensuremath{\\beta} &\\equiv \\textstyle\\sqrt{\\frac{\\ensuremath{1{-}\\MMI}}{\\ensuremath{1{-}\\MMO}}} \\approx 1\n             \\\\\n  \\ensuremath{\\psi_\\text{R}} &\\equiv \\ensuremath{\\psi_\\text{O}} + \\psi_G - \\ensuremath{\\psi_\\text{I}}\n\\end{align}\nThe phasing factor $\\ensuremath{\\psi_\\text{R}}$ shows that the unknown mismatch phasings combine to a single unknown overall phase. This overall phase determines the extent to which the separate beam mismatches of $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$ coherently stack or cancel with each-other. The factor $\\ensuremath{\\alpha}$ is the total squeezer LG0 to readout LG0 coupling factor for the effective mode mismatch of the full system, specifically when the interferometer reflection $\\ensuremath{\\mathfrak{r}}(\\Omega) = -1$. As an effective mismatch, it can be related back to the diagonal elements of \\cref{eq:U_def} to give an effective mismatch loss on reflection, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{R}}$.\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{R}}} &= 1 - \\textstyle |\\ensuremath{\\alpha}|^2 \\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} - 4\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}\\cos(\\ensuremath{\\psi_\\text{R}})\n         \\label{eq:MMR_def}\n\\end{align}\nThis effective mismatch loss becomes apparent after computing the full system efficiency $\\eta(\\Omega)$ (\\cref{eq:eta_passive}) using $\\ensuremath{\\mathfrak{h}}_1$ and $\\ensuremath{\\mathfrak{h}}_2$.\n\\begin{align}\n  \\eta_\\ensuremath{\\text{R}}(\\Omega)\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &=\n    \\left( \\ensuremath{1{-}\\MMO} \\right)\\frac{\\gamma_\\ensuremath{\\text{A}}^2 + \\left(\\ensuremath{1{-}\\MMR}\\right)\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\label{eq:effective_MMR_loss2}\n                  \\\\\n  \\eta_\\ensuremath{\\text{R}}(\\Omega)\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_1}\n  &=\n    \\left( \\ensuremath{1{-}\\MMO} \\right)\\frac{\\Omega^2 + \\left(\\ensuremath{1{-}\\MMR}\\right)\\gamma_\\ensuremath{\\text{A}}^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\label{eq:effective_MMR_loss1}\n\\end{align}\nFor the double cavity system of LIGO, \\cref{fig:data_Q} is presented using the loss rather than efficiency. To relate to the measurement, the loss attributable to mode mismatch is then written\n\\begin{align}\n  \\Lambda_\\ensuremath{\\Upsilon}(\\Omega) \\equiv 1 - \\eta_\\ensuremath{\\text{R}}\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &\\approx\n    \\ensuremath{\\Upsilon_{\\text{O}}} + \\frac{\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\\ensuremath{\\Upsilon_{\\text{R}}}\n    \\label{eq:Lambda_MM}\n\\end{align}\nMode mismatches between the squeezer and OMC were directly measured during the LIGO squeezer installation to be $2\\%-4\\%$, and mismatches from the squeezer and interferometer were indirectly measured but are expected to be of a similar level. The large factors in \\cref{eq:MMR_def} indicate that the independent mismatch measurements are compatible with the observed frequency dependence and levels of the losses to squeezing. The effective mismatch loss $\\ensuremath{\\Upsilon_{\\text{R}}}$ has the following bounds with respect to the independent mismatch measurements.\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{R}}} &\\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when } \\ensuremath{\\Upsilon_{\\text{O}}} = 0\n                          \\\\\n  0 \\le \\ensuremath{\\Upsilon_{\\text{R}}} &\\le 8\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when } \\ensuremath{\\Upsilon_{\\text{I}}} = \\ensuremath{\\Upsilon_{\\text{O}}}\n                          \\\\\n  \\ensuremath{\\Upsilon_{\\text{R}}} &\\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when averaged over } \\ensuremath{\\psi_\\text{R}}\n                         \\label{eq:TMM_bound_avg}\n\\end{align}\nIt is worth noting here how the realistic interferometer differs from this simple two-mode model. The primary key difference is that real mismatch occurs with more transverse modes. Expanding this matrix model to include more modes primarily adds more $\\cos(\\ensuremath{\\psi_\\text{R}})$-type factors to the last term of \\cref{eq:MMR_def}. These factors will tend to average coherent additive mismatch between the squeezer and the OMC away, leaving only the squeezer to interferometer terms. Additionally, not only is there beam parameter mismatch from imperfect beam-matching telescopes, but there is also some amount of misalignment, statically or in RMS drift. Mismatch into modes of different order picks up different factors of $\\ensuremath{\\psi_\\text{G}}$. Together, including more modes leaves the bounds above intact, but makes \\cref{eq:TMM_bound_avg} more representative given the expanded dimensionality of mismatch-space to average away $\\cos(\\ensuremath{\\psi_\\text{R}})$.\n\nThe other notable difference in realistic instruments is that the high order modes pick up small phase shifts of reflection, as the cavities are not perfectly out of resonance at all high order modes. This corresponds to $\\ensuremath{\\mathfrak{r}}_{\\text{hom}} \\ne 1$. The signal recycling mirror is sufficiently low transmissivity that the finesse is low and, even when off-resonance, higher order modes pick up a small but slowly varying phase shift. This has the property of mixing the frequency dependent losses resulting from $\\ensuremath{\\mathfrak{h}}_1$ and $\\ensuremath{\\mathfrak{h}}_2$, resulting in a slightly more varied frequency-dependence that is captured in the full model of \\cref{sec:matrix_coupled_cavity}.\n\n\nWhile the phasing of the mismatch, $\\ensuremath{\\psi_\\text{R}}$, is not directly measurable, it manifests in an observable way. It adds to the complex phase of $\\ensuremath{\\alpha}$ to cause a slight rotation of the squeezing phase, making the cavity appear as if it is detuned. the frequency dependence and magnitude of this rotation is given by (c.f. \\cref{eq:theta_passive}),\n\\begin{align}\n  \\theta_\\ensuremath{\\Upsilon}(\\Omega) \\equiv \\theta\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &\\approx\n    \\frac{-\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}2\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}\\sin(\\ensuremath{\\psi_\\text{R}})\n\\end{align}\nwhich adds to the rotation from cavity length detuning \\cref{eq:double_cav_theta}. The addition of this term with $\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}} \\ensuremath{\\Upsilon_{\\text{O}}}}$ unknown confounds the ability to use the data of \\cref{fig:data_Q} to constrain $\\ensuremath{\\psi_\\text{R}}$. There is a small discrepancy between the length-detuning induced optical spring observed in the interferometer calibration \\cite{Cahillane-PRD17-CalibrationUncertainty, Sun-CQG20-CharacterizationSystematic} and the detuning inferred from the data. The additional mismatch phase shift helps explain that such a discrepancy is possible, but the two should be studied in more detail. Note that the small Gouy phase shift from $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}$ can be significant for this small detuning effect. The expression above is primarily provided to indicate the magnitude of variation as a function of $\\sin(\\ensuremath{\\psi_\\text{R}})$, so that future observations can better constrain $\\ensuremath{\\psi_\\text{R}}$ by comparing squeezing measurements of $\\theta(\\Omega)$ with calibration measurements of the optical spring arising from $\\delta_\\ensuremath{\\text{S}}$.\n\nThe asymmetric contribution of $\\ensuremath{\\alpha}$ in \\cref{eq:h2_TMM} also causes mode mismatch to contribute to optical dephasing, $\\Xi(\\Omega)$ (c.f. \\cref{eq:Xi_passive}). The dependence on $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}, \\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{R}}$ is complex and does not have single dominating contributions, so an analytic expression is not computed here. Using the exact models of \\cref{sec:matrix_coupled_cavity}, that mimic the datasets of \\cref{sec:experiment} give a contribution of $\\sqrt{\\Xi}$ that peaks at $\\gamma_\\ensuremath{\\text{A}}$ and is 10-20\\text{mRad} for the Livingston LLO model, and 10-50mRad for the Hanford LHO model, with a range due to imperfect knowledge of the mismatch parameters.\n\nThe transverse mismatch calculations so far use the parameters $\\ensuremath{\\Upsilon_{\\text{O}}}$, which is directly measurable, and $\\ensuremath{\\Upsilon_{\\text{I}}}$, which is independent, but $\\ensuremath{\\Upsilon_{\\text{I}}}$ can not easily be measured using invasive direct measurements due to the fragile operating state of the GW interferometer. Another mismatch parameter exists for the signal beam traveling with the Michelson fringe-offset light. This beam experiences a separate mode matching efficiency, $\\ensuremath{\\Upsilon_{\\text{F}}}$, denoting the mismatch loss between the interferometer and the OMC. $\\ensuremath{\\Upsilon_{\\text{F}}}$ can be calculated from the original parameters by following the red signal path depicted in \\cref{fig:SQZ_mm_chain}.\n\\begin{align}\ne^{i\\phi_\\ensuremath{\\text{F}}}\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{F}}}, \\ensuremath{\\psi_\\text{S}})\n  &=\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{O}}}, \\ensuremath{\\psi_\\text{O}})\n\\mat{G}\n\\mat{U}^\\dagger(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})\n\\end{align}\nExpanding this form results in the following relations\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{F}}} &\\approx \\ensuremath{\\Upsilon_{\\text{O}}} + \\ensuremath{\\Upsilon_{\\text{I}}} - 2\\sqrt{\\ensuremath{\\Upsilon_{\\text{O}}}\\ensuremath{\\Upsilon_{\\text{I}}}}\\cos(\\ensuremath{\\psi_\\text{R}})\n         \\\\\n   \\ensuremath{\\Upsilon_{\\text{F}}} &\\approx \\ensuremath{\\Upsilon_{\\text{R}}}\/2 + \\ensuremath{\\Upsilon_{\\text{O}}} - \\ensuremath{\\Upsilon_{\\text{I}}}\n\\end{align}\nExperimentally, $\\ensuremath{\\Upsilon_{\\text{F}}}$ can be determined or estimated more directly than $\\ensuremath{\\Upsilon_{\\text{I}}}$ by using signal fields from the arms, though can be confused with projection loss when the local oscillator readout angle $\\xi\\ne 0$ (c.f. \\cref{eq:homodyne_observable}). These formulas provide the set of relations to estimate each of the mode mismatch parameters from the others, and potentially the overall mismatch phase $\\ensuremath{\\psi_\\text{R}}$ as well. These relations are calculated using the assumptions of this section: the two-mode approximation and that $\\ensuremath{\\mathfrak{r}}_{\\text{hom}} \\simeq 1$.\n\nTogether, the relations of this section give insight in to how the physical mismatch parameters, $\\ensuremath{\\Upsilon_{\\text{I}}}$, $\\ensuremath{\\Upsilon_{\\text{O}}}$ and $\\ensuremath{\\Upsilon_{\\text{F}}}$ contribute to squeezing degradations. $\\ensuremath{\\Upsilon_{\\text{R}}}$ is a new form of effective mismatch parameter that is directly measurable from squeezing data, using the analysis of \\cref{sec:experiment}. It indicates how squeezing changes with frequency due to \\crefrange{eq:effective_MMR_loss2}{eq:effective_MMR_loss1}. Together, the complex, coherent interactions of transverse modal mixing on squeezed state can be concisely characterized in cavity-enhanced interferometers.\n\n\\subsection{Implications for Frequency Dependent Squeezing}\nThis analysis of the transverse mismatch applies to the reflection of squeezing off of any form of cavity. Namely, the detuned filter cavity for frequency-dependent rotation of squeezing in the LIGO A+ upgrade. This cavity will be installed on the input, $\\Tmat{H}_\\ensuremath{\\text{I}}$ section of the squeezing transformation sequence. The filter cavity mismatch loss $\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}$ will behave analogously to $\\ensuremath{\\Upsilon_{\\text{I}}}$, introducing losses of ${\\sim}4\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}$ at frequencies resonating in the cavity. The mismatch loss adds to those caused by the internal round-trip cavity loss $\\Lambda_{\\ensuremath{\\text{fc}}}$, creating the effective loss $\\Lambda_{\\ensuremath{\\text{FC}}} \\approx 4\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}} + 4\\lambda_{\\ensuremath{\\text{FC}}}\/\\gamma_{\\ensuremath{\\text{FC}}}$ using \\cref{eq:cavity_relations}.\n\nThe intra-cavity losses then set the scale for how much transverse mismatch is allowable before mismatch dominates the squeezing degradation, $\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}} < \\Lambda_\\ensuremath{\\text{fc}} \/ T_\\ensuremath{\\text{fc}}$. More importantly, they add to the dephasing from the detuned cavity, by creating an effective $\\lambda'_\\ensuremath{\\text{FC}} = \\lambda_{\\ensuremath{\\text{FC}}} + \\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}\\gamma_{\\ensuremath{\\text{FC}}}$ which can be used in \\cref{eq:detuning_dephasing}. The dephasing will set the limit to the allowable injected squeezing $e^{\\pm 2r}$ level as it introduces anti-squeezing at critical frequencies in the spectrum for astrophysics.\n\n\\section{Conclusions}\n\nBefore this work, the squeezing level in the LIGO interferometers was routinely estimated using primarily high-frequency measurements. This was done to utilize a frequency band where the classical noise contributions were small, while also giving a large bandwidth over which to improve the $\\Delta F \\Delta t$ statistical error in noise estimates. In doing so, LIGO recorded a biased view of the state of squeezing performance between the two instruments. The data analysis of this work has revealed several critical features to better understand and ultimately improve the quantum noise in LIGO.\n\nFirst, it indicates that the two sites have similar optical losses in their injection and readout components, as seen from the low-frequency losses of \\cref{fig:data_Q}. There is still a small excess of losses over the predictions given in \\cite{Tse-PRL19-QuantumEnhancedAdvanced}, but substantially less than implied when estimating the losses using high frequency observations. The most culpable loss components in the LIGO interferometers are being upgraded for the next observing run.\n\nSecond, this data analysis indicates that squeezing is degraded particularly at high frequencies, and the modeling and derivations provide the mechanism of transverse optical mode mismatch, external to the cavities, as a plausible physical explanation. This will be addressed in LIGO through the addition of active wavefront control to better match the beam profiles between the squeezer's parametric amplifier, new filter cavity installation, interferometer, and output mode cleaner.\n\nThird, the quantum radiation pressure noise is now not only measured, but employed as a diagnostic tool along with squeezing. QRPN indicates that the effective local oscillator angle in the Michelson fringe offset light at LLO is a specific, nonzero, value. This indicates that to power up the detector further, while maintaining a constant level of fringe light, the angle will grow larger and cause more pronounced degradation of the sensitivity by projecting out of the signal's quadrature. Ultimately, the LO angle should become configurable using balanced homodyne detection, another planned upgrade as part of ``A+''.\n\nFinally, this work carefully derives useful formula to manipulate the quantum squeezing response metrics. These are useful to reason and rationalize the interactions of squeezing with ever more complex detectors, both for gravitational-wave interferometers, and more generally as squeezing-enhanced optical metrology becomes more commonplace. The design of a future generation of gravitational wave detectors must be optimized specifically to maintain exceptional levels of squeezing compared to today. The quantum response metrics derived in this paper will aid that design work by simplifying our interpretation of squeezing with simulations. With these diagnostics and the data from observing run 3, LIGO is now better prepared to install and characterize frequency dependent squeezing in its ``A+'' upgrade not as a demonstration, but for stable, long-term improvement of the quantum enhanced observatories to detect astrophysical events.\n\n\\begin{acknowledgments}\n LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation, and operates under Cooperative Agreement No. PHY-0757058. Advanced LIGO was built under Grant No. PHY-0823459. The authors gratefully acknowledge the National Science Foundation Graduate Research Fellowship under Grant No. 1122374. \n\\end{acknowledgments}\n\n\\input{fdep_main.bbl}\n\n\n\n\n\n\\section{Introduction}\nThe third observing run of the global gravitational wave network has not only produced a plethora of varied and unique astrophysics events \\cite{Abbott-AAP20-PopulationProperties, Abbott-AAP20-GWTC2Compact}, it has defined a milestone in quantum metrology: that the LIGO, VIRGO and GEO600 observatories are now all reliably improving their scientific output by incorporating squeezed quantum states \\cite{Tse-PRL19-QuantumEnhancedAdvanced, VirgoCollaboration-PRL19-IncreasingAstrophysical, Lough-PRL21-FirstDemonstration}. This marks the transition where optical squeezing, a widely researched, emerging quantum technology, has become an essential component producing new observational capability.\n\nFor advanced LIGO, observing run three provides the first peek into the future of quantum enhanced interferometry, revealing challenges and puzzles to be solved in the pursuit of ever more squeezing for ever greater observational range. Studying quantum noise in the LIGO interferometers is not simple. The audio-band data from the detectors contains background noise from many optical, mechanical and thermal sources, which must be isolated from the purely quantum contribution that responds to squeezing. All the while, the interferometers incorporate optical cavities, auxiliary optical fields, kg-scale suspended optics, and radiation pressure forces. The background noise and operational stability of the LIGO detectors is profoundly improved in observing run three \\cite{Buikema-PRD20-SensitivityPerformance}, enabling new precision observations of the interactions between squeezed states and the complex optomechanical detectors.\n\nQuantum radiation pressure noise (QRPN) is the most prominent new observation from squeezing \\cite{Yu-N20-QuantumCorrelations, Acernese-PRL20-QuantumBackaction}. QRPN results from the coupling of photon momentum from the amplitude quadrature of the light into the phase quadrature, as radiation force fluctuation integrates into mirror displacement uncertainty. When vacuum states enter the interferometer, rather than squeezed states, QRPN imposes the so-called standard quantum limit\n\\cite{Braginsky-S80-QuantumNondemolition, Braginsky-92-QuantumMeasurement, Braginsky-RMP96-QuantumNondemolition},\nbounding the performance of GW interferometers. Because the QRPN coupling between quadratures is coherent, squeezed states allow the SQL to be surpassed\n\\cite{Kimble-PRD01-ConversionConventional, Yu-N20-QuantumCorrelations}.\nBoth surpassing the SQL and increasing the observing range is possible by using a frequency-dependent squeezing (FDS) source implemented with a quantum filter cavity\n\\cite{Kimble-PRD01-ConversionConventional, McCuller-PRL20-FrequencyDependentSqueezing, Zhao-PRL20-FrequencyDependentSqueezed, Oelker-PRL16-AudioBandFrequencyDependent, Chelkowski-PRA05-ExperimentalCharacterization, Whittle-PRD20-OptimalDetuning, Khalili-PRD10-OptimalConfigurations, Evans-PRD13-RealisticFilter, Kwee-PRD14-DecoherenceDegradation}. LIGO is including such a source in the next observing run as part of its ``A+'' upgrade\\cite{Whittle-PRD20-OptimalDetuning, McCuller-PRL20-FrequencyDependentSqueezing}. To best utilize its filter cavity squeezing source, the frequency-dependence of LIGO's quantum response must be precisely understood.\n\nDegradations to squeezing from optical loss and ``phase noise'' fluctuations of the squeezing angle are also prominently observed in LIGO. Whereas QRPN's correlations cause frequency dependent effects, loss and phase noise are typically described as causing frequency independent, broadband changes to the quantum noise spectrum. This work analyzes the quantum response of both LIGO interferometers to injected squeezed states, indicating that QRPN and broadband degradations, taken independently, are insufficient to fully describe the observed quantum response to squeezing.\n\nThe first sections of this work expand the response and degradation model of squeezing to examine and explain the LIGO quantum noise data by decomposing it into independent, frequency-dependent parameters. The latter sections relate the parameter decomposition back to interferometer models, to navigate how squeezing interacts with cavities that have internal losses, transverse-mode selectivity, and radiation pressure interactions. The spectra at LIGO are explained using a set of broadly applicable analytical expressions, without the need for elaborate and specific computer simulations. The analytical models elucidate the physical basis of LIGO's squeezed state degradations, prioritizing transverse-mode quality using wavefront control of external relay optics\n\\cite{Cao-OEO20-EnhancingDynamic, Cao-AOA20-HighDynamic, Perreca-PRD20-AnalysisVisualization} to further improve quantum noise. This analysis also demonstrates the use of squeezing as a diagnostic tool\\cite{Mikhailov-PRA06-NoninvasiveMeasurements}, examining not only the cavities but also the radiation pressure interaction. These diagnostics show further evidence of the benefit of balanced homodyne detection \\cite{Fritschel-OEO14-BalancedHomodyne}, another planned component of the ``A+'' upgrade. The description of squeezing in this work expands the modeling of degradations in filter cavities \\cite{Kwee-PRD14-DecoherenceDegradation}, explicitly defining an intrinsic, non-statistical, form of dephasing. Finally, the derivations of the quantum response metrics in sec. \\ref{sec:derivations} show how to better utilize internal information inside interferometer simulations, simplifying the analysis of squeezing degradations for current and future gravitational wave detectors.\n\n\\section{Squeezing Response Metrics}\n\\label{sec:metrics}\n\nTo introduce the frequency-dependent squeezing metrics, it is worthwhile to first describe the metrics used for standard optical squeezing generated from an optical parametric amplifier (OPA), omitting any interferometer. For optical parametric amplifiers, the squeezing level is determined by three parameters. The first is the normalized nonlinear gain, $y$, which sets the squeezing level and scales from 0 for no squeezing to 1 for maximal squeezing at the threshold of amplifier oscillation. For LIGO, $y$ is determined from a calibration measurement of the parametric amplification \\cite{Xiao-PRL87-PrecisionMeasurement, Aoki-OEO06-Squeezing946nm, Takeno-OEO07-ObservationDB, Khalaidovski-CQG12-LongtermStable, Schnabel-OC04-SqueezedLight, Dwyer-OEO13-SqueezedQuadrature}. The second parameter is the optical efficiency $\\eta$ of states from their generation in the cavity all the way to their observation at readout. Losses that degrade squeezed states are indicated by $\\eta < 1$. Finally, there is the squeezing phase angle, $\\phi$, which determines the optical field quadrature with reduced noise and the quadrature with the noise increase mandated by Heisenberg uncertainty, anti-squeezing. By correlating the optical quadratures, variations in $\\phi$ continuously rotate between squeezing and anti-squeezing. These parameters relate to the observable noise as:\n\\begin{align}\n  N(\\phi) &= \\left( 1 - \\frac{4 \\eta y}{(1 + y)^2} \\right)\\cos^2(\\phi) + \\left( 1 + \\frac{4 \\eta y}{(1 - y)^2} \\right)\\sin^2(\\phi)\n\\end{align}\nThe noise, $N(\\phi)$, can be interpreted as the variance of a single homodyne observation of a single squeezed state, but for a continuous timeseries of measurements, $N$ can be considered as a power spectral density, relative to the density of shot-noise. Using relative noise units, $N=1$ corresponds to observing vacuum states rather than squeezing. While the nonlinear gain parameter $y$ may be physically measured and is common in experimental squeezing literature, theoretical work more commonly builds states from the squeezing operator, parameterized by $r$, which constructs an ideal, ``pure'' squeezed state that adjusts the noise power by $e^{\\pm 2r}$. State decoherence due to optical efficiency is then incorporated as a separate, secondary process. This is formally related to the previous expression using:\n\\begin{align}\n  N(\\phi)\n  &= \\eta\\left(\n    e^{-2r}\\cos^2(\\phi) + e^{+2r}\\sin^2(\\phi)\n    \\right) + \\left(1 - \\eta\\right)\n    \\label{eq:etaSQZ_basic1}\n  \\\\\n  e^{-2r}\n  &=\n    1 - \\frac{4y}{(1 + y)^2} \\label{eq:OPA_y_SQZ}\n    , \\hspace{2.5em} e^{+2r} = 1 + \\frac{4y}{(1 - y)^2}\n \n\\end{align}\nIn experiments, the squeezing angle drifts due to path length fluctuations and pump noise in the amplifier, but is monitored using additional coherent fields at shifted frequencies and stabilized by feedback control. This stabilization is imperfect, resulting in a root-mean-square (RMS) phase noise,  $\\ensuremath{\\phi^2_{\\text{rms}}}$, that mixes squeezing and antisqueezing. Using $\\hat{\\phi}$ to represent the statistical distribution of the squeezing angle, and $E[\\cdot]$ the expectation operation, phase noise can be incorporated as a tertiary process given the expectation values:\n\\begin{align}\n \\ensuremath{\\phi^2_{\\text{rms}}} &= E\\left[\\sin^2(\\delta \\hat{\\phi})\\right]\n             &\n \\ensuremath{\\phi} &= E\\left[\\hat{\\phi}\\right]\n  &\n\\delta \\hat{\\phi} &= \\hat{\\phi} - \\ensuremath{\\phi}\n\\end{align}\nresulting in the ensemble average noise $\\overline{N}$, relative shot noise.\n\\begin{align}\n  \\overline{N}(\\phi) &= E\\left[N(\\ensuremath{\\phi} + \\delta \\hat{ \\phi})\\right]\n              \\\\\n \n   &= \\eta\\left(1 - \\ensuremath{\\phi^2_{\\text{rms}}}\\right)\\left(e^{-2r}\\cos^2(\\ensuremath{\\phi}) + e^{+2r}\\sin^2(\\ensuremath{\\phi}) \\right)\n     \\nonumber\\\\ &\\hspace{1em}\n                   + \\eta\\ensuremath{\\phi^2_{\\text{rms}}} \\left(e^{+2r}\\cos^2(\\ensuremath{\\phi}) + e^{-2r}\\sin^2(\\ensuremath{\\phi}) \\right) + (1 - \\eta)\n    \\label{eq:etaSQZ_basic2}\n\\end{align}\nAgain, the relative noise $\\overline{N}$ is computed as a single value here, but represents a power spectral density that is experimentally measured at many frequencies. These equations, as they are typically used, represent a change to the quantum noise that is constant across all measured frequencies. Notably, the $\\ensuremath{\\phi^2_{\\text{rms}}}$ phase noise term, which caps at $1\/2$, enters as a weighting factor that averages the anti-squeezing noise increase with squeezing noise reduction, while $\\eta$ mixes squeezing with standard vacuum.\n\nIncorporating an interferometer such as LIGO requires extending these equations to handle frequency-dependent effects. The equations must include terms to represent multiple sources of loss entering before, during, and after the interferometer, as well as terms for the frequency-dependent scaling of the quantum noise due to QRPN and the interferometer's suspended mechanics. The extension of the metrics is described by the following equations and parameters:\n\\begin{align}\n  N(\\Omega)\n  &\\equiv \\Gamma(\\Omega) \\cdot \\Big( \\eta(\\Omega) S(\\Omega) + \\Lambda_\\ensuremath{\\text{IRO}}(\\Omega) \\Big) \n    \\label{eq:metric_N}\n  \\\\\n \n \n \n \n \n \n \n \n  S(\\Omega)\n  &\\equiv S_{\\!{-}}\\cos^2\\Big(\\ensuremath{\\phi} - \\theta(\\Omega)\\Big) + S_{\\!{+}} \\sin^2\\Big(\\ensuremath{\\phi} - \\theta(\\Omega)\\Big)\n    \\\\\n  S_{\\!\\pm} &\\equiv \\big(1 - \\Xi'(\\Omega)\\big)e^{\\pm2r} + \\Xi'(\\Omega) e^{\\mp2r}\n              \\label{eq:metric_S}\n  \\\\\n  \\Lambda_\\ensuremath{\\text{IRO}}(\\Omega) &\\equiv (1 - \\eta_\\ensuremath{\\text{I}})\\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}} + \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}}) + (1 - \\eta_\\ensuremath{\\text{O}}) \/ \\Gamma\n    \\label{eq:metric_Lambda}\n\\end{align}\nThese metrics are composed of the following variables:\n\\renewcommand{\\descriptionlabel}[1]{\\hspace{\\labelsep}{#1}:}\n\\begin{description}[noitemsep, nolistsep]\n \\item[$N(\\Omega)$] the power spectrum of quantum noise in the readout, relative to the vacuum power spectral density, $\\hbar \\omega \/ 2$, of broadband shot noise.\n \\item[$\\Gamma(\\Omega)$] The quantum noise gain of the interferometer optomechanics. While $N(\\Omega)$ is relative shot-noise, QRPN causes interferometers without injected squeezing to exceed shot noise at low frequencies, resulting in $\\Gamma > 1$. For optical systems with $\\Gamma\\ne1$, the system cannot be passive, and must apply internal squeezing\/antisqueezing to the optical fields.\n \\item[$e^{2r}, e^{-2r}$] The ``pure'' injected squeezing and anti-squeezing level, before including any degradations. This level is computed for optical parametric amplifier squeezers using \\cref{eq:OPA_y_SQZ}.\n \\item[$S_{\\!-}, S_{\\!+}$] The minimum and maximum relative noise change from squeezing at any squeezing angle, ignoring losses.\n \\item[$S(\\Omega)$] The potentially observable injected squeezing level, before applying losses or noise gain.\n  \\item[$\\ensuremath{\\phi}$] The frequency independent squeezing angle chosen between the source and readout. This is usually stabilized with a co-propagating coherent control field and feedback system.\n  \\item[$\\theta(\\Omega)$] the squeezing angle rotation due to the propagation through intervening optical system. In a GW interferometer, this can be due to a combination of cavity dispersion and optomechanical effects. Quantum filter cavities target this term to create frequency dependent squeeze rotation.\n  \\item[$\\eta_\\ensuremath{\\text{I}}(\\Omega)$, $\\eta_\\ensuremath{\\text{O}}(\\Omega)$, $\\eta_\\ensuremath{\\text{R}}(\\Omega)$] The individually budgeted transmission efficiencies of the squeezed field at input, reflection and output paths of the interferometer. $1 - \\eta_{\\text{I,R,O}}$ indicates optical power lost in that component.\n  \\item[$\\eta(\\Omega)$] The collective transmission efficiency of the squeezed field. This is usually the product of the efficiencies in each path, $\\eta = \\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}$, but can deviate from this when $\\Gamma \\ne 1$ and interferometer losses affect both $\\Gamma$ and $\\eta_\\ensuremath{\\text{R}}$.\n  \\item[$\\Lambda_\\ensuremath{\\text{IRO}}(\\Omega)$] The total transmission loss over the squeezing path that contaminates injected squeezed states with standard vacuum. When $\\Gamma \\approx 1$, then $\\Lambda_\\ensuremath{\\text{IRO}} \\approx 1 - \\eta$.\n  \\item[$\\Xi'(\\Omega)$] This is a squeezing-level dependent decoherence mechanism called dephasing. It\n    incorporates both statistical $\\ensuremath{\\phi^2_{\\text{rms}}}$ phase fluctuations and the fundamental degradation arises from optical losses with unbalanced cavities, denoted $\\Xi(\\Omega)$. It can also arise from QRPN with structural or viscous mechanical damping. \\Cref{sec:effective_dephasing} shows how to incorporate fundamental dephasing  $\\Xi(\\Omega)$, standard phase uncertainty, $\\ensuremath{\\phi^2_{\\text{rms}}}$, and cavity tuning fluctuations, $\\ensuremath{\\theta^2_{\\text{rms}}}(\\Omega)$, into $\\Xi'(\\Omega)$ to make a total effective dephasing factor. When small, these factors sum to approximate the effective total $\\Xi'$\n\\end{description}\n\nAfter the data analysis of the next section, these quantum response metrics are derived in \\cref{sec:derivations}. These squeezing metrics indicate three principle degradation mechanisms, all frequency-dependent. These are losses, where $\\Lambda_\\ensuremath{\\text{IRO}}(\\Omega)\\approx 1 {-} \\eta(\\Omega) > 0$; Mis-phasing, from $\\phi{-}\\theta(\\Omega)\\ne 0$; and de-phasing, $\\Xi(\\Omega) > 0$.\n\nThe interaction of squeezing with quantum radiation pressure noise is described within these terms. Broadband Squeezing naively forces a trade-off between increased measurement precision and increased quantum back-action. When squeezing is applied in the phase quadrature, it results in anti-squeezing of the amplitude quadrature. The amplitude quadrature then pushes the mirrors and increases QRPN; thus, the process of reducing imprecision seemingly increases back-action. In other terms, QRPN causes the interferometer's ``effective'' observed quadrature\\footnote{The observed quadrature in this context is with respect to the injected quantum states, be they squeezed or vacuum. It is dependent on the quadrature of the interferometer's homodyne readout, but does not refer specifically to it. The effective observed quadrature also does not refer to the specific quadrature that the interferometer signal is modulated into.} to transition from the phase quadrature at high frequencies to the amplitude quadtrature at low frequencies. In the context of these metrics, the observation quadrature is captured in the derivation of $\\theta(\\Omega)$. The associated back-action trade-off can be considered a mis-phasing degradation, allowing the SQL to be surpassed using the quantum quadrature correlations introduced by varying the squeezing angle\\cite{Yu-N20-QuantumCorrelations}. Frequency dependent squeezing, viewed as a modification of the squeezing source, can be considered as making $\\phi(\\Omega)$ frequency-dependent, tracking $\\theta(\\Omega)$. Alternatively, it can be viewed as a modification of the interferometer, to maintain $\\theta(\\Omega) \\approx 0$. While a quantum filter cavity is not explicitly treated in this work, the derivations of \\cref{sec:derivations} are setup to be able to include a filter cavity as a modification to the input path of the interferometer.\n\nWhile mis-phasing can be compensated using quantum filter cavities, the other two degradations are fundamental. For squeezed states, they establish the noise limit:\n\\begin{align}\n  N(\\Omega) &\\ge \\Gamma{\\cdot}\\left(2\\eta\\sqrt{\\Xi'(1-\\Xi')} + \\Lambda_\\ensuremath{\\text{IRO}}\\right),\n  &\n  e^{-2r} &= \\sqrt{\\Xi'(\\Omega)}\n  \\label{eq:N_limit}\n\\end{align}\nSetting the squeezing level as $\\sqrt{\\Xi'}$ solves for the optimal noise given the dephasing. Squeezing is then further degraded from losses, producing the noise limit. Notably, the optimal squeezing is generally frequency-dependent due to $\\Xi(\\Omega)$, indicating that for typical broadband squeezing sources, this bound cannot always be saturated at all frequencies.\n\n\\subsection{Ideal Interferometer Response}\n\\label{sec:ideal_IFO_metrics}\nBefore analyzing quantum noise data to utilize the squeezing metrics of \\crefrange{eq:metric_N}{eq:metric_Lambda}, it is worthwhile to first review the quantum noise features expected in the LIGO detector noise spectra\\cite{Kimble-PRD01-ConversionConventional, Aasi-CQG15-AdvancedLIGO}, under ideal conditions and without accounting for realistic effects present in the interferometer. The derivations later will then extend how the well-established equations below generalize to incorporate increasingly complex interferometer effects, both by extracting features from matrix-valued simulation models, as well as by extracting features from scalar boundary-value equations for cavities.\n\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_IFO, \n  label={SQZ_mm_IFO},\n  caption={\n    This simplified diagram of the interferometer layout shows the propagation of the source laser (solid red) and squeezed beam (dashed burgundy). At (a), the squeezed beam is sourced from a parametric amplifier cavity and circulated to the interferometer with a Faraday isolator. At (b), the squeezing field reflects from the interferometer. Depending on the frequency and transverse beam profile, the states partially transit the interferometer cavities, but also partially reflect promptly. The squeezing that enters the interferometer symmetrically is beam split inside the signal recycling cavity, coherently resonates in both arms, and recombines again at the beamsplitter, effectively experiencing the two branches as a single linear coupled cavity. Injected at a different port, the red laser field carries substantial laser power and is symmetrically split to pump the arm cavities. Differential length signals are sourced by modulating the circulating pump field, creating a phase-quadrature field that resonates in the same effective linear cavity as the squeezing. The signal is emitted at (b) follows with the reflected squeezing. The transverse beam profile (mode) of the signal and squeezing is then selected using the output mode cleaning cavity at (c). Ultimately, the signal and noise are read as timeseries in photodetectors at (d). This effect of coherent interference between prompt and cavity-circulated squeezing from this sequence is formulated, measured, and analyzed in the following sections.\n},\n}\n\n\nOther than shot noise imprecision, the dominant quantum effect in gravitational wave interferometers arises from radiation pressure noise. In an ideal, on-resonance interferometer, this noise is characterized by the interaction strength $\\ensuremath{\\mathcal{K}}(\\Omega)$ that correlates amplitude fluctuations entering the interferometer to phase fluctuations that are detected along with the signal. $\\ensuremath{\\mathcal{K}}$ is generated from the circulating arm power $P_\\ensuremath{\\text{A}}$ creating force noise that drives the mechanical susceptibility $\\chi(\\Omega)$. The susceptibility relates force to displacement on each of the four identical mirrors of mass $m$ in the GW arm cavities. The QRPN effect is enhanced by optical cavity gain $g(\\Omega)$ which resonantly enhances quantum fields entering the arm cavities and signal fields leaving them.\n\\begin{align}\n  \\mathcal{K}(\\Omega)\n  &= 16k\\frac{P_\\ensuremath{\\text{A}}}{c} g^2(\\Omega)\\chi(\\Omega),\n    &\n  g(\\Omega) &= \\frac{\\sqrt{{\\gamma_\\ensuremath{\\text{A}} c}\/{L_{\\ensuremath{\\text{a}}}}}}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n              \\label{eq:optical_gain_g}\n\\end{align}\nHere, $k$ is the wavenumber of the interferometer laser and $c$ the speed of light.\nThe arm cavity gain $g(\\Omega)$ is a function of the signal bandwidth $\\gamma_\\ensuremath{\\text{A}}$, derived later, and the interferometer arm length $L_\\ensuremath{\\text{a}}$. Unlike in past works, this expression of $\\ensuremath{\\mathcal{K}}(\\Omega)$ here is kept complex, holding the phase shift that arises from the interferometer cavity transfer function. The phase of $\\ensuremath{\\mathcal{K}}(\\Omega)$ is useful for later generalizations. $\\ensuremath{\\mathcal{K}}(\\Omega)$ adds the amplitude quadtrature noise power to the phase quadrature fluctuations directly reflected from the interferometer, setting the noise gain $\\Gamma(\\Omega)$\n\\begin{align}\n  \\Gamma(\\Omega) &= 1 + |\\mathcal{K}(\\Omega)|^2\n  &\n  \\theta(\\Omega) &= \\arctan(|\\ensuremath{\\mathcal{K}}(\\Omega)|)\n                   \\label{eq:GammaTheta_standard}\n\\end{align}\nThe relationship between $\\Gamma(\\Omega)$ and $\\theta(\\Omega)$ from $\\ensuremath{\\mathcal{K}}(\\Omega)$ is stated above as reference, but it will more appropriately handle the complex $\\ensuremath{\\mathcal{K}}(\\Omega)$ when it is derived later. The value $|\\ensuremath{\\mathcal{K}}(\\Omega_\\ensuremath{\\text{sql}})| \\equiv 1$ defines the crossover frequency $\\Omega_\\ensuremath{\\text{sql}}$ between noise contributions from shotnoise imprecision and QRPN, corresponding to $\\Gamma(\\Omega_\\ensuremath{\\text{sql}})= 2$ and $\\theta(\\Omega_\\ensuremath{\\text{sql}})=-45^\\circ$. For the $\\chi(\\Omega)$ susceptibility of a free test mass, the factor $\\ensuremath{\\mathcal{K}}(\\Omega)$ can be expressed using only frequency scales.\n\\begin{align}\n  \\mathcal{K}(\\Omega)\n  &= -\\frac{\\Omega^2_\\ensuremath{\\text{sql}}}{\\Omega^2}\\left( \\frac{\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\\right)^2,\n  &&\\text{given}&\n  \\chi(\\Omega) &\\equiv \\frac{-1}{m \\Omega^2}\n    \\label{eq:Kchi_standard}\n\\end{align}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/dual_d.pdf}\n\t\\caption{\n    This figure plots the total quantum and classical noise measured in the LIGO detectors in displacement amplitude spectral density units. The black trace plots a reference measurement of the total noise without injected squeezing at 0.25Hz resolution over 1.5Hr integration for LLO and 1.1Hr for LHO. The orange shotnoise measurement shows the displacement calibration, $\\sqrt{G(\\Omega)}$, in amplitude density units. Subtracting the shotnoise level from the reference yields the gray datapoints, which have been rebinned using a median statistic applied after the subtraction and with a logararithmic bin spacing. The subtraction primarily shows the classical noise but also contains QRPN. Multiple measurements are taken at varied squeezing angles, with 5 of 12 plotted for Livingston (LLO) and 5 of 34 plotted for Hanford (LHO), using the same median rebinning method as the gray subtraction. The variation in the data errorbars results from the binning span of each datapoint, $\\Delta F$, and the measurement integration time, $\\Delta T$. The measured spectra error relative to the total noise and proportional to $1\/\\sqrt{\\Delta F \\Delta T}$. The squeezing angle of $-3.9^\\circ$ and $1.4^\\circ$ datasets at LLO used ${\\sim}1$Hr integration, and the remainder used 15 min each. The squeezing angle $4.5^\\circ$ dataset at LHO used ${\\sim}1$Hr integration, while all others use 2 minutes each. The squeezing level $e^{\\pm2r}$ is constant over all angles, but different between the two sites. This accounts for the difference in the yellow, ${\\sim}30^{\\circ}$, dataset at each site.\n  }\n\t\\label{fig:data_h}\n\\end{figure*}\n\nFrequency independent losses are applied to squeezing before and after the interferometer using $\\eta=\\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{R}}\\eta_\\ensuremath{\\text{O}}$ where $\\eta_\\ensuremath{\\text{I}} < 1$, $\\eta_\\ensuremath{\\text{O}} < 1$. The ideal interferometer assumption of the formulas above enforce $\\eta_\\ensuremath{\\text{R}}=1$. Phase noise in squeezing is included in this ideal interferometer case using $\\Xi' = \\ensuremath{\\phi^2_{\\text{rms}}}$.\n\nThe above expressions relate the optical noise $N(\\Omega)$ of \\cref{eq:metric_N} to past models of the quantum strain sensitivity of GW interferometers\\cite{Kimble-PRD01-ConversionConventional, Buonanno-PRD01-QuantumNoise, Buonanno-PRD03-ScalingLaw}. Since $N(\\Omega)$ is relative to shot-noise, it must then be converted to strain or displacement using the optical cavity gain $g(\\Omega)$, by how it affects the GW signal through the calibration factor $G(\\Omega)$. This factor $G(\\Omega)$ relates strain modulations to optical field phase modulations in units of optical power.\n\\begin{align}\n  \\text{PSD}_{\\text{strain}}(\\Omega) &= G(\\Omega)N(\\Omega),\n                                       &\n  G(\\Omega) &= \\frac{\\hbar c}{\\eta_\\ensuremath{\\text{O}} L_{\\ensuremath{\\text{a}}}^2|g(\\Omega)|^{2} k P_\\ensuremath{\\text{A}} }\n              \\label{eq:optical_calibration_G}\n\\end{align}\nTogether, these relations allow one to succinctly calculate the effect of squeezing on the strain power spectrum in the case of an ideal interferometer. These factors and the calculations behind them will be revisited as non-idealities are introduced.\n\n\\section{Experimental Analysis and Results}\n\\label{sec:experiment}\n\nA goal of this paper is to use the squeezing response metrics of \\crefrange{eq:metric_N}{eq:metric_Lambda} to relate measurements of the instrument's noise spectrum to the parameters of the squeezer system, namely its degradations due to loss $1-\\eta$, radiation pressure from mis-phasing $\\phi{-}\\theta(\\Omega)$, and dephasings $\\Xi'(\\Omega)$. This section presents measurements from the LIGO interferometers that are best described using the established frequency-dependent metrics. The measurements then motivate the remaining discussion of the paper that construct simple interferometer models to describe this data in the context of the metrics. This section refers to and relates to the later sections to provide early experimental motivation for the discussions that follow. The reader may prefer instead to skip this section and first understand the models before returning to see their application to experimental data.\n\nThe main complexity in analyzing the LIGO data is that the detectors have additional classical noises, preventing a direct measurement of $N(\\Omega)$. The many frequency-dependent squeezing parameters must also be appropriately disentangled. To address both of these issues, the unknown squeezing parameters are fit simultaneously across multiple squeezing measurements. The classical noise contribution is determined by taking a reference dataset where the squeezer is disabled, such that $S(\\Omega)=1$, and then subtracting it from the datasets where squeezing is injected.\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/dual_Q.pdf}\n  \\caption{\n    This figure shows the data of \\cref{fig:data_h} processed as per \\cref{sec:experiment} for each LIGO site. The processing subtracts away the classical noise determined from the unsqueezed reference dataset. The top panels show the relative noise change $Q_k(\\Omega_i)$ of \\cref{eq:Q_processing} computed using $\\Gamma_k(\\Omega_i)$ from the exact interferometer model of \\cref{sec:matrix_coupled_cavity} using the parameters of \\cref{tab:LIGO_params}. The top panel includes dots with errorbars for the processed data and lines for the best-fit $Q_k(\\Omega_i)$. The middle panel shows the best-fit frequency dependent loss as data points, with errorbars propagated through the fit. For LLO, two sets of loss datapoints are shown, corresponding to interferometer models with different readout angles $\\zeta$.\n    The loss plots also show $1{-}\\eta(\\Omega)$ as computed from the exact matrix model, along with a phenomenological fit against the model of \\cref{eq:Lambda_MM} of \\cref{sec:modeling_TMM}. The phenomenological fit assumes frequency independent losses from the input and output squeezing path with a frequency-dependent addition attributed to transverse mismatch. The bottom panels show the frequency-dependent fit to the observed squeezing angle $\\theta_k(\\Omega_i)$, using the convention of $\\theta(2\\pi{\\cdot}3\\text{kHz}) = 0$. It also plots $\\theta(\\Omega)$ as computed using the exact matrix model. For the LLO data, the $\\zeta=0^\\circ$ model is typically assumed for Michelson-like interferometers such as LIGO; However, the model at that readout angle implies losses at low frequencies that are not favored by the $\\eta(\\Omega)$ models explored in this paper. Alternatively, the $\\zeta\\approx -13^\\circ$ model is consistent with both the fitted losses and the fitted squeezing angles.\n  }\n  \\label{fig:data_Q}\n\\end{figure*}\n\nRepresentative strain spectra from the LIGO Livingston (LLO) and LIGO Hanford (LHO) observatory datasets are plotted in \\cref{fig:data_h}. The Livingston dataset is also reported in \\cite{Yu-N20-QuantumCorrelations}, which details the assumptions and error propagation for the classical noise components and calibration. Only statistical uncertainty is considered in this analysis, in order to propagate error to the parameter fits. The strain spectra of \\cref{fig:data_h} include a reference dataset where the squeezer is disabled, shown in black and at the highest frequency resolution. Additionally, the shotnoise ($N=1$) is plotted in orange, indicating the calibration $\\sqrt{G(\\Omega)}$ of \\cref{eq:optical_calibration_G}. The gray subtraction curve depicts the total classical noise contribution summed with the radiation pressure noise $G(\\Omega)\\ensuremath{\\mathcal{K}}^2(\\Omega)$. The gray dataset can equivalently be computed using a cross correlation of the two physical photodetectors at the interferometer readout\\cite{martynov-pra17-quantumcorrelation}. The equivalence of subtraction and cross correlation is used to precisely experimentally determine the shot-noise scale $G(\\Omega)$ from the displacement-calibrated data.\n\n\\subsection{Analysis}\n\nEach squeezing measurement, indexed by $k$, is indicated by $M_{\\text{ref}, k}(\\Omega_i)$, with a value at each frequency indexed by $i$. The reference dataset is denoted $M_{\\text{ref}}(\\Omega_i)$. The two are subtracted to cancel the stationary classical noise component. The calibration $G(\\Omega)$ is removed to result in the differential quantum noise measurement $D_k(\\Omega_i)$.\n\\begin{align}\n  D_k(\\Omega_i)\n  &\\equiv\n    \\frac{M_{\\text{sqz}, k}(\\Omega_i) - M_{\\text{ref}}(\\Omega_i)}{G(\\Omega_i)}\n    \\label{eq:D_processing}\n\\end{align}\nFor these datasets, the squeezing level $e^{\\pm 2r}$, is held constant and independently measured using the nonlinear gain technique\\cite{Dwyer-OEO13-SqueezedQuadrature} to derive $y$ of \\cref{eq:OPA_y_SQZ}. Each differential data $D_k(\\Omega_i)$ is taken at some squeezing angle $\\phi_k$, which is either fit (LLO) or derived from independent measurements (LHO). The parameters $\\eta_i$ and squeezing rotation $\\theta_i$ are independent at every frequency $\\Omega_i$ but fit simultaneously. All $\\phi_k$ are also fit simultaneously across all datasets. Nonlinear least squares fitting was performed using the Nelder-Mead simplex algorithm \\cite{Gao-COA12-ImplementingNelderMead} implemented in SciPy \\cite{Virtanen-NM20-SciPyFundamental}. The residual minimized by least squares fitting is\n\\begin{align}\n  \\mathcal{R} &= \\sum_{\\substack{i=0\\\\k=0}}^{\\mathcal{N}} \\left( \\frac{D_k(\\Omega_i) - \\overline{D}_k(\\Omega_i)}{\\Delta D_k(\\Omega_i)}\\right)^2\n\\end{align}\nThe measurement statistical uncertainty $\\Delta D$, dominated by the statistical uncertainty in power-spectrum estimation, was propagated through the datasets per \\cite{Yu-N20-QuantumCorrelations}. $\\overline{D}_k(\\Omega_i)$ is the model of the data that is a function of the fit parameters, $\\eta_i, \\theta_i, \\phi_k$ as well as independently measured parameters such as $e^{-2r}$. $\\Xi'(\\Omega_i)$ is not fit using this data since the squeezing level $e^{2r}$ is not varied across the datasets. This is discussed below. These given fit parameters affect are propagated through the squeezing metric functions create a model of this particular differential quantum noise measurement.\n\\begin{align}\n  \\overline{S}_k(\\Omega_i) &\\equiv\n  e^{-2r}\\cos^2\\Big(\\ensuremath{\\phi}_k - \\theta_i\\Big) + e^{+2r}\\sin^2\\Big(\\ensuremath{\\phi}_k - \\theta_i\\Big)\n  \\\\\n  \\overline{D}_k(\\Omega_i)\n  &\\equiv \\left.N(\\Omega_i)\\right|_{S = \\overline{S}_k(\\Omega_i)} - \\left.N(\\Omega_i)\\right|_{S = 1}\n\\end{align}\nWhich simplifies to\n\\begin{align}\n  \\overline{D}_k(\\Omega_i)\n  &= \\left(\\overline{S}_k(\\Omega_i) - 1\\right) \\eta_i \\Gamma(\\Omega_i)\n\\end{align}\nNotably, the individual efficiencies $\\eta_\\ensuremath{\\text{I}}, \\eta_\\ensuremath{\\text{R}}, \\eta_\\ensuremath{\\text{O}}$ cannot be individually measured and only the ``total'' efficiency $\\eta(\\Omega)$ is measurable using this differential method, where the classical noise is subtracted using a reference dataset with squeezing disabled. Additionally, the optical efficiency $\\eta$ can only be inferred given some knowledge or assumption of $\\Gamma(\\Omega)$. In effect, the product $\\eta \\Gamma$ is the primary measurable quantity, rather than its decomposition into separate $\\eta$ and $\\Gamma$ terms; However, for the purposes of modeling, decomposing the two is conceptually useful. Furthermore, to characterize physical losses, the efficiency $\\eta$ or loss $\\Lambda_\\ensuremath{\\text{IRO}}\\approx 1 {-} \\eta$ is easier to plot and interpret than the product $\\eta\\Gamma$.\n\nFor these reasons, the differential data $D_k(\\Omega_i)$ is further processed, creating the measurement $Q_k(\\Omega_i)$ with a form similar to \\cref{eq:etaSQZ_basic1}\n\\begin{align}\n  Q_k(\\Omega_i)\n  &\\equiv \n    \\frac{D_k}{\\Gamma(\\Omega_i)} + 1 \\approx S_k(\\Omega_i)\\eta_i + (1 - \\eta_i) + \\Delta Q\n    \\label{eq:Q_processing}\n\\end{align}\nThe LIGO squeezing data expressed in dB's of $Q_k(\\Omega_i)$ are plotted in the upper panels of \\cref{fig:data_Q}. The data and error bars are in discrete points, while the parameter fits to $Q_k$ using $\\eta_i$, $\\theta_i$ and $\\phi_k$ are the solid lines between the data points. The spectra in each set are calculated using the Welch method a median statistic at each frequency to average all of the frames through the integration time. This prevents biases due to instrumental glitches adding non-stationary classical noise. This technique is detailed in \\cite{Yu-N20-QuantumCorrelations}.\n\nAfter computing $Q_k(\\Omega_i)$ at full frequency resolution, the data is further rebinned to have logararithmic spacing by taking a median of the data points within the frequency range of each bin. This rebinning greatly improves the statistical uncertainty at high frequencies, where many points are collected. At lower frequencies, the relative error benefits less from binning; however, both the LLO and the LHO datasets use a long integration time for their reference measurement and at least one of the squeezing angle measurements. Using the median removes narrow-band lines visible in the strain spectra of \\cref{fig:data_h}. Fitting combines the few long-integration, low-error datasets with many short-integration, high-error sets at many variations of the operating parameters. The few low-error datasets reduce the absolute uncertainty in the resulting fit parameters, whereas the many variations reduce co-varying error that would otherwise result from modeling parameter degeneracies.\n\nThe relative statistical error in each bin of the original PSD $M_k(\\Omega_i)$ is approximately $(\\Delta F \\Delta T)^{-{1}\/{2}}$ given the integration time $\\Delta T$ of 2 minutes to 1 hour and bin-width $\\Delta F$ of 0.25Hz. This relative error is converted to absolute error and propagated through the processing steps of \\crefrange{eq:D_processing}{eq:Q_processing}. At low frequencies, the classical noise contribution to each $M_k$ is larger than the quantum noise. Although it is subtracted away to create $D_k(\\Omega_i)$, the classical noise increases the absolute error, and, along with less rebinning, results in the larger relative errors at low-frequency in \\cref{fig:data_Q}. After fitting the squeezing parameters, the Hessian of the reduced chi-square is computed from the Jacobian of the fit residuals with respect to the parameters. This Hessian represents the Fisher information, and the diagonals of its inverse provides the variances indicated by the plotted loss and angle parameter error bars.\n\nFor the LHO data, the fit parameters $\\phi_k$ are determined by mapping the demodulation angles of its coherent control feedback system\\cite{Tse-PRL19-QuantumEnhancedAdvanced, Dooley-OEO15-PhaseControl, Vahlbruch-PRL06-CoherentControl, Chelkowski-PRA07-CoherentControl} back to the squeezing angle. That mapping has 3 unknown parameters, an offset in demodulation angle, an offset in squeezing angle, and a nonlinear compression parameter, all of which are fit simultaneously in all datasets. This $\\phi_k$ mapping was not performed on the LLO data, as some systematic errors in the demodulation angle records bias the results. Despite fitting more independent parameters, the longer integration time of the LLO data gives it sufficiently low statistical uncertainty at frequencies below $\\Omega_\\ensuremath{\\text{sql}}$ that the model and parameter degeneracy between $\\phi_k$, $\\theta_i$ and $\\eta_i$ is not an issue.\n\n\\subsection{Results}\n\nThe middle panels of \\cref{fig:data_Q} show the fits to $\\eta_i$, though plotted as loss $1{-}\\eta_i$ to represent $\\Lambda_\\ensuremath{\\text{IRO}}$. Both datasets additionally include a red loss model curve fit, assembled using the equations in \\cref{sec:modeling}. The orange exact model curves use \\cref{sec:matrix_coupled_cavity}. The data and model curve fit shows a variation in the efficiency, where losses increase from low to high frequencies. This increase in loss can be attributed either to losses within the signal recyling cavity of the interferometer, or to a coherent effect resulting from transverse Gaussian beam parameter mismatch between the squeezer and interferometer cavities. At low frequencies, the optical efficiency is similar between the two LIGO sites, indicating that frequency independent component to the loss are consistent between the implementations at both LIGO sites. The differing high-frequency losses can reasonably be ascribed to variations in the optical beam telescopes of the squeezing system and are analyzed in \\cref{sec:modeling_TMM}.\n\nThe LLO middle panel of \\cref{fig:data_Q} shows two separate inferred loss $1 - \\eta_i$ datasets. These differ in their underlying model of $\\Gamma(\\Omega_i)$. The following section \\ref{sec:derivations} discusses how variations in $\\Gamma$ arise and describes the local oscillator angle $\\zeta$. The $\\zeta = 0$ data reflects the standard, ideal radiation pressure noise model of \\crefrange{eq:Kchi_standard}{eq:GammaTheta_standard}. This model is disfavored given the frequency dependency of $\\eta(\\Omega_i)$ derived using optical cavity models later in this paper. The $\\zeta = -13^\\circ$ model presents an alternative that is compatible with models of the optical efficiency. The need for this alternative indicates that squeezing metrics must account for variations in interferometer noise gain $\\Gamma$. Physically, these variations arise from the readout angle adjusting the prevalence of radiation pressure versus pondermotive squeezing. The $\\zeta = -13^\\circ$ model results in a smaller noise gain $\\Gamma$ at $40 \\text{Hz}$ than does the $\\zeta = 0^\\circ$ model. Since the lower $\\Gamma$ model is favored, this dataset provides some, moderate, evidence that LLO currently benefits from the quantum correlations introduced by the mirrors near $\\Omega_{\\text{SQL}}$, while experiencing lessened sensitivity elsewhere.\n\nThis data demonstrates that the readout angle has an effect on the interferometer sensitivity and the optimal local oscillator is not necessarily $\\xi = 0$ due to radiation pressure.  The quantum benefit of decreased $\\Gamma$ from the readout angle $\\xi$ is a method to achieve sub-SQL performance that is an alternative to injecting squeezing. Like squeezing, it has a frequency dependent enhancement known as the ``variational readout'' technique \\cite{Kimble-PRD01-ConversionConventional, Khalili-PRD07-QuantumVariational}, that a sensitivity increase from lowering $\\Gamma$ while minimizing the sensitivity decrease of the frequency-independent form. For LLO, the reduced sensitivity from $\\xi \\ne 0$ masquerades as a $5\\%$ loss of signal power, but does not actually affect the $\\eta$ or $\\Lambda_\\ensuremath{\\text{IRO}}$ contributions to the squeezing level.\n\nThe bottom panels of \\cref{fig:data_Q} show the fits of $\\theta_i$ of each dataset. The magnitude of $e^{\\pm 2r}$ provides a ``lever arm'' in the variation of $S_k(\\Omega_i)$ that strongly constrains the $\\phi_k{-}\\theta_i$ effective squeezing angle. These leveraged constraints result in small errorbars to the fitted $\\theta_i$. The LLO data are plotted with two models of the $\\theta_i$ based on the assumed local oscillator angle $\\zeta$. The $\\zeta=0^\\circ$ model follows the standard radiation pressure model of \\cref{eq:GammaTheta_standard} at low frequencies and includes a filter-cavity type rotation around the interferometer cavity bandwidth $\\gamma \\approx 2\\pi \\cdot 450\\text{ Hz}$. This rotation is modeled in \\cref{sec:modeling_cavities}. The $\\zeta=-13^{\\circ}$ model is computed using the coupled cavity model of \\cref{sec:matrix_coupled_cavity} and internally includes a weak optical spring effect along with the shifted readout angle $\\zeta$. Together, these effects modify the effective squeezing angle $\\theta$ away from \\cref{eq:GammaTheta_standard} at low frequencies, and agree well with the dataset. This agreement provides further evidence of the reduced radiation pressure noise gain $\\Gamma(\\Omega)$ in LLO that results from the effective LO readout angle $\\zeta$. A nonzero readout angle $\\zeta$ is reasonable to expect due to unequal optical losses in the LIGO interferometer arm cavities. The arm mismatch results in imperfect subtraction of the fringe-light amplitude quadrature at the beamsplitter, creating a static field that adds to the phase-quadrature light created from the Michelson offset and results in $\\zeta \\ne 0$. Past diagnostic measurements\nconclude that some power in the readout diodes must be in the amplitude quadrature, but until now could not determine the sign.\n\nAlthough the squeezing angle parameters $\\phi_k$ and $\\theta_i$ are fit, the frequency-dependent dephasing parameter $\\Xi_i$ cannot be reliably determined from these datasets given the accuracy to which $e^{\\pm 2r}$ is measured. Additionally, the squeezing level $e^{\\pm 2r}$ is not varied in this data, nor is it sufficiently large to resolve an influence from $\\Xi(\\Omega) < 10^{-3}$. This $\\sqrt{\\Xi} \\approx \\ensuremath{\\phi_{\\text{rms}}}$ is expected from independent measurements of phase jitter that propagate through the coherent control scheme of the squeezer system\\cite{Tse-PRL19-QuantumEnhancedAdvanced}. A large source of optically induced $\\Xi$ is not expected has the interferometer cavities are not sufficiently detuned. Measurements of the squeezing system indicate $\\ensuremath{\\phi_{\\text{rms}}} \\lesssim 30\\text{ mRad}$. Future LIGO measurements should include additional datasets that vary $r$ along a third indexing axis $j$ and should increase the injected squeezing level $e^{2r} > 30$ to measure, or at least bound, $\\Xi$ and its frequency-independent contribution $\\ensuremath{\\phi^2_{\\text{rms}}}$. The model fits described above are consistent with the data while assuming $\\Xi = \\ensuremath{\\phi^2_{\\text{rms}}} \\equiv 0$.\n\n\\section{Decomposition Derivation}\n\\label{sec:derivations}\n\nThe factors $\\eta(\\Omega), \\theta(\\Omega)$ and $\\Xi(\\Omega)$ from \\cref{sec:metrics} each describe an independent way for squeezing to degrade. $\\Gamma(\\Omega)$ indicates how the quantum noise scales above or below the shot noise level from squeezing and from quantum radiation pressure within the interferometer. They represent a natural extension of standard squeezing metrics that incorporates frequency dependence, and, as scalar functions, they are simple to plot and to relate with experimental measurements. This section delves into their derivation by employing matrices in the two photon formalism \\cite{Caves-PRA85-NewFormalism, Schumaker-PRA85-NewFormalism} to represent the operations of squeezing, adding loss, shifting the squeezing phase, reflecting from the interferometer, and final projection of the quantum state into the interferometer readout. The derived formulas can be used in frequency-domain simulation tools that compute noise spectra using matrix methods, so that the quantum response metrics can be provided in addition to opaquely propagating squeezing to an simulation result of $N(\\Omega)$. \n\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_cavities, \n  label=SQZ_mm_cavities, \n  caption={\n    The two-photon transformation matrices experienced by squeezing through the sequence of \\cref{fig:SQZ_mm_IFO}. The effective linear coupled cavity, including the optomechanical effect of radiation pressure, is collected and computed into the transformation $\\Tmat{H}_R$. The middle cavity is the signal recycling cavity and the rightmost cavity represents the coherent combination of both arms. Each cavity adds losses from each mirror. For simplicity, these are collected into round-trip cavity loss contributions, $\\Lambda_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{s}}}$, and $\\Lambda_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{s}}}$ that inject standard optical vacuum into the cavities, circulating and transforming into the loss terms $\\Tmat{T}_{\\ensuremath{\\text{R}},\\ensuremath{{\\,\\mu}}}$ while lowering the efficiency $\\eta_\\ensuremath{\\text{R}}$. Transformations of the squeezing at the input and output are included with the terms, $\\Tmat{H}_{\\ensuremath{\\text{I}}}$, $\\Tmat{H}_{\\ensuremath{\\text{O}}}$ and any additive vacuum contributions,  $\\Tmat{T}_{\\ensuremath{\\text{I}},\\ensuremath{{\\,\\mu}}}$, $\\Tmat{T}_{\\ensuremath{\\text{O}},\\ensuremath{{\\,\\mu}}}$.\n  },\n}\n\nTwo-photon matrices are an established method to represent transformations of the optical phase space of Guassian states in an input-output Heisenberg representation of the instrument\\cite{Danilishin-LRR12-QuantumMeasurement}. They are concise yet rigorous when measuring noise spectra from squeezed states using the quantum measurement process of Homodyne readout. Section II of \\cite{Danilishin-LRR19-AdvancedQuantum} provides a review of their usage in the context of gravitational-wave interferometers. Here, two-photon matrices are indicated by doublestruck-bold lettering, and are given strictly in the amplitude\/phase quadrature basis.\n\nEach matrix represents the transformation of the optical phase space of a single optical ``mode'' as it propagates through each physical element towards the readout. The term ``mode'' refers to a basis vector in a linear decomposition of optical field the transverse optical plane of many physical ports\\footnote{Simulations also typically span multiple optical frequencies, but this is not treated here.}. Each plane is further decomposed into transverse spatial modes using a Hermite or Laguerre-Gaussian basis. In this decomposition, each optical mode is indexed by the placeholder $\\mu$ and acts as a continuous transmission channel for optical quantum states. The phase space transformations of these continuous optical states is indexed by time or, more conveniently, frequency. Optical losses and mixing from transverse mismatch behave like beamsplitter operations, serving to couple multiple input modes, generally carrying vacuum states, to the mode of the readout where states are measured.\n\nThe mode of the injected squeezed states, and their specific transformations during beam propagation, must be distinguished from all of the lossy elements that couple in vacuum states. The squeezed states experience a sequence of transformations by the input elements, interferometer, and output elements, denoted $\\Tmat{H}_\\ensuremath{\\text{I}}(\\Omega)$,  $\\Tmat{H}_\\ensuremath{\\text{R}}(\\Omega)$,  $\\Tmat{H}_\\ensuremath{\\text{O}}(\\Omega)$. This sequence multiplies to formulate the total squeeze path propagation $\\Tmat{H}$. \n\\begin{align}\n  \\Tmat{H}(\\Omega) &= \n\\Tmat{H}_{\\text{O}}\\Tmat{H}_{\\text{R}}\\Tmat{H}_{\\text{I}}\n\\end{align}\nLossy optical paths mix the squeezed states with additional standard vacuum states. These are collected into sets of transformation matrices corresponding to each individual loss source, $\\{\\Tmat{T}_\\mu\\}$. See \\cref{fig:SQZ_mm_cavities}. The sets are grouped by their location along the squeezing path where the lossy element is incorporated. The beamsplitter-like operation that couples each loss is given by a $\\Tmat{\\Lambda}_\\mu$, indexed by its location and source along the squeezing path. Loss transformations $\\Tmat{\\Lambda}_\\mu$ are generally frequency-independent. $\\Tmat{\\Lambda}_{\\text{R}, i}$ are an exception, as they occur within the cavities of the interferometer and include some cavity response. The vacuum states associated with each loss then propagate along with the squeezed states and experience the remaining transformations that act on squeezing.\n\\begin{align}\n  \\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{H}_{\\text{O}}\\Tmat{H}_{\\text{R}}\\Tmat{\\Lambda}_{\\text{I},\\ensuremath{{\\,\\mu}}}\n    \\\\\n  \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{H}_{\\text{O}}\\Tmat{\\Lambda}_{\\text{R},\\ensuremath{{\\,\\mu}}}\n    \\\\\n  \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}(\\Omega)\n  &= \n\\Tmat{\\Lambda}_{\\text{O},\\ensuremath{{\\,\\mu}}}\n  \\\\\n  \\left\\{\\Tmat{T}\\right\\} &= \\left\\{\\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}; \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}; \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}  \\right\\}\n\\end{align}\nTogether, all of the transformations of $\\Tmat{H}$ and  $\\{\\Tmat{T}\\}$ define the output states at the readout of the interferometer in terms of the input states entering through the squeezer and loss elements. The two quadrature observables of the optical states are given with the convention $\\hat{q}$ being the amplitude quadrature and $\\hat{p}$ being the phase, and they are indexed to distinguish their input port and transverse mode.\n\\begin{align}\n  \\begin{bmatrix}\\hat{q}_{\\text{out}}(\\Omega) \\\\ \\hat{p}_{\\text{out}}(\\Omega)\\end{bmatrix}\n  &=\n    \\Tmat{H} \\begin{bmatrix}\\hat{q}_{\\text{in}}(\\Omega) \\\\ \\hat{p}_{\\text{in}}(\\Omega)\\end{bmatrix} + \n  \\sum_{\\Tmat{T}_\\ensuremath{{\\,\\mu}}\\in \\left\\{\\Tmat{T}\\right\\}} \\Tmat{T}_\\ensuremath{{\\,\\mu}} \\begin{bmatrix}\\hat{q}_\\ensuremath{{\\,\\mu}}(\\Omega) \\\\ \\hat{p}_\\ensuremath{{\\,\\mu}}(\\Omega)\\end{bmatrix}\n  \\label{eq:qp_out}\n\\end{align}\nThe two-photon matrices $\\Tmat{H}$ and $\\Tmat{T}_\\ensuremath{{\\,\\mu}}$ must preserve commutation relations, namely $[\\hat{q}_\\text{out}, \\hat{p}_\\text{out}] = [\\hat{q}_\\ensuremath{{\\,\\mu}}, \\hat{p}_\\ensuremath{{\\,\\mu}}] = i\\hbar$. In doing so, the matrices ensure that losses within $\\Tmat{H}$ couple ancillary vacuum states that degrade squeezing.\n\nThe readout carries a continuous coherent optical field known as the ``local oscillator'' and the output states are read using homodyne readout. The phase of the local oscillator, $\\zeta$, defines the observed quadrature, $\\hat{m}$, for the homodyne measurement. Gravitational Wave interferometers typically use a ``Michelson offset''\\cite{Fricke-CQG12-DCReadout, Hild-CQG09-DCreadoutSignalrecycled, Ward-CQG08-DcReadout} in the paths adjacent their beamsplitter to operate slightly off of dark fringe. This offset couples a small portion of their pump carrier light to their output as the local oscillator field. This is a form of homodyne readout that fixes $\\zeta$ to measure in the phase quadrature, defined here to be when $\\zeta=0$. Imperfect interference at the beamsplitter can couple some amplitude quadrature and shift $\\zeta$ away from $0$. Balanced homodyne readout is an alternative implementation proposed for LIGO's ``A+'' upgrade and will allow $\\zeta$ to be freely chosen\\cite{Fritschel-OEO14-BalancedHomodyne}. Regardless of the implementation, the homodyne observable is $\\hat{m}$,\n\\begin{align}\n  \\hat{m} &=\n  \\ensuremath{\\Tvec{v}^{\\dagger}}\\begin{bmatrix}\\hat{q}_{\\text{out}}(\\Omega) \\\\ \\hat{p}_{\\text{out}}(\\Omega)\\end{bmatrix}\n           \n  &\n  \\ensuremath{\\Tvec{v}^{\\dagger}}(\\zeta)\n  &=\n    \\begin{bmatrix}\n      \\sin(\\zeta) & \\cos(\\zeta)\n    \\end{bmatrix}\n \\label{eq:homodyne_observable}\n\\end{align}\nHomodyne readout enforces a symmetrized expectation operator, denoted here with the subscript HR, for all measurements of the optical quantum states.\nFurther details of the measurement process are beyond the scope of this work, but the following quadratic expectations arise when computing the noise spectrum and are sufficient to simplify the homodyne expectation values of $\\hat{m}$.\n\\begin{align}\n  1 &= \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}^2}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}^2}_{\\text{HR}},\n      \\label{eq:vacuum_expectations}\n  \\hspace{1em}\n  0 = \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}\\hat{p}_\\ensuremath{{\\,\\mu}}}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}\\hat{q}_\\ensuremath{{\\,\\mu}}}_{\\text{HR}}\n      \\\\\n  0 &= \\braket{\\hat{q}_\\ensuremath{{\\,\\mu}}\\hat{q}_\\ensuremath{{\\,\\nu}}}_{\\text{HR}} = \\braket{\\hat{p}_\\ensuremath{{\\,\\mu}}\\hat{p}_\\ensuremath{{\\,\\nu}}}_{\\text{HR}} \\text{  for } \\nu \\ne \\mu\n      \\label{eq:vacuum_expectations2}\n\\end{align}\nAs a result of these expectations, the vector norm suffices to evaluate noise power using this matrix formalism.\nThe addition of squeezing can be seen either as a modification of the input states $\\hat{q}_{\\text{in}}$, $\\hat{q}_{\\text{in}}$, which violate \\cref{eq:vacuum_expectations}. This work uses the alternative picture, where an additional squeezing transformation occurring at the very start of the squeezing path $\\Tmat{H}$ that acts on $\\hat{q}_{\\text{in}}$, $\\hat{q}_{\\text{in}}$ that are also vacuum states. The squeezing transformation is defined by the squeezing level $r$ and the squeezing angle $\\phi$, which act via the matrices:\n\\begin{align}\n  \\Tmat{R}(\\phi)\n  &\\equiv\n  \\begin{bmatrix}\n    \\cos(\\phi) & {-}\\sin(\\phi)\\\\\n    \\sin(\\phi) & \\phantom{-}\\cos(\\phi)\n  \\end{bmatrix}\n &\n  \\Tmat{S}(r) &\\equiv\n  \\begin{bmatrix}\n    e^r & 0\\\\\n    0 & e^{-r}\n  \\end{bmatrix}\n\\end{align}\nWhen added to the squeezing path, the resulting quantum noise is calculated from the observable $\\hat{m}$.\n\\begin{align}\n  N(\\Omega) &= \\Braket{\\hat{m}^\\dagger\\hat{m}}_{\\text{HR}} = \n              \\left| \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}\\Tmat{R}(\\phi) \\Tmat{S}(r) \\right|^2 + \n\\sum_{\\Tmat{T}_\\ensuremath{{\\,\\mu}}\\in \\left\\{\\Tmat{T}\\right\\}} \\left| \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_\\ensuremath{{\\,\\mu}} \\right|^2\n                             \\label{eq:N_no_sqz_derivation}\n\\end{align}\nThe first term of which is one of the factors in \\cref{eq:metric_N}\n\\begin{align}\n  \\eta(\\Omega)\\cdot S(\\Omega, \\phi) \\cdot \\Gamma(\\Omega, \\zeta)\n \n \n \n \n \n   &= \\left|\\ensuremath{\\Tvec{v}^{\\dagger}}\n     \\Tmat{H}\\Tmat{R}(\\phi)\n     \\Tmat{S}(r)\n     \\right|^2\n    \\label{eq:decomposition_relation}\n\\end{align}\nAt this point, the factors can be separated because: $\\Tmat{R}\\Tmat{S}$ determines the factor $S(\\Omega, \\phi)$; $\\Tmat{H}$ has been ``reduced'' by loss, indicating when $\\eta(\\Omega) < 1$; and the benchmark noise level is defined by $\\Gamma(\\Omega)$, contained in the interferometer's optomechanical element $\\Tmat{H}_{\\text{R}}$.\n\nTo distinguish these terms, further manipulations are necessary. The first is to examine just the vector $\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}$ to determine how the later term $\\Tmat{R}\\Tmat{S}$ results in $S(\\Omega)$. Basis vectors for the two quadrature observables are defined, and the local oscillator is represented using them.\n\\begin{align}\n  \\ensuremath{\\Tvec{v}^{\\dagger}}(\\zeta)\n  &=\n  \\Tvec{e}_p^\\dagger\\Tmat{R}(\\zeta)\n  &\n    \\Tvec{e}_q &= \n    \\begin{bmatrix}\n      1 \\\\ 0\n    \\end{bmatrix}\n  &\n    \\Tvec{e}_p &= \n    \\begin{bmatrix}\n      0 \\\\ 1\n    \\end{bmatrix}\n\\end{align}\nThe basis vectors then allow the vector norm to be split into its two components $\\ensuremath{m_q}$ and $\\ensuremath{m_p}$, defining the \\textit{observed noise quadrature}.\n\\begin{align}\n  \\ensuremath{m_q}(\\Omega) &= \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H} \\Tvec{e}_q & \\ensuremath{m_p}(\\Omega) &= \\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H} \\Tvec{e}_p\n\\end{align}\nThe vector $\\vec{m}$ contains the magnitude and angle of a projection of the quantum state $\\hat{q}_{\\text{in}}$, $\\hat{p}_{\\text{in}}$ at each frequency, but it also contains the complex phase shift from propagation delay in the interferometer and squeezing path. This later phase contribution does not affect noise calculations, but must be properly handled. Projecting it away requires maintaining phase information, and this is why the optomechanical factor $\\ensuremath{\\mathcal{K}}$ is complex in this work.\n\nThe squeezing angle rotation $\\Tmat{R}(\\phi)$ can be viewed through its left-multiplication, applying a rotation to the observed noise quadtrature rather than to the squeezing. In this picture, the angle $\\phi$ can align the observed quadtrature with either the squeezing or anti-squeezing quadrature. The rotation needed to do so determines $\\theta(\\Omega)$, again with the caveat that both $m_q$ and $m_p$ are complex. Their common phase carries the delay information, but their differential phase causes dephasing. In short, differential phase forces $\\vec{m}$ to project into both quadratures at any rotation $\\Tmat{R}(\\phi)$. This has the effect of always adding anti-squeezing to squeezing and vice-versa, resulting in the factor $\\Xi(\\Omega)$. The relations are fully derived in \\cref{sec:phase_noise_composition} using a singular value decomposition to identify the principle noise axes. It leads to the expressions\n\\begin{align}\n  \\theta(\\Omega) &\\approx -\\arctan\\left(\\Re\\!\\left\\{{\\frac{\\ensuremath{m_q}}{\\ensuremath{m_p}}}\\right\\}\\right)\n                   \\label{eq:theta_calculation}\n  \\\\\n  \\Xi(\\Omega) &= \\frac{1}{2} -  \\sqrt{\n        \\frac{\\left(|\\ensuremath{m_q}|^2 - |\\ensuremath{m_p}|^2\\right)^2 + 4 \\Re\\left\\{ \\ensuremath{m_q}\\ensuremath{\\conj{m}_p} \\right\\}}\n        {4\\left(|\\ensuremath{m_q}|^2 + |\\ensuremath{m_p}|^2\\right)^2}\n        } \n                   \\label{eq:Xi_calculation}\n\\end{align}\nThe observation vector $\\vec{m}$, and \\cref{eq:theta_calculation} generalizes the observed noise quadrature description of quantum radiation pressure noise. With it, the observed quadrature angle $\\theta(\\Omega)$ may be computed for any readout angle $\\zeta$ and for more complex interferometers $\\Tmat{H}_\\ensuremath{\\text{R}}$. The ideal interferometer example is demonstrated in \\cref{sec:ideal_IFO_example}\n\nThe phase and magnitude of of the previous argument allows one to determine $S(\\Omega)$ from the form of $\\Tmat{S}$ applied to $\\vec{m}\\Tmat{R}(\\phi)$. Factoring $S$ away, the magnitude of $\\vec{m}$ carries the efficiency of transmitting the squeezed state, along with the noise gain applied to it.\n\\begin{align}\n  \\eta(\\Omega) \\cdot \\Gamma(\\Omega, \\zeta)\n  &=\n    |\\ensuremath{m_q}|^2 + |\\ensuremath{m_p}|^2\n    \\label{eq:eta_gamma}\n\\end{align}\n$\\Gamma(\\Omega)$ expresses the total noise from the interferometer when squeezing is not applied, applying radiation pressure or optomechanical squeezing to both the squeezing path vacuum and internally loss-sourced vacuum. $\\eta \\Gamma$ is affected by all losses, but some of them affect $\\Gamma(\\Omega)$ as well. Using squeezing or a coherent field to probe $\\Tmat{H}$ always measures the product $\\eta \\Gamma$, so the noise gain factor $\\Gamma$ serves primarily as a benchmark. As a benchmark, it relates the dependence of $N(\\Omega)$ to $S(\\Omega)$ and separates the scaling by the efficiency $\\eta$ so that the physical losses may be determined. For this reason, there is freedom to define $\\Gamma$ to make it as independent from the losses as possible, so that it best serves as a benchmark. Here, it is defined using the simulated knowledge of the total noise from the interferometer elements alone:\n\\begin{align}\n \\Gamma(\\Omega) \n  &= \n\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{H}_{\\text{R}}\\right|^2 + \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{\\Lambda}_{\\text{R},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\label{eq:gamma_HR}\n\\end{align}\n$\\eta$ is then determined by dividing \\cref{eq:eta_gamma} by \\cref{eq:gamma_HR}. Under this definition of $\\Gamma$, $\\eta \\propto \\eta_\\ensuremath{\\text{I}}$ and $\\eta \\propto \\eta_\\ensuremath{\\text{O}}$. Losses within the interferometer affect $\\Gamma(\\Omega)$ slightly, and  $\\eta \\propto \\eta_\\ensuremath{\\text{R}}$ is only approximate. \\Cref{sec:radiation_pressure_calc} gives an example of how losses affect $\\eta$ and $\\Gamma$. The primary alternative definition is to use $\\Gamma = N\\big|_{S=1}$, but this definition makes $\\eta_\\ensuremath{\\text{O}}$ both less physically intuitive and also sensitive to interferometer parameters.\n\nSubtracting $\\eta\\Gamma$ from \\cref{eq:N_no_sqz_derivation} and factorizing by the optical paths provides the definition of the remaining efficiency terms.\n\\begin{align}\n  (1 - \\eta_\\ensuremath{\\text{O}})\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{O},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\\\\n  \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}}) \\Gamma\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{R},\\ensuremath{{\\,\\mu}}}\\right|^2\n    \\\\\n  \\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}(1 - \\eta_\\ensuremath{\\text{I}}) \\Gamma\n  &= \n  \\sum_{i}\\left|\\ensuremath{\\Tvec{v}^{\\dagger}} \\Tmat{T}_{\\text{I},\\ensuremath{{\\,\\mu}}}\\right|^2\n\\end{align}\nWhich add together to create the loss term in \\cref{eq:metric_N}.\n\\begin{align}\n    \\Lambda_\\ensuremath{\\text{IRO}}\\Gamma &= \\eta_\\ensuremath{\\text{O}}\\eta_\\ensuremath{\\text{R}}(1 - \\eta_\\ensuremath{\\text{I}})\\Gamma\n    + \\eta_\\ensuremath{\\text{O}}(1 - \\eta_\\ensuremath{\\text{R}})\\Gamma\n    + (1 - \\eta_\\ensuremath{\\text{O}})\n \n \n \n \n \n \n \n\\end{align}\n\n\\subsection{Ideal Interferometer Example}\n\\label{sec:ideal_IFO_example}\n\nThe derivations are now extended to recreate and generalize the ideal noise model of \\cref{sec:ideal_IFO_metrics}, using \\cref{eq:Kchi_standard} for $\\ensuremath{\\mathcal{K}}$. The two-photon matrix corresponding to the interferometer in \\cref{fig:SQZ_mm_cavities} is given below for the lossless interferometer that is perfectly on resonance.\n\\begin{align}\n  \\Tmat{H}_\\ensuremath{\\text{R}}(\\Omega) &\\simeq\n  \\begin{bmatrix}\n    \\ensuremath{\\mathfrak{r}}(\\Omega) & 0\\\\\n    \\ensuremath{\\mathcal{K}}(\\Omega) & \\ensuremath{\\mathfrak{r}}(\\Omega)\n  \\end{bmatrix},\n  &\n  \\ensuremath{\\mathfrak{r}}(\\Omega) &\\simeq\n\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega},\n                 &\\Tmat{\\Lambda}_\\ensuremath{\\text{R}} &= \\Tmat{0}\n \n \n \n \n \n \n \n \n \n \n \n \n \n\\end{align}\nIn the ideal lossless case, the input and output paths also have perfect efficiency $\\eta_\\ensuremath{\\text{I}} \\simeq 1$ with $\\Tmat{H}_\\ensuremath{\\text{I}}(\\Omega) = \\eta_\\ensuremath{\\text{I}}\\Tmat{1},\\; \\Tmat{\\Lambda}_\\ensuremath{\\text{I}}(\\Omega) = \\sqrt{1-\\eta_\\ensuremath{\\text{I}}}\\Tmat{1}$ and similarly for the output. These can be used to compute $\\Tmat{H}$ and $\\vec{m}$.\n\\begin{align}\n  m_q &= \\cos(\\zeta)\\ensuremath{\\mathcal{K}}(\\Omega) + \\sin(\\zeta)\\ensuremath{\\mathfrak{r}}(\\Omega)\n  ,&\n  m_p &= \\cos(\\zeta)\\ensuremath{\\mathfrak{r}}(\\Omega)\n\\end{align}\nThe equations above maintain the correct phase information for this ideal case analysis. Interestingly, $\\ensuremath{\\mathcal{K}}$ and $\\ensuremath{\\mathfrak{r}}(\\Omega)$ have different magnitude responses resulting from different factors of $\\gamma_\\ensuremath{\\text{A}} \\pm i\\Omega$, yet their phase response is the same. This Kramers-Kronig coincedence ensures $\\Xi(\\Omega)=0$ as long as the $\\chi(\\Omega)$ contribution to $\\ensuremath{\\mathcal{K}}(\\Omega)$ is purely real. Thus, lossy mechanics will cause QRPN to dephase injected squeezing. This will not happen to any meaningful level for LIGO, but is noteworthy for optomechanics experiments operating on mechanical resonance.\n\nThe $\\vec{m}$ above also includes the effect of the readout angle. For $\\zeta=0$, it recovers \\crefrange{eq:optical_gain_g}{eq:Kchi_standard}. More generally, it gives\n\\begin{align}\n  \\Gamma(\\Omega) &= |\\cos(\\zeta)\\ensuremath{\\mathcal{K}}(\\Omega)|^2 + \\sin(2\\zeta)|\\ensuremath{\\mathcal{K}}(\\Omega)| + 1\n  \\\\\n  \\theta(\\Omega) &= \\arctan\\big(|\\ensuremath{\\mathcal{K}}(\\Omega)| - \\tan(\\zeta)\\big)\n\\end{align}\nThe exact expressions above can be simplified to better relate them to the LIGO data. Firstly, the squeezing angle is modified to be 0 at high frequencies, to match the conventions of the data. This modified angle is $\\theta'(\\Omega) = \\theta(\\Omega) - \\theta(\\Omega{\\gg}\\gamma_\\ensuremath{\\text{A}})$. Secondly, small shifts of the homodyne angle are linearized.\n\\begin{align}\n  \\Gamma'(\\Omega) &\\approx\n                  \n                  \n                   \\big(1 - |K(\\Omega)|\\big)^2 + 2\\big(1 + \\zeta\\big)|K(\\Omega)|\n \n  \\\\\n  \\theta'(\\Omega) &\\approx \\arctan\\big(|\\ensuremath{\\mathcal{K}}(\\Omega)|) - \\zeta \\frac{|\\ensuremath{\\mathcal{K}}(\\Omega)|^2}{1 + |\\ensuremath{\\mathcal{K}}(\\Omega)|^2}\n\\end{align}\nThe linearized $\\Gamma'$ shows that , when $\\zeta=-13^\\circ=-0.23$, for frequencies near $\\Omega_\\ensuremath{\\text{sql}}$, $\\ensuremath{\\mathcal{K}}(\\Omega_\\ensuremath{\\text{sql}}) \\simeq 1$, the interferometer quantum noise is reduced by about 23\\% with respect to a nominal $\\xi=0$ readout. This change is shown in the blue vs. grey plotted data for the Livingston loss plot in \\cref{fig:data_Q} of $1-\\eta$. There, $\\eta$ changes as the $\\Gamma$ model changes since only $\\eta\\Gamma$ can be measured due to subtracting an unsqueezed reference dataset. The $23\\%$ noise reduction corresponds to approximately 1dB improvement from pondermotive quantum correlations. The angle formula above indicates that for frequencies $\\Omega\\lesssim\\Omega_\\ensuremath{\\text{sql}}$, the local oscillator also adds some additional shift to $\\theta$ at low frequency, which is also observed in the LLO angle fits.\n\nThis analysis gives an example of how the derivations of this section are applied to extend the existing ideal interferometer models towards the real instruments. Exact models including more optical physics are yet more analytically opaque, but give a more complete complete picture if implemented numerically. \\Cref{sec:matrix_coupled_cavity} shows the full matrix solution, including the cavities, to recover these equations while also handling cavity length offset detunings. It also includes transverse modal mismatch in its description. \\Cref{sec:radiation_pressure_calc} gives the minimal extension of this ideal lossless interferometer to incorporate transverse mismatch, showing how the noise gain, $\\Gamma$,  and rotation angle $\\theta$ change specifically from mismatch. In particular, it shows that relating a measurement of $\\Omega_\\ensuremath{\\text{sql}}$ using squeezing back to the arm power $P_\\ensuremath{\\text{A}}$ using \\cref{eq:optical_gain_g} and  \\cref{eq:Kchi_standard} is biased by transverse mismatch.\n\n\\section{Cavity Modeling and Metrics}\n\\label{sec:modeling}\n\nThe previous section derives the general form of the squeezing metrics using matrices of the two photon formalism.\nFor passive systems, the optical transfer function, $\\ensuremath{\\mathfrak{h}}(\\Omega)$, given at every sideband frequency, is sufficient to characterize the response to externally-supplied squeezing. The conceptual simplification and restriction to using only transfer functions is useful for interferometer modeling. Transfer functions, being complex scalar functions, are suitable for analytic calculations of cavity response and can be decomposed into rational function forms to inspect the rational roots, zeros and poles, and the overall gain of the response.\n\nThis section analyzes the coupled cavity system of the interferometer, depicted in \\cref{fig:SQZ_mm_cavities}, through its decomposition into roots. More complicated transverse modal simulations analyze the frequency response of the interferometer cavities for each optical mode to every other mode. Modal simulations thus output a matrix of transfer functions, $\\mat{H}(\\Omega)$, which is difficult to analytically manipulate, but \\cref{sec:modeling_TMM} shows how it can be projected back to a single scalar transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$ and further simplified into the squeezing metrics.\n\nThe transfer function techniques of this section elucidate new squeezing results by avoiding the combined complexity of both two-photon and modal vector spaces. The full generality of two-photon matrices is only required for active systems that introduce internal squeezing, parametric gain or radiation pressure. Passive systems have the property that $\\hat{q}_\\text{out}$, $\\hat{p}_\\text{out}$ also obey the expectations of \\cref{eq:vacuum_expectations,eq:vacuum_expectations2}. Following the notation of \\cref{sec:derivations}, this results in the following condition.\n\\begin{align}\n  \\Tmat{1} &= \\Tmat{H}\\Tmat{H}^\\dagger\n  +\\sum_{\\ensuremath{{\\,\\mu}}}\\Tmat{T}_{\\ensuremath{{\\,\\mu}}}\\Tmat{T}^\\dagger_{\\ensuremath{{\\,\\mu}}}\n             \\label{eq:passivity_condition}\n\\end{align}\nAdditionally, $\\Gamma = 1$ is implied by that condition.\nWithout parametric gain, photons at upper and lower sideband frequencies are never correlated by a passive system. By the passivity condition and manipulations between sideband and quadrature basis, \\cref{sec:passive_derivations} derives the squeezing metrics purely in terms of the transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$.\n\\begin{align}\n  \\theta(\\Omega)\n  &=\n    \\big(\\arg\\big(\\ensuremath{\\mathfrak{h}}(+\\Omega)\\big) + \\arg\\big(\\ensuremath{\\mathfrak{h}}(-\\Omega)\\big)\\big)\/2\n    \\label{eq:theta_passive}\n  \\\\\n  \\eta(\\Omega)\n  &=\n    \\big( |\\ensuremath{\\mathfrak{h}}(+\\Omega)|^2 + |\\ensuremath{\\mathfrak{h}}(-\\Omega)|^2\\big)\/2\n    \\label{eq:eta_passive}\n  \\\\\n  \\Xi(\\Omega)\n  &=\n    \\big( |\\ensuremath{\\mathfrak{h}}(+\\Omega)| - |\\ensuremath{\\mathfrak{h}}(-\\Omega)| \\big)^2\/4\\eta\n    \\label{eq:Xi_passive}\n\\end{align}\nQuantum filter cavities are a method to use an entirely optical system to reduce the radiation pressure associated with squeezed light\\cite{McCuller-PRL20-FrequencyDependentSqueezing, Zhao-PRL20-FrequencyDependentSqueezed}. They are passive cavities, and provide a useful example to study these squeezing metric formulas.\nThe first of these, \\cref{eq:theta_passive}, is a well-established formula for the filter cavity design. It indicates that for cavities with an asymmetric phase response, usually due to being off-resonance or ``detuned'', that the squeezing field picks up a frequency-dependent quadrature rotation. Such a rotation applied in $\\Tmat{H}_\\ensuremath{\\text{I}}$ can be generated by a cavity with transfer function $\\ensuremath{\\mathfrak{h}}_\\ensuremath{\\text{I}}(\\Omega)$ before the interferometer. This cavity rotation compensates the $\\theta(\\Omega)$ due to $\\Tmat{H}_\\ensuremath{\\text{R}}$. Together, the product $\\Tmat{H}_\\ensuremath{\\text{R}}\\Tmat{H}_\\ensuremath{\\text{I}}$ has $\\theta(\\Omega) = 0$, allowing a single choice of squeezing angle $\\phi$ to optimize $N(\\Omega)$ at all frequencies.\n\nThe formulas \\cref{eq:eta_passive} and \\cref{eq:Xi_passive} indicate how losses represented in a transfer function translate to loss-like and dephasing degradations from cavity reflections. For filter cavities, these degradations are investigated in \\cite{Kwee-PRD14-DecoherenceDegradation}, but this new factorization into scalar functions clarifies the discussion. The efficiency $\\eta(\\Omega)$ behaves as expected, an average of the loss in each sideband. The form of $\\Xi(\\Omega)$ is less expected, showing how the combination of loss and detuning in filter cavities creates noise that scales with the squeezing level. A simple picture for the dephasing effect is that when optical quadratures are squeezed, the noise power in both upper and lower sidebands is strictly increased. The sideband correlations allow the increased noise to subtract away for squeezed quadrature measurements but to add for measurements in the anti-squeezed quadrature. The asymmetric losses of detuned cavities preserve the noise increase on one sideband, while degrading the correlations. This ruins the subtraction for the squeezed quadrature and introduces $\\Xi(\\Omega) > 0$. This source of noise is $e^{\\pm r}$ squeezing level dependent but entirely unrelated to fluctuations of the squeezing phase $\\ensuremath{\\phi_{\\text{rms}}}$.\n\n\\subsection{Single Cavity Model for Interferometers}\n\\label{sec:modeling_cavities}\n\nThis section analyzes the effect of the interferometer cavities on squeezing. It starts by considering an interferometer with only one cavity - either in the Michelson arms or from a mirror at the output port, but not both. It represents the first generation of GW detectors. This single cavity scenario is also similar to a quantum filter cavity, in the regime of small detuning\\cite{Komori-PRD20-DemonstrationAmplitude, Corbitt-PRD04-OpticalCavities, Khalili-PRD08-IncreasingFuture}. Advanced LIGO uses a coupled cavity system, depicted in \\cref{fig:SQZ_mm_cavities}, and the transfer function equations for the reflection from the resonant sideband extraction cavity is extended in the next subsection to include the loss and detuning of the additional cavity.\n\nA single cavity operated near resonance may be described using the scale parameters of the cavity bandwidth $\\gamma_\\ensuremath{\\text{A}}$, loss rate $\\lambda_\\ensuremath{\\text{A}}$ and detuning frequency $\\delta_\\ensuremath{\\text{A}}$, which are computed from the physical parameters of the mirror transmissivity $T_\\ensuremath{\\text{a}}$, round-trip loss $\\Lambda_\\ensuremath{\\text{a}}$, cavity length $L_\\ensuremath{\\text{a}}$, and microscopic length detuning $\\Delta L_\\ensuremath{\\text{a}}$.\n\\begin{align}\n  \\gamma_\\ensuremath{\\text{A}}  &= \\frac{c T_\\ensuremath{\\text{a}}}{4 L_\\ensuremath{\\text{a}}}\n  & \n  \\lambda_\\ensuremath{\\text{A}} &= \\frac{c \\Lambda_\\ensuremath{\\text{a}}}{4 L_\\ensuremath{\\text{a}}}\n  &\n  \\delta_\\ensuremath{\\text{A}}  &= -ck\\frac{\\Delta L_\\ensuremath{\\text{a}}}{L_\\ensuremath{\\text{a}}}\n                 \\label{eq:cavity_relations}\n\\end{align}\nThese relations are accurate in the high-finesse limit $T_\\ensuremath{\\text{a}} \\ll 1$, and combine to give the transfer function of the frequency-dependent cavity reflection.\n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_1(\\Omega)\n \n \n  &\\approx\n    -\\frac{(\\gamma_\\ensuremath{\\text{A}} - \\lambda_\\ensuremath{\\text{A}}) - i(\\Omega - \\delta_\\ensuremath{\\text{A}})}{(\\gamma_\\ensuremath{\\text{A}} + \\lambda_\\ensuremath{\\text{A}}) + i(\\Omega - \\delta_\\ensuremath{\\text{A}})}\n    \\label{eq:single_cavity}\n\\end{align}\nNotably, the sign of the reflectivity for a high-finesse cavity on resonance $\\ensuremath{\\mathfrak{r}}_1(\\Omega \\ll \\gamma_\\ensuremath{\\text{A}})=-1$,\nbut outside of resonance $\\ensuremath{\\mathfrak{r}}_1(\\Omega \\gg \\gamma_\\ensuremath{\\text{A}})=1$. This sign determines constructive or destructive interference in transverse mismatch loss analyzed in the next section. The internal losses of the cavity $\\Lambda_\\ensuremath{\\text{a}}$ become cavity-enhanced in the reflection, causing squeezing to experience losses of $\\Lambda_\\ensuremath{\\text{A}}$.\n\\begin{align}\n  \\Lambda_\\ensuremath{\\text{A}}\n  &\\equiv 1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1\\\\ |\\Omega| \\ll \\gamma_\\ensuremath{\\text{A}}}}\n \n \n \n  \\approx \\frac{4\\lambda_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}}\n                            \\approx \\frac{4\\Lambda_\\ensuremath{\\text{a}}}{T_\\ensuremath{\\text{a}}}\n\\end{align}\nFurthermore, detuning the cavity off of resonance causes a rotation of reflected squeezing. For small detunings, the rotation can be approximated.\n\\begin{align}\n  \\theta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1\\\\ k\\Delta L_\\ensuremath{\\text{a}} \\ll T_\\ensuremath{\\text{a}}}}\n  &\\approx\n  \\frac{2\\delta_\\ensuremath{\\text{A}}\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\approx\n   -k \\Delta L_\\ensuremath{\\text{a}}\\frac{8}{T_\\ensuremath{\\text{a}}}\\frac{\\gamma_\\ensuremath{\\text{A}}^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n\\end{align}\nFluctuations in $\\Delta L_\\ensuremath{\\text{a}}$ or $\\delta_\\ensuremath{\\text{A}}$ lead to a phase noise analogous to $\\ensuremath{\\phi_{\\text{rms}}}$, but with the frequency dependence from the above equation\\cite{Kwee-PRD14-DecoherenceDegradation}. Additionally, losses in the cavity lead to intrinsic dephasing $\\Xi(\\Omega)$, calculated below. This calculation is valid at any detuning $\\delta_\\ensuremath{\\text{A}}$, even those larger than the cavity width $\\gamma_\\ensuremath{\\text{A}}$. Its validity only requires being in the overcoupled cavity regime, where losses $\\lambda_\\ensuremath{\\text{A}} \\lesssim \\gamma_\\ensuremath{\\text{A}}\/2$.\n\\begin{align}\n  \\Xi(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_1}}\n  &\\approx\n    \\left(\n  \\frac{4\\gamma_\\ensuremath{\\text{A}}\\lambda_\\ensuremath{\\text{A}}\\delta_\\ensuremath{\\text{A}}\\Omega}{\\left( \\gamma_\\ensuremath{\\text{A}}^2 + (\\Omega - \\delta_\\ensuremath{\\text{A}})^2 \\right)\\left( \\gamma_\\ensuremath{\\text{A}}^2 +  (\\Omega + \\delta_\\ensuremath{\\text{A}})^2 \\right)}\\right)^2\n\\end{align}\nWhen plotted, this expression for $\\Xi(\\Omega)$ has a Lorentzian-like profile, with a peak at $\\Omega_{\\Xi\\text{max}}$. Above $|\\delta_\\ensuremath{\\text{A}}| \\gtrsim \\gamma_\\ensuremath{\\text{A}}$, where the cavity resonance acts entirely either on upper or lower sidebands, the peak dephasing reaches a maximum. At small detunings, $|\\delta_\\ensuremath{\\text{A}}| \\lesssim \\gamma_\\ensuremath{\\text{A}}$, the sideband loss asymmetry scales with the detuning.\n\\begin{align}\n  \\Omega_{\\Xi{\\text{max}}} &\\approx \\sqrt{\\gamma^2_\\ensuremath{\\text{A}}\/4 + \\delta^2 },\n  & \\Xi_{\\text{max}} \n  &\\approx \n    \\frac{\\lambda^2_\\ensuremath{\\text{A}}}{\\gamma^2_\\ensuremath{\\text{A}}}{\\cdot}\\frac{8\\delta^2_\\ensuremath{\\text{A}}}{5\\gamma^2_\\ensuremath{\\text{A}} + 8\\delta^2_\\ensuremath{\\text{A}}}\n    \\label{eq:detuning_dephasing}\n\\end{align}\nThis single cavity model is also useful for analyzing quantum filter cavities and, like the $\\Xi$ metric itself, these peak values have not been calculated in past frequency-dependent squeezing work. Conventional squeezing phase uncertainty, $\\ensuremath{\\phi_{\\text{rms}}}$, can be cast into the units RMS radians of phase deviation, leading to the noise suppression limit for squeezing $S(\\Omega)\\ge 2\\ensuremath{\\phi_{\\text{rms}}}$, by \\cref{eq:N_limit}. For highly detuned cavities such as quantum filter cavities, $\\sqrt{\\Xi(\\Omega)} \\approx \\Lambda_\\ensuremath{\\text{fc}} \/ T_\\ensuremath{\\text{fc}}$. Using the parameters of the A+ filter cavity \\cite{Whittle-PRD20-OptimalDetuning}, $\\Lambda_{\\ensuremath{\\text{fc}}}{\\approx}60\\text{ppm}$ and $T_{\\ensuremath{\\text{fc}}} {=} 1000 \\text{ppm}$ indicates that optical dephasing is of order $60\\text{mRad}$. For an optimal filter cavity with low losses\\cite{Whittle-PRD20-OptimalDetuning}, this dephasing maximum occurs at $\\Omega_{\\Xi{\\text{max}}} = \\sqrt{5\/8}\\Omega_{\\text{SQL}}$. This level of dephasing is commensurate with or even exceeds the expected residual phase uncertainty $\\ensuremath{\\phi_{\\text{rms}}} < 30\\text{mRad}$.\n\nOptical dephasing from the LIGO interferometer cavities is not expected to be large for as they are stably operated on-resonance; however, detuned configurations of LIGO\\cite{Ganapathy-PRD21-TuningAdvanced} are limited by dephasing from the unbalanced response and optical losses in the signal recycling cavity.\n\n\\subsection{Double Cavity Model for Interferometers}\n\nFor interferometers using resonant sideband extraction, like LIGO, the arm cavities have a length $L_\\ensuremath{\\text{a}}$, an input transmissivity of $T_\\ensuremath{\\text{a}}$, and are each on resonance to store circulating laser power. The signal recycling cavity (SRC) has a length $L_\\ensuremath{\\text{s}}$ and a signal recycling mirror (SRM) of transmissivity $T_\\ensuremath{\\text{s}}$. The SRM forms a cavity with respect to the arm input mirror that resonantly increases the effective transmissivity experienced by the arm cavities to be larger than $T_\\ensuremath{\\text{a}}$, broadening the signal bandwidth. While the SRC is resonant with respect to the arm input mirror, it is anti-resonant with respect to the arm cavity, due to the negative sign of \\cref{eq:single_cavity}. The anti-resonance leads to the opposite sign in the reflection transfer function below, \\cref{eq:double_cavity}. The discrepancy in resonance vs. anti-resonance viewpoints is why the signal recycling cavity is also called the signal extraction cavity in GW literature.\n\nThe coupled cavity forms two bandwidth scales for the system, $\\gamma_\\ensuremath{\\text{A}}$, the modified effective arm bandwidth, and $\\gamma_\\ensuremath{\\text{S}}$, the bandwidth of the signal recycling cavity. The arm and signal cavities have their respective round-trip losses $\\Lambda_\\ensuremath{\\text{a}}$ and $\\Lambda_\\ensuremath{\\text{s}}$, as well as length detunings $\\Delta L_\\ensuremath{\\text{a}}$, $\\Delta L_\\ensuremath{\\text{s}}$. In practice, the arm length detuning is expected to be negligible to maximize the power storage, but the signal recycling cavity detuning can be varied by modifying a bias in the control system that stabilizes $\\Delta L_\\ensuremath{\\text{s}}$.\n\nThe scale parameters for the cavity transfer function are approximated from the physical parameters:\n\\begin{align}\n  u_\\ensuremath{\\text{a}} &= 1 - \\sqrt{1 - T_\\ensuremath{\\text{a}}}\n  &\n  u_\\ensuremath{\\text{s}} &= 1 - \\sqrt{1 - T_\\ensuremath{\\text{s}}}\n           \\label{eq:u_factors}\n  \\\\\n  \\gamma_\\ensuremath{\\text{A}}  &= \\frac{c u_\\ensuremath{\\text{a}}}{2 L_\\ensuremath{\\text{a}}} \\cdot\\frac{2 - u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{s}}}\n  & \n    \\gamma_\\ensuremath{\\text{S}}   &= \\frac{c u_\\ensuremath{\\text{s}}}{2 L_\\ensuremath{\\text{s}}}\n                    \\label{eq:double_cavity_gamma}\n  \\\\\n  \\lambda_\\ensuremath{\\text{A}} &= \\frac{c}{L_\\ensuremath{\\text{a}}} \\left(\\frac{\\Lambda_\\ensuremath{\\text{a}}}{4} - \\frac{u_\\ensuremath{\\text{a}} \\Lambda_\\ensuremath{\\text{s}}}{2 u_\\ensuremath{\\text{s}}^2} + \\frac{u_\\ensuremath{\\text{a}} \\Lambda_\\ensuremath{\\text{s}}}{4 u_\\ensuremath{\\text{s}}}\\right)\n  &\n    \\lambda_\\ensuremath{\\text{S}}  &= \\frac{c\\Lambda_\\ensuremath{\\text{s}}}{4L_\\ensuremath{\\text{s}}}\\left(1 - \\frac{u_\\ensuremath{\\text{s}}}{2} \\right)\n  \\\\\n  \\delta_\\ensuremath{\\text{A}} &= -ck\\frac{\\Delta L_\\ensuremath{\\text{a}}}{L_\\ensuremath{\\text{a}}}  - \\frac{\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{S}}}\\delta_\\ensuremath{\\text{S}}\n  &\n  \\delta_\\ensuremath{\\text{S}} &= -ck\\frac{\\Delta L_\\ensuremath{\\text{s}}}{L_\\ensuremath{\\text{s}}} \n           \\label{eq:delta_factors}\n\\end{align}\nThese approximations are valid for the LIGO mirror parameters, see \\cref{tab:LIGO_params}, and model the loss and detuning to {\\color{red} $5\\%$} accuracy. They are derived in \\cref{sec:scalar_coupled_cavity} from Taylor expansions, solving roots, and selectively removing terms. Expanding in the $u$ factors of \\cref{eq:u_factors} gives lower error than expanding in transmissivity or reflectivity factors directly, due to the low effective finesse of the coupled cavity system and the high transmissivity of the SRM.\n\\begin{table}\n\\centering\n\\caption{Parameters of LIGO for data fitting and modeling}\n\\begin{ruledtabular}\n\\begin{tabular}{l|c|cc}\nParameter & Symbol & LLO Value & LHO Value \\\\\n\\hline\narm input transmissivity & $T_\\ensuremath{\\text{a}}$&0.0148 & 0.0142 \\\\\narm length & $L_\\ensuremath{\\text{a}}$ & \\multicolumn{2}{c}{3995 m}\\\\\narm round-trip loss & $\\Lambda_{\\ensuremath{\\text{a}}}$ & \\multicolumn{2}{c}{${\\sim}80$ppm}\\\\\nSRM transmission& $T_\\ensuremath{\\text{s}}$&\\multicolumn{2}{c}{0.325}  \\\\\nSRC length & $L_\\ensuremath{\\text{s}}$ &\\multicolumn{2}{c}{55 m}\\\\\nSRC round-trip loss & $\\Lambda_\\ensuremath{\\text{s}}$ & \\multicolumn{2}{c}{$\\lesssim 3000$ppm}\\\\\n  \\hline\nMirror mass& $m$ &  \\multicolumn{2}{c}{39.9kg}\\\\\nArm power& $P_\\ensuremath{\\text{A}}$ & $200{\\pm} 10\\text{ kW}$  &  $190{\\pm}10\\text{ kW}$ \\\\\nQRPN crossover& $\\Omega_{\\ensuremath{\\text{sql}}}$ & $2\\pi\\cdot 33$ Hz & $2\\pi\\cdot 30$ Hz \\\\\n  \\hline\narm signal band& $\\gamma_\\ensuremath{\\text{A}}$& $2\\pi\\cdot450$ Hz &  $2\\pi\\cdot410$ Hz \\\\\nSRC band& $\\gamma_\\ensuremath{\\text{S}}$& \\multicolumn{2}{c}{$2\\pi\\cdot80$kHz} \\\\\nArm length detuning & $\\Delta L_{\\ensuremath{\\text{a}}}$ & \\multicolumn{2}{c}{0nm}\\\\\nSRC length detuning & $\\Delta L_{\\ensuremath{\\text{s}}}$ & 1.02nm & 1.23nm\\\\\n  \\hline\narm resonant loss& $\\Lambda_\\ensuremath{\\text{A}}$ & \\multicolumn{2}{c}{$\\lesssim 2000$PPM}\\\\\nSRC resonant loss& $\\Lambda_\\ensuremath{\\text{S}}$ & \\multicolumn{2}{c}{${\\sim} 1\\%$ to $3\\%$}\\\\\narm\/SRC detuning& $\\delta_{\\ensuremath{\\text{A}}}$ & $2\\pi\\cdot 10.1$Hz & $2\\pi\\cdot 11.2$Hz \\\\\n  \\hline\nInjected squeezing & $e^{\\pm 2r}$ & {${\\pm}9.7\\text{ dB}$}  &  {${\\pm}8.7\\text{ dB}$} \\\\\nSQZ-OMC mismatch& $\\Upsilon_{\\ensuremath{\\text{O}}}$ & $2\\%$ & $4\\%$ \\\\\nReflection mismatch (fit)& $\\Upsilon_{\\ensuremath{\\text{R}}}$ & $12\\%$ & $35\\%$ \\\\\nAdditional SQZ loss (fit)& $\\Lambda_{\\text{IO}}=1{-}\\eta_\\ensuremath{\\text{I}}\\eta_\\ensuremath{\\text{O}}$ & $31\\%$ & $34\\%$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tab:LIGO_params}\n\\end{table}\nThe scale factors result in the following reflectivity transfer function.\n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_2(\\Omega)\n  &=\n    \\frac{(\\gamma_\\ensuremath{\\text{A}} - \\lambda_\\ensuremath{\\text{A}}) - i(\\Omega - \\delta_\\ensuremath{\\text{A}})}{(\\gamma_\\ensuremath{\\text{A}} + \\lambda_\\ensuremath{\\text{A}}) + i(\\Omega - \\delta_\\ensuremath{\\text{A}})}\n   \n    {\\cdot}\\frac{(\\gamma_\\ensuremath{\\text{S}} - \\lambda_\\ensuremath{\\text{S}}) - i(\\Omega - \\delta_\\ensuremath{\\text{S}})}{(\\gamma_\\ensuremath{\\text{S}} + \\lambda_\\ensuremath{\\text{S}}) + i(\\Omega - \\delta_\\ensuremath{\\text{S}})}\n    \\label{eq:double_cavity}\n\\end{align}\nNotably, this reflectivity is $\\ensuremath{\\mathfrak{r}}_2(\\pm\\Omega \\ll \\gamma)=1$ and $\\ensuremath{\\mathfrak{r}}_2(\\gamma_\\ensuremath{\\text{A}} \\ll \\pm\\Omega \\ll \\gamma_\\ensuremath{\\text{S}})=-1$ which has an opposite overall sign to that of single cavity interferometers. On reflection, the squeezing field experiences different cavity enhanced losses depending on the frequency.\n\\begin{align}\n  \\Lambda_{\\ensuremath{\\text{S}}} &\\equiv \n1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ \\gamma_\\ensuremath{\\text{A}} \\ll |\\Omega| \\ll \\gamma_\\ensuremath{\\text{S}}}}\n&&\\hspace{-3em}\\approx \\frac{2 - u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{s}}}\\Lambda_\\ensuremath{\\text{s}}\n  \\\\\n  \\Lambda_{\\ensuremath{\\text{A}}} &\\equiv \n1 - \\eta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ |\\Omega| \\ll \\gamma_\\ensuremath{\\text{A}}}}\n&&\\hspace{-3em}\\approx \\frac{4\\lambda_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}}+\\Lambda_\\ensuremath{\\text{s}} \\approx \\frac{u_\\ensuremath{\\text{s}}}{u_\\ensuremath{\\text{a}}}\\Lambda_\\ensuremath{\\text{a}}\n\\end{align}\nThe dataset of \\cref{sec:experiment} shows frequency dependent losses, where the loss increases $12\\%$ for LLO and $33\\%$ for LHO. Assuming the losses result from the equations above, this corresponds to round-trip losses in the LIGO signal recycling cavities, $\\Lambda_\\ensuremath{\\text{s}}$, of $1.1\\%$ to $3.2\\%$, which is not realistic. Most mechanisms that introduce loss in the SRC would also introduce it into the power recycling cavity in an obvious manner. The current power recycling factors exclude this possibility, and independent measurements of $\\gamma_\\ensuremath{\\text{A}}$ bound $\\Lambda_\\ensuremath{\\text{s}}$ losses to ${\\le} 3000$ppm. The next section investigates how transverse mismatch can result in this level of observed losses.\n\nIn addition to the losses, \\cref{eq:double_cavity} can be used to determine the cavity-induced squeeze state rotation from the detuning of the signal recycling cavity.\n\\begin{align}\n  \\theta(\\Omega)\\bigg|_{\\substack{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{r}}_2\\\\ \\delta_\\ensuremath{\\text{S}} \\ll \\gamma_\\ensuremath{\\text{S}} \\\\ \\Delta L_\\ensuremath{\\text{a}} = 0}}\n  &\\approx\n  \\frac{2\\delta_\\ensuremath{\\text{A}}\\gamma_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    +\n  \\frac{2\\delta_\\ensuremath{\\text{S}}\\gamma_\\ensuremath{\\text{S}}}{\\gamma_\\ensuremath{\\text{S}}^2 + \\Omega^2}\n \n \n \n \n \n \n \n \n    \\\\\n  &\\approx\n    k\\Delta L_\\ensuremath{\\text{s}}\\frac{4}{u_\\ensuremath{\\text{s}}}\n    \\left(\n    \\frac{\\gamma^2_\\ensuremath{\\text{S}}}{\\gamma_\\ensuremath{\\text{S}}^2 + \\Omega^2}\n    -\n    \\frac{\\gamma^2_\\ensuremath{\\text{A}}}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n \\right)\n    \\label{eq:double_cav_theta}\n \n\\end{align}\nThis indicates the surpising result that detuning the SRC length does not affect the squeezing within the effective\narm bandwidth to first order. Instead, it adds the squeezing rotation in the middle band above the arm bandwidth but below the SRC bandwidth. In the data analysis of \\cref{sec:experiment} and \\cref{fig:data_Q}, the convention for $\\theta(\\Omega)$ is set to be 0 at ``high'' frequencies in this intermediate cavity band, in which case it appears to cause a rotation around $\\gamma_\\ensuremath{\\text{A}}$. This convention used for the data corresponds to omitting the first, $\\gamma_\\ensuremath{\\text{S}}$-scaled term of \\cref{eq:double_cav_theta}.\n\n\\section{Transverse Mismatch Model}\n\\label{sec:modeling_TMM}\n\nSqueezing, as it is typically implemented for GW interferometers, modifies the quantum states in a single optical mode. For LIGO, this mode is the fundamental Gaussian beam resonating in the parametric amplifier cavity serving as the squeezed state source. The cavity geometry establishes a specific complex Gaussian beam parameter that defines a modal basis decomposition into Hermite Gaussian (HG) or Laguerre Guassian (LG) modes. That basis is transformed and redefined during the beam propagation through free space and through telescope lenses on its way to and from the interferometer. The cavities of the interferometer each define their own resonating beam parameters and respective HG or LG basis of optical modes.\n\nIn practice, the telescopes propagating the squeezed beam to and from the interferometer imperfectly match the complex beam parameters, so basis transformations must occur that mix the optical modes. The mismatch of complex beam parameters is called here ``transverse mismatch''. Non-fundamental HG or LG transverse modes do not enter the OPA cavity, and so carry standard vacuum rather than squeezing. Basis mixing from transverse mismatch thus leads to losses; however, unlike typical losses such basis transformations are coherent and unitary, which leads to the constructive and destructive interference effects studied in this section.\n\nThe interferometer transfer function $\\ensuremath{\\mathfrak{h}}(\\Omega)$ is a single scalar function representing the frequency dependence of the squeezing channel from source to readout, but the optical fields physically have many more channels. The cavities visited by the squeezed states each have a transfer function matrix in their local basis, given by $\\mat{H}_{\\ensuremath{\\text{I}}}$, $\\mat{H}_{\\ensuremath{\\text{R}}}$, $\\mat{H}_{\\ensuremath{\\text{O}}}$ for the squeezing input, interferometer reflection, and system output respectively. The diagonals of these matrices indicate the frequency response during traversal for every transverse optical mode. The off-diagonals represent the coupling response between modes that result from scattering and optical wavefront errors.\n\nBetween the cavities, $\\mat{U}$ matrices represent the basis transformations due to transverse mismatch. Here, $\\vec{e}_{\\text{sqz}}$, $\\vec{e}_{\\text{read}}$ are basis vectors for projecting from the single optical mode of the emitted squeezed states and to the single mode of the optical homodyne readout defined by its local oscillator field.\n\\begin{align}\n\\ensuremath{\\mathfrak{h}}(\\Omega) &= {\\vec{e}_{\\text{read}}}^{\\,\\dagger} \\mat{H}(\\Omega)\\vec{e}_{\\text{sqz}}\n                  \\label{eq:TF_projection}\n  \\\\\n\\mat{H}(\\Omega) &=\\mat{H}_\\ensuremath{\\text{O}}\\mat{U}_{\\ensuremath{\\text{O}}, \\ensuremath{\\text{R}}}\\mat{H}_\\ensuremath{\\text{R}}\\mat{U}_{\\ensuremath{\\text{R}}, \\ensuremath{\\text{I}}}\\mat{H}_\\ensuremath{\\text{I}}\n                  \\label{eq:TF_matrix}\n\\end{align}\n\\Cref{eq:TF_projection} and \\cref{eq:TF_matrix} give the general, basis independent, form to compose the effective transfer function for the squeezed field using a multi-modal simulation of a passive interferometer. This is complicated in the general case, but the following analysis develops a simpler, though general, model for how transverse mismatch manifests as squeezing losses.\n\nTransverse mismatch is often physically measured as a loss of coupling efficiency, $\\ensuremath{\\Upsilon}$, of an external Gaussian beam to a cavity measured as a change in optical power. Realistically, more than two transverse modes are necessary to maintain realistic and unitary basis transformations, but, for small mismatches of complex beam parameters, $\\Upsilon < 10\\%$. In this case, only the two lowest modes in the Laguerre-Gauss basis have significant cross-coupling. For low losses, the fundamental Gaussian mode, LG0, loses most of its power to the radially symmetric LG1 mode, assuming low astigmatism and omitting azimuthal indices. This motivates the following simplistic two-mode model to analyze the effect of losses on $\\ensuremath{\\mathfrak{h}}(\\Omega)$. In this model $\\mat{U}$ gives the unitary, though not perfectly physical, basis transformation:\n\\begin{align}\n  \\mat{U}(\\ensuremath{\\Upsilon}, \\psi, \\phi)\n  &\\equiv\n    e^{i\\phi}\\begin{bmatrix}\n    \\sqrt{\\ensuremath{1{-}\\MM}} & -e^{i\\psi}\\sqrt{\\ensuremath{\\Upsilon}}\n    \\\\\n    e^{-i\\psi}\\sqrt{\\ensuremath{\\Upsilon}} & \\sqrt{\\ensuremath{1{-}\\MM}}\n  \\end{bmatrix}\n                           \\label{eq:U_def}\n\\end{align}\nThis unitary transformation includes two unknown phase parameters. The first, $\\psi$, is the phase of the mismatch, which characterizes whether beam size error or wavefront phasing error dominates the overlap integral of the external LG0 and cavity LG1 modes. The second, $\\phi$, is the mismatch phase error from the external LG0 to the cavity LG0. The $\\phi$ term is included above to fully express the unitary freedom of $\\mat{U}$, but is indistinguishable from path length offsets, physically controlled to be $0$, and ignored in further expressions.\n\\autofiguresvgTEX{\n  folder=.\/figures\/, \n  file=SQZ_mm_chain, \n  caption={\n    Propagation of the squeezed beam and unsqueezed higher order transverse beam modes from source to readout. The stages (a)-(d) correspond to the components in \\cref{fig:SQZ_mm_IFO}, depicting the matrix math of \\crefrange{eq:H_refl}{eq:h_chain}. $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ represents the transverse mismatch loss of the squeezing to interferometer, and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$ is the mismatch of the squeezing to readout via the output mode cleaner. These mismatches cause beamsplitter-like mixing between the LG0 and LG1+ modes through \\cref{eq:U_def}. $\\psi_\\ensuremath{\\text{I}}$,  $\\psi_\\ensuremath{\\text{O}}$,  $\\psi_{\\text{G}}$ are unmeasured phasing terms of the interferometer and output mismatch and of the Gouy-phase advance from the beam propagating to the output mode cleaner.\n  },\n  label=SQZ_mm_chain,\n}\n\nIn the case of a GW interferometer with an output mode cleaner, there are two mode matching efficiencies expressed as individually measurable parameters. The first is the coupling efficiency (in power) and phasing associated between the squeezer and interferometer $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}, \\ensuremath{\\psi_\\text{I}}$. The second are parameters for efficiency and phasing between the squeezer and output mode cleaner, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}, \\ensuremath{\\psi_\\text{O}}$, which defines the mode of the interferometer's homodyne readout. Both cases represent a basis change from the Laguerre-Gauss modes of the squeezer OPA cavity into the basis of each respective cavity. In constructing $\\mat{H}(\\Omega)$, however, the squeezing is transformed to the interferometer basis, reflects, and then transforms back to the squeezing basis. This corresponds to the operations of \\cref{fig:SQZ_mm_chain}. There are also parameters to express the coupling efficiency and phase, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{F}}, \\ensuremath{\\psi_\\text{F}}$, between the interferometer cavity and the OMC cavity. The $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{F}}$ parameter is less natural to analyze squeezing is not independent from  $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$. It is considered at the end of this section, as it can also be independently measured.\n\n\\cref{fig:SQZ_mm_chain} is implemented into \\cref{eq:TF_matrix} through this simplistic two-mode representation by assuming that the interferometer reflection transfer function $\\ensuremath{\\mathfrak{r}}(\\Omega)$ applies to the LG0 mode in the interferometer basis. The LG1 mode picks up the reflection transfer function $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}$, which is approximately ${\\sim}1$ due to high order modes being non-resonant in the interferometer cavities and thus directly reflecting. %\n\\begin{align}\n  \\mat{H}_\\ensuremath{\\text{R}} &= \n  \\begin{bmatrix}\n    \\ensuremath{\\mathfrak{r}}(\\Omega) & 0\n    \\\\\n    0 & \\ensuremath{\\mathfrak{r}_{\\text{hom}}}(\\Omega)\n  \\end{bmatrix},\n  &\n  \\mat{G} &= \n    \\begin{bmatrix}\n    1 & 0\n    \\\\\n    0 & e^{i\\psi_G}\n  \\end{bmatrix}\n        \\label{eq:H_refl}\n        \\\\\n  \\ensuremath{\\mathfrak{r}}(\\Omega) &= \\ensuremath{\\mathfrak{r}}_2(\\Omega) \\text{ or } \\ensuremath{\\mathfrak{r}}_1(\\Omega)\n  &\n  \\ensuremath{\\mathfrak{r}_{\\text{hom}}}(\\Omega) &= e^{i\\theta_{\\text{hom}}} \\approx 1\n\\end{align}\nThe reflection term $\\ensuremath{\\mathfrak{r}}(\\Omega)$ can use either the single, \\cref{eq:single_cavity}, or double,  \\cref{eq:double_cavity}, cavity forms. LIGO, using resonant sideband extraction, uses $\\ensuremath{\\mathfrak{r}}_2(\\Omega)$. Frequencies where the reflection takes a negative sign will be shown to experience destructive interference from modal basis changes, increasing squeezing losses. The $\\mat{G}$ matrix includes a phasing factor due to additional Gouy phase of higher-order-modes. This factor is degenerate with the mismatch phasings $\\psi_\\ensuremath{\\text{I}}$ and $\\psi_\\ensuremath{\\text{O}}$ in observable effects. These matrices are composed per \\cref{fig:SQZ_mm_chain} to formulate the overall transfer function of the squeezed field.\n\\begin{align}\n  \\mat{H}(\\Omega) &= \n\\underbrace{\\mat{U}(\\ensuremath{\\Upsilon_{\\text{O}}}, \\ensuremath{\\psi_\\text{O}})\n\\mat{G}\n\\mat{U}^\\dagger(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})}_{\\mat{U}_{\\text{O,R}}}\n\\mat{H}_\\ensuremath{\\text{R}}\n                    \\underbrace{\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})\n                    }_{\\mat{U}_{\\text{R,I}}}\n                    \\label{eq:H_chain}\n                    \\\\\n  \\ensuremath{\\mathfrak{h}}(\\Omega) &= \n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}^T\n  \\mat{H}(\\Omega)\n  \\begin{bmatrix}\n    1 \\\\ 0\n  \\end{bmatrix}\n                    \\label{eq:h_chain}\n  ,\n  \\text{   and using }\n\\mat{H}_\\ensuremath{\\text{O}} = \\mat{H}_\\ensuremath{\\text{I}} = \\mat{1}\n\\end{align}\nIgnoring intra-cavity losses and detunings, the two reflection forms $\\ensuremath{\\mathfrak{r}}_1$, $\\ensuremath{\\mathfrak{r}}_2$ can be simplified to give their respective transfer functions $\\ensuremath{\\mathfrak{h}}_1$, $\\ensuremath{\\mathfrak{h}}_2$.\n\nFor quantum noise below $\\Omega < \\gamma_\\ensuremath{\\text{S}}$, the double cavity reflectivity $\\ensuremath{\\mathfrak{r}}_2(\\Omega)$ behaves like a single cavity, using the $\\gamma_\\ensuremath{\\text{A}}$ of \\cref{eq:double_cavity_gamma} and with the opposite reflection sign as \\cref{eq:single_cavity}. \n\\begin{align}\n  \\ensuremath{\\mathfrak{r}}_2(\\Omega) &\\approx +\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n  &\\Rightarrow&&\n  \\ensuremath{\\mathfrak{h}}_2(\\Omega) &= \\sqrt{\\textstyle \\ensuremath{1{-}\\MMO}}\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\ensuremath{\\alpha}\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n                   \\label{eq:h2_TMM}\n                  \\\\\n  \\ensuremath{\\mathfrak{r}}_1(\\Omega) &= -\\frac{\\gamma_\\ensuremath{\\text{A}} - i\\Omega}{\\gamma_\\ensuremath{\\text{A}} + i\\Omega}\n  &\\Rightarrow&&\n  \\ensuremath{\\mathfrak{h}}_1(\\Omega) &= \\sqrt{\\textstyle \\ensuremath{1{-}\\MMO}}\\frac{i\\Omega - \\ensuremath{\\alpha}\\gamma_\\ensuremath{\\text{A}}}{i\\Omega + \\gamma_\\ensuremath{\\text{A}}}\n\\end{align}\nUsing the factor\n\\begin{align}\n  \\ensuremath{\\alpha} &\\equiv 1 - 2\\ensuremath{\\Upsilon_{\\text{I}}} + 2\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}e^{i\\ensuremath{\\psi_\\text{R}}}\n             \\intertext{where:}\n  \\ensuremath{\\beta} &\\equiv \\textstyle\\sqrt{\\frac{\\ensuremath{1{-}\\MMI}}{\\ensuremath{1{-}\\MMO}}} \\approx 1\n             \\\\\n  \\ensuremath{\\psi_\\text{R}} &\\equiv \\ensuremath{\\psi_\\text{O}} + \\psi_G - \\ensuremath{\\psi_\\text{I}}\n\\end{align}\nThe phasing factor $\\ensuremath{\\psi_\\text{R}}$ shows that the unknown mismatch phasings combine to a single unknown overall phase. This overall phase determines the extent to which the separate beam mismatches of $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{I}}$ and $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{O}}$ coherently stack or cancel with each-other. The factor $\\ensuremath{\\alpha}$ is the total squeezer LG0 to readout LG0 coupling factor for the effective mode mismatch of the full system, specifically when the interferometer reflection $\\ensuremath{\\mathfrak{r}}(\\Omega) = -1$. As an effective mismatch, it can be related back to the diagonal elements of \\cref{eq:U_def} to give an effective mismatch loss on reflection, $\\ensuremath{\\Upsilon}_\\ensuremath{\\text{R}}$.\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{R}}} &= 1 - \\textstyle |\\ensuremath{\\alpha}|^2 \\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} - 4\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}\\cos(\\ensuremath{\\psi_\\text{R}})\n         \\label{eq:MMR_def}\n\\end{align}\nThis effective mismatch loss becomes apparent after computing the full system efficiency $\\eta(\\Omega)$ (\\cref{eq:eta_passive}) using $\\ensuremath{\\mathfrak{h}}_1$ and $\\ensuremath{\\mathfrak{h}}_2$.\n\\begin{align}\n  \\eta_\\ensuremath{\\text{R}}(\\Omega)\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &=\n    \\left( \\ensuremath{1{-}\\MMO} \\right)\\frac{\\gamma_\\ensuremath{\\text{A}}^2 + \\left(\\ensuremath{1{-}\\MMR}\\right)\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\label{eq:effective_MMR_loss2}\n                  \\\\\n  \\eta_\\ensuremath{\\text{R}}(\\Omega)\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_1}\n  &=\n    \\left( \\ensuremath{1{-}\\MMO} \\right)\\frac{\\Omega^2 + \\left(\\ensuremath{1{-}\\MMR}\\right)\\gamma_\\ensuremath{\\text{A}}^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\n    \\label{eq:effective_MMR_loss1}\n\\end{align}\nFor the double cavity system of LIGO, \\cref{fig:data_Q} is presented using the loss rather than efficiency. To relate to the measurement, the loss attributable to mode mismatch is then written\n\\begin{align}\n  \\Lambda_\\ensuremath{\\Upsilon}(\\Omega) \\equiv 1 - \\eta_\\ensuremath{\\text{R}}\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &\\approx\n    \\ensuremath{\\Upsilon_{\\text{O}}} + \\frac{\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}\\ensuremath{\\Upsilon_{\\text{R}}}\n    \\label{eq:Lambda_MM}\n\\end{align}\nMode mismatches between the squeezer and OMC were directly measured during the LIGO squeezer installation to be $2\\%-4\\%$, and mismatches from the squeezer and interferometer were indirectly measured but are expected to be of a similar level. The large factors in \\cref{eq:MMR_def} indicate that the independent mismatch measurements are compatible with the observed frequency dependence and levels of the losses to squeezing. The effective mismatch loss $\\ensuremath{\\Upsilon_{\\text{R}}}$ has the following bounds with respect to the independent mismatch measurements.\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{R}}} &\\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when } \\ensuremath{\\Upsilon_{\\text{O}}} = 0\n                          \\\\\n  0 \\le \\ensuremath{\\Upsilon_{\\text{R}}} &\\le 8\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when } \\ensuremath{\\Upsilon_{\\text{I}}} = \\ensuremath{\\Upsilon_{\\text{O}}}\n                          \\\\\n  \\ensuremath{\\Upsilon_{\\text{R}}} &\\approx 4\\ensuremath{\\Upsilon_{\\text{I}}} &&\\text{ when averaged over } \\ensuremath{\\psi_\\text{R}}\n                         \\label{eq:TMM_bound_avg}\n\\end{align}\nIt is worth noting here how the realistic interferometer differs from this simple two-mode model. The primary key difference is that real mismatch occurs with more transverse modes. Expanding this matrix model to include more modes primarily adds more $\\cos(\\ensuremath{\\psi_\\text{R}})$-type factors to the last term of \\cref{eq:MMR_def}. These factors will tend to average coherent additive mismatch between the squeezer and the OMC away, leaving only the squeezer to interferometer terms. Additionally, not only is there beam parameter mismatch from imperfect beam-matching telescopes, but there is also some amount of misalignment, statically or in RMS drift. Mismatch into modes of different order picks up different factors of $\\ensuremath{\\psi_\\text{G}}$. Together, including more modes leaves the bounds above intact, but makes \\cref{eq:TMM_bound_avg} more representative given the expanded dimensionality of mismatch-space to average away $\\cos(\\ensuremath{\\psi_\\text{R}})$.\n\nThe other notable difference in realistic instruments is that the high order modes pick up small phase shifts of reflection, as the cavities are not perfectly out of resonance at all high order modes. This corresponds to $\\ensuremath{\\mathfrak{r}}_{\\text{hom}} \\ne 1$. The signal recycling mirror is sufficiently low transmissivity that the finesse is low and, even when off-resonance, higher order modes pick up a small but slowly varying phase shift. This has the property of mixing the frequency dependent losses resulting from $\\ensuremath{\\mathfrak{h}}_1$ and $\\ensuremath{\\mathfrak{h}}_2$, resulting in a slightly more varied frequency-dependence that is captured in the full model of \\cref{sec:matrix_coupled_cavity}.\n\n\nWhile the phasing of the mismatch, $\\ensuremath{\\psi_\\text{R}}$, is not directly measurable, it manifests in an observable way. It adds to the complex phase of $\\ensuremath{\\alpha}$ to cause a slight rotation of the squeezing phase, making the cavity appear as if it is detuned. the frequency dependence and magnitude of this rotation is given by (c.f. \\cref{eq:theta_passive}),\n\\begin{align}\n  \\theta_\\ensuremath{\\Upsilon}(\\Omega) \\equiv \\theta\\bigg|_{\\ensuremath{\\mathfrak{h}} = \\ensuremath{\\mathfrak{h}}_2}\n  &\\approx\n    \\frac{-\\Omega^2}{\\gamma_\\ensuremath{\\text{A}}^2 + \\Omega^2}2\\ensuremath{\\beta}\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}}\\ensuremath{\\Upsilon_{\\text{O}}}}\\sin(\\ensuremath{\\psi_\\text{R}})\n\\end{align}\nwhich adds to the rotation from cavity length detuning \\cref{eq:double_cav_theta}. The addition of this term with $\\sqrt{\\ensuremath{\\Upsilon_{\\text{I}}} \\ensuremath{\\Upsilon_{\\text{O}}}}$ unknown confounds the ability to use the data of \\cref{fig:data_Q} to constrain $\\ensuremath{\\psi_\\text{R}}$. There is a small discrepancy between the length-detuning induced optical spring observed in the interferometer calibration \\cite{Cahillane-PRD17-CalibrationUncertainty, Sun-CQG20-CharacterizationSystematic} and the detuning inferred from the data. The additional mismatch phase shift helps explain that such a discrepancy is possible, but the two should be studied in more detail. Note that the small Gouy phase shift from $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}$ can be significant for this small detuning effect. The expression above is primarily provided to indicate the magnitude of variation as a function of $\\sin(\\ensuremath{\\psi_\\text{R}})$, so that future observations can better constrain $\\ensuremath{\\psi_\\text{R}}$ by comparing squeezing measurements of $\\theta(\\Omega)$ with calibration measurements of the optical spring arising from $\\delta_\\ensuremath{\\text{S}}$.\n\nThe asymmetric contribution of $\\ensuremath{\\alpha}$ in \\cref{eq:h2_TMM} also causes mode mismatch to contribute to optical dephasing, $\\Xi(\\Omega)$ (c.f. \\cref{eq:Xi_passive}). The dependence on $\\ensuremath{\\mathfrak{r}}_{\\text{hom}}, \\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{R}}$ is complex and does not have single dominating contributions, so an analytic expression is not computed here. Using the exact models of \\cref{sec:matrix_coupled_cavity}, that mimic the datasets of \\cref{sec:experiment} give a contribution of $\\sqrt{\\Xi}$ that peaks at $\\gamma_\\ensuremath{\\text{A}}$ and is 10-20\\text{mRad} for the Livingston LLO model, and 10-50mRad for the Hanford LHO model, with a range due to imperfect knowledge of the mismatch parameters.\n\nThe transverse mismatch calculations so far use the parameters $\\ensuremath{\\Upsilon_{\\text{O}}}$, which is directly measurable, and $\\ensuremath{\\Upsilon_{\\text{I}}}$, which is independent, but $\\ensuremath{\\Upsilon_{\\text{I}}}$ can not easily be measured using invasive direct measurements due to the fragile operating state of the GW interferometer. Another mismatch parameter exists for the signal beam traveling with the Michelson fringe-offset light. This beam experiences a separate mode matching efficiency, $\\ensuremath{\\Upsilon_{\\text{F}}}$, denoting the mismatch loss between the interferometer and the OMC. $\\ensuremath{\\Upsilon_{\\text{F}}}$ can be calculated from the original parameters by following the red signal path depicted in \\cref{fig:SQZ_mm_chain}.\n\\begin{align}\ne^{i\\phi_\\ensuremath{\\text{F}}}\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{F}}}, \\ensuremath{\\psi_\\text{S}})\n  &=\n\\mat{U}(\\ensuremath{\\Upsilon_{\\text{O}}}, \\ensuremath{\\psi_\\text{O}})\n\\mat{G}\n\\mat{U}^\\dagger(\\ensuremath{\\Upsilon_{\\text{I}}}, \\ensuremath{\\psi_\\text{I}})\n\\end{align}\nExpanding this form results in the following relations\n\\begin{align}\n  \\ensuremath{\\Upsilon_{\\text{F}}} &\\approx \\ensuremath{\\Upsilon_{\\text{O}}} + \\ensuremath{\\Upsilon_{\\text{I}}} - 2\\sqrt{\\ensuremath{\\Upsilon_{\\text{O}}}\\ensuremath{\\Upsilon_{\\text{I}}}}\\cos(\\ensuremath{\\psi_\\text{R}})\n         \\\\\n   \\ensuremath{\\Upsilon_{\\text{F}}} &\\approx \\ensuremath{\\Upsilon_{\\text{R}}}\/2 + \\ensuremath{\\Upsilon_{\\text{O}}} - \\ensuremath{\\Upsilon_{\\text{I}}}\n\\end{align}\nExperimentally, $\\ensuremath{\\Upsilon_{\\text{F}}}$ can be determined or estimated more directly than $\\ensuremath{\\Upsilon_{\\text{I}}}$ by using signal fields from the arms, though can be confused with projection loss when the local oscillator readout angle $\\xi\\ne 0$ (c.f. \\cref{eq:homodyne_observable}). These formulas provide the set of relations to estimate each of the mode mismatch parameters from the others, and potentially the overall mismatch phase $\\ensuremath{\\psi_\\text{R}}$ as well. These relations are calculated using the assumptions of this section: the two-mode approximation and that $\\ensuremath{\\mathfrak{r}}_{\\text{hom}} \\simeq 1$.\n\nTogether, the relations of this section give insight in to how the physical mismatch parameters, $\\ensuremath{\\Upsilon_{\\text{I}}}$, $\\ensuremath{\\Upsilon_{\\text{O}}}$ and $\\ensuremath{\\Upsilon_{\\text{F}}}$ contribute to squeezing degradations. $\\ensuremath{\\Upsilon_{\\text{R}}}$ is a new form of effective mismatch parameter that is directly measurable from squeezing data, using the analysis of \\cref{sec:experiment}. It indicates how squeezing changes with frequency due to \\crefrange{eq:effective_MMR_loss2}{eq:effective_MMR_loss1}. Together, the complex, coherent interactions of transverse modal mixing on squeezed state can be concisely characterized in cavity-enhanced interferometers.\n\n\\subsection{Implications for Frequency Dependent Squeezing}\nThis analysis of the transverse mismatch applies to the reflection of squeezing off of any form of cavity. Namely, the detuned filter cavity for frequency-dependent rotation of squeezing in the LIGO A+ upgrade. This cavity will be installed on the input, $\\Tmat{H}_\\ensuremath{\\text{I}}$ section of the squeezing transformation sequence. The filter cavity mismatch loss $\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}$ will behave analogously to $\\ensuremath{\\Upsilon_{\\text{I}}}$, introducing losses of ${\\sim}4\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}$ at frequencies resonating in the cavity. The mismatch loss adds to those caused by the internal round-trip cavity loss $\\Lambda_{\\ensuremath{\\text{fc}}}$, creating the effective loss $\\Lambda_{\\ensuremath{\\text{FC}}} \\approx 4\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}} + 4\\lambda_{\\ensuremath{\\text{FC}}}\/\\gamma_{\\ensuremath{\\text{FC}}}$ using \\cref{eq:cavity_relations}.\n\nThe intra-cavity losses then set the scale for how much transverse mismatch is allowable before mismatch dominates the squeezing degradation, $\\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}} < \\Lambda_\\ensuremath{\\text{fc}} \/ T_\\ensuremath{\\text{fc}}$. More importantly, they add to the dephasing from the detuned cavity, by creating an effective $\\lambda'_\\ensuremath{\\text{FC}} = \\lambda_{\\ensuremath{\\text{FC}}} + \\ensuremath{\\Upsilon}_{\\ensuremath{\\text{fc}}}\\gamma_{\\ensuremath{\\text{FC}}}$ which can be used in \\cref{eq:detuning_dephasing}. The dephasing will set the limit to the allowable injected squeezing $e^{\\pm 2r}$ level as it introduces anti-squeezing at critical frequencies in the spectrum for astrophysics.\n\n\\section{Conclusions}\n\nBefore this work, the squeezing level in the LIGO interferometers was routinely estimated using primarily high-frequency measurements. This was done to utilize a frequency band where the classical noise contributions were small, while also giving a large bandwidth over which to improve the $\\Delta F \\Delta t$ statistical error in noise estimates. In doing so, LIGO recorded a biased view of the state of squeezing performance between the two instruments. The data analysis of this work has revealed several critical features to better understand and ultimately improve the quantum noise in LIGO.\n\nFirst, it indicates that the two sites have similar optical losses in their injection and readout components, as seen from the low-frequency losses of \\cref{fig:data_Q}. There is still a small excess of losses over the predictions given in \\cite{Tse-PRL19-QuantumEnhancedAdvanced}, but substantially less than implied when estimating the losses using high frequency observations. The most culpable loss components in the LIGO interferometers are being upgraded for the next observing run.\n\nSecond, this data analysis indicates that squeezing is degraded particularly at high frequencies, and the modeling and derivations provide the mechanism of transverse optical mode mismatch, external to the cavities, as a plausible physical explanation. This will be addressed in LIGO through the addition of active wavefront control to better match the beam profiles between the squeezer's parametric amplifier, new filter cavity installation, interferometer, and output mode cleaner.\n\nThird, the quantum radiation pressure noise is now not only measured, but employed as a diagnostic tool along with squeezing. QRPN indicates that the effective local oscillator angle in the Michelson fringe offset light at LLO is a specific, nonzero, value. This indicates that to power up the detector further, while maintaining a constant level of fringe light, the angle will grow larger and cause more pronounced degradation of the sensitivity by projecting out of the signal's quadrature. Ultimately, the LO angle should become configurable using balanced homodyne detection, another planned upgrade as part of ``A+''.\n\nFinally, this work carefully derives useful formula to manipulate the quantum squeezing response metrics. These are useful to reason and rationalize the interactions of squeezing with ever more complex detectors, both for gravitational-wave interferometers, and more generally as squeezing-enhanced optical metrology becomes more commonplace. The design of a future generation of gravitational wave detectors must be optimized specifically to maintain exceptional levels of squeezing compared to today. The quantum response metrics derived in this paper will aid that design work by simplifying our interpretation of squeezing with simulations. With these diagnostics and the data from observing run 3, LIGO is now better prepared to install and characterize frequency dependent squeezing in its ``A+'' upgrade not as a demonstration, but for stable, long-term improvement of the quantum enhanced observatories to detect astrophysical events.\n\n\\begin{acknowledgments}\n LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation, and operates under Cooperative Agreement No. PHY-0757058. Advanced LIGO was built under Grant No. PHY-0823459. The authors gratefully acknowledge the National Science Foundation Graduate Research Fellowship under Grant No. 1122374. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe $S$ matrix describing electron scattering at ultrarelativistic pointlike \ncharges was shown to be determined by the \ngauge phase leading to the Dirac equation represented in the temporal gauge\n\\cite{Eichmann}. \nWe found that it naturally exhibits the same form as the well known eikonal\nexpression, as is expected by Lorentz invariance. \n\nThe gauge phase leading to the temporal gauge reads\n\\[\n\\phi(x)=e^{-i\\int_{-\\infty}^{t}A_0(x)dt'}\n\\]\nThe consideration of asymptotic states corresponds to sending the upper\nbound of the integral to infinity. \nThe infinite time integral  over the scalar part of the\nelectromagnetic potential in the exponent \nhas to be understood as the principal value of the integral. \nIt can be decomposed into a finite term and an infinite quantity,\nexpressing the familiar divergence of phases in Coulomb scattering. The\ninfinite term can be removed by a gauge transformation which converts the\nCoulomb boundary conditions of the original problem into a modified \nshort-range potential allowing for asymptotic plane wave solutions. \n\nThe ultrarelativistic limit of the gauge transformed potential $A_0'(x)$ \nreads \\cite{Aichelburg,Baltz3,Segev1,Eichmann}\n\\begin{equation}\n\\label{potlim}\n\\lim_{\\gamma \\to \\infty} A_0'(x) = Z\\alpha \\delta (z-t) \\ln x_\\perp^2 \n\\end{equation}\nand hence we obtain for the $S$ operator in coordinate space \n($\\hat{\\gamma}_-=\\hat{\\gamma}_0-\\hat{\\gamma}_3$ is the Dirac matrix structure of\nthe interaction) \n\\begin{equation}\n\\label{sopcoo}\nS=e^{-iZ\\alpha \\ln x_\\perp^2 }\\hat{\\gamma}_-\n\\end{equation}\nThe obtained $S$ operator is a\nunitary operatorb due to its conformity to the gauge phase. \nIt agrees with the first term of the Magnus expansion of   \nthe time-evolution-operator \\cite{Magnus}, \nsince the considered gauge-transformed \ninteraction was assumed to be compressed to infinitely short times. \n\nThis result was proven to be of completely\nperturbative nature \\cite{Eichmann}.\nNote, however, \nthat the perturbative derivation did not require the explicit \nFourier transform of the transverse part of (\\ref{potlim}),\nwhich is an\nill defined object. \nFor that reason the deduction of the small-coupling limit \n($Z\\alpha\\to 0$) of (\\ref{sopcoo}) in momentum space \nmust be treated with special care. \n\nIn a rigorous distributional sense \nit can be defined as the {\\it weak limit} $\\lambda\\to 0$\n\\cite{Ferrari}\n\\begin{equation} \n\\label{ftlog}\n\\int d^2x_\\perp e^{-i\\vec{k}_\\perp\\vec{x}_\\perp}\\ln x_\\perp^2\n=\\lim_{\\lambda \\to 0}4\\pi\\left(\\frac{-1}{k_\\perp^2+ \\lambda^2}-\\pi\\delta^2\n(k_\\perp)\\ln\\left(\\frac{\\lambda^2}{\\mu^2}\\right)\\right)\n\\end{equation}\nwith $\\lambda =\n\\omega\/\\gamma$, $\\mu=2\/e^C$.\n'Weak limit' means \nthat the limit $\\lambda\\to 0$ \nhas to be taken after having integrated the expression with a test function. \nThe second term on the RHS arises from the gauge transformation applied to\nthe potential and thus accounts for the Coulomb distortions. \n\nDiscarding the second term and taking the limit $\\lambda\\to 0$ \ndirectly would \naccidentally yield an expression for the \nFourier transform of (\\ref{potlim}), being identical to \nthe Fourier transform of the ungauged\npotential in the limit $\\gamma\\to \\infty$ \n\\[                                      \n\\int d^4x \\;e^{ikx}A_0=-(2\\pi)^2 Z \\alpha \\delta (k_-)\\frac{2}{k^2_\\perp}\n\\]\n\nThis error is made if one intends to extract the correct small-coupling\nlimit from a naive Taylor expansion \nof the Fourier transformed $T$ matrix \nwhose linear term reads\n\\begin{equation}\n\\label{ttaylor}\n\\lim_{Z\\alpha\\to 0}T(k)\\approx (2\\pi)^2\\delta (k_-) iZ \\alpha\n\\frac{2}{k^2_\\perp} \\overline{u}(p')\\hat{\\gamma}_-u(p)\n\\end{equation}\nHere $u(p)$ is the electron unit spinor,\n$p$ and $p'$ are the initial and final  momenta of the electron,\nrepectively, $k=p'-p$.\n\nSince, however,  Taylor expansion and Fourier transformation \ndo not commute in this case, \nthe Taylor expansion of the Fourier transformed $T$ matrix for this purpose\nis not justified.\\footnote{Note, that with the above mentioned\ndistribution-theoretical precautions, it is possible to obtain the correct\nresult via Taylor expansion \\cite{Grignani}.}\nThe correct small-coupling limit of the scattering amplitude in\nmomentum space can thus not be found by a naive \nTaylor expansion of the Fourier transformed $T$ matrix and does not agree \nwith first-order\nperturbation theory. According to (\\ref{ftlog}) this is simply \nbased on the fact, that \nthe gauge transformed potential correctly\naccounts for Coulomb boundary conditions. \n\nIn the following \nwe want to investigate, how the correct treatment of Coulomb boundary\nconditions in all orders of \nperturbation theory influences the cross section of\nthe scattering process. \n\n\n\n\n\n\n\n\n\\section{Implications on the cross section}\n\nWe consider the exact amplitude for electron scattering at\nan ultrarelativistic pointlike charge, moving in $+z$ direction. \nWe state it in terms of the invariant squared momentum transfer \n$t\\approx-k_\\perp^2$ for\n$\\gamma\\to \\infty$\\footnote{Note the striking\nsimilarity between (\\ref{ultraamp}) and the nonrelativistic (Schr\\\"odinger) \namplitude\n\\[\nf(\\theta)=-\\frac{1}{2k^2\\sin^2\\frac{\\theta}{2}}\\frac{\\Gamma\n\\left(1+\\frac{i}{k}\\right)}{\\Gamma\n\\left(1-\\frac{i}{k}\\right)}e^{-\\frac{i}{k}\\ln \\sin^2 \\frac{\\theta}{2} }\n\\]\nwith the squared momentum transfer being proportional to $\\sin^2 \\theta \/2$.\n}\n\\begin{eqnarray}\n\\label{ultraamp}\nA&=& 2\\pi\\delta (k_-) F_{p',p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\n\\overline{u}(p')\\hat{\\gamma}_-u(p)\\nonumber \\\\\n&=&-i8 \\pi^2 Z\\alpha\n\\delta (k_-)\\frac{1}{t}\n\\frac{\\Gamma(-i\\alpha Z)}{\\Gamma(i\\alpha Z)}\ne^{-iZ\\alpha \\ln (-t\/4)}\\overline{u}(p')\\hat{\\gamma}_-u(p)\n\\end{eqnarray}\n$F_{p',p}$ abbreviates the Fourier transform with respect to the transverse\ncoordinates, taken at the momentum \n$\\vec{k}_\\perp=(\\vec{p'}_\\perp-\\vec{p}_\\perp)$.\nThe cross section for this scattering process is found to be exactly the Mott\nformula for Coulomb scattering of ultrarelativistic electrons \nat a static source, \nLorentz-transformed into the electron's rest frame. \nSuch kind of agreement between the exact result and the first order\nperturbation theory is also found in the nonrelativistic case, known\nas one of the peculiarities of the Coulomb field.\n\nThe well established \neikonalization of the scattering amplitude and thus the \nreduction to Mott's result imply, that in the high-energy limit \nthe electron and the positron Coulomb scattering cross section\nbecome identical. This behaviour of the cross section at\nultrarelativistic energies \nis required by the Pomeranchuk theorem \\cite{Itzykson}. \n\nOne can draw an analogy to pomeron exchange in hadron physics: \nThe scattering process can be described in terms of the single exchange of\nan ''effective photon`` according to the modified potential\n\\begin{equation}\n\\label{modpot}\nV_0(x)= V_3(x)\n=\\delta (z-t)\\left(\\left(\\frac{1}{x_\\perp}\\right)^{2iZ\\alpha}-1\\right)\n\\end{equation}\n\nIn the field of\ntwo ultrarelativistic colliding pointlike nuclei, \nthe exact scattering amplitude \nwas shown to retain the structure of the\nsecond-order perturbative result, due to the causal decoupling property\n\\cite{Eichmann}. \nEach interaction can be described by the modified potential (\\ref{modpot}).\nIn the following we consider the symmetric collision of two \nions with charge $Ze$, the extention to asymmetric collisions is trivial. \n\nAccounting for both time orderings, the amplitude reads\n\\begin{eqnarray}\nA^{tot}_{p'p}&=&\\int \\frac{d^2k_\\perp}{(2\\pi)^2}\nF_{k,p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\nF_{p',k}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\ne^{i(\\vec{p'}_\\perp-\\vec{k}_\\perp)\\cdot \\vec{b}}\n\\nonumber \\\\\n&&\\left(\\overline{u}(p')\\frac{-\\hat{\\vec{\\alpha}}_\\perp \n\\cdot \\vec{k}_\\perp + \n\\gamma_0\nm}{p'_+ p_- - {k}_\\perp^2 -m^2 +i\\epsilon} \n\\hat{\\gamma}_+u(p)\\right.\n\\nonumber\\\\\n\\label{Ttot}\n&&\\left.\\overline{u}(p')\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot (\\vec{p}_\\perp +\\vec{p'}_\\perp -\\vec{k}_\\perp) +\n\\gamma_0\nm}{p'_- p_+ -\n(\\vec{p}_\\perp +\\vec{p'}_\\perp -{k}_\\perp)^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_-u(p)\\right)\n\\end{eqnarray}\nHere the trajectory of one ion is shifted by the impact\nparameter $\\vec{b}$, which is accounted for by the factor\n$e^{i(\\vec{p'}_\\perp-\\vec{k}_\\perp)\\cdot \\vec{b}}$.\n\nWe now use the crossing invariance of the amplitude to apply the\nobtained result to electron-positron pair production. The initial electron\nfour momentum $p$ has then to be replaced by the negative final positron\nmomentum $p\\to -p^p$. The final electron momentum will be\ndenoted by $p'=p^e$.\nWith (\\ref{Ttot}) we obtain for the pair production probability\n\\begin{eqnarray}\n\\frac{d\\sigma}{d^2b} &=&  |A^{tot}_{p'p}|^2\n\\frac{md^3p^e}{(2\\pi)^3E^e}\\frac{md^3p^p}{(2\\pi)^3E^p}\n\\nonumber \\\\\n&=&\n\\frac{md^3p^e}{(2\\pi)^3E^e}\\frac{md^3p^p}{(2\\pi)^3E^p}\n\\int \\frac{d^2k_\\perp}{(2\\pi)^2}\\int \\frac{d^2k'_\\perp}{(2\\pi)^2}\nF_{k,-p^p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\nF_{p^e,k}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\\nonumber\\\\\n&&F^\\ast_{k',-p^p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\nF^\\ast_{p^e,k'}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)\ne^{i(\\vec{k'}_\\perp-\\vec{k}_\\perp)\\cdot \\vec{b}}\\nonumber\n\\\\\n&&\\left(\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot \\vec{k}_\\perp +\n\\gamma_0\nm}{-p^e_+ p^p_- - {k}_\\perp^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_+u(-p^p)\\right.\\nonumber\\\\\n&&\\left.\n+\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot (-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -\\vec{k}_\\perp) +\n\\gamma_0\nm}{-p^e_- p^p_+ -\n(-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -{k}_\\perp)^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_-u(-p^p)\\right)\\nonumber\\\\\n&&\\times\\left(\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot \\vec{k'}_\\perp +\n\\gamma_0\nm}{-p^e_+ p^p_- - {k'}_\\perp^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_+u(-p^p)\\right.\\nonumber\\\\\n\\label{csb}\n&&\\left.\n+\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot (-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -\\vec{k'}_\\perp) +\n\\gamma_0\nm}{-p^e_- p^p_+ -\n(-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -{k'}_\\perp)^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_-u(-p^p)\\right)^\\ast\n\\end{eqnarray}\n\nThe integration over the impact parameter yields the pair production cross\nsection. \nDue to the $\\delta^2(\\vec{k'}_\\perp-\\vec{k}_\\perp)$-function occuring in \nthe $\\vec{b}$ integration, \na further momentum integral can be perfomed, leaving \n\\begin{eqnarray}\nd\\sigma&=&\\frac{md^3p^e}{(2\\pi)^3E^e}\\frac{md^3p^p}{(2\\pi)^3E^p}\\int\n\\frac{d^2k_\\perp}{(2\\pi)^2}\n|F_{k,-p^p}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)|^2\n|F_{p^e,k}(e^{-iZ\\alpha \\ln x_\\perp^2}-1)|^2\\nonumber\n\\\\\n&&\\left|\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot \\vec{k}_\\perp +\n\\gamma_0\nm}{-p^e_+ p^p_- - {k}_\\perp^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_+u(-p^p)\\right.\\nonumber\\\\\n\\label{cs}\n&&\\left.+\\overline{u}(p^e)\\frac{-\\hat{\\vec{\\alpha}}_\\perp\n\\cdot (-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -\\vec{k}_\\perp) +\n\\gamma_0\nm}{-p^e_- p^p_+ -\n(-\\vec{p^p}_\\perp +\\vec{p^e}_\\perp -{k}_\\perp)^2 -m^2 +i\\epsilon}\n\\hat{\\gamma}_-u(-p^p)\\right|^2\n\\end{eqnarray}\nThus, upon integration over the whole impact parameter plane, \nthe phases in the individual scattering amplitudes (see (\\ref{ultraamp})) \ncancel. Consequently, in the limit $\\gamma\\to \\infty$ the \ncross section is found to reduce \nto the lowest-order\nperturbation theory, \nthe two-photon result. \nThis behaviour does not only naturally explain \\cite{Segev2} \nthe experimentally \nobserved quadratic\ndependence on the target's and the projectile's charge \n\\cite{Vane}, \nbut also implies,\nthat no asymmetries should occur in \nelectron and positron spectra.  \n\nEquation (\\ref{cs}) is strictly valid only for pointlike ions. \nThe focus on\nelectromagnetic reactions in peripheral heavy-ion collisions \nimplies a restricted range of impact parameters with a lower bound \n$b=r_A+r_B$, $r_A$ and $r_B$ being the radii of the ions.\nTherefore \nin experiments which are triggered on peripheral collisions, effects of the \nCoulomb distortions \ndescribed \nby the phase in (\\ref{ultraamp}) will be visible. \n\nThe eikonal approximation (and thus the cross section) is known to become\nenergy-independent in the ultrarelativistic limit \\cite{Cheng-Wu}. \nThis dependence  \nis restored by accounting for the correct transverse \nmomentum range, which is restricted by the validity of (\\ref{potlim}). This\ncondition reads \\cite{Baltz,Eichmann}\n\\begin{equation}\n\\label{applcond}\n|\\vec{k}_\\perp|\\gg \\frac{\\omega}{\\gamma}\n\\end{equation}\nSuch a low energy cut off is also necessary to cure the divergence in\n(\\ref{Ttot}). \n\n\\section{Equivalent Photon Approximation}\nWe intend to study the behaviour of the cross section,\nboth impact parameter dependent and impact parameter integrated, in the\nWeizs\\\"acker-Williams method of equivalent photons. \nThis approximation uses the similarity between the fields of a fast\nmoving charge and a swarm of real photons moving in beam direction. It\napproximately correstponds to the first order Born approximation in the\ntemporal gauge: Only the transverse part of the interaction is considered -- \nthe longitudinal part is suppressed by $1\/\\gamma^2$ -- and the vertex\nfunction is evaluated on-shell at $k^2=0$, i.e. for an assumed real photon.\nRewriting the exact  \ncross section in terms of the real photon cross section, the\nwhole information about the scattering potential, which can be the retarded\nCoulomb potential or the modified potential (\\ref{modpot}), respectively, is\nthen contained in the distribution function of the equivalent photons\n$n(\\omega)$. \nRoughly speaking, this photon distribution function is determined by the\nsquared absolut value of the Fourier-transformed potential (in temporal\ngauge).\nThe obvious \nadvantage of casting the exchange of effective photons according to\n(\\ref{modpot}) in the Weizs\\\"acker-Williams form is, that \nany difference between the second-order\nperturbative result and the exact calculation will be solely generated by \ndifferences between the equivalent photon distributions.\n\nThe Weizs\\\"acker-Williams approximation is applicable if the exchanged\nmomentum meets the following conditions \\cite{LandauIV} \n\\begin{equation}\n\\label{wwbed1}\n\\frac{\\omega}{\\gamma^2}\\ll |\\vec{k}_\\perp|\\ll m\n\\end{equation}\nand\n\\begin{equation}\n\\label{wwbed2}\n|\\vec{k}_\\perp|\\ll \\omega \\ll m \\gamma\n\\end{equation}\nThe upper bounds mainly stem from the requirement, that $|k^2|$\n is negligible compared to $2p\\cdot k \\ge m^2$, such that the intermediate\n propagators of the scattered particles \ncan be approximated by those describing the interaction with real photons. \nThe particle's rest mass in (\\ref{wwbed1}) is however \na conservative upper\nbound and the equivalent photon method is not strictly invalid for $|k^2|\n\\sim m^2$. \nNote, that for the approximative\ncalculations in \\cite{Bottcher}, the transverse\nmass of the scattered particle was taken as the upper bound for\n(\\ref{wwbed1}).\n\n\nSince the exact amplitude takes the eikonal form, we must point out the\nfollowing: \nThe expansion of the ultrarelativistic\nscattering amplitude in powers of the transferred\nmomentum yields, as the leading term, the eikonal expression (describing the\nminimal deflection from the initial straight-line trajectory)\n\\cite{Abarbanel,Sugar}. Its\nperturbation-theoretical derivation requires that the\nquadratic terms $k^2$ are negligible \nrelative to the terms $2p_i\\cdot k$ in the denominators\nof the propagators, where $k$ is any partial sum of the internal momenta\n\\cite{Levy}.\nThe exact validity of the eikonal formula at infinite energies \ntherefore shows, that the \ntransferred momentum $|k^2|$ does not exceed $m^2$ (irrespective of the\nvalue\nof $m$). \nThis agrees with the theoretical observation, that at high energies\nparticles\nare predominantly scattered into a cone with opening angle $\\theta \\sim 1\/\n\\gamma$, corresponding to momentum transfers $|k^2|\\sim m^2$. The main\ncontributions to the cross section are thus expected from spatial distances\nlarger or equal the Compton wavelength of the particle.\\footnote{From the\nasymmetry of electron and positron spectra\nproduced in $S$(200 GeV\/n)+Au collisions , the\nmean transverse distance from the target ion was deduced to be approximately\ntwo Compton wavelengths \\cite{Vane}. The collision energy corresponds to\n$\\gamma \\approx 10$ in the center of speed system.}\n\nMoreover, the longitudinal part of the interaction vanishes identically in\nthe\nlimit $\\gamma\\to \\infty$.\nHence, the applicability conditions of the Weizs\\\"acker-Williams\napproximation\nare trivially fulfilled in the limit $\\gamma\\to \\infty$.\\footnote{Just as\nthe\neikonal formula, the Weizs\\\"acker-Williams approximation can be viewed as\nthe leading term of an expansion in powers of $k^2\/m^2$ \\cite{Brodsky}. The\nvalidity of the eikonal expression then automatically implies the validity\nof\nthe Weizs\\\"acker-Williams method.}\n\nWe have the freedom to apply this method to the interaction of the electron\nwith both nuclei, \ngiving two possibilities (see Figure \\ref{fig01})\n\\begin{figure}[h]\n\\begin{center}\n\\begin{minipage}{16cm}\n\\parbox{7cm}{\\psfig{figure=comptww.eps,width=6cm}}\\hspace{1cm}\n\\parbox{7cm}{\\psfig{figure=bremsww.eps,width=6cm}}\n\\end{minipage}\n\\end{center}\n\\caption{\\label{fig01}The two possible distinct processes, that can be used\nto describe electron-positron production in heavy-ion collision. One or both  \nultrarelativistic ions can be replaced by an equivalent photon distribution.\nIf the bremsstrahlung process b) is calculated in one ion's rest frame, the\nelectron must be assumed ultrarelativistic, to yield agreement with\na).}\n\\end{figure}\n\nThe two possible calculation schemes (Figure \\ref{fig01})\nagree, since in b) \nthe bremsstrahlung emission and the scattering at the external\npotential decouple. This is due to the fact, that the region in which the\nultrarelativistic electron ''feels`` the external field is assumed to be pointlike and\nany frequency of the emitted photon is ''soft`` compared to the timescale of\nthe scattering.\nCoulomb effects arise, if one explicitly accounts for the finite\ninteraction time, either in the scattering process by correcting the eikonal\nformula\n\\label{eikexp} \nor by keeping the eikonal amplitude for the scattering process\nbut assuming a Rutherford-deflected trajectory for the photon emission\n\\cite{LandauIV}. Corrections to the eikonal formula account for higher\norders of e.g. the Magnus expansion \\cite{Magnus},\nwhich is an expansion in the interaction time $\\tau$\naround the\ninstantaneous interaction $\\tau\\sim 1\/\\gamma\\to 0$.\nIn general these Coulomb effects vanish, if the\nenergy of the emitted photon is too small to resolve details of the\nscattering process, and the recoil of the electron is negligible.  \n\nTo apply the Weizs\\\"acker-Williams method to the bremsstrahlung photon, \nthe recoil of the\nbremsstrahlung photon\nmust, however, be assumed\nnegligible. \nThe small momentum\ntransfer \nis in turn ensured by the\neikonalization of the scattering process. \n\nThe equivalent single-photon distributions $n_{A\/B}(\\omega)$ \nof the ions $A$\nand $B$, are determined from the effective potential\n(\\ref{modpot}). \nThe photon distribution reads \n\\begin{equation}\nn(\\omega)=\\frac{1}{4\\pi^2 \\alpha \\omega} \n\\int_{\\omega\/\\gamma}^{m} k_\\perp dk_\\perp \n\\left|k_\\perp\\pi\nZ\\alpha\\left(\\frac{4}{k_\\perp^2}\\right)^{1-iZ\\alpha}\\frac{\\Gamma(-i\\alpha\nZ)}{\\Gamma(i\\alpha Z)}\\right|^2=\n\\frac{2Z^2 \\alpha}{\\pi\\omega}\n\\ln\\left(\\frac{m\\gamma}{\\omega}\\right)\n\\end{equation}\nThe lower bound of the integral is taken from the condition\n(\\ref{applcond}).\nThe upper bound, the electron rest mass, is imposed by (\\ref{wwbed1}).\nThe prefactor arises from properly rewriting the cross\nsection (\\ref{cs}) in terms of the real photon cross section (i.e. the\nCompton cross section) and photon\ndistribution functions. \n\nThis photon distribution \ncoincides with the equivalent-photon distribution obtained from the\nCoulomb potential to logarithmic accuracy \\cite{LandauIV} \nand is not changed by Coulomb\neffects. \n\n\n\n\n\n\n\n\n\n\n\\section{Impact parameter dependent cross section}\nThe impact parameter dependent equivalent photon method for the exact\ncalculation, using the modified potential (\\ref{modpot}), \ncan be derived similarly to\n\\cite{Vidovic}. \nTo this end \nwe have to modify the integrands in (\\ref{csb}) such that they  \naccount for the limited momentum range ((\\ref{applcond}) and\n(\\ref{wwbed1})). \n\n\nThe cut off of low transverse momenta according to (\\ref{applcond}) can be\nachieved by the following replacement in \n(\\ref{ultraamp})\\footnote{Note, that for the\nSchr\\\"odinger case the exact validity of the eikonal formula \ncan be proven for a certain off shell domain of the \nmomentum transfer for the whole energy plane \\cite{Banyai}.} \n\\begin{equation}\n\\label{text}\nt=-k_\\perp^2 \\to -\\frac{\\omega^2}{\\gamma^2}-k_\\perp^2\n\\end{equation}\nThis substitution suppresses small transverse momenta less strongly than the\nstrict cut off at $k_\\perp=\\omega \/\\gamma$. It assumes the sufficient\naccuracy of the classical\neikonal amplitude for\n$1\/\\gamma$ in the near vicinity of $1\/\\gamma =0$, which is guaranteed by the\npossibility of continuous extentions of the eikonal formula towards finite\n$\\gamma$ and large $t$. \nA Yukawa-type damping of transverse distances corresponding to the cut off\ntransverse momenta yields additional terms that change the character of the\namplitude significantly and can not be motivated physically. \n\nIn accordance with the exact validity of the eikonal formula, the physical\nsituation is such, that the transferred momenta are restricted by  the\ncondition $|k^2|\\ll m^2$. They are, however, \nnaturally cut off, if one introduces \na form factor to account\nfor the finite extent of the nuclei. \nThus, large momenta \nhave to be cut off at $k_\\perp\\approx 1\/r_>$, where $r_>$ is the larger \nvalue of either \nthe nuclear radius or the Compton wavelength of the scattered\nparticle \\cite{Jackson}. In this respect, the electron is an\nexception, since all other particles have Compton wavelengths smaller or\ncomparable to the nuclear size.\nTo present the calculations in a uniform manner, we use the form factor of\nthe nucleus to cut off the large momenta. \n\nThe impact parameter dependent cross section for particle production,\ndescribed in the equivalent photon method reads \\cite{Vidovic}\n\\begin{equation}\n\\label{csbww}\n\\frac{d\\sigma}{d^2b}=\\int d\\omega_1 \\int d\\omega_2\n\\left[n_\\|(\\omega_1,\\omega_2,\\vec{b})\n\\sigma^{\\gamma\\gamma}_\\|(\\omega_1,\\omega_2) +\nn_\\perp(\\omega_1,\\omega_2,\\vec{b})\n\\sigma^{\\gamma\\gamma}_\\perp(\\omega_1,\\omega_2)\\right]\n\\end{equation}\nwith the two-photon distribution functions\n$n_{\\|\/\\perp}(\\omega_1,\\omega_2,\\vec{b})$. The elementary two-photon cross\nsections and the two-photon distribution functions explicitly account for\nthe parallel or orthogonal orientation of the photon-polarizations, denoted\nby the indices $\\|$ and $\\perp$, respectively. \nSince the integration over the impact\nparameter plane implies an averaging over the photon polarizations, the\nexplicit occurrence of the photon polarizations in the impact parameter\ndependent cross section is expected.\nThe functions\n$n_{\\|\/\\perp}(\\omega_1,\\omega_2,\\vec{b})$ can be expressed in terms of\nsingle-photon distribution functions $n(\\omega,b)$, \ndepending on the transverse separation:\n\\begin{eqnarray}\n\\label{n2par}\nn_\\|(\\omega_1,\\omega_2,\\vec{b})&=&\\int d^2x_\\perp\nn(\\omega_1,\\vec{x}_\\perp-\\vec{b})\\,n(\\omega_2,\\vec{x}_\\perp)\n\\left(\\frac{(\\vec{x}_\\perp-\\vec{b})\\cdot\n\\vec{x}_\\perp}{|\\vec{x}_\\perp-\\vec{b}||\\vec{x}_\\perp|}\\right)\\\\\n\\label{n2senk}\nn_\\perp(\\omega_1,\\omega_2,\\vec{b})&=&\\int d^2x_\\perp\nn(\\omega_1,\\vec{x}_\\perp-\\vec{b})\\,n(\\omega_2,\\vec{x}_\\perp)\n\\left(\\frac{(\\vec{x}_\\perp-\\vec{b})\\times \n\\vec{x}_\\perp}{|\\vec{x}_\\perp-\\vec{b}||\\vec{x}_\\perp|}\\right)\n\\end{eqnarray}\nwith\n\\begin{equation} \n\\label{photdist}\nn(\\omega,b)=\\frac{Z^2\\alpha}{\\pi^2\\omega}\\left|\\int_0^\\infty dk_\\perp k_\\perp^2\n\\frac{F(k_\\perp^2+\\omega^2\/\\gamma^2)}{(k_\\perp^2+\\omega^2\n\/\\gamma^2)^{1-iZ\\alpha}}J_1(bk_\\perp)\\right|^2\n\\end{equation}\n$J_1$ is a Bessel function. The function $F$ denotes the chosen form\nfactor of the nucleus. \n\nFor a pointlike charge ($F\\equiv 1$), the photon distribution function can\nbe calculated analytically. We obtain for the Coulomb potential and the\nmodified potential \n\\begin{equation}\n\\label{photdistpt}\nn(\\omega,b)=\\left\\{\\begin{array}{ll}\n\\displaystyle \\frac{Z^2\\alpha}{\\pi^2}\\frac{\\omega}{\\gamma^2}\\left[K_1\\left(\\frac{\\omega\nb}{\\gamma}\\right)\\right]^2\\;\\;\\;&{\\rm retarded\\;Coulomb\\;potential}\\\\\n&\\\\\n\\displaystyle\n\\frac{Z^2\\alpha}{\\pi^2}\\frac{\\omega}{\\gamma^2}\\left[\\frac{K_{1+iZ\\alpha}(\\omega\nb\/\\gamma)}{\\Gamma(1-iZ\\alpha)}\\right]^2&{\\rm\nmodified\\;potential\\;(\\ref{modpot})}\n\\end{array}\\right.\n\\end{equation}\n$K_\\nu$ is a modified Bessel function. For small arguments of the Bessel\nfunction one can use the asymptotic expression \\cite{Abramowitz}\n\\begin{equation}\n\\label{kasym}\nK_\\nu (z)\\sim \\frac{1}{2}\n\\Gamma (\\nu )\n(\\frac{1}{2} z)^{-\\nu }\n\\end{equation}\nTherefore, for $\\omega b \\ll \\gamma$ and assumed\npoint-like charges the photon\ndistribution functions (\\ref{photdistpt}) nearly completely agree. \n\\newpage \n\\begin{figure}[h]\n\\centerline{\\psfig{figure=photvert.eps,width=17cm}}\n\\caption{\\label{fig02}The single-photon distribution function for various \nphoton energies (indicated in the plot) as\na function of the transverse distance from the ultrarelativistic charge. The\ncalculations are done for lead ions ($Z=82$) at\nLHC energies ($\\gamma\\approx 3000$). }\n\\end{figure}\n\nWe have numerically evaluated the photon distribution function\n(\\ref{photdist}) for an extended nucleus, using a gaussian form factor\n$F(Q^2)=e^{-Q^2\/(2Q_0^2)}$ with $Q_0=60MeV$ which describes the $Pb$ nucleus \n\\cite{Drees}. \nFigure \\ref{fig02} shows a comparison between the photon distribution\nfunctions for both, point like nuclei ($F\\equiv 1$) and extended nuclei\nusin either \nthe\nretarded Coulomb potential or the modified potential (\\ref{modpot}),\nwhich represents the exact calculation. \nAt small distances up to a few multiples\nof the nuclear radius, the modified potential gives a smaller number of\nequivalent \nphotons than the pure Coulomb potential. For large distances or large photon\nenergies (\\ref{kasym}) looses its validity and the modified potential gives\na larger number of photons than the Coulomb potential (see Figure\n\\ref{fig03}). \n\nThe significance of these results can be assessed \nin single-photon induced processes,\nsuch as the electromagnetic dissociation of nuclei in peripheral heavy-ion\ncollisions. In the Weizs\\\"acker-Williams approach \nthe dissociation probability at a given impact parameter is\ncalculated by integrating \nthe product of the measured \nphoton-nucleus dissociation cross section of the target,\n$\\sigma_T^\\gamma(\\omega)$, \nand the\nequivalent photon distribution, $n_P(\\omega,b)$, \nof the projectile (\\ref{photdist}) \nover the photon energy \\cite{Vidovic2,Norbury0}\n\\[ \nP(b)=\\int d\\omega \\,n_P(\\omega,b)\\, \\sigma_T^\\gamma(\\omega)\n\\]\nAs shown in \\cite{Vidovic2} for $Pb+Pb$ collisions at LHC energies, \nthe dissociation \nprobability violates unitarity at small impact parameters up to 25 fm. At\nsmall impact parameters, however, the photon distribution of the modified\npotential is suppressed relative to that of the pure Coulomb potential and\nyields a reduction of the probabiltiy, whereas at large impact parameters,\nthe dissociation probability is enhanced. The reduction visible in Figure\n\\ref{fig02} taken alone is too small to cure the unitarity violation.\nPossibly this can be achieved if in addition \nmulti-phonon excitations \\cite{Bertulani,Norbury1} are taken into account,  \nwhich also reduce the dissociation  probability at small impact parameters. \n\\begin{figure}[hbtp]\n\\centerline{\\psfig{figure=ratio.eps,width=17cm}}\n\\caption{\\label{fig03}The ratio of the equivalent photon numbers of the\npure Coulomb potential, $n_C(\\omega,x_\\perp)$, and the photon numbers of the\neffective potential, $n_{eff}(\\omega,x_\\perp)$. At small transverse\ndistances, one finds a deviation of up to $70\\%$ for small photon energies.\nFar outside the nucleus, the photon distribution functions are determined by\n(\\protect\\ref{photdistpt}). For large distances one asymptotically finds a\ndeviation in the order of $40\\%$, independent of the photon energy.}\n\\end{figure}\n\n\nThe impact parameter dependent two-photon distribution functions also have\nto be corrected according to (\\ref{photdist}), thus correcting the\npair production probability (\\ref{csbww}), i.e. (\\ref{csb}), for Coulomb\neffects. Due to the complicated convolution of the single-photon\ndistribution functions in (\\ref{n2par}) and (\\ref{n2senk}), effects on the\ntwo-photon distribution function are not obvious. For large distances\n$x_\\perp \\gg b$, however, one directly finds an enhancement of the\nequivalent two-photon numbers. \n\n\n\n\n\n\\section{Summary}\nIt was shown by either \nsumming the perturbation series \\cite{Eichmann} or by\nmatching plane waves at the delta function potential on the light front\n\\cite{Segev1,Baltz,Eichmann}, that the eikonal expression for the scattering\namplitude becomes exact in the ultrarelativistic limit ($\\gamma\\to \\infty$).\nThis allows to neglect the squared momentum transfer $k^2$ \nrelative to the term $2p_i\\cdot k$ in the denominator of the propagator of the\nscattered particle. As a consequence \nthe applicability\nconditions of the Weizs\\\"acker-Williams method are fulfilled automatically \n-- irrespective of the mass of the scattered particle. \n\nFurthermore, the exact validity of the eikonal formula for ultrarelativistic\nscattering processes confirm the Pomeranchuk theorem,\nstating that\nthe cross sections for antiparticle and particle scattering at a given\ntarget become identical in the ultrarelativistic limit. In analogy to\npomeron exchange in hadron physics, on can describe the exact interaction as\nthe exchange of an effective photon, according to a modified, effective\npotential given by (\\ref{modpot}). \nThe cross section, as a peculiarity of the Coulomb interaction, becomes\nidentical to the Mott result. The exact \npair production cross section in the field\nof two ultrarelativistic colliding (pointlike) ions also reduces to the\nsecond order perturbative result \\cite{Baltz} which was evaluated in \n\\cite{Bottcher}. This allows for two\nconclusions: i) The production rate scales with the square of the target and\nthe projectile charge \\cite{Vane,Segev2} ii) Asymmetries in the electron and\npositron spectra should not occur. \n\nNote, however, that the presented formalism is valid only if the produced\nparticles are fas with respect to both nuclei. \nTherefore, the observed \\cite{Vane} asymmetry at small electron and \npositron momenta remains unaffected by these considerations. \n\nWe applied the Weizs\\\"acker-Williams approach to pair production using the\nmodified potential (\\ref{modpot}), correctly accounting for the Coulomb\nboundary conditions. The impact parameter dependent single-photon\ndistribution calculated with the modified potential shows \ndeviations from the equivalent photon distribution function obtained from\nthe retarded Coulomb potential in order of up to $70\\%$ at small separations \nand approximately $40\\%$ at large separations from the ion. \n\nIn combination with multi-phonon excitation, the correct treatment of\nCoulomb distortions possibly solves the problem of unitarity violation in\nphotodissociation processes in ultrarelativistic heavy-ion collisions. \n\nThe pair production probability is also subject to changes due to the\nmodified photon numbers at given impact parameters and photon energies. \nThe perturbation theoretical probability, as calculated here, rather\nrepresents the average number of produced pairs and exceeds unity at\nsufficiently small\nimpact parameters. The ''true`` pair\nproduction probability has to be corrected by the vacuum-to-vacuum\namplitude, which in turn can be calculated from the perturbative pair\nproduction probability \\cite{Best,Rhoades-Brown}. This nontrivial influence\non the pair production cross section is subject of further studies. \n\n\\section*{Acknowledgements}\nThis work was supported by\n{\\it Deutsche Forschungsgemeinschaft} DFG within the project Gr-243\/44-2.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\\subsection{Motivation and previous work}\n\n\\IEEEPARstart{D}{riven} by the continuously increasing demands for high system throughput, low latency, ultra reliability, improved fairness and near-instant connectivity, \\gls{5G} wireless communication networks are being standardized \\cite{Shafi17} while, at the same time, insights and innovations from industry and academia are paving the road for the coming of the \\gls{6G} \\cite{David18}. As stated by Marzetta \\emph{et al.} in \\cite[Chapter 1]{Marzetta16}, there are three basic pillars at the physical layer that can be used to sustain the spectral and energy efficiencies that these networks are expected to provide: (i) employing massive \\gls{MIMO}, (ii) using \\gls{UDN} deployments, and (iii) exploiting new frequency bands.\n\nMassive \\gls{MIMO} systems, equipped with a large number of antenna elements, are intended to be used as \\gls{MU-MIMO} arrangements in which the number of antenna elements at each \\gls{AP} is much larger than the number of \\glspl{MS} simultaneously served over the same time\/frequency resources. The operation of massive \\gls{MIMO} schemes is based on the availability of \\gls{CSI} acquired through \\gls{TDD} operation and the use of \\gls{UL} pilot signals. Such a setting allows for very high spectral and energy efficiencies using simple linear signal processing in the form of conjugate beamforming or \\gls{ZF} \\cite{Marzetta10,Marzetta16}.\n\nIn \\glspl{UDN}, a large number of \\glspl{AP} deployed within a given coverage area cooperate to jointly transmit\/receive to\/from a (relatively) reduced number of \\glspl{MS} thanks to the availability of high-performance low-latency fronthaul links connecting the \\glspl{AP} to a central coordinating node. Coordination among \\glspl{AP} can effectively control (or even eliminate) intercellular interference in an approach that was first referred to as network \\gls{MIMO} \\cite{Karakayali06,Gesbert10}, later led to the concept of \\gls{CoMP} transmission \\cite{Irmer11} and, more recently, to that of \\gls{C-RAN} \\cite{Checko15}. In a \\gls{C-RAN}, the \\glspl{AP}, which are treated as a distributed \\gls{MIMO} system, are connected to a cloud-computing based \\gls{CPU} in charge, among many others, of the baseband processing tasks of all \\glspl{AP}. Conceptually similar to the \\gls{C-RAN} architecture, but explicitly relying on assumptions specific of the massive \\gls{MIMO} regime, distributed massive \\gls{MIMO}-based \\glspl{UDN} have been recently termed as \\emph{cell-free massive \\gls{MIMO}} networks \\cite{Ngo15,Ngo17}. In these networks, a massive number of \\glspl{AP} connected to a \\gls{CPU} are distributed across the coverage area and, as in the cellular collocated massive \\gls{MIMO} schemes, exploit the channel hardening and favorable propagation properties to coherently serve a large number of \\glspl{MS} over the same time\/frequency resources. Typically using simple linear signal processing schemes, they are claimed to provide uniformly good \\gls{QoS} to the whole set of served \\glspl{MS} irrespective of their particular location in the coverage area.\n\nSince the microwave radio spectrum (from 300 MHz to 6 GHz) is highly congested, the use of massive antenna systems and network densification alone may not be sufficient to meet the \\gls{QoS} demands in next generation wireless communications networks. Thus, another promising physical layer solution that is expected to play a pivotal role in \\gls{5G} and beyond \\gls{5G} communication systems is to increase the available spectrum by exploring new less-congested frequency bands. In particular, there has been a growing interest in exploiting the so-called \\gls{mmWave} bands \\cite{Rappaport13,Boccardi14,Akdeniz14,Rappaport17}. The available spectrum at these frequencies is orders of magnitude higher than that available at the microwave bands and, moreover, the very small wavelengths of \\glspl{mmWave}, combined with the technological advances in low-power CMOS \\gls{RF} miniaturization, allow for the integration of a large number of antenna elements into small form factors. Large antenna arrays can then be used to effectively implement \\gls{mmWave} massive \\gls{MIMO} schemes (see, for instance, \\cite{Gao18,Busari18} and references therein) that, with appropriate beamforming, can more than compensate for the orders-of-magnitude increase in free-space path-loss produced by the use of higher frequencies.\n\nThe performance of cell-free massive \\gls{MIMO} using conventional sub-6 GHz frequency bands and assuming infinite-capacity fronthaul links has been extensively studied in, for instance, \\cite{Ngo17,Nayebi17,Nguyen17,Ngo18}.\nCell-free massive \\gls{MIMO} networks using capacity-constrained fronthaul links have also been considered in \\cite{Bashar18b,Boroujerdi18} but assuming, again, the use of fully digital precoders in conventional sub-6 GHz frequency bands.\nSub-6 GHz massive \\gls{MIMO} systems are often assumed to implement a fully-digital baseband signal processing requiring a dedicated \\gls{RF} chain for each antenna element. The present status of \\gls{mmWave} technology, however, characterized by high-power consumption levels and high production costs, precludes the fully-digital implementation of massive \\gls{MIMO} architectures, and typically forces \\gls{mmWave} systems to rely on hybrid digital-analog signal processing architectures. In these hybrid transceiver architectures, a large antenna array connects to a limited number of \\gls{RF} chains via high-dimensional \\gls{RF} precoders, typically implemented using analog phase shifters and\/or analog switches, and low-dimensional baseband digital precoders are then used at the output of the \\gls{RF} chains \\cite{Ayach14,Gao16,Molisch17}. The network of phase shifters connecting the array of antennas to the \\gls{RF} chains determines whether the structure is fully or partially connected \\cite{Park17TWC}. Thus, the assumptions, methods and analytical expressions in \\cite{Ngo17,Nayebi17,Nguyen17,Ngo18,Bashar18b,Boroujerdi18} cannot by applied directly when assuming the use of \\gls{mmWave} frequency bands. Despite its evident potential, as far as we know, besides \\cite{Alonzo17,Alonzo18} there is no other research work on cell-free \\gls{mmWave} massive \\gls{MIMO} systems and, furthermore, the authors of these works did not face one of the main challenges in the implementation of cooperative \\glspl{UDN}, that is, the fact that these systems require of a substantial information exchange between the \\glspl{AP} and the \\gls{CPU} via capacity-constrained fronthaul links. Moreover, they also considered the use of oversimplified \\gls{mmWave} channel models and \\gls{RF} precoding stages, without constraining the available number of \\gls{RF}-chains at each \\gls{AP}.\n\n\\subsection{Aim and contributions}\n\nMotivated by the above considerations, our main aim in this paper is to address the design and performance evaluation of realistic cell-free \\gls{mmWave} massive \\gls{MIMO} systems using hybrid precoders and assuming the availability of capacity-constrained fronthaul links connecting the \\glspl{AP} and the \\gls{CPU}. The main contributions of our work can be summarized as follows:\n\n\\begin{itemize}[noitemsep,wide=0pt, leftmargin=\\dimexpr\\labelwidth + 2\\labelsep\\relax]\n\n\\item The performance of both the \\gls{DL} and \\gls{UL} of cell-free \\gls{mmWave} massive \\gls{MIMO} systems is considered with particular emphasis on the per-user rate, rather than the system sum-rate, by posing max-min fairness resource allocation problems that take into account the effects of imperfect channel estimation, power control, non-orthogonality of pilot sequences, and fronthaul capacity constraints. Instead of assuming the use of rather simple uniform quantization processes when forwarding information on the capacity-constrained fronthauls, the proposed optimization problems assume the use of large-block lattice quantization codes able to approximate a Gaussian quantization noise distribution. Optimal solutions to these problems are proposed that combine the use of block coordinate descent methods with sequential linear programs\n\n\\item A hybrid beamforming implementation is proposed where the \\gls{RF} high-dimensionality phase shifter-based precoding\/decoding stage is based on large-scale second-order statistics of the propagation channel, and hence does not need the estimation of high-dimensionality instantaneous \\gls{CSI}. The low-dimensionality baseband \\gls{MU-MIMO} precoding\/decoding stage can then be easily implemented by standard signal processing schemes using small-scale estimated \\gls{CSI}. As will be shown in the numerical results section, such a reduced complexity hybrid precoding scheme, when combined with appropriate user selection, performs very well in the fronthaul capacity-constrained \\gls{UDN} \\gls{mmWave}-based scenarios under consideration.\n\n\\item A suboptimal pilot allocation strategy is proposed that, based on the idea of clustering by dissimilarity, avoids the computational complexity of the optimal pilot allocation scheme. The performance of the proposed \\emph{dissimilarity cluster-based pilot assignment algorithm} is compared with that of both the \\emph{pure random pilot allocation approach} and the \\emph{balanced random pilot strategy}.\n\n\\item For those cases in which the number of active \\glspl{MS} in the network is greater than the number of available \\gls{RF} chains at a particular \\gls{AP}, a \\gls{MS} selection algorithm is proposed that aims at maximizing the minimum average sum-energy (i.e., Frobenius norm) of the equivalent channel between the \\glspl{AP} and any of the active \\glspl{MS}, constrained by the fact that each \\gls{AP} can only beamform to a number of \\glspl{MS} less or equal than the number of available \\gls{RF} chains.\n\n\\end{itemize}\n\n\\subsection{Paper organization and notational remarks}\n\nThe remainder of this paper is organized as follows. In Section \\ref{sec:System_model} the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} system is introduced. Different subsections are devoted to the description of the channel model, the large-scale and small-scale training phases, the channel estimation process, and the \\gls{DL} and \\gls{UL} payload transmission phases. The achievable \\gls{DL} and \\gls{UL} rates are presented in Section \\ref{sec:Achievable_rates} and further developed in Appendices \\ref{app:Appendix_1} and \\ref{app:Appendix_2}. Section \\ref{sec:fronthaul_capacity} is dedicated to the calculation of the capacity consumption of both the \\gls{DL} and \\gls{UL} fronthaul links. The pilot assignment, power allocation and quantization optimization processes are described in Sections \\ref{sec:pilot_assignment} and \\ref{sec:power allocation_quantization}. Numerical results and discussions are provided in Section \\ref{sec:numerical_results} and, finally, concluding remarks are summarized in Section \\ref{sec:Conclusion}.\n\n\\emph{Notation}: Vectors and matrices are denoted by lower-case and upper-case boldface symbols. The $q$-dimensional identity matrix is represented by $\\boldsymbol{I}_q$. The operator $\\det(\\bs{X})$ represents the determinant of matrix $\\bs{X}$, $\\tr(\\bs{X})$ denotes its trace, $\\|\\bs{X}\\|_F$ is its Frobenius norm, whereas $\\bs{X}^{-1}$, $\\bs{X}^T$, $\\bs{X}^*$ and $\\bs{X}^H$ denote its inverse, transpose, conjugate and conjugate transpose (also known as Hermitian), respectively. With a slight abuse of notation, the operator $\\diag(\\boldsymbol{x})$ is used to denote a diagonal matrix with the entries of vector $\\boldsymbol{x}$ on its main diagonal, and the operator $\\diag(\\boldsymbol{X})$ is used to denote a vector containing the entries in the main diagonal of matrix $\\bs{X}$. The expectation operator is denoted by $\\mathbb{E}\\{\\cdot\\}$. Finally, $\\mathcal{CN}(\\bs{m},\\bs{R})$ denotes a circularly symmetric complex Gaussian vector distributions with mean $\\bs{m}$ and covariance $\\bs{R}$, $\\mathcal{N}(0,\\sigma^2)$ denotes a real valued zero-mean Gaussian random variable with standard deviation $\\sigma$, and $\\mathcal{U}[a,b]$ represents a random variable uniformly distributed in the range $[a,b]$.\n\n\\section{System model}\n\\label{sec:System_model}\n\nLet us consider a cell-free massive \\gls{MIMO} system where a \\gls{CPU} coordinates the communication between $M$ \\glspl{AP} and $K$ single-antenna \\glspl{MS} randomly distributed in a large area. Each of the \\glspl{AP} communicates with the \\gls{CPU} via error-free fronthaul links with \\gls{DL} and \\gls{UL} capacities ${C_F}_d$ and ${C_F}_u$, respectively. Baseband processing of the transmitted\/received signals is performed at the \\gls{CPU}, while the \\gls{RF} operations are carried out at the \\glspl{AP}. Each \\gls{AP} is equipped with an array of $N > K$ antennas and $L \\leq N$ \\gls{RF} chains. A fully-connected architecture is considered where each \\gls{RF} chain is connected to the whole set of antenna elements using $N$ analog phase shifters. Without loss of essential generality, it is assumed in this paper that the number of active \\gls{RF} chains at each of the \\glspl{AP} in the network is equal to $L_A=\\min\\{K,L\\}$. That is, if $K \\leq L$, all \\glspl{AP} in the cell-free network provide service to the whole set of \\glspl{MS} and if $K > L$, instead, each \\gls{AP} can only provide service to $L$ out of the $K$ \\glspl{MS} in the network and, thus, an algorithm must be devised to decide which are the \\glspl{MS} to be beamformed by each of the \\glspl{AP}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7.8cm]{TDD_frame}\n  \\caption{Allocation of the samples in large-scale and short-scale coherence intervals.}\\label{fig:TDD_frame}\n\\end{figure}\n\nThe propagation channels linking the \\glspl{AP} to the \\glspl{MS} are typically characterized by small-scale parameters that are (almost) static over a coherence time-frequency interval of $\\tau_c$ time-frequency samples (see \\cite[Chapter 2]{Marzetta16}), and large-scale parameters (i.e., path loss propagation losses and covariance matrices) that can be safely assumed to be static over a time-frequency interval ${\\tau_L}_c \\gg \\tau_c$. As shown in the following subsections, these channel characteristics can be leveraged to simplify both the channel estimation and the precoding\/combining processes. In particular, \\gls{DL} and \\gls{UL} transmissions between \\glspl{AP} and \\glspl{MS} are organized in a half-duplex \\gls{TDD} operation whereby each coherence interval is split into three phases, namely, the \\gls{UL} training phase, the \\gls{DL} payload data transmission phase and the \\gls{UL} payload data transmission phase, and every \\emph{large-scale coherence interval} ${\\tau_L}_c$ the system performs an estimation of the large-scale parameters of the channel (see Fig. \\ref{fig:TDD_frame}). In the \\gls{UL} training phase, all \\glspl{MS} transmit \\gls{UL} training orthogonal pilots allowing the \\glspl{AP} to estimate the propagation channels to every \\gls{MS} in the network\\footnote{Note that channel reciprocity can be exploited in \\gls{TDD} systems and therefore only \\gls{UL} pilots need to be transmitted.}. Subsequently, these channel estimates are used to detect the signals transmitted from the \\glspl{MS} in the \\gls{UL} payload data transmission phase and to compute the precoding filters governing the \\gls{DL} payload data transmission. Not shown are guard intervals between \\gls{UL} and \\gls{DL} transmissions.\n\n\\subsection{Channel Model}\n\\label{subsec:Channel_model}\n\nMmWave propagation is characterized by very high distance-based propagation losses that lead to sparse scattering multipath propagation. Furthermore, the use of mmWave transmitters and receivers with large tightly-packet antenna arrays results in high antenna correlation levels. These characteristics make most of the statistical channel models used in conventional sub-6 GHz \\gls{MIMO} research work inaccurate when dealing with mmWave scenarios. Thus, a modified version of the discrete-time narrowband clustered channel model proposed by Akdeniz \\emph{et al.} in \\cite{Akdeniz14} and further extended by Samimi and Rappaport in \\cite{Samimi14} will be used in this paper to capture the peculiarities of mmWave channels.\n\nThe link between the $m$th \\gls{AP} and the $k$th \\gls{MS} will be considered to be in one out of three possible conditions: outage, \\gls{LOS} or \\gls{NLOS} with probabilities:\n\\begin{subequations}\n\\begin{equation}\n   p_{out}(d_{mk})=\\max\\left(0,1-e^{-a_{out}d_{mk}+b_{out}}\\right),\n\\end{equation}\n\\begin{equation}\n   p_{\\gls{LOS}}(d_{mk})=\\left(1-p_{out}(d_{mk})\\right) e^{-a_{\\gls{LOS}}d_{mk}},\n\\end{equation}\n\\begin{equation}\n   p_{\\gls{NLOS}}(d_{mk})=1-p_{out}(d_{mk})-p_{\\gls{LOS}}(d_{mk}),\n\\end{equation}\n\\end{subequations}\nrespectively, where $d_{mk}$ is the distance (in meters) between the \\gls{AP} and the \\gls{MS}, and, according to \\cite[Table I]{Akdeniz14}, $1\/a_{out}=30$~m, $b_{out}=5.2$, and $1\/a_{\\gls{LOS}}=67.1$~m. Those links that are in outage will be characterized with infinite propagation losses, while for the links that are not in outage, the propagation losses will be characterized using a standard linear model with shadowing as\n\\begin{equation}\n   \\PL(d_{mk})[dB]=\\alpha+10 \\beta \\log_{10}(d_{mk})+\\chi_{mk},\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are the least square fits of floating intercept and slope and depend on the carrier frequency and on whether the link is in \\gls{LOS} or \\gls{NLOS} (see \\cite[Table I]{Akdeniz14}), and $\\chi_{mk}$ denotes the large-scale shadow fading component, which is modelled as a zero mean spatially correlated normal random variable with standard deviation $\\sigma_\\chi$ (again, see \\cite[Table I]{Akdeniz14} to obtain the typical values of $\\sigma_\\chi$ for \\gls{LOS} and \\gls{NLOS} links) whose spatial correlation model is described in \\cite[(54)-(55)]{Ngo17}.\n\nThe \\gls{UL} channel vector $\\bs{h}_{mk} \\in \\mathbb{C}^{N \\times 1}$ between \\gls{MS} $k$ and \\gls{AP} $m$ will be modelled as the sum of the contributions of $C_{mk}$ scattering clusters, each contributing $P_{mk}$ propagation paths as\n\\begin{equation}\n   \\bs{h}_{mk}=\\sum_{c=1}^{C_{mk}}\\sum_{p=1}^{P_{mk}} \\alpha_{mk,cp} \\bs{a}\\left(\\theta_{mk,cp},\\phi_{mk,cp}\\right),\n\\end{equation}\nwhere $\\alpha_{mk,cp}$ is the complex small-scale fading gain on the $p$th path of cluster $c$, and $\\bs{a}\\left(\\theta_{mk,cp},\\phi_{mk,cp}\\right)$ represents the \\gls{AP} normalized array response vector at the azimuth and elevation angles $\\theta_{mk,cp}$ and $\\phi_{mk,cp}$, respectively. These angles, as stated by Akdeniz \\emph{et al.} in \\cite[Section III.E]{Akdeniz14} can be generated as wrapped Gaussians around the cluster central angles with standard deviation given by the \\gls{rms} angular spreads for the cluster. The azimuth cluster central angles are uniformly distributed in the range $[-\\pi,\\pi]$ and the elevation cluster central angles are set to the \\gls{LOS} elevation angle. Moreover, the cluster \\gls{rms} angular spreads are exponentially distributed with a mean equal to $1\/\\lambda_{\\gls{rms}}$ that depends on the carrier frequency and on whether we are considering the azimuth or elevation directions (see \\cite[Table I]{Akdeniz14}). The number of clusters is distributed as a random variable of the form\n\\begin{equation}\n   C_{mk} \\sim \\max\\left\\{\\text{Poisson}(\\sigma_C),1\\right\\},\n\\end{equation}\nwhere $\\sigma_C$ is set to the empirical mean of $C_{mk}$. The small-scale fading gains are distributed as\n\\begin{equation}\n   \\alpha_{mk,cp} \\sim \\mathcal{CN}\\left(0,\\gamma_{mk,c}10^{-\\PL(d_{mk})\/10}\\right),\n\\end{equation}\nwhere the cluster $c$ is assumed to contribute with a fraction of power given by\n\\begin{equation}\n   \\gamma_{mk,c}=\\frac{N \\gamma'_{mk,c}}{P_{mk}\\sum_{j=1}^{C_{mk}} \\gamma'_{mk,j}},\n\\end{equation}\nwith\n\\begin{equation}\n   \\gamma'_{mk,j}=U_{mk,j}^{r_\\tau-1} 10^{Z_{mk,j}\/10},\n\\end{equation}\n$U_{mk,j} \\sim \\mathcal{U}[0,1]$, $Z_{mk,j} \\sim \\mathcal{N}(0,\\zeta^2)$, and the constants $r_\\tau$ and $\\zeta^2$ being treated as model parameters (see \\cite[Table I]{Akdeniz14}).\n\nAlthough the small-scale fading gains $\\alpha_{mk,cp}$ are assumed to be static throughout the coherence interval and then change independently (i.e., block fading), the spatial covariance matrices\n\\begin{equation}\n\\begin{split}\n   \\bs{R}_{mk}=&\\mathbb{E}\\left\\{\\bs{h}_{mk} \\bs{h}_{mk}^H \\right\\} \\\\\n              =&10^{-\\PL(d_{mk})\/10}\\sum_{c=1}^{C_{mk}}\\gamma_{mk,c} \\\\\n               &\\times \\sum_{p=1}^{P_{mk}} \\bs{a}\\left(\\theta_{mk,cp},\\phi_{mk,cp}\\right)\\bs{a}^H\\left(\\theta_{mk,cp},\\phi_{mk,cp}\\right),\n\\end{split}\n\\end{equation}\nare assumed to vary at a much smaller pace (i.e., ${\\tau_L}_c \\gg \\tau_c$).\n\n\\subsection{Large-scale training phase}\n\\label{subsec:large-scale-training}\n\n\\subsubsection{\\gls{RF} precoder\/combiner design}\n\nIn order to exploit the \\gls{UL}\/\\gls{DL} channel reciprocity using the \\gls{TDD} frame structure shown in Fig. \\ref{fig:TDD_frame}, it is assumed in this paper that the $N \\times L_A$ \\gls{RF} matrix $\\bs{W}_m^{RF}$, describing the effects of the active analog phase shifters at the $m$th \\gls{AP}, is common to the \\gls{DL} (\\gls{RF} precoding phase) and \\gls{UL} (\\gls{RF} combining phase). Furthermore, denoting by $\\mathcal{K}_m=\\left\\{\\kappa_{m 1}, \\ldots, \\kappa_{m L_A}\\right\\}$ the set of $L_A$ \\glspl{MS} beamformed by the $m$th \\gls{AP}, it is assumed that $\\bs{W}_m^{RF}$ is a function of only the spatial channel covariance matrices $\\left\\{\\bs{R}_{mk}\\right\\}_{k\\in\\mathcal{K}_m}$, known at the $m$th \\gls{AP} through spatial channel covariance estimation for hybrid analog-digital \\gls{MIMO} precoding architectures (see e.g. \\cite{Adhikary14,Mendez15,Park16,Park17}).\n\nUsing eigen-decomposition, the covariance matrix of the propagation channel linking \\gls{MS} k and \\gls{AP} $m$ can be expressed as $\\bs{R}_{mk}=\\bs{U}_{mk}\\bs{\\Lambda}_{mk}\\bs{U}_{mk}^H$, where $\\bs{\\Lambda}_{mk}=\\diag\\left(\\left[\\lambda_{mk,1}\\,\\ldots\\,\\lambda_{mk,r_{mk}}\\right]\\right)$ contains the $r_{mk}$ non-null eigenvalues of $\\bs{R}_{mk}$, and $\\bs{U}_{mk}$ is the $N \\times r_{mk}$ matrix of the corresponding eigenvectors. Hence, assuming the use of (constrained) statistical eigen beamforming \\cite{Park17TSC,Mai18}, the analog \\gls{RF} precoder\/combiner can be designed as\n\\begin{equation}\n\\begin{split}\n   \\bs{W}_m^{RF}=&\\begin{bmatrix}\n                    \\bs{w}_{m \\kappa_{m 1}}^{RF} & \\ldots & \\bs{w}_{m \\kappa_{m L_A}}^{RF}\n                 \\end{bmatrix}\\\\\n                =&\\begin{bmatrix}\n                    e^{-j\\angle\\bs{u}_{m \\kappa_{m 1},\\max}} & \\ldots & e^{-j\\angle\\bs{u}_{m \\kappa_{m L_A},\\max}}\n                 \\end{bmatrix},\n\\end{split}\n\\end{equation}\nwhere $\\bs{u}_{mk,\\max}$ is the dominant eigenvector of $\\bs{R}_{mk}$ associated to the maximum eigenvalue $\\lambda_{mk,\\max}$, and the function $\\angle\\bs{x}$ returns the phase angles, in radians, for each element of the complex vector $\\bs{x}$. Note that using the \\gls{RF} precoding\/combining matrix, the equivalent channel vector between \\gls{MS} k and \\gls{AP} $m$, including the \\gls{RF} precoding\/decoding matrix, is defined as\n\\begin{equation}\n   \\bs{g}_{mk} = {\\bs{W}_m^{RF}}^T \\bs{h}_{mk}\\ \\in\\ \\mathbb{C}^{L_A \\times 1},\n\\end{equation}\nwhose dimension is much less than the number of antennas of the massive \\gls{MIMO} array used at the $m$th \\gls{AP}, thus largely simplifying the small-scale training phase.\n\n\\subsubsection{Selection of \\glspl{MS} to beamform from each \\gls{AP}}\n\nAs previously stated, in those highly probable cases in which the number of active \\glspl{MS} in the network is greater than the number of available \\gls{RF} chains at each \\gls{AP} (i.e., $K > L$), the $m$th \\gls{AP}, with $m\\in\\{1,\\ldots,M\\}$, can only beamform to a group of $L$ out of the $K$ \\glspl{MS} in the network, which are indexed by the set $\\mathcal{K}_m=\\left\\{\\kappa_{m1},\\ldots,\\kappa_{mL}\\right\\}$. As the \\gls{RF} beamforming matrices at the \\glspl{AP} are a function of only the large-scale spatial channel covariance matrices and are common to both the \\gls{UL} and the \\gls{DL}, the selection of the sets of \\glspl{MS} to beamform from each \\gls{AP} must also be based only on the available large-scale \\gls{CSI}. Inspired by the Frobenius norm-based suboptimal user selection algorithm proposed by Shen \\emph{et al.} in \\cite{Shen06}, a selection algorithm is proposed that aims at maximizing the sum of the average energy (i.e., average Frobenius norm) of the equivalent channels (including the corresponding beamformer) between the $M$ \\glspl{AP} and the $K$ \\glspl{MS} with the constraints that, first, the minimum average energy of the equivalent channel between the $M$ \\glspl{AP} and any of the active \\glspl{MS} must be maximized and, second, that each \\gls{AP} can only beamform to $L$ \\glspl{MS}. Note that this optimization problem, which tends to provide some degree of (average) max-min fairness among \\glspl{MS}, can be efficiently solved by using an iterative reverse-delete algorithm (similar to that used in graph theory to obtain a minimum spanning tree from a given connected, edge-weighted graph). In particular, at the beginning of the $i$th iteration of the algorithm the cell-free network is represented by a very simple edge-weighted directed graph with $M$ source nodes and $K$ sink nodes, where the $m$th source node, representing the $m$th \\gls{AP}, is connected to a group $\\mathcal{K}_m^{(i)}$ of sink nodes, representing the active \\glspl{MS} beamformed by the $m$th \\gls{AP}. The connection (edge) between the $m$th source node and the $l$th sink node in $\\mathcal{K}_m^{(i)}$ is weighted by the average Frobenius norm of the equivalent channel linking the $m$th \\gls{AP} and \\gls{MS} $l\\in \\mathcal{K}_m^{(i)}$, that can be obtained as\n\\begin{equation}\n   \\xi_{ml}=\\mathbb{E}\\left\\{\\left\\|{\\bs{w}_{ml}^{RF}}^T\\bs{h}_{ml}\\right\\|_F^2\\right\\}={\\bs{w}_{ml}^{RF}}^T \\bs{R}_{ml} \\bs{w}_{ml}^{RF}.\n\\end{equation}\nThe average sum energy of the equivalent channels between the $M$ \\glspl{AP} and \\gls{MS} $k$ at the beginning of the $i$th iteration is\n\\begin{equation}\n   \\mathcal{E}_k^{(i)} = \\sum_{m\\in\\mathcal{M}_k^{(i)}} \\xi_{mk},\n\\end{equation}\nwhere $\\mathcal{M}_k^{(i)}$ is the set of \\glspl{AP} beamforming to \\gls{MS} $k$ at the beginning of the $i$th iteration. During this iteration, the reverse-delete algorithm removes the edge (i.e., the \\gls{RF} chain and associated beamformer) that, first, goes out of one of those \\glspl{AP} still beamforming to more than $L$ \\glspl{MS} and, second, has the minimum weight maximizing the minimum average sum energy after removal. The algorithm begins with a fully connected graph and stops when all \\glspl{AP} beamform to exactly $L$ \\glspl{MS}. Hence, note that $M(K-L)$ iterations are needed to select the sets $\\mathcal{K}_m$ for $m\\in\\{1,\\ldots,M\\}$.\n\n\\subsection{Small-scale training phase}\n\\label{subsec:small-scale-training}\nCommunication in any coherence interval of a \\gls{TDD}-based massive \\gls{MIMO} system invariably starts with the \\glspl{MS} sending the pilot sequences to allow the channel to be estimated at the \\glspl{AP}. Let $\\tau_p$ denote the \\gls{UL} training phase duration (measured in samples on a time-frequency grid) per coherence interval. During the \\gls{UL} training phase, all $K$ \\glspl{MS} simultaneously transmit pilot sequences of $\\tau_p$ samples to the \\glspl{AP} and thus, the $L_A \\times \\tau_p$ received \\gls{UL} signal matrix at the $m$th \\gls{AP} is given by\n\\begin{equation}\n   {\\bs{Y}_p}_m=\\sqrt{\\tau_p P_p}\\sum_{k'=1}^K \\bs{g}_{mk'} \\bs{\\varphi}_{k'}^T+{\\bs{N}_p}_m,\n\\end{equation}\nwhere $P_p$ is the transmit power of each pilot symbol, $\\bs{\\varphi}_k$ denotes the $\\tau_p\\times 1$ training sequence assigned to \\gls{MS} $k$, with $\\|\\bs{\\varphi}_k\\|_F^2=1$, and ${\\bs{N}_p}_m$ is an $L_A\\times\\tau_p$ matrix of i.i.d. additive noise samples with each entry distributed as\\footnote{Note that in the \\gls{UL} of a fully-connected hybrid beamforming architecture each reception chain is composed of $N$ antenna elements, each connected to a low-noise amplifier (LNA) characterized by a power gain $G_{\\textrm{LNA}}$ and a noise temperature $T_{\\text{LNA}}$. Each of the $N$ LNAs feeds an analog passive phase shifter characterized by an insertion loss $L_{\\text{PS}}$. The outputs of the $N$ phase shifters are introduced to a power combiner whose insertion losses are typically proportional to the number of inputs, that is, $L_{\\text{PC}}=N L_{\\text{PC}_{in}}$. Finally, the output of the power combiner is introduced to an $\\gls{RF}$ chain characterized by a power gain $G_{\\text{RF}}$ and a noise temperature $T_{\\text{RF}}$. Thus, the equivalent noise temperature of each receive chain can be obtained as $T_u=N \\left(T_0+T_{\\text{LNA}}+\\frac{T_0(L_{\\text{PS}}L_{\\text{PC}_{in}}-1)}{G_{\\text{LNA}}}+\\frac{T_{\\text{RF}}L_{\\text{PS}}L_{\\text{PC}_{in}}}{G_{\\text{LNA}}}\\right)$.}  $\\mathcal{CN}(0,\\sigma_u^2(N))$. Ideally, training sequences should be chosen to be mutually orthogonal, however, since in most practical scenarios it holds that $K>\\tau_p$, a given training sequence is assigned to more than one \\gls{MS}, thus resulting in the so-called pilot contamination, a widely studied phenomenon in the context of collocated massive \\gls{MIMO} systems \\cite{Elijah16}.\n\n\\subsection{Channel estimation}\n\\label{channel_estimation}\n\nChannel estimation is known to play a central role in the performance of massive \\gls{MIMO} schemes \\cite{Lu14} and also in the specific context of cell-free architectures \\cite{Ngo17}.\nThe \\gls{MMSE} estimation filter for the channel between the $k$th active \\gls{MS} and the $m$th \\gls{AP} can be calculated as\n\\begin{equation}\n\\begin{split}\n   \\bs{D}_{mk}&=\\arg\\min_{\\bs{D}}\\mathbb{E}\\left\\{\\left\\|\\bs{g}_{mk}-\\bs{D} {\\bs{Y}_p}_m \\bs{\\varphi}_k^*\\right\\|^2\\right\\} \\\\\n                          &= \\sqrt{\\tau_p P_p} \\bs{R}_{mk}^{RF} \\bs{Q}_{mk}^{-1},\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\n   \\bs{R}_{mk}^{RF} = \\mathbb{E}\\left\\{\\bs{g}_{mk}  \\bs{g}_{mk}^H\\right\\}={\\bs{W}_m^{RF}}^T  \\bs{R}_{mk} {\\bs{W}_m^{RF}}^*,\n\\end{equation}\nand\n\\begin{equation}\n   \\bs{Q}_{mk}=\\tau_p P_p\\sum_{k'=1}^K \\bs{R}_{mk'}^{RF} \\left|\\bs{\\varphi}_{k'}^T \\bs{\\varphi}_k^*\\right|^2 + \\sigma_u^2(N) \\bs{I}_{L_A}.\n\\end{equation}\nHence, the corresponding estimated channel vector can be expressed as\n\\begin{equation}\n   \\hat{\\bs{g}}_{mk}=\\bs{D}_{mk} {\\bs{Y}_p}_m \\bs{\\varphi}_k^* = \\sqrt{\\tau_p P_p} \\bs{R}_{mk}^{RF} \\bs{Q}_{mk}^{-1} {\\bs{Y}_p}_m \\bs{\\varphi}_k^*.\n\\end{equation}\nThe \\gls{MMSE} channel vector estimates can be shown to be distributed as ${\\hat{\\bs{g}}}_{mk} \\sim \\mathcal{CN}\\left(\\bs{0},{\\hat{\\bs{R}}}_{mk}^{RF}\\right)$, where\n\\begin{equation}\n   \\hat{\\bs{R}}_{mk}^{RF}\\triangleq \\tau_p P_p \\bs{R}_{mk}^{RF}\\bs{Q}_{mk}^{-1}{\\bs{R}_{mk}^{RF}}^H.\n\\end{equation}\nFurthermore, the channel vector $\\bs{g}_{mk}$ can be decomposed as $\\bs{g}_{mk}=\\hat{\\bs{g}}_{mk}+\\tilde{\\bs{g}}_{mk}$, where $\\tilde{\\bs{g}}_{mk}$ is the \\gls{MMSE} channel estimation error, which is statistically independent of both $\\bs{g}_{mk}$ and $\\hat{\\bs{g}}_{mk}$.\n\n\\subsection{Downlink payload data transmission}\n\nLet us define $\\bs{s}_d=\\left[{s_d}_1 \\ldots {s_d}_K\\right]^T$ as the $K \\times 1$ vector of symbols jointly (cooperatively) transmitted from the \\glspl{AP} to the \\glspl{MS}, such that $E\\left\\{\\bs{s}_d\\bs{s}_d^H\\right\\}=\\bs{I}_K$. Let us also define $\\bs{x}_m=\\mathcal{P}_m\\left(\\bs{s}_d\\right)$ as the $N \\times 1$ vector of signals transmitted from the $m$th \\gls{AP}, where $\\mathcal{P}_m\\left(\\bs{s}_d\\right)$ is used to denote the mathematical operations (linear and\/or non-linear) used to obtain $\\bs{x}_m$ from $\\bs{s}_d$. Note that this vector must comply with a power constraint $\\mathbb{E}\\left\\{\\left\\|\\bs{x}_m\\right\\|_F^2\\right\\}\\leq \\overline{P}_m$, where $\\overline{P}_m$ is the maximum average transmit power available at \\gls{AP} $m$. Using this notation, the signal received by \\gls{MS} $k$ can be expressed as\n\\begin{equation}\n   {y_d}_k=\\sum_{m=1}^M \\bs{h}_{mk}^T \\bs{x}_m + {n_d}_k,\n\\label{eq:yk}\n\\end{equation}\nwhere ${n_d}_k \\sim \\mathcal{CN}(0,\\sigma_d^2)$ is the Gaussian noise sample at \\gls{MS} $k$. The vector $\\bs{y}_d = \\left[{y_d}_1\\, \\ldots\\, {y_d}_K\\right]^T$ containing the signals received by the $K$ scheduled \\glspl{MS} in the network can then be expressed as\n\\begin{equation}\n   \\bs{y}_d=\\sum_{m=1}^M \\bs{H}_m^T \\bs{x}_m + \\bs{n}_d,\n\\end{equation}\nwhere $\\bs{H}_m=\\left[\\bs{h}_{m1}\\, \\ldots\\, \\bs{h}_{mK}\\right]$ and $\\bs{n}_d=\\left[{n_d}_1\\, \\ldots\\, {n_d}_K\\right]^T$.\n\nThe mathematical operations that symbol vector $\\bs{s}_d$ undergoes before being transmitted, generically represented as $\\bs{x}_m = \\mathcal{P}_m(\\bs{s}_d)$, for all $m \\in \\{1,\\ldots,M\\}$, include, first, a baseband precoding task at the \\gls{CPU}, second, a compressing process of all or part of the data that must be sent from the \\gls{CPU} to the \\glspl{AP} through the fronthaul links and, third, an \\gls{RF} precoding task at each of the \\glspl{AP}. Let us denote by ${\\mathcal{Q}_d}_m(\\bs{x})$ and ${\\mathcal{Q}_d}_m^{-1}(\\bs{x})$ the quantization and unquantization mathematical operations performed by the \\gls{CAP}-based \\gls{CPU}-\\gls{AP} functional split on a vector of signal samples $\\bs{x}$ to be transmitted by the $m$th \\gls{AP}. Due to the distortion introduced by the quantization\/unquantization processes, we have that \\cite{Zamir96,Femenias18}\n\\begin{equation}\n   \\hat{\\mathcal{Q}}_{dm}(\\bs{x})\\triangleq {\\mathcal{Q}_d}_m^{-1}({\\mathcal{Q}_d}_m(\\bs{x}))=\\bs{x}+{\\bs{q}_d}_m,\n\\end{equation}\nwhere ${\\bs{q}_d}_m$ is the quantization noise vector, which is assumed to be statistically distributed as ${\\bs{q}_d}_m \\sim \\mathcal{CN}\\left(\\bs{0},\\sigma_{q_{dm}}^2 \\bs{I}\\right)$. As shown by Zamir \\emph{et al.} in \\cite{Zamir96}, this assumption is supported by the fact that large-block lattice quantization codes are able to approximate a Gaussian quantization noise distribution. Thus, the mathematical operations describing the \\gls{CPU}-\\gls{AP} functional split considered in this paper can be summarized as\n\\begin{equation}\n\\begin{split}\n   \\bs{x}_m=\\mathcal{P}_m(\\bs{s}_d)&=\\bs{W}_m^{RF} \\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon}^{1\/2} \\bs{s}_d\\right) \\\\\n                                   &=\\bs{W}_m^{RF} \\left(\\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon}^{1\/2} \\bs{s}_d + {\\bs{q}_d}_m\\right),\n\\end{split}\n\\label{eq:bsxm}\n\\end{equation}\nwhere $\\bs{W}_d^{BB} = \\left[{\\bs{W}_{d\\,1}^{BB}}^T\\ \\ldots\\ {\\bs{W}_{d\\,M}^{BB}}^T\\right]^T\\ \\in\\ \\mathbb{C}^{M L_A \\times K}$, with $\\bs{W}_{d\\,m}^{BB}=\\left[\\bs{w}_{dm1}^{BB}\\ \\ldots\\ \\bs{w}_{dmK}^{BB}\\right]\\ \\in\\ \\mathbb{C}^{L_A \\times K}$ denoting the baseband precoding matrix affecting the signal transmitted by the $m$th \\gls{AP}, and $\\bs{\\Upsilon}=\\diag\\left([\\upsilon_1\\,\\ldots\\,\\upsilon_K]\\right)$ is a $K \\times K$ diagonal matrix containing the power control coefficients in its main diagonal, which are chosen to satisfy the following necessary power constraint at the $m$th \\gls{AP}\n\\begin{equation}\n\\begin{split}\n   \\mathbb{E}\\left\\{\\left\\|\\bs{x}_m\\right\\|_F^2\\right\\} &= \\sum_{k=1}^K \\upsilon_k \\theta_{mk}^{BB\/RF}+{\\sigma_q^2}_{dm} \\left\\|\\bs{W}_m^{RF}\\right\\|_F^2 \\\\\n                                                        &= \\sum_{k=1}^K \\upsilon_k \\theta_{mk}^{BB\/RF} + {\\sigma_q^2}_{dm} L_A N \\leq \\overline{P}_m,\n\\end{split}\n\\label{eq:Power_constraint}\n\\end{equation}\nwhere we have used the definition\n\\begin{equation}\n   \\theta_{mk}^{BB\/RF}=\\mathbb{E}\\left\\{\\left\\|\\bs{W}_m^{RF} \\bs{w}_{dmk}^{BB}\\right\\|_F^2\\right\\}.\n\\end{equation}\n\nUsing the proposed hybrid \\gls{CAP} approach, the signal received by the $K$ \\glspl{MS} can be rewritten as\n\\begin{equation}\n\\begin{split}\n   \\bs{y}_d=&\\sum_{m=1}^M \\bs{H}_m^T \\bs{W}_m^{RF} \\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon}^{1\/2} \\bs{s}_d \\\\\n            &+ \\sum_{m=1}^M \\bs{H}_m^T \\bs{W}_m^{RF} {\\bs{q}_d}_m + \\bs{n}_d \\\\\n           =&\\ \\bs{G}^T \\bs{W}_d^{BB}\\bs{\\Upsilon}^{1\/2} \\bs{s}_d + \\bs{\\eta}_d,\n\\end{split}\n\\label{eq:yd}\n\\end{equation}\nwhere $\\bs{G}=[\\bs{G}_1^T\\,\\ldots\\,\\bs{G}_M^T]^T$, with $\\bs{G}_m={\\bs{W}_m^{RF}}^T \\bs{H}_m$, representing the equivalent \\gls{MIMO} channel matrix between\nthe $K$ \\glspl{MS} and the $M$ \\glspl{AP}, including the RF precoding\/decoding matrices, and\n\\begin{equation}\n   \\bs{\\eta}_d  =\\bs{G}^T \\bs{q}_d + \\bs{n}_d,\n\\end{equation}\nwith $\\bs{q}_d=[{\\bs{q}_d}_1^T \\ldots {\\bs{q}_d}_M^T]^T$, includes the thermal noise as well as the quantization noise samples received from all the \\glspl{AP} in the network. Now, using the classical \\gls{ZF} \\gls{MU-MIMO} baseband precoder to harness the spatial multiplexing, we have that\n\\begin{equation}\n   \\bs{W}_d^{BB}=\\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1}\n\\end{equation}\nor, equivalently,\n\\begin{equation}\n   \\bs{W}_{d\\,m}^{BB} = \\hat{\\bs{G}}_m^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1}\\ \\forall m,\n\\end{equation}\nwhere we have assumed that $\\bs{G}=\\hat{\\bs{G}}+\\tilde{\\bs{G}}$ and $\\bs{G}_m=\\hat{\\bs{G}}_m+\\tilde{\\bs{G}}_m$. Consequently, the signal received by the $k$th \\gls{MS} can be expressed as\n\\begin{equation}\n\\begin{split}\n   {y_d}_k = &\\bs{g}_k^T \\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d + {\\eta_d}_k \\\\\n           = &\\left(\\hat{\\bs{g}}_k^T + \\tilde{\\bs{g}}_k^T\\right) \\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d + {\\eta_d}_k \\\\\n           = &\\sqrt{\\upsilon_k} {s_d}_k + \\tilde{\\bs{g}}_k^T \\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d + {\\eta_d}_k\n\\end{split}\n\\label{eq:ydk}\n\\end{equation}\nwhere ${\\eta_d}_k = \\bs{g}_k^T \\bs{q}_d + {n_d}_k$. The first term denotes the useful received signal, the second term contains the interference terms due to the use of imperfect \\gls{CSI} (pilot contamination), and the third term encompass both the quantification and thermal noise samples.\n\n\\subsection{Uplink payload data transmission}\n\nIn the \\gls{UL}, the vector of received signals at the output of the $L_A$ \\gls{RF} chains (including the \\gls{RF} phase shifters) of the $m$th \\gls{AP} is given by\n\\begin{equation}\n\\begin{split}\n   {\\bs{r}_u}_m=&\\sqrt{P_u}\\sum_{k'=1}^K \\bs{g}_{mk'} \\sqrt{\\omega_{k'}} {s_u}_{k'} + {\\bs{n}_u}_m \\\\\n               =&\\sqrt{P_u}\\bs{G}_m \\bs{\\Omega}^{1\/2} \\bs{s}_u + {\\bs{n}_u}_m,\n\\end{split}\n\\end{equation}\nwhere $P_u$ is the maximum average \\gls{UL} transmit power available at any of the active \\glspl{MS}, $\\bs{s}_u=[{s_u}_1\\,\\ldots\\,{s_u}_K]^T$ denotes the vector of symbols transmitted by the $K$ active \\gls{MS}, $\\bs{\\Omega}=\\diag([\\omega_1\\,\\ldots\\,\\omega_K])$, with $0 \\leq \\omega_k \\leq 1$, is a matrix containing the power control coefficients used at the \\glspl{MS}, and ${\\bs{n}_u}_m \\sim \\mathcal{CN}(\\bs{0},\\sigma_u^2(N) \\bs{I}_{L_A})$ is the vector of additive thermal noise samples at the output of the $L_A$ \\gls{RF} chains of the $m$th \\gls{AP}. The received vector of signals at each of the \\glspl{AP} in the network is quantized and forwarded to the \\gls{CPU} via the \\gls{UL} fronthaul links, where they are unquantized and jointly processed using a set of baseband combining vectors. Using a similar approach to that employed to model the \\gls{DL} transmission, the received vector of (unquantized) samples from the $m$th \\gls{AP} can be expressed as\n\\begin{equation}\n   {\\bs{z}_u}_m=\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)={\\bs{r}_u}_m + {\\bs{q}_u}_m,\n\\end{equation}\nwhere ${\\bs{q}_u}_m$ is the quantization noise vector, which is assumed to be statistically distributed as ${\\bs{q}_u}_m \\sim \\mathcal{CN}\\left(\\bs{0},\\sigma_{q_{u m}}^2 \\bs{I}_{L_A}\\right)$. Now, assuming the use of \\gls{ZF} \\gls{MIMO} detection, the \\gls{CPU} uses the detection matrix\n\\begin{equation}\n   \\bs{W}_u^{BB}=\\left(\\hat{\\bs{G}}^H \\hat{\\bs{G}}\\right)^{-1}\\hat{\\bs{G}}^H = {\\bs{W}_d^{BB}}^T\n\\end{equation}\nor, equivalently\n\\begin{equation}\n   \\bs{W}_{u m}^{BB}=\\left(\\hat{\\bs{G}}^H \\hat{\\bs{G}}\\right)^{-1}\\hat{\\bs{G}}_m^H = {\\bs{W}_{d m}^{BB}}^T,\\ \\forall m,\n\\end{equation}\nto jointly process the vector $\\bs{z}_u=\\left[{\\bs{z}_u}_1^T\\,\\ldots\\,{\\bs{z}_u}_M^T\\right]^T$ and obtain the vector of detected samples\n\\begin{equation}\n\\begin{split}\n   \\bs{y}_u=&\\bs{W}_u^{BB}\\bs{z}_u=\\sqrt{P_u}\\bs{W}_u^{BB} \\bs{G} \\bs{\\Omega}^{1\/2} \\bs{s}_u + \\bs{\\eta}_u \\\\\n           =&\\sqrt{P_u} \\bs{\\Omega}^{1\/2} \\bs{s}_u + \\sqrt{P_u}\\bs{W}_u^{BB} \\tilde{\\bs{G}} \\bs{\\Omega}^{1\/2} \\bs{s}_u + \\bs{\\eta}_u,\n\\end{split}\n\\end{equation}\nwhere $\\bs{\\eta}_u=\\bs{W}_u^{BB}\\left(\\bs{q}_u + \\bs{n}_u\\right)$. Again, the first term denotes the useful received signal, the second term contains the interference terms due to the use of imperfect \\gls{CSI}, and the third term includes both the quantification and thermal noise samples. The detected sample corresponding to the symbol transmitted by the $k$th \\gls{MS} can then be obtained as\n\\begin{equation}\n   {y_u}_k =\\sqrt{P_u} \\omega_k^{1\/2} {s_u}_k + \\sqrt{P_u}\\left[\\bs{W}_u^{BB} \\tilde{\\bs{G}} \\bs{\\Omega}^{1\/2} \\bs{s}_u\\right]_k + {\\eta_u}_k,\n\\label{eq:yuk}\n\\end{equation}\nwhere $[\\bs{x}]_k$ denotes the $k$th entry of vector $\\bs{x}$.\n\n\\section{Achievable rates}\n\\label{sec:Achievable_rates}\n\nAnalysis techniques similar to those applied, for instance, in \\cite{Hassibi03,Yang13,Interdonato16,Marzetta16,Ngo17,Nayebi17}, are used in this section to derive \\gls{DL} and \\gls{UL} achievable rates. In particular, the sum of the second and third terms on the \\gls{RHS} of \\eqref{eq:ydk}, for the \\gls{DL} case, and \\eqref{eq:yuk}, for the \\gls{UL} case, are treated as \\emph{effective noise}. The additive terms constituting the \\emph{effective noise} are, in both \\gls{DL} and \\gls{UL} cases, mutually uncorrelated, and uncorrelated with ${s_d}_k$ and ${s_u}_k$, respectively. Therefore, both the desired signal and the so-called \\emph{effective noise} are uncorrelated. Now, recalling the fact that uncorrelated Gaussian noise represents the worst case, from a capacity point of view, and that the complex-valued fast fading random variables characterizing the propagation channels between different pairs of \\gls{AP}-\\gls{MS} connections are independent, the \\gls{DL} and \\gls{UL} achievable rates (measured in bits per second per Hertz) for \\gls{MS} $k$ can be obtained as stated in the following theorems:\n\\begin{theorem}[Downlink achievable rate] An achievable rate of \\gls{MS} k using the analog precoders $\\bs{W}_m^{RF}$, for all $m\\in\\{1,\\ldots,M\\}$, and the \\gls{ZF} baseband precoder $\\bs{W}_d^{BB} = \\hat{\\bs{G}}^*\\left(\\hat{\\bs{G}}^T \\hat{\\bs{G}}^*\\right)^{-1}$ is ${R_d}_k=\\log_2\\left(1+{\\SINR_d}_k\\right)$, with\n\\begin{equation}\n   {\\SINR_d}_k=\\frac{\\upsilon_k}{\\sum_{k'=1}^K \\upsilon_{k'} \\varpi_{kk'} + \\sigma_{\\eta_{dk}}^2},\n   \\label{eq:SINRdk}\n\\end{equation}\nwhere\n\\begin{equation}\n   \\sigma_{\\eta_{dk}}^2=\\sum_{m=1}^M {\\sigma_q^2}_{dm} \\tr\\left(\\bs{R}_{mk}^{RF}\\right) + \\sigma_d^2,\n\\end{equation}\nand\n\\begin{equation}\n   \\varpi_{kk'}=\\left[\\diag\\left(\\mathbb{E}\\left\\{{\\bs{W}_d^{BB}}^H \\tilde{\\bs{g}}_k^* \\tilde{\\bs{g}}_k^T\\bs{W}_d^{BB}\\right\\}\\right)\\right]_{k'}.\n\\end{equation}\n\\label{theo:theorem_DL}\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix \\ref{app:Appendix_1}.\n\\end{proof}\n\n\\begin{theorem}[Uplink achievable rate] An achievable \\gls{UL} rate for the $k$th \\gls{MS} in the Cell-Free Massive \\gls{MIMO} system with limited capacity fronthaul links and using \\gls{ZF} \\gls{MIMO} detection, for any $M$, $N$ and $K$, is given by ${R_u}_k=\\log_2\\left(1+{\\SINR_u}_k\\right)$, with\n\\begin{equation}\n   {\\SINR_u}_k=\\frac{P_u \\omega_k}{P_u \\sum_{k'=1}^K \\omega_{k'} \\delta_{kk'} + \\sigma_{\\eta_{uk}}^2},\n   \\label{eq:SINRuk}\n\\end{equation}\nwhere\n\\begin{equation}\n   \\delta_{kk'}=\\left[\\diag\\left(\\mathbb{E}\\left\\{\\tilde{\\bs{G}}^H{\\bs{w}_{uk}^{BB}}^H \\bs{w}_{uk}^{BB} \\tilde{\\bs{G}}\\right\\}\\right)\\right]_{k'}\n\\end{equation}\nwith $\\bs{w}_{uk}^{BB}$ denoting the $k$th row of $\\bs{W}_u^{BB}$, or, equivalently,\n\\begin{equation}\n   \\delta_{kk'}=\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\tilde{\\bs{g}}_{k'} \\tilde{\\bs{g}}_{k'}^H{\\bs{W}_u^{BB}}^H\\right\\}\\right)\\right]_k,\n\\end{equation}\nand\n\\begin{equation}\n   \\sigma_{\\eta_{uk}}^2=\\sum_{m=1}^M\\left({\\sigma_q^2}_{um} +\\sigma_u^2(N)\\right){\\nu_u}_{mk},\n\\end{equation}\nwith\n\\begin{equation}\n   {\\nu_u}_{mk}=\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_{u\\,m}^{BB} {\\bs{W}_{u\\,m}^{BB}}^H\\right\\}\\right)\\right]_k.\n\\end{equation}\n\\label{theo:theorem_UL}\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix \\ref{app:Appendix_2}.\n\\end{proof}\n\n\\section{Fronthaul capacity consumption}\n\\label{sec:fronthaul_capacity}\n\nThe \\gls{DL} quantization process performed at the $m$th \\gls{AP} can be expressed as\n\\begin{equation}\n   \\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)=\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d + {\\bs{q}_d}_m.\n\\end{equation}\nFrom standard random coding arguments \\cite{Cover06}, vector $\\bs{s}_d$ can be safely assumed to be distributed as $\\bs{s}_d\\sim\\mathcal{CN}(0,\\bs{I}_K)$ and thus, the quantized vector $\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)$ is distributed as $\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)\\sim\\mathcal{CN}\\left(\\bs{0},\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon} {\\bs{W}_{d\\,m}^{BB}}^H+{\\sigma_q^2}_{dm}\\bs{I}_{L_A}\\right)$. Furthermore, as the differential entropy of a vector $\\bs{x}\\sim\\mathcal{CN}(\\bs{\\omega},\\bs{\\Theta})$ is given by $\\mathcal{H}(\\bs{x})=\\log\\det(\\pi e \\bs{\\Theta})$ \\cite{Cover06}, the required average rate to transfer the quantized vector $\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)$ on the corresponding \\gls{DL} fronthaul link can be obtained as (in bps\/Hz)\n\\begin{equation}\n\\begin{split}\n   \\hat{C}_{dm} &= \\mathbb{E}\\left\\{I\\left(\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right);\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)\\right\\} \\\\\n                &= \\mathbb{E}\\left\\{\\mathcal{H}\\left(\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)\\right)\\right\\} \\\\\n                &\\quad-\\mathbb{E}\\left\\{\\mathcal{H}\\left(\\hat{\\mathcal{Q}}_{dm}\\left(\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right)\\bigr|\\bs{W}_{d\\,m}^{BB}\\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right) \\right\\} \\\\\n                &= \\mathbb{E}\\left\\{\\log_2\\det\\left(\\frac{1}{{\\sigma_q^2}_{dm}} \\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon} {\\bs{W}_{d\\,m}^{BB}}^H + \\bs{I}_{L_A}\\right)\\right\\},\n\\end{split}\n\\end{equation}\nwhere $I(\\hat{\\bs{x}};\\bs{x})$ is used to denote the mutual information between vectors $\\hat{\\bs{x}}$ and $\\bs{x}$, and $\\mathcal{H}(\\hat{\\bs{x}}|\\bs{x})$ is the differential entropy of $\\hat{\\bs{x}}$ conditioned on $\\bs{x}$. Since the determinant is a log-concave function on the set of positive semidefinite matrices, it follows from Jensen's inequality that\n\\begin{equation}\n\\begin{split}\n   \\hat{C}_{dm} &\\leq \\log_2\\det\\left(\\frac{1}{{\\sigma_q^2}_{dm}} \\mathbb{E}\\left\\{\\bs{W}_{d\\,m}^{BB} \\bs{\\Upsilon} {\\bs{W}_{d\\,m}^{BB}}^H\\right\\} + \\bs{I}_{L_A}\\right) \\\\\n                &= \\log_2\\det\\left(\\frac{1}{{\\sigma_q^2}_{dm}} \\sum_{k=1}^K \\upsilon_k \\bs{R}_{mk}^{BB} + \\bs{I}_{L_A}\\right),\n\\end{split}\n\\end{equation}\nwhere $\\bs{R}_{mk}^{BB} = \\mathbb{E}\\left\\{\\bs{w}_{mk}^{BB} {\\bs{w}_{mk}^{BB}}^H\\right\\}$.\n\nAnalogously, the \\gls{UL} quantization process performed at the $m$th \\gls{AP} is given by $\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)={\\bs{r}_u}_m + {\\bs{q}_u}_m$. Thus, using arguments similar to those used in the \\gls{DL} case, the required average rate to transfer the quantized vector $\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)$ on the corresponding \\gls{UL} fronthaul link can be upper bounded as (in bps\/Hz)\n\\begin{equation}\n\\begin{split}\n   &\\hat{C}_{um} = \\mathbb{E}\\left\\{I\\left(\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right);{\\bs{r}_u}_m\\right)\\right\\} \\\\\n   &\\ = \\mathbb{E}\\left\\{\\mathcal{H}\\left(\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)\\right)\\right\\}-\\mathbb{E}\\left\\{\\mathcal{H}\\left(\\hat{\\mathcal{Q}}_{um}\\left({\\bs{r}_u}_m\\right)\\bigr|{\\bs{r}_u}_m\\right) \\right\\} \\\\\n   &\\ \\leq \\log_2\\det\\left(\\frac{P_u}{{\\sigma_q^2}_{um}} \\sum_{k=1}^K \\omega_k \\bs{R}_{mk}^{RF} + \\left(\\frac{\\sigma_u^2(N)}{{\\sigma_q^2}_{um}} +1\\right)\\bs{I}_{L_A}\\right).\n\\end{split}\n\\end{equation}\n\n\n\n\\section{Pilot assignment}\n\\label{sec:pilot_assignment}\n\nTo warrant an appropriate system performance, the \\gls{RRM} unit must efficiently manage both the pilot assignment and the \\gls{UL} and \\gls{DL} power control. As the pilots are not power controlled, pilot assignment and power control can be conducted independently. Since the length of the pilot sequences is limited to $\\tau_p$, there only exist $\\tau_p$ orthogonal pilot sequences. In a network with $K\\leq \\tau_p$ \\glspl{MS}, an optimal pilot assignment strategy simply allocates $K$ orthogonal pilots to the $K$ \\glspl{MS}. The real pilot assignment problem arises when $K > \\tau_p$. In this case, fully orthogonal pilot assignment is no longer possible and hence, other pilot assignment strategies must be devised.\n\nOn the one hand, designing an optimal pilot assignment strategy aiming at maximizing the minimum rate allocated to the active \\glspl{MS} in the network is a very difficult combinatorial problem, computationally unmanageable in most network setups of practical interest \\cite{Ngo17}. On the other hand, using straightforward strategies such as, for instance, the pure \\gls{RPA} scheme \\cite{Ahmadi16}, where each \\gls{MS} is randomly assigned one pilot sequence out of the set of $\\tau_p$ orthogonal pilot sequences, or the \\gls{BRPA} scheme, where each \\gls{MS} is allocated a pilot sequence that is sequentially and cyclically selected from the ordered set of available orthogonal pilots, provides poor performance results. In order to avoid the computational complexity of the optimal strategies while improving the performance of the baseline \\gls{RPA} or \\gls{BRPA} approaches, a suboptimal solution is proposed in this paper that is based on the idea of \\emph{clustering by dissimilarity}. This suboptimal approach, that will be termed as the \\gls{DCPA} strategy, is motivated by the following key observation:\n\n\\textbf{Key observation}: \\emph{In those scenarios where $K > \\tau_p$, cell-free communication is severely impaired whenever \\glspl{MS} showing very similar large-scale propagation patterns to the set of \\glspl{AP} (that is, \\glspl{MS} typically located nearby) are allocated the same pilot sequence. In this case, the inter-\\gls{MS} interference leads to very poor channel estimates at all \\glspl{AP} and, eventually, to low \\glspl{SINR}.}\n\nThe clustering algorithm proposed in this work basically ensures that pilot sequences are only reused by \\glspl{MS} showing \\emph{dissimilar} large-scale propagation patterns to the \\glspl{AP} (that is, \\glspl{MS} typically located sufficiently apart). Two key aspects regarding the clustering operation are thus, on the one hand, to decide which should be the large-scale propagation pattern that ought to be used to represent a given \\gls{MS} and, on the other hand, to decide what metric should be used to measure \\emph{similarity} among the large-scale propagation patterns characterizing different \\glspl{MS}. To this end, and resting upon the premise that the \\gls{CPU} has perfect knowledge of the large-scale gains, let $\\bs{\\xi}_k=\\left[\\xi_{1k}\\,\\ldots\\,\\xi_{Mk}\\right]^T$ denote the $M\\times 1$ vector containing the average Frobenius norms of the equivalent channels linking the $k$th \\gls{MS} to all $M$ \\glspl{AP} in the cell-free network. Vector $\\bs{\\xi}_k$ can be considered as an effective \\emph{fingerprint} characterizing the location of \\gls{MS} $k$. Now, although no single definition of a similarity measure exists, the so-called \\emph{cosine similarity} measure is one of the most commonly used similarity metrics when dealing with real-valued vectors. Hence, as the \\emph{fingerprint} vectors characterizing the different \\glspl{MS} are non-negative real-valued, the cosine similarity measure between two \\emph{fingerprint} vectors $\\bs{\\xi}_k$ and $\\bs{\\xi}_{k'}$, defined as\n\\begin{equation}\n   f_D\\left(\\bs{\\xi}_k, \\bs{\\xi}_{k'}\\right)=\\frac{\\bs{\\xi}_k^T\\bs{\\xi}_{k'}}{\\|\\bs{\\xi}_k\\|_2 \\|\\bs{\\xi}_{k'} \\|_2},\n\\end{equation}\nwill be used as a proper similarity metric in our work. The resulting similarity values range from 0, meaning orthogonality (perfect dissimilarity), to 1, meaning exact match (perfect similarity).\n\nThe proposed \\gls{DCPA} algorithm proceeds as follows. In a first step, it calculates the fingerprint of an imaginary \\gls{MS} centroid, defined as\n\\begin{equation}\n   \\bs{\\xi}_C=\\frac{1}{K}\\sum_{k=1}^K \\bs{\\xi}_k.\n\\end{equation}\nThen, it moves onward to the calculation of the cosine similarity measures among the fingerprint vectors characterizing the $K$ \\glspl{MS} in the network and the fingerprint of the centroid, that is, the algorithm proceeds to the calculation of $f_D\\left(\\bs{\\xi}_k, \\bs{\\xi}_C\\right)$, for all $k\\in\\{1,\\ldots,K\\}$. The \\glspl{MS} are then sorted in descending order of similarity with the centroid, that is, the algorithm obtains the ordered set of subindices $\\mathcal{O}=\\left\\{o_1, o_2, \\ldots, o_K\\right\\}$, such that $f_D\\left(\\bs{\\xi}_{o_1}, \\bs{\\xi}_C\\right) \\leq f_D\\left(\\bs{\\xi}_{o_2}, \\bs{\\xi}_C\\right)\\leq \\cdots \\leq f_D\\left(\\bs{\\xi}_{o_K}, \\bs{\\xi}_C\\right)$. Once the \\glspl{MS} have been sorted, the algorithm constructs $\\tau_p$ clusters of \\glspl{MS}, namely $\\mathcal{K}_1, \\ldots, \\mathcal{K}_{\\tau_p}$, with\n\\begin{equation}\n\\begin{split}\n   \\mathcal{K}_t=&\\mathcal{O}\\left(t:\\tau_p:K\\right) \\\\\n                =&\\left\\{o_t, o_{t+\\tau_p}, o_{t+2 \\tau_p},\\ldots\\right\\},\\ \\forall t\\in\\{1,\\ldots,\\tau_p\\},\n\\end{split}\n\\end{equation}\nand all \\glspl{MS} in cluster $\\mathcal{K}_t$, which are located far from each other, are allocated the same pilot code $\\bs{\\varphi}_t$. Note that the application of this algorithm ensures that, as far as it is possible, two \\glspl{MS} having similar large-scale propagation fingerprints are allocated different pilot codes and, thus, they do not interfere to each other during the \\gls{UL} channel estimation process. In other words, it aims at minimizing the residual interuser interference terms in both \\eqref{eq:ydk} and \\eqref{eq:yuk}.\n\n\\section{Max-min power allocation and optimal quantization}\n\\label{sec:power allocation_quantization}\n\n\\subsection{Downlink power control and quantization}\n\nIn line with previous research works on cell-free architectures \\cite{Ngo15,Ngo17,Nayebi17,Bashar18b}, our aim in this subsection is to find the power control coefficients $\\upsilon_k$, for all $k\\in\\{1,\\ldots,K\\}$, and the quantization noise variances ${\\sigma_q^2}_{dm}$, for all $m\\in\\{1,\\ldots,M\\}$, that maximize the minimum of the achievable \\gls{DL} rates of all \\glspl{MS} while satisfying the average transmit power and \\gls{DL} fronthaul capacity constraints at each \\gls{AP}. Mathematically, this optimization problem can be formulated as\n\\begin{equation}\n\\begin{split}\n   &\\max_{\\substack{\\bs{\\Upsilon} \\succeq 0 \\\\ {\\bs{\\sigma}_q}_d \\succeq 0}}\\ \\min_{k\\in\\{1,\\ldots,K\\}} \\frac{\\upsilon_k}{\\sum_{k'=1}^K \\upsilon_{k'} \\varpi_{kk'} + \\sigma_{\\eta_{dk}}^2} \\\\\n   &\\textrm{s.t. } \\sum_{k=1}^K \\upsilon_k \\theta_{mk}^{BB\/RF} \\leq \\overline{P}_m - {\\sigma_q^2}_{dm} L_A N,\\,\\forall\\,m, \\\\\n   &\\phantom{\\textrm{s.t. }} \\log_2\\det\\left(\\sum_{k=1}^K \\frac{\\upsilon_k}{{\\sigma_q^2}_{dm}} \\bs{R}_{mk}^{BB} + \\bs{I}_{L_A}\\right) \\leq {C_F}_d,\\,\\forall\\,m,\n\\end{split}\n\\label{eq:opt_problem_DL}\n\\end{equation}\nwhere we have used the definition ${\\bs{\\sigma}_q}_d=[{\\sigma_q}_{d1}\\,\\ldots\\,{\\sigma_q}_{dM}]^T$.\n\nOptimization problem \\eqref{eq:opt_problem_DL} is characterized by continuous objective and constraint functions of interdependent block variables, namely, $\\bs{\\Upsilon}$ and ${\\bs{\\sigma}_q}_d$.  A widely used approach for solving optimization problems of this class is the so-called \\gls{BCD} method. This is an iterative optimization approach that, at each iteration and in a cyclic order, optimizes one of the blocks while the remaining variables are held fixed \\cite{Tseng01,Beck13}. Convergence of the \\gls{BCD} method is ensured whenever each of the subproblems to be optimized in each iteration can be exactly solved to its unique optimal solution.\n\nThe first important fact to note is that, given a power allocation matrix $\\bs{\\Upsilon}^{(i-1)}$ obtained at the $(i-1)$th iteration, and as the achievable user rates monotonically increase with the capacity of the fronthaul links between the \\glspl{AP} and the \\gls{CPU}, the optimal solution for the acceptable fronthaul quantization noise in the $i$th iteration is achieved when the fronthaul capacity constraints are satisfied with equality, that is, when\n\\begin{equation}\n   \\det\\left(\\sum_{k=1}^K \\frac{\\upsilon_k^{(i-1)}}{{\\sigma_q^2}_{dm}^{(i)}} \\bs{R}_{mk}^{BB} + \\bs{I}_{L_A}\\right) = 2^{{C_F}_d},\\,\\forall\\,m.\n   \\label{eq:trasc_funct}\n\\end{equation}\nNote that ${\\sigma_q^2}_{dm}^{(i)}$ cannot be expressed in a closed-form algebraic expression as it only admits a solution in the form of a transcendental function\n\\begin{equation}\n   {\\sigma_q^2}_{dm}^{(i)}=F_d\\left(\\bs{\\Upsilon}^{(i-1)}, \\left\\{\\bs{R}_{mk}^{BB}\\right\\}_{k=1}^K, {C_F}_d\\right)\n\\end{equation}\nthat can be numerically solved by applying mathematical software tools to \\eqref{eq:trasc_funct}.\n\nOnce the optimal block of variables ${\\bs{\\sigma}_q}_d^{(i)}$ have been obtained, the optimization problem in \\eqref{eq:opt_problem_DL} can be rewritten in terms of the power allocation matrix $\\bs{\\Upsilon}^{(i)}$ as\n\\begin{equation}\n\\begin{split}\n   &\\max_{\\bs{\\Upsilon}^{(i)} \\succeq 0}\\ \\min_{k\\in\\{1,\\ldots,K\\}} \\frac{\\upsilon_k^{(i)}}{\\displaystyle{\\small{\\sum_{k'=1}^K} \\upsilon_{k'}^{(i)} \\gamma_{kk'} + \\small{\\sum_{m=1}^M} {\\sigma_q^2}_{dm}^{(i)} \\tr\\left(\\bs{R}_{mk}^{RF}\\right) + \\sigma_d^2}} \\\\\n   &\\textrm{s.t. } \\sum_{k=1}^K \\upsilon_k^{(i)} \\theta_{mk}^{BB\/RF} \\leq \\overline{P}_m - N L_A {\\sigma_q^2}_{dm}^{(i)},\\,\\forall\\,m.\n\\end{split}\n\\label{eq:opt_problem_DL_GOPA}\n\\end{equation}\nNote that this is a convergent quasi-linear optimization problem that can be solved using conventional standard convex optimization methods \\cite{Ngo17,Nayebi17}.\n\n\n\n\\subsection{Uplink power control and quantization}\n\nIn this subsection we aim at finding the power control coefficients $\\omega_k$, for all $k\\in\\{1,\\ldots,K\\}$, and quantization noise variances ${\\sigma_q^2}_{um}$, for all $m\\in\\{1,\\ldots,M\\}$, that maximize the minimum of the achievable ulink rates of all \\glspl{MS} while satisfying the power control coefficient constraints at each \\gls{MS} and the \\gls{UL} fronthaul capacity constraints at each \\gls{AP}. This optimization problem can be formulated as\n\\begin{equation}\n\\begin{split}\n   &\\max_{\\substack{\\bs{\\omega} \\succeq 0 \\\\ {\\bs{\\sigma}_q}_u \\succeq 0}}\\ \\min_{k\\in\\{1,\\ldots,K\\}} \\frac{P_u \\omega_k}{P_u \\sum_{k'=1}^K \\omega_{k'} \\delta_{kk'} + \\sigma_{\\eta_{uk}}^2} \\\\\n   &\\textrm{s.t. } 0 \\leq \\omega_k \\leq 1,\\,\\forall\\,k, \\\\\n   &\\phantom{\\textrm{s.t. }} \\det\\left(\\frac{P_u}{{\\sigma_q^2}_{um}} \\sum_{k=1}^K \\omega_k \\bs{R}_{mk}^{RF} + \\vartheta_m\\bs{I}_{L_A}\\right) \\leq 2^{{C_F}_u},\\,\\forall\\,m,\n\\end{split}\n\\label{eq:opt_problem_UL}\n\\end{equation}\nwhere ${\\bs{\\sigma}_q}_u=[{\\sigma_q}_{u1}\\,\\ldots\\,{\\sigma_q}_{uM}]^T$, and we have used the definition $\\vartheta_m= 1+\\sigma_u^2(N)\/{\\sigma_q^2}_{um}$. As for the \\gls{DL} case, problem \\eqref{eq:opt_problem_UL} admits the use of the \\gls{BCD} method where, in each iteration, the nonconvex transcendental function ${\\sigma_q^2}_{um}=F_u\\left(\\bs{\\Omega}, \\left\\{\\bs{R}_{mk}^{RF}\\right\\}_{k=1}^K, P_u, {C_F}_u\\right)$ is approximated by a constant calculated using the power allocation vector obtained in the previous iteration of the algorithm. That is, in the $i$th iteration of the \\gls{UL} optimal power allocation approach, the algorithm solves the optimization problem\n\\begin{equation}\n\\begin{split}\n   &\\max_{\\bs{\\Omega}^{(i)} \\succeq 0}\\ \\min_{k\\in\\{1,\\ldots,K\\}} \\frac{P_u \\omega_k^{(i)}}{P_u \\sum_{k'=1}^K \\omega_{k'}^{(i)} \\delta_{kk'} + {\\sigma_{\\eta_{uk}}^{\\,2(i)}}}, \\\\\n   &\\textrm{s.t. } 0 \\leq \\omega_k \\leq 1,\\,\\forall\\,k,\n\\end{split}\n\\label{eq:opt_problem_UL_SOPA}\n\\end{equation}\nwhere ${\\sigma_q^2}_{um}^{(i)}=F_u\\left(\\bs{\\Omega}^{(i-1)}, \\left\\{\\bs{R}_{mk}^{RF}\\right\\}_{k=1}^K, P_u, {C_F}_u\\right)$. Note that, again, this is a convergent quasi-linear optimization problem that can be solved using conventional convex optimization methods \\cite{Ngo17,Nayebi17}.\n\n\\section{Numerical results}\n\\label{sec:numerical_results}\n\nIn this section, simulation results are obtained in order to quantitatively study the performance of the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} network with constrained-capacity fronthaul links. In particular, we demonstrate the impact of using different pilot allocation strategies, the effects of modifying the capacity of the fronthaul links and the \\gls{RF} infrastructure at the \\glspl{AP}, and the repercussion of changing the density of \\glspl{AP} per area unit. For simplicity of exposition, and without loss of essential generality, a cell-free scenario is considered where the $M$ \\glspl{AP} and $K$ \\glspl{MS} are uniformly distributed at random within a square coverage area of size $D \\times D$~$m^2$. As described in subsection \\ref{subsec:Channel_model}, a modified version of the discrete-time narrowband clustered channel model proposed by Akdeniz \\emph{et al.} in \\cite{Akdeniz14} is used in the performance evaluation. The parameters necessary to implement this channel model can be found in \\cite[Table I]{Akdeniz14}. Furthermore, similar to what was done by Ngo \\emph{et al.} in \\cite{Ngo17}, a shadow fading spatial correlation model with two components is also considered (see \\cite[eqs. (54) and (55)]{Ngo17}) where the decorrelation distance is set to $d_{decorr}=50$~m and the parameter $\\delta$ is set to 0.5. Default parameters used to set-up the simulation scenarios under evaluation in the following subsections are summarized in Table \\ref{tab:parameters}.\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.2}\n\\caption{\\small Summary of default simulation parameters}\n\\label{tab:parameters}\n\\centering\n\\begin{tabular}{l|c}\n\\hline\n\\bfseries Parameters & \\bfseries Value\\\\\n\\hline\n\\footnotesize {Carrier frequency: $f_0$} & \\footnotesize {28 GHz}\\\\\n\\footnotesize {Bandwidth: $B$} & \\footnotesize {20 MHz}\\\\\n\\footnotesize {Side of the square coverage area: $D$} & \\footnotesize {200 m}\\\\\n\\footnotesize {AP antenna height: $h_{AP}$} & \\footnotesize {15 m}\\\\\n\\footnotesize {MS antenna height: $h_{MS}$} & \\footnotesize {1.65 m}\\\\\n\\footnotesize {Noise figure at the MS: ${NF}_{MS}$} & \\footnotesize {9 dB} \\\\\n\\footnotesize {Noise figure of the LNA at the AP: ${NF}_{LNA}$} & \\footnotesize {1.6 dB} \\\\\n\\footnotesize {Gain of the LNA at the AP: $G_{LNA}$} & \\footnotesize {22 dB} \\\\\n\\footnotesize {Attenuation of the phase splitters at the AP: $L_{PS}$} & \\footnotesize {3 dB}\\\\\n\\footnotesize {Attenuation of the power combiner at the AP: ${L_{PC}}_{in}$} & \\footnotesize {3 dB}\\\\\n\\footnotesize {Noise figure of the RF chain at the AP: $NF_{RF}$} & \\footnotesize {7 dB}\\\\\n\\footnotesize {Available average power at the AP: $\\overline{P}_m$} & \\footnotesize {200 mW}\\\\\n\\footnotesize {Available average power at the MS: $P_u=P_p$} & \\footnotesize {100 mW}\\\\\n\\footnotesize {Coherence interval length: $\\tau_c$} & \\footnotesize {200 samples}\\\\\n\\footnotesize {Training phase length: $\\tau_p$} & \\footnotesize {15 samples}\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[t!]\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_var_Pilot_Alloc_min_rate}\n        \\label{fig:DL_var_Pilot_Alloc_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_var_Pilot_Alloc_min_rate}\n        \\label{fig:UL_var_Pilot_Alloc_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of active \\glspl{MS} for different pilot allocation strategies ($N=64$ antennas, $L=8$ \\gls{RF} chains, ${C_F}_d={C_F}_u=64$ bit\/s\/Hz).}\n    \\label{fig:var_Pilot_Alloc_min_rate}\n\\end{figure}\n\n\\subsection{Impact of the pilot allocation process}\n\nOur aim in this subsection is to benchmark the performance of the proposed large-scale \\gls{CSI}-aware \\gls{DCPA} strategy against both the pure \\gls{RPA} and the \\gls{BRPA} schemes. Accordingly, the average max-min rate per user \\emph{versus} the number of active \\glspl{MS} is presented in Fig. \\ref{fig:var_Pilot_Alloc_min_rate} for each of these pilot allocation strategies and for both the \\gls{DL} and the \\gls{UL}. All results have been obtained assuming the default system parameters described in Table \\ref{tab:parameters}, the use of $L=8$ \\gls{RF} chains fully connected to uniform linear antenna arrays with $N=64$ antenna elements, and fronthaul links with a capacity of ${C_F}_d={C_F}_u=64$ bit\/s\/Hz. The first important result to note from Fig. \\ref{fig:var_Pilot_Alloc_min_rate} is that the pure \\gls{RPA} scheme is clearly outperformed by both the \\gls{BRPA} and the \\gls{DCPA} strategies irrespective of the of active \\glspl{MS} in the network. In fact, the \\gls{RPA} scheme cannot guarantee neither the absence of pilot reuse, even for those cases in which $K\\leq\\tau_p$ (in this setup, $\\tau_p=15$ time\/frequency samples), nor the possibility of having pilots that are allocated to a high number of \\glspl{MS} and\/or to \\glspl{MS} exhibiting very similar large-scale propagation patterns to the \\glspl{AP}. Therefore, the higher the number of active \\glspl{MS}, the higher the probability of having one or more users suffering from high levels of pilot contamination, with the consequent reduction of the achievable max-min user rate. If we turn our attention to results provided by the \\gls{BRPA} and \\gls{DCPA} strategies, two disjoint operation regions can be distinguished. In the first one, comprising the scenarios in which $K\\leq\\tau_p$, both approaches allocate orthogonal pilots to the users (absence of pilot contamination) and thus naturally provide the same performance. In the second one, however, comprising the scenarios in which $K>\\tau_p$, pilots have to be reused and, as a consequence, pilot contamination appears (note the rather abrupt performance drop when going from $K\\leq\\tau_p$ to $K>\\tau_p$). In these scenarios, based on a smart exploitation of the available large-scale \\gls{CSI}, the proposed \\gls{DCPA} approach reduces the amount of pilot contamination experienced by the worst users in the network and it clearly improves the achievable max-min user rates provided by the channel-unaware \\gls{BRPA} scheme.\n\nAnother result that is worth emphasizing, since it will repeatedly appear in the following subsections, is that, although in scenarios with high-capacity fronthaul links the achievable max-min \\gls{DL} user rate is higher than that provided in the \\gls{UL}, as the number of active users in the network increases, the performance obtained in both the \\gls{DL} and the \\gls{UL} tend to become increasingly similar. This behavior can be easily deduced from the analysis of the \\gls{SINR} expressions in \\eqref{eq:SINRdk} and \\eqref{eq:SINRuk}. As the number of active \\glspl{MS} in the cell-free network increases, provided that it is greater than $\\tau_p$, the term in the denominator corresponding to the residual interuser interference due to pilot contamination becomes increasingly dominant in comparison to the quantification and thermal noise terms, eventually reaching the point where they can be considered virtually negligible. Under these conditions, and since the pre-coding filters used on both links are identical, the \\gls{DL} and the \\gls{UL} experience similar \\gls{SINR} values and, therefore, tend to provide the same achievable max-min rate per user, except for small differences that can be attributed to, on the one and, the dissimilar amount of quantified information that has to be conveyed through the corresponding fronthaul links and, on the other hand, disparities among the thermal noise powers experienced at both the \\glspl{AP} and the \\glspl{MS}.\n\n\\begin{figure}[t!]\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_varCFd_min_rate}\n        \\label{fig:DL_varCFd_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_varCFu_min_rate}\n        \\label{fig:DL_varCFu_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of active \\glspl{MS} for different values of the fronthaul capacities ($N=64$ antennas, $L=8$ \\gls{RF} chains, \\gls{DCPA}).}\n    \\label{fig:varCF_min_rate}\n\\end{figure}\n\n\\subsection{Modifying the capacity of the fronthaul links and the \\gls{RF} infrastructure at the \\glspl{AP}}\n\nThe max-min achievable rate per user is plotted in Fig. \\ref{fig:varCF_min_rate} against the number of active \\glspl{MS} in the network, assuming the use of fronthaul links with different constraining capacities equal to 16, 32, 64 and 256 bit\/s\/Hz (for the network setups under consideration, using fronthaul links with a capacity of 256 bit\/s\/Hz is virtually equivalent to using infinite-capacity fronthauls). As expected, results show that increasing the fronthaul capacity is always beneficial if the main aim is to increase the achievable max-min user rate. Nevertheless, it is worth stressing that, keeping all the other parameters constant, the marginal increment of performance produced by each new increment of the fronthaul capacity suffers from the law of diminishing returns, especially for network setups with a high number of active \\glspl{MS}. That is, although the performance increase produced by doubling the fronthaul capacity from 16 bit\/s\/Hz to 32 bit\/s\/Hz, or even from 32 bit\/s\/Hz to 64 bit\/s\/Hz, can be justifiable, increasing the fronthaul capacity beyond 64 bit\/s\/Hz does not seem to be reasonable from the point of view of increasing the achievable performance of the system under the considered network setups. As observed in the previous subsection, in cell-free \\gls{mmWave} massive \\gls{MIMO} networks using high-capacity fronthaul links, the achievable max-min \\gls{DL} user rate is always slightly higher than that achieved in the \\gls{UL} irrespective of the number of active \\glspl{MS}. In scenarios with low-capacity fronthaul links and a large number of active \\glspl{MS}, however, the quantization noise experienced in the \\gls{DL} is higher than its \\gls{UL} counterpart and thus, the achievable per-user rate in the \\gls{UL} is slightly higher that than supplied in the \\gls{DL}.\n\n\\begin{figure}[t!]\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_varN_min_rate}\n        \\label{fig:DL_varN_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_varN_min_rate}\n        \\label{fig:DL_varN_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of active \\glspl{MS} for different values of the number of antennas at the \\glspl{AP} ($L=8$ \\gls{RF} chains, ${C_F}_d={C_F}_u=64$ bit\/s\/Hz, \\gls{DCPA}).}\n    \\label{fig:varN_min_rate}\n\\end{figure}\n\n\\begin{figure}[t!]\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_varL_min_rate}\n        \\label{fig:DL_varL_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_varL_min_rate}\n        \\label{fig:DL_varL_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of active \\glspl{MS} for different values of the number of \\gls{RF} chains at the \\glspl{AP} ($N=64$ antennas, ${C_F}_d={C_F}_u=64$ bit\/s\/Hz, \\gls{DCPA}).}\n    \\label{fig:varL_min_rate}\n\\end{figure}\n\nTo understand how the \\gls{RF} infrastructure used at the \\glspl{AP} influences the performance of the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} system under constrained-capacity fronthaul links, Figs. \\ref{fig:varN_min_rate} and \\ref{fig:varL_min_rate} show the achievable max-min user rate against the number of active \\glspl{MS} assuming the use of uniform linear antenna arrays with different number of elements and fully-connected analog \\gls{RF} precoders with different number of \\gls{RF} chains, respectively. In particular, results presented in Fig. \\ref{fig:varN_min_rate} have been obtained assuming the use of an analog precoder with $L=8$ \\gls{RF} chains fully-connected to a linear uniform antenna array with $N=8$, 16, 32, 64 or 128 antenna elements, whereas results presented in Fig. \\ref{fig:varL_min_rate} have been obtained assuming the use of $L=2$, 4, 8 or 16 \\gls{RF} chains fully-connected to a linear uniform antenna array with $N=64$ antenna elements. The first conclusion we may draw when looking at the results presented in Fig. \\ref{fig:varN_min_rate} is that, irrespective of the number of active \\glspl{MS} in the cell-free network, increasing the number of antenna elements at the \\glspl{AP} in scenarios with high capacity fronthaul links (${C_F}_d={C_F}_u=64$ bit\/s\/Hz), although moderate and subject to the law of diminishing returns, always produces an increase in the achievable max-min user rate. As shown in Fig. \\ref{fig:varL_min_rate}, in contrast, the impact produced by an increase in the number of \\gls{RF} chains at the \\glspl{AP} depends on the number of active \\glspl{MS} in the network. In particular, when the number of active users is high, the interuser interference term due to pilot contamination (imperfect \\gls{CSI}) dominates the factors in the denominator of the \\gls{SINR} (i.e., makes the quantization and thermal noises negligible) and thus, increasing the number of \\gls{RF} chains is always beneficial when trying to increase the achievable max-min user rate. When the number of active users in the network is low, however, the quantization noise, which is an increasing function of $L$, is not negligible anymore when compared to the interuser interference term (recall that this term is null when the number of active \\glspl{MS} is less than or equal to $\\tau_p$) and thus, increasing the number of \\gls{RF} chains at the \\glspl{AP} can be clearly disadvantageous.\n\n\\begin{figure}\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_FixK_varN_varCFd_min_rate}\n        \\label{fig:DL_FixK_varN_varCFd_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_FixK_varN_varCFu_min_rate}\n        \\label{fig:UL_FixK_varN_varCFu_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of antennas at the \\glspl{AP} for different values of the fronthaul capacities ($K=20$ users, $L=8$ \\gls{RF} chains, \\gls{DCPA}).}\n    \\label{fig:FixK_varN_varCF_min_rate}\n\\end{figure}\n\n\\begin{figure}\n\n    \\centering\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{DL_FixK_varL_varCFd_min_rate}\n        \\label{fig:DL_FixK_varL_varCFd_min_rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.24\\textwidth}\n        \\centering\n        \\includegraphics[height=7cm]{UL_FixK_varL_varCFu_min_rate}\n        \\label{fig:UL_FixK_varL_varCFu_min_rate}\n    \\end{subfigure}\n\n    \\caption{\\small Average max-min rate per user \\emph{versus} the number of \\gls{RF} chains at the \\glspl{AP} for different values of the fronthaul capacities ($K=20$ users, $N=64$ antennas, \\gls{DCPA}).}\n    \\label{fig:FixK_varL_varCF_min_rate}\n\\end{figure}\n\n\\begin{figure*}\n\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[height=5.9cm]{DL_CFR_VarM}\n        \\label{fig:DL_CFR_VarM}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[height=5.9cm]{UL_CFR_VarM}\n        \\label{fig:UL_CFR_VarM}\n    \\end{subfigure}\n    \\caption{\\small CDF of the \\gls{DL} and \\gls{UL} achievable max-min rate per user for different values of the number of \\glspl{AP} and active \\glspl{MS} in the cell-free network ($N=64$ antennas, $L=8$ \\gls{RF} chains, ${C_F}_d={C_F}_u=64$ bit\/s\/Hz, \\gls{DCPA}).}\n    \\label{fig:CFR_VarM}\n\\end{figure*}\n\nResults presented in Figs. \\ref{fig:varCF_min_rate}, \\ref{fig:varN_min_rate} and \\ref{fig:varL_min_rate} were obtained assuming high-capacity fronthaul links with ${C_F}_d={C_F}_u=64$ bit\/s\/Hz. However, the amount of quantized data that has to be conveyed from (to) the \\gls{CPU} to (from) the \\glspl{AP} in the \\gls{DL} (\\gls{UL}) depends on the number of antennas and \\gls{RF} chains at the \\glspl{AP} (see Section \\ref{sec:fronthaul_capacity}). Thus, in order to deepen in the study of the impact the \\gls{RF} infrastructure may have on the achievable performance of the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} system under constrained-capacity fronthaul links, the average max-min user rate is plotted in Figs. \\ref{fig:FixK_varN_varCF_min_rate} and \\ref{fig:FixK_varL_varCF_min_rate} against the number of antenna elements and \\gls{RF} chains, respectively, for different values of the fronthaul capacities and assuming a fixed number of $K=20$ active \\glspl{MS} in the network. In network setups using very high capacity fronthaul links (i.e., ${C_F}_d={C_F}_u=256$ bit\/s\/Hz), increasing the number of antenna elements $N$ and\/or the number of \\gls{RF} chains $L$ (up to $L=K$) is always beneficial as, in this case, the noise introduced by the quantization process is negligible and the system can take full advantage of the increased \\gls{RF} resources. As the capacity of the fronthaul links decreases, however, the amount of noise introduced by the quantization process increases with both $N$ and $L$ and, therefore, a situation arises where the potential performance improvement provided by the increase of $N$ and\/or $L$ is compromised by the performance reduction due to fronthaul capacity constraints. On the one hand, it can be observed in Fig. \\ref{fig:FixK_varN_varCF_min_rate} that, for fixed numbers of users and \\gls{RF} chains, there is a certain fronthaul capacity constraint value (near 24 bit\/s\/Hz in the setup used in this experiment) under which increasing the number of antenna elements at the array is counterproductive. On the other hand, results presented in Fig. \\ref{fig:FixK_varL_varCF_min_rate} show that, for fixed numbers of users and antenna elements at the arrays, there is always an optimal number of \\gls{RF} chains to be deployed (or activated) at the \\glspl{AP} that is dependent on the capacity of the fronthaul links. In particular, for the network setups under consideration, the optimal number of \\gls{RF} chains is equal to $L=10$, 4, and 1 when using fronthaul links with a capacity of 64 bit\/s\/Hz, 32 bit\/s\/Hz and less than 24 bit\/s\/Hz, respectively.\n\n\\subsection{Impact of the density of \\glspl{AP}}\n\nWith the aim of evaluating the impact the density of \\glspl{AP} per area unit may have on the performance of the proposed cell-free \\gls{mmWave} massive \\gls{MIMO} system, Fig. \\ref{fig:CFR_VarM} represents the \\gls{CDF} of the \\gls{DL} and \\gls{UL} achievable max-min user rate for different values of the number of \\glspl{AP} in the network. It has been assumed in these experiments a fixed number of active \\glspl{MS} equal to either $K=25$ or $K=8$ \\glspl{MS}, the use of $L=8$ \\gls{RF} chains fully-connected to a linear uniform antenna array with $N=64$ antenna elements, and the use of \\gls{DL} and \\gls{UL} fronthaul links with a capacity ${C_F}_d={C_F}_u=64$ bit\/s\/Hz. As expected, cell-free massive \\gls{MIMO} scenarios with a high density of \\glspl{AP} per area unit significantly outperform those with a low density of \\glspl{AP} per area unit in both median and 95\\%-likely achievable per-user rate performance. However, the achievable max-min user rate increase due to increasing the number of \\glspl{AP} in the network is, again, subject to the law of diminishing returns. For instance, in scenarios with $K=25$ \\glspl{MS}, the 95\\%-likely achievable user rate is equal to 2.55, 4.33, 6.11 and 6.50 bit\/s\/Hz for cell-frre massive \\gls{MIMO} networks with $M=25$, 50, 100 and 200 \\glspl{AP}, respectively. That is, doubling the number of \\glspl{AP} per area unit does not result in doubling the 95\\%-likely achievable user rate. Similar conclusions can be drawn when looking at either the median or the average achievable user rates.\n\nAs was observed in results presented in previous subsections for high-capacity fronthaul setups, when the number of active users in the system is low, the achievable max-min rate values in the \\gls{DL} are slightly higher than those achievable in the \\gls{UL}. Instead, when the number of active users increases, the achievable max-min user rates are virtually identical in both the \\gls{DL} and the \\gls{UL}. Also, note that the dispersion of the achievable max-min user rates around the median tends to diminish as the density of \\glspl{AP} increases. That is, cell-free massive \\gls{MIMO} networks with a high density of \\glspl{AP} per area unit tend to offer max-min achievable rates that suffer little variations irrespective of the location of the \\glspl{AP} (i.e, irrespective of the scenario under evaluation).\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nA novel analytical framework for the performance analysis of cell-free \\gls{mmWave} massive \\gls{MIMO} networks has been introduced in this paper. The proposed framework considers the use of low-complexity hybrid precoders\/decoders where the \\gls{RF} high-dimensionality phase shifter-based precoding\/decoding stage is based on large-scale second-order channel statistics, while the low-dimensionality baseband multiuser \\gls{MIMO} precoding\/decoding stage can be easily implemented by standard ZF signal processing schemes using small-scale estimated \\gls{CSI}. Furthermore, it also takes into account the impact of using capacity-constrained fronthaul links that assume the use of large-block lattice quantization codes able to approximate a Gaussian quantization noise distribution, which constitutes an upper bound to the performance attained under any practical quantization scheme. Max-min power allocation and fronthaul quantization optimization problems have been posed thanks to the development of mathematically tractable expressions for both the per-user achievable rates and the fronthaul capacity consumption. These optimization problems have been solved by combining the use of block coordinate descent methods with sequential linear optimization programs. Results have shown that the proposed \\gls{DCPA} suboptimal pilot allocation strategy, which is based on the idea of clustering by dissimilarity, overcomes the computational burden of the optimal small-scale CSI-based pilot allocation scheme while clearly outperforming the pure random and balanced random schemes. It has also been shown that, although increasing the fronthaul capacity and\/or the density of \\glspl{AP} per area unit is always beneficial from the point of view of the achievable max-min user rate, the marginal increment of performance produced by each new increment of these parameters suffers from the law of diminishing returns, especially for network setups with a high number of active \\glspl{MS}. Moreover, simulation results indicate that, as the capacity of the fronthaul links decreases, the potential performance improvement provided by the increase of the number of antenna elements $N$ and\/or the number of \\gls{RF} chains $L$ is compromised by the performance reduction due to the corresponding increase of the fronthaul quantization noise. In particular, for fixed numbers of users and \\gls{RF} chains, there is a certain fronthaul capacity constraint value (near 24 bit\/s\/Hz in the setups under consideration) under which increasing the number of antenna elements at the array is counterproductive. Similarly, for fixed numbers of users and antenna elements at the arrays, there is always an optimal number of \\gls{RF} chains to be deployed (or activated) at the \\glspl{AP} that is dependent on the capacity of the fronthaul links. For future work, it would be interesting to develop low-complexity pilot- and power-allocation techniques specifically designed to maximize the energy efficiency of cell-free \\gls{mmWave} massive \\gls{MIMO} networks considering both the fronthaul capacity constraints and the fronthaul power consumption. It would also be interesting to explore the use of partially-connected \\gls{RF} precoding\/decoding architectures and the implementation of baseband \\gls{MU-MIMO} precoding\/decoding other than the \\gls{ZF} scheme.\n\n\\appendices\n\n\\section{Proof of Theorem \\ref{theo:theorem_DL}}\n\\label{app:Appendix_1}\n\nFollowing an approach similar to that proposed by Nayebi \\emph{et al.} in \\cite{Nayebi17}, the signal received by the $k$th \\gls{MS} in \\eqref{eq:ydk} can be rewritten as ${y_d}_k={y_d}_{k\\,0} + {y_d}_{k\\,1} + {y_d}_{k\\,2} + {n_d}_k$, where the useful, interuser interference, and quantization noise terms can be expressed as ${y_d}_{k\\,0}=\\sqrt{\\upsilon_k} {s_d}_k$, ${y_d}_{k\\,1}=\\tilde{\\bs{g}}_k^T \\bs{W}_d^{BB} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d$, and ${y_d}_{k\\,2} = \\bs{g}_k^T \\bs{q}_d=\\sum_{m=1}^M \\bs{g}_{km}^T {\\bs{q}_d}_m$, respectively. Now, considering that data symbols, quantization noise, thermal noise, and channel-related coefficients are mutually independent, the terms ${y_d}_{k\\,0}$, ${y_d}_{k\\,1}$, ${y_d}_{k\\,2}$ and ${n_d}_k$ are mutually uncorrelated and thus, based on the worst-case uncorrelated additive noise \\cite{Hassibi03}, the achievable \\gls{DL} rate for user $k$ is lower bounded by ${R_d}_k=\\log_2\\left(1+{\\SINR_d}_k\\right)$, with\n\\begin{equation*}\n   {\\SINR_d}_k=\\frac{\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,0}\\right|^2\\right\\}}{\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,1}\\right|^2\\right\\}+\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,2}\\right|^2\\right\\}+\\sigma_d^2},\n\\end{equation*}\nwhere $\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,0}\\right|^2\\right\\}=\\upsilon_k$,\n\\begin{equation*}\n\\begin{split}\n   &\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,1}\\right|^2\\right\\}=\\mathbb{E}\\left\\{\\bs{s}_d^H\\bs{\\Upsilon}^{1\/2} {\\bs{W}_d^{BB}}^H \\tilde{\\bs{g}}_k^* \\tilde{\\bs{g}}_k^T \\bs{W}_d^{BB} \\bs{\\Upsilon}^{1\/2}\\bs{s}_d\\right\\} \\\\\n   &\\qquad=\\tr\\left(\\bs{\\Upsilon} \\mathbb{E}\\left\\{{\\bs{W}_d^{BB}}^H \\tilde{\\bs{g}}_k^* \\tilde{\\bs{g}}_k^T \\bs{W}_d^{BB}\\right\\}\\right) \\\\\n   &\\qquad=\\sum_{k'=1}^K \\upsilon_{k'} \\left[\\diag\\left(\\mathbb{E}\\left\\{{\\bs{W}_d^{BB}}^H \\tilde{\\bs{g}}_k^* \\tilde{\\bs{g}}_k^T \\bs{W}_d^{BB}\\right\\}\\right)\\right]_{k'},\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n   \\mathbb{E}\\left\\{\\left|{y_d}_{k\\,2}\\right|^2\\right\\}&=\\sum_{m=1}^M \\mathbb{E}\\left\\{{\\bs{q}_d}_m^H \\bs{g}_{km}^* \\bs{g}_{km}^T {\\bs{q}_d}_m\\right\\} \\\\\n                                                       &=\\sum_{m=1}^M {\\sigma_q^2}_{dm} \\tr\\left(\\bs{R}_{mk}^{RF}\\right).\n\\end{split}\n\\end{equation*}\n\n\n\\section{Proof of Theorem \\ref{theo:theorem_UL}}\n\\label{app:Appendix_2}\n\nThe detected signal at the \\gls{CPU} corresponding to the symbol transmitted by the $k$th \\gls{MS} in \\eqref{eq:yuk} can be rewritten as ${y_u}_k={y_u}_{k\\,0} + {y_u}_{k\\,1} + {y_u}_{k\\,2} + {y_u}_{k\\,3}$, where the useful, interuser interference, quantization noise and thermal noise terms can be expressed as ${y_u}_{k\\,0}=\\sqrt{P_u} \\sqrt{\\omega_k} {s_u}_k$, ${y_u}_{k\\,1}=\\sqrt{P_u}\\left[\\bs{W}_u^{BB} \\tilde{\\bs{G}} \\bs{\\Omega}^{1\/2} \\bs{s}_u\\right]_k$, ${y_u}_{k\\,2} = \\left[\\bs{W}_u^{BB} \\bs{q}_u \\right]_k$, and ${y_u}_{k\\,3} = \\left[\\bs{W}_u^{BB} \\bs{n}_u\\right]_k$, respectively. As in the \\gls{DL}, since data symbols, quantization noise, thermal noise, and channel-related coefficients are mutually independent, the terms ${y_u}_{k\\,0}$, ${y_u}_{k\\,1}$, ${y_d}_{k\\,2}$ and ${y_d}_{k\\,3}$ are mutually uncorrelated and thus, based on the worst-case uncorrelated additive noise \\cite{Hassibi03}, the achievable \\gls{UL} rate for user $k$ is lower bounded by ${R_u}_k=\\log_2\\left(1+{\\SINR_u}_k\\right)$, with\n\\begin{equation*}\n   {\\SINR_u}_k=\\frac{\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,0}\\right|^2\\right\\}}{\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,1}\\right|^2\\right\\}+\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,2}\\right|^2\\right\\}+\\mathbb{E}\\left\\{\\left|{y_d}_{k\\,3}\\right|^2\\right\\}},\n\\end{equation*}\nwhere $\\mathbb{E}\\left\\{\\left|{y_u}_{k\\,0}\\right|^2\\right\\}=P_u \\omega_k$,\n\\begin{equation*}\n\\begin{split}\n   &\\mathbb{E}\\left\\{\\left|{y_u}_{k\\,1}\\right|^2\\right\\}=P_u\\mathbb{E}\\left\\{\\bs{s}_u^H\\bs{\\Omega}^{1\/2} \\tilde{\\bs{G}}^H {\\bs{w}_{uk}^{BB}}^H \\bs{w}_{uk}^{BB} \\tilde{\\bs{G}}\\bs{\\Omega}^{1\/2} \\bs{s}_u\\right\\} \\\\\n   &\\qquad=P_u \\tr\\left(\\bs{\\Omega} \\mathbb{E}\\left\\{\\tilde{\\bs{G}}^H {\\bs{w}_{uk}^{BB}}^H \\bs{w}_{uk}^{BB} \\tilde{\\bs{G}}\\right\\}\\right) \\\\\n   &\\qquad=P_u \\sum_{k'=1}^K \\omega_{k'} \\left[\\diag\\left(\\mathbb{E}\\left\\{\\tilde{\\bs{G}}^H {\\bs{w}_{uk}^{BB}}^H \\bs{w}_{uk}^{BB} \\tilde{\\bs{G}}\\right\\}\\right)\\right]_{k'},\n\\end{split}\n\\end{equation*}\nwith $\\bs{w}_{uk}^{BB}$ denoting the $k$th row of $\\bs{W}_u^{BB}$, or, equivalently,\n\\begin{equation*}\n\\begin{split}\n   &\\mathbb{E}\\left\\{\\left|{y_u}_{k\\,1}\\right|^2\\right\\} = P_u\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\tilde{\\bs{G}} \\bs{\\Omega} \\tilde{\\bs{G}}^H {\\bs{W}_u^{BB}}^H \\right\\}\\right)\\right]_k \\\\\n   &\\qquad=P_u \\sum_{k'=1}^K \\omega_{k'} \\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\tilde{\\bs{g}}_{k'} \\tilde{\\bs{g}}_{k'}^H {\\bs{W}_u^{BB}}^H \\right\\}\\right)\\right]_k,\n\\end{split}\n\\end{equation*}\nand, finally,\n\\begin{equation*}\n\\begin{split}\n   &\\mathbb{E}\\left\\{\\left|{y_u}_{k\\,2}\\right|^2\\right\\}=\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\bs{q}_u \\bs{q}_u^H {\\bs{W}_u^{BB}}^H\\right\\}\\right)\\right]_k \\\\\n   &\\qquad=\\sum_{m=1}^M \\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_u^{BB} \\bs{q}_{um} \\bs{q}_{um}^H {\\bs{W}_u^{BB}}^H\\right\\}\\right)\\right]_k \\\\\n   &\\qquad=\\sum_{m=1}^M {\\sigma_q^2}_{um}\\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_{u\\,m}^{BB} {\\bs{W}_{u\\,m}^{BB}}^H\\right\\}\\right)\\right]_k,\n\\end{split}\n\\end{equation*}\nand, analogously,\n\\begin{equation*}\n\\begin{split}\n   \\mathbb{E}\\left\\{\\left|{y_d}_{k\\,3}\\right|^2\\right\\}=\\sigma_u^2(N)\\sum_{m=1}^M \\left[\\diag\\left(\\mathbb{E}\\left\\{\\bs{W}_{u\\,m}^{BB} {\\bs{W}_{u\\,m}^{BB}}^H\\right\\}\\right)\\right]_k.\n\\end{split}\n\\end{equation*}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent Speech enhancement aims at separating speech from background interference signals. Mainstream deep learning-based methods learn to predict the speech signal in a supervised manner, as shown in Figure \\ref{fig0} (a). Most prior works operate in the time-frequency (T-F) domain by predicting a mask between noisy and clean spectra \\cite{wang2014IRM, williamson2015complex} or directly predicting the clean spectrum \\cite{xu2013experimental, tan2018convolutional}. Some methods operate in the time domain by estimating speech signals from raw-waveform noisy signals in an end-to-end way \\cite{fu2017raw, pascual2017segan, pandey2019tcnn}. These methods have considerably improved the quality of enhanced speech compared with traditional signal processing based schemes. However, speech distortion or residual noise can often be observed in the enhanced speech, showing that there are still correlations between predicted speech and the residual signal by subtracting enhanced speech from noisy signal.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{illustration_1}\n    \\caption{Illustration of different methods. (a) Most existing deep-learning-based methods directly model speech. (b) Most traditional methods predict speech with noise estimate. (c) Our method simultaneously models speech and noise with information interaction.}\n    \\label{fig0}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{overview_network_structure_v2}\n    \\caption{Overall network structure of SN-Net.}\n    \\label{fig1}\n\\end{figure*}\n\nInstead of only predicting speech and ignoring the characteristics of background noises, traditional signal processing and modeling based methods mostly take the other way (see Figure \\ref{fig0} (b)), i.e. estimating noise or building noise models for speech enhancement \\cite{boll1979suppression, hendriks2010noisePSD, wang2017model, wilson2008NMF1, moham2013NMF2}. Some model-based methods instead model both speech and noise \\cite{srinivasan2006Codebook, srinivasan2005Codebook}, possibly with alternate model update. However, they typically cannot generalize well when prior noise assumption cannot be met or the interference signal is not structured. In deep-learning-based methods, two recent attempts \\cite{odelowo2017noise, odelowo2018study} focus on directly predicting noise considering that noise is dominant in low-SNR conditions. However, the benefit is limited.\n\n\nThe remaining correlation between predicted speech and noise motivates us to explore the information flow between speech and noise estimations, as shown in Figure \\ref{fig0} (c). Since speech-related information exists in predicted noise, and vice versa, adding information communication between them may help to recover some missing components and remove undesired information from each other. In this paper, we propose a two-branch convolutional neural network, namely SN-Net, to simultaneously predict speech and noise signals. Between them are information interaction modules, by which noise or speech related information are extracted from the noise branch and added back to speech features to counteract the undesired noise part or recover the missing speech, and vice versa. In this way, the discrimination capability is largely enhanced. The two branches share the same network structure, which is an encoder-decoder-based model with several residual-convolution-and-attention (RA) blocks in between for separation. Motivated by the success of self-attention technique in machine translation and computer vision tasks \\cite{vaswani2017attention, wang2018nonlocal}, we propose to combine temporal self-attention and frequency-wise self-attention parallelly inside each RA block for capturing global dependency along temporal and frequency dimensions in a separable way.\n\nOur main contributions are summarized as follows.\n\n\\begin{itemize}\n    \\item We propose to simultaneously model speech and noise in a two-branch deep neural network and introduce information flow between them. In this way, speech part is enhanced while residual noise is suppressed for speech estimation, and vice versa.\n    \\item We propose a RA block for feature extraction. Separable self-attention is utilized in this block to globally capture the temporal and frequency dependencies.\n    \\item We validate the superiority of proposed scheme in an ablation study and comparison with state-of-the-art algorithms on two public datasets. Moreover, we extend our method to speaker separation, which also shows great performance. These results demonstrate the superiority and potential of the proposed method.\n\\end{itemize}\n\n\\section{Related Work}\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{overview_Encoder-Decoder}\n    \\includegraphics[width=0.4\\textwidth]{attention_structure_in_Decoder}\n    \\caption{(a) Encoder-decoder structure. The dashed arrow denotes the separation module using RA blocks. (b) Detailed structure of the gated block inside the decoder.}\n    \\label{fig2}\n\\end{figure*}\n\n\\subsection{Deep Learning-based Speech Enhancement}\n\n\\noindent Deep learning-based methods mainly study how to build a speech model. According to the adopted signal domain, these methods can be classified into two categories. Time-Frequency (T-F) domain methods take T-F representation, either complex or log power spectrum of the magnitude, as input. They typically estimate a real or complex ratio mask for each T-F bin to map noisy spectra to speech spectra \\cite{williamson2015complex, wang2014IRM, choi2019phase} or directly predict the speech representation \\cite{xu2013experimental, tan2018convolutional}. Time-domain methods take waveform as input and typically extract a hidden representation of the raw waveform through an encoder and reconstruct an enhanced version from that \\cite{fu2017raw, pascual2017segan, pandey2019tcnn}. Although these methods have shown great improvements over traditional methods, they only focus on modeling speech and neglect the importance of understanding noise characteristics.\n\n\\subsection{Noise-Aware Speech Enhancement}\n\n\\noindent Noise information is often considered in traditional signal processing based methods \\cite{boll1979suppression, hendriks2010noisePSD, wang2017model} with prior distribution assumptions for speech and noise. However, it is a challenging task to estimate the noise power spectral density for non-stationary noises and thus mostly stationary noise is assumed. They are unsuitable in generalization to low SNR and non-stationary noise conditions. Instead, some model-based methods build models for speech and noise and show more promising results, e.g., codebook \\cite{srinivasan2006Codebook, srinivasan2005Codebook} and nonnegative matrix factorization (NMF) \\cite{wilson2008NMF1, moham2013NMF2} based methods. However, they either need prior knowledge of the noise type \\cite{srinivasan2006Codebook, srinivasan2005Codebook} or are only effective for structured noise \\cite{wilson2008NMF1, moham2013NMF2}; therefore their generalization capability is limited.\n\nDeep learning-based methods can better generalize to various noise conditions. There are also some attempts on incorporating noise information, for example, by adding constraints to loss functions \\cite{fan2019noise, xu2020using, xia2020weighted} or by directly predicting noise instead of speech \\cite{odelowo2017noise, odelowo2018study}. The former does not model noise at all and the characteristics of noise are not exploited. The latter loses the speech information and show even worse quality than corresponding speech prediction method in low SNR and unseen noise conditions. A more relevant work utilizes two deep auto encoders (DAEs) to estimate speech and noise \\cite{sun2015unseen} . It first trains a DAE for speech spectrum reconstruction and then introduces another DAE to model noise with the constraint that the sum of outputs of the two DAEs is equal to the noisy spectrum.\n\nDifferent from aforementioned approaches, we proposed a two-branch CNN to predict speech and noise simultaneously and introduce interaction modules at several intermediate layers to make them benefit from each other. Such a paradigm makes it suitable for speaker separation as well. \n\n\\subsection{Two-Branch Neural Networks}\n\n\\noindent Two-branch neural networks have been explored in various tasks for capturing cross-modality information \\cite{nam2017dual, image-text} or different levels of information \\cite{simonyan2014two, wang2020dual}. For speech enhancement, a two-branch modeling is proposed to predict the amplitude and phase of the enhanced signal, respectively \\cite{yin2020phasen}. In this paper, we aim to exploit the two correlated tasks, i.e. speech and noise estimations and explicitly modeling them in an interactive two-branch framework for better discrimination.\n\n\\subsection{Self-Attention Model}\n\n\\noindent Self-attention mechanism has been widely used in many tasks, e.g., machine translation \\cite{vaswani2017attention}, image generation \\cite{zhang2019self} and video question answering \\cite{li2019beyond}. For video, spatio-temporal attention is also considered to exploit long-term dependency along both spatial and temporal dimensions \\cite{wu2019video}. Recently, speech-related tasks have also benefited from self-attention, e.g., speech recognition \\cite{salazar2019ctc} and speech enhancement \\cite{kim2020tgsa, koizumi2020speech}. In these works, self-attention is applied along the temporal dimension only, neglecting the global dependency inside each frame. Motivated by the spatio-temporal attention in video-related tasks, we propose to employ both frequency-wise and temporal self-attention to better capture dependencies along different dimensions. Such an attention is employed in both speech and noise branches for simultaneous modeling the two signals.\n\n\\section{Proposed Method}\n\n\\subsection{Overview}\n\n\\noindent Figure \\ref{fig1} shows the overall network structure of SN-Net. The input is the complex T-F spectrum computed by short-time Fourier transform (STFT), denoted as $ X^I\\in R^{T\\times F\\times 2} $, where $ T $ is the number of frames and $ F $ is the number of frequency bins. There are two branches in SN-Net, one of which predicts speech and the other predicts noise. They share the same network structure but have separate network parameters. Each branch is an encoder-decoder based structure, with several RA blocks inserted inbetween them. In this way, it is capable of simultaneously mining the potential of different components of the noisy signal. Between the two branches are interaction modules designed to transform and share information. After each branch gets its output, a merge branch is employed to adaptively combine the two outputs to generate the final enhanced speech.\n\n\\subsection{Encoder and Decoder}\n\n\\noindent As shown in Figure \\ref{fig2} (a), the encoder has three 2-D convolutional layers, each with a kernel size of (3, 5). The stride is (1, 1) for the first layer and (1, 2) for the following two. The channel numbers are 16, 32, 64, respectively. As a result, the output feature of the encoder is $ \\mathcal{F}^E_k\\in \\mathbb{R}^{T\\times F'\\times C} $, where $ F'=\\frac{F}{4}$, $C=64 $ and $ k\\in \\left\\{S,N\\right\\} $. $ S $ and $ N $ denote speech and noise branches, respectively. For simplicity, the subscript $ k $ will be ignored in the following.\n\nThe decoder consists of three gated blocks followed by one 2-D convolutional layer, which reconstructs the output $ \\mathcal{F}^D\\in \\mathbb{R}^{T\\times F\\times 2} $. As shown in Figure \\ref{fig2} (b), the gated block learns a multiplicative mask on corresponding feature from the encoder, aiming to suppress its undesired part. The masked encoder feature is then concatenated with the deconvolutional feature and fed into another 2-D convolutional layer to generate the residual representation. After three gated blocks, the final convolutional layer learns the amplitude gain and the phase for reconstruction, similar to that in \\cite{choi2019phase}. The kernel size for all 2-D deconvolutional layers is (3,5). The stride is (1,2) for the first two gated blocks and (1,1) for the last one. The channel numbers are 64, 32, 16, respectively. All the 2-D convolutional layers in the decoder have a kernel size of (1,1), a stride of (1,1) and a channel number the same as that of their inputs.\n\nAll the convolutional layers in the encoder and the decoder are followed by a batch normalization (BN) and a parametric ReLU (PReLU). No down-sampling is performed along the temporal dimension to preserve the temporal resolution.\n\n\\subsection{RA Block}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{RA_block}\n    \\caption{Structure of the RA block.}\n    \\label{fig3}\n\\end{figure}\n\n\\noindent The RA block is designed to extract features and perform separation for both speech and noise branches. It is challenging because of the diversities of noise types and the difference between speech and noises. We employ the separable self-attention (SSA) technique to capture the global dependencies along temporal and frequency dimensions, respectively. It is intuitive to use attention for these two dimensions as humans tend to put more attention to some parts of an audio signal (e.g., speech) while less to the surrounding part (e.g., noise) and they perceive differently on different frequencies. When it comes to the speech-noise network in SN-Net, the SSA modules in speech and noise branches perceive signals differently, which will be demonstrated in the ablation study section afterwards. \n\nIn SN-Net, there are four RA blocks between the encoder and the decoder. Each block consists of two residual blocks and a SSA module, as shown in Figure \\ref{fig3}, capturing both local and global dependencies inside the signal. Each residual block has two 2-D convolutional layers with a kernel size of (5,7), a stride of (1,1) and the same number of channels as their inputs. The output feature of two residual blocks $ \\mathcal{F}^{Res}_i\\in \\mathbb{R}^{T\\times F'\\times C} $ ($ i\\in \\left\\{1,2,3,4\\right\\} $ represents the $ i^{th} $ RA block and will be ignored in the following) is fed parallelly into temporal self-attention and frequency-wise self-attention blocks. These two attention blocks produce the outputs $ \\mathcal{F}^{Temp}\\in \\mathbb{R}^{T\\times F'\\times C} $ and $ \\mathcal{F}^{Freq}\\in \\mathbb{R}^{T\\times F'\\times C} $. The three features $\\mathcal{F}^{Res} $, $ \\mathcal{F}^{Temp} $ and $ \\mathcal{F}^{Freq} $ are then concatenated and fed into a 2-D convolutional layer to generate the block output $ \\mathcal{F}^{RA}\\in \\mathbb{R}^{T\\times F'\\times C} $, used in the interaction module.\n\nFor self-attention, we employ the scaled dot-product self-attention here. Considering the computational complexity, channels are reduced by half inside SSA. The temporal self-attention can be represented as\n\\begin{equation}\n\\begin{aligned} \n    \\mathcal{F}_t^{k}&=Reshape^t(Conv(\\mathcal{F}^{Res})), k\\in \\left\\{K, Q, V\\right\\},\\\\\n    SA^{t}&=Softmax(\\mathcal{F}_t^{Q}\\cdot (\\mathcal{F}_t^{K})^T\/\\sqrt{\\frac{C}{2}\\times F'})\\cdot \\mathcal{F}_t^{V},\\\\\n    \\mathcal{F}^{Temp}&=\\mathcal{F}^{Res}+Conv(Reshape^{t*}(SA^{t})),\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{F}_t^{k} \\in \\mathbb{R}^{T\\times (\\frac{C}{2}\\times F')}$, $SA^{t}\\in \\mathbb{R}^{T\\times (\\frac{C}{2}\\times F')}$ and $\\mathcal{F}^{Temp} \\in \\mathbb{R}^{T\\times F'\\times C}$, respectively. $(\\cdot)$ denotes matrix multiplication. $Reshape^t(\\cdot)$ denotes a tensor reshape from $\\mathbb{R}^{T\\times F'\\times \\frac{C}{2}}$ to $\\mathbb{R}^{T\\times (\\frac{C}{2}\\times F')}$ and $Reshape^{t*}(\\cdot)$ is the opposite. The frequency-wise self-attention is given by\n\\begin{equation}\n\\begin{aligned} \n    \\mathcal{F}_f^{k}&=Reshape^f(Conv(\\mathcal{F}^{Res})), k\\in \\left\\{K, Q, V\\right\\},\\\\\n    SA^{f}&=Softmax(\\mathcal{F}_f^{Q}\\cdot (\\mathcal{F}_f^{K})^T\/\\sqrt{\\frac{C}{2}\\times T})\\cdot \\mathcal{F}_f^{V},\\\\\n    \\mathcal{F}^{Freq}&=\\mathcal{F}^{Res}+Conv(Reshape^{f*}(SA^{f})),\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{F}_f^{k}\\in \\mathbb{R}^{F'\\times (\\frac{C}{2}\\times T)}$, $SA^{f}\\in \\mathbb{R}^{F'\\times(\\frac{C}{2}\\times T)}$ and $\\mathcal{F}^{Freq}\\in \\mathbb{R}^{T\\times F'\\times C}$, respectively. $Reshape^f(\\cdot)$ reshapes a tensor from $\\mathbb{R}^{T\\times F'\\times \\frac{C}{2}}$ to $\\mathbb{R}^{F'\\times (\\frac{C}{2}\\times T)}$.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.33\\textwidth]{Interaction_block}\n    \\caption{Structure of the interaction module.}\n    \\label{fig4}\n\\end{figure}\n\nIn the above equations, $ Conv $ denotes a convolutional layer followed by BN and PReLU. All the convolutional layers have a kernel size of (1,1) and a stride of (1,1). \n\n\\subsection{Interaction Module}\n\n\\noindent In SN-Net, the speech and noise branches share the same input signal, which suggests that the internal features of two branches are correlated. In light of this, we propose an interaction module to exchange information between the branches. With this block, information transformed from the noise branch is expected to enhance the speech part and counteract the noise features inside the speech branch, and vice versa. We will show in ablation study afterwards that this module plays a key role in simultaneously modeling the speech and noises. \n\nThe structure of the interaction module is shown in Figure \\ref{fig4}. Taking speech branch as an example, feature from the noise branch $ \\mathcal{F}_N^{RA} $ is first concatenated with that from the speech branch $ \\mathcal{F}_S^{RA}$. They are then fed into a 2-D convolutional layer to generate a multiplicative mask $ \\mathcal{M}^N $, predicting the suppressed and preserved areas of $ \\mathcal{F}_N^{RA} $. A residual representation $ \\mathcal{H}^{N2S} $ is then obtained by multiplying $ \\mathcal{M}^N $ with $ \\mathcal{F}_N^{RA} $ elementally. Finally, the block adds $ \\mathcal{F}_S^{RA} $ and $ \\mathcal{H}^{N2S} $ to get a \"filtered\" version of the speech feature, which will be fed into the next RA block. The process is given by\n\\begin{equation}\n\\begin{aligned} \n    \\mathcal{F}_{S_{out}}^{RA}&=\\mathcal{F}_S^{RA}+\\mathcal{F}_N^{RA}*Mask(\\mathcal{F}_N^{RA},\\mathcal{F}_S^{RA}),\\\\\n    \\mathcal{F}_{N_{out}}^{RA}&=\\mathcal{F}_N^{RA}+\\mathcal{F}_S^{RA}*Mask(\\mathcal{F}_S^{RA},\\mathcal{F}_N^{RA}),\n\\end{aligned}\n\\end{equation}\nwhere $Mask(\\cdot)$ is short for concatenation, convolution and sigmoid operations. $(*)$ denotes element-wise multiplication.\n\n\\subsection{Merge Branch}\n\n\\noindent After reconstructing the speech and noise signals in two branches, a merge module is further employed to combine the two outputs. This is done in the time domain to achieve the cross-domain benefit \\cite{Kim2018MDPhD}. The two decoder outputs are transformed to time-domain and overlapped framed representation using the same window length as the STFT we use, resulting in $ \\tilde{s}\\in \\mathbb{R}^{T\\times K} $ and $ \\tilde{n}\\in \\mathbb{R}^{T\\times K} $, where $K$ is the frame size. These two representations are stacked with the noisy waveform $ x $ and fed into the merge branch. The merge network uses a 2-D convolutional layer, followed by an temporal self-attention block to capture global temporal dependency and two other convolutional layers to learn an element-wise mask $ m\\in \\mathbb{R}^{T\\times K} $. The kernel size of all three convolutional layers is (3,7) and the channel number is 3, 3, 1, respectively. BN and PReLU are used after each convolutional layer except the last one. Sigmoid activation is used in the last layer. Finally, the 2D enhanced signal is obtained by\n\\begin{equation}\n    \\hat{s} = m\\times \\tilde{s} + (1-m)\\times (x-\\tilde{n}).\n\\end{equation}\nThe 1D signal is reconstructed from $\\hat{s}$ after overlap and add.\n\n\\section{Experiments}\n\n\\subsection{Datasets}\n\n\\noindent Three public datasets are used in our experiments.\n\n\\textbf{DNS Challenge} The DNS challenge \\cite{reddy2020interspeech} at Interspeech 2020 provides a large dataset for training. It includes 500 hours clean speech across 2150 speakers collected from Librivox and 60000 noise clips from Audioset \\cite{gemmeke2017audio} and Freesound with 150 classes. For training, we synthesized 500 hours noisy samples with SNR levels of -5dB, 0dB, 5dB, 10dB and 15dB. For evaluation, we use 150 synthetic noisy samples without reverberation inside the test set, whose SNR levels are randomly distributed between 0 dB and 20 dB.\n\n\\textbf{Voice Bank + DEMAND} This is a small dataset created by Valentini-Botinhao et al. \\cite{valentini2016investigating}. Clean speech clips are collected from the Voice Bank corpus \\cite{veaux2013voice} with 28 speakers for training and another 2 unseen speakers for test. Ten noise types with two artificially generated and eight real recordings from DEMAND \\cite{thiemann2013diverse} are used for training. Five other noise types from DEMAND are chosen for the test, without overlapping with the training set. The SNR values are 0dB, 5dB, 15dB and 20dB for training and 2.5dB, 7.5dB, 12.5dB and 17.5dB for test. \n\n\\textbf{TIMIT Corpus} This dataset is used for our speaker separation experiment. It contains recordings of 630 speakers, each reading 10 sentences and there are 462 speakers in the training set and 168 speakers in the test set. Two sentences from different speakers are mixed with random SNRs to generate mixture utterances. Shorter sentences are zero padded to match the size of longer ones. In total, the training set includes 4620 sentences and the test set 1680 sentences.\n\n\\begin{table}[t]\n    \\centering\n    \\caption{Ablation study on DNS Challenge dataset}\n    \\label{table1}\n    \\begin{tabular}{lrcc}\n        \\hline\n        \\multicolumn{2}{l}{\\textbf{Models}} & \\textbf{SDR(dB)} & \\textbf{PESQ}\\\\\n        \\hline\n        \\multicolumn{2}{l}{Noisy} & 9.09 & 1.58\\\\\n        \\hline\n        \\multicolumn{2}{l}{Speech branch w\/o SSA (baseline)} & 18.06 & 3.05\\\\\n        \\multicolumn{2}{l}{Speech branch} & 18.75 & 3.28\\\\\n        \\hline\n        \\multicolumn{2}{l}{SN-Net w\/o interaction} & 19.04 & 3.29 \\\\\n        \\multicolumn{2}{l}{SN-Net} & \\textbf{19.52} & \\textbf{3.39} \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\\subsection{Evaluation Metrics}\n\n\\noindent To evaluate the quality of the enhanced speech, the following objective measures are used. Higher scores indicate better quality.\n\n\\begin{itemize}\n    \\item SSNR: Segmental SNR.\n    \\item SDR \\cite{vincent2006performance}: Signal-to-distortion ratio.\n    \\item PESQ \\cite{rec2005p}: Perceptual evaluation of speech quality, using the wide-band version recommended in ITU-T P.862.2 (from -0.5 to 4.5).\n    \\item CSIG \\cite{hu2007evaluation}: Mean opinion score (MOS) prediction of the signal distortion (from 1 to 5).\n    \\item CBAK \\cite{hu2007evaluation}: MOS prediction of the intrusiveness of background noises (from 1 to 5).\n    \\item COVL \\cite{hu2007evaluation}: MOS prediction of the overall effect (from 1 to 5).\n\\end{itemize}\n\n\\subsection{Implementation Details}\n\\subsubsection{Input}\nAll signals are resampled to 16kHz and clipped to 2 seconds long. We take the STFT complex spectrum as input, with a Hann window of length 20ms, a hop length of 10ms and a DFT length of 320. \n\n\\subsubsection{Loss Function}\nThe loss function includes three terms, i.e. $ \\mathcal{L}= \\mathcal{L}_{Speech}+\\alpha\\mathcal{L}_{Noise}+\\beta\\mathcal{L}_{Merge} $, where $ \\mathcal{L}_{Speech} $, $ \\mathcal{L}_{Noise} $ and $ \\mathcal{L}_{Merge}$ represent the loss of three branches, respectively. $ \\alpha$ and $\\beta $ are weighting factors balancing among the three. All terms use a mean-squre-error (MSE) loss on the power-law compressed STFT spectrum \\cite{ephrat2018looking}. An inverse STFT and forward STFT are conducted on speech and noise branches before calculating the loss to ensure STFT consistency as that in \\cite{wisdom2019differentiable}.\n\n\\subsubsection{Training}\nThe proposed algorithm is implemented in TensorFlow. We use adam optimizer with a learning rate of 0.0002. All the layers are initialized with Xavier initialization. The training is conducted in two stages. The speech and noise branches are jointly trained first with the loss weight $ \\alpha=1 $ and $ \\beta=0 $. Then the merge branch is trained with the parameters of previous two fixed, using only the loss $\\mathcal{L}_{Merge}$. We train both stages for 60 epochs for DNS Challenge and 400 epochs for Voice Bank + DEMAND dataset. The batch size for all experiments is set to 32, unless otherwise specified. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{Visualization_1_v2}\n    \\caption{Log-scale feature visualization for the fourth interaction module. (a) Input feature of speech branch. (b) Transformed feature from noise to speech branch. (c) Output feature of speech branch. (d) Input feature of noise branch. (e) Transformed feature from speech to noise branch. (f) Output feature of noise branch. Three channels with the highest activities are visualized here.}\n    \\label{fig7}\n\\end{figure}\n\n\\subsection{Ablation Study}\n\\subsubsection{Objective Quality}\nWe first evaluate the effectiveness of different parts of the proposed SN-Net based on the DNS Challenge dataset. As shown in Table \\ref{table1}, we take the speech branch without SSA as the baseline. After adding SSA to the single-branch model, we observe a 0.69 dB gain on SDR and 0.23 on PESQ. By comparing \"Speech branch\" with \"SN-Net w\/o interaction\", we can see that when no interaction is employed, adding another branch with merge module at the output only marginally improves the SDR by 0.29 dB and no improvement on PESQ. After introducing the information flow, it evidently improves the SDR by 0.77 dB and PESQ by 0.11 compared to single branch. These results verify the effectiveness of the proposed RA and interaction modules for simultaneously modeling speech and noises.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{SSA_visualization1_v5}\n    \\caption{Visualization of temporal self-attention matrices from different RA blocks. (a) Speech branch. (b) Noise branch. Each matrix is linearly scaled to $[0, 1]$.}.\n    \\label{fig5}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{SSA_visualization2_v5}\n    \\caption{Visualization of frequency-wise self-attention matrices from different RA blocks. (a) Speech branch. (b) Noise branch. Each matrix is linearly scaled to $[0, 1]$.}\n    \\label{fig6}\n\\end{figure}\n\n\\subsubsection{Visualization of Information Flow}\nIn order to further understand how the interaction module works, we visualize the input feature, the output and the feature transformed from the other branch of this module in Figure \\ref{fig7}. An audio signal corrupted by white noises is used for illustration, whose spectrum is shown in the first column.\n\nThe transformed feature shown in Figure \\ref{fig7} (b) is learned from the feature in (d) and added to the feature in (a), resulting in the output feature of speech branch in (c) and vice versa. Comparing (a) and (c), we can see that the speech area is better separated with noise after interaction. For noise branch, the speech part is mostly removed in (f) compared with (d). These results show that the interaction module indeed helps the simultaneous speech and noise modeling with better separation capabilities. In terms of interchanged information, the undesired speech part in (d) is counteracted by features learned from the speech branch (e.g., the second channel of the noise branch) and the undesired noise part in (a) is suppressed by features learned from the noise branch (e.g., the third channel of the speech branch). These observations comply with our previous analysis.\n\n\\subsubsection{Visualization of Separable Self-Attention}\nWe further visualize the attention matrix to explore what it has learned. Figure \\ref{fig5} shows the temporal self-attention matrix inside different RA blocks for the same audio signal as that in Figure \\ref{fig7}. From (a) and (b), we can see that besides the diagonal line, each frame shows strong attentiveness to other frames and speech and noise branches behave differently for each RA module. This is reasonable as the two branches model different signals and their focus differs. For noise branch, the attention goes from local to global as the network goes deeper. The noise branch shows wider attentiveness than the speech branch as white noises spread in all frames while speech signal occurs only at some time.\n\nFigure \\ref{fig6} shows the frequency-wise self-attention matrix for the same audio signal. For speech branch, the focus goes from low-frequency area to full frequencies and from local to global, showing that as the network goes deeper, the frequency-wise self-attention tends to capture global dependency along the frequency dimension. For noise branch, all four RA blocks show a local attention as white noises have a constant power spectral density.\n\n\\subsection{Comparison with the State-of-the-Art}\n\n\\subsubsection{Speech Enhancement}\n\\begin{table}[t]\n    \\centering\n    \\small\n    \\caption{Quality comparisons on Voice Bank + DEMAND}\n    \\label{table2}\n    \\begin{tabular}{lccccc}\n        \\hline\n        \\textbf{Methods} & \\textbf{SSNR} & \\textbf{PESQ} & \\textbf{CSIG} & \\textbf{CBAK} & \\textbf{COVL}\\\\\n        \\hline\n        Noisy & 1.68 & 1.97 & 3.35 & 2.44 & 2.63\\\\\n        \\hline\n        SEGAN & 7.73 & 2.16 & 3.48 & 2.94 & 2.80\\\\\n        MMSE-GAN & - & 2.53 & 3.80 & 3.12 & 3.14\\\\\n        \\hline\n        PHASEN & \\textbf{10.18} & 2.99 & 4.21 & 3.55 & 3.62\\\\\n        \\hline\n        Koizumi et al. & - & 2.99 & 4.15 & 3.42 & 3.57\\\\\n        \\hline\n        Ours & 9.83 & \\textbf{3.12} & \\textbf{4.39} & \\textbf{3.60} & \\textbf{3.77} \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\n\\begin{table}[t]\n    \\centering\n    \\caption{Quality comparisons on DNS Challenge}\n    \\label{table3}\n    \\begin{tabular}{lccc}\n        \\hline\n        \\textbf{Methods} & \\textbf{SDR(dB)} & \\textbf{PESQ} \\\\\n        \\hline\n        Noisy & 9.09 & 1.58 \\\\\n        \\hline\n        TCNN & 16.86 & 2.34  \\\\\n        TCNN-L & 16.58 & 2.78 \\\\\n        \\hline\n        Conv-TasNet-SNR & - & 2.73 \\\\\n        \\hline\n        DTLN & 16.54 & 2.34 \\\\\n        \\hline\n        MultiScale+ & - & 2.71 \\\\\n        \\hline\n        PoCoNet & - & 2.75 \\\\\n        \\hline\n        Ours & \\textbf{19.52} & \\textbf{3.39} \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\\noindent Table \\ref{table2} shows the comparisons with state-of-the-art methods on Voice Bank + DEMAND. SEGAN \\cite{pascual2017segan} and MMSE-GAN \\cite{soni2018time} are two GAN-based methods. PHASEN \\cite{yin2020phasen} is a two-branch T-F domain approach where one branch predicts the amplitude and the other predicts the phase. Koizumi et al. \\cite{koizumi2020speech} is a multi-head self-attention based method. Our method outperforms all of them in almost all metrics. The large improvements on PESQ, CSIG and COVL indicate that our method preserves better speech quality. \n\nTable \\ref{table3} shows the comparison with state-of-the-art methods on DNS Challenge dataset. TCNN \\cite{pandey2019tcnn} is a time-domain low-latency approach. We implemented two versions of it.  \"TCNN\" is exactly the same as described in the paper and \"TCNN-L\" is the long-latency version using the same T-F domain loss function as ours. Conv-TasNet-SNR \\cite{koyama2020exploring} and DTLN \\cite{westhausen2020dual} are real-time approaches. MultiScale+ \\cite{choi2020phase} and PoCoNet \\cite{isik2020poconet} are non-real-time methods, among which the PoCoNet took 1st place in the 2020 DNS challenge's Non-Real-Time track. Since narrow-band PESQ number was reported in the DTLN paper, we used the released model\\footnote{https:\/\/github.com\/breizhn\/DTLN} to generate the enhanced speech and compute the metrics. For other methods, we use the numbers reported in their papers. Our method outperforms all of them by a large margin.\n\n\\begin{table}[t]\n    \\centering\n    \\small\n    \\caption{Two-speaker speech separation on TIMIT}\n    \\label{table4}\n    \\begin{tabular}{lcc}\n        \\hline\n        \\textbf{Methods} & \\textbf{SDRi(dB)} & \\textbf{PESQ}\\tablefootnote{Note that Conv-TasNet outputs 8khz audios. We use narrow-band PESQ here instead of wide-band. Accordingly, we downsample audios to 8khz for our method to match this evaluation.}\\\\\n        \\hline\n        Conv-TasNet & 7.57 & 2.14\\\\\n        Ours & \\textbf{8.39} & \\textbf{2.50} \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\\subsubsection{Extension to Speaker Separation}\n\n\\noindent As SN-Net can simultaneously model two signals, it is natural to extend it for speaker separation task. The merge branch is removed as two outputs are needed. Permutation invariant training \\cite{yu2017permutation} is employed during training to avoid the permutation problem. We conduct the two-speaker separation experiment based on the TIMIT corpus. The batch size is set to 16. For comparison, we train a non-causal version of Conv-TasNet \\cite{luo2019conv}, the state-of-the-art method, using the released code\\footnote{https:\/\/github.com\/kaituoxu\/Conv-TasNet}.\n\nThe results are shown in Table \\ref{table4}. We use SDR improvement (SDRi) and PESQ for evaluation. Our method achieves a considerable gain on PESQ by 0.36 and SDRi by 0.82 dB, compared with Conv-TasNet. This suggests that our method is not limited to specific tasks and has the potential to extract different additive parts from a mixture signal.\n\n\\section{Conclusion}\n\n\\noindent We propose a novel two-branch convolutional neural network to interactively modeling speech and noises for speech enhancement. Particularly, an interaction between two branches is proposed to leverage information learned from the other branch to enhance the target signal modeling. This interaction makes the simultaneous modeling of two signals feasible and effective. Moreover, we design a sophisticated RA block for feature extraction of both branches, which can accommodate the diversities across speech and various noise signals. Evaluations verify the effectiveness of these modules and our method significantly outperforms the state-of-the-art. The two-signal simultaneous modeling paradigm makes it applicable to speaker separation as well. \n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuperconductivity in V$_3$Si has been studied since 1953\\cite{Har53a} and was considered to be of  conventional s-wave, single-band nature for most of the time. It was only recently that, in the wake of MgB$_2$, V$_3$Si was placed on the list of potential two-band superconductors to explain unconventional experimental results. In particular, its superfluid density was reported  \\cite{Nef05a,Kog09a} not only to deviate strongly from the single-band BCS behavior but to match a two-band model, whose interband coupling strength is all but negligible. These conclusions were backed by infrared spectroscopy results and by calculations of the Fermi surface, which was reported to be crossed by several electronic bands \\cite{Per10a}. In contrast, a single-band description worked well for the field dependence of the specific-heat and the thermal conductivity \\cite{Boa03a}. Furthermore, the field dependence of the magnetic penetration depth and of the vortex core size, determined by muon-spin rotation, were regarded as single-band behavior \\cite{Son04b}, and in \\cite{Gur04a} the temperature dependence of the specific heat was reported to be conventional. We conclude that we face a confusing situation, which we wished to clarify by further experiments.\n\nV$_3$Si is a member of the A15 superconductors, whose crystal structure changes from cubic at room temperature to tetragonal in the superconducting state \\cite{Bat64a}. It is a type II superconductor with a Ginzburg-Landau parameter of about 20, a transition temperature of 16 - 17\\,K, and an upper critical field of around 20\\,T at 0\\,K \\cite{Orl79a}; the anisotropy of the superconducting properties is marginal \\cite{Khl99a}. The vortex distribution may change from a hexagonal to a cubic lattice as the magnetic field increases \\cite{Yet99a}.\n\nIn this article we will present new experimental results capable of probing a possible two-band state of V$_3$Si. In the next section, the experimental details and the evaluation methods will be introduced. We will start with the basic characterization of the sample, go on with the measurements of the magnetic moment, and will then show how the reversible magnetization was obtained and fitted using the Ginzburg-Landau model. Finally, the direct determination of the lower critical field will be explained. In the third section, we will present the results. First, we will summarize how two-band effects show up in MgB$_2$ and will then compare this with our findings on V$_3$Si. Potential modifications \\cite{Zeh13a} of the field dependencies will be discussed by analyzing the reversible magnetization, those of the temperature dependencies by analyzing the superfluid density and the lower critical field. We will report differences between literature data and our results and present possible reasons. In the final section, we will summarize and once again explain why we believe that V$_3$Si is a single-band superconductor.  \n\n\\section{Experiment and evaluation \\label{sec:experiment}}\n\nA V$_3$Si single crystal was cut into two pieces of equal sizes of about $3 \\times 0.55 \\times 0.55$ mm$^3$. Both samples, named VA and VB, became superconducting below 16.7\\,K with a transition width of 0.2\\,K and had a residual resistance ratio of 33.\n\nUsing a SQUID magnetometer, we measured the magnetic moment of sample VB at temperatures from 2 to 16\\,K in 1\\,K steps as a function of applied magnetic field from 0 to 7\\,T. Sample VA was analyzed with a non-commercial rotating sample magnetometer, where the sample is glued on the rim of a circular plate, which rotates at a frequency of 15\\,Hz. During one rotation the sample passes four pick-up coils, where it induces electrical voltages proportional to its magnetic moment. For details about the instrument and how the magnetic moment is determined, the reader is referred to \\cite{Eis11a}. The main advantage of the rotating sample magnetometer is its fitting into a cryostat with a 15\\,T magnet. Accordingly, magnetization loops up to 15\\,T were recorded at temperatures of 5.2, 7, 9, 11, 13, and 15\\,K.\n\nThe resulting magnetization loops, either measured in the SQUID or in the rotating sample magnetometer, revealed reversible behavior over most of the field range. Irreversible effects, caused by flux-pinning, emerged merely at low fields and became dominant near 0\\,T. For instance, the critical current density at 5.2\\,K, evaluated using the methods presented in \\cite{Zeh09a}, decreased with field from some 10$^9$\\,Am$^{-2}$ at 0.1\\,T to $5 \\times 10^7$\\,Am$^{-2}$ at 1\\,T and to a negligible value at and above 2\\,T. Accordingly, more than 85 per cent of the loop fell into the reversible regime. The irreversible effects became slightly larger at lower temperatures but considerably smaller at higher temperatures. The critical currents, which are proportional to the hysteresis width of the magnetization loops, were found to be somewhat larger in the SQUID than in the rotating sample magnetometer, yet the reversible parts agreed well.        \n\nAs already mentioned, irreversibility appeared merely at low fields, but even there, knowing the magnetization as a function of increasing, $M(H_{\\rm a}^+$), and decreasing applied field, $M(H_{\\rm a}^-$), allowed us to determine the reversible part via $M_{\\rm r}(H_{\\rm a})$ $\\simeq$ 0.5[$M(H_{\\rm a}^+)$ + $M(H_{\\rm a}^-)$], where $H_{\\rm a}^+$ and $H_{\\rm a}^-$ refer to the same applied field $H_{\\rm a}$. This procedure gives reliable results as long as the hysteresis width is not much larger than the corresponding reversible signal. To get $M_{\\rm r}(B)$ from $M_{\\rm r}(H_{\\rm a})$, we calculated the magnetic induction via $B = \\mu_0(H - D M_{\\rm r} + M_{\\rm r})$, with $\\mu_0$ = $4 \\pi \\times 10^{-7}$\\,NA$^{-2}$, $H$ the applied field corrected by the field induced by the macroscopic currents \\cite{Zeh09a}, and $D$ the numerically calculated demagnetization factor of the sample in the Meissner state. \n\nThe next step was to compare the reversible magnetization, acquired from experiment, with theory. The theoretical magnetization curves of a single-band superconductor were taken from approximate equations of the Ginzburg-Landau theory, provided by Brandt \\cite{Bra03a}. According to this paper \\cite{Bra03a} the approximation errors should be less than two per cent, for the Ginzburg-Landau parameter of V$_3$Si is large. The Ginzburg-Landau curves depend on just two parameters, namely the upper critical field, at which the magnetization becomes zero, and the Ginzburg-Landau parameter, which determines the shape of the curve. The well-known Ginzburg-Landau relations \\cite{Tin75a} allowed us to calculate further quantities, such as the lower and the thermodynamic critical field, the coherence length, the magnetic penetration depth, and the superfluid density.  \n\nThe lower critical field was additionally determined by measuring the field at which vortices start to penetrate the sample. After having minimized the stray fields in the SQUID cryostat, we cooled the sample below its transition temperature in zero field. Then we enhanced the applied field stepwise but interrupted each step by measuring the corresponding remanent magnetic moment in zero field. The remanent moment should vanish at low fields and start to increase above the lower critical field, where vortices are created and pinned in the sample.\n\n\n\n\\section{Results and discussion}\n\nIn this section we will analyze whether superconductivity in V$_3$Si is more reliably described by a single or a two-band model. If V$_3$Si is a two-band superconductor, we expect some properties to deviate from the single-band BCS behavior \\cite{Zeh13a}. Possible effects on the field dependence will be discussed via the reversible magnetization and possible effects on the temperature dependence via the superfluid density. The results will be compared with the behavior of MgB$_2$, a well-known two-band material. We will see that the single-band models provide a fair description of our results in V$_3$Si.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[clip, width = 8.5cm]{Fig1.eps}\n   \\caption{\\label{MgB2GLFit} The reversible magnetization of MgB$_2$ as a function of the magnetic induction. The open circles show the experimental data for the applied field oriented parallel ($H \\parallel c$) or perpendicular ($H \\parallel ab$) to the uniaxial sample axis at 10, 20, and 30\\,K (the reduced temperatures are 0.26, 0.51, and 0.77). The solid lines are fits of the single-band Ginzburg-Landau model to the experimental data over the whole field range, while the broken lines are such fits either to the low or to the high field part of the experimental data, reflecting  the two-band nature of MgB$_2$. For details see \\cite{Zeh04b}.}\n\\end{figure}\n\nWe start by summarizing some two-band effects of MgB$_2$ \\cite{Eis07a}. This material consists of the near-isotropic $\\pi$-band and the anisotropic $\\sigma$-band, having similar electronic densities of states. Due to interband coupling, the gaps are expected to close at the same field. Yet the superconducting properties of the $\\pi$-band are heavily suppressed above a particular field, which is commonly called the upper critical field of the $\\pi$-band and whose value is about a third of that of the $\\sigma$-band. Accordingly, the field dependence of several superconducting properties deviates significantly from the single-band behavior. This is illustrated in figure~\\ref{MgB2GLFit}, where the reversible magnetization of an MgB$_2$ single crystal, indicated by open circles, is shown as a function of the magnetic induction. The solid lines present the single-band Ginzburg-Landau fits. Barring the 30\\,K results, the agreement between theory and experiment is poor and the differences are not merely of quantitative but also of qualitative nature. In contrast, the curves can be nicely fitted by {\\it two} single-band Ginzburg-Landau curves, as shown by the broken lines in the diagrams. One is adjusted to the low and the other to the high field region of the experiment, thus reflecting the two bands with their different upper critical fields \\cite{Zeh04b}. Applying the methods to different field orientations reveals the different anisotropies of the two bands. Similar effects were observed in NbSe$_2$ \\cite{Zeh10a}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[clip, width = 8.5cm]{Fig2.eps}\n   \\caption{\\label{GLFit} The magnetization of V$_3$Si as a function of magnetic induction. The open circles  show the reversible data at 15, 13, 11, 9, 7, and 5.2\\,K (the reduced temperatures are 0.90, 0.78, 0.66, 0.54, 0.42, and 0.31) and the solid lines the corresponding Ginzburg-Landau model fits. The solid lines of the insets show the irreversible data.}\n\\end{figure}\n\nWe are now prepared to shift our focus to V$_3$Si. Figure~\\ref{GLFit} shows the magnetization as a function of the magnetic induction at temperatures of about 15, 13, 11, 9, 7, and 5\\,K. The open circles indicate the reversible magnetization acquired from the rotating sample magnetometer measurements of a V$_3$Si single crystal and the solid lines the single-band Ginzburg-Landau behavior adjusted to the experimental data. In the insets, the solid lines indicate the irreversible magnetization, though only  at low fields, where a significant hysteresis shows up. At not too low temperatures, say about 9 - 15\\,K, we consider the agreement between experiment and single-band theory very good. As the temperature is reduced, the differences between theory and experiment get larger,  becoming apparent at low fields, for we adjusted the fits mainly to the high field regions, where the experiments reveal the reversible data directly. \n\nNext, we will analyze the quality of the fits in more detail. First, we determined the areas under the reversible magnetization curves, which are proportional to the condensation energies. The ratio of the condensation energy obtained from the Ginzburg-Landau fit to that from the experimental curves is considered a sensible measure for the fit quality. In MgB$_2$, the differences in the condensation energies of the high-field fit and the experimental data were some 18 per cent at 30\\,K and 30 per cent at 10 and 20\\,K. In V$_3$Si, we found much smaller deviations, namely 3 - 8 per cent at 9 - 15\\,K and 10 - 12 per cent at 7 and 5.2\\,K. So, even the low-temperature fits of V$_3$Si agree with the experimental data better than any single-band fit of MgB$_2$.\n\nThere are several possible reasons for the larger deviations between theory and experiment in V$_3$Si at low temperatures. To begin with, as shown in figure~\\ref{GLFit}, the low-field irreversible magnetization becomes large at low temperatures, thus enhancing possible errors in the corresponding reversible data. The insets of figure~\\ref{GLFit}, however, illustrate that the irreversible magnetization does not become much larger than the reversible data even at low temperatures and small fields, and hence the errors in calculating the reversible magnetization are not serious. On the other hand, we are faced with the shortcomings of the Ginzburg-Landau theory. As this theory is derived from BCS theory in the vicinity of the transition temperature, the potential errors grow when we go to lower temperatures. Yet the theory has been successfully applied to evaluating and describing experimental data at high and low temperatures, as a function of temperature and as a function of field in a large number of publications. In particular, using adjustable parameters instead of the microscopic BCS ones apparently extends the applicability to much lower temperatures (e.g. \\cite{Sil12a}). Watanabe et al.~\\cite{Wat05a} calculated the field dependence of the reversible magnetization of an s-wave system using the Eilenberger equations, which hold at arbitrary temperatures. We verified that the Ginzburg-Landau model with a Ginzburg-Landau parameter of 49 reproduces the Eilenberger curve of \\cite{Wat05a} close to the transition temperature (see also \\cite{Bel12a}). Reducing the temperature makes the Eilenberger curves slightly steeper at low fields and slightly flatter at high fields. This is basically what we find for the experimental curves in figure~\\ref{GLFit}, namely reducing the temperature makes the experimental curves slightly steeper at low fields and slightly flatter at high fields compared with the Ginzburg-Landau fit. Accordingly, the qualitative deviations between our experiments and the fits are just as expected when we assume that the Eilenberger model describes experiment at all temperatures. The deviations from the Ginzburg-Landau model, assessed via the condensation energies, agree even quantitatively, i.e., we found some 12 per cent for the experimental data and 14 per cent for the Eilenberger curves at a reduced temperature of about 0.3. Granted, the Eilenberger calculations and our experimental data refer to systems with different Ginzburg-Landau parameters ($\\kappa$), but both systems belong to the high-$\\kappa$ regime and hence should behave similarly. To conclude, the changes of the Eilenberger curves are not substantial when the temperature is lowered, which affirms the usefulness of our approach. Still, we do not expect an exact description at the lowest temperatures of figure~\\ref{GLFit}.\n\nIn contrast to V$_3$Si, presented in figure~\\ref{GLFit}, the experimental data of MgB$_2$, presented in figure~\\ref{MgB2GLFit}, also qualitatively disagree with the single-band models. In particular, we found the high-field fit to lead to a much smaller lower critical field, i.e. the magnetization value at $B = 0$\\,T, than obtained from experiment via extrapolation of the low-field data, while in V$_3$Si the two curves result in almost the same lower critical fields.\n\nSummarizing, we consider the agreement between single-band theory and experiment in V$_3$Si as good as can be expected in view of the uncertainties within both the theoretical and the experimental data. Accordingly, we conclude that the reversible magnetization of V$_3$Si supports the single-band scenario.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[clip, width = 8.5cm]{Fig3.eps}\n   \\caption{\\label{Bc2} The upper critical field, presented in the left panel, and the Ginzburg-Landau parameter, presented in the right panel, of V$_3$Si as a function of temperature. The open circles show experimental data acquired from the rotating sample magnetometer and the full circles experimental data from the SQUID measurements. The upper critical field follows the clean-limit BCS behavior, depicted by the solid line, leading to 22.7\\,T at 0\\,K. The Ginzburg-Landau parameter decreases roughly linearly from about 24 to 19 as the temperature increases from 0\\,K to the transition temperature.}\n\\end{figure}\n\nFigure~\\ref{Bc2} presents the upper critical field and the Ginzburg-Landau parameter of V$_3$Si. The open circles indicate results from the rotating sample and the full circles those from the SQUID magnetometry. The upper critical field has been evaluated by adjusting the Ginzburg-Landau model merely to the high-field regime of the reversible magnetization and may thus slightly deviate from the results obtained from full range fits. The solid line presents the clean-limit single-band BCS behavior \\cite{Hel66a}, is in excellent agreement with our experimental data, and leads to 22.7\\,T at 0\\,K, which matches literature data well \\cite{Orl79a}. In contrast to many two-band materials, no clear upward curvature near the transition temperature appears \\cite{Shu98a,Zeh02a,Mun12a}. The Ginzburg-Landau parameter, taken from fits over the whole field range and presented in the right panel, decreases quite linearly from about 24 at 0\\,K to 19 at the transition temperature, a behavior that is also close to the single-band BCS prediction \\cite{Hel66a}. Calculating further properties employing the Ginzburg-Landau relations \\cite{Tin75a} resulted in about 90\\,nm for the magnetic penetration depth, 4\\,nm for the coherence length, 0.6\\,T for the thermodynamic critical field, and 0.07\\,T for the lower critical field at 0\\,K. The critical lengths are close to literature data \\cite{Son04b,Yet05a}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[clip, width = 8.5cm]{Fig4.eps}\n   \\caption{\\label{SFLD} The superfluid density as a function of reduced temperature (left panel) and the reduced lower critical field as a function of temperature (right panel). In the left panel, the open circles show our experimental result on V$_3$Si, obtained from the reversible magnetization, which is in good agreement with the single-band BCS behavior, indicated by the solid line. The broken and the dot-dashed lines show the two-band-like superfluid density of V$_3$Si reported in Refs.~\\cite{Nef05a} and \\cite{Kog09a} schematically; the full diamonds illustrate the two-band behavior of MgB$_2$ \\cite{Zeh04b}. In the right panel, the open circles show the reduced lower critical field of our V$_3$Si single crystal, evaluated from the reversible magnetization, which is all but identical to the superfluid density of the left panel. The open diamonds and the full squares are obtained from direct measurements of the first vortex-penetration field as indicated by the arrows in the inset. The symbols in the inset present the square root of the remanent magnetic moment as a function of applied field at 7\\,K; the solid line is a linear fit to the high-field data.}\n\\end{figure}\n\nWe proceed by analyzing the temperature dependent effects by means of the superfluid density. Having evaluated the upper critical field, $B_{\\rm c2}$, and the Ginzburg-Landau parameter, $\\kappa$, via the above fit procedure, we acquired the magnetic penetration depth, $\\lambda$, by using the Ginzburg-Landau relations, and the superfluid density, $\\rho_{\\rm s}$, via $\\rho_{\\rm s}$(T) = [$\\lambda$(0\\,K) \/ $\\lambda$(T)]$^2$, with $\\lambda = \\kappa \\xi$, $\\xi^2 = \\Phi_0 \/ (2 \\pi B_{\\rm c2})$, and $\\Phi_0 \\simeq 2.07$ $\\times 10^{-15}$\\,Vs. To assess the penetration depth at 0\\,K, we used the SQUID measurements, where the temperature could be reduced to 2\\,K, but the magnetic fields were limited to 7\\,T. We therefore took the inter- and extrapolated upper critical fields from the rotating sample magnetometer results for fitting the SQUID data, so that only the Ginzburg-Landau parameter remained to be determined. To justify the use of these data, we verified that the reversible curves from SQUID and rotating sample magnetometer agree in the overlapping field and temperature range.\n\nThe left panel of figure~\\ref{SFLD} presents the superfluid density of V$_3$Si, indicated by open circles, as a function of reduced temperature. The solid line shows the expected behavior of a single-band BCS superconductor, which is close to our experimental data. Figure~\\ref{SFLD} shows also the superfluid density of MgB$_2$, indicated by full symbols, as an example for two-band superconductivity. In comparison with V$_3$Si, the MgB$_2$ curve decreases much faster at low temperatures and then becomes almost linear as the temperature increases. To explain this behavior, we need to consider that a two-band superconductor has two distinct energy gaps. The smaller gap reduces the excitation energy on the corresponding part of the Fermi surface and hence makes the superfluid density decrease more rapidly at small temperatures. This may lead to a near-linear behavior at intermediate temperatures, as found in several two-band materials, such as MgB$_2$, NbSe$_2$, and the iron-based superconductors \\cite{Man02a,Fle07a,Mar09b}. We conclude that also the superfluid density of V$_3$Si supports the single-band scenario.\n\nWe now come back to the two-band scenario of V$_3$Si proposed in Refs.~\\cite{Nef05a} and \\cite{Kog09a}, based on measurements of the superfluid density. The broken line in figure~\\ref{SFLD} shows the result presented in \\cite{Nef05a}, which was obtained by measuring the microwave surface impedance of a single crystal. There is no doubt that this curve is totally different from our result, that is, it decreases more rapidly at low temperatures and has a sharp kink at about 0.6\\,$T_{\\rm c}$. Analyzing their data with a two-band model, the authors found distinctly different energy gaps, though similar intraband coupling strengths for the two bands and, as indicated by the sharp kink at intermediate temperatures, almost negligible interband coupling. The superfluid density published by Kogan et al.~\\cite{Kog09a}, shown schematically in figure~\\ref{SFLD} by the dot-dashed line, was measured by a tunnel diode resonator technique and is quite different from that of \\cite{Nef05a}, yet it also reflects a two-band scenario with similar intraband and near-negligible interband coupling strengths. \n\nWhat are the possible reasons for the differences between those literature \\cite{Nef05a,Kog09a} and our data? Two points are obvious. First, while we evaluated the penetration depth from fits to the reversible magnetization over the whole field range, save for very small fields, where the experimental data are not available or not reliable, the authors of Refs.~\\cite{Nef05a} and \\cite{Kog09a} evaluated their data solely at very low fields. Thus, a second band, but one with a very small upper critical field, would resolve the contradictions. The second point is that while our method probes the bulk, the methods of Refs.~\\cite{Nef05a} and \\cite{Kog09a} probe the sample surface. Thus, surface irregularities or inhomogeneities that may change the properties on the surface would also resolve the contradictions. Such a statement, however, remains a speculation, for we know nothing about the surfaces of the samples used in the studies. Diener et al.~\\cite{Die09a} faced that problem in MgCNi$_3$. Acquiring data with the same method as used in \\cite{Kog09a} resulted in a penetration depth behavior similar to the broken curves of figure~\\ref{SFLD}, while acquiring the results from measurements of the lower critical field resulted in a BCS-like behavior. The authors suggested that those differences might be caused by inhomogeneities at the sample surface and hence considered the BCS-like behavior correct. \n\nWe also determined the superfluid density at very low magnetic fields,  namely by measuring the lower critical field directly. But measuring the lower critical field directly is anything but trivial. To begin with, we can merely determine the field at which flux lines start to penetrate into the sample. This is accomplished by recording the remanent magnetic moment of the sample in zero field as a function of the maximum applied field, as described at the end of Sec.~\\ref{sec:experiment}. Unfortunately, this first penetration field is usually not simply connected with the lower critical field via a single-valued demagnetization factor. First, sample edges give rise to very high stray fields, which may surpass the lower critical field and hence enforce the creation of vortices at very low applied fields, but we do not know well how the local induction is related to the applied field in such a configuration. Second, surface irregularities would affect the creation of vortices, while clean surfaces may induce additional barriers. \n\nSo, how can we acquire useful results from that procedure? To begin with, we are mainly interested in the relative temperature dependence of the lower critical field and not so much in its absolute value. Second, we aligned the longest length of our sample, the size of which is $3 \\times 0.55 \\times 0.55$ mm$^3$, parallel to the applied field, so that the geometry effects were as small as possible and assumed those geometry effects to be temperature independent. As expected, when we increased the applied field, the remanent magnetic moment remained zero at the beginning, then started to rise at a slope that became gradually steeper, indicating that first vortices had been formed and pinned in the sample, and eventually increased quadratically with field (see inset of figure~\\ref{SFLD}). Assuming a constant current density parallel to the sample borders and ignoring the stray fields revealed this quadratic behavior also in calculations (cf. with \\cite{Mos91a}). We thus consider the extrapolated onset of the quadratic behavior (cf. with inset of figure~\\ref{SFLD}) as a reliable assessment of the lowest field where vortices parallel to the applied field were in the sample. \n\nWe determined both the smallest field at which we observed a slope in the remanent magnetic moment and the field obtained from extrapolating the quadratic part to zero (see inset of figure~\\ref{SFLD}). The temperature dependence of both was found in good agreement with the superfluid density acquired from the Ginzburg-Landau fits, as shown in the right panel of figure~\\ref{SFLD}. Note that Ginzburg-Landau theory predicts the same temperature dependence for the superfluid density and the lower critical field when the changes in the logarithm of the Ginzburg-Landau parameter can be ignored, as is the case in our sample. To assess the absolute lower critical field values, we multiplied the penetration fields with the factor $(1-D)^{-1}$ $\\simeq$ 1.15, where $D \\simeq 0.13$ is the averaged demagnetization factor. In comparison to the lower critical fields from the Ginzburg-Landau fits, leading to 68\\,mT at 0\\,K, this led to lower values by some 10 per cent, namely 60\\,mT at 0\\,K, from evaluating the onset fields, and to larger values by 40 per cent, namely 95\\,mT at 0\\,K, from evaluating the fields where the quadratic behavior started. We consider these differences to be reliable. Finally, we conclude that also at very low magnetic fields, the temperature dependence shows no indication of a two-band behavior.\n\nHaving presented clear support for the single-band behavior, we may ask if our results rule out the two-band scenario completely. This is not the case, for our experiments would not detect a second band if its contribution to the measured quantities were marginal or if its superconducting properties were all but identical to the first band. These scenarios, however, would be different from those proposed in literature \\cite{Nef05a,Kog09a}. Which properties would a hypothetical second band have in our sample? Above, we have discussed the fit quality of the reversible magnetization data in terms of the ratio of the condensation energies from the fit and from experiment. In MgB$_2$ the differences, which are some 30 per cent at low temperatures, are ascribed to the two-band scenario. Accordingly, 70 per cent of the condensation energy are induced by the $\\sigma$-band, which is the band probed by the high-field fits \\cite{Zeh04b}, and 30 per cent by the second band, the $\\pi$-band, {\\it and} interband coupling effects, in rough agreement with \\cite{Eis05a}. In V$_3$Si, the deviations are much smaller and we have shown that they can qualitatively and quantitatively be explained by the imperfections of the Ginzburg-Landau model. Nevertheless, we analyzed the results assuming the deviations to be caused by a hypothetical second band. For example, at 9\\,K the fit error was 5 per cent and hence the second band would contribute to the condensation energy ($E_{\\rm c}$) by {\\it less} than 5 per cent, for these 5 per cent include the interband contributions. Employing BCS theory, where $E_{\\rm c}$ = $0.5 N \\Delta^2$, and assuming both bands to have the same density of states ($N$) would result in a gap ($\\Delta$) ratio larger than 4.4 : 1 (1 : $\\sqrt{5\/95}$), which would be quite large and much larger than the ratios proposed in Refs.~\\cite{Nef05a,Kog09a}, namely 1.4 - 1.8 : 1. \n\nStrong interband effects would also mask two-band effects \\cite{Nic05b}, though it should be noticed that the two-band scenario for V$_3$Si \\cite{Nef05a, Kog09a} came along with a negligible interband coupling strength and the high purity of our single crystals makes strong interband impurity-scattering unlikely. \n\n\\section{Summary and conclusion}\n\nLet us sum up what we have learned from testing V$_3$Si for two-band superconductivity. We have shown that the field dependence of the reversible magnetization matches the single-band Ginzburg-Landau theory reliably well. The differences between the experimental data and the theoretical fits grow as the temperature is reduced but remain small and can qualitatively and quantitatively be explained by the imperfections of the Ginzburg-Landau model. We have also found that the temperature dependence of the superfluid density, determined with different methods at different magnetic fields, follows single-band BCS behavior. Accordingly, all our results support a single-band scenario for V$_3$Si. \n\n\\section*{Acknowledgments}\nThis work was supported by the Austrian Science Fund under Contract No. 21194\nand 23996.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA margin loan is a loan made by a brokerage house to a client that allows the customer to buy stocks on credit. The term margin itself refers to the difference between the market value of the shares purchased and the amount borrowed from the broker.\nAs a margin loan gives customer more to invest, there is a potential for bigger returns. Of course, this also magnifies the potential for losses if the investments perform poorly \\citep{fortune2000,fortune2001,ricke2003}. In order to restrain the level of speculation and to protect investors and brokers,\nthe United States Congress gave the Federal Reserve System the power to control margin requirements in 1934. The initial margin requirement was changed 22 times between 1934 and 1974, and it has been fixed at $50\\%$ since 1974 \\citep{fortune2000}.\n\n\\citet{galbraith} and \\citet{hsieh1990} indicated that low margin requirement was one of the major contributing factors which led to the Stock Market Crash of 1929.\nShiller claimed in a {\\it Wall Street Journal} article that the stock market crash of 1987 and the stock market boom in the late 1990s led to calls for the return to an active margin policy, which sets a minimum equity\nposition on the date of a credit-financed security transaction \\citep{shiller2000,ricke2003}. On the other hand, Bartlett argued that although the adjustment of margin policy can not impact the volatility of stock market, brokers should set their own house margins to react to the changes of stock market, which can protect themselves from customer defaults \\citep{shiller2000}. The purpose of this paper is to set an active margin system for margin loans traded in Chinese market.\n\n\nThe margin system in Chinese market is composed of initial margin requirement, maintenance margin requirement and mandatory liquidation, which is the line of defence for brokers against the risk associated with the transactions of margin loans. Pursuant to the regulations of the Shanghai Stock Exchange (SSE) and the Shenzhen Stock Exchange (SZSE), the initial and maintenance margin ratios are $50\\%$ and $130\\%$ respectively, which is called the required margin system in this paper.\nHowever, \\citet{argiriou2009} pointed out that although supervising authorities give recommendations or limits for margin requirements, there is no information on how these recommendations are derived. Therefore a risk measurement based on conditional probability is adopted in this paper, which is used to set margin system for Chinese market with a specified target of risk control.\n\nThe probability that broker yields a negative return after mandatory liquidation under the condition that a margin call has been issued is called the conditional probability of negative return (CPNR) in this paper. CPNR is used to measure the risk faced by brokers, which does not only take into account the possibility of investor's default, but also emphasize on the stop-loss function of mandatory liquidation \\citep{huang2010}.  A simple margin loan is considered in this paper, where an amount of cash and a portion of the stock to be purchased are used as collateral. An active margin system is given for a particular margin account, such that the value of CPNR is less than or equal to the desired value of broker. The margin system deduced from the constraint of CPNR is called the deduced margin system in this paper.\n\n\n126 stocks listed on the SSE are used to compare the performances of the required and the deduced margin systems, where 200 margin loans with a 30-day period are constructed for each stock. It is found that the average number of margin calls is reduced by $59.8\\%$ under the deduced margin system  with respect to the required margin system, while the average costs under the two margin systems are similar. Those empirical results indicate that CPNR can be used to measure the risk associated with margin loans, and the proposed margin system is an operational method for brokers to set their own margin requirements.\n\nThe remainder of this paper is organized as follows. The margin requirements in Chinese market and the risk measure are introduced in Section 2. Section 3 is devoted to the realization of the risk measurement under a Markov chain model. An active margin system is given for the loans using cash and stock as collateral in Section 4. The empirical investigations of Chinese margin loan market are given in Section 5. The conclusions and discussions are given in Section 6.\n\n\n\\section{Risk measure for Chinese market}\n\n\\subsection{Margin system of Chinese market}\n\nPursuant to the regulations of the SSE and the SZSE, an amount of initial margin should be deposited to the broker on the date of margin trading, which can be composed of cash, the underlying stock, and other marginable securities listed by the exchanges. The ratio between the market value of the initial margin and the market value of the purchased stock is called the initial margin ratio, which must be larger than or equal to $50\\%$. The purchased stock and the initial margin are kept in the account of collaterals. The ratio between the market value of the collaterals and the market value of the loan is called the maintenance margin ratio, which must be larger than or equal to $130\\%$.\n\nWhen the value of maintenance margin ratio is below the required level, a margin call will be issued by broker, which informs the customer to increase the margin that he has deposited or close out his position. If he does none of\nthese,  broker can sell the collateral to meet the margin\ncall, which is called mandatory liquidation in Chinese market.\n\n\n\\subsection{Maintenance margin and margin call}\n\nA relative simple margin loan is considered in this paper, where one share of stock to be purchased by the loan, and the initial margin is composed of an amount of cash and a portion of the stock to be purchased. Denote the amount of cash as $Q_0$, and the proportion of the stock as $\\delta$, where $Q_0 \\ge 0$, and $0 \\le \\delta \\le 1$. The initial margin ratio on the date of margin trading is\n\\begin{equation}\\label{initial_ratio}\nm_0 = \\frac{Q_0 + \\delta P_0}{P_0},\n\\end{equation}\nwhere $P_i$ is the market price of the purchased stock on the $i$th trading day, $i=0,1,2,\\cdots,T$, and $T$ is the maturity date of the loan. And the maintenance margin ratio on the date of trading is\n\\begin{equation}\nw_0 = \\frac{m_0 P_0 + P_0}{P_0}= m_0 + 1,\n\\end{equation}\nwhich should be larger than or equal to the required level. Those observations give the following proposition.\n\\begin{proposition}\\label{adequate}\nSuppose the initial margin ratio on the date of margin trading  is $m_0$, then\n\\begin{equation*}\nm_0 + 1 \\ge w,\n\\end{equation*}\nwhere $w$ is the required level of maintenance margin ratio. Otherwise  the initial margin is inadequate, and the customer should deposit more margin to the broker.\n\\end{proposition}\n\nThe cash in the initial margin can be divided into two parts on the $i$th day, one of which is the required margin $\\Sigma_i$, and the other is the remaining margin $L_i$, i.e.\n\\begin{equation}\nQ_0 \\left( 1+r \\right)^{i} = \\Sigma_i + L_i,\n\\end{equation}\nwhere $i=1,2,\\cdots,T$. $\\Sigma_i$ and $L_i$ satisfy the following proposition before the first margin call.\n\\begin{proposition}\nSuppose the single day loan rate is $R$, and the riskless interest rate is $r$ per day, then the amount of required margin $\\Sigma_i$ and remaining margin $L_i$ are\n\\begin{eqnarray}\n\\Sigma_i & = & w P_0 \\left( 1+R \\right)^{i} - (1+\\delta) P_i, \\\\\nL_i      & = & Q_0 \\left( 1+r \\right)^{i} -  w P_0 \\left( 1+R \\right)^{i} + (1+\\delta) P_i.\n\\end{eqnarray}\n\\end{proposition}\n\n\\begin{remark}\nWhen the amount of remaining margin is less than zero, a margin call should be issued. The time of margin call is determined by the amount of cash in the initial margin, the amount of the loan at time zero, the proportion of the stock in the initial margin, the two interest rates, and the dynamics of the stock prices during the period of the loan. The required initial margin ratio is not one of the key factors which determine the time of margin call.\n\\end{remark}\n\n\n\\subsection{Conditional probability of negative return}\n\nLet $\\tau$ denote  the time of first margin call, i.e.\n\\begin{equation}\n\\tau = \\min \\left\\{ i \\in \\left\\{1,2,\\cdots,T  \\right\\}: L_i \\le 0 \\right\\}.\n\\end{equation}\nThe condition of first margin call can be written as\n\\begin{equation}\nC = \\left\\{\n\\left(1+\\delta\\right)P_{\\tau} \\le wP_0\\left(1+R\\right)^{\\tau} -Q_{0} \\left(1+r\\right)^{\\tau}\n\\right\\}.\n\\end{equation}\nIt is supposed that the customer will default whenever he receives the first margin call in this paper, and the mandatory liquidation is exercised in the next trading day within the period of the loan.\nLet $\\tau^*$ denote the time of mandatory liquidation, which satisfies\n\\begin{equation}\n\\tau^* = \\min \\left\\{ \\tau+1, T \\right\\}.\n\\end{equation}\n\nThe return of the margin loan after mandatory liquidation is\n\\begin{equation}\nQ_0 \\left( 1+r \\right)^{\\tau^*} + \\left( 1+\\delta \\right)P_{\\tau^*}\n   - P_0 \\left( 1+R \\right)^{\\tau^*},\n\\end{equation}\nand the condition of negative return is\n\\begin{equation}\nN = \\left\\{\n  \\left( 1+\\delta \\right)P_{\\tau^*} \\le\n     P_0 \\left( 1+R \\right)^{\\tau^*}\n      - Q_0 \\left( 1+r \\right)^{\\tau^*}\n \\right\\}.\n\\end{equation}\nThe conditional probability of negative return (CPNR) can be written as\n\\begin{equation}\n\\text{CPNR} = \\mathrm{Prob}\\left\\{ N \\mid C\\right\\},\n\\end{equation}\nwhere $\\mathrm{Prob} \\left\\{ N \\mid C \\right\\}$ is the conditional probability of $N$ given $C$.  The value of CPNR is calculated under a Markov chain model in the following sections.\n\n\\section{Markov chain and CPNR}\n\n\\subsection{Construction of Markov chain}\n\nA Markov chain is a random process that jumps from one state to another, whose next state depends only on the present state. In order to construct a Markov model, the\ndaily closing stock prices are used to estimate the state space and the transition probabilities in this paper. The observed stock prices are sorted in order of increasing, and every g different prices are regarded as one state. Suppose there are n elements in the state space, denoted as $\\{s_1,s_2,$ $\\cdots,$ $s_n\\}$, the probability of transition from the ith  to the jth state\nafter one step is estimated by\n\\begin{equation}\n\\hat{p}_{ij}(1)= \\frac{f_{ij}}{f_{i\\cdot}},\n\\end{equation}\nwhere $f_{ij}$ is the observed number of transitions from the ith to the jth state  after one step, and $f_{i\\cdot}=\\sum_{j=1}^{n} f_{ij}$ is the total number of stock prices falling into the ith state.\n\nThe one step transition matrix can be estimated by\n\\begin{displaymath}\n\\mathbf{P(1)} = \\left( \\begin{array}{ccc}\n\\hat{p}_{11}(1) & \\hat{p}_{12}(1) & \\ldots \\quad  \\hat{p}_{1n}(1)\\\\\n\\hat{p}_{21}(1) & \\hat{p}_{22}(1) & \\ldots \\quad  \\hat{p}_{2n}(1)\\\\\n\\vdots & \\vdots & \\vdots\\\\\n\\hat{p}_{n1}(1) & \\hat{p}_{n2}(1) & \\ldots \\quad  \\hat{p}_{nn}(1)\\\\\n\\end{array} \\right).\n\\end{displaymath}\nDenote $\\mathbf{P(n)}$ the n-step transition matrix, which can be estimated by \\begin{equation}\n\\mathbf{P(n)}=\\mathbf{P^n(1)},\n\\end{equation}\nwhere the chain has been supposed to be stationary.\n\n\\subsection{Stopping time and CPNR}\n\nThe loan rate $R$ and the riskless interest rate $r$ are set to be the same for simplicity, and the condition of margin call is rewritten as\n\\begin{equation}\n C = \\left\\{  \\left( 1 +\\delta \\right) P_{\\tau} \\le\n    \\left( wP_0 - Q_0 \\right) \\left( 1 + r \\right)^{\\tau}  \\right\\},\n\\end{equation}\nwhere $r$ is the riskless interest rate. The condition of negative return is also rewritten as\n\\begin{equation}\nN = \\left\\{ \\left( 1 + \\delta \\right) P_{\\tau^*} \\le\n   \\left( P_0 - Q_0 \\right) \\left( 1+r \\right)^{\\tau^*} \\right\\}.\n\\end{equation}\nAnd CPNR is rewritten as\n\\begin{equation}\n\\text{CPNR} = \\mathrm{Prob} \\left\\{ N \\mid C\\right\\} = \\frac{\\mathrm{Prob} \\left\\{NC\\right\\}}{\\mathrm{Prob} \\left\\{ C \\right\\}},\n\\end{equation}\nprovided that the denominator is non zero. On the other hand, when the denominator is zero, the value of CPNR is also zero following its  definition.\nIn order to calculate the value of CPNR,  the numerator and denominator should be calculated\nrespectively.\n\nLet\n\\begin{equation}\nD_i= \\left\\{ \\left( 1+\\delta \\right) P_{i} < \\left( w P_0 - Q_0 \\right) \\left(  1+r \\right)^{i}  \\right\\},\n\\end{equation}\nwhere $i=1,2,\\cdots,T$. The value of $\\mathrm{Prob} \\left\\{ C \\right\\}$ can be calculated with the following proposition.\n\\begin{proposition}\\label{propostion1}\nDenote $C_t$ the event that broker issues the first margin call on the $t$th day, and the stock price at time zero $P_0$ is belong to the $h$th state, then\n\\begin{equation}\n\\mathrm{Prob}\\left\\{ C \\right\\} = \\sum_{t=1}^{T} \\mathrm{Prob} \\left\\{ C_t \\right\\},\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathrm{Prob}(C_t)=\\mathrm{Prob}(\\overline{D}_1)\n\\mathrm{Prob}(\\overline{D}_2|\\overline{D}_1)\\cdots{\\mathrm{Prob}(\\overline{D}_{t-1}|\n\\overline{D}_{t-2})}\\mathrm{Prob}({D_{t}}|\\overline{D}_{t-1}),\n\\end{equation}\nand\n\\begin{equation}\n\\left\\{ \\begin{aligned}\n         \\mathrm{Prob}(\\overline{D}_1)&=1-\\sum_{i=1}^{k_1}\\hat{p}_{hi}(1),\\\\\n         \\mathrm{Prob}(D_m|\\overline{D}_{m-1})&\n         =\\frac{\\sum_{i=k_{m-1}+1}^n\n                  \\sum_{j=1}^{k_m}\\hat{p}_{hi}(m-1)\n                   \\hat{p}_{ij}(1)}{\\sum_{i=k_{m-1}+1}^{n}\n                    \\hat{p}_{hi}(m-1)},\\\\\n  \\mathrm{Prob}(\\overline{D}_m|\\overline{D}_{m-1}) &\n         = 1-\\mathrm{Prob}(D_m|\\overline{D}_{m-1}),\\\\\n         \\end{aligned} \\right.\n                          \\end{equation}\n$m=2,\\cdots,t$, and $k_m$ is the largest state index which satisfies\n\\begin{equation}\n k_m = \\max \\left\\{ k \\in \\left\\{ 1,2,\\cdots,n \\right\\} : (1+\\delta)q_{k} < (wP_0-Q_0)(1+r)^{m} \\right\\},\n\\end{equation}\nwhere $q_k$ is the representative price level for the kth state.\n\\end{proposition}\n\n\\begin{proposition}\\label{propostion2}\nDenote $C_t$ the event that broker issues the first margin call on the $t$th day, and the stock price at time zero $P_0$ is belong to the $h$th state,  then the probability that the two events N and C both occur is given by\n\\begin{equation}\n\\mathrm{Prob}\\left\\{NC\\right\\}=\\sum_{t=1}^T \\mathrm{Prob}(NC_t),\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n         \\mathrm{Prob}(NC_t)= &\\mathrm{Prob}\n         (\\overline{D}_1)\\mathrm{Prob}(\\overline{D}_2|\\overline{D}_1)\n         \\cdots \\\\\n         &\n         {\\mathrm{Prob}(\\overline{D}_{t-1}|\n          \\overline{D}_{t-2})}\\mathrm{Prob}({D_{t}}|\\overline{D}_{t-1})\n          \\mathrm{Prob}(N|D_t),\n          \\end{aligned}\n\\end{equation}\nand\n\\begin{eqnarray}\n\\mathrm{Prob}(N|D_t)=\n\\left\\{\n \\begin{aligned}\n           & \\frac{\\sum_{j=1}^{k_t}\\sum_{l=1}^{a_t}\\hat{p}_{hj}(t)\\hat{p}_{jl}(1)}\n                  {\\sum_{j=1}^{k_t}\\hat{p}_{hj}(t)},\n                  & 1\\le t \\le T-1, \\\\\n         & \\frac{\\sum_{j=1}^{a_{T}}\\hat{p}_{hj}(T)}{\\sum_{l=1}^{k_{T}}\\hat{p}_{hl}(T)}, &\n          t=T,\n \\end{aligned}\n\\right.\n\\end{eqnarray}\nwhere $a_{t}$ is the largest state index which satisfies\n\\begin{equation}\n a_{t}=\\max\n \\left\\{\n k \\in \\left\\{ 1,2,\\cdots,n  \\right\\}: \\left(1 + \\delta \\right)  q_k < (P_0-Q_0)(1+r)^{t}\n \\right\\},\n\\end{equation}\nwhere $q_k$ is the representative price level for the kth state.\n\\end{proposition}\n\n\\begin{remark}\nThe value of CPNR can be calculated with Proposition \\ref{propostion1} and \\ref{propostion2} in a recursive manner. The representative price level is chosen to be the smallest one in a state.\n\\end{remark}\n\n\\section{Active margin system}\n\n\\subsection{Individualized maintenance margin ratio}\n\nThe required margin system is an invariable system, where the initial and maintenance margin ratios are unchangeable with respect to the dynamics of stock prices. However, the value of CPNR is determined by the current stock price $P_0$, the amount of cash  $Q_0$, the proportion of stock $\\delta$, and the dynamics of stock prices during the period of loan. In order to control the risk faced by brokers below the desired level, the margin system should be different for different customers with individualized initial margins.\n\nFor a fixed value of CPNR, such as $\\text{CPNR}=0.05$,  the least maintenance margin ratio\nwhich satisfies the constraint of CPNR can be used as the required maintenance margin ratio for the customer with a particular $Q_0$ and $\\delta$. Proposition \\ref{adequate} can be used to determine whether the initial margin is adequate on the date of  transaction. The resulted margin system is an active system, which changes with respect to the dynamics of stock price $P_0$, the customer's individualized $Q_0$ and $\\delta$. And the risk faced by brokers is controlled below the desired value of CPNR.\n\n\\subsection{Deduced margin system}\n\nIn order to give an example of margin systems deduced from the constraint of CPNR, a simple margin loan is considered in this paper, where one share of stock to be purchased using an amount of cash $Q_0$ and a portion of the same stock $\\delta$ as the collateral.  The margin system can be described by the triple $\\left(m,\\delta,w\\right)$, where $m$ is the initial margin ratio given by equation (\\ref{initial_ratio}), and $w$ is the individualized maintenance margin ratio for $m$ and $\\delta$.\n\nLet\n\\begin{eqnarray}\n m & \\in &  M= \\left\\{ 0.01 k, k=0,1,2,\\cdots, 80 \\right\\},   \\nonumber \\\\\n \\delta & \\in &  D= \\left\\{ 0.01 k, k=0,1,2,\\cdots, 80 \\right\\}, \\nonumber \\\\\n w & \\in &  W= \\left\\{ 0.01 k, k=100,101,102,\\cdots, 200 \\right\\}. \\nonumber\n\\end{eqnarray}\nThe value of CPNR for a particular $\\left(m,\\delta,w\\right)$ is denoted as $\\text{CPNR}\\left( m,\\delta,w \\right)$. For a fixed value of $\\alpha$, let\n\\begin{equation}\n\\text{MDW}_{\\alpha} = \\left\\{ \\left( m,\\delta,w \\right) : m\\in M, \\delta \\in D, w \\in W, 1+m \\ge w, \\text{CPNR}\\left( m,\\delta,w \\right) \\le \\alpha \\right\\},\n\\end{equation}\nwhich is called the indifference set of margin systems. An optimal margin system $(m^*,\\delta^*,w^*)$ is chosen from this set by a least squares method, which solves the following problem\n\\begin{eqnarray*}\n& &\\min_{\\left( m,\\delta,w \\right)} \\sum_{i=1}^{q}(m_i-m)^{2}+(\\delta_i-\\delta)^{2}+(w_i-w)^{2}, \\nonumber \\\\\n& &\\text{s.t.} \\left( m,\\delta,w \\right) \\in \\mathrm{MDW_{\\alpha}},\n\\end{eqnarray*}\nwhere $q$ is the number of elements in $\\mathrm{MDW_{\\alpha}}$. The resulted margin system is called the deduced margin system in this paper.\n\n\\begin{table}\n\\caption{Quantile analysis of the initial margin ratios under the deduced margin system.\n}\\label{initialtable}\n\\centering{\n\\begin{tabular*}{0.9\\textwidth}{@{\\extracolsep{\\fill}}lllllllr}\n\\hline\n\t\\textit{}&\t\\multicolumn{3}{c}{} & \\multicolumn{4}{c}{quantiles}\t \\\\\t\n\\cline{5-8}\t\t\t\t\t\t\n\t\\textit{Statistics}&\tmin\t&\tmax\t&\tmean\t&\t0.70\t&\t0.80\t&\t 0.90\t&\t0.95\t \\\\ \\hline\n\\textit{minimum} \t&   0.36   & 0.54    &0.46   & 0.50   & 0.52  &  0.52   & 0.52\t \\\\\n\\textit{maximum } \t&0.56    &0.76    &0.63    &0.64   & 0.64   & 0.70   & 0.71 \t \\\\\n\\textit{mean }\t &0.50 & 0.58  &0.55   & 0.56  &  0.56  &0.57  &0.57\\\\\n\\textit{quantiles}\t\t\\\\\t\n0.20     &0.42    &0.56    &0.53    &0.54    &0.54    &0.54    &0.56\\\\\n0.30     &0.44    &0.56    &0.54    &0.54    &0.56    &0.56    &0.56\\\\\n0.40     &0.44    &0.58    &0.55    &0.56    &0.56    &0.56    &0.56\\\\\n0.50     &0.46    &0.58    &0.55    &0.56    &0.56    &0.56    &0.56\\\\\n0.60     &0.48    &0.58    &0.56    &0.56    &0.56    &0.58    &0.58\\\\\n0.70     &0.51    &0.58    &0.57    &0.58    &0.58    &0.58    &0.58\\\\\n0.80     &0.54    &0.61    &0.57    &0.58    &0.58    &0.59    &0.60\\\\\n0.90     &0.56    &0.76    &0.59    &0.59    &0.60    &0.60    &0.62\\\\\n0.95     &0.56    &0.76    &0.60    &0.60    &0.61    &0.62    &0.64\\\\\n\\hline\n\\end{tabular*}\n}\n\\end{table}\n\n\\begin{table}[h]\n\\caption{Quantile analysis of the maintenance margin ratios under the deduced margin system.\n}\\label{mainteancetable}\n\\centering{\n\\begin{tabular*}{0.9\\textwidth}{@{\\extracolsep{\\fill}}lllllllr}\n\\hline\n\t\t\\textit{ }\t&\t\\multicolumn{3}{c}{} & \\multicolumn{4}{c}{quantiles}\t \\\\ \\cline{5-8}\t\t\t\t\t\t\n\t\\textit{Statistics }&\tmin\t&\tmax\t&\tmean\t&\t0.70\t&\t0.80\t&\t 0.90\t&\t0.95\t \\\\\n\\hline\n\\textit{minimum} \t&   1.02    &1.26    &1.06    &1.11    &1.15    &1.16   & 1.17\t \\\\\n\\textit{maximum } \t&1.02   & 1.45  &  1.30   & 1.35   & 1.36  &  1.38    &1.39\t\\\\\n\\textit{mean }\t &1.02    &1.34  &1.18   & 1.22 & 1.24  &1.25 & 1.27\\\\\n\\textit{quantiles}\t&\t\t&\t\t&\t\t&\t\t&\t\t&\t\t&\t\\\\\t\n 0.20    &1.02    &1.32    &1.12    &1.19    &1.20    &1.22    &1.25\\\\\n 0.30    &1.02    &1.33    &1.14    &1.21    &1.22    &1.25    &1.26\\\\\n 0.40    &1.02    &1.35    &1.17    &1.21    &1.23    &1.25    &1.28\\\\\n 0.50    &1.02    &1.35    &1.19    &1.23    &1.25    &1.27    &1.29\\\\\n 0.60    &1.02    &1.35    &1.20    &1.24    &1.26    &1.29    &1.30\\\\\n 0.70    &1.02    &1.36    &1.22    &1.27    &1.28    &1.31    &1.31\\\\\n 0.80    &1.02    &1.37    &1.24    &1.28    &1.30    &1.32    &1.33\\\\\n 0.90    &1.02    &1.39    &1.26    &1.29    &1.32    &1.33    &1.35\\\\\n 0.95   &1.02    &1.40    &1.27    &1.31    &1.33    &1.35    &1.36\\\\\n\\hline\n\\end{tabular*}\n}\n\\end{table}\n\n\n\n\\begin{table}[h]\n\\caption{Quantile analysis of the proportions of stock under the deduced margin system.}\\label{proportion}\n\\centering{\n\\begin{tabular*}{0.9\\textwidth}{@{\\extracolsep{\\fill}}lllllllr}\n\\hline\n\t\t&\t\\multicolumn{3}{c}{} & \\multicolumn{4}{c}{quantiles}\t \\\\\n\t\t\t\t\t\\cline{5-8}\t\t\t\n\t\\textit{Statistics}&\tmin\t&\tmax\t&\tmean\t&\t0.70\t&\t0.80\t&\t 0.90\t&\t0.95\t \\\\ \\hline\n\\textit{minimum} \t&    0.08  &  0.50     &0.39   &  0.45   &  0.45   &  0.46    & 0.46\t\\\\\n\\textit{maximum } \t&0.54  &  0.75  &  0.63  &  0.67  &  0.70  &  0.71  &  0.74\t\\\\\n\\textit{mean }\t &0.41   & 0.56  &  0.51  &  0.52   & 0.52  &  0.53   & 0.53\\\\\n\\textit{quantiles}\t&\t\t&\t\t&\t\t&\t\t&\t\t&\t\t&\t\\\\\t\n 0.20   &0.30 &   0.52   & 0.48  &  0.50    &0.50  &  0.50   & 0.50\\\\\n 0.30   & 0.33   & 0.52   & 0.49   & 0.50 &   0.50  &  0.50   & 0.52\\\\\n 0.40   &0.38 &   0.54  &  0.50 &   0.50  &  0.52  &  0.52  &  0.52\\\\\n 0.50   &0.44  &  0.56    &0.51    &0.52  &  0.52  &  0.52   & 0.52\\\\\n 0.60   &0.46 &   0.56  &  0.52   & 0.52  &  0.52  &  0.53  &  0.54\\\\\n 0.70   &0.48  &  0.61   & 0.53    &0.54   & 0.54 &   0.54  &  0.54\\\\\n 0.80   &0.50  &  0.62  &  0.54  &  0.54 &   0.54  &  0.56   & 0.56\\\\\n 0.90   &0.52  &  0.65   & 0.56   & 0.56  &  0.58   & 0.58  &  0.60\\\\\n 0.95  &0.52  &  0.70   & 0.57 &   0.58  &  0.59 &   0.61  &  0.63\n\\\\\n\\hline\n\\end{tabular*}\n}\n\\end{table}\n\n\\section{Empirical analysis of Chinese market}\n\n\\subsection{Design of out-of-sample test}\n\nAn out-of-sample test is designed in this paper to compare the performances of the required and the deduced margin systems. The value of CPNR is fixed at 0.05, and 800 price data before the current date are used to construct the Markov chain model, where every 25 different prices are regarded as one state. The period of the loans is chosen to be 30 days, and 200 loans are constructed for each stock under consideration. The sample size needed in this paper is at least 1030, and 137 stocks listed on the SSE are investigated, where the data between 26 June 2006 and 18 October 2010 are involved.\n\nThe frequency of negative returns are calculated for each stock, and those stocks whose frequencies are less than or equal to 0.05 are said to  pass the out-of-sample test. There are 126 stocks which pass the test under the deduced margin system, and the remainder can not pass the test even under the required margin system, due to the abnormal movements of stock prices. The following analysis is limited to those 126 stocks which have passed the test.\n\n\\subsection{Empirical results}\n\nThe initial and maintenance margin ratios under the deduced margin system are reported in Table \\ref{initialtable} and \\ref{mainteancetable}, where every statistic has 126 observations, which are calculated from the 200 observed ratios of each stock. The mean of those observed initial margin ratios is $55\\%$, and the mean of those maintenance margin ratios is $118\\%$.\n\nThe proportion of stock in the initial margin is\n\\begin{equation}\np=\\frac{\\delta P_0}{\\delta P_0 + Q_0},\n\\end{equation}\nwhich is calculated for every loan of each stock under the deduced margin system, and the results are reported in Table \\ref{proportion}. The mean of those observed proportions is $51\\%$.\n\n\n\\begin{table}\n\\caption{Numbers of margin calls under the required and the deduced margin systems.}\\label{callnumber}\n\\centering{\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}lllllllllr}\n\\hline\n\t&\t\\multicolumn{3}{c}{}\t& \\multicolumn{6}{c}{quantiles}\t\\\\\n \\cline{5-10}\t\n\t&\tmin\t&\tmax\t&\tmean\t&\t0.30 \t&\t0.50 \t&\t0.80 \t&\t0.90 \t &\t 0.95 \t&\t0.99 \t\\\\\n\\hline\n\\textit{Required}\t&\t5.00 &100.00  &45.79  &34.00 &46.00   &63.00 &  71.00 &77.00 &  88.00 \t \\\\\n\\textit{Deduced}\t&\t0.00  &  83.00  & 18.41  &2.00 & 16.00   &33.00 &  42.00 &  48.00 &  63.00 \t \\\\\n\\hline\n\\end{tabular*}\n}\n\\end{table}\n\nFrequent margin calls are regarded as one of the sources of instability in stock markets. The number of margin calls is calculated for each stock  among 200 loans  in this paper, which is the total number of loans issued at least one call during the period of  the loan. The observed results are reported in Table \\ref{callnumber}.\nThe average number of margin calls  is  $45.79$ under the required margin system, and its counterpart under the deduced margin system is $18.41$, which is reduced by $59.8\\%$ with respect to the former. The stock market under the deduced margin system is more stable than the one under the required margin system.\n\n\\begin{table}[h]\n\\caption{Costs of margin loans under the deduced and the required margin systems.}\\label{loancost}\n\\centering{\n\\begin{tabular*}{0.9\\textwidth}{@{\\extracolsep{\\fill}}llllllllr}\n\\hline\n\t&\t\\multicolumn{3}{c}{}\t& \\multicolumn{4}{c}{quantiles}\t& \\\\\n \\cline{5-8}\n\\textit{Statistics}&\tmin\t&\tmax\t&\tmean\t&\t0.70 \t&\t0.80 \t&\t 0.90 \t &\t0.95 \t & \tRD \\\\\n\\hline\n\\textit{minimum}\t&\t1.71   &67.94    &5.94    &6.48    &7.35    &9.34   &11.36   &0.04\t\t\t \\\\\n\t&\t1.47   &62.91    &5.60   & 6.06 &   7.11   & 8.47  & 10.88\t\\\\\n\\textit{maximum}\t&\t2.45   &97.75  & 12.24  & 13.38 &  16.18  & 20.55  & 23.50  &-0.25\t\t\t \\\\\n\t&\t2.97 & 117.97   &15.19  & 17.78 &  19.79 &  24.30   &31.35\t\\\\\n\\textit{mean}\t&\t2.18  & 83.05  &  8.76 &   9.76   &11.40   &14.36  & 17.39 & -0.04\t \\\\\n\t&\t2.02   &79.12  &  8.75  & 10.00  & 11.41  & 14.71 &  18.21 \t\\\\\n\\textit{quantiles}\t&\t\t&\t\t&\t\t&\t\t&\t\t&\t\t&\t\t&\t\t \\\\\n0.20    &1.85   &73.85   & 7.31   & 8.15   & 9.48   &12.17  & 14.01  & 0.12\\\\\n       &1.69   &67.11    &6.71    &7.33   &8.75   &11.04   &12.51\\\\\n0.30   & 1.90   &75.57    &7.74   & 8.77   &10.40  & 12.64  & 15.75   &0.06\\\\\n      & 1.72   &70.53    &7.31    &8.08    &9.84   &11.97   &14.80\\\\\n0.40    &2.02  &77.25   & 8.26   & 9.34   &10.98  & 13.37  & 16.50   &0.04\\\\\n       &1.93   &74.45    &7.87    &8.63   &10.20   &12.52  &15.82\\\\\n0.50   & 2.12  & 86.03   & 8.83   & 9.95  & 11.52   &14.08   &17.02  &0.00\\\\\n       &2.06   &80.18    &8.33    &9.04   &10.98   &13.93   &17.02\\\\\n0.60    &2.18  & 88.91   & 9.24   &10.18  & 12.00  & 15.22  & 18.70  & 0.08\\\\\n       &2.08   &82.80    &8.73    &9.58   &11.47   &14.32   &17.39\\\\\n0.70    &2.26   &90.56   & 9.63   &10.60   &12.67  & 15.81  & 19.01  & 0.06\\\\\n       &2.09   &84.25    &9.28   &10.14   &12.19   &15.83   &17.97\\\\\n0.80    &2.29  & 91.70   &10.07   &11.10  & 13.25   &16.45   &19.42  &-0.09\\\\\n       &2.12   &85.45   &10.45   &11.80   &13.86   &18.90   &21.31\\\\\n0.90    &2.37  & 93.24  & 10.67  & 11.77  & 14.20  & 17.58  & 20.09  &-0.22\\\\\n       &2.15   &87.21   &12.33   &14.55   &16.50   &21.53   &25.92\\\\\n0.95   &2.41   &94.45   &11.12  & 12.12   &14.57   &18.37   &20.92  &-0.21\\\\\n   &2.18  &111.19   &13.49  & 15.54   &17.60   &22.62   &26.61\\\\\n\\hline\n\\end{tabular*}\n}\n\\end{table}\n\nThe cost of a margin loan is the terminal time value of the initial margin and the additional capital deposited to the broker to meet all of the margin calls during the period of the loan. Pursuant to the regulations of the SSE and the SZSE, the additional capital must make the maintenance margin ratio reach $150\\%$, if there is a margin call. And it only need to make the maintenance margin ratio reach $w$ under the deduced margin system in this paper.\nThe costs under the deduced and the required margin systems are reported in Table\n\\ref{loancost}.  The first line of each statistic are the observations under the deduced margin system, and the second line are observed under the required margin system. And the last column of this table are the relatively difference between those two $95\\%$ quantiles for the same statistic  respectively. The average cost under the deduced margin system is $8.76$, which is similar to its counterpart under the required margin system.\n\n\n\\section{Conclusions and discussions}\n\nAn active margin system is proposed for the margin loans using cash and stock as collateral in this paper.  Margin system can be adjusted actively according to the changes of market factors with the help of the probability of negative return (CPNR). Empirical investigations are used to compare the performances of the deduced and the required margin systems with 126 stocks listed on the SSE. It is found that the number of margin calls under the deduced margin system is significantly less than its counterpart under the required margin system, where the average costs of margin loans are similar under the two types of margin systems. Those observations indicate that CPNR can be used to measure the risk faced by brokers, and the deduced margin system   is beneficial to the stability of stock market.\n\nThe performance of individualized maintenance margin is investigated through the deduced margin system,  which is chosen from the indifference set of margin systems by a least squares method. On the other hand, margin system can be set for a particular customer with a specified initial margin. The performance of such margin system is not discussed in this paper, which will be left for further research.\n\n\\section*{Acknowledgments}\n {Guanghui Huang is supported by the Fundamental Research Funds for the Central Universities of China under Grant CDJZR10 100 007.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Abstract}\n\n\n\n\n\\section{Introduction}\\label{sec:intro}\nHandwritten text has served for centuries as our primary mean of communication and cornerstone of our education and culture, and often is considered a form of art~\\cite{Robinson2007}. It has been shown to be beneficial in tasks such as note-taking~\\cite{Mueller2014}, reading in conjunction with writing~\\cite{Sellen2003} and may have a positive impact on short- and long-term memory \\cite{Berninger2012}. However, despite progress in character recognition~\\cite{Liwicki2007}, fully digital text remains easier to process, search and manipulate than handwritten text which has lead to a dominance of typed text.\n\nIn this paper we explore novel ways to combine the benefits of digital ink with the versatility and efficiency of digital text, by making it editable via disentanglement of style and content. Digital ink and stylus-based interfaces have been of continued interest to HCI research (e.g., \\cite{Hammond2015, Hinckley2014, Riche2017, Xia2017, Zitnick2013}).\nHowever, to process digital ink one has typically to resort to optical character recognition (OCR) techniques (e.g.,~\\cite{EvernoteOCR2017}) thus invariably losing the personalized aspect of written text.\nIn contrast, our approach is capable of maintaining the author's original style, thus allowing for a seamless transition between handwritten and digital text.\nOur approach is capable of synthesizing handwritten text, taking either a sequence of digital ink or ASCII characters as input.\nThis is a challenging problem: while each user has a unique handwriting style \\cite{Srihari2002,Zhang2003}, the parameters that determine style are not well defined.\nMoreover, handwriting style is not fixed but changes temporally based on context, writing speed and other factors~\\cite{Srihari2002}.\nHence so far it has been elusive to algorithmically recreate style faithfully, while being able to control content. A comprehensive approach to handwriting synthesis must be able to maintain \\emph{global} style while preserving \\emph{local} variability and context (e.g., many users mix cursive and disconnected styles dynamically).\n\nEmbracing this challenge we contribute a novel generative deep neural network architecture for the conditional synthesis of digital ink. The model is capable of capturing and reproducing local variability of handwriting and can mimic different user styles with high-fidelity.\nImportantly the model provides full control over the content of the synthetic sequences, enabling processing and editing of digital ink at the word level.\nThe main technical contribution stems from the ability to disentangle latent factors that influence visual appearance and content from each other.\nThis is achieved via an architecture that combines autoregressive models with a learned latent-space representation, modeling temporal and time-invariant categorical aspects (i.e., character information)\n\nMore precisely we propose an architecture comprising of a recurrent variational autoencoder \\cite{Chung2015,Kingma2013} in combination with two latent distributions that allow for \\emph{conditional} synthesis (i.e., provide control over style and content) alongside a novel training and sampling algorithm.\nThe system has been trained on segmented samples of handwritten text collected from 294 authors, and takes into account both temporal and stylistic aspects of handwriting. Further, it is capable of synthesizing novel sequences from typed-text, transferring styles from one user to another, editing digital ink at the word level and thus enables compelling application scenarios including spell-checking and auto-correction of digital ink.\n\nWe characterize the performance of the architecture via a thorough technical evaluation, and assess its efficacy and utility via a preliminary experimental evaluation of the interactive scenarios we implemented. Further, we contribute a new dataset that enhances the IAM On-Line Handwriting Database (IAM-OnDB) \\cite{Liwicki2005} and includes handwritten text collected from 294 authors with character level annotations. Finally, we plan to release an open-source implementation of our model. \n\n\\section{Related work}\\label{sec:rw}\nOur work touches upon various subjects including HCI (e.g., \\cite{Hammond2015, Perteneder2015, Riche2017,Xia2017}), handwriting analysis~\\cite{Pulver1972} and machine learning (e.g., \\cite{Elarian2014, Graves2013, Kingma2013,Plamondon2000}).\n\n\\subsection{Understanding Handwriting}\n\\label{section:rw:handwriting-analysis-for-font-design}\nResearch into the recognition of handwritten text has led to drastic accuracy improvements~\\cite{Elarian2014,Plamondon2000} and such technology can now be found in mainstream UIs (e.g., Windows, Android, iOS). However, converting digital ink into ASCII characters removes individual style.\nUnderstanding what exactly constitutes style has been the subject of much research to inform font design~\\cite{Burgert2002, Drucker1995,Noordzij05} and the related understanding of human reading has served as a source of inspiration for the modern parametric-font systems~\\cite{Hussain1999,Knuth1986,Shamir1998}. Nonetheless no equivalent parametric model of handwritten style exists and analysis and description of style remains an inexact science~\\cite{Noordzij05,Pulver1972}. We propose to learn a latent representation of style and to leverage it in a generative model of user specific text.\n\n\\subsection{Pen-based interaction}\nGiven the naturalness of the medium~\\cite{Sellen2003, Cherubini2007}, pen-based interfaces have seen enduring interest in both the graphics and HCI literature~\\cite{Sutherland2016}. Ever since Ivan Sutherland's Sketchpad~\\cite{Sutherland1963} researchers have explored sensing and input techniques for small screens~\\cite{Kienzle2013, Yoon2013}, tablets~\\cite{Hinckley2014,Pfeuffer2017} and whiteboards~\\cite{Mynatt1999,Perteneder2015,Xia2017} and have proposed ways of integrating paper with digital media \\cite{Haller2010, Brandl2010}. Furthermore many domain specific applications have been proposed. For instance, manipulation of hand-drawn diagrams \\cite{Arvo2005appearance} and geometric shapes \\cite{Arvo2000fluid}, note-taking (e.g., NiCEBook~\\cite{Brandl2010}), sharing of notes (e.g., NotePals~\\cite{Davis1999}), browsing and annotation of multimedia content~\\cite{Weibel2012}, including digital documents~\\cite{Yoon2014} using a stylus. Others have explored creation, management and annotation of handwritten notes on large screen displays~\\cite{Perteneder2015,Xia2017}. Typically such approaches do not convert ink into characters to preserve individual style. Our work enables new interactive possibilities by making digital ink editable and interchangeable with a character representation, allowing for advanced processing, searching and editing.\n\n\\subsection{Handwriting Beautification}\n\\label{section:rw:handwriting-beautification}\nZitnick~\\cite{Zitnick2013} proposes a method for beautification of digital ink by exploiting the smoothing effect of geometric averaging of multiple instances of the same stroke. While generating convincing results, this method requires several samples of the same text for a single user. A supervised machine learning method to remove slope and slant from handwritten text and to normalize it's size~\\cite{Espana-Boquera2011} has been proposed. Zanibbi et al \\cite{Zanibbi2001aiding} introduce a tool to improve legibility of handwritten equations by applying style-preserving morphs on user-drawn symbols.\nLu et al.~\\cite{Lu2012} propose to learn style features from trained artist, and to subsequently transfer the strokes of a different writer to the learnt style and therefore inherently remove the original style.\nText beautification is one of the potential applications for our work, however the proposed model only requires a single sequence as input, retains the global style of the author when beautifying and can generate novel (i.e., from ASCII characters) sequences in that style.\n\n\n\\subsection{Handwriting Synthesis}\n\\label{section:rw:handwriting-synthesis}\nA large body of work is dedicated to the synthesis of handwritten text (for a comprehensive survey see ~\\cite{Elarian2014}).\nAttempts have been made to formalize plausible biological models of the processes underlying handwriting~\\cite{Hinton2005} or by learning sequences of motion primitives~\\cite{Williams2007}. Such approaches primarily validate bio-inspired models but do not produce convincing sequences. In \\cite{Bhattacharya2017sigma, Plamondon2014recent} sigma-lognormal models are proposed to synthesize handwriting samples by parameterizing rapid human movements and hence reflecting writer's fine motor control capability. The model can naturally synthesize variances of a given sample, but it lacks control of the content.\n\nRealistic handwritten characters such as Japanese Kanji or individual digits can be synthesized from learned statistical models of stroke similarity~\\cite{Chang2012} or control point positions (requiring characters to be converted to splines)~\\cite{Wang2002}. Follow-up work has proposed methods that connect such synthetic characters~\\cite{Chen2015,Wang2005} using a ligature model. Haines et al.~\\cite{Haines2016} take character-segmented images of a single author's writing and attempts to replicate the style via dynamic programming. These approaches either ignore style entirely or learn to imitate a single reference style from a large corpus of data. In contrast, our method learns first how to separate content and style and then is capable of transferring style from a single sample to arbitrary text, providing much more flexibility.\n\nKingma and Welling~\\cite{Kingma2013} propose a variational auto-encoder architecture for manifold learning and generative modelling. The authors demonstrate synthesis of single character digits via manipulation of two, abstract latent variables but the method can not be used for conditional synthesis.\nThe work most closely related to ours proposes a long short-term memory recurrent (LSTM) neural network to generate complex sequences with long-range structure such as handwritten text~\\cite{Graves2013}. The work demonstrates synthesis of handwritten text in specific styles limited to samples in the training set, and its model has no notion of disentangling content from style.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Method}\\label{sec:method}\nTo make digital ink fully editable, one has to overcome a number of technical problems such as character recognition and synthesis of realistic handwriting. None is more important though than the disentanglement of \\textit{style} and \\textit{content}. Each author has a unique style of handwriting~\\cite{Srihari2002}, but at the same time, they also display a lot of intra-variability, such as mixing connected and disconnected styles, variance in usage of glyphs, character spacing and slanting (see \\figref{fig:problem}). Hence, it is hard to define or predict the appearance of a character, as often its appearance is strongly influenced by its content.\n\nIn this paper, we propose a data-driven model capable of disentangling handwritten text into their content and style components, necessary to enable editing and synthesis of novel handwritten samples in a user-specified style. The key idea underlying our approach is to treat style and content as two separate latent random variables (\\figref{fig:factorization}-a). While the \\textit{content} component is defined as the set of alphanumeric characters and punctuation marks, the \\textit{style} term is an abstraction of the factors defining appearance. It is learned by the model and projected into a continues-valued latent space. One can make use of content and style variables to \\emph{edit} either style, content or both, or one can generate entirely new samples (\\figref{fig:factorization}-b).\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.85\\columnwidth]{figs\/problem-setting}\n\t\\caption{Example of intra-author variation that leads to entanglement of style and content, making conditional synthesis of realistic digital ink very challenging.}\n\t\\label{fig:problem}\n\\end{figure}\n\nWe treat digital ink as a sequence of temporarily ordered strokes where each stroke consists of $(u,v)$ pen-coordinates on the device screen and corresponding \\textit{pen-up} events. While the pen-coordinates are integer values bounded by screen resolution of the device, \\textit{pen-up} takes value $1$ when the pen is lifted off the screen and $0$, otherwise. A handwriting sample is formally defined as $\\mathbf{x} = \\{x_t\\}_{t=1}^{T}$ where $x_t$ is a stroke and $T$ is total number of strokes. Moreover, we label each stroke $x_t$ with character $y_t$, end-of-character $eoc_t$ and beginning-of-word $bow_t$ labels. $y_t$ specifies which character a stroke $x_t$ belongs to. Both $eoc_t$ and $bow_t$ are binary-valued and set to $1$ if $x_t$ correspond to the last stroke of a character sequence or the first stroke of a new word, respectively (\\figref{fig:stroke-sample}).\n\n\nWe propose a novel autoregressive neural network (NN) architecture that contains continuous and categorical latent random variables. Here the continuous latent variable which captures the appearance properties is modeled by an isotropic Normal distribution (\\figref{fig:model-diagram} (green)). Whereas the content information is captured via a Gaussian Mixture Model (GMM), where each character in the dataset is represented by an isotropic Gaussian (shown in \\figref{fig:model-diagram} (blue)). We train the model by reconstructing a given handwritten sample $\\mathbf{x}$ (\\figref{fig:factorization}-c). Handwriting is inherently a temporal domain and require exploiting long-range dependencies. Hence, we leverage recurrent neural network (RNN) cells and operate in the stroke level $x_t$. Moreover, we make use of and predict $y_t$, $eoc_t$, $bow_t$ in auxiliary tasks such as controlling the word spacing, character segmentation and recognition. (\\figref{fig:model-diagram}).\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth, trim={20pt 230pt 360pt 20pt},clip]{figs\/method_factorization}\n\t\\caption{High-level representation of our approach. $\\mathbf{x}, \\mathbf{z}$ and $\\mathbf{\\pi}$ are random variables corresponding to handwritten text, \\textit{style} and \\textit{content}, respectively. (a) A given handwritten sample can be decomposed into \\textit{style} and \\textit{content} components. (b) Similarly, a sample can be synthesized using \\textit{style} and \\textit{content} components. (c) Our model learns inferring and using latent variables by reconstructing handwriting samples. $g^{inp}$ and $g^{out}$ are feed-forward networks projecting the input into an intermediate representation and predicting outputs, respectively.}\n\t\\label{fig:factorization}\n\\end{figure}\n\nThe proposed architecture which we call conditional variational recurrent neural network (C-VRNN~) builds on prior work on variational autoencoders (VAE) \\cite{Kingma2013} and its recurrent variant, variational recurrent neural networks (VRNN) \\cite{Chung2015}. While VAEs only work with non-temporal data, VRNN can reconstruct and synthesize timeseries, albeit without conditioning, providing no control over the generated content. In contrast, our model synthesizes realistic handwriting with natural variation and conditioned on a user-specified content, enabling a number of compelling applications (\\figref{fig:teaser}).\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.85\\columnwidth, trim={25pt 400pt 680pt 20pt},clip]{figs\/stroke_sample_large}\n\t\\caption{Discretization of digital handwriting. A handwriting sample (top) is represented by a sequence of temporarily ordered strokes. Yellow and green nodes illustrate sampled strokes. The green nodes correspond to \\textit{pen-up} events.}\n\t\\label{fig:stroke-sample}\n\\end{figure}\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[width=0.99\\linewidth,trim={5pt 220pt 850pt 50pt},clip]{figs\/model_diagram_v2.pdf}\n\t\\caption{Schematic overview of our handwriting model in training (a) and sampling phases (b), operating at the stroke-level. Subscripts denote time-step $t$. Superscripts correspond to layer names such as \\textit{input}, \\textit{latent} and \\textit{output} layers or the distributions of the random variables such as $z_t^q \\sim q(z_t|x_t)$ and $z_t^p \\sim p(z_t|x_t)$. ($\\tau$ and $h$) An RNN cell and its output. ($g$) A multi-layer feed-forward neural network. (Arrows) Information flow color-coded with respect to source. (Colored circles) Latent random variables. Outgoing arrows represent a sample of the random variable. (Green branch) Gaussian latent space capturing style related information along with latent RNN ($\\tau_t^{latent}$) cell at individual time-steps $t$. (Blue branch) Categorical and GMM random variables capturing content information. (Small black nodes) An auxiliary node for concatenation of incoming nodes.}\n\t\\label{fig:model-diagram}\n\\end{figure*}\n\n\n\\iffalse\n\\subsection{Training time}\nHere we briefly discuss the flow of information through the model (\\figref{fig:model}, left) at training time, from inputs to outputs.\n\\begin{description}[style=unboxed,leftmargin=0cm,noitemsep]\n\\item[Step 1:]  Using the input stroke $x_t$ we update the intermediate representation $h_t^{input}$ of the input $LSTM$ cell (grey).\n\\item[Step 2:] The current input layer state $h_t^{input}$ is combined with the state from the previous step $h_{t-1}^{latent}$ of the cell in the latent layer, to attain a character probability vector $\\pi_t$ (equivalent to the character prediction $\\hat{y}_t$).\n\\item[Step 3:] Using character information $\\pi_t$, we select the corresponding component from the categorical latent distribution and sample $z_t^{gmm}$, a continuous representation of the character.\n\\item[Step 4:] The input state $h_t^{input}$, previous latent state $h_{t-1}^{latent}$ and beginning-of-word information $bow_t$ are furthermore used to calculate the parameters of an isotropic Gaussian, and to sample $z_t^{q}$ (green) from it.\n\\item[Step 5:] An $LSTM$ at the output layer receives $h_{t-1}^{latent}$, $z_t^{q}$ and $z_t^{gmm}$ and reconstructs the inputs $x_t$, $bow_t$ and predicts $eoc_t$.\n\\item[Step 6:] $h_{t-1}^{latent}$ is also used to calculate prior distributions $z_t^p$ and $\\pi_t^p$ for the next step. $z_t^p, \\pi_t^p$ are isotropic Gaussian and multinomial distributions, respectively. Note that we minimize the Kullback-Leibler divergence between pairs of $(z_t^q, z_t^p)$ and $(\\pi_t^q, \\pi_t^p)$, regularizing the network and learning predictive priors used at sampling time.\n\\item[Step 7:] Finally we update the third $LSTM$ cell, responsible for latent space transitions between subsequent steps, via the latent samples $z_t^{q}$ and $z_t^{gmm}$ and input representation $h_{t-1}^{input}$.\n\\end{description}\n\nThe auxiliary predictions for $eoc_t$ and $bow_t$ are needed to keep track which character is currently synthesized (stroke-by-stoke) and when to leave space between consecutive words.\n\n\\subsection{Sampling time}\nThe procedure to synthesize ink sequences is relatively straightforward. Assuming an initial latent state $h_{t-1}^{latent}$ and text (ASCII) as input:\n\\begin{description}[style=unboxed,leftmargin=0cm,noitemsep]\n\t\\item[Step 1:] Calculate parameters of the prior distribution and sample $z_t^p$.\n\t\\item[Step 2:] Retrieve the GMM component corresponding to $\\pi_t$ and sample $z_t^{gmm}$.\n\t\\item[Step 3:] Synthesize a stroke $\\hat{x}_t$ by feeding $h_{t-1}^{latent}$, $z_t^p$ and $z_t^{gmm}$ to the output $LSTM$ cell.\n\t\\item[Step 4:] Use the synthetic sample $\\hat{x}_t$ and update the internal state of the input $LSTM$ cell.\n\t\\item[Step 5:] Update the latent $LSTM$ cell with $z_t^p$, $z_t^{gmm}$ and $h_t^{input}$.\n\\end{description}\nThis procedure is repeated iteratively, adding segments to the stroke sequence until the text input is consumed (i.e., the entire text has been synthesized).\n\nFrom this procedures it should be clear that the content information is captured via the GMM latent random variable $z_t^{gmm}$, and the remaining style information flows through the Gaussian random variable $z_t^{q}$ and the $LSTM$ cell states across subsequent time steps.\n\nBefore detailing the individual components of our architecture we briefly review some background that is necessary to follow our derivation. A final paper would include the technical details, included here for completeness, in the appendix.\n\\fi\n\n\\subsection{Background}\nMulti-layer recurrent neural networks (RNN) \\cite{Graves2013} and variational RNN (VRNN) \\cite{Chung2015} are most related to our work. We briefly recap these and highlight differences. In our notation superscripts correspond to layer information such as \\textit{input}, \\textit{latent} or \\textit{output} while subscripts denote the time-step $t$. Moreover, we drop parametrization for the sake of brevity, and therefore readers should assume that all probability distributions are modeled using neural networks.\n\n\\subsubsection{Recurrent Neural Networks}\nRNNs model variable length input sequences $\\mathbf{x} = (x_1, x_2, \\cdots, x_T)$ by predicting the next time step $x_{t+1}$ given the current $x_t$. The probability of a sequence $\\mathbf{x}$ is given by\n\\begin{align}\n\\begin{split}\n\\label{eq:probability_rnn}\np(\\mathbf{x}) &= \\prod_{t=1}^T p(x_{t+1} | x_t), \\\\\np(x_{t+1} | x_t) &= g^{out}(h_{t}) \\\\\nh_t &= \\tau(x_t, h_{t-1}) ,\n\\end{split}\n\\end{align}\nwhere $\\tau$ is a \\emph{deterministic} transition function of an RNN cell, updating the internal cell state $h$. Note that $x_{t+1}$ implicitly depends on all inputs until step $t+1$ through the  cell state $h$.\n\nThe function $g^{out}$ maps the hidden state to a probability distribution. In vanilla RNNs (e.g., LSTM, GRU) $g^{out}$ is the only source of variability. To express the natural randomness in the data, the output function $g^{out}$ typically parametrizes a statistical distribution (e.g., Bernoulli, Normal, GMM). The output is then calculated by sampling from this distribution. Both functions $\\tau$ and $g^{out}$ are approximated by optimizing neural network parameters via maximizing the log-likelihood:\n\\begin{equation}\n\\mathcal{L}_{rnn}(\\mathbf{x}) = log p(\\mathbf{x}) = \\sum_{t=1}^T log p(x_{t+1} | x_t)\n\\label{eq:rnn_log_likelihood}\n\\end{equation}\n\nMulti-layered LSTMs with a GMM output distribution have been used for handwriting modeling \\cite{Graves2013}. While capable of conditional synthesis, they can not disentangle style from content due to the lack of latent random variables.\n\n\\subsubsection{Variational Recurrent Neural Networks} VRNNs \\cite{Chung2015} modify the deterministic $\\tau$ transition function by introducing a latent random variable $\\mathbf{z} = (z_1, z_2, \\cdots, z_T)$ increasing the expressive power of the model and to better capture variability in the data by modeling\n\\begin{align}\n\\begin{split}\n\\label{eq:probability_vrnn}\np(\\mathbf{x}, \\mathbf{z}) &= p(\\mathbf{x}|\\mathbf{z})p(\\mathbf{z}), \\\\\np(\\mathbf{x}, \\mathbf{z}) &= \\prod_{t=1}^T p(x_{t} | z_{t}) p(z_{t}), \\\\\np(x_{t} | z_{t}) &= g^{out}(z_t, h_{t-1}), \\\\\np(z_{t}) &= g^{p,z}(h_{t-1}), \\\\\nh_t &= \\tau(x_t, z_t, h_{t-1}),\n\\end{split}\n\\end{align}\nwhere $g^{p,z}$ is a multi-layer feed forward network parameterizing the prior distribution $p(z_{t})$ and the latent variable $\\mathbf{z}$ enforces the model to project the data variability on the prior distribution $p(\\mathbf{z})$. Note that $x_t$ still depends on the previous steps, albeit implicitly through the internal state $h_{t-1}$.\n\nAt each time step the latent random variable $\\mathbf{z}$ is modeled as isotropic Normal distribution $z_t \\sim \\mathcal{N}(\\mu_t, \\sigma_tI)$. The transition function $\\tau$ takes samples $z_t$ as input, introducing a new source of variability\n\nSince we do not have access to the true distributions at training time, the posterior $p(\\mathbf{z}|\\mathbf{x})$ is intractable and hence makes the marginal likelihood, i.e., the objective, $p(\\mathbf{x})$ also intractable. Instead, an approximate posterior distribution $q(\\mathbf{z}|\\mathbf{x})$ is employed, imitating the true posterior $p(\\mathbf{z}|\\mathbf{x})$ \\cite{Kingma2013}, where $q(\\mathbf{z}|\\mathbf{x})$ is an isotropic Normal distribution and parametrized by a neural network $g^{q,z}$ as follows:\n\\begin{equation}\nq(z_{t} | x_{t}) = g^{q,z}(x_t, h_{t-1})\n\\label{eq:vrnn_q}\n\\end{equation}\nThe model parameters are optimized by jointly maximizing a variational lower bound:\n\\begin{equation}\nlog p(\\mathbf{x}) \\geq \\mathbb{E}_{q(z_t|x_t)} \\sum_{t=1}^T log p(x_t | z_t) - KL(q(z_t|x_t) || p(z_t)),\n\\label{eq:vrnn_lower_bound}\n\\end{equation}\n\nwhere $KL(q || p)$ is the Kullback-Leibler divergence (non-similarity) between distributions $q$ and $p$. The first term in loss (\\ref{eq:vrnn_lower_bound}) ensures that the sample $x_t$ is reconstructed given the latent sample $z_t \\sim q(z_{t} | x_{t})$ while the $KL$ term minimizes the discrepancy between our approximate posterior and prior distributions so that we can use the prior $z_t \\sim p(z_t)$ for synthesis later. Moreover, the $q(\\mathbf{z} | \\mathbf{x})$ network enables inferring latent properties of a given sample, providing interesting applications. For example, a handwriting sample can be projected into the latent space $z$ and reconstructed with different slant.\n\nA plain auto-encoder architecture learns to faithfully reconstruct input samples, the latent term $\\mathbf{z}$ transforms the architecture into a fully generative model. Note that the $KL$-term never becomes $0$ due to the different amount of input information to $g^{p,z}$ and $g^{q,z}$. Hence, the $KL$-term enforces the model to capture the common information in the latent space $z$.\n\n\\subsection{Conditional Variational Recurrent Neural Network}\nWhile multi-layer RNNs and VRNNs have appealing properties, neither is directly capable of full conditional handwriting synthesis. For example, one can synthesize a given text in a given style by using RNNs but samples will lack natural variability. Or one can generate high quality novel samples with VRNNs. However, VRNNs lack control over \\emph{what} is written. Neither model have inference networks to decouple style and content, which lies at the core of our work.\n\nWe overcome this issue by introducing a new set of latent random variables, $\\mathbf{z},\\mathbf{\\pi}$, capturing style and content of handwriting samples. More precisely our new model describes the data as being generated by two latent variables $\\mathbf{z}$ and $\\mathbf{\\pi}$ (\\figref{fig:factorization}) such that\n\\begin{align}\n\\begin{split}\n\\label{eq:probability_cvrnn}\np(\\mathbf{x}, \\mathbf{z}, \\mathbf{\\pi}) &= p(\\mathbf{x}|\\mathbf{z},\\mathbf{\\pi})p(\\mathbf{z})p(\\mathbf{\\pi}), \\\\\np(\\mathbf{x}, \\mathbf{z}, \\mathbf{\\pi}) &= \\prod_{t=1}^T p(x_{t} | z_{t}) p(z_{t}) p(\\pi_{t}), \\\\\np(x_{t} | z_{t}, \\pi_{t}) &= g^{out}(z_t, \\pi_{t}), \\\\\np(z_{t}) &= g^{p,z}(h_{t-1}^{latent}), \\\\\np(\\pi_{t}) &= g^{p,\\pi}(h_{t-1}^{latent}), \\\\\nh_t^{latent} &= \\tau^{latent}(x_t, z_t, \\pi_{t}, h_{t-1}^{latent}), \\\\\nq(z_{t} | x_{t}) &= g^{q,z}(x_t, h_{t-1}^{latent}),\n\\end{split}\n\\end{align}\n\nwhere $p(\\pi_t)$ is a $K$-dimensional multinomial distribution specifying the characters that are synthesized.\n\nSimilar to VRNNs, we introduce an approximate inference distribution $q(\\mathbf{\\pi} | \\mathbf{x})$ for the categorical latent variable:\n\\begin{equation}\n\\begin{split}\nq(\\pi_t | x_{t}) &= g^{q,\\pi}(x_t, h_{t-1}^{latent})\n\\label{eq:cvrnn_pi_q}\n\\end{split}\n\\end{equation}\nSince we aim to decouple style and content in handwriting, we assume that the approximate distribution has a factorized form $q(z_t, \\pi_t|x_t) = q(z_t|x_t)q(\\pi_t|x_t)$. Both $q(\\mathbf{\\pi} | \\mathbf{x})$ and $q(\\mathbf{z} | \\mathbf{x})$ are used to infer content and style components of a given sample $\\mathbf{x}$ as described earlier.\n\nWe optimize the following variational lower bound:\n\\begin{align}\n\\begin{split}\nlog p(\\mathbf{x}) \\geq \\mathcal{L}_{lb}(\\cdot) = \\mathbb{E}_{q(z_t, \\pi_t|x_t)} \\sum_{t=1}^T log p(x_t | z_t, \\pi_t) \\\\ - KL(q(z_t|x_t) || p(z_t)) - KL(q(\\pi_t|x_t) || p(\\pi_t)),\n\\label{eq:cvrnn_lower_bound}\n\\end{split}\n\\end{align}\nwhere the first term ensures that the input stroke is reconstructed by using its latent samples. We model the output by using bivariate Gaussian and Bernoulli distributions for $2D$-pixel coordinates and binary \\textit{pen-up} events, respectively.\n\nNote that our output function $g^{out}$ does not employ the internal cell state $h$. By using only the latent variables $z$ and $\\pi$ for synthesis, we aim to enforce the model to capture the patterns only in the latent variables $z$ and $\\pi$.\n\n\n\n\\subsection{High Quality Digital Ink Synthesis}\nThe C-VRNN~ architecture as discussed so far enables the crucial component of separating continuous components from categorical aspects (i.e., characters) which potentially would be sufficient to conditionally synthesize individual characters. However, to fully address the entire handwriting task several extension to control important aspects such as word-spacing and to improve quality of the predictions are necessary.\n\n\\subsubsection{Character classification loss} Although we assume that the latent random variables $\\mathbf{z}$ and $\\mathbf{\\pi}$ capture style and content information, respectively, and make a conditional independence assumption, in practice full disentanglement is an ambiguous task. Since we essentially ask the model to learn by itself what style and what content are we found  further guidance at training time to be necessary.\n\nTo prevent divergence during training we make use of character labels at training time and add an additional cross-entropy classification loss $\\mathcal{L}_{classification}$ on the content component $q(\\pi_t|x_t)$.\n\n\\subsubsection{GMM latent space} Conditioning generative models is typically done via one-hot encoded labels. While we could directly use samples from $q(\\pi_t|x_t)$, we prefer using a continuous representation. We hypothesize and experimentally validate (see \\figref{fig:gmm-vs-one-hot}) that the synthesis model can shape the latent space with respect to the loss caused by the content aspect.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\columnwidth]{figs\/gmm_vs_one_hot_vector.png}\n\t\\caption{(top, green) Input samples used to infer style. (middle, red) Synthetic samples of a model with $\\mathbf{\\pi}$ only. They are generated using one-hot-encoded character labels, causing problems with \\textit{pen-up} events and with character placement. (bottom, blue) Synthetic samples of our model \\emph{with} GMM latent space.}\n\t\\label{fig:gmm-vs-one-hot}\n\\end{figure}\n\nFor this purpose we use a Gaussian mixture model where each character in $K$ is represented by an isotropic Gaussian\n\\begin{align}\n\\begin{split}\np(\\varphi_t) = \\sum_{k=1}^K \\pi_{t,k} \\mathcal{N}(\\varphi_t | \\mu_k, \\sigma_k)   ,\n\\label{eq:gmm_latent}\n\\end{split}\n\\end{align}\nwhere $\\mathcal{N}(\\varphi_t | \\mu_k, \\sigma_k)$ is the probability of sampling from the corresponding mixture component $k$. $\\pi$ corresponds to the content variable in \\refequ{eq:cvrnn_pi_q} which is here interpreted as weight of the mixture components. This means that we use $q(\\pi_t|x_t)$ to select a particular Gaussian component  for a given stroke sample $x_t$. We then sample $\\varphi_t$ from the $k$-th Gaussian component and apply the ``re-parametrization trick'' \\cite{Kingma2013,Gurumurthy2017} so that the gradients can flow through the random variables, enabling the learning of GMM parameters via standard backpropagation.\n\\begin{align}\n\\begin{split}\n\\varphi_t = \\mu_k + \\sigma_k\\epsilon,\n\\label{eq:gmm_reparametrization_trick}\n\\end{split}\n\\end{align}\nwhere $\\epsilon \\sim \\mathcal{N}(0,1)$. Our continuous content representation results in similar letters being located closer in the latent space while dissimilar letters or infrequent symbols being pushed away. This effect is visualized in \\figref{fig:gmm_latent_space}.\n\nImportantly the GMM parameters are sampled from a time-invariant distribution. That  is they remain the same for all data samples and across time steps of a given input $\\mathbf{x}$, whereas $z_t$ is dynamic and employs new parameters per time step. For each Gaussian component in $\\varphi$, we initialize $\\mu_k$, $1 \\leq k \\leq K$, randomly by using a uniform distribution $\\mathcal{U}(-1,1)$ and $\\sigma_k$ with a fixed value of $1$. The GMM components are trained alongside the other network parameters. \n\nIn order to increase model convergence speed and to improve results, we use ground truth character labels during training. More precisely, the GMM components are selected by using the real labels $y$ instead of predictions of the inference network $q(\\pi_t|x_t)$. Instead $q(\\pi_t|x_t)$ is trained only by using the classification loss $\\mathcal{L}_{classification}$ and not affected by the gradients of GMM with respect to $\\pi_t$.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.85\\columnwidth]{figs\/gmm_tsne_500.png}\n\t\\caption{Illustration of our GMM latent space $\\varphi^{gmm}$. We select a small subset of our alphabet and draw $500$ samples from corresponding GMM components. We use the t-sne algorithm \\protect\\cite{maaten2008visualizing} to visualize $32$-dimensional samples in $2$D space. Note that the t-sne algorithm finds an arbitrary placement and hence the positioning does not reflect the true latent space. Nevertheless, letters form separate clusters.}\n\t\\label{fig:gmm_latent_space}\n\\end{figure}\n\n\\subsubsection{Word spacing and character limits}\nAt sampling time the model needs to automatically infer word spacing and which character to synthesize (these are a priori unknown). In order to control when to leave a space between words or when to start synthesizing the next character, we introduce two additional signals during training, namely \\textit{eoc} and \\textit{bow} signaling the \\emph{end} of a character and \\emph{beginning} of a word respectively. These labels are attained from ground truth character level segmentations (see Dataset section).\n\nThe \\textit{bow} signal is fed as input to the output function $g^{out}$ and the output distribution of our handwriting synthesis takes the following form:\n\\begin{align}\n\\begin{split}\np(x_{t} | z_{t}, \\pi_{t}) &= g^{out}(z_t, \\varphi_t, bow_t),\n\\label{eq:z_sow_input}\n\\end{split}\n\\end{align}\nforcing the model to learn when to leave empty space at training and sampling time.\n\nThe \\textit{eoc} signal, on the other hand, is provided to the model at training so that it can predict when to stop synthesizing a given character. It is included in the loss function in the form of Bernoulli log-likelihood $\\mathcal{L}_{eoc}$. Along with the reconstruction of the input stroke $x_t$, the $eoc_t$ label is predicted.\n\n\\begin{algorithm}[t]\n\t\\caption{Training \\\\ The flow of information through the model at training time, from inputs to outputs. The model is presented in \\figref{fig:model-diagram}-b color coded components in comments.} \\label{alg:training}\n\t\\begin{algorithmic}[1]\n\t\t\\Require{}\n\t\t\\Statex Strokes $\\mathbf{x} = \\{x_t\\}_{t=1}^{T}$\n\t\t\\Statex Labels ($\\mathbf{y, eoc, bow}) = \\{(y_t, eoc_t, bow_t)\\}_{t=1}^{T}$\n\t\t\\Statex $h_0^{inp} = h_0^{latent} = \\mathbf{0}$\n\t\t\\Ensure{}\n\t\t\\Statex Reconstructed strokes $\\mathbf{\\hat{x}}$, predicted $\\mathbf{\\hat{y}}$ and $\\mathbf{\\hat{eoc}}$ labels.\n\t\t\\State $t \\gets 1$\n\t\t\\While{$t= \\le T$}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Update $h_t^{inp} \\gets \\tau^{inp}(x_t, h_{t-1}^{inp})$ \\Comment{grey} \\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Sample $\\pi_t^q \\sim q(\\pi_t | x_{t})$ using \\refequ{eq:pi_t_q} (equivalent to the character prediction $\\hat{y}_t$), \\Comment{blue} \\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Select the corresponding GMM component and draw a content sample $\\varphi_t$ using \\refequ{eq:gmm_reparametrization_trick}, \\Comment{blue} \\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Estimate parameters of the isotropic Gaussian $q(z_{t} | x_{t})$ using \\refequ{eq:z_t_q} and sample $z_t^q \\sim q(z_{t} | x_{t})$, \\Comment{green} \\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Using $\\varphi_t$, $z_t^q$ and $bow_t$, reconstruct the stroke $\\hat{x_t}$ and predict $\\widehat{eoc_t}$, \\Comment{yellow}\\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Estimate isotropic Gaussian and Multinomial distribution parameters of prior latent variables $z_t^p$ and $\\pi_t^p$ by using \\refequs{eq:z_t_p}{eq:pi_t_p}, respectively, \\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Update $h_t^{latent} \\gets \\tau^{latent}(h_t^{inp}, z_t^q, \\varphi_t, h_{t-1}^{latent})$, \\strut}\n\t\t\\State $t \\gets t + 1$\n\t\t\\EndWhile\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Evaluate \\refequ{eq:total_loss} and update model parameters. \\strut}\n\t\\end{algorithmic}\n\\end{algorithm}\n\\vspace{-0em}\n\\subsubsection{Input RNN cell}\nOur model consists of two LSTM cells in the latent and at the input layers. Note that the latent cell is originally contributing via the transition function $\\tau^{latent}$ in \\refequ{eq:probability_cvrnn}. Using an additional cell at the input layer increases model capacity (similar to multi-layered RNNs) and adds a new transition function $\\tau^{inp}$. Thus the synthesis model can capture and modulate temporal patterns at the input levels. Intuitively this is motivated by the strong temporal consistency in handwriting where the previous letter influences the appearance of the current (cf. \\figref{fig:problem}).\n\nWe now use a temporal representation $h_t^{inp}$ of the input strokes $x_t$. With the cumulative modifications, our C-VRNN~ architecture becomes\n\\begin{align}\n\\label{eq:x_generation}\np(x_{t} | z_{t}, \\pi_{t}) &= g^{out}(z_t, \\varphi_t, bow_t), \\\\\n\\label{eq:z_t_p}\nz_t^p \\sim p(z_{t}) &= g^{p,z}(h_{t-1}^{latent}), \\\\\n\\label{eq:pi_t_p}\n\\pi_{t}^p \\sim p(\\pi_{t}) &= g^{p,\\pi}(h_{t-1}^{latent}), \\\\\n\\label{eq:h_t_inp}\nh_t^{inp} &= \\tau^{inp}(x_t, h_{t-1}^{inp}), \\\\\n\\label{eq:z_t_q}\nz_t^q \\sim q(z_{t} | x_{t}) &= g^{q,z}(h_t^{inp}, h_{t-1}^{latent}), \\\\\n\\label{eq:pi_t_q}\n\\pi_t^q \\sim q(\\pi_t | x_{t}) &= g^{q,\\pi}(h_t^{inp}, h_{t-1}^{latent}), \\\\\n\\label{eq:h_t_latent}\nh_t^{latent} &= \\tau^{latent}(h_t^{inp}, z_t, \\varphi_t, h_{t-1}^{latent}).\n\\end{align}\n\nFinally we train our handwriting model by using algorithm (\\ref{alg:training}) and optimizing the following loss:\n\\begin{align}\n\\begin{split}\n\\mathcal{L}(\\cdot) = \\mathcal{L}_{lb} + \\mathcal{L}_{classification} + \\mathcal{L}_{eoc}.\n\\label{eq:total_loss}\n\\end{split}\n\\end{align}\n\nIn our style transfer applications, by following the steps $1-11$ of algorithm (\\ref{alg:training}), we first feed the model with a reference sample and get the internal state of the latent LSTM cell $h^{latent}$ carrying style information. The sampling algorithm (\\ref{alg:sampling}) to generate new samples is then initialized with this $h^{latent}$.\n\nWe implement our model in Tensorflow \\cite{Abadi2016tensorflow}. The model and training details are provided in the Appendix. Code and dataset for learning and reproducing our results can be found at \\url{https:\/\/ait.ethz.ch\/projects\/2018\/deepwriting\/}.\n\n\\begin{algorithm}[t]\n\t\\caption{Sampling \\\\ The procedure to \\textit{synthesize} novel handwriting sequences by conditioning on content and style. The model is presented in \\figref{fig:model-diagram}-c and color coded components in comments.} \\label{alg:sampling}\n\t\\begin{algorithmic}[1]\n\t\t\\Require{}\n\t\t\\Statex Sequence of characters $\\mathbf{y} = \\{y_n\\}_{n=1}^{N}$ to be synthesized.\n\t\t\\Statex $h_0^{latent}$ is zero-initialized or inferred from another sample.\n\t\t\\Statex Probability threshold $\\epsilon$ for switching to the next character.\n\t\t\\Ensure{}\n\t\t\\Statex Generated strokes $\\mathbf{\\hat{x}}$ of the given text $\\mathbf{y}$.\n\t\t\\State $t \\gets 1$, $n \\gets 1$.\n\t\t\\While{$n \\le N$}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Estimate parameters of style prior $p(z_t)$ using \\refequ{eq:z_t_p} and sample $z_t^p$. \\Comment{green} \\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{$\\pi_t^p \\gets y_n$, select the corresponding GMM component and draw a content sample. $\\varphi_t$ using \\refequ{eq:gmm_reparametrization_trick}, \\Comment{blue} \\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{$bow_t \\gets 1$ if it is the first stroke of a new word, $0$, otherwise. \\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{Using $\\varphi_t$, $z_t^p$ and $bow_t$, synthesize a new stroke $\\hat{x_t}$ and predict $\\widehat{eoc_t}$, \\Comment{yellow}\\strut}\n\t\t\\State \\parbox[t]{\\dimexpr\\linewidth-\\algorithmicindent}{$n \\gets n + 1$ if $\\widehat{eoc}_t > \\epsilon$, \\Comment{next character} \\strut}\n\t\t\\State $t \\gets t + 1$\n\t\t\\EndWhile\n\t\\end{algorithmic}\n\\end{algorithm}\n\\vspace{-0em}\n\\subsection{Character recognition}\nDisentanglement of content requires character recognition of handwritten input. Our model uses the latent variable $\\mathbf{\\pi}$ to infer character labels which is fed by the input LSTM cell. In our experiments we observe that bidirectional recurrent neural networks (BiRNN) \\cite{schuster1997bidirectional} perform significantly better than standard LSTM models in content recognition task ($60\\%$ against $96\\%$ validation accuracy). BiRNNs have access to future steps by processing the input sequence from both directions. However, they also require that the entire input sequence must be available at once. This inherent constraint makes it difficult to fully integrate them into our training and sampling architecture where the input data is available one step at a time.\n\nDue to their desirable performance we train a separate BiRNN model to classify input samples. At sampling time we use its predictions to guide disentanglement. Note that technically $q(\\mathbf{\\pi} | \\mathbf{x})$ also infers character labels but the BiRNN's accuracy improves the quality of synthesis. We leave full integration of high accuracy character recognition for future work.\n\n\\section{The Handwriting Dataset}\\label{sec:dataset}\nMachine learning models such as ours rely on the availability of annotated training data. Prior work mostly relied on the IAM On-Line Handwriting Database (IAM-OnDB) \\cite{Liwicki2005}. However, this was captured with a low-resolution and low-accuracy digital whiteboard and only contains annotations at the sequence level. To disentangle content and style, more fine-grained annotations are needed. Hence, we contribute a novel dataset of handwritten text with character level annotations. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.8\\columnwidth]{figs\/dataset_levels-v2}\\\\ \\hspace{.5cm}\n\\includegraphics[width=.8\\columnwidth]{figs\/samples-dataset-divided}\n\\caption{ (Top) Our dataset offers different level of annotation, including sentence-wide annotation (A), as well as fine-grained segmentation at word (B) and character (C) level.\n(Bottom) Samples from our dataset, with colour-coded character segmentation. Different styles are available, including challenging styles to segment (e.g., joined-up cursive, right).}\n\\label{fig:dataset}\n\\end{figure}\n\nThe proposed dataset accumulates the IAM-OnDB dataset with newly collected samples. At time of writing, the unified dataset contains data from 294 unique authors, for a total of 85181 word instances (writer median 292 words) and 406956 handwritten characters (writer median 1349 characters), with further statistics reported in Table~\\ref{table:dataset_stats}. The data is stored using the JSON data-interchange format, which makes it easy to distribute and use in a variety of programming environments.\n\n\\begin{table}[h!]\n\t\\centering\n\t\\small\n\t\\begin{tabular}{l|l|l|l|}\n\t\t\\cline{2-4}\n\t\t&  IAM-OnDB &  Ours &  \\textbf{Unified}  \\\\ \\hline\n\t\t\\multicolumn{1}{|l|}{Avg. Age (SD) } & 24.84 ($\\pm$ 6.2) & 23.55 ($\\pm$ 5.7)  & \\textbf{24.85 ($\\pm$ 6.19)}  \\\\ \\hline\n\t\t\\multicolumn{1}{|l|}{Females \\%} & 34.00 & 32.63  & \\textbf{33.55}  \\\\ \\hline\n\t\t\\multicolumn{1}{|l|}{Right-handed \\%} & 91.50 & 96.84  & \\textbf{93.22}  \\\\ \\hline\n\t\t\\multicolumn{1}{|l|}{\\# sentences} & 11242 & 63182  & \\textbf{17560}  \\\\ \\hline\n\t\t\\multicolumn{1}{|l|}{\\# unique words} & 11059 & 6418  & \\textbf{12718}  \\\\ \\hline\n\t\t\\multicolumn{1}{|l|}{\\# word instances} & 59141 & 26040 &  \\textbf{85181} \\\\ \\hline\n\t\t\\multicolumn{1}{|l|}{\\# characters} & 262981 & 143975 & \\textbf{406956} \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{Data statistics. In bold, the final \\textit{unified} dataset.}\n\t\\label{table:dataset_stats}\n\\end{table}\n\\vspace{-0em}\n\nThe large number of subjects contained in our dataset allows to capture substantially large variation in styles, crucial to perform any handwriting-related learning task, since handwriting typically exhibits large variation both inter and intra subjects. Furthermore, the dataset contains samples from different digitization mechanisms and should hence be useful in learning models that are robust to the exact type of input.\n\nWe developed a web tool to collect samples from 94 authors (see \\figref{fig:dataset_collection_interface}). Inline with IAM-OnDB we asked each subject to write extracts of the Lancaster-Oslo-Bergen (LOB) text corpus \\cite{Stig1986} using an iPad Pro. Besides stylus information, we recorded age, gender, handedness and native language of each author. The data is again segmented at the character level and misspelled or unreadable samples have been removed.\n\nSamples from 200 authors in the dataset stem from the IAM-OnDB dataset and were acquired using a smart whiteboard.\nThe data provide stylus information (i.e., x-y coordinates, pen events and timestamps) as well as transcription of the written text, on a per-line basis (\\figref{fig:dataset}, A). We purged 21 authors from the original data due to low-quality samples or missing annotations. Furthermore, to improve the granularity of annotations we process the remaining samples, segmenting them down to the character level, obtaining ASCII labels for each character (\\figref{fig:dataset}, B and C). For segmentation we used a commercial tool~\\cite{MyScript2016} and manually cleaned-up the results.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.9\\columnwidth]{figs\/collection-interface-annotated-clipped.png}\n\\caption{Data collection interface. Users are presented with text to write (A) in the scrollable collection canvas (B). A progress bar (C) informs the user on the status. A writer can save, reject or correct samples using the toolbox buttons (D).}\n\\label{fig:dataset_collection_interface}\n\\end{figure}\n\n\n\n\n\\section{Application Scenarios}\\label{sec:apps}\nBy disentangling content from style, our approach makes digital ink truly editable. This allows the generation of novel writing in user-defined styles and, similarly to typed text, of seamless editing of handwritten text. Further, it enables a wide range of exciting application scenarios, of which we discuss proof-of-concept implementations.\n\n\\subsection{Conditional Handwriting Generation}\n\nTo illustrate the capability to synthesize \\emph{novel} text in a user-specific style we have implemented an interface that allows user to type text, browse a database of handwritten samples from different authors and to generate novel handwriting. The novel sequence takes the content from the typed text and matches the style to a single input sample (cf. video). This could be directly embedded in existing note-taking applications to generate personalized handwritten notes from typed text or email clients could turn typed text into handwritten, personalized letters. For demonstration we have synthesized extracts of this paper's abstract in three different styles (see~\\figref{fig:interface-synthesis}).\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\columnwidth]{figs\/abstract-many-styles-extracts}\n\\caption{Handwritten text synthesized from the paper abstract. Each sentence is ``written'' in the style of a different author. For full abstract, see Appendix.}\n\\label{fig:interface-synthesis}\n\\end{figure}\n\n\\subsection{Content Preserving Style Transfer}\n\n\n\nOur model can furthermore transfer \\emph{existing} handwritten samples to \\emph{novel} styles, thus preserving their content while changing their appearance. We implemented an interactive tablet application that allows users to recast their own handwriting into a selected style (see \\figref{fig:interface-transfer} for results and video figure for interface walk through). After scribbling on the canvas and selecting an author's handwriting sample, users see their strokes morphed to that style in real-time. Such solution could be beneficial for a variety of domains. For example, artist and comic authors could include specific handwritten lettering in their work, or preserving style during localization to a foreign language.\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[width=0.8\\columnwidth,trim={0pt 950pt 0pt 0pt},clip]{figs\/transfer-results}\n\\caption{Style transfer. The input sequence (top) is transferred to a selected reference style (black ink, dotted outlines). The results (blue ink, solid outline) preserve the input content, and its appearance matches the reference style.}\n\\label{fig:interface-transfer}\n\\vspace{-7pt}\n\\end{figure}\n\n\\paragraph{Beautification}\nWhen using the users own input style as target style, our model re-generates smoother versions of the original strokes, while maintaining natural variability and diversity. Thus obtaining an averaging effect that suppresses local noise and preserves global style features. The resulting strokes are then beautified (see \\figref{fig:exp_result} and video), in line with previous work that solely relied on token averaging for beautification (e.g., \\cite{Zitnick2013}) or denoising (e.g., \\cite{Buades2005}).\n\n\\subsection{Word-level Editing}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{figs\/edit-vert}\n\t\\caption{Our model allows editing of handwritten text at the word level. A) Handwriting is recognized, with each individual word fully editable. B) Edited words are synthesized and embedded in the original text, preserving the style.}\n\t\\label{fig:edit}\n\\end{figure}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{figs\/spell-check-vert.png}\n\t\\caption{Our spell-checking interface. A) Spelling and grammar mistakes are detected and highlighted directly on the handwriting. Alternative spelling is offered (red box). B) Corrected words are synthesized and embedded in the original text (blue ink), preserving the writer style. }\n\t\\label{fig:interface-spellcheck}\n\\end{figure}\n\nAt the core of our technique lies the ability to edit digital ink at the same level of fidelity as typed text, allowing users to change, delete or replace individual words. \\figref{fig:edit} illustrates a simple prototype allowing users to edit handwritten content, while preserving the original style when re-synthesizing it.\nOur model recognizes individual words and characters and renders them as (editable) overlays. The user may select individual words, change the content, and regenerate the digital ink reflecting the edits while maintaining a coherent visual appearance. We see many applications, for example note taking apps, which require frequent edits but currently do not allow for this without loosing visual appearance.\n\n\\paragraph{Handwriting Spell-checking and Correction}\n\n\n\nA further application of the ability to edit digital ink at the word level is the possibility to spell-check and correct handwritten text. As a proof of concept, we implemented a functional handwriting spell-checker that can analyze digital ink, detect spelling mistakes, and correct the written samples by synthesizing the corrected sentence in the original style (see \\figref{fig:interface-spellcheck} and video figure). For the implementation we rely on existing spell-checking APIs, feeding recognized characters into it and re-rendering the retrieved corrections.\n\n\\section{Preliminary User Evaluation}\\label{sec:evaluation}\n\nSo far we have introduced our neural network architecture and have evaluated its capability to synthesize digital ink. We now shift our focus on initially evaluating users' perception and the usability of our method. To this end, we conducted a preliminary user study gathering quantitative and qualitative data on two separate tasks. Throughout the experiment, 10 subjects ($M=27.9$; $SD=3.34$; 3 female) from our institution evaluated our model using an iPad Pro and Apple Pencil.\n\n\\paragraph{Handwriting Beautification}\nThe first part of our experiment evaluates text beautification. Users were asked to compare their original handwriting with its beautified counterpart. Specifically, we asked our subjects to repeatedly write extracts from the LOB corpus \\cite{Stig1986}, for a total of 12 trials each. In each trial the participant copied down the sample and we beautified the strokes with the results being shown side-by-side (see \\figref{fig:exp_result}, top). Users were then asked to rate the aesthetics of their own script (Q: \\textit{I find my own handwriting aesthetically pleasing}) and the beautified version (Q: \\textit{I find the beautified handwriting aesthetically pleasing}), using a 5-point Likert scale. Importantly these were treated as independent questions (i.e., users were allowed to like both).\n\n\\paragraph{Handwriting Spell-Checking}\nIn the second task we evaluate the spell-checking utility (see \\figref{fig:interface-spellcheck}). We randomly sampled from the LOB corpus and perturbed individual words such that they contained spelling mistakes. Participants then used our tool to correct the written text (while maintaining it's style), and subsequently were asked to fill in a standard System Usability Scale (SUS) questionnaire and take part in an exit interview.\n\n\\paragraph{Results}\n\nOur results, summarized in~\\figref{fig:exp_result} (bottom), indicate that users' reception of our technique is overall positive. The beautified strokes were on average rated higher ($M=3.65$, 95\\% CI [3.33-3.97]) with non overlapping confidence intervals. The SUS results further supports this trend, with our system scoring positively ($SUS=85$). Following the analysis technique suggested in~\\cite{Lewis2009}, our system can be classified as Rank A, indicating that users are likely to recommend it to others.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{figs\/exp_interface-wider} \\\\\n\t\\includegraphics[width=\\columnwidth]{figs\/exp_res_flat.jpg}\n\t\\caption{Task 1. Top: Experimental Interface. Participants input on the left; beautified version shown on the right. Bottom: Confidence interval plot on a 5-point Likert scale.}\n\t\\label{fig:exp_result}\n\\end{figure}\n\nThe above results are also echoed by participants' comments during the exit interviews (e.g., \\textit{I have never seen anything like this}, and \\textit{Finally others can read my notes.}). Furthermore, some suggested additional applications that would naturally fit our model capabilities (e.g., \\textit{This would be very useful to correct bad or illegible handwriting}, \\textit{I can see this used a lot in education, especially when teaching how to write to kids} and \\textit{This would be perfect for note taking, as one could go back in their notes and remove mistakes, abbreviations and so on}). Interestingly, the ability to preserve style while editing content were mentioned frequently as the most valued feature of our approach (e.g., \\textit{Having a spell-checker for my own handwriting feels like writing personalized text messages!}).\n\n\\section{Conclusion and Future Work}\\label{sec:discussion}\nWe have proposed a novel approach for making digital ink editable, which we call conditional variational recurrent neural networks (C-VRNN~). At the core of our method lies a deep NN architecture that disentangles content from style. The key idea underlying our approach is to treat style and content as two separate latent random variables with distributions learned during training. At sampling time one can then draw from the content and style component to \\emph{edit} either style, content or both, or one can \\emph{generate} entirely new samples. These learned statistical distributions make traditional approaches to NN training intractable, due to the lack of access to the true distributions. Moreover, to produce realistic samples of digital ink the model needs to perform auxiliary tasks, such as controlling the spacing in between words, character segmentation and recognition. Furthermore, we have build a variety of proof-of-concept applications, including conditional synthesis and editing of digital ink at the word level. Initial user feedback, while preliminary, indicates that users are largely positive about the capability to edit digital ink - in one's own handwriting or in the style of another author.\n\nTo enable the community to build on our work we release our implementation as open-source.\nFinally, we have contributed a compound dataset, consolidating existing and newly collected handwritten text into a single corpus, annotated to the character level. Data and code are publicly available \\footnote{\\url{https:\/\/ait.ethz.ch\/projects\/2018\/deepwriting\/}}.\n\nWhile our model can create both disconnected and connected (cursive) styles, its performance is currently better for the former, simpler case. This also applies to most state-of-the-art character recognition techniques, and we leave extending our method to fully support cursive script for future work. Further, we are planning to integrate our currently auxiliary character recognition network into the proposed architecture. One interesting direction in this respect would be the inclusion of a full language model. Finally, and in part inspired by initial feedback, we believe that the underlying technology bears a lot of potential for research in other application domains dealing with time-series, such as motion data (e.g., animation, graphics) or sketches and drawings (e.g., arts and education).\n\n\\section{Appendix}\n\\label{sec:appendix}\n\n\\subsection{Alphabet}\nWe use the following numbers, letters and punctuation symbols in our alphabet: \\\\\n$0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJK \\\\ LMNOPQRSTUVWXYZ'.,-()\/$\n\\subsection{Data Preprocessing}\n\\begin{enumerate}\n\\item In order to speed up training we split handwriting samples that have more than $300$ strokes into shorter sequences by using $eoc$ labels so that the stroke sequences of letters remain undivided. We create $705$ validation and $34577$ training samples with average sequence length 261.012 ($\\pm$ 85.1).\n\\item Pixel coordinates ($u_0$, $v_0$) of the first stroke is subtracted from the sequence such that each sample starts at the origin $(0,0)$. \n\\item By subtracting $x_{t}$ from $x_{t+1}$, we calculate the changes in 2D-coordinate space and use relative values in training. Note that the $pen-up$ events remained intact.\n\\item Finally, we apply zero-mean unit-variance normalization on 2D-coordinates by calculating \\textit{mean} and \\textit{std} statistics on the whole training data.\n\\end{enumerate}\n\n\\subsection{Network Configuration}\nBoth $\\tau^{inp}$ and $\\tau^{latent}$ are LSTM cells with $512$ units. Similarly, feed-forward networks $g^*$ consists of $1$-layer with $512$ units and \\textit{ReLu} activation function. We use $32$-dimensional isotropic Gaussian distributions for latent variables $z^p$, $z^q$ and for each GMM component. \n\nOur BiRNN classifier consists of $3$-layer bidirectional LSTM cells with $512$ units. A $1$-layer fully connected network with $256$ units and \\textit{ReLu} activation function takes BiRNN representations and outputs class probabilities. \n\n\\subsection{Training}\nWe use ADAM optimizer with default parameters. Learning rate is initialized with $0.001$ and decayed exponentially with a rate of $0.96$ after every $1000$ mini-batches. We train our model for $200$ epochs by using a mini-batch size of $64$.\n\n\\subsection{Additional Results}\nAs an additional result, we have synthesized the entire abstract from this paper using a different style per sentence. The result is shown in \\figref{fig:interface-synthesis-full}, and illustrates how our model is able to generate a large variety of styles.\n\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.99\\linewidth,trim={5pt 250pt 0pt 0pt},clip]{figs\/abstract-many-styles-v2-dotted}\n\t\\caption{Handwritten text synthesized from the abstract of this paper. Each sentence is ``written'' in the style of a different author.}\n\t\\label{fig:interface-synthesis-full}\n\\end{figure}\n\\section{Acknowledgements}\\label{sec:ack}\nThis work was supported in parts by the ERC grant OPTINT\n(StG-2016-717054). We thank all the participants for their time and efforts in taking part in our experiments.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n    Cellular informations are stored in the genome; cellular\nfunctions, on the other hand, are carried out by biochemistry.  The\ncommunications within and between cells are based on biochemical\nreactions involves enzyme molecules \\cite{rob_book}.  In particular,\nreversible chemical modification of enzymes, via phosphorylation-\ndephosphorylation cycle (PdPC), is one of the most widely used\nmechanisms in cellular signaling and regulation\n\\cite{fischer_kreb_55}.  In its simplest, canonical form, an enzyme\n$E$ can be phosphorylated to become $E^*$, and then dephosphorylated\nto be back to $E$.  These two reactions are catalyzed by their\nrespective enzymes called kinase and phosphatase.  Biologically, the\n$E$ form of the enzyme is inactive while the $E^*$ form is active.\nHence, a change in the concentation of the kinase (or of the\nphosphatase) leads to a change in the activity of the enzyme\n\\cite{qian_arpc_07,qian_book}.\n\n    Mathematical modeling of the PdPC kinetics began\nin the late 1970s \\cite{stadt_chock_77}.  In 1981,\nGoldbeter and Koshland discovered a surprising result:\nIf both kinase and phosphatase, acting as enzymes for\ntheir respective reactions, are operating under a saturated\ncondition, then the change of $E\\rightarrow E^*$ can be\nextreme sharp in response to a change in the kinase\nconcentration.  They called this phenomenon zeroth-order\nultrasensitivity \\cite{gk_pnas_81}.  Before that work,\nbiochemists had believed a sharp, sigmoidal transition is\nthe signature of only enzyme systems with multiple\nsubunits or monomer with  multiple binding sites;\nsigmoidal shaped transition is associated with equilibrium\ncooperativity.  What had not been fully appreciated until recent\nyears is that the PdPC is not a closed equilibrium system, but\nin fact an open-chemical system \\cite{qian_arpc_07}.\n\n\n    There is now a sizable literature on\nultrasensitive PdPC and related biochemical reactions.\nSee \\cite{gunawardena09,huang09} for two very recent reports.\nBut all the studies, as far as we know, have been\nlimited to the steady state analysis, with the exception\nof \\cite{wang_02}.   The present\npaper reports a study of the time-dependent behavior\nof ultrasensitive PdPC.  We discover that in addition\nto the sharp transition, the kinetic of ultrasensitive PdPC\nexhibit a critical slowdown, i.e., the time to reach\nsteady state tends to infinit when the system\napproaches the the critical transition point.\nIn a separated study, we have also discovered that\nat the critical point of transition, the ultrasensitive\nPdPC exhibits large fluctuations \\cite{qian_cooper_08,qian_bpc_03}.\n\n    Putting all these together, this is a reminiscent\nof the critical phenonemon\nin physics \\cite{domb_book,stanley_rmp_99}.  The classical\nphase transition problems in condensed matter physics,\nhowever, are equilibrium phenomena.  The ultrasensitivity\ndiscussed in the present work is a nonequilibrium\nphenomenon.  PdPC is a driven chemical reaction system\n\\cite{qian_arpc_07}.\n\n\n\n\n\\section{Phosphorylation-Dephosphorylation Cycle Kinetics}\n\nThe canonical kinetic system for the PdPC is\n\\cite{gk_pnas_81,bnk_04,gunawardena}\n\\begin{equation}\n\\label{the_rxn}\n    E + K \\overset{k_{11}}{\\underset{k_{-11}}\n        {\\rightleftharpoons}} EK\n    \\overset{k_{12}}{\\longrightarrow} E^* + K, \\ \\ \\\n    E^* + P \\overset{k_{21}}{\\underset{k_{-21}}\n        {\\rightleftharpoons}} E^*P\n    \\overset{k_{22}}{\\longrightarrow} E + P,\n\\end{equation}\nin which $K$ and $P$ represent kinase and phosphatase.\nWe follow the Michaelis-Menten approach to model\nthe enzyme reactions \\cite{murray_book,qian_book}.\nIf the amount of kinase is significantlly smaller\nthan that of $E$, and the amount of phosphatase is\nsignificantly smaller than that of $E^*$, then according\nto the Michaelis-Menten theory of enzyme kinetics, we\nhave \\cite{murray_book,qian_book}\n\\begin{equation}\n\\label{the_eq}\n    \\frac{dc_{E^*}}{d\\tau} = \\frac{V_1 c_E }{K_{M1}+c_E}\n    - \\frac{V_2 c_{E^*}}{K_{M2}+c_{E^*}},\n\\end{equation}\nin which the total concentration $c_E+c_{E^*}=c_T$ is\na constant.  Here $c_E$ and $c_{E^*}$ are the\nconcentrations of $E$ and $E^*$ respectively.\nThe parameters in Eq. (\\ref{the_eq}) are related to\nthe rate constants in (\\ref{the_rxn}):\n\\begin{equation}\n    K_{Mi} = \\frac{k_{-i1}+k_{i2}}{k_{i1}}, \\ \\ (i=1,2); \\ \\\n    V_1 = k_{12}[K]_{tot}, \\ \\ V_2 = k_{22}[P]_{tot},\n\\end{equation}\nin which $[K]_{tot}$ and $[P]_{tot}$ are the total concentrations\nof kinase and phosphatase, respectively.\n\n    Equation (\\ref{the_eq}) can be rigorously\njustified via singular perturbation theory \\cite{murray_book,qian_book}.\nIt is valid for all times, as long as one can neglects the extremely\nfast kinetics.  Eq. (\\ref{the_eq}) in fact is exact for finding the\nsteady state of the reaction system in (\\ref{the_rxn}). The\napproximation comes from $c_E+c_{E^*}=c_T$ since it neglects the\nconcentrations of $EK$ and $E^*P$.\n\n\n     We introduce nondimensionalized variables\n\\begin{equation}\n    x = \\frac{c_{E^*}}{c_T}, \\ \\ t = \\frac{V_2}{c_T}\\tau,\n\\end{equation}\nthen we have\n\\begin{equation}\n\\label{the_eq1}\n    \\frac{dx}{dt} = \\frac{\\theta (1-x)}\n            {K_1+1-x}\n        - \\frac{x}{K_2+x},\n\\end{equation}\nwhere\n\\begin{equation}\n   \\theta = \\frac{V_1}{V_2}=\\frac{k_{12}[K]_{tot}}{k_{22}[P]_{tot}}, \\ \\\n    K_1 = \\frac{K_{M1}}{c_T}, \\ \\\n    K_2 = \\frac{K_{M2}}{c_T}.\n\\end{equation}\nEq. (\\ref{the_eq1}) is the starting point of our\ncurrent study.   The steady state $x^*$ as a function of\n$\\theta$ and $K$s is well understood \\cite{gk_pnas_81,qian_book}.\nWe investigate its time-dependent behavior.\n\n\n\\section{A Simple Mathematical Model for Ultrasensitivity}\n\n    The two terms on the right-hand-side of Eq. (\\ref{the_eq1})\nrepresent the {\\em Michaelis-Menten rate law} of enzyme\ncatalyzed reactions, for the kinase and the phosphatase,\nrespectively.  Taking the term $r=x\/(K_2+x)$ as example, it\nhas the celebrated ``hyperbolic'' saturation form, also known\nas double reciprocal since $1\/r$ is a linear function of $1\/x$\n\\cite{qian_book}.  Therefore, the rate $r$ as a function of\nthe substrate concentration $x$ is linear, i.e., first-order,\nwhen $K_2\\gg x$, but zeroth order when $K_2\\ll x$.\nWhen an enzyme is operating under zeroth order condition, the\ncatalyzed reaction rate is independent of the substrate\nconcentration.\n\n    The important discovery of Goldbeter and Koshland\n\\cite{gk_pnas_81} is that when both kinase and phosphatase\nin (\\ref{the_rxn}) are operating under zeroth order condition,\nthe steady state concentration of phosphorylated enzyme, i.e.,\n$x$ has a very sharp response to the $\\theta$, the activation\nparameter.\n\n    One can solve the steady state of Eq. (\\ref{the_eq1}),\nas a root of a quadratic equation, then let $K_1$ and $K_2\\rightarrow\n0$.  This is the standard method of attack.  However, if one lets\n$K_1=K_2=0$, one immediately obtains\n\\begin{equation}\n\\label{singular_eq}\n    \\frac{\\theta (1-x)}{1-x} = \\frac{x}{x}.\n\\end{equation}\nBeing mathematically careful with Eq. (\\ref{singular_eq}),\none has\n\\begin{equation}\n    x=0; \\ \\textrm{ or } \\\n    x=1; \\  \\textrm{ or} \\ \\\n    \\theta = 1, \\  \\textrm{ if } x\\neq 0, x\\neq 1.\n\\end{equation}\nNoting that $x$ is a monotonic increasing function of $\\theta$ in\nEq. (\\ref{the_eq1}), we have steady state $x$ as a function\nof $\\theta$ given in Fig. \\ref{fig1}A.\n\n\n\\begin{figure}[t]\n\\[\n\\includegraphics[height=5in,angle=-90]{fig1.eps}\n\\]\n\\caption{The simple zeroth-order, ultrasensitivity transition in a\nnutshell: $x$ represents in the fraction of enzyme being\nphosphorylated; and $\\theta$ is the ratio of the activities of the\nkinase and the phosphatase. (A) The activation curve, steady state\n$x$ as a function of $\\theta$, shows a sharp transition. (B) The\ntime-dependent $x(t)$ is a linear function of time $t$, not\nexponential as usually expected from first-order chemical kinetics.\nIt reaches steady state in finite time $t_{\\infty}$. (C) The time to\nreach the steady state, $t_{\\infty}$, exhibit a critical slowdown\nnear the midpoint of the transition, when $\\theta=1$. (D) summarizes\nthe key mathematical results associated with Eq. \\ref{unper_eq}. }\n\\label{fig1}\n\\end{figure}\n\n    This way of setting $K_1=K_2=0$ is in fact the\nspirit of {\\em singular perturbation}\nmethod \\cite{murray_book}.  Following the same\napproach and treating $K_1=K_2=\\epsilon$ as a\nsmall parameter, we have the unperturbed\nequation\n\\begin{equation}\n\\label{unper_eq}\n    \\frac{dx}{dt} = \\theta - 1,  \\ \\ \\\n    (0< x< 1)\n\\end{equation}\nThe time-dependent solution to the equation is\n\\begin{equation}\n\\label{the_td_sol}\n    x(t) = (\\theta-1)t + x(0), \\ \\\n    \\textrm{ for } \\ \\ 0\\le x\\le 1.\n\\end{equation}\nThis is shown in Fig. \\ref{fig1}B.\nThe most interesting insight from this simple\nsolution is that the time reaching the steady state,\n$t_{\\infty}$, is finite when $\\theta\\neq 1$.\nFurthermore, the time\n\\begin{equation}\n\\label{the_time}\n    t_{\\infty} = \\left\\{\\begin{array}{lc}\n        \\frac{x(0)}{1-\\theta} & \\theta < 1,\n        \\\\[7pt]\n        \\frac{1-x(0)}{\\theta-1}  & \\theta > 1.\n\\end{array}\\right.\n\\end{equation}\nThis is shown in Fig. \\ref{fig1}C.\n\n    The $\\theta=1$ is known as the mid-point of\ntransition from unphosphorylated $E$ to the phosphorylated $E^*$.\nIt is the ``critical point'' of the transition.  A\nshape transition like in Fig. \\ref{fig1}A is usually identified as\ncooperative phase transition in condensed matter physics.   The\nresult on the time in Eq. (\\ref{the_time}) and Fig. \\ref{fig1}C\nfurther collaborate this analogue: One of the characteristics of\nphase transition is the ``critical slowdown'' in its dynamics\n\\cite{stanley_rmp_99}.   In the theory of phase transition, critical\nslowdown and shape transition are also intimately related to large\nfluctuations \\cite{stanley_rmp_99}.  This last aspect of PdPC has\nbeen discussed in \\cite{qian_bpc_03,qian_cooper_08}.\n\n    The easiness and transparency of Eq. (\\ref{unper_eq})\nmakes it very attractive as a simple model for ultrasensitivity. The\nremaining of the paper is to provide a more rigorous justification\nof the results obtained from Eq. (\\ref{unper_eq}), summarized in\nFig. \\ref{fig1}D, in terms of a perturbation expansion approach.\nMore specifically, we shall try to provide perturbation corrections\nto the results in Figs. \\ref{fig1}B and \\ref{fig1}C.\n\n\\section{Exact Solution to PdPC Kinetics Eq. (\\ref{the_eq1})}\n\n    To justify the above very simple model for ultrasensitivity,\none can in fact solve exactly the nonlinear differential equation\nin (\\ref{the_eq1}).  For simplicity, we shall assume $K_1=K_2=K$.\nThe Eq. (\\ref{the_eq1}) can be re-written as\n\\begin{equation}\n\\label{the_eq2}\n    \\frac{dx}{dt} = \\frac{(1-\\theta)x^2+(\\theta-\\theta K\n        -K -1)x+\\theta K}{(K+1-x)(K+x)},\n\\end{equation}\nthe numerator of whose right-hand-side has two roots\n$x_1$ and $x_2$:\n\\begin{equation}\n    x_{1,2} = \\frac{-\\theta+\\theta K+K+1\\pm\\sqrt{\n    (1+2K)(\\theta-1)^2+K^2(\\theta+1)^2}}{2(1-\\theta)}.\n\\end{equation}\nOne of the $x_1$ and $x_2$ is $\\in [0,1]$, and it is the\nsteady state of the Eq. (\\ref{the_eq1}).  Let us denote it\nby $x^*$,  then the other one is $x' = \\theta K\/[x^*(1-\\theta)]$.\nIf $0\\leq\\theta\\leq 1$, $x_2\\in [0,1]$, and it is the steady state of\nthe Eq. (\\ref{the_eq1}). We also see that $K\\rightarrow\n0,x_2\\rightarrow 0$. If $\\theta\\geq 1$, $x_1\\in [0,1]$, and we have\n$K\\rightarrow 0,x_1\\rightarrow 1$.\n\n    Applying separation of variables and the method of\npartial fraction to Eq. (\\ref{the_eq2}), we have the solution\nto the Eq. (\\ref{the_eq1})\n\\begin{equation}\n    t =\\frac{1}{1-\\theta}\n    \\left\\{ x(0)-x(t) + \\frac{K}{1-\\theta}\\ln\\left[\n        \\left(\\frac{x(t)-x^*}{x(0)-x^*}\\right)^A\n        \\left(\\frac{x(t)-x'}{x(0)-x'}\\right)^B \\right]\n        \\right\\}.\n\\label{the_sol}\n\\end{equation}\nin which,\n\\begin{eqnarray*}\n&&  A = \\frac{1+K-\\theta K - (1+\\theta)x^*}{x^*-x'},\n\\\\[7pt]\n&&  B = -\\frac{1+K-\\theta K- (1+\\theta)x'}{x^*-x'}.\n\\end{eqnarray*}\n\n\n    From Eq. (\\ref{the_sol}), it is immediately clear that\nif $K=0$, then $x = (\\theta-1)t + x(0)$.  This is the result in\nEq. (\\ref{the_td_sol}).  Furthermore, we can compute the ``time to\nsteady state''.   If $K\\rightarrow 0$, $t\\rightarrow (x_0-x(t))\/(1-\\theta)$.\nHence for $x(t_{\\infty})=0$, $t_{\\infty}=\\frac{x(0)}{1-\\theta}$,\nand for $x(t_{\\infty})=1$, $t_{\\infty}=\\frac{x(0)-1}{1-\\theta}$.\nThese are the results in Eq. (\\ref{the_time}).\n\n    For $K\\neq 0$, mathematically exponential decay takes\ninfinite time to actually reach the steady state.  To give a\nmeasure of the time, however, we consider the time from\n$x(0)=x^*+0.01$ to $x(t_{\\infty})=x^*+0.001$, for $0\\leq \\theta< 1$.\nSimilarly, for $\\theta > 1$, we consider the time from $x(0)=x^*-0.01$\nto $x(t_{\\infty})=x^*-0.001$.\n\n    Fig. \\ref{fig2} shows the $t_{\\infty}$ as a function\nof $\\theta$, with various $K$ values.   It is shown that with\nsufficiently small $K$, the result in Eq. (\\ref{the_time}) is indeed\nvalid; There is a critical slowdown at the $\\theta=1$ for zeroth\norder ultrasensitivity with small $K$.\n\n\n\\begin{figure}[t]\n\\[\n\\includegraphics[height=4in,width=3.25in,angle=-90]{fig2.eps}\n\\]\n\\caption{Time to reach steady state ($x^*)$, defined as from initial\nvalue $x(0)=x^*\\pm 0.01$ to $x(t_{\\infty})=x^*\\pm 0.001$. $\\pm$\ncorresponds to $\\theta<1$ and $>1$, respectively. The values for $K$\nare 10 (open circles), 1 (open squares), 0.1 (filled circles), 0.01\n(filled triangle), and 0.001 (filled square).  The solid line is\nfrom Eq. (\\ref{the_time}).} \\label{fig2}\n\\end{figure}\n\n    We can also consider the time-dependent solution given\nin Eq. (\\ref{the_sol}), $x(t)$.  Fig. \\ref{fig1}B\nshows simple linear functions of time, each of which reaches steady\nstate in finite time.   To better understand this observation,\nFig. \\ref{fig3} shows that the $x(t)$\nasymptotically approaches to Eq. (\\ref{the_td_sol})\nwith diminishing $K$.\n\n\n\n\\begin{figure}[t]\n\\[\n\\includegraphics[height=3.5in,angle=-90]{fig3.eps}\n\\]\n\\caption{Time dependence of PdPC kinetics.  $x$ represents\nthe fraction of substrate enzyme being phosphorylated ($c_{E^*}\/c_T)$.\nAll curves have initial value $x(0)=0.6$ and $\\theta=0.7$.\nDifferent curves correspond to\ndifferent $K$ values: solid lines from top to bottom\ncorrespond to $K = 1$, 0.1, 0.05, and 0.01, respectively.\nThe dashed line is from Eq. (\\ref{the_td_sol}). With\ndiminishing $K$, the curves asymptotically approaches to\nthe dashed line. }\n\\label{fig3}\n\\end{figure}\n\n\n\n\n\\section{Discussion}\n\n\tIn developing a mathematical model for a scientific\nproblem, one should always follow Occam's Razor which states, in\nAlbert Einstein's words ``as simple as possible, but not simpler.''\nThe present paper contains two results:  First, it proposes\nan extremely simple equation for the zeroth-order PdPC, and provides a\nmathematical proof that the simple model captures all the essential\nfeatures of the well-accepted, but more complex model.\nSecond, we analyzed the time-dependent behavior of zeroth-order\nultrasensitivity and demonstrated a critical slowdown at the\ntransition point; Thus it is a critical phenomenon.\n\n    Critical phenomena are the emergent properties of\ncooperative processes \\cite{domb_book,hct,stanley_rmp_99}.\nCalling the zeroth-order\nultrasensitivity a critical phenomenon, thus, begs an answer to\nthe question ``what is the cooperativity in ultrasensitivity''?\nAnswer to this deeper question is outside the scope of the\npresent paper; we refer the readers to two recent papers\non the so-called {\\em temporal cooperativity}\n\\cite{qian_cooper_08,gehao_08}. As a matter of fact, a\nnonequilibrium critical phenomenon in a driven system\nemerging from a cooperativity in {\\em time} seems to be a\nnovel concept offered by the present mathematical model of the\nbiological system.\n\n\\section{Acknowledgement}\n\n\tWe thank Professors Chao Tang, Zhi-Xin Wang\nand Jianhua Xing for interesting discussions.\nThe first author is a receipant of a scholarship from the\nChina Scholarship Council.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\setcounter{equation}{0}\n\nReconstruction of an image from its Radon projections is the \ncentral theme in x-ray tomography and has spectacular \napplications in medical imaging. Mathematically the problem \nis to find a good approximation to a function based on a finite\ncollection of its Radon projections (see, for example, \n\\cite{He,KS,N}).\n\nThe main topic of this paper is the connection between the \nRadon transform and the orthogonal expansion of the function\non a unit ball. This connection was initiated in the classical \npaper \\cite{Co} with an inversion formula of the Radon transform \nbased on spherical harmonic expansions. The relation between \nthe Radon transform of a function, supported on the unit ball, \nand its orthogonal expansion was studied or used in \n\\cite{Da,DaG,LS,Lo, Ma, Marr}, among others (see \\cite{N} for \nfurther references). The papers \\cite{Da, DaG, Lo} studied also the \nsingular value decomposition (SVD) of the Radon transform \nusing an orthogonal basis. Since then SVD has become an \nimportant tool for studying the stability of the inversion \nproblem, the resolution of the reconstruction, and the\nincomplete data problem; see, for example, \\cite{CB, DaG, \nLo2, Maa, N}. The truncated SVD also provides an algorithm\nfor reconstruction of images. Because of the complicated \nformulas involved in the orthogonal or SVD expansions (see, \nfor example, \\cite{Da,Lo,N}), the algorithms did not seem to \nbe used in practical applications.\n\nRecently a new reconstruction algorithm was proposed in \n\\cite{X05} and further studied in \\cite{XT,XTC}. The new \nalgorithm is called OPED, as it is based on orthogonal \npolynomial expansion on the unit disk. The algorithm\nreproduces polynomials of high degrees and allows a fast \nimplementation (\\cite{XT}). The numerical tests shows \nthat the algorithm is fast, stable, and produces high quality\nimages (\\cite{XT, XTC}. The key ingredient  for deriving the \nalgorithm is the following formula for the  partial sum \n$S_{2m} f$ of the orthogonal expansion of $f$ on the \nunit disk, \n\\begin{equation}\\label{1stS2m}\n   S_{2m} f(x,y) = \\frac{1}{2m+1}\\sum_{\\nu=0}^{2m} \n       \\int_{-1}^1 {\\mathcal R}_{\\phi_\\nu} f(t)\n  \\Phi_{2m}(t, x \\cos \\phi_\\nu+y \\sin \\phi_\\nu)dt,\n\\end{equation}\nwhere  $\\phi_\\nu = \\frac{2 \\nu \\pi}{2m+1}$ and \n${\\mathcal R}_\\theta f(t)$ is the Radon projection on the line \n$x \\cos \\theta +  y \\sin \\theta = t$ (see Section 3). It turns\nout that there is a natural extension of this formula to the \nunit ball of higher dimension, which shows that the \northogonal polynomial expansion of $f$ can be expressed \nin terms of the Radon transforms and allows us to extend\nthe OPED algorithm in the unit ball of ${\\mathbb R}^d$.  Furthermore, \nthere is a close relation between SVD and the extension of \nthe formula \\eqref{1stS2m}. In fact, they can be brought \ntogether by the use of a compact formula of the reproducing \nkernel of orthogonal polynomials in \\cite{X99}.  The \northogonal expansion on the unit ball has been studied \nrecently in \\cite{X99,X05a},  which can be used, in particular, \nto derive the uniform convergence of the algorithms. \n\nThe purpose of this paper is two folds. Firstly we will\nclarify the relation between orthogonal expansion on the ball\nand the Radon projections and derive the extension of the \nOPED algorithm in higher dimensions. Secondly, we will \nexplain the connection between SVD of the Radon transform\nand orthogonal expansions. In particular, we shall show that \nusing truncated SVD to reconstruct the image is the same \nas using OPED algorithm. \n\nThe paper is organized as follows. The following section \ncontains a succinct account of the basic results on \northogonal polynomials on the unit ball. The orthogonal \nexpansions in terms of the Radon projections is developed \nin Section 3. The extension of the OPED algorithms and \na convergence result are given in Section 4. Finally, the \nSVD of the Radon transform is discussed in Section 5. \n\n\\section{Preliminaries on orthogonal polynomials}\n\\setcounter{equation}{0}\n\nLet $B^d:= \\{x: \\|x\\| \\le 1\\}$ and $S^{d-1}:=\\{x:\\|x\\| =1\\}$ be\nthe unit ball and the unit sphere of ${\\mathbb R}^d$, respectively. We \ndenote the surface area of $S^{d-1}$ by $\\sigma_d$ and the \nvolume of $B^d$ by $b_d$. Then\n$$\n    \\sigma_d = \\frac{2 \\pi^{d\/2}}{\\Gamma(d\/2)} \n         \\qquad \\hbox{and} \\qquad \n    b_d = \\frac{\\sigma_d}{d} = \\frac{\\pi^{d\/2}}{ \\Gamma((d+2)\/2)}.\n$$\n\n{\\it Inner product on the ball.} \nFor later discussion let us introduce a weight function $W_\\mu$ on \nthe unit ball, \n$$\n    W_\\mu(x) = (1-\\|x\\|^2)^{\\mu -1\/2}, \\qquad x\\in B^d. \n$$\nThe inner product on the unit ball is denoted by \n$$\n \\langle f, g \\rangle_{L^2(B^d)}  = \n      a_{\\mu} \\int_{B^d} f(x) g(x) W_\\mu(x) dx \n$$\nwhere $a_{\\mu}$ is the normalization constant of $W_\\mu$, \nthat is, $a_{\\mu} = 1\/\\int_{B^d} W_\\mu(x) dx$. For $\\mu =1\/2$, \nit is the unit weight (Lebesgue measure) and $a_\\mu$ is equal to \n$b_d^{-1}$. We will mainly work with the Lebesgue measure, so \nthe inner product $\\langle f, g \\rangle_{L^2(B^d)}$ should be \nregarded as with $\\mu = 1\/2$ unless specified otherwise.\n\n\\medskip\n\n{\\it Polynomial spaces.} Let $\\Pi_n^d$ denote the space of \npolynomials of degree $n$ in \n$d$ variables. We say that $P \\in \\Pi_n^d$ is an orthogonal \npolynomial on $B^d$ if $\\langle P, Q \\rangle_{L^2(B^d)} =0$ for \nall $Q \\in \\Pi_{n-1}^d$. Let ${\\mathcal V}_n^d$ denote the space of \northogonal polynomials. It is well-known that \n$$\n  \\dim \\Pi_n^d = \\binom{n+d}{n} \\qquad\\hbox{and}\\qquad\n  \\dim {\\mathcal V}_n^d =\\binom{n+d-1}{n}.\n$$\nSeveral explicit orthonormal bases of ${\\mathcal V}_n^d$ are known\n(see, for example, \\cite{DX}). We will need one given in terms\nof the Jacobi polynomials and spherical harmonics.\n\n\\medskip\n\n{\\it Jacobi polynomials.} The $k$-th Jacobi polynomial is \ndentoed by $P_k^{({\\alpha},{\\beta})}$ and they satisfy the orthogonal \nrelation (\\cite{Sz})\n\\begin{align}\\label{Jacobi}\n&  c_{{\\alpha},{\\beta}} \\int_{-1}^1 P_k^{({\\alpha},{\\beta})}(t)P_l^{({\\alpha},{\\beta})}(t) w_{{\\alpha},{\\beta}}(t)dt \\\\\n& \\qquad\\qquad  = \n   \\frac{({\\alpha}+1)_k({\\beta}+1)_k({\\alpha}+{\\beta}+k+1)}{k!({\\alpha}+{\\beta}+2)_k({\\alpha}+{\\beta}+2k+1)} \\delta_{k,l}\n   :=h_k^{({\\alpha},{\\beta})} \\delta_{k,l}, \\notag\n\\end{align}\nwhere $w_{{\\alpha},{\\beta}}(t)=  (1-t)^{\\alpha}(1+t)^{\\beta}$,  $c_{{\\alpha},{\\beta}}$ is the \nnormalization constant of $w_{{\\alpha},{\\beta}}$, \n$$\n   [ c_{{\\alpha},{\\beta}} ]^{-1} = \\int_{-1}^1 w_{{\\alpha},{\\beta}}(t) dt = 2^{{\\alpha}+{\\beta}+1}\n        \\frac{\\Gamma({\\alpha}+1)\\Gamma({\\beta}+1)}{\\Gamma({\\alpha}+{\\beta}+2)},\n$$ \nand the notation $(a)_k : = a (a+1) \\cdots (a+k-1)$ denotes the \nshifted factorial (Pochhammer symbol).  From \\eqref{Jacobi} \nthe orthonormal Jacobi polynomials are given by $\np_n^{({\\alpha},{\\beta})}(t):= [h_n^{({\\alpha},{\\beta})}]^{-1\/2} P_n^{({\\alpha},{\\beta})}(t)$. \n\n\\medskip\n\n{\\it Gegenbauer polynomials and Chebyshev polynomials.}\nWhen $\\alpha = \\beta = \\lambda -1\/2$, the Jacobi polynomials\nbecome the Gegenbauer polynomials, usually denoted by \n$C_k^\\lambda$ and normalized by \n\\begin{equation} \\label{Gegen}\n c_\\lambda \\int_{-1}^1 C_k^{{\\lambda}}(t)C_l^{{\\lambda}}(t) (1-t^2)^{{\\lambda}-1\/2}dt \n   = \\frac{{\\lambda} (2{\\lambda})_k}{(k+{\\lambda})k!} \\delta_{k,l}: = h_k^{(\\lambda)} \\delta_{k,l}. \n\\end{equation}\nwhere $c_{\\lambda} = \\Gamma(1\/2) \\Gamma({\\lambda} + 1\/2)\/ \\Gamma({\\lambda} +1)$.\nWhen ${\\lambda} =1$ and ${\\lambda}=0$, $C_k^{\\lambda}(t)$ becomes the Chebyshev \npolynomial of the second kind, $U_k(t)$, and the first kind, \n$T_k(t)$, respectively, and \n\\begin{equation} \\label{cheby}\n    U_k(t) = \\frac{\\sin (k+1){\\theta}}{\\sin {\\theta}} \\qquad \\hbox{and} \\qquad \n     T_k(t) = \\cos k {\\theta}, \\quad\\hbox{where}\\quad t = \\cos {\\theta}. \n\\end{equation}\n\n\\medskip\n\n{\\it Spherical harmonics.} These are defined as the restriction \nof the homogeneous harmonic polynomials on the sphere. Let \n${\\mathcal H}_n^d$ denote the space of spherical harmonics of degree \n$n$ in $d$ variables. It is known that \n$$\n  \\dim {\\mathcal H}_n^d = \\binom{n+d-1}{n}  - \\binom{n+d-3}{n}. \n$$\nLet $\\{Y_{k,n}: 1 \\le k \\le \\dim {\\mathcal H}_n^d\\}$ denote an orthonormal\nbasis of ${\\mathcal H}_n^d$. Then \n$$\n \\sigma_d^{-1} \\int_{S^{d-1}} Y_{k,n}(\\xi) Y_{l,n}(\\xi)d\\omega(\\xi) \n      = \\delta_{k,l}, \\qquad 1 \\le k,l \\le \\dim{\\mathcal H}_n^d.\n$$\nWe emphasis that $Y_{k,n}(x)$ are in fact homogeneous \npolynomials in $\\Pi_n^d$. \n\n\\medskip\n\n{\\it An orthonormal basis for ${\\mathcal V}_n^d$.} We give the basis for \ninner product defined in terms of $W_\\mu(x)$. Setting $\\mu =1\/2$\ngives the basis for the Lebesgue measure. Let $Y_{j,m}$ be\nan orthonormal basis for ${\\mathcal H}_m^d$. Define\n\\begin{equation} \\label{OPbasis}\nf_{k,j}^n(x) = [h_{n,k}]^{-1} \n  p_k^{(\\mu -\\frac{1}{2},n-2k+\\frac{d-2}{2})}(2\\|x\\|^2 -1) \n         Y_{j,n-2k}(x), \n\\end{equation}\nwhere \n$$\n   [h_{n,k}]^2 :=\\frac{\\Gamma(\\mu + \\frac{d+1}{2})\n      \\Gamma(n-2k + \\frac{d}{2})}{\\Gamma(\\frac{d}{2})\n        \\Gamma(n-2k+\\mu+\\frac{d+1}{2})}. \n$$\nThen the set $\\{f_{k,j}^n: 1 \\le j \\le \\dim {\\mathcal H}_{n-2k}^d, \n  0 \\le 2 k \\le n\\}$ is an orthonormal basis for ${\\mathcal V}_n^d$;\nthat is, $f_{k,j}^n \\in {\\mathcal V}_n^d$ and \n$ \\langle f_{k,j}^n, f_{k',j'}^{n} \\rangle_{L^2(B^d)} = \n       \\delta_{k,k'}\\delta_{j,j'}$ (see \\cite[p. 39]{DX}).\n  \n\\medskip\n\n{\\it Reproducing kernel of ${\\mathcal V}_n^d$}. The reproducing kernel\n$P_n(\\cdot,\\cdot)$ of ${\\mathcal V}_n^d$ satisfies\n\\begin{equation}\\label{reprod0}\n   a_\\mu \\int_{B^d} f(y) P_n(x,y) W_\\mu(y)dy = f(x),\n        \\qquad \\forall f\\in {\\mathcal V}_n^d.\n\\end{equation}\nLet $\\{P_k^n:  1 \\le k \\le \\dim {\\mathcal V}_n^d\\}$ denote any orthonormal \nbasis of ${\\mathcal V}_n^d$.  Then\n$$\nP_n(x,y) = \\sum_{k=1}^{N_n} P_k^n(x) P_k^n(y), \\qquad \n           N_n = \\dim {\\mathcal V}_n^d. \n$$\nThe definition of $P_n(\\cdot, \\cdot)$, however, is independent \nof the particular choice of bases. In particular, we can take the \northonormal basis in \\eqref{reprod} and get \n\\begin{equation}\\label{reprod}\n P_n(x,y) = \\sum_{0 \\le 2k \\le n} \\sum_{j=1}^{\\dim {\\mathcal H}_{n-2k}^d}\n       f_{k,j}^n(x) f_{k,j}^n(y).\n\\end{equation}\nThe reproducing kernel satisfies a compact formula that will \nplay a fundamental role in our  study; it is given by (\\cite{X99})\n$$\n P_n(x,y) = \\frac{n+{\\lambda}}{{\\lambda}} c_{\\mu-\\frac{1}{2}}\n   \\int_{-1}^1C_n^{\\lambda} (\\langle x, y\\rangle + \n      \\sqrt{1-\\|x\\|^2}\\sqrt{1-\\|y\\|^2} \\,s) (1-s^2)^{\\mu-1} ds\n$$\nwhere $\\lambda = \\mu + \\frac{d-1}{2}$, $\\langle \\cdot, \\cdot\n\\rangle$ is the Euclidean inner product in ${\\mathbb R}^d$, and \n$c_\\lambda$ is defined in \\eqref{Gegen}. In particular, it \nimplies that\n\\begin{equation}\\label{reprod2}\n    P_n(x,\\xi) = \\frac{n+{\\lambda}}{{\\lambda}} C_n^{\\lambda} (\\langle x, \\xi\\rangle),\n         \\qquad \\xi \\in S^{d-1}, \\quad x \\in B^d. \n\\end{equation}\n\n\\medskip\n\n{\\it Orthogonal expansions on $B^d$.} If $\\{P_k^n: 1 \\le k \n\\le N_n\\}$, $N_n=\\dim {\\mathcal V}_n^d$, is an orthonormal basis of \n${\\mathcal V}_n^d$, then the standard Hilbert space theory states that \nthere is an orthogonal expansion\n$$\nf  = \\sum_{k=0}^\\infty \n  \\sum_{k=1}^{N_n} \\langle f, P_k^n\\rangle_{L^2(B^d)}       \n   P_k^n, \\qquad \\forall f \\in L^2(B^d). \n$$ \nLet $\\operatorname{proj}_k: L^2(B^d) \\mapsto {\\mathcal V}_n^d$ denote the projection\noperator. Using the reproducing kernel,  the orthogonal expansion \ncan be stated as \n\\begin{equation} \\label{OrthExpand}\n   f = \\sum_{k=0}^\\infty \\operatorname{proj}_k f, \\qquad \\hbox{where} \\quad\n        \\operatorname{proj}_k f = a_\\mu \\int_{B^d} f(y) P_n(x,y) W_\\mu(y) dy,\n\\end{equation}\nwhich is independent of the particular choices of the bases of \n${\\mathcal V}_n^d$. \n\n\n\\section{Radon Transform and Orthogonal Polynomial Expansion}\n\\setcounter{equation}{0}\n\nLet $f \\in L^1$ be a real valued function. For $\\xi \\in S^{d-1}$ \nand $t \\in {\\mathbb R}$, the Radon transform of $f$ is defined as\n$$\n    {\\mathcal R} f(\\xi, t) := \\int_{\\langle \\xi, x\\rangle = t} f(x) dx\n         = \\int_{\\xi^\\perp} f (t \\xi + y )dy,\n$$\nwhere the integral is over a hyperplane of $(d-1)$-dimension \nperpendicular to $\\xi$ and with minimum distance $t$ to the origin. \nMore general definition on other spaces or manifolds can be found\nin \\cite{Hel}. For properties of Radon transforms we refer to \n\\cite{Hel,N}.  We assume \nthat $f$ has compact support in $B^d$, so that the\nintegral above should be understood as over $B^d \\cap \n\\{x: \\langle \\xi, x\\rangle = t\\}$. In particular, for $\\xi \\in S^{d-1}$,\nlet $Q_\\xi$ denote an orthogonal matrix whose first row is $\\xi$\nand let $B^d(r)$ denote the ball of radius $r$ in ${\\mathbb R}^d$; then \na change of variables $x \\mapsto (t, y)Q_\\xi$ shows that \n\\begin{align} \\label{Radon}\n {\\mathcal R} f(\\xi,t) & = \\int_{B^{d-1}(\\sqrt{1-t^2})} f((t, y) Q_\\xi)dy \\\\\n & = (1-t^2)^{\\frac{d-1}{2}} \\int_{B^{d-1}} f((t, \\sqrt{1-t^2} y) Q_\\xi)dy. \n  \\notag\n\\end{align}\nSince $\\langle (t,y)Q_\\xi, \\xi \\rangle = t$, an immediate consequence \nof \\eqref{Radon} is the following identity, \n\\begin{equation} \\label{IntRf}\n   \\int_{B^d} f(x) g(\\langle x,\\xi \\rangle) dx = \n        \\int_{-1}^1 {\\mathcal R} f(\\xi,t) g(t) dt, \\qquad \\xi \\in S^{d-1},\n\\end{equation}\nwhenever both integrals make sense. The definition of ${\\mathcal R} f$ also \nimplies that \n\\begin{equation} \\label{eq:RfEven}\n     {\\mathcal R} f(- \\xi, -t) = {\\mathcal R} f(\\xi, t), \\qquad \\xi \\in S^{d-1}, \\quad t\\in {\\mathbb R}.\n\\end{equation}\nFor fixed $\\xi$ and $t$, we also call ${\\mathcal R} f(\\xi,t)$ a Radon \nprojection.  The essential problem for x-ray imaging is to find a \ngood approximation to the function $f$ based on a given data \nset of its Rdaon projections. \n\nWe now derive the orthogonal expansion of $f$ in terms of \nRadon projections. The following proposition plays a key role. \n\n\\begin{prop} \\label{Reprod}\nFor $x,y \\in B^d$, the reproducing kernel $P_n(\\cdot,\\cdot)$ \nsatisfies\n\\begin{equation}\\label{r_kernel}\n    P_n(x,y) = \\frac{n+d\/2}{d\/2}  \\sigma_d^{-1} \\int_{S^{d-1}}\n         C_n^{d\/2}(\\langle x,\\xi\\rangle) C_n^{d\/2}(\\langle y,\\xi\\rangle)\n          d\\omega(\\xi). \n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nFrom the explicit formula of $f_{k,j}^n$ at \\eqref{OPbasis} with \n$\\mu =1\/2$, we deduce that \n\\begin{equation} \\label{f_xi}\n      f_{k,j}^n(\\xi) = H_n Y_{j,n-2k}(\\xi), \\qquad \\xi \\in S^{d-1},   \n\\end{equation}\nwhere, using the fact that $p_k^{(0,{\\beta})}(t) = [h_k^{(0,{\\beta})}]^{-1\/2} \nP_k^{(0,{\\beta})}(t)$, $P_k^{(0,{\\beta})}(1) = 1$, and the formula of \n$h_k^{({\\alpha},{\\beta})}$ in \\eqref{Jacobi}, we have\n\\begin{equation} \\label{Hk}\n H_n  =  [h_{n,k}]^{-1} \n  p_k^{(0,n-2k+\\frac{d-2}{2})}(1) = \\sqrt{\\frac{n+d\/2}{d\/2}},\n\\end{equation}\nindependent of $k$. Consequently, integrating over $S^{d-1}$\nwe get\n$$\n   \\sigma_{d}^{-1} \\int_{S^{d-1}} f_{k,j}^n(\\xi) f_{k',j'}^{n}(\\xi) \n       d\\omega(\\xi) = H_n^2 \\delta_{j,j'}\\delta_{k,k'}\n         = \\frac{n+d\/2}{d\/2} \\delta_{j,j'}\\delta_{k,k'}.\n$$\nMultiplying the above equation by $f_{k,j}^n(x)$ and \n$f_{k',j'}^n(y)$ and summing over all $j, j', k, k'$, the stated\nresult follows from \\eqref{reprod} and \\eqref{reprod2}.\n\\end{proof}\n\n\\begin{thm} \\label{thm:3.2}\nFor $n \\ge 0$,\n$$\n  \\operatorname{proj}_n f(x) = \\frac{n+d\/2}{d\/2} \\sigma_d^{-1} \\int_{S^{d-1}}\n       b_d^{-1} \\int_{-1}^1 {\\mathcal R} f(\\xi,t) C_n^{d\/2} (t) dt \\,\n            C_n^{d\/2}(\\langle x,\\xi\\rangle) d \\omega(\\xi). \n$$\nIn particular, for $f \\in L^2(B^d)$, \n\\begin{equation} \\label{expand}\n   f = \\sum_{n=0}^\\infty \\frac{n+d\/2}{d\/2} \\sigma_d^{-1} \\int_{S^{d-1}}\n       b_d^{-1} \\int_{-1}^1 {\\mathcal R} f(\\xi,t) C_n^{d\/2} (t) dt \\,\n            C_n^{d\/2}(\\langle \\cdot,\\xi\\rangle) d \\omega(\\xi). \n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nBy the formula \\eqref{reprod0} with $\\mu =1\/2$ and the \nformula \\eqref{r_kernel} of $P_n(\\cdot,\\cdot)$ we have \n\\begin{align*}\n     \\operatorname{proj}_n f(x) & =  b_d^{-1} \\int_{B^d} f(y) P_n(x,y) dy \\\\\n   &   =       \\frac{n+d\/2}{d\/2}  \\sigma_d^{-1} \\int_{S^{d-1}}\n     b_d^{-1} \\int_{B^d} f(y) C_n^{d\/2}(\\langle y,\\xi\\rangle) dy\n      C_n^{d\/2}(\\langle x,\\xi\\rangle)   d\\omega(\\xi).  \n\\end{align*}\nThe identity \\eqref{IntRf} shows that the inner integral is\n\\begin{equation} \\label{eq:Int1}\n    \\int_{B^d} f(y)C_n^{d\/2}(\\langle y,\\xi\\rangle) dy \n     = \\int_{-1}^1 {\\mathcal R} f(\\xi,t) C_n^{d\/2} (t) dt, \n\\end{equation}\nso that the stated formula follows. \n\\end{proof}\n\nThe formula \\eqref{expand} as stated here has already appeared \nin \\cite{P} in a study of the approximation by ridge functions. See\nalso \\cite{BG} for the case of $d =2$.  Although spherical harmonics\nexpansions for $d=2$ was used in the classical work of \\cite{Co}, its \ncompact form in \\eqref{expand} is quite recent and not used for \nreconstructing images from Radon data until recently (\\cite{X05}). \nIt should also be noted that for $d > 2$, the Gegenbauer polynomials\nand spherical harmonics were used for constructing Radon transforms \nalready in \\cite{Lo}. \n\nLet us mention that there does not seem to be an analogous \nformula for the more general case of orthogonal expansion with\nrespect to $W_\\mu$. In fact, in the general case, the formula \n\\eqref{OPbasis} gives \n$$\n   f_{k,j}^n(\\xi) = H_{n,k} Y_{j,n-2k}(\\xi),  \\qquad \\xi \\in S^{d-1},\n$$\nwhere \n\\begin{equation} \\label{Hnk}\n     H_{n,k} := \\frac{(\\mu+1\/2)_k (\\mu+\\frac{d-1}{2})_{n-k}\n          (n+\\mu+\\frac{d-1}{2}) }\n                   {k! (\\frac{d}{2})_{n-k} (\\mu + \\frac{d-1}{2})},\n\\end{equation}\nwhich depends on both $n$ and $k$ (comparing with \\eqref{Hk}),\nso that Proposition \\ref{Reprod} with $C_n^{d\/2}$ replaced by\n$C_n^{\\mu + \\frac{d-1}{2}}$ does not hold. \n\nLet $S_n f$ denote the partial sum operator of the \northogonal expansion \\eqref{OrthExpand}, \n\\begin{equation} \\label{Snf}\n      S_n f(x) = \\sum_{k=0}^n \\operatorname{proj}_k f(x).  \n\\end{equation}  \nEvidently, the expansion \\eqref{OrthExpand} holds in the sense\nthat $S_n f \\to f$ in $L^2(B^d)$ norm.  \n\n\\begin{cor}\nLet $S_n$ be the partial sum operator defined in \\eqref{Snf}.\nThen \n\\begin{equation} \\label{eq:Sn}\n S_n f(x) = \\sigma_d^{-1} \\int_{S^{d-1}}b_d^{-1} \n \\int_{-1}^1 {\\mathcal R} f(\\xi,t) \\Phi_n( t, \\langle x,\\xi\\rangle) dt d \\omega(\\xi). \n\\end{equation}\nwhere \n\\begin{equation} \\label{eq:Phi}\n   \\Phi_n(t, u) : = \\sum_{k=0}^n \\frac{k+d\/2}{d\/2} \n            C_k^{d\/2} (t) C_k^{d\/2}(u).  \n\\end{equation}\n\\end{cor}\n\nA cubature formula on $S^{d-1}$ of degree $M$ is a discrete \nsum such that \n\\begin{equation} \\label{cubature}\n   \\sigma_d^{-1}  \\int_{S^{d-1}} f(\\xi) d\\omega(\\xi) = \n     \\sum_{\\nu=1}^N \\lambda_\\nu f(\\xi_\\nu), \n        \\qquad f \\in \\Pi_M(S^{d-1}), \n\\end{equation}\nwhere $\\Pi_M(S^{d-1})$ is the space of spherical polynomials,\nthat is, the space of $\\Pi_M^d$ restricted on $S^{d-1}$. If all \n$\\lambda_k$ are positive, the cubature is called {\\it positive}.  \nWe call a polynomial $P \\in \\Pi_M^d$ {\\it even} if it satisfies \n$P(x) = P(-x)$ for all $x \\in {\\mathbb R}^d$. The cubature formula \n\\eqref{cubature} is called {\\it symmetric}, if it is exact for all \neven polynomials in $\\Pi_M(S^{d-1})$. \n\n\\begin{prop}\nSuppose \\eqref{cubature}  is a symmetric cubature formula of \ndegree $2 n$. Then \n\\begin{equation} \\label{SnSemiDis}\n   S_n f(x) = \\sum_{\\nu=1}^{N} \\lambda_\\nu b_d^{-1} \n    \\int_{-1}^1 {\\mathcal R} f(\\xi_\\nu,t) \\Phi_n(t, \\langle x,\\xi_\\nu\\rangle ) dt. \n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nThe equation \\eqref{eq:Int1} shows that $P_x(\\xi):=\n\\int_{-1}^1 {\\mathcal R} f(\\xi,t) \\Phi_n(\\xi,t;x)dt $ is a \npolynomial of degree at most $2n$ in $\\xi$. Furthermore, \nusing the fact that ${\\mathcal R} f(-\\xi,-t) = {\\mathcal R} f(\\xi,t)$, it is easy to see \nthat $P_x$ is even, so that the cubature formula on $S^{d-1}$\nis exact when applied to $P_x(\\xi)$.\n\\end{proof}\n\nWe consider some special cases of lower dimensions below. \n\n\\medskip\\noindent\n{\\bf The case d=2.}  For $\\xi \\in S^1$ we write $\\xi = (\\cos {\\theta}, \\sin {\\theta})$\nand we shall write ${\\mathcal R}_{\\theta} f(t)$, ${\\theta} \\in [0,2\\pi]$, instead of \n${\\mathcal R} f(\\xi,t)$. Since $b_2 = \\pi$ and the following cubature formula\n$$\n \\frac{1}{2\\pi}\\int_{S^1} f(\\xi) d\\omega(\\xi) = \\frac{1}{n+1} \n     \\sum_{\\nu =0}^n f(\\xi_\\nu),\n \\qquad \\xi_\\nu = (\\cos \\tfrac{\\nu \\pi}{n+1}, \\sin \\tfrac{\\nu \\pi}{n+1})\n$$\nis symmetric and of degree $2n$, we conclude that \n\\begin{equation}\\label{d2Sn}\n   S_n f(x) = \\frac{1}{n+1}\\sum_{\\nu=0}^{n} \n    \\int_{-1}^1 {\\mathcal R}_{{\\theta}_\\nu} f(t)\n  \\Phi_n(t, x_1 \\cos {\\theta}_\\nu+x_2 \\sin {\\theta}_\\nu)dt \n\\end{equation}\nwhere ${\\theta}_\\nu = \\tfrac{\\nu\\pi}{n+1}$ and $\\Phi_n$ is \\eqref{eq:Phi} for\n$d =2$,\n$$\n   \\Phi_n(t, u) = \\sum_{k=0}^n (k+1) U_k(t) U_k(u).\n$$\nThis formula can be found implicitly in \\cite{LS} (see (5.9), (4.3) \nand (3.7) there).\nIn the case of $n = 2m$, we can use the elementary relations \n$$\n\\cos \\tfrac{(2\\nu + 1)\\pi}{2m+1}  = - \\cos \\tfrac{(2(\\nu+m) \\pi}{2m+1}, \n\\quad \\qquad\n\\sin \\tfrac{(2\\nu + 1)\\pi}{2m+1}  = - \\sin \\tfrac{(2(\\nu+m) \\pi}{2m+1} \n$$\nand the fact that ${\\mathcal R}({\\theta} + \\pi, -t) = {\\mathcal R}({\\theta}, t)$ to rewrite \n\\eqref{d2Sn} as\n\\begin{equation}\\label{d2Sn2}\n   S_{2m} f(x) = \\frac{1}{2m+1}\\sum_{\\nu=0}^{2m} \n       \\int_{-1}^1 {\\mathcal R}_{\\phi_\\nu} f(t)\n  \\Phi_{2m}(t, x_1 \\cos \\phi_\\nu+x_2 \\sin \\phi_\\nu)dt\n\\end{equation}\nwhere $\\phi_\\nu =\\tfrac{2\\nu\\pi}{2m+1}$. This is the formula \n\\eqref{1stS2m} proved in \\cite{X05} from which the OPED \nalgorithms are derived.  \n\\qed\n\n\\medskip\\noindent\n{\\bf The case d=3.}  For $\\xi \\in S^2$ we use the spherical \ncoordinate\n$$\n\\xi = (\\sin \\phi \\sin {\\theta}, \\sin \\phi \\cos {\\theta}, \\cos \\phi), \\qquad \n      0\\le \\phi \\le \\pi, 0 \\le {\\theta} \\le 2 \\pi.\n$$\nSeveral explicit cubature formulas on the sphere are known, \nsee, for example, \\cite{My, St}. Let $t_{k} = \\cos {\\theta}_k$,\n$k =0, 1, \\ldots,n$, denote the zeros of the Legendre polynomial\nof degree $n+1$ and $\\lambda_{k}$ be the corresponding \nweights of the Legendre-Gaussian quadrature formula. Let \n$$\n\\xi_{k,\\nu} = (\\sin \\tfrac{\\nu \\pi}{n+1} \\sin {\\theta}_k, \n   \\cos \\tfrac{\\nu}{n+1} \\sin {\\theta}_k, \\cos {\\theta}_k), \\qquad 0 \\le k, \\nu \\le n \n$$\nThen the product type cubature formula \n$$\n  \\frac{1}{4 \\pi} \\int_{S^2} f(\\xi)d\\omega(\\xi) = \\frac{1}{n+1}\n       \\sum_{k=0}^n \\lambda_k \\sum_{\\nu=0}^n\n          f(\\xi_{k,\\nu})  \n$$\nis symmetric and of degree $2n$. Consequently, we have \n\\begin{equation}\\label{d3Sn}\n   S_n f(x) = \\frac{1}{n+1} \\sum_{k=0}^n \\lambda_k\n     \\sum_{\\nu=0}^{n} \\int_{-1}^1 {\\mathcal R} f (\\xi_{k,\\nu}, t)\n     \\Phi_n(t, \\langle  \\xi_{k,\\nu},x\\rangle) dt,  \n\\end{equation}\nwhere $\\Phi_n$ is the function \\eqref{eq:Phi} for $d=3$.  \\qed \n\\medskip\n \nThe formula of $S_n f$ in terms of Radon projections allows \nus to give an approximation to $f$ based on finite Radon \nprojections. The convergence of $S_n f$ to $f$ holds in $L^2$\nnorm but does not hold in the uniform norm in general. In fact, \nit is known that \\cite{X01}\n\\begin{equation}\\label{SnNorm}\n    \\|S_n\\|_\\infty = {\\mathcal O}(n^{\\frac{d-1}{2}}), \\qquad d \\ge 2,  \n\\end{equation}\nwhere $\\|\\cdot\\|_\\infty$ is the operator norm of $S_n$ in $C(B^d)$,\nand $A_n = {\\mathcal O}(B_n)$ means $c_1 A_n \\le B_n \\le c_2 A_n$ for \ntwo constants $c_1$ and $c_2$ independent of $n$. There is,\nhowever, a simple construction that gives a better convergence \nresult. \n\nLet $\\eta$ be a $C^{d+2}({\\mathbb R})$ function such that $\\eta(t) \\ge 0$,\n$\\eta(t) =1$ for $0 \\le t \\le 1$ and $\\eta$ has compact support on \n$[0,2]$. Define \n\\begin{equation}\\label{Sn_eta}\n  S_n^\\eta f (x) : = \\sum_{k=0}^{2n} \n      \\eta \\left(\\frac{k}{n}\\right) \\operatorname{proj}_k f(x). \n\\end{equation} \nThe operator $S_n^\\eta$ satisfies the following properties \n\\cite{X05a}: \n\n\\begin{prop} Let $f \\in L^p(B^d)$, $1 \\le p < \\infty$ or $f \\in C(B^d)$\nfor $p = \\infty$. Then\n\\begin{enumerate}\n\\item  $S_n^\\eta f  =  f$ if $f \\in \\Pi_n$;   \n\\item  for $ n \\in {\\mathbb N}$, $\\| \\eta_nf \\|_p \\le c \\|f\\|_p$\n\\item  for $ n \\in {\\mathbb N}$, \n$ \n    \\|f - \\eta_n \\|_p \\le c E_n(f)_p : = \\inf_{p \\in \\Pi_n^d} \\|f - p\\|_p.\n$\n\\end{enumerate}\n\\end{prop}\n\nAs $S_n^\\eta f$ is a polynomial of degree $2n$, the last property\nshows that, up to a constant multiple, it is close to the polynomial\nof the best approximation to $f$. Since $\\operatorname{proj}_n f$ can be written in \nterms of Radon projections, so can $S_n^\\eta f$.  \n\n \n\n\\section{OPED algorithms for reconstruction of images}\n\\setcounter{equation}{0}\n\nThe essential problem in computerized tomography is to \nfind a good approximation to the function $f$ based on a\nset of discrete Radon data. The expression \\eqref{SnSemiDis} \nallows us to derive such an approximation by a simple \nquadrature formula on $[-1,1]$. Because of \\eqref{Radon},\nwe choose the quadrature formula to be of the form\n\\begin{equation} \\label{quadrat}\n   c_{d\/2}  \\int_{-1}^1 f(t) (1-t^2)^{\\frac{d-1}{2}} dt = \n      \\sum_{j=0}^n w_j f(t_j),\n\\end{equation}\nwhere $c_{d\/2}$ is defined as in \\eqref{Gegen},  and assume \nthat it is exact for polynomials of degree $M$. In particular, \nwe can choose the Gaussian quadrature, for which\n$t_j= t_{j,n}$, $0\\le j \\le n$, are zeros of the Gegenbauer \npolynomial $C_{n+1}^{d\/2}(t)$ and $w_j$ are all positive\nand given by explicit formula (see \\cite{Sz}). The Gaussian\nquadrature formula is exact for polynomials of degree up\nto $2n +1$. \n\n\\begin{prop}\nLet \\eqref{cubature} be a positive symmetric cubature formula \nof degree $2n$ and \\eqref{quadrat} be the Gaussian quadrature \nformula. Define\n\\begin{equation} \\label{CAn}\n   {\\mathcal A}_n f(x) = b_d^{-1} \\sum_{\\nu =1}^N \\lambda_\\mu\n       \\sum_{j=0}^n w_j  {\\mathcal R} f(\\xi_\\nu, t_j) \n         \\Phi_n(t_j,\\langle x,\\xi_\\nu\\rangle). \n\\end{equation}\nThen ${\\mathcal A}_n f$ preserves polynomials of degree $n$, that is,\n${\\mathcal A}_n f = f$ whenever $f \\in {\\mathcal P}_n^d$. \n\\end{prop}\n\n\\begin{proof}\nWe start from \\eqref{SnSemiDis}. If $f$ is a polynomial of \ndegree at most $n$ then, by \\eqref{Radon}, \n$(1-t^2)^{- \\frac{d-1}{2}} {\\mathcal R} f(\\xi_\\nu, t)$ is a polynomial \nof degree $n$. As $\\Phi_n (t, \\langle x,\\xi_\\nu\\rangle)$ is \na polynomial of degree $n$ in $t$ and the Gaussian quadrature\nformula is of degree $2n+1$, the fact that ${\\mathcal A}_n f = f$ \nfollows.  \n\\end{proof}\n\nThe functions ${\\mathcal A}_n f$ are obtained from the orthogonal partial\nsums $S_n f$ of $f$ by applying the Gaussian quadrature formula. They provide a sequence of approximation to $f$ based on the \nset of discrete Radon data\n$$\n  \\{ {\\mathcal R} f(\\xi_\\nu, t_j): \\quad 1 \\le \\nu \\le N, 0 \\le j \\le n\\}.\n$$\nIn other word, ${\\mathcal A}_n$ provides an algorithm for reconstruction \nof images from the Radon data. We will show that ${\\mathcal A}_n f$ \nconverges to $f$ uniformly if $f$ is smooth enough. First, however,\nwe consider some special cases. \n\n\\medskip\\noindent\n{\\bf The case d=2.} In this case we can start from the formula \nof $S_{2m}$ at \\eqref{d2Sn2}. The Gaussian quadrature \nformula is \n$$\n  \\frac{1}{\\pi} \\int_{-1}^1 f(t) \\sqrt{1-t^2} dt = \\frac{1}{2m+1}\n      \\sum_{j=1}^{2m} \\sin^2 \\psi_j f(\\cos {\\theta}_j), \n          \\qquad {\\theta}_j =  \\frac{j \\pi}{2m+1},\n$$ \nwhich leads to the OPED algorithm of type II,\n\\begin{equation} \\label{2AnU}\n   {\\mathcal A}_{2m} f(x) = \\sum_{\\nu =0}^{2m} \\sum_{j=1}^{2m}\n       {\\mathcal R}_{\\phi_\\nu} f(\\cos {\\theta}_j) T_{j,\\nu}(x), \n\\end{equation}\nwhere\n$$\n T_{j,\\nu}(x) = \\frac{1}{(2m+1)^2} \\sum_{k=0}^{2m} (k+1) \n   \\sin ((k+1) {\\theta}_j) U_k(x_1 \\cos \\phi_\\nu + x_2 \\sin \\phi_\\nu).\n$$\nThe OPED of type II is closely related to an algorithm in \\cite{BO},\nwhere the connection to orthogonal polynomial expansion was \nnot considered. The formation of the lines on which the Radon \nprojections  take place is often refereed to as scanning geometry, \nas it determines how the object being examined is scanned by \nthe x-rays.  We can use the Gaussian quadrature formula for the \nChebyshev polynomials of the first kind, \n$$\n \\frac{1}{\\pi} \\int_{-1}^1f(t) \\frac{dt}{\\sqrt{1-t^2}} = \\frac{1}{2m+1}\n      \\sum_{k=0}^{2m} f(\\cos \\psi_j), \\qquad \n         \\psi_j = \\frac{(j+\\frac{1}{2})\\pi}{2m+1}, \n$$\nto discretize the integral in \\eqref{d2Sn2} by applying it to the\nintegrant multiplied by $1-t^2$,  leading to the OPED\nalgorithm of type I with a different scanning geometry, which \nhas the same formula as \\eqref{2AnU} except that ${\\theta}_j$ need\nto be replaced by $\\psi_j$ and the summation on $j$ starts \nfrom $j =0$. We refer to \\cite{XTC} for the discussions of these \ntwo scanning geometries and their implementation in practical \nproblems. \n\nBoth types of these two OPED algorithms work well in our \nnumerical testing (\\cite{XT, XTC}).  It should be mentioned \nthat the explicit formula of  $U_n(t)$ in \\eqref{cheby} permits a \nfast implementation of the OPED algorithm, which uses fast \nFourier sine transform and an interpolation step (\\cite{XT}). \n\\qed\n\n\\medskip\\noindent\n{\\bf The case $d=3$.} In this case we can start from the formula\nof $S_n f$ at \\eqref{d3Sn}. We apply the Gaussian quadrature \nformula\n$$\n \\frac{3}{4} \\int_{-1}^1f(t) (1-t^2) dt =\n    \\sum_{j=0}^n w_j f(t_j), \n$$\nwhere $t_j$, $0 \\le j \\le n$, are zeors of $C_n^{3\/2}(t)$. We can \nalso apply the Gaussian quadrature formula for the Lebesgue\nmeasure. This leads to a three dimensional OPED algorithm, \n\\begin{equation} \\label{A_3d}\n {\\mathcal A}_n f(x) =  \\frac{1}{n+1} \\sum_{k=0}^n \\lambda_k\n     \\sum_{\\nu=0}^{n} \\sum_{j=0}^n w_j {\\mathcal R} f (\\xi_{k,\\nu}, t_j)\n     \\Phi_n(t_j, \\langle  \\xi_{k,\\nu},x\\rangle).\n\\end{equation}\nThe Radon data used in \\eqref{A_3d} are integrals over \nplanes $\\langle x, \\xi_{k,\\nu}\\rangle = t_j$. Such data can be\napproximated by integrals over lines.  \\qed\n\n\\medskip\n\nFor $d=3$, one can uses multiple 2D slices to reconstruct image\non a cylindrical domain, as proposed in \\cite{X05}. An interesting\nquestion is to see which of these two algorithms are more suitable\nfor the 3D reconstruction. \n\nNext we consider the convergence of ${\\mathcal A}_n f$ in the uniform\nnorm on $B^d$. \n\n\\begin{thm}\nThe uniform norm of the operator ${\\mathcal A}_n$ is given by\n\\begin{equation}\\label{A_norm}\n    \\|{\\mathcal A}\\|_\\infty = \\sup_{x \\in B^d} \\Lambda_n(x), \\quad \n     \\Lambda_n(x) = \n   \\sum_{\\nu =1}^N {\\lambda}_\\nu \\sum_{j=0}^n w_j (1-t_j^2)^{\\frac{d-1}{2}} \n      \\left |  \\Phi_n(t_j, \\langle x,\\xi_\\nu\\rangle \\right) |.  \n\\end{equation}\nFurthermore, there is a constant $c$ independent of $n$, such that \n\\begin{equation} \\label{CAnNorm}\n   \\|{\\mathcal A} \\|_\\infty \\le c\\, n^{2d}. \n\\end{equation}\nIn particular, if $f$ is smooth enough then ${\\mathcal A}_{2n}f$ converges to \n$f$ uniformly on $B^d$.\n\\end{thm}\n\n\\begin{proof}\nTo estimate the norm of ${\\mathcal A}_n$, we first observe\nthat \n$$\n   \\left |  (1-t^2)^{- \\frac{d-1}{2}} {\\mathcal R} f(\\xi_\\nu, t)\\right |\n        \\le b_{d-1} \\|f\\|_\\infty\n$$\nfrom which it follows that \n$$\n   \\| {\\mathcal A}_n f \\|_\\infty \\le  \\|f\\|_\\infty\n   \\sum_{\\nu =1}^N {\\lambda}_\\nu \\sum_{j=0}^n \n            w_j (1-t_j^2)^{\\frac{d-1}{2}} \n   \\left | \\Phi_n(t_j, \\langle x,\\xi_\\nu\\rangle) \\right |, \n$$\nsince $b_{d-1} b_d^{-1} = c_{d\/2}$. Taking the maximum over\n$B^d$ shows that $\\|{\\mathcal A}\\|_\\infty$ is bounded by the right hand\nside of \\eqref{A_norm}. To prove the equal sign, we construct\na function $f_\\varepsilon$ for each $\\varepsilon > 0$ such \nthat $\\|f_\\varepsilon\\|_\\infty =1$ and $\\|{\\mathcal A} f_\\varepsilon\\|_\\infty\n \\ge \\max_{x\\in B^d} \\Lambda_n(x)- c \\varepsilon$. A moment of \nreflection shows that the construction can be carried out easily; \nsee \\cite{X05} for one special case of $d =2$. \n \nTo prove \\eqref{CAnNorm} we use \\eqref{eq:Phi} and the fact \nthat $|C_n^\\lambda(t)|\n\\le C_n^{\\lambda}(1) = \\binom{n+2{\\lambda} -1}{n} = {\\mathcal O}(n^{2{\\lambda}-1})$,\nwhich implies that \n$$\n |\\Phi_n(\\xi,t)| \\le  \\sum_{k=0}^n \\frac{k+d\/2}{d\/2} [C_n^{d\/2}(1)]^2\n    \\le c \\sum_{k=0}^n \\frac{k+d\/2}{d\/2} k^{2d -2}\n    \\le c \\, n^{2d}. \n$$\nSince ${\\lambda}_\\mu$ and $w_j$ are all positive and, as the cubature \nand the quadrature are exactly for constant function,\n$\\sum_{\\nu =1}^N {\\lambda}_\\nu =1$ and $ \\sum_{j=0}^n w_j =1$, we\nconclude that $\\|{\\mathcal A}_n\\| \\le c\\, n^{2d}$.  If $f \\in C^{2d}$, then \nthe fact that ${\\mathcal A}_n p = p$ for $p \\in \\Pi_n^d$ and the triangle \ninequality shows that \n$$\n   \\|{\\mathcal A}_n f- f\\|_\\infty \\le (1 + \\|{\\mathcal A}\\|_\\infty) E_n (f)_\\infty\n          \\le c \\,n^{2d} E_n (f)_\\infty.\n$$\nIt is shown in \\cite{X05} that $E_n(f) \\le c n^{-2r} \\|{\\mathcal D}^r f\\|$,\nwhere ${\\mathcal D}$ is a second order differential operator, so that \nthe convergence of ${\\mathcal A}_n f$ for functions smooth \nenough follows.\n\\end{proof}\n\nWe should point out that the estimate \\eqref{CAnNorm} is \na rough upper bound, the actual norm should be smaller. In \nfact, in the case of $d =2$, the norm of ${\\mathcal A}_{2m}$ at \n\\eqref{2AnU} was estimated in \\cite{X05} to be \n$$\n   \\|{\\mathcal A}_{2m}\\|_\\infty \\sim m \\log (m+1),\n$$\nwhich is sharp and is just slightly worse than the estimate \n\\eqref{SnNorm} of the norm of the partial sum operator $S_n$\nfrom which ${\\mathcal A}_{2m}$ is obtained. The proof of such a \nsharp estimate is rather involved and requires detail knowledge\nof the zeros and weights of the quadrature and cubature\nformulas. On the other hand, a result in \\cite{Su} shows that \nthe norm of any projection operator from $C(B^d)$ to $\\Pi_n^d$ \nis at least ${\\mathcal O}(n^{\\frac{d-1}{2}})$ for $d \\ge 2$. As ${\\mathcal A}_n$ in \n\\eqref{CAn} is in fact a projection operator, its norm cannot \nbe bounded. We expect that the norm is in the order of \n${\\mathcal O}(n^{d\/2})$ multiplied by a log factor. \n\nIt should be mentioned that other polynomial based algorithms\nmay have better approximation property (\\cite{Ma, MN}. However, \nthe polynomial preserving property seems to be an important\ncharacteristic of OPED and using the partial sum allows also fast\nimplementation of the algorithm. The numerical tests show that\nOPED works very well even for step functions such as \nLogan-Sheff head phantom \\cite{XT,XTC}. \n\n \n\n\\section{Singular value decomposition of the Radon transform}\n\\setcounter{equation}{0}\n\nLet $A: H \\mapsto K$ be a linear continuous operator, where \n$H$ and $K$ are Hilbert spaces. Let $\\{f_k\\}_{k \\ge 0}$ and \n$\\{g_k\\}_{k \\ge 0}$ be orthonormal systems with respect to the \ninner product $\\langle \\cdot,\\cdot\\rangle_H$ in  $H$ and \n$\\langle \\cdot,\\cdot \\rangle_K$ in $K$, respectively. The singular \nvalue decomposition of $A$ is a representation\n\\begin{equation}\\label{Af}\n    A f = \\sum_{k=1}^\\infty \\gamma_k \\langle f,f_k\\rangle_H g_k,\n\\end{equation}\nwhere ${\\gamma}_k$ are the singular values of $A$. Let $A^*$ be\nthe adjoint of $A$. Then\n\\begin{equation}\\label{A*f}\n   A^* g = \\sum_{k=1}^\\infty {\\gamma}_k \\langle g,g_k \\rangle_K f_k. \n\\end{equation}\nEvidently $A f_k = {\\gamma}_k g_k$ and $A^*g_k = {\\gamma}_k f_k$.\nFurthermore,  the generalized inverse of $A$ is given by\n\\begin{equation}\\label{A+f}\n  A^+ g = \\sum_{k=0}^\\infty  {\\gamma}_k^{-1} \\langle f,f_k \\rangle_H  g_k.\n\\end{equation}\n\nThe singular value decomposition of the Radon transform \nwas developed in \\cite{Da, Lo} (see also \\cite{N}). Let \n$Z = S^{d-1} \\times [-1,1]$ and $w(t) = \\sqrt{1-t^2}$, and\ndenote by $L^2(Z, w^{1-d})$ the space of Lebesgue integrable \nfunctions \n$$\n L^2(Z, w^{1-d}): = \\{ g:  g(-\\xi,-t) = g(\\xi,t), \\quad \n        \\|g\\|_{L^2(Z)} < \\infty\\}, \n$$   \nwhere $\\|g\\|_{L^2(Z)}^2 =  \\langle g, g \\rangle_{L^2(Z)}$ and \nthe inner product is defined by\n$$\n    \\langle f, g \\rangle_{L^2(Z)} : = c_{d\/2} \\int_{-1}^1 \n           \\sigma_d^{-1} \\int_{S^{d-1}} f(\\xi,t)g(\\xi,t) d\\omega(\\xi) \n             (1- t^2)^{\\frac{1-d}{2}} dt,\n$$\nin which $c_{d\/2}$ is defined as in \\eqref{Gegen}.  Then it is \nknown (see, for example, \\cite{N}) that \n$$\n    {\\mathcal R}: L^2(B^d) \\mapsto L^2 (Z, w^{1-d}) \n$$\nis continuous. An orthonormal basis of $L^2(Z, w^{1-d})$ is \nreadily available.\n\n\\begin{prop}\nLet $\\{Y_{j,m}: 1 \\le j \\le \\dim {\\mathcal H}_m^d\\}$ denote an \northogomal basis of ${\\mathcal H}_m^d$ and define \n\\begin{equation} \\label{gkj}\n   g_{k,j}^n (\\xi,t) =  [h_n^{(d\/2)}]^{-1\/2}\n            (1-t^2)^{\\frac{d-1}{2}} C_n^{d\/2}(t) Y_{j,n-2k}(\\xi),\n\\end{equation}\nwhere $h_n^{(d\/2)}$ is defined in \\eqref{Gegen}. Then the functions \n$\\{g_{k,j}^n: 0 \\le 2k \\le n, 1 \\le j \\le \\dim{\\mathcal H}_{n-2k}^d\\}$ forms an\northogonomral basis for $L^2(Z,w^{1-d})$. \n\\end{prop}\n\n\\begin{proof}\nIt is straightforward to verify that $\\{g_{k,j}^n\\}$ form an orthonormal \nsystem of  $L^2(Z,w^{1-d})$.  Let $g \\in L^2(Z,w^{1-d})$. Then \n$w^{2d-2} g$ can be expanded in terms of the product orthonomal \nbasis  $\\{[h_n^{(d\/2)}]^{-1\/2} C_n^{d\/2}(t) Y_{j, n-m}(\\xi): 0 \\le m \\le n,\n  0 \\le j \\le \\dim{\\mathcal H}_{n-m}^d\\}$ of $L^2(Z, w^{d-1})$. The condition \n $g(-\\xi, -t) = g(\\xi,t)$ shows that the coefficients of the expansion \nare zero whenever $m$ is odd, so that we can assume \n$m = 2k$ and the expansion is uniquely determined.  \n\\end{proof}\n\nUsing $f_{k,j}^n$ in \\eqref{OPbasis} and $g_{k,j}^n$ \\eqref{gkj}, the \nsingular value decomposition of the Radon transform at \n\\eqref{Af}, \\eqref{A*f} and \\eqref{A+f} become the following: \n\n\\begin{thm} \nAssume $f$ is in the Schwartz space. The singular decomposition of \n${\\mathcal R} f$ is \n\\begin{align} \\label{Rf}\n  {\\mathcal R} f  = \\sum_{n=0}^\\infty {\\gamma}_n \\sum_{0\\le 2k \\le n} \n        \\sum_{j=0}^{M_{n-2k}}\n      \\langle f, f_{k,j}^n \\rangle_{L^2(B^d)} g_{k,j}^n \n\\end{align} \nwhere $M_m= \\dim {\\mathcal H}_{m}^d$, $c_{d\/2}$ is defined at  \n\\eqref{Gegen}; and \n\\begin{align}\\label{R*f}\n  {\\mathcal R}^* g  = \\sum_{n=0}^\\infty {\\gamma}_n \\sum_{0\\le 2k \\le n} \n        \\sum_{j=0}^{M_{n-2k}}\n      \\langle g, g_{k,j}^n \\rangle_{L^2(Z)} f_{k,j}^n.  \n\\end{align} \nFurthermore, \n\\begin{equation}\\label{R+f}\n   f(x) =  \\sum_{n=0}^\\infty {\\gamma}_n^{-1} \\sum_{0\\le 2k \\le n} \n        \\sum_{j=0}^{M_{n-2k}} \n      \\langle g, g_{k,j}^n \\rangle_{L^2(Z)} f_{k,j}^n.\n\\end{equation} \n\\end{thm}\n\nThese equations are the realization of \\eqref{Af}, \\eqref{A*f} and\n\\eqref{A+f} for the Radon transform. They are exactly the SVD \nderived in \\cite{Da, Lo}, once the difference in notations is accounted \nfor. \n\nBelow we derive the singular value decomposition using our notation \nhere. We need a proposition that goes back to \\cite{Marr} when $d=2$. \n\n\\begin{prop}\nLet $P \\in {\\mathcal V}_n^d$. Then for each $t \\in [-1,1]$ and $\\xi \\in \nS^{d-1}$, \n\\begin{equation} \\label{R-OP}\n   {\\mathcal R} P(\\xi, t) = b_{d-1} (1-t^2)^{\\frac{d-1}{2}} \n       \\frac{C_n^{d\/2}(t)}{C_n^{d\/2}(1)} P(\\xi). \n\\end{equation}\nIn particular, the above formula applies to harmonic polynomials of \ndegree $n$. \n\\end{prop}\n\n\\begin{proof}\nLet $Q_\\xi$ be an orthogonal matrix whose first row is $\\xi$. \nThen \\eqref{Radon} shows that\n$$\n   {\\mathcal R} P(\\xi,t) = (1-t^2)^{\\frac{d-1}{2}} \\int_{B^{d-1}} \n                            P( (t, \\sqrt{1-t^2}\\, y)Q_\\xi)dy.  \n$$      \nThe integral is a polynomial of $t$ since an odd power of \n$\\sqrt{1-t^2}$ is always companioned by $y^\\alpha$ with\n$|\\alpha|$ being odd, which has integral zero. Therefore,\n$g(t) = (1-t^2)^{-\\frac{d-1}{2}} {\\mathcal R} f(\\xi,t)$ is of degree $k$\nin $t$. Furthermore, the integral shows that \n$$\ng(1) = \\operatorname{vol} (B^{d-1}) P(\\xi) =  b_{d-1} P(\\xi).   \n$$\nIf $g_j \\in \\Pi_j^d$ for $0 \\le j \\le n-1$, then the equation \n\\eqref{IntRf} and the fact that $P \\in {\\mathcal V}_n^d$ lead to \n$$\n  \\int_{-1}^1g(t)  g_j(t) (1-t^2)^{\\frac{d-1}{2}} dt \n    = \\int_{B^d} P(x) g_j(\\langle x,\\xi \\rangle)dx =0,\n$$\nwhich shows immediately that the polynomial $g(t)$ is an\northogonal polynomial with respect to $(1-t^2)^{\\frac{d-1}{2}}$\non $[-1,1]$, that is, \n$$\n    g(t) = (1-t^2)^{-\\frac{d-1}{2}} {\\mathcal R} f(\\xi,t) = a\\, C_n^{d\/2}(t).\n$$\nSetting $t =1$ determines the constant $a$ and completes the\nproof. Finally, \\eqref{OPbasis} with $k=0$ show that harmonic\npolynomials of degree $n$ are in ${\\mathcal V}_n^d$.\n\\end{proof}\n\n\\begin{cor}\nLet $f_{k,j}^n$ be the orthonormal basis of ${\\mathcal V}_n^d$ given\nin \\eqref{OPbasis}. Then\n$$\n  {\\mathcal R} f_{k,j}^n(\\xi,t) = \\gamma_n g_{k,j}^n(\\xi,t),\n$$\nwhere the singular values $\\gamma_n$ of ${\\mathcal R} f$ are \ngiven by\n\\begin{equation} \\label{svd}\n       {\\gamma}_n =  b_{d-1} \\sqrt{n!\/(d)_n}.     \n\\end{equation}\n\\end{cor} \n\n\\begin{proof}\nUsing \\eqref{f_xi} and \\eqref{gkj}, the equation \\eqref{R-OP}\nshows \n$$\n {\\mathcal R} f_{k,j}^n(\\xi, t) = b_{d-1} [h_n^{(d\/2)}]^{1\/2}\n    (1-t^2)^{\\frac{d-1}{2}}\n      \\frac{C_n^{d\/2}(t)}{C_n^{d\/2}(1)} = {\\gamma}_n  g_{k,j}^n(\\xi,t), \n$$\nwhere ${\\gamma}_n = b_{d-1} [h_n^{(d\/2)}]^{-1\/2} H_n \/C_n^{(d\/2)}(1)$,\nwhich can be simplified by using \\eqref{Gegen}, \\eqref{Hk} and \nthe fact that $C_n^{(d\/2)}(1) = (d)_n\/n!$.\n\\end{proof}\n\n\\begin{thm} \nThe singular decomposition of ${\\mathcal R} f$ satisfies\n\\begin{align} \\label{Rf2}\n  {\\mathcal R} f=c_{d\/2} (1-s^2)^{\\frac{d-1}{2}} \n             \\sum_{n=0}^\\infty  \\left[h_n^{(d\/2)}\\right]^{-1} \n                     \\int_{B^d} f(x) C_n^{d\/2}(\\langle x,\\xi \\rangle) dx \\,\n                  C_n^{d\/2}(t),     \n\\end{align} \nwhere $M_m= \\dim {\\mathcal H}_{m}^d$, $c_{d\/2}$ is defined at  \n\\eqref{Gegen}; and \n\\begin{align}\\label{R*f2}\n  {\\mathcal R}^* g = c\\,\n            \\sum_{n=0}^\\infty {\\gamma}_n  \\left[h_n^{(d\/2)}\\right]^{-1} \n                         \\int_{-1}^1 \\int_{S^{d-1}} {\\mathcal R} f(\\xi,t) \n                     C_n^{d\/2}(\\langle x,\\xi \\rangle) C_n^{d\/2}(t)\n                       d\\omega(\\xi) dt.      \n\\end{align} \nwhere $c =  c_{d\/2} b_{d-1} \\sigma_d^{-1}$. \n\\end{thm}\n\n\\begin{proof}\nTo prove \\eqref{Rf2}, we note that by \\eqref{f_xi},  \n\\begin{equation}\\label{fg=ff}\n f_{k,j}^n(x) g_{k,j}^n(\\xi, t) = [h_n^{(d\/2)}]^{-1\/2} H_n^{-1}\n    (1-t^2)^{\\frac{d-1}{2}} C_n^{d\/2}(t) f_{k,j}^n(x) \n          f_{k,j}^n(\\xi). \n\\end{equation}\nSince the constants are independent of $k$ and $j$, we can use\n\\eqref{reprod2} to write the summations in $k$ and $j$ of \n\\eqref{Rf2} in a compact form. Collecting constants and using \n\\eqref{Hk}, \\eqref{svd} and \\eqref{Gegen}, we easily verify that \n$$\n   {\\gamma}_n [h_n^{(d\/2)}]^{-1\/2}  H_n^{-1} f_{k,j}^n(x) \\frac{n+d\/2}{d\/2}\n     = b_{d-1}  [h_n^{(d\/2)}]^{-1}. \n$$\nFinally we note that $b_{d-1} b_d^{-1} = c_{d\/2}$.  The proof of\n\\eqref{R*f2} is similar. \n\\end{proof}\n\nIt is worth to comment that the two expressions \\eqref{Rf2} and\n\\eqref{R*f2}\nare independent of the choice of\northonormal bases, and the equation \\eqref{reprod2} implies \nthat we can deduce the SVD from them using any \northonormal basis. In \\cite{Da, Lo}, the SVD in terms of \northogonal basis with respect to $W_\\mu$ is derived. In these \nmore general cases, however, the simple analogue of the \nsecond equations of \\eqref{Rf} and \\eqref{R*f} do not hold. The \nreason again lies in the fact that the constant in \\eqref{Hnk}\ndepends on $k$. \n\nFinally, by \\eqref{A+f}, the truncation of the expansion of $f$ \nbecomes \n\\begin{equation*}\n   S_n^*f (x) =  \\sum_{m=0}^n {\\gamma}_m^{-1} \\sum_{0\\le 2k \\le m} \n        \\sum_{j=0}^{M_{m-2k}} \n      \\langle g, g_{k,j}^m \\rangle_{L^2(Z)} f_{k,j}^m.\n\\end{equation*}\nJust as in the equations \\eqref{Rf} and \\eqref{R*f}, we can \nuse \\eqref{fg=ff} and \\eqref{reprod2} to derive a compact\nformula. The formula, however, is exactly $S_nf$.\nAs a consequence, we see the truncated SVD algorithm\nagrees with that formula \\eqref{eq:Sn}. Hence, truncated SVD \ncan be effectively implemented by using the OPED algorithm.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is well known that the efficient market price tends to follow the characteristics of Brownian Motion that strictly depends on stochastic nature\\cite{merton1975optimum}. Thus predicting the information about the short-term return of a financial portfolio is known to be difficult, but there are few exceptions. Firstly, there is a widely held belief among traders that it is possible for some computational deterministic patterns to emerge for short period of time. This belief, combined with econophysics, led the creation of algorithmic trading funds such as Prediction Company and Renaissance Technologies. Secondly, there is a concept of rational bubble that illustrates the rational behavior of people who invest in markets that clearly creating an economic bubble\\cite{diba1988theory}. \n\nWith these two facts in mind, it is possible to imagine a model with reasonable assumptions to predict rational bubble's key behaviors, such as the stock market crash after a bubble economy.\nWhile there are more sophisticated way of prediction such as log-periodic power law, there seems to be no clear white-box model that also have high prediction abilities. In this paper, by using some reasonable assumptions about the investors, a white-box model about the stock market state will be derived. The model will then be applied to two different cases, the Dow Jones Index and Bitcoin market prices to assess the ability of the model.\n\n\\subsection{Previous Study on Rational Bubble and its Predictions}\n\nTheories and methodologies of complex systems helped studies and individuals to make attempts to solve the prediction problems. Log-periodic power law(LPPL) is one of them\\cite{sornette2009stock}, illustrating the dynamics of two conflicting group of investors, value investors and trend followers\\cite{ide2002oscillatory}. There are some successful Implementation cases of this model as well, demonstated by Yan, Woodarda, Sornette\\cite{yan2010diagnosis}. \n\n\n\\section{Model and its Assumptions}\n\n\\subsection{Definition of Rational Bubble}\n\nA rational bubble in this paper's context is the state of a market that the majority of the new investors are investing because of the past market price or any major index. For example, if investors of NASDAQ invest in NASDAQ only because the past NASDAQ index was rising, it is called a rational bubble in this paper. There is more formerly known definition about this particular word. However for the convenience of understanding the model's key points, the definition will be little vague for now.\n\n\\subsection{Assumptions}\n\nBecause investing in markets with rational bubble is caused by the price change, these investors are only interested in price or similar indices. This causes the feedback loop in the market, as a self-fulfilling enthusiasm caused by the investors themselves\\cite{yan2010diagnosis}. There, if the price point $p_\\alpha$ where average return of the investors is same as the market return is higher than the current market price point, it is somewhat true that most investors' portfolio is showing less return than the market presents. This is because not all investors invested in the lowest price point, making the average return of the portfolios grow slower than the market return during the bubble even though the market price change is same to every investor. Thus, if the market price exceeds price point $p_\\alpha$, the investors expectation for return caused by the superficial market return is not satisfied. This makes the investors behave more irrationally, or doubt the market, as the root assumption of their investment is not met. This potential tendency could cause massive market crash just with somewhat minor price change, causing the doubt to become a real action. All of this potential threat is possible after the minimum price point where market return is same as the average return of the investors.\n\n\n\\subsection{Mathematical Derivation of the Model}\n\nFor convenience, it is assumed that every investors invest in the market portfolio. If a certain price point is $p_x$ and the price before the bubble is $p_0$ and assuming that investors of the bubble market started investing after the start of the bubble, the average return at price point $p_x$ of every single stock in a continuous form is as follows.\n\n$$E(R) = {\\int_{p_0}^{p_x} R_p V_p \\, d p \\over \\int_{p_0}^{p_x} V_p \\, d p} $$\n\n$R_p$ is the current return of a market portfolio in interest which is bought at price $p$, and $V_p$ is the volume of a market portfolio in interest which is bought at price $p$.\nAs for the volume, the trading volume will be used for approximation. Also, as the exact data of trading volume could not be used because the discrete nature of trading volume, the trading volume will be approximated by a regression line which shows the relationship between price and trading volume. The formula of the regression line is as follows.\n\n$$v = a p + b = {r_{v p} S(v)\\over S(p)}  p + \\left(E(v) - {r S(v)\\over S(p)} E(p)\\right)$$\n\n$v$ is volume of trades, $p$ is the price, $S(v)$ and $S(p)$ are the standard deviation of $v$ and $p$,  $E(v)$ and $E(p)$ are the average of $v$ and $p$, and $r_{vp}$ is the correlation coefficient of the two variable $v$ and $p$. By using this regression line, the model could be simplified to the following form.\n\n$$E(R) = {\\int_{p_0}^{p_x} {p_x \\over p} (a p +b) \\, d p \\over\\int_{p_0}^{p_x} ap + b \\, d p } = {a p_x(p_x - p_0 ) + p_x b \\ln \\left({p_x\\over p_0}\\right) \\over 0.5 a p_x^2 - 0.5 a p_0^2 + b(p_x - p_0)}$$\n\n\nThe resulting average return is in $1+R$ format. As the stability of the market is removed when the average return of the investors becomes substantially less than the market return during the bubble, The minimal price point where market becomes unstable will be the price where market return $R_M(p_x)$ is\nsame as the average return $E(R)$. In other words, we are finding for $p_\\alpha$ where\n\n$$M_R(p_\\alpha) = {p_\\alpha \\over p_0} = E(R) ={a p_\\alpha(p_\\alpha - p_0 ) + p_\\alpha b \\ln \\left({p_\\alpha\\over p_0}\\right) \\over 0.5 a p_\\alpha^2 - 0.5 a p_0^2 + b(p_\\alpha - p_0)} .$$Lastly, the average return $E(R)$ is compared with the current market return $M_R$ for further evaluation.\n\n\n\n\\section{Practical Implementation}\n\nAs for any model, practical implementation is as important as its thorough mathematical structure. Designing an implementation trial for this model is quite difficult task. Because the model is only designed for assessing markets in some form of the rational bubble, it is hard to determine the sample of the market data that fits the terminology \"rational bubble\". Also, the trial should be capable of assessing specificity as well. To satisfy all of this requirements, testing for every possible time span of 50 trading days had been considered to be the best option. The model's performance is assessed visually. Also, an implementation trial include non-traditional markets as well, for testing the generalization capabilities of the model. \n\nUnlike the clarity of the model's assumptions and structure, most of the model's components are not set for practical usage. Therefore, some carefully designed methodology should be used for implementation of the model. Although the specific methodology that been used for this paper's trial is confidential, some of the key problems will be introduced, and the trial's result will be shown and discussed. \n\n\\subsection{Data Preparation}\n\nFor the stock market data, the Dow Jones Index from January 1999 to August of 2021 is been used, and for the non-traditional market data, the Bitcoin market price data from January 2016 to August of 2021 is been used.\n\n\\subsection{Problems with Implementation}\n\nThere are some problems to solve before applying the model to the real world data. First, the regression line between trade volume and price should be calculated without problematic issues like assigning discrete data of volume to the continuous interval of price. Also, as the model includes logarithm, some solvers are incapable of solving the equation $M_R(p_\\alpha) = E(R)$ for $p_\\alpha$ .\n\nHowever these problems could be solved by simple practical solutions. The method that used for this trial is confidential. \n\n\\subsection{Implementation Results and Discussion}\n\nWhen applied to Dow Jones, the model showed significant prediction about 2008 financial crisis and 2020 stock market crash, which are located at around index 2100 and 5300. In Other cases, the model come out with many contradicting results during short time span or show weak predictions. However, the model seems to consider the time span mid 2020 to mid 2021 as the precursor of another market crash. This might means that the model's specificity is lower than expected, or just the error occurred while dealing with wider price gap. Other explanation is that the stock market is really in the rational bubble, but the bubble is sustained by the stimulus plan of US government.\nAlso, the model seems to also consider the short time span after the market crash as the market crash itself. However this error can be easily adjusted by human evaluator.\n\n\\begin{center}\n\n\\begin{figure}[h]\n\n\\includegraphics[width=16cm]{model.png}\n\\caption{Application of the model to Dow Jones Index. Black line indicates Dow Jones index, and blue color line over or under the black line indicates the model's prediction of price change direction.}\n\\label{fig:mesh2}\n\\end{figure}\n\n\\end{center}\n\n\nAs for the bitcoin market, the model successfully predicted two major crashes, each in 2017 and 2021. \n\n\n\\begin{center}\n\\begin{figure}[h]\n\\includegraphics[width=16cm]{model_bit.png}\n\n\\caption{Application of the model to Bitcoin makert price. Black line indicates Dow Jones index, and blue color line over or under the black line indicates the model's prediction of price change direction.}\n\\label{fig:mesh1}\n\\end{figure}\n\\end{center}\n\n\n\\section{Conclusion}\n\nWhile having some implementation issues and prediction errors, the model presented in this paper suggest the logical explanation and prediction for irrational economic behaviors. Although its capability of clear prediction is limited at this point, we expect to improve the model and its implementation methodology and use the model as the potential index for understanding the irrational nature of our economy.\n\n\\vskip 0.2in\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nTheories of fully developed turbulence rely on the energy conservation in inviscid flows to derive the scaling laws of a turbulent system. However, energy is not the only quadratic invariant of the three dimensional (3D) incompressible Euler equation, there is also the helicity\n\\begin{equation}\nH =\\int{d^3 x \\, \\textbf{v}(\\textbf{x}) \\cdot \\textbf{w}(\\textbf{x})},\n\\end{equation}\nwhere $\\textbf{v}(\\textbf{x},t)$ is the velocity field and $\\textbf{w}(\\textbf{x},t)=\\nabla \\times \\textbf{v}$ is the vorticity field \\citep{Moffatt69}.\nHelical motions are observed in a wide variety of geophysical flows, such as supercell convective storms which seem to owe their long life and stability to the helical nature of their circulation \\citep{lilly86}. In astrophysics it is well known that helicity plays a key role in the generation of large scale magnetic fields through dynamo action \\citep{pouquet76, moffat78}. In hydrodynamic isotropic and homogeneous turbulence, recent studies of flows with net helicity confirmed that there is a joint cascade of energy an helicity to smaller scales \\citep{Borue, Chen03, Gomez}, as expected from theoretical arguments \\citep{Brissaud}. In the case of non-helical flows, velocity and vorticity tend to align locally creating patches of strong positive and negative helicities. In helical flows, helicity still fluctuates strongly around its mean value, both in space and time, creating locallized patches with helicity much larger or smaller than the mean. As helicity is not positive definite, the study of its turbulent fluctuations is difficult and has received less attention. The scaling laws followed by helical fluctuations are unclear, and few comparisons are available between helical and non-helical flows to understand their impact on the energy cascade. It has been proposed that helical structures may have an impact on the dynamics of turbulent flows as helical regions have the velocity and vorticity aligned, with a resulting quenching of the non-linear term in the Navier-Stokes equation \\citep{Tsinober}. Intermittency in the helicity flux has also been recently studied in numerical simulations \\citep{Chen203}, and the geometry of helical structures was considered using wavelet decompositions \\citep{Farge} and Minkowski functionals in the magnetohydrodynamic case \\citep{anvar07}.\n\nIn this work we use the the cancellation exponent \\citep{Ott92} to study the fast fluctuations of helicity in turbulent helical and non-helical flows. The cancellation exponent was introduced to study fast changes in sign of fields on arbitrarily small scales, and is based on the study of the inertial range scaling of a generalized partition function built on a signed measure. Under some conditions, it was shown to be related to the fractal dimension of structures in the flow and to the Hold\\\"er exponent \\citep{Sreenivasan}. It was used to characterize fluctuations in hydrodynamic turbulence and magnetohydrodynamic dynamos \\citep{Ott92}, as well as in two dimensional magnetohydrodynamic turbulence \\citep{Pouquet02, Jonathan}. In solar wind observations it was used to show that magnetic helicity is sign singular \\citep{Bruno97}. However, to the best of our knowledge, no studies of helical fluctuations in hydrodynamic turbulence have been done using the cancellation exponent. We analyze data stemming from direct numerical simulations (DNSs) of the Navier-Stokes equation in a three dimensional periodic box at large resolutions (up to $1024^3$ grid points) with different external mechanical forcings. Both helical and non-helical flows are considered. We find that helicity fluctuations are sign singular in both cases, and the scaling of helicity fluctuations seems to be independent of the net helicity of the flow. Furthermore, we obtain a relation between the helicity cancellation exponent, the fractal dimension of helical structures and the first order scaling exponent of helicity.\n\n\\section{The cancellation exponent}\nThe cancellation exponent \\citep{Ott92} was introduced to characterize the behavior of measures that take both positive and negative values, and to quantify a form of singularity where changes in sign occur on arbitrarily small scales. We can introduce the cancellation exponent for the helicity considering a hierarchy ${Q_{i}(l)}$ of disjoint subsets of size $l$ covering the entire domain occupied by the fluid $Q(L)$ of size $L$. For each scale $l$, a signed measure of helicity is introduced as\n\\begin{equation}\n\\mu_{i}(l)=\\int_{Q_{i}(l)}{d^3 x \\; H(\\textbf{x})}\\bigg\/\\int_{Q(L)}{d^3 x \\; |H(\\textbf{x})|},\n\\label{eq:mu}\n\\end{equation}\nwhere $H(\\textbf{x}) = \\textbf{v}(\\textbf{x}) \\cdot \\textbf{w}(\\textbf{x})$ is the helicity density such that the total helicity is $H=\\int d^3x H(\\textbf{x})$. Since the normalization factor in Eq. (\\ref{eq:mu}) is the integral of the absolute value of $H$ over the entire domain, the signed measure is bounded betwen $1$ and $-1$ and can be interpreted as the difference between two probability measures, one for the positive component and  another for the negative component of the helicity density. To study cancellations at a given length scale, we build the partition function\n\\begin{equation}\n\\chi(l)=\\sum_{Q_{i}(l)}{|\\mu_{i}(l)|}.\n\\label{eq:1}\n\\end{equation}\nIn the inertial range, the scaling law followed by the cancellations can be studied by fitting\n\\begin{equation}\n\\chi(l)\\sim l ^{-\\kappa},\n\\label{eq:chi}\n\\end{equation}\nwhere $\\kappa$ is the cancellation exponent. This exponent represents a quantitative measure of the cancellation efficiency. If such a scaling law exists and its range increases as the inertial range increases, the signed measure is called sign singular. In this case, changes in sign occur everywhere and in any scale considered in the limit of infinite Reynolds number. On the other hand, for a smooth field $\\kappa=0$.\n\n\\section{Code and numerical simulations}\nThe data we use for the analysis stems from DNS of the incompressible Navier-Stokes equation with constant mass density,\n\\begin{equation}\n\\frac{\\partial \\textbf{v}}{\\partial t}+\\textbf{v}\\cdot \\bigtriangledown \\textbf{v}=-\\bigtriangledown p+\\nu\\bigtriangledown^{2}\\textbf{v} + \\textbf{f},\n\\label{eq:Navier-Stokes}\n\\end{equation}\n\\begin{equation}\n\\bigtriangledown \\cdot \\textbf{v}=0.\n\\label{eq:Incompressibility}\n\\end{equation}\nwhere $\\textbf{v}$ is the velocity field, $p$ is the pressure (divided by the mass density), $\\nu$ is the kinematic viscosity, and $\\textbf{f}$ represents an external force that drives the turbulence. \n\nFor the analysis, we consider six numerical simulations at different Reynolds numbers and spatial resolutions, with different forcing functions that inject either energy or both energy and helicity. The simulations are listed in Table \\ref{tab_Tabla1} and described in more detail in a previous paper \\citep{Alexakis06}. Spatial resolutions range from $256^3$ to $1024^3$ grid points, with Reynolds numbers $R_e = UL\/\\nu$ (with $U$ the rms velocity and $L$ the integral scale) ranging form $\\approx 675$ to $\\approx 6200$, and Taylor Reynolds numbers $R_e = U\\lambda\/\\nu$ (where $\\lambda$ is the Taylor scale) ranging from $\\approx 300$ to $\\approx 1100$. Two forcing functions were considered: Taylor-Green (TG) forcing, which injects no helicity in the flow (although helicity fluctuations around zero develop as a result of nonlinearities in the Navier-Stokes equation), and Arn'old-Beltrami-Childress (ABC) forcing, which injects maximum (positive) helicity in the flow. The forcings were applied respectively in the shells in Fourier space with wavenumber $k_F=2$ and 3. All runs were continued for over ten turnover times after reaching the turbulent steady state. As a measure of the helicity content of each flow, we computed a relative helicity $\\rho_{H} = H\/(k_F E)$ (where $E$ is the total energy) averaged in time over the turbulent steady state of each run (see Table \\ref{tab_Tabla1}).\n\n\\begin{table}\n\\begin{center}\n\\begin{minipage}{7cm}\n\\begin{tabular}{@{}cccccccc@{}}\nRun & $N$  & $\\textbf{F}$ & $k_{F}$ & $\\rho_H$& $\\nu$ &$R_{e}$ & $R_{\\lambda} $    \\\\[3pt]\nT1 &$256$ & $TG$  & $2$ & $0.03$ & $2 \\times 10^{-3}$   &$675$  &  $300$ \\\\\nT2 &$512$ & $TG$  & $2$ & $0.01$ & $1.5 \\times 10^{-3}$ &$875$  &  $350$ \\\\\nT3 &$1024$& $TG$  & $2$ & $0.03$ & $3\\times 10^{-4}$    &$3950$ &  $800$ \\\\\nA1 &$256$ & $ABC$ & $3$ & $0.79$ & $2 \\times 10^{-3}$   &$820$  &  $360$ \\\\\nA2 &$512$ & $ABC$ & $3$ & $0.81$ & $2.5 \\times 10^{-4}$ &$2520$ &  $670$ \\\\\nA3 &$1024$& $ABC$ & $3$ & $0.83$ & $2.5 \\times 10^{-4}$ &$6200$ &  $1100$\n\\end{tabular}\n\\end{minipage}\n\\end{center}\n{\\caption[]{Runs used for the analysis. $N$ is the linear grid resolution, $\\textbf{F}$ is the forcing, either Taylor-Green (TG) or Arn'old-Beltrami-Childress (ABC), $k_{F}$ is the forcing wave number, $\\rho_H$ is the relative helicity, $\\nu$ is the kinematic viscosity, $R_{e}$ is the Reynolds number, and $R_{\\lambda}$ is the Reynolds number based on the Taylor scale.}\n\\label{tab_Tabla1}}\n\\end{table}\n\n\\section{Results and analysis}\nTo study scaling laws of helicity fluctuations in the inertial range of helical and non-helical flows, we performed a signed measure analysis and computed the cancellation exponent $\\kappa$ for all runs. Following Eq. (\\ref{eq:chi}), this can be done by computing $\\chi(l)$ and fitting $\\chi(l) \\sim l^{-\\kappa}$ in the inertial range.\n\nAlthough a detailed analysis of the simulations can be found in \\citet{Alexakis06}, in Fig. \\ref{fig:espectros} we show as a reference the energy spectrum of the two simulations at the larger Reynolds numbers attained (runs A3 and T3). The spectra present a short inertial range followed by a bottleneck, as is usual in simulations of hydrodynamic turbulence. The runs also have a range with approximately constant energy flux, as well as a range of scales consistent with Kolmogorov 4\/5 law when third-order structure functions are computed. As usual, these ranges have slightly different extensions depending on the quantity studied, and it will be shown that the cancellation exponent shows a scaling range comprised betwen $l\\approx 0.08$ and $\\approx 0.7$, which correspond roughly to wavenumbers $k\\approx 8$ and $\\approx 80$, which is wider than but includes the inertial range of both runs. At the lower resolution, the inertial range in the energy spectrum and flux, or in the third-order structure function, is shorter and harder to identify.\n\\begin {figure}\n\\begin{center}\n\\includegraphics[width=14.0cm]{fig1.pdf}\n\\caption {Energy spectrum for the A3 (left) and T3 (right) runs. Kolmogorov scaling is indicated as a reference.}\n\\end{center}\n\\label{fig:espectros}\n\\end{figure}\n\nFig. \\ref{fig:cancelacionp} (left) shows the signed measure of helicity as a function of the scale for runs T1, T2, and T3 from top to bottom (non-helical, with increasing Reynolds numbers). As the Reynolds number increases, a wider scaling range can be identified. The values obtained for the cancellation exponent are $\\kappa = 0.92 \\pm 0.09$, $0.75 \\pm 0.03$, and $0.73\\pm0.01$ respectively for runs T1, T2, and T3. The values obtained and the dependence with Reynolds number support the idea that in the limit of vanishing viscosity, helicity fluctuations are sign singular, i.e., that fast changes in sign of helicity occur everywhere in arbitrarily small scales in the flow. This result is consistent with helicity cascading towards small scales, and with observations of the helicity distribution being highly intermittent \\citep{Chen203} (albeit the previous studies are only for flows with net helicity).\n\\begin {figure}\n\\begin{center}\n\\includegraphics[width=6.5cm]{fig2a.pdf}\n\\includegraphics[width=6.5cm]{fig2b.pdf}\n\\end{center}\n\\caption {Signed measure of helicity as a function of the scale for runs with increasing resolution, from top to bottom: TG forcing (left), and ABC forcing (right) after subtracting the mean value of helicity.  Slopes with the cancellation exponent are given as a reference.}\n\\label{fig:cancelacionp}\n\\end{figure}\n\nAs mentioned earlier, the flows with ABC forcing are helical. For such flows the direct cascade of helicity and its intermittency has been studied before using spectral quantities. Although helicity fluctuations develop in these flows, given the predominant sign of helicity, fluctuations occur around the mean value and less changes in sign take place. Indeed, if the signed measure of helicity and the cancellation exponent is computed for run A3, we obtain $\\kappa = 0.26 \\pm 0.04$. This smaller value of the exponent is consistent with the less cancellations, and the larger error is associated to the fact that larger fluctuations are observed in the signed measure as less events whith changes of sign are available. However, we are interested in helicity fluctuations around the mean value, and to correctly consider these fluctuations the mean value of helicity $H$ was subtracted from the helicity density in runs A1, A2, and A3 prior to the analysis. The resulting signed measures are also shown in Fig. \\ref{fig:cancelacionp} (right). In this case we also observe scaling of $\\chi(l)$ expanding over a wider range of scales as the Reynolds number increases, indicating that also for helical flows helicity fluctuations are sign singular.\n\nThe results obtained for the helicity cancellations in helical and non-helical runs are similar. The cancellation exponents are $\\kappa = 0.89 \\pm 0.07$, $0.80 \\pm 0.03$, and $0.72\\pm0.01$ respectively for runs A1, A2, and A3. At the highest resolution three ranges can be identified in both the A3 and T3 runs. At the smaller scales, viscous effects dominate and the flow is smooth. This results in less changes of sign in helicity and a shallower distribution of the signed measure. The slope tends to zero, as expected for a smooth flow, and the signed measure tends to one for the smallest resolved scale. At intermediate scales an inertial range is observed, and at larger scales the results are affected by the external forcing. In this scales, large fluctuations in the value of $\\chi(l)$ are observed as a result. As mentioned before, the inertial range of $\\chi(l)$ is wider than the inertial range observed in the energy spectrum. This is consistent with observations of the helicity spectrum decaying slower than the energy spectrum near the dissipative range in helical turbulence \\citep{Alexakis06}. \n\nThe cancellation exponent obtained in all runs as a function of the Taylor based Reynolds number is shown in Fig. \\ref{fig:kappa} (a). For both the helical and non-helical runs $\\kappa$ decreases with the Reynolds number, and for the largest Reynolds numbers studied the value of $\\kappa$ seems to saturate and is the same within error bars for both helical and non-helical flows. Note that because of the different forcing functions, the helical and non-helical runs have slightly different Reynolds numbers even when the same grid resolution is used. The similarities between the scaling laws of the helical and non-helical flows suggests that statistics of the helicity fluctuations are the same in both cases. This is further confirmed by a histogram of the helicity fluctuations around its mean value for the T3 and A3 runs. As Fig. \\ref{fig:kappa} (b) shows, turbulent fluctuations of helicity are similar, with small differences on the tails. \n\n\\begin {figure}\n\\begin{center}\n\\includegraphics[width=6.5cm]{fig3a.pdf}\n\\includegraphics[width=6.5cm]{fig3b.pdf}\n\\end{center}\n\\caption {(a) Cancellation exponent $\\kappa$ as a function of the Taylor based Reynolds numbers for the helical (triangles) and the non-helical (diamonds). (b) Histogram of helicity fluctuations for the T3 (solid) and A3 (dashed) runs.}\n\\label{fig:kappa}.\n\\end{figure}\n\nHelicity fluctuations are sign singular. The fast oscillations in sign point to a highly intermittent quantity that fluctuates rapidly in localized structures at arbitrarily small scales for infinite Reynolds numbers. As mentioned in the introduction, the cancellation exponent can be related to the geometry of such structures, and under some conditions these relations can be rigorously derived \\citep{Sreenivasan}. The relations can be found also using simple geometrical arguments. In the following, we extend the analysis of \\citet{Pouquet02} for the current density in magnetohydrodynamics to consider the case of kinetic helicity in hydrodynamic turbulence. We assume the helicity is spatially correlated (although not necessarily locally smoth) in $D$ dimensions and uncorrelated in $3-D$ dimensions, where $3$ follows from the dimension of the space. With this choice, a correlated helicity distribution has $D=3$, and a completely uncorrelated helicity distribution has $D=0$. For intermediate cases, we can estimate the signed measure of helicity as follows\n\\begin{eqnarray}\n\\chi(l) = \\sum_{Q_{i}(l)}{\\left|\\int_{Q_i(l)}{d^3 x \\, H({\\bf x})} \\, \\bigg{\/}\\nonumber \n    \\int_{Q(L)}{d^3 x \\, |H({\\bf x})|}\\right|}\\\\  \n           \\sim \\frac{1}{L^3\\langle H^{2}\\rangle^{1\/2}}\n            \\left(\\frac{l}{L} \\right)^3\\left|\\int_{Q(l)}{d^3x H(\\textbf{x})}\\right|, \n\\label{eq:fractal}\n\\end{eqnarray}\nwhere we used homogeneity to replace the sum over all subsets $Q_{i}(l)$ by $(L\/l)^3$ (the total number of terms in the sum) times the integral over a generic box $Q(l)$ of size $l$. Since the absolute values in the denominator prevent any cancellation, the integral below was approximated by the characteristic value $L^{3}H_{rms}$ where $H_{rms}=\\langle H^{2}\\rangle^{1\/2}$ is the rms value of the helicity density.\n\nIn the inertial range, and still under homogeneity asumptions, the helicity follows some scaling law which can be represented by helicity structure functions $\\left\\langle \\delta H(s)\\right\\rangle \\sim s^{h}$, where $s=|\\textbf{s}|$ represents a lineal increment. We can then split the integral over the domain $Q(l)$ in integrals over subdomains with volume $\\lambda^3$ such that they separate the contributions from domains where the helicity is correlated of domains where the helicity is uncorrelated in space. Then, the remaining integral in Eq. (\\ref{eq:fractal}) can be estimated as \\citep{Pouquet02}\n\\begin{equation}\n\\left|\\int_{Q(l)}{d^3x \\; H(\\textbf{x})}\\right| \\sim\n\\langle H^{2}\\rangle^{1\/2} \\int_{Q(l)} d^Ds \\left( \\frac{s}{\\lambda}\\right)^{h}\\int_{Q(l)} d^{3-D}s .\n\\label{eq:contri}\n\\end{equation}\nThe uncorrelated dimensions give a value proportional to the square root of their volume $(l\/ \\lambda)^{(3-D)\/2}$. The correlated dimensions give a contribution proportional to $(l\/\\lambda)^{h+D}$, wich for $h=0$ (locally smoth helicity) is proporcional to their volume. Replacing this results in Eq. (\\ref{eq:fractal}), we get that the signed measure of helicity $\\chi(l)$ scales as\n\\begin{equation}\n\\chi(l) \\sim l^{-\\left(\\frac{3-D}{2}-h\\right)}. \n\\label{eq:final}\n\\end{equation}\n\n\\begin {figure}\n\\begin{center}\n\\includegraphics[width=14.0cm]{fig4.pdf}\n\\end{center}\n\\caption {Contour levels of helicity density in a slice of the T3 run. Only $1\/4$ of the slice is shown. Left: contours with $H({\\textbf x})=0$, i.e., places where cancellations take place. Right: contours with $H({\\textbf x})=12$. Note how contour levels of zero helicity fill the space, while white patches are observed in the contour levels of non-zero helicity, as well as filamentary structures.}\n\\label{fig:filamentos}\n\\end{figure}\n\nThus the cancellation exponent $\\kappa$, the fractal dimension $D$, and the scaling exponent $h$ are related through\n\\begin{equation}\n\\kappa=\\frac{3-D}{2}-h.\n\\label{eq:rel}\n\\end{equation}\nA completely smooth field ($D=3$) has $\\kappa=h=1$, in agreement with the definition of $\\kappa$ and of the first order (H\\\"older) scaling exponent $h$ for the field (see \\citet{Bruno97}). In a turbulent flow, the global helicity is observed to follow Kolmogorov scaling, and therefore we can assume $h=1\/3$ \\citep{kurien, Alexakis06} . Then from the value of $\\kappa=0.73\\pm0.01$ it follows that helical structures are filamentary with fractal dimension $D=0.873\\pm0.005$, in good agreement with previous observations of helical vortex filaments \\citep{Tsinober, Farge, Chen203, Alexakis06}. As an illustration, Fig.\\ref{fig:filamentos} shows a slice of the helicity density in the T3 run. Filamentary structures can be identified, specially for the contour levels corresponding to large concentration of helicity. Structures less elongated were confirmed to correspond to filaments that cross the slice perpendicularly. \n\n\\section{Conclusions}\nSigned measures of helicity were calculated for turbulent flows with different Reynolds numbers and helicity contents. For both helical and non-helical flows cancellation exponents were calculated, obtaining a value compatible with $\\kappa=0.73\\pm0.01$ for helicity fluctuations for both cases at the largest Reynolds number considered. The scaling range increases with the Reynolds number, and the value of $\\kappa$ obtained indicates helicity is sign singular in the limit of infinite Reynolds number. Simple geometrical arguments were used to link the cancellation exponent $\\kappa$ to the fractal dimension $D$ of the structures. To do this we assumed a single scaling exponent $h$ for the lineal helical increments. Assuming $h=1\/3$, a value consistent with numerical results, we obtain that helicity fluctuations are filamentary with fractal dimension of $D=0.873\\pm0.005$, also in agreement with observations.\n\n\\begin{acknowledgments}\nThe authors acknowledge support from Grants No. UBACYT X468\/08 and PICT-2007-02211. PDM acknowledges support from the Carrera del Investigador Cient\\'{\\i}fico of CONICET.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis paper continues the theory of dispersive blow up which was \ninitiated and developed  in \\cite{BS1} and \\cite{BS2}.  The present contribution is especially relevant to nonlinear \nSchr\\\"odinger-type equations, and includes theory for the Davey-Stewartson and the Gross-Pitaevskii equations. The work on \n Schr\\\"odinger equations substantially extends the results already available in \\cite{BS2}.  \n\nDispersive blow up of wave equations is a phenomenon of focusing of \nsmooth initial disturbances with finite-mass (or, finite-energy, depending on the physical context) that \nrelies upon the dispersion relation guaranteeing that, in the\nlinear regime, different\nwavelengths propagate at different speeds.   This is especially the \ncase for models wherein the linear dispersion is unbounded, so that energy can\nbe moved around at arbitrarily high speeds, but even bounded dispersion\ncan exhibit this type of singularity formation.  \n\nTo be more concrete, consider the Cauchy problem \nfor the linear (free) Schr\\\"odinger equation\n\\begin{equation}\\label{eq:linSch}\ni \\partial_t u + \\Delta u = 0, \\quad u\\big|_{t=0} = u_0(x),\n\\end{equation}\nwhere $x\\in {\\mathbb R}^n$ for some $n\\in {\\mathbb N}$.  For $u_0\\in L^2({\\mathbb R}^n)$, elementary \nFourier analysis shows the solution to this initial-value problem is\n\\begin{equation}\\label{eq:formula}\nu(x,t) = e^{i t \\Delta} u_0(x):= \\frac{1}{(2\\pi)^n} \\int_{{\\mathbb R}^n} e^{-i t |\\xi|^2} \\widehat u_0(\\xi) e^{i \\xi\\cdot x} d\\xi.\n\\end{equation}\nHere, $\\widehat u_0$ denotes the Fourier transformed initial data, {\\it viz}.  \n\\[\n\\mathcal Fu_0(\\xi) \\equiv\\widehat u_0(\\xi)= \\int_{{\\mathbb R}^n} u_0(x) e^{-i \\xi \\cdot x} \\, dx.\n\\]\nThe corresponding inverse Fourier transform will be denoted by $\\mathcal F^{-1}$. From \\eqref{eq:formula}, it is immediately inferred that for any $s \\in {\\mathbb R}$, solutions lie in $C({\\mathbb R};H^s)$ \nwhenever $u_0$ lies in the $L^2$-based Sobolev space $H^s$.  \nMoreover,  the evolution  preserves all these Sobolev-space   \nnorms, which is to say \n\\[\\| u(\\cdot, t) \\|_{H^s({\\mathbb R}^n)} = \\| u_0 \\|_{H^s(R^n)} \\] \nfor $t\\in {\\mathbb R}$.   In certain applications of this model, the case $s=0$ in the last formula\n corresponds to \nconservation of total mass in the underlying physical system.\n\nHowever, in Theorem 2.1 of \\cite{BS2}, it was shown that for any given  point $(x_*, t_*)\\in  {\\mathbb R}^{n}\\times {\\mathbb R}_+$, \nthere exists initial data  $u_0\\in C^\\infty({\\mathbb R}^n)\\cap L^2({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ \nsuch that the solution $u(x,t)$ of the corresponding initial-value problem \\eqref{eq:linSch} \nfor the free Schr\\\"odinger equation  is continuous on ${\\mathbb R}^n\\times {\\mathbb R}_+ \\setminus \\{(x_*, t_*)\\}$, but\n\\[\n\\lim _{(x,t)\\in {\\mathbb R}^{n}\\times {\\mathbb R}_+\\to (x_*, t_*)} | u(x,t)| = + \\infty.\n\\] \nThis fact is referred to as (finite-time) dispersive blow up and will \nsometimes be abbreviated  DBU in the following. \nThe analogous phenomena also appears in other linear dispersive equations, such \nas the linear  Korteweg-de Vries equation \\cite{BBM} and the  free surface \nwater waves system \nlinearized around the rest  state \\cite{BS2}.\n\n\nAt first sight, one would expect that nonlinear terms would destroy \ndispersive blow up.  What is a little surprising is that even the \ninclusion of physically relevant nonlinearities in various models \nof wave propagation does not prevent dispersive blow up.  Indeed, \ntheory shows in some important cases that initial data \nleading  to this focusing singularity under the linear evolution continues to \nblow up in exactly the same way when nonlinear terms are \nincluded.  \nIn \\cite{BS1}, this was shown to be \ntrue for the Korteweg-de Vries equation, a model for shallow water waves and other \nsimple wave phenomena.  This result and\nanalogous dispersive blow up theory in \\cite{BS2} for  \nsolutions of the one-dimensional nonlinear Schr\\\"odinger equations,\n\\begin{equation}\n\\label{**}\ni \\partial_t u + \\partial^2_{x} u \\pm |u|^pu=0, \\quad u\\big|_{t=0} = u_0(x),\n\\end{equation}\nwhere $x\\in {\\mathbb R}$ and $p \\in (0,3)$,\n lead to the \nspeculation that such focusing might be one road to the formation of rogue waves \nin shallow and deep water and in nonlinear optics   \n(see \\cite{D, DGE, KPS, SRKJ}).  \n\nThe analysis put forward in \\cite{BS1} and \\cite{BS2} revolves around providing bounds on\nthe nonlinear terms in a Duhamel representation of the \nevolution. Because the phenomenon is due to the linear terms in the equation,\ndata of arbitrarily small size will still exhibit dispersive blow up, and\nindeed it can be organized to happen arbitrarily quickly.   This \nemphasizes the linear aspect of these singularities and differentiates\nit from the  blow up that occurs for some of the same\nmodels when the nonlinear term is focusing and sufficiently strong (see \\cite{SuSu} for a general overview\n of this aspect of Schr\\\"odinger equations). Moreover, even though the theory begins by showing that there are specific initial data that lead to \ndispersive blow up, the result is in fact self-improving.  Dispersive blow up\n continues to hold if this special initial data is subjected to a smooth perturbation. \n The theory further implies that there is $C^\\infty$ initial data with compact \nsupport which can be taken as small as we like that will, in finite time, become large \nin a neighborhood of a prescribed spatial point (see Remark 3.5 for more details). \nDispersive blow up thereby also serves to demonstrate ill-posedness of the considered models in $L^\\infty$--spaces.  \n\n\n\nThe aim of the present work is to generalize the results mentioned above in several respects. \nMost importantly, the dispersive blow up that in \\cite{BS2} was obtained for \\eqref{**}  will be shown to\nhold true of nonlinear Schr\\\"odinger \nequations in {\\it all dimensions} $n \\geq 1$ and for \nthe whole range of nonlinearities $p\\geq \\lfloor\\frac{n}{2}\\rfloor$, with or without a (possibly unbounded) real-valued\npotential.  Here and in the following, for $\\mu \\in {\\mathbb R}$, the quantity $ \\lfloor\\mu \\rfloor$ is\n the greatest integer less than or equal\nto $\\mu$.  Higher-order Schr\\\"odinger equations are also countenanced.  Our theory relies \nespecially on the results of Cazenave and Weissler established in \\cite{CW}. \nIn addition to  Schr\\\"odinger equations, dispersive \n blow up is proved for  Gross-Pitaevskii equations \n with non-trivial boundary conditions at infinity and for the Davey-Stewartson systems.\n\nAs a by-product of our analysis, a sharp global smoothing effect is obtained \nfor the nonlinear integral term in \nthe  equation derived from \\eqref{**} by use of Duhamel's formula.\n\nThe paper proceeds as follows:   Section 2 is concerned with some preliminaries \nwhich are mostly linear in nature.  Dispersive blow up for nonlinear Schr\\\"odinger \nequations is tackled in Section 3. Section 4 deals with the sharp \nglobal smoothing property mentioned earlier.  This latter theory, in addition to being \nof interest in itself, is used to complete the analysis in Section 3.   \nDispersive blow up for the Davey-Stewartson \nsystems then follows more or less as a corollary to the results in Sections 2 and 3.\n  The Gross-Pitaevskii equation takes center stage in \nSection 5 whilst higher-order Schr\\\"odinger equations are studied in Section 6.  \n\n\n\n\n\n\\section{Mathematical preliminaries} \\label{sec:prelim}\n\nIn this section, a review of the basic idea behind dispersive blow up \nis provided in the context of nonlinear\nSchr\\\"odinger equations.  Parts of the currently available\n theory for the linear Schr\\\"odinger group \nare also recalled in preparation for the analysis in Section 3.   \n\n\n\\subsection{Dispersive blow up in linear Schr\\\"odinger equations} \\label{sec:lin}\n\nTo understand the appearance of dispersive blow up in the solution of \\eqref{eq:linSch},\nstart by explicitly computing the inverse Fourier transformation in \\eqref{eq:formula} to\nsee that the free Schr\\\"odinger group admits the \nrepresentation \n\\begin{equation}\\label{eq:group}\nu(x,t)  = \\frac{1}{(4i \\pi t)^{n\/2}} \\int_{{\\mathbb R}^n} e^{i\\frac{|x-y|^2}{4t}} u_0(y)\\, dy,\\quad \\text{for $t\\not =0.$}\n\\end{equation}\nThis representation formula is the starting point of the following lemma.\n\\begin{lemma}\\label{lem:IC}\nLet $\\alpha \\in {\\mathbb R}$, $q\\in {\\mathbb R}^n$ and \n\\begin{equation*} \nu_0(x) = \\frac{e^{- i  \\alpha |x - q|^2}}{(1+|x|^2)^m}\\ ,\\quad \\text{with $\\frac{n}{4} < m \\le \\frac{n}{2}$.}\n\\end{equation*}\nThen, $u_0 \\in C^\\infty({\\mathbb R}^n)\\cap L^2({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ and the associated global in-time solution \n$u \\in C({\\mathbb R}; L^2({\\mathbb R}^n))$ of \\eqref{eq:linSch} has the following properties.\n\\begin{enumerate}\n\\item At the point $(x_*,t_*)=(q,\\frac{1}{4\\alpha})$, the solution $u$ in \\eqref{eq:group} blows up, \nwhich is to say, \n\\[\n\\lim _{(x,t)\\in {\\mathbb R}^{n+1}\\to (x_*,t_*)} | u(x,t)| = + \\infty,\n\\]\n\\item it is a continuous function of $(x,t)$ on  $\\, {\\mathbb R}^n\\times {\\mathbb R} \\setminus \\{t_*\\}$ and\n\\item  $u(x,t_0)$ is a continuous function of $x \\in {\\mathbb R}^n \\setminus \\{x_*\\}$.\n\\end{enumerate} \n\\end{lemma}\n\\begin{proof}   First note that $u_0 \\in C^\\infty({\\mathbb R}^n)\\cap L^2({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$, that \n$u \\in C({\\mathbb R}; L^2({\\mathbb R}^n))$ and that the $L^2$--norm of $u$ is constant in view of mass conservation. On the other hand, \n evaluating \\eqref{eq:group} at $t=\\frac{1}{4\\alpha}$ for this particular $u_0$   gives\n\\[\nu\\left(x, \\frac{1}{4\\alpha}\\right)=\\left( \\frac{\\alpha}{i\\pi} \\right)^{n\/2} e^{i \\alpha (|x|^2-|q|^2)} \\int_{{\\mathbb R}^n} e^{-2i \\alpha y\\cdot (x-q)} \\frac{dy}{(1+|y|^2)^m}.\n\\]\nThus at $x=q$, it transpires that \n\\[\n\\left| u\\left(q, \\frac{1}{4\\alpha}\\right) \\right | =  \\left( \\frac{\\alpha}{\\pi} \\right)^{n\/2}   \\int_{{\\mathbb R}^n}  \\frac{dy}{(1+|y|^2)^m}  = +\\infty\n\\]\nprovided $m \\le \\frac{n}{2}$.  Assertions (ii) and (iii) can then be proved by the same arguments as in the proof of \\cite[Theorem 2.1]{BS2}. \nIn this endeavor, it is useful to note that $(1+x^2)^{-m}$ is closely related to the inverse Fourier transform of the modified Bessel functions $K_\\nu(|x|)$, where \n$\\nu = \\frac{n}{2}-m$. \n\\end{proof}\n\nIn other words,  for any given $q\\in{\\mathbb R}^n, \\alpha \\in {\\mathbb R}$, we have\nconstructed an explicit family of bounded smooth initial data (with finite mass) for which the solution of \nthe free Schr\\\"odinger equation \\eqref{eq:linSch} exhibits dispersive blow up at  the point $(x_*, t_*) = (q, \\frac{1}{4\\alpha})$ in space and time.   This result can be immediately generalized in \nvarious ways.  The  following  sequence of remarks indicates some of them.  \n\n\n\\begin{remark}\\label{amp}\nThe same argument shows that any initial data of the form\n\\[\nu_0(x) = e^{- i  \\alpha |x - q|^2} a(x),\n\\]\nwith an amplitude $a\\in C^\\infty({\\mathbb R}^n)\\cap L^2({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ but $a\\not \\in L^1({\\mathbb R}^n)$ \nwill exhibit   dispersive blow up. \nUsing the superposition principle,  one can construct initial data which yield \ndispersive blow up at any\ncountably many isolated points in space-time ${\\mathbb R}^n \\times {\\mathbb R}$. In addition, multiplying $u_0$ by $\\delta$\nwith  $0<\\delta \\ll1$, allows for initial \ndata which are arbitrarily small, but which nevertheless blow up at $(x_*, t_*) $. \nBy  a suitable spatial truncation of $u_0$, one can also construct small, \nsmooth, bounded initial data with finite mass,  all of whose derivatives also have finite \n$L^2$--norm,  such that the corresponding solution $u$ remains smooth \nbut achieves arbitrarily large values at a given  point in space-time (see \\cite{BS2} for more details). \n\\end{remark}\n\n\n\n\n\n\\begin{remark}\\label{frac}\nIt was also proven in \\cite{BS2} that dispersive blow up  holds true for the  class \n\\begin{equation*}\\label{fracS}\n  \\displaystyle i\\partial_tu+(-\\Delta)^{\\frac{a}{2}}u=0, \\quad 0<a<1,\n\\end{equation*}\nof fractional \nSchr\\\" odinger equations in ${\\mathbb R}^n\\times {\\mathbb R}^+$.\nHowever, observe that, in contrast to the classical Schr\\\"{o}dinger equation, \nthe phase velocity for \\eqref{fracS} becomes arbitrarily large in the long wave limit but is bounded (and actually tends to zero) in the short wave limit.\n\\end{remark}\n\nA natural extension of DBU for linear  Schr\\\"odinger equations is the\ninitial-value problem  in the presence of \nexternal potentials $V(x,t)\\in {\\mathbb R}$.  While we are not \ngoing to deal here with this issue in full generality, we note that an \nimmediate consequence of Remark \\ref{amp} is the appearance of dispersive blow up for Schr\\\"odinger equations with a Stark potential, {\\it i.e.}\n\\begin{equation}\\label{eq:stark}\ni \\partial_t v + \\Delta v - (E\\cdot x)v = 0, \\quad\\,\\, v\\big|_{t=0} = v_0(x),\n\\end{equation}\nwhere $E\\in {\\mathbb R}^n$. This equation models electromagnetic wave propagation in a constant electric field. Solutions of \\eqref{eq:stark} \nare connected to solutions of the free Schr\\\"odinger equation through the Avron-Herbst formula \\cite{AH}. Indeed, it is easy to check that if $v$ solves \n\\eqref{eq:stark}, then \n\\[\nu(x,t) = v\\left(x +  t^2  E, t\\right) e^{-i tE\\cdot x -i  \\frac{t^3}{3}|E|^2},\n\\]\nsolves the linear problem \\eqref{eq:linSch} with the same initial data. Thus, if the  $u$ \nthat solves the free Schr\\\"odinger equation \nexhibits dispersive blow up at a given $(x_*,t_*)$, then so does the solution $v$ of \\eqref{eq:stark} at the point $(x_*+2t_*^2E,t_*)$.\n\n\nAn analogous result is also true in the case of linear Schr\\\"odinger equations with \nisotropic quadratic potentials, {\\it viz.}\n\\begin{equation}\\label{eq:linharm}\ni\\partial_tu + \\Delta u \\pm \\omega^2 |x|^2 u=0, \\quad u\\big|_{t=0} = u_0(x),\n\\end{equation}\nwhere $\\omega \\in {\\mathbb R}$. The two signs correspond, respectively,\n to attractive ($-$) and repulsive ($+$) potentials. \nFollowing \\cite{Ca}, we find that if $v$ solves \\eqref{eq:linharm} with attractive harmonic potential, then \n\\[\nu(x,t) = \\frac{1}{(1+(2\\omega t)^2)^{n\/4}} \\, v \\left(\\frac{\\arctan (2\\omega t) }{\\omega}, \\frac{x}{\\sqrt{1+ \n(2\\omega t) ^2}}\\right) e^{ i  \\frac{ 2\\omega^2  x^2 t}{(1+(2\\omega t)^2}},\n\\]\nsolves \\eqref{eq:linSch}. Thus, dispersive blow up for $u$ again implies dispersive blow up for $v$, although at a \nshifted point in  space-time.\nA similar formula is available in the repulsive case (see \\cite{Ca2}). \n\n\\begin{remark}  Alternatively, one can prove dispersive blow up \nfor  linear Schr\\\"odinger equations with quadratic potentials \nusing (generalized) Mehler formulas for the associated Schr\\\"odinger group.\nThe  Mehler formulas for solutions are \n\\[\nu(x,t)=e^{-in\\frac{\\pi}{4} \\text{sgn}\\; t} \\left|\\frac{\\omega}{2\\pi \\sin \\omega t}\\right|^{\\frac{n}{2}}\\int_{{\\mathbb R}^n} e^{\\frac{i\\omega}{\\sin \\omega t}(\\frac{|x|^2+|y|^2}{2} \\cos \\omega t-x\\cdot y)}\\, u_0(y)\\, dy,\n\\]\n for \\eqref{eq:linharm} in the attractive situation, while in the repulsive case, one has\n\\[\nu(x,t)=e^{-in\\frac{\\pi}{4} \\text{sgn}\\; t} \\left|\\frac{\\omega}{2\\pi \\sinh \\omega t}\\right|^{\\frac{n}{2}}\\int_{{\\mathbb R}^n} e^{\\frac{i\\omega}{\\sinh \\omega t}(\\frac{|x|^2+|y|^2}{2} \\cosh \\omega t-x\\cdot y)}\\, u_0(y) \\, dy\n\\]\n(see  \\cite{Ca2} again).  In the attractive case, Mehler's formula is valid for $|t|<\\frac{\\pi}{2\\omega}$, while in the repulsive case it makes sense for all $t>0$. \nGeneralizations of these formulas are available and allow one to infer that dispersive blow up\ncan occur in the presence  of  anisotropic \nquadratic potentials (see \\cite{Ca2}).\n\\end{remark}\n\nWe close this subsection by noting, that at least for $n=1$, it is easy to show that dispersive blow up is stable under the influence of a \nrather general class of external (real-valued) potentials $V\\in C({\\mathbb R}; L^2({\\mathbb R}))$. To this end, consider the linear\nSchr\\\"odinger equation\n\\begin{equation}\\label{eq:linVSch}\ni \\partial_t u + \\partial^2_{x} u - V(x,t) u=0, \\quad u\\big|_{t=0} = u_0(x),\n\\end{equation}\nwhich which can rewritten, using Duhamel's formula, as\n\\begin{equation}\\label{eq:Vduhamel}\nu(x,t) = e^{i t \\partial_x^2} u_0(x)- i \\int_0^t e^{i (t-s) \\partial _x^2} V(s,x) u(x,s) \\, ds =: e^{i t \\Delta} u_0(x) - i I_{V} (x,t) .\n\\end{equation}\nIn view of \\eqref{eq:group}, we have  formally that\n\\begin{equation}\\label{eq:I}\nI_V(x,t)=\\frac{1}{(4i \\pi t)^{1\/2}} \\int_0^t\\int_{{\\mathbb R}^n}\\frac{1}{(t-s)^{\\frac{1}{2}}}\\exp \\left(i\\frac{|x-y|^2}{4(t-s)}\\right) V(s,y)u(y,s) \\, dy \\, ds.\n\\end{equation}\nNow, assume that the first term on the right-hand side of \\eqref{eq:Vduhamel} exhibits dispersive blow up at some $(x_*, t_*)$.   \nThen, it suffices to show that $I_V(x,t)$ is continuous for $(x,t)\\in {\\mathbb R}^n\\times [0,T]$, for some $T>t_*$ to conclude that the solution of the \\eqref{eq:linVSch} \nexhibits dispersive blow up at the same point $(x_*, t_*)$.  If it is established that $I_V(x,t)$ \nis locally bounded as a function of $x$ and $t$ in the range ${\\mathbb R}^n\\times [0,T]$ for some\n$T > t_*$, then Lebesgue's dominated convergence \ntheorem will imply the desired continuity.  \nTo show local boundedness,  first apply the Cauchy-Schwartz inequality to find\n\\begin{equation}\\label{bound}\n| I_V(x,t) | \\leq \\frac{1}{(4\\pi |t|)^{1\/2}} \\int_0^t \\frac{1}{|t-s|^{\\frac{1}{2}}} \\, \\| V(\\cdot, s) \\|_{L^2} \\| \\| u_0 \\|_{L^2} \\, ds,\n\\end{equation}\nwhere conservation of mass, $\\| u(\\cdot, t)\\|_{L^2} = \\|u_0 \\|_{L^2}$, has been used. \nDue to our assumption on $V$ \nand the fact that $t\\mapsto t^{-1\/2}$ is locally integrable, the right-hand side of \\eqref{bound} is finite and we are done.\n\\vspace{.1cm}\n\nA similar argument will be used preently in the study of DBU for the\n nonlinear Schr\\\"odinger equations in dimension $n=1$. \nIt is clear, however, that in general dimensions $n>1$ a more refined analysis is needed.\n\n\n\n\n\\subsection{Smoothing properties of the free Schr\\\"odinger group}\\label{sec:smooth}\n\nIn this subsection, some  technical results on the smoothing properties of the \nfree Schr\\\"odinger group $S(t) = e^{i t \\Delta}$ are reviewed. They will find use\nin Section 4.   \n\nFirst, recall the notion of admissible index-pairs.\n\\begin{definition} \\label{def:adm} The pair $(p,q)$ is called {\\it admissible} if\n\\begin{equation*}\n\\frac{2}{q}=\\frac{n}{2}-\\frac{n}{p}, \\quad \\text{and} \\ \n    \\left\\lbrace\n    \\begin{array}{l}\n    2\\leq p<\\frac{2n}{n-2},\\ \\text{for} \\, n\\geq 3, \\\\\n    2\\leq p<+\\infty,\\ \\text{if} \\ n=2,\\\\\n    2\\leq p\\leq +\\infty,\\ \\text{if} \\ n=1.\n    \\end{array}\\right.\n\\end{equation*}\n\\end{definition} \n\nFrom now on, for any index $r>0$,  its H\\\"older dual is denoted $r'$, {\\it i.e.} $\\frac{1}{r}+\\frac{1}{r'} = 1$.\nThe well known Strichartz estimates for the Schr\\\"odinger group $S(t) =  e^{i t \\Delta}$ are recounted \nin the next lemma (see \\cite{Caz, LP} for more details).\n\n\\begin{lemma} \\label{lem:Strich}\nIf $(p,q)$ is admissible, then the group $\\{ e^{i t \\Delta} \\}_{t\\in {\\mathbb R}}$ satisfies\n\\[\n\\left(\\int_{-\\infty}^\\infty \\big\\| e^{i t \\Delta} f  \\big\\|^q_{L^p({\\mathbb R}^n)} \\, d t\\right)^{\\frac{1}{q}} \\leq C \\big\\| f \\big\\|_{L^2({\\mathbb R}^n)} \n\\]\nand \n\\[\n \\left(\\int_{-\\infty}^\\infty \\Big \\| \\int_{0}^t e^{i (t-s) \\Delta} g(\\cdot, s)  \\, ds \\Big \\|^q_{L^p({\\mathbb R}^n)} \\, d t\\right)^{\\frac{1}{q}} \\! \\leq \n\\, C  \\left( \\int_{-\\infty}^\\infty \\big\\| g(\\cdot, t)  \\big\\|^{q'}_{L^{p'}({\\mathbb R}^n)} \\, \nd t \\right)^{\\frac{1}{q'}}\\!, \n\\]\nwhere $C = C(p,n)$.\n\\end{lemma}\nThe estimates stated above can be interpreted as global smoothing properties of the free Schr\\\"odinger group $S(t)$. \nIn addition to that, $S(t)$ is known to also induce local smoothing effects, some of which are collected in the following lemma (for proofs, see \\cite[Chapter 4]{LP}).\nFor $1 \\leq j \\leq n$, denote the so-called homogenous derivatives of order $s>0$ by\n\\begin{equation}\\label{eq:homD}\n\\begin{split}\n&D^s_{x_j} f(x):= \\mathcal F^{-1}  \\big(|\\xi_j|^s \\widehat f(\\xi)\\big)(x) \\quad \\text{and, for $n=1$,} \\\\\n&D^s f(x):= \\mathcal F^{-1}  \\big(|\\xi|^s \\widehat f(\\xi)\\big)(x).\n \\end{split}\n\\end{equation}\n\\begin{lemma} \\label{lem:smooth} \nIf $n=1$ and $f \\in L^2({\\mathbb R})$, then\n\\[ \n\\sup_{x\\in {\\mathbb R}} \\ \\int_{\\mathbb R}  \\big| D_{x}^{1\/2}e^{it \\partial^2_x } f (x) \\big|^2 \\, dt \\leq C\\|f\\|^2_{L^2({\\mathbb R})}.\\]\nLet $n\\ge 2$. Then, for all $j\\in \\lbrace 1,\\dots ,n\\rbrace$ and $f \\in L^2({\\mathbb R}^n)$, \n\\[\n\\sup_{x_j\\in {\\mathbb R}} \\ \\int_{{\\mathbb R}^n}  \\big| D_{x_j}^{1\/2}e^{it\\Delta} f (x)\\big|^2 dx_1\\dots dx_{j-1}dx_{j+1}\\dots dx_n  dt  \\leq C\\|f\\|^2_{L^2({\\mathbb R}^n)}.\n\\]\n\\end{lemma}\n\nHelpful inequalities  involving the\nSchr\\\"odinger maximal function\n\\[\nS_T^*f(x):= \\sup_{0\\leq t\\leq T} |e^{it\\Delta}f(x)|\n\\]\nare derived in  \\cite {RV} and  \\cite{V}.  They are reported in the next lemma.\n\\begin{lemma}\\label{lem:B}  The inequality\n\\begin{equation}\\label{max}\n\\| S_T^*f\\|_{L^q({\\mathbb R}^n)} \\leq C_T \\|f\\|_{H^\\sigma({\\mathbb R}^n)}\n\\end{equation}\nholds if either\n\\begin{equation}\\label{5.2}\nn=1\\quad \\text{with}\\quad\\begin{cases}\nq>2\\quad\\text{and}\\quad \\sigma \\geq \\max\\lbrace \\frac{1}{q}, \\frac{1}{2}-\\frac{1}{q}\\rbrace, \n \\\\\nq=2\\quad\\text{and}\\quad \\sigma >\\frac{1}{2},\n\\end{cases}\n\\end{equation}\nor  \n\\begin{equation}\\label{5.2bis}\nn>1\\quad \\text{with}\\quad\\begin{cases}\nq\\in (2+\\frac{4}{(n+1)},\\infty)\\quad\\text{and}\\quad \\sigma >n(\\frac{1}{2}-\\frac{1}{q}),\\\\\nq\\in[2,2+\\frac{4}{(n+1)}]\\quad\\text{and}\\quad \\sigma >\\frac{3}{q}-\\frac{1}{2}.\n\\end{cases}\n\\end{equation}\n\\end{lemma}\n\n\nWith these results at hand, attention is turned to establishing\n dispersive blow up for nonlinear Schr\\\"odinger equations.\n\n\n\n\n\n\\section{Dispersive blow up for nonlinear Schr\\\"odinger equations }\\label{sec:NLS}\n\nIn this section, the initial-value problem\n\\begin{equation}\\label{eq:NLS}\ni\\partial_t u+\\Delta u\\pm |u|^p u=0,\\quad u\\big|_{t=0}=u_0(x),\n\\end {equation}\nfor the nonlinear Schr\\\"odinger equation is considered.  Here,  $x\\in {\\mathbb R}^n$ and $p>0$ is not necessarily an integer. \nFinite-time dispersive blow up was established for $n=1$ and $p\\in (0,3)$ in \\cite{BS2}. \nOur strategy to improve upon this result relies upon the theory developed in  \\cite{CW}, where the Cauchy problem \\eqref{eq:NLS} \nwas studied for $u_0\\in H^s({\\mathbb R}^n)$ for various values of $s$.  \n\n\n\\subsection{Local well-posedness in $H^s$}\\label{sec:LWP}\n\nFor $1\\le r < \\infty$ and $s>0$, define\n\\[\nH^{s,r}({\\mathbb R}^n)=\\Big\\{ f\\in L^r({\\mathbb R}^n): \\mathcal F^{-1}\\big[ (1+|\\xi|^2)^{s\/2} \\widehat f(\\xi) \n \\big] \n\\in L^r({\\mathbb R}^n) \\Big\\}.\n\\] \nThese are the  standard Bessel potential spaces. According to \\cite{St}, which uses the notation $L^{s,r}$ instead of $H^{s,r}$, these spaces may be characterized in the following manner.\nLet $s\\in (0,1)$ and $\\frac{2n}{(n+2s)}<r<\\infty$. Then\n$f\\in H^{s,r}({\\mathbb R}^n)$ if and only if $f\\in L^r({\\mathbb R}^n)$ and \n\\begin{equation}\\label{eq:frac}\n\\mathcal D^s f(x)=\\left(\\int_{{\\mathbb R}^d} \\frac{|f(x)-f(y)|^2}{|x-y|^{n+2s}} \\, dy\\right)^{1\/2}\\in L^r({\\mathbb R}^n).\n\\end{equation}\nThe space $H^{s,r}({\\mathbb R}^n) \\equiv (I-\\Delta)^{-s\/2}L^r({\\mathbb R}^n)$ is equipped with the norm \n \\begin{align*}\n \\|f\\|_{H^{s,r}({\\mathbb R}^n)}=\\|(I-\\Delta)^{s\/2}f\\|_{L^r({\\mathbb R}^n)} =  & \\ \\|f\\|_{L^r({\\mathbb R}^n)}+\\|\\mathcal D^s f\\|_{L^r({\\mathbb R}^n)} \\\\\n \\simeq  & \\ \\|f\\|_{L^r({\\mathbb R}^n)}+\\| D^s f\\|_{L^r({\\mathbb R}^n)},\n\\end{align*} \nwhere $D^s$ is the homogenous derivative defined in \\eqref{eq:homD}. \nObserve that, for $r=2$, $H^{s,2}({\\mathbb R}^n)\\equiv H^{s}({\\mathbb R}^n)$, the usual $L^2$--based Hilbert space. \nIn addition, a straightforward calculation reveals that \n\\begin{equation}\\label{3}\n\\|\\mathcal D^s f\\|_{L^2({\\mathbb R}^n)}=c_n\\||\\xi |^s \\widehat f \\, \\|_{L^2({\\mathbb R}^n)} \\equiv c_n \\| D^s f\\|_{L^2({\\mathbb R}^n)},\n\\end{equation} \nwith $c_n$  a constant depending only on the dimension $n$.  \nIf $(q,r)$ is an admissible pair as defined in Section 2, then the space \n\\begin{equation} \\label{solutionspace}\nW^T_{s,n}\\, = \\, C\\big([0,T];H^s({\\mathbb R}^n)\\big)\\cap L^q\\big([0,T|;H^{s,r}({\\mathbb R}^n)\\big) \n\\end{equation}\nwill also appear.   Of course, these spaces depend  on the admissible pair $(q,r)$, \nbut this dependence is suppressed in the notation.  \nThe following local well-posedness theorem established in \\cite{CW} makes use of both the\nBessel-potential spaces and the latter, spatial-temporal spaces.\n\\begin{proposition}\\label{prop:CW}\nLet $s>s_{p,n}=\\frac{n}{2}-\\frac{2}{p},$ $s>0,$ with $s$ otherwise arbitrary if $p$ is \nan even integer, $s < p+1$ if $p$ is an odd integer and  \n$\\lfloor s \\rfloor <p$ if $p$ is not an integer. For given initial data $u_0\\in H^s({\\mathbb R}^n)$, \n\\begin{enumerate}\n\\item there exist $T=T(\\|u_0\\|_{H^s})>0$ and a unique solution\n\\[\nu\\in C\\big([0,T];H^s({\\mathbb R}^n)\\big)\\cap L^q\\big([0,T|;H^{s,r}({\\mathbb R}^n)\\big)\\equiv W^T_{s,n}\n\\]\nfor all  pairs $(q, r)$ admissible in the sense of Definition \\ref{def:adm}, \n and\n\\item the local existence time $T=T(\\|u_0\\|_{H^s}) \\to +\\infty$, as $\\|u_0\\|_{H^s({\\mathbb R}^n)}\\to 0.$\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{remark}The proof of this result follows  from a fixed point argument based on the Strichartz estimates displayed in Lemma \\ref{lem:Strich}.  Recall that \nthe notation $\\lfloor s \\rfloor$ connotes the largest integer less than or equal to $s$. \\end{remark}\n\n\nTo apply Proposition \\ref{prop:CW} in our context, we need to show that the class of initial data constructed in Section \\ref{sec:lin} \n(yielding dispersive blow up for the free Schr\\\"odinger evolution) admits  Sobolev \nclass regularity that will allow the use of Lemma \\ref{prop:CW}. \n Via scaling and translation, dispersive blow up can be achieved at any point $(x_*, t_*)$ in \nspace-time, so without loss of generality, fix $(x_*, t_*) = (0, 1)$ and focus upon the \ninitial data\n\\begin{equation}\\label{eq:IC}\nu_0(x) = \\frac{e^{- 4 i |x |^2}}{(1+|x|^2)^m}\\ ,\\quad \\text{with $\\frac{n}{4} < m \\le \\frac{n}{2}$.}\n\\end{equation}\nThis initial value $u_0$ has Sobolev regularity explained in the next lemma.\n\\begin{lemma}\\label{lem:linear}\nLet  $u_0$ be as depicted  in \\eqref{eq:IC}. Then\n$u_0\\in H^s({\\mathbb R}^n)$ if \\;$2m>s+\\frac{n}{2}$. In particular, if $m=\\frac{n}{2},$   \nthen $u_0\\in H^s({\\mathbb R}^n)$ for any $ s\\in (0,\\frac{n}{2})$, whereas if $m=\\frac{n}{4}^+,$ then  $s \\in (0,0^+)$.\n\\end{lemma}\n\n\\begin{proof} Consider first the case $0<s<1$.  \nFor $s$ in this range, Propositions 1 and 2 in \\cite{NP} provide the inequalities \n\\[\n| \\mathcal D^s e^{i|x|^2}|\\leq c_n(1+|x|^s)\n\\]\nand\n\\[\n\\|\\mathcal D^s(fg)\\|_{L^2({\\mathbb R}^n)}\\leq \\|f\\mathcal D^s g\\|_{L^2({\\mathbb R}^n)}+\\|g\\mathcal D^s f\\|_{L^2({\\mathbb R}^n)},\n\\]\nwhere $\\mathcal D^s$ is defined in \\eqref{eq:frac}.\nCombining these estimates with identity \\eqref{3} and using interpolation, one arrives \nat the inequality\n\\begin{align*}\n&\\, \\left\\Vert \\mathcal D^s \\left(\\frac{e^{i|x|^2}}{(1+|x|^2)^m}\\right)\\right\\Vert_{L^2({\\mathbb R}^n)} \\\\ \n &\\, \\leq  \\left\\Vert\\frac{1}{(1+|x|^2)^m}\\mathcal D^s(e^{i|x|^2})\\right\\Vert _{L^2({\\mathbb R}^n)}+\\left\\Vert \\mathcal D^s\\left(\\frac{1}{(1+|x|^2)^m}\\right)\\right\\Vert_{L^2({\\mathbb R}^n)}\\\\\n&\\, \\leq c_n \\left\\Vert \\frac{1}{(1+|x|^2)^m}\\right\\Vert_{L^2({\\mathbb R}^n)}+c_n \\left\\Vert \\frac{|x|^s}{(1+|x|^2)^m}\\right\\Vert_{L^2({\\mathbb R}^n)}\\\\\n&\\quad \\  +\\left\\Vert \\frac{1}{(1+|x|^2)^m}\\right\\Vert_{L^2({\\mathbb R}^n)}^{1-s} \\left\\Vert D\\left(\\frac{1}{(1+|x|^2)^m}\\right)\\right\\Vert^s_{L^2({\\mathbb R}^n)}.\n \\end{align*}\nThe right-hand side of this inequality is finite if and only if $2m-s>\\frac{n}{2}.$  \n\nIf $s$ is a positive integer, the result follows from Leibnitz' rule. \nIf instead, $s = s' +k$ where $0 < s' < 1$, then simply apply the above analysis \nto the $k^{\\rm th}$--deriviative $u_0^{(k)}$.    \n\\end{proof}\n\nNotice that if $m = \\frac{n}{2}$, we can certainly choose the value $s$ in the interval $(\\frac{n}{2}-\\frac{2}{p},\\frac{n}{2}]$ where Proposition  \\ref{prop:CW}  applies.   \n\n\n\n\n\\subsection{Proof of dispersive blow up for nonlinear Schr\\\"odinger equations}\\label{sec:DBUNLS}\n\nHere is the detailed statement of dispersive blow up for the initial-value problem \\eqref{eq:NLS}.\nIn this theorem, the \nLebesgue index $p\\geq \\lfloor \\frac{n}{2} \\rfloor$ if $p$ is not an even integer.\n\n\\begin{theorem}\\label{thm:main}\nGiven $t_{*}>0$ and $x_*\\in {\\mathbb R}^n,$ for any $s\\in (\\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}]$,\n there are initial data  $u_0\\in H^s({\\mathbb R}^n)\\cap L^{\\infty}({\\mathbb R}^n)\\cap C^{\\infty}({\\mathbb R}^n)$ \nand a $T = T(\\|u_0\\|_{H^s})>t_*$\n such that \n\\begin{enumerate}   \n\\item the  initial-value problem   \\eqref{eq:NLS} has a unique solution $u$ \nin the class described in Proposition   \\ref{prop:CW} which is defined \nat least on  the time interval $[0,T]$,   and\n\\item $u$ exhibits dispersive blow up, which is to say,\n$$ \\lim_{{(x,t)\\in {\\mathbb R}^n\\times [0,T] \\to (x_*,t_*)}} |u(x,t)|=+\\infty.$$\n\\item Moreover, $u$ is a continuous function of $(x,t)$ on ${\\mathbb R}^n\\times( [0,T])\\setminus \\lbrace t_*\\rbrace)$ and\n\\item $u(\\cdot,t_*)$ is a continuous function of $x$ on ${\\mathbb R}^n\\setminus \\lbrace x_* \\rbrace.$\n\\end{enumerate}\n\\end{theorem}\n\nThis theorem extends the results of \\cite{BS2} to the cases where $n\\geq 2$ and $p\\geq 3.$ \nNotice that the nonlinearity $y \\mapsto |y|^py$ is smooth when $p$ is an even integer.  \nOtherwise, it has finite regularity and hence the restriction on $p$ in those cases.\n\n\n\\begin{proof} The proof is provided in detail  for $p>0$ in the case $n=1$ and, when $n\\geq 2$, \nfor the case $p=2k$,\n$k$ a positive integer.  It \nwill be clear from the argument that the result extends to the case of \n $p\\geq \\lfloor \\frac{n}{2} \\rfloor$ if $p$ is not an even integer. \n\nAs already mentioned, we may assume that the  dispersive blow up for the \nfree Schr\\\"odinger group $S(t)$ \noccurs at $x_* = 0$ and $t_*=1$. Note that the same is true for initial data of the form $\\delta u_0$, where $u_0$ is \nas in \\eqref{eq:IC} with $m = \\frac{n}{2}$, say, $\\delta > 0$  arbitrary and $s$ satisfying \nthe conditions in Proposition \\ref{prop:CW}. \nIn view of  part (3) of the latter proposition,  the local existence time $T^* = T(\\| \\delta u_0\\|_{H^s})>0$ can be made arbitrarily large by choosing $\\delta$  sufficiently small \nand hence we can always achieve $T^*>1 = t_* $.\\\\\n\n{\\it Step 1.}  Take as initial \ndata $\\delta u_0$, where $u_0$ is as in \\eqref{eq:IC} with $m = \\frac{n}{2}$.  Let $s \\in (\\frac{n}{2} - \\frac{2}{p}, \\frac{n}{2}]$ with $p \\geq \\lfloor\\frac{n}{2}\\rfloor$.   Then \n$s$ satisfies the conditions of Proposition \\ref{prop:CW}.  As noted above, by choosing \n$\\delta$ small enough, we can be sure that the solution $u$ of \\eqref{eq:NLS} \nemanating from $u_0$ \nexists and is unique in $C([0,T]:H^s)$ where $T > t_* = 1$.   \n\nDuhamel's formula allows us to represent $u$ in the form\n\\begin{equation}\\label{eq:duhamel}\nu(x,t) = e^{i t \\Delta} u_0(x) \\pm i \\int_0^t e^{i (t-s) \\Delta} |u(x,s)|^p u(x,s) \\, ds =: e^{i t \\Delta} u_0(x) \\pm i I (x,t) ,\n\\end{equation}\nwhere, at least formally, $I(x,t)$ can be written as the double integral\n\\begin{equation}\\label{eq:I}\nI(x,t)=\\frac{1}{(4i \\pi t)^{n\/2}} \\int_0^t\\int_{{\\mathbb R}^n}\\frac{1}{(t-s)^{\\frac{n}{2}}}\\exp \\left(i\\frac{|x-y|^2}{4(t-s)}\\right) |u(y,s)|^p  u(y,s) \\, dy \\, ds.\n\\end{equation}\nThe first term on the right-hand side of \\eqref{eq:duhamel} \n exhibits dispersive blow up at $(x_*, t_*) = (0,1)$ on account of the choice of $u_0$.   \n If it turns out that  $I$ is continuous for $(x,t)\\in {\\mathbb R}^n\\times [0,T]$, then it is \nimmediately concluded that \\eqref{eq:duhamel} (and thus \\eqref{eq:NLS}) exhibits \ndispersive blow up at the same point $(x_*, t_*)=(0,1)$.  To show that $I(x,t)$ \nis continuous, it suffices to prove that it is locally bounded as a function of $x$ and $t$, \nsince then  Lebesgue's dominated convergence theorem will imply $I$ is continuous on ${\\mathbb R}^n\\times [0,T]$. \\\\\n\n{\\it Step 2.}  Consider  $n=1$ first, since  a more direct proof can be made \nin this case.  \nThe  initial data is \n\\[\nu_0(x)= \\, \\frac{\\delta e^{-i4 x^2}}{(1+x^2)^{\\frac{1}{2}}},\\quad x\\in {\\mathbb R}.\n\\] \nThe function $u_0$ lies in $H^s({\\mathbb R})$ for any $s$ in the range  $0\\leq s<\\frac{1}{2}.$ \nProposition \\ref{prop:CW} then provides a local in time solution $u \\in C([0,T]; H^s({\\mathbb R}))$ to the Cauchy problem \\eqref{eq:NLS} provided that \n\\[\n0<s<\\frac{1}{2} \\quad \\,\\, \\text{and} \\,\\, \\quad 0<p\\leq \\frac{4}{1-2s}.\n\\]\nAs mentioned already, $T$ may be taken larger than 1 by choosing $\\delta$ small.  By Sobolev imbedding, $H^s({\\mathbb R})\\subset L^{r+1}({\\mathbb R})$ if $\\frac{1}{r+1}\\geq \\frac{1}{2}-s.$\nHence, for  $s=\\frac{1}{2}-\\varepsilon$, where $\\varepsilon>0$ is fixed and small, it is inferred  \nthat $u\\in C([0,T]; L^{r+1}({\\mathbb R}))$ for all $r$ in the range \n\\[\n0<r\\le \\min \\left\\{\\frac{2}{\\varepsilon}, \\frac{1-\\varepsilon}{\\varepsilon}\\right\\}=\\frac{1-\\varepsilon}{\\varepsilon}. \n\\]\nAs $\\epsilon > 0$ was arbitrary, it  follows that  \n$u(\\cdot, t)\\in L^{r+1}({\\mathbb R})$, for  $r\\geq 1$ arbitrarily large. \nIn consequence, $I(x,t)$ is locally bounded.  \nIndeed, using H\\\"older's inequality, it is seen that\n\\begin{align*}\n|I(x,t)| \\le &\\ \\int_0^t \\frac{1}{(t-s)^{1\/2}} \\, \\big\\| u(\\cdot, s)\\big\\|_{L^{p+1}}^{p+1} \\, ds \\\\\n\\le &\\  \\left(\\int_0^t \\frac{1}{(t-s)^{\\gamma\/2}} \\, ds \\right)^{\\frac{1}{\\gamma} }\n \\left(\\int_0^t \\big\\| u(\\cdot, s)\\big\\|_{L^{p+1}}^{\\gamma' (p+1)} \\, ds \\right)^{\\frac{1}{\\gamma'} }\n\\end{align*}\nwhere $\\frac{1}{\\gamma}+\\frac{1}{\\gamma'}=1$ and $\\gamma \\in (1,2)$. Having in mind that $u\\in C([0,T]; L^{p+1}({\\mathbb R}))$, it transpires that\n\\[ |I(x,t)| \\le C T^{\\frac{1}{\\gamma'}} \\sup_{t\\in[0,T]} \\| u(\\cdot, t) \\|_{L^{p+1}}^{\\gamma'(p+1)},\n\\]\nfor all $x \\in {\\mathbb R}$, which concludes the proof in the case $n=1$.\\\\\n\n\n\n{\\it Step 3.} In  case $n\\ge 2$,  the strategy employed \nabove no longer works beause the factor $t\\mapsto t^{-n\/2}$ appearing \nin the representation formula \\eqref{eq:I} is no longer locally integrable. However, it will be shown \nin Proposition \\ref{prop:key} below that the double integral $I$ is in fact half a derivative \nsmoother than one would naively expect. To make use of this result,  choose initial data $u_0$ \nof the form \\eqref{eq:IC} with $m \\in (\\frac{n}{2}-\\frac{1}{4p},\\frac{n}{2}]$, where \n$p\\ge \\lfloor\\frac{n}{2}\\rfloor$.\nIt is immediately inferred from Lemma \\ref{lem:linear} that \n\\[u_0\\in H^s({\\mathbb R}^n)\\quad \\text{for any} \\,\\, s\\in \\Big( \\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}\\Big].\\]\nNotice that \n\\[\\frac{n}{2}-\\frac{1}{2p}>\\frac{n}{2}-\\frac{2}{p}=s_{p,n}\\] \nso that Proposition \\ref{prop:CW} applies and therefore $u\\in W^T_{s,n}$ (see \\eqref{solutionspace})\nsatisfies the integral form \\eqref{eq:duhamel} of the nonlinear Schr\\\"odinger equation.  \nProposition \\ref{prop:key} below then shows  that \n\\[\nI \\in C([0,T]; H^{s+1\/2}({\\mathbb R}^n)),\n\\]\nand hence, for $s\\in (\\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}],$ one has\n\\[\nI \\in C({\\mathbb R}^n\\times [0,T])\\cap L^\\infty({\\mathbb R}^n\\times [0,T]).\n\\]\nThe proof is complete.\n\\end{proof}\n\n\n\n\n\\begin{remark} It was shown in \\cite[Theorem 2.1]{BS2} that dispersive blow up results \nare stable under smooth and localized perturbations of the data.  That is to say,  \nif they hold for data $u_0,$ then they also hold for data $u_0+w$ where, for instance, $w \\in H^\\infty({\\mathbb R}^n).$   In particular the data \nleading to DBU do {\\it not} need to be radially symmetric. The same is true for Theorem \\ref{thm:main}.\nThe proofs of these results consists of writing the equation satisfied by $w$ and showing\nthat it has bounded, continuous solutions.  The details follow exactly the argument \ngiven already in \\cite{BS2}.  \n\nIn addition, there is  a kind of density of initial data leading to dispersive blow up.  More \nprecisely,  given $u_0\\in H^s({\\mathbb R}^n)$ with $s>n\/2$ and $\\epsilon >0,$ there exists\n$\\phi\\in H^r({\\mathbb R}^n), $ $r\\in \\big(\\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}\\big]$ with \n\\begin{equation}\n  \\label{density}\n    \\|u_0-\\phi \\|_{H^r({\\mathbb R}^n)}<\\epsilon,\n\\end{equation}\nsuch that the initial data $\\phi$ leads to dispersive blow up in the sense of Theorem \\ref{thm:main}. Indeed, it suffices to take \n$\\phi=u_0+\\delta v_0,$ where $v_0$ leads to dispersive blow up and  $\\delta >0$ is small enough \nthat \\eqref{density} holds. A combination of Theorem \\ref{thm:main} and \n Proposition \\ref{prop:CW} then \nimplies the above assertion.\n\\end{remark}\n\n\n\n\n\n\\section{Global smoothing of the Duhamel term and applications}\\label{sec:global}\n\nIn this section, the proof of Theorem \\ref{thm:main} is completed by showing the Duhamel \nterm in the integral representation \\eqref{eq:duhamel} of the solution  of the initial-value problem \\eqref{eq:NLS}\nis smoother than is the linear term involving only the initial data.  In fact, several\ndifferent results of smoothing by the Duhamel term will be developed, though \nthe first one is enough for the dispersive blow up result in Section 3.  \n\n\\subsection{Smoothing by half a derivative}\n\nThe following proposition\nsuffices to complete the proof \nof Theorem \\ref{thm:main}.  \n\n\\begin{proposition}\\label{prop:key}\nLet $u_0\\in H^s({\\mathbb R}^n),$\\;$s>\\frac{n}{2}-\\frac{1}{2p}$ with $p\\ge 1$ \nand $\\lfloor p+1\\rfloor \\geq s + \\frac12$ if $p$ is not an even integer.\nLet  $u\\in W^T_{s,n}$ be the solution of \\eqref{eq:NLS} satisfying\n\\[\nu(x,t)=e^{it\\Delta}u_0(x)\\pm i \\int_0^te^{i(t-s)\\Delta} |u(x,s)|^pu(x,s)ds=:e^{it\\Delta}u_0(x) \\pm i I(x,t).\n\\]\nThen $I \\in C([0,T]; H^{s+\\frac12}({\\mathbb R}^n)).$ \n\\end{proposition}\nIn other words, the integral term $I$ is \\lq\\lq smoother\" than the free propagator $e^{it\\Delta}u_0$ by \nhalf a derivative.\nThis is  the key point needed in the proof of Theorem \\ref{thm:main} for $n\\ge 2$.   In the \nspecial case wherein\nthe nonlinearity $|u|^pu$ is smooth, so when $p = 2k, k$ an integer,  \nProposition \\ref{prop:keybis} will show that  the Duhamel term is\nalmost one derivative smoother than one would expect.\n\n\\begin{remark}  The fact that\nthe nonlinear integral term in  Duhamel's formula is smoother than the linear one\nin certain circumstances has been used in other  works \non nonlinear dispersive equations. \nFor example, in \\cite{LS} it was employed to give a different \nproof of some of the results obtained in \\cite{BS1}. \nIn \\cite{Bou}, this smoothing effect was applied to deduce  global \nwell-posedness below the regularity index provided by the conservation laws of \nmass and energy.  So far as we are aware, however, the result stated in Proposition \n\\ref{prop:key} has not\npreviously been explicitly written down.\n \\end{remark}\n\n\n\n\n\n\n\n\\begin{proof} The details are provided for the case $p=2k, k\\in {\\mathbb N}.$ \nIt will be clear that the arguments extend to the case \nwhere $p\\geq 1$ is not an even integer, but $p\\ge \\lfloor\\frac{n}{2}\\rfloor$ and $n\\ge2$. \\\\\n\n{\\it Step 1.} In the first step, useful estimates on $e^{it\\Delta}u_0$ are derived. \nStart by fixing a $j\\in \\lbrace 1,\\dots,n\\rbrace$ and noticing that\n\\begin{equation}\\label{compA}\n\\sup_{0\\leq t\\leq T}\\left\\{ \\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}\\Big\\{|e^{it\\Delta} u_0(x)|\\Big\\} \\right\\} \\lesssim \n\\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}\\left\\{|e^{it(x_j)\\Delta}u_0(x)|\\right\\}\n\\end{equation}\nfor some $t(x_j)\\in [0,T]$.  For  $q\\ge 2$ and $s>\\frac{(n-1)}{q}$, it follows by Sobolev embedding that\n\\begin{equation}\\label{comp}\n\\begin{aligned}\n\\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}|e^{it(x_j)\\Delta}u_0(x)| & \\, \\lesssim  \n\\big\\|e^{it(x_j)\\Delta}u_0\\big\\|_{H^{s,q}_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}\\left({\\mathbb R}^{n-1}\\right)} \\\\\n&\\, =c\\big\\|e^{it(x_j)\\Delta}J^s_{\\hat \\jmath}u_0\\big\\|_{L^q_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}\\big({\\mathbb R}^{n-1}\\big)}\\\\\n&\\, \\lesssim \\big\\|\\sup_t |e^{it\\Delta}J^s_{\\hat \\jmath}u_0|\\;\n\\big\\|_{L^q_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n} \\left({\\mathbb R}^{n-1}\\right)}\n\\end{aligned}\n\\end{equation}\nwhere here and in the following, \n\\[\nJ^s_{\\hat \\jmath }=(1-(\\partial^2_{x_1}+\\dots+\\partial^2_{x_{j-1}}\n+\\partial^2_{x_{j+1}}+\\dots+\\partial^2_{x_n}))^{s\/2}\n\\]\nis defined via the associated Fourier symbol and the subscripts on the function spaces \nindicate which variables participate in the norm. \nUsing \\eqref{compA} and \\eqref{comp} together with \nthe estimates on the Schr\\\"odinger maximal function given in Lemma \\ref{lem:B}, \nthere appears the inequality\n\\begin{equation}\\label{ouf}\n\\begin{aligned}\n&\\big\\|e^{it\\Delta}u_0\\big\\|_{L^q_{x_j}({\\mathbb R};L^\\infty_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n t}({\\mathbb R}^{n}\\times[0,T]))}\\\\\n&\\hspace{1cm}= \\Big\\|\\sup_{0\\leq t\\leq T}\\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n}|e^{it\\Delta}u_0|\\;\\Big\\|_{L^q_{x_j}({\\mathbb R})}\\\\\n&\\hspace{1cm} \\leq C_T\\Big\\|\\sup_{0\\leq t\\leq T}|e^{it\\Delta}J^s_{\\hat \\jmath}u_0|\\;\\Big\\|_{L^q({\\mathbb R}^n)}\\\\\n&\\hspace{1cm} \\lesssim \\big\\|u_0\\big\\|_{H^{\\sigma+s}({\\mathbb R}^n)},\n\\end{aligned}\n\\end{equation}\nwith $q\\geq 2$, $s>\\frac{n-1}{q}$ and $\\sigma>0$ as specified in Lemma \\ref{lem:B}. \nThe inequality \\eqref{ouf}, together with the local smoothing estimates stated in Lemma \\ref{lem:smooth}, \nreveal that\n\\begin{equation}\\label{ouf2}\n\\big\\|D^{\\theta\/2}_{x_j}e^{it\\Delta} u_0\\big\\|_{L_{x_j}^{q\/(1-\\theta)}({\\mathbb R};L^{2\/\\theta}_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_nt}({\\mathbb R}^{n-1}\\times[0,T]))} \n\\leq c_T\\big\\|u_0\\big\\|_{H^{(1-\\theta)(\\sigma+s)}({\\mathbb R}^n)},\n\\end{equation}\nwhere $\\theta\\in[0,1]$, $q\\geq 2$, $s>\\frac{n-1}{q}$ and $\\sigma>0$ as before. \\\\\n\n\n{\\it Step 2.}  To bound $\\|D_{x_j}^{s+1\/2} I\\|_{L^\\infty_t([0,T]:L^2({\\mathbb R}^n))}$ above,  let  $j\\in \\lbrace 1,\\dots , n\\rbrace$, $p=2k$ and write\n\\begin{equation}\n\\label{tterm}\n\\begin{aligned}\n&\\Big \\|D_{x_j}^{s+1\/2}  \\int_0^t e^{i(t-s)\\Delta}(|u|^{2k}u)(s) \\, ds \\Big \\|_{L^\\infty([0,T];L^2({\\mathbb R}^n))}\\\\\n&\\hspace{1cm} \\leq \\, c\\big\\|D_{x_j}^{s+1\/2} (|u|^{2k} u)\\big\\|_{L^1([0,T];L^2({\\mathbb R}^n))}\\\\\n&\\hspace{1cm} \\leq \\, cT^{1\/2}\\big\\|D_{x_j}^{s+1\/2} (|u|^{2k} u)\\big\\|_{L^2([0,T]\\times {\\mathbb R}^n)}.\n\\end{aligned}\n\\end{equation}\n To estimate the right-hand side of \\eqref{tterm}, the calculus of inequalities involving fractional derivatives \nderived  in \\cite{KPV93} is helpful.  More precisely,  the following inequality, \nwhich is a particular case of those proved in \n \\cite[Theorem A.8]{KPV93}, will be used.   Let $\\alpha\\in (0,1), \\,\\alpha_1,\\alpha_2\\in[0,\\alpha]$ with $\\alpha=\\alpha_1+\\alpha_2$ and  let $p_1,p_2,q_1,q_2\\in [2,\\infty)$ be such that\n \\[\\frac{1}{2}=\\frac{1}{p_1}+\\frac{1}{p_2}=\\frac{1}{q_1}+\\frac{1}{q_2}.\\] Then\n \\begin{equation}\n \\label{inequality}\n \\begin{aligned}\n &\\big\\| D_{x_j}^{\\alpha}(fg)-fD_{x_j}^{\\alpha}f-g D_{x_j}^{\\alpha}f  \\big\\|_{L^2_{x_j}({\\mathbb R};L^2(Q))}\\\\\n& \\hspace{.7cm}\\leq c\\big\\| D_{x_j}^{\\alpha_1}f\\big\\|_{L^{p_1}_{x_j}({\\mathbb R};L^{q_1}(Q))}   \\big\\|D_{x_j}^{\\alpha_2}g\\big\\|_{L^{p_2}_{x_j}({\\mathbb R};L^{q_2}(Q))},\n \\end{aligned}\n \\end{equation}\n where $Q={\\mathbb R}^{n-1}\\times [0,T]$. \n\nTo illustrate the use of the inequality \\eqref{inequality} in estimating the right-hand side of \\eqref{tterm}, \nassume without loss of generality that $s+\\frac{1}{2}=1+\\alpha$ with  $\\alpha\\in (0,1)$. \nThus, omitting the domains of integration ${\\mathbb R}$ and $Q$,\n\\begin{equation}\n\\label{001}\n\\begin{aligned}\n\\big\\| D_{x_j}^{s+1\/2}(fg)\\big\\|_{L^2_{x_j}L^2}& = \\big\\| D_{x_j}^{1+\\alpha}(fg)\\big\\|_{L^2_{x_j}L^2} \\simeq \\big\\| D_{x_j}^{\\alpha}\\partial_{x_j}(fg) \\big\\|_{L^2_{x_j}L^2}\\\\\n&\\leq c\\Big( \\big\\| D_{x_j}^{\\alpha}(f\\partial_{x_j}g)\\big\\|_{L^2_{x_j}L^2}\n\\big\\| D_{x_j}^{\\alpha}(\\partial_{x_j}f g)\\big\\|_{L^2_{x_j}L^2}\\Big).\n\\end{aligned}\n\\end{equation}\nBy symmetry it suffices to consider only one of the terms \non the right-hand side of \\eqref{001}. From  \\eqref{inequality}, with $\\alpha_1=0$, \nthere obtains\n\\begin{equation*}\n\\label{002}\n\\begin{aligned}\n\\big\\| D_{x_j}^{\\alpha}(g\\partial_{x_j}f)\\big\\|_{L^2_{x_j}L^2}& \\leq  \\big\\| D_{x_j}^{\\alpha}(g\\partial_{x_j}f)-g D_{x_j}^{\\alpha}\\partial_{x_j}f - \\partial_{x_j}fD_{x_j}^{\\alpha}g \\big\\|_{L^2_{x_j}L^2}\\\\\n& \\quad +\\big\\| g D_{x_j}^{\\alpha}\\partial_{x_j}f\\big\\| _{L^2_{x_j}L^2} + \\big\\|  \\partial_{x_j}f D_{x_j}^{\\alpha}g\\big\\| _{L^2_{x_j}L^2}\\\\\n&\\leq \\big\\| g D_{x_j}^{\\alpha}\\partial_{x_j}f \\big\\| _{L^2_{x_j}L^2} + c\\big\\| \\partial_{x_j}f\\big\\|_{L^{p_1}_{x_j}L^{q_1}}   \\big\\|D_{x_j}^{\\alpha_2}g\\big\\|_{L^{p_2}_{x_j}L^{q_2}}\n\\end{aligned}\n\\end{equation*}\nwith $p_1,p_2,q_1, q_2$ restricted as above.\n\n\nUsing the latter inequality to continue the inequality \\eqref{tterm} yields\n\\begin{equation}\\label{KPV}\n\\begin{aligned}\n&\\big\\|D_{x_j}^{s+1\/2} I\\big\\|_{L^\\infty_t([0,T]:L^2({\\mathbb R}^n))}  \\\\ \n &\\hspace{.8cm} \\leq  \\ cT^{1\/2} \\big\\|D_{x_j}^{s+1\/2} (|u|^{2k} u)\\big\\|_{L^2([0,T]\\times {\\mathbb R}^n)}\\\\\n&\\hspace{.8cm} \\leq \\ cT^{1\/2}\\Big ( \\big\\|u\\big\\|^{2k}_{L^{4k}_{x_j}({\\mathbb R}; L^\\infty_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))} \\\\\n& \\hspace{1.5cm} \\times \\big\\|D_{x_j}^{s+1\/2} u\\big\\|_{L^{\\infty}_{x_j}({\\mathbb R}; L^2_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))}+R\\Big),\n\\end{aligned}\n\\end{equation}\nwhere the remainder $R$ includes only estimates for terms involving powers of $u$,  $\\partial_{x_j}u$ and $D_{x_j}^{\\alpha}u$ . These are straightforwardly bounded above \n by use of \\eqref{ouf2}. In fact, a bound for them is an interpolation between the first two terms \non the right-hand side of \\eqref{KPV}. It therefore remains to bound only the terms\n\\begin{equation}\\label{eq:A}\n \\big\\|u\\big\\|^{2k}_{L^{4k}_{x_j}({\\mathbb R}; L^\\infty_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:B}\n\\big\\|D_{x_j}^{s+1\/2} u\\big\\|_{L^{\\infty}_{x_j}({\\mathbb R}; L^2_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))},\n\\end{equation}\n$j = 1, \\cdots ,n$, appearing in \\eqref{KPV}.\\\\\n\n{\\it Step 3.} To bound the quantity appearing in  \\eqref{eq:A}, first note that \\eqref{ouf} implies\n\\begin{equation*}\n\\big \\|\\sup_{0\\leq t\\leq T}\\quad\\sup_{x_1\\dots x_{j-1}x_{j+1}\\dots x_n }|e^{it\\Delta}u_0|\\;\\big \\|_{L^{4k}_{x_j}({\\mathbb R})}\n\\lesssim \\big\\|\\sup_{0\\leq t\\leq T} |e^{it\\Delta} J^s_{\\hat \\jmath} u_0|\\;\\big\\|_{L^{4k}({\\mathbb R}^n)},\n\\end{equation*}\nwith $s>\\frac{(n-1)}{4k}$. This estimate can be extended using Lemma \\ref{lem:B} by \nobserving that \n\\begin{equation*}\\label{une}\n\\begin{aligned}\n\\Big\\|\\sup_{0\\leq t\\leq T} |e^{it\\Delta} J^s_{\\hat \\jmath} u_0|\\;\\Big\\|_{L^{4k}({\\mathbb R}^n)} \n\\leq \\big\\| J^s_{\\hat \\jmath} u_0\\big\\|_{H^{\\sigma}({\\mathbb R}^n)}\\lesssim \\big\\|u_0\\big\\|_{H^{\\sigma+s}({\\mathbb R}^n)}=\\big\\|u_0\\big\\|_{H^{\\bar{s}}({\\mathbb R}^n)},\n\\end{aligned}\n\\end{equation*}\nwhere\n\\[\n\\bar{s} =s+\\sigma >\\frac{n-1}{4k}+n\\Big(\\frac{1}{2}-\\frac{1}{4k}\\Big)=\\frac{n}{2}-\\frac{1}{4k}.\n\\]\nInserting this inequality in the Duhamel representation \\eqref{eq:duhamel} with \n\\[\ns\\in\\Big(\\frac{n}{2}-\\frac{1}{4k},\\frac{n}{2}\\Big]=\\Big(\\frac{n}{2}-\\frac{1}{2p},\\frac{n}{2}\\Big],\n\\]\nit follows that\n\\begin{equation*}\n\\big\\|u\\big\\|_{L^{4k}_{x_j}({\\mathbb R};L^\\infty_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_n t}({\\mathbb R}^{n-1}\\times[0,T]))}\\leq C\\big\\|u_0\\big\\|_{s,2}+\n\\big\\|J^s(|u|^pu)\\big\\|_{L^1_t([0,T]; L^2({\\mathbb R}^n))}.\n\\end{equation*}\nSince $p=2k$, use of  a fractional Leibniz rule (see \\cite{KP}) implies that\n\\begin{equation}\\label{KaPo}\n\\big\\|J^s(|u|^{2k}u)\\big\\|_{L^1([0,T]; L^2({\\mathbb R}^n))}\\leq c\\big\\|u\\big\\|^{2k}_{L^{2k}([0,T];L^\\infty({\\mathbb R}^n))}\\big\\|J^su\\big\\|_{L^\\infty([0,T];L^2({\\mathbb R}^n))}.\n\\end{equation}\n\nIf it was known that \n\\begin{equation}\\label{enfin}\n\\|u\\|_{L^{2k}([0,T];L^\\infty({\\mathbb R}^n)))}\\leq cT^{\\theta}\\|J^su\\|_{L^q([0,T];L^r({\\mathbb R}^n))}\n\\end{equation}\nfor some $\\theta>0$ and for some admissible Strichartz pair $(r,q)$, then the sequence \nof inequalities could be closed. To obtain \\eqref{enfin},  recall that \n\\[\ns>\\frac{n}{2}-\\frac{1}{4k}=\\frac{2kn-1}{4k}=\\frac{n}{4kn\/(2kn-1)},\n\\]\nso we can \ntake \n\\[\nr=\\frac{4kn}{(2kn-1)}<\\frac{2n}{(n-2)}, \\quad \\text{if $n\\geq 3$}, \n\\]\n(the cases $n=1$ or $2$ are immediate) and $q=8k.$ By Sobolev embedding, the inequality  \\eqref{enfin} \nis seen to hold with $\\theta=\\frac{3}{8k}$. In summary, all the terms in  \n  \\eqref{eq:A} are shown to be bounded.\\\\\n\n{\\it Step 4.} Finally, attention is turned to terms of the form appearing in \\eqref{eq:B}. \n The local smoothing estimate enunciated in Lemma \\ref{lem:smooth} together \nwith  Duhamel's formula imply that  \n\\begin{eqnarray*}\n&\\big\\|D_{x_j}^{s+1\/2} u\\big\\|_{L^{\\infty}_{x_j}({\\mathbb R}; L^2_{x_1\\cdot \\cdot x_{j-1}x_{j+1}\\cdot \\cdot x_n,t}({\\mathbb R}^{n-1}\\times[0,T]))}\n\\\\\n&\\hspace{.8cm}   \\leq \\big\\|u_0\\big\\|_{H^{s}({\\mathbb R}^n)}+\\big\\|J^s(|u|^{2k}u)\\big\\|_{L^1([0,T];L^2({\\mathbb R}^n))}.\n\\end{eqnarray*}\nThe right-hand side was already estimated in \\eqref{KaPo}--\\eqref{enfin}. \nBecause of \\eqref{KPV}, this shows that  there exists a $C=C(T,n)>0$ such that\n\\[\n\\big\\|D_{x_j}^{s+1\/2} I\\big\\|_{L^\\infty([0,T]:L^2({\\mathbb R}^n))} \\leq  C.\n\\]\nSumming these estimates over $j$ for  $j=1\\cdots,n$  yields the result advertised in \nthe proposition.   \n\nFinally, we remark that in the case where  $p$ is not an even integer, the restriction \non $p$ are necessary and one needs \nto supplement  the Leibnitz-type inequality \\eqref{inequality} with\nthe chain rule for fractional derivatives adduced in the Appendix of \\cite{KPV93}. \n\\end{proof}\n\n\n \n \n\n\n\\subsection{An even stronger smoothing property}\n\nFor large $s$ and higher values of $p$, a stronger smoothing \n result than  that established in Proposition \\ref{prop:key} holds.\n\\begin{proposition}\\label{prop:keybis}\nLet $u_0\\in H^s({\\mathbb R}^n), s > \\frac{n}{2} - \\frac{1}{2p}$ with $p\\geq 2$ and \n$\\lfloor p+1\\rfloor \\geq s + \\frac{1}{2}$ if $p$ is not an even integer.  \nUnder these hypothses, it follows that for any $\\varepsilon > 0$,  \n\\[\nI\\in C([0,T];H^{s+1-\\varepsilon}({\\mathbb R}^n)),\n\\]\n where the  notation is taken from Proposition \\ref{prop:key} \n\\end{proposition}\n\\begin{remark}The loss of $\\varepsilon$ in the regularity of $I$ is needed to obtain a factor \n$T^{\\delta(\\varepsilon)},  \\delta(\\varepsilon)>0$, on the right-hand side of the inequalities below which \nallows them to be closed. \nIt can be recovered by assuming that the data $u_0$ is small enough in $H^s({\\mathbb R}^n).$\n\\end{remark}\nThe proof of Proposition {\\ref{prop:keybis} uses the following smoothing estimate, which is a direct consequence of \nLemma \\ref{lem:smooth}, a duality argument and the  Christ-Kiselev lemma \\cite{CK}. For a proof, see \\cite[Chapter 4]{LP}.\n\n\\begin{lemma}\\label{lem:Kis} For any $n\\in {\\mathbb N}$, the inequality\n\\[\n\\left\\|D^{1\/2}_{x_j}\\int_0^te^{i(t-s)\\Delta}f(\\cdot,s) \\, ds\\right\\|_{L^\\infty({\\mathbb R};L^2({\\mathbb R}^n))}\n\\leq C\\big\\|\\mathcal H_j f \\big\\|_{L^1_{x_j}({\\mathbb R};L^2_{x_1...x_{j-1}x_{j+1}...x_n t}({\\mathbb R}^n))}\n\\]\nholds, where $\\mathcal H_j$ denotes the {\\it Hilbert transform} in the $j$-th variable, which \nis to say, \n\\[\n\\mathcal H_j f(x):=-i \\, \\mathcal F^{-1} \\Big ( \\text{{\\rm sign}}(\\xi_j) \\widehat f(\\xi) \\Big)(x) .\n\\]\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Proposition \\ref{prop:keybis}] \nThe proof is similar to that of Proposition \\ref{prop:key} and hence, we \nonly sketch the main differences. \n\nFirst consider the case of  data $u_0\\in H^s({\\mathbb R}^n)$\nwhich is small, so that all the norms involved are indeed \\lq\\lq small\". \nWe want to show that the integral term $I$ in Duhamel's formula is one order smoother in the Sobolev scale $C([0,T];H^s({\\mathbb R}^n))$ \nthan the the free propagation $e^{it\\Delta} u_0.$ To this end, apply Lemma \\ref{lem:Kis} together with the commutator estimate in \n\\cite[Theorem A.13]{KPV93} to write\n\\begin{equation}\\label{eq:L3}\n\\begin{aligned}\n&\\ \\big\\|\\mathcal H_j D^{s+1}_{x_j} I \\big\\|_{L^\\infty([0,T];L^2({\\mathbb R}^n))} \\\\ \n&\\hspace{.8cm} \\lesssim \n\\big\\| D^{s+1\/2}_{x_j}(|u|^{2k}u) \\big\\|_{L^1_{x_j}({\\mathbb R};L^2_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_n t}({\\mathbb R}^{n-1}\\times[0,T] ))}\\\\ \n& \\hspace{.8cm} \\lesssim\\Big (\\|u\\big\\|^{2k}_{L^{2k}_{x_j}({\\mathbb R};L^2_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_n t}({\\mathbb R}^{n-1}\\times[0,T] ))} \\\\\n&\\hspace{1.3cm} \\times \\|D_{x_j}^{s+1\/2}u\\|_{L^{\\infty}_{x_j}({\\mathbb R};L^2_{x_1\\cdot\\cdot x_{j-1}x_{j+1}\\cdot\\cdot x_n t}({\\mathbb R}^{n-1}\\times[0,T] ))}+R\\Big).\n\\end{aligned}\n\\end{equation}\nTo estimate the  two explicit quantities on the right-hand side of the last inequality,  \none uses arguments similar to those given in the proof of Proposition \\ref{prop:key}. \nThe estimates for the remainder terms represented by $R$ then follow by interpolation \nof the previous estimates. Since the terms on the right-hand side of \\eqref{eq:L3} are \nquadratic and each factor is small, one can close the estimate and get the desired result, but only provided that $u_0$ is sufficiently small.\n\nFor data $u_0\\in H^s({\\mathbb R}^n)$ of arbitrary size one gives up $\\varepsilon$-amount of spatial\nsmoothing for a little temporal smoothing, thereby obtaining the \n factor  $T^{\\delta(\\epsilon)}$, $\\delta(\\varepsilon)>0$  on the  right-hand side.  The \nright-hand side of the estimate then has lower homogeneity than the left side \nand the proof proceeds.\n\\end{proof}\n\n\n\n\\subsection{Extension to the case of non-elliptic Schr\\\"odinger equations} The results above extend to the case of non-elliptic, non-degenerate, nonlinear Schr\\\"odinger equations of the form\n\\begin{equation}\\label{eq:hypNLS}\ni\\partial_t u+\\Delta_{\\rm H} u\\pm |u|^p u=0,\\quad u\\big|_{t=0}=u_0(x),\n\\end{equation}\nwhere \n\\[\n\\Delta_{\\rm H} := \\partial^2_{x_1} + \\dots  \\partial^2_{x_j} -  \\partial^2_{x_{j+1}}\\dots -  \\partial^2_{x_n}.\n\\]\n\\begin{proposition}\\label{prop:hypNLS}\nThe result of Theorem \\ref{thm:main} also holds  for the initial-value \nproblem delineated in  \\eqref{eq:hypNLS} .\n\\end{proposition}\n\\begin{proof} Remark first that for initial data  of the form \n\\begin{equation}\n\\label{data-hyp}\n\\widetilde u_0(x)=\n\\frac{e^{-i\\alpha\\big((x_1-q_1)^2+\\dots+(x_j-q_j)^2-(x_{j+1}-q_{j+1})^2-\\dots-(x_n-q_n)^2\\big)}}{(1+|x|^2)^m},\n\\end{equation}\nwith $n\/4<m\\leq n\/2$, the solution of the  initial-value problem associated to \nthe linearization of \\eqref{eq:hypNLS} around the rest state, \ni.e., \\[u(x,t) =e^{it\\Delta_H}\\widetilde u_0(x),\\] \nsatisfies the conclusions of Lemma \\ref{lem:IC}, and in particular, blows up \n at the point $ \\widetilde q=(q_1,\\dots,q_j,-q_{j+1},\\dots,-q_n)$.\n\n\nNext,  notice  that the the local well-posedness result stated in Proposition \\ref{prop:CW}, is solely \nbased on Strichartz estimates, which are exactly the same for the two groups $\\{e^{it \\Delta} \\}_{t\\in {\\mathbb R}}$ and \n$\\{e^{it {\\Delta_{\\rm H}}} \\}_{t\\in {\\mathbb R}}$ ({\\it cf.} \\cite{GS}). \nIn other words, Lemma \\ref{lem:Strich} also holds in the  non-elliptic case.\n Together with Sobolev embeddings, this yields a unique solution $u\\in W_{s,n}^{T}$ to \\eqref{eq:hypNLS}, \nby the same arguments as in \\cite{CW}. Furthermore, the local smoothing estimates stated in Lemma \\ref{lem:smooth} \ncarry over to the  non-elliptic situation.   \n\nFinally, for the boundedness of the associated maximal function \n(see Lemma \\ref{lem:B}), we point to the \ninequality\n\\begin{equation}\n\\label{max-non-ellip}\n\\Big \\| \\sup_{0\\leq t\\leq T} \\big |e^{it(\\partial^2_{x_1} - \n\\partial^2_{x_2})}f \\big| \\,                              \\Big \\|_{L^4({\\mathbb R}^n)} \\leq C_T \\Big\\| D_x^{1\/2} f \\Big\\|_{L^2({\\mathbb R}^2)},\n\\end{equation}\nproved in \\cite[Theorem 2.6]{RVV} and observe that the same argument used there\n to establish \\eqref{max-non-ellip} shows that\n\\[\n\\Big \\| \\sup_{0\\leq t\\leq T}  \\big|e^{it \\Delta_{\\rm H} }f \\big| \\, \\Big\\|_{L^4({\\mathbb R}^n)} \\leq \nC_T \\Big\\| D_x^{n\/4} f  \\Big\\|_{L^2({\\mathbb R}^n)},\n\\]\nwhich is the desired estimate. The result then follows along the same lines as given in the proof of Theorem \\ref{thm:main}.\n\\end{proof}\n\nAn important consequence of this is the possibility of dispersive blow up for the Davey-Stewartson system in the \\lq\\lq elliptic\/elliptic\" or \\lq\\lq hyperbolic\/elliptic\" cases (see \\cite{GS} for more details \nabout  these systems, in particular for theory of local well-posedness).  This system arose originally \nas an approximate description of surface gravity-capillary waves in shallow water, but has \nother applications as well.   \n\n\\begin{corollary}\nFor  $\\alpha$ and $\\beta$ real and non-zero, consider the Davey-Stewartson system\n\\begin{equation}\n\\label{DS-ivp}\n\\left \\{ \n\\begin{aligned} \n&\\ i\\partial_t u\\pm \\partial^2_{x_1}u + \\partial^2_{x_2} u= \\alpha |u|^2 u + \\beta u \\partial_{x_1}\\phi ,\\;\\;\\;\\;\\;\\;\\;\\;\\;x=(x_1,x_2)\\in {\\mathbb R}^2, \\\\\n& \\ \\Delta \\phi = \\partial_{x_1} |u|^2.\n\\end{aligned}\n\\right.\n\\end{equation}\nThen there exist initial values $u_0\\in H^s({\\mathbb R}^2)\\cap L^{\\infty}({\\mathbb R}^2)\\cap C^{\\infty}({\\mathbb R}^2)$ with $s\\in (\\frac{1}{2},1]$ \nsuch that the solution $u$ of \\eqref{DS-ivp} with the initial condition $u(x,0)=u_0(x)$ exhibits dispersive blow up.\n\\end{corollary}\n\n\\begin{proof}\nRewrite the system \\eqref{DS-ivp} as the single equation\n\\begin{equation*}\ni\\partial_t u\\pm \\partial^2_{x_1}u + \\partial^2_{x_2} u= \\alpha |u|^2 u - \\beta u \\partial_{x_1}^2(-\\Delta)^{-1}(|u|^2),\n\\end{equation*}\nin the usual way.   The latter equation has the  equivalent form\n\\begin{equation*}\ni\\partial_t u\\pm \\partial^2_{x_1}u + \\partial^2_{x_2} u= \\alpha |u|^2 u - \\beta u R_1R_1 (|u|^2),\n\\end{equation*}\nwhere\n\\begin{equation*}\nR_1 f(x_1,x_2):= \\mathcal F^{-1}\\left(i\\frac{\\xi_1}{|\\xi|}\\widehat f(\\xi_1,\\xi_2)\\right)(x_1,x_2)\n\\end{equation*}\n denotes the $1$-Riesz transform in ${\\mathbb R}^2$\n\nThe $L^p$-continuity of the Riesz transform, \nimplies that the result in Proposition \\ref{prop:CW} and the argument entailed in the proof of\n Proposition \\ref{prop:key} still hold. Hence,  Theorem \n\\ref{thm:main} extends to solutions of the system \\eqref{DS-ivp}. \\end{proof}\n\n\n\n\n\n\\section{Dispersive blow up in the Gross-Pitaevskii  equation}\\label{sec:GP}\n\nIn this section, the discussion is moved to the initial-value problem \n\\begin{equation} \\label{eq:GP} \ni \\partial_t \\psi + \\Delta \\psi+ (1-|\\psi|^2)\\psi=0, \\quad \\psi\\big|_{t=0} = \\psi_0(x)\n\\end{equation}\nfor the Gross-Pitaevskii  equation.  \nHere $(x,t) \\in {\\mathbb R}^n\\times {\\mathbb R}$ and $\\psi$ is subject to the boundary condition \n\\begin{equation}\\label{eq:bc}\n\\lim_{|x|\\to \\infty} \\psi(x,t) = 1, \\quad \\text{for all $t\\in {\\mathbb R}$.}\n\\end{equation}\nThe Gross-Pitaevskii equation arises, for example, in the description of Bose-Einstein condensates, superfluid helium $\\text{He}^2$ and, \nin one spatial dimension, as a model for light propagation in a fiber \noptics cable (see for instance the survey article \\cite{BGS3} and other articles in the same volume). \n\\begin{remark} An important conserved quantity of \\eqref{eq:GP} is the Ginzburg-Landau energy, defined by\n\\[\nE(\\psi) = \\frac{1}{2} \\int_{{\\mathbb R}^n} |\\nabla \\psi|^2 \\, dx + \\frac{1}{4} \\int_{{\\mathbb R}^n} (1 - |\\psi|^2)^2 \\, dx.\n\\]\nThis invariant points to a natural energy space for the Gross-Pitaevskii equation, namely \n\\[\n\\mathcal{E}({\\mathbb R}^n) = \\{ \\psi \\in H^1_{\\text{loc}}({\\mathbb R}^n),  E(\\psi) < + \\infty \\}\n\\]\n (see \\cite{Ge1, Ge2} for more details).\n\\end{remark}\n\n\n\n\n\n\\subsection{A reformulation of the Gross-Pitaevskii Equation}\n\nDue to the non-zero boundary condition \\eqref{eq:bc} at infinity, the \nideas that worked in the earlier sections do not apply directly to show \ndispersive blow up.    \nTo prove dispersive blow up for the Gross-Pitaevskii equation, make the change \n \\[\\psi(t,x) =1+u(t,x),\\] \nof the dependent variable so that $u\\in L^2({\\mathbb R}^n)$ describes the deviation from the steady state. In terms of $u$, the \nGross-Pitaevskii equation \\eqref{eq:GP} becomes\n\\begin{equation}\\label{bife}\ni\\partial_t u +\\Delta u-2 \\mathrm{Re} \\,u=F(u) , \\quad  u \\big|_{t=0} = \\psi_0(x)-1,\n\\end{equation} \nwhere \n\\[\nF(u) = u^2+2|u|^2+|u|^2u.\n\\]\nEven for this  transformed initial-value problem \\eqref{bife}, dispersive blow up\n does not fall directly to the general lines of \nargument proposed earlier.  This is due to the appearance of \nthe slightly odd, ${\\mathbb R}$-linear term $-2 \\mathrm{Re}\\,  u$, which at least\n in principle might cancel out the effect of dispersive blow up stemming from the Laplacian. \n\nThis obstacle will be surmounted by using the further reformulation  developed \nby Gustafson, Nakanishi and Tsai in \\cite{GNT1, GNT2}. \nFollowing their lead,  introduce the\nFourier multipliers\n\\begin{equation}  \\label{operators}\nA:=  \\sqrt{-\\Delta(2-\\Delta)} \\quad {\\rm and} \\quad  B:= \\sqrt{-\\Delta(2-\\Delta)^{-1}}\\, .\n\\end{equation}\nThese operators satisfy $A = -\\Delta B^{-1} = (2- \\Delta)B=\\sqrt{-\\Delta(2-\\Delta)}$. Next,  define the ${\\mathbb R}$-linear operator\n\\[\n\\Upsilon u := B\\, \\mathrm{Re}\\, u + i \\,\\mathrm{Im} \\, u\\equiv B u_1 + i u_2, \n\\]\nwhere $u = u_1 + iu_2$.   \nOne checks (see \\cite{GNT1, GNT2}) that the left-hand side of the equation in \\eqref{bife} \ncan be written in the form\n\\begin{equation}\\label{eq:identity}\ni\\partial_t u +\\Delta u-2 \\mathrm{Re} \\,u =  \\Upsilon (i\\partial_t  - A ) \\Upsilon^{-1}u.\n\\end{equation}\nDenote by $v$ the function\n\\[\nv:=\\Upsilon^{-1}u \\equiv \\Upsilon^{-1}(u_1+iu_2)=B^{-1}u_1+iu_2\n\\] \nand rewrite \\eqref{bife} as\n\\begin{equation}\\label{eq:reGP}\ni \\partial_t v - A v = \\Upsilon^{-1}F(u), \\quad v\\big|_{t=0} = \\Upsilon^{-1} (\\psi_0-1).\n\\end{equation}\nThe associated free evolution is \n\\[\nw(x,t)=e^{-itA} v_0 (x) := \\frac{1}{(2\\pi)^n} \\int_{{\\mathbb R}^n} e^{it\\sqrt{|\\xi|^2(|\\xi|^2+2)}} \\widehat u_0(\\xi) e^{i \\xi\\cdot x} \\, d\\xi,\n\\]\nwhich can be represented as follows.\n\\begin{lemma}\\label{lem:decomp}\nFor any $f\\in L^2({\\mathbb R}^n)$ and any $t\\not =0$, the group $\\{e^{-itA}\\}_{t\\in {\\mathbb R}}$ has the representation\n\\[e^{-itA} f(x) = (G(\\cdot , t) \\ast f)(x),\\] where the kernel is $G =  G_1 + G_2$ with\n\\[\nG_1(x,t)   =\\frac{e^{it}}{(4\\pi it)^{n\/2}}\\, e^{\\frac{i|x|^2}{4t}}\\quad and \\quad G_2(x,t) := \\frac{e^{it} }{(2\\pi)^n} \\int_{{\\mathbb R}^n} \\kappa(\\xi, t) e^{it |\\xi|^2} e^{ix\\cdot \\xi}d\\xi.\n\\]\nHere, the convolution is with respect to the spatial variable over ${\\mathbb R}^n$ \nand the kernel  $\\kappa $ is \n\\[\n\\kappa(\\xi, t)=2i\\sin \\Big(\\frac{t}{2}r(\\xi)\\Big)e^{i\\frac{t}{2} r(\\xi)}\n\\]\nwith \n\\[\nr(\\xi)= \\frac{-2|\\xi|^2}{\\left(\\sqrt{|\\xi|^2(|\\xi|^2+2)}+|\\xi|^2\\right)^2}\\sim O(|\\xi|^{-2}), \\ \\text{as $|\\xi|\\to \\infty$.}\n\\]\nMoreover, $r$ lies in $C_{\\rm b}({\\mathbb R}^n)$ and is smooth away from the origin.\n\\end{lemma}\n\nNote that $G_1$ is the usual Schr\\\"odinger group multiplied by $e^{it}$.  Remark also that, as $|\\xi|\\to \\infty$, $\\kappa(\\cdot, t)$ decays to zero like $|\\xi| ^{-2}$, \nuniformly on compact time-intervals. \n\n\\begin{proof} \nThe convolution kernel    $G$ is\n\\[G(x,t)=\\mathcal F^{-1}\\left(e^{it \\sqrt{(|\\xi|^2(|\\xi|^2+2)}}\\right)(x,t).\\]\nObserve that \n\\[\n\\sqrt{(|\\xi|^2(|\\xi|^2+2)}=|\\xi|^2+a(\\xi),\n\\]\nwhere \n\\[\na(\\xi)=\\frac{2|\\xi|^2}{\\sqrt{(|\\xi|^2(|\\xi|^2+2)}+|\\xi|^2}=1-\\frac{2|\\xi|^2}{\\left(\\sqrt{(|\\xi|^2(|\\xi|^2+2)}+|\\xi|^2\\right)^2}=1+r(\\xi).\\]\n Consequently, it follows that\n \\[\n e^{it \\sqrt{(|\\xi|^2(|\\xi|^2+2)}}=e^{it|\\xi|^2}e^{it}e^{itr(\\xi)}=e^{it}e^{it|\\xi|^2}(1+f_t(|\\xi|)),\n \\]\nwhere\n\\[\nf_t(|\\xi|)=2i\\sin \\Big(\\frac{t}{2}r(\\xi)\\Big)e^{it\\frac {r(\\xi)}{2}}\\]\nis continuous, smooth on ${\\mathbb R}^n$ and\ndecays to zero like $|\\xi| ^{-2}$, as $|\\xi|\\to \\infty$, uniformly on compact time intervals in $(0,\\infty)$,  since\n$r({|\\xi|})$ does so.\nThis allows  the propagator in Fourier space to be written as\n\\[\n\\widehat G(\\xi, t)= e^{it\\sqrt{|\\xi|^2(|\\xi|^2+2)}}=e^{it}e^{it|\\xi|^2}(1+\\kappa(\\xi, t))\n\\]\nwith $\\kappa$ as above. \n\\end{proof}\n\nWith this representation in hand, dispersive blow up for the evolutionary group $\\{e^{-itA}\\}_{t\\in {\\mathbb R}}$ \ncan be established for at least  spatial dimensions less than or equal to three. \n\n\\begin{lemma}\\label{lem:DBUv}\nLet $n\\leq 3$  and suppose given $x_*\\in {\\mathbb R}^n$, $t_*>0$ and $s \\in [0,\\frac{n}{2})$.  Then \nthere exist initial data $v_0\\in C^\\infty({\\mathbb R}^n)\\cap H^s({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ such that\n\\[\nw(\\cdot, t) = e^{-itA} v_0 \\in H^s({\\mathbb R}^n)\n\\]\nexhibits dispersive blow up at $(x_*, t_*)$.\n\\end{lemma}\n\\begin{proof} Choose $(x_*,t_*) = (0,\\frac{1}{4})$ without loss of generality. \n\tAs in Lemma \\ref{lem:linear}, let \n\\begin{equation} \\label{initdata}\nv_0(x)=\\frac{e^{-i|x|^2}}{(1+|x|^2)^m}, \\quad \\text{with  $\\frac{n}{4}<m\\leq \\frac{n}{2}$.}\n\\end{equation}\nUsing Lemma \\ref{lem:decomp},  it is found that \n\\[\ne^{-itA} v_0(x) = (G_1(\\cdot , t) \\ast v_0)(x) +(G_2(\\cdot , t) \\ast v_0)(x).\n\\]\nIn view of the calculations in Section \\ref{sec:lin} it is inferred \nthat the first term on the right-hand side will exhibit dispersive blow up at $(0,\\frac{1}{4})$.\nIt thus suffices to show that  $G_2(\\cdot,t)\\ast v_0\\in C_{\\rm b}( {\\mathbb R}^n)$, uniformly on compact time-intervals. \n\nThe kernel $G_2(\\cdot ,t)\\in L^2({\\mathbb R}^n)$ for $n\\leq 3$, due to the decay properties of $r(\\xi)$. Since $v_0\\in L^2({\\mathbb R}^n)$ by \nconstruction,  the product $\\widehat G_2 (\\cdot , t) \\widehat v_0\\in L^1({\\mathbb R}^n)$, uniformly on compact time-intervals.  The Riemann-Lebesgue lemma then implies the desired result.\n\\end{proof}\n\nNote that even though $e^{-it A}$ represents the linear evolution operator associated to \\eqref{eq:reGP}, it includes part of the nonlinearity of the \noriginal Gross-Pitaevskii equation \\eqref{eq:GP}, as can be seen from \\eqref{eq:identity}.\n\n\n\n\n\\subsection{Proof of dispersive blow up for Gross-Pitaevskii} \n\n\nLemma \\ref{lem:DBUv} together with the results of Section \\ref{sec:NLS} now allow us to prove \ndispersive blow up for equation \\eqref{eq:reGP} and, consequently, also for the original \nGross-Pitaevskii equation (in physically relevant dimensions $n = 1,2,3$). \nTo effect a proof, the following lemma on the  mapping properties of $B$ and its inverse will be used.\n\\begin{lemma}\\label{mapping}\nThe operator $B$ in \\eqref{operators} may be written in the form\n\\[B={\\rm Id}+B_1, \\quad B^{-1}={\\rm Id} +B_2,\\] \nwhere $B_1$, $B_2\\in \\mathcal L(H^s({\\mathbb R}^n),H^{s+2}({\\mathbb R}^n))$, for all $s\\in {\\mathbb R}.$ \n\\end{lemma}\n\n\\begin{proof}\nThe proof is a simple computation on the level of the Fourier symbols $b_1$ and $b_2$ associated to $B_1$ and $B_2.$ For example, the symbol $b_1$ is\n\\[\nb_1(\\xi)=-\\frac{2}{(2+|\\xi|^2)( \\sqrt{|\\xi|^2(2+|\\xi|^2)^{-1}}+1)}.\n\\]\nFrom this formula, one sees that $b_1$ is  in fact uniformly bounded and \nis $O(|\\xi|^{-2})$ as $|\\xi| \\to \\infty$. Similarly, one finds\n\\[b_2(\\xi)=\\frac{2}{(2+|\\xi|^2)\\sqrt{|\\xi|^2(2+|\\xi|^2)^{-1}}[1+(|\\xi|^2(2+|\\xi|^2)^{-1}]^{1\/2}},\\]\nfrom which one infers the assertion about $B_2$.\n\\end{proof}\n\n\nIn particular, since $\\Upsilon^{-1}u=B^{-1}u_1+iu_2$, the last lemma\n implies that, for all $s\\in {\\mathbb R}, \\Upsilon$ and  $\\Upsilon ^{-1}$ both lie in \n$\\mathcal L(H^s({\\mathbb R}^n),H^{s}({\\mathbb R}^n))$. \n\nHere is  the main result of this section.\n\n\\begin{theorem} \\label{DBU:v}\nLet $n\\leq 3$, $x_*\\in {\\mathbb R}^n$, $t_*>0$ and $s \\in (0,\\frac{n}{2})$ be given.  Then\nthere exist initial data $v_0\\in C^\\infty({\\mathbb R}^n)\\cap H^s({\\mathbb R}^n)\\cap L^\\infty({\\mathbb R}^n)$ \nsuch that the corresponding solution $v\\in C([0,T], H^s({\\mathbb R}^n))$ of \\eqref{eq:reGP} exhibits dispersive blow up at $(x_*,t_*).$\n\n\\end{theorem}\n\\begin{proof} Take $v_0$ in the form  \\eqref{initdata} in the proof of Lemma \\ref{lem:DBUv}. \nBy an approprite choice of $m$, Lemma \\ref{lem:linear},  guarantees that $v_0$ lies \nin $H^s({\\mathbb R}^n)$.  \nNext, in view of Lemma \\ref{mapping}, the corresponding $u_0=\\Upsilon^{-1} v_0\\equiv B^{-1}\\text{Re} \\, v_0 + i \\text{Im} \\, v_0$ has the same Sobolev regularity as $v_0$. \nFor such a $u_0$, the Cazenave-Weissler theory can be applied to the Gross-Pitaevskii equation \\eqref{bife}, yielding a local in-time solution $u\\in C([0,T], H^s({\\mathbb R}^n))$ for the specified\n $s<\\frac{n}{2}$, which  \n leads to a solution $v\\equiv \\Upsilon u \\in C([0,T], H^s({\\mathbb R}^n))$ of \\eqref{eq:reGP}. \nConsequently,  we can use the strategy followed for the usual \nnonlinear Schr\\\"odinger equation. \nTo this end, write \\begin{equation}\\label{eq:duhamelv}\nv(x,t)=e^{-it A}v_0 (x)+ i \\int_0^te^{-i(t-s)A}  \\Upsilon^{-1}F(u(x,s)) \\, ds,\n\\end{equation}\nwhere the nonlinearity is explicitly given by \n\\[\n \\Upsilon^{-1}F(u) =  B^{-1} (3 u_1^2+u_2^2+|u|^2u_1)+iu_2(2u_1+|u|^2),\n\\]\nsince $u=u_1+iu_2$. As before, the first term on the right-hand side exhibits dispersive blow up at $(x_*, t_*)=(0,\\frac{1}{4})$ provided $n<4$. \nThe advertised result will be in hand when the second term is known to be uniformly bounded.\n\nThe integral term in \\eqref {eq:duhamelv} splits into $I_1+I_2$ where\n\\[\nI_j(x,t)=\\int_0^t G_j(\\cdot, t-s)\\ast \\Upsilon^{-1}F(u(\\cdot,s)) \\, ds,\\quad j=1,2,\n\\]\ncorresponding to $G=G_1+G_2$.\nThe fact that $I_2$ is a bounded continuous function can be concluded by the same argument \nthat prevailed in the proof \nof Lemma \\ref{lem:DBUv}. For $n<4$, $G_2(\\cdot, t)\\in L^2({\\mathbb R}^n)$, and so is $\\Upsilon^{-1} F(u)$ \nbecause\n $u\\in H^s({\\mathbb R}^n)$ and $\\Upsilon^{-1}\\in \\mathcal L(H^s({\\mathbb R}^n),H^{s}({\\mathbb R}^n))$.  \n\nOn the other hand, $I_1$ is given by\n\\[\n I_1(x,t)=\\int_0^t G_1(\\cdot, t-s)\\ast \\big( B^{-1}[3u_1^2+u_2^2+|u|^2u_1]+u_2(2u_1+|u|^2)\\big) (\\cdot,s) \\, ds.\n \\]\nInasmuch as  $G_1$ is, up to a multiplicative constant, the fundamental solution of the usual linear Schr\\\"{o}dinger equation \n(see Lemma \\ref{lem:decomp}), the double  integral $I_1$ \nis of the same form (up the the appearance of $B^{-1}$) as the usual nonlinearity in the Duhamel representation of the nonlinear Schr\\\"odinger equation.\nIn view of the mapping properties of $B^{-1}$, \nthe desired result of boundedness and continuity \nthen follows from the corresponding proof for the usual nonlinear Schr\\\"odinger  equation \ngiven in Sections \\ref{sec:DBUNLS} and \\ref{sec:global}.\n\\end{proof}\n\n\n\n\n\n\n\nAs a corollary, we infer the appearance of dispersive blow up for the original \nGross-Pitaevskii equation \\eqref{eq:GP} in physically relevant dimensions.\n\n\n\\begin{corollary} Let $n\\leq 3$. Given $x_*\\in {\\mathbb R}^n$, $t_*>0$, there exist smooth and bounded initial data $\\psi_0 \\in C_{\\rm b}({\\mathbb R}^n)$ with \n$\\psi_0-1\\in L^2({\\mathbb R}^n)$, such that the solution $\\psi$ exhibits \ndispersive blow up at $(x_*, t_*)$.\n\\end{corollary}\n\n\n\\begin{proof}  To establish dispersive blow up for \\eqref{eq:GP},  the results for $v$ \nneed to be transferred back to the\nvariable $\\psi$.   In search of such a conclusion, note that the initial data in \\eqref{eq:reGP} \nis given by\n\\[\nv_0(x) = \\Upsilon^{-1} (\\psi_0-1) = B^{-1}(\\mathrm{Re} \\, \\psi_0 -1) + i \\, \\mathrm{Im} \\, \\psi_0.\n\\]\nThe specific choice $v_0(x)= \\frac{e^{-i|x|^2}}{(1+|x|^2)^m}$ then corresponds to the  initial data \n\\[\n\\psi_0 (x) = 1+ B \\, \\mathrm{Re} \\, v_0(x)  + i \\, \\mathrm{Im} \\, v_0(x) = 1+ B \\left(\\frac{\\cos |x|^2}{(1+|x|^2)^m} \\right) +  \\frac{i \\sin |x|^2}{(1+|x|^2)^m} \n\\]\nfor \\eqref{eq:GP}.  Since $B:L^2({\\mathbb R}^n)\\to L^2({\\mathbb R}^n)$, it is clear from the last formula \nthat $\\psi_0 - 1 \\in L^2({\\mathbb R}^n)$.  By Lemma \\ref{mapping}, $B={\\rm Id}+ B_1$ with  $B_1\\in \\mathcal L(L^2({\\mathbb R}^n), H^2({\\mathbb R}^n)) $ and $H^2({\\mathbb R}^n)\\subset C_{\\rm b}({\\mathbb R}^n)$ for $n\\leq 3,$ so \nit is concluded that $\\psi_0\\in C_{\\rm b}({\\mathbb R}^n).$\n\nNow, if $v=v_1+iv_2$ is a solution of  \\eqref{eq:reGP} with the dispersive blow up \nproperty at a point $(x_*,t_*)$ provided by Theorem \\ref{DBU:v}, \nthe corresponding $u =u_1+iu_2$ is given by\n\\[u_1=B^{-1}v_1=(I+B_2)v_1,\\quad u_2=v_2.\\]\nSince $B_2$ is a smoothing operator, $u,$  and thus $\\psi=1+u$ satisfies the DBU property at the same point $(x_*,t_*).$  \n\\end{proof}\n\n\n\n\\section{Higher-order nonlinear Schr\\\"odinger equations} \\label{sec:higherNLS}\n\nIn this final section, we indicate how  results of dispersive blow up can be extended \nto higher-order nonlinear Schr\\\"odinger equations.  It is  mathematically \nnatural to inquire whether or not higher-order terms destroy dispersive blow up, \nbut the practical motivation for considering such an extension is perhaps even more \ntelling.   In nonlinear optics, third and fourth-order Schr\\\"odinger-type \nequations frequently appear in the description of various  wave phenomena. \nIn particular, the analysis of optical rogue-wave formation has been based on higher-order \nnonlinear Schr\\\"odinger equations  (see, for example, \\cite{D, DGE, Mu, T1}).\n\nAs the ideas and even much of the technical detail mirror closely what has gone\nbefore, we content ourselves with admittedly sketchy indications of how the theory\nis developed.   The one point which would require serious new effort has to do \nwith an appropriate generalization of the Cazenave--Weissler theory in \\cite{CW} \nto a higher-order setting.  This is not attempted here, but is deserving of \nfurther investigation at a later stage.   \n\n\n\n\n\\subsection{Fourth-order nonlinear Schr\\\"odinger equation}\nIn this subsection,  initial-value problems for \nfourth-order nonlinear  Schr\\\"{o}dinger equations of the form \n\\begin{equation}\\label{eq:4th}\ni\\partial_t u +\\alpha \\Delta u+\\beta \\Delta ^2 u+\\lambda |u|^pu=0, \\quad u\\big|_{t=0}=u_0(x),\n\\end{equation}\n are considered.  Here, the parameters  $\\alpha, \\beta, \\lambda$ are real constants, \n with $\\beta \\neq 0$.  \nTheory for this initial-value problem can be found, for example, in \\cite{FES, P} and in \nthe references cited in these works.   \nIf $\\alpha =0$, the partial differential equation\n is often referred to as the  bi-harmonic NLS equation  (see, e.g., \\cite{BFM}). \n A simple scaling allows us to assume $\\beta=1$ and to consider only the values \n$\\alpha \\in  \\lbrace 0, -1, +1\\rbrace$, though time may need to be reversed.\n\nTo establish dispersive blow up for \\eqref{eq:4th},  the dispersive properties \nof the associated linear equation\n\\begin{equation}\\label{eq:lin4th}\ni \\partial_t u -\\alpha  \\Delta u+ \\Delta^2 u=0, \\quad \\alpha \\in \\lbrace 0, -1, +1\\rbrace\n\\end{equation}\nare helpful, just as for the lower-order cases.  As should be  clear from the preceding theory, \n the possibility of dispersive blow up for \\eqref{eq:lin4th} is  linked to \nthe dispersive  properties of the fundamental solution \n\\begin{equation}\\label{4disp}\n\\Sigma_\\alpha (x,t)= \\frac{1}{(2\\pi)^{n\/2} } \\int_{{\\mathbb R}^n} e^{it(|\\xi|^4+\\alpha |\\xi|^2)+ix\\cdot \\xi} \\, dx\n\\end{equation}\nof \\eqref{eq:lin4th}, which have in fact been established already in \\cite{BAKS}.\n\n\\begin{lemma}\\label{4dec}\nLet $\\Sigma_\\alpha$ be as in \\eqref{4disp} and $\\mu \\in {\\mathbb N}^n$a multi-index. \n\\begin{enumerate}\n\\item If $\\alpha  =0$, there exists a $C>0$ such that for $x \\in {\\mathbb R}^n$ and $t> 0$,\n$$\n|\\partial^\\mu \\Sigma_0(x,t)|\\leq Ct^{-(n+|\\mu|)\/4}\\left(1+\\frac{|x|}{t^{1\/4}}\\right)^{-(n-|\\mu|)\/3}.\n$$\n\\item \nFor $t > 0$ and  either $t\\leq 1$ or $|x|\\geq t$,  there exists a $C>0$ such that\n$$\n|\\partial^\\mu \\Sigma_\\alpha(x,t)|\\leq \nCt^{-(n+|\\mu|)\/4}\\left(1+\\frac{|x|}{t^{1\/4}}\\right)^{-(n-|\\mu|)\/3}\n$$\nfor $\\alpha = \\pm 1$.\n\\end{enumerate}\n\\end{lemma}\n\nStrichartz estimates then follow pretty much directly from Lemma \\ref{4dec}. \n In some detail, we say that \nthe pair $(q,r)$ is admissible for the fourth-order Schr\\\"odinger group $\\{e^{it (\\Delta^2-\\alpha \\Delta)}\\}_{t\\in {\\mathbb R}}$ if\n\\begin{equation}\\label{eq:adm}\n\\frac{1}{q}=\\frac{n}{4}\\left(\\frac{1}{2}-\\frac{1}{r}\\right),\n\\end{equation}\nfor $2\\leq r\\leq \\frac{2n}{n-2}$ if $n \\geq 3$, respectively, $2\\leq r\\leq \\infty$ if $n=1$ \nand  $2\\leq r<\\infty$ if $n=2$. \nUsing this, one has the following estimates, which are the fourth-order counterpart to the ones \nreported in Lemma \\ref{lem:Strich}. In what follows, $T=+\\infty$ when \n$\\alpha =0$ and $T$ is any nonnegative number when $\\alpha= \\pm 1.$\n \n\\begin{lemma}[\\cite{BAKS}]\n\\label{Str}\nLet $(q, r)$ be admissible in the sense of \\eqref{eq:adm}. Then there exists $c=c(n, r,T)$ such that\n\\[  \\big \\|  e^{it (\\Delta^2-\\alpha \\Delta)} f \\big \\|_{L^q((-T,T);L^r({\\mathbb R}^n))}\\leq c\\|f \\|_{L^2({\\mathbb R}^n)}. \\]\nThe linear operator \n\\[\n\\Phi f= \\int_0^t e^{i(t-s)(\\Delta^2-\\alpha \\Delta)}f(s)ds\n\\]\nis bounded in the sense that \n\\[ \\| \\Phi f\\|_{L^q((-T,T);L^r({\\mathbb R}^n))}\\leq c\\|f\\|_{L^{q'}((-T,T);L^{r'}({\\mathbb R}^n))}, \\]\nwhere $\\frac{1}{q}+\\frac{1}{q'}=1$ and  $\\frac{1}{r}+\\frac{1}{r'}=1.$ \n\\end{lemma}\n\nIf we assume for the moment that dispersive blow up holds true for the linear model \\eqref{eq:lin4th}, then the \nStrichartz estimates above are already sufficient to prove dispersive blow up for the nonlinear equation \\eqref{eq:4th} in the physically relevant dimensions $n\\leq 3$. \n\n\n\\begin{proposition}\nLet $n\\leq 3$ and $\\alpha \\in \\lbrace 0, -1, +1\\rbrace$. Assume that the linear fourth-order equation \\eqref{eq:lin4th} exhibits dispersive blow up \nat some  point $(x_*, t_*)$ in space-time. Then, for $p < \\frac{8}{n}-1$, so does the fourth-order \ninitial-value problem \\eqref{eq:4th}.\n\\end{proposition}\n\n\\begin{proof}\nThe proof follows closely the one given in \\cite{BS2} for the one-dimensional, second-order   nonlinear Schr\\\"{o}dinger equation. \nIn particular, the fact that the dispersive estimate in Lemma \\ref{4dec} of the fundamental solution $\\Sigma_\\alpha$ has temporal behavior that goes like  $t\\mapsto t^{-n\/4}$, which is locally integrable for $n<4$, is a key point in the proof. \n\nThe Duhamel representation of \\eqref{eq:4th} is given by\n\\begin{align*}\nu(x,t) = & \\, e^{i t (\\Delta^2-\\alpha \\Delta)} u_0(x) + i \\lambda \\int_0^t e^{i(t-s)(\\Delta^2-\\alpha \\Delta)} |u(x,s)|^p u(x,s) \\, ds \\\\\n= & \\,  e^{i t (\\Delta^2-\\alpha \\Delta)} u_0(x) + i \\lambda I (x,t).\n\\end{align*}\nThe first term on the right-hand side exhibits dispersive blow up by assumption.  \nTo prove that the integral term \nis continuous and bounded, notice that  \n\\[\n|I(x,t) |\\leq  C\\int_0^t\\int_{{\\mathbb R}^n} \\frac{1}{(t-s)^{n\/4}}|u|^{p+1}(x-y,t-s)\\, ds\\, dy \n\\]\nusing Lemma \\ref{4dec}. Applying H\\\"{o}lder's inequality, with a $\\gamma\\in(0,4\/n)$ to be determined presently, it is found that\n\\[\n|I(t,x) | \\leq\\left(\\int_{\\mathbb R} \\frac{ds}{(t-s)^{n\\gamma\/4}}\\right)^{1\/\\gamma}\\left(\\int_{\\mathbb R}\\|u(\\cdot,s)\\|_{p+1}^{\\gamma'(p+1)} ds\\right)^{1\/\\gamma'},\n\\]\nwith $\\frac{1}{\\gamma}+\\frac{1}{\\gamma'}=1$. \nChoose an admissible Strichartz pair in the range \\eqref{eq:adm} as follows.\nTake $r=p+1$ so that $q=\\frac{8(p+1)}{n(p-1)}.$ The condition $\\gamma'(p+1)\\leq q$ then yields\n\\[\\gamma'=\\frac{\\gamma}{\\gamma -1} \\leq  \\frac{8}{n(p-1)}.\\] \nCombined with the condition $\\gamma <\\frac{4}{n}$, one obtains\n\\[\np < \\frac{8}{n} \\left( 1 - \\frac{1}{\\gamma}\\right) = \\frac{8}{n} -1,\n\\]\nand the assertion is proved.\n\\end{proof}\n\n\\begin{remark}\nThe strategy deployed in this proof does not yield the optimal range of exponents $p$ nor is \nit valid for $n \\geq 4$.  This is because \nit is based only on Strichartz estimates \nand because $t\\mapsto t^{-n\/4}$ is locally integrable only  for $n<4$. \nTo extend the proof to higher dimensions $n > 3$, and to more general \nnonlinearities $p>0$, one could argue as in the proof of Proposition \\ref{prop:key}.  However,\nan essential ingredient in our argument was the Cazenave-Weissler result \nrecounted in  Proposition \\ref{prop:CW}.\nThus, to carry out this line of reasoning successfully, we would need \n the analog of the Cazenave-Weissler results  in the case of fourth-order equations, as well as  \nthe corresponding smoothing estimates for the Duhamel term established in Section \\ref{sec:global}. These tasks will be the goal of an upcoming work.\n\\end{remark}\n\nTo close the analysis, it is still required to prove that the linear fourth-order equation \\eqref{eq:lin4th} does exhibit dispersive blow up. \nTo this end, we will need a more precise description of the decay of the fundamental solution. \nThis will only be carried out in the one-dimensional case,\n though the result in higher spatial dimensions does not require new ideas, \njust more complex calculations.\n\n\\begin{proposition}\\label{dbu4th}\nLet $n=1$. Given $(x^*,t^*)\\in {\\mathbb R}\\times ({\\mathbb R}\\setminus \\lbrace 0\\rbrace)$, there exists $u_0 \\in C^\\infty({\\mathbb R})\\cap L^2({\\mathbb R})\\cap L^\\infty({\\mathbb R})$ such that the \nsolution $u \\in C_{\\rm b}({\\mathbb R}; L^2({\\mathbb R}))$ to \\eqref{eq:lin4th} has the following properties.\n\\begin{enumerate}\n\\item The solution $u$ blows up at $(x_*,t_*)$ which is to say\n\\[\n\\lim _{(x,t)\\in {\\mathbb R}\\to (x_*, t_*)} | u(x,t)| = + \\infty.\n\\]\n\\item The function $u$ is continuous on $\\big\\{(x,t)$ on ${\\mathbb R}\\times {\\mathbb R} \\setminus \\{t_*\\}\\big\\}$.\n\\item  The solution $u(\\cdot, t_*)$ is a continuous function on ${\\mathbb R} \\setminus \\{x_*\\}$.\n\\end{enumerate} \n\\end{proposition}\n\n\\begin{proof} For $n=1$, it is readily checked that the fundamental solution of \n\\begin{equation}\\label{4th1D}\ni \\partial_t u_t-\\alpha \\partial_x^2 u + \\partial_x^4u =0\n\\end{equation}\nis given by\n\\[\n\\Sigma_\\alpha(x,t)=\\frac{1}{(4t)^{1\/4}}B\\left(2\\alpha t^{1\/2},\\frac{x}{(4t)^{1\/4}}\\right)\n\\]\nwhere  \n\\begin{equation*}\\label{pear}\nB(x,y)=\\frac{1}{\\pi} \\int_{\\mathbb R} e^{i\\big(\\frac{1}{4}s^4+\\frac{1}{2}xs^2+ys\\big)}\\, ds\n\\end{equation*}\n is the  Pearcey integral. The Pearcey integral is a smooth and bounded function of $(x,y) \\in {\\mathbb R}^2$ which decays in both variables thusly: \n\\begin{equation*}\\label{asPe}\n|B(x,y)|\\leq c\\big(1+y^2+|x|^3\\big)^{-1\/18}\\Big(1+(1+y^2+|x|^3)^{-5\/9}|(3y)^2+(2x)^3|\\Big)^{-1\/4}\n\\end{equation*}\n(see, e.g., \\cite{BAKS}).\nTo establish dispersive blow up for \\eqref{4th1D}, an asymptotic expansion of the Pearcey integral \nwith respect to the second variable $y$ is helpful. Such an expansion has been established in \\cite{Pa}, Formulas (2.14) and (5.12).\n\\footnote{Note the rotation of coordinates in those formulas as compared to the way they \nare written here.} These results imply that\n\\begin{itemize}\n\\item[(i)] in the case $\\alpha =0$, one has\n\\begin{equation*}\\label{Pe0}\nB(0,y)=C_1y^{-1\/3}+C_2 y^{-5\/3}+O(|y|^{-3}),\n\\end{equation*}\n as $|y|\\to +\\infty$, uniformly for bounded values of $x$, where $C_1$ and $C_2$ are nonzero complex constants and\n\\item[(ii)] when $\\alpha \\neq0$, \n\\begin{align*}\\label{Pe2}\nB(x,y)=2^{1\/6}e^{-i\\pi\/24}y^{-1\/3}e^{-i x^2\/6}\n\\exp\\Big(-\\frac{3i}{4^{4\/3}}y^{4\/3}+\\frac{i}{4^{2\/3}}xy^{-2\/3}]\\Big)\\times \\\\\n\\left(1+\\frac{4^{-1\/3}}{3}xy^{-2\/3}(1-\\frac{i}{9}x^2)+O(|y|^{-4\/3})\\right)\n\\end{align*}\nfor $|y|\\to +\\infty$ and bounded $x$.\n\\end{itemize}\n   \n\nWith these results in hand, an argument can be mounted that mimics\n the case of the usual Schr\\\"odinger group. Without loss of generality, take it \nthat  $(x_*,t_*) =(0,\\frac{1}{4})$ and consider in \\eqref{4th1D} the initial data \n\\[\nu_0(x)=\\frac{\\Sigma_0(-x,\\frac{1}{4})}{(1+x^2)^m}=\\frac{B(0,-x)}{(1+x^2)^m},\\quad \\frac{1}{12}<m\\leq \\frac{1}{6},\n\\]\nif $\\alpha =0,$ and\n\\[\nu_0(x)=\\frac{\\Sigma_\\alpha(-x,\\frac{1}{4})}{(1+x^2)^m}=\\frac{B(\\alpha,-x)}{(1+x^2)^m},\\quad \\frac{1}{12}<m\\leq \\frac{1}{6},\n\\]if $\\alpha =\\pm 1$.  \n\nThe asymptotics of the Pearcey integral can then be brought to bear to prove Proposition \\ref{dbu4th}.  In particular, one uses these asymptotics to show that the solution of the linear problem blows \nup at $(0,\\frac14)$, that it is continuous on the \ncomplement of  this point and that the relevant Duhamel integral is everywhere continuous. \nWe pass over the details.\n\\end{proof}\n\nThe analogous result for general dimensions $n\\in {\\mathbb N}$ can be established \nby passing to radial coordinates and reducing it to the one-dimensional case (as in \\cite{BAKS} where the first term in the asymptotic expansion is given).\n\n\n\n\n\n\n\\subsection{Third order NLS in one dimension}\n\nFinally, we briefly discuss the situation for a third-order nonlinear Schro\\\"odinger \nequation in $n=1$ dimensions. Consider  the initial=value problem\n\\begin{equation}\\label{HNLS}\n\\partial_t u +i\\alpha \\partial^2_{x} u +\\beta \\partial^3_{x} u +i\\gamma |u|^2u=0, \\quad u\\big|_{t=0}=u_0(x),\n\\end{equation}\nwhere $x\\in {\\mathbb R}$, and $\\alpha, \\beta, \\gamma \\in {\\mathbb R} \\setminus \\{ 0 \\}$ are  given parameters. \nThis is a  model problem that arises in optical wave propagation (see \\cite{Mu, T1}). \n\nEquation \\eqref{HNLS} appears similar to the well known Korteweg-de Vries equation, hence the \nappearance of dispersive blow up can be established along the same lines as was \npursued in \\cite{BS1} for this latter equation.  Indeed, \nthe associated linear equation,\n\\begin{equation}\\label{linHLS}\n\\partial_t  u +i\\alpha \\partial^2_{x}  u  +\\beta \\partial^3_{x}  u =0,\n\\end{equation}\nadmits a fundamental solution of the form\n\\[\n\\Lambda(x,t) = \\frac{1}{2\\pi} \\int_{\\mathbb R} e^{i\\beta \\xi^3 - i\\alpha \\xi^2 +ix \\cdot \\xi} \\, d\\xi.\n\\]\nThe expansion \n\\[\n\\left(\\xi+\\frac{\\alpha}{3\\beta}\\right )^3=\\xi^3+\\frac{\\alpha}{\\beta} \\xi^2+\\frac{\\alpha^2 \\xi}{3\\beta^2}+\\frac{\\alpha^3}{27 \\beta^3}\n\\]\nallows us to write $\\Lambda$ in the form\n\\[\n\\Lambda(x,t)=\\frac{1}{2\\pi (t\\beta)^{1\/3}}\\exp \\left(\\frac{4it\\alpha^3}{27 \\beta^2}\\right)\\exp\\left(\\frac{-i\\alpha x}{2\\beta^2}\\right)\\text{Ai}\\left(\\frac{1}{t^{1\/3} \\beta^{1\/3}}\n\\Big(x-\\frac{\\alpha ^2}{3\\beta}t\\Big)\\right).\n\\]\nHere, $\\text{Ai}$ denotes the well-known the Airy function defined, for example, as \nthe improper Riemann integral \n\\[\n\\text{Ai}(z)=\\int_{{\\mathbb R}}e^{i(\\xi^3+iz\\xi)}\\, d\\xi.\n\\]\nIn \\cite{BS1}, the dispersive properties of the Airy function are the basis for establishing dispersive blow up for the Korteweg-de Vries equation.  \nFollowing these ideas, dispersive blow up for \\eqref{linHLS} at $(x_*,t_*)=(0,1)$ is easily obtained by taking initial data of the form\n\\[\nu_0(x)=\\frac{A(-x)}{(1+x^2)^m}, \\quad \\text{with $\\frac{1}{8}<m\\leq \\frac{1}{4}$}\n\\]\nand\n\\[\nA(x)=\\text{Ai}\\left (\\frac{\\alpha^2+x}{3\\beta^{4\/3}}\\right ).\n\\]\nThe proof of dispersive blow up for  this third-order  Schr\\\"odinger equation \ncan then be accomplished just as in \\cite{BS1}. \nIndeed, the proof is  easier since, contrary to the Korteweg-de Vries equation, \nthe Duhamel representation of \\eqref{HNLS} does \nnot involve any spatial derivatives of the dependent variable.   Consequently \nit can be shown to be bounded by using only  Strichartz estimates for \nthe linearized Korteweg-de Vries equation (see \\cite{BAKS2, Bo}).\n\n\n\\begin{remark}  Combining  the results of the foregoing sections allows one to deduce\n  dispersive blow up for \nnonlinear Schr\\\"odinger-type equations with anisotropic dispersion, such as\n\\[\ni \\partial_t u+\\alpha \\Delta u +i\\beta \\partial^3_{x_1} u+\\gamma \\partial^4_{x_1}u+|u|^pu=0,\n\\]\nwhere $\\alpha, \\beta, \\gamma\\in {\\mathbb R} \\setminus \\{ 0 \\}$. The Cauchy problem for this equation \nhas been studied in \\cite{Bo} (see \nalso \\cite{FES}).\n\\end{remark}\n\n\\begin{merci}\nWe tender thanks to an anonymous referee for her\/his {\\it very} helpful remarks.\n\\end{merci}\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \n\nIn robot motor control, developing responsive control policies that adapt to unforeseen environments is crucial to task success. These changes and unexpected situations can be intrinsic or extrinsic, such as robot damage, motor failure, varying friction, and external force disturbances. For robot locomotion, traditional approaches of planning and control require expert knowledge and accurate dynamics models and constraints of both the robot and the environment \\cite{fankhauser2018robust, chatzinikolaidis2020}, which are all subject to unforeseeable changes that are difficult to know beforehand. Moreover, even using data-efficient learning techniques such as Bayesian optimization to tune decision variables and control parameters, it can only achieve adaptation on a trial-by-trial basis \\cite{rai2018bayesian} and also require extensive computation \\cite{yuan2019} which is not able to respond to changes on the fly.\n\nRecent advances in Reinforcement-Learning (RL) lead to algorithms achieving human-like or animal-level performance in a range of difficult control tasks. Model-free RL can perform global search of control parameters and obtain globally optimal gaits while combined with walking pattern generation \\cite{dallali2012global}. Also, an RL-based feedback policy can achieve human-like bipedal walking by imitating human motion capture data \\cite{yang2020learning}. With simulation training including actuator properties, a model-free RL scheme can train different locomotion policies separately and deploy on a real quadrupedal robot \\cite{hwangbo2019learning}. By using a multi-expert learning, an hierarchical RL architecture can learn to fuse multiple motor skills and generate multimodal locomotion coherently on a real quadruped \\cite{yang2020multi}. However, in general, model-free RL algorithms have limited sample efficiency, resulting in long training time to produce viable policies. For example, it took a model-free RL algorithm 83 hours to achieve human performance on the Atari game suite, compared to 15 minutes for a human~\\cite{hessel_rainbow_2017}. Similarly, AlphaStar~\\cite{alphastarblog} used 200 years of equivalent real-time to reach expert human performance playing Starcraft II. \n\nOn the other hand, model-based approaches can achieve comparatively high performance while being more sample-efficient by several orders of magnitude, converging faster than model-free approaches for locomotion tasks~\\cite{Chua}. To deploy robots in the real-world, online adaptation to changes in the environment are required as not all conditions can be considered by pre-trained policies, such as drastic changes of environments or robots by amputation. Hence, meta-learning, or learning to learn is a novel and more promising approach for solving such generic adaptations. Model-based meta-RL has been used in the real-world to adapt the control of a six-leg millirobot to different floor conditions~\\cite{GrBAL}. A model-based meta-RL algorithm, FAMLE, was used in the real-world on a Minitaur quadruped, where a latent black-box context vector encoded different environment conditions \\cite{FAMLE}. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{external_force.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{slippery_floor.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n     \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{blocked_motor.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{missing_leg.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\vspace{-2mm}\n    \\caption{Adaptive and robust locomotion against uncertainties: (a) external force, (b) slippery ground, (c) faulty motors, and (d) leg amputation.}\n    \\label{fig:quadruped}\n    \\vspace{-5mm}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{Schematic_meta.png}\n    \\caption{Schematic view of the model-based meta-RL control framework for legged locomotion.}\n    \\label{fig:schematic}\n\\end{figure*}\n\n\n\nOur proposed method has made new improvements that require no prior knowledge of specific gaits. For example, FAMLE relies on sinusoidal gaits and therefore needs to optimize the amplitudes and phases of sinusoidal patterns by the model predictive control at a low frequency of 0.5Hz. In our work, we directly sample in the joint space at a much higher frequency at 50Hz, and we further improve the sampling process by specifying constraints on velocity, acceleration and jerk of the desired joint trajectories. Our study extensively validated the capability of adaptation in simulated test scenarios with large variations in floor friction, external forces or unexpected damage to joints.\n\nBased on the interaction model, our method allows changing the reward function online and therefore is able to modify the behavior of the robot. For example, the learned controller can track a variable forward velocity, even it has been trained on a fixed desired velocity. Likelihood estimation with condition latent vectors allows the meta-model to adapt to already seen conditions. Meta-training should allow \"on the fly\" optimization to better adapt to the current unknown condition.\n\nIn this paper, we present an improved model-based meta-RL approach to quadruped locomotion that is capable of online adaption to changing environments, as shown in Fig.~\\ref{fig:quadruped}. The main contributions of this work are:\n\\begin{enumerate}\n    \\item The proposed algorithm is capable of learning from scratch and requires no prior knowledge of the type of gaits, such as periodic phases of leg movements. \n    \\item Our methods introduces and applies hard constraints of velocity, acceleration and jerk on the sampled actions during the search process. \n    \\item The capability and robustness of online adaption to changes in both the robot and the environment, such as external force disturbances, varying frictions, faulty motors and leg amputation. \n\\end{enumerate}\n\n\nThe remainder of the paper is organized as follows. We outline the background in Section~\\ref{sec:Background} and related work in Section~\\ref{sec:RL}. In Section~\\ref{sec:method}, we elaborate the methodology and technical details on the model-based RL algorithm and the improvements by meta-learning. Section~\\ref{sec:exp} presents extensive simulation validations, results and analysis. Finally, we conclude and suggest future work in Section~\\ref{sec:Conclusion}.\n\n\n\\section{Background}\n\\label{sec:Background}\nThis section presents the preliminaries of RL, Model Predictive Control (MPC) and meta-learning. \n\\subsection{Reinforcement Learning}\n\\label{sec:RL}\n\nIn reinforcement learning, the agent learns to solve a task in an unknown environment $\\mathrm{E}$, defined by a Markov decision process $(\\mathrm{S},\\mathrm{s}_0, \\mathrm{A},\\mathrm{f}_\\mathrm{E},\\mathrm{r})$ where $\\mathrm{S}$ is the set of continuous states of the environment $\\mathrm{E}$, $\\mathrm{s}_0$ the initial state, $\\mathrm{A}$ the set of continuous actions the agent can perform in the environment, $\\mathrm{f}_\\mathrm{E}: \\mathrm{S}\\times \\mathrm{A}\\times\\mathrm{S} \\rightarrow \\mathbb{R}$ the probabilistic transition function and $\\mathrm{r}: \\mathrm{S}\\times \\mathrm{A} \\rightarrow \\mathbb{R}$ the reward function. \n\nThe goal of the agent is to learn a policy $\\Pi_{\\theta}: \\mathrm{S} \\rightarrow \\mathrm{A}$, parameterized by $\\theta$, which decides which action to perform given the current state $\\mathrm{s}_\\mathrm{t}$ to \\textit{maximize the long-term reward} $\\mathrm{R}(\\mathrm{t})=\\underset{\\mathrm{i}=\\mathrm{t}}{\\overset{\\mathrm{t}+\\mathrm{H}}{\\sum}}\\gamma^{\\mathrm{i}-\\mathrm{t}}\\mathrm{r}(\\mathrm{s}_\\mathrm{i},\\Pi(\\mathrm{s}_\\mathrm{i}))$, where $\\mathrm{H}$ is the horizon and $\\gamma$ is the discount factor. \n\nModel-free RL focuses in directly learning such a policy, whereas model-based RL focuses on learning a model of the transition function -- the transition of states given the current state and actions  -- which can be used to train the policy with fictive transitions or with model predictive control.\n\n\n\\subsection{Model Predictive Control}\n\\label{sec:MPC}\nGiven the current state $\\mathrm{s}_\\mathrm{t}$ and a horizon $\\mathrm{H}$, Model Predictive Control (MPC) uses a forward model of dynamics to select an action sequence $\\mathrm{a}_{\\mathrm{t}:\\mathrm{t}+\\mathrm{H}}$ which maximizes the predicted cumulative reward $\\mathrm{R}(\\mathrm{t})$. The agent performs the first action $\\mathrm{a}_\\mathrm{t}$ from the action sequence and collects the resulting state $\\mathrm{s}_{\\mathrm{t}+1}$. The MPC then repeats such optimization and allows the agent to alleviate the possible error in the model prediction. Compared to model-free RL, we can change the reward \\textit{online} to control the agent's behavior using model-based RL in an MPC fashion. \n\n\\subsection{Meta-Learning}\n\\label{sec:ML}\nWe use meta-learning to train an agent to solve several tasks, where the neural network learns to adapt to several varying conditions, such as different floor frictions, the presence of external disturbances or having a damaged motor. For the neural network model, the initial set of weights $\\theta^*$ must be found, such that only a small number of gradient descent steps with little collected data in a unknown environment can produce effective adaptations. \n\n\\section{Related Work}\n\\label{sec:RL}\n\\subsection{Model-Free Deep Reinforcement Learning}\nProximal Policy Optimization~\\cite{PPO} has been used to train a model-free controller to control Minitaur in simulation, producing trotting and gallop gaits \\cite{SimToReal} on the real robot. Soft Actor-Critic (SAC)~\\cite{SAC} has also been used to train on the real Minitaur robot within two hours~\\cite{LearningToWalk, ha_learning_2020}. This limitation of sampling-efficiency motivates us to focus on model-based RL. \n\n\\subsection{Model-Based Deep Reinforcement Learning}\nThere are three types of model-based RL: learning to predict the expected return from a starting state distribution, for example using Bayesian Optimization~\\cite{BayesianOptimisation}; learning to predict the outcome from a given starting state and given policy~\\cite{noveltySearch, MapElite, IMGEP}; and learning to model the transition function using a forward dynamical model. Here, we use the third type of model.\n\nForward models of dynamics are either deterministic or probabilistic, where deterministic models can be linear models~\\cite{levineLinearModel} or neural networks~\\cite{WorldModel}, and probabilistic models estimate uncertainty for modeling stochastic environments or estimating the long-term prediction uncertainty. Gaussian Processes~\\cite{Pilco, Multi-dex} or Bayesian neural networks~\\cite{Gal2015DropoutB} can be used to scale the abilities of Gaussian Processes models to higher dimensional environments. \n\nFor locomotion tasks, model-based RL with a forward model can have the same performance as model-free methods, while requiring at least an order of magnitude less samples~\\cite{Chua}. An ensemble of feed-forward neural networks is used to model the forward dynamics of the environment with uncertainty estimation. MPC uses this uncertainty estimation to formulate a more robust control which alleviates early overfitting model-based RL. The same method has been used with meta-learning to adapt the control of a 6-leg real millirobot to different floors \\cite{GrBAL}. \n\n\\subsection{MPC and Meta-Learning}\nSeveral optimization methods have been used for model-based RL, for example, Model Predictive Path Integral~\\cite{MPPI}, random shooting \\cite{LearningToWalk} or Cross-entropy method~\\cite{CEM, Chua}. We use random shooting for the simplicity, easy parallelism and proven performances on real robots~\\cite{GrBAL}.  \n\nThere are two main methods: a meta-learner model outputs the set of initial weights  $\\theta^*$ of the learner \\cite{learningToLearn}, or $\\theta^*$ is optimized using a meta-loss, it can be gradient-descent~\\cite{MAML} or evolutionary strategies~\\cite{ES-ML}. Gaussian processes have been used~\\cite{ml-gp} but only for low dimension environments. Meta-RL has been used with model-free RL~\\cite{noRML}, model-based RL~\\cite{GrBAL} or a mix of both~\\cite{M3PO}. For model-based RL, gradient based meta-learning was shown to be more data-efficient, resulting in a better and faster adaptation~\\cite{GrBAL}. Hence, we use gradient-based meta-learning. \n\nFor increasing generalization and adaptation to unseen condition of the environment, an adversarial loss has been used \\cite{adMRL}. Other methods employ context variables \\cite{context-variable}, bias transformation \\cite{bias-transformation}, or condition latent vector~\\cite{FAMLE}, to learn different input of sub-parts of the model for different condition, and then adapt this sub-parts to the current condition. \n\n\n\\section{Methodology}\n\\label{sec:method}\nThis section presents details of the model-based RL algorithm as discussed in Section~\\ref{sec:MBRL} and meta-learning algorithm Section~\\ref{sec:meta-learning}. We highlight our improvements which results in new robot capability of robust and versatile walking without a predefined, parameterized gait.\n\n\\subsection{Model-Based Reinforcement-Learning algorithm}\n\\label{sec:MBRL}\nThe model-based RL algorithm runs at 50Hz, sending desired actions to PD controllers running at 250Hz to generate torques for physics simulation. The algorithm is composed of two main parts: the forward dynamics model, and MPC. Fig.~\\ref{fig:schematic} illustrates the schematics of the control framework. \n\n\\subsubsection{The Forward Model of Dynamics}\nWe use a fully-connected feed-forward neural network, with two hidden layers of 256 units using a ReLU (Rectified Linear Unit) activation function. It takes the concatenation of the current state and action ($\\mathrm{s}_{\\mathrm{t}},\\mathrm{a}_\\mathrm{t}$) as input, and learns to predict the difference in the resulting state: $\\Delta \\mathrm{s}_\\mathrm{t}= \\mathrm{s}_\\mathrm{t+1}-\\mathrm{s}_{\\mathrm{t}}$, which is a standard means to get the prediction $\\mathrm{s}_\\mathrm{t+1} = \\mathrm{s}_{\\mathrm{t}} + \\Delta \\mathrm{s}_\\mathrm{t}$. \n\nThe model parameter $\\theta$ is the set of weights of the connections between the units. It is optimized using the gradient-based optimizer Adam~\\cite{adam} on a dataset of triplet $\\mathrm{D}\\in S\\times A \\times S$ using mean squared error as loss function. We depict details of the model-based RL algorithm in Algorithm~\\ref{algo:MBRL}.\n\nCompared to the work in~\\cite{FAMLE} for the Minitaur, our study formulate the state space as: the angular joints positions and velocities, the base orientation angles and angular rates, and the linear base velocities. The addition of the angular velocity of the base is the key of our success for controlling the robot at 50Hz. In contrast, only Euler angles and angular rates of the base in the horizontal plane were used in~\\cite{FAMLE} to control the Minitaur gait parameters at a much lower frequency of 0.5Hz. \n\n\\begin{algorithm}[t]\n\\caption{Model-based reinforcement learning algorithm \\newline Input: Environment $\\mathrm{E}$.}\n\\label{algo:MBRL}\n\\begin{algorithmic}\n\\STATE $\\mathrm{D}$ = $\\mathrm{N}_{init}$ episodes in $\\mathrm{E}$ with a random controller\n\\STATE $\\theta=$ random weights\n\\FOR{$\\mathrm{N}_\\text{training}$ episodes}\n    \\STATE Train $\\mathrm{M}_\\theta$ on $\\mathrm{D}$ using Adam\n    \\STATE $\\mathrm{s}_0=$ $\\mathrm{E}$.reset()\n    \\FOR{$\\mathrm{t}=0$ to $\\mathrm{T}-1$ steps}\n        \\STATE $\\mathrm{a}_\\mathrm{t}=$ MPC($\\mathrm{M}_\\theta$, $\\mathrm{s}_{t}$, $\\mathrm{a}_{\\mathrm{t}-3:\\mathrm{t}-1}$)\n        \\STATE $\\mathrm{s}_\\mathrm{t+1}=$ $\\mathrm{E}$.step($\\mathrm{a}_\\mathrm{t}$)\n        \\STATE $\\mathrm{D}=\\mathrm{D}\\cup \\{( \\mathrm{s}_{\\mathrm{t}}, \\mathrm{a}_\\mathrm{t}, \\mathrm{s}_\\mathrm{t+1})\\}$\n    \\ENDFOR\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Model Predictive Control}\nThe method of random shooting is implemented which is suitable for parallel computing, and the algorithm is detailed in Algorithm~\\ref{algo:MPC}. At each time step, $\\mathrm{N}_\\text{pop}$ action sequences of length $\\mathrm{H}$ are sampled. Each sequence is evaluated starting with the current state, using the model to estimate the corresponding state trajectory. From these trajectories, long term reward is computed and the action with the highest estimated reward is selected.\n\nReal actuators have inherent limitations in velocity, acceleration and jerk. Instead of uniformly sampling desired joint angles within the limits, continuity constraints are used, where each desired joint state of the sequence is sampled using previous joint positions to ensure velocities, accelerations and jerks are smooth and bellow their respective limits. \n\nAs the improvement to the previous work~\\cite{FAMLE}, we enforced physical constraints during sampling of actions: $\\mathrm{q}_{min}\\le\\mathrm{q}\\le\\mathrm{q}_{max}$, $|\\mathrm{A}|\\le \\mathrm{A}_{max}$, $|\\mathrm{V}|\\le \\mathrm{V}_{max}$ and $|\\mathrm{J}|\\le \\mathrm{J}_{max}$, where $\\mathrm{q}$, $\\mathrm{V}$, $\\mathrm{A}$ and $\\mathrm{J}$ are the desired joint angle, velocity, acceleration and jerk, respectively. The limits of velocity, acceleration and jerk are the soft constraints for the smoothness and continuity of actions. For safety reasons, regarding the joint position limits, we further imposed hard constraints of sampled actions on $\\mathrm{q}$ to avoid hitting the physical limit of joint movements. \n\nThis improvements on sampling enforces the MPC in a more suitable subspace. During training, it increased the distance traveled compared to the default condition during a 10s episode by an order of 2: from $1.64\\pm0.12$m (without), to $3.16\\pm0.12$m (with), where p-value $\\le$ 0.001 on 20 episodes. It also reduces the observed jerk by an order of 5: from $1.3\\times {10}^{4} \\pm 260$ rad\/$\\mathrm{s}^3$ (without) to $2.4 \\times 10^3 \\pm 60$ rad\/$\\mathrm{s}^3$ (with), where p-value $\\le$ 0.001 on 20 episodes.\n\n\\begin{algorithm}[t]\n\\caption{MPC algorithm \\newline Inputs: A model $\\mathrm{M}_\\theta$, initial state $\\mathrm{s}_0$, past actions $\\mathrm{a}_{-3:-1}$, horizon $\\mathrm{H}$ and discount factor $\\gamma$.}\n\\label{algo:MPC}\n\\begin{algorithmic}\n\\STATE Sample $\\mathrm{a}_{1:\\mathrm{H}}^{1:\\mathrm{N}_\\text{pop}}$ using $\\mathrm{a}_{-3:-1}$ for continuity\n\\FOR{$\\mathrm{i}=1$ to $\\mathrm{N}$}\n    \\STATE $\\mathrm{R}^{\\mathrm{i}} = 0$\n\\ENDFOR\n\\FOR{$\\mathrm{t}=1$ to $\\mathrm{H}$ steps}\n    \\FORALL{$\\mathrm{i}=1$ to $\\mathrm{N}_\\text{pop}$ samples}\n        \\STATE $\\mathrm{s}^{\\mathrm{i}}_\\mathrm{t+1}=\\mathrm{M}_\\theta(a_{\\mathrm{t}}^{\\mathrm{i}}, \\mathrm{s}_{\\mathrm{t}})$\n        \\STATE $\\mathrm{R}^\\mathrm{i} = \\mathrm{R}^\\mathrm{i} + \\gamma ^{\\mathrm{t}-1}\\mathrm{r}(\\mathrm{s}^{\\mathrm{i}}_{\\mathrm{t}}, \\mathrm{a}^\\mathrm{i}_{\\mathrm{t}})$\n    \\ENDFOR\n\\ENDFOR\n\\STATE $\\mathrm{i}^*=\\underset{\\mathrm{i}}{\\mathrm{argmax}}$ $\\mathrm{R}^\\mathrm{i}$\n\\RETURN $\\mathrm{a}_1^{\\mathrm{i}^*}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Meta-learning algorithm}\n\\label{sec:meta-learning}\nBefore meta-training, an expert is trained for each training condition using the proposed model-based RL algorithm to collect its training data. To adapt the model to each condition $\\mathrm{i}$, a specific latent vector $\\mathcal{C}_\\mathrm{i}$ is optimized during meta-learning using the regression loss on the data of the corresponding condition. This vector of fixed dimensions is then given to the input layer, alongside the current state and action when the condition $\\mathrm{i}$ is selected. We use a first order meta-learning called Reptile~\\cite{reptile}, which is composed of two phases: meta-training (Algorithm~\\ref{algo:meta_training}) and meta-adaptation (Algorithm~\\ref{algo:meta_adaptation}). \n\n\\subsubsection{Meta-training}\n\n\\begin{algorithm}[t]\n\\caption{Meta-learning training, called once before adaptation \\newline Inputs: $\\mathrm{N}$ datasets $\\mathrm{D}_{0:\\mathrm{N}-1}$ from $\\mathrm{N}$ different conditions.}\n\\label{algo:meta_training}\n\\begin{algorithmic}\n\\STATE $\\theta^*$ = random weights\n\\FOR{$\\mathrm{i}=0$ to $\\mathrm{N}-1$}\n    \\STATE $\\mathcal{C}_{\\mathrm{i}}$ = random vector\n\\ENDFOR\n\\FOR{$\\mathrm{n}_{outer}=0$ to $\\mathrm{N}_{outer}-1$}\n    \\STATE $\\mathrm{i} = \\mathrm{n}_{outer} \\mod \\mathrm{N}$\n    \\STATE $\\theta = \\theta^*$\n    \\FOR{$\\mathrm{n}_{inner}=1$ to $\\mathrm{N}_{inner}$}\n        \\STATE $\\mathcal{L} = \\tfrac{1}{|\\mathrm{D}_{\\mathrm{i}}|}\\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in \\mathrm{D}_\\mathrm{i}}{\\sum} (\\mathrm{M}_\\theta(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_{\\mathrm{i}})-\\mathrm{s}')^2$ \n        \\STATE $\\theta= \\theta - \\alpha \\nabla_\\theta \\mathcal{L}$ \n        \\STATE $\\mathcal{C}_{\\mathrm{i}}= \\mathcal{C}_{\\mathrm{i}} - \\alpha \\nabla_{\\mathcal{C}_{\\mathrm{i}}} \\mathcal{L}$ \n    \\ENDFOR\n    \\STATE $\\theta^* = \\theta^* + (1-\\tfrac{\\mathrm{n}_{outer}}{\\mathrm{N}_{outer}})\\beta (\\theta-\\theta^*)$ \n\\ENDFOR\n\\RETURN $\\theta^*$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n\\caption{Meta-learning adaptation, called at each step \\newline Inputs: a meta-trained set of weights $\\theta^*$, a list of learned $\\mathrm{N}$ condition latent vectors $\\mathcal{C}_{1:\\mathrm{N}}$, a dataset $\\mathrm{D}$ of the past $\\mathrm{K}$ steps.}\n\\label{algo:meta_adaptation}\n\\begin{algorithmic}\n\n\\FOR{$\\mathrm{i}=1$ to $\\mathrm{N}$}\n    \\STATE $\\mathcal{L}_\\mathrm{i} = \\tfrac{1}{|\\mathrm{D}|}\\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in \\mathrm{D}}{\\sum} (\\mathrm{M}_\\theta^*(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_\\mathrm{i})-\\mathrm{s}')^2$\n\\ENDFOR\n\\STATE $i^*= \\underset{\\mathrm{i}}{\\operatorname{argmin}} \\mathcal{L}_i$\n\\STATE $\\theta = \\theta^*$\n\\FOR{$n_{inner}=1$ to $\\mathrm{N}_{inner}$}\n    \\STATE $\\mathcal{L} = \\tfrac{1}{|\\mathrm{D}|} \\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in D}{\\sum} (M_\\theta(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_{\\mathrm{i^*}})-\\mathrm{s}')^2$\n    \\STATE $\\theta = \\theta - \\alpha \\nabla_\\theta \\mathcal{L}$ \n    \\STATE $\\mathcal{C}_{\\mathrm{i^*}} = \\mathcal{C}_{\\mathrm{i^*}} - \\alpha \\nabla_{\\mathcal{C}_{\\mathrm{i^*}}} \\mathcal{L}$ \n\\ENDFOR\n\\RETURN $\\theta$, $\\mathcal{C}_{\\mathrm{i^*}}$\n\\end{algorithmic}\n\\end{algorithm}\n\nThe initial set of weights $\\theta^*$ and each condition latent vector $\\mathcal{C}_\\mathrm{i}$ are optimized for adaptation. Meta-training is separated into two nested loops. In the inner loop, one training dataset $\\mathrm{D}_{\\mathrm{i}}$ and its corresponding condition latent vector $\\mathcal{C}_\\mathrm{i}$ are selected. The model weights $\\theta$ are initialized to $\\theta^*$ and Adam~\\cite{adam} optimize both of them for the regression loss of the current dataset $\\mathrm{D}_{\\mathrm{i}}$.\n\nIn the outer loop, $\\theta^*$ is optimized by taking a small step, with a linearly decreasing schedule, towards the optimized weights of the inner-loop. This allows $\\theta^*$ to converge to a nearby point (in the euclidean sense) to the optimal set of weights of each training condition. We detail the algorithm in Algorithm~\\ref{algo:meta_training}.\n\n\n\\subsubsection{Meta-adaptation}\nAt each time step, we select the most likely training condition using the previous $\\mathrm{K}$ time steps, each condition latent vector and the set of weights $\\theta^*$. \nWe then optimize the corresponding latent vector and the set of model weights, starting from $\\theta^*$, using the same optimization procedure as the inner loop but with the past $\\mathrm{K}$ steps. We detail the algorithm in Algorithm~\\ref{algo:meta_adaptation}.\n\nAfter the set of weights and the condition latent vector are optimized for the current condition, we use the MPC to select the optimal action to apply, and then new state information is collected, and the whole meta-adaptation iterates. This procedure allows any changes in the condition to be detected, and therefore the agent can adapt accordingly. \n\n\\subsection{Limitations}\nApart from the standard classical robot control of tuning PD gains and joints limits, the proposed method still requires fine-tuning of reward function, model architecture, hyper-parameters for the meta-learning and the adaptation. The use of MPC instead of a neural network policy has a trade-off between real-time computation and performance, i.e. MPC performs better in terms of adaption but requires more computation needed from the sampling procedure. \n\n\n\\begin{figure}[t]\n    \\centering\n      \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{force_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{slip_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n        \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{damage_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{tripod_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{Walking in presence of large uncertainties: (a) constant external force disturbance, (b) low-friction slippery ground, (c) faulty motors, and (d) with one missing leg.}\n    \n    \\vspace{-0mm}\n    \\label{fig:snapshots}\n\\end{figure}\n\n\n\\section{Results}\n\\label{sec:exp}\nWe used a custom version of the robot model (adapted from the open source SpotMicro robot~\\cite{kim_2019}) in PyBullet simulation to validate our method. Here, we first present the learning capability of the model-based RL algorithm on SpotMicro with a first adaption to a sequence of different conditions (Section~\\ref{sec:general_results}). We further validate the adaptation capability of the proposed meta-learning under various fixed frictions (Section~\\ref{sec:fixed_friction}) and time-varying, decreasing friction (Section~\\ref{sec:decreasing_friction}). \n\n\n\\subsection{Overview}\n\\label{sec:general_results}\n\n\\begin{figure}[t]\n    \\centering\n      \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_forward.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_static.png}\n      \\caption{}\n    \\end{subfigure}\n        \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_backward.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{Walking with a continuously changing velocity from 0.5m\/s to -0.2m\/s: (a) forward, (b) static, and (c) backward.}\n    \\vspace{-0mm}\n    \\label{fig:varying_velocity_snapshots}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{only_left_side_joints.png} \n    \\caption{Measured joints trajectories of SpotMicro from a 20s test scenario where the robot or environment changed every 5s: default friction $\\mu=0.8$ (\\textcolor{YellowGreen}{\\textbf{green}}), blocked right hip pitch joint (\\textcolor{Lavender}{\\textbf{red}}), slippery ground (\\textcolor{Dandelion}{\\textbf{yellow}}) and constant external push (\\textcolor{Bittersweet}{\\textbf{brown}}).}\n    \\vspace{-2mm}\n    \\label{fig:joints_trajectories}\n\\end{figure*}\n\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\nExpert\\textbackslash{}Condition & Default           & Slippery         & Lateral Force     & Damaged Motor      \\\\ \\hline\nDefault                         & 100\\%, 3.2$\\pm$ 0.2 & 20\\%, 0.7$\\pm$ 0.7 & 40\\%, 2.0$\\pm$ 1.0  & 100\\%, 0.7$\\pm$ 0.2  \\\\ \\hline\nSlippery                        & 30\\%, 1.3$\\pm$ 0.5  & 90\\%, 2.4$\\pm$ 0.4 & 10\\%, 0.7$\\pm$ 0.4  & 100\\%, 0.4$\\pm$  0.1 \\\\ \\hline\nLateral Force                   & 0\\%                & 0\\%                & 70\\%, 2.3 $\\pm$ 0.7 & 90\\%, 0.4$\\pm$ 0.1   \\\\ \\hline\nDamage Motor                    & 90\\%, 0.5$\\pm$ 0.2  & 70\\%, 0.3$\\pm$ 0.2 & 30\\%, 0.6$\\pm$ 0.3  & 100\\%, 2.5$\\pm$ 0.1  \\\\ \\hline\nMeta-Trained                    & 70\\%, 3.0$\\pm$ 0.6  & 80\\%, 2.4$\\pm$ 0.7 & 40\\%, 2.0$\\pm$ 1.3  & 100\\%, 2.7$\\pm$ 0.1  \\\\ \\hline\n\\end{tabular}\n\\caption{Success rate and average distance of travel for 10 episodes from different expert and the meta-trained model.}\n\\label{tab:success_rates}\n\\end{table*}\n\nWe trained the expert model with a default condition of friction $\\mu=0.8$, and this resulted default controller for walking is robust to perturbations, withstanding several pushes of 10N for 0.2s. After 300 of 10s-episodes which produced training data in the given condition, the quadruped was able to walk on slippery ground (friction coefficient $\\mu = 0.2$), against external forces or with a blocked motor or a missing\/amputated leg. Tab.~\\ref{tab:success_rates} shows the comparison between experts and meta-trained models under different conditions.\n\nUsing the proposed meta-learning method, the agent was able to adapt to four different conditions: default, fixed front-right hip motor, slippery ground and external forces. It traveled an average distance of $5.1\\pm1.5$m compared to a default expert which traveled $2.6\\pm0.8$m (averaged over 20 episodes). The joint trajectories from these test scenarios are shown in Fig.~\\ref{fig:joints_trajectories}.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{varying_velocity.png}\n    \\caption{Online tracking of continuous and variable walking velocity using the model trained only at the fixed velocity of 0.5m\/s, while the desired velocity in the reward function changes from from 0.5m\/s to -0.2m\/s during the test scenario.}\n    \\vspace{-2mm}\n    \\label{fig:varying_velocity}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{random_velocity.png}\n    \\vspace{-4mm}\n    \\caption{Online tracking of discrete and variable walking velocity using the model trained only at the fixed velocity of 0.5m\/s, while every 2 s the desired velocity is randomly sampled within 0.5m\/s and -0.2m\/s during the test scenario.}\n    \\vspace{-0mm}\n    \\label{fig:random_velocity}\n\\end{figure}\n\n\nThe controller can achieve variable walking speed, despite being trained with only at a constant desired forward velocity, we can command different desired velocities online continuously, as shown in Fig.~\\ref{fig:varying_velocity_snapshots} and Fig.~\\ref{fig:varying_velocity}. Moreover, the trained expert controller can also generate continuous control actions to track discrete, discontinuous commanded velocities (see Fig. \\ref{fig:random_velocity}). \n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{All_on_each_friction.png}\n    \\vspace{-0mm}\n    \\caption{Distance traveled for the full range of frictions from 3 expert models and the meta-trained model.}\n    \\vspace{-2mm}\n    \\label{fig:fixed_friction_adaptation}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{changing_friction_experts_VS_meta-learning_0.04s.png}\n    \\vspace{-0mm}\n    \\caption{Distance traveled with decreasing friction from the 3 expert models and the meta-trained model (p-value: $*<0.05$, $**<0.01$ and $***<0.001$). }\n    \\vspace{-2mm}\n    \\label{fig:changing_friction_adaptation}\n\\end{figure}\n\n\n\\subsection{Ground with Constant Friction}\n\\label{sec:fixed_friction}\nWe evaluated the adaptation capability using the meta-trained model and compared it to experts over the full range of different frictions (0.1 to 0.8 with 0.05 increments). We first trained 5 sets of experts for frictions 0.2, 0.4 and 0.6, using 300 10s-episodes, i.e., 50 minutes of data. Then we meta-trained 5 meta-models to adapt to these 3 frictions, each using one set of experts data, with the purpose that they could adapt to the full range afterwards.\n\nEach set of these 5 models was evaluated for each friction with 4 10s-episodes, so this gives 20 evaluations per expert and 20 evaluations for the meta-learning. The meta-trained models outperformed the experts on the full range of frictions, see Fig.~\\ref{fig:fixed_friction_adaptation}. As expected, each expert had its best performance when the friction constant is around its trained value.  \n\n\n\\subsection{Ground with Decreasing Friction}\n\\label{sec:decreasing_friction}j\n\nAs a comparison, we used the same experts and meta-model and benchmarked thei adaptation capability on a ground with continuously decreasing friction. We evaluated each set of models with 4 10s-episodes where friction coefficient $\\mu$ started at 0.8 and linearly decreased to 0.1. This gave 20 evaluations per expert and 20 evaluations for meta-learning.\n\nThe meta-trained models demonstrated better walking performance and traversed farther (3.38m) than the experts (3.07m), using a t-test with a p-value under $0.05$, see Fig.~\\ref{fig:changing_friction_adaptation}. In Fig.~\\ref{fig:velocity}, the curves and shaded areas are the means and standard deviations of the velocity, respectively. Snapshots of the walking gait using the meta-trained model are shown in Fig.~\\ref{fig:snapshots}, more details of walking performance can be seen in \\href{https:\/\/youtu.be\/s9OWhxorVmc}{the video here}. \n\n\nAdditionally, Fig.~\\ref{fig:condition_estimation_changing_friction} depicts the estimated condition at each time step using the past 0.1s (i.e. 5 time steps). At the beginning of the episode, when friction was higher, the model estimated a friction of 0.6 to be more likely (from 0.8 down to 0.5, i.e., 0-4.5s), then switched to a friction of 0.4 (from 0.5 down to 0.3, i.e., 4.5-7s), and finished by estimating a more likely friction of 0.2 (from 0.3 down to 0.1, i.e., 7-10s).\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{velocity_decreasing_frictions_all.png}\n    \\vspace{-4mm}\n    \\caption{Forward walking velocity with decreasing friction from 3 expert models and the meta-trained model.}\n    \\vspace{-0mm}\n    \\label{fig:velocity}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{likelyhood_estimation_0.1s.png}\n    \\vspace{-4mm}\n    \\caption{Likelihood of each pre-trained condition using online meta-adaptation in presence of decreasing friction.}\n    \\vspace{-0mm}\n    \\label{fig:condition_estimation_changing_friction}\n\\end{figure}\n\n\\subsection{Discussion}\nThe results validate the efficiency and effectiveness of the meta-learning method to detect the most probable current condition and adapt accordingly to different ground friction coefficients. The online adaptive walking using the proposed meta-learning outperformed the specialized experts which were specifically trained on specific frictions. This demonstrated the capability of meta-learning to incorporate knowledge from all training data.\n\nWe have also pushed the extreme test case in terms of unseen hardware failures. We specifically designed a case for meta-training where a motor of one leg of the quadruped was blocked at a fixed joint position (emulated actuator failures), and tested if meta-learning can adapt to the damage from a different leg faster than learning from scratch. Our investigation showed that the meta-adaptation is not able to adapt to such changes on a different leg, and we hypothesize that a second-order meta-learning algorithm, e.g. Model-Agnostic Meta-Learning~\\cite{MAML}, may be a better solution for such an extreme case.\n\n\\section{Conclusion and future work}\n\\label{sec:Conclusion}\nBased on the past work of model-based meta-RL~\\cite{FAMLE}, we have made contributions to improve the algorithm for adaptive and robust quadruped locomotion in changing robot dynamics (motor failure and amputation) and varying environmental constraints (time-varying friction, external pushes). In physics-based simulation, we have demonstrated this method can learn quadrupedal walking without using a periodic gait signal \\cite{FAMLE} or a phase vector \\cite{yang2020learning}. Instead, by updating an interaction model of the robot and environment and applying the optimal control actions, a walking gait is naturally generated as the outcome of maximizing the task reward. We further validated the capability of our proposed framework in adapting to different conditions such as robot damage, changing friction, and external force disturbances. \n\nFuture work will apply this method on the real SpotMicro robot, and identify potential issues of sim2real transfer which will be addressed by new solutions for meta-model and more effective search of model predictive control procedure. We hypothesize the current meta-learning algorithm is efficient enough for multi-task learning, which can be further studied. Also, a second-order meta-learning algorithm~\\cite{MAML} can be Incorporated could potentially achieve better adaptation to novel situations. \n\n\n\\section{Acknowledgement}\nThis work has been supported by EPSRC UK Robotics and Artificial Intelligence Hub for Offshore Energy Asset Integrity Management (EP\/R026173\/1).\n\n\n\\bibliographystyle{IEEEtran}\n\\balance\n\n\\section{Introduction}\n \n\nIn robot motor control, developing responsive control policies that adapt to unforeseen environments is crucial to task success. These changes and unexpected situations can be intrinsic or extrinsic, such as robot damage, motor failure, varying friction, and external force disturbances. For robot locomotion, traditional approaches of planning and control require expert knowledge and accurate dynamics models and constraints of both the robot and the environment \\cite{fankhauser2018robust, chatzinikolaidis2020}, which are all subject to unforeseeable changes that are difficult to know beforehand. Moreover, even using data-efficient learning techniques such as Bayesian optimization to tune decision variables and control parameters, it can only achieve adaptation on a trial-by-trial basis \\cite{rai2018bayesian} and also require extensive computation \\cite{yuan2019} which is not able to respond to changes on the fly.\n\nRecent advances in Reinforcement-Learning (RL) lead to algorithms achieving human-like or animal-level performance in a range of difficult control tasks. Model-free RL can perform global search of control parameters and obtain globally optimal gaits while combined with walking pattern generation \\cite{dallali2012global}. Also, an RL-based feedback policy can achieve human-like bipedal walking by imitating human motion capture data \\cite{yang2020learning}. With simulation training including actuator properties, a model-free RL scheme can train different locomotion policies separately and deploy on a real quadrupedal robot \\cite{hwangbo2019learning}. By using a multi-expert learning, an hierarchical RL architecture can learn to fuse multiple motor skills and generate multimodal locomotion coherently on a real quadruped \\cite{yang2020multi}. However, in general, model-free RL algorithms have limited sample efficiency, resulting in long training time to produce viable policies. For example, it took a model-free RL algorithm 83 hours to achieve human performance on the Atari game suite, compared to 15 minutes for a human~\\cite{hessel_rainbow_2017}. Similarly, AlphaStar~\\cite{alphastarblog} used 200 years of equivalent real-time to reach expert human performance playing Starcraft II. \n\nOn the other hand, model-based approaches can achieve comparatively high performance while being more sample-efficient by several orders of magnitude, converging faster than model-free approaches for locomotion tasks~\\cite{Chua}. To deploy robots in the real-world, online adaptation to changes in the environment are required as not all conditions can be considered by pre-trained policies, such as drastic changes of environments or robots by amputation. Hence, meta-learning, or learning to learn is a novel and more promising approach for solving such generic adaptations. Model-based meta-RL has been used in the real-world to adapt the control of a six-leg millirobot to different floor conditions~\\cite{GrBAL}. A model-based meta-RL algorithm, FAMLE, was used in the real-world on a Minitaur quadruped, where a latent black-box context vector encoded different environment conditions \\cite{FAMLE}. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{external_force.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{slippery_floor.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n     \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{blocked_motor.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{.2\\textwidth}\n      \\centering\n      \\includegraphics[width=\\textwidth]{missing_leg.png}\\hfill\n      \\caption{}\n    \\end{subfigure}\n    \\vspace{-2mm}\n    \\caption{Adaptive and robust locomotion against uncertainties: (a) external force, (b) slippery ground, (c) faulty motors, and (d) leg amputation.}\n    \\label{fig:quadruped}\n    \\vspace{-5mm}\n\\end{figure}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{Schematic_meta.png}\n    \\caption{Schematic view of the model-based meta-RL control framework for legged locomotion.}\n    \\label{fig:schematic}\n\\end{figure*}\n\n\n\nOur proposed method has made new improvements that require no prior knowledge of specific gaits. For example, FAMLE relies on sinusoidal gaits and therefore needs to optimize the amplitudes and phases of sinusoidal patterns by the model predictive control at a low frequency of 0.5Hz. In our work, we directly sample in the joint space at a much higher frequency at 50Hz, and we further improve the sampling process by specifying constraints on velocity, acceleration and jerk of the desired joint trajectories. Our study extensively validated the capability of adaptation in simulated test scenarios with large variations in floor friction, external forces or unexpected damage to joints.\n\nBased on the interaction model, our method allows changing the reward function online and therefore is able to modify the behavior of the robot. For example, the learned controller can track a variable forward velocity, even it has been trained on a fixed desired velocity. Likelihood estimation with condition latent vectors allows the meta-model to adapt to already seen conditions. Meta-training should allow \"on the fly\" optimization to better adapt to the current unknown condition.\n\nIn this paper, we present an improved model-based meta-RL approach to quadruped locomotion that is capable of online adaption to changing environments, as shown in Fig.~\\ref{fig:quadruped}. The main contributions of this work are:\n\\begin{enumerate}\n    \\item The proposed algorithm is capable of learning from scratch and requires no prior knowledge of the type of gaits, such as periodic phases of leg movements. \n    \\item Our methods introduces and applies hard constraints of velocity, acceleration and jerk on the sampled actions during the search process. \n    \\item The capability and robustness of online adaption to changes in both the robot and the environment, such as external force disturbances, varying frictions, faulty motors and leg amputation. \n\\end{enumerate}\n\n\nThe remainder of the paper is organized as follows. We outline the background in Section~\\ref{sec:Background} and related work in Section~\\ref{sec:RW}. In Section~\\ref{sec:method}, we elaborate the methodology and technical details on the model-based RL algorithm and the improvements by meta-learning. Section~\\ref{sec:exp} presents extensive simulation validations, results and analysis. Finally, we conclude and suggest future work in Section~\\ref{sec:Conclusion}.\n\n\n\\section{Background}\n\\label{sec:Background}\nThis section presents the preliminaries of RL, Model Predictive Control (MPC) and meta-learning. \n\\subsection{Reinforcement Learning}\n\\label{sec:RL}\n\nIn reinforcement learning, the agent learns to solve a task in an unknown environment $\\mathrm{E}$, defined by a Markov decision process $(\\mathrm{S},\\mathrm{s}_0, \\mathrm{A},\\mathrm{f}_\\mathrm{E},\\mathrm{r})$ where $\\mathrm{S}$ is the set of continuous states of the environment $\\mathrm{E}$, $\\mathrm{s}_0$ the initial state, $\\mathrm{A}$ the set of continuous actions the agent can perform in the environment, $\\mathrm{f}_\\mathrm{E}: \\mathrm{S}\\times \\mathrm{A}\\times\\mathrm{S} \\rightarrow \\mathbb{R}$ the probabilistic transition function and $\\mathrm{r}: \\mathrm{S}\\times \\mathrm{A} \\rightarrow \\mathbb{R}$ the reward function. \n\nThe goal of the agent is to learn a policy $\\Pi_{\\theta}: \\mathrm{S} \\rightarrow \\mathrm{A}$, parameterized by $\\theta$, which decides which action to perform given the current state $\\mathrm{s}_\\mathrm{t}$ to \\textit{maximize the long-term reward} $\\mathrm{R}(\\mathrm{t})=\\underset{\\mathrm{i}=\\mathrm{t}}{\\overset{\\mathrm{t}+\\mathrm{H}}{\\sum}}\\gamma^{\\mathrm{i}-\\mathrm{t}}\\mathrm{r}(\\mathrm{s}_\\mathrm{i},\\Pi(\\mathrm{s}_\\mathrm{i}))$, where $\\mathrm{H}$ is the horizon and $\\gamma$ is the discount factor. \n\nModel-free RL focuses in directly learning such a policy, whereas model-based RL focuses on learning a model of the transition function -- the transition of states given the current state and actions  -- which can be used to train the policy with fictive transitions or with model predictive control.\n\n\n\\subsection{Model Predictive Control}\n\\label{sec:MPC}\nGiven the current state $\\mathrm{s}_\\mathrm{t}$ and a horizon $\\mathrm{H}$, Model Predictive Control (MPC) uses a forward model of dynamics to select an action sequence $\\mathrm{a}_{\\mathrm{t}:\\mathrm{t}+\\mathrm{H}}$ which maximizes the predicted cumulative reward $\\mathrm{R}(\\mathrm{t})$. The agent performs the first action $\\mathrm{a}_\\mathrm{t}$ from the action sequence and collects the resulting state $\\mathrm{s}_{\\mathrm{t}+1}$. The MPC then repeats such optimization and allows the agent to alleviate the possible error in the model prediction. Compared to model-free RL, we can change the reward \\textit{online} to control the agent's behavior using model-based RL in an MPC fashion. \n\n\\subsection{Meta-Learning}\n\\label{sec:ML}\nWe use meta-learning to train an agent to solve several tasks, where the neural network learns to adapt to several varying conditions, such as different floor frictions, the presence of external disturbances or having a damaged motor. For the neural network model, the initial set of weights $\\theta^*$ must be found, such that only a small number of gradient descent steps with little collected data in a unknown environment can produce effective adaptations. \n\n\\section{Related Work}\n\\label{sec:RW}\n\\subsection{Model-Free Deep Reinforcement Learning}\nProximal Policy Optimization~\\cite{PPO} has been used to train a model-free controller to control Minitaur in simulation, producing trotting and gallop gaits \\cite{SimToReal} on the real robot. Soft Actor-Critic (SAC)~\\cite{SAC} has also been used to train on the real Minitaur robot within two hours~\\cite{LearningToWalk, ha_learning_2020}. This limitation of sampling-efficiency motivates us to focus on model-based RL. \n\n\\subsection{Model-Based Deep Reinforcement Learning}\nThere are three types of model-based RL: learning to predict the expected return from a starting state distribution, for example using Bayesian Optimization~\\cite{BayesianOptimisation}; learning to predict the outcome from a given starting state and given policy~\\cite{noveltySearch, MapElite, IMGEP}; and learning to model the transition function using a forward dynamical model. Here, we use the third type of model.\n\nForward models of dynamics are either deterministic or probabilistic, where deterministic models can be linear models~\\cite{levineLinearModel} or neural networks~\\cite{WorldModel}, and probabilistic models estimate uncertainty for modeling stochastic environments or estimating the long-term prediction uncertainty. Gaussian Processes~\\cite{Pilco, Multi-dex} or Bayesian neural networks~\\cite{Gal2015DropoutB} can be used to scale the abilities of Gaussian Processes models to higher dimensional environments. \n\nFor locomotion tasks, model-based RL with a forward model can have the same performance as model-free methods, while requiring at least an order of magnitude less samples~\\cite{Chua}. An ensemble of feed-forward neural networks is used to model the forward dynamics of the environment with uncertainty estimation. MPC uses this uncertainty estimation to formulate a more robust control which alleviates early overfitting model-based RL. The same method has been used with meta-learning to adapt the control of a 6-leg real millirobot to different floors \\cite{GrBAL}. \n\n\\subsection{MPC and Meta-Learning}\nSeveral optimization methods have been used for model-based RL, for example, Model Predictive Path Integral~\\cite{MPPI}, random shooting \\cite{LearningToWalk} or Cross-entropy method~\\cite{CEM, Chua}. We use random shooting for the simplicity, easy parallelism and proven performances on real robots~\\cite{GrBAL}.  \n\nThere are two main methods: a meta-learner model outputs the set of initial weights  $\\theta^*$ of the learner \\cite{learningToLearn}, or $\\theta^*$ is optimized using a meta-loss, it can be gradient-descent~\\cite{MAML} or evolutionary strategies~\\cite{ES-ML}. Gaussian processes have been used~\\cite{ml-gp} but only for low dimension environments. Meta-RL has been used with model-free RL~\\cite{noRML}, model-based RL~\\cite{GrBAL} or a mix of both~\\cite{M3PO}. For model-based RL, gradient based meta-learning was shown to be more data-efficient, resulting in a better and faster adaptation~\\cite{GrBAL}. Hence, we use gradient-based meta-learning. \n\nFor increasing generalization and adaptation to unseen condition of the environment, an adversarial loss has been used \\cite{adMRL}. Other methods employ context variables \\cite{context-variable}, bias transformation \\cite{bias-transformation}, or condition latent vector~\\cite{FAMLE}, to learn different input of sub-parts of the model for different condition, and then adapt this sub-parts to the current condition. \n\n\n\\section{Methodology}\n\\label{sec:method}\nThis section presents details of the model-based RL algorithm as discussed in Section~\\ref{sec:MBRL} and meta-learning algorithm Section~\\ref{sec:meta-learning}. We highlight our improvements which results in new robot capability of robust and versatile walking without a predefined, parameterized gait.\n\n\\subsection{Model-Based Reinforcement-Learning algorithm}\n\\label{sec:MBRL}\nThe model-based RL algorithm runs at 50Hz, sending desired actions to PD controllers running at 250Hz to generate torques for physics simulation. The algorithm is composed of two main parts: the forward dynamics model, and MPC. Fig.~\\ref{fig:schematic} illustrates the schematics of the control framework. \n\n\\subsubsection{The Forward Model of Dynamics}\nWe use a fully-connected feed-forward neural network, with two hidden layers of 256 units using a ReLU (Rectified Linear Unit) activation function. It takes the concatenation of the current state and action ($\\mathrm{s}_{\\mathrm{t}},\\mathrm{a}_\\mathrm{t}$) as input, and learns to predict the difference in the resulting state: $\\Delta \\mathrm{s}_\\mathrm{t}= \\mathrm{s}_\\mathrm{t+1}-\\mathrm{s}_{\\mathrm{t}}$, which is a standard means to get the prediction $\\mathrm{s}_\\mathrm{t+1} = \\mathrm{s}_{\\mathrm{t}} + \\Delta \\mathrm{s}_\\mathrm{t}$. \n\nThe model parameter $\\theta$ is the set of weights of the connections between the units. It is optimized using the gradient-based optimizer Adam~\\cite{adam} on a dataset of triplet $\\mathrm{D}\\in S\\times A \\times S$ using mean squared error as loss function. We depict details of the model-based RL algorithm in Algorithm~\\ref{algo:MBRL}.\n\nCompared to the work in~\\cite{FAMLE} for the Minitaur, our study formulate the state space as: the angular joints positions and velocities, the base orientation angles and angular rates, and the linear base velocities. The addition of the angular velocity of the base is the key of our success for controlling the robot at 50Hz. In contrast, only Euler angles and angular rates of the base in the horizontal plane were used in~\\cite{FAMLE} to control the Minitaur gait parameters at a much lower frequency of 0.5Hz. \n\n\\begin{algorithm}[t]\n\\caption{Model-based reinforcement learning algorithm \\newline Input: Environment $\\mathrm{E}$.}\n\\label{algo:MBRL}\n\\begin{algorithmic}\n\\STATE $\\mathrm{D}$ = $\\mathrm{N}_{init}$ episodes in $\\mathrm{E}$ with a random controller\n\\STATE $\\theta=$ random weights\n\\FOR{$\\mathrm{N}_\\text{training}$ episodes}\n    \\STATE Train $\\mathrm{M}_\\theta$ on $\\mathrm{D}$ using Adam\n    \\STATE $\\mathrm{s}_0=$ $\\mathrm{E}$.reset()\n    \\FOR{$\\mathrm{t}=0$ to $\\mathrm{T}-1$ steps}\n        \\STATE $\\mathrm{a}_\\mathrm{t}=$ MPC($\\mathrm{M}_\\theta$, $\\mathrm{s}_{t}$, $\\mathrm{a}_{\\mathrm{t}-3:\\mathrm{t}-1}$)\n        \\STATE $\\mathrm{s}_\\mathrm{t+1}=$ $\\mathrm{E}$.step($\\mathrm{a}_\\mathrm{t}$)\n        \\STATE $\\mathrm{D}=\\mathrm{D}\\cup \\{( \\mathrm{s}_{\\mathrm{t}}, \\mathrm{a}_\\mathrm{t}, \\mathrm{s}_\\mathrm{t+1})\\}$\n    \\ENDFOR\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Model Predictive Control}\nThe method of random shooting is implemented which is suitable for parallel computing, and the algorithm is detailed in Algorithm~\\ref{algo:MPC}. At each time step, $\\mathrm{N}_\\text{pop}$ action sequences of length $\\mathrm{H}$ are sampled. Each sequence is evaluated starting with the current state, using the model to estimate the corresponding state trajectory. From these trajectories, long term reward is computed and the action with the highest estimated reward is selected.\n\nReal actuators have inherent limitations in velocity, acceleration and jerk. Instead of uniformly sampling desired joint angles within the limits, continuity constraints are used, where each desired joint state of the sequence is sampled using previous joint positions to ensure velocities, accelerations and jerks are smooth and bellow their respective limits. \n\nAs the improvement to the previous work~\\cite{FAMLE}, we enforced physical constraints during sampling of actions: $\\mathrm{q}_{min}\\le\\mathrm{q}\\le\\mathrm{q}_{max}$, $|\\mathrm{A}|\\le \\mathrm{A}_{max}$, $|\\mathrm{V}|\\le \\mathrm{V}_{max}$ and $|\\mathrm{J}|\\le \\mathrm{J}_{max}$, where $\\mathrm{q}$, $\\mathrm{V}$, $\\mathrm{A}$ and $\\mathrm{J}$ are the desired joint angle, velocity, acceleration and jerk, respectively. The limits of velocity, acceleration and jerk are the soft constraints for the smoothness and continuity of actions. For safety reasons, regarding the joint position limits, we further imposed hard constraints of sampled actions on $\\mathrm{q}$ to avoid hitting the physical limit of joint movements. \n\nThis improvements on sampling enforces the MPC in a more suitable subspace. During training, it increased the distance traveled compared to the default condition during a 10s episode by an order of 2: from $1.64\\pm0.12$m (without), to $3.16\\pm0.12$m (with), where p-value $\\le$ 0.001 on 20 episodes. It also reduces the observed jerk by an order of 5: from $1.3\\times {10}^{4} \\pm 260$ rad\/$\\mathrm{s}^3$ (without) to $2.4 \\times 10^3 \\pm 60$ rad\/$\\mathrm{s}^3$ (with), where p-value $\\le$ 0.001 on 20 episodes.\n\n\\begin{algorithm}[t]\n\\caption{MPC algorithm \\newline Inputs: A model $\\mathrm{M}_\\theta$, initial state $\\mathrm{s}_0$, past actions $\\mathrm{a}_{-3:-1}$, horizon $\\mathrm{H}$ and discount factor $\\gamma$.}\n\\label{algo:MPC}\n\\begin{algorithmic}\n\\STATE Sample $\\mathrm{a}_{1:\\mathrm{H}}^{1:\\mathrm{N}_\\text{pop}}$ using $\\mathrm{a}_{-3:-1}$ for continuity\n\\FOR{$\\mathrm{i}=1$ to $\\mathrm{N}$}\n    \\STATE $\\mathrm{R}^{\\mathrm{i}} = 0$\n\\ENDFOR\n\\FOR{$\\mathrm{t}=1$ to $\\mathrm{H}$ steps}\n    \\FORALL{$\\mathrm{i}=1$ to $\\mathrm{N}_\\text{pop}$ samples}\n        \\STATE $\\mathrm{s}^{\\mathrm{i}}_\\mathrm{t+1}=\\mathrm{M}_\\theta(a_{\\mathrm{t}}^{\\mathrm{i}}, \\mathrm{s}_{\\mathrm{t}})$\n        \\STATE $\\mathrm{R}^\\mathrm{i} = \\mathrm{R}^\\mathrm{i} + \\gamma ^{\\mathrm{t}-1}\\mathrm{r}(\\mathrm{s}^{\\mathrm{i}}_{\\mathrm{t}}, \\mathrm{a}^\\mathrm{i}_{\\mathrm{t}})$\n    \\ENDFOR\n\\ENDFOR\n\\STATE $\\mathrm{i}^*=\\underset{\\mathrm{i}}{\\mathrm{argmax}}$ $\\mathrm{R}^\\mathrm{i}$\n\\RETURN $\\mathrm{a}_1^{\\mathrm{i}^*}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Meta-learning algorithm}\n\\label{sec:meta-learning}\nBefore meta-training, an expert is trained for each training condition using the proposed model-based RL algorithm to collect its training data. To adapt the model to each condition $\\mathrm{i}$, a specific latent vector $\\mathcal{C}_\\mathrm{i}$ is optimized during meta-learning using the regression loss on the data of the corresponding condition. This vector of fixed dimensions is then given to the input layer, alongside the current state and action when the condition $\\mathrm{i}$ is selected. We use a first order meta-learning called Reptile~\\cite{reptile}, which is composed of two phases: meta-training (Algorithm~\\ref{algo:meta_training}) and meta-adaptation (Algorithm~\\ref{algo:meta_adaptation}). \n\n\\subsubsection{Meta-training}\n\n\\begin{algorithm}[t]\n\\caption{Meta-learning training, called once before adaptation \\newline Inputs: $\\mathrm{N}$ datasets $\\mathrm{D}_{0:\\mathrm{N}-1}$ from $\\mathrm{N}$ different conditions.}\n\\label{algo:meta_training}\n\\begin{algorithmic}\n\\STATE $\\theta^*$ = random weights\n\\FOR{$\\mathrm{i}=0$ to $\\mathrm{N}-1$}\n    \\STATE $\\mathcal{C}_{\\mathrm{i}}$ = random vector\n\\ENDFOR\n\\FOR{$\\mathrm{n}_{outer}=0$ to $\\mathrm{N}_{outer}-1$}\n    \\STATE $\\mathrm{i} = \\mathrm{n}_{outer} \\mod \\mathrm{N}$\n    \\STATE $\\theta = \\theta^*$\n    \\FOR{$\\mathrm{n}_{inner}=1$ to $\\mathrm{N}_{inner}$}\n        \\STATE $\\mathcal{L} = \\tfrac{1}{|\\mathrm{D}_{\\mathrm{i}}|}\\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in \\mathrm{D}_\\mathrm{i}}{\\sum} (\\mathrm{M}_\\theta(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_{\\mathrm{i}})-\\mathrm{s}')^2$ \n        \\STATE $\\theta= \\theta - \\alpha \\nabla_\\theta \\mathcal{L}$ \n        \\STATE $\\mathcal{C}_{\\mathrm{i}}= \\mathcal{C}_{\\mathrm{i}} - \\alpha \\nabla_{\\mathcal{C}_{\\mathrm{i}}} \\mathcal{L}$ \n    \\ENDFOR\n    \\STATE $\\theta^* = \\theta^* + (1-\\tfrac{\\mathrm{n}_{outer}}{\\mathrm{N}_{outer}})\\beta (\\theta-\\theta^*)$ \n\\ENDFOR\n\\RETURN $\\theta^*$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n\\caption{Meta-learning adaptation, called at each step \\newline Inputs: a meta-trained set of weights $\\theta^*$, a list of learned $\\mathrm{N}$ condition latent vectors $\\mathcal{C}_{1:\\mathrm{N}}$, a dataset $\\mathrm{D}$ of the past $\\mathrm{K}$ steps.}\n\\label{algo:meta_adaptation}\n\\begin{algorithmic}\n\n\\FOR{$\\mathrm{i}=1$ to $\\mathrm{N}$}\n    \\STATE $\\mathcal{L}_\\mathrm{i} = \\tfrac{1}{|\\mathrm{D}|}\\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in \\mathrm{D}}{\\sum} (\\mathrm{M}_\\theta^*(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_\\mathrm{i})-\\mathrm{s}')^2$\n\\ENDFOR\n\\STATE $i^*= \\underset{\\mathrm{i}}{\\operatorname{argmin}} \\mathcal{L}_i$\n\\STATE $\\theta = \\theta^*$\n\\FOR{$n_{inner}=1$ to $\\mathrm{N}_{inner}$}\n    \\STATE $\\mathcal{L} = \\tfrac{1}{|\\mathrm{D}|} \\underset{(\\mathrm{s},\\mathrm{a},\\mathrm{s}')\\in D}{\\sum} (M_\\theta(\\mathrm{s},\\mathrm{a}, \\mathcal{C}_{\\mathrm{i^*}})-\\mathrm{s}')^2$\n    \\STATE $\\theta = \\theta - \\alpha \\nabla_\\theta \\mathcal{L}$ \n    \\STATE $\\mathcal{C}_{\\mathrm{i^*}} = \\mathcal{C}_{\\mathrm{i^*}} - \\alpha \\nabla_{\\mathcal{C}_{\\mathrm{i^*}}} \\mathcal{L}$ \n\\ENDFOR\n\\RETURN $\\theta$, $\\mathcal{C}_{\\mathrm{i^*}}$\n\\end{algorithmic}\n\\end{algorithm}\n\nThe initial set of weights $\\theta^*$ and each condition latent vector $\\mathcal{C}_\\mathrm{i}$ are optimized for adaptation. Meta-training is separated into two nested loops. In the inner loop, one training dataset $\\mathrm{D}_{\\mathrm{i}}$ and its corresponding condition latent vector $\\mathcal{C}_\\mathrm{i}$ are selected. The model weights $\\theta$ are initialized to $\\theta^*$ and Adam~\\cite{adam} optimize both of them for the regression loss of the current dataset $\\mathrm{D}_{\\mathrm{i}}$.\n\nIn the outer loop, $\\theta^*$ is optimized by taking a small step, with a linearly decreasing schedule, towards the optimized weights of the inner-loop. This allows $\\theta^*$ to converge to a nearby point (in the euclidean sense) to the optimal set of weights of each training condition. We detail the algorithm in Algorithm~\\ref{algo:meta_training}.\n\n\n\\subsubsection{Meta-adaptation}\nAt each time step, we select the most likely training condition using the previous $\\mathrm{K}$ time steps, each condition latent vector and the set of weights $\\theta^*$. \nWe then optimize the corresponding latent vector and the set of model weights, starting from $\\theta^*$, using the same optimization procedure as the inner loop but with the past $\\mathrm{K}$ steps. We detail the algorithm in Algorithm~\\ref{algo:meta_adaptation}.\n\nAfter the set of weights and the condition latent vector are optimized for the current condition, we use the MPC to select the optimal action to apply, and then new state information is collected, and the whole meta-adaptation iterates. This procedure allows any changes in the condition to be detected, and therefore the agent can adapt accordingly. \n\n\\subsection{Limitations}\nApart from the standard classical robot control of tuning PD gains and joints limits, the proposed method still requires fine-tuning of reward function, model architecture, hyper-parameters for the meta-learning and the adaptation. The use of MPC instead of a neural network policy has a trade-off between real-time computation and performance, i.e. MPC performs better in terms of adaption but requires more computation needed from the sampling procedure. \n\n\n\\begin{figure}[t]\n    \\centering\n      \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{force_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{slip_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n        \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{damage_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{tripod_snapshots.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{Walking in presence of large uncertainties: (a) constant external force disturbance, (b) low-friction slippery ground, (c) faulty motors, and (d) with one missing leg.}\n    \n    \\vspace{-0mm}\n    \\label{fig:snapshots}\n\\end{figure}\n\n\n\\section{Results}\n\\label{sec:exp}\nWe used a custom version of the robot model (adapted from the open source SpotMicro robot~\\cite{kim_2019}) in PyBullet simulation to validate our method. Here, we first present the learning capability of the model-based RL algorithm on SpotMicro with a first adaption to a sequence of different conditions (Section~\\ref{sec:general_results}). We further validate the adaptation capability of the proposed meta-learning under various fixed frictions (Section~\\ref{sec:fixed_friction}) and time-varying, decreasing friction (Section~\\ref{sec:decreasing_friction}). \n\n\n\\subsection{Overview}\n\\label{sec:general_results}\n\n\\begin{figure}[t]\n    \\centering\n      \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_forward.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_static.png}\n      \\caption{}\n    \\end{subfigure}\n        \\begin{subfigure}{\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{velocity_snapshots_backward.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{Walking with a continuously changing velocity from 0.5m\/s to -0.2m\/s: (a) forward, (b) static, and (c) backward.}\n    \\vspace{-0mm}\n    \\label{fig:varying_velocity_snapshots}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{only_left_side_joints.png} \n    \\caption{Measured joints trajectories of SpotMicro from a 20s test scenario where the robot or environment changed every 5s: default friction $\\mu=0.8$ (\\textcolor{YellowGreen}{\\textbf{green}}), blocked right hip pitch joint (\\textcolor{Lavender}{\\textbf{red}}), slippery ground (\\textcolor{Dandelion}{\\textbf{yellow}}) and constant external push (\\textcolor{Bittersweet}{\\textbf{brown}}).}\n    \\vspace{-2mm}\n    \\label{fig:joints_trajectories}\n\\end{figure*}\n\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\nExpert\\textbackslash{}Condition & Default           & Slippery         & Lateral Force     & Damaged Motor      \\\\ \\hline\nDefault                         & 100\\%, 3.2$\\pm$ 0.2 & 20\\%, 0.7$\\pm$ 0.7 & 40\\%, 2.0$\\pm$ 1.0  & 100\\%, 0.7$\\pm$ 0.2  \\\\ \\hline\nSlippery                        & 30\\%, 1.3$\\pm$ 0.5  & 90\\%, 2.4$\\pm$ 0.4 & 10\\%, 0.7$\\pm$ 0.4  & 100\\%, 0.4$\\pm$  0.1 \\\\ \\hline\nLateral Force                   & 0\\%                & 0\\%                & 70\\%, 2.3 $\\pm$ 0.7 & 90\\%, 0.4$\\pm$ 0.1   \\\\ \\hline\nDamage Motor                    & 90\\%, 0.5$\\pm$ 0.2  & 70\\%, 0.3$\\pm$ 0.2 & 30\\%, 0.6$\\pm$ 0.3  & 100\\%, 2.5$\\pm$ 0.1  \\\\ \\hline\nMeta-Trained                    & 70\\%, 3.0$\\pm$ 0.6  & 80\\%, 2.4$\\pm$ 0.7 & 40\\%, 2.0$\\pm$ 1.3  & 100\\%, 2.7$\\pm$ 0.1  \\\\ \\hline\n\\end{tabular}\n\\caption{Success rate and average distance of travel for 10 episodes from different expert and the meta-trained model.}\n\\label{tab:success_rates}\n\\end{table*}\n\nWe trained the expert model with a default condition of friction $\\mu=0.8$, and this resulted default controller for walking is robust to perturbations, withstanding several pushes of 10N for 0.2s. After 300 of 10s-episodes which produced training data in the given condition, the quadruped was able to walk on slippery ground (friction coefficient $\\mu = 0.2$), against external forces or with a blocked motor or a missing\/amputated leg. Tab.~\\ref{tab:success_rates} shows the comparison between experts and meta-trained models under different conditions.\n\nUsing the proposed meta-learning method, the agent was able to adapt to four different conditions: default, fixed front-right hip motor, slippery ground and external forces. It traveled an average distance of $5.1\\pm1.5$m compared to a default expert which traveled $2.6\\pm0.8$m (averaged over 20 episodes). The joint trajectories from these test scenarios are shown in Fig.~\\ref{fig:joints_trajectories}.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{varying_velocity.png}\n    \\caption{Online tracking of continuous and variable walking velocity using the model trained only at the fixed velocity of 0.5m\/s, while the desired velocity in the reward function changes from from 0.5m\/s to -0.2m\/s during the test scenario.}\n    \\vspace{-2mm}\n    \\label{fig:varying_velocity}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{random_velocity.png}\n    \\vspace{-4mm}\n    \\caption{Online tracking of discrete and variable walking velocity using the model trained only at the fixed velocity of 0.5m\/s, while every 2 s the desired velocity is randomly sampled within 0.5m\/s and -0.2m\/s during the test scenario.}\n    \\vspace{-0mm}\n    \\label{fig:random_velocity}\n\\end{figure}\n\n\nThe controller can achieve variable walking speed, despite being trained with only at a constant desired forward velocity, we can command different desired velocities online continuously, as shown in Fig.~\\ref{fig:varying_velocity_snapshots} and Fig.~\\ref{fig:varying_velocity}. Moreover, the trained expert controller can also generate continuous control actions to track discrete, discontinuous commanded velocities (see Fig. \\ref{fig:random_velocity}). \n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{All_on_each_friction.png}\n    \\vspace{-0mm}\n    \\caption{Distance traveled for the full range of frictions from 3 expert models and the meta-trained model.}\n    \\vspace{-2mm}\n    \\label{fig:fixed_friction_adaptation}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{changing_friction_experts_VS_meta-learning_0.04s.png}\n    \\vspace{-0mm}\n    \\caption{Distance traveled with decreasing friction from the 3 expert models and the meta-trained model (p-value: $*<0.05$, $**<0.01$ and $***<0.001$). }\n    \\vspace{-2mm}\n    \\label{fig:changing_friction_adaptation}\n\\end{figure}\n\n\n\\subsection{Ground with Constant Friction}\n\\label{sec:fixed_friction}\nWe evaluated the adaptation capability using the meta-trained model and compared it to experts over the full range of different frictions (0.1 to 0.8 with 0.05 increments). We first trained 5 sets of experts for frictions 0.2, 0.4 and 0.6, using 300 10s-episodes, i.e., 50 minutes of data. Then we meta-trained 5 meta-models to adapt to these 3 frictions, each using one set of experts data, with the purpose that they could adapt to the full range afterwards.\n\nEach set of these 5 models was evaluated for each friction with 4 10s-episodes, so this gives 20 evaluations per expert and 20 evaluations for the meta-learning. The meta-trained models outperformed the experts on the full range of frictions, see Fig.~\\ref{fig:fixed_friction_adaptation}. As expected, each expert had its best performance when the friction constant is around its trained value.  \n\n\n\\subsection{Ground with Decreasing Friction}\n\\label{sec:decreasing_friction}j\n\nAs a comparison, we used the same experts and meta-model and benchmarked thei adaptation capability on a ground with continuously decreasing friction. We evaluated each set of models with 4 10s-episodes where friction coefficient $\\mu$ started at 0.8 and linearly decreased to 0.1. This gave 20 evaluations per expert and 20 evaluations for meta-learning.\n\nThe meta-trained models demonstrated better walking performance and traversed farther (3.38m) than the experts (3.07m), using a t-test with a p-value under $0.05$, see Fig.~\\ref{fig:changing_friction_adaptation}. In Fig.~\\ref{fig:velocity}, the curves and shaded areas are the means and standard deviations of the velocity, respectively. Snapshots of the walking gait using the meta-trained model are shown in Fig.~\\ref{fig:snapshots}, more details of walking performance can be seen in \\href{https:\/\/youtu.be\/s9OWhxorVmc}{the video here}. \n\n\nAdditionally, Fig.~\\ref{fig:condition_estimation_changing_friction} depicts the estimated condition at each time step using the past 0.1s (i.e. 5 time steps). At the beginning of the episode, when friction was higher, the model estimated a friction of 0.6 to be more likely (from 0.8 down to 0.5, i.e., 0-4.5s), then switched to a friction of 0.4 (from 0.5 down to 0.3, i.e., 4.5-7s), and finished by estimating a more likely friction of 0.2 (from 0.3 down to 0.1, i.e., 7-10s).\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{velocity_decreasing_frictions_all.png}\n    \\vspace{-4mm}\n    \\caption{Forward walking velocity with decreasing friction from 3 expert models and the meta-trained model.}\n    \\vspace{-0mm}\n    \\label{fig:velocity}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{likelyhood_estimation_0.1s.png}\n    \\vspace{-4mm}\n    \\caption{Likelihood of each pre-trained condition using online meta-adaptation in presence of decreasing friction.}\n    \\vspace{-0mm}\n    \\label{fig:condition_estimation_changing_friction}\n\\end{figure}\n\n\\subsection{Discussion}\nThe results validate the efficiency and effectiveness of the meta-learning method to detect the most probable current condition and adapt accordingly to different ground friction coefficients. The online adaptive walking using the proposed meta-learning outperformed the specialized experts which were specifically trained on specific frictions. This demonstrated the capability of meta-learning to incorporate knowledge from all training data.\n\nWe have also pushed the extreme test case in terms of unseen hardware failures. We specifically designed a case for meta-training where a motor of one leg of the quadruped was blocked at a fixed joint position (emulated actuator failures), and tested if meta-learning can adapt to the damage from a different leg faster than learning from scratch. Our investigation showed that the meta-adaptation is not able to adapt to such changes on a different leg, and we hypothesize that a second-order meta-learning algorithm, e.g. Model-Agnostic Meta-Learning~\\cite{MAML}, may be a better solution for such an extreme case.\n\n\\section{Conclusion and future work}\n\\label{sec:Conclusion}\nBased on the past work of model-based meta-RL~\\cite{FAMLE}, we have made contributions to improve the algorithm for adaptive and robust quadruped locomotion in changing robot dynamics (motor failure and amputation) and varying environmental constraints (time-varying friction, external pushes). In physics-based simulation, we have demonstrated this method can learn quadrupedal walking without using a periodic gait signal \\cite{FAMLE} or a phase vector \\cite{yang2020learning}. Instead, by updating an interaction model of the robot and environment and applying the optimal control actions, a walking gait is naturally generated as the outcome of maximizing the task reward. We further validated the capability of our proposed framework in adapting to different conditions such as robot damage, changing friction, and external force disturbances. \n\nFuture work will apply this method on the real SpotMicro robot, and identify potential issues of sim2real transfer which will be addressed by new solutions for meta-model and more effective search of model predictive control procedure. We hypothesize the current meta-learning algorithm is efficient enough for multi-task learning, which can be further studied. Also, a second-order meta-learning algorithm~\\cite{MAML} can be Incorporated could potentially achieve better adaptation to novel situations. \n\n\n\\section{Acknowledgement}\nThis work has been supported by EPSRC UK Robotics and Artificial Intelligence Hub for Offshore Energy Asset Integrity Management (EP\/R026173\/1).\n\n\n\\bibliographystyle{IEEEtran}\n\\balance\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figure\/Task-v0.1.pdf}\n    \\caption{\\textbf{Solving inductive reasoning tasks grounded in perception.} We consider tasks as inferring a simple formula with noisy perceptual inputs such as images. Solving this problem involves jointly learning to perceive and parse input into symbolic form (e.g., 3 numbers), and reason inductively  to recover programs that explain each arithmetic task.}\n    \\label{fig:setting}\n\\end{figure}\n\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{figure\/framework-v2.3.pdf}\n    \\caption{\\textbf{The framework of our model (ROAP) }for the neurosymbolic program synthesis problem. Given I\/O pairs $\\{(x, y)\\}$ for each task, \\textbf{\\rom{1}.}~our model first uses the symbol grounder $g$ to learn symbolic representations $s$ for high-dimensional perceptual input $x$. \\textbf{\\rom{2}.}~The program synthesizer $r$ then uses the symbol-output pairs as a program specification to infer the program for the task, $q(z| r(\\{s, y\\}))$. \\textbf{\\rom{3}.}~The differentiable interpreter finally executes the program on grounded symbols to predict the outputs $\\hat y = \\denotation{z}(s)$. \\textbf{\\rom{4}.}~The whole model is trained end-to-end using a reconstruction loss of outputs and a program length regularizer.} \n    \\label{fig:framework}\n\\end{figure}\n\nWe seek steps toward AI systems that learn to symbolically process perceptual input.\nConsider, for example, a system which learns to infer the 3D structure of objects: starting from pixels, it must infer low-level  symbols (curves, parts), and then organize them according to symbolic relationships (symmetry, part repetitions, part hierarchy).\nOr, consider a system which learns to control a moving object that navigates around obstacles: starting from sensory data (lidar, RGBD), it must first parse the world (into objects, proximities, freespace), and then compute trajectories using high-level computations (PID controllers, etc.).\nSimilar perceptual-symbolic problems arise when e.g. learning structured world models from pixels, or inferring instructions from natural language.\nWe frame such tasks as \\textbf{neurosymbolic program synthesis}: learning neural components that extract symbols from perception, and synthesizing programs to further process those symbols with more complex computations.\n\nIn this work, we study neurosymbolic program synthesis on a small-scale domain involving both perception and programmatic reasoning. Our ultimate goal is to develop general methods that could, we hope, apply to challenging neurosymbolic tasks like those previously mentioned.\nWe take the stance that symbols should be grounded in perception, and that symbol processing should be implemented by learnable program-like representations.\nHowever, we also propose that rather than hand-code a preordained set of primitive symbols, AI systems should autonomously learn to carve the perceptual world into their own discretization.\nWhat constitutes a `symbol' may vary across domain and across datasets, and can be hard for human engineers to anticipate. By jointly learning the symbols, as well as synthesizing the programs that operate on them, we hope to side-step the pitfalls associated with hand-engineered representations.\n\nConcretely, we consider regression problems mapping perceptual input $x$ to output $y$.\nThe input $x$ is first processed by a `symbol grounder' neural net $g(\\cdot )$, which produces intermediate symbolic output $s=g(x)$.\nThe symbols in $s$ are then transformed by a synthesized program $z$ in order to predict the output $\\hat{y}=\\denotation{z}(s)$. We learn $g(\\cdot )$ and $z$ by encouraging the predicted $\\hat{y}$ to be close to the ground-truth output $y$.\n\nThis setup serves as a basic model of the kinds of neurosymbolic program synthesis we seek to target.\nAlthough simple to write down, the setup is difficult to train:\nthe grounder $g(\\cdot )$ is a neural network (with unknown weights), \nwhile the program $z$ is a discrete program (with unknown structure).\nTo the best of our knowledge, there do not exist stable and scalable training procedures for such mixed discrete\/continuous search problems, especially when $g(\\cdot )$ has many parameters, and when $z$ comes from a combinatorially large space of possible programs.\nAt its core, we have a chicken-and-egg problem: if we had the program, then we could train $g(\\cdot )$ with gradient descent to predict an element of $\\denotation{z}^{-1}(y)$;\nif we had the neural net, we could synthesize the program $z$ using a variety of prior  search methods~\\cite{shah2020learning, 10.1145\/2491956.2462174} on the input-output example $g(x)\\mapsto y$.\n\nBroadly speaking, prior works have taken three approaches for addressing this mixed discrete\/continuous problem.\nFirst, the most popular strategy is to relax the discrete space of programs $z$ and train the entire system end-to-end through backpropagation~\\cite{evans2018learning, graves2014neural}.\nSecond, the complement of that approach has also been tried: discretizing the weight-space of the $g$ network and solving for both the program and the network weights using SAT~\\cite{EVANS2021103521}.\nLast, alternating optimization frameworks which successively update continuous weights and discrete structure, in the style of Expectation-Maximization, have been tried~\\cite{mws}.\n\nWe conjecture that relaxation and end-to-end backprop is likely to yield the best results, at least in the near future, because it allows one to build on the success and infrastructure of deep learning.\nTo that end, we combine three techniques to ameliorate the optimization difficulties associated with mixed discrete\/continuous problems, at least for this setting:\\\\\n\\textbf{1. Multitasking with amortized inference:} We consider a multitask setting with hundreds of interrelated programs sharing the same symbol grounding. Because they share the same symbol grounding, learning $g$ is easier, as the many tasks `triangulate' a good symbolic basis. Because the tasks share structure, we can make discrete program synthesis for $z$ tractable by \\emph{amortizing}~\\cite{gershman2014amortized} program search by training an auxiliary recognition (inference) model.\nThis `recognition model' learns to predict programs conditioned on input-output specifications.\\\\\n\\textbf{2. Overparameterization:} We `overparameterize' the space of programs: for example, even if we only expect a 4-line program, we search over the space of 20-line programs. Although this would be problematic for combinatorial search, gradient descent is well-known to benefit from overparameterized optimization landscapes~\\cite{frankle2018lottery}, and this benefit carries over to continuous relaxations of program spaces, as first noted by terpret~\\cite{gaunt2016terpret}. \\\\\n\\textbf{3. Regularization:} We encourage shorter program solutions by designing program-length regularizer compatible with continuous program relaxations. We find this aids convergence. It also combats the code explosion that overparameterization naively leads to, giving shorter and more interpretable programs.\n\nOur contribution is the combination of these techniques--\\textbf{R}egularize, \\textbf{O}verparameterize, \\textbf{A}mortize, for \\textbf{P}rograms--and so we call our approach ROAP (pronounced rope).\n\n\n\n\n\\section{Related Work}\nNeurosymbolic programming is a growing area that seeks to engineer learning and inference methods for hybrid program\/neural architectures~\\cite{PGL-049}, and our work is a special case of this broad framework.\nSpecifically, we tackle inductive program synthesis~\\cite{gulwani2017program}--synthesizing programs from input-output examples--but where the inputs are high-dimensional and must be preprocessed by a neural network into symbolic form.\nPrior works in this setting assume a hand-engineered inventory of basic symbols~\\cite{ellis2017learning}, while others backpropagate through differentiable programs to jointly train network weights and program structure~\\cite{ gaunt2017differentiable}.\nMultitasking is a known strategy for this setting~\\cite{valkov2018houdini}.\nTraining a neural network to help guide search for discrete programs (amortized inference) is standard~\\cite{devlin2017robustfill, 10.1145\/3453483.3454080}, and we extend that idea to continuous relaxations of program spaces.\n\nThe difficulties of gradient descent over relaxed program spaces is well known, to the extent that it has been dubbed the so-called `terpret problem'~\\cite{gaunt2016terpret}.\nUnfortunately, such an approach is the most straightforward way of training neural net$\\to$program architectures. From a technical perspective, our work hopes to make progress on the `terpret problem', thereby unlocking scalable and reliable training of this class of neurosymbolic programs.\n\n\n\\section{Methods}\n\\label{sec: methods}\n\\subsection{Relaxing spaces of programs}\nWe consider straight-line code: sequences of variable assignments, each of which introduces a new variable and computes its value by applying an operator to variables that were previously computed on earlier lines.\nWe write $z$ to mean the syntactic structure of a straight-line program, which is represented by a triple $(z^O, z^L, z^R)$ of binary 2D arrays encoding the syntactic structure of the program.\nGiven this discrete \\emph{sketch}~\\cite{solar2006combinatorial} of possible programs, we define a denotation operator $\\denotation{z}$ (Figure~\\ref{denotation}).\nGiven input-output examples $x\\mapsto y$, \nsingle-task neurosymbolic synthesis corresponds to\n\\begin{align}\ng(\\cdot ), z&=\\argmin_{\\substack{g(\\cdot ), z\\\\z\\text{'s entries }\\in \\left\\{ 0, 1 \\right\\}}}%\n\\left( y-\\denotation{z}(g(x)) \\right)^2\\label{map}\n\\end{align}\nThe constraint that the entries of $z$ lie in $\\left\\{ 0, 1 \\right\\}$ precludes end-to-end backpropagation.\nBecause our denotation operator is actually a smooth function of $z$, we can relax the above optimization problem by saying the entries of $z$ lie in $[ 0, 1 ]$:\n\\begin{align}\ng(\\cdot ), z&=\\argmin_{\\substack{g(\\cdot ), z\\\\z\\text{'s entries }\\in \\left[ 0, 1 \\right]}}%\n\\left( y-\\denotation{z}(g(x)) \\right)^2\\label{badrelax}\n\\end{align}\nUnfortunately the above objective is problematic: generically there is no reason for gradient descent to converge to a discrete $z$.\nWhile one can incorporate sparsity-favoring regularizers on $z$, doing so causes the probability of convergence to fall off rapidly as the target program length increases~\\cite{evans2018learning, gaunt2016terpret}, suggesting that $z$ only converges to a discrete program if it is randomly initialized close to a correct solution.\n\n\\paragraph{Overparameterization} With this setup so far, we can compactly describe the first of our three techniques: {Overparameterization}.\nWe simply expand the number of possible lines of code in the straight-line sketch, as illustrated in Figure~\\ref{fig:interpreter}, and expect that it will help convergence of the relaxed problem by providing a benign optimization landscape~\\cite{hardt2016identity, du2019gradient}.\n\n\n\n\\begin{figure*}\n\\begin{align*}\n\\denotation{z}(s)&=\\text{Exec}(z, s, L+V)\\text{\\phantom{test}\\emph{execute program and extract output on line }}L+V\\\\\n\\text{Exec}(z, s, l)&=s_l\\text{, whenever }l\\leq V\\text{\\phantom{test}\\emph{load symbolic variables as first lines of code. We have V variables}}\\\\\n\\text{Exec}(z, s, l)&=\\sum_o z^O_{lo}\\times F_o\\left( s, \\quad\\sum_{1\\leq a<l }z^L_{la}\\times\\text{Exec}(z, s, a), \\quad\\sum_{1\\leq b<l }z^R_{lb}\\times\\text{Exec}(z, s, b) \\right)\\text{, whenever }l> V\\\\\n&\\text{where }z\\text{ is a tuple of }(z^O, z^L, z^R)\\\\\nF_1(s, A, B)&=A+B\\text{\\phantom{test}\\emph{add}}\\\\\nF_2(s, A, B)&=A-B\\text{\\phantom{test}\\emph{subtraction}}\\\\\nF_3(s, A, B)&=A\\times B\\text{\\phantom{test}\\emph{multiplication}}\\\\\nF_4(s, A, B)&=A\\text{\\phantom{testttetst}\\emph{no-op\/skip connection}}\n\\end{align*}\n\\caption{\\textbf{Differentiable execution model} for a program sketch containing $L$ lines of code. $z$ parametrizes the program via a triple of 2-dimensional arrays $(z^O, z^L, z^R)$ containing values from 0-1. If $z^O_{lo}=1$, then line $l$ of the program computes its value by executing operator $o$.\nIf $z^L_{la}=1$, then line $l$ of the program gets its left argument for the operator from line $a$.\nIf $z^R_{lb}=1$, then line $l$ of the program gets its rights argument for the operator from line $b$.\nThe first $V$ lines of the program evaluate to variables stored in the symbolic representation, and we assume that there are $V$ such variables and $L$ lines of code that follow.}\\label{denotation}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figure\/interpreter-v2.1.pdf}\n    \\caption{\\textbf{Illustration of the differentiable execution model and overparameterization.} As depth $l$ increases, there are exponentially more ways to implement the same program using the straight-line-code execution model. }\n    \\label{fig:interpreter}\n\\end{figure}\n\n\\subsection{Probabilistic Framing}\nTo alleviate the problems of optimizing objective in Eq~\\ref{badrelax}, we lift our problem statement to an amortized multitask setting and now give a conditional variational autoencoder~\\cite{kingma2013auto} framing.\nWe treat the program $z$ as a latent variable which, given the input $x$, generates the output $y$ through $p(y|x, z)$.\nAs is standard, we introduce an auxiliary variational distribution $\\ensuremath{q}\\left( {z|y, x} \\right)$ to bound the marginal likelihood $p(y|x)$.\nBecause $\\ensuremath{q}\\left( {z|y, x} \\right)$ predicts programs conditioned on input-outputs, we can think of training \\ensuremath{q}~as amortizing the cost of predicting programs given specifications.\nMathematically we seek to maximize $p(y|x)$:\n\\begin{align}\np(y|x)&=\\sum_{z} p(y|x, z) p(z)\\nonumber \\\\\n&\\geq \\expect{\\ensuremath{q}}{\\log \\frac{p(y|x, z)p(z)}{\\ensuremath{q}(z|y, x)}}\\text{, ELBo}\\nonumber \\\\\n&= \\expect{\\ensuremath{q}}{\\log p(y|x, z)}+\\expect{\\ensuremath{q}}{\\log p( z)}+\\entropy{\\ensuremath{q}}\\nonumber \\\\\n&\\geq  \\underbrace{\\expect{\\ensuremath{q}}{\\log p(y|x, z)}}_\\text{reconstruction}+\\underbrace{\\expect{\\ensuremath{q}}{\\log p( z)}}_\\text{regularizer}\\label{variational}\n\\end{align}\nAbove we have dropped the term which encourages max-entropy variational approximations ($\\entropy{\\ensuremath{q}}$), partly for  convenience, and partly because we are concerned with maximum a posteriori (MAP) inference, for which a single point estimate of the posterior suffices.\n\nThe reconstruction term corresponds to the squared loss in Equation~\\ref{map}, given a suitable definition of $p(y|x, z)$.\\footnote{The negative log likelihood of a normal distribution corresponds to squared loss up to an additive constant}\nBecause our latent variable $z$ is discrete, we use a neural network $r(y, s)$ to predict the parameters of multinomial distributions over the entries of $z$:\n\\begin{align}\np(y|x, z)&=\\mathcal{N}\\left( y| \\mu=\\denotation{z}(s) \\right; \\sigma=\\sigma^t)\\\\\n\\ensuremath{q}{\\left( z|y, x \\right)}&=\\prod_{l}\\text{Multinomial}(z^{O}_l|r(y, s)^{O}_l)\\nonumber\\\\\\nonumber&\\phantom{\\prod_l}\\times\\text{Multinomial}(z^{R}_l|r(y, s)^{R}_l)\\\\&\\phantom{\\prod_l}\\times\\text{Multinomial}(z^{L}_l|r(y, s)^{L}_l)\\label{eq:multinomial}\n\\end{align}\nwhere $\\sigma^t$ is the standard deviation of $y$ for task $t$. Recall $s=g(x)$, and see Figure~\\ref{denotation} for the structure of $z$.\n\nWe optimize Equation~\\ref{variational} w.r.t. the parameters of $g$ and $r$, which are the only neural networks in our setup.\nWe use the reparameterization trick to obtain stable gradients through the sampling process for optimization~\\cite{kingma2013auto}. Typically, we use the gumbel softmax trick for each discrete multinomial distribution as defined in Equation~\\ref{eq:multinomial}.\n\n\nThis probabilistic framing ties together \\textbf{Amortization} (by training $\\ensuremath{q}$) and multitasking, by assuming we have a corpus of $\\left\\{ (x_n, y_n) \\right\\}$ pairs, for which we seek to maximize  $\\sum_n \\log p(y_n|x_n)$ using stochastic gradient descent.\n\\textbf{Regularization} enters through a suitable definition of the prior over programs, $p(z)$:\n\\begin{align}\np(z)&\\propto \\exp\\left( -\\lambda \\cdot \\text{length}(z) \\right)\n\\end{align}\nwhere $\\lambda$ is a regularization coefficient and $\\text{length}(z)$ is the length of the program encoded by $z$.\nProgram length can be recursively computed from $z$ in a manner analogous to how $z$ is recursively executed.\n\n\n\n\\subsubsection{Model structure of the program synthesizer}\n\\label{sec: program synthesizer}\nThe program synthesizer predicts the distribution of programs given output and grounded symbols $z\\sim q(\\cdot| r(y, s))$. One single symbol-output pair leaves much ambiguity for program inference. Therefore, we use multiple symbol-output pairs (together as a set) from the same task to predict the program distribution as $z\\sim q(\\cdot| r(\\{y^i, s^i\\})$.\nNoticing that this is a mapping from a set of vectors $\\{(y^{i}, s^i)\\}$ to one vector $z$, we adopt PointNet~\\cite{qi2017pointnet} to build the model. (The descriptions and justification about PointNet are stated in Appendix~\\ref{sec:pointnet}).\n\nOur program synthesizer model first extract features for the properties of the outputs $y^{i}$, the relationships between the $j$-th symbols $s_j^{i}$ and outputs, and the properties of the whole symbol-output pairs, using PointNets:\n\\begin{eqnarray*}\nh_{\\text{out}} & = & \\max_{i} \\text{MLP}_{\\text{out}}\\left(y^{i}\\right),\\\\\nh_{\\text{rel}}[j] & = & \\max_{i} \\text{MLP}_{\\text{rel}}\\left(\\left[y^{i}; s^{i}_j\\right]\\right), ~\\forall j \\in \\{1, 2, \\cdots, V\\},\\\\\nh_{\\text{io}} & = & \\max_{i} \\text{MLP}_{\\text{io}}\\left(\\left[y^{i}; s^{i}\\right]\\right),\n\\end{eqnarray*}\nwhere MLP$_\\text{out}$, MLP$_{\\text{rel}}$, and MLP$_{\\text{io}}$ are three multi-layer perceptron networks, $\\left[~\\cdot~;~\\cdot~\\right]$ means concatenation of the components, $V$ is the dimensionality of the symbolic representations $s$, and $s^{i}_j$ is the $j$-th dimensional element of $s^{i}$.\n\nThe features are then concatenated together to predict the distribution of the programs $z$ as:\n\\begin{equation*}\n    z \\sim \n    \\ensuremath{q}\\left(\\cdot| \\text{MLP}_\\text{prog}\\left(\\left[h_\\text{out}; h_\\text{io}; h_\\text{rel}[0]; h_\\text{rel}[1]; \\cdots; h_\\text{rel}[V]\\right]\\right)\\right),\n\\end{equation*}\nwhere MLP$_\\text{prog}$ is another MLP network for predicting the program distributions according to the features.\n\nIntuitively, our models can learn informative hints for synthesizing programs, such as the outputs $y^{i}$ are never prime, the outputs $y^{i}$ are always bigger than the second symbols $s^{i}_2$, and the outputs $y^{i}$ are always equal to the summation of all symbols $\\sum_{j=1}^V s^{i}_j$.\n\n\\section{Experimental Results}\nWe seek to answer the following questions in our experiments: (1) Is it feasible to jointly learn the symbolic representations from high-dimensional perceptual inputs and synthesize the programs that operate on them (Sec~\\ref{sec:exp-results-main})?  (2) Are the techniques we introduce beneficial (Sec~\\ref{sec:exp-results-ablation})? Specifically, what are the effects of multitasking with amortized inference, overparametrization, and regularization? (3) Can the generalizability of models be improved by learning interpretable symbolic representations with the programmatic prior (Sec~\\ref{sec:exp-results-generalize})?\n\n\\subsection{Experimental Setup}\n\\paragraph{Task and Dataset} As illustrated in Figure~\\ref{fig:setting}, in the neurosymbolic program synthesis problem, we are given dataset containing samples as  I\/O pairs, $\\left(x^{t,i}_1, x^{t,i}_2, \\cdots, x^{t,i}_V, y^{t,i} \\right)$, where $t, i$ index tasks and samples within tasks, respectively, and we assume the I\/O pairs are generated using gold standard symbolic representations, $s^{*,t,i}_j = g^*(x^{t,i}_j)$, and gold standard programs\n$z^{*, t}$ as \n$y^{t,i}=\\llbracket z^{*,t}\\rrbracket\\left(s^{*,t,i}_1, s^{*,t,i}_2, \\cdots, s^{*,t,i}_V\\right)$.\nWe want to train models to jointly infer the symbolic representations $s^{t,i}_j=g(x^{t,i}_j)$ and the program $\\hat z^t=\\arg\\max_z$ $q(z|r(\\{s^{t,i}_j, y^{t,i}_j\\})$ only given distant supervision via the I\/O pairs.\nSuccess means that the symbolic representations are close to the gold standard ones, $s^{t,i}_j\\simeq s^{*,t,i}_j$, and the synthesized programs are equivalent to the gold standard ones, $\\llbracket \\hat z^t\\rrbracket(s) = \\llbracket z^{*,t}\\rrbracket(s), \\forall s\\in \\mathbb{R}^V$, as shown in Figure~\\ref{fig:framework}. \n\n\n\nWe build the dataset by firstly drawing gold standard symbols, $s_j^{*, \\cdot, i}$, uniformly from $\\{1, 2, \\cdots, 10\\}$ for all $i\\le N$ and $j\\le V$, where $N$ is the number of I\/O pairs for each task and $V=3$ is the number of symbols for each input. The placeholder $\\cdot$ denotes that we share the same set of symbols $s_j^{*, \\cdot, i}$ and later inputs $x_j^{\\cdot, i}$ for all tasks $t$, for computational efficiency issues. The perceptual inputs $x_j^{\\cdot, i}$ are then drawn randomly using the CIFAR-10 dataset~\\cite{krizhevsky2009learning}. Specifically, CIFAR-10 dataset contains images in 10 classes, $\\{(\\text{img}^{k}, l^{k})\\}$, where $\\text{img}^{k}\\in \\mathbb{R}^{32\\times 32\\times 3}$ are images and $l^{k}\\in\\{1, 2, \\cdots, 10\\}$ are their labels. The perceptual inputs $x_j^{ \\cdot, i}$ are then drawn uniformly from all images $\\text{img}^{k}$ with label $l^{k}$ equal to $s_j^{*, \\cdot, i}$. The programs $z^{*,t}$ are drawn uniformly from all the semantically distinct arithmetic formulas with at most 3 variables and 3 operators, such as $s_1$, $s_2+s_3$, and $(s_1+s_3)\\times(s_1-s_3)$. The outputs $y^{t, i}$ are then calculated by executing programs on the corresponding inputs as $y^{t,i}=\\denotation{z^{*,t}}(s^{*,\\cdot, i})$, for all $t\\le T$, where $T=500$ is the number of tasks. \n\n\\paragraph{Evaluation Metrics} We evaluate the abilities of models on grounding symbols and synthesizing programs using the following metrics:\n\\begin{itemize}\n    \\item Program success rate: the ratio that the synthesized programs $\\hat z^t$ are equivalent to the gold standard programs $z^{*,t}$. In practice, we empirically evaluate the equivalence of programs using the samples in the dataset, i.e., $\\sum_{t\\le T} \\mathbf{1}(\\denotation{\\hat z^t}(s^{*, \\cdot, i})=y^{t,i}, \\forall i\\le N)~\/~T.$\n   \n    \\item Symbolic loss: the mean squared error (MSE) between the grounded symbols $s^{ \\cdot, i}_j$ and the gold standard symbols $s^{*, \\cdot, i}_j$, i.e., $\\frac{1}{NV}{\\sum_{i\\le N}\\sum_{j\\le V} \\left(g(x^{\\cdot, i}_j)-s^{*, \\cdot, i}_j\\right)^2}$.\n   \n    \\item Test loss: the mean squared error of applying the whole model to predict the outputs, i.e.,\\\\ $\\frac{1}{TN}{\\sum_{t\\le T}\\sum_{i\\le N} \\left(\\denotation{\\hat z^t}(g(x^{t, i}))-y^{t, i}\\right)^2}$.\n\\end{itemize}\nNote that, although we are evaluating models using the gold standard symbols $s^{*, \\cdot, (i)}_j$, they are not available during training and we must learn to ground them in order to achieve good performances on all metrics. Meanwhile, we use different images and random symbolic inputs for generating the training dataset and the testing dataset in order to evaluate models' robustness against the perceptual and sampling noises (CIFAR-10 dataset is split into 50000 samples for training and 10000 samples for testing). All the models and metrics are evaluated on the testing dataset.\n\n\\vspace{2.5pt}\nWe also introduce another metric for evaluating the improvements of models' generalizability from learning interpretable symbolic representations with the programmatic prior. In particular, for a new task $t'$ , we can directly apply our learned symbol grounder $g$ to get the symbolic representation $s^{t', i}_j$ of high-dimensional images $x^{t', i}_j$, and synthesize programs using the symbol-output pairs. To evaluate the out-of-distribution generalizability of such models, we train them on dataset with symbols smaller or equal to 5, $s^{t', i}_j\\le 5, \\forall i,j$, but test them on dataset with symbols larger than 5, $s^{t', i}_j> 5, \\forall i,j$. The evaluation metric is then\n\\begin{itemize}\n    \\item Generalization loss: the mean squared error of applying the model trained on smaller digits to predict outputs on larger digits, i.e., $\\frac{1}{N}{\\sum_{i\\le N} \\left(\\denotation{\\hat z^t'}(s^{t', i})-y^{t', i}\\right)^2}$.\n\\end{itemize}\nNotice that when given the learned symbol grounder, the neurosymbolic program synthesis problem reduces to the classical symbolic program synthesis problem. Any classical algorithms can be applied. In practice, we use the bottom-up enumerative search~\\cite{udupa2013transit} to find the program for the new task. \n\n\\paragraph{Implementation Details} The framework of our models is illustrated in Figure~\\ref{fig:framework} and described in Sec~\\ref{sec: methods}. We apply the standard 18-layer ResNet~\\cite{he2016deep}, which is a classical convolutional neural network, to build the symbol grounder\\footnote{The output dimensionality of the last layer of ResNet is changed to predict symbols instead of image labels.}. All MLPs used in the program synthesizer model have two hidden layers of size 256. Layer normalization~\\cite{ba2016layer} is applied before each activation function, ReLU, to improve the training stability~\\footnote{BatchNorm~\\cite{ioffe2015batch} can not be easily applied in the multi-tasking setting.}. The program sketch contains $L=30$ lines of code. \n\nGiven the large and complex search space of models, there are non-negligible probabilities that a run of the experiment would fail with loss=infinity and gradients equal to nan due to the numerical issues\\footnote{For example, when the symbols are predicted as 10 and the currently synthesized program is $s_0^{20}$, it is easy for the models' outputs to be overflow and fail the training}. To tackle this numerical issue, we instead squash the range of symbols from $\\{1, 2, \\cdots, 10\\}$ to $\\{0.1, 0.2, \\cdots, 1.0\\}$ by dividing by ten.\nAlso, for each model, we run the experiments for multiple times ($\\sim 3$) and report the best one. We select the best learned model using the mean square error of applying the learned model, $g(\\cdot), \\hat z$, to predict outputs in the training dataset. \n\n\nWe train models using the Adam~\\cite{kingma2014adam} optimizer with a learning rate equal to 3e-4 and $\\epsilon=$\\text{1e-5} for 20 epochs. The dataset contains 1e6 I\/O pairs for each task for training and 1000 I\/O pairs for each task for testing. The program length regularizer is not applied until epoch 10 with ratio $\\lambda=$2e-4. The temperature for gumbel softmax is set to 1 in the beginning and changed to 3 from epoch 15 to minimize the error gap from the continuous approximations of programs near the end of training.\n\n\n\\paragraph{Baselines} To analyze the effects of the programmatic prior and the other techniques as described in Section~\\ref{sec: methods}, we compare our model with the following baselines:\n\\begin{itemize}\n    \\item without program structure: These models use neural networks (typically MLPs) instead of programs to predict outputs from symbols as $\\hat y = \\text{MLP}(s)$. We implement two types of models to incorporate the task information and report the best of them: (1) disjoint MLPs for each task, i.e., $\\hat y^t=\\text{MLP}^t(s)$; and (2) one-hot embedding of the task id as features, i.e., $\\hat y^t=\\text{MLP}(\\left[ s;\\text{one-hot}(t)\\right])$.\n    \\item without amortized inference: These models drop the program synthesizer $r$ and instead take $z^t$ as free independent parameters. Therefore, there will be no information shared by amortized inference among the tasks.\n    \\item without gumbel softmax: These models replace gumbel softmax with softmax. So, there is no stochastic sampling, $z\\sim q(z|r(y,s))$, in these models.\n    \\item with Syntax-Tree: These models use the syntax tree execution model instead of straight-line code. More details of the syntax tree execution model are stated in Appendix~\\ref{sec:syntax-tree}. Briefly speaking, the syntax tree execution model has fewer optimal solutions in its solution space and, therefore, is less overparameterized.  \n    \\item with Depth-$D$: The program sketch contains $L=D$ lines of code, instead of $L=30$. Larger depths correspond to more overparameterized models.\n\\end{itemize}\n\n\\subsection{Joint Learning of Symbols and Programs}\n\\label{sec:exp-results-main}\n\nWe demonstrate that it is feasible to jointly learn interpretable symbolic representations and synthesize programs that operate on them using our model. As shown in Table~\\ref{tab:main}, our model successfully learns the symbolic representations and correctly synthesizes the programs for most tasks. The symbolic loss, 0.00086, is negligible for distinguishing symbols in the set $\\{0.1, 0.2, \\cdots, 1.0\\}$. And we correctly find the program for 459 of 500 tasks, which corresponds to the program success rate as 91.8\\%.\n\n\\begin{table*}[h]\n    \\centering\n   \n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline \n         & program success rate & symbolic loss & test loss & generalization loss\\\\\n    \\hline \n     ROAP (our model) & \\textbf{459}~\/~500 &  \\textbf{0.00086} & \\textbf{0.11} & \\textbf{0.07}\\\\\n     \\hline\n     w\/o program, i.e., CNN+MLP & ~~~0~\/~500 &  0.95 & \\textbf{0.11} & 1.03 \\\\\n     w\/o amortized inference & 136~\/~500 &  0.059 & 0.20 & 0.19\\\\\n     w\/o gumbel-softmax & ~~~8~\/~500 &  0.52 & 0.33 & 4.40 \\\\\n     w\/ Syntax-Tree & 345~\/~500 &  0.04 & 0.16 & 0.22 \\\\\n     w\/ depth=10 & ~58~\/~500 & 0.90 & 0.14 & 2.12\\\\\n     w\/ depth=3 & ~56~\/~500 & 1.10 & 0.16 & 2.08 \\\\\n     \\hline\n    \\end{tabular}\n    \\caption{\\textbf{The performances of models for the neurosymbolic program synthesis problem.} Our model (ROAP) successfully learned the interpretable symbolic representations (negligible symbolic loss) and the programs that operate on them (high program success rate). The learned model can predict outputs from input images with small errors on both within-distribution test dataset (test loss) and out-of-distribution test dataset with unseen larger digits (generalization loss). All techniques, including the programmatic prior, multi-tasking with amortized inference, and overparameterization, are beneficial in terms of the joint learning performances by comparing our model with the baselines.}\n   \n    \\label{tab:main}\n\\end{table*}\n\n\\subsubsection{Ablation Studies}\n\\label{sec:exp-results-ablation}\n\nWe analyze the effectiveness of techniques by performing the ablation studies and comparing our model with baselines. The experimental results show that all of our techniques, including multi-tasking with amortized inference, overparameterization, gumbel-softmax, and the program length regularizer, improve performances of models in the neurosymbolic program synthesis problem.\n\n\\paragraph{Effectiveness of multi-tasking with amortized inference} As shown in Table~\\ref{tab:main}, amortized inference with multi-tasks largely improves the performance of program synthesis  and then the learning of interpretable symbolic representations. Without amortized inference, the model can only synthesize programs correctly for 136\/500 tasks. The symbolic loss is also non-negligible as 0.059. It is because amortized inference can share information while synthesizing programs for multiple tasks and, therefore, alleviates the difficulties of program synthesis and then joint learning. More correctly synthesized programs can drive better symbolic representation learning by triangulating a good symbolic basis. Better symbolic representations, in return, enable even better program synthesis by providing more accurate program inputs. We also compare the performances of joint learning in one task, i.e., $T=1$, and 500 tasks in Table~\\ref{tab:multi-task}. We run each experiment more than 10 times, each time with randomly drawn programs, and calculate the success rate metric as the ratio of runs that learn the symbolic representations successfully, i.e., symbolic loss $\\le$ 0.05. We also report the best symbolic loss of both settings in Table~\\ref{tab:multi-task}. The results suggest that multi-tasking can largely improves the success rate of joint learning and also their performances.\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{|c|c|c|}\n    \\hline\n         & success rate & symbolic loss \\\\\n         \\hline \n    500 tasks & \\textbf{33.3\\%} & \\textbf{0.00086} \\\\\n    1 task & 10.5\\% & 0.039 \\\\\n    \\hline \n    \\end{tabular}\n    \\caption{\\textbf{Effects of multitask learning for joint learning.}}\n    \\label{tab:multi-task}\n\\end{table}\n\n\\paragraph{Effectiveness of overparameterization} Baseline models, with Syntax-Tree, with depth=10, and with depth=3, are all less overparameterized than our model. As listed in Table~\\ref{tab:main}, our overparameterized model largely outperforms these baselines in terms of all metrics. These results suggest that overparameterization helps improve the performances of joint learning, probably by providing more global optimal solutions in the solution space and avoiding converging to local optimums. We further modify the degrees of overparameterization by changing the depth $L$ of program sketches and report their performances for tasks with different difficulties in Figure~\\ref{fig:overparameterization}. The results first show that more overparameterized models perform better than less overparameterized models. Moreover, less overparameterized models perform better on easier tasks, i.e., when the relative overparameterization degree (the size of paragram sketch versus the size of correct programs) becomes larger, which also demonstrates the effectiveness of overparameterization.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{figure\/depth-effect.pdf}\n    \\caption{\\textbf{Overparameterization is necessary for robust program recovery.} Programs get depth corresponds to the number of possible lines of code: increasing sketch depth increases overparameterization. In each cell, we show the percentage of programs of a given size successfully synthesized}\n    \\label{fig:overparameterization}\n\\end{figure}\n\n\n\\begin{figure*}[h]\n    \\centering\n   \n   \n   \n    \\includegraphics[width=0.38\\textwidth]{figure\/reg-effect.pdf}\n    \\quad\\quad\n    \\includegraphics[width=0.38\\textwidth]{figure\/reg-effect-lengths.pdf}\n    \\caption{\\textbf{Effects of the program length prior on model convergence.} The regularizer is applied starting from epoch 10. Left: number of program synthesis tasks solved during training. Right: average program length during training. On successful runs (blue),\n    the regularizer maintains performance while dramatically decreasing program length. On `salvageable' runs (orange) the regularizer pushes the model into successful solutions, but not always (green)}\n    \\label{fig:reg-effect}\n\\end{figure*}\n\n\n\\paragraph{Effectiveness of gumbel softmax} As shown in Table~\\ref{tab:main}, models without gumbel softmax perform much worse than ours. It is because the stochastic sampling noises introduced by gumbel softmax help models to explore a larger solution space and then find a better converged point while using the local optimizers such as Adam.\n\n\n\\paragraph{Effectiveness of program length prior} We demonstrate the effects of our program length regularizer in three scenarios in Figure~\\ref{fig:reg-effect}. When the model has already learned good symbols and program synthesizers (blue lines), applying the program length prior will not hurt the models' performances while can largely improve the models' interpretability by inducing synthesizing shorter programs. The averaged length of synthesized programs reduces from 21.6 to 9.6 after applying the regularization, while the program success rate increases from 431 to 455 \/ 500. For runs with poor performances, applying the program length regularization can largely boost the performances of models by providing more programmatic prior for salvageable runs (the orange lines), but not always (the green lines). Specifically, for salvageable runs, the program success rate increases from 4 to 412 \/ 500 and the symbolic loss decreases from 1.23 to 0.066.\n\n\n\\subsection{Out-of-distribution Generalizability by Programs}\n\\label{sec:exp-results-generalize}\nWe finally demonstrate the improvements of out-of-distribution generalizability of models from the programmatic prior using the generalization loss metric. The generalization loss metric evaluates the models' performances on unseen symbols (0.6 to 1.0) that are larger than the symbols (0.1 to 0.5) used during training for a new task. As shown in Table~\\ref{tab:main}, without the programmatic prior, models cannot learn the correct symbolic representations and synthesize the correct programs even though the training and test losses are already low enough. Because of the large flexibility of neural networks, models without programmatic prior can easily overfit and achieve low training losses without learning the correct representations and programs. On the other hand, our model with the programmatic prior can successfully learn the symbolic representations and programs while still achieving the same test loss. More importantly, due to the generalizability of programs, our model can generalize to unseen domains (I\/O pairs with larger digits) and achieve generalization loss as low as 0.07 (compared with 1.03 for models without the programmatic prior).\n\n\n\n\\section{Conclusion}\nOur goal is to make progress on basic neurosymbolic problems: starting from perception, and absent symbol-level supervision, how can we discover grounded symbols and the symbolic routines which manipulate them?\nOur experiments examine a small-scale arithmetic domain, and assumed the premise that end-to-end deep learning could be made to work here.\nEven in this simplistic domain, joint perceptual learning and inductive reasoning proved difficult without additional tricks.\n\nMany directions remain open.\nOne low hanging fruit is pretraining the perceptual frontend $g(\\cdot )$ with self supervised representation learning.\nBut we see the main open question as whether a pure gradient-guided search is actually superior to a hybrid alternating maximization scheme, or a pure symbolic search.\n\nTo what extent do the ROAP tricks generalize? This is an empirical question, and we are actively exploring new application areas to answer exactly that question.\nBut we are optimistic of the general applicability of the ROAP recipe, because we believe that each trick can be motivated from first principles:\n(1) Multitask learning through amortization allows sharing of statistical strength from easy tasks to hard tasks, which should facilitate bootstrap learning.\nMultitasking also constrains the symbolic representation from different directions, `triangulating' the correct symbols for the systems to be using.\n(2) Overparameterization is necessary for reliable neural network convergence, and unsurprisingly also necessary for gradient-guided program search.\n(3) Regularizing program length has solid theoretical underpinnings~\\cite{solomonoff1964formal}.\nAlthough our main demonstration is small-scale, we hope that the ideas we pilot could apply more broadly.\n\n\\section{Broader Impact}\n\nJoint learning of symbolic representations and symbolic programs that operate on them, once achieved, can largely improve models' interpretability and generalizability due to the programmatic prior, while still maintaining scalability to handle noisy high-dimensional perceptual data using neural networks with gradient descent. Human interventions and safety constraints\/verification can be further introduced through the programmatic prior. Therefore, efforts on enabling joint learning of symbols and programs, such as ours, can lead to more usages of machine learning in high-stakes and complex environments such as the medical settings, road, and commerce, where sensory inputs are noisy and high-dimensional, and interpretability and safety are critical. Joint inference of symbols and programs can also help scientific discovery by searching for interpretable and generalizable features to explain nature. On the other hand, interpretable feature extraction and convenient human intervention can be exploited to collect and analyze personal information and for malicious rumor spreading.\n\n\n\n\n\n\\section{Introduction}\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.48\\textwidth]{figure\/Task-v0.1.pdf}\n\t\t\\caption{\\textbf{Solving inductive reasoning tasks grounded in perception.} We consider tasks as inferring a simple formula with noisy perceptual inputs such as images. Solving this problem involves jointly learning to perceive and parse input into symbolic form (e.g., 3 numbers), and reason inductively  to recover programs that explain each arithmetic task.}\n\t\t\\label{fig:setting}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=0.49\\textwidth]{figure\/framework-v2.3.pdf}\n\t\t\\caption{\\textbf{The framework of our model (ROAP) }for the neurosymbolic program synthesis problem. Given I\/O pairs $\\{(x, y)\\}$ for each task, \\textbf{\\rom{1}.}~our model first uses the symbol grounder $g$ to learn symbolic representations $s$ for high-dimensional perceptual input $x$. \\textbf{\\rom{2}.}~The program synthesizer $r$ then uses the symbol-output pairs as a program specification to infer the program for the task, $q(z| r(\\{s, y\\}))$. \\textbf{\\rom{3}.}~The differentiable interpreter finally executes the program on grounded symbols to predict the outputs $\\hat y = \\denotation{z}(s)$. \\textbf{\\rom{4}.}~The whole model is trained end-to-end using a reconstruction loss of outputs and a program length regularizer.} \n\t\t\\label{fig:framework}\n\t\\end{figure}\n\t\n\tWe seek steps toward AI systems that learn to symbolically process perceptual input.\n\tConsider, for example, a system which learns to infer the 3D structure of objects: starting from pixels, it must infer low-level  symbols (curves, parts), and then organize them according to symbolic relationships (symmetry, part repetitions, part hierarchy).\n\tOr, consider a system which learns to control a moving object that navigates around obstacles: starting from sensory data (lidar, RGBD), it must first parse the world (into objects, proximities, freespace), and then compute trajectories using high-level computations (PID controllers, etc.).\n\tSimilar perceptual-symbolic problems arise when e.g. learning structured world models from pixels, or inferring instructions from natural language.\n\tWe frame such tasks as \\textbf{neurosymbolic program synthesis}: learning neural components that extract symbols from perception, and synthesizing programs to further process those symbols with more complex computations.\n\t\n\tIn this work, we study neurosymbolic program synthesis on a small-scale domain involving both perception and programmatic reasoning. Our ultimate goal is to develop general methods that could, we hope, apply to challenging neurosymbolic tasks like those previously mentioned.\n\n\tWe take the stance that symbols should be grounded in perception, and that symbol processing should be implemented by learnable program-like representations.\n\tHowever, we also propose that rather than hand-code a preordained set of primitive symbols, AI systems should autonomously learn to carve the perceptual world into their own discretization.\n\tWhat constitutes a `symbol' may vary across domain and across datasets, and can be hard for human engineers to anticipate. By jointly learning the symbols, as well as synthesizing the programs that operate on them, we hope to side-step the pitfalls associated with hand-engineered representations.\n\t\n\tConcretely, we consider regression problems mapping perceptual input $x$ to output $y$.\n\tThe input $x$ is first processed by a `symbol grounder' neural net $g(\\cdot )$, which produces intermediate symbolic output $s=g(x)$.\n\n\tThe symbols in $s$ are then transformed by a synthesized program $z$ in order to predict the output $\\hat{y}=\\denotation{z}(s)$. We learn $g(\\cdot )$ and $z$ by encouraging the predicted $\\hat{y}$ to be close to the ground-truth output $y$.\n\t\n\tThis setup serves as a basic model of the kinds of neurosymbolic program synthesis we seek to target.\n\tAlthough simple to write down, the setup is difficult to train:\n\tthe grounder $g(\\cdot )$ is a neural network (with unknown weights), \n\twhile the program $z$ is a discrete program (with unknown structure).\n\tTo the best of our knowledge, there do not exist stable and scalable training procedures for such mixed discrete\/continuous search problems, especially when $g(\\cdot )$ has many parameters, and when $z$ comes from a combinatorially large space of possible programs.\n\tAt its core, we have a chicken-and-egg problem: if we had the program, then we could train $g(\\cdot )$ with gradient descent to predict an element of $\\denotation{z}^{-1}(y)$;\n\tif we had the neural net, we could synthesize the program $z$ using a variety of prior  search methods~\\cite{shah2020learning, 10.1145\/2491956.2462174} on the input-output example $g(x)\\mapsto y$.\n\t\n\tBroadly speaking, prior works have taken three approaches for addressing this mixed discrete\/continuous problem.\n\tFirst, the most popular strategy is to relax the discrete space of programs $z$ and train the entire system end-to-end through backpropagation~\\cite{evans2018learning, graves2014neural}.\n\tSecond, the complement of that approach has also been tried: discretizing the weight-space of the $g$ network and solving for both the program and the network weights using SAT~\\cite{EVANS2021103521}.\n\tLast, alternating optimization frameworks which successively update continuous weights and discrete structure, in the style of Expectation-Maximization, have been tried~\\cite{mws}.\n\t\n\tWe conjecture that relaxation and end-to-end backprop is likely to yield the best results, at least in the near future, because it allows one to build on the success and infrastructure of deep learning.\n\tTo that end, we combine three techniques to ameliorate the optimization difficulties associated with mixed discrete\/continuous problems, at least for this setting:\\\\\n\t\\textbf{1. Multitasking with amortized inference:} We consider a multitask setting with hundreds of interrelated programs sharing the same symbol grounding. Because they share the same symbol grounding, learning $g$ is easier, as the many tasks `triangulate' a good symbolic basis. Because the tasks share structure, we can make discrete program synthesis for $z$ tractable by \\emph{amortizing}~\\cite{gershman2014amortized} program search by training an auxiliary recognition (inference) model.\n\tThis `recognition model' learns to predict programs conditioned on input-output specifications.\\\\\n\t\\textbf{2. Overparameterization:} We `overparameterize' the space of programs: for example, even if we only expect a 4-line program, we search over the space of 20-line programs. Although this would be problematic for combinatorial search, gradient descent is well-known to benefit from overparameterized optimization landscapes~\\cite{frankle2018lottery}, and this benefit carries over to continuous relaxations of program spaces, as first noted by terpret~\\cite{gaunt2016terpret}. \\\\\n\t\\textbf{3. Regularization:} We encourage shorter program solutions by designing program-length regularizer compatible with continuous program relaxations. We find this aids convergence. It also combats the code explosion that overparameterization naively leads to, giving shorter and more interpretable programs.\n\t\n\tOur contribution is the combination of these techniques--\\textbf{R}egularize, \\textbf{O}verparameterize, \\textbf{A}mortize, for \\textbf{P}rograms--and so we call our approach ROAP (pronounced rope).\n\t\n\t\n\n\t\n\n\t\n\t\\section{Related Work}\n\tNeurosymbolic programming is a growing area that seeks to engineer learning and inference methods for hybrid program\/neural architectures~\\cite{PGL-049}, and our work is a special case of this broad framework.\n\tSpecifically, we tackle inductive program synthesis~\\cite{gulwani2017program}--synthesizing programs from input-output examples--but where the inputs are high-dimensional and must be preprocessed by a neural network into symbolic form.\n\tPrior works in this setting assume a hand-engineered inventory of basic symbols~\\cite{ellis2017learning}, while others backpropagate through differentiable programs to jointly train network weights and program structure~\\cite{ gaunt2017differentiable}.\n\tMultitasking is a known strategy for this setting~\\cite{valkov2018houdini}.\n\tTraining a neural network to help guide search for discrete programs (amortized inference) is standard~\\cite{devlin2017robustfill, 10.1145\/3453483.3454080}, and we extend that idea to continuous relaxations of program spaces. \n\t\n\tThe difficulties of gradient descent over relaxed program spaces is well known, to the extent that it has been dubbed the so-called `terpret problem'~\\cite{gaunt2016terpret}.\n\tUnfortunately, such an approach is the most straightforward way of training neural net$\\to$program architectures. From a technical perspective, our work hopes to make progress on the `terpret problem', thereby unlocking scalable and reliable training of this class of neurosymbolic programs.\n\t\n\tMore fundamentally, our efforts connect to the body of work on the `symbol grounding problem'~\\cite{harnad1990symbol}:\n\tHow does a system learn to `ground' abstract symbols (e.g., numbers) in terms of their high-dimensional perceptual counterparts (e.g., images of digits)?\n\tThis problem is especially difficult absent strong supervision on the meaning of each abstract symbol~\\cite{chang2020assessing}, and here we consider a distantly supervised setting.\n\tPrior works consider a variety of techniques to address symbol grounding~\\cite{topan2021techniques}.\n\tWe hope that the complimentary methods we introduce can help chip away at a small part of the symbol grounding problem.\n\n\t\n\t\\section{Methods}\n\t\\label{sec: methods}\n\t\\subsection{Relaxing Spaces of Programs}\n\tWe consider straight-line code: sequences of variable assignments, each of which introduces a new variable and computes its value by applying an operator to variables that were previously computed on earlier lines.\n\tWe write $z$ to mean the syntactic structure of a straight-line program, which is represented by a triple $(z^O, z^L, z^R)$ of binary 2D arrays encoding the syntactic structure of the program.\n\tGiven this discrete \\emph{sketch}~\\cite{solar2006combinatorial} of possible programs, we define a denotation operator $\\denotation{z}$ (Figure~\\ref{denotation}).\n\tGiven input-output examples $x\\mapsto y$, \n\tsingle-task neurosymbolic synthesis corresponds to\n\t\\begin{align}\n\t\tg(\\cdot ), z&=\\argmin_{\\substack{g(\\cdot ), z\\\\z\\text{'s entries }\\in \\left\\{ 0, 1 \\right\\}}}%\n\t\t\\left( y-\\denotation{z}(g(x)) \\right)^2\\label{map}\n\t\\end{align}\n\tThe constraint that the entries of $z$ lie in $\\left\\{ 0, 1 \\right\\}$ precludes end-to-end backpropagation.\n\tBecause our denotation operator is actually a smooth function of $z$, we can relax the above optimization problem by saying the entries of $z$ lie in $[ 0, 1 ]$:\n\t\\begin{align}\n\t\tg(\\cdot ), z&=\\argmin_{\\substack{g(\\cdot ), z\\\\z\\text{'s entries }\\in \\left[ 0, 1 \\right]}}%\n\t\t\\left( y-\\denotation{z}(g(x)) \\right)^2\\label{badrelax}\n\t\\end{align}\n\tUnfortunately the above objective is problematic: generically there is no reason for gradient descent to converge to a discrete $z$.\n\tWhile one can incorporate sparsity-favoring regularizers on $z$, doing so causes the probability of convergence to fall off rapidly as the target program length increases~\\cite{evans2018learning, gaunt2016terpret}, suggesting that $z$ only converges to a discrete program if it is randomly initialized close to a correct solution.\n\t\n\t\\paragraph{Overparameterization}  With this setup so far, we can compactly describe the first of our three techniques: {Overparameterization}.\n\tWe simply expand the number of possible lines of code in the straight-line sketch, as illustrated in Figure~\\ref{fig:interpreter}, and expect that it will help convergence of the relaxed problem by providing a benign optimization landscape~\\cite{hardt2016identity, du2019gradient}.\n\t\n\t\n\t\\begin{figure*}\n\t\t\\begin{align*}\n\t\t\t\\hspace{2cm} \\denotation{z}(s)&=\\text{Exec}(z, s, L+V)\\text{\\phantom{test}\\emph{execute program and extract output on line }}L+V\\\\\n\t\t\t\\text{Exec}(z, s, l)&=s_l\\text{, whenever }l\\leq V\\text{\\phantom{test}\\emph{load symbolic variables as first lines of code. We have V variables}}\\\\\n\t\t\t\\text{Exec}(z, s, l)&=\\sum_o z^O_{lo}\\times F_o\\left( s, \\quad\\sum_{1\\leq a<l }z^L_{la}\\times\\text{Exec}(z, s, a), \\quad\\sum_{1\\leq b<l }z^R_{lb}\\times\\text{Exec}(z, s, b) \\right)\\text{, whenever }l> V\\\\\n\t\t\t&\\text{where }z\\text{ is a tuple of }(z^O, z^L, z^R)\\\\\n\t\t\n\t\t\tF_1(s, A, B)&=A+B\\text{\\phantom{test}\\emph{add}}\\\\\n\t\t\tF_2(s, A, B)&=A-B\\text{\\phantom{test}\\emph{subtraction}}\\\\\n\t\t\tF_3(s, A, B)&=A\\times B\\text{\\phantom{test}\\emph{multiplication}}\\\\\n\t\t\tF_4(s, A, B)&=A\\text{\\phantom{testttetst}\\emph{no-op\/skip connection}}\n\t\t\n\t\t\n\t\t\\end{align*}\n\t\t\\caption{\\textbf{Differentiable execution model} for a program sketch containing $L$ lines of code. $z$ parametrizes the program via a triple of 2-dimensional arrays $(z^O, z^L, z^R)$ containing values from 0-1. If $z^O_{lo}=1$, then line $l$ of the program computes its value by executing operator $o$.\n\t\t\tIf $z^L_{la}=1$, then line $l$ of the program gets its left argument for the operator from line $a$.\n\t\t\tIf $z^R_{lb}=1$, then line $l$ of the program gets its rights argument for the operator from line $b$.\n\t\t\n\t\t\tThe first $V$ lines of the program evaluate to variables stored in the symbolic representation, and we assume that there are $V$ such variables and $L$ lines of code that follow. }\\label{denotation}\n\t\\end{figure*}\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.4\\textwidth]{figure\/interpreter-v2.1.pdf}\n\t\t\\caption{\\textbf{Illustration of the differentiable execution model and overparameterization.} As depth $l$ increases, there are exponentially more ways to implement the same program using the straight-line-code execution model. }\n\t\t\\label{fig:interpreter}\n\t\\end{figure}\n\t\n\t\\subsection{Probabilistic Framing}\n\tTo alleviate the problems of optimizing objective in Eq~\\ref{badrelax}, we lift our problem statement to an amortized multitask setting and now give a conditional variational autoencoder~\\cite{kingma2013auto} framing.\n\tWe treat the program $z$ as a latent variable which, given the input $x$, generates the output $y$ through $p(y|x, z)$.\n\tAs is standard, we introduce an auxiliary variational distribution $\\ensuremath{q}\\left( {z|y, x} \\right)$ to bound the marginal likelihood $p(y|x)$.\n\tBecause $\\ensuremath{q}\\left( {z|y, x} \\right)$ predicts programs conditioned on input-outputs, we can think of training \\ensuremath{q}~as amortizing the cost of predicting programs given specifications.\n\tMathematically we seek to maximize $p(y|x)$:\n\t\\begin{align}\n\t\tp(y|x)&=\\sum_{z} p(y|x, z) p(z)\\nonumber \\\\\n\t\t&\\geq \\expect{\\ensuremath{q}}{\\log \\frac{p(y|x, z)p(z)}{\\ensuremath{q}(z|y, x)}}\\text{, ELBo}\\nonumber \\\\\n\t\t&= \\expect{\\ensuremath{q}}{\\log p(y|x, z)}+\\expect{\\ensuremath{q}}{\\log p( z)}+\\entropy{\\ensuremath{q}}\\nonumber \\\\\n\t\t&\\geq  \\underbrace{\\expect{\\ensuremath{q}}{\\log p(y|x, z)}}_\\text{reconstruction}+\\underbrace{\\expect{\\ensuremath{q}}{\\log p( z)}}_\\text{regularizer}\\label{variational}\n\t\\end{align}\n\tAbove we have dropped the term which encourages max-entropy variational approximations ($\\entropy{\\ensuremath{q}}$), partly for  convenience, and partly because we are concerned with maximum a posteriori (MAP) inference, for which a single point estimate of the posterior suffices.\n\t\n\tThe reconstruction term corresponds to the squared loss in Equation~\\ref{map}, given a suitable definition of $p(y|x, z)$.\\footnote{The negative log likelihood of a normal distribution corresponds to squared loss up to an additive constant}\n\tBecause our latent variable $z$ is discrete, we use a neural network $r(y, s)$ to predict the parameters of multinomial distributions over the entries of $z$:\n\t\\begin{align}\n\t\tp(y|x, z)&=\\mathcal{N}\\left( y| \\mu=\\denotation{z}(s) ; \\sigma=\\sigma^t\\right)\\\\\n\t\t\\ensuremath{q}{\\left( z|y, x \\right)}&=\\prod_{l}\\text{Multinomial}(z^{O}_l|r(y, s)^{O}_l)\\nonumber\\\\\\nonumber&\\phantom{\\prod_l}\\times\\text{Multinomial}(z^{R}_l|r(y, s)^{R}_l)\\\\&\\phantom{\\prod_l}\\times\\text{Multinomial}(z^{L}_l|r(y, s)^{L}_l)\\label{eq:multinomial}\n\t\\end{align}\n\twhere $\\sigma^t$ is the standard deviation of $y$ for task $t$. Recall $s=g(x)$, and see Figure~\\ref{denotation} for the structure of $z$.\n\t\n\tWe optimize Equation~\\ref{variational} w.r.t. the parameters of $g$ and $r$, which are the only neural networks in our setup.\n\tWe use the reparameterization trick to obtain stable gradients through the sampling process for optimization~\\cite{kingma2013auto}. Typically, we use the gumbel softmax trick for each discrete multinomial distribution as defined in Equation~\\ref{eq:multinomial}.\n\t\n\n\n\t\n\tThis probabilistic framing ties together \\textbf{Amortization} (by training $\\ensuremath{q}$) and multitasking, by assuming we have a corpus of $\\left\\{ (x_n, y_n) \\right\\}$ pairs, for which we seek to maximize  $\\sum_n \\log p(y_n|x_n)$ using stochastic gradient descent.\n\t\\textbf{Regularization} enters through a suitable definition of the prior over programs, $p(z)$:\n\t\\begin{align}\n\t\tp(z)&\\propto \\exp\\left( -\\lambda \\cdot \\text{length}(z) \\right)\n\t\\end{align}\n\twhere $\\lambda$ is a regularization coefficient and $\\text{length}(z)$ is the length of the program encoded by $z$.\n\tProgram length can be recursively computed from $z$ in a manner analogous to how $z$ is recursively executed.\n\t\n\t\n\t\n\t\\paragraph{Model Structure of the Program Synthesizer}\n\t\\label{sec: program synthesizer}\n\tThe program synthesizer predicts the distribution of programs given output and grounded symbols $z\\sim q(\\cdot| r(y, s))$. One single symbol-output pair leaves much ambiguity for program inference. Therefore, we use multiple symbol-output pairs (together as a set) from the same task to predict the program distribution as $z\\sim q(\\cdot| r(\\{y^i, s^i\\})$.\n\tNoticing that this is a mapping from a set of vectors $\\{(y^{i}, s^i)\\}$ to one vector $z$, we adopt PointNet~\\cite{qi2017pointnet} to build the model. (The descriptions and justification about PointNet are stated in Appendix~\\ref{sec:pointnet}).\n\t\n\tOur program synthesizer model first extract features for the properties of the outputs $y^{i}$, the relationships between the $j$-th symbols $s_j^{i}$ and outputs, and the properties of the whole symbol-output pairs, using PointNets:\n\t\\begin{eqnarray*}\n\t\th_{\\text{out}} & = & \\max_{i} \\text{MLP}_{\\text{out}}\\left(y^{i}\\right),\\\\\n\t\th_{\\text{rel}}[j] & = & \\max_{i} \\text{MLP}_{\\text{rel}}\\left(\\left[y^{i}; s^{i}_j\\right]\\right), ~\\forall j \\in \\{1, 2, \\cdots, V\\},\\\\\n\t\th_{\\text{io}} & = & \\max_{i} \\text{MLP}_{\\text{io}}\\left(\\left[y^{i}; s^{i}\\right]\\right),\n\t\\end{eqnarray*}\n\twhere MLP$_\\text{out}$, MLP$_{\\text{rel}}$, and MLP$_{\\text{io}}$ are three multi-layer perceptron networks, $\\left[~\\cdot~;~\\cdot~\\right]$ means concatenation of the components, $V$ is the dimensionality of the symbolic representations $s$, and $s^{i}_j$ is the $j$-th dimensional element of $s^{i}$.\n\t\n\tThe features are then concatenated together to predict the distribution of the programs $z$ as:\n\t\\begin{equation*}\n\t\tz \\sim \n\t\t\\ensuremath{q}\\left(\\cdot| \\text{MLP}_\\text{prog}\\left(\\left[h_\\text{out}; h_\\text{io}; h_\\text{rel}[0]; h_\\text{rel}[1]; \\cdots; h_\\text{rel}[V]\\right]\\right)\\right),\n\t\\end{equation*}\n\twhere MLP$_\\text{prog}$ is another MLP network for predicting the program distributions according to the features.\n\t\n\tIntuitively, our models can learn informative hints for synthesizing programs, such as the outputs $y^{i}$ are never prime, the outputs $y^{i}$ are always bigger than the second symbols $s^{i}_2$, and the outputs $y^{i}$ are always equal to the summation of all symbols $\\sum_{j=1}^V s^{i}_j$.\n\t\n\t\\section{Experimental Results}\n\tWe seek to answer the following questions in our experiments: (1) Is it feasible to jointly learn the symbolic representations from high-dimensional perceptual inputs and synthesize the programs that operate on them (Sec~\\ref{sec:exp-results-main})?  (2) Are the techniques we introduce beneficial (Sec~\\ref{sec:exp-results-ablation})? Specifically, what are the effects of multitasking with amortized inference, overparametrization, and regularization? (3) Can the generalizability of models be improved by learning interpretable symbolic representations with the programmatic prior (Sec~\\ref{sec:exp-results-generalize})?\n\t\n\t\\subsection{Experimental Setup}\n\t\\paragraph{Task and Dataset} As illustrated in Figure~\\ref{fig:setting}, in the neurosymbolic program synthesis problem, we are given dataset containing samples as  I\/O pairs, $\\left(x^{t,i}_1, x^{t,i}_2, \\cdots, x^{t,i}_V, y^{t,i} \\right)$, where $t, i$ index tasks and samples within tasks, respectively, and we assume the I\/O pairs are generated using gold standard symbolic representations, $s^{*,t,i}_j = g^*(x^{t,i}_j)$, and gold standard programs\n\t$z^{*, t}$ as \n\t$y^{t,i}=\\llbracket z^{*,t}\\rrbracket\\left(s^{*,t,i}_1, s^{*,t,i}_2, \\cdots, s^{*,t,i}_V\\right)$.\n\tWe want to train models to jointly infer the symbolic representations $s^{t,i}_j=g(x^{t,i}_j)$ and the program $\\hat z^t=\\arg\\max_z$ $q(z|r(\\{s^{t,i}_j, y^{t,i}_j\\})$ only given distant supervision via the I\/O pairs.\n\tSuccess means that the symbolic representations are close to the gold standard ones, $s^{t,i}_j\\simeq s^{*,t,i}_j$, and the synthesized programs are equivalent to the gold standard ones, $\\llbracket \\hat z^t\\rrbracket(s) = \\llbracket z^{*,t}\\rrbracket(s), \\forall s\\in \\mathbb{R}^V$, as shown in Figure~\\ref{fig:framework}. \n\t\n\n\t\n\n\t\n\tWe build the dataset by firstly drawing gold standard symbols, $s_j^{*, \\cdot, i}$, uniformly from $\\{1, 2, \\cdots, 10\\}$ for all $i\\le N$ and $j\\le V$, where $N$ is the number of I\/O pairs for each task and $V=3$ is the number of symbols for each input. The placeholder $\\cdot$ denotes that we share the same set of symbols $s_j^{*, \\cdot, i}$ and later inputs $x_j^{\\cdot, i}$ for all tasks $t$, for computational efficiency issues. The perceptual inputs $x_j^{\\cdot, i}$ are then drawn randomly using the CIFAR-10 dataset~\\cite{krizhevsky2009learning}. Specifically, CIFAR-10 dataset contains images in 10 classes, $\\{(\\text{img}^{k}, l^{k})\\}$, where $\\text{img}^{k}\\in \\mathbb{R}^{32\\times 32\\times 3}$ are images and $l^{k}\\in\\{1, 2, \\cdots, 10\\}$ are their labels. The perceptual inputs $x_j^{ \\cdot, i}$ are then drawn uniformly from all images $\\text{img}^{k}$ with label $l^{k}$ equal to $s_j^{*, \\cdot, i}$. The programs $z^{*,t}$ are drawn uniformly from all the semantically distinct arithmetic formulas with at most 3 variables and 3 operators, such as $s_1$, $s_2+s_3$, and $(s_1+s_3)\\times(s_1-s_3)$. The outputs $y^{t, i}$ are then calculated by executing programs on the corresponding inputs as $y^{t,i}=\\denotation{z^{*,t}}(s^{*,\\cdot, i})$, for all $t\\le T$, where $T=500$ is the number of tasks. \n\n\t\n\t\\paragraph{Evaluation Metrics} We evaluate the abilities of models on grounding symbols and synthesizing programs using the following metrics:\n\t\\begin{itemize}\n\t\t\\item Program success rate: the ratio that the synthesized programs $\\hat z^t$ are equivalent to the gold standard programs $z^{*,t}$. In practice, we empirically evaluate the equivalence of programs using the samples in the dataset, i.e., $\\sum_{t\\le T} \\mathbf{1}(\\denotation{\\hat z^t}(s^{*, \\cdot, i})=y^{t,i}, \\forall i\\le N)~\/~T.$\n\t\n\t\t\\item Symbolic loss: the mean squared error (MSE) between the grounded symbols $s^{ \\cdot, i}_j$ and the gold standard symbols $s^{*, \\cdot, i}_j$, i.e., $\\frac{1}{NV}{\\sum_{i\\le N}\\sum_{j\\le V} \\left(g(x^{\\cdot, i}_j)-s^{*, \\cdot, i}_j\\right)^2}$.\n\t\n\t\t\\item Test loss: the mean squared error of applying the whole model to predict the outputs, i.e.,\\\\ $\\frac{1}{TN}{\\sum_{t\\le T}\\sum_{i\\le N} \\left(\\denotation{\\hat z^t}(g(x^{t, i}))-y^{t, i}\\right)^2}$.\n\t\\end{itemize}\n\tNote that, although we are evaluating models using the gold standard symbols $s^{*, \\cdot, (i)}_j$, they are not available during training and we must learn to ground them in order to achieve good performances on all metrics. Meanwhile, we use different images and random symbolic inputs for generating the training dataset and the testing dataset in order to evaluate models' robustness against the perceptual and sampling noises (CIFAR-10 dataset is split into 50000 samples for training and 10000 samples for testing). All the models and metrics are evaluated on the testing dataset.\n\t\n\t\\vspace{2.5pt}\n\tWe also introduce another metric for evaluating the improvements of models' generalizability from learning interpretable symbolic representations with the programmatic prior. In particular, for a new task $t'$ , we can directly apply our learned symbol grounder $g$ to get the symbolic representation $s^{t', i}_j$ of high-dimensional images $x^{t', i}_j$, and synthesize programs using the symbol-output pairs. To evaluate the out-of-distribution generalizability of such models, we train them on dataset with symbols smaller or equal to 5, $s^{t', i}_j\\le 5, \\forall i,j$, but test them on dataset with symbols larger than 5, $s^{t', i}_j> 5, \\forall i,j$. The evaluation metric is then\n\t\\begin{itemize}\n\t\t\\item Generalization loss: the mean squared error of applying the model trained on smaller digits to predict outputs on larger digits, i.e., $\\frac{1}{N}{\\sum_{i\\le N} \\left(\\denotation{{{}\\hat{z}^t}'}(s^{t', i})-y^{t', i}\\right)^2}$.\n\t\\end{itemize}\n\tNotice that when given the learned symbol grounder, the neurosymbolic program synthesis problem reduces to the classical symbolic program synthesis problem. Any classical algorithms can be applied. In practice, we use the bottom-up enumerative search~\\cite{udupa2013transit} to find the program for the new task. \n\t\n\t\\paragraph{Implementation Details} The framework of our models is illustrated in Figure~\\ref{fig:framework} and described in Sec~\\ref{sec: methods}. We apply the standard 18-layer ResNet~\\cite{he2016deep}, which is a classical convolutional neural network, to build the symbol grounder\\footnote{The output dimensionality of the last layer of ResNet is changed to predict symbols instead of image labels.}. All MLPs used in the program synthesizer model have two hidden layers of size 256. Layer normalization~\\cite{ba2016layer} is applied before each activation function, ReLU, to improve the training stability~\\footnote{BatchNorm~\\cite{ioffe2015batch} can not be easily applied in the multi-tasking setting.}. The program sketch contains $L=30$ lines of code. \n\n\t\n\tGiven the large and complex search space of models, there are non-negligible probabilities that a run of the experiment would fail with loss=infinity and gradients equal to nan due to the numerical issues\\footnote{For example, when the symbols are predicted as 10 and the currently synthesized program is $s_0^{20}$, it is easy for the models' outputs to be overflow and fail the training}. To tackle this numerical issue, we instead squash the range of symbols from $\\{1, 2, \\cdots, 10\\}$ to $\\{0.1, 0.2, \\cdots, 1.0\\}$ by dividing by ten.\n\tAlso, for each model, we run the experiments for multiple times ($\\sim 3$) and report the best one. We select the best learned model using the mean square error of applying the learned model, $g(\\cdot), \\hat z$, to predict outputs in the training dataset. \n\n\t\n\n\t\n\tWe train models using the Adam~\\cite{kingma2014adam} optimizer with a learning rate equal to 3e-4 and $\\epsilon=$\\text{1e-5} for 20 epochs. The dataset contains 1e6 I\/O pairs for each task for training and 1000 I\/O pairs for each task for testing. The program length regularizer is not applied until epoch 10 with ratio $\\lambda=$2e-4. The temperature for gumbel softmax is set to 1 in the beginning and changed to 3 from epoch 15 to minimize the error gap from the continuous approximations of programs near the end of training.\n\t\n\t\n\t\\paragraph{Baselines} To analyze the effects of the programmatic prior and the other techniques as described in Section~\\ref{sec: methods}, we compare our model with the following baselines:\n\t\\begin{itemize}\n\t\t\\item without program structure: These models use neural networks (typically MLPs) instead of programs to predict outputs from symbols as $\\hat y = \\text{MLP}(s)$. We implement two types of models to incorporate the task information and report the best of them: (1) disjoint MLPs for each task, i.e., $\\hat y^t=\\text{MLP}^t(s)$; and (2) one-hot embedding of the task id as features, i.e., $\\hat y^t=\\text{MLP}(\\left[ s;\\text{one-hot}(t)\\right])$.\n\t\t\\item without amortized inference: These models drop the program synthesizer $r$ and instead take $z^t$ as free independent parameters. Therefore, there will be no information shared by amortized inference among the tasks.\n\t\t\\item without gumbel softmax: These models replace gumbel softmax with softmax. So, there is no stochastic sampling, $z\\sim q(z|r(y,s))$, in these models.\n\t\t\\item with Syntax-Tree: These models use the syntax tree execution model instead of straight-line code. More details of the syntax tree execution model are stated in Appendix~\\ref{sec:syntax-tree}. Briefly speaking, the syntax tree execution model has fewer optimal solutions in its solution space and, therefore, is less overparameterized.  \n\t\t\\item with Depth-$D$: The program sketch contains $L=D$ lines of code, instead of $L=30$. Larger depths correspond to more overparameterized models.\n\t\\end{itemize}\n\t\n\t\\subsection{Joint Learning of Symbols and Programs}\n\t\\label{sec:exp-results-main}\n\t\n\tWe demonstrate that it is feasible to jointly learn interpretable symbolic representations and synthesize programs that operate on them using our model. As shown in Table~\\ref{tab:main}, our model successfully learns the symbolic representations and correctly synthesizes the programs for most tasks. The symbolic loss, 0.00086, is negligible for distinguishing symbols in the set $\\{0.1, 0.2, \\cdots, 1.0\\}$. And we correctly find the program for 459 of 500 tasks, which corresponds to the program success rate as 91.8\\%.\n\t\n\t\\begin{table*}[h]\n\t\t\\centering\n\t\t\\caption{\\textbf{The performances of models for the neurosymbolic program synthesis problem.} Our model (ROAP) successfully learned the interpretable symbolic representations (negligible symbolic loss) and the programs that operate on them (high program success rate). The learned model can predict outputs from input images with small errors on both within-distribution test dataset (test loss) and out-of-distribution test dataset with unseen larger digits (generalization loss). All techniques, including the programmatic prior, multi-tasking with amortized inference, and overparameterization, are beneficial in terms of the joint learning performances by comparing our model with the baselines. }\n\t\n\t\t\\label{tab:main}\n\t\n\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\\hline \n\t\t\t& program success rate & symbolic loss & test loss & generalization loss\\\\\n\t\t\t\\hline \n\t\t\tROAP (our model) & \\textbf{459}~\/~500 &  \\textbf{0.00086} & \\textbf{0.11} & \\textbf{0.07}\\\\\n\t\t\t\\hline\n\t\t\tw\/o program, i.e., CNN+MLP & ~~~0~\/~500 &  0.95 & \\textbf{0.11} & 1.03 \\\\\n\t\t\tw\/o amortized inference & 136~\/~500 &  0.059 & 0.20 & 0.19\\\\\n\t\t\tw\/o gumbel-softmax & ~~~8~\/~500 &  0.52 & 0.33 & 4.40 \\\\\n\t\t\tw\/ Syntax-Tree & 345~\/~500 &  0.04 & 0.16 & 0.22 \\\\\n\t\t\tw\/ depth=10 & ~58~\/~500 & 0.90 & 0.14 & 2.12\\\\\n\t\t\tw\/ depth=3 & ~56~\/~500 & 1.10 & 0.16 & 2.08 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{table*}\n\t\n\t\\subsubsection{Ablation Studies}\n\t\\label{sec:exp-results-ablation}\n\t\n\tWe analyze the effectiveness of techniques by performing the ablation studies and comparing our model with baselines. The experimental results show that all of our techniques, including multi-tasking with amortized inference, overparameterization, gumbel-softmax, and the program length regularizer, improve performances of models in the neurosymbolic program synthesis problem.\n\t\n\t\\paragraph{Effectiveness of multi-tasking with amortized inference} As shown in Table~\\ref{tab:main}, amortized inference with multi-tasks largely improves the performance of program synthesis  and then the learning of interpretable symbolic representations. Without amortized inference, the model can only synthesize programs correctly for 136\/500 tasks. The symbolic loss is also non-negligible as 0.059. It is because amortized inference can share information while synthesizing programs for multiple tasks and, therefore, alleviates the difficulties of program synthesis and then joint learning. More correctly synthesized programs can drive better symbolic representation learning by triangulating a good symbolic basis. Better symbolic representations, in return, enable even better program synthesis by providing more accurate program inputs. We also compare the performances of joint learning in one task, i.e., $T=1$, and 500 tasks in Table~\\ref{tab:multi-task}. We run each experiment more than 10 times, each time with randomly drawn programs, and calculate the success rate metric as the ratio of runs that learn the symbolic representations successfully, i.e., symbolic loss $\\le$ 0.05. We also report the best symbolic loss of both settings in Table~\\ref{tab:multi-task}. The results suggest that multi-tasking can largely improves the success rate of joint learning and also their performances.\n\t\n\t\\begin{table}\n\t\t\\centering\n\t\t\\caption{\\textbf{Effects of multitask learning for joint learning.}}\n\t\t\\label{tab:multi-task}\n\t\t\\begin{tabular}{|c|c|c|}\n\t\t\t\\hline\n\t\t\t& success rate & symbolic loss \\\\\n\t\t\t\\hline \n\t\t\t500 tasks & \\textbf{33.3\\%} & \\textbf{0.00086} \\\\\n\t\t\t1 task & 10.5\\% & 0.039 \\\\\n\t\t\t\\hline \n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\paragraph{Effectiveness of overparameterization} Baseline models, with Syntax-Tree, with depth=10, and with depth=3, are all less overparameterized than our model. As listed in Table~\\ref{tab:main}, our overparameterized model largely outperforms these baselines in terms of all metrics. These results suggest that overparameterization helps improve the performances of joint learning, probably by providing more global optimal solutions in the solution space and avoiding converging to local optimums. We further modify the degrees of overparameterization by changing the depth $L$ of program sketches and report their performances for tasks with different difficulties in Figure~\\ref{fig:overparameterization}. The results first show that more overparameterized models perform better than less overparameterized models. Moreover, less overparameterized models perform better on easier tasks, i.e., when the relative overparameterization degree (the size of paragram sketch versus the size of correct programs) becomes larger, which also demonstrates the effectiveness of overparameterization.\n\t\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.3\\textwidth]{figure\/depth-effect.pdf}\n\t\t\\caption{\\textbf{Overparameterization is necessary for robust program recovery.} Programs get depth corresponds to the number of possible lines of code: increasing sketch depth increases overparameterization. In each cell, we show the percentage of programs of a given size successfully synthesized}\n\t\t\\label{fig:overparameterization}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure*}[h]\n\t\t\\centering\n\t\n\t\n\t\n\t\t\\includegraphics[width=0.38\\textwidth]{figure\/reg-effect.pdf}\n\t\t\\quad\\quad\n\t\t\\includegraphics[width=0.38\\textwidth]{figure\/reg-effect-lengths.pdf}\n\t\t\\caption{\\textbf{Effects of the program length prior on model convergence.} The regularizer is applied starting from epoch 10. Left: number of program synthesis tasks solved during training. Right: average program length during training. On successful runs (blue),\n\t\t\tthe regularizer maintains performance while dramatically decreasing program length. On `salvageable' runs (orange) the regularizer pushes the model into successful solutions, but not always (green)}\n\t\t\\label{fig:reg-effect}\n\t\\end{figure*}\n\t\n\t\n\t\\paragraph{Effectiveness of gumbel softmax} As shown in Table~\\ref{tab:main}, models without gumbel softmax perform much worse than ours. It is because the stochastic sampling noises introduced by gumbel softmax help models to explore a larger solution space and then find a better converged point while using the local optimizers such as Adam.\n\t\n\t\n\t\\paragraph{Effectiveness of program length prior} We demonstrate the effects of our program length regularizer in three scenarios in Figure~\\ref{fig:reg-effect}. When the model has already learned good symbols and program synthesizers (blue lines), applying the program length prior will not hurt the models' performances while can largely improve the models' interpretability by inducing synthesizing shorter programs. The averaged length of synthesized programs reduces from 21.6 to 9.6 after applying the regularization, while the program success rate increases from 431 to 455 \/ 500. For runs with poor performances, applying the program length regularization can largely boost the performances of models by providing more programmatic prior for salvageable runs (the orange lines), but not always (the green lines). Specifically, for salvageable runs, the program success rate increases from 4 to 412 \/ 500 and the symbolic loss decreases from 1.23 to 0.066.\n\t\n\t\n\t\\subsection{Out-of-Distribution Generalizability by Programs}\n\t\\label{sec:exp-results-generalize}\n\t\n\t\n\tWe demonstrate the improvements of out-of-distribution generalizability of models from the programmatic prior using the generalization loss metric. The generalization loss metric evaluates the models' performances on unseen symbols (0.6 to 1.0) that are larger than the symbols (0.1 to 0.5) used during training for a \\emph{new} task, having already seen those symbols used in \\emph{old} tasks. As shown in Table~\\ref{tab:main}, without the programmatic prior, models cannot learn the correct symbolic representations and synthesize the correct programs even though training and test losses are low. Because of the large flexibility of neural networks, models without a programmatic prior can easily overfit and achieve low training losses without learning the correct representations and programs. On the other hand, our model with the programmatic prior can successfully learn the symbolic representations and programs while still achieving the same test loss. The symbol grounder is shared among multiple tasks so that it can learn to ground unseen symbols for the new task. More importantly, due to the generalizability of programs, our programmatic model can generalize to unseen domains (I\/O pairs with larger digits) and achieve generalization loss as low as 0.07 (compared with 1.03 for models without the programmatic prior).\n\t\n\t\n\t\\section{Conclusion}\n\tOur goal is to make progress on basic neurosymbolic problems: starting from perception, and absent symbol-level supervision, how can we discover grounded symbols and the symbolic routines which manipulate them?\n\tOur experiments examine a small-scale arithmetic domain, and assumed the premise that end-to-end deep learning could be made to work here.\n\tEven in this simplistic domain, joint perceptual learning and inductive reasoning proved difficult without additional tricks.\n\t\n\tMany directions remain open.\n\tOne low hanging fruit is pretraining the perceptual frontend $g(\\cdot )$ with self supervised representation learning.\n\tBut we see the main open question as whether a pure gradient-guided search is actually superior to a hybrid alternating maximization scheme, or a pure symbolic search.\n\t\n\tTo what extent do the ROAP tricks generalize? This is an empirical question, and we are actively exploring new application areas to answer exactly that question.\n\tBut we are optimistic of the general applicability of the ROAP recipe, because we believe that each trick can be motivated from first principles:\n\t(1) Multitask learning through amortization allows sharing of statistical strength from easy tasks to hard tasks, which should facilitate bootstrap learning.\n\tMultitasking also constrains the symbolic representation from different directions, `triangulating' the correct symbols for the systems to be using.\n\t(2) Overparameterization is necessary for reliable neural network convergence, and unsurprisingly also necessary for gradient-guided program search.\n\t(3) Regularizing program length has solid theoretical underpinnings~\\cite{solomonoff1964formal}.\n\tAlthough our main demonstration is small-scale, we hope that the ideas we pilot could apply more broadly.\n\t\n\t\\begin{figure*}[h]\n\t\t\\begin{align*}\n\t\t\t\\hspace{2.5cm}\n\t\t\t\\denotation{z}(s)&=\\text{Exec}(z, s, 0)\\text{\\phantom{test}\\emph{execute program and extract output on the root node with }}d=0\\\\\n\t\t\t\\text{Exec}(z, s, path)&=\\sum_o z^{{path}}\\times F_o\\left( s, \\quad\\text{Exec}(z, s, path.0), \\quad\\text{Exec}(z, s, path.1) \\right)\\text{, whenever len}(path)<D\\\\\n\t\t\t\\text{Exec}(z, s, path)&=\\sum_{1\\le j\\le V}z^{path}_j\\times s_j\\text{, whenever len}(path)=D \\text{\\phantom{test}\\emph{load symbolic variables in leaf nodes}}\\\\\n\t\t\t&\\text{where }z\\text{ is a tuple of }(z^0, z^{0.0}, z^{0.1}, z^{0.0.0}, z^{0.0.1}, \\cdots)\\\\\n\t\t\n\t\t\tF_1(s, A, B)&=A+B\\text{\\phantom{test}\\emph{add}}\\\\\n\t\t\tF_2(s, A, B)&=A-B\\text{\\phantom{test}\\emph{subtraction}}\\\\\n\t\t\tF_3(s, A, B)&=A\\times B\\text{\\phantom{test}\\emph{multiplication}}\\\\\n\t\t\tF_4(s, A, B)&=A\\text{\\phantom{testttetst}\\emph{no-op\/skip connection}}\n\t\t\n\t\t\n\t\t\\end{align*}\n\t\t\\caption{The syntax-tree execution model for a program sketch with a depth-$D$ syntax tree. $z$ parametrizes the program by specifying the operator for all internal nodes in the syntax tree, $z^{path}$ when len$(path)<D$, and specifying the symbolic variables for all leaf nodes, $z^{path}$ when len$(path)=D$.}\n\t\t\\label{fig:syntax-tree}\n\t\\end{figure*}\n\t\n\t\n\n\t\\section{Broader Impact}\n\t\n\tJoint learning of symbolic representations and symbolic programs that operate on them, once achieved, can largely improve models' interpretability and generalizability due to the programmatic prior, while still maintaining scalability to handle noisy high-dimensional perceptual data using neural networks with gradient descent. Human interventions and safety constraints\/verification can be further introduced through the programmatic prior. Therefore, efforts on enabling joint learning of symbols and programs, such as ours, can lead to more usages of machine learning in high-stakes and complex environments such as the medical settings, road, and commerce, where sensory inputs are noisy and high-dimensional, and interpretability and safety are critical. Joint inference of symbols and programs can also help scientific discovery by searching for interpretable and generalizable features to explain nature. On the other hand, interpretable feature extraction and convenient human intervention can be exploited to collect and analyze personal information and for malicious rumor spreading.\n\t\n\t\n\n\n\n\n\n\t\\begin{acks}\n\t\tWe are grateful for helpful discussions with Owen Lewis about overparameterization.\n\t\\end{acks}\n\t\n\t\n\t\n\n\n\t\n\t\\begin{figure}[]\n\t\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{figure\/syntax-tree.pdf}\n\t\t\\caption{An example of the syntax trees for $y=s_1\\times s_2 + s_3$. The symbolic variables are loaded in the leaf nodes and the operators are assigned in the internal nodes.  }\n\t\t\\label{fig:syntax-tree-example}\n\t\\end{figure}\n\t\n\t\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe past one hundred years have witnessed incredible advancements in artificial intelligence (AI)~\\cite{stone2022artificial,littman2022gathering}. These advancements are integrating cyberspace (CS), physical space (PS), and social space(SS), into the cyber-physical-social-system(CPSS), which extraordinarily expands the territories of human civilizations~\\cite{wang2010emergence} extraordinarily. However, AI \\textit{per se} is not enough to establish trust in the CPSS~\\cite{jacovi2021formalizing,jan2020ai}, which is the cornerstone of prosperity in every civilized society. Blockchain, also coined as \\textit{decentralized AI}, has the potential to empower AI to be more trustworthy by creating a decentralized trust of privacy, security, and audit-ability~\\cite{harris2019decentralized,adel2022decentralizing,hussain2021artificial}. However, systematic studies on the design principle of blockchain as a trust engine for the CPSS are still absent. If we could decipher the philosophy of blockchain, we would build the infrastructure of a better digital world. In this article, we provide an initiative for seeking the design principle of blockchain for the betterment of human civilization. Unlike the existing systemization of knowledge (SoK) on blockchain that focuses on a specific discipline purpose or topic of interest, ours aims at opening an intellectual conversation beyond boundaries for the ultimate goal of a better digital world, namely, the SoK of SoKs. Specifically, we hope to initiate the answers to three questions for the past, the current, and the future of blockchain design.\n\\begin{enumerate}\n    \\item {\\em The past:} Does blockchain live up to its original design principles as a distributed database? \n    \\item {\\em The current:} How do the current blockchain literature, industry practices, and global standards address and develop the design principle of blockchain?\n    \\item {\\em The future:} What are the gaps between the current design of blockchain and the design principle of a trust engine for a truly intelligent world?\n\\end{enumerate}\n\nIn \\Cref{sec: the past}, we examine the performance of blockchain according to its original design principle as a distributed database. We present the controversial answers in a dialog style of debate. In \\Cref{sec: the current}, we analyze the current research and practice of blockchain principles by investigating the current SoKs of emerging blockchain literature, the white papers that provide technique cornerstones for blockchain technology in the real world, and the discussions on global standards for blockchain. We identify a taxonomy of blockchain literature into the seven categories of privacy and security, scalability, decentralization, applicability, governance and regulation, system design, and cross-chain interoperability. Integrating the AI of natural language processing (NLP)~\\cite{bird2009natural} methods, we find that the current blockchain design principles, in both research and practice, are more centered around the first category of privacy and security and the fourth category of applicability. Future scholars, practitioners, and policy-makers have vast opportunities in the other much less exploited categories and the synthesis at the interface of multiple categories. In \\Cref{sec: the future}, we envision the future of blockchain technology by pointing out the gaps between the current design of blockchain and the design principle of a trust engine for a truly intelligent world. By providing counterexamples, we question the possibility of developing a plausible solution of singularity to the gaps without crossing the current boundaries of domain expertise.\n\\par\nSages in both the West and East have been craving an ideal society. In the \\textit{Book of Rites}, one of the Confucian (551\u2013479 B.C.E.) classics~\\cite{sep-confucius}, the \\textit{Great Unity} is a Chinese vision of an ideal world in which men of virtue and ability rule. In ancient Greece, Plato~\\cite{sep-plato} (429\u2013347 B.C.E.) envisioned an ideal city-state ruled by a philosopher-king of justice in the \\textit{Republic}, a Socratic dialog. By such a coincidence, both Confucius and Plato place the foundation of an ideal society on cultivating or selecting the men of virtue to be the administrators whom people can trust. Nevertheless, the existence of a benevolent ruler might not be guaranteed. In contrast, blockchain enables a new way of trust creation empowered by computer intelligence beyond human ethics. However, the current blockchain technology is evidently not a panacea for doubts about human trust. Then, why \"throw the baby out with the bathwater?\" Existing mechanisms for cultivating human trust have created economic prosperity and human civilizations for thousands of years. How then can we integrate blockchain technology into the existing governance, economics, and ethics to create an ideal society in the new era of CPSS? We initiate an open dialog of synthetic solutions. \n\\section{The past}\n\\label{sec: the past}\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{figs\/Sunshine_and_ChatGPT3.png}\n  \\caption{A Conversation Between Sunshine and Chat GPT-3 on 2022\/12\/30}\n  \\label{fig:chatGPT3}\n\\end{figure}\n\nWhat was blockchain originally designed for? How does blockchain live up to the design principle of its original purpose? \\citet{sherman2019origins} introduce the origin of blockchain for establishing a distributed database trusted by mutually suspicious groups. \\citet{narayanan2016bitcoin, haber1990time},and~\\citet{bayer1993improving} further elaborate on the follow-up development of blockchain technology for a series of desired properties such as decentralization, immutability, transparency, security, and efficiency. \\Cref{fig:chatGPT3} shows the answer returned by the Generative Pre-trained Transformer (ChatGPT),~\\url{https:\/\/openai.com\/blog\/chatgpt\/}, a chatbot launched by OpenAI in November 2022, to the question of \"what is the design principle of blockchain?\" The answer accurately matches elaborations provided in earlier blockchain literature. We now examine the performance of blockchain on each principle in a dialog style of debate in the spirit of Confucius's \\textit{the Analects}~\\cite{ames2010analects} and Plato's \\textit{the Republic}~\\cite{plato2005republic}. We find the answer to be controversial. Moreover, the desired principles might either conflict with each other or lead to other undesired consequences.  \n\\subsection*{Question 1: Does blockchain live up to its promise of decentralization?}\n\\subsubsection{Pros}\nYes, blockchain is decentralized. Compared with centralized ledgers, blockchain is not subject to the single point failure of a centralized entity~\\cite{puthal2018blockchain}. Blockchain records and validates data by a consensus protocol involving decentralized entities, which has the fault tolerance of minorities~\\cite{zhang2022blockchain}. \n\\subsubsection{Cons}\nNo, blockchain is not decentralized. \\citet{sai2021taxonomy} provide a taxonomy of blockchain centralization in 13 aspects. For example, \\citet{sai2021taxonomy} show that the top four mining pools in Ethereum (Bitcoin) consist of 63.04\\% (50.36\\%) of the hashing power, which is enough to conduct a successful malicious attack. Moreover, \\citet{zhang2022sok} and the references therein show that the usage of blockchain at the application layer has the trend of conversing to centralization.   \n\n\\subsection*{Question 2: Does blockchain live up to its promise of immutability?}\n\\subsubsection{Pros}\nYes, blockchain is immutable. ~\\citet{narayanan2016bitcoin} elaborate on the difficulty of manipulating recorded data due to the chain structure of Merkle trees. \n\\subsubsection{Cons}\nNo, blockchain is mutable. \\citet{hofmann2017immutability} show that the immutability of blockchain can be breached by various attacks. \\citet{politou2019blockchain} even address the conflicts between blockchain immutability and the new European data protection regulation on the right to be forgotten (RtbF), according to which individuals have the right to delete their personal data under certain conditions. \n\n\\subsection*{Question 3: Does blockchain live up to its promise of transparency?}\n\n\\subsubsection{Pros}\nYes, blockchain is transparent. ~\\cite{monrat2019survey} explains that all the transactions on blockchain are recorded and auditable by each replica node in the network. \n\\subsubsection{Cons}\nNo, blockchain is not transparent. \\citet{sai2021taxonomy} demonstrates the centralization in node operation, which is required to audit the blockchain data due to technique barriers. \\citet{feng2019survey} even point out the transparency of blockchain as a negative feature and a threat to user privacy. \n\n\\subsection*{Question 4: Does blockchain live up to its promise of security?}\n\\subsubsection{Pros}\nYes, blockchain is secure. \\citet{zhang2019security} address the blockchain security property of consistency and integrity achieved by the main techniques of consensus algorithms and hash-chained storage. \n\\subsubsection{Cons}\nNo, blockchain is not secure. \\citet{lin2017survey} evidence a long list of real attacks on blockchain together with an analysis of potential security issues. \\citet{budish2022economic} even establishes a conclusion that the mechanism supporting blockchain security is either vulnerable or not worth the economic cost. \n\n\\subsection*{Question 5: Does blockchain live up to its promise of efficiency?}\n\\subsubsection{Pros}\nYes, blockchain is efficient. \\citet{wang2019blockchain} analyze how blockchain improves efficiency in human interactions by automating and streamlining business operations using smart contracts. \n\\subsubsection{Cons}\nNo, blockchain is not efficient. Current major blockchains can only sustain tens of transactions per second (TPS), which is not comparable to the centralized platforms, which have a throughput of thousands of TPS~\\cite{chauhan2018blockchain}.\n\n\n\\section{The current}\n\\label{sec: the current}\nHow do the current blockchain literature, industry practices, and global standards address and develop the design principle of blockchain? We collect all the systemization of knowledge (SoK) papers related to facets of blockchain or surveys and review articles that satisfy the defining features of SoKs. We refer the readers to the JSys website, \\url{https:\/\/www.jsys.org\/type_SoK\/}, for background information on SoKs and the Oakland Conference GitHub site, \\url{https:\/\/oaklandsok.github.io\/}, for the histories of SoKs in computer security and privacy literature. We present a taxonomy of the collected SoKs in \\Cref{tab1} and \\Cref{tab2}, broken into seven categories based on the topic of research questions: \n\\begin{itemize}\n    \\item {\\em Category 1: Privacy and Security.} We include literature that answers questions about private information protection and attacks on blockchain systems. We refer the readers to \\citet{sep-privacy} for a philosophical symposium on \\textit{privacy} and \\citet{bishop2003computer} for the computer science insights on \\textit{security}.\n    \\item {\\em Category 2: Scalability.} We include literature that answers questions related to the scalable deployment of blockchain, usually measured in throughput and quality of service. We refer the readers to \\citet{jogalekar2000evaluating} for the definition and measurement of scalability in the distributed system literature. \n    \\item {\\em Category 3: Decentralization.} We include the literature that answers questions related to the designed and realized decentralization of blockchain. We refer the readers to \\citet{zhang2022sok} for an interdisciplinary synthesis of a taxonomy of blockchain decentralization. \n    \\item {\\em Category 4: Applicability.} We include literature that answers questions related to the application of blockchain to solve social and economic issues in various of human interactions. \\Cref{tab2} shows that the major applications are in financial services~\\cite{zetzsche2020decentralized,chen2020blockchain,harvey2021defi}. \n    \\item {\\em Category 5: Governance and Regulation.} We include literature that answers questions related to utilizing blockchain as a new way of governance, the process of interaction of an organized society, and its interactions with existing governance solutions, including corporate, government, and nongovernment organizations (NGOs). We refer the readers to \\cite{bevir2012governance} for a review of existing governance solutions, including various forms of social coordination and patterns of rules.    \n    \\item {\\em Category 6: System design.} We include literature that answers questions related to designing blockchain as an advancement of existing computer systems. We refer the readers to \\citet{saltzer2009principles} for the design principles of computer systems. \n    \\item {\\em Category 7: Cross-chain and Interoperability.} We include literature that answers questions related to communication, cooperation, and integration among blockchain systems, other computer systems, and human societies. We refer the readers to \\citet{wegner1996interoperability} and \\citet{leal2019interoperability} for the importance and assessment of interoperability among information systems. \n\\end{itemize}\n\n\\Cref{tab6} lists the top 28 blockchain projects, excluding applications on the blockchain system and cross-chain solutions ranked by the market value of their native currency retrieved from Coinmarketcap, \\url{https:\/\/coinmarketcap.com\/}, on Dec. 27, 2022. We further collect the white papers that provide the technical foundations for the 28 projects. We then conduct a text analysis applying natural language processing (NLP) methods to produce a word cloud and bigram networks of the titles and abstracts of the SoKs and white papers. The results are presented in \\Cref{tab4}, \\Cref{tab5}, \\Cref{tab7}, \\Cref{tab8}, \\Cref{tab9}, \\Cref{fig:teaser}, \\Cref{fig:whitepaper_bigram_title}, \\Cref{fig:wordcloud_abstract}, \\Cref{fig:whitepaper_wordcloud_abstract}, \\Cref{fig:whitepaper_wordcloud_title}, \\Cref{fig:bigram_title}, \\Cref{fig:bigram_abstract}, and \\Cref{fig:whitepaper_bigram_abstract}, \\Cref{fig:whitepaper_bigram_title}. The word cloud distinguishes the word frequency in these documents through font size, and the bigram represents the coappearance of two words in sequential order ranking by counts in the tables and in the network figures. We also further research the emerging documents of blockchain standard development and papers discussing blockchain standards by working groups and institutions globally.  \n\\par\n\\textbf{Data and Code Availability}: We open source the data and code for replication and future research on the GitHub: \\url{https:\/\/github.com\/sunshineluyao\/design-principle-blockchain}. \n    \nWe find that the current blockchain design principles, in both research and practice, are more centered around the first category of privacy and security and the fourth category of applicability. Future scholars, practitioners, and policy-makers have vast opportunities in the other much less exploited categories and the synthesis at the interface of multiple categories.\n\\section{The future}\n\\label{sec: the future}\nWhat are the gaps between the current design of blockchain and the design principle of a trust engine for a truly intelligent world? In this section, we do not intend to provide a comprehensive answer. Instead, we question the possibility of developing a plausible solution of singularity to the gaps without crossing the current boundaries of domain expertise by providing counterexamples.\n\n\\subsection*{Case Study 1: \\\\ How can we elaborate on the design principle of privacy and security for a better society?}\nThe scientific community of distributed systems and cryptography has made great milestones in blockchain development by achieving the anonymity of private information and the audibility of public transactions. However, privacy and security are the means but not necessarily the ends for a better society of human prosperity. Through history, critiques~\\cite{sep-privacy} have addressed privacy as the source of criminal activities, economic inefficiency, and the abuse of minorities. \\citet{cong2022crypto} identify that wash trading accounts for approximately 70\\% of the total cryptocurrency trading volumes. \\citet{cong2022anatomy} further provide a taxonomy of crimes enabled by the anonymity feature on blockchain. Abundant literature in behavioral science~\\cite{dawson2018study} establishes a connection between anonymity and abusive behavior. However, none of those negative effects on human behavior are considered in the original design principles of blockchain. How can we redesign privacy and security to cultivate human cooperation better and minimize abusive behavior? Behavior scientists have the expertise to contribute to this discussion. \n\n\\subsection*{Case Study 2: \\\\ How can we elaborate on the design principle of decentralization for a better society?}\nMost top-ranked blockchain projects aim to create a permissionless system that is more inclusive, democratic, and decentralized than those that have been presented previously. However, what is required to satisfy the permission for the permissionless blockchains? You must have access to the internet, afford the computer system requirements to run a replica node, and master the engineering skills needed to participate in the network. Unfortunately, according to World in Data, \\url{https:\/\/ourworldindata.org\/internet}, in most parts of our world, less than half of the population has access to the internet, let alone the higher requirements of computer system and engineering skills. Moreover, \\citet{ao2022decentralized} and the references therein evidence most of the crypto transactions are executed via centralized exchanges but not in the decentralized form of on-chain peer-to-peer transactions. \\citet{cong2022inclusion} further address the inclusion issues of the current Web3 economy due to technology barriers and lack of inclusion consideration in the original design principle of blockchain. The famous Irish playwright Bernard Shaw said, \"Revolutions have never lightened the burden of tyranny. They have only shifted it to another shoulder\"~\\cite{tiliouine2016state}. To direct blockchain development to avoid the fate stated by Bernard Shaw and serve a more decentralized society---of the people, by the people, and for the people---anthropologists, social scientists, and economists are able to contribute to designing a society of democracy, diversity, and inclusion. \n\n\\subsection*{Case Study 3: \\\\ How can we elaborate on the design principle of efficiency for a better society?}\n\nThe current solutions for efficiency or scalability on blockchain focus on improving transaction throughput and automating business processes. However, the pursuit of efficiency in an isolated system of short-sighted consideration might lead to inefficiency for the society as a whole with regard to long-term sustainability. For example, the current scalability solutions in general consider the blockchain system only in isolation but ignore the negative externality of the high-energy consumption design to the outside world~\\cite{truby2018decarbonizing}. Moreover, \\citet{grimmelmann2019all} notes that although smart contracts automate business processes, the current design ignores the ambiguity in the semantics of the programming language. Thus, a temporary convenience of the smart contract operation might lead to costly disputes in unforeseen scenarios when developers and users disagree. Policy-makers and lawyers can contribute to reconsidering the blockchain design to improve the efficiency of negotiations among different stakeholders and our society as a whole. \n\n\n\\subsection*{A Call for Collaboration}\n\\citet{kranzberg1986technology} stated the following in his 1985 address as president of the Society for the History of Technology (SHOT): \"Technology is neither good nor bad, nor is it neutral.\" blockchain is a powerful technology that brings about many new possibilities. How can we redesign blockchain for a truly intelligent world? To address this questions, we call for a joint endeavor from all disciplines. \n\n\\bibliographystyle{ACM-Reference-Format}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n        Gravitational lensing by galaxies is of interest both because\nit provides constraints on the mass distribution of the lensing galaxy,\nand because it allows the study of galaxies at higher redshifts than\nwould otherwise be possible because of the magnification of the lensed\nobject. The Hubble Space Telescope (HST) has proven to be very useful for\ngravitational lensing studies because of its ability to obtain high\nresolution images of faint objects. This utility has been demonstrated\nboth by studies of previously identified lensing candidates\n(e.g.\\ FSC 10214+4724, Eisenhardt et al.\\ 1996), and the discovery of new\ncandidate lenses (e.g. Ratnatunga et al.\\ 1995).\n\n        The HST images of the Hubble Deep Field (HDF) provide a rich\nground for gravitational lens searches because of the combination of\ndeep exposures in multiple colors and the high spatial resolution of \nthe HST (Williams et al.\\ 1996). A visual\ninspection of the field reveals several possible candidate lensing\nsystems (e.g.\\ Hogg et al.\\ 1996). These include a configuration of \nfour compact galaxies with similar blue colors, separated by \n$\\ {\\raise-0.5ex\\hbox{$\\buildrel<\\over\\sim$}}\\  1\\farcs0$ (J123641+621204, or in our notation L4.1), \na red ``elliptical'' with a blue ``arc'' offset by about $1\\farcs8$ \nto the southwest and a potential counterimage on the other side \n(J123652+621227, or L3.1), and a red ``elliptical'' with a\nblue ``arc'' offset by about $0\\farcs9$ to the south (J123656+621221, or L3.2).\n\n        Spectroscopy of the constituent objects is critical for\ndetermining the nature of these candidate gravitational lens\nsystems. Therefore, as part of a modest program of spectroscopy\nof galaxies in the HDF, we used the Keck telescope to obtain low-resolution\nspectra of objects in these candidate lenses with the aim of determining\ntheir redshifts and spectral characteristics. The observations\nthemselves are described in $\\S$ 2, the results of the analysis of the\ndata in $\\S$ 3, and the implications of these results are discussed in $\\S$ 4.\n\n\\section {Observations}\n\n        Spectra of objects in the HDF were obtained on 1996 March 15 (UT)\nwith the Keck 10-m telescope and the Low-Resolution Imaging Spectrograph\n(LRIS, Oke et al.\\ 1995). The LRIS slit-mask facility allowed \nus to obtain spectra of many objects simultaneously. We designed \ntwo slitmasks, each of which contained about 20 objects in the HDF itself \nand its flanking fields. The slitlets were $0\\farcs7$ wide. The spectra \nwere obtained using the 300 l\/mm grating, which provided wavelength \ncoverage from about 5000\\AA\\ through 9500\\AA\\, with the exact coverage\ndependent on the location of the slitlet, and with declining sensitivity \nat the red end of this range.\nWith the 24$\\mu$m pixels of the Tektronix CCD, the wavelength scale is \napproximately 2.4\\AA$\/$pixel and the spatial scale is $0\\farcs214\/$pixel.\n\n        In selecting our objects, we specifically targeted the three sets\nof objects in the HDF which we found to be the most likely lensed systems.\nTrue-color images of these sets of objects and the orientation of our\nslit positions is shown in Figure 1 (Plate 1).\nThe positions, magnitudes, and colors of the objects in these candidate\nlens systems are given in Table 1.\nThe position angles of the two masks were chosen so that the slitlets\npassed through\nboth the candidate lens and lensed object for the two systems in\nwhich both are identifiable. This corresponds to a position angle\nof $19^{\\circ}$ (PA19) for L3.2 and $61^{\\circ}$ (PA61) for L3.1.\nA slitlet was placed across the system of four similar objects\n(L4.1) at both position angles.\n\n        Our observations consist of 80 minutes of integration\ntime with the PA61 mask and 60 minutes for the PA19 mask, each of which\nwas divided into two exposures.\nFrom images taken during the alignment procedure, we estimate the\nseeing was about $0\\farcs9$.\nThe reduction of these data is described in detail in\nMoustakas et al. (1996). Briefly, after standard CCD reduction techniques,\nthe two exposures were combined using bad pixel maps to eliminate cosmic\nrays. The combined image was checked by eye for the few cosmic rays which\nescaped detection in the first pass. The combined image was then transformed\nto linear wavelength scale using arc lamps observed after each exposure.\n\n\nOne-dimensional spectra were then extracted from the two-dimensional\ndata. The location on the slitlet of each extraction was based on\nthe observed location of flux. In all cases, this corresponds to the\nexpected location of a target object. The one-dimensional spectrum\nwas smoothed with a gaussian of $\\sigma = 3.0$\\AA\\\/, which is the\napproximate instrumental resolution. No further smoothing was performed.\nA relative flux calibration\nof the spectra was made using a spectrophotometric standard observed\nwith LRIS in long-slit mode on an earlier observing run with a very\nsimilar setup (kindly provided by H. Spinrad and D. Stern). Although\nthis is clearly far from ideal for flux calibration, the result is\nconsistent with a previously obtained spectrum of one of our targets.\nThe resulting one-dimensional spectra are described\nin detail in the following section.\n\n\n\\section {Analysis}\n\n\\subsection {L4.1 (J123641+621204)}\n\n        The extracted spectra of this set of objects are shown\nin Figure 2, along with a spectrum obtained for the same\nobjects with a wider slit by Steidel et al. (1996, hereafter SGDA).\nThis figure indicates that our spectra confirm SGDA's general\nidentification of Ly$\\alpha$ emission at $z \\simeq 3.22$, and \npossibly their identification of SiIV absorption at the same redshift.\nThis redshift is consistent with the colors\nof the objects given in Table 1, as the redshifted Lyman break\nprovides a natural explanation for the very red F300W $-$ F450W\ncolor. In fact, the objects were selected by SGDA\nto be candidate $z \\sim 3$ galaxies on the basis of their colors.\n\n\tA closer look at our spectra shows that at both of our position\nangles, the emission lines have two distinct components. Based on\nLy$\\alpha$, the redshifts of these components are $z = 3.220$ \nand $z=3.209$, corresponding to a rest frame velocity difference \nof 800 $\\,{\\rm km\\,s^{\\scriptscriptstyle -1}}$.  Moreover, our two-dimensional spectra indicate that\nthe different velocity components are at different spatial locations. \nThis is shown in Figure 3 (Plate 2), in which the Ly$\\alpha$ lines are \nat the same location or only slightly offset along the slit for \nPA61, but are separated by several pixels for PA19. \nA correlation between velocity and position had been tentatively suggested\nby SGDA.\n{\\it The detection of a different spatial\nlocation for the different velocity components rules out \ngravitational lensing as the cause of this compact configuration\nof high redshift objects}. \n\n\tWith our two position angles, we can attempt to identify\nwhich of the four objects is responsible for each feature.\nBecause the emission lines are at roughly the same position on the\nPA61 slitlet, the likely candidates are the northern and southern objects, \nas these are the only two at roughly the same position along the slit \nat this angle. This tentative identification appears to be confirmed by\nthe offset observed on the PA19 slitlet, which is exactly that\nexpected if the two emission components are from these two objects.\nHowever, as a caution we note the seeing was only marginal for\nresolving this system, and a firm identification of individual\nspatial components with velocity components awaits spectral data\nwith better spatial information. \n\n\tWe also note that both redshift components appear to have \nNV emission which is strong relative to Ly$\\alpha$, and which is redshifted\ncompared to its expected location given the $z$ determined from Ly$\\alpha$.\nStrong NV emission is also present in SGDA's spectrum, as is the\noffset of the peak emission to slightly higher redshift than that\nof the corresponding Ly$\\alpha$ feature.\nThe relative strength of the high ionization NV line is \nsuggestive of an AGN (e.g.\\ Kinney et al. 1993). \nHowever, the redward peak of the NV feature and the\npossible evidence for some P-Cygni absorption blueward of the expected\nline center suggest that there may also be a massive star component \nto the NV emission. Therefore, this system appears to be composed\nof two or more star-forming galaxies, possibly with AGN, separated \nby roughly 10 kpc and 800 $\\,{\\rm km\\,s^{\\scriptscriptstyle -1}}$.\n\n\\subsection{L3.1 (J123652+621227)}\n\n\tFrom west to east, our slitlet for this system crossed the \n``arc'', the central ``elliptical'' and then the ``counterimage''\n(see Figure 1). An emission line at $5301$\\AA\\\/ at\nthe position of the counterimage is the only emission feature\ndetected along this slitlet. A one-dimensional spectrum\nextracted at the position of the counterimage is shown in Figure 4.\nThe two-dimensional spectrum in the wavelength region around \nthe detected emission line is shown in Figure 3 (Plate 2).\nThe emission feature is visible on each separate exposure.\nA faint red continuum at the position of the elliptical \ncan be seen on the two-dimensional spectrum, but the signal\nis insufficient for any analysis. An even weaker blue continuum\nis visible on the two-dimensional image at the position of the\narc. Most importantly, no emission line is seen in the arc spectrum \nin the range from $5000$\\AA\\ to $7200$\\AA\\\/. \n\n\tThe presence of an emission line at the position of the\ncounterimage and the absence of any corresponding line at the\nposition of the arc is evidence against a lensing origin for\nthis system. However, the emission feature is not overwhelmingly\nstrong, so it is interesting to consider other data. The colors \nof the objects given in Table 1 provide independent evidence \nthat the arc feature and counterimage are not lensed images\nof a single object. Specifically, the counterimage has a \nsignificantly bluer F450W$-$F606W color than the arc. \nAlthough slightly different colors are possible for different\narc features (Hogg et al.\\ 1996), the combination of the colors \nand our spectral data favor the more straightforward interpretation \nthat the arc and counterimage are different galaxies.\n\n\tThe colors can also help in the identification of the single \nemission line of the counterimage. The colors show that the counterimage\nhas a flat spectrum from F814W through F450W, and then probably reddens \ninto the F300W bandpass. The flat spectral shape is characteristic of\nthe star-forming galaxies, and the red F300W$-$F450W color, if real,\nsuggests that $z \\ {\\raise-0.5ex\\hbox{$\\buildrel>\\over\\sim$}}\\  2.5$, as a result of the Lyman break redshifting \nthrough the F300W bandpass (e.g.\\ SDGA). \nUnfortunately, the faintness of the counterimage means \nthat the absence of a detection only requires F300W$-$F450W $> 0.19$ \nat the $3\\sigma$ level, and F300W$-$F450W $> 1.39$ at the $1\\sigma$ level.\nThe colors do not clearly establish the presence or absence of a Lyman\nbreak. However, it is difficult to find examples of lower redshift \ngalaxies with flat spectra from F814W through F450W and a significantly\nredder F300W$-$F450W color.\n\n\tGuided by the colors, a natural identification for the single\nstrong emission line is Ly$\\alpha$. This identification places the object\nat $z=3.36$. We find no other emission lines in the spectrum, which may\nbe taken as support for this identification, since we might expect to \nsee [OIII]~4959, 5007 and H$\\beta$ lines if the observed line at 5301\\AA\\ is\nidentified as [OII]~3727 . If the Ly$\\alpha$ identification is correct,\nthen the red magnitude of AB$_{606+814} = 27.2$ corresponds to a \nrest frame luminosity of about $L_{1600} = 4.8 \\times 10^{39}~\n{\\rm ergs}~ {\\rm s^{-1}} {\\rm \\AA^{-1}}$ for $q_{0} = 0.5$,\nand $1.5 \\times 10^{40}~\n{\\rm ergs}~ {\\rm s^{-1}} {\\rm \\AA^{-1}}$ for $q_{0} = 0.05$,\nwith H$_{0} = 75$  $\\,{\\rm km\\,s^{\\scriptscriptstyle -1}}$ Mpc$^{-1}$ in both cases.\nFor comparison, a star formation rate of 1 \\hbox{$\\thinspace M_{\\odot}$} ${\\rm yr}^{-1}$\nwith a Salpeter IMF gives produces roughly $L_{1600} = 10^{40}~ \n{\\rm ergs}~ {\\rm s^{-1}} {\\rm \\AA^{-1}}$ (e.g. Leitherer, Robert, \\& Heckman\n1995). \n\n\\subsection {L3.2 (J123656+6212210)}\n\n\tThe extracted spectrum of this pair of objects is shown\nin Figure 5. Because the separation between the red ``elliptical'' \nand the blue ``arc'' is only about $0\\farcs9$, the objects are not \nclearly spatially resolved in our two-dimensional spectrum. Therefore,\nthe extracted spectrum plotted in Figure 5 is a composite of both\nobjects. The composite nature of this spectrum complicates the\nidentification of the lines and the determination of the redshift\nof the objects.\n\n\tIn the one-dimensional spectrum, we tentatively identify a \nnumber of absorption features and a pair of emission lines at\n5650\\AA\\, and 5664\\AA\\\/. It is natural to associate the absorption \nfeatures and most of the continuum with the elliptical, because\nof its red colors and brighter magnitude over the range of wavelengths\ncovered by our spectrum. Similarly, it is natural to identify the\npossible emission lines with the arc, which has blue colors indicative \nof a nearly flat spectrum ($f_{\\nu} \\propto \\nu^{0}$) across the range \nof bandpasses. The continuation of the flat spectrum through the\nF300W bandpass also puts an upper limit of about 2.5 on the redshift\nof the arc, because of the absence of a Lyman break. Moreover,\nthe absence of any strong single emission line in such a blue\nspectrum (indicative of a starburst or nuclear activity), suggests\nthat the redshift is greater than about one, or we would expect to\nsee lines from either [OII] or [OIII] and H$\\beta$. \nThus, the combination of the colors and the spectrum suggests that\nthe arc is at $1.0 \\ {\\raise-0.5ex\\hbox{$\\buildrel<\\over\\sim$}}\\  z \\ {\\raise-0.5ex\\hbox{$\\buildrel<\\over\\sim$}}\\  2.5$. Given this range, our most\nprobable identification of the pair of emission lines is the MgII \ndoublet at 2798\\AA\\\/, which would place the arc at a redshift of 1.02.\nThis redshift is clearly very tentative.\n\n\tAdopting a similar approach of first constraining the range\nof reasonable redshifts from the colors, we find that the elliptical \ngalaxy has colors which are consistent with those of an elliptical at\na redshift of roughly 0.8. This redshift estimate may not be unique,\nas it might also be possible to fit the colors with a redshift around\n3.5. However, in the high redshift \ncase, the object is much brighter than an ordinary galaxy, while at \n$z = 0.8$, the luminosity of the object is close to $L_{*}$. \nDetailed photometric redshift analyses by other groups also favor\nthe lower redshift, with Cowie (1996) finding $z \\simeq 1.1$ and\nthe approach of Lanzetta, Yahil, \\& Fern\\'andez-Soto (1996) \ngiving $z \\simeq 0.72$ (Yahil 1996). At either\nredshift, we have failed to convincingly match the observed absorption \nlines with prominent stellar and interstellar absorption lines.\nFor the reasons given above, a lower redshift appears to be favored,\nbut this hypothesis remains unproven by our spectroscopy.\nTherefore, lensing remains a viable possibility in this case,\nwith our favored redshift being slightly greater than one for\nthe arc and somewhat less than one for the elliptical. However,\nfurther spectroscopy is required to confirm or reject this hypothesis.\n\n\\section{Discussion}\n\n\tOur primary result is that gravitational lensing is unlikely\nto be the cause of the configuration of two of the three candidate \nlensing systems we have studied. Specifically, we find that the system \nL4.1 has two distinct velocity components at slightly different \npositions. We have also found some evidence against gravitational \nlensing in the L3.1 system, which was identified by Hogg et al.\\ (1996) \nas the most probable candidate for lensing in the HDF.\nIn the case of the L3.2 system our data are consistent with lensing, \nbut are inconclusive.  It is therefore plausible that there \nis no more than one multiply imaging lensing system among the \napproximately 750 objects in the HDF to a limiting magnitude of roughly \nF814W$_{AB} < 27$ for the observed magnitudes of the sources.\nAlthough we have not obtained spectra for every object in the HDF, \nwe targeted the three systems which appeared to us to be the most\nlikely lensing cases, and two of these three have been similarly \nidentified by other groups (e.g.\\ Hogg et al.\\ 1996).\nTherefore, we consider it unlikely that \nthere is a large population of undiscovered multiple lensing systems\nwithin the magnitude limit given above.\n\n\tWe can estimate the observed frequency of lensing by combining \nthe number of lenses determined above with the number of potential\nsources in the HDF. As a starting point we adopt the number of objects\nwith F814W$_{AB} < 27$, which is similar to the magnitude of the fainter\ncandidates we studied. This limit gives roughly 750 objects which\nare potential lens sources. The true number of potential sources is\nlikely to be greater, since this number does not account for \nmagnification of fainter galaxies.\nTaking our observational limit of one lensing system, we derive\na rough lensing rate of around 0.000-0.002; this \nestimate clearly has large uncertainties due to the small\nnumbers involved. \n\n\tEven with the large uncertainties, it is difficult\nto reconcile these observations with the high lensing rates\nexpected in some cosmological models. In particular, cosmological\nmodels with large $\\Lambda$ in which the large numbers of faint \nblue galaxies are a normal evolving galaxy population\nappear to be inconsistent with the low rate of multiply\nimaging lensing of very faint galaxies.\nAlthough a detailed model of lensing of galaxies in the HDF is not \nyet available, it is notable that the frequency of multiply imaging \nlensing derived above is about an order of magnitude lower than that \nobserved for quasars, which itself places tight constraints on the \ncosmological constant (e.g.\\ Kochanek 1996 and references therein).\nTherefore, a high $\\Lambda$, passive evolution model for faint galaxies \nis only viable if there is some combination of effects which reduce \nthe observed frequency of galaxy lensing by more than an order of \nmagnitude compared to quasars. A factor of several difference is\nexpected from the lower magnification bias likely for galaxies\nrelative to quasars. However, the faint galaxies are observed to be \ncompact and are expected in these models to be at rather high redshift,\nboth of which make it hard to reduce the observed \nlensing rate relative to quasars by more than an order of magnitude.\nWith better \nstatistics, it may be possible to constrain other models for the faint\nblue galaxies. For example, if the lensing rate for the faint\nblue galaxies can be shown to be less than that for quasars and radio\nsources, the simplest explanation would be that the faint blue galaxies\nare at modest redshifts compared to these other source populations.\n\n\\acknowledgments\n\nThis work is based in part on observations obtained at the W.M. Keck\nObservatory, which is operated jointly by the California Institute\nof Technology and the University of California, and would not have been\npossible without the Keck telescope and the staff who operate it.\nWe also acknowledge Bob Williams and the HDF team for carrying out\nthe survey and making the data immediately available.\nWe thank Chuck Steidel for providing his spectrum of J123641+621204 in\nadvance of publication, and Drew Phillips for providing his software\nfor mask design and alignment, both of which were very useful. Aaron Barth,\nJoe Silk, and Hy Spinrad, provided useful comments during the progress \nof this work, and Len Cowie and Amos Yahil provided photometric redshifts\nin advance of publication. The suggestions of an anonymous referee improved\nthe final paper. This research is supported by NSF grant AST92-21540.\nS.E.Z. acknowledges support from NASA through grant\nnumber HF-1055.01-93A awarded by the Space Telescope Science Institute,\nwhich is operated by the Association of Universities for Research in\nAstronomy, Inc., for NASA under contract NAS5-26555.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuiver is a directed graph with multiply edges and loops allowed.\nLet $Q$ = finite quiver with $k$ vertices and $l$ arrows.\nQuiver representation $M$ is a collection of vector spaces and linear maps, attached to each vertex\nand arrow of $Q$:\n\\begin{equation}\nM=\\{V_1,\\dots,V_k\\}\\cup \\{M_{v,i,j}\\}, \\quad \\mathrm{dim}(V_i)=d_i,\\quad M_{v,i,j}=Mat(d_i\\times d_j,\\mathrm{k}): V_i \\rightarrow V_j\n\\end{equation}\nQuiver grassmannian $Gr_{\\underline{e}}(Q,M)$ of dimension vector $\\underline{e}=(e_1,\\dots,e_k)$ contains all\n$\\underline{e}$-dimensional subspaces of $M$, compatible with the underlying linear maps.\n\nFix an ordered basis $\\mathcal{B}=\\{1,2,\\dots,d_1,d_1+1,\\dots,d_l\\}$, $I\\subset B$ for all $V_i$s. Quiver relations (QR)  are\nof the form \\cite{4}:\n\\begin{equation}\\label{QR}\n\\sum_{s',t'} (-1)^{\\epsilon(s',t')}M_{v,s',t'} \\ \\Delta_s^{s'} \\ \\Delta_{t'}^t \n- \\ \\sum_{s'} (-1)^{\\epsilon(s',t)} M_{v,s',t} \\ \\Delta_{s}^{s'} \\ \\Delta_I = 0, \\quad I\\subset\\mathcal{B}, \\quad |I|=e\n\\end{equation}\n\\textbf{Define $Fset$ as a collection of all non-vanishing $\\Delta^i_j$'s in \\ref{QR}.}\nOur aim is to generate polynomials which are sums of $GL$ characters with coefficients given by the embedding\n\\begin{equation}\\label{Emb}\nGr_{\\overline(e)}(Q,M)\\xrightarrow{\\phi_1} Gr(e,d) \\xrightarrow{\\phi_2} \\mathbb{P}^{e+d-1}, \\quad e=\\sum e_i, d=\\sum d_i\n\\end{equation}\nDefine 2 kind of invariants:\n\\begin{equation}\nP(Q,s,\\mathbf{M})=\\sum\\limits_{R}\\Delta_R\\chi_R(\\overline{t})\n\\end{equation}\nwith $\\Delta=$ homogeneous coordinate on the image of embedding \\ref{Emb}, and\n\\begin{equation}\n\\tilde{P}(Q,s,\\mathbf{M})=\\sum\\limits_{R'}\\tilde{\\Delta}_{R'}\\chi_R(\\overline{t})\n\\end{equation}\nwith $\\tilde{\\Delta}\\in Fset$ only (PR are not taken into part).\nNote that $P,P'$ are inhomogeneous in $R,R'$ (may contain Young diagrams of different size).\n\\\\\n\\textbf{Proposition 1.} \\emph{In case of $\\chi_{\\lambda}=s_{\\lambda}$ polynomials $P(Q,s,\\mathbf{M})$ form a basis of Symm algebra.} This seems to be obvious, but not the only one possible choice of $\\chi$.\n\\\\\nWrite $\\mathbf{M}$ as block matrix:\n\\begin{equation}\n\\mathbf{M}= \\left[ \\begin {array}{cccc} M_{{1,1}}&M_{{1,2}}&M_{{1,3}}&\\dots\n\\\\\\noalign{\\medskip}M_{{2,1}}&M_{{2,2}}&M_{{2,3}}&\n\\\\\\noalign{\\medskip}M_{{3,1}}&M_{{3,2}}&M_{{3,3}}&\n\\\\\\noalign{\\medskip}\\vdots& & &\\ddots\\end {array} \\right] \n\\end{equation}\nwith each $M_{i,j}$ corresponding to $(i,j)$ linear map.\n\n\\section{Computation of $P,\\tilde{P}$. Quiver relations as vanishing minors}\n\n\\subsection{$Q=A_2$, $M=\\{\\mathbb{C},\\mathbb{C},\\mu=Mat(2,2,\\mathbb{C})\\},\\overline{e}=(1,1)$}\n$I=\\{2,4\\}$:\nOne non-trivial equation,  $\\#(+,-)=(2,2)$\n\\begin{equation}\nF \\left( v,2,3 \\right)=\\mu_{{v,1,4}}\\Delta_{{ \\left\\{ 1,4 \\right\\} }}\\Delta_{{ \\left\\{ 2,3\n \\right\\} }}+\\mu_{{v,2,4}}\\Delta_{{ \\left\\{ 2,4 \\right\\} }}\\Delta_{{\n \\left\\{ 2,3 \\right\\} }}-\\mu_{{v,1,3}}\\Delta_{{ \\left\\{ 1,4 \\right\\} }\n}\\Delta_{{ \\left\\{ 2,4 \\right\\} }}-\\mu_{{v,2,3}}{\\Delta_{{ \\left\\{ 2,4\n \\right\\} }}}^{2}=0\n\\end{equation}\nOr:\n\\begin{equation}\nF \\left( v,2,3 \\right)=\\left| \\begin {array}{ccc} \\mu_{{v,1,4}}&0&\\Delta_{{ \\left\\{ 2,4\n \\right\\} }}\\\\\\noalign{\\medskip}\\mu_{{v,2,4}}&\\Delta_{{ \\left\\{ 1,4\n \\right\\} }}&{\\frac {\\mu_{{v,2,3}}{\\Delta_{{ \\left\\{ 2,4 \\right\\} }}}^\n{2}}{\\mu_{{v,1,4}}\\Delta_{{ \\left\\{ 2,3 \\right\\} }}}}\n\\\\\\noalign{\\medskip}\\mu_{{v,1,3}}&\\Delta_{{ \\left\\{ 2,3 \\right\\} }}&\n\\Delta_{{ \\left\\{ 2,3 \\right\\} }}\\end {array} \\right|=\n\\left| \\begin {array}{ccc} \\mu_{{v,1,4}}&\\Delta_{{ \\left\\{ 2,4\n \\right\\} }}&\\Delta_{{ \\left\\{ 2,4 \\right\\} }}\\\\\\noalign{\\medskip}-\\mu\n_{{v,2,4}}&\\Delta_{{ \\left\\{ 1,4 \\right\\} }}&-{\\frac {\\mu_{{v,2,3}}\n\\Delta_{{ \\left\\{ 2,4 \\right\\} }}}{\\mu_{{v,1,3}}}}\\\\\\noalign{\\medskip}\n\\mu_{{v,1,3}}&0&\\Delta_{{ \\left\\{ 2,3 \\right\\} }}\\end {array} \\right| \n =0\n\\end{equation}\n\\begin{equation}\n,\\quad \\Delta_{1,2}=0,\n\\quad \\Delta_{3,4}=0\n\\end{equation}\nNote that if we inverse the arrow, our equation will change:\n\\begin{equation}\nF \\left( v,4,1 \\right) =\\mu'  _{{v,3,2}}\n\\Delta_{{ \\left\\{ 2,3 \\right\\} }}\\Delta_{{ \\left\\{ 1,4 \\right\\} }}+\n\\mu'  _{{v,4,2}}\\Delta_{{ \\left\\{ 2,4\n \\right\\} }}\\Delta_{{ \\left\\{ 1,4 \\right\\} }}-\\mu'\n  _{{v,3,1}}\\Delta_{{ \\left\\{ 2,3 \\right\\} }}\\Delta_{{\n \\left\\{ 2,4 \\right\\} }}-\\mu'  _{{v,4,1}\n}{\\Delta_{{ \\left\\{ 2,4 \\right\\} }}}^{2}=0,\n\\end{equation}\nwhere $\\mu'$ is the inverse map.\nNow fix $\\mu= \\left( \\begin {array}{cc} p&1-p\\\\\\noalign{\\medskip}1-p&p\\end {array}\n \\right) $ -- quantum permutation matrix from https:\/\/arxiv.org\/abs\/1109.4888.\nCombining this with the only Plu\"cker identity \n\\begin{equation}\n\\Delta_{{ \\left\\{ 1,2 \\right\\} }}\\Delta_{{ \\left\\{ 3,4 \\right\\} }}-\n\\Delta_{{ \\left\\{ 1,3 \\right\\} }}\\Delta_{{ \\left\\{ 2,4 \\right\\} }}+\n\\Delta_{{ \\left\\{ 1,4 \\right\\} }}\\Delta_{{ \\left\\{ 2,3 \\right\\} }}=0\n\\end{equation}\nwe obtain our polynomial invariant:\n\\begin{equation}\n\\boxed{P(A_2,4,\\mu)=Qh_{{1}}-{\\frac {Q \\left( -Wp-Q+pQ \\right) h_{{2}}}{W-Wp+pQ}}+W\n \\left( {h_{{1}}}^{2}-h_{{2}}h_{{0}} \\right) -{\\frac { \\left( -Wp-Q+pQ\n \\right) W \\left( h_{{1}}h_{{2}}-h_{{3}}h_{{0}} \\right) }{W-Wp+pQ}}}\n\\end{equation}\nwhere $Q,W$ are arbitrary complex numbers.\n\n\\subsection{$M=\\{\\mathbb{C}^2,\\mathbb{C}^2,\\mu=Mat(4,4,\\mathbb{C})\\},\\overline{e}=(2,2)$}\n$I={3,4,7,8}$: We have 2x2 non-trivial quiver equations () with $\\#(+,-)=(4,5),(6,3)$\nOne of them ()\n\\begin{equation}\nF(v,3,5)=\n \\left| \\begin {array}{ccc} \\mu_{{v,1,7}}+\\mu_{{v,2,7}}+\\mu_{{v,2,8}}&\n\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}&\\Delta_{{ \\left\\{ 3,4,5,7\n \\right\\} }}+2\\,\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}\n\\\\\\noalign{\\medskip}\\mu_{{v,3,8}}-\\mu_{{v,3,7}}-\\mu_{{v,1,5}}&\\Delta_{\n{ \\left\\{ 1,4,7,8 \\right\\} }}+2\\,\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}\n&0\\\\\\noalign{\\medskip}-\\mu_{{v,1,8}}+\\mu_{{v,2,5}}+{\\frac {\\mu_{{v,3,5\n}}\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}}{\\Delta_{{ \\left\\{ 2,4,7,8\n \\right\\} }}}}&\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}+\\Delta_{{\n \\left\\{ 1,4,7,8 \\right\\} }}&2\\,\\Delta_{{ \\left\\{ 3,4,5,8 \\right\\} }}+\n\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}\\end {array} \\right|=0\n\\end{equation}\n\\[\n\\mu_{{v,2,8}}\\Delta_{{ \\left\\{ 2,4,7,8 \\right(\\} }}\\Delta_\n{{ \\left\\{ 3,4,5,7 \\right\\} }}+\\mu_{{v,3,8}}\\Delta_{{ \\left\\{ 3,4,7,8\n \\right\\} }}\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}-\\mu_{{v,1,5}}\\Delta_\n{{ \\left\\{ 1,4,7,8 \\right\\} }}\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}\n\\]\n\\[ -\n\\mu_{{v,2,5}}\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}\\Delta_{{ \\left\\{ 3,\n4,7,8 \\right\\} }}-\\mu_{{v,3,5}}{\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}}\n^{2}\n\\]\nCan we rewrite them in a determinantal form? \n\n{cont.}\n\\begin{equation}\nF_(v,4,5)= \\left| \\begin {array}{ccc} \\mu_{{v,4,7}}&m_{2;{1,2}}&\\Delta_\n{{ \\left\\{ 3,4,5,7 \\right\\} }}\\\\\\noalign{\\medskip}0&\\Delta_{{ \\left\\{ \n3,4,7,8 \\right\\} }}&0\\\\\\noalign{\\medskip}-\\mu_{{v,4,8}}&m_{2;{3,2}}&\n\\Delta_{{ \\left\\{ 3,4,5,8 \\right\\} }}\\end {array} \\right| + \\left| \n\\begin {array}{ccc} \\mu_{{v,2,5}}&0&\\Delta_{{ \\left\\{ 3,4,5,8\n \\right\\} }}\\\\\\noalign{\\medskip}-\\mu_{{v,1,7}}&\\Delta_{{ \\left\\{ 2,3,7\n,8 \\right\\} }}&0\\\\\\noalign{\\medskip}\\mu_{{v,2,7}}&\\Delta_{{ \\left\\{ 1,\n3,7,8 \\right\\} }}&\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}\\end {array}\n \\right| +\n\\end{equation}\n\n\\[\n+ \\left| \\begin {array}{ccc} -\\mu_{{v,1,8}}&\\Delta_{{\n \\left\\{ 2,3,7,8 \\right\\} }}&\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}\n\\\\\\noalign{\\medskip}\\mu_{{v,2,8}}&\\Delta_{{ \\left\\{ 1,3,7,8 \\right\\} }\n}&0\\\\\\noalign{\\medskip}-\\mu_{{v,1,5}}&-{\\frac {\\mu_{{v,4,5}}\\Delta_{{\n \\left\\{ 3,4,7,8 \\right\\} }}}{\\mu_{{v,2,8}}}}&\\Delta_{{ \\left\\{ 3,4,5,\n7 \\right\\} }}\\end {array} \\right|=0.\n\\]\n\n{Polynomial $P(A_2,4,Id)$}\ndefine $P(Q, n, M) =$ linear combination of GL characters with coefficients running through the Fset.\n\\textbf{this turns out to be a special base of Symm.}\nThis case gives\n\\begin{equation}\nFset=\\left( \\begin {array}{cccccccc} &&\\Delta_{{ \\left\\{ 1,4,7,8\n \\right\\} }}&\\Delta_{{ \\left\\{ 1,3,7,8 \\right\\} }}&&&&\n\\\\\\noalign{\\medskip}&&\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}&\\Delta_{\n{ \\left\\{ 2,3,7,8 \\right\\} }}&&&&\\\\\\noalign{\\medskip}&&\\Delta_{{\n \\left\\{ 3,4,7,8 \\right\\} }}&&&&&\\\\\\noalign{\\medskip}&&&\\Delta\n_{{ \\left\\{ 3,4,7,8 \\right\\} }}&&&&\\\\\\noalign{\\medskip}&&&&&\n&\\Delta_{{ \\left\\{ 1,4,7,8 \\right\\} }}&-\\Delta_{{ \\left\\{ 1,3,7,8\n \\right\\} }}\\\\\\noalign{\\medskip}&&&&&&\\Delta_{{ \\left\\{ 2,4,7,8\n \\right\\} }}&-\\Delta_{{ \\left\\{ 2,3,7,8 \\right\\} }}\n\\\\\\noalign{\\medskip}&&&&&&\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}&\n\\\\\\noalign{\\medskip}&&&&&&&\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }\n}\\end {array} \\right)\n\\end{equation}\n \nNow we can write down the explicit form of $P(A_2,4,Id)$:\n\\begin{equation}\n{\\it Fset}=[\\Delta_{{ \\left\\{ 1,3,7,8 \\right\\} }},\\Delta_{{ \\left\\{ 1,\n4,7,8 \\right\\} }},\\Delta_{{ \\left\\{ 2,3,7,8 \\right\\} }},\\Delta_{{\n \\left\\{ 2,4,7,8 \\right\\} }},\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }},\n\\Delta_{{ \\left\\{ 3,4,5,8 \\right\\} }},\\Delta_{{ \\left\\{ 3,4,6,7\n \\right\\} }},\\Delta_{{ \\left\\{ 3,4,6,8 \\right\\} }},\\Delta_{{ \\left\\{ 3\n,4,7,8 \\right\\} }}]\n\\end{equation}\nAssuming all free parameters $=\\pm1$, we get the following solution:\n\\begin{equation}\n{\\it Fset}=[1, 1, 1, -1, -1, 1, -1, -1, 1]\n\\end{equation}\n\nNow choose $\\mu=\\mu(p,q)$. Solving QR's independently, we get:\n\\begin{equation}\n\\Delta_{{ \\left\\{ 1,3,7,8 \\right\\} }}=-\\frac{1}{pq}(\\Delta_{{ \\left\\{ 3,4,5,7\n \\right\\} }}-2\\,\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}q-\\Delta_{{\n \\left\\{ 1,4,7,8 \\right\\} }}p-\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}+p\n\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}+\n\\end{equation}\n\\[\n+\\Delta_{{ \\left\\{ 1,4,7,8\n \\right\\} }}pq+\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}q-p\\Delta_{{\n \\left\\{ 2,4,7,8 \\right\\} }}q-\\Delta_{{ \\left\\{ 2,3,7,8 \\right\\} }}q+p\n\\Delta_{{ \\left\\{ 2,3,7,8 \\right\\} }}q)\n\\]\n\\begin{equation}\n\\Delta_{{ \\left\\{ 3,4,5,8 \\right\\} }}=-\\frac{1}{q}(\\Delta_{{ \\left\\{ 3,4,5,7\n \\right\\} }}-\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}q-\\Delta_{{ \\left\\{ \n1,4,7,8 \\right\\} }}p-\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}+p\\Delta_{{\n \\left\\{ 2,4,7,8 \\right\\} }})\n\\end{equation}\n\\begin{equation}\n\\Delta_{{ \\left\\{ 3,4,6,7 \\right\\} }}=-\\frac{1}{p(2q-1)}(2\\,p\\Delta_{{ \\left\\{ 2,3,7,8\n \\right\\} }}q+2\\,\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}pq-2\\,p\\Delta_{{\n \\left\\{ 2,4,7,8 \\right\\} }}q-p\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}+2\n\\,p\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}-\n\\end{equation}\n\\[\n-2\\,\\Delta_{{ \\left\\{ 3,4,5,7\n \\right\\} }}q+\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}q-\\Delta_{{\n \\left\\{ 2,3,7,8 \\right\\} }}q+\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}-\n\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }})\n\\]\n\\begin{equation}\n\\Delta_{\\{3, 4, 6, 8\\}}=\\frac{1}{pq(2q-1)}(\\Delta_{{ \\left\\{ 3,4,6,8 \\right\\} }}=2\\,p\\Delta_{{ \\left\\{ 2,4,7,8\n \\right\\} }}-2\\,\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}{q}^{2}p-2\\,\n\\Delta_{{ \\left\\{ 1,4,7,8 \\right\\} }}{p}^{2}q+2\\,{p}^{2}\\Delta_{{\n \\left\\{ 2,4,7,8 \\right\\} }}q+2\\,\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}\n{q}^{2}p-\n\\end{equation}\n\\[\n-\n2\\,\\Delta_{{ \\left\\{ 2,3,7,8 \\right\\} }}{q}^{2}p-p\\Delta_{{\n \\left\\{ 3,4,5,7 \\right\\} }}+\n\\Delta_{{ \\left\\{ 1,4,7,8 \\right\\} }}{p}^\n{2}-{p}^{2}\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}+3\\,\\Delta_{{ \\left\\{ \n3,4,5,7 \\right\\} }}pq-4\\,p\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}q+\n\\]\n\\[\n+2\\,\n\\Delta_{{ \\left\\{ 1,4,7,8 \\right\\} }}pq+2\\,p\\Delta_{{ \\left\\{ 2,3,7,8\n \\right\\} }}q+2\\,\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}q-\\Delta_{{\n \\left\\{ 2,3,7,8 \\right\\} }}q+\n \\]\n \\[ \n \\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}-\n\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}+2\\,\\Delta_{{ \\left\\{ 3,4,5,7\n \\right\\} }}{q}^{2}-\n \\]\n \\[\n -\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }}{q}^{2}+\n\\Delta_{{ \\left\\{ 2,3,7,8 \\right\\} }}{q}^{2}-\\Delta_{{ \\left\\{ 1,4,7,8\n \\right\\} }}p-3\\,\\Delta_{{ \\left\\{ 3,4,5,7 \\right\\} }}q)\n \\]\nAssuming $\\left\\{ \\Delta_{{ \\left\\{ 1,4,7,8 \\right\\} }},\\Delta_{{ \\left\\{ 2,3,7\n,8 \\right\\} }},\\Delta_{{ \\left\\{ 2,4,7,8 \\right\\} }},\\Delta_{{\n \\left\\{ 3,4,5,7 \\right\\} }},\\Delta_{{ \\left\\{ 3,4,7,8 \\right\\} }}\n \\right\\}=\\{1\\}$, we can try to resolve all remaining (Plu\"cker) relations.\nIn this case\n\\begin{equation}\n{\\it Fset}=\\left[-{\\frac {2\\,q-qp}{qp}},1,1,1,1,1,-{\\frac {2\\,qp-2\\,q+p}{p\n \\left( -1+2\\,q \\right) }},{\\frac {-2\\,q+3\\,qp-2\\,{q}^{2}p+2\\,{q}^{2}}\n{pq \\left( -1+2\\,q \\right) }},1\\right]\n\\end{equation} \nNow we're almost ready to write our target polynomial. The only thing left is to convert\nthe indices ${i_1,i_2,\\dots,i_s}$ to the corresponding Young diagram. In order to do this,\nexpand the following determinant by Cauchy-Binet formula:\nWe have:\n\\begin{equation}\n\\Delta_{\\{1,3,7,8\\}}\\simeq\\left| \\begin {array}{cccc} h_{{2}}&h_{{4}}&h_{{5}}&h_{{6}}\n\\\\\\noalign{\\medskip}h_{{1}}&h_{{3}}&h_{{4}}&h_{{5}}\n\\\\\\noalign{\\medskip}0&h_{{2}}&h_{{3}}&h_{{4}}\\\\\\noalign{\\medskip}0&h_{\n{1}}&h_{{2}}&h_{{3}}\\end {array} \\right|=s_{\\tiny\\yng(2,3,3,3)},  \n\\Delta_{\\{1,4,7,8\\}}\\simeq \n\\left| \\begin {array}{cccc} h_{{2}}&h_{{3}}&h_{{5}}&h_{{6}}\n\\\\\\noalign{\\medskip}h_{{1}}&h_{{2}}&h_{{4}}&h_{{5}}\n\\\\\\noalign{\\medskip}0&h_{{1}}&h_{{3}}&h_{{4}}\\\\\\noalign{\\medskip}0&0&h\n_{{2}}&h_{{3}}\\end {array} \\right|=s_{\\tiny\\yng(2,2,3,3)}, \\Delta_{\\{2,3,7,8\\}}\\simeq s_{\\tiny\\yng(1,3,3,3)},\n\\end{equation}\n\\[\n\\Delta_{\\{2,4,7,8\\}}\\simeq s_{\\tiny\\yng(1,2,3,3)}, \\Delta_{\\{3,4,5,7\\}}\\simeq s_{\\tiny\\yng(1,1,4,5)},\n\\Delta_{\\{3,4,5,8\\}}\\simeq s_{\\tiny\\yng(1,1,4,4)},\\Delta_{\\{3,4,6,7\\}}\\simeq s_{\\tiny\\yng(1,1,3,5)},\n\\Delta_{\\{3,4,6,8\\}}\\simeq s_{\\tiny\\yng(1,1,3,4)}, \\Delta_{\\{3,4,7,8\\}}\\simeq s_{\\tiny\\yng(1,1,3,3)}.\n\\]\n\n\\subsection{$Q=A_2,d_i=6,e_i=3, \\mu=\\sigma(p,q,6)$}\nTake 2 vertices $v_1,v_2$ with $V_1=V_2=\\mathbb{C}^6$, \n\\begin{equation}\nI=\\left\\{\\underbrace{ 4,5,6},\\underbrace{10,11,12} \\right\\} \n\\end{equation}\nRecall that each $\\Delta$ is a $(6,6)$ complex minor.\nThe following minors are supposed to vanish:\n\\begin{equation}\n\\{ \\Delta_{{ \\left\\{ 1,4,5,6,10,11 \\right\\} }},\\Delta_{{\n \\left\\{ 1,4,5,6,10,12 \\right\\} }},\\Delta_{{ \\left\\{ 1,4,5,6,11,12\n \\right\\} }},\\Delta_{{ \\left\\{ 2,4,5,6,10,11 \\right\\} }},\\Delta_{{\n \\left\\{ 2,4,5,6,10,12 \\right\\} }},\n\\end{equation}\n\\[\n\\Delta_{{ \\left\\{ 2,4,5,6,11,12\n \\right\\} }},\\Delta_{{ \\left\\{ 3,4,5,6,10,11 \\right\\} }},\\Delta_{{\n \\left\\{ 3,4,5,6,10,12 \\right\\} }},\\Delta_{{ \\left\\{ 3,4,5,6,11,12\n \\right\\} }} \\} =\\{0\\}\n\\]\nTotal $\\#QR=3^3$ . (3 for each $i in \\{4,5,6\\}$: $F(v,i,j)$).\nFor arbitrary $\\mu$ these relations look really huge!\n\nThe choice $\\mu=Id(6)$ kills most of the $Fset$ entries, so we can write \n$Fset$ as $(12,12)$ sparse matrix with $\\{0,\\pm1\\}$ entries of the shape\n(empty space is filled with zeroes):\n\\begin{verbatim}\nFset=Matrix([seq(seq(Fset[1+i .. 12+i], i = 12*j), j = 0 .. 11)])\n\\end{verbatim}\n\\begin{equation}\nFset={\\tiny{ \\left( \\begin {array}{cccccccccccc} &&&1&-1&1&&&&&&\n\\\\\\noalign{\\medskip}&&&1&-1&1&&&&&&\\\\\\noalign{\\medskip}&&&\n1&-1&1&&&&&&\\\\\\noalign{\\medskip}&&&1&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&1&&&&&&&\\\\\\noalign{\\medskip}&&&\n&&1&&&&&&\\\\\\noalign{\\medskip}&&&&&&&&&1&1&1\n\\\\\\noalign{\\medskip}&&&&&&&&&1&1&1\\\\\\noalign{\\medskip}&&&\n&&&&&&1&1&1\\\\\\noalign{\\medskip}&&&&&&&&&1&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&1&\\\\\\noalign{\\medskip}&&&\n&&&&&&&&1\\end {array} \\right)}}\n\\end{equation}\nFinally, convert all survived $Fset$ entries to corresponding Young diagrams to get out\ntarget polynomial:\n\\begin{equation}\n\\tilde{P}(A_2,6,Id)=s_{\\tiny\\yng(2,2,2,4,4,4)}-s_{\\tiny\\yng(2,2,3,4,4,4)}+s_{\\tiny\\yng(2,2,4,4,4,4)}+\ns_{\\tiny\\yng(1,2,2,4,4,4)}-s_{\\tiny\\yng(1,2,3,4,4,4)}+s_{\\tiny\\yng(1,2,4,4,4,4)}+\\dots,\n\\end{equation}\ntotal $\\#(0,+,-)=(120,21,3)$\n\n\\subsection{$Q=A_3,d_i=2,\\mu_{i=1..2}=Mat(2,2,\\mathbb{C}),\\overline{e}=(1,1,1)\\}$}\nNow let's add one more vertex:\n\\begin{equation}\nQ=A_3:\\quad \\circ \\xleftarrow{\\mu_{1}} \\circ \\xleftarrow{\\mu_2} \\circ, \\quad \\mu_i\n\\end{equation}\n\\begin{equation}\nGr_{(1,1,1)}(A_3,M)=\n\\end{equation}\nChoose $I=\\{2,4,6\\}$, so now we have the embedding\n\\begin{equation}\nGr_{(1,1,1)}(A_3,M)\\longrightarrow Gr(3,6) \\longrightarrow \\mathbb{P}^{6+3-1}\n\\end{equation}\nFor $v=1$ we have the following non-trivial quiver relations:\n\\begin{equation}\nF(1,2,3)=\\mu_{{1,1,4}}\\Delta_{{ \\left\\{ 1,4,6 \\right\\} }}\\Delta_{{ \\left\\{ 2,3,\n6 \\right\\} }}+\\mu_{{1,2,4}}\\Delta_{{ \\left\\{ 2,4,6 \\right\\} }}\\Delta_{\n{ \\left\\{ 2,3,6 \\right\\} }}-\\mu_{{1,1,3}}\\Delta_{{ \\left\\{ 1,4,6\n \\right\\} }}\\Delta_{{ \\left\\{ 2,4,6 \\right\\} }}-\\mu_{{1,2,3}}{\\Delta_{\n{ \\left\\{ 2,4,6 \\right\\} }}}^{2}=0\n\\end{equation}\nNote that if $\\mu=Id$, the solution looks as follows:\n\\begin{equation}\nFset= \\left( \\begin {array}{cccccc} &\\Delta_{{ \\left\\{ 1,4,6 \\right\\} }}&\n&\\Delta_{{ \\left\\{ 1,2,6 \\right\\} }}&&\\Delta_{{ \\left\\{ 1,2,4\n \\right\\} }}\\\\\\noalign{\\medskip}&1&&&&\\\\\\noalign{\\medskip}&&&\n\\Delta_{{ \\left\\{ 1,4,6 \\right\\} }}&&\\\\\\noalign{\\medskip}&&&1&&\n\\\\\\noalign{\\medskip}&&&&&\\Delta_{{ \\left\\{ 1,4,6 \\right\\} }}\n\\\\\\noalign{\\medskip}&&&&&1\\end {array} \\right),\n\\end{equation}\nwhere all empty cells are zeroes.\n\n\\subsection{$Q=A_3, d_i=4, \\mu_1=\\sigma(p,q,4),\\mu_2=Id$}\nInitial data:\n\\begin{equation}\n\\mathcal{B}=\\{1,2,\\dots,12\\}, I=\\{3,4,7,8,11,12\\}\n\\end{equation}\n\\begin{equation}\n \\mu_{{1}}= \\left(\\begin {array}{cccc} p&1-p&0&0\n\\\\\\noalign{\\medskip}1-p&p&0&0\\\\\\noalign{\\medskip}0&0&q&1-q\n\\\\\\noalign{\\medskip}0&0&1-q&q\\end {array} \\right) ,\\mu_{{2}}= \\left( \n\\begin {array}{cccc} 1&0&0&0\\\\\\noalign{\\medskip}0&1&0&0\n\\\\\\noalign{\\medskip}0&0&1&0\\\\\\noalign{\\medskip}0&0&0&1\\end {array}\n \\right)\n\\end{equation}\nThis case reads\n\\begin{equation}\n{\\it Fset'}=\\left\\{1,\\Delta_{{ \\left\\{ 2,3,4,7,8,11 \\right\\} }},\\Delta_{{\n \\left\\{ 2,3,4,7,8,12 \\right\\} }},\\Delta_{{ \\left\\{ 3,4,7,8,9,11\n \\right\\} }},\\Delta_{{ \\left\\{ 3,4,7,8,10,11 \\right\\} }},-{\\frac {-q+2\n\\,qp}{q \\left( -1+2\\,p \\right) }},-{\\frac {2\\,{p}^{2}q-qp}{qp \\left( -\n1+2\\,p \\right) }}\\right\\}\n\\end{equation}\n\n\n\\section{Diagrammatic notation and exact formulas for invariants}\n\n\\textbf{$\\tilde{P}(A_n,2k,\\sigma)$:}\n\\begin{equation}\n\\tilde{P}(A_2,4,\\sigma)={\\frac { \\left( -2\\,q-qp \\right)  \\left( {h_{{3}}}^{3}h_{{2}}-{h_{{3}}\n}^{2}h_{{1}}h_{{4}}-2\\,h_{{3}}h_{{4}}{h_{{2}}}^{2}+2\\,{h_{{4}}}^{2}h_{\n{2}}h_{{1}}-h_{{4}}h_{{5}}{h_{{1}}}^{2}+h_{{5}}{h_{{2}}}^{3}-h_{{6}}{h\n_{{2}}}^{2}h_{{1}}+h_{{6}}h_{{3}}{h_{{1}}}^{2} \\right) }{qp}}+\n\\end{equation}\n\\[\n+{h_{{3}}\n}^{2}{h_{{2}}}^{2}-h_{{5}}h_{{3}}{h_{{1}}}^{2}-h_{{4}}{h_{{2}}}^{3}+2\n\\,h_{{5}}{h_{{2}}}^{2}h_{{1}} \n-h_{{6}}h_{{2}}{h_{{1}}}^{2}-2\\,h_{{3}}h_{{4}}h_{{2}}h_{{1}}+2\\,{h_{{4\n}}}^{2}{h_{{1}}}^{2}+{h_{{3}}}^{2}h_{{2}}h_{{1}}-h_{{3}}h_{{4}}{h_{{1}\n}}^{2}+\n\\]\n\\[\n{\\frac { \\left( -2\\,qp-2\\,q+p \\right)  \\left( h_{{5}}h_{{3}}{h_{{1}}}^\n{2}-h_{{6}}h_{{2}}{h_{{1}}}^{2} \\right) }{p \\left( 2\\,q-1 \\right) }}+{\n\\frac { \\left( -2\\,q+3\\,qp-2\\,{q}^{2}p+2\\,{q}^{2} \\right)  \\left( h_{{\n3}}h_{{4}}{h_{{1}}}^{2}-h_{{5}}h_{{2}}{h_{{1}}}^{2} \\right) }{pq\n \\left( 2\\,q-1 \\right) }}-\n\\]\n\\[\n-h_{{4}}{h_{{2}}}^{2}h_{{1}}+h_{{5}}h_{{2}}{h_{{1}}}^{2}+h_{{4}}\nh_{{5}}{h_{{1}}}^{2}-h_{{6}}h_{{3}}{h_{{1}}}^{2}+{h_{{3}}}^{2}{h_{{1}}}^{2}-h_{{4}}h_{{2}}{h_{{1}}}^{2}\n\\]\n\nWhat is the asymptotic formula for $\\#+,-$ in $Fset(Id,m)$, when $m\\to \\infty$ ? The first few terms are\nshown on the figure:\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=0.3]{signs}\n\\caption{Convergence of $\\{\\frac{\\#+}{\\#-}\\}$ sequence, $A_2$ quiver, $\\mu=Id,\\mathrm{dim}(M)=i$}\n\\end{figure}\n$Fset$ matrix arrangement for $A_4$ quiver with $d_i=4,\\mu_i=Id$:\n\\begin{equation}\nFset={ \\left( \\begin {array}{cccccccccccccccc} && x_{1}& x_{2}&&&\nx_{3}& x_{4}&&& x_{5}& x_{6}&&& x_{7}& x_{8}\n\\\\\\noalign{\\medskip}&& x_{1}& x_{11}&&& x_{12}& x_{13}&&\n& x_{14}& x_{15}&&& x_{16}& x_{17}\\\\\\noalign{\\medskip}&&1&\n&&&&&&&&&&&&\\\\\\noalign{\\medskip}&&&1&&&&&&&&&&\n&&\\\\\\noalign{\\medskip}&&&&&& x_{1}&- x_{2}&&& x_{18}&\n x_{19}&&& x_{2}& x_{21}\\\\\\noalign{\\medskip}&&&&&& \nx_{1}&- x_{11}&&& x_{22}& x_{23}&&& x_{24}& x_{25}\n\\\\\\noalign{\\medskip}&&&&&&1&&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&1&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&& x_{1}& x_{2}&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&& x_{1}& x_{11}&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&1&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&&1&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&&&&& x_{1}&- x_{2}\n\\\\\\noalign{\\medskip}&&&&&&&&&&&&&& x_{1}&- x_{11}\n\\\\\\noalign{\\medskip}&&&&&&&&&&&&&&1&\n\\\\\\noalign{\\medskip}&&&&&&&&&&&&&&&1\\end {array}\n \\right)}\n\\end{equation}\nwhere $x_j=\\Delta_{I(j)}$. Compare this to $A_2,d_i=8,\\mu_i=Id$ case:\n\\begin{equation}\n\\tiny\n{\\left( \\begin {array}{cccccccccccccccc} &&&&x_{{1}}&-x_{{2}}&x_{{\n3}}&-x_{{4}}&&&&&&&&\\\\\\noalign{\\medskip}&&&&x_{{5}}&x_{{6}\n}&x_{{7}}&x_{{8}}&&&&&&&&\\\\\\noalign{\\medskip}&&&&x_{{9}}&x\n_{{1}}&x_{{11}}&x_{{12}}&&&&&&&&\\\\\\noalign{\\medskip}&&&&x\n_{{13}}&x_{{14}}&x_{{15}}&x_{{16}}&&&&&&&&\\\\\\noalign{\\medskip}\n&&&&1&&&&&&&&&&&\\\\\\noalign{\\medskip}&&&&&1&&&&&\n&&&&&\\\\\\noalign{\\medskip}&&&&&&1&&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&1&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&&&x_{{1}}&x_{{2}}&x_{{3}}&x_\n{{4}}\\\\\\noalign{\\medskip}&&&&&&&&&&&&x_{{5}}&-x_{{6}}&x_{{\n7}}&-x_{{8}}\\\\\\noalign{\\medskip}&&&&&&&&&&&&x_{{9}}&-x_{{\n1}}&x_{{11}}&-x_{{12}}\\\\\\noalign{\\medskip}&&&&&&&&&&&&x_{\n{13}}&-x_{{14}}&x_{{15}}&-x_{{16}}\\\\\\noalign{\\medskip}&&&&&&&&\n&&&&1&&&\\\\\\noalign{\\medskip}&&&&&&&&&&&&&1&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&&&&&1&\n\\\\\\noalign{\\medskip}&&&&&&&&&&&&&&&1\\end {array}\n \\right)}\n\\end{equation}\n\n\\underline{Totally positive} $Fset$ arrangement for $A_4 \\times {\\mathrm{mutations}},d_i=6,\\mu=Id$:\n\\begin{equation}\n \\left( \\begin {array}{cccccccccccccccccccccccc} &&&1&\n&1&&&&1&1&1&&&&1&1&1&&&&1&1&1\n\\\\\\noalign{\\medskip}&&&1&&1&&&&1&1&1&&\n&&1&1&1&&&&1&1&1\\\\\\noalign{\\medskip}&&\n&1&&1&&&&1&1&1&&&&1&1&1&&&&1&1\n&1\\\\\\noalign{\\medskip}&&&1&&&&&&&\n&&&&&&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&1&&&&&&\n&&&&&&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&1&&&&&\n&&&&&&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&1&\n&1&&&&1&1&1&&&&1&1&1\\\\\\noalign{\\medskip}\n&&&&&&&&&1&&1&&&&1\n&1&1&&&&1&1&1\\\\\\noalign{\\medskip}&&&&&\n&&&&1&&1&&&&1&1&1&&&&1&1&1\n\\\\\\noalign{\\medskip}&&&&&&&&&1&\n&&&&&&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n1&&&&&&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&1&&&&&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&1&&1&&&&1&1&1\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&1&&1&&&&1&1&1\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&1&&1&&&&1&1&1\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&1&&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&&1&&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&&&1&&&&&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&&&&&&&1&&1\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&&&&&&&1&&1\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&&&&&&&1&&1\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&&&&&&&1&&\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&&&&&&&&1&\n\\\\\\noalign{\\medskip}&&&&&&&&&&\n&&&&&&&&&&&&&1\n\\end {array} \\right)\n\\end{equation}\n\\section{\u2022}\n\\scalebox{0.5}{\\parbox{\\linewidth}{\n\\begin{equation}\n\\begin{aligned}\n(2,2)=& \\left( s_{\\tiny\\yng(2,2)}+s_{\\tiny\\yng(1,3)} \\right) \\Delta_{\\tiny\\yng(2,2)} & jsh=\\{{\\tiny\\yng(1)}\\} \\\\\n(3,4)=&\\left( s_{\\tiny\\yng(2,2,3,3,5,5)}+s_{\\tiny\\yng(1,1,4,4,5,5)}+s_{\\tiny\\yng(1,1,3,3,6,6)} \\right) \n\\Delta_{\\tiny\\yng(1,1,3,3,6,6)}+ \\left( s_{\\tiny\\yng(1,2,3,3,5,5)}+s_{\\tiny\\yng(1,1,3,4,5,5)}+s_{\n\\tiny\\yng(1,1,3,3,5,6)} \\right) \\Delta_{\\tiny\\yng(1,2,3,3,5,5)} & jsh=\\{{\\tiny\\yng(1),\\yng(1,1)}\\} \\\\\n(4,6)=& \n \\left( s_{\\tiny\\yng(1,1,2,4,4,4,7,7,7,10,10,10)}+s_{\\tiny\\yng(1,1,1,4,4,5,7,7,7,10,10,10)}\n+s_{\\tiny\\yng(1,1,1,4,4,4,7,7,8,10,10,10)}+s_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,10,11)}\n \\right) \\Delta_{\\tiny\\yng(1,1,1,4,4,5,7,7,7,10,10,10)}+ \\left( s_{\\tiny\\yng(1,1,1,4,4,7,7,7,7,10,10,10)}+\ns_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,10,13)}+s_{\\tiny\\yng(1,1,1,4,4,4,7,7,10,10,10,10)}+\ns_{\\tiny\\yng(1,1,4,4,4,4,7,7,7,10,10,10)} \\right) \\Delta_{\\tiny\\yng(1,1,1,4,4,7,7,7,7,10,10,10)}+ \\\\\n&\n\\left( s_{\\tiny\\yng(1,2,2,4,4,4,7,7,7,10,10,10)}+s_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,11,11)}+\ns_{\\tiny\\yng(1,1,1,4,5,5,7,7,7,10,10,10)}+s_{\\tiny\\yng(1,1,1,4,4,4,7,8,8,10,10,10)} \\right) \n\\Delta_{\\tiny\\yng(1,1,1,4,5,5,7,7,7,10,10,10)}+\n \\left( s_{\\tiny\\yng(1,1,1,4,4,4,7,8,10,10,10,10)}+s_{\\tiny\\yng(1,2,4,4,4,4,7,7,7,10,10,10)}+\ns_{\\tiny\\yng(1,1,1,4,5,7,7,7,7,10,10,10)}+s_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,11,13)} \\right)\n \\Delta_{\\tiny\\yng(1,1,1,4,5,7,7,7,7,10,10,10)}+ \\\\\n &\n \\left( s_{\\tiny\\yng(2,2,2,4,4,4,7,7,7,10,10,10)}+\ns_{\\tiny\\yng(1,1,1,4,4,4,8,8,8,10,10,10)}+s_{\\tiny\\yng(1,1,1,5,5,5,7,7,7,10,10,10)}+\ns_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,11,11,11)} \\right) \\Delta_{\\tiny\\yng(1,1,1,5,5,5,7,7,7,10,10,10)}+ \n\\left( s_{\\tiny\\yng(2,2,4,4,4,4,7,7,7,10,10,10)}+s_{\\tiny\\yng(1,1,1,5,5,7,7,7,7,10,10,10)}+\ns_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,11,11,13)}+s_{\\tiny\\yng(1,1,1,4,4,4,8,8,10,10,10,10)} \\right)\n \\Delta_{\\tiny\\yng(1,1,1,5,5,7,7,7,7,10,10,10)}\n& jsh=\\{{\\tiny\\yng(1),\\yng(1,1),\\yng(1,3),\\yng(3),\\yng(1,1,1),\\yng(1,1,3)}\\} \\\\\n(5,8)=&\n\\left( s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,10,13,13,13,13,17,17,17,17)}+\ns_{\\tiny\\yng(1,1,1,2,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17)}+\ns_{\\tiny\\yng(1,1,1,1,5,5,5,6,9,9,9,9,13,13,13,13,17,17,17,17)}+\ns_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,14,17,17,17,17)}+\ns_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,18)} \\right) \n\\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,14,17,17,17,17)}+\\dots \n& jsh=\\{{\\tiny\\yng(1),\\yng(1,1),\\dots,\\yng(1,1,1,1),\\yng(1,1,1,3)}\\}\n\\\\\n\\end{aligned}\n\\end{equation}\n}}\n\n\\scalebox{.2}{\n\\tiny\\yng(1,1,1,2,2,5,7,7,7,7,7,7,13,13,13,13,13,13,19,19,19,19,19,19,25,25,25,25,25,25,31,31,31,31,31,31,37,37,37,37,37,37,43,43,43,43,43,43,49,49,49,49,49,49,55,55,55,55,55,55,61,61,61,61,61,61,67,67,67,67,67,67)}\n\n$\\frac{1}{2}(\\tilde{P}_{+}-\\tilde{P}_{-})$ for cyclic 6-vertex graph:\n\n\\scalebox{0.5}{\\parbox{\\linewidth}{\n\\begin{equation}\n \\left( -s_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,10,10,13,13,16,16,16,16)}+\ns_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,10,13,13,13,13,16,16,16)} \\right)\n \\Delta_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,10,13,13,13,13,16,16,16)}+ \n\\left( s_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,11,13,13,13,13,16,16,16)}-\ns_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,10,10,13,14,16,16,16,16)} \\right)\n \\Delta_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,11,13,13,13,13,16,16,16)}+\n \\left( -s_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,10,10,10,14,14,16,16,16,16)}+\ns_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,11,11,13,13,13,13,16,16,16)} \\right) \n\\Delta_{\\tiny\\yng(1,1,1,4,4,4,7,7,7,11,11,13,13,13,13,16,16,16)}\n\\end{equation}\n}}\n\\newpage\n\\textbf{Lemma 1.} \\emph{Asymptotic formula for $\\tilde{P}_{\\mathrm{max}}(Q_{\\mathrm{univ}}(n),n,Id)$:}\n\\begin{itemize}\n\\item \\emph{$m \\times m$ ``cubic ladder'' decomposition $\\simeq \\Delta$-summands in $\\tilde{P_{I}}$}\n\\item \\emph{each $\\Delta$-summand consists of Young diagrams factorized by the ``ladder-block'' addition}\n\\end{itemize}\n\nThe formula:\n\\begin{equation}\n\\boxed{\\tilde{P}_{\\mathrm{max}}(Q(m),n,Id)=\\sum\\limits_{i=1..m\\cdot\\mathrm{cl}\\left(\\frac{m}{2}\\right)}\\sum\\limits_{j=1..n}s_{\\lambda(i,j)}\\Delta_{\\lambda(i,*)},}\n\\end{equation}\n\n$\\frac{1}{2}(\\tilde{P}_{+}-\\tilde{P}_{-})$ for cyclic 8-vertex graph:\n\n\\scalebox{0.22}{\\parbox{\\linewidth}{\n\\begin{equation}\n\\begin{aligned}\n\\left( -s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,18,18,18,20,21,21,21,21,25,25,25,25,29,29,29,29)}+s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17,22,22,22,24,25,25,25,25,29,29,29,29)} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,15,17,17,17,17,21,21,21,21,25,25,25,25,29,29,29,29)}+ \\left( s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17,21,21,21,21,26,26,26,28,29,29,29,29)}-s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,18,18,18,18,21,21,21,21,25,25,25,25,29,29,29,29)} \\right) \n\\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,17,17,17,17,17,21,21,21,21,25,25,25,25,29,29,29,29)}+\n \\left( s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17,21,21,21,24,25,25,25,25,29,29,29,29)}-s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,20,21,21,21,21,25,25,25,25,29,29,29,29)} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,14,15,17,17,17,17,21,21,21,21,25,25,25,25,29,29,29,29)}+\n  \\left( s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17,21,21,21,21,25,25,25,28,29,29,29,29)}-\n s_{\\tiny\\yng(\n1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,18,21,21,21,21,25,25,25,25,29,29,29,29)} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,14,17,17,17,17,17,21,21,21,21,25,25,25,25,29,29,29,29)}+\n&\n\\\\\n \\left( s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17,21,21,22,24,25,25,25,25,29,29,29,29)}-s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,18,20,21,21,21,21,25,25,25,25,29,29,29,29)} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,14,14,15,17,17,17,17,21,21,21,21,25,25,25,25,29,29,29,29)}+ \\left( \ns_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17,21,21,21,21,25,25,26,28,29,29,29,29)}-s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,18,18,21,21,21,21,25,25,25,25,29,29,29,29)} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,14,14,17,17,17,17,17,21,21,21,21,25,25,25,25,29,29,29,29)}+\n \\left( s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17,21,22,22,24,25,25,25,25,29,29,29,29)}-s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,18,18,20,21,21,21,21,25,25,25,25,29,29,29,29)} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,14,14,14,15,17,17,17,17,21,21,21,21,25,25,25,25,29,29,29,29)}+\n \\left( s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,17,17,17,21,21,21,21,25,26,26,28,29,29,29,29)}-s_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,13,13,13,13,17,18,18,18,21,21,21,21,25,25,25,25,29,29,29,29)}\n \\right) \\Delta_{\\tiny\\yng(1,1,1,1,5,5,5,5,9,9,9,9,14,14,14,17,17,17,17,17,21,21,21,21,25,25,25,25,29,29,29,29)}\n\\end{aligned}\n\\end{equation}\n}}\n\\newpage\n\\section{Re-evaluation of $\\tilde{P}$ for Coxeter quivers. ``Vertex at $\\infty$''}\nLet $Q$ be a quiver with $m$ vertices and all possible arrows (but only 2 opposite arrows for each pair of vertices\nis allowed, so the direction $v=\\pm 1$) .\nChoose $n,e\\in \\mathbb{N}$ and the integer-valued vectors\n\\begin{equation}\nB=\\{1,\\dots,m\\cdot{}n\\}, I=\\{jn-i+1\\}_{i=1..[n\/2],j=1..m}\\subset B,\n|I|=e,\n\\end{equation}\nwhere $[x] \\text{ is the integer part of } x$.\nFix the representation $R(Q)$: vertices $\\sim V_i(\\mathbb{Z}^n)$, arrows $\\sim$ transposition matrices $M_{v,i,j}=\\sigma(\\lambda_{i,j})^{\\mathrm{sign}(v)}$. \n\nEach $\\sigma(\\lambda_{i,j})$ is a $(n \\times n)$ transposition matrix, partitioned by $\\lambda_{i,j}$ for the $(i,j)$-th arrow.\nWe fix \\emph{the particular choice} $\\boxed{R=\\Sigma_0}$:\n\\begin{equation}\n(i,j)\\rightarrow\\lambda_{i,j}: (1,4)\\rightarrow {\\tiny\\yng(1,2,3,4)}, (1,3)\\rightarrow{\\tiny\\yng(1,2,3,3)},  (1,2)\\rightarrow{\\tiny\\yng(1,2,2,2)},  (2,4)\\rightarrow{\\tiny\\yng(2,2,3,4)}, (2,3)\\rightarrow{\\tiny\\yng(2,2,3,3)},\\dots\n\\end{equation}\nAssume also $\\mathrm{dim(Gr}(Q,R))=\\boxed{\\overline{e}=\\left[\\frac{1}{2}\\overline{n}\\right]}$ for the corresponding quiver grassmannian \\cite{2,4}, $e_i=e_j\\ \\forall i,j$.\nFor instance, the case $(m,n)=(4,4)$ is given by\n\\begin{equation}\n\\mathbf{M}_{+}={\\tiny  \\left( \\begin {array}{cccc} 0& \\left[ \\begin {array}{cccc} 1&0&0&0\n\\\\\\noalign{\\medskip}0&0&1&0\\\\\\noalign{\\medskip}0&0&0&1\n\\\\\\noalign{\\medskip}0&1&0&0\\end {array} \\right] & \\left[ \n\\begin {array}{cccc} 1&0&0&0\\\\\\noalign{\\medskip}0&1&0&0\n\\\\\\noalign{\\medskip}0&0&0&1\\\\\\noalign{\\medskip}0&0&1&0\\end {array}\n \\right] & \\left[ \\begin {array}{cccc} 1&0&0&0\\\\\\noalign{\\medskip}0&1&0\n&0\\\\\\noalign{\\medskip}0&0&1&0\\\\\\noalign{\\medskip}0&0&0&1\\end {array}\n \\right] \\\\\\noalign{\\medskip}0&0& \\left[ \\begin {array}{cccc} 0&1&0&0\n\\\\\\noalign{\\medskip}1&0&0&0\\\\\\noalign{\\medskip}0&0&0&1\n\\\\\\noalign{\\medskip}0&0&1&0\\end {array} \\right] & \\left[ \n\\begin {array}{cccc} 0&1&0&0\\\\\\noalign{\\medskip}1&0&0&0\n\\\\\\noalign{\\medskip}0&0&1&0\\\\\\noalign{\\medskip}0&0&0&1\\end {array}\n \\right] \\\\\\noalign{\\medskip}0&0&0& \\left[ \\begin {array}{cccc} 0&1&0&0\n\\\\\\noalign{\\medskip}0&0&1&0\\\\\\noalign{\\medskip}1&0&0&0\n\\\\\\noalign{\\medskip}0&0&0&1\\end {array} \\right] \\\\\\noalign{\\medskip}0&0\n&0&0\\end {array} \\right)\n }\n\\end{equation}\nand $\\mathbf{M}_{-}$ equals by-block inverse of $\\mathbf{M}_{+}$.\nChoose some path $x\\in \\mathcal{P}(Q)$ and solve the quiver relations with vanishing conditions \\cite{4}:\n\\begin{equation}\n\\begin{aligned}\n\\quad &\\sum_{s',t'} (-1)_{\\epsilon(s',t')}M_{v,s',t'} \\ \\Delta_s^{s'} \\ \\Delta_{t'}^t \n- \\ \\sum_{s'} (-1)_{\\epsilon(s',t)} M_{v,s',t} \\ \\Delta_{s}^{s'} \\ \\Delta_I = 0, \\\\\n& \\Delta_J=0 \\text{ for some non-proper } J\\text{'s}\n\\end{aligned}\n\\end{equation}\nDefine symmetric polynomials\n\\begin{equation}\n\\boxed{\\tilde{P}_R(x)=\\sum\\limits_{i,j}\\Delta_{\\mu_{i,i}}s_{\\mu_{i,j}}(p_1,p_2,\\dots)},\n\\end{equation}\nwhere $\\mu_{i,j}$ is a partition derived from the Cauchy-Binet expansion for each $\\Delta$'s, $s_{\\mu}$ is a Schur function.\n\n\\textbf{Prop 1.}\n\\emph{The map $f:x \\mapsto \\tilde{P}_R(x)$ is a homomorphism on a ``proper'' subset $S\\subseteq\\mathcal{P}$, s.t.\nfor any $x,y \\in S$ }\n\\begin{equation}\n\\tilde{P}_R(x\\cdot y)=\\frac{1}{2}\\sum_{\\mu,\\eta,\\nu}\\left(\\mathrm{coeff}(\\tilde{P}_R(x),\\Delta_{\\mu})-\\mathrm{coeff}(\\tilde{P}_R(y),\\Delta_{\\eta})\\right)\\Delta_{\\nu}\\simeq_{|_{Res(\\Delta)}}\\frac{1}{2}\\left( \\tilde{P}_R(x)-\\tilde{P}_R(y) \\right)\n\\end{equation}\n\\underline{Note:} this construction is very sensitive to the choice of $R$ (if we swap the transposition matrices, $S$ will change).\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=0.35]{Screenshot1.png}\n\\caption{Examples of ``good'' (left) and ``bad'' (right) combinations of paths in $S\\subseteq\\mathcal{P}$\nfor $(m,n)=(10,10)$.\n``Good'' means the image of $f$ is not empty}\n\\end{figure}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=0.3]{Screenshot2.png}\n\\caption{Examples of a non-trivial loops \\emph{spiral type 1} (Coxeter odd type $\\tilde{A}_n$) for $(m,n)=(6,6),(8,8),(10,10),\\dots$}\n\\end{figure}\n\n\\textbf{Lemma 2.} \\emph{If $x\\in\\mathcal{P}(Q)$ is of spiral type 1, then }\n\\begin{equation}\n\\tilde{P}_{\\pm}(x)\\equiv 0,\n\\end{equation}\n\\emph{where ``$\\pm$'' means that we take a doubled path with $v=\\pm1$}.\n\\emph{This lemma can be improved if we take all irreducible subsets of the set of all arrows of $Q$, on which our polynomial vanishes.}\nFor instance, each minimal null-set for $4$-vertex quiver is trivial,\n$6$-vertex: $\\{(1,6),(3,4),(1,5),(3,6)\\}$ - is not minimal, but irreducible\n(recall that in this case orientation does not matter),\n$8$-vertex: $\\{(2,7),(3,6),(2,6),(3,5)\\}$,\n$12$-vertex: $\\{(5,8),(3,10),(4,9),(4,8),(5,7),(3,9)\\}$ and so on. \nLet's write the corresponding symmetric polynomials explicitly:\n\\begin{equation}\n\\hat{P}_{i}=\\left.\\frac{\\tilde{P}_{+}(x_i)-\\tilde{P}_{-}(x_i)}{2}\\right|_{\\lambda_{j,k}\\mapsto \\mathcal{Y}(\\lambda_{j,k})}, \\quad \\mathcal{Y}(l_i)=\\emptyset,\\mathcal{Y}(\\lambda>l_i)=[\\lambda-l_i]\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\hat{P}_4=&0\n\\\\\n\\hat{P}_6=&\n \\left( s_{\\tiny\\yng(1,1,3)_2}-s_{\\tiny\\yng(1,1,3)_1} \\right) \\Delta_{\\tiny\\yng(2)_3}+\n \\left(s_{\\tiny\\yng(3)_2}- s_{\\tiny\\yng(3)_1}\\right) \\Delta_{\\tiny\\yng(1,2)_3}+ \\left( s_\n{\\tiny\\yng(1,3)_2}-s_{\\tiny\\yng(1,3)_1} \\right) \\Delta_{\\tiny\\yng(1,1,2)_3}\n\\\\\n\\hat{P}_8=&\n\\left( s_{\\tiny\\yng(1,1,1,3)_3}-s_{\\tiny\\yng(1,1,1,3)_2} \\right) \\Delta_{\\tiny\\yng(2)_4}+\n\\left( s_{\\tiny\\yng(1,1,1,1)_3}-s_{\\tiny\\yng(1,1,1,3)_1} \\right) \\Delta_{\\tiny\\yng(4)_4}+\n\\left( -s_{\\tiny\\yng(3)_2}+s_{\\tiny\\yng(3)_3} \\right) \\Delta_{\\tiny\\yng(1,2)_4}+\n\\\\ & +\n\\left( s_{\\tiny\\yng(1)_3}-s_{\\tiny\\yng(3)_1} \\right) \\Delta_{\\tiny\\yng(1,4)_4}+ \\left( -s_{\\tiny\\yng(1,3)_2}+\ns_{\\tiny\\yng(1,3)_3} \\right) \\Delta_{\\tiny\\yng(1,1,2)_4}+ \\left( s_{\\tiny\\yng(1,1)_3}-s_{\\tiny\\yng(1,3)_1} \\right) \\Delta_{\\tiny\\yng(1,1,4)_4}+\n\\\\ & +\n\\left( s_{\\tiny\\yng(1,1,3)_3}-s_{\\tiny\\yng(1,1,3)_2} \\right) \\Delta_{\\tiny\\yng(1,1,1,2)_4}+ \\left( s_{\\tiny\\yng(1,1,1)_3}-s_{\\tiny\\yng(1,1,3)_1} \\right) \\Delta_{\\tiny\\yng(1,1,1,4)_4}\n\\\\\n\\hat{P}_{10}=&\n \\left( -s_{\\tiny\\yng(3)_3}+s_{\\tiny\\yng(3)_4} \\right) \\Delta_{\\tiny\\yng(2)_9}+ \\left( -s_{\\tiny\\yng(5)_1}+\n s_{\\tiny\\yng(5)_2}+s_{\\tiny\\yng(5)_4}-s_{\\tiny\\yng(5)_3} \\right) \\Delta_{\\tiny\\yng(4)_9}+\n \\\\&+\n \\left( s_{\\tiny\\yng(4)_1}-s_{\\tiny\\yng(2)_3} \\right) \\Delta_{\\tiny\\yng(5)_9}+ \\left( -s_{\\tiny\\yng(1,3)_3}+\n s_{\\tiny\\yng(1,3)_4} \\right) \\Delta_{\\tiny\\yng(1,2)_9}+ \\left( s_{\\tiny\\yng(1,5)_2}-s_{\\tiny\\yng(1,5)_3}- \\right.\n \\\\ &\n\\left.-s_{\\tiny\\yng(1,5)_1}+s_{\\tiny\\yng(1,5)_4} \\right)\n \\Delta_{\\tiny\\yng(1,4)_9}+\n \\left( s_{\\tiny\\yng(1,4)_1}-s_{\\tiny\\yng(1,2)_3} \\right) \\Delta_{\\tiny\\yng(1,5)_9}\n+ \\left( s_{\\tiny\\yng(1,1,3)_4}-s_{\\tiny\\yng(1,1,3)_3} \\right) \\Delta_{\\tiny\\yng(1,1,2)_9}+\n\\\\ & +\n  \\left( -s_{\\tiny\\yng(1,1,5)_3}+s_{\\tiny\\yng(1,1,5)_2}+s_{\\tiny\\yng(1,1,5)_4}-s_{\\tiny\\yng(1,1,5)_1}\n  \\right) \\Delta_{\\tiny\\yng(1,1,4)_9}+ \\left( s_{\\tiny\\yng(1,1,4)_1}-s_{\\tiny\\yng(1,1,2)_3}\n \\right) \\Delta_{\\tiny\\yng(1,1,5)_9}+\n\\\\ & + \n  \\left( -s_{\\tiny\\yng(1,1,1,3)_3}+s_{\\tiny\\yng(1,1,1,3)_4}\n \\right) \\Delta_{\\tiny\\yng(1,1,1,2)_9} \n  \\left( -s_{\\tiny\\yng(1,1,1,5)_1}+s_{\\tiny\\yng(1,1,1,5)_4}-\ns_{\\tiny\\yng(1,1,1,5)_3}+s_{\\tiny\\yng(1,1,1,5)_2} \\right) \\Delta_{\\tiny\\yng(1,1,1,4)_9}+\n\\\\ & +\n \\left( s_{\\tiny\\yng(1,1,1,4)_1}-s_{\\tiny\\yng(1,1,1,2)_3} \\right) \\Delta_{\\tiny\\yng(1,1,1,5)_9}+\n \\left( s_{\\tiny\\yng(1,1,1,1,3)_4}-s_{\\tiny\\yng(1,1,1,1,3)_3} \\right) \n\\Delta_{\\tiny\\yng(1,1,1,1,2)_9}+ \n\\\\ & +\n\\left( s_{\\tiny\\yng(1,1,1,1,5)_4}+s_{\\tiny\\yng(1,1,1,1,5)_2}-\ns_{\\tiny\\yng(1,1,1,1,5)_3}-s_{\\tiny\\yng(1,1,1,1,5)_1} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,4)_9}+\n \\left( s_{\\tiny\\yng(1,1,1,1,4)_1}-s_{\\tiny\\yng(1,1,1,1,2)_3} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,5)_9}\n\\\\\n\\hat{P}_{12}= &\n\\left( s_{\\tiny\\yng(3)_5}-s_{\\tiny\\yng(3)_4} \\right) \\Delta_{\\tiny\\yng(2)_{11}}+\n \\left( s_{\\tiny\\yng(5)_3}-s_{\\tiny\\yng(5)_4}+s_{\\tiny\\yng(5)_5}-s_{\\tiny\\yng(5)_2} \\right) \\Delta_{\\tiny\\yng(4)_{11}}+\n \\left( s_{\\tiny\\yng(1,3)_5}-s_{\\tiny\\yng(1,3)_4} \\right) \\Delta_{\\tiny\\yng(1,2)_{11}}+\n\\\\ & +\n \\left( s_{\\tiny\\yng(1,5)_5}-s_{\\tiny\\yng(1,5)_2}+s_{\\tiny\\yng(1,5)_3}-s_{\\tiny\\yng(1,5)_4} \\right) \\Delta_{\\tiny\\yng(1,4)_{11}}+\n \\left( -s_{\\tiny\\yng(1,1,3)_4}+s_{\\tiny\\yng(1,1,3)_5} \\right) \\Delta_{\\tiny\\yng(1,1,2)_{11}}+\n\\\\ & +\n \\left( -s_{\\tiny\\yng(1,1,5)_4}-s_{\\tiny\\yng(1,1,5)_2}+s_{\\tiny\\yng(1,1,5)_5}+s_{\\tiny\\yng(1,1,5)_3} \\right) \\Delta_{\\tiny\\yng(1,1,4)_{11}}+\n \\left( -s_{\\tiny\\yng(1,1,1,3)_4}+s_{\\tiny\\yng(1,1,1,3)_5} \\right) \\Delta_{\\tiny\\yng(1,1,1,2)_{11}}+\n\\\\ & +\n \\left( -s_{\\tiny\\yng(1,1,1,1,5)_4}+s_{\\tiny\\yng(1,1,1,1,5)_5}+s_{\\tiny\\yng(1,1,1,1,5)_3}-s_{\\tiny\\yng(1,1,1,1,5)_2} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,4)_{11}}+ \\left( s_{\\tiny\\yng(1,1,1,1,1,3)_5}-s_{\\tiny\\yng(1,1,1,1,1,3)_4} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,1,2)_{11}}+\n\\\\ & + \n \\left( -s_{\\tiny\\yng(1,1,1,1,1,5)_2}-s_{\\tiny\\yng(1,1,1,1,1,5)_4}+s_{\\tiny\\yng(1,1,1,1,1,5)_5}+s_{\\tiny\\yng(1,1,1,1,1,5)_3} \\right)\n\\Delta_{\\tiny\\yng(1,1,1,1,1,4)_{11}}+\\dots\n\\end{aligned}\n\\end{equation}\nAnother (stand-alone) example:\n\\begin{equation}\n-I\\hat{P}_6(spiral_2)=s_{\\tiny\\yng(1,1,2)_2}+s_{\\tiny\\yng(1,2)_0}+s_{\\tiny\\yng(1,1,2)_4}-s_{\\tiny\\yng(1,1,2)_3}-s_{\\tiny\\yng(1,2)_5}-s_{\\tiny\\yng(2)_1}\n\\end{equation}\nLet $(m,n)=(6,6)$; consider the following two paths and their concatenation:\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=1]{drawing2}\n\\end{figure}\n{\\small{\n\\begin{equation}\n\\begin{aligned}\n\\tilde{P}(Q_1)= &\\left( s_{\\tiny\\yng(1)_0}+s_{\\tiny\\yng(1,1)_3}+s_{\\tiny\\yng(1)_5}+s_{\\tiny\\yng(2)_2} \\right) \\Delta_{\\tiny\\yng(1)_5}+ \\left( -s_{\\tiny\\yng(2)_0}+s_{\\tiny\\yng(1,2)_3}+s_{\\tiny\\yng(2)_5}-s_{\\tiny\\yng(3)_2} \\right) \\Delta_{\\tiny\\yng(2)_5}+\n\\\\ & +\n \\left( s_{\\tiny\\yng(1)_2}+s_{\\tiny\\yng(3)_0}+s_{\\tiny\\yng(3)_5}+s_{\\tiny\\yng(1,3)_3} \\right) \\Delta_{\\tiny\\yng(3)_5}+ \\left( s_{\\tiny\\yng(1,1)_5}+s_{\\tiny\\yng(1,1)_0}+s_{\\tiny\\yng(1,1,1)_3}+s_{\\tiny\\yng(1,2)_2} \\right) \\Delta_{\\tiny\\yng(1,1)_5}+\n\\\\ & + \n  \\left( -s_{\\tiny\\yng(1,3)_2}+s_{\\tiny\\yng(1,1,2)_3}+s_{\\tiny\\yng(1,2)_5}-s_{\\tiny\\yng(1,2)_0} \\right) \\Delta_{\\tiny\\yng(1,2)_5}+ \\left( s_{\\tiny\\yng(1,3)_0}+s_{\\tiny\\yng(1,1)_2}+s_{\\tiny\\yng(1,3)_5}+s_{\\tiny\\yng(1,1,3)_3} \\right) \\Delta_{\\tiny\\yng(1,3)_5}+\n\\\\ & +  \n   \\left( s_{\\tiny\\yng(1,1,1)_5}+s_{\\tiny\\yng(1,1,2)_2}+s_{\\tiny\\yng(1,1,1)_0}+s_{\\tiny\\yng(1)_3} \\right) \\Delta_{\\tiny\\yng(1,1,1)_5}+ \\left( s_{\\tiny\\yng(2)_3}-s_{\\tiny\\yng(1,1,3)_2}+s_{\\tiny\\yng(1,1,2)_5}-s_{\\tiny\\yng(1,1,2)_0} \\right) \\Delta_{\\tiny\\yng(1,1,2)_5}+\n\\\\ & +   \n    \\left( s_{\\tiny\\yng(1,1,3)_5}+s_{\\tiny\\yng(3)_3}+s_{\\tiny\\yng(1,1,1)_2}+s_{\\tiny\\yng(1,1,3)_0} \\right) \\Delta_{\\tiny\\yng(1,1,3)_5}\n    \\quad \\text{\\# 5 is the longest arc}\n\\\\\n\\tilde{P}(Q_2)= &\n\\left( s_{\\tiny\\yng(1)_4}+s_{\\tiny\\yng(1)_5}-s_{\\tiny\\yng(2)_2}+s_{\\tiny\\yng(1)_1} \\right) \\Delta_{\\tiny\\yng(1)_4}+ \\left( -s_{\\tiny\\yng(2)_5}-s_{\\tiny\\yng(3)_1}+s_{\\tiny\\yng(3)_2}+s_{\\tiny\\yng(2)_4} \\right) \\Delta_{\\tiny\\yng(2)_4}+\n\\\\ & +\n \\left( -s_{\\tiny\\yng(3)_5}+s_{\\tiny\\yng(1)_2}+s_{\\tiny\\yng(3)_4}+s_{\\tiny\\yng(2)_1} \\right) \\Delta_{\\tiny\\yng(3)_4}+ \\left( -s_{\\tiny\\yng(1,1,2)_2}+s_{\\tiny\\yng(1,1,1)_1}+s_{\\tiny\\yng(1,1)_4}+s_{\\tiny\\yng(1,1,1)_5} \\right) \\Delta_{\\tiny\\yng(1,1)_4}+\n\\\\ & + \n  \\left( -s_{\\tiny\\yng(1,1,2)_5}+s_{\\tiny\\yng(1,1,3)_2}+s_{\\tiny\\yng(1,2)_4}-s_{\\tiny\\yng(1,1,3)_1} \\right) \\Delta_{\\tiny\\yng(1,2)_4}+ \\left( s_{\\tiny\\yng(1,1,1)_2}-s_{\\tiny\\yng(1,1,3)_5}+s_{\\tiny\\yng(1,1,2)_1}+s_{\\tiny\\yng(1,3)_4} \\right) \\Delta_{\\tiny\\yng(1,3)_4}+\n\\\\ & +\n   \\left( s_{\\tiny\\yng(1,1)_1}+s_{\\tiny\\yng(1,1)_5}+s_{\\tiny\\yng(1,1,1)_4}-s_{\\tiny\\yng(1,2)_2} \\right) \\Delta_{\\tiny\\yng(1,1,1)_4}+ \\left( -s_{\\tiny\\yng(1,3)_1}-s_{\\tiny\\yng(1,2)_5}+s_{\\tiny\\yng(1,1,2)_4}+s_{\\tiny\\yng(1,3)_2} \\right) \\Delta_{\\tiny\\yng(1,1,2)_4}+\n\\\\ & +\n\\left( s_{\\tiny\\yng(1,1)_2}+s_{\\tiny\\yng(1,2)_1}+s_{\\tiny\\yng(1,1,3)_4}-s_{\\tiny\\yng(1,3)_5} \\right) \\Delta_{\\tiny\\yng(1,1,3)_4}\n\\quad \\text{\\# 4 is the longest arc}\n\\\\\n\\tilde{P}(Q_{1,2})= &\n\\left( -s_{\\tiny\\yng(1,1,3)_2}+s_{\\tiny\\yng(1,1,3)_1}-s_{\\tiny\\yng(1,2)_4}-s_{\\tiny\\yng(1,1,2)_0}+s_{\\tiny\\yng(2)_3}+s_{\\tiny\\yng(1,1,2)_5} \\right)\n \\Delta_{\\tiny\\yng(2)_3}+\n  \\left( s_{\\tiny\\yng(2)_5}+s_{\\tiny\\yng(3)_1}-s_{\\tiny\\yng(3)_2}-s_{\\tiny\\yng(2)_4}-s_{\\tiny\\yng(2)_0}+s_{\\tiny\\yng(1,2)_3} \\right) \\Delta_{\\tiny\\yng(1,2)_3}+ \n\\\\ &  +  \n\\left( s_{\\tiny\\yng(1,2)_5}-s_{\\tiny\\yng(1,2)_0}-s_{\\tiny\\yng(1,1,2)_4}+s_{\\tiny\\yng(1,1,2)_3}+s_{\\tiny\\yng(1,3)_1}-s_{\\tiny\\yng(1,3)_2} \\right) \\Delta_{\\tiny\\yng(1,1,2)_3}=\n\\\\ = & \\frac{1}{2}\\left(\\mathrm{coeff}(P(Q_1),\\Delta_{\\tiny\\yng(1,1,2)_5})-\\mathrm{coeff}(P(Q_2),\\Delta_{\\tiny\\yng(1,2)_4})\\right)\\Delta_{\\tiny\\yng(2)_3}\n+ \\frac{1}{2}\\left(\\mathrm{coeff}(P(Q_1),\\Delta_{\\tiny\\yng(2)_5})-\\mathrm{coeff}(P(Q_2),\\Delta_{\\tiny\\yng(2)_4})\\right)\\Delta_{\\tiny\\yng(1,2)_3}+\n\\\\ & +\n\\frac{1}{2}\\left(\\mathrm{coeff}(P(Q_1),\\Delta_{\\tiny\\yng(1,2)_5})-\\mathrm{coeff}(P(Q_2),\\Delta_{\\tiny\\yng(1,1,2)_4})\\right)\\Delta_{\\tiny\\yng(1,1,2)_3}\n= \\frac{1}{2}\\left(P(Q_1)-\\mathrm{permut}(P(Q_2),\\Delta_{*})\\right)\n\\\\\n\\end{aligned}\n\\end{equation}\n}}\n(This is a particular example related to the proposition 1).\n\\newpage\nThe next goal is to build infinite series like this\n\\begin{equation}\nZ_R(S)= \\frac{1}{2k}\\sum\\limits_{i,j}\\left(\\tilde{P}_R(x_i)-\\tilde{P}_R(x_j)\\right),\n\\end{equation}\nwhere $S= \\{x_1,x_2,\\dots,x_i,\\dots\\}\\subseteq \\mathcal{P}(Q_{n\\to\\infty}) \\ \\text{ is a proper union of loops}$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[scale=0.5]{Screenshot3.png}\n\\caption{Concatenation of $x_i\\in S$ for $R=\\Sigma_0$}\n\\end{figure}\nOf course it can be rewritten in terms of Schur function expansion\n\\begin{equation}\nZ_R(S)=\\sum\\limits_{\\lambda}g_{\\lambda}(\\Delta)s_{\\lambda}(p_1,p_2,\\dots),\n\\end{equation}\nor, in our \\emph{``ladder notation''}, over all $\\Gamma$-type \\emph{weighted} Young diagrams,\nusing the $\\mathcal{Y}$ operator\n\\begin{equation}\nZ_R(S)=\\sum\\limits_{\\lambda_w}g_{\\lambda_w}(\\Delta)\\tilde{s}_{\\lambda_w}(p_1,p_2,\\dots), \\quad w\\in\\mathbb{N},\n\\quad \\lambda_w=\\mathcal{Y}(\\lambda)\n\\end{equation}\nTake $S'=\\{x_1,x_2\\}\\subset \\mathcal{P}(Q_{10}), (m,n)=(10,10)$.\nIn this case the double-path polynomial is non-trivial (!)\n\\begin{equation}\n\\begin{aligned}\n\\tilde{P}_{\\pm}(S')= &\n\\left( s_{\\tiny\\yng(1)_2}+s_{\\tiny\\yng(1,5)_8}+s_{\\tiny\\yng(1)_6}+s_{\\tiny\\yng(1)_3}+\ns_{\\tiny\\yng(1,5)_9}+s_{\\tiny\\yng(1)_1}+s_{\\tiny\\yng(1)_7}+s_{\\tiny\\yng(1,1)_5}+s_{\\tiny\\yng(1,1)_4} \\right)\n \\Delta_{\\tiny\\yng(1)_6}+\n\\\\ & + \n  \\left( s_{\\tiny\\yng(1,1)_3}+s_{\\tiny\\yng(1,1)_2}+s_{\\tiny\\yng(1,1,1)_5}+\ns_{\\tiny\\yng(1,1)_1}+s_{\\tiny\\yng(1,1,5)_8}+s_{\\tiny\\yng(1,1,1)_4}+s_{\\tiny\\yng(1,1,1)_6}+\ns_{\\tiny\\yng(1,1)_7}+s_{\\tiny\\yng(1,1,5)_9} \\right) \\Delta_{\\tiny\\yng(1,1,1)_6}+\n\\\\ & + \n \\left( s_{\\tiny\\yng(1,1,1,1)_5}+\ns_{\\tiny\\yng(1,1,1,1)_6}+s_{\\tiny\\yng(1,1,1)_2}+s_{\\tiny\\yng(1,1,1,1)_4}+s_{\\tiny\\yng(1,1,1)_3}+\ns_{\\tiny\\yng(1,1,1,5)_9}+s_{\\tiny\\yng(1,1,1,1,5)_8}+s_{\\tiny\\yng(1,1,1,1)_7}+\ns_{\\tiny\\yng(1,1,1)_1} \\right) \\Delta_{\\tiny\\yng(1,1,1,1)_6}+\n\\\\ & + \n \\left( s_{\\tiny\\yng(1,1,1,1)_3}+\ns_{\\tiny\\yng(1,1,1,1,5)_9}+s_{\\tiny\\yng(1,1,1,1)_2}+s_{\\tiny\\yng(1,1,1,1,1)_7}+\ns_{\\tiny\\yng(1,1,1,1,1)_5}+s_{\\tiny\\yng(1,1,1,1,1)_4}+s_{\\tiny\\yng(1,1,1,5)_8}+s_{\\tiny\\yng(1,1,1,1)_1}+\ns_{\\tiny\\yng(1,1,1,1,1)_6} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,1)_6}+\n\\\\ & + \ns_{\\tiny\\yng(1,1,1,1,1)_3}+\ns_{\\tiny\\yng(1,1,1,1,1)_1}+s_{\\tiny\\yng(1,1)_6}+s_{\\tiny\\yng(5)_9}+s_{\\tiny\\yng(1,1,1,1,1)_2}+\ns_{\\tiny\\yng(1,1,1)_7}+s_{\\tiny\\yng(5)_8}+s_{\\tiny\\yng(1)_5}\n\\end{aligned}\n\\end{equation}\n(Here we assumed $\\Delta_{{\\tiny{[6, 7, 8, 9, 10, 16, 17, 18, 19, 20,\\dots, 96, 97, 98, 99, 100]}}} = 1$ to simplify the formula).\nIts configuration is drawn on the figure \\ref{fig1} (right).\n\\begin{figure}[h!]\\label{fig1}\n\\centering\n\\includegraphics[scale=0.4]{Screenshot4.png}\n\\caption{Concatenation of $x_1,x_2\\in S'$ for $R=\\Sigma_0$: compatible case}\n\\end{figure}\nNow let's extend the picture for $x_3$:\nThe problem is that in this case $(m,n)=(16,16)$ the image is empty, so we should add some\nisolated points to the graph \\ref{fig1} (left).\n\\newpage\nLet\n\\begin{equation}\nX=\\{G_{i_1}^{(j_1)},G_{i_2}^{(j_2)},\\dots,G_{i_s}^{(j_s)}\\}^k \\in \\mathcal{P}\n\\end{equation}\nbe an element of path algebra of $Q_{\\mathrm{univ}}(k)$, built as concatenation normal loops $G_{i_1},G_{i_2}$,\neach one shifted on the $j$-th position. \nFor example, consider\n3-union $\\{G_4^{(0)},G_6^{(3)},G_{8}^{(2)}\\}^{10}$. QR has unique solution;\nfurthermore, we take $\\Delta_{\\tiny\\yng(1,1,1)_7}=1$ (denominator) to homogenise the resulting formula.\nHere is the associated quiver:\n\\begin{equation}\nQ(X)=\\begin{xy}\n 0;<1pt,0pt>:<0pt,-1pt>:: \n(0,53) *+{0} =\"0\",\n(53,53) *+{1} =\"1\",\n(53,146) *+{2} =\"2\",\n(53,0) *+{3} =\"3\",\n(220,0) *+{4} =\"4\",\n(220,146) *+{5} =\"5\",\n(293,53) *+{6} =\"6\",\n(147,0) *+{7} =\"7\",\n(147,53) *+{8} =\"8\",\n(147,146) *+{9} =\"9\",\n\"2\", {\\ar\"0\"},\n\"0\", {\\ar\"3\"},\n\"1\", {\\ar\"2\"},\n\"3\", {\\ar\"1\"},\n\"8\", {\\ar\"2\"},\n\"2\", {\\ar\"9\"},\n\"7\", {\\ar\"3\"},\n\"3\", {\\ar\"8\"},\n\"6\", {\\ar\"4\"},\n\"4\", {\\ar\"7\"},\n\"8\", {\\ar\"4\"},\n\"5\", {\\ar\"6\"},\n\"8\", {\\ar\"5\"},\n\"9\", {\\ar\"5\"},\n\\end{xy}\n\\end{equation}\n\\begin{figure}[h!]\\label{fig1}\n\\centering\n\\includegraphics[scale=0.5]{Screenshot6.png} \\\\\n\\includegraphics[scale=0.5]{Screenshot7.png}\n\\caption{Non-trivial concatenation of loops $X=\\{G_4^{(0)},G_6^{(3)},G_{8}^{(2)}\\}^{10}$ and\n$X'=\\{G_4^{(0)},G_6^{(3)},G_{8}^{(2)},\\underline{G_{10}^{(2)}}\\}^{10}$ (only the largest arcs are shown)}\n\\end{figure}\nThen $\\tilde{P}_{\\pm}(X)$ become homogeneous in $\\Delta$:\n\\begin{equation}\n\\begin{aligned}\n\\tilde{P}_{\\pm}(X)=& \\left( s_{\\tiny\\yng(1,5)_{9}}+s_{\\tiny\\yng(1)_{0}}+s_{\\tiny\\yng(1,1)_{5}}+s_{\\tiny\\yng(1)_{1}}+s_{\\tiny\\yng(1)_{2}}+\ns_{\\tiny\\yng(1,1)_{4}}+s_{\\tiny\\yng(1)_{6}}+s_{\\tiny\\yng(1)_{3}}+s_{\\tiny\\yng(1,5)_{8}}+s_{\\tiny\\yng(1)_{7}} \\right) \n\\Delta_{\\tiny\\yng(1)_{6}}+\n\\\\ &\n+ \\left( s_{\\tiny\\yng(1,1)_{0}}+s_{\\tiny\\yng(1,1,5)_{8}}+s_{\\tiny\\yng(1,1)_{3}}+\ns_{\\tiny\\yng(1,1)_{1}}+s_{\\tiny\\yng(1,1,1)_{6}}+s_{\\tiny\\yng(1,1,1)_{5}}+s_{\\tiny\\yng(1,1)_{2}}+s_{\\tiny\\yng(1,1,1)_{4}}+\ns_{\\tiny\\yng(1,1)_{7}}+s_{\\tiny\\yng(1,1,5)_{9}} \\right) \\Delta_{\\tiny\\yng(1,1,1)_{6}}+\n\\\\ &\n+ \\left( s_{\\tiny\\yng(1,1,1)_{2}}+\ns_{\\tiny\\yng(1,1,1,1)_{7}}+s_{\\tiny\\yng(1,1,1,1)_{6}}+s_{\\tiny\\yng(1,1,1)_{3}}+s_{\\tiny\\yng(1,1,1,1)_{5}}+\ns_{\\tiny\\yng(1,1,1,1)_{4}}+s_{\\tiny\\yng(1,1,1,5)_{9}}+s_{\\tiny\\yng(1,1,1)_{0}}+s_{\\tiny\\yng(1,1,1)_{1}}+\ns_{\\tiny\\yng(1,1,1,1,5)_{8}} \\right) \\Delta_{\\tiny\\yng(1,1,1,1)_{6}}+\n\\\\ &\n+ \\left( s_{\\tiny\\yng(1,1,1,1,5)_{9}}+\ns_{\\tiny\\yng(1,1,1,5)_{8}}+s_{\\tiny\\yng(1,1,1,1)_{1}}+s_{\\tiny\\yng(1,1,1,1,1)_{7}}+s_{\\tiny\\yng(1,1,1,1,1)_{5}}+\ns_{\\tiny\\yng(1,1,1,1,1)_{6}}+s_{\\tiny\\yng(1,1,1,1)_{0}}+s_{\\tiny\\yng(1,1,1,1)_{2}}+s_{\\tiny\\yng(1,1,1,1)_{3}}+\ns_{\\tiny\\yng(1,1,1,1,1)_{4}} \\right) \\Delta_{\\tiny\\yng(1,1,1,1,1)_{6}}+\n\\\\ &\n+ s_{\\tiny\\yng(1,1,1,1,1)_{0}}+\ns_{\\tiny\\yng(1,1,1,1,1)_{1}}+s_{\\tiny\\yng(5)_{8}}+s_{\\tiny\\yng(1)_{5}}+s_{\\tiny\\yng(1,1,1,1,1)_{3}}+s_{\\tiny\\yng(5)_{9}}+\ns_{\\tiny\\yng(1)_{4}}+s_{\\tiny\\yng(1,1)_{6}}+s_{\\tiny\\yng(1,1,1,1,1)_{2}}\n\\end{aligned}\n\\end{equation}\nRemark: we can see \\underline{only antisymmetric coefficients} $\\Delta$ here! To extend this case for $G_{10}$\none should exclude $G_{8}^{(2)}$.\n\nThe resulting picture is that \\underline{each loop can be scaled separately}, such that increasing $k,s$ simultaneously\ndoes not change the consistency of QR. Our next aim is to investigate asymptotical properties, derived from\n$\\{G_{i_1}^{(j_1)},G_{i_2}^{(j_2)},\\dots,G_{i_s}^{(j_s)}\\}^k$ when $k,s\\to\\infty$.\nThis is of further development.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA domain ${\\Omega}\\subset{{\\rr}^n}$ is said to be a Sobolev $(p, q)$-extension domain for $1\\leq q\\leq p\\leq\\infty$, if there exists a bounded extension operator\n$$ E:W^{1,p}({\\Omega})\\to W^{1,q}({{\\rr}^n}),$$\nsuch that for every $u\\in W^{1, p}({\\Omega})$, we have $E(u)\\in W^{1, q}({{\\rr}^n})$ with \n\\[\\|E(u)\\|_{W^{1, q}({{\\rr}^n})}\\leq C\\|u\\|_{W^{1,p}({\\Omega})}\\]\nfor a constant $C$ independent of $u$. The smallest constant in the inequality above is denoted by $\\|E\\|$. In \\cite{C61, Stein}, Calder\\'on and Stein proved that if $\\Omega\\subset\\mathbb R^n$ is a Lipschitz domain, then there exists a bounded linear extension operator $E: W^{k, p}(\\Omega)\\to W^{k, p}(\\mathbb R^n)$, for all $k\\geq 1$ and $1\\leq p\\leq \\infty$. Here $W^{k, p}({\\Omega})$ is the class of those $L^p$-integrable functions whose weak derivatives up to order $k$ belong to $L^p(\\Omega)$. In \\cite{J81}, Jones introduced the notion of $(\\varepsilon,\\delta)$-domains which are generalizations of Lipschitz domains. He proved that, for every $(\\varepsilon,\\delta)$-domain, there exists a bounded linear extension operator $E: W^{k, p}(\\Omega)\\to W^{k, p}(\\mathbb R^n)$, for all $k\\geq 1$  and $1\\leq p\\leq\\infty$. This has motivated the search for geometric characterizations for Sobolev extension domains. A geometric characterization of simply connected planar Sobolev $(2,2)$-extension domains was obtained in \\cite{VGL79}. By a more recent sequence of results in \\cite{pekkaJFA, KRZ1, KRZ2, ShJFA}, we understand the geometry of simply connected planar Sobolev $(p, p)$-extension domains, for all $1\\leq p\\leq\\infty$. Furthermore, by \\cite{ShZ16}, geometric characterizations of planar simply connected extension domains are also known in the case of homogeneous Sobolev spaces $L^{k, p}(\\Omega)$ for $2<p<\\infty$. Here $L^{k, p}({\\Omega})$ is the seminormed space of those locally integrable functions whose $k$-th order distributional partial derivatives belong to $L^p({\\Omega})$. However, no characterizations are available in the general setting. \n\nIn this paper, we consider Sobolev extension properties for a class of quasiconvex Euclidean domains with only a single singular boundary point.  A domain ${\\Omega}\\subset{{\\rr}^n}$ is called quasiconvex, if for every $x, y\\in{\\Omega}$ there exists a curve $\\gamma_{x,y}\\subset{\\Omega}$ connecting $x, y$ with $${\\rm length}(\\gamma_{x, y})\\leq C|x-y|$$ for a constant $C$ independent of $x$ and $y$. We study the outward cuspidal domains defined by setting\n\\begin{equation}\\label{cusp}\n{\\Omega}^n_\\psi:=\\left\\{z=(t, x)\\in (0, 1]\\times{\\mathbb R}^{n-1}; |x|<\\psi(t) \\right\\}\\cup\\{z=(t, x)\\in[1, 2)\\times{\\mathbb R}^{n-1}; |x|<\\psi(1)\\},\n\\end{equation}\nwhere $\\psi\\colon (0,1]\\to (0,\\infty)$ is a left-continuous and increasing function. (Left-continuity is required just to ensure ${\\Omega}^n_\\psi$ to be open. The term ``increasing'' is used in the non-strict sense.) If the left-continuous and increasing function $\\psi$ is Lipschitz, ${\\Omega}_\\psi^n$ is called a Lipschitz outward cuspidal domain. This kind of model domains have been widely studied, see Maz'ya and Poborchi's monograph \\cite{Mazya} and references therein. From now on, every left-continuous and increasing function $\\psi:(0, 1]\\to(0, \\infty)$ will be called a cuspidal function. Our class of domains ${\\Omega}_\\psi^n$ was introduced in \\cite{SPJZ}. It was shown in \\cite{SPJZ} that for an arbitrary cuspidal function $\\psi$, the Sobolev space $W^{1,p}({\\Omega}_\\psi^n)$ coincides with the Haj\\l{}asz-Sobolev space $M^{1, p}({\\Omega}_\\psi^n)$ for all $1<p\\leq\\infty$. See \\cite{Pitor} for the definition of the Haj\\l{}asz-Sobolev space $M^{1, p}({\\Omega})$. \n\\begin{figure}[h!tbp]\n\\centering\n\\includegraphics[width=0.5\\textwidth]\n{Omega.png}\\label{fig:cusps}\n\\caption{An outward cuspidal domain ${\\Omega}_\\psi^n$}\n\\end{figure}\nReturning to the problem of geometric characterization for Sobolev extension domains, one may ask a natural question:\n\\begin{center}\n\\textbf{For which cuspidal functions $\\psi$, the outward cuspidal domain ${\\Omega}_\\psi^n$ is a Sobolev $(p, q)$-extension domain for given $1\\leq q\\leq p\\leq\\infty$?}\n\\end{center}\nThe first observation is that ${\\Omega}_\\psi^n$ is a Sobolev $(\\infty, \\infty)$-extension domain, for every cuspidal function $\\psi$. This follows from a result in \\cite{hkt} since ${\\Omega}_\\psi^n$ is always quasiconvex for an arbitrary cuspidal function $\\psi$.  \n\nIn \\cite{Mazya1, Mazya2, Mazya}, for certain Lipschitz cuspidal functions $\\psi$, Maz'ya and Poborchi used integrability conditions on $\\psi$ to characterize the Sobolev extension property for ${\\Omega}_\\psi^n$. By transferring an outward cuspidal domain onto a Lipschitz outward cuspidal domain via a global bi-Lipschitz transformation, we obtain a more general version of their result. \n\\begin{theorem}\\label{prop:mazya}\nLet $\\psi:(0, 1]\\to(0, \\infty)$ be a cuspidal function such that $\\psi\\big|_{(0, 1]}>0$ and the function $\\psi(t)\/t$ is nondecreasing on $(0, 1]$ with $\\lim_{t\\to0}\\psi(t)\/t=0$. \nWe have following statements.\\\\\n{\\textbf{$(1):$}} If \n\\begin{equation}\\label{eq:inc1}\n\\int_0^1\\left(\\frac{t^s}{\\psi(t)}\\right)^{\\frac{n}{s-1}}\\frac{dt}{t}<\\infty, \n\\end{equation}\nthen there exists a bounded linear extension operator $E_1$ from $W^{1,p}({\\Omega}^n_\\psi)$ to $W^{1, q}({{\\rr}^n})$ whenever $\\frac{1+(n-1)s}{n}\\leq p<\\infty$ and $1\\leq q\\leq\\frac{np}{1+(n-1)s}$, and a bounded linear extension operator $E_2$ from $W^{1, p}({\\Omega}^n_\\psi)$ to $W^{1, q}({{\\rr}^n})$ whenever $\\frac{1+(n-1)s}{2+(n-2)s}\\leq p<\\infty$ and $1\\leq q\\leq\\frac{(1+(n-1)s)p}{1+(n-1)s+(s-1)p}$.\\\\\n{\\textbf{$(2):$}} If  \n\\begin{equation}\\label{eq:inc2}\n\\int_0^1\\left(\\frac{t^s}{\\psi(t)}\\right)^{\\frac{n}{s-1}}\\left|\\log\\left(\\frac{\\psi(t)}{t}\\right)\\right|^{-\\alpha}\\frac{dt}{t}<\\infty\n\\end{equation}\nwith $\\alpha=\\frac{(n-2)p}{p+1-n}$, then there exists a bounded linear extension operator $E_3$ from $W^{1, p}({\\Omega}^n_\\psi)$ to $W^{1, q}({{\\rr}^n})$ whenever $\\frac{(n-1)^2s+(n-1)}{n}\\leq p<\\infty$ and $1\\leq q\\leq n-1$.\n\nFurthermore, under the doubling condition\n\\begin{equation}\\label{eq:doubling}\n \\psi(2t)\\leq C\\psi(t),\\ {\\rm for}\\ t\\in\\left(0, \\frac{1}{2}\\right),\n \\end{equation}\non $\\psi$, the statements converse to $(1)$ and $(2)$ also hold.\n\\end{theorem}\n\nGiven $1<s<\\infty$, we refer to the domain ${\\Omega}_\\psi^n$ with $\\psi(t)=t^s$ by ${\\Omega}_{t^s}^n$. Let $n\\geq 3$. Theorem \\ref{prop:mazya} yields that ${\\Omega}^n_{t^s}$ is a Sobolev $(p, q)$-extension domain, whenever $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p\\leq\\infty$. Our bi-Lipschitz transformation method allows us to extend this result from the case of $t^s$ to arbitrary cuspidal functions. This result can be regarded as a limit case of Theorem \\ref{prop:mazya} and also of theorems by Maz'ya and Poborchi in \\cite{Mazya}.\n\\begin{theorem}\\label{thm:main}\nLet $3\\leq n<\\infty$ and $\\psi:(0, 1]\\to(0, \\infty)$ be a cuspidal function. Then the corresponding outward cuspidal domain ${\\Omega}^n_\\psi$ is a Sobolev $(p, q)$-extension domain, whenever $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p\\leq\\infty$. \n\\end{theorem}\n\nThe sharpness part of Theorem \\ref{prop:mazya} also yields the sharpness of Theorem \\ref{thm:main}.\n\\begin{proposition}\\label{proposition}\nFor arbitrary $n-1\\leq p<\\infty$, there exists $1<s_1<\\infty$ such that ${\\Omega}_{t^s}^n$ is not a Sobolev $(p, n-1)$-extension domain when $s>s_1$. For arbitrary $1\\leq q<n-1$ and $q\\leq p<(n-1)q\/(n-1-q)$, there exists $1<s_2<\\infty$ such that ${\\Omega}_{t^s}^n$ is not a Sobolev $(p, q)$-extension domain when $s>s_2$.\n\\end{proposition}\n\n \nThe paper is organized as follows. Section $2$ contains definitions and preliminary results. Section $3$ contains proofs of all results presented above. Section $4$ contains some further discussion and a conjecture.\n\\section{Definitions and Preliminaries}\nIn this note, ${\\Omega}\\subset\\ensuremath{\\mathbb{R}}^n$ is always a bounded domain. $C$ will refer to constants that depend on various parameters and may differ even in a chian of inequalities. \nThe Euclidean distance between points $x,y\\in{{\\rr}^n}$ is denoted by $|x-y|$. The open $n$-dimensional ball of radius $r$ centered at the point $x$ is denoted by $B^{n}(x,r)$. \n\n Let us give the definition of the Sobolev space $W^{1,p}({\\Omega})$.\n\\begin{defn}\\label{Sobolev}\nWe define the first order Sobolev space $W^{1,p}({\\Omega})$, $1\\leq p\\leq \\infty$, as the set \n\\begin{equation}\n\\left\\{\\, u\\in{L^{p}(\\boz)} ; \\nabla u\\in L^p({\\Omega};{{\\rr}^n})\\, \\right\\}\\, .\\nonumber\n\\end{equation}\nHere $\\nabla u=\\left(\\frac{\\partial u}{\\partial x_1}\\, ,\\, \\dots \\, ,\\,\\frac{\\partial u}{\\partial x_n}\\right)$ is the weak (or distributional) gradient of the integrable function $u$. \n\\end{defn}\nThe Sobolev space $W^{1,p}({\\Omega})$ is equipped with the norm:\n\\begin{equation}\n\\|u\\|_{{W^{1,p}(\\boz)}}=\\|u\\|_{L^p({\\Omega})}+\\||\\nabla u|\\|_{L^p({\\Omega})}\\nonumber\n\\end{equation}\nfor $1\\leq p\\leq\\infty$, where $\\|f\\|_{L^p({\\Omega})}$ denotes the usual $L^p$-norm for $p \\in [1,\\infty]$. Let us give the definition of Sobolev extension domains.\n\\begin{defn}\\label{de:extdo}\nLet $1\\leq q\\leq p\\leq\\infty$. A bounded domain ${\\Omega}\\subset{{\\rr}^n}$ is said to be a Sobolev $(p, q)$-extension domain, if there is a bounded extension operator $E$ from $W^{1,p}({\\Omega})$ to $W^{1,q}({{\\rr}^n})$ such that, for every $u\\in W^{1,p}({\\Omega})$, there exists a function $E(u)\\in W^{1,q}({{\\rr}^n})$ with $E(u)\\big|_{\\Omega}\\equiv u$ and \n\\[\\|E(u)\\|_{W^{1,q}({{\\rr}^n})}\\leq C\\|u\\|_{W^{1,p}({\\Omega})}\\]\nfor a positive constant $C$ independent of $u$.\n\\end{defn}\n\nAn outward cuspidal domain ${\\Omega}_\\psi^n$ has a singular point on the boundary. However, it still has some nice geometric properties. For example, it satisfies the following $segment\\ condition$.\n\\begin{defn}\\label{defn:segment}\nWe say that a domain ${\\Omega}\\subset{{\\rr}^n}$ satisfies the segment condition if every $x\\in\\partial{\\Omega}$ has a neighborhood $U_x$ and a nonzero vector $y_x$ such that if $z\\in\\overline{\\Omega}\\cap U_x$, then $z+ty_x\\in{\\Omega}$ for $0<t<1$.\n\\end{defn}\nThe following lemma tells us that Sobolev functions on a domain with the segment condition can be approximated by globally smooth functions. See \\cite[Theorem 3.22]{adams}.\n\\begin{lemma}\\label{lem:density}\nIf a domain ${\\Omega}\\subset{{\\rr}^n}$ satisfies the segment condition, then the set of restrictions to ${\\Omega}$ of functions in $C_o^\\infty({{\\rr}^n})$ is dense in $W^{1,p}({\\Omega})$ for $1\\leq p<\\infty$. In short, $C_o^\\infty({{\\rr}^n})\\cap W^{1,p}({\\Omega})$ is dense in $W^{1,p}({\\Omega})$ for $1\\leq p<\\infty$.\n\\end{lemma}\n\n\\section{Proofs of Theorem \\ref{prop:mazya}, Theorem \\ref{thm:main} and Proposition \\ref{proposition}}\nAs mentioned in the introduction, every outward cuspidal domain ${\\Omega}_\\psi^n$ is a Sobolev $(\\infty, \\infty)$-extension domain. We show that every outward cuspidal domain is globally bi-Lipschitz equivalent to a Lipschitz outward cuspidal domain. For the proof of this result, we first introduce a Lipschitz cuspidal function $\\hat\\psi$ generated by a given cuspidal function $\\psi$.\n\\begin{lemma}\\label{le:LipcF}\nLet $\\psi:(0, 1]\\to(0, \\infty)$ be an arbitrary cuspidal function. Then the following claims hold.\\\\\n$(1):$ For every $0<\\hat t<1$, there exists a unique pair $(t_{\\hat t}, r_{\\hat t})$ with $0<t_{\\hat t}<1$, $\\psi(t_{\\hat t})\\leq r_{\\hat t}\\leq\\lim_{s\\to t_{\\hat t}^+}\\psi(s)$ and $t_{\\hat t}+r_{\\hat t}=(1+\\psi(1))\\hat t$.\\\\\n$(2):$ The function $\\hat\\psi:(0, 1]\\to(0, \\infty)$ defined by setting \n\\[\\hat\\psi(\\hat t)=r_{\\hat t}\\ {\\rm for\\ every}\\ \\hat t\\in(0, 1]\\]\nis a Lipschitz cuspidal function.\n\\end{lemma}\n\\begin{figure}[h!tbp]\n\\centering\n\\includegraphics[width=0.4\\textwidth]\n{bi-Lipschitz.png}\\label{fig:Lip}\n\\caption{Bi-Lipschitz transformations}\n\\end{figure}\n\\begin{proof}\nLet $(t, r)\\in(0, 1]\\times(0, \\psi(1))$ be a pair of positive numbers such that for every $x\\in{\\mathbb R}^{n-1}$ with $|x|=r$, we have $(t, x)\\in\\partial{\\Omega}^n_\\psi$. Define a function $T$ on $\\partial{\\Omega}^n_\\psi$ by setting\n\\[T(t, x):=t+|x|\\]\nfor every $(t, x)\\in\\partial{\\Omega}^n_\\psi$. Since $\\psi$ is left-continuous and increasing on $(0, 1]$, we always have $T(t_1,x_1)<T(t_2,x_2)$ for every $(t_1, x_1), (t_2, x_2)\\in\\partial{\\Omega}^n_\\psi$ with $0<t_1<t_2<1$. By the same reason, we have  $T(t, x_1)<T(t, x_2)$ for every $(t, x_1), (t, x_2)\\in\\partial{\\Omega}^n_\\psi$ with $0<t<1$ and $$\\psi(t)\\leq|x_1|<|x_2|\\leq\\lim_{s\\to t^+}\\psi(s).$$ Hence, for every $0<\\hat t<1$, there exists a unique pair $(t_{\\hat t}, r_{\\hat t})$ with $0<t_{\\hat t}<1$, $\\psi(t_{\\hat t})\\leq r_{\\hat t}\\leq\\lim_{s\\to t_{\\hat t}^+}\\psi(s)$ and $t_{\\hat t}+r_{\\hat t}=(1+\\psi(1))\\hat t$. Now, let us show $\\hat\\psi$ is a Lipschitz cuspidal function. By the definition, it is easy to see $\\hat\\psi$ is increasing. Hence, it suffices to show that it is Lipschitz. Define a measurable subset $A\\subset(0, 1]$ by setting\n\\[A:=\\{\\hat t\\in(0, 1]: r_{\\hat t}\\geq t_{\\hat t}\\}.\\] \nLet $\\hat t_1, \\hat t_2\\in(0, 1]$ be arbitrary. According to their locations, we divide the following argument into three cases. First, let us assume $\\hat t_1, \\hat t_2\\in A$. Since $\\hat t\\sim r_{\\hat t}$ for every $\\hat t\\in A$, by the definition, we have \n\\begin{equation}\\label{eq:In1}\n|\\hat\\psi(\\hat t_1)-\\hat\\psi(\\hat t_2)|=|r_{\\hat t_1}-r_{\\hat t_2}|\\leq C|\\hat t_1-\\hat t_2|.\n\\end{equation}\nNext, we assume $\\hat t_1, \\hat t_2\\in (0, 1]\\setminus A$. For every $\\hat t\\in(0, 1]\\setminus A$, we have $\\hat t\\sim t_{\\hat t}$. Hence, the triangle inequality implies\n\\begin{equation}\\label{eq:In2}\n|\\hat\\psi(\\hat t_1)-\\hat\\psi(\\hat t_2)|=|r_{\\hat t_1}-r_{\\hat t_2}|\\leq C|\\hat t_1-\\hat t_2|+C|t_{\\hat t_1}-t_{\\hat t_2}|\\leq C|\\hat t_1-\\hat t_2|.\n\\end{equation}\nFinally, we assume $\\hat t_1\\in A$ and $\\hat t_2\\in (0, 1]\\setminus A$. Since both $t_{\\hat t}$ and $r_{\\hat t}$ are continuous with respect to $\\hat t$, there exists $\\hat s\\in(0, 1]$ between $\\hat t_1$ and $\\hat t_2$ with $t_{\\hat s}=r_{\\hat s}$. By (\\ref{eq:In1}) and (\\ref{eq:In2}), the triangle inequality implies\n\\begin{equation}\\label{eq:In3}\n|\\hat\\psi(\\hat t_1)-\\hat\\psi(\\hat t_2)|\\leq|\\hat\\psi(\\hat t_1)-\\hat\\psi(\\hat s)|+|\\hat\\psi(\\hat s)-\\hat\\psi(\\hat t_2)|\\leq C|\\hat t_1-\\hat s|+C|\\hat s-\\hat t_2|\\leq C|\\hat t_1-\\hat t_2|.\n\\end{equation}\nBy combining (\\ref{eq:In1}), (\\ref{eq:In2}) and (\\ref{eq:In3}), we conclude that $\\hat\\psi$ is Lipschitz.\n\\end{proof}\nWe are ready to prove the following proposition which claims that an arbitrary outward cuspidal domain is globally bi-Lipschitz equivalent to a Lipschitz outward cuspidal domain.\n\\begin{proposition}\\label{prop:bilip}\nFor a given cuspidal function $\\psi$, let $\\hat\\psi$ be the Lipschitz cuspidal function defined in Lemma \\ref{le:LipcF}. Then there exists a global bi-Lipschitz transformation $\\mathcal O:{{\\rr}^n}\\to{{\\rr}^n}$ with $\\mathcal O({\\Omega}^n_\\psi)={\\Omega}^n_{\\hat\\psi}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\psi:(0, 1]\\to(0, \\infty)$ be a cuspidal function. Let $\\hat\\psi:(0, 1]\\to(0, \\infty)$ be the corresponding Lipschitz cuspidal function defined in Lemma \\ref{le:LipcF}. We extend the definition of $\\psi$ to the entire real line ${\\mathbb R}$ by setting $\\psi(t)=0$ for every $t\\leq 0$. Without loss of generality, we may assume $2\\psi(1)<1$. Otherwise, we consider the cuspidal function $\\tilde\\psi$ defined by setting\n\\[\\tilde\\psi(t):=\\frac{\\psi(t)}{2\\psi(1)}\\ {\\rm for}\\ t\\in(-\\infty, 1).\\]\nObviously, ${\\Omega}^n_\\psi$ and ${\\Omega}^n_{\\tilde\\psi}$ are globally bi-Lipschitz equivalent.\n\nWe define\n\\[U_1:=\\left\\{(t, x)\\in (-\\infty, 1+\\psi(1)]\\times{\\mathbb R}^{n-1}: |x|\\leq 1+\\psi(1)-t\\right\\}, U_2:={\\Omega}_\\psi^n\\setminus U_1,\\]\n\\[U_3:=\\left\\{(t, x)\\in(-\\infty, \\infty)\\times{\\mathbb R}^{n-1}:|x|\\geq\\max\\{\\psi(1), 1+\\psi(1)-t\\}\\right\\},\\]\n\\[U_4:=\\left\\{(t, x)\\in[2, \\infty)\\times{\\mathbb R}^{n-1}: |x|\\leq\\psi(1)\\right\\}.\\]\nObviously, we have ${{\\rr}^n}=\\bigcup_{i=1}^4U_i$. \nThen we define a global transformation $\\mathcal O:{{\\rr}^n}\\to{{\\rr}^n}$ by setting \n\\begin{equation}\\label{eq:bilip}\n\\mathcal O(t, x):=\\begin{cases}\n\\left(\\left(\\frac{1}{1+\\psi(1)}\\right)(t+|x|), x\\right), & (t, x)\\in U_1,\\\\\n\\left(\\left(\\frac{t}{1+|x|-\\psi(1)}+\\frac{2(|x|-\\psi(1))}{1+|x|-\\psi(1)}\\right), x\\right), & (t, x)\\in U_2,\\\\\n\\left(t+|x|-\\psi(1), x\\right), & (t, x)\\in U_3,\\\\\n\\left(t, x\\right), & (t, x)\\in U_4\n\\,. \\end{cases}\n\\end{equation}\nThe global homeomorphism $\\mathcal O$ is differentiable almost everywhere and there exists a positive constant $C>1$ such that for almost every $z\\in{{\\rr}^n}$, we have \n\\[\\frac{1}{C}\\leq|D\\mathcal O(z)|\\leq C.\\]\nHence, $\\mathcal O:{{\\rr}^n}\\to{{\\rr}^n}$ is a global bi-Lipschitz transformation. By a simple computation, we obtain \n\\[\\mathcal O(U_1\\cap{\\Omega}^n_{\\psi})=\\{(s, y)\\in(0, 1]\\times{\\mathbb R}^{n-1}: |y|<\\hat\\psi(s)\\}\\]\nand \n\\[\\mathcal O(U_2)=\\{(s, y)\\in(1, 2)\\times{\\mathbb R}^{n-1}:|y|<\\hat\\psi(1)\\}.\\]\nHence $\\mathcal O({\\Omega}^n_\\psi)={\\Omega}^n_{\\hat\\psi}$.\n\\end{proof}\nLet us prove Theorem \\ref{prop:mazya}.\n{\\begin{proof}[Proof of Theorem \\ref{prop:mazya}]\nLet $\\psi:(0, 1]\\to(0, \\infty)$ be a cuspidal function such that $\\psi(t)\/t$ is nondecreasing in $(0, 1]$ with $\\lim_{t\\to 0}\\psi(t)\/t=0$. Let $\\hat\\psi$ be the corresponding Lipschitz cuspidal function, defined as in Lemma \\ref{le:LipcF}. Following the notation from Lemma \\ref{le:LipcF}, for every $\\hat t\\in(0, 1]$, we have \\[t_{\\hat t}+r_{\\hat t}=(1+\\psi(1))\\hat t\\ {\\rm and}\\ \\hat\\psi(\\hat t)=r_{\\hat t}.\\] \nSince $\\psi$ is left-continuous and increasing, for every $\\hat t\\in\\left(0, \\frac{1}{1+\\psi(1)}\\right)$, we have \n\\[\\psi(t_{\\hat t})\\leq r_{\\hat t}<\\psi\\left((1+\\psi(1))\\hat t\\right).\\]\n Hence, we have \n$$\\lim_{\\hat t\\to0}\\frac{\\hat\\psi(\\hat t)}{\\hat t}\\leq C\\lim_{\\hat t\\to 0}\\frac{\\psi(\\hat t)}{\\hat t}=0$$\n and\n \\begin{equation}\\label{eq:100}\n\\frac{\\hat\\psi(\\hat t)}{\\hat t}=\\frac{r_{\\hat t}}{\\hat t}=\\frac{(1+\\psi(1))\\hat t-t_{\\hat t}}{\\hat t}=(1+\\psi(1))\\left(1-\\frac{t_{\\hat t}}{t_{\\hat t}+r_{\\hat t}}\\right).\n\\end{equation}\nBy (\\ref{eq:100}), to show that $\\hat\\psi(\\hat t)\/\\hat t$ is increasing, it suffices to show that \n\\begin{equation}\\label{eq:increasing}\n\\frac{r_{\\hat t_1}}{t_{\\hat t_1}}\\leq\\frac{ r_{\\hat t_2}}{t_{\\hat t_2}}\\ \\ {\\rm for}\\ \\  0<\\hat t_1\\leq\\hat t_2<1.\n\\end{equation}\nSince $\\psi$ is non-decreasing and $\\hat t_1\\leq\\hat t_2$, we always have $r_{\\hat t_1}\\leq r_{\\hat t_2}$. Hence, if $t_{\\hat t_1}=t_{\\hat t_2}$, we immediately obtain inequality (\\ref{eq:increasing}). If $t_{\\hat t_1}<t_{\\hat t_2}$, the fact that $\\psi$ is non-decreasing implies $r_{\\hat t_1}\\leq\\psi(t_{\\hat t_2})\\leq r_{\\hat t_2}$. Since $\\psi$ has at most countably many points of discontinuity, for every $\\epsilon>0$, we can find a point $t_\\epsilon\\in(t_{\\hat t_1}, t_{\\hat t_1}+\\epsilon)$ of continuity of $\\psi$  with $r_{\\hat t_1}\\leq \\psi(t_\\epsilon)$. The fact that $\\psi(t)\/t$ is non-decreasing implies \n\\[\\frac{r_{\\hat t_1}}{t_{\\hat t_1}}\\leq\\lim_{\\epsilon\\to0}\\frac{\\psi(t_\\epsilon)}{t_{\\hat t_1}+\\epsilon}\\leq\\lim_{\\epsilon\\to0}\\frac{\\psi(t_\\epsilon)}{t_\\epsilon}\\leq\\frac{\\psi(t_{\\hat t_2})}{t_{\\hat t_2}}\\leq\\frac{r_{\\hat t_2}}{t_{\\hat t_2}}.\\]\n\n\\textbf{$(1)$:} We show that the assumption that original cuspidal function $\\psi$ satisfies the integrability condition (\\ref{eq:inc1}) implies that the corresponding Lipschitz cuspidal function $\\hat\\psi$ also satisfies the integrability condition (\\ref{eq:inc1}). Define a measurable subset $B\\subset (0, 1]$ by setting \n\\[B:=\\{\\hat t\\in(0, 1]: t_{\\hat t}\\geq r_{\\hat t}\\}.\\]\nThen, for every $\\hat t\\in B$, we have $\\hat t\\sim t_{\\hat t}$ and $\\hat\\psi(\\hat t)=r_{\\hat t}\\geq\\psi(t_{\\hat t})$. Hence, we have \n\\begin{equation}\\label{eq:inc3}\n\\int_B\\left(\\frac{\\hat t^s}{\\hat\\psi(\\hat t)}\\right)^{\\frac{n}{s-1}}\\frac{d\\hat t}{\\hat t}\\leq C\\int_0^1\\left(\\frac{t^s_{\\hat t}}{\\psi(t_{\\hat t})}\\right)^{\\frac{n}{s-1}}\\frac{dt_{\\hat t}}{t_{\\hat t}}<\\infty.\n\\end{equation}\nIf $\\hat t\\in(0, 1]\\setminus B$, we have $\\hat\\psi(\\hat t)=r_{\\hat t}\\sim\\hat t$. Then we have \n\\begin{equation}\\label{eq:inc4}\n\\int_{(0, 1]\\setminus B}\\left(\\frac{\\hat t^s}{\\hat\\psi(\\hat t)}\\right)^{\\frac{n}{s-1}}\\frac{d\\hat t}{\\hat t}\\leq C\\int_0^1\\hat t^{n-1}d\\hat t<\\infty.\n\\end{equation}\nHence, by combining the last two inequalities, we obtain that $\\hat\\psi$ satisfies the integrability condition that\n\\[\\int_0^1\\left(\\frac{\\hat t^s}{\\hat\\psi(\\hat t)}\\right)^{\\frac{n}{s-1}}\\frac{d\\hat t}{\\hat t}<\\infty.\\]\nIn conclusion, we have shown that $\\hat\\psi$ satisfies all assumptions of the theorems due to Maz'ya and Poborchi in \\cite[page 304 and 312]{Mazya}. Hence there exists a bounded linear extension operator $\\widetilde E_1: W^{1, p}({\\Omega}^n_{\\hat\\psi})\\to W^{1, q}({{\\rr}^n})$ whenever $\\frac{1+(n-1)s}{n}\\leq p<\\infty$ and $1\\leq q\\leq\\frac{np}{1+(n-1)s}$. Then, for $\\frac{1+(n-1)s}{n}\\leq p<\\infty$, we define an extension operator $E_1$ on $W^{1,p}({\\Omega}^n_\\psi)$ by setting \n\\[E_1(u)(x):=\\widetilde E_1\\left(u\\circ\\mathcal O^{-1}\\right)(\\mathcal O(x))\\]\nfor every function $u\\in W^{1,p}({\\Omega}^n_\\psi)$ and every $x\\in{{\\rr}^n}$. By the facts that a bi-Lipschitz transformation preserves first order Sobolev spaces and that $\\widetilde E_1: W^{1, p}({\\Omega}^n_{\\hat\\psi})\\to W^{1, q}({{\\rr}^n})$ is a bounded linear extension operator whenever $\\frac{1+(n-1)s}{n}\\leq p<\\infty$ and $1\\leq q\\leq\\frac{np}{1+(n-1)s}$, we obtain that $E_1:W^{1, p}({\\Omega}^n_\\psi)\\to W^{1, q}({{\\rr}^n})$ is also a bounded linear extension operator whenever $\\frac{1+(n-1)s}{n}\\leq p<\\infty$ and $1\\leq q\\leq\\frac{np}{1+(n-1)s}$. The theorem from \\cite[page 312]{Mazya} tells us that there exists a bounded linear extension operator $\\widetilde E_2:W^{1, p}({\\Omega}^n_{\\hat\\psi})\\to W^{1, q}({{\\rr}^n})$ whenever $\\frac{1+(n-1)s}{2+(n-2)s}\\leq p<\\infty$ and $1\\leq q\\leq\\frac{(1+(n-1)s)p}{1+(n-1)s+(s-1)p}$. Then, for every $\\frac{1+(n-1)s}{2+(n-2)s}\\leq p<\\infty$, we define an extension operator $E_2$ on $W^{1, p}({\\Omega}^n_\\psi)$ by setting\n\\[E_2(u)(x):=\\widetilde E_2\\left(u\\circ\\mathcal O^{-1}\\right)(\\mathcal O(x))\\]\nfor every function $u\\in W^{1, p}({\\Omega}^n_\\psi)$ and $x\\in{{\\rr}^n}$. By the same reason as above, $E_2:W^{1, p}({\\Omega}^n_\\psi)\\to W^{1, q}({{\\rr}^n})$ is a bounded linear extension operator whenever $\\frac{1+(n-1)s}{2+(n-2)s}\\leq p<\\infty$ and $1\\leq q\\leq\\frac{(1+(n-1)s)p}{1+(n-1)s+(s-1)p}$. \n\n\\textbf{$(2):$} By an argument similar with the first case $(1)$, we obtain that if the original cuspidal function $\\psi$ satisfies the integrability condition (\\ref{eq:inc2}) then the corresponding Lipschitz cuspidal function $\\hat\\psi$ also satisfies the integrability condition (\\ref{eq:inc2}). Hence, we have shown that $\\hat\\psi$ satisfies all the assumptions of the theorem from \\cite[page 308]{Mazya}. This theorem tells us that there exists a bounded linear extension operator $\\widetilde E_3:W^{1, p}({\\Omega}^n_{\\hat\\psi})\\to W^{1, q}({{\\rr}^n})$ whenever $\\frac{(n-1)^2s+(n-1)}{n}\\leq p<\\infty$ and $1\\leq q\\leq n-1$. For every $\\frac{(n-1)^2s+(n-1)}{n}\\leq p<\\infty$, we define an extension operator $E_3$ on $W^{1, p}({\\Omega}^n_\\psi)$ by setting \\[E_3(u)(x):=\\widetilde E_3\\left(u\\circ\\mathcal O^{-1}\\right)(\\mathcal O(x))\\]\nfor every function $u\\in W^{1, p}({\\Omega}^n_\\psi)$ and $x\\in{{\\rr}^n}$. By the same reason as above, $E_3:W^{1, p}({\\Omega}^n_\\psi)\\to W^{1, q}({{\\rr}^n})$ is a bounded linear extension operator for every $\\frac{(n-1)^2s+(n-1)}{n}\\leq p<\\infty$ and $1\\leq q\\leq n-1$.\n\nNext, let us show the necessity of integrability conditions (\\ref{eq:inc1}) and (\\ref{eq:inc2}). First, let us show that if the doubling condition (\\ref{eq:doubling}) holds for a cuspidal function $\\psi$, it also holds for its corresponding Lipschitz cuspidal function $\\hat\\psi$. Since $\\hat\\psi$ is Lipschitz and increasing, it suffices to show that there exists a positive constant $C>1$ such that for every $\\hat t\\in\\left(0, \\frac{1}{2(1+\\psi(1))}\\right]$, we have \n\\begin{equation}\\label{eq:Doub}\n\\hat\\psi(2\\hat t)\\leq C\\hat\\psi(\\hat t).\n\\end{equation}\nLet $\\hat t\\in\\left(0, \\frac{1}{2(1+\\psi(1))}\\right]$ be arbitrary. There exists a unique pair $(t_{\\hat t}, r_{\\hat t})$ with \n\\begin{equation}\\label{eq:Ineq1}\n\\psi(t_{\\hat t})\\leq r_{\\hat t}\\leq\\lim_{s\\to t_{\\hat t}^+}\\psi(s)\n\\end{equation}\nand \n\\begin{equation}\\label{eq:Ineq2}\nt_{\\hat t}+r_{\\hat t}=(1+\\psi(1))\\hat t.\n\\end{equation}\nMoreover, there exists a unique pair $(t_{2\\hat t}, r_{2\\hat t})$ with \n\\begin{equation}\\label{eq:Ineq3}\n\\psi(t_{2\\hat t})\\leq r_{2\\hat t}\\leq\\lim_{s\\to t^+_{2\\hat t}}\\psi(s)\n\\end{equation}\nand \n\\begin{equation}\\label{eq:Ineq4}\nt_{2\\hat t}+r_{2\\hat t}=(1+\\psi(1))2\\hat t.\n\\end{equation}\nBy the definition in Lemma \\ref{le:LipcF}, we have \n\\[\\hat\\psi(2\\hat t)=r_{2\\hat t}\\ {\\rm and}\\ \\hat\\psi(\\hat t)=r_{\\hat t}.\\]\nIf $r_{2\\hat t}\\leq 2 r_{\\hat t}$, then (\\ref{eq:Doub}) holds with $C=2$. Hence, we assume $r_{2\\hat t}>2r_{\\hat t}$. By (\\ref{eq:Ineq2}) and (\\ref{eq:Ineq4}), we have $t_{2\\hat t}<2 t_{\\hat t}$. Since $\\psi$ is increasing and satisfies inequality (\\ref{eq:doubling}), by (\\ref{eq:Ineq1}) and (\\ref{eq:Ineq3}), we have \n\\[r_{2\\hat t}\\leq\\psi(2t_{\\hat t})\\leq C\\psi(t_{\\hat t})\\leq Cr_{\\hat t}.\\]\nWe have showed inequality (\\ref{eq:Doub}) holds for every $\\hat t\\in\\left(0, \\frac{1}{2(1+\\psi(1))}\\right]$. The fact that ${\\Omega}^n_\\psi$ is globally bi-Lipschitz equivalent to ${\\Omega}^n_{\\hat\\psi}$ implies that ${\\Omega}^n_\\psi$ and ${\\Omega}^n_{\\hat\\psi}$ have the same Sobolev extension properties. By the results due to Maz'ya and Poborch in \\cite[pages 304 and 312]{Mazya}, if ${\\Omega}^n_{\\hat\\psi}$ is a Sobolev $(p, q)$-extension domain with \\[\\left(\\frac{1+(n-1)s}{n}\\leq p<\\infty, 1\\leq q\\leq\\frac{np}{1+(n-1)s}\\right)\\] or \\[\\left(\\frac{1+(n-1)s}{2+(n-2)s}\\leq p<\\infty, 1\\leq q\\leq\\frac{(1+(n-1)s)p}{1+(n-1)s+(s-1)p}\\right),\\] then $\\hat\\psi$ satisfies the integrability condition\n\\[\\int_0^1\\left(\\frac{t^s}{\\hat\\psi(t)}\\right)^{\\frac{n}{s-1}}\\frac{dt}{t}<\\infty.\\] \nBy the theorem from \\cite[page 308]{Mazya}, if ${\\Omega}^n_{\\hat\\psi}$ is a Sobolev $(p, q)$-extension domain with $\\frac{(n-1)^2s+(n-1)}{n}\\leq p<\\infty$ and $1\\leq q\\leq n-1$, then $\\hat\\psi$ satisfies the integrability condition\n\\[\\int_0^1\\left(\\frac{t^s}{\\hat\\psi(t)}\\right)^{\\frac{n}{s-1}}\\left|\\log\\left(\\frac{\\hat\\psi(t)}{t}\\right)\\right|^{-\\alpha}\\frac{dt}{t}<\\infty\\]\nfor $\\alpha=\\frac{(n-2)p}{p+1-n}$. By the definition of $\\hat\\psi$, for every $t\\in(0, 1]$, we always have \n\\[\\psi(t)\\geq\\hat\\psi\\left(\\frac{1}{1+\\psi(1)}t\\right).\\]\nHence, if the corresponding Lipschitz cuspidal function $\\hat\\psi$ satisfies the integrability conditions (\\ref{eq:inc1}) and (\\ref{eq:inc2}), the original cuspidal function $\\psi$ also satisfies them.\n\\end{proof}}\n\nLet us prove Theorem \\ref{thm:main}.\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nAs we mentioned, every outward cuspidal domain is a Sobolev $(\\infty, \\infty)$-extension domain. Hence, it suffices to deal with the case $1\\leq q\\leq p<\\infty$. We first prove the result for Lipschitz outward cuspidal domains and then extend the result to arbitrary outward cuspidal domains via the global bi-Lipschitz equivalence method established in Proposition \\ref{prop:bilip}.\n\nLet $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p<\\infty$ be fixed. Let $\\psi$ be a Lipschitz cuspidal function. We define a cylinder $\\widehat{\\mathsf C}_o$ by setting\n\\begin{equation}\\label{eq:cylin}\n\\widehat{\\mathsf C}_o:=\\{(t, x)\\in[1, 3)\\times{\\mathbb R}^{n-1}: |x|<2\\psi(1)\\}.\n\\end{equation}\nThen we define two sub-cylinders $\\widehat{\\mathsf C}_o^1$ and $\\widehat{\\mathsf C}_o^2$ of $\\widehat{\\mathsf C}_o$ by setting\n\\[\\widehat{\\mathsf C}_o^1:=\\{(t, x)\\in(1, 2)\\times{\\mathbb R}^{n-1}:|x|<2\\psi(1)\\}\\]\nand \n\\[\\widehat{\\mathsf C}_o^2:=\\{(t, x)\\in(2, 3)\\times{\\mathbb R}^{n-1}:|x|<2\\psi(1)\\}.\\]\nWe also define a sub-cylinder $\\mathsf C_o^1$ of $\\widehat{\\mathsf C}_o^1$ by setting \n\\[\\mathsf C_o^1:=\\{(t, x)\\in(1, 2)\\times{\\mathbb R}^{n-1}: |x|<\\psi(1)\\}.\\]\nThen $A_{\\mathsf C_o^1}:=\\widehat{\\mathsf C}_o^1\\setminus\\overline{\\mathsf C}_o^1$ is an annular set. We define a reflection $\\widetilde{\\mathcal R_1}: A_{\\mathsf C_o^1}\\to\\mathsf C_o^1$ by setting\n\\begin{equation}\\label{eq:ref0}\n\\widetilde{\\mathcal R_1}(t, x):=\\left(t, \\left(\\frac{3}{2}\\psi(1)-\\frac{|x|}{2}\\right)\\frac{x}{|x|}\\right).\n\\end{equation}\nThere exists a positive constant $C$ such that for every $(t, x)\\in A_{\\mathsf C_o^1}$, we have \n\\begin{equation}\\label{eq:est1}\n|D\\widetilde{\\mathcal R_1}(t, x)|\\leq C\\ {\\rm and}\\ \\frac{1}{C}\\leq|J_{\\widetilde{\\mathcal R_1}}(t, x)|\\leq C.\n\\end{equation}\nWe also define a cut-off function $\\widetilde L_1$ on the annular set $\\overline{A_{\\mathsf C_o^1}}$ with $\\widetilde L_1\\equiv 0$ on $[1, 2]\\times\\partial B^{n-1}(0, 2\\psi(1))$ and $\\widetilde L_1\\equiv 1$ on $[1, 2]\\times\\partial B^{n-1}(0, \\psi(1))$ by setting \n\\begin{equation}\\label{eq:cutoff1}\n\\widetilde L_1(t, x):=2-\\frac{|x|}{\\psi(1)}.\n\\end{equation}\nThere exists a positive constant $C$, such that for every $(t, x)\\in A_{\\mathsf C_o^1}$, we have \n\\begin{equation}\\label{eq:bond1}\n|\\nabla\\widetilde L_1(t, x)|\\leq C\n\\end{equation}\nNext, we define a reflection $\\widetilde{\\mathcal R_2}:\\widehat{\\mathsf C}_o^2\\to\\widehat{\\mathsf C}_o^1$ by setting \n\\begin{equation}\\label{eq:Ref1}\n\\widetilde{\\mathcal R_2}(t, x):=(t-2, x).\n\\end{equation} \nThere exists a positive constant $C$, such that for every $(t, x)\\in\\widehat{\\mathsf C}_o^2$, we have\n\\begin{equation}\\label{eq:bond2}\n|D\\widetilde{\\mathcal R_2}(t, x)|\\leq C\\ {\\rm and}\\ \\frac{1}{C}\\leq|J_{\\widetilde{\\mathcal R_2}}(t, x)|\\leq C.\n\\end{equation}\nMoreover we define a cut-off function $\\widetilde L_2$ on $\\overline{\\widehat{\\mathsf C}_o^2}$  with $\\widetilde L_2\\equiv 1$ on $\\{2\\}\\times\\overline{B^{n-1}(0, 2\\psi(1))}$ and $\\widetilde L_2\\equiv 0$ on $\\{3\\}\\times\\overline{B^{n-1}(0, 2\\psi(1))}$ by setting\n\\begin{equation}\\label{eq:cutoff2}\n\\widetilde L_2(t, x):=3-t.\n\\end{equation}\nThen, for every $(t, x)\\in\\widehat{\\mathsf C}_o^2$, we have \n\\begin{equation}\\label{eq:bond3}\n|\\nabla\\widetilde L_2(t, x)|\\leq 2.\n\\end{equation}\nNext, we define a double outward cuspidal domain $\\widehat{{\\Omega}_\\psi^n}$ by setting \n\\[\\widehat{{\\Omega}_\\psi^n}:=\\{(t, x)\\in(0, 1]\\times{\\mathbb R}^{n-1}: |x|<2\\psi(t)\\}\\cup\\widehat{\\mathsf C}_o.\\]\nWe will construct a bounded linear extension operator $E$ from $W^{1,p}({\\Omega}_\\psi^n)$ to $W^{1,q}({{\\rr}^n})$ such that for every function $u\\in W^{1.p}({\\Omega}_\\psi^n)$, we have $E(u)=0$ on $\\partial\\widehat{{\\Omega}_\\psi^n}\\setminus\\{0\\}$. We define an annular-type set by setting\n$$A_\\psi^n:=\\{(t, x)\\in(0, 1]\\times{\\mathbb R}^{n-1}: \\psi(t)<|x|<2\\psi(t)\\}.$$\nMoreover we define a reflection $\\mathcal R: A_\\psi^n\\to{\\Omega}_\\psi^n$ by setting \n\\begin{equation}\\label{eq:3ref}\n\\mathcal R(z)=\\mathcal R(t, x):=\\left(t, \\left(\\frac{-|x|}{2}+\\frac{3}{2}\\psi(t)\\right)\\frac{x}{|x|}\\right).\n\\end{equation}\nSince $\\psi$ is Lipschitz, there exists a positive constant $C$ such that for every $z=(t, x)\\in A_\\psi^n$, we have \n\\begin{equation}\\label{eq:Esref}\n|D\\mathcal R(z)|\\leq C\\ {\\rm and}\\ \\frac{1}{C}\\leq\\left|J_{\\mathcal R}(z)\\right|\\leq C.\n\\end{equation}\nWe define a cut-off function $L$ on $A_\\psi^n$ by setting \n\\begin{equation}\\label{eq:Cutoff}\nL(t, x):=\\frac{-|x|}{\\psi(t)}+2.\n\\end{equation}\nSince $\\psi$ is Lipschitz and \n$$\\psi(t)<|x|<2\\psi(t)$$\nfor every point $(t, x)\\in A_\\psi^n$, there exists a positive constant $C$ such that \n\\begin{equation}\\label{eq:HP1}\n\\left|\\nabla L(t, x)\\right|\\leq\\frac{C}{\\psi(t)}\\ {\\rm for\\ almost\\ every}\\ (t, x)\\in A_\\psi^n.\n\\end{equation}\n\nBy Lemma \\ref{lem:density}, $C_o^\\infty({{\\rr}^n})\\cap W^{1, p}({\\Omega}_\\psi^n)$ is dense in $W^{1, p}({\\Omega}_\\psi^n)$. We first define a bounded linear extension operator from the dense subspace $C_o^\\infty({{\\rr}^n})\\cap W^{1,p}({\\Omega}_\\psi^n)$ to $W^{1,q}({{\\rr}^n})$ and then extend it to the full space $W^{1,p}({\\Omega}_\\psi^n)$. Let $u\\in C_o^\\infty({{\\rr}^n})\\cap W^{1,p}({\\Omega}_\\psi^n)$ be fixed. We define an extension $E(u)$ on $\\widehat{{\\Omega}_\\psi^n}$ by setting \n\\begin{equation}\\label{eq:ExtenO}\nE(u)(z):=\\begin{cases}\nu(z), & z\\in\\overline{{\\Omega}_\\psi^n}, \\\\\nL(z)(u\\circ\\mathcal R)(z), & z\\in A_\\psi^n,\\\\\n\\widetilde L_1(z)(u\\circ\\widetilde{\\mathcal R}_1)(z), & z\\in A_{\\mathsf C_o^1}, \\\\\n\\widetilde L_2(z)(E(u)\\circ\\widetilde{\\mathcal R}_2)(z), & z\\in \\widehat{\\mathsf C}_o^2\\, .\n\\end{cases}\n\\end{equation}\nThen, $E(u)$ is continuous on $\\widehat{{\\Omega}_\\psi^n}$ with $E(u)\\equiv 0$ on $\\partial\\widehat{{\\Omega}_\\psi^n}\\setminus\\{0\\}$. Hence, we can simply extend $E(u)$ to be zero outside the domain $\\widehat{{\\Omega}_\\psi^n}$. First, let us estimate the $L^q$-norm of $E(u)$. By the definition of $E(u)$, the H\\\"older inequality implies \n\\begin{equation}\\label{eq:lpno1}\n\\left(\\int_{{\\Omega}_\\psi^n}|E(u)(z)|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nBy (\\ref{eq:bond2}) and the fact that $0\\leq\\widetilde L_2\\leq 1$ on $\\widehat{\\mathsf C}_o^2$, the change of variables formula implies\n\\begin{equation}\\label{eq:lpno2}\n\\int_{\\widehat{\\mathsf C}_o^2}|E(u)(z)|^qdz\\leq C\\int_{\\widehat{\\mathsf C}_o^2}\\left|\\left(E(u)\\circ\\widetilde{\\mathcal R}_2\\right)(z)\\right|^qdz\\leq C \\int_{\\widehat{\\mathsf C}_o^1}|E(u)(z)|^qdz.\n\\end{equation}\nBy (\\ref{eq:est1}) and the fact that $0\\leq\\widetilde L_1\\leq 1$ on $A_{\\mathsf C_o^1}$, the change of variables formula and the H\\\"older inequality imply \n\\begin{multline}\\label{eq:lpno3}\n\\left(\\int_{\\widehat{\\mathsf C}_o^1}|E(u)(z)|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{\\mathsf C_o^1}\\left|E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\\\\n+C\\left(\\int_{A_{\\mathsf C_o^1}}\\left|E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{multline}\nBy combining (\\ref{eq:lpno2}) and (\\ref{eq:lpno3}), we obtain \n\\begin{equation}\\label{eq:lpno4}\n\\left(\\int_{\\widehat{\\mathsf C}_o}|E(u)(z)|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nBy (\\ref{eq:Esref}) and the fact that $0\\leq L(z)\\leq 1$ on $A_\\psi^n$, the change of variables formula and the H\\\"older inequality imply \n\\begin{equation}\\label{eq:Hp1}\n\\left(\\int_{A_\\psi^n}|L(z)(u\\circ\\mathcal R)(z)|^qdz\\right)^{\\frac{1}{q}}\\leq \\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nCombining inequalities (\\ref{eq:lpno1}), (\\ref{eq:lpno4}) and (\\ref{eq:Hp1}), we obtain \n\\begin{equation}\\label{eq:Hp2}\n\\left(\\int_{\\widehat{{\\Omega}_\\psi^n}}|E(u)(z)|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\n\nNow, let us estimate the $L^q$-norm of $\\left|\\nabla E(u)\\right|$. First, the H\\\"older inequality implies\n\\begin{equation}\\label{eq:Whp1}\n\\left(\\int_{{\\Omega}_\\psi^n}|\\nabla E(u)(z)|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|\\nabla u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation} \nThe chain rule implies that for almost every $z\\in A_{\\mathsf C_o^1}$, we have \n\\begin{equation}\\label{eq:spno30}\n\\left|\\nabla E(u)(z)\\right|\\leq\\left|\\nabla\\tilde L_1(z)(u\\circ\\widetilde{\\mathcal R}_1)(z)\\right|+\\left|\\tilde L_1(z)\\nabla(u\\circ\\widetilde{\\mathcal R}_1)(z)\\right|.\n\\end{equation}\nBy (\\ref{eq:est1}) and (\\ref{eq:bond1}), the change of variables formula and the H\\\"older inequality imply\n\\begin{equation}\\label{eq:spno31}\n\\left(\\int_{A_{\\mathsf C_o^1}}\\left|\\nabla\\tilde L_1(z)(u\\circ\\widetilde{\\mathcal R}_1)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}\\left|u(z)\\right|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nBy (\\ref{eq:est1}) and the fact that $0\\leq\\tilde L_1(z)\\leq 1$ for every $z\\in A_{\\mathsf C_o^1}$, the change of variables formula and the H\\\"older inequality imply \n\\begin{equation}\\label{eq:spno32}\n\\left(\\int_{A_{\\mathsf C_o^1}}\\left|\\tilde L_1(z)\\nabla(u\\circ\\widetilde{\\mathcal R}_1)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}\\left|\\nabla u(z)\\right|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nBy combining (\\ref{eq:spno30}), (\\ref{eq:spno31}) and (\\ref{eq:spno32}), we obtain \n\\begin{equation}\\label{eq:spno33}\n\\left(\\int_{A_{\\mathsf C_o^1}}\\left|\\nabla E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}\\left|u(z)\\right|^p+\\left|\\nabla u(z)\\right|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nThe chain rule implies that for almost every $z\\in\\widehat{\\mathsf C}_o^2$, we have \n\\begin{equation}\\label{eq:spno34}\n\\left|\\nabla E(u)(z)\\right|\\leq\\left|\\nabla\\tilde L_2(z)\\left(E(u)\\circ\\widetilde{\\mathcal R}_2\\right)(z)\\right|+\\left|\\tilde L_2(z)\\nabla\\left(E(u)\\circ\\widetilde{\\mathcal R}_2\\right)(z)\\right|.\n\\end{equation}\nBy (\\ref{eq:bond2}), (\\ref{eq:bond3}) and (\\ref{eq:lpno3}), the change of variables formula and the H\\\"older inequality imply\n\\begin{multline}\\label{eq:spno35}\n\\left(\\int_{\\widehat{\\mathsf C}_o^2}\\left|\\nabla\\tilde L_2(z)\\left(E(u)\\circ\\widetilde{\\mathcal R}_2\\right)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{\\widehat{\\mathsf C}_0^1}\\left|E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\\\\n\\leq C\\left(\\int_{{\\Omega}_\\psi^n}\\left|u(z)\\right|^pdz\\right)^{\\frac{1}{p}}.\n\\end{multline}\nBy (\\ref{eq:bond2}) and the fact that $0\\leq\\tilde L_2(z)\\leq 1$ for almost every $z\\in\\widehat{\\mathsf C}_o^2$, the change of variables formula, (\\ref{eq:Whp1}) and (\\ref{eq:spno33}) imply \n\\begin{multline}\\label{eq:spno36}\n\\left(\\int_{\\widehat{\\mathsf C}_o^2}\\left|\\tilde L_2(z)\\nabla\\left(E(u)\\circ\\widetilde{R}_2\\right)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{\\widehat{\\mathsf C}_o^1}\\left|\\nabla E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\\\\n\\leq C\\left(\\int_{A_{\\mathsf C_o^1}}\\left|\\nabla E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}+\\left(\\int_{\\mathsf C_o^1}\\left|\\nabla E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\\\\n\\leq C\\left(\\int_{{\\Omega}_\\psi^n}\\left|u(z)\\right|^p+\\left|\\nabla u(z)\\right|^pdz\\right)^{\\frac{1}{p}}.\n\\end{multline}\nHence, by combining (\\ref{eq:spno34}), (\\ref{eq:spno35}) and (\\ref{eq:spno36}), we obtain \n\\begin{equation}\\label{eq:spno37}\n\\left(\\int_{\\widehat{\\mathsf C}^2_o}\\left|\\nabla E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}\\left|u(z)\\right|^p+\\left|\\nabla u(z)\\right|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nSince $\\mathsf C_o^1\\subset{\\Omega}_\\psi^n$, by combining (\\ref{eq:Whp1}), (\\ref{eq:spno33}) and (\\ref{eq:spno37}), we obtain \n\\begin{equation}\\label{eq:spno38}\n\\left(\\int_{\\widehat{\\mathsf C}_o}\\left|\\nabla E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}\\left|u(z)\\right|^p+\\left|\\nabla u(z)\\right|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nThe chain rule implies that for almost every $z\\in A_\\psi^n$, we have \n\\begin{equation}\\label{eq:Hp3}\n\\left|\\nabla E(u)(z)\\right|\\leq \\left|\\nabla L(z)(u\\circ\\mathcal R)(z)\\right|+\\left|L(z)\\nabla(u\\circ\\mathcal R)(z)\\right|.\n\\end{equation}\nBy (\\ref{eq:Esref}) and the fact $0\\leq L(z)\\leq 1$ on $A_\\psi^n$, the change of variables formula and the H\\\"older inequality yield \n\\begin{equation}\\label{eq:Hp4}\n\\left(\\int_{A_\\psi^n}\\left|L(z)\\nabla(u\\circ\\mathcal R)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq \\left(\\int_{{\\Omega}_\\psi^n}|\\nabla u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation} \nThe H\\\"older inequality implies \n\\begin{equation}\\label{eq:Hp5}\n\\left(\\int_{A_\\psi^n}\\left|\\nabla L(z)(u\\circ\\mathcal R)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq\\left(\\int_{A_\\psi^n}|\\nabla L(z)|^{\\frac{pq}{p-q}}dz\\right)^{\\frac{p-q}{pq}}\\cdot\\left(\\int_{A_\\psi^n}|(u\\circ\\mathcal R)(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nBy (\\ref{eq:Esref}), the change of variables formula implies \n\\begin{equation}\\label{eq:Hp6}\n\\int_{A_\\psi^n}|(u\\circ\\mathcal R)(z)|^pdz\\leq C\\int_{{\\Omega}_\\psi^n}|u(z)|^pdz.\n\\end{equation}\nBy (\\ref{eq:HP1}) and the fact that $\\psi$ is Lipschitz, we have \n\\begin{equation}\\label{eq:Hp7}\n\\int_{A_\\psi^n}|\\nabla L(z)|^{\\frac{pq}{p-q}}\\leq C\\int_0^1\\psi(t)^{n-1-\\frac{pq}{p-q}}dt<C\\int_0^1t^{n-1-\\frac{pq}{p-q}}dt<\\infty\n\\end{equation}\nwhenever $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p<\\infty$. By combining inequalities (\\ref{eq:Hp5}), (\\ref{eq:Hp6}) and (\\ref{eq:Hp7}), we obtain \n\\begin{equation}\\label{eq:Hp8}\n\\left(\\int_{A_\\psi^n}\\left|\\nabla L(z)(u\\circ\\mathcal R)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nBy combining (\\ref{eq:Hp3}), (\\ref{eq:Hp4}) and (\\ref{eq:Hp8}), we obtain \n\\begin{equation}\\label{eq:Hp9}\n\\left(\\int_{A_\\psi^n}\\left|\\nabla E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^p+|\\nabla u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nBy combining inequalities (\\ref{eq:Whp1}), (\\ref{eq:spno37}) and (\\ref{eq:Hp9}), we obtain \n\\begin{equation}\\label{eq:Hp10}\n\\left(\\int_{\\widehat{{\\Omega}_\\psi^n}}|\\nabla E(u)(z)|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^p+|\\nabla u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nFinally, by combining (\\ref{eq:Hp2}) and (\\ref{eq:Hp10}), we obtain the desired norm inequality\n\\begin{equation}\\label{eq:Hp11}\n\\left(\\int_{\\widehat{{\\Omega}_\\psi^n}}\\left|E(u)(z)\\right|^q+\\left|\\nabla E(u)(z)\\right|^qdz\\right)^{\\frac{1}{q}}\\leq C\\left(\\int_{{\\Omega}_\\psi^n}|u(z)|^p+|\\nabla u(z)|^pdz\\right)^{\\frac{1}{p}}.\n\\end{equation}\nHence, the extension operator $E$ defined in (\\ref{eq:ExtenO}) is a bounded linear extension operator from the dense subspace $C_o^\\infty({{\\rr}^n})\\cap W^{1,p}({\\Omega}_\\psi^n)$ to $W^{1,q}({{\\rr}^n})$ whenever $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p<\\infty$. For an arbitrary $u\\in W^{1,p}({\\Omega}_\\psi^n)$, there exists a Cauchy sequence $\\{u_m\\}_{m=1}^\\infty\\subset C_o^\\infty({{\\rr}^n})\\cap W^{1,p}({\\Omega}_\\psi^n)$ which converges to $u$ with respect to the $W^{1,p}$-norm. Since $E$ is a  bounded linear extension operator from $C_o^\\infty({{\\rr}^n})\\cap W^{1,p}({\\Omega}_\\psi^n)$ to $W^{1,q}({{\\rr}^n})$, there exists a subsequence of $\\{u_m\\}$ which converges to $u$ almost everywhere on ${\\Omega}_\\psi^n$ and $\\{E(u_m)\\}$ is also a Cauchy sequence in $W^{1,q}({{\\rr}^n})$ which converges to some function $v\\in W^{1,q}({{\\rr}^n})$. To simplify the notation, we still denote this subsequence by $\\{u_m\\}$. Then, we have $v\\big|_{{\\Omega}_\\psi^n}(z)=u(z)$ for almost every $z\\in{\\Omega}_\\psi^n$ and \n\\[\\|v\\|_{W^{1,q}({{\\rr}^n})}\\leq\\lim_{m\\to\\infty}\\|E(u_m)\\|_{W^{1,q}({{\\rr}^n})}\\leq C\\lim_{m\\to\\infty}\\|u_m\\|_{W^{1,p}({\\Omega}_\\psi^n)}\\leq C\\|u\\|_{W^{1,p}({\\Omega}_\\psi^n)}.\\]\nFurthermore, by picking an extra subsequence if necessary, it follows that $v(z)=E(u)(z)$ for almost every $z\\in{{\\rr}^n}$. Hence, $E$ is a bounded linear extension operator from $W^{1,p}({\\Omega}_\\psi^n)$ to $W^{1,q}({{\\rr}^n})$, whenever $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p<\\infty$. In conclusion, we have proved that a Lipschitz outward cuspidal domain ${\\Omega}_\\psi^n$ is a Sobolev $(p, q)$-extension domain, whenever $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p<\\infty$.\n\nNext, we extend the result to an arbitrary outward cuspidal domain by using global bi-Lipschitz transformations. Let $\\psi:(0, 1]\\to(0, \\infty)$ be an arbitrary cuspidal domain. By Proposition \\ref{prop:bilip}, there exists a Lipschitz cuspidal function $\\hat\\psi$ and a global bi-Lipschitz homeomorphism $\\mathcal O:{{\\rr}^n}\\to{{\\rr}^n}$ with $\\mathcal O({\\Omega}_\\psi^n)={\\Omega}_{\\hat\\psi}^n$. Fix $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p<\\infty$. Let $u\\in W^{1,p}({\\Omega}_\\psi^n)$ be arbitrary. Since $\\mathcal O$ is bi-Lipschitz with $\\mathcal O({\\Omega}_\\psi^n)={\\Omega}_{\\hat\\psi}^n$, we have $u\\circ\\mathcal O^{-1}\\in W^{1,p}({\\Omega}_{\\hat\\psi}^n)$ with \n\\begin{equation}\\label{eq:ineq1}\n\\|u\\circ\\mathcal O^{-1}\\|_{W^{1,p}({\\Omega}_{\\hat\\psi}^n)}\\leq C\\|u\\|_{W^{1, p}({\\Omega}_\\psi^n)}.\n\\end{equation}\nBy the argument above, the Lipschitz outward cuspidal domain ${\\Omega}_{\\hat\\psi}^n$ is a Sobolev $(p, q)$-extension domain, there exists a function $E(u\\circ\\mathcal O^{-1})\\in W^{1,q}({{\\rr}^n})$ with $E(u\\circ\\mathcal O^{-1})\\big|_{{\\Omega}_{\\hat\\psi}^n}\\equiv u\\circ\\mathcal O^{-1}$ and\n\\begin{equation}\\label{eq:ineq2}\n\\|E(u\\circ\\mathcal O^{-1})\\|_{W^{1,q}({{\\rr}^n})}\\leq C\\|u\\circ\\mathcal O^{-1}\\|_{W^{1,p}({\\Omega}_{\\hat\\psi}^n)}.\n\\end{equation}\nBy the fact that $\\mathcal O$ is bi-Lipschitz, we have $(E(u\\circ\\mathcal O^{-1}))\\circ\\mathcal O\\in W^{1,q}({{\\rr}^n})$ with\n\\begin{equation}\\label{eq:ineq3}\n\\|(E(u\\circ\\mathcal O^{-1}))\\circ\\mathcal O\\|_{W^{1,q}({{\\rr}^n})}\\leq C\\|E(u\\circ\\mathcal O^{-1})\\|_{W^{1,q}({{\\rr}^n})}.\n\\end{equation}\nBy the definitions, we have $(E(u\\circ\\mathcal O^{-1}))\\circ\\mathcal O\\big|_{{\\Omega}_\\psi^n}\\equiv u$. By combining (\\ref{eq:ineq1}), (\\ref{eq:ineq2}) and (\\ref{eq:ineq3}), we obtain the desired norm inequality \n\\[\\|(E(u\\circ\\mathcal O^{-1}))\\circ\\mathcal O\\|_{W^{1,q}({{\\rr}^n})}\\leq C\\|u\\|_{W^{1, p}({\\Omega}_\\psi^n)}.\\]\nHence, ${\\Omega}_\\psi^n$ is a Sobolev $(p, q)$-extension domain whenever $1\\leq q<n-1$ and $(n-1)q\/(n-1-q)\\leq p<\\infty$.\n\\end{proof}\n\nLet us prove Proposition \\ref{proposition} which will show the sharpness of Theorem \\ref{thm:main}.\n\\begin{proof}[Proof of Proposition \\ref{proposition}]\nLet $n-1\\leq p<\\infty$. Set \n\\[s_1:=\\frac{np-(n-1)}{(n-1)^2}.\\] \nBy Theorem \\ref{prop:mazya}, ${\\Omega}^n_{t^s}$ is a Sobolev $(p, n-1)$-extension domain if and only if $1\\leq s\\leq s_1$. Let $1\\leq q<n-1$ and $q\\leq p<\\frac{(n-1)q}{n-1-q}$. Set \n\\[s_2:=\\frac{pq+p-q}{pq+(n-1)(q-p)}.\\]\nBy Theorem \\ref{prop:mazya}, ${\\Omega}^n_{t^s}$ is a Sobolev $(p, q)$-extension domain if and only if $1\\leq s<s_2$.\n\\end{proof}\n\n\\section{Further comments}\nIn the monograph \\cite{Mazya} and their papers referred to therein, Maz'ya and Poborchi also dealt with  generalized outward cuspidal domains with Lipschitz base domains. To be more precise, for a bounded Lipschitz domain $U\\in{\\mathbb R}^{n-1}$ with $0\\in U$ and a cuspidal function $\\psi:(0, 1]\\to(0, \\infty)$, the corresponding generalized outward cuspidal domain with the base domain $U$ is defined by setting\n\\begin{equation}\\label{eq:Lipcusp}\nU_\\psi^n:=\\left\\{(t, x)\\in(0, 1]\\times{\\mathbb R}^{n-1}: x\\in \\psi(t)U\\right\\}\\cup\\left\\{(t, x)\\in(1, 2)\\times{\\mathbb R}^{n-1}: x\\in\\psi(1)U\\right\\}.\n\\end{equation} \nWe have only discussed outward cuspidal domains whose base domains are the unit ball. Maz'ya and Poborchi established results for generalized Lipschitz outward cuspidal domains $U_\\psi^n$ with Lipschitz base domains $U\\subset{\\mathbb R}^{n-1}$. Hence, our results in Theorem \\ref{prop:mazya} only extend Maz'ya and Poborchi's results in a special case. For a full extension, one would need to establish a global bi-Lipschitz equivalence of Proposition \\ref{prop:bilip} analog for general outward cuspidal domains $U_\\psi^n$. This appears to be technically challenging but we expect it to be doable.\n\\begin{conjecture}\nFor every generalized outward cuspidal domain $U_\\psi^n$, there exists a Lipschitz cuspidal function $\\hat\\psi$, a Lipschitz base domain $\\hat U$ and a global bi-Lipschitz transformation $\\mathcal O:{\\mathbb R}^n\\to{\\mathbb R}^n$ with $\\mathcal O(U_\\psi^n)=\\hat U_{\\hat\\psi}^n$.\n\\end{conjecture}\n\n \n \n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFunctional inequalities of Dirichlet forms are powerful tools in the study of Markov semigroups and spectral theory of Dirichlet operators, see \\cite{B,D,G,Wbook} for accounts on functional inequalities and applications. To establish a functional inequality, one often needs to verify some conditions on the generator, for instances the Bakry-Emery curvature condition in the diffusion setting or the Lyapunov condition in a\n general setting, see e.g. \\cite{CGWW,GLWN,Wbook}. Since these conditions exclude generators with less regular coefficients, to establish functional inequalities in a more general setting one treats the singularity part as a perturbation. So, it is important to investigate perturbations of functional inequalities.\n\nIn \\cite{BLW}, sharp growth conditions have been presented for perturbations of super Poincar\\'e and  weak Poincar\\'e inequalities in the diffusion setting (i.e.\\ the underlying Dirichlet form is local). Note that these two kinds of functional inequalities are general enough to cover all Poincar\\'e\/Sobolev\/Nash type inequalities, and thus, have a broad range of applications. Recently, explicit sufficient conditions were derived in \\cite{CW1, CW2, MRS, WW} for functional inequalities of stable-like   Dirichlet forms. The aim of this paper is to extend perturbation results derived in \\cite{BLW} to the non-local setting, so that combining with the existing sufficient conditions we are able to establish functional inequalities for more general Dirichlet forms. Due to the lack of the chain rule,  the study of the non-local setting is usually more complicated. Nevertheless, we are able to present some relatively clean perturbation results, which are sharp as illustrated by some examples latter on, and when the range of jumps is finite,  are natural extensions to the corresponding  results derived earlier for diffusion processes.\n\nLet $(E,d)$ be a Polish space equipped with the Borel $\\si$-field $\\F$ and a probability measure $\\mu$. Let $\\B(E)$ be the set of all measurable functions on $E$, and let $\\B_b(E)$ be the set of all bounded elements in $\\B(E).$ Let $q\\in\\B(E\\times E)$ be  non-negative   with $q(x,x)=0, x\\in E,$ such that\n\\begin{equation}\\label{intro-02}\n\\ll:= \\sup_{x\\in E}\\int_{E} \\big(1\\land d(x,y)^2\\big)\nq(x,y) \\mu(\\d y)<\\infty.\\end{equation}Then\n\\begin{equation}\\label{intro-01}\\beg{split}\n&\\GG(f,g)(x):= \\int_{E} (f(x)-f(y))(g(x)-g(y)) q(x,y)\\mu(\\d y),\\ \\ \\forall x \\in E,\\\\\n&f,g\\in \\scr A:= \\big\\{f\\in \\B_b(E):  \\GG(f,f)\\in \\B_b(E)\\big\\}\\end{split}\\end{equation} gives rise to a non-negatively definite bilinear map from $\\A\\times\\A$ to $\\B_b(E).$  We assume that $\\A$ is dense in $L^2(\\mu)$. Then it is standard that the form\n\\beq\\label{intro-03} \\beg{split}\\E(f,g):&= \\mu(\\GG(f,g))\\\\\n &=\\iint_{E\\times E} (f(x)-f(y))(g(x)-g(y))q(x,y)\\mu(\\d y)\\mu(\\d x),\\ \\ f,g\\in \\A\\end{split}\\end{equation}  is closable in $L^2(\\mu)$ and its closure $(\\E,\\D(\\E))$ is a symmetric conservative Dirichlet form,\nsee e.g.\\ \\cite[Example 1.2.6]{FOT}. A typical example of the\nframework is the $\\aa$-stable-like Dirichlet form, where $E=\\R^m$\nwith $d(x,y)=|x-y|$,  and\n$$q(x,y)= \\ff{\\tt q (x,y)}{u(y)|x-y|^{m+\\aa}},\\ \\ \\mu(\\d x)= u(x)\\d x$$ for some $\\aa\\in (0,2)$,   non-negative $\\tt q\\in \\B_b(\\R^m\\times\\R^m)$, and positive $u\\in\\B(\\R^m)$ such that $\\mu(\\d x)$ is a probability measure.  In this case we have $\\A\\supset C_0^2(\\R^m)$ which is dense in $L^2(\\mu)$ for any probability measure $\\mu$ on $\\R^m$.\n\n\nTo investigate perturbations of functional inequalities using growth conditions as in \\cite{BLW}, we fix a point $o\\in E$ and  denote $\\rr(x)=d(o,x), x\\in E.$ Now, for\n a $\\rho$-locally bounded measurable function $V$ on $E$ (i.e. $V$ is bounded on the set $\\{x \\in E: \\ \\rho(x)\\le r\\}$ for all $r>0$)  such that\n $\\mu(\\e^V)=1,$ let $\\mu_V(\\d x) =   \\e^{V(x)} \\mu(\\d x)$.\n Since for every $f \\in \\A$, $\\GG(f,f)$ given by (\\ref{intro-01}) is a bounded measurable function on $E$, we have\n$$\n \\EE_V(f,f):=\\int_{E}\\GG(f,f)(x)\\,\\mu_V(\\d x) <\\infty.\n$$\nAgain by the argument in \\cite[Example 1.2.6]{FOT},  the form\n$$  \\E_V(f,g):= \\mu_V(\\GG(f,g))=\\iint_{E\\times E} (f(x)-f(y))(g(x)-g(y))q(x,y)\\mu(\\d y)\\mu_V(\\d x)$$ defined for $f,g\\in \\A$ is closable in $L^2(\\mu_V)$ and its closure $(\\E_V,\\D(\\E_V))$ is a symmetric conservative Dirichlet form.\n\n\nIn this paper, we shall assume that $\\EE$ satisfies a functional inequality and then search for conditions on $V$ such that $\\EE_V$ satisfies the same type of functional inequality.   In the following two sections,   we   study   perturbations of\nthe super Poincar\\'e inequality and\nthe weak Poincar\\'e inequality respectively.     Each section includes some typical examples   to illustrate the main results. Throughout\nthe paper, we simply denote that  $\\GG(f)=\\GG(f,f),$\n$\\EE(f)=\\EE(f,f)$ and $\\EE_V(f)=\\EE_V(f,f).$\n\n\\section{Perturbations of the  super Poincar\\'{e} inequality}\n\nIn Subsection 2.1 we state two results for perturbations of the super Poincar\\'e inequality using growth conditions and present some examples to illustrate the results. Subsection 2.2 includes proofs of these results. Finally, in Subsection 2.3 we prove that the super Poincar\\'e inequality is stable for perturbations under a variation condition on  the support of $q$, which extends a known result for diffusion processes.\n\n\n\\subsection{Main results and examples}\n\nWe consider the following super Poincar\\'e inequality introduced in \\cite{W00a, W00b}:\n \\beq\\label{sp} \\mu(f^2)\\le r \\EE(f) +\\bb(r)\\mu(|f|)^2,\\ \\ r>0, f\\in \\scr{D}(\\EE),\\end{equation}\n where $\\bb: (0,\\infty)\\to (0,\\infty)$ is a decreasing function.\n Note that if \\eqref{sp} holds for some non-decreasing function $\\bb$, it holds also for the decreasing function\n $\\tt\\bb(r):=\\inf_{s\\in (0,r]} \\bb(s)$ in place of $\\bb(r).$\n\n\nTo establish a super Poincar\\'e inequality for $\\EE_V$, we  need the following   quantities. For any $n\\ge 1$ and $k\\ge 1$, let\n\\beg{equation*}\\beg{split} &K_{n,k}(V)= \\sup_{\\rho(x)\\le n+2}V(x)-\\inf_{\\rho(x)\\le n+k+2}V(x),\\\\\n &J_{n,k}(V)= \\sup_{\\rho(x)\\le n+1}V(x)-2\\inf_{\\rho(x)\\le n+k+2}V(x),\\\\\n &\\vv_{n,k}(V) = \\sup_{m\\ge n} \\Big\\{\\bb^{-1}\\big(1\/[2\\mu(\\rho>m-1)]\\big) \\e^{K_{m,k}(V)}\\Big\\},\n \\end{split}\\end{equation*}\n where\n $\\bb^{-1}(s):=\\inf\\{r>0:\\bb(r)\\le s\\}$ for $s>0$,  with $\\inf\\emptyset :=\\infty$ by convention.\n\n\nWhen the jump is of finite range, i.e. there exists $k_0>0$ such\nthat $q(x,y)=0$ for $d(x,y)> k_0$, we have the following   result\nsimilar to \\cite[Theorem 3.1]{BLW}   for local Dirichlet forms.\n\n\\beg{thm} \\label{finite-sp} Assume that $(\\ref{sp})$ holds and there\nexists $k_0\\ge 1 $ such that $q(x,y)=0$ for  $d(x,y)> k_0$.\n\\begin{itemize}\n\\item[$(1)$] If $\\inf\\limits_{n\\ge1,k\\ge k_0}\\vv_{n,k}(V)=0,$ then the super Poincar\\'{e} inequality\n\\eqref{th-sp-1} holds with\n\\beg{equation*}\\beg{split} \\bb_V(r):=\\inf\\Big\\{&(1+8\\lambda r')\\e^{J_{n,k}(V)}\\bb(s): s>0, r'\\in (0,r], n\\ge 1, k\\ge k_0\\\\\n&\\quad {\\rm such\\ that}\\ 8\\vv_{n,k}(V)+s\\e^{K_{n,k}(V)}\\le \\frac{r'}{2+16\\ll r'} \\Big\\}.\\end{split}\\end{equation*}\n\\item[$(2)$] If $\\inf\\limits_{n\\ge 1,k\\ge k_0}\\vv_{n,k}(V)<\\infty,$ then the defective Poincar\\'{e} inequality\n\\beq\\label{th-sp-2}\\mu_V(f^2)\\le C_1\\EE_V(f) +C_2\\mu_V(|f|)^2,\\ \\ f\\in\\D(\\EE_V)\\end{equation}  holds for some constants $C_1,C_2>0$.\n\\end{itemize}\n\\end{thm}\nWe note that according to \\cite[Proposition 1.3]{RW} (see also\n\\cite[Proposition 4.1.2]{Wbook}), if $\\EE_V$ satisfies the weak\nPoincar\\'e inequality (see Section 4 below) then the defective\nPoincar\\'e inequality (\\ref{th-sp-2}) implies the Poincar\\'e\ninequality\n\\begin{equation*}\\label{P}\\mu_V(f^2)\\le C\\EE_V(f),\\ \\ f\\in \\D(\\EE_V), \\mu_V(f)=0\\end{equation*}\nfor some constant $C>0.$\n\nWhen the jump is of infinite range, we will need additional notation and assumptions to control the uniform norm appearing in the perturbed functional inequalities (see Lemmas \\ref{L3.1} and \\ref{L3.2} below), which is an essentially different feature from the diffusion setting.  For any $n, k\\ge 1$ and $\\delta>1$, let\n\\beg{equation*}\\beg{split}\n &Z_n(V)= \\sup_{\\rho(x)\\le n+1}V(x),\\\\\n  &\\zeta_{n}(V)=\\sup_{m\\ge n} \\Big\\{\\bb^{-1}\\big(1\/[2\\mu(\\rho>m-1)]\\big) \\e^{Z_{m+1}(V)}\\Big\\},\\\\\n &t_{i,n,k}(\\dd,V):= \\bb^{-1}\\Big(\\ff 1 4 \\dd^i\\e^{-J_{n,k}(V)}\\Big).\\end{split}\\end{equation*}\nMoreover, let\n\\beg{equation*}\\beg{split} &\\gg_{n,k}  =\\iint_{\\{d(x,y)>k, \\rr(y)\\ge n-1\\}}q(x,y)\\mu(\\d y)\\mu(\\d x),\\\\\n&\\eta_{n,k} =\\iint_{\\{\\rr(x)>n+k+2, \\rr(y)\\le n+1\\}}q(x,y)\\mu(\\d\ny)\\mu(\\d x).\\end{split}\\end{equation*} By  \\eqref{intro-02}, we see\nthat $\\gg_{n,k}+\\eta_{n,k} \\downarrow 0$ as $n\\uparrow \\infty$ holds\nfor any $k\\ge 1.$\n\n\\noindent We assume\n \\paragraph{(A)} \\ There exist    $\\dd>1$ and   sequences $\\{(n_i,k_i)\\}_{i\\ge 1}\\subset  \\mathbb N^2$   such that $n_i\\uparrow\\infty$ and\\beg{enumerate}\n \\item[(A1)] $\\lim_{i\\to\\infty}(\\vv_{n_i,k_i}(V)+t_{i,n_i,k_i}(\\dd,V)\\e^{K_{n_i,k_i}(V)})=0;$\n \\item[(A2)] $\\sum_{i=1}^\\infty\\big\\{ \\zeta_{n_i}(V)\\gg_{n_i,k_i} +t_{i,n_i,k_i}(\\dd,V)\\e^{Z_{n_i}(V)}\\eta_{n_i,k_i}\\big\\} <\\infty.$\\end{enumerate}\n\n \\\n\n We shall let $I_\\dd$ denote the set of all sequences $\\{(n_i,k_i)\\}\\subset \\mathbb N^2$ such that $n_i\\uparrow \\infty$ and (A1)-(A2) hold. Moreover, for any $r>0$ and $\\{n_i,k_i\\}\\in I_\\dd$,\n let $D(r,\\{(n_i,k_i)\\})$ be the set of $j\\in\\mathbb N$ such that\n \\beg{equation}\\label{A-01}\\sup_{i\\ge j}\\big(8\\vv_{n_i,k_i}(V)+   t_{i,n_i,k_i}(\\dd,V)\n \\e^{K_{n_i,k_i}(V)}\\big)\\le \\ff{1}{64\\ll}\\land \\ff{c(\\dd)r}{16},\\ \\ i\\ge j,\\end{equation} where $ c(\\dd):= \\big(\\ff{\\ss\\dd-1}\\dd\\big)^2,$ and such that\n \\beq\\label{A-02} \\sum_{i=j}^\\infty \\Big(6\\zeta_{n_i}(V)\\gg_{n_i,k_i} +t_{i,n_i,k_i}(\\dd,V)\\e^{Z_{n_i}(V)}\\eta_{n_i,k_i}\\Big)\\dd^{i+2}\\le \\ff{1}{256}.\\end{equation}\n By {\\bf (A)}, we see that for any $r>0$ and $\\{(n_i,k_i)\\}\\in I_\\dd$, the set $D(r,\\{(n_i,k_i)\\})$ is non-empty.\n\n\\beg{thm}\\label{th-sp} Assume that $(\\ref{sp})$ holds.\n\\begin{itemize}\n\\item[$(1)$] If {\\bf\n(A)} is satisfied, then the super Poincar\\'e inequality\n\\begin{equation}\\label{th-sp-1}\\mu_V(f^2)\\le r\\EE_V(f)+\\bb_V(r) \\mu_V(|f|)^2,\\ \\ r>0,\\ \\ f\\in \\scr{D}(\\EE_V)\\end{equation} holds with\n $$\\bb_V(r):= \\inf\\bigg\\{2\\dd^j:\\ \\{(n_i,k_i)\\}\\in I_\\dd,\\ j\\in D(r,\\{(n_i,k_i)\\})\\bigg\\} <\\infty,\\ \\ r>0.$$\n\n\\item[$(2)$] If $(A2)$ is satisfied and $(A1)$ is replaced by the following weaker assumption\n\\item[$(A1')$]$\\limsup_{i\\to\\infty}(\\vv_{n_i,k_i}(V)+t_{i,n_i,k_i}(\\dd,V))<\\infty,$\\newline\nthen the  defective Poincar\\'e inequality \\eqref{th-sp-2} holds for some  $C_1,C_2>0.$\n\\end{itemize}\n \\end{thm}\n\n\nThe following example shows that Theorem \\ref{th-sp} is sharp in some specific situations.\n\n\\begin{exa}\\label{ex-sp-1}\\rm\nLet $E=\\R^m$ with $d(x,y)=|x-y|$, and let   $$q(x, y)=\\frac{(1+|y|)^{m+\\aa}\\log(1+|y|)}{|x-y|^{m+\\alpha}},\\ \\ \\mu(\\d\nx)=\\frac{c_{m,\\aa}\\d x}{(1+|x|)^{m+\\alpha}\\log(1+|x|)},$$   where\n$ \\alpha\\in (0,2)$ and $c_{m,\\aa}>0$ is the normalizing constant such that $\\mu$ is a probability measure. It is easy\nto see that all the assumptions in the introduction for\n$(\\EE,\\scr{D}(\\EE))$ are satisfied. We consider $V$ satisfying\n\\begin{equation}\\label{ex-sp-1-01}\n-s\\varepsilon\\log\\log(e+|x|)-K \\le  V(x)\\le (1-s)\\varepsilon\\log\\log(e+|x|)+K,\\ \\ x\\in \\R^m\n\\end{equation}\nfor some constants $\\varepsilon\\in(0,1]$, $s \\in [0,1]$ and $K \\in \\R$ such that $\\mu(\\e^V)=1.$\n\\begin{itemize}\n\\item[(1)]  If $\\vv<1$ then  (\\ref{th-sp-1}) holds\nwith\n\\begin{equation}\\label{ex-sp-1-02}\n\\beta_V(r)=\\exp\\big(C_1(1+{r}^{-1\/(1-\\vv)})\\big),\n\\end{equation}\nfor some constant $C_1>0$.\n\\item[(2)] $\\beta_V$ in (1) can not be replaced by any essentially smaller functions, i.e.\n when $V(x)=\\vv\\log\\log(\\e+|x|)+K_0$ for some constant $K_0\\in \\R$ such that $\\mu(\\e^V)=1$, the estimate\n\\eqref{ex-sp-1-02} is sharp in the sense that the super Poincar\\'e\ninequality (\\ref{th-sp-1}) does not hold if\n$$\\lim_{r\\to 0}r^{1\/(1-\\vv)}\\log \\bb_V(r)=0.$$\n\\item[(3)] In (1) the constant $K$ can not be replaced by any unbounded positive function, i.e. for\nany increasing function $\\phi: [0,\\infty)\\to [0,\\infty)$ with $\\phi(r)\\uparrow\\infty$ as $r\\uparrow\\infty$, there exists $V$ such that \\begin{equation}\\label{ABC1}\n  \\vv\\log\\log (\\e+|x|)\\le V(x)\\le  \\varepsilon\\log\\log(\\e+|x|)+\\phi(|x|)\n\\end{equation} with $\\mu(\\e^V)<\\infty,$ but     the super Poincar\\'e\ninequality (\\ref{th-sp-1}) with   $\\beta_{V}$\ngiven by (\\ref{ex-sp-1-02}) does not hold for $\\mu_V(\\d x):= \\ff{\\e^{V(x)}\\d x}{\\mu(\\e^V)}.$\n\\item[(4)] If $\\vv=1$, then\n$(\\EE_V,\\scr{D}(\\EE_V))$ satisfies the  Poincar\\'e\ninequality\n\\begin{equation}\\label{ex2.2}\n\\mu_V(f^2)\\le C\\EE_V(f,f)+\\mu_V(f)^2,\\ \\ \\ f \\in \\scr{D}(\\EE_V)\n\\end{equation} with some constant $C>0$.\n\n\n\\end{itemize}\n\\end{exa}\n\n\\begin{proof} As (2) is included in \\cite[Corollary 1.3]{WW}, we only prove (1), (3) and (4).\n\n(a) According to \\cite[Corollary 1.3(3)]{WW}, we know the\nlogarithmic Sobolev inequality holds for $(\\EE,\\scr{D}(\\EE))$, i.e.\\\nthe super Poincar\\'{e} inequality \\eqref{sp} holds for\n$(\\EE,\\scr{D}(\\EE))$ with the rate function\n$\\beta(r)=\\exp\\big(c_1(1+{r}^{-1})\\big)$ for some constant $c_1>0$.\n\nNext, by   (\\ref{ex-sp-1-01}), there exists a constant $c_2>0$\nsuch that for $n$ large enough\n\\begin{equation}\\label{e3.10}\n\\begin{split}\n& K_{n,n}(V)\\le \\vv\\log\\log n+c_2,\\ \\ J_{n,n}(V)\\le (1+s)\\vv \\log\\log n+c_2,\\ \\\\\n& Z_{n}(V)\\le (1-s)\\vv \\log\\log n+c_2,\\ \\ \\zeta_n(V)\\le \\frac{c_2}{\\log^{1-(1-s)\\vv}n},\\ \\ \\vv_{n,n}(V)\\le\n\\frac{c_2}{\\log^{1-\\vv} n},\\\\\n&\\gg_{n,n}\\le \\frac{c_2}{n^{\\alpha}},\\ \\ \\eta_{n,n}\\le \\frac{c_2}{n^{\\alpha}\\log(1+n)}.\n\\end{split}\n\\end{equation}\nFor any $\\delta>1$, taking\n$\\{n_i\\}=\\{\\delta^i\\}_{i=1}^{\\infty}$, and $k_i=n_i$, we\nget that $t_{i,n_i,k_i}(\\dd,V)\\le {c_3}{i}^{-1}$ for $i$ large\nenough and some constant $c_3>0$. So, assumption {\\bf (A)} is\nfulfilled.\n\nMoreover, by (\\ref{A-01}) and (\\ref{A-02}), it is easy to check that\nfor $r>0$ small enough we have\n\\begin{equation*}\n\\Big\\{j \\ge1:  j \\ge c_4r^{-\\frac{1}{1-\\vv}}\\Big\\}\\subseteq D(r,\\{(n_i,k_i)\\}).\n\\end{equation*}\nThen, the   assertion in (1)  follows from Theorem \\ref{th-sp}(1) and \\eqref{e3.10}.\n\n(b) Let $ \\psi(r)= (1+\\log(1+r))\\land \\e^{\\phi(r)}$ for $r\\ge 0.$\nThen $1\\le\\psi\\le \\e^\\phi, \\psi(r)\\uparrow\\infty$ as $r\\uparrow\\infty$ and $\\psi(r)\\le 2\\log r$ for large $r$. Let\n$$V(x)= \\vv\\log\\log (\\e+|x|) +\\log\\psi(|x|).$$ Then $V$ satisfies (\\ref{ABC1}) and $\\mu(\\e^V)<\\infty.$ Up to a normalization constant we may simply assume that $\\mu_V(\\d x)= \\e^{V(x)}\\d x.$\n\nNow, suppose that  (\\ref{th-sp-1}) holds  with $\\bb_V$ given by\n(\\ref{ex-sp-1-02}).\nFor any $n\\ge1$, let $f_n\\in C^\\infty(\\R^d)$ satisfy that\n\\begin{equation*} f_{n}(x)\n\\begin{cases}\n& =0,\\ \\ \\ \\ \\ \\ \\ \\quad \\,\\,|x|\\le n,\\\\\n& \\in [0,1],\\ \\ \\ \\ \\quad n\\le |x|\\le 2n,\\\\\n& =1,\\ \\ \\ \\ \\ \\ \\ \\ \\quad |x|\\ge 2n,\n\\end{cases}\n\\end{equation*} and $|\\nabla f_n|\\le \\frac{2}{n}.$ Then, there exists a constant $c_5>0$ independent of $n$ such that\n\\begin{equation}\\label{AB0}\n\\begin{split}\n\\Gamma(f_n,f_n)(x)&=c_{m,\\aa}\\int\\frac{(f_n(y)-f_n(x))^2}{|y-x|^{m+\\aa}}\\,\\d y\\\\\n&\\le \\frac{4c_{m,\\aa}}{n^2}\\int_{\\{|y-x|\\le n\\}}\\frac{\\d y}{|y-x|^{m+\\alpha-2}}+c_{m,\\aa}\\int_{\\{|y-x|> n\\}}\\frac{\\d y}{|y-x|^{m+\\alpha}}\\\\\n&\\le \\frac{c_5}{n^\\alpha},\\quad n\\ge1.\n\\end{split}\n\\end{equation}\nAccording to the definition of $f_n$ and the increasing property of $\\psi$, there exists $c_6>0$ such that\n\\begin{equation}\\label{proof-e-0100}\\mu_V(f_n^2)\\ge \\frac{c_6  \\psi(n)}{n^{\\aa}\\log^{1-\\vv}n},\\ \\ n\\ge 2.\\end{equation}\nOn the other hand, since $\\psi(r)\\le 2\\log r$ for large $r$, we have\n$$\\mu_V(|\\cdot|>n)\\le c_7 \\int_n^\\infty \\ff{\\psi(r)\\d r}{r^{1+\\aa}\\log^{1-\\vv}r }\\le\n\\ff{c_8\\log^{\\vv} n}{n^\\aa},\\ \\ n\\ge 2.$$ Then\n$$\\mu_V(f_n)^2\\le \\mu_V(f_n^2) \\mu_V(|\\cdot|>n) \\le \\ff{c_8\\log^{\\vv}n}{n^\\aa}\\mu_V(f_n^2),\\ \\ n\\ge 2.$$\nCombining this with (\\ref{th-sp-1}), (\\ref{AB0}) and\n(\\ref{proof-e-0100}), we obtain\n\\beq\\label{AD}1-\\ff{c_8\\log^{\\vv}n}{n^\\aa}\\bb_V(r)\\le\\ff{c_5r}{n^\\aa\\mu_V(f_n^2)}\n\\le \\ff{c_9r\\log^{1-\\vv}n}{\\psi(n)},\\ \\ r>0, n\\ge 2\\end{equation}\nfor some constant $c_9>0.$ Since $\\psi(n)\\le 2\\log n$ for large $n$,\nit is easy to see that\n$$r_n:= \\bb_V^{-1}\\Big(\\ff{n^\\aa}{2c_8\\log^{\\vv}n}\\Big)\\le \\ff{c_{10}}{\\log^{1-\\vv}n}$$ holds for some constant $c_{10}>0$ and large $n$. Therefore, it follows from (\\ref{AD}) that\n$$\\ff 1 2 \\le \\lim_{n\\to\\infty} \\ff{c_9r_n\\log^{1-\\vv}n}{\\psi(n)}=0,$$ which is a contradiction.\n\n\n(c) If $\\vv=1$ in (\\ref{ex-sp-1-01}), the estimate (\\ref{e3.10}) is\nstill true. According to Theorem \\ref{th-sp}(2), the defective\nPoincar\\'e inequality (\\ref{th-sp-2}) holds for $(\\EE_V,\n\\scr{D}(\\EE_V))$. On the other hand, by \\cite[Theorem 1.1(3)]{WW},\nthe weak Poincar\\'e inequality holds for $(\\EE_V, \\scr{D}(\\EE_V))$.\nTherefore, the required Poincar\\'e inequality follows from \\cite[Proposition 1.3]{RW} or\n\\cite[Proposition 4.1.2]{Wbook}. \\end{proof}\n\n\n\nFinally,  we consider an example for finite range of jumps to\nillustrate Theorem \\ref{finite-sp}.\n\n\\begin{exa}\\label{ex2}\\rm\nLet $E=\\R^m$  with $d(x,y)=|x-y|$ and let\n$$q(x, y)=\\frac{\\e^{|y|^\\kk}}{{|x-y|}^{m+\\alpha}}1_{\\{{|x-y|}\\le 1\\}} ,\\ \\  \\mu(\\d x)=c_{m,\\kappa}\\e^{-|x|^{\\kappa}}\\d x $$ for some\nconstants $0<\\alpha<2$,  $\\kappa>1$ and $c_{m,\\kk}\\ge 1$ such that\n$\\mu$ is a probability measure.\n It is easy to see that all the assumptions in the introduction for\n$(\\EE,\\scr{D}(\\EE))$ is satisfied. Consider $V$ satisfying\n\\begin{equation}\\label{ex2.0}\n-C_1(1+|x|^{\\theta-1})-K \\le  V(x)\\le C_1(1+|x|^{\\theta-1})+K,\\quad\nx\\in \\R^m\n\\end{equation}\nfor some constants $\\theta \\in (1,\\kappa]$, $C_1>0$ and $K\\in\\R$\nsuch that $\\mu(\\e^V)=1.$\n\\begin{itemize}\n\\item[(1)] If $\\theta<\\kk,$ then   the super Poincar\\'e inequality holds\nfor $(\\EE_V, \\scr{D}(\\EE_V))$ with\n$$\n\\beta_V(r)=C_2\\exp\\Big[C_2 \\log^{\\frac{\\kappa}{\\kappa-1}}(1+r^{-1})\n\\Big]\n$$\nfor some positive constants $C_2>0.$\n\n\\item[(2)] Let $\\theta=\\kk$. Then if $C_1>0$ is small enough,  then   the Poincar\\'e inequality \\eqref{ex2.2}  holds for some constants $C>0.$\n\\end{itemize}\n\\end{exa}\n\n\\begin{proof} According to \\cite[Example 1.2(2)]{CW2}, we know the super Poincar\\'{e} inequality\n\\eqref{sp} holds for $(\\EE,\\scr{D}(\\EE))$ with\n\\beq\\label{DG0}\\beta(r)=c_1\\exp\\Big[c_1\\log^{\\frac{\\kappa}{\\kappa-1}}(1+\nr^{-1})\\Big]\\end{equation} for some constant $c_1>0$.\n\n(1) By (\\ref{ex2.0}) and the fact that $\\theta<\\kappa$, one can find\nsome constants $c_2, c_3>0$ such that for $n$ large enough\n\\begin{equation*}\n\\begin{split}\n& K_{n,n}(V)\\le c_2n^{\\theta-1},\\ \\ J_{n,n}(V)\\le c_2n^{\\theta-1},\\\n\\ \\vv_{n,n}(V)\\le \\e^{-c_3n^{\\kappa-1}}.\n\\end{split}\n\\end{equation*}\nTherefore, for every $r$ small enough, taking\n$$n=c_4\\log^{\\frac{1}{\\kappa-1}}(1 +{r}^{-1})$$ and\n$$s=\\exp\\big(-c_5\\log(1+{r}^{-1})\\big)$$ for some constants\n$c_5>>c_4>>1$,  we obtain\n\\begin{equation*}\n8\\vv_{n,n}(V)+s\\e^{K_{n,n}(V)}\\le \\frac{r}{2(1+8\\ll r)}.\n\\end{equation*}\nThe first required assertion for super Poincar\\'{e} inequality of\n$(\\EE_V,\\scr{D}(\\EE_V))$ follows from Theorem \\ref{finite-sp}(1) and\nall the estimates above.\n\n(2) Let $\\theta=\\kk$. By \\eqref{ex2.0}, (\\ref{DG0}) and the definition of $\\mu$,    for $n$ large enough we have\n\\begin{equation*}\nK_{n,n}(V)\\le 3C_1n^{\\kappa-1},\\ \\ \\bb^{-1}(\\mu(|x|>n-1)^{-1})\\le\n\\e^{-c_6n^{\\kk-1}},\n\\end{equation*} for some constant $c_6>0$ depending only on $\\bb$ and $\\mu$.\nSo, if $C_1< c_6\/3$, then $\\inf_{n\\ge1}\\vv_{n,n}(V)<\\infty$.\nTherefore,  by Theorem \\ref{finite-sp}(2), the  defective Poincar\\'e\ninequality holds for $(\\EE_V, \\scr{D}(\\EE_V))$. Finally, according\nto \\cite[Proposition 2.6]{CW2}, the weak Poincar\\'e inequality holds\nfor $(\\EE_V, \\scr{D}(\\EE_V))$. Therefore, the Poincar\\'e inequality\nholds for $(\\EE_V, \\scr{D}(\\EE_V))$.\\end{proof}\n\nTo show that in  Example \\ref{ex2}(2) it is essential to assume that\n$C_1>0$ is small, we present below a counterexample inspired by\n\\cite[Proposition 5.1]{BLW}.\n\n\\beg{prp} In the situation of Example $\\ref{ex2}$, let $\\theta=\\kk, \\aa\\in (0,1)$ and $m=1.$  Let\n$$V(x)= K_0+ L \\sum_{n=1}^\\infty 1_{[nH, (n+1)H)}(x)(n+1)^{\\kk-1} \\Big( 2n+1-\\ff{2x}H\\Big),$$\nwhere $H>4, L>\\ff{\\kk H^\\kk}{H-2}$ and $K_0\\in \\R$ are constants\nsuch that $\\mu(\\e^V)=1.$ Then   $(\\ref{ex2.0})$ holds for some\nconstant $C_1>0$ and $K\\in\\R$; however, for any $C>0$,  the\nPoincar\\'e inequality $(\\ref{ex2.2})$ does not hold.\\end{prp}\n\n\\beg{proof}   It suffices to disprove the Poincar\\'e inequality, for which we are going to construct a sequences of functions $\\{f_n\\}\\subset\\A$ such that\n\\begin{equation*}\\label{DPP} \\lim_{n\\to\\infty} \\ff{\\EE_V(f_n)}{{\\rm Var}_{\\mu_V} (f_n)}=0.\\end{equation*}\nFor $n\\ge 1$, let\n\\begin{equation*}\nf_{n}(x)=\\begin{cases}\n& \\frac{\\int_{Hn+1}^{x}\\exp(y^{\\kappa}-V(y))\\d y}\n{\\int_{Hn+1}^{H(n+1)-1}\\exp(y^{\\kappa}-V(y))\\d y},\\ \\ \\ x \\in (Hn+1,H(n+1)-1) ,\\\\\n& \\qquad\\quad1,\\ \\ \\ \\ \\ \\ \\qquad\\qquad \\,\\,\\,\\,x \\in [H(n+1)-1,+\\infty),\\\\\n& \\qquad\\quad0,\\ \\ \\ \\ \\ \\ \\ \\ \\qquad \\qquad\\text{otherwise},\n\\end{cases}\n\\end{equation*}\nwhich is a bounded Lipschitz continuous function on $\\R$, so that $f_n\\in\\A$. In the following calculations $C$ stands for a constant which   varies from line to line but is independent of $n$ (may depend on $H$ or $L$). We simply denote $K_n=L(n+1)^{\\kk-1}$ for $n\\ge 1$, so that\n$$V(x)=K_0+ \\sum_{n=1}^\\infty 1_{[nH, (n+1)H)}(x)K_n\\Big(2n+1-\\ff{2x}H\\Big).$$\n\n(a) Estimate on ${\\rm Var}_{\\mu_V}(f_n).$ For $n$  large enough, since\n$z\\mapsto\\big(z^{\\kappa}-K_{n+1}(2n+3-\\ff{2z}H)\\big)$ is increasing, we have\n\\begin{equation*}\n\\begin{split}\n\\mu_{V}(f_{n}^2)  &\\ge \\mu_{V}\\big(H(n+1)\\le x \\le H(n+2)\\big)\\\\\n&\\ge C\\int_{H(n+1)}^{H(n+2)}\\frac{\\exp\\big(K_{n+1}(2n+3-\\ff{2z}H)-z^{\\kappa} \\big)}{\\kappa z^{\\kappa-1}+\\frac{2K_{n+1}}{H}} \\d\\bigg(z^{\\kappa}-K_{n+1}(2n+3-\\ff{2z}H)\\bigg)\\\\\n&\\ge \\frac{C\\big\\{\\exp\\big(-(H(n+1))^{\\kappa}+K_{n+1}\\big)-\\exp\\big(-(H(n+2))^{\\kappa}-K_{n+1}\\big)\\big\\}}{ (n+2)^{\\kappa-1}}\\\\\n&\\ge \\frac{C\\exp\\big(-(H(n+1))^{\\kappa}+K_{n+1}\\big)}{ (n+2)^{\\kappa-1}}.\n\\end{split}\n\\end{equation*}  Noting that for $n$ large enough,\n\\begin{equation*}\n\\mu_{V}(f_{n})^2=\\mu_{V}(f_{n} 1_{\\{x \\ge n\\}})^2\\le \\mu_{V}(f_{n}^2)\\mu_{V}(x \\ge n)\n\\le \\frac{ \\mu_{V}(f_{n}^2)}{2},\n\\end{equation*} we arrive at\n\\beq\\label{PRO-01} {\\rm Var}_{\\mu_V}(f_n)\\ge\n\\frac{C\\exp\\big(-(H(n+1))^{\\kappa}+K_{n+1}\\big)}{(n+2)^{\\kappa-1}}.\\end{equation}\n\n(b) Estimate on $\\EE_V(f_n)$. Let $$g_{n}(x) =\\bigg(\\int_{Hn+1}^{H(n+1)-1}\\exp\\big(y^{\\kappa}-V(y)\\big)\\d y\\bigg)f_{n}(x).$$\nNoting that\n$g_{n}'(x)\\neq 0$ only when $x \\in (Hn+1,H(n+1)-1)$,\nwe have\n\\begin{equation*}\n\\begin{split}\n\\EE_{V}(g_{n})& \\le C\\int_{Hn}^{H(n+1)}\\bigg(\\int_{\\{|x-y|\\le 1\\}}\n\\frac{|\\int_{x}^{y} g_{n}'(z)\\d z|^2}{{|x-y|}^{1+\\alpha}}\\d y \\bigg)\\exp\\big(-x^{\\kappa}+V(x)\\big) \\d x\\\\\n&=C\\int_{Hn}^{H(n+1)}\\bigg(\\int_{\\{{|x-y|}\\le 1, x \\le y\\}}\n\\frac{|\\int_{x}^{y} g_{n}'(z)\\d z|^2}{{|x-y|}^{1+\\alpha}}\\d y\\bigg)\\exp\\big(-x^{\\kappa}+V(x)\\big) \\d x\\\\\n&\\quad+C\\int_{Hn}^{H(n+1)}\\bigg(\\int_{\\{{|x-y|}\\le 1, x > y\\}}\n\\frac{|\\int_{x}^{y} g_{n}'(z)\\d z|^2}{{|x-y|}^{1+\\alpha}}\\d y \\bigg)\\exp\\big(-x^{\\kappa}+V(x)\\big) \\d x\\\\\n&=:C(I_1+I_2).\n\\end{split}\n\\end{equation*}\nSince\n$g_{n}'(x)=\\exp\\big(x^{\\kappa}-V(x)\\big)$ for $x \\in (Hn+1,H(n+1)-1)$, and since for large $n$ the function $z\\mapsto z^\\kk-V(z)$ is increasing on $(Hn, H(n+1))$,\nby the Cauchy-Schwarz inequality we have, for $x\\in (Hn, H(n+1)-1)$ and $ x\\le y\\le x+1$,\n\\beg{equation*}\\beg{split} & \\frac{|\\int_{x}^{y} g_{n}'(z)\\d z|^2}{{|x-y|}^{1+\\alpha}}  \\\\\n&=  \\frac{|\\int_{x \\vee (Hn+1)}^{y\\wedge (H(n+1)-1)} g_{n}'(z)\\d z|^2}{{|x-y|}^{1+\\alpha}} \\\\\n&\\le \\frac{|\\int_{x \\vee (Hn+1)}^{y\\wedge (H(n+1)-1)}\n\\exp\\big(z^{\\kappa}-V(z)\\big)\\d z|}{{|x-y|}^{1+\\alpha}}\\int_{x \\vee (Hn+1)}^{y\\wedge (H(n+1)-1)} |g_{n}'|^2(z)\\e^{-z^{\\kappa}+V(z)}\\d z \\\\\n&\\le \\frac{\\exp\\big((x+1)^{\\kappa}-V(x+1)\\big)}{{|x-y|}^{\\alpha}}\n\\Big(\\int_{Hn+1}^{H(n+1)-1}|g_{n}'|^2(z)\\e^{-z^{\\kappa}+V(z)}\\d\nz\\Big).\n\\end{split}\\end{equation*} Thus, since $\\aa\\in (0,1)$,\n\\begin{equation*}\n\\begin{split}\nI_1 &\\le\nC\\Big(\\int_{Hn+1}^{H(n+1)-1}|g_{n}'|^2(z)\\exp\\big(-z^{\\kappa}+V(z)\\big)\\d\nz\\Big) \\\\\n&\\qquad\\times\\bigg[\\int_{Hn}^{H(n+1)-1}\\Big(\\int_{\\{|x-y|\\le 1\\}}\n\\frac{{1}}{{|x-y|}^{\\alpha}}\\d y\\Big)\\exp\\Big((x+1)^{\\kappa}-x^{\\kappa}-\nV(x+1)+V(x)\\Big) \\d x\\bigg]\\\\\n&\\le C  \\Big(\\int_{Hn+1}^{H(n+1)-1}\\exp\\big(z^{\\kappa}-V(z)\\big)\\d z\\Big)\n\\exp\\Big(\\kappa (H(n+1))^{\\kappa-1}+\\frac{2K_{n}}{H}\\Big).\n\\end{split}\n\\end{equation*}\n Similarly, we have\n\\begin{equation*}\nI_2 \\le C\\int_{Hn+1}^{H(n+1)-1}\\exp\\big(z^{\\kappa}-V(z)\\big)\\d z.\n\\end{equation*}\nSo, for $n$ large enough,\n\\begin{equation*}\n\\EE_{V}(g_{n}) \\le C\n\\Big(\\int_{Hn+1}^{H(n+1)-1}\\exp\\big(z^{\\kappa}-V(z)\\big)\\d z\\Big)\n\\exp\\bigg(\\kappa (H(n+1))^{\\kappa-1}+\\frac{2K_{n}}{H}\\bigg).\n\\end{equation*}\nMoreover, for large $n$,\n\\begin{equation*}\n\\begin{split}\n& \\int_{Hn+1}^{H(n+1)-1}\\exp\\big(z^{\\kappa}-V(z)\\big)\\,\\d z\\\\\n&=\\int_{Hn+1}^{H(n+1)-1}\\frac{\\exp\\big(z^{\\kappa}-\\frac{2K_{n}(Hn-z)}{H}-K_{n}\\big)}{\\kappa z^{\\kappa-1}+\\frac{2K_{n}}{H}}\n \\d\\Big(z^{\\kappa}-\\frac{2K_{n}(Hn-z)}{H}-K_{n}\\Big)\\\\\n&\\ge \\frac{C\\exp\\big((H(n+1)-1)^{\\kappa}+\\frac{(H-2)K_{n}}{H}\\big)-\\exp\\big((Hn+1)^{\\kappa}-\\frac{(H-2)K_{n}}{H}\\big)}\n{ (n+1)^{\\kappa-1}}\\\\\n&\\ge \\frac{C\\exp\\big((H(n+1)-1)^{\\kappa}+\\frac{(H-2)K_{n}}{H}\\big)}{ (n+1)^{\\kappa-1}},\n\\end{split}\n\\end{equation*}\nTherefore, for large $n$,\n\\beq\\label{PRO-02}\\beg{split}  \\EE_V(f_n)&=\\ff{\\EE_V(g_n)}{(\\int_{Hn+1}^{H(n+1)-1}\\exp\\big(z^{\\kappa}-V(z)\\big)\\d z)^2}\\\\\n&\\le \\ff{C \\exp\\big(\\kappa (H(n+1))^{\\kappa-1}+\\frac{2K_{n}}{H}\\big)}{\\int_{Hn+1}^{H(n+1)-1}\\exp\\big(z^{\\kappa}-V(z)\\big)\\d z}\\\\\n&\\le C(n+1)^{\\kk-1} \\exp\\Big(\\kk(H(n+1))^{\\kk-1} -(H(n+1)-1)^\\kk -\\ff{(H-4)K_n}H\\Big). \\end{split}\\end{equation}\n\n\n(c) Combining (\\ref{PRO-01}) with (\\ref{PRO-02}) and noting that\n$$(H(n+1))^\\kk-(H(n+1)-1)^\\kk\\le\\kk (H(n+1))^{\\kk-1},$$ we obtain\n\\begin{equation*}\n\\begin{split}\n&\\lim_{n\\to\\infty} \\frac{\\EE_{V}(f_{n},f_{n})}{{\\rm Var}_{\\mu_{V}}(f_{n})}\\\\\n&\\le C\\lim_{n\\to\\infty} (n+2)^{2(\\kk-1)} \\exp\\Big(2\\kk(H(n+1))^{\\kk-1} -\\ff{2(H-2)}H K_n\\Big) =0\\end{split}\\end{equation*}\nsince $$L>\\ff{\\kk H^\\kk}{H-2}.$$\n\\end{proof}\n\n\n\n\n\\subsection{Proofs of Theorems \\ref{finite-sp} and \\ref{th-sp}}\n\n\n\nAs in \\cite[Theorem 3.1]{BLW}, we shall adopt a split argument by\nestimating $\\mu(f^21_{\\{\\rr\\ge n\\}})$ and $\\mu(f^21_{\\{\\rr\\le n\\}})$\nrespectively. Unlike in the local setting where the chain rule is\navailable, in the present situation  the uniform norm $\\|f\\|_\\infty$\nwill appear in our estimates when the range of jumps is infinite.\nBelow we simply denote $K_{n,k}=K_{n,k}(V), J_{n,k}=J_{n,k}(V),\nZ_n=Z_n(V),\\vv_{n,k}=\\vv_{n,k}(V), \\zeta_n=\\zeta_n(V)$ and\n$t_{i,n,k}=t_{i,n,k}(\\dd,V).$\n\n\\beg{lem}\\label{L3.1} For any $n\\ge 1$, $k\\ge 1$ and $f\\in \\A$,\n$$\\mu_V(f^21_{\\{\\rr\\ge n\\}})\\le 12 \\vv_{n,k} \\EE_V(f)+ 128\\ll\\vv_{n,k} \\mu_V(f^2)+96 \\zeta_{n} \\gg_{n,k} \\|f\\|_\\infty^2.$$\\end{lem}\n\n\\beg{proof} If $f|_{\\{\\rho\\le n-1\\}}=0$,\n  by the Cauchy-Schwartz inequality we have\n\\begin{equation*}\n\\mu(|f|)^2=\\mu\\big(|f| 1_{\\{\\rho>n-1\\}}\\big)^2\\le \\mu(\\rho>n-1)\\mu(f^2).\n\\end{equation*}\nSubstituting this into  \\eqref{sp}  with $r=\n\\bb^{-1}\\big(\\frac{1}{2\\mu(\\rho>n-1)}\\big)$  we obtain\n\\beq\\label{proof-e-01} \\mu(f^2)\\le 2\\bb^{-1}\n\\big(1\/[2\\mu(\\rho>n-1)]\\big)\\EE(f),\\quad f\\in \\A, \\ f|_{\\{\\rho\\le\nn-1\\}}=0.\\end{equation}\n\nTo apply (\\ref{proof-e-01}) for general $f\\in\\A$, we consider $fl_{n}$ instead of $f$, where $l_{n}:= h_{n}\\circ\\rr$ for\nsome function  $h_n\\in C^\\infty([0,\\infty); [0,\\infty))$   satisfying \\begin{equation*} h_{n}(s)\n\\begin{cases}\n& =1,\\ \\ \\ \\ \\ \\ \\ \\quad \\,\\, n\\le s\\le n+1,\\\\\n& \\in [0,1],\\ \\ \\ \\ n-1\\le s<n \\,\\,\\,\\,{\\textrm{ or}}\\,\\, \\,\\,n+1<s\\le n+2,\\\\\n& =0,\\ \\ \\ \\ \\ \\ \\ \\ \\quad s<n-1\\,\\,\\,\\,{\\textrm\n{or}}\\,\\,\\,\\,s>n+2,\n\\end{cases}\n\\end{equation*}\nand $$\\sup_{s\\in[0,\\infty)}|h'_{n}(s)|\\le 2.$$    It is easy to see that\n$$|l_n(x)-l_n(y)|\\le 2(1\\wedge d(x,y))\\big\\{1_{\\{\\rr(x)\\in (n-2,n+3)\\}}+1_{\\{\\rr(y)\\in (n-1,n+2), \\rr(x)\\notin (n-2,n+3)\\}}\\big\\}.$$ This implies\n\\beg{equation}\\label{proof-e-02}\\beg{split} &\\GG(fl_n)(x) \\\\\n& \\le 2\\int_{E} \\big\\{l_n(y)^2 (f(x)-f(y))^2\n +  f(x)^2   (l_n(x)-l_n(y))^2\\big\\}q(x,y)\\mu(\\d y)\\\\\n&\\le 2 \\int_{\\{ \\rho(y)\\in (n-1,n+2)\\}} (f(x)-f(y))^2 q(x,y)\\mu(\\d y)+8\\ll f(x)^21_{\\{\\rr(x)\\in (n-2,n+3)\\}}\\\\\n  &\\quad + 8f^2(x) 1_{\\{\\rr(x)\\notin(n-2,n+3)\\}}\\int_{\\{\\rr(y)\\in (n-1,n+2)\\}}   q(x,y)\\mu(\\d y). \\end{split}\\end{equation}\nSince $$1_{\\{\\rr(x)\\notin(n-2,n+3)\\}}\\int_{\\{\\rr(y)\\in (n-1,n+2)\\}}   q(x,y)\\mu(\\d y)\\le\n1_{\\{\\rr(x)\\notin(n-2,n+3)\\}}\\int_{\\{d(x,y)>1\\}}   q(x,y)\\mu(\\d y)\\le \\lambda,$$\nwe have $\\GG(fl_n)\\in \\B_b(E)$, and $fl_n \\in \\A$.\n\nLet\n  $$\\dd_{n,k} =  \\e^{K_{n,k}}  \\bb^{-1} \\big( 1\/[2\\mu(\\rho>n-1)]\\big),\\\n  \\theta_{n}:=\\e^{\\sup_{\\rho\\le n+2}V}  \\bb^{-1} \\big( 1\/[2\\mu(\\rho>n-1)]\\big).$$\n Combining\n(\\ref{proof-e-01}) with \\eqref{proof-e-02}, and noting that $\\rr(y)\\in (n-1,n+2)$ and $\\rr(x)\\ge n+k+2$ imply $d(x,y)>k$,\nand\n\\begin{equation*}\n\\begin{split}\n&\\iint_{\\{\\rr(x)>n+k+2, \\rho(y)\\in (n-1,n+2)\\}} (f(x)-f(y))^2 q(x,y)\\mu(\\d y)\\mu(\\d x)\\\\\n&\\le 4 \\|f\\|_\\infty^2 \\iint_{\\{\\rr(x)>n+k+2, \\rr(y) \\in (n-1,n+2),d(x,y)>k\\}}q(x,y)\\mu(\\d y)\\mu(\\d x),\\\\\n\\end{split}\n\\end{equation*}\n we obtain\n\\beq\\label{proof-e-03}\\beg{split}&\\mu_V(f^2l_n^2) \\le\n\\e^{\\sup_{\\rho\\le n+2}V}\\mu(f^2l_n^2)\\\\\n&\\le 2\\e^{\\sup_{\\rho\\le n+2}V}  \\bb^{-1} \\big( 1\/[2\\mu(\\rho>n-1)]\\big)\\EE(fl_n)\\\\\n&\\le 2\\e^{\\sup_{\\rho\\le n+2}V}  \\bb^{-1} \\big( 1\/[2\\mu(\\rho>n-1)]\\big)\\mu(1_{\\{\\rho\\le n+k+2\\}}\\GG(fl_n))\\\\\n&\\quad + 2\\e^{\\sup_{\\rho\\le n+2}V}  \\bb^{-1} \\big( 1\/[2\\mu(\\rho>n-1)]\\big)\\mu(1_{\\{\\rho> n+k+2\\}}\\GG(fl_n))\\\\\n  &\\le   4 \\dd_{n,k}\\iint_{\\{\\rr(x)\\le n+k+2, \\rr(y)\\in (n-1,n+2)\\}}(f(x)-f(y))^2q(x,y)\\mu(\\d y)\\mu_V(\\d x) \\\\\n &\\quad +16\\dd_{n,k}\\ll \\mu_V\\big(1_{\\{\\rr\\in (n-2,n+3)\\}}f^2\\big)\\\\\n &\\quad+16\\dd_{n,k} \\iint_{\\{\\rr(x)\\le n+k+2, \\rr(x) \\notin (n-2,n+3),\\rr(y)\\in (n-1,n+2)\\}}f(x)^2q(x,y)\\mu(\\d y)\\mu_V(\\d x)\\\\\n &\\quad+32 \\theta_{n}\\|f\\|_\\infty^2\\iint_{\\{\\rr(y)\\in (n-1,n+2),d(x,y)>k\\}}q(x,y)\\mu(\\d y)\\mu(\\d x). \\end{split}\\end{equation}\nNoting that $\\vv_{n,k}=\\sup_{m\\ge n} \\dd_{m,k}$, $\\zeta_n=\\sup_{m \\ge n} \\theta_m$, $\\sum_{m=1}^\\infty 1_{\\{\\rr(y)\\in (m-1,m+2)\\}}\\le 3$ and $$1_{\\{\\rr(x)\\le n+k+2, \\rr(x) \\notin (n-2,n+3),\\rr(y)\\in (n-1,n+2)\\}}\n\\le 1_{\\{d(x,y)>1, \\rr(y)\\in (n-1,n+2)\\}},$$ and taking summations in (\\ref{proof-e-03}) from $n$, we arrive at\n \\beg{equation*}\\beg{split}  & \\mu_V(f^2 1_{\\{\\rho\\ge n\\}}) \\le\n\\sum_{m=n}^\\infty \\mu_V(f^2l_m^2)\\\\\n&\\le 12 \\vv_{n,k}\\iint_{E\\times E}(f(x)-f(y))^2q(x,y)\\mu(\\d y)\\mu_V(\\d x)   +80\\vv_{n,k}\\ll \\mu_V(f^2)\\\\\n &\\quad+48\\vv_{n,k}\\iint_{\\{d(x,y)>1\\}}f(x)^2q(x,y)\\mu(\\d y)\\mu_V(\\d x) \\\\\n &\\quad+\n 96\\zeta_n\\|f\\|_\\infty^2\\iint_{\\{\\rr(y)\\ge n-1,d(x,y)>k\\}}q(x,y)\\mu(\\d y)\\mu(\\d x)\\\\\n&\\le 12 \\vv_{n,k} \\EE_V(f)+ 128\\ll\\vv_{n,k} \\mu_V(f^2)+96 \\zeta_{n} \\gg_{n,k} \\|f\\|_\\infty^2.\\end{split}\\end{equation*}\n\\end{proof}\n\n\\beg{lem} \\label{L3.2} For any $n,k\\ge 1$, $s>0$ and $f\\in \\A$,\n$$\\mu_V(f^21_{\\{\\rr\\le n\\}}) \\le  2s \\e^{K_{n,k}}\\EE_V(f) +16\\ll s\\e^{K_{n,k}}\\mu_V(f^2) + 16 s \\e^{Z_{n}}\\eta_{n,k} \\|f\\|_\\infty^2\n +\\bb(s) \\e^{J_{n,k}} \\mu_V(|f|)^2.$$\\end{lem}\n\n\\beg{proof} Let $\\phi_n:[0,\\infty)\\to [0,\\infty)$ be a smooth function\nsuch that\n\\begin{equation*} \\phi_n(s) \\begin{cases}\n& =1,\\ \\ \\ \\ \\ \\ \\ \\quad \\,\\, s\\le n,\\\\\n& \\in [0,1],\\ \\ \\ \\ n\\le s<n+1,\\\\\n& =0,\\ \\ \\ \\ \\ \\ \\ \\ \\quad s>n+1,\n\\end{cases}\n\\end{equation*}\nand $$\\sup_{s \\in [0,\\infty)}|\\phi'_{n}(s)|\\le 2.$$ Set\n$g_n =\\phi_n\\circ\\rho.$ Then $g_n\\in\\A$ and\n$$|g_n(x)-g_n(y)|\\le   2(1\\wedge d(x,y))\\big(1_{\\{\\rr(x)\\le n+1\\}}+1_{\\{\\rr(x)>n+1,\\rr(y)\\le n+1\\}}\\big).$$\nSo, similarly to (\\ref{proof-e-02}) we have\n\\beg{equation*}\\beg{split} \\GG(fg_n)(x) &\\le 2 \\int_{\\{\\rr(y)\\le n+1\\}}(f(x)-f(y))^2q(x,y)\\mu(\\d y) \\\\\n&\\quad + 8\\ll f^2(x)1_{\\{\\rr(x)\\le n+k+2\\}}\n+ 8 f^2(x) 1_{\\{\\rr(x)>n+k+2\\}} \\int_{\\{\\rr(y)\\le n+1\\}} q(x,y) \\mu(\\d y).\n\\end{split}\\end{equation*}\nNote that $\\rr(x)> n+k+2$ and $\\rr(y)\\le n+1$ imply that $d(x,y)>k$,\nand\n\\begin{equation*}\n\\begin{split}\n&\\iint_{\\{\\rr(x)>n+k+2,\\rho(y)\\le n+1\\}} (f(x)-f(y))^2 q(x,y)\\mu(\\d y)\\mu(\\d x)\\\\\n&\\le 4 \\|f\\|_\\infty^2 \\iint_{\\{\\rr(x)>n+k+2, \\rr(y)\\le n+1,d(x,y)>k\\}}q(x,y)\\mu(\\d y)\\mu(\\d x).\n\\end{split}\n\\end{equation*}\nCombining all the estimates above with \\eqref{sp},   we obtain\n\\beg{equation*}\\beg{split} &\\mu_V(f^21_{\\{\\rho\\le n\\}})\\\\\n&  \\le \\mu_V(f^2g_n^2)\n \\le \\e^{\\sup_{\\rho\\le n+1}V}\\mu(f^2g_n^2)\\\\\n&\\le \\e^{\\sup_{\\rho\\le n+1}V}\\Big\\{s\\mu(\\GG(fg_n)) +\\bb(s) \\mu(|fg_n|)^2\\Big\\}\\\\\n&= \\e^{\\sup_{\\rho\\le n+1}V}\\Big\\{s\\mu(1_{\\{\\rho\\le n+k+2\\}}\\GG(fg_n))\n +s\\mu(1_{\\{\\rho> n+k+2\\}}\\GG(fg_n))+\\bb(s) \\mu(|fg_n|)^2\\Big\\}\\\\\n&\\le s\\e^{K_{n,k}} \\mu_V(1_{\\{\\rho\\le n+k+2\\}}\\GG(fg_n))\n  +s\\e^{\\sup_{\\rr\\le n+1} V} \\mu (1_{\\{\\rho> n+k+2\\}}\\GG(fg_n))+\\bb(s)\\e^{J_{n,k}} \\mu_V(|f|)^2 \\\\\n&\\le 2s \\e^{K_{n,k}}\\EE_V(f) +16\\ll s\\e^{K_{n,k}}\\mu_V(f^2) + 16 s\\e^{Z_n} \\eta_{n,k} \\|f\\|_\\infty^2\n +\\bb(s) \\e^{J_{n,k}} \\mu_V(|f|)^2.\\end{split}\\end{equation*}\n\\end{proof}\n\nNow, we are in a position to prove Theorem \\ref{finite-sp}.\n\n\\beg{proof}[Proof of Theorem $\\ref{finite-sp}$] It suffices to prove\nfor $f\\in\\A$.\n\n(1) Since $q(x,y)=0$ for $d(x,y)> k_0$, we have $\\gg_{n,k}=\\eta_{n,k}=0$ for all $k\\ge k_0$ So, by Lemmas \\ref{L3.1} and \\ref{L3.2},\n\\beq\\label{DF}\\beg{split} \\mu_V(f^2)\\le\n&2\\big(6\\vv_{n,k} +  s\\e^{K_{n,k}}\\big)\\EE_V(f) + 16\\ll\\big(8 \\vv_{n,k}\n + s\\e^{K_{n,k}}\\big)\\mu_V(f^2)\\\\\n& +\\bb(s)\\e^{J_{n,k}}\\mu_V(|f|)^2,\\ \\ \\ s>0, n\\ge 1, k\\ge k_0.\\end{split}\\end{equation}  If $\\inf_{n\\ge1,k\\ge k_0}\\vv_{n,k}= 0$, then for any $r'\\in (0,r]$ there exist $s>0$, $n\\ge1$ and $k\\ge k_0$ such that\n$$8\\vv_{n,k} +s \\e^{K_{n,k}}\\le \\ff {r'}{2+16\\ll r'}.$$ Combining this with (\\ref{DF}) we obtain\n$$\\mu_V(f^2)\\le r'\\EE_V(f) + (1+8\\ll r')\\bb(s) \\e^{J_{n,k}}\\mu_V(|f|)^2\\le r\\EE_V(f) + (1+8\\ll r')\\bb(s) \\e^{J_{n,k}}\\mu_V(|f|)^2.$$ This implies\nthe super Poincar\\'e inequality for the desired $\\bb_V$.\n\n(2) If \\beq \\label{AAA} \\inf_{n\\ge1,k\\ge k_0} \\vv_{n,k}<\\ff 1 {128\\ll},\\end{equation} then there exist $n\\ge 1$, $k\\ge k_0$ and $s>0$ such that\n$$16\\ll\\big(8 \\vv_{n,k}\n + s\\e^{K_{n,k}}\\big)<1.$$ Therefore, the defective Poincar\\'e inequality follows from (\\ref{DF}).\n\n  To prove (\\ref{th-sp-2}) without condition  (\\ref{AAA}),    we   follow the approach in the proof of \\cite[Theorem 3.1(2)]{BLW} by making bounded perturbations of $V$.  For any $N\\ge 1$,\nwrite\n$$ V= V_N+ (V\\land N)\\lor (-N),\\ \\\nV_N:=(V-N)1_{\\{V \\ge N\\}}+(V+N)1_{\\{V \\le -N\\}}.$$\nSince $(V\\land N)\\lor (-N)$ is bounded and the defective Poincar\\'e inequality is stable under bounded perturbations of $V$, we only need to prove that when $V$ is unbounded we may find $N\\ge 1$ such that   the defective Poincar\\'e inequality holds for $V_N$ in place of $V$.\nIt is easy to check that for $n$ large enough\n\\begin{equation*}\n\\sup_{\\rho \\le n}V_N\\le\n\\begin{cases}\n& \\big(\\sup_{\\rho \\le n}V\\big)-N,\\ \\ \\ \\ \\text{if}\\ \\ \\sup_{E}V=+\\infty,\\\\\n& \\sup_{\\rho \\le n}V,\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\quad\\ \\text{if}\\ \\\n\\sup_{E}V<+\\infty,\n\\end{cases}\n\\end{equation*} and\n\\begin{equation*}\n\\inf_{\\rho \\le n}V_N\\ge\n\\begin{cases}\n& \\big(\\inf_{\\rho \\le n}V\\big)+N,\\ \\ \\ \\ \\text{if}\\ \\ \\inf_{E}V=-\\infty,\\\\\n& \\inf_{\\rho \\le n}V,\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\quad \\text{if}\\ \\\n\\inf_{E}V>-\\infty.\n\\end{cases}\n\\end{equation*}Thus, for large $n$ we have $K_{n,k}(V_N)\\le K_{n,k}(V)-N$, so that\n\\begin{equation}\\label{proof-e-12}\n\\begin{split}\n&\\inf_{n\\ge1,k\\ge k_0}\\vv_{n,k}(V_N) \\le\n\\e^{-N}\\inf_{n\\ge 1,k\\ge k_0}\\vv_{n,k}(V).\n\\end{split}\n\\end{equation}\nSince $\\inf_{n\\ge 1,k\\ge k_0}\\vv_{n,k}(V)<\\infty,$ we see that (\\ref{AAA}) holds for $V_N$ in place of $V$ when $N$ is  large enough. Therefore, the defective Poincar\\'e inequality holds for $V_N$ in place of $V$ as observed above.\n\\end{proof}\n\n\nNext, we turn to the proof of Theorem $\\ref{th-sp}$. To get rid of the uniform norm included in Lemmas \\ref{L3.1} and \\ref{L3.2}, we  adopt a cut-off\n argument as in the proof of \\cite[Theorem 3.2]{W00a} or \\cite[Theorem 3.3.3]{Wbook}. More precisely,  for $\\dd>1$ in assumption {\\bf (A)} and a  non-negative function $f$, let\n \\begin{equation}\\label{pro-function}f_{\\dd,i}  =(f-\\dd^{\\ff i 2})^+\\land (\\dd^{\\ff{i+1}2} -\\dd^{\\ff i 2}),\\ \\ i\\ge 0.\\end{equation}\n According to \\cite[Lemma 3.3.2]{Wbook}, for $f\\in \\D(\\EE_V)$ and $i,j\\ge 0$, we have $f_{\\dd,i}$, $(f-\\dd^{\\ff j 2})^+$ and $f\\land \\dd^{\\ff j 2}\\in\\D(\\EE_V)$. Moreover,\n \\beq\\label{dri-01} \\sum_{i=j}^\\infty \\EE_V(f_{\\dd,i})\\le \\EE_V((f-\\dd^{\\ff j 2})^+),\\ \\ \\EE_V((f-\\dd^{\\ff j 2})^+)+\\EE_V(f\\land \\dd^{\\ff j 2})\\le \\EE_V(f).\\end{equation}\nWe also have the following lemma.\n\n \\beg{lem}\\label{L3.3} For any non-negative function $f$ and $k\\in \\mathbb Z_+:=\\{0,1,2,\\cdots\\}$,\n \\beq\\label{dri-02} \\sum_{i=k}^\\infty f_{\\dd,i}^2\\ge c(\\dd) \\big({(f-\\dd^{\\ff k 2})^+}\\big)^2.\\end{equation} \\end{lem}\n\n \\beg{proof} We shall simply use $f$ to denote its value at a fixed point.\n If $ f \\le 1$, then both sides in (\\ref{dri-02}) are equal to zero. Assume that $f\\in (\\dd^{\\ff l 2}, \\dd^{\\ff{l+1}2}]$ for some $l\\in \\mathbb Z_+$. If $l\\le k$ then\n $$\\sum_{i=k}^\\infty f_{\\dd,i}^2= \\big({(f-\\dd^{\\ff k 2})^+}\\big)^2 \\ge c(\\dd)\\big({(f-\\dd^{\\ff k 2})^+}\\big)^2. $$ Next, if $l>k$ then\n $$\\sum_{i=k}^\\infty f_{\\dd,i}^2\\ge (\\dd^{\\ff l 2}-\\dd^{\\ff{l-1}2})^2= c(\\dd)\\dd^{l+1} \\ge c(\\dd)f^2.$$\n In conclusion, (\\ref{dri-02}) holds. \\end{proof}\n\n\n\n\\beg{proof}[Proof of Theorem $\\ref{th-sp}$] Since $\\EE_V(|f|,|f|)\\le \\EE_V(f,f)$ for every\n $f \\in \\A$, without loss of generality, we may and do assume that $f\\in\\A$ with $f\\ge 0$ and $\\mu_V(f^2)=1$.\n\n(1) By Lemmas \\ref{L3.1} and \\ref{L3.2}, we have\n  \\beq\\label{proof-e-09}\\beg{split}\\mu_V(f^2)\\le &\n2\\big(6\\vv_{n,k} +  s\\e^{K_{n,k}}\\big)\\EE_V(f) + 16\\ll\\big(8 \\vv_{n,k}+ s\\e^{K_{n,k}}\\big)\\mu_V(f^2)\\\\\n&+\\bb(s)\\e^{J_{n,k}}\\mu_V(|f|)^2+16\\big(6\\zeta_{n}\\gg_{n,k}+s\\e^{Z_{n}}\\eta_{n,k} \\big) \\|f\\|_\\infty^2,\\\n\\ s,k\\ge 1,n \\ge 1.\\end{split}\\end{equation}\n\nNext,  let $r>0$,\n$\\{(n_i,k_i)\\}\\in I_\\dd$, $j\\in D(r,\\{(n_i,k_i)\\})$ be fixed, and let $f_{\\dd,i}$ be   defined by \\eqref{pro-function}. Since\n$\\|f_{\\dd,i}\\|^2_\\infty\\le c(\\delta)\\dd^{i+2}$ and by the\nCauchy-Schwarz inequality\n$$\\mu_V(f_{\\dd,i})^2=\\mu_V(f_{\\dd,i}1_{\\{f\\ge \\delta^{i\/2}\\}})^2\n\\le \\mu_V(f_{\\dd,i}^2)\\mu_V(f^2\\ge \\dd^i)\\le\n\\mu_V(f_{\\dd,i}^2)\\dd^{-i},$$ it follows from (\\ref{A-01}) and\n(\\ref{proof-e-09}) with $n=n_i$ and $s= t_{i,n_i,k_i}$   that\n$$\\mu_V(f_{\\dd,i}^2)\\le \\frac{c(\\dd)r}{8} \\EE_V(f_{\\dd,i})\n+\\ff 1 2 \\mu_V(f_{\\dd,i}^2) + 16c(\\delta)(6\n\\zeta_{n_i}\\gg_{n_i,k_i}+t_{i,n_i,k_i}\\e^{Z_{n_i}}\n\\eta_{n_i,k_i}) \\dd^{i+2},\\ \\ i\\ge j.$$ That is,\n$$\\mu_V(f_{\\dd,i}^2)\\le \\frac{c(\\dd)r}{4} \\EE_V(f_{\\dd,i})\n+ 32c(\\delta)(6 \\zeta_{n_i,k_i}\\gg_{n_i,k_i}+t_{i,n_i,k_i}\n\\e^{Z_{n_i}}\\eta_{n_i,k_i})\\dd^{i+2},\\ \\ i\\ge j.$$ Taking\nsummation over $i\\ge j$ and using (\\ref{dri-01}), (\\ref{dri-02}) and\n(\\ref{A-02}), we obtain \\beq\\label{proof-e-10}\n\\mu_V\\big({(f-\\dd^{\\ff j 2})^+}^2\\big)\\le \\ff r 4 \\EE_V((f-\\dd^{\\ff\nj 2})^+) +\\ff 1 8.\\end{equation} On the other hand, noting that\n$c(\\delta)\\in(0,1)$ and $\\delta>1$, applying (\\ref{proof-e-09}) with\n$n=n_j$ and $s= t_{j,n_j,k_j}$ to $f \\wedge \\dd^{\\frac{j}{2}}$, and\ncombining with (\\ref{A-01}), (\\ref{A-02}), we obtain\n\\beg{equation*}\\beg{split}\n\\mu_V(f^2\\land\\dd^j)&\\le\\frac{c(\\delta)r}{8} \\EE_V(f\\land  \\dd^{\\ff\nj 2}) +\\ff 1 4 \\mu_V(f^2\\wedge\\dd^{ j})+ \\ff {\\dd^j}4 \\mu_V(|f|)^2 \\\\\n&\\quad+\n16( 6 \\vv_{n_j,k_j}\\gg_{n_j,k_j}+t_{j,n_j,k_j}\\e^{Z_{n_j}}\\eta_{n_j,k_j})\\dd^{j }\\\\\n&\\le \\ff r {8} \\EE_V(f\\land  \\dd^{\\ff j 2}) +\\ff 1 2\n\\mu_V(f^2\\wedge\\dd^{ j})+ \\ff {\\dd^j}4 \\mu_V(|f|)^2 + \\frac{1}{16}.\n \\end{split}\\end{equation*} Thus,\n$$\\mu_V(f^2\\land\\dd^j)\\le \\ff r 4 \\EE_V(f\\land  \\dd^{\\ff j 2}) +\\ff{\\dd^j} 2 \\mu_V(|f|)^2 +\\ff 1 8.$$\nCombining this with (\\ref{proof-e-10}) and using (\\ref{dri-01}), we\narrive at\n\\beg{equation*}\\beg{split} 1&= \\mu_V(f^2)\\le \\mu_V\\big\\{((f-\\dd^{\\ff j 2})^++(f\\wedge\n\\dd^{\\ff j 2}))^2\\big\\}\\\\\n&\\le 2 \\mu_V\\Big(\\big({(f-\\dd^{\\ff j 2})^+}\\big)^2\\Big)+ 2  \\mu_V(f^2\\land\\dd^j)\\\\\n&\\le \\ff r 2\\Big(\\EE_V((f-\\dd^{\\ff j 2})^+)+\\EE_V(f\\land\n\\dd^{\\ff j2})\\Big)+ \\ff 1 2 +\\dd^j \\mu_V(|f|)^2\\\\\n&\\le \\ff r 2 \\EE_V(f)\n+\\ff 1 2 +\\dd^j \\mu_V(|f|)^2.\\end{split}\\end{equation*}  Therefore,\n\\beq\\label{DDE} \\mu_V(f^2)\\le r\\EE_V(f) + 2 \\dd^j\\mu_V(|f|)^2\\end{equation}\nholds for all $\\{(n_i,k_i)\\}\\in I_\\dd$ and $j\\in D(r,\\{(n_i,k_i)\\})$. This proves (\\ref{th-sp-1}) for the desired $\\bb_V$.\n\n\n(2) If \\beq\\label{DDF} \\limsup_{i\\to\\infty} (8\\vv_{n_i,k_i}\n+t_{i,n_i,k_i}e^{K_{n_i,k_i}})\\le \\ff 1 {64\\ll},\\end{equation}then there exist a\nconstant $r>0$ and $j\\ge 1$\n  such that\n(\\ref{A-01}) and (\\ref{A-02}) hold, i.e.\\ $j\\in D(r,\\{(n_i,k_i)\\})$.\nSo, the arguments in (1) ensure    (\\ref{DDE}),  and so the\ndefective Poincar\\'e inequality for $C_1=r$ and $C_2= 2\\dd^j.$ Let\n$V_N$ be in the proof of Theorem \\ref{finite-sp}(2). It follows from the proof of Theorem \\ref{finite-sp}(2) that for $n$ large enough\n$K_{n,k}(V_N)\\le K_{n,k}(V)-N$. Since\n$J_{n,k}(V_N)\\le J_{n,k}(V)$ and $\\beta(r)$ is decreasing, we have\n\\begin{equation*}\n\\begin{split}\n&\\limsup_{i \\to\\infty}t_{i,n_i,k_i}(\\dd,V_N)e^{K_{n_i,k_i}(V_N)} \\le\n\\e^{-N}\\limsup_{i\\to\\infty}t_{i,n_i,k_i}(\\dd,V)e^{K_{n_i,k_i}(V)},\n\\end{split}\n\\end{equation*} which combined with \\eqref{proof-e-12} yields that\n$$\\limsup_{i \\to\\infty}\\Big(\\vv_{n_i,k_i}(V_N)+t_{i,n_i,k_i}(\\dd,V_N)e^{K_{n_i,k_i}(V_N)} \\Big)\\le\n\\e^{-N}\\limsup_{i\\to\\infty}\\Big(\\vv_{n_i,k_i}(V)+t_{i,n_i,k_i}(\\dd,V)e^{K_{n_i,k_i}(V)}\\Big).$$\nThen the remainder of the proof is similar to that of Theorem\n\\ref{finite-sp}(2) by using (\\ref{DDF}) instead of (\\ref{AAA}).\n\\end{proof}\n\n\n\\subsection{Perturbations for the super Poincar\\'e inequality under a variation condition\n }\n\n It is well known that in the diffusion case the super Poincar\\'e inequality is stable under Lipschitz perturbations (see \\cite[Proposition 2.6]{W00a}). The aim of this section is to extend this result to the non-local setting using a variation condition on ${\\rm supp}\\,q:=\\{(x,y): q(x,y)>0\\}$.\n\n \\beg{thm}\\label{th-sup} Assume that $(\\ref{sp})$ holds. If there exists a constant $\\kk_1>0$ such that  $\\kk_2:= \\mu(\\e^{-2V})<\\infty$ and\n \\beq\\label{th-sup-1}   |V(x)-V(y)|\\le \\kk_1(1\\land d(x,y)),\\ \\ (x,y)\\in{\\rm supp}\\,q.\\end{equation} Then  $(\\ref{th-sp-1})$ holds for\n $$\\bb_V(s):= \\inf\\Big\\{ 16\\kk_2\\bb(r)^3(4+\\ll \\kk_1^2 s'):\\ s'\\in (0,s], 0<r\\le \\ff{s'\\e^{-\\kk_1}}{4+\\ll \\kk_1^2s'  }\\Big\\},\\ \\ s>0.$$\\end{thm}\n\n \\beg{proof}\nTo prove (\\ref{sp}) for the desired $\\bb_V$, we may and do assume that $V$ is bounded. Indeed, for any $n\\ge 1$ let\n $$V_n= (V\\land n)\\lor (-n) -\\log \\mu(\\e^{(V\\land n)\\lor(-n)}).$$ Then $\\mu(\\e^{V_n})=1$, (\\ref{th-sup-1}) holds for $V_n$ in place of $V$, and\n $$\\lim_{n\\to\\infty} \\mu(\\e^{-2V_n})=\\kk_2.$$ Thus, applying the assertion to the bounded  $V_n$   and letting $n\\to\\infty$, we complete the proof.\n\n Now, let $V$ be bounded and let $f\\in\\A$ with $\\mu_V(|f|)=1$. Take $\\tt f=f\\e^{\\ff V2}.$\n\n By (\\ref{th-sup-1}) we have for every $x,y \\in \\text{supp}q$,\n \\begin{equation*}\n \\begin{split}\n &\\big(1-\\e^{\\frac{V(x)-V(y)}{2}}\\big)^2 \\le \\frac{\\e^{\\kk_1}\\kk_1^2 }{4}\\big(1\\wedge d^2(x,y)\\big),\n \\end{split}\n \\end{equation*}\n hence\n \\beg{equation*}\\beg{split} \\GG(\\tt f)(x)&\\le 2\\int_{E} \\Big\\{\\e^{V(y)}(f(x)-f(y))^2+ f^2(x)(\\e^{\\ff {V(x)}2}-\\e^{\\ff {V(y)}2})^2\\Big\\}q(x,y)\\mu(\\d y)\\\\\n &\\le 2\\e^{V(x)+\\kk_1}\\GG(f)(x) + \\ff{1}2 \\kk_1^2\\ll \\e^{V(x)+\\kk_1}f^2(x).\n\\end{split}\\end{equation*} Since $V$ is bounded,  this implies $\\tt f\\in\\A$. Moreover,    combining this with  (\\ref{sp}), we obtain\n \\beg{equation}\\label{th-sup-2}\\beg{split} \\mu_V(f^2)&= \\mu({\\tt f}^2)\\le r \\mu(\\GG(\\tt f)) +\\bb(r) \\mu(|\\tt f|)^2\\\\\n &\\le 2 r \\e^{\\kk_1}\\EE_V(f) + r\\ff{  \\kk_1^2 }2\\ll \\e^{\\kk_1} \\mu_V(f^2) +\\bb(r) \\mu_V( |f|\\e^{-\\ff V 2})^2.\\end{split}\\end{equation}\n Since $\\mu_V(|f|)=1$, for any $R>0$ we have $\\mu_V(|f|>R)\\le \\ff 1 R$, and hence,\n \\beg{equation*}\\beg{split} \\mu_V(|f|\\e^{-\\ff V 2})^2 &\\le 2 \\mu_V(|f|\\e^{-\\ff V 2}1_{\\{|f|\\le R\\}})^2 + 2 \\mu_V(|f|\\e^{-\\ff V 2}1_{\\{|f|\\ge R\\}})^2\\\\\n &\\le 2 R \\mu_V(|f|^{\\ff 1 2} \\e^{-\\ff V 2})^2 +2 \\mu_V(f^2) \\mu_V(\\e^{-V} 1_{\\{|f|>R\\}})\\\\\n &\\le 2 R \\mu_V(|f|)\\mu_V(\\e^{-V}) + 2\\mu_V(f^2) \\ss{\\mu_V(|f|>R)\\mu_V(\\e^{-2V})}\\\\\n  &\\le 2 R + 2 \\mu_V(f^2)\\ff{\\ss{\\kk_2}}{\\ss R}.\\end{split}\\end{equation*} Taking $R= 16\\kk_2\\bb(r)^2$, we get\n  $$\\bb(r)\\mu_V(|f|\\e^{-\\ff V 2})^2 \\le 32\\kk_2\\bb(r)^3 +\\ff 1 2\\mu_V(f^2).$$\n  Substituting this into \\eqref{th-sup-2}, we arrive at\n   $$\\mu_V(f^2)\\le 4r\\e^{\\kk_1}\\EE_V(f)+ r \\kk_1^2 \\ll \\e^{\\kk_1}\\mu_V(f^2) +64 \\kk_2\\bb(r)^3.$$\n  Therefore,  for any $s'\\in (0,s]$ such that\n  $$r\\le \\ff {s'\\e^{-\\kk_1}}{4 + \\ll \\kk_1^2  s'},$$ we have\n  $$\\mu_V(f^2)\\le s' \\EE_V(f)+ 16\\kk_2\\bb(r)^3(4+\\ll\\kk_1^2 s')\\le s \\EE_V(f) + 16\\kk_2\\bb(r)^3(4+\\ll\\kk_1^2 s').$$\n  This implies the desired super Poincar\\'e inequality.\n \\end{proof}\n\nLet $E=\\R^m$ and $d(x,y)=|x-y|$. If the jump has a finite range, i.e. there is a constant $k\\ge 1$ such that $q(x,y)=0$ for $|x-y|>k$, then (\\ref{th-sup-1}) holds for any Lipschitz function  $V$. Therefore, the above theorem implies that the super Poincar\\'e inequality is stable for  all Lipschitiz perturbations  as is known in the diffusion case. In particular,   since  the defective log-Sobolev inequality\n$$ \\mu(f^2\\log f^2)\\le C_1 \\EE(f)+C_2,\\ \\ f\\in\\D(\\EE),\\mu(f^2)=1$$ holds for some $C_1,C_2>0$ if and only if\n the super Poincar\\'e inequality $(\\ref{sp})$ holds for $\\bb(r)=\\e^{c(1+r^{-1})}$ for some $c>0$, see \\cite[Corllary 1.1]{W00b} for $\\dd=1$,\n we conclude from Theorem \\ref{th-sup} that the defective log-Sobolev inequality is stable under perturbations of Lipschitz functions $V.$  See \\cite[Example 1.2]{CW2} for examples of $\\mu$ and $q$ having finite range of jumps such that the log-Sobolev inequality holds.\n\n\n\n\n\\section{Perturbations for the weak Poincar\\'{e} inequality\n\n\n\nSuppose that the  weak Poincar\\'{e}\ninequality\n\\begin{equation}\\label{wp} \\mu(f^2)\\le \\beta(r)\n\\EE(f,f)+r\\|f\\|_\\infty^2,\\quad r>0, f\\in \\D(\\EE), \\mu(f)=0,\\end{equation}\nholds for some decreasing function $\\beta:(0,\\infty)\\to (0,\\infty).$\nTo derive the weak Poincar\\'e inequality for $\\EE_V$ using growth conditions on $V$, for any $n,k\\ge 1$ let\n\\begin{equation*}\n\\begin{split}\n&\\tt K_{n,k} (V)=\\sup_{\\rho\\le n}V -\\inf_{\\rho \\le n+k+1}V,\\quad\\quad  \\tt Z_n(V)=\\sup_{\\rho\\le n}V,\\\\\n&\\tt\\eta_{n,k} =\\iint_{\\{\\rr(x)> n+k+1, \\rr(y)\\le n+1\\}}q(x,y)\\mu(\\d\ny)\\mu(\\d x),\\ \\ \\tt \\gg_k=\\iint_{\\{d(x,y)>k\\}}q(x,y)\\mu(\\d y)\\mu(\\d\nx).\n\\end{split}\n\\end{equation*} It is clear that $\\tt\\eta_{n,k}\\le  \\tt \\gg_k$. By \\eqref{intro-02} we have $\\tt\\eta_{n,k}\\downarrow 0$ as $n\\uparrow\\infty$ or $k\\uparrow\\infty$.\n\n\n\n\\beg{thm}\\label{th-wp} Assume that the weak Poincar\\'{e} inequality\n\\eqref{wp} holds. If for any $\\vv>0$\n\\beq\\label{th-wp-1}\\inf_{n,k\\ge 1} \\e^{\\tt Z_{n}(V)}\\beta\\big(\\vv \\e^{-\\tt Z_{n}(V)}\\big)\\big(\\tt\\eta_{n,k}\n+\\tt \\gg_k+\\mu(\\rr>n-k)\\big)=0,\\end{equation} then\n \\begin{equation}\\label{th-wp-2} \\mu_V(f^2)\\le\n\\beta_V(r)\\EE_V(f,f)+r\\|f\\|_\\infty^2,\\quad r>0,f\\in \\D(\\EE_V),\n\\mu_V(f)=0\\end{equation} holds for\n\\beg{equation*}\\beg{split} \\beta_V(r):=\\inf\\Big\\{&2\\beta\\Big(\\ff r 8 \\e^{-\\tt Z_{n}(V)}\\Big)\\e^{\\tt K_{n,k}(V)}:\\\n 6\\mu_V(\\rr>n)\\\\\n  &+2\\e^{\\tt Z_{n}(V)}\\beta\\Big(\\ff r 8 \\e^{-\\tt Z_{n}(V)}\\Big)\\big(4\\tt \\eta_{n,k} +\\tt \\gg_k+4\\ll\\mu(\\rr>n-k)\\big)\\le \\ff r 2\\Big\\}<\\infty,\\ \\ r>0.\n \\end{split}\\end{equation*}  \\end{thm}\n\n\\beg{proof} It is easy to see that (\\ref{th-wp-1}) implies $\\beta_V(r)<\\infty$ for $r>0.$ Let $g_n$ be in the proof of Lemma \\ref{L3.2}. Then for any $f\\in\\A$ with $\\mu_V(f)=0$, we have\n$$\\mu_V(fg_{n})^2=\\mu_V(f(1-g_{n}))^2\\le\\|f\\|_\\infty^2 \\mu_V(\\rr>n)^2.$$ Moreover,\n\\beg{equation*}\\beg{split} {\\rm Var}_{\\mu_V}(fg_{n}) &\\le \\mu_V\\big((fg_{n}-\\mu(fg_{n}))^2\\big)\\\\\n&\\le \\e^{\\sup_{\\rr\\le n}V}\\mu\\big(1_{\\{\\rr\\le n\\}}(fg_{n}-\\mu(fg_{n}))^2\\big) +\\mu_V\\big(1_{\\{\\rr>n\\}}(fg_{n}-\\mu(fg_{n}))^2\\big)\\\\\n&\\le \\e^{\\sup_{\\rr\\le n}V}{\\rm Var}_{\\mu}(fg_{n}) +4\\|f\\|_\\infty^2 \\mu_V(\\rr>n).\\end{split}\\end{equation*} Then\n\\beg{equation}\\label{th-wp-3}\\beg{split} \\mu_V(f^2) &\\le {\\rm Var}_{\\mu_V }(fg_{n})+\\mu_V(fg_{n})^2 +\\mu_V(f^21_{\\{\\rr>n\\}})\\\\\n&\\le \\e^{\\sup_{\\rr\\le n} V}{\\rm Var}_{\\mu}(fg_{n}) +6\\|f\\|_\\infty^2 \\mu_V(\\rr>n).\\end{split}\\end{equation}\nOn the other hand, we have\n\\beg{equation*}\\beg{split} \\e^{\\sup_{\\rr\\le n}V} \\EE(fg_{n})\n  \\le &2 \\e^{\\sup_{\\rr\\le n}V}\\iint_{E\\times E} g_{ n}^2(y)(f(x)-f(y))^2q(x,y)\\mu(\\d y)\\mu(\\d x)\\\\\n  & + 2 \\e^{\\sup_{\\rr\\le n}V}\\iint_{E\\times E}f^2(x)(g_{n}(x)-g_{n}(y))^2 q(x,y)\\mu(\\d y)\\mu(\\d x)\\\\\n\\le &2 \\e^{\\tt K_{n,k}(V)}\\EE_V(f) +2\\e^{\\tt Z_n(V)}(4\\tt\\eta_{n,k}+\\tt \\gg_k+4\\ll\\mu(\\rr>n-k))\\|f\\|_\\infty^2,\\end{split}\\end{equation*}\nwhere in the last inequality we made\nthe first integral on the sets $\\{\\rr(x)\\le n+k+1\\}$ and $\\{\\rr(x)>n+k+1\\}$, and the second\nintegral on the sets $\\{\\rr(x)\\le n-k\\}$ and $\\{\\rr(x)>n-k\\}$, and also used the facts that\n\\beg{equation*}\\beg{split} &\\iint_{\\{\\rr(x)> n+k+1\\}}g_n^2(y)(f(x)-f(y))^2\nq(x,y)\\mu(\\d y)\\mu(\\d x) \\le \\tt4\\eta_{n,k} \\|f\\|_\\infty^2,\\\\\n&\\iint_{\\{\\rr(x)\\le n-k\\}}f^2(x)(g_n(x)-g_n(y))^2q(x,y)\\mu(\\d\ny)\\mu(\\d x)\n \\le  \\tt \\gg_k \\|f\\|_\\infty^2,\\\\\n&\\iint_{\\{\\rr(x)> n-k\\}}f^2(x)(g_n(x)-g_n(y))^2q(x,y)\\mu(\\d y)\\mu(\\d\nx)\\le 4\\lambda\\mu(\\rr>n-k)\\|f\\|_\\infty^2.\n\\end{split}\\end{equation*}\nCombining this with (\\ref{th-wp-3}) and \\eqref{wp}, we arrive at\n\\beg{equation*}\\beg{split} \\mu_V(f^2) \\le &2\\beta(s)\\e^{\\tt K_{n,k}(V)} \\EE_V(f)\\\\\n&+ \\|f\\|_\\infty^2\\big\\{6\\mu_V(\\rr>n) + 4s\\e^{\\tt Z_{n}(V)}+\n2\\beta(s)\\e^{\\tt Z_{n}(V)} (4\\tt\\eta_{n,k} + \\tt\n\\gg_k+4\\ll\\mu(\\rr>n-k))\\big\\}.\\end{split}\\end{equation*} So, for any\n$r>0$, let $s= \\ff r 8 \\e^{-\\tt Z_{n}(V)}$. If for some $n,k\\ge 1$\none has\n$$ 6\\mu_V(\\rr>n)\n   +2\\e^{\\tt Z_{n}(V)}\\beta\\Big(\\ff r 8 \\e^{-\\tt Z_{n}(V)}\\Big)\\big(4\\tt \\eta_{n,k} +\\tt \\gg_k+ 4\\ll\\mu(\\rr>n-k)\\big)\\le \\ff r 2,$$ then\n$$\\mu_V(f^2)\\le 2\\beta(s)\\e^{\\tt K_{n,k}(V)} \\EE_V(f)+r\\|f\\|_\\infty^2.$$ Therefore, the proof is finished.\n\\end{proof}\n\n\nTo conclude this section,  we present an example where $(\\EE,\n\\scr{D}(\\EE))$ satisfies the Poincar\\'e inequality, i.e. the weak\nPoincar\\'e inequality \\eqref{wp} holds for a constant function\n$\\bb.$\n\n\\begin{exa}\\label{exa-wp}\\rm Let $E=\\R^m$  with $d(x,y)=|x-y|$.\nLet\n$$q(x,y)=\\ff{(1+|y|)^{m+\\aa}}{|x-y|^{-(m+\\alpha)}},\\ \\  \\mu(\\d x)=\\frac{c_{m,\\aa}\\d\nx}{(1+|x|)^{m+\\alpha}}$$ for some constant $0<\\alpha<2$, where\n$c_{m,\\aa}$ is a normalizing constant such that $\\mu$ is a\nprobability measure. Then $\\A\\supset C_0^2(\\R^m),$ and according to \\cite[Corollary 1.2(1)]{WW}, \\eqref{wp} holds\nfor a constant rate function $\\beta(r)\\equiv \\beta>0.$\n\nNow, let   $V$ be measurable satisfying\n\\begin{equation}\\label{exa-wp-01}\n-s\\varepsilon\\log(1+|x|)-K \\le  V(x)\\le\n(1-s)\\varepsilon\\log(1+|x|)+K,\\quad x\\in \\R^m\n\\end{equation}\nfor some constants $\\varepsilon\\in [0,\\alpha)$, $s \\in [0,1]$ and\n$K\\in \\R$ such that $\\mu(e^V)=1$. Then the weak Poincar\\'{e} inequality \\eqref{th-wp-2} holds\nwith\n\\begin{equation}\\label{exa-wp-02}\n\\beta_V(r)=C\\big(1+r^{-\\varepsilon\/(\\aa-(1-s)\\vv)}\\big)\n\\end{equation}\nfor some constant $C>0$.\n\nMoreover, the assertion is sharp in the following two cases with $s=0$.\n\\beg{enumerate}\\item[(i)] $\\beta_V$ in     \\eqref{exa-wp-02} is sharp, i.e. $\\beta_V$ can not be replaced by any essentially smaller functions:    if\n$$\\lim_{r\\to0} r^{\\vv\/(\\aa-\\vv)}\\beta_V(r) =0,$$ then the weak Poincar\\'{e} inequality \\eqref{th-wp-2} does not hold.\n \\item[(ii)] The constant $K$ can not be replaced by any unbounded functions: for \\begin{equation}\\label{exa-wp-03}\n V(x)=\n \\varepsilon\\log(1+|x|)+ \\phi(|x|)+K_0,\n\\end{equation}\nwhere $\\vv\\in [0,\\aa)$,\n$\\phi:[0,+\\infty)\\rightarrow [0,+\\infty)$\nis an increasing function with $\\phi(r)\\uparrow\\infty$ as $r\\uparrow\\infty$ such that\n$\\mu(e^{\\varepsilon\\log(1+|\\cdot|)+ \\phi(|\\cdot|)})<\\infty$, and $K_0\\in\\R^d$ is such that $\\mu(e^V)=1$, the weak Poincar\\'e inequality\n(\\ref{th-wp-2}) with the rate function $\\beta_{V}$ given by\n(\\ref{exa-wp-02}) does not hold. \\end{enumerate}\n\\end{exa}\n\n\n\\begin{proof} Take $k=\\frac{n}{2}$. Then\n  there is a constant $c_1>0$ such that for $n$ large enough\n\\begin{equation*}\n\\begin{split}\n&\\tt K_{n,\\frac{n}{2}}(V)  \\le \\vv \\log(1+n)+c_1,\\ \\ \\ \\tt Z_n(V)\\le \\vv(1-s) \\log(1+n)+c_1,\\\\\n&\\tt\\eta_{n,\\frac{n}{2}}+\\tt \\gg_{\\frac{n}{2}}  +\\mu\\big(\\rr>\\frac{n}{2}\\big)\\le \\frac{c_1}{n^{\\aa}},\n\\ \\ \\mu_V(\\rr>n)\\le \\frac{ c_1}{n^{\\aa-(1-s)\\vv}}.\n\\end{split}\n\\end{equation*}\nSince $\\vv \\in [0,\\aa)$, we see that (\\ref{th-wp-1}) holds and there exists $c_2>0$ such that\n$$6\\mu_V(\\rr>n)\n   +2\\beta\\e^{\\tt Z_{n}(V)} \\big(4\\tt \\eta_{n,\\frac{n}{2}}+\\tt \\gg_{\\frac{n}{2}} +4\\ll\\mu(\\rr>\\frac{n}{2})\\big)\\le \\ff{c_2}{2n^{\\aa-(1-s)\\vv}},\\ \\ n\\ge 1.$$\nThus, in the definition of $\\beta_V$ for small $r>0$ we may take $n= (\\ff{c_2}r)^{\\ff 1{\\aa- (1-s)\\vv}}$ to get\n $$\\beta_V(r)\\le 2\\beta \\e^{\\tt K_{n,\\frac{n}{2}}(V)}\\le c_3r^{-\\vv\/(\\aa-(1-s)\\vv)}$$ for some constant $c_3>0.$\nTherefore, there exists $C>0$ such that the weak Poincar\\'{e} inequality \\eqref{th-wp-2} holds for   $\\beta_V$ given in\n\\eqref{exa-wp-02}.\n\nIt remains to verify (i) and (ii), where the assertion (i) has been included in \\cite[Corollary 1.2(3)]{WW}. So, it suffices to consider (ii).\n  Now, assume\nthat the weak Poincar\\'e inequality holds for\n$(\\EE_{V},\\scr{D}(\\EE_{V}))$ with the rate function\n$$\n\\beta_V(r)=c_4(1+r^{-\\frac{\\vv}{\\alpha-\\vv}})\n$$ for some constant $c_4>0$. For any $n\\ge1$, let $f_n\\in C^\\infty(\\R^d)$ be in part (b) of the proof of Example \\ref{ex-sp-1}. By (\\ref{AB0}) and noting that $\\mu_V(f_n)^2\\le \\ff 1 2 \\mu_V(f_n^2)$ for large $n$, there exist constants $c_5, c_6>0$ such that for $n$ large enough,\n  $$\\EE_V(f_n)\\le c_5 n^{-\\aa},\\quad \\mu_V(f_n^2)-\\mu_V(f_n)^2\\ge \\frac{c_6\\e^{\\phi(n)}}{n^{\\aa-\\vv}}.$$ Combining these with \\eqref{th-wp-2}, we obtain that there exists $c_7>0$ such that for all $r>0$ and for $n$ large enough,\n  $$\\frac{c_7\\e^{\\phi(n)}}{n^{\\aa-\\vv}}\\le \\frac{\\bb_V(r)}{n^\\aa}+r.$$\n Taking $r=\\frac{c_7}{2}\\frac{\\e^{\\phi(n)}}{n^{\\aa-\\vv}}$ in the inequality above, we arrive at\n\\begin{equation}\\label{exa-wp-04}\\bb_V\\Big(\\frac{c_7}{2}\\frac{\\e^{\\phi(n)}}{n^{\\aa-\\vv}}\\Big) \\ge \\frac{c_7}{2}n^\\vv \\e^{\\phi(n)}.\\end{equation}\n\nSince there is $c_8>0$ such that for $n$ large enough\n$$\\frac{\\e^{\\phi(n)}}{n^{\\aa-\\vv}}\\le c_8\\int_{\\{|x|\\ge n\\}}\\frac{\\e^{\\phi(|x|)}}{(1+|x|)^{m+\\aa-\\vv}}\n\\,\\d x,$$ it holds that $$\\lim_{n\\to \\infty}\\frac{\\e^{\\phi(n)}}{n^{\\aa-\\vv}}=0,$$ which, along with the definition of $\\bb_V$, yields that there is a constant $c_9>0$ such that for $n$ large enough\n$$ \\bb_V\\Big(\\frac{c_7\\e^{\\phi(n)}}{2n^{\\aa-\\vv}}\\Big)\\le c_9n^\\vv \\e^{-\\vv\\phi(n)\/(\\aa-\\vv)},$$\n which is a contradiction to \\eqref{exa-wp-04} since $\\lim_{r\\to\\infty}\\phi(r)=\\infty.$\n Therefore, the weak Poincar\\'e\ninequality does not hold with the rate function\n(\\ref{exa-wp-02}).\n\\end{proof}\n\n\n\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecent experimental progresses in studying the microcavity polaritons have lead to\na huge growth of interest in their collective dynamics\n(see, e.g., review articles~\\cite{keeling-2007,amo-2010} and references therein).\nPolaritons possess an extremely small effective mass which allows their\ncondensation at temperatures much greater than that of ultracold atomic vapors.\nBesides that, parameters of the polariton superfluid can be easily tuned with\nthe use of resonant lasers. However,\npolaritons have a finite lifetime, and to maintain their steady-state population a\ncontinuous pumping is required. Experimentally, it is observed\n\\cite{kasprzak-2006,lai-2007} that above a threshold pumping strength\nan accumulation of low\nenergy polaritons is accompanied by a significant increase of spatial coherence that extends over the entire cloud of\npolaritons which can then be described by a single order parameter (polariton condensate wave function)\nobeying an effective Gross-Pitaevskii equation.\nOn the contrary to the atomic condensate situation, the density of the polariton condensate is not an arbitrary parameter\nanymore. Instead, it is determined by the condition of balance between pumping and dissipation processes. This implies that\nthe non-conservative effects can play a crucial role in the condensate's nonlinear dynamics. For example, the\ngeneration of oblique solitons by the flow of a condensate past a localized obstacle has been observed~\\cite{amo-2011} at\nsubsonic speed which is impossible in the case of conservative atomic condensate. The formation of oblique solitons followed\nby their decay into vortex streets in a non-uniform cloud of polariton condensate has been\nstudied in~\\cite{grosso-2011}. Other geometries are also of great interest. In particular, quasi-1D flow of polariton\ncondensate along ``quantum wires'' was studied experimentally in \\cite{wertz-2010}. In the atomic condensate case\nsuch a flow past an obstacle leads to generation of dispersive shock waves propagating upstream and downstream from the\nobstacle \\cite{hakim-1997,pavloff-2002,radouani-2004}, as it was observed in the experiment~\\cite{ea-2007} and explained theoretically in~\\cite{legk}. However, it is\neasy to see that this theory cannot be applied to the polariton condensate flow which density must be fixed (if\npumping and dissipation are balanced) far enough from the obstacle thus preventing formation of jumps in the flow parameters.\nThis means that in the dissipative case the dispersive shock waves generated by the flow must always be\nattached to the obstacle\nand relax to the steady-state flow far enough from the obstacle. This qualitative difference between wave patterns in\nconservative and dissipative cases makes the theory of dissipative flow much more complicated and this Letter is devoted\nto its development.\n\n\n\\section{Theoretical model}\nSeveral models have been suggested for theoretical description of polariton condensate\n(see, e.g., \\cite{km-2003}). They were based on various generalizations of the\nGross-Pitaevskii (GP) equation. Here we have to take into account the effects of losses and\npumping of polaritons on generation of dispersive shock waves by the flow of condensate\npast an obstacle. To this end, we will use the simple model introduced in\n\\cite{kb-2008} where nonresonant pumping due to stimulated scattering of\npolaritons into the condensate and their linear losses are described by the effective\n``gain'' term $\\partial_t\\psi=\\gamma\\psi$ where $\\psi$ is the polariton condensate wave\nfunction. The saturation of gain is modeled by the nonlinear term\n$\\partial_t\\psi=-\\Gamma|\\psi|^2\\psi$ which brings the condensate density into\nequilibrium with the one of external reservoir, $\\rho\\equiv|\\psi|^2=\\gamma\/\\Gamma$.\nThe dispersion of the lower branch of polaritons in the\neffective mass approximation and the nonlinear interaction due to the\nexciton component  of polaritons lead to the following generalized GP equation\n\\begin{equation}\\label{eq1}\n    i\\psi_t+\\tfrac12\\psi_{xx}-|\\psi|^2\\psi=V(x)\\psi+i(\\gamma-\\Gamma|\\psi|^2)\\psi,\n\\end{equation}\n(written in standard non-dimensional units) where $V(x)$ denotes the potential\nof the obstacle\nand $x$ is the direction of the flow of the condensate. The theory developed below\ncan be generalized to other forms of nonlinear gain as, for\nexample, the model considered in~\\cite{wc-2007},\nand the results remain qualitatively similar. However, to be definite, we shall consider here\nthe model (\\ref{eq1}) for which the description looks especially simple.\n\nIt is easy to see that eq.~(\\ref{eq1}) without potential ($V(x)\\equiv0$) admits a\nplane wave solution which constant amplitude is determined by the gain and\ndamping coefficients,\n\\begin{equation}\\label{eq2}\n    \\psi=ae^{i(u_0x-\\mu t)},\\quad a=\\sqrt{{\\gamma}\/{\\Gamma}},\\quad\n    \\mu={\\gamma}\/{\\Gamma}+\\tfrac12u_0^2,\n\\end{equation}\nwhere $u_0$ denotes the uniform flow velocity of the condensate and\n$\\rho_0={\\gamma}\/{\\Gamma}$ is its density.\nThe analysis of modulation stability of such plane waves with respect to\nharmonic perturbations $\\propto \\exp[i(Kx-\\Omega t)]$ shows that the disturbance\npropagates along the wave (\\ref{eq2}) with the dispersion law\n\\begin{equation}\\label{eq3}\n \\Omega=u_0K-i\\gamma\\pm\\sqrt{K^2(a^2+K^2\/4)-\\gamma^2}.\n\\end{equation}\nThe expression under the square root vanishes at the wave number\n\\begin{equation}\\label{eq4}\n    K_c=a\\left[2(\\sqrt{1+\\Gamma^2}-1)\\right]^{1\/2}\n\\end{equation}\nwhich separates two different regimes of evolution of a harmonic perturbation;\nsee the plots in fig.~\\ref{fig.1}.\nIn the conservative limit $\\gamma,\\,\\Gamma\\to0,\\,\\gamma\/\\Gamma=\\mathrm{const}$ we reproduce\nthe standard Bogoliubov dispersion law with sound velocity $c_s=a=\\sqrt{\\rho_0}$.\nWe shall call this parameter ``sound velocity'' also for small\n$\\gamma,\\,\\Gamma\\ll1$; it corresponds to the almost linear part $\\Omega\\approx c_sK$\nof the dispersion law for $K_c\\approx \\Gamma a \\ll K \\ll a$ (see fig.~\\ref{fig.1}).\nNotice that, as long as repulsive nonlinearity is considered, the plane wave solution\n(\\ref{eq2}) is modulationally stable, since $\\mathrm{Im}(\\Omega)<0$  for any $K$-value.\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.11]{fig1.pdf}}\n\\caption{(Color online) The real and imaginary parts of $\\Omega$\nas functions of $K$ (see eq.~(\\ref{eq3})) are shown in the case  $\\gamma=\\Gamma=0.05$  and  $u_0=0$. }\n\\label{fig.1}\n\\end{figure}\n\nIt is convenient to transform eq.~(\\ref{eq1}) into a hydrodynamic form by means of the substitution\n\\begin{equation}\\label{eq5}\n    \\psi=\\sqrt{\\rho}\\,\\exp\\left(i\\int^x u(x',t)\\upd x'\\right)e^{-i\\mu t}\n\\end{equation}\nwhich yields\n\\begin{equation}\\label{eq6}\n    \\rho_t+(\\rho u)_x=2\\Gamma\\rho(\\rho_0 - \\rho),\n\\end{equation}\n\\begin{equation}\\label{eq7}\nu_t+uu_x+\\rho_x+\\left(\\frac{\\rho_x^2}{8\\rho^2}-\\frac{\\rho_{xx}}{4\\rho}\\right)_x=-V_x.\n\\end{equation}\nThe last term in the left-hand side of (\\ref{eq7}) describes the dispersion effects; the right-hand side of\n(\\ref{eq6}) describes the gain and loss effects in the system; the other terms in\neqs.~(\\ref{eq6}),(\\ref{eq7}) have standard hydrodynamic meaning.\n\n\\section{Hydraulic approximation}\nIn what follows we assume that the size $l$ of the obstacle is much greater than the\n``healing length'' (equal to unity in our non-dimensional variables). In our\nnumerical simulations we will use an obstacle potential in the form\n\\begin{equation}\\label{eq8}\n    V(x)=V_m\\exp(-x^2\/l^2),\\quad l\\gg1.\n\\end{equation}\nAll numerical plots below are obtained for $l=5$.\nWe are interested in stationary patterns with\n$\\rho_t\\equiv0,\\,u_t\\equiv0$ generated by the steady flow of the condensate past a\nlocalized obstacle ($V(x)\\to0$ as $|x|\\to\\infty$), i.e.\n$\\rho$ and $u$ must satisfy the boundary conditions\n\\begin{equation}\\label{eq9}\n    \\rho\\to \\rho_0,\\quad u\\to u_0\\quad \\text{as}\\quad |x|\\to\\infty.\n\\end{equation}\nThen eq.~(\\ref{eq6}) can be reduced to\n\\begin{equation}\\label{eq10}\n    (\\rho u)_x=2\\Gamma\\rho(\\rho_0-\\rho)\n\\end{equation}\nand the stationary eq.~(\\ref{eq7}) can be integrated once to give\n\\begin{equation}\\label{eq10a}\n    \\tfrac12{u^2}+\\rho+\\frac{\\rho_x^2}{8\\rho^2}-\\frac{\\rho_{xx}}{4\\rho}+V(x)=\n    \\tfrac12{u_0^2}+\\rho_0\n\\end{equation}\nFor $l\\gg 1$ it is natural to assume that the wave pattern has the\ncharacteristic length about $l$ and, hence, the dispersive terms in\n(\\ref{eq10a}) having higher order derivatives are negligibly small compared with\nother terms; then we get\n\\begin{equation}\\label{eq11}\n    \\tfrac12{u^2}+\\rho+V(x)=\n    \\tfrac12{u_0^2}+\\rho_0.\n\\end{equation}\nIntegration of eq.~(\\ref{eq10}) over space shows that stationary patterns\nsatisfy the condition (see also \\cite{kb-2008})\n\\begin{equation}\\label{eq12}\n    \\int_{-\\infty}^{\\infty}\\rho(\\rho_0-\\rho)\\upd x=0.\n\\end{equation}\nExcluding $u$ from Eqs.~(\\ref{eq11}) and (\\ref{eq10}) we get the equation\n\\begin{equation}\\label{eq13}\n    \\left(\\rho\\sqrt{u_0^2+2(\\rho_0-\\rho-V(x))}\\right)_x=2\\Gamma\\rho(\\rho_0-\\rho)\n\\end{equation}\nwhich determines the dependence of $\\rho$ on $x$ for a given potential $V(x)$\nprovided the solution satisfies the condition (\\ref{eq9}).\nThis equation represents the so-called hydraulic approximation for our system.\n(It generalizes the well-known Thomas-Fermi approximation to non-zero flow\nvelocity.)\n\nAt the tails of the wave pattern ($|x|\\to\\infty$) we can neglect the potential $V(x)$\nand linearize eq.~(\\ref{eq13}) with respect to small deviations $\\rho-\\rho_0$\nfrom the steady state. This gives the asymptotic behavior\n\\begin{equation}\\label{eq14}\n    |\\rho-\\rho_0|\\propto\\exp\\left(\\frac{2\\Gamma\\rho_0u_0}{\\rho_0-u_0^2}x\\right)\n\\end{equation}\nwhich shows that such tails extending beyond the range of the potential\ncan exist only under the condition\n\\begin{equation}\\label{eq15}\n    \\frac{|\\rho_0-u_0^2|}{\\Gamma\\rho_0u_0}\\gg l.\n\\end{equation}\nThis means that the sound velocity $c_s=\\sqrt{\\rho_0}$ separates two\ndifferent regimes---subsonic ($u_0<c_s$) and supersonic ($u_0>c_s$)---with\ndrastically different properties. We shall discuss them separately.\n\n\\section{Subsonic flow}\nThe hydraulic approximation describes the profiles of the disturbance well enough practically\nin the entire subsonic regime $u_0<\\sqrt{\\rho_0}$ if the velocity is not too close\nto the sound velocity. In fig.~\\ref{fig.2}\nwe compare the direct numerical solution of eq.~(\\ref{eq1}) with its hydraulic approximation\nobtained by the numerical solution of eq.~(\\ref{eq13}); quite good agreement is observed\n(notice that in all simulations we use $\\gamma=\\Gamma=0.05$ that gives\nsound velocity $c_s=1$).\nNaturally, the asymptotic behavior (\\ref{eq14}) is also confirmed in the upstream flow $x<0$. However,\nthere is no linearized solution decaying to zero at $x\\to+\\infty$, hence the transition to the\nasymptotic steady flow in the downstream region $x>0$ has to occur within the range of the\npotential $V(x)$, and this conclusion is confirmed by direct numerical solution.\nIndeed, as one observes in fig.~\\ref{fig.2}, the stationary disturbance has a smooth shape with\na very long monotonically decaying left tail.\nThis tail is attached to the density dip located almost completely within the potential region.\nThe amplitude of the disturbance (i.e. the difference between maximal and minimal density in the\ncondensate) in the subsonic regime monotonically increases with growth of the potential strength.\nIt is convenient to introduce the ``number of particles''\n\\begin{equation}\\label{number}\n    N=\\int_{-\\infty}^{\\infty}[\\rho^{1\/2}(x)-a]^2\\upd x\n\\end{equation}\ndisturbed by the flow.\nThis variable monotonically increases with the strength of potential [fig.~\\ref{fig.3}(a)].\nInterestingly, there exists a critical strength\nof potential $V_m^{cr}$ above\nwhich one cannot find stationary profiles. When $V_m\\to V_m^{cr}$  the tangent\nto $N(V_m)$ becomes vertical. For this strength of potential the amplitude of disturbance is maximal.\nNote that the critical potential strength $V_m^{cr}$ diverges when the incident velocity of the\ncondensate $u_0\\to0$ and it monotonically decreases when $u_0$ increases\n[fig.~\\ref{fig.3}(b)]. We were able to obtain the critical potential strength only for\n$0<u_0<0.8c_s$. For higher velocities $[0.8c_s<u_0<c_s]$ the condensate starts\ndeveloping small oscillations on its right tail. Solutions characterized by different\nnumber of oscillations on their right tail may form continuous families that may be obtained\neven for potentials with $V_m>1$.\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.11]{fig2.pdf}}\n\\caption{(Color online) Comparison of density profile\nat  $u_0=0.6$, $V_m=0.1$, and $\\gamma=\\Gamma= 0.05$ obtained numerically from\neq.~(\\ref{eq1}) (black curve)\nand in the framework of the hydraulic approximation (red curve). The dashed line shows\nplane wave solution (\\ref{eq2}) in absense of potential. }\n\\label{fig.2}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.11]{fig3.pdf}}\n\\caption{(a) Number of particles (\\ref{number}) versus potential strength\nfor subsonic flow with  $u_0=0.6$. The circle corresponds to the parameters used in fig.~\\ref{fig.2}.\n(b) Critical strength of potential\nversus  $u_0$. In all cases  $\\gamma=\\Gamma=0.05$.}\n\\label{fig.3}\n\\end{figure}\n\n\nThe condition (\\ref{eq15}) breaks down when the incident velocity $u_0$ is close to the critical value\n$\\sqrt{\\rho_0}$. In this case the potential $V(x)$ cannot be neglected in the\nhydraulic approximation (\\ref{eq13}. Even more, as $u_0$ approaches the critical value, the upstream\ndensity profile steepens and its slope can become so large that the dispersive terms\nin eq.~(\\ref{eq10a}) can no longer be neglected. As is known, the dispersive effects lead to generation of\noscillations in the regions with fast change of the variables. Hence, for $u_0\\gtrsim \\sqrt{\\rho_0}$\nwe should expect generation of dispersive shock waves and now we proceed to the discussion\nof the supersonic flow.\n\n\\section{Supersonic flow}\nIf $u_0>\\sqrt{\\rho_0}$ and the condition (\\ref{eq15}) is fulfilled, then the linearized solution\n(\\ref{eq14}) describes decaying disturbance in the downstream flow at $x\\to+\\infty$. Thus,\ncontrarily to the subsonic case, the right tail for $x>0$ can extend far beyond the range of\nthe potential.\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.12]{fig4.pdf}}\n\\caption{(Color online) The density distribution for shock wave at $u_0=1.4$,\n$V_m=0.8$,\nand $\\gamma=\\Gamma=0.05$. Red curves show analytical approximations of tails from\neqs.~(\\ref{eq14}) and (\\ref{eq40}). Dashed line shows plane wave solution in the absence of potential. }\n\\label{fig.4}\n\\end{figure}\n\nAs mentioned above, the generation of an oscillatory dispersive shock wave is expected in the upstream flow.\nThis is confirmed by numerical solution of (\\ref{eq1}) with the supersonic boundary conditions;\nsee fig.~\\ref{fig.4}. The dispersive shock waves can be represented as a modulated solution of the undisturbed ($\\Gamma=\\gamma=0,\\,V(x)\\equiv0$)\nequation (\\ref{eq1}) or the system (\\ref{eq6}), (\\ref{eq7}) which can be written in the form (see, e.g., \\cite{kamch2000})\n\\begin{equation}\\label{eq16}\n    \\begin{split}\n    \\rho(x,t)&=\\nu_1+(\\nu_2-\\nu_1)\\mathrm{sn}^2(\\sqrt{\\nu_3-\\nu_1}\\,\\theta,m)\\\\\n    &=\\tfrac14(\\lambda_1-\\lambda_2-\\lambda_3+\\lambda_4)^2+(\\lambda_2-\\lambda_1)(\\lambda_4-\\lambda_3)\\\\\n    &\\times\\mathrm{sn}^2(\\sqrt{(\\lambda_4-\\lambda_2)(\\lambda_3-\\lambda_1)}\\,\\theta,m),\n    \\end{split}\n\\end{equation}\n\\begin{equation}\\label{eq17}\n\\begin{split}\n    u(x,t)&=U-{q}\/{\\rho(x,t)},\\\\\n    q&=-\\tfrac18(\\lambda_1-\\lambda_2-\\lambda_3+\\lambda_4)(\\lambda_1-\\lambda_2+\\lambda_3-\\lambda_4)\\\\\n    &\\times(\\lambda_1+\\lambda_2-\\lambda_3-\\lambda_4),\n    \\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq19}\n    \\theta=x-Ut,\\quad U=\\tfrac12(\\lambda_1+\\lambda_2+\\lambda_3+\\lambda_4)\n\\end{equation}\n$\\nu_1,\\nu_2,\\nu_3$ are related to the Riemann invariants $\\lambda_i,i=1,2,3,4,$ by the formulae\n\\begin{equation}\\label{eq18}\n\\begin{split}\n    \\nu_1&=\\tfrac14(\\lambda_1-\\lambda_2-\\lambda_3+\\lambda_4)^2,\\\\\n    \\nu_2&=\\tfrac14(\\lambda_1-\\lambda_2+\\lambda_3-\\lambda_4)^2,\\\\\n    \\nu_3&=\\tfrac14(\\lambda_1+\\lambda_2-\\lambda_3-\\lambda_4)^2,\n    \\end{split}\n\\end{equation}\nand\n\\begin{equation}\\label{eq20}\n    m=\\frac{\\nu_2-\\nu_1}{\\nu_3-\\nu_1}=\\frac{(\\lambda_2-\\lambda_1)(\\lambda_4-\\lambda_3)}{(\\lambda_3-\\lambda_1)(\\lambda_4-\\lambda_2)}.\n\\end{equation}\nIn the modulated wave the Riemann invariants $\\lambda_i$ become slow functions of $x$ and $t$\nand their evolution is governed by the Whitham equations. They were derived for the case of eq.~(\\ref{eq1})\nwith $\\Gamma=0$ in \\cite{kamch2004} and their generalization to the nonzero $\\Gamma$ is straightforward.\nTherefore we will write down here the final result without its derivation.\n\nIn our stationary case $\\lambda_i$ do not depend on $t$ and the phase velocity $U$ equals to zero,\n\\begin{equation}\\label{eq21}\n    \\lambda_1+\\lambda_2+\\lambda_3+\\lambda_4=0.\n\\end{equation}\nThen the Whitham equations can be written in the form\n\\begin{equation}\\label{eq22}\n    \\frac{\\upd\\lambda_i}{\\upd x}=\\frac2{L}\\cdot\\frac{I_1\\lambda_i+I_2}{\\prod_{m\\neq i}(\\lambda_i-\\lambda_m)},\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq23}\n\\begin{split}\n    I_1&=\\Gamma\\int_{\\nu_1}^{\\nu_2}\\frac{\\nu(\\rho_0-\\nu)}{\\sqrt{\\mathcal{R}(\\nu)}}\\upd\\nu,\\\\\n    I_2&=\\frac{\\Gamma u_0\\rho_0}2\\int_{\\nu_1}^{\\nu_2}\\frac{\\rho_0-\\nu}{\\sqrt{\\mathcal{R}(\\nu)}}\\upd\\nu,\n    \\end{split}\n\\end{equation}\n\\begin{equation}\\label{eq24}\n    \\mathcal{R}=(\\nu-\\nu_1)(\\nu-\\nu_2)(\\nu-\\nu_3)\n\\end{equation}\nand $L$ is the wavelength\n\\begin{equation}\\label{eq25}\n    L=\\frac{2\\mathrm{K}(m)}{\\sqrt{(\\lambda_3-\\lambda_1)(\\lambda_4-\\lambda_2)}},\n\\end{equation}\n$\\mathrm{K}(m)$ being the complete elliptic integral of the first kind.\n\nDue to the special structure of Eqs.~(\\ref{eq22}), the symmetric functions of $\\lambda_i$,\n\\begin{equation}\\label{eq26}\n\\begin{split}\n    s_1&=\\sum_i\\lambda_i,\\quad s_2=\\sum_{i\\neq j}\\lambda_i\\lambda_j,\\quad s_3=\\sum_{i\\neq j\\neq k}\\lambda_i\\lambda_j\\lambda_k,\\\\\n    s_4&=\\lambda_1\\lambda_2\\lambda_3\\lambda_4,\n    \\end{split}\n\\end{equation}\nobey very simple equations\n\\begin{equation}\\label{eq27}\n    \\frac{\\upd s_1}{\\upd x}=0,\\quad \\frac{\\upd s_2}{\\upd x}=0,\\quad \\frac{\\upd s_3}{\\upd x}=\\frac{2I_1}L,\\quad\n    \\frac{\\upd s_4}{\\upd x}=-\\frac{2I_2}L.\n\\end{equation}\nThe condensate density $\\rho$ oscillates in the interval $\\nu_1\\leq \\rho\\leq\\nu_2$, i.e. $\\nu_1$ and $\\nu_2$ are the envelopes\nof the oscillations in the dispersive shock wave. Let us study their asymptotic behavior at $x\\to-\\infty$.\n\nThe asymptotic plane wave corresponds to $\\nu_2=\\nu_1$ (or $\\lambda_2=\\lambda_1$) so that the parameters of the incident condensate\nare expressed in terms of the Riemann invariants as\n\\begin{equation}\\label{eq28}\n    \\rho_0=\\tfrac14(\\lambda_4-\\lambda_3)^2\n\\end{equation}\nand\n\\begin{equation}\\label{eq29}\n    u_0=\\tfrac12(\\lambda_3+\\lambda_4-2\\lambda_1).\n\\end{equation}\nFrom (\\ref{eq21}), (\\ref{eq28}) and (\\ref{eq29}) we find the asymptotic values of the Riemann invariants\n\\begin{equation}\\label{eq30}\n\\begin{split}\n    \\lambda_1&=\\lambda_2=-\\frac{u_0}2,\\\\\n    \\lambda_3&=\\frac{u_0}2-\\sqrt{\\rho_0},\\quad \\lambda_4=\\frac{u_0}2+\\sqrt{\\rho_0}.\n    \\end{split}\n\\end{equation}\nThe periodic solution corresponds to the ordering $\\lambda_1\\leq\\lambda_2\\leq\\lambda_3\\leq\\lambda_4$, so that from $\\lambda_3\\geq\\lambda_2$\nwe find that $u_0\\geq\\sqrt{\\rho_0}$, i.e. the incident flow must be supersonic for the generation of such a stationary shock wave.\nThe wavelength (\\ref{eq25}) of small amplitude oscillations ($m=0$) reduces in the limit $x\\to-\\infty$ to\n\\begin{equation}\\label{eq31}\n    L=\\frac{\\pi}{\\sqrt{u_0^2-\\rho_0}}.\n\\end{equation}\n\nThe asymptotic values of $\\nu_1,\\nu_2,\\nu_3$ at $x\\to-\\infty$ are $\\nu_1(-\\infty)=\\nu_2(-\\infty)=\\rho_0$,\n$\\nu_3(-\\infty)=u_0^2$. We introduce small deviations from these values,\n\\begin{equation}\\label{eq32}\n    \\nu_1=\\rho_0+\\delta \\nu_1,\\quad \\nu_2=\\rho_0+\\delta \\nu_2,\\quad \\nu_3=u_0^2+\\delta\\nu_3,\n\\end{equation}\nand linearize eqs.~(\\ref{eq27}) with respect to $\\delta\\nu_i$ with account of the identities\n\\begin{equation}\\label{eq33}\n\\begin{split}\n    &\\nu_1+\\nu_2+\\nu_3=-2s_2,\\\\\n    &\\nu_1\\nu_2+\\nu_1\\nu_3+\\nu_2\\nu_3=-s_2^2+4s_4,\\quad\n    \\nu_1\\nu_2\\nu_3=s_3^2.\n    \\end{split}\n\\end{equation}\nAs a result we arrive at the equations\n\\begin{equation}\\label{eq34}\n    \\frac{\\upd(\\delta\\nu_1+\\delta\\nu_2)}{\\upd x}=-\\frac{\\upd\\delta\\nu_3}{\\upd x}=\\frac{2\\Gamma \\rho_0u_0}{u_0^2-\\rho_0}(\\delta\\nu_1+\\delta\\nu_2).\n\\end{equation}\nHence we get\n\\begin{equation}\\label{eq35}\n    \\delta\\nu_1+\\delta\\nu_2,\\delta\\nu_3\\propto \\exp\\left(\\frac{2\\Gamma\\rho_0u_0}{u_0^2-\\rho_0}x\\right).\n\\end{equation}\nThese equations suggest that the sum of deviations $\\delta\\nu_1+\\delta\\nu_2$ decays at $x\\to-\\infty$ faster\nthan the deviations $\\delta\\nu_1$ and $\\delta\\nu_2$ separately. If we introduce small deviations of the\nRiemann invariants from their asymptotic values (\\ref{eq30})\n\\begin{equation}\\label{eq36}\n    \\begin{split}\n    \\lambda_1&=-\\frac{u_0}2+\\delta\\lambda_1,\\quad \\lambda_2=-\\frac{u_0}2+\\delta\\lambda_2\\\\\n    \\lambda_3&=\\frac{u_0}2-\\sqrt{\\rho_0}+\\delta\\lambda_3,\\quad \\lambda_4=\\frac{u_0}2+\\sqrt{\\rho_0}+\\delta\\lambda_4,\n    \\end{split}\n\\end{equation}\nthen we find that the asymptotic behavior (\\ref{eq35}) corresponds to $|\\delta\\lambda_3|,|\\delta\\lambda_4|\\ll|\\delta\\lambda_1|,|\\delta\\lambda_2|$\nso that\n\\begin{equation}\\label{eq37}\n\\begin{split}\n    &\\nu_1\\cong\\tfrac14(2\\sqrt{n_0}+\\delta\\lambda_1-\\delta\\lambda_2)^2,\\\\\n    &\\nu_2\\cong\\tfrac14(2\\sqrt{n_0}-\\delta\\lambda_1+\\delta\\lambda_2)^2,\\\\\n    &\\nu_3\\cong\\tfrac14(2u_0-\\delta\\lambda_1-\\delta\\lambda_2)^2,\n    \\end{split}\n\\end{equation}\nand in leading order approximation $\\delta\\nu_1+\\delta\\nu_2=0$ which means that (\\ref{eq35})\ncorresponds to higher order corrections.\n\nThus, in the main approximation we assume that $\\nu_3\\cong u_0^2$ and from (\\ref{eq23}) we obtain the\nfollowing equations for $\\delta\\lambda_1$ and $\\delta\\lambda_2$:\n\\begin{equation}\\label{eq38}\n\\begin{split}\n    \\frac{\\upd\\delta\\lambda_1}{\\upd x}&=\\frac{\\Gamma \\rho_0u_0}{2(u_0^2-\\rho_0)}(\\delta\\lambda_1-\\delta\\lambda_2),\\\\\n    \\frac{\\upd\\delta\\lambda_2}{\\upd x}&=-\\frac{\\Gamma \\rho_0u_0}{2(u_0^2-\\rho_0)}(\\delta\\lambda_1-\\delta\\lambda_2).\n    \\end{split}\n\\end{equation}\nFrom these equations we get\n\\begin{equation}\\label{eq39}\n    \\delta\\lambda_{1,2}\\cong \\pm C\\exp\\left(\\frac{\\Gamma \\rho_0u_0}{u_0^2-\\rho_0}x\\right)\n\\end{equation}\n($C$ is the integration constant),\nand hence the envelopes of oscillations in the dispersive shock decay at leading order approximation as\n\\begin{equation}\\label{eq40}\n    \\delta\\nu_1,\\delta\\nu_2\\propto \\exp\\left(\\frac{\\Gamma \\rho_0u_0}{u_0^2-\\rho_0}x\\right).\n\\end{equation}\nThis is slower than the dependence in (\\ref{eq35}) as it should be.\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.11]{fig5.pdf}}\n\\caption{(a) Number of  particles versus potential strength for supersonic  flow with\n$u_0=1.4$. The circle corresponds to solution in fig.~\\ref{fig.4}.\n(b) Critical strength of potential versus $u_0$. In all cases $\\gamma=\\Gamma=0.05$.  }\n\\label{fig.5}\n\\end{figure}\n\nThese analytical predictions were confirmed by the direct solution of eq.~(\\ref{eq1}).\nAs one can see from fig.~\\ref{fig.4}, a typical shock wave solution may extend far beyond\nthe repulsive potential, both in the positive and negative $x$ directions. Note the excellent\nagreement at $x\\to\\pm\\infty$ between the numerical density and its analytical fit that uses\nthe expressions (\\ref{eq14}) and (\\ref{eq40}). The wavelength of small amplitude oscillations\nestimated numerically\nfor the parameters of the flow shown in fig.~\\ref{fig.4} is equal to $L_{num}\\cong 3.16$ and\nthe analytical formula (\\ref{eq31}) gives $L_{theor}\\cong 3.21$, with an accuracy better\nthan 2\\%.\nIt should be stressed that the solution in fig.~\\ref{fig.4} is fully stationary (that is, it does not change in time),\nin contrast to previously reported one-dimensional shock waves in conservative systems. The oscillating\nleft tail of the dispersive shock wave in the supersonic regime becomes more pronounced with\nincrease of potential strength $V_m$ and incident velocity $u_0$. Just as for the subsonic flow\nthe increase of $V_m$ is accompanied by an increase of the amplitude of the shock wave.\nHowever, in the supersonic regime the amplitude of shock wave may be comparable with the amplitude\nof the unperturbed plane wave and the density may decrease almost to zero in the downstream region,\nespecially for strong potentials. The number of particles in the shock wave increases with\n$V_m$  [fig.~\\ref{fig.5}(a)] and the character of this dependence also points to the existence\nof a critical defect strength beyond which shock waves do not exist.\n\n\nA linear stability analysis performed for stationary solutions shows that wave patterns\nare stable for any strength of potential $V_m$\nup to the critical one as long as the incident velocity is small enough,\n$u_0<0.9c_s$. When $u_0>0.9c_s$, stationary solutions become unstable if the strength\nof potential exceeds certain limiting value $V_m^{cr}$ (the instability of\nthe dispersive shock waves at $V_m>V_m^{cr}$ is accompanied by the development and\nemission of small-scale disturbances on the right tail of the shock wave).\nStill, even shock waves with very long oscillating tails, like the one shown in fig.\\ref{fig.4},\nmay be stable. The critical value of potential increases monotonically with $u_0$ as\nit is shown in fig.~\\ref{fig.5}(b).\n\n\\section{Conclusion}\nIt is instructive to compare our results with those obtained in the case of the flow of a conservative\nfluid past an obstacle. As was shown in \\cite{legk},\nin the conservative case one\ncan distinguish three characteristic ranges of flow velocity---subcritical\n($u_0<u_-$), transcritical ($u_-<u_0<u_+$), and supercritical ($u_0>u_+$)\nwhere the critical values of the velocity are located at opposite sides\nof the value of the sound velocity ($u_-<c_s<u_+$). If the flow velocity\nis subcritical or supercritical, then the disturbance is stationary and has\nthe dimension of the obstacle's size with definite sign of the difference $\\rho-\\rho_0$; it has a form\nof a dip in subcritical region and of a hump in supercritical region.\nIn the present case of non-conservative dynamics\nthe disturbance must have ``dips'' and ``humps'' to be compensated\nin the integral (\\ref{eq12}). Examples\nof such disturbances shown in figs.~\\ref{fig.2} and \\ref{fig.4}  demonstrate\nthis property.\n\nIn the conservative situation the transcritical regime\ncorresponds to non-stationary generation of upstream and\/or downstream dispersive\nshock waves. In the non-conservative situation this regime actually disappears and\ninstead a supersonic flow generates a {\\it stationary dispersive shock wave}\nupstream the obstacle. There are no downstream dispersive shock waves; instead they are\nreplaced by smooth profiles decaying asymptotically to the stationary plane\nwave.\n\nThe above properties of the wave patterns generated by the flow of polariton\ncondensate past an obstacle are derived for\nthe model (\\ref{eq1}).\nWe believe, however, that these properties remain qualitatively\nthe same for other models having stationary plane wave states formed due to balance\nof pumping and dissipation effects. In particular, this theory can find applications\nto a so-called ``superfluid motion of light'' \\cite{pr-1993,bcz-2001,lm-2010}.\n\n\n\n\n\n\\acknowledgments\nWe are grateful to A.~Amo, N.~Berloff, J.~Bloch,\nA.~Bramati, I.~Carusotto, C.~Ciuti, E.~Giacobino, Yu.G.~Gladush, N.~Pavloff, D.~Sanvitto\nfor discussions of superfluidity in the cavity polariton physics.\nWe thank RFBR for partial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Electronic Submission}\n\\label{submission}\n\nSubmission to ICML 2017 will be entirely electronic, via a web site\n(not email).  Information about the submission process and \\LaTeX\\ templates\nare available on the conference web site at:\n\\begin{center}\n\\textbf{\\texttt{http:\/\/icml.cc\/2017\/}}\n\\end{center}\nSend questions about submission and electronic templates to\n\\texttt{icml2017pc@gmail.com}.\n\nThe guidelines below will be enforced for initial submissions and\ncamera-ready copies.  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In particular, reviewers are\nnot required to look a the Supplementary Material when writing their\nreview.\n\n\\subsubsection{Camera-Ready Author Information}\n\\label{final author}\n\nIf a paper is accepted, a final camera-ready copy must be prepared.\nFor camera-ready papers, author information should start 0.3~inches\nbelow the bottom rule surrounding the title. The authors' names should\nappear in 10~point bold type, in a row, separated by white space, and centered. \nAuthor names should not be broken across lines.\nUnbolded superscripted numbers, starting 1, should be used to refer to affiliations. \nAffiliations should be numbered in the order of appearance.  A single footnote block of text should be used to list all the affiliations. (Academic affiliations should list Department, University, City, State\/Region, Country. Similarly for industrial affiliations.)\nEach distinct affiliations should be listed once.  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Turn on the \\texttt{[accepted]} option to the ICML stylefile to see the names rendered. \nAll of the guidelines above are automatically met by the \\LaTeX\\ style file.\n\n\\subsection{Abstract}\n\nThe paper abstract should begin in the left column, 0.4~inches below\nthe final address. The heading `Abstract' should be centered, bold,\nand in 11~point type. The abstract body should use 10~point type, with\na vertical spacing of 11~points, and should be indented 0.25~inches\nmore than normal on left-hand and right-hand margins. Insert\n0.4~inches of blank space after the body. Keep your abstract brief and \nself-contained,\nlimiting it to one paragraph and roughly 4--6 sentences.  Gross violations will require correction at the camera-ready phase.\n\n\\subsection{Partitioning the Text} \n\nYou should organize your paper into sections and paragraphs to help\nreaders place a structure on the material and understand its\ncontributions.\n\n\\subsubsection{Sections and Subsections}\n\nSection headings should be numbered, flush left, and set in 11~pt bold\ntype with the content words capitalized. Leave 0.25~inches of space\nbefore the heading and 0.15~inches after the heading.\n\nSimilarly, subsection headings should be numbered, flush left, and set\nin 10~pt bold type with the content words capitalized. Leave\n0.2~inches of space before the heading and 0.13~inches afterward.\n\nFinally, subsubsection headings should be numbered, flush left, and\nset in 10~pt small caps with the content words capitalized. Leave\n0.18~inches of space before the heading and 0.1~inches after the\nheading. \n\nPlease use no more than three levels of headings.\n\n\\subsubsection{Paragraphs and Footnotes}\n\nWithin each section or subsection, you should further partition the\npaper into paragraphs. Do not indent the first line of a given\nparagraph, but insert a blank line between succeeding ones.\n \nYou can use footnotes\\footnote{For the sake of readability, footnotes\nshould be complete sentences.} to provide readers with additional\ninformation about a topic without interrupting the flow of the paper. \nIndicate footnotes with a number in the text where the point is most\nrelevant. Place the footnote in 9~point type at the bottom of the\ncolumn in which it appears. Precede the first footnote in a column\nwith a horizontal rule of 0.8~inches.\\footnote{Multiple footnotes can\nappear in each column, in the same order as they appear in the text,\nbut spread them across columns and pages if possible.}\n\n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{icml_numpapers}}\n\\caption{Historical locations and number of accepted papers for International\n  Machine Learning Conferences (ICML 1993 -- ICML 2008) and\n  International Workshops on Machine Learning (ML 1988 -- ML\n  1992). At the time this figure was produced, the number of\n  accepted papers for ICML 2008 was unknown and instead estimated.}\n\\label{icml-historical}\n\\end{center}\n\\vskip -0.2in\n\\end{figure} \n\n\\subsection{Figures}\n \nYou may want to include figures in the paper to help readers visualize\nyour approach and your results. Such artwork should be centered,\nlegible, and separated from the text. Lines should be dark and at\nleast 0.5~points thick for purposes of reproduction, and text should\nnot appear on a gray background.\n\nLabel all distinct components of each figure. If the figure takes the\nform of a graph, then give a name for each axis and include a legend\nthat briefly describes each curve. Do not include a title inside the\nfigure; instead, the caption should serve this function.\n\nNumber figures sequentially, placing the figure number and caption\n{\\it after\\\/} the graphics, with at least 0.1~inches of space before\nthe caption and 0.1~inches after it, as in\nFigure~\\ref{icml-historical}.  The figure caption should be set in\n9~point type and centered unless it runs two or more lines, in which\ncase it should be flush left.  You may float figures to the top or\nbottom of a column, and you may set wide figures across both columns\n(use the environment {\\tt figure*} in \\LaTeX), but always place\ntwo-column figures at the top or bottom of the page.\n\n\\subsection{Algorithms}\n\nIf you are using \\LaTeX, please use the ``algorithm'' and ``algorithmic'' \nenvironments to format pseudocode. These require \nthe corresponding stylefiles, algorithm.sty and \nalgorithmic.sty, which are supplied with this package. \nAlgorithm~\\ref{alg:example} shows an example. \n\n\\begin{algorithm}[tb]\n   \\caption{Bubble Sort}\n   \\label{alg:example}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Input:} data $x_i$, size $m$\n   \\REPEAT\n   \\STATE Initialize $noChange = true$.\n   \\FOR{$i=1$ {\\bfseries to} $m-1$}\n   \\IF{$x_i > x_{i+1}$} \n   \\STATE Swap $x_i$ and $x_{i+1}$\n   \\STATE $noChange = false$\n   \\ENDIF\n   \\ENDFOR\n   \\UNTIL{$noChange$ is $true$}\n\\end{algorithmic}\n\\end{algorithm}\n \n\\subsection{Tables} \n \nYou may also want to include tables that summarize material. Like \nfigures, these should be centered, legible, and numbered consecutively. \nHowever, place the title {\\it above\\\/} the table with at least \n0.1~inches of space before the title and the same after it, as in \nTable~\\ref{sample-table}. The table title should be set in 9~point \ntype and centered unless it runs two or more lines, in which case it\nshould be flush left.\n\n\n\\begin{table}[t]\n\\caption{Classification accuracies for naive Bayes and flexible \nBayes on various data sets.}\n\\label{sample-table}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lcccr}\n\\hline\n\\abovespace\\belowspace\nData set & Naive & Flexible & Better? \\\\\n\\hline\n\\abovespace\nBreast    & 95.9$\\pm$ 0.2& 96.7$\\pm$ 0.2& $\\surd$ \\\\\nCleveland & 83.3$\\pm$ 0.6& 80.0$\\pm$ 0.6& $\\times$\\\\\nGlass2    & 61.9$\\pm$ 1.4& 83.8$\\pm$ 0.7& $\\surd$ \\\\\nCredit    & 74.8$\\pm$ 0.5& 78.3$\\pm$ 0.6&         \\\\\nHorse     & 73.3$\\pm$ 0.9& 69.7$\\pm$ 1.0& $\\times$\\\\\nMeta      & 67.1$\\pm$ 0.6& 76.5$\\pm$ 0.5& $\\surd$ \\\\\nPima      & 75.1$\\pm$ 0.6& 73.9$\\pm$ 0.5&         \\\\\n\\belowspace\nVehicle   & 44.9$\\pm$ 0.6& 61.5$\\pm$ 0.4& $\\surd$ \\\\\n\\hline\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\nTables contain textual material that can be typeset, as contrasted \nwith figures, which contain graphical material that must be drawn. \nSpecify the contents of each row and column in the table's topmost\nrow. Again, you may float tables to a column's top or bottom, and set\nwide tables across both columns, but place two-column tables at the\ntop or bottom of the page.\n \n\\subsection{Citations and References} \n\nPlease use APA reference format regardless of your formatter\nor word processor. If you rely on the \\LaTeX\\\/ bibliographic \nfacility, use {\\tt natbib.sty} and {\\tt icml2017.bst} \nincluded in the style-file package to obtain this format.\n\nCitations within the text should include the authors' last names and\nyear. If the authors' names are included in the sentence, place only\nthe year in parentheses, for example when referencing Arthur Samuel's\npioneering work \\yrcite{Samuel59}. Otherwise place the entire\nreference in parentheses with the authors and year separated by a\ncomma \\cite{Samuel59}. List multiple references separated by\nsemicolons \\cite{kearns89,Samuel59,mitchell80}. Use the `et~al.'\nconstruct only for citations with three or more authors or after\nlisting all authors to a publication in an earlier reference \\cite{MachineLearningI}.\n\nAuthors should cite their own work in the third person\nin the initial version of their paper submitted for blind review.\nPlease refer to Section~\\ref{author info} for detailed instructions on how to\ncite your own papers.\n\nUse an unnumbered first-level section heading for the references, and \nuse a hanging indent style, with the first line of the reference flush\nagainst the left margin and subsequent lines indented by 10 points. \nThe references at the end of this document give examples for journal\narticles \\cite{Samuel59}, conference publications \\cite{langley00}, book chapters \\cite{Newell81}, books \\cite{DudaHart2nd}, edited volumes \\cite{MachineLearningI}, \ntechnical reports \\cite{mitchell80}, and dissertations \\cite{kearns89}. \n\nAlphabetize references by the surnames of the first authors, with\nsingle author entries preceding multiple author entries. Order\nreferences for the same authors by year of publication, with the\nearliest first. Make sure that each reference includes all relevant\ninformation (e.g., page numbers).\n\n\\subsection{Software and Data}\n\nWe strongly encourage the publication of software and data with the\ncamera-ready version of the paper whenever appropriate.  This can be\ndone by including a URL in the camera-ready copy.  However, do not\ninclude URLs that reveal your institution or identity in your\nsubmission for review.  Instead, provide an anonymous URL or upload\nthe material as ``Supplementary Material'' into the CMT reviewing\nsystem.  Note that reviewers are not required to look a this material\nwhen writing their review.\n\n\\section*{Acknowledgements} \n \n\\textbf{Do not} include acknowledgements in the initial version of\nthe paper submitted for blind review.\n\nIf a paper is accepted, the final camera-ready version can (and\nprobably should) include acknowledgements. In this case, please\nplace such acknowledgements in an unnumbered section at the\nend of the paper. Typically, this will include thanks to reviewers\nwho gave useful comments, to colleagues who contributed to the ideas, \nand to funding agencies and corporate sponsors that provided financial \nsupport.  \n\n\n\\nocite{langley00}\n\n\n\\section{Introduction}\n\\label{submission}\n\nAutomated driving development has radically changed during the past few years, driven by advances in Artificial Intelligence (AI), and specifically Deep Learning (DL). \nDeveloping an efficient and safe driving policy is in the heart of reaching high level of autonomy in a robot car. Traditional methods are driven with rule based approaches \\cite{le2006review}\\cite{pasquier2001fuzzylot}, while significant advancement in the field is driven by learning approaches \\cite{sallab2016end}\\cite{el2017deep} \\cite{bojarski2016end}. Reinforcement learning is the arm of AI which is concerned with solving the control problem based on learning while interacting with the environment.\n\nOur motivation is to take the work in \\cite{el2017deep} and \\cite{sallab2016end} a step further towards real car deployment. The constraints of such task are: 1) No damage caused by interaction with real environment (which was possible in the game engines world), 2) Sample efficiency, so learning time has to be reasonable 3) Generalization, where the learnt policy of the driving agent should be able to capture the basics of the intuitive driving when deployed in unseen environments.\n\nThere are several approaches to find an efficient driving policy. Optimal control methods based on traditional planning require knowledge of an environment model. While model based approaches are gaining more popularity \\cite{polydoros2017survey}, it is still hard to develop an efficient environment model especially for complex ones like urban and city scenarios. On the other hand, model free approaches \\cite{Watkins1989a} \\cite{sallab2016end} require the definition of a reward function, which only exist in simulated environments and game engines but not in real world.\nImitation learning is another successful approach, which might suffer the risk of de-generalization to unseen environments other than trained on. The most popular algorithms are based on data aggregation, which could be done through some hacks as in \\cite{bojarski2016end}, or formally as in \\cite{ross2011reduction}.\n\nIn this paper we propose a Meta learning framework for automated driving, to enforce transfer learning from one environment to another. Our hypothesis is that Meta learning will help capturing generic features without memorizing one specific environment. One proof of generalization is the visualization of which features the agent has learnt. If the learnt features are quite specific to the training environment, then it means that such agent will fail if deployed in another environment with different features. Under the Meta framework, we propose a novel algorithm; MetaDAgger, which is based on a Meta learning framework to aggregate data across different environments. To ensure generalization, we split the learning agent into two: 1) Meta learner, whose objective is to capture generic features not specific to an environment and 2) Low level learner, whose objective is to drive in a specific environment. The role of Meta learning is to smooth the learning performance across different environments. \n\nThe two learners are trained using Convolutional Neural Networks (ConvNets) on two different data sets. The meta learner is trained on a data set of environments; where  a set of environments are kept for training, while another independent set of environments are kept for testing and are not allowed to alter the learning process. One the other hand, the Low learner is trained on a data set aggregated from each specific environment, in the form of state-action pairs. \nThe link between both learners is done through continual lifelong learning, where the Low learner communicates its learning to the Meta learner whenever a switch from an environment to the other is undergone. In this way the learning is preserved across different environments, which enable the Meta learner to capture general features as proved by the features visualizations. It is worth noting that in \\cite{sallab2016end} an interesting conclusion is that continual learning highly improves the learning curve, which is in line with the proposed metal learning framework proposed in this work.\n\nThe experimental setup is based on The Open Race Car Simulator (TORCS) \\cite{wymann2000torcs}. The Meta dataset is formed of 19 tracks, with 10 training tracks and 9 testing tracks. Low learners are based on Convolutional Neural Networks (ConvNets) for policy learning. The results show significant advantage of MetaDAgger over DAgger on both training and testing tracks. MetaDAgger is not only improving generalization, but also significantly improving the sample efficiency and learning time over DAgger. The objective in each low level task is to keep the central lane. The demonstration in all tracks comes from a tradition PID controller, with access to the position of the ego car with respect to the left and right lanes. The control actions is applied to the steering wheel, with continuous actions output. The input states in all experiments are taken as the raw pixels information as provided by TORCS.\n\nThe rest of the paper is organized as follows; first we discuss the related work, then the Meta learning framework is described followed by the MetaDAgger algorithm. Then the experimental setup is described followed by the discussion of results and visualizations, and finally we conclude.\n\n\n\\section{Related Work}\n\\label{submission}\n\nThe problem of developing a driving policy is essentially formulated under the framework of optimal control. The solution of the optimization problem is easily found under the traditional planning framework only if an environment model exists, a requirement which is difficult to achieve in the real world of high way, urban or city driving. Model based approach is a wide topic that has been recently tackled in \\cite{polydoros2017survey}.\n\nOn the other hand, model free approaches have undergone a huge advancement in the area of human level control, the Deep Q Net is a famous example of which \\cite{MnihKavukcuogluSilverEtAl2015a} \\cite{MnihKavukcuogluSilverEtAl2013a}. The success of model free Reinforcement Learning \\cite{Sutton1988a} \\cite{Watkins1989a} has reached the area Automated Driving \\cite{Karavolos2013a} \\cite{el2017deep} \\cite{sallab2016end}. However, model free approaches require interacting with an environment through actions and rewards scheme. Such interaction usually requires a controlled environment such as simulators and game engines. The reward function is relatively easy acquired in such configuration of game engines. However, it is not clear how such reward function can be obtained in real world without causing too much damage. Moreover, model free approaches are known of their sample inefficiency, which is compensated by huge number of interactions with the simulated environment, which is again infeasible in real world. \n\nAnother learning approach based on human demonstrations has been successfully deployed in automated driving \\cite{bojarski2016end}.  Apprenticeship learning has been tackled in \\cite{AbbeelNg2004} \\cite{NgRussellothers2000a}, under the framework of Inverse Reinforcement Learning (IRL). IRL goal is to deduce the reward function the coach has been trying to achieve or maximize throughout the course of apprenticeship. Such hard goal is achieved on the expense of high complexity algorithm, involving two nested RL loops.\n\nIn order to tackle the problem of sample efficiency without knowing the environment models, supervised learning models are employed, where learning is based on demonstrated behavior to imitate (imitation learning). Learning from demonstrations has been approached as a supervised learning problem for automated driving in \\cite{bojarski2016end}. The risk of such an approach is the lack of generalization, where supervised learning schemes could converge to exact behavior cloning. The challenge of getting such approaches to work lies in the ability to enrich the training data by introducing new unseen states from the demonstration. One solution is data aggregation, where the agent is able to extend his actions to further states than the demonstrated ones.  Some hacks have been proposed in \\cite{bojarski2016end} to achieve data aggregation, where two cameras has been added in addition to the central camera to augment the training states with different situations supported with the correct action to take in each. A more formal approach has been formulated in the Dataset aggregation (DAgger) algorithm \\cite{ross2011reduction}. DAgger algorithm is also extensible to include safety constraints, as in SafeDAgger \\cite{zhang2016query}.\n\nTransfer of experience is an essential component of the human learning process. Transfer of knowledge could be in the form of external coaching, demonstrations or apprenticeship, or it could take the form of self accumulated experience through lifelong, continual learning \\cite{urgen1996simple}. Meta learning \\cite{thrun2012learning} \\cite{urgen1996simple} is another name of continual and transfer learning. Meta learning has also been used to search the best hyper parameters settings \\cite{hochreiter2001learning} \\cite{andrychowicz2016learning}. Continual learning fosters the transfer of experience across the life time of the agent. In addition, Meta learning provides a framework to transfer the knowledge acquired in one task to other tasks as well thanks to the captured information from task to task in the high level or Meta learner \\cite{thrun2012learning}.\n\n\\section{Meta Learning Framework for Automated Driving}\n\\label{submission}\n\nIn this section the Meta Learning framework is presented $Figure$ \\ref{Fig1}. The goal of the framework is to reach smooth and generalized performance over a set of environments\\{${E_{1},E_{2},..,E_{N}}$\\}. The performance is measured with the ability to take driving actions (e.g. steering) in a correct manner defined by the demonstrated behavior. \n\nTo ensure and test generalization, the set of environments are split into training and testing environments; $E_{train}$  and $E_{test}$. The learning algorithm is allowed to access the ground truth actions set of training environments $E_{train}$ , while it is not allowed to access the ground truth actions of the test environments $E_{test}$ . The learning is said to be generalized if it achieves the desired performance on the test environments $E_{test}$. The architecture is based on two learners:\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=120mm]{1}}\n\\caption{\\label{Fig1} Meta Learning Framework for Dataset Aggregation}\n\\label{icml-historical}\n\\end{figure*} \n\n\\subsection{Low Level Learner ($L$)}\n\nThe scope of the Low learner $L$ is to capture the specific features of each environment. The environment in operation $E$ is selected from a pool of training environments $E_{train}$. The low level learner is associated with a training data set $L_{train}$ which is formed due to interaction with an environment in operation $E$.  The training data set $L_{train}$  is formed of a list of state-action pairs \\{${s,a}$\\}$^t$, where t is the index of the interaction time stamp, with t=\\{$0,1,...,T$\\}, where $T$ is the end of interaction episode. An interaction episode is terminated with the $L$ reaching the goal, made a terminating mistake or after certain defined number of steps. The state $s$ is the measurement obtained by probing the environment $E$. The action $a$ is obtained from a demonstration, either from Human or from a reference algorithm.\n\nUnder the framework of Automated Driving, $L$ is trained using supervised learning imitating the demonstrated behavior.\n\n\n\\subsection{Meta Learner ($M$)}\n\nThe goal of the Meta Learner $M$ is to capture general features across all environments \\{${E_{1},E_{2},..,E_{N}}$\\}. With each interaction episode, $M$ communicates the initialization parameters to the Low learner $L$ to start interaction with the in operation environment $E$. After the Low learner finishes an episode, the learnt parameters of $L$ are communicated back to the $M$ to ensure continual learning.\n\nThe Meta Learner $M$ parameters are updated based on aggregated data over different interactions with the training environments $E_{train}$. The result of data set aggregation is called $M_{train}$. The members of $M_{train}$ have the same format of  state-action pairs  \\{${s,a}$\\}$^t$ as $L_{train}$. The training of $M$ is also following a supervised learning scheme using the training data in $M_{train}$.\n\nThe driving policy $\\pi^M$ is obtained as the parameters of the Meta Learner $M$, such that an action at time $t$ is obtained as  . In the context of Neural Networks, $\\pi^M$ is parameterized by the weights parameters of the network.\nThe performance of $\\pi^M$ is tested against a test data set $M_{test}$, which is formed from the set of test environments $E_{test}$. \n\n\n\\section{MetaDAgger Algorithm} \n \nIn this section the MetaDAgger algorithm is described as previous, in the light of the Meta learning framework described in $Figure$ \\ref{Fig1}.\n\n\\begin{algorithm}[tb]\n   \\caption{MetaDAgger}\n   \\label{alg:example}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Loop on $E_{train}$:} \/\/ Data Collection Step\n    \\STATE  \\qquad Collect $L_{train}$ += \\{${s, a_{ref}}\\}^t$ \/\/ $a_{ref}$ is obtained \n    \\STATE  \\qquad from a demonstration or a reference algorithm \n    \\STATE  \\qquad Aggregate $M_{train}$ += $L_{train}$\n    \\STATE Fit a Meta model $M$ based on $M_{train}$\\\\\n    -------------------------------------------------------------------\n   \\STATE {\\bfseries Loop until $N$-iter is reached} \/\/ Data Aggregation Step\n   \\STATE \\qquad { \\textbf{for} each $E$ in $E_{train}$} \\textbf{do}\n   \\STATE \\qquad \\quad Initialize $L$ = $M$ \n   \\STATE \\qquad \\quad Loop for $N_{steps}$\n   \\STATE \\qquad \\qquad $L$ interacts with $E$ to measure current state $s$\n   \\STATE \\qquad \\qquad Execute $a_{L}$ = $\\pi^L$(s), where $\\pi^{L}$ is obtained us-\n   \\STATE \\qquad \\qquad ing the parameters of $L$.\n   \\STATE \\qquad \\qquad \\textbf{if}  {$a_{L} != a_{ref}$}: \/\/Aggregate incorrect actions\n   \\STATE  \\qquad \\qquad \\qquad Aggregate $L_{train}$ += \\{${s, a_{ref}}$\\}$^t$\n    \\STATE \\qquad \\qquad Fit a Low model $L$ based on $L_{train}$\n    \\STATE \\qquad \\quad Save back M=L\n \\STATE Return $M$, $\\pi^M$\n\n\\end{algorithmic}\n\\end{algorithm}\n\nThe algorithm is based on two main steps:\n\n\\subsection{Data Collection}\nDuring data collection the reference demonstration is interacting with the environment to collect reference data. The reference demonstration could be a human or a reference algorithm. The collected data take the shape of state-action \\{${s, a_{ref}}\\}^t$ pairs at each interaction time step $t$, where $a_{ref}$  is the action provided by the demonstration.\nThis data is aggregated into $M_{train}$  to train the initial model parameters $M$, which could be a ConvNet fitting the supervised data.\n\n\n\\subsection{Data Aggregation}\nData aggregation is repeated for a number of iterations $N$-iter $>$ $|E_{train}|$, which means that the aggregation is repeated more than once over all the training environments.\nEvery iteration, $L$ interacts with $E$, starting from the parameters captures in $M$, which ensures continual learning. The parameters of $L$ define the interaction policy $\\pi$ that will be executed. The interaction happens for N-steps, during which data aggregation takes place. The action to be executed is obtained from the policy parameters $\\pi^{L}$ based on the measured state st. The aggregated data $L_{train}$ is then used to retrain $L$. At the end of each episode, the model is saved back to $M$ to ensure lifelong and continual learning for the next environment.\n\nIn order to improve sample efficiency, only the incorrect actions are saved. Hence, only actions that are different from the reference actions are saved. The difference is taken within certain tolerance (say 40\\% error). This is important in two aspects: 1) the aggregated data set is focused on mistakes only, which need to be corrected when $L$ is re-trained, and 2) in real world, we have no access to the reference action $a_{ref}$, however, it is easier to ask a human supervisor to attend and correct only the algorithm mistakes, even before it causes damage, in which case the corrected state-action pair is aggregated and retraining happens to correct it.\n\n\n\n\n\\section{Experimental Setup} \n\n\\subsection{Environment}\nMetaDAgger is designed to be ready for deployment in real car. The algorithm steps ensure data sampling efficiency and zero damage to the experimental car. Moreover, aggregation of incorrect actions ensures easy human supervision requiring the least effort.\n\nAs a first step towards deploying MetaDAgger in real car, we perform our experiments under TORCS game engine. We divide the 19 tracks in TORCS into 10 training tracks ($E_{train}$), and 9 test tracks ($E_{test}$). We use the Gym-TORCS environment \\cite{GYM_TORCS}, with the visual image input \\cite{Visual_TORCS}. Hence, the input state is the raw image pixels (64x64) as provided by the visual TORCS client as shown in $Figure$ \\ref{Fig2}.\nThe actions provided by MetaDAgger are the continuous steering angle values. The supervision signal is obtained using a reference PID controller that can access the position of the car from the two side lanes, which is provided by the TORCS game engine. Although we could use this supervision signal to aggregate all encountered states in the data aggregation step, however, we keep aggregating only the incorrect ones. Since the actions are continuous, mapping the absolute action value to a steering angle is sensitive to small differences. Because of this, we relax the exact matching between the reference and algorithm actions to a certain tolerance, empirically set to 40\\%.\n\n\\begin{figure}\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=50mm]{2}}\n\\caption{\\label{Fig2}TORCS visual input state}\n\\label{icml-historical}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\n\\subsection{Network Architecture and Hyper Parameters}\n\nFor the Low learner $L$ and the Meta learner $M$ we fit the same ConvNet model shown in $Figure$ \\ref{Fig3} , which simplifies copying models back and forth between $M$ and $L$ for continual learning. The kernel sizes are kept to a small size (3x3) due to the small input image size (64x64). Batch normalization is found to significantly improve the learning time \\cite{ioffe2015batch}. Xaviar initialization \\cite{glorot2010understanding} is used to initialize all weights. Dropout of 0.25 is used in the convolution layers, followed by Dropout of 0.5 in the fully connected ones.\nReLU activation is used in all layers, except for the output, where linear activation to produce the continuous output. The loss criterion is Mean Squared Error (MSE) minimization, since our task is a regression one.\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=\\textwidth]{3}}\n\\caption{\\label{Fig3}ConvNet architecture for Low and Meta learners}\n\\label{icml-historical}\n\\end{figure*}\n\n\\subsection{Results}\nWe first evaluate the generalization performance of MetaDAgger. The evaluation is done over the 9 test tracks $M_{test}$. Results are shown in $Figure$ \\ref{Fig4}. The horizontal axis represents the number of data aggregation iterations ($N$-iter), while the vertical axis is the number of steps the car moved without exiting the lane. We should note that 1000 steps mean one complete lap.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{5}}\n\\caption{\\label{Fig4}MetaDAgger performance on test tracks. Most of the tracks are completed just after 0 or 1 iterations of data aggregation.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\n\n\nWe compare the performance of MetaDAgger against traditional DAgger \\cite{ross2011reduction}. Results are shown in $Figure$ \\ref{Fig5}, where the vertical axis represents the number of steps without collision (one complete lap equals 1000 steps), and the horizontal axis represents the test track number. For some of the test tracks, MetaDAgger is able to complete one or two laps. For other tracks, performance is better than DAgger, although no laps are completed.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{4}}\n\\caption{\\label{Fig5}MetaDAgger vs. DAgger. Overall MetaDAgger is better than DAgger. For some tracks, it is still hard even with Meta learning to capture some hard turns.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\nWe analyze the learnt features in case of DAgger and MetaDAgger. We use Grad CAM \\cite{selvaraju2016grad}  visualization technique of the gradient of the last neuron (linear activation for action output), projected back to the input image. The result is a heat map showing which part of the image contributes more to the output. In case of DAgger, the learnt features are more memorizing and capturing the details of the track it is trained on, as shown in $Figure$ \\ref{Fig6}. For example, we can see the most important part is the horizon or the place with the mountain; this is because the theme of this track is a desert one. It is then understood why it is hard for such model to generalize to other tracks with different themes.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{6}}\n\\caption{\\label{Fig6}DAgger visualization. Most of the features represent the horizon or the mountain features.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\n\n\nOn the other hand, when we visualize the filters of MetaDAgger in $Figure$ \\ref{Fig7}, we see that the learnt features are more representing the relevant features to the driving task, like the positions of the lanes. This proves the generality of such model.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{7}}\n\\caption{\\label{Fig7}MetaDAgger filter visualization. Lane features are captured in many cases.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\nWe further evaluate the sample efficiency of MetaDAgger versus DAgger in $Figure$ \\ref{Fig8}. Here we evaluate how many data aggregation iterations are needed in order for the algorithm to complete a lap. An iteration terminates after certain number of steps, a complete lap or a termination condition. In our experiments the number of steps are taken to be 1000 (complete lap), while the termination condition is met when the car is out of the track. In case of DAgger, it takes 4 iterations, while for MetaDAgger it takes only two. Moreover, just after the data collection and behavior cloning step, the algorithm is able to complete 80\\% of the track.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{8}}\n\\caption{\\label{Fig8}Sample efficiency of MetaDAgger vs. DAgger. After only data collection, MetaDAgger is able to complete 80\\% of the lap.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn this paper we presented a framework for generalized imitation learning, based on the principles of Meta learning and data set aggregation. The proposed algorithm MetaDAgger is shown to be able to generalize on unseen test tracks, achieving much less training time and better sample efficiency.  The results on TORCS show significant improvement on both training and testing tracks, supported by visualizations of the generic learnt features. The proposed algorithm is designed to be ready for real car deployment, where the data aggregation step is only limited to correcting the mistakes the algorithm makes in real world.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuppose \n$\\be$ is defined as a family of $C^*$-correspondences $\\{E_1, \\ldots,E_k\\},$ over a $C^*$-algebra $\\mathcal{B}$ along with the unitary\nisomorphisms $t_{i,j}: E_i \\ot E_j \\to E_j \\ot E_i$ ($i>j$) where $k\\in \\mathbb N.$ With the help of these unitary isomorphisms, we may identify $C^*$-correspondence  $\\be ({\\bf n})$ as $E_1^{\\ot^{ n_1}} \\ot \\cdots \\ot E_k^{\\ot^{n_k}},$ for all\n${\\bf n}=(n_1, \\cdots, n_k) \\in \\Nk.$  Define maps $t_{i,i} := \\id_{E_i \\ot E_i}$ and $t_{i,j} := t_{j,i}^{-1}$ when $i<j.$ \n In this case, we say that $\\be$ is a {\\it product system over $\\Nk$} (cf. \\cite{F02}). \n \n A {\\it completely bounded, covariant representation} (CB-representation) (cf. \\cite{S008}) of $\\be$ on a\n\t\tHilbert space $\\mathcal{H}_{V}$ is defined as a tuple $ V:=(\\sigma, V^{(1)}, \\ldots, V^{(k)})$, where $\\sigma$ is a\n\t\trepresentation of  $\\mathcal{B}$ on $B(\\mathcal{ H}_{V}),$ and  $V^{(i)}:E_i \\to B(\\mathcal{H}_{V})$ are  completely bounded linear  maps satisfying\n\t\t\\[ V^{(i)}(a \\xi_i b) = \\sigma(a) V^{(i)}(\\xi_i) \\sigma(b), \\;\\;\\; a,b \\in \\mathcal M, \\xi_i \\in E_i,\\]\n\t\tand satisfying the commutation relation\n\t\t\\begin{equation} \\label{rep} \\wV^{(i)} (I_{E_i} \\ot \\wV^{(j)}) = \\wV^{(j)} (I_{E_j} \\ot \\wV^{(i)}) (t_{i,j} \\ot I_{\\mathcal {H}_{V}})\\end{equation}\n\twhere $1\\leq i,j\\leq k.$ We shall use notations $\\widetilde{V}^{(i)}_l: E_i^{\\ot l}\\otimes \\mathcal {H}_{V}\\to \\mathcal{ H}_{V}$ defined by\n\t\\[ \\widetilde{V}^{(i)}_l (\\xi_1 \\ot \\cdots \\ot \\xi_l \\ot h) := V^{(i)} (\\xi_1) \\cdots V^{(i)}(\\xi_l) h\\]\n\twhere $\\xi_1, \\ldots, \\xi_l \\in E_{i}, h \\in \\mathcal{ H}_{V}$.\n\t\n\t \n\t \n\t \n\t \n\t \n\t The following definition of doubly\n\t commuting isometric representation is a generalization of  doubly $\\Lambda$-commuting row isometries \\cite{GP20} and non-commuting isometries \\cite{MP}.\n\\begin{definition}  \\label{dcom}\n\tA  CB-representation $(\\sigma, V^{(1)}, \\ldots,$ $ V^{(k)})$ of  $\\be$ on  $\\mathcal{H}_{V}$ is said to be {\\rm doubly\n\t\tcommuting isometric representation} (DCI-representation)  (cf. \\cite{S008}) if for each distinct $i,j \\in \\{1,\\ldots,k\\}$ we have\n\t\\begin{equation*}\\label{doubly}\\wV^{(j)^*} \\wV^{(i)} =\n\t(I_{E_j} \\ot \\wV^{(i)})  (t_{i,j} \\ot I_{\\mathcal{ H}_{V}})  (I_{E_i} \\ot \\wV^{(j)^*}),\n\t\\end{equation*} and each $\\widetilde{V}^{(i)}$ is an isometry. \n\\end{definition}\n The {\\rm Fock module} of  $\\mathbb{E},$ is given by\n\t$$\\mathcal{F}(\\mathbb{E}):=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\mathbb{E}(\\mathbf{n}),$$ where the natural left action $\\phi_{\\infty}$  is given as $\\phi_{\\infty}(a)(\\oplus_{\\mathbf{n} \\in \\mathbb{N}_0^k} \\xi_{\\mathbf{n}})=\\oplus_{\\mathbf{n} \\in \\mathbb{N}_0^k} a \\cdot \\xi_{\\mathbf{n}},\\: \\xi_{\\mathbf{n}} \\in \\mathbb{E}(\\mathbf{n}), a \\in \\mathcal{B}.$\n\t Define for $\\mathbf{m}=(m_1, \\cdots, m_k) \\in \\mathbb{N}_0^k $, the map $V_{\\mathbf{m}}: \\mathbb{E}(\\mathbf{m}) \\to B(\\mathcal{H}_{V})$  by $V_{\\mathbf{m}}(\\xi_{\\mathbf{m}})h:=\\widetilde{V}_{\\mathbf{m}}(\\xi_{\\mathbf{m}} \\otimes h), ~ \\xi_{\\mathbf{m}} \\in \\mathbb{E}(\\mathbf{m}), h \\in \\mathcal{H}_{V},$ where $\\widetilde{V}_{\\mathbf{m}}:\\mathbb{E}(\\mathbf{m})\\otimes_{\\sigma} \\mathcal{H}_{V} \\to \\mathcal{H}_{V}$ is defined by\n$$\\widetilde{V}_{\\mathbf{m}}=\\widetilde{V}^{(1)}_{m_1}\\left(I_{E_1^{\\otimes m_1}} \\otimes\\widetilde{V}^{(2)}_{m_2}\\right) \\cdots \\left(I_{E_1^{\\otimes m_1} \\otimes \\cdots \\otimes E_{k-1}^{\\otimes {m_{k-1}}}} \\otimes\\widetilde{V}^{(k)}_{m_k}\\right).$$ \n\nNow we recall the definition of invariant subspace (cf. \\cite{SZ08}) for the covariant representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$.\n\\begin{definition}\t\n\t\\begin{itemize}\n\t\t\\item[$(1)$] Consider a CB-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and suppose  $\\mathcal K$ is a closed subspace of $ \\mathcal{ H}_{V}.$  Then we say $\\mathcal K$ is  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-{\\em invariant}   if it  is $\\sigma(\\mathcal M)$-invariant\n\t\n\t\n\t\tand, is invariant by each operator $V^{(i)}(\\xi_{i}),\\: \\xi_{i} \\in E_{i}, i \\in I_{k}.$  In addition, if  $ \\mathcal{K}^{\\bot}$ is   invariant by $V^{(i)}(\\xi_{i})$ for $\\xi_{i} \\in E_{i}, i \\in I_{k}$ then we say $\\mathcal{K}$ is $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-{\\em reducing}.  Restricting naturally  this representation we get  another representation of $E$ on $\\mathcal{K}$ which will be denoted as $(\\sigma, V^{(1)}, \\dots, V^{(k)})|_{\\mathcal{K}}.$ \n\t\t\n\t\t\\item[$(2)$]  A closed  subspace $\\mathcal{W}$ of $\\mathcal{H}_{V}$ is called {\\em wandering} subspace  for $(\\sigma, V^{(1)}, \\dots, V^{(k)})$, if it is $\\sigma(\\mathcal{B})$-invariant and $\\mathcal{W}\\perp\\wt{V}_{\\mathbf n}(\\mathbb{E}({\\mathbf n} )\\ot \\mathcal{W})$ for every $n \\in \\mathbb{N}_{0}^{k}.$ The representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ has {\\em generating wandering subspace property} (GWS-property) if there is a wandering subspace $\\mathcal{W}$ of $\\mathcal{H}_{V}$ satisfying\n\t\t$$\\mathcal{H}_{V}=\\displaystyle\\bigvee_{\\mathbf n\\in \\mathbb{N}_0^{k}}\\wt{V}_{\\mathbf n}(\\mathbb{E}({\\mathbf  n}) \\ot \\mathcal{W})$$\n\t\tand the corresponding  wandering subspace $\\mathcal{W}$\n\t\tis called generating wandering subspace (GWS). \n\t\\end{itemize}\n\t\n\\end{definition}\n\t\nConsider  pure DCI-representations $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and $(\\mu, T^{(1)}, \\dots, T^{(k)})$ of $\\mathbb{E}$ on the Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively. Let $I_{k}:=\\{1,2,\\dots, k\\}.$ Then using \\cite[Corollary 3.5]{HV21}, we get $$\\mathcal{H}_{V}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_0^k}\\wt{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes \\mathcal{W}_{\\mathcal{H}_{V}} ) \\:\\: \\mbox{and} \\:\\: \\mathcal{H}_{T}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_0^k}\\wt{T}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes \\mathcal{W}_{\\mathcal{H}_{T}} ),$$ where $\\mathcal{W}_{\\mathcal{H}_{V}}$ and $\\mathcal{W}_{\\mathcal{H}_{T}}$ are the generating wandering subspaces for $V$ and $T,$ respectively.\n\nA bounded operator $A: \\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ is called {\\it multi-analytic} (cf. \\cite{HV21}) if it satisfies \n\t\\begin{align*}\n\tAV^{(i)}(\\xi_i)h=T^{(i)}(\\xi_i)Ah \\hspace{0.7cm}\\mbox{and} \\hspace{0.7cm}A \\sigma(a)h= \\mu(a)Ah,\n\t\\end{align*}\n\twhere $\\xi_i \\in E_i, \\: h \\in \\mathcal{H}_{V},$ $ a \\in \\mathcal{B}$ and $ i \\in I_{k}.$ Each such $A$ is uniquely determined by  $\\Theta: \\mathcal{W}_{\\mathcal{H}_{V}} \\to \\mathcal{H}_{T}$ satisfying  $\\Theta \\sigma(a)h=\\mu(a)\\Theta h, h \\in \\mathcal{W}_{\\mathcal{H}_{V}},$ where $ \\Theta= A|_{\\mathcal{W}_{\\mathcal{H}_{V}}}.$ This follows  because for every $\\xi_{\\mathbf{n}} \\in \\mathbb{E}(\\mathbf{n}), h \\in \\mathcal{W}_{\\mathcal{H}_{V}}$ we have $AV_{\\mathbf{n}}(\\xi_{\\mathbf{n}})h=T_{\\mathbf{n}}(\\xi_{\\mathbf{n}})\\Theta h$ and $\\mathcal{H}_{V}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_0^k}\\wt{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes \\mathcal{W}_{\\mathcal{H}_{V}} ).$ Conversely, if we start with $\\Theta: \\mathcal{W}_{\\mathcal{H}_{V}} \\to \\mathcal{H}_{T}\\\\\\left( = \\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_0^k}\\wt{T}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes \\mathcal{W}_{\\mathcal{H}_{T}} ) \\right)$ which  satisfies $\\Theta \\sigma(a)h=\\mu(a)\\Theta h, h \\in \\mathcal{W}_{\\mathcal{H}_{V}},$ then we get $M_{\\Theta}: \\mathcal{H}_{V} \\to \\mathcal{H}_{T}$ defined by the equation $$M_{\\Theta}V_{\\mathbf{n}}(\\xi_{\\mathbf{n}})h=T_{\\mathbf{n}}(\\xi_{\\mathbf{n}})\\Theta h=T_{\\mathbf{n}}(\\xi_{\\mathbf{n}})M_{\\Theta} h, \\:\\:\\: \\xi_{\\mathbf{n}} \\in \\mathbb{E}(\\mathbf{n}), h \\in \\mathcal{W}_{\\mathcal{H}_{V}},$$ which is multi-analytic.\n\tWe shall always consider $\\Theta$ such that $M_{\\Theta}$ is a contraction. It is easy to check that\n\t$$M_{\\Theta}\\left(\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}h_{\\mathbf{n}} \\right)=\\sum_{\\mathbf{n} \\in \\mathbb{N}^k_0} \\wt{T}_{\\mathbf{n}}(I_{\\mathbb{E}(\\mathbf{n})} \\otimes \\Theta)\\wt{V}_{\\mathbf{n}}^*h_n \\hspace{1cm} \\mbox{for}\\: \\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}h_{\\mathbf{n}} \\in \\mathcal{H}_{V}.$$\n\n\n\tThe map $\\Theta,$ as defined above, will be called (cf. \\cite[Proposition 4.4]{HV21}) \n\t\\begin{enumerate}\n\t\t\\item[$(1)$]{\\it outer} if $\\overline{M_{\\Theta}\\mathcal{H}_{V}}=\\mathcal{H}_{T}$ (equivalently, $\\Theta \\mathcal{W}_{\\mathcal{H}_{V}} $ is cyclic for  $T,$ i.e., $$\\bigvee_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\wt{T}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes  \\Theta \\mathcal{W}_{\\mathcal{H}_{V}} )=\\mathcal{H}_{T}).$$ \n\t\t\\item[$(2)$] {\\it inner} if $M_{\\Theta}$ is an isometry, (equivalently, $\\Theta \\mathcal{W}_{\\mathcal{H}_{V}} $ is a wandering subspace for\\\\ $T$  and $\\Theta$ is an isometry. )\n\\end{enumerate}\nIndeed, it follows that $\\Theta$ is inner and  outer if and only if $\\Theta$ is a unitary operator from $ \\mathcal{W}_{\\mathcal{H}_{V}}$ to $\\mathcal{W}_{\\mathcal{H}_{T}}.$\n\n\\begin{definition}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a pure isometric  representation of  $\\mathbb{E}$ on $\\mathcal{H}_{V}$  and $\\mathcal{K}$ be a $\\sigma$-invariant subspace (IS) of $\\mathcal{H}_{V}$, i.e., $\\sigma (b)\\mathcal{ K}\\subseteq\\mathcal{ K}$ for each $b \\in \\mathcal{B}.$ Then  $\\mathcal{K}$ is said to be  {\\rm quotient subspace} (QS) of $\\mathcal{H}_{V}$ if for each $ i\\in I_{k}$, $ \n\\widetilde{V}^{(i)*}\\mathcal{K}\\subseteq E_{i}\\ot \\mathcal{K}.\n$\\end{definition}\n\\begin{remark}\n\t Note that $\\mathcal{K}$ is $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-IS  of $\\mathcal{H}_{V}$  if and only if  $\\mathcal{K}^{\\perp}$  is a QS of $\\mathcal{H}_{V}. $\n\\end{remark}\nIn particular, for $k=1,$ recall that  $(\\sigma , V)$ is said to be {\\it completely contractive}  if $I_{\\mathcal{H}_{V}}-\\widetilde{V}\\widetilde{V}^{*}\\geq 0 $ and {\\it pure (analytic)} if $\\{\\widetilde{V}_{m}^{*}\\}\\to 0$ as $m \\to \\infty,$ with respect to strong operator topology. Now according to Beurling's theorem \\cite{NFS}, $\\mathcal{L}$ is the image of an inner function if $\\mathcal{L}$ is a closed invariant subspace for the shift $M_z$ of the Hardy space $H^2(\\mathbb{D})$. The following  Beurling-Lax-Halmos type theorem \\cite[Theorem 4.5]{HV21} provides a characterization of  QS $\\mathcal{ K}$ of $\\mathcal{H} _V$  which is a generalization  of Popescu's version of Beurling-Lax theorem \\cite[Theorem 2.2]{G89}. \n\\begin{theorem}\\label{hs} Let $(\\sigma, V)$ be a pure isometric representation of  ${E}$ on $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a  closed subspace of $\\mathcal{ H}_{V}$. Then $\\mathcal{K}$ is a QS of $\\mathcal{ H}_{V}$ if and only if there exist a Hilbert space $\\mathcal{W},$ a representation $\\sigma_1$ of $\\mathcal{B}$ on $\\mathcal{W}$ and an isometric multi-analytic operator $M_\\Theta: {\\mathcal{F}(E) \\otimes_{\\sigma_1} \\mathcal{W}} \\to \\mathcal{H}_{V}$ such that $$\\mathcal{K}=\\mathcal{H}_{V}\\ominus M_{\\Theta}(\\mathcal{F}(E) \\otimes_{\\sigma_1} \\mathcal{W}),$$ where  $\\mathcal{W}_{\\mathcal{F}(E) \\otimes \\mathcal{W}}$ is the generating wandering subspace for the induced representation $(\\rho,S)$ of $E$ on $\\mathcal{F}(E) \\otimes \\mathcal{W}_{\\mathcal{ K}}$  induced by $\\sigma|_{\\mathcal{W}_{\\mathcal{ K}}}$, where $\\mathcal{W}_{\\mathcal{ K}}$ is wandering subspace for $(\\sigma, V)|_{\\mathcal{K}}.$ \n\\end{theorem}\n For example if $\\mathcal{ K}$ is a QS of $\\mathcal{H}_{V}$ then the covariant representation  $(\\sigma|_{\\mathcal{ K}},V|_{\\mathcal{ K}}),$  is a pure completely contractive covariant representation of $E$ on $\\mathcal{K}.$ \n\n\n\n\nGenerally a QS  $\\mathcal{K}$ of $\\mathcal{H}_{V}$, where $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ is a  pure DCI-representation, is not necessarily  have the Beurling-type representation \\cite{D89,WR,YR19}. In this article we give a description of QS  $\\mathcal{K}$ of $\\mathcal{H}_{V}$ based on \\cite{BDDS}. For a pure DCI-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ of  $\\mathbb{E}$ on a Hilbert space $\\mathcal{H}_{V},$ a  $\\sigma$-IS $\\mathcal{K}$  is said to be {\\it Beurling quotient subspace} (BQS) of $\\mathcal{H}_{V}$ if there exist a Hilbert space $\\mathcal{H}_W,$ a pure DCI-representation  $(\\pi, W^{(1)}, \\dots, W^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_W$ and an isometric multi-analytic operator $M_\\Theta:{\\mathcal{H}_W} \\to \\mathcal{H}_{V}$ such that\n\\begin{equation*}\n\\mathcal{K}=\\mathcal{H}_{V}\\ominus M_{\\Theta}\\mathcal{H}_W,\n\\end{equation*} where $\\mathcal{W}_{\\mathcal{H}_W}$ is the generating wandering subspace for $(\\pi, W^{(1)}, \\dots, W^{(k)}).$ The following natural question arises: \n  \n\\begin{question}\\label{ques}\nWhich QS of $\\mathcal{H}_{V}$ admits Beurling representation ?\n\\end{question} \n To answer this question, consider a  pure DCI-representation  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and let  $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}.$ Define  $$\\mu(a):=P_{\\mathcal{ K}}\\sigma(a){P_{\\mathcal{K}}}  \\:\\mbox{and} \\:\\:T^{(i)}(\\xi_{i}):=P_{\\mathcal{K}}V^{(i)}(\\xi_{i}){P_{\\mathcal{K}}},$$\nfor $\\xi_{i} \\in E_{i}$ and $a \\in \\mathcal{B},$ i.e.,\n$\\wT^{(i)}:=P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}),$ where $P_{\\mathcal{K}}$ is orthogonal projection of $\\mathcal{H}_{V}$ on $\\mathcal{ K}$  and  $i\\in I_{k}$. Then $(\\mu,T^{(1)},T^{(2)}, \\dots, T^{(k)}) $ is a contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{K}.$ The following theorem  answers  Question \\ref{ques} which is the main result of Section \\ref{2}.\n\\begin{theorem}\\label{Beurling}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a pure DCI-representation of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}.$ Let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}$. Then $\\mathcal{K}$ is a BQS of $\\mathcal{H}_{V}$ if and only if \\begin{equation*}\n\t(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{K}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))=0.\n\t\\end{equation*}\n\\end{theorem}\n\n\nFor $k=2,$  $(\\sigma, V^{(1)}, V^{(2)})$   satisfies {\\it Brehmer-Solel condition} if \\begin{equation*}\n\\Delta_{*}(V):=I_{\\mathcal{H}}-\\widetilde{V}^{(1)}\\widetilde{V}^{(1)*}-\\widetilde{V}^{(2)}\\widetilde{V}^{(2)*}+\\widetilde{V}^{(1)}(I_{E_1} \\ot \\widetilde{V}^{(2)}\\widetilde{V}^{(2)*})\\widetilde{V}^{(1)*}\\geq 0.\n\\end{equation*}\nIn Section \\ref{3}, first we  give a geometrical proof of the regular dilation theorem for pure  completely contractive covariant representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of $\\mathbb{E}$ on $\\mathcal{H}_{V}$ satisfying Brehmer-Solel condition.  Using the dilation theorem \\cite{MV93,CV95,S008,S009}  there exists a QS $\\mathcal{ K}$ of $\\mathcal{F}(\\mathbb{E}) \\ot \\mathcal{ H}$ such that   \\begin{equation}\\label{imp}\n(\\sigma, V^{(1)}, V^{(2)})\\equiv(\\rho|_{\\mathcal{K}}, S^{(1)}|_{\\mathcal{K}}, S^{(2)}|_{\\mathcal{K}}),\n\\end{equation}  where $(\\rho, S^{(1)}, S^{(2)})$ is the induced  representation of $\\mathbb{E}$ on $\\mathcal{F}(\\mathbb{E}) \\ot \\mathcal{ H}.$ But it is not necessary that $\\mathcal{K}$ obtained in this way is a BQS of $\\mathcal{F}(\\be) \\ot \\mathcal{H}.$ This motivates  Section \\ref{3}.\n\n\n\n\n\nConsider a  pure DCI-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and the induced representation \\\\ $(\\rho, S^{(1)}, S^{(2)}, \\dots, S^{(k)})$  of  $\\mathbb{E}$ on the Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}$ respectively, where $\\mathcal{H}$ is a Hilbert space. Consider $M_{\\Theta}: \\mathcal{ H}_{V} \\to\\mathcal{F}(\\mathbb{E})\\ot \\mathcal{ H}   $ be  an isometric multi-analytic operator. Let us denote $\\mathcal{ K}_{\\Theta}= (M_{\\Theta}(\\mathcal{ H}_{V}))^{\\perp}$ and  $\\mathcal{S}_{\\Theta}= M_{\\Theta}(\\mathcal{ H}_{V})$  be BQS and BS, respectively. For each $i \\in I_k$, define  $$\\pi_{\\Theta}(a)=P_{\\mathcal{K}_{\\Theta}}\\rho(a)|_{{\\mathcal{K}_{\\Theta}}}  \\:\\mbox{and} \\:\\:W_{\\Theta}^{(i)}(\\xi_{i})=P_{\\mathcal{K}_{\\Theta}}S^{(i)}(\\xi_{i})|_{{\\mathcal{K}_{\\Theta}}},$$\nfor $\\xi_{i} \\in E_{i}$ and $a \\in \\mathcal{B}.$ Then  $W_{\\Theta }=(\\pi_{\\Theta}, W_{\\Theta}^{(1)}, W_{\\Theta}^{(2)}, \\dots, W_{\\Theta}^{(k)} )$ is a completely contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{K}_{\\Theta}.$\nThe following theorem gives a necessary and sufficient condition when QS become BQS : \\begin{theorem}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a  completely contractive covariant representation of  $\\mathbb{E}$ on a Hilbert space $\\mathcal{H}_{V}.$ Then the following are equivalent:\\begin{enumerate}\n\t\t\\item $(\\sigma, V^{(1)}, \\dots, V^{(k)})\\equiv (\\pi_{\\Theta}, W_{\\Theta}^{(1)}, W_{\\Theta}^{(2)}, \\dots, W_{\\Theta}^{(k)} )$ for some BQS $\\mathcal{K}_{\\Theta}.$\n\t\t\\item  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ is pure and satisfies the Brehmer-Solel condition and \\begin{equation*}\n\t\t(I_{E_{j}}\\ot ( I_{E_{i}\\ot P_{\\mathcal{K}_{\\Theta}}}- \\wV^{(i) *}\\wV^{(i)}))(t_{i,j} \\ot I_{\\mathcal{K}_{\\Theta}})(I_{E_{i}}\\ot  (I_{E_{j}\\ot P_{\\mathcal{K}_{\\Theta}}}- \\wV^{(j) *}\\wV^{(j)}))=0,\n\t\t\\end{equation*} for distinct $i\\neq j.$ \n\t\\end{enumerate}\n\\end{theorem}\nSz.Nagy-Foias \\cite[Chapter VII, Theorem 2.3]{NFS} provides a necessary and sufficient for a nontrivial invariant subspace of the completely non-unitary contraction $V$ on a (separable) Hilbert space, if the characteristic function of $V$ admit a nontrivial factorization. Bercovici \\cite[Chapter 5, Proposition 1.21]{B88} extends this result and gave a description of the invariant subspaces of a functional model. Section \\ref{4} presents a characterization of a invariant subspaces of compression of a pure isometric covariant representation which is a generalization of \\cite[Chapter VII, Theorem 2.3]{NFS}  and \\cite[Chapter 5, Proposition 1.21]{B88}. \\begin{theorem}\n\tLet $(\\sigma, V)$ and $(\\mu, T)$ be pure isometric  representations of $E$ on Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively.  Suppose ${\\mathcal{W}_{\\mathcal{H}_{V}}}$  and ${\\mathcal{W}_{\\mathcal{H}_{T}}}$ are the generating wandering subspace for $(\\sigma, V)$ and $(\\mu,T),$ respectively. Let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator. Then  the covariant representation $(\\pi'_{\\Theta}, W'_{\\Theta})$ has an IS   if and only if there exist a Hilbert space $\\mathcal{H}_{R},$ a pure isometric  representation $(\\nu, R)$ of $E$ on $\\mathcal{H}_{R}$, and isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that  \\begin{equation*}\n\t{\\Theta}={\\Phi}{\\Psi}.\n\t\\end{equation*}\n\\end{theorem} More generally if $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and $(\\mu, T^{(1)}, \\dots, T^{(k)})$  be a DCI-representations of  $\\mathbb{E}$ on the Hilbert space $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively and let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator, then we can factorize $\\Theta$  in terms of inner operators.\n\n Moreover if we take $k=2,$ i.e., $(\\pi'_{\\Theta}, W_{\\Theta}^{'(1)},W_{\\Theta}^{'(2)})$-joint IS alone does not guarantee the factorization of an inner operator and this motivates Section \\ref{4}. Note that any reducing subspace for the covariant representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  is doubly commuting subspace. However the converse need not be true in general. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Beurling Quotient Subspace For The Covariant Representations}\\label{2}\n\n\nIn this section, we prove a necessary and sufficient condition regarding when a QS becomes  BQS for DCI-representations of a product system. For this purpose we need some fundamental lemmas regarding BQS in terms of cross commutators. First we recall a Beurling-type characterization for  the doubly commuting invariant subspaces for DCI-representations.\n\\begin{thm}\\cite[Theorem 4.12]{HV21}\\label{MT5}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a pure DCI-representation of  $\\mathbb{E}$ on the Hilbert space  $\\mathcal{H}_{V}.$  Let $\\mathcal{K}$ be a  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-IS of $\\mathcal{H}_{V}$. Then  $\\mathcal{K}$ is a $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-{\\rm doubly commuting subspace (DCS)}, that is,\n\t$${\\wt{V}|_{\\mathcal{K}}^{(i)}}^{*}{\\wt{V}|_{\\mathcal{K}}^{(j)}}=(I_{E_i} \\otimes {\\wt{V}|_{\\mathcal{K}}^{(j)}})(t_{j,i} \\otimes I_{\\mathcal{K}})(I_{E_j} \\otimes{\\wt{V}|_{\\mathcal{K}}^{(i)}}^{*})$$ where $i,j \\in I_k$ with $ i \\neq j,$   if and only if there exist a Hilbert space $\\mathcal{H}_{T}$, a pure DCI-representation $(\\mu, T^{(1)}, \\dots, T^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_T$ and an isometric multi-analytic operator $M_\\Theta:{\\mathcal{H}_T} \\to \\mathcal{H}_{V}$ such that\n\t$$\\mathcal{K}=M_{\\Theta}\\mathcal{H}_T.$$ $\\mathcal{ K}$ is also called as {\\it Beurling subspace}. Indeed if $\\mathcal{K}=\\mathcal{H}_{V},$ then $\\Theta$ is outer.\n\\end{thm}\n\n \n\\begin{remark}\\label{1} Note that if $\\mathcal{K}$ is invariant subspace of $\\mathcal{ H}_{V},$ then for each $ i \\in I_{k},$ we have ${\\wt{V}|_{\\mathcal{K}}^{(i)}}^{*}=P_{E_{i}\\ot \\mathcal{ K}}{\\wt{V}^{(i)*}}.$ \tFurthermore if $\\mathcal{K}$ is a $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-IS of $\\mathcal{ H}_{V},$ then ${\\wt{V}|_{\\mathcal{K}}^{(i)}}=P_{\\mathcal{ K}}{\\wt{V}|_{\\mathcal{K}}^{(i)}}$ and if $\\mathcal{K}$ is a $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-co-invariant subspace of $\\mathcal{ H}_{V},$ then ${\\wt{V}|_{\\mathcal{K}^\\perp}^{(i)}}^{*}=P_{{E_{j}\\ot \\mathcal{K}^{\\perp}}}\\widetilde{V}^{(i)*}|_{\\mathcal{ K}^{\\perp}}.$\n\\end{remark}\n Consider a completely bounded representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$    of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a subspace of $\\mathcal{ H}_{V}.$  \nFor each $i,j \\in I_{k},$  we denote the cross commutator of ${\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}$ and ${\\wt{V}|_{\\mathcal{K}}^{(i)}}$ by $[{\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}, {\\wt{V}|_{\\mathcal{K}}^{(i)}}]$ and  define it by \\begin{align*}\n&[{\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}, {\\wt{V}|_{\\mathcal{K}}^{(i)}}]:=\t{\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}{\\wt{V}|_{\\mathcal{K}}^{(i)}} - (I_{E_{j}} \\ot {\\wt{V}|_{\\mathcal{K}}^{(i)}})(t_{i,j} \\ot I_{\\mathcal{K}})(I_{E_{i}}\\ot {\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}),\n\\end{align*} where ${\\wt{V}|_{\\mathcal{K}}^{(i)}}:{E_{i}\\ot \\mathcal{ K}} \\to \\mathcal{ H}_{V}$ for each $ i \\in I_{k}.$\n\\begin{remark}\\label{11}\n\tIt follows from Theorem \\ref{MT5}  that  $\\mathcal{K}^{\\perp}$ is a  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-DCS of $\\mathcal{H}_{V}$ if and only if \\begin{align*}\n\t[\\widetilde{R}^{({j})*}, \\widetilde{R}^{(i)}]=0,\\:\\: \\mbox{for all}\\:\\: i\\neq j,\\:\\: i,j \\in I_{k}\n\t\\end{align*} where $\\widetilde{R}^{(i)}:={\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}} .$ In this case, we say that $\\mathcal{ K}$ is a BQS of $\\mathcal{H}_{V}.$ This gives the following lemma.\n\\end{remark}\n\n\\begin{lemma}\\label{Lemma 2.1}\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a pure DCI-representation of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}.$ For each $i, j \\in I_{k}$, define \n \\begin{equation*}\nX_{i,j}:=(I_{E_{j}} \\ot P_{ \\mathcal{K}^{\\perp}} \\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*} P_{\\mathcal{K}^\\perp}).\n\\end{equation*}\nThen $\\mathcal{K}$ is a BQS of $\\mathcal{H}_{V}$ if and only if $X_{i,j}=0$ for  $i \\neq j$.\n\\end{lemma}\t\n \\begin{proof}\n \tFor distinct $i, j\\in I_{k},$ using the DCS condition and Remark \\ref{1}\n  we obtain  \\begin{align*}[\\widetilde{R}^{({j})*}, \\widetilde{R}^{(i)}]&=[{\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*}, {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}}]=\t{\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*}{\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}} - (I_{E_{j}} \\ot {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}})(t_{i,j} \\ot I_{\\mathcal{K}^{\\perp}})(I_{E_{i}}\\ot {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*})\\\\&=P_{{E_{j}\\ot \\mathcal{K}^{\\perp}}}\\widetilde{V}^{(j)*}{\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}}-P_{{E_{j}\\ot \\mathcal{K}^{\\perp}}}(I_{E_{j}} \\ot {\\wt{V}^{(i)}})(t_{i,j} \\ot P_{\\mathcal{K}^{\\perp}})(I_{E_{i}}\\ot {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*})\\\\&=P_{{E_{j}\\ot \\mathcal{K}^{\\perp}}}(I_{E_{j}} \\ot {\\wt{V}^{(i)}})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*}),\n \t\\end{align*}  where $P_{\\mathcal{K}^{\\perp}}$ is an orthogonal projection of $\\mathcal{H}_{V}$ on $\\mathcal{K}^{\\perp}$ and $P_{E_{j} \\ot \\mathcal{K}^{\\perp}}$ is an orthogonal projection of $E_{j} \\ot \\mathcal{H}_{V}$ on $E_{j} \\ot \\mathcal{K}^{\\perp}.$  Hence Remark \\ref{11} gives that $\\mathcal{ K}^{\\perp}$ is a DCS of $\\mathcal{ H}_{V}$  if and only if $${\\big((I_{E_{j}} \\ot P_{ \\mathcal{K}^{\\perp}})(I_{E_{j}}\\ot \\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*})\\big)}|_{E_{i} \\ot \\mathcal{K}^{\\perp}}=0,$$ which means  $X_{i,j}=0.$\\end{proof}\n\n\n\n  Consider a DCI-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ of  $\\mathbb{E}$ on a Hilbert space $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}.$ Define  $$\\mu(a):=\\sigma(a)P_{\\mathcal{K}}  \\:\\mbox{and} \\:\\:T^{(i)}(\\xi_{i}):=P_{\\mathcal{K}}V^{(i)}(\\xi_{i})P_{\\mathcal{K}}, \\:\\:\\: \\text {for each}\\:\\: \\xi_{i} \\in E_{i},a \\in \\mathcal{B}, \\text{and}\\:\\:  i\\in I_{k}$$\n i.e.,\n $\\wT^{(i)}=P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}).$ Then $T=(\\mu,T^{(1)},T^{(2)}, \\dots, T^{(k)}) $ is a completely contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{H}_{V}$ with having the commutant relation (\\ref{rep}). \n \n\n \n Throughout in this section we fix $i,j \\in I_{k}$ and assume that $i\\neq j$ and also $ \n\\mathbf{\\hat{m}}_{i}$ denote multi-indices in $\\mathbb{N}^{k}_{0}$ whose  $i^{th}$ entry is zero. The following lemma will take an important role in to prove main result of this section.\n\\begin{lemma}\\label{2.2}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a DCI-representation of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}.$ Let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}$. Then \\begin{equation*}\n\t[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}).\n\t\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n\tObserve that $\\widetilde{T}^{*}_{\\mathbf{m}}=\\widetilde{V}_{\\mathbf{m}}^{*}P_{\\mathcal{K}}$ and therefore $\\widetilde{T}_{\\mathbf{m}}=P_{\\mathcal{K}}\\widetilde{V}_{\\mathbf{m}}$   for all $\\mathbf{m}=(m_1,m_2,\\dots,m_k)\\in \\mathbb{N}^k_{0}.$  Substituting the values of  $\\wT_{\\mathbf{\\hat{m}}_{i}}^{*}$ and $\\wT^{(i)},$  we get\\begin{align*}\n\t&\t[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_i}^{*}]\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,\\mathbf{\\hat{m}}_{i}}\\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}})-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}} \\widetilde{V}^{(i)})(t_{i,\\mathbf{\\hat{m}}_{i}}\\ot I_{\\mathcal{H}_{V}})( I_{E_{i}}\\ot ((I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i})}}\\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*} P_{\\mathcal{K}}))-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}} \\widetilde{V}^{(i)})(t_{i,\\bf{\\hat{m}}_{i}}\\ot I_{\\mathcal{H}_{V}})( I_{E_{i}}\\ot \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*} P_{\\mathcal{K}})-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*} \\widetilde{V}^{(i)}( I_{E_{i}}\\ot P_{\\mathcal{K}})-(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}[I_{\\mathcal{H}_{V}}-P_{\\mathcal{K}}]\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}),\n\t\\end{align*} which proves the lemma.\\end{proof}\n For each $i\\in I_{k},$ we have  \\begin{align}\\label{positive} 0 &\\leq (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}) \\nonumber\\\\&= (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}(I_{\\mathcal{H}_{V}}-P_{\\mathcal{K}})\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\nonumber\\\\&=  (I_{E_{i}}\\ot P_{\\mathcal{K}})-\n \\wT^{(i) *}\\wT^{(i)} .\n \\end{align} Hence  $(I_{E_{i}}\\ot P_{\\mathcal{K}}-\\wT^{(i) *}\\wT^{(i)})^{\\frac{1}{2}}$ exists and let it be denoted by  $\\Delta{({T|_{\\mathcal{K}}^{(i)}})}.$  \n \n \nThe following lemma  relate $\\Delta{({T|_{\\mathcal{K}}^{(i)}})}$ to $[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}].$  \n\\begin{lemma}\\label{lemma2.5}\n\tLet $\\mathbf{\\hat{m}}_{i} \\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\}.$ Then there exist contractions $ X_{\\mathbf{\\hat{m}}_{i}}:E_{i} \\ot \\mathcal{K} \\to \\mathbb{E}(\\mathbf{\\hat{m}}_{i})\\ot \\mathcal{K}$ and $Y_{\\mathbf{\\hat{m}}_{i}}:  \\mathbb{E}(\\mathbf{\\hat{m}}_{i})\\ot \\mathcal{K} \\to E_{i} \\ot \\mathcal{K} $  such that $[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]= X_{\\mathbf{\\hat{m}}_{i}}\\Delta{({T|_{\\mathcal{K}}^{(i)}})}$ and $[\\wT_{\\mathbf{\\hat{m}}_{i}},\\wT^{(i)*}]=\\Delta{({T|_{\\mathcal{K}}^{(i)}})} Y_{\\mathbf{\\hat{m}}_{i}}.$\n\\end{lemma}\n\\begin{proof}\n\tUsing  Lemma \\ref{2.2} we get \\begin{align*}&\n\\Delta{({T|_{\\mathcal{K}}^{(i)}})}^{2}-[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]^{*}[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]\\\\&=(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})-(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}[I_{\\mathcal{H}_{V}}-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i})}} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}] P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\& =((I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}})[I_{\\mathcal{H}_{V}}-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i})}} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}] ((I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}})^{*}.\n\t\\end{align*} As $\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})$ is a contraction, $\n\tI_{\\mathcal{H}_{V}}-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i})}} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*} \\geq 0,\n\t$ and hence \\begin{equation*}\n\t\\Delta{({T|_{\\mathcal{K}}^{(i)}})}^{2}-[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]^{*}[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}] \\geq 0.\n\t\\end{equation*}That is $[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]^{*}[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]\\leq \t\\Delta{({T|_{\\mathcal{K}}^{(i)}})}^{2}.$ Therefore by Douglas's range inclusion theorem \\cite{D96}, there exists a contraction $ X_{\\mathbf{\\hat{m}}_{i}}:E_{i} \\ot \\mathcal{K} \\to \\mathbb{E}(\\mathbf{\\hat{m}}_{i})\\ot \\mathcal{K}$ such that $ [\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]=X_{\\mathbf{\\hat{m}}_{i}}\\Delta({T|_{\\mathcal{K}}^{(i)}}).$  Since  $ [\\wT^{(i)},\\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]^{*}=[\\wT_{\\mathbf{\\hat{m}}_{i}},\\wT^{(i)*}],$ which proves the second equality.\n\\end{proof}\n\\begin{lemma}\\label{lemma 2.4}\n\t\tConsider a  DCI-representation  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$   of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}$. If \\begin{equation}\\label{hyp}\n\t\t(I_{E_{j}} \\ot \\Delta{({T|_{\\mathcal{K}}^{(i)}})}(t_{i,j} \\ot I_{\\mathcal{H}_{V}})\t(I_{E_{i}} \\ot \\Delta{({T|_{\\mathcal{K}}^{(j)}})}=0,\n\t\t\\end{equation} then, for each $\\mathbf{\\hat{m}}_{i},\\mathbf{\\hat{n}}_{j} \\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\} ,$ \\begin{enumerate}\n\t\t\t\\item $(I_{E_j}\\ot (I_{\\mathbb{E}{(\\mathbf{\\hat{m}_i})}} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*})X_{i,j}(I_{E_i} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}} (I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_j})} \\ot P_{\\mathcal{K}}))=0,\n\t\t\t$\n\t\t\t\\item $ (I_{E_j}\\ot (I_{E_{i}} \\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*})X_{i,j}(I_{E_i} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}} (I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_j})} \\ot P_{\\mathcal{K}}))=0,$ and\n\t\t\t\\item $(I_{E_j}\\ot (I_{\\mathbb{E}{(\\mathbf{\\hat{m}_i})}} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*})X_{i,j} (I_{E_i} \\ot \\widetilde{V}^{(j)}( I_{E_{j}} \\ot P_{\\mathcal{K}}))=0.$\n\t\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof} (1)  By  Lemma \\ref{lemma2.5}, there exist contractions $X_{\\mathbf{\\hat{m}}_{i}}$ and $Y_{\\mathbf{\\hat{n}}_{j}}$  such that $[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]= X_{\\mathbf{\\hat{m}}_{i}}\\Delta{({T|_{\\mathcal{K}}^{(i)}})}$ and $[\\wT_{\\mathbf{\\hat{n}}_{j}},\\wT^{(j)*}]= \\Delta{({T|_{\\mathcal{K}}^{(i)}})} Y_{\\mathbf{\\hat{n}}_{j}}$  and therefore  \\begin{align*}& (I_{E_j} \\ot[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}] )(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot[\\wT^{(j)}, \\wT_{\\mathbf{\\hat{n}}_{j}}^{*}]^{*} )\\\\&=\n(I_{E_j} \\ot X_{\\mathbf{\\hat{m}}_{i}}\\Delta{({T|_{\\mathcal{K}}^{(i)}})} )(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot \\Delta{({T|_{\\mathcal{K}}^{(i)}})} Y_{\\mathbf{\\hat{n}}_{j}})\\\\&=0,\n\\end{align*} here the last equality follows by hypothesis (\\ref{hyp}).\n\nOn the other hand,  by using Lemma \\ref{2.2}, we obtain\n\\begin{align*}&\n(I_{E_j} \\ot[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}] )(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot[\\wT^{(j)}, \\wT_{\\mathbf{\\hat{n}}_{j}}^{*}]^{*} )\\\\&=(I_{E_j} \\ot (I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}_{\\mathbf{\\hat{n}}_{j}}(I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_{j})}} \\ot P_{\\mathcal{K}}))\\\\&=(I_{E_j}\\ot (I_{\\mathbb{E}{(\\mathbf{\\hat{m}_i})}} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*})X_{i,j}(I_{E_i} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}} (I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_{j})}} \\ot P_{\\mathcal{K}})).\n\\end{align*} Thus we get (1).\n\n(2) Using hypothesis ({\\ref{hyp}}), we have \\begin{align*}&\n0=(I_{E_{j}} \\ot ((I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot[\\wT^{(j)}, \\wT_{\\mathbf{\\hat{n}}_{j}}^{*}]^{*} )\\\\&=(I_{E_{j}} \\ot (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}_{\\mathbf{\\hat{n}}_{j}}(I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_{j})}} \\ot P_{\\mathcal{K}}))\\\\&= (I_{E_j}\\ot( I_{E_{i}} \\ot P_{\\mathcal{K}}) \\widetilde{V}^{(i)*})X_{i,j}(I_{E_i} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}}(I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_j})} \\ot P_{\\mathcal{K}})),\n\\end{align*} where the previous equality follows from Lemma \\ref{2.2}. Hence we proved (2).\n\n\n(3) In the similar way we can prove (3) as  we proved (2). \n\\end{proof}\n\nFor the proof of the main theorem first we show that $ \\mathcal{K}$ reduces $\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)},$ for each $ i \\in I_{k}.$ For this it is enough to show that \\begin{equation}\\label{2.5}\n(I_{E_{i}}\\ot P_{\\mathcal{K}})(\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})=(\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(I_{E_{i}}\\ot P_{\\mathcal{K}}), \\:\\:\\:  i\\in I_{k}.\n\\end{equation} \nConsider \\begin{align*}\n(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})&=\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})-(I_{E_{i}}\\ot  P_{\\mathcal{K}^{\\perp}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})-(I_{E_{i}}\\ot  P_{\\mathcal{K}^{\\perp}})\\widetilde{V}^{(i) *}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}),\n\\end{align*} as $(I_{E_{i}}\\ot  P_{\\mathcal{K}^{\\perp}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}= (I_{E_{i}}\\ot  P_{\\mathcal{K}^{\\perp}})\\widetilde{V}^{(i) *}.$  After taking adjoint both sides, we obtain \\begin{equation*}\\label{2.4}\n(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})=(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)},\n\\end{equation*}thus $ \\mathcal{K}$ reduces $\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)},$ for each $ i \\in I_{k}.$\n\\begin{theorem}\\label{Beurling}\nConsider a pure DCI-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$   of  $\\mathbb{E}$ on $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a   QS of $\\mathcal{H}_{V}$. Then $\\mathcal{K}$ is a  BQS of $\\mathcal{H}_{V}$ if and only if \\begin{equation*}\n\t(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))=0.\n\t\\end{equation*}\n\\end{theorem}\n\\begin{proof}\n\tFirst assume  $\\mathcal{K}$ to be a BQS of $\\mathcal{H}_{V}.$  Therefore by using Theorem \\ref{MT5} there exist a Hilbert space $\\mathcal{H}_W,$ a pure DCI-representation  $(\\pi, W^{(1)}, \\dots, W^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_W$ and an isometric multi-analytic operator $M_\\Theta:{\\mathcal{H}_W} \\to \\mathcal{H}_{V}$ such that\n\t$$\\mathcal{K}=\\mathcal{H}_{V}\\ominus M_{\\Theta}\\mathcal{H}_W,$$\n\twhere $\\mathcal{W}_{\\mathcal{H}_W}$ is the generating wandering subspace for $(\\pi, W^{(1)}, \\dots, W^{(k)}).$  \n\tThen $P_{\\mathcal{K}^{\\perp}}=M_{\\Theta}M_{\\Theta}^{*}.$ Using Equations (\\ref{positive}) and (\\ref{2.5}) we get the following equality \\begin{equation}\\label{2.6}\n\t(I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}=(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})= (\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(I_{E_{i}}\\ot P_{\\mathcal{K}}),\n\t\\end{equation} where $ i \\in I_{k}.$ Note that  \\begin{align*}&\n(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))\\\\&=(I_{E_{j}}\\ot  \\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(I_{E_{j}} \\ot I_{E_{i}}\\ot P_{\\mathcal{K}})(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\& =(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{j,i} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})  \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{j,i} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}})),\n\\end{align*} here the last inequality follows by (\\ref{2.5}).\nSince ${\\Theta}$ is an inner operator, $M_{\\Theta}$ is an isometric multi-analytic, i.e., $M_{\\Theta}^{*}M_{\\Theta}=I_{\\mathcal{H}_{W}} $ and \\begin{align}\\label{multi}\nM_{\\Theta}W^{(i)}(\\xi_i)h=V^{(i)}(\\xi_i)M_{\\Theta}h \\hspace{0.7cm}\\mbox{and} \\hspace{0.7cm}M_{\\Theta} \\pi(a)h= \\sigma(a)M_{\\Theta}h,\n\\end{align}\nwhere $\\xi_i \\in E_i, \\: h \\in \\mathcal{H}_{W},$ $ a \\in \\mathcal{B}$ and $i \\in I_{k}.$  Equation (\\ref{multi}) provides that \\begin{equation}\\label{2.7}\n(I_{E_{j}} \\ot M_{\\Theta}^{*})\\widetilde{V}^{(j)*}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot M_{\\Theta})=\\widetilde{W}^{(j)*}\\widetilde{W}^{(i)}.\n\\end{equation}\nNow using  $P_{\\mathcal{K}^{\\perp}}=M_{\\Theta}M_{\\Theta}^{*}$, Equations (\\ref{2.6}) and (\\ref{2.7}), we obtain  \\begin{align*}&\n(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{j,i} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} M_{\\Theta}M_{\\Theta}^{*})\\widetilde{V}^{(j)*}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot M_{\\Theta}M_{\\Theta}^{*}\\widetilde{V}^{(j)}( I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} M_{\\Theta})\\widetilde{W}^{(j)*}\\widetilde{W}^{(i)}(I_{E_{i}}\\ot M_{\\Theta}^{*}\\widetilde{V}^{(j)} (I_{E_{j}}\\ot P_{\\mathcal{K})})\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} M_{\\Theta})(I_{E_{j}} \\ot \\widetilde{W}^{(i)})(t_{j,i}\\ot I_{\\mathcal{H}_{W}})(I_{E_{i}}\\ot\\widetilde{W}^{(j)*})(I_{E_{i}}\\ot M_{\\Theta}^{*}\\widetilde{V}^{(j)} (I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*})(I_{E_{j}}\\ot M_{\\Theta} \\widetilde{W}^{(i)})(t_{j,i}\\ot I_{\\mathcal{H}_{W}})(I_{E_{i}}\\ot\\widetilde{W}^{(j)*} M_{\\Theta}^{*})(I_{E_{i}}\\ot\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot I_{E_{i}}\\ot M_{\\Theta})(t_{j,i}\\ot I_{\\mathcal{H}_{W}})(I_{E_{i}}\\ot(I_{E_{j}}\\ot M_{\\Theta}^{*})(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(t_{i,j}\\ot I_{\\mathcal{H}_{W}})(I_{E_{i}}\\ot I_{E_{j}}\\ot M_{\\Theta} M_{\\Theta}^{*})(I_{E_{i}}\\ot I_{E_{j}}\\ot P_{\\mathcal{K}})=0.\n\\end{align*}\n\n\n\n\n\nFor the converse part, assume \\begin{equation*}\n(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{ K}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))=0.\n\\end{equation*} We have to show that  $\\mathcal{K}$ is a  BQS of $\\mathcal{H}_{V}$. For this we need to show $X_{i,j}=0$ (using Lemma \\ref{Lemma 2.1}). Define $\\mathcal{L}:=\\displaystyle\\bigvee_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K})$. First we observe that $\\mathcal{L}$ is a reducing subspace of $\\mathcal{H}_{V}$. Indeed, one can easily see that $\\mathcal{L}$ is a IS for $(\\sigma, V^{(1)}, \\dots, V^{(k)})$. Now for reducing, only we have to show that $\\widetilde{V}^{(i)*}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K}) \\subseteq  E_{i} \\ot \\mathcal{L}.$ Consider \\begin{align*}\n\\widetilde{V}^{(i)*}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K})&=\\widetilde{V}^{(i)*}\\widetilde{V}^{(i)}_{n_{i}}(I_{E_{i}^{\\ot n_i}}\\ot \\widetilde{V}_{\\mathbf{\\hat n}_{i}})(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K})\\\\&=(I_{E_{i}} \\ot \\widetilde{V}^{(i)}_{n_{i}})(t_{i,n_{i}}\\ot I_{\\mathcal{H}_{V}})(I_{E_{i}^{\\ot n_i}}\\ot \\widetilde{V}^{(i)*})(I_{E_{i}^{\\ot n_i}}\\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{i}})(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K}) \\\\&\\subseteq (I_{E_{i}} \\ot \\widetilde{V}^{(i)}_{n_{i}})(E_{i} \\ot E_{i}^{\\ot n_{i}}\\ot \\mathcal{K})\\subseteq  E_{i} \\ot \\mathcal{L},\n \\end{align*}  where $\\mathbf{e}_{i}$ is multi-indices with $ 1$ in the $i^{th}$ place and zero elsewhere. Hence $\\mathcal{L}$ reduces $(\\sigma, V^{(1)}, \\dots, V^{(k)}).$  Similarly $\\mathcal{L}^{\\perp}$ is also reducing subspace for $(\\sigma, V^{(1)}, \\dots, V^{(k)}).$ \n \n \n Therefore $(\\sigma, V^{(1)}, \\dots, V^{(k)})|_{\\mathcal{L}}$ and $(\\sigma, V^{(1)}, \\dots, V^{(k)})|_{\\mathcal{L}^{\\perp}}$ are  pure DCI-representations  for $i\\in I_{k}.$ Then using \\cite[Corollary 3.5]{HV21} there exist wandering subspaces $\\mathcal{W}$ of $\\mathcal{L}$ and  $\\mathcal{W}_{0}$ of $\\mathcal{L}^{\\perp}$ such that \\begin{equation*}\n\\mathcal{L}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{W})\\:\\:\\text{and}\\:\\: \\mathcal{L}^{\\perp}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{W}_{0}).\n\\end{equation*} Note that \\begin{align*}\n\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{W}_{\\mathcal{H}})=\\mathcal{H}_{V}=\\mathcal{L}\\oplus\\mathcal{L}^{\\perp}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot (\\mathcal{W} \\oplus \\mathcal{W}_{0})).\n\\end{align*} Thus by uniqueness of the generating wandering subspace, we get $\\mathcal{W} \\oplus \\mathcal{W}_{0}=\\mathcal{W}_{\\mathcal{ H}}.$ Since $\\mathcal{K}\\subseteq\\mathcal{L},$ $\\mathcal{L}^{\\perp}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_{0}}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{W}_{0})\\subseteq \\mathcal{K}^{\\perp}.$ Hence, for each $ h \\in \\mathcal{L}^{\\perp}$ \\begin{equation*}\n(I_{E_{i}} \\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}h=(I_{E_{i}} \\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}h=0,\n\\end{equation*} as $ \\mathcal{L}^{\\perp}$ reduces $(\\sigma, V^{(1)}, \\dots, V^{(k)}).$  This proves that \\begin{equation*}\nX_{i,j}|_{E_{i} \\ot {\\mathcal{L}^{\\perp}}}=0.\n\\end{equation*} \n\nNow we only need to show $X_{i,j}|_{E_{i} \\ot {\\mathcal{L}}}=0,$ which is equivalent to show $X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=0$\n for $ \\mathbf{n}\\in \\mathbb{N}^k_{0}.$ Suppose $\\mathbf{n}= \\mathbf0$, then $X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=X_{i,j}(I_{E_{i}} \\ot P_{\\mathcal{K}})=0$ as $X_{i,j}=(I_{E_{j}} \\ot P_{ \\mathcal{K}^{\\perp}})(I_{E_{j}}\\ot \\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*})(I_{E_{i}}\\ot P_{\\mathcal{K}^\\perp}),$ therefore we have to prove $X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=0 $ only for  $ \\mathbf{n}\\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\}.$ Furthermore for each $ h\\in \\mathcal{K}$, $ h_{0} \\in \\mathcal{L}^{\\perp},$ $\\xi_{i}\\in E_{i} $ and $\\eta_{\\mathbf{n}}\\in \\mathbb{E}(\\mathbf{n}), $ we have \\begin{align*}&\n \\langle X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h), \\xi_{j} \\ot h_{0} \\rangle\\\\&=\\langle (I_{E_{j}} \\ot P_{ \\mathcal{K}^{\\perp}} \\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*} P_{\\mathcal{K}^\\perp} \\widetilde{V}_{\\mathbf{n}})(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h),\\xi_{j} \\ot h_{0} \\rangle\\\\&=\\langle (t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*})(I_{E_{i}}\\ot P_{\\mathcal{K}^\\perp})(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h),(I_{E_{j}}\\ot \\widetilde{V}^{(i)*}) \\xi_{j} \\ot h_{0}\\rangle=0.\n \\end{align*}  Thus $X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(E_{i} \\ot \\mathbb{E}(\\mathbf{n}) \\ot \\mathcal{K}) \\perp E_{j} \\ot \\mathcal{L}^{\\perp}.$ Therefore it suffices to show that \\begin{equation*}\nX_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(E_{i} \\ot \\mathbb{E}(\\mathbf{n}) \\ot \\mathcal{K}) \\perp E_{j} \\ot \\mathcal{L},   \n \\end{equation*} which is equivalent to prove that  \\begin{equation*}\n X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(E_{i} \\ot \\mathbb{E}(\\mathbf{n}) \\ot \\mathcal{K}) \\perp E_{j} \\ot \\widetilde{V}_{\\mathbf{m}}(\\mathbb{E}(\\mathbf{m})\\ot \\mathcal{K}),\\:\\:\\mathbf{m},\\mathbf{n}\\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\},\n \\end{equation*}  as $\\text{ran}  X_{i,j}\\subseteq E_{j} \\ot \\mathcal{ K}^{\\perp},$  the above equation is obvious  for $ \\mathbf{m}= \\mathbf0.$ For each $ h\\in \\mathcal{K}$, $\\xi_{i}\\in E_{i}$  and $\\eta_{\\mathbf{n}}\\in \\mathbb{E}(\\mathbf{n}), $ \\begin{align*}&\n \\langle X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h),  I_{E_{j}} \\ot \\widetilde{V}_{\\mathbf{m}}(I_{\\mathbb{E}({\\mathbf{m}})}\\ot P_{\\mathcal{K}})(\\xi_{j}\\ot \\eta_{\\mathbf{m}}\\ot h)\\rangle\\\\&=\\langle (I_{E_{j}} \\ot (I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{m}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h),  (\\xi_{j}\\ot \\eta_{\\mathbf{m}}\\ot h)\\rangle\n \\end{align*} and hence we only need to show that \\begin{equation}\\label{main}\n (I_{E_{j}} \\ot (I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{m}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=0.\n \\end{equation} We start by showing the following equation \\begin{equation}\\label{main2}\n (I_{E_{j}}\\ot\\widetilde{V}^{(i)*})X_{i,j}(I_{E_{i}}\\ot \\widetilde{V}^{(j)})=0.\n \\end{equation}By using hypothesis and Equation (\\ref{positive}) we have \n \\begin{align*}&\n0= (I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))\\\\&=(I_{E_{j}}\\ot  (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot  (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot  \\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)})\\\\&=(I_{E_{j}}\\ot  \\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)})\\\\&= (I_{E_{j}}\\ot\\widetilde{V}^{(i)*})X_{i,j}(I_{E_{i}}\\ot \\widetilde{V}^{(j)}).\n \\end{align*} Now for  $ \\mathbf{m}, \\mathbf{n}\\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\}$ with $m_{i},n_{j}\\neq0$ and using $\\widetilde{V}_{\\mathbf{m}}=\\widetilde{V}^{(i)}(I_{E_{i}} \\ot \\widetilde{V}_{{m_{i}\\mathbf{e}_{i}-\\mathbf{e}_{i}}})(I_{\\mathbb{E}({m_{i}\\mathbf{e}_i})}\\ot \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}})$ and  $\\widetilde{V}_{\\mathbf{n}}=\\widetilde{V}^{(j)}(I_{E_{j}} \\ot \\widetilde{V}_{{n_{j}\\mathbf{e}_{j}-\\mathbf{e}_{j}}})(I_{\\mathbb{E}({n_{j}\\mathbf{e}_{j}})}\\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}})$ in the left hand side of the Equation (\\ref{main}), we obtain \\begin{align*}&\n(I_{E_{j}} \\ot ( I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{m}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}} \\ot (I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}})(I_{E({m_{i}\\mathbf{e}_i})}\\ot \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}} ^{*})(I_{E_{i}} \\ot \\widetilde{V}_{{m_{i}\\mathbf{e}_{i}-\\mathbf{e}_{i}}}^{*})\\widetilde{V}^{(i)*})X_{i,j}\\\\&\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:(I_{E_{i}} \\ot\\widetilde{V}^{(j)}(I_{E_{j}} \\ot \\widetilde{V}_{{n_{j}\\mathbf{e}_{j}-\\mathbf{e}_{j}}})(I_{\\mathbb{E}({n_{j}\\mathbf{e}_{j}})}\\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}})(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=0,\n \\end{align*} here the last equality follows by the Equation (\\ref{main2}). Now we will discuss the remaining cases : $ \\mathbf{m}, \\mathbf{n}\\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\}$ where \n \\begin{enumerate}\n \t\\item $m_{i}=n_{j}=0$, \n \t\\item $m_{i}\\neq 0$ and $n_{j}=0,$ \n \t\\item $m_{i}= 0$, $n_{i}\\neq 0$. \n \\end{enumerate} \n   In first case when we substitute these values in left hand side of the Equation (\\ref{main}), we get \\begin{align*}&(I_{E_{j}} \\ot I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}})(I_{E_{j}} \\ot \\widetilde{V}_{\\mathbf{m}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))\\\\&=\n   (I_{E_{j}} \\ot I_{\\mathbb{E}({\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})(I_{E_{j}} \\ot \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}}(I_{\\mathbb{E}({\\mathbf{\\hat{n}}_{j}})}\\ot P_{\\mathcal{K}}))=0,\n   \\end{align*} by using first part of Lemma \\ref{lemma 2.4}.\n     \n    For the (2), substitute $\\widetilde{V}_{\\mathbf{m}}=\\widetilde{V}^{(i)}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{m}-\\mathbf{e}_{i}})$,  $m_{i}\\neq 0$ and $n_{j}=0$ in left hand side of Equation (\\ref{main}), we get \\begin{align*}&\n   (I_{E_{j}} \\ot I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}} (I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{m}-\\mathbf{e}_{i}}^{*})\\widetilde{V}^{(i)*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{i}}(I_{\\mathbb{E}({\\mathbf{\\hat{n}}_{i}})}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}} \\ot I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{m}-\\mathbf{e}_{i}}^{*}) (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{i}}(I_{\\mathbb{E}({\\mathbf{\\hat{n}}_{i}})}\\ot P_{\\mathcal{K}}))=0,\n    \\end{align*} by using second part of Lemma \\ref{lemma 2.4}. Using the same technique, which is used in (2) and third part of Lemma \\ref{lemma 2.4}, we can easily prove the Equation (\\ref{main})for $m_{i}= 0$ and $n_{i}\\neq 0.$ This completes the proof. Hence $\\mathcal{K}$ is a  BQS of $\\mathcal{H}_{V}.$\n\\end{proof}\n\n\\section{Geometrical approach to isometric dilation}\\label{3}\n\n\t \n\\subsection{Dilation theory for single pure contractive representation} Muhly and Solel in \\cite{MS98} gave the isometric dilation of  completely contractive covariant representation. In this subsection, we construct the isometric dilation of pure completely contractive covariant representation using the geometrical approach based on \\cite{AC14}. First we recall some basic definitions that will be needed.\n\n\nThe following definition of induced representation is a generalization of the multiplication operator $M_z\\otimes I_{\\mathcal{H}}$ on the vector valued Hardy spaces $H^2(\\mathbb D)\\otimes \\mathcal{H}$:\n\\begin{definition}\n\t\\begin{enumerate}\n\t\t\\item For $\\xi \\in E$, we define the {\\it creation operator} $T_{\\xi}$ on $\\mathcal{F}(E)$ by $$T_{\\xi}(\\eta):=\\xi \\otimes \\eta, \\:\\: \\eta \\in E^{\\otimes n}, n \\in \\mathbb N_0.$$\n\t\t\\item  Let $\\pi $ denote a representation of a $C^*$-algebra $\\mathcal{B}$ on a Hilbert space $\\mathcal{H}$. Let $E$ be a $C^*$-correspondence over $\\mathcal M.$ The isometric covariant representation $(\\rho, S)$ of $E$ on the Hilbert space $\\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$ defined as\n\t\t\\begin{align*}\n\t\t\\rho(a):&=\\phi_{\\infty}(a) \\otimes I_{\\mathcal{H}} \\:\\:, a \\in \\mathcal{B}&\\\\\n\t\tS(\\xi):&=T_{\\xi}\\otimes I_{\\mathcal{H}} ,\\:\\: \\xi \\in E.\n\t\t\\end{align*}\n\t\tis called an {\\it induced representation} (cf. \\cite{R74}) (induced  by $\\pi$).\n\t\\end{enumerate}\n\\end{definition}\n\n\n\\begin{definition}\n\tLet $(\\sigma, V)$ and  be the  completely contractive covariant representation of $E$ on  the Hilbert space $\\mathcal{H}_{V}$ and $(\\mu, T)$ be isometric representation of $E$ on $\\mathcal{H}_{T},.$ We say that  $(\\mu,T)$ is an isometric dilation (cf. \\cite {MS98}) of $(\\sigma, V)$ if there exists an isometry $\\Pi: \\mathcal{H}_{V} \\rightarrow \\mathcal{H}_{T}$ such that   $\\Pi \\sigma(a)=\\mu(a)\\Pi, $ and $(I_E \\otimes \\Pi)\\wt{V}^*=\\wt{T}^*\\Pi, $ for $a \\in \\mathcal{B}.$\n\\end{definition}\n\n Consider the following natural question: Which type of covariant representation $(\\sigma, V)$ of $E$ on $\\mathcal{H}_{V}$ is  unitarily equivalent to the restriction of the induced representation  $(\\rho, S)$ of $E$ on  $\\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$? Since $(\\rho, S)$ is completely contractive, $(\\sigma, V)$ is completely contractive representation. Therefore we need to find only an isometry $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$ such that \\begin{align} \\label{sid}\n \\Pi_V \\sigma(a)=\\rho(a)\\Pi_V\\:\\: \\mbox{and} \\:\\: (I_{E} \\ot \\Pi_V)\\widetilde{V}^{*}=\\widetilde{S}^{*}\\Pi_V, \\:\\:\\:a \\in \\mathcal{B} . \n \\end{align}  If such an isometry exists, then the required QS of  $\\mathcal{F}(E)\\ot \\mathcal{H}$ will be $\\Pi_{V}(\\mathcal{ H}_{V}).$\n \n Suppose such an isometry $\\Pi_V$ exists. Observe that  $\\Pi_{V}=\\bigoplus_{n=0}^{\\infty}\\Pi_V^{n},$ where $\\Pi_V^n:\\mathcal{H}_{V}\\to E^{\\ot n}\\ot \\mathcal{H}$ defined by $\\Pi_V^nh=P_{{E^{\\ot n}}\\ot \\mathcal{H}}\\Pi_Vh, h \\in \\mathcal{H}_{V}, n \\in \\mathbb{N}_0.$ Now using (\\ref{sid}), we obtain $\\Pi_V^{n+1}=(I_{E } \\ot \\Pi_V^n)\\widetilde{V}^{*},$ for every $n \\in \\mathbb{N}_{0}.$ Indeed \\begin{align*}\n (I_{E} \\ot \\Pi_V^n)\\widetilde{V}^{*}&=(I_{E}\\ot P_{{E^{\\ot n}}\\ot \\mathcal{H}})(I_{E}\\ot \\Pi_V)\\widetilde{V}^{*} =(I_{E}\\ot P_{{E^{\\ot n}}\\ot \\mathcal{H}})\\widetilde{S}^{*}\\Pi_V\\\\&= P_{{E^{\\ot n+1}}\\ot \\mathcal{H}}\\Pi_V=\\Pi_V^{n+1}.\n \\end{align*} Therefore, we can easily see by induction $ \\Pi_V^{n}=(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}.$ Thus, it's sufficient  to find $ \\Pi_V^{0}$ only,  since the remaining $\\Pi_V^{n}$ will be determined by $ (I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}.$ Since $ \\Pi_V$ is an isometry, we have \\begin{align}\\label{3.1}\n \\|h\\|^{2}=\\| \\Pi_Vh\\|^{2}&=\\sum_{n =0}^{\\infty}\\| \\Pi_V^{n}h\\|^{2}=\\sum_{n =0}^{\\infty}\\|(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}h\\|^{2}\n \\end{align} for all $ h \\in \\mathcal{H}_{V}.$ \n Note that \\begin{align*}\n (I_{E} \\ot \\Pi_{V})\\widetilde{V}^{*}= (I_{E} \\ot\\bigoplus_{n=0}^{\\infty}(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*})\\widetilde{V}^*=\\bigoplus_{n=1}^{\\infty}(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*},\n \\end{align*}\n and since $(I_{E} \\ot \\Pi_{V})$ is an isometry,  we get \\begin{align}\\label{3.2}\n \\|\\widetilde{V}^{*}h\\|^{2}=\\|(I_{E} \\ot \\Pi_{V})\\widetilde{V}^{*}h\\|^{2}=\\sum_{n=1}^{\\infty}\\|(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}h\\|^{2}.\n \\end{align}\n \n \n Finally using Equations (\\ref{3.1}) and (\\ref{3.2}), we have $ \\|\\Pi_V^{0}h\\|^{2}=\\|h\\|^{2}-\\|\\widetilde{V}^{*}h\\|^{2}, \\:\\:h \\in \\mathcal{H}_{V}.$  Hence the simplest choice for   $\\Pi_V^{0}=\\Delta_{*}({V})$ and $\\mathcal{H}=\\mathcal{D}_{ *,V}$ where  $\\Delta_{*}(V) =(I_{\\mathcal{H}_{V}}-\\widetilde{V}\\widetilde{V}^{*})^{\\frac{1}{2}}$ and $\\mathcal{D}_{ *,V}=\\overline{\\Delta_{*}(V) \\mathcal{H}_{V}}.$  Then for $ h \\in \\mathcal{H}_{V}$,\\begin{align*}\n \\| \\Pi_Vh\\|^{2}=\\sum_{n=0}^{\\infty}\\|(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}h\\|^{2}=\\sum_{n=0}^{\\infty}(\\|\\widetilde{V}_{n}h\\|^{2}-\\|\\widetilde{V}_{n+1}h\\|^{2})=\\|h\\|^{2}-\\lim_{n \\rightarrow \\infty}\\|\\widetilde{V}_{n}h\\|^{2}.\n \\end{align*} Hence $\\Pi_{V}$ is  an isometry if and only if $(\\sigma,V)$ is pure. Since  $\\mathcal{K}:=\\Pi_V \\mathcal{H}_{V}$ is IS under $\\widetilde{S}^{*}$, $\\mathcal{K}$ is a QS of $\\mathcal{F}(E)\\ot \\mathcal{H}.$  Therefore $(\\sigma, V)$ is isomorphic to  $(\\rho, S)|_\\mathcal{K},$ equivalently \\begin{equation}\\label{imp1}\n (\\sigma, V)\\equiv(\\rho|_{\\mathcal{K}}, S|_{\\mathcal{K}}).\n \\end{equation} In other words, we can say a pure, completely contractive covariant representation $(\\sigma, V)$ of $E$ on $\\mathcal{H}_{V}$ dilates to an induced representation $(\\rho, S).$ \n  The above discussion, which is based on \\cite[Theorem 3]{AC14} and \\cite[Theorem 2.22.]{V11}, gives a   geometric proof of the following Sz.Nagy-Foias type dilation.\n  \\begin{theorem}\\label{dilation}\n\tLet $(\\sigma, V)$  be a pure completely contractive covariant representation  of $E$ on $\\mathcal{H}_{V}.$ Then $(\\sigma, V)$ is an unitarily equivalent to  the  restriction of the induced representation $(\\rho, S)$ to $\\Pi_V \\mathcal{H}_{V},$ a QS of $\\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$, where $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$, is an isometry.\n\\end{theorem} \nNote that from Theorem \\ref{MT5} it follows that $\\mathcal{K}=\\Pi_V \\mathcal{H}_{V}$ is a BQS of $\\mathcal{F}(E)\\ot \\mathcal{H}$, therefore there exist a Hilbert space $\\mathcal{H}_{T}$, a pure DCI-representations  $(\\mu, T)$ of ${E}$ on $\\mathcal{H}_T$ and an isometric multi-analytic operator $M_\\Theta:{{\\mathcal{H}_{T}}} \\to \\mathcal{F}(E)\\ot \\mathcal{H}$ such that\n$$\\mathcal{K}=\\mathcal{F}(E)\\ot \\mathcal{H}\\ominus M_{\\Theta}\\mathcal{H}_{T}.$$ \n\n\n\\subsection{Regular Dilation}\n  The regular  isometric dilation of completely contractive representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of the product systems $\\be$ over $\\mathbb{N}^k_0$  on the Hilbert space $\\mathcal{H}_{V}$ is discussed in \\cite{S008, S009}. In this subsection, we discuss geometrical construction of dilation in the pure case based on \\cite{AC14}. \n\n\n\nWe start with the  definition  of induced representation  which is a generalization of the multiplication operators $M_{z_i} \\otimes I_{\\mathcal{H}}$ on the vector valued Hardy space $H^2_{\\mathcal{H}}(\\mathbb{D}^k).$  \n\\begin{definition}\n\tLet $\\pi$ be a representation of $\\mathcal{B}$ on  $\\mathcal{H}.$  Define an isometric  covariant representation $(\\rho, S^{(1)},S^{(2)}, \\dots , S^{(k)})$ of $\\mathbb{E}$ on $\\mathcal{F}(\\mathbb{E})\\otimes_{\\pi} \\mathcal{H}$ (cf. \\cite{SZ08}) by  $$\\rho(a)=\\phi_{\\infty}(a) \\otimes I_{\\mathcal{H}}, \\:\\:\\mbox{and}\\:\\: S^{(i)}(\\xi_i)=V_{{\\xi}_i} \\otimes I_{\\mathcal{H}},\\: i\\in I_{k}, \\xi_i \\in E_i, a \\in \\mathcal{B},$$ where $V_{\\xi_i}$  denotes the creation operator on $\\mathcal{F}(\\mathbb{E})$ determined by $\\xi_i,$ that is, $V_{\\xi_i}(\\eta)=\\xi_i \\otimes \\eta,$ where $\\eta \\in \\mathcal{F}(\\mathbb{E})$ and  $\\phi_{\\infty}$  denotes the canonical left action of $\\mathcal{B}$ on $\\mathcal{F}(\\mathbb{E}).$ It is easy to see that the above isometric covariant representation $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ is doubly commuting  and it is called {\\rm induced representation} of $\\mathbb{E}$ induced by $\\pi.$ Any covariant representation of $\\mathbb{E}$ which is isomorphic to $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ is also called an {\\rm induced representation}.\n\\end{definition}\n\\begin{definition}\n\tLet $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$  be the  completely contractive covariant representation of $\\mathbb{E}$ on  the Hilbert space $\\mathcal{H}_{V}$ and  $(\\mu, T^{(1)}, T^{(2)},\\dots T^{(k)})$  isometric representation of $\\mathbb{E}$ on  the Hilbert space $\\mathcal{H}_{T}$ . We say that    $(\\mu, T^{(1)}, T^{(2)},\\dots T^{(k)})$ is an isometric dilation (cf. \\cite{S009}) of $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$  if there exists an isometry $\\Pi: \\mathcal{H}_{V} \\rightarrow \\mathcal{H}_{T}$ such that $\\Pi \\sigma(a)=\\mu(a)\\Pi, $ and $(I_E \\otimes \\Pi)\\wt{V}^{(i)*}=\\wt{T}^{(i)*}\\Pi, $ for $a \\in \\mathcal{B}$ and $i \\in I_{k}.$ \n\\end{definition}\nWe want to find out a covariant representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of $\\mathbb{E}$ on the Hilbert space $\\mathcal{H}_{V} $ such that it is unitarily equivalent to the restriction of the induced representation $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ to QS of $\\mathcal{F}(\\mathbb{E})\\otimes\\mathcal{H}.$  Since   $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ is completely contractive, the representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of $\\mathbb{E}$ on the Hilbert space $\\mathcal{H}_{V}$ is  completely contractive.   \nTherefore, consider a completely contractive covariant representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_{V}.$ We need to prove that there exists a isometry $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(\\mathbb{E})\\otimes_{\\pi}\\mathcal{H}$ such that \n\\begin{align}\\label{dilate}\n&\\Pi_V\\sigma(a)=\\rho(a)\\Pi_V \\:\\:\\:\\:\\mbox{and}\\:\\:\\:\\: (I_{E_{i}} \\ot \\Pi_V)\\widetilde{V}^{(i)*}=\\widetilde{S}^{(i)*}\\Pi_V,  \n\\end{align} where  $a \\in \\mathcal{B}$ and $ i \\in I_{k}.$ If such an isometry exists, then the needed QS of  $\\mathcal{F}(\\mathbb{E})\\ot \\mathcal{H}$ will equal its range, $\\Pi_{V}(\\mathcal{ H}_{V}).$\nSuppose that such an isometry $\\Pi_{V}$ exists. Note that $\\Pi_{V}=\\bigoplus_{\\mathbf{n}\\in\\mathbb{N}_{0}^{k}}\\Pi_V^{\\mathbf{n}},$ where $\\Pi_V^\\mathbf{n}:\\mathcal{H}_{V}\\to \\mathbb{E}({\\mathbf{n}})\\ot \\mathcal{H}$ defined by $\\Pi_V^\\mathbf{n}h=P_{{\\mathbb{E}(\\mathbf{n})}\\ot \\mathcal{H}}\\Pi_Vh, h \\in \\mathcal{H}_{V}.$ Now using Equation(\\ref{dilate}), we obtain \\begin{equation}\\label{regular}\n\\Pi_V^{\\mathbf{n}+\\mathbf{e}_{i}}=(I_{E_{i} } \\ot \\Pi_V^{\\mathbf{n}})\\widetilde{V}^{(i)* }\n\\end{equation} for every $\\mathbf{n} \\in \\mathbb{N}_{0}^{k}$ and $ i \\in I_{k}.$  Indeed \\begin{align*}\n(I_{E_{i} } \\ot \\Pi_V^{\\mathbf{n}})\\widetilde{V}^{(i)* }&=(I_{E _{i}} \\ot P_{{\\mathbb{E}(\\mathbf{n})}\\ot \\mathcal{H}})(I_{E _{i}} \\ot \\Pi_V)\\widetilde{V}^{(i)*} =(I_{E _{i}} \\ot P_{{\\mathbb{E}(\\mathbf{n})}\\ot \\mathcal{H}})\\widetilde{S}^{(i)*}\\Pi_V\\\\&=(I_{E_{i}}\\ot  P_{{\\mathbb{E}(\\mathbf{n})}\\ot \\mathcal{H}})\\Pi_V= P_{{\\mathbb{E}(\\mathbf{n}+\\mathbf{e}_{i})}\\ot \\mathcal{H}}\\Pi_V=\\Pi_V^{\\mathbf{n}+\\mathbf{e}_{i}},\n\\end{align*} where $\\mathbf{e}_{i}:=(0,0,\\dots,1,0,\\dots 0),$ i.e., $1$ at $i^{th}$ place and $0$ otherwise. More generally, for $ {m_{i}}\\in \\mathbb{N}_{0} $ we have \\begin{align*} \n\\Pi_V^{\\mathbf{n}+{m_{i}}\\mathbf{e}_{i}}=(I_{E_{i}^{\\ot m_i} } \\ot \\Pi_V^{\\mathbf{n}})\\widetilde{V}^{(i)* }_{m_i}.\n\\end{align*}\nIt follows  from the  above equality, we obtain \n\\begin{align*}\n\\Pi_V^{\\mathbf{n}}=(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}.\n\\end{align*}\nIndeed, for $\\mathbf{n}=({n_1,n_2,\\dots,n_{k}}) \\in \\mathbb{N}_{0}^{k}$ we have\n\\begin{align*}\n\\Pi_V^{\\mathbf{n}}&=(I_{E_{1}^{\\ot {{n_{1}}}} } \\ot \\Pi_V^{\\mathbf{n}-{n_{1}}\\mathbf{e}_1})\\widetilde{V}^{(1)* }_{{n_{1}}}\\\\&=(I_{E_{1}^{\\ot{{n_1}}} } \\ot (I_{E_{2}^{\\ot {{n_{2}}}}}\\ot\\Pi_V^{\\mathbf{n}-({n_1}\\mathbf{e}_1+{n_{2}}\\mathbf{e}_2)} )\\widetilde{V}^{(2)* }_{{n_{2}}})\\widetilde{V}^{(1)* }_{{n_{1}}}\\\\&=(I_{E_{1}^{\\ot {{n_{1}}}} } \\ot (I_{E_{2}^{\\ot {{n_{2}}}}}\\ot(I_{E_{3}^{\\ot {{n_{3}}}}}\\ot \\Pi_V^{\\mathbf{n}-({n_{1}}\\mathbf{e}_1+{n_{2}}\\mathbf{e}_2+{n_{3}}\\mathbf{e}_3)}) \\widetilde{V}^{(3)* }_{{n_{3}}} )\\widetilde{V}^{(2)* }_{{n_{2}}})\\widetilde{V}^{(1)* }_{{n_{1}}}=\\cdots\\\\&=(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}.\n\\end{align*}Thus we get $ \\Pi_V^{\\mathbf{n}}=(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}.$ Therefore,  it is  sufficient to find  $ \\Pi_V^{\\mathbf{0}}$ only, since the remaining $\\Pi_V^{\\mathbf{n}}$ will be determined by $ (I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}.$ Since $\\Pi_V$ is an isometry, we have \\begin{align}\n\\|h\\|^{2}=\\| \\Pi_Vh\\|^{2}&=\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\| \\Pi_V^{\\mathbf{n}}h\\|^{2}=\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\|(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}h\\|^{2},\n\\end{align} for all $ h \\in \\mathcal{H}_{V}.$ \n\nSuppose $u=\\{u_1,u_2,\\dots, u_r\\}\\subseteq I_{k}$ such that $u_1< u_2< \\dots < u_r.$ Define $\\mathbf{e}(u):=\\mathbf{e}_{u_1}+\\mathbf{e}_{u_2}+\\dots+\\mathbf{e}_{u_r}.$ \nNote that  \\begin{align*}\n(I_{\\mathbb{E}(\\mathbf{e}(u))} \\ot \\Pi_{V})\\widetilde{V}^{*}_{\\mathbf{e}(u)}= (I_{\\mathbb{E}(\\mathbf{e}(u))}  \\ot\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*})\\widetilde{V}^{*}_{\\mathbf{e}(u)}.\n\\end{align*}\n Since $(I_{\\mathbb{E}(\\mathbf{e}(u))} \\ot \\Pi_{V})$ is an isometry, we get \\begin{align}\\label{F}\n\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*}h\\|^{2}=\\|(I_{\\mathbb{E}(\\mathbf{e}(u))} \\ot \\Pi_{V})\\widetilde{V}_{\\mathbf{e}(u)}^{*}h\\|^{2}=\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\|(I_{\\mathbb{E}(\\mathbf{e}(u))} \\ot(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*})\\widetilde{V}^{*}_{\\mathbf{e}(u)}h\\|^{2},\\:\\:\\:h \\in \\mathcal{H}_{V}.\n\\end{align} Denote by $|u|$ the cardinality of $u.$ It follows from Equation ({\\ref{F}}) that \\begin{align*}\n\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*}h\\|^{2}&=\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\left(\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\| (I_{\\mathbb{E}(\\mathbf{e}(u))}   \\ot(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*})\\widetilde{V}^{*}_{\\mathbf{e}(u)}h\\|^{2}\\right)\\\\&=\\sum_{\\mathbf{m} \\in \\mathbb{N}^k_0}\\left({\\sum_{{\\mathbf{n}\\leq\\mathbf{m}},{\\max\\{m_j-n_j\\}\\leq 1}}}(-1)^{|\\mathbf{m}|-|\\mathbf{n}|} \\| (I_{\\mathbb{E}(\\mathbf{m})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{m}}^{*}h\\|^{2}\\right)\\\\&=\\|\\Pi_V^{\\mathbf{0}}h\\|^{2}.\n\\end{align*}\nHence \\begin{align*}\n\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*}h\\|^{2}\\geq 0,\\:\\:\\:\\:h\\in \\mathcal{H}_{V},\n\\end{align*}  equivalently \\begin{align*}\n\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\widetilde{V}_{\\mathbf{e}(u)}\\widetilde{V}_{\\mathbf{e}(u)}^{*}\\geq 0.\n\\end{align*}\nNow, define  $\\Delta_{*}(V)=(\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\widetilde{V}_{\\mathbf{e}(u)}\\widetilde{V}_{\\mathbf{e}(u)}^{*})^{\\frac{1}{2}}$ and $\\mathcal{D}_{*,V}=\\overline{Im\\Delta_*(V)}.$ Therefore we can choose   for $\\mathcal{H}$ is $\\mathcal{D}_{*,V}$ and  $\\Pi_V^{\\mathbf{0}}=\\Delta_{*}(V),$ thus  $ \\Pi_V^{\\mathbf{n}}=(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Delta_{*}(V))\\widetilde{V}_{\\mathbf{n}}^{*}.$ Then for each $ h \\in \\mathcal{H}_{V},$ we obtain \\begin{align*}\n\\| \\Pi_Vh\\|^{2}&=\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\| \\Pi_V^{\\mathbf{n}}h\\|^{2}=\\lim_{p\\rightarrow \\infty}\\sum_{{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}},{\\max n_j\\leq p-1}}\\|(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}h\\|^{2}\\\\&=\\lim_{p \\rightarrow \\infty}\\sum_{{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}},{\\max n_j\\leq p-1}}\\sum_{{u}\\subseteq{I_{k}}}(-1)^{|u|}\\|(I_{\\mathbb{E}(\\mathbf{n})}\\ot\\widetilde{V}^{*}_{\\mathbf{e}(u)} )\\widetilde{V}_{\\mathbf{n}}^{*}h\\|^{2}\\\\&=\\lim_{p \\rightarrow \\infty}\\sum_{{\\mathbf{m} \\in \\mathbb{N}_{0}^{k}},{\\max m_j\\leq p}}\\left( \\sum_{\\mathbf{n}\\leq \\mathbf{m},\\max \\{{m_j-n_j}\\}\\leq 1,{\\max n_j\\leq p-1}}(-1)^{|\\mathbf{m}|-|\\mathbf{n}|}\\|\\widetilde{V}_{\\mathbf{m}}^{*}h\\|^{2}\\right).\n\\end{align*} Observe that \\begin{align*}\n\\sum_{\\mathbf{n}\\leq \\mathbf{m},\\max \\{{m_j-n_j}\\}\\leq 1,{\\max n_j\\leq p-1}}(-1)^{|\\mathbf{m}|-|\\mathbf{n}|}=0,\n\\end{align*} whenever $0< m_i< p,$ for all $i \\in I_{k}.$ If $m_i=0$ or $m_i=p,$ then \\begin{align*}\n\\sum_{\\mathbf{n}\\leq \\mathbf{m},\\max \\{{m_j-n_j}\\}\\leq 1,{\\max n_j\\leq p-1}}(-1)^{|\\mathbf{m}|-|\\mathbf{n}|}=(-1)^{|{\\supp\\mathbf{m}}|}.\n\\end{align*} Therefore\n\\begin{align*}\n\\| \\Pi_Vh\\|^{2}=\\lim_{p \\rightarrow \\infty}\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*p}h\\|^{2}=\\|h\\|^{2}+\\lim_{p \\rightarrow \\infty}\\sum_{\\emptyset\\neq {u\\subseteq I_{k}}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*p}h\\|^{2},\\:\\:\\:\\: h \\in \\mathcal{ H}_{V}.\n\\end{align*}\nHence $\\Pi_{V}$ will be an isometry if $SOT-\\lim_{p \\rightarrow \\infty}\\widetilde{V}^{({j}) *} _p=0$ for $j\\in I_{k}.$ \n\n\nNow we define  the following Brehmer-Solel condition.\n\\begin{definition}\n\tLet $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ be  a pure, completely contractive representation of $\\mathbb{E}$ on $\\mathcal{H}_{V}.$ Then  we say it is satisfy Brehmer-Solel condition (cf. \\cite{S008}) if it satisfies the following condition\\begin{align*}\n\t\\sum_{u \\subseteq I_k}(-1)^{|u|}\\widetilde{V}_{\\mathbf{e}(u)}\\widetilde{V}_{\\mathbf{e}(u)}^{*}\\geq 0.\n\t\\end{align*}\n\\end{definition} \nThe above observations yields the following theorem which is generalization of \\cite[Theorem 4]{AC14}.\n\\begin{theorem}\\label{6}\n\tLet $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$  be a  completely contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{H}_{V},$ satisfying Brehmer-Solel condition. Then there exists  $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(\\mathbb{E})\\otimes_{\\pi}\\mathcal{D}_{ *,V},$ satisfy \\begin{align*}\n\t&\\Pi_V\\sigma(a)=\\rho(a)\\Pi_V \\:\\:\\:\\:\\mbox{and}\\:\\:\\:\\: (I_{E_{i}} \\ot \\Pi_V)\\widetilde{V}^{(i)*}=\\widetilde{S}^{(i)*}\\Pi_V,  \n\t\\end{align*} where  $a \\in \\mathcal{B}$ and $ i \\in I_{k}$  such that \\begin{align*}\n\t\\| \\Pi_Vh\\|^{2}=\\lim_{p \\rightarrow \\infty}\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*p}h\\|^{2}=\\|h\\|^{2}+\\lim_{p \\rightarrow \\infty}\\sum_{\\emptyset\\neq {u\\subseteq I_{k}}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*p}h\\|^{2},\\:\\:\\:\\: h \\in \\mathcal{ H}_{V}.\n\t\\end{align*}\n\\end{theorem}\nIn particular if we take a pure completely contractive covariant representation  then we obtain the following corollary which is  a generalization of \\cite[Theorem 6]{AC14}.\n\\begin{corollary}\\label{dilationN}\n\tLet $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$  be a pure, completely contractive covariant representation of $\\mathbb{E}$ on the Hilbert space $\\mathcal{H}_{V},$ satisfying Brehmer-Solel condition. Then it is an unitarily equivalent to  the  restriction of the induced representation $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ to QS $ \\Pi_V \\mathcal{H}_{V}$  of $\\mathcal{F}(\\mathbb{E})\\otimes\\mathcal{D}_{ *,V}$ where $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(\\mathbb{E})\\otimes_{\\pi}\\mathcal{D}_{ *,V}$ is an isometry. Therefore it dilates the induced representation $(\\rho, S^{(1)}, S^{(2)}, \\dots, S^{(k)}).$ \n\\end{corollary} \n\nIn particular there exists an isometry $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(\\mathbb{E})\\otimes_{\\pi}\\mathcal{H}$ satisfying Equation (\\ref{dilate}).\n Then $\\mathcal{ K}=\\Pi_{V}\\mathcal{ H}_{V}$ is a QS of $\\mathcal{F}(\\mathbb{E})\\ot \\mathcal{H},$ therefore  $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ is isomorphic to  $(P_{\\mathcal{ K}}\\rho, P_{\\mathcal{ K}}S^{(1)}, P_{\\mathcal{ K}}S^{(2)}, \\dots, P_{\\mathcal{ K}}S^{(k)})|_{\\mathcal{K}}.$\n \n \n \n \n \n   Note that if $k=1,$ then $\\mathcal{K}$ will be a BQS under $(\\rho, S)$ by Theorem \\ref{hs}, i.e. there exist a Hilbert space $\\mathcal{ H'}$ and a representation $\\Psi$ and an multi-analytic operator $M_{\\Theta}: \\mathcal{F}(E)\\ot \\mathcal{ H'} \\to \\mathcal{F}(E)\\ot \\mathcal{ H}  $ such that $\\mathcal{ K}_{\\Theta}= (M_{\\Theta}(\\mathcal{F}(E)\\ot_{\\Psi} \\mathcal{ H'}))^{\\perp}$\n \\begin{equation*}\n (\\sigma,V)\\equiv (P_{(M_{\\Theta}(\\mathcal{F}(E)\\ot_{\\Psi} \\mathcal{ H'}))^{\\perp}} \\rho, P_{(M_{\\Theta}(\\mathcal{F}(E)\\ot_{\\Psi} \\mathcal{ H'}))^{\\perp}}S)|_{(M_{\\Theta}(\\mathcal{F}(E)\\ot_{\\Psi} \\mathcal{ H'}))^{\\perp}}.\n \\end{equation*}  The space $\\mathcal{ K}_{\\Theta}$ is said to be {\\it model space} corresponding to $(\\sigma,V).$ Therefore each pure completely contractive covariant representation is unitarily equivalent to compression of the induced representation to the model space. \n\n\n\nLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a  completely contractive covariant representation of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}.$ \nNow in the $k >1$ case, consider $M_{\\Theta}:  \\mathcal{H}_{V} \\to\\mathcal{F}(\\mathbb{E})\\ot \\mathcal{ H}   $ be  an isometric multi-analytic operator. Let  $\\mathcal{ K}_{\\Theta}= (M_{\\Theta}(\\mathcal{H}_{V}))^{\\perp}$ and  $\\mathcal{S}_{\\Theta}= M_{\\Theta}(\\mathcal{H}_{V})$  which are BQS and BS, respectively. For each $i \\in I_k$, define  $$\\pi_{\\Theta}(a)=P_{\\mathcal{K}_{\\Theta}}\\rho(a)|_{{\\mathcal{K}_{\\Theta}}}  \\:\\mbox{and} \\:\\:W_{\\Theta}^{(i)}(\\xi_{i})=P_{\\mathcal{K}_{\\Theta}}S^{(i)}(\\xi_{i})|_{{\\mathcal{K}_{\\Theta}}},$$\nfor $\\xi_{i} \\in E_{i}$ and $a \\in \\mathcal{B}.$ Then  $W_{\\Theta }=(\\pi_{\\Theta}, W_{\\Theta}^{(1)}, W_{\\Theta}^{(2)}, \\dots, W_{\\Theta}^{(k)} )$ is a completely contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{K}_{\\Theta}.$ Indeed, \\begin{align*}\n(\\phi^{(i)}(a)\\ot I_{\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}})\\widetilde{W}_{\\Theta}^{ (i)*}&=(\\phi^{(i)}(a)\\ot I_{\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}})(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{S}^{ (i)*}P_{\\mathcal{K}_{\\Theta}}\\\\&=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})(\\phi^{(i)}(a)\\ot I_{\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}})\\widetilde{S}^{ (i)*}P_{\\mathcal{K}_{\\Theta}}\\\\&=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{S}^{ (i)*}\\rho(a)P_{\\mathcal{K}_{\\Theta}}\\\\&=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{S}^{ (i)*}P_{\\mathcal{K}_{\\Theta}}\\rho(a)P_{\\mathcal{K}_{\\Theta}}=\\widetilde{W}_{ \\Theta}^{(i)*}\\pi_{\\Theta}(a),\n\\end{align*} where the last equality follows by the fact that $\\mathcal{K}_{\\Theta}$ is a QS of ${\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}}.$ Now using the definition of QS we have \\begin{align*}\n\\sum_{{u}\\subseteq{I_{k}}}(-1)^{|u|}\\widetilde{W}_{\\mathbf{e}(u)} \\widetilde{W}_{\\mathbf{e}(u)}^{*}=\\sum_{{u}\\subseteq{I_{k}}}(-1)^{|u|}P_{\\mathcal{K}_\\Theta}\\widetilde{S}_{\\mathbf{e}(u)}\\widetilde{S}_{\\mathbf{e}(u) }^{*}P_{\\mathcal{K}_\\Theta},\n\\end{align*} Therefore $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ satisfies Brehmer-Solel condition if and only if $W_{\\Theta}$  satisfies the  Brehmer-Solel condition whenever they are equivalent. One can now ask which type of covariant representation are unitarily  equivalent to $W_{\\Theta }$ on BQS, the following theorem gives answer to  this question.\n\\begin{theorem}\nLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a  completely contractive covariant representation of  $\\mathbb{E}$ on a Hilbert space $\\mathcal{H}_{V}.$ Then the following are equivalent:\\begin{enumerate}\n\t\\item $(\\sigma, V^{(1)}, \\dots, V^{(k)})\\equiv (\\pi_{\\Theta}, W_{\\Theta}^{(1)}, W_{\\Theta}^{(2)}, \\dots, W_{\\Theta}^{(k)} )$ for some BQS $\\mathcal{K}_{\\Theta}.$\n\t\\item  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ is pure and satisfies the Brehmer-Solel condition and \\begin{equation*}\n\t(I_{E_{j}}\\ot ( I_{E_{i}\\ot P_{\\mathcal{K}_{\\Theta}}}- \\wV^{(i) *}\\wV^{(i)}))(t_{i,j} \\ot I_{\\mathcal{K}_{\\Theta}})(I_{E_{i}}\\ot  (I_{E_{j}\\ot P_{\\mathcal{K}_{\\Theta}}}- \\wV^{(j) *}\\wV^{(j)}))=0,\n\t\\end{equation*} for distinct $i\\neq j.$ \n\t\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\tThe proof  follows from Theorem \\ref{Beurling}  and Corollary \\ref{dilationN}.\n\\end{proof}\n\\section{Factorizations and invariant subspaces}\\label{4}\n \nThis section establishes a relationship between invariant subspaces of compression of a pure isometric covariant representation with factorization of the corresponding inner function.\n \n Let $(\\sigma, V)$ and $(\\mu, T)$ be pure isometric  representations of $E$ on Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively. Let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator.\n Let us denote  $\\mathcal{K}_{\\Theta}=\\mathcal{H}_{T}\\ominus M_{\\Theta} \\mathcal{H}_{V},$ and $ M_{\\Theta} \\mathcal{H}_{V}$  be BQS and Beurling subspace of $\\mathcal{H}_{T}$ corresponding to $\\Theta,$ respectively.\n Define  $$\\pi_{\\Theta}'(a):=P_{\\mathcal{K}_{\\Theta }}\\mu(a)|_{{\\mathcal{K}_{\\Theta}}}  \\:\\mbox{and} \\:\\:W'_{\\Theta}(\\xi)=P_{\\mathcal{K}_{\\Theta}}T(\\xi)|_{{\\mathcal{K}_{\\Theta}}},$$\n for $\\xi \\in E$ and $a \\in \\mathcal{B}.$ Then $(\\pi'_{\\Theta}, W^{\\prime}_{\\Theta})$ is a covariant representation of $E$ on $ \\mathcal{K}_{\\Theta }\\subseteq\\mathcal{H}_{T}.$\n \nThe following theorem  presents a characterization of a invariant subspaces of compression of a pure isometric covariant representation which is a generalization of  Sz.Nagy-Foias \\cite[Chapter VII, Theorem 2.3]{NFS}  and  Bercovici \\cite[Chapter 5, Proposition 1.21]{B88}.\n\\begin{theorem}\\label{Ber}\n\tLet $(\\sigma, V)$ and $(\\mu, T)$ be pure isometric  representations of $E$ on Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively.  Suppose ${\\mathcal{W}_{\\mathcal{H}_{V}}}$  and ${\\mathcal{W}_{\\mathcal{H}_{T}}}$ are the generating wandering subspace for $(\\sigma, V)$ and $(\\mu,T),$ respectively. Let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator. Then  the covariant representation $(\\pi'_{\\Theta}, W'_{\\Theta})$ has an IS   if and only if there exist a Hilbert space $\\mathcal{H}_{R},$ a pure isometric  representation $(\\nu, R)$ of $E$ on $\\mathcal{H}_{R}$, isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that  \\begin{equation*}\n\t{\\Theta}={\\Phi}{\\Psi}.\n\t\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\n\t\n\t\n(\t$\\implies$)\n \tLet $\\mathcal{S}$ be a nontrivial $(\\pi'_{\\Theta}, W'_{\\Theta})$-IS of $\\mathcal{K}_{\\Theta} \\subseteq \\mathcal{H}_{T},$ then $\\mathcal{S}\\oplus M_\\Theta \\mathcal{H}_{V}$ is a $(\\mu, T)$-IS of $\\mathcal{H}_{T}.$ Since $(\\mu, T)$ is pure,  using Theorem \\ref{MT5}, there exist a Hilbert space $\\mathcal{H}_{R}$, a pure isometric  representation $(\\nu, R) $ of $E$ on $\\mathcal{H}_{R}$ and an isometric multi-analytic operator $M_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$ such that \\begin{equation*}\\label{quotient}\n\\mathcal{S}\\oplus M_\\Theta \\mathcal{H}_{V}=M_{\\Phi}{\\mathcal{H}_{R}},\n\t\\end{equation*} where $ \\mathcal{W}_{\\mathcal{H}_{R}}$ is the  generating wandering subspace for $(\\nu,R).$ Note that $M_{\\Theta}\\mathcal{H}_{V}\\subseteq M_{\\Phi}\\mathcal{H}_{R}$ and $\\mathcal{S}=M_{\\Phi}\\mathcal{H}_{R}\\ominus M_{\\Theta}\\mathcal{H}_{V}.$   Then by Douglas's range  inclusion theorem \\cite{D96}, there exists a contraction $Z: \\mathcal{H}_{V} \\to \\mathcal{H}_{R}$ such that \\begin{equation*}\n\tM_{\\Theta}=M_{\\Phi}Z.\n\t\\end{equation*}  Using multi-analytic property of $M_{\\Phi}$ and $M_{\\Theta}$, we obtain \\begin{equation*}\nM_{\\Phi}Z\\widetilde{V}=M_{\\Theta}\\widetilde{V}=\\wT(I_{E}\\ot M_{\\Theta})=\\wT(I_{E}\\ot M_{\\Phi} Z)=M_{\\Phi}\\widetilde{R}(I_{E} \\ot Z).\n\t\\end{equation*} Thus $\\widetilde{R}(I_{E} \\ot Z)=Z\\widetilde{V}$ and  similarly $\\nu(a)Z=Z\\sigma(a),$ for $ a \\in \\mathcal{B},$ i.e.,   $Z$ is multi-analytic, therefore $Z=M_{\\Psi}$, for some inner operator $\\Psi: \\mathcal{W}_{\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}.$ Indeed, ${\\Theta}={\\Phi}{\\Psi}$ and  $M_{\\Phi}$ and $M_{\\Theta}$ are isometries, we deduce\n\t\\begin{equation*}\n\t\\|h\\|=\\|M_{\\Theta}h\\|=\\|M_{\\Phi}M_{\\Psi}h\\|=\\|M_{\\Psi}h\\|,\\:\\:\\: h\\in \\mathcal{ H}_{V}.\n\t\\end{equation*} \n\t\n\t\n($\\impliedby$)\t\nOn the other hand, suppose there exist a Hilbert space $\\mathcal{H}_{R},$ a pure isometric representation $(\\nu, R)$ of $E$ on $\\mathcal{H}_{R}$ and  isometric multi-analytic operators  $M_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ and $M\\Phi: {\\mathcal{H}_R} \\rightarrow  {\\mathcal{H}_T}$ such that   \\begin{equation*}\n\t{\\Theta}={\\Phi}{\\Psi}.\n\t\\end{equation*}  Since  ${\\Theta}={\\Phi}{\\Psi},$ we have $M_{\\Theta}(\\mathcal{H}_V) \\subseteq M_\\Phi(\\mathcal{H}_R).$  Define $\\mathcal{S}:=M_{\\Phi}(\\mathcal{H}_R)\\ominus M_{\\Theta}(\\mathcal{H}_V).$ Clearly $\\mathcal{S}$ is a closed subspace of $\\mathcal{K}_{\\Theta}.$ Also note that $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}=\\mathcal{K}_{\\Phi}.$ \n Since $\\widetilde{W'^{*}_{\\Theta}}=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Theta}},$ and $\\mathcal{K}_{\\Phi}\\subseteq\\mathcal{K}_{\\Theta}$, we have $\\mathcal{K}_{\\Phi}$ is $ \\widetilde{W'^{*}_{\\Theta}} $-IS. Indeed let $ h \\in \\mathcal{K}_{\\Phi}$ \\begin{align*}\n\t \\widetilde{W'^{*}_{\\Theta}}h=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Theta}}h=\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Theta}}h=\\widetilde{T}^{*}h=\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Phi}}h= \\widetilde{W'^{*}_{\\Phi}}h\\in \\mathcal{K}_{\\Phi},\n\t \t\\end{align*} where  $\\widetilde{W'^{*}_{\\Phi}}=(I_{E}\\ot P_{\\mathcal{K}_{\\Phi}})\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Phi}},$ Also in similar way  $\\mathcal{S}^{\\perp}$ is $ \\widetilde{W'^{*}_{\\Theta}} $-IS. That is  $\\mathcal{S}\n\t \t$ is an $(\\pi'_{\\Theta},W'_{\\Theta})$-IS.\n\\end{proof} We are now ready for the multi-variable analog of Theorem \\ref{Ber}. The following theorem is a generalization of \\cite[Theorem 4.4]{BDDS}.\t  \n\\begin{theorem}\\label{44}\nLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and $(\\mu, T^{(1)}, \\dots, T^{(k)})$  be pure DCI-representations of  $\\mathbb{E}$ on Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively. Let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator. Then the following are equivalent:\n\\begin{enumerate}\n\t\\item There exist a Hilbert space $\\mathcal{H}_{R},$ a pure DCI-representation $(\\nu, R^{(1)},\\dots,R^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R}$ and isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that \\begin{equation*}\n\t{\\Theta}={\\Phi}{\\Psi}.\n\t\\end{equation*}\n\t\\item  There exists  $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS $\\mathcal{S}\\subseteq \\mathcal{K}_{\\Theta}$  such that $\\mathcal{S}\\oplus\\mathcal{S}_{\\Theta}$ is a Beurling subspace of $\\mathcal{H}_{T}.$\n\t\\item There exists  $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS $\\mathcal{S}\\subseteq \\mathcal{K}_{\\Theta}$ of $\\mathcal{H}_{T}$ such that \\begin{equation*}\n\t(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}_{\\Theta} \\ominus \\mathcal{S}})- \\widetilde{U}^{(i) *}\\widetilde{U}^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{T}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}_{\\Theta} \\ominus \\mathcal{S}})- \\widetilde{U}^{(j) *}\\widetilde{U}^{(j)}))=0,\n\t\\end{equation*} where  $\\widetilde{U}^{(i)}=P_{\\mathcal{K}_{\\Theta} \\ominus \\mathcal{S}}\\widetilde{W'_{\\Theta}}^{(i)}{_{|_{{E_{i}}\\ot {\\mathcal{K}_{\\Theta} \\ominus \\mathcal{S}}}}},$ for each $i\\in I_{k}.$\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n Let us denote  $\\mathcal{K}_{\\Theta}=\\mathcal{H}_{T}\\ominus M_{\\Theta} \\mathcal{H}_{V},$ and $ M_{\\Theta} \\mathcal{H}_{V}$  be BQS and Beurling subspace of $\\mathcal{H}_{T}$ corresponding to $\\Theta,$ respectively.\n\tFor each $i \\in I_{k}$, define  $$\\pi'_{\\Theta}(a):=P_{\\mathcal{K}_{\\Theta}}\\mu(a)|_{{\\mathcal{K}_{\\Theta}}}  \\:\\mbox{and} \\:\\:W_{\\Theta}^{'(i)}(\\xi_{i}):=P_{\\mathcal{K}_{\\Theta}}T^{(i)}(\\xi_{i})|_{{\\mathcal{K}_{\\Theta}}},$$\n\tfor $\\xi_{i} \\in E_{i}$ and $a \\in \\mathcal{B}.$ Then $(\\pi'_{\\Theta}, W_{\\Theta}^{'(1)}, W_{\\Theta}^{'(2)}, \\dots, W_{\\Theta}^{'(k)} )$ is a covariant representation of $\\mathbb{E}$ on $\\mathcal{K}_{\\Theta} \\subseteq \\mathcal{H}_{T}.$\n\t\n\t\n\t\n\t\n\t\n\t\t$(1) \\implies (2)$ : Suppose there exist  a Hilbert space $\\mathcal{H}_{R},$ a pure DCI-representation $(\\nu, R^{(1)},\\dots,R^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R}$ and  isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that $\t{\\Theta}={\\Phi}{\\Psi}.$ Therefore $M_{\\Theta}\\mathcal{H}_V \\subseteq M_\\Phi\\mathcal{H}_R,$ we can define $\\mathcal{S}:=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V.$ Clearly $\\mathcal{S}$ is a closed subspace of $\\mathcal{K}_{\\Theta}.$ Also note that $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}=\\mathcal{K}_{\\Phi}.$\tSince $\\widetilde{W'_{\\Theta}}^{(i)*}=(I_{E_{i}} \\ot P_{\\mathcal{ K}_{\\Theta}})\\widetilde{T}^{(i)*}P_{\\mathcal{K}_{\\Theta}},$ and $\\mathcal{K}_{\\Phi}\\subseteq\\mathcal{K}_{\\Theta}$, we have $\\mathcal{K}_{\\Phi}$ is $\\pi'_{\\Theta}$ and $ \\widetilde{W'_{\\Theta}}^{(i)*} $-IS for each $i\\in I_{k}.$ Indeed, let $ h \\in \\mathcal{K}_{\\Phi}$ \\begin{align*}\n\t\\widetilde{W'_{\\Theta}}^{(i)*}h=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{T}^{(i)*}P_{\\mathcal{K}_{\\Theta}}h=\\widetilde{T}^{(i)*}P_{\\mathcal{K}_{\\Theta}}h=\\widetilde{T}^{(i)*}h=\\widetilde{T}^{(i)*}P_{\\mathcal{K}_{\\Phi}}h= \\widetilde{W'_{\\Phi}}^{(i)*}h\\in \\mathcal{K}_{\\Phi}.\n\t\\end{align*} Also in similar way  $\\mathcal{S}^{\\perp}$ is $ \\widetilde{W'_{\\Theta}}^{(i)} $-IS. That is  $\\mathcal{S}\n\t$ is an $(\\pi_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS.\n\t\n\t$(2) \\implies (1)$: Let $\\mathcal{S}$ be a $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS of $\\mathcal{H}_{T}$ such that $\\mathcal{S}\\oplus M_\\Theta \\mathcal{H}_{V}$  is a DCS of $\\mathcal{H}_{T}$. Therefore by using Theorem \\ref{MT5}, there exist a pure DCI-representation $(\\nu, R^{(1)},\\dots,R^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R}$ and an isometric multi-analytic operator $M_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$ such that \\begin{equation}\\label{quotient 11}\n\t\\mathcal{S}\\oplus M_\\Theta \\mathcal{H}_{V}=M_{\\Phi}{\\mathcal{H}_{R}}.\n\t\\end{equation} Thus $M_{\\Theta}\\mathcal{H}_{V}\\subseteq M_{\\Phi}\\mathcal{H}_{R},$ and hence by Douglas's range  inclusion theorem, there exists a contraction $Z: \\mathcal{H}_{V} \\to \\mathcal{H}_{R}$ such that \\begin{equation*}\n\tM_{\\Theta}=M_{\\Phi}Z.\n\t\\end{equation*} Using multi-analytic property of $M_{\\Phi}$ and $M_{\\Theta}$, we get \\begin{equation*}\n\tM_{\\Phi}\\widetilde{R}^{(i)}(I_{E} \\ot Z)=\\wT^{(i)}(I_{E}\\ot M_{\\Phi} Z)=\\wT^{(i)}(I_{E}\\ot M_{\\Theta})=M_{\\Theta}\\widetilde{V}^{(i)}=M_{\\Phi}Z\\widetilde{V}^{(i)},\\:\\:\\: \\text{for each}\\:\\: i \\in I_{k}.\n\t\\end{equation*} Thus $\\widetilde{R}^{(i)}(I_{E} \\ot Z)=Z\\widetilde{V}^{(i)}$, similarly $\\nu(a)Z=Z\\sigma(a),$ for $ a \\in \\mathcal{B}.$ Then there exists an inner operator $\\Psi: \\mathcal{W}_{\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that $Z= {M}_{\\Psi}.$ Indeed, \t${\\Theta}={\\Phi}{\\Psi}$ and using  $M_{\\Phi}$ and $M_{\\Theta}$ are isometries, we deduce\n\t\\begin{equation*}\n\t\\|h\\|=\\|M_{\\Theta}h\\|=\\|M_{\\Phi}M_{\\Psi}h\\|=\\|M_{\\Psi}h\\|,\\:\\:\\: h\\in \\mathcal{ H}_{V}.\n\t\\end{equation*}\n\t\n\t\n\t\n\t$(1) \\implies (3)$ : Suppose there exist a pure DCI-representation $(\\nu, R^{(1)},\\dots,R^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R}$ and isometric multi-analytic operators $M_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, $M_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that $\t{\\Theta}={\\Phi}{\\Psi}.$ Define $\\mathcal{S} =M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V$ and then $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}=\\mathcal{K}_{\\Phi}.$  Therefore  $\\mathcal{K}_{\\Phi}$ is a BQS of $\\mathcal{H}_{T}$  using Theorem \\ref{Beurling}, we get  \\begin{equation*}\n\t(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}_{\\Phi}})- \\widetilde{U}^{(i) *}\\widetilde{U}^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{T}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}_{\\Phi}})- \\widetilde{U}^{(j) *}\\widetilde{U}^{(j)}))=0,\n\t\\end{equation*} where  $\\widetilde{U}^{(i)}=P_{\\mathcal{K}_{\\Phi}}\\widetilde{W'_{\\Theta}}^{(i)}{_{|_{{E_{i}}\\ot {\\mathcal{K}_{\\Phi}}}}},$ for each $i\\in I_{k}.$\n\t\n\t\n \n\t\n\t\n\t \n\t\n\t\t$(3) \\implies (2)$: Since $\\widetilde{W'_{\\Theta}}^{(i)}=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{T}^{(i)*}|_{{\\mathcal{K}_{\\Theta}}},$ it follows that $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}$  is a QS of $\\mathcal{H}_{T}$. Therefore by hypothesis and using Theorem \\ref{Beurling}, $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}$  is  a BQS. Now observe that \\begin{align*}\n\t \\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}= \\mathcal{H}_{T}\\cap (M_{\\Theta}(\\mathcal{H}_{V})\\cup \\mathcal{S})^{\\perp}= \\mathcal{H}_{T} \\ominus (M_{\\Theta}(\\mathcal{H}_{V}) \\oplus\\mathcal{S}),\n\t\t\\end{align*} which implies that $\\mathcal{S}\\oplus\\mathcal{S}_{\\Theta}$ is a Beurling submodule of $\\mathcal{H}_{T}.$\n\t\\end{proof}\nIn the above theorem we discussed only non-trivial $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS $\\mathcal{S}\\subseteq \\mathcal{K}_{\\Theta}$ of $\\mathcal{K}_{\\Theta}$. Let us discuss trivial case :\\begin{enumerate}\n \\item First take $\\mathcal{S} =\\{0\\}$. Recall that $\\mathcal{S}=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V$  therefore $M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V=\\{0\\}$, it follows that $M_\\Phi\\mathcal{H}_{R}\\subseteq M_\\Theta\\mathcal{H}_{V}.$ Since \t${\\Theta}={\\Phi}{\\Psi}$, $M_\\Theta\\mathcal{H}_{V}\\subseteq M_\\Phi\\mathcal{H}_{R}.$ Hence \\begin{align*}\n M_\\Phi\\mathcal{H}_{R}=M_\\Theta\\mathcal{H}_{V}= M_{\\Phi}M_{\\Psi}\\mathcal{H}_{V}, \n \\end{align*} we obtain $M_{\\Psi}\\mathcal{H}_{V}=\\mathcal{H}_{R}.$ Thus ${\\Psi}$ is a unitary.\n \\item Now take $\\mathcal{S} =\\mathcal{K}_{\\Theta}$. Therefore \\begin{align*}\n M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V=\\mathcal{K}_{\\Theta}=\\mathcal{H}_{T} \\ominus M_{\\Theta}\\mathcal{H}_{V},\n \\end{align*} it implies that  we obtain $ M_{\\Phi}\\mathcal{H}_R=\\mathcal{H}_{T}.$ Thus ${\\Phi}$ is a unitary.\n\\end{enumerate} Therefore from the above discussion, if $\\mathcal{S}$ is  a non-trivial  $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS of $ \\mathcal{K}_{\\Theta}$ if and only if isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ are not unitary.\nThe following corollary is a generalization of \\cite[Corollary 4.5]{BDDS}.\n\\begin{corollary}\nLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and $(\\mu, T^{(1)}, \\dots, T^{(k)})$  be  pure DCI-representations of  $\\mathbb{E}$ on  Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively.  Let ${M}_\\Theta: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{T}}$ be an isometric multi-analytic operator. Then  isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}},$ coming from Theorem (\\ref{44}), are non-unitary if and only if the following holds: \n\\begin{enumerate}\n\t\\item $\\mathcal{S}$ is a non-trivial $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS of $\\mathcal{K}_{\\Theta}.$ \n\t\\item $\\mathcal{S}$ is not a Beurling subspace of $\\mathcal{H}_{T}.$\n\t\\item  $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}$ does not reduces $(\\mu, T^{(1)}, \\dots, T^{(k)})$.\n\t\\end{enumerate}  \n\\end{corollary}\n\n\n\\begin{proof}\n\tFirst assume isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ are non-unitary.  As by above discussion, if $\\mathcal{S}$ is  a non-trivial  $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS of $ \\mathcal{K}_{\\Theta}$ if and only if  isometric multi-analytic operators ${M}_\\Phi$, ${M}_\\Psi $  are not unitary. This proves (1).\n\t\n\t\n\t\n\t Let if possible $\\mathcal{S}$ is a Beurling subspace of $\\mathcal{H}_{T}$. Therefore by using Theorem \\ref{MT5}, there exist a pure DCI-representation $(\\mu_1, R_1^{(1)},\\dots,R_1^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R_1}$ and an isometric multi-analytic operator $M_{\\Phi_1}: {\\mathcal{H}_{R_1}} \\to {\\mathcal{H}_{T}}$ such that $\\mathcal{S}=M_{\\Phi_1}{\\mathcal{H}_{R_1}}.$ But \\begin{align}\\label{inv}\n\t\\mathcal{S}=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V=M_{\\Phi_1}{\\mathcal{H}_{R_1}},\n\t\\end{align} we have $M_{\\Phi_1}{\\mathcal{H}_{R_1}} \\subseteq M_{\\Phi}\\mathcal{H}_R,$ and hence by Douglas's range  inclusion theorem, there exists a contraction $Z: \\mathcal{H}_{R_1} \\to \\mathcal{H}_{R}$ such that \\begin{equation*}\n\tM_{\\Phi_1}=M_{\\Phi}Z.\n\t\\end{equation*}\n\t\n\tUsing multi-analytic property of $M_{\\Phi}$ and $M_{\\Phi_1}$, we get \\begin{equation*}\nM_{\\Phi}\\widetilde{R}^{(i)}(I_{E} \\ot Z)=\\wT^{(i)}(I_{E}\\ot M_{\\Phi} Z)=\\wT^{(i)}(I_{E}\\ot M_{\\Phi_1})=M_{\\Phi_1}\\widetilde{R}_{1}^{(i)}=M_{\\Phi}Z\\widetilde{R}_{1}^{(i)},\\quad i \\in I_{k}.\n\\end{equation*} Thus $\\widetilde{R}^{(i)}(I_{E} \\ot Z)=Z\\widetilde{R}_{1}^{(i)}$, similarly $\\nu(a)Z=Z\\mu_1(a),$ for $ a \\in \\mathcal{B}.$ Then there exists inner operator $\\Phi_2: \\mathcal{W}_{\\mathcal{H}_{R_1}} \\to \\mathcal{W}_{\\mathcal{H}_{R}}$ such that $Z= {M}_{\\Phi_{2}}.$ Indeed, $\\Phi_1=\\Phi\\Phi_2$ and   $M_{\\Phi}$ and $M_{\\Phi_1}$ are isometries, we deduce\n\t\\begin{equation*}\n\t\\|h\\|=\\|M_{\\Phi_1}h\\|=\\|M_{\\Phi}M_{\\Phi_2}h\\|=\\|M_{\\Phi_2}h\\|,\\:\\:\\: h\\in \\mathcal{ H}_{R_{1}}.\n\t\\end{equation*} By using Equation (\\ref{inv}) \\begin{align*}  \\mathcal{S}=M_{\\Phi_1}{\\mathcal{H}_{R_1}}\n\tM_{\\Phi}(M_{\\Phi_2}{\\mathcal{H}_{R_1}})=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V&=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Phi}M_{\\Psi}\\mathcal{H}_V\\\\&=M_{\\Phi}(\\mathcal{H}_R\\ominus M_{\\Psi}\\mathcal{H}_V),\n\t\\end{align*} we obtain $M_{\\Phi_2}{\\mathcal{H}_{R_1}}=\\mathcal{H}_R\\ominus M_{\\Psi}\\mathcal{H}_V=\\mathcal{K}_{\\Psi}.$ It implies that $\\mathcal{K}_{\\Psi}$ is  $(\\nu, R^{(1)}, \\dots, R^{(k)})$-reducing subspace of $\\mathcal{H}_{R}$ It implies that $\\mathcal{K}_{\\Psi}$ is trivial subspace which means that $\\Psi$ is an unitary. But $\\Psi$ is not unitary operator, thus it is contradiction. Hence  $\\mathcal{S}$ is not a Beurling subspace of $\\mathcal{H}_{T}$. This proves (2). \n\n\n\n\nSuppose  $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}$  reduces $(\\mu, T^{(1)}, \\dots, T^{(k)}).$ Then $(\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S})^{\\perp}= \\mathcal{S}\\oplus \\mathcal{S}_{\\Theta}$ is also reduces $(\\mu, T^{(1)}, \\dots, T^{(k)}).$ On the other hand, By using Equation (\\ref{quotient 11})  $\\mathcal{S}\\oplus \\mathcal{S}_{\\Theta}=M_{\\Phi}\\mathcal{H}_{R},$ it follows that $\\Phi$ is an unitary, which is contradiction as $\\Phi$ is not unitary inner operator.\n\nFor the converse part, suppose $\\mathcal{S}$ is a non-trivial $(\\pi_{\\Theta}, W^{(1)}_{\\Theta},\\dots, W^{(k)}_{\\Theta})$-IS of $\\mathcal{K}_{\\Theta}.$ Since $\\Theta=\\Phi\\Psi$ and $\\Theta$ is non constant, both $\\Phi$ and $\\Psi$ can not be unitaries. It remains to show that $\\Phi$ and $\\Psi$ can not be  isometry operators. Let if possible  $\\Phi\\equiv X_1$ for some non-unitary isometry $X_1$ and $\\Psi$ is not unitary. Then \\begin{align*}\n\\mathcal{S}\\oplus \\mathcal{S}_{\\Theta}=M_{\\Phi}\\mathcal{H}_{R}=X_1\\mathcal{H}_{R}=\\mathcal{H}_{X_1R}\n\\end{align*} and hence $\\mathcal{S}\\oplus \\mathcal{S}_{\\Theta}$ reduces $(\\mu, T^{(1)}, \\dots, T^{(k)}),$ which contradicts to (3). On the other hand, if $\\Psi\\equiv X_2$ for some non-unitary isometry $X_2$ and $\\Phi$ is not unitary. Then\\begin{align*}\n\\mathcal{S}= M_{\\Phi}(\\mathcal{H}_R\\ominus M_{\\Psi}\\mathcal{H}_V)=M_{\\Phi}(\\mathcal{H}_R\\ominus(X_2\\mathcal{H}_V))=M_{\\Phi}\\mathcal{H}_{R-X_{2}V}\n\\end{align*} is a Beurling subspace of $\\mathcal{H}_{T},$ which contradicts to (2). This completes the proof. \n\\end{proof}\n\\subsection*{Acknowledgment}\nAzad Rohilla is supported by a UGC fellowship (File No: 16-6(DEC.2017)\n\/2018(NET\/CSIR)). Shankar Veerabathiran thanks ISI Bangalore for the visiting scientist position. Harsh Trivedi is supported by MATRICS-SERB  \nResearch Grant, File No: MTR\/2021\/000286, by the Science and Engineering Research Board\n(SERB), Department of Science \\& Technology (DST), Government of India. We thank Jaydeb Sarkar for making us aware of reference \\cite{AC14}. We acknowledge the Centre for Mathematical \\& Financial Computing and the DST-FIST grant for the financial support for the computing lab facility under the scheme FIST ( File No: SR\/FST\/MS-I\/2018\/24) at the LNMIIT, Jaipur. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nThe discovery of graphene in 2004 \\cite{Novoselov2014Science} was a significant breakthrough in materials science and since then there has been sustained research interest in graphene and a rush to discover other stable two-dimensional (2D) materials. 2D materials have the thickness of one or a few atomic layers and have markedly different material properties than their bulk counterparts due to the quantum confinement effect. Since graphene, there have been several advances in the field of 2D materials such as the discovery of semiconducting monolayer transition metal dichalcogenides (TMDs) \\cite{2_mak_lee_hone_shan_heinz_2010, 2_wang_kalantar-zadeh_kis_coleman_strano_2012}. The growing number and variety of 2D materials has fuelled interest in the use of 2D materials in the development of novel nanoscale devices.\n\nA recent development is the experimental isolation of a single layer of bulk black phosphorus, also known as phosphorene \\cite{Brent2014CC,Buscema2014NC,Castellanos-Gomez20142DM,Li2014NN,Liu2014AN,Qiao2014NC,Rodin2014PRL,Zhu2014PRL,Reich2014Nature,Churchill2014NN,Koenig2014APL,Fei2014NL,Fei2015PRB,Tran2014PRB,Tran2014PRB2,Guo2014JPCC,Peng2014PRB,Peng2014MRE,Wang2015PRB,Cai2014SR}. Like graphene, phosphorene was first obtained by mechanical exfoliation \\cite{Li2014NN,Liu2014AN}. Liquid exfoliation, which is a scalable process, has also been demonstrated as a possible alternative means of producing phosphorene \\cite{Brent2014CC}. Several experiments have demonstrated that phosphorene is a direct-gap semiconductor and also has a high hole mobility\\cite{Li2014NN,Xia2014NC,Liu2014AN}. These characteristics make phosphorene attractive for use in electronic and optoelectronic devices \\cite{Xia2014NC}. Field effect transistors (FET) based on few-layer phosphorene were shown to have high on\/off ratios \\cite{Li2014NN}. It also has the potential to be used as an anode material in lithium-ion batteries \\cite{Li2015NL}. Another potential application is in thin film excitonic solar cells as phosphorene has a predicted band gap in the visible region \\cite{Dai2014JPCL}. Excitonic solar cells (XSCs) based on some 2D materials, such as MoS$_2$, WSe$_2$, graphene, h-BN, SiC$_2$ and bilayer phosphorene, are potentially seen as the new generation of thin film solar cells\\cite{1_tsai_su_chang_tsai_chen_wu_li_chen_he_2014, 2_pospischil_furchi_mueller_2014,Bernardi2012ACS,Dai2014JPCL,Zhou2013NL,Britnell2013Science}, and they might have higher efficiencies than existing XSCs which typically have less than 10\\% efficiency \\cite{1_green_emery_hishikawa_warta_dunlop_2015}. Till now, despite the limitations of fabrication methodologies for such 2D solar cells, there has been some progress on the fabrication of few-layer heterostructures such as graphene-WS${_2}$\\cite{1_georgiou_jalil_belle_britnell_gorbachev_morozov_kim_gholinia_haigh_makarovsky_2012, 2_tan_avsar_balakrishnan_koon_taychatanapat_ofarrell_watanabe_taniguchi_eda_castro_neto_2014}, graphene-MoS${_2}$\\cite{Britnell2012Science} and phosphorene-MoS${_2}$\\cite{Deng2014ACS}.\n\nIn this paper we study the excited-state properties of monolayer phosphorene, and evaluate the viability of monolayer phosphorene as one building block of of an excitonic solar cell heterostructure. For the other building-block material in the heterostructure, the semiconducting monolayer TMDs, which have been extensively researched, are chosen. The semiconducting TMDs considered here include semiconducting MoS${_2}$, MoSe${_2}$, MoTe${_2}$, WS${_2}$, WSe${_2}$, WTe${_2}$, TiS${_2}$ and ZrS${_2}$. This paper is organized as follows: In \\textbf{Sec. II}, we first introduce the structures of geometrically optimized monolayer phosphorene and computational details. \\textbf{Sec. III} is the main part of results and discussions, which includes three subsections: In \\textbf{SubSec. A}, we show the excited-state properties of monolayer phosphorene, including in quasi-particle band structures and optical spectra. In \\textbf{SubSec. B}, we calculate the excited-state properties of 8 semiconducting TMDs. In \\textbf{SubSec. C}, the band alignment of phosphorene and TMDs is presented. Then the power conversion efficiencies (PCE) of the phosphorene-TMD heterostructures are discussed. In \\textbf{Sec. IV}, we conclude our studies.\n\n\n\n\\section{Structures and COMPUTATIONAL DETAILS}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Fig1.eps}\\\\\n\\caption{(Color online) GW-calculated (solid lines) and HSE06-calculated (dash lines) band structure of monolayer phosphorene. The Fermi level is set on the top of the valance band. The ball and stick model of phosphorene (light color of the upper layer) with primitive cell and the reciprocal lattice with the high symmetry points are shown in inset.}\n\\label{fig1}\n\\end{figure}\n\nPhosphorene has a puckered honeycomb structure as shown in inset of \\textbf{Figure 1}. The underlying lattice is rectangular which leads to anisotropy in the band structure and optical properties. The calculated lattice constants are $a = 4.58$~\\AA~and $b = 3.30$~\\AA~which are consistent with results in literature \\cite{Liu2014AN}. The calculations were performed using the Vienna Ab-initio Simulation Package (VASP) \\cite{16_kresse_furth_1996, 15_kresse_furth_1996, 13_kresse_hafner_1993, 14_kresse_hafner_1994}, and a projector augmented wave (PAW) basis set was used \\cite{1_blochl_1994, 2_kresse_joubert_1999}. Geometry optimisation was done using density function theory (DFT) with the generalised gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional. A $14\\times10\\times1$ $k$-point grid was used for phosphorene while an $11\\times11\\times1$ $k$-point grid was used for TMDs. The geometry was relaxed until the force acting on the atoms was less than 0.01 eV\/atom. To ensure that the interlayer interaction is negligible, the out of plane lattice parameter, which is perpendicular to the plane of the material, was set as at least 15 \\AA.\nBand gaps and band structures were calculated using the GW method. The band gaps\/structures were also calculated through the screened exchange hybrid density functional by Heyd, Scuseria, and Ernzerhof (HSE06) for reference. A high number of empty conductions bands is necessary for the convergence of the absolute band positions\\cite{Liang2013APL}. A total of 1024 bands were used for phosphorene and 1536 bands were used for TMDs. Single shot GW calculations, $G_{0}W_{0}$, were performed on phosphorene to obtain the ground state band energies. For TMDs, one eigenvalue update is performed to obtain the expected direct band gap of trigonal prismatic TMDs \\cite{Shi2013PRB,Cheiwchanchamnangij2012PRB, Ramasubramaniam2012PRB, Qiu2013PRL}. The band structure was then interpolated from Wannier functions rather than evaluated directly at discrete $k$-points. This was done using the WANNIER90 library \\cite{12_mostofi_yates_lee_souza_vanderbilt_marzari_2008} and the VASP2WANNIER interface. The optical gap was calculated by solving the Bethe-Salpeter equation (BSE). The GW method has more computational demands and the band gaps converge with a much smaller number of bands\\cite{Liang2013APL}. Thus to streamline optical calculations, the $G_{0}W_{0}$ calculation was run with 128 bands for phosphorene and 192 bands for TMDs before solving the BSE. This produced the frequency dependent dielectric tensor that was used to calculate the absorption spectrum. By solving the BSE, electron-hole interactions such as excitons are accounted for in the dielectric tensor. For phosphorene the $x$- and $y$-components of the dielectric tensor were treated separately because of the anisotropy but for TMDs, the average value of the $x$- and $y$-components was used due to crystal symmetry.\n\n\n\n\\section{RESULTS AND DISCUSSIONS}\n\\subsection{Excited-state properties of phosphorene}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{Fig2.eps}\\\\\n\\caption{(Color online) GW-calculated (solid lines) and HSE06-calculated (dashed lines) absorption spectrum of phosphorene for \\emph{armchair} polarised light. The energy values of first absorption peak, observed in the experiment\\cite{Li2014NN}, is presented by a black solid circle for reference.}\n\\label{fig2}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Fig3.eps}\\\\\n\\caption{(Color online) (a) The band structures of MoS${_2}$ with a ball and stick model of the trigonal prismatic geometry. (b) The band structures of TiS${_2}$ with a ball and stick model of the octahedral geometry. The reciprocal lattice with high symmetry points is shown in the inset. The Fermi level is set on the top of the valance band.}\n\\label{fig3}\n\\end{figure}\nThe calculated band structure of phosphorene is shown in \\textbf{Figure 1}. The GW calculation predicts a band gap value of 2.15 eV and the gap is approximately located at the $\\Gamma$ point, which is much higher than the DFT-calculated band gap of 0.91 eV \\cite{Qiao2014NC}. The conduction band minimum (CBM) and the valence band maximum (VBM) are not exactly aligned in the GW-calculated band structure, but they are sufficiently close to be considered as a direct band gap \\cite{Tran2014PRB2}. A similar computational approach by Tran \\textit{et al}. predicted a band gap of 2.0 eV and a comparable profile of the band structure \\cite{Tran2014PRB2}. The CBM position is around -4.25 eV with respect the the vacuum level. Besides the GW method, the hybrid density functional HSE06 was used to calculate the band structures and band gap for reference in \\textbf{Fig. 1}. As can be seen, the HSE06-calculated band gap is 0.6 eV lower than the GW-calculated gap, which is in agreement with previous calculations \\cite{Qiao2014NC}.\n\n\nThe optical gap of phosphorene is calculated using the GW-BSE approach and is determined to be 1.6 eV. This is the first optical peak of the absorption spectrum for light polarised along the armchair direction as seen in \\textbf{Fig. 2}. This optical gap is slightly larger than the experimental photoluminescence measurement value of 1.45 eV of phosphorene \\cite{Liu2014AN}, but smaller than the HSE06-calculated value of 1.8 eV. The optical gap of 1.6 eV is much lower than the electronic band gap of 2.15 eV which suggests that significant excitonic effects are present in phosphorene. The exciton binding energy of 0.55 eV in monolayer phosphorene is quite huge. Both the self-energy corrected large electronic gap and small optical gap of phosphorene indicate significant many-electron effect in phosphorene.\n\n\n\\subsection{Excited-state properties of TMDs}\n\n\nUsing a similar approach to the above, the geometry, band positions and band structures of 8 semiconducting TMDs are calculated. A summary of the lattice constants, absolute positions of the valence band maximum (VBM) and conduction band minimum for the different materials is shown in \\textbf{Table 1}. The TMDs are categorised into either trigonal prismatic or octahedral TMDs, both of which have a hexagonal lattice but different coordination of the atoms within the unit cell. The two different structures are shown in the inset of \\textbf{Fig. 3}. For trigonal prismatic TMDs, the lattice constant is largely determined by the size chalcogen atom which increases with atomic number (sulphur to selenium to tellurium). Furthermore, as can be seen, phosphorene has a large exciton binding energy compared to TMDs because of its unique quasi 1D band dispersions.\n\n\n\\begin{table*}\n\\centering\n\\caption{Summary of the lattice constants $a$ and $b$, the band gap $E_g$, valence band maximum $E_{VBM}$, conduction band minimum $E_{CBM}$ and optical gap $E_{opt}$}\\label{tab:res}\n\\begin{tabular}{c c c c c c c c c}\n\\toprule\nMaterial & Lattice & Structure & $a$ (\\AA) & $b$ (\\AA) & $E_g$ (eV) & $E_{VBM}$ (eV) & $E_{CBM}$ (eV) & $E_{opt}$ (eV) \\\\\nPhosphorene & Rectangular & - & 4.58 & 3.30 & 2.15 & -6.40 & -4.25 & 1.6\\\\\nMoS${_2}$  & Hexagonal & Trigonal Prismatic & 3.16 & - & 2.68 & -6.57 & -3.89 & 2.3\\\\\nMoSe${_2}$ & Hexagonal & Trigonal Prismatic & 3.29 & - & 2.39 & -5.97 & -3.58 & 2.2\\\\\nMoTe${_2}$ & Hexagonal & Trigonal Prismatic & 3.52 & - & 1.74 & -5.43 & -3.69 & 1.7\\\\\nWS${_2}$   & Hexagonal & Trigonal Prismatic & 3.16 & - & 2.94 & -6.47 & -3.53 & 2.5\\\\\nWSe${_2}$  & Hexagonal & Trigonal Prismatic & 3.26 & - & 2.70 & -5.84 & -3.14 & 2.6\\\\\nWTe${_2}$  & Hexagonal & Trigonal Prismatic & 3.52 & - & 1.98  & -5.35 & -3.37 & 1.9\\\\\nTiS${_2}$  & Hexagonal & Octahedral & 3.37 & - & 1.88 & -6.80 & -4.88 & -\\\\\nZrS${_2}$  & Hexagonal & Octahedral & 3.58 & - & 2.65 & -7.56 & -4.81 & -\\\\\n\\end{tabular}\n\\end{table*}\n\n\n\n\nThe trigonal prismatic TMDs have a direct band gap the the $K$ point [see \\textbf{Supplementary Materials}]. They have the similar band structures. The band structure of MoS${_2}$ is shown in \\textbf{Fig. 3(a)} as an example. For trigonal prismatic TMDs, the band gap decreases with increasing chalcogen atomic number. The CBM position also decreases with increasing chalcogen atomic number. These trends and the band gap values are consistent with other similar studies of these materials \\cite{Gong2013APL,Liang2013APL,Kang2013APL, Ding2011PBCM,Shi2013PRB}. Octahedral TMDs have an indirect band gap between the $\\Gamma$ and $M$ points [see \\textbf{Supplementary Materials}]. This is also in agreement with band profiles calculated in previous studies. \\cite{Liang2013APL,Gong2013APL,6_ivanovskaya_2005,Li2014RSC,Kang2013APL,Jiang2012JPCC}. The band structure of TiS${_2}$ is shown in \\textbf{Figure 3(b)}.\n\n\n\\subsection{Band offset and PCE of excitonic solar cells}\n\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{Fig4.eps}\\\\\n\\caption{(Color online) Band alignment of phosphorene with semiconducting monolayer TMDs.}\n\\label{fig6}\n\\end{figure}\n\nNote that many important properties and potential device applications of semiconductors are not determined entirely by the band gap only. The band alignment and corresponding band offsets (the relative band-edge energies) of two or more semiconductors are other fundamental\/critical parameters in the design of heterojunction devices \\cite{Wei1998APL,Zhang2000PRL,Jiang2012JPCC,Kang2013APL,Liang2013APL}, for example, the 2D heterostructure devices for photocatalytic water splitting \\cite{Jiang2012JPCC,Kang2013APL,Liang2013APL}, field effect transistors \\cite{Gong2013APL} and \\textit{p-n} diodes\\cite{Deng2014ACS}. Chemical trends of the band offesets provide a useful tool for predicting catalytic ability of TMDs-based heterojunctions. \\textbf{Figure 4} shows the band alignment (using the vacuum level as reference) of phosphorene with 8 semiconducting monolayer TMDs. It can be seen that the CBM of trigonal prismatic TMDs is higher than that of monolayer phosphorene, while the CBM of octahedral TMDs is lower than phosphorene. Thus, trigonal prismatic TMDs could function as the donor whereas octahedral TMDs would function as the acceptor in heterostructure with phosphorene. The optical gap of 8 TMDs is calculated, which is determined from the absorption spectrum. In most cases, the optical gap is slightly lower than the electronic band gap as shown in \\textbf{Table 1}. Notice that because of strong many-electron effects in some 2D materials, such as phosphorene and MoS$_2$, we might not get the accurate band offset parameters in some cases without considering the excited-state effect. For example, the HSE-calculated CBM band energy of monolayer MoS$_2$ is -4.21~eV \\cite{Guo2014JPCC} (-4.25~eV \\cite{Kang2013APL}) and phosphorene is -3.94~eV \\cite{Guo2014JPCC} (-3.92~eV \\cite{Cai2014SR}). Thus, phoshporene (MoS$_2$) is the donor (acceptor) in the phosphorene-MoS$_2$ 2D heterojunction \\cite{Guo2014JPCC} and the predicted PCE is 17.5\\% \\cite{Guo2014JPCC}. However, if we consider many-electron effects, the GW-calculated CBM band energy of monolayer MoS$_2$ is -3.89~eV (-3.74~eV \\cite{Liang2013APL}) and phosphorene is -4.25~eV. Interestingly, phoshporene (MoS$_2$) becomes the acceptor (donor) instead and the predicted PCE is reducing to 10\\%. Regarding to other 2D materials without strong many-electron effects, both HSE and GW can give similar band offset \\cite{Liang2013APL,Kang2013APL}.\n\nA model developed by Scharber et al. for organic solar cells \\cite{1_scharber_2006} and later adapted for exciton based 2D solar cells \\cite{Bernardi2012ACS,Dai2014JPCL} is used to predict the maximum PCE, $\\eta$, based on the fill-factor, $\\beta_{FF}$, open circuit voltage, $V_{oc}$, and short circuit current, $J_{sc}$.\n\\begin{align}\n\\eta = \\frac{\\beta_{FF}V_{oc}J_{sc}}{P_{\\mathrm{solar}}}\n\\end{align}\nwhere $P_{\\mathrm{solar}}$ is the total incident solar power per unit area based on the Air Mass (AM) 1.5 solar spectrum \\cite{1_astm_2012, 2_RReDC_2014}. The fill factor is the ratio of power output at the maximum power point to the product of the open circuit voltage and the short circuit current. The fill factor is estimated to be $0.65$ from literature. The $V_{oc}$, in units of V, and $J_{sc}$, in units of A\/$\\mathrm{m}^2$, are estimated in the limit of 100\\% external quantum efficiency as\n\\begin{align}\nV_{oc} &= \\frac{1}{e}\\left( E_{opt}^d - \\Delta E_{CBM} - 0.3 \\right) \\label{eq:voc}\\\\\nJ_{sc} &= e\\int_{E_{opt}^d}^\\infty \\frac{(\\hbar \\omega)}{\\hbar \\omega} d \\hbar \\omega\n\\end{align}\nwhere $e$ is the elementary charge, $E_{opt}^d$ is the donor optical gap, $\\Delta E_{CBM}$ is the conduction band offset and $P(\\hbar \\omega)$ is AM 1.5 solar spectrum. In equation \\ref{eq:voc}, the constant 0.3 eV is an empirical parameter that estimates losses due to energy conversion kinetics.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Fig5.eps}\\\\\n\\caption{(Color online) Power conversion efficiency of potential excitonic thin film solar cell heterojunctions. The labels indicate the material that complements phosphorene.}\n\\label{fig7}\n\\end{figure}\n\n\n\nUsing this model the TMDs are paired with phosphorene. The material with the lower CBM is the acceptor and the material with the higher CBM is the donor. Phosphorene is the donor when paired with both octahedral TMDs and the acceptor when paired with trigonal prismatic TMDs. The maximum PCE values for these eight heterostructures are marked on \\textbf{Fig. 5}. Of the eight heterostructures, phosphorene-ZrS${_2}$ and MoTe${_2}$-phosphorene have the highest PCE value of 12\\%. This efficiency is higher than that achieved by existing excitonic solar cells, and comparable to the proposed 2D g-SiC$_2$\/GaN (14.2\\%), PCBM\/CBN (10-20\\%)\\cite{Bernardi2012ACS}, and bilayer-phosphorene\/MoS$_2$ (16-18\\%) solar cells.\n\n\\begin{figure} [htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{Fig6.eps}\\\\\n\\caption{(Color online) (a) The band structure of phosphorene under applied 2\\% armchair compressed strain. The Fermi level is set on the top of the valance band. The dashed black lines are the band structure of phosphorene without strain for reference. (b) The schematic diagram of bilayer strained phosphorene-free phophorene heterostructure. The strain of the 1$^{\\textrm{st}}$ layer can be induced by the substrate, whereas the 2$^{\\textrm{nd}}$ layer is unstrained because of the weak van der Waals interaction.}\n\\label{fig7}\n\\end{figure}\nActually, a further observation based on this model is that a solar cell with a phosphorene donor could have maximum PCE values of up to about 20\\% with an appropriate choice for the acceptor. The conduction band offset (CBO) between phosphorene and TiS${_2}$ is 0.63 eV while the CBO between phosphorene and ZrS${_2}$ is 0.56 eV for the two cases here where monolayer phosphorene is the donor. This translates to a significant drop in $V_{oc}$ in the model which results in a lower PCE. In addition to looking out for new materials that have a better band alignment with phosphorene, means of tuning the properties of both the donor and acceptor can be considered. For example, strained phosphorene may be used as the acceptor material because the strain effect is a well-known method to tune the band structure of materials \\cite{Fei2014NL,Fei2015PRB}. The band structures of different strained monolayer phosphorene are shown in \\textbf{Supplementary Materials}. Here, we choose the phosphorene with 2\\% compressed strain (along the armchair direction) as a donor. Its band structure is shown in \\textbf{Fig. 6}. For comparison, the band structure of phosphorene without strain is also presented in the same figure. As can be seen, the position of CMB can be effectively tuned by the strain effect. 2\\% compressed armchair strain can shift the CBM down to the Fermi level around 0.11 eV because the compressed strain enhances the interaction of hybridized $s-p$ orbitals of P atoms, which contribute to the CBM. The calculated PCE value of -2\\% strained-phosphorene\/phosphorene can be around 20\\% as shown by the white solid circle in \\textbf{Fig. 5}. Based on the theoretical studies of mechanical properties of monolayer phosphorene, the mechanical stability of phosphorene can be up to under 30\\% strain \\cite{Wei2014APL,Peng2014PRB}. The 2\\% compressed strain of a monolayer phorphorene can be realized by the substrate effect in the experiment [see Fig. 6(b)]. Meanwhile, there is no strain on the second deposited monolayer phosphorene because of the weak van der Waals interaction between the two phosphorene layers. This can realize the strain\/non-strain phosphorene heterostructure.\n\nAlternatively multilayer structures of phosphorene or TMDs may also be considered to increase the overall power conversion per unit area. Our calculated optical gap of phosphorene is 1.6 eV which is at the edge of the infrared region. Therefore, heterostructures with a phosphorene donor may be considered for multijunction solar cells absorbing photons across the entire visible spectrum. Such cells incorporate several junctions that aim to absorb different portions of the solar spectrum so as to maximise total absorption [See \\textbf{Supplementary Materials}]. Given the anisotropy of phosphorene, the stacking orientation in multilayer structures may be a significant factor \\cite{Dai2014JPCL}.\n\n\n\n\\section{CONCLUSIONS}\nIn conclusion, through GW calculations, the band structures and optical spectra of monolayer phosphorene have been calculated. The electronic gap of 2.15 eV and the optical gap of 1.6 eV are desirable for solar cell applications because of the strong exciton binding energy. When paired with ZrS${_2}$ or MoTe${_2}$ the power conversion efficiency of the excitonic solar cells can be as high as 12\\%. There is further potential to improve the PCE of phosphorene based solar cells substantially by tuning the materials, such as through the strain and multi-stacking effect, to achieve a better band alignment.\n\n\n\\section{ACKNOWLEDGEMENT}\n\n\nAuthors thank Minggang Zeng for his helpful discussion. This project was supported by the Computational Condensed Matter Physics Laboratory, NUS. Computational resources were provided by Centre for Advanced 2D Materials, NUS.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\nModern proton driven accelerator applications, such as neutron spallation sources and high energy hadron colliders, demand increasingly higher beam currents.\nA common solution is to inject \\ensuremath{\\text{H}^-\\ } ions via the charge exchange process, which overcomes the limit imposed by Liouville's theorem in circular machines~\\cite{dimov1996}.\nSuch high beam currents present a challenge for conventional, invasive beam diagnostics, which may be damaged by the high beam powers.\nContinuous online monitoring of the beam parameters also requires diagnostics that has minimal influence on the beam.\nFor this purpose, a laserwire provides an inherently indestructible and essentially non-invasive probe, that replaces the mechanical counterpart, such as a wire or slit.\nThe laser neutralizes a small fraction of the \\ensuremath{\\text{H}^-\\ } beam via the photo-detachment process, generating free electrons and \\ensuremath{\\text{H}^0\\ } atoms.\nScanning the laser transversely across the particle beam and measuring either of the products of the interaction provides information on the beam properties.\\\\\nLaserwire diagnostics for measuring an \\ensuremath{\\text{H}^-\\ } beam were originally built at Los Alamos National Laboratory~\\cite{Cottingame1985, Connolly1992}, and subsequently developed at facilities including the Brookhaven National Laboratory LINAC~\\cite{bnl93}, and the Spallation Neutron Source at Oak Ridge National Laboratory~\\cite{Liu2010,Liu2012}, and more recently considered for future applications such as the Front End Test Stand (FETS) at Rutherford Appleton Laboratory, UK~\\cite{Gabor2005, Gibson2013}.\\\\\nExisting systems typically use lasers with peak powers above the megawatt level and free-space beam transport. Such laser systems need frequent maintenance, can be susceptible to stability and alignment issues, and also require a high level of safety measures. In contrast, the laserwire system presented in this paper has been designed to use a fibre laser with comparatively low pulse power in the kilowatt range. By transporting the laser light in an optical fibre to the interaction region the instrument is compact, has a stable output in terms of spatial and temporal properties and is unaffected by mechanical vibrations or misalignments. This has been demonstrated in \\cite{Liu2013}, with a picosecond laser used to measure the longitudinal beam profile. The instrument developed for the LINAC4 at CERN will be used to measure the transverse beam emittance.\\\\\nLINAC4 is currently being commissioned in the framework of the LHC (Large Hadron Collider) Injector Upgrade (LIU)~\\cite{Hanke_LIU}. It will replace the existing proton accelerator, LINAC2, as the injector to the Proton Synchrotron Booster (PSB). After its completion expected by the end of 2016, the 90\\,m long linac will produce an \\ensuremath{\\text{H}^-\\ } beam with mean current of 40~mA and energy of 160\\,MeV \\cite{TDR2006}. The transverse emittance of the LINAC4 beam must be monitored during operation to optimize injection into the PSB. Since conventional invasive diagnostics are unsuited to continuous online monitoring of the LINAC4 beam, instead, the development of a non-invasive laserwire instrument was proposed~\\cite{Hofmann2013}. To achieve this goal, a collaboration between the CERN Beam Instrumentation group and the UK FETS beam diagnostics team was established~\\cite{Gibson2013}. As the first step in this ongoing development, a laserwire emittance scanner has been prototyped and tested at the 3\\,MeV beam energy during the first commissioning phase of LINAC4.\\\\\nThis paper describes the configuration of the laserwire prototype and presents first results from transverse emittance measurements of the 3\\,MeV beam. The instrument design is first reviewed in Sec.~\\ref{sec:design}, which includes simulation studies that instruct the system requirements. In Sec.~\\ref{sec:exp_setup} the installed setup is detailed and the essential components are characterized in Sec.~\\ref{sec:calibration}. The emittance measurement results are presented and compared with a conventional diagnostic in Sec~\\ref{sec:results}. Finally, the paper provides an overview of future developments.\n\\section{\\label{sec:design}Instrument design}\n\\subsection{\\label{sec:concept}Principle of operation}\nThe operational principle of a laserwire instrument is to cross an accelerated beam of particles with a laser beam, such that the charged particles and photons can interact in a finely controlled overlap region, as shown schematically in Fig.~\\ref{fig:concept}. The fundamental process exploited for \\ensuremath{\\text{H}^-\\ } beam diagnostics is photo-detachment; the weakly bound outer electron in the \\ensuremath{\\text{H}^-\\ } ion is permanently ejected if a sufficiently energetic photon is absorbed. The binding energy for the outermost electron is very low (0.75\\,eV), and so is detached by photons with a wavelength of less than 1.65\\,$\\mu$m. Only a small transverse section of the particle bunch is sampled, as the laser is stepped through a series of y-positions. The distribution of the particles neutralized (\\ensuremath{\\text{H}^0\\ } atoms) at each laser position is measured by a downstream detector, after being separated from the main beam by a bending dipole magnet. Merging each measured \\ensuremath{\\text{H}^0\\ } profile with the vertical laser position provides the necessary information to reconstruct the transverse emittance of the \\ensuremath{\\text{H}^-\\ } beam. In our setup we conducted measurements only the vertical plane. The horizontal plane could be obtained likewise with the laser and detector setup 90$^{\\circ}$ rotated.\\\\\n\n\\begin{figure} [!hb]\n\\begin{center}\n\\includegraphics{Fig01_LW_Principle.eps}\n\\caption{Schematic principle of a laserwire emittance scanner. A fraction of the \\ensuremath{\\text{H}^-\\ } particle beam collides with photons of a focused laser beam. The neutralized \\ensuremath{\\text{H}^0\\ } are detected by a downstream detector after being separated from the main beam by a dipole magnet. By measuring the \\ensuremath{\\text{H}^0\\ } profile at the detector plane for different laser y-positions, the angular spread and ultimately the transverse phase-space can be reconstructed.}\n\\label{fig:concept}\n\\end{center}\n\\end{figure}\nThe dependence of the photo-detachment cross-section on the wavelength of the incident photon~\\citep{detach} is shown in Fig.~\\ref{fig:cross}. The probability of ejecting one electron by interacting with a laser photon in the visible or near infrared region is generally high so the signal produced by the laserwire will be robust even when relatively low power laser pulses are used. However, because of the very low binding energy of the outer electron in the \\ensuremath{\\text{H}^-\\ }, the particles can be easily neutralized also by means of other processes such as black body radiation, magnetic fields or collisions with residual gas atoms, the latter being the dominant source of background \\cite{cheymol2011}. In the next subsection, the processes of photo-detachment and collisions with the residual gases are considered quantitatively to predict the expected ratio between signal and background.\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig02_Cross_Section.eps}\n\\caption{Cross-section of the photo-detachment process of a non-relativistic \\ensuremath{\\text{H}^-\\ } ion. The dashed line indicates the selected laser wavelength for the emittance measurements at 3\\,MeV.}\n\\label{fig:cross}\n\\end{center}\n\\end{figure}\n\\subsection{Theoretical models and simulations}\nA theoretical framework of the laserwire interaction was developed to evaluate the expected performance of the laserwire instrument and establish the range of key parameters required for the laser source and detection system. The theoretical framework consists of a model of the laser photo-detachment signal, a model of the residual gas background and a comparative simulation.\n\\subsubsection{Photo-detachment model}\nThe probability that the outer electron is stripped from an \\ensuremath{\\text{H}^-\\ } ion during exposure to laser radiation is modeled by~\\cite{sns2002}:\n\\begin{equation}\\label{eq:stripping_probability}\n\\mathbb{P}_{Laser}=1-\\exp\\left(-\\sigma(E_{CM})\\rho(x,y,z)t\\right),\n\\end{equation}\nwhere $\\sigma(E_{CM})$ is the photo-detachment cross-section, $\\rho(x,y,z)$ is the laser photon density and $t$ is the time of interaction. The cross-section in~Eq.~\\ref{eq:stripping_probability} depends on the center-of-mass energy, \\begin{equation}\\label{eq:cm_energy}\nE_{CM}=\\gamma E_L\\left(1-\\beta cos\\Theta_L\\right),\n\\end{equation}\nwhich is a function of the laser energy $E_L$, the Lorentz factor  $\\gamma=1\/\\sqrt{1-\\beta^2}$, the relativistic velocity ratio of the \\ensuremath{\\text{H}^-\\ } ions $\\beta=v\/c$, and the angle $\\Theta_L$ between the laser and particle beams. In the case of the 3\\,MeV \\ensuremath{\\text{H}^-\\ } beam at LINAC4, the Lorentz factor is small enough that relativistic corrections have negligible impact on the cross-section. For the final laserwire system that will measure a 160\\,MeV beam, the effective value of the selected laser wavelength will be relativistically shifted close to the peak in the photo-detachment cross-section of Fig.~\\ref{fig:cross}.\n\nThe model simulated the stripping interactions that occur when an input distribution of \\ensuremath{\\text{H}^-\\ } ions cross the laser beam. Equation~\\ref{eq:stripping_probability} was applied to each \\ensuremath{\\text{H}^-\\ } ion, taking into account the time spent by the ion as it traverses the varying photon density of the laser beam. Thus the portion of ions stripped by the laser was determined, as detailed in~\\cite{Hofmann2013}. The simulations were performed for one laser pulse travelling in the $x$-direction that is geometrically centered in $y$ on the \\ensuremath{\\text{H}^-\\ } beam. The trajectories of \\ensuremath{\\text{H}^0\\ } particles produced in this overlap region were propagated 3\\,m downstream to the detector plane and integrated, such that the expected measurable profile of particles could be plotted.\n\\subsubsection{Neutralization due to collisions with residual gas}\nThe probability of ejecting one electron from an \\ensuremath{\\text{H}^-\\ } ion by collision with a stray particle present in the vacuum pipe is\n\\begin{equation} \n\\mathbb{P}_{Gas}=1-e^{-l\/\\lambda}\\ {\\rm with}\\ \\lambda=\\frac{kT}{\\sigma P};\\,\\sigma\\sim\\frac{1}{\\beta^2_{rel}}.\n\\label{eq:gas_strip}\n\\end{equation}\nThis stripping probability $\\mathbb{P}_{Gas}$ is calculated from the interaction length $l$, the beam pipe temperature $T$, and the pressure $P$ of the residual gas, which consists mainly of hydrogen. The dependence of the collisional detachment cross-section, $\\sigma$, on the beam energy was derived from the literature~\\citep{gas_lebt,gas_mev}.\n\\subsubsection{Signal and background simulations}\nThe parameters used to calculate the distribution of photo-detached ions at the detector plane are listed in Tab.~\\ref{tab:sim_laser}.\nOf particular interest was the calculation of the signal produced by a laser with a relatively low peak power (in the kW range) that could be transported in optical fibre. The laser pulse duration was also chosen in a way that the produced signal could be realistically detected and digitized without bandwidth restrictions. The value for the \\ensuremath{\\text{H}^-\\ } beam current and the beam size were set to reflect real conditions of the LINAC4 machine during commissioning.\\\\\nThe simulations for the background distribution in the plane of the \\ensuremath{\\text{H}^0\\ } detector were performed using gas pressures and beam dynamics data from the \\ensuremath{\\text{H}^-\\ } source to the bending magnet, through the accelerator components described in Sec.~\\ref{sec:linac4},\nTab.~\\ref{tab:conditions_backg} summarizes the parameters for the background simulation. $\\mathbb{P}_{Gas}$ was calculated in 2\\,cm intervals along the beam axis and the stripped portion was propagated to the \\ensuremath{\\text{H}^0\\ } detector. \n\\begin{table}[h]\n\\begin{center}\n\\caption{Photo-detachment signal simulation parameters.}\n\\label{tab:sim_laser}\n\\begin{tabular}{l c r}\n\\hline\\hline\nParameter & Value & Units\\\\\n\\hline\nLaser pulse energy          &67          &$\\mu$J \\\\\nLaser pulse length (2$\\sigma$)          &106         &ns \\\\\nLaser waist radius          &76          &$\\mu$m \\\\\nIon beam current            &8.5        &mA \\\\ \nVertical ion beam size      &1.01       &mm\\\\\nDistance from the IP to the detector\t\t&3.4    &m\\\\ \n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Residual gas background simulation parameters.}\n    \\begin{tabular}{lcccr}\n    \\hline\n    \\hline\n    Parameter               &LEBT   &RFQ    &Chopper    &Units\\\\\n    \\hline\n\tLength                  & 2.0   & 3.0   & 3.7   & m  \\\\\n    Residual gas pressure   & 20    & 10    & 1     & 10$^{-10}$\\,bar  \\\\\n    Min. aperture radius    & 50    & 3.54  & 6.2   & mm  \\\\\n    Mean pulse current      & 13    & 11    & 8.5   & mA  \\\\\n    \\hline\n    \\hline\n    \\end{tabular}%\n  \\label{tab:conditions_backg}%\n\\end{table}\nThe results of both the simulations of signal and background are summarized in Fig.~\\ref{fig:laser_backg}. \n\\begin{figure}[!b]\n\\begin{center}\n\\includegraphics{Fig03_Signal_Bkg_Simulation.eps}\n\\caption{Flux of \\ensuremath{\\text{H}^0\\ }  arriving at the detection plane from laser stripping in the center of the \\ensuremath{\\text{H}^-\\ } beam - in average while laser pulse duration - (dots); \\ensuremath{\\text{H}^0\\ } flux from the residual gas background - permanently while macro-pulse - (dashed line).}\n\\label{fig:laser_backg}\n\\end{center}\n\\end{figure}\nThe dotted curve in the plot shows the resulting \\ensuremath{\\text{H}^0\\ } density in the detector plane produced by a laser pulse crossing the  3\\,MeV \\ensuremath{\\text{H}^-\\ } beam center - in average while laser pulse duration - and the dashed curve corresponds to the flux of \\ensuremath{\\text{H}^0\\ } produced by residual gas interaction. Due to the time structure of both signals being very different, they can be separated in frequency domain without difficulty. In fact, being the signal produced only during the interaction with the pulsed laser, it will have a similar duration, whereas the background, as it is generated all along the beam-pipe, will be composed of \\ensuremath{\\text{H}^0\\ } s with kinetic energies ranging from 45\\,keV to 3\\,MeV and it arrives at the detector plane unbunched and with a duration comparable with the particle macro-pulse. \n\n\\subsection{\\label{sec:requirements}System requirements}\nAnalyzing the simulation result, summarized in Fig~\\ref{fig:laser_backg}, one can derive requirements for the laser and detector system.\\\\ Tab.~\\ref{tab:laser_req} gives an overview of the requirements for the laser system. The wavelength is for our system not very critical, since the cross-section is very broad (see Fig.~\\ref{fig:cross}). Therefore we aimed for 1064\\,nm laser where a wide choice of lasers are commercial available. To reach a signal level which is significant above the background level one needs a laser with peak power in the kilowatt range. To relax the bandwidth requirements for the detector, a laser pulse-length of at least 10\\,ns is required. With an repetition rate of more than 10\\,kHz, synchronized to the LINAC4 timing one can sample the 400\\,$\\mu$s \\ensuremath{\\text{H}^-\\ } pulse multiple times. \\\\\nTo realize a laserwire scanner, a nearly constant diameter of the laser beam which is a factor of 10 smaller than the \\ensuremath{\\text{H}^-\\ } beam diameter is the pre-condition. Taking the \\ensuremath{\\text{H}^-\\ } beam $\\sigma$ of 1...3\\,mm, a laser beam with a beam quality factor ($M^2$) smaller than 3 is needed. E.g. in our simulation we used an laser with $M^2$ = 1.8 and achieved a waist radius of 76\\,$\\mu$m with a Rayleigh length of 6.5\\,mm. \nThis will be discussed more in detail in section~\\ref{sec:inst_setup}\\\\\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Requirements for the laser system}\n\\label{tab:laser_req}\n\\begin{tabular}{lcr}\n\\hline\\hline\nParameter                           &Requirement        &Units\\\\\n\\hline\nWavelength                          & 900 $\\pm$ 300     &nm\\\\\nPulse peak power                    & 1...10            &kW\\\\\nPulse length                        & 10...100          &ns\\\\\nRepetition rate                     & $>$ 10            &kHz\\\\\nSynchronisation to LINAC4           & Required          &--\\\\\t\nBeam quality factor ($M^2$)         & $<$ 3             &--\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nRegarding the \\ensuremath{\\text{H}^0\\ } detector the time resolution is a key-factor. It needs to resolve a laser pulse so that the signal is not dominated by residual gas background but laser stripped \\ensuremath{\\text{H}^0\\ }. A bandwidth of min. $20$\\,MHz is required to record the beam pulse with negligible distortion \\cite{hofmann15}. To achieve a reasonable electrical signal the detector also needs to have internal gain. This requirement follows from simulation results shown in Fig.~\\ref{fig:laser_backg}. As the \\ensuremath{\\text{H}^0\\ } density at the detector will be in the order of 10~$\\ensuremath{\\text{H}^0\\ } ns^{-1}\\,mm^{-2}$, it corresponds to only nanoampere currents.\\\\\nOn the same time, the background flux of approx. 1~$\\ensuremath{\\text{H}^0\\ } ns^{-1}\\,mm^{-2}$ can create significant damage in detector materials integrated over the LINAC4 pulse length of 400\\,$\\mu$s. For this reason the radiation resistance of the detector is also an important factor.\n\\section{\\label{sec:exp_setup}Experimental setup}\n\\subsection{\\label{sec:linac4}Overview of LINAC4}\n\nThe final layout of the LINAC4  will consist of a 45\\,keV \\ensuremath{\\text{H}^-\\ } ion source, a radio frequency quadrupole (RFQ), which accelerates the beam to 3\\,MeV, and a chopper. A drift tube linac (DTL), a cell-coupled drift tube linac (CCDTL) and a Pi-mode structure (PIMS) will be then used to accelerate the beam to its final energy of 160\\,MeV\\,\\cite{l4}. During the LINAC4 construction, the beam commissioning is taking place in stages and at every stage the beam has to be fully characterized. A schematic block diagram of the machine is shown in Fig.~\\ref{fig:l4} and the design parameters of the LINAC4 are summarized in Table\\,\\ref{tab:l4params}.\n\\begin{figure}[!hb]\n\\begin{center}\n\\includegraphics{Fig04_L4_Block_Diagram.eps}\n\\caption{LINAC4 block diagram.}\n\\label{fig:l4}\n\\end{center}\n\\end{figure}\n\\begin{table}[htb!]\n\\caption{\\label{tab:l4params} LINAC4 Parameters}\n\\begin{tabular}{l c c r}\n\\hline \\hline\nParameter             & Symbol             & Value               & Units\\\\\n\\hline\nOverall linac length        \t\t\t& $\\text{L}$            \t\t\t& 90    & m \\\\\nOutput energy               \t\t& $\\text{E}$            \t\t\t& 160   & MeV \\\\       \nBunch frequency       \t\t& f$_{\\text{bunch}}$    \t& 352.2 & MHz \\\\\nPulse length           \t\t& t$_{\\text{pulse}}$    \t\t& 400   & $\\mu$s\\\\\nPulse repetition rate          & t$_{\\text{rep}}$       & 1.2   & s\\\\\nAverage pulse current       \t\t& I$_{\\text{pulse}}$    \t\t& 40    & mA\\\\\nBeam transverse emittance\t\t& $\\varepsilon$         \t\t& 0.4   & $\\pi$~mm~mrad\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\\subsection{\\label{sec:3mev_setup}Diagnostic setup at 3\\,MeV}\nThe laserwire system was integrated into a movable test bench that was used to measure the beam parameters after the chopper, during the commissioning phase at 3\\,MeV. The test bench also contained various diagnostic tools such as beam position monitors, beam current transformers (BCT), slit and grid emittance scanner etc. for full characterisation of the beam at different commissioning stages. The schematic diagram of the test bench is illustrated in Fig.\\,\\ref{fig:test_bench} (right hand side).\\\\\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics*[width=2\\columnwidth]{Fig05_L4LW_Test_Bench.eps}\n\\caption{Layout of the \\SI{3}{MeV} diagnostics test bench as it was installed after the Chopper, including the operational emittance meter (slit and grids) and the laserwire emittance scanner prototype.}\n\\label{fig:test_bench}\n\\end{figure*}\nThe laser and \\ensuremath{\\text{H}^0\\ } monitoring subsystems, which will be described in detail in the next sections, have been installed as close as possible to the slit and wire grids respectively in order to facilitate the comparison between the two methods. The spectrometer dipole in between the slit (and laser) and the wire grids (and \\ensuremath{\\text{H}^0\\ } detector) was designed to steer the beam (during periods not used for slit-grid measurements) into the spectrometer line for absolute and relative energy measurements via horizontal profile measurements at the end of this line. Powering such a dipole allows separating the stripped \\ensuremath{\\text{H}^0\\ } (to be measured in the straight line by the diamond detector) from the un-stripped \\ensuremath{\\text{H}^-\\ } particles and thus performing the laserwire measurements.\\\\\nSimulations of the beam envelope with nominal quadrupole settings resulted in an expected beam size in the plane of the laser of $\\sigma_x = 2.6$\\,mm and $\\sigma_y = 1.4$\\,mm.\n\\subsection{\\label{sec:laser_setup}Laser system and optical layout}\nThe laser is a Q-switched, diode pumped, Yb doped fibre Master Oscillator Power Amplifier (MOPA) manufactured by ``Manlight S.A.S'' (Lannion, France), model: ML-30-PL-R-TKS. The oscillator generates pulses with approximately 80~ns width (FWHM) at wavelength of 1080~nm and a repetition rate selectable between 30 and 100~kHz.\n\nThe fibre amplifier is pumped by a laser diode that can run either in continuous wave (CW) or can be driven by an external signal with a repetition rate of up to 5~kHz and an adjustable duty cycle, which enables the train of amplified pulses to be synchronized to an external source. The maximum average power of the laser system is approximately 28~W when pumped in continuous wave (CW) regime and the maximum pulse peak power about 8.5~kW.\n\nIn order to optimize the duty cycle of the laser, and therefore use a lower average power, the pump diode was synchronized to the LINAC4 macropulse. Also, the pulse duration was set to 1 ms in order to comfortably overlap the laser amplified pulses with the 400 $\\mu$s long macropulses. A detailed overview of synchronization between the laser pulses and the accelerator bunches is shown in Fig.~\\ref{fig:timing}, including a zoomed view of the overlap between the single laser pulse and the ions micro-bunches.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig06_Timing_Diagram.eps}\n\\caption{Timing diagram of the laserwire system.}\n\\label{fig:timing}\n\\end{center}\n\\end{figure}\nThe transport system designed to convey the laser beam to the interaction point is based on a large mode area (LMA) optical fibre. The schematic of the optical system is shown in Fig.\\,\\ref{fig:test_bench} (left hand side). The coupling and focusing optics were mounted on breadboards and enclosed in two separate interlocked boxes. In the coupling box, the output from the Q-switched MOPA fibre-laser is coupled into a 5~m long optical LMA fibre with a core diameter of 20~$\\mu$m. In order to continuously monitor the laser power participating to the photo-neutralization a fast photodiode with sub-nanosecond risetime is set inside the coupling box to pick up a parasitic signal. The laser delivery system is setup just in front of the beam-pipe, which is accessible to the laser via  a anti-reflexion coated vacuum window. The output face of the fibre together with the focus optics are set on a small plate and mounted on a stack of two automated translation stages with 50 mm range and $\\sim$1~$\\mu$m step resolution, one set to scan the laser beam vertically (scanning stage or Y-stage) and a second one to allow longitudinal positioning of the laser focus within the vacuum pipe (longitudinal stage or X-stage). The focus optics consist of collimation lens with 6.24~mm focal length, a remotely controllable variable beam expander that has a range of magnification adjustable between 1x and 8x and a focusing lens with focal length of 500 mm that focuses the laser radius to $\\sim$75~$\\mu$m at the interaction region, which is much smaller than the particle beam size to be scanned.\n\nA secondary optical path is also installed within the focusing box. It is setup in a way that when the Y-stage is set at its lower position, the laser beam is directed towards an optical window that reflects a small portion of light into a CCD camera set on a translation stage so that the laser spatial characteristics can be measured on demand. The larger part of light that is transmitted through this optical window is detected by a photodiode for peak power and pulse duration measurements.\n\n\\subsection{\\label{sec:diamond_setup} Diamond detector system}\nIn section~\\ref{sec:requirements} the requirements for a suitable \\ensuremath{\\text{H}^0\\ } detector were determined. After analysis of different types of detectors we choose a  polycrystalline (pCVD) diamond strip detector for the measurement campaign. This kind of detector is able to measure even single particle events due to its internal gain of about $10^4$ electrons per penetrating \\ensuremath{\\text{H}^0\\ }. The response time in the sub-nanosecond range permits to resolve an arriving laserpulse (FWHM~=~80~ns) with sufficient time resolution. Finally, the radiation hardness of approx. $10^{15}\\,proton\/cm^2$ comparing to $10^{14}\\,proton\/cm^2$ for silicon \\cite{si_vs_dia} is essential.\\\\\nA photo of the detector is shown in Fig.~\\ref{fig:diamond}. The 20~x~20\\,mm, 500\\,$\\mu$m thick pCVD diamond disc is mounted on a ceramics printed circuit board (PCB) while the five 0.2\\,$\\mu$m thick aluminium electrodes on the front side of the detector are bonded to the circuit paths. On the PCBs backside a bias voltage of 500\\,V is applied to readout the charge created in the diamond bulk. Capacitors (1\\,$\\mu$F) in parallel to the detector rapidly recharge the diamond when electrons are read out.\\\\\nThe PCB with the detector is fixed to a fork-shaped support-frame situated inside a 4-way crosschamber. The setup is mounted on a stepping motor which provides vertical movement in a range of 80\\,mm  with 150\\,$\\mu$m resolution.\\\\\nThe signals need to be pre-amplified directly after the vacuum feed-through due to an unexpected low sensitivity of the detector. The effect causing this will be discussed quantitatively in chapter~\\ref{dia_sig}. We were using AC-coupled linear amplifiers with 46\\,dB gain and 100\\,MHz bandwidth. Thereafter the signal from the diamond channels were digitized with a 1 GS\/s oscilloscope.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width = 7cm]{Fig07_Diamond_Detector.png}\n\\caption{pCVD diamond detector with 5 strip electrodes \\cite{cividec}.}\n\\label{fig:diamond}\n\\end{center}\n\\end{figure}\n\\section{\\label{sec:calibration}System characterisation and measurements}\n\\subsection{\\label{sec:inst_setup}Laser characterisation}\nAs the laser beam is used as a probe to scan the particle beam, a full characterisation of its properties is required.\nTo measure the spatial characteristics of the laser beam we used a CCD camera which was set on a motorized translation stage in our lower level diagnostic optical path (see laser delivery system in Fig.~\\ref{fig:test_bench}). We moved the camera in steps of 1~mm and recorded the beam spot size variation along the longitudinal axis. Fig.~\\ref{fig:m2} shows the laser spot size at the focal plane of the focusing lens. The horizontal and vertical laser diameters extracted from the images are plotted along the direction of propagation in Fig.~\\ref{fig:m2}. The two sets of data were fitted with the laser propagation formula for quasi-Gaussian beams,\n\\begin{equation}\n\\label{eq:m2} \nW(z)=W_{0}~\\sqrt{1+(z \\frac{\\lambda~M^{2}}{\\pi~W_{0}^{2}})^{2}}\n\\end{equation}\nwhere W(z) is the laser spot radius (distance from the center of the distribution to the position where the intensity drops by a factor $e^{2}$), $W_{0}$ is the minimum laser waist, $\\lambda$ is the laser wavelength and $M^{2}$ is a factor which describes the quality of the real beam compared to an ideal $TEM_{00}$ Gaussian beam (for which $M^{2}=1$) \\cite{laser_phy}.\nFor our application the results in the vertical plane are by far more important. They define the resolution for the sampling (in analogy to the slit size). The resulting value for $M^{2}_y$ is 2.0, the laser waist $W_{0y}$ at the focal plane of the lens is approximately 75$\\mu$m (corresponding to a $\\sigma$ of 37.5$\\mu$m) and the Rayleigh length is about 8\\,mm. Summarizing the characteristics and comparing to the \\ensuremath{\\text{H}^-\\ } beam the laser beam can be approximated as a thin cylinder with constant diameter ($\\diameter = 150 \\mu$m).\\\\\nRegarding the characterisation of the longitudinal parameters of the laser pulse we measured the pulse shape before and after the fibre propagation with fast photodiodes. Furthermore we measured the average power before and after the fibre to check the coupling and transport efficiency of the fibre transport. The efficiency of the fibre delivery was found to be about 70\\,\\%\\,$\\pm\\,5\\%$ with higher efficiency at lower laser power.\\\\\nFor the emittance measurements the laser beam pulse energy interacting with the \\ensuremath{\\text{H}^-\\ } beam was measured to be 154\\,$\\pm\\,6\\,\\mu$J.\\\\\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics[width=1\\columnwidth]{Fig08_M2_Measurements.eps}\n\\caption{Measured 2$\\sigma$ width of the laser beam focused by a lens with a focal length $f$ = 500~mm. At the bottom left is shown the laser spot image taken at a position of -3mm, where the minimum vertical size is obtained.}\n\\label{fig:m2}\n\\end{center}\n\\end{figure}\n\\subsection{\\label{sec:diamond_signal}Diamond detector signal examination}\n\\label{dia_sig}\nTo test the signal response of the diamond detector, laser and detector were positioned in order to maximize the signal in one specific channel of the diamond detector.\\\\\nFig.~\\ref{fig:detector_pd_signal_trace} shows in the upper plot the amplified laser pulses, interacting with the \\ensuremath{\\text{H}^-\\ } beam, recorded with a photodiode in the coupling box. In the lower plot, the resulting signal of an amplified channel of the diamond detector during a \\ensuremath{\\text{H}^-\\ } beam pulse is displayed. This signal corresponds to one laser position and one diamond position and each segment represents a 1~$\\mu$s time-interval.\\\\\nComparing the continuous laser pulses with the signal on the diamond detector, one finds a clear correlation just between the segments 4 and 24 where the 400~$\\mu$s long \\ensuremath{\\text{H}^-\\ } macropulse can interact with the laserpulses. In this timeframe the \\ensuremath{\\text{H}^0\\ } created by the laser interaction form sharp peaks in the diamond signal. Furthermore the \\ensuremath{\\text{H}^0\\ } background originating by collisions with the residual gas molecules cause an offset on the diamond.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics{Fig09_Detector_Pd_Signal.eps}\n\\caption{Comparison of laser pulses recorded with a photodiode and the amplified signal at the diamond detector.}\n\\label{fig:detector_pd_signal_trace}\n\\end{center}\n\\end{figure}\n\\begin{figure}[!b]\n\\begin{center}\n\\includegraphics{Fig10_Detector_Pd_Signal_Peaks.eps}\n\\caption{Zoom on one of the diamond signals of Fig.~\\ref{fig:detector_pd_signal_trace} (solid - inverted), compared to the corresponding laser pulse (dashed). Dotted trace -- linear fit of the background.}\n\\label{fig:pulses}\n\\end{center}\n\\end{figure}\n\nFig.~\\ref{fig:pulses} shows a zoom to one of the pulses where the diamond signal (solid) and the laser pulse (dashed) are overlaid. The distinct agreement in the shape of the signals demonstrates the linear relation between laser-power and output signal of the diamond detector.\\\\\nTo characterize further the detector response, a comparison was done between the \\ensuremath{\\text{H}^-\\ } beam current recorded shortly behind the laser interaction and the unamplified diamond detector raw signal. Fig.~\\ref{fig:diamond-vs-bct} shows the relation of both signals. The slight increase of the floor of the diamond signal can be explained by the \\ensuremath{\\text{H}^0\\ } atoms created by residual gas collisions. The ratio between the peak signal and baseline value is fairly constant (between 10 and 11). Given the uncertainties of the simulation input (e.g. gas pressure, detector time response etc...), this value compares well to the simulations (see Fig.~\\ref{fig:laser_backg}).\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig11_Det_BCT_Signal.eps}\n\\caption{Unamplified signal from a diamond detector channel sampling the \\ensuremath{\\text{H}^0\\ } particles for a given laser position ($E_{pulse}~=~67~\\mu J; f_{pulse} = 30~kHz$), as function of time along the 3\\,MeV \\ensuremath{\\text{H}^-\\ } beam pulse. This is compared to the beam current measured with a beam current transformer (BCT) downstream the laser station.}\n\\label{fig:diamond-vs-bct}\n\\end{center}\n\\end{figure}\nIn addition the time response of the diamond signal was compared with the BCT signal all along the LINAC4 macropulse. Even though in some cases the signal from the diamond detector reproduced well the beam current evolution~\\cite{linac14}, in most cases, as presented here, the agreement was poor in the second part of the beam pulse.\\\\\nEven though such an occurrence, obviously affecting the overall measurement accuracy, has not been fully understood yet, it can be well related to the implantation of \\SI{3}{\\mega\\eV} protons into the diamond bulk, after approx. 50\\,$\\mu$m of detector material. Simulation results give more than $10^{7}$ protons implanted by the background for one LINAC-pulse. The variation of the electrostatic field inside the diamond could thus lead to a decreased Charge Collection Efficiency (CCE) during the LINAC macropulse. This might also explain the signal amplitude of just a few mV, which is two orders of magnitude lower than expected by calculations not considering the effects of the proton implantation \\cite{hofmann15}. This disturbance is supposed to be negligible for higher beam energies (starting from \\SI{12}{\\mega\\eV}) for which the protons have high probability to traverse and escape the diamond bulk without being stopped. First measurement results at the 12\\,MeV beam show first evidence confirming this hypothesis.\\\\\n\n\n\\section{\\label{sec:results}Profile and emittance measurement results}\n\\subsection{\\label{sec:data}Analysis method}\nTo extract the transverse emittance value from the raw data recorded with the diamond detector an analysis routine was developed and applied to the gathered data. The first action is to define the segments with useful data in the digitized signal (e.g. Seg. 6-23 in Fig.~\\ref{fig:detector_pd_signal_trace}). Hereafter each segment is treated individually. At first a finite impulse response (FIR) low-pass filter (3\\,dB cutoff at 22\\,MHz) is used to suppress high-frequency noise. Now a time-interval is defined, which separates the laserpulse from the background (in Fig.~\\ref{fig:pulses} e.g. 300...700 ns). By fitting linearly the data outside of this window a subtraction of the diamond signal from the background can be conducted. Finally the data in the time-interval of the laserpulse can be integrated.\\\\ \nTo get a mean emittance value along the duration of the LINAC4 macropulse, the integrated values for the selected segments are averaged. The resulting value corresponds to one sample in the transverse phasespace. The referring $y\/y'$ values are calculated by the current detector and laser position.\\\\\nThe RMS transverse emittance can finally be calculated by using these samples in the phasespace as shown in Eq.~\\ref{eq:em_formula}.\n\n\\begin{equation}\\label{eq:em_formula}\n\\epsilon_{y} = \\sqrt{\\langle y^2 \\rangle \\langle y'^2 \\rangle -\\langle yy' \\rangle^2}\n\\end{equation}\nTo extract the vertical beam profile from this data, the values at different $y'$ positions can be integrated for each $y$ coordinate. \n\n\\subsection{\\label{sec:emittance}Profile and emittance reconstruction and comparison with slit~\\&~grid system}\nIn order to measure vertical transverse emittance the following technique has been used. The laser focus was moved vertically across the ion beam with variable step size (0.5~mm at the tails and 0.25~mm around the peak) in order to accurately sample the shape of the distribution. At each laser position, the diamond detector was scanned across the beamlet of neutral particles with steps of 1.81~mm in order to increase the angular resolution. Each detector scan corresponding to certain laser focus position was stored in separate data file. The data from these files then were analysed using the same analysis method as described above. The integrated and averaged signal then was used to plot the phase-space distribution.\\\\\nThe laserwire results were verified with the independent measurements from conventional slit and grid emittance meter. The slit, installed at the same point as the laser IP was scanned across the ion beam in vertical direction. At each vertical position, well defined with respect to the ion beam center, the slit selects a narrow slice of the ion beam. The distribution of the particles transmitted through the slit was measured after 3.4~m drift space by two wire grids~\\cite{slitgrid}. Same as in case of laserwire, by scanning the slit vertically across the ion beam and measuring the profiles of particle beamlets passed through the slit the whole phase-space was reconstructed. Measured phase-space distributions for 3~MeV setup obtained with laserwire and slit and grid methods are presented in Fig.~\\ref{fig:phase_space}.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig12_Phase_Space_3_Mev.eps}\n\\caption{Vertical transverse phase-space of the 3 MeV \\ensuremath{\\text{H}^-\\ } beam measured by laserwire (top) and slit and grid method (bottom).}\n\\label{fig:phase_space}\n\\end{center}\n\\end{figure}\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Comparison of normalized emittance and Twiss parameters obtained with both methods}\n    \\begin{tabular}{lcc}\n    \\hline\n    \\hline\n                                   &Slit-Grid   &Laser-Diamond\\\\\n    \\hline\n\tNorm. emittance [$\\pi$ mm mrad]   & 0.242   & 0.215\\\\\n\tBeta [m]                        & 0.906   & 0.848\\\\\n\tGamma [$m^{-1}$]                & 1.16    & 1.328\\\\\n\tAlpha                           & -0.225  & -0.348\\\\\n    \\hline\n    \\hline\n    \\end{tabular}%\n  \\label{tab:twiss}%\n\\end{table}\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig13_LW_PhaseSpace_Profiles.eps}\n\\caption{Vertical profile of the \\ensuremath{\\text{H}^-\\ } beam measured by laserwire (dotted curve) and slit and grid (dashed curve) methods.}\n\\label{fig:profiles}\n\\end{center}\n\\end{figure}\nAs one can see from the plot, despite the lower spatial resolution of the laserwire data in this first test, the ellipse size and orientation in both pictures is in a good agreement. Vertical position of centroids on both pictures are slightly different. It can be explained by the fact that the exact position of the laser focus and the detector center was not perfectly aligned with respect to the center of the ion beam.\\\\\nTable~\\ref{tab:twiss} shows a comparison of the obtained values for the normalized RMS emittance and the twiss parameters. Both methods show reasonable agreement. The differences can be explained by several effects. The implantation effects of the protons in the diamond bulk might have caused some non-linearities in the detector response. Furthermore for the quadrupol settings used, a small percentage of particles might have missed the diamond detector, as it is smaller than the wire-grid. Finally, the noise suppression excluded all values in phase space which are below a certain threshold corresponding to a percentage of the maximum amplitude. Some dependency on the applied threshold also has an influence on the results shown in Table~\\ref{tab:twiss}.\\\\\nIn order to compare as well the vertical beam profiles, projections of the phase-space distributions were calculated. The vertical profile of the ion beam for both laserwire and slit and grid measurements is presented in Fig.~\\ref{fig:profiles}.\nAs one can see, the beam core is in a very good agreement. Values at the tails are slightly bigger when measured with the laserwire system. Also, it should be noted that the beam profiles obtained with two different methods have a non-Gaussian shape. The fact that the vertical profile of the ion beam has a non-Gaussian shape regardless of the measurement methods indicates that such shape of the beam is caused by the processes of beam generation at the source rather than a measurement error.\n\n\\section{\\label{sec:conclusions}Summary and outlook}\n\nIn the last 2 years the development of a laserwire emittance scanner was accomplished. In a first step simulations were conducted to determine the key parameters for laser and detector subsystems. After characterisation of the identified components, a prototype system was designed and installed at the LINAC4 3\\,MeV testbench.\nIn comparison to conventional systems for emittance measurement, the proposed laserwire system has major advantages. Because of its principle, space charge perturbations are excluded. Its range of application starts at beam energies of MeV and reaches beyond Multi-GeV. Since no mechanical parts intercept the beam, it is a reliable and fully non-destructive method.\\\\\nThe chosen laser system with kilohertz pulse repetition rate allows to to probe the emittance value of the \\ensuremath{\\text{H}^-\\ } beam pulse with microsecond resolution. The fibre based laser system increases the reliability of the laser transport system and reduces greatly its complexity.\\\\\nThanks to its diamond-based detector system with high bandwidth and sensitivity the presented system works with 3 orders of magnitude less laser peak-power than previous laser-based  systems, which marks a rise of the state of the art in this field.\\\\\n\nThe prototype system was successfully commissioned at the LINAC4 3\\,MeV beam. A comparison of phase-space and beam profile with the conventional slit and grid technique showed very good agreement. The twiss parameters and normalized emittance showed reasonable agreement taking into account the systematic error sources for this first prototype. \\\\\nThe following commissioning stages at 50~MeV and 100~MeV at LINAC4 will be used to test a modified system for beam profile measurements by collecting the detached electrons. Subsequently further improvements for the final system design will be conducted. Main targets will be the detector resolution and performance as well as the range of the fibre-based laser transport. Finally an operational system is planned to be installed in the 160\\,MeV area of LINAC4 to monitor constantly the transverse emittance of the \\ensuremath{\\text{H}^-\\ } beam at its future operation.\n\n\\section{Acknowledgements}\nWe want to thank the LINAC4 operations team to give us the opportunity to benchmark our system, the FETS collaboration for lending us their laser, Peter Savage and Richard Epsom for support with design and manufacture of final focus assembly, the BI-group at CERN for their engineering support and Benjamin Cheymol and Francesca Zocca for the slit and grid reference measurements. Moreover we acknowledge the support of the Marie Curie Networks LA3NET and oPAC which are funded by the European Commission under Grant Agreement Number 289191 and 289485.\\\\\nEspecially we would like to acknowledge the contribution of Christoph Gabor to this work and the development of laserwire technologies generally, who sadly passed away before this paper could be published. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\nMany of the results obtained for the blocks of  group algebras are extended to the general situation of $p$-permutation algebras, for instance this is the case of the Brauer pairs that where introduced in this general situation in  \\cite{BrPu}. M. Harris obtained in \\cite{Ha} two category isomorphisms involving generalized Brauer pairs  determined by a $G$-invariant block of  $kN,$ where $N$ is normal in $G.$ We  extend these results to a general class of $p$-permutation algebras. \n\nThroughout the paper $k$ is a field of characteristic $p,$ not necessarily algebraically closed and $N$ is a normal subgroup of the finite group $G.$  Section \\ref{sec2} refers to the definition of saturated triples, while Section \\ref{sec3} offers an extension of a useful result. Section \\ref{sec4} proves the first main result of the paper, that is Theorem \\ref{theorem39}. Sections \\ref{sec5} and \\ref{sec6} gather information on covering points on permutation algebras, that were introduced in \\cite{Co}. Finally, in Section \\ref{sec7} we prove the second main result of the paper, namely Theorem \\ref{theorem63}.\n\nFor basic facts regarding points on $G$-algebras we follow \\cite{Th} and for the language of Brauer pairs in the context of $p$-permutation algebras we refer to \\cite{AsKeOl} and \\cite{BrPu}. We use  methods on lifting idempotents as in \\cite[ Section 3]{Pu}. \n\n\\section{On the definition of saturated triples}\\label{sec2}\nFirst  we collect  without proof some properties about idempotents   of a  $k$-algebra $A,$ as consequences of \\cite[Proposition 3.19]{Pu}.\n\\begin{lem}\\label{Lemma71} Let $A$ be a unital $k$-algebra. Let $e,f$ be two idempotents of $A$.\n\\begin{itemize}\n\\item[(i)] If $e$ is in $Z(A)$ or $f$ is in $Z(A)$  then $ef=fe$  is also an idempotent;\n\\item[(ii)] If $e$ is primitive then $ef=0$ or  there is $a\\in A^*$ with $ef^a=f^ae=e;$\n\\item[(iii)] If $e$ is a primitive idempotent of $A$ and $e\\in Z(A)$ or $f\\in Z(A)$ then either $ef=0$ or $ef=e=fe.$\n\\end{itemize}\n\\end{lem}\n\nThe purpose of the next proposition is to show that  the definition of saturated triple $(A,c,G)$ (see\\cite[Definition 3.2, Part IV]{AsKeOl}) does not require $c$ to be a central idempotent.\n\\begin{prop}\\label{prop22} Let $A$ be a $p$-permutation $G$-algebra and $c$ a primitive idempotent of $A^G$. Let $Q$ be a $p$-subgroup of $G$ and $e$ a block of $A(Q)$. If $e$ is a primitive idempotent of $(A(Q))^{C_G(Q,e)}$ then $(Q,e)$ is a $(A,c,G)$-Brauer pair if and only if $\\operatorname{Br}_Q(b)e=e.$\n\\end{prop}\n\n\\begin{proof}\nIf $(Q,e)$ is a $(A,c,G)$-Brauer pair then $\\operatorname{Br}_Q(b)\\neq 0.$ The $N_G(Q)$-algebra epimorphism $A^Q\\to A(Q)$ gives $\\operatorname{Br}_Q(A^G)=A(Q)^{N_G(Q)},$ hence $\\operatorname{Br}_Q(c)\\in (A(Q))^{C_G(Q,e)}.$ At this point Lemma \\ref{Lemma71} establishes the result.\n\\end{proof}\n\\begin{rem} Recall from  \\cite[Introduction p.92]{KeKuMi} that a triple $(A,c,G)$ is saturated if: $c$ is a central, primitive idempotent of $A^G$ (where $A$ is  a $p$-permutation $G$-algebra); and, for any $(A,c,G)$-Brauer pair $(Q,e)$, the idempotent $e$ is primitive in $A(Q)^{C_G(Q,e)}$. Following carefully the proof of \\cite[Theorem 1.6]{KeKuMi} we noticed that the condition (i) of this theorem, the fact that $c$ is a central idempotent of $A$, is needed in \\cite[Lemma 3.2]{KeKuMi} and \\cite[Lemma 3.4]{KeKuMi}. As a matter of fact in \\cite[Lemma 3.4]{KeKuMi} the authors missed this assumption.  However, by Proposition \\ref{prop22} we can show now that the conclusions of  \\cite[Lemma 3.2]{KeKuMi} and \\cite[Lemma 3.4]{KeKuMi} are true assuming condition (ii) of  \\cite[Theorem 1.6]{KeKuMi}, without assuming that $c$ is central in $A$.\n\\end{rem}\n\\section{Generalized Brauer pairs and $p$-permutation $N$-interior $G$-algebras}\\label{sec3}\nLet $A$ be a $p$-permutation $N$-interior $G$-algebra, where $N\\trianglelefteq G$. Also consider $c\\in A^G$, a primitive idempotent of that algebra.\n\nSimilar to \\cite[Definition 1.6]{BrPu} we recall that\n$(Q,f_Q)$ is an $(A,c,G)$- Brauer pair if $Q$ is a $p$-subgroup of $G$ and $f_Q$ is a block of $A(Q)$ with $\\operatorname{Br}_Q(c)f_Q\\neq 0$. We follow the notations of \\cite{KeKuMi} and \\cite[Part IV]{AsKeOl}.\nNotice  that in an $(A,c,G)$-Brauer pair,  $Q$ is chosen such that $\\operatorname{Br}_Q(c)\\neq 0$. Also $\\operatorname{Br}_Q(c)f_Q$ is an idempotent of $A(Q)^{C_N(Q)}$.\nThe next proposition is an extension of \\cite[Lemma 8.6.4]{Link} and of \\cite[IV, Lemma 3.5]{KeKuMi}, although the proof is similar to those of the mentioned lemmas, for completeness we present it here.\n\\begin{prop}\\label{prop23}\nLet $(Q,f_Q)$ denote an $(A,c,G)$-Brauer pair verifying that $f_Q$ is primitive in $A(Q)^{C_N(Q)}$. Consider a $p$-subgroup $S$ of $G$ and a subgroup $H$ of $G$ with  $C_N(Q)\\leq H\\leq N_G(Q, f_Q)$, such that $Q\\leq S\\leq H.$ Also let $f_S$ be a block of $A(S)$. The following are equivalent:\n\\begin{itemize}\n    \\item[i)] $(S,f_S)$ is an $(A,c,G)$- Brauer pair with $(Q,f_Q)\\leq(S,f_S)$;\n    \\item[ii)] $(S,f_S)$ is an $(A(Q),f_Q,H)$-Brauer pair.\n\\end{itemize}\n\n\\end{prop}\n\\begin{proof}\nSince $A(Q)^H\\subseteq A(Q)^{C_N(Q)}$ it follows that $f_Q$ remains a primitive idempotent in $A(Q)^H$. In \\cite[Theorem 1.8]{BrPu} is stated that $(Q,f_Q)$ is the unique Brauer pair included in $(S,f_S)$ and that $f_Q$ is invariant with respect to the $S$-action. This means $\\operatorname{Br}_S(f_Q)f_S=f_S$.\n\nConversely, by our assumptions we have  $\\operatorname{Br}_Q (c)f_Q=f_Q$ and $\\operatorname{Br}_S (f_Q) f_S =f_S$, see Proposition \\ref{prop22}. Indeed, $\\operatorname{Br}_S:A(Q)^S\\rightarrow A(S)$ is an epimorphism of algebras verifying $\\operatorname{Br}_S((Z(A(Q)^S))\\subseteq Z(A(S))$.\n\nThe commutativity of the diagram\n$$\\begin{xy} \\xymatrix{ A^S\\ar@{->}[rr]^{\\operatorname{Br}_Q}\\ar@{->}[dr]_{\\operatorname{Br}_S} &&A(Q)^S\\ar@{->}[dl]^{\\operatorname{Br}_S}\\\\ &A(S)    } \\end{xy}$$\ngives\n$$\\operatorname{Br}_S(c)f_S=\\operatorname{Br}_S(c)\\operatorname{Br}_S(f_Q)f_S=\\operatorname{Br}_S(\\operatorname{Br}_Q(c))\\cdot \\operatorname{Br}_S(f_Q)f_S = $$\n$$ = \\operatorname{Br}_S(\\operatorname{Br}_Q(c)\\cdot f_Q)f_S = \\operatorname{Br}_S(f_Q)f_S  = f_S$$\nThis shows that $(S,f_S)$ is a $(A,c,G)$-Brauer pair.\n\nApplying again \\cite[Theorem 1.8]{BrPu}, it provides an $S$-invariant block of $A(Q)$, denoted $f_Q'$, which satisfy $(Q, f_Q')\\leq (S,f_S)$ and $\\operatorname{Br}_S(f_Q')f_S = f_S$. If $f_Q\\neq f_Q'$ then $$f_S = f_S \\operatorname{Br}_S(f_Q) = f_S \\operatorname{Br}_S(f_Q)\\operatorname{Br}_S(f_Q')=0,$$ a contradiction.\n\\end {proof}\n\n\\section{Isomorphisms of categories of Brauer pairs} \\label{sec4}\nIn this section we assume that $c$ is a $G$-invariant primitive idempotent of $A^N$ and that $Q$ is a defect group of $c$ in $N$. \n\nFor any defect group $Q$ of $c$ in $N$ there is a defect group $P$ of $c$ in $G$ such that $Q=P\\cap N$. This claim is well known but, for consistency,  we give the justifying details  in the next lines. Indeed, if $Q$ is a defect group in $N$ of $c$, then $c\\in A_Q^N$ and $Q$ is minimal with this property. Similarly, if $R$ is a defect group of $c$ in $G$ then $R$ is minimal with the property $c\\in A_R^G$. Given the decomposition:\n$$A_R^G\\subseteq \\underset{x\\in [N\\setminus {G\/R}]}{\\displaystyle\\sum} A_{N\\cap\\  R^x  }^N,$$\nwe obtain $c\\in A^N_{N\\cap R^x}$, for some $x\\in G$. Since $\\operatorname{Br}_R(c)\\neq 0$ it follows $\\operatorname{Br}_{R^x}(c)\\neq 0$, hence $\\operatorname{Br}_{N\\cap\\ R^x}(c)\\neq 0$. So, there is $y\\in N$ with $Q^y=N\\cap\\ R^x$. It follows $Q=N\\cap\\ R^z,\\ z=xy^{-1}$. We set\n$P =R^z$ and the claim is proved.\n\\begin{rem}\\label{rem32}\nIf we fix a $(A,c,G)$- Brauer pair $(Q,f_Q)$ such that $Q$ is a defect group of $c$ in $N$, the above discussion provides a maximal $(A,c,G)$-Brauer pair $(P,f_P)$ such that\n$$(Q,f_Q)\\leq (P,f_P)^x\\text{ for some } x\\in N.$$\n\\end{rem}\n\\begin{defn}\\label{definition33} Let $(Q,f_Q)$ be a pair as in Remark \\ref{rem32}, such that $f_Q$ remains primitive in $A(Q)^{C_N(Q)}$. Denote $H=N_G(Q,f_Q)$ and define the sets:\n$$F_1:=\\{(R,f_R)|(R,f_R)\\text{ is a }(A,c,G)\\text{-Brauer pair such that}$$\n$$(Q,f_Q)\\leq(R,f_R)\\text{ as }(A,c,G)\\text{-Brauer pairs }\\}$$\n$$F_2:=\\{(R,f_R)|(R,f_R)\\text{ is a }(A(Q),f_Q,H)\\text{-Brauer pair such }$$\n$$\\text{that }(Q,f_Q)\\leq(R,f_R)\\text{ as }(A(Q),f_Q, H)\\text{-Brauer pairs}\\}$$\n\n\\end{defn}\n\\begin{cor}\\label{corollary34}\nAssume $(Q,f_Q)$ is as in Definition \\ref{definition33}. Let $S$ be a $p$-subgroup of $G$ such that $Q\\leq S$ and let $f_S$ be a block of $A(S)$. The following are equivalent:\n\\begin{itemize}\n    \\item[a)] $(S,f_S)\\in F_1;$\n    \\item[b)] $(S,f_S)\\in F_2.$\n\\end{itemize}\n\\end{cor}\n\\begin{proof}\nRemark \\ref{rem32} states that $Q\\trianglelefteq S$. The rest of the proof follows from Proposition \\ref{prop23}.\n\\end{proof}\n\\begin{prop}\n\\label{Proposition35}\nLet $S,T$ be two $p$-subgroups of $G$ and let $f_S,f_T$ denote blocks of $A(S),$ $A(T)$, respectively.\n\nThe following are equivalent:\n\\begin{itemize}\n    \\item[1)] $(S,f_S)\\leq(T,f_T)$ as elements of $F_1$;\n    \\item[2)] $(S,f_S)\\leq(T,f_T)$ as elements of $F_2$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nLet $e$ denote a primitive idempotent  appearing in a pairwise orthogonal  decomposition of $f_T$ into primitive idempotents in $A(T).$ By lifting idempotents, there exists a primitive idempotent $i\\in A^T$  such that $e=\\operatorname{Br}_T(i)$, hence $\\operatorname{Br}_T(i)f_T=\\operatorname{Br}_T(i)$. By our assumption we also have $\\operatorname{Br}_S(i)f_S\\neq 0$. Since $i$ is  primitive in $A^T$, we obtain that $j:=\\operatorname{Br}_Q(i)$ is also a primitive idempotent of $A(Q)^T$. We have\n$$0\\neq \\operatorname{Br}_T(i)f_T=\\operatorname{Br}_T^{A(Q)}(j)f_T.$$\nSimilarly  we obtain $$0\\neq \\operatorname{Br}_S(i)f_S=\\operatorname{Br}_S^{A(Q)}(\\operatorname{Br}_Q(i))f_S.$$\nThis proves that $(S,f_S)\\leq (T,f_T)$ as elements of $F_2$ according to \\cite[Corollary 1.9]{BrPu}.\n\nConversely, considering a similar decomposition of $f_T$ in $A(T)$, we find a primitive idempotent $e$ with $f_Te=f_T$. Let $j$ be such that $\\operatorname{Br}_T^{A(Q)}(j)=e$ and let $i$ be such that $\\operatorname{Br}_T(i)=e$. Then we may assume $i$ and $j$ to be primitive idempotents in $A^T,A(Q)^T$, respectively. Further we get $\\operatorname{Br}_Q(i)=j^a,$ for some unit $a$ in $A(Q)^T.$\nWe obtain:\n$$\\operatorname{Br}_T(i)f_T = \\operatorname{Br}_T^{A(Q)}(j^a)f_T = f_T\\neq 0.$$\nAgain $\\operatorname{Br}_T(i)\\neq 0$ implies $\\operatorname{Br}_S(i)\\neq 0$ and then\n$$0\\neq \\operatorname{Br}_S^{A(Q)}(j^a)f_S=\\operatorname{Br}_S(i)f_S.$$\nThis concludes the proof.\n\\end{proof}\n\\begin{cor}\n\\label{corollary36} Let  $(U,f_U)$ be an $(A,c,G)$-Brauer pair such that $(Q,f_Q)\\leq (U,f_U)$. Then $(U,f_U)$ is a maximal $(A,c,G)$-Brauer pair if and only if it is a maximal $(A(Q),f_Q,H)$-Brauer pair.\n\\end{cor}\n\n\n\nIf $(U,f_U)$ is a maximal $(A,c,G)$-Brauer pair, then $\\mathcal{F}_{(U,f_U)}(A,c,G)$ is a finite category, called fusion system,  introduced in \\cite[Definition 2.3]{KeKuMi}.\nWe will use the same notations for the homomorphisms in  $\\mathcal{F}_{(U,f_U)}(A,c,G)$. If $S,T\\leq U$  then\n$$\\mathrm{Hom}_{\\mathcal{F}_{(U,f_U)}(A,c,G)}(S,T)=\\{c_x: S\\rightarrow T| x\\in G,\\  {}^x(S,f_S)\\leq (T,f_T) \\},$$\nwith $c_x(u)=xux^{-1}$ for any $u\\in S$.\n  The similar notations are used for $\\mathcal{F}_{(U,f_U)}(A(Q),f_Q,H)$. In this case (that is $(U,f_U)$ is maximal), the full subcategory of $\\mathcal{F}_{(U,f_U)}(A,c,G)$  whose elements are those of $F_1$ is denoted by $\\mathcal{C}$. The full subcategory of $\\mathcal{F}_{(U,f_U)}(A(Q),f_Q,H)$  whose elements are those of $F_2$ is is denoted by $\\mathcal{D}$.\n\n\\begin{prop}\n\\label{proposition38}\nLet  $(U,f_U)$ be a maximal  $(A,c,G)$-Brauer pair such that $(Q,f_Q)\\leq (U,f_U)$. If $(S,f_S)\\in F_1$ and $(T,f_T)$ is a Brauer pair of  $\\mathcal{F}_{(U,f_U)}(A,c,G)$ then\nthe set $\\mathrm{Hom}_{\\mathcal{F}_{(U,f_U)}(A,c,G)}(S,T)$ is included in $ \\mathrm{Hom}_{\\mathcal{F}_{(U,f_U)}(A(Q), f_Q,H)}(S,T)$ . \n\\end{prop}\n\\begin{proof}\nSimilarly to the proof of \\cite[Proposition 5]{Ha}.\n\\end{proof}\nBy collecting the results of this section we can prove now the following theorem.\n\\begin{thm}\n\\label{theorem39} The identity functor from $\\mathcal{D}$ to the category $\\mathcal{C}$ provides an isomorphism.\n\\end{thm}\n\n\n\\section{Covering points revisited} \\label{sec5}\nIn this section we assume that our $p$-permutation $G$-algebra $A$ contains $C$ as a direct sumand of $k$-modules, where $C$ is an $N$-interior $G$-algebra, for some normal subgroup $N$ of $G$. We recall the definition of covering idempotents in the general situation, see \\cite[Definition 3.2]{Co}.\n\\begin{defn}\n\\label{definition41} Let $j\\in C^N$ denote a primitive idempotent with defect group $Q$, a $p$-subgroup of $N$. The primitive idempotent $i\\in A^G$ covers $j$ if:\n\\begin{itemize}\n    \\item[$a)$] $i$ admits a defect group $P$ such that $P\\cap N=Q$;\n    \\item[$b)$] there is a primitive idempotent $f$ of $A^N$ with defect group $Q$ such that\n    $$if^a=f^ai=f^a\\text{ and }jf=fj=fi\\text{ for some } a\\in(A^N)^*.$$\n\\end{itemize}\n\\end{defn}\n\\begin{rem}\n\\label{remark42}\nIf $i\\in A^G$ covers $j$ than the entire $(A^G)^*$-conjugacy class of $i$ covers $j$ as in \\cite[Definition 3.2]{Co}. Consequently a point $\\alpha\\subseteq A^G$ covers $j$, if $\\alpha$ contains an idempotent that verifies $a)$ and $b)$ in Definition \\ref{definition41}. Also note that the entire $(A^N)^*$-conjugacy class of $j$ is covered by $\\alpha$.\n\\end{rem}\nOur situation requires  a particular definition of covering idempotents, that is also equivalent to the block covering situation when working with group algebras.\n\\begin{defn}\n\\label{definition43}\nLet $j\\in C^N$ be a primitive idempotent with defect group $Q$ in $N$ and  assume $j$ is $G$-invariant. A point $\\alpha\\in A^G$ covers $j$ if there is $i\\in\\alpha$ and a point $\\epsilon\\subseteq A^N$ with  defect group $Q$ verifying the conditions:\n\\begin{itemize}\n    \\item[$a')$] $ji = ij$;\n    \\item[$b')$] $if = fi = f$ and $jf_1=f_1j=f_1$ for some $f$ and $f_1$ belonging to $\\epsilon$.\n\\end{itemize}\n\\end{defn}\n\\begin{prop}\n\\label{proposition44} Fix a primitive idempotent $j\\in C^N$, admitting a defect group $Q$ and assume $j\\in C^G$. If the point $\\alpha\\subseteq A^G$ covers $j$ as in Definition \\ref{definition43} then $\\alpha$ covers $j$ as in Definition \\ref{definition41}.\n\\end{prop}\n\\begin{proof}\nCondition $b')$ of  Definition \\ref{definition43} implies condition $b)$ of Definition  \\ref{definition41}. We only need to prove that if $a')$ holds then a defect group $P$ of $\\alpha$ gives $P\\cap N = Q$.\\\\\nLet $R$ be a defect group of $\\alpha$. There is $a\\in A^R$ such that\n$$i=Tr_R^G(a)=\\underset{x\\in [N\\setminus G\/R]}{\\sum}Tr_{N\\cap R^x}^N(a^x)\\in\\underset{x\\in[N\\setminus G\/R]}{\\sum}A_{N\\cap R^x}^N$$\nSince $if=fi=f$ the idempotent $f$ belongs to some ideal $A_{N\\cap R^x}^N$, implying $Q^y\\leq N\\cap R^x$ for some $y\\in N$. Then $Q\\leq N\\cap R^g$ for some $g\\in G$. Let $T$ denote a defect group of $j$ as an idempotent of $A^G$, hence $j\\in C_T^G\\subseteq A_T^G$. We also have $ij=ji=i$ in $A^G$, hence  we may assume $R\\leq T$. Next, there is $b\\in C^T$ with\n$$j=Tr_T^G(b)=\\underset{y\\in [N\\setminus G\/T]}{\\sum}Tr_{N\\cap T^y}^N(b^y)\\in\\underset{y\\in[N\\setminus G\/T]}{\\sum}C_{N\\cap T^y}^N$$\nBy our assumption $j\\in C_{N\\cap T^y}^N$ for some $y\\in G$. Since $\\operatorname{Br}_T(j)\\neq 0$ we get $\\operatorname{Br}_{T^y}(j)\\neq 0$ and then $\\operatorname{Br}_{N\\cap T^y}(j)\\neq 0 $. This means $Q^t=N\\cap T^y$ for some $t\\in N$, equivalently $Q = N\\cap T^z$ for some $z\\in G$. At last $$Q\\leq N\\cap R^g\\leq N\\cap T^g = Q^{z^{-1}g},$$ forcing $z^{-1}g\\in N_G(Q)$ and $$Q=N\\cap R^g = N\\cap T^g.$$\n\\end{proof}\n\\begin{cor}\n\\label{corollary45}\nIf $j$ is a primitive idempotent of $C^N$ with defect group $Q$ and $G_j$ is the stabilizer of $j$ in $G$ then there is a defect group $T$ of $j$ in $G_j$ such that $T\\cap N=Q$.\n\\end{cor}\n\\begin{rem}\n\\label{remark46}\nAs it follows from \\cite[Remark 3.1]{Co} there is a point $\\epsilon\\in A^N$ with defect group $Q$ such that $jf = fj = j$ in $A^N$ for some $f\\in \\epsilon$.\nThe justification for introducing  Definition \\ref{definition43} is explained in the next lines. Since $j$ is $G$-fixed we obtain $j\\in C^G\\leq A^G$. The points covering $j$ are some of the $(A^G)^*$-conjugacy classes of primitive idempotents of $A^G$ appearing in a decomposition $j=\\underset{i\\in I}{\\sum}i$ of $j$ into pairwise orthogonal idempotents. All idempotents $i\\in I$ give $ij=ji=i$,  but only  part of them verify $if_1=f_1i=f_1$, for some $f_1\\in\\epsilon$.\n\\end{rem}\n\\begin{cor}\n\\label{corollary47}\nLet $j$ be a primitive idempotent in  $C^N$, admitting a defect group $Q$ and assume that $j$ is $G$-invariant. The point $\\alpha\\subseteq A^G$ covers $j$ if and only if  there exists a point $\\epsilon$ of $N$ on $jAj,$ adimitting with defect group $Q,$    and a primitive idempotent $i\\in\\alpha$ satisfying $i\\in(jAj)^G$ and $if=fi=f$ for some $f\\in\\epsilon$.\n\\end{cor}\n\\begin{proof}\nIf $\\alpha\\subseteq A^G$ covers $j$,  Definition \\ref{definition43} states that there is $i\\in\\alpha$ such that $ij=ji=i$ or, equivalently, $i\\in(jAj)^G$. There is also $\\epsilon'\\subseteq A^N$ with defect group $Q$ such that $if=fi=f$ and $jf_1=f_1j=f_1$ where $f,f_1\\in\\epsilon'$. Then $$jf=jif=if=fj=f,$$ where $f$ and $f_1$ are $(A^N)^*$-conjugates, primitive idempotents of $(jAj)^N$. Let $\\epsilon$ be the point of $(jAj)^N$ which contains $f$. Since $\\epsilon'$ is the unique point of $A^N$ corresponding to $\\epsilon$ via the embedding $$(jAj)^N\\rightarrow A^N$$ and since $f,f_1\\in\\epsilon'$ we must have $f_1\\in\\epsilon$. Clearly $\\epsilon$ has defect group $Q$.\n\nThe converse is immediate by noticing that for any $a\\in((jAj)^N)^*$ the element $b:=a+i-j$ is in $(A^N)^*$ and for any $f\\in\\epsilon$ we have $f^a=f^b$.\n\\end{proof}\n\n\\section{On pairs determined by covering points} \\label{sec6}\nAs in Section \\ref{sec4} we continue with a $p$-permutation $G$-algebra $A$, which contains $C$ as direct summand of $k$-modules. The first term $C$ is an $N$-interior $G$-algebra, for a normal subgroup $N$ of $G$. Let $c$ be a $G$-invariant primitive idempotent of $C^N$. Recall that  a  $( C, c, G)$- Brauer pair $(R,f_R)$ consists of a $p$-subgroup $R$ in $G$ and a block $f_R$ of $C(R)$ such that $\\operatorname{Br}_R(c)f_R\\neq 0$.\n\n\\begin{rem}\n\\label{remark52}\nIf the $p$-subgroup $Q$ of $N$ is a defect group of $c$ and $(Q,f_Q)$ is a $(C,c,G)$- Brauer pair such that $f_Q$ is primitive in $C(Q)^{C_N(Q)}$ then we may also consider the sets $F_1$ and $F_2$ as in Section \\ref{sec3} such that Theorem \\ref{theorem39} still applies.\n\\end{rem}\n\\begin{prop}\n\\label{proposition53}\nLet $\\alpha\\subseteq A^G$ be a point covering $c$. There is a defect group $P$ of $\\alpha$ such that $P\\cap N=Q$ and a block $f_P$ in $C(P)$ such that $(P,f_P)\\in F_1$.\n\\end{prop}\n\\begin{proof}\nBy Proposition \\ref{proposition44} we have a defect group $R$ of $\\alpha$ with $R\\cap N=Q$. We obtain $\\operatorname{Br}_Q(c)f_Q=f_Q$ in $C(Q)^{C_N(R)}$ and then, for any $x\\in N_G(Q)$, we still have $\\operatorname{Br}_Q(c)f_Q^x=f_Q^x$ in $C(Q)$. If we set $H=N_G(Q,f_Q)$ then $s:=\\underset{x\\in N_G(Q)\/H}{\\sum}f_Q^x$ is an idempotent lying in $$C(Q)^{N_G(Q)}\\cap Z(C(Q))$$  and satisfying $$\\operatorname{Br}_Q(c)=\\operatorname{Br}_Q(c)s=s\\  \\operatorname{Br}_Q(c)=s$$ in $C(Q)^{N_G(Q)}$, since $\\operatorname{Br}_Q(c)$ is a primitive idempotent in this algebra. In particular $\\operatorname{Br}_Q(c)$ lies in $$Z(C(Q))\\cap C(Q)^R\\subseteq Z(C(Q)^R),$$ since $Q$ is normal in $R$.\n\n Since $\\alpha$ covers $c$ there is $i\\in\\alpha$ with $ic=i=ci$. Hence $\\operatorname{Br}_R(c)\\neq 0.$ Let $\\operatorname{Br}_R(c)=\\sum f_R$ be a decomposition of $\\operatorname{Br}_R(c)$ into blocks of  $Z(C(R))$. Also consider  $c=\\underset{j\\in J}{\\sum}j,$  a decomposition of $c$ into  pairwise orthogonal primitive idempotents of $C^R$. There is $j\\in J$ giving   $\\operatorname{Br}_R(j)\\neq 0.$  The idempotent $\\operatorname{Br}_R(j)$ is primitive in $C(R)$ and then there is a unique block of $C(R)$, denoted $\\overline{f_R}$, with the properties $0\\neq \\operatorname{Br}_R(j)\\overline{f_R}$ and $0\\neq \\operatorname{Br}_R(c)\\overline{f_R}.$\n\nWe also have $\\operatorname{Br}_Q(j)\\operatorname{Br}_Q(c)\\neq 0$, a relation which determines at least one block, say $f_Q^x$, with $\\operatorname{Br}_Q(j)f_Q^x\\neq 0$. According to \\cite[Corollary 1.9]{BrPu} the blocks $\\overline{f_R}$ of $C(R)$ and $f_Q^x$ of $C(Q)$ must verify $(Q,f_Q^x)\\leq (R,\\overline{f_R})$. Since $x\\in N_G(Q)$ this last relation becomes $(Q,f_Q)\\leq (R^{x^{-1}},\\overline{f_R}^{x^{-1}})$. We set $P:=R^{x^{-1}}$ and $f_P:=\\overline{f_R}^{x^{-1}}$ to conclude the proof.\n\\end{proof}\n\n\n\n\\section{Isomorphism of fusion subcategories} \\label{sec7}\nIn this section $A$ is an interior $G$-algebra  which admits a $G\\times G$-stable $k$-basis.  As before,  $C$ is an $N$-interior $G$-algebra and a direct summand of $A$ as $k$-module. We also assume that $A$ is projective as an $k[N\\times 1]$-module and as an $k[1\\times N]$-module. We keep the assumptions on the block $f_Q.$\n \nLet $X:=\\overline{N}^{\\operatorname{Aut}(Q)}_A(Q)$  denote the extended Brauer quotient, the $N_N(Q)$-interior $N_G(Q)$-algebra constructed in \\cite{PuZh}. Note that, according to \\cite[Proposition 2.2]{CoTo}, there is an action of $N_G(Q)$ on $\\operatorname{Aut}(Q)$. Also let $Y:=\\overline{N}^{\\operatorname{Aut}(Q)}_C(Q)$ denote the extended Brauer quotient constructed in \\cite{CoTo}, associated with the   $N$-interior $G$-algebra $C.$ Notice that $Y$ is an $N_N(Q)$-interior $N_G(Q)$-subalgebra of $X$, in fact $Y$ is a direct summand of $X$ as $k$-modules. Furthermore if $K\\leq \\operatorname{Aut}(Q)$ with $K=N_N(Q)\/C_N(Q)$, then the map\n$$ C(Q)\\otimes_{C_N(Q)}N_N(Q)\\ni \\bar{c}\\otimes n\\mapsto \\overline{cn}\\in \\overline{N}_C^K(Q)$$ is an $N_N(Q)$-interior $N_G(Q)$-algebra isomorphism, making $C(Q)\\otimes_{C_N(Q)}N_N(Q)$ into an $N_N(Q)$-interior $N_G(Q)$-subalgebra of $Y.$ For details see \\cite[Section 2]{CoMa}. Finally if $A=kG=kN\\otimes_NG$ then \\cite[Remark 4.5) 2]{CoMa} states $X=kN_G(Q).$\n\n\\begin{lem}\n\\label{lemma61}\nLet $H = N_G(Q,f_Q)$. Any point of $H$ on $X$ that covers $f_Q$ has a defect group $P$ such that $P\\cap N = Q$ and, there is a block $f_P$ of $C(P)$ such that $(Q,f_Q)\\leq (P,f_P)$ as $(C(Q),f_Q, H)$-Brauer pairs.\n\\end{lem}\n\\begin{proof}\nThe block $f_Q$ of $C(Q)$ remains a primitive idempotent of $C(Q)^L$, for any subgroup $L$ of $H$ containing $QC_N(Q)$. The algebra $C(Q)\\otimes_{C_N(Q)}QC_N(Q)$ is an $H$-acted subalgebra of $Y$, and then $f_Q$ is a primitive idempotent of $(C(Q)\\otimes_{C_N(Q)}QC_N(Q))^{QC_N(Q)}$, according to \\cite[Theorem 3.1]{CoTo}.\n\nWe have $$f_Q\\in C(Q)^{C_N(Q)}=C(Q)_Q^{QC_N(Q)}\\subseteq (C(Q)\\otimes_{C_N(Q)}QC_N(Q))^{QC_N(Q)}_Q$$ and $\\operatorname{Br}_Q^{C(Q)\\otimes_{C_N(Q)}QC_N(Q)}(f_Q)\\neq 0$, implying that $Q$ is a defect group of $f_Q$ in $QC_N(Q)$. Indeed, in fact $\\operatorname{Br}_Q^{C(Q)\\otimes_{C_N(Q)}QC_N(Q)}(f_Q)=f_Q$, since $(C(Q)\\otimes u)(Q)=0$ for any $u\\in Q\\setminus Z(Q)$.\n\nAssuming that for some $u\\in Q\\setminus Z(Q)$ there is an element $\\overline{c}$ belonging to a $N_G(Q)$-stable basis of $C(Q)$ that verifies $(\\overline{c}\\otimes u)^y = \\overline{c}\\otimes u$ for any $y\\in Q$, we obtain $y^{-1}uy=u$ for any $y\\in Q$, a contradiction. Then $$(C(Q)\\otimes_{C_N(Q)}QC_N(Q))(Q)=C(Q),$$ equality which clarifies the last statement. We apply Proposition \\ref{proposition53} for $H,QC_N(Q), X,$ $C(Q)\\otimes_{C_N(Q)}QC_N(Q) $ and $f_Q$, in place of $G,N,A,C$ and $c$ respectively, concluding that for a $p$-subgroup $P$ in $H$  with $P\\cap QC_N(Q)=Q$ and a block $f_P\\in(C(Q)\\otimes_{C_N(Q)}QC_N(Q))(P)$, we obtain $(Q,f_Q)\\leq (P,f_P)$ as\n$(C(Q)\\otimes_{C_N(Q)}QC_N(Q), f_Q,H)$- Brauer pairs. Notice that\n$$(C(Q)\\otimes_{C_N(Q)}QC_N(Q))(P)=(C(Q)\\otimes_{C_N(Q)}QC_N(Q)(Q))(P)=(C(Q))(P)=C(P)$$\nand then\n$(Q,f_Q)\\leq(P,f_P)$ as $(C(Q),f_Q,H)$-Brauer pairs.\n\\end{proof}\n\\begin{prop}\n\\label{proposition62} Any point of $H$ on $X$ with defect group $P$ that covers $f_Q$ determines a point of $G$ on $A$ with defect group $P$ that covers $c$. Moreover, there is a block $f_P$ of $C(P)$ such that $(Q,f_Q)\\leq(P,f_P)$ as $(C(Q), f_Q,H)$-Brauer pairs and as $(C,c,G)$-Brauer pairs.\n\\end{prop}\n\\begin{proof}\nLemma \\ref{lemma61} applies in this situation and yields the $(C(Q),f_Q, H)$- Brauer pair denoted $(P,f_P)$. Theorem \\ref{theorem39} applied for the $G$-algebra $C$ states that $(P,f_P)$ is also a $(C,c,G)$-Brauer pair containing $(Q,f_Q)$. All that remains to prove is the first assertion  of the proposition.\n\nConsider the map\n$$Tr_H^{N_G(Q)}:(f_QX f_Q)^H\\rightarrow(f_Q X f_Q)_H^{N_G(Q)},\\quad a\\mapsto Tr_H^{N_G(Q)}(a).$$\nIt is easily shown that $Tr_H^{N_G(Q)}$ is an algebra isomorphism. Let $\\gamma\\subseteq X^H$ be a point that covers $f_Q$. There is $l\\in (f_QX f_Q)^H\\cap \\gamma$ with defect group $P$ and $P\\cap N=Q$. There is also a primitive idempotent $f$, with defect group $Q$, in $(f_QX f_Q)^{QC_N(Q)}$, giving $lf=fl=f$. Now, the element\n$Tr_H^{N_G(Q)}(l)$ is a primitive idempotent with defect group $P$. Indeed,  otherwise $$Tr_H^{N_G(Q)}(l)\\in\\underset{R\\leq P}{\\sum}(f_QX f_Q)_R^P,$$ and since $f_Q$ is $P$-fixed we get $$l=f_Q\\cdot Tr_H^{N_G(Q)}(l)\\in\\underset{R\\leq P}{\\sum}(f_QX f_Q)_R^P.$$\nThere is $f_1$, a primitive idempotent of  $(f_QX f_Q)^{N_N(Q,f_Q)},$ such that $lf_1=f_1l=f_1$ and $f_1f^a=f^af_1=f^a$ for some $a\\in ((f_QX f_Q)^{QC_N(Q)})^*$. The block $f_Q$ is also a primitive idempotent of $$(C(Q)\\otimes_{C_N(Q)}N_N(Q,f_Q))^{N_N(Q,f_Q)},$$ actually a block of $$C(Q)\\otimes_{C_N(Q)}N_N(Q,f_Q),$$ with defect group $Q$. Since $\\operatorname{Br}_Q(f^a)\\neq 0$ we must have $\\operatorname{Br}_Q(f_1)\\neq 0$. Also $f_1f_Q=f_1$, forcing $f_1$ to be projective relative to $Q$. This shows that $\\gamma$ covers $f_Q$, seen as an $H$-invariant, primitive idempotent of $N_N(Q,f_Q)$ on $C(Q)\\otimes_{C_N(Q)}N_N(Q,f_Q)$.\nNext, we have $f_1Tr_H^{N_G(Q)}(l)=Tr_H^{N_G(Q)}(l)f_1=f_1$. The idempotent\n$Tr_{N_N(Q,f_Q)}^{N_N(Q)}(f_1)$ lies in $(f_QX f_Q)_{N_N(Q,f_Q)}^{N_N(Q)},$\nbeing primitive  with defect group $Q$.\n\nNext, let $T$ denote a set of pairwise orthogonal primitive idempotents of the algebra $(\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_N(Q)}$ with\n$$Tr_{N_N(Q,f_Q)}^{N_N(Q)}(f_1)=\\sum_{t\\in T}t.$$\nLet $W$ denote a set of pairwise orthogonal primitive idempotents of $(\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_G(Q)}$ such that $$Tr_{H}^{N_G(Q)}(l)=\\sum_{w\\in W}w.$$\nThere is at least one idempotent in $T$, say $t$, belonging to a point with defect group $Q$. Then, since\n$$Tr_{H}^{N_G(Q)}(l)Tr_{N_N(Q,f_Q)}^{N_N(Q)}(f_1)=Tr_{N_N(Q,f_Q)}^{N_N(Q)}(f_1)$$\nwe obtain $Tr_{H}^{N_G(Q)}(l)t=t$, equality that assures the existence of an invertible element $a\\in((\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_N(Q)})^*$ such that $t^aw=wt^a=t^a$ for some $w\\in W$.\\\\\nIf $\\operatorname{Br}_P(w)=0$ then $\\operatorname{Br}_Q(w)=0$, forcing $\\operatorname{Br}_Q(t^a)=0$. This contradiction proves that $w$ belongs to a point of $N_G(Q)$ on $X$ that covers $\\operatorname{Br}_Q(c)$.\nThe last statement says that the point of $N_G(Q)$ on $X$ that contains $w$ covers $\\operatorname{Br}_Q(c)$, viewed as a $N_G(Q)$-invariant primitive idempotent of $Y^{N_N(Q)}$. Since $ \\operatorname{Br}_Q(c)\\in C(Q)^{N_N(Q)}$ being also $N_G(Q)$-invariant, $w$ determines a point of $N_G(Q)$ on $A(Q)$ that covers $\\operatorname{Br}_Q(c)$. We have\n$$(\\operatorname{Br}_Q(c) A(Q) \\operatorname{Br}_Q(c))^{N_G(Q)}\\leq (\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_G(Q)},$$\nand according to \\cite[Proposition 3.3]{PuZh} there is $w_1\\in (\\operatorname{Br}_Q(c)A(Q)\\operatorname{Br}_Q(c))^{N_G(Q)}$ such that $w_1=w^{a_1}$ for some unit $a_1$ in $(\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_G(Q)}$.\nThere is also a primitive idempotent $t_1\\in (\\operatorname{Br}_Q(c)A(Q)\\operatorname{Br}_Q(c))^{N_N(Q)}$ such that $$w_1t_1=t_1w_1=t_1, \\quad t_1\\cdot t^{aa_1b}=t^{aa_1b}t_1=t^{aa_1b}$$ for some unit $b$ of $(\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_N(Q)}$, where $w_1$ has defect group $P$, while $t_1$ has defect group $Q$.\nThe point containing $w_1$ covers $\\operatorname{Br}_Q(c)$ and lifts to $\\alpha, $ a point of $G$ on $A$ that covers $c.$ This is a consequence of \\cite[Theorem 3.5]{Co}. Notice that $\\alpha$ verifies the axioms of Definition \\ref{definition41}.\n\nAt last, the  algebra epimorphism\n$$\\operatorname{Br}_Q:(cAc)^G\\rightarrow (\\operatorname{Br}_Q(c) A(Q) \\operatorname{Br}_Q(c))^{N_G(Q)}$$\nproves that $w_1$ lifts to an idempotent of $(cAc)^G$, hence $\\alpha$ also verifies the axioms of Definition \\ref{definition43}.\n\\end{proof}\nLet $\\mathcal{C}(P,f_P)$ denote the full subcategory of $\\mathcal{C}$  consisting of Brauer pairs $(T,f_T)$ such that\n$(T,f_T)\\leq(P,f_P).$ Let $\\mathcal{D}(P,f_P)$ denote the full subcategory of $\\mathcal{D}$ consisting of Brauer pairs $(T,f_T)$ with $(T,f_T)\\leq(P,f_P)$.\nAs a conclusion of the above results we can show the second main result of this paper.\n\\begin{thm}\\label{theorem63}\nThe category $\\mathcal{C}(P,f_P)$ is isomorphic to $\\mathcal{D}(P,f_P)$.\n\\end{thm}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA classical result discovered by Carath\\'{e}odory and Fej\\'{e}r in 1911 \\cite{caratheodory1911zusammenhang} states that, if an $N\\times N$ Hermitian Toeplitz matrix $\\m{T}$ is positive semidefinite (PSD) and has rank $r\\leq N$, then it can be factorized as\n\\equ{\\m{T} = \\m{A}\\m{P}\\m{A}^H, \\label{eq:vanderdecp1}}\nwhere $\\m{P}$ is an $r\\times r$ positive definite diagonal matrix and $\\m{A}$ is an $N\\times r$ Vandermonde matrix whose columns are discrete sinusoidal waves with distinct frequencies. Moreover, such a decomposition is unique if $r<N$. This Vandermonde decomposition result has become important for information and signal processing since the 1970s when it was rediscovered by Pisarenko and used for frequency estimation by interpreting the Toeplitz matrix $\\m{T}$ as the data covariance matrix. The Vandermonde decomposition in \\eqref{eq:vanderdecp1} is therefore also referred to as the Carath\\'{e}odory-Fej\\'{e}r-Pisarenko decomposition. As a result of this rediscovery, a class of methods have been developed for frequency estimation based on the signal subspace of a data covariance estimate, known as the subspace-based methods. Prominent examples are multiple signal classification (MUSIC), estimation of parameters by rotational invariant techniques (ESPRIT) and various variants of them (see the review in \\cite{stoica2005spectral}). Besides, this decomposition result is important in moment theory, operator theory and system theory \\cite{grenander1958toeplitz, georgiou2007caratheodory}. As an example, it can be applied to give a solution to the truncated trigonometric moment problem (a.k.a.~the moment problem on the unit circle given a finite moment sequence) \\cite{akhiezer1965classical}.\n\nIn the past few years, a new class of methods for frequency estimation have been devised, namely the gridless sparse methods (see the review in \\cite{yang2016sparse}), in which the Vandermonde decomposition is evoked and plays an important role. It is well-known that sparse methods for frequency estimation developed in the past two decades exploit the signal sparsity, which arises naturally from the fact that the number of frequencies is small, and attempt to find, among all candidates consistent with the observed data, the solution consisting of the smallest number of frequencies. Since frequency estimation is a highly nonlinear problem and to overcome such nonlinearity, gridding in the continuous frequency domain used to be a standard ingredient of early sparse methods, which transforms approximately the original nonlinear continuous parameter estimation problem as a problem of sparse signal recovery from a linear system of equations (see, e.g., \\cite{gorodnitsky1997sparse,malioutov2005sparse}). The newly developed gridless sparse methods completely avoid gridding, work directly in the continuous domain, and have strong theoretical guarantees. These methods have been developed based on the atomic norm \\cite{chandrasekaran2012convex,candes2013towards,candes2013super, bhaskar2013atomic,tang2012compressed}---a continuous analogue of the $\\ell_1$ norm used in the early sparse methods---and covariance fitting \\cite{yang2015gridless}. A main difficulty of applying these gridless sparse methods underlies in how to solve the nonlinearity problem, which makes the resulting optimization problems nonconvex with respect to the unknown frequencies. To do so, the key is to apply the Vandermonde decomposition of Toeplitz matrices to cast these optimization problems as semidefinite programs (SDP), in which the frequencies are encoded in a PSD Toeplitz matrix, as $\\m{T}$ in \\eqref{eq:vanderdecp1}. Once the SDP is solved, the frequencies are finally retrieved from the Vandermonde decomposition of the solved Toeplitz matrix. Note that the Vandermonde decomposition result has also been generalized to high dimensions and used for multidimensional frequency estimation \\cite{yang2016vandermonde}.\n\nNotice that the frequencies in the Vandermonde decomposition in \\eqref{eq:vanderdecp1} may take any value in the normalized band $\\mbra{0,1}$ (or the unit circle), in which 0 and 1 are identified. This paper is motivated by various practical applications in which the (normalized) frequencies can be known {\\em a priori} to lie in certain frequency bands. For example, when a signal is oversampled by a factor, the frequencies will lie in a band narrowed by the same factor. Due to the path loss effect, the maximum value of the range\/delay, which can be interpreted as a frequency parameter, of a detectable aircraft can be estimated in advance. Similarly, the maximum Doppler frequency can be obtained if the aircraft's characteristic speed can be known. In underwater channel estimation, the frequency parameters of interest can reside in a known small interval \\cite{beygi2015multi}. Similar prior knowledge might also be available given weather observations \\cite{doviak1993doppler}. Therefore, it would be interesting to exploit such prior knowledge in gridless sparse methods for frequency estimation, and by doing so, the estimation accuracy is expected to improve.\n\nThe important role of the Vandermonde decomposition in gridless sparse methods encourages us to incorporate the prior interval knowledge into the decomposition. In other words, we ask the following question: {\\em Can the frequencies in the Vandermonde decomposition of the Toeplitz matrix $\\m{T}$, as in \\eqref{eq:vanderdecp1}, be restricted to lie in a given interval $\\mathcal{I}\\subset\\mbra{0,1}$, instead of the entire domain $\\mbra{0,1}$, under explicit conditions on $\\m{T}$?} In fact, we also want the conditions to be convex due to our interest in optimization problems. The resulting decomposition is referred to as frequency-selective (FS) Vandermonde decomposition. The question asked above is challenging since, by \\eqref{eq:vanderdecp1}, $\\m{T}$ is a highly nonlinear function of the frequencies and it is unclear how to link $\\m{T}$ to a frequency interval $\\mathcal{I}$.\n\nIt is interesting to note that similar questions have been investigated in a class of moment problems known as truncated $K$-moment problems, a.k.a.~truncated moment problems on a semialgebraic set $K$, instead of on an entire domain \\cite{schmudgen1991thek}. When $K$ is in the real or the complex domain, solutions to these problems have been successfully obtained \\cite{curto2000truncated,lasserre2009moments}. To the best of our knowledge, however, the problem is still open when $K$ is defined on the unit circle $\\mbra{0,1}$, which is known as the truncated trigonometric $K$-moment problem. In this paper, we show that the study of the FS Vandermonde decomposition can provide a solution to this open problem.\n\nIn this paper, an affirmative answer is provided to the question asked above. Concretely, it is shown that a PSD Toeplitz matrix $\\m{T}$ admits an FS Vandermonde decomposition on a given interval if and only if $\\m{T}$ satisfies another linear matrix inequality (LMI). Interestingly, this FS Vandermonde decomposition result is linked by duality to the positive real lemma (PRL) for trigonometric polynomials \\cite{dumitrescu2007positive}. The usefulness of the new result is also demonstrated. In the theory of moments, it provides a solution to the truncated trigonometric $K$-moment problem. For frequency estimation with prior interval knowledge, it is used to derive a primal SDP formulation for the atomic norm exploiting the prior knowledge. Numerical examples are also provided.\n\n\\subsection{Related Work}\n\nThis paper extends our conference paper \\cite{yang2016vandermonde_ccc} in which the FS Vandermonde decomposition of Toeplitz matrices was studied. In addition to this, we show in this paper the connection between the FS Vandermonde decomposition and the PRL for trigonometric polynomials. Its applications to the moment theory and frequency estimation are also studied in more detail.\n\nThe problem of frequency estimation with restriction on the frequency band was studied in \\cite{mishra2015spectral,yang2016weighted, chao2016extensions}. In \\cite{mishra2015spectral}, an FS atomic norm formulation (or constrained atomic norm in the language of \\cite{mishra2015spectral}) was proposed and a dual SDP formulation was presented by applying the theory of positive trigonometric polynomials. In contrast to this, we show in this paper that a primal SDP formulation of the FS atomic norm can be obtained by applying the new FS Vandermonde decomposition. In \\cite{yang2016weighted}, the interval prior was interpreted as a prior distribution of the frequencies and a weighted atomic norm approach was then devised that is an approximate but faster implementation of the FS atomic norm. Although the paper \\cite{chao2016extensions} does not provide or imply the FS Vandermonde decomposition result, it obtained independently a primal SDP formulation of the FS atomic norm based on a different technique.\n\nThe paper \\cite{de2015exact} studied the super-resolution problem on semialgebraic sets in the real domain and provided an SDP formulation of the resulting atomic norm. To do so, the key is to apply the moment theory on semialgebraic sets in the real domain (a.k.a. the truncated $K$-moment problem in the real domain). In contrast to this, we provide a first solution to the truncated trigonometric $K$-moment problem and then apply this result to study super-resolution on semi-algebraic sets on the unit circle.\n\n\n\n\\subsection{Notations}\n\n\n\n\nNotations used in this paper are as follows. $\\mathbb{R}$ and $\\mathbb{C}$ denote the set of real and complex numbers, respectively. $\\mathbb{T}\\coloneqq\\mbra{0,1}$ denotes the unit circle, in which 0 and 1 are identified. Boldface letters are reserved for vectors and matrices. $\\abs{\\cdot}$ denotes the amplitude of a scalar or the cardinality of a set. $\\onen{\\cdot}$, $\\twon{\\cdot}$ and $\\frobn{\\cdot}$ denote the\n$\\ell_1$, $\\ell_2$ and Frobenius norms respectively. $\\m{A}^T$ and $\\m{A}^H$ are the matrix transpose and conjugate transpose of $\\m{A}$ respectively. $\\rank\\sbra{\\m{A}}$ denotes the rank and $\\tr\\sbra{\\m{A}}$ is the trace. For PSD matrices $\\m{A}$ and $\\m{B}$, $\\m{A}\\geq\\m{B}$ means that $\\m{A}-\\m{B}$ is PSD. $\\Re$ and $\\Im$ return the real and the imaginary parts of a complex argument respectively.\n\nA Hermitian trigonometric polynomial of degree one is defined as:\n\\equ{g(z) = r_1 z^{-1} + r_0 + r_{-1} z, \\quad r_{-1}=\\overline{r}_1,\\quad r_0\\in\\mathbb{R}, \\label{eq:gz}}\nwhere $z$ is a complex argument and $\\overline{\\cdot}$ denotes the complex conjugate operator. When $\\m{z}$ is on the unit circle, i.e., when $z = e^{i2\\pi f}$, $f\\in\\mathbb{T}$, we write without ambiguity $g(f) \\coloneqq g\\sbra{e^{i2\\pi f}}$. It follows that\n\\equ{g(f) = r_1e^{-i2\\pi f} + r_0 + \\overline{r}_1 e^{i2\\pi f} = r_0 + 2\\Re\\lbra{r_1e^{-i2\\pi f}}, }\nand $g(f)$ is real on $\\mathbb{T}$.\n\nAn $N\\times N$ Toeplitz matrix $\\m{T}\\coloneqq \\m{T}\\sbra{\\m{t}}\\coloneqq \\m{T}\\sbra{N,\\m{t}}$ is formed by using a complex sequence $\\m{t} = \\mbra{t_j}$, $j=1-N,\\dots,N-1$ and defined by $T_{mn} = t_{n-m}$, $0\\leq m,n\\leq N-1$. Given $\\m{t}$ and a degree-1 trigonometric polynomial $g$ as defined in \\eqref{eq:gz}, an $(N-1)\\times (N-1)$ Toeplitz matrix $\\m{T}_{g}\\coloneqq \\m{T}_{g}\\sbra{\\m{t}}\\coloneqq \\m{T}_{g}\\sbra{N,\\m{t}}$ is defined by\n\\equ{\\mbra{T_g}_{mn} = r_1 t_{n-m+1}+ r_0t_{n-m} + r_{-1}t_{n-m-1}, \\label{eq:Tg}}\n$0\\leq m,n\\leq N-2$.\nAlso, let $\\m{a}\\sbra{f} \\coloneqq \\m{a}\\sbra{N,f}\\coloneqq \\mbra{1,e^{i2\\pi f}, \\dots, e^{i2\\pi (N-1)f}}^T$ denote a size-$N$ discrete complex sinusoid with frequency $f\\in\\mathbb{T}$.\n\n\\subsection{Paper Organization}\nThe rest of the paper is organized as follows. Section \\ref{sec:standardVD} introduces the standard Vandermonde decomposition of Toeplitz matrices. Section \\ref{sec:VDint} presents the new FS Vandermonde decomposition. Section \\ref{sec:duality} shows connections between the new result and the theory of trigonometric polynomials. Section \\ref{sec:moment} illustrates its application in the theory of moments. Section \\ref{sec:application} turns to the application in line spectral estimation with prior knowledge. Section \\ref{sec:conclusion} concludes this paper.\n\n\n\\section{Vandermonde Decomposition of Toeplitz Matrices} \\label{sec:standardVD}\n\nThe standard Vandermonde decomposition theorem of Toeplitz matrices \\cite{caratheodory1911zusammenhang,stoica2005spectral} is summarized in this section. Although its proof can be found in, e.g., \\cite{stoica2005spectral}, a new proof, inspired by \\cite{gurvits2002largest}, is provided here which will form the basis of the proof of the FS Vandermonde decomposition given in Section \\ref{sec:VDint}.\n\n\\begin{thm} A Toeplitz matrix $\\m{T}\\in\\mathbb{C}^{N\\times N}$ admits the following $r$-atomic, $r=\\rank\\sbra{\\m{T}}$, Vandermonde decomposition:\n\\equ{\\m{T} = \\sum_{k=1}^r p_k \\m{a}\\sbra{f_k}\\m{a}^H\\sbra{f_k}, \\label{eq:VD}}\nwhere $f_k\\in\\mathbb{T}$, $k=1,\\dots,r$ are distinct and $p_k>0$, if and only if $\\m{T} \\geq \\m{0}$. Moreover, the decomposition is unique if $\\m{T}$ is rank-deficient. \\label{thm:VD}\n\\end{thm}\n\\begin{proof} Suppose that $\\m{T}$ can be written as in \\eqref{eq:VD}, where $p_k>0$, it is evident that $\\m{T}$ is PSD. This completes the `only if' part. We next show the `if' part. To do so, we start with the case of $r=\\rank\\sbra{\\m{T}}\\leq N-1$. Since $\\m{T}\\geq \\m{0}$, there exists $\\m{V} = \\mbra{\\m{v}_1^T,\\dots,\\m{v}_{N}^T}^T\\in\\mathbb{C}^{N\\times r}$ satisfying $\\m{T} = \\m{V}\\m{V}^H$, where $\\m{v}_j\\in\\mathbb{C}^{1\\times r}$, $j=1,\\dots,N$. Let $\\m{V}_U = \\mbra{\\m{v}_1^T,\\dots, \\m{v}_{N-1}^T}^T$ and $\\m{V}_L = \\mbra{\\m{v}_2^T,\\dots, \\m{v}_{N}^T}^T$. By the structure of $\\m{T}$, we have that $\\m{V}_U\\m{V}_U^H = \\m{V}_L\\m{V}_L^H$. By \\cite[Theorem 7.3.11]{horn2012matrix}, there exists an $r\\times r$ unitary matrix $\\m{U}$ satisfying $\\m{V}_L = \\m{V}_U\\m{U}$. It follows that $\\m{v}_j = \\m{v}_1\\m{U}^{j-1}$, $j=2,\\dots,N$ and therefore,\n\\equ{t_j = \\m{v}_1 \\m{U}^{-j}\\m{v}_1^H, \\quad j=1-N,\\dots,N-1. \\label{eq:tjinvU}}\nNote that $\\m{U}$ has the following eigen-decomposition:\n\\equ{\\m{U} = \\widetilde{\\m{U}} \\diag\\sbra{z_1,\\dots,z_r} \\widetilde{\\m{U}}^H, \\label{eqw:UinUz}}\nwhere $\\widetilde{\\m{U}}$ is also an $r\\times r$ unitary matrix and $z_k = e^{i2\\pi f_k}$ with $f_k\\in\\mathbb{T}$, $k=1,\\dots,r$. Insert \\eqref{eqw:UinUz} into \\eqref{eq:tjinvU} and let $p_k = \\abs{\\m{v}_1\\widetilde{\\m{u}}_k}^2>0$, $k=1,\\dots,r$, where $\\widetilde{\\m{u}}_k$ denotes the $k$th column of $\\widetilde{\\m{U}}$. Then we have that\n\\equ{t_j = \\sum_{k=1}^r p_k e^{-i2\\pi j f_k}.}\nUsing the identity above, $\\m{T}$ can be written as in \\eqref{eq:VD}. It is evident that $f_k$, $k=1,\\dots,r$ are distinct since otherwise, $\\rank\\sbra{\\m{T}} < r$, which cannot be true.\n\nWe now consider the case of $r=N$, in which $\\m{T}$ is positive definite. To obtain a decomposition as in \\eqref{eq:VD}, we choose arbitrarily $f_N \\in\\mathbb{T}$ and let $p_N = \\sbra{\\m{a}^H\\sbra{f_N} \\m{T}^{-1} \\m{a}\\sbra{f_N}}^{-1}>0$. After that, we define a new sequence $\\m{t}'=\\mbra{t'_j}, \\;\\abs{j}\\leq N-1$ as:\n\\equ{t'_j = t_j - p_N e^{-i2\\pi j f_N}. \\label{eq:t1j}}\nIt follows that\n\\equ{\\m{T}\\sbra{\\m{t}'}\n= \\m{T} - p_N\\m{a}\\sbra{f_N}\\m{a}^H\\sbra{f_N}. \\label{eq:Tu1}}\nBy the choice of $p_N$, the matrix\n\\equ{\\begin{bmatrix} p_N^{-1} & \\m{a}^H\\sbra{f_N} \\\\ \\m{a}\\sbra{f_N} & \\m{T} \\end{bmatrix} = \\begin{bmatrix} \\m{a}^H\\sbra{f_N}\\m{T}^{-\\frac{1}{2}} \\\\ \\m{T}^{\\frac{1}{2}} \\end{bmatrix}\\begin{bmatrix} \\m{a}^H\\sbra{f_N}\\m{T}^{-\\frac{1}{2}} \\\\ \\m{T}^{\\frac{1}{2}} \\end{bmatrix}^H \\notag}\nis PSD and rank-deficient.\nNotice that $\\m{T}\\sbra{\\m{t}'}$ is the Schur complement of $\\m{T}$ in the above matrix, and therefore\n\\equ{\\m{T}\\sbra{\\m{t}'}\\geq \\m{0}. \\label{eq:Tu1psd}}\nMoreover, it holds that\n\\equ{\\rank\\sbra{\\m{T}\\sbra{\\m{t}'}}<N \\label{eq:Tu1rankd}}\nsince, otherwise, $\\begin{bmatrix} p_N^{-1} & \\m{a}^H\\sbra{f_N} \\\\ \\m{a}\\sbra{f_N} & \\m{T} \\end{bmatrix}$ has full rank.\nCombining \\eqref{eq:Tu1rankd} and\n\\equ{\\rank\\sbra{\\m{T}\\sbra{\\m{t}'}} \\geq \\rank\\sbra{\\m{T}} - \\rank\\sbra{p_N\\m{a}\\sbra{f_N}\\m{a}^H\\sbra{f_N}} = N-1 \\notag}\nresults in\n\\equ{\\rank\\sbra{\\m{T}\\sbra{\\m{t}'}}= N-1.\\label{eq:Tu1rank}}\nFollowing from \\eqref{eq:Tu1psd}, \\eqref{eq:Tu1rank} and the result in the case of $r\\leq N-1$ that we just proved, $\\m{T}\\sbra{\\m{t}'}$ admits a Vandermonde decomposition as in \\eqref{eq:VD} with $r=N-1$. It then follows from \\eqref{eq:Tu1} that $\\m{T}$ admits an $N$-atomic Vandermonde decomposition.\n\nWe finally show the uniqueness in the case of $r\\leq N-1$. Write \\eqref{eq:VD} in matrix form as $\\m{T}=\\m{A}\\sbra{\\m{f}} \\m{P}\\m{A}^H\\sbra{\\m{f}}$, where $\\m{P} = \\diag\\sbra{p_1,\\dots,p_r}$ and $\\m{A}\\sbra{\\m{f}} = \\mbra{\\m{a}\\sbra{f_1},\\dots, \\m{a}\\sbra{f_r}}$. Suppose that $\\m{T}$ has another decomposition: $\\m{T} = \\m{A}\\sbra{\\m{f}'} \\m{P}'\\m{A}^H\\sbra{\\m{f}'}$, in which, similarly, $f'_j\\in\\mathbb{T}$, $j=1,\\dots,r$ are distinct and $p'_j>0$. It is evident that\n\\equ{\\m{A}\\sbra{\\m{f}'} \\m{P}'\\m{A}^H\\sbra{\\m{f}'}=\\m{A}\\sbra{\\m{f}} \\m{P}\\m{A}^H\\sbra{\\m{f}}.}\nTherefore, there exists an $r\\times r$ unitary matrix $\\m{U}'$ satifying $\\m{A}\\sbra{\\m{f}'}\\m{P}'^{\\frac{1}{2}} = \\m{A}\\sbra{\\m{f}} \\m{P}^{\\frac{1}{2}} \\m{U}'$. It follows that\n\\equ{\\m{A}\\sbra{\\m{f}'} = \\m{A}\\sbra{\\m{f}} \\m{P}^{\\frac{1}{2}} \\m{U}' \\m{P}'^{-\\frac{1}{2}}.}\nThis means that for every $j = 1,\\dots,r$, $\\m{a}\\sbra{f'_j}$ lies in the range space spanned by $\\lbra{\\m{a}\\sbra{f_k}}_{k=1}^r$. By the fact that $r\\leq N-1$ and that any $N$ atoms $\\m{a}\\sbra{f_k}$ with distinct $f_k$'s are linearly independent, we have that $f'_j\\in\\lbra{f_k}_{k=1}^r$ and thus the two sets $\\lbra{f'_j}_{j=1}^r$ and $\\lbra{f_k}_{k=1}^r$ are identical. It follows that the two decompositions of $\\m{T}$ are identical.\n\\end{proof}\n\nWe next discuss how to obtain the Vandermonde decomposition, to be specific, how to solve for $f_k$ and $p_k$ in \\eqref{eq:VD}. In fact, a computational approach can be provided based on the proof of Theorem \\ref{thm:VD}. In the case of $r\\leq N-1$, using Cholesky decomposition, we can compute $\\m{V}\\in\\mathbb{C}^{N\\times r}$ satisfying $\\m{T} = \\m{V}\\m{V}^H$. By the arguments of the proof, it is easy to show the following equation:\n\\equ{\\sbra{\\m{V}_U^H\\m{V}_L - z_k \\m{V}_U^H\\m{V}_U}\\widetilde{\\m{u}}_k = 0,}\nfrom which $z_k$ and $\\widetilde{\\m{u}}_k$, $k=1,\\dots,r$ can be computed as the eigenvalues and eigenvectors of the matrix pencil $\\sbra{\\m{V}_U^H\\m{V}_L, \\m{V}_U^H\\m{V}_U}$. Finally, the parameters are obtained as: $f_k = \\frac{1}{2\\pi}\\Im \\ln z_k\\in\\mathbb{T}$ and $p_k = \\abs{\\m{v}_1\\widetilde{\\m{u}}_k}^2$, $k=1,\\dots,r$, where $\\m{v}_1$ is the first row of $\\m{V}$. In the case of $r= N$, $f_N\\in\\mathbb{T}$ can be chosen arbitrarily first, and the rest can be done following from the proof.\n\n\\section{FS Vandermonde Decomposition of Toeplitz Matrices} \\label{sec:VDint}\n\nWe present the FS Vandermonde decomposition result in this section. To encode the interval information into the Vandermonde decomposition, we first construct a trigonometric polynomial that is nonnegative on the interval $\\mathcal{I}$ and negative on its complement. We first clarify some notations. For $f_L\\neq f_H\\in\\mathbb{T}$, if $f_L< f_H$, then $\\mathcal{I}=\\mbra{f_L, f_H}$ denotes a closed interval as usual. Otherwise, we define $\\mathcal{I}=\\mbra{f_L, f_H}\\coloneqq \\mathbb{T}\\backslash \\sbra{f_H, f_L}$. By this definition, we can conveniently deal with the case in which $0$ (or $1$) is an interior point of $\\mathcal{I}$. The trigonometric polynomial, $g$, is defined as:\n\\equ{g(z) = \\frac{1}{z\\sqrt{z_Lz_H}}\\sbra{z-z_L}\\sbra{z-z_H}\\sgn\\sbra{f_H-f_L}, \\label{eq:g1}}\nwhere $z_L \\coloneqq e^{i2\\pi f_L}$, $z_H \\coloneqq e^{i2\\pi f_H}$ and $\\sgn\\sbra{\\cdot}$ is the sign function. With simple derivations, we have\n\\equ{g(z)= r_1 z^{-1} + r_0 + \\overline{r}_1 z, \\label{eq:gz1}}\nwhere\n{\\setlength\\arraycolsep{2pt}\\equa{r_0\n&=& - \\frac{z_L+z_H}{\\sqrt{z_Lz_H}} \\sgn\\sbra{f_H-f_L} \\notag \\\\\n&=& -2\\cos\\mbra{\\pi\\sbra{f_H-f_L}}\\sgn\\sbra{f_H-f_L}, \\label{eq:r0}\\\\ r_1\n&=& \\sqrt{z_Lz_H}\\sgn\\sbra{f_H-f_L} \\notag\\\\\n&=&  e^{i\\pi\\sbra{f_L+f_H}} \\sgn\\sbra{f_H-f_L}.\\label{eq:r1}\n}}It is evident that $g(z)$ is a Hermitian trigonometric polynomial that is real-valued on $\\mathbb{T}$. By the way that $g(z)$ is constructed, we know that $g(z)$ has two single roots $z_L$ and $z_H$, and equivalently, $g(f)$ has two single roots $f_L$ and $f_H$. Therefore, $g(f)$ flips its sign around $f_L$ and $f_H$. Two possibilities are: $g(f)$ is positive on $\\sbra{f_L, f_H}$ and negative on $\\sbra{f_H, f_L}$, or negative on $\\sbra{f_L, f_H}$ and positive on $\\sbra{f_H, f_L}$. To determine which one is true, we check the value at $f=\\frac{1}{2}\\sbra{f_L+f_H}$:\n\\equ{\\begin{split}\n&g\\sbra{\\frac{1}{2}\\sbra{f_L+f_H}} \\\\\n&= r_0 + 2\\Re \\sbra{r_1 e^{-i\\pi \\sbra{f_L+f_H} }} \\\\\n&= \\lbra{2-2\\cos\\mbra{\\pi\\sbra{f_L-f_H}}} \\sgn\\sbra{f_H-f_L}. \\end{split}}\nConsequently, the sign of $g$ at $f=\\frac{1}{2}\\sbra{f_L+f_H}$ is identical to that of $f_H-f_L$, meaning that $g(f)$ is always positive on $\\sbra{f_L, f_H}$ and negative on $\\sbra{f_H, f_L}$ whenever $f_L<f_H$ or $f_L>f_H$.\n\nNow we are ready to present the FS Vandermonde decomposition result, which is summarized in the following theorem.\\footnote{Part of the FS Vandermonde decomposition result was extended to a general form in the recent preprint \\cite{chao2016semidefinite}, which appeared online after our conference paper \\cite{yang2016vandermonde_ccc} was accepted.}\n\n\\begin{thm} Given $\\mathcal{I}\\subset\\mathbb{T}$, a Toeplitz matrix $\\m{T}\\in\\mathbb{C}^{N\\times N}$ admits an FS Vandermonde decomposition, as in \\eqref{eq:VD}, with $f_k\\in\\mathcal{I}$, if and only if\n{\\setlength\\arraycolsep{2pt}\\equa{\\m{T}\n&\\geq& \\m{0}, \\label{eq:Tupsd}\\\\ \\m{T}_g\n&\\geq& \\m{0}, \\label{eq:Tunew}\n}}where $g$ is defined by \\eqref{eq:gz1}-\\eqref{eq:r1} and $\\m{T}_g$ by \\eqref{eq:Tg}.\nMoreover, the decomposition is unique if either $\\m{T}$ or $\\m{T}_g$ is rank-deficient. \\label{thm:BCVD}\n\\end{thm}\n\n\\begin{proof} We first show the ``if'' part. Consider the case of $r\\leq N-1$. It then follows from \\eqref{eq:Tupsd} and Theorem \\ref{thm:VD} that $\\m{T}$ admits a unique Vandermonde decomposition as in \\eqref{eq:VD}. So, it suffices to show $f_k\\in \\mathcal{I}$, $k=1,\\dots,r$ under the additional condition \\eqref{eq:Tunew}. To do so, note by \\eqref{eq:VD} that\n\\equ{t_{n-m} = T_{mn} = \\sum_{k=1}^r p_k e^{i2\\pi (m-n) f_k}. \\label{eq:tlk}}\nIt immediately follows that\n\\equ{\\begin{split}\\mbra{T_g}_{mn}\n&= \\sum_{j=-1}^1 r_j t_{n-m+j} \\\\\n&= \\sum_{j=-1}^1 r_j \\sum_{k=1}^r p_k e^{i2\\pi (m-n-j) f_k}\\\\\n&= \\sum_{k=1}^r p_k e^{i2\\pi (m-n) f_k} \\sum_{j=-1}^1 r_j e^{-i2\\pi j f_k}\\\\\n&= \\sum_{k=1}^r p_k g\\sbra{f_k} e^{i2\\pi (m-n) f_k}, \\end{split} \\label{eq:Tgkl}}\nand hence\n\\equ{\\begin{split}\\m{T}_g\n&= \\sum_{k=1}^r p_k g\\sbra{f_k} \\m{a}\\sbra{N-1,f_k}\\m{a}^H\\sbra{N-1,f_k}\\\\\n&= \\m{A}\\sbra{N-1,\\m{f}}\\diag\\sbra{p_1g\\sbra{f_1},\\dots, p_rg\\sbra{f_r}} \\m{A}^H\\sbra{N-1,\\m{f}}, \\end{split} \\label{eq:T1ueq}}\nwhere $\\m{A}\\sbra{N-1,\\m{f}} \\coloneqq \\mbra{\\m{a}\\sbra{N-1,f_1},\\dots,\\m{a}\\sbra{N-1,f_r}}$ is an $\\sbra{N-1}\\times r$ Vandermonde matrix and $\\diag\\sbra{p_1g\\sbra{f_1},\\dots, p_rg\\sbra{f_r}}$ denotes a diagonal matrix with $p_kg\\sbra{f_k}$, $k=1,\\dots,r$ on the diagonal. Note that $\\m{A}\\sbra{N-1,\\m{f}}$ has full column rank since $r\\leq N-1$. Using \\eqref{eq:T1ueq} and \\eqref{eq:Tunew}, we have that\n\\equ{\\begin{split}\n&\\diag\\sbra{p_1g\\sbra{f_1},\\dots, p_r g\\sbra{f_r}} = \\m{A}^{\\dag}\\sbra{N-1,\\m{f}} \\m{T}_g \\m{A}^{\\dag H}\\sbra{N-1,\\m{f}} \\geq \\m{0}, \\end{split}}\nwhere $\\cdot^{\\dag}$ denotes the matrix pseudo-inverse operator.\nThis means that $p_kg\\sbra{f_k}\\geq 0$, and since $p_k>0$, we have $g\\sbra{f_k}\\geq 0$, $k=1,\\dots,r$. By the property of $g(f)$, finally, we have $f_k\\in\\mathcal{I}$, $k=1,\\dots,r$.\n\nWe next consider the case of $r=N$ in which $\\m{T}$ is positive definite. Let $f_N = f_L$ and $p_N = \\sbra{\\m{a}^H\\sbra{f_N} \\m{T}^{-1} \\m{a}\\sbra{f_N}}^{-1}>0$. Similar to that in the proof of Theorem \\ref{thm:VD}, we define a new sequence $\\m{t}'=\\mbra{t'_j}, \\;\\abs{j}\\leq N-1$ as in \\eqref{eq:t1j}, which therefore satisfies \\eqref{eq:Tu1}, \\eqref{eq:Tu1psd} and \\eqref{eq:Tu1rank}. Moreover, we have\n\\equ{\\begin{split}\\mbra{T_g\\sbra{\\m{t}'}}_{mn}\n&= \\sum_{j=-1}^1 r_j t'_{n-m+j} \\\\\n&= \\mbra{T_g}_{mn} - p_N g(f_N) e^{i2\\pi (m-n) f_N}, \\end{split}}\nand hence\n\\equ{\\m{T}_g\\sbra{\\m{t}'} = \\m{T}_g  - p_Ng\\sbra{f_N} \\m{a}\\sbra{N-1,f_N}\\m{a}^H\\sbra{N-1,f_N}. \\label{eq:Tgt1}}\nBy \\eqref{eq:Tunew} and the fact that $g\\sbra{f_N}=g\\sbra{f_L}=0$, we have\n\\equ{\\m{T}_g\\sbra{\\m{t}'} = \\m{T}_g \\geq \\m{0}. \\label{eq:Tu1newpsd}}\nNow consider $\\m{T}\\sbra{\\m{t}'}$ that satisfies \\eqref{eq:Tu1psd}, \\eqref{eq:Tu1rank} and \\eqref{eq:Tu1newpsd}. Following from the ``if'' part of Theorem \\ref{thm:BCVD} in the case of $r\\leq N-1$ that we just proved, $\\m{T}\\sbra{\\m{t}'}$ admits a unique decomposition as in \\eqref{eq:VD}, with $f_k\\in\\mathcal{I}$, $k=1,\\dots,r=N-1$. Therefore, it follows from \\eqref{eq:Tu1} that\n\\equ{\\m{T} = \\m{T}\\sbra{\\m{t}'} + p_N\\m{a}\\sbra{f_N}\\m{a}^H\\sbra{f_N}}\nhas a decomposition as in \\eqref{eq:VD}, with $f_k\\in\\mathcal{I}$, $k=1,\\dots,r=N$. So we complete the ``if'' part.\n\nThe ``only if'' part can be shown by similar arguments. In particular, given $\\m{T}$ as in \\eqref{eq:VD}, it is evident that \\eqref{eq:Tupsd} holds. Moreover, \\eqref{eq:Tunew} also holds, since we still have \\eqref{eq:T1ueq}, in which $g\\sbra{f_k}\\geq 0$, $k=1,\\dots,r$ by the property of $g$.\n\nWe finally shown the uniqueness under the additional condition that $\\m{T}$ or $\\m{T}_g$ is rank-deficient. When $\\m{T}$ is rank-deficient, this is a direct consequence of Theorem \\ref{thm:VD}. In the other case when $\\m{T}$ has full rank and $\\m{T}_g$ is rank-deficient, note first that there are at least $N$ distinct $f_k$'s in the FS Vandermonde decomposition of $\\m{T}$, since, otherwise, $\\m{T}$ loses rank. We now recall \\eqref{eq:T1ueq}, in which $\\m{A}\\sbra{N-1,\\m{f}}$ has full row rank and $g(f_k)\\geq 0$. To guarantee that $\\m{T}_g$ is rank-deficient, $g(f_k)\\neq 0$ must hold for maximally $N-2$, $f_k$'s and the other $f_k$'s must be either $f_L$ or $f_H$. This means that the decomposition consists of exactly $N$ atoms and two of them are located at $f_L$ and $f_H$. Therefore, the other $N-2$ frequencies are fixed as well, and the FS Vandermonde decomposition is unique.\n\\end{proof}\n\nThe FS Vandermonde decomposition can be computed similarly as the standard Vandermonde decomposition provided that the conditions of Theorem \\ref{thm:BCVD} are satisfied. More concretely, in the case when $\\m{T}$ is rank-deficient, it admits a unique Vandermonde decomposition that can be computed as in Section \\ref{sec:standardVD}. In the case when $\\m{T}$ has full rank, an $N$-atomic decomposition can be computed following from the proof of Theorem \\ref{thm:BCVD}, to be specific, fix $f_N=f_L$ first and compute the other parameters following the proof.\n\nFinally, note that the FS Vandermonde decomposition result can be extended straightforwardly to the multiple frequency band case. Let $K=\\bigcup_{l=1}^J \\mbra{f_{Ll}, f_{Hl}}$, where $\\mbra{f_{Ll}, f_{Hl}}\\subset\\mathbb{T}$, $l=1,\\dots, J\\geq 2$ are disjoint. We have the following corollary of Theorem \\ref{thm:BCVD}, the proof of which is straightforward and thus is omitted.\n\\begin{cor} Given $K = \\bigcup_{l=1}^J \\mbra{f_{Ll}, f_{Hl}}$, a Toeplitz matrix $\\m{T}\\in\\mathbb{C}^{N\\times N}$ admits an FS Vandermonde decomposition, as in \\eqref{eq:VD}, with $f_k\\in K$, if and only if there exist sequences $\\m{t}_l$, $l=1,\\dots,J$ satisfying\n{\\setlength\\arraycolsep{2pt}\\equa{\\sum_{l=1}^J \\m{t}_l\n&=& \\m{t}, \\label{eq:sumTleqT}\\\\  \\m{T}\\sbra{\\m{t}_l}\n&\\geq& \\m{0}, \\label{eq:Tupsdl}\\\\ \\m{T}_{g_l}\\sbra{\\m{t}_l}\n&\\geq& \\m{0}, \\quad l=1,\\dots,J, \\label{eq:Tunewl}\n}}where $g_l$, $l=1,\\dots,J$ are $g$ defined with respect to $\\mbra{f_{Ll}, f_{Hl}}$, respectively. \\label{cor:BCVDM}\n\\end{cor}\n\n\n\\section{Duality} \\label{sec:duality}\nUsing the FS Vandermonde decomposition result presented in the previous section, we can explicitly characterize the cone of Toeplitz matrices admitting such decompositions. Due to the interest in optimization problems, we naturally look at the dual cone, which, as we will see, enables us to link the FS Vandermonde decomposition to the theory of trigonometric polynomials, to be specific, the PRL given in \\cite{davidson2002linear,alkire2002convex} (see also \\cite{dumitrescu2007positive}).\n\nFor a sequence $\\m{t}=\\mbra{t_j}$, $\\abs{j}\\leq N-1$ with $t_{-j} = \\overline{t}_j$, let $\\m{t}_R = \\mbra{\\Re t_{N-1}, \\dots, \\Re t_{1}, \\frac{\\sqrt{2}}{2}t_0, \\Im t_1, \\dots, \\Im t_{N-1}}^T\\in\\mathbb{R}^{2N-1}$ be a representation of $\\m{t}$ in the real domain, where the coefficient $\\frac{\\sqrt{2}}{2}$ for $t_0$ is chosen for convenience. It is obvious that all $N\\times N$ Toeplitz matrices admitting an FS Vandermonde decomposition on a given interval $\\mathcal{I}\\subset\\mathbb{T}$ form a cone that can be identified with\n\\equ{\\begin{split}\n&\\mathcal{K}_{\\text{VDF}} \\coloneqq \\lbra{\\m{t}_R:\\; \\m{T} = \\sum_{k} p_k\\m{a}\\sbra{f_k}\\m{a}^H\\sbra{f_k},\\; p_k\\geq 0, f_k\\in \\mathcal{I}}.\\end{split} \\label{eq:KVDF}}\nDefine\n\\equ{\\begin{split}\\mathcal{K}_{\\text{VDM}}\n&\\coloneqq \\lbra{\\m{t}_R:\\; \\m{T}\\geq\\m{0}, \\; \\m{T}_{g}\\geq \\m{0}}, \\end{split} \\label{eq:KVDM}}\nwhere $g$ is defined in Theorem \\ref{thm:BCVD}. A direct consequence of Theorem \\ref{thm:BCVD} is that\n\\equ{\\mathcal{K}_{\\text{VDF}} = \\mathcal{K}_{\\text{VDM}}. \\label{eq:VDconeeq} }\n\nWe next consider the dual cone of $\\mathcal{K}_{\\text{VDF}}$ defined as \\cite{boyd2004convex}\n\\equ{\\mathcal{K}_{\\text{VDF}}^*\\coloneqq \\lbra{\\m{\\alpha}\\in\\mathbb{R}^{2N-1}:\\; \\m{t}_R^T\\m{\\alpha}\\geq 0 \\text{ for any } \\m{t}_R\\in\\mathcal{K}_{\\text{VDF}}}. \\label{eq:defdualcone}}\nBefore proceeding to the main result of this section, we first introduce some notations. Let\n\\equ{\\mathcal{K}_{\\text{PolF}}\\coloneqq \\lbra{\\m{\\gamma}_R:\\; \\sum_{j=1-N} ^{N-1} \\gamma_j e^{i2\\pi j f} \\geq 0,\\; f\\in \\mathcal{I}} \\label{KPolF}}\ndenote the cone of trigonometric polynomials of order $N-1$ and nonnegative on $\\mathcal{I}$, where $\\m{\\gamma}_R$ is similarly defined as $\\m{t}_R$. Let also $\\m{\\Theta}_j$, $\\abs{j}\\leq N-1$ be an $N\\times N$ elementary Toeplitz matrix with ones on its $j$th diagonal and zeros elsewhere. With respect to $\\m{\\Theta}_j$ and the trigonometric polynomial $g$ defined by \\eqref{eq:gz1}-\\eqref{eq:r1}, we define the $(N-1)\\times (N-1)$ Toeplitz matrix $\\m{\\Theta}_{gj}$, like $\\m{T}_g$ with respect to $\\m{T}$. By definition, it is easy to verify that\n\\setlength\\arraycolsep{2pt}{\\equa{\\m{T}\n&=& \\sum_{j=1-N}^{N-1} \\m{\\Theta}_j t_j, \\label{eq:TinTheta}\\\\ \\m{T}_g\n&=& \\sum_{j=1-N}^{N-1} \\m{\\Theta}_{gj} t_j. \\label{eq:TginThetag}\n}}We also define the cone\n\\equ{\\begin{split} \\mathcal{K}_{\\text{PolM}}\\coloneqq \\Big\\{\\m{\\gamma}_R:\\;\n& \\gamma_{-j} = \\tr\\sbra{\\m{\\Theta}_j \\m{Q}_0} + \\tr\\mbra{ \\m{\\Theta}_{gj} \\m{Q}_1 }, \\\\\n& \\abs{j}\\leq N-1, \\\\\n& \\m{Q}_0\\in\\mathbb{C}^{N\\times N}, \\m{Q}_1\\in\\mathbb{C}^{(N-1)\\times (N-1)},\\\\\n& \\m{Q}_0\\geq\\m{0},\\m{Q}_1\\geq\\m{0} \\Big\\}. \\end{split} \\label{KPolM}}\nThe main result of this section is given in the following theorem.\n\n\\begin{thm} We have the following identities:\n {\\setlength\\arraycolsep{2pt}\\equa{\\mathcal{K}_{\\text{VDF}}^*\n &=& \\mathcal{K}_{\\text{PolF}}, \\label{eq:dualeq1} \\\\ \\mathcal{K}_{\\text{PolM}}^*\n &=& \\mathcal{K}_{\\text{VDM}}. \\label{eq:dualeq2}\n }}Therefore, provided that $\\mathcal{K}_{\\text{VDF}} = \\mathcal{K}_{\\text{VDM}}$ we can conclude that $\\mathcal{K}_{\\text{PolF}} = \\mathcal{K}_{\\text{PolM}}$, and vice versa.\n\\label{thm:dualcone}\n\\end{thm}\n\n\\begin{proof} We first show \\eqref{eq:dualeq1}. Note that $\\m{t}_R\\in\\mathcal{K}_{\\text{VDF}}$ if and only if\n\\equ{t_j = \\sum_{k} p_k e^{-i2\\pi j f_k}, \\quad j = 1-N,\\dots,N-1, \\label{eq:tj_KVDF}}\nwhere $p_k\\geq 0$ and $f_k\\in\\mathcal{I}$. For any $\\m{\\alpha}=\\mbra{\\alpha_{1-N},\\dots,\\alpha_{N-1}}^T\\in\\mathbb{R}^{2N-1}$, we define $\\m{\\gamma}\\in\\mathbb{C}^{2N-1}$ such that $\\gamma_0 = \\sqrt{2}\\alpha_0$, $\\gamma_j = \\alpha_{-j}+i\\alpha_{j}$ and $\\gamma_{-j} = \\alpha_{-j}-i\\alpha_{j}$, $j=1,\\dots,N-1$. It follows that $\\m{\\alpha} = \\m{\\gamma}_R$ and\n\\equ{\\begin{split}\\m{t}_R^T\\m{\\alpha}\n&= \\frac{\\sqrt{2}}{2}t_0\\cdot \\frac{\\sqrt{2}}{2}\\gamma_0 + \\Re \\sum_{j=1}^{N-1}  \\overline{t}_j \\gamma _j\\\\\n&= \\frac{1}{2} \\sum_{j=1-N}^{N-1} \\overline{t}_j\\gamma_j. \\end{split} \\label{eq:realtHgamma1}}\nInserting \\eqref{eq:tj_KVDF} into \\eqref{eq:realtHgamma1}, we have that\n\\equ{\\m{t}_R^T\\m{\\alpha} = \\frac{1}{2} \\sum_{k} p_k \\sum_{j=1-N}^{N-1} \\gamma_j e^{i2\\pi j f_k}. \\label{eq:realtHgamma}}\nBy \\eqref{eq:realtHgamma} and the definition of the dual cone, $\\m{\\alpha} = \\m{\\gamma}_R \\in\\mathcal{K}_{\\text{VDF}}^*$ if and only if the right hand of \\eqref{eq:realtHgamma} is nonnegative for any $p_k\\geq 0$ and any $f_k\\in\\mathcal{I}$. The above condition holds if and only if $h(f) \\coloneqq \\sum_{j=1-N}^{N-1} \\gamma_j e^{i2\\pi j f}$ is nonnegative on $\\mathcal{I}$, or equivalently, $\\m{\\alpha} \\in \\mathcal{K}_{\\text{PolF}}$ by \\eqref{KPolF}.\n\nTo show \\eqref{eq:dualeq2}, we can similarly define $\\m{t}$ for $\\m{\\alpha}\\in\\mathbb{R}^{2N-1}$ such that $\\m{\\alpha} = \\m{t}_R$. It follows that $\\m{T}$ and $\\m{T}_g$ are Hermitian. For any $\\m{\\gamma}_R\\in \\mathcal{K}_{\\text{PolM}}$, which can be expressed as in \\eqref{KPolM}, we have that\n\\equ{\\begin{split}\\m{\\gamma}_R^T\\m{\\alpha}\n&= \\frac{1}{2} \\sum_{j=1-N}^{N-1} \\overline{\\gamma}_j t_j\\\\\n&= \\frac{1}{2} \\sum_{j=1-N}^{N-1} \\gamma_{-j} t_j \\\\\n&= \\frac{1}{2} \\sum_{j=1-N}^{N-1} t_j \\lbra{\\tr\\sbra{\\m{\\Theta}_j \\m{Q}_0} + \\tr\\mbra{ \\m{\\Theta}_{gj} \\m{Q}_1 }}. \\end{split}}\nUsing the identities in \\eqref{eq:TinTheta} and \\eqref{eq:TginThetag}, we have that\n\\equ{\\m{\\gamma}_R^T\\m{\\alpha} = \\frac{1}{2} \\tr\\sbra{\\m{T}\\m{Q}_0} + \\frac{1}{2} \\tr\\sbra{\\m{T}_g \\m{Q}_1}. \\label{eq:equ_dualcone}}\nBy the definition of the dual cone, $\\m{\\alpha} \\in \\mathcal{K}_{\\text{PolM}}^*$ if and only if $\\m{\\gamma}_R^T\\m{\\alpha}\\geq 0$ for any $\\m{\\gamma}_R\\in \\mathcal{K}_{\\text{PolM}}$. Using \\eqref{KPolM} and \\eqref{eq:equ_dualcone}, the above condition holds if and only if $\\tr\\sbra{\\m{T}\\m{Q}_0} + \\tr\\sbra{\\m{T}_g \\m{Q}_1}\\geq 0$ for any $\\m{Q}_0\\geq \\m{0}$ and $\\m{Q}_1\\geq \\m{0}$, which holds if and only if $\\tr\\sbra{\\m{T}\\m{Q}_0}\\geq 0$ for any $\\m{Q}_0\\geq \\m{0}$ and $\\tr\\sbra{\\m{T}_g \\m{Q}_1}\\geq 0$ for any $\\m{Q}_1\\geq \\m{0}$, and is further equivalent to the condition $\\m{T}\\geq \\m{0}$ and $\\m{T}_g\\geq \\m{0}$. The last condition is equivalent to  $\\m{\\alpha} = \\m{t}_R\\in \\mathcal{K}_{\\text{VDM}}$ by \\eqref{eq:KVDM}.\n\nFinally, provided that $\\mathcal{K}_{\\text{VDF}} = \\mathcal{K}_{\\text{VDM}}$ and using \\eqref{eq:dualeq1} and \\eqref{eq:dualeq2}, we have that\n\\equ{\\mathcal{K}_{\\text{PolF}} = \\mathcal{K}_{\\text{VDF}}^* = \\mathcal{K}_{\\text{VDM}}^* = \\mathcal{K}_{\\text{PolM}}^{**}. }\nUsing the identify that $\\mathcal{K}_{\\text{PolM}}^{**} = \\mathcal{K}_{\\text{PolM}}$, which follows from the fact that $\\mathcal{K}_{\\text{PolM}}$ is convex and closed \\cite{boyd2004convex}, we conclude that $\\mathcal{K}_{\\text{PolF}} = \\mathcal{K}_{\\text{PolM}}$.\nBy similar arguments we can also show that $\\mathcal{K}_{\\text{VDF}} = \\mathcal{K}_{\\text{VDM}}$ provided that $\\mathcal{K}_{\\text{PolF}} = \\mathcal{K}_{\\text{PolM}}$.\n\\end{proof}\n\nBy Theorem \\ref{thm:dualcone}, the FS Vandermonde decomposition on $\\mathcal{I}$ is linked via duality to the trigonometric polynomials nonnegative on the same interval. Moreover, the identity that $\\mathcal{K}_{\\text{PolF}} = \\mathcal{K}_{\\text{PolM}}$ provides a matrix form parametrization of the coefficients of these polynomials. In fact, this is exactly the Gram matrix parametrization concluded by the PRL in \\cite{davidson2002linear,alkire2002convex} (see also \\cite{dumitrescu2007positive}). This means that the PRL in \\cite{davidson2002linear,alkire2002convex} can be obtained from the FS Vandermonde decomposition; conversely, the PRL also provides an alternative way to characterize the set of Toeplitz matrices admitting an FS Vandermonde decomposition.\\footnote{Note that Theorem \\ref{thm:BCVD} is stronger in the sense that it concludes that all such Toeplitz matrices always admit a decomposition containing $N$ atoms or less.} Therefore, it will not be surprising that, as we will see, for certain convex optimization problems the two techniques can be applied to give the primal and the dual problems, respectively. But note that there are indeed scenarios in which one technique can be applied while the other cannot. Examples will be provided in the ensuing sections to demonstrate the usefulness of the FS Vandermonde decomposition.\n\n\\begin{rem} The trigonometric polynomial $g(z) = r_{-1}z + r_0 + r_1z^{-1}$ that is nonnegative on $\\mathcal{I}$ and negative on its complement plays an important role in both the FS Vandermonde decomposition of Toeplitz matrices and the Gram matrix parametrization of trigonometric polynomials. It is worth noting that the polynomial defined in the present paper (recall \\eqref{eq:gz1}-\\eqref{eq:r1}) is different from those in \\cite{dumitrescu2007positive,davidson2002linear,alkire2002convex}. As a matter of fact, while the polynomial we define applies uniformly to all intervals $\\mathcal{I}\\in\\mathbb{T}$, certain modifications to the polynomial or additional operations such as sliding the interval have to be taken in \\cite{dumitrescu2007positive, davidson2002linear,alkire2002convex} when $\\mathcal{I}$ contains certain critical points such as $0$ (or $1$) and $\\frac{1}{2}$.\n\\end{rem}\n\n\\section{Application in the Theory of Moments} \\label{sec:moment}\n\n\\subsection{Problem Statement}\nFor a given sequence $t_j$, $\\abs{j}\\leq N-1$ and a given domain $F$, a truncated moment problem entails determining whether there exists a positive Borel measure $\\mu$ on $F$ such that \\cite{akhiezer1965classical}\n\\equ{t_j = \\int_F z^j \\text{d} \\mu\\sbra{z}, \\quad \\abs{j}\\leq N-1. \\label{eq:tj_moment}}\nThe problem is further referred to as a truncated $K$-moment problem if $\\mu$ is constrained to be supported on a semialgebraic set $K\\subset F$, i.e., \\cite{schmudgen1991thek}\n\\equ{\\supp\\sbra{\\mu} \\subset K. \\label{eq:supp_moment}}\nA measure $\\mu$ satisfying \\eqref{eq:tj_moment} is a representing measure for $\\m{t}$; $\\mu$ is a $K$-representing measure if it satisfies \\eqref{eq:tj_moment} and \\eqref{eq:supp_moment}.\n\nThe truncated moment and $K$-moment problems have been solved when $F$ is the real or the complex domain (note that the complex moment problem is defined slightly differently from \\eqref{eq:tj_moment}) \\cite{curto1991recursiveness,curto2000truncated,lasserre2009moments}. The truncated moment problem is also solved when $F$ is the unit circle, known as the truncated trigonometric moment problem \\cite{grenander1958toeplitz,curto1991recursiveness}. In fact, the solution is given by evoking the Vandermonde decomposition of Toeplitz matrices: A representing measure $\\mu$ exists if and only if the Toeplitz matrix $\\m{T}$ formed using $\\m{t}$ admits a Vandermonde decomposition, or equivalently, $\\m{T}\\geq \\m{0}$ by Theorem \\ref{thm:VD}. To the best of our knowledge, however, the truncated trigonometric $K$-moment problem is still open. This section is devoted to a solution to this problem by applying the FS Vandermonde decomposition.\n\nNote that a semialgebraic set $K$ on the unit circle $\\mathbb{T}$ can be identified with the union of finite disjoint subintervals $\\mbra{f_{Ll}, f_{Hl}} \\subset \\mathbb{T}$, $l=1,\\dots,J$. Therefore, the moment problem of interest can be restated as follows. For a given sequence $t_j$, $\\abs{j}\\leq N-1$, the truncated trigonometric $K$-moment problem entails determining whether there exists a $K$-representing measure $\\mu$ on $\\mathbb{T}$ satisfying that\n{\\setlength\\arraycolsep{2pt}\\equa{t_j\n&=& \\int_{\\mathbb{T}} e^{-i2\\pi jf} \\text{d} \\mu\\sbra{f}, \\quad \\abs{j}\\leq N-1, \\label{eq:tj_moment1} \\\\ \\supp\\sbra{\\mu}\n&\\subset& K=\\bigcup_{l=1}^J \\mbra{f_{Ll}, f_{Hl}}\\subset \\mathbb{T}. \\label{eq:supp_moment1}} }\n\n\\subsection{Proposed Solution}\nLet $\\m{T}$ be the $N\\times N$ Toeplitz matrix formed using the moment sequence $t_j$, $\\abs{j}\\leq N-1$. Suppose that an $r$-atomic $K$-representing measure $\\mu$ for $\\m{t}$ exists that satisfies \\eqref{eq:tj_moment1} and \\eqref{eq:supp_moment1}. It follows from \\eqref{eq:supp_moment1} that\n\\equ{\\mu\\sbra{f} = \\sum_{k=1}^r p_k \\delta_{f_k}, \\quad f_k \\in K, \\label{eq:mut}}\nwhere $\\delta_{f}$ is the Dirac delta function and $p_k>0$ denotes the density at $f_k$. Inserting \\eqref{eq:mut} into \\eqref{eq:tj_moment1}, we have that\n\\equ{t_j= \\sum_{k=1}^r p_k e^{-i2\\pi jf_k}, \\quad \\abs{j}\\leq N-1,\\; f_k\\in K.}\nIt follows that\n\\equ{\\m{T} = \\sum_{k=1}^r p_k \\m{a}\\sbra{f_k}\\m{a}^H\\sbra{f_k}, \\quad f_k\\in K.}\nThis means that $\\m{T}$ admits an $r$-atomic FS Vandermonde decomposition on $K$. It is easy to show that the above arguments also hold conversely. So we conclude the following result.\n\\begin{lem} An $r$-atomic $K$-representing measure $\\mu$ for $\\m{t}$ exists if and only if $\\m{T}$ admits an $r$-atomic FS Vandermonde decomposition on $K$. \\label{lem:moment_VD}\n\\end{lem}\n\nWe next provide explicit conditions on $\\m{T}$ by applying Theorem \\ref{thm:BCVD}. In the case when $K$ is a single interval, the following theorem is a direct consequence by combining Lemma \\ref{lem:moment_VD} and Theorem \\ref{thm:BCVD}.\n\n\\begin{thm} Given $K=\\mbra{f_L, f_H}$, an $r$-atomic $K$-representing measure $\\mu$ for $\\m{t}$ exists if and only if \\eqref{eq:Tupsd} and \\eqref{eq:Tunew} hold, where $r=\\rank\\sbra{\\m{T}}$, and $g$ is defined by \\eqref{eq:gz1}-\\eqref{eq:r1}. Moreover, $\\mu$ can be found by computing the FS Vandermonde decomposition of $\\m{T}$ on $K$, and it is unique if $\\m{T}$ or $\\m{T}_g$ is rank-deficient. \\label{thm:momentsolution1}\n\\end{thm}\n\nIn the multiple frequency band case in which $K=\\bigcup_{l=1}^J \\mbra{f_{Ll}, f_{Hl}}$, corresponding to Corollary \\ref{cor:BCVDM}, we have the following corollary of Theorem \\ref{thm:momentsolution1}. The proof is trivial and is omitted.\n\\begin{cor} Given $K = \\bigcup_{l=1}^J \\mbra{f_{Ll}, f_{Hl}}$, a $K$-representing measure $\\mu$ for $\\m{t}$ exists if and only if\nthere exist sequences $\\m{t}_l$, $l=1,\\dots,J$ satisfying \\eqref{eq:sumTleqT}-\\eqref{eq:Tunewl}. \\label{cor:momentsolutionM}\n\\end{cor}\n\nCorollary \\ref{cor:momentsolutionM} provides a numerical approach to finding a $K$-representing measure, if it exists, by solving the following feasibility problem that is a SDP:\n\\equ{\\begin{split}\n&\\text{Find}\\; \\m{t}_l, \\quad l=1,\\dots,J,\\\\\n&\\text{subject to \\eqref{eq:sumTleqT}-\\eqref{eq:Tunewl}}. \\end{split} \\label{eq:feasprob}}\nIf a solution, denoted by $\\m{t}_l^*$, $l=1,\\dots,J$, can be found, then we can find representing measures for $\\m{t}_l^*$ on each corresponding interval by Theorem \\ref{thm:momentsolution1}, the sum of which finally form a $K$-representing measure for $\\m{t}$. If \\eqref{eq:feasprob} is infeasible, then no $K$-representing measure for $\\m{t}$ exists.\n\n\\begin{rem} In the case when $\\m{T}$ has full rank, the representing measure $\\mu$ might not be unique, if it exists. By solving \\eqref{eq:feasprob}, we actually find one among them. In this case the obtained measure $\\mu$ may consist of as large as $NJ$ atoms. To possibly reduce the number of atoms (a.k.a. to simplify the obtained measure), we can find the one minimizing certain convex function of $\\m{t}_l$, $l=1,\\dots,J$, e.g., $\\pm\\tr\\sbra{\\m{T}\\sbra{\\m{t}_1}}$. By doing so, it is expected that certain $\\m{T}\\sbra{\\m{t}_l}$'s are rank-deficient and thus result in a small number of atoms. \\label{rem:simplifymeasure}\n\\end{rem}\n\nFinally, it is interesting to note that the dual problem of \\eqref{eq:feasprob} can be easily obtained using the result in Section \\ref{sec:duality}. Using the cone notations \\eqref{eq:feasprob} can be written as:\n\\equ{\\begin{split}\n&\\text{Find}\\; \\m{t}_{R,l}\\in \\mathcal{K}_{\\text{VDM},l}, \\quad l=1,\\dots,J,\\\\\n&\\text{subject to } \\sum_{l=1}^J \\m{t}_{R,l} = \\m{t}_R, \\end{split} \\label{eq:feasprob2}}\nwhere $\\m{t}_{R,l}\\coloneqq \\mbra{\\m{t}_l}_R$, and $\\mathcal{K}_{\\text{VDM},l}$ denotes $\\mathcal{K}_{\\text{VDM}}$ in \\eqref{eq:KVDM} with $g$ being $g_l$. The Lagrangian function is given by:\n\\equ{\\begin{split}\\mathcal{L}\\sbra{\\m{t}_{R,1},\\dots,\\m{t}_{R,J}, \\m{\\alpha}}\n&= \\sbra{\\sum_{l=1}^J \\m{t}_{R,l} - \\m{t}_R}^T\\m{\\alpha} \\\\\n&= \\sum_{l=1}^J \\m{t}_{R,l}^T\\m{\\alpha} - \\m{t}_R^T\\m{\\alpha}, \\end{split}}\nwhere $\\m{t}_{R,l}\\in \\mathcal{K}_{\\text{VDM},l}$, $l=1,\\dots,J$, and $\\m{\\alpha}$ is the Lagrangian multiplier. Using the knowledge of the dual cone, we have that\n\\equ{\\min_{\\m{t}_{R,l} \\in \\mathcal{K}_{\\text{VDM},l}} \\mathcal{L} = \\left\\{ \\begin{array}{ll} -\\m{t}_R^T\\m{\\alpha}, & \\text{if } \\m{\\alpha}\\in \\mathcal{K}_{\\text{VDM},l}^*, l=1,\\dots,J;\\\\ -\\infty, & \\text{otherwise.} \\end{array} \\right. }\nTherefore, the dual problem is given by:\n\\equ{\\max_{\\m{\\alpha}} \\m{t}_R^T\\m{\\alpha}, \\text{ subject to } \\m{\\alpha}\\in \\bigcap_{l=1}^J \\mathcal{K}_{\\text{PolM},l}, \\label{eq:dualprob}}\nwhere we have used the identity that $\\mathcal{K}_{\\text{VDM},l}^*=\\mathcal{K}_{\\text{PolM},l}$ given by Theorem \\ref{thm:dualcone}. Note that \\eqref{eq:dualprob} can be cast as SDP following from \\eqref{KPolM}.\n\n\\begin{exa} Suppose that the moment sequence $t_j$, $\\abs{j}\\leq N-1$ is generated from its $3$-atomic representing measure\n\\equ{\\mu_1 = 0.7\\delta_{0.1} + 2\\delta_{0.25} + \\delta_{0.7},}\nwhich is plotted in Fig. \\ref{Fig:moment} together with $\\mu_j$, $j=2,\\dots,5$ that will be solved for.\n\\begin{itemize}\n \\item[1)] In the case of $N\\geq 4$, we can form the Toeplitz matrix $\\m{T}$ using $\\m{t}$, having that $\\rank\\sbra{\\m{T}} = 3 < N$. By Theorem \\ref{thm:VD}, $\\mu_1$ is the unique representing measure for $\\m{t}$.\n \\item[2)]Suppose that $N=3$ and $K = \\mbra{0.05, 0.75}$. Since $K$ includes all the frequencies in $\\mu_1$, one representing measure on $K$ has already been given by $\\mu_1$. By the existence of the representing measure, it follows from Theorem \\ref{thm:momentsolution1} that $\\m{T}$ and $\\m{T}_g$ are both PSD. Applying the proposed FS Vandermonde decomposition algorithm to the solution, the following $3$-atomic $K$-representing measure is obtained:\n     \\equ{\\mu_2 =0.4630\\delta_{0.05} + 2.2485\\delta_{0.2383} + 0.9885\\delta_{0.6927}, \\notag}\n     which is somehow similar to $\\mu_1$. Note that the frequency $0.05$ in $\\mu_2$ is nothing but the staring point of $K$, which has been deliberately chosen in the presented decomposition algorithm. Note that\n    \n \\item[3)]Suppose that $N=3$ and $K = \\mbra{0.05, 0.3}\\cup \\mbra{0.65,0.75}$. One representing measure for $\\m{t}$ is also given by $\\mu_1$. To possibly find another one, we solve \\eqref{eq:feasprob} using SDPT3 \\cite{toh1999sdpt3} in Matlab and a solution is successfully found. Applying FS Vandermonde decomposition to the solution, a $6$-atomic $K$-representing measure is given by:\n     \\equ{\\begin{split}\\mu_3\n     &= 0.1825\\delta_{0.05} + 1.2284\\delta_{0.1764}\\\\\n     &\\quad+ 1.2713\\delta_{0.2722}+ 0.1546\\delta_{0.65} \\\\\n     &\\quad + 0.5088\\delta_{0.6917} + 0.3545\\delta_{0.7436}. \\end{split} \\notag}\n     In $\\mu_3$, $0.05$ and $0.65$ are the starting points of the two intervals of $K$. The first three frequencies are located on the first interval and the other three frequencies are on the other interval.\n  \\item[4)] Suppose that $N=3$. We want to check whether one representing measure exists on $K = \\mbra{0.2, 0.3} \\cup \\mbra{0.6, 0.8}$. To do so, we also solve \\eqref{eq:feasprob} and a solution is successfully found. This means that a $K$-representing measure exists for $\\m{t}$ by Corollary \\ref{cor:momentsolutionM}. Applying the FS Vandermonde decomposition, a $6$-atomic $K$-representing measure is given by:\n     \\equ{\\begin{split}\\mu_4\n     &= 1.9614\\delta_{0.2} + 0.1296\\delta_{0.2290}\\\\\n     &\\quad+ 0.4456\\delta_{0.2891}+ 0.2437\\delta_{0.6} \\\\\n     &\\quad + 0.3637\\delta_{0.6467} + 0.5561\\delta_{0.7962}. \\end{split} }\n \\item[5)] With the same settings as in 4), instead of solving \\eqref{eq:feasprob}, we find the one maximizing $\\tr\\sbra{\\m{T}\\sbra{\\m{t}_1}}$ among all feasible representing measures on $K$, following Remark \\ref{rem:simplifymeasure}. The obtained solution $\\sbra{\\m{t}_1^*, \\m{t}_2^*}$ satisfies that $\\rank\\sbra{\\m{T}\\sbra{\\m{t}_1^*}}=\\rank\\sbra{\\m{T}\\sbra{\\m{t}_2^*}}=2<N$, resulting in the following $4$-atomic representing measure:\n     \\equ{\\begin{split} \\mu_5\n      &= 2.0837\\delta_{0.2} + 0.4726\\delta_{0.3}\\\\\n      &\\quad + 0.6218\\delta_{0.6382} + 0.5219\\delta_{0.8}. \\end{split}}\n     Compared to $\\mu_4$, the number of atoms of $\\mu_5$ is reduced.\n \\item[6)] Suppose that $N=3$ and $K = \\mbra{0.2, 0.3} \\cup \\mbra{0.6, 0.75}$. Then \\eqref{eq:feasprob} is infeasible. This means that no $K$-representing measure for $\\m{t}$ exists by Corollary \\ref{cor:momentsolutionM}.\n\\end{itemize}\n\\end{exa}\n\n\n\n\\begin{figure*}\n\\centering\n      \\includegraphics[width=4in]{Fig_moment.pdf}\n\\centering\n\\caption{Solved representing measures $\\mu_j$, $j=2,\\dots,5$ given a moment sequence generated from $\\mu_1$ and a semialgebraic set $K$ (indicated by the line segments on the $x$-axis).} \\label{Fig:moment}\n\\end{figure*}\n\n\n\n\n\\section{Application in Line Spectral Estimation} \\label{sec:application}\n\n\\subsection{Problem Statement}\nLine spectral estimation can be found in wide applications such as communications, radar, sonar, and so on \\cite{stoica2005spectral}. In particular, we have the following data model in the absence of noise:\\footnote{Note that the noisy case can be dealt with similarly with minor modifications on the presented solution. Discussions will be provided later.}\n\\equ{\\m{y}^o = \\sum_{k=1}^r \\m{a}\\sbra{f_k}s_k = \\m{A}\\sbra{\\m{f}}\\m{s}, \\label{eq:model1}}\nwhere $\\m{y}^o\\in\\mathbb{C}^N$ is a uniformly sampled signal (at a Nyquist rate), $f_k\\in\\mathbb{T}$ and $s_k\\in\\mathbb{C}$ are the normalized frequency and the complex amplitude of the $k$th sinusoid respectively, and $r$ is the number of sinusoids. To estimate the frequencies, we are given a part of the entries of $\\m{y}^o$ that form the subvector $\\m{y}_{\\Omega}^o\\in\\mathbb{C}^M$, where $\\Omega$ denotes the set of sampling indexes and is of cardinality $M<N$. This frequency estimation problem is referred to as off-grid\/continuous compressed sensing in \\cite{tang2012compressed} in the sense that we have compressive data as in the pioneering work of compressed sensing \\cite{candes2006robust}, but differently, the frequencies can take any continuous value in $\\mathbb{T}$ as opposed to the discrete setting in \\cite{candes2006robust}.\n\nIn this section, we consider the case when the frequencies are known {\\em a priori} to lie in an interval $\\mathcal{I}\\subset\\mathbb{T}$. Inspired by the recent atomic norm techniques \\cite{candes2013towards,candes2013super, bhaskar2013atomic, tang2012compressed,yang2015gridless}, the paper \\cite{mishra2015spectral} proposed an FS atomic norm approach (or constrained atomic norm in the language of \\cite{mishra2015spectral}) that was shown to achieve better performance than the standard atomic norm by exploiting the prior knowledge. In particular, define the (FS) set of atoms\n\\equ{\\mathcal{A}\\sbra{\\mathcal{I}} \\coloneqq \\lbra{\\m{a}\\sbra{f_k,\\phi_k}=\\m{a}\\sbra{f}\\phi:\\; f\\in\\mathcal{I}, \\;\\abs{\\phi}=1}. \\label{eq:A_I}}\nThe FS atomic norm is the atomic norm induced by $\\mathcal{A}\\sbra{\\mathcal{I}}$:\n\\equ{\\begin{split}\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}} \\coloneqq\n&\\inf_{c_k>0,\\m{a}_k\\in\\mathcal{A}\\sbra{\\mathcal{I}}}\\lbra{\\sum_k c_k:\\; \\m{y} = \\sum_k c_k \\m{a}_k}\\\\ =\n&\\inf_{f_k\\in\\mathcal{I},s_k}\\lbra{\\sum_k \\abs{s_k}:\\; \\m{y} = \\sum_k \\m{a}\\sbra{f_k}s_k}. \\end{split} \\label{eq:atomn_I}}\nThe following FS atomic norm minimization (FS-ANM) problem was proposed in \\cite{mishra2015spectral}:\n\\equ{\\min_{\\m{y}} \\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}, \\text{ subject to } \\m{y}_{\\Omega} = \\m{y}_{\\Omega}^o. \\label{eq:atomnmin_I}}\nThis means that, among all candidates $\\m{y}$ which are consistent with the acquired samples $\\m{y}_{\\Omega}^o$, we find the one $\\m{y}^*$ with the minimum FS atomic norm as the signal estimate, and the frequencies composing $\\m{y}^*$ form the frequency estimates. Note that the noisy case can be dealt with similarly following a standard routine (by replacing the equality constraint in \\eqref{eq:atomnmin_I} by $\\twon{\\m{y}_{\\Omega} - \\m{y}_{\\Omega}^o}\\leq \\eta$ given the upper bound $\\eta$ on the noise energy).\nNote also that \\eqref{eq:A_I}-\\eqref{eq:atomnmin_I} degenerate to the existing standard forms in the case of $\\mathcal{I}=\\mathbb{T}$.\n\nSince the FS atomic norm defined in \\eqref{eq:atomn_I} is inherently semi-infinite programming (SIP), a finite-dimensional formulation of it is required to practically solve \\eqref{eq:atomnmin_I}, which is dealt with in the ensuing section by applying the FS Vandermonde decomposition.\n\n\n\\subsection{SDP Formulation of FS Atomic Norm}\n\nBy applying the FS Vandermonde decomposition, the FS atomic norm is cast as SDP in the following theorem.\n\\begin{thm} It holds that\n\\equ{\\begin{split} \\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}\n=& \\min_{x, \\m{t}} \\frac{1}{2}x + \\frac{1}{2}t_0,\\\\\n& \\text{ subject to } \\begin{bmatrix} x & \\m{y}^H \\\\ \\m{y} & \\m{T} \\end{bmatrix}\\geq \\m{0} \\text{ and } \\m{T}_g\\geq \\m{0}, \\end{split} \\label{eq:atomnDsdp}}\nwhere $g$ is as defined previously.\n \\label{thm:atomnD}\n\\end{thm}\n\n\\begin{proof} Let $F^*$ be the optimal objective value of \\eqref{eq:atomnDsdp}. We need to show that $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}} = F^*$.\n\nWe first show that $F^* \\leq \\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}$. To do so, let $\\m{y} = \\sum_k c_k\\m{a}\\sbra{f_k,\\phi_k}$ be an FS atomic decomposition of $\\m{y}$ on $\\mathcal{I}$. Then let $\\m{t}$ be such that $\\m{T}(\\m{t}) = \\sum_k c_k \\m{a}\\sbra{f_k}\\m{a}^H\\sbra{f_k}$ and $x = \\sum_k c_k$. By Theorem \\ref{thm:BCVD}, we have that $\\m{T}_g\\geq\\m{0}$. Moreover, it holds that\n\\equ{\\begin{bmatrix} x & \\m{y}^H \\\\ \\m{y} & \\m{T} \\end{bmatrix} = \\sum_k c_k\\begin{bmatrix} \\overline{\\phi}_k \\\\ \\m{a}\\sbra{f_k} \\end{bmatrix} \\begin{bmatrix} \\overline{\\phi}_k \\\\ \\m{a}\\sbra{f_k} \\end{bmatrix}^H \\geq \\m{0}.}\nTherefore, $x$ and $\\m{t}$ constructed above form a feasible solution to the problem in \\eqref{eq:atomnDsdp}, at which the objective value equals\n\\equ{\\frac{1}{2}x + \\frac{1}{2}t_0 = \\sum_k c_k.}\nIt follows that $F^* \\leq \\sum_k c_k$. Since the inequality holds for any FS atomic decomposition of $\\m{y}$ on $\\mathcal{I}$, we have that $F^* \\leq \\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}$ by the definition of $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}$.\n\nOn the other hand, suppose that $\\sbra{x^*, \\m{t}^*}$ is an optimal solution to the problem in \\eqref{eq:atomnDsdp}. By the fact that $\\m{T}(\\m{t}^*)\\geq \\m{0}$ and $\\m{T}_g(\\m{t}^*)\\geq \\m{0}$ and applying Theorem \\ref{thm:BCVD}, we have that $\\m{T}\\sbra{\\m{t}^*}$ has an FS Vandermonde decomposition on $\\mathcal{I}$ as in \\eqref{eq:VD} with $\\sbra{r, p_k, f_k}$ denoted by $\\sbra{r^*, p_k^*, f_k^*}$.\nBy the fact that $\\begin{bmatrix} x^* & \\m{y}^H \\\\ \\m{y} & \\m{T}\\sbra{\\m{t}^*} \\end{bmatrix}\\geq\\m{0}$, we have that $\\m{y}$ lies in the range space of $\\m{T}\\sbra{\\m{t}^*}$ and thus has the following FS atomic decomposition:\n\\equ{\\m{y} = \\sum_{k=1}^{r^*} c_k^*\\m{a}\\sbra{f_k^*,\\phi_k^*}, \\quad f_k^*\\in\\mathcal{I}. \\label{eq:yatomdec}}\nMoreover, it holds that\n{\\setlength\\arraycolsep{2pt}\\equa{x^*\n&\\geq& \\m{y}^H \\mbra{\\m{T}\\sbra{\\m{t}^*}}^{\\dag}\\m{y} = \\sum_{k=1}^{r^*} \\frac{c_k^{*2}}{p_k^*},\\\\ t_0^*\n&=& \\sum_{k=1}^{r^*} p_k^*.\n}}It therefore follows that\n\\equ{\\begin{split}F^*\n&= \\frac{1}{2}x^* + \\frac{1}{2}t_0^* \\\\\n&\\geq \\frac{1}{2}\\sum_k \\frac{c_k^{*2}}{p_k^*} + \\frac{1}{2}\\sum_k p_k^* \\\\\n&\\geq \\sum_{k} c_k^*\\\\\n&\\geq \\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}. \\end{split} \\label{eq:Fleqsum} }\nCombining \\eqref{eq:Fleqsum} and the inequality that $F^* \\leq \\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}$ as shown previously, we conclude that $F^* = \\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}$ and complete the proof. At last, it is worth noting that by \\eqref{eq:Fleqsum} it must hold that $p_k^* = c_k^*$ and $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}} = \\sum_{k} c_k^*$. Therefore, the FS atomic decomposition in \\eqref{eq:yatomdec} must achieve the FS atomic norm.\n\\end{proof}\n\n\\begin{rem} Note that the SDP formulation of the FS atomic norm presented in Theorem \\ref{thm:atomnD} can be easily extended to the multiple frequency band case by applying Corollary \\ref{cor:BCVDM}, to be specific, by replacing the constraints in \\eqref{eq:atomnDsdp} resulting from \\eqref{eq:Tupsd} and \\eqref{eq:Tunew} by those in \\eqref{eq:sumTleqT}-\\eqref{eq:Tunewl}. The proof of Theorem \\ref{thm:atomnD} can still be applied in this case with minor modifications. \\label{rem:SDP_MB}\n\\end{rem}\n\nIt immediately follows from Theorem \\ref{thm:atomnD} that \\eqref{eq:atomnmin_I} can be written as the following SDP:\n\\equ{\\begin{split}\n& \\min_{\\m{y}, x, \\m{t}} \\frac{1}{2}x + \\frac{1}{2}t_0,\\\\\n& \\text{ subject to } \\begin{bmatrix} x & \\m{y}^H \\\\ \\m{y} & \\m{T} \\end{bmatrix}\\geq \\m{0}, \\m{T}_g\\geq \\m{0}  \\text{ and } \\m{y}_{\\Omega} = \\m{y}_{\\Omega}^o. \\end{split} \\label{eq:CANM_sdp}}\nNote that \\eqref{eq:CANM_sdp} can be solved using off-the-shelf SDP solvers such as SDPT3. Given its solution, the frequencies can be retrieved from the FS Vandermonde decomposition of $\\m{T}$. Moreover, as in the standard atomic norm method, the Toeplitz matrix $\\m{T}$ in \\eqref{eq:CANM_sdp} can be interpreted as the ``data covariance matrix'' \\cite{yang2015gridless,yang2016vandermonde}. By solving \\eqref{eq:CANM_sdp} we actually fit the data covariance matrix $\\m{T}$ by exploiting its structures, e.g., PSDness (the first constraint), Toeplitz (explicitly imposed) and low rank ($t_0$ in the objective is proportional to the nuclear or trace norm of $\\m{T}$), and its connection to the acquired data $\\m{y}_{\\Omega}^o$ (the first and the last constraints). But different from the standard atomic norm method, more precise knowledge of $\\m{T}$ is exploited in the FS atomic norm method by additionally including the constraint $\\m{T}_g\\geq \\m{0}$.\n\nBefore proceeding to the next subsection, we note that \\eqref{eq:atomnmin_I} was solved by studying its dual in \\cite{mishra2015spectral}. In particular, the dual of \\eqref{eq:atomnmin_I} is given by:\n\\equ{\\max_{\\m{z}} \\Re \\sbra{\\m{y}_{\\Omega}^o \\m{z}_{\\Omega}}, \\text{ subject to } \\norm{\\m{z}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}^*\\leq 1 \\text{ and } \\m{z}_{\\Omega^c}=\\m{0}, \\label{eq:dualprb}}\nwhere $\\Omega^c$ denotes the complement of $\\Omega$ and $\\norm{\\m{z}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}^*$ is the dual FS atomic norm. By the fact that\n\\equ{\\norm{\\m{z}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}^* = \\sup_{\\m{a}\\in\\mathcal{A}\\sbra{\\mathcal{I}}} \\Re\\sbra{\\m{a}^H\\m{z}} = \\sup_{f\\in\\mathcal{I}} \\abs{\\m{a}^H\\sbra{f}\\m{z}}, }\nthe constraint that $\\norm{\\m{z}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}^*\\leq 1$ can be cast as the following:\n\\equ{\\abs{\\m{a}^H\\sbra{f}\\m{z}}\\leq 1 \\text{ for any } f\\in\\mathcal{I}, \\label{eq:absazleq1}}\nwhere\n\\equ{q(f)\\coloneqq \\m{a}^H\\sbra{f}\\m{z} \\label{eq:qf}}\nis referred to as the dual polynomial \\cite{candes2013towards,mishra2015spectral}.\nIt follows that $1-\\abs{q(f)}^2$ is a Hermitian trigonometric polynomial nonnegative on $\\mathcal{I}$ and, by the PRL, admits a Gram matrix parametrization as in \\eqref{KPolM}. With some further derivations that we will omit, it can be shown that \\eqref{eq:absazleq1} holds if and only if the unit polynomial (the right hand side of the inequality in \\eqref{eq:absazleq1}) has the following Gram matrix parametrization:\n\\equ{\\tr\\sbra{\\m{\\Theta}_j \\m{Q}_0} + \\tr\\mbra{ \\m{\\Theta}_{gj} \\m{Q}_1} = \\left\\{ \\begin{array}{ll} 1, & \\text{if } j=0,\\\\ 0, & \\text{otherwise}, \\end{array}\\right. \\label{eq:sdpbrl1}}\nwhere $\\m{Q}_0$ and $\\m{Q}_1$ satisfy\n\\equ{\\begin{bmatrix}1 & \\m{z}^H \\\\ \\m{z} & \\m{Q}_0 \\end{bmatrix} \\geq \\m{0} \\text{ and } \\m{Q}_1\\geq \\m{0}. \\label{eq:sdpbrl2}}\nIn fact, the characterization of \\eqref{eq:absazleq1} using \\eqref{eq:sdpbrl1} and \\eqref{eq:sdpbrl2} is nothing but the result of the bounded real lemma (BRL) for trigonometric polynomials \\cite{davidson2002linear,dumitrescu2007positive}. This can be viewed as a more precise result of the PRL when dealing with bounded polynomials as in \\eqref{eq:absazleq1}. Finally, \\eqref{eq:dualprb} is cast as the following SDP:\n\\equ{\\max_{\\m{z}, \\m{Q}_0,\\m{Q}_1} \\Re \\sbra{\\m{y}_{\\Omega}^o \\m{z}_{\\Omega}}, \\text{ subject to } \\eqref{eq:sdpbrl1}, \\eqref{eq:sdpbrl2} \\text{ and } \\m{z}_{\\Omega^c}=\\m{0}. \\label{eq:dualSDP}}\nWithout surprise, it follows from a standard Lagrangian analysis that \\eqref{eq:dualSDP} is the dual of \\eqref{eq:CANM_sdp} (note that the analysis uses \\eqref{eq:TinTheta} and \\eqref{eq:TginThetag} and will be left to interested readers). Since strong duality holds \\cite{boyd2004convex}, the solution to \\eqref{eq:dualSDP} can be obtained for free when solving \\eqref{eq:CANM_sdp} using a primal-dual algorithm, and vice versa.\n\nIn summary, the FS Vandermonde decomposition can be applied to provide a primal SDP formulation of \\eqref{eq:atomnmin_I}, while the trigonometric polynomial based technique in \\cite{mishra2015spectral} provides a dual SDP formulation. Moreover, the FS Vandermonde decomposition also provides a new method for frequency retrieval. In fact, it is found that the new method results in higher numerical stability, as compared to the root-finding method in \\cite{candes2013towards,mishra2015spectral}. This can be explained as follows. By using the FS Vandermonde decomposition, we can always determine the number of frequencies first by computing $\\rank\\sbra{\\m{T}}$, which can effectively reduce the problem dimension and improve stability. In contrast to this, the root-finding method requires to solve all, up to $2N-2$, roots of the polynomial $1-\\abs{q(f)}^2$, among which appropriate ones (those with unit modulus) are then selected to produce the frequencies.\n\n\n\\subsection{Computational Complexity}\nWe next analyze the computational complexity of the presented FS atomic norm method, to be specific, the complexity of solving the SDP in \\eqref{eq:CANM_sdp}. To do so, we consider the general multiple band case in which, according to Remark \\ref{rem:SDP_MB}, \\eqref{eq:CANM_sdp} becomes:\n\\equ{\\begin{split}\n& \\min_{\\m{y}, x, \\m{t}_l} \\frac{1}{2}x + \\frac{1}{2}\\sum_{l=1}^J t_{l0},\\\\\n& \\text{ subject to } \\begin{bmatrix} x & \\m{y}^H \\\\ \\m{y} & \\sum_{l=1}^J \\m{T}(\\m{t}_l) \\end{bmatrix}\\geq \\m{0}, \\\\\n&\\phantom{\\text{ subject to }} \\m{T}(\\m{t}_l) \\geq \\m{0}, \\; \\m{T}_g(\\m{t}_l)\\geq \\m{0}, l=1,\\dots,J,\\\\\n&\\phantom{\\text{ subject to }} \\m{y}_{\\Omega} = \\m{y}_{\\Omega}^o. \\end{split} \\label{eq:CANM_sdp_MB}}\nEvidently, the SDP in \\eqref{eq:CANM_sdp_MB} has $n=O(JN)$ free variables and $m=2J+1$ LMIs, and the $i$th LMI has size of $k_i\\times k_i$ with $k_i=O(N)$. It follows from \\cite{ben2013lectures} that a primal-dual algorithm for \\eqref{eq:CANM_sdp_MB} has a computational complexity on the order of\n\\equ{\\sbra{1+\\sum_{i=1}^m k_i}^{\\frac{1}{2}} n\\sbra{n^2+n\\sum_{i=1}^m k_i^2 + \\sum_{i=1}^m k_i^3} = O\\sbra{J^{3.5}N^{4.5}}. \\label{eq:complexity}}\nBy arguments similar to those above, the standard atomic norm method in the absence of prior knowledge has a computational complexity of $O\\sbra{N^{4.5}}$. This together with\n\\eqref{eq:complexity} indicates that, with a fixed number of intervals $J$, the presented FS atomic norm method has a complexity higher than the standard atomic norm method by a constant factor and the factor increases with $J$.\n\n\\subsection{Numerical Simulation}\nWe provide a simple illustrative example below to demonstrate the advantage of using the prior knowledge for frequency estimation.\n\n\\begin{exa} Consider a line spectrum composed of $K=3$ frequencies $\\m{f}=[0.22, 0.23, 0.28]^T$ as shown in Fig. \\ref{Fig:LSE}. To estimate\/recover the spectrum, $M=16$ randomly located noiseless samples are acquired among $N=64$ uniform samples. The standard ANM and the FS-ANM methods are implemented using SDPT3 to estimate the line spectrum. In FS-ANM, the prior knowledge that the frequencies lie in $\\mathcal{I}=\\mbra{0.2, 0.3}$ is used. The estimation results are presented in Fig. \\ref{Fig:LSE}. It can be seen that FS-ANM exactly recovers the spectrum but ANM does not. For both ANM and FS-ANM, the recovered frequencies retrieved using the Vandermonde decomposition match the locations at which the dual polynomials have unit magnitude. For FS-ANM the frequencies computed using the FS Vandermonde decomposition have recovery errors on the order of $10^{-10}$ while those computed using the root-finding method have errors on the order of $10^{-6}$.\n\\end{exa}\n\n\\begin{figure}\n\\centering\n  \\subfigure[ANM]{\n    \\label{Fig:ANM}\n    \\includegraphics[width=3in]{Fig_AppLSE_ANM.pdf}}%\n  \\subfigure[FS-ANM]{\n    \\label{Fig:CANM}\n    \\includegraphics[width=3in]{Fig_AppLSE_CANM.pdf}}\n\\centering\n\\caption{Line spectral estimation results of (a) ANM and (b) FS-ANM.} \\label{Fig:LSE}\n\\end{figure}\n\nNote that the presented method can deal with noise with minor modifications, as shown in \\cite{mishra2015spectral}. In the noisy case, a simulation has been included in \\cite{mishra2015spectral} to compare the signal recovery errors of the atomic norm method in cases with and without the prior knowledge. It is shown that ``the prior information formulation yields a higher stability in presence of noise.'' Readers are referred to \\cite[Section VIII-B]{mishra2015spectral} for detail.\n\n\n\n\n\n\\subsection{Extension to FS Atomic $\\ell_0$ Norm}\nIn this subsection, we provide an example in which the FS Vandermonde decomposition result is applicable but the theory of trigonometric polynomials is not. In particular, we study the FS atomic $\\ell_0$ norm defined by:\n\\equ{\\begin{split}\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}},0} \\coloneqq\n&\\inf_{c_k>0,\\m{a}_k\\in\\mathcal{A}\\sbra{\\mathcal{I}}}\\lbra{\\mathcal{K}:\\; \\m{y} = \\sum_{k=1}^{\\mathcal{K}} c_k \\m{a}_k}\\\\ =\n&\\inf_{f_k\\in\\mathcal{I},s_k}\\lbra{\\mathcal{K}:\\; \\m{y} = \\sum_{k=1}^{\\mathcal{K}} \\m{a}\\sbra{f_k}s_k}. \\end{split} \\label{eq:atom0n_I}}\n$\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}},0}$ is of interest since it exploits sparsity to the greatest extent possible, while $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}$ is in fact its convex relaxation. It has been vastly demonstrated in the literature on compressed sensing that improved performance can usually be obtained by solving (or approximately solving) $\\ell_0$ norm based problems (see, e.g., \\cite{candes2008enhancing,andersson2014new,yang2016vandermonde}). More recently, a new trend of frequency estimation is to directly solve the $\\ell_0$ norm based formulations using nonconvex optimization techniques for low rank matrix recovery \\cite{cho2016fast,cai2016fast}. To do so, the key is to formulate the frequency estimation problem in the continuous setting as a matrix rank minimization problem. In the context of the FS atomic $\\ell_0$ norm, the following result can be obtained by applying the FS Vandermonde decomposition.\n\n\\begin{thm} It holds that\n\\equ{\\begin{split} \\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}},0}\n=& \\min_{x, \\m{t}} \\rank\\sbra{\\m{T}},\\\\\n& \\text{ subject to } \\begin{bmatrix} x & \\m{y}^H \\\\ \\m{y} & \\m{T} \\end{bmatrix}\\geq \\m{0} \\text{ and } \\m{T}_g\\geq \\m{0}, \\end{split} \\label{eq:atom0nDsdp}}\nwhere $g$ is as defined previously.\n \\label{thm:atom0nD}\n\\end{thm}\n\n\\begin{proof} The proof is similar to that of Theorem \\ref{thm:atomnD}. At the first step, by applying the FS Vandermonde decomposition, we can construct a feasible solution, as in the proof of Theorem \\ref{thm:atomnD}, to the optimization problem in \\eqref{eq:atom0nDsdp}, which concludes that $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}},0}\\leq r^*$, where $r^*$ denotes the optimal objective value of \\eqref{eq:atom0nDsdp}. At the second step, for any optimal solution that achieves the optimal value $r^*$, we can similarly obtain an $r^*$-atomic FS decomposition of $\\m{y}$, which results in that $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}},0}\\geq r^*$. So we complete the proof.\n\\end{proof}\n\nIt follows from Theorem \\ref{thm:atom0nD} that $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}},0}$ can be cast as a rank minimization problem, while solving (or approximately solving) the resulting optimization problem is beyond the scope of this paper. It is worth noting that, since $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}},0}$ is nonconvex, a trigonometric polynomial based technique, as used for $\\norm{\\m{y}}_{\\mathcal{A}\\sbra{\\mathcal{I}}}$ in \\cite{mishra2015spectral}, cannot be applied in this case to provide a finite-dimensional formulation.\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nIn this paper, the FS Vandermonde decomposition of Toeplitz matrices on a given interval was studied. The new result generalizes the classical Vandermonde decomposition result. It was shown by duality to be connected to the theory of trigonometric polynomials. It was also applied to provide a solution to the classical truncated trigonometric $K$-moment problem and a primal SDP formulation of the recent FS atomic norm for line spectral estimation with prior knowledge.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nContinuous word representation has been a fundamental ingredient to many NLP tasks with the advent of simple and successful approaches such as Word2Vec \\cite{mikolov2013efficient} and GloVe \\cite{pennington2014glove}.\nAlthough it has been verified that they are effective in formulating semantic and syntactic relationship between words, there are some limitations.\nFirst, they are only available to words in pre-defined vocabulary thus prone to the Out-of-Vocabulary(OOV) problem.\nSecond, they cannot utilize subword information at all because they regard word as a basic unit.\nThose problems become more magnified when applying word-based methods to agglutinative languages such as Korean, Japanese, Turkish, and Finnish.\nIn this work, we propose a new model that utilizes syllables as basic components of word representation to alleviate the problems, especially for Korean.\nIn our experiment, we confirm that our model constructs  representation of words which contains a semantic and syntactic relationship between words.\nWe also show that our model can handle OOV problem and capture morphological information without dedicated analysis.\n\n\n\\section{Related Work}\n\nRecent works that utilize subword information to construct word representation could be largely divided into two families: The models that use morphemes as a component and the others taking advantage of characters.\n\n\\vspace{2mm}\n\\noindent\\textbf{Morpheme-based representation models}\n\nA morpheme is the smallest unit of meaning in linguistics.\nTherefore, there are many researches that consider morphemes when building word representations \\cite{luong2013better, botha2014compositional, cotterell2015morphological}.\n\n\\newcite{luong2013better} applies a recursive neural network over morpheme embeddings to obtain word embeddings.\nAlthough morpheme-based models are good at capturing semantics, one major drawback is that most of them require manually annotated data or an explicit morphological analyzer which could introduce unintended errors.\nOur model doesn't need such a preprocessing.\n\n\\begin{figure*}[!htbp]\n\t\\includegraphics[width=1.0\\textwidth,height=200pt]{model}\n\t\\caption{Overall architecture of our model. Each syllable is a $d$-dimensional vector. For a given word `\\begin{CJK}{UTF8}{mj}\uc548\ub155\ud558\uc138\uc694\\end{CJK}' (hello, \\emph{annyeonghaseyo}), we concatenate vectors according to syllable order in word. After passing through the convolutional layer and max pooling layer, word representation is produced. All parameters are jointly trained by Skip-gram scheme.\\label{fig:model}}\n\\end{figure*}\n\n\\vspace{2mm}\n\\noindent\\textbf{Character-based representation models}\n\nRecently, utilizing information from characters has become one of the active NLP research topics.\nOne way to extract knowledge from a sequence of characters is using character n-grams \\cite{wieting2016charagram, bojanowski2016enriching}.\n\n\\newcite{bojanowski2016enriching} suggests an approach based on the Skip-gram model \\cite{mikolov2013efficient}, where the model sums character n-gram vectors to represent a word.\nOn the other hand, there are some approaches \\cite{dos2014deep, ling2015finding, santos2015boosting, zhang2015character, kim2016character, jozefowicz2016exploring, chung2016character} in which word representations are composed of character embeddings via deep neural networks such as convolutional neural networks (CNN) or recurrent neural networks (RNN).\n\n\\newcite{kim2016character} introduces a language model that aggregates subword information through a character-level CNN. \nModels based on characters have shown competitive results on many tasks.\nA problem of character-based models is that characters themselves have no semantic meanings so that models often concentrate on only local syntactic features of words.\nTo avoid the problem, we select syllables which have fine-granularity like a character but has its own meaning in Korean as a basic component of the representation of words.\n\n\\section{Proposed Model}\n\n\\noindent\\textbf{Characteristics of Korean Words}\n\nMorphologically, unlike many other languages, a Korean word (\\emph{Eojeol}) is not just a concatenation of characters.\nIt is constructed by the following hierarchy: a sequence of syllables (\\emph{Eumjeol}) forms a word, and the composition of 2 or 3 characters (\\emph{Jaso}) forms a syllable \\cite{kang1994syllable}.\n\nIn linguistics, Korean language is categorized as an agglutinative language, where each word is made of a set of morphemes.\nTo complete the Korean word (\\emph{Eumjeol}), a root morpheme must be combined with a bound morpheme (\\emph{Josa}), or a postposition (\\emph{Eomi}). \nThis derivation produces about 60 different forms of the similar meaning, which causes the explosion of vocabulary.\nFor the same reason, the number of occurrences of each word is relatively small even with a large corpus, which prevents the model from an efficient learning.\nThus, most of the Korean word representation models use morphemes as an embedding unit, though it requires an additional preprocessing.\nThe problem is that errors coming from an immature morpheme analyzer might be propagated to the word representation model.\nMoreover, a single Korean syllable possess a semantic meaning.\nFor example, the word `\\begin{CJK}{UTF8}{mj}\ub300\ud559\\end{CJK}'(college, \\emph{daehag}) is a composition of `\\begin{CJK}{UTF8}{mj}\ub300\\end{CJK}'(big, or great, \\emph{dae}) and `\\begin{CJK}{UTF8}{mj}\ud559\\end{CJK}'(learn, or a study, \\emph{hag}).\nTherefore, our model regards syllables as embedding units rather than words or morphemes.\nFor instance, the representations of `\\begin{CJK}{UTF8}{mj}\ub098\ub294\\end{CJK}'(I am, \\emph{naneun}), `\\begin{CJK}{UTF8}{mj}\ub098\uc758\\end{CJK}'(my, \\emph{naui}), or `\\begin{CJK}{UTF8}{mj}\ub098\uc5d0\uac8c\\end{CJK}'(to me, \\emph{na-ege}) are constructed by leveraging the same syllable vector `\\begin{CJK}{UTF8}{mj}\ub098\\end{CJK}'(I, \\emph{na}).\n\n\\vspace{5mm}\n\\noindent\\textbf{Syllable-based Representation}\n\nSimilar to \\cite{kim2016character}, let $\\mathcal{S}$ be a set of all Korean syllables.\nWe embed each syllables into $d$-dimensional vector space, so that $Q \\in \\mathbb{R}^{d \\times |\\mathcal{S}|}$ becomes a syllable embedding matrix.\nLet $(s_1, s_2 , ..., s_l)$ denote a word $t \\in V$ which consists of $l$ syllables, $t$ is represented by concatenating syllable vectors as a column vector: $(Qs_1, Qs_2, ..., Qs_l) \\in \\mathbb{R}^{d \\times l}$.\nThen we apply a convolution filter $H \\in \\mathbb{R}^{d \\times w}$ having a width $w$, we get a feature map $f^t \\in \\mathbb{R}^{l-w+1}$.\nFor filters whose widths are more than 1, they need a zero padding when processing words coming from only a single syllable.\n\nIn detail, for the given filter $H$, the feature map can be calculated as follows:\n\\begin{equation}\nf^t_i = \\tanh (\\langle (Qs_i, ..., Qs_{i+w-1}) ,H \\rangle + b)\n\\end{equation}\n\\noindent where $\\langle A, B \\rangle = \\text{tr}(AB^\\intercal)$ denoting Frobenius inner product.\nWe then apply a max pooling $y^t = \\max_i f^t_i$ to extract the most important feature.\nBy using multiple filters, namely $H_1, H_2, ..., H_h$, we get a final representation $y^t = (y^t_1, ..., y^t_h)$ for the word $t$.\n\nFor training, we adopt Skip-gram \\cite{mikolov2013distributed} method with negative sampling so that for a given center word $y^t$, we maximize the log-probability of predicting context word $y^c$.\nWe jointly train syllable embedding matrix and convolution filters all together.\nFigure \\ref{fig:model} shows overall architecture of our model.\n\n\\section{Experiments and Results}\n\n\\noindent\\textbf{Datasets and Baselines}\n\nThe Experiments are performed on a randomly sampled subset of Korean News corpus collected from 2012 to 2014, containing approximately 2.7M tokens, 11k vocabulary, and 1k syllables.\nWe compare our model to the original \\emph{skip-gram model with negative sampling}~\\cite{mikolov2013distributed} as a baseline.\n\n\\vspace{2mm}\n\\noindent\\textbf{Implementation details}\n\nFor all experiments, we use the following common parameters for both our model and baseline.\nWe use vector representations of dimension 320, the size of window is 4 and the negative-sampling parameter is 7.\nWe train over twelve epochs.\nIn our model, the dimension of syllable embedding is 320.\nEmpirically, using filters with size 1\\textasciitilde4 was enough since most of Korean words are composed of 2\\textasciitilde4 syllables\\footnote[1]{About 95\\% of words in a training set had a length less than 5.}.\n\n\\begin{figure}[!htbp]\n  \\includegraphics[width=\\linewidth]{score}\n  \\caption{Test result on translated WordSim353 dataset. It contains similarity and relatedness test and measured by Pearson correlation. Our model outperformed the baseline in similarity task.\\label{fig:score}}\n\\end{figure}\n\n\\subsection{Quantitative Evaluation}\n\nWe use the WordSim353 dataset \\cite{finkelstein2001placing, agirre2009study} for the word similarity and relatedness task.\nAs WordSim353 dataset is an English data, we translated it into Korean.\nThe quality of the word vector representation is evaluated by computing Pearson correlation coefficient between human judgment scores and the cosine similarity between word vectors.\n\nThe graph in Figure \\ref{fig:score} shows that our model outperforms the baseline on WS353-Similarity dataset.\nWe estimated it since a lot of similar words share the same syllable(s) in Korean.\nOn the other hand, on WS353-Relatedness, the performance is not as good in comparison with the similarity task.\nWe presume that leveraging syllables on computing representations can be a noise among related words without common syllables.\n\n\\subsection{Qualitative Evaluation}\n\n\\noindent\\textbf{Out-Of-Vocabulary Test}\n\n\\begin{table}\n  \\centering\n  \\begin{tabular}{c|c}\n    \\toprule\n    Original word & Newly coined word \\\\\n    \\midrule\n    \\makecell{\\begin{CJK}{UTF8}{mj}\uad6c\uae00\\end{CJK}\\\\(Google, \\emph{gugeul})} & \\makecell{\\begin{CJK}{UTF8}{mj}\uad6c\uae00\uc2e0\\end{CJK}\\\\(God google, \\emph{gugeulsin})} \\\\\n    \\makecell{\\begin{CJK}{UTF8}{mj}\uc774\ub4dd\\end{CJK}\\\\(Profit, \\emph{ideug})} & \\makecell{\\begin{CJK}{UTF8}{mj}\uac1c\uc774\ub4dd\\end{CJK}\\\\(Real profit, \\emph{gaeideug})} \\\\\n    \\makecell{\\begin{CJK}{UTF8}{mj}\ud1f4\uadfc\\end{CJK}\\\\(Leave work,\\\\ \\emph{toegeun})} & \\makecell{\\begin{CJK}{UTF8}{mj}\ud1f4\uadfc\uac01\\end{CJK}\\\\(Time to leave work,\\\\ \\emph{toegeungag})} \\\\\n    \\makecell{\\begin{CJK}{UTF8}{mj}\uac24\ub7ed\uc2dc\ub178\ud2b8\\end{CJK}\\\\(Galaxy Note,\\\\ \\emph{gaelleogsinoteu})} & \\makecell{\\begin{CJK}{UTF8}{mj}\uac24\ub178\ud2b8\\end{CJK}\\\\(Gal'Note,\\\\ \\emph{gaelnoteu})} \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{4 newly coined words in Korean which did not appear in training data. Proposed model successfully recognized stem from the original word, and predicted it as the most similar word.\\label{tab:oov}}\n\\end{table}\n\nSince our model uses syllable vectors when computing word representation, it is possible to achieve representation of OOV words by combining syllables.\nTo evaluate the representations of OOV words, we manually chose 4 newly coined words not appear in training data (Table \\ref{tab:oov}).\nThese words were derived from original words.\nFor example, `\\begin{CJK}{UTF8}{mj}\uad6c\uae00\uc2e0\\end{CJK}'(God Google, \\emph{gugeulsin}) is derived from `\\begin{CJK}{UTF8}{mj}\uad6c\uae00\\end{CJK}'(Google, \\emph{gugeul}) and `\\begin{CJK}{UTF8}{mj}\uac24\ub178\ud2b8\\end{CJK}`(Gal'Note, \\emph{gaelnoteu}) is a abbreviation form of `\\begin{CJK}{UTF8}{mj}\uac24\ub7ed\uc2dc\ub178\ud2b8\\end{CJK}'(Galaxy Note, \\emph{gaelleogsinoteu}).\nMorphologically, two of them concatenate additional syllables to the original word, and the other two remove some syllables.\n\nWe examined the nearest neighbor of the representations of OOV words, and confirmed that each original word vector is placed in the nearest distance.\nIt is no wonder since almost every newly coined word keeps the syllables of original word with their positions fixed.\n\n\\vspace{2mm}\n\\noindent\\textbf{Morphological Representation Test}\n\nWe now evaluate our model on language morphology by observing how word representation leverages morphological characteristics.\nAs mentioned above, the process of forming a sentence of Korean is totally different from many other languages.\nIn case of Korean, a word can function in the sentence only if it is combined with the bound morpheme.\nFor example, `\\begin{CJK}{UTF8}{mj}\uc11c\uc6b8\uc744\\end{CJK}'(of Seoul, \\emph{seoul-eul}) is a combination of full morpheme `\\begin{CJK}{UTF8}{mj}\uc11c\uc6b8\\end{CJK}'(Seoul, \\emph{seoul}) + bound morpheme `\\begin{CJK}{UTF8}{mj}\uc744\\end{CJK}'(of, \\emph{eul}).\n\n\\begin{figure}[!ht]\n  \\adjustbox{minipage=2em,valign=c}{\\subcaption{}\\label{sfig:pca-a}}%\n  \\begin{subfigure}[c]{\\dimexpr0.9\\linewidth-1em\\relax}\n    \\centering\n    \\includegraphics[width=\\textwidth,valign=c]{pca-baseline}\n  \\end{subfigure}\n  \\adjustbox{minipage=2em,valign=c}{\\subcaption{}\\label{sfig:pca-b}}%\n  \\begin{subfigure}[c]{\\dimexpr0.9\\linewidth-1em\\relax}\n    \\centering\n    \\includegraphics[width=\\textwidth,valign=c]{pca-ours}\n  \\end{subfigure}\n  \\caption{PCA projections of vector representation of 100 randomly sampled pairs of word. Each pair is composed of a word and the same word with postposition. In (b), our model shows that words forming the discriminative parallel clusters against postposition-combined-words.\\label{fig:pca}}\n\\end{figure}\n\nTo compare how models learn the morphological characteristics, we randomly sampled hundred words and the same words combined with certain postposition(`\\begin{CJK}{UTF8}{mj}\uc744\\end{CJK}', \\emph{eul}) from the training data.\nThe graph in Figure \\ref{fig:pca} shows this result more clearly.\nWe can observe that words forming the discriminative parallel clusters against postposition-combined-words while the baseline doesn't.\n\n\\section{Conclusion}\n\nWe present a syllable-based word representation model experimented with Korean, which is one of morphologically rich languages.\nOur model keeps the characteristics of Skip-gram models, in which word representation learns from context words.\nIt also takes into account the morphological characteristics by sharing parameters between the words that contain common syllables.\nWe demonstrate that our model is competitive on quantitative evaluations.\nFurthermore, we show that the model can handle OOV words, and capture morphological relationships.\nAs a future work, we have a plan to expand our model so that it can utilize overall information extracted from words, morphemes and characters.\n\n\\section*{Acknowledgments}\nThis work was supported by the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2016M3C4A7952587, PF Class Heterogeneous High Performance Computer Development).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMolecular dynamics (MD) is a computationally intensive simulation technique\nwhich is used to gather  insights into the structural,\nthermodynamic and kinetic properties of \nsystems modeled at the atomistic level. MD simulations have been extensively used to study, e.g.,\nbiologically-relevant conformational changes of proteins, their\ninteractions with small molecules, and with other proteins.  The\nproblem of achieving sufficient sampling of the phase space has been\ntackled  by careful software parallelization~\\cite{phillips_scalable_2005,pronk_gromacs_2013} and\naccelerated processors~\\cite{harvey_acemd:_2009}.\nMoreover, the sampling problem can be addressed by\nenhanced sampling\ntechniques, such as umbrella sampling~\\cite{torrie_nonphysical_1977},\nmetadynamics~\\cite{laio_metadynamics:_2008}, steered MD~\\cite{giorgino_high-throughput_2011}, accelerated MD~\\cite{hamelberg_accelerated_2004}, etc.\nThese approaches  \nalter the Hamiltonian or the temperature of the system,  therefore  modifying\nits kinetic properties. However, they allow the  reconstruction of  the free\nenergy landscape.\n\n\n\n\nMost of the biased sampling methodologies require the modeler to\nchoose a few ``important'' \ncollective variables (CV), along which the important dynamics of the\nsystem is assumed to happen. CVs are arbitrary functions of the internal coordinates of the\nsystem; software such as PLUMED~\\cite{tribello_plumed_2014} and\nColvars~\\cite{fiorin_using_2013} allows their specification them \\emph{via}\nconcise symbolic expressions.\nThe choice of CVs is normally done before\nperforming the simulation~\\cite{laio_escaping_2002}. However it is common practice to choose the CVs\nfor a production run  by\n performing an analysis on a preliminary simulation. \nA careful choice of CVs\nis also needed in unbiased molecular dynamics simulations,  in order to\nobtain insight on the relevant processes that possibly occurred during the\ntime evolution. \n\n\n\nIn general,  choosing the CVs that better describe a conformational change, even if this change can be observed by visualizing the MD trajectory, can be far from trivial. \nFor these reasons, there is a need\nfor software interfaces that simplify their iterative definition and\nevaluation~\\cite{giorgino_plumed-gui:_2014,biarnes_metagui._2012}.\n\n\n\\section{Features}\n\nWe here present METAGUI 3, a plug-in providing a graphical user interface\n(GUI) to construct thermodynamic and kinetic models of processes\nsimulated by large-scale MD. The GUI is based on  the well-known MD visualization\nsoftware VMD~\\cite{humphrey_vmd:_1996}; it extends the\nfeatures of an earlier version~\\cite{biarnes_metagui._2012}, exploiting in particular: (a) its ability to toggle between\ntwo representations, structural and free energy, enabling the quick\ninspection of the configurations associated to  specific values of the CVs; (b) a procedure for grouping together similar structures in {\\em microstates}\nbased  only on the collective variables; and (c) a simplified and\nguided workflow for the analysis of metadynamics results.  METAGUI 3\nwas extensively rewritten to provide the following new features,\nwhich will be presented in detail in the next sections:\n\n\\begin{enumerate}\n\\item an interface for defining and adding interactively new collective variables;\n\\item new algorithms for finding the  microstates, namely $k$-medoids~\\cite{kaufman_book_1990} and Daura's algorithm~\\cite{daura_peptide_1999};\n\\item a reorganized graphical interface;\n\\item support for the analysis of unbiased MD trajectories;\n\\item a new clustering method~\\cite{rodriguez_clustering_2014} for identifying kinetic basins.\n\\end{enumerate}\n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.7\\textwidth]{overview}\n  \\caption{Overview of the analysis steps in METAGUI 3. The shaded\n    boxes correspond to the three main sections of the GUI. The microstates,\n    the free energies landscape, and the clusters can be\n    visualized independently after the corresponding step in the\n    analysis. }\n  \\label{fig:overview}\n\\end{figure}\n\n\n\n\n\n\\subsection{The graphical user interface}\n\nUpon starting the tool, the user is presented with the METAGUI 3\nwindow; the three main tasks are shown as tabs, namely \\emph{Define\n  inputs}, \\emph{Analyze} and \\emph{Visualize}. The three sections\ncorrespond to the three logical steps of the analysis workflow shown\nin Figure~\\ref{fig:overview}.\n\nThe \\emph{Define inputs} tab (Figure~\\ref{fig:inputs}) allows to\nspecify the input files, including the topology file of the system\n(i.e.\\ a file in PDB, PSF, or GRO format), and one or more trajectory files.\nFor each trajectory, a file containing the values of collective variables\n(\\emph{COLVAR} files) is required as well as the temperature of the\nrun; if the simulations are biased by metadynamics, the time-varying\npotentials have to be supplied in matching \\emph{HILLS}\nfiles, produced by PLUMED-patched MD engines during the simulation.\nThe part of the plug-in dedicated to the analysis of biased simulations \nis unchanged with respect to the earlier version of METAGUI~\\cite{biarnes_metagui._2012}. \nThe \\emph{Save} and \\emph{Load configuration} buttons allow storing\nand retrieving the whole set-up at once. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{inputs_captioned}\n  \\caption{The \\emph{Define inputs} tab, which allows setting the trajectory files, the system's topology (A) and  the temperatures of the available trajectories (B). The middle section (C) opens the Plumed-GUI collective-variable editor, allowing the run-time definition of unbiased CVs in addition to those originally  present in the \\emph{COLVAR} files. Defined collective variables appear in section (D), either over a white background (CVs in the simulation files) or over a green \nbackground (CVs defined at run-time).}\n  \\label{fig:inputs}\n\\end{figure}\n\n\nActivating the \\emph{Load all} button loads all the simulation data into\nmemory; this includes the molecular topology, all of the trajectory\nframes, and the values of the collective variables. A summary of the\nCVs appears on the lower-hand panel of the GUI, along with any\nadditional CV defined manually at runtime, as discussed in\nSection~\\ref{sec:run-time-definition}.\n\n\nAfter the  input is defined, operations in the \\emph{Analyze} tabs can\ntake place (Figure~\\ref{fig:analyze}). In particular, the first step is\nto group the configurations explored  during the simulations into\n\\emph{microstates}, namely sets of similar structures. Three partitioning algorithms are available, namely\n(a) $k$-medoids with $k$-means++ initial seeding~\\cite{kaufman_book_1990,vassilvitskii2006k} with or without\nsieving~\\cite{shao_clustering_2007}; (b) an implementation of the\nalgorithm by Daura et al.~\\cite{daura_peptide_1999}; and (c) the\nsimple grid partitioning in CV space implemented in the earlier versions of\nMETAGUI.\nThe $k$-medoids correspond to a partitioning of\nthe space in $k$ Voronoi cells. The parameter $k$ directly defines\nthe number of microstates that are obtained.  Daura's\nclustering algorithm~\\cite{daura_peptide_1999} automatically determines the number \nof microstates by finding groups within a cut-off distance, that should be specified by the user.  \nWith respect to the grid partitioning, included also in this version, the use of more \nelaborated partition methods has the advantage of automatically focusing on \nthe populated regions of the CV space. This avoids the exponential \ngrowth with the number of CVs of the  microstates with negligible population.\n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{analyze}\n  \\caption{Two steps of the \\emph{Analysis} procedure: (left) compute\n    microstates via one of the available partitioning procedures; \n    (right) compute the thermodynamics of each microstate via either\n    WHAM (biased trajectories) or Boltzmann\n    inversion of the populations (unbiased).}\n\\label{fig:analyze}\n\\end{figure}\n\n\nOnce microstates have been computed, the tool allows their visualization.\nThe \\emph{Visualize} tab lists the microstates along with their free\nenergy (Figure~\\ref{fig:visualize-microstates}).  Selecting one of them it is possible to visualize the corresponding structures or\nsave them into a PDB file. It is also possible to seamlessly switch between a\ncollective variables space representation of microstates\nand their atomic structure representation (Figure~\\ref{fig:fes});\nthis greatly helps associating  structural rearrangements\nwith collective variable changes.\n\n\n\\subsection{Support for unbiased simulations}\n\nA new feature in METAGUI 3.0 enables the analysis of trajectories resulting from\nboth unbiased and metadynamics-based simulations. It is possible to\nswitch between the two modes by toggling a \\emph{Biased} check-box on\neach trajectory. \nIn  both biased and unbiased simulations the structures are grouped together\ninto a set of microstates, namely sets of structures with similar values of the\ncollective variables. For biased simulations,  free energies \nof the microstates  are then\ncomputed by the\nweighted histogram analysis method (WHAM)~\\cite{roux_calculation_1995,Marinelli_2009}. For unbiased simulations,  free energies are\ncomputed applying the Boltzmann inversion to the populations of the\nmicrostates (visit counts).\nThe tool also allows analyzing simulations for which part of the trajectories are biased and part of the trajectories are not.\n\n\n\\subsection{Run-time definition of collective variables}\\label{sec:run-time-definition}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{plumed-gui-1}\n  \\caption{New collective variables can be defined via the Plumed-GUI\n    interface \\cite{giorgino_plumed-gui:_2014} and  included in the analysis at run-time. In\n    this example, the user defined two runtime CVs in PLUMED 2.0\n    syntax: the distance between the center of mass (\\texttt{COM}) of\n    the heavy atoms of residues 1 and 2; and the angle formed by the\n    COM of residues 1, 2 and 3.}\n\\label{fig:plumed-gui}\n\\end{figure}\n\n\nIn METAGUI 3, PLUMED output can be read after the\nsimulation, in the form of \\emph{COLVAR} files  as described\nabove, as well as recomputed at \\emph{run-time}. This is an important improvement with respect to the previous version of the interface:  the user can now define\ncollective variables in the PLUMED language, and have them evaluated on\nthe loaded trajectories. CVs defined at run-time can either replace or be added to the\nlist of pre-computed CVs.  Definition of CVs at run-time is done\nthrough the Plumed-GUI software (Figure~\\ref{fig:plumed-gui}), which\nprovides a guided environment to prototype CV declarations with\ntemplates and symbolic atom selection keywords~\\cite{giorgino_plumed-gui:_2014}.\nPlumed-GUI is distributed with VMD since version 1.9.2, and can be updated separately\nfrom it.\n\nThe new version of METAGUI described in this work  can process the output of the PLUMED 1.3~\\cite{bonomi_plumed:_2009} and 2.x~\\cite{tribello_plumed_2014}\nengines, making it compatible with a number of different molecular\ndynamics packages like AMBER~\\cite{case_amber_2005}, NAMD~\\cite{phillips_scalable_2005},\nGROMACS~\\cite{pronk_gromacs_2013} and several others.\nThe use of CVs computed at run-time only requires\nPLUMED's \\emph{driver}, a stand-alone\nexecutable which is built independently of MD engines.\n\n\n\n\n\n\n\\subsection{Clustering strategies}\n\nMETAGUI 3 allows quickly detecting the presence of metastable states by applying the \nDensity Peak clustering approach~\\cite{rodriguez_clustering_2014}. The approach is based \non the assumption that cluster centers are surrounded by neighbors with lower local \ndensity and that they are at a relatively large distance from any points with a higher \nlocal density. The method proceeds by computing two quantities for each\npoint: the density $\\rho_i$ and the minimum distance to a point with higher density \n$\\delta_i$. In its simplest formulation, the density is estimated by the number of \nneighbors within a cut-off, although more elaborated definitions can be employed. \nOnce computed, both quantities are plotted in the so-called decision plane. \nCluster centers stand out naturally as outliers on the graph and can be manually picked up.\nAfter the cluster centers have been found, each remaining point is assigned to the \nsame cluster as its nearest neighbor of higher density.\n\nIn the context of molecular dynamics, the quantity $\\rho$ has a direct physical interpretation\nthrough the Boltzmann equation, which relates the logarithm of the probability density with the\nfree energy. Therefore, in the version implemented in METAGUI 3, the density of a microstate\nis computed by $\\rho_i=\\exp\\left(-\\Delta G_i \/ k_BT \\right)$, where $k_B$ is the Boltzmann constant and $T$ is the temperature. This definition leads to realistic\ndensity values even in the case of biased simulations, where standard density calculation \nmethods cannot be employed. The distance between microstates is computed as the Euclidean distance \nin the CVs space. However, to deal with the different ranges of variation of the CVs, it is necessary to\ninclude some kind of normalization. This is done by dividing the coordinates by the bin size in the grid. \nTherefore, the distance can be interpreted as the number of hypercubes between microstates. \nThe clusters obtained by this procedure are a good approximation   of the metastable\nstates normally obtained by Markov State Modeling. Therefore, they can be used \nas a starting point for kinetic analysis.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{visualize_microstates}\n  \\caption{List of microstates obtained from the partitioning step, each\n    listed with the corresponding population (\\emph{Size}),\n    coordinates of the centroid (\\emph{CVs}), free energy\n    ($\\Delta G$), and cluster assignment (available after the\n    Density Peak  clustering step).  }\n\\label{fig:visualize-microstates}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{fes_captioned}\n  \\caption{\n    Free energy surface (FES) plotted with respect to two CVs\n    ($x$ and $y$ axis). Each microstate is shown as a dot, whose color\n    and $z$ coordinate reflect the corresponding $\\Delta G$. Lines\n    join kinetically-communicating microstates.}\n\\label{fig:fes}\n\\end{figure}\n\n\n\\section{A benchmark on a 125 microseconds-long MD trajectory}\n\nWith the aim of illustrating the main features of METAGUI 3, we analyzed an \nunbiased simulation of Villin-headpiece folding~\\cite{Lindorff-Larsen517}. \nThe trajectory is 125 $\\mu$s long, and includes approximately 12 folding events.\nThe collective variables employed were the content of $\\alpha$-helix, \nthe radius of gyration, the backbone dihedral distance to the crystal structure \nand the number of hydrophobic contacts. All of them were computed \nby means of PLUMED~\\cite{tribello_plumed_2014} using the Plumed-GUI\ninterface~\\cite{giorgino_plumed-gui:_2014}. \nThe content of $\\alpha$-helix was computed by means of the \\texttt{ALPHARMSD} keyword~\\cite{pietrucci_collective_2009}, \nusing $r_0=1.0 \\, \\mbox{\\AA}, d_0=0.0 \\, \\mbox{\\AA}, n=6$, and $m=12$. The radius of gyration was \ncomputed by taking into account only the C$\\alpha$ coordinates. For the backbone\ndihedral distance, the reference structure considered was PDB code 2F4K~\\cite{Kubelka2006546}; \nthe dihedrals of terminal residues  were ignored. The number of hydrophobic contacts was computed as\nthe combination of the coordination coordinates between all the possible hydrophobic\nresidues side-chain pairs. Only the heavy atoms were considered and $r_0$ was set to \n3.5 \\r{A}.\n\n\nA total of 400 microstates were found by the $k$-medoids method, reducing \nthe computational time by using the sieving option~\\cite{shao_clustering_2007} with a maximum distance \nmatrix dimension of 15,000. \nThen, the free energy of each microstate was computed \napplying the Boltzmann inversion to their populations. Finally, the free energy\nwells were localized by Density Peak clustering~\\cite{rodriguez_clustering_2014},\nallowing the classification of the microstates in 3 different clusters.\n\nFigure \\ref{fig:proj} summarizes the results of the analysis, showing the projection on \ndifferent CVs of the free energy surface, one using $\\alpha$-helix and the \nradius of gyration (panel A) and the other using the dihedral distance to the \ncrystal structure and the number of hydrophobic contacts (panel C),\nthe clusters partitions of the same projections (panels B and D respectively), \nthe decision plane that origins the basin partition (panel E) and the \nrepresentative structures for each basin (labeled as S1, S2 and S3). \nAs expected, the microstate with minimum free energy (S1) corresponds to\nthe folded state. Moreover, the folding funnel is evident in both the projections. \nS2 and S3 are configurations corresponding to intermediates of the folding process. \nRemarkably, the different \nprojections shown in figure  do not correspond to different calculations of\nthe system, but different visualizations of the same analysis. METAGUI 3  allows \ntoggling between them by changing the checked CVs in the \n\\emph{FES and Basins plot} frame (Figure~\\ref{fig:visualize-microstates}). \nMoreover, by using the \\emph{Pick} function, one can \nvisualize the structures corresponding to a  microstate by simply clicking on \na sphere in the free energy representation. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{Figure_Metagui}\n  \\caption{ METAGUI 3 output of the analysis of a 125 $\\mu$s-long molecular dynamics trajectory\n    of the villin headpiece:\n (A) free energy surface projected on the $\\alpha$-helix and the radius of gyration CVs; \n (B) assignment of the microstates to clusters in the projection of panel A;\n (C) free energy surface projected on the dihedral distance to the\ncrystal structure and the number of maintained hydrophobic contacts;\n (D) assignment of the microstates to clusters in the projection of panel C;\n (E) decision graph that leads to the cluster  partition shown in panes B and D.\n S1, S2 and S3 are the cartoon representation of the representative structures \nin each cluster.\n    }\n\\label{fig:proj}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nThe introduction of graphical tools in the MD analysis workflow can\nspeed-up iterations in the search for appropriate collective variables,\na necessary step towards the data-driven rationalization\nof MD results.  METAGUI~3 aims at providing a quick way to\ncross-reference structural and thermodynamic information, by enabling\nboth the selection of a set of relevant CVs on existing simulations,\nand providing a quick way to define new ones. \n\nThe source code of\nMETAGUI 3 is  available at the URL\n\\url{https:\/\/github.com\/metagui\/metagui3}; it can be freely\nredistributed and\/or modified under the terms of the GNU General\nPublic License (Free Software Foundation) version 3 or later.\n\n\n\n\n\\section{Acknowledgements}\n\nThe code of METAGUI 3.0 includes parts from the earlier METAGUI~2.0\nversion, written by Xevi Biarn\u00e9s, Fabio Pietrucci, Fabrizio Marinelli\nand Alessandro Laio.  Alessandro Laio and Alex Rodriguez \nacknowledge financial support from the grant Associazione Italiana per \nla Ricerca sul Cancro 5 per mille, Rif.\\ 12214.\n\n\n\n\\bibliographystyle{model1-num-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThroughout this paper, we study the \\emph{$(n,k)$-Bernoulli-Laplace model}.  In the model, there are two urns, a \\emph{left} urn and a \\emph{right} urn, each of which contains exactly $n$ balls.  Of the total $2n$ balls contained in both urns, $n$ are colored red and $n$ are colored white.  Starting from a given coloration of $n$ balls in each urn, at each step $k$ balls are selected uniformly at random without replacement from each urn.  The selected balls are then swapped and placed in the opposite urn.  The process then repeats itself.  Letting $X_t$ denote the number of red balls in the left urn after $t$ swaps, the process $(X_t)$ is Markov.  Our main goal is to understand how long it takes for the chain to be within $\\epsilon>0$ of its stationary distribution $\\pi$ in total variation.  Our focus in this paper is on the case when the number of swaps $k$ is of order $n$ where $n\\gg1$.  The main result of this paper partially resolves an open question posed by Eskenazis and Nestoridi in~\\cite{EN}.    \n\n\nOur interest in the $(n,k)$-Bernoulli-Laplace model comes from shuffling large decks of cards.  Mapping the above model to this setting, the deck of cards has size $2n\\gg1$ and at each step of the shuffle we cut the deck into two equal piles of $n$ cards, shuffle each deck independently and perfectly, reassemble the deck and then move the top $k$ cards to the bottom.  This process repeats itself until sufficient mixing is achieved.  From this description, it follows that the $(n,k)$-Bernoulli-Laplace model describes this card shuffling algorithm \\emph{without} the separate step of shuffling each of the smaller decks independently and perfectly at each step.  See~\\cite{NW_19} for further details.  \n\n\\subsection{Preliminaries}  Before discussing existing results in the literature and the results of this paper, we first fix some notation and terminology.  \n\nThroughout, $\\mathcal X= \\{0,1,2,\\ldots, n\\}$ denotes the state space of the $(n,k)$-Bernoulli-Laplace chain $(X_t)$, and $p_t(x,y)$ denotes the associated probability of transitioning from $x\\in\\mathcal X$ to $y\\in \\mathcal X$ in $t$ steps; that is,\n\\begin{align}\n\\label{eqn:trans}\np_t(x,y):= \\mathbf{P}_x \\{ X_t =y \\}\n\\end{align}  \nwhere the subscript in the probability $\\mathbf{P}$ indicates that $X_0=x$.  One can write down the specific formulas for the transitions $p_t(x,y)$ (see~\\cite{EN}) but these formulas are not particularly important in our analysis. Similar to $\\mathbf{P}_{X_0}\\{ \\cdots\\}$, we use an analogous notation for the expectation $\\mathbf{E}$; that is, $\\mathbf{E}_{X_0}\\{ \\cdots \\}$ indicates that $(X_t)$ has initial distribution $X_0$.  For any $A\\subset \\mathcal X$ and $x\\in \\mathcal X$, we let \n\\begin{align}\nP_t(x, A) := \\sum_{y\\in A} p_t(x,y) = \\mathbf{P}_x\\{ X_t \\in A\\}\n\\end{align}\ndenote the probability of transitioning from state $x$ to the set $A$ in $t$ steps.  It is known (see~\\cite{Tai_96}) that $(X_t)$ has a unique stationary distribution $\\pi$ which is hypergeometric; specifically, $\\pi$ satisfies\n\\begin{align}\n\\label{eqn:pidist}\n\\pi(\\{x\\})= \\frac{\\binom{n}{x} \\binom{n}{n-x}}{\\binom{2n}{n}}, \\qquad x\\in \\mathcal X. \n\\end{align}  \nObserve that each of the quantities above depends on the parameter $n\\in \\mathbf{N}$.  Throughout, unless otherwise specified (see, for example, the paragraph below), we will suppress this dependence.    \n\nLet \n\\begin{align}\nd^{(n)}(t):= \\max_{x\\in \\mathcal X} \\| P_t(x, \\, \\cdot \\,) - \\pi (\\, \\cdot \\,) \\|_{TV} = \\tfrac{1}{2}\\max_{x\\in \\mathcal X} \\sum_{y\\in \\mathcal X} | p_t(x,y) - \\pi(y)|,  \n\\end{align} \nand define for $\\epsilon >0$ the \\emph{mixing time} $t_{\\text{mix}}^{(n)} (\\epsilon)$ by \n\\begin{align}\nt_{\\text{mix}}^{(n)}(\\epsilon) = \\min \\{ t\\in \\mathbf{N} \\, : \\, d^{(n)}(t) \\leq \\epsilon \\}. \n\\end{align}\nWe say that the Markov chain $(X_t)$ exhibits \\emph{cutoff} if  \n\\begin{align}\n\\label{eqn:cutoff}\n\\lim_{n\\rightarrow \\infty} \\frac{t_{\\text{mix}}^{(n)}(\\epsilon)}{t_{\\text{mix}}^{(n)}(1-\\epsilon)} =1 \\,\\,\\, \\,\\,\\text{ for all }\\, \\,\\,\\, \\, \\epsilon \\in (0, 1).  \n\\end{align}\nIf the Markov chain $(X_t)$ exhibits cutoff and for every $\\epsilon \\in (0,1)$ there exists a constant $c_\\epsilon$ and a sequence $w_n$ satisfying   \n\\begin{align}\nw_n=o(t_{\\text{mix}}^{(n)}(1\/2)) \\qquad \\text{ and } \\qquad t_{\\text{mix}}^{(n)}(\\epsilon)- t_{\\text{mix}}^{(n)}(1-\\epsilon) \\leq c_\\epsilon w_n   \\quad \\text{ for all }\\quad  n, \n\\end{align}  \nwe say that $(X_t)$ has \\emph{cutoff window} $w_n$.  For other preliminaries concerning mixing times of Markov chains, see~\\cite{LPW_17}.     \n\n\n\\subsection{Previous results and statement of the main result}\nExisting results on mixing  times for the $(n,k)$-Bernoulli-Laplace model largely focus on the case when the number of swaps $k$ is \\emph{much smaller} than the number of balls $n$ in each urn.  The earliest works of Diaconis and Shahshahani~\\cite{DS_81, DS_87}  and Donnely, Floyd and Sudbury~\\cite{DLS_94} treat the case when $k=1$ and establish cutoff in total variation and in the \\emph{separation distance}, respectively.  Diaconis and Shahshahani proved their results by analyzing random walks on Cayley graphs of the symmetric group where edges correspond to transpostions. These results were extended to random walks on distance regular graphs in~\\cite{B_98}.  See also~\\cite{scarabotti1997time} for the Bernoulli-Laplace model with multiple urns in the case when $k=1$.  We refer to~\\cite{school_02} which contains a signed generalization of the model.  \n\nThe case when the number of swaps $k>1$ in the Bernoulli-Laplace two-urn model was first studied by Nestoridi and White~\\cite{NW_19}, where a number of estimates (not all sharp) were deduced for $t_{\\text{mix}}(\\epsilon)$ for general $k$.  These estimates were made sharp in the case when the number of swaps $k$ satisfies $k=o(n)$ in a joint paper of Eskenazis and Nestoridi~\\cite{EN}, ultimately yielding the bound \n\\begin{align}\n\\label{eqn:upperbksmall}\n\\frac{n}{4k}\\log n - \\frac{c(\\epsilon) n}{k}\\leq t_{\\text{mix}}(\\epsilon) \\leq \\frac{n}{4k} \\log n + \\frac{3n}{k}\\log \\log n + O\\Big( \\frac{n}{\\epsilon^4 k}\\Big).   \n\\end{align}\nThus, the model with $k=o(n)$ exhibits cutoff with cutoff window $\\tfrac{n}{k} \\log \\log n$.  Estimates deduced in the case when $k=O(n)$ were not optimal; that is, in~\\cite{NW_19} it was shown that if $k\/n \\rightarrow \\lambda \\in (0,1\/2)$, then \n \\begin{align}\n \\label{eqn:lowb1}\n \\frac{\\log n }{2 |\\log (1-2\\lambda)|} -c(\\epsilon) \\leq t_{\\text{mix}}(\\epsilon) \\leq \\frac{\\log(n\/\\epsilon)}{2\\lambda(1-\\lambda)}.  \n \\end{align} \nMoreover, it was conjectured in~\\cite{EN} that the lower bound in~\\eqref{eqn:lowb1} is sharp.  In general, it was left there as an open problem to determine the mixing time of the $(n,k)$-Bernoulli-Laplace model when $k\/n\\rightarrow \\lambda \\in (0,1\/2)$.   In this paper, we make progress on solving this problem under \\emph{reasonable} assumptions on the convergence rate $k\/n\\rightarrow \\lambda \\in (0,1\/2)$, which we now describe.\n\n\n    \n\\begin{Assumption}\n\\label{assump:1}\nThe parameter $k=k(n)$ in the $(n,k)$-Bernoulli-Laplace model satisfies the following conditions:\n\\begin{itemize}\n\\item[\\textbf{(c0)}] $k\/n\\rightarrow \\lambda \\in (0, 1\/2)$ as $n\\rightarrow \\infty.$  \n\\item[\\textbf{(c1)}] There exists $\\delta \\in (0,1\/2)$ for which $k\/n\\in (0, \\delta)$ for all $n$.\n\\item[\\textbf{(c2)}] $\\Delta_n :=\\tfrac{k}{n}- \\lambda$ satisfies the asymptotic condition\n\\begin{align}\n\\Delta_n = o(1\/\\log n) \\,\\,\\,\\, \\text{ as } \\,\\,\\,\\, n\\rightarrow \\infty. \n\\end{align}\n\\end{itemize}\n\n\\begin{Remark}\n\\label{rem:1}\nAlthough \\cref{assump:1} is not explicitly employed in the paper~\\cite{NW_19}, to the best of our knowledge it seems that one needs to impose some condition like the one above to make the asymptotic formulas previously used in~\\cite{NW_19} to deduce a lower bound on $t_{\\text{mix}}^{(n)}(\\epsilon)$.  For further information on this point, we refer the reader to~\\cref{rem:2}.   \n\\end{Remark}\n\n\\end{Assumption}\n\n\n\nThroughout, using the notation in \\cref{assump:1}, we define \n\\begin{align}\n\\label{def:times}\ns_n = \\lambda^{-1} \\log \\log n \\qquad \\text{ and } \\qquad t_n = \\frac{\\log n}{2| \\log (1-2\\lambda)|}.\n\\end{align}\nOur main result is the following:\n\\begin{Theorem}\n\\label{thm:main}\nSuppose that in the $(n,k)$-Bernoulli-Laplace chain, the parameter $k$ satisfies \\cref{assump:1}.  \n For any $\\epsilon \\in (0,1)$, there exists a constants $c(\\epsilon), N(\\epsilon)>0$ depending only on $\\epsilon$ such that \n\\begin{align}\n\\label{eqn:upperb}\nt_n - c(\\epsilon) \\leq t_{\\text{mix}}^{(n)}(\\epsilon) \\leq t_n + 3s_n + 1 \n\\end{align}\nfor all $n\\geq N(\\epsilon)$. \nIn particular, the $(n,k)$-Bernoulli-Laplace chain with parameter $k$ satisfying \\cref{assump:1} has mixing time $t_n$ with cutoff window $s_n.$\n\\end{Theorem}\n\n\n\\section{Proof outline }\n\\label{sec:outline}\n\n\n\\subsection{The lower bound}\nThe proof of the lower bound with $k$ satisfying \\cref{assump:1} was first estabished in \\cite{NW_19}.  \nWe present the lower bound argument and fill in some details in \\cref{sec:lowerb} for completeness. \nThe lower bound argument follows the standard idea of finding a set $A\\subseteq \\mathcal X$ where $|P_t (0,A)-\\pi_n(A)|$ is large. In particular, the argument uses the estimate \n\\[\\TV{P_t(0, \\, \\cdot\\,)-\\pi_n(\\, \\cdot \\,)} \\geq \\sup_{A\\subseteq \\mathcal X} |P_t(0,A)-\\pi_n(A)|.\\]\nIt is worth noting that the lower bound argument uses Chebyshev's inequality to bound the $P_t(0,A)$ from above and to bound $\\pi_n(A)$ from below, for a clever choice of $A$. \nThis argument is instructive and provides intuitive support for the lower bound being sharp since the application of Chebychev's inequality fails to bound $P_t(0,A)$ and $\\pi_n(A)$ away from each other for $t\\geq t_n$.\nOur main contribution in this paper is the upper bound~\\eqref{eqn:upperb}. \n\n\\subsection{The upper bound~\\eqref{eqn:upperb}}\n\nWhile the broad structure of the argument giving the upper bound~\\eqref{eqn:upperb} is similar to that used to obtain the upper bound in~\\eqref{eqn:upperbksmall} in the case when $k=o(n)$~\\cite{EN}, the details giving these results are quite different.  This is especially true because the asymptotics are more delicate when $k$ is order $n$ as in~\\cref{assump:1}.  Furthermore, we cannot rely on a known result of Diaconis and Freedman~\\cite{DF_1980} in the last step of the proof.  This was used crucially in~\\cite{EN} to show that once two copies of the chain are sufficiently close, in this case at distance $o(k)$, then in one step the processes are distance $o(1)$ apart with high probability.  This step relies inherently the binomial approximation of the hypergeometric distribution (see, for example,~\\cite{R_01}), which is not valid in our regime of interest since $k$ is order $n$.  The last step, i.e. going from distance $o(k)$ to $o(1)$, is arguably the most involved because we manufacture a \\emph{discrete normal} approximation (in total variation) of the hypergeometric.  This same approximation was also used in~\\cite{LCM_07} to establish a Berry-Esseen theorem for the hypergeometric distribution in nonstandard parameter cases.  For more on Barry-Esseen theorems, we refer the reader to Feller's book~\\cite{F_08}.  All of these steps are carried out in detail in \\cref{sec:osmallrootn} and \\cref{sec:upperbound}.        \n\n\n\n\n\\subsubsection{The coupling}        \nThe proof of the upper bound~\\eqref{eqn:upperb} employs coupling methods.  Specifically, recall that if $\\mu$ and $\\nu$ are probability measures defined on all subsets of $\\mathcal X$, then we can write\n\\begin{align}\n\\| \\mu - \\nu \\|_{TV} = \\inf \\{ \\mathbf{P}\\{ X\\neq Y \\} \\, : \\, (X,Y) \\text{ is a coupling of } \\mu \\text{ and } \\nu \\}.  \n\\end{align}\nHere we recall that $(X,Y)$ is a \\emph{coupling} of the measures $\\mu$ and $\\nu$ if $X$ and $Y$ are random variables on a common probability space $(\\Omega, \\mathcal{F}, \\mathbf{P})$ with $X\\sim \\mu$ and $Y\\sim \\nu$.  Thus in order to achieve the claimed upper bound~\\eqref{eqn:upperb}, we will utilize a particular coupling of two copies of the chain to bound the total variation distance from above. \n\nRecall that in the $(n,k)$-Bernoulli-Laplace model, $X_t$ denotes the number of red balls in the left urn at time $t$.  Thus if $(X_t)$ and $(Y_t)$ are two copies of the $(n,k)$-Bernoulli-Laplace chain, then $X_t$ and $Y_t$ are the number of red balls in two separate left urns at time $t$.  The coupling we use can be described as follows (see also~\\cite{EN, NW_19}):   \n\n\\begin{Coupling}\\label{coupling}\nLet $(X_t)$ and $(Y_t)$ be two copies of the $(n,k)$-Bernoulli-Laplace chain and fix a time $t\\geq 0$.  Given the distributions $X_t$ and $Y_t$ at time $t$, we define a coupling of $X_{t+1}$ and $Y_{t+1}$.  First, label the balls in the two left urns at time $t$ separately from $1$ to $n$ so that each red ball has a smaller label than each white ball. \nFurthermore, in a similar way label all of the balls in the two right urns from $n+1$ to $2n$ so that each red ball has a smaller label than each white ball. \nUniformly and independently select subsets $A\\subseteq \\{1,\\dots,n\\}$ and $B\\subseteq \\{n+1,\\dots, 2n\\}$ with $|A|=|B|=k$.  To obtain $X_{t+1}$ and $Y_{t+1}$,  \nswap the balls indexed by the elements of $A$ in each left urn with the balls in the corresponding right urn with index belonging to $B$.  \\end{Coupling}\n\n \n\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[thick, scale = 1]\n    \n     \n    \\foreach \\x in {0, 3, 7, 10}{\n        \\foreach \\y in {0, 2, 4, 6}{\n        \\draw (\\x -1, \\y) arc [ start angle = 180, end angle = 360, x radius =1, y radius = .25];\n        }\n        \\foreach \\y in {2,6}{\n        \\draw (\\x + 1, \\y) arc [ start angle = 0, end angle = 180, x radius =1, y radius = .25];}\n        \\foreach \\y in {0,4}{\n        \\draw[dashed] (\\x + 1, \\y) arc [ start angle = 0, end angle = 180, x radius =1, y radius = .25];\n        \\draw (\\x - 1, \\y ) -- (\\x - 1, \\y + 2);\n        \\draw (\\x + 1, \\y ) -- (\\x + 1, \\y + 2);\n        \\foreach \\z in {.2,.8, 1.4}{\n            \\draw[fill=white] (\\x -.4, \\y + \\z) circle (.25);}\n        \\foreach \\z in {.5,1.2}{\n            \\draw[fill=white] (\\x +.4, \\y + \\z) circle (.25);}\n            }\n            }\n        \n   \n    \\foreach \\z in {.2, .8, 1.4}{\n        \\draw[fill = ourRed] (-.4, 4 + \\z) circle (.25);}\n    \\foreach \\z in {1.4, .8}{\n        \\draw[fill = ourRed] (2.6, 4 + \\z) circle (.25);}\n    \\foreach \\x in {0,7}{\n        \\draw[fill = ourRed] (\\x - .4, 1.4) circle (.25);}\n    \\draw[fill = ourRed] (6.6, 5.4) circle (.25);\n    \\foreach \\x in {3,10}{\n        \\foreach \\z in {.2, .8, 1.4}{\n            \\draw[fill = ourRed] (\\x - .4, \\z) circle (.25);}\n        \\draw[fill = ourRed] (\\x + .4, 1.2) circle (.25);}\n    \\foreach \\z in {.2, .8, 1.4}{\n        \\draw[fill = ourRed] (9.6, 4 + \\z) circle (.25);}\n    \\draw[fill = ourRed] (10.4, 5.2) circle (.25);\n\n    \n   \n   \n    \\foreach \\y in {0,4}{\n        \\foreach \\z\/\\i in {1.4\/1, .8\/2, .2\/3}{\n            \\draw ( - .4, \\y + \\z) node {\\i};}\n        \\foreach \\z\/\\i in {1.2\/4, .5\/5}{\n            \\draw (.4, \\y + \\z) node {\\i};}\n            }\n    \\foreach \\y in {0,4}{\n            \\foreach \\z\/\\i in {1.4\/6, .8\/7, .2\/8}{\n                \\draw ( 2.6, \\y + \\z) node {\\i};}\n            \\foreach \\z\/\\i in {1.2\/9, .5\/10}{\n                \\draw (3.4, \\y + \\z) node {\\i};}\n                }\n            \n   \n    \\draw node [below] at (1.5, 3.5) {$X_t$};\n    \\draw node [below] at (1.5, -.5) {$Y_t$};\n    \\draw node [below] at (8.5, 3.5) {$X_{t+1}$};\n    \\draw node [below] at (8.5, -.5) {$Y_{t+1}$};\n    \n   \n    \\foreach \\y in {0,4}{\n        \\foreach \\x in {0, 7}{\n            \\draw (\\x, \\y - .5) node(\\x\\y){$L$};\n        }\n        \\foreach \\x in {3, 10}{\n            \\draw (\\x, \\y - .5) node(\\x\\y){$R$};\n        }}\n    \n    \\end{tikzpicture}\n    \\caption{An illustration of \\cref{coupling} with $n=5$, $k=2$ and $A=\\{1,3\\}$ and $B=\\{9,10\\}$.}\n    \\label{fig:1}\n\\end{figure}\n\n\n    \n    \n\n\n\n\n\n Refer to \\cref{fig:1} for an illustration of this coupling.  Importantly, two chains coupled as in \\cref{coupling} have the property that $|X_{t+1}-Y_{t+1}|\\leq |X_t - Y_t|$~\\cite{NW_19}. \nFurthermore, we have the next proposition (see~\\cite[Proposition 2]{EN}) whose argument is a  modification of the \\emph{path coupling theorem} of Bubley and Dyer~\\cite{BD_97}.   \n\n\\begin{Proposition}\\label{prop:couple}\nLet $(X_t)$ and $(Y_t)$ be two instances of the $(n,k)$-Bernoulli-Laplace chain coupled at each time $t\\geq 1$ as in \\cref{coupling} starting from $X_0, Y_0 \\in \\mathcal X$. For $r>0$, let \n\\begin{align}\n\\label{eqn:tcouple}\n\\tau_{\\emph{couple}}(r) := \\min\\{t\\,:\\, |X_t-Y_t|\\leq r\\}.\n\\end{align}\nThen, for every $t\\in \\mathbb{N}$, \n\\[\\mathbf{P}(\\tau_{\\emph{couple}}(r)>t)\\leq \\Big( 1-\\frac{2k(n-k)}{n^2}\\Big)^t\\frac{|X_0-Y_0|}{r}.\\]\n\n\\end{Proposition}\n\n\n\\subsubsection{The steps in the upper bound argument}\nIn \\cref{sec:osmallrootn} and~\\cref{sec:upperbound}, we provide the details for the upper bound~\\eqref{eqn:upperb}.  Here we lay out the argument giving the upper bound in several steps.  \n\n\n\\textbf{Step 1}.  Recalling $t_n$ as in~\\eqref{def:times} and fixing $\\kappa_1\\gg 1$, we will  first prove that with high probability, there exists $t\\leq t_n$ such that \n\\begin{align*}\nX_{t}, Y_{t} \\in (\\tfrac{n}{2}- \\kappa_1 \\sqrt{n}, \\tfrac{n}{2}+ \\kappa_1 \\sqrt{n}). \n\\end{align*} \nAs a consequence, by $t_n$ steps, with high probability (depending on the choice of $\\kappa_1$) the chains have been at distance order $\\sqrt{n}$ apart.   This step will be achieved using the first and second eigenfunctions for the chain, which translates to precise first and second moment estimates for $(X_t)$.      \n\n\\textbf{Step 2}.  Recalling $s_n$ as in~\\eqref{def:times} and fixing $\\kappa_2 , \\kappa_3\\gg 1$, we will show that if $X_0, Y_0 \\in (\\tfrac{n}{2}-\\kappa_2\\sqrt{n},\\tfrac{n}{2}+ \\kappa_2 \\sqrt{n})$, then with high probability (depending on $\\kappa_2, \\kappa_3$) if $X_t, Y_t$ are coupled according to \\cref{coupling} there exists $t \\leq s_n$ so that \n\\begin{align*}\n|X_t - Y_t| \\leq \\frac{\\sqrt{n}}{\\log \\log n} \\qquad \\text{ and } \\qquad X_t, Y_t \\in (\\tfrac{n}{2}- \\kappa_3 \\sqrt{n} (\\log n)^{|P_\\lambda|\/2}, \\tfrac{n}{2}+ \\kappa_3 \\sqrt{n} (\\log n)^{|P_\\lambda|\/2})\\end{align*}\nwhere $P_\\lambda =\\log(1-2\\lambda)  \\tfrac{2}{\\lambda}$.  Thus the chains become closer asymptotically but they may deviate slightly from $\\tfrac{n}{2}$ by more than order $\\sqrt{n}$.  Note that this deviation becomes particularly pronounced when $\\lambda \\approx 1\/2$ by the definition of $P_\\lambda$.     \n\n\\textbf{Step 3}.  In this step, we correct for the deviation in \\textbf{Step 2} from $\\tfrac{n}{2}$ in $2 s_n$ steps again making use of~\\cref{coupling}.  That is, fix $\\kappa_4 \\gg 1$ and suppose $X_0, Y_0$ satisfy\n\\begin{align*}\n|X_0 - Y_0| \\leq \\frac{\\sqrt{n}}{\\log \\log n} \\qquad \\text{ and } \\qquad X_t, Y_t \\in (\\tfrac{n}{2}- \\kappa_3 \\sqrt{n} (\\log n)^{|P_\\lambda|\/2}, \\tfrac{n}{2}+ \\kappa_3 \\sqrt{n} (\\log n)^{|P_\\lambda|\/2}).\\end{align*}\nSupposing that the chains $(X_t)$ and $(Y_t)$ are coupled according to~\\cref{coupling}, we will show that with high probability (depending on $\\kappa_4 \\gg 1$) there exists $t\\leq 2 s_n$ so that \n\\begin{align}\n\\label{eqn:step3}\n|X_t - Y_t| \\leq \\frac{\\sqrt{n}}{\\log \\log n} \\qquad \\text{ and } \\qquad X_t, Y_t \\in (\\tfrac{n}{2}- \\kappa_4 \\sqrt{n} , \\tfrac{n}{2}+ \\kappa_4 \\sqrt{n} ). \\end{align}\n\n\\textbf{Step 4}.  At this point, by using the strong Markov property and combining \\textbf{Step 1}, \\textbf{Step 2} and \\textbf{Step 3}, with high probability there exists $t\\leq t_n + 3 s_n$ such that~\\eqref{eqn:step3} is satisfied regardless of $X_0$, $Y_0$. From this point, we will see that in one additional step the processes are within distance $o(1)$ of each other with high probability.   This step is more involved than the other steps and constitutes its own section (cf. \\cref{sec:upperbound}).  The main reason this step is more involved is that we have to understand the hypergeometric distributions that make up the one-step behavior of the chain within this  particular parameter range on the initial data.  Here, because $k$ is order $n$ instead  of $k=o(n)$, we cannot appeal to an existing result to do so.  Instead, we make use of some asymptotic estimates for the hypergeometric distributions  in~\\cite{LCM_07} to compare the hypergeometrics with what we call \\emph{discrete normal} distributions.  \n\n\n\n\\section{Lower bound}\n\\label{sec:lowerb}\n\n\nHere we establish the lower bound on the mixing time~\\eqref{eqn:upperb}.  First, however, we state some auxiliary results to be used throughout the paper.  For the reader, it may be more valuable to skip the proof of these results on a first pass and proceed directly to the proof of the lower bound.  The proof of the auxiliary results will be given later in~\\cref{sec:aux}. \n\n\n\\subsection{Auxiliary lemmata}\nFor $x\\in \\mathcal X$, define $f_i(x)$, $i=1,2,3$, by \n\\begin{align}\n\\label{eqn:fs}\n    f_1(x)= 1-\\frac{2x}{n}, \\qquad f_2(x) = 1-\\frac{2(2n-1)x}{n^2}+\\frac{2(2n-1)x(x-1)}{n^2(n-1)}, \\,\\,\\, \\text{ and } \\,\\,\\, f_3(x)= 1- \\frac{2x(n-x)}{n^2}.\n\\end{align}\nWe remark that while $f_1$ and $f_2$ are eigenfunctions for the Bernoulli-Laplace model with respective eigenvalues $f_1(k)$ and $f_2(k)$~\\cite{EN}, $f_3(k)$ is contained in the righthand side of the bound in \\cref{prop:couple}.  For further information on the dual Hahn eigenfunctions for the Bernoulli-Laplace chain, we refer the reader to~\\cite{NW_19}.  See also~\\cite{KZ_09} for further reading on polynomial eigenfunctions of Markov chains in general.     \n\nOne can check that $|f_1| \\leq 1$ on $\\mathcal X$ and, after a short exercise optimizing the quadratic $f_2(x)$ on $\\mathcal X$, for $n\\geq 2$ we have  \n\\begin{align}\n\\label{eqn:f2bound}\n1- \\frac{2n-1}{2n-2} \\leq f_2(x) \\leq 1  \\,\\,\\,\\,\\, \\text{ for all } \\,\\,\\, x\\in \\mathcal X.  \n\\end{align} \nWe will also need the following basic structural facts about the functions $f_1$ and $f_2$.  \n\\begin{Lemma}\n\\label{lem:1}\nFor all $t\\geq 0$ and all $X_0\\in \\mathcal X$, we have the identities\n\\begin{align}\n\\label{id:1}\n\\mathbf{E}_{X_0} f_1(X_t) &=\\Big(1-\\frac{2k}{n}\\Big)^t f_1(X_0)\\\\\n\\label{id:2} \\mathbf{E}_{X_0} f_1(X_t)^2& = \\frac{1}{2n-1}+ \\frac{2n-2}{2n-1} f_2(k)^t f_2(X_0).  \n\\end{align}\nFurthermore, let $q\\geq0$ be a constant and suppose $X_0= \\tfrac{n}{2}+ O( \\sqrt{n} (\\log n)^{q\/2} )$ as $n\\rightarrow \\infty$.  Then we have the asymptotic formula \n\\begin{align}\n\\label{eqn:f2as}\nf_2(X_0) = \nO(n^{-1} (\\log n)^q).  \n\\end{align}\n\\end{Lemma}\n\n\n\n\\begin{Lemma}\n\\label{lem:2}\nSuppose that \\cref{assump:1} is satisfied.  Let $t=t(n)\\in \\mathbf{N}$ satisfy $t\\leq t_n$ for all $n$ where $t_n$ is as in~\\eqref{def:times}.  Let $h_1, h_2$ be given by $f_2(k)=1-2h_1(k)$ and $(1-2\\lambda)^2= 1- 2h_2(\\lambda)$, and set $\\Delta_n':= h_1(k)-h_2(\\lambda)$ and $\\Delta_n''= k(n-k)\/n^2 - (\\lambda- \\lambda^2)$.  Then as $n\\rightarrow \\infty$\n\\begin{align}\n\\label{as:id1} f_1(k)^{t} &=(1-2\\lambda)^{t}\\bigg(1 -\\frac{ 2t \\Delta_n}{1-2\\lambda} + O(t^2 \\Delta_n^2) \\bigg),\\\\\n\\label{as:id2}f_2(k)^t &= (1-2\\lambda)^{2t}\\bigg(1 - \\frac{2t \\Delta_n'}{1- 2\\lambda} +O(t^2( \\Delta_n')^2) \\bigg),\\\\\n\\label{as:id3} f_3(k)^t&=  (1-2\\lambda + 2\\lambda^2)^t\\bigg(1- \\frac{2t \\Delta_n''}{1- 2\\lambda} +O(t^2( \\Delta_n'')^2)   \\bigg).\n\\end{align}\nFurthermore, \n\\begin{align}\n\\label{eqn:dtil}\n|\\Delta_n'| \\leq 2 |\\Delta_n| + O(1\/n)   \\qquad \\text{ and } \\qquad |\\Delta_n''| \\leq |\\Delta_n| \n\\end{align}\nfor all $n$. \n\\end{Lemma}\n\n\\begin{Remark}\n\\label{rem:2}\nNote that a similar formula to~\\eqref{as:id1} is employed on~\\cite[p.444]{NW_19} but without an assumption similar to~\\cref{assump:1}.  Such a formula is not valid unless $k\/n\\rightarrow \\lambda \\in (0,1\/2)$ sufficiently fast.     \n\\end{Remark}\n\n\\subsection{Proof of the lower bound}\n\nIf we let $f(x)=\\sqrt{n-1}f_1(x)=\\sqrt{n-1}(1-2x\/n)$, then $f$ is tightly concentrated around its mean with respect to the stationary distribution $\\pi_n$. \nThis is true because $\\pi_n$ has mean $n\/2$ and the bulk of its mass is concentrated around its mean with exponentially decaying tails.  On the other hand, $\\mathbf{E}_0 f(X_t) \\approx (1-2\\lambda)^{-c}$ for $t=t_n -c$ and large fixed $c$. \nFurthermore,  $P_t(0,\\cdot)$ must concentrate values of $f(x)$ tightly around its mean, due to a constant bound on its variance. \nThus, it is intuitive that $P_t(0,\\cdot)$ must assign a significant portion of its mass away from $n\/2$. \nIn short, the bulk of the masses from $\\pi_n$ and $P_t(0,\\cdot)$ do not overlap, providing a lower bound on the total variation distance between these measures.  Note that this proof is presented in~\\cite{NW_19}.  Below, we give the proof in full detail as it gives credence to the claim that the lower bound in~\\eqref{eqn:lowb1} is sharp.    \n\n\\begin{Theorem}\nSuppose that \\cref{assump:1} is satisfied.\nFor all positive $\\epsilon>0$, \n\\[\\TV{P_t(0,\\cdot)-\\pi_n} \\geq 1-\\epsilon\\] \nprovided that $n$ is large enough and \\[t\\leq \\frac{\\log n}{2|\\log(1-2\\lambda)|}-\\frac{\\log(\\sqrt{3} +100)- \\tfrac{1}{2}\\log \\epsilon }{ |\\log (1-2\\lambda)|}.\\]\n\\end{Theorem}\n\n\\begin{proof}\nLet $\\pi$ be the stationary distribution of $(X_t)$ with $n$ suppressed.  \nLet $f(x) = \n\\sqrt{n-1}(1-\\frac{2x}{n})$.  Since $\\pi$ has hypergeometric distribution as in~\\eqref{eqn:pidist}:\n\\[\\mathbf{E}_{\\pi} f(X_t) = \\sqrt{n-1}\\, \\mathbf{E}_{\\pi} f_1(X_t)=0.\\]\nAlso note that \n\\begin{align*}\n    \\text{Var}_{\\pi} f(X_t) &= \\mathbf{E}_{\\pi}f(X_t)^2-(\\mathbf{E}_{\\pi} f(X_t))^2\\\\\n    &= (n-1) \\mathbf{E}_{\\pi} f_1(X_t)^2\\\\\n    &= \\frac{n-1}{2n-1} +(n-1)\\frac{2n-2}{2n-1} f_2(k)^t\\mathbf{E}_{\\pi}f_2(X_t)\n\\end{align*}\nwhere the last equality follows from \\cref{lem:1}.\nSince $f_2(x)$ is an (right) eigenfunction of the chain with eigenvalue $f_2(k)\\neq 1$, it is orthogonal to $\\pi$.\nIn particular, $\\mathbf{E}_{\\pi} f_2(X_t)=0$. \nThus, $\\text{Var}_{\\pi} f(X_t)\\leq 1\/2$. \n\nWe repeat these calculations for the chain started from $0$ for $t$ satisfying \n\\begin{align}\n\\label{eqn:tdef2}\nt= \\frac{\\log n}{2|\\log(1-2\\lambda)|}-c=t_n-c \n\\end{align}\nfor a constant $c$ which we will determine later.  Observe by \\cref{lem:1} and \\cref{lem:2} \n\\begin{align}\\mathbf{E}_{0} f(X_t) = \\sqrt{n-1}\\mathbf{E}_0 f_1(X_t) &=\\sqrt{n-1} (1-\\tfrac{2k}{n})^t \\nonumber\\\\ \\label{eqn:310}\n&=\\sqrt{n-1} (1-2\\lambda)^t(1+o(1))\\\\\n&= \\frac{\\sqrt{n-1}}{\\sqrt{n}} (1-2\\lambda)^{-c}(1+o(1))\\nonumber\\\\ \\label{eqn:311}\n&= (1+o(1))(1-2\\lambda)^{-c}.\n\\end{align}\nSimilarly, combining \\eqref{eqn:310} with \\cref{lem:1} and \\cref{lem:2} we have \n\\begin{align*}\n    \\text{Var}_{0} f(X_t)&= \\mathbf{E}_0f(X_t)^2- (\\mathbf{E}_0f(X_t))^2\\\\\n    &= (n-1) \\left[\\frac{1}{2n-1}+ \\frac{2n-2}{2n-1} f_2(k)^t f_2(0)\\right] - (n-1)(1-2\\lambda)^{2t}(1+o(1))\\\\\n  &= (n-1) \\left[\\frac{1}{2n-1}+\\frac{2n-2}{2n-1}(1-2\\lambda)^{2t}( 1+ o(1))\\right] - (n-1)(1-2\\lambda)^{2t}(1+o(1))  \\\\\n  &= \\frac{n-1}{2n-1} + (n-1) (1-2\\lambda)^{2t}(1+o(1))-(n-1)(1-2\\lambda)^{2t}(1+o(1))\\\\\n  &= \\frac{1}{2}+ o(1) (n-1) (1-2\\lambda)^{2t}= \\frac{1}{2} +o(1) (1-2\\lambda)^{-2c}.\n          \\end{align*}\nIn particular, $\\text{Var}_{0}f(X_t) \\leq 3\/2$ for $n$ sufficiently large. \n\n\n\nWe finish the proof by applying Chebychev's inequality. \nLet \n\\begin{align*}\nA_\\alpha = \\{x\\in \\mathcal X \\, : \\, |f(x)|\\leq \\alpha\\} \\qquad \\text{ and } \\qquad B_{r,c}=\\{x\\in \\mathcal X \\, : \\, |f(x) - \\mathbf{E}_0 f(X_t)|\\leq r\\sqrt{3\/2}\\}\n\\end{align*}\nwhere we recall that $t$ is as in~\\eqref{eqn:tdef2}. \nNotice that \\[\\pi(A_\\alpha)=\\mathbf{P}_\\pi (\\{|f(X_t)|\\leq \\alpha\\})=1-\\mathbf{P}_\\pi (\\{|f(X_t)|>\\alpha\\})\\geq 1- \\frac{1}{2\\alpha^2}\\]\nwhere we used $\\mathbf{P}_\\pi(\\{|f(X_t)|>\\alpha\\})\\leq \\alpha^{-2}\\mathbf{E}_\\pi |f(X_t)|^2 \\leq 1\/(2\\alpha^2)$.  \nOn the other hand, in a similar way we have \n\\[ {P_t}(0,B_{r,c}) = 1-P_t(0, B_{r,c}^{\\text{c}}) =\n1-\\mathbf{P}_0\\{ |f(X_t)-\\mathbf{E}_0f(X_t)|> r\\sqrt{3\/2}\\}\\geq 1-\\frac{1}{r^2}\\]\nfor $n$ sufficiently large.  \nNotice that if $(1-2\\lambda)^{-c}(1+o(1))-r\\sqrt{3\/2}\\geq 100 \\alpha$, then $A_\\alpha$ and $B_{r,c}$ are disjoint.\nTherefore, $\\mu(B_\n{r,c}^{\\text{c}}) \\geq \\mu(A_\\alpha)$ for any measure $\\mu$ on subsets of $\\mathcal X$.\nIn particular, for appropriate choices of $\\alpha, c, r$ we have that $P_t(0,A_\\alpha)\\leq 1\/r^2$. \n\nNow let $\\epsilon>0$ be small but fixed. Choose $\\alpha, r$ such that $\\frac{1}{2\\alpha^2},\\frac{1}{r^2}\\leq \\epsilon\/2$ and pick $c$ such that $(1-2\\lambda)^{-c}-r\\sqrt{3\/2}\\geq 100\\alpha$.  Note that under these constraints, we may choose $$c=\\frac{\\log(\\sqrt{3} +100)- \\tfrac{1}{2}\\log \\epsilon }{ |\\log (1-2\\lambda)|}.$$  \nThen\n\\begin{align*}\n    \\TV{P_t(0,\\,\\cdot\\,)-\\pi (\\, \\cdot \\,)}&\\geq |P_t(0,A_\\alpha)-\\pi(A_\\alpha)|= 1-\\frac{1}{2\\alpha^2}- \\frac{1}{r^2}\\geq 1-\\epsilon. \n\\end{align*}\nThis concludes the proof.\n\\end{proof}\n\n\\begin{Remark}\nNotice that if $c\\leq 0$, then $\\mathbf{E}_{0}f(X_t)=(1+o(1))(1-2\\lambda)^{-c}$ and will quickly approach $0$ as $c\\to -\\infty.$ \nIn particular, $\\mathbf{E}_{0}f(X_t)\\to 0$ as $t$ increases beyond $t_n$. \nThis roughly translates to the claim that $P_t(0,\\cdot)$ and $\\pi_n$ both assign almost all of their mass close to $n\/2$. \nIf this is true, then the total variation distance between $P_t(0,\\cdot)$ and $\\pi_n$ should be small.\nThis provides evidence for believing that the lower bound on the mixing time is sharp.\n\\end{Remark}\n\n\\section{Getting at distance $o(\\sqrt{n})$}\n\\label{sec:osmallrootn}\n\n\n\nThroughout this section, $(X_t)$ and $(Y_t)$ will denote two copies of the Bernoulli-Laplace chain.  The goal of this section is to show that in $3s_n +t_n$ steps (with $s_n, t_n$ as in~\\eqref{def:times}) the chains are distance $o(\\sqrt{n})$ apart with high probability.  The argument establishing this fact is broken into a few pieces based on the size of the initial distance $|X_0-Y_0|$.  First, we see that in $t_n$ steps, the two copies are within distance $O(\\sqrt{n})$ (see~\\cref{Lemma:Orootn} below) with high probability, regardless of their initial states.  Then, starting at distance $O(\\sqrt{n})$ apart, we see that in \\cref{Lemma:orootn} using the path coupling described in \\cref{sec:outline}, the distance is decreased to $o(\\sqrt{n})$ in $s_n$ steps with high probability.  Although the two processes become closer in this second step, there is a possibility that they slightly deviate from the mean  $n\/2$ of the stationary distribution.  Consequently, in \\cref{prop:returntomean} we show that in $2s_n$ steps, the chains return to the $O(\\sqrt{n})$ band about $n\/2$ while maintaining a distance of $o(\\sqrt{n})$ between the two chains.        \n\n\\subsection*{Asymptotic notation}\nBelow, we adopt some further asymptotic notation for convenience of mathematical expression.  In what follows, if $a_n, b_n $ are sequences of nonnegative real numbers, we write $a_n \\lesssim b_n$ if $a_n =O(b_n)$ as $n\\rightarrow \\infty$ and the constant in the asymptotic estimate does \\emph{not} depend on the parameters $\\kappa_i>0$ below.  We will also write $a_n \\cong b_n$ if both $a_n \\lesssim b_n$ and $b_n \\lesssim a_n$.  \n\n\n\\subsection{Getting at distance $O(\\sqrt{n})$}\n\nApplying \\cref{lem:1} and~\\cref{lem:2}, we will now show that running the chains $(X_t)$ and $(Y_t)$ for $t=t_n$ steps brings $X_t$ and $Y_t$ within distance $O(\\sqrt n)$ of each other with high probability. \n\n\\begin{Lemma}\\label{Lemma:Orootn}\nSuppose \\cref{assump:1} is satisfied and   \nfor a fixed constant $\\kappa_1\\in (0,\\infty)$, let \\[\\tau_1(\\kappa_1):=\\min \\left\\{t\\,:\\, X_t,Y_t\\in (\\tfrac{n}{2}-\\kappa_1\\sqrt{n},\\tfrac{n}{2}+\\kappa_1\\sqrt{n}) \\right\\}.\\]\nThen, \\[\\mathbf{P}(\\tau_1(\\kappa_1)>t_n) \\lesssim  \\frac{1}{\\kappa_1^2}.\\]\n \\end{Lemma}\n\\begin{proof}\nBy combining Chebychev's inequality with \\cref{lem:1} and~\\cref{lem:2} we have\n\\begin{align}\n\\nonumber \\mathbf{P}_{X_0}\\left(\\left|X_{t_n}-\\tfrac{n}{2}\\right|>\\kappa_1 \\sqrt{n}\\right)= \\mathbf{P}(f_1(X_{t_n})^2>4\\kappa_1^2\/n)&\\leq \\frac{n}{4 \\kappa_1^2} \\mathbf{E}_{X_0} f_1(X_{t_n})^2\\\\\n\\nonumber &= \\frac{n}{4\\kappa_1^2}[\\tfrac{1}{2n-1} + \\tfrac{2n-2}{2n-1}f_2(k)^{t_n} f_2(X_0)]\\\\\n&\\lesssim \\frac{1}{\\kappa_1^2} + \\frac{n}{\\kappa_1^2}(1-2\\lambda)^{2t_n} \\lesssim \\frac{1}{\\kappa_1^2} \\label{cheby3}  \n\\end{align}\nwhere in the asymptotic inequality~\\eqref{cheby3} we used~\\eqref{eqn:f2bound} to bound $f_2(X_0)$.  \nThus, \n\\[\\mathbf{P}\\left(\\tau_1(\\kappa_1)> t_n\\right)\\leq \\mathbf{P}_{X_0} \\left(|X_{t_n}-\\tfrac{n}{2}|>\\kappa_1\\sqrt{n}\\right)+\\mathbf{P}_{Y_0} \\left(|Y_{t_n}-\\tfrac{n}{2}|>\\kappa_1\\sqrt{n}\\right)\\lesssim \\frac{1}{\\kappa_1^2}\\]\nby the estimate \\eqref{cheby3}. \n\\end{proof}\n\n\nGiven that the chains are sufficiently close to $n\/2$, we want to ensure that they stay near $n\/2$ for a significant window of time.  \\cref{prop:step1} below shows that with high probability, the chains will stay close to the mean for $s_n=\\lambda^{-1}\\log\\log n$ steps after falling within the $\\kappa_1\\sqrt{n}$ range of $n\/2$.  Later, this provides ample time to move the chains closer without deviating too far from the mean of the stationary distribution.\n\n\\begin{Proposition}\\label{prop:step1}\nSuppose that~\\cref{assump:1} is satisfied and that $|X_{0}-\\tfrac{n}{2}|<\\kappa_1 \\sqrt{n}$. Then, for every $r\\in (0, \\infty)$, $s\\in\\mathbf{N}$ we have \n\\[\\mathbf{P}\\Big(\\sup\\limits_{t\\in I_n(s)} |X_t-\\tfrac{n}{2}|>r\\Big)\\lesssim \\frac{n}{r^2}(\\log n)^{|P_\\lambda|},\\]\nwhere $I_n(s)=[s, s+ s_n]$, $P_\\lambda = \\log(1-2\\lambda)\\frac{2}{\\lambda}$ and $s_n= \\lambda^{-1}\\log \\log n$ is as in~\\eqref{def:times}. \n\\end{Proposition}\n\n\\begin{proof}Let $s \\in \\mathbf{N}$.  For $t\\ge 0$, note that $|X_t-\\tfrac{n}{2}|>r$ if and only if $f_1(X_t)=|1-2X_t\/n|>2r\/n$.\nDefine $M_t =f_1(X_t)\/f_1(k)^t$. It follows that $M_t$ is a martingale by~\\cref{lem:1} relation \\eqref{id:1}.  Setting $s_n=  \\lambda^{-1} \\log \\log n$, we thus obtain\n\\begin{align*}\n    \\mathbf{P}\\Big(\\sup\\limits_{t\\in I_n(s)} \\left|X_t-\\tfrac{n}{2}\\right|>r\\Big)=\\mathbf{P}\\Big(\\sup\\limits_{t\\in I_n(s)} |f_1(X_t)|>\\tfrac{2r}{n}\\Big)&=\\mathbf{P}\\Big(\\sup\\limits_{t\\in I_n(s)}f_1(k)^t |M_t|>\\tfrac{2r}{n}\\Big)\\\\\n    &\\leq \\mathbf{P}\\bigg(\\sup\\limits_{t\\in I_n(s)} |M_t|>\\frac{2r}{n}\\frac{1}{f_1(k)^s }\\bigg)\\\\\n    &\\leq \\frac{n^2f_1(k)^{2s}}{4r^2}\\mathbf{E}|M_{s+s_n} |^2,\n\\end{align*}\nwhere on the last inequality we used the fact that $|M_t|^2$ is a submartingale.  Note that\\begin{align*}\n  \\frac{n^2f_1(k)^{2s}}{4r^2}\\mathbf{E}|M_{s+s_n} |^2\n    &=\\frac{n^2}{4r^2}\\frac{1}{f_1(k)^{2s_n}}\\mathbf{E} f_1(X_{s+s_n})^2\\\\\n    &=\\frac{n^2}{4r^2}\\frac{1}{\\left(1-\\frac{2k}{n}\\right)^{2s_n}}\\left(\\frac{1}{2n-1}+\\frac{2n-2}{2n-1}f_2(k)^{s+s_n}f_2(X_0)\\right).  \n\\end{align*}\nwhere on the last line we use \\cref{lem:1} relation~\\eqref{id:2}.  \n\nNow observe  that by \\cref{lem:1}, $f_2(X_0)=O(n^{-1})$ since $|X_0- \\tfrac{n}{2}| \\leq \\kappa_1 \\sqrt{n}$.  Also, using~\\cref{lem:2} with \\cref{assump:1} implies $f_2(k)^{s+s_n}= o(1)$ as\n $n\\rightarrow \\infty$.  Combining the previous two observations and using \\cref{lem:2} again produces\n\\begin{align*}\n    \\frac{n^2}{4r^2}\\frac{1}{\\left(1-\\frac{2k}{n}\\right)^{2s_n}}\\left(\\frac{1}{2n-1}+\\frac{2n-2}{2n-1}f_2(k)^{s+s_n}f_2(X_0)\\right)\n    \\cong \\frac{n}{8r^2}\\frac{1}{\\left(1-\\frac{2k}{n}\\right)^{2s_n}} \\cong \\frac{n}{8r^2}(\\log n)^{|P_\\lambda|}.   \n   \n\\end{align*}\nThus,\n\\[\\mathbf{P}\\bigg(\\sup\\limits_{t\\in I_n(s)} \\left|X_t-\\tfrac{n}{2}\\right|>r\\bigg)\\lesssim \\frac{n}{r^2}(\\log n)^{|P_\\lambda|} ,\\] as claimed.  \n\\end{proof}\n\n\n\\subsection{Getting at distance $o(\\sqrt{n})$ from $O(\\sqrt{n})$}\n\nNow, given that $X_0$ and $Y_0$ are within distance $O(\\sqrt{n})$ of $n\/2$, \\cref{Lemma:orootn} below will bring the two chains $X_t$ and $Y_t$ within distance $o(\\sqrt{n})$ of each other with high probability in a negligible number of steps.  This result is a modified version of~\\cite[Lemma 12]{EN}.  Note, however, that although the distance between the chains decreases, the chains may slightly deviate from the mean.  See the choice of $r_n=\\sqrt{n} (\\log n)^{|P_\\lambda|\/2}$ below in the statement of~\\cref{Lemma:orootn}.   \n\n\n\\begin{Lemma}\\label{Lemma:orootn}\nSuppose that~\\cref{assump:1} is satisfied and that $X_0,Y_0\\in (\\frac{n}{2}-\\kappa_2\\sqrt{n},\\frac{n}{2}+\\kappa_2\\sqrt{n})$ for some $\\kappa_2\\in (0,\\infty)$.  Suppose, furthermore, that the chains $(X_t)$,$(Y_t)$ are coupled according \\cref{coupling}.  For $\\kappa_3\\in (0,\\infty)$, consider the stopping time \n\\[\\tau_3(\\kappa_3):= \\min \\left\\{ t:|X_t-Y_t|\\leq \\frac{\\sqrt{n}}{\\log \\log n}\\quad\\text{ and }\\quad X_t,Y_t\\in \\left(\\tfrac{n}{2}-\\kappa_3r_n,\\tfrac{n}{2}+\\kappa_3r_n\\right )\\right\\}\\]\nwhere $r_n =\\sqrt{n}(\\log n)^{|P_\\lambda|\/2}.$\nThen we have \n\\[\\mathbf{P}\\left(\\tau_3(\\kappa_3)>s_n\\right)\\lesssim \\frac{1}{\\kappa_3^2}\\]\nwhere $s_n= \\lambda^{-1} \\log \\log n$ is as in~\\eqref{def:times}. \n\\end{Lemma}\n\n\n\n\\begin{proof}\nRecall by~\\cref{assump:1}, $k<\\tfrac{n}{2}$.   Also recall that (see~\\eqref{eqn:tcouple})\n\\[\\tau_{\\emph{couple}}\\left(\\frac{\\sqrt{n}}{\\log \\log n}\\right)=\\min\\left\\{t: |X_t-Y_t|\\leq \\frac{\\sqrt{n}}{\\log\\log n}\\right\\}.\\] \nBy definition of $\\tau_3(\\kappa_3)$,\n\\begin{align*}\n\\left\\{\\tau_3(\\kappa_3)>s_n\\right\\}&\\subseteq\\Big\\{\\tau_{\\emph{couple}}\\left(\\tfrac{\\sqrt n}{\\log\\log n}\\right)> s_n\\Big\\} \\cup \\bigg( \\bigcup_{t\\in [0,s_n]}\\left\\{|X_t-\\tfrac{n}{2}|\\vee|Y_t-\\tfrac{n}{2}|>\\kappa_3 r_n \\right\\}\\bigg)  .\\end{align*}\nBy \\cref{prop:couple}, the assumption on $|X_0-Y_0|$ and \\cref{lem:2}, we know that \n\\begin{align*}\n    \\mathbf{P}\\Big(\\tau_{\\emph{couple}}\\left(\\tfrac{\\sqrt{n}}{\\log\\log n}\\right)>s_n\\Big)\n    &\\lesssim (1-2\\lambda +2\\lambda^2)^{s_n}\\frac{2\\kappa_2\\sqrt{n}}{\\tfrac{\\sqrt{n}}{\\log\\log n}}\\\\\n    &= e^{\\log(1-2\\lambda + 2\\lambda^2)\\lambda^{-1}\\log\\log n}2 \\kappa_2\\log\\log n\\\\\n    &=\\frac{2\\kappa_2\\log\\log n}{(\\log n)^{|\\log(1-2\\lambda+2\\lambda^2)|\\lambda^{-1}}} =o(1).  \n\\end{align*}\nApplying \\cref{prop:step1} we get \n\\[\\mathbf{P}\\bigg(\\bigcup_{t\\in [0,s_n]}\\left\\{|X_t-\\tfrac{n}{2}|\\vee|Y_t-\\tfrac{n}{2}|>\\kappa_3r_n )\\right\\}\\bigg)\\lesssim \\frac{n}{\\kappa_3^2 r_n^2}(\\log n)^{|P_\\lambda|}\\lesssim \\frac{1}{\\kappa_3^2}.\\]\nUsing a union bound, we obtain the claimed result.  \n\\end{proof}\n\nRecall that although \\cref{Lemma:orootn} brings $X_t$ and $Y_t$ within distance $o(\\sqrt{n})$ of one another in a negligible amount of time,  it is possible for $X_t$ and $Y_t$ to deviate from $n\/2$ by more than $O(\\sqrt{n})$. \nThe idea in \\cref{prop:returntomean} is that, even if this does happen, by taking another negligible number of steps we can ensure that with high probability the copies will return to the desired distance apart within the desired  $O(\\sqrt{n})$ window about $n\/2$.   \n\n\n\n\\begin{Proposition}\\label{prop:returntomean}\nSuppose \\cref{assump:1} is satisfied and that $(X_t)$ and $(Y_t)$ are coupled as in \\cref{coupling} with \n\\begin{align*}\nX_0,Y_0\\in \\left(\\tfrac{n}{2}-\\kappa_3\\sqrt{n}(\\log n)^{\\frac{|P_{\\lambda}|}{2}},\\tfrac{n}{2}+\\kappa_3\\sqrt{n}(\\log n)^{\\frac{|P_{\\lambda}|}{2}}\\right) \\qquad \\text{ and } \\qquad |X_0-Y_0| \\leq \\frac{\\sqrt{n}}{\\log\\log n}\n\\end{align*}\nfor some $\\kappa_3\\in (0,\\infty)$. \nFor $\\kappa_4\\in (0,\\infty)$, define the stopping time \n\\[\\tau_4(\\kappa_4):= \\min \\left\\{ t:|X_t-Y_t|\\leq \\frac{\\sqrt{n}}{\\log \\log n}\\,\\,\\,\\text{ and }\\,\\,\\, X_t,Y_t\\in \\left(\\tfrac{n}{2}-\\kappa_4\\sqrt{n},\\tfrac{n}{2}+\\kappa_4\\sqrt{n}\\right )\\right\\}.\\]\nThen we have \n\\[\\mathbf{P}\\left(\\tau_4(\\kappa_4)>2s_n \\right)\\lesssim \\frac{1}{\\kappa_4^2}\\]\nwhere $s_n$ is as in~\\eqref{def:times}.  \n\\end{Proposition}\n\n\\begin{proof}\nBy construction of the \\cref{coupling}, $t\\mapsto |X_t -Y_t|$ is monotone decreasing.  Hence, by the hypothesis on the initial condition we have \\[|X_t-Y_t|\\leq |X_0-Y_0| \\leq \\frac{\\sqrt{n}}{\\log \\log n}\\]\nfor all $t\\geq 0$.  \nIn particular, \\[\\tau_4(\\kappa_4)= \\min \\left\\{ t: X_t,Y_t\\in \\left(\\frac{n}{2}-\\kappa_4\\sqrt{n},\\frac{n}{2}+\\kappa_4\\sqrt{n}\\right )\\right\\}.\\]\nUsing \\cref{lem:1} with~\\cref{lem:2} gives\n\\begin{align}\n   \\nonumber \\mathbf{E}_{X_0} f_1(X_{2s_n})^2 &= \\frac{1}{2n-1} + \\frac{2n-2}{2n-1} f_2(k)^{2s_n} f_2(X_0)\\\\\n  \\label{eqn:cong1} & \\cong \\frac{1}{2n-1} + \\frac{2n-2}{2n-1} (1-2\\lambda)^{4s_n} f_2(X_0).  \n    \\end{align}\n    By hypothesis on $X_0$, it follows by~\\cref{lem:2} that\n    \\begin{align*}\n    f_2(X_0) = O\\Big( \\frac{(\\log n)^{|P_\\lambda|}}{n}\\Big).  \n    \\end{align*}\n    Plugging this into~\\eqref{eqn:cong1} implies \n    \\begin{align*}\n   \\mathbf{E}_{X_0} f_1(X_{2s_n})^2 &\\lesssim \\frac{1}{2n-1} + \\frac{2n-2}{2n-1} O(n^{-1} (\\log n)^{- |P_\\lambda|})\\\\\n   & = O(n^{-1})  \n   \\end{align*}   \nwith the same estimate holding true for $Y_{2s_n}$.  \n\n\nCombining these estimates with the monotonicity of the coupling along with a union bound and Chebychev's inequality then gives \n\\begin{align*}\n\\mathbf{P}_{X_0,Y_0}(\\tau_4(\\kappa_4)>2 s_n)&\\leq \\mathbf{P}_{X_0,Y_0}\\left(|X_{2s_n}-\\tfrac{n}{2}|\\vee |Y_{2s_n}-\\tfrac{n}{2}|> \\kappa_4\\sqrt{n}\\right)\\\\\n&=\\mathbf{P}_{X_0,Y_0}\\left(|f_1(X_{2s_n})|\\vee |f_2(Y_{2s_n}) |> 2\\kappa_4\/\\sqrt{n}\\right) \\lesssim  \\frac{1}{\\kappa_4^2},  \n\\end{align*}\nfinishing the proof.  \n\\end{proof}\n\n\n\n\\section{The upper bound on the mixing time}\n\\label{sec:upperbound}\n\n\nIf $X$ is a random variable, then we let $\\mu_X$ denote its distribution.  The goal of this section is to show that if $X_0-Y_0=o(\\sqrt{n})$ and $X_0=\\tfrac{n}{2}+O(\\sqrt{n})$, $Y_0=\\tfrac{n}{2}+O(\\sqrt{n})$ as $n\\rightarrow \\infty$, then \n\\begin{align}\n\\label{eqn:TVo1}\n\\TV{P_1(X_0, \\, \\cdot \\,) - P_1(Y_0, \\, \\cdot \\,) }=\\TV{\\mu_{X_1}-\\mu_{Y_1}} = o(1) \\,\\, \\text{ as }\\,\\, n\\rightarrow \\infty.  \n\\end{align}\nCombining this with the results of the previous section ultimately yields the desired upper bound on the mixing time.    \n\nOur method to show~\\eqref{eqn:TVo1} is quite different than the method used in~\\cite{EN}.  To see why,   \nlet $\\text{Hyper}(j, \\ell, m)$ denote the hypergeometric distribution of $m$ objects selected without replacement from a total of $j$ objects, $\\ell$ of which are of type 1 and $j-\\ell$ are of type 2.  Also, let $\\text{Bin}(j, r)$ denote the binomial distribution of $j$ trials with success probability $r$.  Observe that we can write \n\\begin{align}\nX_1-X_0= H_1 - H_0 \\qquad \\text{ and } \\qquad Y_1 - Y_0 = H_3 - H_2 \n\\end{align}\nwhere $H_1\\sim \\text{Hyper}(n,n-X_0, k)$ and $H_0\\sim \\text{Hyper}(n, X_0, k)$ while $H_3 \\sim \\text{Hyper}(n,n-Y_0, k)$ and $H_2\\sim \\text{Hyper}(n, Y_0, k)$.  Thus we wish to estimate\n\\begin{align}\\label{eqn:hyperrep}\n\\TV{\\mu_{X_1}-\\mu_{Y_1}}= \\TV{ \\mu_{X_0+ H_1 - H_0} -\\mu_{Y_0 +H_3- H_2} }.  \n\\end{align} \n\nIn order to estimate~\\eqref{eqn:hyperrep}, we employ a large $n$ comparison of the hypergeometric distributions $H_i$, $i=0,1,2,3$, in the total variation distance.  Because $k\/n\\rightarrow \\lambda \\in (0,1\/2)$ instead of $k=o(n)$ as $n\\rightarrow \\infty$, we cannot appeal to the result of Diaconis and Freedman~\\cite{DF_80}, which in particular shows that if $k=o(n)$, then \n\\begin{align}\n\\label{eqn:hyperb}\n\\TV{\\mu_{H_i} -\\mu_B}\\lesssim \\frac{k}{n} + \\sqrt{\\frac{k}{n}}\n\\end{align}\nwhere $B\\sim \\text{Bin}(k, 1\/2)$.  Using normal approximation of the binomial, the desired result~\\eqref{eqn:TVo1} follows~\\cite[Lemma 16]{EN} in the case when $k=o(n)$ by using the triangle inequality several times in~\\eqref{eqn:hyperrep}.       \n\nNote that in our case, the righthand side of~\\eqref{eqn:hyperb} is order $1$.  Hence, instead of using a binomial and then normal approximation, we manufacture a discrete version of the normal that approximates the $H_i$'s in total variation.  Combining this approximation with the triangle inequality a few times in~\\eqref{eqn:hyperrep}, we will then be able to conclude~\\eqref{eqn:TVo1}.        \n\n\\subsection{The discrete normal distribution}\nIn what follows, we will use $\\phi :\\mathbf{R}\\rightarrow \\mathbf{R}$ to denote the probability density function of the standard normal on $\\mathbf{R}$; that is,\n\\begin{align}\n\\phi(x) = \\frac{e^{-\\frac{x^2}{2}}}{\\sqrt{2\\pi}}, \\qquad x\\in \\mathbf{R}. \n\\end{align}\n\n\\begin{Definition}\nWe say that a random variable $Z$  distributed on the integers $\\mathbf{Z}$ has a \\emph{discrete normal distribution} with parameters $x\\in \\mathbf{R}$, $s>0$ and finite set $\\mathcal{S}  \\subset \\mathbf{Z}$, denoted by $Z \\sim \\text{dN}(x, s, \\mathcal{S})$, if \n\\begin{align*}\n\\mathbf{P}(Z= j)  = \n\\begin{cases}\n\\frac{1}{\\mathcal{N}}\\frac{\\phi \\big( \\frac{j-x}{s}\\big)}{s} & \\text{ if } \\,\\, j \\in \\mathcal{S}\\\\\n0 & \\text{ if } j \\notin \\mathcal{S}\n\\end{cases}\n\\end{align*}\nwhere $\\mathcal{N}:= \\sum_{j\\in \\mathcal{S}} \\frac{\\phi \\big( \\frac{j-x}{s}\\big)}{s}$ is the normalization constant.  \\end{Definition}\n\n\\begin{Remark}\nDepending on the choice of parameters $x\\in \\mathbf{R}$, $s>0$  and finite set $\\mathcal{S}$, the name \\emph{discrete normal} above is a bit of a misnomer.  Indeed, unless properly shifted and scaled with a large enough set $\\mathcal{S}$, a random variable $Z\\sim \\text{dN}(x,s, \\mathcal{S})$ may have little to do with the usual normal distribution.  All of the discrete normals used below, however, will indeed be reminiscent of a normal distribution due to the particular choices of $x,s$ and $\\mathcal{S}$.     \n\\end{Remark}\n\nIn order to setup and prove the next result, set $\\mathcal X_k= \\{0,1,2,\\ldots, k\\}$, fix $\\ell \\in \\mathcal X$ and define parameters\n\\begin{align}\n\\label{eqn:not1}\np:=\\frac{\\ell}{n},\\, \\,\\, f:= \\frac{k}{n},\\, \\,\\,q:=1-p, \\,\\,\\, \\sigma:= \\sqrt{k pq (1-f)}.\n\\end{align}\nFurthermore, for $j\\in \\mathcal X_k$ set\n\\begin{align}\n\\label{eqn:not2}\n x_j := \\frac{j - kp}{\\sqrt{k p q}} \\qquad \\text{ and } \\qquad \\tilde{x}_j := \\frac{x_j}{\\sqrt{1-f}}= \\frac{j-kp}{\\sigma}.  \n\\end{align}\nWe first need a technical lemma concerning the behavior of the normalization constant for a particular discrete normal. \n\\begin{Lemma}\n\\label{lem:normalization}\nGiven the choice of parameters in~\\eqref{eqn:not1} and~\\eqref{eqn:not2}, suppose \\cref{assump:1} is satisfied and that $\\ell=n\/2+O(\\sqrt{n})$.  Then $$\\mathcal{N}_n:=\\sum_{j\\in \\mathcal X_k}\\frac{\\phi(\\tilde{x}_j)}{\\sigma}=1 +O(n^{-1\/2})\\,\\,\\, \\text{ as }\\,\\,\\, n\\rightarrow \\infty.$$ \n\\end{Lemma}\n\n\\begin{proof}\nThe proof of this result follows by integral comparison. Note that \n\\begin{align*}\n\\mathcal{N}_n &\\leq \\frac{1}{\\sqrt{2\\pi \\sigma^2}}+ \\sum_{j=0}^{\\lfloor kp\\rfloor-1}\\frac{ e^{-\\frac{1}{2}\\big(\\frac{j-kp}{\\sigma} \\big)^2}}{\\sqrt{2\\pi \\sigma^2}} + \\sum_{j=\\lfloor kp\\rfloor +1}^k \\frac{ e^{-\\frac{1}{2}\\big(\\frac{j-kp}{\\sigma} \\big)^2}}{\\sqrt{2\\pi \\sigma^2}}\\\\\n& \\leq \\int_\\mathbf{R} \\frac{ e^{-\\frac{1}{2}\\big(\\frac{x-kp}{\\sigma} \\big)^2}}{\\sqrt{2\\pi \\sigma^2}} \\, dx + \\frac{1}{\\sqrt{2\\pi \\sigma^2}}=1 +   \\frac{1}{\\sqrt{2\\pi \\sigma^2}}.   \n\\end{align*} \nGiven the asymptotic behavior of $\\sigma$, this finishes the proof of the upper bound.  To obtain the lower bound, note that \n\\begin{align*}\n\\mathcal{N}_n &\\geq  \\sum_{j=1}^{\\lfloor kp\\rfloor }\\frac{ e^{-\\frac{1}{2}\\big(\\frac{j-kp}{\\sigma} \\big)^2}}{\\sqrt{2\\pi \\sigma^2}} + \\sum_{j=\\lfloor kp\\rfloor}^k \\frac{ e^{-\\frac{1}{2}\\big(\\frac{j-kp}{\\sigma} \\big)^2}}{\\sqrt{2\\pi \\sigma^2}} - \\frac{1}{\\sqrt{2\\pi \\sigma^2}} \\\\\n& \\geq \\int_0^k \\frac{ e^{-\\frac{1}{2}\\big(\\frac{x-kp}{\\sigma} \\big)^2}}{\\sqrt{2\\pi \\sigma^2}} \\, dx  - \\frac{1}{\\sqrt{2\\pi \\sigma^2}}\\\\\n&= 1- \\int_{-\\infty}^{-kp\/\\sigma} \\phi(x) \\, dx - \\int_{kq\/\\sigma}^\\infty \\phi(x) \\, dx - \\frac{1}{\\sqrt{2\\pi \\sigma^2}}. \\end{align*}\nNow if $\\mathcal{Z}$ is a standard normal random variable on $\\mathbf{R}$, we note that \n\\begin{align*}\n\\mathcal{N}_n &\\geq 1- \\int_{-\\infty}^{-kp\/\\sigma} \\phi(x) \\, dx - \\int_{kq\/\\sigma}^\\infty \\phi(x) \\, dx - \\frac{1}{\\sqrt{2\\pi \\sigma^2}} \\\\\n&=1- \\mathbf{P}(\\mathcal{Z}\\leq -k p\/\\sigma) -\\mathbf{P}(\\mathcal{Z} \\geq kq\/\\sigma)- \\frac{1}{\\sqrt{2\\pi \\sigma^2}}. \n\\end{align*}\nBy Chebychev's inequality on the square $\\mathcal{Z}^2$, we see that both $\\mathbf{P}(\\mathcal{Z}\\leq -kp\/\\sigma)$ and $\\mathbf{P}(\\mathcal{Z} \\geq kq\/\\sigma)$ are order $1\/n$ as $\\ell=n\/2 + O(\\sqrt{n})$. We thus conclude the result.  \n\\end{proof}\n\n\nBefore stating the main comparison lemma, we will use a result from \\cite{LCM_07}.\nThough the following lemma is not stated explicitly in \\cite{LCM_07}, the proof of it is contained within the proof of \\cite[Theorem~1]{LCM_07}.\nIn particular, we will restate the assumptions that are required for \\cite[Equations~(4.28)-(4.29)]{LCM_07}.\n\n\\begin{Lemma}\\label{StatLemma}\nSuppose $0<f<1$, $0<p<1$, and $6(kp\\wedge kq)\\geq 1$ where $f= k\/n,$ $p= \\ell \/ n$, and $q= 1-p$. \nLet \\[L = \\inf\\{j\\in \\mathbf{Z}_+ :\\tilde x_j\\geq -\\delta \\sigma\\} \\quad \\text{and}\\quad J_x = \\lfloor kp - x\\sigma \\rfloor\\] for $x\\in \\mathbf{R}$. \nFor any $\\delta \\in (0,1\/2]$ and $x\\in [-\\delta \\sigma, 0]$, we have \n\\[ \\sum_{j=L}^{J_x}|\\mathbf{P}(H=j)-\\mu(j)|\\leq \\frac{C}{\\sigma (1-f)}[(1+x^2)\\exp(-0.07x^2).\\]\n\\end{Lemma}\n\nWe will use the previous lemma in the next result, which provides the critical comparison.  \n\n\\begin{Proposition}\\label{hyper}\nGiven the choice of parameters in~\\eqref{eqn:not1} and~\\eqref{eqn:not2}, suppose \\cref{assump:1} is satisfied.  Let $H\\sim \\text{\\emph{Hyper}}(n, \\ell,k)$ and $Z \\sim \\text{\\emph{dN}}(kp, \\sigma, \\mathcal X_k)$ with $\\ell= n\/2+O(\\sqrt{n})$.  Then \n\\begin{align}\n\\TV{\\mu_H-\\mu_Z}= O(n^{-1\/2}) \\,\\,\\text{ as } \\,\\, n \\rightarrow \\infty. \n\\end{align}    \n\\end{Proposition}\n\n\n\\begin{proof}\nUsing the notation above in~\\eqref{eqn:not1} and~\\eqref{eqn:not2}, we first introduce some further notation that helps connect with the setup in~\\cite{LCM_07}.  Let  \n\\begin{align*}\n\\bar{f}= f\\wedge (1-f), \\,\\,  a:= \\frac{\\bar{f} +4}{4(1-\\bar{f})}, \\,\\, \\, \\delta := \\frac{1}{10(a\\vee 2)} \n\\end{align*}\nand define\n\\[L=\\inf\\{ j \\in \\mathcal X_k \\,: \\,\\tilde x_j\\geq -\\delta\\sigma\\} \\qquad\\text{and} \\qquad R=\\sup\\{j \\in \\mathcal X_k \\,:\\, \\tilde x_j\\leq \\delta\\sigma\\}.\\]\nSetting $\\mu(j)= \\mathbf{P}(Z=j)$, we find that \n\\begin{align*}\n    2\\TV{\\mu_H-\\mu_Z}&=\\sum_{j=0}^{L-1}|\\mathbf{P}(H=j)-\\mu(j))|+\\sum_{j=L}^{R}|\\mathbf{P}(H=j)-\\mu(j)|+\\sum_{\\ell=R+1}^{k}|\\mathbf{P}(H=j)- \\mu(j)|\\\\\n    & \\leq \\sum_{j=0}^{L-1}(\\mathbf{P}(H=j)+\\mu(j))+\\sum_{j=L}^{R}|\\mathbf{P}(H=j)-\\mu(j)|+\\sum_{\\ell=R+1}^{k}(\\mathbf{P}(H=j)+\\mu(j))\\\\\n    & \\leq \\sum_{j=L}^{R}|\\mathbf{P}(H=j)-\\mu(j)|+ \\frac{2k e^{-\\delta^2 \\sigma^2}}{\\mathcal{N}_n\\sqrt{2\\pi\\sigma}} + \\mathbf{P}(H < L) + \\mathbf{P}(H > R).\n    \\end{align*}\n   \n   \n   \n   \n    Using the well known tail bound in \\cite{Hoeffding_1963}, we arrive at \n    \\begin{align*}\n    P(H<L) + P(H>R) \\leq 2e^{-2k\\left(\\delta pq(1-f)\\right)^2}\\lesssim \\frac{1}{\\sqrt{n}}. \n    \\end{align*}\n    Furthermore, it follows from symmetry, \\cref{lem:normalization}, and \\cref{StatLemma} that\n    \\begin{align*}\n     \\sum_{j=L}^{R}|\\mathbf{P}(H=j)-\\mu(j)| \\leq2\\sum_{j=L}^{\\lfloor kp\\rfloor}\\bigg|\\mathbf{P}(H=j)-\\frac{\\phi((j-kp)\/\\sigma)}{\\sigma}\\bigg| + O(n^{-1\/2})\\leq \\frac{C}{\\sigma} \\lesssim \\frac{1}{\\sqrt{n}}.  \n    \\end{align*}\n    This concludes the proof.\n\\end{proof}\n\n\n\\subsection{Using the hypergeometric comparison}\nLet us now combine the previous result with~\\eqref{eqn:hyperrep} in order to see what we have left to estimate.  To this end, let $Z_i \\sim \\text{dN}(k p_i, \\sigma_i, \\mathcal X_k), i=0,1,2,3$, with\n\\begin{align}\n\\label{eqn:not3}\n\\ell_0= X_0, \\,\\,\\, \\ell_1=n-X_0, \\,\\,\\, \\ell_2= Y_0, \\,\\,\\, \\ell_3=n-Y_0, \\,\\,\\, p_i=\\ell_i\/n,\n\\end{align}  \nand \n\\begin{align}\n\\label{eqn:not4}\n\\sigma_i = \\sqrt{kp_i (1-p_i) (1-k\/n)}, i=0,1,2,3. \n\\end{align}\nNote that if $\\eta= X_0-Y_0$, then by the triangle inequality and properties of the total variation distance of random variables distributed on $\\mathbf{Z}$\n\\begin{align*}\n\\TV{\\mu_{X_1}- \\mu_{Y_1}} &= \\TV{\\mu_{X_0+ H_1-H_0}- \\mu_{Y_0+ H_3-H_2} }\\\\\n    & \\leq \\TV{\\mu_{X_0 + H_1-H_0} - \\mu_{X_0+ Z_1-Z_0} } + \\TV{\\mu_{X_0 +Z_1 -Z_0}- \\mu_{Y_0 +Z_3-Z_2}}\\\\\n    & \\qquad + \\TV{\\mu_{Y_0 +Z_3 -Z_2}- \\mu_{Y_0 +H_3-H_2}}\\\\\n    &= \\TV{\\mu_{H_1-H_0} - \\mu_{Z_1-Z_0} } + \\TV{\\mu_{\\eta+Z_1 -Z_0}- \\mu_{Z_3-Z_2}}\\\\\n    & \\qquad + \\TV{\\mu_{Z_3 -Z_2}- \\mu_{H_3-H_2}}\\\\\n    & \\leq \\sum_{i=0}^3\\TV{\\mu_{H_i}- \\mu_{Z_i}} + \\| \\mu_{\\eta + Z_1}- \\mu_{Z_3} \\|_{TV} + \\|\\mu_{Z_0}- \\mu_{Z_2} \\|_{TV}.  \n    \\end{align*}\n    By \\cref{hyper}, \n    \\begin{align*}\n    \\sum_{i=0}^3\\TV{\\mu_{H_i}- \\mu_{Z_i}} = O(n^{-1\/2}) \n        \\end{align*}\n   assuming $\\eta=X_0-Y_0=o(\\sqrt{n})$ and $X_0= \\tfrac{n}{2}+O(\\sqrt{n}), Y_0= \\tfrac{n}{2}+O(\\sqrt{n})$.     \nThus, we must bound \n\\begin{align*}\n\\| \\mu_{\\eta + Z_1}- \\mu_{Z_3} \\|_{TV} + \\|\\mu_{Z_0}- \\mu_{Z_2} \\|_{TV}\n\\end{align*}\nunder the same assumptions.  \n\\begin{Lemma}\\label{normal}\nSuppose that \\cref{assump:1} is satisfied and consider the parameters $\\ell_i$, $p_i$, $\\sigma_i$ as in \\eqref{eqn:not3} and the random variables $Z_i \\sim \\text{\\emph{dN}}(kp_i, \\sigma_i, \\mathcal X_k)$.  If $\\eta=X_0-Y_0=o(\\sqrt{n})$ and $X_0=n\/2+ O(n^{1\/2})$, $Y_0= n\/2+O(n^{1\/2})$, then as $n\\rightarrow \\infty$ \n\\begin{align}\n\\label{eqn:Zs}\n\\| \\mu_{\\eta + Z_1}- \\mu_{Z_3} \\|_{TV} + \\|\\mu_{Z_0}- \\mu_{Z_2} \\|_{TV} =o(1).\n\\end{align} \n\\end{Lemma}\n\n\\begin{proof}\nNote that it suffices to show that $\\TV{\\mu_{\\eta+Z_1} - \\mu_{Z_3}}\\rightarrow 0$ as $n\\rightarrow \\infty$.   Let $\\epsilon >0$, $\\mathcal{Y}_\\eta := \\mathcal X_k \\cap (\\mathcal X_k + \\eta)$ and $\\mathcal{N}_{n,1}$ and $\\mathcal{N}_{n,3}$ denote the respective normalization constants for $Z_1$ and $Z_3$.  Fixing a constant $K>0$ to be determined momentarily, let $J_n(K)=[\\tfrac{k}{2}-K \\sqrt{n}, \\tfrac{k}{2}+ K \\sqrt{n}]$.  Note we can write\n\\begin{align*}\n2 \\TV{\\mu_{\\eta+Z_1} - \\mu_{Z_3}}&= \\sum_{j\\in \\mathbf{Z}} | \\mathbf{P}(Z_1 = j- \\eta) - \\mathbf{P}(Z_3=j) | \\\\\n&= \\sum_{j \\in \\mathcal{Y}_\\eta\\cap J_n(K)} \\bigg|\\frac{\\phi\\big(\\frac{j-\\eta-kp_1}{\\sigma_1} \\big)}{\\mathcal{N}_{n,1}\\sigma_1 }- \\frac{\\phi\\big(\\frac{j-kp_3}{\\sigma_3} \\big)}{\\mathcal{N}_{n,3}\\sigma_3 }\\bigg|\\\\\n&\\qquad +  \\sum_{j \\in \\mathcal{Y}_\\eta\\cap J_n(K)^c} \\bigg|\\frac{\\phi\\big(\\frac{j-\\eta-kp_1}{\\sigma_1} \\big)}{\\mathcal{N}_{n,1}\\sigma_1 }- \\frac{\\phi\\big(\\frac{j-kp_3}{\\sigma_3} \\big)}{\\mathcal{N}_{n,3}\\sigma_3 }\\bigg|\\\\\n&\\qquad +  \\sum_{j\\in \\mathcal X_k - (\\mathcal X_k +\\eta)} \\mathbf{P}(Z_3=j) +  \\sum_{j\\in (\\mathcal X_k +\\eta)-\\mathcal X_k} \\mathbf{P}(Z_1=j)\\\\\n&=: T_1 + T_2 + T_3 +T_4.    \n\\end{align*}     \nWe next show how to estimate $T_3$.  The term $T_4$ can be done analogously, so we omit those details.  Observe that if $\\eta >0$ and $j\\in \\mathcal X_k - (\\mathcal X_k + \\eta)$, then $j \\leq \\eta-1$.  Also if $\\eta \\leq 0$ and $j \\in \\mathcal X_k -(\\mathcal X_k + \\eta)$, then $j \\geq k+\\eta+1$.  Now since $\\eta=o(\\sqrt{k})$\n\\begin{align*}\nT_3 &\\leq \\sum_{j=0}^{|\\eta|} \\mathbf{P}(Z_3=j) + \\sum_{j=k+1-|\\eta|}^k \\mathbf{P}(Z_3 = j)\\\\\n& \\lesssim \\frac{(| \\eta|+ 1)}{\\sigma_3} \\bigg(\\phi\\bigg( \\frac{|\\eta|-kp_3}{\\sigma_3}\\bigg)+ \\phi\\bigg( \\frac{k+1 -|\\eta| - kp_3}{\\sigma_3}\\bigg) \\bigg) \\lesssim e^{-\\epsilon' n}\n\\end{align*}   \nfor some constant $ \\epsilon' >0$ independent of $n$.    \n\nWe next estimate $T_2$.  Note that, by using integral comparison and that $\\ell_i= n\/2 + O(n^{1\/2})$ and $\\eta=o(n^{1\/2})$, it follows that there exists a constant $C>0$ independent of $K,n$ such that for all $K$ and $n$ large enough\n\\begin{align*} \nT_2 \\leq C \\int_{-\\infty}^{-K\/2} \\phi(x) \\, dx + C \\int_{K\/2}^\\infty \\phi(x) \\, dx.   \n\\end{align*}\nThus pick $K>0$ large enough so that $T_2<\\epsilon\/2$ for all $n$ large enough.  \n\nTurning finally to $T_1$, first note that since $\\eta=X_0-Y_0=o(n^{1\/2})$, $X_0=n\/2+ O(n^{1\/2})$ and $Y_0= n\/2+O(n^{1\/2})$\n\\begin{align*}\n\\frac{1}{\\sigma_1}- \\frac{1}{\\sigma_3}&= \\frac{\\eta + \\frac{Y_0^2-X_0^2}{n}}{\\sqrt{X_0 Y_0(1-X_0\/n)(1-Y_0\/n)(1-\\frac{k}{n})}} \\frac{\\sqrt{n\/k}}{\\sqrt{X_0(1-X_0\/n)}+ \\sqrt{Y_0(1-Y_0\/n)}}\\\\\n&= o(n^{-1}).   \n\\end{align*}\nThus combining this with \\cref{lem:normalization} produces \n\\begin{align*}\nT_1 &= \\sum_{j \\in \\mathcal{Y}_\\eta\\cap J_n(K)} \\bigg|\\frac{\\phi\\big(\\frac{j-\\eta-k p_1}{\\sigma_1} \\big)}{\\sigma_1 }- \\frac{\\phi\\big(\\frac{j-k p_3}{\\sigma_3} \\big)}{\\sigma_3 }\\bigg| + O(n^{-1\/2})\\\\\n&= \\sum_{j \\in \\mathcal{Y}_\\eta\\cap J_n(K)} \\bigg|\\frac{\\phi\\big(\\frac{j-\\eta-kp_1}{\\sigma_1} \\big)}{\\sigma_1 }- \\frac{\\phi\\big(\\frac{j-kp_3}{\\sigma_3} \\big)}{\\sigma_1 }\\bigg| + o(1)\\\\\n&=: T_1' + o(1).\\end{align*}\nNote that for $j\\in \\mathcal X_k \\cap J_n(K)$, both $(j-\\eta-kp_1)\/\\sigma_1$ and $(j-kp_3)\/\\sigma_3$ are bounded in $n$.  Hence for $T_1'$ we may write for some constant $C>0$ \n\\begin{align*}\nT_1' &=  \\sum_{j \\in \\mathcal{Y}_\\eta\\cap J_n(K)} \\bigg|\\frac{\\phi\\big(\\frac{j-\\eta-kp_1}{\\sigma_1} \\big)}{\\sigma_1 }- \\frac{\\phi\\big(\\frac{j-kp_3}{\\sigma_3} \\big)}{\\sigma_1 }\\bigg| \\\\\n&= \\frac{1}{\\sqrt{2\\pi \\sigma_1^2}} \\sum_{j\\in \\mathcal{Y}_\\eta \\cap J_n(K)} e^{-\\frac{(j-\\eta-k p_1)^2}{2\\sigma_1^2}}\\bigg| 1- e^{- \\big(\\frac{(j-kp_3)^2}{2\\sigma_3^2}-\\frac{(j-\\eta-kp_1)^2}{2\\sigma_1^2}\\big) }\\bigg| \\\\\n& \\leq  \\frac{C}{\\sqrt{2\\pi \\sigma_1^2}}  \\sum_{j\\in \\mathcal{Y}_\\eta \\cap J_n(K)} e^{-\\frac{(j-\\eta-kp_1)^2}{2\\sigma_1^2}}\\bigg| \\frac{j-kp_3}{\\sigma_3}- \\frac{j-\\eta - kp_1}{\\sigma_1}\\bigg|\\\\\n& =   \\frac{C}{\\sqrt{2\\pi \\sigma_1^2}}  \\sum_{j\\in \\mathcal{Y}_\\eta \\cap J_n(K)} e^{-\\frac{(j-\\eta-k p_1)^2}{2\\sigma_1^2}}\\bigg| \\frac{j-k p_3}{\\sigma_1}- \\frac{j-\\eta - k p_1}{\\sigma_1}\\bigg| + O(n^{-1\/2})\\\\\n&\\leq  \\frac{C}{\\sqrt{2\\pi \\sigma_1^2}}  \\sum_{j\\in \\mathcal{Y}_\\eta \\cap J_n(K)} e^{-\\frac{(j-\\eta-k p_1)^2}{2\\sigma_1^2}} \\frac{2\\eta}{\\sigma_1} + O(n^{-1\/2})= o(1)\n\\end{align*}\nas $\\eta= o(n^{1\/2})$ and $\\sigma= O(n^{1\/2})$.  This completes the proof since by choosing $N=N(K)>0$ large enough we have \n\\begin{align}   \n\\TV{\\mu_{X_1} - \\mu_{Y_1} } < \\epsilon\n\\end{align}\nfor all $n\\geq N$.  \n\n\\end{proof}\n\n\\subsection{Upper bound}\n\nWe now use the previous estimates to conclude the upper bound in~\\eqref{eqn:upperb}.\n\n\\begin{Theorem}\nLet $\\epsilon >0$.  For the $(n,k)$-Bernoulli-Laplace model under \\cref{assump:1}, we have \n\\[t_{mix}(\\epsilon)\\leq \\frac{\\log(n)}{2|\\log(1-2\\lambda)|} + 3\\lambda^{-1}\\log\\log n +1\\] for large enough $n$. \n\\end{Theorem}\n\nIn order to setup the proof, we recall the definitions of $\\tau_1(\\kappa_1), \\tau_3(\\kappa_3)$ and $\\tau_4(\\kappa_4)$:\n\\begin{align*}\n\\tau_1(\\kappa_1)&:=\\min \\left\\{t\\,:\\, X_t,Y_t\\in (\\tfrac{n}{2}-\\kappa_1\\sqrt{n},\\tfrac{n}{2}+\\kappa_1\\sqrt{n}) \\right\\},\\\\\n\\tau_3(\\kappa_3)&:= \\min \\left\\{ t:|X_t-Y_t|\\leq \\frac{\\sqrt{n}}{\\log \\log n}\\quad\\text{ and }\\quad X_t,Y_t\\in \\left(\\tfrac{n}{2}-\\kappa_3r_n,\\tfrac{n}{2}+\\kappa_3r_n\\right )\\right\\},\\\\\n\\tau_4(\\kappa_4)&:= \\min \\left\\{ t:|X_t-Y_t|\\leq \\frac{\\sqrt{n}}{\\log \\log n}\\,\\,\\,\\text{ and }\\,\\,\\, X_t,Y_t\\in \\left(\\tfrac{n}{2}-\\kappa_4\\sqrt{n},\\tfrac{n}{2}+\\kappa_4\\sqrt{n}\\right )\\right\\},\n\\end{align*}\nwhere $r_n =\\sqrt{n}(\\log n)^{|P_\\lambda|\/2}$.   \n\n\n\\begin{proof}\nBelow, for simplicity of expression, we will suppress the $\\kappa_i$'s in $\\tau_i(\\kappa_i)$.  For $t< \\tau_4$, we couple the two chains $(X_t)$ and $(Y_t)$ according to \\cref{coupling}.  For $t\\geq \\tau_4$, we pick the optimal coupling; that is, the coupling $(X_t, Y_t)$ of $\\mu_{X_t}$ and $\\mu_{Y_t}$ such that $\\TV{\\mu_{X_t}- \\mu_{Y_t}} = \\mathbf{P}\\{X_t \\neq Y_t\\}=\\mathbf{P}\\{ |X_t - Y_t | \\geq 1 \\}$.    \nNote by \\cref{Lemma:Orootn} and the Strong Markov Property:  \n\\begin{align*}\n    \\mathbf{P}(\\tau_4 >t_n+3s_n \\,|\\,X_0,Y_0)&=\\mathbf{P}(\\tau_4 >t_n+3s_n, \\tau_1> t_n \\,|\\,X_0,Y_0)+\\mathbf{P}(\\tau_4 >t_n+3s_n, \\tau_1\\leq  t_n \\,|\\,X_0,Y_0)\\\\\n    &\\lesssim \\frac{1}{\\kappa_1^2} + \\mathbf{P}(\\tau_4 >t_n+3s_n, \\tau_1\\leq  t_n|X_0,Y_0)\\\\\n    &= \\frac{1}{\\kappa_1^2} + \\mathbf{E}\\E(\\textbf{1}\\{\\tau_4 >t_n+3s_n\\}\\textbf{1}\\{ \\tau_1\\leq  t_n\\}|\\mathcal F_{\\tau_1})\\\\\n    &= \\frac{1}{\\kappa_1^2} + \\mathbf{E}(\\textbf{1}\\{\\tau_1\\leq  t_n\\}\\mathbf{P} (\\tau_4 >t_n+3s_n|\\mathcal F_{\\tau_1}))\\\\\n    &\\leq \\frac{1}{\\kappa_1^2} + \\mathbf{E}\\mathbf{P} (\\tau_4 >3s_n\\,|\\, X_{\\tau_1},Y_{\\tau_1}).\n\\end{align*}\nContinuing in this way, we obtain by \\cref{Lemma:orootn}\n\\begin{align*}\n     \\mathbf{E} \\mathbf{P} (\\tau_4 >3s_n\\,|\\,X_{\\tau_1},Y_{\\tau_1})&= \\mathbf{E} \\mathbf{P} (\\tau_4 >3s_n, \\tau_3>s_n \\,|\\,X_{\\tau_1},Y_{\\tau_1})+ \\mathbf{E}\\mathbf{P} (\\tau_4 >3s_n,\\tau_3\\leq s_n\\,|\\,X_{\\tau_1},Y_{\\tau_1})\\\\\n    &\\lesssim \\frac{1}{\\kappa_3^2} + \\mathbf{E}\\mathbf{P} (\\tau_4 >3s_n,\\tau_3\\leq s_n\\,|\\,X_{\\tau_1},Y_{\\tau_1})\\\\\n    &= \\frac{1}{\\kappa_3^2} + \\mathbf{E}\\E (\\textbf{1}\\{\\tau_4 >3s_n\\}\\textbf{1}\\{\\tau_3\\leq s_n\\}\\,|\\,\\mathcal{F}_{\\tau_3})\\\\\n    &= \\frac{1}{\\kappa_3^2} + \\mathbf{E}(\\textbf{1}\\{\\tau_3\\leq s_n\\}\\mathbf{P}(\\tau_4> 3s_n| \\mathcal F_{\\tau_3}))\\\\\n    &\\leq \\frac{1}{\\kappa_3^2} + \\mathbf{E}\\mathbf{P}(\\tau_4>2s_n| X_{\\tau_3}, Y_{\\tau_3})\\lesssim \\frac{1}{\\kappa_3^2}+\\frac{1}{\\kappa_4^3}.\n\\end{align*}\nCombining the above with \\cref{normal} and setting $t= t_n+ 3s_n + 1$ gives\n\\begin{align*}\n\\TV{\\mu_{X_t}- \\mu_{Y_t} }& \\leq \\mathbf{P} ( |X_t - Y_t | \\geq 1 ) \\\\\n&= \\mathbf{P} ( |X_t - Y_t | \\geq 1, \\, \\tau_4 > t_n + 3s_n ) + \\mathbf{P} ( |X_t - Y_t | \\geq 1, \\, \\tau_4 \\leq t_n + 3s_n )\\\\\n& \\lesssim \\frac{1}{\\kappa_1^2} + \\frac{1}{\\kappa_3^2}+ \\frac{1}{\\kappa_4^2} +  \\mathbf{P} ( |X_t - Y_t | \\geq 1, \\, \\tau_4 \\leq t_n + 3s_n )\\\\\n&\\leq \\frac{1}{\\kappa_1^2} + \\frac{1}{\\kappa_3^2}+ \\frac{1}{\\kappa_4^2} + \\mathbf{E} \\mathbf{P} ( |X_1 - Y_1| \\geq 1 \\, | \\, X_{\\tau_4}, Y_{\\tau_4} )\\\\\n&=\\frac{1}{\\kappa_1^2} + \\frac{1}{\\kappa_3^2}+ \\frac{1}{\\kappa_4^2} + o(1). \\end{align*}\nNote that picking the $\\kappa_i$'s and $n$ large enough finishes the proof. \n\\end{proof}\n\n\n\n\\section{Proof of the auxiliary results}\n\\label{sec:aux}\nIn this section, we prove \\cref{lem:1} and \\cref{lem:2}.   \n\n\\begin{proof}[Proof of~\\cref{lem:1}]\nLet $(\\mathcal{F}_t)_{t\\geq 0}$ denote a filtration of $\\sigma$-fields to which $(X_t)$ is adapted. We start with the proof of~\\eqref{id:1} and~\\eqref{id:2}.  These identities are contained in the proof of \\cite[Lemma 3]{EN}, but we provide the details for completeness.  For~\\eqref{id:1}, since $f_1(x)$ is an eigenfunction with eigenvalue $f_1(k)=1-2k\/n$ (cf.~\\cite{DS_87, EN}), we obtain for $t\\geq 1$\n\\begin{align*}\n\\mathbf{E}_{X_0} f_1(X_t) = \\mathbf{E}_{X_0} \\mathbf{E}_{X_0}[ f_1(X_t) | \\mathcal{F}_{t-1}] = \\mathbf{E}_{X_0} \\mathbf{E}_{X_{t-1}}[ f_1(X_1)] = (1-\\tfrac{2k}{n}) \\mathbf{E}_{X_0} f_1(X_{t-1}).  \n\\end{align*}   \nRepeating the process above produces~\\eqref{id:1}.  In order to obtain~\\eqref{id:2}, by a direct calculation it follows that\n\\begin{align*}\nf_1(x)^2= \\frac{1}{2n-1} + \\frac{2n-2}{2n-1} f_2(x).  \n\\end{align*}\nThus following the same eigenvalue procedure as above but $f_2$ playing the role of $f_1$ we obtain~\\eqref{id:2}.  \n\nIn order to establish~\\eqref{eqn:f2as}, let $q\\geq 0$ and $X_0= \\tfrac{n}{2}+ \\zeta$ where $\\zeta$ is $O(\\sqrt{n} (\\log n)^{q\/2})$.  Then\n\\begin{align*}\nf_2(X_0) &= \\bigg(1+ \\frac{2n-1}{2n-2} - \\frac{2n-1}{n}\\bigg) + \\zeta\\bigg(\\frac{2(2n-1)}{n(n-1)}- \\frac{2(2n-1)}{n^2} \\bigg) + \\zeta^2\\frac{2(2n-1)}{n^2(n-1)} -  \\frac{2n-1}{n(n-1)}\\\\\n&\\qquad \\qquad - \\zeta \\frac{2(2n-1)}{n^2(n-1)}\\\\\n&= \\zeta^2\\frac{2(2n-1)}{n^2(n-1)} +O(n^{-1}) = O(n^{-1} (\\log n)^q),\n\\end{align*}  \nas claimed.  \n\\end{proof}\n\n\n\\begin{proof}[Proof of~\\cref{lem:2}]\nWe first prove~\\eqref{as:id1}.  Letting $g: (-\\infty, 1\/2)\\rightarrow \\mathbf{R}$ be given by $g(x)= \\log (1-2x)$, observe that \n\\begin{align}\n\\label{eqn:as1}\nf_1(k)^t = (1-2\\lambda)^t \\frac{(1-2k\/n)^t}{(1-2\\lambda)^t} &=(1-2\\lambda)^t \\exp ( t[g(k\/n) - g(\\lambda)])\\\\\n\\nonumber &=: (1-2\\lambda)^t \\exp( t \\Delta(g)).  \n\\end{align} \nTaylor's formula then implies\n\\begin{align*}\n\\Delta(g)=g(k\/n) -g(\\lambda) = -\\frac{2\\Delta_n}{1-2\\lambda} + \\Delta_n^2 \\frac{g''(\\xi)}{2} \n\\end{align*} \nfor some $\\xi $ in between $k\/n$ and $\\lambda$.  Employing \\cref{assump:1} (c1) then gives\n\\begin{align*}\n \\Delta_n^2\\frac{g''(\\xi)}{2!} = O( \\Delta_n^2) .\n \\end{align*} \n Plugging this into~\\eqref{eqn:as1} using $t\\leq t_n$ and \\cref{assump:1}(c2) then implies\n \\begin{align*}\nf_1(k)^t &= (1-2\\lambda)^t \\sum_{j=0}^\\infty \\frac{(t \\Delta(g))^j}{j!}\\\\\n &= (1-2\\lambda)^t(1- 2t \\Delta_n\/(1-2\\lambda) + O(t^2 \\Delta_n^2)),\n \\end{align*}\n which is~\\eqref{as:id1}.  \n \n In order to obtain~\\eqref{as:id2}, we follow a similar reasoning to the one used to arrive at~\\eqref{as:id1} to see that\n \\begin{align*}\n f_2(k)^t = (1-2\\lambda)^{2t} \\frac{f_2(k)^t}{(1-2\\lambda)^{2t}} = (1- 2\\lambda)^{2t} \\exp\\{t (g(h_1(k))- g(h_2(\\lambda )))\\}. \n \\end{align*}   \nA short calculation shows that\n \\begin{align*}\n \\Delta_n'= h_1(k)-h_2(\\lambda) =2 \\Delta_n (1-\\tfrac{k}{n}- \\lambda) +O(1\/n),  \n \\end{align*} \n yielding the first part of~\\eqref{eqn:dtil}.  Following the same reasoning as above yields~\\eqref{as:id2}. \n \n The proof of~\\eqref{as:id3} and the second part of~\\eqref{eqn:dtil} are similar, so we omit the details.  \n \n\\end{proof}\n\n\n\n\n\n\\bibliographystyle{plainurl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\adjincludegraphics[Clip=0 0 0 {0.404\\height}, width=0.8\\linewidth]{images\/detector.jpg}\n\t\\caption{The central detector of JUNO. An acrylic sphere with a diameter of 35.4 meters filled with 20 kt of liquid scintillator. The detector contains \\num{17612} large (20 inches) PMTs and \\num{25600} small PMTs (3 inches).}\n\t\\label{fig:detector}\n\\end{figure}\n\nJUNO is a neutrino observatory under construction in southern China. Its physical program covers a wide range of problems~\n\\cite{JUNO}. The main goals are to determine the neutrino mass ordering and to accurately measure the parameters of neutrino oscillations $\\sin^2{\\theta_{12}}, \\Delta m_{21}^2, \\Delta m^{2}_{31}$. JUNO will detect reactor neutrinos from the Yangjiang and Taishan nuclear power plants. Simultaneously JUNO will be able to observe neutrinos from supernovae, atmospheric neutrinos, solar neutrinos and geoneutrinos.\n\nFigure~\\ref{fig:detector} shows the detector design. The detector is a transparent acrylic sphere with a diameter of 35.4 meters that is located underground in a cylindrical water pool. The sphere is filled with 20 kt of liquid scintillator. The detector is equipped with a huge number of photo-multiplier tubes (PMTs) of two types: \\num{17612} large PMTs (20 inches) and \\num{25600} small PMTs (3 inches). Neutrinos, which are produced in nuclear reactors, interact with the protons of the scintillator in the detector via the inverse beta-decay (IBD) channel: $\\overline \\nu_{e} + p \\rightarrow e^{+} + n$. The scintillator then produces visible light upon the interaction of the ejected positron with the media. The amount of emitted photons is tightly related to the neutrino energy. The neutron, after some time, is captured by a hydrogen atom of liquid scintillator, producing 2.2 MeV de-excitation gammas. Thus, the time coincidence of signals from the positron and the neutron makes it possible to separate the event from backgrounds. The information collected by PMTs is used for estimation of the neutrino energy.\n\nTo resolve the neutrino mass ordering the energy resolution must be $\\sigma \\leqslant  3\\%$ at 1 MeV, which is very close to the statistical limit corresponding to the light yield in JUNO, about 1300 detected photons (hits) at 1~MeV. The energy nonlinearity uncertainty should be < 1\\%~\\cite{JUNO}.\n\nMachine Learning (ML) methods are very popular in science today, including high energy physics, in particular, neutrino experiments~\\cite{Psihas} and collider experiments~\\cite{Guest}. We use ML approach for energy reconstruction in the JUNO experiment. Our problem is a regression supervised learning problem. The data (time and charge information) collected by PMTs is used as input for supervised training of ML model. Earlier we demonstrated that the ML approach can have the quality required for the JUNO experiment on our data and also has the advantage of speed and ease of application~\\cite{CNNs}.\n\nIn this work we use Boosted Decision Trees (BDT)~\\cite{Friedman} for energy reconstruction in the energy range of 0\u201310 MeV covering the region of interest for IBD events from reactor electron antineutrinos. Compared to~\\cite{CNNs} we designed and studied new features and achieved much better resolution with BDT, which is now comparable to the resolution of more complex models.\n\n\\section{Data description}\n\\label{DataDesc}\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[width=0.49\\textwidth]{images\/event_vis.pdf}\\label{fig:data_vis_ch}}\n\t\\subfloat[]{\\includegraphics[width=0.49\\textwidth]{images\/event_vis_ht.pdf}\\label{fig:data_vis_ht}}\n\t\\caption{Example of an event seen by 17612 large ($20''$) PMTs for a positron event of 5.5 MeV. Only fired PMTs are shown. In fig (a) color represents the accumulated charge in PMTs: yellow points show the channels with more hits, red points \u2014 the channels with fewer hits. In fig (b) color indicates PMT activation time --- darker blue color shows an earlier activation. The primary vertex is shown by the gray sphere.}\n\t\\label{fig:data_vis}\n\\end{figure}\n\nThe dataset is generated by the full detector Monte Carlo method using the official JUNO software \\cite{Huang}. The detector simulation is based on the Geant4 framework~\\cite{Allison} with the geometry implemented in detail. The train and test datasets are described as follows:\n\\begin{enumerate}\n\t\\item \\textbf{Training dataset} consists of 5 million events, uniformly distributed in kinetic energy from 0 to 10 MeV and in the volume of the central detector (in liquid scintillator).\n\t\n\t\\item \\textbf{Testing dataset} consists of subsets with discrete kinetic energies of 0 MeV, 0.1 MeV, 0.2 MeV, ..., 1 MeV, 2 MeV, ..., 10 MeV. Each subset contains about 10 thousand events. This dataset is used to estimate performance after the end of training.\n\\end{enumerate}\nOur data have four configurations: 1) without electronics effects; \n2) taking into account the transit time spread (TTS) of PMTs;\n3) taking into account the dark noise (DN) of PMTs; and 4) taking into account both effects.\nTTS occurs due to the stochasticity of the path of photo-electron from the photo-cathode to the anode and effectively smears the time information. DN effect gives spontaneous hits on PMTs. In further TTS and DN are always enabled if not specified otherwise.\n\nFigure~\\ref{fig:data_vis} illustrates an example of accumulated charge in the PMT channels (left) and the evolution in time of the same signal in terms of the first hit time distribution (right).\n\n\\section{Boosted Decision Trees}\n\\label{BDT}\n\nBDT is the an ensemble model, where a simple and quickly learning Decision Tree (DT) model is used as the base algorithm. DTs in BDT are trained sequentially. Each subsequent DT is trained to correct errors of previous DTs in the ensemble. \nIn this work we use the XGBRegressor implementation of BDT from the XGBoost library~\\cite{XGB}.\n\nDT is built recursively starting from the root node, splitting the source set into two subsets (left and right) based on the values of the input features. To build a tree, we need a principle based on which we will split the original set of objects into subsets. XGBRegressor uses Gain maximization to splitting input data into subsets.\n\nIn XGBoost the objective function contains a two parts: the training loss and the regularization term: \n\\begin{equation}\\label{eq:obj}\n\\mathcal{L}(\\phi) = \\sum_{i}{l \\left(\\hat y_{i}, y_{i} \\right)} + \\sum_{k} {\\Omega (f_k)}\n\\end{equation}\nA tree is penalized if the sum of the norm of values in its leaves is very large. Therefore, the regularization term is introduced here as follows:\n\\begin{equation}\n\\Omega(f) = \\gamma T + \\frac{1}{2}\\lambda \\sum^{T}_{j=0}{\\omega^2_j},\n\\end{equation}\nwhere $T$ is the number of leaves, $\\omega_j$ are values in the leaves, $\\gamma$ and $\\lambda$ are numerical parameters of the regularization. In~\\cite{XGB} the authors showed that the optimization of an objective function \\eqref{eq:obj} reduces to maximizing of the Gain. And Gain is defined as:\n\\begin{equation}\n{\\rm Gain} = \\frac{1}{2}\\left[ \\frac{G_l^2}{H_l^2 + \\lambda} + \\frac{G_r^2}{H_r^2 + \\lambda} - \\frac{(G_l + G_r)^2}{H^2_l + H^2_l + \\lambda}\\right]  - \\gamma,\n\\end{equation}\nwhere $G$, $H$ are the corresponding sums of the first and second derivatives of the objective function for a given partition and the indices $l$ and $r$ mean the left and right partition.\n\n\\section{Feature Engineering}\n\\label{FeatEng}\n\nThe basic features for the energy reconstruction are the following aggregated features: \n\\begin{enumerate}\n\t\\item Total number of detected photo-electrons (hits): \\texttt{nHits}. \n\t\n\tIn the first approximation, the total number of hits is proportional to the event energy.\n\t\\item Coordinate components of the center of charge:\n\t\\begin{equation}\n\t(x_{cc},\\ y_{cc},\\ z_{cc}) = \\mathbf{r}_{\\rm cc} = \\frac{\\sum_i^{N_{\\rm PMTs}} \\mathbf{r}_{{\\rm PMT}_i} n_{{\\rm p.e.}, i}}{\\sum_i^{N_{\\rm PMTs}}n_{{\\rm p.e.}, i}},\n\t\\end{equation}\n\tand its radial component: \n\t\\begin{equation}\n\tR_{cc} = \\sqrt{x_{cc}^2 + y_{cc}^2 + z_{cc}^2}.\n\t\\end{equation}\n\t\n\tCoordinate components of the center of charge are rough approximations of the location of the energy deposition. These features are important for energy reconstruction since the number of hits depends on the location of the energy deposition.\n\t\\item Coordinate components of the center of first hit time:\n\t\\begin{equation}\n\t(x_{cht},\\ y_{cht},\\ z_{cht}) = \\mathbf{r}_{\\rm cht} = \\frac{1}{\\sum_i^{N_{\\rm PMTs}} \\frac{1} {t_{{\\rm ht},i} + c}} \\sum_i^{N_{\\rm PMTs}} \\frac{\\mathbf{r}_{{\\rm PMT}_i}} {t_{{\\rm ht},i} + c},\n\t\\end{equation}\n\tand its radial component: \n\t\\begin{equation}\n\t    R_{cht} = \\sqrt{x_{cht}^2 + y_{cht}^2 + z_{cht}^2}.\n\t\\end{equation}\n\tHere the constant $c$ is required to avoid division by zero. \n\tThese features bring extra information on the location of the energy deposition.\n\t\\item Mean and standard deviation of the first hit time distributions: \\texttt{ht\\_mean}, \\texttt{ht\\_std}.\n\\end{enumerate}\nFor ML models including Boosted Decision Trees, it is often useful to engineer new features from the existing features~\\cite{Heaton}. We use the following extra synthetic features:\n\\begin{gather}\n\t\\gamma_{z}^{cc} = \\frac{z_{cc}}{\\sqrt{x_{cc}^2 + y_{cc}^2}},\\\n\t\\gamma_{y}^{cc} = \\frac{y_{cc}}{\\sqrt{x_{cc}^2 + z_{cc}^2}},\\ \n\t\\gamma_{x}^{cc} = \\frac{x_{cc}}{\\sqrt{z_{cc}^2 + y_{cc}^2}}; \\\\ \n\t\\theta_{cc} = \\arctan{\\frac{\\sqrt{x_{cc}^2 + y_{cc}^2}}{z_{cc}}},\\ \n\t\\phi_{cc} = \\arctan{\\frac{y_{cc}}{x_{cc}}}; \\\\\n\tJ_{cc} = R_{cc}^2 \\cdot \\sin{\\theta_{cc}},\\  \n\t\\rho_{cc} = \\sqrt{x_{cc}^2 + y_{cc}^2}.\n\\end{gather}\nAnd some trigonometric functions of angles $\\theta_{cc}$, $\\phi_{cc}$: $ \\sin{\\theta_{cc}},\\ \\cos{\\theta_{cc}},\\ \\sin{\\phi_{cc}},\\ \\cos{\\phi_{cc}}$. We also use 11 similar features for the center of first hit time.\n\nIn addition, we prepare five more features related to the location of the PMT received the maximum number of photo-electrons: \\texttt{x\\_max}, \\texttt{y\\_max}, \\texttt{z\\_max}, \\texttt{theta\\_max}, \\texttt{phi\\_max}, the maximum number of photons on PMT \\texttt{npe\\_max}, and the average number of photons on PMTs \\texttt{npe\\_mean}. \n\nAlso we added the following features: \\texttt{entries1}, \\texttt{entries2}. Here, \\texttt{entries1} is the percentage of PMTs with only 1~hit, \\texttt{entries2} --- with 2~hits. And the one more feature is \\texttt{nPMTs} --- the total number of fired PMTs.\n\nNow let's take a closer look at the first hit time distribution. Consider what fraction of fired PMTs received at least one photon depending on time, which is, in essence, the cumulative distribution function (CDF) of the first hit time distribution. Figure~\\ref{fig:cdf_and_pdf_ht} illustrates an example of a 7~MeV event. The entire event typically lasts for about \\num{1000}~ns, but the majority of photo-electrons are recorded by the PMTs in the first hundred nanoseconds and then mainly dark hits are recorded. \n\nIn Figure~\\ref{fig:cdf_and_pdf_ht} one also can see that at the beginning of events there is a short period of time $\\Delta t$ during which the photons have not reached the PMTs and only dark noise is recorded.\n\nFigure~\\ref{fig:cdf_and_pdf_ht_diff_R} shows how the CDFs for the events with the energy of 7~MeV change depending on their location, closer to the edge (large R) or closer to the center of the detector (small R).\nIn the case where R is quite small, it takes more time for the photons to be detected by PMTs and also it takes more time to ``saturate''.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\subfloat[Example for a specific R.]{\\includegraphics[width=0.49\\textwidth]{images\/cdf_and_pdf_ht.pdf}\\label{fig:cdf_and_pdf_ht}}\n\t\\subfloat[Example for different R.]{\\includegraphics[width=0.49\\textwidth]{images\/cdf_and_pdf_ht_diff_R.pdf}\\label{fig:cdf_and_pdf_ht_diff_R}}\n\t\\caption{Examples of CDFs and PDFs (bottom right) of the first hit time distribution for events with energy equal to 7~MeV.}\n\t\\label{fig:ht_profiles}\n\\end{figure}\n\nFinally, the idea is to simply decompose the entire curve into a set of percentiles, and then select those that suit best for energy reconstruction. X\\%-percentile indicates how long it takes to register X\\% of the first PMT hits. We use the following set of percentiles: \\{1\\%, 2\\%, ..., 10\\%, 15\\%, ..., 90\\%, 91\\%, ..., 99\\%\\}.\n\n\\section{Selection of event time window}\n\\label{EventTimeWinSel}\n\nOne can also see in Figure~\\ref{fig:ht_profiles} that the signal hits arrive within the first few hundred nanoseconds, while the dark hits form quasi-constant pedestal. Therefore, a time window can be selected, based on the data, in a way to contain mainly signal events. For this purpose we trained Boosted Decision Trees model with different bounds of window: \\{75ns, 125ns, 175ns, 250ns, 500ns, 750ns, 1500ns\\} and always started from $t=0$. It was trained on a 200k dataset and using all new features. \n\nTable~\\ref{tab:bounds_results} shows the Mean Absolute Error (MAE), the Root Mean Squared Error (RMSE), Mean Absolute Percentage Error (MAPE) on the test dataset for different window bounds. The best one was found to be 500~ns, however all the windows longer than 175~ns showed similar performance. \n\n\\begin{table}[!htb]\n\t\\centering\n\t\\caption{MAE, RMSE, MAPE metrics for BDT models for different window bounds.}\n\t\\label{tab:bounds_results}\n\t\\begin{tabular}{llll}\n\t\t\\hline\n\t\tBound, ns& MAE, MeV & RMSE, MeV & MAPE, \\% \\\\\n\t\t\\hline\n\t\t75 & 0.1019 & 0.1664 & 3.705 \\\\\n\t\t125 & 0.0562 & 0.0781 & 1.858 \\\\\n\t\t175 & 0.0505 & 0.0698 & 1.662 \\\\\n\t\t250 & 0.0487 & 0.0676 & 1.594 \\\\\n\t\t\\textbf{500} & \\textbf{0.0477} & \\textbf{0.0660} & \\textbf{1.569} \\\\\n\t\t750 & 0.0480 & 0.0664 & 1.585 \\\\\n\t\t1500 & 0.0479 & 0.0662 & 1.589 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\section{Feature selection}\n\\label{FeatSel}\n\nFinally, we have a large set of features, but many of them are highly correlated, so we expect that a small set of them contain all the information and provide a performance close to the best possible one. Thus, the next task is to get a subset of the most informative features from the available set of all 78 features. For this purpose we use a dataset with 1M events.\n\nTo select the most informative features, we use the following algorithm. First, we train a model on all features and computed RMSE on the validation dataset with 150k events. Then we take an empty list and start populating it with features. On each step we pick a feature which provide the best improvement of the model in terms of RMSE calculated on the validation dataset and put it to the end of the list. We continue while this RMSE value differs from the RMSE value for the model trained on all features by more than $\\varepsilon$, chosen to be 0.0002. This procedure results to the following set of features (sorted by importance): \\[\\texttt{nHits},\\  \\texttt{ht\\_20p},\\ \\texttt{jacob\\_cc},\\ \\texttt{ht\\_2p},\\ \\texttt{ht\\_35p},\\ \\texttt{R\\_cc},\\ \\texttt{ht\\_75p}\\]\nNot surprisingly, for the energy reconstruction, the most informative feature is the total number of hits \\texttt{nHits}, because its strongly correlated with energy, but at the same time it is hard to interpret the order of the rest features. The subset of the selected features contains \\texttt{jacob\\_cc} and \\texttt{R\\_cc}, which bring the spatial information allowing to recover the non-uniformity of detector response. \nWe checked simpler combinations of features for the center of charge position. It turned out that the combination of \\texttt{rho\\_cc} and \\texttt{R\\_cc} gives the same result as \\texttt{jacob\\_cc} and \\texttt{R\\_cc}, so we have chosen them as they are more intuitive. Our final set of features is: \\[\\texttt{nHits},\\  \\texttt{ht\\_20p},\\ \\texttt{rho\\_cc},\\ \\texttt{ht\\_2p},\\ \\texttt{ht\\_35p},\\ \\texttt{R\\_cc},\\ \\texttt{ht\\_75p}\\]\nFigure~\\ref{fig:percentiles_vis} illustrates the selected percentiles of the CDFs for an event with energy of 7~MeV and for different radial positions. As one can see \\texttt{ht\\_2p} contains information about the beginning of the event, that is about the moment when the number of photons PMT hits begins to grow sharply. The remaining percentiles contain information about the shape of the CDF curve and help us to separate one curve from another. The 75\\% percentile is close to the moment of ``saturation'' of the CDF curve.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{images\/percentiles_vis.pdf}\n\t\\caption{The selected percentiles of the CDFs of the first hit time distributions for a 7~MeV event for different radial positions: 2\\% (top left), 20\\% (top right), 35\\% (bottom left) and 75\\% (bottom right).}\n\t\\label{fig:percentiles_vis}\n\\end{figure} \n\n\\section{Results}\n\\label{Res}\n\nTo evaluate the quality of the model, we use two metrics: resolution and bias. These metrics are obtained as a result of the Gaussian fit of $E_{\\rm pred} - E_{\\rm true}$ distribution. The resolution is defined as $\\sigma \/ E_{\\rm true}$ and the bias --- as $\\mu \/ E_{\\rm true}$, where $\\sigma$ and $\\mu$ are the standard deviation and the mean of the Gaussian distribution respectively. The performance is shown dependent on the so called visible energy, i.e.\\ the maximal energy that can be converted into light: $E_{\\rm vis} = E_{\\rm kin} + 1.022 \\text{ MeV}$. This procedure is described in more details in~\\cite{CNNs}.\n\nFigure~\\ref{fig:res_and_bias_diff_sizes_of_datasets} illustrates the results for BDT models trained on datasets that contain different amount of events: 100k, 1M, 5M. We obtained that 1M events can provide the best possible accuracy of the model, providing only a little improvement compared to 100k events. This illustrates that fast learning is one of the advantages of the BDT model: one can get an acceptable quality already on a relatively small number of events in the dataset.\n\nFigure~\\ref{fig:res_and_bias_diff_options} shows a comparison for the BDT model trained on the 5M dataset for different options: without TTS \\& DN, with TTS only, with DN only, with TTS \\& DN. One can see that DN worsens the resolution, TTS --- almost does not.\n\nA comparison of the BDT model with other more complex deep learning models (ResNet, VGG, and GNN)~\\cite{CNNs} is shown in Figure~\\ref{fig:res_and_bias_all_models}. All the models are trained on the dataset with 5M events. We can see that the performance of the energy reconstruction with BDT model is practically similar to the complex deep learning models. At the same time the computations required for training and prediction are much faster due to the minimalistic nature of BDT.\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\subfloat[Comparision of different sizes of datasets.]{\\includegraphics[width=0.48\\textwidth]{images\/res_and_bias_diff_sizes_of_datasets}\\label{fig:res_and_bias_diff_sizes_of_datasets}}\n    \\hspace{1em\n\t\\subfloat[Comparison of different TTS \\& DN options for a dataset with 5M events.]{\\includegraphics[width=0.48\\textwidth]{images\/res_and_bias_diff_options}\\label{fig:res_and_bias_diff_options}}\n\t\\caption{Results of the energy reconstruction for the BDT model: resolution (upper panel) and bias (lower panel). Note that the first point corresponds to 1.122 MeV.}\n\t\\label{fig:results_bdt}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=0.7\\textwidth]{images\/res_and_bias_all_models}\n\t\\caption{Energy reconstruction performance: resolution (upper panel) and bias (lower panel) obtained with BDT, ResNet-J, VGG-J and GNN-J models. The plots are offset along X-axis within $\\pm$0.06 MeV for better readability. Note that the first point corresponds to 1.122 MeV.}\n\t\\label{fig:res_and_bias_all_models}\n\\end{figure} \n\n\\section{Summary}\n\\label{Sum}\n\nIn this work we have presented the use of Boosted Decision Trees for energy reconstruction in the JUNO experiment in the relevant energy range. We have designed and investigated a large set of features and have selected a small subset providing the performance nearly equal the one obtained with the full set of features. Using such a minimalistic and fast model as BDT we achieved a performance similar to the one of more complex models like ResNet, VGG, GNN.\n\n\\section*{Acknowledgements}\n\\begin{acknowledgement}\nWe are immensely grateful to Yury Malyshkin for his invaluable contribution to this work. We would like to thank Weidong Li, Jiaheng Zou, Tao Lin, Ziyan Deng, Guofu Cao and Miao Yu for their tremendous contribution to the development of JUNO offline software and to Xiaomei Zhang and Jo\\~ao Pedro Athayde Marcondes de Andr\\'e for production of the MC samples. We are grateful to N.~Kutovskiy, N.~Balashov for providing an extensive IT support and computing resources of JINR cloud services~\\cite{Baranov}.\nFedor Ratnikov is supported by the Russian Science Foundation under grant agreement \\textnumero{17-72-20127}.\n\\end{acknowledgement}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nBlood is a multi-component fluid primarily consisting of dispersed red blood cells (RBCs), white blood cells (WBCs) and platelets, in plasma. Normal levels of the volume fraction of RBCs (hematocrit) range from 35$\\%$ to 50$\\%$ \\citep{reinhart2016optimum}. Consequently, the rheological behaviour of RBCs strongly affect blood properties which significantly change when the hematocrit level varies for instance in hemodilution-increased plasma\/anemia-reduced RBC \\citep{nader2019blood}. Particle aggregation strongly depends on hematocrit and can lead to large fluctuations of shear rate locally which effects blood viscosity \\citep{mehri2018red}. This non-Newtonian shear-thinning rheological behaviour of blood can have a significant impact on blood flow \\cite{yin2013multiple}. This manifests that a single fluid approach with uniform properties for numerically investigating cardiovascular diseases such as atherosclerosis might fail to model the underpinning phenomena due to particulate motion.\n\nA multiphase computational fluid dynamics (CFD) model of blood flow can be used for the accurate resolution of locally varying distribution of RBC within a vessel and the haemodynamic consequences of such build up. The majority of continuum multiphase CFD studies for multi-component blood flow modelling are based on Eulerian-Eulerian approaches where a simple transport equation for the volume fraction of each phase is solved. These Eulerian approaches offer the opportunity of calculating the forces exerted on the fluids by solving for momentum for each phase allowing for studying inter-phase mass and momentum transfer. \nSome studies choose to couple the Eulerian-Eulerian framework with implicit methods for interface capturing, leading to a more accurate calculation of the volume fraction for each phase \\cite{kim2016computational}. These are usually based on the volume of fluid (VOF) method \\cite{Hirt1981}. \nVOF is one of the most commonly used implicit methods which combines simplicity and mass conservation by default. Given a sufficient grid resolution, the VOF approach models the larger scales and bulk fluid-fluid interfaces or for cases where interdispersion between two fluids can be neglected \\cite{Prosperetti2009}.\nThe correct calculation of the interface curvature and conservation of mass for the RBC component while maintaining numerical stability in regions where volume fraction exhibits sharp changes are extremely important for accurate multiphase CFD models for numerically investigating the behaviour of RBC and plasma in cardiovascular diseases such as atherosclerosis.\n{\\color{black} Increasing the accuracy of interface capturing for the larger scales is a key point for increasing overall accuracy of the multiphase analysis. \nFor addressing these challenges here, we deploy more advanced implicit methods rather than the traditional VOF, while guaranteeing that mass is conserved.}\n\nIn this paper we propose an improved numerical approach for the calculation of the volume fraction of each phase for modelling the RBC distribution. For this purpose, we employ a coupled VOF and level set method that we have recently developed and validated for different multiphase flow cases in \\cite{LyrasLee2021a} and has been proven to give a more accurate interface description compared to other state-of-the-art VOF-based approaches.\nBlood is modelled as a mixture and the interface of the two phases is captured using one of the most attractive features of the level set e.g a sharp interface representation. The formulation leads to a multi-scale approach that combines the advantages of both level set for the larger scales, and Eulerian-Eulerian two-fluid approach which can be used to model the characteristics of the mixture into the under-resolved (sub-grid) scales.  \n\nThe presented methodology is tested for the study of partially occluded arteries due to a stenotic obstruction of eccentric shape. Although these lesions account for the majority of lesions among patient population \\cite{hausmann1994lumen, waller1989eccentric} they are given little attention in literature from numerical works. Contrary to symmetric stenosis models, asymmetry not only offers a more realistic representation of diseased arteries, but the generated secondary flow patterns alter the flow downstream at the end of the stenosis, resulting in a fundamentally different haemodynamics compared to the idealised axisymmetric stenoses \\cite{kaazempur2005mass}.\n{\\color{black} In addition to eccentricity, the length of a lesion, $L_{s}$ is another important factor that also affects blood flow \\cite{LyrasLee2021c}}. Although some studies for discrete lesions with $L_{s} \\le 10mm$ --type A in the American College of Cardiology\/American Heart Association (ACC\/AHA) classification of coronary lesions, have been studied \\cite{gholipour2018, zhao2021}, there are no detailed studies for tubular coronary lesions (type B, with $ 10mm \\le L_{s} \\le 20mm$) which involve a moderate risk, higher than in discrete lesions \\cite{kushner2009}.\nTo the best knowledge of the authors, the effects of hematocrit and stenosis on the RBCs transport and its effect on the blood flow in tubular coronary lesions (with $ 10mm \\le L_{s} \\le 20mm$, $L_{s}$= lesion length) have not been previously reported elaborately. \nIn this study, the blood flow patterns and particulate aggregation {\\color{black}in coronary arteries with a tubular lesion} are numerically studied using the developed multiphase solver.\n\n{\\color{black}Owing to the complexity of the multiphase flow phenomena and their dependence on the lesion geometry, apart from the eccentricity and length of a lesion, the hematocrit is also studied for investigating its role on the local haemodynamic parameters such as the velocity and the WSS on the endothelial cell membrane in the coronary artery.}\nAlthough the effects of the hematocrit in the viscosity have been previously studied \\cite{benis1970effect, ccinar1999effect}, many open questions still remain regarding the local haemodynamic factors, such as the wall shear stress, red blood cell buildup and velocity \\cite{jung2006hemodynamic, kim2008multiphase, baskurt2011red, lucker2015dynamic, ling2021numerical}. These important parameters which vary alongside the curved vessel and during the cardiac cycle \\citep{Thim2012, Han2016} are the primary focus of this research here.\nFirst, the impact of hematocrit and stenosis severity on the calculated pressure drop $\\Delta p$ is studied.  \nCalculating $\\Delta p$ along a lesion is extremely important since it is typically used for characterising the significance of the stenosis through the calculation of Fractional Flow Reserve (FFR). FFR is a haemodynamic index, based on invasive measurement of the pressure ratio distal and proximal to a given lesion which is employed for guiding revascularisation.  \nSince in the particular case of tubular lesions, there are no numerical studies that evaluate the effects of the RBC levels on $\\Delta p$, the velocity and red blood cell distribution, we include some tests together with comparisons with single phase and reduced order models.\n\nComparisons with single-phase flow demonstrate the differences in pressure when considering the mixture.\nSingle-phase simulations over-predict the pressure drop for the same stenosis severity compared to the multiphase simulations. The model here generally predicted higher RBC buildup in the area of maximum curvature and the end of the stenosis compared to the upstream location.\nWSS region was found to vary with respect to the local RBC buildup, in the area of maximum curvature of the vessel. Secondary flow patterns and vessel geometry significantly influence the RBC buildup, primarily because of the prolonged particulate residence times. \nThese observations could have significant implications towards better understanding of the haemodynamic parameters that might have an impact on the development of cardiovascular diseases such as atherosclerosis in tubular coronary lesions.\n \n\\section{Materials and Methods}\n\\label{sec:Euler-Euler framework}\nThe open-source code OpenFOAM \\cite{weller1998} is used for the simulations presented. The numerical implementation here has used as its basis the solver multiphaseEulerFoam solver \\cite{wardle2013hybrid} and enables simultaneous modelling of complex multiphase flows with different flow regimes (i.e. both dispersed and segregated flows). \nThe solver uses a coupled level set (LS) and VOF interface capturing method to model the large fluid-fluid interfaces, whereas the Euler multi-fluid approach is used for modelling the dispersed flow. The two approaches are coupled by a simplified switching term, which is defined for each phase pair \\cite{wardle2013hybrid}. \nThe solver is a combination of two multiphase solvers i.e. a coupled VOF-LS and an Eulerian-Eulerian solver with additional sub-models to model the forces involved.  \nIn addition, the solver includes models for phase change (e.g. cavitation) which is deactivated here.  \nAlthough the methodology can be used to simulate multiple fluids, here we choose to present it with two fluids. This is an Eulerian-Eulerian two-fluid model which regards RBCs as suspended particles dispersed in plasma, which has been shown that it accurately describes the blood flow \\cite{yin2013multiple}. \nFollowing other researchers, the effect of other components such as platelets was considered here to be insignificant, since they exhibit volume fraction values which are less than 1$\\%$     \n\\cite{jung2006hemodynamic}.\nThe multi-fluid model equations for incompressible, isothermal flow are given by a set of mass and momentum equations for each phase k,\n\n\\begin{equation}\n\\frac{\\partial (\\rho_k \\alpha_k u_k)}{\\partial t}+ (\\rho_k \\alpha_k u_k) \\cdot \\nabla(u_k)=-\\alpha_k \\nabla p + \\nabla \\cdot (\\mu_k \\alpha_k \\nabla u_k) + \\rho_k \\alpha_k \\mathbf{g} + \\mathbf{F}_{D,k}+ \\mathbf{F}_{s,k} + \\mathbf{F}_{vm}\n\\label{momentumEqn}\n\\end{equation}\nwhere the density, phase fraction, and velocity for phase $k$ are given by $\\rho_k$, $\\alpha$, $u_k$. The $\\rho_k \\alpha_k \\mathbf{g}$ is the gravitational force and the other two interfacial forces in Eq.\\ref{momentumEqn} are the drag force $\\mathbf{F}_{D,k}$ and the surface tension force $\\mathbf{F}_{s,k}$. For selected phase pairs where the VOF method is used to resolve the interface, an additional compression term is added to their phase transport equation. The method here includes the level set method for capturing the interface of each pair of fluids.\nThe drag term $\\mathbf{F}_{D,k}$ is given by \n\\begin{equation}\n\\mathbf{F}_{D,k} = \\frac{3}{4}\\rho_c \\alpha_c \\alpha_d C_D \\frac{|\\mathbf{u}_d - \\mathbf{u}_c|(\\mathbf{u}_d - \\mathbf{u}_c)}{d_d}= \\alpha_c \\alpha_d K (u_d - u_c)\n\\end{equation}\nwhere the subscripts $c$ and $d$ denote the continuous (plasma) and dispersed phase (RBC) values respectively and $K$ is \n\\begin{equation}\nK = \\frac{3}{4} \\rho_c C_D\\frac{|\\mathbf{u}_d - \\mathbf{u}_c|}{d_d}\n\\end{equation} \nAmong the various models for calculating the drag coefficient $C_D$ available in OpenFOAM, the commonly used model of Schiller and Naumann \\cite{schiller1933drag} is employed here. In this model, the drag coefficient is a function of the Reynolds number $Re$ according to\n\\begin{equation}\nC_D = \n\\begin{cases}\n\\frac{24(1+0.15Re^{0.683})}{Re} & \\text{, } Re \\leq 1000 \\\\\n0.44 & \\text{, } Re > 1000  \n\\end{cases}\n\\end{equation}\nwhere the Reynolds number is calculated based on the relative velocity of the two phases, the dispersed phase diameter and the kinematic viscosity $\\nu$ according to\n\\begin{equation}\nRe = \\frac{|\\mathbf{u}_d - \\mathbf{u}_c |d_d}{\\nu_c}\n\\end{equation}\nCalculation of the drag coefficient can be done by the independent calculation with each phase as the dispersed phase and the overall drag coefficient applied to the momentum equations taken as the volume fraction weighted average of the two values. \nThis blended scheme is a useful approximation for flows with regions in which either phase becomes the primary phase.\nThe virtual mass effect accounts here for the acceleration of the particulate phase (RBC) relative to the plasma. The virtual mass forces $F_{vm,c},F_{vm,d}$ are calculated for both phases as\n\\begin{equation}\n\\mathbf{F}_{vm,c} = -\\mathbf{F}_{vm,d} = 0.5 \\alpha_d \\rho_c \\left( \\frac{\\text{d}u_c}{\\text{d}t} -\\frac{\\text{d}u_d}{\\text{d}t}\\right)\n\\end{equation}\nThe lift force acts on the RBCs due to the developed velocity gradients in the plasma which has its own velocity field. The lift forces are calculated as \n\\begin{equation}\n\\mathbf{F}_{lift,c} = -\\mathbf{F}_{lift,d} = -0.5 \\alpha_d \\rho_c \\left(\\mathbf{u}_c -\\mathbf{u}_d\\right) \\times (\\nabla \\times\\mathbf{u}_c ) \n\\end{equation}\nA constant droplet diameter size, defined independently for each phase is used in this framework with \nthe RBCs considered as spherical particles with a diameter of 8 $\\mu$m\n\\cite{jung2006hemodynamic}.\n\n\\label{sec:Level set method}\n\\subsection{Marker function for level set}\nThe level set function is used here to describe the interface dynamics of the fluids. For simplicity, the case of two fluids is presented. The level set function $\\psi$ is defined to be a signed distance function calculated from the distance at the interface, $\\Gamma$ that separates the two fluids \n\\begin{equation}\n|\\psi| = min(\\vec{x}-\\vec{x_{\\Gamma}})\n\\label{psiDefinition}\n\\end{equation} \nA given point $\\vec{x}$ can have either a positive or a negative distance which defines whether it belongs in one fluid or the other. The interface that separates the two fluids is then the set of points that have a zero-distance (zero-level). \nA Heaviside function $H(\\psi,t)$ is used to calculate the smoothed volume fraction that depends on the values of $\\psi >0$. Here, the following expression is chosen \\cite{Olsson2005},\n\\begin{equation}\nH(\\psi) \n= \\begin{cases}\n1 & \\text{\\quad if in fluid 1}  \\\\\n\\frac{1}{2}\\left[ 1+\\frac{\\psi}{\\epsilon}+\\frac{1}{\\pi}sin(\\frac{\\pi \\psi}{\\epsilon})\\right] \n & \\text{\\quad at the interface} \\\\\n0 & \\text{\\quad if in fluid 2}  \n\\end{cases}\n\\label{HEqn}\n\\end{equation}  \nIn order to conserve mass, the equation for $H$ needs to be satisfied \n\\begin{equation}\n\\frac{\\partial H(\\psi,t)}{\\partial t}+ \\mathbf{u} \\cdot \\nabla{ H(\\psi,t)} =0\n\\label{rhoHEqn}\n\\end{equation}\nThen $H(\\psi,t)$ resembles a smoothed volume fraction $\\alpha_c$ of one fluid and 1-$H(\\psi,t)$ for the other fluid. The total mass with respect to time is then $\\int_{V}^{}\\rho_i H(\\psi)dV$.\nThe level set function according to Eq.\\ref{psiDefinition} is a distance function that is defined wherever an interface exists. The distance function can be advected using \n\\begin{equation}\n\\frac{\\partial \\psi}{\\partial t}+ \\mathbf{u} \\cdot \\nabla{\\psi} =0\n\\label{psiEqn}\n\\end{equation}\nEq.\\ref{psiEqn} is solved first and the distance function is advected. The level set is then re-initialised for allowing $\\psi$ to remain a signed distance function. For this, the Eikonal equation $|\\nabla \\psi| = 1$ is solved. \n\\begin{equation}\n\\frac{\\partial \\psi_{d}}{\\partial \n\\tau}=\\textrm{sgn}(\\psi)(1-|\\nabla \\psi_{d}|)\n\\label{reinitGeneric}\n\\end{equation}\nThe solutions of Eq.~\\eqref{reinitGeneric} have the same zero-level as $\\psi$ and are distance functions. \nDenoting with $\\tilde{\\psi}_d$ e.g. $\\tilde{\\psi}_d=\\psi^{n+1}_d$ the new value of $\\psi$ after solving Eq.~\\eqref{reinitGeneric} an additional step for satisfying mass conservation is performed. \\cite{LyrasLee2021a}\nThe correction denoted here with $\\hat{\\psi}$ is added to $\\tilde{\\psi}_d$ so that the following is satisfied for an appropriate $\\psi$ \\citep{Kees2011}\nThe following reaction-diffusion equation is solved \n\\begin{equation}\nH(\\psi_d + \\hat{\\psi}) = \\alpha_c + \\lambda \\Delta \\psi_d \n\\label{massConservationCorr}\n\\end{equation}\nwhere $\\lambda$ is a parameter that is a function of the cell size and is generally taken here $\\approx 0.5(\\Omega)^{1\/3}$, where $\\Omega$ is the volume of the cell. \nEq.~\\eqref{massConservationCorr} is solved with an implicit solver with the laplacian term using an unbounded, second-order conservative scheme using linear interpolation. The resulting $\\hat{\\psi}$ value leads to the final level set value  \n\\begin{equation}\n\\psi^{n+1} = \\psi^{n+1}_d + \\hat{\\psi}\n\\end{equation}\nThis new value satisfies the mass conservation since $H(\\psi^{n+1}) = \\alpha_c$ and preserves on of the most attractive features of LS methods, a sharp interface profile wherein an accurate curvature value is obtained and is directly used in the momentum equation Eq.\\eqref{momentumEqn}. \n\nThe method avoids the assumption made by similar approaches for the interface position that is not located at the iso-surface contour $\\alpha$-0.5 as in \\citep{Kunkelmann2010, Albadawi2013, Dianat2017, Lyras2020}. Compared to other previous works, the level set function is not reconstructed only from the advected volume fraction \\citep{Albadawi2013, Lyras2020} but neither is based on specifically designed schemes for re-initialisation \\cite{Lyras2020, Hartmann2010} that allows for simpler and faster calculations.\n\n\\subsection{Switching between segregated and dispersed flow regimes}\nFirst the transport equation for obtaining an initial approximation of the liquid volume fraction is solved. The equation includes an interface compression term $\\mathbf{u}_c$ and is given as\n\\begin{equation}\n\\frac{\\partial \\alpha_c}{\\partial t}+ u_k \\cdot \\nabla(\\alpha_c) + \\nabla \\cdot (u_c \\alpha_c (1-\\alpha_c))=0\n\\end{equation}\nwhere the velocity $u_c$ is applied normally to the interface to compress the volume fraction field and maintain a sharp interface. The $\\alpha_c (1-\\alpha_c)$ term ensures the term is only active\nin the interface region. The value for the artificial interface compression velocity is given by\n\n\\begin{equation}\nu_c = min (C_{\\alpha}|u|,max(|u|))\\frac{\\nabla \\alpha_c}{|\\nabla \\alpha_c|}\n\\end{equation}\nThe $\\frac{\\nabla \\alpha_c}{|\\nabla \\alpha_c|}$ term is the interface unit normal vector\nfor the direction of the applied compression velocity.   \nThe coefficient $C_{\\alpha}$ is used for controlling the interfacial compression, and when considered to be $\\in [0,1]$ it is simply a binary coefficient which switches interface sharpening on\/off when it is equal to 1\/0 respectively. \nWhen $C_{\\alpha}=0$ for a given pair of phase, no interface compression is imposed resulting in phase dispersion. When $C_{\\alpha}=1$, sharp interface capturing is applied and the flow is treated with VOF.\n\nThe switch between approaches is done dynamically imposing the interface sharpening in those areas of the domain that the $C_{\\alpha}$ disctates. This allows for a unified method which allows simulation of complex flows with any combination of regimes ranging from fully dispersed to fully segregated. The switch is done based on evaluating the magnitude of the gradient of the volume fraction ($\\gamma$), assuming that when the gradient of the volume fraction becomes smaller than a defined cut-off value ($\\gamma_{cut-off}$) phase dispersion occurs and $C_{\\alpha}=0$. The parameter $\\gamma$ is then defined by\\cite{wardle2013hybrid}\n\\begin{equation}\n\\gamma = \\frac{\\nabla \\alpha_c}{max(\\nabla \\alpha_c)}\n\\end{equation}\nThus, when $\\gamma \\geq \\gamma_{cut-off}$ the value of $C_{\\alpha}$ is set to 1. The value of $\\gamma_{cut-off}$ is set to 0.4. \n\n\\section{Results}\n\\label{sec:results}\nThe numerical tests here included an eccentric stenosis with a stenosis severity $S_{D}$ that varied from $50\\%$ to $70\\%$. Two cases of different hematocrit levels are used, one with RBC=35$\\%$ and another one with RBC=45$\\%$. The length of the stenosis $L_{s}=14 mm$ and the non-constricted diameter of the artery was $D=4mm$, shown in Fig.\\ref{fig:geoms_bc2}. \n\\begin{figure}[h]\n \\vspace{6pt}\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\centering\n  \\includegraphics[width=1.\\linewidth]{Figs\/domain.eps}\n  \\label{fig:sub1}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\centering\n  \\includegraphics[width=1.\\linewidth]{Figs\/BC2.eps}\n  \\label{fig:sub2}\n\\end{subfigure}\n \\caption{Domain and inlet used here for the numerical tests.}\n    \\label{fig:geoms_bc2}\n\\end{figure}\nFor the inlet velocity, a pulsating unsteady blood flow with a period of $T=1s$ was imposed and is shown in Fig.\\ref{fig:geoms_bc2}. The flow upstream the stenosis was assumed to be fully developed downstream with a Neumann boundary condition considered at the outlet. A zero slip velocity was employed for the RBCs and plasma at the lesion walls. The inlet profiles for velocity and RBC were considered to be uniform across the inlet cross section similar to other similar works\\cite{jung2006hemodynamic}. \n\nThe multiphase CFD results for pressure drop were compared with the results obtained from two reduced-order models (ROMs). \nThe model of Young and Tsai\\cite{young1973steady,young1973unsteady} is a trans-stenotic pressure drop model developed for general vascular stenosis. The model uses some empirical parameters obtained from in vitro experiments. \nThe model of Lyras and Lee \\cite{LyrasLee2021b} was developed based on analysis of the phenomena that contribute the most to pressure reduction in coronary arteries e.g. the viscous and the inertial effects, turbulence and unsteady flow. \nThese models treat the stenosis as one-dimensional fluid flow through an area-varying tube. Although these models provide a simplified expression for pressure drop, they have been shown to be to predict $\\Delta p$  with reasonably small difference compared to CFD and experiments.\\cite{stergiopulos1992,LyrasLee2021b} Together with the results from the reduced models, the results here are compared to the case of single-phase CFD simulations with laminar and Reynolds-Averaged Navier-Stokes(RANS) simulations are considered for the tests here. The $k-\\omega-$SST model \\cite{menter1993} is used here and has been shown to be an appropriate choise for the flows considered here \\cite{khair2015}. \n   \n\\begin{figure}[h]\n    \\vspace{6pt}\n    \\centering\n    \\includegraphics[scale=0.18]{Figs\/ecc_0_5_bc2.eps}\n       \\centering\n    \\caption{Pressure drop inside the vessel for different RBC and for 50$\\%$ stenosis severity. Results are compared with the reduced-order models and the single phase flow approach.}\n    \\label{fig:pressure_50pc}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\vspace{6pt}\n    \\centering\n    \\includegraphics[scale=0.18]{Figs\/ecc_0_7_bc2.eps}\n       \\centering\n    \\caption{Pressure drop inside the vessel for different RBC and for 70$\\%$ stenosis severity. Results are compared with the reduced-order models and the single phase flow approach.}\n    \\label{fig:pressure_70pc}\n\\end{figure}\n\nThe results for $\\Delta p$ for $S_D = 50\\%$ are shown first in Fig.\\ref{fig:pressure_50pc} and for $S_D = 70\\%$ in Fig.\\ref{fig:pressure_70pc} for both RBC cases and ROMs.\nThe results for velocity, shear stresses and RBC are shown here for three different locations, at the beginning of the stenosis (L1), at the position at the maximum curvature at x=24mm (L2), and the end of the stenosis (L3). \nFig.\\ref{fig:ux_wss} shows the velocity distribution at the radial direction and the x=L2 position and the shear stresses with respect to time.\nThe normalised velocity for the two different cases for different RBC and for 50-70$\\%$ stenosis severity are shown in Fig.\\ref{fig:velocity_captions}. The results are compared at the position L2 for the two radial directions Y-Z perpendicular to the main X flow direction.\nIn Fig.\\ref{fig:rbc_captions} the distribution of the red blood cells for the different cases for different RBC and for 50-70$\\%$ stenosis severity are presented. Results are compared for the three different positions L1, L2, L3.\nIn the region of maximum curvature (L2), the volume fraction of RBC generally increased with the WSS.\nFig.\\ref{fig:wss_captions} shows the shear stresses for the different cases for different RBC and for the two cases of 50, RBC=35$\\%$ and 70, RBC=45$\\%$. Results are compared for the two different positions L2, L3.\n\n\\begin{figure}[H]\n \\vspace{6pt}\n\\centering\n\\begin{subfigure}{.58\\textwidth}\n  \\centering\n  \\includegraphics[width=1.\\linewidth]{Figs\/ux_middle.eps}\n  \\label{fig:sub1}\n\\end{subfigure}%\n\\begin{subfigure}{.58\\textwidth}\n  \\centering\n  \\includegraphics[width=1.\\linewidth]{Figs\/wall_shear_stresses.eps}\n  \\label{fig:sub2}\n\\end{subfigure}\n\\caption{Velocity distribution at the radial direction and the x=L2 position (on the left). The shear stresses are shown with respect to time at the point of maximum wall shear stress at x=L2 (on the right).}\n\\label{fig:ux_wss}\n\\end{figure} \n\n\\begin{figure}[H]\n    \\vspace{6pt}\n    \\centering\n    \\includegraphics[scale=0.21]{Figs\/Fig5.eps}\n       \\centering\n    \\caption{Normalised velocity for the two different cases for different RBC and for 50-70$\\%$ stenosis severity. Results are compared at the position L2 for the two radial directions Y-Z perpendicular to the main X flow direction.}\n    \\label{fig:velocity_captions}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\vspace{6pt}\n    \\centering\n    \\includegraphics[scale=0.2]{Figs\/Fig6.eps}\n       \\centering\n    \\caption{Red blood cell levels for the different cases for different RBC and for 50-70$\\%$ stenosis severity. Results are compared for the three different positions L1, L2, L3.}\n    \\label{fig:rbc_captions}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\vspace{6pt}\n    \\centering\n    \\includegraphics[scale=0.16]{Figs\/Fig7.eps}\n       \\centering\n    \\caption{Shear stresses for the different cases for different RBC and for the two cases of 50, RBC=35$\\%$ and 70, RBC=45$\\%$. Results are compared for the two different positions L2, L3.}\n    \\label{fig:wss_captions}\n\\end{figure}\n\n\\section{Discussion} \n\\label{sec:discussion}\nAlthough the drag and pressure forces are the ones that primarily influence the motion of the RBCs \\cite{kim2016computational}, it is not clear in the literature which if the other forces have a significant impact on the haemodynamics \\cite{jung2006hemodynamic, wu2015study}. For this reason the virtual mass and lift forces are also included in the simulations together with the drag forces for describing the RBC-plasma interactions. \nConsidering the effects of gravity and the main forces here is paramount for the successful simulation of the problem, the absence of which would otherwise contribute to large deviations in the RBC distribution.\n\nThe hematocrit is a key variable that determines the red blood cells aggregation which effects blood viscosity. Increasing red blood cells aggregation and consequently the viscosity, results in higher pressure drop values as shown in Fig.\\ref{fig:pressure_50pc}. \n\nThe pressure obtained with higher RBC was closer to the sigle-phase results. Since both CFD approaches use the same numerical algorithm for solving Navier-Stokes (presure-based, PIMPLE algorithm and same numerical schemes for the discritisation), this difference is attributed to the presence of RBC and the different forces included and the calculated shear stresses \\cite{xu2013large}. \nThe results here are in agreement with the experiments of Freidoonimehr et al.,\\cite{freidoonimehr2018effect} who studied lesions with much shorter stenoses ($L_{s}\/D \\leq 1 $) than the ones considered here. \n\nThe effect of hematocrit is more evident for the lesions with a higher stenosis severity as shown in Fig.\\ref{fig:pressure_70pc}. For all cases, the pressure drop increases for higher $S_D$ which is expected and in accordance to previous studies and the ROM \\cite{young1973unsteady, LyrasLee2021b}.\nIn all simulations the flow pattern changes due to the varying cross sectional area. Although the differences between the lesions with the same RBC are not significant upstream the stenosis, the flow regime is influenced by $S_{D}$ downstream (positions L2 and L3). \nAn increase in the fluid velocity passing the stenosis results in the observed pressure drop which is intensified more in the tests of $S_{D}=70\\%$. \nDue to the abrupt increase in the velocity, the formation of turbulent structures and flow seperation grow after the constriction leading to a more significant drop in pressure across the constrictions as observed in previous studies \\cite{gould1978}.\n\nThe radial velocity profiles in Fig.\\ref{fig:ux_wss} were parabolic for all cases and the effect of the hematocrit was shown to have a lesser effect on the radial velocity compared to the stenosis severity.  \nThe magnitude of the axial velocity were higher for both lesions with $S_D=70\\%$ with a peak velocity magnitude of 1.6 m\/s for RBC=45$\\%$ and 0.95 m\/s for $S_D=70\\%$ for RBC=45$\\%$ during the systole cycle. \nOur results of the velocity profile are qualitatively in agreement with the ones in Jung et al.\\cite{jung2006hemodynamic} who also observed similar flat profiles at the radial direction of the vessel. \n\nAt the L2 position the velocity profile has a parabolic-shape profile. This shape differs from the profiles reported in the work of \\cite{buradi2019effect} who observed a plug-shaped flow for axisymmetric lesions of various stenosis severities including the ones studied here. \nThe reason for this difference in the profile with the results here, is that in the case of the axisymmetric idealised lesions, the RBCs move towards the centreline of the vessel whereas in this study, due to eccentricity, the red blood cells exhibit a core with high RBC levels at the centreline, but also at regions near the wall at the lower part of the vessel.\n\nThe shear stresses observed with the single-phase and the two-phase approaches have the same trend with the inlet velocity. For the same stenosis, the single phase approach predicted higher values for shear stress (Fig.\\ref{fig:ux_wss}-right).  \nThe WSS when the cross-sectional areas is smaller becomes higher, having high shear rates near the vessel wall.\nThe non-Newtonian shear thinning is responsible for this difference, and is related to a lower shear rate in the case the flow is considered to be multiphase. \nThe approach described here treats the RBCs (which mostly contribute to the haemodynamic forces) as a shear-thinning fluid with a viscosity which is a function of the volume fraction \\cite{brooks1970interactions}. \n\nIn this study it is assumed that the red blood cells are deformable and non-spherical. Following previous  studies the dispersed phase consists of rigid spherical particles without considering the effects of the normal stress effects \\cite{massoudi2012modeling}. The stress tensor for the RBCs is,\n\\begin{equation}\n\\tau_{ij,d} = k_1I + k_2S{ij,d}\n\\end{equation}\n\\begin{equation}\nk_1 = -p \\alpha_{d}\n\\end{equation}\n\\begin{equation}\nk_2(\\alpha_{d},\\dot{\\gamma})= \\left[\\mu_{\\infty} + (\\mu_{0}-\\mu_{\\infty})\\frac{1+ln(1+k\\dot{\\gamma})}{1+k\\dot{\\gamma}}(\\alpha_{d}+\\alpha^{2}_{d})\\right]\n\\end{equation}\n\\begin{equation}\n\\dot{\\gamma} = [2tr(S^{2}_{ij,d})]^{1\/2}\n\\end{equation}\nwhere $\\tau_{ij,d}$ is the partial stress tensor for the RBC with a corresponding shear rate $S_{ij,d}$. The constant $k_2$ accounts for the shear-thinning effect of RBCs, calculated by \\cite{yeleswarapu1996evaluation}, $\\dot{\\gamma}$ is the RBC shear rate, $\\mu_{\\infty}$ is an asymptotic viscosity for the case of an infinite shear rate, $\\mu_{0}$ is the viscosity for the case of zero shear rate and $k$ is a parameter that describes the shear thinning aspects of the RBCs.\n\nThe shear stresses at the L2 point in Fig.\\ref{fig:ux_wss} were also higher for higher RBC concentration as shown in Fig.\\ref{fig:rbc_captions} for the same stenosis, and higher than those of the single-phase model.    \nDepending on the region considered inside the vessel, WSS varied with the L3 giving lower WSS in the downflow region in Fig.\\ref{fig:wss_captions} compared to L2. In this downflow position, the high shear stresses regions formed in the areas of higher axial velocities and low RBC (shown in Fig.\\ref{fig:rbc_captions}). The low viscosity in those areas might be also a parameter that contributes to the formation of these low-RBC regions as observed in \\cite{jung2006hemodynamic}.  \nFor higher initial RBC levels from 35$\\%$ to 45$\\%$, the relative RBC volume fraction generally increased in the downflow location L3 but WSS decreased near the wall, while preserving higher values in the location L2 for both cases for different stenosis severity.\n\nThe curvature-induced secondary flow profiles exhibited vortices in the flow as shown in Fig.\\ref{fig:velocity_captions}, with a recirculation pattern that varied during the cardiac cycle.\nAs shown in Fig.\\ref{fig:ux_wss} high velocity gradients were observed at the near wall regions with a zero gradient closer to the centre.\nLarge secondary recirculating flows were observed in all cases here with similar patterns for the different stenosis severity and initial RBC level (Fig.\\ref{fig:velocity_captions}). These recirculation zones might contribute to large pressure fluctuations locally, and lead the migration of the RBCs towards locations other than the centreline \\cite{zhao2011shear}.\n\nFig.\\ref{fig:rbc_captions} presents the distribution of the volume fraction of the red blood cells on different radial cut planes for the various severities considered when the blood flow was at maximum, i.e., at the beginning of diastole (t\/T=0.05s). \n\nInside the curved part of the vessel, the RBCs mitigate towards the centreline as a result of the shear-induced particulate deposition. This is more pronounced for a higher degree of stenosis and inside the maximum curvature position, since further downstream the end of the stenosis, the RBC flow pattern becomes more complicated under the influence of the secondary flow and recirculation flow motions.\n\nAt the upstream position of the stenosis (L1) a lesser RBC migration is observed with similar relative profiles for all the vessel models studied here.\nOn the other hand, for all the vessels, low RBC volume fraction values were generally obtained at the downstream location (L3) compared to L2, where lower WSS led to a RBC migration.\n\nAreas with lower WSS such as areas with flow recirculation might potentially form atherosclerotic plaque \\cite{ku1985pulsatile}. The RBC patterns at the downstream position were less uniform than the ones in the other locations, due the impact of turbulence, the impact of which, increases in more stenosed vessels, and was also reported in \\cite{mahalingam2016numerical}.\n\nLocal fluctuations in the red blood cell distribution at the maximum curvature and downstream locations together with the changes in viscocity might be a factor that could potentially lead to a limited intake of oxygen at the arterial endothelial surface causing hypoxia \\cite{ogawa1990effect}. Furthermore, nitric oxide (NO) is captured by red blood cells and scavenged inside through hemoglobin. Hence, the changes in the RBC distribution due to the particulated deposition might lead to changes in the NO transport by the red blood cell hemoglobin which has been reported to cause atherogenesis and atherosclerosis \\cite{premont2020role, jeney2014red}.\n\nThe WSS is a significant haemodynamic parameter that is usually considered to study the platelet and vascular behaviour in blood flow through vessels. Areas with high WSS might have increased risk for diseases such as thrombosis. Areas with low or oscillating WSS which are more evident at the lesion models with stenosis severity of 70$\\%$ were observed as shown in Fig.\\ref{fig:wss_captions}.\n\nThe oscillatory patterns together with the secondary flow that occurs inside the stenosis and especially at regions with high curvature, have a significant contribution to the RBC deposition in the lesion. These changes downstream the maximum curvature position can potentially lead to a reduced WSS. In the areas of lower WSS, the endothelium might be more easily compromised by other particulates (monocytes and proteins) which might cause thrombosis and atherogenesis \\cite{melder1995selectin, rajendran2013vascular}. \n\n\\section{Conclusion} \nThe paper here presents a novel study for the impact of hematocrit for calculating pressure drop through eccentric tubular lesions considering the most important fluid forces such as lift, drag, virtual, gravity, which has not been previously presented in the literature. \n{\\color{black} Our results showed that} pressure drop is not strongly affected from single phase, and neither is mean velocity {\\color{black} thus confirming single phase modelling is adequate for estimating these variables. This implies the validity of the majority of previous studies.} On the other hand, particle buildup significantly changes which is only revealed by the multiphase approach. \nThe observed migration of red blood cells and their interactions at the surface of endothelial cells are usually ignored in single-phase algorithms. With the present approach, considering additional blood components, one may get a more accurate view of the wall shear stress and particulate buildup related to cardiovascular diseases such as atherogenesis.\nHydrodynamic diffusion of RBCs occurs downstream the stenosis and is more intensive for higher stenosis severity changing the RBC distribution and WSS.\n\nFuture studies will include the effect of these extra particulates in the model since the model has no limitations in the number of phases considered. Moreover, although a lot of efford has been made to consider a plethora of different sub-models for the occuring physical phenomena, the present work could be further improved without being limited to the assumption of spherical RBC. Options for allowing nonspherical particles using the shape factor concept \\cite{jung2006hemodynamic} could provide more insights for the haemodynamic parameters studied here. \n\n\\section*{Conflict of Interest} \nThe authors declare no conflict of interest.\n \n\\section*{Acknowledgment}\nThis work was supported by Heart Research UK [RG2670\/18\/21] and the Wellcome\/EPSRC Centre for Medical Engineering [WT 203148\/Z\/16\/Z]. \n\n\\section*{References}\n\n\\bibliographystyle{model6-num-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\nIn many statistical regression problems, one has\nto deal with problems where the available data are insufficient to provide a robust regression.\nIf conducting regression directly in such problems, one often risks of overfitting or\nbeing incorrectly regularized. In either case, the resulting regression model\nmay lose its prediction accuracy.\nExtracting and selecting the important features or eliminating the redundant ones is a key step\nto avoid overfitting and improve the robustness of the regression task~\\cite{fukumizu2004dimensionality}.\nThe feature extraction and selection thus constitutes of identifying a low dimensional subspace of the predictors $\\-x$ which\nretains the statistical relationship between $\\-x$ and the response $y$, i.e. a supervised dimension reduction problem.\nMathematically such problems are often posed as to estimate the central dimension reduction~(DR) subspace~\\cite{cook2005sufficient}.\nA very popular class of methods estimate this central subspace by considering the statistics of the predictors $\\-x$ conditional on\nthe response $y$, and such methods include the sliced inverse regression (SIR) proposed in\nthe seminal work~\\cite{li1991sliced}, the sliced average variance estimation~\\cite{cook1991sliced,dennis2000save}, and many of their variants,\ne.g. \\cite{cook2005sufficient,li2007directional,zhu2007kernel,li2008sliced,xia2002adaptive,li2009dimension,ma2012semiparametric,tan2018convex,lin2019sparse}.\nSome of the extensions and variants have been developed specifically for machine learning\nproblems, e.g., \\cite{fukumizu2012gradient,wu2009localized,kim2008dimensionality}.\nThe literature in this topic is vast and we refer to \\cite{ma2013review,li2018sufficient} for a more comprehensive overview\nof the subject.\nIt should be noted that most of the aforementioned methods adopt nonparametric formulation without assuming any specific relation between $\\-x$ and $y$. \nAs will be shown in the examples, the nonparametric approaches may not work well for\nthe problems with very small number of data, which is considered in the present work. \nTo this end, an alternative type of methods is to assume a parametric model of the likelihood function $p(y|\\-x)$,\nand then compute the reduced dimensions with maximum likelihood estimation~\\cite{cook2011ldr,cook2009likelihood}, or in a Bayesian formulation~\\cite{mao2010supervised,reich2011sufficient}.\nA main disadvantage of the parametric models is that they may be lack of the flexibility to \naccurately characterize of the relation between the predictors $\\-x$ and the response $y$.  \n\n \nIn this work we present a  method  incorporating the SIR\/SAVE type of methods with the model base ones, \nto make them more effective for small data problems.  \nIn particular we remain in the SIR\/SAVE framework to identify the DR space.\nAs one can see,  many  works in this class focus on the question:\nwhat statistical information of the conditional distribution $\\pi(\\-x|y)$ should one use to obtain the DR subspace?\nFor example, SIR makes use of the expectation of $\\pi(\\-x|y)$ to identify the DR directions, SAVE utilizes\nthe variance of it, and the method in \\cite{yin2003estimating} is based on the third moments.\nIn this work  we consider a different aspect of the problem: how to obtain the conditional distribution $\\pi(\\-x|y)$\n when the data set is small?\nIn SIR and SAVE, the conditional moments are approximately estimated by slicing the data~\\cite{li1991sliced}.\nAs will be demonstrated with numerical examples, the slicing strategy does not perform well if\nwe have a very small data set.\nThe main purpose of the work is to address the problem of computing the conditional distribution $\\pi(\\-x|y)$.\nIn particular we present a Bayesian formulation which can provide not only the first or second moments,\nbut the full conditional distribution $\\pi(\\-x|y)$, and once the conditional distribution is available one can use any desired methods to estimate the DR subspace based on the conditional distribution.\nJust like \\cite{cook2011ldr,cook2009likelihood}, our method also involves constructing the likelihood function $\\pi(y|\\-x)$ from data, but a main difference here is that\n we characterize the likelihood function with a nonparametric Gaussian Process (GP) model~\\cite{williams2006gaussian}, \n which may provide more flexibility than a parametric model.\nOnce the likelihood function is available, we can compute the posterior distribution $\\pi(\\-x|y)$ from the likelihood function\nand a desired  distribution of $\\-x$.\nIn this work we  choose to mainly use the first order moment of the conditional distribution (following SIR) to demonstrate the method, while noting that the method can be easily extended to other conditional moments.\nIt is important to note here that, while the conditional distribution $\\pi(\\-x|y)$ is computed in a Bayesian fashion,\nthe core of the method (i.e. the estimation of the DR subspace) remains frequentist, \nand so it is fundamentally different from the\n methods~\\cite{tokdar2010bayesian,mao2010supervised,reich2011sufficient} that do estimate \nthe DR subspace with a Bayesian formulation (e.g., imposing a prior on the DR subspace). \n\n\n\nTo summarize, the main contribution of the work is to propose a GP based Bayesian formulation to compute the conditional distribution \n$\\pi(\\-x|y)$ for any value of $y$ in the SIR\/SAVE framework, and by doing so it avoids slicing the samples,\nwhich makes it particularly effective for problems with very small numbers of data.\n\nThe rest of this paper is organized as follows. In section \\ref{sec:sliced} we set up a formulation of dimension reduction and go through the basic idea of the classic dimension reduction approaches SIR and SAVE. The Bayesian inverse regression and \nthe Bayesian average variance estimation are introduced explicitly in Section \\ref{sec:bir}, including a Bayesian formulation\nfor computing $\\pi(\\-x|y)$, the GP model used, and complete algorithms to draw samples from $\\pi(\\-x|y)$. In section \\ref{sec:examples} we provide several numerical examples. Section \\ref{sec:conclusion} offers concluding remarks.\n\n\n\n\\section{Dimension reduction and the sliced methods}\\label{sec:sliced}\n\\subsection{Problem setup}\n\nWe consider a generic supervised dimension reduction problem.\tLet $\\-x$ be a $p$-dimensional random variable defined on $R^p$ following a distribution $\\pi_0(\\-x)$,\nand suppose that we are interested in a scalar function of $\\-x$, which ideally can be written as,\n\t\\begin{equation}\n\t{y}=f(\\-b_1^T\\-x, \\-b_2^T\\-x,..., \\-b^T_K\\-x,\\epsilon), \\label{1}\n\t\\end{equation}\n\t where $\\-b_k$ for $k=1...K$ are some $p$-dimensional vectors,  and $\\epsilon$ is small noise independent of $\\-x$.\n\t It should be clear that, when this model holds, the projection of the $p$-dimension variable $\\-x$  onto the $k$ dimensional\n\tsubspace of $R^p$ spanned by $\\{\\-b_1,...,\\-b_K\\}$, captures all the information of $\\-x$ with respect to $y$, and if $K<p$, we can achieve the goal of data reduction by estimating the coefficients $\\{\\-b_k\\}_{k=1}^K$.\n\t In practice, both the explicit expression of $f$ and the coefficients $\\{\\-b_k\\}_{k=1}^K$ are unknown, and instead we\n\t have a set of data pairs $\\{(\\-x_j,y_j)\\}_{j=1}^n$ drawn from the joint distribution $\\pi(\\-x,y)$ defined by $\\pi_0$ and Eq.~\\eqref{1}.\n\t Finding a set of $\\{\\-b_k\\}_{k=1}^K$ that satisfy the Eq.~\\eqref{1} from the given data set\n\t $\\{(\\-x_j,y_j)\\}_{j=1}^n$ is the task of supervised dimension reduction.\n\tIn what follows we shall refer to the\n\tcoefficients $\\{\\-b_k\\}_{k=1}^K$ as  dimension-reduction (DR) directions, and the linear space $B$ spanned by the $\\{\\-b_k\\}_{k=1}^K$ as the DR subspace.\nFor a more formal and generic description of the DR problem~(in the Central DR Subspace and Sufficient Dimension Reduction framework)  we refer to \\cite{cook2005sufficient}.\n\n\n\\subsection{Sliced inverse regression}\n\nThe SIR approach~\\cite{li1991sliced} estimates the DR directions based on the idea of inverse regression (IR). In contrast to the forward regression\n\t\t$E({y}\\,|\\, \\-x)$, IR regresses each coordinate of $\\-x$ against $y$. Thus as $y$ varies, $E(\\-x\\,|\\,{y})$ draws a curve in $R^p$ along the\n\t\t${y}$ coordinate, whose center is located at $E(E(\\-x\\,|\\,{y}))=E(\\-x)$.\n\t\tFor simplicity we shall assume that throughout this section $\\-x$ is a standardized random variable:\n\t\tnamely $E(\\-x)=0$ and $\\mathrm{Cov}(\\-x)= I$.\n\t\n\t\tUnder the following condition the IR curve $E(\\-x\\,|\\,y)$ is contained in the DR subspace $B$~\\cite{li1991sliced}:\n\t\\begin{condition} For any $\\beta\\in R^p$, the conditional expectation $E(\\beta^T\\-x\\,|\\,\\-b^T_1\\-x,...,\\-b^T_K\\-x)$ is linear in\n\t$\\-b^T_1\\-x,..., \\-b_K^T\\-x$. \\label{cond:1}\n\t\\end{condition}\n\tThis condition is satisfied when the distribution of $\\-x$ is elliptically symmetric~\\cite{li1991sliced}.\n\tAn important implication of this property is that the covariance matrix $\\mathrm{Cov}[E(\\-x\\,|\\,y)]$ is degenerated in any direction orthogonal to the DR subspace $B$. We see, therefore, that the eigenvectors associated with the largest $K$ eigenvalues of $\\mbox{Cov}[E(\\-x\\,|\\,y)]$ are the DR directions. So the key of estimating the DR direction is to obtain the covariance of the conditional expectation of the data,\n\t$\\mbox{Cov}[E(\\-x\\,|\\,{y})]$.\n\nOne of the most popular approaches to estimate the covariance $\\mbox{Cov}[E(\\-x|y_j)]$ is SIR. Simply put, SIR\n produces a crude estimate of $E(\\-x|y)$,  by slicing the data ${(\\-x_1,y_1),...,(\\-x_n,y_n)}$ into $H$ partitions according to the value\nof ${y}_j$ and then estimating $E(\\-x\\,|\\,y\\in I_h),h=1,...,H$\n\t using the data inside the interval $I_h$ for each $h=1,...,H$.\n\t  Finally one use the $H$ samples to compute an estimate of the covariance matrix $\\mbox{Cov}[E(\\-x|y)]$.\n\t\tA complete SIR scheme is described as follows:\n\t\\begin{enumerate}\n\t\\item Divide range of $y$ into $H$ slices, $I_1,...,I_H$.\n\tLet the proportion of the ${y}_j$ that falls in slice $I_h$ be $\\hat{p}_h$, i.e.,\n\t\\[\\hat{p}_h = \\frac1n\\sum_{j=1}^n\\delta_h({y}_j),\\]\n\twhere $\\delta_h(\\text{y}_j)$ takes the values 0 or 1 depending on whether $y_j$ falls into the $h$th slice $I_h$ or not.\n\t\\item Within each slice, compute the sample mean of the ${\\-x}_j$'s, denoted by $\\hat{\\-x}_h(h=1,...,H)$:\n\t\\[\\hat{\\-x}_h = \\frac1{(n\\hat{p}_h)}\\sum_{\\text{y}_j\\in I_h}{\\-x_j}.\\]\n   \\item\n\tCompute the weighted covariance matrix\n\t\\[\\hat{C}=\\sum_{h=1}^{H}\\hat{p}_h\\hat{\\-x}_h\\hat{\\-x}_h^T.\\]\n    \\item Perform eigenvalue decomposition of $\\hat{C}$, and return\n\tthe eigenvectors associated with the $k$ largest eigenvectors as the estimated DR directions $\\hat{\\-b}_1,...,\n\t\\hat{\\-b}_K$.\n\t\t\\end{enumerate}\n\t\t\n\t\tAs is mentioned in Section~\\ref{sec:intro}, the slicing treatment is often not sufficiently accurate when the data set is small,\n\tand in what follows\twe shall provide an alternative  to compute the covariance matrix.\n\n\t\\subsection{Sliced average variance estimation}\n        The SAVE method extract the DR directions from the variance of $\\pi(\\-x|y)$,\n       \n \nand by doing so it is able to recover the information that could be overlooked by SIR because of symmetries in the forward regression \nfunction~\\cite{dennis2000save}. Let the columns of $\\-b$ form a basis for the DR space. To use SAVE, we need to assume the following two conditions~\\cite{dennis2000save}:\n        \\begin{enumerate}\n            \\item $E(\\-x|\\-B^T\\-x) ]$ is linear in $\\-B^T\\-x$,\n            \\item $\\mbox{Var}(\\-x|\\-B^T\\-x) $ is a constant,\n        \\end{enumerate}\n        where $\\-B$ is any basis matrix of $R^p$.\n        The conditions hold when $\\-x$ is normally distributed although normality is not necessary.\n \n  \n  \n  Under these two conditions, one can derive that  \n\t$$\\mbox{span}\\{I_p-E(\\mbox{Cov}[{\\-x|y}])\\}$$ is a DR space~\\cite{dennis2000save},\n  which is the basis for SAVE.  A complete SAVE scheme is as follows:\n    \\begin{enumerate}\n    \\item Divide range of $y$ into $H$ slices, $I_1,...,I_H$.\n\tLet the proportion of the ${y}_j$ that falls in slice $I_h$ be $\\hat{p}_h$, i.e.,\n\t\\[\\hat{p}_h = \\frac1n\\sum_{j=1}^n\\delta_h({y}_j),\\]\n\twhere $\\delta_h({y}_j)$ takes the values 0 or 1 depending on whether $y_j$ falls into the $h$th slice $I_h$ or not.\n    \\item Within each slice, compute the sample covariance matrix of the ${\\-x}_j$'s, denoted by $\\hat{M}_h(h=1,...,H)$:\n          \\begin{equation}\n          \\hat{M}_h = \\sum_{\\text{y}_j\\in I_h}{\\-x_j\\-x_j'}.\n          \\end{equation}\n    \\item The $j$-th sample SAVE DR direction can now be constructed by perform eigenvalue decomposition on the following matrix , and return the eigenvectiors associated with the $k$ largest eigenvectors:\n           \\begin{equation}\n           \\hat{C} = \\sum_{h=1}^{H}\\hat{p}_h{(I-\\hat{M}_h)^2}\n           \\end{equation}\n    \\end{enumerate}\n\t\t\n\n\n\n\\section{Bayesian inverse regression}\\label{sec:bir}\n\n\\subsection{Bayesian formulation for $\\pi(\\-x|y)$}\nRecall that in the SIR framework, a key step is to compute the covariance $\\mbox{Cov}[E(\\-x|y)]$.\nA natural choice to estimate the covariance $\\mbox{Cov}[E(\\-x\\,|\\,{y})]$ is to use the sample covariance of the\ndata points,\n\\begin{equation}\n\\hat{C}= \\frac1{n-1} \\sum_{j=1}^n (\\hat{\\-x}_j - \\bar{\\-x})(\\hat{\\-x}_j - \\bar{\\-x})^T,\n\\quad \\bar{\\-x} = \\frac1n\\sum_{j=1}^n \\hat{\\-x}_j, \\label{e:cov}\n\\end{equation}\nwhere $\\hat{\\-x}_j$ is an estimate of $E(\\-x|y_j)$ for all $j=1...n$,\nand $(y_1,...,y_n)$ are the data points.\nNext we need to compute   $\\hat{\\-x}_j$, the estimate of $E(\\-x|y_j)$, and we\npropose to do so in a Bayesian framework.\nNamely we formulate the problem as to compute the posterior distribution:\n\\begin{equation}\n\\pi(\\-x|y) \\propto {\\pi(y|\\-x)\\pi(\\-x)},\\label{e:postdis}\n\\end{equation}\nwhere $\\pi(y|\\-x)$ is the likelihood function and $\\pi(\\-x)$ is the prior of $\\-x$.\n\nWe consider the prior distribution $\\pi(\\-x)$ first.\nTo start we note that the choice of prior does not affect the DR subspace as this subspace structure\nlies in the function $f(\\-x,\\epsilon)$ in Eq.~\\eqref{1} rather than the distribution of $\\-x$.\nAs such, in principle one may use any prior distribution that satisfies the conditions required by SIR\/SAVE.\nHowever,  if the chosen $\\pi(\\-x)$ is too different from $\\pi_0$, the GP model constructed from the data (following $\\pi_0(\\-x)$)\nmay not be accurate for the samples drawn according to $\\pi(\\-x)$, which in turn may hurt the accuracy of the posterior\n$\\pi(\\-x|y)$.\nTo this end, one should choose the prior to be $\\pi_0$ or close to it. \nWe consider the following three cases.\nFirst in certain problems, especially those where the data are generated from computer models,\nthe distribution $\\pi_0(\\-x)$ may be known in advance.  \nSecondly for most problems where $\\pi_0$ is not available in advance, a natural choice is\nto perform a crude density estimation for the data $\\{\\-x_j\\}_{j=1}^n$ and use the estimated density as the prior.\nFor example, one may use Gaussian mixtures~\\cite{mclachlan2004finite} or a simple Gaussian to estimate the prior distribution from the data $\\{\\-x_j\\}_{j=1}^n$.\nFinally, for problems where estimating the density of $\\-x$ are particularly challenging, \n we can just use the original data points\n $\\{\\-x_j\\}_{j=1}^n$, and in this case the prior is simply $\\pi_0$.\n\n\n\\subsection{The GP regression}\nThe next step is to construct the likelihood function $\\pi(y|\\-x)$ from data,\nwhich, as mentioned earlier, is done by using the GP regression model.\n\n\nSimply speaking the GP regression performs a nonparametric regression in a Bayesian framework~\\cite{williams2006gaussian}.\nThe main idea of the GP method is to assumes that the function of interest $f(\\-x,\\epsilon)$ is a realization from a Gaussian random field,\nwhose mean is $\\mu(\\-x)$ and covariance is specified by\na kernel function $k(\\-x,\\-x')$, namely,\n\\[ \\mathrm{Cov}[f(\\-x),f(\\-x')] = k(\\-x,\\-x'). \\]\nThe kernel $k(\\-x,\\-x')$ is positive semidefinite and bounded.\n\nNow given the data points $\\{(\\-x_j,y_j)\\}_{j=1}^n$,\nwe want to predict the value of $y$ at a new point $\\-x$.\nNow we let $\\-X := \\left[\\-x_1, \\ldots, \\-x_n\\right]$,\nand $\\-Y =[y_1,\\ldots, y_{n}]$.\nUnder the GP assumption,  it is easy to see that the joint distribution of $(\\-Y,\\,y)$ is Gaussian,\n\\begin{equation}\n\\left[ \\begin{array}{c}\n         \\-Y \\\\\n        y \\end{array} \\right] \\sim \\@N\\left(\\begin{array}{c}\n         \\mu(\\-X) \\\\\n        \\mu(\\-x) \\end{array},\n\t\t\t\t\\left[\n\t\t\t\t\\begin{array}{ll}\n         K(\\-X,\\-X)+\\sigma_n^2I &K(\\-X,\\-x) \\\\\n        K(\\-x,\\-X) &K(\\-x,\\-x) \\end{array}\\right]\n\t\t\t\t\\right) , \\label{e:jointdis}\n\\end{equation}\nwhere $\\sigma_n^2$ is the variance of observation noise, $I$ is\nan identity matrix, and the notation $K(\\-A,\\-B)$ denotes the matrix of the covariance evaluated at all pairs of points in set $\\-A$ and in set $\\-B$ using\nthe kernel function $k(\\cdot,\\cdot)$.\n\nIt follows immediately from Eq.~\\eqref{e:jointdis} that  the conditional distribution $\\pi_{GP}(y|\\-x,\\-X,\\-Y)$  is also Gaussian:\n\\begin{subequations}\n\\label{e:gp}\n  \\begin{equation}\n\t\\pi_{GP}(y|\\-x,\\-X,\\-Y) =\\mathcal{N}(\\mu_\\mathrm{pos}, \\sigma^2_\\mathrm{pos}),\n\t\\end{equation}\nwhere the posterior mean and variance are,\n\\begin{align*}\n&\\mu_\\mathrm{pos}(\\-x)=\\mu(\\-x)+k(\\-x,\\-X)(k(\\-X,\\-X)+\\sigma_n^2I)^{-1}(\\-Y-\\mu(\\-x)),\\\\\n&\\sigma^2_\\mathrm{pos} = k(\\-x,\\-x)-k(\\-x,\\-X)(k(\\-X,\\-X)+\\sigma_n^2I)^{-1}k(\\-X,\\-x).\n\\end{align*}\n\\end{subequations}\nThere are also a number of technical issues in the GP model, such as choosing the kernel function and determining the hyperparameters.\nFor detailed discussion of these matters, we refer the readers to~\\cite{williams2006gaussian}.\nIn what follows we shall use the GP posterior as\nthe likelihood function, i.e., letting $\\pi(y|\\-x) =\\pi_{GP}(y|\\-x,\\-X,\\-Y)$.\n\n\n\n\n\\subsection{Computing the posterior mean}\\label{sec:comppost}\n\n\n\nOnce we obtain the likelihood function and the prior, a straightforward idea is to draw samples\nfrom the posterior distribution~\\eqref{e:postdis} with the Markov chain Monte Carlo (MCMC) simulation.\nAn alternative strategy is to sample from $\\pi(\\-x)$ in an importance sampling (IS) formulation.\nNamely suppose that we draw a set of samples $\\{\\-x_i\\}_{i=1}^{n_{MC}}$ from the prior distribution $\\pi(\\-x)$,\nand for each $\\-x_i$ we can compute the weight $$w_i=\\pi(y|\\-x_i).$$\nFinally the weights $w_1,...,w_{n_{MC}}$ are normalized so that\n$\\sum_{i=1}^{n_{MC}} w_i=1$ (if these samples are drawn\nwith MCMC, then $w_i=1\/n_{MC}$ for all $i=1...{n_{MC}}$).\nWe thus obtain obtain a set of weighted samples\n$\\{(\\-x_i,w_i)\\}_{i=1}^{n_{MC}}$ drawn from the posterior $\\pi(\\-x|y)$.\nNow let $\\{(\\-x_i,w_i)\\}_{i=1}^{n_{MC}}$ be a set samples draw from the posterior, and\nwe can estimate $E(\\-x|y)$ as\n\\begin{equation}\n\\hat{\\-x}=\\sum_{i=1}^{n_{MC}} w_i\\-x_i.\n\\end{equation}\nWe repeat this procedure for each $y_j$ for $j=1...n$, and then use Eq.~\\eqref{e:cov} to compute $\\mbox{Cov}[E(\\-x|y)]$.\nSince we use a Bayesian method to estimate $E(\\-x|y)$, we refer to proposed method as\nBayesian inverse regression (BIR).\nSimilarly the samples can also be used to estimate the conditional covariance $\\mbox{Cov}[\\-x|y]$ in SAVE,\nand the resulting method is termed as Bayesian average variance estimation (BAVE). \nAs is discussed earlier, the key of BIR\/BAVE is  essentially provides a means to\ndraw samples from the conditional distribution $\\pi(\\-x|y)$ without slicing the data, and its application is not limited\nto estimate $E(\\-x|y)$ or $\\mbox{Cov}[\\-x|y]$, and it is possible to make use of the conditional distribution in a different manner.\nFinally we present the BIR algorithm in Alg.~\\ref{alg:bir}\nand BAVE in Alg.~\\ref{alg:bir2}.\n\n\\begin{algorithm}[t]\n\\caption{The Bayesian inverse regression algorithm with MCMC}\\label{alg:bir}\n\\let\\-=\\mathbf\n\\begin{algorithmic}[1]\n\\Require{$\\{(\\-x_j,y_j)\\}_{j=1}^n$, $n_{MC}$, $\\pi(\\-x)$}\n\\Ensure{The estimated DR directions: $\\hat{\\-b}_1,\\,...,\\,\\hat{\\-b}_K$}\n\\State Construct the GP model from data $\\{(\\-x_j,y_j)\\}_{j=1}^n$: $\\pi_{GP}(y|\\-x,\\-X,\\-Y)$;\n\n\\For{$j=1$ to $n$}\n\n \n\t\n\t\t\\State Draw $n_{MC}$ samples from $\\pi_{GP}(y_j|\\-x,\\-X,\\-Y)\\pi(\\-x)$: $\\{\\-x_i\\}_{i=1}^{n_{MC}}$;\n\t\t\t\t\t\\State Compute\n$\\hat{\\-x}_j=\\frac1{n_{MC}}\\sum_{i=1}^{n_{MC}}\\-x_i$;\n\\EndFor\n\\State Compute $\\hat{C}$ using Eq.~\\eqref{e:cov} and $\\{\\hat{\\-x}_j\\}_{j=1}^n$;\n\\State Perform eigenvalue decomposition of $\\hat{C}$;\n\\State Return the eigenvectors associated with the $k$ largest eigenvalues as $\\hat{\\-b}_1,...,\\hat{\\-b}_K$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n\\caption{The Bayesian average variance estimation algorithm with MCMC}\\label{alg:bir2}\n\\let\\-=\\mathbf\n\\begin{algorithmic}[1]\n\\Require{$\\{(\\-x_j,y_j)\\}_{j=1}^n$,$n_{MC}$,$\\pi(\\-x)$}\n\\Ensure{The estimated DR directions: $\\hat{\\-b}_1,\\,...,\\,\\hat{\\-b}_K$}\n\\State Construct the GP model from data $\\{(\\-x_j,y_j)\\}_{j=1}^n$: $\\pi_{GP}(y|\\-x,\\-X,\\-Y)$;\n\n\\For{$j=1$ to $n$}\n\\State Draw $n_{MC}$ samples from $\\pi_{GP}(y_j|\\-x,\\-X,\\-Y)\\pi(\\-x)$: $\\{\\-x_i\\}_{i=1}^{n_{MC}}$;\n\t\t\t\t\\State Compute\n$\\hat{\\-x}=\\frac1{n_{MC}}\\sum_{i=1}^{n_{MC}}\\-x_i$;\n\t\t\t\t\t\\State Compute\n$\\hat{M}_j=\\frac1{n_{MC}-1}\\sum_{i=1}^n (\\-x_i-\\hat{\\-x}) (\\-x_i-\\hat{\\-x})^T$;\n\\EndFor\n\\State Compute $\\hat{C} = \\frac{1}{n}\\sum_{j=1}^{n}(I_p-\\hat{M}_j)^2$;\n\\State Perform eigenvalue decomposition of $\\hat{C}$;\n\\State Return the eigenvectors associated with the $k$ largest eigenvalues as $\\hat{\\-b}_1,...,\\hat{\\-b}_K$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\begin{remark}\nIt is important to reinstate here that, the BIR\/BAVE methods only use\nthe Bayes' formula to compute the conditional distribution $\\pi(\\-x|y)$, and the DR methods themselves are\n \\emph{frequentist}. \n\\end{remark}\n\n\\begin{remark}\nA key step in the proposed method is to construct the likelihood $\\pi(y|\\-x)$ with GP.\nIt is well known that GP may not perform well as a regression model for high dimensional problems. \nNevertheless, as demonstrate by the examples, \nwhile it is unable to provide accurate regression results, the resulting GP model \nare often adequate for the dimension reduction purposes.  \nMoreover, as is stated earlier, in this work we focus on problems \nwith modestly high dimensionality (less than 100) and a very limited number of data (hundreds or less). \n\\end{remark}\n\n\\begin{remark}\nAnother issue that should be mentioned here is how to select the number of the reduced dimensions;\nsince BIR is also a method based on the eigenvalue decomposition of $\\mbox{Cov}[E(\\-x|y)]$,\nthe methods used in \\cite{li1991sliced} and related works, e.g., \\cite{ferre1998determining}, can be used directly here.\n\\end{remark}\n\n\n\\section{Numerical examples}\n\\label{sec:examples}\n\n\nIn this section we compare the performance of the proposed BIR\/BAVE method with a number of common methods: SIR, SAVE, \nlikelihood-based DR~(LDR)~\\cite{cook2011ldr}, the Localized SIR (LSIR), \nin three mathematical and two real-data examples.\nThe first example uses data simulated from a mathematical function, with which we want to exam the scalability of the\nmethods with respect to the dimensionality of the problem.\nThe second one is also a mathematical example, and with this example we compare the performance of different methods\naffected by the non-ellipticity of the distribution of $\\-x$.\nThe third example is used specifically to compare the two second moment methods: SAVE and BAVE. \nOur last two examples are based on real data, in which we compare the performance of different methods in the small data situation.\nIn the GP model used in all the examples, we set the prior mean $\\mu(\\-x)=0$, and choose the Automatic Relevance Determination (ARD) squared\nexponential kernel~\\cite{williams2006gaussian}:\n\\begin{equation}\nk(\\-x,\\-x')=\\sigma^2_0\\exp(-\\frac12\\sum_{i=1}^p\\frac{(x_i-x'_i)^2}{\\lambda_i^2}),\n\\end{equation}\n where the hyperparameters $\\sigma_0$, $\\lambda_1\\,...,\\lambda_d$, and the $\\sigma_n$ are determined by\nmaximum likelihood estimation~\\cite{williams2006gaussian}.\nIn all the examples except the one in Section~\\ref{sec:rosenbrock}, the prior is obtained by fitting \na Gaussian distribution to the data, while for the example in Section~\\ref{sec:rosenbrock}, we assume that the distribution $\\pi_0$ is known, which is used as the prior.\nIn addition, in all the examples,  \n10000 MCMC samples are used to represent the conditional distribution $\\pi(\\-x|y)$ in \nthe BIR and BAVE methods. \n\n\n\\begin{figure*}\n \n \\subfigure[]{ \\includegraphics[width=0.33\\linewidth]{figs\/eg1_1}\n\t\\includegraphics[width=0.33\\linewidth]{figs\/eg1_2}\n  \\includegraphics[width=0.33\\linewidth]{figs\/eg1_3}}\n\n\t\n\n\t\\subfigure[]{\\includegraphics[width=0.33\\linewidth]{figs\/eg2_1}\n\t\\includegraphics[width=0.33\\linewidth]{figs\/eg2_2}\n  \\includegraphics[width=0.33\\linewidth]{figs\/eg2_3}}\n\t\\caption{The $R^2$ accuracy of the DR subspace (left), \n\t the first DR direction $\\-b_1$ (center) and the second DR direction\n$\\-b_2$\t(right), all plotted against the dimensionality.\n(a) results for function~\\eqref{e:fun1}; (b) results for function~\\eqref{e:fun2}.} \\label{f:eg1}\n\\end{figure*}\n\n\\subsection{Mathematical examples with increasing dimensions}\nFirst we consider a $d$-dimensional problem where $\\-x$ follows a standard normal distribution.\nThe data are simulated from the following functions:\n\\begin{subequations}\n\\begin{align}\n\\quad f(\\-x,\\epsilon) &= x_1(x_2+x_3)+0.5\\epsilon,\\label{e:fun1}\\\\\n\\quad f(\\-x,\\epsilon) &=\\frac{x_1+x_2+x_3}{0.5+(x_4+x_5)^2}+0.1\\epsilon,\\label{e:fun2}\n\\end{align}\n\\end{subequations}\nwhere $\\epsilon\\sim N(0,1)$.\nBoth problems have two DR directions.\nIn the regression content, a well known limitation of the GP method is that it can not handle high dimension, and so here we want to test the scalability of the BIR method with respect to dimensionality.\nTo do so we perform experiments for various dimensions: $d=10,\\,20,\\,30,\\,40,\\, 50$, where  we set the number of data points\nto be $n=5d$, i.e., growing linear with respect to dimensionality.\nTo evaluate the performance of the methods, we use the $R^2$ metric of accuracy used in \\cite{li1991sliced}\n\n\nto measure the accuracy of the DR subspace and the DR directions.\n\n\n\nWe repeat all the tests for $100$ times and report the average.\nSpecifically, we show the $R^2$-accuracy of the DR subspace $B$ and the two DR directions in\nFigs.~\\ref{f:eg1}. We can see that the BIR method has the best performance in all the tests in the two examples,\nexcept one situation: $d=10$ for function~\\eqref{e:fun2}.\nThe $R^2$ accuracy for each DR direction provide more information on the results.\nNamely, for Function~\\ref{e:fun1}, BIR performs better than all the other methods in both of the directions.\nFor function~\\ref{e:fun2}, the accuracy of BIR is slightly lower than than SIR and LSIR for the first direction, but it achieves significantly higher accuracy on the second dimension than all the other ones.\nFinally we want to note here that as the dimensionality increases, the performance of BIR does not decay evidently, suggesting\nthat the method can handle rather high dimensional problems.\n\n\\subsection{Mathematical examples with non Gaussian distributions}\\label{sec:rosenbrock}\nIn our second example, we want to test the performance of the methods when the distribution of $\\-x$ is strongly non-Gaussian.\n We assume $\\-x$ is a 10-dimensional variable and the data are generated as follows.\nFirst let $\\-u=(u_1,u_2)$ follow a two-dimensional standard normal distribution. We then perform the following transform:\n\\begin{equation}\nx_1=u_1,\\quad x_2 = u_1-bu_1^2,\n\\end{equation}\nwhere $b\\geq0$. Here by varying parameter $b$ one can control how different the distribution of $\\-x$ is from Gaussian.\nData of $y$ are generated from $\\-u$, and so the transformation used to generating $\\-x$ does not affect the data of $y$.\nIn this example we use the following two functions to generate $y$:\n\\begin{subequations}\n\\begin{align}\n\\quad y &= \\frac{u_1}{0.5+(u_2+1.5)^2}+0.5\\epsilon,\\label{e:fun3}\\\\\n\\quad y & = \\sin(5\\pi u_1) + u_2^2 +0.1\\epsilon,\\label{e:fun4}\n\\end{align}\n\\end{subequations}\nwhere $\\epsilon\\sim N(0,1)$.\nIn this test, we choose five different values of $b$: $b=0,\\,5,\\,10,\\,15,\\,20$ with sample size $n=100$, and we show the scatter plots\nof the data points for all these cases in Fig.~\\ref{f:rosenbrock},\nwhere we can see that the resulting data points move apart from Gaussian as $b$ increases.\nWe plot the $R^2$ accuracy against the value of $b$ in Figs.~\\ref{f:eg2} for both functions.\nFrom the figures we can see that for function~\\ref{e:fun3}, BIR clearly outperforms all the other methods for all the values of $b$,\nand for function~\\ref{e:fun4}, the BIR also has the best performance in all the cases,\nwith LDR being about the same at $b=10$ and $20$.\n\\begin{figure*}\n \n \\centerline{ \\includegraphics[width=1\\linewidth]{figs\/rosenbrock2}}\n\t\\caption{The scatter plots of $(x_1,x_2)$ for different values of $b$.} \\label{f:rosenbrock}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\n\\centerline{ \\includegraphics[width=0.43\\linewidth]{figs\/eg3_1}\n\n \n\t\\includegraphics[width=0.43\\linewidth]{figs\/eg4_1v2}}\n\n \n\t\\caption{The $R^2$-accuracy of the DR subspace plotted as a\n\tfunction of $b$,  for function~\\eqref{e:fun3} (left) and function~\\eqref{e:fun4} (right) respectively.} \\label{f:eg2}\n\\end{figure*}\n\n\\subsection{Mathematical example for BAVE}\nWe now consider a mathematical example which requires to consider the 2nd moments. \nLet $\\-x$ be a 20 dimensional random variable following standard normal distribution, and  let \n$$y =  x_1^2 + 0.1\\epsilon,$$ \nwhere noise $\\epsilon \\sim \\mathcal{N}(0,1)$. It is easy to verify that $E(\\-x|y) = 0$, which implies that the first moment based approach, i.e.,\n\t\tSIR, does not apply to this problem.\n\t\t\n\t\tWe conduct numerical experiments with six different sample sizes: 30, 40, 60, 80, 100 and 120, \n\tand for each sample size,\twe randomly generate 100 sets of data.\n\tWith each set of data, we estimate the DR direction with SIR, LDR, SAVE and BAVE.\n\tThe $R^2$ accuracy of the DR direction obtained by each method, averaged over the 100 trials, is\n\tshown in Fig.~\\ref{f:bave}. As expected, SIR fails completely for this example -- \n\tits resulting $R^2$ accuracy is near zero, regardless of the sample size. \n\tThe results of LDR are better than SIR but the overall accuracy remains quite low (less than $0.4$) even when the sample size\n\treaches 120. \n\tOn the other hand,  the performance of SAVE and BAVE increases notably as the sample size increases,\n\twhile for each sample size, the results of BAVE are considerably better than those of SAVE, \n\tsuggesting that BAVE performs considerably better than SAVE for this small dataset problem. \n\t\n\n\\begin{figure*}\n \n \\centerline{ \\includegraphics[width=.65\\linewidth]{figs\/bave}}\n\t\\caption{The $R^2$ accuracy of the DR direction computed with different same sizes.} \\label{f:bave}\n\\end{figure*}\n\n\n\n\n\\subsection{Death rate prediction}\n    \\begin{table*}[h]\n\t\\centering\n\t    \\begin{tabular}{lcccccc\n\t\n\t\t\\hline\n\t\t\\hline\n\t\t{}{Methods} & $n = 15 $ & $n = 20 $ & $n = 25 $ & $n = 30 $  & $n = 35 $ &  $n = 40$    \\\\ \\hline\n\t\tw\/o DR & $.1832$ & $ .0855$ & $ .0551$ & $ \\mathbf{.0460}$ & $ \\mathbf{.0425} $ & $ \\mathbf{.0380}$ \\cr\n\t\t & $\\left(.2013\\right) $ & $ \\left(.0502\\right) $ & $ \\left(.0216\\right)$ & $ \\left( .0171\\right)$ & $ \\left(.0161\\right) $ & $\\left(.0089\\right)$ \\\\\\hline\n\t\tLDR   & -    & $.0823$      & $.0569$    & $.0490$         & $.0444$ & $.0383$      \\cr\n\t\t& (-)      & $\\left(.0518 \\right) $ &    $\\left(.0207\\right) $ & $\\left(.0184\\right) $ & $\\left(.0173\\right) $ & $\\left(.0089\\right)$\\\\ \\hline\n\t\tSIR   & $.4403$    & $.0982$ & $.0653$ & $.0548$ & $.0525$      & $.0430$ \\cr\n\t\t& $\\left(1.2417\\right )$ & $\\left(.0769\\right) $ & $\\left(.0310\\right) $& $\\left(.0216\\right) $ & $\\left(.0217\\right) $  & $\\left(.0108\\right)$\\\\ \\hline\n\t\tLSIR & - & $.0876$ & $.0648$ & $.0557$ & $.0485$ & $.0429$ \\cr\n\t\t & (-)       & $\\left(.0461 \\right) $ &    $\\left(.0252\\right) $ & $\\left(.0224\\right) $ & $\\left(.0174\\right) $ & $\\left(.0100\\right)$\\\\\\hline\n\t\tBIR & $ \\mathbf{.0484}$ & $\\mathbf{.0451}$ & $\\mathbf{.0465}$ & $ .0481$ & $ .0466$ & $ .0468$ \\cr\n\t\t& $\\left(.0110\\right)$ & $\\left(.0104\\right)$ & $\\left(.0111\\right)$ & $\\left(.0105\\right)$ & $\\left(.0114\\right)$& $\\left(.0126\\right)$\\\\\n\t\t\n\t\t \\hline\\hline\n\t    \\end{tabular}\t\n\t\t\t\\medskip\n\t\t\t\n\t\t\t\\caption{The mean and the standard deviation (in parenthesis) of  MRRE for Example 3. The best results are marked in\n\t\t\t{bold}.} \\label{tb:eg3}\n    \\end{table*}\n\t\t    \\begin{table}[h]\n    \t\\centering\n    \t\n    \t\\begin{tabular}{p{2cm}p{2cm}p{2cm}p{2cm}p{2cm}p{2cm}p{2cm}}\n    \t\n    \t\t\\hline\n    \t\t\\hline\n    \t\tMethods & min&  max\\\\ \\hline\n\t\t\t\tw\/o DR & $.0289$ & $.3374$ \\\\\n    \t\tLDR & $.0281$ & $.7913$ \\\\\n    \t\tSIR & $.0362$ & $.2534$ \\\\\n    \t\tLSIR & $.0380$ & $.2350$\\\\\n    \t\tBIR &   $\\mathbf{.0247}$ &$\\mathbf{ .1183}$ \\\\\n    \t\t \\hline\\hline\n    \t\\end{tabular}\t\t\t\\medskip\n\t\t\t\n\t\t\t\\caption{The minimal and maximal relative regression error (RRE) in the 100 trials\n\t\t\twith 20 data points for the death rate example. The best results are marked in\n\t\t\t{bold}.} \\label{tb:eg32}\n\n    \\end{table}\nThe example considered in this section  is to use pollution and related factors to predict the death rate~\\cite{mcdonald1973instabilities,chatterjee2015regression}. This is a regression problem with 15 predictors and 60 data points and we choose this example to test how the methods perform with very small number of data. We first apply the DR  methods to select one feature (we have conducted tests with  2 and 3 features which does not improve the regression accuracy,\nand so we omit those results here) and then construct a standard linear regression model of the data in the reduced dimension.\nAs a comparison, we also perform the regression directly without DR.\nTo test the methods with different numbers of  data, we perform\nthe experiments with $15$, $20$, $25$, $30$, $35$, $40$ data points randomly selected from the data set and\nanother randomly selected 20 data points used as the test set.\nIn each experiment we can compute the mean relative regression error (MRRE) using the data in the test set.\nSpecifically, suppose $\\{(\\-x_i,y_i)\\}_{i=1}^{n_t}$ is the training set and $f_r(\\cdot)$ is the regression model, the MRRE is computed as,\n\\[\n\\mathrm{MRRE} = \\frac1{n_t}\\sum_{i=1}^{n_t} \\frac{|y_i-f_r(\\-x_i)|}{y_i}.\n\\]\nWe repeat all the experiments 100 times, and  compute the mean and the standard deviation\nof the obtained MRRE, which is shown in Table~\\ref{tb:eg3}.\nFirst we observe that for $n=40$ all the methods can achieve rather good accuracy;\nas $n$ decrease, the results of all the other methods become evidently worse,\nwhile that of BIR remains quite stable, suggesting that the BIR is especially effective\nin the small data case.\nIt should be noted that for $n=15$ LDR and LSIR fail to produce reasonable results due to numerical instability,\nand so we omit the results here.\nMore importantly it can be seen from the table that starting from $n=30$,\nthe regression without DR actually has the best performance,\nsuggesting that implementing DR is only necessary when the number of data points is below 30.\nIn all the cases DR is genuinely needed, i.e., $n<25$, the BIR method performs significantly better than all other methods.\nTo further analyze the performance, we also compute the minimal and the maximal relative regression errors (RRE)\nfor the 20 data-point case, and present the results in Table~\\ref{tb:eg32}.\nOnce again, we can see that the BIR method has the best results in both the minimal and the maximal cases.\n\n\n  \\begin{table*}[htbp!]\n\t\t\n\n\t    \\begin{tabular}{lccccccccc\n\t\n\t\t\\hline\n\t\n\t\t\\hline\n\t\t{sample size} & $20 $ & $30 $ & $40 $ & $ 50 $  & $60 $ &  $70$ & $80$ & $90$& $100$    \\\\ \\hline\n\t\tw\/o DR & $.377$ & $ .254$ & $ .210$ & $ .203$ & $ .181 $ & $ .173$& $ .173$ & $ .169$ & $ .170$ \\cr\n\t\t & $\\left(.153\\right) $ & $ \\left(.072\\right) $ & $ \\left(.035\\right)$ & $ \\left( .0393\\right)$ & $ \\left(.025\\right) $ &$\\left(.026\\right)$ & $\\left(.022\\right)$ & $ \\left(.022\\right) $ & $\\left(.022\\right)$  \\\\\\hline\n\t\tLDR   &$.394$ &$.262$& $.209$& $.198$ & $.178$ & $.173$ & $.175$&$.172$ & $.171$ \\cr\n\t\t& $\\left(.175\\right) $      & $\\left(.097 \\right) $ &    $\\left(.035\\right) $ & $\\left(.033\\right) $ & $\\left(.028\\right) $ & $\\left(.031\\right)$& $\\left(.024\\right)$& $\\left(.027\\right)$& $\\left(.024\\right)$\\\\ \\hline\n\t\tSIR   & $.497$& $.284$ & $.2358$ & $.217$ & $.199$& $.192$& $.193$& $.189$ & $.187$ \\cr\n\t\t& $\\left(.224\\right )$ & $\\left(.093\\right) $ & $\\left(.049\\right) $& $\\left(.044\\right) $ & $\\left(.035\\right) $  & $\\left(.034\\right)$& $\\left(.032\\right) $ & $\\left(.030\\right) $  & $\\left(.029\\right)$\\\\ \\hline\n\t\tLSIR & $.489$ & $.284$ & $.225$ & $.216$ & $.194$ & $.189$& $.190$ & $.183$ & $.178$ \\cr\n\t\t & $\\left(.210\\right) $      & $\\left(.079 \\right) $ &$\\left(.043\\right) $ & $\\left(.040\\right) $ & $\\left(.032\\right) $ & $\\left(.034\\right)$& $\\left(.029\\right) $ & $\\left(.033\\right) $ & $\\left(.027\\right)$\\\\\\hline\n\t\tBIR & $ \\mathbf{.188}$ & $.\\mathbf{184}$ & $\\mathbf{.178}$ & $ \\mathbf{.178}$ & $\\mathbf{ .167}$ & $\\mathbf{ .164}$& $\\mathbf{.167}$ & $ \\mathbf{.165}$ & $\\mathbf{.162}$ \\cr\n\t\t& $\\left(.034\\right)$ & $\\left(.034\\right)$ & $\\left(.031\\right)$ & $\\left(.029\\right)$ & $\\left(.025\\right)$& $\\left(.027\\right)$& $\\left(.021\\right)$ & $\\left(.024\\right)$& $\\left(.023\\right)$\\\\\n\t\t \\hline\n\t    \\end{tabular}\n\t\t\t\\medskip\n\t\t\t\n\t\t\t\\caption{The mean and the standard deviation (in parenthesis) of  MRRE for Example 4. The best results are marked in\n\t\t\t{bold}.} \\label{tb:eg4}\n    \\end{table*}\n\t\n\n\n\n\\begin{table}[htbp!]\n\t\\centering\n\t\\begin{tabular}{p{2cm}p{2cm}p{2cm}p{2cm}p{2cm}p{2cm}p{2cm}}\n\t\n\t\t\\hline\n\t\t\\hline\n\t\tMethods& min&  max\\\\ \\hline\n\t\tw\/o DR  & $.1555$ & $1.273$ \\\\\t\n\t\t\tLDR &$.170$ & $1.134$ \\\\\n\t\tSIR &$.169$& $1.874$ \\\\\n\t\tLSIR & $.217$ & $1.146$\\\\\n\t\t\tBIR & $\\mathbf{.113}$& $\\mathbf{0.287}$\n\t\t\\\\ \\hline\\hline\n\t\\end{tabular}\n\t\\caption{The minimal and maximal relative regression error (RRE) in the 100 trials\n\t\t\twith 20 data points for the automobile price example. The best results are marked in\n\t\t\t{bold}.}\n\t\\label{tb:eg42}\n\\end{table}\n\n\\subsection{Automobile data set}\nOur last example is the automobile data set in the UCI Machine Learning Repository~\\cite{Dua:2019}.\nThe original data set contains 205 instances described by 26 attributes including 16 continuous and 10 categorical. We preprocess the data set in the following way:\nwe neglect the 10 categorical attributes, and  remove the instances with missing values, yielding a data set with 159 instances and 16 attributes.\nWe select one of the 16 attributes as the response and the others as the predictors:\nspecifically we want to predict the price of an automobile based the other 15 attributes of it.\nIn this problem we first select one feature using the DR methods, and then perform a linear regression with the selected feature.\nJust like the previous example, we want to examine the performance of the DR methods in the small-data setting, i.e., a setting\nwhere direct regression can not provide accurate results.\nTo do so, we conduct the experiments with $n=10,\\,20,\\,...,90,100$ randomly selected samples\nand another 50 random samples used as the test set for all the cases.\nWe repeat each experiment 100 times, and compute the MRRE each time.\nThe mean and the standard deviation of the MRRE results are reported in Table~\\ref{tb:eg4}.\nFrom the data given in Table~\\ref{tb:eg4}, we obtain rather similar conclusions  as those of Example 3.\nNamely, the BIR method has the best MRRE of all the four methods used.\nIn Table~\\ref{tb:eg42}, we show the minimal and the maximal RRE for the 20 data-point case,\nand just like the results in Example~3, we find that\n the BIR method has the smallest RRE in both the minimal and the maximal cases.\n\n\n\n\n\n\\section{Conclusions}\\label{sec:conclusion}\n\nWe consider dimension reduction problems for regression and we propose a Bayesian\napproach for computing the conditional distribution $\\pi(\\-x|y)$ and perform the dimension reduction.\nThe method construct the likelihood function from the data with a GP regression model\nand MCMC to generate samples from the conditional distribution $\\pi(\\-x|y)$.\nNumerical examples demonstrate that the proposed method is particularly effective for problems with very small data set.\nWe reinstate here that, due to the use of GP model, BIR does not apply to problems with very high dimensions.\nRather, we expect BIR can be useful for problems with moderately high dimensions,\nand a very limited amount of data.\n\nWe believe the method can be useful in many real world applications. \nFor example, in many high dimensional inverse problems and data assimilation problems,\none the data can only be informative on a small number of dimensions~\\cite{cui2014likelihood,solonen2016dimension,zahm2018certified}.\nA method that utilizes the DR methods to identify such data informed dimensions is currently under investigation. \nOn the other hand, in certain problems gradient information is available,\nand DR methods which takes advantages of the gradient information\nhave also been developed, e.g.~\\cite{fukumizu2012gradient,constantine2015active,lam2018multifidelity}.\nIn this case, we expect that the gradient information can also be used to enhance the performance of the BIR method,\nvia, for example, Gradient-Enhanced Kriging~\\cite{morris1993bayesian}, and we plan to investigate this problem in the future.\n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect:1}\nEver since Einstein developed his theory of general relativity, group-theoretical methods have played\nan important role in deriving new solutions of the Einstein equations and understanding their\nproperties \\cite{CUP2009,SA1,SA2,Carmeli}, and also in investigating the asymptotic structure of \nspace-time \\cite{Pen62,Pen63,Pen64,Pen65,Pen67}. In particular, shortly after that \nBondi, Metzner and Sachs laid the foundations of the asymptotic\nsymmetry group (hereafter referred to as BMS) of asymptotically flat spacetime \n\\cite{Bondi62,Sachs62,Sachs1}, McCarthy elucidated several\nfeatures of this group and its representations \\cite{MC1,MC2,MC3,MC4,MC5}. For example, unlike the infinite-dimensional\nrepresentations of the Lorentz group, that allow for particles of arbitrary spin, a result first\nobtained by Majorana \\cite{Majo32}, McCarthy proved that the infinite-dimensional representations of the\nBMS group only allow for discrete values of the spin of elementary particles \\cite{MC1}.\nFurthermore, over the last few years, the BMS group has been found to lead to new perspectives on\nclassical gravitational scattering \\cite{ST2014,HPS1,HPS2} and on the problem of black-hole evaporation in\nquantum gravity \\cite{HPS1}. From a more mathematical point of view, all of this adds evidence in\nfavour of the pseudo-group structure of the functional equations of classical and quantum\nphysics being able to improve our understanding of the fundamental laws of nature. \n\nWithin this conceptual framework, our review aims at introducing the general reader, who is not\nnecessarily a general relativist, to the modern way of understanding the BMS group and its applications.\nFor this purpose, we begin by recalling that the\nimportance of the concept of energy within a physical theory, if introduced correctly, arises from \nthe fact that it is a conserved quantity in time and hence a very useful tool. \nThus, in general relativity one of the most interesting questions is related to the meaning of gravitational energy.\\\\\nStarting from any vector $J^a$ that satisfies a local conservation equation, that can be put in the form\n\\begin{equation}\n\\label{eqn:0}\n\\nabla_aJ^a=0,\n\\end{equation}\none can deduce an integral conservation law which states that the integral over the boundary \n$\\partial\\mathscr{D}$ of some compact region $\\mathscr{D}$ of the flux of the vector $J^a$ across this \nboundary necessarily vanishes. In fact, using Gauss' theorem we have\n\\begin{equation}\n\\label{eqn:0.1}\n\\int_{\\partial\\mathscr{D}} J^ad\\sigma_a=\\int_{\\mathscr{D}}\\nabla_aJ^adv=0.\n\\end{equation}\nNow we know that in General Relativity the energy-momentum tensor $T_{ab}$  satisfies the local conservation law\n\\begin{equation}\n\\label{eqn:1}\n\\nabla_aT^{ab}=0,\n\\end{equation}\nwhich follows directly from the Einstein field equations. However from \\eqref{eqn:1} we cannot deduce any integral conservation law. This is because in this case the geometric object to integrate over a 4-volume (as on the \nright-hand side of \\eqref{eqn:0.1}) would be a vector and we can not take the sum of two vectors at different \npoints of a manifold. This picture is ameliorated if space-time possesses symmetries, \ni.e. Killing vectors. If $K^a$ is a Killing vector,\n\\begin{equation*}\n\\nabla_{(a}K_{b)}=0,\n\\end{equation*}\nwe may build the vector \n\\begin{equation*}\nP^a=T^{ab}K_b,\n\\end{equation*}\nthat satisfies \\eqref{eqn:0}, since\n\\begin{equation*}\n\\nabla_a P^a=\\left(\\nabla_aT^{ab}\\right)K_b+T^{ab}\\nabla_aK_b=0.\n\\end{equation*}\nThe second term vanishes because $T^{ab}$ is symmetric and so $T^{ab}\\nabla_aK_b=T^{ab}\\nabla_{(a}K_{b)}=0$.\nTherefore the presence of Killing vectors for the metric leads to an integral conservation law. In  \nflat Minkowski space-time we know that there are 10 Killing vectors:\n\\begin{equation*}\n\\textbf{L}_{\\alpha}=\\frac{\\partial}{\\partial x^{\\alpha}},\\hspace{2.2cm}(\\alpha=0,1,2,3)\n\\end{equation*}\n\\begin{equation*}\n\\textbf{M}_{\\alpha\\beta}=e_{\\alpha}x^{\\alpha}\\frac{\\partial}{\\partial x^{\\beta}}\n-e_{\\beta}x^{\\beta}\\frac{\\partial}{\\partial x^{\\alpha}},\\hspace{2cm}\n(\\mathrm{no}\\hspace{1.3mm}\\mathrm{summation};\\alpha,\\beta =0,1,2,3)\n\\end{equation*}\nwhere $e_{\\alpha}$ is +1 if $\\alpha=0$ and -1 if $\\alpha=1,2,3$. The first four generate space-time translations \nand the second six \\lq rotations\\rq\\hspace{0.1mm} in space-time (these are just the usual ten generators of \nthe inhomogeneous Lorentz group). One may use them to define ten vectors $P^a_{\\alpha}$ and \n$P^a_{\\alpha\\beta}$ which will obey \\eqref{eqn:0}. We can think of $\\textbf{P}_0$ as representing the flow of \nenergy, and $\\textbf{P}_1$, $\\textbf{P}_2$ and $\\textbf{P}_3$ as the flow of the three components of linear \nmomentum. The $\\textbf{P}_{\\alpha\\beta}$ can be interpreted as the flow of angular momentum. If the metric is \nnot flat there will not, in general, be any Killing vectors. It is worth noting that the diffeomorphism group \nhas, for historical reasons, frequently been invoked as a possible substitute for the Poincar\\'e group for a \ngeneric space-time. However, it is not really useful in this context, being much too large and preserving only \nthe differentiable structure of the space-time manifold rather than any of its physical more important properties.\n\\\\ However, one could introduce in a suitable neighbourhood of a point $q$ normal coordinates $\\{x^a\\}$ so that \nthe components $g_{ab}$ of the metric are $e_a\\delta_{ab}$ (no summation) and that the components of \n$\\Gamma^a{}_{bc}$ are zero at $q$. One may take a neighbourhood $\\mathscr{D}$ of $q$ in which $g_{ab}$ and \n$\\Gamma^a{}_{bc}$ differ from their values at $q$ by an arbitrary small amount. Then \n$\\nabla_{(a}L_{\\alpha\\hspace{1mm}b)}$ and $\\nabla_{(a}M_{\\alpha\\beta\\hspace{1mm}b)}$ will not exactly vanish in \n$\\mathscr{D}$, but will in this neighbourhood differ from zero by an arbitrary small amount. Thus\\begin{equation*}\n\\int_{\\partial\\mathscr{D}}P^b_{\\alpha}d\\sigma_b\\hspace{1cm}\\mathrm{and}\\hspace{1cm}\n\\int_{\\partial\\mathscr{D}}P^b_{\\alpha\\beta}d\\sigma_b\n\\end{equation*}\nwill still be zero in the first approximation.\nHence the best we can get from \\eqref{eqn:1} is an approximate integral conservation law, if we integrate \nover a region whose typical dimensions are very small compared with the radii of curvature involved in \n$R_{abcd}$. We can interpret this by regarding the space-time curvature as giving a non-local contribution \nto the energy-momentum, that has to be considered in order to obtain a correct integral conservation law.\\\\\nFrom the above discussion we deduce that no exact symmetries can be found for a generic space-time. \nHowever, if we turn to the concept of asymptotic symmetries and we apply it to asymptotically flat space-times, \nwe will see that the picture is not so bad and that we can still talk about the Poincar\\'e group. The basic \nidea, developed in the remainder of the paper, is that, since we are taking into account asymptotically \nflat space-times, we may expect that by going to \\lq infinity\\rq\\hspace{0.1mm} one might acquire the Killing \nvectors necessarily for stating integral conservation laws. \n\\section{Bondi-Sachs coordinates and Boundary Conditions} \n\\label{sect:2}\nConsider the Minkowski metric\n\\begin{equation*}\ng=\\eta_{ab}dx^a\\otimes dx^b=dt\\otimes dt-dx\\otimes dx-dy\\otimes dy-dz\\otimes dz.\n\\end{equation*}\nWe introduce new coordinates \n\\begin{equation}\n\\label{eqn:135}\nu=t-r,\\hspace{1cm}r\\cos\\theta=z,\\hspace{1cm}r\\sin\\theta e^{i\\phi}=x+iy,\n\\end{equation}\nin terms of which the Minkowski metric takes the form\n\\begin{equation}\n\\label{eqn:137}\ng=du\\otimes du+du\\otimes dr+dr\\otimes du-r^2(d\\theta\\otimes d\\theta+\\sin^2\\theta d\\phi\\otimes d\\phi),\n\\end{equation}\nwhich can also be written as\n\\begin{equation}\n\\label{eqn:139}\nds^2=du\\otimes du+du\\otimes dr+dr\\otimes du-r^2q_{AB}dx^A\\otimes dx^B, \n\\end{equation}\nwhere \n\\begin{equation*}\nq_{AB}=\\left(\\begin{matrix} 1& 0\\\\ 0& \\sin^2\\theta\\end{matrix}\\right),\\hspace{0.5cm}A,B,...=2,3.\n\\end{equation*}\nNote that $q_{AB}$ represents the metric on the unit $2$-sphere. \nThe coordinate $u$ is called \\textit{retarded time}.\\\\\nWe proceed to the interpretation of the coordinates \\eqref{eqn:135}. The hypersurfaces given by the equation \n$u=\\mathrm{const}$ are null hypersurfaces, since their normal co-vector $k_a=\\nabla_a u$ is null. They are \neverywhere tangent to the light-cone. Note that it is a peculiar property of null hypersurfaces that their \nnormal direction is also tangent to the hypersurface. The coordinate $r$ is such that the area of the surface \nelement $u=\\mathrm{const}$, $r=\\mathrm{const}$ is $r^2\\sin\\theta d\\theta d\\phi$. Define a ray as the line with \ntangent $k^a=g^{ab}\\nabla_bu$. Then the scalars $\\theta$ and $\\phi$ are constant along each ray.\\\\\nNow we would like to introduce for a generic metric tensor a set of coordinates $(u,r,x^A)$ which has the same \nproperties as the ones of \\eqref{eqn:135}. These coordinates are known as \\textit{Bondi-Sachs coordinates} \n\\cite{Bondi62,Sachs62,Sachs1}. The hypersurfaces $u=\\mathrm{const}$ are null, i.e. the normal co-vector \n$k_a=\\nabla_au$ satisfies $g^{ab}(\\nabla_au)(\\nabla_bu)=0$, so that $g^{uu}=0$, and the corresponding \nfuture-pointing vector $k^a=g^{ab}\\nabla_bu$ is tangent to the null rays. Two angular coordinates $x^A$, \nwith $A,B,...=2,3$, are constant along the null rays, i.e. $k^a\\nabla_ax^A=g^{ab}(\\nabla_au)\\nabla_bx^A=0$, \nso that $g^{uA}=0$. The coordinate $r$, which varies along the null rays, is chosen to be an areal coordinate \nsuch that $\\mathrm{det}[g_{AB}]=r^4\\mathrm{det}[q_{AB}]$, where $q_{AB}$ is the unit sphere metric associated \nwith the angular coordinates $x^A$, e.g. $q_{AB}=\\mathrm{diag}(1,\\sin^2\\theta)$ for standard spherical coordinates \n$x^A=(\\theta,\\phi)$. The contravariant components $g^{ab}$ and covariant components $g_{ab}$ are related by \n$g^{ac}g_{cb}=\\delta^a_b$, which in particular implies $g_{rr}=0$ (from $\\delta_{ur}=0$) and $g_{rA}=0$ \n(from $\\delta_{uA}=0$). See Figure \\ref{fig:3.1}.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.52]{tronco.eps}\n\\caption{The Bondi-Sachs coordinate system. The coordinates $u$, $r$, and $\\phi$ and the vector $k^a$ are shown \nin the hypersurface $\\theta=\\mathrm{const}$.}\n\\label{fig:3.1}\n\\end{center}\n\\end{figure}\n\\\\It can be shown \\cite{Bondi62} that the metric tensor takes the form \n\\begin{eqnarray}\n\\label{eqn:136}\ng&=& g_{ab}dx^a\\otimes dx^b=e^{2\\beta}\\frac{V}{r}du\\otimes du+e^{2\\beta}(du\\otimes dr+dr\\otimes du)\n\\nonumber \\\\\n&+& g_{AB}(dx^A-U^Adu)\\otimes(dx^B-U^Bdu),\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{eqn:168}\ng_{AB}=r^2h_{AB},\\hspace{1cm}\\mathrm{det}[h_{AB}]=h(x^A).\n\\end{equation}\nUsing Jacobi's formula for the derivative of a determinant for a generic matrix $g_{\\mu\\nu}$,\n\\begin{equation*}\n \\partial_{\\rho}\\mathrm{det}[g_{\\mu\\nu}]=\\partial _{\\rho}g=gg^{\\mu\\nu}\\partial_{\\rho}g_{\\mu\\nu},\n\\end{equation*}\nwe have from the second of \\eqref{eqn:168}\n\\begin{equation}\n\\label{eqn:169}\n\\partial_u h=0\\Rightarrow h^{AB}\\partial_uh_{AB}=0,\\hspace{1cm}\\partial_r h=0\\Rightarrow h^{AB}\\partial_rh_{AB}=0.\n\\end{equation}\nWe also have\n\\begin{equation*}\ng^{ur}=e^{-2\\beta},\\hspace{0.5cm}g^{rr}=-\\frac{V}{r}e^{-2\\beta},\\hspace{0.5cm}g^{rA}\n=U^Ae^{-2\\beta},\\hspace{0.5cm}g^{AB}=-\\frac{1}{r^2}h^{AB}.\n\\end{equation*}\nA suitable representation for $h_{AB}$ is the following:\\\\\n\\begin{equation}\n\\label{eqn:141}\nh_{AB}=\\left(\\begin{matrix}\\cosh2\\delta e^{2\\gamma} & \\sin\\theta\\sinh2\\delta \\\\ \\\\ \\\\\\sin\\theta\\sinh2\\delta \n& \\sin^2\\theta\\cosh2\\delta e^{-2\\gamma}\\end{matrix}\\right)\\Rightarrow \\mathrm{det}[h_{AB}]=\\sin^2\\theta.\n\\end{equation}\nHere $V$, $\\beta$, $U^A$, $\\gamma$ and $\\delta$ are any six functions of the coordinates. The form \n\\eqref{eqn:136} holds if and only if $(u,r,\\theta,\\phi)$ have the properties stated above. Note that this \nform differs from the original form of Sachs \\cite{Sachs62} by the transformation $\\gamma\\rightarrow(\\gamma+\\delta)\/2$ \nand $\\delta\\rightarrow(\\gamma-\\delta)\/2$. The original axisymmetric Bondi metric \\cite{Bondi62} with rotational \nsymmetry in the $\\phi$-direction was characterized by $\\delta=U^{\\phi}=0$ and $\\gamma=\\gamma(u,r,\\theta)$, \nresulting in a metric with reflection symmetry $\\phi\\rightarrow-\\phi$ so that it is not suitable for describing \nan axisymmetric rotating star. \\\\\nThe next step is to write down the Einstein vacuum field equations in the above coordinate system in order to find \nthe equations that rule the evolution of the six arbitrary functions on which the metric depends. \nAs shown in \\cite{Sachs62} or \\cite{BSF} the Einstein vacuum field equations\n\\begin{equation*}\nG_{ab}=R_{ab}-\\frac{1}{2}Rg_{ab}=0,\n\\end{equation*}\nseparate into the \\textit{Hypersurface equations},\n\\begin{equation*}\nG^u_a=0,\n\\end{equation*}\nand the \\textit{Evolution equations},\n\\begin{equation*}\nG_{AB}-\\frac{1}{2}g_{AB}g^{CD}G_{CD}=0.\n\\end{equation*}\nThe former determines $\\beta$ along the null rays ($G^u_r=0$), $U^A$ ($G^u_A=0$) and $V$ ($G^u_u=0$), \nwhile the latter gives informations about the retarded time derivatives of the two degrees of freedom \ncontained in $h_{AB}$. Usually one requires the following conditions:\n\\begin{enumerate}\n\\item For any choice of $u$ one can take the limit $r\\rightarrow\\infty$ along each ray;\n\\item For some choice of $\\theta$ and $\\phi$ and the above choice of $u$ the metric \\eqref{eqn:136} \nshould approach the Minkowski metric \\eqref{eqn:137}, i.e. \n\\begin{equation}\n\\label{eqn:138}\n\\lim_{r\\to\\infty}\\beta=\\lim_{r\\to\\infty}U^A=0,\\hspace{0.5cm}\\lim_{r\\to\\infty}\\frac{V}{r}\n=1,\\hspace{0.5cm}\\lim_{r\\to\\infty}h_{AB}=q_{AB}.\n\\end{equation}\nNote that these conditions, as pointed out in \\cite{Sachs62}, are rather unsatisfactory from a geometrical \npoint of view. They will be completely justified later, using the method of the conformal structure, introduced by Penrose;\n\\item Over the coordinate ranges $u_0\\leq u\\leq u_1$, $r_0\\leq r\\leq\\infty$, $0\\leq\\theta\\leq\\pi$ \nand $0\\leq\\phi\\leq 2\\pi$ all metric functions can be expanded in series of $r^{-1}$. \n\\end{enumerate}\nUsing the Einstein equations with these assumptions it can be shown \\cite{Sachs62,BSF} that the \nfollowing asymptotic behaviours hold:\n\\begin{subequations}\n\\label{eqn:145}\n\\begin{align}\n&V=r-2M+O(r^{-1}),\\\\\n&h_{AB}=q_{AB}+\\frac{c_{AB}}{r}+O(r^{-2}),\\\\\n&\\beta=-\\frac{c^{AB}c_{AB}}{32r^2}+O(r^{-3}),\\\\\n&U^A=-\\frac{D_{B}c^{AB}}{2r^2}+O(r^{-3}),\n\\end{align}\n\\end{subequations}\ni.e. the metric \\eqref{eqn:136} admits the asymptotic expansion\n\\begin{eqnarray}\n\\label{eqn:140}\ng&=& du\\otimes du+du\\otimes dr+dr\\otimes du-r^2q_{AB}dx^A\\otimes dx^B\n\\nonumber \\\\\n&-& \\frac{2m_{_{B}}}{r}du\\otimes du-\\frac{c^{AB}c_{AB}}{4r^2}(du\\otimes dr+dr\\otimes du)\n\\nonumber \\\\\n&-& rc_{AB}dx^A\\otimes dx^B-\\frac{D_Fc^{F}_{A}}{2}(du\\otimes dx^A+dx^A\\otimes du)+...\n\\end{eqnarray}\nHere the function $m_{_{B}}=m_{_{B}}(u,\\theta,\\phi)$ is called the \\textit{Bondi mass aspect}, $c_{AB}=c_{AB}(u,\\theta,\\phi)$ \nrepresents the $O(r^{-1})$ correction to $h_{AB}$ and $D_A$ is the covariant derivative with respect to the metric \non the unit $2$-sphere, $q_{AB}$ \\cite{Haw17}. Capital letters A, B,... can be raised and lowered with respect \nto $q_{AB}$. In carrying out the $1\/r$ expansion of the field equations the covariant derivative ${\\cal D}_A$ corresponding \nto the metric $h_{AB}$ is related to the covariant derivative $D_A$ \ncorresponding to the unit sphere metric $q_{AB}$ by \n\\begin{subequations}\n\\label{eqn:160}\n\\begin{equation}\n{\\cal D}_AV^B=D_AV^B+C^B{}_{AE}V^E,\n\\end{equation}\nwhere\n\\begin{equation}\nC^B{}_{AE}=\\frac{1}{2r}q^{BF}\\Bigr(D_Ac_{FE}+D_Ec_{FA}-Dc_{AE}\\Bigr)\n+O(r^{-2}).\n\\end{equation}\n\\end{subequations}\nThis property will be useful later.\n\\begin{defn}$\\\\ $\n\\label{defn:AF1}\nA space-time $(\\mathscr{M},g)$ is \\textit{asymptotically flat} if the metric tensor $g$ and its components \nsatisfy the conditions \\eqref{eqn:145} and \\eqref{eqn:140}. These conditions are often \nreferred to as \\textit{boundary conditions}.\n\\end{defn} \n\\section{Bondi-Metzner-Sachs group}\n\\label{sect:3}\nIn this section our purpose is to find the coordinate transformations which preserve the asymptotic \nflatness condition. In other words we want to find the asymptotic isometry group of the metric \\eqref{eqn:136} and we must demand some conditions to hold in order for the coordinate conventions and boundary conditions \nto remain invariant. It is clear that, from \\eqref{eqn:140}, the corresponding changes suffered from the metric \nmust therefore obey certain fall-off conditions, i.e.\n\\begin{equation}\n\\label{eqn:146}\n\\delta g_{rr}=0,\\hspace{1cm}\\delta g_{rA}=0,\\hspace{1cm} g^{AB}\\delta g_{AB}=0.\n\\end{equation} \nand \n\\begin{subequations}\n\\label{eqn:181}\n\\begin{equation}\n\\label{eqn:147}\n\\delta g_{uu}=O(r^{-1}),\\hspace{1cm}\\delta g_{uA}=O(1),\n\\end{equation} \n\\begin{equation}\n\\label{eqn:180}\n\\delta g_{ur}=O(r^{-2}),\\hspace{1cm}\\delta g_{AB}=O(r).\n\\end{equation} \n\\end{subequations}\nThe third of \\eqref{eqn:146} expresses the fact that we \\textit{don't} want the angular metric $g_{AB}$ to undergo \nany conformal rescaling under the transformation. However a generalization which includes conformal rescalings \nof $g_{AB}$ can be found in \\cite{Barn2010}.\\\\\nWe know that the infinitesimal change $\\delta g_{ab}$ in the metric tensor is given by the Lie derivative of \nthe metric along the $\\xi^a$ direction, $\\xi^a$ being the generator of the transformation of coordinates:\n\\begin{equation}\n\\label{eqn:148}\n\\delta g_{ab}=-\\nabla_a\\xi_b-\\nabla_b\\xi_a.\n\\end{equation}\nClearly the vector $\\xi^a$ obeys \\textit{Killing's equation},\n\\begin{equation*}\n\\nabla_a\\xi_b+\\nabla_b\\xi_a=0,\n\\end{equation*}\nif and only if the corresponding transformations are isometries. What we want to solve now is an \n\\textit{asymptotic Killing's equation}, obtained putting together \\eqref{eqn:146} and \\eqref{eqn:181} \nwith \\eqref{eqn:148}. We get from the first of \\eqref{eqn:146} \n\\begin{equation*}\n\\nabla_r\\xi_r=\\partial_r\\xi_r-\\Gamma^{u}{}_{rr}\\xi_u-\\Gamma^{r}{}_{rr}\\xi_r-\\Gamma^{A}{}_{rr}\\xi_A=0,\n\\end{equation*}\nand using the Christoffel symbols given in \\ref{C}, we get \n\\begin{equation*}\n\\partial_r\\xi_r=2\\partial_r\\beta,\n\\end{equation*}\nand hence\n\\begin{equation}\n\\label{eqn:143}\n\\xi_r=f(u,x^A)e^{2\\beta},\n\\end{equation}\nwhere $f$ is a suitably differentiable function of its arguments.\\\\\nFrom the second of \\eqref{eqn:146} we obtain\n\\begin{equation*}\n\\nabla_r\\xi_A+\\nabla_A\\xi_r=\\partial_r\\xi_A+\\partial_A\\xi_r-2\\Gamma^{u}{}_{rA}\n\\xi_u-2\\Gamma^{r}{}_{rA}\\xi_r-2\\Gamma^{B}{}_{rA}\\xi_B=0,\n\\end{equation*}\nand thus, using \\eqref{eqn:143} we get \n\\begin{equation*}\n\\partial_r\\xi_A-r^2h_{AB}f\\left(\\partial_rU^B\\right)-\\frac{2\\xi_A}{r}\n-\\left(\\partial_rh_{AC}\\right)h^{BC}\\xi_B=-\\left(\\partial_Af\\right)e^{2\\beta},\n\\end{equation*}\nand after some manipulation\n\\begin{equation*}\n\\partial_r\\left(\\xi_Bg^{BD}+fU^D\\right)=-e^{2\\beta}g^{AD}\\left(\\partial_Af\\right),\n\\end{equation*}\nwhich leads to \n\\begin{equation*}\n\\xi_A=-h_{DA}f^Dr^2+fU^Dh_{DA}r^2+r^2h_{DA}\\left(\\partial_Bf\\right)\\int_r^{\\infty}\\frac{e^{2\\beta}h^{BD}}{r'^2}dr'\n\\end{equation*}\n\\begin{equation}\n\\label{eqn:144}\n=-f_Ar^2+fU_Ar^2+I_Ar^2+O(r),\n\\end{equation}\nwhere \n\\begin{equation*}\nI^D(u,r,x^A)=(\\partial_Bf)\\int_r^{\\infty}\\frac{e^{2\\beta}h^{BD}}{r'^2}dr'=\\frac{\\partial^Df}{r}+O(r^{-2}),\n\\end{equation*}\nwhere $f^D$ are suitably differentiable functions of their arguments and the indices A, B etc. \nare raised and lowered with respect to the metric $q_{AB}$.\\\\ We can solve algebraically \nthe third equation in \\eqref{eqn:146} to obtain $\\xi_u$:\n\\begin{equation*}\n\\xi_u=-\\frac{e^{2\\beta}}{2r}\\left(-\\partial_a\\xi_B+\\Gamma^{r}{}_{AB}\\xi_r+\\Gamma^{C}{}_{AB}\\xi_C\\right)h^{AB}.\n\\end{equation*}\nWorking with Christoffel symbols we get the following expression for $\\xi_u$:\n\\begin{eqnarray}\n\\label{eqn:149}\n\\xi_u &=& -\\frac{e^{2\\beta}r}{4}\\partial_D\\left(h_{AB}f^D\\right)h^{AB}+\\frac{e^{2\\beta}r}{2}\n\\left(\\partial_Af\\right)U^{A}-\\frac{e^{2\\beta}r}{4}\\partial_D\\left(h_{AB}I^D\\right)h^{AB}\n\\nonumber \\\\\n&+& e^{2\\beta}\\frac{V}{r}+r^2h_{AB}(U^Af^B-r^2U^AU^Bf-r^2U^AI^B).\n\\end{eqnarray}\nNow equations \\eqref{eqn:181} can be used to give constraints on the arbitrary functions \n$f$ and $f^A$. From the second of \\eqref{eqn:180} we get\n\\begin{equation*}\n\\nabla_A\\xi_B+\\nabla_B\\xi_A=\\partial_A\\xi_B+\\partial_B\\xi_A-2\\Gamma^{u}{}_{AB}\n\\xi_u-2\\Gamma^{r}{}_{AB}\\xi_r-2\\Gamma^C{}_{AB}\\xi_C=O(r).\n\\end{equation*}\nUsing asymptotic expansions \\eqref{eqn:145}, taking the order $r^2$ of the previous equation and putting it equal to zero we get\n\\begin{equation*}\n-\\partial_Af_B-\\partial_Bf_A+\\frac{1}{2}q_{AB}\\partial_D\\left(q_{CE}f^D\\right)q^{CE}+q^{CD}\n\\left(\\partial_Aq_{DB}+\\partial_Bq_{DA}-\\partial_Dq_{AB}\\right)f_C=0,\n\\end{equation*}\nthus\n\\begin{equation*}\n-\\partial_Af_B+\\gamma^{C}{}_{AB}f_C-\\partial_Af_B+\\gamma^{C}_{AB}f_C=-\\frac{1}{2}q_{AB}\\partial_D\\left(q_{CE}\\right)q^{CE},\n\\end{equation*}\nwhere $\\gamma^A{}_{BC}$ are the Christoffel symbols with respect to the metric on the unit sphere $q_{AB}$. We eventually get\n\\begin{equation}\n\\label{eqn:150}\nD_Af_B+D_Bf_A=\\frac{1}{2}q_{AB}\\partial_D\\left(q_{CE}f^D\\right)q^{CE}.\n\\end{equation}\nand hence\n\\begin{equation*}\nD_Af_B+D_Bf_A=f^D\\frac{1}{2}q_{AB}\\left(\\partial_Dq_{CE}\\right)q^{CE}+\\left(\\partial_Df^D\\right)q_{AB}=q_{AB}D_Cf^C.\n\\end{equation*}\nThus $f^B$ are the conformal Killing vectors of the unit 2-sphere metric $q_{AB}$.\\\\\nFrom the second of \\eqref{eqn:147} we get \n\\begin{equation*}\n\\nabla_u\\xi_A+\\nabla_A\\xi_u=\\partial_u\\xi_A+\\partial_A\\xi_u-2\\Gamma^{u}{}_{Au}\n\\xi_u-2\\Gamma^{r}{}_{Au}\\xi_r-2\\Gamma^{B}{}_{Au}\\xi_B=O(1).\n\\end{equation*}\nPutting the order $r^2$ of this equation equal to zero we obtain\n\\begin{equation}\n\\label{eqn:151}\n\\partial_uf_A=0. \n\\end{equation}\nFrom the first of \\eqref{eqn:180} we get\n\\begin{equation*}\n\\nabla_u\\xi_r+\\nabla_r\\xi_u=\\partial_u\\xi_r+\\partial_r\\xi_u-2\\Gamma^{u}_{ur}\n\\xi_u-2\\Gamma^{r}_{ur}\\xi_r-2\\Gamma^{A}_{ur}\t\\xi_A=O(r^{-2}).\n\\end{equation*}\nPutting the term of order $r^0$ of the previous equation equal to zero we get \n\\begin{equation}\n\\label{eqn:152}\n\\partial_uf=\\frac{1}{4}\\partial_D\\left(q_{AB}f^D\\right)q^{AB}.\n\\end{equation}\nPutting all the results together we have\n\\begin{subequations}\n\\label{eqn:153}\n\\begin{equation}\n\\partial_uf_A=0\\Rightarrow f_A=f_A(x^B),\\end{equation}\n\\begin{align}\nD_Af_B+D_Bf_A=2q_{AB}\\partial_uf\\Rightarrow \\left\\{\\begin{matrix} \\partial^2_uf=0,&\\\\\n\\partial_uf=\\frac{1}{2}D_Af^A.&\\end{matrix}\\right.\n\\end{align}\n\\end{subequations}\nWe get for $f$ the following expansion\n\\begin{equation}\nf=\\alpha+\\frac{u}{2}D_Af^A,\n\\end{equation}\nwhere $\\alpha$ is a suitably differentiable function of $x^A$.\\\\\nConsider now\n\\begin{equation*}\n\\xi^a=g^{ab}\\xi_b,\n\\end{equation*}\nfrom which we get \n\\begin{equation}\n\\label{eqn:154}\n\\xi^u=f=\\alpha+\\frac{u}{2}D_Af^A,\n\\end{equation}\n\\begin{equation}\n\\label{eqn:155}\n\\xi^A=f^A-I^A=f^A-\\frac{D^A\\alpha}{r}-u\\frac{D^AD_Cf^C}{2r}+O(r^{-2}),\n\\end{equation}\n\\begin{eqnarray}\n\\label{eqn:161}\n\\xi^r &=& -\\frac{r}{2}\\left[D_A\\xi^A-U^A\\partial_Af\\right]=-\\frac{r}{2}D_{C}\\xi^C+O(r^{-1})\n\\nonumber \\\\\n&=& -\\frac{r}{2}D_Cf^C+\\frac{D_CD^C\\alpha}{2}+u\\frac{D_CD^CD_Af^A}{4}+O(r^{-1}).\n\\end{eqnarray}\nThe second equality in \\eqref{eqn:161} follows from \\eqref{eqn:160} and from \n\\begin{equation*}\nq^{AB}c_{AB}=0,\n\\end{equation*}\nwhich follows from satisfying at order $r^{-2}$ the second of \\eqref{eqn:169} in the form \n\\begin{equation*}\n0=h^{AB}\\partial_rh_{AB}=[q^{AB}-\\frac{c^{AB}}{r^2}+O(r^{-3})][-\\frac{c_{AB}}{r^2}+O(r^{-3})].\n\\end{equation*}\nAs $r\\rightarrow\\infty$  \\eqref{eqn:154} and \\eqref{eqn:155} become, respectively\n\\begin{align}\n\\label{eqn:156}\n&\\xi^u=\\alpha+\\frac{u}{2}D_Af^A,\\\\\n\\label{eqn:157}\n&\\xi^A=f^A.\n\\end{align}\nFinally we can state that the asymptotic Killing vector is of the form \\begin{equation}\n\\label{eqn:158}\n\\xi=\\xi^a\\partial_a=\\left[\\alpha(x^C)+\\frac{u}{2}D_Af^A(x^C)\\right]\\partial_u+f^A(x^C)\\partial_A,\n\\end{equation}\nwhere $\\alpha$ is arbitrary and $f^A$ are the conformal Killing vectors of the metric of the unit sphere. In order to fix ideas, set $x^A=(\\theta,\\phi)$.\nIt is clear then that $\\theta$ and $\\phi$ undergo a finite conformal transformation, i.e. \n\\begin{subequations}\n\\label{eqn:159}\n\\begin{equation}\n\\label{eqn:182}\n\\theta\\rightarrow\\theta'=F(\\theta,\\phi),\n\\end{equation}\n\\begin{equation}\n\\label{eqn:183}\n\\phi\\rightarrow\\phi'=G(\\theta,\\phi),\n\\end{equation}\nfor which\n\\begin{equation*}\nd\\theta'^2+\\sin^2\\theta' d\\phi'^2=K^2(\\theta,\\phi)(d\\theta^2+\\sin^2\\theta d\\phi^2),\n\\end{equation*}\nand hence\n\\begin{equation}\n\\label{eqn:171}\nK^4=J^{2}(\\theta,\\phi;\\theta',\\phi')\\sin^2\\theta\\left(\\sin\\theta'\\right)^{-2},\\hspace{0.5cm}J\n=\\mathrm{det}\\left(\\begin{matrix}\\frac{\\partial F}{\\partial\\theta} & \\frac{\\partial F}\n{\\partial\\phi}\\\\\\frac{\\partial G}{\\partial\\theta} & \\frac{\\partial G}{\\partial\\phi}\n\\end{matrix}\\right).\n\\end{equation}\nBy definition of conformal Killing vector we also have\n\\begin{equation}\n\\label{eqn:214}\nK^2=e^{D_Af^A}.\n\\end{equation}\nThe finite form of the transformation of the coordinate $u$ is given, as can be easily checked, by \n\\begin{equation}\n\\label{eqn:184}\nu\\rightarrow u'=K[u+\\alpha(\\theta,\\phi)].\n\\end{equation}\n\\end{subequations}\n\\begin{defn}$\\\\ $\n\\label{defn:BMS}\nThe transformations \\eqref{eqn:159} are called \\textit{BMS (Bondi-Metzner-Sachs) transformations}, and \nare the set of diffeomorphisms which leave the asymptotic form of the metric of \nan asymptotically flat space-time unchanged.\n\\end{defn}\nThe BMS transformations form a group. In fact, as is known, the conformal transformations form a group, so that \n$F$, $G$, and $K$ have all the necessary properties. Thus, one must only check the fact that if one \ncarries out two transformation \\eqref{eqn:184} the corresponding $\\alpha$ for the product is again \na suitably differentiable function of $\\theta$ and $\\phi$. If\n\\begin{equation*}\nu_1\\rightarrow u_2=K_{12}[u_1+\\alpha_{12}]\n\\end{equation*} \nand \n\\begin{equation*}\nu_2\\rightarrow u_3=K_{23}[u_2+\\alpha_{23}]\n\\end{equation*}\nthen we have \n\\begin{equation*}\nu_1\\rightarrow u_3=K_{13}[u_1+\\alpha_{13}],\\hspace{0.5cm}K_{13}=K_{12}K_{23},\\hspace{0.5cm}\n\\alpha_{13}=\\alpha_{12}+\\frac{\\alpha_{23}}{K_{12}}.\n\\end{equation*}\nSince $\\alpha_{13}$ is a suitably differentiable function it follows that \n\\begin{prop}$\\\\ $\nThe BMS transformations form a group, denoted with $\\mathscr{B}$.\n\\end{prop}\n\\begin{defn}$\\\\ $\nThe BMS transformations for which the determinant $J$, defined in \\eqref{eqn:171}, \nis positive form the \\textit{proper} subgroup of the BMS group.\n\\end{defn}\nIn the remainder we will omit the word \\lq proper\\rq, even if all of our considerations will regard this component of $\\mathscr{B}$.\n\\begin{oss}$\\\\ $\nNote that the $r$ coordinate too may be involved in the BMS group of transformations, but such a \ntransformation is somewhat arbitrary since it depends on the precise type of radial coordinate used and \nit is not relevant to the structure of the group. Clearly the BMS group is infinite-dimensional since \nthe transformations depend upon a suitably differentiable function $\\alpha(\\theta,\\phi)$.\n\\end{oss}\n\\section{Conformal Infinity}\n\\label{sect:4}\nIn this part of the work the notion of conformal infinity, originally introduced by Penrose, is developed. \nThe idea is that if the space-time is considered from the point of view of its conformal structure only, \n\\lq points at infinity\\rq\\hspace{0.1mm} can be treated on the same basis as finite points. This can be \ndone completing the space-time manifold to a highly symmetrical conformal manifold by the addition of \na null cone at infinity, called $\\mathscr{I}$. We want to construct \\cite{Pen62,Pen63,Pen64,Pen67}, \nstarting from the \\lq physical space-time\\rq\\hspace{0.1mm} $(\\mathscr{\\tilde M},\\tilde{g})$, another \n\\lq unphysical space-time\\rq\\hspace{0.1mm} $(\\mathscr{M},g)$ with boundary $\\mathscr{I}\n=\\mathscr{\\partial M}$ \\cite{Lang}, such that $\\mathscr{\\tilde M}$ is conformally equivalent to the \ninterior of $\\mathscr{M}$ with $g_{ab}=\\Omega^2\\tilde g_{ab}$, given an appropriate function $\\Omega$. \nThe two metrics $\\tilde g_{ab}$ and $g_{ab}$ define on $\\mathscr{\\tilde{M}}$ the same null-cone structure. \nThe function $\\Omega$ has to vanish on $\\mathscr{I}$, so that the physical metric would have to be infinite \non it and cannot be extended. The boundary $\\mathscr{I}$ can be thought as being at infinity, in the sense \nthat any affine parameter in the metric $\\tilde{g}$ on a null geodesic in $\\mathscr{M}$ attains unboundedly \nlarge values near $\\mathscr{I}$. This is because if we consider an affinely parametrized null geodesic \n$\\gamma$ in the unphysical space-time $(\\mathscr{M},g)$ with affine parameter $\\lambda$, whose equation is\\begin{equation*}\n\\frac{d^2x^a}{d\\lambda^2}+\\Gamma^a{}_{bc}\\frac{dx^b}{d\\lambda}\\frac{dx^c}{d\\lambda}=0,\n\\end{equation*}\nit is easy to see that the corresponding geodesic $\\tilde{\\gamma}$ in the physical space-time \n$(\\tilde{\\mathscr{M}},\\tilde{g})$ with affine parameter $\\tilde{\\lambda}(\\lambda)$ is solution of the equation \n\\begin{equation*}\n\\frac{d^2x^a}{d\\tilde{\\lambda}^2}+\\tilde{\\Gamma}^a{}_{bc}\\frac{dx^b}{d\\tilde{\\lambda}}\n\\frac{dx^c}{d\\tilde{\\lambda}}=-\\frac{1}{\\tilde{\\lambda}'}\\left(\\frac{\\tilde{\\lambda}''}\n{\\tilde{\\lambda}'}+2\\frac{\\Omega'}{\\Omega}\\right)\\frac{dx^a}{d\\tilde{\\lambda}},\n\\end{equation*} \nwhere $'$ denotes a $\\lambda$ derivative. If we want the parameter $\\tilde{\\lambda}$ to be affine the \nright-hand side of the above equation must vanish, and hence we must have\n\\begin{equation*}\n\\frac{d\\tilde{\\lambda}}{d\\lambda}=\\frac{c}{\\Omega^2},\n\\end{equation*}\nwhere $c$ is an arbitrary constant. Since $\\Omega=0$ on $\\mathscr{I}$, $\\tilde{\\lambda}$ diverges and hence \n$\\tilde{\\gamma}$ never reaches $\\mathscr{I}$, which apparently really is at infinity. Thus, from the point \nof view of the physical metric, the new points (i.e. those on $\\mathscr{I}$) are infinitely distant from \ntheir neighbours and hence, physically, they represent \\lq points at infinity\\rq .\\\\ The advantage in studying \nthe space-time $(\\mathscr{M},g)$ instead of $(\\tilde{\\mathscr{M}},\\tilde{g})$ is that the infinity of the \nlatter gets represented by a finite hypersurface $\\mathscr{I}$ and the asymptotic properties of the fields \ndefined on it can be investigated by studying $\\mathscr{I}$ and the  behaviour of such fields on $\\mathscr{I}$. \\\\\nHowever, there is a large freedom for the choice of the function $\\Omega$. Anyway, it turns out \\cite{Penrin2} \nfrom general considerations that an appropriate behaviour for $\\Omega$ is that it should approach zero \n(both in the past and in the future) like the reciprocal of an affine parameter $\\lambda$ on a null geodesic \nof the space-time considered ($\\lambda\\Omega\\rightarrow \\mathrm{constant}$ as $\\lambda\\rightarrow\\pm\\infty$).\n\\\\Consider physical Minkowski space-time in spherical polar coordinates\n\\begin{equation}\n\\tilde{g}=dt\\otimes dt -dr\\otimes dr-r^2\\Sigma_2,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eqn:sfer}\n\\Sigma_2=d\\theta\\otimes d\\theta+\\sin^2\\theta d\\phi\\otimes d\\phi.\n\\end{equation}\nIntroduce now the standard retarded and advanced null coordinates $(t,r)\\rightarrow(u,v)$ defined by\n\\begin{equation*}\nu=t-r,\\hspace{1cm}v=t+r,\\hspace{1cm} v\\geq u.\n\\end{equation*}\nThe coordinates $u$ and $v$ serve as affine parameters into the past and into the future of \nnull geodesics of Minkowski space-time.\\\\\nThe metric tensor becomes\n\\begin{equation*}\n\\tilde{g}=\\frac{1}{2}(du\\otimes dv+dv\\otimes du)-\\frac{1}{4}(v-u)^2\\Sigma_2.\n\\end{equation*}\nConsider now the unphysical metric\n\\begin{equation*}\ng=\\Omega^2\\tilde{g},\n\\end{equation*}\nwith the choice\n\\begin{equation*}\n\\Omega^2=\\frac{4}{(1+u^2)(1+v^2)}.\n\\end{equation*}\nNote that for $u,v\\rightarrow\\pm\\infty$  we have $\\Omega u$, $\\Omega v\\rightarrow \\mathrm{constant}$, \nas pointed out before.\\\\ Now to interpret this metric it is convenient to introduce new coordinates\n\\begin{equation*}\nu=\\tan p,\\hspace{1cm}v=\\tan q,\\hspace{1cm}-\\frac{\\pi}{2}<p\\leq q<\\frac{\\pi}{2},\n\\end{equation*}\nsuch that we have\n\\begin{equation}\n\\label{eqn:2}\ng=2(dp\\otimes dq+dq\\otimes dp)-\\sin^2(p-q)\\Sigma_2.\n\\end{equation}\nIt is possible to bring the metric \\eqref{eqn:2} in a more familiar form by setting\n\\begin{equation*}\nt'=q+p,\\hspace{0.8cm}r'=q-p,\\hspace{0.8cm}-\\pi<t'<\\pi,\\hspace{0.8cm}-\\pi<t'-r'<\\pi,\\hspace{0.8cm}0<r'<\\pi,\n\\end{equation*}\nfrom which follows \n\\begin{equation}\n\\label{eqn:3}\ng=dt'\\otimes dt'-dr'\\otimes dr'-\\sin^2 (r')\\Sigma_2.\n\\end{equation}\nIt is worth noting that the metric \\eqref{eqn:3} is that of \\textit{Einstein's static universe}, $\\mathscr{E}$, \nthe cylinder obtained as product between the real line and the 3-sphere, $S^3\\times\\mathbb{R}$. However, the \nmanifold $\\mathscr{M}$ represents just a finite portion of such a cylinder.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.29]{cilindro.eps}\n\\caption{The cylinder $\\mathscr{E}=S^3\\times\\mathbb{R}$, of which $\\mathscr{M}$ is just a finite portion, \ndelimited by $\\mathscr{I^+}$, $\\mathscr{I^-}$, $i^+$, $i^-$ and $i^0$.\nWe note that the $(\\theta,\\phi)$ coordinates are suppressed, so that each point \nrepresents a 2-sphere of radius $\\sin r'$.}\n\\label{fig:2.3}\n\\end{center}\n\\end{figure}\n\\\\The metric \\eqref{eqn:2} is defined at $q=\\pi\/2$ and $p=-\\pi\/2$: those values correspond to the infinity \nof $\\mathscr{\\tilde M}$ and therefore they represent the hypersurface $\\mathscr{I}$. Hence we have defined \na conformal structure on  $\\mathscr{M}$, whose coordinates are free to move in the range  \n$-\\pi\/2\\leq p\\leq q\\leq\\pi\/2$. The boundary is given by $p=-\\pi\/2$ or $q=\\pi\/2$ and the interior of \n$\\mathscr{M}$ is conformally equivalent to Minkowski space-time.\\\\\nWe introduce the following points in $\\mathscr{M}$:\n\\begin{itemize}\n\\item $i^+$, called \\textit{future timelike infinity} given by the limits $t\\pm r\\rightarrow\\infty$, \n$u,v\\rightarrow\\infty$, $p,q\\rightarrow\\frac{\\pi}{2}$, $t'\\rightarrow\\pi$, $r'\\rightarrow 0$. \nAll the images in $\\mathscr{M}$ of timelike geodesics terminate at this point;\n\\item $i^-$, called \\textit{past timelike infinity} given by the limits $t\\pm r\\rightarrow -\\infty$, \n$u,v\\rightarrow-\\infty$, $p,q\\rightarrow-\\frac{\\pi}{2}$, $t'\\rightarrow -\\pi$, $r'\\rightarrow 0$.  \nAll the images in $\\mathscr{M}$ of timelike geodesics originate at this point;\n\\item $i^0$, called \\textit{spacelike infinity} given by the limits $t\\pm r\\rightarrow\\pm \\infty$, \n$u\\rightarrow-\\infty$, $v\\rightarrow\\infty$, $p\\rightarrow-\\frac{\\pi}{2}$, $q\\rightarrow\\frac{\\pi}{2}$, \n$t'\\rightarrow0$, $r'\\rightarrow \\pi$. All spacelike geodesics originate and terminate at this point.\n\\end{itemize}\nWe also introduce the following hypersurfaces in $\\mathscr{M}$:\n\\begin{itemize}\n\\item $\\mathscr{I^+}$, called \\textit{future null infinity}, is the null hypersurface where all the \noutgoing null geodesics terminate and is obtained in the following way. Null outgoing geodesics are \ndescribed by $t=r+c$, with $c$ finite constant, from which $u=t-r=c$ and $v=t+r=2t-c$. Taking the limit \n$t\\rightarrow\\infty$ we get $u=c$ and $v=\\infty$, hence $q=\\pi\/2$ and $p=\\tan^{-1}c=p_0$ with \n$-\\pi\/2< p_0<\\pi\/2$. In $(t',r')$ coordinates $t'=\\pi\/2+p_0$ and $r'=\\pi\/2-p_0$. As $p_0$ runs in its \nrange of values this is a point moving on the segment connecting $i^+$ and $i^0$. All outgoing null \ngeodesics terminate on this segment, described by the equation $t'=\\pi-r'$.\n\\item $\\mathscr{I^-}$, called \\textit{past null infinity}, is the hypersurface form which all null \ningoing geodesics originate. It can be shown that this is given by the region $p=-\\pi\/2$ and \n$-\\pi\/2<q_0<\\pi\/2$ and is described, in terms of $(t',r')$ coordinates, by the segment of equation \n$t'=\\pi+r'$ connecting $i^{-}$ and $i^{0}$.\n\\end{itemize}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.45]{Penrose.eps}\n\\caption{A Penrose diagram for $\\mathscr{M}$, using $(t',r')$ coordinates.} \n\\label{fig:2.3.1}\n\\end{center}\n\\end{figure}\nPutting \n\\begin{equation*}\nf^{\\pm}(t',r')=t'\\pm r'-\\pi,\n\\end{equation*} \nthe two equations defining the hypersurfaces $\\mathscr{I}^+$ and $\\mathscr{I}^-$ are \n\\begin{equation*}\nf^{\\pm}(t',r')=0,\n\\end{equation*}\nrespectively. \nThe normal co-vectors to $\\mathscr{I}^+$ and $\\mathscr{I}^-$ are \n\\begin{equation*}\nn^{\\pm}_a=\\frac{\\partial f^{\\pm}}{\\partial x^a}=(1,\\pm1,0,0).\n\\end{equation*}\nSince $g^{ab}n^{\\pm}_an^{\\pm}_b=0$ it follows that $\\mathscr{I}^+$ and $\\mathscr{I}^-$ are null hypersurfaces.\\\\\nAt this stage, we can build some useful representation of the space-time $\\mathscr{M}$. One of them is depicting \n$\\mathscr{M}$ as a portion of the cylinder $\\mathscr{E}=S^3\\times E^1$, see Figure \\ref{fig:2.3}. Another one is a portion of the plane in $(t',r')$ coordinates, that is an example of Penrose diagram. Each \npoint of the Penrose diagram represents a sphere $S^2$, and radial null geodesics are represented by \nstraight lines at $\\pm 45^{\\circ}$, see Figure \\ref{fig:2.3.1}\n\\\\One more representation for Minkowski space-time is furnished by Figure \\ref{fig:2.4}.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.43]{dueconi.eps}\n\\caption{This is another useful way of depicting $\\mathscr{M}$ as the interior of two cones joined base to base. \nThis picture however is not conformally accurate: in fact $i^0$ appears as an equatorial region \nwhereas it should be a point.}\n\\label{fig:2.4}\n\\end{center}\n\\end{figure}\n\\\\ \\\\We note here that for $\\mathscr{M}$ the points $i^+$, $i^-$ and $i^0$ are regular and that $\\mathscr{I^+}$ \nand $\\mathscr{I^-}$ both have $S^2\\times\\mathbb{R}$ topology. Furthermore, the boundary of $\\mathscr{M}$ \nis given by $\\mathscr{I}=\\mathscr{I^+}\\cup\\mathscr{I^-}\\cup i^+\\cup i^-\\cup i^0$.\n\\\\Consider now Schwarzschild space-time, with metric\n\\begin{equation}\n\\label{eqn:4}\n\\tilde{g}=dt\\otimes dt\\left(1-\\frac{2m}{r}\\right)-dr\\otimes dr\\left(1-\\frac{2m}{r}\\right)^{-1}-r^2\\Sigma_2.\n\\end{equation}\nIntroducing $(u,w)$ coordinates as \\begin{equation}\n\\label{eqn:5}\nu=t-\\left[r+2m\\ln\\left(\\frac{r}{2m}-1\\right)\\right],\\hspace{1cm}w=1\/r,\n\\end{equation}\nwe have\n\\begin{equation}\n\\label{eqn:5.1}\n\\tilde{g}=du\\otimes du\\left(1-2mw\\right)-(du\\otimes dw+dw\\otimes du)\\frac{1}{w^2}-\\frac{1}{w^2}\\Sigma_2.\n\\end{equation}\nThe first of \\eqref{eqn:5} is just the null retarded coordinate, corresponding to a null outgoing geodesic. \nNote that the coordinate $r^*=r+2m\\ln\\left(r\/2m-1\\right)$ in \\eqref{eqn:5} is the usual Wheeler-Regge \n\\lq tortoise coordinate\\rq\\hspace{0.1mm} introduced in \\cite{Wheel}.\nConsider now the unphysical metric\n\\begin{equation*}\ng=\\Omega^2d\\tilde{g},\\hspace{1cm}\\Omega=w,\n\\end{equation*}\n\\begin{equation}\n\\label{eqn:6}\ng=w^2(1-2mw)du\\otimes du-(du\\otimes dw+dw\\otimes du)-\\Sigma_2.\n\\end{equation}\nSchwarzschild space-time, $\\mathscr{\\tilde M}$, is given by $0<w<1\/2m$ because $2m<r<\\infty$. We remark \nthat the Schwarzschild solution can be easily extended beyond the event horizon, i.e. $0<r<\\infty$ and \n$0<w<\\infty$ because the apparent singular point $r=2m$ of the metric \\eqref{eqn:4} is just a coordinate \nsingularity and not a physical one, as can be noticed from \\eqref{eqn:5.1}. The metric \\eqref{eqn:6} is \ndefined for $w=0$ (i.e. $r=\\infty$) and hence for $\\mathscr{M}$  we may take the range $0\\leq w<1\/2m$, \nsuch that the hypersurface $\\mathscr{I^+}$ is given by $\\Omega=w=0$. \\\\ Re-expressing \\eqref{eqn:6} in \nterms of a null advanced coordinate\n\\begin{equation*}\nv=u+2r+4m\\ln\\left(\\frac{r}{2m}-1\\right),\n\\end{equation*}\ncorresponding to a null ingoing geodesic we get\n\\begin{equation}\n\\label{eqn:7}\ng=w^2(1-2mw)dv\\otimes dv+(dv\\otimes dw+dw\\otimes dv)-\\Sigma_2.\n\\end{equation}\nBy doing this it is now possible to introduce $\\mathscr{I^-}$ as the hypersurface of $\\mathscr{M}$ described \nby \\eqref{eqn:7} for $w=0$. It is easy to check that the hypersurfaces $\\mathscr{I}^+$ and $\\mathscr{I}^-$, \ngiven by the equations $f^{\\pm}(w)=w=0$ are again null hypersurfaces. \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.51]{Schwarzschild.eps}\n\\caption{Null infinity for Schwarzschild space-time. Note that $w=0$ corresponds both to $\\mathscr{I^+}$ \nand $\\mathscr{I^-}$. The points $i^{\\pm}$ and $i^0$ are singular and have been deleted.}\n\\label{fig:2.5}\n\\end{center}\n\\end{figure}\n\\\\The main difference between the Minkowski space-time case emerges from the fact that the points $i^+$, $i^-$ \nand $i^0$ in the Schwarzschild case are not regular, as could be deduced by the study of the eigenvalues of \nthe Weyl tensor. However, it should not be surprising that $i^+$ and $i^-$ turn out to be singular, since the \nsource generating the gravitational field becomes concentrated at these points, at the two ends of its history. \nThus, we will omit $i^+$, $i^-$ and $i^0$ from the definition of $\\mathscr{I}$, that will be just  \n$\\mathscr{I}=\\mathscr{I^-}\\cup\\mathscr{I^+}$. We have two disjoint boundary null hypersurfaces $\\mathscr{I^-}$ \nand $\\mathscr{I^+}$ each of which is a cylinder with topology $S^2\\times\\mathbb{R}$. These null hypersurfaces \nare generated by rays (given by $\\theta$,$\\phi=$constant, $w=0$) whose tangents are normals to the hypersurfaces. \nThese rays may be taken to be the $\\mathbb{R}'\\mathrm{s}$ of the topological product $S^2\\times\\mathbb{R}$. An useful representation of the Schwarzschild space-time is furnished by Figure \\ref{fig:2.5}.\\\\\nTake now into account a space-time $(\\mathscr{\\tilde M},\\tilde g)$ with metric tensor \\cite{Pen67,Penrin2}\n\\begin{equation}\n\\label{eqn:7.1}\n\\tilde g=r^{-2}Adr\\otimes dr+B_i(dx^i\\otimes dr+dr\\otimes dx^i)+r^2C_{ij}dx^i\\otimes dx^j,\n\\end{equation}\nwith $A$, $B_i$ and $C_{ij}$ sufficiently differentiable functions (say $C^3$) of $x^{\\mu}$, with $x^0=r^{-1}$, \non the hypersurface $\\mathscr{I}$ defined by $x^0=0$ and in its neighbourhood. If the determinant \n\\begin{equation*}\n\\mathrm{det}\\left(\\begin{matrix}A & B_i \\\\ B_j & C_{ij}\\end{matrix}\\right)\n\\end{equation*}\ndoes not vanish, the space-time $(\\mathscr{M},g)$ with metric $g=\\Omega^2\\tilde{g}$, being  $\\Omega=r^{-1}$,\n\\begin{equation*}\ng=Adx^0\\otimes dx^0-B_i(dx^i\\otimes dx^0+dx^0\\otimes dx^i)+C_{ij}dx^i\\otimes dx^j\n\\end{equation*}\nis regular on $\\mathscr{I}$. It is clear that Schwarzschild space-time is just a particular case of this \nmore general situation described by \\eqref{eqn:7.1}. Furthermore, this metric includes all\nmetrics of Bondi-Sachs type and describes a situation where there is an isolated source (with \nasymptotic flatness) and outgoing gravitational radiation. Hence a regularity assumption for $\\mathscr{I}$ \nseems a not unreasonable one to impose if we wish to study asymptotically flat space-times and allow the \npossibility of gravitational radiation. In such situations, therefore, we expect a future-null conformal \ninfinity $\\mathscr{I}$ to exist. The choice made for $\\Omega$ possesses the important property that its \ngradient at $\\mathscr{I}$, $\\partial\\Omega\/\\partial x^{\\mu}=(1,0,0,0)$, is not vanishing and hence defines \na normal direction to $\\mathscr{I}$ ($\\mathscr{I}$ being described by the equation $\\Omega=0$).\\\\\nRoughly speaking, to say that a space-time is asymptotically flat means that its infinity is \n\\lq similar\\rq\\hspace{0.1mm} in some way to the Minkowski one. As a consequence we may expect the \nconformal structure at infinity of an asymptotically flat space-time \nto be similar to the one found for the Minkowski case.\\\\\nWith those ideas in mind we may now proceed to a rigorous definition of asymptotic simplicity for a \nspace-time. However we must also bear in mind that asymptotic flatness is, by itself, a mathematical idealization, and hence mathematical convenience and elegance constitute, by themselves, important criteria for selecting the appropriate idealization.\n\\begin{defn} $\\\\ $\n\\label{defn:AS}\nA space-time $(\\mathscr{\\tilde M},\\tilde g)$ is \\textit{$k$-asymptotically simple} if some $C^{k+1}$ \nsmooth manifold-with-boundary $\\mathscr{M}$, with metric $g$ and smooth boundary \n$\\mathscr{I}=\\mathscr{\\partial M}$ exists such that:\n\\begin{enumerate}\n\\item $\\mathscr{\\tilde M}$ is an open sub-manifold of $\\mathscr{M}$; \n\\item there exists a real-valued and positive function $\\Omega>0$, that is $C^k$ throughout $\\mathscr{M}$, \nsuch that $g_{ab}=\\Omega^2\\tilde g_{ab}$ on $\\mathscr{\\tilde{M}}$;\n\\item $\\Omega=0$ and $\\nabla_a\\Omega\\neq 0$ on $\\mathscr{I}$;\n\\item every null geodesic on $\\mathscr{M}$ has two endpoints on $\\mathscr{I}$.\n\\end{enumerate}\nThe space-time $(\\mathscr{\\tilde M},\\tilde g)$ is called \\textit{physical space-time}, while \n$(\\mathscr{M},g)$ is the \\textit{unphysical space-time}.\n\\end{defn}\n\\begin{defn}{\\cite{HawEll}}$\\\\ $\n\\label{defn:EAS}\nA space-time $(\\mathscr{\\tilde M},\\tilde g)$ is \\textit{$k$-asymptotically empty and simple} if it is \n$k$-asymptotically simple and if satisfies the additional condition\n\\begin{enumerate}[start=5]\n\\item $\\tilde{R}_{ab}=0$ on an open neighbourhood of $\\mathscr{I}$ in $\\mathscr{M}$ (this condition can \nbe modified to allow for the existence of electromagnetic radiation near $\\mathscr{I}$).\n\\end{enumerate}\n\\end{defn}\n\\begin{oss}$\\\\ $\nNote that there are many different definitions of asymptotic simplicity. We used here the one which is \ndue to \\cite{Penrin2}, but others which slightly differ from this are conceivable \\cite{Pen67,HawEll,Stew}.\n\\end{oss}\n\\begin{oss}$\\\\ $\nNote that, although the extended manifold $\\mathscr{M}$ and its metric are called \\lq unphysical\\rq, \nthere is nothing unphysical in this construction. The boundary of $\\mathscr{\\tilde{M}}$ in $\\mathscr{M}$ is \nuniquely determined by the conformal structure of $\\mathscr{\\tilde{M}}$ and, therefore, it is just \nas physical as $\\mathscr{\\tilde{M}}$. \n\\end{oss}\nNow we try to justify the previous assumptions.\\\\\nClearly with $1.$, $2.$ and $3.$ we mean to build $\\mathscr{I}$ as the null infinity of \n$(\\mathscr{\\tilde M},\\tilde{g})$, using the results obtained in the Minkowski case, with which it must share some \nproperties. Condition $4.$ ensures that the whole of null infinity is included in $\\mathscr{I}$. \nFurthermore, null geodesics in $\\mathscr{\\tilde M}$ correspond to null geodesics in $\\mathscr{M}$ because \nconformal transformations map null vectors to null vectors: the concept of null geodesic is conformally invariant. \nThus, we deduce that past and future infinity of any null geodesic in  \n$\\mathscr{\\tilde M}$ are points of $\\mathscr{I}$. Condition $5.$ ensures that the physical Ricci curvature \n$\\tilde{R}_{ab}$ vanishes in the asymptotic region far away from the source of the gravitational field. \nFinally note how the points $i^+$, $i^-$ and $i^0$ are ruled out from the definition of $\\mathscr{I}$, since \n$\\mathscr{I}$ is not a smooth manifold at these points. \nNow we briefly summarize some of the properties of an asymptotically simple space-time, under the assumption that \nthe vacuum Einstein equations hold and hence the cosmological constant equals zero.\n\\begin{itemize}\n\\item \\textit{$\\mathscr{I}$ is a null hypersurface}\\\\\nThis is because of condition $5.$ and condition $3.$. In fact it is easy to see that the Ricci scalar $R$ of \nthe metric $g_{ab}$ is related to the Ricci scalar $\\tilde{R}$ of the metric $\\tilde{g}_{ab}$ by \n\\begin{equation*}\n\\tilde{R}=\\Omega^{-2}R -6\\Omega^{-1}g^{cd}\\nabla_c\\nabla_d\\Omega+3\\Omega^{-2}g^{cd}\\nabla_c\\Omega\\nabla_d\\Omega,\n\\end{equation*}\nand hence, by multiplying both members by $\\Omega^2$, and by evaluating this equation on $\\mathscr{I}$ where \n$\\Omega=0$, it follows that $g^{cd}\\nabla_c\\Omega\\nabla_d\\Omega=0$. By condition $3.$, since \n$\\nabla_c\\Omega\\neq 0$, it follows that \n$g^{cd}\\nabla_c\\Omega\\nabla_d\\Omega =0$ and thus $\\nabla_c\\Omega$, the normal vector to $\\mathscr{I}$, is \nnull and, by definition, $\\mathscr{I}$ is a null hypersurface;\\\\\n\\item \\textit{$\\mathscr{I}$ is shear-free}\\\\\n$R_{ab}$ is related to $\\tilde{R}_{ab}$ by \n\\begin{equation*}\n\\tilde{R}_{ab}=R_{ab}-2\\Omega^{-1}\\nabla_a\\nabla_b\\Omega-g_{ab}(\\Omega^{-1}\\nabla_c\\nabla^c\\Omega\n-3\\Omega^{-2}\\nabla_c\\Omega\\nabla^c\\Omega).\n\\end{equation*}\nSince $\\nabla_c\\Omega$ is null and $R_{ab}$ is defined on $\t\\mathscr{I}^+$, if condition $5.$ holds, \nthe previous equation on $\\mathscr{I}^+$ leads to\n\\begin{equation*}\n2\\nabla_a\\nabla_b\\Omega+g_{ab}\\nabla_c\\nabla^c\\Omega=0. \n\\end{equation*}\nContracting with $g^{ab}$ it gives \n\\begin{equation*}\n\\nabla_c\\nabla^c\\Omega=0\\Rightarrow\\nabla_a\\nabla_b\\Omega=0.\n\\end{equation*}\nHence the normal vector to $\\mathscr{I}$ is divergence- and shear-free;\\\\\n\\item \\textit{$\\mathscr{I}^+$ has two connected components, $\\mathscr{I}^+$ and $\\mathscr{I}^-$, \neach of which has topology $S^2\\times\\mathbb{R}$}\\\\\nThe first proof of this theorem, involving sophisticated arguments, is due to \\cite{Pen65}. However, \nas remarked in \\cite{NewRP}, the arguments carried out by Penrose are incorrect, and a more rigorous \nproof can be found in \\cite{Ger71} or in \\cite{HawEll}. The significance of this property lies in the \nfact that the structure of the conformal infinity found for Minkowski space-time is that \nof any asymptotically simple space-time.\n\\end{itemize}\nAt this stage we must make a clarification. In fact we must point out that condition $4.$ is difficult to verify in practice and is not even satisfied by some space-times \nthat we would like to classify as asymptotically flat. As an example, for Schwarzschild space-time, \nit is known that there exist null circular orbits with radius $3m$, and hence do not terminate on \n$\\mathscr{I^+}$. For these reasons condition $4.$ is often too strong and gets replaced by a weaker \none that brings to the notion of weakly asymptotically simple space-time. \n\\begin{defn} $\\\\ $\n\\label{defn:WAS}\nA space-time ($\\mathscr{\\tilde M},\\tilde g)$ is \\textit{weakly asymptotically simple} if there exists an \nasymptotically simple space-time $(\\mathscr{\\tilde M'},\\tilde g')$ with associated unphysical space-time \n$(\\mathscr{M'},g')$, such that for a neighbourhood  $\\mathscr{H'}$ of $\\mathscr{I'}$ in $\\mathscr{M'}$, \nthe region $\\mathscr{\\tilde M'}\\cap\\mathscr{H'}$ is isometric to a similar neighbourhood \n$\\mathscr{\\tilde H}$ of $\\mathscr{\\tilde M}$.\n\\end{defn}\nIn this way a weakly asymptotically simple space-time possesses the same properties of the conformal \ninfinity of an asymptotically simple one, but the null geodesics do not necessary reach it because it \nmay have other infinities as well. Such space-times are essentially required to be isometric to an \nasymptotically simple space-time in a neighbourhood of $\\mathscr{I}$. \n\\begin{oss}$\\\\ $\nNote that the definition \\ref{defn:AF1} of asymptotic flatness seems to be completely different from \nthat of asymptotic simplicity \\ref{defn:AS} and weak asymptotic simplcity \\ref{defn:EAS}. However, \nthe two approaches are equivalent, as shown in \\cite{New62,NewUn}, since they lead to the same asymptotic \nproperties, using two different ways. It is worth noting that the conformal method introduced by \nPenrose represents a \\lq natural evolution\\rq\\hspace{0.1mm} of the previous one, being more geometrical.\n\\end{oss}\n\\section{Symmetries on $\\mathscr{I}$}\n\\label{sect:5}\nThe geometrical approach to asymptotic flatness, discussed in the previous section, affords us a \nmuch more vivid picture of the significance of the BMS group.\\\\\nThe idea is that, by adjoining to the physical space-time $(\\tilde{\\mathscr{M}},\\tilde{g})$ an appropriate \nconformal boundary $\\mathscr{I}$, as done in Sect. \\ref{sect:4}, we may obtain the asymptotic symmetries as \nconformal transformations of the boundary, the boundary having a much better chance of having a meaningful \nsymmetry group than $\\mathscr{\\tilde{M}}$. \\\\ We start by making an example to better understand the nature \nof the problem, which is due to \\cite{Pen72}. Consider Minkowski space-time with standard coordinates \n$(t,x,y,z)$, the metric being given by\n\\begin{equation*}\ng=\\eta_{ab}dx^a\\otimes dx^b=dt\\otimes dt-dx\\otimes dx-dy\\otimes dy-dz\\otimes dz,\n\\end{equation*} \nand consider the null cone $\\mathscr{N}$ through the origin, given by the equation \n\\begin{equation}\n\\label{eqn:162}\nt^2-x^2-y^2-z^2=0.\n\\end{equation}\nThe generators of $\\mathscr{N}$ are the null rays through the origin, given by\n\\begin{equation*}\nt:x:y:z=\\mathrm{const},\n\\end{equation*}\nwith $t,x,y,z$ satisfying \\eqref{eqn:162}. Let us consider $S^2$ to be the section of $\\mathscr{N}$ by \nthe spacelike 3-plane $t=1$. Then there exists a (1-1)-correspondence between the generators of $\\mathscr{N}$ \nand the points of $S^2$ (i.e. that given by the intersections of the generators with $t=1$). We may regard \n$S^2$ as a realization of the space of generators of $\\mathscr{N}$. However, we could have used any \nother cross-section $\\hat{S}^2$ of $\\mathscr{N}$ to represent this space. The important point is to \nrealize that the map which carries any one such cross-section into another, with points on the same \ngenerator of $\\mathscr{N}$ corresponding to one another, is a conformal map. The situation is \nreported in Figure \\ref{fig:6.1}.:\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.3]{generators.eps}\n\\caption{The generators of the null cone $\\mathscr{N}$ establish a 1-1 map between any two \ncross-sections of $\\mathscr{N}$.}\n\\label{fig:6.1}\n\\end{center}\n\\end{figure}\n\\\\The above mentioned map being conformal, the space of generators of $\\mathscr{N}$ may itself be \nassigned a conformal structure, i.e. that of any of these sections. To see that the map is conformal \nwe may re-express the induced metric on $\\mathscr{N}$ in the form\n\\begin{equation}\n\\label{eqn:163}\ng_{_{\\mathscr{N}}}=-r^2\\gamma_{\\alpha\\beta}(x^{\\gamma})dx^{\\alpha}\\otimes dx^{\\beta}+0\\cdot dr\\otimes dr,\n\\end{equation}\nwhere $x^{\\alpha}$ and $r$ are coordinates on $\\mathscr{N}$, the generators being given by the coordinate \nlines $x^{\\alpha}=\\mathrm{const}$ (the term \\lq$0$\\rq\\hspace{0.1mm} takes into account that, the surface \n$\\mathscr{N}$ being null, its induced metric is degenerate, i.e. with vanishing determinant). There exist \nobviously many ways of attaining the form \\eqref{eqn:163}. One is to use ordinary spherical coordinates \nfor Minkowski space-time, giving $g_{_{\\mathscr{N}}}=-r^2(d\\theta\\otimes d\\theta+\\sin^2\\theta d\\phi\\otimes d\\phi)\n+0\\cdot dr\\otimes dr$. Since a cross-section of $\\mathscr{N}$ is given by specifying $r$ as function of \n$x^{\\alpha}$ it is clear that any two cross-sections give conformally related metrics, being mapped to one \nanother by the generators of $\\mathscr{N}$. It is now obvious that many other cone-like null surfaces will \nshare this property of $\\mathscr{N}$, provided their metrics can be put in the form \\eqref{eqn:163}. \nNow if we suppose here to deal with an empty asymptotically simple space-time $(\\mathscr{\\tilde{M}},\\tilde{g})$ \n(according to definition \\ref{defn:EAS}, with associated unphysical space-time $(\\mathscr{M},g)$) \nwe know that, if $\\mathscr{I}$ is null,  it has the important property to be shear-free, as discussed in \nSect. \\ref{sect:4}. Physically, the shear-free nature of the generators of $\\mathscr{I}$ tells us that \nsmall shapes are preserved as we follow these generators along $\\mathscr{I}$. Hence any diffeomorphism \nwhich maps each null generator of $\\mathscr{I}^+$ into itself is a conformal transformation for any metric \non $\\mathscr{I}^+$. That is to say, if we take any two cross-sections $S_1$ and $S_2$ of $\\mathscr{I}^+$ \nor $\\mathscr{I}^-$, then the correspondence between $S_1$ and $S_2$ established by the generators is a \nconformal one. This is exactly the same situation we encountered in the example with $\\mathscr{N}$. \nWe have the following\n\\begin{prop}$\\\\ $\n\\label{prop:gen}\nIf $\\mathscr{I}$ is null, then any two cross-sections of $\\mathscr{I}^{\\pm}$ are mapped to one another \nconformally by the generators of $\\mathscr{I}^{\\pm}$. \n\\end{prop}\nIn Sect. \\ref{sect:4} we have discussed that the topology of $\\mathscr{I}^{\\pm}$ is $S^2\\times\\mathbb{R}$, \nwhere the $\\mathbb{R}$ factor may be taken as the null-geodesic generator $\\mathscr{I}^{\\pm}$. Hence these \ngenerators, by proposition \\ref{prop:gen}, establish a conformal mapping between any two $S^2$ cross-sections \nof $\\mathscr{I}^{\\pm}$, these sections being of course conformal spheres. It is a theorem that any conformal \n2-surface with the topology of a sphere $S^2$ is conformal to the unit 2-sphere in Euclidean 3-space. \nThus we can assume without loss of generality, that the conformal factor $\\Omega$ has been chosen so that \nsome cross-section $S$ has unphysical squared line element $-ds^2$ of a unit 2-sphere. Given one choice of \n$\\Omega$, we can always make a new choice $\\Omega'=\\Theta\\Omega$ which again has the property of vanishing at \n$\\mathscr{I}$ with non-zero gradient there. The factor $\\Theta$ has to be an arbitrary smooth positive function \non $\\mathscr{I}$ and can be chosen to rescale the metric on $\\mathscr{I}$ as we please. It is worth noting \nthat the shear-free condition can be saved by the change $\\Omega'=\\Theta\\Omega$, as discussed in \\cite{Stew}. \nThis property can be interpreted as a \\lq gauge freedom\\rq\\hspace{0.1mm} in the choice of the conformal \nfactor $\\Omega$. We can use this freedom to set the metric of a continuous sequence of cross-sections \nalong the generators equal to that of $S$. Hence, in spherical polar coordinates the induced metric \non $\\mathscr{I}^{+}$ is \n\\begin{equation}\n\\label{eqn:164}\ng_{_{\\mathscr{I^+}}}=d\\theta\\otimes d\\theta+\\sin^2\\theta d\\phi\\otimes d\\phi+0\\cdot du\\otimes du,\n\\end{equation}\nwhere $u$ is a retarded time coordinate, i.e. a parameter defined along each generator increasing monotonically \nwith time from $-\\infty$ to $+\\infty$, the corresponding form with an advanced time coordinate $v$ in place \nof $u$ holding for $\\mathscr{I}^{-}$. The surfaces $u=\\mathrm{const}$ are cross-sections of $\\mathscr{I}^+$, \neach of which has the metric of a unit 2-sphere, as is clear from \\eqref{eqn:164}. \\\\\nFrom the above discussion it follows that the metric on $\\mathscr{I}^+$ belongs to an equivalence class of \nmetrics, two elements being equivalent if they are conformally related one to the other. Hence, the form \nof the metric \\eqref{eqn:164} is just one element of this equivalence class that we have chosen as \nrepresentative. Let us consider the group of conformal transformations of $\\mathscr{I}^{+}$, i.e. the \ngroup of transformations which conformally preserve the metric \\eqref{eqn:164}. It is clear that any smooth \ntransformation which maps each generator into itself will be allowable:\n\\begin{equation}\n\\label{eqn:165}\nu\\rightarrow u'=F(u,\\theta,\\phi),\n\\end{equation}\nwith $F$ smooth on the whole $\\mathscr{I}^+$ and $\\partial F\/\\partial u >0$, since it has to map the whole \nrange for $u$ to itself, for any $\\theta$ and $\\phi$. In addition, we can allow conformal transformations \nof the $(\\theta,\\phi)$-sphere into itself. These transformations can be regarded as those of the \ncompactified complex plane $\\mathbb{C}\\cup\\{\\zeta=\\infty\\}$ into itself. \nIntroducing the complex stereographic coordinate\n\\begin{equation*}\n\\zeta=e^{i\\phi}\\cot\\frac{\\theta}{2},\n\\end{equation*}\nwe have that \\eqref{eqn:164} may be written as \n\\begin{equation}\n\\label{eqn:167}\ng_{_{\\mathscr{I^+}}}=\\frac{2(d\\zeta \\otimes d\\bar{\\zeta}+d\\bar{\\zeta}\\otimes d\\zeta)}\n{(1+\\zeta\\bar{\\zeta})^2}+0\\cdot du\\otimes du.\n\\end{equation}\nThen the most general conformal transformation of the compactified plane is given by\n\\begin{equation}\n\\label{eqn:166}\n\\zeta\\rightarrow\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d},\n\\end{equation}\nwhere $a$,$b$,$c$,$d\\in\\mathbb{C}$, that can be normalized to satisfy $ad-bc=1$.\n\\begin{oss}$\\\\ $\nSince conformal transformations can be equivalently expressed in terms of $x^A$ or $\\zeta$ coordinates, \nin the remainder we will use both of them, depending on the convenience.\n\\end{oss}\nThe particular functional form of the transformations in \\eqref{eqn:166} results from the request that \nthey must be diffeomorphisms of the compactified plane $\\mathbb{C}\\cup\\{\\zeta=\\infty\\}$  into itself. \nHence the transformations must have at least one pole, at $\\zeta^*$ say, corresponding to the point \nthat is mapped to the north pole $F(\\zeta^*)=\\infty$ and at least one zero, at $\\zeta^{**}$ say, \ncorresponding to the point that is mapped to the south pole $F(\\zeta^{**})=0$. Thus, the transformations \nmust be some rational complex function where the roots of the numerator and the denominator correspond \nto the points that are mapped to the south and the north pole, respectively. Since the transformation \nmust be injective there must be one, and only one, point that is mapped to the south pole, and also \nexactly one other point that is mapped to the north pole. This requires that both numerator and denominator \nbe linear functions of $\\zeta$. Requiring this map to be surjective finally imposes that the complex \nnumbers $a,b,c,d$ in \\eqref{eqn:166} must satisfy $ad-bc\\neq0$ (all of these parameters can be appropriately \nrescaled to get $ad-bc=1$ leaving the transformation unchanged). It is worth remarking that in pure \nmathematics these transformations were studied by Poincar\\'e and other authors when they developed the \ntheory of what are nowadays called automorphic functions, i.e. meromorphic functions such that \n$f(z)=f((az+b)\/(cz+d))$ \\cite{PO1912}. It is easy to see that these transformations contain:\n\\begin{itemize}\n\\item Translations $\\zeta\\rightarrow \\zeta'=\\zeta+b,\\hspace{1cm}b\\in\\mathbb{C}$;\n\\item Rotations $\\zeta\\rightarrow \\zeta'=e^{i\\theta}\\zeta,\\hspace{1cm}\\theta\\in\\mathbb{R}$;\n\\item Dilations $\\zeta\\rightarrow \\zeta'=e^{-\\chi}\\zeta,\\hspace{1cm}\\chi\\in\\mathbb{R}$;\n\\item Special transformations $\\zeta\\rightarrow \\zeta'=\n-\\displaystyle{\\frac{b^2}{\\zeta^2}},\\hspace{1cm}b\\in\\mathbb{C}$;\n\\end{itemize}\nAny transformation of the form \\eqref{eqn:166} can be obtained as the composition of a special transformation, \na translation, a rotation and a dilation.\\\\\nUsually, transformations \\eqref{eqn:166} are referred to as the \\textit{conformal group} (in two dimensions), \nthe \\textit{projective linear group}, the \\textit{M\\\"{o}bius transformations} or the \\textit{fractional linear \ntransformations}, and is denoted by PSL$(2,\\mathbb{C})\\cong\\mathrm{SL}(2,\\mathbb{C})\/\\mathbb{Z}_2$ \n(as will be discussed in the next section). Under these transformations we have\n\\begin{equation*}\n\\frac{2(d\\zeta'\\otimes d\\bar{\\zeta'}+d\\bar{\\zeta'}\\otimes d\\zeta')}{(1+\\zeta'\\bar{\\zeta'})^2}\n=K^2(\\zeta,\\bar{\\zeta})\\frac{2(d\\zeta\\otimes d\\bar{\\zeta}+d\\bar{\\zeta}\\otimes d\\zeta)}\n{(1+\\zeta\\bar{\\zeta})^2}\\Rightarrow g'_{_{\\mathscr{I}^+}}=K^2g_{_{\\mathscr{I}^+}},\n\\end{equation*}\nwith \n\\begin{equation}\n\\label{eqn:170}\nK(\\zeta,\\bar{\\zeta})=\\frac{1+\\zeta\\bar{\\zeta}}{(a\\zeta+b)(\\bar{a}\\bar{\\zeta}+\\bar{b})\n+(c\\zeta+d)(\\bar{c}\\bar{\\zeta}+\\bar{d})}.\n\\end{equation}\nIt can be shown that transformations \\eqref{eqn:166} are equivalent to \\eqref{eqn:182} and \\eqref{eqn:183}, \nand that the conformal factor $K$ in \\eqref{eqn:170} is the same as one in \\eqref{eqn:171}, \nexpressed in terms of the $(\\theta,\\phi)$ variables.\\\\\n\\begin{defn}$\\\\ $\nThe group of transformations\n\\begin{subequations}\n\\begin{align}\n\\label{eqn:172}\n&\\zeta\\rightarrow\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d},\\\\\n\\label{eqn:172a}\n&u\\rightarrow u'=F(u,\\theta,\\phi),\n\\end{align}\n\\end{subequations}\nwith $ad-bc=1$ and with $F$ smooth and $\\partial F\/\\partial u>0$ is the \\textit{Newman-Unti (NU) group}.\n\\end{defn}\n\\begin{oss}$\\\\ $\nNote that \\eqref{eqn:172} are the non-reflective conformal transformations of the $S^2$-space of \ngenerators of $\\mathscr{I}^+$ (the conformal structure being defined equivalently by one of its cross-sections), \nwhile \\eqref{eqn:172a}, when \\eqref{eqn:172} is the identity ($a=d=1$, $b=c=0$), give the general non-reflective \nsmooth transformations of the generators to  themselves.\n\\end{oss}\nThe conformal metric \\eqref{eqn:167} is considered to be part of the universal intrinsic structure of \n$\\mathscr{I}^+$ (universal, in the sense that any space-time which is asymptotically simple and vacuum near \n$\\mathscr{I}$ has a $\\mathscr{I}^+$ metric and similarly a $\\mathscr{I}^-$ metric which is conformal to \n\\eqref{eqn:167}). Hence the NU group can be regarded as the group of non-reflective transformations of \n$\\mathscr{I}^+$ preserving its intrinsic (degenerate) conformal metric \\cite{Pen72,Pen82}.\\\\\nHowever, the NU group is different from the BMS group, the former being larger than the latter. In fact the NU \ngroup allows a greater freedom in the function $F$, while in the BMS group $F$ is constrained to be of the \nform \\eqref{eqn:184}. Thus, we want to be somehow able to reduce this freedom, assigning a further geometric \nstructure to $\\mathscr{I}^+$, the preservation of which will furnish the BMS group, restricting exactly the form of $F$ to be the one of \n\\eqref{eqn:184}. This additional structure is referred to as the \\textit{strong conformal geometry} \n\\cite{Penrin2,Pen72}. The most direct way to specify this structure is the following. Consider a \nreplacement of the conformal factor,\n\\begin{equation}\n\\label{eqn:173}\n\\Omega\\rightarrow\\Omega'=\\Theta\\Omega.\n\\end{equation}\nWe choose the function $\\Theta$ to be smooth and positive on $\\mathscr{M}$ and nowhere vanishing on \n$\\mathscr{I}^+$. Under \\eqref{eqn:173} the metric transforms as \n\\begin{equation*}\ng_{ab}\\rightarrow g'_{ab}=\\Theta^2g_{ab},\\hspace{1cm}g^{ab}\\rightarrow g'^{ab}=\\Theta^{-2}g^{ab},\n\\end{equation*}\nand the normal co-vector to $\\mathscr{I}^{+}$ as\n\\begin{equation*}\nN_a=-\\nabla_a\\Omega\\rightarrow N'_a=-\\nabla'_a\\Omega'=-\\nabla_a\\Omega'=-\\Omega\\nabla_a\\Theta\n-\\Theta\\nabla_a\\Omega\\approx\\Theta N_a,\n\\end{equation*}\nwhile the vector\n\\begin{equation*}\nN^a=g^{ab}\\partial_b\\Omega\\rightarrow N'^a=g'^{ab}N'_b\\approx\\Theta^{-1}N^a,\n\\end{equation*}\nwhere we introduced the \\lq weak equality\\rq\\hspace{0.1mm} symbol $\\approx$. Considering two fields \n$\\psi^{...}_{...}$ and $\\phi^{...}_{...}$, saying that \n\\begin{equation}\n\\label{eqn:80}\n\\psi^{...}_{...}\\approx\\phi^{...}_{...}\n\\end{equation}\nmeans that $\\psi^{...}_{...}-\\phi^{...}_{...}=0$ on $\\mathscr{I}$. \nThe line element $dl$ of $\\mathscr{I}^+$ rescales according to \n\\begin{equation}\n\\label{eqn:176}\ndl\\rightarrow dl'=\\Theta dl.\n\\end{equation}\nHaving done any allowable choice of the conformal factor $\\Omega$, through the function $\\Theta$, and hence \nsome specific choice of the metric $dl$ for cross-sections of $\\mathscr{I}^+$, then it is defined, from \n$N_a=-\\nabla_a\\Omega$, a precise scaling for parameters $u$ on the generators of $\\mathscr{I}^+$, fixed by\n\\begin{equation*}\n\\frac{\\partial}{\\partial u}=N^a\\nabla_a,\\hspace{0.7cm}\\mathrm{i.e.}\\hspace{0.7cm}N^a\\nabla_a u=1.\n\\end{equation*}\nUnder \\eqref{eqn:173} we see that to keep the scaling of the parameters $u$ along the generators fixed we must choose\n\\begin{equation}\n\\label{eqn:175}\ndu\\rightarrow du'=\\Theta du, \n\\end{equation}\nso that \n\\begin{equation*}\nN^a\\nabla_a u\\rightarrow N'^a\\nabla'_au'=N'^a\\nabla_au'=N'^a\\frac{\\partial u'}{\\partial x^a}\n=\\Theta^{-1}\\Theta N^a\\nabla_a u=1.\n\\end{equation*}\nAll the parameters $u$, linked by \\eqref{eqn:175}, scale in the same way along the generators of $\\mathscr{I}^+$. \nFrom \\eqref{eqn:176} and \\eqref{eqn:175} we see that the ratio\n\\begin{equation}\n\\label{eqn:177}\ndl:du\n\\end{equation}\nremains invariant and it is independent of the choice of the conformal factor $\\Omega$. It is the invariant \nstructure provided by \\eqref{eqn:177} that can be taken to define the strong conformal geometry. To better \nreformulate this invariance we introduce the concept of \\textit{null angle} \\cite{Pen63,Penrin2,Pen72,BMS75}. \nConsider two non-null tangent directions at a point $P$ of $\\mathscr{I}^+$. Let $[X]$ and $[Y]$ be such \ndirections. If no linear combination of $X\\in [X]$ and $Y\\in [Y]$ is the null tangent direction at $P$, \nthen the angle between $[X]$ and $[Y]$ is defined by the metric \\eqref{eqn:164}. However, if the null tangent \ndirection at $P$ is contained in the plane spanned by $[X]$ and $[Y]$, then the angle between $[X]$ and $[Y]$ \nalways vanishes. To see this choose $X\\in [X]$, $Y\\in [Y]$ and $N\\in [N]$ ($[N]$ being the null direction \ntangent at $P$), such that $Y=X+N$. Then since $N$ is null we have, using the metric $g$ on $\\mathscr{I}^+$ \ngiven in \\eqref{eqn:164} (and hence any other one of its equivalence class)\n\\begin{equation*}\n0=g(N,X)=g(Y-X,X)=g(Y,X)-g(X,X),\n\\end{equation*}\nand \n\\begin{equation*}\n0=g(N,Y)=g(Y-X,Y)=g(Y,Y)-g(X,Y),\n\\end{equation*}\nfrom which the angle $\\theta$ between $[X]$ and $[Y]$, given by \n\\begin{equation*}\n\\cos\\theta=\\frac{g(X,Y)}{\\sqrt{g(X,X)g(Y,Y)}}=1,\n\\end{equation*}\nvanishes. However, if we require the strong conformal geometry structure to hold and hence the invariance of \nthe ratio \\eqref{eqn:177} we can numerically define the null angle $\\nu$ between two tangent directions at \na point $P$ of $\\mathscr{I}^{+}$ by \n\\begin{equation}\n\\label{eqn:178}\n\\nu=\\frac{\\delta u}{\\delta l},\n\\end{equation}\nwhere the infinitesimal increments $\\delta u$ and $\\delta l$ are as indicated in Figure \\ref{fig:6.3} \\cite{Penrin2}.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.4]{nullangle.eps}\n\\caption{A null angle $\\nu$ on $\\mathscr{I}^+$, given by $\\nu=\\delta u\/\\delta l$, is defined between a pair \nof directions on $\\mathscr{I}^+$ whose span contains the null normal direction to $\\mathscr{I}^+$.}\n\\label{fig:6.3}\n\\end{center}\n\\end{figure}\n\\\\By virtue of the strong conformal geometry, under change of the conformal factor for the metric of $\\mathscr{I}^+$, null angles \nremain invariant. For further insights about the strong conformal geometry and the interpretation of null angles we suggest to read \\cite{Pen72} or \\cite{BMS75}. \\\\A transformation of $\\mathscr{I}^+$ to itself which \npreserves angles and null angles, i.e. that respects the strong conformal geometry structure, must have \nthe effect that any expansion (or contraction) of the spatial distances $dl$ is accompanied by an equal \nexpansion (or contraction) of the scaling of the special $u$ parameters. The allowed transformations\nhave the form \\eqref{eqn:172}, where function $F$ must now have the  precise form that allows the ratio \n$du:dl$ to remain invariant. Under the transformation \\eqref{eqn:173} we have, as seen, that the sphere \nof the cross-section of $\\mathscr{I}^+$ undergoes a conformal mapping, i.e.\n\\begin{equation*}\ndl\\rightarrow dl'=\\Theta dl.\n\\end{equation*}\nSince $\\Theta$ is the conformal factor of the transformation, it depends only on $\\theta$ and $\\phi$ or, \nequivalently, on $\\zeta$ and $\\bar{\\zeta}$ and must have the form given in \\eqref{eqn:170}. \nWe must therefore also have \n\\begin{equation*}\ndu\\rightarrow du'=\\Theta du, \n\\end{equation*}\nIntegrating we get\n\\begin{equation*}\nu\\rightarrow u'=\\Theta[u+\\alpha(\\zeta,\\bar{\\zeta})],\n\\end{equation*}\nwhere $\\Theta$ assumes the form\n\\begin{equation*}\n\\Theta(\\zeta,\\bar{\\zeta})=\\frac{1+\\zeta\\bar{\\zeta}}{(a\\zeta+b)(\\bar{a}\\bar{\\zeta}+\\bar{b})\n+(c\\zeta+d)(\\bar{c}\\bar{\\zeta}+\\bar{d})},\n\\end{equation*}\nwith $a,b,c,d\\in\\mathbb{C}$ and $ad-bc=1$. By virtue of definition \\ref{defn:BMS} we have obtained the following\n\\begin{prop}$\\\\ $\nThe group of conformal transformations of $\\mathscr{I}^+$ which preserve the strong conformal geometry, i.e. \nboth angles and null angles, is the BMS group.\n\\end{prop}\n\\begin{oss}$\\\\ $\nConformal transformations, and hence the NU group, always preserve finite angles, but null angles, i.e. \nangles between tangent vectors of which $N^a$ is a linear combination, are preserved by the BMS group only. \n\\end{oss}\nThe general form of a BMS transformation is thus\n\\begin{subequations}\n\\begin{equation}\n\\label{eqn:200}\n\\zeta\\rightarrow\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d},\n\\end{equation}\n\\begin{equation}\n\\label{eqn:201}\nu\\rightarrow u'=\\frac{(1+\\zeta\\bar{\\zeta})[u+\\alpha(\\zeta,\\bar{\\zeta})]}{(a\\zeta+b)(\\bar{a}\\bar{\\zeta}\n+\\bar{b})+(c\\zeta+d)(\\bar{c}\\bar{\\zeta}+\\bar{d})},\n\\end{equation}\n\\end{subequations}\n\\\\with $a,b,c,d\\in\\mathbb{C}$ and $ad-bc=1$. Clearly the BMS group is a subgroup of the NU group, the \nfunction $F$ having its form fixed. However it is still an infinite-dimensional function-space group.\n\\section{Structure of the BMS group}\n\\label{sect:6}\nWe discuss first the BMS transformations obtained by setting $\\alpha=0$,\n\\begin{equation}\n\\label{eqn:220}\nu\\rightarrow u'=Ku,\\hspace{1cm}\\zeta\\rightarrow\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d},\n\\end{equation}\nwith \n\\begin{equation*}\nK(\\zeta,\\bar{\\zeta})=\\frac{1+\\zeta\\bar{\\zeta}}{(a\\zeta+b)(\\bar{a}\\bar{\\zeta}+\\bar{b})\n+(c\\zeta+d)(\\bar{c}\\bar{\\zeta}+\\bar{d})},\n\\end{equation*}\n\\\\i.e. a rescaling for $u$ and a transformation of the group $\\mathrm{PSL}(2,\\mathbb{C})$ for $\\zeta$. \nAny of these transformations is specified by the 4 constants $a,b,c,d\\in\\mathbb{C}$, satisfying $ad-bc=1$. \nHence there are only 3 independent complex parameters, i.e. 6 independent real parameters. \nAny element $f\\in \\mathrm{PSL}(2,\\mathbb{C})$ reads as\n\\begin{equation*}\nf=\\frac{a\\zeta+b}{c\\zeta+d}\\equiv\\{a,b,c,d\\}.\n\\end{equation*}\nNote that under simultaneous change $a\\rightarrow-a$, $b\\rightarrow-b$, $c\\rightarrow-c$, \n$d\\rightarrow-d$ any element $f\\in\\mathrm{PSL}(2,\\mathbb{C})$ remains unaffected, i.e.\n\\begin{equation}\n\\label{eqn:203}\nf=\\{a,b,c,d\\}=\\{-a,-b,-c,-d\\}.\n\\end{equation}\nNow take into account the group $\\mathrm{SL}(2,\\mathbb{C})$ of $(2\\times 2)$ complex matrices \n$Q$ with $\\mathrm{det}[Q]=1$:\n\\begin{equation*}\nQ=\\left(\\begin{matrix}A& B\\\\ C& D\n\\end{matrix}\n\\right),\\hspace{1cm}\\mathrm{det}[Q]=AD-BC=1,\\hspace{1cm}A,B,C,D\\in\\mathbb{C}.\n\\end{equation*} \nClearly the dimension of $\\mathrm{SL}(2,\\mathbb{C})$ is 6. Hence we can consider a map $\\tilde{\\varphi}$, \nbetween $\\mathrm{PSL}(2,\\mathbb{C})$ and $\\mathrm{SL}(2,\\mathbb{C})$ defined by\n\\begin{equation}\n\\label{eqn:202}\n\\tilde{\\varphi}:f=\\{a,b,c,d\\}\\in \\mathrm{PSL}(2,\\mathbb{C})\\longrightarrow\\tilde{\\varphi}(f)\n=\\left(\\begin{matrix}a & b\\\\\nc & d\\end{matrix}\\right)\\in\\mathrm{SL}(2,\\mathbb{C}).\n\\end{equation}\nIt is easy to show that, since the group operation of $\\mathrm{PSL}(2,\\mathbb{C})$ is the function \ncomposition $\\circ$, given $f=\\{a,b,c,d\\}$ and $g=\\{a',b',c',d'\\}\\in\\mathrm{PSL}(2,\\mathbb{C})$ we have\n\\begin{equation*}\nf\\circ g=\\{aa'+bc',ab'+bd',ca'+dc',cb'+dd'\\}\\in\\mathrm{PSL}(2,\\mathbb{C}).\n\\end{equation*}\nThen taking the images of $f$ and $g$ through $\\tilde{\\varphi}$,\n\\begin{equation*}\n\\tilde{\\varphi}(f)=\\left(\\begin{matrix}a & b\\\\\nc & d\\end{matrix}\\right),\\hspace{1cm}\\tilde{\\varphi}(g)=\\left(\\begin{matrix}a' & b'\\\\\nc' & d'\\end{matrix}\\right),\n\\end{equation*}\nwe have, since the operation in $\\mathrm{SL}(2,\\mathbb{C})$ is the ordinary matrix product,\n\\begin{equation}\n\\label{eqn:204}\n\\tilde{\\varphi}(f)\\cdot\\tilde{\\varphi}(g)=\\left(\\begin{matrix}aa'+bc' &ab'+bd'\\\\\nca'+dc' & cb'+dd'\\end{matrix}\\right)=\\tilde{\\varphi}(g\\circ f).\n\\end{equation}\nNote that, by virtue of \\eqref{eqn:203}, to the same element $f$ there correspond, through $\\tilde{\\varphi}$, \ntwo different elements, $\\tilde{\\varphi}(f)$ and $-\\tilde{\\varphi}(f)$. If we consider now the map $\\varphi$:\n\\begin{equation*}\n\\varphi:f=\\{a,b,c,d\\}\\in \\mathrm{PSL}(2,\\mathbb{C})\\longrightarrow\\varphi(f)=\\left(\\begin{matrix}a & b\\\\\nc & d\\end{matrix}\\right)\\in\\mathrm{SL}(2,\\mathbb{C})\/\\mathbb{Z}_2.\n\\end{equation*}\nit is clear that property \\eqref{eqn:204} holds for $\\varphi$ as well. This map is a group isomorphism, \n$\\varphi(f)$ and $-\\varphi(f)$ being now identified in $\\mathrm{SL}(2,\\mathbb{C})\/\\mathbb{Z}_2$. We can state\n\\begin{equation}\n\\label{eqn:205}\n\\mathrm{PSL}(2,\\mathbb{C})\\cong\\mathrm{SL}(2,\\mathbb{C})\/\\mathbb{Z}_2.\n\\end{equation}\nThe group $\\mathrm{SL}(2,\\mathbb{C})$ is the double covering of $\\mathrm{PSL}(2,\\mathbb{C})$. \nFurthermore, it is a well known result that\n\\begin{equation}\n\\label{eqn:210}\n\\mathscr{L}\\cong\\mathrm{SL}(2,\\mathbb{C})\/\\mathbb{Z}_2,\n\\end{equation}\nwhere $\\mathscr{L}$ is the connected component of the Lorentz group. Thus\n\\begin{equation}\n\\label{eqn:211}\n\\mathrm{PSL}(2,\\mathbb{C})\\cong \\mathscr{L}.\n\\end{equation}\nWe have the following\n\\begin{prop}$\\\\ $\nThe connected component of the Lorentz group is isomorphic with the subgroup $\\mathrm{PSL}(2,\\mathbb{C})$ of the BMS group.\n\\end{prop}\nTo make this isomorphism explicit take an element $f=\\{a,b,c,d\\}\\in\\mathrm{PSL}(2,\\mathbb{C})$ and through \n$\\varphi$ assign to it an element of $\\mathrm{SL}(2,\\mathbb{C})$,\n\\begin{equation*}\n\\varphi(f)=\\left(\\begin{matrix}a & b\\\\ c & d\n\\end{matrix}\\right)\\in\\mathrm{SL}(2,\\mathbb{C}),\\hspace{1cm}ad-bc=1.\n\\end{equation*}\nThen, using the isomorphism of \\eqref{eqn:210} it can be shown with lengthy calculations \\cite{Oblak16} \nthat to $\\varphi(f)$ there corresponds an element of $\\mathscr{L}$ given by the matrix\n\\begin{equation}\n\\label{eqn:iso}\n\\Lambda(f)=\n\\end{equation}\n\\begin{equation*}\n\\left(\\begin{matrix}\\frac{1}{2}\\left(|a|^2+|b|^2+|c|^2+|d|^2\\right) & -\\mathrm{Re}\\left\\{a\\bar{b}+c\\bar{d}\\right\\} \n& \\mathrm{Im}\\left\\{a\\bar{b}+c\\bar{d}\\right\\} & \\frac{1}{2}\\left(|a|^2-|b|^2+|c|^2-|d|^2\\right) \\\\ \n-\\mathrm{Re}\\left\\{\\bar{a}c+\\bar{b}d\\right\\} & \\mathrm{Re}\\left\\{\\bar{a}d+\\bar{b}c\\right\\} \n& -\\mathrm{Im}\\left\\{a\\bar{d}-b\\bar{c}\\right\\} & - \\mathrm{Re}\\left\\{\\bar{a}c+\\bar{b}d\\right\\}\\\\\n\\mathrm{Im}\\left\\{\\bar{a}c+\\bar{b}d\\right\\} & -\\mathrm{Im}\\left\\{\\bar{a}d+\\bar{b}c\\right\\} \n& \\mathrm{Re}\\left\\{a\\bar{d}-b\\bar{c}\\right\\} &  \\mathrm{Im}\\left\\{\\bar{a}c+\\bar{b}d\\right\\}\\\\\n\\frac{1}{2}\\left(|a|^2+|b|^2-|c|^2-|d|^2\\right) & -\\mathrm{Re}\\left\\{a\\bar{b}-c\\bar{d}\\right\\} \n& \\mathrm{Im}\\left\\{a\\bar{b}-c\\bar{d}\\right\\} & \\frac{1}{2}\\left(|a|^2-|b|^2-|c|^2+|d|^2\\right)\n\\end{matrix}\\right).\n\\end{equation*}\\\\\nAt this stage, the relation between the Lorentz group and the sphere appears as a mere coincidence. \nIn particular, since the original Lorentz group is defined by its linear action on a four-dimensional space, \nthere is no reason for it to have anything to do with certain non-linear transformations of a two-dimensional \nmanifold such as the sphere. However, it can be shown that this is not accidental. Following \\cite{Oblak16}, \nwe can suppose to perform a Lorentz transformation in Minkowski space-time \nequipped with standard coordinates $(t,x,y,z)$, i.e.\n\\begin{equation*}\nx'^{\\mu}=\\Lambda^{\\mu}_{\\nu}x^{\\nu},\\hspace{1cm}\\Lambda\\in \\mathscr{L}.\n\\end{equation*} \nWe may introduce Bondi coordinates $(u,r,x^A)$ for Minkowski space-time as done in \\eqref{eqn:135}. \nThen if we evaluate the limit for large values of the radial coordinate $r=\\sqrt{x^2+y^2+z^2}$ \nkeeping the value of $u=t-r$ fixed (i.e. on $\\mathscr{I}$) and use the isomorphism \\eqref{eqn:210} \nand hence \\eqref{eqn:iso} we obtain the following behaviour:\n\\begin{equation*}\n\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d}+O(r^{-1}).\n\\end{equation*}\nwhere\n$\\zeta=e^{i\\phi}\\cot\\frac{\\theta}{2}$. Furthermore it can be checked that both $u$ and $r$, under \nthe effect of a Lorentz transformation on $\\mathscr{I}$, undergo an angle-dependent rescaling. Hence \nwe have obtained a fundamental result: Lorentz transformations acting on $\\mathscr{I}$, expressed in \nterms of the parameters $a,b,c,d$ coincide with conformal transformations of $\\mathrm{PSL}(2,\\mathbb{C})$. \nSince asymptotically flat space-times have the same structure of a Minkowski space-time at infinity, \nthis argument can be extended to all of them too. In the remainder we will use $\\mathscr{L}$ to describe \nthe group structure of $\\mathscr{B}$, the isomorphism with $\\mathrm{PSL}(2,\\mathbb{C})$ being implicit.\\\\\nNow we turn to analyse the transformations which involve a non-vanishing $\\alpha(\\theta,\\phi)$. \n\\begin{defn}$\\\\ $\n\\label{defn:suptra}\nThe Abelian subgroup of BMS transformations for which\n\\begin{equation}\n\\label{eqn:225}\n\\theta'=\\theta,\\hspace{0.5cm}\\phi'=\\phi,\\hspace{0.5cm}u'=u+\\alpha(\\theta,\\phi),\n\\end{equation}\nis called \\textit{supertranslation subgroup} and is denoted by $\\mathscr{S}$.\n\\end{defn}\nUnder such a transformation the system of null hypersurfaces $u=\\mathrm{const}$ is transformed into another \nsystem of null hypersurfaces $u'=\\mathrm{const}$.  \\\\\nTo proceed further in the analysis of the structure of the BMS group we need to recall the concepts of \n\\textit{right} and \\textit{left} \\textit{cosets} and, hence, that of \\textit{normal subgroup} \\cite{Algebra}. \nConsider a group $G$ and a subgroup $H$ of $G$. Introduce in $G$ the equivalence relation \n$\\sim$ defined, for $g$, $a\\in G$, as\n\\begin{equation*}\ng\\sim a \\Longleftrightarrow ag^{-1}\\in H\\Longleftrightarrow a\\in Hg.\n\\end{equation*}\nIt is easy to verify that the previous relation is reflexive, symmetric and transitive. \n\\begin{defn}$\\\\ $\nThe equivalence class with respect to $\\sim$ is called \\textit{right coset of $H$ in $G$ with respect to $g$} \nand is denoted by $Hg$:\n\\begin{equation*}\n[g]=\\{hg : h\\in H\\}= Hg.\n\\end{equation*}\n\\end{defn}\nSimilarly, the \\textit{left coset of $H$ in $G$ with respect to $g$} can be introduced as\n\\begin{equation*}\n[g]^*=\\{gh : h\\in H\\}=gH.\n\\end{equation*}\nIn general, right and left cosets are different sets.\n\\begin{defn}$\\\\ $\nA subgroup  $N$ of $G$ which defines a unique partition,\n\\begin{equation*}\n[g]=[g]^*\\Longleftrightarrow gN=Ng\\hspace{0.5cm}\\forall g \\in G\n\\end{equation*}\nis called \\textit{normal subgroup of} $G$. \n\\end{defn}\nClearly it follows that for every $n\\in N$ and $g\\in G$ the product $gNg^{-1}\\subseteq N$. Note that every group \n$G$ possesses normal subgroups, since $G$ and the identity are normal subgroups. \\\\\nNow consider for a general subgroup $H$ of $G$ the \\textit{quotient group} (or \\textit{factor group}) $G\/H$, defined as\n\\begin{equation*}\nG\/H=\\{[g] : g\\in G\\}.\n\\end{equation*}\nIf $H$ is normal the elements of $G\/H$ are, indistinctly, the right and left cosets. Furthermore, \nunder this hypothesis, the set $G\/H$ can be equipped with a group structure in a natural way by \ndefining the product $\\ast$:\n\\begin{align*}\n\\ast : &G\/H\\times G\/H\\longrightarrow G\/H \\\\\n&gH\\ast g'H \\equiv gg'H,\n\\end{align*}\ni.e., \n\\begin{equation*}\n[g]\\ast [g']\\equiv [gg'].\n\\end{equation*}\nIt can be shown that $G\/H$ equipped with the product $\\ast$ satisfies the group axioms. We are now ready to \ndiscuss further the BMS properties.\\\\\nAny element $b$ of $\\mathscr{B}$ can be written as\n\\begin{equation*}\nb=(\\Lambda,\\alpha).\n\\end{equation*}\nNote that with this nomenclature any element $\\Lambda$ of $\\mathscr{L}$ (or, equivalently, of \n$\\mathrm{PSL}(2,\\mathbb{C})$) can be written as $\\Lambda=(\\Lambda,0)$ and any element $s$ of $\\mathscr{S}$ \nas $s=(\\mathbb{I},\\alpha)$, where $\\mathbb{I}$ denotes the identity in $\\mathscr{L}$.\\\\ \nThe action of $b$ on the variables $(\\zeta,u)$ is \n\\begin{equation*}\nb(\\zeta,u)=(f(\\zeta),K[u+\\alpha(\\zeta,\\bar{\\zeta})]),\n\\end{equation*}\nwhere $f$ is the element of $\\mathrm{PSL}(2,\\mathbb{C})$ which corresponds to $\\Lambda$ through the above \ndiscussed isomorphism and $K$ is its conformal factor.\\\\\nIt is easy to show that, with this notation, we have for the inverse of $b$:\n\\begin{equation}\n\\label{eqn:208}\nb^{-1}=(\\Lambda^{-1},-K\\alpha).\n\\end{equation}\nOn considering an element $s=(\\mathbb{I},\\beta)$ of $\\mathscr{S}$ we have\n\\begin{equation*}\nbsb^{-1}(\\zeta,u)=(\\zeta,u+K\\beta(\\zeta,\\bar{\\zeta}))=s'(\\zeta,u),\n\\end{equation*}\nwith \n\\begin{equation*}\ns'=(\\mathbb{I},K\\beta)\\in\\mathscr{S}.\n\\end{equation*}\nFrom the above discussion we have the following\n\\begin{prop}$\\\\ $\n\\label{prop:BMS1}\nThe supertranslations $\\mathscr{S}$ form an Abelian normal, infinite-parameter, subgroup of the BMS group:\n\\begin{equation*}\nb\\mathscr{S}b^{-1}=\\mathscr{S}\\hspace{0.5cm}\\mathrm{for}\\hspace{1.2mm} \\mathrm{all}\\hspace{1.2mm} b\\in\\mathscr{B}.\n\\end{equation*}\n\\end{prop}\nUnder the assumption that the function $\\alpha$ is twice differentiable, we can expand it into spherical harmonics as\n\\begin{equation}\n\\label{eqn:206}\n\\alpha(\\theta,\\phi)=\\sum_{l=0}^{\\infty}\\sum_{m=-l}^{l}\\alpha_{l,m}Y_{l,m}(\\theta,\\phi),\\hspace{1cm}\\alpha_{l,-m}\n=(-1)^m\\bar{\\alpha}_{l,m}.\n\\end{equation}\n\\begin{defn}$\\\\ $\nIf in decomposition \\eqref{eqn:206} $\\alpha_{l,m}=0$ for $l>2$, i.e.\n\\begin{equation}\n\\label{eqn:207}\n\\alpha\\equiv\\alpha_t=\\epsilon_0+\\epsilon_1\\sin\\theta\\cos\\phi+\\epsilon_2\\sin\\theta\\sin\\phi+\\epsilon_3\\cos\\theta,\n\\end{equation}\nthen the supertranslations reduce to a special case, called \\textit{translation} subgroup, denoted by \n$\\mathscr{T}$, with just four independent parameters $\\epsilon_0,...,\\epsilon_3$. \n\\end{defn}\nIt is easy to show that $\\zeta=e^{i\\phi}\\cot\\frac{\\theta}{2}$ implies\n\\begin{equation*}\n\\cos\\phi=\\frac{\\zeta+\\bar{\\zeta}}{2\\sqrt{\\zeta\\bar{\\zeta}}},\\hspace{1cm}\\sin\\phi\n=\\frac{i(\\bar{\\zeta}-\\zeta)}{2\\sqrt{\\zeta\\bar{\\zeta}}},\n\\end{equation*}\n\\begin{equation*}\n\\cos\\theta=\\frac{\\zeta\\bar{\\zeta}-1}{1+\\zeta\\bar{\\zeta}},\\hspace{1cm}\\sin\\theta\n=\\frac{2\\sqrt{\\zeta\\bar{\\zeta}}}{1+\\zeta\\bar{\\zeta}}.\n\\end{equation*}\nThen equation \\eqref{eqn:207} becomes\n\\begin{equation*}\n\\alpha_t=\\epsilon_0+\\epsilon_1\\frac{\\zeta+\\bar{\\zeta}}{1+\\zeta\\bar{\\zeta}}+\\epsilon_2\n\\frac{(i\\zeta-i\\bar{\\zeta})}{1+\\zeta\\bar{\\zeta}}+\\epsilon_3\\frac{\\zeta\\bar{\\zeta}-1}{1+\\zeta\\bar{\\zeta}}\n\\end{equation*}\n\\begin{equation*}\n=\\frac{A+B\\zeta+\\bar{B}\\bar{\\zeta}+C\\zeta\\bar{\\zeta}}{1+\\zeta\\bar{\\zeta}},\n\\end{equation*}\nwith $A$ and $C$ real. \nHence in terms of $\\zeta$ and $\\bar{\\zeta}$ a translation is\n\\begin{equation*}\nu=t-r\\rightarrow u'=u+\\frac{A+B\\zeta+\\bar{B}\\bar{\\zeta}+C\\zeta\\bar{\\zeta}}{1+\\zeta\\bar{\\zeta}},\n\\end{equation*}\n\\begin{equation*}\n\\zeta\\rightarrow\\zeta'=\\zeta.\n\\end{equation*}\nIf we let $t,x,y,z$ be Cartesian coordinates in Minkowski space-time, it is easy to see that\n\\begin{equation*}\nZ^2\\zeta=\\frac{(x+iy)(1-z\/r)}{4r},\\hspace{1cm}x=r(\\zeta+\\bar{\\zeta})Z,\n\\end{equation*}\n\\begin{equation*}\ny=-ir(\\zeta-\\bar{\\zeta})Z,\\hspace{1cm}z=r(\\zeta\\bar{\\zeta}-1)Z,\n\\end{equation*}\nwhere $Z=1\/(1+\\zeta\\bar{\\zeta})$. Now if we perform a translation\n\\begin{equation*}\nt\\rightarrow t'=t+a,\\hspace{0.5cm}x\\rightarrow x'=x+b,\\hspace{0.5cm}y\\rightarrow y'=y+c,\n\\hspace{0.5cm}z\\rightarrow z'=z+d,\n\\end{equation*}\nit is easy to get \n\\begin{align*}\n&u=t-r\\rightarrow u'=u+Z(A+B\\zeta+\\bar{B}\\bar{\\zeta}+C\\zeta\\bar{\\zeta})+O(r^{-1}),\\\\\n&\\zeta\\rightarrow\\zeta'=\\zeta+O(r^{-1}).\n\\end{align*}\nwith $A=a+d$, $B=b-ic$ and $C=a-d$. Thus, the nomenclature \\lq translation\\rq\\hspace{0.1mm} is consistent \nwith that for the space-time translations in Minkowski space-time. In fact we have just shown that any \ntranslation in the ordinary sense induces a translation (i.e. an element of $\\mathscr{T}$) on $\\mathscr{I}^+$.\\\\\nIt is easy to verify that for any $b=(\\Lambda,\\alpha)\\in\\mathscr{B}$ and for any \n$t=(\\mathbb{I},\\alpha_t)\\in\\mathscr{T}$ we have \n\\begin{equation*}\nbtb^{-1}(\\zeta,u)=(\\zeta,u+K\\alpha_t)=t'(\\zeta,u),\n\\end{equation*}\nwith \n\\begin{equation*}\nt'=(\\mathbb{I},K\\alpha_t(\\zeta,\\bar{\\zeta}))\\in\\mathscr{T}.\n\\end{equation*}\nNote that it is not obvious that $K\\alpha_t$ is still a function of $\\theta$ and $\\phi$ containing only \nzeroth- and first-order harmonics. A proof of this result will be given in Sect. \\ref{sect:8}. \nOn taking for the moment this result for true, the following proposition holds:\n\\begin{prop}$\\\\ $\nThe translations $\\mathscr{T}$ form a normal four-dimensional subgroup of $\\mathscr{B}$:\n\\begin{equation*}\nb\\mathscr{T}b^{-1}=\\mathscr{T}\\hspace{0.5cm}\\mathrm{for}\\hspace{1.2mm} \\mathrm{all}\\hspace{1.2mm} b\\in\\mathscr{B},\n\\end{equation*}\nand clearly\n\\begin{equation*}\ns\\mathscr{T}s^{-1}=\\mathscr{T}\\hspace{0.5cm}\\mathrm{for}\\hspace{1.2mm} \\mathrm{all}\\hspace{1.2mm} s\\in\\mathscr{S}.\n\\end{equation*}\n\\end{prop}\nWe have the following inclusion relations:\n\\begin{equation*}\n\\mathscr{T}\\subset\\mathscr{S}\\subset\\mathscr{B}.\n\\end{equation*}\nThe next step will be to investigate the group structure of $\\mathscr{B}$. It is easy to show \nthat for any $b\\in\\mathscr{B}$ there exists a unique $\\Lambda\\in\\mathscr{L}$ and $s\\in\\mathscr{S}$ \nsuch that $b=\\Lambda s$. In fact given \n\\begin{equation*}\n\\Lambda=(\\Lambda,0)\\in\\mathscr{L},\\hspace{1cm}s=(\\mathbb{I},\\alpha)\\in\\mathscr{S},\n\\end{equation*}\nwe have that\n\\begin{equation*}\n\\Lambda s(\\zeta,u)=\\Lambda(\\zeta,u+\\alpha)=(f(\\zeta),K[u+\\alpha(\\zeta,\\bar{\\zeta})])=b(\\zeta,u)\n\\end{equation*}\nwith $b=(\\Lambda,\\alpha)$. The uniqueness results from the observation that $\\mathscr{L}\\cap\\mathscr{S}=\\{e\\}$ where \n$e=(\\mathbb{I},0)$ is the identity in $\\mathscr{B}$, since if $fs=f's'$, then $f'^{-1}f=s's^{-1}\\in\\mathscr{L}\\cap\\mathscr{S}$ \nimplying $f'=f$ and $s'=s$. Hence we have\n\\begin{equation}\n\\label{eqn:233}\n\\mathscr{B}=\\mathscr{L}\\mathscr{S}.\n\\end{equation}\nFurthermore, the supertranslations $\\mathscr{S}$ form an (Abelian) normal subgroup of $\\mathscr{B}$, according \nto \\ref{prop:BMS1}. Thus we can already state that, by definition of semi-direct product, \n\\begin{prop}$\\\\ $\nThe BMS group is a semi-direct product of the conformal group of the unit $2$-sphere with the \nsupertranslations group, i.e.\n\\begin{equation*}\n\\mathscr{B}=\\mathscr{L}\\rtimes\\mathscr{S}.\n\\end{equation*}\n\\end{prop}\nWe can say more by specifying an action of $\\mathscr{L}$ on $\\mathscr{S}$ and, hence, by specifying a product rule \nfor two elements of $\\mathscr{B}$. Let $\\mathscr{S}$ be the vector space of real functions on the Riemann sphere. \nLet $\\sigma$ be a smooth right action of $\\mathscr{L}$ on $\\mathscr{S}$ defined as \n\\begin{subequations}\n\\label{eqn:212}\n\\begin{equation}\n\\sigma:(\\Lambda,\\alpha)\\in\\mathscr{L}\\times\\mathscr{S}\\longrightarrow\\sigma_{\\Lambda}(\\alpha)\n\\equiv \\alpha\\Lambda\\in\\mathscr{S}\n\\end{equation}\nsuch that \n\\begin{equation}\n\\alpha(\\zeta,\\bar{\\zeta})\\Lambda=K^{-1}\\alpha(f(\\zeta),\\bar{f}(\\bar{\\zeta})),\n\\end{equation}\n\\end{subequations}\nwhere $K$ is the conformal factor associated with $f$, the element of $\\mathrm{PSL}(2,\\mathbb{C})$ that \ncorresponds to $\\Lambda$. Then it is easy to verify that the composition law for the elements of $\\mathscr{B}$ is \n\\begin{equation*}\nb_1\\cdot b_2=(\\Lambda_1,\\alpha_1)\\cdot(\\Lambda_2,\\alpha_2)=(\\Lambda_1\\cdot\\Lambda_2,\\alpha_2+\\alpha_1f_2).\n\\end{equation*}\nNote that the inverse of $b$ in \\eqref{eqn:208} may be written as\n\\begin{equation*}\nb^{-1}=(\\Lambda^{-1},\\left[\\alpha \\Lambda^{-1}\\right]^{-1}).\n\\end{equation*}\nThus the BMS group is the \\textit{right} semi-direct product \\cite{Col14} of $\\mathscr{L}$ with $\\mathscr{S}$ \nunder the action $\\sigma$, i.e.\n\\begin{equation}\n\\label{eqn:209}\n\\mathscr{B}=\\mathscr{L}\\rtimes_{\\sigma}\\mathscr{S}.\n\\end{equation}\nHistorically, this semi-direct product structure was realized by \\cite{Cant}. Succesively this idea was developed \nby \\cite{GerBMS} who gave an incorrect formula for the action \\eqref{eqn:212}. Eventually the mistake was amended \nby \\cite{MC2}, who defined a good action to describe the semi-direct product structure of the BMS group. \nHowever, the idea used here of giving a right action and hence of describing the BMS group as a \\textit{right} \nsemi-direct product was not developed by any of these authors and it is an original contribution of our work.\\\\\nFurthermore, it can be shown that from the above discussion it follows that, by virtue \nof the first isomorphism theorem,\n\\begin{equation}\n\\label{eqn:234}\n\\mathscr{L}\\cong\\mathscr{B}\/\\mathscr{S},\n\\end{equation}\ni.e. $\\mathscr{L}$ is the factor group of $\\mathscr{B}$ with respect to its normal subgroup $\\mathscr{S}$. \n\\begin{oss}$\\\\ $\nNote that the structure of the BMS group is similar to that of the Poincar\\'e group, denoted by $\\mathscr{P}$. \nIn fact the Poincar\\'e group can be expressed as the semi-direct product of the connected component \nof the Lorentz group $\\mathscr{L}$ and the translations group $\\mathscr{T}$, the former being the factor group of \n$\\mathscr{P}$ with respect to the latter, i.e. $\\mathscr{L}\\cong\\mathscr{P}\/\\mathscr{T}$. The action of \n$\\mathscr{L}$ on $\\mathscr{T}$ is the \\lq natural\\rq\\hspace{0.1mm} one, i.e. the usual multiplication of \nan element $\\Lambda\\in \\mathscr{L}$ with a vector $b\\in\\mathscr{T}$.\n\\end{oss}\n\\begin{thm}$\\\\ $\nIf $N'$ is a 4-dimensional normal subgroup of $\\mathscr{B}$, then $N'$ is contained in $\\mathscr{S}$.\n\\end{thm}\n\\begin{proof}\nConsider the image $N'\/\\mathscr{S}$ of $N'$ under the homomorphism $\\mathscr{B}\\rightarrow\\mathscr{B}\/\\mathscr{S}$. \nSince $N'$ by hypothesis is a normal subgroup of $\\mathscr{B}$, $N'\/\\mathscr{S}$ is a normal subgroup of \n$\\mathscr{B}\/\\mathscr{S}$ and hence, by proposition \\ref{prop:BMS1}, a subgroup of the connected component \nof the Lorentz group $\\mathscr{L}$. However, the only normal subgroups of $\\mathscr{L}$ are $\\mathscr{L}$ \nitself and the identity $e$ of $\\mathscr{L}$. Then $N'$ must be $6$-dimensional, contrary to hypothesis. \nTherefore $N'\/\\mathscr{S}=e$; $N'$ is thus contained in $\\mathscr{S}$. \n\\end{proof}\n\\section{BMS Lie Algebra}\n\\label{sect:7}\nIn this section we are going to investigate the Lie Algebra of the BMS group. At first we consider the \ngenerators of $\\mathrm{PSL}(2,\\mathbb{C})$. For an infinitesimal conformal transformation we know \nthat the $x^A$ coordinates change as\n\\begin{equation*}\nx^A\\rightarrow x'^A=x^A+f^A,\n\\end{equation*}\nwhere $f^A$ is a conformal Killing vector of the unit 2-sphere. Furthermore, from \n\\eqref{eqn:220} and taking into account \\eqref{eqn:214} an infinitesimal transformation for $u$ reads as\n\\begin{equation*}\nu\\rightarrow u'=Ku=e^{\\frac{1}{2}D_Af^A}u\\simeq u+\\frac{u}{2}D_Af^A.\n\\end{equation*}\nThus, the generator of transformation \\eqref{eqn:220} is\n\\begin{equation}\n\\label{eqn:221}\n\\xi_R=f^A\\partial_A+\\frac{u}{2}D_Af^A\\partial_u.\n\\end{equation}\nTo see how their Lie algebra closes, consider the Lie bracket of two of them, $\\xi_{R_1}$ and $\\xi_{R_2}$:\n\\begin{equation*}\n[\\xi_{R_1},\\xi_{R_2}]=\\left[f_1^A\\partial_A+\\frac{u}{2}D_Bf_1^B\\partial_u,f_2^C\\partial_C\n+\\frac{u}{2}D_Cf_2^C\\partial_u\\right]\n\\end{equation*}\n\\begin{equation*}\n=(f_1^A\\partial_Af_2^C-f_2^A\\partial_Af_1^C)\\partial_C+\\frac{u}{2}(f_1^A\\partial_AD_Cf_2^C\n-f_2^C\\partial_CD_Bf_1^B)\\partial_u.\n\\end{equation*}\nAfter some calculation the term proportional to $\\partial_u$ becomes\n\\begin{equation*}\n\\frac{u}{2}D_C(f_1^A\\partial_Af_2^C-f_2^A\\partial_Af_1^C)+\\frac{u}{2}(\\partial_A\\Gamma^{C}{}_{CB})\n(f_1^Af_2^B-f_1^Bf_2^A).\n\\end{equation*}\nThe last term in the previous equation vanishes since it can be shown by direct calculation \nthat for the metric $q_{AB}$\n\\begin{equation*}\n\\partial_A\\Gamma^{C}{}_{CB}=-\\frac{1}{\\sin^2\\theta}\\delta^{\\theta}_{A}\\delta_B^{\\theta},\n\\end{equation*}\nand hence\n\\begin{equation*}\n\\left(\\partial_A\\Gamma^{C}{}_{CB}\\right)(f_1^Af_2^B-f_1^Bf_2^A)=-\\frac{1}{\\sin^2\\theta}(f_1^{\\theta}f_2^{\\theta}\n-f_1^{\\theta}f_2^{\\theta})=0.\n\\end{equation*}\nFinally we get \n\\begin{subequations}\n\\label{eqn:222}\n\\begin{equation}\n\\label{eqn:223}\n[\\xi_{R_1},\\xi_{R_2}]=\\xi_{\\hat{R}}=\\hat{f}^A\\partial_A+\\frac{u}{2}D_A\\hat{f}^A\\partial_u\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eqn:224}\n\\hat{f}^A=f_1^B\\partial_Bf_2^A-f_2^B\\partial_Bf_1^A.\n\\end{equation}\n\\end{subequations}\nWe take now into account the generators of supertranslations. It is clear from \\eqref{eqn:225} that these are\n\\begin{equation*}\n\\xi_T=\\alpha\\partial_u.\n\\end{equation*}\nwhere $\\alpha$ is an arbitrary function of $\\theta$ and $\\phi$. It follows that the Lie bracket of two \ngenerators, $\\xi_{T_1}$ and $\\xi_{T_2}$ vanish, i.e.\n\\begin{equation}\n\\label{eqn:228}\n[\\xi_{T_1},\\xi_{T_2}]=0,\n\\end{equation}\nthat is just a restatement that the supertranslation group is Abelian. The only thing left to do is to \ncalculate the Lie bracket of $\\xi_{R}$ and $\\xi_{T}$. It is easy to see that \n\\begin{subequations}\n\\label{eqn:226}\n\\begin{equation}\n\\label{eqn:227}\n[\\xi_R,\\xi_T]=[f^A\\partial_A+\\frac{u}{2}D_Bf^B\\partial_u,\\alpha\\partial_u]\n=\\xi_{\\hat{T}}=\\hat{\\alpha}\\partial_u,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eqn:228}\n\\hat{\\alpha}=f^A\\partial_A\\alpha-\\frac{\\alpha}{2}D_Bf^B.\n\\end{equation}\n\\end{subequations}\nIf we consider now $\\xi$ as defined in \\eqref{eqn:158} it turns out that $\\xi=\\xi_R+\\xi_T$. \nFrom the above discussions one obtains that\n\\begin{eqnarray}\n\\label{eqn:229}\n[\\xi_1,\\xi_2]&=& [\\xi_{R_1},\\xi_{R_2}]+[\\xi_{R_1},\\xi_{T_2}]+[\\xi_{T_1},\\xi_{R_2}]\n\\nonumber \\\\\n&=& \\hat{f}^A\\partial_A+\\frac{u}{2}D_A\\hat{f}^A\\partial_u+(\\hat{\\alpha}_{2}-\\hat{\\alpha}_1)\\partial_u,\n\\end{eqnarray}\nwith \n\\begin{equation*}\n\\hat{\\alpha}_{2}=f^A_1\\partial_A\\alpha_2-\\frac{\\alpha_2}{2}D_Bf^B_1,\\hspace{1cm}\\hat{\\alpha}_{1}\n=f^A_2\\partial_A\\alpha_1-\\frac{\\alpha_1}{2}D_Bf^B_2.\n\\end{equation*}\nTo sum up, the Lie algebra of the BMS group, $\\mathfrak{bms_4}$, is\n\\begin{align*}\n&[\\xi_{R_1},\\xi_{R_2}]=\\xi_{\\hat{R}},\\hspace{0.5cm}\\mathrm{with}\\hspace{0.5cm}\\hat{f}^A\n=f_1^B\\partial_Bf_2^A-f_2^B\\partial_Bf_1^A; \\\\\n&[\\xi_{T_1},\\xi_{T_2}]=0;\\\\\n&[\\xi_R,\\xi_T]=\\xi_{\\hat{T}},\\hspace{0.5cm}\\mathrm{with}\\hspace{0.5cm}\\hat{\\alpha}\n=f^A\\partial_A\\alpha-\\frac{\\alpha}{2}D_Bf^B.\n\\end{align*}\nSince, as shown in the previous section, the BMS group is a semi-direct product, it follows that the \nBMS Lie algebra, $\\mathfrak{bms_4}$, should be taken to be the semi-direct sum of the Lie algebra of \nconformal Killing vectors $X=f^A\\partial_A$ of the Riemann sphere, denoted by $\\mathfrak{so(3,1)}$ (since it \ncan be taken to be the algebra of $\\mathscr{L}$) with that of the functions $\\alpha(x^A)$ on the Riemann \nsphere, which we denote by $\\mathscr{S}$, the supertranslation group being Abelian. Given an element \n$X=f^A\\partial_A\\in\\mathfrak{so(3,1)}$ ($f^A$ being a generator of conformal transformations in \\eqref{eqn:157}) \nwe know that the exponential map associated to it, $e^{X}$, is an element of $\\mathscr{L}$. \nThen consider the 1-parameter group of transformations in $\\mathscr{S}$ defined as\n\\begin{equation}\n\\label{eqn:219}\n\\sigma_{e^{tX}}(\\alpha)=\\alpha e^{tX},\n\\end{equation}\nwhere $\\sigma$ is that of \\eqref{eqn:212}. Consider the map\n\\begin{equation*}\n\\Sigma:f^A\\partial_A\\in\\mathfrak{so(3,1)}\\longrightarrow\\Sigma_{f^A\\partial_A}\\in\\mathrm{End}\\mathscr{S},\n\\end{equation*}\nsuch that \n\\begin{equation*}\n\\Sigma_{f^A\\partial_A}:\\alpha\\in\\mathscr{S}\\longrightarrow\\Sigma_{f^A\\partial_A}(\\alpha)=\\frac{d}{dt}\t\n\\left.(\\sigma_{e^{tf^A\\partial_A}}(\\alpha))\\right|_{t=0}\\in\\mathscr{S}.\n\\end{equation*}\n\\\\\nNote that $\\Sigma_{f^A\\partial_A}(\\alpha)$ is the infinitesimal generator of \\eqref{eqn:219}. \nHence \\cite{OblakBMS} we have \n\\begin{equation}\n\\label{eqn:213}\n\\mathfrak{bms_4}=\\mathfrak{so(3,1)}\\oplus_{\\Sigma}\\mathscr{S}.\n\\end{equation}\nThe Lie algebra $\\mathfrak{bms_4}$ is determined by three arbitrary functions $f^A$ and $\\alpha$ on the circle. \nThus, defining $X=f^A\\partial_A$ and labelling elements of \\eqref{eqn:213} as pairs $(X,\\alpha)$, \nwe know that the Lie bracket in $\\mathfrak{so(3,1)}\\oplus_{\\Sigma}\\mathscr{S}$ are\n\\begin{equation}\n\\label{eqn:215}\n[(X_1,\\alpha_1),(X_2,\\alpha_2)]=([X_1,X_2],\\Sigma_{f_1^A\\partial_A}(\\alpha_2)-\\Sigma_{f^B_2\\partial_B}(\\alpha_1)).\n\\end{equation}\nEquation \\eqref{eqn:215} follows from the fact that $\\mathscr{S}$ is Abelian, otherwise there would be an extra \nterm involving the commutator of the two elements $\\alpha_1$ and $\\alpha_2$.\nSince we have, using \\eqref{eqn:212} and \\eqref{eqn:214}\n\\begin{equation*}\n\\Sigma_{f^A\\partial_A}(\\alpha)(x^B)=\\frac{d}{dt}\\left.(K^{-1}_{e^{tf^A\\partial_A}}\\alpha\n(e^{tf^C\\partial_C}x^B))\\right|_{t=0}=\\frac{d}{dt}\\left.(e^{-\\frac{1}{2}tD_Af^A}\n\\alpha(e^{tf^C\\partial_C}x^B))\\right|_{t=0}\n\\end{equation*}\n\\begin{equation*}\n=-\\frac{\\alpha}{2}D_Af^A+ f^B\\partial_B\\alpha,\n\\end{equation*}\nthen \\eqref{eqn:215} may be written as\n\\begin{subequations}\n\\label{eqn:230}\n\\begin{equation}\n\\label{eqn:231}\n[(X_1,\\alpha_1),(X_2,\\alpha_2)]=(\\hat{X},\\hat{\\alpha}),\n\\end{equation}\nwith \n\\begin{equation}\n\\label{eqn:232}\n\\hat{f}^A=f_1^B\\partial_Bf_2^A-f_2^B\\partial_Bf_1^A,\\hspace{1cm}\\hat{\\alpha}=f_1^B\\partial_B\\alpha_2\n-\\frac{\\alpha_2}{2}D_Af^A_1-(1\\leftrightarrow2),\n\\end{equation}\n\\end{subequations}\nas remarked in \\cite{Barn2010,Barn2010a}.\n\\begin{oss}$\\\\ $\nNote that this result, obtained from the theory of semi-direct product of groups and their Lie algebra, \nis in complete agreement with that obtained just by looking at the generators, expressed in \\eqref{eqn:229}. \nNote also that $\\hat{f}^A$ of \\eqref{eqn:232} coincides with that of \\eqref{eqn:224} and that \n$\\hat{\\alpha}=\\hat{\\alpha}_2-\\hat{\\alpha}_1$.\n\\end{oss}\nDepending on the space of functions under consideration, there are many options which define what is actually \nmeant by $\\mathfrak{bms_4}$. The approach that will be followed in this work is originally due to \n\\cite{Sachs1} and successively amended by \\cite{Ant91}. Another approach, based on the Virasoro algebra, \ncan be found in \\cite{Barn2010}. \\\\\nIn general, we consider any $S$-dimensional Lie transformation group of a $R$-dimensional space. Let \nthe coordinates of the space be $y^{\\alpha}$ $(\\alpha=1,..,R)$ and the parameters of the group be \n$z^{\\mu}$ $(\\mu=1,...,S)$, where $z^{\\mu}=0$ is the identity of the group. Then the transformations have the form \n\\begin{equation*}\ny'^{\\alpha}=f^{\\alpha}(y^{\\beta};z^{\\mu}),\\hspace{1cm}\\mathrm{where}\\hspace{1cm}f^{\\alpha}(y^{\\beta};0)=y^{\\alpha}.\n\\end{equation*}\nThe functions $f^{\\alpha}$ are assumed to be twice differentiable. The $S$ generators of the group are \nthe vector fields given by\n\\begin{equation}\n\\label{eqn:217}\nP_{\\mu}=\\left.\\frac{\\partial f^{\\alpha}}{\\partial z^{\\mu}}\\right|_{z^{\\mu}=0}\\frac{\\partial}{\\partial y^{\\alpha}}.\n\\end{equation}\nApplying these ideas to the BMS group,with the Sachs notation \\cite{Sachs1}, one finds for the supertranslations, using the expansion \\eqref{eqn:206}:\n\\begin{equation*}\nP_{l,m}=Y_{l,m}(\\theta,\\phi)\\frac{\\partial}{\\partial u},\\hspace{1cm}P_{l,m}=(-1)^m\\bar{P}_{l,-m},\n\\end{equation*}\nand hence\n\\begin{equation*}\n[P_{l,m},P_{n,r}]=0,\n\\end{equation*}\ni.e. two supertranslations commute. \\\\\nTo find the generators of conformal transformations we have to be careful. We know that any conformal \ntransformation has the form \n\\begin{equation*}\n\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d},\\hspace{1cm}\\zeta=e^{i\\phi}\\cot\\frac{\\theta}{2}.\n\\end{equation*}\nBy direct calculations one obtains that the following equations hold for $\\theta'$, $\\phi'$ and $u'$:\n\\begin{subequations}\n\\label{eqn:216}\n\\begin{align}\n&\\theta'=2\\arctan\\left[\\left(\\frac{|c\\zeta+d|^2}{|a\\zeta+b|^2}\\right)^{1\/2}\\right],\\\\ \\nonumber\\\\\n&\\phi'=\\arctan\\left[\\frac{\\mathrm{Im}\\{(a\\zeta+b)(\\bar{c}\\bar{\\zeta}+\\bar{d})\\}}\n{\\mathrm{Re}\\{(a\\zeta+b)(\\bar{c}\\bar{\\zeta}+\\bar{d})\\}}\\right],\\\\ \\nonumber\\\\\n&u'=\\frac{1+|\\zeta|^2}{|a\\zeta+b|^2+|c\\zeta+d|^2}u.\n\\end{align}\n\\end{subequations}\nOn denoting by $x$ the parameter of the transformation it is clear that $a,b,c,d$ are functions of $x$ such \nthat $a(0)=d(0)=1$ and $c(0)=b(0)=0$. We have to apply \\eqref{eqn:217} to \\eqref{eqn:216}. \nIt is easy to verify that \n\\begin{align*}\n\\left.\\frac{d\\theta'}{dx}\\right|_{x=0}&=\\frac{\\cos^3\\frac{\\theta}{2}}{\\sin\\frac{\\theta}{2}}\n\\frac{d}{dx}\\left.\\left[\\frac{|c|^2|\\zeta|^2+|d|^2+c\\bar{d}\\zeta+\\bar{c}d\\bar{\\zeta}}\n{|a|^2|\\zeta|^2+|b|^2+a\\bar{b}\\zeta+\\bar{a}b\\bar{\\zeta}}\\right]\\right|_{x=0},\\\\ \\\\\n\\left.\\frac{d\\phi'}{dx}\\right|_{x=0}&=\\cos^2\\phi\\frac{d}{dx}\\left.\\left[\\frac\n{\\mathrm{Im}\\{a\\bar{c}|\\zeta|^2+a\\bar{d}\\zeta+b\\bar{c}\\bar{\\zeta}+b\\bar{d}\\}}\n{\\mathrm{Re}\\{a\\bar{c}|\\zeta|^2+a\\bar{d}\\zeta+b\\bar{c}\\bar{\\zeta}+b\\bar{d}\\}}\\right]\\right|_{x=0},\\\\ \\\\\n\\left.\\frac{du'}{dx}\\right|_{x=0}&=\\sin^2\\frac{\\theta}{2}\\frac{d}{dx}\\left.\\left[(|a|^2+|c|^2)|\\zeta|^2\n+(a\\bar{b}+c\\bar{d})\\zeta+(b\\bar{a}+d\\bar{c})\\bar{\\zeta}+|b|^2+|d|^2\\right]\\right|_{x=0}u.\n\\end{align*}\nThese equations hold in general for any conformal transformation. We choose now to work with Lorentz \ntransformations, and thus to use Lorentz generators $L_i$ and $R_i$ of rotations and boosts, respectively. \nTo know the coefficients $a,b,c,d$ corresponding to a Lorentz transformation we need to use the isomorphism \n\\eqref{eqn:210}. Any rotation of an angle $\\varphi$ about an axis $\\hat{n}$ and any boost of rapidity \n$\\chi$ about an axis $\\hat{e}$ can be performed by using a $\\mathrm{SL}(2,\\mathbb{C})$ matrix given by\n\\begin{subequations}\n\\label{eqn:236}\n\\begin{align}\n&U_{\\hat{n}}(\\varphi)=e^{\\frac{i}{2}\\varphi\\hat{n}\\cdot\\vec{\\sigma}}=\\mathbb{I}\\cos\\frac{\\varphi}{2}\n+i\\hat{n}\\cdot\\vec{\\sigma}\\sin\\frac{\\varphi}{2},\\\\\n&H_{\\hat{e}}(\\chi)=e^{\\frac{1}{2}\\chi\\hat{e}\\cdot\\vec{\\sigma}}=\\mathbb{I}\\cosh\\frac{\\chi}{2}\n+\\hat{e}\\cdot\\vec{\\sigma}\\sinh\\frac{\\chi}{2},\n\\end{align}\n\\end{subequations}\nrespectively, where $\\vec{\\sigma}=(\\sigma_x,\\sigma_y,\\sigma_z)$ are the Pauli matrices. The parameter \n$x$ of the two transformations is $\\varphi$ and $\\chi$, respectively. After some calculations we find that \nthe vector fields that generate the transformations are \n\\begin{align}\nL^{23}=L_x&=-\\sin\\phi\\frac{\\partial}{\\partial\\theta}-\\cot\\theta\\cos\\phi\\frac{\\partial}{\\partial \\phi},\\\\\nL^{13}=L_y&=-\\cos\\phi\\frac{\\partial}{\\partial\\theta}+\\cot\\theta\\sin\\phi\\frac{\\partial}{\\partial \\phi},\\\\\nL^{12}=L_z&=\\frac{\\partial}{\\partial \\phi},\\\\\nL^{10}=R_x&=\\cos\\theta\\cos\\phi\\frac{\\partial}{\\partial\\theta}-\\frac{\\sin\\phi}{\\sin\\theta}\\frac{\\partial}{\\partial\\phi}\n-u\\sin\\theta\\cos\\phi\\frac{\\partial}{\\partial u},\\\\\nL^{20}=R_y&=-\\cos\\theta\\sin\\phi\\frac{\\partial}{\\partial\\theta}-\\frac{\\cos\\phi}{\\sin\\theta}\\frac{\\partial}{\\partial\\phi}\n+u\\sin\\theta\\sin\\phi\\frac{\\partial}{\\partial u},\\\\\nL^{30}=R_z&=-\\sin\\theta\\frac{\\partial}{\\partial \\theta}-u\\cos\\theta\\frac{\\partial}{\\partial u}.\n\\end{align}\nNote that rotations are characterized by $K=1$. The $\\{P_{l,m}\\}$\nand $\\{L^{ab}\\}$ form a complete set of linearly independent vector fields for the Lie algebra \n$\\mathfrak{bms}_4$. We can find now the commutators\n\\begin{equation*}\n[L^{ab},L^{cd}]=\\eta^{ac}L^{bd}+\\eta^{bd}L^{ac}-\\eta^{ad}L^{bc}-\\eta^{bc}L^{ad},\n\\end{equation*}\n\\begin{equation*}\n[L_i,L_j]=\\epsilon_{ijk}L_k,\\hspace{1cm}[R_i,R_j]=-\\epsilon_{ijk}L_k,\\hspace{1cm}[L_i,R_j]=-\\epsilon_{ijk}R_k,\n\\end{equation*}\nwhere $\\eta^{ab}=\\mathrm{diag}(1,-1,-1,-1)$ and $\\epsilon_{ijk}$ is the Levi-Civita symbol. Note that we have \njust obtained the classical Lorentz algebra. Furthermore, it is easy to derive the following commutator:\n\\begin{equation}\n\\label{eqn:218}\n\\left[L^{ab},\\alpha(\\theta,\\phi)\\frac{\\partial}{\\partial u}\\right]=\\left[L^{ab}\\alpha(\\theta,\\phi)\n-\\alpha(\\theta,\\phi)W(L^{ab})\\right]\\frac{\\partial}{\\partial u},\n\\end{equation}\nwhere $W(L^{ab})$ is defined by the relation \n\\begin{equation*}\n\\frac{\\partial}{\\partial u}(L^{ab}f)=L^{ab}\\frac{\\partial f}{\\partial u}+W(L^{ab})\\frac{\\partial f}{\\partial u},\n\\end{equation*}\nfor arbitrary $f(u)$. \\\\\nFor convenience we introduce the raising and lowering operators,\n\\begin{align*}\nL^{\\pm}&=L_y\\pm iL_x=-e^{\\pm i\\phi}\\left(\\frac{\\partial}{\\partial \\theta}\n\\pm i\\cot\\theta\\frac{\\partial}{\\partial \\phi}\\right),\\\\\nR^{\\pm}&=R_x\\mp i R_y=e^{\\pm i\\phi}\\left(\\cos\\theta\\frac{\\partial}{\\partial \\theta}\\pm\\frac{i}{\\sin\\theta}\n\\frac{\\partial}{\\partial \\phi}-u\\sin\\theta\\frac{\\partial}{\\partial u}\\right), \n\\end{align*}\nin terms of which, using equation \\eqref{eqn:218}, we give the commutation relations \nwith the generators of supertranslations:\n\\begin{align*}\n&[L_z,P_{l,m}]=imP_{l,m},\\\\ \\nonumber\\\\\n&[L^+,P_{l,m}]=-\\sqrt{(l-m)(l+m+1)}P_{l,m+1},\\\\ \\nonumber\\\\\n&[L^-,P_{l,m}]=\\sqrt{(l+m)(l-m+1)}P_{l,m-1},\n\\end{align*}\n\\begin{align*}\n&[R_z,P_{l,m}]=-(l-1)\\sqrt{\\frac{(l+m+1)(l-m+2)}{(2l+1)(2l+3)}}P_{l+1,m}\\\\\n&+(l+2)\\sqrt{\\frac{(l+m)(l-m)}{4l^2-1}}P_{l-1,m},\\\\ \\\\\n&[R^+,P_{l,m}]=(l-1)\\sqrt{\\frac{(l+m+1)(l+m+2)}{(2l+1)(2l+3)}}P_{l+1,m+1}\\\\\n&+(l+2)\\sqrt{\\frac{(l-m-1)(l-m)}{4l^2-1}}P_{l-1,m+1},\\\\ \\\\\n&[R^-,P_{l,m}]=-(l-1)\\sqrt{\\frac{(l-m+1)(l-m+2)}{(2l+1)(2l+3)}}P_{l+1,m-1}\\\\\n&-(l+2)\\sqrt{\\frac{(l+m-1)(l+m)}{4l^2-1}}P_{l-1,m-1}.\n\\end{align*}\nThe form of the commutation relations shows that the BMS algebra is the semi-direct sum of the Lorentz algebra \n$\\mathfrak{so(3,1)}$ with the infinite Lie algebra $\\mathscr{T}$, as remarked before.\n\\section{Good and bad cuts}\n\\label{sect:8}\nWe begin this section by citing a remarkable result obtained by Sachs.\n\\begin{thm}{\\cite{Sachs1}}$\\\\ $\n\\label{thm:transl}\nThe only $4$-dimensional normal subgroup of the BMS group is the translation group.\n\\end{thm}\nTheorem \\ref{thm:transl} characterizes translations uniquely: the translation normal subgroup of the BMS group \nis singled out by its group-theoretic properties. Since we have shown that the translations $\\mathscr{T}$ are \nthe BMS transformations induced on $\\mathscr{I}^+$ by translations in Minkowski space-time, theorem \n\\ref{thm:transl} makes it possible for  us to define the asymptotic translations \nof a general asymptotically flat space-time \nas the BMS elements belonging to this normal subgroup. However a similar procedure for $\\mathscr{L}$, \ni.e. rotations and boosts, fails. Thus, as we will discuss in this section, there are several problems in \nidentifying the Poincar\\'e group as a subgroup of $\\mathscr{B}$.\\\\\nThe Poincar\\'e group is the symmetry group of flat space-time, hence it might have been thought that a suitably \nasymptotically flat space-time should, in some appropriate sense, have the Poincar\\'e group as an asymptotic \nsymmetry group. Instead, it turns out that in general we seem only to obtain the BMS group (which has  \nthe unpleasant feature of being an infinite-dimensional group) as the asymptotic symmetry group of an \nasymptotically flat space-time. \\\\\nTo better understand the nature of this problem we revert to Minkowski space-time and see how the Poincar\\'e \ngroup arises in that case as a subgroup of the BMS group. The BMS group was defined as the group of \ntransformations which conformally preserves the induced metric on $\\mathscr{I}^+$ and the strong conformal \ngeometry. However, the BMS group is much larger than the Poincar\\'e group and thus the former must preserve \nless structure on $\\mathscr{I}^+$ than does the latter. The preservation of this additional structure, \nin the case of Minkowski space-time, should allow us to restrict the BMS transformations to Poincar\\'e \ntransformations, since we know that $\\mathscr{P}$ in that case \\textit{is} a subgroup of $\\mathscr{B}$.\\\\\nIn Minkowski space-time a null hypersurface is said to be a \\textit{good cone} if it is the future light cone \nof some point, and a \\textit{bad cone} if its generators do not meet at a point. Consequently we define a \n\\textit{good cross-section}, often called a \\textit{good cut}, a cross-section of $\\mathscr{I^+}$ which is \nthe intersection of a future light cone of some point and the null hypersurface $\\mathscr{I}^+$. \nA \\textit{bad cut} is, on the other hand, the intersection of $\\mathscr{I}^+$ with some null hypersurface \nwhich does not come together cleanly at a single vertex. The situation is represented in Figure \\ref{fig:6.4}.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.4]{goodcut.eps}\n\\caption{A good cross-section of $\\mathscr{I}^+$ for Minkowski space-time is one arising as the intersection \nof $\\mathscr{I}^+$ with the future light cone of a point.}\n\\label{fig:6.4}\n\\end{center}\n\\end{figure}\n\\medskip \\medskip \\\\ Using Bondi-Sachs coordinates the Minkowski metric tensor takes the form\n\\begin{equation*}\ng=du\\otimes du+du\\otimes dr+dr\\otimes du-r^2(d\\theta\\otimes d\\theta+\\sin^2\\theta d\\phi\\otimes d\\phi).\n\\end{equation*}\nWe see that each cut of $\\mathscr{I}^+$ given by $u=t-r=\\mathrm{const}$ is a good cut, since it arises from \nthe future light cone of a point on the origin-axis $r=0$. In particular, all the good cuts can be obtained \nfrom the one given by $u=0$ by means of a space-time translation. Hence, as discussed in Sect.\\ref{sect:6} we \nobtain that every good cut can be expressed in the form\n\\begin{equation*}\nu=\\frac{A+B\\zeta+\\bar{B}\\bar{\\zeta}+C\\zeta\\bar{\\zeta}}{1+\\zeta\\bar{\\zeta}}=\\left(\\frac{A+C}{2}\\right)+\\left(\\frac{C-A}{2}\\right)\\cos\\theta\n\\end{equation*}\n\\begin{equation}\n\\label{eqn:235}\n+\\left(\\frac{B+\\bar{B}}{2}\\right)\n\\sin\\theta\\cos\\phi+i\\left(\\frac{B-\\bar{B}}{2}\\right)\\sin\\theta\\sin\\phi,\n\\end{equation}\n$A,C$ being real and $B$ being complex. Hence the equations describing good cuts are given, generally, \nby setting $u$ equal to a function of $\\theta$ and $\\phi$ which consists only of zeroth- \nand first-order spherical harmonics.\\\\\nThe effect of a transformation of the connected component of the Lorentz group is to leave invariant the \nparticular good cut $u=0$. Such a transformation is\n\\begin{subequations}\n\\label{eqn:237}\n\\begin{align}\n&\\zeta\\rightarrow\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d},\\\\\n&u\\rightarrow u'=\\frac{1+\\zeta\\bar{\\zeta}}{|a\\zeta+b|^2+|c\\zeta+d|^2}u,\n\\end{align}\n\\end{subequations}\nwith $a,b,c,d\\in\\mathbb{C}$ such that $ad-bc=1$. Note that transformations \\eqref{eqn:237} preserve the \nfunctional form of good cuts given in \\eqref{eqn:235}. In fact we have, applying \\eqref{eqn:237}, that\n\\begin{equation*}\nu'=\\left(\\frac{A+B\\zeta+\\bar{B}\\bar{\\zeta}+C\\zeta\\bar{\\zeta}}{1+\\zeta\\bar{\\zeta}}\\right)\n\\frac{1+\\zeta\\bar{\\zeta}}{|a\\zeta+b|^2+|c\\zeta+d|^2},\n\\end{equation*}\nwhere $\\zeta$ and $\\bar{\\zeta}$ have now to be expressed as functions of $\\zeta'$ and $\\bar{\\zeta}'$. \nIt is straightforward to show that\n\\begin{equation*}\nu'=\\frac{A'+B'\\zeta'+\\bar{B}'\\bar{\\zeta}'+C'\\zeta'\\bar{\\zeta}'}{1+\\zeta'\\bar{\\zeta}'},\n\\end{equation*}\nwhere \n\\begin{align*}\n&A'=A|a|^2-Bb\\bar{a}-\\bar{B}\\bar{b}a+C|b|^2,\\\\\n&B'=Bd\\bar{a}+\\bar{B}\\bar{b}c-A\\bar{a}c-Cd\\bar{b}\t,\\\\\n&C'=A|c|^2-B\\bar{c}d-\\bar{B}c\\bar{d}+C|d|^2.\n\\end{align*}\nFor example, if we perform a boost in the $z$ direction we have from \\eqref{eqn:236} \n$a=e^{\\chi\/2},d=e^{-\\chi\/2},c=b=0$. Hence we get $A'=e^{\\chi}A,B'=B,C'=e^{-\\chi}C$. \\\\\nNow it is clear that the general BMS transformation which maps good cuts into good cuts must obtain the particular \ngood cut $u=0$ from some other good cut. We can therefore express the BMS transformation as the composition \nof a translation which maps this other good cut into $u=0$, with a Lorentz transformation which leaves \n$u=0$ invariant. Thus, the BMS transformation is\n\\begin{subequations}\n\\label{eqn:238}\n\\begin{align}\n&\\zeta\\rightarrow\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d},\\\\\n&u\\rightarrow u'=\\left(\\frac{1+\\zeta\\bar{\\zeta}}{|a\\zeta+b|^2+|c\\zeta+d|^2}\\right)\\left(u+\\frac{A+B\\zeta\n+\\bar{B}\\bar{\\zeta}+C}{1+\\zeta\\bar{\\zeta}}\\right).\n\\end{align}\n\\end{subequations}\nThese BMS transformations form a $10$-real-parameters group. This is exactly the Poincar\\'e group of Minkowski \nspace-time, being a composition of a Lorentz transformation and a translation. We have obtained the following \n\\begin{prop}$\\\\ $\nThe Poincar\\'e group $\\mathscr{P}$ is the group of transformations which maps good cuts into good \ncuts in Minkowski space-time.\n\\end{prop}\nHowever, there are many other subgroups of the BMS group which can be expressed in the form \\eqref{eqn:238} and \nwhich are therefore isomorphic with the Poincar\\'e group. In fact $\\mathscr{L}$ is not a normal subgroup of the \nBMS group since for any $b=(\\Lambda,\\alpha)\\in\\mathscr{B}$ and for any $\\Lambda'=(\\Lambda',0)\\in\\mathscr{L}$ \nthe product $b\\Lambda'b^{-1}$ is not necessarily an element of $\\mathscr{L}$, as can be easily verified, and \nhence $\\mathscr{L}$ does not get canonically singled out, occurring only as a factor group of $\\mathscr{B}$ \nby the infinite-parameter Abelian group of supertranslations $\\mathscr{S}$. In particular, Lorentz transformations \ndo not commute with supertranslations and if we take any supertranslation $s$ and consider the group \n$\\mathscr{L}'=s\\mathscr{L}s^{-1}$ then it is a subgroup of the BMS group which is distinct from $\\mathscr{L}$ \nbut still isomorphic, and thus equivalent, to it. Explicitly, having fixed a supertranslation $s=(\\mathbb{I},\\alpha)$, \na transformation of $\\mathscr{L}'$ reads as\n\\begin{subequations}\n\\label{eqn:239}\n\\begin{align}\n&\\zeta\\rightarrow\\zeta'=\\frac{a\\zeta+b}{c\\zeta+d},\\\\\n&u\\rightarrow u'=\\left(\\frac{1+\\zeta\\bar{\\zeta}}{|a\\zeta+b|^2+|c\\zeta+d|^2}\\right)\\left(u-\\alpha\\right)+\\alpha.\n\\end{align}\n\\end{subequations}\nIf we start with the good cut described by the equation $u=0$, which is left invariant by $\\mathscr{L}$, and \nperform the supertranslation $s$, we obtain a new (bad) cut given by $u=\\alpha$. This is the cut which is left \ninvariant by \\eqref{eqn:239} and hence by $\\mathscr{L}'$. Hence $\\mathscr{L}'$ maps bad cuts into bad cuts.\nIt follows that if we conjugate the whole Poincar\\'e group $\\mathscr{P}$ of \\eqref{eqn:238} with respect to \nany supertranslation $s$ which is not a translation obtaining $\\mathscr{P}'=s\\mathscr{P}s^{-1}$ we get a \ndistinct subgroup of $\\mathscr{B}$, but isomorphic and completely equivalent to $\\mathscr{P}$, which maps \nbad cuts into bad cuts. Of course, for a general $s$, $\\mathscr{P'}$ and $\\mathscr{P}$ have only the \ntranslations $\\mathscr{T}$ in common. There exist \\textit{many} subgroups of $\\mathscr{B}$ which are \nisomorphic with $\\mathscr{P}$ and hence the Poincar\\'e group \\textit{is not} a subgroup of the BMS group \nin a canonical way. However we have just seen that in Minkowski space-time, if we require the group of \ntransformations to preserve the conformal nature of $\\mathscr{I}^+$ and the strong conformal geometry, together \nwith the property of mapping good cuts into good cuts, just one of the several copies of the \nPoincar\\'e group gets singled out.\n\\begin{oss}$\\\\ $\nNote that, again, the situation is similar to what happens for $\\mathscr{L}$ within $\\mathscr{P}$. In fact \n$\\mathscr{L}$ does not arise naturally as a subgroup of $\\mathscr{P}$, since if we form the group \n$\\mathscr{L}'=t\\mathscr{L}t^{-1}$, where $t$ is a translation, it is a different subgroup of $\\mathscr{P}$ \nbut isomorphic to $\\mathscr{L}$. We can say, since the commutator of a Lorentz transformation and a \ntranslation is a translation, that $\\mathscr{L}$, as a subgroup of $\\mathscr{P}$, depends on the choice \nof an arbitrary origin in Minkowski space-time. \n\\end{oss}\nWe turn now to the case when the space-time is asymptotically flat. The difficulty here is that there seems \nto be no suitable family of cuts that can properly take over the role of Minkowskian good cuts. This means \nthat, although the translation elements of $\\mathscr{B}$ are canonically singled out, there is no canonical \nconcept of a \\lq supertranslation-free\\rq\\hspace{0.1mm} Lorentz transformation. Hence the notion of a \n\\lq pure translation\\rq\\hspace{0.1mm} still makes sense, but that of \\lq pure rotation\\rq\\hspace{0.1mm} \nor \\lq pure boost\\rq\\hspace{0.1mm} does not. However, as remarked by \\cite{Lusanna2015,Lusanna2001} it can be shown that the use of appropriate \\lq post-Minkowskian\\rq\\hspace{0.1mm} boundary conditions in the 3+1 makes it possible to get rid of the supertranslations and to single out the asymptotic Poincar\\'e ADM group.\\\\\nThe most obvious generalization, for an asymptotically flat space-time, of the Minkowskian definition \nof a good cut, i.e. the intersection of future light cone of a point with $\\mathscr{I}^+$, is totally \ninappropriate. One first reason is that there are many perfectly reasonable asymptotically flat space-times \nin which no cuts of $\\mathscr{I}^+$ at all would arise in this way, e.g. \\cite{Pen72}. Even if we \nrestrict attention only to asymptotically flat space-times which do contain a reasonable number of good cuts \nof this kind, we are not likely to obtain any of the BMS transformations (apart from the identity) which \nmaps this system of cuts into itself. The difficulty lies in the fact that the detailed irregularities of \nthe interior of the space-time would be reflected in the definition of \\lq goodness\\rq\\hspace{0.1mm} \nof a cut. In other words, the light-cone cuts are far more complicated than those \\eqref{eqn:235} of flat space. \\\\\nHowever, there is a more satisfactory way to characterize good cuts of $\\mathscr{I}^+$, based on the shear \nof null hypersurfaces intersecting $\\mathscr{I}^+$. Suppose now that the (physical) space-time under consideration \ncontains a null curve $\\mu$ of a null geodesic congruence $\\mathscr{C}$ affinely parametrized by $\\tilde{r}$. \nIt can be shown \\cite{Penrin2} that the physical shear of the null hypersurfaces generated by $\\mathscr{C}$ \nhas the following asymptotic behaviour for large values of $\\tilde{r}$:\n\\begin{equation*}\n\\tilde{\\sigma}=\\frac{\\tilde{\\sigma}^0}{\\tilde{r}^2}+O(\\tilde{r}^{-4}),\n\\end{equation*}\nwhere $\\tilde{\\sigma}^0$ is called \\textit{asymptotic shear}. In the case of a flat space-time the vanishing of \nthe asymptotic shear implies the vanishing of the whole shear, but if we turn to the case of asymptotically \nflat space-times that may contain matter, although the leading term of the asymptotic behaviour of \n$\\tilde{\\sigma}$ does not change, the vanishing of $\\tilde{\\sigma}^0$ does not imply the vanishing of \n$\\tilde{\\sigma}$ \\cite{New2012}.\\\\\nIt can be shown \\cite{GRG} that if we consider the unphysical space-time obtained with conformal factor \n$\\Omega=\\tilde{r}^{-1}$, the shear transforms as\n\\begin{equation*}\n\\sigma=\\Omega^{-2}\\tilde{\\sigma},\n\\end{equation*}\nand hence\n\\begin{equation*}\n\\left.\\sigma\\right|_{_{\\mathscr{I}^+}}=\\tilde{\\sigma}^0.\n\\end{equation*}\nIn Minkowski space-time the good cones are characterized locally by the fact that the null rays generating them \npossess no shear and it can be shown that the cuts of $\\mathscr{I}^+$ which we defined earlier are precisely the \nones arising from the intersection of $\\mathscr{I}^+$ with null hypersurfaces characterized by $\\tilde{\\sigma}^0=0$. \nThus, a definition of \\lq goodness\\rq\\hspace{0.1mm} is provided, for Minkowski space-time, which refers only \nto quantities defined asymptotically. \\\\\nWe would like to extend this definition of good cut to asymptotically flat space-times too. We could say that \nsome cut is a good cut if its complex shear equals zero (since on $\\mathscr{I}^+$ we have $\\sigma=\\tilde{\\sigma}^0$). We cannot, however, define good cones simply by requiring $\\tilde{\\sigma}^0=0$. In many cases it is not possible \nto arrange $\\tilde{\\sigma}^0=0$ for all values of $\\theta$ and $\\phi$. But even in cases where it is possible \nwe have another problem, which is due to the presence of gravitational radiation. To make this point clear, \nwe cite now some important results regarding the relation between asymptotic shear and gravitational radiation \nwhich are basically due to \\cite{Bondi62,Sachs62,NewUn}. A first result is that $\\tilde{\\sigma}^0$ forms part \nof the initial data on $u=0$ used to determine the space-time asymptotically. Furthermore, it turns out that\n\\begin{equation*}\n\\frac{\\partial \\tilde{\\sigma}^0}{\\partial u}=-\\bar{N}\n\\end{equation*}\nwhere $N$ is the \\textit{Bondi news function} and that the rate of energy-momentum loss due to gravitational \nradiation through a hypersurface $\\mathscr{S}$ which spans some two-dimensional cross-section \n$S$ of $\\mathscr{I}^+$ is\n\\begin{equation}\n\\label{eqn:244}\n\\frac{dP^a}{du}=-\\frac{1}{4\\pi}\\int W^a|N|^2dS,\n\\end{equation}\nwhere \n\\begin{equation*}\nW^0=1,\\hspace{0.5cm}W^1=\\sin\\theta\\cos\\phi,\\hspace{0.5cm}W^2=\\sin\\theta\\sin\\phi,\\hspace{0.5cm}W^3=\\cos\\theta.\n\\end{equation*}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.37]{enmom.eps}\n\\caption{The outgoing gravitational radiation through two hypersurfaces $\\mathscr{S}$ and $\\mathscr{S}'$ with \nassociated cross-sections $S$ and $S'$ respectively.}\n\\label{fig:6.5}\n\\end{center}\n\\end{figure}\n\\\\Note that $S$ is topologically a sphere $S^2$ and can always be transformed, by the introduction of a suitable \nconformal factor, into a metric sphere of unit radius. Hence $dS$ can be taken to be\t\n\\begin{equation*}\ndS=\\sin\\theta d\\theta d\\phi.\n\\end{equation*}\nThus, the squared modulus of $N$ represents the flux of energy-momentum of the outgoing gravitational radiation. \n\\\\The time component of \\eqref{eqn:244} gives the famous \\textit{Bondi-Sachs mass-loss formula}:\n\\begin{equation}\n\\label{eqn:243}\n\\frac{dm_{_{B}}}{du}=-\\frac{1}{4\\pi}\\int |N|^2dS\\leq 0.\n\\end{equation}\nAn useful representation of the situation is furnished by Figure \\ref{fig:6.5}. The positivity of the integrand in \\eqref{eqn:243} shows that if a system emits gravitational waves, i.e. if \nthere is news, then its Bondi mass must decrease. If there is no news, i.e. $N=0$, the Bondi mass is constant. \nThe reason we get a mass loss rather than a mass gain is simply that all we have said has been applied \nto $\\mathscr{I}^+$ instead of $\\mathscr{I}^-$.\\\\\nFrom the above discussion it follows that, if the cut $S$ is given by $u=0$ and is shear-free, the cross-sections \n$u=\\mathrm{const}$, which are translations of $\\mathscr{I}^+$, will not be shear-free in the presence of \ngravitational radiation. In other words, if $\\tilde{\\sigma}^0=0$ for one value of $u$, we will generally \nhave $\\tilde{\\sigma}^0\\neq 0$ for a later value of $u$, i.e. \\lq goodness\\rq\\hspace{0.1mm} would not be \ninvariant under translation. However, a difficulty arises even more serious than this. Since to specify \na cut we just need to specify the value of $u$ on each generator of $\\mathscr{I}^+$, the freedom in choosing \na cut is one real number per point of the cut. On the other hand the quantity $\\tilde{\\sigma}^0$ is \ncomplex, its vanishing therefore, representing two real numbers per point of the section. We briefly \ndiscuss, without going into details \\cite{NewPen66} how to solve this problem. The first step is to define \nthe magnetic and the electric part of $\\tilde{\\sigma}^0$:\n\\begin{equation*}\n\\tilde{\\sigma}^0=\\tilde{\\sigma}^0_e+\\tilde{\\sigma}^0_m\n\\end{equation*}\nIt can be shown that for each hypersurface $u=\\mathrm{const}$ the splitting of $\\tilde{\\sigma}^0$ into \nits electric and magnetic part is invariant under BMS conformal transformation, i.e. Lorentz transformations \nwhich leave the surface $u=\\mathrm{0}$ invariant. Furthermore, in the Minkowski case, the asymptotic \nshear $\\tilde{\\sigma}^0=\\tilde{\\sigma}^0_e+\\tilde{\\sigma}^0_m$ behaves, when $u\\rightarrow-\\infty$ as\n\\begin{subequations}\n\\label{eqn:250}\n\\begin{align}\n&\\tilde{\\sigma}^{0}_e(u,\\theta,\\phi)\\xrightarrow[u \\rightarrow -\\infty]{}S_e(\\theta,\\phi),\\\\\n&\\tilde{\\sigma}^{0}_m(u,\\theta,\\phi)\\xrightarrow[u \\rightarrow -\\infty]{}0,\n\\end{align}\n\\end{subequations}\nwith $S_e(\\theta,\\phi)$ purely electric and independent of $u$. The magnetic part $\\tilde{\\sigma}^0_m(u,\\theta,\\phi)$, as \n$u\\rightarrow-\\infty$ vanishes, i.e. $S_m(\\theta,\\phi)=0$.\\\\\nOn the basis of what happens in the Minkowski case we wish to impose a physical restriction on the behaviour of \n$\\tilde{\\sigma}^0$ as $u\\rightarrow-\\infty$ for a generic space-time. Although no actual cuts of $\\mathscr{I}^+$ \nmay be shear-free, it is reasonable to expect that in the limit $u\\rightarrow-\\infty$ on $\\mathscr{I}^+$, \nsuch cuts will exist. Requiring that this limiting shear-free cuts be mapped into one another, we can actually \nrestrict the BMS transformations to obtain a canonically defined subgroup of the BMS group, which is isomorphic \nto the Poincar\\'e group. The Poincar\\'e group which emerges in this way, by virtue of the considerations we have \ndeveloped on the gravitational radiation, may be thought of as that which has relevance to the remote past, \n\\textit{before} all the gravitational radiation has been emitted. In analogy with \\eqref{eqn:250} we require that\n\\begin{equation}\n\\label{eqn:248}\n\\tilde{\\sigma}^0(u,\\theta,\\phi)\\longrightarrow S(\\theta,\\phi).\n\\end{equation}\nIf the analogy with the Minkowski theory can be trusted, we would expect $S(\\theta,\\phi)$ to be purely electric. \nHowever, it is not essential since it will be possible to extract the Poincar\\'e group only on the basis of \n\\eqref{eqn:248}. Hence we will treat the case in which $S(\\theta,\\phi)$ could have a magnetic part too, i.e. \n$S(\\theta,\\phi)=S_e(\\theta,\\phi)+S_m(\\theta,\\phi)$ with $S_m(\\theta,\\phi)\\neq 0$. \\\\\nIt can be shown \\cite{Sachs62} that, under a conformal BMS transformation, the asymptotic shear transforms as\n\\begin{equation}\t\n\\label{eqn:249}\n\\tilde{\\sigma}^0{}'(u,\\theta,\\phi)=K^{-1}\\left[\\tilde{\\sigma}^0(u,\\theta,\\phi)+\\frac{1}{2}\\eth^2\\alpha(\\theta,\\phi)\\right],\n\\end{equation}\nwhere $\\eth$ is a differential operator, whose properties can be found in \\cite{New67}. It has to be remarked \nthat $\\tilde{\\sigma}^0{}'(u,\\theta,\\phi)$ refers to the asymptotic shear of the hypersurface $u'=\\mathrm{const}$ \nof the transformed coordinate system evaluated at $(u,\\theta,\\phi)$. The complete transformation \n$\\tilde{\\sigma}^0{}'(u',\\theta',\\phi')$ is more complicated. Applying \\eqref{eqn:248} to \\eqref{eqn:249} gives\n\\begin{align*}\n&S'_e(\\theta,\\phi)=K^{-1}\\left[S_e(\\theta,\\phi)-\\frac{1}{2}\\eth^2\\alpha(\\theta,\\phi)\\right],\\\\\n&S'_m(\\theta,\\phi)=K^{-1}S_m(\\theta,\\phi),\n\\end{align*}\nsince $\\alpha(\\theta,\\phi)$ is real and the magnetic part is imaginary. It is always possible to set\n\\begin{equation*}\nS_e(\\theta,\\phi)=\\eth^2G(\\theta,\\phi),\n\\end{equation*}\nfor some real $G(\\theta,\\phi)$. Since $\\alpha(\\theta,\\phi)$ can be chosen arbitrarily on the sphere, it follows \nthat a BMS transformations for which $\\alpha(\\theta,\\phi)=2G(\\theta,\\phi)$ imposes $S'_e(\\theta,\\phi)=0$. \nThus, we have introduced coordinate conditions for which $S_e(\\theta,\\phi)=0$ at $u=-\\infty$. Now the BMS \ntransformations which preserve the condition $S_e(\\theta,\\phi)=0$ are those for which\n\\begin{equation*}\n\\eth^2\\alpha(\\theta,\\phi)=0.\n\\end{equation*}\nIt can be easily shown that this condition restricts $\\alpha(\\theta,\\phi)$ to be of the form \\eqref{eqn:207} \nand thus the allowed supertranslations are simply the translations. The Lorentz transformations, given by \n$\\alpha(\\theta,\\phi)=0$ do not spoil the coordinate conventions. We have finally obtained the following: \n\\begin{thm}$\\\\ $\nThe group of asymptotic isometries of an asymptotically flat space-time which preserves the condition \n$S_e(\\theta,\\phi)=0$ at $u=-\\infty$ is isomorphic with the Poincar\\'e group $\\mathscr{P}$.\n\\end{thm}\n\\begin{oss}\nWe could have carried out the same arguments by taking the limit $u\\rightarrow +\\infty$, and it would \nhave been an independent choice. Thus, in a similar way, we could have extracted another Poincar\\'e \ngroup which has relevance to the remote future, i.e. \\textit{after} all the gravitational radiation has \nbeen emitted. There seems to be no reason to believe that these two Poincar\\'e groups will be the same, in general. \n\\end{oss}\n\\section{BMS transformations and gravitational scattering}\n\\label{sect:9}\nThe ground is now ready for considering a very recent application of BMS transformations,\ni.e. the discovery by Strominger \\cite{ST2014} that there exist BMS transformations acting non-trivially on\noutgoing gravitational scattering data while preserving the intrinsic structure at future null\ninfinity. His analysis begins with the local expression of a generic Lorentzian metric in\nretarded Bondi coordinates, that we know from Sect. \\ref{sect:2}, and in particular\nwith the asymptotic expansion of the metric about future null infinity (where $r=\\infty$),\nreading as (cf. \\eqref{eqn:140})\n\\begin{eqnarray}\ng &= &du \\otimes du +(du \\otimes dr + dr \\otimes du)\n-r^{2}\\gamma_{z {\\bar z}}(dz \\otimes d{\\bar z}+d {\\bar z} \\otimes dz)\n \\nonumber \\\\\n&- &2{m_{B}^+\\over r}du \\otimes du - r C_{zz}dz \\otimes dz \n-r C_{{\\bar z}{\\bar z}}d{\\bar z}\\otimes d{\\bar z} \n\\nonumber \\\\ \n&+ &U_{z}(dz \\otimes d{\\bar z}+d{\\bar z} \\otimes dz)\n+U_{{\\bar z}}(du \\otimes d{\\bar z}+d{\\bar z} \\otimes du)\n+...,\n\\label{(9.1)}\n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\gamma_{z{\\bar z}} \\equiv {2 \\over (1+z{\\bar z})}, \\;\nU_{z} \\equiv -{1 \\over 2}D^{z}C_{zz},\n\\label{(9.2)}\n\\end{equation}\n$D^{z}$ denoting covariant differentiation with respect to the $2$-sphere metric \n$\\gamma_{z{\\bar z}}$. The vector fields that \ngenerate BMS transformations are of two types: there are $6$ with an\nasymptotic $\\mathrm{SL}(2,\\mathbb{C})$ Lie bracket algebra, and an infinite number of commuting supertranslations.\nGlobal $\\mathrm{SL}(2,\\mathbb{C})$ conformal transformations are generated on $\\mathscr{I}^{+}$ by the real part of\nthe complex vector fields (cf. Sect. \\ref{sect:7})\n\\begin{eqnarray}\n{}^{(+)}\\zeta^{a}\\partial_{a}&=&\\left(1-{u \\over 2r}\\right){}^{(+)}\\zeta^{z}\\partial_{z}\n-\\left(1+{u \\over r}\\right){D_{z}{}^{(+)}\\zeta^{z}r \\over 2}\\partial_{r}\n\\nonumber \\\\&-&{u \\over 2r}\\gamma^{z {\\bar z}}D_{z}^{2}{}^{(+)}\\zeta^{z}\\partial_{{\\bar z}}\n+{u \\over 2}D_{z}{}^{(+)}\\zeta^{z}\\partial_{u}\n+{\\rm O}(r^{-2}),\n\\label{(9.3)}\n\\end{eqnarray}\nwhere ${}^{(+)}\\zeta^{z}=(1,z,z^{2},i,iz,iz^{2})$ \\cite{ST2014}. The action\nof the Lie derivative operator yields \\cite{ST2014}\n\\begin{align}\n&{\\cal L}_{{}^{(+)}\\zeta}C_{zz}={}^{(+)}\\zeta^{z}\\partial_{z}C_{zz}\n+2 \\partial_{z}{}^{(+)}\\zeta^{z}C_{zz}+{D_{z}{}^{(+)}\\zeta^{z}\\over 2}\n(u \\partial_{u}-1)C_{zz},\n\\label{(9.4)} \\\\\n&{\\cal L}_{{{}^{(+)}\\bar \\zeta}}C_{zz}={{}^{(+)}\\bar \\zeta}^{{\\bar z}}\\partial_{{\\bar z}}C_{zz}\n+{D_{{\\bar z}}{{}^{(+)}\\bar \\zeta}^{{\\bar z}}\\over 2}(u \\partial_{u}-1)C_{zz},\n\\label{(9.5)} \\\\\n&{\\cal L}_{{}^{(+)}\\zeta}m_B^+=\\left({}^{(+)}\\zeta^{z}\\partial_{z}+{u \\over 2}D_{z}{}^{(+)}\\zeta^{z}\\partial_{u}\n+{3 \\over 2}D_{z}{}^{(+)}\\zeta^{z}\\right)m_{B}^+ \\nonumber \\\\+&{u \\over 2}\\partial_{u}\n(U_{z}{}^{(+)}\\zeta^{z}-U^{z}D_{z}^{2}{}^{(+)}\\zeta^{z}).\n\\label{(9.6)}\n\\end{align}\n\nThe $\\mathrm{SL}(2,\\mathbb{C})$ transformations on $\\mathscr{I}^{-}$ are generated by the complex vector \nfields (cf. (9.3) and replace $u$ by $-v$ therein)\n\\begin{eqnarray}\n{}^{(-)}\\zeta^{a}\\partial_{a} &=&\\left(1+{v \\over 2r}\\right){}^{(-)}\\zeta^{z}\\partial_{z}\n-\\left(1-{v \\over r}\\right){D_{z}{}^{(-)}\\zeta^{z}r \\over 2}\\partial_{r}\n\\nonumber \\\\&+&{v \\over 2r}\\gamma^{z {\\bar z}}D_{z}^{2}{}^{(-)}\\zeta^{z}\\partial_{{\\bar z}}\n+{v \\over 2}D_{z}{}^{(-)}\\zeta^{z}\\partial_{v}\n+{\\rm O}(r^{-2}),\n\\label{(9.7)}\n\\end{eqnarray}\nand the associated Lie derivative acts according to\n\\begin{align}\n&{\\cal L}_{{}^{(-)}\\zeta}D_{zz}={}^{(-)}\\zeta^{z}\\partial_{z}D_{zz}\n+2 \\partial_{z}{}^{(-)}\\zeta^{z}D_{zz}+{D_{z}{}^{(-)}\\zeta^{z}\\over 2}\n(v \\partial_{v}-1)D_{zz},\n\\label{(9.8)} \\\\\n&{\\cal L}_{{{}^{(-)}\\bar \\zeta}}D_{zz}={{{}^{(-)}\\bar \\zeta}}^{{\\bar z}}\\partial_{{\\bar z}}D_{zz}\n+{D_{{\\bar z}}{{}^{(-)}\\bar \\zeta}^{{\\bar z}}\\over 2}(v \\partial_{v}-1)D_{zz},\n\\label{(9.9)} \\\\\n&{\\cal L}_{{}^{(-)}\\zeta}m_{B}^{-}=\\left({}^{(-)}\\zeta^{z}\\partial_{z}+{v \\over 2}D_{z}{}^{(-)}\\zeta^{z}\\partial_{v}\n+{3 \\over 2}D_{z}{}^{(-)}\\zeta^{z}\\right)m_{B}^{-} \\nonumber \\\\&-{v \\over 2}\\partial_{v}\n(V_{z}{}^{(-)}\\zeta^{z}-V^{z}D_{z}^{2}{}^{(-)}\\zeta^{z}).\n\\label{(9.10)}\n\\end{align}\nThe supertranslations on $\\mathscr{I}^{+}$ are generated by the vector fields\n\\begin{equation}\n{}^{(+)}f \\partial_{u}-{1 \\over r}\\Bigr(D^{{\\bar z}}{}^{(+)}f \\partial_{{\\bar z}}\n+D^{z}{}^{(+)}f \\partial_{z}\\Bigr)+D^{z}D_{z}{}^{(+)}f \\partial_{r}, \\;\n{}^{(+)}f={}^{(+)}f(z,{\\bar z}),\n\\label{(9.11)}\n\\end{equation}\nand one finds the Lie derivatives\n\\begin{align}\n&{\\cal L}_{{}^{(+)}f}C_{zz}={}^{(+)}f \\partial_{u}C_{zz}-2D_{z}^{2}{}^{(+)}f,\n\\label{(9.12)}\\\\\n&{\\cal L}_{{}^{(+)}f}U_{z}={}^{(+)}f \\partial_{u}U_{z}\n-{1 \\over 2}D^{z}{}^{(+)}f \\partial_{u}C_{zz}+D^{z}D_{z}^{2}{}^{(+)}f.\n\\label{(9.13)}\n\\end{align}\n\nSupertranslations on past null infinity $\\mathscr{I}^{-}$ are also of interest, and are\ngenerated by the vector fields\n\\begin{equation}\n{}^{(-)}f\\partial_{v}+{1 \\over r}\\Bigr(D^{{\\bar z}}{}^{(-)}f\\partial_{{\\bar z}}\n+D^{z}{}^{(-)}f\\partial_{z} \\Bigr)-D^{z}D_{z}{}^{(-)}f\\partial_{r},\n\\label{(9.14)}\n\\end{equation}\nthe Lie derivatives along which read as \\cite{ST2014}\n\\begin{align}\n&{\\cal L}_{{}^{(-)}f}D_{zz}={}^{(-)}f\\partial_{v}D_{zz}+2D_{z}^{2}{}^{(-)}f,\n\\label{(9.15)}\\\\\n&{\\cal L}_{{}^{(-)}f}V_{z}={}^{(-)}f\\partial_{u}V_{z}+{1 \\over 2}D^{z}{}^{(-)}f\\partial_{u} D_{zz}\n+D^{z}D_{z}^{2}{}^{(-)}f.\n\\label{(9.16)}\n\\end{align}\nWe are now going to outline the connection between BMS transformations on\n$\\mathscr{I}^{+}$ and $\\mathscr{I}^{-}$, denoted by ${\\rm BMS}^{+}$ and\n${\\rm BMS}^{-}$, respectively.\n\n\\subsection{The Christodoulou-Klainerman space-times}\n\nAfter the singularity theorems of Penrose, Hawking and Geroch \n\\cite{CS1,CS2,CS3,CS4,CS5,CS6,CS7,CS8,CS9,CS10,CS11}, it was thought for a long\ntime that singularities are a generic property of general relativity, if the energy-momentum \ntensor obeys suitable conditions. However, an outstanding piece of work of Christodoulou and\nKlainerman \\cite{CK1993} proved that there exist asymptotically flat space-times that are\ngeodesically complete and hence singularity-free in classical theory.\n\nChristodoulou and Klainerman studied asymptotically flat initial data in the center-of-mass \nframe on a maximal spacelike slice for which the Bach tensor\n$$\n\\varepsilon^{ijk} \\; { }^{(3)}D_{j} \\; { }^{(3)}G_{kl}\n$$\nof the induced $3$-metric decays at least as $r^{-{7 \\over 2}}$ at spatial infinity,\nwhile the extrinsic curvature decays like $r^{-{5 \\over 2}}$. This implies in particular\nthat, in normal coordinates about infinity, the leading part of the $3$-metric takes the\nSchwarzschild form, while the correction decays like $r^{-{3 \\over 2}}$. Remarkably, all such\ninitial data, when supplemented by a global smallness condition, give rise to a geodesically\ncomplete solution of the Einstein equations. Since such a smallness condition holds in a\nfinite neighbourhood of Minkowski space-time, the result obtained is said to be the global\nnon-linear stability of Minkowski space-time \\cite{CK1993}.\n \nAs we know from Sect. \\ref{sect:8}, the gravitational radiation flux is proportional to the square of\nthe Bondi news $N_{zz}$ which, for any finite energy data, vanishes on the boundaries\n$\\partial_{+}\\mathscr{I}^{+}$ and $\\partial_{-}\\mathscr{I}^{+}$ of future null infinity, i.e.\n\\begin{equation}\n\\left . N_{zz} \\right|_{\\partial_{\\pm} \\mathscr{I}^{+}}=0.\n\\label{(9.17)}\n\\end{equation}\nFor Christodoulou-Klainerman space-times the falloff property for $u \\rightarrow \\pm \\infty$ is\n\\begin{equation}\nN_{zz}(u) \\sim |u|^{-{3 \\over 2}},\n\\label{(9.18)}\n\\end{equation}\nor even faster than this. With the coordinates used in \\eqref{(9.1)}, the Weyl curvature component\n$\\Psi_{2}^{0}$ reads as\n\\begin{eqnarray}\n\\Psi_{2}^{0}(u,z,{\\bar z})&=&-\\lim_{r \\to \\infty} \\Bigr(r C_{uzr{\\bar z}} \\; \\gamma^{z{\\bar z}}\\Bigr) \\nonumber \\\\\n&=&-m_{B}^++{1 \\over 4}C^{zz}N_{zz}\n-{1 \\over 2}\\gamma^{z {\\bar z}}\\Bigr(\\partial_{\\bar z}U_{z}\n-\\partial_{z}U_{\\bar z}\\Bigr).\n\\label{(9.19)}\n\\end{eqnarray}\nFor center-of-mass Christodoulou-Klainerman spaces at $r=\\infty,u=\\infty$ one finds\n\\begin{equation}\n\\left . \\Psi_{2}^{0}\\right |_{\\partial_{+}\\mathscr{I}^{+}}=0,\n\\label{(9.20)}\n\\end{equation}\nwhile at $r=\\infty,u=-\\infty$\n\\begin{equation}\n\\left . \\Psi_{2}^{0}\\right |_{\\partial_{-}\\mathscr{I}^{+}}=-M,\n\\label{(9.21)}\n\\end{equation}\nthe ratio ${M \\over G}$ being the Arnowitt-Deser-Misner mass. The real part of these formulae, \njointly with \\eqref{(9.17)}, imply for the mass aspect the following boundary conditions:\n\\begin{align}\n&\\left . m_{B}^+ \\right |_{\\partial_{+}\\mathscr{I}^{+}}=0,\n\\label{(9.22)}\\\\\n&\\left . m_{B}^+ \\right |_{\\partial_{-}\\mathscr{I}^{+}}=M.\n\\label{(9.23)}\n\\end{align}\nFrom the imaginary part of \\eqref{(9.19)} one obtains \n\\begin{equation}\n\\Bigr[\\partial{{\\bar z}}U_{z}-\\partial_{z}U_{{\\bar z}}\\Bigr]_{\\partial_{\\pm}\\mathscr{I}^{+}}=0.\n\\label{(9.24)}\n\\end{equation}\nA similar procedure on past null infinity yields the boundary conditions\n\\begin{equation}\n\\left . m_{B}^{-} \\right |_{\\partial_{-}\\mathscr{I}^{-}}\n=\\Bigr[\\partial_{{\\bar z}}V_{z}-\\partial_{z}V_{{\\bar z}}\\Bigr]_{\\partial_{\\pm}\\mathscr{I}^{-}}\n= \\left . \\partial_{v}D_{zz} \\right|_{\\partial_{\\pm}\\mathscr{I}^{-}}=0,\n\\label{(9.25)}\n\\end{equation}\nand\n\\begin{equation}\n\\left . m_{B}^{-} \\right |_{\\partial_{+}\\mathscr{I}^{-}}=M.\n\\label{(9.26)}\n\\end{equation}\nThe falloff of the Bondi news, here denoted by $M_{zz}$, is\n\\begin{equation}\nM_{zz} \\equiv \\partial_{v}D_{zz} \\sim |v|^{-{3 \\over 2}},\n\\label{(9.27)}\n\\end{equation}\nor even faster.\n\nNote that, while it seems reasonable that the total integrated mass should decay to zero\nat the future boundary of future null infinity for a sufficiently weak gravitational disturbance, \nit is not a priori clear that the unintegrated mass aspect function $m_{B}$ should itself\napproach $0$. Furthermore, the news tensor might decay more slowly near the future and past\nboundaries of null infinity and still provides a finite value for the total energy flux\n\\cite{ST2014}. Strominger has also considered the possible coupling to any kind of massless\nmatter that dissipates at late times on $\\mathscr{I}^{+}$ or early times on\n$\\mathscr{I}^{-}$. He imposes the boundary conditions \\eqref{(9.20)}-\\eqref{(9.26)} as restriction on\nthe desired solutions of Einstein's equations.\n\n\\subsection{Link among future and past null infinity near spacelike infinity}\n\nThe future null infinity data $m_{B}^+$ and $C_{zz}$ are related by the constraint \\cite{ST2014}\n\\begin{equation}\n\\partial_{u}m_{B}^+=-{1 \\over 2}\\partial_{u}\\Bigr[D^{z}U_{z}+D^{\\bar z}U_{\\bar z}\\Bigr]-T_{uu},\n\\label{(9.28)}\n\\end{equation}\nwhere $T_{uu}$ is the total outgoing radiation energy flux for gravity plus matter at \nfuture null infinity, whose explicit form is \\cite{ST2014}\n\\begin{equation}\nT_{uu}={1 \\over 4}N_{zz}N^{zz}+4 \\pi G \\lim_{r \\to \\infty}\n\\Bigr[r^{2}T_{uu}^{M}\\Bigr].\n\\label{(9.29)}\n\\end{equation}\n\nA classical gravitational scattering problem consists of a set of final data on future\nnull infinity which evolve from some initial data on past null infinity. On going from a\nCristodoulou-Klainerman space-time to these data one encounters a physical ambiguity\nunder ${\\rm BMS}^{+} \\times {\\rm BMS}^{-}$ transformations. In particular one has to\nevaluate, once the Bondi news and matter radiation flux are given, the leading terms $m_{B}$\nand $C_{zz}$ of the metric in the coordinates used in \\eqref{(9.1)} everywhere on future null infinity.\nIn order to achieve this one integrates equation \\eqref{(9.28)} and the equation\n\\begin{equation}\n\\partial_{u}C_{zz}=N_{zz}\n\\label{(9.30)}\n\\end{equation}\nalong a null generator of future null infinity, but initial conditions at the past\nboundary of $\\mathscr{I}^{+}$ are required. For Christodoulou-Klainerman space-times one\nhas indeed the initial condition \\eqref{(9.23)}. Under a $\\mathrm{SL}(2,\\mathbb{C})$ transformation on future null\ninfinity, which forces us to abandon the center-of-mass frame, the restriction of $m_{B}$\nto $\\partial_{-}\\mathscr{I}^{+}$ transforms according to \\cite{ST2014}\n\\begin{equation}\n{\\cal L}_{{}^{(+)}{\\zeta}} \\left . m_{B}^+ \\right |_{\\partial_{-}\\mathscr{I}^{+}}\n=\\left({}^{(+)}\\zeta^{z}\\partial_{z}+{3 \\over 2}D_{z}{}^{(+)}\\zeta^{z}\\right)\n\\left . m_{B}^+ \\right |_{\\partial_{-}\\mathscr{I}^{+}},\n\\label{(9.31)}\n\\end{equation}\nand hence is no longer constant.\n\nAs far as the initial value of $C_{zz}$ is concerned, this may be any solution of\nequation \\eqref{(9.24)}, which can be re-expressed in the form\n\\begin{equation}\n\\Bigr[D_{\\bar z}^{2}C_{zz}-D_{z}^{2}C_{{\\bar z}{\\bar z}}\\Bigr]_{\\partial_{-}\\mathscr{I}^{+}}=0.\n\\label{(9.32)}\n\\end{equation}\nIf $C$ denotes some real-valued function, the general integral of equation \\eqref{(9.32)} reads as\n\\begin{equation}\nC_{zz}(-\\infty,z,{\\bar z})=D_{z}^{2}C.\n\\label{(9.33)}\n\\end{equation}\nThis implies that\n\\begin{equation}\nU_{z}=-{1 \\over 2}\\partial_{z}(C+D_{z}D^{z}C).\n\\label{(9.34)}\n\\end{equation}\n\nBy proceeding along similar lines on past null infinity, the mass aspect $m_{B}^{-}$ is\ndetermined by the constraint \\cite{ST2014} \n\\begin{equation}\n\\partial_{v}m_{B}^{-}={1 \\over 2}\\partial_{v}\n\\Bigr[D^{z}V_{z}+D^{\\bar z}V_{\\bar z}\\Bigr]+T_{vv},\n\\label{(9.35)}\n\\end{equation}\nwhere $T_{vv}$ denotes the total incoming radiation flux at past null infinity. This can be solved\nby imposing data at $\\partial_{+}\\mathscr{I}^{-}$ and integrating until \n$\\partial_{-}\\mathscr{I}^{-}$ is reached. One can assume the boundary condition \\eqref{(9.26)}. The\nmass aspect has the transformation property\n\\begin{equation}\n\\left . {\\cal L}_{{}^{(-)}\\zeta}m_{B}^{-} \\right |_{\\partial_{+}\\mathscr{I}^{-}}\n=\\left({}^{(-)}\\zeta^{z}\\partial_{z}+{3 \\over 2}D_{z}{}^{(-)}\\zeta^{z}\\right)\n\\left . m_{B}^{-} \\right |_{\\partial_{+}\\mathscr{I}^{-}} .\n\\label{(9.36)}\n\\end{equation}\nMoreover, for some real-valued function $F$, one can take\n\\begin{equation}\nD_{zz}(-\\infty,z,{\\bar z})=D_{z}^{2}F.\n\\label{(9.37)}\n\\end{equation}\nThe next task is to relate the initial data at $\\partial_{+}\\mathscr{I}^{-}$ to those\nat $\\partial_{-}\\mathscr{I}^{+}$. For this purpose, one has first to relate the points\non $\\partial_{+}\\mathscr{I}^{-}$ to the points on $\\partial_{-}\\mathscr{I}^{+}$. \nThe desired relation derives from the conformal infinity picture in which spacelike\ninfinity is a point and future (resp. past) null infinity is its future (resp. past) lightcone.\nNull generators of null infinity then go from past null infinity to future null infinity\nthrough spacelike infinity. These generators provide the necessary identification: all points\non the same generator are labeled by the same coordinate $(z,{\\bar z})$.\n\nThe ordinary Lorentz transformations of Minkowski space-time are generated by vector fields\nof the form \\eqref{(9.3)} and \\eqref{(9.7)}, restricted by the condition\n\\begin{equation}\n{}^{(+)}\\zeta^{z}={}^{(-)}\\zeta^{z}.\n\\label{(9.38)}\n\\end{equation}\nThis condition is defined only after having identified points on\n$\\partial_{+}\\mathscr{I}^{-}$ and $\\partial_{-}\\mathscr{I}^{+}$. Such Lorentz transformations \npreserve the condition\n\\begin{equation}\n\\left . m_{B}^+ \\right |_{\\partial_{-}\\mathscr{I}^{+}}(z,{\\bar z})\n=\\left . m_{B}^+ \\right |_{\\partial_{+}\\mathscr{I}^{-}}(z,{\\bar z}),\n\\label{(9.39)}\n\\end{equation}\ndespite the fact that both mass aspects change under boosts.\n\nFurthermore, one has to relate the initial data for $C_{zz}$ at $\\partial_{-}\\mathscr{I}^{+}$\nto those for $D_{zz}$ at $\\partial_{+}\\mathscr{I}^{-}$ in a way compatible with \\eqref{(9.38)}. \nThis is obtained by imposing\n\\begin{equation}\nD_{zz}(\\infty,z,{\\bar z})=-C_{zz}(-\\infty,z,{\\bar z}).\n\\label{(9.40)}\n\\end{equation}\nThe equations for $U_{z}$ (see \\eqref{(9.2)}) and $V_{z}$ then imply\n\\begin{equation}\n\\left . V_{z} \\right |_{\\partial_{+}\\mathscr{I}^{-}}\n=\\left . U_{z} \\right |_{\\partial_{-}\\mathscr{I}^{+}}.\n\\label{(9.41)}\n\\end{equation}\nThis simple but non-trivial condition unifies the separate BMS symmetries on future and past\nnull infinity into a single one acting on both parts of null infinity, so that\n\\begin{equation}\n{}^{(-)}f(z,{\\bar z})={}^{(+)}f(z,{\\bar z}).\n\\label{(9.42)}\n\\end{equation}\nIf ${}^{(+)}f$ and ${}^{(-)}f$, besides being equal, are constant, it means we deal with global time\ntranslations. The three constant spatial translations are given by the formula\n\\begin{equation}\nf={(1-z{\\bar z})\\over (1+z{\\bar z})}.\n\\label{(9.43)}\n\\end{equation}\nGenerators that obey \\eqref{(9.38)} and \\eqref{(9.42)} are the diagonal generators of \n${\\rm BMS}^{+} \\times {\\rm BMS}^{-}$. Strominger has stressed \\cite{ST2014} that the\npossibility of linking data on future and past null infinity depends crucially on the\nproperty of Christodoulou-Klainerman space-times, and such a link is not always possible\nfor more general types of asymptotically flat space-times. For example, if the news tensor\nobeys the ${1 \\over |u|}$ fall off, $C_{zz}$ diverges logarithmically, and the boundary\ncondition \\eqref{(9.40)} cannot be imposed.\n\n\\subsection{Symmetries of classical scattering}\n\nTo sum up, one starts from a Christodoulou-Klainerman space-time in the center-of-mass frame,\njointly with conformally invariant radiative data $N_{zz}$ and $M_{zz}$ on null infinity.\nIn order to be able to regard this as a scattering solution, one has to construct both the\ninitial data on past null infinity and the final data on future null infinity. For this purpose,\none assumes that (9.39) is fulfilled with a boundary value $M$, while\n\\begin{equation}\n\\left . C_{zz} \\right |_{\\partial_{-}\\mathscr{I}^{+}}(z,{\\bar z})\n=\\left . D_{zz} \\right |_{\\partial_{+}\\mathscr{I}^{-}}(z,{\\bar z})=0,\n\\label{(9.44)}\n\\end{equation}\nand one performs integration of the constraint equations on future and past null infinity\naway from spacelike infinity. Hence one obtains final data $(C_{zz},m_{B}^+)$ that solve the\nscattering problem in correspondence to the initial data $(D_{zz},m_{B}^{-})$.\n\nStrominger has also stressed that, for every Christodoulou-Klainerman space, one can\ngenerate an infinite-parameter family of solutions to the scattering problem. Some of these\ndiffer by Poincar\\'e transformations, while others differ by supertranslations.\n\n\\section{Recent developments and concluding remarks}\n\nThe latest applications of all concepts and properties described so far have to do with\nblack-hole physics in quantum gravity. Indeed, supertranslations transform the Minkowski\nvacuum into a physically inequivalent zero-energy vacuum. This means that the vacuum is not\ninvariant, and hence supertranslation symmetry is spontaneously broken. But if the \nquantum gravity vacuum is not unique, it becomes conceivable that the final vacuum state\nmight get correlated with the thermal Hawking radiation in such a way that no loss of\nunitarity ever occurs \\cite{HPS1}. \n\nMoreover, Bondi-Metzner-Sachs transformations are \ndiffeomorphisms that change the physical state (e.g. a supertranslation maps a stationary\nblack hole to a physically inequivalent one). During black hole evaporation, supertranslation\ncharge is radiated through null infinity, and since this charge is conserved, the sum of\nthe black hole and radiated supertranslation charge is fixed at all times. This implies\nthat black holes carry soft hair (i.e. additional data, besides mass, charge and\nangular momentum), and if the black hole evaporates completely, the net supertranslation\ncharge in the outgoing radiation must be conserved. This leads in turn to correlations \nbetween the early- and late-time Hawking radiation \\cite{HPS1}.  \n\nIn subsequent work \\cite{HPS2}, the same authors have proved that black hole spacetimes in \nclassical general relativity are characterized by an infinite head of supertranslation hair, in\naddition to Arnowitt-Deser-Misner mass $M$, linear momentum $\\vec P$, angular momentum\n$\\vec J$ and boost charge $\\vec K$. Classical superrotation charges measured at infinity lead to\ndistinct black holes. Solutions with supertranslation hair are diffeomorphic to \nSchwarzschild spacetime, and a black hole can be supertranslated by throwing in an asymmetric\nshock wave \\cite{HPS2}.\n\nSome interesting recent work inspired by Refs. \\cite{ST2014,HPS1,HPS2} is the one by\nAsorey and coauthors in Ref. \\cite{Asorey}. These authors study infrared transformations\nin local quantum physics. Their observables are smeared by test functions, at first vanishing\nat infinity. They prove that the equations of motion can be seen as constraints, that generate\nthe group of space- and time-dependent gauge transformations. Infrared non-trivial effects are\nthen obtained by allowing for test functions that do not vanish at infinity. The resulting\nextended operators generate a larger group, and the quotient of the two groups gives rise\nto the so-called superselection sectors, that distinguish among the various infrared sectors.\nThe BMS group plays the role of changing the superselection sector.\n\nAs far as black hole theory is concerned, we find it worth bringing to the attention of the reader\nthe work in Ref. \\cite{Bousso}, where the authors have provided a derivation and an observable\nrealization of the algebra imposed by Strominger on unobservable boundary degrees of freedom. The\nconservation laws associated with asymptotic symmetries are found to arise from free propagation \nof infrared modes. In their opinion, this adds evidence in favour of soft charges failing to constrain\nthe hard scattering problem, and hence not being relevant for the so-called black hole\ninformation paradox.\n\nOn reverting now to classical theory, we stress again the new perspective resulting from the\nwork in Refs. \\cite{ST2014,HPS2}: the very existence of a scattering problem from past to future\nnull infinity in general relativity makes it necessary to consider an infinite number of conserved\nsupertranslation and superrotation charges. This property was not discovered for several decades,\npossibly because many research lines focused on space-times where the peeling property and\nthe Penrose conformal infinity can be exploited. Thanks to the work of Christodoulou and \nKlainerman \\cite{CK1993}, it is by now well known that generic space-times in a finite\nneighbourhood of Minkowski lie precisely in the region where it is possible to define an infinite\nnumber of finite, non-vanishing and conserved supertranslation and superrotation charges. \nThe work in Ref. \\cite{HPS2} has studied a family of space-times whose asymptotic forms lie in\nthe same region and possess the conserved charges, although they are not necessarily in a\nsmall neighbourhood of Minkowski and may well contain black holes in the interior.\nFurther work on the rigorous theory of supertranslation and superrotation charges is likely\nto shed new light on the mathematical structures of space-time \n\\cite{ES1992} and their physical implications.\n\n\\section*{acknowledgments}\nThe authors are grateful to the Dipartimento di Fisica ``Ettore Pancini'' of Federico II University for\nhospitality and support. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\part{Introduction}\n\n\n\\section{Introduction}\n \n\n \n\\subsection{Basic definitions}\n \n\n \n\nGiven two sequences $a_{n}>0$ and $b_{n}=\\bar{b}_{n}$ where $n\\in {\\Bbb Z}_{+}=\\{0,1,\\ldots \\}$, \n a \n Jacobi matrix is defined by an equality\n  \\begin{equation}\n{\\cal J} = \n\\begin{pmatrix}\n b_{0}&a_{0}& 0&0&0&\\cdots \\\\\n a_{0}&b_{1}&a_{1}&0&0&\\cdots \\\\\n  0&a_{1}&b_{2}&a_{2}&0&\\cdots \\\\\n  0&0&a_{2}&b_{3}&a_{3}&\\cdots \\\\\n  \\vdots&\\vdots&\\vdots&\\ddots&\\ddots&\\ddots\n\\end{pmatrix} .\n\\label{eq:ZP+}\\end{equation}\nLet     $u= (u_{0}, u_{1}, \\ldots)^\\top =: (u_{n})$ be a column. \n  Then \n\\begin{equation}\n( {\\cal J} u) _{0} =  b_{0} u_{0}+ a_{0} u_{1} \\quad \\mbox{and}\\quad\n( {\\cal J} u) _{n} = a_{n-1} u_{n-1}+b_{n} u_{n}+ a_{n} u_{n+1} \\quad \\mbox{for}\\quad n\\geq 1.  \n \\label{eq:ZP+1}\\end{equation}\n \n  \n     \n\n\nLet us consider a second-order difference equation  \n\\begin{equation}\n  a_{n-1} u_{n-1}+b_{n} u_{n}+ a_{n} u_{n+1} =z u_{n}, \\quad n\\geq 1,  \n  \\label{eq:Jy}\\end{equation}\n  associated with the operator $\\cal J$. Clearly,   values  $u_{0}$ and $u_{1 }$  determine uniquely a solution   $u_{n}$ of  equation \\eqref{eq:Jy}. In particular, \nthe   solutions $p_{n}(z)$ and $q_{n}(z)$  are distinguished by  boundary conditions\n \\begin{equation}\np_{0} (z)= 1,\\quad  q_{0} (z)= 0, \n  \\label{eq:Pz}\\end{equation}\n  and\n  \\begin{equation}\np_1 (z)= (z-b_{0} )\/a_{0} , \\quad  q_1 (z)= 1\/a_{0} .\n  \\label{eq:Qz}\\end{equation}\n  It is easy to see that $p_{n}(z)$ is a  polynomial of degree $n$ and $q_{n}(z)$ is a  polynomial of degree $n-1 $.    Let us set $p(z)= (p_{0}(z),p_{1}(z),\\ldots)^\\top$, $q(z)= (q_{0}(z),q_{1}(z),\\ldots)^\\top$\n  and    $e_{0}= (1,0,\\ldots)^\\top$.  Then  \n  \\begin{equation}\n  {\\cal J} p(z) =z p(z) \\quad \\mbox{and} \\quad   {\\cal J} q(z) =z q(z)  +e_{0}.\n  \\label{eq:PQzz}\\end{equation}\n  \n   Let $u= ( u_{n} )$ and $v=(v_{n})$ be two solutions of equation \\eqref{eq:Jy}. A direct calculation shows that their Wronskian\n  \\begin{equation}\n\\{ u, v  \\} : = a_{n}  (u_{n}  v_{n+1}-u_{n+1}  v_{n})\n\\label{eq:Wr}\\end{equation}\ndoes not depend on $n\\in{\\Bbb Z}_{+}$. Clearly, the Wronskian $\\{ u, v  \\} =0$ if and only if the solutions $u$ and $v$ are proportional.  Note that $\\{p,q \\}=1$ so that\n  \\begin{equation}\n p_{n}(z) q_{n+1}(z)- p_{n+1}(z) q_{n}(z)=a_n^{-1}, \\quad \\forall n\\in {\\Bbb Z}_{+}.\n  \\label{eq:PQW}\\end{equation}\n \n \n\n      We consider Jacobi operators $J$ acting    in \n the space $\\ell^2 ({\\Bbb Z}_{+})$ with the scalar product   $\\langle u, v \\rangle=\\sum_{n\\in {\\Bbb Z}_{+}}u_{n} \\bar{v}_{n}$.\n Let $\\cal D  $ be a dense in  $\\ell^2 ({\\Bbb Z}_{+})$ set of vectors $u $ with only a  finite number of non-zero components  $u_{n}$.\n  The minimal Jacobi operator $J_{\\rm min}$ is  defined   by the equality $J_{\\rm min} u= {\\cal J} u$ on domain \n $  {\\cal D} (J_{\\rm min}) =\\cal D$.\n  It extends to a bounded operator defined on the whole space $\\ell^2 ({\\Bbb Z}_{+})$ \n  if and only if the corresponding recurrence coefficients $( a_{n} )\\in \\ell^\\infty ({\\Bbb Z}_{+}) $ and $ ( b_{n} )\\in \\ell^\\infty ({\\Bbb Z}_{+}) $.  In the general case, we introduce also\n the maximal  operator $J_{\\rm max}$   by the   same  formula $J_{\\rm max} u= {\\cal J} u$ on a set $ {\\cal D} (J_{\\rm max})$  of all vectors $u \\in \\ell^2 ({\\Bbb Z}_{+})$ such that ${\\cal J}u \\in \\ell^2 ({\\Bbb Z}_{+})$. \n    The operator $J_{\\rm min} $ is\nsymmetric  in the space $\\ell^2 ({\\Bbb Z}_{+})$, and its adjoint operator  $J^*_{\\rm min} = J_{\\rm max}$.\n\nThe operator $J_{\\rm max}$  is not of course symmetric.  \n     For all $u, v \\in {\\cal D} (J_{\\rm max})$, we have an identity  (the Green formula)\n       \\begin{equation}\n\\langle {\\cal J} u,v \\rangle - \\langle u,  {\\cal J} v \\rangle = \\lim_{n\\to\\infty}  a_{n } (u_{n +1}\\bar{v}_{n }- u_{n }\\bar{v}_{n +1})\n\\label{eq:qres4}\\end{equation}\nwhere the limit in the right-hand side exists.\n Indeed, a direct calculation shows that\n      \\[\n\\sum_{m=0}^n ( {\\cal J} u)_{m}\\bar{v}_{m} -\\sum_{m=0}^n u_{m} ( {\\cal J} \\bar{v})_{m} =     a_{n } (u_{n +1}\\bar{v}_{n }- u_{n }\\bar{v}_{n +1}).\n\\]\nPassing here to the limit $n\\to\\infty$ and using that ${\\cal J} u  \\in \\ell^2 ({\\Bbb Z}_{+})$ for $u \\in {\\cal D} (J_{\\rm max})$, we obtain \\eqref{eq:qres4}.\n \n\n\n  \n  \n\n\n \n The comprehensive presentation of the theory of self-adjoint Jacobi operators can be found in  the books \\cite{AKH} (Chapters~I and II), \\cite{Ber} (Chapter~VII), \\cite{Schm} (Part~I)  and the survey \\cite{Simon}.\n\n  \\subsection{Determinacy versus indeterminacy}\n  \n    Since the operator $J_{\\rm min}  $  commutes with the complex conjugation,  its deficiency indices \n   \\[\n   d_{\\pm}: =\\dim\\ker (J_{\\rm max}-z I),\\quad \\pm \\Im z>0,\n   \\]\n     are equal, i.e. $d_{+} =d_{-}=:d $, and, so, $J_{\\rm min}  $ admits self-adjoint extensions.  Here and below $I$ is the identity operator. For an arbitrary $z\\in{\\Bbb C}$,\n       let us consider the equation ${\\cal J}u=zu$. By definition \\eqref{eq:ZP+1}, its solutions  are given by the formula $u=u_{0}p(z)$. They belong to $ {\\cal D} (J_{\\rm max})$ if and only if $p (z) \\in  \\ell^2 ({\\Bbb Z}_{+})$.  Therefore $d=0$ if $p (z)\\not\\in  \\ell^2 ({\\Bbb Z}_{+})$  for $\\Im z\\neq 0$; otherwise $d=1$.  \n     \n      This dichotomy can also be stated in terms of the Weyl limit point\/circle theory and in terms of the determinacy\/indeterminacy  of the associated moment problem.  Recall that,\n   similarly to differential equations,   difference equation \\eqref{eq:Jy} always has a non-trivial solution in $\\ell^2 ({\\Bbb Z}_{+})$ for $\\Im z \\neq 0$. This solution is either unique (up to a constant factor) or all solutions of \\eqref{eq:Jy} belong to $\\ell^2 ({\\Bbb Z}_{+})$.\nThe first instance is known as the limit point (LP) case and the second one --  as the limit circle (LC) case.   In the LP case $p (z)\\not\\in  \\ell^2 ({\\Bbb Z}_{+})$   and in the LC case $p (z) \\in  \\ell^2 ({\\Bbb Z}_{+})$  for $\\Im z\\neq 0$. Thus, the operator $J_{\\rm min}$ is essentially self-adjoint if and only if the LP case occurs.\n \n\n\n \n\n \n  \n     \n \n\n   \n\n  \n  \n  \n  \n       \n \n    \n \n\n\n \n  \nAs is well known (see, for example, Theorem~6.16 in \\cite{Schm}),  in the LC case\ninclusions\n  \\begin{equation}\n  p(z)\\in  \\ell^2 ({\\Bbb Z}_{+}) \\quad \\mbox{and} \\quad    q (z)\\in  \\ell^2 ({\\Bbb Z}_{+})  \n  \\label{eq:PQz}\\end{equation}\nhold true for all $z\\in {\\Bbb C}$.  It now follows from \\eqref{eq:PQW} that  in the LC case  necessarily  \n    \\begin{equation}\n\\sum_{n=0}^\\infty a_{n}^{-1}<\\infty.\n\\label{eq:noCarl}\\end{equation}\nEquivalently, this fact can be stated   \n  as a sufficient condition (known as the Carleman condition) \n     \\begin{equation}\n\\sum_{n=0}^\\infty a_{n}^{-1}=\\infty\n\\label{eq:Carl}\\end{equation}\nfor the essential self-adjointness of the operator $J_{\\rm min}$. \n\nIn general,  the essential self-adjointness of  operators $J_{\\rm min}$ is determined by both sets of coefficients $(a_{n})$ and $(b_{n})$. This is briefly discussed in Sect.~3. Here we only note that, even in the case $(b_{n})=0$,\ncondition \\eqref{eq:Carl} is not necessary for the essential self-adjointness of   $J_{\\rm min}$. However Theorem~1.5 in Chaper~VII of the book \\cite{Ber} shows that this is   true under an additional assumption \n  $a_{n-1}a_{n+1}\\leq a_{n}^2$\n \n(for large $n$). Without   this assumption, the operators  $J_{\\rm min}$ may be  essentially self-adjoint even for the coefficients $(a_{n})$ satisfying  conditions \\eqref{eq:noCarl}  and $(b_{n})=0$.\n (see \\cite{Kost}).\n \n We also note a link with the  Hamburger moment problem\n      \\begin{equation}\ns_{n} = \\int_{-\\infty}^\\infty \\lambda^n d\\rho(\\lambda).\n\\label{eq:moment}\\end{equation}\nRecall that these equations for a nonnegative  measure $d\\rho(\\lambda)$ with infinite support have a solution if and only if the sequence $(s_{n})$ is positive definite, i.e., \n    \\begin{equation}\n\\sum_{n,m\\in{\\Bbb Z}_{+}}s_{n+m}\\xi_{m}\\bar{\\xi_{n}}>0\\quad \\forall \\xi=(\\xi_{n})\\in {\\cal D}, \\;\\xi\\neq 0.\n\\label{eq:moment1}\\end{equation}\nThis problem is called determinate if the measure satisfying \n\\eqref{eq:moment} is unique. Otherwise it  is called indeterminate.\n\nFor a Jacobi matrix \\eqref{eq:ZP+}, we  set\n$    s_{n}=({\\cal J}^n e_{0}, e_{0})$.\n Then condition \\eqref{eq:moment1} is satisfied. If  $J$ is an arbitrary self-adjoint extension of the operator $J_{\\rm min}$  and $E_{J}(\\lambda)$ is its spectral family,  then the     measure $d\\rho_{J}(\\lambda)= d(E_{J}(\\lambda)e_{0}, e_{0})$ satisfies equation \\eqref{eq:moment}.\n  It turns out (see Theorem~2 in \\cite{Simon})  that  moment problem \\eqref{eq:moment} is   determinate  if and only if the operator $J_{\\rm min}$ is     essentially  self-adjoint.\n\nWe also note that the  essential  self-adjointness of an operator $J_{\\rm min}$ is determined by a behavior of its coefficients $a_{n}$ and $b_{n}$ for large $n$ only. Indeed, if  $( \\tilde{a}_{n}-a_{n})\\in \\ell^\\infty ({\\Bbb Z}_{+}) $  and $( \\tilde{b}_{n}- b_{n})\\in \\ell^\\infty ({\\Bbb Z}_{+}) $, then the operator $\\widetilde{J}_{\\rm min}-J_{\\rm min}$ is bounded so that the deficiency indices of the operators    $\\widetilde{J}_{\\rm min}$ and $J_{\\rm min}$ are the same.\n\n\n  \n    \\subsection{Self-adjoint extensions}\n    \n  Recall that\n\\[\n\\clos J_{\\min}= J_{\\min}^{**}  = J_{\\max}^{*}. \n\\]\n  In this paper we are interested\n      in the LC case where the minimal Jacobi operators  $J_{\\rm min}$  are not essentially self-adjoint so that      \\begin{equation}\n\\clos J_{\\rm min}\\neq J_{\\rm max}=J_{\\rm min}^*.\n  \\label{eq:cl}\\end{equation}\n \n   \n \n  \n Let us set\n   \\begin{equation}\n{\\sf D }:={\\cal D}(\\clos J_{\\rm min}).\n  \\label{eq:MIN}\\end{equation}\n  For a vector $h\\in\\ell^2 ({\\Bbb Z}_{+})$, we denote by $\\{h\\}$ the one dimensional subspace of $\\ell^2 ({\\Bbb Z}_{+})$ spanned by the vector $h$. The symbol $\\dotplus$ denotes the direct sum of subspaces. The following description  (see Lemma~6.22 and Theorem~6.23 in \\cite{Schm}  or Theorem~2.6 in  \\cite{Simon}) of self-adjoint extensions of the operators $J_{\\rm min}$ is a modification of von Neumann formulas adapted to Jacobi operators. \n  \n   \\begin{proposition}\\label{Neum}\n    Let relation \\eqref{eq:cl} be satisfied. Choose some $\\zeta\\in {\\Bbb R}$.\nThen \n\\begin{equation}\n{\\cal D}(J_{\\rm max})= {\\sf D }\\dotplus \\{p(\\zeta)\\}\\dotplus \\{q(\\zeta)\\}.\n  \\label{eq:Neum}\\end{equation}\n  All self-adjoint extensions $J_{t}$ of the operator $J_{\\rm min}$ are parametrized by numbers $t\\in{\\Bbb R}$ and\n  $t=\\infty$. The domains of the operators $J_{t}$ are determined by the equalities\n\\begin{equation}\n{\\cal D}(J_t)= {\\sf D }\\dotplus \\{t p(\\zeta) +q(\\zeta)\\}, \\quad t\\in{\\Bbb R},\n    \\label{eq:Neum1}\\end{equation}\n  and\n\\begin{equation}\n{\\cal D}(J_\\infty)= {\\sf D }\\dotplus \\{ p(\\zeta)  \\}.\n  \\label{eq:Neum2}\\end{equation}\n \\end{proposition} \n \n Below we use Proposition~\\ref{Neum}  for $\\zeta=0$ only.\n   \n Our main goal is to find an explicit formula for the resolvents $R_{t} (z)= (J_{t}-zI)^{-1}$  of   self-adjoint operators $J_{t}$.  To that end, we define, for all $z\\in{\\Bbb C}$, a bounded operator ${\\cal R} (z)$ in the  space $\\ell^2 ({\\Bbb Z}_{+})$  by an equality\n        \\begin{equation}\n ( {\\cal R} (z)h)_{n} =  q_{n}(z) \\sum_{m=0}^n  p_{m} (z) h_{m}+   p_{n}(z)   \\sum_{m=n+1}^\\infty  q_{m} (z) h_{m}.\\label{eq:RR11}\\end{equation} \n Actually, the operator ${\\cal R} (z)$ belongs to the Hilbert-Schmidt class.\nWe prove (see  Theorem~\\ref{res}) that, in  a natural sense,  ${\\cal R} (z)$ can be considered as a quasiresolvent of the operator $J_{\\rm max}$. A representation for the resolvents $R_{t} (z)$ is obtained in Theorem~\\ref{RES}.\n \n Another goal of the paper is to find an alternative description of self-adjoint extensions  of the operator $J_{\\rm min}$ in terms of an asymptotic behavior of solutions $u_{n}$ of the Jacobi equation  \\eqref{eq:Jy} for $n\\to\\infty$.  This is discussed in Section~3.\n \n     \n \n\\section{Resolvents of self-adjoint Jacobi operators}\n\n\n In this section we only suppose that a minimal Jacobi operator $J_{\\rm min}$ is not essentially self-adjoint so that we are in the LC case.\n \n\n \n      \\subsection{Quasiresolvent}\n \n \n\nRecall that in the LC  case\n inclusions \\eqref{eq:PQz}  are satisfied for all $z\\in {\\Bbb C}$.    Let us define an operator ${\\cal R}  (z)$ playing the role of the resolvent of the operator $J_{\\rm max}$ by   equality \\eqref{eq:RR11}.  In particular, it follows from \\eqref{eq:Pz}, \\eqref{eq:Qz} that  \n  \\begin{equation}\n  ({\\cal R} (z)h)_{0}=\\langle h, q(\\bar{z})\\rangle \n  \\label{eq:r01}\\end{equation} \nand\n \\begin{equation}\n   ({\\cal R} (z)h)_{1}= h_{0}  a_{0}^{-1}+\\langle h, q(\\bar{z})\\rangle  (z-b_{0})a_{0}^{-1} \n\\label{eq:r02}\\end{equation} \nfor all  $h\\in {\\ell}^2  ({\\Bbb Z}_{+})$. \n In view of inclusions \\eqref{eq:PQz} the operators ${\\cal R}  (z)$ are bounded in the space $\\ell^2 ({\\Bbb Z}_{+})$ for all $z\\in {\\Bbb C}$.  Note also that ${\\cal R} (z)^*={\\cal R} (\\bar{z})$.\n\n\nA proof of the following statement is  close to the construction (see, e.g., Lemma~5.1 in \\cite{JLR}) of the resolvents for  essentially self-adjoint Jacobi operators.    \n   \n    \\begin{theorem}\\label{res}\n    Let relation \\eqref{eq:cl} be satisfied.\n For all $z \\in {\\Bbb C}$, we have\n  \\begin{equation}\n{\\cal R}  (z): \\ell^2 ({\\Bbb Z}_{+})\\to  {\\cal D} (J_{\\rm max} )\n\\label{eq:qres}\\end{equation}\nand\n  \\begin{equation}\n(J_{\\rm max} -zI) {\\cal R}  (z)=I.\n\\label{eq:qres1}\\end{equation}\n   \\end{theorem} \n \n  \\begin{pf} \n  Recall that an operator ${\\cal J}$  was defined by equalities \\eqref{eq:ZP+1}. We will  check that \n    \\begin{equation}\n  (  ({\\cal J}-z I) {\\cal R} (z)h)_{n}=h_{n} \n    \\label{eq:RRm}\\end{equation}\nfor all $n \\in {\\Bbb Z}_{+}$ and $h=\\{h_{n}\\}\\in \\ell^2 ({\\Bbb Z}_{+})$.  For $n=0$, we have\n\\[\n(  ({\\cal J}-z I) {\\cal R} (z)h)_0= (b_{0}-z)({\\cal R} (z)h)_0  + a_{0}({\\cal R} (z)h)_1=h_{0}\n\\]\naccording to formulas \\eqref{eq:r01} and \\eqref{eq:r02}. For $n\\geq 1$, we rewrite definition  \\eqref{eq:RR11}   as\n \\[\n ( {\\cal R} (z)h)_{n} =  q_{n} (z) x_{n}  (z) +   p_{n} (z)   y_{n} (z)\n \\]\nwhere\n   \\begin{equation}\nx_{n}(z) =\\sum_{m=0}^n  p_{m} (z) h_{m},\\quad    y_{n}(z) =\\sum_{m=n+1}^\\infty  q_{m} (z) h_{m}.\n\\label{eq:RR2}\\end{equation} \n  It now follows from definition \\eqref{eq:ZP+1}     that\n   \\begin{multline}\n ( ({\\cal J} -z  I) {\\cal R} (z) h)_{n} =  a_{n-1}\\big(q_{n-1}x_{n-1}+p_{n-1} y_{n-1} \\big)\n  \\\\\n  + (b_{n}-z) \\big(q_{n}x_{n}+ p_{n} y_{n}  \\big)+ a_{n}\\big(q_{n+1}x_{n+1} +p_{n+1} y_{n+1}    \\big), \\quad n\\geq 1.\n\\label{eq:RR3}\n\\end{multline}\n According to \\eqref{eq:RR2}  we have\n\\[\nq_{n-1}x_{n-1}+p_{n-1} y_{n-1}=q_{n-1}(x_{n}-p_{n} h_{n})+  p_{n-1}(y_{n}+q_{n} h_{n}) \n\\]\nand\n \\begin{multline*}\nq_{n+1} x_{n+1} +p_{n+1} y_{n+1} =\nq_{n+1}(x_{n}  +p_{n+1}h_{n+1}) \n\\\\\n +  p_{n+1} (y_{n} - q_{n+1}h_{n+1})=\nq_{n+1}x_{n}  +  p_{n+1}y_{n} .\n\\end{multline*}\n Substituting these expressions into the right-hand side of \\eqref{eq:RR3},  we see that\n \\begin{multline*}\n  \\big( ({\\cal J} -z I) {\\cal R} (z) h\\big)_{n} =  a_{n-1}\\Big(q_{n-1}(x_{n}- p_{n} h_{n})+  p_{n-1}(y_{n}+q_{n} h_{n}) \\Big)\n  \\\\\n  + (b_{n}-z) \\big(q_{n}x_{n}+ p_{n} y_{n}  \\big)+ a_{n}\\big(q_{n+1}x_{n}  +  p_{n+1}y_{n}    \\big).\n\\end{multline*}\nLet  us now collect together all terms containing $x_{n}$, $y_{n}$ and $h_{n}$.  Then\n \\begin{multline}\n  \\big( ({\\cal J} -z I) {\\cal R} (z) h\\big)_{n} = \\Big(a_{n-1}q_{n-1}+  (b_{n}-z) q_{n} + a_{n} q_{n+1}\\Big) x_{n}\n  \\\\\n    + \\Big(a_{n-1}p_{n-1}+  (b_{n}-z)p_{n} + a_{n}p_{n+1}\\Big) y_{n}+a_{n-1}\\big( -p_{n} q_{n-1}+  p_{n-1} q_{n}\\big) h_{n}  .\n\\label{eq:RR3A}\n\\end{multline}\nThe coefficients at $x_{n}$ and $y_{n}$ equal zero by virtue of   Jacobi equation \\eqref{eq:Jy}  for $( q_{n} )$ and $(p_{n})$, respectively. Since $\\{p,q \\} =1$, the right-hand side of \n\\eqref{eq:RR3A} equals $ h_{n}$.   \nThis proves \\eqref{eq:RRm}  whence $   ({\\cal J}-z I) {\\cal R} (z)h=h $. In particular, we see that ${\\cal R}  (z) h\\in {\\cal D} (J_{\\rm max})  $ so that ${\\cal J}$ can be replaced here by $J_{\\rm max}$.\nThis yields both \\eqref{eq:qres} and \\eqref{eq:qres1}. \n    \\end{pf}\n    \n    \\begin{remark}\\label{res3}\n    In definition  \\eqref{eq:RR11},\n    one can replace   $q_{n}(z)$ by the polynomials $\\widetilde{q}_{n}(z)= q_{n}(z)+ c p_{n}(z)$ for an arbitrary  $c\\in {\\Bbb C}$. Then $\\widetilde{\\cal R} (z)= {\\cal R} (z) +c \\langle \\cdot, p(\\bar{z})\\rangle p(z)$ and formulas \\eqref{eq:qres}, \\eqref{eq:qres1}  remain true.\n      \\end{remark} \n     \n           \n    \n    Since $p(z)$ is a unique (up to a constant factor)  solution of the homogeneous equation $(J_{\\rm max} -zI) u =0$, we can also state \n    \n     \\begin{corollary}\\label{res1}\n     All solutions of the equation\n       \\[\n(J_{\\rm max} -zI) u =h \\quad \\mbox{where}  \\quad z\\in {\\Bbb C} \\quad \\mbox{and}\\quad h\\in \\ell^2 ({\\Bbb Z}_{+})\n\\]\n  for $u\\in {\\cal D} (J_{\\rm max} )$ are  given by the formula\n \\begin{equation}\n  u = \\Gamma p(z)+ {\\cal R}  (z) h \\quad \\mbox{for some} \\quad \\Gamma=\\Gamma (z;h) \\in{\\Bbb C}.\n\\label{eq:qres3}\\end{equation}\n   \\end{corollary} \n   \n\n   \n\n \n \n\n   \n    \n       \nAn  asymptotic relation for $ ( {\\cal R} (z)h)_{n}$ is a direct consequence of definition  \\eqref{eq:RR11}\n   and condition  \\eqref{eq:PQz}:\n      \\begin{equation}\n ( {\\cal R} (z)h)_{n} =  q_{n}(z) \\langle    h, p (\\bar{z}) \\rangle+   o(|p_{n} (z)| +|q_{n}(z)| )\\quad {\\rm as}\\quad n\\to\\infty.\n \\label{eq:Ras}\\end{equation}\n This relation   can be supplemented by the following result.\n \n     \\begin{proposition}\\label{AS}\n    For all $z\\in {\\Bbb C}$ and  all $h\\in \\ell^2 ({\\Bbb Z}_{+})$, we have\n    \\begin{equation}\n u:= {\\cal R} (z)h - q (z) \\langle    h, p (\\bar{z}) \\rangle\\in {\\sf D}.\n \\label{eq:Ras1}\\end{equation}\n        \\end{proposition} \n        \n              \\begin{pf}\n              Let first  $h\\in {\\cal D}$. Then $ ( {\\cal R} (z)h)_{n} =  q_{n}(z) \\langle    h, p (\\bar{z}) \\rangle$ for sufficiently large $n$ so that $u\\in {\\cal D}\\subset {\\sf D}$. \n              \n              \nLet now $h$ be an arbitrary vector in  $  \\ell^2 ({\\Bbb Z}_{+})$.     Observe that $ u \\in {\\sf D}$ if and only if there exists a sequence $u^{(k)}\\in {\\cal D}$ such that \n                \\begin{equation}\nu^{(k)}\\to u \\quad \\mbox{and} \\quad {\\cal J}u^{(k)}\\to {\\cal J}u \\quad \\mbox{as} \\quad k\\to\\infty.\n \\label{eq:Ras2}\\end{equation}\n            Let us take any sequence $h^{(k)}\\in {\\cal D}$ such that   $h^{(k)}\\to h$ and set\n                 \\[\n u^{(k)}= {\\cal R} (z)h^{(k)} - q (z) \\langle    h^{(k)}, p (\\bar{z}) \\rangle .\n \\]\n Then $u^{(k)} \\in {\\cal D}$ and  $u^{(k)}\\to u$ as $k\\to\\infty$ because the operator $ {\\cal R} (z)$ is bounded.  It follows from equalities  \\eqref{eq:PQzz}  and \\eqref{eq:qres1}  that\n \\[\n({\\cal J}  -z) u^{(k)}=  h^{(k)} - e_{0} \\langle    h^{(k)}, p (\\bar{z}) \\rangle \\to h - e_{0} \\langle    h , p (\\bar{z} ) \\rangle =({\\cal J}  -z) u \n\\]\nas $k\\to\\infty$. This proves relations \\eqref{eq:Ras2} whence $u\\in {\\sf D}$. \n                              \\end{pf}\n                              \n                              In view of Proposition~\\ref{AS}  representation \\eqref{eq:qres3}  is consistent with formula  \\eqref{eq:Neum}.\n      \n          \\subsection{Main result}\n          \n          Let us first find a link between the polynomials $p(z)$, $q(z)$ for an arbitrary  $z \\in {\\Bbb C}$ and  the polynomials $p(0)$, $q(0)$.  We always suppose that\n               condition \\eqref{eq:cl} is  satisfied which implies inclusions \\eqref{eq:PQz} for all $z \\in {\\Bbb C}$.\n          \n          \\begin{lemma}\\label{PQ}\n For all $z \\in {\\Bbb C}$, we have\n  \\begin{equation}\np(z) =\\big( 1-z \\langle p(z), q(0)\\rangle \\big) p(0) + z {\\cal R}(0) p(z)\n\\label{eq:PP}\\end{equation}\nand\n  \\begin{equation}\nq(z) =    - z \\langle q(z), q(0)\\rangle  p(0) + q(0)+ z {\\cal R}(0) q(z).\n\\label{eq:QQ}\\end{equation}\n \\end{lemma}\n \n   \\begin{pf}\n   To prove \\eqref{eq:PP}, we set\n    \\begin{equation}\n   u=p(z) -z {\\cal R}(0) p(z)\n\\label{eq:PP1}\\end{equation}\nso that \n    \\begin{equation}\nu_{0}= 1-z \\langle p(z), q(0)\\rangle\n\\label{eq:PP2}\\end{equation} \n  according to  equality   \\eqref{eq:r01} (for $z=0$).\n   It follows from relation \\eqref{eq:qres1}  that ${\\cal J} u={\\cal J} p(z) -z p(z) =0$ whence $u=u_{0} p(0)$. \n   Now definition \\eqref{eq:PP1} implies that $ p(z)= u_{0} p(0)+ z {\\cal R}(0) p(z)$. It remains to use \\eqref{eq:PP2}.\n    \n   \n   The proof of \\eqref{eq:QQ}  is quite similar.  We now set\n    \\begin{equation}\n   v=q(z) - q(0) -z {\\cal R}(0) q(z)\n\\label{eq:QQ1}\\end{equation}\nso that  $v_{0}= -z \\langle q(z), q(0)\\rangle$.     It again follows from relations  \\eqref{eq:PQzz} and  \\eqref{eq:qres1} \nthat  ${\\cal J} v= {\\cal J} q(z)- {\\cal J} q(0)- z q(z)=0$  whence $v=v_{0} p(0)= - z \\langle q(z), q(0)\\rangle  p(0)$. Therefore \n\\eqref{eq:QQ} is a direct consequence of definition \\eqref{eq:QQ1}. \n\\end{pf}\n \n Putting together Lemma~\\ref{PQ}  with Proposition~\\ref{AS} (for $z=0$), we can also state the following result.\n \n   \\begin{lemma}\\label{PQ+} \n For all $z \\in {\\Bbb C}$, we have\n  \\begin{equation}\np(z) -\\big( 1-z \\langle p(z), q(0)\\rangle \\big) p(0) - z \\langle p(z), p(0)\\rangle  q(0) \\in {\\sf D}\n\\label{eq:PP+}\\end{equation}\nand\n  \\begin{equation}\nq(z) + z \\langle q(z), q(0)\\rangle  p(0) - \\big( 1+ z \\langle q(z), p(0)\\rangle \\big)   q(0)     \\in {\\sf D}.\n\\label{eq:QQ+}\\end{equation}\n \\end{lemma}\n\n          \n          Now it easy to construct     resolvents  of   self-adjoint extensions $J_{t}$ of the operators $J_{\\rm min}$.  Recall that domains of the operators $J_{t}$ are defined by relations \\eqref{eq:Neum1}  and \\eqref{eq:Neum2}.\n     \n     \\begin{theorem}\\label{RES}\n          Let assumption \\eqref{eq:cl} hold.\n     For all $z\\in {\\Bbb C}$ with $\\Im z\\neq 0$ and all $h\\in\\ell^2 ({\\Bbb Z}_{+})$, the resolvent $R_t (z)= (J_t-zI)^{-1}$ of the operator $J_t$ is given by an  equality \n           \\begin{equation}\nR_{t} (z) h = \\gamma_{t} (z) \\langle h,  p(\\bar{z})\\rangle p(z)+ {\\cal R}(z)h\n      \\label{eq:RES1}\\end{equation}\n      where  \n           \\begin{equation}\n\\gamma_{t}(z)= \\frac{z \\langle q(z), q(0)\\rangle+\\big( 1+ z \\langle q(z), p(0)\\rangle \\big)t } \n{1- z \\langle p(z), q(0)\\rangle -z \\langle p(z), p(0)\\rangle t }, \\quad t\\in {\\Bbb R},\n    \\label{eq:RES2}\\end{equation}\n    and\n            \\begin{equation}\n\\gamma_{\\infty}(z)= - \\frac{ 1+ z \\langle q(z), p(0)\\rangle } \n{z \\langle p(z), p(0)\\rangle   }.\n    \\label{eq:RES3}\\end{equation}\n    \\end{theorem} \n    \n    \\begin{pf}\n    According to Theorem~\\ref{res}  and Corollary ~\\ref{res1}  a vector $u=R_{t}h$ is given by equality \\eqref{eq:qres3}  where \n  $ \\Gamma= \\Gamma_{t}(z;h)$ is a bounded linear functional of $h\\in \\ell^2 ({\\Bbb Z}_{+})$ so that\n  $\\Gamma_{t}(z;h)= \\langle h, \\tilde{p_{t}}(z)\\rangle$\n   for some vector $ \\tilde{p}_{t} (z)\\in \\ell^2 ({\\Bbb Z}_{+})$.\n   Since $R_{t} (z)^*=R_{t} (\\bar{z})$ and ${\\cal R} (z)^*={\\cal R}  (\\bar{z})$,  we see that \n   \\[\n   \\langle h, \\tilde{p_{t}}(\\bar{z})\\rangle p(\\bar{z}) =\\langle h,  p (z)\\rangle \\tilde{p_{t}}(z)\n   \\]\n   for all $h\\in\\ell^2 ({\\Bbb Z}_{+})$. It follows that $\\tilde{p_{t}}(z)= \\overline{\\gamma_{t} (z)}p(\\bar{z})$ for some $\\gamma_{t} (z) \\in{\\Bbb C}$. This yields representation \\eqref{eq:RES1} where the constant \n$\\gamma_{t} (z) $ is determined by the condition $R_{t} (z) h\\in {\\cal D} (J_{t})$.\nIn view of relations  \\eqref{eq:RES1}  and  \\eqref{eq:Ras1}, this is equivalent to the condition\n      \\begin{equation}\nR_{t} (z) h -\\langle h,  p(\\bar{z})\\rangle \\big( \\gamma_{t} (z)  p(z) + q(z)\\big)\\in{\\sf D}.\n      \\label{eq:RES4}\\end{equation}\n     \n   Proposition~\\ref{Neum} where $\\zeta=0$ means that\n       \\begin{equation}\nR_{t} (z) h - X  \\big( t  p(0) + q(0)\\big)\\in{\\sf D} \\quad \\mbox{if} \\quad t\\in {\\Bbb R}\n \\quad \\mbox{and} \\quad R_\\infty (z) h - X  p(0)  \\in{\\sf D}\n      \\label{eq:REA}\\end{equation}\n      for some number $X=X_{t}(z)\\in {\\Bbb C}$.  Comparing  \\eqref{eq:RES4} and \\eqref{eq:REA}, we see that\n      \\begin{equation}\n\\gamma_{t} (z)  p(z) + q(z) - Y  \\big( t  p(0) + q(0)\\big)\\in{\\sf D} \\quad \\mbox{if} \\quad t\\in {\\Bbb D}\n      \\label{eq:REA1}\\end{equation}\n and\n       \\begin{equation}\n \\gamma_\\infty (z)  p(z) + q(z) - Y  p(0)  \\in{\\sf D}\n      \\label{eq:REA2}\\end{equation}\n         for some number $Y\\in {\\Bbb C}$. \n      \n     Using     Lemma~\\ref{PQ+}, we see that \n       inclusion \\eqref{eq:REA1} is  equivalent to an equality\n\\begin{multline*}\n  \\gamma_{t} (z) (\\big( 1-z \\langle p(z), q(0)\\rangle \\big) p(0)  +z \\langle p(z), p(0)\\rangle q(0)\\Big)\n\\\\\n + \\Big(- z \\langle q(z), q(0)\\rangle p(0) + \\big( 1+ z \\langle q(z), p(0)\\rangle \\big)   q(0)   \\Big) = Y (t p(0) + q(0)) .\n \\end{multline*}\n  Comparing here the coefficients at $p(0)$ and $q(0)$, we obtain an equation\n      \\[\n      \\frac{  \\gamma_{t} (z)  ( 1-z \\langle p(z), q(0)\\rangle )- z \\langle q(z), q(0)\\rangle }\n      {\\gamma_{t} (z) z \\langle p(z), p(0)\\rangle + 1+ z \\langle q(z), p(0)\\rangle } =t.\n      \\]\n      Solving this equation with respect to $\\gamma_{t} (z) $, we arrive at formula  \\eqref{eq:RES2}.\n         Similarly, substituting expressions \\eqref{eq:PP+} and \\eqref{eq:QQ+} into the left-hand side of \\eqref{eq:REA2}, we see that inclusion \\eqref{eq:REA2} holds true  if and only if the coefficient at $q(0)$   equals zero. This yields formula  \\eqref{eq:RES3}.\n        \\end {pf}\n        \n        \n          We emphasize that, for different $t$, the resolvents  $R_{t}(z)$ of the operators $J_{t}$ differ from each other only by the coefficient $\\gamma_{t}(z)$ at the rank one operator $\\langle\\cdot, p(\\bar{z})\\rangle p(z)$.\n   Observe also that\n    $  \\overline{\\gamma_{t}(z)}= \\gamma_t (\\bar{z})$.\n      \n     \n     Since  $\\langle {\\cal R}(z) e_{0}, e_{0}\\rangle=0$, we see that $\\langle R_{t}(z) e_{0}, e_{0}\\rangle=\\gamma_{t} (z)$. Thus Theorem~\\ref{RES}  implies the classical Nevanlinna representation obtained in \\cite{Nevan}  for the Cauchy-Stieltjes transform of the spectral measures $d\\rho_{t} (\\lambda)= d (E_{t} (\\lambda) e_{0}, e_{0})$ of the operators $J_{t}$.\n     \n     \\begin{corollary}\\label{RESc}\n     For all $z\\in {\\Bbb C}$ with $\\Im z\\neq 0$,  we have\n     \\[\n     \\int_{-\\infty}^\\infty\n (\\lambda-z)^{-1} d\\rho_{t}(\\lambda)= \\gamma_{t} (z),\n \\]\n where $\\gamma_{t} (z)$ is given by equalities \\eqref{eq:RES2}  or  \\eqref{eq:RES3}.\n    \\end{corollary} \n\nSince the functions $ \\langle p(z), p(0)\\rangle  $ and $ \\langle p(z), q(0)\\rangle  $  are entire, it follows from \\eqref{eq:RES2}  and  \\eqref{eq:RES3} that the spectra of the operators $J_{t}$  and $J_{\\infty}$ are discrete. The eigenvalues of the operators $J_{t}$ are given by the equation \n   $z \\langle p(z), p(0)\\rangle t + z \\langle p(z), q(0)\\rangle =1$ if $t\\in{\\Bbb R}$ and  by the equation \n   $z \\langle p(z), p(0)\\rangle =0$ if $t=\\infty$. This result is also due to R.~Nevanlinna.\n   \n     \\begin{corollary}\\label{RESb}\n     If $z\\in {\\Bbb C}$  is  not an eigenvalue of the operator $J_{t}$, then its resolvent $R_{t}  (z)$ is in the Hilbert-Schmidt class.\n    \\end{corollary} \n    \n     \\begin{remark}\\label{ze}\n     Theorem~\\ref{RES} remains true if one  uses an arbitrary real point $\\zeta$ in parametrization of self-adjoint extensions of the operator $J_{\\rm min}$ in Proposition~\\ref{Neum}. In this case $p(0)$ and $q(0)$ in formulas \n     \\eqref{eq:RES2}  and  \\eqref{eq:RES3}  should be replaced by $p(\\zeta)$ and $q(\\zeta)$; the factor $z$ should be replaced by $z-\\zeta$. \n       \\end{remark}  \n       \n       Finally, let us compare resolvent formulas in the LP and LC cases. In the LP case the resolvent $R(z)$ of the Jacobi operator $J=\\clos J_{\\rm min}$   is given (see, e.g., Lemma~5.1 in \\cite{JLR}) by the relation\n        \\begin{equation}\n ( R (z)h)_{n} = \\frac{1}{\\{p(z), g(z)\\}} \\Big(g_{n}(z) \\sum_{m=0}^n  p_{m} (z) h_{m}+   p_{n}(z)   \\sum_{m=n+1}^\\infty  g_{m} (z) h_{m}\\Big)\n \\label{eq:R-LP}\\end{equation}\n where $g(z)$ is a unique (up to a constant factor) solution of equation \\eqref{eq:Jy}  belonging to $\\ell^2 ({\\Bbb Z}_{+})$. It can be chosen in a form $g(z)= q(z)+ w(z) p(z)$ where $w(z)$ is known as the Weyl function. Substituting this expression into \\eqref{eq:R-LP}, we see that {\\it formally}\n   \\begin{equation}\n  R (z)  = w(z)\\langle\\cdot,p(\\bar{z}\\rangle p(z) + {\\cal R} (z)\n \\label{eq:R-LP1}\\end{equation}\n where $ {\\cal R} (z)$ is given by equality  \\eqref{eq:RR11}. This relation looks algebraically similar to \\eqref{eq:RES1}  where $\\gamma_{t}(z)$ plays the role of the Weyl function $w(z)$. Note, however, that in the LP case \n $w(z)$ is determined uniquely by the condition $q(z)+ w(z) p(z)\\in \\ell^2 ({\\Bbb Z}_{+})$ while in the LC case \n $\\gamma_{t}(z)$ depends on the choice of a self-adjoint extension of the operator $  J_{\\rm min}$.  We also emphasize that  relation \\eqref{eq:R-LP1} is only formal  because $p(z)$ and $q(z)$ are not in $\\ell^2 ({\\Bbb Z}_{+})$.\n       \n           \n\n \n       \n  \\section{Asymptotic behavior of solutions of the Jacobi equation}\n  \n  In this section we consider a specific class of Jacobi operators $J_{\\rm min}$ that are not essentially   essentially self-adjoint.  We   describe  all their self-adjoint extensions $J$  in terms of asymptotic behavior of solutions $u_{n}$ of the Jacobi  equation  \\eqref{eq:Jy} for $n\\to\\infty$.  Then we find a representation for the resolvents of the operators $J$ in terms of asymptotic coefficients in formulas for $u_{n}$. Our construction here is independent of Proposition~\\ref{Neum}  (or the   von Neumann formulas) and of the results of Sect.~2.2.\n  \n  \\subsection{Setting  the problem}\n  \n    \n             To motivate the construction below, let us compare descriptions of self-adjoint extensions of minimal symmetric operators for differential and Jacobi operators. Take, for example, an operator $H_{\\rm min}=-d^2\/ d x^2$ with domain ${\\cal D} (H_{\\rm min})= C_{0}^\\infty ({\\Bbb R}_{+})$ in the space $L^2 ({\\Bbb R}_{+})$.  Of course, its adjoint operator $H_{\\rm min}^*= H_{\\rm max}$ as well as self-adjoint extensions can be described by the von Neumann formulas. It is, however, essentially more convenient  to define these operators in terms of boundary conditions at the point $x=0$.  Indeed, the set ${\\cal D} (H_{\\rm max})$ coincides with the Sobolev class ${\\sf H}^2  ({\\Bbb R}_{+})$. Functions $f\\in{\\sf H}^2  ({\\Bbb R}_{+})$  have boundary values $f(0)$ and $f' (0)$.\n        The domain  ${\\cal D} (\\clos H_{\\rm min})$ consists of $f\\in{\\sf H}^2  ({\\Bbb R}_{+})$ such that $f(0)=f' (0)=0$, and all self-adjoint extensions of $H_{\\rm min}$ are determined by  boundary conditions\n          \\begin{equation}\nf' (0)=h f(0) \\;  \\mbox{where} \\; h\\in  {\\Bbb R} \\quad \\mbox{or} \\quad f(0)=0.\n   \\label{eq:ABX3b}\\end{equation}\n   \n   We are looking for a similar description for self-adjoint extensions of the Jacobi operator $J_{\\rm min}$. \n   The starting point of our construction is an asymptotic formula\n     \\begin{equation}\nu_{n} = a_{n}^{-1\/2}  \\big(s_{+}   e^{i\\varphi_{n}} + s_{-}  e^{-i \\varphi_{n}}+ o(1)\\big)      , \\quad n\\to\\infty,\n\\label{eq:gen}\\end{equation} \nfor vectors $u\\in\\{ u_{n}\\}\\in {\\cal D} (J_{\\rm max})$.  \n Here the   phases $\\varphi_{n} $ are given by an explicit formula (see \\eqref{eq:Grf}, below).    The asymptotic coefficients $s_+= s_+(u)$ and $s_-= s_- (u)$ in \\eqref{eq:gen} depend on $u$.  They play the role of the boundary values $f(0)$ and $f' (0)$ for functions $f\\in{\\sf H}^2  ({\\Bbb R}_{+})$, and we are looking for an analogue of the mapping\n     \\begin{equation}\nf\\mapsto(f(0), f' (0))\n\\label{eq:f}\\end{equation}\nof the set  ${\\cal D} (H_{\\rm max})$ onto ${\\Bbb C}^2$.\n   \n     \n \n \n  \n   \n\nOur  first goal   is to describe all self-adjoint extensions   of  operators  $J_{\\rm min}$ in terms of asymptotic coefficients $s_\\pm$ in formula \\eqref{eq:gen}. It turns out that such extensions   ${\\bf J}_{\\omega}$  can be  parametrized by complex numbers $\\omega\\in{\\Bbb T}\\subset {\\Bbb C}$, and the domain  $ {\\cal D} (  {\\bf J}_\\omega)$ of ${\\bf J}_{\\omega}$ consists of elements $u\\in {\\cal D} (    J_{\\max})$  satisfying  the condition    \n     \\begin{equation}\n   s_{+} (u) =\\omega s_{-} (u) , \\quad |\\omega|=1 .\n      \\label{eq:gen2}\\end{equation}\n      This formula plays the role of \\eqref{eq:ABX3b}.\n      \n      \n      \n   \n          \n         \n         \n    \n   \n   Another goal of this section is to find an explicit formula for the resolvents ${\\bf R}_{\\omega} (z)=({\\bf J}_{\\omega}-zI)^{-1}$ of the operators ${\\bf J}_{\\omega}$. Of course this formula has the same structure as \\eqref{eq:RES1},\n      \\begin{equation}\n{\\bf R}_{\\omega} (z) h = \\boldsymbol{\\gamma}_{\\omega} (z) \\langle h,  p(\\bar{z})\\rangle p(z)+ {\\cal R}(z)h,\n      \\label{eq:ABY1}\\end{equation}\n   but the coefficient\n        \\begin{equation}\n\\boldsymbol{\\gamma}_{\\omega}  (z)= -\\frac{s_{+}(q(z))-\\omega s_{-}(q(z))} {s_{+}(p(z))-\\omega s_{-}(p(z))}\n    \\label{eq:ABY2}\\end{equation}\n    is  expressed in terms of the asymptotic coefficients in \\eqref{eq:gen}. \n    \n    \\subsection{Jost solutions}\n    \n    We here suppose that the coefficients $a_{n}\\to\\infty$ as $n\\to\\infty$  sufficiently rapidly so that condition \\eqref{eq:noCarl} is satisfied (otherwise the operator $J_{\\rm min}$ is essentially self-adjoint).  We also assume that  diagonal elements $b_{n}$ are small compared to $a_{n}$. Note that the operator $J_{\\rm min}$ is essentially self-adjoint if $b_{n}$ are large compared to $a_{n}$. The critical case where $b_{n}$ and $a_{n}$ are of the same order was considered in \\cite{crit-nC}. It is out of the scope of the present paper.\n    \n    \n      From analytic point of view, we rely on  the results of paper \\cite{nCarl}  where  so called Jost solutions $f_{n}^{(\\pm)}(z)$ were distinguished by their asymptotics as $n\\to\\infty$.  Their construction requires some assumptions on the recurrence coefficients.  In addition to \\eqref{eq:noCarl}, we assume that\n      \\begin{equation}\n   -\\frac{b_{n}}{2\\sqrt{a_{n-1}a_{n}} }=:\\beta_{n}\\to  \\beta_{\\infty}  \\; \\mbox{where}\\;  |\\beta_{\\infty}| < 1 \\;\\mbox{as}\\;  n\\to\\infty.\n\\label{eq:Haupt}\\end{equation}\nWe also require   certain regularity of the behavior of the sequences $a_{n}$ and $\\beta_{n}$.    Let us set\n   \\[\n\\alpha_{n}=\\sqrt{\\frac{a_{n+1}}{a_{n }}}, \\quad k_{n} = \\frac{  \\alpha_{n-1}} {  \\alpha_{n}} =\\frac{a_{n}  } {\\sqrt{a_{n-1} a_{n+1}} } .\n\\]\nWith respect to $a_{n}$, we assume that\n\\begin{equation}\n \\{k_{n} -1 \\}\\in \\ell^1 ({\\Bbb Z}_{+} ).\n\\label{eq:Gr6b}\\end{equation} \nWith respect to  the ratios $\\beta_{n}$, we assume that\n     \\begin{equation}\n  \\{\\beta_{n}' \\}\\in \\ell^1 ({\\Bbb Z}_{+} ).\n\\label{eq:Gr8}\\end{equation}\nIt is shown in \\cite{nCarl}, Lemma~2.1, that\nunder assumption \\eqref{eq:Gr6b} there exists a finite limit\n \\[\n\\lim_{n\\to\\infty}  \\alpha_{n} = : \\alpha_\\infty\\quad \\mbox{and} \\quad\\alpha_\\infty\\geq 1.\n\\]\n \n\n\nLet us  set  \n \\begin{equation}\n \\theta_{n}=\\arccos\\beta_n \\quad\\mbox{and}\\quad\n\\varphi_{n}= \\sum_{m=0}^{n-1} \\theta_{m}, \\quad n\\geq 1,\n\\label{eq:Grf}\\end{equation}\nwhere the sum is restricted to $m$ such that $|\\beta_{m}|\\leq 1$.  Note that\n\\[\n\\varphi_{n}= n\\arccos\\beta_\\infty + o(n)\\quad \\mbox{as}\\quad n\\to\\infty.\n\\]\n   \n   \n \n \n\n\n\n  \n    \\begin{theorem}\\label{GSS}\\cite[Theorem~4.1]{nCarl}\n      Let  assumptions  \\eqref{eq:noCarl},  \\eqref{eq:Haupt}    as well as  \\eqref{eq:Gr6b},  \\eqref{eq:Gr8} be satisfied.\n   Then, for all $z\\in{\\Bbb C} $,   equation \\eqref{eq:Jy} has  solutions $f^{(\\pm)}( z ) = (f_{n}^{(\\pm)}( z ))$ with asymptotics\n    \\begin{equation}\nf_{n}^{(\\pm)}(  z )  = \\frac{1}{\\sqrt{a_{n}} } e^{\\pm i \\varphi_{n}} \\big(1 + o(1)\\big) , \\quad n\\to \\infty.\n\\label{eq:A22G}\\end{equation} \nThe Wronskian \\eqref{eq:Wr} of these solutions equals\n   \\begin{equation}\n\\{ f^{(+)}(  z ), f^{(-)}(  z ) \\}=  -2i \\alpha_{\\infty}^{-1} \\sqrt {1-\\beta_{\\infty}^2}\\neq 0,\n\\label{eq:A2CG1}\\end{equation}\nso that they    are linearly independent.\nFor all $n\\in {\\Bbb Z}_{+}$, the functions $f_{n}^{(\\pm)} ( z )$ are entire functions of $z\\in  \\Bbb C$.\n \\end{theorem}\n \n We emphasize that the right-hand side of \\eqref{eq:A22G} depends on $z$ only through the remainder $o(1)$.  Note also that\n    \\begin{equation}\nf_{n}^{(-)}(  z )  = \\overline{f_{n}^{(+)}(  \\bar{z} )}.\n\\label{eq:f+-}\\end{equation}\n \n   \\begin{corollary}\\label{GSSc}\n   Since $f^{(\\pm)}(z)\\in \\ell^2 ({\\Bbb Z}_{+})$, under the assumptions of Theorem~\\ref{GSS} the operator $J_{\\rm min}$ is not essentially self-adjoint. It has deficiency indices $(1,1)$.\n \\end{corollary}\n \n  \\begin{remark}\\label{s-adj}\n  If the limit in \\eqref{eq:Haupt}  exists but $|\\beta_{\\infty}|>1$, then,  as shown in \\cite{nCarl} (see Theorem~4.11 and Corollary~4.12), the operator $J_{\\rm min}$ is   essentially self-adjoint. In the critical case $|\\beta_{\\infty}|=1$  its essential self-adjointness depends (see \\cite{crit-nC}) on details of a behavior of the coefficients $a_{n}$ and $b_{n}$ for $\\to\\infty$.\n \\end{remark}\n\n \n \n   \n       \\begin{example}\\label{GKy}\n        Both  conditions  \\eqref{eq:noCarl} and \\eqref{eq:Gr6b}\n are   satisfied for $a_{n}=   n^p$ where   $p>1$ and for $a_{n}=   x^{ n^q}$ where $x>1$, $q< 1$.  In these cases $\\alpha_\\infty=1$. For $a_{n}=  x^{ n}$,  conditions  \\eqref{eq:noCarl} and \\eqref{eq:Gr6b}\n are   also satisfied but    $\\alpha_\\infty=\\sqrt{x}$. On the  contrary, condition  \\eqref{eq:Gr6b} fails if $a_{n}= x^{ n^q}$  with $q>1$.\n      \\end{example}\n \n  \\subsection{Orthogonal polynomials}\n\nAn arbitrary solution   of the Jacobi equation  \\eqref{eq:Jy} is a linear combination of the Jost solutions $f_{n}^{(+)}(  z )$ and $f_{n}^{(-)}(  z )$. In particular, this is true for polynomials $ p_{n}(z) $ and $ q_{n}(z) $ so that   \n  \\begin{equation}\np_{n}(z)= \\sigma_{+} (z)f_{n}^{(+)}(  z ) + \\sigma_{-} (z) f_{n}^{(-)}(  z )\n\\label{eq:LC}\\end{equation}\nand\n  \\begin{equation}\nq_{n}(z)= \\tau_{+} (z)f_{n}^{(+)}(  z ) + \\tau_{-} (z) f_{n}^{(-)}(  z ),\n\\label{eq:LCq}\\end{equation}\nwhere the coefficients $ \\sigma_{\\pm} (z)$ and $ \\tau_{\\pm} (z)$\n can be expressed via the Wronskians:\n   \\[\n  \\sigma_{+}(z)= \\alpha_{\\infty} \\frac{ \\{p(z), f^{(-)}(  z )\\}} {2i \\sqrt {1-\\beta^2_{\\infty}}}, \\quad\n  \\sigma_{-}(z)= - \\alpha_{\\infty} \\frac{ \\{p(z), f^{(+)}(  z )\\}}{2i \\sqrt {1-\\beta^2_{\\infty}}} \n\\]\nand \n \\[\n  \\tau_{+}(z)= \\alpha_{\\infty} \\frac{ \\{q(z), f^{(-)}(  z ) \\} } {2i \\sqrt {1-\\beta^2_{\\infty}}}, \\quad\n  \\tau_{-}(z)= - \\alpha_{\\infty} \\frac{\\{ q(z), f^{(+)}(  z )\\} }{2i \\sqrt {1-\\beta^2_{\\infty}}} .\n\\]\n \n \nObserve that\n\\[\n\\sigma_{-} (z)= \\overline{\\sigma_{+}(\\bar{z})}\\quad \\mbox{and}\\quad \\tau_{-} (z)= \\overline{\\tau_{+}(\\bar{z})}\n\\]\n because $p_{n}(z)= \\overline{p_{n}(\\bar{z})}$, $q_{n}(z)= \\overline{q_{n}(\\bar{z})}$  and $ f_{n}^{(\\pm)}(  z )$ satisfy   \\eqref{eq:f+-}. \nOf course, all coefficients $ \\sigma_{\\pm}(z)$ and $ \\tau_{\\pm}(z)$  are  entire functions of $z$.  \n\n \nAccording to \\eqref{eq:LC}  and \\eqref{eq:LCq}   the following result is a direct consequence of Theorem~\\ref{GSS}. \n\n \\begin{theorem}\\label{LC}\n    Under the assumptions of Theorem~\\ref{GSS} the orthogonal polynomials $p_{n}(z)$ and $q_{n}(z)$\n have   asymptotics, as $n\\to\\infty$,\n  \\begin{equation}\np_{n} (z)= a_{n} ^{-1\/2} \\big(\\sigma_{+} (z)e^{ i \\varphi_{n}} + \\sigma_{-}(z) e^{- i \\varphi_{n}}  + o(1)\\big)      \n\\label{eq:LC1P}\\end{equation} \nand\n \\begin{equation}\nq_{n} (z)= a_{n} ^{-1\/2} \\big(\\tau_{+} (z)e^{ i \\varphi_{n}} + \\tau_{-} (z) e^{-i \\varphi_{n}}+ o(1)\\big)   .\n\\label{eq:LC2q}\\end{equation} \n   \\end{theorem}\n\n    In view of conditions \\eqref{eq:Pz}  and \\eqref{eq:Qz} the Wronskian $\\{ p(  z ), q(z)\\} =1$. On the other hand, we can calculate this Wronskian using relations   \\eqref{eq:A2CG1} and \\eqref{eq:LC}, \\eqref{eq:LCq}. This yields an identity\n   \\begin{equation}\n  2i \\alpha_{\\infty}^{-1} \\sqrt {1-\\beta_{\\infty}^2} (\\sigma_{+} (z) \\tau_{-} (z) -\\sigma_{-} (z)\\tau_{+} (z))=1,\\quad \\forall z\\in{\\Bbb C}.\n\\label{eq:Wro}\\end{equation}\nWe also note an identity  \n  \\begin{equation}\n|\\sigma_{+} (z)|^{2} - |\\sigma_{-} (z)|^{2}  =\\Im z \\: \\alpha_{\\infty}  (1-\\beta_{\\infty}^2)^{-1\/2}  \\sum_{n=0}^{\\infty}  |p_{n} (z)|^{2}\n \\label{eq:LC2p}\\end{equation} \n  established in Theorem~4.4 of \\cite{nCarl}.\n\nNext, we extend  asymptotic formulas of Theorem~\\ref{LC} to all vectors in $ {\\cal D}(J_{\\rm max})$.  Recall that the number $\\Gamma (z; h)$ was defined by formula \\eqref{eq:qres3}.\n\n  \\begin{theorem}\\label{LCR}\n  Let $u= (u_{n})\\in{\\cal D}(J_{\\rm max})$. \n  Choose some $z\\in {\\Bbb C}$.\n     Under the assumptions of Theorem~\\ref{GSS} a  sequence $ u_{n} $ has asymptotics\n  \\begin{equation}\nu_{n} = a_{n} ^{-1\/2} \\big(s_{+}   e^{i \\varphi_{n}} +s_{-}  e^{-i \\varphi_{n}}+ o(1)\\big)      , \\quad n\\to\\infty,\n\\label{eq:LC2}\\end{equation} \nwhere the  coefficients $s_{\\pm} = s_{\\pm} (u) $ can be constructed by relations\n \\begin{equation}\n  \\begin{split}\ns_{+}(u)=  \\Gamma(z; ({\\cal J}-zI)u) \\sigma_{+} (z)+ \\langle ( {\\cal J} -zI) u, p(\\bar{z})\\rangle \\tau_{+} (z),\n  \\\\\n s_{-}(u)= \\Gamma(z; ({\\cal J}-zI)u) \\sigma_{-} (z)+ \\langle ({\\cal J}-zI) u, p(\\bar{z})\\rangle \\tau_{-} (z).\n  \\end{split}\n  \\label{eq:LC31}\\end{equation}  \n\n  Conversely, for arbitrary $ s_{+},s_{-}  \\in {\\Bbb C} $, there exists a vector  \n$u \\in{\\cal D}(J_{\\rm max})$ such that    asymptotics \\eqref{eq:LC2}  holds.\n \\end{theorem}\n \n\n \n   \\begin{pf}\n   According to Corollary~\\ref{res1}  a vector   $u \\in{\\cal D}(J_{\\rm max})$  admits representation  \\eqref{eq:qres3} where the operator ${\\cal R} (z)$  is defined by equality \\eqref{eq:RR11}.\n      In view of   relation \\eqref{eq:Ras}  and asymptotics \\eqref{eq:LC1P}, \\eqref{eq:LC2q} we have\n    \\begin{equation}\n ( {\\cal R} (z)h)_{n} =    a_{n} ^{-1\/2}  \\big( \\tau_{+} (z)e^{i \\varphi_{n}} + \\tau_{-} (z) e^{-i \\varphi_{n}} \\big) \\langle h, p(\\bar{z})\\rangle     + o(a_{n} ^{-1\/2}), \\quad n\\to\\infty,\n\\label{eq:RR1r}\\end{equation}\n  for all vectors $h \\in \\ell^2 ({\\Bbb Z}_{+})$.  Therefore it follows from \\eqref{eq:qres3}  that\n    \\begin{multline*}\nu_{n} = a_{n} ^{-1\/2}  \\Gamma(z; ({\\cal J}-zI)u)  \\big(\\sigma_{+} (z) e^{i \\varphi_{n}} + \\sigma_{-} (z)e^{-i \\varphi_{n}}\\big) \n\\\\\n+ a_{n} ^{-1\/2} \\big(\\tau_{+} (z) e^{i \\varphi_{n}} + \\tau_{-} (z)e^{-i \\varphi_{n}}\\big) \\langle  ( {\\cal J}-zI) u, p(\\bar{z})\\rangle    + o(a_{n} ^{-1\/2})   \n \\end{multline*} \nas $ n\\to\\infty$. This yields  relation \\eqref{eq:LC2} with the coefficients $s_{\\pm}$ defined by \\eqref{eq:LC31}.\n\n\n\nConversely, given $ s_{+} $ and $ s_{-} $ and fixing some $z\\in{\\Bbb C}$, we consider   a system of equations \n \\begin{equation}\n  \\begin{split}\ns_{+}=  \\Gamma\\sigma_{+} (z)+ \\langle h , p(\\bar{z})\\rangle \\tau_{+} (z),\n  \\\\\n s_{-} = \\Gamma \\sigma_{-} (z)+ \\langle h , p(\\bar{z})\\rangle \\tau_{-} (z).\n  \\end{split}\n  \\label{eq:LC3x}\\end{equation}  \n   for  $\\Gamma$ and $ \\langle h, p(\\bar{z})\\rangle $. According to \\eqref{eq:Wro} the determinant  of this system is not zero so that  $\\Gamma$ and $ \\langle h, p(\\bar{z})\\rangle $ are uniquely determined by  $s_{+} $ and $ s_{-} $.\n Then we take any $h$ such that its scalar product with $ p(\\bar{z})$ equals the found value of  $ \\langle h, p(\\bar{z})\\rangle$.\n  Finally, we define $u$ by formula  \\eqref{eq:qres3}. Asymptotics as $n\\to\\infty$ of $p_{n}(z)$   and $( {\\cal R} (z)h)_{n} $ are given by formulas \\eqref{eq:LC1P} and \\eqref{eq:RR1r}, respectively. In view of equations \\eqref{eq:LC3x} this leads to asymptotics \\eqref{eq:LC2}.\n       \\end{pf}\n       \n      \n       \n      Theorem~\\ref{LCR}  yields a mapping $ {\\cal D}(J_{\\rm max})\\to{\\Bbb C}^2$ defined by the formula\n       \\begin{equation}\nu\\mapsto (s_{+}  (u) , s_{-}(u)).\n\\label{eq:mapping}\\end{equation}\n      The construction of      Theorem~\\ref{LCR} depends on the choice of $z\\in{\\Bbb C}$, but this mapping is defined intrinsically.  In particular, we can set $z=0$ in all formulas of  Theorem~\\ref{LCR}.  Note that mapping \\eqref{eq:mapping} is surjective.\n\n  \nEvidently,   \\eqref{eq:mapping} plays the role of mapping \\eqref{eq:f}  for the differential operator $-d^2\/ dx^2$ in the space $L^2 ({\\Bbb R}_{+})$ and formula \\eqref{eq:qres4} plays the role of the integration-by-parts formula\n\\[\n\\int_{0}^\\infty f'' (x) \\overline{g} (x) dx-\\int_{0}^\\infty f (x) \\overline{g''} (x) dx= f(0) \\bar{g}' (0)- f'(0) \\bar{g} (0).\n\\]\n \n     Under the assumptions of Theorem~\\ref{GSS} the right-hand side of \\eqref{eq:qres4} can be expressed  in terms of the coefficients $s_{+}$ and $s_{-}$.\n\n \n\n\n \\begin{proposition}\\label{resAS}\n     For all $u, v \\in {\\cal D} (J_{\\rm max})$, we have an identity\n    \\begin{equation}\n\\langle J_{\\rm max}u,v \\rangle - \\langle u, J_{\\rm max}v \\rangle = 2 i \\alpha_{\\infty}^{-1} \\sqrt{ 1-\\beta_{\\infty}^2}\\big( s_{+}(u) \\overline{s_{+}(v) }- s_{-}(u) \\overline{s_{-}(v) }\\big)  .\n\\label{eq:ABS}\\end{equation}\n \\end{proposition} \n   \n    \\begin{pf}\n    It follows from formula \\eqref{eq:LC2}  that\n       \\begin{multline*}\n\\sqrt{a_{n+1}  a_{n } } (u_{n +1}\\bar{v}_{n }- u_{n }\\bar{v}_{n +1})= \\big(s_{+}(u)   e^{ i \\varphi_{n+1}} + s_{-} (u)  e^{-i\\varphi_{n+1}}\\big)\\big(\\overline{s_{+}(v) }  e^{-i \\varphi_{n}} + \\overline{ s_{-} (v)}  e^{ i \\varphi_{n}}\\big)\n\\\\\n-  \\big(s_{+}(u)   e^{i \\varphi_{n}} + s_{-} (u)  e^{-i \\varphi_{n}}\\big)\\big(\\overline{s_{+}(v) }  e^{- i \\varphi_{n+1}} + \\overline{ s_{-} (v)}  e^{ i  \\varphi_{n+1}}\\big) + o(1)\n\\\\\n=\\big( s_{+}(u) \\overline{s_{+}(v) }-s_{-}(u) \\overline{s_{-}(v) }\\big) (e^{i\\theta_{n}} -e^{-i\\theta_{n}}) + o(1).\n \\end{multline*}\nPassing here to the limit $n\\to\\infty$ and using equality \\eqref{eq:qres4}, we obtain identity \\eqref{eq:ABS}.\n       \\end{pf}\n       \n    We can now characterize set \\eqref{eq:MIN}.\n       \n     \n    \n    \n       \n       \\begin{proposition}\\label{ABS}\n       A vector\n    $v  \\in {\\cal D} (J_{\\rm max})$ belongs to     $ {\\sf D}  $ if and only if\n    $v_{n}= o(a_{n}^{-1\/2})$, that is, \n     \\begin{equation}\n    s_{+} (v) = s_{-} (v) =0.\n      \\label{eq:ABS4}\\end{equation}\n    \\end{proposition} \n     \n     \\begin{pf}    \n     A vector $v$ belongs to     $ {\\cal D} ( J_{\\max}^{*})$ if and only if\n       \\begin{equation}\n    \\langle J_{\\max}u,v \\rangle= \\langle u, J_{\\max}v \\rangle\n  \\label{eq:ABS2}\\end{equation}\n    for all   $u  \\in {\\cal D} (J_{\\rm max})$.  According to Proposition~\\ref{resAS}  equality \\eqref{eq:ABS2} is equivalent to\n     \\begin{equation}\n s_{+}(u) \\overline{s_{+}(v) }- s_{-}(u) \\overline{s_{-}(v) }=0.\n  \\label{eq:ABS3}\\end{equation}\n  This  is of course true if \\eqref{eq:ABS4} is satisfied. Conversely, if  \\eqref{eq:ABS3} is satisfied for all  $u\\in {\\cal D} ( J_{\\max})$, we use that according to Theorem~\\ref{LCR}  the numbers $ s_{+}(u)$ and $s_{-}(u)$  are arbitrary. This implies \\eqref{eq:ABS4}.\n         \\end{pf}       \n    \nThis result shows that  \\eqref{eq:mapping} considered as a mapping of the factor space  $  {\\cal D} (J_{\\rm max})\/ {\\sf D}$ onto ${\\Bbb C}^2$ is injective.\n  \n    \\subsection{Self-adjoint extensions}\n    \n    All self-adjoint extensions ${\\bf J}_{\\omega} $ of the operator $J_{\\min}$ are now parametrized by complex numbers $\\omega\\in{\\Bbb T}\\subset {\\Bbb C}$.  Let a set $ {\\cal D} (  {\\bf J}_\\omega)\\subset {\\cal D} (    J_{\\max})$ of vectors $u$  be distinguished by  condition     \\eqref{eq:gen2}.\n    \n\n    \n         \n      \\begin{theorem}\\label{ABX}\n           Let the assumptions of Theorem~\\ref{GSS}  be satisfied. Then\n     for all $\\omega\\in{\\Bbb T}$, the operators ${\\bf J}_{\\omega}$ are  self-adjoint. Conversely, every   operator $J$ such that \n     \\begin{equation}\n  J_{\\min} \\subset  J =J^*\\subset J_{\\max} \n      \\label{eq:ABX2}\\end{equation}\n      equals ${\\bf J}_{\\omega}$ for some $\\omega\\in{\\Bbb T}$.\n    \\end{theorem} \n    \n    \\begin{pf}\n    We proceed from Proposition~\\ref{resAS}.   If $u,v \\in  {\\cal D} (  {\\bf J}_\\omega)$,  it follows from condition \\eqref{eq:gen2} that   $s_{+}(u) \\overline{s_{+}(v) }= s_{-}(u) \\overline{s_{-}(v) }$. Therefore   according to equality \\eqref{eq:ABS} $({\\bf J}_{\\omega}u,v)= (u, {\\bf J}_{\\omega}v)$ whence ${\\bf J}_{\\omega}\\subset {\\bf J}_{\\omega}^*$. If $ v\\in {\\cal D} (  {\\bf J}_\\omega^*)$, then \n   $\\langle {\\bf J}_{\\omega}u,v \\rangle= \\langle u, {\\bf J}_{\\omega}^* v \\rangle $\n    for all $ u\\in {\\cal D} (  {\\bf J}_\\omega)$ so that in view of \\eqref{eq:ABS} equality \\eqref{eq:ABS3} is satisfied.\n    Therefore $s_{-}(u) (\\omega\\overline{s_{+}(v) }- \\overline{s_{-}(v) })=0$. Since $s_{-}(u)$ is arbitrary, we see  that  $ \\omega\\overline{s_{+}(v) }- \\overline{s_{-}(v) }=0$, and hence $ v\\in {\\cal D} (  {\\bf J}_\\omega)$.\n    \n    Suppose that an operator $J$  satisfies conditions  \\eqref{eq:ABX2}.  Since  $J$ is symmetric, it follows from  Proposition~\\ref{resAS} that  equality   \\eqref{eq:ABS3} equality is true  for all $ u, v\\in {\\cal D} (  J )$.  Setting here $u=v$, we see that $|  s_{+}(v)|=|   s_{-}(v)|$.   There exists a vector $v_{0} \\in {\\cal D} (  J )$ such that\n    $s_{-}(v_{0})\\neq 0$ because  $J\\neq \\clos J_{\\rm min}$. Let us set $\\omega= s_{+}(v_{0})\/ s_{-}(v_{0})$. Then $|\\omega|=1$  and relation  \\eqref{eq:gen2}  is a direct consequence of \\eqref{eq:ABS3}.\n        \\end{pf}\n        \n      \n   \n       \\subsection{Resolvent}\n     \n     Now it easy to construct the   resolvent  of the operator ${\\bf J}_\\omega$  defined in the previous subsection.\n     We previously note that, by definition \\eqref{eq:LC2}, \n     \\[\n     s_{\\pm} (p(z))= \\sigma_{\\pm} (z) \\quad \\mbox{and} \\quad      s_{\\pm} (q(z))= \\tau_{\\pm} (z).\n     \\]\n     \n     \\begin{theorem}\\label{ABX1}\n            Let the assumptions of Theorem~\\ref{GSS}  be satisfied. Then\n     for all $z\\in {\\Bbb C}$ with $\\Im z\\neq 0$ and all $h\\in\\ell^2 ({\\Bbb Z}_{+})$, the resolvent ${\\bf R}_{\\omega} (z)= ({\\bf J}_{\\omega}-zI)^{-1}$ of the operator ${\\bf J}_\\omega$ is given by   equality \\eqref{eq:ABY1} \n      where $\\boldsymbol{\\gamma}_{\\omega}(z)$ is defined by formula  \\eqref{eq:ABY2}, that is,\n           \\begin{equation}\n\\boldsymbol{\\gamma}_{\\omega}(z)= -\\frac{\\tau_{+}(z)-\\omega \\tau_{-}(z)} {\\sigma_{+}(z)-\\omega \\sigma_{-}(z)}.\n    \\label{eq:gam}\\end{equation}\n    \\end{theorem} \n    \n         \\begin{pf}\n           According to      Corollary~\\ref{res1} a vector  $u={\\bf R}_{\\omega}(z) h$ is given by formula \\eqref{eq:qres3} where the coefficient $\\Gamma$ is determined by condition \n            \\eqref{eq:gen2}. It follows from Theorem~\\ref{LCR}  than the components $u_{n}$ of $u$ have  asymptotics \\eqref{eq:LC2}  with the coefficients $s_{\\pm}$ defined by relations \n            \\eqref{eq:LC3x}.  Thus, \n          $u\\in {\\cal D}({\\bf J}_{\\omega})$ if and only if  \n    \\[\n    \\Gamma \\sigma_{+} (z) + \\tau_{+} (z) \\langle h,  p(\\bar{z})\\rangle=\\omega \\big( \\Gamma \\sigma_{-} (z) + \\tau_{-} (z)\\langle h,  p(\\bar{z})\\rangle\\big)\n    \\]\n    whence\n    \\[\n\\Gamma= -\\frac{\\tau_{+}(z)-\\omega \\tau_{-}(z)} {\\sigma_{+}(z)-\\omega \\sigma_{-}(z)}\\: \\langle h,  p(\\bar{z})\\rangle.\n      \\]\nSubstituting this expression into \\eqref{eq:qres3}, we arrive at  formulas\n    \\eqref{eq:ABY1},   \\eqref{eq:ABY2}.\n          \\end{pf}\n          \n   \n          \n          It follows from formula \\eqref{eq:gam} that the spectrum of the operator ${\\bf J}_{\\omega}$ consists of the points  $z$ where\n          \\begin{equation}\n          \\sigma_{+}(z)-\\omega \\sigma_{-}(z)=0.\n         \\label{eq:ABY3}\\end{equation}\n                 Since the functions $\\sigma_{+} (z)$ and $\\sigma_{-} (z)$  are analytic, the set of such points $z$ is discrete. Moreover,  according to \\eqref{eq:LC2p} $\\sigma_{+} (z)\\neq \\sigma_{-} (z)$ if $\\Im z\\neq 0$, and therefore zeros $z$ of equation \\eqref{eq:ABY3} lie on the real axis.  This results has  of course to be expected since $z$ are eigenvalues of the self-adjoint operator ${\\bf J}_{\\omega}$.  We finally note that discreteness of the spectrum of the operators ${\\bf J}_{\\omega}$ is quite natural because their domains ${\\cal D}({\\bf J}_{\\omega})$ are distinguished by boundary conditions at the point $n=0$ \n and for $n\\to\\infty$. Therefore  ${\\bf J}_{\\omega}$ acquire some features of regular operators.\n       \n\n     \n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{New approach to greedy vector quantization}\n\n\t\\section{Introduction}\n\tLet $d \\geq 1$, $r \\in (0, + \\infty)$ and $L^r_{\\mathbb{R}^d}(\\mathbb{P})$ (or simply  $L^r (\\mathbb{P})$) the set of $d$-dimensional random variables $X$ defined on the probability space $(\\Omega,\\mathcal{A},\\mathbb{P})$ such that $\\mathbb{E}\\| X\\|^r<+\\infty$ where $\\|.\\|$ denotes any norm on $\\mathbb{R}^d$. We denote $P=\\mathbb{P}_X$ the probability distribution of $X$.\n\tOptimal vector quantization is a technique derived from signal processing, initially devised to optimally discretize a continuous (stationary) signal for its transmission. Originally developed in the $1950s$ (see \\cite{gegr98}), it was introduced as a cubature formula for numerical integration in the early $1990s$ (see \\cite{Pages98}) and for approximation of conditional expectations in the early $2000s$ for financial applications (see \\cite{BaPaPr01,BaPa03}). Its goal is to find the best approximation of a continuous probability distribution by a discrete one, or in other words, the best approximation of a multidimensional random vector $X$ by a random variable $Y$ taking at most a finite number $n$ of values. \\\\\n\tLet $\\Gamma=\\{x_1,\\ldots,x_n\\}$ be a $d$-dimensional grid of size $n$. The idea is to approximate $X$ by $q(X)$, where $q$ is a Borel function defined on $\\mathbb{R}^d$ and having values in $\\Gamma$.\n\n\tIf we consider, for $q$, the nearest neighbor projection $\\pi_{\\Gamma}:\\mathbb{R}^d \\rightarrow \\Gamma$ defined by \n\t$$\\pi_{\\Gamma} (\\xi) =\\ds \\sum_{i=1}^{n} x_i \\mathds{1}_{W_i(\\Gamma)}(\\xi),$$\n\twhere \\begin{equation}\n\t\\label{Voronoicells}\n\tW_{i}(\\Gamma)  \\subset\\{\\xi \\in \\mathbb{R}^d : \\|\\xi-x_i\\|\\leq  \\min_{j\\neq i} \\|\\xi-x_j\\|\\}, \\qquad i =1,\\ldots,n,\n\t\\end{equation} is the Vorono\\\"i partition induced by $\\Gamma$, then the Vorono\\\"i quantization of $X$ is defined by\n\t\\begin{equation} \n\t\\widehat{X}^{\\Gamma} =\\pi_{\\Gamma}(X) := \\sum_{i=1}^{n}x_i \\mathds{1}_{W_{i}(\\Gamma)}(X). \n\t\\end{equation}\n\tWe will denote, most of the times, $\\widehat{X}$ instead of $\\widehat{X}^\\Gamma$ when there is no need for specifications. The $L^r$-quantization error associated to the grid $\\Gamma$ is defined, for every $r\\in (0,+\\infty)$, by \n\t\\begin{equation} \n\t\\label{quanterror}\n\te_r(\\Gamma,X)  =  \\|X-\\pi_\\Gamma(X)\\|_r = \\|X-\\widehat{X}^{\\Gamma}\\|_r = \\left \\|\\min_{1\\leq i \\leq n} |X - x_i|\\right \\|_r\n\t\\end{equation}\n\twhere $\\|.\\|_r$ denotes the $L^r(\\mathbb{P})$-norm (or quasi-norm if $0<r<1$). Consequently, the optimal quantization problem comes down to finding the grid $\\Gamma$ that minimizes this error. \n\n\n\n\n\tIt has been shown (see \\cite{GraLu00,Pages15,Pages18}) that this problem admits a solution and that the quantization error converges to $0$ when the size $n$ goes to $+\\infty$. The rate of convergence is given by two well known results exposed in the following theorem. \n\t\\begin{thm} \n\t\t\\label{Zadoretpierce}\n\t\t\\noindent $(a)$ {\\rm Zador's Theorem (see \\cite{Zador82})} :  \n\t\tLet $X \\in L_{\\mathbb{R}^d}^{r+\\eta}(\\mathbb{P})$, $\\eta>0$, \n\t\twith distribution $P$ such that \n\t\t$dP(\\xi)= \\varphi(\\xi) d \\lambda_d(\\xi)+ d\\nu(\\xi)$. \n\t\tThen, \n\t\t$$\\ds\\lim_{n\\rightarrow +\\infty} n^{\\frac{1}{d}} e_{r,n}(X) = \\tilde{J}_{r,d} \\|\\varphi\\|^{\\frac{1}{r}}_{L^{\\frac{r}{r+d}}(\\lambda_d)}$$\n\t\twhere $\\tilde{J}_{r,d} = \\ds \\inf_{n \\geq 1} n^{\\frac{1}{d}} e_{r,n}(U([0,1]^d)) \\in (0,+ \\infty)$.  \\\\\n\t\n\t\n\t\t\\noindent $(b)$ {\\rm Extended Pierce's Lemma (see \\cite{LuPa08})}:\n\t\tLet $r,\\eta >0 $. There exists a constant $\\kappa_{d,r,\\eta} \\in (0,+\\infty)$  such that, \n\t\t$$\\forall n\\geq 1,\\quad e_{r,n}(X) \\leq \\kappa_{d,r,\\eta} \\sigma_{r+\\eta} (X) n^{- \\frac{1}{d}}$$\n\t\twhere, for every $r \\in (0, + \\infty), \\, \\sigma_r(X) = \\ds \\inf_{a \\in \\mathbb{R}^d} \\|X-a\\|_r$ is the $L^r$-standard deviation of $X$.\n\\end{thm}\nHowever, the numerical implementation of multidimensional optimal quantizers requires the computation of grids of size $N \\times d$ which becomes too expensive\nwhen $N$ or $d$ increase. Hence, there is a need to provide a sub-optimal solution\nto the quantization problem which is easier to handle and whose convergence rate remains similar (or comparable) to that induced by optimal quantizers.\nA so-called greedy version of optimal vector quantization has been developed in \\cite{LuPa15}. It consists this time in building a {\\it sequence} of points $(a_n)_{n \\geq 1}$ in $\\mathbb{R}^d$ which is recursively optimal step by step, in the sense that it minimizes the $L^r$-quantization error at each iteration. This means that, having the first $n$ points $a^{(n)}=\\{a_1,\\ldots,a_n\\}$ for $n\\geq 1$, we add, at the $(n+1)$-th step, the point $a_{n+1}$ solution to\n\\begin{equation}\n\\label{greedydef}\n\\qquad a_{n+1} \\in \\mbox{argmin}_{\\xi \\in \\mathbb{R}^d} \\, e_r(a^{(n)} \\cup \\{\\xi\\}, X),\n\\end{equation}  \nnoting that $a^{(0)} = \\varnothing$, so that {\\em $a_1$ is simply an\/the $L^r $-median of the distribution $P$ of $X$}. \nThe sequence $(a_n)_{n \\geq 1}$ is called an $L^r$-optimal greedy quantization sequence for $X$ or its distribution $P$. The idea to design such an optimal sequence, which will hopefully produce  quantizers with a rate-optimal behavior as $n$ goes to infnity, is very natural and may be compared to sequences with low discrepancy in Quasi-Monte Carlo methods when working on the unit cube $[0,1]^d$. \nIn fact, such sequences have already been investigated in an $L^1$-setting for compactly supported distributions $P$ as a model of short term experiment planning versus long term experiment planning represented by regular quantization at a given level $n$ (see \\cite{BBSS09}) and, then, in \\cite{LuPa15} where the authors investigated more deeply this greedy version of vector quantization for $L^r$-random vectors taking values in $\\mathbb{R}^d$. They showed that the problem~$(\\ref{greedydef})$ admits at least one solution $(a_n)_{n \\geq 1}$ when $X$ is an $\\mathbb{R}^d$-valued random vector (the existence of such sequences can be proved in Banach spaces but, in this paper, we will only focus on $\\mathbb{R}^d$). This sequence may not be unique since greedy quantization depends on the symmetry of the distribution (consider for example the $\\mathcal{N}(0,1)$ distribution). However, note that, if the norm $\\|.\\|$ is strictly convex and $r>1$, then the $L^r$-median is unique. They also showed that the $L^r$-quantization error converges to $0$ when $n$ goes to infinity\nand, if $\\mbox{supp}(P)$ contains at least $n$ elements, then the sequence $a^{(n)}$ lies in the convex hull of supp($P$), $e_r (a^{(k)} , X)$ is decreasing w.r.t. $k \\in \\{1,\\ldots,n\\}$ and $P\\left( \\{\\xi \\in \\mathbb{R}^d: \\|\\xi -a_n\\| < \\min_{1 \\leq i \\leq n } \\|\\xi-a_i\\|\\} \\right) > 0$. The proof of these results (see Propositions $2.1$ and $2.2$ in \\cite{LuPa15}) are based on micro-macro inequalities given in \\cite{mismatch08}. Moreover, the authors showed in \\cite{LuPa15} that these sequences have an optimal rate of convergence to zero, compared to optimal quantizers, and that they satisfy the distortion mismatch problem, i.e. the property that the optimal rate of $L^r$-quantizers holds for $L^s$-quantizers for $s>r$. The proofs were based on the integrability of the $b$-maximal functions associated to an $L^r$-optimal greedy quantization sequence $(a_n)_{n \\geq 1}$ given by \n\\begin{equation}\n\\label{psib}\n\\forall \\xi \\in \\mathbb{R}^d, \\quad \\Psi_b(\\xi)=\\sup_{n \\in \\mathbb{N}}\\frac{\\lambda_d\\left(B(\\xi,b \\, \\mbox{dist}(\\xi,a^{(n)}))\\right)}{P\\left(B(\\xi,b \\, \\mbox{dist}(\\xi,a^{(n)}))\\right)}.\n\\end{equation}\nIn this paper, we will extend those rate of convergence and distortion mismatch results to a much larger class of functions. Instead of maximal functions, we will rely on a new micro-macro inequality involving an auxiliary probability distribution $\\nu$ on $\\mathbb{R}^d$. When this distribution $\\nu$ satisfies an appropriate control on balls, with respect to an $L^r$-median $a_1$ of $P$, defined later in section~$\\ref{rateoptimality}$, we will show that the rate of convergence of the $L^r$-quantization error of greedy sequences is $\\mathcal{O}(n^{-\\frac1d})$, just like the optimal quantizers. Furthermore, considering appropriate auxiliary distributions $\\nu$ satisfying this control allows us to obtain Pierce type, and hybrid Zador-Pierce type, $L^r$-rate optimality results of the error quantization, instead of only Zador type results as given in \\cite{LuPa15}.\\\\\n\nA very important field of applications is to use these greedy sequences instead of $n$-optimal quantizers in quantization-based numerical integration schemes. In fact, the size of the grids used in these procedures is large in a way that the RAM storing of the quantization tree may exceed the storage capacity of the computing device. So, using greedy quantization sequences will dramatically reduce this drawback, especially since we will show that they behave similarly to optimal quantizers in terms of convergence rate. The computation of greedy quantizers is performed by algorithms, detailed in \\cite{LuPa215}, allowing also the computation of the weights $(p_i^n)_{1 \\leq i \\leq n}$ of the Vorono\\\"i cells of the sequence $a^{(n)}$. Theses quantities are  mandatory for the greedy quantization-based numerical integration to approximate an integral $I$ of a function $f$ on $\\mathbb{R}^d$ by the cubature formula \n$$I(f) \\approx \\sum_{i=1}^n p_i^n f\\big(a_i^{(n)}\\big).$$  \nCompared to other methods of numerical approximation, such as quasi-Monte Carlo methods (QMC), the quantization-based methods present an advantage in terms of convergence rate, since QMC, for example, is known to induce a convergence rate of $\\mathcal{O}\\Big(\\frac{\\log n}{n^{\\frac1d}} \\Big)$ when integrating Lipschitz functions (see \\cite{Proinov88}) while quantization-based numerical integration produces an $\\mathcal{O}\\big(n^{-\\frac1d}\\big)$ rate (see \\cite{Pages18}). However, it seems to have a drawback which is the computation of the non-uniform weights $(p_i^n)_{1 \\leq i \\leq n}$, unlike the uniform weights in QMC (equal to $\\frac1n$). In this paper, we expose how the recursive character of greedy quantization provides several improvements to the algorithm, making it more advantageous. Moreover, this character induces the implementation of a recursive formula for numerical integration, that can replace the usual cubature formula, reducing the time and cost of the computations. This recursive formula will be introduced first in the one-dimensional case, and then extended to the multi-dimensional case for product greedy quantization sequences, computed from one-dimensional sequences, used to reduce the cost of implementations while always preserving the recursive character.\\\\  \n\n\n\nThe paper is organized as follows. We first show that greedy quantization sequences can be rate optimal just like the optimal quantizers in section~\\ref{rateoptimality} where we extend the results already presented in \\cite{LuPa15} and we give Pierce type results. Likewise, the distortion mismatch problem will be solved and extended in section~\\ref{distortionmismatch}. In section~\\ref{algorithmics}, we present the improvements we can apply to the algorithm of designing the greedy sequences, as well as the new approach for greedy quantization-based numerical integration. Numerical examples will illustrate and confirm the advantages brought by this new approach in section~\\ref{numericalexample}. Finally, section~\\ref{numericalproperties} is devoted to some numerical conclusions about further properties of greedy quantization sequences such as the sub-optimality, the convergence of empirical measures, the stationarity (or quasi-stationarity) and the discrepancy, to see to what extent greedy sequences can be close to optimality.\n\n\n\\iffalse\n\\section{Existence of optimal greedy quantization sequences}\nWe consider a random variable $X$ taking values in $\\mathbb{R}^d$ and with distribution $P=\\mathbb{P}_X$. Optimal greedy quantization sequences $a^{(n)}$ of $X$ exist, the corresponding quantization error $e_r (a^{(n)} , X)$ is decreasing as a function of the number of points $n$ and converges towards $0$ when $n$ goes to infinity. Moreover, this sequence lies in the convex hull of supp($\\mathbb{P}$). These results were established and demonstrated in $\\cite{LuPa15}$. We recall, in the following, these results and then give lower and upper bounds of the increments of the $L^r$-optimal greedy quantization errors.\nThe existence of such $L^r$-optimal greedy quantization sequences in Banach spaces can be proved but in this paper we will study only the case when $X$ takes values in $\\mathbb{R}^d$.\n\\begin{prop}\n\tLet $r \\in (0,+\\infty)$. If $X \\in L^r (\\mathbb{P})$ and takes values in $\\mathbb{R}^d$, then~$(\\ref{greedydef})$ admits at least one solution $(a_n)_{n \\geq 1}$ where $a_1$ is the $L^r$-median of $X$. This sequences is an $L^r$-optimal greedy quantization sequence for $X$.\n\\end{prop} \nNote that if $\\|.\\|$ is strictly convex and $r>1$, then the $L^r$-median is unique.\n\\iffalse\n\\begin{proof}\n\tFor $n=1$, $\\mbox{argmin}_{y \\in E} e_r(\\{y\\},X) \\neq \\varnothing$ since $\\{y\\}$ is the $1$-optimal quantizer of $X$. \\\\\n\tLet $ n \\geq 1$ and $f$ a function on $E$ taking values in $\\mathbb{R}^d$ such that $f(y)=D_{r,n+1}^X(a_1,\\ldots,a_n,y)$, namely, the distortion function of the sequence $(a_1,\\ldots,a_n,y)$ associated to $X$.\\\\\t\n\tWe consider the topology $\\tau_0$ on $E$ generated by the system $\\{B(x, s)^c ;\\; x\\in F, s>0\\}$ and the product topology $\\tau$ on $E^{n+1}$. The family $\\{B(x, s);\\; x \\in F, s > 0\\}$ being a compact system in $E$, it\n\tfollows from the Alexander subbase theorem (see \\cite{kelly55}, theorems $5$ and $6$) that $E$ is $\\tau_0$-quasi-compact, that is, every $\\tau_0$-open cover of $E$ has a finite subcover. Consequently, $E^{n+1}$ is $\\tau$-quasi-compact. Now, the proof amounts to showing that the distortion function $D_{r,n+1}^X$ is $\\tau$-lower semicontinuous.\\\\\n\tFor every $x \\in F$ and $a \\in E$ , set $d(x, a) := \\min_{1 \\leq i \\leq n} \\|x - a_i\\|$. Then\n\t$$\\{a \\in E^{n+1} ;\\;  d(x,.)^r \\leq c\\}= \\bigcup_{i=1}^{n+1} \\{a \\in E^{n+1} ;\\; a_i \\in B(x, c^{\\frac{1}{r}})\\}$$\n\tis $\\tau$ -closed for every $c \\geq 0$. Hence, $a \\rightarrow d(x, a)^r$ is $\\tau$-lower semicontinuous. In turn, any\tconvex combination of such functions is $\\tau$-lower semicontinuous as well. This implies that $D_{r,n+1}^X$ is $\\tau$ -lower semicontinuous.\t\t\n\tSo, $f$ is also lower semicontinuous and, consequently, reaches a minimum on $E$ so that $\\mbox{argmin}_{y \\in E} e_r(\\{a_1, \\ldots,a_n\\} \\cup \\{y\\}, X) \\neq \\varnothing$. Hence, the existence of a $L^r$-optimal greedy sequence of size $n$.\n\t\\hfill $\\square$\n\\end{proof}\n\\fi\n\\begin{prop}\n\t\\label{converreur}\n\tLet $(a_n)_{n \\geq 1}$ be an $L^r$-optimal greedy quantization sequence for $X$ and $a^{(n)} = \\{a_1 ,\\ldots, a_n \\}$. Then\n\t$$\\lim_{n \\rightarrow +\\infty} e_r (a^{(n)} , X) = 0.$$\n\tMoreover, if $\\mbox{card} ( \\mbox{supp}(P)) \\geq n$, then $e_r (a^{(k)} , X)$ is decreasing w.r.t. $k \\in \\{1,\\ldots,n\\}$, $$P (C_{a_n}(a^{(n)})) > 0 \\quad \\mbox{ where }C_{a_n}(a^{(n)})= \\{\\xi \\in \\mathbb{R}^d; |\\xi -a_n| < \\min_{1 \\leq i \\leq n } |\\xi-a_i|\\}$$  \n\tand $\\{a_n\\} \\in C_{r,1}(P(.|C),E)$ \n\tfor every Borel set $C \\subset E$ with $C_{a_n} (a^{(n)}) \\subset C \\subset W_{a_n} (a^{(n)} )$.\n\\end{prop}\n\\begin{cor}\n\tLet $X$ be a random vector taking values in $\\mathbb{R}^d$ and $(a_n)_{n \\geq 1}$ be an $L^r$-optimal greedy quantization sequence for $X$. If $\\mbox{card}( \\mbox{supp}(P )) \\geq n $, then\n\t$$a_n \\in \\overline{\\mbox{conv}} (\\mbox{supp}(P)).$$ \n\\end{cor}\nFor the proof of the previous results, we refer to \\cite{LuPa15} (Propositions $2.1$ and $2.2$). The second proposition is based on micro-macro inequalities given in \\cite{mismatch08}.\\\\\n\nWe recall that $L^r$-optimal greedy sequences may not be unique since greedy quantization depends on the symmetry of the distribution (consider for example the $\\mathcal{N}(0,1)$ distribution).\n\\fi\n\\section{Rate optimality: Universal non-asymptotic bounds}\n\\label{rateoptimality}\nIn \\cite{LuPa15}, the authors presented the rate optimality of $L^r$-greedy quantizers in the sense of Zador's theorem based on the integrability of the $b$-maximal function $\\Psi_b(\\xi)$ defined by~$(\\ref{psib})$. Here, we present Pierce type non-asymptotic estimates relying on micro-macro inequalities applied to a certain class of auxiliary probability distributions $\\nu$. Different specifications of $\\nu$ lead to various versions of Pierce's Lemma. \\\\\nIn all this section, we denote $V_d=\\lambda_d\\big(B(0,1)\\big)$ w.r.t. the norm $\\|\\cdot\\|$.\nWe recall, first, a micro-macro inequality that will be be used to prove the first result.\n\\iffalse\n\\begin{prop}[Micro-macro inequality]\nLet $\\Gamma \\subset \\mathbb{R}^d$ be a finite quantizer of a random variable $X$ with distribution $P$. Then, for every $c \\in (0,\\frac12)$ and $y \\in \\mathbb{R}^d$,\n\\begin{equation}\n\\label{micromacro}\ne_r(\\Gamma,P)^r - e_r(\\Gamma \\cup \\{y\\},P)^r \n\\geq ((1-c)^r-c^r) P \\left(B\\left(y, cd\\left(y,\\Gamma\\right)  \\right) \\right) d\\left(y,\\Gamma\\right)^r. \n\\end{equation}\n\\end{prop}\n\\begin{proof}\n\tLet $\\Gamma_1=\\Gamma \\cup \\{y\\}$. We have $B(y, cd(y,\\Gamma))\\subset W_{y}(\\Gamma_1)$, where $ W_{y}(\\Gamma_1)$ is the Vorono\\\"i cell associated to $\\Gamma_1$ of centroid $y$, so that for every $x \\in B(y, cd(y,\\Gamma))$,\n\t$$d(x,\\Gamma) \\geq d(y,\\Gamma)=\\|x-y\\| \\geq (1-c)d(y,\\Gamma).$$\n\tConsequenty,\n\t\\begin{align*}\n\te_r(\\Gamma,P)^r - e_r(\\Gamma \\cup \\{y\\},P)^r & = \\int_{\\mathbb{R}^d} \\left(d(x,\\Gamma)^r -d(x,\\Gamma_1)^r\\right)dP(x)\\\\\n\t& \\geq \\int_{W_y(\\Gamma_1)} \\left(d(x,\\Gamma)^r-\\|x-y\\|^r \\right)dP(x)\\\\\n\t& \\geq \\int_{B(y, cd(y,\\Gamma))} ((1-c)^r-c^r) d(y,\\Gamma)^r dP(x)\\\\\n\t& =((1-c)^r-c^r)P\\left(B(y, cd(y,\\Gamma)) \\right) d(y,\\Gamma)^r.\n\t\\end{align*}\n\t\\hfill $\\square$\n\\end{proof}\n\\fi\n\\begin{prop}\n\t\\label{inc}\n\tAssume $\\int\\|x\\|^rdP(x)<+\\infty$. Then, for every probability distribution $\\nu$ on $(\\mathbb{R}^d,\\mathcal{B}(\\mathbb{R}^d))$, every $c \\in (0, \\tfrac 12)$ and every $n \\geq 1$\n\t\\begin{align*}\n\te_r(a^{(n)},P)^r - e_r(a^{(n+1)},P)^r \n\t \\geq \\frac{(1-c)^r-c^r}{(c+1)^r} \\int \\nu \\left(B\\left(x, \\frac{c}{c+1}d\\left(x,a^{(n)}\\right)  \\right) \\right) d\\left(x,a^{(n)}\\right)^r dP(x). \n\t\\end{align*}\n\\end{prop}\n\\begin{proof}\n\t\\noindent \\textsc{\\bf Step 1:} Micro-macro inequality\\\\\n\tLet $\\Gamma \\subset \\mathbb{R}^d$ be a finite quantizer of a random variable $X$ with distribution $P$ and $\\Gamma_1=\\Gamma \\cup \\{y\\}$, $y \\in \\mathbb{R}^d$. For every $c \\in (0, \\tfrac 12)$, we have $B(y, cd(y,\\Gamma))\\subset W_{y}(\\Gamma_1)$, where $ W_{y}(\\Gamma_1)$ is the Vorono\\\"i cell associated\n\n\tcentroid $y$ form a Voronoi partition induced by $\\Gamma_1$, as defined by~$(\\ref{Voronoicells})$. Hence,  for every $x \\in B(y, cd(y,\\Gamma))$,\n\t$d(x,\\Gamma) \\geq d(y,\\Gamma)-\\|x-y\\| \\geq (1-c)d(y,\\Gamma).$ \n\tConsequently,\n\t\\begin{align*}\n\te_r(\\Gamma,P)^r - e_r(\\Gamma \\cup \\{y\\},P)^r & = \\int_{\\mathbb{R}^d} \\left(d(x,\\Gamma)^r -d(x,\\Gamma_1)^r\\right)dP(x)\\\\\n\t& \\geq \\int_{W_y(\\Gamma_1)} \\left(d(x,\\Gamma)^r-\\|x-y\\|^r \\right)dP(x)\\\\\n\t& \\geq \\int_{B(y, cd(y,\\Gamma))} ((1-c)^r-c^r) d(y,\\Gamma)^r dP(x).\n\t\\end{align*}\n\tFinally, we obtain the micro-macro inequality\n\t\\begin{equation}\n\t\\label{micromacro}\n\te_r(\\Gamma,P)^r - e_r(\\Gamma \\cup \\{y\\},P)^r \n\t\\geq ((1-c)^r-c^r) P \\left(B\\left(y, cd\\left(y,\\Gamma\\right)  \\right) \\right) d\\left(y,\\Gamma\\right)^r. \n\t\\end{equation}\n\t\\textsc{\\bf Step 2:} \n\tBased on the micro-macro inequality~$(\\ref{micromacro})$, we have for every $ c\\in \\left(0, \\frac{1}{2} \\right)$ and every $ y \\in \\mathbb{R}^d,$\n\t$$e_r(a^{(n)},P)^r- e_r(a^{(n)}\\cup \\{y\\},P)^r \\geq \\left((1-c)^r-c^r\\right)P\\left(B\\left(y,cd(y,a^{(n)})\\right)\\right) d(y,a^{(n)})^r .$$\n\tSince $e_r(a^{(n+1)},P)\\leq e_r(a^{(n)} \\cup \\{y\\},P)$ for every $y \\in \\mathbb{R}^d$,\n\t$$e_r(a^{(n)},P)^r- e_r(a^{(n+1)},P)^r \\geq \\left((1-c)^r-c^r\\right)P\\left(B\\left(y,cd(y,a^{(n)})\\right)\\right) d(y,a^{(n)})^r .$$\n\tWe integrate this inequality with respect to $\\nu$ to obtain \n\t$$e_r(a^{(n)},P)^r- e_r(a^{(n+1)},P)^r  \\geq ((1-c)^r-c^r)\\int P\\left(B\\left(y,cd(y,a^{(n)})\\right)\\right) d(y,a^{(n)})^rd\\nu(y).$$\n\tNow, we consider the closed sets \n\t$$F_1=\\left\\{ (x,y) \\in (\\mathbb{R}^d)^2: \\|x-y\\|\\leq cd(y,a^{(n)}) \\right\\} \\quad \\mbox{and} \\quad F_2=\\left\\{ (x,y) \\in (\\mathbb{R}^d)^2: \\|x-y\\|\\leq \\frac{c}{c+1}d(x,a^{(n)}) \\right\\}.$$\n\tWe notice that \n\t$$F_2 \\subset F_1 \\cap \\left\\{(x,y) \\in (\\mathbb{R}^d)^2: d(y,a^{(n)}) \\geq \\frac{1}{c+1} d(x,a^{(n)}) \\right\\},$$\n\tIn fact, for $(x,y) \\in F_2$, \n\t$$d(y,a^{(n)}) \\geq d(x,a^{(n)})-\\|x-y\\| \\geq d(x,a^{(n)})-\\frac{c}{c+1}d(x,a^{(n)}) \\geq \\frac{1}{c+1} d(x,a^{(n)})$$\n\tand \n\t$$\\|x-y\\| \\leq \\frac{c}{c+1} d(x,a^{(n)}) \\leq c d(y,a^{(n)}).$$\n\tThen, \n\t\\begin{align*}\n\t\\int P(B(y,cd(y,a^{(n)}))) d(y,a^{(n)})^rd\\nu(y) &=  \\int \\int \\mathds{1}_{F_1}(x,y) d(y,a^{(n)})^r d\\nu(y) dP(x) \\\\\n\t& \\geq \\frac{1}{(c+1)^r} \\int \\int \\mathds{1}_{F_2}(x,y) d(x,a^{(n)})^r d\\nu(y) dP(x) \\\\\n\t& =\\frac{1}{(c+1)^r} \\int \\nu \\left(B\\left(x, \\frac{c}{c+1}d\\left(x,a^{(n)}\\right)  \\right) \\right) d(x,a^{(n)})^r dP(x).\n\t\\end{align*}\n\t\\hfill $\\square$\n\\end{proof}\n\nIn order to prove the rate optimality of the greedy quantization sequences and obtain a non-asymptotic Pierce type result, we will consider auxiliary probability distributions $\\nu$ satisfying the following control on balls with respect to an $L^r$-median $a_1$ of $P$: for every $\\varepsilon \\in (0,\\ve_0)$, for some $\\ve_0\\!\\in (0, 1]$, there exists a Borel function $g_{\\varepsilon}: \\mathbb{R}^d \\rightarrow [0,+\\infty)$ such that, for every $x \\in \\mbox{supp}(P)$ and every $t \\in [0,\\varepsilon \\|x-a_1\\|]$, \n\\begin{equation}\n\\label{criterenu}\n  \\nu(B(x,t)) \\geq g_{\\varepsilon}(x) V_d t^d.\n\\end{equation}\nOf course, this condition is of interest only if the set $\\{g_\\ve >0\\}$ is sufficiently large. Note that $a_1 \\in a^{(n)}$ for every $n \\geq 1$ by construction of the greedy quantization sequence. We begin by a technical lemma which will be used in the proof of the next proposition.\n\\begin{lem}\n\t\\label{lemmab31}\n\tLet $C, \\rho \\in (0,+\\infty)$ be some real constants and $(x_n)_{n \\geq 1}$ be a non-negative sequence satisfying, for every $n \\geq 1$,\n\t\\begin{equation}\n\t\\label{lemb31}\n\t\tx_{n+1} \\leq x_n -Cx_n^{1+\\rho}.\n\t\\end{equation}\n\t Then for every $n \\geq 1$,\n\t$$(n-1)^{\\frac{1}{\\rho}}x_n \\leq \\left(\\frac{1}{C\\rho} \\right)^{\\frac{1}{\\rho}}.$$\n\\end{lem}\n\\begin{proof}\n\tWe rely on the following Bernoulli inequalities, for every $x \\geq -1$,\n\t$$\n\t(1+x)^{\\rho} \\geq 1+\\rho x, \\quad \\mbox{ if } \\rho \\geq 1, \\qquad \\mbox{ and } \\qquad (1+x)^{\\rho} \\leq 1+\\rho x, \\quad \\mbox{ if } 0 < \\rho < 1.\n\t$$\n\tThese inequalities can be obtained by studying the function $f$ defined for every $ x \\in (-1,+\\infty)$ by $f(x)= (1+x)^{\\rho} - (1+\\rho x) $. Assuming that $(x_n)_{n \\geq 1}$ is non-increasing and that $x_n >0$ for every $n \\geq 1$, it follows from~$(\\ref{lemb31})$ that\n\t$$\n\t\\frac{1}{x_{n+1}^{\\rho}} \\geq \\frac{1}{x_n^{\\rho}(1-C\\, x_n^{\\rho})^{\\rho}} \\geq \\frac{1}{x_n^{\\rho}}(1+C\\, x_n^{\\rho})^{\\rho}.\n\t$$\nIf $\\rho \\geq 1$, the Bernoulli inequalities imply \n$\\frac{1}{x_{n+1}^{\\rho}} \\geq \\frac{1}{x_n^{\\rho}}(1+C\\, \\rho \\, x_n^{\\rho})=\\frac{1}{x_n^{\\rho}} +C \\rho.$ By induction, one obtains\n$$\n\\frac{1}{x_{n}^{\\rho}} \\geq \\frac{1}{x_{1}^{\\rho}} +(n-1)C \\rho \\geq (n-1)C \\rho\n$$\nto deduce the result easily. If $0 < \\rho< 1$, then $-C \\rho x_n^{\\rho} \\geq -1$ for every $n \\geq 1$, and the result is deduced by using the Bernoulli inequality and then reasoning by induction. \n\\hfill $\\square$\t\n\\end{proof}\n\\begin{prop}\n\t\\label{mainthm}\n\tLet $P$ be such that $\\int_{\\mathbb{R}^d} \\|x\\|^rdP(x)<+\\infty$. For any distribution $\\nu$ and Borel function $g_{\\varepsilon} : \\mathbb{R}^d\\rightarrow \\mathbb{R}_+$, $\\varepsilon \\in (0,\\frac13)$, satisfying~$(\\ref{criterenu})$,\n\t\\begin{equation}\n\t\\label{gepsilongeneral}\n\t\\forall n \\geq 2, \\quad e_r(a^{(n)},P) \\leq \\varphi_r(\\varepsilon)^{-\\frac1d} V_d^{-\\frac1d} \\left(\\frac rd \\right)^{\\frac1d}\\left(\\int g_{\\varepsilon}^{-\\frac rd} dP  \\right)^{\\frac1r} (n-1)^{-\\frac1d}\n\t\\end{equation}\n\twhere $\\ds \\varphi_r(u)=\\left(\\frac{1}{3^r}-u^r\\right)u^d$.\n\n\\end{prop}\n\\begin{proof}\n\tWe may assume that $\\int g_\\ve^{-\\frac rd}dP<+\\infty$. Assume $c \\in (0 , \\frac{\\varepsilon}{1-\\varepsilon}] \\cap (0,\\frac12)$ so that $\\frac{c}{c+1} \\leq \\varepsilon$. Moreover $d(x,a^{(n)}) \\leq d(x,a_1)$ since $a_1 \\in a^{(n)}$. Consequently, for any such $c$, $\\ds \\frac{c}{c+1} \\, d(x,a^{(n)}) \\leq \\varepsilon \\|x-a_1\\|$ so that, by~$(\\ref{criterenu})$, there exists a function $g_{\\varepsilon}$ such that \n\t$$\\nu \\left(B\\left(x, \\frac{c}{c+1} \\,d\\big(x,a^{(n)}\\big)  \\right) \\right) \\geq V_d \\, \\left(\\frac{c}{c+1} \\right)^{d} d(x,a^{(n)})^d\\, g_{\\varepsilon}(x).$$\n\tThen, noting that $ \\frac{(1-c)^r-c^r}{(1+c)^r} \\geq \\frac{1}{3^r} -\\big( \\frac{c}{c+1}\\big)^r \\; >0$, since $c \\in (0,\\tfrac 12)$, Proposition~$\\ref{inc}$ implies that\n\t\\begin{equation}\n\t\\label{equ1}\n\te_r(a^{(n)},P)^r - e_r(a^{(n+1)},P)^r  \n\t\\geq V_d \\, \\varphi_r\\left(\\frac{c}{c+1} \\right) \\int g_{\\varepsilon}(x) d(x,a^{(n)})^{d+r} dP(x)\n\t\\end{equation}  \n\twhere $\\ds \\varphi_r(u)=\\left(\\frac{1}{3^r}-u^r\\right)u^d, \\; u \\in (0,\\tfrac13)$.\n\tApplying the reverse H\\\"older inequality with the conjugate H\\\"older exponents $p =-\\frac rd$ and $q=\\frac{r}{r+d}$ yields\n\t\\begin{align*}\n\t\te_r(a^{(n)},P)^r - e_r(a^{(n+1)},P)^r \n\t\t & \\geq V_d \\, \\varphi_r\\left(\\frac{c}{c+1} \\right) \\left(\\int g_{\\varepsilon }(x)^{-\\frac rd} dP(x) \\right)^{-\\frac dr} \\left(\\int d(x,a^{(n)})^r dP(x) \\right)^{1+\\frac dr} \\\\\n\t\t & \\geq V_d \\, \\varphi_r\\left(\\frac{c}{c+1} \\right) \\left(\\int g_{\\varepsilon} (x)^{-\\frac rd} dP(x) \\right)^{-\\frac dr} \\left(e_r(a^{(n)},P)^r \\right)^{1+\\frac dr} .\n\t\\end{align*}\n\tThen, applying lemma~\\ref{lemmab31} to the sequence $x_n=e_r(a^{(n)},P)^r $ with $C=V_d \\, \\varphi_r\\left(\\frac{c}{c+1} \\right) \\, \\left(\\int g_{\\varepsilon} (x)^{-\\frac rd} dP(x) \\right)^{-\\frac dr}$ and $\\rho=\\frac dr$, one obtains, for every $c \\in (0,\\tfrac12)$,\n\t$$e_r(a^{(n)},\\mathbb{P}) \\leq V_d^{-\\frac 1d} \\left(\\frac rd\\right)^{\\frac 1d} \\varphi_r\\left(\\frac{c}{c+1} \\right)^{-\\frac 1d} \\left(\\int g_{\\varepsilon}^{-\\frac rd} dP  \\right)^{\\frac1r} (n-1)^{-\\frac1d}.$$\n\tSince in most applications $\\varepsilon \\mapsto \\left(\\int g_{\\varepsilon}^{-\\frac rd} dP  \\right)^{\\frac1r}$ is increasing on $(0,1\/3)$, we are led to study $\\varphi_r\\left( \\frac{c}{c+1}\\right)^{-\\frac 1d}$ subject to the constraint $c \\in \\big(0,\\frac{\\varepsilon}{1-\\varepsilon}\\big] \\cap \\big(0,\\frac12\\big)$.\n\t$\\varphi_r$ is increasing in the neighborhood of $0$ and $\\varphi_r(0)=0$, so, one has, for every $\\varepsilon \\in (0,\\frac 13)$ small enough,\n\t$\\varphi_r\\left( \\frac{c}{c+1}\\right) \\leq \\varphi_r(\\varepsilon), \\mbox{ for } c \\in (0,\\tfrac{\\varepsilon}{1-\\varepsilon}].$ \n\tThis leads to specify $c$ as  \n\t$ c=\\frac{\\varepsilon}{1-\\varepsilon} \\mbox{, so that}  \\frac{c}{c+1}=\\varepsilon,$ \n\t to finally deduce the result.\n\t\n\n\t\\hfill $\\square$\\\\\n\\end{proof}\n\n\nBy specifying the measure $\\nu$ and the function $g_{\\varepsilon}$, we will obtain two first natural versions of the Pierce Lemma.\n\n\\begin{thm}[Pierce's Lemma]\n\t\\label{Piercetype}\n\t$(a)$ Assume $\\int_{\\mathbb{R}^d} \\|x\\|^r dP(x) < +\\infty$. Let $\\delta >0$. Then $e_r(a_1,P)=\\sigma_r(P)$ and\n\t\t$$\\forall n \\geq 2, \\qquad e_r(a^{(n)},P) \\leq \\kappa_{d,\\delta,r}^{\\text{Greedy,Pierce}} \\sigma_{r+\\delta}(P) (n-1)^{-\\frac1d}$$\n\t\twhere $\\ds \\kappa_{d,\\delta,r}^{\\text{Greedy, Pierce}} \\leq V_d^{-\\frac1d} \\left(\\frac rd \\right)^{\\frac1d} \\left(\\left(\\frac{\\delta}{r} \\right)^{\\frac{r}{r+\\delta}}+\\left(\\frac{r}{\\delta}\\right)^{\\frac{\\delta}{r+\\delta}} \\right)^{1+\\tfrac{\\delta}{r}} \\left( \\int_{\\mathbb{R}^d} (\\|x\\| \\vee 1)^{-d-\\frac{d\\delta}{r}}dx \\right)^{\\frac 1d}\\, \\min_{\\varepsilon \\in (0,\\tfrac 13)} (1+\\varepsilon)\\varphi_r(\\varepsilon)^{-\\frac 1d}$.\\\\\n\t\t$(b)$ Assume $\\int_{\\mathbb{R}^d} \\|x\\|^r dP(x) < +\\infty$. Let $\\delta >0$. Then\n\t\t$$\\forall n \\geq 2, \\; e_r(a^{(n)}, P) \\leq \\kappa_{d,r,\\delta}^{\\text{Greedy}} \\left(\\int \\left( \\|x-a_1\\| \\vee 1 \\right) ^r \\left( \\log (\\|x-a_1\\| \\vee e) \\right)^{\\frac{r}{d}+\\delta} dP(x) \\right)^{\\frac{1}{r}}(n-1) ^{-\\frac{1}{d}}$$ \n\t\twhere $\n\t\t\\kappa_{d,r,\\delta}^{\\text{Greedy}} \\leq  V_d^{-\\frac1d}\\left(\\frac rd \\right)^{\\frac 1d}  \\min_{\\varepsilon \\in (0,\\frac13)} (1+\\varepsilon) \\varepsilon^{\\frac 1d +\\frac{\\delta}{r}} \\varphi_r(\\varepsilon)^{-\\frac 1d}.   \\left(\\int \\frac{dx}{(1 \\vee \\|x\\|)^{d}\\left(\\log(\\|x\\| \\vee e) \\right)^{1+\\frac{d\\delta}{r}}} \\right)^{\\frac 1d}.$\\\\\n\t\tIn particular, if \n\t\t$\\int_{\\mathbb{R}^d} \\|x\\|^r (\\log^+\\! \\|x\\|)^{\\frac{r}{d}+\\delta} dP(x) < +\\infty$, then\n\t\t$$\\lim \\sup_n n^{\\frac{1}{d}}\\sup \\{e_r(a^{(n)},P): (a_n) \\, L^r\\mbox{-optimal greedy sequence for } P \\} < +\\infty.$$\n\\end{thm}\n\\begin{proof} \n\t$(a)$ Let $\\delta>0$ be fixed. We set $\\nu(dx)=\\gamma_{r,\\delta}(x) \\lambda_d(dx)$ where\n\t\t$$\\gamma_{r,\\delta}(x)=\\frac{K_{\\delta,r}}{(1 \\vee \\|x-a_1\\|)^{d(1+\\frac {\\delta}{r})}}\\;  \\qquad \\mbox{with} \\qquad  K_{\\delta,r}= \\left(\\int \\frac{dx}{(1 \\vee \\|x\\|)^{d(1+\\frac {\\delta}{r})}} \\right)^{-1} <+\\infty$$\n\t\tis a probability density with respect to the Lebesgue measure on $\\mathbb{R}^d$.\\\\\n\t\tLet $\\varepsilon \\in (0,1)$ and $t >0$. For every $x \\in \\mathbb{R}^d$ such that $\\varepsilon \\|x-a_1\\| \\geq t $ and every $y \\in B(x,t)$,\n\t\t$\\|y-a_1\\| \\leq \\|y-x\\|+\\|x-a_1\\| \\leq (1+\\varepsilon) \\|x-a_1\\|$ so that \n\t\t$$\\nu(B(x,t)) \\geq \\frac{K_{\\delta,r} V_d \\,t^d}{\\left(1 \\vee \\left[(1+\\varepsilon)\\|x-a_1\\|\\right]\\right)^{d(1+\\frac {\\delta}{r})}}.$$\n\t\tHence,~$(\\ref{criterenu})$ is verified with $$g_{\\varepsilon}(x)=\\frac{K_{\\delta,r}}{\\left(1 \\vee \\left[(1+\\varepsilon)\\|x-a_1\\|\\right]\\right)^{d(1+\\frac {\\delta}{r})}},$$\n\t\tso we can apply Proposition~$\\ref{mainthm}$.  We have\n\t\t$$\\int g_{\\varepsilon}(x)^{-\\frac rd} dP(x) \\leq K_{\\delta,r}^{-\\frac rd} \\int \\left(1 \\vee (1+\\varepsilon)\\|x-a_1\\| \\right)^{r+\\delta} dP(x)$$\n\t\tso that, applying $L^{r+\\delta}$-Minkowski inequality, one obtains\t\n\t\t$$\\left(\\int g_{\\varepsilon}(x)^{-\\frac rd} dP(x) \\right)^{\\frac 1r} \\leq K_{\\delta,r}^{-\\frac 1d} \\left(1 + (1+\\varepsilon)\\sigma_{r+\\delta} \\right)^{1+\\tfrac{\\delta}{r}}.$$\n\t\tConsequently, by Proposition~$\\ref{mainthm}$, for $\\ve \\!\\in (0,1\/3)$,\n\t\t\\begin{align}\n\t\t\\label{equ1b}\n\t\te_r(a^{(n)},P) \\leq  V_d^{-\\frac1d} \\left(\\frac rd \\right)^{\\frac1d} K_{\\delta,r}^{-\\frac1d}\\left(1 + (1+\\varepsilon)\\sigma_{r+\\delta}\\right)^{1+\\tfrac{\\delta}{r}} \\varphi_r(\\varepsilon)^{-\\frac1d}(n-1)^{-\\frac1d}\n\t\t\\end{align}\n\t\t\n\t\tNow, we introduce an equivariance argument. For $\\lambda>0$, let $X_{\\lambda}:=\\lambda(X-a_1)+a_1$ and $(a_{\\lambda,n})_{n\\geq 1}:=(\\lambda(a_n-a_1)+a_1)_{n \\geq 1}$. It is clear that $(a_{\\lambda,n})_{n \\geq 1}$ is an $L^r$-optimal greedy sequence for $X_{\\lambda}$ and $e_r(a^{(n)},X)=\\frac{1}{\\lambda}e_r(a_{\\lambda}^{(n)},X_{\\lambda})$. \n\t\n\n\n\n\t\tPlugging this in inequality~$(\\ref{equ1})$ yields \n\t\t\\begin{align*}\n\t\te_r(a^{(n)},P) \\leq & V_d^{-\\frac1d} \\left(\\frac rd \\right)^{\\frac1d} K_{\\delta,r}^{-\\frac1d}\\, \\frac{1}{\\lambda}\\, \\left(1 + (1+\\varepsilon)\\,\\lambda\\,\\sigma_{r+\\delta}\\right)^{1+\\tfrac{\\delta}{r}} \\varphi_r(\\varepsilon)^{-\\frac1d}(n-1)^{-\\frac1d}\\\\\n\t\t\\leq & V_d^{-\\frac1d} \\left(\\frac rd \\right)^{\\frac1d} K_{\\delta,r}^{-\\frac1d} \\left(\\lambda^{-\\frac{r}{\\delta+r}} + (1+\\varepsilon)\\,\\lambda^{\\frac{\\delta}{\\delta+r}}\\,\\sigma_{r+\\delta} \\right)^{1+\\tfrac{\\delta}{r}}\\varphi_r(\\varepsilon)^{-\\frac1d}(n-1)^{-\\frac1d}.\n\t\t\\end{align*}\n\t\tFinally, one deduces the result by setting $\\ds \\lambda=\\frac{r}{\\delta} \\frac{1}{(1+\\varepsilon)\\sigma_{r+\\delta}}$. \\\\\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t$(b)$ Let $\\delta>0$ be fixed. We set $\\nu(dx)=\\gamma_{r,\\delta}(x) \\lambda_d(dx)$ where\n\t\t$$\\gamma_{r,\\delta}(x)=\\frac{K_{\\delta,r}}{(1 \\vee \\|x-a_1\\|)^{d}\\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{1+\\frac{d\\delta}{r}}},$$ with $K_{\\delta,r}= \\left(\\int \\frac{dx}{(1 \\vee \\|x\\|)^{d}\\left(\\log(\\|x\\| \\vee e) \\right)^{1+\\frac{d\\delta}{r}}} \\right)^{-1} <+\\infty,$\n\t\tis a probability density with respect to the Lebesgue measure on $\\mathbb{R}^d$.\\\\\n\t\t\n\t\tLet $\\varepsilon \\in (0,1)$ and $t >0$. For every $x \\in \\mathbb{R}^d$ such that $\\varepsilon \\|x-a_1\\| \\geq t $ and every $y \\in B(x,t)$,\n\t\t$\\|y-a_1\\| \\leq \\|y-x\\|+\\|x-a_1\\| \\leq (1+\\varepsilon) \\|x-a_1\\|$ so that \n\t\t\\begin{align*}\n\t\t\\nu(B(x,t)) \\geq & \\frac{K_{\\delta,r}V_d t^d}{(1 \\vee (1+\\varepsilon)\\|x-a_1\\|)^{d}\\left(\\log((1+\\varepsilon)\\|x-a_1\\| \\vee e) \\right)^{1+\\frac{d\\delta}{r}}}\\\\\n\t\t\\geq & \\frac{K_{\\delta,r}V_d t^d}{(1+\\varepsilon)^{d}\\, \\varepsilon^{1+\\frac{d\\delta}{r}}\\, (1 \\vee \\|x-a_1\\|)^{d}\\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{1+\\frac{d\\delta}{r}}}\n\t\t\\end{align*}\n\t\tsince $\\log(1+\\varepsilon) \\leq \\varepsilon.$ Hence,~$(\\ref{criterenu})$ is verified with $$g_{\\varepsilon}(x)=\\frac{K_{\\delta,r}}{(1+\\varepsilon)^{d}\\, \\varepsilon^{1+\\frac{d\\delta}{r}}\\, (1 \\vee \\|x-a_1\\|)^{d}\\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{1+\\frac{d\\delta}{r}}},$$\n\t\tso we can apply proposition~$\\ref{mainthm}$.  We have\n\t\t$$\\left(\\int g_{\\varepsilon}(x)^{-\\frac rd} dP(x)\\right)^{\\frac 1r} \\leq K_{\\delta,r}^{-\\frac 1d} (1+\\varepsilon) \\varepsilon^{\\frac 1d +\\frac{\\delta}{r}} \\left( \\int \\left(1 \\vee \\|x-a_1\\| \\right)^{r} \\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\delta+\\frac rd}dP(x)\\right)^{\\frac1r}.$$\n\t\tConsequently, one applies Proposition~$\\ref{mainthm}$ to deduce the first part.\n\t\tFor the second part of the proposition, we  start by noticing that \n\t\t$$\\left(1 \\vee \\|x-a_1\\| \\right)^{r} \\leq \\left(1 + \\|x\\|+\\|a_1\\| \\right)^{r}\\leq 2^{(r-1)_+}\\left(\\|x\\|^r+(1+\\|a_1\\|)^r \\right)$$\n\t\tand \n\t\t\\begin{equation*}\n\t\t\\log(\\|x-a_1\\| \\vee e) \\leq \\log(\\|x\\| \\vee e)+ \\frac{\\|a_1\\|\\vee e}{\\|x\\| \\vee e} \\leq \\log_+ \\|x\\|+1+\\frac{\\|a_1\\|\\vee e}{e}\n\t\t\\end{equation*}\n\t\twhere $\\log_+ u =\\log u \\mathds{1}_{u\\geq 1}$, \n\t\tso that\n\t\t\\begin{align*}\n\t\t\\left(1 \\vee \\|x-a_1\\| \\right)^{r} \\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\delta+\\frac rd}  \n\t\n\t\t\\leq  & \\; 2^{(r-1)_++(\\frac rd+\\delta-1)_+} \\left( \\|x\\|^r \\log_+\\|x\\|^{\\frac rd+\\delta}+A_2\\|x\\|^r\\right.\\\\\n\t\t& \\left. +A_1\\log_+\\|x\\|^{\\frac rd+\\delta}+A_1A_2 \\right)\n\t\t\\end{align*}\n\t\twhere $A_1= (1+\\|a_1\\|)^r$ and $A_2=\\left(1+\\frac{\\|a_1\\|\\vee e}{e}\\right)^{\\frac rd+\\delta}$.\n\t\tSince $\\log\\|x\\|^{\\frac rd+\\delta}=\\frac 1r \\left(\\frac rd+\\delta \\right) \\log\\|x\\|^r$, then $\\log_+\\|x\\|^{\\frac rd+\\delta}=\\frac 1r \\left(\\frac rd+\\delta \\right) \\log_+\\|x\\|^r$. Moreover, $\\log_+\\|x\\|^r \\leq \\|x\\|^r-1$ if $\\|x\\|^r \\geq 1$ and equal to zero otherwise so \n\t\t$$\\log_+\\|x\\|^{\\frac rd+\\delta} \\leq \\frac 1r \\left(\\frac rd+\\delta \\right) (\\|x\\|^r-1)_+ \\leq \\frac 1r \\left(\\frac rd+\\delta \\right) (1+\\|x\\|^r). $$\n\t\tConsequently,\n\t\t$$\\left(1 \\vee \\|x-a_1\\| \\right)^{r} \\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\delta+\\frac rd}  \\leq 2^{(r-1)_++(\\frac rd+\\delta-1)_+} \\left( \\|x\\|^r \\log_+\\|x\\|^{\\frac rd+\\delta}+A'_1\\|x\\|^r+A'_2 \\right)$$\n\t\twhere \n\t\t$A'_1=A_2+\\frac 1r \\left(\\frac rd+\\delta \\right)A_2$ and $A'_2=\\frac 1r \\left(\\frac rd+\\delta \\right)A_1+A_1A_2$. \n\t\tThe result is deduced from the fact that $\\sup \\{\\|a_1\\|: a_1 \\! \\in  {\\rm argmin}_{\\xi\\in \\mathbb{R}^d}e_r(\\{\\xi\\},P) < +\\infty$ (see~\\cite[Lemma 2.2]{GraLu00}) and $\\kappa_{d,r,\\delta}$ does not depend on $a_1$.\n\t\\hfill $\\square$\n\\end{proof}\n\\begin{rmq}\nOne checks that $\\varphi_r$ attains its maximum at $\\frac 13 \\left( \\frac{d}{d+r}\\right)^{\\frac 1r}$ on $(0,\\tfrac13)$, so one concludes that\n\n\n\n\t$\\min_{\\varepsilon \\in (0,\\frac 13)}(1+\\varepsilon)\\varphi_r \\left( \\varepsilon \\right)^{-\\frac 1d} \\leq \\left(1+\\frac13\\left(\\frac{d}{d+r} \\right)^{\\frac 1r}\\right)3^{\\frac rd+1} \\left(1+\\frac dr \\right)^{\\frac 1d}\\left(1+\\frac rd \\right)^{\\frac 1r}$\n\tand \\\\\n\n\t$\\min_{\\varepsilon \\in (0,\\frac 13)}(1+\\varepsilon)\\varepsilon^{\\frac 1d +\\frac{\\delta}{r}}\\varphi_r \\left( \\varepsilon \\right)^{-\\frac 1d} \\leq\n\t\\left(1+\\frac13\\left(\\frac{d}{d+r} \\right)^{\\frac 1r}\\right)\n\t3^{1+\\frac{r-1}{d}-\\frac{\\delta}{r}} \\left(1+\\frac dr \\right)^{\\frac 1d-\\frac1r}\\left(1+\\frac rd \\right)^{\\frac 1r}.$\n\t\n\\end{rmq}\n\nAt this stage, one can wonder if it is possible to have a kind of hybrid Zador-Pierce result where, if $P=h.\\lambda_d$, one has  $$e_r(a^{(n)},P)\\leq C \\|h\\|_{\\frac{d}{d+r}} n^{\\frac 1d}$$ for some real constant $C$. To this end, we have to consider \n$$\\nu=\\frac{h^{\\frac{d}{d+r}}}{\\int h^{\\frac{d}{d+r}}d\\lambda_d}.\\lambda_d.$$\nThis is related to the following local growth control condition of densities.\n\\begin{defi}\n\tLet $A \\subset \\mathbb{R}^d$. A function $f:\\mathbb{R}^d \\rightarrow \\mathbb{R}_+$ is said to be  almost radial non-increasing  on A w.r.t. $a \\in A$ if there exists a norm $\\|.\\|_0$ on $\\mathbb{R}^d$ and real constant\n\n\t$ M\\!\\in (0,1]$ such that\n\t\\begin{equation}\n\t\\label{radialtails}\n\t\\forall x\\! \\in \n\n\tA\\setminus\\{a\\}, \\quad f_{|B_{\\|.\\|_0}(a,\\|x-a\\|_0) \\cap (A\\setminus\\{a\\})}\n\n\t\\geq Mf(x).  \n\t\\end{equation}\n\tIf~$(\\ref{radialtails})$ holds for $M=1$, then $f$ is called radial non-increasing on $A$ w.r.t. $a$. \n\\end{defi}\n\n\\begin{rmq}\n\t\\noindent $(a)$~$(\\ref{radialtails})$ reads $f(y) \\geq M f(x)$ for all $x,y \\! \\in A\\setminus\\{a\\}$\n\n\tfor which $\\|y-a\\|_0 \\leq \\|x-a\\|_0$.\n\t\n\t\n\t\n\t\t\n\t\t\\smallskip\n\t\t\\noindent $(b)$ If $f$ is radial non-increasing on $\\mathbb{R}^d$ w.r.t. $a \\in \\mathbb{R}^d$ with parameter $\\|.\\|_0$,\n\t\n\t\tthen there exists a non-increasing measurable function $g:(0, +\\infty) \\rightarrow \\mathbb{R}_+$ satisfying $f(x) = g(\\|x -a\\|_0 )$ for every $x\\neq a$.\n\t\t\n\t\t\\smallskip\n\t\n\t\t\\noindent $(c)$ From a practical point of view, many classes of distributions satisfy~$(\\ref{radialtails})$, e.g. the $d$-dimensional normal distribution $\\mathcal{N}(m,\\sigma_d)$ for which one considers $h(y)=\\frac{1}{(2\\pi)^{\\frac d2}\\mbox{\\rm det}(\\sigma_d)^{\\frac12}}e^{-\\frac{y^2}{2}}$ and density $f(x)= h(\\|x-m\\|_0)$ where $\\|x\\|_0=\\|\\sigma_d^{-\\frac12}x\\|$, and the family of distributions defined by $f(x) \\propto \\|x\\|^c e^{-a\\|x\\|^b}$, for every $x \\in \\mathbb{R}^d, a,b >0$ and $c>-d$, for which one considers $h(u)=u^ce^{-au^b}$. In the one dimensional case, we can mention the Gamma distribution, the Weibull distributions, the Pareto distributions and the log-normal distributions.\n\\end{rmq}\n\\iffalse\n\\begin{exemple}\nFrom a practical point of view, many classes of distributions satisfy~$(\\ref{radialtails})$, e.g. the $d$-dimensional normal distribution $\\mathcal{N}(m,\\sigma_d)$ for which one considers $h(y)=\\frac{1}{(2\\pi)^{\\frac d2}\\mbox{det}(\\sigma_d)^{\\frac12}}e^{-\\frac{y^2}{2}}$ and $\\|x\\|_0=\\|\\sigma_d^{-\\frac12}(x-m)\\|$, and the family of distributions defined by $f(x) \\propto \\|x\\|^c e^{-a\\|x\\|^b}$, for every $x \\in \\mathbb{R}^d, a,b >0$ and $c>-d$, for which one considers $h(u)=u^ce^{-au^b}$.\nIn the one dimensional case, we can mention the Gamma distribution, the Weibull distributions, the Pareto distributions and the log-normal distributions.\n\\end{exemple}\n\\fi\n\\begin{thm}\n\t\\label{zadorpierce}\n\t\tAssume $P=h. \\lambda_d$ with $h \\in L^{\\frac{d}{d+r}}(\\lambda_d)$ and  $\\int_{\\mathbb{R}^d} \\|x\\|^r dP(x) <+\\infty$. Let $a_1$ denote the $L^r$-median of $P$. Assume that $\\mbox{supp}(P) \\subset A$ and $a_1 \\in A$ for some $A$ star-shaped and peakless with respect to $a_1$ in the sense that\n\t\t\\begin{equation}\n\t\t\\label{peakless}\n\t\t\\mathfrak{p}(A,\\|.-a_1\\|) := \\inf \\left\\{ \\frac{\\lambda_d(B(x,t)\\cap A)}{\\lambda_d(B(x,t))}; x \\in A,0<t\\le  \\|x-a_1\\| \\right\\} >0.\n\t\t\\end{equation}\n\t\tAssume $h$ is almost radial non-increasing on $A$ with respect to $a_1$ in the sense of~$(\\ref{radialtails})$.\n\t\n\t\tThen,\n\t\n\t\t$$ \n\t\t\\forall\\, n\\ge 2,\\quad e_r(a^{(n)},P) \\leq \\kappa_{d,r,M,C_0,\\mathfrak{p}(A,\\|.-a_1\\|)}^{\\text{G,Z,P}} \\;\\|h\\|_{L^{\\frac{d}{d+r}}(\\lambda_d)}^{\\frac{1}{r}} (n-1)^{-\\frac{1}{d}},\n\t\t$$\t\n\t\twhere\n\t\t$\\kappa_{d,r,M,C_0,\\mathfrak{p}(A,\\|.-a_1\\|)}^{\\text{G,Z,P}} \\leq  \\frac{2C_0^2\\,r^{\\frac 1d}\\,}{d^{\\frac1d}M^{d+r}V_d^{\\frac1d}\\mathfrak{p}(A,\\|.-a_1\\|)^{\\frac1d}} \\min_{\\varepsilon \\in (0,\\frac13)} \\varphi_r(\\varepsilon)^{-\\frac 1d}.$\n\\end{thm}\n\\begin{rmq}\\label{rk:2.9}\n$(a)$ If $A=\\mathbb{R}^d$, then $\\mathfrak{p}(A,\\|.-a\\|)=1$ for every $a \\in \\mathbb{R}^d$.\n\n\\smallskip\n\\noindent  $(b)$ The most typical unbounded sets satisfying~$(\\ref{peakless})$ are convex cones that is cones  $K \\subset \\mathbb{R}^d$  of vertex $0$ with $0 \\in K$ ($K\\neq \\varnothing$) and such that $\\lambda x \\in K$ for every $x \\in K$ and $\\lambda \\geq 0$. For such convex cones $K$  with $\\lambda_d(K) >0$, we even have that the lower bound \n\t\t$$\n\t\t\\mathfrak{p}(K) := \\inf \\left\\{\\frac{\\lambda_d(B(x,t)\\cap K)}{\\lambda_d(B(x,t))}; x \\!\\in K,\\,t>0 \\right\\} = \\frac{\\lambda_d\\big(B(0,1)\\cap K) \\big)}{V_d}>0.\n\t\t$$\n\t\tThus if $K= \\mathbb{R}_+^d$, then $\\mathfrak{p}(K)= 2^{-d}$.\n\\end{rmq}\n\\noindent The proof of theorem~\\ref{zadorpierce} is based on the following lemma.\n\\begin{lem}\n\t\\label{lem2}\n\tLet $\\nu=f . \\lambda_d$ be a probability measure on $\\mathbb{R}^d$ where $f$ is almost radial non-increasing \n\n\ton $A\\!\\in \\mathcal{B}(\\mathbb{R}^d)$ w.r.t. $a_1\\!\\in A$, $A$ being star-shaped relative to $a_1$ and  satisfying~$(\\ref{peakless})$.\n\n\tThen, for every $x\\! \\in A$ and   positive $ t\\!\\in (0,  \\|x-a_1\\|]$,\n\t$$\\nu(B(x,t))\\geq M \\mathfrak{p}(A,\\|.-a_1\\|)(2C_0^2)^{-d} V_d f(x) t^d$$\n\twhere $C_0 \\in [1,+\\infty)$ satisfies, for every $x \\in \\mathbb{R}^d$, $\\ds \\frac{1}{C_0}\\|x\\|_0 \\leq \\|x\\| \\leq C_0 \\|x\\|_0$.\n\n\\end{lem}\n\\begin{proof}\n\tFor every $x \\in A$ and $t > 0$, \n\t$$\\nu(B(x,t))=\\int_{B(x,t)} fd\\lambda_d \\geq \\int_{B(x,t)\\cap A \\cap \\{f \\geq Mf(x)\\} } fd\\lambda_d \\;\\geq \\; Mf(x)\\lambda_d\\big( B(x,t)\\cap A \\cap \\{f \\geq Mf(x)\\} \\big)$$\n\n\tand\n\t$$B(x,t)\\cap (A\\setminus\\{a_1\\}) \\cap B_{\\|.\\|_0}(a_1,\\|x-a_1\\|_0) \\subset B(x,t)\\cap A \\cap \\{f \\geq Mf(x)\\}.$$\n\tNow, assume $0<t \\le \\|x-a_1\\|\\le C_0\\|x-1\\|_0$. Setting $x':=\\left(1-\\frac{t}{2C_0 \\|x-a_1\\|_0} \\right)x+\\frac{t}{2C_0\\|x-a_1\\|_0}a_1 \\in A$ (since $A$ is star-shaped with respect to $a_1$), we notice that, for $y \\in B\\left(x',\\frac{t}{2C_0^2} \\right) \\subset B_{\\|.\\|_0}\\left(x',\\frac{t}{2C_0} \\right)$, \n\t$$\n\t\\|y-x\\| \\leq \\|y-x'\\|+C_0\\|x'-x\\|_0 \\leq \\frac{t}{2C_0^2}+C_0 \\left\\|\\frac{t}{2C_0\\|x-a_1\\|_0}(x-a_1) \\right\\|_0 =\\frac{t}{2C_0^2}+\\frac{t}{2} \\leq t\n\t$$\n\tand\n\t$$\n\t\\|y-a_1\\|_0  \\leq \\|y-x'\\|_0+\\|x'-a_1\\|_0 \\leq \\frac{t}{2C_0}+ \\left\\|\\left(1-\\frac{t}{2C_0\\|x-a_1\\|_0}\\right)(x-a_1) \\right\\|_0 =\\|x-a_1\\|_0,\n\t$$\n\tso that,\n\t$B\\left(x',\\frac{t}{2C_0^2} \\right) \\subset B(x,t) \\cap B_{\\|.\\|_0}(a_1,\\|x-a_1\\|_0).$ \n\tConsequently,\n\t$$\\nu(B(x,t)) \\geq Mf(x) \\lambda_d \\left(B\\left(x',\\frac{t}{2C_0^2} \\right)\\cap A \\right).$$\n\tMoreover, \n\n\t$\\frac{t}{2C_0^2}\\leq \\frac{t}{2} \\leq \\frac{1}{2} \\|x-a_1\\| \\leq \\|x'-a_1\\|.$ Hence, we have \n\t$$\\lambda_d \\left(B\\left(x',\\frac{t}{2C_0^2} \\right)\\cap A \\right) \\geq \\mathfrak{p}(A,\\|.-a_1\\|) \\lambda_d \\left(B\\left(x',\\frac{t}{2C_0^2} \\right) \\right) =\\mathfrak{p}(A,\\|.-a_1\\|)(2C_0^2)^{-d}t^d \\lambda_d(B(0,1)).$$ \n\t\\hfill $\\square$\n\\end{proof}\n\\begin{refproof}\n\tConsider \n\t$$ \n\t\\nu=h_r .\\lambda_d:=\\frac{h^{\\frac{d}{d+r}}}{\\int h^{\\frac{d}{d+r}}d\\lambda_d}.\\lambda_d.\n\t$$ \n\tNotice that $h_r$ is alsmost radial non-increasing on $A$ w.r.t. $a_1$ with parameter $M^{\\frac{d}{d+r}}$ so that Lemma~\\ref{lem2} yields for every $x\\!\\in A$ and $t\\!\\in [0, \\|x-a_1\\|]$\n\t\\[\n\t\\nu\\big( B(x,t)\\big)\\ge M^{\\frac{d}{d+r}}\\mathfrak{p}(A,\\|\\cdot-a_1\\|) (2C_0^2)^{-d}V_d h_r(x)t^d.\n\t\\]\n\tConsequently, using that \n\t\\[\n\t\\int_{\\mathbb{R}^d} h_r^{-\\frac rd} dP = \\|h\\|_{L^{\\frac{d}{d+r}}(\\lambda_d)},\n\t\\]\n\tthe assertion follows from Proposition~\\ref{mainthm}.\n\t\\end{refproof}\n\\begin{rmq}\n\tNote that, by applying H\\\"older inequality with the conjugate exponents $p=1+\\frac rd$ and $q =1+\\frac dr$, one has\n\t$$\n\t\\int_{\\mathbb{R}^d} h(\\xi)^{\\frac{d}{d+r}}d\\xi \\leq \\left(\\int_{\\mathbb{R}^d} h(\\xi) (1 \\vee |\\xi|)^{r+\\delta}d\\xi \\right)^{\\frac{d}{d+r}}\\left( \\int_{\\mathbb{R}^d} \\frac{d\\xi}{(1 \\vee |\\xi|)^{d(1+\\frac{\\delta}{r})}}\\right)^{\\frac{r}{d+r}}.\n\t$$\n\tConsequently, since $\\ds \\int_{\\mathbb{R}^d} \\frac{d\\xi}{(1 \\vee |\\xi|)^{d(1+\\frac{\\delta}{r})}} < +\\infty$, one deduces that \n\t$\\|h\\|_{\\frac{d}{d+r}}^{\\frac 1r} \\asymp \\sigma_{r+\\delta}^{1+\\frac{\\delta}{r}}.$\n\\end{rmq}\n\\noindent We note that Zador theorem implies \n$\\liminf_n n^{\\frac{1}{d}}e_r(a^{(n)},P) \\geq \\liminf_n n^{\\frac{1}{d}} e_{r,n}(P,\\mathbb{R}^d)\\geq Q_r(P)^{\\frac{1}{r}}.$ \nThe next proposition may appear as a refinement of Pierce's Lemma and Theorem~$\\ref{zadorpierce}$ in the sense that it gives a lower convergence rate for the discrete derivative of the quantization error, that is its increment.\n\\begin{prop}\n\tAssume $\\int_{\\mathbb{R}^d} \\|x\\|^r dP(x) <+\\infty$. Then, \n \t$$\\liminf_n n^{1+\\frac{r}{d}}\\min_{1\\leq i \\leq n} \\left(e_r(a^{(i)},P)^r-e_r(a^{(i+1)},P)^r \\right) >0.$$\n\\end{prop}\n\\begin{proof}\n\tWe start by choosing $N>0$ such that $P(B(0,N))>0$. Proposition~$\\ref{inc}$ yields, for every probability measure $\\nu $ on $\\mathbb{R}^d$, for every $n \\geq n_0$ and $c \\in \\left(0,\\frac{1}{2}\\right)$, \n\t\\begin{align*}\n\te_r(a^{(n)},P)^r & - e_r(a^{(n+1)} ,P)^r \\\\\n\t& \\geq \\frac{(1-c)^r-c^r}{(c+1)^r} \\int_{B(0,N)\\cap supp(P)} \\nu \\left(B\\left(x, \\frac{c}{c+1}d\\left(x,a^{(n)}\\right)  \\right) \\right) d\\left(x,a^{(n)}\\right)^r dP.\n\t\\end{align*}\n\tWe choose $\\nu=\\mathcal{U}(B(0,N))$. Then, for every $x \\in B(0,N), \\, t \\leq N$ and $x'=\\left( 1-\\frac{t}{2N}\\right)x$, one has \n\t$B\\left( x',\\frac{t}{2}\\right) \\subset B(x,t) \\cap B(0,N)$\n\tsince, for every $y \\in B\\left( x',\\frac{t}{2}\\right)$,\n\t$$\\|y-x\\|\\leq \\|y-x'\\|+\\|x'-x\\| \\leq \\frac{t}{2}+\\frac{t}{2N}\\|x\\| \\leq t$$\n\tand $$\\|y\\|\\leq \\|y-x'\\|+\\|x'\\|\\leq \\frac{t}{2}+\\left( 1-\\frac{t}{2N}\\right)\\|x\\|\\leq \\frac{t}{2}+\\left( 1-\\frac{t}{2N}\\right)N\\leq N.$$\n\tConsequently, $$\\nu(B(x,t)) \\geq \\frac{\\lambda_d(B(x',\\frac{t}{2}))}{\\lambda_d(B(0,N))}=(2N)^{-d}t^d.$$\n\tMoreover, we denote $C:=\\sup_{n \\geq 1} \\max_{x \\in B(0,N) \\cap supp(P)} d(x,a^{(n)})$ which is finite because $a^{(n)} \\in \\overline{conv}(supp(P))$.\n\tConsequently, for every $n \\geq n_0$ and every $c \\in \\left(0,\\frac{1}{2} \\right)$ such that $\\ds \\frac{c}{c+1}C \\leq N$,\n\t\\begin{align*}\n\te_r(a^{(n)},P)^r  - e_r(a^{(n+1)} ,P)^r &\\geq  \\frac{(1-c)^r-c^r}{(c+1)^r} \\left(\\frac{c}{c+1}\\right)^d (2N)^{-d} \\int_{B(0,N)} d(x,a^{(n)})^{d+r}dP(x)\\\\\n\t& \\geq \\frac{(1-c)^r-c^r}{(c+1)^r} \\left(\\frac{c}{c+1}\\right)^d (2N)^{-d} P(B(0,N)) e_{d+r}^{d+r}(a^{(n)},P(.|B(0,N))).\n\t\\end{align*}  \n\tNow, using that $\\left(e_{d+r}^{d+r}\\left(a^{(n)},P(.|B(0,N))\\right)\\right)_{n \\geq 1}$ is nonincreasing and relying on Zador's theorem, we deduce\n\t$$\\liminf_n n^{1+\\frac{r}{d}}\\min_{1\\leq i \\leq n} \\left(e_r(a^{(i)},P)^r-e_r(a^{(i+1)},P)^r \\right)>0.$$\n\t\\hfill $\\square$\n\\end{proof}\n\\begin{rmq}\n\tFor every $m,n \\in \\mathbb{N}$, if we denote $W_b(a^{(n)})$ the Vorono\\\"i cell associated to the sequence $a^{(n)}$ of centroid $b \\in a^{(n)}$ and use the fact that $e_r(a^{(n+1)},X) \\leq e_r(a^{(n)}\\cup \\{b\\},X)$ for every $b \\in \\mathbb{R}^d$, we deduce\n\t\\begin{align*}\n\te_r(a^{(n)},X)^r-e_r(a^{(n+m)},X)^r =& \\sum_{b \\in a^{(n)}} \\int_{W_b(a^{(n+m)})} \\left( d(x,a^{(n)})^r-\\|x-b\\|^r \\right) dP \\\\\n\t&+\\sum_{b \\in a^{(n+m)}\\setminus a^{(n)}} \\int_{W_b(a^{(n+m)})} \\left( d(x,a^{(n)})^r-\\|x-b\\|^r \\right) dP\\\\\n\t& = \\sum_{b \\in a^{(n+m)}\\setminus a^{(n)}} \\int_{W_b(a^{(n+m)})} \\left( d(x,a^{(n)})^r-d(x,a^{(n)} \\cup \\{b\\})^r \\right) dP \\\\\n\n\t& \\leq m \\, e_r(a^{(n)},X)^r-e_r(a^{(n+1)},X)^r.\n\t\\end{align*}\nConsequently, considering $n=i$ and knowing that $l \\rightarrow e_r(a^{(l)},X)$ is non-increasing, one has\n\t$$\\min_{1 \\leq i \\leq n}\\left(e_r(a^{(i)},X)^r-e_r(a^{(i+1)},X)^r \\right) \\geq \\frac{1}{m} \\left(e_r(a^{(n)},X)^r-e_r(a^{(m)},X)^r \\right).$$\t\n\\end{rmq}\n\\section{Distortion mismatch}\n\\label{distortionmismatch}\nWe address now the problem of distortion mismatch, i.e. the property that the rate optimal decay property of $L^r$-quantizers remains true for $L^s(P)$-quantization error for $s \\in (0, +\\infty)$. This problem was originally investigated in \\cite{mismatch08} for optimal quantizers. If $s \\leq r$, the monotonicity of the $L^s$-norm as a function of $s$ ensures that any $L^r$-optimal greedy sequence remains $L^s$-rate optimal for  the $L^s$-norm. The challenge is when $s$ is larger than $r$. The problem is solved in \\cite{LuPa15} for $s \\in (0,+\\infty)$ relying on an integrability assumption of the $b$-maximal function $\\Psi_b$. However, we give an additional nonasymptotic result for $s \\in (r,d+r)$ in the following theorem, in the same settings as for Theorem~$\\ref{mainthm}$, considering auxiliary probability distributions $\\nu$ satisfying~$(\\ref{criterenu})$.\n\\begin{thm}\n\t\\label{mismatch}\n\t Let $P$ be such that $\\int_{\\mathbb{R}^d} \\|x\\|^rdP(x)<+\\infty$. Let $s \\in (r,d+r)$.\n\t\n\t Let $(a_n)$ be an $L^r$-optimal greedy sequence for $P$. For any distribution $\\nu$ and Borel function $g_{\\varepsilon} : \\mathbb{R}^d\\rightarrow \\mathbb{R}_+$, $\\varepsilon \\in (0,\\frac13)$, satisfying~$(\\ref{criterenu})$, for every $n \\geq 3$,\n\t\\begin{equation*}\n\te_s\\big(a^{(n)},P\\big) \\leq \\kappa_{d,r,\\varepsilon}^{\\text{Greedy}} \\left(\\int g_{\\varepsilon}^{-\\frac{s}{d+r-s}}dP \\right)^{\\frac{d+r-s}{s(d+r)}}\\left(\\int g_{\\varepsilon}^{-\\frac{r}{d}} dP \\right)^{\\frac{1}{d+r}}(n-2)^{-\\frac{1}{d}}\n\t\\end{equation*}\n\twhere\n\t$\n\t\\kappa_{d,r,\\varepsilon}^{\\text{Greedy}} = 2^{\\frac{1}{d}}\\,\\,\\left(\\frac rd \\right)^{\\frac{r}{d(d+r)}}V_d^{-\\frac1d} \\varphi_r(\\varepsilon)^{-\\frac1d}.$\n\\end{thm} \n\\begin{proof}\n\tWe assume $\\tfrac{1}{g_\\ve} \\in L^{\\tfrac{s}{d+r-s}}(P)$ so that $\\tfrac{1}{g_\\ve}\\in L^{\\frac{r}{d}}(P)$ since $\\ds \\tfrac{s}{d+r-s} \\geq \\tfrac{s}{d} \\geq \\tfrac{r}{d}$. Inequality~$(\\ref{equ1})$ from the proof of Proposition~$\\ref{mainthm}$ still holds, i.e.\n\t\\begin{equation*}\n\te_r(a^{(n)},P)^r  - e_r(a^{(n+1)} ,P)^r \n\t\\geq C \\int g_{\\varepsilon}(x) d\\left(x,a^{(n)}\\right)^{d+r} dP(x). \n\t\\end{equation*}\n\twith, for every $c\\!\\in (0, \\tfrac{\\ve}{1-\\ve}]\\cap (0,1\/2)$, $C=V_d\\, \\varphi_r\\left(\\frac{c}{c+1}\\right)$ where $\\varphi_r(u)=\\left(\\tfrac {1}{3^r} -u^r\\right)u^d$. The reverse H\\\"older inequality applied with $p=\\frac{s}{d+r} \\in (0,1)$ and $q =-\\frac{s}{d+r-s} \\in (-\\infty,0)$ yields that\n\n\n\t$$e_r(a^{(n)},P)^r - e_r(a^{(n+1)} ,P)^r \\geq C_1 e_s\\big(a^{(n)},P\\big)^{d+r}$$\n\twhere\n\t$C_1=C\\; \\big(\\int g_{\\varepsilon}^{-\\frac{s}{d+r-s}}dP \\big)^{-\\frac{d+r-s}{s}}.$ Hence, knowing that $k \\mapsto e_s\\big(a^{(k)},P\\big)$ is non-increasing and summing between $n$ and $2n-1$, we obtain for $n\\geq 1$\n\t\\begin{align*}\n\tn\\, e_s(a^{(2n-1)},P)^{d+r} \\leq \\sum_{k=n}^{2n-1} e_s\\big(a^{(k)},P\\big)^{d+r} \\leq \\frac{1}{C_1} \\sum_{k=n}^{2n-1} e_r(a^{(k)},P)^r - e_r(a^{(k+1)} ,P)^r \\leq \\frac{1}{C_1} e_r(a^{(n)},P)^r.\t\n\t\\end{align*}\n\tFinally, since $ 2 \\left\\lceil \\frac{n}{2} \\right\\rceil -1 \\leq n$, we have $e_s\\big(a^{(n)},P\\big)\\leq  e_s\\left(a^{2 \\left\\lceil \\frac{n}{2} \\right\\rceil -1},P\\right)$ and we derive that \n\t$$\n\t\\frac{n}{2} e_s\\big(a^{(n)},P\\big)^{d+r} \\leq \\left\\lceil \\frac{n}{2} \\right\\rceil e_s\\big(a^{(n)},P\\big)^{d+r} \\leq  \\left\\lceil \\frac{n}{2} \\right\\rceil e_s\\left(a^{2 \\left\\lceil \\frac{n}{2} \\right\\rceil -1},P\\right)^{d+r} \\leq \\frac{1}{C_1}e_r\\left(a^{ \\left\\lceil \\frac{n}{2} \\right\\rceil },P\\right)^{r}.\n\t$$\n\tConsequently, plugging in $C_1$,\n\t\\begin{align*}\n\te_s\\big(a^{(n)},P\\big)  &\\leq  \\left(\\frac{2}{C_1}\\right)^{\\frac{1}{d+r}} n^{-\\frac{1}{d+r}} e_r\\left(a^{ \\left\\lceil \\frac{n}{2} \\right\\rceil },P\\right)^{\\frac{r}{d+r}}\\\\\n\t & =  2^{\\frac{1}{d+r}} V_d^{-\\frac{1}{d+r}} \\varphi_r\\left(\\frac{c}{c+1}\\right)^{-\\frac{1}{d+r}} \\left(\\int g_{\\varepsilon}^{-\\frac{s}{d+r-s}}dP \\right)^{\\frac{d+r-s}{s(d+r)}}\tn^{-\\frac{1}{d+r}} e_r\\left(a^{ \\left\\lceil \\frac{n}{2} \\right\\rceil },P\\right)^{\\frac{r}{d+r}}.\n\t\\end{align*} \n\tConsequently, one can deduce from Proposition~$\\ref{mainthm}$, for $n\\ge 3$,\n\t\\begin{align*}\n\te_s\\big(a^{(n)},P\\big)  \\leq &  2^{\\frac{1}{d}} V_d^{-\\frac{1}{d}} \\left(\\frac rd\\right)^{\\frac{r}{d(d+r)}} \\left(\\int g_{\\varepsilon}^{-\\frac{s}{d+r-s}}dP \\right)^{\\frac{d+r-s}{s(d+r)}}\\left(\\int g_{\\varepsilon}^{-\\frac rd}dP \\right)^{\\frac{1}{d+r}} \\varphi_r\\left(\\frac{c}{c+1}\\right)^{-\\frac{1}{d}} (n-2)^{-\\frac1d}.\n\n\t\\end{align*} \n\tHence, the result is owed to the fact that $\\varphi_r\\left(\\frac{c}{c+1}\\right) \\leq \\varphi_r\\left(\\varepsilon\\right)$ for $c \\in (0,\\frac{\\varepsilon}{1-\\varepsilon}]$.\n\t\\hfill $\\square$\n\\end{proof}\n\\begin{cor}\n\tLet $s \\in (r,d+r)$. Assume , for $\\delta >0$, \n\t\\begin{equation}\n\t\\label{assumption}\n\t\\int \\|x\\|^{\\frac{ds}{d+r-s}} (\\log^+\\! \\|x\\|)^{\\frac{s}{d+r-s}+\\delta} dP(x) < +\\infty\n\t\\end{equation}\n\tthen\n\t$$\n\t\\lim \\sup_n n^{\\frac{1}{d}}\\sup \\big\\{e_s\\big(a^{(n)},P\\big): (a_n) L^r\\mbox{-optimal greedy sequence for } P \\big\\} < +\\infty.\n\t$$\n\\end{cor}\n\\begin{proof} The proof is divided in two steps.\n\n\\smallskip\n\\noindent {\\sc Step 1:} Let $\\delta>0$ be fixed and $\\beta=1+\\frac{(d+r-s)\\delta}{s}$. Just as in the proof of Theorem~\\ref{Piercetype}(b), we set $\\nu(dx)=\\gamma_{r,\\delta}(x) \\lambda_d(dx)$ where  \n\t$$\\gamma_{r,\\delta}(x)=\\frac{K_{\\delta,r}}{(1 \\vee \\|x-a_1\\|)^{d}\\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\beta}}, \\mbox{ with }   K_{\\delta,r}= \\left(\\int \\frac{dx}{(1 \\vee \\|x-a_1\\|)^{d}\\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\beta}} \\right)^{-1},$$\n\tis a probability density with respect to the Lebesgue measure on $\\mathbb{R}^d$. \n\tThe density $\\gamma_{r,\\delta}$ is radial non-increasing on the whole $\\mathbb{R}^d$ w.r.t. $a_1$ (and $\\|\\cdot\\|_0= \\|\\cdot\\|$) so that $\\mathfrak{p}(\\|\\cdot-a_1\\|)=1$ by Remark~\\ref{rk:2.9}$(a)$ and, in turn, Lemma~\\ref{lem2} yields for every $x\\!\\in \\mathbb{R}^d$ and $t\\le \\|x-a_1\\|$\n\t\\[\n\t\\nu\\big( B(x,t)  \\big)\\ge 2^{-d} V_d \\gamma_{r, \\delta}((x) t^d.\n\t\\]\n\tConsequently, Theorem~$\\ref{mismatch}$ yields, for $n\\ge 3$,\n\t\\begin{align*}\n\te_s\\big(a^{(n)},P\\big) \\leq & C_{d,r,\\delta} \\left( \\int \\left(1 \\vee \\|x-a_1\\| \\right)^{r} \\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\beta \\frac rd}dP(x)\\right)^{\\frac {1}{d+r}} \\\\\n\t& \\times \\left( \\int \\left(1 \\vee \\|x-a_1\\| \\right)^{\\frac{sd}{d+r-s}} \\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\delta+\\frac{ s}{d+r-s}}dP(x)\\right)^{\\frac {d+r-s}{s(d+r)}} (n-2)^{-\\frac 1d}\n\t\\end{align*}\n\twhere \n\t$\n\tC_{d,r,\\delta}  \\leq 2^{1+\\frac 1d} V_d^{-\\frac1d} \\left( \\frac rd \\right)^{\\frac{r}{d(d+r)}}K_{\\delta,r}^{-\\frac 1d} \\min_{\\varepsilon \\in (0,\\frac 13)} (1+\\varepsilon)^d \\varepsilon^{\\frac{\\beta}{d}} \\varphi_r(\\varepsilon)^{-\\frac 1d}.\n\t$\\\\\n\t\\textsc{Step 2:}\n\tJust as in the proof of Theorem~\\ref{Piercetype}(b), we have \n\t$$\\left(1 \\vee \\|x-a_1\\| \\right)^{r} \\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\beta \\frac rd}  \\leq 2^{(r-1)_++(\\beta \\frac rd-1)_+} \\left( \\|x\\|^r \\log_+\\|x\\|^{\\beta \\frac rd}+A_1\\|x\\|^r+A_2 \\right)$$\n\tand\n\t\\begin{align*}\n\t\\left(1 \\vee \\|x-a_1\\| \\right)^{\\frac{sd}{d+r-s}} \\left(\\log(\\|x-a_1\\| \\vee e) \\right)^{\\delta+\\frac{ s}{d+r-s}} \\leq &2^{(\\frac{ds}{d+r-s}-1)_++(\\delta+ \\frac{s}{d+r-s}-1)_+} \\\\\n\t& \\times \\left( \\|x\\|^{\\frac{ds}{d+r-s}} \\log_+\\|x\\|^{\\delta+\\frac{s}{d+r-s}}+B_1\\|x\\|^r+B_2 \\right)\n\t\\end{align*}\n\twhere \n\t$A_1,A_2,B_1$ and $B_2$ are constants depending only on $r,d,s,\\delta$ and $a_1$. \n\tSince, $\\frac{s}{d+r-s} \\geq \\frac rd$, one has $\\frac{ds}{d+r-s}>r$ and $\\delta+\\frac{s}{d+r-s}\\geq \\beta \\frac rd$, so that the two above quantities are finite (by assumption~$(\\ref{assumption})$).\n\tThe result is deduced from the fact that $\\sup \\big\\{\\|a_1\\|: a_1\\! \\in {\\rm argmin}_{\\xi\\in \\mathbb{R}^d} e_r(\\{\\xi\\}, P) \\big\\} < \\infty$. \\hfill $\\square$\n\\end{proof}\n\\section{Algorithmics}\n\\label{algorithmics}\nAn important application of quantization is numerical integration. Let us consider the quadratic case $r=2$ and an $L^2$-optimal greedy quantization sequence $a^{(n)}$ for a random variable $X$ with distribution $\\mathbb{P}_X=P$. Since we know that $e_2(a^{(n)},X)=\\|X-a^{(n)}\\|_2$ converges to $0$ when $n$ goes to infinity, this means that $a^{(n)}$ converges towards $X$ in $L^2$ and hence in distribution. So, denoting $\\big(W_i(a^{(n)})\\big)_{1 \\leq i \\leq n}$ the Vorono\\\"i diagram corresponding to $a^{(n)}$, one can approximate $\\mathbb{E}[f(X)]$, for every continuous function $f:\\mathbb{R}^d \\rightarrow \\mathbb{R}$, by the following cubature formula\n\\begin{equation}\n\\label{quadratureformula}\nI(f):=\\mathbb{E}[f(X)]\\approx \\sum_{i=1}^{n}p_i^n \\, f(a_i^{(n)})\n\\end{equation}\nwhere, for every $i \\in \\{1,\\ldots,n\\}$, $p_i^n=P\\big(X \\in W_i(a^{(n)}) \\big)$ represents the weight of the $i^{th}$ Vorono\\\"i cell corresponding to the greedy quantization sequence $a^{(n)}=\\{a_1^{(n)},\\ldots,a_n^{(n)}\\}$. When the function $f$ satisfies certain regularities, one can establish error bounds for this quantization-based cubature formula, we refer to \\cite{Pages18} for more details.\nFor example, if $f$ is $[f]_{\\text{Lip}}$-Lipschitz continuous, one has \n$$\\left|\\sum_{i=1}^{n}p_i^n f(a_i^{(n)})-\\mathbb{E}[f(X)]\\right| \\leq [f]_{\\text{Lip}} \\; e_r(a^{(n)},X).$$\nso one can approximate $\\mathbb{E}[f(X)]$ with an $\\mathcal{O}(n^{-\\frac 1d})$ rate of convergence.\n\nWhen working on the unit cube $[0, 1]^d$, it is natural to compare an optimal greedy sequence of the uniform distribution $\\mathcal{U}([0,1]^d)$ and a uniformly distributed sequence with low discrepancy used in the quasi-Monte Carlo method (QMC). A $[0,1]^d$-valued sequence $\\xi=(\\xi_n)_{n \\geq 1}$ is uniformly distributed if $ \\mu_n=\\frac1n \\sum_{k=1}^n\\delta_{\\xi_k}$ converges weakly to $\\lambda_{d_{ \\mid [0,1]^d }}$ (where $\\lambda_d$ denotes the Lebesgue measure on $(\\mathbb{R}^d,\\mathcal{B}(\\mathbb{R}^d))$). It is well known (see \\cite{KuiNied} for example) that $(\\xi_n)_{n \\geq 1}$ is uniformly distributed if and only if \n$$D_n^*(\\xi)=\\sup_{u \\in [0,1]^d} \\left|\\frac{1}{n} \\sum_{i=1}^{n} \\mathds{1}_{\\xi_i \\in [0,u]^d} -\\lambda_d([0,u]^d) \\right| \\; \\rightarrow \\; 0 \\quad \\mbox{ as } \\quad n \\rightarrow +\\infty.$$\nThe above modulus is known as the star-discrepancy of $\\xi$ at order $n$ and can be defined, for fixed $n \\in \\mathbb{N}$, for any $n$-tuple $(\\xi_1,\\ldots,\\xi_n)$ whose components $\\xi_k$ lie in $[0,1]^d$. There exists many sequences (Halton, Kakutani, Faure, Niederreiter, Sobol', see \\cite{BoLe94,Pages18} for example) achieving a $\\mathcal{O}\\left(\\frac{(\\log n)^d}{n} \\right)$ rate of decay for their star-discrepancy and it is a commonly shared conjecture that this rate is optimal, such sequences are called sequences with low discrepancy.  \nBy a standard so-called Hammersley argument, one shows that if a $[0,1]^{d-1}$-valued sequence $\\zeta=(\\zeta_n)_{n \\geq 1}$ has low discrepancy i.e. there exists a real constant $C(\\zeta) \\in (0,+\\infty)$ such that \n$D_n^*(\\zeta) \\leq C(\\zeta)\\frac{(\\log n)^{d-1}}{n}$, for every $n \\geq 1$,\nthen, for every $n \\geq 1$, the $[0,1]^d$-valued $n$-tuple $\\left((\\zeta_k,\\frac{k}{n}) \\right)_{1\\leq k \\leq n}$ satisfies \n$$D_n^*\\left( \\left((\\zeta_k,\\frac{k}{n}) \\right)_{1\\leq k \\leq n}\\right) \\leq C(\\zeta)\\frac{(\\log n)^{d-1}}{n}.$$\nThe QMC method finds its gain in the following error bound for numerical integration. Let $(\\xi_1,\\ldots,\\xi_n)$ be a fixed $n$-tuple in $([0,1]^d)^n$, then, for every $f:[0,1]^d \\rightarrow \\mathbb{R}$ with finite variation (in the Hardy and Krause sense, see \\cite{Nied93} or in the measure sense see \\cite{BoLe94, Pages18}), \n\\begin{equation}\n\\label{koksmahlawka}\n\\left|\\frac1n \\sum_{i=1}^n f(\\xi_k) -\\int_{[0,1]^d}f(u)du \\right| \\leq V(f) D_n^*(\\xi_1,\\ldots,\\xi_n).\n\\end{equation}\nwhere $V(f)$ denotes the (finite) variation of $f$. So, for this class of functions, an $\\mathcal{O}\\left(\\frac{(\\log n)^{d-1}}{n} \\right)$ or $\\mathcal{O}\\left(\\frac{(\\log n)^{d}}{n} \\right)$ rate of convergence can be achieved depending on the composition of the sequence. However, the class of functions with finite variation becomes sparser in the space of functions defined from $[0,1]^d$ to $\\mathbb{R}$ and it seems natural to evaluate the performance of the low-discrepancy sequences or $n$-tuples on a more natural space of test functions like the Lipschitz functions. This is the purpose of Pro\\\"inov's theorem reproduced below.\n\\begin{thm}(Proinov, see \\cite{Proinov88})\n\t\\label{proinov}\n\tLet $(\\mathbb{R}^d,\\|.\\|_{\\infty})$. Let $\\xi=(\\xi_1, \\ldots, \\xi_n)$ a sequence of $[0,1]^d$. For every continuous function $f:[0,1]^d\\rightarrow (\\mathbb{R},|.|_{\\infty})$, we define the uniform continuity modulus of $f$ by \n\t$w(f, \\delta)=\\sup_{\\xi, \\xi^{'} \\in [0,1]^d, |\\xi -\\xi^{'}|_{\\infty} \\leq \\delta} |f(\\xi)-f(\\xi^{'})|$\n\twhere $|u|_{\\infty}=\\max_{1 \\leq i \\leq d} |u_i|$ if $u=(u_1,\\ldots,u_d)$.\n\tThen, for every $n \\geq 1$,\n\t$$\\ds \\left|  \\frac1n \\sum_{i=1}^{n}f(\\xi_i) - \\int_{[0,1]^d}f(x)dx\\right| \\leq C_d \\, w(f, D_n^*(\\xi)^{\\frac{1}{d}}),$$\n\twhere $C_d$ is a constant lower than $4$ and depending only on the dimension $d$. \\\\\n\tIn particular, if $f$ is $[f]_{\\text{Lip}}-$Lipschitz and $\\xi$ has low discrepancy, one has\n\t$$\\ds \\left| \\frac1n \\sum_{i=1}^{n} f(\\xi_i) - \\int_{[0,1]^d}f(x)dx\\right| \\leq C_d \\,[f]_{Lip} D_n^*(\\xi)^{\\frac{1}{d}} \\leq C_d \\,[f]_{Lip} \\frac{\\log\\, n}{n^{\\frac{1}{d}}}.$$\n\\end{thm}\nThis suggests that, at least for a commonly encountered class of regular functions, the curse of dimensionality is more severe with QMC than with quantization due to the extra $(\\log n)^{1-\\frac 1d}$ factor in QMC. This is the price paid by QMC for considering uniform weights $p_i=\\frac1n, i=1,\\ldots,n.$\n\nWith greedy quantization sequences, we will show that it is possible to keep the $n^{-\\frac1d}$ rate of decay for numerical integration but also keep the asset of a sequence which is a recursive formula for cubatures.\\iffalse: if we set $\\mu_n(f)=\\frac1n \\sum_{i=1}^n f(\\xi_k)$, one has \n$$\\mu_{n+1}(f)=\\frac{n}{n+1}\\mu_n(f)+\\frac{f(\\xi_{n+1})}{n+1}=\\mu_n(f)+\\frac{f(\\xi_{n+1})-\\mu_n(f)}{n+1}.$$\nIn fact, the recursive character of greedy quantization induces some improvements to the computation of the weights of the Voronoi cells and the interpoint inertia at each iteration, reducing the cost of the computations and allowing to handle them better in this purpose. \n\\fi\n\\subsection{Optimization of the algorithm and the numerical integration in the $1$-dimensional case}\nQuadratic optimal greedy quantization sequences are obtained by implementing algorithms such as Lloyd's I algorithm, also known as $k$-means algorithm, or the Competitive Learning Vector Quantization (CLVQ) algorithm, which is a stochastic gradient descent algorithm associated to the distortion function. We refer to \\cite{LuPa215} (an extended version of \\cite{LuPa15} on ArXiv) where greedy variants of these procedures are explained in detail. According to Lloyd's algorithm, the construction of the sequences is recursive in the sense that, at the iteration $n$, we add one point $a_n$ to $\\{a_1, \\ldots, a_{n-1}\\}$, and we denote $\\{a_1^{(n)}, \\ldots, a_n^{(n)}\\}$ an increasing reordering of $\\{a_1, \\ldots, a_n\\}$ where the new added point is denoted by $a_{i_0}^{(n)}$.\\\\\nSince the other points are frozen, we can notice that the local inter-point inertia $\\sigma_i^2$  defined by\n\\begin{equation}\n\\sigma_i^2:= \\ds \\int_{a_i^{(n-1)}}^{a_{i+\\frac{1}{2}}^{(n-1)}} |a_i^{(n-1)}-\\xi|^2 P(d\\xi) \\, + \\,  \\int_{a_{i+\\frac{1}{2}}^{(n-1)}}^{a_{i+1}^{(n-1)}} |a_{i+1}^{(n)-1}-\\xi|^2 P(d\\xi), \\quad i=0,\\ldots, n-1\n\\end{equation}\n(where\n$a_0^{(n-1)} =-\\infty$,  \n$a_{n}^{(n-1)}=+\\infty$ and \n$  a_{i+\\frac{1}{2}} ^{(n-1)}= \\frac{a_i^{(n-1)}+ a_{i+1}^{(n-1)}}{2}$ with $a_{\\frac{1}{2}}^{(n-1)}=-\\infty ,  \\quad a_{n-\\frac{1}{2}}^{(n-1)} = +\\infty$)\nremains untouched for every $ i \\in  \\{0, \\ldots, n-1\\}$ except $\\sigma_{i_0}^2$ (the inertia between the point $a_{i_0}^{(n)}$ added at the $n$-th iteration and the following point) and $\\sigma_{i_0-1}^2$ (the inertia between  $a_{i_0}^{(n)}$  and the preceding point). \nThus, at each iteration, the computation of $n$ inertia can be reduced to the computation of only $2$, thereby reducing the cost of the procedure. Likewise, the weights $p_i^n=P(W_i(a^{(n)}))$ of the Vorono\\\"i cells remain mostly unaffected. The only cells that change from one step to another are the cell $W_{i_0}(a^{(n)})$ having for centroid the new point $a_{i_0}^{(n)}$ and the two neighboring cells $W_{i_0-1}(a^{(n)})$ and $W_{i_0+1}(a^{(n)})$. Thus, the online computation of cell weights just needs $3$ calculations instead of $n$ (or $2$ in case the added point is the first or last point in the reordered sequence).\nThe utility of the weights of the Vorono\\\"i cells is featured in the numerical integration allowing to approximate $\\mathbb{E}[f(X)]$ for $f:\\mathbb{R}^d \\rightarrow \\mathbb{R}$ by the quadrature formula~$(\\ref{quadratureformula})$ using the reordered sequence $a^{(n)}$. Thus, based on the fact that only $3$ Vorono\\\"i cells are modified at each iteration, one can deduce an iterative formula for the approximation of $I(f)$ by $I_n(f)$, requiring the storage of only $2$ weights and $2$ indices, as follows\n\\begin{eqnarray}\n\\label{caliteratifd1}\n\\ds I_{n}(f) &=& I_{n-1}(f) - p_{-}^{n}f(a_{i_0-1}^{(n)})-p_{+}^{n}f(a_{i_0+1}^{(n)})+(p_{+}^{n}+p_{-}^{n})f(a_{i_0}^{(n)}) \\nonumber \\\\\n&=& I_{n-1}(f)- p_{-}^{n}(f(a_{i_0-1}^{(n)})-f(a_{i_0}^{(n)}))-p_{+}^{n}(f(a_{i_0+1}^{(n)})-f(a_{i_0}^{(n)})),\n\\end{eqnarray}\nwhere\n\\begin{enumerate}\n\t\\item[$\\bullet$] $\\ds a_{i_0}^{(n)}$ is the point added to the greedy sequence at the $n$-th iteration, in other words, it is the point $a_{n}$, \n\t\\item [$\\bullet$] $\\ds a_{i_0-1}^{(n)}$ and $a_{i_0+1}^{(n)}$ are the points  lower and greater than $ a_{i_0}^{(n)}$, i.e. $ a_{i_0-1}^{(n)}<a_{i_0}^{(n)}<  a_{i_0+1}^{(n)} $,\n\t\\item[$\\bullet$] \\begin{equation}\n\t\\label{poidsinfetsup}\n\tp_{-}^{n} = P\\big(\\big[a_{i_0-\\frac{1}{2}}^{(n)}, a_{\\text{mil}}^{(n)}\\big]\\big) \\qquad \\mbox{and} \\qquad p_{+}^{n} = P\\big(\\big[a_{\\text{mil}}^{(n)},a_{i_0+\\frac{1}{2}}^{(n)}\\big]\\big).\n\t\\end{equation}\n\twhere \n\t$\\ds a_{i_0 \\pm \\frac{1}{2}}^{(n)}=\\frac{a_{i_0}^{(n)}+a_{i_0 \\pm 1}^{(n)}}{2}$  and  $\\ds a_{\\text{mil}}^{(n)}=\\frac{a_{i_0 + 1}^{(n)}+a_{i_0 - 1}^{(n)}}{2}$, with $a_0=-\\infty$ and $a_{n}=+\\infty$.\n\\end{enumerate}\nPractically, this numerical iterative method can be applied without storing the whole ordered greedy quantization sequence nor computing the weights of the Vorono\\\"i cells, which could appear as significant drawbacks for quantization. Instead, it requires the possession of $2$ indices of $2$ particular points of the non-ordered greedy quantization sequence and $2$ weights. In fact, one can start by determining the indices $\\overline{n}$ and $\\underline{n}$ of the points preceding and following $a_{n}$ in the ordered sequence, in other words, the points in the non-ordered sequence corresponding to $a_{i_0-1}^{(n)}$ and $a_{i_0+1}^{(n)}$. Then, we will be able to compute the weights $p_{-}^{n}$ et $p_{+}^{n}$ to finally proceed with the iterative approximation of $I(f)$ according to~$(\\ref{caliteratifd1})$.\n\\subsection{Product greedy quantization ($d>1$)}\n\\label{dimensiond}\nIn higher dimensions, greedy quantization has always the recursive properties, so it gets interesting to apply the same numerical improvements as in the one-dimensional case. However, the construction of multidimensional greedy quantization sequences is complex and expensive since it relies on complicated stochastic optimization algorithms. As an alternative, one can use one-dimensional greedy quantization grids as tools to obtain multidimensional greedy quantization sequences in some cases.\n\\subsubsection{How to build multi-dimensional greedy product grids}\nMultidimensional greedy quantization sequences can be obtained as a result of the tensor product of one-dimensional sequences, when the target law is a tensor product of its independent marginal laws. These grids are, of course, not optimal nor asymptotically optimal but they allow to approach the multidimensional law.\\\\\nLet $X_1,\\ldots, X_d$ be  $d$ independent $L^2$-random variables taking values in $\\mathbb{R}$ with respective distributions $\\mu_1,\\ldots, \\mu_d$ and $a^{1,(n_1)}, \\ldots, a^{d,(n_d)}$ the corresponding greedy quantization sequences. By computing the tensor product of the $d$ one-dimensional greedy sequences of the laws $\\mu_1,\\ldots,\\mu_d$, we obtain the $d$-dimensional greedy quantization grid $a^{1,(n_1)} \\otimes \\ldots \\otimes a^{d,(n_d)}$ of the product law $\\mu=\\mu_1 \\otimes \\ldots \\otimes \\mu_d$, given by $\\big(a_{\\underline{j}}^{(n)}\\big)_{1 \\leq \\underline{j}\\leq n}=\\big(a_{j_1}^{1,(n_1)}, \\ldots, a_{j_d}^{d,(n_d)}\\big)_{1 \\leq j_1 \\leq n_1, \\ldots, 1\\leq j_d \\leq n_d}$ of size $\\ds n=\\prod_{i=1}^{d}n_i$. The corresponding quantization error is given by \n\\begin{equation}\n\\label{proderror}\ne_r(a^{1,(n_1)} \\otimes \\ldots \\otimes a^{d,(n_d)}, X_1 \\otimes \\ldots \\otimes X_d)^r = \\sum_{k=1}^{d}e_r(a^{k,(n_k)},X_k)^r.\n\\end{equation}\nMoreover, the weights $p_{\\underline{j}}^{(n)}$ of the $d$-dimensional Vorono\\\"i cells $\\big(W_{\\underline{j}}\\big(a^{(n)}\\big)\\big)_{1 \\leq \\underline{j} \\leq n}$ can be computed from the weights  $p^{k,n_k} ,\\, k=1,\\ldots,d$, of the Vorono\\\"i cells $\\big(W^{k,n_k}_{i}\\big(a^{k,(n_k)}\\big)\\big)_{1 \\leq i \\leq n_k}$ of each one-dimensional greedy quantization sequence, via\n$$\\ds p_{\\underline{j}}=p^{1,n_1}_{j_1} \\times \\ldots \\times p_{j_d}^{d,n_d} \\qquad \\forall j_k \\in \\{1, \\ldots,n_k\\}, \\forall k \\in \\{1,\\ldots,d\\}, \\, \\forall \\underline{j} \\in \\{1,\\ldots,n\\}.$$\nThe implementation of $d$-dimensional grids is not a point-by-point implementation. In fact, at each iteration $n$, $a^{(n)}$ is obtained from $a^{1,(n_1)},\\ldots, a^{d,(n_d)}$, keeping in mind that $n_1 \\times \\ldots \\times n_d =n$. Having the $d$ one-dimensional sequences, one must add a point to one one-dimensional sequence, generating this way several points of the multidimensional sequence. At this step, we must choose between $d$ possibilities: adding one point to only one sequence $a^{k,(n_k)}$ among the $d$ marginal sequences, obtaining $\\ds a^{(n_1\\times\\ldots\\times n_{k-1}\\times(n_k+1)\\times n_{k+1} \\times \\ldots  \\times n_d)}$. These $d$ cases are not similar since each one produces a different error quantization. So, the implementation is not a random procedure. To make the right decision, one must compute in each case, using~$(\\ref{proderror})$, the quantization error $E_k$ obtained if we add a point to $a^{k,(n_k)}$ for a $k \\in \\{1,\\ldots,d\\}$. In other words, we compute, for $k=1,\\ldots,d$\n$$\nE_k  = e_r(a^{k,(n_k+1)},\\mu_k)^r+\\sum_{l\\in\\{1,\\ldots, d\\}\\setminus \\{k\\}}e_r(a^{l,(n_l)},\\mu_l)^r.\n$$\nThen, one choses the index $i$ such that $\\ds E_{i}=\\min_{1\\leq k \\leq d}E_k$ and, so, one adds a point to the sequence $a^{i,(n_{i})}$ and obtains the grid $\\ds a^{(n_1\\times\\ldots\\times n_{i-1}\\times(n_{i}+1)\\times n_{i+1} \\times \\ldots  \\times n_d)}$.\\\\\nWe note that if the marginal laws $\\mu_1,\\ldots ,\\mu_d$ are identical, this step is not necessary and the choice of the sequence to which a point is added, at each iteration, is systematically done in a periodic manner.\n\\subsubsection{Numerical integration}\nSimilarly to the $1$-dimensional case, the majority of the Vorono\\\"i cells do not change while passing from an iteration $n$ to an iteration $n+1$.\nAt the $n$-th iteration, having $n_1 \\times \\ldots \\times n_d$ points in the sequence, one adds a new point to $a^{(i,n_i)}$. Hence, we will have $n_1\\times\\ldots\\times n_{i-1}\\times n_{i+1}\\times \\ldots \\times n_d$ new created cells having for centroids the new points added to the $d$-dimensional sequence $a^{(n)}$, and another $2(n_1\\times\\ldots\\times n_{i-1}\\times n_{i+1}\\times \\ldots \\times n_d)$ modified cells, corresponding to all the neighboring cells of the new added cells. In total, there is $3(n_1\\times\\ldots\\times n_{i-1}\\times n_{i+1} \\times \\ldots \\times n_d)$ new Vorono\\\"i cells, while the rest of the cells remain unchanged. \nThis leads to an iterative formula for quantization-based numerical integration (where the same principle as in the one dimensional case is applied)\nas follows\n\\begin{align}\n\\label{calculiteratifd}\n\\ds I_{n+1}(f) & =I_n(f)-p_{-}^{i,n_i+1}\\left(\\sum_{\\substack{j_k=1 \\\\ k\\in \\{1,\\ldots,d\\} \\setminus\\{i\\}}}^{n_k} \\prod_{\\substack{k=1,\\ldots,d \\\\ k\\neq i}}p_{j_k}^{k,(n_k)} \\left( f(a_{j_1}^{1,(n_1)},\\ldots,a_{i_0-1}^{i,(n_i+1)},\\ldots,a_{j_d}^{d,(n_d)}) \\right. \\right. \\nonumber \\\\\n& \\left. \\left. -f(a_{j_1}^{1,(n_1)},\\ldots,a_{i_0}^{i,(n_i+1)},\\ldots, a_{j_d}^{d,(n_d)}) \\right) \\vphantom{(\\sum_{\\substack{j_k=1 \\\\ k\\in \\{1,\\ldots,d\\} \\setminus\\{i\\}}}^{n_k}} \\right) \\nonumber \\\\\n& -p_{+}^{i,n_i+1}\\left( \\sum_{\\substack{j_k=1 \\\\ k\\in \\{1,\\ldots,d\\} \\setminus\\{i\\}}}^{n_k } \\prod_{\\substack{k=1,\\ldots,d \\\\ k\\neq i}}p_{j_k}^{k,(n_k)} \\left( f(a_{j_1}^{1,(n_1)},\\ldots,a_{i_0+1}^{i,(n_i+1)},\\ldots,a_{j_d}^{d,(n_d)}) \\right. \\right.  \\nonumber \\\\\n& \\left. \\left. -f(a_{j_1}^{1,(n_1)},\\ldots,a_{i_0}^{i,(n_i+1)},\\ldots, a_{j_d}^{d,(n_d)}) \\right) \\vphantom{(\\sum_{\\substack{j_k=1 \\\\ k\\in \\{1,\\ldots,d\\} \\setminus\\{i\\}}}^{n_k}}\\right) \n\\end{align}\nNote that in the $d$-dimensional case, the use of the weights $p^{k,(n_k)}$ for $k \\in \\{1,\\ldots,d\\} \\setminus \\{i\\}$ of the Vorono\\\"i cells of the other marginal sequences obtained at the previous iteration is essential, as well as the use the ordered one-dimensional greedy sequences $a^{k,(n_k)}$ for $k \\in \\{1,\\ldots,d\\} \\setminus \\{i\\}$.\n\\section{Numerical applications and examples}\n\\label{numericalexample}\n\\subsection{Greedy quantization sequences for Gaussian distribution via Box-M\\\"uller}\n\\label{boxmuller}\nThe Box-M\\\"uller method allows to generate a random vector with normal distribution $\\mathcal{N}(0,I_2)$, actually two independent one-dimensional random variables $Z_1$ and $Z_2$ with distribution $\\mathcal{N}(0,1)$ by considering two independent random variables $E$ and $U$ with respective distributions $\\mathcal{E}(1)$ and $\\mathcal{U}([0,1])$. Then, $2E \\sim \\mathcal{E}(\\frac{1}{2})$ and $2 \\pi U \\sim \\mathcal{U}([0,2\\pi])$, so, the two variables\n$$Z_1=\\sqrt{2E}\\, \\cos (2\\pi U)\\qquad \\mbox{et} \\qquad Z_2=\\sqrt{2E}\\, \\sin(2 \\pi U)$$\nare independent and with normal distribution $\\mathcal{N}(0,1)$. \\\\\nIn order to apply greedy properties, we use greedy quantization sequences $\\varepsilon^{(n_1)}$ and $u^{(n_2)}$ of respective distributions $\\mathcal{E}(1)$ and $\\mathcal{U}[0,1]$ to design two $\\mathcal{N}(0,1)$-distributed independent sequences $z_1^{(n)}$ et $z_2^{(n)}$, of size $n=n_1 \\times n_2$, via the previous formulas so we can get a greedy sequence $z^{(n)}$ of the two-dimensional normal distribution $\\mathcal{N}(0,I_2)$. \nThe procedure is implemented as described in section~$\\ref{dimensiond}$. At each iteration, we must choose the one-dimensional distribution to which we should add a point. Thus, we compute the error induced if we add a point to $u^{(n_2)}$\n$$E_u = e_2\\left(u^{(n_2+1)},\\mathcal{U}[0,2\\pi]\\right)^2+e_2\\left(\\varepsilon^{(n_1)},\\mathcal{E}\\left(\\frac{1}{2}\\right)\\right)^2\n= 4\\pi^2 e_2\\left(u^{(n_2+1)},\\mathcal{U}[0,1]\\right)^2+4e_2\\left(\\varepsilon^{(n_1)},\\mathcal{E}\\left(1\\right)\\right)^2,$$\nand the error induces if we add a point to $\\varepsilon^{(n_1)}$\n$$E_{\\varepsilon}  =e_2\\left(u^{(n_2)},\\mathcal{U}[0,2\\pi]\\right)^2+e_2\\left(\\varepsilon^{(n_1+1)},\\mathcal{E}\\left(\\frac{1}{2}\\right)\\right)^2=  4\\pi^2e_2\\left(u^{(n_2)},\\mathcal{U}[0,1]\\right)^2+4e_2\\left(\\varepsilon^{(n_1+1)},\\mathcal{E}\\left(1\\right)\\right)^2,$$\nand we add a point to $u^{(n_2)}$ if $E_u<E_{\\varepsilon}$ and a point to $\\varepsilon^{(n_1)}$ if $E_{\\varepsilon}<E_u$. \\\\\n \nTo design sequences in dimension $d >2$, one uses several couples $(E_i,U_i)$ to get several pairs $(Z_i,Z_j)$ and use the wanted number of $(Z_k)_{k}$ to obtain multidimensional sequences.\nIn figure~$\\ref{BM3}$, we compare two greedy quantization sequences of the distribution $\\mathcal{N}(0,I_3)$ of size $N=15^3$, one is obtained using the Box-M\\\"uller method based on two greedy exponential sequences $\\mathcal{E}(1)$ and two greedy uniform sequences $\\mathcal{U}([0,1])$, and the other obtained by greedy product quantization based on $3$ one-dimensional Gaussian greedy sequences. The weights of the Vorono\\\"i cells in both cases are represented by a color scale (growing from blue to red) and we observe that the weights of the cells in the center have the highest values and those values decrease as long as we sweep away to the borders, as expected for a Gaussian distribution.\\\\\nWe should also note that, even if the greedy product quantization of a Normal distribution takes the shape of a cube (which is unusual fo such distribution), the low values of the Vorono\\\"i weights at the edges of this cube allow to consider such a sequence as a valid approximation of the Gaussian distribution. \n\\begin{figure}\n\t\\begin{center}\n\t\t\n\t\t\\begin{tabular}{cc}\n\t\t\t\\includegraphics[width=.4\\textwidth,height=.25\\textheight,angle=0]{grille_norm3_961.pdf}&\n\t\t\t\\hspace{1cm}\n\t\t\t\\includegraphics[width=.4\\textwidth,height=.25\\textheight,angle=0]{grille_norm3_15_3_poids.pdf}\\\\\n\t\t\\end{tabular}\n\t\\end{center} \n\n\t\\caption{Greedy quantization sequences of the distribution $\\mathcal{N}(0,I_3)$ of size $N=15^3$ designed by Box M\\\"uller method (left) and greedy product quantization (right).}\n\t\\label{BM3}\n\\end{figure}\n\n\\iffalse\n\\subsection{Pricing of Best-of-Call Vanilla European option in dimension $d=2$}\nWe consider a $2$-dimensional risk-neutral correlated Black-Scholes model for $2$ risky assets $X_1$ et $X_2$ satisfying, for $i \\in \\{1;2\\}$\n$$dX_i=X_{i,t} \\left(rdt +\\sigma_idW_{i,t}\\right), \\qquad X_{i,0}=x_{i,0}.$$\nwhere $r$ is the interest rate and $\\sigma_i$ the volatility of $X_i$. $W=(W_1,W_2)$ represents a correlated $2$-dimensional Brownian motion, i.e.\n$$W_{2,t}=\\rho W_{1,t} +\\sqrt{1-\\rho^2} \\widetilde{W}_{2,t},$$\nwhere $(W_1,\\widetilde{W_2})$ is a standard $2$-dimensional Brownian motion.\\\\\nThen, one has, for every $t \\in [0,T]$\n$$X_{i,t}=X_{i,0} \\exp\\left((r-\\frac{\\sigma_i^2}{2})t +\\sigma_i W_{i,t}\\right), \\quad i=1,2.$$   \nA European Vanilla option of maturity $T>0$ is an option associated to the European payoff $h_T=h(X_T)$ which depends only on $X_T$. We consider a complete market where the value of the option at time $0$ is given by \n$$V_0=e^{-rT} \\mathbb{E}(h(X_T)).$$\nWe consider a Best-of-Call with strike price $K$, this means the payoff is given by\n$$h_T=\\left( \\max(X_{1,T}-X_{2,T})-K \\right)_{+}.$$\n\nOur aim is to compute $V_0$ relying on the greedy quantization sequences and using the new technique of numerical integration introduced in the previous sections, to show their effectiveness and their importance. At this point, we need to write the payoff as a function of $2$ independent random variables $(Z_1,Z_2)$ with standard normal distribution:\n\\begin{align*}\n\\ds e^{-rT}h_T &= \\phi(Z_1,Z_2)\\\\\n& = \\left( \\mbox{max} \\left(x_{1,0}e^{-\\frac{\\sigma_1^2}{2}T+\\sigma_1 \\sqrt{T}Z_1} , x_{2,0}e^{-\\frac{\\sigma_2^2}{2}T +\\sigma_2\\sqrt{T}(\\rho Z_1 +\\sqrt{1-\\rho^2}Z_2)}\\right)-Ke^{-rT}\\right)_+\n\\end{align*}\nWe use $2$-dimensional greedy quantization sequences $z=(z^{1,(n)},z^{2,(n)})$ of the gaussian distribution $\\mathcal{N}(0,I_2)$ obtained by greedy product quantization on the one hand, and by the Box-M\\\"uller algorithm on the other hand. And, we approximate \n\\begin{align}\n\\label{V0}\nV_0 &=e^{-rT}\\mathbb{E}[(\\max(X_1,X_2)-K)_+] \\nonumber \\\\ \n& =\\mathbb{E}[\\phi(Z_1,Z_2)].\n\\end{align}\nWe estimate the expectation~$(\\ref{V0})$ first by a quadrature formula according to~$(\\ref{quadratureformula})$\n\\begin{equation*}\n\\ds V_0 =  \\sum_{j_1=1}^{n} \\sum_{j_2=1}^{n}  \\phi(z_{j_1}^{1,(n)},z_{j_2}^{2,(n)}) p_{j_1}^{1,(n)} p_{j_2}^{2,(n)}.\n\\end{equation*}\nwhere $(z^{1,(n)},z^{2,(n)})$ is obtained by the Box-M\\\"uller algorithm, explained in section~$(\\ref{boxmuller})$, relying on a one-dimensional exponential greedy sequence $\\varepsilon^{(n_1)}, \\, n_1=16$ and a one-dimensional uniform greedy sequence $u^{(n_2)}, \\, n_2=64$. \n\nAlso, we estimate it by the recursive formula~$(\\ref{calculiteratifd})$ for $d=2$ for the sequence obtained by greedy product quantization implemented using two one-dimensional normal greedy sequences of size $32$ each. \\\\\nSo, in both cases we obtain sequences of size $1\\,024$. The parameters are \n$$r=0,1 \\; , \\; \\sigma_i=0,2 \\; ,\\; \\rho=0,5 \\; ,\\; X_{i,0}=100 \\; ,\\; T=1 \\; \\mbox{and} \\; K=100.$$\n\nThe reference price is given by a large Monte Carlo simulation of size $M=10^5$. We compare both results in figure~$\\ref{VanillaBoC}$ for $n\\in \\{1,\\ldots,1\\,024\\}$ where we represent, in a logarithmic scale, the error induced by each method as a function of the number of points varying between $1$ and $1\\,024$. \n\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=.75\\textwidth,height=.35\\textheight,angle=0]{comp_BM_QPIt.pdf}\n\t\\end{center} \n\t\\caption{Estimation of a European Vanilla Best-of-Call $V_0$ in a Black-Scholes model computed using $2$-dimensional greedy normal quantization sequences obtained by Box-M\\\"uller via quadrature formula and by greedy product quantization via a recursive formula for $n=1, \\ldots, 1\\, 000$ (logarithmic scale).}\n\t\\label{VanillaBoC}\n\\end{figure}\n\n\\fi\n\\subsection{Pricing of a $3$-dimensional basket of European call options }\nWe consider a Call option on a basket of $3$ positive risky assets, with strike price $K$ and maturity $T$, with payoff\n$h_T=\\left( \\sum_{i=1}^{3} w_i X_i -K \\right)_+$\nwhere $(X_1,X_2 ,X_3)$ represent the prices of the $3$ traded assets of the market and $w_i$ are positive weights such that $ \\sum_{i=1}^{3}w_i=1$.\nWe consider a $3$-dimensional correlated Black-Scholes model where the prices of the assets are given by\n$$dX_i=X_{i,t} \\left(rdt +\\sigma_idW_{i,t}\\right), \\qquad X_{i,0}=x_{i,0}.$$\nwhere $r$ is the interest rate, $\\sigma_i$ the volatility of $X_i$ and the $(W_i)_i$ represent a correlated $3$-dimensional Brownian motion, i.e. $(W_i,W_j)=\\rho_{ij}t.$\nThen, one has for every $i\\in \\{1,2,3\\}$\n$$\\ds X_i=X_{0,i} e^{(r-\\frac{\\sigma_i^2}{2})t+\\sum_{j=1}^{q}\\sigma_{ij} W_{j,t}}, \\quad t \\in [0,T].$$\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=.7\\textwidth,height=.3\\textheight,angle=0]{comp_basketdim3_BM_QPQ_QPI.pdf}\n\t\\end{center} \n\t\\vspace{-1cm}\n\t\\caption{Errors induced by the pricing of a $3$-dimensional basket of call options $V_0$ in a Black-Scholes model computed using $3$-dimensional greedy normal quantization sequences obtained by Box-M\\\"uller via quadrature formula and by greedy product quantization via a recursive formula and via a quadrature formula for $n=1, \\ldots, 30\\, 000$ (logarithmic scale).}\n\t\\label{basketdim3}\n\\end{figure}\nOur aim is to compute $$V_0=e^{-rT} \\mathbb{E}[h_T(X_T)]=e^{-rT} \\mathbb{E}[h_T(X_1,X_2,X_3)]$$ relying on the greedy quantization sequences and using the recursive formula for numerical integration introduced in the previous sections. \\\\\nFirst, we estimate $V_0$ by a quadrature formula according to~$(\\ref{quadratureformula})$\n\\begin{equation*}\n\\ds V_0 =  \\sum_{j_1=1}^{n} \\sum_{j_2=1}^{n} \\sum_{j_3=1}^{n} \\phi(z_{j_1}^{1,(n)},z_{j_2}^{2,(n)},z_{j_3}^{3,(n)}) p_{j_1}^{1,(n)} p_{j_2}^{2,(n)} p_{j_3}^{3,(n)}.\n\\end{equation*}\nwhere $(z^{1,(n)},z^{2,(n)},z^{3,(n)})$ is a $3$-dimensional greedy quantization sequences of the Gaussian distribution $\\mathcal{N}(0,I_3)$ obtained, on one hand, by the Box-M\\\"uller algorithm relying on $2$ one-dimensional exponential greedy sequence $\\varepsilon^{(n_1)}, \\, n_1=16$ and $2$ one-dimensional uniform greedy sequence $u^{(n_2)}, \\, n_2=15$ and, on the other hand, by greedy product quantization of $3$ one-dimensional sequences of size $32$ each. Then, we estimate $V_0$ by the recursive formula~$(\\ref{calculiteratifd})$ for $d=3$ using the greedy product quantization sequence. \nWe obtain sequences of size $32\\, 000$ and we consider \n$$r=0.1 \\; , \\; \\sigma_i=0.3 \\; ,\\; X_{i,0}=100 \\; ,\\; T=1 \\; \\mbox{and} \\; K=100.$$\nMoreover, we consider a Brownian motion such that $\\rho_{1,1}=\\rho_{1,2}=\\rho_{1,3}=0.5$ and all the others $\\rho_{i,j}$'s are equal to $0$. The reference price is given by a large Monte Carlo simulation with control variate of size $M=2.10^7$. We consider the control variate \n$$k_T=\\left(e^{\\sum_{i=1}^3 w_i\\log(X_i)}-K\\right)_+$$\nwhich is positive and lower than $h_T$ owing to the convexity of the exponential. Since $e^{-rT}\\mathbb{E} k_t$ has a normal distribution with mean $(r-\\frac12\\sum_{i=1}^3 w_i \\sigma_i^2)T$ and variance $w^t \\sigma \\sigma^t w T$, it admits a closed form given by \n\\begin{equation*}\ne^{-rT}\\mathbb{E} k_t=\\mbox{Call}_{\\mbox{BS}}\\left(\\prod_{i=1}^3 X_{i,0}^{w_i} e^{-\\frac12 T (\\sum_{i=1}^3 w_i \\sigma_i^2-w^t \\sigma \\sigma^t w)}, K,r,\\sqrt{w^t \\sigma \\sigma^t w},T \\right).\n\\end{equation*}\n\\begin{table}\n\t\\begin{center}\n\t\t\\begin{tabular}{|l|l|l|l|}\n\t\t\t\\hline\n\t\t\t$n$         &  BM    &    GPQ     &     GPI  \\\\      \n\t\t\t\\hline\n\t\t\t$100$   & $1.72$  & $1.68$   &  $1.84$          \\\\\n\t\t\t\\hline\n\t\t\t$1\\, 000$     & $0.07$  & $0.42$ & $0.42$       \\\\\n\t\t\t\\hline\n\t\t\t$8\\, 000$  &  $0.04$  &  $0.08$  &  $0.08$       \\\\\n\t\t\t\\hline \n\t\t\t$15\\, 000$ & $0.07$ & $0.08$ & $0.08$\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Errors of the approximation of a $3$-dimensional basket of call options $V_0$ in a Black-Scholes model by Box-M\\\"uller with quadrature formula (BM), greedy product quantization with quadrature formula (GPQ) and greedy product quantization with recursive formula (GPI) for different number of points $n$.} \n\t\\label{tabbasketdim3}\n\\end{table}\nWe compare the three methods in figure~$\\ref{basketdim3}$ where we represent, in a logarithmic scale, the error induced by each method as a function of the number of points varying between $1$ and $32\\,000$ and in table~\\ref{tabbasketdim3} where we expose the errors obtained by each method for some particular number of points.\nThe recursive numerical integration gives the same results as the quadrature formula-based numerical integration making quantization-based numerical integration less expensive and more advantageous by reducing the cost in time and storage. Moreover, one deduces that the Box-M\\\"uller algorithm is more accurate than the greedy product quantization. This can be explained by the fact that Box-M\\\"uller sequences fill the space in a way that resembles more to the normal distribution, we can notice a kind of ball different than the cube observed when implementing greedy product sequences (see figure~\\ref{BM3}).\n\\section{Further properties and numerical remarks}\n\\label{numericalproperties}\nIn this section, we present, based on numerical experiments, some properties of the one-dimensional quadratic greedy quantization sequences. \nWe recall that $a^{(n)}=\\{a_1^{(n)},\\ldots,a_n^{(n)}\\}$ denotes the reordered greedy sequence of the $n$ first elements $\\{a_1,\\ldots,a_n\\}$ of $(a_n)_{n \\geq 1}$.\n\\subsection{Sub-optimality of greedy quantization sequences}\nThe implementation of a greedy quantization sequence $(a_n)_{n\\geq 1}$ of a distribution $P$ and the computation of the corresponding weights $p_i^{n}$ of the Vorono\\\"i cells $W_i(a^{(n)})$ for $i \\in \\{1,\\ldots, n\\}$ defined by~$(\\ref{Voronoicells})$ is, in general, not optimal. However, numerical implementations and graphs representing \n$a_i \\mapsto p_i^n=P(X \\in W_i(a^{(n)}))$\nfor different number of points $n$ show that, for certain distributions, the weights of the Vorono\\\"i cells converge towards the density curve of the corresponding distribution when the greedy sequence has a certain number of points.\\\\\nFor the normal distribution, \nthis is observed when the size of the sequence is equal to $n=2^k-1$, for every integer $k \\geq 1$. So, we can say that the greedy quantization sequence is sub-optimal since the subsequence \n\\begin{equation}\n\\label{suiteoptnorm}\n\\ds \\alpha^{(n)}= \\alpha^{(2^k-1)} \\, \\mbox{t.q.} \\quad  n=2^k-1,\\, k \\in \\mathbb{N}^*\n\\end{equation}\nis itself optimal.\nRegarding the uniform distribution on $[0,1]$, we can check that there exists $2$ sub-optimal sequences of the greedy sequence defined by\n\\begin{multicols}{2}\\noindent\n\\[\n\\label{suiteopt1unif}\n\\left\\{\n\\begin{array}{lr}\n\\alpha_0=3, \\\\\n\\alpha_n=2\\alpha_{n-1}+1  & \\mbox{if } n \\equiv 1 \\pmod 3,\\\\\n\\alpha_n=2(\\alpha_{n-1}-2)+1 &  \\mbox{if }n \\equiv 2 \\pmod 3,\\\\\n\\alpha_n=2(\\alpha_{n-1}+2)+1  & \\mbox{if } n \\equiv 0 \\pmod 3.\n\\end{array}\n\\right.\n\\\n\\\n\\label{suiteopt2unif}\n\\left\\{\n\\begin{array}{lr}\n\\alpha_0=11, \\\\\n\\alpha_n=2\\alpha_{n-1}+1  & \\mbox{if }n \\equiv 1 \\pmod 3,\\\\\n\\alpha_n=2(\\alpha_{n-1}-2)+1  & \\mbox{if } n \\equiv 2 \\pmod 3,\\\\\n\\alpha_n=2(\\alpha_{n-1}+2)+1  & \\mbox{if }n \\equiv 0 \\pmod 3.\n\\end{array}\n\\right.\n\\\n\\end{multicols}\nThese results explain, in a certain way, the cycloid aspect of the graphs of the quantization error, the points at which the quantization error reaches its minimum correspond to the optimal subsequences defined above\n\nSome results for the normal distribution are represented in figure~$\\ref{weightssuboptimal}$ where we observe the unimodal weights for $n=255=2^8-1$\nand non-unimodal weights for $n=400$.\n\\begin{figure}\n\t\\begin{center}\n\t\t\\vspace{-2cm}\n\t\t\\begin{tabular}{cc}\n\t\t\t\\includegraphics[width=.4\\textwidth,height=.4\\textheight,angle=0]{poids_norm_greedy_2551.pdf}&\n\t\t\t\\hspace{0.5cm}\n\t\t\t\\includegraphics[width=.4\\textwidth,height=.4\\textheight,angle=0]{poids_norm_greedy_4001.pdf}\\\\\n\n\t\t\n\t\t\n\t\t\\end{tabular}\n\t\\end{center} \n\t\\vspace{-2.5cm}\n\t\\caption{Representation of $a_i \\mapsto p_i^n$ where $(p_i^n)_{1 \\leq i \\leq n}$ denote the Vorono\\\"i weights corresponding to the greedy quantization sequence of the normal distribution $\\mathcal{N}(0,1)$ implemented by Lloyd's algorithm for $n=255$ (left), $n=400$ (right).}\n\n\t\\label{weightssuboptimal}\n\\end{figure}\n\nSimilarly, the greedy quantization sequence of the Laplace distribution with parameters $0$ and $1$ admits optimal subsequences taking the form $a^{(2^k-1)}, k \\in \\mathbb{N}^*$. These observations allow to conjecture the sub-optimality of such subsequences for symmetrical distributions around $0$.\n\\subsection{Convergence of standard and weighted empirical measures}\nThe existence of suboptimal greedy quantization sequences, as detailed previously, gives a motivation to dig deeper and study the empirical measures associated to a greedy quantization sequences.\nIn fact, sequences of asymptotically optimal $n$-quantizers $(\\Gamma_n)_{n \\geq 1}$, of an absolutely continuous distribution $P$ w.r.t. the Lebesgue measure with density $f$, satisfy some empirical measure convergence theorems established in \\cite{GraLu00} (see theorem $7.5$ p. $96$) and \\cite{Delattre04} and recalled below, where\n$$\\ds \\widehat{P}_n=\\frac{1}{n} \\sum_{i=1}^{n} \\delta_{x_i^{n}} \\qquad \\mbox{ and } \\qquad  \\widetilde{P}_n=\\sum_{i=1}^{n} P(W_i(\\Gamma_n)) \\delta_{x_i^{n}}$$ \ndesignate, respectively, the empirical standard measure and the empirical weighted measure associated to $\\Gamma_n=\\{x_1^n,\\ldots,x_n^n\\}$\n\\begin{thm}\n\tAssume $P$ is absolutely continuous w.r.t the Lebesgue measure on $\\mathbb{R}^d$ with density $f$. Let $\\Gamma_n$ be an asymptotically optimal $n$-quantizer of $P$. Then, denoting $C=\\big(\\int_{\\mathbb{R}} f^{\\frac{d}{d+p}}(u) du \\big) ^{-1}$, one has\n\t\\begin{equation}\n\t\\label{convmesemp}\n\t \\widetilde{P}_n \\underset{n\\rightarrow +\\infty}{\\Rightarrow} P \\qquad \\mbox{and} \\qquad \\widehat{P}_n \\underset{n\\rightarrow +\\infty}{\\Rightarrow} C\\, f^{\\frac{d}{p+d}}(u) du.\n\t\\end{equation}\n\n\\end{thm}\nWe hope to obtain such results for greedy sequences or, at least, for sub-optimal greedy sequences defined in the previous section. To this end, we {\\em ``divide''} the two limits mentioned in~$(\\ref{convmesemp})$, along the sequence $(W_i(a^{(n)}))_{1 \\leq i \\leq n}$ of the Vorono\\\"i weights associated to $(a_n)_{n \\geq 1}$, and we obtain \n$n P_l(W_i(a^{(n)}))  \\simeq \\frac{1}{C} f^{1-\\frac{d}{d+p}}(a_i^{(n)})\n= C^{-1} f^{\\frac{p}{d+p}}(a_i^{(n)}). $ \nHence, for all $i \\in \\{1, \\ldots , n\\}$, the limiting measure of the Vorono\\\"i cells of the greedy sequence is given by \n\\begin{equation}\n\\label{poidslimite}\nP_l(W_i(a^{(n)})) \\simeq \\frac{f^{\\frac{p}{d+p}}(a_i^{(n)})}{C n}\n\\end{equation}\nIn other words, if the greedy sequences satisfy the convergence of the empirical measures, then the weights of the Vorono\\\"i cells, computed by\n\\begin{equation}\n\\label{poidsnormal}\n\\ds P(W_i(a^{(n)}))= F_{P}\\left(a_{i+\\frac{1}{2}}^{(n)}\\right)-F_{P}\\left(a_{i-\\frac{1}{2}}^{(n)}\\right),\n\\end{equation}\nwhere $ a_{i+\\frac{1}{2}}^{(n)}=\\frac{a_i^{(n)}+a_{i+1}^{(n)}}{2}$ for $1 \\leq i \\leq n-1$ and $F_{P}$ is the c.d.f of $P$, must converge to the limit weights $P_l(W_i(a^{(N)}))$ given in~$(\\ref{poidslimite})$. \n\nNumerical experiments were established for the normal distribution $\\mathcal{N}(0,1)$, uniform distribution $\\mathcal{U}([0,1])$, exponential distribution $\\mathcal{E}(1)$ and Laplace distribution $\\mathcal{L}(0,1)$. We observe positive results in the four cases, the weights of the Vorono\\\"i cells computed online get closer to the limit weights $P_l$ when $n$ increases. For the gaussian distribution, we observe a more important convergence for the subsequences $a^{(2^k-1)}$ (as predicted). We present, in figure~\\ref{convempiricalmeasure} the obtained results for the exponential distribution where we compare the weights~$(\\ref{poidsnormal})$ (blue) and the limit weights~$(\\ref{poidslimite})$ (red) for different number of points $n$.\n\\begin{figure}\n\t\\begin{center}\n\t\n\t\t\\begin{tabular}{cc}\n\t\t\t\\includegraphics[width=.35\\textwidth,height=.25\\textheight,angle=0]{expo_poidstest_645.pdf} &\n\t\t\t\\hspace{1cm}\n\t\t\t\\includegraphics[width=.35\\textwidth,height=.25\\textheight,angle=0]{expo_poidstest_1379.pdf}\n\t\t\\end{tabular}\n\t\\end{center} \n\t\\vspace{-0.5cm}\n\t\\caption{Comparison of the weights of the Vorono\\\"i cells computed online (blue) to the limit weights of the cells (red) for the exponential distribution $\\mathcal{E}(1)$ for $n=645$ (left) and $n=1\\,379$ (right).}\n\t\\label{convempiricalmeasure}\n\\end{figure}\n\\subsection{Stationarity and $\\rho$-quasistationarity}\nAn interesting question is to see if the greedy sequences are stationary i.e. satisfy\n$$a_i^{(n)}=\\mathbb{E}(X|X \\in W_i(a^{(n)})), i=1, \\ldots,n,$$\nor can be close to stationarity, a property shared by quadratic optimal quantizers. We compute the error $ \\|\\hat{X}^{a^{(n)}} -\\mathbb{E}(X|\\hat{X}^{a^{(n)}})\\|_1$ under the standard empirical measure $ \\widehat{P}_n= \\frac{1}{n} \\sum_{i=1}^{n} \\delta_{a_i^{(n)}}$, i.e.\n\\begin{equation}\n\\label{aqs}\n\\sum_{i=1}^{n}\\left| a_i^{(n)} - \\mathbb{E}(X|X \\in W_i(a^{(n)}))\\right|= \\sum_{i=1}^{n}\\left| a_i^{(n)}-\\ds\\frac{\\int_{W_i(a^{(n)})}\\xi dP(\\xi)}{P(W_i(a{(^n)}))}\\right|.\n\\end{equation}\nand we hope to observe a convergence to $0$, when $n$ increases. But, numerical experiments, conducted for several probability distributions, show that $a^{(n)}$ cannot be stationary, in the sense of~$(\\ref{aqs})$. In fact, we will show, for a specific class of distributions, that greedy quantization sequences are not stationary (in the sense of~$(\\ref{aqs})$), except when $n\\in \\{1,3\\}$. We will use the following result given in \\cite{Kieffer}.\n\\begin{thm} \\label{kieffer} (J.C. Kieffer)\n\tLet $d=1$ and $\\mu$ a probability distribution with log-concave density. Then, there exists a unique stationary quantizer of $\\mu$.\n\\end{thm}\n\n\\begin{prop}\n\t\\label{stationary}\n\tLet $X$ be a random variable with distribution $P$ which is symmetric and unimodal (log-concave density) and $a^{(n)}$ a corresponding greedy quantization sequence. Then, for every $n \\in \\mathbb{N}\\setminus \\{1,3\\}$, the sequence $a^{(n)}$ is not stationary. \n\\end{prop}\n\\begin{proof}   \n\tWe suppose that $\\mathbb{E}[X]=0$ (symmetric around $0$). If it is not the case, a translation gives the same results. We will detail the proof in $3$ cases\\\\ \n\t$\\rhd$ {\\bf For $n=3$:} \n\t\tSince $\\mathbb{E}[X]=0$, the first point is $a_1=0$. A second point is given by \n\t\t$$a_2 =  {\\rm argmin}_{a \\in \\mathbb{R}} \\mathbb{E}X^2 \\wedge (X-a)^2 =\\big\\{ \\nabla_{a_2} \\,  \\mathbb{E}X^2 \\wedge (X-a_2)^2=\\int_{W_2(a^{(i)})}(\\xi - a_2)dP(\\xi)=0\\big\\}.$$\n\t\tHence, $a_2=\\frac{\\int_{W_2(a^{(n)})}\\xi dP(\\xi)}{P(W_2(a^{(n)}))}$ is stationary. The third point is $a_3=-a_2$ by symmetry of $P$ so $a_3$ is also stationary. Finally, $a_1=0$ is also stationary since\n\t\t$\\int_{a_2\/2}^{a_3\/2}\\xi dP(\\xi)=\\int_{a_2\/2}^{-a_2\/2}\\xi dP(\\xi)=0$. Consequently, the sequence $a^{(3)}=\\{-a_2;a_1;a_2\\}$ is stationary.\\\\% and, by theorem~$\\ref{kieffer}$, $a^{(3)}$ is equal to the $3$-optimal quantizer of $P$.\\\\\n\t\t$\\rhd$ {\\bf For $n=2k$ even:}\n\t\tSince $P$ is unimodal, the stationary quantizer is unique, let $x^{(n)}$ be this quantizer, which is the $n$-optimal quantizer of $P$ because we know it is stationary. The symmetry of $P$ lets us know that the quantizer $(x_{n+1-l}^{(n)})_{1 \\leq l \\leq n}$ of $P$ is also stationary, so, for every $l \\in \\{1, \\ldots,n\\}$, $x_l^{(n)}+x_{n+1-l}^{(n)}=0.$ \n\t\tSince $n=2k$ is even, we have, in particular, \n\t\t$$x_{k}^{(n)}=-x_{n+1-k}^{(n)}=-x^{(n)}_{n+1-\\frac{n}{2}}=-x^{(n)}_{k+1},$$\n\t\tso,\n\t\t$x_k^{(n)}<0<x_{k+1}^{(n)}$\n\t\tand, since, $x_k^{(n)}$ et $x_{k+1}^{(n)}$ are two consecutive terms of the grid, we deduce that $0$ is not an element of $x^{(n)}$, and hence can not be a point of a stationary quantizer. Consequently, the greedy sequence starting at $a_1=0$ can not be stationary.\\\\\t\t\n\t\t$\\rhd$ { \\bf For $n=2k+1$ odd:}\n\t\tWe consider the greedy non-stationary sequence  $a^{(2k)}$ of even size. There exists, at least, one non-stationary Vorono\\\"i cell $W_i(a^{(2k)})$. Its symmetric cell $W_{2k+1-i}(a^{(2k)})$ will also be non-stationary because $P$ is symmetric. So, we have at least $2$ non stationary Vorono\\\"i cells. While building the sequence $a^{(2k+1)}$, we add a new point which will be in one of the Vorono\\\"i cells without modifying the others. If the new point is added in one of the non-stationary cells, we know that the second one will remain untouched, having, at least, one non-stationary cell in $a^{(2k+1)}$. And, if the new point is not in these cells, then they will remain untouched and there will be, at least, $2$ non-stationary cells in $a^{(2k+1)}$.\t\t\\hfill $\\square$\\\\\n\\end{proof}\n\nHowever, further different numerical observations show that most greedy quantization sequences satisfy a certain criteria that we can call $\\rho$-quasi-stationarity. This criterion approaches to the stationary character verified by optimal quantizers and  can be defined, for $r \\in \\{1,2\\}$ and $\\rho \\in [0,1]$, by\n\n\\begin{equation}\n\\label{rhoaqs}\n \\|\\widehat{X}^{a^{(n)}} -\\mathbb{E}(X|\\widehat{X}^{a^{(n)}})\\|_r= o(\\| \\widehat{X}^{a^{(n)}}-X\\|_{1+\\rho}^{1+\\rho}), \\quad \\mbox{or} \\quad \\frac{ \\|\\widehat{X}^{a^{(n)}} -\\mathbb{E}(X|\\widehat{X}^{a^{(n)}})\\|_r}{\\| \\widehat{X}^{a^{(n)}}-X\\|_{1+\\rho}^{1+\\rho}} \\underset{n \\rightarrow +\\infty}{\\longrightarrow} 0.\n \\end{equation}\nIt is satisfied by greedy sequences for $\\rho$ lower than certain optimal values $\\rho_l$ depending on the distribution $P$ and on the value of $r$. We expose, in table~$\\ref{rhooptimal}$, these values of $\\rho_l$ for $r=1$ and $r=2$ for the normal, uniform or Gaussian distribution.\n\\begin{table}\n\t\\begin{center}\n\t\t\\begin{tabular}{|l|c|c|r|}\n\t\t\t\\hline\n\t\t\t& $\\mathcal{N}(0,1)$ & $\\mathcal{U}([0,1])$ & $\\mathcal{E}(1)$ \\\\ \n\t\t\t\\hline\n\t\t\t$r=1$    &      $\\rho_l= 0.92$   &     $\\ds \\rho_l=\\frac{3}{4}$   & $\\ds \\rho_l=\\frac{2}{3} $ \\\\\n\t\t\t\\hline\n\t\t\t$r=2$     &  $\\rho_l=0.47$       &     $\\ds \\rho_l=\\frac{3}{8}$   &  $\\ds \\rho_l=\\frac{1}{3}$\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\n\t\\end{center}\n\t\\caption{Values of optimal $\\rho_l$ for different distributions and for $r \\in \\{1;2\\}$.} \n\t\\label{rhooptimal}\n\\end{table}\nThis property is important because it brings improvements to quantization-based numerical integration. The error induced by this integration using standard cubature formula for functions with $\\rho$-H\\\"older gradient is bounded by \n$$|\\mathbb{E}f(X)-\\mathbb{E}f(\\widehat{X}^{a^{(n)}})|  \\leq \\frac{1}{1+\\rho}[\\nabla f]_{\\rho} \\|X-\\widehat{X}^{a^{(n)}}\\|_{1+\\rho}^{1+\\rho}$$\nwhile the classical error bound is given by (see \\cite{Pages15})\n$$|\\mathbb{E} f(X)-\\mathbb{E} f(\\widehat{X}^{a^{(n)}})| \\leq [f]_{\\text{Lip}} \\|X-\\widehat{X}^{a^{(n)}}\\|_p.$$\nIn fact, if $\\rho \\in [0,1]$ and $f$ is a continuous function with $\\rho$-h\\\"older gradient with H\\\"older coefficient $[\\nabla  f]_{\\rho}$, we have \n\\begin{align*}\n\\mathbb{E}f(X)-\\mathbb{E}f(\\widehat{X}^{a^{(n)}}) \\leq & \\,  \\mathbb{E}\\big(\\nabla f(\\widehat{X}^{a^{(n)}})|X-\\widehat{X}^{a^{(n)}}\\big) \\\\\n& + \\mathbb{E}\\left[  \\int_0^1 \\left( \\nabla f\\left( \\widehat{X}^{a^{(n)}}+t(X-\\widehat{X}^{a^{(n)}})\\right)-\\nabla f(\\widehat{X}^{a^{(n)}})| X-\\widehat{X}^{a^{(n)}}\\right)dt \\right].\n\\end{align*}\nSince,\n\\begin{align*}\n\\int_0^1 \\left( \\nabla f\\left(  \\widehat{X}^{a^{(n)}}+t(X-\\widehat{X}^{a^{(n)}})\\right)-\\nabla f( \\widehat{X}^{a^{(n)}})| X-\\widehat{X}^{a^{(n)}}\\right)dt \n\\leq [\\nabla f]_{\\rho} \\mathbb{E}|X-\\widehat{X}^{a^{(n)}}|^{1+\\rho}\\int_0^1 t^{1+\\rho} dt, \n\\end{align*}\nand \n\\begin{align*}\n\\mathbb{E}(\\nabla f(\\widehat{X}^{a^{(n)}})|X-\\widehat{X}^{a^{(n)}})  =\\mathbb{E}(\\nabla f(\\widehat{X}^{a^{(n)}})|X)-\\mathbb{E}(\\nabla f(\\widehat{X}^{a^{(n)}})|\\widehat{X}^{a^{(n)}}) \\nonumber = \\mathbb{E}\\left( \\nabla f(\\widehat{X}^{a^{(n)}})|\\mathbb{E}(X|\\widehat{X}^{a^{(n)}})-\\widehat{X}^{a^{(n)}}\\right),\n\\end{align*}\n we have\n\\begin{equation*}\n|\\mathbb{E}f(X)-\\mathbb{E}f(\\widehat{X}^{a^{(n)}})| \\leq \\|\\nabla f(\\widehat{X}^{a^{(n)}})\\|_2 \\|\\mathbb{E}(X|\\widehat{X}^{a^{(n)}})-\\widehat{X}^{a^{(n)}}\\|_2 +  \\frac{1}{1+\\rho}[\\nabla f]_{\\rho} \\|X-\\widehat{X}^{a^{(n)}}\\|_{1+\\rho}^{1+\\rho} .\n\\end{equation*}\nHence, if~$(\\ref{rhoaqs})$ is satisfied, then on can conclude that\n\\begin{equation}\n\\label{bornerho}\n\\limsup_{n} \\frac{|\\mathbb{E}f(X)-\\mathbb{E}f(\\widehat{X}^{a^{(n)}})|}{ \\|X-\\widehat{X}^{a^{(n)}}\\|_{1+\\rho}^{1+\\rho}} \\leq \\frac{1}{1+\\rho}[\\nabla f]_{\\rho}\n\\end{equation}\nand hence the gain in the quantization-based numerical integration error bounds.\n\\subsection{Discrepancy of greedy sequences}\nThe comparison established, in the beginning of section~\\ref{algorithmics}, between greedy quantization-based numerical integration and quasi-Monte Carlo methods, showing a gain of $\\log(n)$-factor with greedy quantization in terms of convergence rate, drives us to build a relation, based on Pro\\\"inov's Theorem~$\\ref{proinov}$, between the error quantization and the discrepancy. In fact, for every $n$-tuple $\\Xi=(\\xi_1,\\ldots,\\xi_n) \\in [0,1]^n$, noticing that a Lipschitz function $f$ has always a finite variation \nand considering the function $f:u \\rightarrow \\min_{1 \\leq i \\leq n}|u-\\xi_i|$ which is $1$-Lipschitz (since $\\left| \\min_i a_i -\\min_i b_i  \\right| \\leq \\max_i |a_i-b_i|)$ and satisfies $f(\\xi_i)=0$ for every $i \\in \\{1, \\ldots ,n\\}$ and $\\ds \\int_0^1 f(u)du =e_1(X,\\mathcal{U}([0,1]))$, one applies the Koksma-Hlawka inequality ($\\ref{koksmahlawka}$) to $f$ to deduce that\n\\begin{equation}\n\\label{lien1}\ne_1(\\Xi,\\mathcal{U}([0,1])) \\leq D_n^*(\\Xi).\n\\end{equation}\nThis motivates us to study the discrepancy of greedy sequences hoping that they can be comparable to low discrepancy sequences. We compute the discrepancy of greedy quantization sequences, for $d \\in \\{1,2,3\\}$, using formulas given in \\cite{Doerr14} (theorems $1$, $2$ and $3$) and recalled below.\n\\begin{thm}\n\t$(a)$ Let $\\Xi=(\\xi_i)_{1 \\leq i \\leq n}$ be a sequence in $[0,1]$ and assume $\\xi_1 \\leq \\ldots \\leq \\xi_n$. Then, for every $n \\geq 1$\n\t\t\\begin{equation}\n\t\tD_n^*(\\Xi)=\\max_{1 \\leq i \\leq n} \\max \\left\\{\\frac in -\\xi_i\\, ,\\, \\xi_i -\\frac{i-1}{n} \\right\\}=\\frac{1}{2n} +\\max_{1 \\leq i \\leq n} \\left|\\xi_i-\\frac{2i-1}{2n} \\right|.\n\t\t\\end{equation}\n\t$(b)$ Let $\\Xi=(\\xi_i)_{1 \\leq i \\leq n}$ be a sequence in $[0,1]^2$ where each $\\xi_i$ have two components $(x^1_i,x_i^2)$. Assume $x_1^1 \\leq \\ldots \\leq x_n^1$. For every $i \\in \\{1,\\ldots,n\\}$, we consider $(\\xi_{i,0},\\ldots,\\xi_{i,i+1})$ an increasing reordering of $(0,x_1^2,\\ldots,x_i^2,1)$. Then,\n\t\t\\begin{equation}\n\t\tD_n^*(\\Xi)=\\max_{1 \\leq i \\leq n} \\max_{0 \\leq k \\leq i} \\max \\left\\{\\frac kn -x^1_i \\xi_{i,k}\\, ,\\, x^1_{i+1}\\xi_{i,k+1} -\\frac{k}{n} \\right\\}.\n\t\t\\end{equation}\n\t$(c)$ Let $\\Xi=(\\xi_i)_{1 \\leq i \\leq n}$ be a sequence in $[0,1]^3$ where each $\\xi_i$ have three components $(x^1_i,x_i^2,x_i^3)$. Assume $x_1^1 \\leq \\ldots \\leq x_n^1$. For every $i \\in \\{1,\\ldots,n\\}$, we consider $(\\xi_{i,0},\\ldots,\\xi_{i,i+1})$ an increasing reordering of $(0,x_1^2,\\ldots,x_i^2,1)$.\tFor a fixed $i$ and $k \\in \\{1,\\ldots,i\\}$, we consider $\\{\\eta_{i,k,0},\\ldots,\\eta_{i,k,k+1}\\}$ an increasing reordering of $\\{0,x_1^3,\\ldots,x^3_k,1\\}$. Then,\n\t\t\\begin{equation}\n\t\tD_n^*(\\Xi)=\\max_{1 \\leq i \\leq n} \\max_{0 \\leq k \\leq i} \\max_{0 \\leq l \\leq k} \\max \\left\\{\\frac ln -x^1_i \\xi_{i,k} \\eta_{i,k,l}\\, ,\\, x^1_{i+1}\\xi_{i,k+1} \\eta_{i,k,l+1} -\\frac{l}{n} \\right\\}.\n\t\t\\end{equation}\n\\end{thm}\nNumerical results show that, in the one-dimensional case, greedy sequences can be used as a low discrepancy sequence. But, when $d$ becomes larger than $1$, the situation becomes less convincing. In fact, if we use pure greedy sequences designed by implementing Lloyd's algorithm, the discrepancy of these sequences and that of low discrepancy sequences (Niederreiter sequences for example) are comparable and the results are not so bad, but the problem that arises is the complexity of the computations\nmaking greedy sequences less practical. On the other hand, if we use the greedy product multidimensional grids to solve this problems, the computation will be less expensive but the numerical experiments show that there is no gain in terms of discrepancy. Figure~$\\ref{discrepancy}$ shows a comparison of the discrepancy of a Niederreiter sequence in dimension $2$ to that of a product greedy quantization sequence of $\\mathcal{U}([0,1]^2)$ on the one hand, and to that of pure greedy quantization sequence of $\\mathcal{U}([0,1]^2)$ implemented by Lloyd's algorithm on the other hand, emphasizing the conclusions made. \\\\\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{tabular}{cc}\n\t\t\t\\includegraphics[width=8cm,height=5.8cm,angle=0]{discdim2_greedyprodvsniedvraie.pdf} \n\t\t\t\\includegraphics[width=7.7cm,height=5.7cm,angle=0]{disc_greedy_nied_vraies_2.pdf}\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\vspace{-0.5cm}\n\t\\caption{Comparisons of the star discrepancy of the Niederreiter sequence to a greedy product quantization sequence of the uniform distribution $\\mathcal{U}([0,1]^2)$ (left) and to a pure greedy quantization sequence (right) for $d=2$.}\n\t\\label{discrepancy}\n\\end{figure}\nThe positive results obtained in the one-dimensional case encourage us to try and manipulate low discrepancy sequences, such as Van der Corput sequences, in order to be able to use them as greedy quantization sequences. In other words, we will assign to them a Vorono\\\"i diagram, compute the weights of the corresponding Vorono\\\"i cells instead of considering uniform weights and observe the impact the may bring to numerical integration. To this end, we consider a basic example where we compute the price of a European call $$C_0=\\mathbb{E}[(X_T-K)_{+}]$$ for a maturity $T$ and a strike price $K$ where the price of the asset $X_t$ at a time $t$ is given by\n$$X_t=x_0e^{(r-\\frac{\\sigma^2}{2})t +\\sigma \\sqrt{t} Z_t}$$\nwhere $r$ is the interest rate, $\\sigma$ the volatility and $(Z_t)_{0 \\leq t \\leq T}$ is an i.i.d. sequence of random variables with distribution $\\mathcal{N}(0,1)$. We compute the price of this European call via a classical quadrature formula using the new weights $p_i^n$ assigned to the VdC sequence instead of uniform weights. We consider \n$$T=1 , \\, K=9, \\, x_0=10, \\, \\mu=0.06, \\, \\sigma =0.1.$$\nThe exact price is approximately equal to $1.5429$ due to the closed formula known in the Black-Scholes case. In figure~\\ref{intnumVdC}, we compute the error induced by this approximation and we compare it to the one obtained by a classical quasi-Monte Carlo method (i.e. where we use the uniform weights of a VdC sequence) and to the one obtained by a quantization-based numerical integration quadrature formula using a greedy quantization sequence of the $\\mathcal{U}([0,1])$-distribution. We observe that the procedure using the greedy quantization sequence converges faster than the ones using the Van der Corput sequence. Consequently, one can say that greedy sequences are more advantageous than low discrepancy sequences, even if we assign to them non-uniform weights.  \\\\\n\n\\begin{figure}\n\t\\begin{center}\n\t\\vspace{-0.5cm}\n\t\t\\includegraphics[width=11cm,height=7cm,angle=0]{comp_callBS_dim1_QG_QMC_QMCpond.pdf}\n\t\\end{center} \n\t\\vspace{-1cm}\n\t\\caption{Price of a European call in a Black-Scholes model via a usual QMC method (blue), greedy quantization-based quadrature formula (red) and quadrature formula using VdC sequence with non-uniform weights (logarithmic scale).}\n\t\\label{intnumVdC}\n\\end{figure}\n\\smallskip\n\\small\n\\noindent {\\em Acknowledgments.} The authors would like to express a sincere gratitude to, Dr. Rami El Haddad, the co-advisor of R. El Nmeir, for his help and advice during this work. Also, they would like to acknowledge the National Council for Scientific Research of Lebanon (CNRS-L) for granting a doctoral fellowship to Rancy El Nmeir, in a joint program with Agence Universitaire de la Francophonie of the Middle East and the research council of Saint-Joseph University of Beirut.\n\\small \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\nAlso by David Rieff\n\n_The Reproach of Hunger: Food, Justice, and Money in the Twenty-First Century_ (2015)\n\n_Swimming in a Sea of Death: A Son's Memoir_ (2008)\n\n_At the Point of a Gun: Democratic Dreams and Armed Intervention_ (2005)\n\n_A Bed for the Night: Humanitarianism in Crisis_ (2002)\n\n_Slaughterhouse: Bosnia and the Failure of the West_ (1995)\n\n_The Exile: Cuba in the Heart of Miami_ (1993)\n\n_Los Angeles: Capital of the Third World_ (1991)\n\n_Going to Miami: Exiles, Tourists, and Refugees in the New America_ (1987)\n\n_Texas Boots_ (with Sharon DeLano) (1981)\n\nPublished with assistance from the foundation established in memory of Amasa Stone Mather of the Class of 1907, Yale College.\n\nCopyright \u00a9 2016 by David Rieff.\n\n\"The End and the Beginning\" and \"Reality Demands\" from _Miracle Fair,_ by Wis\u0142awa Szymborska, translated by Joanna Trzeciak.   \nCopyright \u00a9 2001 by Joanna Trzeciak. Used by permission of W. W. Norton & Company, Inc.\n\nAll rights reserved.\n\nThis book may not be reproduced, in whole or in part, including illustrations, in any form (beyond that copying permitted by Sections 107 and 108 of the U.S. Copyright Law and except by reviewers for the public press), without written permission from the publishers.\n\nYale University Press books may be purchased in quantity for educational, business, or promotional use. For information, please e-mail sales.press@yale.edu (U.S. office) or sales@yaleup.co.uk (U.K. office).\n\nSet in Minion type by IDS Infotech, Ltd.\n\nPrinted in the United States of America.\n\nLibrary of Congress Control Number: 2015954282\n\nISBN 978-0-300-18279-8 (cloth : alk. paper)\n\nA catalogue record for this book is available from the British Library.\n\nThis paper meets the requirements of ANSI\/NISO Z39.48\u20131992 (Permanence of Paper).\n\n10 9 8 7 6 5 4 3 2 1\n_This book is for Dasantha Pillay_.\nToo long a sacrifice can make a stone of the heart.\n\n\u2014W. B. YEATS\nContents\n\n_Acknowledgments_\n\nONE Footprints in the Sands of Time, and All That\n\nTWO Must We Deform the Past in Order to Preserve It?\n\nTHREE What Is Collective Memory Actually Good For?\n\nFOUR The Victory of Memory over History\n\nFIVE Forgiveness and Forgetting\n\nSIX The Memory of Wounds and Other Safe Harbors\n\nSEVEN _Amor Fati_\n\nEIGHT Against Remembrance\nAcknowledgments\n\nThroughout its gestation, this book has had more and better friends already than any writer could sensibly hope for. It also has a slightly checkered history. In 2009, Louise Adler and Elise Berg at the University of Melbourne Press were kind enough to invite me to write a polemic on political memory, which they published two years later under the title _Against Remembrance. In Praise of Forgetting_ builds on the work I did then, and so I want to thank Louise, Elise, and their colleagues \"in the name\" of both books.\n\nIn recent years, I have begun spending as much time as I can manage in Ireland. But being a Hibernophile hardly qualifies me as an expert in Irish history and politics, and on those questions I have had the good fortune to have been the beneficiary of the learning and insight of Rosemary Byrne, Kevin O'Rourke, Cormac \u00d3 Gr\u00e1da, Tom Arnold, Paul Durcan, Denis Staunton, and John Banville in Dublin, and Jim Fahy in Galway. They are of course in no way, shape, or form responsible for the uses to which I have put that learning.\n\nThe same disclaimers apply to the \"tutorial\" on Jewish history, including on Yosef Yerushalmi's work, that my old and true friend Leon Wieseltier has been trying, with what I suspect he would say has been mixed success, to give me for decades now. They also apply to two newer friends, R. R. Reno in New York and Fr. Bernard Treacy in Dublin, who, though their views seem to me to diverge on a number of important issues of interpretation, have both taught me much about the Catholic understanding of the relationship between history and memory. They will be the best judges of the extent to which I have understood them properly, and, to reiterate, any errors I have made are mine alone.\n\nSince the days when I was his student at Amherst College in what now seems almost like another geological era, and _was_ almost forty-five years ago, I have benefited from the learning and friendship of Norman Birnbaum. If I have gotten L\u00f6with, Halbwachs, Renan, and other thinkers on whom I have relied in this book even partly right, this is as much Norman's doing as mine, even if, all these years later, T\u00f6nnies still defeats me.\n\nAnd I would have been defeated in the writing of this book without the extraordinary help I have received during the period in which I was researching it from Megan Campisi, and, during the fact checking after it was finished, from Megan and from Elisa Matula.\n\nFinally, I owe the fact that _In Praise of Forgetting_ exists at all to Steve Wasserman, my editor at Yale University Press, whose gift to me it was to make it possible for me to have another bite at the apple of memory and forgetting. Steve and I have known each other practically our entire adult lives. We were young together, middle-aged together, and now we are growing old together. Given that there is no cure for _that,_ I can't think of a better friend with whom to have shared and still be sharing the ride.\nONE\n\nFootprints in the Sands of Time, and All That\n\nLawrence Binyon's poem \"For the Fallen\" was first published in the _London Times_ on September 21, 1914, six weeks after the Great War had begun. It is sometimes suggested that Binyon, who was a distinguished art historian as well as a poet (he was the British Museum's Keeper of Oriental Prints and Drawings when the war began), wrote the poem in despair over how many had already died and how many more were being condemned to the same fate. But there is no basis for such a reading. Binyon simply could not have known this, if for no other reason than that it was not till the end of the First Battle of Ypres two months later, an engagement at which the majority of Britain's prewar professional army was either killed or wounded, that people at home began to realize just how terrible a toll the war promised to exact.\n\nIn reality, \"For the Fallen\" is a classic patriotic poem, far closer in spirit to Horace's \"Dulce et decorum est pro patria mori\" (It is sweet and fitting to die for one's country)\u2014an injunction that actually had been graven into the wall of the chapel of the Royal Military Academy at Sandhurst in 1913\u2014than to the work of the great British soldier-poets such as Wilfred Owen, who would appropriate the motto for one of his finest poems, but only in order to call it \"the old lie.\"\n\nThat such prescience regarding what was to come was unavailable to Binyon weeks into the war hardly dishonors him. Too old to serve in the trenches, in 1916 he nonetheless would volunteer for duty as a hospital orderly on the Western Front\u2014no mean commitment. And Binyon's poem has endured. As I write this, 101 years after the First Battle of Ypres, \"For the Fallen\" remains the quasi-official poem of remembrance, without which virtually no memorial ceremony for the dead of both World War I and World War II in Britain, Canada, Australia, or New Zealand is considered complete. Its fourth and best-known stanza reads:\n\nThey shall grow not old, as we that are left grow old:\n\nAge shall not weary them, nor the years condemn.\n\nAt the going down of the sun and in the morning\n\nWe will remember them.\n\nIn Australia, where the memory of Australians' sacrifice during the First World War, above all during the Dardanelles campaign against the Turks in 1915, played an extraordinarily important role in the forging of the modern Australian state,1 \"For the Fallen\" is now known as \"The Ode of Remembrance.\" And at many Anzac Day ceremonies, after the fourth stanza is declaimed, it is customary for those present to respond with the words \"Lest we forget,\" as if to the invocation at a church service, which in a sense, of course, it is. In doing so, the participants meld the Binyon poem with Rudyard Kipling's far greater poem \"Recessional,\" from which the line \"Lest we forget\" is taken, a line that, twice repeated, concludes each of its stanzas, as in this, the best known of them:\n\nFar-called, our navies melt away;\n\nOn dune and headland sinks the fire:\n\nLo, all our pomp of yesterday\n\nIs one with Nineveh and Tyre!\n\nJudge of the Nations, spare us yet,\n\nLest we forget\u2014lest we forget!\n\nAs is so often the case in his work, Kipling had a far more complicated and pessimistic view of the world, above all of the fate of nations, than the memory of him either among his critics or his (sadly diminished number of) admirers would lead one to believe. Although advanced in inverted terms, since the poet is in effect appealing for divine intercession for a deferral of what he knows to be the foreordained outcome, \"Lest we forget\" is a mournful acknowledgment that such forgetting is inevitable\u2014both on our own parts as individuals and with regard to us after we are gone. Implicit in Kipling's line is a far more terrifying one: \"When we forget.\" In this, \"Recessional\" echoes the chilling words of Ecclesiastes 1:11: \"There is no remembrance of former things; neither shall there be any remembrance of later things that are to come with those that shall come after.\" And more proximately, Kipling's poem is a gloss\u2014explicitly so, at one point\u2014on Shelley's \"Ozymandias\" and its unflinching meditation on the ephemeral nature of even the most monumental creations and martial accomplishments of human beings and the societies to which they have belonged.\n\nDeep down, we all know this to be true, however difficult it is to imagine how we could live our daily lives with even a bare minimum of serenity or success without at least acting as if we believed otherwise. For to fully make Ecclesiastes 1:11 our true north would be to live as if we were already dead. And this, a few saints and mystics excepted, and surely their capacities in this regard are inseparable from their disengagement from the world, we cannot do\u2014\"No man can stare for long at death or the sun,\" as La Rochefoucauld said\u2014nor, even were it possible, is there any moral or ethical imperative for our doing so. It is probably true that were we to imagine those of our loved ones whom we will survive only as dust, we might be more loving, more forgiving, more willing to put their wishes and desires before our own. But to do this, we would have to be able to live our lives as if we had already experienced the whole of them, which is a contradiction in terms. For as Kierkegaard said, \"Life must be lived forward, but can only be understood backward.\"\n\nA successful society based on such a focus is still harder to imagine. For while such a society might be more \"truthful\" in some absolute sense, fostering or conserving a moral conscience within it would be impossible. Even to care about the present would be challenge enough, let alone finding a way to care about either the past or the future. It may be that \"the ultimate truth is fearless,\" as the Buddhist sage Trungpa Rinpoche once put it. But in thinking about history, this particular truth is far more likely to terrify and paralyze than encourage and inspire. As the German philosopher Karl L\u00f6with put it in his now largely forgotten masterpiece _The Meaning of History_ (1947), \"To ask earnestly the question of the ultimate meaning of history takes one's breath away. It transports us into a vacuum which only hope and faith can fill.\"\n\nBut if we have no other practical choice than to try to live our lives and contribute to our societies as if Ecclesiastes were wrong, this should not manumit us from mustering the courage, at least from time to time, to contemplate the ultimate meaninglessness of history, and to recognize that, to paraphrase Trotsky's quip about war, \"You may not be interested in the geological record, but the geological record is interested in you.\" And even if we radically narrow the frame, excluding not only evolutionary or geological time but also the approximately 192,000 years that elapsed between the emergence in Africa 200,000 years ago of anatomically modern _Homo sapiens_ and the advent of proto-writing (usually ideograms) in the sixth millennium before the Common Era\u2014if not, far more pertinently, the emergence of writing and thus of recorded history 2,000 years later in the fourth millennium BCE\u2014there still can be no reprieve from the reality that sooner or later every human accomplishment, like every human being, will be forgotten. It is this that Kipling is inviting his readers to recognize in \"Recessional\": sooner or later our own polities will vanish just as surely as did Nineveh and Tyre.\n\nWhat possible historical basis could there be for thinking otherwise? The world of states we live in today has existed for only a comparatively small fraction of recorded history, and a great deal even of that history has already been forgotten by everyone except historians. This applies not only to states with very long histories but even to those with very short ones, of which there are in fact a great many. Italy and Germany in the form they exist today date back only to the nineteenth century, while many of the independent African states that emerged out of the ruins of empire in the mid-twentieth century did so with borders that more often than not were the product of British and French colonial mapmakers and bore little resemblance to the polities that existed before the European imperial conquest of the continent. The modern United States from its beginnings as a collection of colonies to the present day is a little more than four hundred years old. The First Fleet sailed into Botany Bay at the beginning of 1788, which means that the modern Australia has existed for a little more than half that time. Even France, which has been a state in the modern sense for far longer than any other in Europe, can be said to have coalesced into the form it takes today only about six hundred years ago at the earliest. And as Theodore Zeldin has shown in his magisterial _History of France, 1848\u20131945,_ the sense of collective French national identity began as a self-conception largely restricted to the ruling classes and became widely shared by the population as a whole only at the time of the French Revolution; it was not wholly generalized until well into the nineteenth century.\n\nGiven how much older China and India are as continuous civilizations, the picture is somewhat different in Asia. But we still cannot speak of a unified Indian state until the advent of the Mughal Empire in the sixteenth century. And even China, by far the oldest state in world history, did not become a unified country until the beginning of the Qin dynasty in the last part of the third century before the Common Era. Again, this may be a very long time in historical terms, but in terms of geological time it is little more than the blink of an eye. We may choose to ignore this, and consider the question exclusively in the context of historical time. Even so, on what basis other than the narcissism of the living or a reckless disregard for history and logic could anyone seriously suggest that even the most coherent and solid of the states that now exist will still be around in anything like the same form in another thousand years, or two, or three?\n\nThe reality is that no intelligent person believes anything of the sort. Whereas a believer in one of the great religions might insist that Christianity, Islam, Judaism, and Hinduism are true, and therefore cannot be \"forgotten\" in the conventional sense of the term (or, at least, can always be \"retrieved\"),2 in the secular world, in the deepest sense, it is inconceivable that the political order that characterizes our own era\u2014which, notwithstanding globalization, is still, broadly speaking, one of nation-states\u2014could be immortal, however long-lived it may prove to be. This is not simply a matter of elementary reasoning. For what history actually shows is that throughout all of recorded history, every society without a single exception has proven to be every bit as mortal as individual human beings. To try to think otherwise is a fool's errand. Buddhism, whether it is a religion or not in the conventional sense, is almost certainly the only philosophical system that teaches its adherents that clinging to the past, like clinging to the self, is a forlorn illusion.\n\nThis is not to deny that there are good and sufficient reasons for living inside such a consoling conceit. Buddhism promises liberation, but the release from suffering it offers requires a degree of self-abnegation that few modern people have the fortitude to contemplate, let alone successfully practice. So, with apologies to Freud, the illusion that nothing that really matters to we who live now will ever be forgotten by our posterity through the millennia does indeed have a future. For if, despite the consoling fictions offered up by a Kantian perspective that illusion, identified as such or not, today dominates the moral imagination of so many of the most scrupulous and ethically conscientious among us, truth and morality can at times be incommensurable, then the same can and must be said about reality and necessity. As the moral philosopher Bernard Williams once quipped, \"The will is as free as it needs to be.\" Reality may demand that we acknowledge the certainty that all nations and civilizations will eventually vanish just as surely as they arose. But how to reconcile what reality demands with the fact that for most people to live without remission in its shadow would be paralyzing, a slippery slope from knowledge to impotent despair. As Kipling wrote in his memoir, _Something of Myself,_ \"Every nation, like every individual, walks in a vain show\u2014else it could not live with itself.\" Instead, as he put it in his poem \"Cities and Thrones and Powers,\" a work every bit as despairing as \"Recessional\":\n\nThis season's Daffodil,\n\nShe never hears,\n\nWhat change, what chance, what chill,\n\nCut down last year's;\n\nBut with bold countenance,\n\nAnd knowledge small,\n\nEsteems her seven days' continuance\n\nTo be perpetual.\n\nPerhaps it is consolation enough if we can believe, even while acknowledging that history has no intrinsic meaning, that it still possesses another kind of meaning, one derived from the way human beings order their experience of it and their aspirations for how it might be better ordered in the present and in the future, thus infusing it with significance and, of course, passing it along to their posterity. But how to reconcile ourselves to the reality that even such constructed meanings are mortal, and take on board the fact that sooner or later the past will recede in importance before it is lost entirely? For here the personal oblivion we call death and the societal oblivion we call forgetting are two sides of the same coin.\n\n\"Birth was the death of him,\" wrote Beckett, and it's as applicable to civilizations as it is to individuals. But just as it is possible for at least some people to shift the center of their perceptions from their own personal fate\u2014nonbeing\u2014to an other-centered focus (and this is probably the only way those of us who find the prospect of our own extinction unbearable can keep from being driven mad by the knowledge of it), there is no reason why the same shifting should not be possible with respect to our collective fate. And if this is even partly right, then the mortality of societies and civilizations need not be regarded solely as a calamity. For a world in which everything endured through all eternity in a form recognizable to those of us living now all the way to the end of humanity's lifespan as a species is as unimaginable as the fantasy of personal immortality, which also posits such an outcome.\n\nAnd might not a Freudian detect an essential connection between the brute facts of our personal and collective transience, our ephemerality (in its denotative meaning of \"lasting for a markedly brief time\"), and the need to see in human life a level of meaning that an individual endowed with absolute rationality would see through in an instant, discerning instead our inevitable biological fate? Freud understood that too much rationality so gravely undermines group life that past a certain threshold no society can survive it. And regardless of whether we completely accept Freud's generalization, we hardly need a profound insight into human nature to understand that to remain productive\u2014and possibly even to stay entirely sane\u2014we human beings need to behave _as if_ the era in which we are fated to live and die and, after we have been extinguished forever, a relatively short period of the future to come about whose essential characteristics we feel that the present allows us to foresee would be recognizable to us were we to be resurrected at some more far-off point in the future.\n\nMost science-fiction writing and television- and moviemaking incarnates this conceit. For although their stories are usually set in the future, far more often than not they are actually transpositions of events taking place today that reflect the hopes and fears of the present age. Sometimes, in what might be called the hopeful, _Star Trek_ model, they extrapolate from our accomplishments and our desires; at other times\u2014the _Planet of the Apes_ model\u2014they mirror our follies and our fears. The Los Angeles of _Blade Runner,_ to cite an example of the second type, is far more reflective of a late-twentieth-century white American's dystopic view of the world that exists today, one increasingly made up of people of color, than it is an imagining of the future.\n\nCertainly, few novelists or scriptwriters working in the genre have conceived of their project in the way that the British science-fiction writer Olaf Stapledon did when he wrote that his work demanded \"a detachment from all private, all social, all racial ends\" and contained within it \"a kind of piety toward fate.\" And in his novels _Last and First Men: A Story of the Near and Far Future_ (1930) and its sequel, _Star Maker_ (1937), Stapledon indeed tried to encompass nothing less than the evolution of the human species\u2014\"this brief music that is man,\" as he described it at the conclusion of the first book\u2014from the twentieth century to two billion years into the future.\n\nBut Stapledon is an outlier. As a general rule, even in science fiction the historical reach of the imagination, whether backward or forward, is limited. And if imaginative literature has a shelf life, so too does historical memory. Take Australia and New Zealand, the two countries where Binyon's poem remains most resonant. As I write this in 2015, the Remembrance Day ceremonies that have been held since the end of World War I almost certainly make not just historical but ethical sense to a majority of the citizens of both countries. In doing so, they are trying to ensure that those whose ancestors fought in that war continue to be honored, while the recent immigrants to Australia and New Zealand, most of them non-European, who make up an increasing proportion of the citizens of both nations, are, depending on one's political views, invited or pressured to assimilate themselves into the national sense of belonging that these ceremonies serve to instill and subsequently to confirm.\n\nIn other countries, though, analogous celebrations of national sacrifice have fallen into desuetude. Anyone who has driven through the small towns of Massachusetts, Connecticut, or Pennsylvania that were founded before the American Civil War will have seen the monuments to the local men who lost their lives serving in the Union Army, while monuments to the Confederate war dead are all but ubiquitous in towns of similar longevity throughout the South. Between the end of the war in 1865 and the centennial of its beginning in 1961, such historical landmarks served as the focal points of widespread remembrance ceremonies. As with every project focused on historical remembrance, whether the goal in any given instance is to forge it, sustain or affirm it, challenge it, reconstruct it, or replace it, there was nothing innocent about these commemorations. As Caroline E. Janney puts it in her remarkable _Remembering the Civil War: Reunion and the Limits of Reconciliation,_ \"As early as 1865, the veterans and civilians who survived the four bloody years of war were acutely aware that people were actively shaping what should be remembered\u2014and omitted\u2014from the historical record. . . . What individuals and communities elected to tell of the war held enormous potential for staking claims of authority and power.\"\n\nAnnual observances honoring the dead were initiated by Confederate war veterans in 1866. Two years later, in 1868, the Union veterans organization, the Grand Army of the Republic, followed suit. On both sides, women played a key role. In contemporary France, the debate over how the country's colonial past should be commemorated is often referred to as the \"memory wars.\" A strong argument can be made that in the French context the expression is excessive. But there can be no doubt that between the end of the American Civil War and the end of the nineteenth century, the metaphor was an entirely appropriate way of describing the incompatible, and on both sides immensely bitter, accounts northerners and (white) southerners put forward as to why the war had been fought. Given the fact that the South had been defeated, it is more than a little surprising that these \"dueling\" memories could be maintained. After all, when a war ends with a crushing victory by one side, as that one certainly did, the victory confers the power unilaterally to shape the collective memory of the conflict\u2014a power that the victors nearly always exercise, as the American, British, French, and Russian occupiers did in post\u2013World War II Germany and the Tutsi-led Rwandan Patriotic Front did in post-genocide Rwanda. Historically, it is only when there is no clear winner that both sides may be able to sustain their own incompatible memories. An example of this is the Bosnia-Herzegovina that came into being after the Dayton Peace Agreement of 1995 brought the war to an end.\n\nIt is for this reason that many historians of the Civil War and of Reconstruction have argued that, as Janney summarizes it, \"White northerners eventually capitulated to Confederate memory.\" Given that from the end of Reconstruction in the 1870s white northerners clearly capitulated to southern segregationism, this would seem beyond debate. And while Janney is adamant that at least for the late nineteenth century the historians' claim is too broad, she readily concedes that it did become the case in the twentieth century. By the 1930s, she writes, white Americans had become \"increasingly receptive to the Lost [i.e., Confederate] Cause.\" She points to the emblematic case of the extraordinary commercial success of the film _Gone with the Wind,_ noting that \"Americans could not get enough of the romantic epic depicting white southern resolve in the face of defeat.\"\n\nBut even assuming Janney is correct about the timing, what was quickly forgotten by virtually every white American, northerner and southerner alike, was that the first celebration of what eventually would become the national holiday first known as Decoration Day and now called Memorial Day took place on May 1, 1865, and was organized by African American freedmen in Charleston, South Carolina, in honor of the 257 Union soldiers who had died in captivity in the local racecourse that the Confederates had used as a prison camp during the war, and whom the freedmen had earlier reburied properly. The event drew more than ten thousand participants (including a brigade of Union soldiers with three African American units, the 54th Massachusetts and the 34th and 104th Colored Troops), as well as drawing wide attention in the national press at the time\u2014a correspondent for the _New York Tribune_ wrote that the event was \"a procession of mourners and friends [such] as South Carolina and the United States never saw before.\" But for post-Reconstruction white America, the event might just as well never have taken place. It was only black Americans who, as Janney puts it, \"continued to offer up a counter-memory of the Civil War and Reconstruction\" completely at odds with what the African American historian Dorothy Sterling despairingly and angrily described in 1961 as a false picture of the conflict, one in which \"brave brother [fought] against brother, with both separately but equally righteous in their causes.\"\n\nIn the early twenty-first century, Memorial Day commemorations themselves largely serve as occasions for a generalized, watered-down patriotism from which the Civil War and the reasons it was fought are increasingly excluded, in favor of honoring the dead of all U.S. wars, but also as the occasion for a three-day holiday weekend during which both the most important American automobile race, the Indianapolis 500, and an important golf championship, the Memorial, are held. It is not that the Civil War no longer resonates in the United States. In a country in which the wound of racism has never been fully staunched, that would be impossible. But it is not a Memorial Day now leached of all its historical specificity but rather the Confederate battle flag that still flies in several southern state capitals and is incorporated into several of these states' flags that keeps the memory of the war alive. That they are there at all is the product of a brilliantly executed political campaign largely orchestrated between the end of Reconstruction and 1915 by Confederate veterans' groups, and even more importantly, as Caroline Janney has shown in another fine study, _Burying the Dead but Not the Past: Ladies' Memorial Associations and the Lost Cause,_ by organizations of white southern women dedicated, in Janney's words, to \"perpetuating nostalgia for the Confederate past, or for what has come to be known as the Lost Cause.\"\n\nCivil rights groups have been warning since the 1950s that continuing to fly these flags was not only immoral but dangerous, and called for their removal. Their worst fears were realized when it became clear that the Confederate battle flag had been an inspiration to the white-supremacist terrorist Dylann Roof, who in June 2015 walked into the evening service at the Emanuel African Methodist Episcopal Church in Charleston, one of the oldest AME churches in the United States, and opened fire, killing nine people, including Senator Clementa Pinckney, the senior pastor. In the aftermath of the massacre, even white southerners who had previously defended official displays of the flag were forced to bow to public pressure to put an end to its use, and in the (particularly egregious) case of South Carolina, finally to stop flying it from atop the state capitol.\n\nIn the aftermath of Roof's acts, the question of why the Civil War had been fought became, for the first time in decades, part of the mainstream debate.3 Many whites, northerners and southerners alike, who had seemed impervious to the proposition that the Confederacy was not a noble lost cause and that celebrating the memory of a secession that had taken place in order to preserve slavery was anything but harmless, began to reconsider their views when confronted by the undeniable fact that the racism that had been the Confederacy's raison d'\u00eatre still lived on in people like Roof. If any good news could be found, it was that, however diminished its importance in the American imagination it may have been 150 years after it had ended, the remembrance of the Civil War remained at least somewhat accessible to many if not perhaps to most Americans.\n\nBut imagine as a thought experiment that the focus of Dylann Roof's racist hatred had been Native Americans rather than blacks, and that he had struck at a church on a reservation somewhere in Massachusetts. Even assuming that the shock would have been as great, there could have been no similar reservoir of historical memory to draw upon that, whether accurately or inaccurately (an important question in its own right but not relevant here), could have linked the crimes of the past with the crimes of the present. The reasons for this would have had nothing to do with history and everything to do with what Americans remember and what they have forgotten. Yet so searing and cataclysmic had been King Philip's War of 1675\u201376, also known as Metacom's Rebellion, that in its immediate aftermath, and for a century thereafter, it would have been inconceivable for either the settlers of the Plymouth Bay Colony or the native Wampanoag and Narragansett nations of Massachusetts and Rhode Island that, in the hoariest trope of the language of remembrance, \"anyone would ever forget\" it. For in the two years during which it raged, the war almost succeeded in destroying the Plymouth Colony, and at least for a time put an end to the European conquest of the northeastern part of what would become the United States.\n\nOn a per capita basis, King Philip's War was the bloodiest in American history. Most of the surviving Native American warriors were executed, and their wives and children transported to the West Indies and sold as slaves. By 1830 the Native American population of Massachusetts and Rhode Island had decreased by about 60\u201380 percent. Settler losses were far smaller but large enough for two eminent historians of the war to describe it as having posed \"a real threat to the colony's continued prosperity, perhaps even to its survival.\" And yet, professional historians aside, King Philip's War is almost never talked about, even by Americans familiar with the Trail of Tears, President Jackson's ethnic cleansing of the Seminoles, Cherokee, Chickasaw, Choctaw, and Muscogee, and with the genocidal wars the U.S. army conducted against Native American nations of the Great Plains and the Southwest.\n\nWhat this shows is that the historical importance of an event in its own time and in the decades that follow offers no guarantee that it will be remembered in the next century, let alone for many centuries after. A case can certainly be made that reminding the Americans of today of what happened during King Philip's War might be salutary morally. But even assuming that it would be possible to do this, at some point in history even events as terrible as that conflict will be forgotten, if only, to put it starkly, to make room for the memory of other, less distant events, just as those alive today must sooner or later die to make room for those yet to be born. And whether one regrets this or not, forgetting and being forgotten are what must happen sooner or later. Would even the most historically minded among us claim that there is a civic duty or a moral imperative to hold services of remembrance annually in honor of the Norman and Anglo-Saxon dead of the Battle of Hastings of 1066 or to mourn the sacrifice of those who fell at the Battles of Sekigahara in 1600 and Osaka Castle in 1615 that led to the establishment of the Tokugawa shogunate in Japan?\n\nAt least those battles are still recalled, however superficially and sentimentally, by most educated people in Britain and Japan, respectively, although whether there is any practical use in doing so, for non\u2013history buffs and those who live near where these battles took place (where commemorations still generate dependable streams of tourist income), is not as self-evident as the reigning pieties about remembrance seem to insist. In any case, the farther one goes back in time, say to the battles of the Chu-Han War in China (206\u2013202 BCE) or to the Battle of Salamis between the Athenians and the Persians (480 BCE), the more questionable any moral justification for commemorating such events becomes. And yet these battles were as critical in their time, as firmly entrenched in the minds and hearts of those who lived through them and for many generations to come, as defining moments for their civilizations, as the destruction of the World Trade Center on September 11, 2001, is for so many people today, Americans and non-Americans alike.\n\nThis does not mean that there is no interest in studying the Battle of Salamis; indeed, one can argue plausibly that, because the second half of the twentieth century and the first two decades of the twenty-first have been something of a golden age for historical scholarship, more is known today about the wars between the Greek city-states and Persia that culminated with the Athenians' victory at Salamis than was understood in Europe during the eighteenth and nineteenth centuries when, culturally, Hellenism provided a central model for poets and philosophers (and to some extent architects) from Schelling and Goethe to Byron and, though to a lesser extent, Matthew Arnold. The German veneration was so extreme that in 1935, the British scholar Eliza Butler could write a book she matter-of-factly and with little fear of contradiction from her peers titled _The Tyranny of Greece over Germany_. The last iteration of this was Heidegger's contention that the only truly philosophical languages were classical Greek and German.\n\nAnother version of the eighteenth- and nineteenth-century European elite's obsession with Greece and Rome was its centrality to the self-conception of the British Empire. As David Gilmour put it in his fine study _The Ruling Caste: Imperial Lives in the Victorian Raj,_ \"British officials [in India] had begun to think of themselves as Romans by the end of the eighteenth century.\" And as Gilbert Murray, the Regius Professor of Greek at Oxford between 1908 and 1924, wrote, \"At home, England is Greek. In the Empire she is Roman,\" a sentiment that Kipling echoes in a number of his best stories, notably \"A Centurion of the Thirtieth,\" where the Roman officers and legionaries are clearly transpositions of the civil servants and soldiers of his beloved Raj. Even when their empire had ended, many members of the British elite told themselves that the United Kingdom could continue to enact a global role by playing Greece to the American empire's Rome\u2014a fantasy that Christopher Hitchens brilliantly anatomized in his _Blood, Class, and Nostalgia: Anglo-American Ironies_.\n\nThis was almost certainly the last contemporary appropriation of classical Greek and Roman history.4 Though there are other causes, to be sure, the eclipse of Latin and Greek as school subjects in Europe and North America has all but ensured the rupture since one cannot be tyrannized by a past with whose cultural and historical references one is entirely unfamiliar. The Greco-Roman past lasted far longer than did the collective memories of most other historical epochs. But the 2,500 years between the Battle of Salamis and the early twenty-first century have finally put an end to it. We can still admire the _Iliad_ and Cicero, the Pantheon and the Acropolis, and be inspired by their force and beauty, but only in a form stripped of any authority over our moral and political imaginations. As vehicles for our mythmaking about ourselves, they are as good as lost to us.\n\nBut what a run! In most cases, the connection to earlier political orders, and to the art and thought they produce, is severed far more quickly. Think of Rubens's portraits of the seventeenth-century English nobility; in their own time, these were clearly understood as glorifications of the Stuart dynasty and as apologias in pictorial form for the divine right of kings. Today, the political and moral beliefs and understandings that are at the heart of Rubens's work, informing his way of seeing, have no hold over us precisely because he was so much a man of his time\u2014indeed, in many ways its biographer. We have only to contrast him with Turner, whom art critic John Berger once described as \"a man alone, surrounded by implacable and indifferent forces.\" Turner is still accessible to us not because he was a better painter than Rubens but rather because, unlike Rubens, he exists outside his own time, thanks to the logic of his work, thus allowing even those who are indifferent to the tradition from which he developed to understand him as their contemporary.\n\nOf course, most painters whose work still matters to us are more in the mold of Rubens than of Turner. And as long as the past remains legible to the present, we may have preferences but we do not have to choose. But what happens when the past, even the quite recent past, becomes illegible? To take another example from the history of painting, Picasso was interested in Vel\u00e1zquez, who died 220 years before he was born. But in the early twenty-first century, young painters are rarely interested even in their predecessors of 50 years before them, let alone those separated by centuries. And once the fundamental disconnection of the processes by which the past of an artistic, or political, or ethical tradition is transmitted down through the generations has been achieved, it is difficult to see how it can ever be successfully undone or redressed. Admittedly, this is not a matter of forgetting in the proper sense of the word: one cannot forget what one never knew. As the philosopher and sociologist Theodor Adorno puts it in _Minima Moralia,_ \"Just as voluntary memory and utter oblivion always belonged together, organized fame and remembrance lead ineluctably to nothingness.\"\n\n1. By the 1920s, Anzac Day, which was first celebrated in both Australia and New Zealand on April 25, 1916, to commemorate the first anniversary of the landing on the Gallipoli peninsula that marked the beginning of that ill-fated campaign, had eclipsed Empire Day, which was held every May 24, Queen Victoria's birthday.\n\n2. Buddhism is a special case to the degree that it insists on the impermanence of all things, another way of saying it insists that nothing lasts, in the memory or anywhere else.\n\n3. So did the issue of the laxity of U.S. gun laws, but even those optimistic about the use of the Confederate flag finally being ended were under no illusions that American gun laws were going to change one iota as a result of the slaughter.\n\n4. The American empire has other myths (above all that it is not an empire); the modern-day Chinese empire will have still others, though as yet these are still largely on the drawing board.\nTWO\n\nMust We Deform the Past in Order to Preserve It?\n\nEverything will indeed be forgotten sooner or later, but as the gyrations that have taken place between 1865 and the early twenty-first century in the way the American Civil War has been remembered painfully illustrate, it turns out that it is actually quite easy for nations or groups to \"revise\" and \"rewrite\" their collective memories. Since, as the great French historian Jacques Le Goff once remarked, \"memory only seeks to rescue the past in order to serve the present and the future,\" it is hardly surprising that exercises in collective historical remembrance far more closely resemble myth on one side and political propaganda on the other than they do history, at least as that is understood as an academic discipline\u2014the kind of history that when done properly is always critical and whose insights, though they may from time to time be deemed useful to society as a whole, were not set out to instruct. In contrast, historical remembrance is generally considered valuable insofar as it is of service to society.\n\nOur understanding of memory as a social construct goes back to the 1920s and derives in large measure from the pioneering work of the French sociologist Maurice Halbwachs, the most gifted of \u00c9mile Durkheim's disciples.1 Halbwachs would almost certainly have found the late-modern claim that everything, from human sexuality to our understanding of our historical traditions, is socially constructed both unoriginal and largely self-evident. What mainly interested him was the reconstruction rather than the deconstruction of how societies \"remember.\" More specifically, in the words of the American sociologist Lewis Coser, Halbwachs understood collective memory as \"a reconstruction of the past in the light of the present.\" Halb-wachs derived much of his theory on group memory formation from the British neurologist Henry Head's studies of World War I veterans who had suffered head wounds. \"What [someone with aphasia] lacks,\" Halbwachs wrote, \"is less memories themselves than the framework in which to situate them.\" Obviously there is no such thing physiologically as collective memory, but what Halbwachs understood was that this does not make it any less of a sociological reality in which, to use the historian Horst M\u00f6ller's gloss on Halbwachs's argument, \"collective consciousness applied to memory results in _collective_ memory,\" memories that, in turn, \"mold social groups, generations, and nations and constitute _identity_.\"\n\nHalbwachs was keenly aware of how costly the process could be. For while, as he put it, \"The various groups that compose society are capable at every moment of reconstructing their past,\" at the same time that \"they reconstruct it, they deform it.\" But for all its brilliance and originality, Halbwachs's work never sought to address why, both politically and morally, some of these \"reconstructions\" and \"rewritings\" have proven to be harmful or at least far riskier than others at given points in history but then later evolve or are transformed (the mechanisms that make this happen are rarely straightforward) into phenomena that pose little or no danger to the societies and polities to which they were once so toxic.\n\nTo say that collective memory is a social construct, however, tells us little or nothing about the moral character of such remembrance. The proof of this, as Le Goff has pointed out, is that even in the present golden age of collective memory, commemorations of the past were never more common than in Hitler's Germany and Mussolini's Italy (one might add Lenin and Stalin's Russia and Mao's China). And certainly if Coser's rendering of Halbwachs's thesis is right, and memory needs \"continuous feeding from collective sources and is sustained by social and moral props,\" then if the morality of those sources is tainted, the memories being created or solidified will be poisonous as well.\n\nThese are extreme cases and as such are almost certainly less instructive than they may appear (though doubtless they perform a pedagogic service for those who continue to believe that remembrance is always positive). The more difficult cases are those in which a particular set of \"constructed\" collective memories serves to foment or exacerbate anger and conflict at one moment in time and then, a few generations later, is not only seen as harmless but is enlisted in the diametrically opposed mission of palliating the culture of grievance that appealing to such forms of remembrance once nurtured. Irish history provides a particularly illuminating case study of the uses and misuses of the past in the construction, reconstruction, amendment, and transformation of the collective memory.\n\nThat the mythical Ireland still to be found in the frozen republicanism of a declining portion of the Irish diaspora and in the clich\u00e9s of the Irish National Tourist Board never really existed\u2014any more than the mythical France of baguettes and reason, the mythical United States as humankind's last best hope, or the mythical China as the only significant civilization in the world existed\u2014should not need stating. But what to make of what the Irish historian John M. Regan has called the foundation-myth of the Irish state that venerated Irish history as \"the immemorial struggle against English misrule, . . . eulogized physical-force and honored the pieties of separatist republican-nationalism . . . [and] was rooted in quasi-religious representations enveloping the republican rebellion or 'rising' of Easter 1916?\" While obviously many Irish people never believed anything of the sort, and many so-called revisionist Irish historians tried to modulate and modify it, between Irish independence in 1921 and the so-called Celtic Tiger economic boom of the late 1990s and early 2000s followed by the crash of 2008,2 the dominant strain of collective memory in Ireland (and even more strongly in the Irish diaspora) nonetheless remained anchored in what the Irish literary critic Declan Kilberd has called \"the old morality-tale of Holy Ireland versus Perfidious Albion.\" This was confirmed institutionally by the political domination over Irish life for the first seventy-five years of the state by one political party, Fianna F\u00e1il, that saw itself as the inheritor of the \"spirit\" of Easter 1916.\n\nIt was this mix of Fenianism and Catholicism, along with a particular variant of Irish cultural nationalism with its emphasis on the Irish language incarnated by one of the leaders of the Easter Rising, Patrick Pearse, and the rejection of Irish political parties seeking Home Rule for Ireland by peaceful means, that would for most of the history of the Republic of Ireland become what the Irish historian R. F. Foster has called the country's \"received wisdom\" about its own history. In reality, though, what Foster once described as \"Pearse's particular ideology of blood-sacrifice and mystical Catholicism\" represented only one strain of nineteenth- and early-twentieth-century Irish nationalism, and a narrower strain of the Irish cultural revival between the late 1880s and the beginning of the First World War. As Foster has shown in his _Vivid Faces: The Revolutionary Generation in Ireland, 1890\u20131923,_ many held views that could not have been farther from Pearse's, and were committed to what Foster has described elsewhere as \"an opening out of attitudes, a modernization of nationalism, an exploration of cultural diversity, [and] a questioning of too-readily-received forms of authority in public and indeed private life.\" Foster even speculates at the end of _Vivid Faces_ that for many members of the revolutionary generation, the revolution they made \"may not have been the revolution they intended, or wanted\"\u2014a historical fate hardly unique to Irish revolutionaries.\n\nEven leaving aside the ways that the central cultural and social questions such as religious faith and sexuality were framed at the time in Ireland and focusing solely on politics, the consensus among Irish historians today is that the shift of the center of gravity of Irish nationalism from parliamentary politics to the Romantic cultural nationalism of the Gaelic Revival and the armed resistance of the Irish Volunteers, the military wing of the Irish Republican Brotherhood, occurred in large measure because of the political failure of Irish constitutional nationalism, in which the best option for Ireland was seen as Home Rule within the British state system. Pearse might still have denounced the Home Rulers' leader in 1916, John Redmond, for \"untruth and blasphemy . . . the mumblings and gibberings of lost souls,\" but had not Redmond's predecessor as leader of the Irish Parliamentary Party, Charles Stewart Parnell, been brought down by sexual scandal in 1890, the collective memory in Ireland of what was at the heart of Irish nationalism might well have been different.\n\nInstead, however, the idea that there had been a symbiotic relationship between the political and military goals of the Easter Rising and the cultural ambitions of the Gaelic Revival, and that taken together these represented the collective goal of the people of Ireland, came to be so central to the collective understanding of Irish nationalism that in _Ulysses_ James Joyce could invent a nationalist conversation in Barney Kiernan's pub in which the politics are overwhelmingly cultural and revolve around \"Irish sports and shoneen games the like of lawn tennis and about hurley and putting the stone and racy of the soil and building up a nation once again.\"\n\nNone of this is \"historical.\" No one in the milieu Joyce drew on for nationalist characters in _Ulysses_ imagined that it was possible to re-create the clan-based Irish nation that had existed before Cromwell's slaughter of the Irish or, going still farther back, Henry VIII's unseating of the Fitzgerald dynasty. Nor would Joyce's republicans have wanted to, for that matter. What was of concern to them, as has generally been the case for cultural nationalists whether in Ireland or elsewhere, was the trauma of the Irish nation's destruction and the heady prospect of being able to breathe life back into it. At first glance, the rhetoric accompanying this appears to be highly specific, as when, to return to Joyce's archetypal rendering, \"Irish\" sports such as hurley are described as the antitheses of shoneen ones (the word is a derogatory description of Irish people who prefer English attitudes and styles). But in historical terms, these memories of the past are as imprecise and, in some cases, as anachronistic as they are impassioned. Crucially, there is absolutely no room for ambiguity. And yet as Foster has said about Irish history (and as with \"his\" revolutionaries' disenchantment with their revolution, this is true of a number of other nations as well), its ambiguities \"are, in many ways, the most distinctive thing about it.\"\n\nBut as Foster himself certainly knows, even if he is right, that is beside the point in the sense that the essence of historical remembrance consists of identification and psychological proximity rather than historical accuracy, let alone historical nuance and depth. Perhaps this is what led Foster in one of his best essays, \"Theme-Parks and Histories,\" to declare that he had some sympathy for the Northern Irish critic Edna Longley's suggestion that the next commemoration of Irish history \"might take the form of raising a monument to Amnesia, and forgetting where we put it.\" As a historian, Foster wrote, \"I have to be rather shocked by the idea. But as an Irishman I am rather attracted to it.\"\n\nIrony aside, the question of whether historical remembrance is constructed, imagined, manufactured out of whole cloth, or willed into being is of profound importance to professional historians\u2014not least because, as M\u00f6ller has observed, they \"know that myths and legends can be as historically important and politically potent as reality, if they happen to match a society's expectations at a given point in time.\" But as the best historians realize, the more important collective memory is for a society, the less likely they themselves are to be heeded. And this is leaving aside the question Claude L\u00e9vi-Strauss was inviting the profession to consider when he asked whether \"when we try to do scientific history, do we really do something scientific, or do we too remain astride our own mythology?\"\n\nWhat both professional historians and the general public agree on is that, as the nineteenth-century (nationalist) French historian Ernest Renan put it, nations are founded \"on a rich legacy of memories.\" Or, in the sociologist Dominique Schnapper's more modern formulation, \"For individuals and peoples alike, memory is the predicate of self.\" In Renan's case, although he was himself a superb archival researcher, a master of both written and material sources, he did not call for such memories to be based on the best available historical research. To the contrary, he was adamantly opposed to letting the historical chips fall where they may. \"Forgetting,\" he wrote in _What Is a Nation?_ (1882), \"and I would even say historical error, is an essential factor in the creation of a nation.\"\n\nAnd far from welcoming \"the progress of historical studies,\" Renan insisted that it often posed \"a danger to [the sense of] nationhood.\" Taken out of context, this may seem more romantic than it actually was. For Renan was under no illusions: \"Nations,\" he wrote, \"are not something eternal. They had their beginnings and they will end.\" (Presciently, he foresaw the eventual replacement of nation-states such as France and Germany by what he called a \"European Confederation.\") Given this eventual fate, and given that Renan understood the continued existence of a nation as being based on what he called a \"daily plebiscite\"\u2014the risk of disintegration that it faces from its beginning to its end\u2014Renan insisted that, if called upon to choose, a nation would be well advised to opt for myth, codified in collective remembrance, over history.\n\nIt is a tribute to Renan's influence that his contention that a nation is \"a large-scale solidarity\" remains, whether implicitly or explicitly, the dominant view not only of most countries' elites but of ordinary citizens as well. Were this not the case, other than invoking the old Marxist dogma of \"false consciousness,\" it would make no sense that the increasingly common attempts to debunk or even modify collective national myths have so often provoked such alarm. An extreme example of this, in the instance that shared myths guaranteed democracy and shattering them represented an existential danger to it, was the suggestion in 1971 by the Irish political scientist Brian Farrell that \"once the myth of the state is questioned extra-constitutional methods become valid.\" A less apocalyptic iteration of this view is that while coherent and persuasive collective memory may be able to be formed with little regard for historical accuracy (a case can be made that, indeed, such indifference is a sine qua non of such projects), too much critical history can undermine the consensus over what societies need to recall and what they need to forget if their cohesion and their citizens' sense of mutual solidarity and, probably more important, allegiance to the state is to be maintained. Or, to put it another way, can a sense of national belonging endure if pride in a nation's history is undermined by the wide dissemination of those elements of the historical record Renan was convinced needed to be ignored or forgotten?\n\nWe are far removed from scholarly debate here. For not only are we not talking about history, we are not even talking about memory in any proper sense, but rather about morality, ideology, and, often, the intellectual raw materials for cultural and political mobilizations. To insist on the point is not to assert that the claims of collective memory are equally powerful at all times in all places, or are necessarily always compelling even in countries such as Ireland where the authority of nationalism has run so deep for so long. Writing of his own childhood in the 1930s and the early to mid-1940s, for example, the Irish writer John McGahern could affirm that \"the 1916 Rising was not considered to be of any great importance in the country I grew up in.\" Because Easter 1916 had occurred comparatively recently, he explained, \"it probably was too close in time for the comfort of mythmaking.\"\n\nBut at other times, the need for the security that such mythmaking affords can seem desperately important. For example, large numbers of people in the United States today care passionately about whether American schoolchildren should continue to be taught, as they have been since the founding of the republic, that Columbus was the heroic discoverer of America or whether instead, as at least some U.S. schoolchildren are now being taught, he should be portrayed as a brutal and amoral servant of the Spanish Empire intent on despoiling a continent whose location he could not even correctly identify.\n\nIt might be more convenient for those who favor the revisionist version if those who insist that the traditional view of Columbus must continue to prevail were know-nothings who were either unable or unwilling to face historical reality. But the matter is hardly so simple, and there is plenty of anecdotal evidence that suggests that what most worries those who favor the retention of the traditional curriculum is the possibility (some would say the inevitability) that if the old consensus is repudiated, not only will Americans no longer understand how to take pride in their nation's origins, many will be tempted to repudiate its present as well. In a country of immigrants, an increasing proportion of whom have arrived within the past decade and still maintain ties with their nations of origin via the Internet that would have been impossible even half a century earlier, such concerns are defensible. Nor are they restricted to the United States. The early twenty-first century has turned out to be one of the great periods of global mass migration in modern history. Given this unanticipated version of the Americanization of at least the rich world, it should come as no surprise that similar \"education wars\" are raging all over the Global North.\n\nThe case of Australia is especially illuminating. During the late 1990s and early 2000s, John Howard's conservative government made a concerted effort to blunt what it viewed as the inroads of a leftist \"multicultural\" view of the country's history, one that the Australian center right and right believed wildly overemphasized Australia's mistreatment of its indigenous peoples. By accentuating the crimes and failures of the Australian state and Australian society over its positive accomplishments, conservatives believed that the multiculturalists were depriving new immigrants, the majority of whom now came from Asia and the Middle East rather than, as in the past, Europe, of the chance to be integrated into the Australian national \"family.\" The multiculturalists, Howard claimed, had espoused a \"black armband\" view of history, and even their putative accomplishments\u2014above all in recognizing the sufferings and celebrating the accomplishments of marginalized populations, especially the country's Aboriginal peoples\u2014sooner or later would be dwarfed by the social discord they would engender and the national disunity they would sow.\n\nWe do not have to accept Howard's prescriptions to recognize that some prominent Australian multiculturalists have left themselves open to the charges he leveled at them; political positions aside, their indifference to, not to say disdain for, the historical record can be breathtaking. An emblematic figure in this is the University of Melbourne academic Chris Healy. He has called for new histories to be written of the story of Eliza Fraser\u2014a European woman who, after having been shipwrecked along the Queensland coast in 1836, lived with the Ngulungbara people for some years before being found by a settler search party. Healy's proposal might seem reasonable were it not for the fact that he went on to insist that these new works would need to \"eschew a desire to rescue an authentic Eliza Fraser.\" And Healy was anything but the only prominent Australian multiculturalist to treat history with such disdain.\n\nHowever, it is important not to exaggerate, as so many conservatives, and by no means only in Australia, often do, the existential dangers to our societies that this crass form of multiculturalism poses. Bitter as these history wars often are, the experience of the past century in the United States, Canada, Australia, and western Europe shows that not only do all such cultural conflicts end at some point but that, whether explicitly or implicitly, they invariably end in some form of compromise in which elements of the older and newer conceptions of the national past are joined. In any case, even were such \"historic compromises\" to prove impossible to arrive at, it is an illusion to imagine that collective historical myths remain fixed for more than a few generations; viewed from a longer perspective, they eventually mutate, often out of all recognition from where they started. As Frances FitzGerald put it in _America Revised,_ her history of American history textbooks from the nineteenth century to the late 1970s (when her book was published), \"These works are consensus documents . . . themselves a part of history in that they reflect the concerns, the conventional wisdom, and even the fads of the age that produced them.\"\n\nAnd at some point, the reigning pieties, whatever they are, begin to ring hollow. In 1918, when Lytton Strachey published _Eminent Victorians,_ his series of debunking portraits of four of the great iconic figures of Victorian England\u2014General Charles \"Chinese\" Gordon, Florence Nightingale, Henry Cardinal Manning, and Thomas Arnold\u2014the historical facts he uncovered were scarcely new, nor were his arguments particularly convincing from a scholarly perspective. To the contrary, in the words of the old Oxbridge joke, \"what was true wasn't new, and what was new wasn't true.\" If the book found a receptive audience, it was largely because, in the last days of the First World War, the myths and memorial pieties of the prewar era had been completely undermined and discredited by the slaughter on the Western Front and in the Middle East, at Gallipoli and Kut, and the British public was ready for a radical revision of what, with apologies to those involved in the education wars, might be called the established canon of collective memory.\n\nIn his introduction to _The Invention of Tradition,_ the pathbreaking book he edited with Terrence Ranger in 1983, Eric Hobsbawm wrote of what he called \"new traditions\" being established \"when a rapid transformation of society weakens or destroys the social pattern for which 'old' traditions had been designed.\" One of the most startling cases described in the book is that of the Highland tradition in Scotland, particularly in its sartorial expressions such as the kilt and \"clan tartans.\" The former appears to have been invented in the 1720s by an English Quaker from Lancashire, Thomas Rawlinson, while the latter did not exist in their modern form until the 1750s. _The Invention of Tradition_ focuses on Britain and the British Empire, but as Hobsbawm pointed out in his introduction, similar phenomena are to be found in many if not most countries. The white southern women who began forming Ladies' Memorial Associations soon after the end of the Civil War are one example of this. As Caroline Janney has put it, these associations \"succeeded in creating memorial tributes and traditions that intensified existing emotional attachment to the Confederate cause.\"\n\nThere can be little doubt that the efforts of the southern women's organizations were crucial to creating collective memories among white southerners that were extremely damaging to the nation both morally and politically. Yet when Horst M\u00f6ller, writing of Europe at the end of the Second World War, insists that both \"the individual memories of individual Europeans and the collective memory of nations could not remain as they had been,\" we sympathize with his view. Whether we can call the collective memory of the white American South an instance of the abuse of memory and the transformation of post-1945 European remembrance an example of memory's appropriate use (as the French philosopher Tzvetan Todorov, who used _The Abuses of Memory_ as the title for a fascinating small book on the subject, presumably would) is a key question, and its answer is far less morally self-evident than is commonly supposed.\n\nHow does collective memory function? To understand this, the political scientist Benedict Anderson's template of the \"imagined community\" is invaluable. The distinctions that need to be drawn between how history and memory operate and the different ends they serve are by themselves insufficient. What the study of the engineering of traditions and the model of the national as an imagined community help clarify is the extent to which collective historical remembrance falls between history and memory, in a sense using both instrumentally without being a great respecter of either.\n\nWhatever its purposes, the authority of collective memory depends, as Renan understood, on our not inquiring too insistently about its factuality and not worrying overmuch about its contingency, but instead allowing ourselves to be swept away by a strong emotion dressed up in the motley of historical fact. Typologically, it matters little whether the feeling in question is one of solidarity, of mourning, of love of one's own nation or disenchantment with it, or of hatred for another's nation or envy of it. Where remembrance is concerned, it would seem that it is Nietzsche who has the last word: \"There are no facts, only interpretations.\"\n\nWhat are the dangers here? The most obvious is that, too often, the suspension of disbelief required to keep faith with such collective self-conceptions has proved to be far more costly humanly and politically than is commonly assumed. The Oxford historian Timothy Garton Ash once asserted confidently in an interview that just as \"a person without a memory is a child,\" so \"a national or any political community without memory is likely to be childish.\" But it is anything but self-evident that this is true. Empirically there is much to support the contrary argument: in many parts of the world it is not the relinquishing but the holding on to memories that seems to make societies childish. And in societies in which there is a real risk of fragmentation or worse, invoking certain memories can sometimes resemble nothing so much as the proverbial yelling of \"Fire!\" in a crowded theater.\n\nThe crucial point is this: we do not have to deny the value of memory to insist that the historical record (the verifiable one, not the mythopoeic one) does not justify the moral free pass that remembrance is usually accorded today. Collective historical memory, and the forms of remembrance that are its most common expression, are neither factual, nor proportional, nor stable. To be sure, were the political implications of this largely positive or, failing that, at least largely neutral, then arguing for a more skeptical view of remembrance would be both disrespectful to all those people to whom it provides strength and solace, and unnecessary. But is this the case?\n\nPresumably, Timothy Garton Ash would insist that it is. \"Memory,\" he has said, \"is a vital part of building a European identity,\" and he has been critical of the inroads that what he has described as the American understanding of history as something relatively unimportant have made into the central and eastern European view of its significance. But the strength of the U.S. model, and the problematic character of the one Garton Ash has championed, have been brutally exposed by the hostility with which the Central and Eastern European countries to which he refers have reacted to the arrival of more than one million refugees and economic migrants from Syria, Iraq, Afghanistan, sub-Saharan Africa, and Bangladesh and Pakistan in 2015, and to the very real prospect that they are the \"advance guard\" of migrant flows that will soon increase to many millions. And however politically incorrect it may be to point this out, if memory is an essential element to building a European identity, then the arrival of masses of people who share none of these memories, and, more to the point, bring with them extra-European memories of their own, will unquestionably make the project Garton Ash espouses at the very least a great deal more difficult in the near and medium term, and may perhaps force a radical rethinking by Europeans of the role of memory in the definition of what it means to be European in the twenty-first century.\n\nIn contrast, an Australian or a New Zealander, each coming from a society that, on balance, has made a successful transition from monocultural to multicultural societies, might understandably react to my notes of caution by demanding to know what could be wrong with wearing a poppy in one's lapel on Anzac Day or Remembrance Day. Part of the reply to this is that nothing is wrong with it: it is an entirely appropriate and decent thing to do precisely because people in Australia and New Zealand do not wear poppies in their lapels on all the other days of the year, and that the significance of the gesture would be very different if they did. Such blanket displays are scarcely unknown. Think for example of the significance in the United States of wearing an American flag lapel pin. Whatever the original intention behind wearing it was, such pins rapidly became emblems of the American right, ubiquitous on Fox News though rarely seen on the other major networks. But the gesture so resonated with the U.S. public that only a few years later not wearing the pins left a politician open to the charge of being unpatriotic, and as a result many liberal American senators and representatives started wearing them as well to rebuff such charges. This group included Barack Obama, who only began to do so when in 2007, while still a U.S. senator, he launched his campaign for the presidency.\n\nMemory can be used as a political litmus test, for good causes and bad ones alike, in much the same way as a lapel pin bearing the image of the national flag (the practice is now no longer exclusively American) is used. If we assume that this is the case, then the question for those who insist, as Garton Ash does, on the link between weak collective memories and weak collective identities, is whether this is always a bad thing and if so, why? It should go without saying that there are no easy or definitive answers to this. Nietzsche wrote, \"The question of the degree to which life requires the service of history at all is one of the supreme questions and concerns in regard to the health of a man, a people, or a culture.\" To offer two extreme examples, one can agree with M\u00f6ller that \"the culture of memory [in Germany after 1945] is one of the great moral, political and societal achievements of the Federal Republic of Germany,\" as has been demonstrated most recently by the humane welcome Angela Merkel's conservative government in Berlin has afforded to the masses of refugees and migrants for whom Germany stands as a modern-day promised land. But would one not say, and not just in light of the slaughter at the Emanuel African Methodist Episcopal Church in Charleston, that the culture of memory of many white southerners from the Confederates' surrender at Appomattox in 1865 to the present day is one of the great moral, political, and societal failures of the United States?\n\nThe example of the American South is anything but unique. I remember going to Belgrade in 1993 to visit Vuk Dra\u0161kovi\u0107, the Serb nationalist politician and writer who was then leading the mass opposition against the Slobodan Milo\u0161evi\u0107 regime, and had drawn liberal as well as ultra-nationalist support in Serbia for his cause. As I was leaving his office, my head still ringing with Dra\u0161kovi\u0107's romantic paeans of praise for the Chetnik leader Dra\u017ea Mihailovi\u0107, one of his young aides pressed a folded bit of paper into my hand. It turned out to be blank except for a date: 1453\u2014the year Orthodox Constantinople fell to the Muslim Ottomans. Friends of mine who had worked in the former Yugoslavia during the Croatian and Bosnian wars had similar experiences in Zagreb and Sarajevo, though the dates in question were different. It certainly seemed as if the \"sores of history,\" as the Irish writer Hubert Butler once called them, remained unhealed more than half a millennium later\u2014at least in the desperate, degraded atmosphere of that time and place.\n\nThere is a broader and more consequential lesson to be learned from this. It is that far too often collective historical memory as understood and deployed by communities, peoples, and nations\u2014which, again, is always selective, more often than not self-serving, and historically anything but unimpeachable\u2014has led to war rather than peace, to rancor and ressentiment (which increasingly appears to be the defining emotion of our age) rather than reconciliation, and to the determination to exact revenge rather than commit to the hard work of forgiveness.\n\nThis is what happened in the American South after 1865 and, while diminished, is still happening today; it blighted the former Yugoslavia in the 1990s. Today, it is endemic in Israel-Palestine, in Iraq and Syria, in the Hindu nationalist populism of India's Bharatiya Janata Party, which the government of Narendra Modi is systematically institutionalizing, and among jihadis and Islamists both in the Muslim world and in the Muslim diaspora in western Europe, North America, and Australia. To insist on this is emphatically not to suggest that there is an easy solution. On the contrary, it is probable that human beings' need for community, already compelling in times of peace and plenty, comes to feel like a psychic and moral necessity in troubled times. But at least let there be no turning a blind eye to the high price societies have paid and are continuing to pay for the solace of remembrance.\n\nHistory is not a menu. You can't have the solidarity that a national myth helps form and sustain without the self-absorption, nor can you have the pride without the fear. Nor, despite all the loose talk about globalization, is the universal ethical community that in his fine study _The Ethics of Memory_ the Israeli philosopher Avishai Margalit has called for\u2014here echoing Kant's vision of a worldwide civil society administering universal rights\u2014anything other than a pipe dream, at least for now.\n\nMargalit is certainly correct in concluding that if we could create what he calls a shared moral memory based on some generally accepted moral minima, there could be a globalization of conscience on a par in the ethical sphere with the globalization of capitalism and of migration. Yet he does not satisfactorily address the problem raised by the scholar of cultural reporting Susie Linfield of what such collective remembering by \"humanity as a whole\" would mean. \"Why, and how,\" she asks, should \"[a victim] of the Rwandan genocide remember those of the Gulag? And why should a comfortable 25-year-old citizen of, say, Amsterdam\u2014or Cairo or Beijing\u2014remember either?\" Margalit himself concedes that creating and legitimating the shared moral memory he is calling for will be extraordinarily difficult. But having set out the daunting character of the task, he then in effect falls back on the old German idealist view that holds that if something should be done, if it is truly a moral imperative, then it must be done\u2014the world as \"cosmological idea,\" to deploy another Kantian formulation. This injunction can be summarized in the formula \"ought implies can.\"\n\nIn a time in which universal values seem to be under assault everywhere, even as, paradoxically, our era is increasingly defined by mass migrations on a scale not seen in generations and from much wider geographical sources than seen in centuries, this scarcely seems likely. There is even a whiff of eschatology about it: \"Alle Menschen werden Br\u00fcder\" (all men become brothers), The End. Even more problematically, Margalit's vision of the future is too close to the old fantasy of world government\u2014a moral analogue to Esperanto that is admirable, but when all is said and done is a waste of hope. As the philosopher and historian Leszek Kolakowski once remarked, \"We can imagine a universal brotherhood of wolves but not of humans, since the needs of wolves are limited and definable and therefore conceivably satisfied, whereas human needs have no boundaries we could delineate.\"\n\nFor better or worse, how we weigh the human need for remembrance against remembrance's dangers must take place in the context of how we live now and who we are as human beings. \"In the terrifying time in which we live and create,\" wrote the exemplary historian of Judaism Yosef Yerushalmi, \"eternity is not our immediate concern.\" Nor do we live in a world that, even were it to be transformed into a less terrifying place (which in the age of global warming and Daesh [also known as ISIS] hardly seems likely), ever will be converted to idealistic Kantian absolutism. Our challenge is different: to keep our bearings in a world in which, too often for it to be dismissed as a limiting case, values have proven to be incommensurable.\n\nAn obvious example of this is the often though by no means always inimical relationship in the aftermath of savage wars among three virtuous goals\u2014truth, justice, and peace. This is especially relevant to the increasing number of conflicts in which there is no clear victor. In such circumstances, often what is needed is appeasement in its denotative sense of \"soothing\" or \"bringing peace,\" though since the word has become so identified with Neville Chamberlain and \u00c9douard Daladier in Munich in 1938 caving in to Hitler over Czechoslovakia, another term would have to be found. To achieve such appeasement will require some version of what the British philosopher John Gray has described as a modus vivendi among civilizations, cultures, religions, and nations. Far from political remembrance being always a moral imperative, then, there will be times when such remembrance is what stands in the way.\n\n1. Halbwachs had a long and distinguished academic career. In 1944, he was awarded the chair in social psychology at the Coll\u00e8ge de France. A few months later, he was arrested by the Gestapo and died of hunger and dysentery in Buchenwald in 1945. Jorge Sempr\u00fan, a fellow prisoner with him there, wrote an extraordinary account of Halbwachs's death in his memoir _Literature or Life_.\n\n2. As Fintan O'Toole, a fierce critic of the Ireland of the boom, has pointed out, it at least liberated the Irish from the crushing weight of historical memory.\nTHREE\n\nWhat Is Collective Memory Actually Good For?\n\nOld men forget, as Shakespeare has the king say in _Henry V_. But left to their own devices, societies forget too, and far more quickly than they used to. This \"acceleration of history,\" as the French historian Daniel Hal\u00e9vy dubbed it, has been recognized since at least the latter part of the eighteenth century, and is the product both of the scientific revolutions of the previous century and of the intellectual and philosophical revolution that was the Enlightenment that culminated in the American and French Revolutions. In both its original and its subsequent forms, conceiving of history as something that accelerates, or even\u2014as two of the central figures of the French Revolution, Condorcet and Robespierre, believed, for all that divided them\u2014that can be _made_ to accelerate, is inseparable from the idea of progress, which itself is a concept that historically only slightly predates the Enlightenment. Indeed, the historian Hugh Trevor-Roper once described the idea of progress as a seventeenth-century heresy.\n\nIn the orthodox Christian view, as Trevor-Roper noted, progress is simply not possible. History has a trajectory all right, running from the Creation to the Last Judgment, but as the French historian Fr\u00e9d\u00e9ric Rouvillois put it in his pathbreaking _Invention of Progress, 1680 to 1730,_ any history of human progress, particularly if one agrees with Hal\u00e9vy that such progress is constantly accelerating, is antithetical to the \"linearity of the history of Salvation.\" That, after all, is the point the preacher in Ecclesiastes 1:2 is making when he declares, \"Vanity of vanities, all is vanity.\" In Rouvillois's words, for the believer waiting for the end of days when God will judge if he or she is to be damned or saved, \"progress is _impossible,_ most importantly because it is _useless_.\" As it says in John 12:31: \"Now is the judgment of this world, now shall the prince of this world be cast out.\"\n\nBut one does not have to be a believer to find the idea of progress far-fetched. What for lack of better terms, and for all their imprecision and oversimplification, we call premodern or traditional societies may be better off during a given historical period than they were in a previous one\u2014for example, because of good weather that leads to fuller harvests, or following success in war. But they may also find themselves worse off, should, say, they suffer famine (which they would have been likely to do given that it is a catastrophe that had afflicted all societies for all of recorded time until the second half of the twentieth century, when for the first time in human history they began to wane) or military defeat. Since the Enlightenment, the essence of the idea of progress has been that however difficult it may be to achieve in any given realm of human activity or of the human condition, it is at once inevitable and inexorable, rather than contingent and reversible, as it is in traditional societies. This is because, as Rouvillois writes at the end of his study, \"Once one has adopted an optimistic conception of human history and one has affirmed the _reality_ of progress, then one draws from one's sense that it is _necessary_ the confidence that it will thus be _perpetual_.\"\n\nTo insist on this point is not to claim that things never change in traditional cultures, the majority of which have been peasant cultures. Obviously they do, but usually so slowly that these changes are imperceptible to each generation. But the slowness of the change is not so much because such societies are governed by custom and precedent since, after all, this is also the purpose that, once successfully established, invented traditions are meant to serve. Rather, it is because, as Hobsbawm emphasized in his essay on the invention of tradition, there is so little demand in traditional societies for rapid change\u2014again, except in extremis, as in cases where famine drives people from the countryside to the city, where they find that the older traditions are no longer fit for the purpose, and thus have to be either radically modified or jettisoned entirely.\n\nThe link between Hal\u00e9vy's \"acceleration\" and Hobsbawm's \"invention of tradition\" is that, as Hobsbawm put it in 1983, \"[Large and rapid] changes have been particularly significant in the past 200 years, and it is therefore reasonable to expect these instant formalizations of new traditions to cluster during this period.\" Once the stable arrangements that began to come apart at the time of the Industrial Revolution had eroded past a certain point, they could never again be reconstituted. But few human beings can live without any traditions to which to adhere. As a result, when any real continuity with the past became impossible, that past had to be reimagined and reconstructed both for the present's and for the future's sake.\n\nIn the past fifty years, the acceleration has been particularly extreme in the hard sciences and technology, certainly, but also in the arts. The philosopher Karl Popper's dictum that the essential difference between the sciences and the humanities is that in the former propositions are empirically falsifiable whereas in the latter they are not is over-broad. It is true that historically innovation has played a more important role in Western art than it did in African, Indian, Southeast Asian, or Chinese art. Nonetheless, the divorce of the art of the present from the art of the past is all but absolute, and in this sense seems more radical than previous moments of rupture. The mental stance of the contemporary artist strikingly parallels that of the scientist in that the art of the past is deemed to be as irrelevant to the art of the present as pre-Copernican physics is deemed to be to string theory or trepanning to brain surgery.\n\nThe scientists and technologists are on firmer ground. One does not have to endorse the techno-utopianism of a Ray Kurzweil, or share Bill and Melinda Gates's conviction, which has informed all the work of their foundation\u2014by far the richest in the history of philanthropy\u2014that disease, hunger, and poverty will be all but ended or at least radically reduced thanks to technological innovation, to appreciate how much the world has changed, and how much that change continues to accelerate.1 In the early twenty-first century, and in every domain except our identities as mortal, biological beings (though Ray Kurzweil and other like-minded cyber-utopians in Silicon Valley would reject even that caveat), the transformation continues. For the first time in human history, a majority of the world's population lives in cities instead of in the countryside. The free movement of capital is taken for granted, as is the alternative, borderless geography of cyberspace. And all the while, the unprecedented movement of poor people from the Global South toward the Global North, which is an accelerator of history if ever there was one, and which may be able to be managed but certainly cannot be stopped, is hollowing out national identities even as it makes a mockery of national borders.\n\nAnd if what in Spanish is called _mestizaje_ (the term has cultural and historical connotations that makes it far more resonant than its English translation \"the mixing of the races\") is proceeding apace, so is global cultural and linguistic homogenization. Not a year goes by without at least ten small languages dying out, while small cultures are being steadily subsumed into a few dominant ones, largely articulated in Chinese, Spanish, English, Arabic, Hindi, Bengali, and to some extent French and Portuguese. Climate change, even if the rise in global temperatures is held to a (comparatively) manageable range of between 2 and 2.5 degrees Celsius rather than a catastrophic 3.5- to 5-degree range, will transform cultures as surely as it will trans-form the environment. And scientists are now convinced that the earth has entered the sixth wave of mass extinctions of plants and animals in the past half-billion years and predict that a quarter or more of all land species will no longer exist by the year 2050. In short, the historical stresses that led to the accelerated manufacture of traditions that Hobsbawm and Ranger identified are only a foretaste of those the world is subject to now, let alone what we are likely to confront in the future.\n\nNarrowing the frame to politics, the acceleration of history template is particularly useful when applied to those periods when empires fall and new polities rise or are revived, or when the demography of a given society is altered so dramatically that emphasizing a historical consensus that is no longer shared by a preponderance of the state's citizens no longer makes sense. Historically, this question has been especially bedeviling to settler societies such as the United States, Canada, and Australia, though the recent huge immigration to western European countries has blurred the differences between settler and nonsettler, virtually guaranteeing that most of the nations of the European Union will soon be confronting similar challenges. In the United States, for example, when the overwhelming majority of Americans were of European origin, what even in the age of the European colonial empires at their zenith in global terms was certainly a disproportionate emphasis on the history of Europe was defensible. But in the increasingly de-Europeanized United States of today it is not. The problem, of course, is what to replace it with.\n\nTimothy Garton Ash has suggested that a possible approach to the challenges facing a European Union now expanded to twenty-seven countries is not to try to forge a common collective memory, for, as he rightly observed, one would \"have to do extreme violence to the historical truth\" in order to do so. Instead, Garton Ash proposed defining European identity as \"nations that are coming from very different histories but aspiring toward shared goals.\" Yet as Garton Ash undoubtedly understands as well as anyone, \"shared goals\" are hard enough to define in good times, and these are anything but good times in Europe. The Greek crisis alone, but also the rise of populist parties on both the left (Syriza in Greece, Podemos in Spain, and others) and on the right (the Front National in France and the Danish People's Party, for example), the inability of European governments to agree on how to craft a common policy on the resettlement of the latest waves of migrants, and the cultural, political, and economic panic, whether justified or not (probably both, in my view), that the refugee exodus has caused offer ample evidence that the era in which serious thinkers such as Garton Ash and J\u00fcrgen Habermas could continue to believe that the humane vision of statesmen such as Jean Monnet, Robert Schuman, Paul-Henri Spaak, and Emile No\u00ebl for Europe and its vocation in the world still enjoyed a preponderance of support from the European public may well be ending, if it has not ended already.\n\nBut let us assume that Habermas's anxieties that the European project is in the process of being transformed into the opposite of what its founders had intended, and that the world's \"first transnational democracy [risks becoming] an especially effective, because disguised, arrangement for exercising a kind of post-democratic rule,\" are misplaced. And let us hope that enough hearts and heads are swayed by Garton Ash's fierce and eloquent response to the reaction of the vice president of the European Commission to Syriza's election victory in January 2015 (\"We don't change policies depending on elections\") when he wrote, \"Given the choice between democracy and a paternalistic, top-down, Euro-Leninist version of European integration, I will choose democracy every time.\" Even assuming such a positive outcome, the success of the European project will remain in doubt.\n\nThe reasons for this have been brilliantly analyzed by the French political scientist Zaki La\u00efdi in his _A World Without Meaning: The Crisis of Meaning in International Politics_. \"If in Western societies,\" he writes, \"the social and political forces which claim to be the paradigm of the transformation are in crisis, it is precisely because identity transmission and social transformation seem disconnected. Transmission is thought old-fashioned (nationalism) and transformation destructive (globalization), whereas transmission ought to be made positive and transformation protective.\"\n\nIf anything, La\u00efdi understates the problem, especially that of transmission, since today even to speak of a common European, or, for that matter, American or Australian, identity is problematic. This is at least in part because, as the Dutch academic Joep Leerssen has pointed out, \"Historical investigation has turned from victors and triumphant elites to the downtrodden, the persecuted, the victimized.\" Instead of commemorations of shared trauma in what Leerssen calls \"the Binyon\/Tennyson mode of transmuting loss and grief into edification and catharsis,\" remembrance largely inhabits a \"'never again' mode.\" Garton Ash's prescription for Europeans to explore \"the diversity of our memories\" in order to arrive at shared goals simply takes insufficient account of the degree to which the new politics of commemoration is grounded in what Leerssen calls the \"experience of oppression, of defeat, of injustice and grievance\" and informed by the idea that \"the lesson to be drawn from past injustice is one of vigilance and assertiveness.\" Meanwhile, those who continue to want the previous consensus to prevail are furious at what they see as disloyalty and ingratitude, particularly on the part of immigrant populations and sexual minorities. In our new era of ressentiment, the idea of a shared anything seems increasingly out of reach.\n\nBut even if the divide were narrower between collective memory in the service of national unity and collective memory in the service of those victimized by the nation in question who await at least an acknowledgment and often an apology for what they or their ancestors have suffered, the new synthesis of the two, no matter how satisfactory or unsatisfactory to both sides, will be as mortal as any other set of political arrangements and cultural compromises. If there is a puzzle in all this, it is that anyone could believe that political arrangements could be other than transitory, no matter how potent they might seem at a particular moment in history. In a fascinating short memoir called \"Where Statues Go to Die,\" the historian of the British Empire David Cannadine illustrated the fragility and contingency of even the most grandiose emblems of bygone empires by describing how during a visit to India in 2003 he had come face to face with the inglorious fate of colonial monuments. He was taken to a large open space on the edge of New Delhi, where in colonial times British viceroys had held their great ceremonial gatherings known as \"durbars.\" What Cannadine found was a \"neglected, overgrown, obscure piece of ground\" that conveyed the message that \"earthly power is transient, and that imperial dominion is ephemeral.\"\n\nIn the remotest corner of the site, he describes being led to \"a most astonishing scene: a dozen immense statues, rising up from the bushes and the brambles, like the chessmen arrayed for that terrifying contest towards the end of the first Harry Potter film.\" They were statues of George V, emperor of India, and a number of British viceroys, \"placed by the British in New Delhi to be permanent monuments to men whose lives and deeds they deemed worthy of everlasting commemoration.\" But India had moved on\u2014just as virtually every other former European colony, French, Dutch, Portuguese, or Spanish, where such statues once occupied pride of place has moved on. A brilliant Cartier-Bresson photograph taken in 1949 shows the official portraits of colonial officials being removed like so much rubbish from the governor's mansion in Jakarta shortly after Indonesia gained its independence from Holland. The journalist and writer Ryszard Kapu\u015bci\u0144ski witnessed something similar in Angola in 1975. As he recounts it, the Portuguese colonists all but stripped their houses, crating up the contents to be loaded onto freighters that would take them from Luanda back to Portugal. But the colonists paid no heed to the colonial statuary, and soldiers of the victorious MPLA (Popular Movement for the Liberation of Angola) removed them from their plinths and carted them unceremoniously to a field much like the one David Cannadine had seen on the outskirts of New Delhi.\n\nSuch high, iconoclastic moments, when all the grand symbols of colonial rule reverted to what they had been when they were still being cast at the foundry\u2014bits of hewn and polished stone now representing something barely intelligible\u2014are hardly restricted to the public monuments and official portraits of European colonial officials. After 1989, many of the same sorts of scenes were played out throughout the Soviet Union, and again, a few years later, in the Balkans.2 Somehow I doubt that Lenin or Stalin ever imagined, any more than Lord Curzon did, that his images would be transformed from symbols of domination to kitsch in a matter of decades.3 And yet that is precisely what happened in Russia, to the point that the English-language _Moscow Times_ could run a story in 1996 that set out the going prices in the \"antique\" shops along the Old Arbat for such objects as busts of Lenin and \"a large plate in good condition of Stalin and Voroshilov at work.\"\n\nDespite the comic turns these episodes can take, the underlying reality is deadly serious. For we are back with Kipling once more. He would certainly not have been surprised by what Cannadine saw in that field on the outskirts of New Delhi. After all, for Kipling the mortality of every empire was as certain as our own individual mortality (and possibly even more so since, at the end of his life the poet had the uneasy consolation in believing in spirits as well as in Jesus Christ). Where empire was concerned, though, there could be no appeal. As it says in Psalms 146:3\u20134: \"Put not your trust in princes, nor in the son of man, in whom there is no help. His breath goeth forth, he returneth to his earth; in that very day his thoughts perish.\"\n\nThat through all of this\u2014\"action and suffering, power and pride, sin and death,\" as Karl L\u00f6with put it\u2014God endured could offer no consolation to Kipling. He well understood that God is mainly outside history, intervening only at its beginning with the act of love that is the Creation, and at its end, the Apocalypse, which Leszek Kolakowski once described as the \"never-ending framework of Jesus' preaching.\" Eschatology, not the future in this world, was what concerned many of the best known of the early Christian martyrs, such as Thecla, who vowed to remain a virgin and finally fled into the mountains, where she preached the Gospel until her death at ninety. Or Maximilla, whose love of God impelled her to pray for Him to intercede so that she would no longer have to have sex with her husband, Aegeates. How could these women have committed themselves to relinquishing all their property and have been so ambivalent even about having children had they not seen themselves as being betrothed to the world to come?\n\nFor the rest, game, set, and match to Shelley:\n\n\"My name is Ozymandias, King of Kings:\n\nLook on my works, ye Mighty, and despair!\"\n\nNothing beside remains. Round the decay\n\nOf that colossal wreck, boundless and bare\n\nThe lone and level sands stretch far away.\n\nAlthough he, too, despaired, Kipling still chose to write about forgetting in the conditional: \"Lest we forget!\" He was right to do so, for had he done anything else, his appeal for remembrance would have been rendered all but meaningless (imagine what Shelley would have done with the theme of \"Recessional\"). And remembrance is emphatically not meaningless except in the cognitively and probably ethically useless framework of eternity. Human beings are drawn to ceremony and, whether they are believers in a religion or not, to piety as well.4 This is why, without falling into a subspecies of ancestor worship, surely there is something impious, or at the least ethically impoverished, about forgetting the sacrifices and the sufferings of those who came before. If those who died in battle or gave their lives for their beliefs are not remembered, how can their acts have any meaning? And for their sacrifices, read all sacrifice. In Dryden's great phrase, such a forgetting truly would be \"the untuning of the sky.\"\n\n\"Imagine . . . a world without memories,\" runs the tagline on the website of the government-backed \"Australian Memory of the World Register,\" part of a global project on memory organized by UNESCO. The effort is less doom-laden than these words might lead us to believe, and largely involves the preservation of historical documents and the recording of oral histories. But that does not make the admonition on the home page any less emblematic of the nonsense that so much of the discussion about memory seems to produce. Quite simply, the world does not have memories; nor do nations; nor do groups of people. Individuals remember, full stop. Yet in the early twenty-first century, collective memory is often spoken of as if it were indeed on a par with individual, which is to say genuine, memory, and not infrequently, though almost never explicitly, as if it morally outranked it.\n\nTo paraphrase the progressive leftist intellectual Randolph Bourne's sardonic remark about war, is collective memory now the health of state? Reading the literature on remembrance, one might very well think so. The fact that individuals forget, whether through the sad cognitive deficits that come with age or, conversely, some happy remission in their private life (such as fading of the memory of the lover who broke one's heart), is not thought to pose a threat to society as a whole. In contrast, a collective failure of remembrance is often presented as if it were an invitation to moral or political disaster. The paradigmatic contemporary expression of this is the commonly voiced assertion that shirking our moral obligation to remember the Shoah is for all intents and purposes to strike a blow against what Thomas Jefferson called \"the decent opinions of mankind.\" A person who did so might not be an \"assassin of memory,\" the term that the historian Pierre Vidal-Naquet rightly affixed to Holocaust deniers such as Robert Faurisson. But he or she has committed a grave ethical solecism just the same.\n\nThis sense underpins the views of Avishai Margalit. For him, the issue is not a matter of Jewish particularism, though because the Jews are a religion that became a people rather than a people who became a religion (the formulation is Harold Bloom's), the question of that particularism, and, more broadly, of a Jewish approach to history that is in many ways unique, must be addressed if one hopes to get to the heart of the matter. Karl L\u00f6with thought that because Christians were not a historical people\u2014\"their solidarity [was] merely one of faith\"\u2014it was impossible to apply a purely Christian interpretation to the historical destiny of Christian peoples. But he argued that the destiny of the Jewish people _was_ a subject for Jewish interpretation, presumably up to and including the Shoah. Margalit, though, is interested in global ethical minima: that is, in arriving at a standard that should command not just one people's or even one civilization's allegiance but humanity's as a whole. For he believes\u2014and to be clear, not as a matter of Jewish particularism\u2014that there are certain \"moral nightmares,\" as he calls them, the Shoah first and foremost, that must remain in our collective memories because they are \"striking examples of radical evil and crimes against humanity, such as enslavement, deportations of civilian populations, and mass exterminations.\"\n\nIn effect, what Margalit is saying is that the need to be alert to radical evil (he borrows the term from Kant but imbues it with a somewhat different meaning) imposes on humanity as a collectivity the prudential requirement of constructing a moral memory that can be universally understood, shared, transmitted, and defended. To do anything less would be to leave humanity itself vulnerable to what Margalit describes as \"the effort of [the forces of radical evil] to undermine morality itself by, among other means, rewriting the past and controlling collective memory.\"\n\nIn this, as in his arguments in favor of forgiveness but against forgetting, Margalit follows the philosopher Paul Ricoeur, who wrote, \"We must remember because remembering is a moral duty. We owe a debt to the victims. . . . By remembering and telling, we . . . prevent forgetfulness from killing the victims twice.\" But however counterintuitive my argument may seem, and however much one honors the moral seriousness of those like Ricoeur, Margalit, Todorov, Yerushalmi, and Vidal-Naquet who have advanced various, and not entirely congruent versions of this position, what if they are wrong? What if, over the long term, forgetfulness is inevitable, while even in the comparatively short term the memory of an instance of radical evil, up to and including the Shoah itself, does nothing to protect society from future instances of it?\n\nShortly before his death in July 2015 at the age of 106, Sir Nicholas Winton, who had traveled to Prague on the eve of the Second World War and managed to rescue 669 Jewish schoolchildren and bring them safely back to Britain, gave an interview to Stephen Sackur of the BBC. Sackur asked Winton what lessons he drew from the past. From the look of surprise on Sackur's face, he probably did not expect the answer he received. Winton was categorical. \"What good does concentrating on the past do us,\" he demanded; \"Who has ever learned anything by concentrating on the past?\" Winton's question is worth the asking, as is that of the cofounder of the Annales School, the French historian Marc Bloch, who concluded his eloquent review _On Collective Memory_ by commenting that Maurice Halbwachs \"pushes us to reflect on the conditions of the historical development of humanity; indeed, what would this development look like without 'collective memory'?\"\n\nWhat if the collective memory of a nation, which Margalit himself concedes has been defined as a society that nourishes, Renan-like, a common delusion about its ancestry, is not just wildly overrated as a measure of that society's coherence, and not just ultimately futile (the message at the heart of Kipling's \"Recessional\"), but often actively dangerous? And what if, instead of heralding the end of meaning, a decent measure of communal forgetting is actually the sine qua non of a peaceful and decent society, while remembering is the politically, socially, and morally risky pursuit? Or, to put it somewhat differently, what if the past can provide no satisfactory meaning, no matter how generously and inclusively, as in Timothy Garton Ash's prescription, it is interpreted? In short, what if, at least in some places and on some historical occasions, the human and societal cost of the moral demand to remember is too high to be worth paying?\n\nAlmost no one believes this anymore. Marx did, when he wrote, \"The social revolution of the nineteenth century cannot draw its poetry from the past, but only from the future. . . . In order to arrive at its own content, the revolution of the nineteenth century must let the dead bury their dead.\" And Nietzsche did, though he did not do so consistently, even when he praised \"the art and power of _forgetting_ and of enclosing oneself within a bounded _horizon_.\" But the contemporary consensus bends entirely in the opposite direction. In fairness, the most brilliant writing in defense of remembrance, and here Margalit and Todorov stand out, offers a number of caveats about how remembrance can be misused. Todorov has been especially firm in this regard, observing that \"the path between the sacralization and the banalization of the past, between serving one's own self-interest and giving moral lessons to others, can seem narrow,\" before adding, in an echo of Galileo's \"eppur si muove,\" \"And yet it is there.\"\n\nTo put it kindly, such subtleties are not often in evidence in most arguments in support of collective memory as a moral and social imperative. All too many of them seem to take as their point of departure George Santayana's far too celebrated and, insofar as it was meant as a generalization, demonstrably false injunction, \"Those who cannot remember the past are condemned to repeat it.\"5 This is the view that has become the conventional wisdom today, and the conviction that memory is a species of morality now stands as one of more unassailable pieties of our age. All of this operates at least as much under the sign of what Todorov has called moral correctness as it does under that of political correctness. To remember is to be responsible\u2014to truth, to history, to one's faith, to one's country, to the traditions of one's own people or gender or sexuality (in this last instance what is usually meant is a group's sufferings, the history of its oppression). Anything less is an act of irresponsibility that threatens to undermine both one's community and, in our therapeutic age, oneself as well. And even to question this consensus is to disturb what Tony Judt, writing in praise of Hannah Arendt, described as \"the easy peace of received opinion.\"\n\n1. Scientific progress is no myth. But as John Gray has written, \"The myth is that the progress achieved in science and technology can occur in ethics, politics, or, more simply, civilization.\"\n\n2. These could sometimes be comic. In Tirana, shortly after the fall of the Albanian regime, I visited a foundry most of whose work during the entire Communist period had been casting statues of party leaders, above all of Enver Hoxha, the country's dictator. When I met the works manager, though, he seemed remarkably cheerful. It turned out that the foundry was as busy as ever... making statues of Mother Teresa.\n\n3. A strong case can be made that when such iconoclastic moments do _not_ occur, a successful transition from one kind of society to another is less likely to be achieved. Had the federal authorities in Washington, D.C., been willing after Reconstruction to stop the widespread reemergence and official display throughout the South of Confederate flags and other symbols, the ideology of the Lost Cause might not have become dominant in the way that it did.\n\n4. As John Gray has pointed out, the resurgence of religion in the twenty-first century is a global phenomenon. If anything, it may be atheism that proves to be the historical parenthesis.\n\n5. To which Michael Herr noted in _Dispatches_ the \"little history joke\" that \"those who remember the past are condemned to repeat it too.\"\nFOUR\n\nThe Victory of Memory over History\n\nThe instauration of memory as an indispensable public good, and its corollary, the damning of forgetting as a form of civic nihilism, depends on the idea, as Avishai Margalit posits it, that humanity \"can be shaped into a community of memory.\" Margalit understands full well how difficult this will be to achieve and recognizes that it has never yet happened. He concedes, \"It is hard to form effective institutions that will store such memories and diffuse them.\" And he points out, though without giving it the emphasis it deserves, the additional danger of what he calls \"biased salience,\" in which events from \"the so-called First World, or the technologically developed world, are likely to be more salient to us than comparable events in the Third World.\" Margalit does not discuss the relation between the continued dominance in the twenty-first century of the global media by Western communications companies.1 Had he done so it would only have strengthened his argument, since the fact that, as he rightly points out, Kosovo is \"better remembered\" than Rwanda, is largely the result of the imbalance between the coverage of the former and of the latter.\n\nMargalit is certainly right to worry about \"a false moral superiority\" being attached to atrocities that take place in Europe as opposed to those that occur elsewhere. Margalit also is right to argue that Nazi crimes \"are glaring examples of what morality requires us to remember.\" And yet, as Tony Judt pointed out in 2008, considerable evidence already exists that in many parts of the world the Shoah is losing (if it has not already lost) what he called \"its universal resonance.\" \"Moral admonitions from Auschwitz that loom huge on the memory screens of Europeans,\" he observed, \"are quite invisible to [many] Asians and Africans.\"\n\nBut even if one believes that Judt was mistaken about the particular case of the Shoah, Margalit's broader call for a shared moral memory for humankind remains problematic. Margalit himself conceded that he was \"unclear in my mind as to how to go about creating such a memory.\" In the end, he is forced to fall back on little more than a combination of hope in the face of even the small likelihood of this hope being fulfilled\u2014Margalit cites Romans 4:18, wherein Abraham is described as a man \"who against hope believed in hope\"\u2014and his neo-Kantian resolve that _ought_ implies _can_. He is hardly alone among philosophers in his confidence that, as the French philosopher Vladimir Jank\u00e9l\u00e9vitch once put it, \"Morality always has the last word.\" But since elsewhere in his book Margalit admits that \"we live with insufficient sources to justify our ethics and morality,\" we might have thought this would have given him pause before he went so far in raising what the historian and poet Robert Conquest with some asperity once described as \"the dragons of expectation.\"\n\nThis is not to say that Margalit is wrong to claim the existence of an ethics of memory. But in making the case for it, he glosses over the possibility that the reality of remembrance, and, indeed, of forgetting, which is almost of equal concern to him, might be fundamentally and ineradicably political, as Renan had understood it to be. Even when he speaks of humanity, Margalit employs a universalizing language that for the most part offers a view of the world in which there are individuals, some good, some evil, and there is the human race, as if when all were said and done there would be no need to grapple with the fact that ours is a world in which different groups of people see the world in irreconcilable or at least incommensurable ways and are almost certain to go on doing so. Even if one does not share the French social theorist Serge Latouche's anxieties about efforts to instill what he has called \"the confidence trick of a bogus universality,\" surely in the early twenty-first century and for the foreseeable future the most pressing problem is not finding a way to instill a universal code of ethics but figuring out how to stave off the worst by arriving at John Gray's modus vivendi, in which, yes, we will have to agree to disagree on what needs to be remembered and how, and, as much if not more important, on what needs to be forgotten.\n\nLike Todorov, whose _Abuses of Memory_ has an epigraph from Jacques Le Goff enjoining his readers to \"ensure that collective memory contributes to the liberation rather than the enslavement of mankind,\" Margalit is painfully aware that collective memory can be abused. What does not seem to command his attention, let alone excite his skepticism, though, is the extent to which the concept permits virtually anybody and everybody, whatever their moral, political, or social views, to subscribe to it. Today, fascists and multiculturalists, servants of the state and revolutionaries committed to bringing the state to its knees, elites and counterelites unite in paying homage to \"The Duty of Memory.\" We need only Google the French equivalent, _devoir de m\u00e9moire,_ to understand how protean the term has become. The Association Devoir de M\u00e9moire describes itself on its website as \"a pageant association representing the Canadian troops [in] World War II who worked for the liberation of Europe.\" Les \"Oubli\u00e9-e-s\" de la M\u00e9moire is a national memorial organization that works to educate the public about the deportation of homosexuals during the Nazi occupation of France; it campaigns in both France and elsewhere for the recognition of what happened. Two right-wing presidents of France in succession, Jacques Chirac and Nicolas Sarkozy, on one side of the ideological divide, and, on the other, a well-known activist group on the hard left that calls itself Duties [plural] of Memory, have invoked the \"duty of memory\" with regard to the slave trade.\n\nThe French case is extreme in the sense that, in the words of the French historian Pierre Nora, whose pioneering research and many books on the subject established him as the foremost student of the role collective historical memory has played and continues to play in French society, \"'Memory' has taken on a meaning so broad and all-inclusive that it tends to be used purely and simply as a substitute for 'history' and to put the study of history in the service of memory.\" The takeover of history by memory is also the takeover of history by politics. The result in practice, if not necessarily in theory, has taken us far from Margalit's world of ethical obligations and moral minima. Instead, we have entered a world in which the essential function of collective memory is one of legitimizing a particular worldview and political and social agenda, and delegitimizing those of one's ideological opponents.\n\nRenan would have loathed what Nora called the \"democratization\" of history, which he argued provided the ethical basis for subordinating history to collective memory. But far from undermining Renan's view, the shift Nora identified vindicates it in the sense that the creation of a given group identity is dependent on what of and how the past is remembered. What has changed, and this would indeed have profoundly worried Renan, is that whereas until the 1970s collective memory was largely a state monopoly, particularly of those elements of the state responsible for education and war\u2014a \"memorial regime of national unity,\" as the French political scientist Johann Michel has described it\u2014now it is up for grabs, with ethnic, religious, and sexual minorities challenging traditional mainstream accounts and seeking to modify them if not to transform them entirely.\n\nDoes any common ground remain? The answer seems to be, \"More than we might think.\" For no matter how bitterly the two sides differ both over _what_ should be remembered and _how_ this remembrance should be commemorated, they agree that not to remember would be far worse and constitute what Jank\u00e9l\u00e9vitch called a \"shameful amnesia.\" In 2008, early in his presidential term, Nicolas Sarkozy invoked the \"duty of memory\" when he ordered that all French schoolchildren entering their last year of primary school study the life story of one of the eleven thousand Jewish schoolchildren deported by the Nazis and murdered in the camps. His successor, Fran\u00e7ois Hollande, who was then leader of the Socialist opposition, wholeheartedly endorsed Sarkozy's initiative, declaring that \"any time that it is possible to transmit that which the duty of memory demands, it must be done.\" For their part, groups demanding that France \"remember\" the crimes of its colonial past and the affronts committed against groups within France itself that were excluded from the national story\u2014even, as some campaigning associations have done, offer an apology, or reparations, whether symbolic or substantive\u2014would endorse Hollande's formula, though the uses they would make of it are completely different.\n\nIn any case, let us assume for the sake of argument that the \"memorialists,\" whatever their ideological inclinations, are correct in insisting, with Jank\u00e9l\u00e9vitch, that it is both morally shameful to forget the horrors of the past and morally uplifting to remember or, better still, to recuperate that past. Does it then follow that these are the last words that need to be said on the subject? The answer to that question has to be no\u2014a qualified no, to be sure, but no just the same.\n\nTo be clear, I am not arguing that it is always wrong to insist on remembrance as a moral imperative. When a historical crime or tragedy has been covered up, even one that occurred long before anyone now living was born, or if the history books tell lies or a partial truth about what occurred, or even if the realities of what happened have simply become muddied, no matter whether maliciously or out of ignorance, lifting the veil about what took place is almost always something to be welcomed. The Armenian genocide is an obvious case in point; another would be the massacres perpetrated by British and French imperial forces throughout the colonial period. The same could be said about war crimes and other mass atrocities, especially if they happened in living memory. There, if a practical possibility exists not only of establishing an honest record of what was done but also of bringing the perpetrators to justice, in principle it should be done. One example of such a crime is the massacre of Muslims at Srebrenica, another the women and girls, most of them Korean and Chinese, who between 1932 and 1945 were forced by the Imperial Japanese Army into sexual slavery.\n\nBut even in such cases, things are sometimes more complicated morally than they may first appear. Most people, for example, would regard the warrant issued in 1998 by the Spanish magistrate Baltasar Garz\u00f3n for the arrest of the former Chilean military dictator Augusto Pinochet as a long-overdue blow for justice. But many Chileans, including a substantial number of those who welcomed Garz\u00f3n's action, also believe that had his order been issued in 1990, at the time Pinochet left office, he might have refused to relinquish power or that, had he done so, the Chilean military, which at the time was still loyal to him, would have stayed in its barracks. Under those circumstances, how can we be certain that the democratic transition would have gone forward? Assuming that the arrest might have offered at least a serious risk to that transition, would standing for the truth, or, yes, upholding the demands of justice, still have been worth it? Surely there would be nothing dishonorable in answering no to that question.\n\nTo raise this possibility is emphatically not to suggest that the answer should usually be no. To the contrary, more often than not it should be yes. When Jank\u00e9l\u00e9vitch denounced \"shameful amnesia,\" for example, he was responding to those who were criticizing the French government's 1987 prosecution of Klaus Barbie for crimes against humanity\u2014the first such case ever to be brought in France. Barbie was the former head of the Gestapo in German-occupied Lyon, and his many atrocities included personally torturing a number of prisoners to death, as well as responsibility for the deaths of thousands more, including the French Resistance leader Jean Moulin. Jank\u00e9l\u00e9vitch was not alone in his condemnation. It was those who opposed the trial, among other reasons on the grounds that stirring up the past could only be destructive, that Pierre Vidal-Naquet denounced in _Assassins of Memory_.\n\nJank\u00e9l\u00e9vitch and Vidal-Naquet's fears were warranted, as Adolf Hitler's own words, recorded in the transcripts later published as his \"table talk,\" should have been enough to demonstrate. On August 22, 1939, Hitler had declared that Germany could and would exterminate the Poles \"mercilessly and without compassion.\" The international community would not object, he said, because \"Who, after all, speaks today of the annihilation of the Armenians?\"2 The accusation, fortunately not as common today as it was in the 1950s and 1960s, that Nietzsche's philosophy was one of the inspirations of Nazi ideology is a calumny. But Hitler's comment is uncomfortably close to Nietzsche's stark reminder \"Whichever interpretation prevails at a given time is a function of power, not of truth.\"\n\nThe claim that to remember is in and of itself a moral act derives much of its force from its ambition to prove Nietzsche and Hitler wrong. The judging and execution or imprisonment of Nazi officials and concentration camp guards can never be comprehensive, and therefore cannot be put forward as a sufficient reckoning with the horror of what had occurred. In any case, even at its best, beyond assigning guilt and punishing the guilty, justice can only establish facts and at times provide the survivors and the relatives of those who did not survive with a measure of vindication and release. But as Elie Wiesel put it, \"Justice without memory is incomplete justice, false and unjust. [For to] forget would be an absolute injustice in the same way that Auschwitz was the absolute crime. To forget would be the enemy's final triumph.\" In this Wiesel echoes, though in universalist rather than in purely Jewish terms, the philosopher and rabbi Emil Fackenheim's injunction that if the Shoah is ever forgotten, Hitler will not only have triumphed at Auschwitz but will have won a posthumous victory as well.\n\nWhat Nietzsche said about power and truth is usually understood as a cynical comment on the way in which the powerful misrepresent the truth. But although history is replete with examples that vindicate Nietzsche's claim, unless you believe that everything states do is invariably malign, in principle there is no reason why this power cannot be made to serve moral ends instead of immoral ones. Some states, notably France, have tried to enlist the law into the service of historical truth. In practice, the most important expression of this has been a series of legislative measures aimed at banning the denial or questioning in a public context of the reality of some (though not all) of the worst crimes in human history. The first of what came to be known as \"memorial laws\" was the _loi Gayssot_ (1990), which criminalized all forms of such \"revisionism\" with regard to the Shoah. In 2001, a law was passed in the French parliament that recognized the reality of the Armenian genocide. In 2003 another piece of legislation, the _loi Taubira,_ recognized the slave trade as a crime against humanity. And in 2005, it became an offense to deny that the Armenian genocide had taken place.3\n\nThere has since been some spillover from these French initiatives to the European Union as a whole. In 2007, a proposal in front of the European parliament was tabled that would have made punishable by a term of imprisonment an expansive array of \"denialism.\" These included (the explanatory notes in parentheses are mine): \"genocides\" (plural), \"war crimes of a racist character [ _sic_ ] and crimes against humanity,\" \"gross banalization\" (that is, saying these crimes are not especially important or deserving of special status, as some French lawyers defending Nazi war criminals like Klaus Barbie had done), and even \"complicity\" in that banalization (no matter when the crimes in question had occurred and what political, administrative, or judicial authority had determined them to be established as historical facts).\n\nUnsurprisingly, the consensus among professional historians has been to fiercely oppose such blanket prohibitions. They have not done this out of na\u00efvet\u00e9; after all, historians know in intimate detail the ways in which states have tried to cover up shameful episodes in their history. What most objected to was the recourse to law in the effort to combat \"denialism.\" As Pierre Nora put it at the time the loi Taubira was passed: \"Victims and orphans [of the victims] were before our eyes, and the authors of these abominations very much alive. With [the passage of the] _Loi Taubira_ we reached back five or six centuries, and with the Armenians to crimes in which France played no role. What about the Vend\u00e9e? . . . The Albigensians, the Cathars, the Crusades?\" Nora concluded, \"On the model of the _Loi Gayssot,_ we are creating a system that can only constrain research and paralyze teachers.\"\n\nWhile I share this view, it is unlikely that the series of laws Nora viewed with such dread posed a danger to historical truth of the same order of magnitude as the tendency of most governments (and probably all at one time or another) to seek to cover up not only the historical skeletons in their own closet but also those of other states, movements, or institutions with which they find themselves in sympathy. And when a government is committed to that course, the consequences for those who do try to bring what happened to light can be severe. The case of Chelsea Manning, the U.S. soldier who revealed the details of a massacre committed by her comrades in Iraq and was sentenced to thirty-five years in prison as a result, is only one of a myriad of contemporary illustrations of this.\n\nThe case of the Irish essayist Hubert Butler is illuminating in this regard. Butler was a member of the Irish Republic's Protestant minority. He was sent to an English public school and went on to attend an English university. Butler graduated in 1922 and returned to Ireland. Four years later, he left again, and for almost fifteen years traveled extensively in China, the United States, Egypt, the Soviet Union, Yugoslavia (where he spent three years), and central Europe. In 1938 and in 1939, he worked with a Quaker group in Vienna helping Jews escape post-Anschluss Austria, impelled to do so, as he would later recall, after hearing an anti-Semitic statement by the Irish parliamentarian Oliver J. Flanagan. \"I was as Irish as Oliver Flanagan,\" Butler later recalled, \"and I was determined that Jewish refugees should come to Ireland.\" After Vienna, Butler went to London, where he worked briefly for the British Colonial Service. But following his father's death in 1941, he returned to Ireland, moving into the family house in his native County Kilkenny, where he would live for the next fifty years until his own death in 1991 at the age of ninety.\n\nImmediately after the end of the Second World War, Butler went back to Yugoslavia, where he tried to investigate the wartime campaign by the Nazi-installed Croatian fascist regime of Ante Pavelic\u00b4 to convert the almost 2.9 million Eastern Orthodox Serbs of Croatia and Bosnia-Herzegovina to Roman Catholicism, murdering many of those who would not renounce their faith. When he got back to Ireland, he gave a talk on Radio E\u00edreann on Yugoslavia, and, at a time when the coverage of Yugoslavia in the Irish press was focused on the victorious Tito dictatorship's persecution of religion, Butler felt honor-bound to also bring attention to what he described as \"the more terrible Catholic persecution which had preceded it.\" In the polemics that followed, Butler was excoriated, above all for having suggested that the then archbishop of Zagreb, Cardinal Stepinac, had been complicit in the forcible conversions, which, Pavelic\u00b4 and Stepinac's defenders in Ireland insisted, could never have occurred in the first place, since in the words of Count O'Brien, the editor of Ireland's leading Catholic weekly, _The Standard,_ \"the Catholic Church has always insisted that conversion must be from the heart.\"\n\nThe denunciations of Butler did not stop there. Count O'Brien published a book defending Stepinac, complete with an effusive foreword by the archbishop of Dublin. Father R. S. Devane, a well-known Jesuit of the day of whom, early in his career, it was said that he \"had been known to confiscate British publications from unwilling newsagents in Limerick,\" took up the cry that there had been no forcible conversions. For his part, James Dillon, the Irish minister of agriculture in the first interparty government of 1948\u201351, who later became the leader of one of Ireland's two principal parties, Fine Gael, advised a group of Irish law students to model themselves on figures such as Stepinac, Pavelic\u00b4, and Cardinal Mindszenty of Hungary\u2014all men, Dillon said, who had \"gallantly defended freedom of thought and conscience.\" As Butler observed, \"Those who knew Yugoslavia were aghast, for Pavelic\u00b4 . . . was the Yugoslav counterpart of Himmler.\"\n\nButler describes the furor in an extraordinary essay, \"The Sub-Prefect Should Have Held His Tongue.\" Despite the firestorm around him, however, Butler was determined to do nothing of the sort. In 1952, he was invited to attend a public meeting in Dublin of the Foreign Affairs Association where Count O'Brien was presenting a lecture titled \"Yugoslavia\u2014the Pattern of Persecution.\" At the end of the talk, Butler rose to rebut what he called its \"crude simplifications.\" As he would later write, \"I had spoken only a few sentences when a stately figure rose from among the audience and walked out.\" The figure in question was the papal nuncio. The meeting was hastily brought to a close, and the next day, the headline in one newspaper read, \"Pope's Envoy Walks Out. Government to Discuss Insult to Nuncio.\" As the Irish novelist John Banville puts it in his fine introduction to _The Invader Wore Slippers,_ a collection of Butler's European essays, the upshot was that Butler was \"forced into internal exile.\"\n\nFrom the onset of the controversy, Butler seems to have been fully aware of the personal risks he was running. But, as he wrote, he felt he had no choice in the matter, especially because he was an Irish Protestant. Banville makes the essential point that, despite the range of his interests and the variety of his experiences, as a writer Butler was very much a localist, and cites Butler's reflection that even when his essays \"appear to be about Russia or Greece or Spain or Yugoslavia, they are really about Ireland.\" For that reason, the controversy fell too close to home for Butler to turn away from it. For while he steadfastly insisted throughout his life that he was \"an Irish Nationalist,\" Butler was convinced that for all their mistakes, arrogance, and other derelictions, the demonization of the Irish Protestant community in the de facto clerical state that was \u00c9amon de Valera's Ireland was a gross falsification of history. Because of what he saw around him, Butler was unmovable. \"If we agreed,\" he wrote, \"that history should be falsified in Croatia in the interests of Catholic piety, how could we protest when our own history was similarly distorted?\"\n\nIn the early 1950s, unlike in the early twenty-first century, there was still a vital distinction to be drawn between celebrity and notoriety. Butler paid dearly for his determination to set the record straight. As an Irishman of his time, he understood better than most what the cost was likely to be of tearing the scabs off such deep and as yet unhealed historical wounds. But once the controversy had subsided, Butler, who was the subtlest of writers whatever his topic, never again confronted these questions head-on. From what he did write, however, it seems evident that for him the basis of any decent society had to be a politics of truth\u2014one in which even the most inconvenient, unwelcome, or, to use an expression much favored by generations of engineers of human souls, right and left, religious and secular, \"unhelpful\" facts needed to be aired. As Butler put it, \"If you suppress a fact because it is awkward, you will next be asked to contradict it.\"\n\nThis statement is that most old-fashioned of things: a noble sentiment. But as Butler himself surely would have understood, the question of historical memory is a more vexed one; such binary conceptions as truth versus lie and the concealed versus the revealed get us less far than it is commonly assumed they do, and certainly nowhere near as far as we need to go. To repeat: what do we actually mean by historical remembrance and collective memory?\n\nHere is what they cannot be: they cannot be what individuals remember. As any good lawyer or police investigator will tell you, the longer the period that elapses following an accident or a crime, the less accurate and reliable the testimony of a victim or witness is likely to be. In the case of individuals, at least _some_ accurate memory may remain. In the case of the historical memory of an event in history, we usually mean the collective remembrance of people who did not themselves actually live through it but rather had it passed down to them through family stories, or, likelier still in this era of acceleration, through intermediaries such as the state, above all in schools and public commemorations, or through associations, some of which commemorate versions of events that oppose or at least modify the official accounts.\n\nWhen those \"remembering\" reach this point, can we be said to be talking about memory at all? For this is not just a flawed transmission, it is an impossible one. The verb _to remember_ simply cannot be conjugated in the plural except when in reference to those who lived through what they commemorate. It is impossible to speak of a people's collective memory in the same way that we speak of individual memory: it is a metaphor meant to interpret reality and carries with it all the risks inherent in the metaphoric understanding of the world. And it is equally absurd to speak of a people's collective guilt4 for the Shoah or for the Rwandan genocide _in the same way_ that we speak of individuals' guilt for their crimes during these horrors. In her essay \"Organized Guilt and Universal Responsibility,\" Hannah Arendt called for \"a sharper dividing line between political (collective) responsibility on one side, and moral and\/or legal (personal) guilt on the other.\" This is the problem that lies at the heart of the effort of Arendt's friend the philosopher Karl Jaspers in his _The Question of German Guilt_ (1961) to think through whether the German people could be held to have been collectively responsible for the crimes of the Nazis. Jaspers insisted on the distinction between moral guilt based on what one has done and moral guilt based on who one is, and on the necessity not to conflate the two. But today, where collective memory is concerned, conflating the two is precisely what we are constantly being instructed that morality and ethics require us to do. Relying as it does on highly questionable notions of collective consciousness, it is a dubious demand intellectually and, insofar as it manumits those who believe themselves to have been wronged from distinguishing between those who have actually wronged them and those who did nothing, or did not do enough, to prevent that wrong from occurring, a dangerous one socially and politically, no matter how well intended.\n\n1. The success of Al-Jazeera has not changed the basic terms of reference, the emblem of this being the fact that the network, in order to appear global and not \"Arab\" or \"Third World,\" has hired a disproportionate number of veterans of major Western television channels as presenters on their English-language news shows.\n\n2. Tzvetan Todorov has pointed out, though, that there is a rich irony in Hitler's speaking of \"everyone\" having forgotten an event he was himself recalling. Todorov cites a similar phrase of Stalin's about Peter the Great's murder of the Boyars.\n\n3. The _loi Gayssot_ is named for the Communist deputy Jean-Claude Gayssot, the _loi Taubira_ for the Socialist deputy Christiane Taubira, who would later become minister of justice during Fran\u00e7ois Hollande's presidency.\n\n4. As opposed to their collective responsibility, for which a strong case can indeed be made.\nFIVE\n\nForgiveness and Forgetting\n\nKlaus Barbie's guilt was never really in doubt. Even his sinister, charismatic defense lawyer, Jacques Verg\u00e8s, never claimed otherwise, insisting instead that the Nazis' crimes had been no different in kind either morally, or, more important, legally under the terms of the statute under which Barbie was being tried, from those committed by the European colonial empires. In _Assassins of Memory,_ Pierre Vidal-Naquet ably refuted some, though not all, of Verg\u00e8s's claims.1 But for Vidal-Naquet, the greatest danger to the understanding of the Shoah came not from such Stakhanovites of moral equivalence as Verg\u00e8s, but rather from the \"revisionists\" who were determined to negate the reality of the Nazis' extermination of European Jewry. Vidal-Naquet did not underestimate what he described as \"the tension, not to say [at times the] opposition\" between history and memory. \"History's mode of selection,\" he wrote, \"functions differently from that of memory and forgetting.\" But the revisionists, by denying that the Shoah had ever happened, Vidal-Naquet argued, were attempting to sever the connections linking an anguished Jewish community that was still in mourning from its own past\u2014in effect, trying to murder its memories.\n\nAssuming that Yosef Yerushalmi was largely correct when, in his extraordinary _Zakhor: Jewish History and Jewish Memory,_ he argued, \"Only in Israel and nowhere else is the injunction to remember felt as a religious imperative to an entire people,\" then the affront Vidal-Naquet anatomizes is a particularly grave one. The issue of the Jewish people's in at least some way unique relation to memory is especially acute. But although in the title essay of his book Vidal-Naquet rightly emphasized the malign effectiveness, in a Jewish context, of assaulting group memory of the Shoah, he was adamant that the fundamental issue was \"not one of sentiments but of truth.\" This claim is incontestable. Vidal-Naquet also spoke of the need for a history of Nazism's crimes to take into account \"the transformations of memory.\" Here he seems to be referring to the attempts by the revisionists to distort and if possible annul those memories. However, there is another sense in which the transformation of memory can be understood, and it is both more puzzling and more problematic than the heroic but far more morally clear-cut struggle to which Vidal-Naquet devoted much of his life.\n\nTo put it starkly, by 2035 and probably much sooner, not a single German or non-German collaborationist perpetrator of Nazi crimes is likely to be alive since the oldest person in the world to have lived through the period will have been ten years old at the end of the Second World War. By 2045, if there are any survivors among the victims of those atrocities, they will be more than a hundred years old. When that day comes, the role of historical memory with regard to the Shoah will have a very different resonance from the one that has seemed so crucial to Vidal-Naquet and other like-minded historians and philosophers. An Emil Fackenheim or a Yosef Yerushalmi would almost certainly have disputed this, with regard to the Jewish people at any rate, on the assumption that, in Yerushalmi's words, the Jews could succeed in retaining \"a commonality of values that would enable [them] to transform history into memory.\"2 But it is by no means obvious that this is correct, even in the case of the Jews, this despite the fact that, in the United States at least, a 2013 Pew poll showed that an increasing number of Jews defined their Jewishness more in terms of remembering the Holocaust than of being part of a Jewish community. Surely we are obliged to ask this question: At some point in time, will not Nazi atrocities, collaboration, even the Shoah itself become what the German historian Norbert Frei somewhat regretfully categorized as, \"in scholarly terms, 'plain' history\"?\n\nWe have not reached this point as yet, in 2015. In Europe and North America the Shoah will probably for the foreseeable future continue to be considered what Tony Judt called a \"universal reference,\" presented, somewhat contradictorily, both as \"a singular crime, an evil never matched before or since,\" and as \"an example and a warning.\" And on an individual level, in light of the fact that the best available psychological evidence strongly suggests that the trauma survivors suffer from is passed along for two, three, or even four generations, the salience of the Shoah is unlikely to diminish a great deal for some time, if not universally then, again, at least in Europe and North America. Nonetheless, sooner or later the brute reality of the passage of time all but guarantees that a new set of difficulties will arise\u2014one to which the mantra that, no matter how much time has elapsed, to remember a given event must always remain a defining moral imperative for any society, even assuming it is correct, does not fully respond.\n\nA similar transition, which has in large measure been provoked by analogous actuarial realities, has already taken place in the case of the Armenian genocide. In 2015, even the youngest murderers among the Turks and Kurds who carried out the slaughter have themselves died as well. In the unlikely event that any of the Armenian survivors are still living, they are more than a hundred years old. And just as in the case of the Shoah, when the last survivors are gone who could answer questions face to face or simply roll up their sleeves to reveal the concentration camp tattoos on their arms, what will\u2014indeed, what can\u2014the remembrance of the Armenian genocide consist of?3 In other words, to use the title that Leon Wieseltier gave to his fine essay on the opening of the U.S. Holocaust Museum, in the case of both of these catastrophes, what happens \"After Memory\"?\n\nThere are conflicting answers to this question. On one side, there is Avishai Margalit's response, which is that the death of all those who witnessed the moral nightmare does not change by one iota the obligation of the living to protect the morality radical evil seeks to undermine. And on the other side there is Tony Judt's far more pessimistic suggestion: \"Maybe all our museums and memorials and obligatory school trips today are not a sign that we are ready to _remember_ but an indication that we feel we have done our penance and can now begin to let go and _forget,_ leaving the stones to remember for us.\" Judt recalled that during a visit to Berlin's Memorial to the Murdered Jews of Europe, he saw \"bored schoolchildren on an obligatory outing [playing] hide-and-seek among the stones.\" And he argued, \"When we ransack the past for political profit\u2014selecting the bits that can serve our purposes and recruiting history to teach opportunistic moral lessons\u2014we get bad morality _and_ bad history.\" To which one should add: we also get kitsch.\n\nEven when done well, commemoration almost always skates precariously close to kitsch. One might wish that the Holocaust were an exception in this regard, and that it will always, in Leon Wieseltier's phrase, \"press upon the souls of all who learn of it.\" But it is not, much as we might wish otherwise. This is a distinct problem, not to be confused with the fact that since 1945 the Shoah has regularly been employed to serve political agendas, the most obvious, as Judt emphasized, being to justify more or less any policy of the State of Israel with regard to its neighbors or to its Arab minority. But even when the remembrance of the Shoah is innocent of such subtexts, it has still been smothered in kitsch as Milan Kundera once defined it: all answers being \"given in advance and [precluding] any questions.\" Again, it is understandable to hope that people will be moved by an act of collective remembrance. And it is often, though not always, right to insist that they have a moral duty to remember. Where such acts become kitsch is when people take the fact that they are moved as a reason to think better of themselves.\n\nIt is unfortunate that a prime example of the instauration of this kind of kitsch remembrance is the U.S. National Holocaust Museum itself\u2014the largest and best-known memorial to the Shoah in the world other than the Yad Vashem Memorial Museum and Center in Israel. To be sure, much of what is in the museum is as heartbreakingly far from kitsch as it is possible to get\u2014above all, what Wieseltier called \"the objects, the stuff, the things of the persecutions and the murders,\" when he rightly described the Holocaust Museum as \"a kind of reliquary.\" But these exhibits and films, photographs, and documents are bracketed by two extraordinarily kitschy pieces of set dressing. As one first enters the museum and before one has seen a single image or artifact of either Nazi atrocity or Jewish martyrdom, one must first walk by the serried battle flags of the U.S. Army divisions that liberated some of the concentration camps (there are no British or Russian standards, even though a great many of the museum's exhibits concern Bergen-Belsen, liberated by the British, and Auschwitz, liberated by the Soviets). And as one leaves the last room of the museum, the final exhibit one sees contains a series of images of David Ben-Gurion proclaiming the independence of the State of Israel, and, beyond them at the exit, a column of tan sandstone that is simply identified as having come from Jerusalem.\n\nOne can only hope that in addition to the American triumphalism and what even by the most generous of interpretations is a highly partisan pro-Israeli view of the creation of the state as the existential remediation of the Nazis' war of extermination against the Jews, the intention here was to palliate what, apart from the part of the exhibit devoted to the Danes' rescue of most of their country's Jewish population, is the pure horror of what the museum contains by beginning and ending on an uplifting note. The impulse is an understandable one. But it is also both a historical and a moral solecism that perfectly illustrates Judt's admonition that the result is both bad history and bad morality.\n\nThe current emphasis both in Israel and in the Jewish diaspora that is exemplified by the museum's last exhibit and that presents the Jewish state's moral legitimacy as inextricably bound up with the Shoah seems to me an indefensible justification of the Zionist project _in Zionist terms,_ at least in the long run. It is both ahistorical, since obviously Zionist-inspired Jewish immigration to Palestine far predates the Shoah, and morally dubious, since the Palestinians bear no responsibility for what the Nazis did. As a matter of history, though not of morality, what a Zionist would be on firmer ground claiming is that at the heart of the Zionist project itself, secular and religious alike, is the conviction that the land of Israel with Jerusalem as its capital is not just the historic but the spiritual home of the Jewish people, who in all their wanderings never relinquished what the Israeli writer Yoram Kaniuk once called their mystical deed to it. In this sense, at least, it is surely fair to say that whatever the justice or injustice of this claim, without the preservation of Jewish collective memory over the centuries the establishment of the modern State of Israel would have been far more difficult. As he did so often, Yosef Yerushalmi got to the heart of the matter when he wrote, \"Jewish historiography can never substitute for Jewish memory.\"\n\nTo say this is not to imply that Zionism is concerned only with historical continuity, whether (to the extent that the two are distinguishable) real or \"invented.\" But it does not augur well for what the remembrance of the Shoah will become after it has passed into history in Norbert Frei's sense that the first exhibit of the museum dedicated to commemorating it is in reality little more than an ostentatious display of American nationalism and that the last is kitsch Zionist theodicy pure and simple. But unsettling and unseemly as they are, neither such American narcissism nor the Jewish communitarianism that Vidal-Naquet in his preface to _Assassins of Memory_ declares he is determined to transcend tells the whole story. To the contrary, Holocaust memorials and museums are attempts to keep faith with two moral imperatives: honoring and remembering those who died and, by reminding as many people as possible of the murder of European Jewry, helping individuals and societies alike become more resistant to such evils, and perhaps even to prevent them from recurring in the present or in the future.\n\nThese matters are delicate, as they should be, and if we take such questions on we have a moral obligation to proceed with great caution. But about the argument that the memory of the Shoah is likely to have a deterrent effect\u2014the view encapsulated in the injunction \"Never Again\"\u2014there simply is no way of avoiding the conclusion that this is magical thinking, and of a fairly extreme kind. I am reminded again of Sir Nicholas Winton's remark that no one ever truly learned anything from the past. Yes, \"Never Again\" is a noble sentiment. But unless one subscribes to one of the cruder forms of progress narratives, be they religious or secular, there is no reason to suppose that an increase in the amount of remembrance will so transform the world that genocide will be consigned to humanity's barbarous past. This is where the contemporary heirs and assigns of Santayana go wrong: we never repeat the past, at least not in the way he was suggesting we did. To imagine otherwise is to leach both the past and the present of their specific gravity. Auschwitz did not inoculate us against East Pakistan in 1971, or East Pakistan against Cambodia under the Khmer Rouge, or Cambodia under the Khmer Rouge against Hutu Power in Rwanda in 1994.\n\nIn contrast, establishing the historical truth about a great crime while those who committed it and those who were or at least knew its victims are alive often not only should but also can be done (as opposed to cases where doing so ought but, contra Kant, often _can't_ be done). But such efforts require the investigators to think like historians, investigating the facts and letting the chips fall where they may. If politicians subsequently use the findings of Truth Commissions to their own ends, as they have done in South Africa and in Argentina, to name two of the most obvious examples, this is a price well worth paying. Yerushalmi was doubtless correct to emphasize the greater importance of memory over historiography in the Jewish tradition. But in investigating occluded truths from the past, surely it is history that must be the senior partner and memory the junior one, at least if the goal is, as it should be, to amass the facts necessary to establish an unimpeachable historical record\u2014something that collective memory, which, as even most of its staunchest advocates concede, involves \"editing\" the past to further the needs of the present, rarely if ever does well.\n\nIn the aftermath of the fall of a dictatorship, assembling a reliable and comprehensive record is always important. In the case of apartheid South Africa, the Chile of General Pinochet, or the Argentina of General Leopoldo Galtieri, it was especially urgent because when they were at the height of their power these regimes had taken pains to commit their crimes in secret, just as the Nazis had. And also like the Nazis, they had attempted to cover up the remaining traces of what they had done once they realized they might be ousted from power. There need not even be a change of regime. Klaus Barbie's trial did not unveil much about the crimes the Nazis committed during their occupation of France that was not already widely known. But such was not the case when Maurice Papon finally came to trial before a French court in 1998. Papon had been secretary general for police in Bordeaux under Vichy, and had played a key role in the deportation of sixteen hundred Jews from that city to German concentration camps. After the war, however, far from being treated as the war criminal he was, Papon was seamlessly reintegrated into the French state, and during both the Fourth and the Fifth Republics he served in a series of important posts: prefect in Algeria during the Algerian War of Independence, chief of police of Paris, and finally budget minister during Val\u00e9ry Giscard d'Estaing's presidency.\n\nPapon's trial, in contrast to that of Barbie, did have some of the same morally emancipatory effect in France that the Truth and Reconciliation processes in South Africa and Latin America have had. This may seem surprising to those unversed in the history of post\u2013World War II France, but it took decades for the truth to come out about what had happened between the country's defeat in 1940 and its liberation in 1944. Indeed, until the pathbreaking book on Vichy by the American historian Robert O. Paxton appeared in a French edition in 1973, there had been great unwillingness in France to acknowledge the fact that from a strictly legal perspective, it was P\u00e9tain in Vichy and not De Gaulle in London who had been the legitimate leader of his country. And it was not until the 1981 screening on French television of Marcel Ophuls's film _The Sorrow and the Pity,_ which had been made in 1969 but banned from the airwaves for twelve years, that the consensual silence about the extent and enthusiasm of French collaboration with the Nazis began to lift. (Hara Kazuo's 1987 documentary, _The Emperor's Naked Army Marches On,_ which broke with the established notion of the Japanese as the \"victims\" in World War II, had a somewhat similar effect in Japan.) And it lifted slowly. When Ophuls's film was first shown, French schoolchildren were still being taught that the main current in German-occupied and Vichy France alike had been the Resistance. Airing the truth provoked consternation, denial, and above all anger that such divisive facts had been allowed to come to light. The Gaullist party denounced the film as unpatriotic. At the time, many French people, in all probability a majority, agreed.4\n\nThe fact that their objectors' fears turned out to have been groundless has sometimes been held up by those who call for more of what the American ethicist Jeffrey Blustein has described as \"doing justice to the past\" as evidence that not only is getting to the truth about the past a moral imperative but that those who fear its negative effects have far less cause to do so than they suppose. In _The Moral Demands of Memory,_ Blustein gives a nuanced account of the role an apology can play. When made properly, he argues, it can create new terms of reference between the \"responsible party\" and the \"aggrieved party,\" with the result that \"the past no longer _means_ what it meant before the apology.\" This is a claim so sweeping it makes the Renaissance alchemists who believed they could turn base metals into gold seem almost modest by comparison, though Blustein qualifies it by saying that the longer the delay in issuing such an apology, the more the affront of the original injustice is likely to be compounded. Following the argument of Adam Nossiter's _Algeria Hotel: France, Memory, and the Second World War,_ Blustein points specifically to the fact that it took fifty years, from 1945 to 1995, for a French president, Jacques Chirac, to apologize to French Jews on behalf of the French people, thus assuming national responsibility for their persecution. But for Blustein, however welcome, that apology failed to \"repair the injustice because it [did] not address the fact that the apology was so long in coming.\"\n\nAs in the case of the distinction that can be drawn between the effect on Chilean democracy that indicting Pinochet at the time he relinquished power would have had and doing so eight years later had, however, delay is not always the moral solecism it is often claimed to be. The reality, however unpalatable, is that collective remembrance has not always been a salutary goad to peace and reconciliation, nor has the failure to remember, or, more precisely, in Blustein's term \"properly\" remember, an injustice that a particular group has suffered been toxic to their societies. To the contrary, at numerous times and in numerous places, remembrance has provided the toxic adhesive that was needed to cement old grudges and conflicting martyrologies, as it did in Northern Ireland and in the Balkans for generations, if not for centuries.\n\nThe question arises: Despite the overwhelming consensus to the contrary, does not the historical record\u2014again contra Kant\u2014in the world as it is and not the world as philosophers have claimed it should be and might one day become justify asking whether in some places and at some moments in history what has ensured the health of societies and individuals alike has been not their capacity for remembering but their ability to forget? What I propose is not replacing a _bien-pensant_ fairy tale about memory with a _mal-pensant_ cautionary tale about forgetting. Nor do I suggest that, even if I am right about the uses of such forgetting, it should take place in the immediate aftermath of a great crime or while its perpetrators are still at large. Leaving the needs of history aside, these are moments when commonsense morality and the minimal requirements of justice weigh strongly in favor of remembrance. There are certainly also times when relations between states can be improved and much bitterness removed when a state that has committed a crime against another state acknowledges its culpability. And the same is also the case when the crimes being committed are by a state against its own people.\n\nThe history of the relations between Poland and Russia is instructive in this regard. For generations, there was a bitter joke in Poland: \"Who does a Pole kill first, a German or a Russian?\" \"A German, of course; duty before pleasure.\" Today, Russo-Polish relations are again strained by the Russian Federation's actions in Ukraine. Nonetheless, because in 2010 the Russian parliament finally acknowledged that Russia, not Nazi Germany, was guilty of the mass murder of twenty-two thousand Polish officers in the Katyn forest in western Russia in 1940, in another generation that joke will almost certainly make little or no sense to contemporary citizens of Poland, Germany, or Russia. To put it starkly, we should never underestimate the power of an official apology or deny that memory can be the catalyst for it.\n\nThere are at least some intimations in Maurice Halbwachs's work that he considered memory to be strongly linked to the hope for progress. It is a connection that Avishai Margalit makes explicitly in _The Ethics of Memory_ when he writes, \"Even remembering the gloomiest of memories is a hopeful project [because] it ultimately rejects the pessimist[ic] thought that all will be forgotten, as expressed by Ecclesiastes.\" Margalit is not just calling for remembrance for remembrance's sake. Quite the contrary, he believes there to be both an absolute moral obligation to remember and, at times and depending on the case, lessons to be taught and learned by doing so. Assume for the sake of argument that Margalit is right about the obligation: does this mean that he is also correct in his estimation of the enduring pedagogic value of collective memory? Tony Judt was far closer to the mark when he wrote, \"The trouble with lessons, as the Gryphon observed, is that they really do lessen from day to day.\"\n\nEventually, there comes a time when the need to get to the truth should no longer be assumed to trump all other considerations. Kant thought that no right action could ever have a wrongful element. Perhaps it is because I spent fifteen years observing and writing about what for lack of a better term we call humanitarian emergencies, which are almost invariably situations in which (and this is very much a best-case scenario) even when relief groups are overwhelmingly doing good they are also doing some harm,5 but I confess I do not see how this could ever be true. Almost as incomprehensible is the neo-Kantianism of the international human rights establishment, which refuses to entertain the possibility that when they call for justice, above all an end to impunity, the long-term consequences may prove to have had abidingly negative effects,6 though they will sometimes concede that in the short run there might be negative as well as positive effects, particularly in war zones while fighting still rages.\n\nIt is not that international human rights activists are unaware of how terrible late-twentieth- and early-twenty-first-century wars can be. The present age is one in which most hot wars are one form or another of civil or insurrectionary war within states, or, as in Iraq, to paraphrase Clausewitz, the continuation of terrorism by other means, a situation that seems to make the Geneva Conventions and the other elements of international humanitarian law, most of which were originally devised in the age of interstate war, less and less fit for their purposes. Human rights workers on the ground know this as well as or better than anyone. But the majority of them insist\u2014whatever some may believe privately\u2014that they must proceed on the basis that without justice there can be no lasting peace. One explanation for their absolutism in this regard is that it derives from the fact that the human rights movement is first and foremost grounded in law; its proponents have imbibed not just Kant's idealism but another element in the Kantian worldview that holds that the imperative of justice \"outranks\" all other moral claims.\n\nEmpirically, this is highly debatable, as the case of Bosnia illustrates. From a human rights perspective, the 1995 Dayton Peace Agreement that ended the Bosnian War was an unjust peace that let Slobodan Milo\u0161evi\u0107, the principal architect of the death of Yugoslavia, along with the army and militia commanders and Serb nationalist politicians who had served him, off the hook both politically and judicially; it was only after another Balkan War, this time in Kosovo in 1999, that the great powers involved in the conflict decided to put an end to the impunity Milo\u0161evi\u0107 had enjoyed since Dayton. And the human rights establishment was correct: it _was_ an unjust settlement. But for many of us, who, whether as aid workers or journalists, had seen the horror of the Balkan wars at firsthand, almost any peace, no matter how unfair, was infinitely preferable to the seemingly endless infliction of death, suffering, and humiliation.\n\nThose who, with Fackenheim and Yerushalmi, believe forgetting to be a moral catastrophe, or at a minimum the halfway house toward it, often invest remembrance with the same moral authority that justice has for the international human rights movement. And the two views overlap, as when in the essay \"Reflections on Forgetting,\" in his _Zakhor,_ Yerushalmi asked whether it were possible that the \"antonym of 'forgetting' is not 'remembering,' but 'justice'?\" Even if he was right, however morally attractive his formulation, it does little to invalidate the empirical claim that in the world as it is, peace and justice can sometimes be inimical to each other. This reality should be cause for mourning: it certainly was for the victims of a war such as the relatives and friends of the eight thousand Bosnian men and boys massacred at Srebrenica, where the failure to secure justice for them has inflicted a moral affront in addition to all that they have already suffered.\n\nStill, those who insist that there can be no lasting peace without justice7 blind themselves to reality. The sad fact is that history is replete with outcomes that provided the first while denying the second. When General Pinochet stepped down in 1990, for instance, thus clearing the way for Chile's return to democracy, it was clear that justice had been not done. But the demand for democracy seemed more compelling to more Chileans who had opposed Pinochet (if not necessarily to all the families of Pinochet's victims) than did the demand for justice. At the time, what appeared to be a grant of immunity to Pinochet seemed like a price worth paying. That a measure of justice was finally done eight years later when Judge Garz\u00f3n handed down his indictment cannot change the earlier injustice. But put the case that Pinochet had never been indicted and arrested. In a hundred years, how many Chileans are likely to look back on the transition from dictatorship to democracy in Chile in the 1990s and conclude that impunity for Pinochet had been an intolerable price to pay for their country's freedom?\n\nHere it may be useful to invoke Pierre Nora's distinction between the imperatives of memory and the imperatives of history, and to add to it the contingencies of politics that so often seem to hover over both. \"Memory is life,\" Nora writes, \"borne by living societies founded in its name. It remains in permanent evolution, open to the dialectic of remembering and forgetting, unconscious of its successive deformations. . . . Insofar as it is affective and magical, [it] only accommodates those facts that suit it.\" My imaginary Chilean of a hundred years hence would fit comfortably into Nora's rubric of memory. In contrast, that Chilean would fit far less easily into the rubric of history, which Nora defines as \"the reconstruction, always problematic and incomplete, of what is no longer.\" For Nora, the relationship between history and memory is dialectical: \"Memory instills remembrance within the sacred; history, always prosaic, releases it again.\"\n\nAs a complement to Nora's evocation of the sacred, I would add that collective memory often also functions as an escape and an idyll, providing a moral warrant for nostalgia\u2014an extremely problematic emotion ethically, not least because, to reverse Freud's conclusion about mourning, deference to reality _never_ gains the day. The Cuban-American writer Orlando Ricardo Menes was making a related point when he wrote, \"Idyllic memories are a jeweled noose.\" He knew what he was talking about: the Cuban exile community in the United States to which Menes belongs provides a textbook case of the way nostalgia and self-absorption (the other cardinal vice of the exiled and the scorned), however understandable a community's resorting to them may be, also often serve as a prophylactic against common sense, political or otherwise.\n\nBut Cuban Americans are hardly alone in their self-imposed predicament; at various points in their history, the Irish, the Jews, the Armenians, and the Tamils have been equally trapped in their own particular versions of what the writer Svetlana Boym has called \"the dictatorship of nostalgia.\"\n\n1. Vidal-Naquet himself concedes that because the Lyon court's definition of crimes against humanity, which was far broader than that of the Nuremberg Tribunal, seemed to provide the scope for Verg\u00e8s to invoke the systematic torture used by the French during the Algerian War of Independence as a parallel to the crimes of the Gestapo, the advocate's argument could not be dismissed out of hand. And the distinction Vidal-Naquet makes between the French army acting in defiance of French law and Gestapo officers like Barbie acting in accordance with Nazi law seems to treat too lightly the parallels between imperialism and Nazism that Hannah Arendt pointed to in _The Origins of Totalitarianism_.\n\n2. Yerushalmi, who was painfully candid about his \"terror of forgetting,\" was by no means confident this would happen. \"What has long been called the crisis of historiography,\" he wrote in his bleak coda to _Zakhor_ , \"Reflections on Forgetting,\" \"is but a reflection of the crisis of our culture, of our spiritual life.\"\n\n3. Even in this age of cyberspace, it is difficult to see how valuable initiatives like the one now being undertaken by the Illinois Holocaust Museum and Education Center, which is recording survivors' testimony and then presenting holographic representations of these interviews in which visitors ask questions and, thanks to an algorithm, survivors appear to be responding directly to them, can stand in for the survivors themselves.\n\n4. An Ipsos poll conducted at the time of the Barbie trial showed that a majority of those surveyed wished it were not being held.\n\n5. Although many such groups routinely deploy the line from the Hippocratic Oath, \"First do no harm,\" what they almost always strive for in practice is \"Minimize harm as much as you can.\"\n\n6. While it is exasperating and holier-than-thou, it seems more like self-promotion than an intellectual solecism when major human rights groups call for stopping a war without having any idea of how to do so, as, for instance, when Kenneth Roth, the head of Human Rights Watch, tweeted on July 5, 2015, \"Want to stop flood of refugees? Curb atrocities at source: Syria, Afghanistan, Somalia, Sudan.\"\n\n7. As opposed to the entirely defensible claim that there can be no decent society without justice, which I often feel is what human rights activists actually mean.\nSIX\n\nThe Memory of Wounds and Other Safe Harbors\n\n\"It is possible that there is no other memory than the memory of wounds.\" If we accept Czes\u0142aw Mi\u0142osz's formulation, we must acknowledge the danger that sacralizing collective remembrance is likely to lead to even graver distortions of historical reality than the use of collective memory in the service of the self-glorification of the state, or, for that matter, from the efforts rich and powerful individuals have made throughout history to assert their preeminence by ensuring they will be remembered. At least states have often been straightforward about the pragmatic political nature of the commemorative project. A good example of this, which Jacques Le Goff cited in his essay \"Memory: Written and Figured,\" is the report on the bill in the French parliament that in 1880 reestablished the July 14 Bastille Day national holiday, invented during the Revolution but banned by Napoleon. The _rapporteur_ had stated matter-of-factly that the main purpose of a celebration of this type was to remind the people \"of the memories that are linked to the existing political institution,\" which, he asserted, \"is a necessity that all governments have recognized and put into practice.\"\n\nAnd as both Le Goff and his friend and colleague, the French classical historian Paul Veyne, have pointed out, the phenomenon of rich philanthropists, in Veyne's words, sacrificing \"part of their wealth to ensure the memory of their role\" dates back to Greco-Roman times. To a great extent, the practice continues largely unchanged twenty-five hundred years later, especially in the United States and Canada, where at least since the rise of private philanthropy during the Gilded Age of the late nineteenth century, an increasing number of hospital buildings and hospital centers, as well as business schools, libraries, residence halls, and other university buildings, along with theaters, concert halls, and other cultural centers, have borne the names of the philanthropists who underwrote them, just as tennis and golf stars wear the logos of their corporate sponsors on their clothes when they compete. This goes on even though philanthropists certainly know\u2014shades once more of Shelley's \"Ozymandias\"\u2014that sooner or later their names may be removed from, say, a building when it needs to be refurbished and the price the philanthropist willing to underwrite the new construction is likely to demand will be the replacement of the original donor's name with his or her own. Even here, there is a depoliticized echo of classical times, when after an emperor's death the Roman Senate often removed his name from a building or a monument that during his lifetime the senators had named after him.1\n\nBut the effects of instilling collective memory based on a sense of national or individual greatness differ significantly from those of memory anchored in a sense of personal and collective injury, whether physical, legal (discrimination), or cultural or psychological (exclusion), or in a sense of shared suffering. Indeed, while both typologies involve collective memory, the difference between the former and the latter may be summed up as that which exists between the memory of a victory and the memory of a defeat. The former may offend or annoy because of its triumphalism, in the way that British and Belgian commemorations of the hundredth anniversary of the Battle of Waterloo irked the French, or, more properly, a segment of the French political and cultural establishment, or that African American veterans of the Second World War were rightly offended at the failure of U.S. memorial celebrations to pay tribute to their role. But whether such complaints were worth taking seriously, as was the case with the black U.S. veterans, or were of no consequence, as with the Waterloo centennial, to the extent there were any at all, their harmful effects were comparatively small. In contrast, it is difficult to imagine how the Serb nationalist obsession with the Battle of Kosovo Polje in 1389 could have been more destructive to everyone affected by it, including, of course, the nationalists themselves.\n\nAvishai Margalit confronts this problem head-on in _The Ethics of Memory_. Why, he asks, did the memory of Kosovo Polje wreak such havoc in the former Yugoslavia in the 1990s, whereas the memory of the Battle of Hastings of 1066 that sealed the Norman conquest of England has had no such harmful effects? Margalit's question is better than his answer. He seeks to rebut the claim of the American political philosopher Russell Hardin, who himself evoked the Kosovo Polje\u2013Hastings dichotomy to buttress his own argument that \"the assertion of historical memory contributes to mystification rather than explanation or understanding.\" But in order to do so, Margalit resorts to the claim that, historically, Hastings was a less dreadful and haunting event for the English than Kosovo Polje had been for the Serbs. According to Margalit, the former was \"more a war of succession . . . [and] besides, mostly good things happened in England after the battle,\" while the latter had \"marked the fall of a glorious Serbian independent state . . . [and] the inception of a vassal state for four hundred years under the Ottomans, an alien force with a different religion.\"\n\nMargalit's historical claims are unconvincing in and of themselves. To say that mostly good things happened in England after Hastings is, to put it charitably, an extreme form of the already highly questionable Whig interpretation of British history in which the country's past is asserted to have been an inevitable progression toward ever greater liberty and constitutional government. And the Battle of Hastings, far from being comparable to, say, the peaceful ascension to the British throne of William of Orange in 1688 (the so-called Glorious Revolution), led to the virtually complete destitution of the defeated Saxon nobility by the Norman invaders so that within twenty years 95 percent of the land south of the River Tees in the north of England had passed into Norman hands. A highly romanticized, but by no means entirely inaccurate account of the tensions between Normans and Saxons long after the Conquest can be found in Sir Walter Scott's novel _Ivanhoe_ and, far more dubiously, in some of the popular legends about Robin Hood that represent him as a leader of resistance against Anglo-Norman rule.\n\nBut even though Margalit's irenic rendering of British history cannot be defended, Margalit nonetheless offers, along with Todorov, the most lucid early-twenty-first-century arguments in favor of the idea that societies and the individual human beings of which they are composed have a general duty to remember but no equivalent general duty to forget. By this Margalit means that memory plays a crucial role in allowing us to understand not just where we came from but who we are. As a consequence, unless we are mystics who have chosen to withdraw from the world, we have a duty both to ourselves and to our societies to remember: \"Who we are depends on our not forgetting things that happened and that are important in our lives.\" Perhaps if Margalit believed that it was possible to forget _voluntarily,_ he would not have been so categorical. But he does not. Philip Roth's inspired phrase \"Remember to forget\" is for Margalit a contradiction in terms. Instead, Margalit likens the cognitive and social systems of remembrance and forgetting to the essential functional difference between the somatic and the autonomic nervous systems\u2014the muscles controlled by the former work \"on the demand\" of the brain, while the latter do not. \"I can voluntarily think of a white elephant,\" he writes, \"but I cannot follow the instruction not to think of a white elephant.\" From this he draws the conclusion that \"forgetting cannot be voluntary.\"\n\nBut here, whereas Margalit remarks early in _The Ethics of Memory_ that he shares \"Wittgenstein's belief that the first philosophical move should be to loosen the grip of metaphor,\" he seems trapped by the limitations of the powerful neurological metaphor that he puts forward to buttress his critique of forgetting. For while it is obviously true that an individual cannot choose to _forget_ in the literal sense, it is equally true that societies cannot choose to _remember_ in the literal sense. And that is because unlike individual memory, collective memory is a metaphor too, as Yerushalmi conceded when he wrote, \"The 'memory of a people' is a psychological metaphor.\" Margalit acknowledges this when he writes, \"There is nothing natural about shared memory, and nothing natural about the groups that are the natural candidates for being communities of memory. . . . They are all, in the jargon of today, social constructs.\" It is Margalit himself, then, and not Philip Roth who has made the fundamental error. Because if communities of memory and the collective memories they \"decide\" to share are social constructs\u2014as of course they are\u2014then in and of itself it is no more unnatural, immoral, or impossible to posit the feasibility of a socially constructed community of forgetting than one of remembrance. Neither is less artificial, or, more to the point, less imaginable than the other. This does not mean that one cannot favor remembrance over forgetting on moral grounds, but it does make specious the claim that the former is more \"real\" than the latter.\n\nIn his \"Reflections on Forgetting,\" Yerushalmi describes being struck when a friend sent him a report of a _Le Monde_ poll conducted at the time of the Barbie trial in which respondents were asked: \"Of the following two words, 'forgetting' or 'justice,' which is the one that best characterizes your attitude toward the events of this period of the war and the Occupation?\" As we have seen, this led Yerushalmi to a new formulation: Was it possible \"that the journalists have stumbled across something more important than they perhaps realized . . . [and that the] antonym of 'forgetting' is not 'remembering' but 'justice'?\" Margalit also seems to view forgetting as a kind of injustice, though he is more concerned that forgiveness come before forgetting than that forgetting never take place. \"I maintain,\" he writes, \"that what is needed for successful forgiveness is not forgetting the wrong done but rather overcoming the resentment that accompanies it.\"\n\nLet us accept that Margalit is right when he asserts not only that this process can happen but that it has already happened in the past and will happen again.2 But as Margalit certainly knows as well as anyone, there also have been numerous times in history when this has not happened\u2014that is, when societies or groups within societies could not overcome their resentment, when they could not forgive\u2014and that too will recur.3 His schema has much to offer with regard to how societies should best respond in such cases, and, indeed, how they can prevent the resentment from fueling a new conflict while the search for an accommodation of some kind continues. To cite one example, this was the challenge facing those attempting to negotiate a peace agreement in Northern Ireland throughout the decade leading up to the Easter Sunday accords of April 1998 that finally largely brought the country's long war to a close. And Jacques Le Goff's hope that (collective) memory will aid the liberation rather than the enslavement of humanity is relevant here. But while Le Goff's sentiment is one that no sane human being could fail to wish were always the case, it too begs the question of what happens when it isn't. Are we to wait for people to come to their senses? If so, we may be in for a very long wait indeed. Or is it possible that this is where forgetting without the preamble of forgiveness or the promise of justice has a value that even thinkers as brilliant as Le Goff, Margalit, Todorov, and Yerushalmi have not been able to bring themselves to grant?\n\nI do not claim that forgetting would be an appropriate response in cases where justice or forgiveness (or both) are a realistic alternative, as in many cases, including some grave and seemingly intractable ones, they will be. But the ultimate metric here should not be the ideal but the probable, or at least the feasible. Bismarck's celebrated remark that no one should look too closely at the making of sausages or laws surely applies even more forcefully to peace settlements. When it is possible, by all means let societies remember, provided of course\u2014and this is a very big caveat indeed, and one those convinced that remembrance is a moral imperative consistently underestimate\u2014remembering does not engender further horrors. But when it is not possible, then, to paraphrase the slogan of the anti\u2013Vietnam War movement of the late 1960s, it may be time to give forgetting a chance, which is another way of saying that it is time to give politics a chance and idealism a rest.\n\nAnd idealism it most emphatically is. Take, for example, Todorov's observation that for historical remembrance to have enduring value and be of enduring help to the societies that undertake it, it must lead to \"a [generalizable] principle of justice, a political ideal, or a moral rule [that must be] legitimate in and of themselves [ _sic_ ] and not because they derive from a memory that is dear to us.\" On the basis of what historical evidence, or, indeed, of any principle other than hope, should we conclude that this is where the exercise of collective memory is likely to lead us? And since, in contrast, we know that historically memory has been and continues to be toxic in many parts of the world, why put our ethical trust in it, no matter how morally desirable such an outcome might prove to be?\n\nWe do, after all, already understand the risks. Like Margalit, Mi\u0142osz was being metaphorical when he mused about whether human beings had any memory other than the memory of wounds. But there have certainly been periods in the history of every nation, and probably every community, in which making such a claim has required no poetic license. Again, the case of Northern Ireland stands out, since there, until the late 1990s, as one Ulster poet put it, the country got \"martyrs when it needed men.\" Those were the days when remembrance was hatred's forge and forgiveness was not on offer. Is it not, then, at least worth considering, if only as a thought experiment, whether it might have been better for everyone in the six counties of Northern Ireland if they had found a way to forget the wrongs of the past, whether real or imagined, and whether accurately or inaccurately \"remembered\"?\n\nLike any counterfactual, the question is one that can have no definitive answer. But what we know all too well is the damage collective memory can do. The Irish writer and politician Conor Cruise O'Brien once observed of a particularly dark and despair-inducing period of the conflict in Northern Ireland that during the secret negotiations in which, as a member of Garet Fitzgerald's government in Dublin, he had been involved, just as it seemed as if Republicans and Unionists might finally, painstakingly be coming close to an agreement, a representative at the talks from one of the two sides would remember one of great militant songs\u2014the IRA's \"Rising of the Moon\" or the Ulster Volunteer Force's \"Sash My Father Wore\"\u2014and whatever hope had existed would soon vanish. Cruise O'Brien was much given to embellishment, but in this instance he didn't need to. Those who may be tempted to doubt the potency of those songs might recall that \"the rising of the moon\" were the last five words of the final entry in the prison diary kept by the Provisional IRA leader Bobby Sands before he died on the sixty-sixth day of the hunger strike he had organized and led among the Republican prisoners in the Long Kesh (Maze) prison in County Down, Northern Ireland, in 1981. \"They won't break me,\" Sands had written of the prison authorities, \"because the desire for freedom, and for the freedom of the Irish people, is in my heart. The day will dawn when all the people of Ireland will have the desire for freedom to show. It is then we'll see the rising of the moon.\"\n\nThere are also echoes in Sands's diaries of _The King's Threshold,_ a play about a hunger striker set in the mythical past that Yeats had written in 1904, but which, as Sands surely knew, Yeats had rewritten in 1920 in the aftermath of the death on hunger strike of the imprisoned Sinn F\u00e9in lord mayor of Cork, Terence MacSwiney, himself a playwright. Sands's act also seemed born of the same conviction that had led Patrick Pearse to confidently insist that even if the Easter Rising ended in failure, and he himself was killed by the British (he was), his and his comrades' deaths would be a \"blood sacrifice\" that would stir the whole Irish people and lead to their rising up and finally seizing their independence. Of Sands's hunger strike and eventual death, Seamus Heaney, Ireland's greatest poet since Yeats, himself originally from County Derry in Northern Ireland, subsequently wrote: \"I was highly aware of the propaganda aspect of the hunger strike and cautious about being enlisted. There was realpolitik at work; but at the same time, you knew you were witnessing something like a sacred drama.\"\n\nAs Heaney knew well, sacred drama is the antithesis of any decent politics. For once the sacred has been invoked, there can be no compromise with one's adversaries, only their unconditional surrender. To the extent that this can still be called politics, it is a politics of totalitarianism. Yeats might write in his poem \"September 1913,\" \"Romantic Ireland's dead and gone; it's with O'Leary in the grave,\" but, in reality, as Yeats would realize in the aftermath of the Easter Rising and sum up in another poem, \"Easter, 1916,\" with the words \"a terrible beauty is born,\" romantic Ireland survived the Fenian John O'Leary, financial officer of the Irish Republican Brotherhood and editor of _The Irish People,_ as it would survive Pearse and Easter Sunday, 1916, and MacSwiney, and Sands too. It was only the deed of the very different Easter Sunday, in 1998, that finally put romantic Ireland in its grave. That Ireland had lived and battled in poetry, whereas whatever else has gone wrong in the Ireland of the twenty-first century, the continuation of peace in the North amply demonstrates the truth of a former governor of New York State, Mario Cuomo, who said that one campaigns in poetry but governs in prose.\n\nPoetry, which is among other things the native language of myth, facilitates long memories, while prose, at least in the Cuomo sense, helps shorten them. For as John Kenneth Galbraith admonished us, and as anyone with direct experience of practical politics knows, \"Nothing is so admirable in politics as a short memory.\" Of course, myths, like the proverbial wires, can become crossed, allowing the legends of the distant past and the ideological needs of the present to blur into each other. Take as an example \"The Dying C\u00fachulainn,\" the statue of the hero of the Irish medieval epic the _T\u00e1in B\u00f3 C\u00faailnge,_ which stands inside the General Post Office in Dublin. The statue was made in 1911 by the Irish sculptor Oliver Sheppard, a leading figure in the Celtic Revival and an ardent nationalist. But it is there not as a tribute to ancient Ireland; it was installed in 1935 at \u00c9amon de Valera's personal request as a memorial to the Easter Rising. Samuel Beckett treats these contradictions venomously in his novel _Murphy,_ published only three years later, in which the character Neary says that he wanted to \"engage with the arse of the statue of C\u00fachulainn, the ancient Irish hero, patron saint of pure ignorance and crass violence,\" by banging his head against it.\n\nThe current commonplace condemnation of contemporary life holds that because of a surfeit of technological stimulation we have shorter and shorter attention spans\u2014but if true this has done nothing to shorten our collective memories of grievance. Whether the United States or France was the first out of the gate in this is an open question, but in the early twenty-first century there are few democratic societies that are not embroiled in their own memory wars, which have become arenas for competing, or, at the very least, for competing matyrologies. The American sociologist Jeffrey Olick has described the shift from earlier forms of official commemoration as one in which governments now \"commemorate failures as well as triumphs,\" while \"social movements and other identity groups turn to 'repressed' histories as sources of their cohesion and as justification for their programs.\" For Olick, Max Weber's idea of a new \"theodicy of disprivilege\" lies at the heart of the shift.\n\nWhere remembrance is concerned, despite impassioned assertions that truths previously concealed have been brought out into the open and need to be acknowledged, the question of historical accuracy rarely seems as crucial as does the group solidarity such remembrance is meant to engender. That this politics is anchored not just in ressentiment\u2014the point, following Nietzsche, the philosopher Max Scheler, and Weber, that Olick emphasizes\u2014but in radical subjectivity should come as no surprise. Moreover, it is a psychological truism that an individual's effort to recover his or her own memories, whether readily available or repressed, when done properly and seriously in a therapeutic context (Freud's _Durcharbeiten_ ) can be healing. Unfortunately, this has led to the psychological pop-culture commonplace that to be able to remember a traumatic experience is the necessary first step in coming to terms with it. And the same is now thought to be the case with the collective memories of social groups. This is what my late father, Philip Rieff, meant when he wrote of \"the triumph of the therapeutic.\"\n\nA shrewd defense of the therapeutic view has been offered by the American psychiatrist Janet Baird, who has argued that both individuals' traumatic memories and the collective historical memories of groups, for all the obvious differences in the ways these are formed, retained, and transmitted, \"retain the quality of 'now,' rather than receding into the subjective past.\" Baird adds that where collective historical memory is concerned, social stress seems to awaken and activate \"the historical memory in [such] a way that the protagonists of the past become resurrected in the 'now.\"' Although what Baird describes is undoubtedly true clinically, what may be constructive for a therapist treating a patient's individual trauma could be highly dangerous politically when nations, peoples, or social groups act on their collective traumas. In this sense, to extend the psychotherapeutic metaphor, they are not treated, they self-medicate. Part of the reason for this is that while individuals' memories are often distorted (and in extreme cases, partly invented or even false), they are undeniably real in the sense that they derive from actual lived experience. In contrast, once the transmission of collective memories continues for more than three or four generations, it can no longer be called memory _other than_ metaphorically. To use two storied examples, Irish men and women today do not \"remember\" the Great Famine of 1847, nor does an observant Jew praying in the synagogue \"remember\" Jerusalem.\n\nEven far more benign examples of this are no less paradoxical. \"Je me souviens\" (I remember), the official motto of Qu\u00e9bec, is one such case. The phrase was the coinage of the architect Eug\u00e8ne-\u00c9tienne Tach\u00e9, the son of a former prime minister of United Canada and one of the architects of Canada's Confederation of 1867, which is to say of its birth as a modern nation. Tach\u00e9 had been commissioned to design Qu\u00e9bec's new parliament building, and he appears to have decided, completely on his own, to add \"je me souviens\" to the coat of arms granted to the province by the British crown in 1868. Although some mid-twentieth-century Qu\u00e9bec nationalists claimed that Tach\u00e9 had had a separatist meaning in mind, there is no convincing evidence for this, as there is for the watchword \"n\u00f4tre ma\u00eetre le pass\u00e9\" (our master the past), which was an avowedly separatist rallying cry when it was coined in 1936 by Father Lionel Groulx, one of the founders of modern Qu\u00e9becois nationalism.\n\nExamples like Tach\u00e9's are more the exception than the rule, as the history of Qu\u00e9bec after the so-called Quiet Revolution of the 1960s amply demonstrates. The Quiet Revolution put an end to the mixture of opportunism and clerical conservatism that had marked the politics of the province during the long premiership of Maurice Duplessis, which was, in significant ways, as close as North America has come to a homegrown Franco- or Salazar-style corporatism. It is true that Qu\u00e9becois nationalism in the post-Duplessis era for the most part followed the more conventional pattern, in which the invocation of collective historical memory is used to accentuate differences rather than, as Tach\u00e9 seems to have attempted to do, to bridge them. But the irony is that Qu\u00e9becois nationalism after 1965 emphasized Qu\u00e9bec's distinctness while rupturing much of what linked the province's present with much (some would even say most) of its historical past. Where Qu\u00e9bec had traditionally been politically conservative, the new nationalist intellectuals identified themselves overwhelmingly with the left; and where Qu\u00e9bec had been religious, in the main, they were nonbelievers. In short, the claim that Qu\u00e9bec had always been a distinct society and should have become and one day would be an independent country coexisted with the repudiation of nearly everything, apart from the French language, that historically has made French Canada unique.\n\nAnd yet contemporary Qu\u00e9bec is as obsessed with commemorations of its past as the rest of the world, though the province's collective memory is now being rewritten, however grudgingly, to include what in Canada are called the First Peoples, and to focus specifically on the struggles of the ruled, rather than the rulers, and above all on women. But while the content of the debate about what should be remembered varies from country to country, the general pattern remains largely the same. The paradox is that while what Pierre Nora has rightly called the memory industry has expanded to the point that it has begun to seem like a demonstration of the second law of thermodynamics, with the appetite and quest for memories becoming ever more dispersed with time, schoolchildren in virtually all developed countries know less and less about contemporary politics, world geography, or history. And what little history they do know is not history in the proper sense of the term but remembrance. The danger should be obvious: whatever the pitfalls of history in the traditional sense, it is not narcissistic, which is another way of saying it is genuinely concerned with the past, fully taking on board the understanding that the English novelist L. P. Hartley encapsulated in the first line of _The Go-Between_ (1953): \"The past is a foreign country: they do things differently there.\" In contrast, remembrance is about self-love, and self-recognition, which means more often than not that it is little more than the present in drag.\n\n\"What is the Ninth Symphony,\" Karl Kraus asked, \"compared to a pop tune played by a hurdy-gurdy and a memory?\" That is remembrance in a nutshell. At best, it is a consolation or an ego boost, while at worst it is a wallowing, no matter whether in past triumphs or past injuries and traumas. In contrast, history is hard, and the better the history, the more demanding and outward-looking it is. Never mind the Ninth Symphony: think Berg or Schnittke. Instantly gratifying (even if the form that gratification takes is anger, bitterness, or ressentiment),4 the overvaluing of collective memory and the undervaluing of history is a perfect fit with the spirit of an age that is itself dominated by instant gratification. Add what the Australian writer Robert Hughes called \"the culture of complaint\" to a self-absorption that is one of the constitutive elements of Guy Debord's \"society of the spectacle\" in multicultural drag (\"all that was once directly lived has become mere representation\" was how Debord formulated it). On the other side of the ideological divide, throw in the old simplifications of nationalism, which, as Ernest Gellner pointed out in the 1970s, had always been opportunistically selective in what it chose to pay obeisance to, and all the sentimentality, complacency, and kitsch nostalgia that goes with it. The result? The world as pop song\u2014and one that is more akin to the Latvian entry in the Eurovision contest than to Leonard Cohen's verities. If the memories are not actually there\u2014and not to belabor the obvious, in the literal sense at least, they never are\u2014teach the myth.\n\nIs much of this harmless? Of course it is. But it is dangerous enough of the time, especially for peoples who, for a wide variety of historical, religious, and cultural reasons, are highly prone or at least vulnerable to self-mythologization. And we could legitimately go farther and assert that the world would be a better place if, instead of being convinced that collective memory (rightly understood, to be sure, as Margalit and Todorov would correctly insist) should be a moral imperative for us, we instead chose to forget. The last thing I want to do is replace one hypercategorical prescription with its opposite, let alone issue a jeremiad on the order of Yerushalmi's when he insisted that \"in the world in which we live it is no longer merely a question of the decay of collective memory and the declining consciousness of the past, but of the aggressive rape of whatever memory remains.\" What can be demonstrated is the considerable extent to which many of the arguments for collective memory are in their essence profoundly anti- or at least post-political in orientation in much the way the human rights movement's arguments are. To put this in the form of two questions: If humankind is or at least can become, as Margalit would have it, a moral community, what are that community's politics? and If that community is above or outside politics, can it be a community worthy of the name?\n\nGiven that the human rights project shares with the duty of memory project the same law-based view of morality, these parallels should come as no surprise. In addition, both worldviews are absolutist, whereas the essence of democratic politics is compromise\u2014that most effective of prophylactics against fanaticism. And even though everything will be eventually forgotten in the fullness of geological time, the conceit of collective memory is that in theory at least it can be renewed forever. Such renewal, far from ensuring justice, is a formula for unending grievance and vendetta.\n\n1. In 2015, for example, the billionaire philanthropist David Geffen donated $100 million to refurbish the concert space at New York's Lincoln Center then named Avery Fisher Hall, after its original benefactor, on condition that it be renamed David Geffen Hall. In accepting his offer, Lincoln Center was obliged to pay the Fisher family $15 million.\n\n2. To some extent, though not as greatly as the received wisdom outside the country would lead one to believe since it had and still has many opponents, this can be said to have occurred in the case of the Truth and Reconciliation Commission in South Africa.\n\n3. See, for example, the fury with which Bosnian crowds greeted the arrival of the Serbian prime minister, Aleksandar Vu\u010di\u0107, at the twentieth-anniversary ceremonies in 2015 marking the Srebrenica massacre.\n\n4. And even ressentiment is a complicated business. Jeffrey Olick, who sees this clearly, offers this insight from the radical American political theorist Wendy Brown: \"Identity politics may be partly configured by a peculiarly shaped and peculiarly disguised form of class resentment that is displaced onto discourses of injustice other than class, but a resentment, like all resentments, that retains the real or imagined holdings of its reviled subject as objects of desire.\"\nSEVEN\n\n_Amor Fati_\n\nWith the possible exception of the Jews, for whom questions of law, tradition, memory, and custom are notoriously difficult to disentangle, collective historical memory is no respecter of the past. Indeed, historically, most of the claims to continuity that are made for it are at best partly specious and often utterly so. This is not simply a matter of inaccuracy, willful or otherwise, of the type one encounters in the many contemporary television miniseries that attempt to re-create a past historical era\u2014Showtime's _The Tudors,_ say, or HBO's _Rome,_ about the latter of which the classicist Mary Beard remarked that for her the great pleasure of watching it was seeing how many historical howlers could be crammed into a given episode. Such entertainments rarely if ever challenge the conventional order, and usually reflect it\u2014think of the Hollywood films of the 1930s such as _Lives of a Bengal Lancer_ and _Gunga Din,_ whose hymns to colonialism Bertolt Brecht skewered so mordantly in the diaries he kept during his Los Angeles exile. In contrast, when states, political parties, and social groups appeal to collective historical memory, their motives are far from trivial.\n\nUntil well into the second half of the twentieth century, the goal of such appeals was almost invariably to foster national unity, along the same lines that Renan the historian had anatomized and Renan the nationalist had prescribed. It would be comforting to believe that damnable regimes have been more given to this practice than decent ones, which may help account for the fact that many discussions, including Todorov's and Margalit's, that attempt to validate the concept of collective memory but frankly acknowledge how often it is misused,1 highlight Hitler's and Stalin's manipulations of the past, as in the case of Stalin's attempts to mobilize the Russian people to resist the German invasion by appealing to the memory of the medieval prince of Novgorod, Alexander Nevsky, who had repelled a previous German invasion when he defeated the Teutonic Knights at the Battle of the Ice in 1242. But the reality is that such efforts to mobilize and manipulate collective memory or manufacture it have been made by regimes and political parties of virtually every type.\n\nThere have even been times when rival political movements have vied for \"ownership\" of a particular historical figure who is thought to incarnate the nation. A case in point was Joan of Arc in nineteenth-century France. Between 1841, when a biography of her appeared written by the influential historian Jules Michelet, and the early twentieth century, the left and the right in France both claimed her as their symbol even though their portraits of the Maid of Orleans were incompatible. For the right, she was seen as the emblem of France's determination to repel foreign invaders, while for the largely anticlerical French left\u2014and in this they were faithful to the portrait the militant anticlerical Michelet had presented of her\u2014she was a victim of the church that had condemned her to be burnt at the stake. Once the Catholic Church beatified her in 1909 (she was then canonized in 1920), the left could no longer credibly claim her as one of their own. Yet the \"memory\" of Joan of Arc continued to be contested; it became a rallying point for the right, first for the extreme conservative Catholic movement, the Action fran\u00e7aise, and the Vichy government during the Second World War over which the movement exerted considerable influence, then, beginning in the late 1980s, when its leader Jean-Marie Le Pen made his first significant electoral breakthrough, for the French ultra-right party, the Front National, which commemorates Joan of Arc every May 1, not coincidentally the date of the left's most important annual holiday.\n\nPresumably not only Tzvetan Todorov and Avishai Margalit but almost anyone who was reasonably literate historically would view skeptically the French right's manipulation of the figure of Joan of Arc. That effort to inculcate a \"collective memory\" to suggest that just as she incarnated France's struggle against the English foreign invaders of her time, so too does today's Front National, this time against Muslims and other immigrants that the party wishes to persuade the French people are the invaders of today, represent a gross distortion of history. Yet typologically the right's manipulation of Joan of Arc is no different from and no more inaccurate than the determined efforts of the impeccably social democratic Scottish National Party to appropriate the figure of William Wallace, the late-twelfth-century nobleman who was an early leader of medieval Scotland's wars of independence, for its own ideological and electoral ends.\n\nIf anything, the William Wallace that the SNP held out as a model for emulation for Scottish voters bears even less resemblance to the historical figure than does the Joan of Arc touted by the Front National. We probably have Hollywood to thank for this: the SNP shamelessly capitalized on Mel Gibson's preposterous biopic of Wallace, _Braveheart,_ using the launch of the film in Scotland in 1995 to jump-start a massive recruitment drive for the party. Volunteers handed out leaflets to filmgoers as they left cinemas all over Scotland that read in part: \"You've seen the movie\u2014Now face the reality. . . . Today, it's not just bravehearts who choose independence, it's also wise heads.\" The juxtaposition was patently absurd, and yet the SNP's then vice president, Paul Scott, seemed to have no problem drafting into his party's cause the figure of a Scottish minor nobleman about whom, apart from his military campaign of 1297\u201398 and the ghastly details of his public execution by the English in 1305, virtually nothing is known. \"In modern terms,\" Scott told an interviewer, \"the desires of civic nationalism are exactly the same [as those of Wallace].\"\n\nThere is no _inherent_ difference between the use the Front National has made of Joan of Arc and that which the SNP has made of William Wallace. Yet given the contrast between the furious response to the former and the comparatively relaxed reception of the latter, one would be hard-pressed not to conclude that the objection is not to the manipulation of history but to the fact that it is the Front National doing the manipulating. As the old psychoanalytic joke would have it, \"When the right person does the wrong thing, it's right; when the wrong person does the right thing, it's wrong.\" To be sure, at some point in their lives, almost all adults have had to confront the contingency of their convictions. But it hardly seems intellectually credible to claim that the Front National's manipulation of Joan of Arc is an abuse of collective memory while the SNP's of William Wallace is not. And one wonders whether, had the left's equally historically unsustainable claims regarding Joan of Arc prevailed in twentieth-century France, the same bien-pensant severities would have been brought to bear on them.\n\nNo one who has written about collective memory has been more alert to the dangers of its abuse than Tzvetan Todorov. In 2010, for example, he returned from a trip to Argentina, where he had visited the sites commemorating the victims of the military dictatorship, including the former Navy Mechanics School in Buenos Aires\u2014the so-called ESMA, where hundreds were tortured to death\u2014and wrote a highly critical account of them in a column in the Spanish newspaper _El Pa\u00eds_. \"Memory,\" he wrote, \"is subjective . . . which is why it can be used by a [particular] group as a means of gaining or reinforcing its political position.\" And yet Todorov has continued to insist that, while \"a society needs history, not just memory,\" when done properly historical remembrance provides exemplary instances of more general categories, thus serving \"as a model for understanding new situations involving different [historical] actors.\" And for him, this is at least \"potentially liberating.\" Todorov freely concedes that \"not every lesson is a good one\" (to this extent, at least, echoing Margalit's vision of humanity as potentially an ethical community of memory), but he nonetheless insists that each can be evaluated on its merits \"with the help of universal, rational criteria that sustain [genuine] human dialogue.\"\n\nBut is he right to believe this, or is he instead imputing to human beings a far greater degree both of rationality and of solidarity than we as a species possess? To insist on the point is not the same as claiming that human beings are too irrational and self-interested to learn from the past, or are incapable of applying what they have learned (and not in the reductive, Santayana-like sense, with which Todorov has never sympathized). Rather, it is to say that, contra Margalit, to ask whether collective memory exists is, in an essential sense, to ask the wrong question. Of course collective memory exists, _but only metaphorically,_ which makes it subject to numerous distortions that should put the claims for its importance morally and ethically under severe strain. In his lovely book on metaphors in literature, Denis Donoghue writes, \"We normally\u2014and justly\u2014speak of metaphor as an irruption of desire, specifically the desire to transform life by reinterpreting it, giving it a different story. . . . It expresses one's desire to be free, and to replace the given world by an imagined world of one's devising.\"\n\nIs the case of history so far removed from that of literature as to be irrelevant to it? That hardly seems likely. No matter what the arena of human consciousness and thought, to apply in the name of the duty to remember an essentially metaphoric understanding of the past to the present seems far more likely to elicit unreason than reason, if on no other basis than that, in Leon Wieseltier's stark phrase, \"The mind cannot do without the imagination, but the imagination can do without the mind.\"\n\nIf this is right, then the contradictions inherent in any definition of collective memory and not, as Ricoeur, Margalit, Todorov, and others have argued, its abuse are what make it difficult for human beings to avoid falling for or indeed propagating such historical travesties as the Front National's adoption of Joan of Arc or the Scottish National Party's of William Wallace. It is true that the metaphoric essence of collective memory does indeed free it, and allows it, as in Donoghue's sketch of it in the context of literature, to replace the given world with an imagined world of one's own devising. But it is the freedom of the permanent adolescent. And as is the case with many adolescents, remembrance too often proceeds as if gravitationally drawn to suffering, conflict, and sacrifice.\n\nRenan saw all this clearly, and unlike the partisans of collective memory of the early twenty-first century, he viewed its emphasis on suffering as its essence rather than as a distortion or abuse of that essence: \"More valuable than common customs posts and frontiers conforming to strategic ideas is the fact of sharing, in the past, a glorious heritage and regrets, and of having, in the future [a shared] program to put into effect, or the fact of having suffered, enjoyed, and hoped together. These are the kinds of things that can be understood in spite of differences of race and language. I spoke just now of 'having suffered together' and, indeed, suffering in common unites more than joy does. Where national memories are concerned, griefs are of more value than triumphs, for they impose duties and require a common effort.\"\n\nFrom the vantage point of 2015, in an era when the grand narrative of the state has been or is in the process of being challenged, overhauled, and in some cases replaced entirely by often competing agendas of those mistreated, oppressed, and excluded by states and by dominant classes and races, what needs to be added is that these agendas too are not just strengthened by grief, as Renan understood, but sustained by the sense of traumatized victimhood on the part of both the individuals and the collectivities concerned. Most of the time, this is harmless. But not all of the time, and those are the instances when it needs to be kept in mind that there are few phenomena more uncontrollable socially and, hence, more dangerous politically than a people or a social group that believes itself to be a victim. This was what Auden was talking about in \"September 1, 1939,\" when he pointed to \"What all schoolchildren learn\/Those to whom evil is done\/Do evil in return.\"\n\nOne proof of this can be found in the fact that in the minds of their perpetrators, virtually every great crime of the twentieth century has been committed in an atmosphere of fear and with the justification of self-defense, whether reactive or preemptive, that is to say, of \"us or them.\" The Turks thought the Armenians were a Russian fifth column; Stalin thought the Kulaks were subverting his program of agricultural collectivization on which the future of the Russian food supply depended; and the Nazis conceived of the Jews as being the moral equivalent of a lethal microbe and of having been responsible for Germany's defeat in the First World War. Despite all the promises of \"Never Again\" put forward in the wake of the Seond World War, in the postwar world, existential fears have driven other groups to commit mass murder in the name of self-preservation, the most terrible of which in terms of the proportion of the population that was killed was the Rwandan genocide.\n\nIf faced squarely, what these events and the motivations behind them should serve to refute is the belief that sometime in the twentieth century humanity turned an existential corner.2 No matter how many ceremonies of commemoration we hold, no matter how many museums to the Shoah or the Great Irish Famine we build, no matter how many laws we pass along the lines of the French parliament's criminalization of the denial of the Armenian genocide, and no matter how radically we revise the school curricula in countries that grew rich off the slave trade in order to acknowledge its horrors, we are bound to fall short of the millenarian goals that we delude ourselves are now within our reach. However seductive the thought might be, to imagine otherwise is not going to help us remember \"better\" and more usefully in an ethical or social sense. Despite Todorov's injunction that it is possible to \"make use of the lessons of past injustices to fight against those taking place in the present,\" the effect in at least some extreme situations is more likely to be to de-historicize these events, thus leaching them of their specificity. In Rwanda after the genocide, a P\u00e8re Blanc who had lived through it was asked by a journalist whether witnessing it had not caused him to lose his faith in God. His immediate reply was to explain with great conviction why it had not. But then, after a short pause, he added almost matter-of-factly that it _had_ destroyed forever his faith in human beings. \"The devil is an optimist,\" Karl Kraus wrote, \"if he thinks he can make people meaner.\"\n\nI am not prescribing moral Alzheimer's here. Self-evidently, to be wholly without memory would be to be without a world. Nor am I arguing against the determination for a group to memorialize its dead or demand acknowledgment of its sufferings from those who inflicted them or who stood by and did nothing to prevent it. That would be as morally obtuse as demanding of traumatized individuals or family that they forget their loved ones or, in the case of physical abuse or deep psychic wound, to go on covering up what went on in the past behind closed doors. To do so would be to counsel a species of moral and psychological self-mutilation of tragic proportions. On the other hand, too much forgetting is hardly the only risk. There is also too much remembering, and in the early twenty-first century, when people throughout the world but in the Global North in particular, are, in Todorov's words, \"obsessed by a new cult, that of memory,\" despite Yerushalmi's impassioned arguments to the contrary, the latter seems to have become a far greater risk than the former.\n\nHyperthymesia is a rare medical condition that has been defined as being marked by \"unusual autobiographical remembering.\" The medical journal _Neurocase: The Neural Basis of Cognition_ identifies its two main characteristics: first that a person spends \"an abnormally large amount of time thinking about his or her personal past,\" and second that the person \"has an extraordinary capacity to recall specific events from [his or her] personal past.\" It is similar, though not identical, to a case documented by the mid-twentieth-century Russian neurologist Aleksandr R. Luria in his _The Mind of the Mnemonist_ of a man who could forget only by an act of will. Luria described this as the inability to engage in what is sometimes called \"ordinary forgetting.\"\n\nIf we are at all skeptical about the contemporary elevation of remembrance and deprecation of forgetting, these can come to seem like nothing so much as hyperthymesia writ large. But even many people who would not go that far would probably agree that the cult of remembrance has become something of a fetish. In his magisterial final work, _Memory, History, Forgetting,_ Paul Ricoeur argued that there could be no art of forgetting in the way that, since the Rosicrucians at least, there has been understood to be an art of memory. For Ricoeur, as for Margalit, what needed to be cultivated was not forgetting but forgiving. And yet even Ricoeur seems to have worried about the contemporary obsession with memory.\n\nHe was right to do so. For remembrance, however important a role it may and often does play in the life of groups, and whatever moral and ethical demands it not only responds to but often can fulfill, carries with it political and social risks that at times also have an existential character. This can be serious during wars or social and political crises, particularly when these are intercommunal or religiously inspired. In such situations, the danger is not Yerushalmi's \"terror of forgetting\" but rather the terror of remembering too well, too vividly.\n\nSuppose, for the sake of argument, that Yerushalmi was correct, and that forgetting destroys not just the link with the past in the obvious formal sense but also the commonality of values that should link the living with their ancestors and their own descendants with them. Even this by no means rules out the possibility that the price of remembering, at least in certain political circumstances and at certain social and historical conjunctures, might still be too high. These are the cases, small in number, no doubt, but high in the potential for human suffering, in which it is possible that whereas forgetting does an injustice to the past, remembering does an injustice to the present. On such occasions, when collective memory condemns communities to feel the pain of their historical wounds and the bitterness of their historical grievances\u2014and all communities have such wounds, whether at a given point in history they are oppressors or the oppressed3\u2014it is not the duty to remember but a duty to forget that should be honored. And despite the eloquent arguments Ricoeur deployed against such a parallelism, if there can be a will to remember, why, if only in extremis, can there not also be a will to forget?\n\nIn these situations, at least, is it possible to state with confidence which is worse, remembering or forgetting? There can be no categorical answer. If the prospect exists of \"curing war,\" as Einstein put it in 1931 in his celebrated exchange of letters with Freud, then Margalit's championing of forgiveness without forgetting might itself be sufficient. But as Freud said in his answer to Einstein, \"There is no likelihood of our being able to suppress humanity's aggressive tendencies.\" And if Freud was right, then it is at least possible that forgetting, for all the sacrifices it imposes (and, to be clear, these can be terrible indeed), may be the only safe response\u2014and as such should be a cause for a measure of relief, rather than consternation. There are many historical examples of such forgetting taking place far sooner than might reasonably have been expected. As an illustration, when General De Gaulle had his historic change of heart and decided that France would have to accede to Algerian independence, one of his advisers is said to have protested, exclaiming, \"So much blood has been shed.\" To which De Gaulle answered, \"Nothing dries quicker than blood\"\u2014a reply that exemplifies Nietzsche's idea that what he called \"active forgetting\" is an important attribute of a man of power.\n\nTo put the dilemma even more bluntly, remembrance may be the ally of justice, but, despite the conventional wisdom of the human rights movement, it is no reliable friend to peace, whereas forgetting can and at times has played such a role. An example of this is the so-called _Pacto del olvido_ (Pact of Forgetting) between the right and the left that, while never formalized, was essential to the political settlement that restored democracy in Spain in the 1970s after Franco's death. In an important sense, the democratic transition came on the wings both of rewriting and of forgetting. The myriad avenues and boulevards that had been named after Franco himself or his prominent subordinates following the fascists' victory in 1939 were renamed. But instead of replacing them with the names of Republican heroes and martyrs\u2014Juan Negr\u00edn, Francisco Largo Caballero, General Jos\u00e9 Miaja\u2014the Spanish leaders chose to use names from the royal past: the street in Madrid named after Gonzalo Queipo de Llano, one of Franco's most important generals and, briefly, his rival, became Calle del Principe de Vergara, and so on. It was a practice that was later institutionalized in the Law of Historical Memory passed by the Spanish parliament in 2007. And in a sense, even that law could reasonably be construed as being one of \"historical forgetting\" because, save for a few exceptions, it ordered the authorities to remove monuments, plaques, and street names that, in the words of the statute, \"exalted the Civil War or the repression under the dictatorship\" between 1936 and 1975.\n\nThe justification for the Pacto del olvido was similar to the one advanced in Chile when it was decided not to prosecute General Pinochet when he relinquished power: it was meant to placate Franco's loyalists, of whom there were many, including, crucially, a number inside the armed forces, at a time when the right's willingness even to acquiesce to the transition was anything but assured\u2014a reality that the attempted coup in 1981, more than six years after Franco's death, led by a Guardia Civil colonel, Antonio Tejero, demonstrated all too painfully. From the start, the pact had many detractors, not just on the left. And even a substantial number of those who did not oppose it in principle thought that it would not succeed unless accompanied by a South African\u2013 or Argentine-style Truth Commission. But as has so often been the case where human rights are concerned, it eventually fell to a magistrate to try to initiate through judicial procedures what the politicians continued to steadfastly refuse to contemplate. In 2008, Judge Baltasar Garz\u00f3n, the same magistrate who had issued the arrest warrant for Pinochet, opened an investigation into the deaths of the 114,000 people estimated to have been murdered by the fascist side both during the Civil War itself and in the subsequent decades of Franco's rule. Garz\u00f3n also demanded that nineteen mass gravesites be opened and the bodies exhumed.\n\nGarz\u00f3n's efforts were immensely controversial in Spain, not only because many Spaniards were still convinced that the Pacto del olvido had worked, but also because the country's 1977 Amnesty Law holds that murders and atrocities committed by either side during the Civil War that could be categorized as having had what the statute calls \"political intention\" were sheltered from prosecution. Garz\u00f3n denied that he had exceeded his authority. \"Any amnesty law,\" he argued, \"that seeks to whitewash a crime against humanity is invalid in law.\" His many supporters in Spain, the most ardent of whom belonged to the Association for the Recovery of Historical Memory, agreed and did a great deal to sway Spanish public opinion in favor of what he was trying to do. And even though, in the end, higher courts not only overruled Garz\u00f3n but went on to suspend him from the judiciary (in 2014 he became one of the lead attorneys representing the founder of Wikileaks, Julian Assange), his supporters have never wavered in their conviction that Garz\u00f3n's actions represented the only ethically licit response. This was summed up by the rhetorical question that has appeared intermittently on the Association's website: \"Why have the authors of the Constitution left my uncle in a ditch?\"\n\nGarz\u00f3n's views are strikingly similar to those of Elie Wiesel, who believes that finding out the truth about what happened to the victims of Franco is both a moral and a legal obligation, binding for society as a whole, no matter what backroom deals politicians may have made in the 1970s. The general tendency among human rights activists, including members of the judiciary such as Garz\u00f3n, has been to present law and morality as inseparable, at least in cases when the matter under consideration is clearly within the jurisdiction of a court. And because most of them assume that justice is the essential prerequisite for lasting peace, they tend to downplay when they don't categorically dismiss the risk of any negative political and social consequences flowing from their actions. But in the event that such consequences do occur, their stance has generally been that it is the politicians' responsibility, and not theirs, to sort them out.\n\nIt would be dishonest to focus on whether there might be times when remembrance could be thought of as either not yet being helpful (to peace, to reconciliation) or, alternatively, as having outlived its usefulness, without acknowledging the many instances in which forgetting, too, may have a lifespan, and at times quite a short one. This is a point that the Association for the Recovery of Historical Memory made repeatedly in its campaign in support of what Garz\u00f3n was trying to do. From an analytical point of view, furthermore, the group made a valid point when it argued that \"the Amnesty law was key to moving from an atrocious dictatorship to democracy, and for years benefitted from wide popular support. But in this decade [the 2000s], the victims turned to a government of the Left so that there will no longer be impunity for the crimes against humanity [committed during the Civil War and under the Franco dictatorship].\"\n\nThe Association was also probably right when it claimed that twenty-first-century Spain no longer needs the Pacto del olvido, just as when _The Sorrow and the Pity_ finally aired on French television it soon became clear that France had changed sufficiently that the truth about what had happened during the Occupation caused no grievous harm to the country's moral or historical ecology. But in both France and Spain, for all the difficulties they face\u2014including terrorism and, in France's case, the continued expeditionary campaigns in the Sahel related to it\u2014the major wars and great crimes are almost certainly behind them.4 What this means is that the risks that come with remembrance are probably manageable, even if the rewards may not prove to be as great as the activists, international lawyers, and human rights campaigners both in NGOs and in the universities routinely claim. This is not where the dangers of remembrance lie; except for terrorists\u2014which in France and Spain nowadays almost invariably means jihadis\u2014no one in either country is likely to kill or to die because of what has been forgotten or failed to have been forgotten, or what has been remembered or has failed to be remembered. In many parts of the world, however, killing and dying are exactly the stakes, and it is with regard to those places that the issue of whether we should stop praising remembrance and start praising forgetting is most acute.\n\nThe places to which this has applied in the very recent past or applies now are glaringly obvious: the Balkans, Israel-Palestine (and much of the rest of the Islamic Middle East), Ireland. In other places, it is less a question of \"forgetfulness now\" as of the realization that at some point in the future, whether that moment comes relatively quickly or is deferred for a long time, the victories, defeats, wounds, and grudges being commemorated would be better let go. That list would include, for starters, Sri Lanka, Colombia, Ukraine. And at or at least near the top, it would also include the United States and the memory of the attacks of September 11, 2001.\n\n1. For Margalit, \"The distinction needed... is between the illusion _of a_ collective memory and [the] illusions within collective memory,\" while Todorov argues that one way to distinguish between its proper and improper use is whether \"the actions that have been claimed to be based on it have been for the better or the worse.\" My own view, though, is that this criterion doesn't get Todorov nearly as far as he seems to think in establishing a viable set of criteria for making the correct choice.\n\n2. This view, which is demonstrably utopian, now also informs the mainstream relief and development world, where the consensus view is that for the first time in history it is not only possible but likely that by the middle of the twenty-first century extreme poverty and hunger will have all but disappeared.\n\n3. If the Rwandan genocide and its aftermath should have taught us anything, it is how quickly such roles can flip and how easily yesterday's victimizers become today's victims.\n\n4. To be clear, this claim is comparative, not an assertion that these states are not now committing crimes or will not do so in the future.\nEIGHT\n\nAgainst Remembrance\n\nOn September 11, 2011, the tenth anniversary of the attacks that destroyed the Twin Towers of the World Trade Center in Lower Manhattan, the official memorial was dedicated at Ground Zero. Designed by the architect Michael Arad and the landscape architect Peter Walker and titled \"Reflecting Absence,\" the memorial is a little under eight acres in size and consists of two sunken reflecting pools, each surrounded by an enormous waterfall, the largest humanmade waterfalls in North America. The names of the 2,983 people who died on September 11, 2001, and in the failed 1993 attempt to destroy the towers are etched on the bronze panels edging the memorial pools. The closing sentence of the memorial's mission statement reads, \"May the lives remembered, the deeds recognized, and the spirit reawakened be eternal beacons, which reaffirm respect for life, strengthen our resolve to preserve freedom, and inspire an end to hatred, ignorance, and intolerance.\"\n\nThese are unexceptional sentiments. A memorial is a place for solidarity rather than subtlety, deference rather than criticism, piety rather than revisionism. But in affirming that remembrance is humanly necessary, we must not pretend that it is ever completely innocent, not even when, in Todorov's sense (and not allowing for this, in my view, is one of the most serious limitations to his argument), it does not seem to involve any abuse of memory and when its promises to offer an exemplary lesson for both the present and the future seem credible. In the case of the 9\/11 memorial, its mission statement poses more questions than it answers. For although there is nothing morally problematic about remembering the fallen and honoring the heroism of the first responders, the call to \"strengthen our resolve to preserve freedom\" is anything but an innocent piety. To the contrary, it bears echoes of President George W. Bush's speech to a joint session of Congress nine days after the attacks in which he argued that they had occurred because the terrorists \"hate our freedoms\u2014freedom of religion, our freedom of speech, our freedom to vote and assemble and disagree with each other.\"\n\nEven those who have accepted Bush's account, despite its failure to acknowledge the possibility that it was America's actions globally rather than the American way of life that the jihadis hated, presumably would grant that the president was making a political claim in the broad sense. That the opening of the memorial marked an event that is seared into the lives and consciousness of most Americans should not obscure the fact that the ghost at the banquet of all public commemorations is always politics, whether for the Renanist purpose of mobilizing national solidarity and what might be called memorial unity, as the ceremonies at the 9\/11 memorial were clearly meant to do, or for the purpose of serving as a vehicle for undermining the version of history a state propagates, one that, through the creation of some form of what is sometimes called countermemory, initiates, then speeds up, and finally helps institutionalize whatever social transformations a particular activist group has been campaigning for.\n\nIn the specific case of 9\/11, it is important not to exaggerate. Whatever meaning historians eventually assign to the attacks (assuming a consensus is ever arrived at, which hardly can be taken as a given), it is highly unlikely that commemorations of them will harm America as a society, even if Americans are unlikely to learn much from them either, any more than one does from eulogies at a funeral. In an important sense, for a great many of the relatives and friends of those who died on that day, remembrance has afforded a measure of recognition and consolation, if not of closure\u2014one of the more malign and corrosive psychological fantasies of the age. Commemorations are not generally valued for their ability to shed light on the truth. And that is entirely appropriate.1 The problem is that such piety both nourishes illusions about how long human beings can remember, and, far more seriously, puts considerations of the grave political consequences it can engender out of bounds. And yet, in light of the dismal fact that there have been many occasions in the past when remembrance has been the incubator of a determination of a defeated people or group to secure vengeance, no matter how long it takes or what the human cost of doing so will be\u2014think of Ireland, Bosnia, Kosovo\u2014such deference to collective grief and trauma, not just humanly understandable but honorable as it generally is, can cost nations and societies dearly, at a price that may be exacted for generations.\n\nCommemorations of national tragedies such as the September 11 attacks are also occasions for the affirmation of the wholly illogical belief that events that quite rightly seem central to us today will be as or almost as important to our descendants long after those of us who lived through them are dust. This assumption is not only almost certainly false; it also carries risk, especially for rich societies such as those of the United States, Canada, and the nations of the European Union. In these countries, there is a growing tendency to conflate desire with fate, wish with reality, and most relevantly in terms of collective memory, commemoration, and group remembrance, the present with eternity. Among the other effects of this is a reluctance to acknowledge and draw the difficult lessons from not just individual human transience but societal, national, and civilizational transience as well. For hopes and desires almost invariably have been poor guides where history is concerned. To insist on this is not to say that on the tenth anniversary of the attacks, those who participated should have forsworn such illusions, however indefensible they are intellectually. In that specific case the alternative would have been to state that sooner or later our descendants will forget about them, and that would have been as unbearable as it was both cruel and pointless.\n\nWhat is open to question is not whether, as Kipling and Shelley knew, the most monumental accomplishments and deepest sorrows of human beings will be forgotten over the long run, but rather where the temporal outer limits of a society's capacity for remembrance, commemoration, celebration, and mourning are. In June 1940, as he tried to rally his people for what he called the Battle of Britain, Winston Churchill said, \"If the British Empire and its Commonwealth last for a thousand years, men will still say, 'This was their finest hour.\"' What is remarkable about his speech in terms of collective memory is that even at his most floridly rhetorical, Churchill, one of the most intransigent defenders of the British Empire ever to have lived, was nonetheless a good enough student of history neither to imagine nor to suggest that his country's imperium could last more than a millennium. If anything, that _If_ at the beginning of his sentence seemed to suggest that he thought that its reign would be considerably shorter. And lest it be forgotten, even Adolf Hitler spoke of a \"thousand-year Reich,\" not of an eternal one.\n\nIn the American context, compare the use of the memory of Pearl Harbor, first by Franklin D. Roosevelt in the immediate aftermath of the Japanese attack on the U.S. fleet in Hawaii, and then by George W. Bush in the wake of 9\/11, and consider the difference between what people in FDR's time and people sixty years later expected to be remembered. FDR referred to December 7, 1941, the day the attack took place, as \"a date which will live in infamy.\" Worth noting is that even in this formulation, Roosevelt sensibly did not say that the date would live in infamy for a thousand years, let alone _forever_. He knew better than that; he was speaking for his listeners in the here and now.\n\nThere is little doubt that on December 7, 1951, ten years after Pearl Harbor, the date continued to live in infamy for most Americans, just as 9\/11 does in 2015, as I write. But the history of the memory of Pearl Harbor offers instruction in how transient the vivid remembrance of even fairly recent historical traumas can be. Anyone with half a heart and no countervailing political agenda or allegiance who goes to the Pearl Harbor memorial today will almost certainly be immensely moved, perhaps even to anger as well as grief. But how many twenty-first-century Americans still remember the 1,177 American sailors killed on the U.S.S. _Arizona_ that day, most of whose remains lie directly under the memorial? And before long, those who lived through the attack, or were old enough at the time to know it had occurred, will have died. At that point, Leon Wieseltier's point about the U.S. National Holocaust Museum\u2014that before long visitors would come when the _living_ memory of Shoah had disappeared\u2014will apply to the Pearl Harbor memorial as well.\n\nAnd what about FDR's prediction that the Japanese attack would \"live in infamy\"? During the 1950s many Americans refused to buy German cars, while anger at the Japanese remained palpable. It is tempting to argue that reconciliation with Germany took place because the Germans themselves acknowledged their guilt and, in important ways, created a state whose essence was the antithesis of Nazi Germany. But a similar process of reconciliation followed by forgetting took place vis-\u00e0-vis Japan, even though there was no German-style contrition there. Indeed, the extent to which the Japanese persist in refusing to acknowledge the crimes they committed between the start of the Second Sino-Japanese War in 1937 and Japan's surrender in 1945 remains a profound source of anger to China and Korea, whose people were the principal victims. Yet almost no one in the United States thinks angry thoughts about the Japanese every December 7 (or on any other day of the year for that matter). And the scenario of an American refusing to contribute to a fund to help the survivors of the tsunami that struck Japan in 2011 on the grounds of what the Japanese had done at Pearl Harbor is ludicrous. Yet it would have been anything but ridiculous on December 7, 1951. To the contrary, in analogous circumstances then, such a rancorous response would not have surprised anyone.\n\nThe Second World War is over, not only in reality but also in people's hearts, just as all wars must end, including those that seem interminable, as is the case of the so-called Long War against the jihadis of the early twenty-first century. This truth can only offer cold comfort while a conflict rages on, however, and that is where the risks that remembrance will prolong a war come in. Will remembrance have this effect in the case of the commemorations of 9\/11? It is too early to say. In 2015, fourteen years after the attacks on the towers, and with even the \"end of the beginning\" of the Long War, to use the words Churchill applied to the British victory over the Germans at El Alamein in 1942, not even in sight, there can be no question of either forgetting or forgiveness. But if it is far too early to move toward either, surely it is anything but premature to ask whether peace will ever come without our society being open to both. For even the work of mourning, essential as it is, must eventually end if life is to go on. The South African writer Nadine Gordimer once remarked that she believed writers should write as if they were already dead. On a certain level, asking people to forgo remembrance, and possibly even embrace forgetting, is also to ask them to behave as if they were already dead. We are back to Mi\u0142osz's adage about there being no other memory but the memory of wounds.\n\nPerhaps these memories are too precious for human beings to give up. For societies, especially societies and groups that either feel themselves to be under existential threat or want to impose their own religion, or values, or territorial demands on their neighbors, the possibility may be still more remote, particularly if the collective memories, however dubious historically, seem to fuel these efforts. Consider, for example, the use Daesh, al-Qaeda, and other jihadi groups, and, for that matter, many mainstream Islamic clerics, from Indonesia to the suburbs of Paris, have made of the words \"Crusade\" and \"Crusader.\" As the Cambridge social historian Paul Connerton, arguably Halbwachs's most gifted inheritor, has pointed out, \"Medieval Muslim historians did not share with the medieval European Christians the sense of witnessing a great struggle between Islam and Christendom for control of the Holy Land.\" Connerton added that the words \"Crusade\" and \"Crusader\" never appear in the Muslim chronicles and other historical writing of the time; instead they use the terms \"Franks\" or \"infidels.\" But according to Connerton, beginning sometime in the nineteenth century \"an expanding body of Arabic historical writing has taken the Crusades as its theme,\" with the term becoming \"a code word for the malign intentions of the Western powers . . . culminating in the foundation of the State of Israel.\" On Connerton's reading, at least, one of the effects of each of the Arab-Israeli wars has been to galvanize further studies of the Crusades.\n\nThe Crusaders as proto-Zionists! It may not be history, but it offers a textbook case of the development and then of the deployment of political collective memory in the service of large-scale solidarity\u2014yet another iteration of those \"griefs requiring common efforts\" that Renan advanced as the sine qua non of nation-building. The fact that virtually nothing in the contemporaneous Arab writing about the Crusades supports the Arab world's collective memory of those griefs is neither here nor there. That the myth fills a need, and subsequently can be manufactured convincingly enough to captivate and inspire those to whom it is directed, is what matters. Think of it as the transformation of the wound into the weapon.\n\nThe renowned British historian of the Crusades Jonathan Riley-Smith believes that the construction of what remains the consensus collective memory in the Muslim world of the Crusades began in the late nineteenth century when Sultan Abdulhamid II described the European imperial powers' conquests of Ottoman lands as a new crusade. In 1915 it seemed appropriate to name the new university in Jerusalem after Saladin, who, as Riley-Smith puts it, began to be praised by Arab nationalist writers for having undone the first European seizure of the Holy Land when, at the Battle of Hattin in 1187, he put an end to the nearly century-old Christian Kingdom of Jerusalem that had been established in the aftermath of the First Crusade in 1099. By the 1980s, this association between the Kingdom of Jerusalem and the modern State of Israel had come to seem so self-evident to many Arab artists and intellectuals that the Palestinian poet Mahmoud Darwish could write of the Israeli siege of Beirut in 1982 as being the work of \"leftover Crusaders . . . taking their revenge for all medieval history.\"\n\nLess than two months after the September 11 attacks, Osama bin Laden recorded a speech in which he described the U.S. invasion of Afghanistan that was only then just beginning as linked to \"a long series of crusader wars against the Islamic world.\" These had not only happened in the immediate post\u2013World War I period in which, as he described it, \"the whole Islamic world fell under the crusader banner\u2014under the British, French, and Italian governments.\" For bin Laden, these efforts at conquest had taken place without respite throughout the twentieth century and included Russia's wars in Chechnya and the actions of \"the crusader Australian forces [who landed] on Indonesian shores . . . to separate East Timor, which is part of the Islamic world.\"\n\nThe late twentieth and early twenty-first centuries in the Islamic world have been a graveyard of many forms of rationality, but most notably of skepticism. And in the context of piety and ressentiment now running rampant in the ummah, it seems inconceivable that at least a large number, though of course not all, of those who watched bin Laden's speech on social media or, for that matter, read Darwish's words, found themselves \"remembering\" this crusader \"past,\" in which Balian of Ibelin (c. 1143\u20131193), the great Christian knight of the Kingdom of Jerusalem; John III Sobieski (1629\u20131696), the Polish king who lifted the Ottoman siege of Vienna in 1683; Ferdinand de Lesseps (1805\u20131894), the French developer of the Suez Canal; Field Marshal Edward Allenby (1861\u20131936), the British commander whose army captured Jerusalem from the Ottomans in 1917; David Ben-Gurion (1886\u20131973), the main founder of the State of Israel; Boris Yeltsin (1931\u20132007), the president of the Russian Federation during the first Chechen War; and John Howard (born 1939), the Australian prime minister who ordered Australia's intervention in East Timor in 1999, all fuse together to become leaders of the same now millennium-old crusade to subjugate the Islamic world.\n\nThat this is a manipulation of history of the grossest kind and is in fact an _anti_ -historical exercise of the contemporary political imagination should be obvious. But that bin Laden's understanding is accepted as history throughout the Islamic world should be equally clear. Tzevtan Todorov would be within his intellectual rights to insist that the bin Laden view exemplifies the abuse of collective memory. And yet a principal element of his argument, one that in some ways is the key to his claim to distinguish between remembrance's proper use and its abuse, would be easy to adopt for someone wanting to make the case for the truth of bin Laden's account. Recall how Todorov distinguished between what he called \"literal\" memory and \"exemplary\" memory, in which the former, \"particularly when pushed to extremes, carr[ies] [great] risks,\" while the latter, \"potentially liberating,\" offers an exemplum from which \"one draws a lesson [and in which the past offers a principle of action for the present].\"\n\nSurely an advocate of bin Laden's viewpoint would insist that this is exactly what al-Qaeda's leader was doing when, to use part of Todorov's description of what exemplary memory can achieve, he took instances from the past \"as a model for understanding new situations, with different participants,\" thus opening a particular memory \"to analogy and generalization.\" For wasn't this why bin Laden annexed the contemporary assault (as he saw it) to the Crusades and generalized from that analogy? And when Todorov writes that one of the attributes of exemplary memory is to make it possible to \"make use of the lessons of [past] injustices to combat those that are occurring today, to leave the self and go toward the other,\" could not a bin Laden supporter legitimately claim that this was precisely his goal as well and muster as evidence for this the incontrovertible fact that the Western colonial powers did dominate the Islamic world for centuries\u2014a domination that to a considerable extent the American empire, informal though it is, has tried to maintain?\n\nTodorov knows this, and, immediately after laying out the distinction between the literal and the exemplary, he is quick to emphasize that \"self-evidently not all lessons are good ones.\" But his idea that we will be able to distinguish good ones from bad ones through \"universal rational criteria\" hardly provides the prophylactic against the \"bad lesson\" that he seems to think or at least hope they do. Human beings are not as rational as this argument supposes. More important, _pace_ John Gray, incommensurability is often a more reliable guide than universality to the realities of our world. It is also vital to fully take on board the degree to which, as Wieseltier has put it, \"memory has become our mysticism, so great that it is the generation of authenticity that it confers. It would be too strong to call it a hoax, but it is certainly a kind of trick.\" And therein lies the problem. Wieseltier is right to add that \"the effect [of the trick] evaporates in the sunlight of critical history.\" But critical anything has always been and always will be a minority taste, while the appetite for the mystical and the authentic is as voracious as it is universal, which is what Karl Kraus was getting at when he contrasted Beethoven to the hurdy-gurdy and the memory. This is the unpalatable truth that even minds as fine and deep as Todorov's and Margalit's don't seem able to accept about collective memory when they offer their nuanced and admirably contingent claims for what, at its best and in many contexts, it undoubtedly can achieve. For when in history has the mystical _ever_ played second fiddle to the critical?\n\nIt certainly is not common in political systems, or, for that matter, within social movements of any kind, and becomes ever less likely the more extreme such systems are. The political theorist Karl Deutsch once quipped (the remark is sometimes wrongly attributed to Renan), \"A nation is a group of people united by a mistaken view of the past and a hatred of their neighbors.\" He was exaggerating. But it is no exaggeration to say that collective memory is the ideal delivery system for such mistaken views. The fact that in most of the rich world state-centric, top-down, triumphalist collective memory has ceded a great deal of ground to the claims of what Jeffrey Olick has called \"the politics of regret\" does not change this. To the contrary, Renan's idea that what bound a nation together was the \"spiritual principle\" has proved to work equally well for groups demanding recognition and redress in this age of destabilized national identities. The odds are stacked in favor of the mystical, and it makes no more sense to bet against this than it does for a gambler to bet against the house in a casino.\n\nWieseltier once warned that nationalist politics grounded in collective memory can \"destroy the empirical attitude that is necessary for the responsible use of power.\" It is an insight that events in the Middle East, that proving ground for the _irresponsible_ use of power, seem to confirm every day. To take only one of myriad examples, when Israeli forces encircled Beirut in 1982, Israel's then prime minister, Menachem Begin, announced that the Israeli Defense Forces had the \"Nazis surrounded in their bunker,\" even though it was Yasser Arafat and Fatah that were trapped in the Lebanese capital. It was a paradigmatic example of what happens when collective memory born of trauma finds political and, above all, military expression. Margalit may be correct to insist that this need not be the result of collective memory, but even he concedes that collective memory is _usually_ manipulated. If this is the case, surely what collective memory brings about\u2014and not only in Israel-Palestine\u2014is more important than what it might accomplish in an ideal situation.\n\nIsrael offers a florid illustration of how disastrously collective memory can deform a society. The settler movement routinely appeals to a version of biblical history that is as great a distortion of that history as the Islamist fantasy about the supposed continuities between the medieval Kingdom of Jerusalem and the modern State of Israel. At the entrance to the settler outpost of Givat Assaf on the West Bank, a placard reads, \"We have come back home.\" In an interview, Benny Gal, one of the settlement's leaders, insisted, \"On this exact spot, 3,800 years ago, the land of Israel was promised to the Hebrew people.\" Shani Simkovitz, the head of Gush Etzion, the settlement movement's principal \"philanthropy,\" echoed Gal's claim: \"More than three thousand years ago our fathers gave us a land, which is not Rome, it is not New York, but this: the Jewish land.\"\n\nEven when it is secular, mainstream Zionist collective memory is often as mystical and as much of a manipulation of history as these views. Consider the simultaneous mythologizing and politicization of archaeology in Israel that has now reached the point where scholarship and state-building have come to seem like two sides of the same coin. Writing in 1981, Amos Elon observed that Israeli archaeologists were \"not merely digging for knowledge and objects, but for the reassurance of roots, which they find in the ancient Israelite remains scattered throughout the country.\" He added, \"The student of nationalism and archaeology will be tempted to take note of the apparent cathartic effects of both disciplines.\"\n\nNowhere has this been more evident than in the use of the ruins of the fortress of Masada, which were excavated in the early 1960s by Yigael Yadin, the retired IDF chief-of-staff turned archaeologist. It was at Masada that the Jewish Zealots who had risen in revolt against Roman rule in the year 70 of the Common Era made their last stand and where they eventually committed mass suicide (the story is chronicled by Flavius Josephus in _The Jewish War_ ). Soon after Yadin's excavations had been completed, fledgling soldiers in the Israeli military's armored corps began to be brought to the site for their passing-out parades. There, along with the standard ceremonies that accompany the end of basic training in any army, the graduates would chant, \"Masada will never fall again.\" As Elon pointed out, such \"historical\" evocations, which he called ceremonies \"staged by secular moderns over the graves of ancient religious zealots,\" were in reality completely ahistorical. \"The zealots of Masada,\" he wrote, \"would no doubt have opposed modern Israel's Westernized and secular character just as they opposed the Romanized Jews of their time.\"\n\nElon was thinking within the framework of critical history. He is skeptical of group memory, above all because he knows it to be unreliable. In contrast, Yigael Yadin, who in one of his last interviews spoke approvingly of what he called Israel's \"Masada Complex,\" can be legitimately interpreted as having devoted his archaeological career to restoring what the critic Harold Bloom, in his foreword to Yerushalmi's _Zakhor,_ called \"the lost coherence of Jewish memory at its strongest, which was messianic and therefore redemptive.\" Here is Yadin in Masada in 1963, addressing an IDF armored corps graduation ceremony: \"When Napoleon stood among his troops next to the pyramids of Egypt, he declared: 'Four thousand years of history look down upon you.' But what would he not have given to be able to say to his men: 'Four thousand years of your own history look down upon you.'\"\n\nFour thousand years of history. How can Wieseltier's empirical attitude, necessary for the responsible exercise of power, compete with that? Let us not pretend that such exhortations are not better for group cohesion than Elon's critique and, more generally, his ambivalence. For if history teaches us anything, it is that in politics as in war, human beings are not hard-wired for ambivalence; they respond to loyalty and certainty. And just as Renan had argued in \"What Is a Nation,\" to the extent these can be strengthened by collective remembrance, it is of no importance whether the memories in question are historically accurate or if, instead, they are inventions of purely modern manufacture.\n\nOnly one important twentieth-century European thinker, the German political philosopher Ulrich Beck, has directly confronted this problem. In a number of his works, Beck suggested that it might be possible to replace the \"national grandeur\" template for collective memory, which he called \"methodological nationalism,\" with some form of national \"shared ambivalence\" about the past, though he is less clear than he might be on how this would work in practice. France is a good example, for it remains the \"capital\" of the memory industry, the avant-garde of what Pierre Nora has dubbed a \"universally guilty conscience.\" Presumably Beck would have supported a school syllabus that would be attentive to both the Enlightenment and the slave trade, the French Revolution and the Algerian War. But he never explained how studying history in all its ambivalence, which, as Nora has warned, runs the risk of leading to a wholesale criminalization of France's past, would be a better guide for young French people than focusing on the triumphant national mythmaking of the past as their predecessors did until at least the early 1970s. No, there is no going back, but the multiculturalist mirror image of the older pedagogy does not seem to be working very well either.\n\nYosef Yerushalmi thought that the fundamental problem with the modern age was that without some form of commanding authority, which he thought of in terms of what in the Jewish tradition is called _halakhah,_ and what less parochially might be thought of as moral law, people no longer knew what needed to be remembered and what could safely be forgotten. But however moving, Yerushalmi's veneration of memory as the guarantor of tradition, and his horror at the possibility that in the modern era this chain of tradition is in the process of being broken on the wheel of forgetfulness, calls out to a moral structure under sentence of death. For despite the talk of the centrality of identity politics to modern self-understanding, the kind of fixed identity that Yerushalmi deemed essential for the successful transmission of a community's traditions through the generations has become ever more difficult to maintain. Instead, in Marx's grand phrase, \"All that's solid melts into air.\"\n\nBut if Yerushalmi's fears were warranted, and any real continuity between past, present, and future has been broken, replaced by collective memories of the past that are no more real than the invented traditions whose study Hobsbawm and Ranger pioneered, then surely the time has come to scrutinize our inherited pieties about both remembrance and forgetting. A good place to start might be the Edict of Nantes, issued by Henri IV in 1598 to bring to an end to the wars of religion in France. Henri quite simply forbade all his subjects, Catholic and Protestant alike, to remember. \"The memory of all things that took place on one side or the other from March 1585 [forward] . . . ,\" the edict decreed, \"and in all of the preceding troubles, will remain extinguished, and treated as something that did not take place.\" Would it have worked? Could such bitterness really have been annealed by royal fiat? Since Henri was assassinated in 1610 by a Catholic fanatic opposed to the edict, which itself was eventually repealed, we can never know. But is it not conceivable that were our societies to expend even a fraction of the energy on forgetting that they now do on remembering, and if it were accepted that in certain political circumstances at least the moral imperative might be Nietzsche's \"active forgetting,\" not Santayana's \"those who forget the past are condemned to repeat it,\" peace in some of the worst places in the world might actually be a step closer?\n\nAs a reporter during the Bosnian War, which was in large measure a slaughter fueled by collective memory, or, more precisely, by the inability to forget, I used to carry with me increasingly creased and faded copies of two poems by the Polish poet Wis\u0142awa Szymborska. In both \"The End and the Beginning\" and \"Reality Demands,\" that most humane and antidogmatic of poets, a woman who once said that her favorite phrase had become \"I don't know,\" certainly understood the moral imperative of forgetting. She had lived through Poland's agonies under Germans and Russians alike. For her, as for the majority of her generation, the soil of her nation's countryside and the paving stones of its cities were drenched in blood, suffused with memories of the most tragic, unbearable, and destructive character. And yet, Szymborska concluded \"The End and the Beginning\" with these words:\n\nThose who knew\n\nwhat was going on here\n\nmust make way\n\nfor those who know little.\n\nAnd less than little.\n\nAnd finally as little as nothing.\n\nIn the grass that has overgrown\n\ncauses and effects,\n\nsomeone must be stretched out\n\nblades of grass in his mouth\n\ngazing at the clouds.\n\nIn \"Reality Demands\" she went farther:\n\nReality demands\n\nwe also state the following:\n\nlife goes on.\n\nIt does so near Cannae and Borodino,\n\nAt Kosovo Polje and Guernica.\n\nWhat Szymborska articulates in both poems is the ethical imperative of forgetting so that life can go on\u2014as it must. And she is right to do so. For everything must end, including the work of mourning and with it Mi\u0142osz's memory of wounds. Otherwise the blood never dries, the end of a great love becomes the end of love itself, and, as they used to say in Ireland, long after the quarrel has stopped making any sense, the memory of the grudge endures. Those who insist on the centrality of forgiveness are right up to a point. But forgiving is not enough because it can never escape its own contingency. \"I do not speak either of vengeance or of forgiveness,\" Borges wrote. \"Forgetting is the only vengeance and the only forgiveness.\" Perhaps he went too far. But without at least the option of forgetting, we would be wounded monsters, unforgiving and unforgiven . . . and, assuming that we have been paying attention, inconsolable.\n\n1. The Latin phrase _De mortuis nil nisi bonum_ , \"Of the dead speak only the good,\" has often been lampooned with the quip _De mortuis nil nisi bunkum_ , but this is wrong. Candor during a commemoration is not admirable but rather childish and self-serving.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n**The Big Book of World Chess Championships**\n**Andre Schulz**\n\n**The Big Book of World Chess Championships**\n\n46 Title Fights \u2013 from Steinitz to Carlsen\n\n**New in Chess 2016**\n_Rolf, we are missing you_\n\u00a9 2016 New In Chess\n\nTranslated from _Das gro\u00dfe Buch der Schach-Weltmeisterschaften_ (Schulz, New in Chess 2015) by Ian Adams\n\nPublished by New In Chess, Alkmaar, The Netherlands\n\nwww.newinchess.com\n\nAll rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission from the publisher.\n\nPhotos: New in Chess archives\n\nCover design: Ron van Roon\n\nProduction: Harald Keilhack\n\nSupervision: Peter Boel\n\nProofreading: Ian Adams, Harald Keilhack\n\nHave you found any errors in this book?\n\nPlease send your remarks to editors@newinchess.com. We will collect all relevant corrections on the Errata page of our website www.newinchess.com and implement them in a possible next edition.\n\nISBN: 978-90-5691-635-0\n**Contents**\n\nPreface\n\nIntroduction\n\n**Part I \u2013 The age of private World Championships**\n\n1. There can only be one: Steinitz vs. Zukertort 1886\n\n2. Fighting the Russian Bear: Steinitz vs. Chigorin 1889\n\n3. 'Mephisto' in person: Steinitz vs. Gunsberg 1890\n\n4. Tarrasch has no time: Steinitz vs. Chigorin 1892\n\n5. The triumph of defence: Steinitz vs. Lasker 1894\n\n6. The annoying return match: Lasker vs. Steinitz 1896\/97\n\n7. Attacker against defender: Lasker vs. Marshall 1907\n\n8. Two top German players: Lasker vs. Tarrasch 1908\n\n9. The fateful tenth game: Lasker vs. Schlechter 1910\n\n10. Either Lasker won or Janowski lost: Lasker vs. Janowski 1910\n\n11. Hot games in Havana: Lasker vs. Capablanca 1921\n\n12. Friends become enemies: Capablanca vs. Alekhine 1927\n\n13. Duel of the cellmates: Alekhine vs. Bogoljubow 1929\n\n14. Chess WCh in Nazi Germany: Alekhine vs. Bogoljubow 1934\n\n15. Professor against alcoholic: Alekhine vs. Euwe 1935\n\n16. The title was only on loan: Euwe vs. Alekhine 1937\n\n**Part II \u2013 The Soviet era**\n\n17. The victory of Soviet chess: WCh tournament The Hague\/Moscow 1948\n\n18. Was he obliged to lose the 23rd game? Botvinnik vs. Bronstein 1951\n\n19. With great ease: Botvinnik vs. Smyslov 1954\n\n20. Tactics missed: Botvinnik vs. Smyslov 1957\n\n21. The revenge: Smyslov vs. Botvinnik 1958\n\n22. The magician from Riga: Botvinnik vs. Tal 1960\n\n23. Against Tal and a clairvoyant: Tal vs. Botvinnik 1961\n\n24. Cheating in Cura\u00e7ao? Botvinnik vs. Petrosian 1963\n\n25. The obstreperous pupil: Petrosian vs. Spassky 1966\n\n26. Fischer mates himself: Petrosian vs. Spassky 1969\n\n**Part III \u2013 The new era**\n\n27. The match of the century: Spassky vs. Fischer 1972\n\n28. Fischer doesn't appear: Karpov becomes World Champion without a fight\n\n29. Political thriller in Baguio: Karpov vs. Kortchnoi 1978\n\n30. Heaps of electronic devices: Karpov vs. Kortchnoi 1981\n\n31. The abandoned match: Karpov vs. Kasparov 1984\/85\n\n32. Perestroika against the establishment: Karpov vs. Kasparov 1985\n\n33. Return match under protest: Kasparov vs. Karpov 1986\n\n34. A traitor in the camp? Kasparov vs. Karpov 1987\n\n35. Under Russian colours: Kasparov vs. Karpov 1990\n\n**Part IV \u2013 The time of schism**\n\n36. Excluding FIDE: Kasparov vs. Short 1993\n\n37. Doors bang in the World Trade Center: Kasparov vs. Anand 1995\n\n38. To victory with the 'Berlin wall': Kasparov vs. Kramnik 2004\n\n39. The title match in a tobacco factory: Kramnik vs. Leko 2004\n\n**Part V \u2013 Reunification and what followed**\n\n40. 'Toiletgate' in Elista: Kramnik vs. Topalov 2006\n\n41. World Champion the second time around: WCh tournament Mexico City 2007\n\n42. 74 years later, back to Germany: Anand vs. Kramnik 2008\n\n43. Blackout in Sofia: Anand vs. Topalov 2010\n\n44. Chess is art: Anand vs. Gelfand 2012\n\n45. The high-flyer from Norway: Anand vs. Carlsen 2013\n\n46. Chess blindness in Sochi: Carlsen vs. Anand 2014\n\nAll (classical) World Champions\n\nAll (classical) World Championships\n\nVenues for classical World Championships\n\nGlossary\n\nBibliography\n**Symbols**\n\n**The chessboard with its coordinates:**\n\n\u2654 King\n\n\u2655 Queen\n\n\u2656 Rook\n\n\u2657 Bishop\n\n\u2658 Knight\n\nx capturing\n\n\\+ check\n\n# checkmate\n\n! good move\n\n!! excellent move\n\n!? interesting move\n\n?! dubious move\n\n? bad move\n\n?? blunder\n\n= balanced position\n\n+\u2013 White has a decisive advantage\n\n\u2013+ Black has a decisive advantage\n\n  White to move\n\n  Black to move\n**Preface**\n\nThere are numerous ways to relate to chess. There are many people who do not even play chess but derive pleasure from, perhaps, collecting beautiful chess sets, chess books or stamps which are related to chess. Chess can be played solely as a pastime or as a competitive sport.\n\nThe history of such chess contests stretches far back into the past. But at the latest from the end of the 19th century matches were played between the best players in the world for the 'World Chess Championship'. After this the history of such World Championships did not always run in a straight line, but the tradition has lasted until the present day.\n\nCountless books have been written about the individual World Championships, in which, as was natural, the games of the matches were at the focal point of contemplation. So far, however, there have scarcely been any comparative studies of the various World Championships. This book is intended to plug that gap.\n\nHere it is not the games which are in the foreground and also not the course of the contests, but rather what was happening beside the board: what were the venues and what were the circumstances for the World Championship encounters? Under what conditions and according to what rules were they played? What plots were hatched before and during the competitions? Some of the matches turned into real psychological warfare and from time to time lifelong enmity was a further result of the matches.\n\nWhen describing the struggles beside the board, I have tried to limit myself to the portrayal of the facts such as they have been published in the sources which were available to me. As I did so, I did not want to take sides for or against any participant or to influence the forming of the reader's own opinion. Should any participant or person mentioned in these pages feel that he or she has been wrongly or unjustly portrayed, then that has happened purely due to a lack of ability on my part, it is in no way a question of bad faith.\n\nIn the past, most reflections on World Chess Championships focussed above all on the players who were contesting the matches. Their biographies, which have been presented here in compact form, offer an insight into their era and the then prevailing living conditions. The best chess players in the world were born in different countries and into differing social backgrounds. Many began their life in poverty and earned a certain material security through their knowledge of chess. Others were born into well-off houses and died in misery. Wilhelm Steinitz, the first World Champion, was born in pitiful circumstances and eked out the whole of his life on the verges of total poverty. Nowadays the World Chess Champion becomes a millionaire.\n\nBut actually many other chess lovers have made perhaps an even greater contribution than the World Champions themselves, and they have done so through their efforts to bring about the matches. First and foremost, one must mention the patrons and the sponsors who provided the prize money and who assumed the costs of the staging of the contests. The players were supported by seconds, whose work has often not received sufficient recognition. The arbiters assured that the course of the match followed the rules.\n\nSo, in this description of the history of World Championships I have attempted to name as many as possible of those who took part directly or indirectly. Unfortunately many hard-working chess lovers who also participated in the organisation of the World Championships were never named in the sources. In addition I have taken pains to describe the numerous links between the world of chess and the world at large in order to demonstrate how much the game of chess and its outstanding connoisseurs are to be understood as a component of our culture.\n\nNevertheless, a book about the World Chess Championships totally without games would be something of a rarity and therefore I have chosen from each World Championship a single game, added to it contemporary and also more recent comments and checked the variations which have been given with strong up-to-date chess programs and engines such as _Houdini_ or _Stockfish_. Where necessary I have added further variations and explanations according to my own understanding and from time to time I have corrected mistakes in old analysis with the help of the chess engines. This also allowed many an interesting discovery to be made, since many of the WCh games had no longer been looked at in depth for many years and now and then appear somewhat different in the light of present-day computer analysis.\n\nI hope to demonstrate with this description of the history of the World Chess Championships that the game of chess has many more sides to offer than the presentation of the games and that the struggles for first place in the world ranking list of chess were far removed from happening simply at the board. Chess lovers who take an interest in the history of their sport will hopefully find a few stories which are new to them. It would please me even more if many a reader who has as yet had little contact with the game of chess could perhaps be bitten by the chess bug as a result of this book.\n\nMay I thank Johannes Fischer for moral support, proof reading, motivational help and access to his library; may I also thank Michael Dombrowsky, who made some rare books available to me. Rolf Gehrke and particularly Thomas Stark have been of great assistance to me with numerous comments and proof reading.\n\n_Hamburg, June 2015_\n\n_Andre Schulz_\n**Introduction**\n\nThere is no doubt that chess is a very special game. It was invented around 500 A.D. in India, first of all as a game for four people \u2013 chaturanga. In the 6th century an Indian ambassador brought the game as a present from his king Divsaraman to the Persian court of Chosraus I. The word chess (from the Persian shah = king) is a reminder of its Persian past. Even back then in Persia the game fascinated all those who came into contact with it. After the Arabs conquered Persia, many of them too were infected by the 'chess virus'. There soon arose a literature with pretty chess puzzles and even already professional players. Via the Arabs the game of chess spread as far as Europe, following two routes. Via Spain and Italy it reached the countries of south, central and western Europe and the game came to Russia through the Caucasus.\n\nAs time went by, the rules and the strength of the pieces would change. In Europe the Persian _vizir_ became a woman, the queen who stood by the side of the medieval king. The king now also placed his trust in the support of the church (the bishops), the nobility (the knights) and his castles (though properly called rooks in English, their form is that of a tower). The front line of the army is composed of pawns (for which the German word 'Bauern' means 'peasants'). They too have their role to play in the structure of medieval society. And every pawn can even be promoted to an officer, if it can reach the back rank of the opposing side of the board. But it also has to be resigned to being condemned as a 'pawn sacrifice' for the good of the rest of society.\n\nSince the 15th century chess manuals have also been published in Europe. Today the total number of books published on chess is estimated at over 100 000 titles. Many a collector has tried to get hold of all of them. Chess became a fixed component of European culture. Chess pieces proved to be an invitation to artists to portray them according to the tastes of their day and age.\n\nThere were soon especially smart players who astounded spectators with their skills. And there was always one of them who was reckoned to be the best of all. Unlike in many other arts, it is easy to find out in chess who the better player is: in a game or a match between them. Or in a competition which imitates the form of the medieval tournament. However things are not quite so bloody in chess. In a tournament several players meet each other in a k.-o. system or in the form of an all-play-all. In a match it is man against man.\n\nAnd thus were born the matches for the World Championship, without this concept even having existed at the start. In the middle of the 19th century the idea of a 'World Champion' gradually emerged in common parlance. And then there were suddenly two players who each claimed to be that World Champion: Johannes Zukertort and Wilhelm Steinitz. A match to decide the question was required. So in 1886 the first World Chess Championship was held. Thereafter the winners held the title as their private property and only ventured it in matches for good prize money.\n\nIn 1946 the last of these 'private World Champions' died \u2013 Alexander Alekhine. Then the World Chess Federation (F\u00e9d\u00e9ration Internationale des \u00c9checs, FIDE) took over the organisation of the World Championship and brought in fixed rules for qualification and World Championship matches. Since there was no title defender, FIDE organised their first World Championship in 1948 in the form of a tournament. It was won by the Soviet Russian Mikhail Botvinnik, who then went on several times to defend his title successfully or to recover it in return matches, before he was replaced by another Soviet player.\n\nSoviet domination continued until 1972 when it was broken by the US American Robert Fischer \u2013 though only temporarily. Fischer won the title from Spassky and then disappeared from the scene. Therefore Anatoly Karpov became World Champion, and later lost the title to Kasparov. The FIDE system lasted until 1993. Then Nigel Short and Garry Kasparov 'hijacked' the World Championship title in their dispute with the then FIDE president Florencio Campomanes and again treated it more or less like private property. FIDE organised its own World Championship but it did not meet with general recognition, also because the successor to Campomanes, Kirsan Ilyumzhinov, simultaneously president of the autonomous Russian republic of Kalmykia, threw overboard the once neat organisation and rules for the staging of the World Championships. The FIDE World Championship was now played in a k.-o. system with 128 participants and this system opened the barn door to chance.\n\nKasparov and his successor Vladimir Kramnik, however, hung on to the old system of matches. Finally in 2006, in a match, overshadowed by scandal, between Kramnik and Veselin Topalov, we had a 'reunification' of the World Championships. Kramnik defended his title successfully against Topalov, but then lost it in a WCh tournament in 2007 to Viswanathan Anand. The latter defended his title in matches in 2008 against Kramnik, in 2010 against Topalov and in 2012 against Gelfand.\n\nIn 2013 the Indian player had to appear again, this time against the young Norwegian Magnus Carlsen \u2013 the most difficult opponent of them all. For Magnus Carlsen had risen like a rocket to the top of the world ranking list and in doing so had surpassed the highest rating held until then, the record which had been held for many years by Garry Kasparov. In fact Anand was unable to match the pressure and the energy of the challenger. In the match in his home city in India Anand lost three games without winning a single one. The new World Champion was from Norway: Magnus Carlsen. In the very next year he had to defend his title. It turned out to be a re-run of his match against Anand, though this time the roles were reversed, since Carlsen was now defending the title and Anand was the challenger. Carlsen also managed this task in majestic fashion.\n**Part I \u2212 The age of private World Championships**\n\nAfter the game of chess reached Europe in various ways from 900 A.D., it spread at first in noble courts and in the 13th century knowledge of it was even counted among the seven knightly virtues. The game underwent several reforms and soon also became the object of theoretical examination. It also gradually spread amongst the upper middle classes and took its place among the favourite occupations in the coffee houses. Around 1700 the Scottish scholar Alexander Cunningham of Block was considered the best player in the world. His admirers included amongst others Gottfried Wilhelm Leibniz. In the middle of the 18th century the Caf\u00e9 de la R\u00e9gence in Paris became the chess centre of the world and players like Legall de Kermeur and Fran\u00e7ois-Andr\u00e9 Danican Philidor took over the mantle of the best players in the world.\n\nThe concept of a World Champion arose, chosen in practical terms by general agreement. Matches decided who was the better of two players. And the one who was able to defeat everyone else was the best player in the world. Then there were suddenly two players who at the same time both laid claim to being the best player in the world: Wilhelm Steinitz and Johannes Zukertort. A match for the World Championship had to be held. Steinitz won and Zukertort, whose constitution was weak, broke down as a result.\n\nThereafter Steinitz chose his own next challengers and only appeared against opponents who could come up with a financial stake. In 1894 the German Emanuel Lasker won the title of World Champion and held on to it for 27 years, partly because of the lack of activity in the world of chess during and after the First World War. Lasker lost his title in 1921 to the Cuban Jos\u00e9 Raul Capablanca, who proceeded to establish a really high hurdle: the challenger had to come up with 10 000 dollars of a stake. It took six years till Alexander Alekhine, the Russian player then living in France, managed this with the help of patrons. Alekhine won the title and then demanded that Capablanca put up exactly the same sum for the right to a return match. However, Capablanca was unable to raise the required amount and thus begged in vain for years for his return match. Previously friends, they now became enemies who at the end could not bear even to be in the same room together. Instead Alekhine entered the lists \u2013 moreover for less money than he had demanded from Capablanca \u2013 twice against the Germano-Russian Efim Bogoljubow and against the young Dutchman Max Euwe. Euwe was a clear outsider, but surprisingly he won the match and was then the World Champion, but only for two years. The mathematics teacher and man of honour accorded Alekhine a return match, without demanding any preconditions of him. And Alekhine recovered his title.\n\nThe start of the Second World War brought international tournament chess to an end. However play continued in the various countries. World Chess Champion Alekhine lived with his Jewish wife in the sphere of influence of the German Reich and had to play where its rulers wanted. At the end of the war that took him to Spain and Portugal. Soon after the end of the war he died in mysterious circumstances, alone and impoverished in Estoril. With him the time of the private World Championships also came to an end.\n**1. There can only be one**\n\n**The first official World Championship 1886:  \nWilhelm Steinitz against Johannes Zukertort**\n\nWilhelm Steinitz was born on the 14th May 1836, the ninth of 13 children in the Jewish ghetto ('Josefstadt'), in number six of the fifth district of the Bohemian capital, Prague. His father was a tailor and teacher of the Talmud, Josef Salomon Steinitz (1789-1868), his mother Anna Steinitz, n\u00e9e Torschowa (Germanised to Torscha, 1802-1845). Their son received the Jewish name Wolf, which later became Wilhelm.\n\nThe family lived in poverty. Steinitz' four younger siblings died in childhood, as did two older siblings. Steinitz himself was born with a club foot and all his life required the aid of a crutch. Steinitz was small in stature, no taller than 1.50 metres. After the death of his mother in 1845 Steinitz' father remarried and with his second wife he had another child at the age of 61.\n\nWilhelm Steinitz (1836-1900)\n\nSteinitz learned chess at the age of 12 from a friend of his father, or according to other sources from a school friend. He is supposed to have carved his first chess pieces himself and used a piece of chequered cloth as a chess board.\n\nThere are various commonly found versions of Steinitz' schooling. According to many sources Steinitz is said to have gone to the Jewish school in the ghetto and there have received instruction in Hebrew grammar and in biblical studies. Amongst his ancestors there were some scholars of the Talmud and Steinitz too, according to the wishes of his parents, was to become such a scholar, and attend the 'Yeshiva', the high school for the study of the Talmud. After he refused to give in to these desires, a break with his parents is supposed to have occurred.\n\nAccording to another version Steinitz attended the Volksschule in Prague where he attracted attention on account of his gift for mathematics. Because of his son's poor health, the father had intended him for a secular job, whereas Wilhelm was aiming to study mathematics. In this version too, it came to a break with his family. His father's second marriage and perhaps a bad relationship with his step-mother might also have played a part.\n\nIn 1849 the obligation for Jews to live in the ghetto was rescinded. Jews were now allowed, once they had requested a passport, to move freely throughout the whole city of Prague and the lands of the Danube monarchy. In 1850 the Prague ghetto was totally done away with. In the same year, at the age of 15, Wilhelm Steinitz left his family and apparently lived for a time on the streets of Prague. It is only after 1855 that a new place of abode is recorded. Without the support of his family Steinitz did not have the means to attend a secondary school and kept his head above water with odd jobs. He worked as a clerk and office worker in various small businesses in and around Prague.\n\nBut apparently at this time Steinitz was already a regular visitor to the chess caf\u00e9s of Prague, such as the 'Caf\u00e9 Wien'. According to numerous sources, he was even then the best chess player in Prague, but there is no written evidence of this. In 1853 Steinitz got to know Josef Popper, who later became known under the pseudonym Lynkeus as a writer and social reformer, but also as the author of technical treatises. Popper came from Kollin and his biography is pretty much like that of Steinitz. With his help, Steinitz caught up in his studies in the 'Lesehalle der deutschen Studenten' (or 'reading room for German students'). The two of them embarked on a lifelong friendship.\n\nIn 1858, on the urging of his friend, Steinitz went to Vienna and began his study of mathematics at the Polytechnic Institute, for which he had at first to pass a two year long preparatory course. In Vienna, Steinitz quickly made contact with the local chess scene, including the lawyer Phillip Meitner, father of the physicist Lise Meitner and one of the best chess players in Vienna and the doctor Carl Cohn.\n\nHis very first appearance in the Caf\u00e9 Rebhuhn had caused a sensation. When Steinitz showed an interest in the chess players and their games, he was asked whether he knew anything about the game. Yes, was his answer, he could even play blindfold. This claim was promptly put to the test by two players, and after Steinitz had defeated both of them without having sight of the board he had already made a name for himself in the Vienna chess scene. He received further chess instruction from the imperial councillor Carl Hamppe, a finance official and one of the best players in Vienna.\n\nAt the start Steinitz financed his studies with journalistic work, parliamentary reporting for the _Constitutionelle \u00d6sterreichische Zeitung_. But he soon had to abandon this activity in view of his weak eyesight. Then Steinitz began to play chess for money and to give blindfold chess exhibitions in the coffee houses of Vienna, for example the Caf\u00e9 Romer, the Caf\u00e9 L'Express, the Caf\u00e9 Central or the Caf\u00e9 Rebhuhn (behind the 'Graben', not far from St Stephen's Cathedral) \u2013 the latter being at that time the seat of the 'Wiener Schachgesellschaft' (Vienna Chess Society).\n\nSince he could not meet the fees for his studies in this way and was having health problems with his lungs and eyes, Steinitz stopped his studies in 1858, but from time to time continued to attend lectures with his friend Popper, for example in 1861\/62 the lectures by Ernst Mach on _Investigative methods in physics_ and _The principles of mechanics and mechanical physics in their historical development_. Influenced by these, Steinitz later began to apply scientific principles to chess and in doing so founded chess theory.\n\nFrom 1860 on Steinitz was active as a professional chess player, playing matches for high stakes and in doing so coming into contact with a series of patrons. Among his opponents were the banker Gustav Leopold Ritter von Epstein and the railway builder and inventor Josef Schulhof. The latter's second hobby was shooting. Amongst other things he invented the repeating rifle.\n\nOne of Steinitz' pupils was the young Baron Albert Salomon Anselm Rothschild, the youngest son of Anselm Salomon Freiherr von Rothschild and Charlotte von Rothschild. Albert Rothschild studied in Bonn, completed his banking education in Hamburg and in 1874 after the death of his father he took over the Rothschild Bank in Vienna. With a fortune estimated at a billion crowns, Rothschild was considered to be the richest man in Europe. All his life Rothschild remained a passionate chess player and was frequently active as a chess patron. In 1872 he took office as the president of the Wiener Schachgesellschaft, and also from 1897 that of its successor the Wiener Schachklub, at the helm of which he remained until his death in 1911.\n\nIn 1861 Steinitz won the chess championship of Vienna with 30 victories, three draws and only one defeat. In 1862 he took part in the second great London tournament (after that of 1851), his first international tournament, as the official representative of Austria. He was financially supported by the Vienna Chess Society, namely by the Viennese banker and entrepreneur Eduard von Todesco (in his biography of Steinitz Landsberger names a banker 'Tedesco'). Steinitz won for his sixth place five pounds, in today's money the equivalent of 364 pounds.\n\nAfter the tournament he remained in London, at that time the centre of the chess world, and there too earned his living by playing chess for money. In 1863\/64 Steinitz won a series of matches, among others against Joseph Henry Blackburne (8:2). In 1866 in London Steinitz defeated Adolf Anderssen, who was until then considered the best chess player in the world, by 8:6 \u2013 with no draws. From then on Steinitz considered himself to be 'World Champion'. He went on to win more matches, e.g. against Bird, Blackburne and Zukertort. All in all between 1863 and 1894 Steinitz won 27 of his 29 matches.\n\nWhen Steinitz was teaching chess in Cambridge in 1869 one of his students there was Lord Randolph Churchill, later father of Winston Churchill. Some years later, in 1880, Lord Churchill invited Steinitz to his estate Blenheim Palace in Woodstock and also played some chess there with him. Winston Churchill was then six years old. It is not known whether Steinitz met the future British prime minister on that occasion.\n\nAfter his tournament victory in Vienna in 1873 Steinitz, with the exception of a match against Blackburne (7:0), withdrew from tournament chess for nine years and worked instead as a chess journalist and theoretician. He now earned his money with a chess column in the magazine for country living and sports, _The Field_ , a column which he wrote from 1873 till 1882, and in addition with playing for money in chess caf\u00e9s. However, Steinitz then quarrelled with the publisher of _The Field_ , who finally dismissed him and closed the chess column. Steinitz' belligerence was known to all. Thus he was at times banned from a series of London coffee houses, including the famous 'Simpson's-in-the-Strand'. The chess column in _The Field_ was later continued by Leopold Hoffer and Johannes Zukertort.\n\nIn 1882 Steinitz brought to an end his abstinence from tournament chess and took part in the tournament in Vienna (sharing first place with Szymon Winawer). In 1883 in London he came in second, three points behind Zukertort. In the same year Steinitz accepted an invitation to the USA. From there he took part in a year-long public feud with his successor in _The Field_ , Leopold Hoffer; this became known in the history of chess as the 'Steinitz-Hoffer-Ink-War'. In the USA Steinitz then published from 1885 on his own chess magazine, _The International Chess Magazine_.\n\nJohannes Hermann Zukertort was born on the 7th September 1842 in Lublin, Russian Poland. His father was a Jew who had converted to evangelical Christianity, Jakub Zukertort (then still spelled Cukiertordt), and who after baptism took the Christian name Bogomil. His mother was the latter's second wife and was also newly baptised as Paulina Zukertort, born in Heilbronn. Johann Herrmann had a total of eight siblings, two of whom died in infancy. At first the family lived in the Ulica Krakowskie Przedmiescie in Lublin and then moved to the Ulica Namiestnikowska (now Ulica Narutowicza) No. 293.\n\n**Johannes H. Zukertort (1842-1888)**\n\nZukertort's father worked as a missionary for the 'London Society', which ostensibly sought to convert Jews to Christianity. For a time the family lived in Warsaw in a missionary house of the 'London Society' in the Leszno Stra\u00dfe. In summer 1850 it moved to Piotrkow Trybunalski (Petrikau), a county town close to Lodz. Zukertort spent the first two years of his schooldays there. In 1854 the 'London Society' fell into conflict with the Evangelical Church which finally brought about a ban on the Society and the expulsion of their members. The organisation was accused of being a front for espionage.\n\nIn February 1855 the Zukertorts had to leave Russia. The family moved to Breslau. At Easter 1861 Zukertort sat his Abitur there at the Maria-Magdalena-Gymnasium and in the same year began his study of medicine. Zukertort was matriculated in the medical faculty of the University of Breslau till the summer of 1866. In 1867 he was struck from the list of students on account of poor attendance and he left the university without taking either intermediate or final exams.\n\nAt the age of 16 Zukertort had learned to play chess from a schoolfellow. He bought his first chess set second hand for 30 Pfennig at a fair. After Zukertort had joined the 'Akademischer Schachklub' of Breslau in 1861, he met amongst others Adolf Anderssen, whose student he became. Other members of the club were Samuel Mieses (1841-1884), an uncle of Jacques Mieses, and Jakob Rosanes, professor of mathematics at the University of Breslau. By the very next year Zukertort was considered the second best player in Breslau after Anderssen. According to his own accounts, Zukertort played no less than 6000 games against Anderssen during his time in Breslau.\n\nZukertort was described by his contemporaries as multi-talented: he is supposed to have spoken ten languages and to have had a phenomenal memory. He was moreover musically gifted, supposed to have been an excellent pianist, practised fencing and riding and to have earned money from time to time as a music critic for a well respected publication in Breslau. However, there are also doubts about these accounts, most of which come from the pen of an English chess lover whose actual source was probably Zukertort himself, who perhaps desired after his arrival in England to spruce up his biography with a few facts which were at that date hard to check up on.\n\nIn 1867 Zukertort went to Berlin and became until 1871 editor of the _Neue Berliner Schachzeitung_. During the Austro-Prussian War of 1866 and the Franco-Prussian War of 1870\/71 Zukertort was in each case active in the Prussian medical corps. During his time in Berlin there was a series of blindfold simultaneous exhibitions with the help of which Zukertort sought to popularise chess in Berlin. For the time, his 16 opponents set a record.\n\nIn 1869 Zukertort and his _Neue Berliner Schachzeitung_ got into a dispute with the composer of chess problems and chess publicist Johannes Minckwitz. The latter had in the _Leipziger Schachzeitung_ (later _Deutsche Schachzeitung_ ) published a negative review of a collection of chess problems published by Zukertort and in doing so hurt Zukertort's pride. The dispute was carried out in public over several issues of the two chess magazines. It was probably also Minckwitz who prevented the publication of a tournament book composed by Zukertort for the West German Chess Congress held in Krefeld in 1871.\n\nIt was possibly as a result of this quarrel that the publication of the _Neue Berliner Schachzeitung_ was very suddenly stopped in 1871 by its publisher J. Springer and Zukertort sacked. Since Zukertort had now lost his material support in Berlin, he accepted the invitation of some English chess patrons who offered him 20 guineas travel expenses, and in 1872 he moved to England. These people saw in Zukertort a possible rival to the not particularly popular Steinitz. In that same year in London Zukertort played a first match against Wilhelm Steinitz, which however he lost clearly by 3:9. In 1873 Zukertort published a series of articles in the _Westminster Papers_ , collaborated with the _City of London Chess Magazine_ and founded in 1879, together with Leopold Hoffer, who bore the organisational and financial risk whereas Zukertort took care of the chess analyses, the magazine _The Chess Monthly_. Their collaboration lasted until 1888.\n\nZukertort became a firm part of the London chess scene, was a member of the St. George's Chess Club and was elected to an honorary membership of the City of London Chess Club. In the 1870s and 1880s Zukertort participated successfully in a series of international tournaments, winning among others the tournament in Cologne 1877 and the Paris tournament of 1878. Moreover, he was victorious with 22 points out of 26 games in the strong London tournament of 1883, three points ahead of Steinitz.\n\nZukertort now considered himself the best chess player in the world and thus entered into competition with Wilhelm Steinitz, who had also claimed this distinction for himself since his match victory over Adolf Anderssen. The rivalry between the two 'World Champions' had become more pointed as a result of the famous 'Ink War', a journalistic controversy involving Steinitz in _The Field_ and Zukertort in _The Chess Monthly_ , which had originally begun simply over a few pieces of chess analysis.\n\nSo finally the idea was mooted that a match for the 'World Championship in Chess' should be held. Immediately after Zukertort's victory in the London tournament Steinitz offered off his own bat a match, which according to his suggestion should be for a stake of 200 pounds, or even more, and consist of eight to ten games. Steinitz proposed dates between October 1883 and January 1884. However Zukertort declined the dates through his second on account of other supposed commitments and also did not agree to an immediate start of the match.\n\nBut behind his refusal there were apparently health reasons. Zukertort's doctor absolutely advised his patient against any further match or tournament. When Zukertort was in the USA in 1884 for a simultaneous tour, Steinitz renewed his offer, which was however once again declined by Zukertort. After a piece of journalistic provocation, published by Steinitz in his new magazine _The International Chess Magazine_ , Zukertort finally reacted with his own offer of a match, which he made public in March 1885 in _The Chess Monthly_.\n\nThis first official match for the World Chess Championship took place from the 11th January till the 29th March 1886 in the USA and was agreed to be for ten wins. In the event of 9:9 the defender should retain his title. The match games were played in various chess clubs in New York, St. Louis and New Orleans and financed by local chess patrons.\n\nPreceding the match there was a long disagreement concerning the match conditions, which could only be conducted indirectly since Zukertort and Steinitz still only communicated with each other in writing or via seconds after their mutual insults of the 'Ink War'. There was a dispute about the number of wins which were required for victory in the match, the venues and especially all the financial matters. The thinking time was also subject to discussion. Use was made of the then new chess double-clock, which had first been employed for the control of thinking time at the tournament in London in 1883. Steinitz had once previously played with a mechanical clock, in 1866 against Anderssen. Agreement was finally reached on a thinking time of two hours for the first 30 and one hour for the next 15 moves and so on. After the first 30 moves a pause of two hours was allowed. Steinitz had originally demanded a slower tempo, Zukertort would have preferred to play with the thinking time which is still frequently chosen, that is two hours for 40 moves.\n\nThe prize fund was composed of a stake from each player of 2000 dollars put up by their personal patrons, their so-called 'backers'. The victor was to receive the total sum of 4000 dollars as his prize. Zukertort however was assured by the organisers of 750 dollars compensation for loss of earnings in the event of a loss. Steinitz' second was at first the secretary of the New York Manhattan Chess Club, Gustav Simonson, later Thomas Frere. Zukertort named James Innes Minchin, the secretary of the London St. George's Chess Club. He was later replaced by Charles M\u00f6hle. The negotiations about the match conditions were carried out by Frere and Minchin. The contract was finally signed by the players on the 29th December 1885.\n\nThe match began on the 11th January 1886 in New York in the hall of Cartier's Academy, No. 80, Fifth Avenue, organised by the Manhattan Chess Club, which itself had raised 1000 dollars for the match. Play was to continue at this first venue until three wins had been achieved. Steinitz and Zukertort played on the same board and with the same pieces as Paul Morphy and Louis Paulsen had used in the same venue in 1857 at the 1st USA Congress. Morphy had won that match.\n\nThe match for the World Championship was advertised with posters and billboards. More spectators came to the games than could be accommodated in the hall. The moves were transmitted live by telegraph to various American chess clubs and to London. In New York Zukertort went into a 4:1 lead as Steinitz missed some winning positions. In an interview with the New York Tribune he explained his bad start in these terms: 'I could not sleep then and my nerves were strongly affected. The first game which I won in New York was considered brilliant and well played, however between that game and the next one I spent seventeen hours working on literary and analytical material for my _International Chess Magazine_ instead of resting and taking exercise in the open air.'\n\nAfter a break of twelve days the second stage was in St. Louis in the 'Chess, Checkers and Whist Club _Harmonie_ ' (at the intersection of Olive and 8th Streets), which was to be the host until the next three wins had been achieved. Steinitz won games six and seven and the eighth was drawn. The ninth game also went to Steinitz, who had thus drawn level in the match. Since from the point of view of the organisation it had not been expected that three wins would be achieved so quickly in St. Louis, the players still had some time before the move to New Orleans. The two of them enjoyed playing whist together in the club.\n\nThe final venue for the first WCh match in history was the New Orleans Chess, Checkers and Whist Club (Baronne Street). Zukertort, who suffered from heart disease, was no longer fit for the effort required, as his doctor had previously feared, and he lost six games, winning only one. After a total of 20 games Steinitz had won the required number of ten games, with a final result of 12\u00bd:7\u00bd.\n\n **Steinitz \u2013 Zukertort**\n\nNew Orleans, 12th game  \n3rd March 1886  \nRuy Lopez (C67)\n\n**1.e4 e5 2. \u2658f3 \u2658c6 3.\u2657b5 \u2658f6 4.0-0 \u2658xe4**\n\nThis variation of the Ruy Lopez, the so-called 'Berlin Defence', was really popular in the 19th century, but later lost in popularity compared to the 3...a6 variation. The Berlin Defence celebrated a renaissance after Vladimir Kramnik employed it in 2000 in his WCh match against Garry Kasparov and the title defender was unable to win a single game against it. It is very popular in present day tournament praxis and is regularly employed by many top players.\n\n**5. \u2656e1**\n\nNowadays a very popular alternative is the sequence 5.d4 \u2658d6 6.\u2657xc6 dxc6 7.dxe5 \u2658f5 8.\u2655xd8+ \u2654xd8 etc., with a queenless middlegame which is typical of this variation.\n\n**5... \u2658d6 6.\u2658xe5 \u2657e7**\n\nThis avoids the trap 6...\u2658xb5 7.\u2658xc6+, and White wins the queen. In the 4th match game Zukertort had chosen instead of the text move 6...\u2658xe5 7.\u2656xe5+ \u2657e7 and had later won the game.\n\n**7. \u2657xc6**\n\n7.\u2657d3 0-0 8.\u2658c3 \u2658xe5 9.\u2656xe5 was up for discussion in the 6th, 10th and also the 14th games.\n\n**7...dxc6**\n\nBlack has a structural disadvantage on account of the doubled pawns on the c-file, but there is sufficient compensation for this in the form of the bishop pair. 7...bxc6 8.d4 leads instead to a rather immobile pawn centre for Black. In addition the \u2657c8 is hindered in its development.\n\n**8. \u2655e2 \u2657e6**\n\nNot 8...0\u20130? on account of 9.\u2658xc6 bxc6 10.\u2655xe7 \u2656e8 11.\u2655xd8 and White wins. Though desirable, 11...\u2656xe1# is not possible because the black rook is pinned.\n\n**9.d3**\n\nAfter 9.d4 \u2658f5 10.c3 0-0 11.\u2657f4 Black would have the option of resolving his doubled pawns with 11...c5 12.dxc5 \u2657xc5 13.\u2658d2=.\n\n**9... \u2658f5**\n\nPreparing castling. After the immediate 9...0-0 Black probably feared the tactical blow 10.\u2658xf7. But what follows would not be risk-free for White either: 10... \u2656xf7 11.\u2655xe6 \u2658f5 and White must now be careful: 12.\u2658c3? (12.\u2657e3 is better, but after 12...\u2657h4 Black has enough counterplay for his pawn) 12...\u2658d4\u2013+.\n\n**10. \u2658d2**\n\n10.c3 would have been better, is the opinion of Johannes Minckwitz in his contemporary tournament book on the 1st World Chess Championship.\n\n**10...0-0**\n\nInstead of the move in the game, it was well worth considering 10...\u2658d4!? with an attack on the white queen and the c2-pawn. After 11.\u2655d1 Black has gained time for his development.\n\n**11.c3 \u2656e8 12.\u2658e4 \u2655d5 13.\u2657f4 \u2656ad8 14.d4 \u2658d6 15.\u2658c5 \u2657c8 16.\u2658cd3**\n\nUp till here Steinitz had used one hour of his time.\n\n**16...f6 17. \u2658b4**\n\nAfter 17.\u2658f3 according to Emil Schallopp 17...\u2657g4 would be annoying.\n\n**17... \u2655b5 18.\u2655xb5 \u2658xb5**\n\n18...cxb5 was good: 19.\u2658ed3 c6 with a solid position for Black on the queenside.\n\n**19. \u2658ed3 \u2657f5**\n\nHere Zukertort had used up his first hour of thinking time. Instead of the move in the game Minckwitz suggested 19...a5! intending 20.\u2658c2 \u2657f5 21.\u2656ad1 and then 21...a4 with advantage for Black. After 21...c5 White's problems with the positioning of the two knights become even more obvious.\n\n**20.a4! \u2658d6**\n\nSteinitz suggests 20...a5! as better: 21.axb5 axb4 22.\u2658xb4! (not 22.bxc6? \u2657xd3 23.cxb7 \u2656b8 24.\u2656a8 \u2654f7\u2013+) 22...\u2657xb4 23.cxb4 cxb5=.\n\n**21.a5 \u2658b5?!**\n\nThis move was criticised by contemporary commentators and 21...a6 suggested in its place. White then has only a slight initiative. 22.\u2658c5 is simply followed by 22...\u2657c8.\n\n**22.a6 \u2657xd3 23.\u2658xd3 b6 24.\u2656e3 \u2654f7 25.\u2656ae1 \u2656d7?**\n\nIntending...\u2656ed8,...\u2657f8 and...c6-c5. Now, however, Black loses the c6-pawn and gets into a bad position. So the correct move was 25...\u2657d6, which both Steinitz and Minckwitz recommended as better.\n\n**26. \u2658b4! g5 27.\u2657g3 f5 28.f4**\n\nThere was the even stronger 28.\u2658xc6! f4 29.\u2658e5+ \u2654e6 30.\u2657xf4 gxf4 31.\u2656h3 and White wins according to Steinitz.\n\n**28...c5 29. \u2658c6 cxd4 30.cxd4 \u2654f8**\n\nOf course not 30...\u2658xd4? 31.\u2658e5+ +\u2013.\n\n**31. \u2656e5**\n\n31.\u2657f2!? gxf4 32.\u2656e6 and Black is in zugzwang.\n\n**31... \u2658xd4 32.\u2658xd4!**\n\nPrecisely played, since after the forced liquidation 32.\u2656xe7 \u2656dxe7 33.\u2656xe7 \u2656xe7 34.\u2658xd4 gxf4 35.\u2657xf4 \u2656e4 36.\u2657h6+ \u2654e7 37.\u2658b5 the ending is unclear.\n\n**32... \u2656xd4 33.\u2656xf5+ \u2654g7**\n\n33...\u2654g8 34.\u2656xg5+ \u2657xg5 35.\u2656xe8+ and White collects the black pawns on the queenside with \u2656a8 and \u2656c8 and wins easily (Steinitz).\n\n**34.fxg5**\n\nAfter 34.\u2656xg5+ \u2657xg5 35.\u2656xe8 \u2657xf4, however, according to Steinitz a win would still have required a tiresome length of time.\n\n**34... \u2657c5**\n\n**35. \u2656xc5!**\n\nBut not 35.\u2656xe8 on account of 35... \u2656d1#. After 35.\u2654f1 \u2656xe1+ 36.\u2654xe1 \u2654g6 White would only have a minimal advantage.\n\n**35... \u2656xe1+ 36.\u2657xe1 bxc5 37.\u2657c3**\n\nThe point of the little combination. White wins back the exchange and the pawn ending is won.\n\n**37... \u2654g6 38.\u2657xd4 cxd4 39.h4 \u2654f5 40.\u2654f2 \u2654e4**\n\nBut 40...\u2654g4 is followed by 41.g3+\u2013 with the plan: 42.\u2654e2 \u2654g3 43.h5 and 44.g6.\n\n**41. \u2654e2 c5 42.b3 \u2654e5 43.\u2654d3 \u2654f4 44.b4**\n\nBlack resigned. Time used:\n\nSteinitz: 2 hours 39 minutes\n\nZukertort: 1 hour 35 minutes\n\nZukertort, who at the end of his life suffered from numerous illnesses, including arteriosclerosis, narrowing of the coronary arteries, rheumatism and kidney problems, died only two years after the WCh contest, on the 20th June 1888, of a brain haemorrhage.\n\nThe day before his death Zukertort had been in the best of spirits in the British Chess Club (37 King's Street, Covent Garden). Later he went to Simpson's Divan, where he played a game against Mr Sylvain Meyer. After approximately 25 minutes Zukertort suddenly slumped at the board and in doing so knocked a few pieces from the table, which he was not in a state to be able to pick up. Zukertort was brought back to the British Chess Club, since he was known there. Since his condition did not improve, he was finally taken at half past two in the morning to Charing Cross Hospital. Zukertort died at ten in the morning, at the age of 45.\n\nOn the 26th June 1888, at 10.30, in the presence of approximately 20 chess lovers, including Bird, Hoffer and Gunsberg, he was buried in the Brompton Cemetery in London. As time went by the location of his grave was forgotten. In March 2011, however, the English grandmaster Stuart Conquest discovered in the cemetery the totally decayed grave of Zukertort. With the help of the 'Polish Heritage Society' in England and some chess lovers Conquest collected 2000 pounds and had the grave restored. On the 26th June 2012 it was revealed to the public with its new headstone.\n**2. Fighting the Russian Bear**\n\n**The World Championship 1889:  \n_Wilhelm Steinitz against Mikhail Chigorin_**\n\nMikhail Chigorin was born on the 12th November 1850 in Gatchina, just 50 kilometres from St. Petersburg. His father worked in a gunpowder factory. Chigorin's parents died young and so he grew up in the orphanage of Gatchina. At 16 he learned to play chess, but he only began to take a more intensive interest in the game at the age of 23. He mostly spent his time playing in the Caf\u00e9 Domenika in St. Petersburg. There in 1873 he got to know Emanuel Schiffers (who was of German extraction), a former tutor who now made his living from chess and who became Chigorin's first chess teacher. In 1875 on a visit to St. Petersburg, Szymon Winawer recognised the talent of Chigorin and encouraged him.\n\n**Mikhail Chigorin (1850-1908)**\n\nChigorin was soon scoring his first successes; in 1879 he won the Russian national championship in St. Petersburg. In 1881 he took part in his first international tournament in the Berlin Chess Congress and shared third place with Winawer. After that Chigorin travelled around in Europe and America and played in other chess tournaments. Thus in 1889 he won the 6th American Chess Congress in New York.\n\nMoreover Chigorin also wrote a chess column in a Russian daily newspaper and was the publisher of the chess journal _Schachmatny Listok_ , later _Schachmatny Vestrik_. At home, as his daughter remembered later, he is said to have spent many days exclusively sitting at his desk composing articles on chess or attending to his extensive correspondence regarding chess. Many of the letters which reached Chigorin from all over the world had as an address simply 'Chigorin, Russia' \u2013 that was enough for a letter to reach him.\n\nContemporaries describe Chigorin as a tall, bearded man, who could appear very fierce in difficult situations over the chess board. His daughter characterised him as nervy and impatient. A non-smoker, in chess tournaments and matches he hated having to sit in the smoke of his opponents' cigars. Between rounds however, he liked to drink 'vodka to the point of oblivion' (Schonberg). As Jacques Mieses once jokingly remarked, Chigorin's style of life as far as meals were concerned was marked by great regularity: 'breakfast at eight, lunch at twelve, dinner at seven. And of course that meant: breakfast at eight in the evening whenever he got up, lunch at midnight and dinner at seven in the morning.'\n\nWhen Steinitz was giving a simultaneous exhibition in Havana in 1887, the local chess club offered to organise for him a World Championship match, with a considerable amount of prize money. Steinitz was asked about a possible opponent and named Mikhail Chigorin, against whom Steinitz had lost twice in the London tournament of 1883. The match was played in Havana from the 20th January till the 24th February 1889 and set for a maximum of 20 games.\n\nIt was also a contest between different views of how the game should be played. Steinitz wrote after the match: 'Here a young player of the old school was up against an old player of the new school.' Whilst Chigorin was a representative of the era of Romantic chess, in which one sought to overwhelm the opponent tactically in a more or less well prepared sacrificial attack, Steinitz had founded a new positional school and represented the thesis that the better plan for the game, based on specific strategic characteristics of the position, would win through in the end.\n\nThe match games were very hard fought. Only the 17th and final game ended as a draw. The previous 16 games were all won by one side or the other. Chigorin had the better start and after seven games was still leading 4:3. Then Steinitz won three games in succession. Chigorin was unable to catch up. The final result was 10:6 in wins. The prize money for the title defender was 1150 dollars.\n\n **Steinitz \u2013 Chigorin**\n\nHavana, 10th game  \n8th February 1889  \nChigorin Defence (D07)\n\n**1. \u2658f3 d5 2.d4 \u2657g4**\n\nAfter any other move White continues 3.c4 and the game transposes to the Queen's Gambit.\n\n**3.c4**\n\nIn the second match game, 3.\u2658e5 \u2657h5 4.\u2655d3!? was played instead. Steinitz won after 38 moves.\n\n**3... \u2658c6**\n\nThe development of the \u2658b8 in front of the c-pawn in the Queen's Gambit Declined is characteristic of Chigorin. The variation 1.d4 d5 2.c4 \u2658c6 \u2013 that is the original move order of the line \u2013 is thus called in his honour the 'Chigorin Defence'. In modern tournament praxis the Russian grandmaster Alexander Morozevich has adopted the ideas of his compatriot and shown that the latter's opening is still very playable even more than 100 years later.\n\n**4.e3**\n\nThe alternative is 4.cxd5 \u2657xf3 5.gxf3 \u2655xd5 6.e3 e5 7.\u2658c3 \u2657b4 8.\u2657d2 \u2657xc3 9.bxc3 \u2655d6 with decent play for Black.\n\n**4...e5**\n\nThe rapid advance of the e-pawn is one point of the early development of the knight to c6. Now one of the main variations of the Chigorin Defence has arisen.\n\n**5. \u2655b3!**\n\nThe principled move at this point. White must play energetically. The continuation 5.dxe5 dxc4 offers absolutely no advantage to White and 5.\u2657e2 e4 is even favourable for Black. In the event of 5.cxd5 \u2655xd5 6.\u2658c3 \u2657b4 7.\u2657e2 exd4 8.exd4 \u2657xf3 9.\u2657xf3 \u2655c4 a position known from the Scotch Gambit would arise by transposition and which arises in that original move order after 1.e4 e5 2.\u2658f3 \u2658c6 3.d4 exd4 4.c3 d5. Here too Black has complete equality.\n\n**5... \u2657xf3 6.gxf3**\n\nNot 6.\u2655xb7? with an attack on the knight, which can evade the said attack with 6...\u2658b4 and then threaten a fork on c2.\n\n**6...exd4?!**\n\nPerhaps not the best continuation. Later in the 14th game Chigorin chose 6...\u2658ge7.\n\n**7.cxd5 \u2658e5**\n\nAfter 7...\u2658b4 8.e4 d3 9.\u2657xd3 \u2658xd3+ 10.\u2655xd3 White retains a sound extra pawn.\n\n**8.exd4 \u2658d7 9.\u2658c3!?**\n\nIt was also worth considering 9.\u2655xb7. After 9...\u2655e7+ 10.\u2657e3 \u2655b4+ 11.\u2655xb4 \u2657xb4+ 12.\u2658c3 \u2658gf6 White retains a material advantage.\n\n**9... \u2655e7+ 10.\u2657e3 \u2655b4 11.\u2655c2!**\n\nAfter 11.\u2655xb4 \u2657xb4 12.a3 \u2657d6 13.\u2658e4 \u2658b6 Black recovers the d5-pawn.\n\n**11... \u2658gf6**\n\nAfter 11...0-0-0 12.0-0-0 \u2658b6 follows 13.d6! and after 13...\u2657xd6 14.\u2658b5 \u2654b8 15.\u2654b1 White threatens 16.\u2657d2 \u2655a4 17.b3 winning the queen.\n\n**12. \u2657b5**\n\nAnother strong move here was 12.0-0-0!? intending 13.\u2658b5.\n\n**12... \u2656d8 13.0-0-0 a6**\n\nTime used by Black so far: one hour.\n\n**14. \u2657a4 \u2657e7 15.\u2656hg1 g6**\n\nKingside castling 15...0-0 was not possible, on account of 16.\u2657h6 \u2658e8 17.\u2655f5 \u2658df6 18.\u2656xg7+! \u2658xg7 19.\u2656g1 and White wins.\n\n**16. \u2657h6!**\n\nThis prevents Black from castling. With his king in the middle Black will now have severe problems.\n\n**16...b5 17. \u2657b3**\n\nTime used by White so far: one hour.\n\n**17... \u2658b6 18.\u2656ge1 \u2654d7 19.\u2657f4**\n\nThe threat is 20.\u2657xc7 and 21.\u2658xb5+.\n\n**19... \u2656c8 20.a3 \u2655a5**\n\n**21. \u2657g5!+\u2013**\n\nAnd 21.\u2656xe7+ would also have won: after 21...\u2654xe7 22.\u2657g5 \u2658bd7 23.\u2658e4 \u2655b6 24.\u2658xf6 \u2658xf6 25.\u2655c6 \u2655xc6+ 26.dxc6 \u2656he8 27.\u2656d3 Black loses the knight. The ending is won for White.\n\n**21... \u2658g8**\n\nOr 21...\u2658bxd5 22.\u2658xd5 \u2658xd5 23.\u2657xe7 \u2658xe7 24.\u2656xe7+ \u2654xe7 25.\u2655c5+ \u2654e8 26.\u2655e5+ \u2654d7 27.\u2655d5+ \u2654e7 28.\u2655xf7+ \u2654d8 29.\u2657e6 followed by mate on d7, or 21...\u2658fxd5 22.\u2657xe7+\u2013.\n\n**22. \u2657xe7 \u2658xe7 23.\u2658e4 \u2656b8**\n\n23...\u2658exd5 24.\u2658c5+ \u2654c6 25.\u2657xd5+ \u2658xd5 26.\u2658b3+ +\u2013.\n\n**24. \u2658f6+ \u2654d8 25.\u2656xe7 \u2654xe7 26.\u2655xc7+ \u2658d7**\n\n26...\u2654xf6 27.\u2655e5#, 26...\u2654f8 27.\u2655xb8+ \u2654g7 28.\u2658h5+ gxh5 29.\u2656g1+ \u2654f6 (29...\u2654h6 30.\u2655f4#) 30.\u2655e5#.\n\n**27. \u2655xa5**\n\nBlack resigned. Time used:\n\nSteinitz: 1 hour 48 minutes\n\nChigorin: 1 hour 53 minutes\n**3. 'Mephisto' in person**\n\n**The World Championship 1890:  \n_Wilhelm Steinitz against Isidor Gunsberg_**\n\nIsidor Arthur Gunsberg was born on the 2nd November 1854 to a Jewish family in Pest, nowadays a district of Budapest. His father came from Russian Poland. When he was eight years old, in 1862, the family moved to London. In 1866 he travelled with his father to Paris, where he went to the Caf\u00e9 de la R\u00e9gence and impressed the players there with his chess skills.\n\nIn London in 1879 he met Charles Godfrey G\u00fcmpel. The latter was a manufacturer of custom made prostheses and had started in 1870 to construct a chess automaton, which he completed around 1876. He called it Mephisto, because in its human form the automaton was modelled on that character.\n\nIsidor Gunsberg (1854-1930)\n\nIn 1878 G\u00fcmpel with his Mephisto took part in and won a tournament of the 'County Chess Association'. The Irish master George MacDonnell had previously refused to play against the automaton and withdrawn from the tournament. G\u00fcmpel then installed Mephisto in his house and challenged the best players in London to play against it. Unlike the chess Turk of Wolfgang von Kempelen or Ajeeb, Mephisto was worked at a distance with the help of electro-magnetic technology by a cable from a neighbouring room. This was done from 1879 to 1889 by Isidor Gunsberg, who was then replaced by Jean Taubenhaus. In 1883 Chigorin played against Mephisto in London, well aware that the latter was being operated by Gunsberg, and he lost.\n\nIn 1889 G\u00fcmpel travelled with his Mephisto to the World Exhibition in Paris, in order to present it there. After that it was dismantled and all traces of the chess automaton were lost. But G\u00fcmpel had also given consideration to the possibility of constructing a proper chess automaton which would be worked with a system of punch cards. But after he calculated that 500 000 billion people would have to spend their whole life making the punch cards so that every eventuality would be covered, G\u00fcmpel dropped his plan.\n\nWhen he was employed by G\u00fcmpel, Gunsberg abandoned his training as a tobacco dealer and decided to become a chess professional. In the 1880s he won a series of tournaments (Hamburg 1885, Bradford 1888, London 1888) and matches (against Bird by 5\u00bd:2\u00bd and against Blackburne by 8:5). In order to find a challenger for Steinitz, the Manhattan Chess Club organised in 1889 a candidates' tournament with 20 participants in the form of a double round all-play-all, in which a game had to be replayed if it ended in a draw. Miksa White, who moreover soon after retired from chess to pursue his banking career with Rothschild's, and Mikhail Chigorin shared first place, but neither was interested in a match against Steinitz. Gunsberg was third and was prepared to accept. After Gunsberg had drawn a match against Chigorin in 1890 (9:9), Steinitz accepted the challenge.\n\nThe match for the World Championship between Wilhelm Steinitz and Isidor Gunsberg was staged from the 9th December 1890 till 22nd January 1891 in New York and organised by the Manhattan Chess Club. The prize fund was to be put up by stakes found by the players with the help of patrons. After Gunsberg was unable to manage the minimum stake, Steinitz went back on his original demands and played for less money. The match was for 20 games.\n\nThe arbiter of the match was Professor Isaac Leopold Rice, president of the Manhattan Chess Club and a well-known chess patron beyond the borders of the USA. Rice actually came from Wachenheim in Bavaria, where he had been born on the 22nd February 1855. However, the family emigrated to the USA before Rice was six years old. In his new homeland Rice first attended the Central High School in Philadelphia. In 1880 he got the degree of Bachelor of Law from Columbia Law School. In 1902 he was awarded an honorary doctorate by Bates College. In 1885 Rice married Julia Hynemann Barnett. The couple had six children.\n\nTo begin with Rice worked as a lecturer at the Columbia Law School and till 1886 as the dean of the School of Political Science. After that he specialised in the law regarding railways and reorganised various US railway companies. In 1887 he founded the 'Electric Vehicle Company' (for the production of electric cars), then the 'Electric Storage Battery Company' (for the production of accumulators), finally in 1899 the 'Electric Boat Company', which built for the US navy in the First World War 85 submarines and 722 submarine chasers. The 'Electric Boat Company' later became the 'General Dynamics Corporation', nowadays one of the world's biggest producers of fighter aircraft.\n\nIn his New York house, 'Villa Julia', built in 1890 on Riverside Drive at the corner of 89th Street, Rice had installed a chess room in the cellar, which was excavated underground into the cliff and reached by a lift. In 1909 he sold the house for 600 000 dollars to the Schinasi brothers, two tobacco manufacturers. The house still exists today and since 1954 has been in the possession of a Talmud high school (Yeshiva).\n\nProfessor Rice publicised a special version of the Kieseritzky Gambit, the unsound 'Rice Gambit', and sponsored some thematic tournaments and matches with this opening. For example, Lasker and Chigorin played a thematic match a few years later (in 1903 in Brighton) in which all the games started with the Rice Gambit. Lasker was also otherwise closely linked with Rice at the start of the new century and even acted from 1906 as secretary of a 'Rice Society' founded by Rice with the aim of furthering the popularity of the eponymous gambit.\n\nBut at the time of the WCh match between Steinitz and Gunsberg Lasker's star was not yet on the rise. In any case Leopold Rice was in a positive sense a chess fanatic, to whom the masters of the era owed a good deal materially speaking, and they certainly gladly overlooked his little whims and vanities when it came to 'his' gambit. Rice died on the 2nd November 1915.\n\nThe games of the third match for the World Championship were played from 13.30 till 17.00 and then after a pause continued from 19.00 till 22.30. Gunsberg decided the fourth, fifth, twelfth and sixteenth games in his favour. After the fifth game he was leading 3:2. Steinitz won the second, sixth, seventh, tenth, thirteenth and eighteenth games. His strong end spurt had allowed Steinitz to successfully defend his title once more. The winner received a prize of 3000 dollars, Gunsberg as the loser 1000 dollars.\n\n **Steinitz \u2013 Gunsberg**\n\nNew York, 7th game  \n22nd December 1890  \nQueen's Gambit Accepted (D24)\n\n**1.d4 d5 2.c4 dxc4 3. \u2658f3**\n\nThis move was in its day an idea of Joseph Henry Blackburne. White prevents the counterthrust...e7-e5. In the 5th game 3.e3 e5 4.dxe5 \u2655xd1+ 5.\u2654xd1 \u2658c6 6.\u2657xc4 \u2658xe5 7.\u2657b5+ c6. was played instead. Black was already better and went on to win.\n\n**3... \u2658f6**\n\nThe attempt to hang on to the pawn on c4 with 3...b5 does not lead to success, for example 4.a4 c6 5.e3 (or 5.axb5 cxb5 6.b3 with advantage to White) 5...\u2657d7 6.\u2658e5 e6 7.axb5 cxb5 8.\u2655f3 1-0, Blackburne-M. Fleissig, Vienna 1873.\n\n**4.e3 e6**\n\nAlso playable is 4...\u2657g4.\n\n**5. \u2657xc4 \u2657b4+?!**\n\nThis move is not very useful here. Nowadays 5...c5 or 5...a6 is usual.\n\n**6. \u2658c3 0-0 7.0-0 b6**\n\nAfter 7...c5!? Black could also immediately take up the struggle for the centre.\n\n**8. \u2658e5**\n\nIntending f2-f3.\n\n**8... \u2657b7 9.\u2655b3 \u2657xc3?!**\n\nThe surrender of the bishop pair was not absolutely necessary. After 9...\u2657e7 the strike 10.\u2657xe6 leads after 10...fxe6 11.\u2655xe6+ \u2654h8 12.\u2658f7+ \u2656xf7 13.\u2655xf7 to an unclear position. But 9...\u2655e7!? looks solid, e.g.: 10.a3 \u2657d6 11.f4 c5 etc.\n\n**10.bxc3 \u2657d5**\n\nBlack is afraid of the strike on f7 or e6, but this move weakens the c6-square after the exchange of the bishop. The alternative was 10...c5 11.f3 \u2657d5.\n\n**11. \u2657xd5 exd5 12.\u2657a3 \u2656e8 13.c4 c5!**\n\nAfter 13...c6 14.\u2656ac1 White obtains strong pressure on the c-file, whilst Black has worries about the development of the \u2658b8.\n\n**14. \u2656ac1 \u2658e4?**\n\nThe correct way was 14...cxd4!? 15.exd4 \u2658a6. After 16.\u2655a4 \u2658c7 17.\u2656fe1 \u2658e6= Black has counterplay against d4. Another option was 16...\u2658c5!?. After 17.dxc5 \u2656xe5 18.cxd5 bxc5 19.\u2656xc5 \u2656xd5 20.\u2656xd5 \u2655xd5 21.\u2657b2 the game would have been level. On the other hand the move in the game weakens d5.\n\n**15. \u2656fd1 cxd4 16.exd4 f6?**\n\nThis is refuted tactically. In the event of 16...dxc4 17.\u2655xc4 Black has major worries about his f7-pawn. Perhaps the return of the knight to f6 was still the best.\n\n**17.cxd5! fxe5 18.d6+ \u2654h8 19.\u2655d5?!**\n\nThings were clearer after 19.\u2655f3 \u2658xf2 20.\u2655xf2 exd4 21.\u2655xd4+\u2013.\n\n**19... \u2658xf2**\n\nAfter 19...exd4 20.\u2655xa8 \u2658c3 then 21.\u2656e1 \u2658e2+ 22.\u2656xe2 \u2656xe2 23.d7+\u2013 wins.\n\n**20. \u2656d2**\n\nAfter the 'normal move' 20.\u2654xf2 Black also continues with 20...\u2658d7. White will lose time after this with the retreat of the king to g1. Instead of the move in the game White could also easily play 20.\u2656f1 and after 20...\u2658d3 21.\u2656cd1 \u2658d7 22.\u2656xd3 e4 23.\u2656g3 he is winning. On the other hand 20.\u2655xa8 \u2658xd1 21.\u2656xd1 exd4 22.\u2655xa7 h6 is less favourable and leaves White only a slight advantage.\n\n**20... \u2658d7**\n\nIt was worth trying 20...\u2658h3+!? 21.gxh3 \u2658d7 with perhaps better chances of a draw than in the game.\n\n**21. \u2656xf2 \u2658f6**\n\nBut 21...\u2656c8 does not change much: 22.\u2656cf1+\u2013.\n\n**22. \u2656xf6+\u2013 gxf6**\n\nAfter 22...\u2655xf6 there follows 23.d7 \u2656f8 (23...\u2656ed8 24.dxe5+\u2013) 24.\u2657xf8 \u2656xf8 25.\u2655f3 \u2655d8 26.\u2655xf8+ \u2655xf8 27.dxe5+\u2013. On the other hand 23.dxe5? would be inaccurate: 23...\u2655xe5 24.\u2655xe5 \u2656xe5 25.d7. Black now holds the draw with 25...\u2656g8 26.\u2656c8 \u2656d5 27.\u2656c7 \u2656d8 29.\u2657e7 \u26565xd7.\n\n**23.d7**\n\nAnother powerful move was 23.\u2656c7, intending 24.\u2655f7+\u2013.\n\n**23... \u2656g8 24.dxe5**\n\nAlso strong was 24.\u2655e6, intending 25.\u2657e7+\u2013.\n\n**24... \u2656g5**\n\nOr 24...fxe5 25.\u2657d6+\u2013.\n\n**25. \u2655xa8!**\n\nOr 25.\u2657e7+\u2013.\n\n**25... \u2655xa8 26.\u2656c8+ \u2656g8 27.\u2656xa8 \u2656xa8 28.e6**\n\nBlack resigned.\n\nTime used:\n\nSteinitz: 1 hour 5 minutes\n\nGunsberg: 1 hour 35 minutes\n\nIn 1900 Gunsberg shared second place with James Mason behind Richard Teichmann in the moreover not all that strong London tournament. After some less well placed forays in Monte Carlo 1901 and 1902, as well as in the 13th DSB Congress in Hanover 1902, he was once more third in London in 1904. The preliminary round of the St. Petersburg tournament of 1914 saw the now 60 year old come in last with only a single point (two draws).\n\nAs well as taking part in tournaments Gunsberg was active as a chess journalist for several newspapers, where at the end it was not he but his wife who wrote the articles which appeared under his name, as well as being a tournament organiser (Ostende 1906). During his life Gunsberg married three times. He probably married his first wife Jane in 1879 in London. She died in 1891 at the age of 38. Gunsberg had three sons with her. In 1893 Gunsberg married Miriam Clarke, with whom he had two daughters. His second wife also died young, at the age of 39. In 1897 Gunsberg married for the third and last time in Lambeth. With his third wife Agnes Jane Ramage he had a daughter. It was only in 1908 that Gunsberg was accorded British citizenship.\n\nIn 1916 Gunsberg took to court _Associated Newspapers_ and the _Chess News Agency_ (or _Evening News_ ) for libel, after they made the claim that his chess column in the _Daily Telegraph_ , which he had been writing for 26 years, was seriously flawed. The reason for this accusation was that many of the chess problems published by Gunsberg were unsound, containing more than one solution. The British high court found in favour of Gunsberg and accepted his argument that chess problems quite frequently had more than one solution and that the column could not therefore be regarded as 'full of blunders'. Gunsberg had called numerous witnesses in his support. Gunsberg was accorded by the court damages to the tune of 250 pounds. In 1923, however, Gunsberg had to file for personal bankruptcy. On the 2nd May 1930 he died in London.\n**4. Tarrasch has no time**\n\n**The World Championship 1892:  \n_Wilhelm Steinitz against Mikhail Chigorin_**\n\nSteinitz had actually chosen Siegbert Tarrasch for another title defence, once more in Havana. But the latter declined and based his refusal on the long journey to get there and his professional commitments as a doctor. Thus Chigorin received for a second time the privilege of being allowed to challenge the World Champion. As well as Havana, St. Petersburg also thereupon made an offer, but Steinitz stuck with Havana as the venue. The location for the games in Havana was the Centro Asturiano Club. The match was set for 10 wins with a maximum of 20 games. In the event of 10:10 (including draws) play should continue until one of the players had managed 10 victories. The stake consisted of 2000 dollars.\n\nThe match was a hard-fought affair. Chigorin won the first game and after the 19th game he was still in front by 8:7; Steinitz equalised with the 20th game and with a victory in the 21st game took the lead in this match for the first time. In the 23rd game Chigorin was clearly winning, but blundered the game in a single move.\n\n **Chigorin \u2013 Steinitz**\n\nHavana, 23rd game  \n28th February 1892  \nKing's Gambit (C34)\n\n**1.e4 e5 2.f4**\n\nThe King's Gambit was during the Romantic movement in chess one of the most popular openings. Later analyses showed that the gambit is not totally sound and it lost in popularity.\n\n**2...exf4 3. \u2658f3**\n\nThere is also the very playable 3.\u2657c4 after which the tempting 3...\u2655h4+ is not so good because Black loses a further tempo with the queen, after 4.\u2654f1 d6 5.\u2658f3.\n\n**3... \u2658f6**\n\nIn its day a rather extravagant continuation. A more principled reply is the main variation with 3...g5 4.h4 g4 5.\u2658e5 d6 6.\u2658xg4 \u2658f6 etc. There is also the equally popular counter-strike in the centre with 3...d5.\n\n**4.e5**\n\nAfter 4.\u2658c3 there comes 4...d5, e.g.: 5.exd5 \u2658xd5 6.\u2658xd5 \u2655xd5 7.d4 \u2657g4 8.\u2657xf4 \u2658c6 9.\u2657e2 0-0-0 with good play for Black.\n\n**4... \u2658h5**\n\nThat was the idea. The knight on the edge protects the f4-pawn, without Black having to compromise his pawn structure with a move like...g7-g5.\n\n**5. \u2657e2**\n\nThe alternative is the immediate 5.d4.\n\n**5...g6 6.d4 \u2657g7 7.0-0 d6 8.\u2658c3 0-0 9.\u2658e1**\n\n9.\u2658d5 dxe5 10.dxe5 \u2658c6 11.\u2658xf4 \u2655xd1 12.\u2656xd1 \u2658xf4 13.\u2657xf4 \u2657e6=. Chigorin suggested 9.exd6! as an impro-vement: 9...cxd6 (9...\u2655xd6 10.\u2658e5) 10.\u2658e1! \u2658c6! 11.\u2657xh5 \u2657xd4+ 12.\u2654h1 with compensation according to Chigorin.\n\n**9...dxe5 10. \u2657xh5**\n\nA more precise way is 10.dxe5 \u2655xd1 11.\u2657xd1 \u2658c6 (11...\u2657xe5 12.\u2657xh5 gxh5 13.\u2658f3) 12.\u2657xh5 gxh5 13.\u2658d5 and the game remains open.\n\n**10...gxh5 11.dxe5 \u2655xd1 12.\u2658xd1 \u2658c6**\n\nIn comparison to the note to move 10, here the knight is on d1 instead of being actively posted on d5. Black now has the freer game with his bishop pair. Moreover the white pawn on e5 is weak.\n\n**13. \u2657xf4 \u2657f5?!**\n\nSimply 13...\u2658xe5 14.\u2658e3 \u2657e6 was better and Black has a clear advantage according to Chigorin.\n\n**14. \u2658e3 \u2657e4 15.\u2658f3**\n\nIt was also worth considering 15.\u2658d3!?.\n\n**15... \u2656fe8**\n\n**16. \u2658g5 \u2657g6 17.\u2658d5 \u2657xe5 18.\u2658xc7 \u2657xc7 19.\u2657xc7 \u2656ac8 20.\u2657g3**\n\n20.\u2657d6!?.\n\n**20... \u2658d4 21.c3 \u2658e2+ 22.\u2654f2 h4?**\n\nThis throws away his advantage. The correct move was 22...\u2658xg3 23.hxg3 (23.\u2654xg3 \u2656e2) 23...\u2656e5 24.\u2658f3 \u2656b5 winning a pawn and leaving Black with a clear advantage.\n\n**23. \u2657d6**\n\n**23... \u2658d4?**\n\nBlack wants to mate, only his combination has a big hole in it. After 23...\u2656cd8 White's advantage is only minimal.\n\n**24.cxd4 \u2656c2+ 25.\u2654g1 \u2656ee2**\n\nThe penetration of the rooks to the second rank should decide the game, however...\n\n**26. \u2656ae1 \u2656xg2+ 27.\u2654h1**\n\n... and the \u2657d6 protects the h2-square and the \u2658g5 and the \u2656e1 prevent the \u2657g6 from reaching e4 with decisive effect.\n\n**27... \u2654g7**\n\nThe exchange of rooks 27...\u2656ge2 28.\u2656xe2 \u2656xe2 leads after 29.\u2654g1 \u2656xb2 30.a3 h6 31.\u2658h3 \u2657e4 32.\u2656f4 to a won position for White.\n\n**28. \u2656e8**\n\nThis threatens mate with 29.\u2657f8+ \u2654g8 30.\u2657h6#.\n\n**28...f5?!**\n\nPerhaps 28...\u2657d3!? 29.\u2656xf7+ \u2654g6 was more tenacious, intending 30.\u2656g8+ \u2654h5 31.\u2658h3 \u2656xg8 32.\u2658f4+ \u2654g5 33.\u2658xd3 and White's victory is not yet home and dry.\n\n**29. \u2658e6+?!**\n\nThe correct idea, but the wrong execution of it. 29.\u2656e7+ won on the spot: 29...\u2654h6 (29...\u2654f6 30.\u2658e4#) 30.\u2657f4 \u2654h5 31.\u2658e6 \u2654g4 32.\u2656xb7.\n\n**29... \u2654f6 30.\u2656e7 \u2656ge2**\n\nAfter 30...\u2656xh2+ 31.\u2657xh2 \u2654xe7 32.d5 \u2656e2 (32...\u2656xb2 33.\u2656e1+\u2013) 33.\u2658d4 \u2656xb2 34.d6+ \u2654d7 35.\u2656e1 White's win is also clear.\n\n**31.d5 \u2656cd2**\n\n**32. \u2657b4??**\n\nA blatant case of chess blindness. In a winning position, White allows himself to be mated. 32.\u2656xb7 intending 32... \u2656xd5 33.\u2658f4 and White wins.\n\n**32... \u2656xh2+**\n\nWhite resigned.\n\nThis was the tenth win for the World Champion and the end of the match. Had Chigorin won the game, who knows how the match would then have finished. Therefore Wilhelm Steinitz had once more defended his title, though only just. It was, however, to be the last time that he managed to do so.\n\nMikhail Chigorin (1850-1908)\n\nMikhail Chigorin died in 1908 of advanced cancer. In 1907 he had, though already seriously ill, taken part in the tournament in Karlsbad. Before the match Steinitz had already felt the heavy blows of fate. In 1888 his daughter had died. In May 1892, moreover, he lost his wife Caroline, n\u00e9e Golder. She died of hepatitis. He finally had to close his _International Chess Magazine_ , because it was simply losing money. Shortly after the match against Chigorin Steinitz married for a second time, the 28 year younger Elisabeth, with whom he had two more children.\n**5. The triumph of defence**\n\n**The World Championship 1894:  \n_Wilhelm Steinitz against Emanuel Lasker_**\n\nEmanuel Lasker hailed from the Pomeranian town of Berlinchen, which nowadays belongs to Poland and is called Barlinek. He was born on the 24th December 1868 as the youngest son of the teacher and cantor Aron Michaelis Lasker and his wife Rosalie Israelssohn. The family was distantly related to the Prussian politician Eduard Lasker and also to the chess and go player Edward Lasker, who emigrated to the USA in 1914. The latter's niece, Anita Lasker-Wallfisch, later became famous with her auto-biography _Inherit the truth, 1939-1945: The documented experiences of a survivor of Auschwitz and Belsen_ as 'the cellist of Auschwitz'.\n\nEmanuel Lasker (1868-1941)\n\nThe origins of the Lasker family lie in the Polish town of Lask. The sons and grandsons of the rabbi Abraham Meier Hindels who lived there adopted the name of Lasker at the start of the second half of the 18th century after they moved to other localities. Emanuel Lasker had an eight year older brother Berthold (born 1.12.1860), who was also an excellent chess player, and two older sisters, Theophila (also named as Charfila Hedwig, born 1.6.1862) and Amalia (Amalia Thekla, born 16.2.1865). Theophila was later a victim of the persecution of the Jews by the Nazis and died on the 6th April 1943 in the Sobibor concentration camp.\n\nIn 1879 his parents sent Emanuel Lasker to his brother in Berlin, in order to continue his schooling in the latter's charge. Lasker then learned to play chess from Berthold and together with his brother he was soon a regular visitor to the chess caf\u00e9s, of which there were very many in Berlin, such as the well-known Caf\u00e9 Royal, the Caf\u00e9 Kaiserhof, the Caf\u00e9 Bauer, the Romanische Caf\u00e9 on the Kurf\u00fcrstendamm \u2013 a well-known meeting place for artists in Berlin, in which chess was also played \u2013, the Caf\u00e9 Kerkau owned by the World Billiards Champion and chess lover Hugo Kerkau in the Friedrichstra\u00dfe 59\/60 on the corner with the Leipziger Stra\u00dfe (from 1901 the meeting place for the Berlin Chess Society which was founded in 1827, and which then continued from 1921 under the name of the Caf\u00e9 Zielka, later from 1929 the Moka Efti), the Caf\u00e9 Victoria, the Caf\u00e9 K\u00f6nig, the Caf\u00e9 des Westens and also the 'tea halls' in the Oranienburger Stra\u00dfe.\n\nIn the caf\u00e9s chess was played for money, which certainly came in useful for the Lasker brothers who came from a materially not very well endowed family. But apparently the owner of the Caf\u00e9 Royal banned the two Laskers, giving as a reason: 'two Laskers are too many', for the two Laskers did not leave the other guests any hope of winning from time to time.\n\nAfter the parents had learned that their youngest son was frequenting the chess caf\u00e9s with his older brother instead of worrying about his education, they set him up in another area of Berlin. When this too did not help matters, they sent Emanuel to a secondary school in Landsberg an der Warthe (nowadays: Gorzow Wielkopolski), in the province of Brandenburg, where Lasker then took his final school exam in 1888. Here too he was not completely cut off from chess, since his mathematics teacher Dr. Georg Kewitsch was the president of the local chess club and the strongest player in the area. As an underage schoolboy, Lasker was officially not supposed to enter the chess club, but he played 'secret' matches against its members.\n\nBerthold Lasker finished his medical studies and settled in 1893 in Elberfeld, which nowadays is part of the city of Wuppertal, as doctor and obstetrician. There he got to know Else Sch\u00fcler and married her on the 15th January 1894. The couple moved to Berlin-Charlottenburg (16 Br\u00fcckenstra\u00dfe). There Else Lasker-Sch\u00fcler first took lessons in drawing and then was soon moving freely in the literary circles of Berlin. She became the best known Bohemian of Berlin under the name of 'wild Else'. She achieved some literary recognition first as a poet with her unconventional poems and later as a dramatist with her expressionist play _Die Wupper_.\n\nHowever her marriage with Berthold Lasker rapidly failed, partly because the latter's Polish-Silesian-Jewish family was always a bit of a mystery to her. On the 11th April 1903 they were officially divorced. Her son, who was born in 1899 (and she denied that he was Berthold Lasker's), died young in 1927 of phthsis. Even during her marriage to Berthold Lasker Else Lasker-Sch\u00fcler saw many other men and she was the lover, for example, of the authors Ernst Toller, Theodor D\u00e4ubler and Hannes Heinz Ewers as well as the painters Franz Marc and George Grosz. For a long time she was in love with Gottfried Benn, to whom she gave the nickname 'King Giselher', and she got remarried to the musician and publisher Herwarth Walden, with whom she edited the expressionist magazine _Der Sturm_. However, she never moved in with the latter, but lived in furnished rooms and boarding houses.\n\nIn the meantime Berthold Lasker ran a flourishing dermatological clinic on the Alexanderplatz. He still also occasionally played chess in the Caf\u00e9 Bauer, but only against friends, not for stakes any more. Carl Ahues evaluated Berthold Lasker's playing strength as hardly any weaker than that of his brother Emanuel.\n\nIn 1889 Emanuel Lasker started to study mathematics in Berlin, and he continued in 1890 in G\u00f6ttingen. In the same year in Breslau he played in his first international chess tournament. Also in 1890 he won, together with his brother, a tournament in Berlin and in Liverpool defeated Henry Edward Bird in a match by 8\u00bd:3\u00bd. In 1891 Lasker decided to become a professional chess player and on account of the better prospects for chess players he moved to London. He achieved numerous successes in tournaments and in matches, including against Joseph Blackburne (8:2). In England, moreover, he edited the chess journal _London Chess Fortnightly_.\n\nOn the 28th September 1892 Lasker took ship on the _Southampton_ and travelled to New York. After some successes in the USA \u2013 in 1893 he won for example the championship of the Manhattan Chess Club, in which Harry Pillsbury also participated, by 13:0 \u2013 and in Canada he found sponsors for a WCh match against Steinitz.\n\nThe system for financing the World Chess Championships still worked along the lines of betting on horses. The players acquired patrons, the so-called 'backers', who were prepared to bet on the result of the match. At the end the persons who won the wager gave the victorious player half of their winnings. It took several months till Lasker had found enough 'backers'. Finally 3000 dollars prize money had been collected, to be divided in the ratio of 2250 dollars for the winner, 750 dollars for the loser. Originally the hopes had been for a purse of 5000 dollars. Steinitz finally accepted a smaller stake and Lasker and Steinitz signed on the 5th March 1894 a contract in which the detailed arrangements for the match were set out.\n\nThe match for the World Championship finally took place from the 15th March till the 26th May 1894 in New York, Philadelphia and Montreal, and was set for ten wins. In the event of defeat, Steinitz had secured the right to a return match. Thinking time was to be two hours for the first 30 moves and one hour for each further 15 moves. Although Steinitz was 30 years older than was the 26 year old Lasker at the time of the match, people were absolutely confident that the title defender would be victorious in this match too.\n\nAt the start in the New York Union Square Hotel Lasker won the first and third games, Steinitz the second and fourth. Next came two draws. Then, still in New York, the challenger decided the seventh and eighth games in his favour and before they moved to Philadelphia he was leading by 4:2. There Lasker won all three games and extended his lead to 7:2. Now the final stage began in Montreal, Steinitz seemed a beaten man. However, after a draw in the twelfth game the title defender won the thirteenth and fourteenth games and closed the score to 7:4. After that the match was interrupted for a week. After the pause Lasker won the fifteenth and sixteenth games and with the score at 9:4 in wins he then only required one victory to win the title. Steinitz won game seventeen, but with his victory in the eighteenth game Emanuel Lasker became the second World Chess Champion. After the final game of the match Steinitz stood up and shouted: 'Three cheers for the new World Champion!'\n\n **Lasker \u2013 Steinitz**\n\nNew York, 7th game  \n3th April 1894  \nRuy Lopez\n\n**1.e4 e5 2. \u2658f3 \u2658c6 3.\u2657b5 d6**\n\nThis variation of the Ruy Lopez was named after the first World Champion, but is considered today to be somewhat passive and is thus no longer particularly popular.\n\n**4.d4**\n\nFrequently 4.0-0 is played first.\n\n**4... \u2657d7**\n\nAnother plan is 4...exd4 5.\u2658xd4 \u2657d7 6.\u2658c3 \u2658f6 7.0-0 \u2657e7 8.\u2656e1 0-0. Black has a passive, but solid position.\n\n**5. \u2658c3 \u2658ge7**\n\nA more usual move is 5...\u2658f6.\n\n**6. \u2657e3**\n\nLasker commented on the course of the game so far in these words: 'With move 5 the game started along new lines, since I completely changed my plan of attack in order to extract more advantage from the position. The most important difference is the early queenside castling in order to attack on the kingside.' Steinitz: 'A new variation so as to play for an attack.' In his first three games as White Lasker had on each occasion continued 6.\u2657c4 \u2658xd4 7.\u2658xd4 exd4 8.\u2655xd4 \u2658c6 9.\u2655e3.\n\n**6... \u2658g6 7.\u2655d2 \u2657e7 8.0-0-0 a6 9.\u2657e2 exd4 10.\u2658xd4 \u2658xd4 11.\u2655xd4**\n\nBy castling queenside White has opened for himself the option of an attack on the king. Moreover, White has something of an advantage in space. For his part Black can point to a solid position.\n\n**11... \u2657f6 12.\u2655d2 \u2657c6 13.\u2658d5 0-0**\n\nThe moves 13...\u2657xd5 14.\u2655xd5 would bring the white queen into a very central position, from which it could also quickly swing over to the kingside.\n\n**14.g4**\n\nLasker: 'It appears that with this move I was asking too much of my attack. A quiet move like 14.f3 would have bolstered my position.' 14.h4 was also worth considering. After 14...\u2657xh4 15.g3 \u2657f6 16.f4 White then has the semi-open h-file at his disposal for the attack.\n\n**14... \u2656e8 15.g5**\n\nSteinitz: 'This advance is premature. He obviously overlooked Black's continuation. 15.f3 was better.'\n\n**15... \u2657xd5 16.\u2655xd5**\n\nSteinitz: 'This is a bad move, which should lose; 16.exd5 was correct.' After 16.exd5 then 16...\u2656xe3 (16...\u2657e7 17.h4 with an attack, 16...\u2657xg5 17.\u2657xg5 \u2656xe2 18.\u2657xd8 \u2656xd2 19.\u2656xd2 \u2656xd8 with advantage to White) 17.fxe3 \u2657xg5 holds things level.\n\n**16... \u2656e5 17.\u2655d2?!**\n\nThis allows the subsequent combination. After 17.\u2655xb7 \u2657xg5 18.\u2657xg5 \u2656xg5 19.\u2656hg1 Black only has a slight advantage.\n\n**17... \u2657xg5 18.f4**\n\nSteinitz: 'This leads to the loss of a second pawn, but it is the best chance for White, to mount an attack.'\n\n**18... \u2656xe4 19.fxg5 \u2655e7**\n\nThe point \u2013 both bishops are hanging.\n\n**20. \u2656df1**\n\nLasker: 'Steinitz has won two pawns with a prettily laid trap. I could have recovered one with 20.\u2657f3. But I preferred to keep on attacking.' After 20.\u2657f3 \u2656xe3 21.\u2656he1 \u2656e5 22.\u2657xb7 \u2656b8 23.\u2657xa6 \u2655xg5 24.\u2655xg5 \u2656xg5 White is a pawn down in the endgame and has to struggle for a draw.\n\n**20... \u2656xe3 21.\u2657c4**\n\nWhite is now two pawns down, but his pieces are more actively posted and thus White has compensation for the disadvantage in material.\n\n**21... \u2658h8?!**\n\nLasker: 'Black apparently underestimated the future course of events or else he would have played 21...\u2656f8.' After 21...\u2656f8 22.h4 (22.\u2656e1 \u2656xe1+ 23.\u2656xe1 \u2658e5\u2013+) there follows 22...b5 23.h5 (23.\u2657d3 \u2655e5 24.h5 \u2658f4\u2013+) 23...bxc4 24.hxg6 \u2655xg5 with advantage to Black.\n\n**22.h4 c6 23.g6!**\n\nLasker: 'He was probably not expecting my 23rd move. When I offered my third pawn he could not find a good way to accept the sacrifice, because after it I would force the opening of the h-file.' Steinitz: 'After some reflection Black could have worked out that he can accept the sacrifice without any danger: 23...hxg6, followed by...g6-g5, if White advances his h-pawn.'\n\n**23...d5?**\n\nHe had to play 23...hxg6 24.h5 gxh5 (24...g5, as suggested by Steinitz, is met by 25.h6, and Black is facing the same problems as in the game) 25.\u2656xh5 \u2656e5. After 26.\u2656xh8+ (26.\u2656xe5 \u2655xe5 27.\u2654b1 \u2656e8 with a major advantage to Black) 26...\u2654xh8 27.\u2655h2+ \u2654g8 28.\u2657xf7+ \u2655xf7 29.\u2656xf7 \u2654xf7 30.\u2655f4+ White perhaps has drawing chances with the help of a perpetual check.\n\n**24.gxh7+ \u2654xh7 25.\u2657d3+ \u2654g8 26.h5 \u2656e8 27.h6 g6**\n\n27...gxh6 28.\u2656fg1+ \u2658g6 29.\u2657xg6 fxg6 30.\u2656xg6+ \u2654f7 31.\u2656gg1 looks dangerous.\n\n**28.h7+ \u2654g7 29.\u2654b1**\n\n29.\u2655h2!? \u2655g5 30.\u2654b1 \u2656g3=.\n\n**29... \u2655e5 30.a3**\n\nWhite needs an airhole. For example, not 30.\u2655h2 \u2656e1+ and then mate.\n\n**30...c5 31. \u2655f2 c4**\n\n31...f5!? intending 32.\u2655h4 \u2658f7.\n\n**32. \u2655h4**\n\nThe threat is mate on h6. Steinitz: 'At first glance it looks as though White could win with 32.\u2657xg6. But that is not the case after 32...fxg6 33.\u2655h4 \u2658f7 34.h8\u2655\\+ \u2656xh8 35.\u2656xf7+ \u2654xf7 36.\u2655xh8 \u2655xh8 etc.'\n\n**32...f6**\n\nLasker: 'I was expecting 32...\u2654f8, after which 33.\u2657f5 would have given me good chances of a draw, because the bishop cannot be taken on account of 34.\u2656hg1. Instead my opponent chose 32...f6, which was rather risky. After that I got a very strong position and Black ought to be warned to stop playing for a win. At the end Black's chances of winning had as good as disappeared, even if White did not already have the best of things.'\n\n**33. \u2657f5 \u2654f7 34.\u2656hg1**\n\n**34...gxf5**\n\nSteinitz: '34...g5 was better.' After 34... \u2654g7 35.\u2655g4 g5 36.\u2655h5 c3 37.\u2656xg5+ fxg5 38.\u2655xg5+ \u2658g6 39.h8\u2655\\+ \u2654xh8 40.\u2655xg6 \u2656e1+ a draw is the most probable end to the game.\n\n**35. \u2655h5+ \u2654e7 36.\u2656g8 \u2654d6**\n\nA possible move was 36...\u2654d7. After 37.\u2656xf5 \u2655e6 38.\u2656xe8 \u2655xe8 39.\u2655g4 White has sufficient activity in exchange for the material he has lost.\n\n**37. \u2656xf5 \u2655e6 38.\u2656xe8 \u2655xe8 39.\u2656xf6+ \u2654c5**\n\nSteinitz: 'Now 39...\u2654c7 was much better.'\n\n**40. \u2655h6 \u2656e7 41.\u2655h2**\n\nThe threat is mate with 42.\u2655d6. However, 41.\u2655d2 was even stronger, with the threat of 42.b4 and mate.\n\n**41... \u2655d7**\n\nLasker: 'Steinitz, who was still playing for a win, makes a blunder here, losing his queen and knight or rook, and has to resign soon afterwards.'\n\nSteinitz: 'A catastrophe: Black was in time trouble. A better move was 41... \u2656e6'.\n\n**42. \u2655g1+ d4 43.\u2655g5+ \u2655d5 44.\u2656f5 \u2655xf5 45.\u2655xf5+ \u2654d6 46.\u2655f6+**\n\nBlack resigned.\n\nCurt von Bardeleben characterised the play of the new World Champion in the _Deutsche Schachzeitung_ (Berlin 1894) in these words:\n\n'He is far more skilful in defence than in attack, to which he only turns when he perceives a quite clear and decisive advantage or else when he finds himself in such a disadvantageous situation that purely passive defence no longer offers any prospects.'\n\nA week after the end of the match, on the 2nd June 1894, Steinitz transmitted to Lasker his challenge for a return match. Steinitz was of the opinion that he had only lost the match on account of bad health and that the next time things would look different.\n**6. The annoying return match**\n\n**The World Championship 1896\/97:  \n_Emanuel Lasker against Wilhelm Steinitz_**\n\nIn the agreed conditions for the WCh contest between Steinitz and Lasker the title defender had insisted on a contractual right to a return match in the event of his being defeated. It was even agreed that the return match should take place in the same year. Now that the return match was a real issue, Lasker explained that as he would first be going on a journey around the world the further negotiations as to the match conditions would be conducted by his second W.M. de Visser. Lasker would only be personally available after his return. Steinitz suggested March 1895 as the start of the return match. Lasker replied that his commitments did not permit a start any earlier than October 1895.\n\nSteinitz was annoyed by this answer and then simply considered himself to still be World Champion, since Lasker had not stuck to the agreement. So before his own departure to the Hastings tournament of 1895 he sent Lasker a letter, in which he stated in no uncertain terms: 'Your breaking of the agreement as to a return match is a clear admission of inferiority on your part; and since you, World Champion for approxi-mately 28 days, wish right from the start to dictate to me, World Champion for almost 28 years, humiliating conditions, I feel that I am justified in accusing you of a default, in line with the motto: Honi soit qui mal y pense.' ('Shame on him who thinks evil of it', the motto of the British Order of the Garter).\n\nThus Lasker and Steinitz did not at first meet in their return match but in some tournaments. The first took place in Hastings in 1895. The surprise winner there was neither Lasker nor Steinitz, but the 23 year old US American Harry Pillsbury. Chigorin came in second, and only then Lasker. Steinitz only managed sixth place. Lasker was able to win the subsequent four man tournament in St. Petersburg in 1895\/96. The other three players were in order Steinitz, Pillsbury and Chigorin. After three of the total of six rounds, Pillsbury had again been leading the field. But in round four he was defeated by Lasker in a spectacular game, which Lasker himself considered as one of the best in his career. Pillsbury then collapsed completely and did not win a single other game in this tournament.\n\nThe next, today we would say 'super-tournament', took place in the summer of 1896 in Nuremberg. Lasker again won, this time with 13\u00bd out of 18, ahead of Geza Maroczy (12\u00bd), Siegbert Tarrasch and Harry Pillsbury (each on 12). Steinitz was sixth. During the tournament Lasker was defeated by Harry Pillsbury, Dawid Janowski and in addition Rudolf 'Reszo' Charousek, who was making his international debut in Nuremberg. Lasker was impressed by the latter's play and even saw in the young Hungarian a possible future challenger. But Charousek died of tuberculosis in 1900 at the age of only 26. In his novel _Der Golem_ , first published in 1913\/14, Gustav Meyrink provided Charousek with a literary memorial in the form of the chess playing student of medicine 'Innozenz Charousek'. Nor did Pillsbury live to an old age. In 1906 he died at the age of 33 of progressive paralysis brought on by syphilis.\n\nAfter his success in Nuremberg Lasker then finally agreed with Steinitz to set the due return match for the end of 1896. The return match between Lasker and Steinitz for the World Championship then took place from 7th November 1896 till 1st January 1897 in the main hall of the Moscow Medical Society (Great Dmitrovka, today 32 Pushkin Street), not far from the Bolshoi Theatre. The prize money of 3000 roubles was raised by Moscow patrons, including the millionaire M. N. Bostanshoglo. 2000 were destined for the winner, 1000 for the loser. Lasker and Steinitz also agreed a bet of 500 roubles. The match was set for ten wins.\n\nIn the meantime Steinitz' health was bad and he was only a shadow of his former self. Therefore the match was very one-sided. Lasker immediately won the first four games. During the fourth game Steinitz held a bag of ice cubes against his head to assuage his continual headaches. For game five Steinitz changed his opening move from 1.e4 to 1.d4. The fifth game was drawn, but Lasker won also the sixth game. After that there was a pause of ten days. Steinitz managed to draw the next three games. In game ten Lasker won from a completely level endgame. Another pause of six days was taken. After the resumption Lasker also won the eleventh game and the score was 7:0 in wins. After a further pause of a week, nevertheless, Steinitz managed wins in games 12 and 13. However, games 14 and 16 again went to Lasker. Now a renewed pause of 12 days was called. In the subsequent 17th game Lasker scored his tenth victory and in doing so defended his title.\n\n **Steinitz \u2013 Lasker**\n\nMoscow, 3rd game  \n14th November 1896  \nItalian Game (C54)\n\n**1.e4 e5 2. \u2658f3 \u2658c6 3.\u2657c4 \u2657c5**\n\nThe Italian Game. The Two Knights Defence is reached after 3...\u2658f6 with some sharp variations after 4.\u2658g5.\n\n**4.c3**\n\nThe Evans Gambit with 4.b4 \u2657xb4 5.c3 etc. was thoroughly tested in the WCh match between Steinitz and Chigorin. There Steinitz had the black pieces in each case.\n\n**4... \u2658f6 5.d4**\n\nNowadays the quiet plan with 5.d3 d6 6.0-0 0-0 7.\u2657b3 a6 8.\u2658bd2 etc. is also popular.\n\n**5...exd4 6.cxd4 \u2657b4+ 7.\u2658c3**\n\nThe alternative is 7.\u2657d2.\n\n**7... \u2658xe4 8.0-0 \u2657xc3 9.bxc3 d5**\n\nThe best. Praxis has shown that Black has the better prospects after this move.\n\n**10. \u2657a3**\n\nWhite sacrifices a piece, in order to prevent Black from castling. 10.\u2657d3 is not so risky.\n\n**10...dxc4 11. \u2656e1 \u2657e6**\n\nA pragmatic decision. Black returns the piece and obtains the slightly better position.\n\nIn the 1st game of the match the same opening had appeared on the board. Lasker had then chosen 11...f5 and subsequently had to defend very carefully. It continued 12.\u2658d2 \u2654f7 13.\u2658xe4 fxe4 14.\u2656xe4 \u2655f6 15.\u2655e2 \u2657f5 16.\u2655xc4+ \u2654g6 17.\u2656e3, and White has compensation for the piece he has sacrificed.\n\nLasker did win the game, but preferred to choose an uncomplicated continuation here.\n\n**12. \u2656xe4 \u2655d5 13.\u2655e2 0-0-0 14.\u2658e5 \u2656he8 15.\u2658xc6 \u2655xc6 16.\u2656e1 \u2656g8**\n\nStepping out of the pin on the e-file. But the immediate 16...b6 was also good.\n\n**17. \u2656e5 b6 18.\u2657c1**\n\nThe bishop is looking for new tasks. But here it is difficult for White to strengthen his position.\n\n**18...g5!?**\n\nLasker now wants to attack himself and gives up a pawn in order to open files. The sacrifice also hinders the planned activation of the bishop to f4.\n\n**19. \u2656xg5 \u2656xg5 20.\u2657xg5 \u2656g8 21.f4 \u2657d5**\n\nThe attack on the g2-pawn forces White into a further loosening of his king position.\n\n**22.g3 \u2654b7**\n\nBlack has the superior position and is in absolutely no hurry.\n\n**23.h3 \u2655b5**\n\nPreparing to exchange the bishop and queen on the a8-h1 diagonal.\n\n**24. \u2654h2 \u2656g6 25.\u2655c2 f6 26.\u2657h4 \u2657c6 27.g4 \u2655d5**\n\nNow White must always keep an eye out for the threat of mate on g2.\n\n**28. \u2655f2**\n\nWhite should perhaps risk 28.f5, e.g.: 28...\u2656g8 29.\u2657xf6 h5 30.\u2655e2 hxg4 31.hxg4 \u2655d7 32.\u2657e7 \u2655d5 33.\u2657f6 and White's position is full of holes. But there is no direct way to a win for Black in sight.\n\n**28...h5 29.g5?**\n\nAfter this mistake the game is lost. 29.f5!? was the last chance.\n\n**29...fxg5 30. \u2657xg5 h4 31.\u2656f1 \u2656g8 32.\u2655d2 a5 33.a4 \u2656e8 34.f5**\n\nBlack meets 34.\u2657xh4 with the pretty 34...\u2656e3! and 35.\u2657g3 (35.\u2655xe3 \u2655g2#) is followed by 35...\u2656xc3\u2013+.\n\n**34... \u2656g8 35.\u2656e1**\n\n35.\u2656g1 \u2656xg5 36.\u2655xg5 \u2655d6+ \u2013+.\n\n**35... \u2655xf5 36.\u2656e5 \u2655f3 37.d5 \u2655g3+ 38.\u2654h1 \u2655xe5 39.dxc6+ \u2654xc6**\n\nWhite resigned.\n\n'Why did I lose as comprehensively as that?', Steinitz explained to his readers in the New York _Sun_ on the 17th December 1896: 'Because Lasker is the greatest master of the game of chess whom I have ever encountered, probably the greatest of all who have ever lived'. Steinitz did not use the excuse of his state of health: 'During a match a chess master has just as little right to be ill as does a general on the battlefield'.\n\nNor did Emanuel Schiffers, who attended the match in Moscow as a spectator, see the reasons for Steinitz' defeat in the state of the latter's health and he wrote in _Schachmatny jurnal_ (No. 1, January 1897): 'Steinitz' games clearly show that his play has in no way become weaker. He played just as Steinitz also played previously. All it would have taken was for some other player than Lasker to try to challenge Steinitz \u2013 he would have felt the whole impact and power of the latter's play'. Schiffers was well placed to know this, after all he had lost a match against Steinitz by 4:6 (with one draw) in Rostov in spring 1896.\n\n**Wilhelm Steinitz (1836-1900)**\n\nNevertheless, the health factor cannot be rejected out of hand. In Moscow Steinitz was admitted to a mental facility on the instigation of his secretary. The first World Champion actually believed that he could telephone simply by the power of thought. When she saw him standing at the window waiting for a reply to his telepathic call, his secretary called the doctor. Steinitz was then held against his will for 40 days in the Moscow 'Korsakov Clinic'. After treatment he returned via Vienna to New York and on the way played in a few tournaments. In February and March 1900 he again received psychiatric treatment in New York, since he believed that he was emitting electro-magnetic currents with the help of which he could move the chess pieces without touching them.\n\nOn the 12th August 1900 he died of heart failure totally impoverished in New York State Asylum on Ward's Island, nowadays the 'Manhattan Psychiatric Center'. Steinitz had made a written note of his income for the last two years of his life: no more than 250 dollars. From the clinic he wrote to a friend in Vienna: 'Like all lunatics I am convinced that the doctors here are madder than I am'. In September he was buried in a pauper's grave. The German Press Club paid for the re-interment in the Evergreen Cemetery in Brooklyn (Bethel Slope, Grave No. 5892). The inscription on his gravestone is in German: 'Hier ruht in Frieden'.\n**7. Attacker against defender**\n\n**The World Championship 1907:  \n_Emanuel Lasker against Frank Marshall_**\n\nAfter his successful return match against Steinitz of 1896\/97 no fewer than ten years would pass before Lasker once more defended his title. After the Steinitz match Lasker took a three year break from tournaments, during which time he continued his mathematical studies in Heidelberg and Berlin. At the tournaments in London in 1899 and Paris in 1900 Lasker then celebrated a brilliant comeback and won both tournaments majestically. From 1899, moreover, he took on the editing of the _Deutsche Schachzeitung_ , which he continued until 1904.\n\nIn 1900 in Erlangen Lasker took his Dr. phil. (Mathematics) with his thesis _\u00dcber Reihen auf der Convergenzgrenze_ ('On Series at Convergence Boundaries'). With this in his pocket, Lasker was planning an academic career, but his applications in Germany, Manchester and at the Columbia University of New York were all refused.\n\nDuring his stays in Berlin Lasker was always a regular in the chess caf\u00e9s. One episode, which supposedly occurred in the Caf\u00e9 Kaiserhof, was a favourite story. In it there was an enthusiastic chess lover who played very badly but who loved to play against the World Champion, 'because the latter played such classical chess'. The Russian master Benjamin Blumenfeld, who was studying law in Berlin, was an eyewitness to this encounter and reported: 'And Lasker really played \"classical chess\", for example: 1.e2-e3 2.\u2658g1-e2 3.\u2658e2-g3 4.\u2656h1-g1 5.\u2658g3-h1. And Lasker continued to regroup all the other pieces like this, so that after 20-25 moves all the pieces on the first rank had changed places, which gave rise to the most curious things. Lasker accompanied every move with \"strategic\" and \"philosophical\" comments, which he put forward with the greatest seriousness. These games between the best and the worst chess players in the world always attracted crowds of spectators and regularly ended in general amusement and hilarious laughter.'\n\nIn 1902 Lasker moved to New York because he thought that he would have better professional options there as a mathematician. In 1905 he there published the paper _Zur Theorie der Moduln and Ideale_ ('On the theory of modules and ideals'). Since Lasker's efforts to find an academic post were not crowned with success, he now turned to philosophy and published in 1907 in both English and German the paper _Struggle_ ('Kampf').\n\nIn the same year Lasker defended his title for the first time since 1897 \u2013 in a very one-sided match. Previously Lasker had declined an offer from the St. Petersburg chess club to play a WCh match against Chigorin and also one from the Moscow chess club to defend his title against Tarrasch. A match against Tarrasch proposed for the autumn of 1904 had to be called off because Tarrasch claimed to have injured himself while skating and considered himself unfit to play the match against Lasker in such circumstances. A planned WCh match against Geza Maroczy failed. Maroczy and Lasker had met in April 1906 in the New York 'Rice Chess Club' and firmly agreed to the match. It should actually have begun on 15th October 1906, but Maroczy could not raise the stake and so the match fell through.\n\nFrank Marshall was born on the 10th August 1877 in New York as one of the five sons of Alfred George and Sarah Marshall. His father came from England, his mother was of Scots-Irish descent. The family lived in a flat in 'Hell's Kitchen', the district inhabited by Irish born immigrants in New York, on Eighth Avenue and 50th Street. The houses were later demolished to make space for the second Madison Square Garden.\n\nWhen Marshall was eight years old the family moved to Montreal, Canada. There he learned chess at the age of ten from his father, who was also a really good chess player. After recognising his son's talent, he took him to the Hope Coffee House so as to find stronger opponents. There he met the Irish born William Henry Kraus Pollock (1859-1896), who spent the last four years of his life in Montreal and had represented Canada in the 1895 tournament in Hastings. Another strong player in Montreal was A. T. Davison. At 13 Frank Marshall was already the best chess player of Montreal. At 15 he won the championship of the Montreal Chess Club. In 1893 Marshall played in a simultaneous exhibition which Wilhelm Steinitz gave in Montreal and was defeated in 26 moves. Steinitz nevertheless praised his opponent in a newspaper interview: 'I have never had such a young player who gave me so much difficulty'.\n\nAt this point at the latest, chess became the focal point in Marshall's life. This was possibly the reason why his education suffered. But perhaps a congenital weakness in reading and writing was the reason why Marshall made a remarkably high number of basic spelling errors in all that he wrote. Nevertheless in Canada Marshall acquired a decent knowledge of the French language and such knowledge of German that in an encounter with Edward Lasker in 1910 he was able to analyse in German.\n\nAt 19 Marshall returned to the USA, back to Brooklyn, which at that time was still an independent neighbouring town to New York, and joined the Manhattan Chess Club (105 E. 22nd Street) and also the Brooklyn Chess Club. At 22 he won the championship of both clubs.\n\nThe two clubs then financed in 1899 Marshall's participation in the London tournament, which took place in May of that year in the 'Royal Aquarium'. However, his name was not well enough known to gain him entry to the A-tournament, in which a series of great players of the time, including Lasker, were participating. Marshall played instead in the main tournament, won it and received 70 pounds prize money.\n\nInternational recognition came to Marshall when at the tournament in Paris in 1900, played on the occasion of the world exhibition in the 'Grand Cercle' on the Boulevard Montmartre, out of 17 participants he shared third place with Maroczy on 12 points, behind Lasker and Pillsbury. The tournament had an unusual rule. In the event of a draw the game was replayed with reversed colours. It was only if there was a second draw that the point was shared, which happened only seven times out of 136 encounters; otherwise the result of the second game counted. It was noted as a special sensation that Marshall, as yet almost unknown in Europe, had defeated the World Champion Lasker. This was Lasker's only defeat in the Paris tournament. Lasker had only one draw against Chigorin, winning all his other games.\n\nAt the subsequent tournament in Munich Lasker and also Marshall, who was not himself playing in the tournament, joined a chess players union conceived by the English master Amos Burn; among its founding members there were also Joseph Blackburne, Isidor Gunsberg, Richard Teichmann, Siegbert Tarrasch, Mikhail Chigorin, Emanuel Schiffers, Carl Schlechter, Georg Marco and Max 'Miksa' White. The union was intended to represent the interests of professional and semi-professional players, but, however, achieved little.\n\nFrank Marshall (1877-1944)\n\nIn summer 1903 Marshall already submitted for the first time a challenge to Lasker, but he could not meet the World Champion's financial conditions. Lasker demanded a stake of 5000 dollars. In 1904 Marshall won the US national championship but did not accept the title, however, because the supposedly strongest player in the USA, Harry Pillsbury, had not taken part on account of illness. Pillsbury died two years later. In the same year Marshall was victorious in the Cambridge Springs tournament, without defeat and two points ahead of Janowski and Lasker, and he renewed his challenge to Lasker \u2013 once more unsuccessfully. Lasker now demanded for the first time a penalty which he was to receive if Marshall could not put up the stake or if other conditions were not fulfilled.\n\nIn Europe Marshall played in 1905 in the tournaments of Monte Carlo, Ostende, Barmen and Scheveningen \u2013 winning the latter, which was not, however, a particularly strong one. To prove his case for a WCh match against Lasker, Marshall first played matches against the other two possible challengers, Janowski and Tarrasch. In Paris in 1905 he defeated Janowski (8:4 in wins, four draws), but in the same year lost to Tarrasch in Nuremberg, by the absolutely clear score of 1:8 in wins (with eight draws). Moreover, after this match Marshall and Tarrasch tried to enforce copyright on the scores of their games. The games were not made publicly accessible, only being published in a book about the match. In 1906 Marshall then won in superior fashion the 15th Congress of the German Chess Federation in Nuremberg and in doing so eclipsed Janowski and Tarrasch.\n\nAfter the matches against Tarrasch and Maroczy had failed, Lasker finally accepted the challenge from Frank Marshall, on one hand because a match against the US American in the USA, where Lasker was living at that time, was the easiest to finance, on the other so as to indirectly compete with Tarrasch, who in 1905 had played against and defeated Marshall. Lasker's ambition was to win against Marshall by a better score than Tarrasch had managed. To make the match possible, there being great interest in it in the USA, Lasker reduced his financial demands.\n\nThe match between Emanuel Lasker and Frank Marshall for the World Championship was staged from 26th January till 8th April 1907 in chess clubs in New York (games one to six and game 15), Philadelphia (games seven to nine), Baltimore (game 10), Chicago (game 11) and Memphis (games 12 to 14), and was set for eight wins. The prize money was 1000 dollars. Lasker demanded in addition 250 dollars appearance money per game, or 500 dollars for a package of three games and had offered the match in advance to various clubs in the USA.\n\nThe two rivals could not have been more different, either in appearance or in their understanding of chess. Lasker was small in stature and somewhat stocky, and coolly objective in his play. Marshall on the other hand was almost two metres tall, in other ways too an imposing presence, and in chess a pure Romantic who loved to attack.\n\nBut both had one preference in common \u2013 they liked to smoke cigars. Lasker's cigars, however, or so some of his opponents claimed, gave off an unpleasant smell. In one of his tournaments Lasker is supposed to have on one occasion received from an unknown joker the present of a cigar which was laced with opium. For that reason, after that he only smoked his own cigars which his wife passed to him during the games.\n\nAgainst Frank Marshall Lasker certainly didn't smoke any opium-spiked cigars, because he was wide awake in all the games. In the first game against Marshall, Lasker, with the black pieces, sacrificed a piece. That was actually what the US spectators expected of their idol Marshall. Lasker won the game with his superior technical skills in the endgame and immediately seized the psychological initiative in the match. Marshall subsequently gave up 1.e4 and from then on played only 1.d4. Lasker then chose the Queen's Gambit Declined, whilst Lasker's 1.e4 was invariably followed by the French Defence.\n\nLasker won the first three games and after four subsequent draws the eighth game too. Next came three further draws, then Lasker won four games in succession for a final result of 8:0 (with seven draws). Lasker had thus majestically defended his title and in doing so also topped Tarrasch's result against Marshall.\n\n **Marshall \u2013 Lasker**\n\nNew York, 3rd game  \n31st January 1907  \nQueen's Gambit Declined (D55)\n\n**1.d4 d5 2.c4 e6 3. \u2658c3 \u2658f6 4.\u2657g5 \u2657e7 5.e3 \u2658e4**\n\nThis knight sortie is characteristic of the Lasker Variation of the Queen's Gambit Declined. The modern version is 5...0-0 6.\u2658f3 h6 7.\u2657h4 \u2658e4 8.\u2657xe7 \u2655xe7.\n\n**6. \u2657xe7 \u2655xe7 7.\u2657d3 \u2658xc3 8.bxc3 \u2658d7**\n\nAnother idea is 8...dxc4 9.\u2657xc4 b6.\n\n**9. \u2658f3 0-0 10.0-0 \u2656d8 11.\u2655c2 \u2658f8**\n\nFrom here the knight reliably defends the h7-square and can hardly come under attack.\n\n**12. \u2658e5 c5 13.\u2656ab1 \u2655c7**\n\nSo as to be able to play b6 (Tarrasch). The immediate 13...b6? is followed by 14.\u2658c6.\n\n**14. \u2655b3 b6 15.cxd5 exd5 16.\u2655a4?**\n\nTarrasch: 'This was the last point at which to get a good game with 16.c4, e.g.: 16...cxd4 17.exd4 \u2657e6 (Tarrasch was not taking 17...\u2658e6! into account) 18.\u2656bc1 \u2656ac8 (18...\u2655d6 19.c5) 19.cxd5 \u2655xc1 20.dxe6 \u2655c7? 21.exf7+ \u2654h8 22.\u2657xh7. Even if Black's moves in this variation are not quite forced, this continuation does give some idea of the strength of the white position after the move c4.'\n\n**16... \u2657b7 17.\u2655d1 \u2656d6**\n\nNice prophylaxis. Black anticipates the queen sortie to g4 and clears the way for the other rook, which would otherwise be shut in after...\u2657b7-c8. In addition, from d6 the rook can neatly swing over to the kingside.\n\n**18. \u2655g4 \u2656e8 19.\u2655g3**\n\nIntending \u2658g4-f6+.\n\n**19... \u2656de6 20.\u2657f5 \u26566e7**\n\n'The white pieces are almost all much more aggressively placed than the corresponding black ones; compare the position of the two knights, the two bishops, the two queens. But the black pieces are all lurking dangerously in ambush and the black position nowhere offers a point to attack. The game is apparently level, but in my view Black is better and that principally on account of the c5-pawn.' (Tarrasch).\n\n**21.f4?**\n\n'The position becomes untenable after this move, which makes the e-pawn backward and weakens the e4-point.' (Tarrasch)\n\n**21... \u2657c8 22.\u2657xc8 \u2656xc8**\n\n'Not 22...\u2655xc8 after which 23.f5 could cramp Black's play.' (Tarrasch) After 23... f6 24.\u2658g4 the threat is again 25.\u2658xf6+ but after 24...\u2658d7 Black is not worse.\n\n**23. \u2655f3**\n\n23.f5 f6 24.\u2658d3 \u2655xg3 25.hxg3 \u2656xe3 with advantage to Black.\n\n**23... \u2655d6 24.\u2656fc1 \u2656ec7 25.h3**\n\n'Marshall is embarrassed for moves.' (Tarrasch)\n\n**25...h6**\n\nSo as to bring the knight via h7 to e4. In playing this Lasker turns down 25...cxd4 26.exd4 f6 27.\u2658d3 (27.\u2658g4 \u2658e6) 27... \u2656xc3 28.\u2656xc3 \u2656xc3 winning a pawn.\n\n**26. \u2654h2**\n\n26.\u2656d1 was better.\n\n**26... \u2658h7**\n\n26...cxd4 27.exd4 f6, and after 28.\u2658g4 again 28...\u2658e6 was possible.\n\n**27. \u2655h5 \u2658f6 28.\u2655f5**\n\n'A genuine Marshall trap, which Lasker probably did not spot at once. Otherwise he would probably not have taken right away on d4, since this does not lead to immediate success.' (Tarrasch)\n\n**28...cxd4**\n\n28...\u2655e7!?.\n\n**29.exd4**\n\n**29... \u2658e4?!**\n\nThe 'Marshall trap' consisted of the tempting 29...\u2656xc3?. However that would have led to 30.\u2655xc8+! \u2656xc8 31.\u2656xc8+ \u2654h7 32.\u2656h8+! \u2654xh8 33.\u2658xf7+ +\u2013. Objectively the best was 29...g6 30.\u2655d3 (30.\u2658xg6? achieves nothing: 30...fxg6 31.\u2655xg6+ \u2656g7 32.\u2655xh6? \u2658g4+ \u2013+) 30...\u2655a3 31.\u2656c2 \u2658e4 32.\u2656b3 \u2655d6 with advantage to Black.\n\n**30. \u2658xf7**\n\nThis is probably what Lasker overlooked. He makes a virtue out of necessity, as Tarrasch remarked, and with the subsequent exchange sacrifice obtains an attack which should be enough for a draw, but no more than that.\n\n**30... \u2656xf7 31.\u2655xc8+ \u2656f8 32.\u2655b7 \u2655xf4+ 33.\u2654g1**\n\nAfter 33.\u2654h1 there is for example 33...\u2658g3+ 34.\u2654g1 \u2658e2+ \u2013+ winning.\n\n**33... \u2655e3+ 34.\u2654h2**\n\n34.\u2654h1 \u2658f2+ 35.\u2654h2 \u2655f4+ 36.\u2654g1 (36.g3 \u2655f5\u2013+) 36...\u2658xh3+ \u2013+ (Tarrasch).\n\n**34... \u2655g3+ 35.\u2654g1 \u2658d2**\n\nThis threatens...\u2658f3+.\n\n**36. \u2655xd5+**\n\n36.\u2656f1 \u2655e3+ \u2013+.\n\n**36... \u2654h8 37.\u2654h1**\n\n**37... \u2658f3! 38.gxf3 \u2655xh3+ 39.\u2654g1 \u2655g3+ 40.\u2654h1 \u2656f4**\n\n**41. \u2655d8+**\n\n'With this and with the following move Marshall probably believed that he would not only save the draw, but even win. This is, however, a mighty error.' (Tarrasch) The correct way was 41.\u2655h5 \u2656h4+ 42.\u2655xh4 \u2655xh4+ 43.\u2654g2 with a draw (Tarrasch).\n\n**41... \u2654h7 42.\u2656f1?**\n\n42.\u2656c2 \u2656h4+ (after the better 42... \u2655xf3+ 43.\u2656g2 \u2655e4 44.\u2654h2 \u2656h4+ 45.\u2655xh4 \u2655xh4+ 46.\u2654g1 Black gets another pawn and is better) 43.\u2655xh4 \u2655xh4+ 44.\u2654g2= (Tarrasch).\n\n**42... \u2656f5 43.\u2655e8 \u2655h4+**\n\nWhite resigned in view of 44.\u2654g2 \u2656g5#.\n\nIn 1904 at the age of 27 Marshall had married the ten year younger Carrie D. Krauss, daughter of a Brooklyn businessman. On the 28th December 1905 their son Frank Rice Marshall was born.\n\nAt the outbreak of the First World War Marshall was among those taking part in the tournament for the 19th DSB Congress in Mannheim. Whilst the Russians, among them the future World Champion Alekhine, were interned on the 3rd August 1914, Marshall received 375 marks compensation and as a citizen of the neutral USA was allowed to depart. In adventurous circumstances he took the train to Amsterdam, a journey \u2013 normally a seven hour journey \u2013 which at the start of the war now took 39 hours. In the Netherlands he found a ship which took him to the USA.\n\nIn 1918 Frank Marshall founded his own chess club, which found a prestigious address as 'Frank Marshall's Chess Divan' firstly in the back room of a restaurant called 'Keen's Chop House' (70 W. 36 Street, near Broadway). A year's membership cost 10 dollars, life membership only 25 dollars. After a short time, still in 1918, now renamed the 'Marshall Chess Club' it moved into 118 W. 49th Street. Two years later the club found yet another domicile, this time 146 W. 4th Street.\n\nIn 1925, after a long period of forced abstinence because of the events of the war, Marshall once more set out on a European journey. He took part in the tournament in Baden-Baden \u2013 the first international tournament in Germany after the war \u2013 and shared fifth place, which earned him prize money of 400 marks. The winner here was Alexander Alekhine. In Marienbad Marshall reached a shared third place with Carlos Torre. Aaron Nimzowitsch and Akiba Rubinstein shared the first prize. Marshall received the brilliancy prize for his game against Yates \u2013 a silver cigarette case. Marshall finally took part in the tournament in Moscow in 1925, the first tournament in Russia after the war, and came in fourth. There followed a series of other tournaments, in the USA and in Europe.\n\nIn 1929 the USA and the rest of the world were hit by the big Wall Street crash, the so-called 'Black Monday', which obliterated between September and the middle of November 1929 approximately 50 billion dollars of total US share values of 80 billion dollars. At this point Marshall was accompanying his friend Nardus, a wealthy art dealer and chess patron, on a journey to Tunis. Whilst Marshall himself was not affected by the crash, US chess life was very much hit by it. The chess patrons and sponsors had lost their fortunes. Many members of the Marshall Chess Club could no longer pay their subscriptions and resigned. The club, which at this point was in a building on 135 West 12th Street, was obliged to move out, but with the help of a benefactor got a new place to stay in 23 West 10th Street.\n\nIn 1930 Marshall represented the USA, alternating with Isaac Kashdan on first board at the Chess Olympiad in Hamburg. The board order was not fixed and could be changed from one round to the next. In the first round Marshall met an old acquaintance, the artist Marcel Duchamp. In the meantime the Frenchman had completely given up his career as an artist to become a chess professional. Marshall and Duchamp knew each other well, since Duchamp had been a member of the Marshall Chess Club for many years. Now Duchamp was playing in the French national team against Marshall. The game between Marshall and Duchamp ended in a draw. At the end of the Olympiad Marshall's impressive final score was 12\u00bd:4\u00bd.\n\nAt the following Chess Olympiad in Prague 1931 the USA (Isaac Kashdan, Frank Marshall, Arthur William Dake, Israel Albert Horowitz, Herman Steiner) won the gold medal. Two years later the US team \u2013 this time with Isaac Kashdan, Frank Marshall, Reuben Fine, Arthur William Dake, Albert Simonson \u2013 repeated their success at the Chess Olympiad in Folkestone. Originally Chicago wanted to hold the tournament, but could not find the costs and had to withdraw its bid.\n\nThe Chess Olympiad in Warsaw 1935 was also won by the USA, with Marshall again playing on board two, this time behind Reuben Fine. In addition he had taken on the task of team captain. In the match against Austria Marshall left the playing hall for a short time and realised to his surprise on returning that all three of his team mates had disappeared. They had drawn their games after only one hour. Marshall was infuriated and personally took care of getting the winning point against Gr\u00fcnfeld. In the evening he buttonholed his colleagues in 'individual conversations'.\n\nAt the next Chess Olympiad in 1937 in Stockholm \u2013 the USA did not participate in the 1936 Olympiad in Munich \u2013 Marshall played on fourth board behind Samuel Reshevsky, Reuben Fine and Isaac Kashdan. The reserve was Israel Albert Horowitz. The USA won the gold medal for the fourth time.\n\nIn 1941 the chess life of the Marshall Chess Club was hard hit by the entry of the USA into the Second World War. Many members were called up, including Marshall's son. In March 1942 Jos\u00e9 Raul Capablanca suffered his second stroke in the Marshall Chess Club and died of its consequences on the 8th March. Two and a half years later, on the 9th November 1944, Marshall crossed the Hudson River on a ferry on his way to Jersey City, where he intended to spend an evening at bingo. On Van Vorst Street, according to Andrew Soltis in his biography of Marshall, the WCh challenger of 1907 collapsed after a heart attack. The police were called but arrived too late to do anything. Frank Marshall was cremated on the 13th November in Fresh Pond Crematorium on Long Island.\n\nDuring his career Marshall wrote ten chess books, of which the best known is _My Fifty Years in Chess_. It was actually compiled on Marshall's behalf by prolific writer Fred Reinfeld on the basis of the latter's hand-written notes and annotations.\n**8. Two top German players**\n\n**The World Championship 1908:  \n_Emanuel Lasker against Siegbert Tarrasch_**\n\nSiegbert Tarrasch was born on the 5th March 1862, the son of Jewish parents in Breslau, a ground floor flat in the house on 16 Herrenstra\u00dfe. His father was the grain merchant Moritz Tarrasch, his mother Philippine Tarrasch, n\u00e9e Grabower. He had a brother who was four years older, Georg Tarrasch. Later, in 1909, Siegbert Tarrasch converted to the evangelical Lutheran church.\n\nLike Steinitz, Tarrasch was also handicapped from birth with a club foot. As soon as he had learned to read he worked through his father's library with great enthusiasm. As a school pupil Tarrasch later always felt the need to finish every task as the best in the class. From 1867 the Jewish primary school and then from 1870 the Elisabeth-Gymnasium, the same school in Breslau in which Adolf Anderssen had been a pupil. In 1880 Tarrasch passed his Abitur here. In the meantime his parents had separated \u2013 the father had emigrated to the USA, where he died in 1873 in New York.\n\nSiegbert Tarrasch (1862-1934)\n\nTarrasch then studied medicine in Berlin at the Royal Friedrich-Wilhelm University and was supported in this endeavour by the family of his mother. In 1882 Tarrasch continued his studies in Halle. At Easter 1885 Tarrasch passed the final exam of his medical studies at the age of 23. He then made efforts to improve his knowledge in a position in a hospital. Since this was not a success, he took a position which had become vacant as a doctor in Geroldsgr\u00fcn (the district of Hof, Upper Franconia). In 1887 he married Anna Rosalie Rudolf and moved to Nuremberg, where he opened a practice first in 2 Obere Kanalstra\u00dfe, then in 64 F\u00fcrther Stra\u00dfe.\n\nTarrasch had started his serious involvement in chess at the age of 15 after a fellow pupil had loaned him the _Praktische Schachb\u00fcchlein_ by Alphons von Breda. Tarrasch made rapid progress and in 1878 became a member of the chess club founded by Anderssen in Bayrischer Hof (on the Zwingerplatz). At the age of seventeen he met the pupil of Anderssen Fritz Riemann and played two games with him. After Tarrasch had become all too active in the chess club, he was urgently advised by the president of the club (a teacher in his school) that he would do better to distance himself from the life of the club and get back to working more for the school. Tarrasch followed this advice only insofar that he no longer went to the chess club, but became a regular visitor to the coffee house Fischer & Busch and met fellow pupils there to play chess.\n\nDuring his studies in Berlin, at the start of the 1880s, Tarrasch joined the Berlin chess club on the recommendation of his uncle Max Grabower. Other members there included Berthold Lasker, Emanuel Lasker's older brother. Tarrasch soon struck up a close friendship with Berthold Lasker. The two took part in 1881 in the main tournament of the 2nd DSB Congress in Berlin. In his masterpiece _300 chess games_ Tarrasch wrote about Berthold Lasker: 'Among my most frequent opponents over the board belonged (...) above all my friend Berthold Lasker, a very inspired chess player, whose strength unfortunately never received what it deserved in a tournament as a result of his nervous disposition.'\n\nIn 1881\/82 Tarrasch also met Emanuel Lasker for the first time. In 1887 Tarrasch could still even give the latter a knight odds in two games in the Berlin 'tea halls' and yet win one game and draw the other. Two further games without odds then ended, however, in a win for each side, as Lasker later reported proudly in an interview. Tarrasch celebrated his first great success as a chess player in 1883 in Nuremberg. Soon afterwards he was considered to be one of the best players in Germany. In 1892 he declined to play a match against Emanuel Lasker, because in Tarrasch's opinion the latter still had no successes to which he could point, but only two years later he had to admit that in the meantime Lasker had dethroned Steinitz as World Champion.\n\nTarrasch himself had in 1890 stupidly rejected an invitation from the chess club of Havana to play against Steinitz for the World Championship, since for him it was absolutely out of the question to abandon his medical practice for a quarter of a year on account of a chess match in Cuba.\n\nAt the end of the 1890s Tarrasch celebrated his greatest successes with victories in the tournaments in Manchester 1890, Dresden 1892, Leipzig 1894, Vienna 1898 and Breslau 1898. It is suggested that Tarrasch avoided Lasker. The two rarely played in the same tournament. In 1898 Tarrasch won the Imperial Jubilee Tournament in Vienna. After a tournament pause of three and a half years Tarrasch took part in the tournament in Monte Carlo 1902, but only came in fifth. Lasker had again withdrawn his acceptance when he heard that Tarrasch would be playing. So Lasker was also avoiding Tarrasch.\n\nNevertheless, in 1903 it did come to negotiations between Lasker and Tarrasch, which finally led to an agreement. The match for the World Championship would start in September or October 1904 and would be played to eight wins. Drawn games would not be counted. Both sides had to provide a stake of 8000 marks. In January 1904, however, Tarrasch suffered according to his own account an accident while skating, which 'for a certain time rendered him unfit for a match', and turned in March of the year to Lasker, in order to achieve a postponement of the match for a year. Lasker declined to accord a postponement and declared the previously agreed contract to be invalid.\n\nHis contemporaries already supposed that Tarrasch had been unable to bring together the unusually high stake of 8000 marks in time and had thought up the story of the accident in order to achieve a postponement without loss of face. 'There were probably few chess lovers who were ready to believe that the World Championship was played for with the legs rather than with the head', was the ironic comment about the apparent accident by Milan Vidmar. Lasker pointed to the right that other masters also had to challenge him and explained that in principle he remained open to a fresh challenge from Tarrasch \u2013 though only at the conditions already negotiated.\n\nAfter winning his match against Marshall in Nuremberg in the autumn of 1905 8:1 in wins (with eight draws), Tarrasch again raised the question of a WCh match against Lasker, but without formally challenging the title defender. He wrote: '... that after my latest and perhaps greatest performance I need recognise no one in the world of chess as superior to me. I am prepared to play a match against Lasker under cheap conditions, but I will not challenge him; that is for the person of lesser renown and lesser successes. But for 20 years my successes have been at least equal to his; when I challenged him 2 years ago, that was a faux pas on my part.' Lasker's reaction was of course one of annoyance and in his own _Lasker's Chess Magazine_ he published the following characterisation of Tarrasch:\n\n'Tarrasch's strength, or weakness, if you prefer, is his pronounced amour-propre. Without it he would have become only a very mediocre chess player. But through this unusual gift he has become a giant. His egotism is so great that he had to distinguish himself in some field or other. The game of chess offered him the most suitable field and what he loves in chess is principally only his own games of chess. He has written two chess books and is now writing a third, all about himself alone, his wins, his career. His writing is very witty and entertaining, but his naive self-worship often clouds his judgement of people and things, yes even of chess positions. In the whole world there is no game played by anyone \u2013 apart from Dr. Tarrasch himself \u2013 in which he cannot find a mistake or a quicker way to win or an improvement.'\n\nAt the DSB Congress in Nuremberg in 1906 Tarrasch could only reach the middle of the field, whilst Marshall won the tournament in superior style. This was not the only reason why Tarrasch had such a bad memory of the 1906 Nuremberg tournament. The well-known chess journalist Leopold Hoffer had suggested a change in the rules as far as overstepping the time limit. Hoffer was probably one of the most influential chess journalists in the whole history of chess. Together with Zukertort he was the publisher of _Chess Monthly_ (1879-1896) and moreover for many years the author of the chess column in the magazine _The Field_. Hoffer suggested that overstepping the usual thinking time (one hour for 15 moves) should not mean as usual the loss of the game but should attract a 'more lenient' punishment, specifically a fine. The suggestion was adopted and it was fixed that a player who exceeded the established time limit had to pay a fine of a mark for every minute over. In his game against Salwe the application of this rule punished Tarrasch in two ways: first he lost the game in an annoying way and then he also had to pay 95 marks for having gone over the time limit.\n\nIn spite of the propaganda war the negotiations between Tarrasch and Lasker as to a match for the World Championship were resumed, though not directly but via the president of the German Chess Federation, Dr. Rudolf Gebhardt. After some tough to-ing and fro-ing the match between Lasker and Tarrasch finally took place in 1908. Tarrasch was however, at this point in time already 46 years old and had passed his zenith.\n\nLasker demanded for the WCh match first of all an honorarium of 15 000 marks, but then contented himself with 7500 marks (according to today's purchasing power that is the equivalent of approximately 35000 to 40000 euros). Tarrasch did not insist on an honorarium so as to enable the match to actually take place. Moreover a prize fund of 6500 marks was made available by the German Chess Federation, of which 4000 marks would go to the winner, 2500 to the loser. The venues were D\u00fcsseldorf and Munich, where the greatest share of the funds had been raised (500 marks in D\u00fcsseldorf, 6000 marks in Munich) for the match with the help of patrons. Before the match both players had to come up with a deposit of 2000 marks, which in the case of the non-appearance of one player would be paid to his opponent.\n\nThe WCh match began on the 17th August 1908 in D\u00fcsseldorf and was continued on the 1st September 1908 in Munich. The venue in the metropolis on the Rhine was the 'Kunstpalast', a large new Renaissance style building of 1902 in the Kaiser Wilhelm Park. The Kunstpalast later received a new facade and is nowadays known as the 'Kunstpalast Museum' (4-5 Ehrenhof) and is part of the D\u00fcsseldorf museum mile. In Munich games five to eight and thirteen to sixteen were played in the museum club in the Palais Portia (12 Kardinal Faulhaber Stra\u00dfe, previously 12 Promenadestra\u00dfe). The palace was in 1731 a present from the Electoral Prince of Bavaria Karl Albrecht to his mistress Gr\u00e4fin Josepha Topor-Morawitzka. The latter later married Prince Porcia (or Portia), after whom the palace was then named. In 1819 the building fell into the possession of the Munich Literary Society, which used it for cultural purposes. In the Second World War the palace was bombed, with only the facade left standing. The house now belongs to the HypoVereinsbank. Because of the large throng of spectators and the lack of space in the Palais Portia after the eighth game there was a temporary change (9th to 12th games) to the Alte Rathaus on the Marienplatz. Thinking time was one hour for every 15 moves. No more than six hours were played on any one day.\n\nFor the first time in the history of the World Chess Championships a player was supported by masters acting as his seconds. Lasker engaged as his helpers in the preparation for the match the Austrian Heinrich Wolf and the Russian Simon Alapin. The latter also gave on the 27th August 1908 in the restaurant of the Kunstpalast a blindfold simultaneous exhibition for the entertainment of the spectators. In addition Lasker was also supported locally by his brother Berthold. Berthold Lasker also worked the demonstration board in a neighbouring room together with Rudolf Spielmann. Jacques Mieses, who was commenting on the games, also sat there. Tarrasch was supported in Munich by his son Fritz Tarrasch. But Tarrasch and Lasker had in addition agreed not to make use of outside help in analysing adjourned games.\n\nThe match took on a really one-sided character. Lasker won the first two games. Tarrasch was able to shorten the lead in the third game, but Lasker then decided the fourth game in his favour. After that the players moved to Munich. Tarrasch blamed his arrears of 1:3 on the unfavourable 'maritime climate' in the Rhineland. In other tournaments, however, Tarrasch had been a notoriously late starter. After the change of venue Lasker also won the fifth game. Next came a draw, then the seventh game went to Lasker, meaning a score of 5:1 in wins. Games eight and nine ended in draws. In the second half of the match Tarrasch held on somewhat better, winning the tenth and twelfth games, but here too Lasker was on top with wins in the eleventh, thirteenth and sixteenth games. It finished as a clear 8:3 for Lasker with five draws.\n\nThe match for the World Chess Championship between the two German world-class players was well supported by the German public. For the 12th game in the Munich town hall more than 1000 spectators were present. In the Palais Portia many of the tournament days saw as many as 2400. Lasker speaks in his tournament book of a total of 30 000 spectators for all the games.\n\n **Tarrasch \u2013 Lasker**\n\nMunich, 16th game  \n30th September 1908  \nFour Knights Game (C49)\n\n**1.e4 e5 2. \u2658f3 \u2658c6 3.\u2657b5 \u2658f6**\n\n3...a6 later became very popular.\n\n**4. \u2658c3**\n\nAfter 4.0-0 the so-called 'Berlin Defence' is very popular: 4...\u2658xe4 5.d4 \u2658d6 6.\u2657xc6 dxc6 7.dxe5 \u2658f5 8.\u2655xd8+ \u2654xd8.\n\n**4... \u2657b4**\n\nNow a variation has arisen by transposition of moves which is listed under the Four Knights Game.\n\n**5.0-0 0-0 6.d3 d6 7. \u2657g5**\n\nHere 7.\u2658e2 is also played.\n\n**7... \u2657e6**\n\nIn his notes to the game Lasker considered this to be a novelty, but this is not quite correct. Before this game, the move had actually been seen only a few times. However, 7...\u2657xc3 8.bxc3 and then for example: 8...\u2655e7 9.\u2656e1 \u2658d8 10.d4 \u2658e6.\n\n**8.d4**\n\nIn a previous game obviously not known to Lasker there followed 8.\u2658d5 \u2657xd5 9.exd5 \u2658e7, Janowski-Caro, Vienna 1898.\n\n**8...exd4 9. \u2658xd4 h6 10.\u2657h4**\n\n'Tarrasch scorns playing to win a pawn with 10.\u2657xf6 \u2655xf6 11.\u2658xc6 bxc6 12.\u2657xc6 \u2656ab8 13.\u2655d3 \u2657xc3, since the white pawns on the queenside would have been weak on account of 14.bxc3 \u2656b2 and the doubling of the rooks on the b-file.' (Lasker)\n\n**10... \u2658e5**\n\n'I could also simply have continued with 10...\u2657xc3 11.bxc3 g5 12.\u2657g3 \u2658xe4 13.\u2658xc6 bxc6 14.\u2657xc6 \u2658xg3 15.fxg3 \u2656b8, but the complications after the text move seemed to me very promising.' (Lasker)\n\n**11.f4**\n\n'This move was necessary because Black was threatening to obtain strong play with...\u2658g6.' (Lasker). The tame 11.\u2658d5 \u2657xd5 12.exd5 \u2657c5 13.c3 \u2658g6 14.\u2657xf6 \u2655xf6 15.\u2658f3 leads to a level game.\n\n**11... \u2657c5**\n\n'What arises is a difficult position. Now 11...\u2658g6 would have been followed by 12.\u2657xf6.' (Lasker): 12...\u2655xf6 13.f5 \u2657c5 14.\u2658ce2 and White wins a piece.\n\n**12. \u2657xf6?**\n\nAfter this, White gets into difficulties. Lasker commented: 'The knight cannot be taken on account of 12.fxe5 dxe5 13.\u2658ce2 exd4 14.\u2654h1 \u2657e7 with good play for Black. If White immediately plays 12.\u2658ce2 then 12...\u2657g4 13.fxe5 \u2657xe2 14.\u2655xe2 dxe5, which would once again be favourable to Black.' It was, however, well worth considering 12.\u2654h1!?. After 12...\u2658g6 13.\u2658xe6 fxe6 14.\u2657g3 c6 15.\u2657d3 White has slightly better prospects in view of the bishop pair.\n\n**12... \u2655xf6 13.fxe5?**\n\nIt was better to relieve the pin on the knight with 13.\u2654h1. After 13...\u2658g4!? 14.\u2658xe6 \u2655h4 Black has an attack.\n\n**13... \u2655xe5 14.\u2658ce2 \u2657g4 15.\u2656f3**\n\n'The best move. 15.\u2656f5 would have failed to 15...\u2657xe2.' (Lasker)\n\n**15... \u2657xf3**\n\n'15...\u2655xe4 would also have been strong since 16.\u2656f4 \u2655xf4 17.\u2658xf4 \u2657xd1 18.\u2656xd1 allows Black the slight superiority of rook and two pawns against bishop and knight.' (Lasker)\n\n**16.gxf3 f5**\n\nAnother good move was 16...d5!? 17.exd5 \u2655xd5 18.c3 (18.\u2654g2 \u2656ad8\u2013+) 18...c6 19.\u2657a4 \u2655xf3 with a strong position for Black.\n\n**17. \u2655d3**\n\n'The only good move. After 17.exf5 \u2656xf5 18.\u2654h1 \u2656h5 19.\u2655g1 \u2656f8 20.\u2656f1 \u2657xd4 21.\u2658xd4 \u2656f4 22.c3 \u2656xd4 23.cxd4 \u2655xb5 Black would have had a winning game.' (Lasker)\n\n**17...c6 18. \u2657c4+ \u2654h8**\n\n**19. \u2654h1?**\n\nA blunder, which, however, remained unnoticed. A better try was 19.exf5 d5 20.\u2657b3 \u2656ae8 21.\u2654h1 with only a slight advantage to Black.\n\n**19...b5?**\n\n'The only manoeuvre which promised success. The bishop must be forced away from the protection of the e4-point.' (Lasker) In his comment Lasker overlooked the win of a piece by 19...fxe4 20.\u2655xe4 (20.fxe4 \u2656f2 21.\u2658f3 \u2655h5 22.\u2658f4 \u2655g4\u2013+) 20... \u2657xd4 21.\u2655xd4 (21.\u2658xd4 \u2656f4) 21...\u2655xd4 22.\u2658xd4 \u2656f4 23.c3 c5\u2013+.\n\n**20. \u2657b3 fxe4 21.\u2655xe4 \u2655xe4**\n\n'21...\u2656ae8 22.\u2655xe5 \u2656xe5 23.c3 \u2656fe8 would also have been strong.' (Lasker)\n\n**22.fxe4 \u2656ae8 23.\u2658xc6**\n\n'If White had played 23.\u2658e6, then 23... \u2656f6 would have followed. After 24.\u2658xc5 then 24...dxc5 threatens c4.' (Lasker). After 24.\u26582d4 \u2657xd4 (or 24...\u2657b6 25.a4 bxa4 26.\u2657xa4 \u2656fxe6 27.\u2658xe6 \u2656xe6 28.\u2657xc6=) 25.\u2658xd4 \u2656xe4 26.\u2658xc6 there is, however, hardly any trace of an advantage for Black to be seen.\n\n**23... \u2656xe4 24.\u2658g3 \u2656ee8 25.\u2656d1**\n\n'I consider this move to be a mistake. It would have been better to play 25.a4.\u2656af1 is forced sooner or later in view of the threat...g6,...h5,...h4 and a doubling of the rooks on the second rank. So \u2656ad1 is the waste of a tempo.' (Lasker)\n\n**25... \u2656f2**\n\n'This threatens...h6-h5-h4,...\u26568e2' (Lasker).\n\n**26. \u2658d4?**\n\n'An unbelievable mistake which spoils the otherwise good impression the game makes. White's situation is very difficult, since after 26.\u2656f1 Black could continue his attack with either 26...\u2656d2 or gain a pawn and a strong position with 26...\u2656xf1+ 27.\u2658xf1 \u2656e1 28.\u2654g2 \u2656b1 29.\u2658d2 \u2656xb2 \u2013 threatening...a7-a5. 26.a4 was probably the only correct move. We were both in time trouble and uncommonly exhausted.' (Lasker)\n\n**26... \u2657xd4**\n\nWhite resigned in view of 27.\u2656xd4 \u2656e1+ and then mate.\n\nThe relationship between Lasker and Tarrasch was tense from the start and did not improve with the passing of time either \u2013 on the contrary. At the opening ceremony for the World Championship in 1908, according to Tarrasch biographer Wolfgang Kamm, at a specially scheduled 'reconciliation meeting', Tarrasch on catching sight of Lasker literally clicked his heels with the words: 'For you, Herr Lasker, I have only three words \u2013 check and mate!'\n\nBut after the end of the match Tarrasch became somewhat more conciliatory. He allowed himself to be photographed with Lasker, complaining however with something of a wink but with real background reason about Lasker's clouds of smoke during the games: 'Lasker's cigars are gas producing, they remove the oxygen, so how could I win... it was not his play which did it for me, but his cigars.' Moreover during the games in D\u00fcsseldorf Lasker had been warned, because he had been pointedly rustling newspapers when it was his opponent's move.\n\nTwo years after the WCh match against Lasker, at the great 17th DSB Congress of 1910 in Hamburg, Tarrasch was the talk of the tournament because he spoke against the inclusion of the English player Frederick Yates in the A-tournament, because according to Tarrasch the latter had not the slightest playing credentials to show for being included in such a strong tournament. And in fact Yates came in a distant last. He only won a single game \u2013 against Tarrasch.\n\nTarrasch's private life was overshadowed by a series of blows of fate. In 1896 his four year old daughter Hedwig died of meningitis. In 1911 his marriage went on the rocks and in their common flat he even had a dividing wall built. In the following year his wife Rosa moved to Bamberg with their 15 year old daughter Eva, in 1918 to their daughter Grete in Regensburg. In 1924 their marriage was dissolved. Tarrasch himself moved in 1914 to Munich (22 Rheinstra\u00dfe, on the first floor on the right). Tarrasch's youngest son Paul committed suicide at the age of twenty in Hamburg from lovesickness. He died on the 9th September 1912 in the Eppendorf hospital. The oldest son Friedrich Max Tarrasch was fatally wounded on 14th May 1915 at Le Veaux F\u00e9ry, near Verdun. Tarrasch's second oldest son Hans Richard died as a result of an accident in 1916. He was run over by a tramcar in Munich. This was also perhaps a suicide, because Tarrasch's grandson Rudolf Gall reported that Hans Richard Tarrasch had 'thrown himself in front of the tramcar.' On the 4th July 1918 Tarrasch's son-in-law Hanns Bolz, the husband of his daughter Grete, also died from the consequences of a war wound. At the age of 62, Tarrasch married for a second time, the 30 years younger Gertrude Schr\u00f6der. In 1933 the couple separated after nine years of marriage.\n\nWith the coming to power of the Nazis Tarrasch was also exposed to increasing enmity on account of his Jewish ancestry. On the 17th February 1934 Siegbert Tarrasch died in the hospital in Schwabing from pneumonia. On the 19th February he was buried in Munich's Nordfriedhof (plot 128).\n\nTarrasch had a great influence on the game of chess as a theoretician. His three books, _Three hundred chess games_ (1895), _Die moderne Schachpartie_ (1912) and _The game of chess_ (1931) achieved widespread circulation. His manual 'The game of chess' is still equally relevant until this day and is the chess book with the greatest number of printings in Germany. His insight into the game of chess was: 'Like music, like love, chess has the power to make men happy.'\n**9. The fateful tenth game**\n\n**The World Championship 1910:  \n_Emanuel Lasker against Carl Schlechter_**\n\nCarl Schlechter was born into a Catholic family in Vienna on 2nd March 1874. He was the only child of Adalbert and Marie Schlechter, n\u00e9e Rieger. He learned chess at the age of 13. His chess teacher was the Hungarian doctor, journalist and problem composer Dr. Samuel Gold who lived in Vienna. Schlechter also later for his whole life composed two and three movers. In 1890 for the first time the _Deutsches Wochen-schach_ published a two mover by Schlechter.\n\nAfter school Schlechter completed an apprenticeship as a typesetter before going to the commercial college. After completing his education he worked temporarily in a trading house before he began to occupy himself exclusively with chess.\n\n**Carl Schlechter (1874-1918)**\n\nAfter 1883 Schlechter took part in over 50 tournaments and won, sometimes jointly, 1900 in Munich (12th DSB Congress), 1904 in Coburg (14th DSB Congress), 1906 in Ostende, 1906 in Stockholm, 1908 in Vienna, 1908 in Prague and 1910 in Hamburg (17th DSB Congress). He won the Trebitsch Memorial in Vienna three times (1910, 1911, 1912). Together with Arthur Kaufmann and Hugo F\u00e4hndrich, Schlechter continued the Vienna chess school founded by Miksa White and earned his living as a professional chess player. But he was also fascinated by the game of dominoes, which he often played with F\u00e4hndrich in the New Vienna chess club.\n\nDuring his career Schlechter also contested a number of matches, including ones against Georg Marco, Dawid Janowski and Siegbert Tarrasch, of which most ended in draws. Schlechter's drawing ratio of 50% was unusually high for the age. He had the reputation of an extremely fair player, who never exploited the ailments of his opponent to his own advantage. In the novel _Carl Haffner's love of the draw_ (1998) Thomas Glavinic has presented Schlechter's life in literary form. The statistician Jeff Sonas puts Carl Schlechter as number four for the year 1910 in the historical world ranking list he has calculated for that time, behind Emanuel Lasker, Akiba Rubinstein and Jos\u00e9 Raul Capablanca.\n\nFrom 1892 to the end of his life Schlechter ran the chess column in the _Wiener Allgemeine Sportzeitung_. Moreover from 1899 he was the publisher and editor of the _Deutsche Schachzeitung_. Between 1912 and 1916 Schlechter edited the eighth edition of the _Handbuch des Schachspiels_ by Bilguer with its 1000 pages and during that time he no longer played in tournaments.\n\nAfter his victory in the tournament in Ostende in 1906 Schlechter challenged Lasker for the first time to a WCh match. However, at this time Lasker already had in front of him challenges from Marshall and Tarrasch, which he first 'disposed of' with his matches in 1907 against Marshall and in 1908 against Tarrasch. After Lasker's match against Tarrasch, Schlechter repeated his challenge to Lasker and the World Champion now accepted it. In December 1908 Schlechter travelled to Berlin to discuss the details with Lasker. The date which they had in mind for the match was the end of 1909.\n\nLasker first suggested the following conditions: the match should comprise 30 games. The winner should have a lead of two points. In the event of a tie, there should be a tie-break. The stake was to be the same as for Lasker's match against Marshall, i.e. 1000 dollars for the winner. Lasker wanted to take care of the organisation and claimed in return for that sole copyright of the games. Schlechter agreed with everything.\n\nBut obtaining sponsors for this match proved more difficult than had been thought and thus in 1909 Lasker issued an open appeal 'to the world of chess', which was published in various chess magazines \u2013 however without success. Negotiations with chess lovers in St. Petersburg who were interested in staging the final part of the match also came to no result, although the players had reduced their appearance money from 1000 to 800 marks pro game.\n\nAfter this experience Lasker reduced the length of the match to 15 games and wanted to stage it in three parts in Vienna, Berlin and London. The arbiter was to decide whether a lead of one or two points was required for victory. Since Lasker could not find an organiser in London, the match was finally even shortened to ten games and the venues limited to Vienna and Berlin. The idea that the winner had to have a lead of one or two points was also abandoned. It was instead decided: 'Whoever wins the majority of games is victorious and wins the title of World Champion. In the event of a draw, the arbiter has to make the decision about who has the title.'\n\nInstead the match finally took place from the 7th January to the 10th February 1910, half in Vienna, where 3000 crowns were available for the prize fund, and half in Berlin, where 2000 marks had been collected. After four draws Schlechter went into the lead with a win in the fifth game. Thereafter they moved to Berlin. There the match was played in the 'Grand Hotel de Rome' (Unter den Linden). Not till the tenth and last game could Lasker equalise and he retained the title with a score of 5:5.\n\nSince, despite being in the lead, Schlechter had allegedly played riskily for a win in the last game, there was speculation about what might have been the reason for that. Many authors, such as Garry Kasparov, took the opinion that according to the match conditions Schlechter was obliged to have two points of a lead in order to decide the title match in his favour. On the other hand, there is a statement by Lasker, who two days before the tenth game was quoted in the New York _Evening Post_ as follows: 'The match with Schlechter is nearing its end and it appears probable that for the first time in my life I shall be the loser. If that should happen, a good man will have won the World Championship.'\n\nEdward Winter even surmises that Schlechter lost on purpose in order to avoid the return match which had been agreed and for which he would not have had the financial means. Perhaps it was Lasker who previously gave the reasons for Schlechter's fear of victory. In his _Lasker's Chess Magazine_ three years before he had written about Schlechter: 'The Austrian Schlechter has the ability that would enable him to compete... but he has only the ability \u2014 and nothing more. He is a man who likes a quiet life and has so little of the devil about him that he could not be wooed to take anything coveted by someone else.'\n\nThe tenth game which was so decisive for the outcome of the match has been analysed by generations of chess lovers, not always objectively and absolutely accused of mistakes. The contention that Schlechter played riskily for a win, can hardly be supported by the course of the game. Moreover, the game is extremely complex and was incapable of being fully explained without appropriate present day computer support. A detailed analysis, taking previous comments into consideration, was given by Robert H\u00fcbner in 1999. The following version of the game contains a summary of previous comments:\n\n **Lasker \u2013 Schlechter**\n\nBerlin, 10th game  \n8th February 1910  \nQueen's Gambit, Slav Defence (D94)\n\n**1.d4 d5 2.c4 c6 3. \u2658f3 \u2658f6 4.e3**\n\nIn 1910 this plan had not been much researched. Today's version of the Queen's Gambit, Slav Defence is reached after 4.\u2658c3 dxc4 5.a4 \u2657f5, with this plan, however, first becoming fashionable in the 1920s.\n\n**4...g6**\n\nOther options are 4...\u2657g4 or 4...\u2657f5. Today's popular Meran Variation is reached after 4...e6 5.\u2658c3 \u2658bd7 6.\u2657d3 dxc4 7.\u2657xc4 b5. In 1910 this plan, however, had not yet been 'discovered'.\n\n**5. \u2658c3 \u2657g7 6.\u2657d3 0-0 7.\u2655c2**\n\n'7.0-0 and 7.\u2655b3 are usual. In fact Black can now equalise effortlessly.' (H\u00fcbner)\n\n**7... \u2658a6**\n\n'In my view the strongest continuation' (H\u00fcbner). The alternatives were 7...\u2658bd7 (Schlechter) or 7...dxc4 8.\u2657xc4 c5 (8...\u2657f5=) 9.dxc5 \u2658bd7= (Euwe).\n\n**8.a3 dxc4**\n\nThe surrender of the centre was criticised by Tarrasch. He recommended 8...\u2658c7 intending...\u2658e8-d6 and...\u2657f5. 'The advance 8...c5 leads to excellent play for Black.' (H\u00fcbner)\n\n**9. \u2657xc4 b5?!**\n\nThis weakens the pawns on the queenside. 9...\u2658c7!? 10.0-0 (10.e4 \u2657g4=) 10...\u2657e6= (Euwe) or 9...c5!? 10.d5 \u2658c7 (H\u00fcbner).\n\n**10. \u2657d3 b4 11.\u2658a4**\n\nAfter 11.\u2657xa6 \u2657xa6 12.axb4 \u2655c8 13.\u2658e2 \u2658d5 14.\u2657d2 \u2656b8 Black takes over the initiative.\n\n**11...bxa3 12.bxa3**\n\nA less good option was 12.\u2655xc6? \u2658b4 13.\u2655c3 (13.\u2655xa8 \u2658xd3+ followed by 14...\u2655c7 and 15...\u2657a6, trapping the queen, is favourable for Black), since after 13...\u2658xd3+ 14.\u2655xd3 axb2 15.\u2657xb2 \u2657f5 Black is better with his bishop pair, lead in development and passed pawn on the queenside.\n\n**12... \u2657b7 13.\u2656b1 \u2655c7 14.\u2658e5!?**\n\nIntending \u2657xa6 and \u2655xc6. However, 14.0-0 (H\u00fcbner) was simpler.\n\n**14... \u2658h5**\n\n**15.g4?!**\n\nBranded a mistake by Tarrasch. But Lasker, who absolutely had to win this game, is aiming for complications. H\u00fcbner recommended instead: 15.0-0 \u2657xe5 16.dxe5 \u2658g7 17.e4 \u2658e6 18.\u2657e3 \u2656fd8 19.\u2655e2 \u2658b8 20.\u2657c4+\u2013, 'Black is lost, with no hope of saving the game.'\n\n**15... \u2657xe5 16.gxh5?!**\n\nThere was the objectively superior 16.dxe5 \u2658g7 17.0-0 and White retains something of an advantage.\n\n**16... \u2657g7 17.hxg6 hxg6 18.\u2655c4**\n\nWith the threats of 19.\u2657xg6 and 19.\u2656xb7 followed by 20.\u2655xa6. Tarrasch recommended instead 18.f4.\n\n**18... \u2657c8?!**\n\nAfter 18...c5 Lasker gives 19.\u2656g1 \u2655xh2 20.\u2656xg6 \u2655h1+ (20...\u2658c7!?, H\u00fcbner, 21.\u2656g3 (21.\u2656xb7!? \u2655h1+ 22.\u2654e2 \u2655xb7 23.dxc5) 21...\u2657a6 22.\u2655c2 \u2657xd3 23.\u2655xd3 cxd4=) 21.\u2654d2 as advantageous for White.\n\n**19. \u2656g1**\n\nSchlechter now wanted to meet 19.\u2657xg6 with 19...\u2657e6, after which the position is complicated and unclear. If 19.\u2657d2, then 19...e5! as recommended by Euwe leads to a level game.\n\n**19... \u2655a5+?!**\n\nAfter 19...\u2655xh2 20.\u2656xg6 the best is 20...\u2658c7, with a complicated position. H\u00fcbner suggested 19...e6!? and then 20.f4 (20.\u2656g3 \u2656b8) 20...\u2656b8 21.\u2656xb8 \u2658xb8 22.\u2658c5 \u2655e7 with counterplay.\n\n**20. \u2657d2 \u2655d5 21.\u2656c1 \u2657b7 22.\u2655c2**\n\nIntending 23.\u2656xg6 fxg6 24.\u2657c4+\u2013 and 23.\u2657e4 (Lasker, Schlechter). H\u00fcbner recommended instead 22.h4!? or 22.\u2658c3!? with chances of a win for White.\n\n**22... \u2655h5**\n\n**23. \u2657xg6?**\n\nWins a pawn but allows Black counterplay. Lasker indicated 23.\u2655b3 as better: 23...\u2656ab8 24.\u2656xg6 c5 25.\u2656g3! etc.\n\n**23... \u2655xh2**\n\n23...fxg6 is followed by 24.\u2655b3+ \u2654h7 25.\u2655xb7; after 25...\u2655h4 26.\u2656f1 \u2656xf2 27.\u2656xf2 \u2656f8 28.\u2655xa6 \u2656xf2 29.\u2655d3 e5 White is a piece up, but Black still has some counterplay against the white king in the centre.\n\n**24. \u2656f1 fxg6 25.\u2655b3+ \u2656f7 26.\u2655xb7 \u2656af8! 27.\u2655b3**\n\n27.\u2655xa6? \u2656xf2 28.\u2656xf2 \u2656xf2 and there is the threat of mate with 29...\u2655g1#. After 27.f4 there follows 27...\u2655h4+ and Black can give perpetual check: 28.\u2654d1 \u2655g4+ 29.\u2654c2 \u2655f5+ 30.\u2654d1, since the white king cannot go on to the b-file on account of...\u2656b8.\n\n**27... \u2654h8 28.f4**\n\n**28...g5!?**\n\nAfter 28...\u2655g3+ 29.\u2654d1 \u2656xf4 30.\u2656xf4 \u2656xf4 Lasker wanted to continue with 31.\u2654c2.\n\nIt was also worth considering 28...e5, then: 29.dxe5 \u2656d7 30.\u2655b2 \u2655g3+ and Black has enough play to hold the draw.\n\n**29. \u2655d3**\n\n29.\u2656xc6 \u2655g3+ 30.\u2654d1 (30.\u2654e2? \u2655g2+) 30...\u2656xf4 31.\u2656h1+ \u2656h4 tends to be favourable for Black, since White cannot allow 32.\u2656xh4+ gxh4 33.\u2656xa6 \u2655f3+ 34.\u2654c2 h3. The passed pawn cannot be stopped.\n\n**29...gxf4**\n\n29...\u2655g3+ 30.\u2654d1 \u2656xf4 31.\u2654c2 \u2656xf1 32.\u2656xf1 \u2656xf1 33.\u2655xf1 \u2658c7 34.\u2655h1+ \u2654g8 35.\u2655xc6 \u2655d6 leads to a level endgame.\n\n**30.exf4**\n\n30.\u2655xa6? fails to 30...fxe3\u2013+.\n\n**30... \u2655h4+**\n\nAlso playable was 30...\u2658c7 31.\u2655f3 (31.\u2656xc6 \u2658d5 32.\u2655e4 \u2658xf4=) 31...\u2655h4+ 32.\u2654d1 \u2654g8 33.\u2656xc6 \u2658b5 34.\u2656h1 and after 34...\u2655g5! Black holds on.\n\n**31. \u2654e2**\n\n31.\u2654d1 \u2655g4+ 32.\u2654c2 \u2655f5=.\n\n**31... \u2655h2+ 32.\u2656f2 \u2655h5+ 33.\u2656f3**\n\n33.\u2654e3? \u2656xf4! (Lasker).\n\n**33... \u2658c7 34.\u2656xc6 \u2658b5**\n\nTarrasch considered Black to have an advantage, but this is too pessimistic a way of looking at things. In reality the chances are level. 34...\u2658d5 was also playable.\n\n**35. \u2656c4**\n\n35.\u2656c5 fails to 35...\u2658xd4+ 36.\u2655xd4 \u2655xf3+ 37.\u2654xf3 \u2657xd4 with advantage to Black. According to H\u00fcbner 35.\u2654e1 \u2655h1+ 36.\u2655f1 \u2655xf1+ 37.\u2656xf1 \u2658xd4 38.\u2656c7 leads to a draw.\n\n**35... \u2656xf4**\n\nBlack had other options at his disposal here, for example:\n\na) 35...e5 36.dxe5 \u2657xe5 37.\u2654e1 (Capablanca);\n\nb) 35...\u2658d6 36.\u2656c5 \u2658f5 (H\u00fcbner) or\n\nc) 35...\u2656d8 36.\u2654e1! \u2655h1+ 37.\u2655f1 (Minev), in each case with equality.\n\n**36. \u2657xf4 \u2656xf4 37.\u2656c8+ \u2657f8 38.\u2654f2**\n\nH\u00fcbner suggested 38.\u2656d8 as an improvement for White, since after the text move Black can force a draw.\n\n**38... \u2655h2+**\n\nInstead, 38...\u2655h4+ 39.\u2654g2 (but not 39.\u2654e2? \u2658xd4+ \u2013+) 39...\u2655g4+ 40.\u2656g3! \u2655xc8 41.\u2655g6!+\u2013 (Tarrasch, Schlechter).\n\n**39. \u2654e1**\n\n**39... \u2655h1+? +\u2013**\n\nBlack could instead force a draw here with 39...\u2655h4+ 40.\u2654d2! \u2655h2+ 41.\u2654e1 \u2655h4+ and perpetual check. And a draw in this last game would have been enough for Schlechter to win the match and become World Champion. If White plays anything other than 40.\u2654d2, he will be at a disadvantage:\n\na) 40.\u2656g3? \u2655h1+ 41.\u2654d2 \u2656f2+ \u2013+;\n\nb) 40.\u2654d1? \u2655h1+ 41.\u2654e2 \u2656xf3 42.\u2655xf3 \u2658xd4+ \u2013+;\n\nc) 40.\u2654f1 \u2655h3+ 41.\u2654f2 \u2656xf3+ 42.\u2655xf3 \u2655xc8 43.\u2655h5+ \u2654g8 44.\u2655xb5 \u2657g7 with advantage to Black.\n\n**40. \u2656f1 \u2655h4+ 41.\u2654d2 \u2656xf1**\n\n41...\u2656xd4? appears to win the queen, but White gets in first: 42.\u2656cxf8+ \u2654g7 43.\u26561f7+ \u2654h6 44.\u2656h8+ \u2654g5 45.\u2656g8+ \u2654h6 46.\u2656h7# (Blackburne).\n\n**42. \u2655xf1 \u2655xd4+ 43.\u2655d3 \u2655f2+**\n\nAfter the exchange of queens 43...\u2655xd3+ 44.\u2654xd3 \u2654g7 45.\u2658c5+\u2013 (Barcza) White wins with his exchange up.\n\n**44. \u2654d1 \u2658d6**\n\nIn the event of 44...\u2655g1+ 45.\u2654c2 \u2658d4+ 46.\u2654b2 \u2655g2+ 47.\u2654a1 \u2655h1+ 48.\u2654a2 \u2655g2+ White ends the series of checks with 49.\u2658b2+\u2013 (Lasker).\n\n**45. \u2656c5?! \u2657h6 46.\u2656d5 \u2654g8**\n\n46...\u2655a2 was suggested by Schlechter. But there then follows 47.\u2656h5 \u2655a1+ 48.\u2654c2 \u2655a2+ 49.\u2658b2 \u2655e6 50.\u2655d4+ \u2654h7 51.\u2656e5 and 'White is winning'. (H\u00fcbner)\n\n**47. \u2658c5 \u2655g1+?!**\n\n'This drives the white king on to better squares.' (H\u00fcbner) The waiting move 47...\u2655g2 was better.\n\n**48. \u2654c2**\n\n**48... \u2655c1+?!**\n\n48...\u2655f2+ 49.\u2654b3 \u2657g7 50.\u2656g5 \u2655f7+ 51.\u2655d5 \u2655xd5+ 52.\u2656xd5 and 'In my opinion White will win'. (H\u00fcbner)\n\n**49. \u2654b3 \u2657g7 50.\u2658e6 \u2655b2+ 51.\u2654a4 \u2654f7 52.\u2658xg7 \u2655xg7 53.\u2655b3 \u2654e8 54.\u2655b8+ \u2654f7 55.\u2655xa7**\n\nAn exchange ahead and with the passed pawn on the a-file, winning the game is now easy for White.\n\n**55... \u2655g4+ 56.\u2655d4 \u2655d7+ 57.\u2654b3 \u2655b7+ 58.\u2654a2 \u2655c6 59.\u2655d3 \u2654e6 60.\u2656g5 \u2654d7 61.\u2656e5 \u2655g2+ 62.\u2656e2 \u2655g4 63.\u2656d2 \u2655a4 64.\u2655f5+ \u2654c7 65.\u2655c2+ \u2655xc2+ 66.\u2656xc2+ \u2654b7 67.\u2656e2 \u2658c8 68.\u2654b3 \u2654c6 69.\u2656c2+ \u2654b7 70.\u2654b4 \u2658a7 71.\u2654c5**\n\nAnd Black resigned.\n\nCarl Schlechter died only a few years after the match against Lasker. In the summer of 1914 the First World War had broken out and in Austria-Hungary, just as in the German empire, the food supply situation had continued to deteriorate as a result of the British continental blockade since 1916. In January 1918 the bread ration for normal citizens (workers in heavy industry received more) was reduced from 1260 to 630 grammes per week \u2013 too little for survival. Diseases stemming from malnutrition such as tuberculosis were the consequence. Schlechter was living on an estate close to Baden and did not understand, as others did, how to get help in the increasing distress. Whereas many a chess player, such as Albin or Marco, practically lived in his chess club and somehow managed to get by, Schlechter did not complain but starved in silence.\n\nUnlike in Vienna the supply situation in Budapest was still reasonably good and so chess lovers invited Schlechter to a tournament in the Hungarian capital, though, however, he was only able to finish in last place as a result of his poor health. On the 23rd December Schlechter wanted to leave Budapest to celebrate Christmas with his mother in Vienna. He was, however, robbed at the Budapest station and had to return to the chess club in order to borrow money. On the 26th December he set out again but collapsed while still in Budapest. He was taken to a hospital and died there that same night, on the 27th December, possibly from pneumonia exacerbated by his debilitated state. \n**10. Either Lasker won or Janowski lost**\n\n**The World Championship 1910:  \n_Emanuel Lasker against Dawid Janowski_**\n\nDawid Janowski was born on the 25th Mai 1868 in Wolkowysk, near Grodno, in Russian Poland. The place now belongs to Belarus. The family later moved to Lodz and finally at the beginning of the 1880s to Warsaw. In Warsaw, Dawid Janowski, together with his brother, went to the local chess club. In 1886 he is supposed to have moved first to Paris, but after that also to have lived for a time in Berlin and New York.\n\n**Dawid Janowski (1868-1927)**\n\nBut from approximately 1890 Janowski made his name as a strong player in the famous Caf\u00e9 de la R\u00e9gence in Paris. Janowski favoured a sharp attacking style and, playing like this, took first place or shared first place in the tournaments of Monte Carlo 1901, Hanover 1902, Vienna 1902 and Barmen 1905. With his quick and aggressive style of play he was particularly successful against the older players such as Steinitz, Chigorin or Blackburne. Against the younger ones like Tarrasch or Rubinstein, however, he had negative scores. His results against Lasker and Capablanca were even pretty catastrophic. Nevertheless, Capablanca admired Janowski's attacking skills. In the endgame, however, he had particularly great weaknesses. Whenever Janowski lost he always found excuses for it. So he was not particularly popular with other players, because he would play on for a long time in completely lost positions in the hope that on the way to victory his opponent would perhaps still blunder.\n\nJanowski was also a gambler away from the chess board. He lost many a prize at roulette in the casino. Thus in 1901 at the tournament in Monte Carlo he won 5000 francs as victor, but he then promptly lost the whole sum at roulette in the casino. The casino generously paid for his return ticket to his home in Paris. At the tournament at the same place in the following year Janowski was third, but once again gambled this prize money away at the roulette table. Janowski's passion for gambling was very well known to his colleagues, of course. They often even gave him money to bet on their behalf. Janowski often even won at the start, but like all gambling addicts he never managed to stop at the right moment but kept on playing until all the money had been lost.\n\nAfter the London tournament of 1899 \u2013 with 15 players in an enormous hall in the Royal Westminster Aquarium \u2013 in which Janowski finished joint second on the same number of points as Maroczy and Pillsbury behind Lasker, he issued a first challenge to Lasker. Lasker accepted in principle and sent Janowski his conditions: a stake of 400 pounds sterling (10 000 Swiss francs), the match should start between the 15th January and March 1900 and be decided by eight wins. Janowski, however, wanted to start in February 1900 at the earliest and set the match to ten wins. Lasker insisted, however, on his conditions without any change and so the negotiations failed. But then with the help of his sponsor, the millionaire Leo Nardus, Janowski played between 1909 and 1910 a total of three matches against Lasker, one of which as a match for the World Championship.\n\nLeonardus Nardus, born 1868 in Utrecht as the son of the antiquities dealer Emanuel Salomon and his wife Alida Ballen, was an artist, art dealer, collector and world traveller. In 1889 he went to Argentina looking for gold. In 1894 he then settled in the USA as an art dealer. Around the turn of the century Nardus returned to Europe and then lived in the majority in France. In the meantime Nardus had himself developed some skills as a painter. He undertook numerous journeys to Spain, Egypt, Algeria and Tunisia. He later settled in Tunisia. Nardus became active as a sponsor for Dawid Janowski, but was also a close friend of Frank Marshall, whom he had got to know in the USA. Nardus never became active as a chess player, at least none of his games have been handed down, but he was very successful in another sport. In the Summer Olympic Games of 1912 Nardus won a bronze medal for fencing in the Dutch team.\n\nBefore Nardus gave the money for a WCh match, a test match was agreed between Lasker and Janowski. This took place in 1909 between the 12th and 21st May in Nardus' villa in Suresnes near Paris, but was played over only four games and ended 2:2. Nardus provided 2500 francs, of which 750 francs each were paid as appearance money and 1000 francs were to go to the winner. After the draw the prize money was finally shared. There followed in the same year a match of ten games, for which Nardus made 6000 francs available and the 'Hotel Regina and Excelsior' 1000 francs. Here too appearance money was paid for the individual games. The prize for the winner consisted of 2000 francs. The match was played from 18th October till 10th November in the 'Grand Cercle' on the Boulevard Montmartre. Lasker was a clear winner with 7:1 in won games.\n\nIn some sources the second match was regarded as a WCh match, but the chess historian Edward Winter argued that this match was not described in any of the contemporary chess magazines as a match for the World Championship. In the New York _Evening Post_ of the 23rd October it was even specifically pointed out that the match between Lasker and Janowski was not a match for the World Championship but that it served as preparation for the forthcoming WCh match between Lasker and Schlechter. One of the reasons for the heavy defeat was moreover that Janowski kept on playing for a win in some clearly drawn positions \u2013 and then even lost them.\n\nThere was then a further match, this time really for the title of World Champion. Nardus put up 5000 francs for the prize fund. The Berliner Schachgesellschaft was enlisted as a co-sponsor. This third match between Lasker and Janowski was played out to eight wins from the 8th November till the 8th December 1910. The first ten games were to be played in Berlin, in the newly erected three-storey 'Kerkau Palast' (48 Behrenstra\u00dfe). The Kerkau Palace was named after its owner, the professional Billiards World Champion Hugo Kerkau, and had no fewer than 50 billiard tables on offer. Chess was, however, also well thought of at Kerkau's. The subsequent games were to be played in Paris.\n\nAt the official start of the match on the 5th November, it became apparent that Lasker had agreed different playing times with Janowski than he had with the Berlin organiser. Lasker and Janowski had agreed on a playing time from 14.00 to 20.00. The Berlin organisers had reckoned on play being from 16.30-19.30 and then from 21.00 to 23.00. But they finally compromised on a playing session of 16.00 to 20.00. The first game was played on the 8th November 1910. Lasker won the first, the fourth, the fifth, and then also the seventh to eleventh games. The chess publicist Georg Marco commented ironically on the course of the match with the words 'Either Lasker won or Janowski lost'. Since the score after the tenth game was already 7:0, they did without the move to Paris and finished the match in Berlin.\n\nThere were very few reports in the press about the match, since Lasker had claimed the right to the notation of the games for himself and royalties for printing them. Marco openly criticised the World Champion for this: 'With his demands Dr. Lasker has almost completely eliminated the participation of the public at large. He has condemned to silence those people who would have praised and admired his art and challenged others to contradict him.'\n\nAfter the end of the match Janowski did not seem very affected by his heavy loss and was already speaking of challenging Lasker again. Janowski was even convinced that he had outplayed Lasker in almost all the games and that the World Champion could actually not play chess. The way he moved around with the pieces had disgusted him. Thus it had been enough for Janowski to reach a winning position and after that he had lost interest in the games. Speaking to Edward Lasker Janowski gave the following evaluation: 'Your namesake plays such stupid chess that I simply cannot look at the board while he is thinking.' At the start of the match Janowski had discovered in Berlin a gambling club in which he spent his evenings playing roulette during the contest.\n\n **Janowski \u2013 Lasker**\n\nBerlin, 10th game  \n6th December 1910  \nPirc Defence (B07)\n\n**1.d4 d6**\n\nAfter always playing 1...d5 in the previous games Lasker tried something new here.\n\n**2.e4 e5**\n\nNowadays 2...\u2658f6 3.\u2658c3 is usually inserted: 3...e5 4.dxe5 dxe5 5.\u2655xd8+ \u2654xd8 etc.\n\n**3.dxe5**\n\nThe early exchange of queens suits Lasker, since it draws the fangs of the attacking player Janowski. The loss of the right to castle is of hardly any importance here. It is inexplicable that Janowski did not instead maintain the tension with 3.\u2658f3.\n\n**3...dxe5 4. \u2655xd8+ \u2654xd8 5.\u2658f3 \u2657d6**\n\nA less flexible option is 5...\u2658c6, after which a position from the Scotch Game arises, in which White has good practical success. White can continue with 6.\u2657c4 or 6.\u2657b5.\n\n**6. \u2658c3**\n\nAfter 6.\u2657c4 Black has the amazing 6... \u2657e6 at his disposal. The doubled pawns in the centre after 7.\u2657xe6 fxe6 are very useful for Black, since he controls a lot of squares. 8.\u2658g5 \u2654e7 now achieves nothing for White.\n\n**6... \u2657e6 7.\u2657e3 \u2658f6 8.0-0-0**\n\nThreatening \u2658xe5, but Black easily wards it off.\n\n**8... \u2658g4**\n\nAlso worth considering was 8...\u2654e7.\n\n**9. \u2657g5+**\n\nWhite could also allow the bishop on e3 to be taken: 9.\u2657e2 \u2658d7 10.h3 \u2658xe3 11.fxe3=.\n\n**9...f6**\n\nOr 9...\u2654c8!? 10.\u2657h4 \u2657c5 11.\u2656d2 f6 12.h3 \u2658h6 13.g4 \u2658f7=.\n\n**10. \u2657h4 \u2658d7 11.h3**\n\nAfter 11.\u2658b5 \u2657c5 12.\u2656d2 c6 13.\u2658c3 \u2654c7 14.\u2657g3 \u2656hd8 Black is in no way worse.\n\n**11... \u2658h6 12.\u2658b5 \u2658f7 13.\u2658xd6 cxd6**\n\nLasker gratefully accepts the invitation to the opening of the c-file. 13...\u2658xd6 14.\u2658d2 \u2654e7= was also equally possible.\n\n**14. \u2658d2 \u2656c8 15.b3 \u2654e7 16.\u2654b2 \u2656c7 17.c4 \u2656hc8 18.\u2657d3?**\n\nThis gifts Black a tempo, which the latter immediately makes use of to become active. 18.f3 was better.\n\n**18... \u2658c5 19.\u2657e2**\n\n**19...b5 20.f3**\n\n20.cxb5 g5 21.\u2657g3 \u2658xe4 22.\u2658xe4 \u2656c2+ \u2013+.\n\n**20...bxc4 21. \u2657xc4 a5 22.\u2657f2 a4 23.\u2657xc5 \u2656xc5 24.\u2656c1**\n\nAfter 24.\u2657xe6 Black invades with the rook: 24...\u2656c2+ 25.\u2654b1 \u2654xe6 26.bxa4 \u2654d7 (otherwise \u2656c1 is possible) 27.\u2656hg1 \u2658d8 28.\u2658b3 \u2658e6, but he cannot cash in on it.\n\n**24...axb3 25.axb3 \u2658d8 26.\u2656a1 \u2658c6 27.\u2656hc1 \u2658d4**\n\nLasker has successfully improved the position of his knight.\n\n**28. \u2656a7+ \u26568c7 29.\u2656ca1 \u2654d7 30.\u2656xc7+ \u2656xc7 31.\u2656a6 \u2654e7 32.\u2656b6**\n\n32.g4 is followed by 32...\u2657xc4 33.bxc4 g6 34.\u2654c3 \u2654d7 with a slightly freer game for Black.\n\n**32...f5**\n\n**33. \u2654b1!?**\n\nAfter 33.exf5 White keeps things level.\n\n**33... \u2657xc4 34.\u2658xc4**\n\n34.bxc4 weakens the c-pawn.\n\n**34...fxe4 35.fxe4 \u2658xb3 36.\u2658xd6 \u2658d2+ 37.\u2654a2**\n\nAfter 37.\u2654b2 Black forces the rook ending and in it wins a pawn: 37...\u2658c4+ 38.\u2658xc4 \u2656xc4 39.\u2656b7+ \u2654f6 40.\u2656b6+ \u2654g5 41.\u2656b7 \u2654h6 42.g4 \u2656xe4.\n\n**37...g6!**\n\nThis prevents the knight from retreating via f5. Now the threat is, for example,...\u2656d7, after which the \u2658d6 must abandon the protection of the e4-pawn.\n\n**38.h4?!**\n\n38.\u2656a6!? is followed by 38...\u2656c3 39.\u2654b2 \u2656g3 with advantage to Black.\n\n**38... \u2656d7**\n\n38...\u2656c3 39.\u2654b2 \u2656g3 40.h5 \u2656xg2 41.\u2654c3 gxh5 42.\u2658f5+ is not very profitable for Black here.\n\n**39. \u2658b5 \u2658xe4 40.\u2654b3 \u2658f6 41.\u2656c6 \u2656d3+ 42.\u2656c3 \u2656xc3+ 43.\u2658xc3 \u2654e6**\n\nBlack is a pawn up and has the more active king.\n\n**44. \u2654c2 \u2654f5 45.\u2654d3 \u2654g4 46.\u2654e3 \u2654g3 47.h5 gxh5**\n\nSimpler than 47...\u2658xh5 48.\u2654e4 \u2654xg2 49.\u2654xe5 and White can still fight for the draw.\n\n**48. \u2658b5 h4 49.\u2658d6 \u2654xg2 50.\u2658f5 h3 51.\u2658h4+ \u2654g3 52.\u2658f5+ \u2654g4**\n\nWhite resigned.\n\nIn 1914 Janowski played in the 19th DSB Congress in Mannheim. During the tournament the First World War broke out and Janowski was interned, as were Alekhine, Bogoljubow and other Russians. After his release to Switzerland he moved to the USA, but he was no longer able to emulate his earlier successes.\n\nOn the 19th December 1926 Janowski travelled to Hy\u00e8res (France), to take part in a tournament which was to start there on the 24th January 1927. Janowski arrived with a heavy 'cold' which was, however, then diagnosed by a local doctor as the final stages of tuberculosis. On the 15th January 1927 Janowski died, only 56 years old, alone and completely penniless. A benefactor paid for his burial and had a gravestone erected with an inscription from the 11th century Persian poet Omar Khayyam:\n\n'This is the only truth: we are pieces in the mysterious game of chess played by God. He places us, holds us, drives us forward and then throws us one after the other into the box of nothingness'.\n**11. Hot games in Havana**\n\n**The World Championship 1921:  \n_Emanuel Lasker against Jos\u00e9 Raul Capablanca_**\n\nThe best players in the world were invited in 1911 to the tournament of San Sebastian, organised by Jacques Mieses: a total of 15 masters who had all won at least one tournament. Only Emanuel Lasker was missing. Ossip Bernstein had championed the invitation to Jos\u00e9 Raul Capablanca. The young Cuban had abruptly drawn attention to himself and his ability with a clear victory in a match against Frank Marshall. To general surprise Capablanca then immediately won the unofficial candidates' tournament in San Sebastian ahead of Akiba Rubinstein and Milan Vidmar.\n\n**Jos\u00e9 Raul Capablanca (1888-1942)**\n\nAfter his tournament victory Capablanca sent on the 26th October 1911 a challenge to Lasker. The latter, however, demanded in his conditions for the match that the challenger should have to win the WCh match by at least two clear points in order to acquire the title of World Champion. Capablanca declined this as 'unfair', whereupon Lasker made it public that Capablanca had insulted him and broke off all future contact. In 1913 Lasker then agreed with Rubinstein a match for the World Championship for the following year. On account of the outbreak of the First World War, however, the WCh contest between Lasker and Rubinstein never took place.\n\nJos\u00e9 Capablanca, actually Jos\u00e9 Raul Capablanca y Graupera, born on the 19th November 1888 in Havana the son of a Spanish colonial official, was a so-called 'chess prodigy'. He is said to have learned chess at the early age of four, apparently simply by watching. At twelve he defeated the Cuban national champion Juan Corzo in a match by 4:3 with six draws. After his final school exams Capablanca studied chemistry and sport at the New York Columbia University. After one term, however, his bursary was withdrawn because he was too involved with chess and too little with his studies.\n\n1906 saw his first encounter with Emanuel Lasker in New York Manhattan Chess Club. Both Lasker and Capablanca were playing in a rapid chess tournament with 20 seconds thinking time per move. Of the 32 participants it was these two who qualified for the final \u2013 and Capablanca won. In a match in 1909 the Cuban defeated Frank Marshall by 8:1 with 14 draws.\n\nThe Cuban state gave Capablanca a post in the diplomatic service in 1913 and at the same time liberated him for all tournaments. This made Capablanca practically a state sponsored professional. After Lasker had refused Capablanca's challenge, the two met in the St. Petersburg tournament of 1914. Capablanca won the main tournament, however Lasker overtook him in the subsequent final for the five best-placed players.\n\nThe St. Petersburg tournament was played in a spectacular setting. Between the two parts of the tournament a banquet was held during which the Baltic-Russian jeweller Peter Carl Faberg\u00e9 presented each of the players with a gilded wineglass. Sergei Prokofiev, himself a great chess lover, was playing in the music room. Prokofiev later compared Lasker with Bach and Capablanca with Mozart. The organisers of the tournament aimed to take in a lot of money with the help of entry fees. They charged five roubles for entrance to the tournament hall and only two roubles for entrance to the room with the demonstration boards. On the first day they took in 800 roubles (roubles and dollars were of equivalent value at that time), but nevertheless for the preparation of the tournament they had to rely on personal donations from the Czar to the extent of 1000 roubles, because the tournament involved high costs. Lasker, for example, received 250 roubles appearance money per game. At the end of the tournament Lasker and Capablanca buried their differences, discussed the possibility of a WCh match and shortly thereafter in Berlin played a blitz match which Capablanca won by 6\u00bd:3\u00bd. But then the First World War broke off at a stroke all international chess contacts.\n\nAfter the end of hostilities, which so lastingly shattered Europe, the Dutch Chess Federation took the initiative in 1919 to set in motion once more the negotiations for a WCh match between Lasker and Capablanca. Lasker was, however, no longer in the first flush of youth. In 1918 the World Champion had already celebrated his 50th birthday. On the 23rd January 1920 the two players met in The Hague and signed an agreement for a match for the World Championship. It was to take place where the best offer would be made. As Capablanca was already in negotiations with some possible organiser, for example in Havana, to everyone's surprise Lasker gave out a statement in which he announced his withdrawal. Lasker explained he wanted to transfer his title without a match to the most worthy successor, Capablanca. Lasker's explanation was his reaction to a critical letter, though Lasker was overlooking that this letter was a whole year late in reaching him and had absolutely no more validity.\n\nIn any case Capablanca did not want to win the title without a contest and so travelled back to Europe to meet Lasker in The Hague and to come to a new agreement. He had previously told him about a lucrative offer from Havana. Lasker and Capablanca then signed in September 1920 a new contract. After the Cuban had left, Lasker suddenly demanded an advance payment, according to Capablanca's account. Neither the Cuban nor the sponsors of the match wanted to accept this. Capablanca again returned to The Hague to meet Lasker for the third time.\n\nIn the long run the two players agreed to a compromise. For Lasker the most important argument for playing this match was probably the fee. There was in prospect for the match a total of 20 000 dollars. This was for that time a relatively sensationally high amount of money, of which Lasker was to receive 11000 dollars independent of the result of the match. Like many other Germans Lasker had invested his fortune in war loans before the war and lost it all, so he was therefore thankful for that sort of income, and in addition in the form of hard currency. In Germany in the meantime inflation had started gnawing away at the value of the mark. Lasker insisted, however, in view of his previous resignation, that in his match against Capablanca he considered himself to be the challenger.\n\nThe match between Emanuel Lasker and Jos\u00e9 Raul Capablanca for the World Championship finally took place from the 18th March till the 28th April in Havana and was arranged for 24 games (in his Lasker biography Jacques Hannak says 30 games). If the score was 12:12 the title defender was to remain World Champion. The start of the match was originally set for the 15th January, but since on world markets the price of sugar (Cuba's main source of income at that time) was continuing to fall, that meant greater difficulties than had been foreseen in the gathering of the prize fund. At the end of January the prize money was finally ready and was later, after the fifth game, raised by a further 5000 dollars by the bureau of tourism, which was to be divided in a ratio of 3:2 between winner and loser.\n\nLasker embarked on the 16th February 1921 on the _Hollandia_ in Rotterdam and stepped on to Cuban soil on the 7th March. He was accompanied by his wife Martha (n\u00e9e Bamberger), whom he had married in 1911 after the death of her first husband Emil Cohn. Before her marriage to Lasker, Martha Cohn had been active as an author, had written poetic texts and satirical articles and published under the pseudonym L. Marco numerous short stories. She later gave up this activity and supported her husband. The Lasker couple took up residence in the hotel 'Trocha del Vedado'. This was Lasker's third visit to Cuba during his career. Back at the start of the 1890s he had looked for patrons in Havana for a WCh match against Steinitz. In 1906 he had been for a second time a guest in Cuba.\n\nFrom the second game on the match was played 'in a corner' (Martha Lasker) of the large building of the fashionable 'Gran Casino de la Playa', whilst close by in the main rooms there was gambling and dancing. The opening game was played in the building of the Union Chess Club. In the hall of the casino an enormous demonstration board had been hung up to which the moves were transferred. From there the journalists present transmitted the moves by telex to their papers. The games were played on the same table and with the same pieces which had also been used for the second match between Chigorin and Steinitz, Havana, 1892. Capablanca later received the table and pieces as a present from the Havana Chess Club. Today they can be found together with other objects from his estate in the sports museum in Havana.\n\nThe games were all played from 21.00 till 01.00. Lasker drew the first four games. In the fifth game he spoiled a good position and lost. In the next four games Lasker tried to restore equality and Capablanca to hold on to his lead. These four games also ended as draws. In the tenth game Lasker once more obtained a good position with certain winning chances. But he spoiled this too and had to accept his second defeat. Thereafter Lasker had internally given up. Capablanca won the eleventh and after two draws also the fourteenth game. Lasker now suggested continuing the match somewhere with a milder climate, for example in New York or Philadelphia, but Capablanca declined to do so. Lasker thereupon resigned the match prematurely at a score of 4:0 (with ten draws).\n\n **Capablanca \u2013 Lasker**\n\nHavana, 11th game  \n13th April 1921  \nQueen's Gambit Declined (D66)\n\n**1.d4 d5 2. \u2658f3 e6 3.c4 \u2658f6 4.\u2657g5 \u2658bd7 5.e3 \u2657e7 6.\u2658c3 0-0 7.\u2656c1**\n\nThis is one of the basic positions for the Queen's Gambit Declined.\n\n**7... \u2656e8**\n\nA more modern treatment is 7...c6 8.\u2657d3 dxc4 9.\u2657xc4 \u2658d5 10.\u2657xe7 \u2655xe7.\n\n**8. \u2655c2 c6 9.\u2657d3 dxc4 10.\u2657xc4 \u2658d5 11.\u2657xe7 \u2656xe7?!**\n\nExtravagant and bad. The rook is misplaced here. The usual move is 11...\u2655xe7.\n\n**12.0-0 \u2658f8**\n\nKasparov gives 12...\u2658xc3 13.\u2655xc3 b6 as better.\n\n**13. \u2656fd1**\n\n13.\u2658e5!?.\n\n**13... \u2657d7**\n\nThe development of the light-squared bishop is one of the main problems in the Queen's Gambit Declined. Here Black is for the moment stuck with his 'bad bishop'. Perhaps 13...b6 14.a3 \u2657b7 was a better option.\n\n**14.e4 \u2658b6**\n\nA more natural move was 14...\u2658xc3.\n\n**15. \u2657f1 \u2656c8 16.b4 \u2657e8**\n\n'The defensive position is characteristic of Steinitz, with most of the pieces on the back two ranks. There are no weak points but the black position suffers from a lack of space and the chance to manoeuvre the pieces.' (Capablanca)\n\n**17. \u2655b3 \u2656ec7 18.a4 \u2658g6 19.a5**\n\nForces another black piece into passivity.\n\n**19... \u2658d7 20.e5 b6 21.\u2658e4 \u2656b8**\n\nHere 21...bxa5 22.bxa5 \u2656b8 23.\u2655a3 \u2656cb7 was more active. But here too there is no question as to White's advantage.\n\n**22. \u2655c3**\n\nKasparov recommends the immediate 22.\u2655a3, since after 22...\u2658f4 23.\u2658d6 \u2658d5 White does not lose a tempo because of the attack on the queen.\n\n**22... \u2658f4 23.\u2658d6 \u2658d5 24.\u2655a3 f6**\n\nL\u00f6wenfisch and Panov gave 24...\u2655e7 as an improvement here, so as to be able to take on e8 with the rook: 25.\u2657c4 (after 25.\u2658xe8 \u2656xe8 White's advantage would have disappeared) 25...\u2658f8 26.\u2657a2 f6 etc.\n\n**25. \u2658xe8 \u2655xe8 26.exf6 gxf6**\n\n26...\u26587xf6 27.\u2658e5 with advantage to White.\n\n**27.b5**\n\n'The exposed position of the black monarch invites an attack on the king. But before White acts on this, he must first liquidate his queenside pawns, so as to dispose of any possible weakness. If the two pawns disappear White can devote his full attention to the kingside and does not have to worry about the other flank.' (Capablanca)\n\n**27... \u2656bc8**\n\n27...c5 would according to Capablanca be followed by 28.dxc5 (a better try is 28.axb6 axb6 29.\u2657c4, Panov) 28...bxc5 (Kasparov pointed to 28...\u2658xc5 as a better option: 29.\u2658d4 \u2656bc8 'with an unclear position.') 29.\u2657c4.\n\n**28.bxc6**\n\n28.\u2656xc6 \u2656xc6 29.bxc6 \u2656xc6 30.axb6 axb6 31.\u2656e1 is the position in the game, but is the more precise move order should 28...\u2658b8 actually be an improvement.\n\n**28... \u2656xc6**\n\nThere was also the interesting 28...\u2658b8!? e.g.: 29.\u2657c4 (29.\u2656c2 \u2658xc6 30.axb6 axb6 31.\u2656dc1 \u2655d7=) 29...\u2655d8 30.\u2656e1 \u2656xc6 31.\u2657xd5 \u2655xd5 32.axb6 axb6 with only a slight advantage for White.\n\n**29. \u2656xc6 \u2656xc6 30.axb6 axb6 31.\u2656e1 \u2655c8**\n\nPossibly it was worth considering 31... \u2655f8!?, in order to get some relief with an exchange of queens: 32.\u2655b3 \u2655b4.\n\n**32. \u2658d2 \u2658f8**\n\nLasker gave 32...\u2656c3 33.\u2655a1 \u2658f8 34.\u2658e4 \u2656c7 as more accurate.\n\n**33. \u2658e4 \u2655d8 34.h4 \u2656c7**\n\n34...f5 is followed by 35.\u2655g3+ \u2654h8 36.\u2655e5+ \u2654g8 37.\u2657b5 with advantage to White. The best move here according to Capablanca was 34...h6 intending...f6-f5.\n\n**35. \u2655b3 \u2656g7 36.g3 \u2656a7 37.\u2657c4 \u2656a5**\n\nThis avoids the trap 37...h6 38.\u2657xd5 exd5 39.\u2655xd5+ \u2655xd5 40.\u2658xf6+ +\u2013.\n\n**38. \u2658c3 \u2658xc3 39.\u2655xc3 \u2654f7 40.\u2655e3 \u2655d6 41.\u2655e4 \u2656a4?**\n\n'Suicide' (Lasker). 41...\u2656a7 was better. After 42.d5 e5 43.\u2657f1 White retains his advantage.\n\n**42. \u2655b7+ \u2654g6**\n\n42...\u2655e7 43.\u2655c6 \u2656a7 44.d5+\u2013.\n\n**43. \u2655c8?**\n\nKasparov pointed out the following pretty forced win: 43.h5+ \u2654h6 (43... \u2654xh5 44.\u2655g7 threatens \u2657e2#) 44.\u2655f7 \u2655d8 45.\u2657d3 \u2656xd4 46.\u2656xe6 and if 46... \u2656xd3, then 47.\u2656xf6+ \u2654g5 48.\u2655g7+ \u2658g6 49.\u2656xg6+ hxg6 50.\u2655xg6#.\n\n**43... \u2655b4?**\n\n43...\u2654h6 is more resilient: 44.\u2657xe6 \u2656xd4 45.\u2655c1+ \u2654g7 and White is far from having a win.\n\n**44. \u2656c1+\u2013 \u2655e7 45.\u2657d3+ \u2654h6**\n\n45...f5 46.\u2656c7+\u2013.\n\n**46. \u2656c7 \u2656a1+ 47.\u2654g2 \u2655d6 48.\u2655xf8+**\n\nBlack resigned.\n\nThe match ended very harmoniously in the final meeting of the two players and the organisers. Later too Lasker and Capablanca would always meet each other with mutual respect. In his little book _Mein Wettkampf mit Capablanca_ (Berlin 1921) Lasker gives the adverse circumstances as the reason for his defeat. After his return to Germany Lasker, according to Hannak, indicated he needed a rest cure of several months in the spa at Karlsbad in order to recover from the strain. Capablanca contradicted this representation of the facts in a letter which was printed in the _British Chess Magazine_ of October 1922 (pp. 376-380).\n\nAfter the loss of the World Champion title Lasker still notched up some outstanding successes. Above all his results at the tournaments of New York 1924 and Moscow 1925, where on both occasions he left Capablanca in his wake, prove that despite his age Lasker was not yet on the scrapheap.\n\nThe tournament in Moscow was above all on account of the enormous public interest the greatest of the year 1925. For the staging of the tournament the organisers received 30000 roubles from the communist government and invited 21 masters, ten local ones and ten foreigners and in addition the one who was born in Russia and lived in Germany, Efim Bogoljubow.\n\nOn the 9th November the opening of the tournament was celebrated in the 'Blue Hall' of the House of Trade Unions. The tournament itself was then played in the 'Fountain Hall' of the Hotel Metropol. The degree of interest surprised even the organisers. At the start approximately 500 spectators visited the tournament every day, and soon there were more than a thousand chess enthusiasts. Outside up to 50000 followed the events on a huge demo board. The foreign chess masters were f\u00eated and honoured like pop stars. Moscow's 'chess fever' of 1925 inspired the Russian director Vsevolod Pudovkin to shoot the film _Shakhmatnaya goryachka_ (English: 'Chess fever', 1925).\n\nAs well as chess Lasker now devoted himself more and more to the game of go, in which he soon rivalled Germanys leading go-player Felix Dueball, to bridge and also to poker. However, Lasker was not such a great master at the game of bridge, remarked Milan Vidmar in his classic _Goldene Schachzeiten_ , and he is also said not to have been an agreeable playing partner. Lasker apparently had problems losing gracefully. Moreover Lasker invented his own board game, Lasca. He published a series of books, including his _Manual of chess_ and together with his brother Berthold the expressionist drama _Vom Menschen die Geschichte_ , which, however, was never staged.\n\nAt the start of the 1920s or perhaps earlier Lasker met Albert Einstein in the house of the Germano-Polish author Alexander Moszkowski and became friends with the physicist. He later spoke of this encounter during a lecture in the USA: 'Einstein was aware that I had raised a certain objection to his theory, but he was not keen to enter into a discussion about it. My thesis is that \"lim c = \u221e\", whilst he maintained that c is a finite constant'. On the same occasion Lasker also met Walther Rathenau and discussed philosophical questions with the German foreign minister who was later the victim of an assassination in 1922.\n\nLasker was also among other things a regular visitor to the Romanisches Caf\u00e9 on the Kurf\u00fcrstendamm, the best known meeting place for artists in Berlin. The director Geza von Cziffra later reminisced about the figures to be seen in the Romanisches Caf\u00e9, whom he divided into 'swimmers' \u2013 those who were able to make a living from their work \u2013 and 'non-swimmers'. And in addition there were on the balcony the 'moon dwellers': 'the chess players who had been rapt from this world, and among them no less a figure than the World Chess Champion Lasker'. The director and actor Fritz Kortner even had a mask made with Lasker's features.\n\nAt the end of the 1920s Lasker was still undertaking numerous journeys in Europe. After the seizure of power by the Nazis and the start of the persecution of the Jews, he left Germany with his wife and at first lived in the Netherlands. After a year he moved to London, running several chess columns there and giving simultaneous exhibitions. In 1934 he participated in the tournament in Zurich. In 1935, at the invitation of the Moscow Academy of Sciences, he went to the Soviet Union and there took part in the strong tournaments in Moscow in 1935 and 1936. In the meantime at home in Germany his flat had been smashed up, as Nikolai Krylenko later reported in a letter in which he begged Stalin to accord Lasker the right to stay in the USSR. The request was granted and Lasker received in 1935 a flat in Moscow's Great Spasso Glinichtchevsky Lane No. 8. The flat in the centre of Moscow soon became the meeting point for many Russian players and the presence of the former World Champion was an inspiration for the chess life in Moscow.\n\nLasker's last international tournament was Nottingham 1936. In the following year in Moscow he wrote the story _Wie Wanja Meister wurde_ ('How Vanya became a master'), which was only published in 1973 in Russian (in an edition of 75000 copies!) and even published in German in 2001. Lasker and his wife travelled in 1937 to New York, apparently to visit Martha Lasker's daughter from her first marriage, Charlotte Cohn. Because Martha Lasker became ill there, that was the official explanation, the couple remained in New York, where the Laskers lived in 610 West 139th Street, Apartment 3C. Nevertheless, the emigration to the USA had probably already been planned by the Laskers in Moscow, since before they set off they had packed up all their possessions and taken them with them to New York.\n\nIn 1937 the political climate in Moscow had changed dramatically, Some wellknown persons from the Moscow chess scene had already been arrested. Finally even Nikolai Krylenko, People's Commissar for Justice and Advocate General (famous quotation: 'It is not enough to have executed the guilty. It is only when a few innocent people have been shot that the people is impressed.'), chess partner of Lenin and founder of the chess boom in the USSR, was arrested and in 1938, like many other comrades of Lenin, executed on the orders of Stalin. In 1938 Lasker was also deprived of his German citizenship in Nazi Germany.\n\nIn 1940 Lasker was taken ill and was taken to the New York Mount Sinai Hospital. On the 11th January 1941 he died and was buried in the Beth Olom cemetery in Queens. From 2008 Lasker has been the only chess player to belong to the 'Hall of Fame' of German sport.\n**12. Friends become enemies**\n\n**The World Championship 1927:  \n_Jos\u00e9 Raul Capablanca against Alexander Alekhine_**\n\nAlexander Alekhine was born on the 13th October 1892 (according to the Gregorian calendar) in Moscow. His father Alexander Ivanovich Alekhine belonged to the nobility and was a member of the Duma; his mother Anissia Ivanovna Alekhine (n\u00e9e Prokhorova), was the daughter of an industrialist. Alekhine grew up in great luxury in Moscow. His father, a regular visitor to the Riviera, is supposed to have lost on one occasion a million roubles in the casino of Monte Carlo.\n\n**Alexander Alekhine (1892-1946)**\n\nAlekhine first learned to play chess at the age of seven from his mother. His father and his four year older brother Alexei then gave him more tips about the game. Alekhine, moreover, had a sister, Varvara (Barbara), who later became an actress. When Alekhine was nine years old, Harry Nelson Pillsbury visited Moscow in order to give a blindfold simultaneous display. Alekhine himself was too young to be allowed to the event but his brother played in it and even scored a draw. Alekhine's father made sure that as well as his secondary school education his son also learned German and French at an early age. After secondary school Alekhine continued his education at the 'Provedenie' in St. Petersburg, a law faculty for nobles who were aiming at a career in the service of the state.\n\nAlekhine's progress in chess continued to be less meteoric than that of other top players. Modest successes in many tournaments were interspersed by failures. But what was noticeable was his fantastic memory, which moved Georg Marco to the statement: 'A youth in years, an old man in knowledge.' At the 17th DSB Congress in Hamburg (1910) Alekhine was already taking part at the age of 17, but for a time had to be carried into the tournament hall because he could not walk on account of a serious swelling of the lymphatic vessels of his ankles.\n\nFrom 1913 Alekhine's play finally acquired a certain amount of stability. He finally had a sensational success to celebrate in 1914 at the grandmaster tournament of St. Petersburg, where, though the only non-grandmaster, he took third place behind Lasker and Capablanca. The concept of a 'grandmaster', which had occasionally been used previously, became common after the tournament. Those masters who had already won an important tournament were described as grandmasters. During the tournament Alekhine became friends with Capablanca and analysed games together with the Cuban.\n\nCapablanca was already considered after his victory in San Sebastian as a possible challenger to World Champion Lasker. Nobody could have suspected that it would take more than ten years till they finally managed to play a match. And also the friendly relationship between Capablanca and Alekhine would later change dramatically.\n\nIn the middle of July 1914 Alekhine travelled to the 19th DSB Congress in Mannheim. When the seventh round was being played, on the 28th July 1914, Austria-Hungary, as its reply to the assassination of the successor to the Austrian throne Franz Ferdinand and his wife in Sarajevo, declared war to Serbia. Within the shortest possible space of time the crisis escalated into the great war into which almost all the states of Europe were drawn. On the 1st August 1914 at 19.00, after the expiry of an ultimatum, Germany declared war on Russia. The tournament was suspended and Alekhine was interned together with ten other Russians \u2013 Efim Bogoljubow, Fyodor Bohatyrchuk, Alexander Flamberg, N. Koppelman, Boris Malyutin, Ilia Rabinovich, Peter Romanovsky, Peter Saburov, Alexei Selesniev and Samuel Weinstein.\n\nDuring his captivity in Germany he was held in various places \u2013 Mannheim, Ludwigshafen, Rastatt and Baden-Baden. The longest period of time was spent in Rastatt in a cell with Efim Bogoljubow, Ilia Rabinovich and 'a certain Weinstein' (Alekhine), who later, according to Alekhine, became active as 'chess watchdog' for the Commissar for Justice Nikolai Krylenko. There were neither books nor newspapers and so Alekhine and Bogoljubow spent their time playing blindfold chess, because the Russians did not have a chess set either.\n\nSince after an examination the Germans considered Alekhine unfit for military service, on the 14th September 1914 he was released from detention along with Bohatyrchuk, Saburov and Koppelman and sent to Switzerland. From there Alekhine made his way with Bohatyrchuk to Genoa, the destination for many stranded Russians. Of course, during their long wait for a ship to take them back to Russia, the two Russian chess masters spent their time playing chess. In the middle of October Alekhine and Bohatyrchuk got back to Russia by ship via Genoa, Gibraltar, London, Stockholm, Finland, St. Petersburg \u2013 now re-baptised Petrograd. Bogoljubow on the other hand became friendly with the daughter of a prison warder and remained in Germany after the war too.\n\nAfter his return to Russia Alekhine married the Russian Baroness Anna von Severgin, in order to legitimise their daughter Valentina, born in 1913. It was Anna von Severgin's second marriage, her first husband having died at the start of the First World War. But the marriage with Alexander Alekhine ended in divorce a few months later. Valentina Alekhine later lived in Austria and died around 1985 in Vienna.\n\nAlekhine joined the Red Cross, took part in the war as a Red Cross helper and was wounded in 1916, receiving severe contusions to his back. He spent several months confined to his bed in a convent hospital in Tarnopol. Alekhine's father died around this time after spending more than a year as a prisoner-of-war in a German camp. Alekhin's mother had already died in 1913, mentally deranged as a result of her continuing use of narcotics.\n\nAfter his recovery Alekhine worked for the new communist authorities in Odessa as an investigator, but was one day denounced as a noble and put into a common prison. Alekhine escaped the execution of his death sentence only because the Ukrainian chess master Yakov Vilner knew Alekhine and turned to the Ukrainian political commissar Christian Rakovsky (or according to another versions the agricultural commissar Dimitry Manuilsky) and asked for help. The latter pardoned Alekhine and arranged for his release. In yet another version of this story, one day Leo Trotsky, himself a good chess player, appeared in Alekhine's cell and is said to have played with him a game in which the stake was Alekhine's life \u2013 and Alekhine won.\n\nAfter his return to Moscow Alekhine enrolled in 1919 in the film college in order to become an actor, but gave up this enterprise in the first year. In November 1920 Alekhine was once more questioned by the secret police in Moscow on account of supposed anti-revolutionary views. The accusation was, however, again dropped at the start of 1921.\n\nIn 1920 during his work as a translator for the Comintern (Communist International) Alekhine had made the acquaintance of the then 41 year old Swiss author and avowed communist Annaliese R\u00fcegg. During the '3rd International' she was on a lecture tour of Russia. The couple married on the 15th March 1921 and since Anneliese R\u00fcegg had good contacts in the Bolshevik leadership, even being personally known to Lenin, she obtained an exit visa for her husband. The approval was granted by Lenin's intimate friend Karl Radek, with the justification: 'Even if Alekhine is also a counter-revolutionary, he is a major chess genius. This gift can only find expression outside of Russia.'\n\nAlso in 1921 Alekhine's son Alexander was born, who later became a well-known handball referee in Switzerland. Alekhine's marriage to Annaliese R\u00fcegg ended in divorce in 1926. After her death in 1934 Alekhine arranged for his son in Switzerland to get a place in a boarding school and put him in the care of the Swiss master and FIDE delegate Erwin Voellmy. Alekhine went Paris and according to his own accounts there earned the title of Doctor of Law in 1925. He was then living with the widow of the Russian general V. Vassiliev, Nadezhda Fabritzky, and lived with her in Paris in the Rue de la Croix-Nivert. His new partner was also clearly older than Alekhine. Apparently, however, the couple had not yet married. During this time Alekhine earned his living with simultaneous and blindfold displays.\n\nHis tournament successes in the early 1920s included in addition to victories in The Hague 1921 and Hastings 1922 and second place in London 1922 (behind Capablanca) a shared second place in Bad Pistyan, a joint first place in Karlsbad 1923, a third place in New York 1924 and a clear victory at the tournament in Baden-Baden 1925 with 16 points and 1\u00bd points of a lead ahead of Rubinstein. In 1926 Alekhine just won a match against Max Euwe 5\u00bd:4\u00bd. In Semmering in the same year he was second behind Rudolf Spielmann, in the subsequent tournament in Dresden once more second, this time behind Aaron Nimzowitsch. After his tournament victory in The Hague in 1921 Alekhine had already addressed a first challenge to Capablanca, who had just succeeded Lasker as World Champion.\n\nAfter winning his World Championship match against Lasker, Capablanca had underlined his position as the best chess player in the world with his tournament win in London 1922. The new World Champion now explained that he would accept a challenge from any player as long as certain conditions were fulfilled. Capablanca agreed in London with the leading players of the tournaments, namely Alexander Alekhine, Akiba Rubinstein, Efim Bogoljubow, Richard R\u00e9ti, Geza Maroczy, Milan Vidmar and Savielly Tartakower, that the rules for World Chess Championship matches should be unified and formalised. After lively debate these eight players put together the 'London Rules' and signed on the 9th August 1922 the so-called 'London Agreement'.\n\nAccording to it WCh matches should now be set for six wins. The time control should be two and a half hours for 40 moves. The World Champion had to defend his title within a year of receiving a challenge. The challenger, however, had to be a recognised master. In addition the challenger had to come up with a prize fund of 10 000 dollars, of which the title defender was to receive 20% as an appearance fee. The remainder was to be divided 60:40 between the winner and the loser. In addition provision had to be made for the travel costs of both players. Further conditions were specified in a total of 21 points. The 'London Rules' were published for example in the _American Chess Bulletin_ (December 1923, pp. 185-186).\n\nAs far as his chess career was concerned, Capablanca was far less active in the first half of the 1920s than Alekhine. He played in the big tournaments in New York 1924 and Moscow 1925 and on both occasions had to concede to Lasker. The London Rules, with their demands for prize money of 10000 dollars, turned out to be an almost insurmountable hurdle for the masters of that time. Nobody managed to come up with that high sum. Actually Akiba Rubinstein was considered the legitimate challenger on the basis of his successes, but the Polish master could not raise the money either.\n\nIn November 1923 Alekhine reminded Capablanca that Rubinstein's deadline for raising the money for the match would come to an end on the 1st January 1924 and that it would then be his turn to be the challenger. Moreover, Alekhine set out on a five month publicity tour of South America during which he gave a series of blindfold exhibitions. The tournament in New York 1924, however, represented for Alekhine a certain setback, since in it he was unable to keep pace with Lasker and Capablanca, though he was third. Alekhine was not invited to the first big tournament in the Soviet Union, in 1925 in Moscow.\n\nIn February 1927 there was a re-run of the tournament in New York. Lasker, the majestic victor of 1924, declined an invitation because in the first tournament he had felt himself disadvantaged in his game against Capablanca on account of a clock which was not working correctly. Moreover a public quarrel had arisen about it between the former World Champion and the head organiser of the tournament Norbert Lederer, a prominent New York lawyer, the secretary of the Marshall Chess Club, and a good friend of Capablanca. This time Capablanca won the tournament.\n\nOutwith the tournament Alekhine, Capablanca and organisers from Argentina agreed to stage a match between Capablanca and Alekhine for the World Championship in Buenos Aires. This WCh match was then the first, and remained the only, one which was played according to the 'London Rules' agreed in 1922, thus with its six wins and being unlimited in the total number of games. In many sources it has been reported that Capablanca would only have needed a score of 5:5 to retain the title. Whether this is actually the case or not has so far not been completely cleared up.\n\nWith its 10000 dollars the prize fund met exactly the specifications of the London Agreement. The organisers were the 'Club Argentino de Ajedrez' in Buenos Aires, which had made a financial outlay of a total of 40000 pesos for the organisation. With its 34 games this became the then longest match in history. It was not surpassed until 1984\/85 when Karpov and Kasparov would see their legendary match of 48 games in Moscow broken off without a winner.\n\nPlay took place in Buenos Aires from 16th September till 27th November 1927, that is to say a time of over two months. With the exception of the first and third games, curiously all the games opened with the Queen's Gambit Declined. To the great surprise of the world of chess Alekhine took the lead with a victory in the first game:\n\n **Capablanca \u2013 Alekhine**\n\nBuenos Aires, 11th game  \n16th September 1927  \nFrench Defence (C01)\n\n**1.e4 e6 2.d4 d5 3. \u2658c3 \u2657b4**\n\nThis move characterises the Winawer Variation of the French Defence.\n\n**4.exd5**\n\nA more ambitious move is 4.e5, but this advance did not become fashionable till later. The Exchange Variation 4.exd5 is nowadays regarded as tame, but is not without its venom. Capablanca, however, plays really cautiously here.\n\n**4...exd5 5. \u2657d3 \u2658c6 6.\u2658ge2 \u2658ge7 7.0-0 \u2657f5 8.\u2657xf5 \u2658xf5 9.\u2655d3 \u2655d7**\n\nAfter 9...\u2657xc3? White can take with the knight since 10.\u2658xc3 \u2658fxd4 11.\u2656e1+ \u2658e6 12.\u2658xd5 would be unpleasant for Black (Rolf Schwarz).\n\n**10. \u2658d1**\n\nWhite would like to exchange the active \u2658f5, but that costs time. Alekhine recommended 10.\u2657f4 0-0-0 with equality.\n\n**10...0-0 11. \u2658e3 \u2658xe3 12.\u2657xe3**\n\n'The white minor pieces are now blocking the important e-file. That is the convincing proof of a failed opening strategy', commented Alekhine in over-exaggerated fashion.\n\n**12... \u2656fe8 13.\u2658f4?!**\n\n13.\u2657f4 \u2656e4 14.c3 \u2657d6 15.\u2657xd6 \u2655xd6 16.\u2658g3 is equal (Alekhine); Euwe recommended 13.\u2658g3, also with equality.\n\n**13... \u2657d6 14.\u2656fe1 \u2658b4 15.\u2655b3**\n\n15.\u2655d2 would according to Alekhine be followed by 15...\u2655f5 16.\u2656ec1 h5! with a slight advantage to Black.\n\n**15... \u2655f5 16.\u2656ac1?**\n\nSlightly careless play. Capablanca gave 16.\u2658d3 \u2658xd3 17.\u2655xd3 \u2655xd3 18.cxd3 as level, but Alekhine saw an advantage for Black after 18...\u2657b4 19.\u2656ec1 c6, intending...a7-a5.\n\n**16... \u2658xc2! 17.\u2656xc2**\n\n17.\u2655xc2 \u2655xc2 18.\u2656xc2 \u2657xf4 changes nothing since the \u2657e3 is pinned here too.\n\n**17... \u2655xf4**\n\n'This possibility was obviously over-looked by Capablanca on his 16th move.' (Alekhine)\n\n**18.g3 \u2655f5**\n\nBlack is clearly better. Also interesting according to Alekhine was 18...\u2655f3!? 19.\u2655xb7 h5 20.\u2655b5 h4 21.\u2655e2 \u2655f5.\n\n**19. \u2656ce2 b6 20.\u2655b5 h5**\n\nIntending...h5-h4-h3, after which...\u2655f3-g2# is in the air.\n\n**21.h4 \u2656e4**\n\n**22. \u2657d2**\n\nAfter 22.a3? there is a win after 22... \u2656xh4 23.gxh4 \u2655g4+ 24.\u2654h1 \u2655h3+ 25.\u2654g1 \u2655h2+ 26.\u2654f1 \u2655h1# and 22.\u2655d3 is followed by 22...\u2655f3 (with the threat 23...\u2656xh4, 24...\u2655g4+, 25...\u2655h3+ and mate on h2) 23.\u2656c2 \u2657xg3 24.fxg3 \u2655xg3+ 25.\u2654f1 \u2656ae8 26.\u2655d1 \u2655h3+ 27.\u2656g2 \u2656xe3 28.\u2656xe3 \u2656xe3\u2013+.\n\n**22... \u2656xd4**\n\n22...\u2656ae8 was better: 23.\u2655xe8+ \u2656xe8 24.\u2656xe8+ \u2654h7 with advantage to Black according to Alekhine.\n\n**23. \u2657c3 \u2656d3?!**\n\n23...\u2656c4? was inaccurate on account of 24.\u2656e5! \u2657xe5 25.\u2655xd5 with counterplay. But 23...\u2656g4! intending 24...\u2657c5 and some pressure was also strong.\n\n**24. \u2657e5 \u2656d8 25.\u2657xd6 \u2656xd6**\n\n25...cxd6 26.\u2655c6!.\n\n**26. \u2656e5 \u2655f3**\n\n26...\u2655g6? 27.\u2656g5!.\n\n**27. \u2656xh5! \u2655xh5**\n\nThis avoids the trap 27...\u2656e6?? 28.\u2655e8+! \u2656xe8 29.\u2656xe8#.\n\n**28. \u2656e8+**\n\nIt was worth considering 28.\u2655xd3!?. After 28...\u2656e6 29.\u2656xe6 fxe6 30.\u2655a6 d4 31.\u2655xa7 the queen ending would be unclear.\n\n**28... \u2654h7 29.\u2655xd3+ \u2655g6 30.\u2655d1**\n\nAfter 30.\u2655xg6+ \u2654xg6 31.\u2656e7 Black obtains an advantageous rook ending on account of the passed pawn on the d-file. After 30.\u2655f3!? followed by 31.h5 White could, however, obtain counterplay.\n\n**30... \u2656e6!?**\n\n'An interesting idea: Black returns his extra pawn, so as to combine the advantage of the passed d-pawn with a mating attack.' (Alekhine) After 30...d4 the move 31.\u2655f3!? was also good here.\n\n**31. \u2656a8 \u2656e5**\n\n'Intending to place the queen behind the rook and to prepare the pawn formation b6-c5-d4.' (Alekhine). 31...d4! was later suggested by Averbakh as an improvement, since 32.\u2655xd4 fails to 32...\u2656e1+ 33.\u2654h2 \u2655c6\u2013+.\n\n**32. \u2656xa7**\n\nIt was again worth considering the move 32.\u2655f3!?. After 32...\u2655b1+ (32...\u2656f5 33.\u2655d3 \u2655e6 34.\u2656e8=) 33.\u2654g2 \u2656f5 34.\u2655e2 d4 35.b3 the chances are level.\n\n**32...c5**\n\n**33. \u2656d7?!**\n\n'This shortens the suffering.' (Alekhine). Capablanca suggested 33.\u2654g2 as an equalising continuation. 'From this evaluation one can see that, even after the game, Capablanca has not understood his unfortunate situation.' (Alekhine) 33...d4 34.\u2656a3 \u2655e6! 35.\u2655f3 c4\u2013+ (Alekhine). Once again 33.\u2655f3! was the best move. After 33...\u2656f5 34.\u2655d3 d4 35.b3 there can be no question of a win for Black.\n\n**33... \u2655e6 34.\u2655d3+ g6 35.\u2656d8 d4 36.a4**\n\n36.\u2655f3 \u2656f5\u2013+ (36...\u2655xa2?? 37.\u2655f6+\u2013) or 36.\u2654g2 \u2656e2 37.b4 \u2655f6\u2013+.\n\n**36... \u2656e1+**\n\nBut another strong move according to Alekhine was 36...\u2655e7! 37.\u2656b8 \u2655c7 intending 38.\u2656f8 \u2656e1+ 39.\u2654h2 \u2655b7\u2013+.\n\n**37. \u2654g2 \u2655c6+ 38.f3**\n\n38.\u2655f3? \u2656g1+ \u2013+.\n\n**38... \u2656e3 39.\u2655d1 \u2655e6\u2013+ 40.g4**\n\n40.b3 \u2656e2+ 41.\u2654f1 \u2656h2 42.\u2654g1 \u2655h3 and then mate, 40.\u2654f2 \u2655h3 41.\u2655g1 \u2656b3\u2013+.\n\n**40... \u2656e2+ 41.\u2654h3**\n\n41.\u2654f1 \u2656h2 42.\u2654g1 \u2655e5\u2013+ (Albert Becker).\n\n**41... \u2655e3 42.\u2655h1 \u2655f4!**\n\nThe game was adjourned here.\n\n**43.h5 \u2656f2**\n\nAnd Capablanca resigned: 44.hxg6+ \u2654g7\u2013+.\n\nAfter the defeat in the first game, which was completely unexpected for him, Capablanca was so shocked that he took a time-out of three days and sailed out into the Atlantic on a yacht in order to recover from the shock in absolute seclusion.\n\nCapablanca managed to equalise with a win in the third game. The Cuban won the seventh game and went into the lead. In games eleven and twelve Alekhine landed a double blow, which now brought the challenger a 3:2 lead. A series of eight draws followed, before Alekhine increased his lead to 4:2 with his win in game 21. After another series of seven draws it was again Capablanca's turn. Alekhine finally won the 32nd and 34th games. The 34th game was adjourned on the 27th November 1927 on the 82nd move.\n\nOn the following day Capablanca sent a letter in French to the challenger: 'Dear Mr Alekhine! I resign the game. You are now the new World Champion. Accept my congratulations and best wishes. Sincerely, your J. R. Capablanca.' Later in the day Capablanca went to the chess club and also congratulated Alekhine in person. At the closing banquet on the 8th December, at which Alekhine was officially declared the new World Champion, Capablanca, however, was not present.\n\nAlekhine won the match by 6:3 (with 25 draws) and thus became the fourth World Chess Champion. The outcome of the match came as a surprise not only to Capablanca, but also to the whole world of chess. Alekhine later explained his triumph with the supposition that Capablanca had probably previously been too convinced of his victory.\n\nAlekhine returned to Paris as World Champion, had a reception in the Palais Royal and was f\u00eated in the Caf\u00e9 de la Rotonde, the Caf\u00e9 de la R\u00e9gence and in the Russian Club. In the Russian Club he gave a toast, which on the following day was quoted in the newspapers for Russian \u00e9migr\u00e9s. 'May the myth of the invincibility of the Bolsheviks turn out to be just as false as the myth of the invincibility of Capablanca.' The reaction from Moscow was immediate: from now on Alekhine would be regarded as an enemy of the state, announced Nikolai Krylenko.\n\nImmediately after the match Alekhine had reaffirmed in an interview with the newspaper _La Prensa_ that he was ready for a return match with Capablanca \u2013 though only under the same conditions under which the match in Buenos Aires had been played. While still in Buenos Aires Capablanca had suggested limiting the number of games, but Alekhine insisted on unchanged conditions. In 1928 Capablanca brought an offer from Bradley Beach, New Jersey, but the negotiations about the match conditions drew out over years and the once good relationship between the two players deteriorated more and more. At the end they no longer spoke to each other and Alekhine totally avoided playing in the same tournaments as Capablanca. For the tournament in San Remo 1930 Alekhine demanded of the organisers 20000 lira appearance money, or, should Capablanca also be invited, 40000 lira. Therefore Capablanca was not invited.\n\nThere was no return match; instead Alekhine preferred to play against other challengers, firstly against Bogoljubow and after that against Euwe, and was also prepared to deviate from the London Rules in those cases. Capablanca's attempt to play against Euwe for the title, after the latter had defeated Alekhine and become World Champion, also failed. The match, which was planned for 1939, never took place, since Euwe lost his 1937 return match against Alekhine.\n\nCapablanca was considered a chess genius who conducted his games with great ease \u2013 and despite that won most of them. Above all he was hard to defeat. In his adult career he lost just 34 of his tournament games. From the 10th February 1916 (defeated by Oscar Chajes) till the 21st March 1924 (his defeat at the hands of Richard R\u00e9ti) he did not lose a single one out of 63 games (including the WCh match against Lasker) and won 40 of them.\n\nAfter the loss of the title of World Champion Capablanca again participated more often in tournaments so as to emphasise his claim to a return match. In 1928 he won in Budapest and Berlin, in 1929 in Budapest and Barcelona, in 1929\/30 in Hastings, in 1931 in New York. In the same year he won a match against Euwe by 2:0, with eight draws. In 1931 Capablanca withdrew temporarily from tournament chess. In the spring of 1934 at a party given by the Cuban consul in New York the charming Cuban made the acquaintance of his future second wife, Olga Chagodaev, who was born Olga Chubarova in 1898 in the Caucasus. After he divorced in 1937 from his first wife Gloria Simioni y Betancourt the couple married in 1938.\n\nAfter his comeback in 1933\/34 in Hastings (fourth place) Capablanca played a few tournaments with less success. Then, however, in 1936 he won the tournament in Moscow and shared first place in Nottingham (with Botvinnik). For the first time since 1927 Alekhine and Capablanca had again played in the same tournament. At the AVRO-tournament in 1938 Capablanca suffered his first slight stroke. Nevertheless he finished the tournament, only coming in, however, second last.\n\nAt the Chess Olympiad of 1939 in Buenos Aires, during which the Second World War broke out, Capablanca undertook a final attempt to get from Alekhine an agreement for a return match, but once more in vain. When the French and Cuban teams met, both players took a rest day simply not to have to play against each other.\n\nOn the 7th March 1942 a second stroke befell Capablanca as he was kibitzing in the Manhattan Chess Club. He was taken to the Mount Sinai Hospital where he died on the 15th March 1942. The year before Emanuel Lasker had also died in the same hospital. After a public funeral in Havana, Capablanca found his final resting place in the local Colon Cemetery.\n**13. Duel of the cellmates**\n\n**The World Championship 1929:  \n_Alexander Alekhine against Efim Bogoljubow_**\n\nInstead of playing a return match against Capablanca, two years after winning the title Alekhine preferred to play a WCh title match against Efim Bogoljubow. This had already been agreed upon in 1928.\n\n**Efim Bogoljubow (1889-1952)**\n\nEfim Bogoljubow ( _Bogo lyubov_ , the Russian for 'loved by God') was born on the 14th April 1889 in Stanislavchik, near Kiev (the Russian empire, nowadays Ukraine). He first studied theology in a seminary for priests in Kiev and later changed to the Polytechnic Institute, where he began to study agriculture. It was not till he was 15 that Bogoljubow learned to play chess and he began at 18 to take a greater interest in it when he had met a few fellow students who were chess enthusiasts. In 1911 he shared first place in the championship of Kiev. In the same year he was fourth in the master tournament in Warsaw behind Rubinstein, Rotlevi and Flamberg.\n\nIn 1914 Bogoljubow applied to take part in the 19th DSB Congress in Mannheim and received an invitation. The tournament began on 20th July 1914 and was abandoned after the 11th round when the German Empire announced the mobilisation of its forces on 1st August 1914. The declaration of war followed on the same evening at 19.00. Germany and Russia were now at war with each other and the Russian chess players, including Alekhine and Bogoljubow, were interned, firstly in the central police station in Mannheim, then in the military prison of Ludwigshafen, then in Rastatt, where Alekhine and Bogoljubow shared a cell and passed their time with blindfold chess \u2013 they did not have a chess board. Later the Russians were moved to Baden-Baden, then to Triberg.\n\nFour of the Russian chess players, amongst them Alekhine, were released as being unfit for military service and deported to Switzerland. During their internment the remaining Russian players took part in eight tournaments, the first in Baden-Baden 1914, the others in Triberg (1914-1917). Bogoljubow won five of these tournaments.\n\nDuring the time of his internment in Germany Bogoljubow met his wife, the daughter of a teacher, and after the end of the First World War remained in Germany, settling in Triberg. After the end of the war he won the 1919 tournament in Berlin. In 1920 he was narrowly defeated in a match in Stockholm and Gothenburg by Akiba Rubinstein (4:5 and three draws), but won in Gothenburg a match against Aaron Nimzowitsch by 3:1. In 1922 Bogoljubow was victorious in the tournament in Bad Pistyan ahead of Alekhine. In 1923 in Karlsbad he shared first place with Alekhine and Maroczy. In 1924 he was finally convinced by Nikolai Krylenko, the Commissar and initiator of the future Soviet chess boom, to return to Russia, which had now become part of the 'Soviet Union'. Bogoljubow answered the call and won the USSR championships of 1924 and 1925. He finally even took the great Moscow tournament of 1925, the first international chess tournament in the new Soviet Union, ahead of Capablanca and Lasker. In 1926, however, he fled the country, apparently unhappy with the living conditions and political circumstances under the new authorities. Contemporaries later described Bogoljubow as a staunch anti-Bolshevik.\n\nAfter his 'flight' he became in the USSR a non-person \u2013 even the mention of his name was forbidden. Bogoljubow's later opportunism under the Nazi regime further strengthened the negative attitude of the authorities in the Soviet Union towards their prodigal son. According to communist usage, Bogoljubow's name was struck from all lists and even the entries of his tournament successes were removed. Since Bogoljubow harboured the idea of taking part in the tournament in Merano in 1926 \u2013 from now on this was forbidden to him as a Soviet citizen in Mussolini's Italy \u2013 he renounced his Russian citizenship and requested German citizenship. But he did not receive this till 1927.\n\nIn April\/May 1928 and around the turn of the year 1928\/29 Efim Bogoljubow and Max Euwe played two matches in various cities in the Netherlands, which on the initiative of FIDE president Alexander Rueb were described as the 'FIDE championship'. FIDE would have liked to take over the organisation the World Championships and established their 'FIDE champion' \u2013 the description World Champion was avoided if possible \u2013 as the challenger of the World Champion. But finally, Bogoljubow, who won the two matches against Euwe, was denied widespread recognition as the 'FIDE World Champion'. Between the two matches against Euwe Bogoljubow also won the tournament in Bad Kissingen, ahead of Capablanca, Euwe and Rubinstein.\n\nEven before his tournament victory in Bad Kissingen Bogoljubow had sent to Alekhine his challenge, which the World Champion answered positively in a letter of the 11th September 1928, but in doing so drawing attention to the conditions agreed in the London protocol, that is a match to six wins, 10 000 dollars stake to be provided by the challenger, and a time control of two and a half hours for 40 moves. But Bogoljubow obtained two changes. The number of games was fixed at a maximum of 30 and the victor was to be whoever reached 15\u00bd points and won six games. Should the challenger have this score without having managed six wins, the match would be continued until either he had reached the required number of victories or the title defender had equalised. In addition Alekhine contented himself with a fixed fee of 25200 Reichsmarks (plus expenses).\n\nOrganisation of the match was left to Bogoljubow. The latter had contacted various German spas and made to them suggestions as to refinancing. Baden-Baden and Bad Kissingen turned him down. The administrators of the spa in Wiesbaden, on the other hand, were very interested and had previously staged a few chess tournaments. They undertook to provide 12500 Reichsmarks of the total costs, of which 3000 Reichsmarks came as a subsidy from the Prussian ministry of the interior and 500 dollars (= 2100 Reichsmarks) from New York lawyer Herbert R. Limburg, the vicepresident of the Manhattan Chess Club.\n\nThe World Championship match between Alexander Alekhine and Efim Bogoljubow was finally held from 6th September till 12th November in the cities of Wiesbaden (games 1 to 8 and 24 and 25), Heidelberg (games 9 to 11), Berlin (games 12 to 17), The Hague (games 18 to 19 and 23), Rotterdam (game 20) and Amsterdam (games 21 to 22).\n\nAt the start Bogoljubow was able to keep up. Alekhine won the first game, then after two draws Bogoljubow equalised in the fourth game. The title defender won the fifth game, Bogoljubow equalised in the sixth. But then Alekhine won the seventh and eighth games and thus after the first stage in Wiesbaden he was leading by 4:2 in wins. There was then a pause of two weeks to give Alekhine the opportunity to attend the 6th FIDE congress in Venice as the French delegate to FIDE. Of the next three games in Heidelberg Alekhine won the tenth. The ninth and eleventh games ended as draws.\n\nThe following six games were then played in Berlin. Here the magistrate of the city of Berlin had donated 2000 Reichsmarks. A further 1000 Reichsmarks were provided by the owner of the Caf\u00e9 K\u00f6nig (Unter den Linden). The games were played in the Caf\u00e9 K\u00f6nig, in its 'hall of mirrors'. Other spectators could follow the games in two other rooms on large demonstration boards, which were operated by the masters August Babel and Ludwig Rellstab. The match jury was headed by Emanuel Lasker in person.\n\nThere was also an illustrious throng analysing in the press room, namely the Russian master Evgeny Znosko-Borovsky who had emigrated to France, Jacques Mieses, Hans Kmoch, Dr. Jakob Adolf Seitz, Wilhelm Orbach and the chess author Heinrich Ranneforth, editor of the Deutsche Schachzeitung and from 1902 to 1906 vice-president of the German Chess Federation. A few years later Dr. Seitz was among those players who were playing in the Chess Olympiad in Buenos Aires in 1939 and who remained in Argentina when the Second World War broke out in Europe. After the Second World War Dr. Seitz returned to Europe and from then on lived in Switzerland. The Jewish master from Offenbach Wilhelm Orbach, on the other hand, died in 1944 in the Auschwitz concentration camp.\n\nBogoljubow more or less completed the section of the match in Berlin on terms of equality, with wins in the 13th and 14th games, whilst Alekhine took the 12th and 16th. Game 15 was drawn. Alekhine then won the 17th game with an intermediate result of 8:4. 'Alekhine is extremely nervous', reported Dr. Seitz, 'even more so than usual. Already in the opening he spends a great deal of time, almost always ends up in extreme time-trouble and then constantly looks at the clock. When the opponent is thinking, he sometimes walks around the hall with long steps, and then sometimes stops in front of the demonstration board and looks at it attentively, as though hoping to notice something new... Alekhine gave up smoking a long time ago, but Bogoljubow constantly smokes thick 'Havanas' or cheap Swiss cigars.' Then the World Championship continued with two games in The Hague. Bogoljubow won the 18th game, Alekhine the 19th game. After that a game was played in Rotterdam, it was drawn. Alekhine then also won both games in Amsterdam. A further game in The Hague ended in a draw. After that the match returned to Wiesbaden, where the final two games were also drawn. The final score was 15\u00bd:9\u00bd (11:5 in wins) for Alekhine.\n\n **Alekhine \u2013 Bogoljubow**\n\nBerlin, 17th game  \n21st October 1929  \nGr\u00fcnfeld Indian Defence (D70)\n\n**1.d4 \u2658f6 2.c4 g6 3.f3**\n\nAn extravagant way to prepare e2-e4. The idea comes from Fritz S\u00e4misch. The more usual move is 3.\u2658c3 after which Black mainly continues with 3...\u2657g7 (King's Indian) or 3...d5 (Gr\u00fcnfeld Indian). After 4.cxd5 \u2658xd5 5.e4 \u2658xc3 6.bxc3 c5 Black achieves his desired structure.\n\n**3...d5 4.cxd5 \u2658xd5 5.e4**\n\nUnlike with 3.\u2658c3 there is nothing here on c3 for the \u2658d5 to take. That is the idea behind 3.f3. The white centre therefore remains more flexible.\n\n**5... \u2658b6 6.\u2657e3 \u2657g7 7.\u2658c3**\n\nIn 1929 this position was unknown territory. In the WCh match between Anand and Gelfand in 2012 this variation was once again up for discussion.\n\n**7... \u2658c6**\n\nNowadays the more usual way is first 7...0-0 8.\u2655d2 \u2658c6 9.0-0-0 and then 9...e5 or 9...f5.\n\n**8.d5**\n\nWith 8.\u2657b5 White can try to extract some profit from Black's 7th move, e.g.: 8...0-0 9.\u2658ge2 (9.\u2657xc6 bxc6 10.b3 e5 gives White nothing) 9...\u2658a5 10.b3 e5!?. But here too Black has enough counterplay. The main variation today is reached via 8.\u2655d2 0-0 9.\u2658ge2.\n\n**8... \u2658e5 9.\u2657d4**\n\nThe line 9.f4 \u2658g4 10.\u2657b5+ \u2657d7 11.\u2655xg4 \u2657xc3+ 12.bxc3 \u2657xb5 13.\u2657d4 leads to level play, after 13...\u2656f8 (or also 13...f6).\n\n**9...f6?**\n\nThe 'normal' and good move was 9...0-0. After 10.f4\n\na) 10...\u2658ed7 11.\u2657xg7 \u2654xg7 12.\u2655d4+ leads according to Alekhine to a strong attack for White. But Black has, after 12...\u2658f6 13.0-0-0 c6, enough counterplay for equality;\n\nb) 10...\u2658g4 11.\u2657xg7 \u2654xg7 12.\u2655d4+ \u2658f6 13.0-0-0 e6 results in a level game.\n\n**10.f4**\n\nMore accurate was first 10.a4 0-0 11.f4 \u2658f7 12.a5 \u2658d7 13.\u2658f3 with advantage to White.\n\n**10... \u2658f7 11.a4 e5!**\n\nThis counterstrike was not possible after 10.a4.\n\n**12.dxe6**\n\nAfter 12.fxe5 fxe5 13.\u2657e3 0-0 14.a5 \u2658d7 15.\u2658f3 \u2658f6 Black has sufficient counterplay.\n\n**12... \u2657xe6 13.a5 \u2658d7 14.a6 b6 15.\u2657b5 \u2655e7 16.\u2658ge2 c5**\n\n16...0-0-0 is followed by 17.\u2657c6. After 16...0-0 17.0-0 \u2658d6 18.\u2657c6 \u2656ac8 19.f5 gxf5 20.\u2658d5 White also has some pressure.\n\n**17. \u2657f2**\n\nAnother strong move was 17.f5!?, to divert the bishop from the defence of the d5-square and to enable \u2658d5.\n\n**17...0-0-0**\n\n17...0-0 was followed by 18.\u2658d5 \u2655d6 (18...\u2657xd5 19.\u2655xd5 \u2656fd8 20.0-0! \u2658f8 21.\u2655b7 \u2655e6 22.f5! and White wins: 22...gxf5 23.exf5 \u2655xf5 24.\u2657c4) 19.\u2655a4! \u2657xd5 20.\u2656d1 and wins.\n\n**18. \u2655a4?!**\n\nThe queen is aiming at the weak c6- and b7-squares. But it was more accurate to first play 18.\u2658d5! \u2655d6 (18...\u2657xd5 19.\u2655xd5 \u2658b8 20.\u2655a8!+\u2013) and then 19.\u2655a4+\u2013.\n\n**18...f5**\n\nWhite was threatening 19.f5 gxf5 20.exf5 \u2657xf5 21.\u2657d3! intending 21...\u2657xd3 22.\u2655c6+ \u2654b8 23.\u2655b7#.\n\n**19.e5 g5! 20. \u2657c4**\n\nAfter 20.g3 gxf4 21.gxf4 the sacrifice 21...\u2658dxe5! 22.fxe5 \u2657xe5 is more effective than in the game continuation, since the white king is somewhat exposed.\n\n**20... \u2658dxe5**\n\nAfter 20...\u2658b8 Alekhine was planning 21.\u2657xe6+ \u2655xe6 22.0-0 followed by b2-b4, with an attack.\n\n**21. \u2657xe6+ \u2655xe6 22.fxe5 \u2658xe5 23.0-0 \u2655c4**\n\nThe alternative was 23...\u2658d3 with wild complications, though they probably favoured White. Here too the best is 24.b4!.\n\n**24.b4 \u2655xb4**\n\n24...cxb4 is followed by the strong 25.\u2658b5! followed in turn by 26.\u2658xa7 and 27.\u2658c6.\n\n**25. \u2655c2**\n\nIntending \u2656a4 and \u2655xf5.\n\n**25... \u2658d3 26.\u2656fb1 \u2655c4 27.\u2656a4 \u2655e6 28.\u2658b5**\n\n**28... \u2654b8?**\n\nThis loses. The correct way was 28...\u2658xf2 29.\u2654xf2 (29.\u2658xa7+? \u2654b8 30.\u2655xc5 \u2656d1+ \u2013+) 29...\u2654b8 30.\u2658g3 \u2656hf8 31.\u2656a3! 'with a decisive strengthening of the positional pressure.' (Alekhine).\n\nHans M\u00fcller, however, pointed out that after 31...\u2655d5 the 'white win would have been a long, long way away'. Here it was also worth considering 31...c4 32.\u2654g1 \u2656d5 33.\u2656f3 \u2655c6 34.\u2658c3 \u2656d3 with an initiative for Black.\n\n**29. \u2658ed4 \u2655e4**\n\n29...\u2657xd4 30.\u2657xd4 \u2656xd4 31.\u2656xd4 \u2655e3+ 32.\u2654f1 cxd4 33.\u2655c7+ \u2654a8 34.\u2655b7#.\n\n**30. \u2658c3 \u2655e8 31.\u2655xd3 cxd4 32.\u2657xd4**\n\nThe immediate 32.\u2655f3 was also good: 32...\u2656d7 33.\u2656xd4 \u2657xd4 34.\u2657xd4 \u2656g8 35.\u2657xb6 axb6 36.\u2656xb6+ \u2654c7 37.\u2655c6+ \u2654d8 38.\u2656b8+ +\u2013.\n\n**32... \u2655e6 33.\u2655f3 \u2655f7 34.\u2657xb6!**\n\nBlack resigned in view of 34...axb6 35.\u2656xb6+ \u2654c8 36.a7.\n\nAfter the match Bogoljubow expressed himself as follows in an interview: 'No one now has any chances of defeating Alekhine in a match. Nimzowitsch has none at all. And I wouldn't advise Capablanca to play a return match, since after this new encounter his halo will completely fade. During the next four or five years I will follow his victories, and then we will play again. I do not consider myself completely defeated.'\n**14. Chess WCh in Nazi Germany**\n\n**The World Championship 1934:  \n_Alexander Alekhine against Efim Bogoljubow_**\n\nThe really one-sided match of 1929 between Alexander Alekhine and Efim Bogoljubow was followed five years later by a return match. Many commentators have pointed out that Alekhine probably did not pick the strongest of all possible challengers in Bogoljubow. But according to the calculations of the statistician Jeff Sonas, in 1929 on account of his successes Bogoljubow was nevertheless number four in the world ranking list behind Capablanca, Alekhine and Nimzowitsch. In 1934 he was still occupying fifth place behind Alekhine, Isaac Kashdan, Salo Flohr and Max Euwe.\n\nThe second match between Alekhine and Bogoljubow was however received with mutterings in the world of chess of the day. Alekhine's superiority in the WCh match of 1929 had been too clear. On 20th August 1933 Bogoljubow had sent his renewed challenge to Alekhine, but he had to wait six weeks for an answer since at that time Alekhine was in some unknown spot in the USA. Bogoljubow thought he could organise the second match against Alekhine in a German spa, but a change in the spa administration brought these plans to nothing.\n\nBogoljubow was going to drop his plans for the match, but then during a simultaneous display in Karlsruhe he got to know the leader of the chess federation of Baden, the member of parliament Herbert Kraft, who, with the help of the government of Baden, was able to bring onside as hosts and patrons for the match various places in South Germany. The contract between Alekhine and Bogoljubow was signed at the start of 1934. In December 1933 the chess federation of Baden, represented by Herbert Kraft, had already accepted contractually to host 13 games of the match, later it organised the whole match. The costs for the organisation were to have been 40000 Reichsmarks (in today's purchasing power approximately 200000 euros).\n\nThe organisation was entrusted to Th. A. Bergmann and A. Herrmann, who negotiated with the various cities to find hosts. In addition they were supported by Hans Schemm, minister of state and Bavarian minister of culture and Gauleiter of the Ostmark, and also the chess enthusiast and minister of state Hans Frank, at that time Commissioner of justice for Germany. Kraft and Schemm had also campaigned for chess in schools and created the hope that 'young people would become a loyal following for the German game of war', as Bogoljubow formulated it in the vocabulary of the era in his tournament book of 1935. But the match between 'the two Russians on German soil' did not create much enthusiasm throughout the Greater German Chess Federation, which to a large extent had already been captivated by the new national-socialist spirit. After his first title defence against Bogoljubow Alekhine had won in superb fashion the tournament in San Remo 1930, where in 15 games he in fact conceded only two draws. Aaron Nimzowitsch and Akiba Rubinstein were distanced by 3\u00bd respectively 4 points. In Bled 1931 his tournament victory was already clear after the tenth round. After 1933, however, Alekhine began drinking, finally indulging in proper excessive consumption of alcohol, and became more and more ego-centric. Moreover he was a chain-smoker of 80 to 100 cigarettes per day. It is alleged that one of his simultaneous exhibitions had to be broken off after the World Champion had urinated on the floor. At times he was of the opinion that chess games could be won by hypnosis.\n\nThe second match for the World Championship between Alexander Alekhine and Efim Bogoljubow took place from the 1st April till the 14th June 1934 in twelve German towns: Baden-Baden (games 1 to 3), Villingen (games 4 to 5, in the Hotel Blume Post), Freiburg (games 6 to 8), Pforzheim (games 9 to 10), Stuttgart (games 11 and 12), Munich (games 13 to 15), Bayreuth (game 16), Bad Kissingen (games 17 and 18), Nuremberg (games 19 and 20), Karlsruhe (game 21) Mannheim (games 22 and 23) and Berlin (games 25 and 26). The winner was to be the one who had the most points after 30 games. If the score were 15:15 Alekhine was to remain World Champion. The time control was set at two and a half hours for 40 moves. The games were adjourned after 40 moves. Hans M\u00fcller acted as second for Bogoljubow, and for Alekhine Hans Kmoch.\n\nThe opening of the match took place on Holy Saturday 1934 in the main spa hall in Baden-Baden. FIDE president Alexander Rueb supervised the drawing of lots which gave Bogoljubow the white pieces in the first game. The course of the match was even clearer than that of 1929 and it came to an end after 26 games, because Alekhine's lead was so great that he could no longer be overtaken.\n\nIn the first game there was a curious incident when in a rook ending Alekhine claimed a draw on account of a threefold repetition of the position and arbiter Albert Hild granted the claim. Bogoljubow did not disagree and it was only after the game that he noticed that there was no threefold repetition of the position. Analyses showed that the position at which the game was agreed drawn was in fact winning for Bogoljubow. Arbiter Hild was then replaced by B. Bader.\n\nAlekhine won the second, the fourth and the ninth games. In game ten Bogoljubow was able to shorten the lead with a win. Alekhine, however, then won the eleventh, 16th, 17th and 21st games and led by 7:1 with 13 draws. On the 9th May, the rest day in the part of the match played in Munich (games 13 to 15), the two players put on simultaneous displays in the B\u00fcrgerbr\u00e4ukeller in front of 700 spectators against respectively 50 and 51 opponents. In the 23rd and 24th games Bogoljubow was victorious, but in the 25th it was then Alekhine again. With a draw in the 26th game the World Champion again secured the defence of his title. During the match Bogoljubow had wasted a series of good positions. In his tournament book Emanuel Lasker's judgement was that Bogoljubow tired too quickly in complicated positions because of a lack of training.\n\n **Alekhine \u2013 Bogoljubow**\n\nBayreuth, 16th game  \n13th May 1934  \nRuy Lopez (C77)\n\n**1.e4 e5 2. \u2658f3 \u2658c6 3.\u2657b5 a6 4.\u2657a4**\n\nThe more common version of the Exchange Variation is 4.\u2657xc6 dxc6. Here Black still has...f7-f6 at his disposal for the defence of the e5-pawn.\n\n**4... \u2658f6 5.\u2657xc6 dxc6**\n\nIn the Delayed Exchange Variation of the Ruy Lopez White 'makes a present' of a tempo to Black, since unlike in the Exchange Variation after 4.\u2657xc6 the knight is there already on f6. But White is hoping that in this structure the knight is misplaced on f6.\n\n**6. \u2658c3**\n\n6.d3 is more flexible, to develop the knight to d2: 6...\u2657d6 7.\u2658bd2 \u2657e6 8.b3 \u2658d7 9.\u2657b2 c5 10.\u2658c4 f6.\n\n**6... \u2657d6 7.d3**\n\n7.d4 has also been played here.\n\n**7...c5 8.h3 \u2657e6 9.\u2657e3 h6**\n\nBogoljubow suggested 9...b5!? as a possible improvement.\n\n**10.a4 c4**\n\nBlack would like to resolve his doubled pawns. Alternatives were 10...\u2658d7 to bring the knight via b8 to c6, or 10... \u2655d7!? to prepare queenside castling.\n\n**11.d4 exd4 12. \u2657xd4 \u2657b4 13.0-0 c6**\n\nThis weakens the d6-square. 13...0-0 was very playable or also 13...\u2657xc3 14.\u2657xc3 \u2658xe4 15.\u2657xg7 \u2656g8=.\n\n**14.e5 \u2658d5**\n\nAfter 14...\u2657xc3 Alekhine gives 15.exf6! \u2657xd4 16.\u2658xd4 \u2655xf6 17.\u2658xe6 fxe6 18.\u2655h5+ with advantage. After 18...g6 19.\u2655c5 0-0-0 20.\u2655xc4 \u2656d4 the game is, however, more or less level.\n\n**15. \u2658e4 \u2658f4**\n\n15...0-0!? with equality.\n\n**16. \u2657c5**\n\nTo further weaken the d6-square.\n\n**16... \u2657xc5 17.\u2655xd8+ \u2656xd8 18.\u2658xc5 b6!**\n\nBogoljubow considered this a good move, Alekhine thought it a mistake. The title defender saw an advantage for White after 18...\u2656b8 19.\u2658xe6 \u2658xe6 20.a5. White is planning \u2656a4 and \u2658d2. Mieses suggested 18...\u2657c8!? as an alternative to the move in the game.\n\n**19. \u2658b7**\n\nThe line 19.\u2658xa6 \u2657c8 (19...0-0!? with counterplay) 20.\u2658c7+ \u2654d7 21.\u2658a8! \u2658d5 22.a5! bxa5 23.\u2656xa5 \u2657b7 24.\u2656a7 \u2656b8 25.\u2656e1! c5 (a better move is 25...\u2656he8) 26.e6+ results according to Alekhine in a winning position for White.\n\n**19... \u2656d7 20.\u2658d6+ \u2654e7**\n\nThe manoeuvre 20...\u2654d8 21.\u2656fe1 \u2654c7 was cleverer, taking the king out of the range of the white knights.\n\n**21. \u2658d4 \u2657d5 22.g3!?**\n\nAlekhine solves the problem of his g-pawn in radical fashion. Bogoljubow's suggestion 22.f3 is met by Black, according to Alekhine, by 22...g6! which deprives the white knights of the f5-square.\n\n**22... \u2658xh3+ 23.\u2654h2 \u2658g5 24.f4 \u2658e4 25.\u26586f5+ \u2654d8**\n\nPerhaps a better way was 25...\u2654f8 26.\u2658e3 \u2654g8 27.f5 \u2654h7 28.e6 after which Bogoljubow sees an advantage for Black, whilst Alekhine thinks that White has enough compensation for the material disadvantage. After 28...\u2657xe6 29.fxe6 \u2656xd4 30.exf7 \u2656f8 White, however, has to fight for the draw.\n\n**26. \u2658xg7 f6 27.\u2656ad1 \u2654c8**\n\n27...\u2656xg7? 28.\u2658e6+ +\u2013.\n\n**28. \u2658df5 fxe5**\n\nThe alternative was 28...\u2656g8 29.\u2658h5 fxe5 30.fxe5 \u2656e8 (30...\u2657e6!?) 31.\u2658f6 \u2658xf6 32.exf6 \u2656e2+ 33.\u2654h3 with a better endgame for White according to Alekhine: 33...\u2657g2+ (33...\u2657e6 34.g4) 34.\u2654g4 \u2657xf1 35.\u2656xd7 \u2654xd7 36.f7 \u2656e8 37.fxe8\u2655\\+ \u2654xe8 38.c3 with the weakness of h6 (Alekhine). But after 38...\u2657e2+ 39.\u2654f4 h5 Black seems to be able to hold the game.\n\n**29.fxe5**\n\n**29... \u2656g8?**\n\nThis loses.\n\n29...\u2658g5 30.\u2656xd5 (intending e5-e6) 30...\u2656xd5 31.\u2658e7+ \u2654d7 3 2.\u2658xd5 cxd5 33.\u2656f6 \u2654c7 34.e6 \u2654d6 holds the game.\n\n**30.e6! \u2656dxg7 31.\u2658xg7 \u2656xg7**\n\n**32. \u2656xd5! cxd5 33.\u2656f8+ \u2654c7 34.\u2656f7+ \u2654d6**\n\n34...\u2656xf7 35.exf7 and the pawn gets through.\n\n**35. \u2656xg7 \u2654xe6 36.\u2656g6+ \u2654e5**\n\n36...\u2658f6 37.\u2656xh6 d4 38.\u2654g2 b5 39.g4+\u2013.\n\n**37. \u2654g2 b5 38.a5! d4 39.\u2656xa6 b4 40.\u2654f3 c3 41.bxc3 bxc3 42.\u2656e6+ \u2654xe6 43.\u2654xe4**\n\nBlack resigned.\n\nAfter the 16th game, which was played in Bayreuth, Bogoljubow and Nimzowitsch were sitting at dinner in the hotel. At the same time there was a Nazi meeting locally and Bayreuth was full of people in uniforms, including the hotel restaurant. Being a Jew, Nimzowitsch felt extremely uneasy in this environment. Bogoljubow on the other hand was enjoying the situation and, in order to annoy him, kept on suggesting to Nimzowitsch that the (not kosher) pork chops were particularly tasty. Bogoljubow has always been criticised for having been too close to the Nazis. At the tournament of 1936 in Zandvoort Bogoljubow insisted to the Dutch organisers on having a swastika banner at his table as the flag of his country. The organisers thereupon did without any flags.\n\nIn 1938 he is reputed to have joined the NSDAP, probably in order to guarantee his two daughters better prospects of a university place. Some contemporaries, such as his compatriot Bohatyrchuk, describe Bogoljubow on the other hand as unpolitical and even somewhat against the Nazis. During the war, however, Bogoljubow played in tournaments in the 'Generalgouvernement' in Poland headed by Hans Frank and worked in the administration of Krakow as a translator. For that reason FIDE first passed him over in 1950 in the award of grandmaster titles. But in 1951 he was named as a grandmaster.\n\nEfim Bogoljubow\n\nContemporaries describe Bogoljubow as jovial and friendly in character with a somewhat crude sense of humour. Hans Kmoch reported that at the tournament in Zandvoort Bogoljubow had asked the Jewish player Fine straight out whether it had made him happy that the German airship Hindenburg had come to grief in New Jersey. Fine was shocked and left speechless. Later Fine is said to have claimed that Bogoljubow had arranged for it that his potential rivals in Nazi Germany were sent to concentration camps. Fine never gave any answer to the question as to how he knew this.\n\nOn the 18th June 1952 Bogoljubow died in Triberg after a short but serious illness, probably as a result of liver cancer (Kmoch). Shortly before, though visibly marked by the illness, he had taken part in a tournament in Belgrade. Bogoljubow's bestknown legacy apart from his games is his declaration: 'When I have White, I win because I have White. When I have Black, I win because I am Bogoljubow.'\n\nDuring his second match against Bogoljubow there were already negotiations taking place for a match for the World Championship between Alekhine and Euwe.\n**15. Professor against alcoholic**\n\n**The World Championship 1935:  \n_Alexander Alekhine against Max Euwe_**\n\nMachgielis ('Max') Euwe was born on the 20th May 1901 in Watergraafsmeer near Amsterdam, the son of the teacher Cornelis Euwe and his wife Elisabeth. The family lived there in a house at 45 Ringdijk. Max Euwe had an older brother, Willem, and four younger siblings: Annie, Kees, Tiny and Bob. With a teacher's annual salary of 800 guilders the family's material circumstances were such that they just got by. So, in order to save money, their father preferred to walk for long distances rather than travelling by bus or train.\n\n**Max Euwe (1901-1981)**\n\nThe parents, but especially the mother, were both great chess lovers and taught their son Max the game when he was four years old. Max Euwe also learned to read and write before being of school age, meaning that in his primary school, the Van Ostade School in Amsterdam Oud-Zuid, he was allowed to skip the first year. In the meantime the family had moved to the Amsterdam suburb of De Pijp and lived in the Jan Steen Straat.\n\nThe game of chess interested Euwe more and more and so he joined the 'Chess and Draughts Club'. At the age of nine he wanted to play in his first chess tournament. This was held in a bar 'De Ruyter'. A guilder was required as an entrance fee. Since Euwe, however, had only 50 cents on him, he had to postpone participation in his first chess tournament. But in the following year, 1911, he had enough money in his pocket to pay the full entrance fee in 'De Ruyter' and with three wins in three games he won his group. At this time, moreover, football interested the young Euwe more than chess. He played regularly in football tournaments and throughout his life he never lost interest in that sport. At 12 Euwe became a member of the Amsterdam Chess Society. In 1913 Frank Marshall was a guest there and gave a simultaneous. Euwe was only allowed to spectate, but in the following year he defeated Jan Willem Te Kolst\u00e9, when the latter also gave a simultaneous in the chess society.\n\nAfter the First World War a series of chess masters sought out the Netherlands, which had to a large extent emerged undamaged from the war, and better living conditions could be found there than at home. As well as Lasker, regular visitors to the relevant Dutch chess centres (of which the best known was 'De Roode Leeuw', a hotel with a restaurant in Amsterdam, 93-94 Damrak) included Richard R\u00e9ti, Savielly Tartakower and Geza Maroczy. Their presence proved fruitful for the chess life of the Netherlands.\n\nIn 1918 Euwe took his final secondary exam and started on a study of mathematics. In 1926 he took his doctorate under Roland Weitzenb\u00f6ck, who came from Austria and was himself a good chess player. At times Weitzenb\u00f6ck had been president of the chess club in Blaricum where he lived. From 1926 to 1940 and then again after 1945 Euwe worked as a teacher of mathematics in Winterswijk, Rotterdam and at a girls secondary school in Amsterdam. In 1921 he won for the first time the national championship of the Netherlands, a success which he would repeat another eleven times up to 1955.\n\nAfter completing his doctoral thesis Euwe played his first match against Alekhine at the turn of 1926\/27 and narrowly lost by 2:3 with 5 draws. In 1928 he won the 'Amateur World Championship' in The Hague. In the same year he shared third place in Bad Kissingen behind Bogoljubow and Capablanca. In 1930\/31 Euwe won the Hastings tournament ahead of Capablanca. Euwe drew a match against Salo Flohr in Amsterdam and Karlsbad (3:3 with ten draws). At the tournament in Berne he finished second behind Alekhine.\n\nIn 1934 Euwe felt strong enough to challenge Alekhine to a match for the World Championship. Dutch chess lovers founded the 'Nationaal Nederlandsch Comit\u00e9 Wereldkampioenschap Schaken', to raise the prize money for the match against Alekhine. On the 28th May 1934, that is to say before the second match between Alekhine and Bogoljubow was finished, Euwe and Alekhine signed the contract which settled the conditions for the forthcoming match. Before the match Alekhine was considered the clear favourite and even Euwe only saw himself as an outsider.\n\nThe World Championship match between Alexander Alekhine and Max Euwe took place in 13 cities in the Netherlands from 3rd October till 15th December 1935: in Amsterdam, Delft, Rotterdam, Utrecht, Gouda, The Hague, Groningen, Baarn, 's-Hertogenbosch, Eindhoven, Zeist, Ermelo and Zandvoort, and was for 30 games. The time control was two and a half hours for 40 moves. Alekhine was supported by his second Salo Landau. Landau was born in 1903 in Bochnia, Galicia, then Austria-Hungary, came in the First World War via Vienna to the Netherlands and in the 1930s was the best player there after Euwe. After the occupation of the Netherlands by the German army Landau, together with his wife and his daughter, was caught fleeing in 1942 and sent to the Auschwitz concentration camp and then to a work camp near Gr\u00e4ditz (Grodziszcze). From there he managed to have smuggled out a message to his friend Max Euwe, begging the latter for help. Euwe turned to Alekhine, but the latter either could not or would not commit himself for his former second. Landau's wife Susanne and his daughter Henriette Ren\u00e9e were murdered on the 14th October 1944 in Auschwitz. Landau himself died in the same year in Gr\u00e4ditz.\n\nEuwe's official second was Geza Maroczy. Hans Kmoch, who had previously been expected to assume this task, acted instead as principal arbiter for the match. Salo Flohr and Reuben Fine supported Euwe in his preparation. Alekhine received an appearance fee of 10000 guilders, which was raised by the cities or organisers purchasing individual rounds of the match. Enthusiasm for chess in the Netherlands was overwhelming during the match. There were daily reports in the press. Amongst the foreign visitors who were following the games were Capablanca, Lasker, Bernstein, Flohr, the 70 year old Jacques Mieses, who was the correspondent for Reuters, Savielly Tartakower, who had at his command five languages and published daily analyses in the Amsterdam paper _De Telegraaf_ and several other newspapers, and other wellknown masters. The first game was played on the 3rd October 1935 in Amsterdam in the congress hall of the 'Carlton Hotel'. The start of the game was announced for 17.30, but things did not take place with quite the same regard for punctuality. The first move was made by the mayor of Amsterdam. Alekhine won in 30 moves.\n\nThe second game, on the 6th October, was also played in Amsterdam, this time in the 'Militiezaal'. This time Euwe won. The third game, 8th October, was again played in the 'Carlton Hotel' and again went Alekhine's way. The venue for game four on the 10th October was the 'Witte Brug Hotel' in The Hague. In some sources the official name 's-Gravenhage is employed for The Hague. Euwe lost with white and was now trailing 1:3. There followed two draws in the fifth game (12th October, Delft Student Club) and the sixth game (Rotterdam, Hotel Coomans, 15th October).\n\nThe seventh game was played on the 17th October in the Museum for Art and Science in Utrecht. Alekhine, who was reported to have enjoyed his evenings during the match playing bridge and who frequently brought to the games his Siamese cat 'Chess', which he then allowed to snuffle around on the chess table, surprised Euwe with a speculative opening innovation and won the game for a score of 2:5.\n\nOn the 20th October the Amsterdam 'Militiezaal' was once more the scene of the action. This time Euwe came up with a novelty and won the eighth game. The ninth game (22th October) was staged in the assembly hall of the Upper Girls School in the Reijnier Vinkeleskade (Amsterdam). Euwe worked at this school as a mathematics teacher. He again lost as Black to Alekhine straight out of the opening. With a score of 6:3 the match appeared to be decided already. The tenth game followed on the 24th October in the 'Kunstmin Zaal' of the student club 'Ons Genoegen' in Gouda. Alekhine chose the same variation of the Slav Defence to the Queen's Gambit as in the eighth game, came up with a not particularly good novelty and lost this game too.\n\nFor the eleventh game on the 27th October they moved back to the hotel 'De Witte Brug' in The Hague. The game ended in a draw. In game 12 (29th October), played in the chess club of the stock exchange in Amsterdam, Alekhine speculatively sacrificed a pawn, later a piece, and played on for a long time in a lost position before finally resigning. The 13th game was played on the 1st November in the Watergraafsmeer chess club (in the back room of the Caf\u00e9 Van Klaveren, 88-90 Middenweg). Euwe again held the draw with black in this game. On the 2nd November the 14th game followed in the Japanese Room of the Frigge Hotel in Groningen. Alekhine fell badly behind right in the opening and lost once more with black. Euwe had drawn level. Before game 15, arranged for November 5th in the 'Badhotel' in Baarn, Alekhine spoke to Lasker and told him: 'I want to start playing more solidly now, more based on logic and less on foolhardiness.'\n\nAlekhine's intention was as yet nowhere to be seen in game 15. The game was adjourned after an adventurous first part and it was resumed on the following day. Alekhine, whose car broke down on the journey from Amsterdam to Baarn, arrived half an hour late. The game was actually lost for Euwe, but in his analysis the challenger had discovered an interesting defensive plan. Alekhine was taken by surprise by the idea and gave away a possible win. But the World Champion then managed one in the 16th game, played on 7th November in the casino in 's-Hertogenbosch ('Den Bosch'). The 17th game (9th November, Philips Leisure Hall in Eindhoven) and the 18th game (12th November, Amsterdam 'Militiezaal') each came to a rapid end with the sharing of the point. The 19th game (Figi Hotel in Zeist, 14th November) went to Alekhine, thus restoring a two point lead (10\u00bd:8\u00bd). Euwe, however, struck back with victories in games 20 (16th November, Amsterdam 'Militiezaal') and 21 (19th November in the building of the Christelijk Volksbelang, Ermelo) and once more levelled the score. The 21st game was overshadowed by an incident which will be described below.\n\nWas there going to be a sensation? Next came three draws (game 22, on the 24th November in the De Witte Brug Hotel in The Hague, game 23 on the 26th November in the stock exchange chess club in Amsterdam, game 24 on the 28th November in the student club 'Eensgezindheit' in Delft), with Alekhine giving away a possible win in game 24. The World Champion now called on an additional second, the Austrian Ernst Klein, but the latter arrived too late to turn the match in a different direction.\n\nThen Euwe in fact won the 25th game (on the 1st December in an absolutely packed-out 'Militiezaal' in Amsterdam) and the 26th (begun on the 3rd December in the Monopole Hall in Zandvoort, continued on the 4th December in the 'Militiezaal' Amsterdam) and four games before the finish he had a lead of two points. Alekhine shortened this with a win in the 27th game (on the 6th December in the De Witte Brug Hotel, The Hague). There followed two draws (game 28 on the 8th December in the 'Militiezaal' in Amsterdam, game 29 on the 12th December in the same place). In the last game (Bellevue in Amsterdam on the 15th December) Euwe finally conceded a draw in a winning position, thus becoming the new World Champion.\n\nAlekhine's defeat was to a large extent ascribed to his alcoholism. Often enough he drank before his games and even during the games, as everyone could see, for example at the tournament in Zurich 1934. During his match against Euwe Alekhine had probably been drinking (according to Euwe) before game 18 (drawn), certainly before game 21 (Euwe won) and before the 30th game (drawn). But Euwe admitted, perhaps out of politeness and respect for his opponent, that possible a wrong impression might have been created because despite his poor eyesight Alekhine did not wear glasses and therefore had difficulty moving the pieces.\n\nBut in his book 'Caissas Weltreich' Euwe later described how at that time people spelled Alekhine's name (in its German spelling): 'AL as in alcohol, JE as in jenever (Dutch gin), CH as in Champagne and IN as in Ingwerbier (i.e. ginger beer).' For the 21st game Alekhine turned up late and so drunk that everyone could see it. He instigated a fight, swore at Euwe, refused to play and practically had to be forced to start the game. He then played it like a madman. During the game he continually knocked over pieces when making his moves and Euwe left the room as often as possible on account of the strong smell of alcohol. Although Alekhine's consumption of alcohol during the match was obvious \u2013 according to a report by Emil Diemer, during the match he is supposed to have spent 800 guilders on Dutch gin in his hotel over six weeks \u2013 this was at first not mentioned in the reporting of the World Championship. But after the incidents in the 21st game the subject was expatiated upon in detail in the general press, whilst the chess journalists had not picked up on it at all, because for reasons of cost they were following the game by teleprinter in Amsterdam. As a result of the press reports Alekhine even received threatening letters. Euwe thus even suggested playing the next games outside of the Netherlands.\n\nFor the final game in Amsterdam a large hall had been rented, which was totally filled with 5000 spectators. Even more chess enthusiasts were waiting in front of the entrance where a policeman, armed with a travelling chess set and the latest position, prevented access. The 26th game, with which Euwe raised his lead to two points, went down in the history of chess as the 'pearl of Zandvoort':\n\n **Euwe \u2013 Alekhine**\n\nZandvoort, 26th game  \n3rd December 1935  \nDutch Defence (A90)\n\n**1.d4 e6 2.c4**\n\nAfter 2.e4 d5 we would have the French Defence.\n\n**2...f5**\n\nAlekhine was behind in the match by one game and had to take something of a risk. The Dutch Defence allows Black to fight for the initiative.\n\n**3.g3 \u2657b4+ 4.\u2657d2 \u2657e7**\n\nThis loses half a tempo, but the \u2657d2 also has to find a better square.\n\n**5. \u2657g2 \u2658f6 6.\u2658c3 0-0 7.\u2658f3 \u2658e4**\n\nWith 7...d5 Black can aim for the so-called Stonewall setup: 8.0-0 c6 9.\u2657f4 \u2658e4 etc.\n\n**8.0-0 b6**\n\n8...\u2657f6 was played in the 24th game.\n\n**9. \u2655c2**\n\nIt was also worth considering 9.\u2658xe4!? fxe4 10.\u2658e5 or 9.\u2658e5 \u2658xd2 10.\u2655xd2 c6 11.\u2656ad1 intending 11...d5 12.\u2658xc6 \u2658xc6 13.cxd5 or even 9.d5.\n\n**9... \u2657b7**\n\nThis position is also known from the Queen's Indian Defence.\n\n**10. \u2658e5**\n\n10.d5!? is a typical idea here too.\n\n**10... \u2658xc3**\n\n10...d6? would be followed by 11.\u2658xe4! fxe4 (11...dxe5 12.\u2658f6+) 12.\u2657xe4 \u2657xe4 13.\u2655xe4 c6 14.\u2658xc6 \u2658xc6 15.\u2655xc6 and White has won two pawns.\n\n**11. \u2657xc3**\n\nAccording to Euwe 11.\u2657xb7 was not as good: 11...\u2658xe2+ 12.\u2654g2 \u2658xd4 13.\u2655d3 \u2658bc6 14.\u2658xc6 \u2658xc6 (14...dxc6!? 15.\u2657xa8 \u2655xa8 16.\u2655xd4?? c5+ \u2013+) 15.\u2657xa8 \u2655xa8 intending 16.\u2655xd7?? \u2658e5+ \u2013+.\n\n**11... \u2657xg2 12.\u2654xg2 \u2655c8 13.d5 d6 14.\u2658d3 e5 15.\u2654h1 c6 16.\u2655b3 \u2654h8**\n\nWhite was threatening 17.c5 bxc5 18.\u2658xe5 intending 18...dxe5 19.d6+. After 16...c5 17.f4 e4 Euwe was planning 18.\u2658e1 intending \u2658g2-e3 with good play.\n\n**17.f4 e4 18. \u2658b4!**\n\nNow the threat is 19.dxc6 and 20.\u2658d5.\n\n**18...c5 19. \u2658c2 \u2658d7 20.\u2658e3 \u2657f6**\n\nAccording to Flohr 20...\u2658f6 was more prudent.\n\n**21. \u2658xf5!?**\n\nThis came as a surprise to Alekhine. White gives up a piece for three pawns and an attack.\n\n**21... \u2657xc3 22.\u2658xd6 \u2655b8 23.\u2658xe4 \u2657f6 24.\u2658d2**\n\nIntending e2-e4-e5.\n\n**24...g5!**\n\nBlack could also aim for counterplay with 24...b5!?.\n\n**25.e4 gxf4 26.gxf4 \u2657d4 27.e5 \u2655e8 28.e6 \u2656g8**\n\nAfter 28...\u2658f6 29.\u2658f3 \u2658g4 30.\u2658xd4 cxd4 31.\u2655d3 Black also gets counterplay: 31...\u2655h5 32.\u2655xd4+ \u2656f6!.\n\n**29. \u2658f3?!**\n\n29.exd7? would be over-hasty on account of 29...\u2655e2! with a double attack. After 29.\u2655h3! \u2658f6 30.\u2658f3 \u2657xb2 31.\u2656ab1, however, Euwe considered White to have a decisive advantage.\n\n**29... \u2655g6 30.\u2656g1**\n\n30.\u2658g5 \u2658e5! with counterplay.\n\n**30... \u2657xg1 31.\u2656xg1 \u2655f6**\n\nThis move was criticised by contemporary commentators, but does not yet lose the game as was thought at the time. Nevertheless Flohr's suggestion is simpler: 31...\u2655f5!. The difference compared to the move in the game consists of the fact that White cannot now get to h3 with the queen. 32.exd7 (32.\u2658g5 h6!) 32...\u2656xg1+ 33.\u2654xg1 \u2655xd7 34.\u2654f2=.\n\n**32. \u2658g5!**\n\n**32... \u2656g7?**\n\nThis move causes Black difficulties.\n\na) 32...h6? did not help, according to Euwe: 33.\u2658f7+ (33.exd7 is also good) 33...\u2654h7 34.\u2655d3+ \u2656g6 35.\u2658e5! \u2658xe5 (35...\u2658f8 36.e7!) 36.fxe5 \u2655g7 37.d6 intending 38.\u2655xg6+ \u2656xg6 39.\u2656xg6 \u2654xg6 40.d7 and 41.e7+\u2013 (Euwe). And 32...\u2656xg5 33.fxg5 \u2655d4 is followed by 34.\u2655c3+\u2013 (Euwe);\n\nb) 32...\u2656g6! holds the game: 33.exd7 \u2655xf4 34.\u2655c3+ \u2654g8 35.\u2655e1 \u2656xg5 36.\u2656xg5+ \u2655xg5 37.\u2655e8+ \u2654g7 38.\u2655xa8 \u2655c1+ 39.\u2654g2 \u2655d2+ with perpetual check.\n\n**33.exd7 \u2656xd7 34.\u2655e3 \u2656e7 35.\u2658e6 \u2656f8**\n\nIn the event of 35...\u2655xb2 the advance of the d-pawn wins: 36.d6! \u2656ee8 (36...\u2656d7 37.\u2658c7 \u2656f8 38.\u2655e5+ +\u2013) 37.d7 \u2656e7 38.d8\u2655\\+ \u2656xd8 39.\u2658xd8 with a win, according to Euwe, since 39...\u2656xe3?? is followed by 40.\u2658f7#.\n\n**36. \u2655e5 \u2655xe5 37.fxe5 \u2656f5**\n\nA better move according to Euwe was 37...\u2656xe6 38.dxe6 \u2656f5! 39.\u2656e1 \u2654g8 40.\u2656e3! after which Euwe nevertheless saw the endgame as won for White. But it is open to question whether this judgement also really still stands after 40...\u2654f8 41.\u2656a3 a5 42.\u2656b3 \u2656f4.\n\n**38. \u2656e1**\n\nEuwe gave 38.\u2656g5 as better. After 38... \u2656f2, which Euwe did not take into account, White is better but not winning.\n\n**38...h6?!**\n\nEuwe indicated 38...\u2656xe6! as better, but nevertheless held the position to be won after 39.dxe6 \u2654g8 40.\u2656e3!. But 38...\u2654g8 39.\u2656g1+ \u2654h8 (39...\u2654f7 40.\u2658d8+ \u2654f8 41.\u2658c6+\u2013 Euwe) 40.\u2656g5 (40.\u2656e1 \u2654g8=) 40...\u2656f2! holds the game.\n\n**39. \u2658d8 \u2656f2 40.e6 \u2656d2**\n\nThe game was adjourned here. The position is now won for White.\n\n**41. \u2658c6 \u2656e8 42.e7 b5 43.\u2658d8 \u2654g7 44.\u2658b7 \u2654f6 45.\u2656e6+ \u2654g5 46.\u2658d6 \u2656xe7 47.\u2658e4+**\n\n**1-0**\n\nIn 1978 Euwe spoke in an interview about his preparation for the match against Alekhine: 'Just before our match we played in a strong tournament in Zurich, which Alekhine won, but I beat him. I finished second, with 12 out of 15, after losing to Lasker in the first round. In analysing the games, we came to the conclusion that Alekhine's superiority over other masters was his opening knowledge. If he could not get the initiative or some advantage in the opening, he was willing to enter complications to try to muddy the water. So I went to Vienna for a few months to study Becker's files on the openings, which were the most complete and up to date at that time. Besides Kmoch, who was an expert in the openings, I also had Maroczy's help, mostly in the endgame.'\n\nFor Alekhine the result of the match was doubly bitter. At a point when he still had a clear lead in the match, he had contacted Nikolai Krylenko in Moscow and made efforts to get an invitation to the planned tournament in Moscow 1936. This was also supposed to be the attempt to clear things up with his homeland. Alekhine sent to Krylenko, who was also the publisher of the chess magazine _64_ , a message of greeting for publication: 'It is not only as a chess player with many years of experience, but also as a person who is able to appreciate the enormous significance of what has been achieved in all aspects of cultural life in the USSR, that I send my sincere greetings to all the chess players in the USSR on the 18th anniversary of the October Revolution. Al. Alekhine.' However, Alekhine had expected to appear in Moscow as World Champion. Now, however, that he had lost the title, his 'market value' was clearly less and of his own accord he stopped making efforts to obtain an invitation.\n\nThanks to Euwe's success interest in chess grew enormously in the Netherlands. Up until 1934 the _Koninklijke Nederlandse Schaakbond_ (founded in 1873) could count between 3000 and 4000 registered members. After Euwe's victory in the WCh match against Alekhine the number of registered chess players in the Netherlands soared in 1935 to 11 000. During the war years too the number remained generally stable and later, when Euwe almost won the tournament in Groningen 1946, where he finally finished second behind Botvinnik and ahead of Smyslov, the number of registered chess players even rose to 17 000.\n**16. The title was only on loan**\n\n**The World Championship 1937:  \n_Max Euwe against Alexander Alekhine_**\n\nTwo years after Alekhine lost his title to Euwe a return match took place in the Netherlands. Like the first match, it was for 30 games, with the victor needing to have six wins and a score of 15\u00bd points. The match was held from 5th October to the 4th December 1937 in the cities of The Hague, Rotterdam, Amsterdam, Haarlem, Leiden, Groningen and Delft.\n\nUnlike in the first match, this time Euwe was considered the favourite. In his preparation for the return match and for the match itself Alekhine had imposed upon himself absolute abstinence from alcohol. And he controlled the second WCh match against Euwe much better than the first. To be sure, the first and also the fifth games went to Euwe, but Alekhine equalised in the second and sixth games respectively and took the lead with wins in the seventh and eighth games. The third, the fourth and then also the ninth games ended in draws, but then Alekhine got going again with a victory in the tenth game and was then clearly in the lead by three wins.\n\nIn the phase between games 11 and 20 Euwe gradually stabilised the situation and was able to decide the thirteenth and seventeenth games in his favour. During this phase Alekhine had only one win, in the 14th game, and thus the title defender was able to shorten his opponent's lead. However, the former World Champion and present challenger stepped up another gear and won games 21, 22, 24 and 25. At the end Alekhine had recovered his title in very impressive fashion with 10:4 in wins and a score of 15\u00bd:9\u00bd points including the draws.\n\n **Alekhine \u2013 Euwe**\n\nNetherlands, 6th game  \n16th October 1937  \nQueen's Gambit, Slav Defence (D10)\n\n**1.d4 d5 2.c4 c6 3. \u2658c3 dxc4**\n\n3...e5 4.cxd5 cxd5 5.e4! dxe4 6.\u2657b5+ was given by Alekhine as clearly better for White. Later praxis showed, however, that Black has a level game: 6...\u2657d7 7.dxe5 \u2657b4 etc.\n\n3...\u2658f6 4.e3 g6 (the moves 4...e6 and 4...a6 are also very playable) 5.f3 \u2657g7 6.e4 dxe4 7.fxe4 e5! 8.d5 0-0 9.\u2658f3 with somewhat better play for White according to Alekhine. Here too the game tends to be level.\n\n**4.e4 e5**\n\n4...b5 is better: 5.a4 and now the simplest: 5...b4 6.\u2658a2 \u2658f6 7.e5 \u2658d5 8.\u2657xc4 with a solid position for Black.\n\n**5. \u2657xc4 exd4**\n\n5...\u2655xd4 6.\u2655b3 \u2655d7 7.\u2657g5! followed by \u2656d1 is advantageous for White (Alekhine).\n\n**6. \u2658f3**\n\nAmusing tactics, but its objective value is questionable though it was extremely effective in this game.\n\n**6...b5?**\n\nEuwe overlooked the next move. Black would be best going in for 6...dxc3 7.\u2657xf7+ \u2654e7 8.\u2655b3. Then the best was 8...cxb2!! 9.\u2657xb2 \u2655b6! with a complicated position, in which Black's chances are no worse, for example: 10.\u2657a3+ c5 11.\u2657xg8 \u2656xg8 12.\u2657xc5+ \u2655xc5 13.0-0 \u2655h5! (a suggestion of Goncharov in _64_ , 1938) 14.\u2655xg8 \u2657e6 15.\u2655h8 \u2658c6 and the white queen is trapped or 10.\u2657xg8 \u2656xg8 11.\u2655xg8 \u2655b4+ 12.\u2658d2 \u2655xb2 13.\u2656b1 \u2655c2 with advantage to Black.\n\nSurprised by Alekhine's stroke of genius at the board, Euwe was in any case in no position to work out all these complications.\n\n**7. \u2658xb5 \u2657a6**\n\n7...cxb5 8.\u2657d5+\u2013.\n\n**8. \u2655b3 \u2655e7**\n\n8...\u2657xb5? 9.\u2657xf7+ \u2654d7 10.\u2658xd4! (10.\u2657xg8? \u2656xg8 11.\u2655xg8 \u2657b4+ \u2013+) with a strong initiative for White.\n\n**9.0-0 \u2657xb5 10.\u2657xb5 \u2658f6**\n\n10...cxb5 11.\u2655d5+\u2013.\n\n**11. \u2657c4**\n\nWhite is now already winning.\n\n**11... \u2658bd7 12.\u2658xd4!**\n\n12.e5?! \u2658xe5 13.\u2658xe5 \u2655xe5.\n\n**12... \u2656b8 13.\u2655c2 \u2655c5 14.\u2658f5**\n\nThis is stronger than 14.\u2658xc6 \u2656c8 (14...\u2655xc6? 15.\u2657xf7+ +\u2013) 15.\u2655b3 \u2655xc6 16.\u2657xf7+ \u2654d8 17.e5 \u2658xe5 18.\u2657f4 \u2658ed7 19.\u2656ad1 but here too White has a clear advantage.\n\n**14... \u2658e5 15.\u2657f4**\n\nOr 15.\u2658xg7+ \u2654d8 16.\u2656d1+ \u2654c7 17.\u2658f5 \u2655xc4 18.\u2655xc4 \u2658xc4 19.\u2657f4+ +\u2013.\n\n**15... \u2658h5?**\n\n15...\u2658fd7 was more tenacious: 16.\u2656ac1 g6 17.\u2658e3 \u2657g7 18.\u2656fd1+\u2013.\n\n**16. \u2657xf7+!**\n\nAnother good move was 16.\u2657xe5 \u2655xe5 17.\u2655a4 \u2656c8 18.\u2656ad1 \u2658f6 19.\u2655xa7 \u2657e7 20.\u2656fe1+\u2013.\n\n**16... \u2654xf7 17.\u2655xc5 \u2657xc5 18.\u2657xe5 \u2656b5**\n\n18...\u2656be8 19.\u2657d6 \u2657b6 20.e5+\u2013.\n\n**19. \u2657d6 \u2657b6 20.b4 \u2656d8 21.\u2656ad1 c5 22.bxc5 \u2657xc5 23.\u2656d5**\n\nBlack resigned.\n\nAfter his majestic victory in this return match Alekhine somewhat complacently proclaimed: 'I had only loaned Euwe the title for two years.' In the case of a successful defence of the title Euwe had planned to turn over to the world chess federation FIDE the organising of the World Championship. Since, however, he did not manage to do so, the world chess federation, in which the Dutch player would later play such an important role, had to wait until 1948 before it could finally take over the organisation of the World Chess Championship. Many years later, in 1993, it once again slipped out of the hands of FIDE, however, for a period of some years under the presidency of Florencio Campomanes.\n\nAfter winning back the title Alekhine started a new attempt to come to an agreement with the leaders of the USSR and wrote to Nikolai Krylenko: 'It would give me great pleasure to again play a part in the organisation of chess in the USSR. I hope that my mistakes of the past, of which I am now fully conscious, will not represent some insuperable obstacle. I most deeply regret those mistakes to the extent that in recent years my indifference towards the powerful increase of achievements in the USSR has turned to admiration.' This and further letters remained unanswered and, moreover, remained unknown until 1967, when they were published in an article in _Shakhmaty v SSSR_ (No. 9\/1967).\n\nAfter the loss of his title Max Euwe remained regularly active on the international scene until the 1950s; he played as a member of the national team of the Netherlands into the 1960s. In 1938 in Amsterdam, in an _Algemene Vereniging Radio Omroep_ tournament, sponsored by the Dutch radio company and known by the shorter name as the AVRO tournament, he occupied a good place in the middle of the table on the same number of points as Samuel Reshevsky and Alekhine, behind the new rising masters Paul Keres, Reuben Fine and Mikhail Botvinnik and ahead of Capablanca \u2013 who had suffered a slight stroke during the tournament \u2013 and Salo Flohr.\n\nIn 1939 he won the tournament of the 'Koninklijke Nederlandse Schaakbond' (KNSB) on the same number of points as Flohr and Laszlo Szabo. In the same year he only just lost to Keres in a match by 6:5 with three draws. In 1940 in Budapest he won a tournament in honour of his chess trainer Geza Maroczy. One year later in Karlsbad he defeated Efim Bogoljubow in a match organised by the Greater German Chess Federation by 5:2 with three draws. In 1946 in a top class field Euwe took second place behind Botvinnik in the Staunton Memorial in Groningen. In the following year Euwe played in tournaments in Buenos Aires\/La Plata and Mar del Plata (won by Miguel Najdorf) and in 1948 in both New York and in Venice (won by Miguel Najdorf) he occupied fourth place in each case.\n\nAfter the death of Alekhine Euwe was one of the five players to contest the vacant World Championship title in the WCh tournament in The Hague\/Moscow 1948. However, with 4 points out of 20 games he was only a distant last. After that Euwe's name was rarely to be seen in the first class tournaments. Nevertheless in 1950 at the strong tournament in Amsterdam (won by Miguel Najdorf) he still came in sixth. In the candidates' tournament in Zurich 1953, on the other hand, Euwe could only occupy the 14th and last but one place.\n\nEuwe's attitude to Nazi Germany was unsympathetic, but at least not openly hostile. In 1937, as reported in the _Deutsche Schachbl\u00e4tter_ (Number 16\/1937, 15th August 1937) at a reception for the enthusiastic chess lover Reichsminister Hans Frank he described himself as the first 'Germanic World Chess Champion'. After the occupation of his country by the Germans, Euwe had hardly any other option than coming to terms with the occupiers if he did not want to put his life at risk. Euwe did play the match against Bogoljubow, but he declined invitations to tournaments in Frank's Generalgouvernement and to other tournaments in Germany, citing his professional obligations.\n\nDuring the war he had given up his position as a teacher and worked as the president of the Van Amerongen retail chain, later as an insurance statistician. During his time with Van Amerongen he supported the resistance as far as he could with the secret allocation of extra food rations. In 1945 Euwe took up a position again in a girls' school. In 1954 he became professor of computer science and cybernetics and worked as a consultant to the computer firm Remington. In 1971 he was named emeritus professor in Tilburg.\n\nAt the FIDE congress held during the Chess Olympiad in Siegen in 1970 Euwe was elected to be third president of the world chess federation, the successor of Alexander Rueb (1924-1949) and Folke Rogard (1949-1970). He remained in office till 1978. Euwe's goal was an expansion of FIDE and gaining new member nations in the third world. His successors Fridrik Olafsson (1978-1982) and Florencio Campomanes (1982-1995) continued this policy. Because the voting rights of FIDE have never changed since its foundation (each member nation, independent of the number of its own members, had exactly one vote in conferences of delegates and in elections) in the further history of FIDE the small federations from the so-called chess developing countries have more and more often been able to outvote the large federations and have obtained within the world federation a position of power which is absolutely not in line with their sometimes very small numbers of members.\n\nLike many other Western intellectuals, before the Second World War Euwe was fascinated by the ideas of communism. In 1934 he visited the USSR for the first time and in letters to Dutch newspapers he expressed his enthusiasm. During the war Euwe even took Russian lessons from Karel van het Reve. At the opening ceremony of the tournament in Groningen in 1946 he sang together with his daughter a patriotic Russian song. But after the Second World War Euwe's enthusiasm for the Soviet Union declined markedly. During his visit to Moscow in 1948 he asked Botvinnik about old friends from pre-war years and learned that they had all been executed as 'enemies of the people'. What happened during the Budapest uprising of 1956 finally changed Euwe's views about communism and the Soviet Union.\n\nThe politically explosive WCh matches between Boris Spassky and Robert Fischer, 1972, and Anatoly Karpov and Viktor Kortchnoi, 1978, fell during Euwe's time in office. Euwe had made use of his influence as FIDE president to enable Fischer to take part in the interzonal tournament of 1970 in Palma de Mallorca. In 1976 he supported Viktor Kortchnoi when the latter fled from the Soviet Union and made sure that the dissident was able to take part in the candidates' matches, although the Eastern bloc federations had imposed a boycott on Kortchnoi \u2013 no grandmaster from the East bloc would take part in a tournament in which Kortchnoi was playing.\n\nMax Euwe also made a name for himself as the author of numerous chess manuals. Some of these became classics of chess literature, for example _Master against Amateur_ , _Judgement and Planning in Chess_ or _The Middlegame_. In addition to that he published numerous tournament books. Lodewijk Prins collaborated as co-author or ghost writer for a series of these books.\n\nMax Euwe died on the 26th November 1981 of the consequences of a heart attack he had suffered during a trip to Israel and was buried on the 1st December at the Driehuis-Westerveld cemetery. In Amsterdam the Max Euweplein was named after the 5th World Chess Champion. In 1986 in a former prison there they opened the Max Euwe Centre. Euwe's granddaughter Esm\u00e9 Lammers published in 1995 a chess story book for children, _Lang lebe die K\u00f6nigin_ ('Long live the queen'), which was also turned into a film and in which she used two games of her grandfather.\n**Part II \u2212 The Soviet era**\n\nLong before the revolution chess had already been popular in Russia. It was reported of Peter the Great that he regularly played chess. Catharine the Great is also supposed to have been a great lover of the game and it is said that Ivan the Terrible died during a game of chess.\n\nBut after the 1917 revolution chess became a sport of the people under the new authorities in the Soviet Union. Lenin and his comrades, such as Mikhail Frunze, Mikhail Kalinin, Valerian Kuibyshev and quite particularly Nikolai Krylenko, were enthusiastic chess players and played regularly during their time in exile. It was reported of Lenin that he was a bad loser, which is characteristic of many chess players, for example also Karl Marx, who too was very keen on chess and also did not like losing, as Wilhelm Liebknecht reported.\n\nTo be sure, the best Russian players Alexander Alekhine and Efim Bogoljubow had moved abroad, but gradually the Soviet Federation, which had been founded by Krylenko in 1924, managed with the help of an ever-increasing network of trainers to discover and systematically encourage new talents. In 1925, 1935 and 1936 three world class tournaments were held in Moscow. Then the Soviet Union sealed itself off more and more from the rest of the world and nobody could find out what was happening there generally, and also in chess. In 1938 the best Soviet player, Mikhail Botvinnik, played his last tournament in the West, as the Second World War then interrupted international chess life.\n\nWith the death of Alexander Alekhine the time of the private World Championships came to an end. The world chess federation FIDE then took over the organisation of the World Chess Championships and introduced fixed rules for its implementation and the qualification of challengers. With its 700000 members the Soviet Chess Federation played an important role within FIDE.\n\nThe first World Championship after the Second World War was staged in 1948 as a tournament with five players. Mikhail Botvinnik won in superior fashion. He later defended his title against David Bronstein (1951) and Vassily Smyslov (1954), lost it to Smyslov (1957) and Mikhail Tal (1960) and recovered it twice, in 1958 against Smyslov and in 1961 against Tal in return matches. It was only in 1963 that Botvinnik definitively lost the title to another Soviet player, Tigran Petrosian. The series of Soviet World Champions continued: Petrosian was followed in 1969 by Boris Spassky. It was only in 1972 that Robert Fischer broke into the phalanx of Soviet grandmasters, which had not always observed the rules of fair play.\n**17. The victory of Soviet chess**\n\n**The World Championship 1948:  \n_World Championship tournament The Hague\/Moscow_**\n\nAfter Alekhine had recovered his title in the return match against Max Euwe in 1937, a title defence was actually planned for 1938 against the young Czechoslovakian master Salo Flohr. The world chess federation supported this match on the basis of the successes of Flohr at the start and in the middle of the 1930s and the Czech shoe manufacturer Tomas Bata was prepared to finance the match. The occupation of Czechoslovakia by Hitler's Germany put an end to this plan and the Second World War which soon followed then ended all further ideas of WCh matches.\n\nAt the outbreak of war, Alekhine, like most good chess players, was at the Chess Olympiad in Buenos Aires, but unlike a number of other players who preferred to remain in Argentina, he returned to France. After his return he was called up into the French army as a non-commissioned officer and served as a translator. At the time of the capitulation of France Alekhine was in Arcachon. From there he fled ahead of the Germans into the unoccupied zone, to Marseille where he was released from military service. From Marseille he went to Lisbon, in order perhaps to go from there to Cuba and then on to Argentina.\n\nAfter his escape, the Austrian journalist and later Lasker biographer Jacques Hannak, who after the 'Anschluss' linking Austria to Germany was arrested several times from 1938-1939 as a Jew and a Social Democrat and detained in the concentration camps of Dachau and Buchenwald, met Alekhine in March 1941 in Lisbon and reported that the latter had already secured his visa for Cuba and was planning to travel from there to New York and perhaps play a match against Reshevsky.\n\nAlekhine dropped his plan, however, and returned to his wife in France. Grace Wishaar \u2013 her family came from France and was originally called Weishaar \u2013, born in 1877 in New Jersey, was Alekhine's third wife. She had made herself a name in the USA as a painter of miniature portraits. Jack London, for whom she did a portrait of his daughter, was one of her clients. In 1914 she had travelled to Paris for an exhibition of her works, remained in Europe and later married the British owner of a tea plantation. After the latter's death in the early 1930s she was left a wealthy widow and then owned for example the stately castle 'La Chatellenie' in Saint-Aubin-le-Cauf, south-west of Dieppe in Normandy. After the occupation of France it was confiscated by the Nazis and the Alekhines moved to Paris. After the end of the Second World War Wishaar got the castle back and sold it. Nowadays it is used as a hotel, one of the rooms being named after Alekhine. Wishaar also played chess, even took an interest in correspondence chess and had got to know Alekhine when she took part in a simultaneous display he gave in Tokyo. In March 1934 the couple married in Villefranche-sur-Mer near Nice. Wishaar was 16 years older than Alekhine, being 58 at the time of their marriage. In Paris chess circles the marriage was made fun of with the remark that Alekhine had married Philidor's widow.\n\nAfter his return to Paris, Alekhine got in touch with the German authorities. He then played a series of tournaments in the territory of Greater Germany and for some time moved to Krakow and in 1942 to Prague. Krakow was the residence of the chess enthusiast and governor general of occupied Poland Hans Frank, ironically called the 'King of Poland' even by his own Nazi colleagues on account of his high-handed way of administration. Frank organised chess tournaments and founded a chess school, the teachers of which were Bogoljubow and Alekhine.\n\nAlekhine would later dispute the authorship of a six-part anti-Semitic series of articles on 'Jewish and Aryan chess' published in 1941. In 1956 the hand-written originals of the articles were apparently found in his wife's estate.\n\nIn 1943 Alekhine challenged Keres to play him for the World Championship. Keres was the only possible challenger who like Alekhine had remained in the territory of Greater Germany. But Keres declined the offer. On one hand Keres was in a state of depression brought about by the political circumstances, on the other with future developments in mind he did not want to compromise himself as a result of a match with Alekhine: this was the explanation of Keres biographer Walter Heuer for the behaviour of the Estonian.\n\nIn autumn 1943, on the invitation of the Spanish Chess Federation Alekhine went to give a simultaneous display in Madrid. His wife did not receive an exit visa. Alekhine remained in Spain, gave simultaneous displays, played matches and took part in some smaller tournaments. He trained the then Spanish prodigy Arturo Pomar and wrote for him a private manual _Legado_ (Legacy).\n\nIn the confusion towards the end of the Second World War Alekhine was no longer able to get back to France and Paris. He was stuck in the Iberian peninsula, alone and ill. Since the start of the 1940s, Alekhine had been suffering, probably as a result of his many years of alcoholism, from cirrhosis of the liver, duodenitis and arteriosclerosis, as well as bouts of depression. When in the winter of 1942\/43 he had lived in Prague with his wife, he became ill with scarlet fever and only just escaped death. In a rare case of coincidence, in 1929 Alekhine's friend Richard R\u00e9ti had died there of the same disease \u2013 and actually in the same hospital in which Alekhine too was treated.\n\nIn addition, Alekhine was now completely impoverished. To compensate for the lack of a regular income, he more or less kept his head above water with playing chess. In July 1944 Alekhine wrote to the journalist and chess player Juan Fernandez Rua: 'The better part of my life has been spent between two world wars which have devastated Europe. Both wars have ruined me, but with the following difference: at the end of the first war I was 26 years old, with boundless enthusiasm, which is no longer available to me. If one day I write my memoirs \u2013 which is very possible \u2013 then people will recognise that chess represented a small part of my life. It gave me the possibility to strive for something and at the same time convinced me of the senselessness of this striving. Today I continue to play chess because it keeps my mind occupied and prevents me from brooding and lapsing into reminiscences.'\n\nAlekhine's wife also played in chess tournaments. In 1944 she took part in the championships of Paris and won the ladies' championship. In the autumn of 1945 Alekhine travelled to Estoril (Portugal), where his friend Francisco Lupi lived. Since in Estoril things were going from bad to worse for Alekhine economically and in health matters, the chess lovers around Franscisco Lupi wrote a letter to Alekhine's wife: 'Since your husband has been here he has been ill and without means in a desperate situation. He is living off the goodness and mercy of the owners of the guesthouse.' But there was no reply.\n\nAt the end of November Alekhine received an invitation from the British federation to the first post-war tournaments in London and Hastings. After a protest from the US federation, however, the invitation was retracted on account of his 'collaboration' with the Germans. Alekhine thereupon wrote to the organisers a letter which was published at the start of 1946 in the _British Chess Magazine_ and in _Chess:_ 'The fact is that in Germany and in the occupied territories we were under constant surveillance by the Gestapo and were exposed to the threat of being deported to a concentration camp. I played chess in Germany and in the occupied territories in order to assure our livelihood and in doing so also paid the price for the freedom of my wife. Nothing in these articles, which first appeared in 1941 during my stay in Portugal and of which my first knowledge came when they were reproduced in the _Deutsche Schachzeitung_ , comes from me...'.\n\nAt the start of March Alekhine received a telegram from the president of the English Chess Federation, Derbyshire, who informed him that Botvinnik had applied to his federation for a WCh match with Alekhine. This remarkably improved the mood of Alekhine, who had previously been severely depressed. On the 5th March, however, Winston Churchill made his 'iron curtain speech' in Fulton (Missouri) and declared the Soviet Union to be an enemy of the West. After that Alekhine no longer believed that his match would take place.\n\nOn the morning of the 24th March 1946 Alexander Alekhine was found dead by a waiter in his boarding house in Estoril. The circumstances surrounding his death are mysterious. 'Death by asphyxiation', was the finding on the official autopsy report. A piece of meat from his dinner was supposed to have caused his death by asphyxiation. There exists a photo of the dead World Champion, which appears remarkably staged. Alekhine is sitting slumped in his armchair and wearing a coat. In March it will not have been so cold in Portugal that a coat was necessary in a hotel room, even an unheated one. On the table in front of the dead man there are some plates, which look as if they had been hurriedly thrown there. All the plates are empty, so that the World Champion must have been asphyxiated by literally the final bite of his meal if the statement as to the cause of death is correct. To the left in front of the World Champion a chess set has been positioned which also looks as if it had been put there by someone for decorative reasons.\n\nThe Canadian grandmaster Kevin Spraggett, who has chosen to live in Portugal, has looked into the case in detail. 'How can it be that someone suffocates and in doing so does not thrash around, perhaps knocking over the table and the chess board, but instead simply collapses in his armchair', he asks. According to Spraggett, the doctor who produced the death certificate, Dr. Antonio Ferreira, is said to have later said that Alekhine was found dead in the street with a gunshot wound and had been later brought to his hotel room. Official agencies had forced him to compose the death certificate in the form in which it now exists.\n\nThe handwritten death certificate, very much timeworn, has been preserved and is in the possession of a collector. The text reads: 'I was present at the autopsy of Alexander Alekhine, which was held in the faculty of legal medicine in the institute of medicine of the University of Lisbon. Alekhine was found dead in a room in his hotel in Estoril in circumstances which were considered suspicious and which appeared to necessitate an autopsy in order to determine the cause of death. The autopsy concluded that the cause of Alekhine's death was asphyxiation as a result of a piece of meat, apparently part of a meal, becoming stuck in his throat. There were no indications of anything untoward being involved, neither suicide nor murder. There were no other illnesses to which his sudden and unexpected death might have been attributed. Antonio J. Ferreira, M.D.'\n\nWho could have had an interest in the death of the World Champion? The 10th World Champion in the history of chess, Boris Spassky, believes that 'death squads' of the French resistance exacted revenge on collaborators at the end of the Second World War. According to another theory, a branch of the Soviet NKVD was responsible for the murder of Alekhine. When Reuben Fine proposed a WCh tournament with the remaining participants of the AVRO-Tournament 1938 \u2013 but without Capablanca, who had died, and without Alekhine, who had earned himself a bad reputation with his articles about 'Jewish and Aryan chess' \u2013 Mikhail Botvinnik had already challenged Alekhine. At the top of the Soviet Chess Federation, namely around Botvinnik's adversary Boris Weinstein, there was, however, strong opposition to this idea, because Alekhine was regarded as a traitor to the country. Nevertheless, at the start of 1946 the match was starting to take form. A letter from the British federation, which wanted to organise the WCh match in the summer of 1946 in London, was already on its way to Alekhine. According to this theory, the murder of Alekhine would have been the final chance to prevent this match. Alekhine's son was convinced that 'the hand of Moscow' was responsible for the death of his father.\n\nAnother thesis has been advanced by Hans M\u00fcller in his biography _Schachgenie Aljechin_. He quotes a report from the first funeral in Estoril, in which we find: 'It is a civil burial since the conditions for a religious ceremony were not fulfilled' and concludes from this way of putting things and from the circumstance that burial did not take place until 23 days after the death, that Alekhine had committed suicide. But possibly the 'conditions for a religious burial' were not met simply because Alekhine belonged to the Russian Orthodox church which was not represented in Catholic Portugal.\n\nThe burial in Estoril was actually only intended as a provisional measure and it was planned to rebury Alekhine in France. This, however, did not happen until 1956 on the initiative of FIDE. Alekhine's body was exhumed and transferred to the cemetery of Montparnasse in Paris. At the head of the grave a commemorative stone in red granite was erected with on it an image of Alekhine in Carrara marble. The grave is covered with a marble chess board. The inscription on the tombstone reads: 'This memorial stone was erected on 25th March 1956 by FIDE, the world chess federation \u2013 Folke Rogard, president, Sweden, Viacheslav Ragozin, vice-president, Russia, Marcel Berman, vice-president, France, Mikhail Botvinnik, World Champion, USSR, Gian Carlo Dal Verme (Italy), Pierre Dierman (Belgium).' Later his third wife Grace Wishaar was also buried here with Alekhine. She outlived her husband by 10 years. In 1999 the tombstone was destroyed in a hurricane, but later restored with the help of the Russian patron Andrei Filatov.\n\nDuring his time as an active player Alekhine published a series of outstanding tournament books, the best known of which are _The grandmaster tournament of New York 1924_ , _The New York chess tournament of 1927_ and _The international chess tournament of Nottingham 1936_. In addition his biographical works attracted great attention: _My best games 1908-1923_ and _On the way to the World Championship 1923-1927_. Although living abroad, until 1929 Alekhine regularly wrote articles for the Soviet chess magazine _Shakhmaty_ , published by Nikolai Grekov. The magazine was closed by Nikolai Krylenko for that reason among others.\n\nIn the Australian state of Queensland a hill was named after Alekhine, the 500 foot high Mount Alekhine, 50 miles from Townsville. The name was given to it by the Irish goldminer Patrick Joseph Finnerty.\n\nThe history of the modern World Chess Championships then begins in 1948. The world chess federation FIDE (F\u00e9d\u00e9ration Internationale des \u00c9checs) made use of the opportunity to take over the organisation of the World Championships, since now, after the death of Alekhine, there was no legitimate successor. The world chess federation had existed since 1924. In parallel to the summer Olympics of 1924 in Paris a chess tournament had been organised in the Hotel Majestic, during which the general secretary of the French Chess Federation Pierre Vincent announced the founding of the world chess federation with as its founding members Argentina, Belgium, Great Britain, Finland, France, Italy, Yugoslavia, Canada, the Netherlands, Poland, Romania, Switzerland, Spain, Czechoslovakia and Hungary.\n\nIn 1928 FIDE had made a first attempt to involve itself in the organisation of the World Championships when it ran a FIDE World Championship match between Efim Bogoljubow and Max Euwe (3:2). From 1936 to 1938 debates were held within FIDE about the modalities concerning the running of a WCh cycle, without, however, FIDE being able to gain control of it at that point. The outbreak of the Second World War finally put an end to further activity in that direction. At the first FIDE congress after the Second World War in Winterthur (25th till 27th July 1946) the implementation of a World Championship tournament was decided upon, as was a three-year WCh cycle, consisting of qualification tournaments (zonal tournaments and interzonal tournaments), candidates' tournaments or matches and finally the WCh match.\n\nIn 1947, the Soviet Union with its 700000 registered chess players joined FIDE and in doing so added extra weight. The Hague and Moscow were thought of as venues for the WCh tournament. Then an article in a Dutch newspaper created ill-feeling. Soviet players were accused in the Dutch press of pre-arranging results in tournaments \u2013 it would not be the last time in the history of chess that Soviet players would be subjected to such a reproach. As a protest the Soviets considered a boycott of the Netherlands. But finally at the FIDE congress of The Hague (30th July to 2nd August 1947) it was possible to get agreement on the implementation of the original plan, namely playing the tournament in two parts, in the Netherlands and in the Soviet Union.\n\nThe first World Chess Championship after the war, the 17th, finally took place from the 2nd March till the 16th May, including a longish pause for the journey, in The Hague and Moscow. At the previous FIDE congress the organisation had invited the six reputedly best players in the world to take part: Vassily Smyslov (USSR, 26 years old), Paul Keres (USSR, 32 years old), Reuben Fine (USA, 33 years old), Mikhail Botvinnik (USSR, 36 years old), Samuel Reshevsky (USA, 36 years old) and Max Euwe (Netherlands, 46 years old). It had originally been intended to invite the winners of the Staunton Memorial in Groningen and the Treybal Memorial in Prague as participants. The Staunton Memorial was, however, won by Botvinnik who was already invited in any case. The victor in the Prague tournament was Miguel Najdorf, but it was decided that the Prague tournament had been too weak and Najdorf was not nominated for the WCh tournament.\n\nReuben Fine, a psycho-analyst by profession, then declined to participate. It had actually been Fine who had been the first to think of organising a WCh tournament, after Alekhine had rendered himself unworthy in his eyes to be the World Chess Champion because of his series of articles _Jewish and Aryan chess_. Fine wanted to stage the tournament in the USA, but he could neither find sponsors in the USA for a WCh tournament, nor could he come to an agreement with other top players on the format. Various reasons were suggested for his refusal of the invitation: Fine was working on his PhD thesis and was disappointed that the WCh match did not take place in 1947 as originally planned. Also the support he had hoped for from the US Chess Federation was not forthcoming. The federation even voted for Isaac Kashdan or Arnold Denker instead of Fine as the second US participant along with Reshevsky. Finally, according to the chess player and chess journalist Larry Evans later, Fine also feared pre-arranged results among the three Soviet players.\n\nFine was not replaced and so the tournament was held with only five players, with three of them from the Soviet Union, who would be meeting each other in five of the rounds. A particular role among the Soviet players fell to the Estonian Paul Keres. He was born on the 7th January 1916 in Narva. Estonia had belonged to the Russian empire since 1710, culturally it was marked by Baltic-Germans. After 1885, as a result of a campaign of russification, German was replaced by Russian as the official administrative language. In the course of the collapse of the Russian Czarist empire after the revolution Estonia had declared itself independent on 24th February 1918, but only maintained its independence until 1940. Then according to the agreements in the secret supplementary protocol to the Hitler-Stalin pact it was occupied by Soviet Russia. After Germany's attack on the Soviet Union in 1941 Estonia fell into the grip of the Third Reich and was then occupied by the Germans from 1941 to 1944. In the autumn of 1944 the country was once again taken into the possession of the Soviets after the retreat of the German army.\n\nSo during this period Keres had been obliged to change his citizenship several times. In 1935 Keres, who was moreover one of the best tennis players in his country and on one occasion runner-up in the national championship, represented Estonia at the Chess Olympiad in Warsaw. In 1940 he took part in the USSR championships. From 1942 to 1944 Keres played in tournaments in the territory of the German Reich, for example in the General Government of Hans Frank, in 1943 at an event for entertainment for the Wehrmacht. In the light of his activities in German territory, Keres' fate appeared uncertain when the Red Army once again marched into Estonia, but he was able to continue his career in the USSR, even becoming USSR champion in 1947. Since Keres always did particularly badly against Mikhail Botvinnik, the suspicion arose that the Estonian had been obliged to give precedence to Botvinnik.\n\nBotvinnik was the clear favourite for victory. Even before the war he was being treated as the possible successor to Alekhine. Vassily Smyslov was a young player from the new 'Soviet Chess School' and his third place in the tournament of Groningen 1946 behind Botvinnik and Euwe recommended him as a candidate for the WCh tournament. Born in 1911 close to Lodz (Poland), the US American Samuel Reshevsky (actually: Szmul Rzeszewski) had made a name as a 'chess prodigy'. He learned to play the game at four and at eight he was already giving well respected simultaneous displays. In 1920 his parents emigrated to the USA in order to make money from their son's skills. Reshevsky, who had not previously enjoyed proper schooling, took a break in his chess career from 1924 to 1931 to catch up on his education, finishing it in 1934 at the University of Chicago as an accountant. After that he began to take part successfully in international tournaments too.\n\nThe fifth and final participant in the WCh tournament was the Dutch mathematics teacher and the last living World Champion Machgielis (Max) Euwe. Many authors claim that in 1947 Euwe was proclaimed by FIDE as the last living World Champion to be Alekhine's successor but that he was dethroned after one day on the intervention of the Soviet federation. The chess historian Edward Winter has searched for sources for this claim but has been unable to discover any proof. According to Hans Ree the WCh tournament was suggested originally, but then there were practical problems in carrying it out. For that reason the delegates at the FIDE congress in The Hague 1947 discussed whether for the sake of simplicity Euwe ought to be proclaimed World Champion. The latter, so it was argued, could then defend his title against Reshevsky, with the winner then meeting Botvinnik. After the Soviet delegation arrived on the next day and spoke against this suggestion, it was rejected. Euwe later joked that he had been like a 'World Champion for one day'. In any case he was never officially declared to be such.\n\nSome players who were also counted among the strongest in the world at that time were also absent from this tournament for the World Championship. As well as Miguel Najdorf, who according to the historical world ranking lists and Elo ratings established retrospectively by the statistician Jeff Sonas was in 1948 number two in the world behind Mikhail Botvinnik, Gideon Stahlberg (number three) and Isaak Boleslavsky (number four) were not considered. However neither Elo ratings nor world ranking lists were available as a measure to get an exact evaluation of playing strengths.\n\nMax Euwe (with his head bowed down), The Hague 1948\n\nOf the total of 25 rounds in the WCh tournament the first ten were played between the 2nd and 26th March 1948 in The Hague. After that the tournament moved to Moscow and the remaining rounds were played from the 11th April till the 16th May. The time control was 2\u00bd hours for the first 40 moves, after that an hour for each further 16 moves. After five hours the game could be adjourned at the request of either player. The chief arbiter was Milan Vidmar.\n\nThe Soviet delegation in the Netherlands consisted of no less than 25 persons. As well as the three players, Botvinnik, Keres, Smyslov, there were the official seconds Viacheslav Ragozin, Alexander Tolush and Vladimir Alatortsev and Igor Bondarevsky, Salo Flohr and Andor Lilienthal had arrived apparently as 'correspondents' for various Soviet newspapers and magazines. In reality these also had the task of helping the Soviet players, and especially Mikhail Botvinnik, with preparation and analysis of adjourned positions. Alexander Kotov acted as a member of the arbitration panel. Additionally the Soviet team also had a doctor. Botvinnik's wife and his daughter were also present in The Hague. The leader of the Soviet Delegation was the head of the Russian chess section Postnikov. Compared to that the US delegation was extremely slim: Reshevsky had arrived at the tournament quite alone. Shortly before the start of the tournament he was given Lodewijk Prins as his second. Max Euwe was supported by Theo van Scheltinga.\n\nThe 1st March 1948 had been fixed for the start of the tournament. The opening ceremony took place in the presence of the mayor of The Hague, Willem Adriaan Johan Visser. The drawing of lots resulted in Botvinnik having number one, Euwe number two. Smyslov drew number three, Reshevsky number four and Keres number five. Play took place in the great hall of the zoo. Spectators sat at small tables on which they could place the sets they had brought with them.\n\nAfter the first flight of games in which everybody had played everyone else, Botvinnik with 3\u00bd points was already leading the field, a point ahead of Reshevsky. Euwe had lost all his games and was last. In the second flight Botvinnik was less dominant with 2\u00bd points but with his 6 points he had extended his lead over Reshevsky to 1\u00bd points. After this second flight the tournament travelled by rail to Moscow.\n\nAs they entered Russia, the Russian border officials were curious about strange and possibly hostile notes in Euwe's luggage. These were Euwe's opening analyses for the WCh tournament. Botvinnik intervened and with a call to Moscow arranged for Euwe to keep his notes. Nevertheless the Dutch player had to sign a declaration that his notes were not directed against the Soviet Union. Euwe did this and after waiting for a few hours the participants were allowed to continue their journey. Euwe later joked to Botvinnik that his opening preparation could either be used only against Reshevsky, the sole non-Soviet participant in the World Championship apart from Euwe himself, or else it was bad and totally useless.\n\nIn his tournament book Euwe speaks of the great enthusiasm which the grandmasters had encountered at the various steps on their journey to Moscow. At their stop in Berlin the players were the guests of the Soviet military government and gave simultaneous displays against really strong players in the presence of hundreds of spectators. As they continued their voyage, their group was met at every largish station by a delegation of local chess lovers. In Minsk they were received by Gavriil Veresov, the champion of Belarus. While changing in Brest-Litovsk to a train with the correct gauge used in Russia a great ceremony was organised with speeches, music and several hundred visitors, according to Euwe. In Moscow too there was an exceptionally friendly reception when they arrived on the 4th of April, attended by numerous officials, and also a great number of representatives of the press.\n\nThe second part of the tournament was staged in the showpiece hall of columns of the House of the Unions, in the centre of Moscow, where the Marx Prospect intersects with the previous Pushkin Street. In 1993 the street was renamed Great Dimitrovka Street ('Bolshaya Dimitrovka') and the House of the Unions was given the number 1. Built in the 18th century, this was once the seat of the assembly of nobles. The splen-did hall of columns is so-called after the 28 Corinthian columns around it which rise to a height of 9.80 meters. This is where in the days of the USSR the coffins of dead Soviet leaders were laid in state on red carpets, so that the population could take their leave of them. But by the end of the 19th century some of the rooms in the building were also being used for play by a Moscow chess club. Before the war the hall of columns was in 1936 the grand venue for the famous Moscow grandmaster tournament, won by Capablanca ahead of Botvinnik and Lasker.\n\nFor the opening ceremony all 'the heavyweights of Russian chess life' (Euwe) were present. The president of the Soviet department for chess Postnikov made the opening speech.\n\nIn the Moscow part of the World Championship tournament Botvinnik took in the first two flights of games in each case 3 points out of the four games and increased his lead after this fourth flight to 3\u00bd points in front of Reshevsky and Smyslov. At the end he won the tournament with 14 points ahead of Smyslov (11), Reshevsky and Keres (10\u00bd) and Euwe (4).\n\nThere was later some speculation about whether Keres had deliberately lost his games against Botvinnik. Botvinnik emerged victorious from four of the five games against Keres. It was only the last game, when Botvinnik's tournament victory was already secure, that Keres was able to decide in his favour. This thesis has been recently supported amongst others by the Dutch chess publicist Hans Ree. As proof that at the WCh tournament 1948 'things were not above board', Ree points to the game from the third flight in which Keres was a pawn down in a rook ending against Botvinnik. Despite the pawn deficit the ending should finish as a draw, but with the move 40.a4 and the manoeuvre 53.\u2656d3 and 54.\u2656a3 Keres made his rook totally passive and thus decisively worsened his position.\n\nIt is a fact that on account of his 'collaboration' with the Germans during the Second World War Keres found himself under special scrutiny in the Soviet Union. Nevertheless, thanks to the support of the leader of the Estonian communist party Nikolai Karotamm he was allowed to play in foreign tournaments. Keres knew of course that the Soviet leadership was very keen for a Soviet player, specifically Botvinnik or Smyslov, to win the WCh tournament. Keres certainly did not want to be the one who stood in the way of the desired result. Later, during a visit to England in 1962, Keres denied in a conversation with the historian Ken Whyld having lost on purpose against Botvinnik at the WCh tournament. He gave instead as a justification of his four losses to Botvinnik at the WCh tournament that he had always done badly against the future World Champion.\n\nBotvinnik himself also addressed this subject in an interview which appeared in 1991 in the Dutch magazine _Vrij Nederland_. According to his representation of the facts, Stalin himself had even suggested that after the move of the WCh tournament to Moscow Keres and Smyslov should deliberately lose their games against Botvinnik to ensure the latter's victory. Botvinnik would, however, have protested against this suggestion.\n\nWith Botvinnik the title of World Chess Champion had wandered to the Soviet Union and it would stay there for a long time \u2013 without interruption to 1972. As a result of his success the first Soviet World Chess Champion was so well known in his home country that whenever he went to the Bolshoi Theatre, the audience would stand up when he came in.\n\n **Botvinnik \u2013 Euwe**\n\nMoscow, 12th round  \n13th April 1948  \nQueen's Gambit, Semi-Slav Defence (D49)\n\nMax Euwe was the only player in the WCh tournament whom Botvinnik had never previously defeated.\n\n**1.d4 d5 2. \u2658f3 \u2658f6 3.c4 e6 4.\u2658c3 c6 5.e3 \u2658bd7 6.\u2657d3 dxc4 7.\u2657xc4 b5**\n\nBotvinnik liked to play the Meran Variation of the Semi-Slav Defence to the Queen's Gambit \u2013 both with white and with black. The variation received its name after Akiba Rubinstein and also Ernst Gr\u00fcnfeld had some nice successes with black with this idea in the tournament in Meran 1924. It had, however, been employed previously by Ossip Bernstein \u2013 in 1914 against Jos\u00e9 Raul Capablanca \u2013 and also by Euwe himself in 1923. The first game with this variation was even played back in 1906 in Ostende, between Carl Schlechter and Julius Perlis.\n\n**8. \u2657d3 a6**\n\nThe oldest move at this point. Alternatives are 8...\u2657b7 (Wade Variation) and 8...b4 (Lundin Variation).\n\n**9.e4 c5 10.e5**\n\nThe equally sharp alternative is the Reynolds Attack: 10.d5 c4 (the 'modern' move is 10...\u2655c7) 11.dxe6 fxe6 12.\u2657c2 \u2655c7 13.0-0 \u2657b7 14.\u2658g5 (or 14.\u2658d4) with dynamic play. Botvinnik considered this attack more dangerous than 10.e5.\n\n**10...cxd4 11. \u2658xb5 axb5**\n\nWith black Botvinnik himself preferred the older alternative 11...\u2658xe5.\n\n**12.exf6 \u2655b6**\n\nA more popular move nowadays is 12... gxf6. Botvinnik investigated this then very new variation together with his friend Viacheslav Ragozin in the spring of 1939 during their common preparation for the 11th USSR championship.\n\n**13.fxg7 \u2657xg7**\n\nIn his tournament book Euwe considered as the 'normal' variation the continuation 14.\u2655e2 0-0 15.0-0 and then after 15...\u2658c5? 16.\u2657xh7 \u2654xh7 17.\u2658g5 \u2654g6 18.\u2655g4 f5 19.\u2655g3 Black came under a devastating attack in Kottnauer-Kotov, Prague 1947. According to Euwe the correct move was 15...\u2657b7.\n\n**14.0-0 \u2658c5**\n\nLater the choice here was more often 14...\u2657b7, after which White also occupies e5 with 15.\u2656e1 0-0 16.\u2657f4 \u2657d5 17.\u2658e5. But Black retains sufficient counterplay.\n\n**15. \u2657f4 \u2657b7 16.\u2656e1 \u2656d8?**\n\nAfter this move Black falls behind. It was more logical to play 16...\u2657xd3 17.\u2655xd3 \u2657xf3 (17...0-0 fails according to Euwe to 18.\u2658g5) 18.\u2655xf3 0-0 with simplification. But after 19.\u2655g4 White retains something of an advantage since the b5-pawn tends to be weak and Black's king position is not quite secure.\n\n**17. \u2656c1 \u2656d5 18.\u2657e5 \u2657xe5**\n\nIn the event of 18...0-0 White obtains a promising attacking position after 19.\u2657xg7 \u2654xg7 20.\u2658e5 \u2658xd3 21.\u2655xd3, for example: 21...f6 22.\u2656c7+! \u2655xc7 23.\u2655g3+ \u2654h8 (23...\u2654h6 24.\u2658g4+) 24.\u2658g6+ hxg6 25.\u2655xc7 or 21...h6 22.\u2656e4! (Kasparov).\n\n**19. \u2656xe5 \u2656xe5?!**\n\nIn the event of 19...\u2656g8 White has an extra pawn and the better position after 20.\u2656xd5 \u2657xd5 21.\u2657xh7 \u2656g4 22.h3 (Kasparov). For Black it was worth considering 19...\u2658xd3!? with the possible continuation 20.\u2655xd3 \u2655d6 21.\u2656xd5 \u2655xd5 22.\u2655xd4 0-0 and White may be better, but he is still far away from a win.\n\n**20. \u2658xe5 \u2658xd3 21.\u2655xd3 f6**\n\n21...\u2656g8 now fails to 22.\u2655xh7! \u2656xg2+ 23.\u2654f1 and f7 cannot be protected.\n\n**22. \u2655g3!**\n\nAfter 22.\u2658g4 \u2654e7 White's advantage would still be limited according to Kasparov.\n\n**22...fxe5 23. \u2655g7 \u2656f8 24.\u2656c7**\n\nThe threat of mate on e7 can only be warded off with loss of material.\n\n**24... \u2655xc7**\n\nThe alleged way out 24...\u2655d6 25.\u2656xb7 d3 is refuted by 26.\u2656a7 \u2655d8 27.\u2655xh7 (Kasparov): 27...d2 28.\u2655h5+ \u2656f7 29.\u2655xf7#.\n\n**25. \u2655xc7 \u2657d5 26.\u2655xe5 d3 27.\u2655e3 \u2657c4 28.b3 \u2656f7 29.f3! \u2656d7 30.\u2655d2 e5 31.bxc4 bxc4 32.\u2654f2 \u2654f7**\n\nAfter 32...c3 there is a win with 33.\u2655xc3 d2 34.\u2655c8+ \u2654e7 35.\u2655xd7+ \u2654xd7 36.\u2654e2.\n\n**33. \u2654e3 \u2654e6 34.\u2655b4 \u2656c7**\n\n34...d2 35.\u2655xc4+ \u2654e7 36.\u2655b4+ \u2654f7 37.\u2655xd2+\u2013.\n\n**35. \u2654d2 \u2656c6 36.a4**\n\n**1-0**\n\nMikhail Botvinnik was born on the 17th August 1911 into a well-off Jewish family in Kuokkala. After the Russian Revolution Kuokkala belonged to a then independent Finland, but after the final end to the Soviet-Finnish hostilities in 1944 had to be ceded to the Soviet Union and in 1948 was renamed Repino in honour of the Russian painter Ilya Repin born there in the 18th century.\n\n**Mikhail Botvinnik (1911-1995)**\n\nBotvinnik's family had its roots in Belarus, near Minsk, where Botvinnik's grandfather had a farm. Botvinnik's father Moissei Lvovich Botvinnik (1878-1931) left the paternal farm at the age of 27, was at first active in the underground against the czars and then started to work as a dental technician. He moved from Minsk to St. Petersburg where he met his wife, Seraphim Samoilovna Rabinovich (1879-1952).\n\nBotvinnik's mother also came from Belarus, from Krasava, which is nowadays part of Latvia. In 1901 she was exiled to Siberia at the age of 22, on account of membership of the Jewish Workers League, and a second time for two years in 1905 for distribution of subversive writings, when she became politically active in the Menshevik fraction of the Social Democratic Workers Party of Russia. After her return Seraphim Rabinovich worked as a dentist in a metal factory, but gave up her profession after her marriage to Moissei Botvinnik. She and her husband occupied in St. Petersburg a large flat with seven rooms on the 3rd floor in 88 Nevsky Prospect, quite close to the place (86 Nevsky Prospect) where in 1895-96 the great tournament of St. Petersburg with Lasker, Steinitz, Pillsbury and Chigorin had taken place.\n\nMikhail Botvinnik was the middle one of three children. He had a three years older brother, Isaak, and a younger sister, Maria. Up until the Revolution things were very good financially for the family. As well as a cook and a domestic servant the Botvinniks also employed a nanny. After the political upheaval of 1917 this changed dramatically. Like many others the Botvinniks now experienced from time to time great hunger. In 1920 Moissei Botvinnik left his family in order to marry a noblewoman, though he did continue to care for his children and also supported them materially. But when Mikhail Botvinnik's mother became ill the good times were finally over for the family. As a child Mikhail Botvinnik's health was frail, but then on the instructions from a book he took the habit of regular physical exercise every morning, a habit which he kept all of his life. Botvinnik started reading at an early age, preferably classics, often till late into the night, which, he believed, ruined his eyes so that he soon had to wear spectacles with very thick lenses.\n\nMikhail Botvinnik learned chess in the autumn of 1923 at the age of 12 from Leonid Baskin, a school friend of his older brother Isaak who lived in the same house. In his school, School No. 157 in 5 Finnish Alley _(Finski pereulok)_ , Botvinnik then frequently took part in school competitions but at the start was often not as good as the older and more experienced pupils. Amongst Botvinnik's first chess reading material were two bound annual issues of the magazine _Schachmaty Listok_ of 1876 and 1877. He learned all the games by heart.\n\nWhen in 1924 Lasker visited the city, which had been called Petrograd since 1914 and renamed Leningrad in 1924, and gave a simultaneous display in the offices of the tax authorities, Mikhail Botvinnik took part but had to abandon the game prematurely after 15 moves since Lasker was playing very slowly and it would soon be time for the 13 year old to go back home. But in the same year Botvinnik won the championship of school No. 157 and on the 1st June 1924 joined the Petrograd chess society, though he had to cheat a little on the question of his age since the minimum age was 16, and at 13 he was clearly under that limit. He made rapid progress and won his first tournament and 18 roubles prize money.\n\nIn 1925 the World Champion Capablanca, who was playing in the Moscow tournament, gave a simultaneous display against 30 opponents in the Leningrad Philharmonic on one of the rest days. There are various accounts concerning what motivated the World Champion to undertake the train journey from Moscow to Leningrad, nine hours in each direction. As well as a high fee, it might also have been the prospect of a rendezvous with a lady who had set Capablanca's heart on fire during a visit to St. Petersburg in 1913 in his capacity as Cuban diplomat. Organiser Jakov Rochlin had originally intended the 14 year old Botvinnik to be a reserve. Botvinnik's mother, who had actually been rather sceptical about her son's chess activities, nevertheless purchased for him a new jacket for his appearance in the simul against the World Chess Champion. Botvinnik was allowed to play and thanked her with a victory against Capablanca.\n\nAfter that Botvinnik made further progress and was soon one of the best players in Leningrad. At the age of 15 he was selected in 1926 to represent his home city in a match between Stockholm and Leningrad and at 16 Botvinnik reached the category of 'master'. Nevertheless, he felt hindered in his progress by older players, who were jealous of his rise, especially by Piotr Romanovsky, who played an important role in the development of chess in the Soviet Union as a trainer and official. After school Botvinnik started his studies at the Electromechanical Faculty of Leningrad Polytechnic Institute. After two terms he was called up to Novgorod in the summer of 1929 for six weeks of military training for students, which would remain his only military experience.\n\nAt the age of 20, in 1931, Botvinnik won the USSR championship for the first time. In 1934, during a visit to his friend and mentor Jakov Rochlin, who was married to the young soloist from the Leningrad Ballet Valentina Sopuchinoi, the young master made the acquaintance of his future wife Gajane Ananova. She was of Armenian descent, also a ballet dancer working in the Kirov Theatre and at that time only 19 years old. The couple married on the 2nd May 1935. But their only child, their daughter Olga, was not born until 1942.\n\nIn 1935, together with Salo Flohr, Botvinnik obtained first place in the tournament in Moscow. For his performance, in addition to the prize money of 4000 roubles and a raise in his state stipend from 300 to 500 roubles he received from the People's Commissar for Heavy Industry Grigol Ordzhonikidze, who only a few years later at the start of the 'Great Purge' in 1937 either chose to or was obliged to commit suicide, the present of a car. This made Botvinnik one of the first private car owners in the Soviet Union. Someone later wrote that at that time driving around in a private car in the Soviet Union would be like landing in a pedestrian zone in a Ufo nowadays.\n\nThe next year Botvinnik occupied second place behind Capablanca in the next big tournament in Moscow and shortly after won the Nottingham tournament together with Capablanca. Many chess players also fell victim to the Stalinist terror, including even Nikolai Krylenko, the great chess promoter and head of the Soviet federation since 1924, a companion of Lenin. He was arrested in January 1938, condemned in April and shot on the 29th July. Botvinnik himself remained unscathed. The regime needed him. At the AVRO tournament in 1938 in Amsterdam Botvinnik took third place behind Keres and Fine and was able to defeat both Alekhine and Capablanca. There are various representations of the circumstances surrounding the invitation to the AVRO-tournament. In one version it was originally the older Grigory Levenfish who was invited, but according to Kortchnoi Botvinnik had intervened and within the Soviet federation pointed out Levenfish's czarist past. According to Botvinnik he had been personally invited. Levenfish had argued against this, but unsuccessfully.\n\nThe organisers of the AVRO tournament had tried to establish the winner of their tournament as the next challenger, but Alekhine scotched this idea and at the opening ceremony read out a statement in German (a language of which he was a perfect master). In it he pointed out that only the World Champion chose the challenger and that he would only accept a player who could put up the required stake. After the tournament Botvinnik met Alekhine in the latter's room for negotiations and he brought Salo Flohr along as a witness. Alekhine was basically in agreement with a match against Botvinnik and also explained that he would be prepared to play it in Moscow, as long as the London Rules were adhered to. Alekhine demanded two thirds of the 6700 dollars as a winner's prize, but in order to keep things simpler when presenting them to the Soviet authorities, Botvinnik offered him the 6700 dollars as an appearance fee, no matter how the match turned out.\n\nAfter the tournament all the players in it met in order to advise in general terms about the arrangements for future World Championship matches. But curiously, out of the eight players in the tournament there were only ever seven in the room at any one time because Alekhine and Capablanca were then even avoiding being in the same room at the same time and therefore attended the council turn about. The players finally founded a sort of players union. Each of the masters present was to have the right to challenge the World Champion, as long as he was in a position to come up with the required 10 000 dollars prize fund. Only Capablanca was obliged, according to Alekhine's demand, to furnish 18 000 dollars, which was the exact equivalent of the 10 000 dollars in gold which he himself had demanded for the match in 1927.\n\nAfter his return to Moscow Botvinnik immediately informed Mikhail Bulganin, then president of the state bank, about the conversation. Soon afterwards Botvinnik received a telegram from Viacheslav Molotov, in which the latter promised the support of the government. Alekhine then suggested by letter to play half of the WCh match in London, but Botvinnik declined. The outbreak of the Second World War ended the negotiations, but in the autumn of 1945 an interview with Alekhine appeared in the English magazine _Chess_ , in which the World Champion evoked the conversation with Botvinnik and reaffirmed his agreement to appear in a match under the conditions agreed. Botvinnik then received confirmation from the Soviet government that it would make the stake available.\n\nIn March 1946, however, Alekhine died in the circumstances described above. The match never took place. The Second World War had disrupted the whole of international tournament organisation. But in 1946 things got off to a fresh start with the big tournament in Groningen. Botvinnik won it half a point ahead of Euwe.\n**18. Was he obliged to lose the 23 rd game?**\n\n**The World Championship 1951:  \n_Mikhail Botvinnik against David Bronstein_**\n\nMikhail Botvinnik's first challenger after the new rules was David Bronstein, a Ukrainian Jew, son of a mill manager and a doctor. Bronstein was born on the 19th February 1924 in Bila Zerkwa, an industrial area in Oblast Kiev. He learned to play chess aged six from his grandfather. Later Alexander Konstantinopolsky took him under his wing.\n\nIn 1937 Bronstein's father Johonon was arrested during the Stalinist purges and on false testimony was put in a gulag for seven years as an 'enemy of the people'. It has been supposed that the reason for the arrest was that he was related to the family of Leon Trotsky, who was actually called Lev Bronstein. This has not been able to be confirmed by any sources, but it has also not been able to be refuted.\n\nDavid Bronstein (1924-2006)\n\nAt 15 Bronstein came in second in the 1939 championship of Kiev and at 16 second in the Ukrainian championship. At the outbreak of the Second World War Bronstein was deemed unfit for military service on account of his bad eyesight and so spared army service. He then studied for a year in Leningrad, after fleeing Kiev ahead of the advancing Germans.\n\nIn the 1940s Bronstein then developed into one of the strongest players in the Soviet Union. For example, in 1944 he also had a win against Mikhail Botvinnik. In 1948 and 1949 he won the USSR championships. In 1948 Bronstein was victorious in the interzonal tournament of Saltsj\u00f6baden and thus qualified for the candidates' tournament in Budapest in 1950. He won this jointly with Isaak Boleslavsky (12 points each), with whom Bronstein surprisingly and under remarkable circumstances had caught up at the end of the tournament. Two rounds before the finish Boleslavsky had had a clear lead, but he drew the last two games practically without a fight, whereas Bronstein won.\n\nBehind that there was Boris Weinstein, Bronstein's mentor, who was supporting his prot\u00e9g\u00e9 at the tournament as a second. Weinstein was the head of the economic department of the NKVD, under the NKVD head and personal friend of Stalin, Lavrenty Beria, therefore very influential and powerful. In addition Weinstein was head of the NKVD chess club Dynamo, of which Bronstein was also a member of course, as well as the Soviet Chess Federation.\n\nWeinstein and Botvinnik on the other hand had already become bitter enemies before the war as to whether Botvinnik should play a WCh match with the 'traitor' Alexander Alekhine. In reality their mutual antipathy went even further back, to 1929 when they once got into an argument over the evaluation of their adjourned game in a team match. Weinstein had already secretly and successfully thwarted, even against the instructions of Stalin, Botvinnik's desire to play against Alekhine for the World Championship. With the help of his prot\u00e9g\u00e9 Bronstein he now wanted to dethrone Botvinnik as World Champion. To that end Weinstein had systematically promoted Bronstein and for years allowed him to live in his flat. Weinstein had persuaded Boleslavsky to allow Bronstein to catch up. It was obviously not too hard for Boleslavsky to agree to the request of the NKVD officer since he was a close friend of Bronstein.\n\nWeinstein's actual plan was to turn the scheduled WCh match into a tournament of three with title defender Botvinnik and the two challengers on the same number of points, Bronstein and Boleslavsky. The FIDE had little it could do about the intrigues in Moscow, but Botvinnik refused and so a playoff between Bronstein and Boleslavsky was arranged. This took place in a friendly atmosphere in the central chess club for railway workers. It was finally a close victory for Bronstein. Many years later, in 1984, Bronstein married the 22 year younger daughter of his friend, Tatiana Boleslavskaya.\n\nHow little FIDE was involved in the proceedings in Moscow can be seen from the fact that the USSR Chess Federation did not communicate the result of the playoffs to FIDE till 20 days later. Whereas Bronstein had the support of a series of NKVD officers, Botvinnik could count on his links to the top of the party. For example, he had good contacts to foreign minister Viacheslav Molotov. After his victory in the WCh tournament of 1948 Botvinnik had only taken part in a single tournament. Instead, he was working on his doctoral thesis in engineering, which he finally presented under the title _The influence of oscillating current on the vibration in synchronous machine rotors_. He was awarded the doctorate on 28th June 1951, that is to say after his match against Bronstein.\n\nBotvinnik's role model in the professional sense was the Yugoslavian grandmaster Milan Vidmar, who as well as having a chess career was also an outstanding engineer. Botvinnik's chess preparation for the WCh match consisted of the study of openings, of the games of his opponent and of training matches against Ragozin and Smyslov. Bronstein prepared with training matches against Ratmir Kholmov.\n\nIn 1944 Bronstein's father was released from the camp on account of illness and exhaustion, but as a traitor to the people was not allowed to go to Moscow. He did so, however, with the help of a forged passport which his son had got for him by bribery, so that in 1951 he was able to follow the games of WCh match against Botvinnik live.\n\nBefore the start of the match Botvinnik began a discussion with Bronstein about the rules and tried to bring about changes in certain points. This was a quirk of Botvinnik, so as to have a position of dominance or else to work up an aggressive mood against his match opponent with the help of such disputes. He discussed with Bronstein the way a draw was to be offered and demanded that on the adjournment of a game after 40 moves the sealed move should be placed inside two envelopes. It was actual the usual practice to put the sealed move in one envelope which the arbiter kept until the resumption of the game. What Botvinnik wanted to prevent with his change in the rules was any irregularities which might occur with a possibly partisan arbiter. The discussion about the second envelope is said to have lasted a month before Bronstein finally gave in.\n\nThe memorable match between Botvinnik and Bronstein took place from the 15th March till 11th May 1951 in the Tchaikovsky Concert Hall in Moscow, not far from the Mayakovskaya metro station. Adjourned games were continued in the concert hall of the Central Club of the Soviet Army or in the October Hall in the House of the Soviets. The match turned the two players into bitter enemies and after the match they never spoke another word to each other. This, however, happened quite frequently in the history of chess with contestants in WCh matches. The match was set for 24 games. In the event of equality the title defender was to retain his title.\n\nAfter a decision of FIDE at their 1949 congress the federation of the reigning World Champion had the prerogative when it came to staging the World Championship. An official prize fund was also recommended, namely 5000 dollars for the winner of a WCh match and 3000 dollars for the loser. Since for WCh matches in the Soviet Union the sums were paid in roubles, the players received in reality only a fraction of the value of these amounts. Karel Opocensky acted as arbiter, assisted by Gideon Stahlberg. The official second of Botvinnik was Viacheslav Ragozin. Bronstein was supported by Alexander Konstantinopolsky.\n\nAfter four drawn games at the start of the match Bronstein went into the lead with a win in the fifth game, but Botvinnik won the next game and drew level. The seventh game nearly went uncontested to Bronstein thanks to a higher power. After his win in the sixth game, Botvinnik decided to spend the following two rest days with his nine year old daughter in his dacha outside of Moscow in Nikolina Gora. To do so he had to cross the river Moskva. Because of spring high water levels a wooden bailey bridge had already been dismantled as a precaution. Thus the other bank could only be reached by ship.\n\nOn the morning of the seventh day of play Botvinnik realised to his horror that the water level had risen further and brought with it numerous ice floes. For that reason ship traffic had been stopped and Moscow could not be reached. Botvinnik got in telephone communication with the organisers and told them about his problem. When he later returned to the river to re-examine the situation, all the ice floes had disappeared as if by magic and the ships had started ferrying passengers again. Botvinnik was able to head for the playing hall and reached it in time. As he later learned, the ice floes had been jammed by a sharp bend in the river and left the following part of the Moskva ice-free.\n\nThe seventh game again went to Botvinnik, putting him in the lead. After three drawn games Bronstein equalised in the eleventh game. Botvinnik won the twelfth game, the 17th game again went to Bronstein. Then Botvinnik once more took the lead with a win in the 19th game.\n\nThe end of the match turned into a thriller. In game 20 there was another draw. Then Bronstein won the 21st and also the 22nd games and took a lead of 5:4 in wins. Two draws would now have been enough for him to win the title. Bronstein, however, lost the 23rd game, meaning that Botvinnik had equalised. Botvinnik hung on to his title with a draw in the final game.\n\nAccording to Botvinnik after the match Bronstein had at first claimed he had been obliged to think about his imprisoned father during game 23. The push for this claim is supposed to have come from Bronstein's mentor Weinstein, but this was pure nonsense. But soon the rumour circulated that Bronstein had even been forced to lose the 23rd game. Bronstein himself later denied this and wrote: 'A lot of nonsense has been published about this. I was in fact subject to all sorts of psychological pressure during this match, but it was my sole responsibility to deal with it.'\n\nBotvinnik also had some difficulties to overcome and later wrote about the match: 'In the spectators' room, just opposite the stage, there were seats for the KGB, where all the fans from the Dynamo Club would sit. Whenever Bronstein sacrificed something or won a pawn, there was loud applause from there. Bronstein would make a move, disappear like lightning behind the stage, reappear and disappear again. The public laughed, which annoyed me.'\n\n **Botvinnik \u2013 Bronstein**\n\nMoscow, 23rd game  \n8th May 1951  \nGr\u00fcnfeld Indian Defence (D71)\n\n**1.d4 \u2658f6 2.c4 g6 3.g3 c6 4.\u2657g2 d5 5.cxd5 cxd5**\n\nBlack has reacted to the fianchetto of the white bishop to g2 with 4...d5 and a transposition to the Gr\u00fcnfeld Indian Defence. 3...d6 would have led instead to the King's Indian Defence.\n\n**6. \u2658c3**\n\n6.\u2658f3 \u2657g7 7.0-0 0-0 is more usual.\n\n**6... \u2657g7 7.\u2658h3**\n\n7.\u2658f3 is more common, but White wanted to avoid the main lines. (Botvinnik)\n\n**7... \u2657xh3**\n\nSolving in radical fashion the problem with the d5-pawn but also at the cost of the bishop pair.\n\n**8. \u2657xh3 \u2658c6 9.\u2657g2 e6 10.e3 0-0 11.\u2657d2 \u2656c8 12.0-0 \u2658d7**\n\nBronstein loves such 'intermezzo' moves when the pieces retreat and then return to their original positions. (Botvinnik)\n\n**13. \u2658e2 \u2655b6 14.\u2657c3 \u2656fd8 15.\u2658f4 \u2658f6 16.\u2655b3 \u2658e4**\n\n16...\u2655xb3 17.axb3 degrades the white pawn structure, but also opens the a-file for the rook.\n\n**17. \u2655xb6 axb6 18.\u2657e1**\n\nWhite wants to preserve the bishop pair in all cases.\n\n**18... \u2658a5 19.\u2658d3 \u2657f8**\n\nThe game is completely level. Black has something of a lead in development and the initiative on the queenside, White has in the bishop pair an advantage for the forthcoming endgame.\n\n**20.f3 \u2658d6 21.\u2657f2 \u2657h6**\n\n21...\u2656c2 22.\u2656fc1 \u2656dc8 23.\u2656xc2 \u2656xc2 24.\u2656c1 achieves nothing for Black.\n\n**22. \u2656ac1 \u2658ac4 23.\u2656fe1 \u2658a5**\n\n23...\u2658f5!? forces 24.f4 after which the position remains closed.\n\n**24. \u2654f1 \u2657g7 25.g4 \u2658c6 26.b3 \u2658b5 27.\u2654e2 \u2657f8**\n\n28...\u2657a3 followed by...\u2658c3 is now a serious threat.\n\n**28.a4 \u2658c7 29.\u2657g3 \u2658a6 30.\u2657f1 f6 31.\u2656ed1 \u2658a5**\n\nWith his knight manoeuvres Black has loosened up the white structure on the queenside and is threatening to win the b3-pawn. But since Black himself has only immobile doubled pawns on the b-file, winning the white b-pawn is of little use to him.\n\n**32. \u2656xc8 \u2656xc8 33.\u2656c1 \u2656xc1 34.\u2658xc1 \u2657a3 35.\u2654d1**\n\n**35... \u2657xc1?**\n\nAccording to Bronstein 'the worst mistake in the whole match'.\n\nBecause here Botvinnik, as so often in this match, was in time trouble, Bronstein felt the challenge to take the pawn. Black could instead, according to Botvinnik, have maintained equality after 35...\u2654f7.\n\n**36. \u2654xc1 \u2658xb3+ 37.\u2654c2 \u2658a5 38.\u2654c3**\n\nThe bishop pair secures an advantage for White in this endgame \u2013 despite the slight material disadvantage.\n\n**38... \u2654f7 39.e4 f5?**\n\nBronstein later thought: 'I never wanted to analyse this ending, but I suppose that without...f6-f5 it is still drawn.'\n\n**40.gxf5 gxf5 41. \u2657d3 \u2654g6**\n\nThe game was adjourned after this move. All the grandmasters in the press centre were convinced that Botvinnik had sealed 42.\u2658b1 as his move and were of the opinion that that move was winning, but...\n\n**42. \u2657d6**\n\nBotvinnik's plan was to bring the bishops to d6 and b1, exchange on d5 and then win the d5-pawn with \u2657a2. Botvinnik's second had instead spent the whole night analysing the winning move 42.\u2657b1, without Botvinnik letting him know that he had sealed another move. This only became clear to Flohr on the following day and at the same time revealed to him the fact that Botvinnik did not trust him.\n\n**42... \u2658c6**\n\n**43. \u2657b1**\n\nIn the analysis of the adjourned game Bronstein had also looked into 42.\u2657d6, although he was not counting on it being played, and had found a plan against it. After the resumption of the game he was so surprised by 42.\u2657d6 that he forgot his defensive plan.\n\n**43... \u2654f6?**\n\n43...\u2658a7! was Bronstein's defensive idea: 44.exd5 exd5 45.\u2657a2 and now 45...b5 holds the game.\n\n**44. \u2657g3!**\n\n'After a sleepless night I found this decisive move at eight o'clock in the morning.' (Botvinnik)\n\n**44...fxe4**\n\nThe point of White's move is 44... \u2658ab4 45.\u2657e5+ and then: 45...\u2654g6 46.exf5+ exf5 47.\u2657d6 \u2658a6 48.\u2657a2+\u2013.\n\n**45.fxe4 h6 46. \u2657f4 h5 47.exd5 exd5 48.h4 \u2658ab8 49.\u2657g5+ \u2654f7 50.\u2657f5 \u2658a7 51.\u2657f4 \u2658bc6 52.\u2657d3 \u2658c8 53.\u2657e2 \u2654g6 54.\u2657d3+ \u2654f6 55.\u2657e2 \u2654g6 56.\u2657f3 \u26586e7 57.\u2657g5**\n\nBlack resigned.\n\nWhen the players shook hands, many spectators thought they had agreed a draw. In fact, however, Bronstein had resigned. His trainer Konstantinopolsky asked him why he had not tried 57...\u2658c6 58.\u2657xd5 \u2658d6, after which a win for White is at least no longer obvious. But Bronstein had not examined this possibility more closely. Later Smyslov found a win for White: 59.\u2657f3 \u2654f5 60.\u2657c1 b5 61.\u2657xc6 bxc6 62.a5 and the a-pawn queens: 62...\u2654e6 63.a6 \u2658c8 64.\u2657g5 \u2654d5 65.\u2657e7 \u2658b6 66.\u2657c5 \u2658a8 67.a7 \u2654e6 68.\u2654b4 \u2654d7 69.\u2654a5 \u2654c8 70.\u2654a6 \u2658c7+ 71.\u2654b6+\u2013.\n\nBronstein was one of the few USSR grandmasters who in 1976 refused to sign a resolution against Viktor Kortchnoi who had fled the USSR, and for this refusal he was punished with 14 years of refusal of permission to travel to the West. Moreover in 1952 Bronstein had refused to sign a petition against the alleged participants in the so-called 'Doctors' plot', Botvinnik had not. In the 'Doctors' plot' prominent doctors, above all of Jewish descent, were accused of having wanted to poison the political and military leaders of the USSR. The alleged plot was a pretext for anti-Semitic persecution with countless arrests and executions, which had its origins in the paranoia of the seriously ill Stalin. The latter died soon afterwards, on the 5th March 1953.\n**19. With great ease**\n\n**The World Championship 1954:  \n_Mikhail Botvinnik against Vassily Smyslov_**\n\nVassily Smyslov, born in Moscow on the 24th March 1921, learned the game of chess at age six from his father, an engineer who was also a strong chess player and who in 1912 had even beaten Alexander Alekhine in a club tournament of the St. Petersburg chess club. Until the age of 14 Smyslov was trained by his father and improved his understanding of chess by playing through games from his father's chess library of more than 100 volumes. In his training Smyslov's father laid particular weight on the acquisition of a feeling for the position in positions which had few pieces.\n\n**Vassily Smyslov (1921-2010)**\n\nLater Smyslov would amaze the chess world with the ease with which he found the correct squares for his pieces. Moreover, his particular strength lay in the endgame, whereas he preferred to avoid tactical complications involving the calculation of numerous variations \u2013 a weakness, however, which during his career cost him many a (half) point.\n\nAs well as his chess education, Vassily Smyslov also received music lessons. He learned to play the piano and from 1948 took singing lessons from Professor Konstantin Slobin. His idol was Enrico Caruso. He only just missed out on a position as baritone at the Bolshoi Theatre. Smyslov was, moreover, very religious and for that reason never joined the communist party.\n\nAt 14 Vassily Smyslov began taking part in tournaments and soon reached master strength. At the USSR championship of 1940 he came in third ahead of Keres, Boleslavsky and Botvinnik. In 1942 and 1945 he won the Moscow championships. At the USSR championship of 1944 he took second place behind Botvinnik. In 1946 Smyslov was third in his first appearance in the West, in the Staunton Memorial in Groningen. It was because of these successes that Smyslov was invited to the WCh tournament in The Hague and Moscow and he justified this by coming in second behind Botvinnik. At the candidates' tournament in Budapest Smyslov missed victory, trailing Bronstein and Boleslavsky by two points. Bronstein became the challenger to Botvinnik in his place. At the Chess Olympiad of 1952 in Helsinki, the first in which the Soviet Union participated, Smyslov played on board two behind Keres and with his 10\u00bd out of 13 scored the best result on that board. The USSR team took gold ahead of Argentina and Yugoslavia.\n\nVassily Smyslov had already prequalified for the candidates' tournament in Neuhausen (from the 28th August) and Zurich (from the 13th September) 1953 as a result of his third place in the candidates' tournament of 1950 in Budapest. Boleslavsky as joint winner, Keres as fourth and Najdorf as fifth in the tournament enjoyed the same privilege. The victor of the candidates' tournaments, Bronstein, qualified as the defeated player in the previous WCh match. Reshevsky and Euwe were, in addition, also entitled to take part as participants in the WCh tournament of 1948, as were the first players from the interzonal tournament in Saltsj\u00f6baden 1952: Alexander Kotov, Tigran Petrosian, Mark Taimanov, Efim Geller, Yuri Averbakh, Gideon Stahlberg, Laszlo Szabo and Svetozar Gligoric. The tournament later achieved special fame in the world of chess under the name 'Zurich 1953' as the result of a series of tournament books, including the one by David Bronstein, _Zurich International chess tournament 1953_ , considered as particularly successful. In the USSR 300 000 copies were printed. However, only a part of the book was from the pen of the named author, a great part having been written by Bronstein's mentor Boris Weinstein. The latter in turn had made copious use of the tournament book of Miguel Najdorf.\n\nThe winner of the candidates' tournaments, which was carried out in two stages, was Vassily Smyslov, who reached the goal with 18 points and two points of a lead over David Bronstein, Paul Keres and Samuel Reshevsky. Smyslov had nine wins and only one loss \u2013 against Kotov. Alexander Kotov was an employee of the KGB and could take certain liberties. Despite his defeat against Kotov, Smyslov was the next challenger of Botvinnik. That was exactly what had been planned by the Soviet leadership, or the Soviet Committee for Sports. In any case, Samuel Reshevsky had to be prevented from winning the tournament and thus becoming Botvinnik's challenger, reported David Bronstein later in an article in _64_ , which, however, did not appear until 50 years later, in 2001.\n\nThe Soviet delegation in Switzerland was led by Dimitry Postnikov, Igor Bondarevsky and a KGB agent, who together were supposed to ensure that the tournament had the desired result. To that end these three gave the participating Soviet players instructions as to how they were to play against each other. Keres was told to draw with white against Smyslov. He refused, but lost the game. Bronstein was also convinced to draw with white against Smyslov. Geller received the order to defeat Bronstein. In the position he adopts in this article, Smyslov replies that it is scandalous to expose colleagues in this way, without the ex-World Champion contradicting the content of the article. Averbakh and Taimanov denied having known about team orders, but did not deny the possibility that such might have existed.\n\nAfter his successful title defence against Bronstein, Botvinnik was slightly more active than after winning the title in 1948. In 1951 he played in the 19th USSR championship, though he was only fifth. It was won by Paul Keres. In 1952 Botvinnik participated in the tournament in Budapest, where he occupied third place behind Keres and Geller. But then he finished the 20th USSR championship joint first with Mark Taimanov (each on 13\u00bd points) and won the title with a victory in the playoff. In the same year he played a series of short training matches against top Soviet players, defeating Smyslov 2:0. Even a training game against Bronstein has been recorded. However, his preferred sparring partner was Ilya Kan, against whom he was a clear winner in training matches in both 1953 and 1954.\n\nThe World Championship match between Botvinnik and Smyslov took place from 16th March till 13th May 1954 in the Moscow Tchaikovsky Concert Hall and was once more over 24 games. In the event of a draw after 24 games the World Champion would retain his title. Gideon Stahlberg, Karel Opocensky and Harry Golombek were to be the arbiters. After Stahlberg had to withdraw, Laszlo Szabo was nominated on Botvinnik's suggestion.\n\nThe English player Harry Golombek had worked together during the Second World War with other members of the British national team, including C.H.O'D. Alexander, who later became head of the codes department of MI5 and therefore was no longer able to play tournaments in the eastern block, and Stuart Milner-Barry in Bletchley Park as 'code breakers' on deciphering messages from the German encryption machine 'Enigma'. That, however, only became public much later. After the end of his career as a player, Golombek published numerous chess books and was active as an arbiter.\n\nIlya Kan and Vladimir Simagin were appointed official seconds. The players received additional support from their clubs, Botvinnik from the 'Energija', Smyslov from the 'Nauka' sporting society. Before the match Botvinnik had again instigated a discussion about the rules. His suggestion of again putting the sealed move in two envelopes was, however, very quickly accepted by Smyslov. Smyslov would have liked to start the match on the 15th April, but then it would have lasted until June and at that time it was already too hot for a match in Moscow in Botvinnik's view, so he declined the later start to the match.\n\nAfter four games Botvinnik already had a lead of 3:0 in wins, but then Smyslov got into his stride better and took the lead after four wins, three of which in succession in games 9, 10 and 11. Along the way Botvinnik had missed a clear win in the eighth game and lost the 9th game after the opening. Games 12 to 16 were particularly hardfought. Botvinnik won four of them, whilst Smyslov could only win once. So after the 16th game Smyslov was trailing by 7:5. When in the 14th game Botvinnik chose an opening he had never played before and yet Smyslov came up with a novelty, the ever mistrustful Botvinnik after the game accused his second Ilya Kan of having betrayed his opening preparation to Smyslov. It was the last time that Botvinnik would have Kan help him as a second. The challenger then won the 20th and the 23rd games, but he could not achieve more than equality. So Botvinnik remained World Champion, according to the rules then in force a draw was sufficient for the defence of the title.\n\nIn the opinion of the future World Champion Tigran Petrosian, who analysed the match afterwards, Smyslov took greater risks after falling behind at the start and was successful with them. After he took the lead in the 11th game, the challenger, according to Petrosian, then failed to return to his actual safe style.\n\n **Smyslov \u2013 Botvinnik**\n\nMoscow, 15th game  \n17th April 1954  \nSicilian Defence (B25)\n\n**1.e4 c5 2. \u2658c3 \u2658c6 3.g3**\n\nThe main variation of the Closed Variation of the Sicilian Defence is considered rather harmless nowadays. Smyslov, however, liked to play it. Later the more aggressive 3.f4 g6 4.\u2658f3 \u2657g7 5.\u2657c4 (or 5.\u2657b5) became fashionable.\n\n**3...g6 4. \u2657g2 \u2657g7 5.d3 d6**\n\nIn game 13 5...b6 was played, later evaluated as insufficient by Botvinnik, since after 6.\u2658ge2 then 6...d6 is forced, otherwise White would get in d3-d4.\n\n**6. \u2658ge2**\n\nThere are more promising moves in 6.\u2657e3 or 6.f4.\n\n**6...e5**\n\n'The g2-bishop is reliably blocked' (Botvinnik). Here too the main alternative is 6...e6.\n\n**7. \u2658d5**\n\n7.0-0 \u2658ge7 8.f4 0-0 followed by 9...f5 'and Black has no difficulties.' (Botvinnik)\n\n**7... \u2658ge7 8.c3?**\n\n'The root of all the problems.' A better move is 8.\u2658ec3 or 8.\u2658xe7 (Botvinnik).\n\n**8... \u2658xd5 9.exd5 \u2658e7 10.0-0**\n\n'The variation 10.d4 exd4 11.cxd4 cxd4 12.\u2658xd4 0-0 (but not 12...\u2658xd5 13.0-0 \u2658e7 14.\u2658b5 d5 15.\u2657f4) would be favourable for Black, because the effectiveness of the bishop would increase by leaps and bounds.' (Botvinnik)\n\n**10...0-0**\n\nBlack is already slightly better here. The \u2658e2 cuts an especially sorry figure \u2013 for the moment it does not have a square.\n\n**11.f4?**\n\n'After this pseudo-active move White gets a difficult position. The f4-pawn simply limits the effective range of the e2-knight and the c1-bishop.' (Botvinnik) 11.d4 was better.\n\n**11... \u2657d7**\n\nThis provokes h2-h3. If not,...\u2655c8 and...\u2657h3 will follow.\n\n**12.h3**\n\n'The continuation 12.fxe5 dxe5 13.d6 \u2658f5 14.\u2657xb7 \u2656b8 is unpromising for White.' (Botvinnik)\n\n**12... \u2655c7 13.\u2657e3**\n\n13.g4 would be followed by 13...f5. '13.\u2657d2 was more prudent.' (Botvinnik)\n\n**13... \u2656ae8 14.\u2655d2?!**\n\n'14.\u2657f2 had to be played.' (Botvinnik)\n\n**14... \u2658f5 15.\u2657f2 h5**\n\n**16. \u2656ae1**\n\n'Passive.' (Botvinnik) 16.g4 would now be followed by 16...hxg4 17.hxg4 \u2658h6 18.g5 \u2658g4 with a clearly better game for Black, since the white kingside is massively weakened. Botvinnik nevertheless pleaded in his comments on the game in favour of this continuation as the only active option for White.\n\n**16... \u2655d8!**\n\nBlack now successively improves the positioning of his pieces.\n\n**17. \u2654h2**\n\nAn admission that there is nothing White can try. 17.g4 would be followed by 17...hxg4 18.hxg4 \u2658h4 with the exchange of the light-squared bishop: '17.fxe5 \u2657h6 18.\u2655c2 \u2658e3 would be bad.' (Botvinnik)\n\n**17... \u2657h6**\n\nBlack is now threatening...h5-h4, uprooting the f4-pawn.\n\n**18.h4**\n\n18.g4 fails here too 18...hxg4 19.hxg4 \u2658h4 20.g5 \u2658xg2 21.\u2654xg2 exf4 22.gxh6 \u2655g5+ 23.\u2654h2 \u2655xh6+ 24.\u2654g1 \u2655g5+ 25.\u2654h2 \u2654g7 (Botvinnik).\n\n**18... \u2655f6!**\n\nIt is no longer possible to avoid a material disadvantage.\n\n**19. \u2657e4 exf4 20.\u2658xf4 \u2658xh4 21.\u2657e3 \u2658f5**\n\n**22. \u2657xf5**\n\nThe point of Black's combination can be seen after 22.\u2658xh5 gxh5 23.\u2657xf5 (23.\u2657xh6 \u2655xh6\u2013+) 23...\u2657xe3 24.\u2656xe3 \u2655g5! winning the exchange (but 24...\u2657xf5 25.\u2656ef3 \u2656e5 26.d4 cxd4 27.cxd4 \u2656xd5 28.\u2655f2 would be weaker): 25.\u2657xd7 \u2656xe3\u2013+.\n\n**22... \u2655xf5\u2013+ 23.\u2655g2 \u2655g4 24.\u2655e2**\n\n24.c4? g5 25.\u2658h3 \u2655xh3+ 26.\u2655xh3 \u2657xh3 27.\u2654xh3 g4+ winning a piece.\n\n**24... \u2655xe2+ 25.\u2656xe2 \u2656e5 26.\u2656ee1 \u2656fe8 27.\u2657f2 h4 28.\u2656xe5 \u2656xe5 29.d4**\n\n29.c4 \u2657xf4 30.gxf4 \u2656e2 31.\u2654g1 h3 is not much fun either.\n\n**29...hxg3+ 30. \u2654xg3 \u2656g5+ 31.\u2654h2 \u2656f5 32.\u2657e3**\n\n32.\u2658d3 \u2657b5\u2013+.\n\n**32...cxd4 33.cxd4 \u2654h7**\n\nPreparing...g6-g5, which would win a piece. After the immediate 33...g5 White can, however, only delay what is coming with 34.\u2656g1, but not prevent it: 34...\u2654h7 35.\u2658e2 \u2656f3 intending 36.\u2657xg5 \u2656h3+ \u2013+.\n\n**34. \u2656f2 g5 35.\u2658e2 \u2656xf2+ 36.\u2657xf2 f5**\n\n**0-1**\n\nThe ending is hopeless for White.\n\nAs Botvinnik's daughter Olga, 12 years old at the time of the match, later remembered, during the match her father received through the post threatening letters. In one of them he was even threatened with death if he did not lose the match against Smyslov.\n**20. Tactics missed**\n\n**The World Championship 1957:  \n_Mikhail Botvinnik against Vassily Smyslov_**\n\nThree years later there was a re-run of the WCh match of 1954 under the same conditions. Smyslov had once again qualified as challenger after his victory in the candidates' tournament in Amsterdam in 1956. As well as Smyslov, who had prequalified as the loser of the last WCh match, the first nine players from the interzonal tournaments of Gothenburg 1955 had qualified for the candidates' tournament.\n\nAt the FIDE congress of 1955 in Gothenburg Botvinnik had submitted several suggestions. Thus the World Champion made efforts to be allowed to play in the candidates' tournament because he was of the opinion that the qualification cycle conferred an advantage on the challenger for the WCh match, since unlike the inactive title defender he (the challenger) was getting tournament practice.\n\nThe world of chess saw this from exactly the opposing angle: the challenger was tiring himself out through the qualification process and moreover had to disclose a part of his opening repertoire. FIDE president Folke Rogard asked the other players in the candidates' tournament, but they declined Botvinnik's suggestion. In addition Botvinnik wanted to create within FIDE a committee of top players, but FIDE president Folke Rogard showed no interest in this.\n\nBotvinnik was, however, successful with another point. Before the FIDE congress in Gothenburg there was in force a rule according to which a World Champion who lost the title had the possibility of getting it back in a return tournament with the new World Champion and a third player. Botvinnik wanted to eliminate the third player and reduce the right to a return match to a simple match between two players. Botvinnik was supported by Ragozin and the new leader of the Committee for Sports Lev Abramov, with whom Botvinnik was friendly. After some hesitation Folke Rogard and the FIDE delegates gave in on this point. On the other hand, however, the number of players in the candidates' tournament who came from the same country was restricted. This especially disadvantaged the many strong Soviet players, but was also to the advantage of Botvinnik, as it would spare him some of the competition from his own country.\n\nSmyslov won the candidates' tournament in Amsterdam in superior fashion with 11\u00bd points and 1\u00bd points of a lead over Keres. Botvinnik tried to prepare himself for the World Championship with training matches. This time his sparring partner was Yuri Averbakh. Botvinnik went with him to his dacha in Nikolina Gora, 20 km outside of Moscow, where apart from him many other 'heroes of the Soviet Union' had their weekend houses. During the games Botvinnik left a radio on so as to get used to the noise there would be in the tournament hall. Moreover, now 45 years old, he prepared himself with sporting activities \u2013 skating and skiing \u2013, hiking, salt baths and sleeping with a half-open window.\n\nIn some sources it is claimed that the permanently mistrustful Botvinnik, in view of his experiences in the previous match against Smyslov when he believed his opening preparation to have been betrayed by his second Ilya Kan, had done without a second for the following two matches. Other sources name Averbakh and Goldberg as seconds.\n\nThis time too, the WCh was staged in the Tchaikovsky Concert Hall, the largest concert hall in Moscow, and was as had usually been the case set for 24 games. The arbiters were Gideon Stahlberg and Harry Golombek. The programme included simultaneous displays by grandmasters, every Tuesday and Thursday, commentary on the games and chess problems competitions.\n\nAlmost all the spectators in the hall were armed with a plug-in set which they used to analyse the games in progress. When one of the two players made a particularly good move, applause broke out among the spectators. Occasional applause was tolerated, but not any other noises. At the slightest disruption a sign immediately lit up on which was printed 'quiet in the hall'. Even whispering was not allowed. If one of the players felt disturbed by too much noise, he could demand the exclusion of the public. This happened once in this match, during the 12th game, when, after losing the initiative, Botvinnik's nerves failed him.\n\nSmyslov won the opening game, but was then defeated in games four and five. With the sixth game he equalised. The challenger won the eighth and the twelfth games. Game 13 went to Botvinnik. In the 15th game Botvinnik was again heading for a win, but he gave this away with a blunder after the resumption. With wins in games 17 and 20 game Smyslov brought about the final score of 6:3 in wins and became the seventh World Champion.\n\nBotvinnik was very self-critical in his evaluation of his play. 'I was not able to come up with a clear system in the opening. In addition I was not very good at analysing adjourned games and also lacked tenacity. My misfortune came from the fact that I did not see through my rival's tactics. He was always cautiously steering the games along quiet paths and that made his opponent think of a draw which is easy to achieve: you just need to exchange pieces. And Smyslov proposed piece exchanges, however, in such a way that each exchange brought him a slight advantage. By the endgame this advantage had grown considerably and then all the virtuosity of Smyslov's proficiency was revealed.'\n\nSmyslov showed in this match that he had better opening preparation and endurance, whereas Botvinnik always became more and more tired especially after the fourth hour of play.\n\n **Smyslov \u2013 Botvinnik**\n\nMoscow, 20th game  \n23rd April 1957  \nFrench Defence (C18)\n\n**1.e4 e6**\n\nTrailing by 8\u00bd:10\u00bd before this game, Botvinnik had to take risks and if possible avoid a draw. In the 14th and 18th games, however, Botvinnik had only managed to share the points with the French Defence.\n\n**2.d4 d5 3. \u2658c3 \u2657b4**\n\nIn the middle of the 20th century the Winawer Variation was considered the best option for Black in the French Defence after 3.\u2658c3. Later 3...\u2658f6 (Classical Variation) and 3...dxe4 (Rubinstein Variation) came back into fashion more.\n\n**4.e5 c5 5.a3 \u2657xc3+**\n\nIn the first match for the World Championship against Smyslov, in 1954, Botvinnik several times made use of 5...\u2657a5.\n\n**6.bxc3 \u2655c7**\n\nIn an earlier game between the two players Botvinnik here chose 6...\u2658e7 followed by 7.\u2658f3 \u2658bc6 8.\u2657d3 \u2655a5 9.\u2655d2 c4 10.\u2657e2 \u2655a4 (0-1\/42), Smyslov-Botvinnik, Leningrad\/Moscow 1941.\n\n**7. \u2655g4 f6**\n\nA novelty by Botvinnik. Unlike after 7...f5 Black here has the option of taking on e5 later. In the 14th game of the match 7...f5 8.\u2655g3 \u2658e7 was played.\n\n**8. \u2658f3**\n\n8.exf6 does not come into consideration since the move simply helps Black develop. It is followed simply by 8...\u2658xf6.\n\n**8... \u2658c6 9.\u2655g3**\n\nThe queen is better here than on g4, where it could be attacked by the knight or bishop.\n\n**9... \u2655f7**\n\nA necessity, since the queen is tied to the protection of g7 and otherwise the \u2657c8 and the \u2658g8 cannot be developed to d7 or e7. 9...fxe5 10.\u2658xe5 \u2658xe5 11.\u2657f4 intending 11...\u2658f6 12.\u2657b5+ is good for White. After 9...c4 10.a4 \u2655a5 there follows 11.\u2656a2!.\n\n**10.dxc5**\n\nWhite takes the pawn and does not worry about his structure, e.g. the tripled pawns on the c-file.\n\n**10... \u2658ge7 11.\u2657d3 fxe5**\n\nThe game takes another direction after 11...0-0 12.0-0 \u2658f5 (not 12...fxe5? 13.\u2657xh7+ +\u2013) 13.\u2657xf5 (13.\u2655f4 g5!?) 13...exf5 14.exf6 \u2655xf6 15.\u2655d6 \u2657e6 16.\u2657g5 \u2655f7 17.\u2656fe1 \u2656fe8 18.\u2656ab1 h6 19.\u2657d2 \u2656ad8 20.\u2655g3 and White has an advantage here too.\n\n**12. \u2658xe5 \u2658xe5 13.\u2655xe5 0-0 14.0-0 \u2658c6 15.\u2655g3 e5**\n\nAs some compensation for being a pawn down, Black has now set up a strong pawn centre.\n\n**16. \u2657e3 \u2657f5**\n\nIntending to halve the white bishop pair. In the event of 16...e4 17.\u2657e2 \u2657e6 18.\u2656ab1 \u2656ad8 19.f3 the black centre is broken up.\n\n**17. \u2656ab1 \u2657xd3**\n\nThe exchange improves the white structure, however. Botvinnik probably considered 17...\u2656ab8 too passive.\n\n**18.cxd3 \u2656ae8 19.f4**\n\n**19... \u2655c7**\n\nBotvinnik was hoping to get better drawing chances in the endgame. After 19...exf4 20.\u2657xf4 \u2655d7 21.\u2657d6 \u2656xf1+ 22.\u2656xf1 White retains the extra pawn with the better position. 19...e4 was probably better. Teschner was of the opinion in his comments in the _Deutsche Schachzeitung_ (5\/1957), that after it 20.f5! exd3 21.f6 with the double threat of \u2657h6 or fxg7 leads to a decisive attack for White. But 21...\u2655g6 keeps the game level. However, it was worth considering 21.\u2657d4.\n\n**20.fxe5 \u2656xf1+ 21.\u2656xf1 \u2655xe5 22.\u2655xe5 \u2658xe5 23.\u2656d1 \u2654f7 24.h3 \u2658c6 25.\u2657f4 \u2656e7 26.\u2657d6 \u2656d7 27.\u2656f1+**\n\nThis brings the rook to a better square with tempo.\n\n**27... \u2654e6 28.\u2656e1+ \u2654f7 29.\u2654f2 b6 30.\u2656b1 \u2654e6**\n\nOr 30...bxc5 31.\u2657xc5 d4 32.c4 \u2654e6 33.\u2654f3 \u2656f7+ 34.\u2654e4 \u2656f2 35.\u2656e1 \u2654d7 36.g4.\n\n**31. \u2656b5 d4 32.c4 bxc5 33.\u2657h2**\n\nLess committal than 33.\u2657xc5.\n\n**33... \u2656f7+ 34.\u2654e2 \u2656e7**\n\n34...\u2656f5 does not work now: 35.g4 \u2656g5 36.\u2657f4.\n\n**35. \u2656xc5 \u2654d7+**\n\n35...\u2654d6 would have been a nice discovered check, but the \u2657h2 prevents the black king from using that square.\n\n**36. \u2654d2 \u2656e6 37.\u2656g5 g6 38.\u2656d5+ \u2654c8 39.\u2657g1 \u2656f6 40.\u2657xd4 \u2658xd4 41.\u2656xd4 \u2656f2+**\n\nThe game was adjourned here and resigned by Black later without resuming it. 42.\u2654c3 \u2656xg2 43.c5 \u2656g3 44.h4 h6 45.\u2656d6 could have followed and the black pawns on the kingside are immobile: 45...g5? 46.\u2656xh6 g4 47.\u2656g6+\u2013.\n\nBotvinnik had been dethroned as World Champion, but according to the FIDE he had the right to a return match, ironically called the 'Lex Botvinnik' \u2013 a further privilege which gave the title defender an advantage in addition to the 'equal scores rule'.\n**21. The revenge**\n\n**The World Championship 1958:  \n_Vassily Smyslov against Mikhail Botvinnik_**\n\nNot only did the champion keep his title in the event of a draw, but if he lost it he had the right to a return match. From his victory in the WCh tournament in 1948 World Champion Botvinnik, to be precise, had until this time not really won a single WCh match in the purely sporting sense. Against Bronstein in 1951 a 12:12 was enough for him, as it was against Smyslov 1954. In 1957 he even clearly lost the match against the latter and with it the title. Botvinnik retained the right to a return match.\n\nThis privilege was an anachronism, which FIDE had continued on the basis of an old pre-war custom. Wilhelm Steinitz had agreed on a return match with Emanuel Lasker and Euwe had allowed this right to Alekhine. Steinitz did not make use of it, but Alekhine recovered the title in 1937. The right to a return match, however, had in no way been a formal rule before the Second World War. Capablanca, for example, had not negotiated this right ahead of his match against Alekhine and after losing the title had to beg for a return match, without it, however, being granted by Alekhine.\n\nFIDE took over this custom, but did not write it into their rules until 1956. Thus the new World Champion Vassily Smyslov had only a year to enjoy his title of World Champion before having to appear in a return match. Moreover, after winning the title he was struck by a severe blow of fate. His stepson Vladimir Selimanov, the son of his wife Nadezhda and also a gifted chess player, took his own life after returning from the Youth World Championships of 1957 in Toronto. In it Selimanov had 'only' managed fourth place, which was felt among the leaders of Soviet chess to be a disappointment. The reasons for Selimanov's suicide are unknown, since the Smyslovs never spoke about it. Andrew Soltis expressed the supposition that bullying as a reaction to the poor result might have been the cause.\n\nBotvinnik, on his part, was exposed to a certain political pressure since he was being urged to give up the right to a return match so as not to humiliate himself with another and perhaps even more resounding loss.\n\nThe return match between Smyslov and Botvinnik took place in Moscow from the 4th March till 9th May under the same conditions as the previous WCh matches, therefore set for a total of 24 games. At 12:12 the title defender retained his title as usual, but this time it would have been Smyslov. The chief arbiter for the match was Gideon Stahlberg, the second was Harry Golombek. Since the Tchaikovsky Concert Hall was otherwise occupied at the start of the match, the opening ceremony and the first games were held in hall of the 'Sovietskaya' hotel. This could, however, only accommodate 800 spectators.\n\nDuring the opening ceremonies the drawing of lots was held, with Smyslov drawing White for the first game. After the drawing of lots there was first of all a concert and then a short film was shown about the tournament of Moscow 1925. Capablanca had played in it as World Champion and given a simultaneous display in Leningrad on a rest day. The then 14 year old Botvinnik had been one of three players who had managed to defeat Capablanca.\n\nIn this match Botvinnik surprised Smyslov with the Caro-Kann Defence, against which the title defender was obviously not well prepared. This coup brought Botvinnik two points with black in the first three games. Since in the second game he was also successful with white against Smyslov's King's Indian, Botvinnik went straight into a lead of 3:0. Smyslov was not able to make up this backlog. The title defender did win the fifth, the eleventh, the 15th, when Botvinnik overstepped the time limit in a winning position because he thought he had already made 40 moves, the 19th and the 22nd games. Botvinnik, however, stood firm by winning after his three opening games the sixth, the twelfth, the 14th and the 18th games. After game 19 Smyslov fell ill. A bout of flu had turned into pneumonia. The 20th game could only be played after a pause of a week. Finally Botvinnik won a WCh match in the purely sporting sense for the first time, by 12\u00bd:10\u00bd. Smyslov recognised after the match the greater experience of his opponent. All in all, the match lasted for nine weeks, since no more than three games were played per week. Many chess lovers felt this was too long.\n\n **Smyslov \u2013 Botvinnik**\n\nMoscow, 3rd game  \n11th March 1958  \nCaro-Kann Defence (B19)\n\n**1.e4 c6**\n\n'The Caro-Kann defence secures a totally solid position for Black in the opening. Its main defects, which explain the rare appearance of this defence in modern competitions, are the limited options for Black in the struggle for the initiative', was Viktor Kortchnoi's comment on the choice of opening in the bulletin.\n\n**2. \u2658c3 d5 3.d4**\n\nIn the first game 3.\u2658f3 \u2657g4 4.h3 \u2658xf3 5.\u2655xf3 \u2658f6 6.d3 was played, but Smyslov did not obtain any kind of opening advantage.\n\n**3...dxe4 4. \u2658xe4 \u2657f5 5.\u2658g3 \u2657g6 6.h4 h6 7.\u2658f3 \u2658d7 8.\u2657d3**\n\nNowadays 8.h5 \u2657h7 9.\u2657d3 is almost exclusively played here, but at that time the discussion about whether the white pawn was better placed on h4 or h5 had not yet been concluded.\n\n**8... \u2657xd3 9.\u2655xd3 \u2655c7**\n\nThis prevents \u2657f4, but is not absolutely necessary. In the event of 9...e6 10.\u2657f4 Black can reach the same position again with 10...\u2655a5+ 11.\u2657d2 \u2655c7.\n\n**10. \u2657d2 \u2658gf6 11.0-0-0 e6 12.\u2654b1 0-0-0 13.c4 c5 14.\u2657c3 cxd4**\n\n'A premature move, which exposes Black to some difficulties. 14...\u2657d6 15.\u2658e4 \u2658xe4 16.\u2655xe4 \u2658f6 17.\u2655e2 a6, which prepares for the appropriate moment the resolution of the tension in the centre, would have been more prudent.' (Kortchnoi)\n\n**15. \u2658xd4 a6**\n\n'15...\u2658e5 16.\u2655e2 \u2658xc4 17.\u2658b5 \u2655c5 18.\u2657xf6 gxf6 19.\u2656c1 \u2655xb5 20.\u2656xc4+ \u2657c5 21.\u2658e4 would be bad.' (Kortchnoi)\n\n**16. \u2655e2 \u2657d6 17.\u2658e4 \u2658xe4 18.\u2655xe4 \u2658f6**\n\nAfter 18...\u2655xc4 19.\u2656c1 \u2658c5 20.\u2657a5 \u2658xe4 21.\u2656xc4 \u2654b8 White does not play 22.\u2657xd8 on account of the fork 22...\u2658d2+, but 22.\u2658xe6, for example 22...\u2656de8 23.\u2656xe4 \u2656xe6 24.\u2656xe6 fxe6 and he is no worse.\n\n**19. \u2655e2 \u2656d7**\n\nDirected against the manoeuvre \u2658f3-e5.\n\n**20. \u2656c1**\n\n'White's superiority, which consists of the more active piece placement, is incontestable, but it is hardly enough for a win.' (Kortchnoi) If 20.\u2658f3 then 20...\u2656hd8 and now 21.\u2658e5? costs a piece: 21...\u2657xe5 22.\u2657xe5 \u2656xd1+ 23.\u2656xd1 \u2656xd1+ 24.\u2655xd1 \u2655xe5\u2013+.\n\n**20... \u2655c5**\n\nIt was also worth considering 20...\u2657f4!?.\n\n**21. \u2658b3 \u2655f5+ 22.\u2656c2 \u2657c7 23.c5 \u2656d5**\n\n23...\u2655d3!? leads to equality: 24.c6 bxc6 25.\u2655xd3 \u2656xd3 26.\u2658c5 \u2656d5 27.\u2658xa6=.\n\n**24.c6 \u2657b6**\n\nThe dark-squared bishop denies the white knight entry to the c5- and d4-squares, and the black king is henceforth in safety. 24... \u2656hd8 would be weaker: 25.cxb7+ \u2654xb7 26.\u2657d4! \u2656xd4 27.\u2658c5+.' (Kortchnoi)\n\n**25. \u2658d2**\n\nThis threatens 26.\u2658c4 and 27.\u2658e3, which forces the next move.\n\n**25... \u2655d3**\n\n25...\u2655xf2? 26.\u2658c4.\n\n**26. \u2658c4 \u2657c7 27.\u2655xd3 \u2656xd3**\n\n**28. \u2658e5?**\n\n'Such a blunder had never before been seen in Smyslov's play', was Botvinnik's comment on this move. After the 'normal' 28.cxb7+ Smyslov saw no chances of a win and with only 10 minutes left for the remaining 13 moves the title defender made a momentous wrong decision \u2013 28.cxb7+ \u2654xb7 29.\u2657e5 \u2657xe5 30.\u2658xe5 \u2656d5 31.f4 \u2658d7 32.\u2656hc1 \u2658xe5 33.\u2656c7+ \u2654b6 34.fxe5 \u2656f8 35.b4 \u2654b5= (Botvinnik).\n\n**28... \u2656xc3 29.cxb7+ \u2654xb7 30.\u2656xc3 \u2657xe5**\n\nWhite has lost two pieces for the rook and is losing.\n\n**31. \u2656b3+ \u2654a7 32.\u2656c1 \u2656b8**\n\n'The exchange of one of the actively posted rooks is necessary.' (Kortchnoi)\n\n**33. \u2656xb8 \u2654xb8 34.\u2656c4 \u2658d5 35.\u2654c2 h5**\n\nThis fixes the pawns on the kingside. Botvinnik is seeking to turn them into a welcome target for his pieces.' (Kortchnoi)\n\n**36.b4 \u2654b7 37.\u2654b3 \u2657d6 38.a3 \u2657c7 39.\u2656c2 \u2657b6 40.\u2654c4 \u2658f4 41.g3**\n\nThe game was adjourned here.\n\n**41... \u2658h3 42.f3 \u2658g1 43.f4 \u2658f3 44.a4 \u2658d4 45.\u2656d2 \u2658f5 46.a5 \u2657e3 47.\u2656d8**\n\nOf course not 47.\u2656d7+ \u2654c6 48.\u2656xf7? \u2658d6+. (Botvinnik)\n\n**47... \u2657f2 48.b5**\n\n48.\u2654b3!? was more tenacious.\n\n**48... \u2654c7 49.\u2656g8 axb5+ 50.\u2654xb5 \u2657xg3 51.a6 \u2657f2 52.\u2654a5 g6 53.\u2656a8 \u2657e1+ 54.\u2654b5 \u2658d6+ 55.\u2654a4 \u2658c8**\n\n55...\u2657xh4 was also good and if 56.a7 then simply 56...\u2654b7\u2013+.\n\n**56. \u2654b5 \u2657f2 57.\u2654a5 \u2657a7 58.\u2654b5 f6 59.\u2654b4 e5 60.fxe5 fxe5 61.\u2654c3 \u2657b8 62.\u2654d3 \u2658b6 63.a7 \u2658xa8 64.axb8\u2655\\+ \u2654xb8 65.\u2654e4 \u2658b6 66.\u2654xe5 \u2658d7+**\n\n**0-1**\n\nBotvinnik commented on his victory in an interview as follows: 'Before the start of the previous year's match with V. Smyslov I underestimated my opponent. I noticed this mistake too late, during the match when it was too late to do anything about it. I prepared for the return match fully conscious of the responsibility on me and taking properly into account the playing strength of my opponent. Perhaps V. Smyslov repeated my mistake of the previous year? He will be the best person to give an answer to that question!'\n\nDuring the match Botvinnik had complained about growing anti-Semitism, which manifested itself according to Botvinnik's comments in anonymous calls and threats. Before the match, he worked himself into the right mood against Smyslov, with whom he normally got along quite well, by fighting with him about the wish to have a fourth timeout in the case of illness. Until then only three timeouts had been allowed per player. After winning the match Botvinnik complained that on one occasion Smyslov had not turned up personally to resign an adjourned game and also not to resign for the whole match.\n\nAfter the match, chief arbiter Stahlberg spoke in favour of a change in the conditions. The matches were too long in his opinion. Stahlberg suggested a length of 20 games and voted to do away with the weekly rest day, which would only be for the players a greater drain on their energy on account of the analysis and preparation which took place on those days. But Stahlberg was unable to get his suggestions through.\n\nSmyslov was never again able to qualify as challenger, but right up to the end of the 1990s he was taking part in WCh cycles. In 1983 he played a candidates' quarter-final against the top German player Robert H\u00fcbner in Velden in Austria. After 10 games the score was 5:5 (one victory each, eight draws). The match went into extra time, but all four tiebreak games were drawn. It was finally concluded that the match would be decided by lot with the help of a roulette ball. Curiously the first attempt also ended as undecided since the ball landed on zero. Smyslov finally won the lottery. H\u00fcbner had already left as a protest against this way of deciding. In 1984 Smyslov was not defeated till the candidates' final, by the young Garry Kasparov. Even in 1997 Smyslov, by then 76 years old, took part in the FIDE k.-o. candidates' tournament in Groningen, but was eliminated in the first round.\n\nSmyslov was able to maintain considerable playing strength for a long time, which was probably due to his deep understanding of chess and his economical style of play. During his career Smyslov qualified 20 times for the finals of the USSR championships: in 1949 he shared first place with Bronstein, in 1955 with Geller, by whom he was defeated in the playoff. Nine times between 1952 and 1972 he won the gold medal with the USSR team at Chess Olympiads. In 1967 he was awarded the Order of Lenin of the Soviet Union. Smyslov died, shortly after his 89th birthday, on the 27th March 2010 from the effects of heart disease.\n**22. The magician from Riga**\n\n**The World Championship 1960:  \n_Mikhail Botvinnik against Mikhail Tal_**\n\nAt the end of the 1950s there were some strong talents knocking on the doors of those who had already made it. The young Robert Fischer was attracting attention in the USA and in 1958 the 14 year old was the youngest player of all time to win the US championship. In the same year he managed a shared fifth place in the interzonal tournament in Portoroz and by doing so qualified for the candidates' tournament which was played in Bled, Zagreb and Belgrade.\n\nOn the other hand, the winner of the Portoroz interzonal tournament was the 22 year old Mikhail Tal. This was remarkable also for the fact that Tal had had to undergo an operation two weeks before the interzonal. 'He was too weak to carry his case. I did it for him', Yuri Averbakh would later report. Tal also won the subsequent candidates' tournament in Yugoslavia in 1959. It was played between players over four rounds. Tal won with 20 points and 1\u00bd points of a lead over Keres. Against Fischer he won all four of their games.\n\nMikhail Tal (1936-1992)\n\nMikhail Tal was born into an upper middle class Jewish family in Riga on the 9th November 1936, the son of Nechemia and Ida Grigoryevna Tal. His parents were cousins. The rumour that Tal's biological father had been in reality a family friend, 'Uncle Robert', was dismissed by Tal's later wife Angelina to the realm of fairy stories. In 1941 the Tal family had to flee ahead of the advancing Germans and in doing so lost all their goods.\n\nTal was extraordinarily gifted, learned to read at the age of three and as a five year old he could do complicated multiplication sums in his head. As well as Russian he went on to speak fluent English, German, Serbo-Croat and Spanish. But even as a child Tal had to struggle with health problems. His kidneys did not work well and during his life he had to undergo a total of twelve operations. The future World Champion Tigran Petrosian later said about Tal: 'The healthiest of us all is Mikhail Tal. Nobody else would live longer than a year with his illnesses.'\n\nHowever, his weak health did not prevent Tal from living his later life to excess. He drank a lot, but according to Angelina Tal only 'clean' drinks such as vodka, preferably the Russian export brands 'Pshenichnaya', 'Kristall' or 'Stolichnaya', and whisky, never beer, wine or cognac, and was a heavy smoker. The cigarette brands he enjoyed were 'Benson and Hedges' and 'Kent'. A chain smoker, Tal had the habit of often only smoking a quarter of his cigarette and then lighting a new one. Between 1970 and 1975, moreover, he became dependent on morphine.\n\nIn addition to his weak health Tal was also handicapped by ectrodactyly: from birth he had only three fingers on his right hand. The cause of the malformation was, ac-cording to Angelina Tal, that when her husband was absent during her pregnancy Tal's mother Ida had mistakenly injected a dose of potassium chloride into her muscle rather than her vein, which led to complications. Despite this handicap Tal was a really good piano player. Because of his outstanding performance Tal was allowed to jump two classes in school and began his university study of Russian language and literature at the early age of 15. He brought it to a conclusion with a dissertation on the Russian satirists Ilya Ilf and Evgeny Petrov.\n\nWhen his cousin kept on beating him at chess, Tal's pride was hurt and he began to take a more serious interest in the game. Soon after that he became a member of the chess section of the Young Pioneers in Riga. He actually wanted to sign on for the drama group but changed his mind when he saw a door with on it the notice 'Chess section'. Tal thought he would be able to get tips for the games against his cousin.\n\nTal's first chess teacher was Janis Kruzkops. After a few months instruction Tal found himself in the position of being able to beat not only his cousin but also his older brother. In 1949 the Riga chess master and trainer Alexander Koblenz took Tal under his wing. From there on Tal would work with the latter for many decades. Thanks to the training by Koblenz he made lightning progress and in 1954 he was able to defeat for the first time a grandmaster, namely Yuri Averbakh. In 1957 Tal won the USSR championship, the youngest player of all time to do so. FIDE thereupon awarded him the title of international grandmaster, without Tal having previously held the title of international master, as was the custom. In 1958 he was a member of the victorious Soviet team in the Chess Olympiad in Munich.\n\nTal's chess was characterised by brilliant attacking play. He overwhelmed his opponents with 'intuitive' sacrifices, the soundness of which was usually incalculable. His admirers called him the 'magician of Riga'. Fischer would later characterise Tal's play in less reverent terms with the sentence: 'Tal moves his pieces into the centre and then he sacrifices them somewhere'. After the end of 1958 Tal was already, according to the later calculations of the statistician Jeff Sonas, the best player in the world.\n\nIn 1959 Tal met the 19 year old actress and singer Sally Landau and married her. Sally Landau sang among others in the Big Band of the German Jewish jazz musician Eddie Rosner, who was actually called Eduard Adolph Rosner. Fleeing from the Nazis, Rosner had come to the Soviet Union and been very successful there. Soon after the Second World War, however, he found disfavour and was banished to a gulag with other musicians. After his release in 1953 he managed to reform his Big Band to repeat his success. After his popularity declined and the cultural authorities withdrew their support, Rosner made in vain several applications for an exit visa and finally in 1972 fled in disguise into the US embassy in Moscow. From there he was able to travel to Germany, where he died in 1976 in Berlin.\n\nWith Sally Landau Tal had a son, Gera, who later became a doctor like his grandfather. The marriage between Tal and Sally Landau ended, however, in divorce in 1970, after the pair had already been going their separate ways. Little is known about Tal's second marriage, which lasted only a very short time. Tal met his third wife, the eight year younger Angelina Petuchova, for the first time at a simultaneous exhibition, in which she played against him. Since Angelina Petuchova worked for the Riga _Shakhmatya_ magazine \u2013 she was responsible for the correspondence chess column \u2013 and Tal was a regular visitor to the office, they met again. The couple married in 1970, had a daughter Zhanna and remained together until Tal's death.\n\nBefore the candidates' tournament of 1959, which took place in the Yugoslavian cities of Bled, Zagreb and Belgrade, Alexander Koblenz had engaged Yuri Averbakh as a second for Tal. Averbakh first informed Botvinnik about this enquiry. The latter raised no objections, even though it meant that an important training partner of earlier days would no longer be available to him for his preparations for the match.\n\nSo at the World Championship of 1960 the magician now met the strategist Mikhail Botvinnik. The title defender was in no doubt about to whom the sympathies went for this match: 'At that time everyone was rather fed up with me, especially my grandmaster colleagues. How long could one person continue to occupy the chess throne'.\n\nAlexander Koblenz had discovered that Botvinnik's psychological preparation for matches consisted of dictating unacceptable conditions to his opponent. If these were turned down, then he could get into an argument about them and thus create as it were an emotional bogeyman. So Koblenz advised Tal to simply accept all conditions without dispute. Thus it was suggested by the tournament adviser that the toilets be supervised so that a trainer could not speak to a player on his way there. Tal agreed. Botvinnik also demanded that in the event of noise in the tournament hall, the arbiter could move the game to a side room, if one of the seconds requested this. Botvinnik feared outbreaks of cheering by Tal supporters if their idol again came up with one of his unfathomable sacrifices. This rule was actually made use of at a critical moment in the sixth game. And, as in the matches against Smyslov, Botvinnik insisted that in adjourned games the sealed move was written twice in two envelopes.\n\nThe match was played from 15th March till 7th May under the same conditions as had been in force for the previous WCh matches: 24 games, if the match was drawn the defender retained the title, and moreover he had the right to a return match. This privilege was done away with by FIDE in 1959. But the new rule would not enter in force until the next WCh cycle. The chief arbiter was once more Gideon Stahlberg.\n\n**Tal and his trainer Alexander Koblenz during the 1960 match**\n\nBotvinnik was supported by Grigory Goldberg as his second, Tal by his trainer Alexander Koblenz. Koblenz took with him to Moscow a card index of openings, in which 4000 games were recorded, certainly one of the biggest game collections of the day. The venue for the match was the 1100 spectator Pushkin Theatre. Adjourned games were to be continued in the Central Chess Club.\n\nThe match attracted great interest in the USSR and especially in Moscow, which of course was due in appropriate measure to the charisma of the 'young gun' Mikhail Tal. Tickets cost five roubles, which was quite enormous for the day. At that time a flat could be rented for ten roubles per month. Nevertheless, there was a great rush for them. For every game, big crowds gathered in front of the theatre and called for a board. A giant demonstration board was finally installed and the moves executed on it. In other places in Moscow, for example on the Tverskoy Boulevard, demonstration boards were set up. Sometimes spectators in the theatre could not contain themselves with the excitement and from time to time called out loud suggestions for the game: 'Take the knight!'\n\nTal won the first game and after a series of draws the sixth and seventh also for a temporary 3:0 lead. But Tal admitted that this was not because of superior play on his part, but to a large extent because of mistakes made by Botvinnik in time trouble. Botvinnik caught up with wins in the eighth and ninth games. But Tal won the eleventh game. After that the World Champion claimed a pause of two days on account of an indisposition. After game 13 Tal fell ill and a further pause was scheduled. After the resumption of the match Tal won the seventeenth and nineteenth games for a final score of 6:2 in wins (overall result 12\u00bd:8\u00bd).\n\n **Botvinnik \u2013 Tal**\n\nMoscow, 6th game  \n26th March 1960  \nKing's Indian Defence (E69)\n\n**1.c4 \u2658f6 2.\u2658f3 g6 3.g3 \u2657g7 4.\u2657g2 0-0 5.d4**\n\nAlternatively White can also choose in 5.\u2658c3 d6 6.d3 a variation which is listed under the English Opening: 6...e5 7.0-0 \u2658c6 etc.\n\n**5...d6 6. \u2658c3 \u2658bd7 7.0-0 e5**\n\nThe King's Indian Defence suits Tal's enterprising style very well.\n\n**8.e4 c6 9.h3**\n\nIn his first match against Smyslov Botvinnik faced a shipwreck after 9.\u2657e3 \u2658g4 10.\u2657g5 \u2655b6 11.h3 exd4 etc.\n\n**9... \u2655b6 10.d5**\n\n'Our choice of opening turned out to be successful psychologically speaking. Botvinnik avoided all double-edged tactical continuations and immediately closed the centre, hoping to later gain a tempo with the attack on the queen.' (Tal) After 10.\u2656e1 Black can continue with 10...exd4 11.\u2658xd4 \u2658g4.\n\n**10...cxd5 11.cxd5 \u2658c5 12.\u2658e1**\n\nA typical knight manoeuvre in this structure. White is fighting for the c5-square and makes the f-pawn mobile.\n\n**12... \u2657d7 13.\u2658d3 \u2658xd3 14.\u2655xd3 \u2656fc8**\n\nPlayed after 16 minutes thought. Tal forgoes the typical King's Indian flank attack 14...\u2658e8 15.\u2657e3 Tal did not like 16.\u2655e2, because now 16...f5 cannot be carried out without problems on account of 17.exf5. Kasparov therefore suggested as preparation for...f7-f5 the move 16...\u2655e8, which protects the \u2658h5 and after...f7-f5 enables the recapture with the pawn,...gxf5.\n\n**15. \u2656b1 \u2658h5 16.\u2657e3 \u2655b4 17.\u2655e2 \u2656c4**\n\n'17...f5 18.exf5 \u2657xf5 19.\u2656bc1 and White will be gifted the e4-square.' (Kasparov)\n\n**18. \u2656fc1 \u2656ac8 19.\u2654h2**\n\n19.\u2657f3 f5! (Kasparov). An improvement is perhaps 19.\u2657f1!? intending 19...f5 20.a3.\n\n**19...f5 20.exf5 \u2657xf5 21.\u2656a1**\n\nHere it was well worth considering 21.a3 \u2655b3 22.\u2658e4 \u2656c2 23.\u2656xc2 \u2656xc2 24.\u2655d1 and now 24...\u2658f4 25.gxf4 exf4 fails to 26.\u2656c1+\u2013.\n\n**21... \u2658f4!?**\n\n21...\u2658f6!?. (Kasparov)\n\n**22.gxf4 exf4 23. \u2657d2?**\n\nAfter this Black gets a strong attack. A better move was 23.a3 with wild complications, which are not unfavourable for White.\n\n**23... \u2655xb2**\n\nAccording to Kasparov there was the more precise 23...\u2657e5 24.f3 (or 24.\u2657f3 \u2655xb2 25.\u2658d1 \u2655a3 26.\u2656xc4 \u2656xc4 27.\u2657c3 \u2656xc3 28.\u2658xc3 \u2655xc3 with a strong attack for Black) 24...\u2655xb2 25.\u2658d1 \u2655d4 26.\u2656xc4 \u2656xc4 27.\u2656c1 \u2656xc1 28.\u2657xc1 \u2655xd5 29.\u2658f2 and the game is roughly level.\n\n**24. \u2656ab1 f3! 25.\u2656xb2?**\n\nAfter this move Black is proved right. The correct way was 25.\u2657xf3 \u2657xb1 26.\u2656xb1 \u2655c2 and then 27.\u2657e4 (Flohr) or perhaps even better 27.\u2656c1, after which White is better.\n\n**25...fxe2 26. \u2656b3 \u2656d4 27.\u2657e1 \u2657e5+ 28.\u2654g1 \u2657f4**\n\nAfter this move the cheering of the Tal fans in the spectator room was so loud, that the arbiter moved the game to a side room. Tal later gave 28...\u2656xc3 29.\u2656bxc3 \u2656d1 30.\u2656c4 \u2657b2 as even more powerful.\n\n**29. \u2658xe2 \u2656xc1 30.\u2658xd4 \u2656xe1+ 31.\u2657f1 \u2657e4 32.\u2658e2**\n\n32.\u2656xb7 \u2657d3\u2013+.\n\n**32... \u2657e5 33.f4 \u2657f6 34.\u2656xb7 \u2657xd5 35.\u2656c7**\n\n35.\u2656xa7 \u2656xe2 36.\u2657xe2 \u2657d4+ \u2013+.\n\n**35... \u2657xa2 36.\u2656xa7 \u2657c4**\n\n36...\u2656xe2? 37.\u2656a8+ \u2654g7 38.\u2657xe2.\n\n**37. \u2656a8+ \u2654f7 38.\u2656a7+ \u2654e6 39.\u2656a3 d5 40.\u2654f2 \u2657h4+ 41.\u2654g2 \u2654d6 42.\u2658g3 \u2657xg3 43.\u2657xc4 dxc4 44.\u2654xg3 \u2654d5 45.\u2656a7 c3 46.\u2656c7 \u2654d4**\n\nWhite resigned.\n\nOn the 10th May 1960 Mikhail Tal received from the vice-president of FIDE Marcel Berman the gold medal of FIDE World Chess Champions and the laurel wreath.\n\nThe way in which Tal won his games drew a lot of attention. In the German weekly _Die Zeit_ the author and chess player Martin Beheim-Schwarzbach described the match as the victory of the 'player' over the 'thinker'. But the 'younger guy' did not simply catch the 'older guy' unawares. Tal's play was also characterised by deep strategic plans, was the evaluation of Beheim-Schwarzbach.\n\nAs a reward for winning the title Tal received the present of a car, a Volga which at that time was the best model one could have in the USSR. Tal, who had, however, absolutely no idea about technical matters and for example never even wore a watch, gave the car to his older brother Yasha as a present.\n**23. Against Tal and a clairvoyant**\n\n**The World Championship 1961:  \n_Mikhail Tal against Mikhail Botvinnik_**\n\nLatvian chess lovers were very keen on having the return match between Mikhail Tal and Mikhail Botvinnik if not completely, then at least half played in Riga. Botvinnik, however, argued that the return match should take place under as far as possible the same conditions as the previous one. The issue was laid before FIDE president Folke Rogard at the Chess Olympiad of 1960 in Leipzig for a decision. Because of a car breakdown, however, Tal arrived late in Leipzig and in the meantime Botvinnik and Rogard had already agreed to what Botvinnik wanted.\n\nIn addition, before the start of the match Botvinnik had got into a dispute with Nikolai Romanov, the president of the Committee for Sports. Romanov had informed Botvinnik in his office that Tal had reported ill on account of renal colic and that there would possibly be a delay in starting the match. Botvinnik insisted however on a medical certificate delivered by a Moscow doctor. Tal had only presented a certificate from a Latvian doctor. He would therefore have had to travel to Moscow for an examination. Romanov considered this first and foremost as superfluous. But he finally did demand it from Tal, but the latter preferred to accept the originally decided start to the match because he considered this procedure demeaning.\n\nKoblenz later indicated that two weeks before the match Tal had been the victim of a mild heart attack, and despite that he had moreover not given up his 50-60 cigarettes per day. In an interview in 1988 Tal was asked whether looking back over his life there was any decision he regretted. 'Yes, I would probably have been better postponing the return match against Botvinnik', was his reply.\n\nAs preparation for the return match Botvinnik again played training matches, this time against Semyon Furman, who replaced Yuri Averbakh.\n\nThe return match between Mikhail Tal and the now almost 50 year old Mikhail Botvinnik took place from 15th March to 12th May 1961 once more in Moscow, this time in the main hall of the Estrada Theatre (20\/2, Bersenevskaya Embankment). Founded in 1954 the theatre was moved in February 1961 from Mayakovskogo Square to its new home in a large grey complex of buildings by the Moskva river diagonally opposite the Kremlin. During the Stalin era leading comrades had their flats in the same block. Even the general secretary of the Comintern Georgy Dimitrov, who died in mysterious circumstances in Moscow in 1949, or Marshall Mikhail Tukhachevsky, a victim of the 'purges' of 1937, had lived there, as had the German painter and communist Heinrich Vogeler, who was transported to Kazakhstan after Hitler's attack on the Soviet Union and who died there of exhaustion after a year of forced labour. Later, in 1964, Marlene Dietrich gave a song recital in the Estrada Theatre, singing among others Pete Seeger's 'Where have all the flowers gone' and 'Lola' from the film the _Blue Angel_. The theatre is still operational today.\n\nFor the opening game of the World Championship match of 1961 the hall was full to the very last seat with approx. 1500 spectators. The drawing of lots had been done the previous day in the Hotel National in a small ceremony. Ex-World Champion Max Euwe had travelled to Moscow as an honoured guest for the opening ceremony and also watched the first games. One way Botvinnik had prepared for this match was with training matches, this time against Semyon Furman. The games of these training matches were only published much later. The rules for the return match were the same as for the previous WCh matches, with the exception that this time Botvinnik did not bother with his quirk of two envelopes and the double writing down of the sealed move.\n\nThis time the match took a different course from that of the first match between the two players the previous year. The games were very hard fought. Botvinnik won the first game and after Tal had equalised in game two, the third game too. Three draws were followed by a succession of seven decisive games, of which five went to Botvinnik. In the final third of the match Botvinnik won three more games, and Tal two. In the closing phase of the match, which was level, Botvinnik had felt tired, as he reported later in his tournament book.\n\nMoreover, he commented that the well-known hypnotist Wolf Messing had been brought from Latvia to support Tal and had been seated in the room for spectators. In any case, Botvinnik had not noticed him at all during the games on account of his extreme short-sightedness. According to Botvinnik, his bad eyesight had not been taken into account in Riga when the clairvoyant and hypnotist had been hired to go to Moscow.\n\nMessing, born near Warsaw, lived in Germany in the 1930s and had been much talked about as a result of numerous predictions and performances in which he demonstrated telepathic powers. After Hitler had put a price of 200 000 marks on his head on account of an unfavourable prediction \u2013 Messing had prophesied that Germany and the Nazis would be destroyed in the near future \u2013 Messing fled to the Soviet Union. There Stalin took an interest and subjected him to some tests, which Messing however passed. Apparently Messing predicted to Stalin the outcome of the Second World War and in doing so influenced the latter's decision to conclude the pact with Hitler.\n\nFinally Botvinnik \u2013 apparently thanks to his short-sightedness \u2013 won this return match too, and clearly so with 10:5 in wins. During the match Tal was, however, weakened because of acute health problems caused by his kidneys. He himself did not accept this excuse: 'I think I lost because he defeated me. He was excellently prepared for this match. Botvinnik knew my game much better than I did his.'\n\nLater, in his book _The Life and Games of Mikhail Tal_ , Tal again reviewed the match and with a twinkle in his eye he suggested two possible reasons for his defeat: during the 1960 match he and Botvinnik had lived in adjacent rooms in the 'Moscow' hotel. Before the games Tal's second Koblenz cheered up his prot\u00e9g\u00e9 by singing Neapolitan songs. Tal felt motivated by this, while it possibly demoralised Botvinnik next door, conjectured Tal. In any case, for the return match Botvinnik moved into another hotel.\n\nAngelina Tal describes her husband, at least as far as chess was concerned, as really superstitious and so the dethroned World Champion perhaps thought of another possibility. For the eighth game of the return match Tal had finally discovered a 'lucky pencil' with which to write down the moves. But after winning that game Tal forgot the pencil on the table. When he returned for the next game, it had disappeared. 'Perhaps an unknown supporter of Mikhail Moiseevich had taken it', conjectured Tal. Another 'lucky pencil' could not be found.\n\n **Botvinnik \u2013 Tal**\n\nMoscow, 7th game  \n29th March 1961  \nNimzo-Indian Defence (E24)\n\n**1.c4 \u2658f6 2.\u2658c3 e6 3.d4 \u2657b4**\n\nThe Nimzo-Indian Defence was up for discussion several times in this match.\n\n**4.a3**\n\nThe S\u00e4misch Variation: White wants to clarify the position on the queenside straight away. Tal had of course worked out that Botvinnik could again turn to the S\u00e4misch Variation and prepared for that case a rarely played variation. In the 3rd game of the match 4.e3 was played (Rubinstein Variation).\n\n**4... \u2657xc3+ 5.bxc3 b6**\n\nIn the first WCh match between Botvinnik and Tal 5...\u2658e4 was up for discussion in four games.\n\n**6.f3 \u2657a6 7.e4 d5**\n\nThis is the move Tal had prepared. Botvinnik considered it a positional mistake, since White gets rid of his weak c4-pawn.\n\nBlack had good success in practice with 7...\u2658c6.\n\n**8.cxd5 \u2657xf1 9.\u2654xf1**\n\nThe loss of the right to castle is a certain disadvantage to White. On the other hand, he obtains some superiority in the centre.\n\n**9...exd5 10. \u2657g5**\n\nThis was a new move by Botvinnik. Now e4-e5 is always in the air.\n\nTal had instead expected 10.e5 \u2658g8 11.\u2658e2 \u2655d7 12.\u2654f2 \u2658c6 13.\u2656e1 0-0-0 etc.\n\n**10...h6**\n\n'Tal wants to complicate play with a pawn sacrifice. 10...dxe4 11.fxe4 gives White a clear advantage in the centre and the f3-square'. (Botvinnik)\n\n**11. \u2655a4+**\n\n'A little intermediate move which forces Black into a concession.' 11.\u2657xf6 \u2655xf6 12.exd5 0-0 'with a more comfortable game for Black.' (Botvinnik)\n\n**11...c6**\n\n'Double-edged since the white bishop is left and no black knight can now come to c4.' (Botvinnik)\n\n11...\u2655d7 12.\u2655xd7+ \u2658bxd7 13.\u2657xf6 (13.\u2657h4!? dxe4 14.\u2656e1) 13...\u2658xf6 14.e5 \u2658d7 'with a visible advantage for White.' (Botvinnik), 11...\u2658bd7 12.\u2657xf6 \u2655xf6 13.exd5 first wins a pawn.\n\n**12. \u2657h4 dxe4?!**\n\n**13. \u2656e1**\n\n13.fxe4 is followed by 13...g5 14.\u2657g3 \u2658xe4 15.\u2657xb8 \u2655f6+ (15...\u2656xb8? 16.\u2655xc6+ +\u2013) 16.\u2658f3 \u2658d2+ 17.\u2654e2 \u2658xf3 18.\u2656hf1 \u2656xb8 19.\u2656xf3 (19.\u2655xa7 0-0 20.\u2656xf3 \u2655d6 favours Black) 19... \u2655g6 and Black is no worse.\n\n**13...g5 14. \u2657f2**\n\nProtecting the d4-pawn and enabling when appropriate c3-c4. 14.\u2657g3?! \u2655d5= would not be so good.\n\n**14... \u2655e7**\n\nOn 14...\u2655d5!? Botvinnik was planning 15.c4 \u2655e6 16.\u2655c2 ('16.fxe4 \u2658xe4 17.\u2655c2 f5 18.g4 \u2654d8 19.gxf5 \u2655xc4+ 20.\u2655xc4 \u2658d2+ would not be convincing' (Botvinnik)), followed by 17.fxe4 'with a decisive advantage'. (Botvinnik)\n\n**15. \u2658e2**\n\n15.fxe4 \u2658xe4 16.\u2655c2 f5 17.g4 \u2658d7 is unclear.\n\n**15...b5**\n\nOf course not 15...exf3 16.\u2658g3 losing the queen.\n\n**16. \u2655c2**\n\n16.\u2655b3 is met with 16...\u2655e6. After 17.\u2655xe6+ fxe6 Black has solved his problems.\n\n**16... \u2655xa3**\n\n16...e3 17.\u2657xe3 \u2658d5 18.\u2657c1. (Botvinnik)\n\n**17.h4 gxh4?!**\n\n'If 17...g4 then 18.fxe4 and the black knight cannot occupy the g4-square. He had to play 17...\u2656g8 18.hxg5 hxg5 after which, however, the white rook is immediately activated.' (Botvinnik)\n\n**18. \u2657xh4**\n\n'Now the bishop also enters the game with decisive effect.' (Botvinnik)\n\n**18... \u2658bd7 19.\u2658g3 0-0-0 20.\u2658xe4**\n\n**20... \u2656he8**\n\nThere was the better 20...\u2658xe4 21.\u2655xe4 (21.\u2657xd8? \u2658g3+ =) 21...\u2655xc3!? 22.\u2657xd8 \u2656xd8 23.\u2656xh6 \u2658b8 24.\u2656h7 a5 and Black still has some counterplay.\n\n**21. \u2654f2**\n\nAfter 21.\u2658xf6 \u2656xe1+ 22.\u2654xe1 (a better way was 22.\u2657xe1 \u2658xf6 23.\u2655f5+ +\u2013) 22...\u2655a1+ 23.\u2655d1 \u2655xc3+ 24.\u2654f1 \u2658e5 'some complications would still arise'. (Botvinnik)\n\n**21... \u2658xe4+ 22.fxe4 f6 23.\u2656a1**\n\nWith the conquest of the a-file the game comes to a rapid end.\n\n**23... \u2655e7 24.\u2656xa7 \u2655xe4 25.\u2655xe4 \u2656xe4 26.\u2656a8+ \u2658b8 27.\u2657g3 \u2654b7 28.\u2656ha1 \u2656c8 29.\u26568a7+ \u2654b6 30.\u2657xb8 b4**\n\n30...\u2656xb8 31.\u26561a6#.\n\n**31. \u2657d6 bxc3 32.\u2657c5+ \u2654b5 33.\u26561a4**\n\nAnd mate can no longer be avoided. Black resigned.\n\nAs had Vassily Smyslov, so did Mikhail Tal also lose his return match against Botvinnik relatively clearly. Tigran Petrosian later described the World Champion's right to a return match as 'moral disarmament' and tried to explain in this way why the challengers, who had in the previous matches shown such superiority, then after winning the title lost so surprisingly clearly in the return match. The challengers had won the right by victories in exhausting tournaments and elimination matches to challenge the World Champion. They had prepared meticulously for the struggle and won it. But then the return match followed like a prolongation of their struggle and they could not adopt a correct attitude to it. Only one year later they no longer had the energy to put in the same effort as they had before winning the title.\n\nSpassky believed, moreover, that the short period of only one year between the gain of the title and the return match worked very much to the advantage of Botvinnik. Whilst the new World Champion did not recover during the year, Botvinnik was already preparing for the return during the match he lost. Botvinnik himself admitted that the matches brought him a lot of information about his opponents which he made use of in the return match.\n\nAfter losing the title of World Champion Mikhail Tal remained for a long time at the absolute top of world chess with outstanding results in numerous tournaments. He took part in the USSR championships a total of 21 times, and won six times (1957, 1958, 1967, 1972, 1974, 1978), a record which he shares with Botvinnik. On several occasions he reached the WCh candidates' matches, but was never again able to qualify as challenger. From July 1972 to April 1973 Tal played 86 games without defeat (47 wins, 39 draws). Shortly afterwards he improved on this record with a new series of 95 games without defeat (46 win, 49 draws) in the time between 23rd October 1973 and 16th October 1974. Tal's ability was particularly great in blitz chess. In 1988 he won at an age of 51 the Blitz World Championship in Saint John in the final against Vaganian; Kasparov and Karpov had been eliminated in the earlier rounds.\n\nIn addition to his other ailments, Tal suffered at the end of his life from chronic hepatitis C and an equally chronic staphylococcal infection, both of which Tal fought with alcohol consumption. After the break-up of the Soviet Union in 1990 Tal and his family had to leave their flat in Riga, since it and the remainder of the house were restored to their original owners. Tal moved to Germany, where the Bonn chess philanthropist Ernst Eimert took care of him and his family. On the 27th June 1992 Tal died in Moscow of a haemorrhage in his oesophagus. He was buried in the New Jewish Cemetery in Riga. Ernst Eimert continued to look after Tal's family and shortly before his own death he married Angelina Tal so that she would be provided for.\n**24. Cheating in Cura\u00e7ao?**\n\n**The World Championship 1963:  \n_Mikhail Botvinnik against Tigran Petrosian_**\n\nAccording to the calculations of Jeff Sonas, in May 1962 Tigran Petrosian had passed Tal in the world ranking list and had himself occupied the top spot. Tigran Petrosian was of Armenian descent but was born on the 17th June 1929 in Tbilisi, capital of the then Georgian Socialist Soviet Republic, nowadays the capital of Georgia. His father Vartan Petrosian worked there as a caretaker in a home for officers. Tigran had two older siblings, a brother Hmayak and a sister Vartoosh. All three were described as good pupils. In Tbilisi Tigran Petrosian went to the 73rd Armenian School.\n\nPetrosian learned to play chess at age eight when he watched soldiers of the garrison playing, though it took him quite some time to learn the rules of the game. At 12 Petrosian joined the chess section of the pioneer palace of Tbilisi. He became friends with two other chess pupils, one was called Fine, not to be confused with the US grandmaster, the other was Andro Tekiashvili, and began analysing chess games together with them. In the pioneer palace Archil Ebralidze, a believer in the positional teachings of Nimzowitsch and Capablanca, was his first trainer.\n\nAfter the outbreak of the Second World War, hard times started for the citizens of Tbilisi. This was particularly true for the Petrosian family, which was struck by heavy blows of fate. In the winter of 1942 Petrosian's mother died. A great-aunt then moved in with the family and supported Vartan, who was then 70 years old. The children too now had to help earn their living. Tigran Petrosian took up various odd jobs, including that of street sweeper. The work in the open made him severely ill one winter and he was absent from school for six months. As a result of his illness he became hard of hearing, probably as the consequence of delayed otitis. The deafness was a handicap all his life and it was only in later years that he wore a hearing aid. From his meagre income Petrosian bought the manuals of Aaron Nimzowitsch and Rudolf Spielmann. It was particularly Nimzowitsch that he took as his model. Towards the end of the Second World War Petrosian's father also died and the 15 year old Tigran took on his father's job in the officers' quarters. At times he lived in the chess club and slept on a table there.\n\nIn 1946 Petrosian won by 8:6 a match against Genrikh Kasparyan for the Championship of Armenia. Approximately in 1947 Petrosian moved from Tbilisi to the Armenian capital Yerevan, where he was now supported amongst others by Kasparyan and worked in the Spartak Chess Club as a trainer. At the end of 1949 he made the leap to Moscow, after previously making contact with Moscow's Spartak Chess Club. On his arrival in Moscow he was wearing a summer coat, summer shoes, and carrying a few chess books under his arm \u2013 those were all of Petrosian's possessions. In Moscow Petrosian soon started working with Andor Lilienthal.\n\nHe also took some important private decisions: in 1951 he got to know the English teacher and translator Rona Yakovlevna Avinesar and married her the following year. The couple had two sons, Mikhail and Vartan. Rona Petrosian-Yakovlevna later gained great influence in the Soviet Committee for Sports and, on account of the intrigues she was accused of, was not popular with many Soviet players. For Petrosian, however, she was a great support on his way to the top.\n\nIn 1951 Petrosian was second at the USSR championships. He also came in second at the interzonal tournament of 1952 in Saltsj\u00f6baden and thus qualified for the candidates tournament in Zurich 1953, which he finished in fifth place. During the subsequent years Petrosian's play suffered somewhat from too great an eagerness to take draws, which prevented him from achieving even greater successes. On the other hand Petrosian was considered by all the top players as the one who was hardest to beat since he had developed quite particular skills in defence. In 1959 Petrosian won the USSR championship for the first time, a success he was able to repeat in 1961. At the interzonal tournament in Stockholm 1962 he shared second and third place with Efim Geller (15 points each) behind the outstanding Robert Fischer (17\u00bd points out of 22). The first three, and Viktor Kortchnoi, Paul Keres, Miroslav Filip, Pal Benko and Mikhail Tal qualified for the candidates' tournament in Cura\u00e7ao.\n\nThis took the form of a four-round all-play-all and was won by Tigran Petrosian with 17\u00bd points ahead of Efim Geller (17). Tal was hospitalised during the tournament with renal colic and after three rounds withdrew from the tournament.\n\nAfter his outstanding success in the interzonal tournament in Stockholm many considered the 19 year old Bobby Fischer as a future World Champion, but at the candidates' tournament he started with two defeats, against Benko and against Geller, and could only finish in fourth place (14 points). After his return to the USA Fischer gave an interview in _Sports Illustrated_ ('The Russians have fixed world chess'), in which he accused the Soviet players of having agreed the results of their games in order to stop him from winning the tournament. In fact, Petrosian and his friend Geller, but also Keres, who was actually considered to be a sportsman of integrity, had drawn all their twelve games among each other in no more than 19 moves each, thus saving their energies for the games against the other players, especially Fischer. In addition Fischer claimed that Kortchnoi had been directed to lose to the other three Soviets. According to Fischer, moreover, the Soviet players had given each other advice during their games against him.\n\nThis was not the only occasion on which Soviet chess was accused of manipulation, private agreements and other foul play. One grandmaster once formulated it like this: 'Whenever two Soviet players are playing against one another, the result which emerges is always the most useful one.' Yuri Averbakh, who was the head of the Soviet delegation in Cura\u00e7ao, would later explain the short draws thus: the slightly older Keres wanted to save his energy and Petrosian and Geller did not play for a win against each other because they were good friends. Fischer's accusations against Kortchnoi were later rejected by the latter. Kortchnoi, who fled the USSR in 1976, later said that he too had been surprised by the series of short draws and considered that he too had been a victim of this agreement.\n\nWithin the Soviet delegation there were further mutual intrigues. In a game which was decisive for tournament victory between Benko and Keres, Petrosian and Geller offered Benko their help with the analysis after the adjournment of the game. But, according to his own account, Benko declined this and won the game without their help. There was also trouble between the two US Americans, Fischer and the Hungarian born Pal Benko, because the latter claimed the help of the second Arthur Bisguier, although Bisguier, commissioned by the US federation, was only there to support Fischer. 'I will never again take part in such a manipulated tournament', stated Fischer after the tournament and in fact withdrew from top chess for some years.\n\nCura\u00e7ao was also for a long time the last candidates' tournament. After it FIDE introduced the elimination of candidates in matches held on a k.-o. system. At the FIDE congress in Saltsj\u00f6baden (25th August till 5th September 1962) FIDE additionally agreed that draws before move 30 had to be approved by the arbiter. But despite all the accusations and complaints Petrosian was the victor of the candidates' tournament and thereby the challenger to the World Champion Botvinnik.\n\nAfter his victory in the return match against Tal, Botvinnik had explained at an event in Moscow that should a Soviet player win the candidates' tournament he might possible renounce the idea of defending his title. The Soviet Chess Federation prepared for this eventuality by organising a match between the second and third from the candidates' tournament, Keres and Geller, so as to have a match opponent ready for Petrosian if Botvinnik should do so. For Petrosian, however, that meant he either had to delay his preparation until Botvinnik had taken a decision or risk it being in vain.\n\nBotvinnik took his time about deciding. It only became clear in November 1962 that Botvinnik would again be there for a defence of his title. At the end of November 1962 Petrosian had an appointment for an operation to do away with his chronic angina. For that reason he would have liked to delay the start of the match from the 14th March till the 1st April 1963, to have more time for recuperation, but Botvinnik refused a delay because depending on circumstances the match could then have lasted into June, when it could be very hot in Moscow. Since the two players could not agree on a starting date, they called in FIDE president Folke Rogard to arbitrate. He then, like Solomon, set the date for the 23rd March. In the course of the disputes personal relations between the two players, which had previously been completely friendly, had cooled distinctly.\n\nPetrosian had at first retreated with his training partners Boleslavsky and Suetin to a sanatorium to prepare. One month before the start of the match he moved with his little team to a convalescent home of the federation of architects close to Moscow, a place where he had previously cloistered himself before important tournaments and before matches. To toughen himself up physically Petrosian regularly went skiing there.\n\nDuring his preparation Petrosian, who had been living since 1949 in Moscow, also bought his first warm winter coat so as to be armed against disease in the cold Moscow winter. Despite the coldness of the Moscow winters he had never previously owned anything like it. Even in the 1960s Soviet citizens apparently still lacked necessities.\n\nPetrosian had noticed that in the matches which he had won, Botvinnik had usually taken an early lead. So Petrosian wanted above all to avoid falling behind at an early stage. Three weeks before the start of the match Petrosian stopped all chess activities, simply concentrated on sport, took walks, played billiards, read books and in other ways distracted himself before the forthcoming match.\n\nIn his preparation for the match Botvinnik had, as before the return match against Tal in 1961, worked with Semyon Furman, who was considered a proven expert in the openings. He had already quarrelled with his former trainer and friend Grigory Goldberg after the first match against Tal, since Botvinnik had the feeling that Goldberg 'felt drawn' to Tal. In summer 1960 Goldberg sent a letter to Botvinnik and on nine pages set out the conditions under which he was prepared to continue his cooperation with Botvinnik. Botvinnik later explained that he was not prepared to meet a single one of these conditions. Before the match against Petrosian, Botvinnik played a secret training match with Furman, consisting of eight games, which were not published until much later. Furman also belonged to Botvinnik's team during the match.\n\nSo the match for the World Championship between Botvinnik and Petrosian began on the 23rd March 1963 and lasted till the 20th May 1963. The venue was once more Moscow's Estrada Theatre. The rules were essentially the same as had been in effect in previous matches. The match was set for a maximum of 24 games. If one player previously reached 12\u00bd or 13 points, the match would be brought to an end forthwith. The time control as in previous WCh matches was two and a half hours for 40 moves. After adjournment the game continued with a thinking time of one hour for 16 moves. For the analysis of adjourned games the players were only allowed the help of a second who had to have been officially nominated two weeks before. Petrosian named Isaak Boleslavsky as his official second, whereas Botvinnik did without a second.\n\nIn the playing hall there was to be the strictest silence. If there was too much noise the game could be continued with the public excluded. Photographers had to bear in mind that the use of flash was forbidden during the games. In the event of illness a player could postpone for a maximum of three days. Gideon Stahlberg and Harry Golombek were named as arbiters for the match. The drawing of lots for colours was done totally without ceremony on the 20th March 1963 in the hotel room of arbiter Stahlberg in the Hotel Metropol. Max Euwe was present to represent FIDE. On the following day the match was opened in the Estrada Theatre.\n\nUp until game 15 the match was level. Botvinnik won the first and 14th games, when Petrosian's plan of avoiding an early deficit did not work. But the fifth and seventh game went to Petrosian. When Botvinnik took a 'timeout' at the start of the 9th game, Mikhail Tal entertained the spectators in the hall by playing a game against them. Tal was actually otherwise active in the press centre as a correspondent and analysed the games together with the other grandmasters present, including Geller and Kotov.\n\nWith game 15 the match turned in Petrosian's favour. He won that game, then in addition games 18 and 19 too. Before the 20th game, with the score at 11:8, Petrosian reported himself ill with a stomach upset, so that the game had to be postponed for two days. The game then ended in a very quick draw. Botvinnik had accepted his defeat. Two other short draws followed. Then Petrosian had won the match with a score of 12\u00bd:9\u00bd.\n\n **Petrosian \u2013 Botvinnik**\n\nMoscow, 15th game  \n29th April 1963  \nGr\u00fcnfeld Indian Defence (D97)\n\n**1.d4 \u2658f6 2.c4 g6 3.\u2658c3 d5 4.\u2655b3**\n\nAn idea which Botvinnik introduced in the game against Levenfish at the 8th USSR championship 1933. With the insertion of 4.\u2658f3 \u2657g7 the move 5.\u2655b3 leads to the so-called Moscow Variation, one of the main variations against the Gr\u00fcnfeld Indian Defence: 5...dxc4 6.\u2655xc4 0-0 with positions similar to those in the game.\n\n**4...dxc4 5. \u2655xc4 \u2657g7 6.e4 0-0**\n\nThe basic idea of the Gr\u00fcnfeld Indian Defence for Black consists of letting White set up a pawn centre, in order to then attack it with pieces or to attack the flank pawns on the c- and f-files.\n\n**7. \u2657e2 \u2658c6 8.\u2658f3 \u2658d7 9.\u2657e3 \u2658b6 10.\u2655c5**\n\n10.\u2655d3 f5!?.\n\n**10... \u2657g4**\n\n10...f5, to attack the centre from the flank as described, was also worth considering.\n\n**11.d5 \u2658d7 12.\u2655a3**\n\nThe alternative was 12.\u2655c4!?.\n\n**12... \u2657xf3 13.\u2657xf3**\n\n13.dxc6 \u2657xe2 14.\u2658xe2 bxc6 15.\u2656d1 \u2655e8 16.0-0 \u2658b6 leads to a level position.\n\n**13... \u2658d4 14.0-0-0**\n\nMore ambitious than 14.\u2657d1 c5 15.dxc6 bxc6.\n\n**14... \u2658xf3**\n\n'After 14...c5 15.dxc6 \u2658xc6 16.\u2654b1 \u2655c8 17.\u2657e2 White has the bishop pair and the active plan f2-f4 and e4-e5.' (Kasparov)\n\n**15.gxf3 \u2658b6**\n\n15...c6 16.dxc6 bxc6 17.h4 h5 18.f4 e6 19.f5 is advantageous for White, since the black king position is weakened.\n\n**16. \u2655b3 \u2655d7**\n\nAnother possibility was 16...\u2655c8!? to prepare...c7-c6.\n\n**17.h4 h5**\n\nAnother option was 17...c6. After 18.h5 Black does not play 18...cxd5 19.hxg6 hxg6 20.\u2657xb6! axb6 21.\u2658xd5 with advantage to White, but 18...\u2657xc3! 19.\u2655xc3 cxd5 with counterplay.\n\n**18.f4 e6?!**\n\n18...c6!? (Kasparov), so as to get counterplay through the opening of the c-file, offered more prospects of counterplay.\n\n**19.dxe6 \u2655xe6 20.\u2655xe6 fxe6 21.\u2656hg1 \u2654h7 22.\u2658b5 \u2656f7 23.\u2658d4 \u2656e8**\n\nPerhaps a better way was 23...\u2657xd4!? 24.\u2656xd4 \u2656af8. (Kasparov) In the bulletin 24...\u2656e8 was suggested. Kasparov gives 25.\u2656g5 \u2656d7 26.f5 as better for White. It was also worth considering 23...\u2658c4 24.\u2658xe6 \u2657xb2+ 25.\u2654c2 \u2657f6 26.\u2656d5 in any case also with an initiative for White.\n\n**24. \u2658f3 \u2657h6 25.\u2658g5+ \u2657xg5 26.\u2656xg5**\n\n'The black position is strategically lost.' (Kasparov)\n\n**26... \u2658c4?**\n\n26...\u2658c8 was more accurate. Now Black replies to 27.f5? (27.\u2656dg1 \u2656g8 28.b3 \u2658d6 leads to the game) 27...exf5 28.exf5 \u2656xf5 29.\u2656d7+ with 29...\u2656e7=.\n\n**27. \u2656dg1?!**\n\nWhite misses 27.f5! (Konstantinopolsky) 27...exf5 28.exf5 intending 28...\u2656xf5? 29.\u2656d7+ \u2654h8 30.\u2657d4+ \u2658e5 31.f4+\u2013.\n\n**27... \u2656g8 28.\u2654c2! b6**\n\nKonstantinopolsky suggested 28...\u2658d6!? 29.f3 a6 intending...\u2656d7 and...\u2658f7.\n\n**29.b3 \u2658d6 30.f3 \u2656d7**\n\n30...c5!? was more active: 31.a4 \u2658b7 32.\u2654c3 \u2658a5 so as to bring the knight via c6 to d4.\n\n**31. \u26565g2 \u2656dd8?!**\n\nA better move was 31...\u2658b5 32.\u2656d1 \u2658a3+ 33.\u2654c1 \u2656gd8 34.\u2656xd7+ \u2656xd7 35.\u2656d2 \u2656xd2 36.\u2654xd2 c5 and Black would have been freed from most of his worries.\n\n**32.a4!**\n\nPreventing the possibility previously mentioned.\n\n**32... \u2658f7 33.\u2657c1 e5 34.\u2657e3 exf4**\n\nIn the bulletin 34...\u2656df8 or 34...c5!? was recommended instead.\n\n**35. \u2657xf4 \u2656d7?**\n\nThe immediate 35...c5 saves an important tempo. After 36.\u2656d2 \u2656xd2+ 37.\u2654xd2 \u2656d8+ 38.\u2654e2 \u2656d7 39.a5 \u2658d8 40.axb6 axb6 41.\u2656a1 \u2658c6 42.\u2656a6 \u2658d4+ Black is able to fight for the draw.\n\n**36. \u2656d2 \u2656xd2+ 37.\u2654xd2 \u2656d8+ 38.\u2654e2 c5 39.a5! \u2656d7 40.axb6 axb6 41.\u2656a1 \u2654g7**\n\nA better try was 41...\u2658d8 42.\u2656a6 \u2656b7 43.\u2656a8 \u2658c6. The game was adjourned here.\n\n**42. \u2656a6**\n\n42.\u2656a8!?.\n\n**42... \u2656b7 43.\u2656a8 \u2654f6**\n\n43...b5 44.\u2656c8 c4 is followed by 45.b4! with advantage. Black's play on the queenside is finished, since the c-pawn cannot advance without being lost.\n\n**44. \u2656c8 \u2658e5 45.\u2654e3 \u2658d7 46.\u2656c6+ \u2654f7 47.e5 \u2658f8 48.\u2656f6+ \u2654g7 49.\u2654e4**\n\nWhite's pieces are clearly more active.\n\n**49...b5 50. \u2656c6 \u2654f7 51.\u2656xc5**\n\nAfter the loss of the c-pawn the game is finally decided.\n\n**51... \u2658e6 52.\u2656d5 \u2654e7 53.\u2657e3 \u2656b8 54.\u2656d6 b4 55.\u2656a6 \u2656b5 56.\u2656a7+ \u2654e8 57.f4 \u2654f8 58.f5**\n\nBlack resigned on account of 58...gxf5+ 59.\u2654xf5 \u2658g7+ 60.\u2654g6 \u2658e8 (if 60... \u2656xe5 61.\u2656f7+ \u2654e8 then 62.\u2657f4+\u2013) 61.e6 \u2656e5 62.\u2657h6+ \u2654g8 63.\u2656e7+\u2013.\n\nBotvinnik thus lost his title after 15 years \u2013 with two interruptions \u2013 and this time he was not able to win it back. The right to a return match had been done away with. There is no doubt that the great age difference between the two contestants had its part to play. Petrosian was 18 years younger than Botvinnik. As the defeated player in the WCh match Botvinnik would have been entered into the next candidates' matches, but he renounced this right and took no more part in the WCh cycle.\n\nAfter losing the match Botvinnik accorded his recognition to the new World Champion: 'It was very difficult to play against Tigran. The reason is that he has a quite different understanding of positional play. He goes more deeply into the position than is usual and I, a universal player, did not completely understand Tigran's method and the depth of his positional judgement, although I correctly evaluated all the positions'. When on the 20th May 1963, shortly after 17.00, Petrosian's victory became official with a draw in game 22, hundreds of his fans stormed on to the stage of the Estrada Theatre and f\u00eated their idol. On his return to Yerevan Petrosian was welcomed by more than 100000 enthusiastic Armenians and f\u00eated at the central square. Spontaneously a collection was organised for the World Champion. At the end of it over a million roubles had been gathered for the first Armenian World Chess Champion.\n\nAfter the loss of the title Botvinnik went on to play in the 1964 Chess Olympiad in Tel Aviv and in the European team championships in 1965 in Hamburg for the USSR and won the gold medal with his team each time. Moreover Botvinnik continued to take part in various team tournaments in the Soviet Union. In individual tournaments too, he still took excellent places: he was second behind Larsen in Palma de Mallorca 1967 and Monte Carlo 1968, also second behind Geller in Wijk aan Zee 1969. In the Netherlands sponsors would have been prepared to finance a match between Botvinnik and Fischer. Fischer did not take the opportunity. He insisted on a match to six wins, whereas Botvinnik wanted to play a maximum of 18 games.\n\nBotvinnik's final important international tournament was the match USSR against the Rest of the World in 1970 in Belgrade, in which he had four draws against Milan Matulovic. The 'patriarch' was involved with his Botvinnik chess school in the education of many talented young Soviet players. Students from this crucible of talent were the future World Champions Karpov, Kasparov and Kramnik. Botvinnik was also successful as an author. His best known works were _K dostizheniu cheli (Achieving the aim)_ , Moscow 1978, which appeared as an edition of 75000 copies and _Polveka w Skakhmatakh (Half a century of chess)_ , Moscow 1978, with a printing of 100 000 copies.\n\nIn the middle of the 1980s Botvinnik fell into disfavour with the Sports Committee because he had expressed himself critically about the situation in the USSR in an interview with a journalist from a US newspaper. When Mikhail Gorbachov called forth glasnost and perestroika in the Soviet Union, Botvinnik quarrelled with his former model pupil Garry Kasparov on account of differing views about the development of the country. Kasparov welcomed the changes, Botvinnik refused them. His beloved wife Gayane died on the 4th December 1987, but Botvinnik outlived her by more than seven years. With age Botvinnik gradually became blind. He died on the 5th May 1995 at the age of 83. He was buried in the Novodevichi Cemetery (Cemetery of the nine virgins) in Moscow. His final wish was a quiet funeral with no officials and no chess players.\n**25. The obstreperous pupil**\n\n**The World Championship 1966:  \n_Tigran Petrosian against Boris Spassky_**\n\nBoris Spassky, born on the 30th January 1937 in Leningrad, now St. Petersburg, was perhaps the most gifted player of all Soviet chess.\n\n**Boris Spassky (born in 1936)**\n\nHis father Vassily Spassky was a civil engineer and came from a family of priests in the orthodox church who lived in the small town of Tim in the region of Kursk. His mother Petrova Ekaterina Petrovna was the illegitimate daughter of a large landowner from Riadnevo, in the region of Pskov. She grew up with foster parents, relatives of her mother Daria Ivanova. Spassky's parents got married in 1932 in Leningrad and had three children, Boris, his older brother Georgy (born in 1934) and a younger sister Iraida (born in 1944).\n\nDuring the Second World War Spassky, together with his brother and other children from Leningrad, was evacuated to the region of Kirov. In the children's home there he learned chess at the age of five by watching the games of the older children. At the start of 1943, however, Spassky was close to death in this children's home on account of debilitation and hunger. Just at that point his parents managed to escape from the besieged city of Leningrad. Their arrival in Kirov saved their son's life as there was then someone to take care of him. From 1943 to 1946 the family lived in Sverlovka, some 40 kilometres from Moscow. In 1944 the parents divorced. Spassky's mother went back to Leningrad in the summer of 1946, alone and now with three children since her daughter Iraida had also been born. The fatherless family of four lived there in a single room measuring 14 square metres.\n\nOne day Spassky discovered on the Kirov Islands in the Baltic a chess pavilion with glazed chess tables, which entranced him and caught hold of his imagination, and he watched the adults playing. To travel there cost him 15 kopecks per day for his ticket and he had further hardship to endure \u2013 neither Boris nor his brother Georgy had shoes and so they had to go about barefoot. After the pavilion was closed in the autumn, Spassky set about joining a chess club and finally managed at the age of nine to get into the chess section of the Leningrad pioneer palace, which was based in the Anichkov Palace, the former residence of Alexander III.\n\nAt that point he was assigned to the 33 year old trainer Vladimir Zak, who took particular pains with his youngest pupil, and from 1948\/49 even received a monthly state stipend of 120 roubles (Jan van Reek suggests 1200 roubles in his biography, but that would have been much more than other players received), with which he contributed to the living expenses of the family, since his mother, who had for a long time worked as a harvester, was now only considered partially able to work as the result of a back injury. Spassky's seven year younger sister was also a talented player, though of another board game \u2013 Iraida Spasskaya became Soviet draughts champion four times.\n\nAt the age of ten in a simultaneous exhibition in Leningrad Spassky defeated Mikhail Botvinnik. At 11 Spassky gave his own first simultaneous display against 15 opponents in an officers' mess. Boris Spassky now began to train with Alexander Tolush. Tolush, 42 years old when Spassky made his acquaintance, was a player through and through, who in addition to chess also loved billiards, cards and horse racing. He liked alcohol, 'but knew when to stop' (Spassky). His personality, according to Spassky, was that of a landowner who had fallen on hard times. When he was in a bad mood he would say 'How deep...', without finishing the sentence, which would have continued '... the country has sunk'. Tolush had experienced the communist terror and the subsequent executions, without ever speaking about it. Basically they never talked about politics. Spassky, now 15, learned from Tolush not only the principles of chess but also some important things for daily life which he had not known until then, namely how to eat with a knife and fork, to use a handkerchief and to knot a tie. In addition Spassky adopted Tolush's sceptical political views.\n\nAt the first international tournament, at which Spassky participated along with Tolush, he shared 4th to 6th places and in doing so defeated no less than Vassily Smyslov. His trainer won the tournament. During the tournaments the Soviet delegation received a telegram from the Committee for Sports with the instruction that from then on the players should no longer struggle against each other but act in common to prevent Laszlo Szabo, who was leading the field, from winning the tournament. Spassky learned about Soviet dirigisme, which also extended to chess.\n\nSpassky ended his first USSR championship in 1955 with a shared 2nd to 6th place and in doing so qualified for the interzonal tournament in Gothenburg, since this USSR championship was at the same time considered to be a zonal tournament. In the same year he became U20 Junior World Champion. During the Junior World Championship in Antwerp Spassky asked the leader of the Soviet delegation, Commissar E.P. Soloviev, whether it was true that Lenin had had syphilis. The consequence of this naive question was a serious investigation by the Sports Committee. Spassky may well have been pardoned, but from then on he found himself under constant surveillance from the Leningrad KGB. His 6th-9th place at the interzonal tournament allowed Spassky to take part in the candidates' tournament in Amsterdam in 1956. The nineteen year old took third to seventh place. Thanks to his successes he was then allocated for himself, his mother and his siblings a 28 square metre two room flat.\n\nAfter his schooldays Spassky started to study mathematics, but gave that up after two months. After that he studied journalism, which he did not like either. Despite that he finished his studies after five years with a diploma. The subject of his dissertation was the first Russian chess magazine _Shakhmatny Listok_ 1859-1863.\n\nIn 1960 Spassky experienced a serious personal crisis. He fell ill with angina, then kidney stones. The following year saw the break-up of his first marriage with Nadezhda Latynzeva, a fellow student, to whom he had been married since 1959 and with whom he had a daughter. Spassky offered a divorce, but his wife declined and continued to live with Spassky's family in the 28 square metre flat, with the result that a state of war then existed between the couple. An influential admirer of Spassky finally organised a one room flat for the latter's wife so that she could move out with their daughter. At the Students' Olympiad Spassky then lost a spectacular and decisive game against the US American player William Lombardy and as a punishment was reprimanded by the Soviet Committee for Sports.\n\nFinally his trainer Tolush, who had treated Spassky like his own son, turned away from him. Tolush had always protected Spassky through all his problems, but finally lost patience with his prot\u00e9g\u00e9 and resigned his post as trainer. Tolush and Spassky later saw each other again occasionally, but never recovered their former friendly relationship. Tolush died four months before Spassky became World Champion.\n\nIn 1961 Spassky began working with Igor Bondarevsky, who also became a father substitute for him. Spassky, by then 24 years old, moved into Bondarevsky's three room flat. Bondarevsky came from a family of Don Cossacks and had a Turkish grandmother. Unlike Tolush, he had more pedagogical skill in dealing with his obstreperous pupil. But in common with Tolush he had a liking for alcohol. In 1961, under Bondarevsky's care, Spassky won the 29th USSR championship in Baku. On a suggestion from Bondarevsky, Spassky moved in 1963 from Leningrad, where the local KGB section had him under constant surveillance, into a 20 square metre flat close to Moscow.\n\nBoris Spassky and Tigran Petrosian\n\nAt the 1964 interzonal tournament in Amsterdam Spassky was joint first and thus qualified for the candidates' stages, which after the experiences of Cura\u00e7ao 1962 were now being played as matches. Spassky won the candidates quarter-final against Paul Keres, played over ten games in 1965 in Riga, by 6:4. After that, in the same place he defeated Efim Geller in the semi-final by 5\u00bd:2\u00bd. The candidates' final was held in 1965 in Tbilisi. There Spassky met Tal and won by 7:4, winning the final three games. This made Spassky the challenger to the World Champion Petrosian.\n\nThe match for the World Championship between Tigran Petrosian and Boris Spassky took place from the 9th April to the 9th June in Moscow's Estrada Theatre. Play was on Mondays, Wednesdays and Fridays. Adjourned games were resumed in the Central Chess Club. This WCh match was also for a maximum of 24 games. If it was drawn the title defender remained World Champion. The chief arbiter was Alberic O'Kelly de Galway. During the match Spassky shared a room in the Moskva hotel with Bondarevsky, his official second for this match.\n\nThe match began with a series of six draws, before Petrosian took the lead in the seventh game and extended his lead with a victory in the tenth game. Spassky equalised with wins in games 13 and 19. The 20th and the 22nd games once again went to the title defender. Before game 20 Spassky and Bondarevsky had gone for a boat trip with the Smyslov couple and had suffered severe sunburn in the spring sunshine. Bondarevsky recommended a timeout, but Spassky refused and lost the 20th game. Spassky did close up again with victory in game 23. But that was no longer enough \u2013 Petrosian won by 12\u00bd:11\u00bd.\n\n **Petrosian \u2013 Spassky**\n\nMoscow, 10th game  \n2nd May 1966  \nKing's Indian Defence (E66)\n\n**1. \u2658f3 \u2658f6 2.g3 g6 3.c4 \u2657g7 4.\u2657g2 0-0 5.0-0 \u2658c6**\n\nThe classical reply against the Fianchetto System in the King's Indian Defence is reached after 5...d6 6.d4 \u2658bd7 7.\u2658c3 etc.\n\n**6. \u2658c3 d6 7.d4**\n\nAfter 7.d3 e5 a popular line of the English Opening would arise: 8.\u2656b1 a5 9.a3 etc.\n\n**7...a6**\n\nAnd we have reached the Panno Variation of the King's Indian Defence. Other options are 7...e5 or 7...\u2657f5.\n\n**8.d5 \u2658a5**\n\nThat is the idea. The fianchetto of the light-squared bishop to g2 has weakened the c4-pawn, which is now attacked.\n\n**9. \u2658d2 c5 10.\u2655c2 e5**\n\nThe alternative is 10...\u2656b8, used by Spassky for example in his game against Bronstein, Amsterdam\/Leeuwarden 1956.\n\n**11.b3 \u2658g4**\n\n11...\u2656b8 as preparation for the advance...b7-b5 is also still playable here.\n\n**12.e4**\n\nIn the bulletin 12.\u2657b2 f5 13.\u2656ae1!? was suggested.\n\n**12...f5 13.exf5 gxf5 14. \u2658d1!?**\n\n14.\u2657b2! \u2657d7 15.\u2656ae1 b5 16.\u2658d1 with a slight advantage to White. (Bulletin)\n\n**14...b5 15.f3?!**\n\nThis move was criticised because it invites Black to advance his e-pawn. There was the more obvious 15.\u2657b2 \u2656b8 16.f3 \u2658f6 17.\u2657c3 \u2657h6 18.\u2656e1.\n\n**15...e4!?**\n\n'15...\u2658h6 was more prudent, so as to organise one step at time some counterplay on the kingside.' (Suetin)\n\n**16. \u2657b2 exf3 17.\u2657xf3 \u2657xb2 18.\u2655xb2 \u2658e5 19.\u2657e2**\n\n**19...f4!**\n\nThere was also the interesting preparatory move 19...\u2656a7!? intending 20.cxb5 axb5 21.\u2657xb5 f4. Now the game becomes very tactical and very concrete.\n\n**20.gxf4?!**\n\nAfter the better 20.\u2656xf4!? \u2656xf4 21.gxf4 \u2658g6 22.\u2658e4 \u2658xf4 23.\u2658df2 then 23...\u2656a7? would be unfavourable on account of 24.\u2657g4, for example: 24...\u2656g7 25.\u2654h1 bxc4 26.\u2657xc8 \u2655xc8 27.\u2658f6+ \u2654f8 28.\u2658xh7+ \u2656xh7 29.\u2655f6+ \u2654e8 30.\u2656g1 \u2654d7 31.\u2655f5+ +\u2013. Instead, a better way is 23...\u2658xe2+ 24.\u2655xe2 \u2655e7 with slightly better play for White. 'The move in the game is linked to a trap, into which Black immediately falls.' (Suetin)\n\n**20... \u2657h3?**\n\nThere is now the strong threat of 21... \u2656xf4, but White can defend himself. A better try was the immediate 20...\u2656xf4 21.\u2658e3 (21.\u2656xf4 \u2655g5+ =) 21...\u2655g5+ 22.\u2654h1 \u2656xf1+ 23.\u2658exf1 (23.\u2658dxf1 \u2657h3 with an attack) 23...\u2657f5 24.\u2658g3 \u2657g6=.\n\n**21. \u2658e3!**\n\nSpassky had overlooked this move.\n\n**21... \u2657xf1**\n\n21...\u2656xf4 22.\u2656xf4 \u2655g5+ is now followed by 23.\u2656g4! \u2658xg4 24.\u2658xg4 \u2657xg4 25.\u2657xg4 \u2655xg4+ 26.\u2654h1 and after 26...\u2655d4 27.\u2656g1+ \u2654h8 28.\u2655xd4+ cxd4 29.\u2658e4 White has an advantage in the endgame since the d4-pawn is weak.\n\n**22. \u2656xf1 \u2658g6**\n\n22...\u2658d7 23.\u2658e4, also with advantage to White.\n\n**23. \u2657g4 \u2658xf4?**\n\n23...\u2656xf4? 24.\u2657e6+ \u2654f8 25.\u2656xf4+ \u2658xf4 26.\u2655h8+ \u2654e7 27.\u2658f5#. The only defence was 23...\u2655f6! 24.\u2657e6+ \u2654h8 25.\u2655xf6+ \u2656xf6 26.f5 \u2658e5 27.\u2658e4!, though here too White has an advantage.\n\n**24. \u2656xf4!**\n\nAnother good move was 24.\u2658e4+\u2013.\n\n**24... \u2656xf4!?**\n\nAnother attempt was 24...\u2655g5!, after which 25.\u2656xf8+? would be a mistake. After 25...\u2656xf8 both \u2658e3 and \u2657g4 are hanging. The correct way is 25.\u2656e4 \u2655f6 26.\u2655xf6 \u2656xf6 27.\u2656e7 bxc4 28.\u2657e6+ \u2654h8 29.bxc4 and White is winning.\n\n**25. \u2657e6+ \u2656f7**\n\n25...\u2654f8 26.\u2655h8+ \u2654e7 27.\u2655xh7+ \u2654e8 28.\u2655h5+ \u2654e7 29.\u2655g5+ +\u2013.\n\n**26. \u2658e4 \u2655h4**\n\n26...\u2656aa7 27.\u2658f5 \u2655f8 28.\u2655f6 followed by 29.\u2658g5+\u2013.\n\n**27. \u2658xd6 \u2655g5+**\n\n27...\u2655e1+ 28.\u2654g2 \u2655xe3 29.\u2657xf7+ \u2654f8 30.\u2655h8+ \u2654e7 31.\u2658f5+ \u2654xf7 32.\u2655g7+ and 33.\u2658xe3+\u2013.\n\n**28. \u2654h1 \u2656aa7**\n\n28...\u2655xe3 29.\u2657xf7+ \u2654f8 30.\u2655h8+ \u2654e7 31.\u2658f5+ \u2654xf7 32.\u2655g7+ and 33.\u2658xe3.\n\n**29. \u2657xf7+ \u2656xf7 30.\u2655h8+!**\n\nA pretty finish. Black resigned due to 30...\u2654xh8 31.\u2658xf7+.\n**26. Fischer mates himself**\n\n**The World Championship 1969:  \n_Tigran Petrosian against Boris Spassky_**\n\nAs finalists in the previous candidates' matches Boris Spassky and Mikhail Tal were qualified for the candidates' matches of the next WCh cycle. To them were added six qualifiers from the interzonal tournament of Sousse (Tunisia) 1967: Bent Larsen, Viktor Kortchnoi, Efim Geller, Svetozar Gligoric, Lajos Portisch and Samuel Reshevsky.\n\nIn Sousse Robert Fischer took a fresh run at the title of World Champion. After the scandal of the Cura\u00e7ao tournament of 1962 he had taken a pause from tournament play and only in 1965 did he participate in serious tournaments again. In Sousse he led the tournament in superior fashion, but was in permanent strife with the organisers. The American was disturbed by the light, the noise, a photographer who was taking pictures and even by the sparkling of the stones in the chandelier of the ballroom of the Palace Hotel, where the tournament was being played. When the US ambassador to Tunisia Francis Russell came to dinner, Fischer would not allow even him to take a photo. After eight rounds an undefeated Fischer was in the lead ahead of Larsen and Reshevsky.\n\nDuring his abstinence from tournaments Fischer had turned towards a sect in the USA, 'The Church of God', which imposed upon its members a strict code of conduct, including strict observance of the rest day on the Sabbath. Fischer therefore demanded of the organisers that his Saturday games be moved to a different day, but he then again became dissatisfied when he noticed that then he would have difficult games to play on four successive days. He then refused to play his game against Aivars Gipslis, departed and went to Tunis, 80 km away. At the request of the US ambassador and the Tunisian federation he returned and on the following day, Sunday 29th October, arrived five minutes before the elapse of the hour's grace after which he would have had to forfeit his game against Samuel Reshevsky, whom he nevertheless convincingly defeated with the white pieces. Reshevsky's protest against the circumstances surrounding this game was rejected.\n\nIn the next round Fischer defeated Robert Byrne. Since the dispute about the rescheduling of Fischer's games had not yet been resolved, Fischer departed for the second time before his game against Vlastimil Hort. Negotiations again took place about his return, which were continued until 20 minutes before the start of the following round. Fischer was ready to recommence playing in the tournament, but wanted his game to be postponed by an hour, so that he could return from Tunis in time for the start of the game against Bent Larsen. When this was refused he finally quit the tournament.\n\nFischer laconically announced that he first wanted to go to Rome to buy some shoes and then to acquire a few chess books in Germany. Thus Larsen won the tournament.\n\nIn the candidates' matches Spassky again met Geller, this time in the quarter-finals played in Suchumi, and won by 5\u00bd:2\u00bd as in the previous cycle. The semi-final against Bent Larsen was hosted by Malm\u00f6. Spassky also defeated the top Danish player by 5\u00bd:2\u00bd. He also won the final against Kortchnoi, played from 6th to 26th September in the Palace of Culture in Kiev, by a clear 6\u00bd:3\u00bd and was therefore for the second time the challenger of World Champion Petrosian. To prepare for the match against Petrosian Spassky moved with his two seconds Bondarevsky and Nikolai Krogius for three months to a dacha in Dubna, one hour from Moscow, and also stayed there during the match.\n\nThe second match for the World Championship between Petrosian and Spassky was staged from the 14th April till the 16th June 1969 in Moscow's Estrada Theatre once more. Petrosian won the first game with the black pieces in front of a sell-out house, but Spassky took the fourth, fifth and eighth games for a lead for the meantime of 3:1. Petrosian had not been very optimistic going into the match. During the match his confidence level dropped even more, and he finally quarrelled with his two seconds Boleslavsky and Suetin. In the 9th game he chose a different opening from what had been previously discussed with his seconds and should actually have lost the game, but only thanks to some superficial analysis on Spassky's part after the adjournment he managed to get a draw. As the match continued he did completely without any common preparation.\n\nPetrosian then profited from a period of weakness by Spassky and equalised with wins in the 10th and 11th games. Game 14 played an important part in the way the match ended. After the adjournment, Petrosian was winning, but he gave away the win with several inaccuracies. The game ended as a draw. Spassky then won the 17th and 19th games and Petrosian the 20th. Spassky finally won the 21st game for a final score of 6:4 in wins. Spassky was confirmed as the tenth World Champion.\n\n **Spassky \u2013 Petrosian**\n\nMoscow, 19th game  \n4th June 1969  \nSicilian Defence (B94)\n\n**1.e4 c5 2. \u2658f3 d6 3.d4 cxd4 4.\u2658xd4 \u2658f6 5.\u2658c3 a6**\n\nThe Najdorf Variation in the Sicilian is nowadays one of the most popular openings after 1.e4. It mainly leads to sharp tactical play and therefore is not perhaps absolutely suited to Petrosian's style. However, the title defender had previously played the variation on occasions and possibly wanted to play for a win here as he had a deficit of 3:4. In doing so he perhaps thought first and foremost of the Closed Variation, which Spassky liked to play but which is actually harmless.\n\n**6. \u2657g5 \u2658bd7**\n\nFor a long time 6...e6 was more popular, when White mainly continues with 7.f4. Petrosian's move has again become very popular of late.\n\n**7. \u2657c4**\n\nThis move was obligatory when the game was played. White later also played 7.f4 with the offer to transpose to the variations following 6...e6 after 7...e6.\n\n**7... \u2655a5**\n\nAnother possibility was 7...e6. Now-a-days 7...\u2655b6 is also popular.\n\n**8. \u2655d2 h6?!**\n\nThe more usual move is 8...e6 9.0-0-0 b5 10.\u2657b3 \u2657b7 as in H. Lehmann-Petrosian, Munich 1958.\n\n**9. \u2657xf6!**\n\nA novelty. At that time the usual move was the retreat of the bishop to h4. Spassky and Bondarevsky had analysed the position in detail and had come to the conviction that the exchange favours White.\n\n**9... \u2658xf6 10.0-0-0 e6**\n\nAfter 10...e5 11.\u2658f5 \u2657xf5 12.exf5 \u2656c8 13.\u2657b3 \u2657e7 14.\u2656he1 the d5-square is in White's hands. 10...g6 is double-edged: 11.f4 \u2657g7 12.\u2656he1 0-0 13.f5 and White rapidly gets an attack on the kingside.\n\n**11. \u2656he1 \u2657e7**\n\nThis prepares kingside castling. However, the black king soon comes under fire. So 11...\u2657d7! intending...0-0-0, as suggested by Tal and Boleslavsky in the bulletin, was perhaps the better option.\n\n**12.f4 0-0 13. \u2657b3 \u2656e8**\n\nThe advance...b7-b5 does not work at the moment on account of \u2658d4-c6. If 13...\u2657d7 then 14.\u2654b1 and now 14...b5 is unfavourable on account of 15.\u2658d5 \u2655d8 (15...\u2655xd2? 16.\u2658xe7+) 16.\u2658xe7+ \u2655xe7 17.f5 e5 18.\u2658f3 \u2658e8 19.g4 with an attack for White. 13...\u2656d8!? was suggested as an improvement.\n\n**14. \u2654b1 \u2657f8**\n\n14...\u2657d7 is now met with 15.e5 dxe5 16.fxe5 \u2658h7 17.\u2658f5. After 17...\u2657c6 18.\u2658xe7+ \u2656xe7 19.\u2658d5 White is winning.\n\n**15.g4!**\n\nWhite opens the g-file with a pawn sacrifice.\n\n**15... \u2658xg4**\n\nThe alternatives were:\n\na) 15...\u2658d7 16.h4 \u2658c5 17.g5 with a strong attack for White (Geller);\n\nb) 15...\u2657d7 16.\u2656g1 also with a strong attack for White (Bondarevsky);\n\nc) Boleslavsky suggested 15...e5!? as an improvement, for example 16.fxe5 dxe5 17.\u2658f5 \u2657xf5 18.gxf5 \u2656ad8 19.\u2655g2 \u2655c7 20.\u2656xd8 \u2656xd8 21.\u2656g1, but now 21...\u2655c5!? is perhaps playable.\n\n**16. \u2655g2 \u2658f6**\n\n16...e5 17.\u2658f5 \u2657xf5 18.exf5 \u2658f6 19.\u2655xb7 is unfavourable for Black as is 16...\u2655h5 17.\u2656d3 intending \u2656h3.\n\n**17. \u2656g1 \u2657d7?!**\n\n17...\u2655c5!? (Averbakh) perhaps still offers sufficient defensive resources: if 18.\u2658f3 then 18...\u2655c6 19.e5 dxe5 20.fxe5 \u2658e4 21.\u2658xe4 \u2655xe4 22.\u2656d4 \u2655h7. Black holds on and is not worse.\n\n**18.f5 \u2654h8**\n\nHere 18...exf5 was suggested as an improvement. After 19.exf5 b5 then 20.\u2657xf7+! \u2654xf7 21.\u2658d5 with the threats \u2655g6+ and \u2658xf6 wins. But 19...\u2655e5 perhaps offers Black sufficient defensive resources.\n\n**19. \u2656df1**\n\nWith the immediate threat of fxe6 and \u2656xf6. It is already hard to still find a satisfactory defence for Black.\n\n**19... \u2655d8**\n\nIf 19...e5, then simply 20.\u2658de2 and White has a clear advantage, since the f7-pawn cannot be satisfactorily defended. 20...\u2654g8? 21.\u2655g6+\u2013 or 20...\u2656e7 21.\u2658d5+\u2013. After 19...\u2655e5 there comes 20.\u2658f3 \u2655f4 21.\u2655h3 followed by \u2658g5.\n\n**20.fxe6 fxe6**\n\n20...\u2657xe6 21.\u2658xe6 fxe6 22.\u2658e2 intending \u2658f4. (Smyslov)\n\n**21.e5!+\u2013**\n\nClears the e4-square for a knight. After that the black position falls apart in a flash.\n\n**21...dxe5 22. \u2658e4 \u2658h5**\n\n22...\u2658xe4 23.\u2656xf8+ \u2656xf8 24.\u2655xg7#; 22...exd4 23.\u2656xf6 gxf6 24.\u2655g8#.\n\n**23. \u2655g6! exd4**\n\n23...\u2658f4 is followed by 24.\u2656xf4! exf4 25.\u2658f3 \u2655b6 26.\u2656g5!! \u2657c6 (26...\u2655d8 27.\u2658e5 hxg5 28.\u2658f7+ \u2654g8 29.\u2658xd8 \u2656axd8 30.\u2658f6+ \u2654h8 31.\u2655h7#; 26...hxg5 27.\u2658exg5 \u2654g8 28.\u2655h7#) 27.\u2658f6 \u2657e4 28.\u2655xh6+! (Geller) 28...gxh6 29.\u2656g8#.\n\n**24. \u2658g5**\n\n**1-0**\n\n24...hxg5 25.\u2655xh5+ \u2654g8 26.\u2655f7+ \u2654h7 27.\u2656f3 followed by 28.\u2656h3#.\n\nIn the final stage of the game the public in the Estrada Theatre broke into storms of applause, in which even the Armenian fans of Petrosian joined.\n\nFrom 1963 to 1966 Petrosian was the chief editor of the magazine _Shakhmatnaya Moskva_. In 1968 he took over the job at the magazine _64_ and was its editor-in-chief until 1977. At that time _64_ had a circulation of 100 000 copies and was the most read chess magazine in the world. In 1968 Petrosian took his doctorate at the Philosophy Faculty of the University of Moscow. His PhD thesis was entitled _Some logical problems in chess thinking_. After the WCh match against Spassky Petrosian took part in another four WCh cycles. In 1971 he was defeated by Fischer in the candidates' final, in 1974 he lost to Kortchnoi in the semi-final, in 1977 and 1980 also against Kortchnoi in the quarter-final. On the 13th August 1984 Petrosian died in Moscow aged 55 of the consequences of stomach cancer and he was buried in Moscow's Vagankovoer cemetery.\n\nIn Armenia Tigran Petrosian is considered a national hero. His successes inspired the younger generation of players in the country and after the latter became independent the Armenian team was able to twice win the gold medal at the Chess Olympiads. The chess centre in Yerevan is named after Petrosian, and in addition a monument has been raised in the honour of the Armenian World Chess Champion in both Yerevan and Aparan.\n**Part III \u2212 The new era**\n\nRobert Fischer ushered in the start of a new phase in the history of chess. Before the Second World War chess players, even World Champions, found it difficult to live from chess. Steinitz and Lasker died in poverty. Capablanca was employed by the Cuban state and therefore had no financial worries. Alekhine had married rich women, though at the end of the Second World War, separated from his wife, he died in misery. In the USSR, chess grandmasters after the Second World War received at least a regular income as a sort of state professionals. By their successes they documented on behalf of the Soviet leadership the intellectual superiority of the working class. On a visit of Soviet grandmasters to Germany the news magazine _Der Spiegel_ used as the title of a report 'Red move-machines'. The Soviet players lived in simple circumstances, but clearly better than the majority of the population of the USSR.\n\nWith Robert James Fischer chess, of which previously not much notice had been taken by the world public, came once again to centre stage. At the zenith of the Cold War, which was also being fought out in many sporting comparisons between West and East, Fischer was considered the most likely person to be in a position to break Soviet supremacy. Chess became more important and was now worth something. Fischer demanded money and he got it. After Fischer, World Chess Champions no longer had only a good income, they even became rich. In one of the many encounters accompanied by so many escapades, Fischer defeated the Soviet World Champion Spassky and became the first Western World Champion since Euwe. However, after the winning of the World Championship the comet that was Fischer immediately burned out. Fischer disappeared and three years later did not appear to defend his title.\n\nFischer was followed by Karpov, who also understood how to use his title to make cash with advertising contracts in the West. In doing so he did not always sufficiently respect all the rules of his home country. The matches between Karpov and Kortchnoi in 1978 and 1981 were accompanied by considerable political background noise. Kortchnoi, dissatisfied with the preferential treatment enjoyed by Karpov in the Soviet Union, fled to the West and played these matches as a 'dissident' and a critic of the Soviet system. Particularly the match of 1978 turned into a war of nerves. This chess contest between the systems attracted great attention. Karpov, with the whole Soviet apparatus behind him, won twice.\n\nBut then, in the form of Kasparov, there emerged an even more serious challenger. The first assault on the chess throne by the young Kasparov failed, but not completely. The match that never wanted to end of 1984-85 was finally terminated without a winner. But Kasparov could negotiate good conditions for the next time. He had in Heydar Aliev, the future first president of an independent Azerbaijan, a strong ally in the Politburo. Kasparov also showed himself to be a master in the area of propaganda. In the West he sold himself to the press as the representative of perestroika, whilst Karpov appeared to represent the old nomenklatura (establishment). In his second assault Kasparov obtained the title, however he had to defend it several times against Karpov. In the meantime the World Championship matches had long been attracting prizes in the millions.\n**27. The match of the century**\n\n**The World Championship 1972:  \n_Boris Spassky against Robert Fischer_**\n\nRobert James, or Bobby, Fischer was born on the 9th March 1943 in Chicago, Illinois. His mother Regina Fischer, n\u00e9e Wender, had been born in Zurich. Legally his father was Heinz-Gerhardt Fischer, but it seems reasonably certain that his biological father was the Hungarian physicist and friend of Regina Fischer Paul Nemenyi.\n\n**Bobby Fischer (1943-2008)**\n\nHeinz-Gerhardt and Regina Fischer had married in 1933 in Moscow. Both were apparently convinced communists and had probably received in the 1930s in Moscow the formation given by the Comintern (Communist International) to all their activists and agents so as to place them abroad in the struggle for the world revolution. Whilst Heinz-Gerhardt Fischer at first fought in the Spanish Civil War in the international brigades before being sent to Chile where we lose track of him, Regina went back to the USA in 1939. She had grown up in St. Louis but during the economic crisis of the 1930s had first gone to Germany and then to Moscow.\n\nHowever, Moscow did not find much use for Regina Fischer when she lived in the USA. Nevertheless, very much later she occasionally demonstrated against the Vietnam War. On the other hand, Regina Fischer's good links to Moscow proved useful for the chess development of her talented son. Thanks to her good contacts she managed in 1958 to get him an invitation to the Moscow Central Chess Club. The 15 year old Fischer travelled to the Soviet Union with his sister Joan and played blitz in the Central Chess Club against a series of Soviet grandmasters.\n\nIn spite of Regina Fischer's minimal political activity in the USA, the FBI built up a file of 900 pages on her. Regina Fischer had moved with her two children, Robert and Joan, to Brooklyn and worked as a seamstress, a teacher and a nurse \u2013 in Moscow she had started medical studies. She raised the children alone. Fischer's childhood is marked by a tense relationship with his mother, whose character had all the traits of someone with a borderline personality disorder. Regina Fischer was described as litigious and by her neighbours as 'insufferable'. In 1960 the relationship between Regina and her son Robert had become so tense that the mother moved out of the common apartment and left her 17 year old son there alone. Fischer counted that as a relief when his mother left him, although he was completely overwhelmed at having to look after himself.\n\nBobby Fischer learned to play chess at the start of 1949 from his then 11 year old sister Joan. In 1951 he became a member of the Brooklyn Chess Club; later he changed to the Manhattan Chess Club. In 1956 in the Rosenwald Memorial the 13 year old played the so-called 'game of the century' according to Hans Kmoch, in which he defeated the grandmaster Donald Byrne with an amazing queen sacrifice. This was followed by a series of brilliant successes which made him well known in the world of chess, both for them and for his bad manners.\n\nThe chess columnist Heinrich Fraenkel, a Jew of German origin, born in Lissa (the Polish Leszno) and interned during the First World War on the Isle of Man where he learned to play chess, met Fischer on several occasions. He wrote in his book _The pleasures of chess_ published under the pseudonym Assiac (an anagram of the name of the fictional goddess of chess, Caissa): 'At the tournament in Mar del Plata 1959, which was played in a luxury hotel, the 16 year old Fischer, to the annoyance of the organisers, appeared at the evening dinner in baggy, dirty jeans. They thereupon made him the present of a suit, which Fischer accepted without thanks but never wore. Later, however, Fischer made it his habit that in every town where he was playing in a tournament he would buy a suit, where possible an expensive custom made suit, and on each day of the tournament he would wear a different tie with the suit.'\n\nIn 1962 at 19 Fischer was the majestic winner of the interzonal tournament in Stockholm. At the subsequent candidates' tournament in Cura\u00e7ao in 1962 he felt cheated by the Soviet players Petrosian, Geller and Keres, but also he did not play his best chess. Offended, he withdrew from top class chess for three years. At the Chess Olympiad of 1968 in Lugano Fischer appeared with the US team, then complained about the lighting in the tournament hall, about the too noisy public and the intrusive press photographers and left. Fischer's assault on the WCh title began with participation in the interzonal tournament in Sousse, in 1967. But as has already been described, despite being in the lead he kicked himself out of the tournament and withdrew once more for three years from tournament chess.\n\nIn 1970 he celebrated his comeback at the match between the USSR and the Rest of the World and defeated ex-World Champion Petrosian on board two by 3:1. After his return to the tournament arena he won in addition to other tournaments the interzonal tournament of Palma de Mallorca in 1970 and qualified for the candidates' matches. He was only able to play in the interzonal tournament because Pal Benko, third placed in the US championships, renounced his place in Fischer's favour on the initiative of the US federation, because Fischer had not participated in the said US championships which were the qualifying tournament for the interzonal tournament. In return Benko received 2000 dollars compensation from the US federation for giving up his place. It certainly did not conform to the rules, because Benko could not give up his place in favour of a specified replacement. According to the rules the replacement should have been the fourth placed player in the US championship.\n\nIn his book _Chess Duels_ Yasser Seirawan has drawn attention to the fact that the legend of the apparently lone fighter Fischer is a myth. In reality Fischer had all the relevant people from the US federation behind him for support. Leroy Dubeck, president of the USCF from 1969 till 1972, had agreed with the executive director Edmund Edmondson that all the resources of the federation should be gathered for the project 'Fischer plays for the World Championship'. For this purpose they also used all membership subscriptions of the USCF. In addition Fischer had the support of Fred Cramer, also from the federation, as his legal adviser. Amongst the chess players on whose help Fischer could count were the grandmasters Bill Lombardy, Lubomir Kavalek as well as Fischer's close friend the international master Anthony Saidy. A special role was played by Lina Grumette, who was a sort of mother substitute for Fischer and whenever necessary offered him refuge in her house.\n\nGrumette had moved to New York in 1917 at the age of nine. There she came into contact with the game of chess and had the good fortune to be trained by Isaac Kashdan. One of the best women players in the USA, she played in the 1940s in US national championships for women. Grumette later moved with her family to the west coast and in West Hollywood founded the chess club 'The Chess Set' \u2013 actually with the intention of attracting people from show business to the game of chess.\n\nGrumette was a close friend of Jacqueline Piatigorsky (6th November 1911 till 15th July 2012). The daughter of the French banker \u00c9douard Alphonse de Rothschild, her second marriage was to the Ukrainian-Jewish cellist Gregor Piatigorsky. When the Nazis marched into France, they emigrated to the USA and in 1949 had moved to Los Angeles. She was one of the best tennis players in the country, but then she discovered a passion for the game of chess and took lessons from IM Herman Steiner, who came from Austria-Hungary. At the first Women's Chess Olympiad in 1957 in Emmen (Netherlands) Jacqueline Piatigorsky played in the US team.\n\nLina Grumette helped her with the organisation of the two supertournaments which became famous as the 'Piatigorsky Cup'. At the '1st Piatigorsky Cup' in 1963 in the Ambassador Hotel of Los Angeles, first place was shared between Tigran Petrosian and Paul Keres. The '2nd Piatigorsky Cup' was held in 1966 in the Miramar Hotel in Santa Monica. Boris Spassky won it ahead of Robert Fischer. Piatigorsky supported many talented chess players, paid their travel costs and also sponsored the match between Samuel Reshevsky and Robert Fischer in 1961.\n\nThere was great rivalry between the until then leading player of the USA, Samuel Reshevsky, and the up-and-coming Fischer for the position of number one in the USA. The match was funded to the extent of 6000 dollars \u2013 a sensationally high prize for the time \u2013 and arranged for 16 games. At the score of 5\u00bd:5\u00bd the twelfth game was postponed from Saturday to Sunday, because as an orthodox Jew Reshevsky was not allowed to play before sunset on a Sabbath. This had not been taken into account when the schedule had been drawn up. After the game had first been postponed to 13.30 on the Sunday, Jacqueline Piatigorsky then wished to bring forward the start to 11.00, in order to be able to attend a concert by her husband on that evening. Fischer protested and did not turn up for the game, which thereupon was scored as a loss for him. Then he did not appear for the thirteenth game and Reshevsky was declared the winner of the match. The media and the US federation took Reshevsky's side and condemned Fischer's behaviour. Kasparov proposes this event as the starting point for Fischer's later anti-Semitism.\n\nLina Grumette accompanied Fischer to the WCh match in Iceland, but later Fischer turned away from her too, and accused her, just as he did other trusted people before and after, of cooperating with the media and the KGB. Grumette created a foundation for the advancement of chess talent and organised numerous junior tournaments on the west coast until the 1980s. She died on the 21st July 1988 of lung cancer at the age of 80.\n\nIn the candidates' quarter-final Fischer met Mark Taimanov in Vancouver. Taimanov occupies a special place among Soviet grandmasters. Born on the 7th February 1926 he was a child film star in the USSR, after a film production company had looked for a young violinist for the film _Beethoven concert_ and found one in the ten year old Mark Taimanov. After being cast, Taimanov had prepared for the role with a year of intensive violin playing. The film won an international film award and suddenly Taimanov was famous. Annoyed by all the hustle and bustle around him, he started to play chess. In addition he also brought his piano playing up to championship level and together with his wife Lyubov Bruk made up a celebrated piano duo, which in 1970 stood on the threshold of a world-wide career.\n\nTaimanov had qualified for the candidates' rounds with a fifth place in the interzonal tournament of Palma de Mallorca, though a shadow lay over that success. In the final round Taimanov met the Yugoslav Milan Matulovic. The latter arrived 15 minutes late and just like Taimanov spent only just over an hour's thinking time on the 42 moves of the game. As Kortchnoi later revealed, Matulovic had sold the point for 400 dollars to Taimanov, or the Soviet federation.\n\nThe match between Fischer and Taimanov was actually meant to take place in the hall of the University of Vancouver. Fischer insisted on it being moved to the library of the university and spectators being excluded. They agreed on a middle-sized room with 200 spectators. Fischer annihilated Taimanov by 6:0 (after six games!), though the games themselves were not so clear-cut as the result might suggest.\n\nThe match against Fischer destroyed what had been Taimanov's life until then. As a result of the preparation and the match Taimanov's marriage broke up and with it the piano duo Bruk\/Taimanov. The result 0:6 was seen in the USSR as unworthy of a Soviet grandmaster. When on his return Soviet customs found in Taimanov's luggage books by the banned author Alexander Solzhenitsyn, this was a welcome excuse to side-line him. In the Soviet Union the following joke later became current: 'Why was Solzhenitsyn expelled from the Soviet Union? Taimanov's book on the Nimzo-Indian Defence was found in his flat.' Fischer was later described by Taimanov as 'paranoid': 'He firmly believed that the Soviet secret services wanted to get rid of him. The list of his enemies was as follows: Jews, Bolsheviks, KGB.'\n\nAn offer from the city of Denver, Colorado (USA) was accepted for the staging of the semi-final between Fischer and Bent Larsen. Larsen was considered along with Fischer as the best player in the West and in the match 'USSR against the Rest of the World' he had even played on first board ahead of Fischer. The damp and warm summer climate in Denver was however hard for the Dane. It was the hottest summer since 1936 and even the workers in the factories were sent home because it was too hot. The match began on 6th July in the theatre auditorium of a high school. After four games, all won by Fischer, Larsen reported himself ill. The match was continued after a pause and Fischer won two more games and with them the match. All in all Fischer had now won his last 19 games in succession.\n\nFischer now met Tigran Petrosian, who had emerged victorious from the intra-Soviet duel against Viktor Kortchnoi. The match between the two Soviet grandmasters had, however, been fixed, since the Committee for Sports considered Petrosian to have the better chances of stopping Fischer on his way to the World Championship. For his pre-arranged defeat Kortchnoi was rewarded with three invitations from abroad. In the next cycle Kortchnoi and Petrosian again met, Kortchnoi won and one could have the impression that Petrosian had wished to return the present of victory in the match. Kortchnoi, however, denied all pre-arrangements with Petrosian.\n\nThe candidates' final between Fischer and Petrosian was staged in Buenos Aires. With an offer for the prize fund of 12000 dollars \u2013 7500 dollars for the winner, 4500 dollars for the loser \u2013 the Argentinians had outdone the other bidders Belgrade and Athens. 'Fischermania' had also seized Argentina and from nine in the morning there were already queues for the tickets for the games, which started at 17.00. Some 1500 were there for each of the games of this candidates' final.\n\nThe match began on 30th September 1971. Fischer won the first game \u2013 his 20th in succession \u2013 although confronted with a variation prepared in depth by Petrosian's team, but lost the 2nd match game, ending his series of victories. Next came three draws. Fischer then won the sixth and seventh games. Before the eighth game, Petrosian claimed a timeout. His doctor had diagnosed low blood pressure. After the match resumed Petrosian finally collapsed and lost the eighth and ninth games for a final score of 6\u00bd:2\u00bd. Fischer was now the challenger to World Champion Spassky. On his return to the USSR Petrosian the usual post-defeat letter from the 'Committee for Physical Education and Sport', in which it was regretted that 'his delegation had not been up to the tasks required of them'. He was, moreover, relieved of his position as editor-in-chief of the Soviet chess magazine _64_.\n\nFIDE were now looking for organisers for the WCh match and asked for offers before the 1st January 1972. Fischer would have liked to play in the USA but categorically refused any venue in the USSR. By the 1st January there were numerous lucrative offers on the table: Belgrade offered 152000 dollars prize money, Sarajevo 120000 dollars, Buenos Aires and Bled each 100000 dollars, as well as numerous other propositions from the most varied collection of countries in Europe and the Americas, including an offer from Reykjavik to the tune of 125 000 dollars plus 30% of all income from TV broadcasts. The offer of the Icelandic Chess Federation was financially supported by the Icelandic government.\n\nBoth players were now asked for their preferences. Spassky voted for Reykjavik, Amsterdam, Dortmund and Paris. Fischer's list read: Belgrade, Sarajevo, Buenos Aires and Montreal. So there was absolutely no common ground of agreement. The negotiations for Fischer were conducted by Edmund Edmondson, from 1966 till 1975 executive director of the US Chess Federation (USCF), who also functioned as Fischer's manager in 1970 and 1971. Edmondson travelled to Moscow and finally agreed with the Soviets on Reykjavik as the venue, but he had, however, come to the agreement without Fischer, who was still insisting on Belgrade as the venue, followed by the other places on his list.\n\nThen FIDE president Euwe then suggested a compromise, according to which the first half of the match would be played in Belgrade and the second half in Reykjavik. Fischer agreed, the Soviets were at first against but on the insistence of Spassky finally accepted on the 5th March 1972. All involved, representatives of the two venues, FIDE and the players met in Amsterdam to negotiate the details. The contract was signed on the 20th March.\n\nTwo days later a telegram from Fischer arrived, in which he demanded that all the revenues from the WCh match, after the costs had been covered, should go to the players. Euwe then sent an ultimatum to Fischer. He was to agree the contract by the 4thApril. Paul Marshall, a New York lawyer, signified agreement in Fischer's name. Fischer himself said nothing. Meanwhile the organisers in Belgrade had become uneasy and demanded of the Soviet and American federations the posting of a guarantee of 35 000 dollars as security in case the match did not take place. Since the USCF refused, Belgrade withdrew. The president of the Icelandic federation Johann Thorarinsson then offered to stage the whole match. On the 8th May Euwe received a telegram from Fischer's representatives Edmondson and Marshall, in which he was informed that Fischer would play, but under protest.\n\nOn the 21st June the Soviet delegation arrived in Reykjavik. Spassky had Efim Geller, Nikolai Krogius and Ivo Nei as his seconds and occupied suite 730 of the Saga Hotel in Reykjavik. Krogius, also a KGB informer, was to support Spassky especially in psychological matters. Just a few days before the start of the match Fischer had still not arrived, only his entourage, consisting of former USCF president Fred Cramer, the lawyer Andrew Davis, the lawyer Paul Marshall, the journalist Brad Darrach and his only second William Lombardy \u2013 a strong chess player, who was now a priest.\n\nAccording to the contracts the winner was to receive from the total prize fund 78125 dollars, the loser 46 875 dollars (plus 30% of the income from TV and film rights). Fischer now also demanded 30% of the entry fees. Whilst Fischer was being awaited in Iceland and long overdue, the challenger was still in the house of the friend of his youth Anthony Saidy in Douglaston.\n\nOn the 1st July 1972, the day of the opening ceremony, Fischer was still absent. His seat next to Spassky at the opening ceremony in the National Theatre of Reykjavik remained empty. After the opening the US ambassador in Iceland, Theodor Trembley, called Washington and reported what had happened in Reykjavik. He asked for support because this event meant that the Icelandic people had been insulted by a representative of the American people. Henry Kissinger, then security adviser to Richard Nixon, got involved and phoned Fischer. 'America wants you to go there and beat the Russians!', he told him on the telephone.\n\nIn the meantime Fischer's lawyers in Reykjavik were attempting to have Fischer's financial demands met, and to obtain a postponement of the start of the match, in the hope that Fischer would think better of it and still arrive. Their client was suffering from exhaustion and for that reason had not yet come to Reykjavik, was the reason they gave. In order to save the match Max Euwe ordered a postponement of the drawing of lots from the 2nd to the 4th July. Viktor Baturinsky, the leader of the Soviet chess section, thereupon sent furious protests to Euwe, but Spassky had indicated to the FIDE president that he agreed to the postponement.\n\nOn the Monday morning the English millionaire Jim Slater was on his way to work when by chance he heard on his car radio that the start of the match between Spassky and Fischer had been postponed by a few days. Slater, a chess fan, who had already supported the tournaments in Hastings and the English program to foster young chess players, phoned his friend Leonard Barden and told him that he would double the prize fund in Reykjavik from 125000 dollars to 250000 dollars. Barden contacted Fischer's lawyer Paul Marshall. The latter informed his client. Fischer, basically distrustful on account of numerous broken promises he had received throughout his career, did not, however, believe this.\n\nWhat finally decided Fischer to then set out for Iceland is unclear. On the 3rd July he took a plane and arrived early on the 4th in Reykjavik, where he was met by the future FIDE president Fridrik Olafsson. Fischer was at first supposed to occupy a house a little outside Reykjavik, but he preferred to move into the Hotel Loftleidir.\n\nAs a result of Fischer's antics the atmosphere between the organisers, FIDE and the two delegations had in the meantime become extremely strained. Spassky, moreover, found himself under some pressure from Moscow. There they had had enough of Fischer and wanted to drop the match, which would easily have been possible because the postponement of the start was not covered by any rule in the statutes of FIDE. The next unreasonable impertinence for Spassky had been that Fischer did not himself appear for the ceremony of the drawing of lots, but sent his second Lombardy. The Soviets left the ceremony and sent in a protest note, in which they demanded a punishment for Fischer on account of his continual breaking of the rules, and apologies from Euwe and Fischer. To general surprise, Fischer actually sent a written letter of apology to Spassky.\n\nAfter all the negotiations, during which the WCh match was always just on the point of being broken off, finally the lots were drawn on the 7th July and Fischer again made Soviet faces red with anger when he once more kept them waiting by turning up late. The drawing of lots resulted in Spassky having the white pieces in the first game. That first game, however, was postponed till the 11th July on the request of the Soviet side.\n\nFor the match 1000 chairs had been set up for spectators. The mahogany table with its inset chess board had been made specially for the match by Gunnar Magnusson. The marble chess board was made by Thorstein Bjornsson. At the inspection of the playing hall, Fischer found fault with the board because it reflected the light too much. A new one had to be arranged for. For his chair Fischer had flown in from New York a special leather armchair designed by Charles Eames. Fischer had already used this armchair in his match against Petrosian. A normal upholstered wooden chair with two arm rests had been provided for Spassky.\n\nFor the first of the 24 games of the match Fischer appeared slightly late (six minutes). After the exchange of most of the pieces a completely level bishop ending stood on the board. On move 29 Fischer took a pawn on h2 and allowed his bishop to be shut in. What looked like a typical beginners' mistake, and was also commented on in these terms by contemporary grandmasters, was probably a risky attempt to play for a win with a pawn majority on the kingside. Later analyses showed that the game was probably still drawn even after the loss of the piece. After five hours the game was adjourned in a losing position for Fischer.\n\nAfter the very first move of the opening game Fischer had complained to the arbiter Lothar Schmid about the presence of the cameras on the stage and the noise they made. During the game Fischer repeated his complaint several times. The film rights had been granted to the American film maker Chester Fox and, as reported, the players were to receive 30% of the revenues from these film rights. The game was resumed on the following day, but Fischer had to resign after a further 16 moves. He had previously given Schmid an ultimatum that the cameras had to disappear from the stage and not be set up there any more. The organisers found themselves in a dilemma, because they had contractually awarded all the film rights to Chester Fox.\n\nSince the problem had not been solved before the start of the second game on the following day, Fischer did not turn up for the game. While his clock was running \u2013 the rules provided that he had an hour to appear, or else the game would be lost \u2013 hectic negotiations were taking place. Chester Fox finally agreed to remove the cameras from the stage. 40 minutes had elapsed, which the challenger no longer had on his ticking clock. After Fischer received the news of the removal of the cameras, he demanded that his clock be reset. Arbiter Schmid refused this and indicated the rules. Fischer thereupon did not appear for the second game and lost it to a walkover. The match now stood at 2:0 for the title defender and was once more just on the point of being abandoned.\n\nThere followed hectic negotiations as to whether it was lawful to score the game as a loss for Fischer and about what was to happen with the cameras. Measures were taken as to how loud the cameras were and it was established that they actually made no measureable sound. Fischer received a further call from Henry Kissinger, who tried to convince him to continue. Fischer, who in the meantime had booked flights to New York, finally explained that he would appear for the third game, on 16th July 1972, but that it must be played in a separate room behind the stage without cameras. Arbiter Schmid asked Spassky whether he agreed, Spassky did and so the playing table was set up in a room behind the stage.\n\nWhen Fischer arrived, he discovered a small camera, which was to send pictures out into the hall for spectators and shouted at Schmid. Now Spassky also slowly became angry. At a critical point Schmid pushed both players back into their chairs and ordered them: 'Play chess!' They did. After move 40 the game was adjourned in a winning position for Fischer. On the next day Spassky saw the sealed move and then resigned. That evening Spassky sent a letter to Schmid, in which he demanded a return to the stage. He said that the room behind the stage was too noisy on account of the air-conditioning and the noises from outside. Fischer accepted the return to the stage \u2013 as long as no cameras were set up there.\n\nIn the fourth game Spassky obtained an advantage with black, but was not able, however, to convert it into the full point. The fifth game was played by Fischer in the style of Nimzowisch. On the 27th move the title defender overlooked a tactical trick and had to resign. Fischer had drawn level. The sixth game was like a demonstration. As White, Fischer opened with an unusual move for him, 1.c4, and went on to outplay Spassky in the Tartakower Variation of the Queen's Gambit. After his 0:2 start, the challenger, four games later, was now leading 3:2 (with one draw).\n\nIn the meantime the Soviets had arranged for Spassky to receive the same swivel armchair as the one on which Fischer had been sitting the whole time. Nor had Fischer been inactive; he had had the table changed because it was too big for him. In addition he was disturbed by the light reflected from the marble board, which was replaced by a wooden board.\n\nFischer's match strategy clearly consisted of avoiding any opening preparation by using completely different openings from the ones he had always played. But in the seventh game he chose one of his favourite variations, the Poisoned Pawn Variation in the Sicilian Najdorf System, and obtained a good position, but he could not turn it into a win. On the other hand, the eighth game again went to Fischer, who again opened with 1.c4. The decisive mistake was made by Spassky as early as move 19. Spassky now reported ill and a two-day pause was taken. The ninth game was drawn. In the tenth game, with a 4:2 lead to fall back on, Fischer chose for the first time his standard move 1.e4. Spassky replied with the Ruy Lopez Breyer Variation and for a long time kept the game on a level keel. In the ending he was an exchange down (bishop against rook), but in return had a pair of passed pawns and certain chances of a draw. But the title defender was outplayed by Fischer and lost once more.\n\nNumerically the score was 5:2, but in reality Spassky had won once because of a mysterious error by Fischer and once by a walkover \u2013 so he had not yet scored a single full point off his own bat. But this changed with the eleventh game, when Spassky was able to tear Fischer's Poisoned Pawn Variation to bits. Now Fischer had the airconditioning switched off because the humming of the machine disturbed him. During the twelfth game he complained several times about the noise coming from the spectators. From then on, according to his wishes, the seven rows of seating for the spectators had to remain empty. In the 13th game Fischer turned to the rarely played Alekhine Defence and after a total of nine hours play restored the previous margin \u2013 6:3 in wins.\n\nBefore game 14 Spassky once more reported ill and the game was postponed from the Sunday to the Tuesday. This brought about a protest from the US delegation to arbiter Lothar Schmid, because the medical reasons for the timeout for illness had not seemed precise enough for the Americans. A telephone call from Schmid to Cramer disposed of this problem. Previously Fischer had already sent a written complaint about the 'unbearable noise' (Assiac) which a spectator had caused in unwrapping a packet of sweets. From then on, according to instructions from the chief arbiter, only sweets which were already unpacked could be sold.\n\nBefore the 15th game Chester Fox had flown an official US 'subpoena server' to Reykjavik, so as to serve Fischer personally with a complaint according to US law. Chester Fox wanted to sue the challenger for 1.75 million dollars for breach of contract and loss of earnings. The sum was later raised to 2.5 million dollars. The complaint was to be publicly served on him on stage during the 15th game. But the organisers had got wind of it and during game 15 the stage was guarded by police officers who prevented any unauthorised entry.\n\nThe next two games ended in draws, with the 15th game in the main line of the Sicilian Najdorf Variation being particularly hardfought and exciting. Before game 17 Spassky's second Geller issued a press statement, in which the Soviet delegation complained about Fischer's constant antics and continual protests. Moreover, according to them Spassky was possibly being exposed to the effects of electronic measures and chemical substances which were having a negative influence on him. After the 17th game, in which for the first time in his life Fischer played the Pirc Defence and reached a draw, the hall was 'swept for electronic measures and chemical substances'. Apart from two dead flies in the lighting, however, nothing was found.\n\nAlso after the 17th game the Soviets sent Spassky's second Ivo Nei back home. According to the observations of the Soviets, he had been speaking too intensively and too often with the US grandmaster Robert Byrne and had analysed the games of the match with him. Paul Marshall was planning a book about the match with Byrne and Nei as authors. Nei was supposed to provide information from Spassky's camp. After his return Nei was warned and for two years blocked from all tournaments abroad, but before that he sent his analyses for the book to the USA. Games 18 to 20 also ended in draws. The 21st game was adjourned on move 41 in an advantageous position for Fischer. 2500 spectators gathered on the next day for its resumption. But Spassky did not appear for the continuation of the hopeless game. He had previously called arbiter Schmid and resigned the game and the match.\n\nAccording to the later calculations of Jeff Sonas, Fischer had taken the lead in the world ranking list back in August 1966, which in November 1967 he had extended to almost 40 points over Tal. Spassky was only in fourth place, behind Petrosian. In 1970 FIDE then introduced the Elo system for the calculation of playing strength. In it Fischer is credited in 1970 with 2720 Elo points and led the world ranking list ahead of Kortchnoi and Spassky (both on 2670). Two years later, in 1972, Fischer was leading with 2785 Elo and more than 100 points of a lead over Spassky, who had 2660. These figures allow us to recognise the superiority of the American before the WCh match of 1972, even if before that match he had not been able to win a single game against Spassky.\n\n **Spassky \u2013 Fischer**\n\nReykjavik, 3rd game  \n16th July 1972  \nBenoni Defence (A77)\n\nThe situation before the game was as follows: Fischer had lost the first game through a risky attempt to win in a drawn position and had not appeared to the second game, which also went to Spassky as a walkover. Fischer was thus trailing by two games, 0:2.\n\n**1.d4 \u2658f6 2.c4 e6 3.\u2658f3 c5 4.d5 exd5 5.cxd5 d6 6.\u2658c3 g6 7.\u2658d2**\n\nAfter the 'normal' move order 7.e4 \u2657g7 8.\u2657e2 0-0 9.0-0 Black has the extra option of 9...\u2657g4.\n\n**7... \u2658bd7**\n\nThe alternative 7...\u2657g7 8.\u2658c4 (8.e4) 8...0-0 9.\u2657f4 \u2658e8 was known to Spassky from his own praxis (Donner-Spassky, Leiden 1970, \u00bd-\u00bd\/24).\n\n**8.e4**\n\n8.\u2658c4 is followed simply by 8...\u2658b6.\n\n**8... \u2657g7 9.\u2657e2 0-0 10.0-0 \u2656e8 11.\u2655c2**\n\nSpassky plays a move which had rarely been employed up till then.\n\nThe alternative 11.a4 was, for example, known from a previous game of Fischer's: 11.a4 \u2658e5 12.\u2655c2 g5 13.\u2658f3 \u2658xf3+ 14.\u2657xf3 h6 (0-1\/35), Gligoric-Fischer, Palma de Mallorca 1970.\n\n**11... \u2658h5!?**\n\nThis was a novelty by Fischer. Black surprisingly accepts the degrading of his pawn structure on the kingside, but obtains in return the bishop pair. Other moves are 11...\u2658b6, 11...a6 12.a4 \u2658e5 or 11...\u2658e5.\n\n**12. \u2657xh5**\n\nSpassky picks up the gauntlet. 12.g3 was tamer.\n\n**12...gxh5 13. \u2658c4 \u2658e5 14.\u2658e3**\n\nByrne suggested 14.\u2658xe5 \u2657xe5 15.\u2657e3. After 15...\u2655h4 16.g3 \u2655f6 Black has counterplay in the centre and on the kingside. The chances are level.\n\n**14... \u2655h4 15.\u2657d2**\n\nByrne's suggestion 15.f3! bolsters the centre and prevents...\u2658g4.\n\n**15... \u2658g4**\n\nLess committal than the sacrifice 15... \u2658f3+!? 16.gxf3 \u2657e5. After 17.\u2656fc1 Byrne considered the white position to be won, but Black still has a few shots in his locker and White must play very accurately to hold the position: 17...\u2655h3! (threatens mate after 18...\u2657xh2+) 18.\u2658f1 (after other moves Black gets a very strong or decisive attack) 18...\u2655xf3 19.\u2655d1 \u2655h3 (now the threat is 20...\u2657g4) 20.f3 \u2654h8 21.\u2655e2 \u2656g8+ 22.\u2654h1 \u2656g6 (intending 23...\u2657d7 and 24...\u2656ag8), and Black has sufficient compensation for the piece he has sacrificed.\n\n**16. \u2658xg4 hxg4 17.\u2657f4**\n\nIt was worth considering 17.\u2658e2!? instead, for example 17...f5 18.\u2658g3 fxe4 19.\u2656fe1 intending 20.\u2657c3 (Smyslov).\n\n**17... \u2655f6 18.g3?!**\n\nAfter this move it becomes hard to mobilise the white f-pawn \u2013 one reason why White later has difficulties finding active play. Improvements suggested were 18.\u2657g3 h5 19.f3!? (Byrne) 19...h4 20.fxg4 \u2655e7 21.\u2657f4 \u2657xg4 22.\u2656ae1= and 18.\u2655d2 (Smyslov).\n\n**18... \u2657d7**\n\nIntending...b7-b5.\n\n**19.a4 b6 20. \u2656fe1**\n\n20.a5?! would simply be followed by 20...b5.\n\n**20...a6 21. \u2656e2?!**\n\n21.\u2656a3 was better. After 21...b5 (21... \u2655g6!?) 22.axb5 axb5 23.\u2656xa8 \u2656xa8 White has counterplay with 24.e5=.\n\n**21...b5 22. \u2656ae1**\n\nSince the rook has left the back rank, 22.axb5 axb5 23.\u2656xa8 \u2656xa8 24.e5 now fails to 24...\u2656a1+ 25.\u2654g2 dxe5 26.\u2656xe5 (26.\u2657xe5 \u2655h6!) 26...b4 27.\u2658e4 \u2655a6 with the threat of mate on f1.\n\n**22... \u2655g6**\n\nPreventing the planned advance e4-e5 by pinning the pawn.\n\n**23.b3 \u2656e7**\n\nAfter 23...\u2657xc3 24.\u2655xc3 bxa4 25.bxa4 \u2657xa4 26.e5 (Byrne) White gets counterplay.\n\n**24. \u2655d3 \u2656b8**\n\nThe threat is 25...bxa4, followed by 26... \u2656b3 with advantage to Black. (Byrne)\n\n**25.axb5 axb5**\n\nThe threat now is 26...b4 and 27...\u2657b5. 25...\u2657xb5?! 26.\u2658xb5 would instead give away Black's advantage (Byrne) 26...axb5=.\n\n**26.b4**\n\nAfter 26.\u2658d1 the move 26...\u2656be8 hangs on to the advantage. In the event of 26...b4 White has a way out: 27.\u2655a6 \u2657b5 28.\u2655xd6=.\n\n**26...c4**\n\n26...cxb4 27.\u2658a2 achieves nothing for Black.\n\n**27. \u2655d2 \u2656be8 28.\u2656e3**\n\nNot 28.\u2657g5?? \u2657xc3 29.\u2655xc3 \u2655xg5\u2013+.\n\n**28...h5**\n\nAfter 28...\u2657xc3 29.\u2655xc3 \u2656xe4 30.\u2656xe4 \u2656xe4 31.\u2656xe4 \u2655xe4 32.\u2655f6 White would get counterplay.\n\n**29. \u26563e2 \u2654h7**\n\nThe immediate 29...\u2657xc3 was possible (Byrne) 30.\u2655xc3 \u2656xe4 31.\u2656xe4 \u2656xe4 32.\u2656xe4 \u2655xe4, but since there is nothing White can undertake, Black can take the time for this liquidation.\n\n**30. \u2656e3 \u2654g8 31.\u26563e2 \u2657xc3 32.\u2655xc3 \u2656xe4 33.\u2656xe4 \u2656xe4 34.\u2656xe4 \u2655xe4 35.\u2657h6**\n\nThe alternative was 35.\u2655f6. There can follow 35...\u2655b1+ 36.\u2654g2 \u2657f5 (threatening...\u2657e4 then mate) and then for example 37.\u2655g5+ \u2657g6 38.\u2657xd6 f6 39.\u2655xf6 \u2657e4+ 40.f3 \u2655c2+ 41.\u2654f1 \u2655d1+ 42.\u2654f2 \u2655d2+ 43.\u2654f1 \u2657d3+ 44.\u2654g1 \u2655e1+ 45.\u2654g2 \u2655f1# (Alexander). After 35.\u2657xd6 then 35...\u2655xd5 wins, followed by...\u2657c6 (Byrne).\n\n**35... \u2655g6 36.\u2657c1**\n\n36.\u2655e3 \u2654h7 37.\u2657f8 (Alexander) 37...c3 38.\u2655f4 (38.\u2657xd6 \u2655b1+ 39.\u2654g2 \u2657f5) 38...c2 39.\u2657e7 \u2657c8 40.\u2657d8 \u2657b7\u2013+. But 36.\u2657d2 f6 etc. was more tenacious.\n\n**36... \u2655b1 37.\u2654f1?!**\n\n37.\u2655e1 f6 38.\u2657d2 was better.\n\n**37... \u2657f5 38.\u2654e2 \u2655e4+ 39.\u2655e3 \u2655c2+**\n\nNot 39...\u2655xd5?? 40.\u2655g5+ (Byrne) with perpetual check. The ending after 39... \u2655xe3+? 40.\u2654xe3 would be drawn on account of the bishops of opposite colours.\n\n**40. \u2655d2 \u2655b3**\n\n**41. \u2655d4?**\n\nWhite has been on the defensive the whole time. But possibly he was still not lost till this move. After 41.\u2654e1 there is at least no forced win to be seen for Black, for example: 41...c3 42.\u2655g5+ \u2657g6 43.\u2655d8+ \u2654h7 44.\u2655xd6 c2 (the threat is...\u2655c3+ and then mate) 45.\u2655f4 \u2655xd5 46.\u2655d2 \u2655h1+ 47.\u2654e2 \u2655f3+ (not 47...\u2655xh2?? 48.\u2655h6+ \u2654g8 49.\u2657b2 and it is Black who is mated) 48.\u2654e1 \u2655e4+ 49.\u2654f1 and White has set up a sort of fortress position here. In his efforts to attack Black must always be careful not to run into a counter-attack on the dark squares.\n\n**41... \u2657d3+!**\n\n**0-1**\n\n42.\u2654e1 (42.\u2654d2 \u2655c2+ 43.\u2654e1 \u2655xc1#; 42.\u2654e3 \u2655d1! 43.\u2655b2 \u2655f3+ 44.\u2654d4 \u2655e4+ 45.\u2654c3 \u2655e1+! intending 46...\u2655e5#) 42...\u2655xb4+ \u2013+.\n\nFor 24 years the title of World Chess Champion title had been in Soviet hands. Now a US American had conquered it. The organisers were afraid that Fischer would not appear at the closing ceremony either. In fact the new World Champion appeared an hour late, but he did come. Soon after he arrived, Fischer took out a pocket chess set and showed Spassky some analysis of the closing position of their final game.\n\nIn Reykjavik Fischer received a prize of 76123 dollars, and later the same amount from Jim Slater. As the loser of the match Spassky received a total of 93750 dollars. Since at that time in the USSR there were no regulations concerning such high earnings in sporting contests, Spassky was able to keep all the money. Nobody asked any questions. It was only later that it was laid down that Soviet players had to give half of their prize money to the state. The Georgian Soviet tennis player Alexander Metreveli, Wimbledon finalist in 1973, once stated that in the Soviet Union in the 1970s there were only two people who owned a Ford Mustang. One was Metreveli himself, the other Boris Spassky.\n\nBecause of his defeat against Fischer, Spassky was banned by the Soviet Committee for Sports for nine months from all tournaments and his stipend was reduced. In 1974 Spassky was eliminated in the semi-final of the candidates' matches against Anatoly Karpov. This match was exceptionally notable for the fact that Spassky's second for the match against Fischer, Efim Geller, was now suddenly working for Karpov.\n\nIn September 1975 Spassky married \u2013 for the third time \u2013 Marina Sherbacheva, a granddaughter of the former czarist general Dimitry Sherbachev, who emigrated to France after the revolution of 1919. In August 1975 Spassky's flat was broken into under mysterious circumstances and he no longer felt safe in Moscow. In the summer of 1976 the Soviets granted his wish to leave. He moved to France with his wife and lived in the outskirts of Paris, in Meudon.\n\nIn 1977 Spassky also was qualified for the candidates' matches. In the quarter-final he defeated Vlastimil Hort in Reykjavik, in the semi-final in Switzerland Lajos Portisch. In the final he then met Viktor Kortchnoi, who had moved to the West in 1976, and was defeated in a poisonous match, which was conducted in an extremely hostile atmosphere. In the next WCh cycle Spassky also reached the candidates' matches, but was eliminated by Portisch in 1980. In 1985 he played in the candidates' tournament in Montpellier and shared sixth place.\n\nUntil 1990 he played for Solingen in the German Chess Bundesliga, then he gradually withdrew from tournament chess. Nevertheless he was always a greatly respected guest of honour at chess tournaments. In 1992 Spassky accepted an invitation to Yugoslavia and played a 'return' match against Fischer. When Fischer was arrested in 2004 in Japan, Spassky stuck up for his former match opponent in an open letter to President Georges Bush Jr. On the 1st October Spassky suffered a slight stroke at a lecture in San Francisco. On the 23rd September 2010 he fell victim to a severe stroke which paralysed his left side. After quite a long time, which Spassky had passed in rehabilitation, the tenth World Champion fled France in August 2012 and moved to his new partner in Moscow.\n**28. Fischer doesn't appear**\n\n**The World Championship that never was (1975):  \n_Anatoly Karpov becomes World Champion without a fight_**\n\nAnatoly Karpov was born on the 23rd May 1951 in Zlatoust, a city in the South Urals in Chelyabinsk Oblast. His father was Evgeny Karpov, on the staff of a metal working factory in Zlatoust, who at the time of Karpov's birth was just finishing a three year period of further study at the Bauman Moscow State Technical University. He later became mayor. Karpov's mother had given up her post in business administration in order to bring up her son and her daughter Larissa, who was five years older than Anatoly.\n\nSome months after his birth, Karpov fell ill with a stubborn case of whooping cough. When the parents feared that their son might die of it they had him baptised on the insistence of his grandmothers. Karpov recovered from the illness but all his life he remained susceptible to any disease in the ear-nose-throat area.\n\nAnatoly Karpov (born in 1951)\n\nThe family lived in a communal housing development in Zlatoust. During the three years of his study, his father received from his firm a stipend of 700 roubles (approximately worth 225 dollars), but in order to make ends meet Karpov's mother took in sewing, which she did in the evenings and at night. One day in 1953, the family's flat was suddenly searched by the KGB, after the father had been denounced. The accusations were, however, dropped.\n\nKarpov learned chess at the age of four by watching whenever his father played games with friends. The first games of his own were played against other children and youngsters from the district. When he went to school Karpov was already on a level footing with all the other children in the school in chess. His first regular chess partner was Sasha Kolishkin, a fellow pupil of his sister Larissa. At the age of seven Anatoly, nicknamed Tolya, was allowed to join the chess section in his father's factory. And at ten Karpov was already champion of Zlatoust. His first junior tournament outside of Zlatoust was played in Borovichi under, as one can imagine, poor conditions. The young people slept ten to a room. Only limited sanitary facilities were available.\n\nIn the summer of 1963 Karpov was invited to Moscow for a course at the Botvinnik Chess School, where he also met the 'patriarch' personally on the occasion of a clock simultaneous display. Botvinnik was not very impressed by Karpov's understanding of the game, generally only looked at his games superficially and noted: 'The boy has no idea about chess and so there is in no way any future for him as a chess player.'\n\nThe fact that Mikhail Botvinnik was rigorous in his judgements was confirmed later by the future German grandmaster Michael Bezold. When in the 1980s in his chess school Botvinnik was invited to cast his eye over some talented German chess players, for each of the young players he made notes about their strengths and weaknesses in two columns. Bezold's list of weaknesses was a long one, but under the heading of strengths there was only one note: 'Enjoys his chess.'\n\nApart from the negative evaluation, the meeting with Botvinnik was not very encouraging for Karpov either. Botvinnik told the children and young people that he was working on a chess computer, which, when it was finished, would defeat all grandmasters, probably even the World Champion. Professional chess players would then be superfluous. But the young people should not be worried. They would nevertheless find work, and that would be as chess programmers in his computer chess project. Botvinnik's chess computer, however, never got beyond the experimental stage.\n\nKarpov's first chess book was _Basics of chess openings (Kurs debiutov)_ by Vassily Panov. Of the chess books which Karpov has read during his studies, he gives _Selected games of Capablanca (Capablanca: Biographiya i 64 izbrannii partii)_ , also by Vassily Panov, as the one which most influenced him. It was the second chess book which Karpov received. It was a present from his father on his eighth birthday.\n\nKarpov's father took up a post in 1965 in a power plant in Tula, to where the family then moved. Karpov joined the army sports club there and began collecting stamps. His collection of chess related stamps is considered today to be the biggest in the world. In 1966 he was appointed 'Master of sport' which made him financially independent. At the start Karpov was supported with a monthly stipend of 100 roubles, which he received from the army sports club. When he became a grandmaster he got 140 roubles and as a player in the national team 200 roubles per month.\n\nIn 1967 in Groningen, Karpov became at 16 European junior champion ahead of Jan Timman and Andras Adorjan. In 1969 he also won the Youth World Championship. In the meantime he had started working with Semyon Furman, which was a special honour since Furman was much sought after as a trainer. Like all striving players in the Soviet Union before him and like Kasparov after him, Karpov felt held back by others in his advance. He thus believed that Spassky saw him as a future rival and thanks to his authority tried to keep him out of international tournaments.\n\nKarpov started to study mathematics at the State University in Moscow, but broke this off after a year and a half and changed instead to the economics faculty in Leningrad. It is difficult to reconcile mathematics and chess because both demand a great deal of time, was Karpov's later explanation for this step. A further reason for changing cities was that his trainer Furman lived in Leningrad. Karpov later finished his studies with a diploma. The subject of his work was: 'Free time and its economic significance in socialism'.\n\nIn 1970 Karpov became a grandmaster. In the following year he was joint winner with Leonid Stein of the Alekhine Memorial Tournament in Moscow. At the Chess Olympiad in Skopje 1972 Karpov had an excellent result with 13 out of 15. One year later he won along with Viktor Kortchnoi the interzonal tournament in Leningrad and thus qualified for the candidates' matches. The Soviets had employed a little intrigue to eliminate the best two western players, Bent Larsen and Robert H\u00fcbner. They insisted to FIDE that both these players should be placed in the stronger Leningrad interzonal tournament and not in the weaker one in Petropolis.\n\nIn the candidates' matches Karpov defeated Lev Polugaevsky by 5\u00bd:2\u00bd in the quarter-final, though the match was not as clear-cut as the result makes it appear. After three draws Polugaevsky had the initiative in the fourth and fifth games, but missed his winning chances. In the semi-final Karpov eliminated Spassky by 7:4.\n\nIn the candidates' final, set for five wins with a maximum of 24 games and which lasted from September to November 1974, Karpov met Viktor Kortchnoi. The final was seen as an important match in the USSR and thus one of the three venues was the hall of columns in the house of the trade unions. Karpov had available as seconds Geller and Furman, but also unofficially Petrosian. Kortchnoi's seconds were Roman Dzindzichashvili and Viacheslav Osnos. Paul Keres had also offered his help, but Kortchnoi had refused. Before the match, disagreements had arisen between Kortchnoi and Karpov, which the president of the Committee for Sports Sergey Pavlov decided in Karpov's favour. That was not the only reason for Kortchnoi to feel himself at a disadvantage. Karpov won the second and the sixth games, then after a long series of draws finally the 17th game too and was leading by 3:0 in wins. Kortchnoi, however, then managed wins in games 19 and 21. Nevertheless he was not able to achieve equality. Karpov just won by 12\u00bd:11\u00bd and was now the challenger to the World Champion Bobby Fischer.\n\nIn the candidates' final against Karpov, Kortchnoi is supposed to have made use for the first time of the help of a parapsychologist. During the games, Rudolf Zagainov, who was a well known psychologist in Russia in the world of sport, sat in Kortchnoi's box and tried to catch Karpov's line of sight. Karpov then engaged Vladimir Zukhar, with whom he prepared for his match against Fischer.\n\nAfter his victory over Boris Spassky in 1972 in Reykjavik, Fischer had disappeared from public view, but was negotiating with FIDE about the conditions for the forthcoming WCh match. From 1951 to 1972 all WCh matches had been played over 24 games, with the defender retaining his title if the score were 12:12. In 1971 FIDE decided on a change and decreed that from the WCh match of 1975, an unlimited number of games was to be played until one side had scored six wins. Fischer, on the other hand, demanded via his manager Fred Cramer that play should be continued until 10 wins and that the defender should retain his title at 9:9.\n\nAt the FIDE congress of 1974 in Nice (France) FIDE actually agreed Fischer's demand for ten wins, but not the condition that at the score of 9:9 the World Champion should retain his title, since this was seen as too great a disadvantage for the challenger. The number of games, moreover, was fixed at a maximum of 36. Fischer reacted to this decision on the very same day with a telegram, in which he announced that he would not appear for the WCh match of 1975 if his demands were not met in their entirety.\n\nSoon after the congress the Philippines offered to stage the WCh match with the enormous sum of five million dollars as a prize fund. Thereupon there was a power struggle in FIDE between the US federation and the USSR federation over the match conditions. The US federation mobilised some other chess federations and with 30% of the members' votes in March 1975 forced an extraordinary FIDE congress in Bergen aan Zee (Netherlands), in order to vote there to nevertheless change the rules of the WCh match to suit the wishes of Fischer.\n\nThe USSR federation had its leading grandmasters publish an open letter to FIDE in which they spoke out against a change in the rules. On the following day Mikhail Botvinnik published a further open letter, in which he accused FIDE of partisanship in favour of Fischer. At the vote in Bergen the Fischer demands were again rejected by 35:32. The federations of the developing chess countries had voted for the demands, the delegates of the eastern block states and the western states against.\n\nSince right up until the deadline of the 1st April 1975 Fischer had not signified his acceptance to participation in a match against Karpov, on the 3rd April 1975 FIDE president Max Euwe declared Karpov to be World Champion. After winning the World Championship on a walkover Karpov legitimised his title by successful participation in a plethora of tournaments. During his career he reached a total of more than 100 tournament victories.\n\nIn 1976, more by chance than anything, there was a meeting between Fischer and Karpov in Tokyo. Florencio Campomanes, who would have loved to stage the Fischer-Karpov match in the Philippines, had brought this about. Karpov was on a simultaneous exhibition tour in Asia and was invited to a banquet by the president of the Japanese federation. Fischer too 'just happened' to be there. The two players discussed a possible match, not necessarily for the World Championship, and a possible date. Karpov was absolutely up for this match, but Sergey Pavlov, the head of the Committee for Sports, forbade the match; somewhere behind this decision there was the chief ideologist in the Kremlin, Mikhail Suslov, the secretary of the central committee.\n\nDespite that there was a further meeting between Karpov and Fischer in the same year in Montilla (Spain). The president of the Philippines Ferdinand Marcos even sent an official request to the Soviet head of state Leonid Brezhnev and an offer to stage the match at the end of 1977, but the Soviet leadership still said no. There was finally even a third encounter between Karpov and Fischer in 1977 in Washington. Karpov attempted to budge Fischer from his demand to play the match for the World Championship to ten wins, with a victory for the title defender \u2013 whom Fischer considered to be himself \u2013 at 9:9, but in vain.\n\nOn his return to Moscow Karpov met Pavlov and suggested playing the match against Fischer after the WCh cycle which ended in 1978. This time Suslov agreed with the suggestion, but in the meantime contact with Fischer had been lost. In 1977 in Caracas, Karpov again met Edmund Edmondson and Max Euwe in order to negotiate the details for a possible match, but no formula could be found which was acceptable to all three parties. In his book _Karpov, chess genius_ Viktor Baturinsky expresses the opinion that Fischer, on account of a pathological anomaly, found it impossible to return to the chess board, and he thought that FIDE president Euwe was also aware of that.\n\nDuring the preparation for the match with Fischer planned for 1975, Alexander Nikitin was part of Karpov's team of trainers. When Karpov secretly met Fischer, Nikitin informed the Committee for Sports about the meeting. Karpov felt betrayed, dismissed Nikitin and made sure that Nikitin was also sacked as a trainer by the federation. Nikitin later became a trainer for Kasparov and with the help of his new prot\u00e9g\u00e9 dethroned Karpov as World Champion.\n\nBobby Fischer had disappeared from public view after his WCh victory of 1972. To be precise, he was avoiding all possible journalists. As he later explained to his Icelandic friend Einar Einarsson, Fischer's mistrust of all journalists was based on his disappointment with his former confidant Brad Darrach. Before the WCh match of 1972 the latter was in close contact with Fischer and later used his observations with-out any agreement from Fischer for newspaper articles and a book ( _Bobby Fischer vs. The Rest of the World, 1974_ ). Fischer later lost in litigation against Darrach.\n\nFrom time to time, the ex-World Champion now lived in Pasadena and San Francisco. In 1982 he published a 21-page booklet: _I was tortured in the Pasadena jailhouse!_ , in which he complained about his treatment in a Pasadena jail in which he was imprisoned for two days as a result of a mix-up. Later he stayed in Budapest, where he met Peter Leko and the Polgar sisters, Zsuzsa, Zsofia and Judit, and played chess and analysed games with them, without this ever being made public.\n\nThe World Chess Champion who had disappeared became a myth. The future World Champion Anand later found a telling comparison with 'Fischer is our Marilyn Monroe'. There was, for example, speculation about whether Fischer was not much better than Karpov and Kasparov.\n\nFrom time to time Fischer would meet chess lovers, who all reported about his characteristic anti-Semitism. In the autumn of 1990 Fischer popped up at Lothar Schmid's, the arbiter of the 1972 WCh match, in Bamberg. The German grandmaster, whose main profession was co-owner of the Karl-May publishing house, lodged Fischer in Waischenfeld, near Bamberg, in the 'Pulverm\u00fchle' hotel, owned by the Bezold family. He analysed games with Michael Bezold, who later himself became a grandmaster. Fischer believed that all the games and matches between Karpov and Kortchnoi and also between Karpov and Kasparov had been fixed and at that time was working on a manuscript with analyses of the games of these matches which were to prove his supposition.\n\nIn 1992 Fischer suddenly popped up out of the woodwork and in Yugoslavia played a 'return match' against his opponent of 1972, Boris Spassky. The match was financed by the president of the Yugoslavian Jugoscandic Bank, Jezdimir Vasiljevic, a close friend of the Serbian president Slobodan Milosevic, and was, as demanded by Fischer in 1975, to be to ten wins. At a score of 9:9 Fischer would remain the 'undefeated World Champion'. The total prize found was the fabulous sum of five million dollars, of which the winner received 3.65 million dollars. The match began on the 2nd September 1992 on the Montenegran island Sveti Stefan. After eleven games the match moved on the 30th September to the Belgrade 'Sava Business Center'. After game 30 Fischer had won his tenth game and with it the match.\n\nIt later became known that the banker Vasiljevic had robbed the customers of his Jugoscandic Bank of more than 130 million dollars by means of a pyramid scheme. In April 2009 he was arrested in the Netherlands, after seeking asylum there with forged papers, and extradited to Serbia. In Belgrade he had been tried in court in 2007, but Vasiljevic had escaped from remand.\n\nDuring the Yugoslavian civil war, the USA had instituted an economic boycott of Serbia. Before the start of his match against Spassky, Fischer had been specifically warned in a letter from the treasury against playing in Yugoslavia, because in the opinion of the US authorities that would infringe the boycott. At the official press conference before the match and in front of the cameras, the 49 year old Fischer spat on the letter from the treasury and stated that he would not observe the boycott.\n\nAt the time of the match Fischer lodged part of his chess estate, books, scoresheets, handwritten analyses, in two cases with Svetozar Gligoric and never collected them. Later a part of these valuable memorabilia passed into the possession of a collector. It is not known where the remainder of this legacy went after the death of Gligoric in 2012.\n\nAfter the match against Spassky, Fischer again disappeared and from then on lived predominantly in the Philippines, where he had a friend in Eugenio Torre, and in Japan. There he became friends with Miyoko Watai, president of the small Japanese Chess Federation.\n\nAfter the attack on the World Trade Center of the 11th September 2001 Fischer spoke in a radio interview and welcomed the terror attack as the result of American policies. This was greeted with outrage in the USA and in the western press. After the US government rescinded his passport, Fischer was arrested in the airport on the 13th July 2004 when leaving Japan and imprisoned in Ushiko. The USA requested his extradition from Japan on account of his breaking the economic embargo with his 1992 match against Spassky in Yugoslavia and of tax evasion, against which Fischer and his partner Miyoko Watai fought with all the means at their disposal. On the 17th August 2004 the pair married, to improve Fischer's prospects of being allowed to remain in Japan. His Icelandic friends from the time of his match against Spassky in 1972 finally arranged for Fischer to receive Icelandic citizenship in March 2005. Thereupon he was deported from Japan to Iceland.\n\nThe images of the arrival in Iceland of the former World Chess Champion are of an unkempt man who was apparently also in ill health. Fischer lived from then on in Iceland, regularly meeting his Icelandic friends. His time was spent, among other things, with visits to an antiquarian bookshop 'Bokin', which became like a second home for the ex-World Champion.\n\nFrom time to time the 11th World Champion had visitors. Thus, for example, one of his successors on the World Champion's throne, Viswanathan Anand, met Fischer and found him 'astonishingly normal' in conversation. But also the rock musician Patti Smith, three years younger than the World Chess Champion and who like him had grown up in New York, took the opportunity to meet Fischer on the occasion of a concert which she gave on the 5th September 2006 in Reykjavik. She too was able to enjoy a pleasant conversation with the World Chess Champion in exile.\n\nOn the 17th January 2008 Robert James Fischer died at the age of 64 \u2013 one year for every square on the chess board \u2013 in the 'Landspitali' hospital in Reykjavik of kidney failure and was buried in the small cemetery of Laugard\u00e6lakirkja near Selfoss. On Fischer's wishes, only five people were present at the funeral on the 21st January 2008: Fischer's friend Garthar Sverrisson, his wife Krisin, their two children and Fischer's wife Miyoko Watai. The funeral was undertaken by a Catholic priest from Reykjavik. After Fischer's death a dispute flared up about his legacy of some two million dollars. As well as Fischer's widow Miyoko Watai the children of his sister Joan Tart put in a claim, as did the Filipina Marilyn Ong, who claimed that Fischer was the father of her daughter Jinky. An Icelandic court ordered the exhumation and a paternity test. This proved that Fischer was not the father of Jinky Ong.\n**29. Political thriller in Baguio**\n\n**The World Championship 1978:  \n_Anatoly Karpov against Viktor Kortchnoi_**\n\nViktor Kortchnoi was born on the 23rd March (some sources suggest 23rd July) 1931 in Leningrad. His father and his mother both came from the Ukraine, from where, independently of each other, they had fled to Leningrad during the great famine of the 1920s \u2013 a consequence of the forced collectivisation of agriculture which cost six million people their lives.\n\n**Viktor Kortchnoi (born in 1931)**\n\nSome few years after Kortchnoi's birth his parents separated and Kortchnoi grew up with his father and the latter's second wife. The family was extremely poor and lived in a 13 room state flat, which his family had to share with ten other families. Despite this his father gave special attention to the education of his son and paid for a private tutor who taught Kortchnoi German after school. At the age of six Kortchnoi learned the game of chess from his father.\n\nDuring the siege of Leningrad in the Second World War by the German Wehrmacht Kortchnoi remained in the city and survived the starvation caused by the blockade whereas many members of his family died. He saw his father for the last time in 1941; then the latter was killed in an air attack when he was on Lake Ladoga in a barge. When the situation in Leningrad started to become more normal in 1943\/44, Kortchnoi bought his first chess books from an antique shop: Lasker's _Lehrbuch des Schachspiels_ ('Manual of Chess') and Tartakower's _Das entfesselte Schach_.\n\nAt the age of 13 Kortchnoi started going to the chess section of the Leningrad pioneer palace. His first chess teachers there were Andrei Batuev and Abram Model. The latter had already worked with Botvinnik. Kortchnoi later became one of the pupils of Vladimir Zak, who took over the training in the Leningrad pioneer palace in 1945, after his return from the war, and was in charge of it until 1970. Kortchnoi had to give up his wish to become an actor because despite elocution his delivery was not clear enough. So Kortchnoi began to concentrate totally on chess.\n\nWhilst the six years younger Spassky made lightning progress, Kortchnoi's understanding of chess developed somewhat slowly. In 1947 Kortchnoi became junior champion of Leningrad ahead of Ivo Nei and the following year he shared first place in the Soviet junior championships with Nei. At the end of his schooldays in 1948 Kortchnoi started his studies at the History Faculty of the University of Leningrad and finished them six years later with his dissertation on 'The Front National and the communist party of France on the eve of the Second World War'.\n\nIn 1962 his fourth place in the interzonal tournament in Stockholm qualified Kortchnoi for the candidates' tournament in Cura\u00e7ao, in which he only came in fifth, however. Just like Fischer, Kortchnoi felt himself betrayed by the agreed draws between Petrosian, Geller and Keres. During the next WCh cycle but one Kortchnoi reached the candidates' matches with a second place in the interzonal tournament in Sousse and was not defeated until the candidates' final by Boris Spassky. In the next cycle Kortchnoi was seeded to be a candidates' finalist and exited in the semi-final against Petrosian \u2013 according to a secret agreement, because Petrosian was considered to have better chances against Fischer. As we know, this dodge of the Soviets did not bear fruit. Petrosian had no chance against Fischer either. In 1973 Kortchnoi won the interzonal tournament in Leningrad together with Karpov, but was defeated by the latter in the candidates' final of 1974.\n\nAfter the candidates' final Kortchnoi gave Bozidar Kazic an interview, which appeared in the Yugoslavian newspaper _Politika_. Kortchnoi claimed that in his candidates' matches Karpov had not performed particularly well. He also showed some sympathy with Fischer's demands relating to the forthcoming WCh matches. The interview, however, did not go down well with the Soviet Committee for Sports. Kortchnoi was summoned to the office of the Committee for Sports and punished with a one year ban on foreign tournaments. His stipend was reduced from 300 to 200 roubles. In addition he was forbidden from making any statements in public.\n\nFrom then on Kortchnoi turned his mind more and more seriously to the idea of leaving the Soviet Union and wrote a letter to Josip Broz Tito. He asked the Yugoslav head of state to accept him in Yugoslavia, but received no answer. On the occasion of his participation in the IBM tournament in Amsterdam, Kortchnoi then sought political asylum in 1976 in the Netherlands. When taking part in the Hastings tournament in January 1976, Kortchnoi had already made concrete plans for a flight to the West. With the help of his friend Genna Sosonko, a Soviet grandmaster of Jewish heritage, who had emigrated to the Netherlands in 1972, he made contact with Max Euwe. The FIDE president promised his support and took care that despite his flight Kortchnoi was later able to take part in the candidates' matches.\n\nIn the Netherlands Kortchnoi may have been refused asylum on political grounds, but he was granted the right of domicile on humanitarian grounds. The newspapers reported the flight of the top Soviet player in detail. This was how Petra Leeuwerik learned about it. Leeuwerik got in contact with Kortchnoi because she felt the need to help in his struggle against the Soviets. After the Second World War she had been abducted by the Soviets during a visit to Austria and deported to a labour camp near Workuta in North Siberia. The official reason given was espionage. Leeuwerik, who had been born in Vienna, supposed that her contacts with a Catholic student organisation in Leipzig where she was studying was the reason for the abduction. She was not released for the labour camp till ten years later during the Adenauer initiative of 1955.\n\nWhen Kortchnoi's flight became known publicly, the first to congratulate him was Robert Fischer. He sent a telegram with the text: 'Congratulations on this good move!' The following year there was even a meeting with Fischer in Pasadena when Kortchnoi was in the USA on a tour of simultaneous displays. Kortchnoi described the conversation with Fischer as stimulating, even if the latter was already hooked on his idea of a Jewish world conspiracy. He was especially surprised by Fischer's chess memory. Fischer knew practically every grandmaster game. On the following day, however, Kortchnoi received a letter from Fischer in which he was decried as a Soviet agent and then broke off contact with Fischer.\n\nViktor Baturinsky, vice-president of the USSR Chess Federation, formerly public prosecutor and amongst other cases one of the prosecutors in 1963 in the show trial against the US spy Oleg Penkovsky, who had in 1962 passed on information about the planned installation of Soviet rockets on Cuba, organised after Kortchnoi's flight an open letter from leading Soviet grandmasters, in which they condemned the said flight. Only Botvinnik, Spassky, Bronstein and Karpov refused to sign. In addition the Soviets instigated a tournament boycott on the fugitive \u2013 no Soviet grandmaster would participate where Kortchnoi was playing.\n\nThe Soviet grandmasters, however, were obliged to appear in the candidates' matches or else they would have lost them without a fight. In the quarter-final in Il Ciocco (Italy) Kortchnoi met Tigran Petrosian and defeated him by 6\u00bd:5\u00bd. There were tensions during the match when Petrosian claimed that Kortchnoi was in telepathic communication with his seconds, while Kortchnoi speculated that Petrosian was using his hearing aid as a wireless device and receiving signals. In the semi-final in Evian (France) Kortchnoi annihilated Lev Polugaevsky by 8\u00bd:4\u00bd. The word was gradually circulating, especially in Soviet chess circles, that he was 'Viktor, the terrible'.\n\nBelgrade staged the candidates' final between Kortchnoi and Spassky. Though previously friends, the match turned the two of them into enemies. In the meantime Kortchnoi had proclaimed his attack on Karpov and on the World Championship title to be a political mission. The final obstacle between him and Karpov was Spassky. Spassky's realisation after the match was: 'If you play a match against Viktor, you can no longer be friends with him after it.' The match turned into psychological warfare. Spassky complained that the places for spectators in the front rows were occupied by Kortchnoi's people who stared at him so intently that he could no longer concentrate.\n\nKortchnoi won the second, third, seventh, eighth and tenth games. In the tenth game the growing strife between the players escalated and during the game Spassky spent most of his time behind the stage in a rest room. As an official reason for this, he stated that the light on the stage was tiring his eyes. Kortchnoi was annoyed by Spassky's absence and he complained after the game that in the rest room Spassky was using a demonstration board to think about his moves and had it taken away by the arbiter. When the score stood at 7\u00bd:2\u00bd after ten games, Spassky finally managed his first win. After this game Spassky threatened not to appear for the 12th game if the demo board in the rest room were not re-instated.\n\nIn the meanwhile FIDE president Euwe had arrived and calmed the players. The demo board was set up again in such a way that everyone could see it, so that Spassky could not somehow secretly analyse on it as Kortchnoi was insinuating. After Spassky also won the twelfth and 13th games, Kortchnoi now threatened to leave. The suspicion was expressed from within his camp that Kortchnoi was being exposed to radiation which was weakening him. Kortchnoi also remained convinced for a long time after the match that the Soviets had 'programmed' his former team mate and training partner as a medium, and he kept on finding new proofs for this thesis. Even his protectress Petra Leeuwerik saw Kortchnoi falling under the influence of Soviet parapsychological powers in the course of the match. Kortchnoi also went on to lose game 14, for which Spassky had turned up with a silver anti-glare eye-shade, and he then had only one point of a lead. After that Kortchnoi overcame his crisis and drew the next two games.\n\nThe high point of the curious events of this match came in game 17 when Spassky wore in addition to his eye-shade sunglasses and finally divers' goggles. For his part, during the match Kortchnoi had spotted Soviet citizens in the hall with a small black case, which was obviously directing something 'devilish' against him, and described in his book _Anti-Chess_ how, thanks to the help of a group of Swiss parapsychologists who were not even present in Belgrade but working at a distance, he held out against it. So, despite all the 'devilish tricks' Kortchnoi won this and also the next game for a final score of 10\u00bd:7\u00bd.\n\nAccording to Kortchnoi Spassky also believed in parapsychological powers and in his book _Chess is my life_ he quotes a meeting with Spassky in 1986. Spassky apparently then remembered how forces beyond his control during their 1977 match in one specific game moved a knight from d4 to f5. Spassky then thought that the chief arbiter Kazic had been the counterforce, but he later noticed that his trainer Bondarevsky was responsible for it. And his trainer who died in 1984 had apparently worked together with the hypnotist Wolf Messing. Messing had been a fan of Spassky, according to Kortchnoi, and possibly Bondarevsky might have been able to deploy the powers of Messing for his own purposes. This, as Kortchnoi went on to think, must also have been the reason why Spassky stuck so closely to 'old' Bondarevsky, who practically did not have a clue about modern opening theory.\n\nA series of federations or cities had bid for the WCh match between Karpov and Kortchnoi and each offered around a million Swiss francs as a prize fund. Karpov put Hamburg first in his list of preferences, left the second place blank and for the third chose Baguio City. In Kortchnoi's list Graz was in first place, Baguio in second. Germany was out of the question as a venue for Kortchnoi, because Karpov had 'always received a particularly hearty welcome in Germany and moreover that was where he kept his illegal earnings'. In addition, in West Germany, according to Kortchnoi, 'pro-Soviets could not be distinguished from normal inhabitants even by their language'. The initiator of the Hamburg bid was actually Karpov's close friend Helmut Jungwirth, who soon after that, however, developed a business deal which turned out to be very unpleasant for Karpov.\n\nFollowing the lists of preferences, FIDE president Max Euwe decided to award the WCh match to Baguio City in the Philippines, against which both players protested. Euwe gave as his reason for the decision that Baguio City, whose organisers had made an offer of over 585 750 dollars, was the only city mentioned on both lists.\n\nThe chief organiser there was Florencio Campomanes. Campomanes was born on the 22nd February 1927 in Manila. He studied political sciences in the Philippines and in the USA and worked for a doctorate until 1954. Campomanes was one of the best players in his country and had won the national championship in 1956 and 1960. Between 1956 and 1966 he had played for his country in five Chess Olympiads. As a delegate for his chess federation he then began to become active in chess politics in Asia and also within FIDE. On the 17th November 1982 at the FIDE congress during the Chess Olympiad in Lucerne Campomanes won the election for FIDE president on the second round of voting, defeating his predecessor Fridrik Olafsson by 65 votes to 43. The Soviet Chess Federation originally wanted to vote for the Yugoslav candidate Bozidar Kazic, but then received from higher authority the instruction to vote for Campomanes. The Filippino had already been received as an honoured guest in Moscow at the Spartakiad in 1979 and the Olympic Games in 1980.\n\nThe vote of the Soviet federation also pointed out the correct way to vote for all the other east block federations and they followed suit. The three Dutch FIDE employees, including the secretary of many years standing Ineke Bakker, had threatened to resign if Campomanes were elected and did so. Campomanes retained the office of FIDE president till 1995 and was the first FIDE president to take an honorarium for his work for the world chess federation. His annual salary was 150000 Swiss francs. It was during Campomanes' period of office that the WCh match of 1984\/85 was abandoned under circumstances which have not been fully explained. The decay of the World Chess Championships after 1993 into a 'Classical World Championship' and the so-called 'FIDE World Championship' as a result of the dispute with Kasparov and Short is also partly the responsibility of Campomanes.\n\nOn that account, and also because many FIDE delegates considered him corrupt, he was voted out of office in 1995. His successor was the Russo-Kalmykian politician Kirsan Ilyumzhinov, for a long time president of the autonomous Russian Republic of Kalmykia. But Campomanes continued to remain active in FIDE, obtaining votes from Asian countries for his successor, and was elected honorary president.\n\nIn 2003 Campomanes was found guilty in his home country by a court, which was dealing with cases of corruption in the post-Marcos era, of having misappropriated approx. 238000 US dollars. This was money from the ministry of sport of the Philippines which FIDE had intended for the organisation of the Chess Olympiad of Manila (6th-25th June 1992) \u2013 but which never reached it. Campomanes was sentenced to one year and ten months in prison. On account of his advanced age the punishment was changed to probation. Campomanes appealed and obtained a revision of the judgement on account of a technicality. In the book _The KGB plays chess_ Campomanes is accused of having worked for the Soviet KGB. As a quid pro quo for his help with the match between Karpov and Kortchnoi in 1978 the Soviet Union is supposed to have made sure that with the votes of the eastern block and the developing countries he was elected president of FIDE in 1982. In 2007 Campomanes was badly injured in a car accident in Turkey, but recovered. He died on the 3rd May 2010 in the Philippines.\n\nKortchnoi's seconds for the WCh match were, as they had been for the previous candidates' matches, the two English players Michael Stean and Raymond Keene, and in addition Jacob Murey. Petra Leeuwerik acted as delegation leader. Two weeks before the start of the match Kortchnoi's team travelled to Baguio City, which lies approx. 150 kilometres north of Manila on the island of North Luzon, in order to acclimatise in peace.\n\nAhead of the match disagreements had arisen between Kortchnoi and Keene, because, without asking Kortchnoi, Keene had published a book about the match between Kortchnoi and Spassky and had made use of common analysis in it. The book had appeared with the help of David Levy, who was at the same time ironically preparing a book about Karpov under the title _Chess is my life_. An autobiography of Kortchnoi had, however, appeared with exactly the same title. So for the WCh match Kortchnoi had concluded with Keene a contract which explicitly forbade the latter from publishing a book about the match without the permission of Kortchnoi.\n\nThe Soviet delegation was 14 strong and was led by Viktor Baturinsky. In the Karpov team, there were, for example, a doctor, a cook, two 'specialists in biological processes' and the psychologist Vladimir Zukhar. Tal later expressed the opinion that Karpov always needed the help of a psychologist because he was inclined to self-doubt, but never admitted it, and in addition tired quickly both mentally and physically. Kortchnoi guessed that at least six members of the delegation were employed by the KGB. For chess advice Karpov had as his official seconds Yuri Balashov, Igor Zaitsev and Mikhail Tal, who was, however, accredited as a correspondent for _64_. Tal later joked that he had only helped Karpov because if Kortchnoi had won 'chess would have been forbidden in the USSR'. In addition there was also the rumour that Karpov had telephone access to a supercomputer in Moscow.\n\nAccording to his second Adrian Mikhalchishin the KGB assembled in Karpov's analysis room a military bug-proof tent flown in from the USSR. During analysis, moreover, the seconds were not to talk about their analysis in normal chess terms but had to employ a special code. Since it is known that one fears from the opposition exactly the same measures which one uses oneself, it can be supposed that the opinion later expressed by Kortchnoi that he had been monitored during the match, probably did not come from thin air.\n\nThe rules for the encounter had been fixed at the FIDE committee meeting of October 1977 in Caracas. The match was set for six wins, drawn games were not to be counted. In the event of defeat, the title defender, Karpov, was granted the right to a return match, a rule which had been abolished in 1963. The two players in the WCh match were to be allocated all their expenses and the costs for two seconds. The chief arbiter for the match, which started on the 18th July, was Lothar Schmid.\n\nThe psychological warfare, which Kortchnoi had already waged against Petrosian and Spassky, was continued in Baguio. Kortchnoi brought with him his own chair, which at the request of the Soviet delegation was submitted to X-ray in the central hospital of Baguio before the first game. Nothing unusual could be found. Kortchnoi's wish to play under Swiss colours was denied him. Instead he was described on the instigation of the Soviets as 'stateless'. At the opening ceremony for the match, in the presence of the president of the Philippines Ferdinand Marcos and the Soviet ambassador, instead of the Soviet national anthem the 'Internationale', the anthem of the communist party, was played on purpose.\n\nAfter the second game Kortchnoi objected to Karpov being given a yoghurt during the game \u2013 after all, the flavour of the yoghurt might have some meaning. The arbiter Lothar Schmid decided that the flavour of the yoghurt had to be announced beforehand. Kortchnoi also conjectured that Karpov was being given cortisone with the yoghurt, which according to Kortchnoi was the reason why the World Champion was putting on weight during the match instead of losing it, which is what frequently does happen in chess matches.\n\nDuring the games Kortchnoi wore reflective sunglasses, also had a geiger counter installed and threatened to break off the match immediately if the same rays were employed against him which had presumably weakened him during the candidates' final against Spassky. Karpov protested against the wearing of the sunglasses because he felt disturbed by the reflected light. Before the ninth game Lothar Schmid sat down at the playing table with another arbiter, who was wearing sunglasses, as a test, but could not discern any disturbance due to reflected light and rejected the protest. After the seventh game, when Karpov in a clearly better position on the adjournment surprisingly offered a draw on the resumption, Kortchnoi expressed the opinion that the Soviets had monitored his analyses or that there was a leak from his team. Here he suspected Keene, who in fact after every game was sending to London a secret telex for a just as secretly planned book about the match. Every telex was, however, read by Campomanes \u2013 and he was working for the Soviets.\n\nAccording to Karpov, Kortchnoi had in his team a series of psychologists, para-psychologists and hypnotists. These were supposed above all to increase Kortchnoi's belief in his own victory. Since in the candidates' final of 1974 Kortchnoi had already worked with the sports psychologist Rudolf Zagainov, Karpov was prepared and had engaged the psychologist Vladimir Zukhar, the director of the Central School for Psychology in Moscow, who was supposed to not only support Karpov psychologically but also to be employed for 'neutralising the powers' of Kortchnoi's helpers. According to Kortchnoi, Zukhar stared at him constantly during the games in order to disturb and to influence him.\n\nSo in one of the games at the start of the match Petra Leeuwerik took a seat beside Zukhar and tried to disturb him in his 'work' by bumping into and kicking him. Leeuwerik was then, however, prevented from annoying Zukhar by Karpov's fitness trainer Valery Krylov. The tournament jury was called into session and finally decided that Kortchnoi was indeed being disturbed and that should this be repeated the game would be moved to a room without spectators or the person who was creating the disturbance would be removed from the spectators' area.\n\nAfter this incident before the eighth game Karpov refused Kortchnoi the handshake which was written into the rules for such matches. After the first seven games ended in draws, Karpov took the lead with a win in the eighth game. Before the eleventh game Kortchnoi had summoned to neutralise Zukhar's powers the Israeli psychologist Dr Vladimir Berginer, who now, at first unrecognised, took a seat in the spectators' room. Kortchnoi equalised in the eleventh game, which he put down to the powers of Dr Berginer. But Karpov won thirteenth and fourteenth games for a score of 3:1. Kortchnoi's explanation for this was: in the meantime Berginer had been discovered by the Soviets, who then did not let him 'work' in peace. So the Swiss, now ineffective, left after the fourteenth game.\n\nPetra Leeuwerik meanwhile tried to further politicise the match and had Keene translate into English a statement in which she referred to, amongst other things, the invasion of Czechoslovakia by Soviet troops in 1968 and her own abduction into a labour camp. Keene was apparently unhappy with Leeuwerik's statement and tried via the Swiss federation to have her sacked as delegation leader. FIDE president Euwe prohibited this attempt.\n\nBefore game 17, Kortchnoi refused to start play if Dr Zukhar was not moved back in the room. Kortchnoi even threatened to become violent and so Campomanes had the first six rows cleared of spectators, with which Kortchnoi was satisfied. Kortchnoi missed a win in this game and later allowed himself to be mated in a drawn position.\n\nAfter the game the playing hall was examined by nuclear specialists at the demand of the Soviet delegation. The Soviets claimed that Kortchnoi's one-way glasses emitted harmful rays, but the presence of such could not be confirmed. Kortchnoi took his last two timeouts and left Baguio City in the direction of Manila, in order to rest and to give a press conference. Kortchnoi now threatened to break off the match if the problem with Dr Zukhar could not be solved and demanded in the press conference the installation of one-way mirror. A compromise was negotiated between the delegations: Dr Zukhar sat from then on next to the other members of the Soviet delegation. The demand for a one-way glass wall was withdrawn. Kortchnoi no longer wore reflective glasses.\n\nIn the 18th and 20th games Karpov spoiled winning positions. For the 18th there appeared in the spectators' room at the instigation of Kortchnoi two members of the US section of the Indian Ananda-Marga sect, Steven Dwyer and Victoria Shepherd, in conspicuously bright clothing, so as to strengthen Kortchnoi by meditation and neutralise Zukhar. Kortchnoi later pointed out that the couple had sought him out of their own accord in order to help him and had not requested any fee for this. The Soviet delegation protested against their presence. Apparently the two gurus were only free on bail after attacking and murdering an Indian embassy official in the Philippines to draw attention to the arrest of an Ananda-Marga member in India. For this reason they were forbidden entry to the tournament hall after game 20.\n\nIn the 21st game Kortchnoi shortened the lead to 2:4. In the 22nd game Karpov again missed a possible win. But Karpov then increased the lead to 5:2 with a win in game 27. He was now only one game short of a title defence. But then Karpov fell prey to a phase of weak play and lost the 28th, 29th and 31st games. Kortchnoi had levelled a match which most people had already considered all but over. The next victory would decide the match \u2013 one way or the other.\n\nDuring the match Karpov had concluded with the chess computer manufacturer Novag from Hong Kong an advertising contract which earned the World Champion, according to Kortchnoi, half a million dollars. Should he lose the match against Kortchnoi, Karpov was supposed to have planned not to return to the USSR and to have instructed his German agent Helmut Jungwirth to buy him a house in the USA. Helmut Jungwirth, an editor for the broadcaster NDR in Hanover, had produced with Karpov a game 'Television spectators against the World Champion', then become friendly with Karpov and had been helpful to him with a few business deals. Later he is supposed to have embezzled some 1.3 million DM from Karpov's advertising revenues, was sued by Karpov and arrested in 1985. In the subsequent trial, however, a few details about Karpov's business activities were also made public, details which Karpov would certainly have liked to continue to remain 'private'. The World Chess Champion emphatically denied Jungwirth's claim that he had been supposed to buy a house in the USA for Karpov.\n\nAfter the 31st game FIDE president Max Euwe left, since the Chess Olympiad was to begin shortly in Buenos Aires. At the FIDE congress in Buenos Aires Euwe was then no longer a candidate for the office of president. The Icelandic grandmaster Fridrik Olafsson became his successor. In the 32nd game, Dr Zukhar had again taken a seat near the stage, in the fourth row, against the agreement which had been reached. For this game Kortchnoi had prepared in the Pirc Defence a surprise weapon. But Karpov was not surprised, or at least he did not let it show, and moved without taking a long time to think, which again fed Kortchnoi's suspicion that his team was being bugged or that there was a leak \u2013 Kortchnoi again suspected Keene of having passed on his preparation, without saying so publicly at this point. Andrew Soltis noted in his book _Soviet Chess 1917-1991_ that according to a book which was published in 1999 18 KGB officers were tasked with ensuring Kortchnoi's defeat in Baguio City.\n\nKarpov won the 32nd game. It was adjourned in a hopeless position for Kortchnoi, but the latter wanted neither to continue it nor to resign, so that finally Keene resigned the game for him. With it Karpov had defended his title. In a telegram he reported to 'Comrade Leonid Brezhnev' that the match had ended 'with our victory'. According to his own account, Karpov donated 350000 dollars of his prize money to the Committee for Sports for the renovation of the Central Chess Club. The defeated challenger did not attend the closing ceremony. Keene brought him the cheque with his share of the prize money. According to Kortchnoi Keene received from Campomanes 20000 Swiss francs for his 'services' and later, before the match in Meran 1981, is supposed to have worked directly for Karpov. Kortchnoi appealed against the result of the match in front of the International Court in The Hague, since the Soviets had broken the agreement they shad signed concerning Dr Zukhar. After a three year case the complaint was rejected.\n\n **Kortchnoi \u2013 Karpov**\n\nBaguio City, 17th game  \n26th August 1978  \nNimzo-Indian Defence (E47)\n\n**1.c4 \u2658f6 2.\u2658c3 e6 3.d4 \u2657b4 4.e3 0-0 5.\u2657d3 c5**\n\nThe main variation of the Rubinstein System in the Nimzo-Indian Defence is more often reached via the move order 5...d5 6.\u2658f3 c5.\n\n**6.d5!?**\n\nThe challenger had already surprised Karpov in the 7th game with this innovation, an idea of Kortchnoi's second Jacob Murey. More popular moves are 6.\u2658f3 d5 7.0-0 or 6.\u2658ge2.\n\n**6...b5!?**\n\nA pawn sacrifice in the style of the Blumenfeld or Volga Gambit.\n\n**7.dxe6**\n\n7.e4? is not possible on account of 7...bxc4 8.\u2657xc4? \u2658xe4.\n\n**7...fxe6**\n\nPerhaps 7...bxc4!? is better: 8.exf7+ \u2654h8 9.\u2657xc4. (Karpov)\n\n**8.cxb5 a6**\n\nThe 7th game of the match continued as follows: 8...\u2657b7 9.\u2658f3 d5 10.0-0 \u2658bd7.\n\n**9. \u2658ge2 d5 10.0-0 e5**\n\nAt the cost of a pawn Black has constructed an imposing pawn centre. The alternative was 10...axb5 11.\u2657xb5 \u2657a6 12.\u2657xa6 \u2658xa6. (Kasparov) It was also worth considering 10...c4 11.\u2657c2 axb5 12.\u2658xb5 \u2657d7 13.\u2658ec3 \u2655b6 14.a4 \u2658c6.\n\n**11.a3 axb5**\n\nThis move was criticised. Karpov could have reacted instead with a bishop move. 11...\u2657a5 is followed by 12.b4! (Kasparov) 12...cxb4 13.axb4 \u2657xb4 14.\u2655b3 with advantage to White. But there was the better 11...\u2657xc3!? 12.\u2658xc3 (12.bxc3? is not possible here: 12...c4 13.\u2657c2 axb5) 12...c4 13.\u2657e2 axb5 14.\u2658xb5 \u2657a6 15.\u2658c3 d4 16.exd4 exd4 17.\u2657f3 \u2656a7 18.\u2658e2 \u2656d7 'with nice counterplay'. (Kasparov)\n\n**12. \u2657xb5 \u2657xc3 13.bxc3**\n\nThis is the difference from the variation with 11...\u2657xc3. White recaptures with the pawn and thus stops the advance of the black centre.\n\n**13... \u2657a6**\n\n13...c4?! 14.a4 \u2657a6 15.\u2657a3 \u2656f7 16.f4 \u2655b6 17.\u2655d2 with advantage to White. (Karpov) But it was worth considering 13...\u2657b7!?. (Karpov)\n\n**14. \u2656b1 \u2655d6**\n\nAfter14...\u2658bd7 White plays 15.e4 \u2657xb5 16.\u2656xb5, followed by a3-a4, \u2657a3 and f2-f4. (Karpov)\n\n**15.c4 d4**\n\n15...\u2657xb5 16.\u2656xb5 d4 17.\u2658g3 also favours White.\n\n**16. \u2658g3**\n\n16.f4 d3 17.\u2658g3 exf4 18.\u2656xf4 is also favourable for White. (Karpov)\n\n**16... \u2658c6 17.a4**\n\nAfter 17.\u2658f5 \u2655e6 18.\u2657xc6 \u2655xf5 19.e4 \u2655e6 20.\u2657xa8 \u2656xa8 'followed by 21... \u2657xc4 the linked passed pawns offer excellent compensation for the exchange he has lost.' (Kasparov)\n\n**17... \u2658a5?**\n\n17...\u2657b7 was better. (Karpov)\n\n**18. \u2655d3 \u2655e6 19.exd4 cxd4 20.c5 \u2656fc8 21.f4 \u2656xc5**\n\n'It is obvious that I have lost the strategic battle. Now all that I had left was to build on my defensive skills and look for tactical opportunities.' (Karpov)\n\n**22. \u2657xa6**\n\nAfter 22.fxe5 \u2657xb5 23.axb5 \u2655xe5 24.b6 \u2658b7 25.\u2657d2 White also has a winning position.\n\n**22... \u2655xa6 23.\u2655xa6?!**\n\nKortchnoi misses the strong continuation 23.\u2656b8+ \u2654f7 24.\u2656b5 with the threat fxe5+\u2013.\n\n**23... \u2656xa6 24.\u2657a3 \u2656d5 25.\u2658f5 \u2654f7 26.fxe5 \u2656xe5 27.\u2656b5**\n\nAfter 27.\u2658xd4 \u2658c4 the tension has been lifted and the game is drawn. (Karpov) Kasparov, however, sees a continuing advantage for White, for example: 28.\u2656b7+ \u2654g6 29.\u2657f8 \u2658e8 30.\u2656b4.\n\n**27... \u2658c4 28.\u2656b7+**\n\n28.\u2658d6+? \u2656xd6. (Kasparov)\n\n**28... \u2654e6 29.\u2658xd4+ \u2654d5 30.\u2658f3?**\n\nIn time trouble White chooses a liquidation, in which he gets a rook for two knights. A more solid and better way was 30.\u2658c2 \u2656xa4 31.\u2657f8 \u2656e8 (31...\u2658e8 32.\u2658b4+ \u2654e4 33.\u2658c6+\u2013) 32.\u2657xg7 \u2658e4 33.\u2656d1+ \u2654c6 34.\u2656dd7 \u2658a5 35.\u2656bc7+ \u2654b5 36.\u2657d4+\u2013. (Kasparov)\n\n**30... \u2658xa3 31.\u2658xe5 \u2654xe5 32.\u2656e7+ \u2654d4 33.\u2656xg7 \u2658c4 34.\u2656f4+ \u2658e4 35.\u2656d7+ \u2654e3!?**\n\n35...\u2654e5 36.g3 \u2658cd6 (36...\u2656xa4? 37.\u2656e7+ +\u2013) 37.\u2656xh7 \u2656xa4 38.\u2656e7+ \u2654d5 39.g4 \u2656a1+ 40.\u2656f1=.\n\n**36. \u2656f3+ \u2654e2 37.\u2656xh7?**\n\n37.\u2656e7 \u2658cd2 38.\u2656a3 and Black has to fight for the draw. (Kasparov).\n\n**37... \u2658cd2 38.\u2656a3 \u2656c6 39.\u2656a1??**\n\nKortchnoi was certainly not thinking that anything could happen to him here.\n\n**39... \u2658f3+**\n\nAnd White resigned, since he will be mated: 40.gxf3 \u2656g6+ 41.\u2654h1 \u2658f2#.\n**30. Heaps of electronic devices**\n\n**The World Championship 1981:  \n_Anatoly Karpov against Viktor Kortchnoi_**\n\nAs a WCh finalist from the previous cycle Kortchnoi was seeded into the candidates' matches and in the quarter-final in Velden he met, as he had three years previously, Tigran Petrosian. The chemistry between the two players had not improved. Back at the notorious candidates' tournament of 1962 Kortchnoi had felt himself to be a victim of the agreed draws between Petrosian, Geller and Keres. In 1971 it had been agreed, presumably fixed by the Soviet Committee for Sports, that Kortchnoi had to lose his semi-final match against Petrosian, since Petrosian was considered to have the better chances against Fischer.\n\nBefore the match Kortchnoi gave a press conference in which he announced that it should actually have been Timman and not Petrosian playing against him, since Petrosian's win over the Yugoslav Boris Ivkov in the final round of the interzonal tournament in Rio de Janeiro 1979 had been achieved by improper means. It was only this victory which had enabled Petrosian to catch Timman in the classification and thus qualify with Portisch and H\u00fcbner for the candidates' matches.\n\nKortchnoi feared for his life in Velden and demanded the installation of a sheet of armoured glass between the players and the spectators. His second Yasser Seirawan remembered that Kortchnoi then selflessly offered him his comfortable bed in his suite while he himself preferred a couch. Only later did Seirawan realise that in case the Soviets wanted to kill him in his sleep, Kortchnoi preferred not to sleep in his own bed. Petrosian himself accused arbiter Harry Golombek of being partisan and biased. Kortchnoi won the quarter-final against Petrosian by 5\u00bd:3\u00bd.\n\nThe semi-final against Lev Polugaevsky in August 1980 was closer. This was played in a booth of bullet-proof glass in Buenos Aires. After 12 games the score was 6:6. The match went into extra-time. Before game 14 Kortchnoi read an Argentinian newspaper, in which a game from the second semi-final being played at the same time in Abano Terme between Robert H\u00fcbner and Lajos Portisch was printed. Portisch had come up with an interesting novelty in it. Kortchnoi employed the same novelty against Polugaevsky and won the 14th game for a final score of 7\u00bd:6\u00bd.\n\nIn the final in Merano Kortchnoi met Robert H\u00fcbner. The match was to be for 16 games. After six games H\u00fcbner was leading by 3\u00bd:2\u00bd, but in the seventh game he blundered away a whole rook. Finally H\u00fcbner resigned the match early at a score of 3\u00bd:4\u00bd. Two adjourned games, the ninth and the tenth games, were left unfinished. This meant that Kortchnoi was again the challenger to Karpov.\n\nWhen in 1976 Kortchnoi fled the USSR, he left his wife Bella and his son Igor behind. Public appeals to the Soviet leadership to allow his family to leave were unsuccessful. Instead his son was called up into the army and Kortchnoi was afraid that he would be badly treated there. Originally the match between Karpov and Kortchnoi should have begun on the 19th September 1981, but Kortchnoi made use of the public interest to draw attention to the plight of his family and refused to appear as long as his family was kept prisoner in the USSR.\n\nThe new FIDE president Olafsson then delayed the match for a month in the hope that the Soviets would allow Kortchnoi's family to leave in the name of human rights. At a FIDE congress in Atlanta the Soviets protested irately against Olafsson's decision. Since the organisers in Merano were not yet ready to stage the match on the arranged date, finally the 1st October 1981 was fixed as a new starting date. In the meantime, Kortchnoi's son had been sentenced to two and a half years in a labour camp on account of his refusal to do military service.\n\nThe match, again to six wins, finally lasted until the 19th November 1981. The prize fund was 800 000 Swiss francs, which had been raised amongst others by the Savings Bank of Merano as the main sponsor. Merano had been chosen to organise the match thanks to its 'chess working group' around organiser Siegfried Unterberger rather than Las Palmas or Reykjavik. The arbiters of the match were Paul Klein (Ecuador), Gertrude Wagner (Graz, Austria) and Gudmundur Arnlaugsson (Iceland). Svetozar Gligoric (Yugoslavia), Lodewijk Prins (Netherlands) and Alfred Kinzel (Germany) formed the appeals committee.\n\nOriginally Lothar Schmid had been expected instead of Paul Klein as chief arbiter. The grandmaster from Bamberg had already been in charge of the tricky WCh matches of 1972 and 1978. But apparently objections were raised by Kortchnoi's delegation. They had not been in agreement with certain decisions by the German grandmaster during his spell as arbiter in Baguio City.\n\nKortchnoi and his team \u2013 as well as Petra Leeuwerik, Yasser Seirawan and Lev Gutman there were again Michael Stean and in addition as press officer the Russo-Polish chess master and writer Emanuel Stein \u2013 had taken lodging in the Merano spa hotel Palace. The leader of Kortchnoi's delegation was the Swiss lawyer Alban Brodbeck.\n\nKarpov officially lived with his two official seconds Igor Zaitsev and Yuri Balashov and delegation leader Viktor Baturinsky, a colonel and a lawyer, in the Hotel Ritz Stefanie, but also had available for himself and his team a three-storeyed villa with swimming pool. Also present were Mikhail Tal and Lev Polugaevsky. They were officially accredited as journalists but of course provided help for Karpov. In addition Efim Geller and Rafael Vaganian belonged to Karpov's team. Kortchnoi mentioned, moreover, Evgeny Vasiukov as another trainer of Karpov. Igor Zaitsev, Karpov's 'senior trainer', later reported that Karpov was at least from time to time also supported by Elizbar Ubilava, Adrian Mikhalchishin, Vitaly Tseshkovsky, Albert Kapengut, Rudolf Kimelfeld, S. Sepioshkin, Rozenberg and Georgi Borissenko. Semyon Furman, previously for many years the trainer and paternal friend of Karpov, had died in 1978 and was no longer there. Alexander Roshal took on the role as press officer for the Karpov delegation.\n\nBefore travelling to Merano, Karpov had scheduled a final training session with his trainers in Plavinas, a small seaside resort in Latvia. In the Soviet delegation in Merano there was also a series of persons who did not necessarily have anything to do with chess. Before the start of the match the Soviets had sent the organisers a catalogue with 70 demands. In Moscow there was above all interest in the crime rates in Merano, since they were afraid of kidnappings.\n\nUnlike in the match in Baguio City, this time the Soviets had no objections to Kortchnoi playing under the Swiss flag. Kortchnoi may have been living in Switzerland, but he was not a Swiss citizen and so actually had no formal claim to having the Swiss flag placed beside him on the table.\n\nThis time, instead of Viktor Zukhar it was Professor Modest Kabanov who had taken on the task of the psychological care of the World Champion. Kortchnoi had strengthened his two mental supporters from the Ananda-Marga with another person. This time, however, the Soviets took no notice of it.\n\nThe excitement among the approx. 150 journalists present at this point \u2013 all in all 400 journalists had taken out accreditation \u2013 was taken care of before the start of the match this time by an article from _Sovietski Sport_ , of which excerpts were disseminated by the _TASS_ press agency. In the article it was said that Kortchnoi's wife Bella had not actually made an application for an exit visa. In fact only one such application was known from the past when she had requested a visa to visit an uncle of Kortchnoi's in Israel. It had been refused. In addition Kortchnoi had been trying to get a divorce from his wife. He had not officially applied for an exit visa for his wife until just before the WCh. He had also been the one who had talked his son into refusing to do his military service. All the excitement about Kortchnoi's supposedly imprisoned family was now presented according to the Soviet account as having been made up by the WCh challenger.\n\nKortchnoi, who started the match with the white pieces, immediately lost the first game and was also defeated in game number two. The organisers, among them the sponsor and Merano hotel magnate Eisenkeil, feared a rapid end to the match, which they had of course wished to have as a spectacle over several weeks.\n\nAfter the second defeat Kortchnoi, who felt he was being bugged by the KGB, left the hotel and was taken by Petra Leeuwerik to a secret location. As became known later, Kortchnoi moved with the Ananda-Marga disciple Victoria Shepherd, nicknamed 'Didi', to temporary quarters in the nearby village of Algund and practised asceticism with meditation and uncooked vegetarian food. After a draw in the third game, despite his asceticism Kortchnoi also lost the fourth game, which like the others was played in front of approx. 500 spectators in the congress centre of Merano.\n\nKortchnoi himself and his seconds found the challenger's weak play puzzling and so Kortchnoi continued to pursue his conspiracy theories and said, for example, that Karpov had arrived with 40 specialists and three containers \u2013 in a report of the Italian Chess Federation it was even suggested eight containers \u2013 full of secret apparatus. Not even Karpov's wife Irina was allowed into the villa occupied by the Soviets, according to Kortchnoi. After the match the equipment was supposedly removed by night in lorries.\n\nKortchnoi was convinced that the Soviet specialists sitting among the first rows of the spectators, were able not only to record with their equipment every sound and remark, but also to measure his blood pressure and his physical state of health. He suspected that Karpov of having hidden in his hair earphones, through which his trainers could give him hints during the games. The head of the organising committee reported that with their equipment the Soviet delegation had been monitoring noise pollution, radiation pollution, water quality and climatic conditions. The containers of the Soviet delegation had supposedly contained as well as technical equipment Soviet tinned food and apparently also 7000 books. After the fourth game, moreover, Kortchnoi had felt unwell and he also believed the KGB to be responsible for that. He conjectured that KGB agents had exposed him to radiation or tried in some other way to influence his health. Moreover his press officer Emanuel Stein had surprised some Soviet citizens doing something in the hotel rooms of the Kortchnoi delegation and had been stunned by them.\n\nBut in the meantime, open strife had also broken out within Kortchnoi's team. Petra Leeuwerik, the secret delegation leader, accused 'Didi' of having lastingly weakened Kortchnoi with her asceticism. Also in the meanwhile the former fianc\u00e9e of Kortchnoi's son Igor, Natasha Pesikova \u2013 the former Soviet citizen had since been living in exile in the USA \u2013 had arrived in Merano and was causing additional unrest. Tal quipped: 'The board is the only place where he can get some rest.'\n\nKortchnoi won the sixth game, Karpov struck back in the ninth game. The challenger's nervous tension had now got so great that he openly insulted Karpov during the eighth and ninth games and had to be warned several times by the arbiter. In the following games too, there were insults on Kortchnoi's side and mutual protests to arbiter Klein. When in game 12 Kortchnoi felt disturbed by Karpov's rocking in his chair, he grumbled at the World Champion with the words: 'Stop it, you little worm!' After the game, Baturinsky handed Gligoric, the president of the disputes committee, a formal protest from the Soviet delegation. After the checking of the facts of the case, Kortchnoi was officially warned and was threatened with a fine of 12000 Swiss francs should there be a repetition.\n\nKortchnoi caught up in the 13th game, but in game 14 Karpov got his fifth victory. The title defender now only required one win, which he got in game 18.\n\n **Karpov \u2013 Kortchnoi**\n\nMerano, 18th game  \n19th November 1981  \nRuy Lopez (C80)\n\n**1.e4 e5 2. \u2658f3 \u2658c6 3.\u2657b5 a6 4.\u2657a4 \u2658f6 5.0-0 \u2658xe4 6.d4 b5 7.\u2657b3 d5 8.dxe5 \u2657e6**\n\nIn the first World Championship between Karpov and Kortchnoi in 1978, the Open Variation of the Ruy Lopez was up for debate no less than eight times. In this match another four times. Total score 4:2 for Karpov with six draws.\n\n**9. \u2658bd2**\n\nThe alternative moves 9.\u2655e2 and 9.c3 were also used.\n\n**9... \u2658c5**\n\nKortchnoi always preferred this move after 9.\u2658bd2 compared to 9...\u2657e7 or 9...\u2657c5.\n\n**10.c3 d4**\n\nAnother plan is 10...\u2657e7 11.\u2657c2 \u2657g4 12.\u2656e1 \u2655d7.\n\n**11. \u2657xe6 \u2658xe6 12.cxd4 \u2658cxd4 13.a4!**\n\nA novelty by Karpov. The idea came from Polugaevsky. In two previous games Karpov had played 13.\u2658e4. This idea was checked out by Efim Geller and Yuri Razuvaev in Moscow after the head of the Committee for Sports Viktor Ivonin had tasked them with finding for Karpov a novelty in the Open Ruy Lopez.\n\n**13... \u2657e7**\n\nLater 13...\u2657c5, 13...\u2656b8 and also 13... \u2657b4 were tried as replies.\n\n**14. \u2658xd4 \u2658xd4**\n\nPraxis later showed that 14...\u2655xd4 did not offer Black equality either.\n\n**15. \u2658e4**\n\n'This thematic move secures a solid advantage for White with the queens on the board, but after an exchange of queens too.' (Karpov)\n\n**15... \u2658e6**\n\nIn view of the threat 16.axb5 axb5 16.\u2656xa8 Black's knight must give up its position in the centre. However, the knight is not safe on e6 either. Kasparov recommended 15...0-0 16.axb5 \u2658xb5 17.\u2657e3 \u2655c8 as better.\n\n**16. \u2657e3 0-0 17.f4 \u2655xd1**\n\nOther moves are not better: 17...g6 18.\u2655f3 followed by 19.\u2656ad1; 17...f5 18.\u2655b3 (18.exf6 \u2655xd1 19.\u2656fxd1) 18...\u2655c8 19.\u2658c3 c6 20.\u2656fc1 in each case with advantage to White.\n\n**18. \u2656fxd1**\n\n'The time used (only 10 minutes for Karpov compared to 95 minutes for Kortchnoi) speaks volumes and testifies once more to the poor condition of the challenger.' (Hecht)\n\n**18... \u2656fb8?**\n\nBlack should not have allowed the rook to penetrate to the 7th rank. A better way was 18...f5!? 19.exf6 \u2657xf6 (19...gxf6 20.f5 \u2658g7 21.g4 h5, Hecht) 20.\u2658xf6+ gxf6 21.f5 \u2658g7 22.g4 h5 23.h3 \u2656fd8 and Black can still fight for a draw. 18...\u2656ad8!?.\n\n**19. \u2656d7 \u2657f8**\n\n'19...\u2657d8 was more accurate. But after 20.a5 \u2658f8 21.\u2656d3 the black position is anything but pleasant.' (Karpov)\n\n**20.f5 \u2658d8 21.a5**\n\nAnd the planned black counterplay on the queenside has run its course. White first scorns to win the pawn on c7 and plays for control. He has an enormous advantage in space and dominates the centre.\n\n**21... \u2658c6 22.e6 fxe6 23.f6 \u2658e5**\n\n23...\u2656d8 is no better: 24.f7+ \u2654h8 25.\u2656xc7 or 23...\u2656c8 24.\u2656c1 \u2658xa5? 25.\u2657d4 in each case with superiority to White.\n\n**24. \u2656xc7 \u2656c8 25.\u2656ac1 \u2656xc7 26.\u2656xc7 \u2656d8 27.h3**\n\n'There is no reason for haste. The move secures a retreat square for the white king'. (Karpov)\n\n**27...h6**\n\nBlack is running out of moves: 27...\u2656d7 28.\u2656c8 \u2656d1+ 29.\u2654h2 \u2658d7 30.\u2658g5 gxf6 31.\u2658xe6 \u2654f7 32.\u2658xf8 \u2658xf8 33.\u2656a8+\u2013.\n\n**28. \u2656a7 \u2658c4**\n\nAfter 28...\u2656d1+ White wins as follows: 29.\u2654f2 (but not 29.\u2654h2? \u2656e1 with counterplay) 29...\u2656b1 30.\u2657d4 \u2658c6 31.f7+ \u2654h7 32.\u2656a8 \u2658xd4 33.\u2656xf8 \u2656xb2+ 34.\u2654g1 \u2656b1+ 35.\u2654h2 \u2656f1 36.\u2658d6 b4 37.\u2656a8 and the passed pawn costs material.\n\n**29. \u2657b6 \u2656b8**\n\n29...\u2658xb6 30.axb6 \u2656b8 31.\u2654f2 a5 (31...\u2656xb6 32.f7+ \u2654h7 33.\u2656a8 and White wins the bishop) 32.\u2654e3 g6 33.b7+\u2013.\n\n**30. \u2657c5**\n\nAnother good move was 30.f7+ \u2654h7 31.\u2657c7 \u2656c8 32.b3 \u2658e3 33.\u2658d6+\u2013.\n\n**30... \u2657xc5+ 31.\u2658xc5 gxf6 32.b4 \u2656d8 33.\u2656xa6 \u2654f7 34.\u2656a7+ \u2654g6 35.\u2656d7 \u2656e8 36.a6 \u2656a8 37.\u2656b7 \u2654f5 38.\u2656xb5 \u2654e5 39.\u2656b7 \u2654d5 40.\u2656f7 f5 41.\u2656f6**\n\nThe game was adjourned here. Kortchnoi wrote down 41...e5 as his sealed move, but resigned the game on the 20th November without resuming it. That was also the loss of the WCh match. After 41...e5 White has a plethora of winning moves to choose from, e.g.: 42.\u2658a4, 42.\u2658d7 or 42.\u2656xf5+\u2013.\n\nAccording to the reminiscences of Baturinsky, Kortchnoi informed arbiter Klein about his resignation in a handwritten note with the following content: 'I inform you that I am resigning the 18th game and the whole match and I congratulate Karpov and the whole Soviet delegation for their excellent electronic technology. Kortchnoi.' When Baturinsky later requested a photocopy of the note, Klein, however, denied its existence.\n\nOf the total prize money, Karpov received 500 000 Swiss francs, Kortchnoi 300000 Swiss francs. The organisers were content with a total of more than 55000 newspaper articles, which for some time made Merano the focus of world-wide attention. During the match, the composer of musicals Tim Rice visited Merano and was inspired by events there to write his musical _Chess_. The music was composed by the Swedish pop group ABBA.\n\nIn the following candidates' cycle Kortchnoi was eliminated in 1983 in the semi-final by Kasparov. In the dispute between FIDE and the Russian Chess Federation about the venue \u2013 the USSR did not agree with the award of the match to Pasadena \u2013 the match first crashed. After the Soviets belatedly fell into line, Kortchnoi declared himself ready to play the match against Kasparov, though he had actually won it without a contest. It took place at the end of the year in London. In return the Soviets lifted their boycott against Kortchnoi and paid him compensation.\n\nIn the next WCh cycle Kortchnoi also qualified for the candidates' matches with his victory in the interzonal tournament in Zagreb. However, he was defeated in the last 16 by the Icelander Johann Hjartarson. After taking sixth place in the interzonal tournament in Manila 1990 during the next WCh cycle Kortchnoi again reached the candidates' matches, but was defeated in the quarter-final by Jan Timman.\n\nKortchnoi continue to remain one of the most active tournament players, participating in countless individual and team events. In principle he accepted every invitation, giving as a reason: 'If I refuse an invitation, then perhaps I will not be invited the next time.' Until a great age he remained successful in tournaments and not until 2006, at the age of 75, did he accept his 'senior' status and won the World Championship for seniors at his first attempt. During his career he took part from 1960 on in seventeen Chess Olympiads, six times for the USSR and eleven times for Switzerland. In his new country he played for SG Zurich, the oldest still existing chess club in the world. At the end of December 2012 Kortchnoi had to go into a Swiss clinic on account of the consequences of a stroke which he had suffered during 2012 and be treated for heart problems. Thus it was only then that he de facto withdrew from tournament chess. Kortchnoi wrote several autobiographical chess books.\n**31. The abandoned match**\n\n**The World Championship 1984\/85:  \n_Anatoly Karpov against Garry Kasparov_**\n\nGarry Kasparov was born on the 13th April 1963 in Baku. Even the exact time is known: it was a quarter of an hour before midnight. His mother Klara Kasparova was of Armenian descent, his father Kim Weinstein came from a Jewish family with a great musical tradition. Kasparov's grandmother Olga Yulevna was a music teacher, his grandfather Moissei Weinstein, who died in the summer of 1963, was a conductor, composer and violinist. Kasparov's father, on the other hand, was an electrical engineer, whereas the latter's brother and Kasparov's uncle, Leonid Weinstein, kept up the musical tradition and was also a wellknown musician in Azerbaijan.\n\nGarry Kasparov (born in 1963)\n\nKasparov's parents had played some chess in their youth and loved solving the studies which were printed in a local newspaper. At the age of five, Kasparov, who had learned to read at a young age, followed his parents' attempts to solve an endgame study on a chess board. To their surprise he then suggested the correct solution, which motivated his parents to show him the game in more depth.\n\nSince Kasparov already had an excellent memory as a child, he found it easy to learn. Whilst the grandmother advocated a musical education for her grandson, Kasparov's father pleaded for him to become intensively involved with chess. After he came back from work he would regularly train with his son in the evenings. In the summer of 1970 the father fell ill with a cancer of the lymph glands and died the following year at the age of only 39. His mother Klara thereupon moved with her son to her own parents.\n\nKasparov was making good progress in chess and at the age of nine played a simultaneous display against oil workers whose boss was his grandfather on his mother's side. Soon afterwards, after an operation on his appendix, Kasparov gave a blindfold simultaneous display against the doctors in the hospital in which he was being treated. At the age of ten Kasparov was diagnosed with rheumatic carditis, also known as 'rheumatic fever'. For that reason he had to take antibiotics regularly until he was 15.\n\nKasparov's first chess trainer was his uncle Konstantin Grigorian, to whom his father's sister Nelly was married. In 1970 he took Kasparov to the local pioneer palace for further education. His first lessons there were from Rostislav Korsunsky, a youth whom he knew from the neighbourhood. Amongst the talented chess players in Baku at that time were Elmar Magerramov, Mikhail Shur, Alexander Avshalumov, Elena Glatz and Boris Sheinin, whose son Teimur Radjabov would later advance to the top of world chess. Until then grandmaster Vladimir Bagirov and the women's WCh challenger Tatiana Satulovskaya had been the best known players in the city.\n\nKasparov's first proper chess trainer was Oleg Provorotsky, who quickly recognised Kasparov's talent. In 1972 already Kasparov reached the final tournament for the adults blitz championship of Baku and made it into the final, moreover still under the name of his father, Weinstein. It was only later that he would take the name of his mother. In January 1973 Kasparov shared third place in the Baku junior championship, where he was discovered by Alexander Shakarov, the junior trainer of Azerbaijan, who then put him into his country's junior team.\n\nIn August 1973, on the initiative of Alexander Nikitin Kasparov received an invitation to the Botvinnik Chess School in Dubna. After a conversation with then 62 year old 'patriarch' Kasparov was accepted. Botvinnik was especially impressed by his new pupil when the latter pointed out that Fischer's analysis of his game against Botvinnik in Varna 1962 was wrong and that in reality there was no way to show a win. Botvinnik soon after applied to the Committee for Sports and recommended Kasparov for special support. In 1974 Kasparov had the opportunity to follow live a game in the candidates' finals between Anatoly Karpov and Viktor Kortchnoi in Moscow. On his return from the Botvinnik Chess School he stopped off in Moscow with his trainer Nikitin and sat amongst the spectators during the 21st game of the match. In 1975 Kasparov took part in the U18 national championship at the age of 11 and took seventh place out of 36 participants.\n\nIn August 1975 the family council decided that the boy who had until then borne the name of his father Weinstein, should from then on take his mother's name. The initiative came from Alexander Nikitin, who feared that as his prot\u00e9g\u00e9 rose to the top, the name Weinstein would run into the anti-Semitism prevalent everywhere in the USSR, especially at the higher levels.\n\nIn November 1975 Kasparov met Karpov and Kortchnoi in person for the first time. After a junior team championship the two top players gave simultaneous displays against the young people. It was only after long resistance that the 12 year old Kasparov was defeated by the then 24 year old Karpov. Against Kortchnoi Kasparov managed a draw.\n\nIn 1976 Kasparov won the USSR junior championship. In the following year he repeated his success with 8\u00bd out of 9 and two points ahead of the rest of the field. In the same year he was third in the World Cadet Championship in Cagnes-sur-Mer behind Jon Arnason and Jay Whitehead.\n\nThe name of the American Whitehead has almost been forgotten today. Jay Whitehead, only two years older than Kasparov, was a very talented player in his youth. Later he became a member of the Hare Krishna movement and would often turn up at tournaments (he played mostly in the USA) in the typical clothing of that Indian sect. In the late 1980s Whitehead, however, more or less gave up tournament chess. He died of cancer in 2011 only a few days before his 50th birthday. Jon Arnason was not only the first Icelander to win a World Champion title in chess, but even the first Icelander to become World Champion in any sport. Then only 12 years old, Nigel Short also played in the cadet championship, but his play waned as the tournament progressed and he did not get a top placing.\n\nIn 1978 Kasparov qualified for the first time for the USSR championship in Tbilisi, being at 15 the youngest participant, and took ninth place. Tseshkovsky and Tal won the tournament. In the following year, thanks to a recommendation by Botvinnik, Kasparov was sent to his first international tournament in Banja Luka (Yugoslavia). To the surprise of all, especially of the western chess world which had so far not heard much about Kasparov's successes in the USSR, the untitled 16 year old without an Elo rating won the strong tournament with two points of a lead. At the end of 1979 Kasparov came in third in the USSR championship behind Geller and Artur Jussupow. In April 1980 Kasparov took an international tournament in Baku ahead of Alexander Beliavsky, Eduard Gufeld and Adrian Mikhalchishin. In August 1980 Kasparov finally won the World Junior Championship in Dortmund with 10\u00bd out of 13 ahead of Nigel Short (9), Ivan Morovic Fernandez, Adrian Negulescu and Klaus Bischoff (8\u00bd). In the same year he represented his country for the first time in the Chess Olympiad in Malta.\n\nIn January 1981 Kasparov was already number three in Russia with an Elo rating of 2625 and number six in the world. Karpov led both lists with 2690 Elo. In April 1981 Kasparov was invited to the 'Tournament of the Stars', which was played in the Moscow International Trade Centre, and shared second place with Polugaevsky and Smyslov. It was won by Karpov. At the supertournament in Tilburg in October 1981 Kasparov only took eighth place. But shortly afterwards he won, jointly with Lev Psakhis, the USSR championship for the first time in Frunze. In May 1982 he was victorious in the international tournament in Bugojno (Yugoslavia). In September of the same year there followed victory in the interzonal tournament in Moscow, which brought Kasparov the right to take part in the candidates' tournaments. Shortly afterwards, he played in Lucerne in his second Chess Olympiad.\n\nThere are two different ways of recounting the circumstances surrounding Kasparov's rise to the top. Kasparov complained later in numerous books about how greatly he had been handicapped by the leaders of the Soviet federation, which was supposedly under Karpov's control. According to Karpov, however, the young man from Baku was supported as nobody else before him had been. At 13 he had already had Alexander Shakarov provided for him as a trainer and paid by the Azeri federation. At the same time Kasparov was already receiving a stipend of 100 roubles, arranged for, according to Nikitin, by Botvinnik. His mother Klara also received a stipend. In the following year Kasparov already had two full-time trainers at his disposition.\n\nAs soon as he reached 18 Kasparov joined the Communist party. He was, moreover, proud of being a member of the Azeri Komsomol central committee. The Komsomol was the youth organisation of the Communist Party of the Soviet Union. In 1981 Kasparov came into contact with Viktor Litvinov, the head of the Azeri KGB department responsible for sport. Through Litvinov Kasparov reached Heydar Aliyev, the head of the Communist Party of Azerbaijan, who later, after the collapse of the USSR and the declaration of independence, became the first head of state of Azerbaijan. But even in the USSR Aliyev was in an influential position.\n\nDuring the Chess Olympiad the pairings were drawn for the forthcoming candidates' matches. In the quarter-final Kasparov met Alexander Beliavsky, one of the strongest opponents. The match took place from 26th February to 20th March in the conference room of the 22 storey high Moscow Sport Hotel on Leninsky Prospect and was set for 10 games. Kasparov worked with a staff of trainers, namely Alexander Nikitin, Alexander Shakarov, Georgy Timoshenko and Evgeny Vladimirov. For the match against Beliavsky Valery Chekhov was also one of Kasparov's seconds. Kasparov won the second game, but Beliavsky equalised in the fourth game. The fifth game again went to Kasparov, who then also won the eighth and ninth games and with them the match.\n\nKasparov now met Viktor Kortchnoi, who had won in the first round of the candidates against Lajos Portisch. The staging of this match led to one of the numerous scandals in the history of FIDE. The starting date was set for the 1st August 1983. The bids to stage the match had come from Rotterdam with a prize fund of 100000 Swiss francs, Las Palmas with a 25000 Swiss francs prize fund and Pasadena with 100000 Swiss francs. According to FIDE regulations for the candidates' matches, the players could first express their preferences. For the choice of venue FIDE would then first search for agreement or a compromise. Kortchnoi gave only Rotterdam as a choice. On Kasparov's list, which he handed on to the Soviet Committee for Sports, the order was Rotterdam and Las Palmas, 'since Las Palmas would not be chosen in any case on account of the low prize fund, but since they wanted to support it on political grounds', was the way it was explained to him. In any case there were also rumours that Kasparov's mentor Botvinnik had spoken against Pasadena for whatever reason.\n\nShould the players not be able to agree on a place, the FIDE president had according to statute the right to decide on the venue. This was now the case and Campomanes selected Pasadena. Kortchnoi agreed with the choice, but the Soviet Committee for Sports categorically refused Pasadena as a venue. The official reason given was that the security of the Soviet delegation could not be guaranteed there, because Pasadena and Los Angeles were at that time forbidden to Soviet diplomats. This on its own was unbelievable, because the Soviet swimmer Vladimir Salnikov had just broken a record over 800 metres quite close to Pasadena \u2013 so therefore he was allowed to appear there.\n\nKasparov was called to Moscow, where he was informed in the propaganda ministry that 'they would not allow themselves to be dictated to concerning the conditions for sporting encounters' and that the match would not take place in the USA. Kasparov was instructed to reject Pasadena as a venue. It is possible that at that time the boycott of the Olympic Games of Los Angeles 1984 \u2013 the Soviets' reply to the West's boycott of the Moscow Olympics of 1980 \u2013 had already been decided and the boycott of Pasadena as a venue was perhaps a preface to that.\n\nOn the 12th July 1983 there was a meeting in the building of the Committee for Sports, at which in addition to Kasparov and Campomanes Marat Gramov (president of the Soviet Olympic committee), Viktor Ivonin (deputy sports minister), Nikolai Krogius, Vitaly Sevastianov (president of the Soviet Chess Federation 1977-1986), Anatoly Karpov and Vassily Smyslov were present. The Soviets were prepared to allow Smyslov to play in Abu Dhabi, but wanted the match between Kortchnoi and Kasparov to be moved to Rotterdam. During the heated discussion Campomanes insisted on his right as FIDE president to decide the venue and on his decision in favour of Pasadena, and criticised Kasparov for not putting Rotterdam in first place on his list.\n\nAt this point, according to his own account, there grew in Kasparov the idea that he could possibly become the victim of an intrigue. The theory that Anatoly Karpov or higher powers who supported the World Champion were pulling the strings in the background was also backed by Viktor Kortchnoi. 'They would rather see me as World Champion than Kasparov', he is supposed to have said.\n\nThe reason for the choice of Pasadena as the venue by Campomanes could also lie somewhere quite different. The bid was also linked to a 'contribution' of 40000 Swiss francs to Campomanes' 'chess development programme'.\n\nCampomanes soon afterwards declared publicly that Kasparov would have lost the match against Kortchnoi to a walkover should he not appear in Pasadena on the 1st August. Shortly thereafter he postponed the start of the match by five days till the 6th August 1983. When that was reached Kortchnoi sat in the City College in Pasadena and played 1.d4. After the hour's grace for a player to appear which was then usual, Kortchnoi was declared the winner of the match since Kasparov had, as expected, not turned up.\n\nFor the match between Vassily Smyslov and Zoltan Ribli there had been no bids at the start. But shortly before the planned start of the match Campomanes announced Abu Dhabi as the venue, with a prize fund of 25 000 dollars. On the 31st July the organisers in Abu Dhabi, however, withdrew their bid to stage the match because there had been absolutely no reaction from the Soviet Chess Federation to the invitation to Abu Dhabi. Thereupon Zoltan Ribli was announced on the 9th August by Campomanes as the victor of a non-contested match. Both semi-final candidates' matches were thus decided without a contest.\n\nFIDE found itself in a serious crisis, also because the Soviet Committee for Sports were in the meantime seriously considering breaking away from FIDE and organising their own WCh match, in which Karpov would play against a Soviet challenger. At the high point of the crisis Kasparov visited Karpov in the latter's flat on Vspolny Pereulok and sought help. At that time relations between Karpov and Kasparov were obviously still intact. Kasparov was then invited at short notice to a supertournament in Niksic (Yugoslavia) and after an impressive victory with two points of a lead had underlined his claim as a possible challenger to Karpov. On the initiative of Kasparov the participants in the tournament signed a petition to FIDE demanding that the semi-final between Kasparov and Kortchnoi be arranged again.\n\nAt the same time Kasparov started negotiations with Kortchnoi, who declared himself prepared to renounce his right to a walkover and to play a re-arranged match \u2013 under certain conditions. At the end of the tournament Kortchnoi travelled to Niksic to discuss the details. His condition was an end to the tournament boycott imposed on him by the USSR and east block since his flight in 1976. On the 16th September Kasparov and Kortchnoi signed a common declaration, in which they demanded that the match be re-arranged.\n\nWork was also in progress eagerly in the wings. Heydar Aliyev, Azerbaijan's communist party boss, took position for the match still taking place. Kasparov had in addition written to Brezhnev successor Yuri Andropov and begged for support. According to Yuri Averbakh there was within the party a struggle between the supporters of Karpov, led by Mikhail Zimianin, formerly editor of Pravda and now head of propaganda, and Heydar Aliyev as well as the council of ministers, who supported Kasparov. Aliyev won this power struggle according to Averbakh. In negotiations with the Soviet federation Raymond Keene offered to organise a re-run of both previously uncontested matches in London. At the FIDE congress in Manila in October 1983 a letter of apology from the USSR federation was read out and both candidates' matches re-arranged for the autumn of 1983 in London. Zoltan Ribli, Vassily Smyslov's opponent, also agreed to this. Moreover, the USSR federation paid the organisers in Pasadena 50000 dollars and FIDE 100000 dollars for damages and costs. In addition the Soviet federation is supposed to have paid Viktor Kortchnoi 30 000 dollars for having agreed to the re-arrangement of his match against Kasparov.\n\nThe London offer to FIDE for the organisation of both semi-finals included, for example, the following financial details: 130000 Swiss francs as prize fund for the two matches, another 20000 Swiss francs to be used either to increase the prize fund or for other purposes, a further 50 000 Swiss francs for FIDE and 40000 Swiss francs for the FIDE development programme, though to be precise about the latter case it was to be books to the value of the sum mentioned. Campomanes decided on the 27th October in favour of the offer from London and against a bid from the Dutch federation, which was slightly less than the London bid.\n\nThe Kasparov against Kortchnoi match took place in December 1983 and began badly for Kasparov. He lost the first game, and that with the white pieces. In the next four games Kasparov could not do better than draw, with his play as White looking somewhat less than full-blooded after the defeat in the opening game. Finally Kasparov won the sixth and seventh games to take the lead. The ninth and eleventh games also went to Kasparov, who had now emphatically won the match by 7:4. This was not the last candidates' match for the then already 52 year old Kortchnoi. In 1987 Kortchnoi won the interzonal tournament of Zagreb, thus again qualifying for the candidates' round, but then was eliminated by Johann Hjartarson in the last 16.\n\nIf the match against Kortchnoi had already been a war of the generations for the 20 year old Kasparov, then in the candidates' final Kasparov was going to meet someone who was even a decade older \u2013 Vassily Smyslov (63). The World Champion of 1957 had previously won his candidates' quarter-final in Velden against Robert H\u00fcbner by the drawing of lots and the other semi-final in London by 6\u00bd:4\u00bd against Zoltan Ribli.\n\nThe candidates' final was finally played in March and April in the White Hall of the Vilnius Museum of Art (Lithuania). Just after the match Kasparov celebrated his 21st birthday, making him exactly a third of the age of his opponent. For Smyslov, who had won his title six years before Kasparov's birth, it was a sensational success to have made it as far as the final. Kasparov managed to win the third and fourth games as well as the ninth and twelfth for a final score of 8\u00bd:4\u00bd. This made Kasparov the challenger of World Champion Karpov. In the meanwhile he had even overtaken the World Champion in the Elo list (January 1984: 2715 compared to 2705).\n\nThe prospect that Kasparov would take over from Karpov attracted for the subsequent WCh match about 500 journalists. The match began on the 9th September 1984. To prepare Karpov had once more retired to the Latvian spa Pliavinas with his trainers. However, Karpov was not a particularly hard worker and preferred playing blitz and cards against his trainers. During the match Karpov then lived with his team in the former dacha of Marshall Ivan Koniev, on an army site 30 kilometres outside of Moscow. Since Karpov was a member of the army sports club he enjoyed the support of the army. Kasparov was on the other hand a member of Spartak Moscow, the club for Soviet trade.\n\nKarpov's team included Sergey Makarichev, Yuri Balashov, Efim Geller, Tamas Georgadze, Lev Polugaevsky and Adrian Mikhalchishin and as the leader Igor Zaitsev. All the analyses of the other trainers were kept by Zaitsev in a special safe. Rafael Vaganian, who was a friend of Karpov, often dropped by. In Moscow's Hotel Peking Elizbar Ubilava and Mikhail Podgaets were also standing by. These two had specific analytical tasks to fulfil. Only some of the trainers officially belonged to the staff, the others were 'unofficial'. Mikhalchishin, for example, was officially correspondent for a daily newspaper in Lviv.\n\nCare was taken to ensure that it did not become known who all were working for Karpov. On the car trip to the venue the secret additional trainers always lay flat on the rear seats as soon as the car reached the House of The Unions so as not to be seen with Karpov. Polugaevsky did not like supporting Karpov against Kasparov and his visits to the training camp were irregular. Later he declared himself to be ill. Nevertheless Karpov's win in a rook ending in the 27th game was due in a large part to Polugaevsky's analyses. The technical leader of the Karpov team was Alexander Bach. He also procured the food, which came from the military base. Sometimes, however, the fare was really monotonous. Once for a whole week they had ox tongue with peas, as Adrian Mikhalchishin later remembered.\n\nThe challenger too could count on a strong team. Kasparov's most prominent trainer was Mikhail Botvinnik, who regularly advised him during the match. In Mikhail Tal another World Champion was among Kasparov's supporters. Elmar Magerramov did a great deal of the work, above all during the preparation in Baku. After the match, however, he and Kasparov went their separate ways.\n\nBoth players, moreover, relied on the help of 'para-psychologists'. In Kasparov's team there was Tofik Dadashev. Mikhalchishin reported that Karpov was supported by several para-psychologists. Their main task consisted above all in neutralising opposing forces. One of them was actually a sexual therapist with his own practice. Karpov's security chief was KGB lieutenant colonel Vladimir Pistchenko, who appeared once a week and looked after things. Pistchenko had for many years been the KGB companion and minder for Karpov for his visits abroad.\n\nOn the 15th February 1985 after 48 games the match was abandoned without a winner at a score of 5:3 (with 40 draws) for Karpov. The circumstances of the abandonment have so far never been completely clarified. The WCh match was played in the emblematic hall of columns in the House of the Unions. In 1924 Lenin had been laid there in his coffin. In 1936 the Moscow international tournament, which was won by Capablanca ahead of Botvinnik and Lasker, had been played there. This was the hall where Max Euwe had in 1975 declared Anatoly Karpov to be World Champion. After two draws Karpov went into the lead with a win in the third game:\n\n **Karpov \u2013 Kasparov**\n\nMoscow, 3rd game  \n17th September 1984  \nSicilian Defence (B44)\n\n**1.e4 c5 2. \u2658f3 e6 3.d4 cxd4 4.\u2658xd4 \u2658c6**\n\nThis setup has acquired the name of the Taimanov Variation.\n\n**5. \u2658b5**\n\nThe threat of 6.\u2658d6+ forces Black to commit himself.\n\n**5...d6 6.c4 \u2658f6 7.\u26581c3 a6 8.\u2658a3**\n\nThe disadvantage of move 5 is that White has to accept this somewhat unharmonious knight placement.\n\n**8... \u2657e7**\n\nIn the later re-run of the match in 1985 Kasparov surprised Karpov with 8...d5.\n\n**9. \u2657e2 0-0 10.0-0 b6 11.\u2657e3 \u2657b7 12.\u2655b3**\n\nThis position was very well known at the time of the game.\n\n**12... \u2658a5!?**\n\nA novelty by the young challenger. Black reacts to the attack on the b6-pawn with a counter-attack on the e4-pawn (and the white queen!). The usual move is 12...\u2658d7.\n\n**13. \u2655xb6**\n\nAfter the petty 13.\u2655c2 Black would certainly have no problems: 13...\u2656c8 etc.\n\n**13... \u2658xe4 14.\u2658xe4 \u2657xe4 15.\u2655xd8 \u2657xd8**\n\nBlack now threatens the unpleasant 16... \u2657f6. But 15...\u2656fxd8 would be followed by 16.\u2657b6 winning the exchange.\n\n**16. \u2656ad1**\n\n**16...d5?**\n\nKasparov thought for 50 minutes and then sacrificed a pawn, to avoid a passive defence. 16...\u2657f6 would now be followed by 17.b3 \u2657b2 18.\u2658b1 \u2658b7 19.f3 \u2657c6 20.\u2657d4 \u2657xd4+ 21.\u2656xd4 with a better position for White. But 16...\u2657e7 17.\u2658b1 \u2658c6 18.\u2658c3 \u2657g6 19.\u2656d2 \u2656ab8 20.\u2656fd1 \u2658b4 seems to be very playable, or 16...\u2656b8 17.b3 \u2657e7 with level chances.\n\n**17.f3**\n\n17.cxd5 \u2657xd5 would of course be great for Black.\n\n**17... \u2657f5 18.cxd5 exd5 19.\u2656xd5 \u2657e6**\n\nThat was the idea. It turns out, however, that here the tactics work in White's favour, amongst other things also because the \u2658a5 is badly posted on the edge of the board.\n\n**20. \u2656d6**\n\nThe a6-pawn is hanging.\n\n**20... \u2657xa2**\n\n'The correct way was 20...\u2657e7 21.\u2656xa6 \u2656xa6 22.\u2657xa6 \u2656b8 23.\u2657d4 \u2658c6 24.\u2657c3 \u2657c5+ 25.\u2654h1 and only then 25...\u2657xa2.' (Baturinsky), but perhaps 23.\u2658b5 is better: 23...\u2658c4 (23...\u2656a8 24.\u2658c7+\u2013; 23...\u2657xa2? 24.\u2656a1) 24.\u2657d4 \u2658xb2 25.\u2657xb2 \u2656b6 26.\u2658c7 \u2656xb2 27.\u2658xe6 fxe6 28.\u2657c4 with an advantage.\n\n**21. \u2656xa6 \u2656b8?**\n\n21...\u2656xa6 22.\u2657xa6 \u2657e7 23.\u2657e2 \u2656b8 24.\u2656c1 \u2657e6 25.\u2658b5 \u2658b3 26.\u2656d1 \u2656c8 27.\u2658c3 g6 and the white advantage is less than in the game.\n\n**22. \u2657c5**\n\n22.\u2656d1!? intending b2-b4.\n\n**22... \u2656e8 23.\u2657b5 \u2656e6 24.b4 \u2658b7**\n\n'Nor is 24...\u2658b3 25.\u2656xe6 fxe6 any better, nor 24...\u2656xa6 25.\u2657xa6 \u2658b3 26.\u2656e1 threatening mate.' (Baturinsky)\n\n**25. \u2657f2 \u2657e7 26.\u2658c2 \u2657d5 27.\u2656d1 \u2657b3 28.\u2656d7 \u2656d8**\n\n'Nor can the position be saved after 28... \u2657xc2 29.\u2656xe6 fxe6 30.\u2656xe7 \u2658d6 31.\u2657d7 \u2656xb4 32.\u2657c5 \u2656b1+ 33.\u2654f2+\u2013.' (Baturinsky)\n\n**29. \u2656xe6 \u2656xd7 30.\u2656e1 \u2656c7 31.\u2657b6**\n\nBlack resigned due to 31...\u2656xc2 32.\u2656xe7 \u2658d6 33.\u2657c5+\u2013.\n\nOn the 26\/27th September after the sixth game it became 2:0; in the subsequent game on the 28th it was already 3:0. On the 5\/6th October Karpov won the next game for a score of 4:0.\n\nKasparov then dropped his 'up and at 'em' style, which was making absolutely no impression on Karpov, played safe and tried to protract the match with short draws. There followed a long series of 17 draws, with some very boring and uneventful games which to the disappointment of the spectators would be finished after an hour's playing time. Then on the 23rd November Karpov also won the 27th game for 5:0. Now an enormous table was set up in Karpov's dacha for the victory celebration, but that turned out to be premature. On the 14th December 1984 in game 32 Kasparov achieved his first win and shortened the lead to 5:1.\n\nThe never-ending match and its two protagonists were becoming a laughing stock, and not only in chess circles. Nevertheless, the long-lasting match left time for other things. Thus Kasparov managed to have a liaison with the well-known and 16 year older Russian actress Marina Neyolova. The relationship lasted until 1987. Because of this connection many Russian actors found an interest in the World Chess Championship and regularly visited the hall of columns.\n\nThe 33rd and 34th games both ended in draws and were still played in the hall of columns. But then on the death of the defence minister of the USSR, Dmitry F. Ustinov, his bier was put on display in the hall of columns and the match was interrupted for a week. It was continued on the 26th December. There followed 12 more draws, the last in the 45th game, on 23rd January 1985. During this series Karpov missed a possible win in the 41st game.\n\nGradually even in the chess community of Moscow amongst spectators and journalists there was more and more criticism of the match which would not end. Despite the long match Anatoly Karpov was still in good condition. His medical adviser Prof. Gershanovich reported to Dr Helmut Pfleger that until that point Karpov had only lost two kilos, though during matches he normally lost up to six kilos. In the meantime the length of the match, on which nobody had been counting, was causing organisational problems on all sides. Thus the visitor visas they had for Moscow were gradually running out for Kasparov's seconds. Karpov's seconds had tournament obligations. It was not appropriate for the hall of columns to be occupied eternally. Several film teams, who wanted to film there, were waiting for the chess match to finally come to an end.\n\nAlso the costs of the organisation were becoming ever greater, day by day and game by game. In the meantime the suggestion was made that the match should be moved from the hall of columns and continued in the Moscow Sport Hotel, which seemed to rather degrade the WCh match. Karpov made use of his good relationships to 'higher circles' and obtained that the WCh match should for the moment at least be continued in the hall of columns. In a letter to the Committee for Sports he promised that the match would be finished by at the latest the end of January. On the 28th January there was another draw. Two days later, however, Kasparov won his second game and shortened the lead to 5:2. After this game the transfer of the match to the Sport Hotel on the outskirts of Moscow was announced. The move took up a week.\n\nIn the meantime FIDE president Campomanes, who was in Dubai, where, in view of the forthcoming 1986 Chess Olympiad, he had been dealing with the future hosts about the status of the players from Israel, hurried to Moscow. He had obviously been phoned by Svetozar Gligoric on behalf of Alfred Kinzel, who spoke no English, and told about the difficult situation in Moscow. Kinzel was the president of the German Chess Federation and had a very good relationship with Karpov, whom he had driven around Germany for numerous tours giving simultaneous displays. At the match in Moscow he had been acting as chairman of the disputes committee, which would have been active had there been disagreements. Campomanes now wanted to achieve a speedy end to the match through negotiations with the players. On the 29th February Kinzel made a statement: 'At the end of January the FIDE president developed the plan of trying to limit the marathon match through an agreement with the two players.' According to Kasparov the idea was 'to bring to a dignified and cheap finish a protracted struggle, that had become nerve-wracking and exhausting not only for Karpov and myself, but for the whole chess world and the general public.'\n\nThe first was to restrict the match to eight more games. If no decision had then been reached, the match should be abandoned. Kasparov was on the other hand against the immediate termination of the match. Both sides were agreed that a newly scheduled match should then be limited to 24 games. There was no consensus as to how the present match and the points scored in it should be evaluated. Campomanes left on the 2nd February and was kept informed by telephone about the progress of the negotiations. Since these had not yet been concluded, the 48th game was played on the 8th February 1985 in the Sport Hotel. Kasparov won again and drew closer with 3:5.\n\nOn the 11th February Campomanes returned to Moscow. Kasparov took a timeout, then Campomanes decreed a pause in the match. On the 15th February Campomanes called a press conference in the Sport Hotel; it was chaired by the representative of the foreign ministry. Karpov was already on his way to his dacha outside of Moscow when he received a call on the car phone and was informed that Campomanes would terminate the match. Karpov turned back and reached the Sport Hotel where the press conference had begun.\n\nSome days previously he had reached agreement with Campomanes about the conditions of the cancellation, but at the press conference Campomanes made public quite different conditions. The FIDE president let it be known that 'the match was finished without any decision having been reached. A new match will be played from scratch \u2013 zero-zero (...), with the agreement of both players (...). This title match (...) has exhausted the physical and perhaps also the mental reserves not only of the participants but of all those who have been involved with the match.'\n\nSuddenly Karpov stood up and explained that he was ready to continue with the match. Kasparov should also be invited to comment. The representative of the foreign ministry interrupted Karpov and said: 'The decision has been taken.' Then Kasparov spoke up angrily and shouted: 'What is this show all about?' The representative of the foreign ministry then declared the press conference over.\n\nCampomanes, however, continued with his explanation. Finally there was a pause in the press conference. Campomanes appeared an hour and a half later and announced: 'The World Champion accepts the decision of the president, the challenger submits to the decision of the president.' According to Alexander Nikitin, two days previously Kasparov had received a call from the match organiser Piotr Demichev, the minister of culture of the USSR, which called on him to agree to the cancellation and not to try to get help from Heydar Aliyev, since in any case that would serve no purpose.\n\nNikitin also reported that the cancellation of the match had been decided in the central committee of the communist party after game 47. After it Demichev had been instructed to bring about the termination of the match. The leader of Kasparov's delegation, Yuri Mamedov, had been previously advised, according to Nikitin, not to get involved in forthcoming events. Campomanes too in an interview with the Soviet chess magazine _64_ gave hints that the instruction to abandon the match had come from official political circles in the Soviet Union. Campomanes claimed that it was only on the day on which the press conference was held that he had decided to break off the match.\n\nThat is probably not true. Alfred Kinzel had already informed the German Chess Federation on the previous day that he would be leaving Moscow the next day since the WCh match would then be over. Towards the end of the match, the general secretary of the communist party Konstantin Chernenko was dying and the political leadership in Moscow certainly had very different things to worry about than a neverending chess match. In Karpov's opinion, Heydar Aliyev, who was responsible for sport in the Politburo, had been involved with the decision to break off the match. And during the match Kasparov had had a direct line to Aliyev.\n\nThe US grandmaster Lev Alburt, who was born in the Soviet Union, was also of the opinion that Kasparov had had the better political contacts with Aliyev and with these would have been absolutely in a position to go for a breaking off of the match, if he had wanted to. Karpov's supporters on the other hand were placed no higher than in the propaganda bureau.\n\nIt was however discovered by the journalist Yuri Vassiliev that at decisive points Karpov and his supporters had had some influence. In Karpov's opinion, Vassiliev had shown himself in his articles to be very taken with Kasparov's play and had in addition dared on one occasion to make an ironic comment about the World Champion. When he travelled to the World Championship as a correspondent for the most widely circulated newspaper in the world _Trud_ (circulation 18 million), he was informed there that he would not receive accreditation, 'because he was not on the list'. It was pointed out to him that Marat Gramov, the head of the Committee for Sports, was behind this decision. When Vassiliev complained about his problem to Kasparov's mother, she pulled the levers which were available to her and Kasparov. A phone call to an aide of Heydar Aliyev was sufficient to arrange accreditation for the journalist immediately. But, according to the explanations of Vladimir Popov in _The KGB plays chess_ , during the match the 11th section of the 5th KGB department of the USSR bugged and recorded all the telephone conversations of Kasparov, his trainers and even his mother, with the approval of Philip Bobkov, the depute director of KGB, and was therefore well informed about Kasparov's preparation.\n\nIn this match a special role was played for Kasparov by the psychologist and 'clairvoyant' Tofik Dadashev. He had apparently been sent to Moscow by the communist party of Azerbaijan to provide psychological support to Kasparov when the score was 0:4. Dadashev won Kasparov's trust and managed to stabilise the challenger psychologically. Even Karpov believed in the powers of the psychologist and later considered the effect he had among the spectators to be the reason for him not having found the courage to play sharply for a win when the score was 5:0, which would perhaps have cost him one or two defeats but probably also gained him the sixth victory. Dadashev also helped Kasparov later during the re-run of the match in Leningrad.\n\nBefore the fourth KK-match in Sevilla in 1987 Karpov had won over Dadashev according to Kasparov's account, and brought him into his own team. Kasparov felt betrayed. According to Karpov, however, Dadashev never belonged to his team. In an interview with the news magazine _Der Spiegel_ Karpov said that Dadashev had sought him out and then is said to have reported which of Karpov's mental weaknesses were known to Kasparov. After the abandoned match Kasparov began a propaganda offensive and gave several interviews, including to _Der Spiegel_ , in which he explained how badly disadvantaged he had been during his World Championships match against Karpov.\n**32. Perestroika against the establishment**\n\n**The World Championship 1985:  \n_Anatoly Karpov against Garry Kasparov_**\n\nAt the FIDE congress, from 24th-31st August 1985 in Graz, only three days before the start of the second WCh match between Karpov and Kasparov, the general assembly approved in retrospect the termination of the first match by the president Florencio Campomanes. After 48 games neither player had achieved the six wins required for victory and at the time the match was abandoned it did not look as if there was likely to be any change in that position. In addition the rules newly formulated in Tunis 1985 by the executive committee for the staging of World Championships were adopted. From now on 24 was the fixed number of games, so as from then on to avoid marathon matches like that between Karpov and Kasparov.\n\nOther changes in the rules, however, would in Kasparov's opinion give a clear advantage to the reigning World Champion Karpov for the forthcoming match. The title defender was to retain his title if the match was drawn after 24 games. Moreover, he was also accorded a return match should he lose the actual match. This right to a return match, once known as the 'Lex Botvinnik', had been done away with before Botvinnik's match against Tigran Petrosian in 1963, but had been brought back again in 1977. Since Karpov had won his matches of 1978 and 1981, it had not been used. At the FIDE congress of Graz in 1985 that right was again done away with. But since the WCh match of 1985 was officially the continuation of the abandoned match of 1984\/85, then \u2013 according to the reasoning of FIDE's leadership \u2013 the same rules as in the first match had to apply, and in them the right to a return was still valid.\n\nShould he also lose the return match, Karpov was seeded straight into the final of the following WCh cycle to play whoever had eliminated all his opponents till then in the matches. Previously all other dethroned World Champions had also been qualifiers for the subsequent candidates' matches but had had to start their matches in the quarter-finals. So it would only take Karpov one victorious match to return as the challenger the World Championship. This was an extraordinary special rule.\n\nAccording to his own account, Kasparov only learned about the changes to the rules, including the right to a return match, two days before the start of the re-run of his match against Karpov. In a further change to the regulations FIDE also laid claim to one percent of the prize fund for each game which was drawn. The background to this was the long series of draws in the 1984\/85 match and perhaps the idea of motivating the players to produce more decisive games.\n\nThe arbiters for the match were decided, differently from previous procedures, by Campomanes at the FIDE congress in Graz: Andrey Malchev (Bulgaria) and Vladas Mikenas (Soviet Union) as chief arbiters, Lembit Vachessaar (Soviet Union) and Lodewijk Prins (Netherlands) deputies. Previously it had been usual for the players themselves to make suggestions for the arbiters' positions. Lothar Schmid had thus been named by both players. But since Campomanes delayed the decision for such a long time, Schmid was finally unavailable to accept for professional reasons.\n\nLondon, Moscow and Marseille had put in bids to be the venue for the WCh match. The bids with their prize fund on offer were kept in sealed envelopes in the FIDE offices in Lucerne and were ceremonially opened on the 1st May. London and Moscow had each offered approximately a million Swiss francs. Marseille's offer of 1.6 million Swiss francs was clearly higher. Nevertheless the match was awarded to Moscow. Campomanes justified this decision by saying that other criteria than the amount of the prize fund had a role to play, for example the general conditions, public interest in the venue and the wishes of the players.\n\nTo prepare for the match Kasparov played training matches against Robert H\u00fcbner (sponsored by _Der Spiegel_ and held in Hamburg, 4\u00bd:1\u00bd) and Ulf Andersson (in Belgrade, 4:2). In Hamburg, moreover, he defeated in a simultaneous display 32 chess computers by 32:0. Karpov won the double-round OHRA tournament in Amsterdam undefeated.\n\nThe repeat match began on 3rd September 1985, this time in the venerable Tchaikovsky Concert Hall. The tickets for it were in great demand and cost 15 roubles. Viktor Baturinsky took on the job of Karpov's delegation leader.\n\nIn the meantime, there had been a change in the power structure in Moscow. Mikhail Gorbachov was the new general secretary of the communist party of the Soviet Union and had presented his programme for reform. His foreign minister Eduard Shevardnadze greeted Kasparov before the match, but not Karpov, which the augurs in the Soviet empire saw as a paradigm shift by the new Soviet leadership. But the title defender was not badly treated, since Adrian Mikhalchishin later reminisced:\n\n'All the seconds lived in the wonderful dacha of Marshall Ivan Koniev, a famous Soviet veteran from the days of the Second World War: the legendary Efim Geller, Igor Zaitsev, Valery Salov, Sergey Makarichev, Viktor Cheliandinov and I. An hour before the start of each game Tolya had a 15-minute session with his psychologist, a man from Odessa, who was very pleasant \u2013 he made a fantastic fish soup! He began with his mental preparation and through the wall we could hear him shouting in the other room: \"Kill him! Destroy him! Bring him to his knees!\" We almost died laughing. Geller quipped that if the world heavyweight boxing champion happened to come into the room now, Tolya would rip him to pieces!'\n\nAs well as the trainers named, Evgeny Sveshnikov visited Karpov's camp on and off and tried to persuade the World Champion to combat Kasparov's Sicilian with 2.c3. In addition, Yuri Balashov was also a member of the team and Alexander Roshal as _TASS_ correspondent. Kasparov's head of delegation was once more Yuri Mamedov, his mother Klara Kasparova took on the duties of his manager. From the chess point of view Kasparov was supported by Georgy Timoshenko and Alexander Nikitin as seconds. Andras Adorjan did not receive an entry visa for Moscow.\n\nAt the start of the match Karpov was suffering from a bout of flu and immediately lost the first game. The second was adjourned in what was for him a hopeless position, but on the resumption the title defender was able to hold the game. Otherwise he would have fallen behind decisively right at the beginning. After another draw Karpov won the fourth and fifth games and went into the lead. But the challenger then obtained wins in the eleventh, 16th and 19th games, after which he was the one in the lead by 4:2. Karpov closed the gap with a victory in the 22nd game, but Kasparov finally won the 24th game for a final score of 13:11, counting the draws.\n\n **Kasparov \u2013 Karpov**\n\nMoscow, 11th game  \n1st October 1985  \nNimzo-Indian Defence (E21)\n\n**1.d4 \u2658f6 2.c4 e6 3.\u2658c3 \u2657b4 4.\u2658f3**\n\nThis move had been prepared by Kasparov for this match as his reply to the Nimzo-Indian Defence.\n\n**4...0-0**\n\nIn the 1st game Karpov experienced a shipwreck with the variation 4...c5 5.g3 \u2658e4 6.\u2655d3 \u2655a5 7.\u2655xe4 \u2657xc3+ 8.\u2657d2 \u2657xd2+ 9.\u2658xd2 \u2655b6 10.dxc5 \u2655xb2 11.\u2656b1 \u2655c3 12.\u2655d3 \u2655xd3 13.exd3.\n\n**5. \u2657g5 c5**\n\nIn the 7th game of the match 5...d6 6.e3 \u2658bd7 7.\u2655c2 b6 was played.\n\n**6.e3 cxd4 7.exd4 h6 8. \u2657h4 d5 9.\u2656c1**\n\nThe alternative was 9.\u2657d3.\n\n**9...dxc4**\n\nA novelty by Karpov. In an earlier game with this variation between Taimanov and Parma, Tbilisi 1973, 9...\u2658c6 10.cxd5 exd5 11.\u2657e2 was played.\n\n**10. \u2657xc4 \u2658c6 11.0-0 \u2657e7 12.\u2656e1 b6**\n\nNow a typical isolani position has arisen, in which White is already well developed and Black has weakened himself slightly with...h7-h6 on the kingside. Nevertheless the game is balanced. Karpov now develops his queenside.\n\n**13.a3**\n\nA standard plan in this sort of position: White brings his bishop to b1, the queen to d3, and forces Black into further weaknesses on account of the threat of mate on h7. Moreover the thrust d4-d5 is always in the offing.\n\n**13... \u2657b7 14.\u2657g3**\n\nBlack would react to 14.\u2657a2 \u2656c8 15.\u2657b1 with 15...\u2658h5.\n\n**14... \u2656c8**\n\nNow 14...\u2658h5 would be met by the central pawn advance 15.d5 exd5 16.\u2658xd5 \u2658xg3 17.hxg3 and White's pieces are very actively placed.\n\n**15. \u2657a2 \u2657d6**\n\n15...\u2658h5 16.d5.\n\n**16.d5 \u2658xd5 17.\u2658xd5 \u2657xg3 18.hxg3 exd5 19.\u2657xd5 \u2655f6**\n\nBlack attacks the b2-pawn and threatens...\u2656fd8. After 19...\u2656c7!? the white queen lift to the kingside would not have been possible: 20.\u2656c2 \u2656d7 21.\u2656d2 \u2658a5 22.\u2657xb7 \u2656xd2 23.\u2655xd2 \u2658xb7=.\n\n**20. \u2655a4 \u2656fd8 21.\u2656cd1 \u2656d7**\n\nBlack would like to relieve his position by doubling rooks and exchanging them on the d-file. For a moment, however, the \u2656d7 is unprotected. 21...\u2656d6 was safer.\n\n**22. \u2655g4**\n\n**22... \u2656cd8??**\n\nThis was the plan.\n\nWith his stereotyped reaction, however, Karpov overlooks another slightly hidden defect in his position \u2013 his weak back rank. After 22...\u2656d6 everything was still OK for Black.\n\n**23. \u2655xd7! \u2656xd7 24.\u2656e8+ \u2654h7 25.\u2657e4+**\n\nBlack resigned: 25...g6 26.\u2656xd7 and Black loses further material: 26...\u2657a6 27.\u2657xc6 \u2657c4 (27...\u2655xc6 28.\u2656xf7#) 28.\u2658e5+\u2013.\n\nThe match ended on the 9th November 1985. After ten years Karpov had been replaced as World Champion. In two WCh matches the players had sat opposite each other over the board for 325 hours and played 72 games. For his victory Kasparov received 696000 Swiss francs prize money, and as the loser Karpov 520000 Swiss francs.\n\nDuring the match, on 30th September 1985, an article appeared in _Der Spiegel_ magazine which reported that Karpov's former friend, the NDR editor Helmut Jungwirth, had embezzled some 450000 dollars from Karpov. What was involved was a fee which Karpov had received from the chess computer firm Novag for his advertising work and which Helmut Jungwirth had been managing for Karpov along with other advertising revenue in Germany. Jungwirth had perhaps counted on Karpov not having declared this income at home and not therefore wanting to take proceedings against him. Karpov, however, sued Jungwirth before a German court. The latter was legally convicted of embezzlement and deceit and paid 800000 marks to Karpov.\n\nKarpov said he spent the money for the promotion of chess in the USSR. _Der Spiegel_ suggested that he had been forced to do so by the authorities in the USSR, since he would otherwise have been threatened with a prosecution on account of currency offences. Karpov denied this. After the publication of the article in _Der Spiegel_ , Karpov lost three games out of nine and did not win a single one between the 1st and 24th October. Karpov found the article to be extremely discrediting and later stated in an interview with _Der Spiegel_ that the contribution had been deliberately published during the WCh match in order to disturb him.\n**33. Return match under protest**\n\n**The World Championship 1986:  \n_Garry Kasparov against Anatoly Karpov_**\n\nAfter winning the World Championship match against Karpov, Kasparov played a match against Jan Timman (15th to 22nd December, Hilversum), who at that time was considered number three in the world. Kasparov won 3:1 with two draws. Immediately after the match Kasparov began a media offensive and gave a press conference, in which he spoke out against the planned return match against Karpov \u2013 the start having been fixed by Campomanes to be very soon, on the 10th February. Moreover he demanded the deposition of Campomanes as FIDE president. In Lincoln Lucena Kasparov himself offered a new candidate for the office, in conjunction with Raymond Keene.\n\nCampomanes had been chosen in earlier elections with the votes of the USSR and its satellites and those of the chess developing countries. The western countries were in the main against the Filipino. Keene, till then a follower of Campomanes, hoped that with the help of Kasparov they could win over the USSR federation and the federations of the eastern block countries for the new leadership duo at the next presidential election. The Brazilian Lucena could count on the votes from the developing countries. Kasparov was supported by the European Chess Union (ECU), which had been founded at the FIDE congress in Graz 1985 and whose first president, the Swede Rolf Litorin, requested FIDE in a letter to cancel the return match. Timman, Larsen, Short, Ljubojevic, Najdorf, Seirawan and other western grandmasters supported the initiative by Kasparov. In a letter to the USSR Chess Federation the new World Champion Kasparov asked his federation to support him in his position and to speak out against the return match, but in vain. The leadership of the Soviet federation continued, to Kasparov's way of thinking, to support Karpov.\n\nFIDE, then based in Lucerne, had in the meantime reacted to Kasparov's demands for the cancellation of the match and threated to depose him as World Champion if he would not play. Kasparov was requested to explain in writing by the 7th January 1986 whether he would appear for the return match, an ultimatum which according to Kasparov was not covered by any rule since the venue had not even been fixed yet. Leningrad made a bid to organise it with a symbolic offer of a million Swiss francs, of which, however, the players would only receive 72000, and moreover that would be in roubles, and London offered 1.8 million Swiss francs. On the designated deadline of 13th January no decision had yet been taken.\n\nKasparov had the impression that the decision was being deliberately drawn out and that Karpov was active behind the scenes with the intention of having the match fixed for a date at which Kasparov's trainers would not be available to him. Both were part of the army and thus subject to more or less arbitrary orders or releases for training and tournaments. On the 18th January Kasparov once more declared that he would not play. Instead he suggested a three-man tournament with Kasparov, Karpov and the winner of the candidates' matches. At a meeting of the Soviet Chess Federation on 22nd January, Kasparov and Karpov finally came to an agreement according to which the return match was to take place in the summer of 1986, and according to the wishes of both players in Leningrad.\n\nKasparov thought that by giving in he had gained Karpov and the Soviet federation as allies against Campomanes. Later he had to accept that Campomanes had previously come to a different agreement with Vassily Gavrilin, the vice-president of the Soviet Committee for Sports, and David Anderton, the representative of the British Chess Federation, about the staging of the match: it would take place half in London and half in Leningrad. In May 1986 Kasparov played a match in Basel with Anthony Miles, one of the strongest western players of that time, and annihilated him by 5\u00bd:\u00bd.\n\nIn his match with Karpov, in addition to his regular trainers Alexander Nikitin, Josif Dorfman, Evgeny Vladimirov and Georgy Timoshenko, Kasparov was also supported by Mikhail Gurevich and Elmar Magerramov as seconds. Also in the delegation which travelled with him to London, moreover, were Seyavush Eganov representing the Soviet Republic of Azerbaijan and Viktor Litvinov as a representative of the KGB. Kasparov had hired as his manager the Englishman Andrew Page, who rented a house near Kensington Gardens for the Kasparov delegation. Karpov's official seconds were Igor Zaitsev and Sergey Makarichev.\n\nThe host for the match, the first WCh match between two Soviet grandmasters to take place outside of the USSR, was the London Park Lane Hotel. The games were played in the main ballroom of the hotel. The arbiter was Lothar Schmid. Even the British prime minister Margaret Thatcher attended the opening ceremony of the WCh match on the 27th July. That ceremony was a lavish production by Tim Rice, whose musical _Chess_ was being performed parallel to the WCh match in London's Prince Edward Theatre. The ceremony was paid for by the Duncan Lawrie banking house. Costs of 100000 pounds or so were mentioned.\n\nOn each of the match days a long queue of chess lovers formed in front of the Park Lane Hotel to get entry tickets. The games were broadcast electronically to a monitor, a new technique for the time, developed by David Levy and Kevin O'Connell with their firm 'Intelligent Chess Software'.\n\nAfter three draws to start with Kasparov was able to decide the fourth game in his favour with the white pieces. Karpov promptly equalised in the fifth game. The sixth and seventh games were again drawn. Then Kasparov won the eighth game in convincing fashion: ten moves before the time control Karpov overstepped the time limit in a lost position.\n\n **Kasparov \u2013 Karpov**\n\nLondon, 8th gamee  \n15th August 1986  \nQueen's Gambit Declined (D35)\n\n**1.d4 d5 2.c4 e6 3. \u2658c3 \u2657e7**\n\nAn idea of the Russian grandmaster Vladimir Alatortsev. After the exchange on d5 Black avoids the development of the white bishop to g5. The more usual move is 3...\u2658f6. White can then continue with 4.cxd5 exd5 5.\u2657g5.\n\n**4.cxd5 exd5 5. \u2657f4 \u2658f6**\n\nIn the 7th game of the match the same opening was played with reversed colours. There then followed 5...c6 6.\u2655c2 g6 7.e3 \u2657f5 etc.\n\n**6.e3 0-0 7. \u2657d3 c5**\n\n7...c6 is also often played.\n\n**8. \u2658f3 \u2658c6 9.0-0 \u2657g4 10.dxc5 \u2657xc5 11.h3**\n\nThere is no way to prevent the black thrust...d5-d4. So White secures at least the bishop pair.\n\n**11... \u2657xf3**\n\nA novelty by Karpov. Other moves are worse: 11...\u2657h5? 12.g4 \u2657g6 13.\u2657xg6 hxg6 14.g5 \u2658h5 15.\u2655xd5 winning a pawn or 11...\u2657e6 and White would get control of the d4-square: 12.\u2658b5!? a6 13.\u2658bd4.\n\n**12. \u2655xf3 d4 13.\u2658e4**\n\n'The attempt to win a pawn with 13.exd4 \u2658xd4 14.\u2655xb7 leads to exchanges which are unfavourable for White: 14...\u2658e6 15.\u2656ad1 \u2658xf4 16.\u2657xh7+ \u2654xh7 17.\u2656xd8 \u2656axd8.' (Nikitin)\n\n**13... \u2657e7**\n\n'The natural 13...\u2658xe4 14.\u2657xe4 dxe3 brings about after 15.\u2655h5 exf2+ 16.\u2654h1 f5 17.\u2657xf5 g6 18.\u2657xg6 hxg6 19.\u2655xg6+ \u2654h8 20.\u2656ad1! complications which Garry evaluated as being to his advantage: 20...\u2657d4 (20...\u2655f6 21.\u2655h5+ \u2654g8 22.\u2656d5+\u2013) 21.\u2655h5+ \u2654g7 22.\u2655g4+ \u2654h8 23.\u2657e3 \u2657xe3 24.\u2656xd8 \u2656axd8 25.\u2655h4+ \u2654g7 26.\u2655g3+ +\u2013 winning the bishop.' (Nikitin)\n\n**14. \u2656ad1 \u2655a5!**\n\nAfter 14...\u2655b6 15.\u2657d6 \u2658d5 16.\u2655f5 \u2657xd6 17.\u2658xd6 \u2658f6 18.\u2655g5!? (intending \u2658f5) 18...\u2658e7 19.\u2658c4 White is clearly better.\n\n**15. \u2658g3 dxe3 16.fxe3!**\n\n**16... \u2655xa2!?**\n\n'The primitive 16...g6 17.\u2657h6 \u2656fe8 is unable to prevent 18.\u2658f5.' (Nikitin) 18.\u2658e4 is even better.\n\n**17. \u2658f5 \u2655e6 18.\u2657h6**\n\nStronger than 18.\u2658xe7+ \u2658xe7 (18... \u2655xe7 19.\u2657d6) 19.\u2655xb7.\n\n**18... \u2658e8 19.\u2655h5 g6 20.\u2655g4 \u2658e5**\n\n**21. \u2655g3**\n\n'It would be objectively stronger to play 21.\u2658xe7+ \u2655xe7 22.\u2657xf8 \u2654xf8 23.\u2655f4, but then Black would have clarity and a stable position. White wanted to maintain the tension and the uncertainty, since Karpov was beginning to think about each move for a long time.' (Nikitin)\n\n**21... \u2657f6**\n\nKarpov now had only 14 minutes left on his clock for the remaining moves.\n\n**22. \u2657b5?!**\n\n'White tries to hang on to his dwindling initiative.' (Nikitin) 22.\u2658d4 \u2655e7 23.\u2657xf8 \u2654xf8 offers White a slight advantage. Even if the move was not objectively the best, it is effective. Karpov used up a further ten minutes thinking time on the next five moves.\n\n**22... \u2658g7 23.\u2657xg7 \u2657xg7 24.\u2656d6 \u2655b3 25.\u2658xg7 \u2655xb5**\n\nBlack has stabilised his position with clever defence and is now trying, despite time trouble, to play for a win with his extra pawn. White is not interested in sharing the points either.\n\n**26. \u2658f5 \u2656ad8**\n\nBlack scorns the draw after 26...f6 27.\u2658h6+ \u2654g7 28.\u2658f5+ \u2654g8 29.\u2658h6+. Karpov now has only four minutes on the clock for the remaining 14 moves.\n\n**27. \u2656f6**\n\nInstead of this White could also have forced a draw here with 27.\u2656xd8 \u2656xd8 28.\u2655g5 \u2656d7 29.\u2658h6+ \u2654g7 30.\u2658f5+ \u2654g8 31.\u2658h6+.\n\n**27... \u2656d2**\n\nAccording to Nikitin 27...\u2654h8 was safer: 28.\u2658d4 \u2655c5 29.\u2658e6 fxe6 30.\u2656xf8+ \u2656xf8 31.\u2656xf8+ \u2655xf8 32.\u2655xe5+ \u2654g8 33.\u2655xe6+ \u2655f7=.\n\n**28. \u2655g5 \u2655xb2**\n\n28...\u2654h8!? 29.\u2655h6 \u2656g8 30.\u2658e7 with approximate equality according to Nikitin.\n\n**29. \u2654h1**\n\n**29... \u2654h8??**\n\n29...\u2656d7 30.\u2658h6+ \u2654g7 31.\u26566f4 \u2658c6 would have led to a draw.\n\n**30. \u2658d4 \u2656xd4 31.\u2655xe5**\n\n**1-0**\n\nBefore playing his 31st move Karpov overstepped the time limit.\n\nAfter 31...\u2656d2 32.\u2655e7 \u2656dd8 33.\u2656xf7 \u2656xf7 34.\u2656xf7 \u2654g8 White wins by advancing the e-pawn: 35.e4 \u2655c1+ 36.\u2654h2 \u2655h6 37.e5 \u2656f8 38.e6 g5 39.\u2656xf8+ \u2655xf8 40.\u2655xg5+ \u2654h8 (40...\u2655g7 41.\u2655d8+ \u2655f8 42.e7+\u2013) 41.e7 \u2655e8 42.h4 h5 43.g4 hxg4 44.h5 \u2654h7 45.\u2655g6+. 'White's win is obvious.' (Nikitin)\n\nThis was followed by four draws. At a score of 6\u00bd:5\u00bd after half of the programmed 24 games the World Championship then moved to Leningrad, now St. Petersburg. After game 12 Kasparov had sat down with his seconds because he wondered why his lavish opening preparation with its many surprises was making absolutely no impression on Karpov and was ineffective. Kasparov expressed the suspicion that there was perhaps a leak in his team and that information was getting out.\n\nKasparov and Karpov flew from London to Leningrad in the same plane and spent their time playing cards together. On landing in Leningrad the World Champion was collected from the airport only by a normal limousine with Baku licence plates, as Kasparov later remembered, whereas Karpov was driven to his lodgings in a luxury limousine with a police escort. Both delegations moved into quarters near each other on the Kamenny Island in the north of today's St. Petersburg. Normally contestants in World Championships live far apart so as to avoid each other. Karpov had, however, decided to live close to Kasparov, which the latter's delegation considered highly suspicious.\n\nThe games in Leningrad were played in the Concert Hall of the Hotel Leningrad. The match was actually due to restart on the 2nd September 1986, but, backed up by a medical certificate, Karpov requested an additional timeout of two days, so that the match could only be continued on the 5th September.\n\nThe first game of the Leningrad half of the match, game 13, also ended in a draw. Then Kasparov won the spectacular 14th game against Karpov's Ruy Lopez Zaitsev Variation and thus increased his lead to two points. Kasparov now took a timeout to enjoy his win. The following, 15th, game ended in a draw. In game 16 Kasparov went after the same variation and once more won brilliantly after an intensive tactical battle. Eight games before the finish Kasparov already looked like a safe winner. But then the title defender lost three games in succession: the 17th, 18th and 19th. Suddenly Karpov was on level terms.\n\nIn London Kasparov had already wondered why Karpov was apparently not in the least surprised by his choice of the Gr\u00fcnfeld Indian Defence and moreover appeared to be perfectly prepared for it. Kasparov had gained the impression that Karpov was not prepared for the objectively best moves, but for the variations analysed by Kasparov's team. More and more Kasparov's suspicion grew, as he later reported, that his analyses were being passed on to Karpov. At first Timoshenko was suspected, but he was no longer involved in Leningrad. There the three defeats in a row finally fed Kasparov's conviction that there was something wrong. Litvinov and Nikitin now told Kasparov that Vladimirov, who had been Kasparov's closest collaborator in the past five years, had continually made copies of all common analyses. An hour before the 18th game, moreover, Vladimirov could not be found at short notice.\n\nTo be quite sure that nobody was communicating with the outside world, all the telephones belonging to the team members were cut off from the net, including a second 'secret' telephone belonging to Vladimirov. The latter, confronted with the suspicions against him, was outraged and explained that he wanted to leave the team. Nobody talked him into staying and so Vladimirov went home, but before doing so he had to sign a declaration admitting the copying of analyses for his own purposes.\n\nKasparov later admitted that he had had no proof of any treachery by Vladimirov, but was nevertheless of the opinion that pressure could perhaps have been put on Vladimirov. Vladimirov, as a member of the army, had in any case belonged to the same sports club as Karpov \u2013 the Central Army Sports Club. Karpov was later asked in a _Spiegel_ interview about the Vladimirov affair and stated that at no point had he been in contact with him, far less had he hired him. Vladimirov himself indicated in interviews that Kasparov had never been able to show any proof for his accusations and explained what had happened by Kasparov's paranoia after three defeats in a row. Kasparov needed something on which to blame the disaster \u2013 apart from himself. Finally he found in Vladimirov an alleged guilty party.\n\nAfter three wins in succession Karpov did not take advantage of the moment to perhaps inflict a fourth defeat on the psychologically ailing Kasparov, but took a timeout before game 20. In line with his conspiracy theory Kasparov later pointed out that perhaps Karpov had now lost important help with his analytical work, (naturally Vladimirov was meant but the name was never mentioned), and needed extra time for his preparation.\n\nThe 20th game was a short draw. The 21st game, in which Karpov had white, also ended without a winner, though only after a long struggle. The 22nd game finally went to Kasparov, who defeated Karpov in the Queen's Gambit. Kasparov, who as title defender only needed a 12:12, then only required a single draw from the remaining two games to win the match. He achieved this in the 23rd game. With it Kasparov had held on to his title. The 24th game was only played to determine the ratio for the division of the prize money.\n**34. A traitor in the camp?**\n\n**The World Championship 1987:  \n_Garry Kasparov against Anatoly Karpov_**\n\nAfter his victory in the return match against Karpov, Kasparov wrote together with Donald Trelford, the editor-in chief of the _Sunday Observer_ , the book _Child of Change_ , his autobiography.\n\nAt the end of October 1986 Kasparov again sought to gain influence on the Soviet Chess Federation for it to support at the forthcoming FIDE congress in Dubai his candidates Lucena and Keene. But without success. The USSR federation announced before the election its support for Campomanes. The announcement made sure that all the other eastern block countries voted for Campomanes. In addition the Arab states were for the reigning FIDE president, as well as all the third world federations who had travelled to Dubai for the Chess Olympiad at the costs of the organisers. In view of the fact that he had no chance, Kasparov's candidate Lucena withdrew before the vote. With his re-election the status of Campomanes underwent a change: for the first time in the history of FIDE the president now received an honorarium for his work.\n\nSince Kasparov was unable to force Campomanes from his office as FIDE president, he founded in Dubai, together with other top grandmasters, the 'Grandmaster Chess Association' (GMA). The seven founder members were: Garry Kasparov, Anatoly Karpov, Jan Timman, Ljubomir Ljubojevic, John Nunn, Yasser Seirawan and Lajos Portisch. The first thing they did was to call into life a series of tournaments, the World Cup. The sponsor of the GMA and its World Cup, or the contact to the sponsors, was the Dutch tele-communications manager Bessel Kok, director of SWIFT, a firm which developed communication systems for banks. FIDE itself had lost standing with its president Campomanes and was hardly an adequate interlocutor for potential sponsors any more. According to Kasparov too much sponsor cash intended for FIDE projects was ending up in the pockets of Campomanes. Discussions with FIDE to give professional chess players via the GMA more influence in the world chess federation soon ran into sand, however.\n\nIn December 1986 Kasparov won the OHRA tournament in Brussels. For the first time in the space of three years he did not have to play a match against Karpov and had more time for tournaments. After the OHRA tournament he played two clock simuls: in Hamburg he defeated the Bundesliga team of the Hamburg Chess Club by 7:1 and in doing so took his revenge for a 3\u00bd:4\u00bd defeat in 1985 at the hands of the same team. In another clock simul he beat the Swiss national team by 5\u00bd:\u00bd.\n\nIn February\/March 1987 in the candidates' 'superfinal' in Linares Karpov defeated Andrei Sokolov by 7\u00bd:3\u00bd, thus becoming the challenger of Kasparov in the next WCh match. Doha (United Arab Emirates) had put in a bid to organise the WCh match. In view of the close contacts the Arab federations had to Campomanes, Doha was out of the question for Kasparov as a venue, whilst Karpov would have gladly played there. Instead, Kasparov's first choice was Seattle. But finally the staging of the WCh was awarded to Sevilla (Spain), where interesting events were already being planned in view of the forthcoming world exhibition ExPO 92. The 10th October 1987 was fixed for the start of the match.\n\nAfter the departure of his previous seconds Timoshenko and Vladimirov, the trainers remaining to Kasparov were Alexander Shakarov, Alexander Nikitin and Josif Dorfman. For his preparation for the latest match against Karpov, Kasparov now also brought on board Zurab Azmaiparashvili and Sergey Dolmatov.\n\nOn the 1st June 1987 Kasparov received, according to his notes, a call from Alexander Feldman, whom he did not then know. The latter asked for a meeting and at it he reported that he had in the past carried out certain tasks for Karpov but that he was now disenchanted with his former boss. According to Feldman, Josif Dorfman, who had been hired as a second by Kasparov shortly before the first match in 1984\/85, then wanted to sell his insider information for betting purposes. That was how the contact was made to Feldman, whom Dorfman knew from their youth. Between the fourth and eleventh games he said Dorfman had received on each occasion 150-200 roubles from Feldman for information about openings, sealed moves and analysis of adjourned games. But Kasparov believed that Dorfman's information had, without the latter knowing it, also been passed from Feldman to Karpov. But when in game 11 Karpov played 1.\u2658f3, Dorfman realised that Karpov had access to his information and he then demanded more money from Feldman. Karpov, who was already leading 4:0, did not see, in Kasparov's view, any need for this and thus ended the connection.\n\nAdrian Mikhalchishin, who in the first two matches between Karpov and Kasparov had acted as a trainer for Karpov, later stated no information of any sort from the Kasparov camp had ever reached Karpov. The link between Dorfman and the Russian gambling mafia was totally plausible, according to Mikhalchishin, but the part of the story which suggested a flow of information from the Kasparov camp towards Karpov was a consequence of the paranoia which had built up in Kasparov's camp during the match and from which the latter had not been able to free himself since then. In reality there had even been a leak in the Karpov camp, because the wife of one of Karpov's trainers had passed on details of preparation.\n\nIn the meantime Dorfman left the team and took part in the USSR championship in Tashkent. He returned for the 32nd game and then, according to Kasparov, is supposed to have made contact with Feldman again. After the first match was abandoned and the date for a restart set for September 1985, Feldman is supposed to have sought out Dorfman again in his home town of Lviv in order to make arrangements for the next match. When they could not come to an agreement, Dorfman anxiously turned on 25th September 1985 to the head of Kasparov's delegation, Gennady Rzaev, who was at the same time the head of the Azerbaijani department of sport, and told him that he had received a telephone death threat. He revealed to him part of his contacts with Feldman. Rzaev then suggested that at his next meeting with Feldman Dorfman should explain that he was ready to cooperate, but that he should demand the allocation of a flat in Moscow in return. Even with the best of contacts this ought to be impossible to arrange in the short time available, was Rzaev's calculation. In the meantime the match would long since have been finished and the whole business taken care of.\n\nTo general surprise in the Kasparov team, on the 30th September Feldman showed Dorfman a letter from the Soviet Committee for Sports to the head of Moscow administration A.I. Kostenko, in which Kostenko was requested to be helpful to Dorfman when he moved to Moscow. Kostenko had written an instruction on the letter and signed it. Rzaev brought this business to the attention of Litvinov, though not till the 1st October. The latter realised that in this case higher authorities were obviously involved and turned to the KGB offices in Azerbaijan. KGB central in Moscow was informed via the local communist party department. The whole business was clas-sified top secret, since had it been made public it could have caused a scandal and the termination of the match.\n\nKasparov at first wanted to sack Dorfman on account of these events, but Litvinov advised keeping him on as a second so as not to weaken the team. Michael Gurevich had also been offered money for information by Feldman. In Kasparov's view Karpov must have known about the actions of the Feldman group. In addition, according to Kasparov, the fingerprints of the KGB had been all over it. In an interview with the German news magazine _Der Spiegel_ , Kasparov had moreover in 1988 expressed the opinion that the discrediting of his second Dorfman by Feldman had been an attempt from outside to sow discord in his team.\n\nIn Sevilla, Kasparov lived with his mother and Soviet security people in a luxury villa with a covered swimming pool and tennis court which had been rented by his manager Andrew Page, whereas his seconds were lodged elsewhere and only visited Kasparov on a day-to-day basis for common preparation and analysis. Every day at ten o'clock there was a conference; that was followed by the division of the tasks. Whilst Kasparov then kept himself fit by playing tennis, the seconds got down to the work.\n\nKarpov, who had got married for the second time just before the match, to the girlfriend of his youth Natalia Bulanowa, had also rented a villa and had moved into it with his whole team. That comprised as well as his seconds his head of delegation Saryapin, and also as his bodyguard and head of security Karpov's trusted friend, KGB lieutenant-colonel Vladimir Pistchenko. Karpov had a direct line to Moscow laid into his villa, where, as was supposed, a second analysis team was kept available to him.\n\nThe match was played in the 'Lope de Vega Theatre' in the centre of Sevilla. Following the recent rules for WCh contests the match was to be to a maximum of 24 games. At a score of 12:12 the World Champion was to retain his title. All the games started at 16.30. The Dutchman Geurt Gijssen was chosen as chief arbiter. On the 10th October the opening ceremony took place and lots were drawn for the colours. Karpov drew the white pieces for the starting game.\n\nBefore the games the players were brought from their residences in white luxury cars and used different entrances to go into the theatre, which had been renovated for the event; whoever had the white pieces on any particular day came in through the main entrance, whereas the player with black used the side entrance. The first rows of seats in the 831 seat theatre \u2013 which was sold out for the first time for the seventh game \u2013 were reserved for the players' entourages and for guests of honour. Monitors were installed outside of the hall showing the position on the board by video signal.\n\nThe games were analysed by commentators in a side room. These alternated and included Boris Spassky, Vassily Smyslov, Mikhail Tal, Viktor Kortchnoi, Ljubomir Ljubojevic, Ulf Andersson, Tamas Georgadze, Gennady Sosonko, Jose Fernandez and Roman Toran, who at the time was vice-president of FIDE. The final phase of the match was commented on by the former women's World Champion Maya Chiburdanidze.\n\nThe first game was played on the 12th October. Kasparov met Karpov's 1.d4 with the Gr\u00fcnfeld Indian Defence, his main weapon which he had also prepared for this match. The game was drawn. In the second game Kasparov came up with a slight surprise, 1.c4. But Karpov showed himself well up to the task. On move 9 he came up with a novelty (9...e3), which he had already prepared in 1981 for the match against Kortchnoi. Kasparov thought about it for 83 minutes before finally continuing with 10.d3. Both players got into time trouble and had only three minutes left on the clock for the final 15 moves. On the 26th move Kasparov replied immediately but forgot to press his clock. His time ran out almost completely since Karpov, for understandable reasons, did not make a move. When Kasparov realised, he had only 20 seconds left for the remaining 14 moves, played them at blitz, made two bad mistakes and resigned.\n\nIn the third game Kasparov again held the draw with his Gr\u00fcnfeld Indian Defence. In the fourth game he repeated the opening from the second game, but Karpov avoided a theoretical discussion, chose a different move from his strong innovation of the second game and lost: the match was all square again. In the fifth game Karpov then served up a new idea in the main variation of the Gr\u00fcnfeld Indian Defence. Kasparov achieved a good position after a difficult opening, but spoiled things and lost. Karpov was leading 3:2. The sixth game ended in a draw. In the seventh game Karpov repeated his variation from the fifth game, which has since been called the Sevilla Variation. After a long struggle the game ended with a sharing of the point on move 79. Kasparov equalised to 4:4 with his win in the eighth game, again with the English Opening. The ninth game again saw the Sevilla Variation and was again drawn. The tenth also ended without a winner. This time Kasparov played 1.e4, but achieved nothing against Karpov's Caro-Kann Defence. In the 11th game the Sevilla Variation was back on the board. But Kasparov obtained an advantage after a mistake by Karpov and won. The World Champion was in the lead for the first time.\n\nGames 12 to 15 were all undecided, though the draw in the 15th game only came about after some turbulent 'negotiations'. After the adjournment of the game, Karpov had his confidant Pistchenko call arbiter Gijssen the following day, Saturday 21st November 1987, at 13.00 to announce that he was offering a draw. Gijssen then called Kasparov, but only the Spanish housekeeper was at home. Half an hour later, it was announced that the World Champion had lain down and wanted on no account to be disturbed before 15.00. When Karpov learned this he became furious and then indicated to Gijssen that he wanted to continue the game. The arbiter pointed out to him that once a draw had been offered that offer could not be withdrawn.\n\nAt 15.00 Kasparov called on Gijssen. When he heard that Karpov wanted to withdraw his offer of a draw, the World Champion also became furious and he too wanted to continue the game, but unshaven and wearing a T-shirt. He finally accepted the offer of a draw, but turned up in the theatre for the resumption, just in case. Karpov came too and sat among the spectators, armed with a ball-point pen, ready to play if necessary. The arbiter, however, declared the game a draw. In the 16th game, Kasparov had the white pieces again. In the English Opening the title defender fell behind, however, and lost: Karpov had levelled the scores. Games 17 to 22 ended without a winner. 'Goalless draws', was Bronstein's comment on the match. At a score of 11 to 11 two games were left. If Karpov wanted to recover the title, he had to win one of them.\n\nIn the 23rd game the challenger had White and opened with 1.c4. The game was adjourned after move 40. On the resumption Kasparov made on move 50 a blunder which cost him the game. Suddenly, one game before the finish, the World Champion was trailing and absolutely had to win the final game in order to hang on to the title with a score of 12:12. After a long game with a series of mistakes on both sides Kasparov was able to master this difficult task and only just hold on to his title.\n\nOf a total prize fund of 2280000 Swiss francs the two Soviet citizens each received 137000 francs plus an equivalent sum in roubles. The players had to pass on 80% of the prize money to the Soviet Committee for Sports. FIDE received 700000 francs, of which 300 000 were spent on organisational matters.\n\n **Kasparov \u2013 Karpov**\n\nSevilla, 24th game  \n18th December 1987  \nR\u00e9ti Opening (A14)\n\n**1.c4 e6 2. \u2658f3 \u2658f6 3.g3 d5 4.b3**\n\nWith this move Kasparov is aiming for a closed game and avoids early simplifications.\n\n**4... \u2657e7 5.\u2657g2 0-0 6.0-0 b6**\n\nThe alternative is 6...c5 7.\u2657b2 \u2658c6 etc.\n\n**7. \u2657b2 \u2657b7 8.e3 \u2658bd7**\n\n'The strongest move is 8...c5'. (Kasparov) White's best continuation then is 9.\u2658c3 \u2658c6 10.cxd5 \u2658xd5 (10...exd5 11.d4) 11.\u2658xd5 \u2655xd5 (11...exd5 12.d4) 12.d4!.\n\n**9. \u2658c3**\n\n'9.\u2655e2 is probably more accurate.' (Kasparov)\n\n**9... \u2658e4 10.\u2658e2!?**\n\nTo avoid the exchange of the knight and simplifications.\n\n**10...a5**\n\nKarpov is expecting 11.d3. After 10... \u2657f6 there would be 11.d4!? c5 12.\u2658f4.\n\n**11.d3 \u2657f6**\n\nThat was the idea.\n\n**12. \u2655c2 \u2657xb2 13.\u2655xb2 \u2658d6**\n\n'13...\u2655f6?! 14.\u2655c2 \u2658ec5 15.cxd5 (15.d4? dxc4) 15...exd5 16.\u2658f4.' (Kasparov)\n\n**14.cxd5 \u2657xd5**\n\n'14...exd5!? 15.d4 c5 16.dxc5 bxc5 17.\u2656fd1 with some pressure for White.' (Taimanov)\n\n**15.d4**\n\n'There was also the attractive 15.\u2658f4 \u2657b7 16.d4 c5 17.\u2656ac1 \u2655e7 (17...cxd4 18.\u2656fd1!) 18.\u2658d3 \u2656ac8 (18...\u2658e4 19.\u2658fe5) 19.dxc5 with active play.' (Kasparov)\n\n**15...c5 16. \u2656fd1 \u2656c8**\n\n'At first glance the black position looks really safe. But Karpov has not yet reached full equality. His queenside structure is not the best and there are too many pieces on the d-file.' (Taimanov)\n\n**17. \u2658f4 \u2657xf3**\n\n'17...\u2657e4? is worse: 18.dxc5 \u2658xc5 19.\u2655e5 \u2658cb7 20.\u2658h5+\u2013' (bulletin) or 17... c4?! 18.\u2658xd5 exd5 19.bxc4 dxc4 20.a4! 'with advantage to White.' (bulletin)\n\n**18. \u2657xf3 \u2655e7 19.\u2656ac1 \u2656fd8 20.dxc5 \u2658xc5 21.b4!**\n\nTo make b6 a target.\n\n**21...axb4**\n\n'21...\u2658ce4 22.\u2656xc8 \u2656xc8 23.\u2655d4 would be good for White.' (Taimanov)\n\n**22. \u2655xb4 \u2655a7 23.a3 \u2658f5**\n\n23...\u2658e8=. (Kasparov)\n\n**24. \u2656b1 \u2656xd1+ 25.\u2656xd1 \u2655c7**\n\n25...\u2655a5 26.\u2656c1.\n\n**26. \u2658d3!**\n\n'Intending to exchange as many pieces as possible on c5, so as to enter the endgame with a distant passed pawn on the a-file. In what followed Karpov overestimated the dangers and became nervous.' (Kasparov)\n\n**26...h6?!**\n\nAfter this move Karpov had only ten minutes left on the clock. 26...g6 was a better airhole: 27.\u2654g2 \u2658xd3 28.\u2656xd3 \u2656d8 29.\u2656c3 \u2655d6 with a level game according to Kasparov.\n\n**27. \u2656c1 \u2658e7**\n\nAfter 27...\u2658d6 28.\u2658e5 f6 Kasparov wanted to continue with 29.\u2658g6! \u2658f7 30.h4.\n\n**28. \u2655b5 \u2658f5**\n\n'After 28...\u2655a7 29.\u2656c3 \u2658f5 30.\u2658e5 \u2658d6 31.\u2655b4 Black also remains under pressure.' (Kasparov)\n\n**29.a4 \u2658d6 30.\u2655b1 \u2655a7**\n\n'Or 30...\u2655d8 31.\u2658xc5 bxc5 32.\u2656d1 with advantage to White.' (Kasparov)\n\n**31. \u2658e5!**\n\n'In Karpov's time trouble Kasparov creates problems. The threat is 32.\u2658c6.' (Taimanov)\n\n**31... \u2658xa4?**\n\nHe had to play 31...\u2655xa4 32.\u2655xb6 \u2655a3! and Black holds on, or else 31...\u2658f5!? 32.\u2658c6 \u2655xa4 33.\u2655xb6 \u2658d3 34.\u2656f1 \u2655a3 35.\u2657e4 \u2654h8=.\n\n**32. \u2656xc8+ \u2658xc8**\n\nThis is the decisive moment in the whole match. Kasparov has a big advantage, but gives it away, but Karpov does not take the opportunity to get equality and a probable draw.\n\n**33. \u2655d1?**\n\nAfter 33.\u2655b5! White would have obtained a winning position. The threat of penetrating on e8 is much stronger than the move in the game, which threatens \u2655d8+, because from e8 it is also attacking f7. There is no satisfactory defence for Black after that: 33...\u2658d6 34.\u2655c6+\u2013, 33...\u2654f8 34.\u2658c6 \u2655a8 35.\u2655d3! g6 36.\u2655d4+\u2013 or 33...\u2654h7 34.\u2655e8 \u2658d6 35.\u2655d8 \u2658f5 36.\u2658c6 \u2655b7 37.g4 \u2658h4 38.\u2657e4+ g6 39.\u2658e7 and wins.\n\n**33... \u2658e7?**\n\nBlack missed 33...\u2658c5! which keeps an eye on the d7-square: 34.\u2655d8+ \u2654h7 35.\u2654g2 f6! 36.\u2658c6 \u2655d7 37.\u2655xd7 \u2658xd7 38.\u2658d8 \u2658c5 39.\u2658xe6! \u2658xe6 40.\u2657g4=. And 35.\u2657d1? would also be bad for White: 35...f5! 36.\u2655xc8 \u2655a1.\n\n**34. \u2655d8+**\n\n'34.\u2657h5 \u2658c5 35.\u2657xf7+ \u2654h7 36.h4 \u2655c7 37.\u2655a1 with a strong attack.' (Kasparov)\n\n**34... \u2654h7 35.\u2658xf7**\n\nThreatening mate beginning with 36.\u2655h8+.\n\n**35... \u2658g6 36.\u2655e8**\n\n36.\u2658d6? \u2655e7 37.\u2655a8 \u2658c5 38.\u2658e4 \u2658e5.\n\n**36... \u2655e7**\n\n36...\u2658c5? 37.\u2657h5 \u2655a1+ 38.\u2654g2 \u2655f6 39.h4!+\u2013.\n\n**37. \u2655xa4 \u2655xf7 38.\u2657e4**\n\nDespite the inaccuracies White is still in the driving seat.\n\n**38... \u2654g8 39.\u2655b5**\n\n'After 39.\u2657xg6 \u2655xg6 40.\u2655b3 \u2655g4 41.\u2655xb6 \u2655d1+ 42.\u2654g2 \u2655d5+ 43.f3 \u2655a2+ 44.\u2654h3 the queen ending could go on forever.' (Kasparov)\n\n**39... \u2658f8 40.\u2655xb6 \u2655f6 41.\u2655b5 \u2655e7 42.\u2654g2**\n\nThe sealed move. In analysis Kasparov and his team were unable to find a clear way to a win, but of course White has excellent practical chances.\n\n**42...g6?**\n\n42...g5 43.f4! \u2655f6 and 42...\u2655f6 43.h4 g5 44.h5! were suggested as better. White has an advantage but is not necessarily winning.\n\n**43. \u2655a5 \u2655g7 44.\u2655c5 \u2655f7 45.h4 h5?**\n\nThis weakens g6 and h5. 45...\u2654g7 was better.\n\n**46. \u2655c6 \u2655e7 47.\u2657d3 \u2655f7**\n\nOf course, all Black can do is wait and see.\n\n**48. \u2655d6 \u2654g7 49.e4 \u2654g8 50.\u2657c4 \u2654g7 51.\u2655e5+ \u2654g8**\n\nAfter 51...\u2655f6 52.\u2655xf6+ \u2654xf6 53.f4 e5 54.\u2654f3 \u2658d7 55.\u2654e3 \u2658c5 56.\u2657d5 White wins with the king march \u2654d2-c3-c4.\n\n**52. \u2655d6 \u2654g7 53.\u2657b5 \u2654g8 54.\u2657c6 \u2655a7 55.\u2655b4! \u2655c7 56.\u2655b7! \u2655d8**\n\n56...\u2655xb7 57.\u2657xb7 e5 58.f4+\u2013. The minor piece ending is won.\n\n**57.e5!+\u2013**\n\nPractically puts Black in zugzwang.\n\n**57... \u2655a5 58.\u2657e8 \u2655c5 59.\u2655f7+ \u2654h8 60.\u2657a4 \u2655d5+ 61.\u2654h2 \u2655c5**\n\n61...\u2658h7 62.\u2657c2 \u2655xe5 63.\u2655e8+ \u2654g7 64.\u2655xg6+ +\u2013.\n\n**62. \u2657b3 \u2655c8 63.\u2657d1 \u2655c5 64.\u2654g2**\n\nKarpov resigned.\n\nAfter for example 64...\u2655b4 65.\u2657f3 \u2655c5 66.\u2657e4 \u2655b4 White must, however, still be careful, since the thoughtless 67.\u2657xg6?? \u2658xg6 68.\u2655xg6 \u2655b7+ 69.\u2654h2 is surprisingly followed by 69...\u2655g2+!! and stalemate is unavoid-able. Nor does 69.\u2654f1 help on account of 69...\u2655g2+ 70.\u2654e2 \u2655xf2+ 71.\u2654d3 and now 71... \u2655c2+ 72.\u2654xc2 forces the stalemate.\n\nSo the correct move is 67.f3! and after 67...\u2655d2+ 68.\u2654h3 \u2655b4 then 69.\u2657xg6 \u2658xg6 70.\u2655xg6 now works because White reacts to 70...\u2655xh4+ with 71.\u2654g2!. And now there is no stalemate left in the position, because the black h-pawn can move.\n**35. Under Russian colours**\n\n**The World Championship 1990:  \n_Garry Kasparov against Anatoly Karpov_**\n\nThe cities of Lyon (France) and Wellington (New Zealand) had bid to be the venue for the World Championship of 1990. In New Zealand the match was to take place as part of the celebrations of the 150th anniversary of the country. The bids were submitted to the FIDE congress of 1989 in Mayag\u00fcez (Puerto Rico).\n\nAlso up for discussion at this meeting of FIDE was how the organisation of chess professionals, the Grandmaster Association (GMA), could cooperate with FIDE in the question of the organisation of the World Chess Championships. However, FIDE president Campomanes largely ignored the GMA representatives Bessel Kok, Mikhail Botvinnik, Jan Timman and Lothar Schmid. In view of the unclear situation Wellington finally withdrew its bid and so all that was left to FIDE was the offer from Lyon. Kasparov organised another sponsor in New York, the entrepreneur, film producer and billionaire Ted Field, and finally agreement was reached to hold the match half in New York and half in Lyon.\n\nMeanwhile, in Kasparov's home country of Azerbaijan pogroms had started against the Armenian part of the population. Kasparov's family and many of his friends no longer felt safe in Baku and took refuge first of all in the resort of Zagulba, 40 kilometres from Baku. But there too, the situation soon became volatile. At the instigation of the chairman of the USSR Chess Federation Popov a plane was sent to Baku to fly out the Kasparov group. On the 17th January 1990 the World Champion left his homeland for ever with his family, friends and their families.\n\nKasparov, who had always previously held his training camp in Zagulba, now had to organise his preparations afresh. At the time there were disagreements with his trainer of many years, Nikitin, who thereupon ended his cooperation with Kasparov. Kasparov completed a first section of his preparation together with Alexander Beliavsky in March-April of the year on the outskirts of Moscow. In the middle of June, Kasparov set up his training camp in Murcia (Spain), preparing himself there with his regular seconds Mikhail Gurevich, Sergey Dolmatov and Zurab Azmaiparashvili. In addition, this time Gia Giorgadze and Kasparov's previous trainer Alexander Shakarov were part of the team. During the preparation, moreover, Kasparov played a training match against Lev Psakhis, whom he defeated 5:1. In September of the year there followed a third training session in the USA. Kasparov had rented a villa there to the north-east of New York on the island Martha's Vineyard. Alexander Beliavsky was also present.\n\nAs the loser of the previous match Karpov was seeded into the quarter-finals of the candidates' matches and had qualified as challenger with victories over Johann Hjartarson (3\u00bd:1\u00bd), Artur Jussupow (4\u00bd:3\u00bd) and Jan Timman (6\u00bd:2\u00bd). For the WCh match against Kasparov, Karpov put his trust in his official seconds Lajos Portisch and Ron Henley. In addition he was supported by Igor Zaitsev, Mikhail Podgaets, Alexander Kharitonov and Alexey Kuzmin. Karpov's head of delegation was Nikolai Krogius. In this match Karpov was supported psychologically by Rudolf Zagainov, who had worked for Kortchnoi in the candidates' final of 1974. Karpov also hired him later for his candidates' match against Nigel Short in 1992. Shortly before the start of the match in New York Karpov and his team occupied a roomy house in 63rd Street in East Side Manhattan, but during the match they moved to a hotel because it was too noisy in East Side Manhattan. Kasparov resided with his team only a few blocks away on the 17th floor of the Regency Hotel.\n\nAs previously fixed, the WCh match was played from 8th October to 20th December 1990, half in New York (till the 12th game on 7th November) and in Lyon (from the 13th game on 24th November). The maximum number of games was to be 24. The prize fund consisted of three million dollars, of which 5\/8 went to the winner. The chief arbiter was Geurt Gijssen.\n\nThe venue in New York was the Hudson Theatre, not far from Broadway and Times Square. The tickets cost between ten dollars (reduced rate) and 100 dollars. In spite of the high prices the theatre was well filled during the games with several hundred spectators. The press centre was on the eighth floor of the Macklowe Hotel. Among the approx. 300 accredited journalists were many well-known names such as Miguel Najdorf, Samuel Reshevsky, Mikhail Tal and Efim Geller. The press officers for the organisation were Yasser Seirawan and Larry Christiansen.\n\nFIDE president Florencio Campomanes was not present, being as he was at that time in Moscow so as to gain support for his re-election at the subsequent FIDE presidential election. The latter was held at the FIDE congress during the Chess Olympiad in Novi Sad (16th November till 4th December 1990) and Campomanes was once more re-elected, coming in front of the Puerto Rican Narcisco Rabell Mendez and the Spaniard Roman Toran, who was supported by Karpov. Campomanes had suggested as vice-president the Yugoslav grandmaster and publisher of the _Chess Informant_ Alexander Matanovic and in doing so secured the support of the organisers of the Chess Olympiad. He had also, moreover, arranged paid posts as arbiters at the Chess Olympiad for a series of delegates.\n\nThe influential Soviet Committee for Sports had in its session before the Chess Olympiad actually decided not to support Campomanes this time. However, there was no consensus within the committee, and so before the election the Soviet delegate Alexander Bach announced on Yugoslav television that the Soviets would support Campomanes, which many federations from the eastern block took as an indication of which way to vote.\n\nBefore the match Kasparov had explained that he wanted to play under the Russian, not the Soviet flag, and gave voice to this decision in a common press conference in the Macklowe Hotel. Karpov commented on this with the words: 'I consider myself more of a Russian than Kasparov, but it does not matter to me under which flag he plays', but he then later protested against Kasparov's wish, since it went against FIDE rules. These laid down that a player was obliged to play under the flag of the state of which he was a citizen. But in 1990 there was not yet a Russian state. By a decision of the arbitration committee the flags were then removed from the table. After this decision, however, Kasparov appeared with a Russian flag on his lapel.\n\nAt the opening press conference Karpov surprisingly suggested that in the event of a 12:12 draw they should play on until one player won a game. Kasparov refused this, however, pointing out that Karpov was the challenger and so had to defeat him.\n\nThe psychological warfare had, according to Kasparov, already started on the 14th September when in the weekly magazine _Literaturnaya Rossiya_ an interview with his previous confidant Tofik Dadashev appeared under the headline 'How I helped Kasparov'. Moreover, Kasparov felt attacked by the way he was portrayed in Karpov's new book _Sestra moia Kaissa_ ('My sister Caissa'). The book had appeared in Russian and English just at the right time for the match in 1990 and was presented by Karpov in New York on the day of the press conference. Whereas in his book Karpov sees himself as a fighter for justice and democracy, Kasparov is described as 'cowardly' and as someone who distorts things. The book was Karpov's reply to Kasparov's _Politische Partie_ , in which the things had been portrayed exactly the other way round: Karpov was portrayed in it as the representative of the old unfair Soviet system, whereas Kasparov saw himself as the representative of glasnost and perestroika.\n\nOn the 7th October the opening ceremony was held as a charity event in the Waldorf-Astoria Hotel. 700 guests turned up and paid 1500 dollars each for entry. The proceeds benefitted the organisation 'Chess in the schools'. The drawing of lots for colours during this event gave Karpov the white pieces for the first game \u2013 for the fourth time in the total of five matches between these two players, as Kasparov pointed out in his report on the match.\n\nIn the first game, in an excellent position Karpov missed the best continuation and had to content himself with a draw. Kasparov then took the lead in the second game with white against Karpov's Ruy Lopez Zaitsev Variation, Karpov equalised in the seventh game after Kasparov gave away the game with a blunder. The other ten games played in New York were drawn, though some of them took a very exciting course. More than once Kasparov had to trust to luck to get a draw from bad positions.\n\nThere were more decided games after the move to Lyon, where the match was continued on 24th November 1990 in the Palais des Congr\u00e8s. In addition to their share of the prize fund, the French organisers had put up a valuable special prize for the winner of the match: a trophy created by the Paris jeweller Korloff, made of gold, platinum and diamonds, in which two 'Ks' were interwoven.\n\nAfter three draws, of which game 14 especially was praised as outstanding, Kasparov won the 16th and 18th games and Karpov the 17th. The 16th game, which Kasparov had opened with the until then little played Scotch Game, was adjourned twice and not won by Kasparov until move 106. It is the longest decided World Championship game in the history of chess. In the 19th game Kasparov offered a draw in a clearly better position. The World Champion justified this with a general state of exhaustion he had reached, which led to him no longer evaluating the position correctly.\n\nThe supposition for the arrangement was that the players wanted to keep the match exciting for as long as possible and had arranged this with the organisers. A further possibility was considered, namely that the players were aiming for a score of 12:12, because this would be the most profitable with the bookmakers since both players had perhaps placed bets there via middle-men.\n\nA win in the 20th game saw Kasparov increase his lead to two points. He was once more able to defeat Karpov's Ruy Lopez Zaitsev Variation. In game 23 Karpov again shortened the lead, which further fed the rumours that 12:12 had been pre-arranged. In the 24th and final game, played on New Year's Eve, Karpov was losing when Kasparov offered him a draw. Karpov accepted and Kasparov had once more defended his title. Looking back on it, Kasparov was not satisfied with his play during this World Championship match. Many games were marked by mistakes and in a series of games the title defender let slip a possible win.\n\n **Kasparov \u2013 Karpov**\n\nLyon, 20th game  \n15th December 1990  \nRuy Lopez (C92)\n\n**1.e4 e5**\n\nIn this match with black Karpov trusted only the open games with 1...e5, but suffered several defeats against the Ruy Lopez and the Scotch.\n\n**2. \u2658f3 \u2658c6 3.\u2657b5**\n\nIn the 14th and 16th games Kasparov chose the Scotch Game 3.d4, which until then had hardly ever been played at the top level.\n\n**3...a6 4. \u2657a4 \u2658f6 5.0-0 \u2657e7 6.\u2656e1 b5 7.\u2657b3 d6 8.c3 0-0 9.h3 \u2657b7 10.d4 \u2656e8**\n\nAt that time a very modern interpretation of the Closed Variation of the Ruy Lopez, this was an idea of Karpov's trainer Igor Zaitsev and is named after him.\n\n**11. \u2658bd2**\n\n11.\u2658g5 \u2656f8 12.\u2658f3 \u2656e8 13.\u2658g5 could bring about a draw through a threefold repetition of the position. But of course that is not the intention of the player with white.\n\n**11... \u2657f8 12.a4 h6 13.\u2657c2 exd4 14.cxd4 \u2658b4 15.\u2657b1 c5**\n\nIn game 2 of the match 15...bxa4 16.\u2656xa4 a5 etc. was played. White won.\n\n**16.d5 \u2658d7 17.\u2656a3 f5 18.\u2656ae3!?**\n\nKasparov seeks to improve upon his play from the 4th game, where he had continued with 18.exf5. The idea comes from Kasparov's training session with Alexander Beliavsky.\n\n**18... \u2658f6 19.\u2658h2**\n\nThe prepared novelty. 19.exf5 \u2656xe3 20.\u2656xe3 \u2658bxd5 with an unclear position.\n\n**19... \u2654h8 20.b3**\n\nWhite prepares the development of his bishop to b2.\n\n**20...bxa4**\n\nIf Black, after 20...fxe4 21.\u2658xe4, takes the pawn on d5, no matter which way, White obtains a strong attack: 21...\u2658fxd5 22.\u2656g3 \u2658f6 23.\u2658xf6 \u2655xf6 24.\u2657d2 \u2656xe1+ 25.\u2657xe1 \u2655f7 26.\u2657c3 \u2658d5 27.\u2657a1 with a strong attack (Kasparov) or 21...\u2657xd5 22.\u2658xf6 \u2656xe3 23.\u2656xe3 \u2655xf6 24.\u2658g4! \u2655f4 25.axb5 axb5 26.\u2657b2+\u2013. And 21...\u2658bxd5? is also unfavourable for Black: 22.\u2658xf6 \u2656xe3 (22...\u2658xe3 23.\u2655d3+\u2013) 23.\u2656xe3 \u2658xf6 24.\u2658g4 d5 25.\u2655c2 \u2658e4 26.f3. Gurevich suggested 20...c4 21.bxc4 bxc4 22.\u2658xc4 (22.\u2657b2!?) 22...fxe4, with a complicated position.\n\n**21.bxa4 c4?**\n\nNow 21...fxe4 22.\u2658xe4 was the better choice, since here 22...\u2657xd5 appears playable: 23.\u2658xf6 \u2656xe3 24.\u2656xe3 \u2655xf6 and then 25.\u2658g4 (25.\u2657d2 \u2656b8 is unclear) is followed by 25...\u2655a1 intending 26.\u2657d2 \u2657a2 (Kasparov). On the other hand 22...\u2658fxd5?! would be risky: 23.\u2656g3 with a strong attacking position.\n\n**22. \u2657b2! fxe4 23.\u2658xe4 \u2658fxd5**\n\n23...\u2658bxd5 would be worse on account of 24.\u2658xf6 \u2656xe3 (24...\u2658xe3 25.\u2655h5 gxf6 26.\u2655g6+\u2013) 25.\u2656xe3 gxf6 (25...\u2658xf6 26.\u2655c2+\u2013, 25...\u2658xe3 26.\u2655h5+\u2013) 26.\u2656g3+\u2013.\n\n**24. \u2656g3**\n\nHere 24.\u2658xd6 would have been a letdown: 24...\u2658xe3 25.\u2658f7+ \u2654g8 26.\u2658xd8 \u2658xd1 27.\u2656xe8 \u2658xb2 28.\u2657g6 \u2657d5 and Black has a winning position. After 24.\u2655h5 Black obtains counterplay with 24...c3!.\n\n**24... \u2656e6!**\n\nKarpov still finds a good defensive resource in a complicated position. 24...\u2658d3 would be worse: 25.\u2657xd3 cxd3 26.\u2655h5 with an attack.\n\n**25. \u2658g4?**\n\n'A mistake. After it Black can equalise.' (Kasparov) 25.\u2658f3 was more accurate so as to get to g5 or d4 with the knight.\n\n**25... \u2655e8?**\n\nKarpov does not take the chance which was on offer. The correct move was 25...\u2658d3!? (Spassky), for example 26.\u2657xd3 cxd3 and now 27.\u2655d2 \u2655e7 28.\u2658xh6 \u2656xe4 29.\u2656xe4 \u2655xe4 30.\u2658f7+ \u2654g8 31.\u2658h6+ leads to a draw by perpetual check. 25...\u2655d7!? (Sch\u00fcssler) was also worth considering.\n\n**26. \u2658xh6! c3**\n\nThe advance of the c-pawn does not bring the hoped for relief. 26...\u2656xh6 is followed by 27.\u2658xd6 and then 27... \u2655h5 is relatively the best: 28.\u2656g5 \u2655xd1 29.\u2658f7+ \u2654g8 30.\u2658xh6+ \u2654h8 31.\u2656xd1 c3 32.\u2658f7+ \u2654g8 33.\u2657g6 and then 33...cxb2 is followed by mate with 34.\u2656h5 b1\u2655 35.\u2656h8#.\n\n**27. \u2658f5 cxb2 28.\u2655g4**\n\nWhite now has an overwhelming attacking position. Black has no defence.\n\n**28... \u2657c8**\n\n28...\u2656c8 29.\u2654h2 intending 30.\u2658g5+\u2013.\n\n**29. \u2655h4+ \u2656h6**\n\n29...\u2654g8 30.\u2654h2 intending 31.\u2658g5+\u2013.\n\n**30. \u2658xh6 gxh6 31.\u2654h2!**\n\nNot 31.\u2658f6? \u2655xe1+ 32.\u2654h2 \u2657e6\u2013+, nor 31.\u2658xd6? \u2655xe1+ 32.\u2654h2 \u2655e6. Another strong move was 31.\u2656e2+\u2013.\n\n**31... \u2655e5**\n\n31...\u2656a7 is followed by mate: 32.\u2658f6 \u2655f7 33.\u2656e8 \u2658xf6 34.\u2655xh6+ \u2658h7 35.\u2655xh7+ \u2655xh7 36.\u2656xf8+ \u2655g8 37.\u2656fxg8#.\n\n**32. \u2658g5 \u2655f6 33.\u2656e8! \u2657f5**\n\n**34. \u2655xh6+**\n\nA quicker way was 34.\u2658f7+! \u2655xf7 35.\u2655xh6+ \u2657h7 36.\u2656xa8 followed by 37.\u2656xf8+.\n\n**34... \u2655xh6 35.\u2658f7+ \u2654h7 36.\u2657xf5+ \u2655g6 37.\u2657xg6+**\n\n37.\u2656xg6! \u2658e7 38.\u2656xe7!+\u2013.\n\n**37... \u2654g7 38.\u2656xa8 \u2657e7 39.\u2656b8 a5 40.\u2657e4+ \u2654xf7 41.\u2657xd5+**\n\nBlack resigned.\n\nOn the 2nd January 1991 at the closing ceremony, Kasparov was hailed as the title defender and received his cheque and the Korloff trophy, which was reputed to be worth a million dollars. He sold it to the Kalmykian millionaire Kirsan Ilyumzhinov, the future FIDE president, and donated the cash to a foundation for the support of Armenian refugees from Azerbaijan. According to Kasparov, his second Zurab Azmaiparashvili is supposed to have received an offer before the match of 100000 roubles to betray Kasparov's opening preparation. This offer came, as did one to Josif Dorfman in a previous match, from the Russian gambling mafia, which wanted to make money by betting on the outsider \u2013 and that was Karpov.\n**Part IV \u2212 The time of schism**\n\nAfter the WCh cycle of 1990-1993 Nigel Short became the challenger \u2013 for the first time since Robert Fischer a challenger who had not been born in the Soviet Union. The struggle of a player from the West had to be worth a lot of money to sponsors, Short believed, but after the numerous matches between Kasparov and Karpov the interest in the World Chess Championship had cooled off somewhat. The result achieved by FIDE in their negotiations for a prize fund seemed too little to Short and he talked Kasparov into running the WCh match without any participation by FIDE and finding by themselves organisers, who were prepared to come up with more prize money. The exclusion of FIDE, which always diverted a considerable share of the offer by the organisers for its own purposes, automatically increased the share of the players.\n\nFIDE president Florencio Campomanes reacted to the decision by Kasparov and Short with some hectic measures. Firstly he removed the World Championship title from Kasparov; then he threw the two 'dissidents' out of FIDE, which meant, for example, that their games were no longer evaluated for the FIDE Elo world ranking list. Finally Campomanes organised a counter World Championship with the two players who had gone furthest in the candidates' matches behind Short, namely Karpov and Jan Timman.\n\nIn doing this Campomanes created a split in the chess world, which now had to live with two competing World Championships. Kasparov and Short founded a new organisation for the marketing of the World Chess Championships, the _Professional Chess Association_ (PCA). After the match against Short, which Kasparov won clearly, he organised with the help of his sponsor Intel a candidates' tournament and matches in competition with FIDE. The victor of this series was Viswanathan Anand, who, however, lost his WCh match against Kasparov in New York in 1995. FIDE also completed a complete WCh cycle, though without the best player in the world by far, Kasparov.\n\nIn 1996 Campomanes was obliged to retire as FIDE president and his successor was the president of the autonomous Russian Republic of Kalmykia Kirsan Ilyumzhinov. The latter completely got rid of the previous three year WCh cycle with its interzonal tournaments, candidates' matches and WCh match and replaced it with one big WCh tournament in knock-out mode with 128 players. But the top players gradually refused to recognise this mode in which form on the day and chance played too great a role. Between 1997 and 2004 five such WCh tournaments were played in k.-o. format.\n\nIntel soon quit as sponsor of the PCA and attempts by Kasparov to find a replacement sponsor failed. Between 1998 and 2000 Kasparov made efforts to find a somehow credible challenger to play against him. Finally Kramnik was designated as challenger. In the WCh match in London 2000 Kasparov could not find a way to deal with his opponent and was dethroned as World Champion. Without Kasparov, however, the claim to be sole 'Classical World Champion' compared to the FIDE k.-o. World Champion became harder to sustain. In the meantime there had been attempts to do away with the split between the World Championships. In 2002 during a tournament in Prague there was a meeting between the two camps and other important people on the chess scene and a plan was worked out for the reunification of the World Championships.\n**36. Excluding FIDE**\n\n**The World Championship 1993:  \n_Garry Kasparov against Nigel Short_**\n\nNigel Short was born on the 1st June 1965 in Leigh (England). His parents were David and Jean Short; they had two other sons, Martin and Jonathan. Nigel Short grew up in Atherton where he attended St Philip's Primary School on Bolton Old Road. He then went to Bolton School and Leigh College.\n\n**Nigel Short (born in 1965)**\n\nHe learned to play chess at the age of six by looking on when his father was actually explaining the rules to Nigel's older brother Martin. Nigel understood the particularities of the game at enormous speed and was soon able to beat his father regularly. His hunger for chess, which was in any case great, was further sharpened by the attention grabbing WCh match between Spassky and Fischer, Reykjavik 1972, for the British daily papers gave regular reports on it on their front pages.\n\nNigel's mother now wanted to enrol her seven year old son in a chess club, but the Bolton Chess Club did not accept him on account of his young age. Through the mediation of his headmaster Nigel then received lessons from Richard Furness, president of the Manchester Chess Association and chess teacher at a neighbouring school. Furness then took Nigel Short for the first time to his chess club, the Golborne Chess Club.\n\nOn Furness' suggestion Nigel Short took part in the 'Liverpool Schools Chess Congress' over Easter 1973 and in his first tournament won four out of ten games. At ten Nigel Short achieved a win against Viktor Kortchnoi in a simultaneous display. In 1977 at the age of twelve he became the youngest player in a British championship. Two years later he shared first place in the British championship in Chester with John Nunn and Robert Bellin. After taking part in the Hastings tournament of 1979\/80 Short received the title of International Master, being at that point the youngest player of all time to receive that title and breaking the record set by Bobby Fischer in 1958. At the Junior World Championship of 1980 in Dortmund, 15 year old Short was second behind Kasparov. In 1984 he became an international grandmaster at 19 and was the youngest grandmaster in the world at that time.\n\nIn the interzonal tournament of Biel 1985 Short was the first Briton to qualify for the candidates' rounds but at the candidates' tournament in Montpellier only reached tenth place. In the next WCh cycle Short again qualified from the interzonal tournament in Subotica 1987 for the candidates' rounds, which this time were held in match format. He defeated Gyula Sax, but was then eliminated by his fellow countryman Jonathan Speelman. With third place in the interzonal tournament of 1990 in Manila Short once more reached the candidates' matches. In the last sixteen he defeated Boris Gelfand and this time got through the quarter-final against Jonathan Speelman after a playoff. His victory over Anatoly Karpov (4:2 in wins with four draws) in the semi-final ended the latter's participation in WCh and candidates' matches which had been unbroken since 1974.\n\nThe semi-finals took place in Linares in the Hotel Anibal, which belonged to the patron and organiser Luis Rentero, who had offered for both matches together a prize fund of 300000 Swiss francs. Short, who was supported in Linares by his second Lubomir Kavalek, and Karpov, who had Vladimir Epishin and Mikhail Podgaets with him as seconds, were amusingly put up in neighbouring rooms in the Hotel Anibal with only thin Spanish walls between them, so that Short, frightened about being heard in the neighbouring room, always whispered when he was analysing with Kavalek. In the second semi-final Jan Timman got through against Artur Jussupow, making it clear that the next challenger would be a player from the West.\n\nIn 1988, when it looked as though that would never be the case, the chess computer firm Hegener and Glaser (manufacturers of the _Mephisto_ chess computer) had as an advertising gag offered a bonus of a million Swiss francs for the first western challenger. Now that things were becoming serious for Hegener and Glaser, the Munich chess computer manufacturer tried to wriggle out of this old promise, giving as a justification that the break-up of the USSR had altered the conditions. However, a law suit by Short and Timman ensured that the promise of a bonus remained valid. In the final in San Lorenzo de El Escorial (near Madrid, Spain), with a prize fund of 300000 Swiss francs, Short defeated Timman (5:3 in wins and five draws) and thus became the first non-Soviet WCh challenger since Fischer.\n\nLos Angeles and a consortium around the businessman Jim McKay had put in an offer of four million dollars to stage the match between Kasparov and Short. Kasparov was closely linked to this consortium. After the unrest in Los Angeles in reaction to the mistreatment of the Afro-American Rodney King by four white policemen ('LA riots') the offer from Los Angeles was, however, withdrawn. FIDE collected the deposit of 400000 dollars and re-opened the bidding process. By 8th February FIDE had two bids, one from Santiago de Compostela of a million Swiss francs, the minimum bid for World Championships, and one from the Yugoslav banker Jezdemir Vasiljevic and his bank Yugoscandic of five million dollars. Vasiljevic had already financed the return match between Fischer and Spassky in 1992. But at this point Yugoslavia was still under a UN embargo. To get round this, Vasiljevic proposed to hold the WCh match in Bucharest. FIDE declined the offer, however, in view of the UN sanctions. Soon afterwards Vasiljevic fled to Israel and took with him a series of deposits of his bank's customers, including a part of the prize money of Bobby Fischer from the 1992 match.\n\nFIDE then extended the tendering process till the 22nd February 1993. David Anderton, for many years the director of the British Chess Federation, then formed a consortium of donors with the intention of bringing the match to Manchester. Short expected an offer of around three million pounds sterling, but just before the deadline for the tenders Manchester made it known that the total of the bid would 'only' be 2.185 million Swiss francs. On the 22nd February FIDE opened all the bids which had until then been in sealed envelopes. As well as the offer from Manchester a 'London Chess Group' had put up a prize fund of 2.3 million Swiss francs. In addition there was a bid from _Channel Four_ of 2.6 million Swiss francs, though this was only in conjunction with the cession of all TV rights. But the official offer from Manchester was now 2.538 million Swiss francs. Apparently this had been improved upon at short notice with knowledge of the offer from London.\n\nAccording to the rules of FIDE the players now had to be asked for their opinion, but at that time Short was on a trip to Greece, where he had boarded a ferry on the 22nd February and could not be contacted for two days. On 23rd February FIDE announced that 'after consultations with both players' the WCh match would be awarded to Manchester. In reality Campomanes had only spoken to Kasparov, who did not find Manchester ideal but in principle declared himself willing to play there. When Short learned on the 24th February from his confidant Dominic Lawson of the awarding of the match to Manchester with a prize fund he considered disappointing, he was very angry about not having been informed beforehand. Short phoned Kasparov, who at that time was in Linares, and suggested organising the match 'inhouse' and without FIDE. 'I have been waiting for this moment for eight years', was Kasparov's reaction.\n\nOn 26th February Kasparov and Short put out a common press statement, in which they announced they would be playing the match outwith FIDE and in addition made known the founding of the Professional Chess Association (PCA). 10% of the prize money was to go to the new organisation. FIDE replied on the same day, pointing out that only they held the rights to the World Championship in chess and that all measures would be taken to defend those rights. Kasparov and Short now requested all interested parties to submit bids by the 19th March for the 'Professional World Chess Championship'. It was announced that the offers were to be opened \u2013 unlike in FIDE's usual way \u2013 publicly on 22nd March in Simpson's-in-the-Strand.\n\nIn the meantime, Campomanes had got in touch with Kasparov and had tried to get the World Champion back on board the FIDE ship. In London on the 16th March there was a secret meeting, of which Short knew nothing, between Kasparov, his London manager Andrew Page and his New York lawyer Bob Rice and Campomanes and Tony Ingham, the organiser of the Manchester bid. Ingham raised the offer of prize money to 1.6 million pounds sterling. The negotiations soon became known.\n\nShort and Lawson then feared that Kasparov would strike separate deals with the Manchester group and finally let the PCA match drop. Keene, who in the meantime was already trying together with the _Times_ to organise the match in London, warned Kasparov that the press conference in Simpson's-in-the-Strand could if necessary be held without him and that the WCh match would then fall into two halves. Finally Kasparov's negotiations with Campomanes broke down, because Kasparov demanded the transfer of the rights to the World Champion title to him and in addition the reimbursement of the 400 000 dollars deposit from the Los Angeles group. Campomanes was not ready to concede either of these. The _Times_ had in the meantime raised their original prize fund by 300000 pounds so as to equal the 1.6 million pounds bid by the Manchester group.\n\nAfter the failure of the negotiations with Kasparov, Campomanes continued to try to drive a wedge between the two players, by playing to Short a cassette with notes of the negotiations. It was possible to hear how Kasparov explained the way he would convince Short to nevertheless play the match through FIDE, after the offers on the table had possibly been disappointing for Short.\n\nOn the 22nd March the bids were opened. Of the five submitted, two were superior to the bids collected by FIDE. The London Chess Group around Matthew Patten offered two million pounds sterling, the _Times_ , together with the TV broadcaster _Teleworld_ , 1.7 million pounds. In addition the otherwise usual deduction of 25% of the prize fund for FIDE did not have to be paid and instead benefitted the players. When the bids were presented in detail, however, the London Chess Group could not present any guarantees and Kasparov expressed doubts about the seriousness of the bid. The following day, the _Times_ announced without consultation with the PCA, that it had been accepted. The London Chess Group thereupon withdrew its bid.\n\nFor their decision to play the WCh match without FIDE, Kasparov and especially Short were much criticised by the chess world. FIDE, or its president Campomanes, reacted on the 23rd March and deposed Kasparov as World Champion and Short as challenger and at the same time had the players' names removed from the Elo list.\n\nShort prepared for the WCh match with his trainer Lubomir Kavalek in the latter's home town in Reston, Virginia (USA). One day Patrick Wolff called and asked to speak to Short. The US grandmaster, son of the well-known philosophy professor Robert Paul Wolff, named some of the highly secret variations which Short intended to play against Kasparov, for example the \u2657c4 Variation against the Sicilian Najdorf, and offered his help. Short was furious. Kavalek was then able to find the leak quickly. Kavalek had spoken about it with his friend and fellow countryman, grandmaster Lubomir Ftacnik, and Ftacnik had then passed on the details to Patrick Wolff, who was a specialist in this variation. In addition, in a simultaneous display in 1988 Wolff had defeated Kasparov in only 25 moves. On the very next day Ftacnik faxed a written apology to Short.\n\nKasparov too was offered help \u2013 as Short found out, from his compatriot Anthony Miles. Kasparov declined it and preferred to put his trust in the help of Sergey Makarichev, Zurab Azmaiparashvili and Alexander Beliavsky. One month before the start of the match Short engaged Jonathan Speelman as a second, and on the 13th August Robert H\u00fcbner too.\n\nThe opening ceremony for the WCh match was held on Friday, the 3rd September 1993 in Simpson's-in-the-Strand. The drawing of lots decided that Kasparov would start the first game with white. One day before the start of the match Short had his suite in the Savoy Hotel, apart from the rooms of his seconds, searched for bugs.\n\nCampomanes had already warned Kavalek long beforehand that Kasparov could demand the help of the KGB at any time. Short, however, was more anxious that journalists would try to get exclusive information this way. Shortly before the first game the chess enthusiast and friend of Kavalek, Czechoslovakian film director Milos Forman, turned up and wished Short luck.\n\nThe venue for the match was the Savoy Theatre on Trafalgar Square, quite close to the legendary chess caf\u00e9 Simpson's-in-the-Strand. The WCh match was set for 24 games. The arbiters were Yuri Averbakh and the Spaniard Carlos Falcon.\n\nIn the first game Short, with the black pieces, offered the Marshall Gambit in the Ruy Lopez. As time passed, Kasparov obtained a clear winning position after a mistake by Short, but then gave it away. Finally both players were in time trouble. Kasparov offered a draw on move 38, Short declined without noticing that he had only ten seconds left on his clock, and overstepped the time limit on his 39th move in a level position. In the second game Short replied to Kasparov's Najdorf Variation with the then most popular move 6.\u2657g5. Kasparov chose a side variation and once more avoided any possible preparation. The game ended in a draw. In the third game Short started with the black pieces an attack on the king which is typical of the Ruy Lopez. Kasparov defended inaccurately and thus allowed Short real winning chances, but the challenger missed the best opportunity and landed up in a lost endgame \u2013 Kasparov took a 2:0 lead (in wins).\n\nAfter the game there was an argument between Short and his second Kavalek, who in Short's opinion had so far in London hardly taken any part in the analytical work or even done any. Short now had had enough of his second, for whose work he had paid 125000 dollars, and sent him home.\n\nIn the fourth game Kasparov picked up the gauntlet. He replied to Short's 6.\u2657g5 in the Najdorf Variation with his standard weapon, the Poisoned Pawn Variation \u2013 thus for the first time in this match going into a theoretical duel \u2013 and won the game. Backed up by a 3:0 lead, Kasparov changed in the fifth game to 1.d4. Short was able to surprise the World Champion with a variation in the Nimzo-Indian Defence which had rarely been played up till then, but it brought him no more than a draw. In the following game Short took Kasparov on with the latter's own weapon, the Sozin Variation as his reply to Kasparov's Najdorf System. In a sold-out Savoy Theatre, in front of 1000 spectators, Short pressed Kasparov hard in an attack on the king. But the English player could not manage his first whole point here either. The game was finally drawn. Nevertheless, his fighting spirit earned him a long burst of applause.\n\nThe seventh game brought the World Champion his fourth victory and Kasparov fulfilled his own prediction when he had commented on the forthcoming match with the words: 'His name is Short, and it will be short.' Kasparov had gone back to 1.e4 and inflicted on Short in the third Ruy Lopez of the match the third defeat. A guest of honour for this game was Lady Diana, Princess of Wales, who before the game had wished Nigel Short luck.\n\nBefore the eighth game Robert H\u00fcbner presented Short with a new attacking idea in the Sozin Variation. After a tactically very complicated game Short reached a winning position after an attack on the king, but at the decisive moment he once more missed the best continuation \u2013 again the game ended with what from the English player's point of view must have been a disappointing draw. The course of the next game was also very unfortunate for the challenger. Kasparov repeated the opening from the fifth game, brought out an innovation and was finally two pawns up in a rook ending. After an inaccuracy Short could have reached a drawn position, as Jonathan Speelman discovered immediately after the game, but the challenger over-looked this possibility and conceded his fifth defeat in this match.\n\nBut that was not yet the end of Short's losing streak. In the tenth game, after a positional queen sacrifice in the Sozin Variation he again reached a clear winning position, which he squandered in time trouble just like the previous ones. The game was drawn. There followed three more draws till Kasparov scored his sixth win in the 15th game in the Exchange Variation of the Queen's Gambit. In game 16 the challenger finally had a success and with the help of the Sozin Variation notched up his first win in this match.\n\nAfter the next four games had ended in draws, the WCh match came to its premature finish at the score of 12\u00bd:7\u00bd (6:1 wins) on the 21st October 1993. For the sake of the public, the two players occupied the following match days with a four game rapid chess match, which Kasparov won by 4:0.\n\n **Kasparov \u2013 Short**\n\nLondon, 9th game  \n25th September 1993  \nNimzo-Indian Defence (E35)\n\n**1.d4 \u2658f6 2.c4 e6 3.\u2658c3 \u2657b4 4.\u2655c2 d5 5.cxd5 exd5 6.\u2657g5 h6 7.\u2657h4**\n\nThe alternative is 7.\u2657xf6 \u2655xf6 8.a3 \u2657xc3+ 9.\u2655xc3 0\u20130 10.e3 c6 11.\u2658f3 \u2657f5 etc.\n\n**7...c5 8.dxc5 g5**\n\n8...0-0 9.e3 \u2658bd7 etc. is less sharp.\n\n**9. \u2657g3 \u2658e4 10.e3 \u2655a5**\n\nShort's novelty in this line.\n\n**11. \u2658ge2!**\n\nKasparov's prepared improvement upon the 5th game. In it 11.\u2657e5 0-0 12.\u2657d3 \u2658c6 was played and the game was drawn by move 18.\n\n**11... \u2657f5 12.\u2657e5**\n\nHere at the latest Short had to stand on his own feet, since in his preparation he had not taken this move into account.\n\n**12...0-0**\n\nPlayed after 41 minutes of thought. Kasparov has prepared a tactically explosive variation, which can hardly be calculated in all its subtleties over the board without preparation. The discovered attacks by the knight were not so good: 12...\u2658g3? 13.\u2655b3 \u2658xh1 14.\u2657xh8+\u2013 (Ftacnik), or 12...\u2658xf2? 13.\u2655xf5 \u2658xh1 14.\u2655c8+ \u2655d8 15.\u2655xd8+ \u2654xd8 16.\u2657xh8+\u2013 (Ftacnik), and also 12...\u2658xc3 13.\u2655xf5 \u2658e4+ (13...\u2658xe2+ 14.\u2654xe2 0-0 15.\u2654f3 \u2658c6 16.\u2657f6 \u2656fe8 17.\u2657d3+\u2013) 14.\u2658c3 0\u20130 (14...\u2658xc3? 15.\u2655c8+ \u2655d8 16.\u2655xd8+ \u2654xd8 17.bxc3+\u2013) 15.\u2657d3 (Ftacnik) favours White.\n\n**13. \u2658d4 \u2657g6?!**\n\nThis move turns out to be inferior. Later 13...\u2658xc3 14.\u2658xf5 \u2658e4+ and 13...\u2656e8 were tried, both being very playable for Black.\n\n**14. \u2658b3 \u2658xc3**\n\n14...\u2655d8 15.\u2657d3 gives White good play, but 14...\u2655a4!? seems playable, though after 15.\u2657d3 \u2658d7 16.\u2657d6 \u2657xc3+ 17.bxc3 \u2656fe8 18.h4 White has the initiative.\n\n**15. \u2657xc3!**\n\n15.\u2658xa5 achieves nothing for White: 15...\u2658a4+ 16.\u2657c3 \u2657xc3+ 17.\u2655xc3 \u2658xc3 18.bxc3 \u2656c8 19.\u2658xb7 \u2658d7 with level chances.\n\n**15... \u2657xc2 16.\u2658xa5 \u2657xc3+**\n\nHere Short had used up 90 minutes of his thinking time, Kasparov only 28 minutes. In the press room Short's pessimistic seconds considered the game to be already strategically lost here. If 16...\u2657xc5 then 17.\u2658xb7 with advantage to White.\n\n**17.bxc3 b6 18. \u2654d2!?**\n\nThe 'normal' way was 18.\u2658b3!? or 18.\u2656c1!?.\n\n**18...bxa5**\n\nBut it was worth considering 18...\u2657g6!? 19.cxb6 axb6 20.\u2658b3 \u2658c6 21.a4 with advantage to White, but Black has counterplay on the a- and c-files.\n\n**19. \u2654xc2 \u2656c8 20.h4**\n\nWith the help of his prepared variation Kasparov has achieved a psychological effect and after Short's mistake on move 13 also an advantage on the board. But the game is still far from won.\n\n**20... \u2658d7 21.hxg5 \u2658xc5!?**\n\nSeeking counterplay. In the event of 21...hxg5 22.\u2656h5 f6 23.\u2657d3 \u2656xc5 24.\u2656ah1 White has a clear advantage.\n\n**22.gxh6 \u2658e4 23.c4 \u2658xf2**\n\nAfter 23...dxc4? 24.\u2656h4 \u2658xf2 25.\u2657xc4 White would have solved all his problems.\n\n**24. \u2656h4 f5 25.\u2656d4 dxc4 26.\u2657xc4+ \u2654h7 27.\u2656f1 \u2658g4 28.\u2654d2 \u2656ab8 29.\u2656xf5 \u2656b2+ 30.\u2654d3 \u2656xg2 31.\u2657e6 \u2656c7 32.\u2656xa5?**\n\nHere there was the strong manoeuvre 32.\u2656h5 \u2658f6 33.\u2657f5+ \u2654h8 34.h7!, because now Black cannot allow the back rank check with promotion: 34...\u2656e7 35.\u2656d6+\u2013.\n\n**32... \u2658f2+?**\n\nAfter this the knight goes astray. A more tenacious try was 32...\u2658xh6 33.\u2656d7+ \u2656g7 34.\u2656xc7 \u2656xc7 35.e4 and White still has some work to do.\n\n**33. \u2654e2+\u2013 \u2656h2 34.\u2654f3 \u2658h1 35.\u2656d7+**\n\n35.\u2656a6 intending 36.\u2657f5+ was just as strong.\n\n**35... \u2656xd7 36.\u2657xd7 \u2654xh6 37.\u2656xa7 \u2654g5 38.\u2656a5+ \u2654f6 39.\u2657c6 \u2656c2 40.\u2656f5+ \u2654e7**\n\n40...\u2654xf5? 41.\u2657e4+ would be an easy win.\n\n**41. \u2657d5 \u2654d6 42.\u2656h5 \u2656d2**\n\n**43. \u2656xh1?**\n\nWhite gives away a good part of his advantage. There was a win after 43.a4! \u2656xd5 44.\u2656xd5+ \u2654xd5 45.e4+ \u2654e5 46.a5, as the knight is trapped.\n\n**43... \u2656xd5 44.a4 \u2656a5 45.\u2656a1**\n\n45.\u2656d1+ \u2654e5 46.\u2656d4 changes nothing.\n\n**45... \u2654e5 46.e4?**\n\nGives away the e4-pawn. The correct way was 46.\u2654e2 \u2654e4 47.\u2656a3 \u2654d5 48.\u2654d3 \u2654c5 49.e4 \u2654c6 50.\u2656c3+ \u2654d6 51.\u2656c4 \u2656a8 52.\u2654c3 \u2654d7 53.\u2654b4 and wins. As a result of this game, this endgame position achieved some notoriety, because despite being two pawns down Black can save the game.\n\n**46... \u2654e6?**\n\nAfter a long defensive struggle Black misses his best \u2013 and last \u2013 chance. 46...\u2656c5! forces a draw: 47.a5 \u2656c3+ 48.\u2654g4 \u2654xe4 49.a6 \u2656c8 50.a7 \u2656a8 51.\u2656a5 \u2654d4 52.\u2654f5 \u2654c4 53.\u2654e6 \u2654b4 54.\u2656a1 \u2654c5 55.\u2654d7 \u2654b6 56.\u2656b1+ \u2654c5 57.\u2656b7 \u2656h8=, or 47.\u2656a3 \u2656c4 48.a5 \u2656xe4 49.a6 \u2656f4+ 50.\u2654e3 \u2656f8 51.\u2656a5+ \u2654d6 52.a7 \u2656a8=.\n\n**47. \u2654e3 \u2654d6 48.\u2654d4 \u2654d7 49.\u2654c4 \u2654c6 50.\u2654b4 \u2656e5 51.\u2656c1+ \u2654b6 52.\u2656c4**\n\nBlack resigned.\n\nThe win is now simple. White plays his king to d4 and then advances the e-pawn, whereas the black king is cut off.\n\nVery much later Kasparov considered his decision to play the WCh match against Short outside the auspices of FIDE as the worst decision of his whole chess career. 'I should have agreed to play in Manchester \u2013 and only then, after defending the title of FIDE champion, thought about setting up a Professional Chess Association. I would have saved myself a mass of nervous energy, and the chess world would have retained a single champion.'\n**37. Doors bang in the World Trade Center**\n\n**The World Championship 1995:  \n_Garry Kasparov against Viswanathan Anand_**\n\nViswanathan Anand was born on the 11th December 1969 in Madras (Chennai). His first name is Anand, his family name Viswanathan, after his father. When he first played tournaments in Europe and the Europeans saw his name, which was written according to Indian custom in the order Viswanathan Anand, they thought Viswanathan was his first name, Anand the family name. Then a short form was developed for his somewhat complicated name, 'Vishy'.\n\nViswanathan Anand (born in 1969)\n\nAnand learned to play chess at roughly six years old from his mother, in whose family the game of chess occupied an important place. In 1979 his father, manager of the railway firm Southern Railways, was working in Manila. The World Championship between Karpov and Kortchnoi had taken place there the previous year and set in motion a certain chess boom. Thus in Philippine television there were regular chess programs with chess exercises. Whilst Anand was in school, his mother noted down these exercises for him and Anand solved them in the afternoon when he was back from school. The solutions could be sent to the TV station and there were prizes for the correct solutions, for example chess books. As the months went by Anand won a heap of these prizes.\n\nAfter the return to India he began to play in tournaments and in 1983 won the Indian U16 and U19 junior championships. After a fourth place in the adult championship he was selected for the Indian national team in 1984 and at the age of 14 played in the Chess Olympiad in Thessaloniki. In 1986 he became India's youngest ever national champion. In 1987 Anand won the U20 Junior World Championship. Anand's play at that time was characterised by enormous speed. He did not appear to think about his moves at all, but saw at first glance where he had to place the pieces. Often he only used a quarter of an hour out of his thinking time for the whole game.\n\nIn 1990 he came in fourth of 64 participants at the interzonal tournament in Manila, played over 13 rounds in the Swiss System, behind Vassily Ivanchuk, Boris Gelfand and Nigel Short, and qualified for the candidates' matches. In the last 16 in Madras he defeated Alexey Dreev (4\u00bd:1\u00bd), but went out in the quarter-final in Brussels in 1991 against Anatoly Karpov by 3\u00bd:4\u00bd. In 1993 Anand again qualified for the FIDE candidates' matches with his tenth place in the interzonal tournament in Biel.\n\nBut at the same time he also qualified with his first place in the PCA tournament for the candidates' matches in the new Professional Chess Association (PCA) founded by Kasparov in competition to FIDE, so that in 1994 Anand took part in both WCh cycles in parallel. Whereas he was eliminated from the FIDE cycle, after a victory over Artur Jussupow, by Gata Kamsky in a playoff, he managed in the PCA cycle to defeat one after the other Oleg Romanishin (5:2, in New York), Michael Adams (5\u00bd:1\u00bd, in Linares) and Gata Kamsky (6\u00bd:4\u00bd, in Las Palmas) and in doing so qualify as challenger to Garry Kasparov.\n\nThe World Championship between Kasparov and Anand was actually supposed to take place in Germany. Organisers from Dortmund had applied to Kasparov for the right to stage the match during the Moscow Chess Olympiad of 1994. The venue discussed was the Harenberg City Centre, headquarters of the Harenberg publishing house, which was a very successful publisher of a series of chronicles. But the discussions with Harenberg broke down and moreover the German federation, which remained loyal to FIDE, threatened a boycott of the World Championship match. So the Dortmund organisers withdrew their bid at the start of 1995.\n\nAs a replacement, the WCh match was then supposed to be played in Cologne. The chess computer firm Novag made available some money as a start to the financing of it and the PCA hoped that Intel would come up with the rest of the sponsor money, which, however, turned out to be a misapprehension. The PCA manager Bob Rice, a New York lawyer with Milbank, Tweed, Hadley & McCloy, then charged the marketing firm 'Fairway', which was active in golf, to find sponsors for a World Chess Championship match in Germany, but this agency's efforts were not crowned with success.\n\nThen in June 1994, the mayor of New York, Rudolph Giuliani, came up to Kasparov during a rapid chess tournament organised by the PCA and suggested holding the match on the 107th floor of the World Trade Center. The latter was due to be reopened in September 1995 after being severely damaged on the 26th February 1993 by a bomb attack motivated by fundamental islamists, in which six people died and 1000 were injured.\n\nThus the match was transferred at very short notice from the originally planned venue in Germany to the USA, to New York. Anand did not learn about the change till very late on and had even booked his hotel rooms in Cologne long after the match had been transferred to New York. Also it was not until very shortly before the start of the match, in July or August, that Bob Rice announced that the prize fund had been reduced from the originally planned 1.5 million dollars to 1.35 million dollars. The difference was supposed to be employed for 'organisational costs'. 'The organisation bordered on the laughable', commented Anand later and remembered that there were a lot of breakdowns and not much information. 'Shortly before the start of the match we also learned that we had to write a regular column for _USA Today_.' The whole business and the incompetence of the PCA organisers weighed heavily on Anand and at the end of the match he was simply glad that it was all over, he wrote.\n\nAnand was left with six months to prepare before the match, though within that time he also still had some tournament obligations to fulfil. As well as Elizbar Ubilava, his regular trainer, and Artur Jussupow, who had already helped him against Kamsky, Anand engaged as another second Jonathan Speelman, who in 1993 had been Short's second in the latter's WCh match against Kasparov, and Patrick Wolff, a specialist in the Sicilian. Wolff later ended his career as a professional chess player and started a hedge fund together with the German born former PayPal CEO Peter Thiel. According to Kasparov Anand also received help from Mark Dvoretsky and a series of 'invisible' players, who would have liked to see Kasparov tumbled.\n\nSince Kasparov had a wide repertoire, Anand's team had a powerful amount of preparation to do. Kasparov had moved in July and August with his seconds Alexander Shakarov, Yuri Dokhoian and Evgeny Pigusov to a training camp on the Croatian coast. For a week Vladimir Kramnik also supported the Kasparov team.\n\nIn New York Kasparov's manager Michael Khodarkovsky took on the task of press officer. The match then began officially on the 10th September 1995 with the opening ceremony and the drawing of lots for colours, done by New York's mayor Giuliani, and lasted until the 10th October. The games were played in a glass booth specially built for the match, but it turned out to be far less soundproof than the organisers had perhaps thought. In the course of the match the two players were often disturbed by the noise which penetrated from the spectators' room \u2013 there were seats for 200 spectators.\n\nThe match was for 20 games, played according to a fixed schedule on Mondays, Tuesdays, Thursdays and Fridays without the possibility of additional rest days. If it were drawn, Kasparov would retain the title. The time control was two hours for 40 moves, a further hour for 20 moves and another half hour for the remainder of the game. The entry price for the spectators was 15 dollars per game for entry to the foyer only and 75 dollars for those who wanted sit directly within range of the players so as to experience the game live. After the games the players were available for a press conference.\n\nCarol Jarecki was named as the arbiter. Jarecki would later also take on the not always easy role as arbiter for the spectacular match between the IBM computer _Deep Blue_ and Kasparov which followed two years later. Kasparov lost this 1997 match against the machine and thus became the first World Champion to lose a match against a computer. Carol Jarecki, whose origins were in Germany, came to chess via her son John and personally flew him, since she had a pilot's licence, to various tournaments in the USA. John Jarecki took part between 1980 and 1984 in the Chess Olympiads in La Valetta, Lucerne and Thessaloniki and was second board after the painter William (Bill) Hook for the British Virgin Islands. Born in 1969, John Jarecki was only eleven years old when he first played in a Chess Olympiad and was considered a major talent. His mother was second reserve in 1980 for the British Virgin Islands \u2013 though she did not play \u2013 and since then represented the British Virgin Islands as delegate to FIDE. In the USA she also became active as a tournament organiser.\n\nBefore Carol Jarecki appeared on the chess scene, she had already made a name for herself with her husband, the Heidelberg physician Dr Richard Jarecki, in Europe's casinos. The Jareckis had developed a system for roulette based on tables with wheels which were 'off' as far as the bias was concerned. At the start of the 1970s the couple several times broke the bank at San Remo, meaning that on each occasion the casino had to be temporarily closed, and they won in the casinos of San Remo and Monte Carlo a total of more than five million German marks. Richard's brother Heinrich (Henry) Jarecki, however, became even richer. He also studied in Heidelberg, moved to the USA and practised as a psychiatrist. Then he founded a firm which dealt on the stock exchange with precious metals and later became successful as a film producer. Henry Jarecki acquired two of the British Virgin Islands and in 2009\/2010 donated no less than 60 million dollars to the University of Heidelberg.\n\nThe first game of the WCh match between Kasparov and Anand was played on the 11th September (that is exactly six years to the day before the horrendous attack on the World Trade Center) on the 107th floor of the south tower of the Center. Giuliani started the game with the symbolic first move. There is the anecdote that the mayor played 1.c4, but that Anand then took the move back and opened the game with 1.e4.\n\nThe match started with eight draws, with the challenger missing a win in the 3rd game. Anand then somewhat surprisingly took the lead when in game 9 he managed to crack Kasparov's Scheveningen Variation. But Kasparov immediately equalised in the next game with a deeply prepared variation against Anand's Open Variation in the Ruy Lopez. For the first twenty moves the title defender required only five minutes. He could hardly contain his emotions at the board, groaning and breathing deeply. After every move he made, Kasparov leaped up, left the room and slammed the door behind him. His inappropriate behaviour soon became a subject of conversation among all the journalists in the press centre. Kasparov, however, was so happy with his prepared variation that he was unable to control his emotions. In the next game he surprised Anand with the Dragon Variation. He offered a draw on move 19, Anand declined but gave away the game with a blunder on the 30th move. Kasparov then also won the thirteenth and fourteenth games for a final result of 4:1 in wins.\n\nAnand later gave a certain lack of match experience as the reason for his defeat. To be sure, he had already played a series of matches before, but had not been able to imagine how tense the situation would be in a match for the World Championship. Whereas Kasparov made no serious mistakes, that happened to Anand in the thirteenth and fourteenth games.\n\nWhat Anand politely never mentioned later, was Kasparov's inappropriate behaviour during some games. After losing the 9th game Kasparov banged down on the board the moves of his preparation in the tenth game and slammed the door behind him whenever he left the room.\n\nBut Anand processed the match he had lost really well. It was not to be the Indian's last attack on the World Championship title.\n\n **Anand \u2013 Kasparov**\n\nNew York, 13th game  \n2nd October 1995  \nSicilian Defence (B77)\n\n**1.e4 c5 2. \u2658f3 d6 3.d4 cxd4 4.\u2658xd4 \u2658f6 5.\u2658c3 g6**\n\nIn the 11th game of the match Kasparov had surprised Anand with the Dragon Variation. Anand had certainly absolutely not been thinking of that defence, since Kasparov had never before played it in a serious game with black and neither did any of his seconds, Pigusov, Dokhoian or Kramnik, have the 'Dragon' in their repertoire.\n\n**6. \u2657e3 \u2657g7 7.\u2655d2**\n\nA slight deviation from the normal move order of the Yugoslav Attack, but Kasparov does not go into it. The usual move order is 7.f3 0-0 8.\u2655d2.\n\n**7... \u2658c6**\n\nIt was worth considering 7...\u2658g4 8.\u2657g5 h6 9.\u2657h4.\n\n**8.f3 0-0 9. \u2657c4 \u2657d7 10.h4 h5 11.\u2657b3 \u2656c8 12.\u2658xc6?!**\n\nA rarely chosen move at this point and not a particularly good one. The only reason for it was probably to get Kasparov out of his prepared variations. The usual way is 12.0-0-0 \u2658e5.\n\n**12...bxc6**\n\nIn exchange for the c-file which has now been shut again, Black gets control over the d5-square and the semi-open b-file.\n\n**13. \u2657h6 c5**\n\nThis causes some distress to the \u2657b3.\n\n**14. \u2657c4 \u2655b6 15.\u2657xg7 \u2654xg7 16.b3 \u2657e6 17.\u2658d5 \u2657xd5 18.exd5 e5**\n\nAfter 18...e6 19.0-0-0 there is nothing better than 19...e5.\n\n**19.dxe6?**\n\nAccording to Kasparov, until then Anand had been playing against the e7-pawn. Now, since the weak pawn has turned into a strong one, Anand reacted impulsively \u2013 and wrongly. Opening the position with the white king in the centre cannot be good on grounds of principle. The correct move was 19.0-0-0 with a level game.\n\n**19...d5 20. \u2657e2**\n\n**20...c4**\n\n'For the first time I prevented with one move both forms of castling.' (Kasparov)\n\n**21.c3?**\n\nThis move loses. A better one was 21.\u2656d1 c3 (21...\u2656ce8!?) 22.\u2655d4 fxe6 (Kasparov) 23.\u2655xb6 axb6=. 21.0-0-0? cxb3 22.axb3 \u2655xb3\u2013+.\n\n**21... \u2656ce8!**\n\nNot quite an obvious move. With the white king in the centre the opening of the e-file is decisive.\n\n**22.bxc4**\n\n22.exf7 \u2656xf7 23.\u2656f1 (23.\u2654d1 \u2656fe7 24.\u2656e1 d4 25.cxd4 \u2658d5\u2013+) 23...\u2656fe7 24.\u2656f2 \u2656xe2+ 25.\u2656xe2 \u2655g1# (Kasparov), 22.\u2656b1 \u2656xe6 23.\u2655d4 \u2656fe8 24.\u2656b2 \u2656e3 25.\u2654f2 \u2655e6 26.\u2657f1 \u2656e1\u2013+.\n\n**22... \u2656xe6 23.\u2654f1**\n\n23.cxd5 \u2656e5 (better than 23...\u2658xd5 24.\u2655xd5 \u2655b2 25.0-0 \u2656xe2 26.\u2655g5 \u2656fe8) 24.f4 \u2656xd5 25.\u2655c2 \u2656e8\u2013+.\n\n**23... \u2656fe8 24.\u2657d3 dxc4 25.\u2657xc4**\n\n**25... \u2658e4!**\n\nWhite resigned on account of 26.fxe4 (26.\u2655e1 \u2656d6 27.fxe4 \u2656f6+ 28.\u2654e2 \u2656xe4+ \u2013+) 26...\u2656f6+ 27.\u2654e1 \u2656xe4+ 28.\u2657e2 \u2655f2+ 29.\u2654d1 \u2656xe2\u2013+.\n\nAt the end of 1994 Kasparov's PCA had concluded an agreement with Campomanes and FIDE about a reunification of the World Championships. According to it the winner of the PCA match between Kasparov and Anand was to meet the winner of the FIDE cycle \u2013 won in 1996 by Karpov against Gata Kamsky \u2013 in a reunification match.\n\nBut when Karpov could only finish last in the supertournament in Las Palmas, whereas Kasparov won the tournament, no sponsors could be found to finance this match. The balance of power between the two perpetual rivals seemed to have shifted too far. Moreover, Kirsan Ilyumzhinov soon replaced Campomanes as FIDE president and proposed a new format for the staging of the World Championship \u2013 the k.-o. system.\n**38. To victory with the 'Berlin wall'**\n\n**The World Championship 2000:  \n_Garry Kasparov against Vladimir Kramnik_**\n\nVladimir Kramnik was born on the 25th June 1975 in Tuapse, a town on the Black Sea, not far from Krasnodar and the seaside resort of Sochi. His father Boris Petrovich Kramnik was a painter and sculptor, his mother Irina Fedorovna a music teacher. Vladimir Kramnik has one brother, Evgeny.\n\n**Vladimir Kramnik (born in 1975)**\n\nKramnik learned chess from his parents at the age of four. At five he was already a member of the chess section of the local pioneer palace. His first chess trainers there were Orest Averkin and Alexei Ossachuk. At the early age of seven Kramnik won the adult championship of Tuapse. For his eleventh birthday his parents offered their gifted son an invitation to the Botvinnik-Kasparov Chess School, where it was mostly Botvinnik who still taught chess to the country's talented young players and Kasparov from time to time played simuls.\n\nHis first international tournament was the U14 Junior World Championship of 1989 in Aguadilla (Puerto Rico), where he came in second behind Veselin Topalov. In 1991 he won the U18 Junior World Championship in Guarapuava (Brazil). After some tournament successes Kramnik was called up in 1992 by Kasparov for the Chess Olympiad in Manila into the new Russian national team, where as well as the team gold medal, he also took gold for the best result on board four (8\u00bd out of 9).\n\nIn 1993 Kramnik qualified for the two competing WCh cycles of FIDE and the PCA with seventh place at the FIDE interzonal tournament in Biel 1993 and fourth place at the PCA tournament in Groningen.\n\nIn the FIDE candidates' matches he defeated in the last sixteen in Wijk aan Zee (Netherlands) 1994 Leonid Yudasin (4\u00bd:2\u00bd), but was then eliminated in the quarter-final in Sanghi Nagar (India) by Boris Gelfand (3\u00bd:4\u00bd). In the PCA candidates' matches the quarter-final in New York against Gata Kamsky was already as far as he went (1\u00bd:4\u00bd). In the subsequent years Kramnik achieved a series of further successes and moved up into third place in the world ranking list. In 2000 he was finally able to play against Kasparov for the WCh title.\n\nThe WCh match of 2000 between Garry Kasparov and Vladimir Kramnik had an eventful pre-history. After Kasparov agreed with Intel's competitor IBM to play a match under tournament conditions against the IBM computer _Deep Blue_ , Intel soon cancelled its sponsorship contract with the PCA, which soon after that quietly dissolved itself as an organisation. Without the Intel sponsorship funds Kasparov had difficulties organising a WCh cycle. In 1998 he decided as a replacement that those players who had been best placed after him in the Linares tournament should play a match for the right to challenge him. This mini cycle was to be organised and financed by the recently founded 'World Chess Council' (WCC). As a total prize fund Luis Rentero, co-founder with Kasparov of the WCC, and in addition organiser and sponsor of the Linares tournaments, offered 2.1 million dollars, of which 200000 dollars were to go to the loser of the 'candidates' finals' and 1.9 million dollars to be shared between the players of the WCh match in a ratio of 65% to 35% for winner and loser (50% to 50% for a draw).\n\nThe winner of the Linares tournament was Anand, but he had signed a contract with FIDE, which in his opinion excluded any participation in Kasparov's WCh cycle. Also, after his experiences with the organisation of the WCh match of 1995 the Indian player certainly was not greatly interested in this challenge. So the qualifiers were Alexei Shirov, who came in second \u2013 ahead of Kasparov \u2013 and Vladimir Kramnik who was fourth. The match between Kramnik and Shirov was played from 24th May to 4th June 1998 in Cazorla (Spain) and was for ten games. Kramnik was considered the favourite, but Shirov won the match early by 5\u00bd:3\u00bd (2:0 in wins).\n\nBut then there were difficulties realising the plans for the staging of the WCh match. This had been planned for 18 games and was supposed to take place half in Sevilla and half in Linares. Kasparov had been working on purely verbal promises he had been given by Rentero. The latter had contacts in the socialist government of Andalusia from where \u2013 though also only verbally \u2013 he had received agreement for the financing of the WCh match. Half of its budget, two million dollars, was supposed to come from the Ministry for Tourism and Sport. After elections which took place at that time, however, the socialists had to cede this ministry to the nationalists \u2013 and the latter had absolutely no interest in a World Chess Championship. As a protest Rentero resigned from the town council of Linares.\n\nThe Swiss banker Dr William Wirth, also a member of the WCC, and Kasparov's manager Owen Williams tried to save the match in negotiations with possible donors \u2013 but without succes.\n\nShirov felt betrayed about the WCh match, has never since spoken to Kasparov, and was later even annoyed with Kramnik when the latter, despite his defeat in the match against Shirov of 1998, appeared in 2000 as the challenger to Kasparov. Kasparov saw himself being put under pressure in his legitimacy as World Champion by the organ-isational disaster of the 'World Chess Council' and so reaffirmed in a declaration that he continued to be the rightful World Champion and that anyone wishing to become World Champion in his turn had to defeat him in a match.\n\nAt the start of 1999 Kasparov finally gave up his efforts to find a sponsor for a WCh match against Shirov. Instead he now tried with the help of Bessel Kok to get a sponsor for a WCh match against Anand. To that purpose Bessel Kok engaged the Canadian promoter Serge Grimaux. The match was to be in October 1999 with a prize fund envisaged at three million dollars. In return Anand gave up his participation in the FIDE World Championship. But no sponsor was found for the match, so it was postponed until 2000.\n\nIn the meantime Kasparov had turned to Raymond Keene to help him with the organisation of a WCh match. Keene had previously been involved in the organisation of the WCh match of Kasparov against Karpov in 1986 in London, and in addition with the organisation of the match between Kasparov and Short in 1993. Together with his brother-in-law David Levy he ran the firm 'Mind Sports', which from time to time staged competitions in sports involving mental skills in the form of the 'Mind Sports Olympiad'. After Kasparov had entrusted him with the organisation of a future WCh match, Keene founded without the knowledge of his partner Levy the competing firm 'Braingames Network PLC', which then became more or less the owners of the rights to the WCh match in chess.\n\nLevy later accused his brother-in-law of having illegally employed for that purpose 50 000 pounds sterling from the deposits of the firm 'Mind Sports'. Levy later made public further remarkable transactions and fictitious business dealings, which Keene, his partners Don Morris, David Massey and his businessman friend Alan Lubin were supposed to have been involved in with 'Brain Games Asia', 'Giloberg Finance Ltd' as well as a letterbox company registered in the USA 'RTG Ventures' and other letterbox companies such as 'X Border Corporate Services' or 'Far East Challenges plc'. The aim of these transactions, in Levy's opinion, was to interest investors in 'Braingames' and then to divert their payments to the pockets of the managing directors. Thus the company is supposed to have paid out 675 000 to its directors when it was already heading for bankruptcy.\n\nThe actual business idea of Braingames, however, consisted of broadcasting the forthcoming WCh match with Kasparov live over the Internet with new streaming technology in the form of pay-per-view. Keene first dealt with Anand, who was number two on the Elo list after Kasparov, but the conditions of the contract could not be agreed. Finally, according to Keene, the negotiations broke down because Anand demanded a guarantee of 300 000 dollars before the start of the match. For that reason they turned to the next player on the Elo list. That was Vladimir Kramnik, who in 1998 had lost in his candidates' match against Shirov.\n\nThe match for the World Championship between Kasparov and Kramnik was played from 8th October till 2nd November 2000 in London under the title _Braingames World Chess Championships_ and set for 16 games. The time controls were 2 hours for 40 moves, 1 hour for the next 20 moves and 30 minutes for the remainder of the game with an increment of 10 seconds per move for the final part of the game. It was held in the 'Riverside Studios', a former cinema in the London district of Hammersmith, not far from Hammersmith Bridge. The prize fund was 2 million dollars.\n\nKramnik and his team occupied a house on the Thames in the London district of Chiswick, which he rented for a total of six weeks. The team included Miguel Illescas, Joel Lautier \u2013 the French player was one of the few who managed to beat Kasparov during his career \u2013, Evgeny Bareev, John Adrian Tremayne Rodd, 3rd Baron Rennell \u2013 a former Scottish rugby player, who took on several tasks, for example fitness trainer and bodyguard \u2013 and Illescas' uncle Antonio as cook. Kramnik had also prepared before the match with the help of Boris Gelfand, Alexander Morozevich, Peter Svidler and Sergey Dolmatov and during the match remained in contact with these players by e-mail. Thus Kramnik's win in the tenth game is said to be based essentially on the analytical work of Boris Gelfand.\n\nAfterwards Kasparov complained about Gelfand, with whom he had prepared in 1998 for his possible match against Shirov. Scarcely two years later Gelfand would then be working for his opponent Kramnik with the knowledge of Kasparov's analyses, was the complaint of Kasparov, who described this as unethical behaviour. Gelfand rejected the accusations. Most of his time with Kasparov had been spent analysing variations of the Sicilian. He also said it was not true that he had been working with Kramnik at a distance. All that was correct was that he had showed Kramnik a game by Laszlo Hazai, which became a model for Kramnik's victory in the tenth game. In the press conference after the tenth game Kramnik had specifically thanked Gelfand for it.\n\nIn London, Kasparov put his trust in his regular trainer Yuri Dokhoian and moreover added Mikhail Kobalia and Andrei Kharlov to his team as seconds.\n\nKramnik surprised Kasparov in this match with his choice of the Berlin Defence, a variation of the Ruy Lopez, in which there is an early exchange of queens leaving Black with a slightly passive, but very solid position. At the time of the match this variation was considered antiquated and was not very popular. But Kramnik's second Joel Lautier had already had some totally positive experiences with it. Kasparov could not find an antidote to this opening during the match. In the 13th game, trailing by 0:2, he even gave a draw in 14 moves in his exasperation.\n\nKramnik won the second game with white against Kasparov's main weapon against 1.d4, the Gr\u00fcnfeld Indian Defence, and then also disarmed the World Champion with the black pieces. Kasparov, whose strength was always based on deep opening preparation, moved over to less deeply analysed openings and was finally unsuccessful with these, though he also missed some good chances. In the tenth game he lost very quickly in the Nimzo-Indian Defence, since he was not knowledgeable enough in this area and was surprised by Kramnik's choice of a rarely played move. All the other games were drawn and the final score was 8\u00bd:6\u00bd.\n\nBefore the fifth game, there was a slight provocation coming from Kasparov, when he demanded via the organisers that Kramnik should be accompanied when he visited the toilet and that the booth should not be closed so that he did not consult forbidden written notes on the opening. Apparently, so the story goes among chess grandmasters, such little cribs with opening variations were not at all unusual even in matches for the World Championship between Karpov and Kasparov, though in a sort of unspoken gentlemen's agreement between both of them there had been mutual toleration of the practice. Kramnik, who arrived ten minutes late for the game, was informed about Kasparov's desire on his way to his seat, agreed but demanded that the same procedure should also apply to Kasparov.\n\nWith his victory Kramnik became Kasparov's successor as 'Classical World Champion' in the tradition which had begun with Wilhelm Steinitz. He was the 14th World Champion.\n\n **Kramnik \u2013 Kasparov**\n\nLondon, 10th game  \n24th October 2000  \nNimzo-Indian Defence (E54)\n\n**1.d4 \u2658f6 2.c4 e6 3.\u2658c3 \u2657b4 4.e3 0-0 5.\u2657d3 d5 6.\u2658f3 c5 7.0-0 cxd4**\n\nThe more popular alternative is 7...\u2658c6 8.a3 \u2657xc3 9.bxc3 dxc4 10.\u2657xc4 \u2655c7 etc.\n\n**8.exd4 dxc4 9. \u2657xc4 b6**\n\nThe variation was a favourite of Karpov in the 1970s, but is perhaps not so well suited to Kasparov's active style.\n\n**10. \u2657g5 \u2657b7 11.\u2656e1 \u2658bd7 12.\u2656c1 \u2656c8 13.\u2655b3!?**\n\nAn aggressive continuation, which Kasparov obviously did not know.\n\n**13... \u2657e7?!**\n\n13...\u2657xc3 is considered safer: 14.\u2656xc3 h6 15.\u2657h4 and Black has had good experiences with 15...\u2655e8!?. And 15...\u2657d5 16.\u2657xd5 \u2656xc3 17.\u2655xc3 \u2658xd5 18.\u2657xd8 \u2658xc3 19.\u2657xb6 \u2658xa2 20.\u2657xa7 \u2656a8 21.\u2657c5 \u2658xc5 22.dxc5 \u2656c8 23.\u2656e5 \u2658b4 24.\u2658e1 \u2658a6 is also enough for equality.\n\n**14. \u2657xf6! \u2658xf6**\n\nAfter 14...\u2657xf6 15.\u2658b5 \u2657xf3 16.\u2655xf3 a6 17.\u2658d6 Black also has problems: 17...\u2656c7 18.\u2658xf7!?.\n\n**15. \u2657xe6!? fxe6?**\n\nLater, the cold-blooded 15...\u2656c7! was found: 16.\u2658g5 (16.\u2657c4 \u2657xf3 17.gxf3 \u2657d6=) 16...\u2657d6 17.\u2658b5 \u2656e7 18.\u2658xd6 \u2655xd6 19.\u2657c4 \u2655f4 20.\u2658f3 \u2657xf3 21.\u2655xf3 \u2656xe1+ 22.\u2656xe1 \u2655xd4=, Markosian-Afromeev, Tula 2000 (\u00bd-\u00bd\/37).\n\n**16. \u2655xe6+ \u2654h8 17.\u2655xe7 \u2657xf3 18.gxf3 \u2655xd4 19.\u2658b5!**\n\nBlack has major difficulties.\n\n**19... \u2655xb2**\n\nAll this had already been played and was also known to Kramnik after the tip from Gelfand. In the preceding game 19...\u2655f4 was played: 20.\u2656xc8 \u2656xc8 21.\u2658d6 \u2655xf3?! 22.\u2658xc8 \u2655g4+ 23.\u2654f1 \u2655h3+ 24.\u2654e2 \u2655xc8 25.\u2654d2+\u2013, Hazai\u2013Danielsen, Valby 1994 (1-0\/32).\n\n**20. \u2656xc8 \u2656xc8 21.\u2658d6 \u2656b8**\n\nNor is 21...\u2656a8 any better: 22.\u2658e8! (22.\u2658f7+ \u2654g8 is not so clear) 22...\u2658g8 23.\u2655d7! and now simply the advance of the f-pawn to f6 is decisive, whilst all three black pieces have hardly any moves.\n\n**22. \u2658f7+ \u2654g8 23.\u2655e6**\n\nWhite has a winning position.\n\n**23... \u2656f8**\n\nOr 23...h5 24.\u2658g5+ \u2654h8 25.\u2655f5 \u2655c3 (25...h4 26.\u2656e4+\u2013), and now 26.\u2656e6 decides.\n\n**24. \u2658d8+ \u2654h8 25.\u2655e7**\n\nBlack resigned.\n\nKasparov explained his defeat in the match as follows: 'My preparations went wrong and my opponent had better preparation. In the end I was just very tired. I've had my best results in the last two years and when you win so many times you begin to believe you're invincible but I will now make changes based on what I have learned in this match.'\n\nKasparov admitted that his match strategy had not worked and that he had then failed to change it in time. His second Dokhoian had warned him earlier that Kramnik would possibly aim for anodyne positions with few pieces. He had been gripped by a sort of panic when in the second game he had suffered his first defeat in the match. After the loss of the tenth game he had already inwardly thrown in the towel. A psychological mistake. At that point at the latest he should have stopped the battle around preparation and instead simply played chess.\n\nIt had occurred to Kramnik in a conversation after the 14th game how tired Kasparov already was at that point. After this game, Kasparov had seemed completely absent-minded. According to Kramnik, after four to five hours playing time his energy appeared to have dried up.\n\nImmediately after his defeat Kasparov was already demanding a return match, something which he had not had written into the contract before, presumably because Kasparov had not remotely considered that he could lose this match: 'It is his decision but I hope that he lives up to the same moral standard as I have and defends his title against the strongest opponent', was the argument of the dethroned World Champion. Kramnik replied to these demands evasively and never later allowed Kasparov a return match.\n\nThe rights for the staging of the 'Classical World Championship' were now in the hands of Keene's company 'Braingames', which soon afterwards went bankrupt after having previously sold the rights to a private London TV company by the name of _Einstein TV_. During the World Championship Braingames had failed with its business plan of selling the live broadcasts of the WCh games in a pay-per-view system over the Internet. There were few subscribers. It turned out, moreover, that the moves of a game of chess cannot be protected by copyright, even as a live broadcast. Whereas Braingames wanted cash from the spectators for following the moves on its official WCh web page, British newspapers broadcast the games for free on their web pages. So Keene had the Scottish chess journalist John Henderson thrown out of the spectators' room by security staff because he was transmitting the moves to the outside on his mobile phone. However, the attempt to legally prevent the taking down of the moves failed.\n\nFor Kasparov it had been relatively easy after the split of the World Championships to give legitimacy to his claim to be the 'real' World Chess Champion compared to the FIDE title holder. He had, after all, won the title fairly and squarely under the conditions imposed by FIDE and for the whole time he had remained number one in the world ranking list. For Kramnik it was much harder to prove his legitimate right to the title, which was also due to the fact that after winning the said title he had motivational problems, hardly scored any successes and took early draws in a large number of games. His status as World Champion was therefore called into question on various fronts.\n\nGarry Kasparov made efforts after losing the match to somehow get into a WCh match, for example as part of a 'process of reunification', but he failed and that perhaps explains his decision to announce after the 2005 tournament in Linares his withdrawal from tournament chess. He went into politics instead and became active in Russia as an opposition politician against the reigning Russian head of state Vladimir Putin.\n\nAfter Putin took harsh measures against the opposition in his country, as of June 2013 Kasparov no longer returned to Russia out of fears for his safety. In March 2014 he was awarded Croatian citizenship personally by the president of Croatia Zoran Milanovic. During the Yugoslavian civil war Kasparov had been a strong advocate for an independent Croatia and he had had for a long time a second home in Croatia. Kasparov's mother Klara remained on the other hand in Moscow.\n\nAfter withdrawing from Russian politics Kasparov again concentrated more on his activities in chess politics. After having supported Anatoly Karpov as a candidate in opposition to Kirsan Ilyumzhinov in the FIDE-presidential elections during the Chess Olympiad in Khanty-Mansiysk in 2010, Kasparov entered the lists himself against Ilyumzhinov four years later at the Chess Olympiad in Troms\u00f8, but he lost just as clearly as Karpov had done before him.\n**39. The title match in a tobacco factory**\n\n**The World Championship 2004:  \n_Vladimir Kramnik against Peter Leko_**\n\nPeter Leko was born on the 8th September 1979 in Subotica (Serbia). When he was two years old his parents separated and his mother moved with her two sons to Szeged (Hungary). Leko learned chess at the age of seven from his father and made lightning progress. His first trainer was Tibor Karolyi, with whom he worked from 1989. In 1991 Leko won the U12 World Junior Championship in Mamaia (Romania). In 1992 he became U14 European Junior Champion. Leko was considered a chess prodigy. From 1993 on he worked with Andras Adorjan as his trainer.\n\nAt 14 Leko had already the title of International Grandmaster and was at that point the youngest grandmaster of all time. Soon after that he was invited to top tournaments, where he did hold his own but did not manage any outstanding successes. Through these games against the best players in the world Leko developed a very safe but also draw-prone style, which however handicapped him in his later career since he became risk-averse. Nevertheless, he was soon to be found among the top ten players in the world and moved up to fourth place in the world ranking list.\n\nLeko did less well in FIDE's k.-o.-tournaments. At the 1997 FIDE candidates' tournament he was surprisingly eliminated in the very first round by Roman Slobodjan. In 1999 at FIDE's k.-o. WCh he lost in round two against Sergei Movsesian. At FIDE's 2000 k.-o. WCh in New Delhi, the second round was once more his departure point. He lost to Alexander Khalifman by 3\u00bd:4\u00bd after a tie-break. At the 2001 k.-o. World Championship in Moscow he again made a quick exit in the second round, this time against Ashot Anastasian.\n\nLeko, who in addition to English also speaks fluent German, was also supported in Germany and received a sponsorship contract from the energy producer RWE. The sports manager Carsten Hensel, who was active around the Dortmund Chess Days, took Leko under his wing and became his manager. Leko now regularly took part in the Dortmund tournaments.\n\nThe pre-history to the WCh match between Vladimir Kramnik and Peter Leko is at least as extensive as that to the match between Kasparov and Kramnik. After Kramnik's victory over Kasparov in the WCh match in London there was movement in the business of a possible reunification of the two World Championship systems. On the initiative of the Dutch communications manager Bessel Kok, who had already founded with Kasparov back in 1986 the GMA, a union of professional players, and who at that time was working in Prague, there was a meeting in May 2002 in Prague during the rapid chess tournament called the Eurotel Trophy, of a majority of the leading figures in chess, and deliberations as to how it would be possible to bring the two World Championships back together.\n\nThe following plan was finally agreed: as for the classical World Championship, the Dortmund tournament of 2002 would be played as a candidates' tournament. The winner would play a match against Vladimir Kramnik. In the FIDE domain, Kasparov, who as leader of the world ranking list had brought himself back into play through Ilyumzhinov, would play a match against the FIDE World Champion, who at that time was Ruslan Ponomariov. The winners of the two matches were to play a reunification match, the victor of which would then be the single World Chess Champion recognised by all.\n\nHowever, FIDE and especially its president Kirsan Ilyumzhinov did not fulfil their obligations. Kasparov's match against the FIDE World Champion was arranged four times and cancelled four times. With his retirement in March 2005, Kasparov took his leave, also frustrated by the Prague reunification plan.\n\nThe causes of the postponements differed in their nature. There are on the whole only suppositions about the reasons since FIDE has made little known about what happened. Originally the match was to be played in summer 2003 in Buenos Aires, then it was postponed and finally cancelled, because according to FIDE the financing required to stage it was not there. In Argentine chess circles it was said that Ilyumzhinov had been handling a weapons deal as an emissary of the Russian government. The chess match was to have been financed with funds provided for this, or was somehow connected with this deal. But then the weapons deal collapsed and with it the match in Buenos Aires.\n\nThen FIDE announced Yalta as the new venue for the match. The date was first scheduled for June 2003, then September 2003 (starting on the 18th September). But then the Kasparov-Ponomariov match was called off at extremely short notice even for FIDE on the 29th August. FIDE gave the reason for that as being insuperable differences with Ruslan Ponomariov concerning the contractual details.\n\nPonomariov denied this version. The FIDE World Champion stated that he had signed an agreement with FIDE on the 12th August 2003, but admitted that he had not signed an English version of the contract because promises made to him during the negotiations had not been included and because he had not understood all the ways things had been formulated in the English version. Therefore he had asked for a Russian version of the text of the contract, which he would then sign. Moreover Ponomariov complained in an open letter that during the negotiations FIDE continually put him under pressure so as to disqualify him and replace him if he did not agree to the conditions. FIDE representatives had in addition given false information to the media and the public about the state of the negotiations.\n\nPossibly the true reason for the cancellation of the match is to be found somewhere completely different. According to Ukrainian sources, the Russian president Vladimir Putin had pressed Kirsan Ilyumzhinov to cancel the match. Perhaps Putin wanted to punish Kasparov for having become politically active against him in the opposition party 'Committee 2008'. In return Ilyumzhinov is said to have been granted another mandate as president of Kalmykia.\n\nAfter the cancellation FIDE then announced that, failing that, Kasparov would play against the next FIDE World Champion the eliminatory match for the reunification. If the theory of a politically motivated cancellation of the Ponomariov-Kasparov-matches is true, then, however, Ilyumzhinov cannot have been particularly interested in bringing about a match between Kasparov and whoever. In 2004 Rustam Kasimdzhanov won the FIDE k.-o. tournament in Tripoli and became the new FIDE World Champion. FIDE announced a match between Kasparov and Kasimdzhanov for January 2005. It was to be hosted by Dubai and the prize fund was to be 1.2 million dollars. But this match too never happened. The whole reunification plan in its original form was finally untenable when Kasparov withdrew from tournament chess in March 2005. Moreover, Kasparov himself never believed that Putin had been responsible for the cancellation of his match against Ponomariov.\n\nThe group around Kramnik, on the other hand, fulfilled its obligations, though not without difficulties. One of the hurdles turned out to be the question of rights. Kasparov had transferred the rights to the 'Classical World Chess Championship' to Raymond Keene's Braingames Network plc and when signing his contract for the WCh match in London Kramnik had also tied himself to Braingames. Thus Kramnik did become World Champion but he had no marketing rights attached to his title. Via a company called 'Intellectual Leisure Ltd', which held shares in Braingames and which was bought up by the company 'Einstein Group plc', the marketing rights wandered to the Einstein Group, a private multimedia and TV company which claimed as its business sector TV productions on scientific subjects, but also mental sports events and such like.\n\nFounded in 1999, the firm was quoted on the Stock Exchange from March 2000. The company proudly announced that in 2002 Einstein productions were being fed into TV sets in 15 countries via the Pay-TV broadcaster Sky. For the 'Braingames World Chess Championships' Einstein was planning live broadcasting of the games on the Internet, but also on television. In April 2002 Einstein managing director Steve Timmins met the Parisian chess patron Nahed Ojjeh, for whose NAO Chess Club Vladimir Kramnik played. Together they made new suggestions as to how the reunification process should be carried out in FIDE.\n\nIn the summer of 2002 the Dortmund grandmaster tournament was then played as a candidates' tournament for the 'Classical World Championship'. The tournament consisted of two double-round preliminary groups followed by k.-o. rounds. Group A was won by Alexei Shirov on tie-break ahead of Veselin Topalov (both on 4 points), Boris Gelfand (2\u00bd) and Christopher Lutz (1\u00bd). In the B-group the winner was Evgeny Bareev (4) in front of Peter Leko (3\u00bd), Michael Adams (2\u00bd) and Alexander Morozevich (2). In the semi-final Peter Leko defeated Alexei Shirov and then also won the final against Veselin Topalov. This made the Hungarian the challenger to Vladimir Kramnik.\n\nDuring 2002 the Einstein Group became financially lop-sided. In February 2003 Einstein announced that a million pounds had to be taken out as a loan in order to settle outstanding bills and salaries. Soon afterwards the company became insolvent. The rights to the 'Classical World Championship' had now apparently been acquired by Kramnik himself. The clarification of the question of the rights had, however, cost some time.\n\nIn any case, someone to stage and sponsor the match was soon found. Christian Burger, owner and managing director of the Swiss cigar manufacturers Burger&S\u00f6hne (founded in 1864), had read in the _Financial Times Deutschland_ , a sister-paper of the British _Financial Times_ founded in 2000 and shut down in December 2012, some interesting articles about the young chess stars Sergey Karjakin and Alexandra Kosteniuk and thus hit on the idea to deploy chess as a marketing vehicle for his cigar brands. Burger&S\u00f6hne in 1988 purchased the Dannemann brand 1999, acquiring in Tessin the 'Fabbrica Tabacchi Brissago SA'.\n\nBurger had then set up in 2003 the 'Centro Dannemann' in the spacious building in Brissago directly on the shore of the Lake Maggiore. On the ground floor expensive cigars were produced, whereas on the roomy first floor of the building cultural and sports events were held. Chess fitted in very well with this in Christian Burger's view. In January 2003 the Centro Dannemann staged in this lovely setting \u2013 from the enormous windows there is a view over Lake Maggiore and the picturesque mountains on the opposite bank \u2013 a match between Karjakin and Kosteniuk. That brought into action Carsten Hensel, who in the meantime had taken over the management for Vladimir Kramnik. After a call to Christian Burger, the latter was prepared to stage the WCh match between Kramnik and Leko in his Centro Dannemann and to offer a total prize fund of a million Swiss francs.\n\nThe match was played from the 25th September till the 18th October 2004 as the 'Classical World Chess Championship 2004' in the Centro Dannemann in Brissago and set for 14 games. In the event of a draw the title defender was to remain World Champion. Kramnik's team had seen the departure of Joel Lautier, who was nevertheless present as president of the players' organisation ACP (Association of Chess Professionals). He was replaced by Peter Svidler in Kramnik's team. Leko's second was Vladimir Akopian. The match was played in a remarkably friendly atmosphere, since both players had as their manager Carsten Hensel, who was from Dortmund and so came, as it were, practically from the same 'stable'.\n\nThe venue, Brissago, is a peaceful little place in Italian Switzerland, on the west bank of Lake Maggiore, the last Swiss town before the border with Italy. Towards the north lie the chic resorts of Ascona and Locarno. The mountains come right down to the road along the bank. Although the town is hard to reach, some spectators did come to Brissago to watch the games live. Especially at weekends the spectators' room, which could hold perhaps 200 spectators, was well filled. Dannemann broadcast the games live to the Internet, a technology which at the time was new and expensive, and in addition took a video stream from ChessBase of the press conference with the players after the game. It was possible to follow both free on the Dannemann website.\n\nDuring the match there were two sorts of games: those in which chess was played and which were hard fought, and those which quickly ended in draws. There had apparently been an internal agreement with the organiser that play should last for at least two hours, so that even the short draws in the second game (18 moves), third game (23 moves), sixth game (20 moves), seventh game (21 moves), ninth game (16 moves) and eleventh game (17 moves) were only finished after the aforementioned two hours.\n\nThe match began, however, with a bang, since Kramnik won the very first game, and that with the black pieces. He had secured for himself a safe position in the Petroff Defence with a novelty. Leko played for a win but was caught out by Kramnik's counterplay. In the fifth game Leko surprised Kramnik by choosing for the first time 1.d4 as his opening move. From the Queen's Gambit he reached straight out of the opening a slightly better endgame, which Black normally draws, but Kramnik did not manage to do so.\n\nIn the 8th game Kramnik's preparation against Leko's Ruy Lopez Marshall Variation went fundamentally wrong. He followed a sharp and forced variation, which he had found with the help of a computer. It then turned out that Kramnik's seconds Bareev and Svidler had not checked the variation deeply enough. Kramnik followed the computer variation, then in the game, instead of drawing by repetition of moves, sacrificed the queen in the belief that he was winning, and lost because, without preparation, Leko evaluated the position on the board correctly and played the best moves. Leko thus took the lead. In the twelfth game the challenger achieved a promising position with black and to the astonishment of the spectators offered a draw with his strong 35th move. Possibly this is where the Hungarian missed the chance of becoming World Champion, since he was clearly better in the final position.\n\nKramnik now absolutely had to win one of the last two games in order to level the match, which according to the match rules would be enough for him to have defended his title. In game 13 Leko held the draw with white. The 14th game was Kramnik's best performance in the match and his win secured the required overall draw.\n\n **Kramnik \u2013 Leko**\n\nBrissago, 14th game  \n18th October 2004  \nCaro-Kann Defence (B12)\n\n**1.e4 c6**\n\nAfter the eighth game, which Leko had won in fortunate circumstances, the Hungarian no longer trusted his Ruy Lopez Marshall Gambit and had changed to the Caro-Kann Defence, but he had certain problems with it.\n\n**2.d4 d5 3.e5 \u2657f5**\n\nRecently 3...c5 has also been played more often.\n\n**4.h4**\n\nThere is the much played and sharper 4.\u2658c3 e6 5.g4 \u2657g6 6.\u2658ge2. Nigel Short made quick development with 4.\u2658f3 e6 5.\u2657e2 popular. The immediate 4.g4 is also played: 4...\u2657d7 (4...\u2657e4!?) etc.\n\n**4...h6**\n\nAfter the main move 4...h5 White has the choice between 5.\u2658e2 e6 6.\u2658g3 and 5.c4.\n\n**5.g4 \u2657d7 6.\u2658d2**\n\nA novelty by Kramnik. The development of the knight to b3 was more frequently tried after this game, even in other lines of the Advance Variation of the Caro-Kann Defence.\n\n**6...c5**\n\n6...e6 7.\u2658b3 c5 is a transposition of moves.\n\n**7.dxc5**\n\nAfter 7.c3 \u2658c6 8.\u2658b3 cxd4 9.cxd4 e6 Black has nice play against the weak d4-pawn.\n\n**7...e6**\n\nRainer Knaak suggested 7...\u2658c6 as an improvement for Black, so as to provoke 8.f4, after which White's position is somewhat loose. 7...\u2655c7 was also worth considering according to Knaak.\n\n**8. \u2658b3 \u2657xc5 9.\u2658xc5 \u2655a5+ 10.c3 \u2655xc5 11.\u2658f3 \u2658e7 12.\u2657d3**\n\n12.h5!? looks more logical.\n\n**12... \u2658bc6**\n\nThe threat is...d5-d4, but also...\u2658e7-g6.\n\n**13. \u2657e3 \u2655a5 14.\u2655d2!**\n\nThis takes the sting out of...d5-d4. 14.\u2655c2?! is unfavourable on account of 14...\u2656c8. (Knaak)\n\n**14... \u2658g6**\n\nBlack could play 14...d4!? as a pawn sacrifice, so as to get hold of the d5-square for a knight. However, after 15.cxd4 \u2658b4 16.0-0 (16.a3? \u2658xd3+ \u2013+) 16...\u2657b5 17.\u2657xb5+ \u2655xb5 18.\u2656fc1 \u2656d8 19.a3 \u2658bd5 Black's counterplay probably is not sufficient to compensate for the disadvantage in material.\n\n**15. \u2657d4**\n\n15.\u2657xg6?! fxg6 opens the f-file for Black.\n\n**15... \u2658xd4**\n\nIt was also worth considering 15...\u2655c7 16.\u2655e3 \u2658xd4 17.\u2655xd4 \u2655a5=.\n\n**16.cxd4 \u2655xd2+**\n\n16...\u2655b6= was also playable.\n\n**17. \u2654xd2**\n\nIn the decisive last game, which he did have to win, it was relatively bold of Kramnik to allow so many simplifications till this endgame arose. However, Kramnik also has particular strengths in the endgame.\n\n**17... \u2658f4?!**\n\nThis move was later criticised. The alternative was 17...\u2658e7 intending 18...\u2658c6.\n\n**18. \u2656ac1! h5**\n\nIn the event of 18...\u2658xd3 19.\u2654xd3 \u2657b5+ 20.\u2654e3 \u2654d7 21.h5 \u2656ac8 White plays 22.\u2658h2!. (Knaak) After 22...\u2656xc1 23.\u2656xc1 however 23...\u2656g8!? is an interesting idea: 24.f4 g6 with counterplay. 23...\u2656c8? 24.\u2656xc8 \u2654xc8 25.f4 \u2654d7 26.g5 is on the other hand good for White, since the knight is better than the bishop here.\n\n**19. \u2656hg1 \u2657c6**\n\n19...\u2658h3 20.\u2656g3 \u2658xf2 21.gxh5 is good for White. After 19...\u2656c8 20.\u2656xc8+ \u2657xc8 21.gxh5 \u2658xh5 22.\u2654e3 Black is also facing a difficult defence.\n\n**20.gxh5 \u2658xh5 21.b4**\n\nWhite has the initiative, but the question is whether it is enough for a win.\n\n**21...a6 22.a4 \u2654d8?**\n\nAfter this the white advantage takes on clear outlines. White also has strong pressure after 22...\u2657xa4 23.\u2656c7 \u2657c6 24.\u2658g5. But there was the more tenacious 22...\u2654e7!?, since after 23.b5 axb5 24.axb5 \u2657d7 25.\u2656c7 b6 26.\u2654e3 \u2656a3 27.\u2656b7 \u2656c8 Black gets counterplay.\n\n**23. \u2658g5 \u2657e8 24.b5 \u2658f4**\n\nAfter 24...axb5 25.\u2657xb5 too, White is in the driving seat.\n\n**25.b6! \u2658xd3**\n\nOr 25...\u2656c8 26.\u2656xc8+ \u2654xc8 27.\u2656c1+ \u2654b8 28.\u2656c7+\u2013. (Knaak)\n\n**26. \u2654xd3 \u2656c8 27.\u2656xc8+ \u2654xc8 28.\u2656c1+ \u2657c6 29.\u2658xf7 \u2656xh4 30.\u2658d6+ \u2654d8 31.\u2656g1 \u2656h3+ 32.\u2654e2 \u2656a3 33.\u2656xg7 \u2656xa4 34.f4 \u2656a2+**\n\n34...\u2656xd4 35.f5! exf5 36.e6 \u2656e4+ 37.\u2658xe4 fxe4 38.\u2656c7+\u2013.\n\n**35. \u2654f3**\n\nWhite sets off into the black camp with his king, in order to mate the black king. But there was also the good, though less spectacular 35.\u2654d3 \u2656a3+ 36.\u2654c2 \u2656a2+ 37.\u2654b3 \u2656a1 (37...\u2656f2 38.f5 exf5 39.\u2658xb7+ \u2657xb7 40.\u2656xb7 \u2656e2 41.\u2656f7+\u2013) 38.\u2658xb7+ \u2657xb7 39.\u2656xb7 \u2654c8 40.\u2656c7+ \u2654b8 41.\u2654b4+\u2013.\n\n**35... \u2656a3+**\n\n35...\u2656a1 36.f5+\u2013.\n\n**36. \u2654g4 \u2656d3**\n\n36...\u2656a1 37.\u2658xb7+ \u2657xb7 38.\u2656xb7 \u2656g1+ 39.\u2654h5 \u2654c8 40.\u2656c7+ \u2654b8 41.f5+\u2013.\n\n**37.f5 \u2656xd4+ 38.\u2654g5 exf5 39.\u2654f6 \u2656g4 40.\u2656c7 \u2656h4 41.\u2658f7+**\n\n**1-0**\n\n41...\u2654e8 42.\u2656c8+ \u2654d7 43.\u2656d8#.\n\nThis left Kramnik World Champion. After the match it became known that the title defender suffered from a rheumatic disease which had strongly handicapped him during the match and required medical treatment. This explains the partly muted play of the title defender in some games. Leko, however, as he stated later in an interview, had known nothing about Kramnik's illness.\n**Part V \u2212 Reunification and what followed**\n\nFollowing on from the Prague reunification plan of 2002, FIDE and Kramnik had fulfilled their obligations and now all that had to happen was the match between the two World Champions, the 'Classical World Champion' in the tradition of Steinitz, Lasker and their successors, and the FIDE World Champion \u2013 who was, after his victory in the FIDE WCh tournament in San Luis in 2005, Veselin Topalov \u2013 and that would reunite the word of chess.\n\nKirsan Ilyumzhinov and FIDE were still making a fuss about actually holding the reunification match, but apparently the still influential Russian leadership circles exerted pressure on the FIDE president and in 2006 in Elista the long awaited match actually came to pass. It was then held in scandalous circumstances and at times was on the brink of being abandoned. Kramnik won and was then recognised on all sides as the World Chess Champion. After 13 years the schism had finally come to an end.\n\nThere followed in 2007 a WCh tournament which was won by Viswanathan Anand. The new World Champion then defended his title in matches in 2008 against Kramnik, in 2010 against Topalov and in 2012 against Gelfand. Whereas Anand won convincingly against Kramnik, his victory over Topalov was much more arduous. Against Gelfand the Indian had to wait until the playoff to get through.\n\nIn 2012 Anand's opponent could already have been Magnus Carlsen. But the Norwegian had withdrawn from the WCh cycle, because he was unhappy with the scheduling by FIDE. In the meanwhile Carlsen had made his way to first place in the world ranking list, in which he soon led by a big margin. Anand on the other hand was less successful with his play and fell back in the world ranking list. At the candidates' tournament in London in 2013 Carlsen was victorious, though only just on tiebreak score compared to Vladimir Kramnik. That confirmed Carlsen as the new challenger to Anand.\n**40. 'Toiletgate' in Elista**\n\n**The World Championship 2006:  \n_Vladimir Kramnik against Veselin Topalov_**\n\nVeselin Topalov was born on the 15th March 1975 in the city of Russe in the north of Bulgaria, the son of an economist and a doctor. His childhood has been described as difficult, without there being any more information than that. He learned to play chess from his father at the age of eight and made rapid progress. His first trainers were Dimitar Sinabov, and from 1987 Petko Atanasov. When Topalov was twelve, Silvio Danailov started to look after him and he has been his manager until the present day.\n\n**Veselin Topalov (born in 1975)**\n\nIn 1989 Topalov won the U14 Junior World Championship in Aguadilla (Puerto Rico). In 1992 Danailov and Topalov went to Spain, where Topalov played in countless tournaments. After good tournament successes Topalov achieved fourth place in the July world ranking list of 1996 on the same number of points as Anand (Elo 2750). After 1998, however, he first fell back under 2700, but in 2002 recovered his place in the top ten. In the same year he reached the final of the candidates' tournament in Dortmund, but lost against Leko.\n\nAfter Topalov had, between 2002 and 2004, stagnated in the world ranking list around 2740, a series of tournament successes then helped him to a lightning climb in that ranking list. At the FIDE WCh of 2004 in Tripoli Topalov scored 9\u00bd points in 10 games and was not defeated until the semi-final against Rustam Kasimdzhanov after a rapid chess tiebreak. After Kasparov's withdrawal Topalov took first place in the world ranking list (Elo 2804) in April 2006 and held it till January 2007 (Elo 2783) with his highest rating being in October 2006 (Elo 2813). After January 2007 he continually fell down the list and lost approx. 40 points.\n\nIn October 2005 Topalov sailed through the FIDE WCh tournament in San Luis (Argentina). 'Topalov surpassed his opponents in all aspects of the game: he was better prepared in the opening. He played faster, hardly calculating at all, and in general he didn't make any bad mistakes. Veselin didn't get tired in long games and didn't have any doubts when he made decisions', was how Bareev and Levitov characterised the Bulgarian's play.\n\nThis made Topalov FIDE's candidate for a possible reunification match against the World Champion in 'classical chess', Vladimir Kramnik. During the tournament in San Luis, Kramnik's manager Carsten Hensel had in a conversation with Topalov's manager Danailov had already reached an agreement in principle about a reunification match between Kramnik and Topalov. The recently founded Dortmund firm Universal Event Promotion (UEP), with the Germano-Russian metals dealer Josef Resch, a friend of Kramnik, wanted to stage the match. UEP offered a prize fund of a million dollars, 500000 dollars for each of the two players. The match was to be for 14 games from 25th November till 17th December 2006 under the title 'World Chess Match of the Champions' in Bonn's Art and Exhibition Hall.\n\nBefore it came to the signing of the contract in November 2005, however, FIDE intervened and prevented this reunification match from happening under the control of UEP. Danailov gave a reason in a statement in which he made it clear that he did not recognise Kramnik as the World Champion and moreover explained that he had come to no agreement with the UEP, had signed nothing and invited Kramnik to challenge the 'reigning World Champion' Topalov, if he wanted to play him for the World Championship. Carsten Hensel pointed out in his reply that both FIDE and Silvio Danailov as well as Veselin Topalov had in various agreements explicitly recognised the status of Vladimir Kramnik as 'World Champion in classical chess'. FIDE emphasised that the reunification match had to take place under their aegis.\n\nThe positions of the two parties appeared to have hardened. Then however, Alexander Zhukov, deputy prime minister of the Russian government and at the same time president of the Russian Chess Federation, got involved. An admirer of Kramnik, he insisted on the holding of the reunification match. Since at this point Kirsan Ilyumzhinov was faced with two decisions, on one hand his re-election as FIDE president during the congress at the Chess Olympiad in Torino in 2006, and on the other his confirmation as president of the Russian Republic of Kalmykia, and that in both cases he was reliant on help from Moscow, it was advisable for him to follow the wishes of Zhukov.\n\nOn Good Friday 2006 (14th April) Ilyumzhinov publicly and surprisingly announced the holding of the reunification match from the 21st September till the 13th October. There had been some negotiations behind the scenes preceding this. The particularly difficult subject was the official way of handling the status of the two players. In no way did Kramnik wish to appear as the 'challenger' of Topalov. Topalov insisted on his status as the 'true' World Champion and did not see the match as that of two players on the same level. Agreement was finally reached that this would be a match for the title of World Champion between two top players. The concepts of 'FIDE World Champion', 'Classical World Champion' and 'challenger' were avoided in the announcement of the match. Both players had agreed that after this WCh match all future World Championships from then on should take place under the aegis of FIDE. Kramnik had in addition declared himself willing, should he win, to take part in the WCh tournament planned by FIDE for 2007, although he had always been in favour of the match format as the appropriate one for the WCh. Moreover, Kramnik had the right to a return match included: should he as the winner of the match against Topalov take part in the WCh tournament but not win it, he would be allowed a return match against the new World Champion.\n\nThe reunification match then took place, as announced, from 21st September till 13th October 2006 in Elista and was set for twelve games. The winner should be whoever reached 6\u00bd points. In the event of a draw there were to be four rapid chess games, if it was still a draw then two blitz games and finally a so-called 'sudden death' blitz game. The time control was that of classical tournament chess, two hours for 40 moves, one hour for the next 20 moves and 15 minutes for the remainder plus 30 seconds time increment for each move in this final part of the game.\n\nThe prize money of one million dollars was to be divided equally between Kramnik and Topalov independently of the result. However, only the winner would qualify for the WCh tournament for the following year. The loser was out.\n\nThe chief arbiter was Geurt Gijssen. The members of the appeals committee were to be FIDE vice-presidents Georgios Makropoulos, Zurab Azmaiparashvili and Jorge Vega. The appeals committee functions as a jury for disputes, which in most cases is only named for symbolic reasons and usually does not have much to do. However, it costs the organiser quite a lot of money. 6000 dollars for the chairman and 4000 dollars for the other two members are the usual honoraria. The FIDE uses these committee nominations to reward the loyalty of their partisans at the presidential elections. In this WCh match, however, the jury was to have a special role to play. Kramnik's manager Carsten Hensel later admitted that he had not paid enough attention to its composition.\n\nZurab Azmaiparashvili, especially, does not enjoy the best of reputations in the world of chess, since he has been involved in a series of scandals. The Georgian grandmaster was, for example, openly accused of having fixed with three other players a tournament in his favour in 1995 in Strumica. It was a six-round tournament and Azmaiparashvili scored 16 points out of 18 games. This result enabled the former Kasparov second to clearly improve his Elo rating. Azmaiparashvili, however, has always denied the accusation of game fixing. In a game at the European championship of 2003 he took back a move, which is against the rules. After his opponent accepted the taking back of the move, Azmaiparashvili won the game and even became European champion.\n\nIn 2004 at the closing ceremony of the Chess Olympiad in Calvia (Spain) he became involved in a brawl with the Spanish police and was taken into police custody for two days. The cause for the scrimmage was a misunderstanding. Azmaiparashvili had wanted to draw attention to the fact that the awarding of the so-called Nona Gaprindashvili prize \u2013 for the nation which has the best average team ranking over the men's and women's events \u2013 had been forgotten and so he stormed on to the stage, but was physically restrained by the security staff. In the same year, however, the Georgian women players Lela Javakhishvili and Ana Matnadze complained that in connection with the Women's World Championship Azmaiparashvili had behaved to them 'in a hostile and intimidating manner, using inappropriate and vulgar language.'\n\nAnd there was no way that Azmaiparashvili was not partisan, since financial interests linked him to Topalov's manager Silvio Danailov. Azmaiparashvili was a trainer of the top Azeri player Teimour Radjabov. After it became clear that Topalov would win the WCh tournament of San Luis in 2005 after his outstanding first series of games (six wins and one draw against the absolute top of world chess), Silvio Danailov reached in 'negotiations' with FIDE a short-term change in the rules for the World Championship of the world chess federation. In fact they were sitting over drinks in the bar one evening in San Luis when Danailov came out with his suggestion and the FIDE representatives immediately agreed, according to observers. Danailov's suggestion was that any player with an Elo rating over 2700 should have the right to challenge the World Champion without any further qualification. What lay behind this suggestion was an offer from the financially strong Azeri Chess Federation to Topalov to play a WCh match against the then Azeri number one Teimour Radjabov in April 2007 in Baku. Topalov was to receive an appearance fee of a million dollars. As the trainer of Radjabov, Zurab Azmaiparashvili was involved in this piece of 'business'. If, however, Topalov did not win the match against Kramnik, then of course the match against Radjabov would not take place.\n\nKramnik's team in Elista included this time, as well as Miguel Illescas, Sergey Rublevsky and Alexander Motylev as seconds. Valery Krylov was responsible for Kramnik's physical fitness. Kramnik's manager Carsten Hensel acted as delegation leader. In addition to his constant companions Silvio Danailov and Ivan Cheparinov, Topalov arrived with Francisco Vallejo Pons and Alexander Onischuk as extra seconds. Moreover, he had brought six other people with him: a masseur, a press secretary, a bodyguard, put at his disposition by the Bulgarian president, an official from the Bulgarian embassy in Moscow and two computer experts, whose principal task was to ensure that the Bulgarians were not bugged in Russia.\n\nTopalov began the match in too up-beat a fashion and over-estimated his own strengths compared to Kramnik. In advance Topalov and Danailov had let it be known that there could be absolutely no doubt about who was going to win the match. In the first game the Bulgarian reached a good position, miscalculated however and lost. In the second game he overlooked a forced mate, missed several more winning chances, spoiled things first towards a draw and then also to the point of losing the game.\n\nOver a short distance of twelve games a two point deficit is quite a disadvantage. So Danailov decided to open up a battleground on a second front. Someone had informed him that whenever Kramnik retired to his rest room between moves he also frequently visited the toilet which was there. The players each had their own rest room with its own toilet. Whereas there was surveillance of the rest room with a video camera, that did not apply to the toilet for understandable reasons. The images from the video cameras in the two rest rooms were displayed on two monitors on the arbiter's table, so that the arbiter was also able to observe the players in their rest rooms. The images from the cameras were recorded on tapes.\n\nAfter the second game Silvio Danailov handed to the organisational head Valeri Bobaev a note of protest and complained about the match conditions. For example, he demanded to view the recorded tapes. After the fourth game the Topalov team were given the tapes to view by the appeals committee, something for which there was no justification in the rules. Danailov extrapolated from them the number of times Kramnik visited the toilet during the games and from these fairly frequent visits developed the accusation that in the toilet Kramnik was getting advice from forbidden help, i.e. a computer. He then formulated an official protest.\n\nNormally the appeals committee should have rejected this protest, after all the number of visits to the toilet is in no way subject to any rule. Instead, it had Kramnik's private toilet closed after the fourth game. Kramnik's head of delegation Hensel was informed of this on the evening of the 28th September, but, so as not to disturb Kramnik in his preparation, he only passed this on to him the following morning. When Kramnik learned of the decision he reacted very impulsively, took as his standpoint that this decision was not in accordance with the contracts which had been agreed and as a protest did not appear for the fifth game. On account of his non-appearance the game was scored as a win for Topalov.\n\nKramnik contested the legality of this measure and threatened to abandon the match \u2013 which would certainly have suited Danailov and Topalov very nicely \u2013 and to make a complaint to FIDE at the end of the match. The appeals committee showed a letter signed by Ilyumzhinov in which the latter accepted the decisions of the appeals committee. As it later turned out, this letter was a forgery; a signature stamp of Ilyumzhinov had been used. On the 29th September Danailov had handed in a further protest, in which he declared the decision of the appeals committee \u2013 the closing of the private toilet \u2013 to be insufficient, and demanded that Kramnik be accompanied by another person when he visited the toilet.\n\nAfter the fifth game did not take place, the appeals committee gave a press conference and tried to justify its decision. Kramnik then appeared and had an agitated discussion with Makropoulos, accused the committee together with Danailov of criminal machinations and finally said: 'I did not sign a contract to appear in a reality show. What is happening here is contrary to all ethical norms and an invasion of my privacy.'\n\nFIDE president Kirsan Ilyumzhinov had actually opened in his Kalmykian capital the reunification match which had been long awaited by the whole world of chess, but had then left to attend one of the meetings of provincial governors with Vladimir Putin in Sochi. In his absence the match got out of control. When events were made public on the internet, Putin sent him back home with the comment 'Put some order in that chess match back in your home.' After hurrying back to Elista, the FIDE president tried to pour oil on troubled waters and to save his reunification match. Ilyumzhinov replaced Makropoulos and Azmaiparashvili by Boris Kutin and Faik Hasanov and explained that the match would continue with the score at 3:2 for Kramnik.\n\nThe latter, who in the meantime had understood the intentions of Danailov, Topalov and Azmaiparashvili, finally decided to continue the match under protest. Things continued in Elista. After the sixth and seventh games had ended in draws, Topalov won the eighth and ninth games, then Kramnik equalised with a victory in the tenth game. The eleventh and twelfth games were once more drawn and a playoff had to decide matters. This was won by Kramnik by 2\u00bd:1\u00bd and with it he became the World Champion who was 'recognised by everyone'.\n\nWhy Kramnik visited the toilet so often was never explained. During the match his manager Carsten Hensel gave as a reason for this that Kramnik liked to walk up and down \u2013 not a very plausible explanation, since the room was not big enough for that. Then the story was that Kramnik was smoking secretly in there and did not want this to become known, since he had promised his wife to stop. But perhaps Kramnik had simply been going there for the usual reasons \u2013 many a chess player has found that when playing chess nerves have an effect on the bladder. If that were the case, he certainly did not have to offer any explanation.\n\n **Topalov \u2013 Kramnik**\n\nElista, 2nd game  \n24th September 2006  \nQueen's Gambit, Slav Defence (D19)\n\nIn the first game of the match Topalov had finished by losing from a promising position. Now in the second game with the white pieces he showed that his motivation was still high. It turned into an extremely complicated game, which can hardly be evaluated in all its subtleties without a computer. Moreover, fresh analysis with modern programs shows that in 2006 even the chess programs were not good enough to evaluate this complex game correctly.\n\n**1.d4 d5 2.c4 c6 3. \u2658c3 \u2658f6 4.\u2658f3 dxc4 5.a4 \u2657f5**\n\nThis safe version of the Slav Defence to the Queen's Gambit is in Kramnik's repertoire, but he had not used it for a long time before this game.\n\n**6.e3**\n\nIn the sixth game of the match 6.\u2658e5 was played: 6...e6 7.f3 c5 8.e4 \u2657g6 9.\u2657e3 cxd4 10.\u2655xd4 \u2655xd4 11.\u2657xd4 etc., \u00bd-\u00bd\/31.\n\n**6...e6 7. \u2657xc4 \u2657b4 8.0-0 \u2658bd7 9.\u2655e2 \u2657g6 10.e4 0-0 11.\u2657d3 \u2657h5 12.e5**\n\nThe alternative is 12.\u2657f4.\n\n**12... \u2658d5 13.\u2658xd5 cxd5 14.\u2655e3 \u2657g6**\n\nAnother playable move is 14...\u2657e7.\n\n**15. \u2658g5 \u2656e8 16.f4 \u2657xd3**\n\nA new move at the time of the game. Till then 16...\u2658f8 was known.\n\n**17. \u2655xd3 f5**\n\n17...\u2658f8? 18.f5, 17...g6!? 18.g4!?.\n\n**18. \u2657e3**\n\n18.\u2658xe6!? is probably too speculative: 18...\u2656xe6 19.\u2655xf5 \u2656c6 20.\u2655g4 \u2654h8 21.f5 \u2656c4. But there was the more aggressive immediate 18.g4 with wild complications which appear favourable for White.\n\n**18... \u2658f8 19.\u2654h1**\n\n19.g4 \u2655d7 is now less dangerous for Black.\n\n**19... \u2656c8**\n\nAfter 19...h6 or 19...\u2657e7! Black always has to reckon with 20.\u2658xe6 \u2658xe6 21.\u2655xf5.\n\n**20.g4 \u2655d7**\n\nThe consequences of 20...h6 21.\u2658xe6 \u2656xe6 22.gxf5 \u2656ec6 23.f6 gxf6 24.\u2656g1+ \u2654h8 are not clear. But Black seems to be able to hold on.\n\n**21. \u2656g1 \u2657e7**\n\n**22. \u2658f3**\n\nHere too, according to Marin, 22.\u2658xe6!? was well worth considering: 22...\u2655xe6 (22...\u2658xe6? 23.gxf5 \u2658c7? 24.f6 +\u2013) 23.gxf5 with a dangerous initiative for White.\n\n**22... \u2656c4 23.\u2656g2 fxg4 24.\u2656xg4 \u2656xa4 25.\u2656ag1 g6 26.h4 \u2656b4 27.h5 \u2655b5 28.\u2655c2**\n\nNot 28.hxg6? \u2655xd3 29.gxh7+ \u2654xh7 30.\u2656g7+ \u2654h6 and the attack is beaten off.\n\n**28... \u2656xb2 29.hxg6! h5**\n\n29...\u2656xc2? 30.gxh7+ \u2654xh7 31.\u2656g7+ \u2654h6 32.f5+ \u2654h5 33.f6+\u2013 leads to mate.\n\n**30.g7! hxg4**\n\n30...\u2658h7 31.\u2655g6+\u2013.\n\n**31.gxf8 \u2655+**\n\n**31... \u2657xf8?**\n\nSince Kramnik judged the alternative to be completely hopeless, he chose this move, which should lead to a loss. In any case he overlooked a hidden resource to the recapture with the king: after 31... \u2654xf8 32.\u2655g6 \u2655e2 33.\u2655xg4 apparently an unstoppable mate is threatened on g7, but the amazing move 33...\u2657g5! holds the position: White can take the bishop neither with the knight (pinned) nor with the pawn (this blocks the g-file). After 34.\u2656e1 \u2655c2 35.\u2655xg5 then 35...\u2656e7 holds the balance.\n\n**32. \u2655g6+?**\n\nInstead of this move, 32.\u2656xg4+ \u2657g7 33.\u2655c7 won immediately. The mate cannot be prevented: 33...\u2655f1+ 34.\u2658g1+\u2013.\n\n**32... \u2657g7 33.f5**\n\nBut White continues to have a strong attack. 33.\u2658g5 leads, however, after 33...\u2656e7 34.\u2655h7+ \u2654f8 35.\u2655g6 \u2655e2 (35...\u2654g8 36.\u2656xg4) 36.\u2658h7+ \u2654g8 37.\u2658f6+ only to perpetual check.\n\n**33... \u2656e7**\n\nAfter 33...exf5? White wins: 34.\u2658g5! \u2655c6 35.\u2655f7+ \u2654h8 36.e6+\u2013. (Marin)\n\n**34.f6 \u2655e2 35.\u2655xg4 \u2656f7 36.\u2656c1?**\n\n36.\u2655h5! was strong, intending 37.\u2656g3 and 38.fxg7. White keeps up the attack.\n\n**36... \u2656c2 37.\u2656xc2 \u2655d1+**\n\nThere was the simpler 37...\u2655xc2 38.\u2658g5 \u2655b1+ 39.\u2654g2 \u2655c2+ with equality.\n\n**38. \u2654g2 \u2655xc2+ 39.\u2654g3 \u2655e4?**\n\n39...\u2655f5! leads instead to a draw: 40.\u2655xf5 exf5 41.\u2658g5 \u2656c7 42.\u2654f4 \u2656c3! 43.fxg7 \u2654xg7 44.\u2658e6+ \u2654g6 45.\u2654f3 \u2656b3=.\n\n**40. \u2657f4?**\n\nInstead, after 40.\u2655xe4 dxe4 41.\u2658g5 White gets a winning endgame, which is, however, hard to calculate out in advance, for example: 41...\u2657h6 42.\u2658xf7 \u2657xe3 43.\u2658d8 a5 44.d5!+\u2013 or 43...\u2657xd4? 44.f7+ \u2654f8 45.\u2658xe6+ \u2654xf7 46.\u2658xd4 a5 47.\u2654f4 a4 48.\u2658b5+\u2013. (Marin) White also wins after 41...\u2657f8 42.\u2658xe6 a5 43.\u2658xf8 44.d5 \u2654e8 45.\u2654f4 a4 46.\u2657d4.\n\n**40... \u2655f5 41.\u2655xf5 exf5 42.\u2657g5?**\n\nAfter this the game tips in Black's favour. After 42.fxg7 \u2656xg7+ 43.\u2654f2 Black's chances are only slightly better. The best was perhaps 42.\u2658g5!.\n\n**42...a5**\n\nNow White suddenly has to fight for the draw.\n\n**43. \u2654f4**\n\n43.fxg7 \u2656xg7 44.e6 a4 45.\u2654f4 no longer suffices for White: 45...a3 46.\u2658d2 a2 47.\u2658b3 \u2656h7\u2013+.\n\n**43...a4 44. \u2654xf5 a3 45.\u2657c1 \u2657f8 46.e6 \u2656c7**\n\n**47. \u2657xa3?**\n\nA better way is 47.e7! \u2657xe7 48.fxe7 \u2656xe7 49.\u2657xa3 \u2656e3 50.\u2657d6 \u2656xf3+ 51.\u2654e5 \u2656b3 52.\u2654xd5 with practical drawing chances.\n\n**47... \u2657xa3 48.\u2654e5 \u2656c1**\n\nHe had better chances of a win after 48... \u2656c3!? 49.\u2658g5 \u2656g3 50.\u2658f7 \u2656g6 51.\u2654f5 \u2656g1.\n\n**49. \u2658g5?**\n\nA more tenacious try was 49.\u2654xd5.\n\n**49... \u2656f1 50.e7 \u2656e1+ 51.\u2654xd5 \u2657xe7 52.fxe7 \u2656xe7 53.\u2654d6 \u2656e1?!**\n\nAccording to John Nunn, who has analysed the game with 'Tablebases', an infallible endgame database, here only 53...\u2656e3 leads to a win by force: 54.d5 \u2654f8 55.\u2654d7 b5 56.\u2658e6+ \u2654g8 57.d6 b4 58.\u2658c5 \u2654f7 59.\u2654c6 \u2656c3 60.\u2654b5 b3 61.\u2658a4 \u2656c2 62.d7 \u2654e7\u2013+.\n\n**54.d5 \u2654f8**\n\n**55. \u2658e6+?**\n\nAccording to the endgame database 55.\u2654d7 instead holds the draw for White. 55...b5 56.\u2658e6+ \u2654f7 (56...\u2654g8 57.d6 b4 58.\u2658c5=) 57.\u2658d8+ \u2654f6 58.\u2658c6 \u2656b1 59.\u2654d6! b4 60.\u2654c5!=. (Marin)\n\n**55... \u2654e8 56.\u2658c7+ \u2654d8 57.\u2658e6+ \u2654c8 58.\u2654e7 \u2656h1!**\n\nBut not 58...b5? 59.d6 \u2656d1 60.\u2658c5 b4 61.d7+ \u2654c7 62.\u2658a6+ (or 62.\u2658e6+) with a draw. (Marin)\n\n**59. \u2658g5?**\n\nA more resilient move was 59.\u2654d6.\n\n**59...b5 60.d6 \u2656d1 61.\u2658e6 b4 62.\u2658c5 \u2656e1+ 63.\u2654f6 \u2656e3**\n\nWhite resigned.\n\nThe schism within the World Championships was now finally brought to an end after 13 years with this match. But the scandalous circumstances which had surrounded the playing of this WCh match once again split the world of chess into two camps. Whilst some condemned the behaviour of Topalov and his manager Danailov as 'dirty tricks', others believed the account of the Bulgarians, namely that Kramnik had cheated in the toilet. Danailov sought to prove this after the match by having the ceiling tiles of Kramnik's toilet removed and finding some cables behind them including a network cable. The reunification match, planned as a brilliant finish to a long and painful process, finished by going down in the history of chess as 'Toiletgate'. The international press did in fact deal extensively with this World Chess Championship, but principally from the point of view of the scandal.\n\nAfter the match Silvio Danailov threatened FIDE with a lawsuit for millions on account of the circumstances surrounding the match in Elista and what in his view were the unsatisfactory decisions by FIDE. By doing so he later obtained some special rights for his prot\u00e9g\u00e9 Veselin Topalov for the next WCh cycle. It was agreed that Topalov should get the right to a return match if Kramnik should win the forthcoming WCh tournament in Mexico. If that should not be the case, then Topalov should play in a candidates' final against the World Cup- winner from Khanty-Mansiysk, in order to qualify via this route as the challenger for the next WCh match.\n**41. World Champion the second time around**\n\n**The World Championship 2007:  \n_World Championship tournament in Mexico City_**\n\nFIDE had actually decided to return to the match format for the World Championships. But before that was possible it had contractual obligations to fulfil. Namely, before FIDE had come to an agreement about the format with Kramnik, it had already fixed in 2005 that the World Championship of 2007 would also be played in tournament format and awarded the tournament to Mexico City.\n\n**Viswanathan Anand (born in 1969)**\n\nFIDE had already staged in the middle of 2007 (26th May till 14th June 2007) two rounds of candidates' matches in the k.-o. system in Elista for qualification purposes. The candidates had qualified for these via the FIDE World Cup 2005 or their places in the world ranking list.\n\nBefore these candidates' matches could finally be held, there was for a long time discussion about the modalities. FIDE had difficulties finding an organiser for the candidates' matches and thus also considered at their executive committee meeting of 22nd and 23rd September 2006 in Elista a possible candidates' tournament with sixteen players. But there were protests from some players who had already got into match mode and had prepared for the players who had been nominated for the matches. Thus Boris Gelfand protested against the changed plans with an open letter. FIDE fell into line and the originally planned format was reinstated.\n\nDue to the lack of other bidders, FIDE president Ilyumzhinov himself stood in as the organiser and staged the matches in Elista, the capital city of his autonomous Russian province of Kalmykia.\n\nThe candidates' matches were each for six games, with tiebreak games at reduced thinking time in the event of draws. The qualifiers from the candidates' matches for the WCh tournament were Levon Aronian, Peter Leko, Boris Gelfand and Alexander Grischuk. Viswanathan Anand, Peter Svidler and Alexander Morozevich had prequalified due to their placing in the WCh tournament of 2005. To them was added Vladimir Kramnik as the reigning World Champion.\n\nOriginally it was only supposed to be the first two from the WCh tournament of San Luis who would go on to the next WCh tournament, but during the tournament itself FIDE changed the rules at short notice and decided that the first four players from San Luis should take part in Mexico. The idea behind it was at that point to bind as many players as possible to FIDE contractually and thus to reduce the possible number of challengers for the 'other World Champion' Kramnik.\n\nThe World Championship tournament in Mexico was played from 12th till 30th September 2007 as a double-round all-play-all. The time control was the classical one with 2 hours for 40 moves, 1 hour for the next 20 moves and 15 minutes for the remainder of the game with an increment of 30 seconds per move. It was hosted by the five star Sheraton Centro Hist\u00f3rico Hotel & Convention Centre. The head organiser Jorge Saggiante had put up a total prize fund of 1.3 million dollars, of which as usual 20% was diverted to FIDE. Anand won four games in the tournament and drew the remaining ten games, which brought him victory in the tournament (9 points) and the title of World Champion. Compared to the scandal-ridden World Championship match between Kramnik and Topalov the tournament ran virtually harmoniously. The players were welcomed with great enthusiasm by the Mexicans.\n\nThere were, however, victims to the street crime of Mexico City. Hans-Walter Schmitt, for many years the organiser of the Frankfurt and Mainz Chess Classic and who had travelled there to support his friend Viswanathan Anand, was attacked on the street, but engaged in a violent scuffle with the muggers.\n\nOn the other hand, the Mexicans too had reason to complain. Zurab Azmaiparashvili, the FIDE representative at the event, was expelled from the Sheraton Hotel on account of an incident. But since he was obliged to observe the games for FIDE, the Georgian received permission to go to and leave the playing venue in the hotel along a specifically designated route. FIDE swept the business under the carpet so that it did not become public. Since a change of government in 2012 Azmaiparashvili for a time held the rank of a 'deputy minister' in the Georgian ministry of sport. In 2014 in Troms\u00f8 he was elected president of the European Chess Union and replaced Silvio Danailov, who had held the office from 2012 to 2014.\n\n **Anand \u2013 Morozevich**\n\nMexico City, 11th round  \n25th September 2007  \nSicilian Defence (B90)\n\n**1.e4 c5 2. \u2658f3 d6 3.d4 cxd4 4.\u2658xd4 \u2658f6 5.\u2658c3 a6**\n\nThe Sicilian Najdorf Variation is one of the most popular openings in modern chess.\n\n**6.f3**\n\nThis leads into the English Attack. Other options are 6.\u2657e2, 6.\u2657c4, 6.f4, 6.\u2657g5, which usually gives rise to very sharp play, and 6.\u2657e3, which should also lead to the English Attack.\n\n**6...e5**\n\nThis double step of the pawn is the basic idea of the Najdorf Variation. The alternative 6...e6 is classified as belonging to the Scheveningen Variation. With 6... \u2655b6 Black could try to exploit White's move order, i.e. 6.f3 instead of 6.\u2657e3.\n\n**7. \u2658b3 \u2657e6 8.\u2657e3 \u2658bd7**\n\n8...\u2657e7 is more popular.\n\n**9.g4 \u2658b6**\n\n9...b5 is also playable.\n\n**10.g5 \u2658h5 11.\u2655d2 \u2656c8 12.0-0-0 \u2657e7 13.\u2656g1 0-0 14.\u2654b1 \u2655c7 15.\u2655f2**\n\nThis attacks the knight on b6 so as to conquer the d5-square. The position has been well researched theoretically and had already occurred in numerous games.\n\n**15... \u2658c4 16.\u2657xc4 \u2657xc4 17.\u2658d5 \u2657xd5 18.\u2656xd5 f5 19.gxf6 \u2656xf6**\n\nNot 19...\u2658xf6 20.\u2656d3 \u2658xe4? on account of 21.\u2655g2 winning a piece.\n\n**20. \u2655e2 \u2658f4 21.\u2657xf4 \u2656xf4 22.\u2656d3 \u2655d7 23.\u2658c1 \u2656cf8 24.a3!**\n\nWhite makes room for the knight, which is to be brought via a2-c3 to d5.\n\n**24... \u2654h8**\n\nIt was worth considering 24...\u2655e6!? 25.\u2658a2 \u2655f7 26.\u2656f1 (26.\u2656g3 \u2657h4 27.\u2656h3 \u2655e6 28.\u2655g2 \u2657e1!?) 26...d5=.\n\n**25. \u2658a2 \u2655h3 26.\u2656g3 \u2655h5**\n\nHere Black has some counterplay against the f3-pawn, but White can easily neutralise it.\n\n**27. \u2655g2 \u2656h4 28.h3 \u2655h6 29.\u2656b3**\n\nHere White's advantage in space makes itself felt. White can easily bring his pieces from one flank to the other, whilst in Black's case the d6-pawn prevents such manoeuvres along the 6th rank.\n\n**29...b5**\n\n29...\u2656b8 30.\u2658b4 (the threat is \u2658xa6) 30...\u2656e8 31.\u2658d5 with a good position for White.\n\n**30. \u2658b4 \u2656h5 31.\u2655f1**\n\nThe immediate 31.\u2658xa6 is followed by 31...\u2657h4 32.\u2656g4 \u2657d8 and then not 33.\u2655f1 on account of 33...\u2656xh3 34.\u2655xb5 \u2656h1+ 35.\u2654a2 \u2655c1 and now Black is in front.\n\n**31... \u2656h4**\n\nThis threatens...\u2656xe4. But 31...\u2657h4 is followed by 32.\u2656g1 \u2657d8 33.\u2658xa6 \u2656xh3 34.\u2655xb5 with an advantage for White.\n\n**32. \u2655g2 \u2656h5 33.\u2658xa6 \u2657h4 34.\u2656g4 \u2657f6 35.\u2655e2**\n\n35.h4 was not bad: 35...\u2657xh4 36.\u2656xb5 with advantage to White.\n\n**35... \u2656xh3 36.\u2656xb5 \u2657d8**\n\nBlack abstains from 36...\u2656h1+ 37.\u2654a2 \u2655c1 38.\u2656g2 h5!? with good counterplay.\n\n**37. \u2656b8 \u2655f6 38.\u2658b4 \u2656xf3 39.\u2658d5**\n\n39.\u2656g1!? \u2656f2 40.\u2655d1 (intending \u2658c6) 40...\u2655e6 41.\u2655d3 and White is ready to advance the a-pawn.\n\n**39... \u2655f7 40.\u2655a6 h5 41.\u2656g2 h4?**\n\nBlack would like to advance the h-pawn as quickly as possible and gives up his two central pawns. There was, however, a better move: 41...\u2655e6.\n\n**42. \u2655xd6 \u2657e7 43.\u2655xe5**\n\nWhite now has in addition to his passed pawn on the queenside another one in the centre.\n\n**43... \u2656xb8 44.\u2655xb8+ \u2654h7 45.\u2655c7 \u2657f8?**\n\nThere was the more tenacious 45...\u2656f1+ 46.\u2654a2 \u2655f3 47.\u2656h2 \u2657g5, though there was also an advantage for White.\n\n**46. \u2655xf7 \u2656xf7**\n\nThe exchange of queens helps White.\n\n**47. \u2656g4 \u2656f1+**\n\n47...g5 48.\u2656xg5 \u2656f1+ 49.\u2654a2 \u2654h6 50.\u2656g2 \u2654h5 51.e5 h3 52.\u2656e2+\u2013.\n\n**48. \u2654a2 \u2656h1 49.e5**\n\nPerhaps the a-pawn was the faster: 49.a4 \u2654h6 50.a5 g5 51.a6 \u2657c5 52.b4 \u2657f2 53.\u2656g2 \u2657a7 54.b5 h3 55.\u2656g3 \u2657b8 56.\u2658e3+\u2013.\n\n**49... \u2657c5?**\n\nA waste of time. 49...h3 was better.\n\n**50.e6 \u2654h6 51.\u2656c4**\n\nAnother good move was 51.b4.\n\n**51...h3**\n\n51...\u2657f8 52.\u2656c8 \u2656f1 53.e7 \u2657xe7 54.\u2658xe7 \u2654h7 55.b4 \u2656f7 56.\u2658d5 g5 57.\u2656c3+\u2013.\n\n**52. \u2656xc5 h2**\n\nNow the threat is...\u2656e1 and...h2-h1\u2655 and Black wins.\n\n**53. \u2658e3! \u2656a1+**\n\n53...\u2656g1 54.e7 h1\u2655 55.e8\u2655+\u2013 with an enormous material advantage for White. Or 53...\u2656e1 54.\u2658g4+ +\u2013.\n\n**54. \u2654xa1 h1\u2655+**\n\nBlack is now actually the first to get a new queen, but cannot prevent White from also promoting his pawn.\n\n**55. \u2654a2 \u2655e4 56.\u2656e5!**\n\nThis trick allows White to secure the promotion of his pawn: 56...\u2655xe5 57.\u2658g4+. Black resigned.\n**42. 74 years later, back to Germany**\n\n**The World Championship 2008:  \n_Viswanathan Anand against Vladimir Kramnik_**\n\nAccording to the agreements which FIDE had made with Kramnik during the reunification match, Kramnik, who had not won the WCh tournament of Mexico City, had been accorded the right to a return match against the World Champion Viswanathan Anand. The staging of this was entrusted to Josef Resch's company 'Universal Event Promotion' (UEP), behind which there was essentially the organisational team of the Dortmund grandmaster tournaments, financially guaranteed by the patron Josef Resch. The UEP had already bid for the staging of the reunification match but had been turned down for not quite comprehensible grounds.\n\nThe UEP had an influential supporter in the chess enthusiast and minister-president of the federal state of Nordrhein-Westfalen Peer Steinbr\u00fcck, future federal finance minister (from November 2005). Steinbr\u00fcck had always wanted to bring a chess WCh to Nordrhein-Westfalen. He finally became the patron of the match. Moreover, with their partner the Federal Art and Exhibition Hall in Bonn they could point to an excellent venue. The venue had already seen a first general rehearsal on 5th May 2005 when Peer Steinbr\u00fcck played an exhibition match against Vladimir Kramnik in its auditorium \u2013 and did quite well in it. After winning the reunification match Kramnik appeared from 25th November till 5th December in the said Federal Art and Exhibition Hall in a man-against-machine match against the program _Deep Fritz_. Kramnik lost by 2:4 the match which was so excellently staged by the UEP and which caused a great amount of media hype in Germany and attracted numerous spectators.\n\nThe sponsors of the WCh match between Anand and Kramnik, the Essen energy firm Evonik and the Russian energy producer Gazprom, put up a prize fund of 1.5 million euros. After the 'FIDE tax' had been taken off, there still remained for each of the players the sum of 600000 euros. The match, set for 12 games, was played from 14th October till 2nd November 2008 in the forum (auditorium) of the Art and Exhibition Hall and was the first World Chess Championship in Germany since 1934 (Alekhine against Bogoljubow). If it was drawn after twelve games there was to be a playoff of four rapid chess games with 25 minutes plus 10 seconds\/move per player. If things were still level, then two blitz games with 5 minutes plus 10 seconds\/move. And finally it would have come down to a 'sudden death' game.\n\nAfter the scandalous match in Elista the organisers of the match in Bonn explicitly adopted special security measures which were intended to make any form of cheating impossible. Thus the stage was separated from the spectators' area by a one-way curtain to prevent any eye-contact between spectators and players. The use of mobile telephones in the spectators' room was forbidden. The games, moreover, were subject to a time delay before being broadcast. This delay was, however, abandoned after five games because various newspapers, which had included the live broadcast of the games on their web pages, were unhappy with the delay.\n\nThe project and program leader for the Forum of the Art and Exhibition Hall Stephan Andreae had arranged an attractive event around the World Championship match under the title 'Hinter den Spiegeln'. The title is a play on Lewis Carroll's second children's book _Through the looking glass and what Alice found there_ , which is based on the game of chess. Despite the very high entry price of 35 euros per day, the rows of spectators in the 400 seat auditorium were always well filled and especially at the weekends there was a big rush. On weekend match days at the end of the contest it was even necessary to turn away several hundred interested persons at the entrance because there were no more seats left.\n\nFor the presentation both in the venue and on the internet the UEP had prepared a series of innovations. Thus the games were commented on live in a commentary room by, in turn, grandmasters Klaus Bischoff, Helmut Pfleger and Artur Jussupow. On every match day various prominent guests were introduced in panel discussions. In a dedicated VIP room there was also live commentary in a club-like atmosphere. Immediately after the games the two players attended a press conference in which they answered the questions of spectators and journalists and explained the games. To make things more attractive the UEP hired models who stood in the background at the start of the games or at the press conferences and held up banners with the names of the sponsors.\n\nFor internet spectators the UEP together with its Dutch partner firm DGT (Digital Game Technology) had developed for the presentation of the games a new technology called Foidos. For a fee one could follow the games, including live commentary in several languages and video streams. However, they could not find enough subscribers for the system and the joint venture soon went bankrupt after the WCh.\n\nAs well as his regular second Peter Heine Nielsen, Anand had put together for the match a team of helpers, the composition of which, however, did not become known until the match was over. It included: Rustam Kasimdzhanov, Surya Shekhar Ganguly and Radoslaw Wojtaszek. Anand had already started to prepare for the match at the end of 2007. For example, he played training games in Madrid with Magnus Carlsen, against whom he tried out his surprise weapon 1.d4. Kramnik was supported during the match by Sergey Rublevsky, Laurent Fressinet and Peter Leko.\n\nTo surprise Kramnik, with the exception of the final game Anand opened all of his games in this match as White with the opening move which he had rarely ever used otherwise, 1.d4. After draws in the first two games, Anand won the third with the black pieces with the help of the Meran Variation. After another sharing of the point, he did this again in the fifth game, with an almost identical opening. It was only on move 15 that he deviated from the previous game. In the sixth game the title defender hit out with the white pieces and won against Kramnik's Nimzo-Indian Defence. Anand then had a clear lead at the half-way mark of 3:0 \u2013 the match was already as good as over.\n\nWith a win in the tenth game Kramnik was able to shorten the deficit, but a draw in the 11th game secured victory in the match for Anand with one game to play. This made Anand the first player to have won or defended the World Championships in all three formats \u2013 in the FIDE k.-o. system in 2000 in New Delhi and Teheran, in the all-play-all tournament in 2007 in Mexico-City and now also in the match against Kramnik.\n\n **Kramnik \u2013 Anand**\n\nBonn, 3rd game  \n17th October 2008  \nQueen's Gambit, Semi-Slav Defence (D49)\n\n**1.d4 d5 2.c4 c6 3. \u2658f3 \u2658f6 4.\u2658c3 e6 5.e3**\n\nAfter 5.\u2657g5 some complicated systems can arise if Black, with or without the insertion of 5...h6 6.\u2657h4, takes the pawn on c4.\n\n**5... \u2658bd7 6.\u2657d3 dxc4 7.\u2657xc4 b5**\n\nThis manoeuvre characterises the Meran Variation, which leads to very dynamic play.\n\n**8. \u2657d3 a6**\n\nAlternatives are 8...\u2657b7 and 8...b4. Recently 8...\u2657d6 has also become popular.\n\n**9.e4 c5 10.e5**\n\n10.d5 would be the Reynolds Attack.\n\n**10...cxd4 11. \u2658xb5 axb5**\n\n11...\u2658xe5 is also playable: 12.\u2658xe5 axb5 13.\u2657xb5+ \u2657d7.\n\n**12.exf6 gxf6 13.0-0**\n\nBut not 13.\u2657xb5?? \u2655a5+ \u2013+.\n\n**13... \u2655b6 14.\u2655e2**\n\n14.\u2657e4 \u2657b7 15.\u2657xb7 \u2655xb7 16.\u2658xd4 \u2656g8 with counterplay. (Anand)\n\n**14... \u2657b7!?**\n\nBlack gives up the b5-pawn in return for rapid development. Until then a more popular line was 14...b4 15.\u2656d1 \u2657c5 etc.\n\n**15. \u2657xb5!**\n\nThe principled move. But 15.\u2657f4!? is also possible in order to use the bishop on g3 as extra protection on the g-file, 15...b4 etc.\n\n**15... \u2657d6!?**\n\nIn the 5th match game Anand deviated here and won with this variation for the second time: 15...\u2656g8 16.\u2657f4 \u2657d6 17.\u2657g3 f5 18.\u2656fc1 f4 19.\u2657h4 \u2657e7 etc. 0-1\/35.\n\n**16. \u2656d1**\n\nAfter 16.\u2658xd4!? \u2655xd4 17.\u2656d1 Black has the spectacular 17...\u2657xh2+! at his disposal, but after 18.\u2654xh2 \u2655h4+ 19.\u2654g1 \u2657xg2! 20.\u2657xd7+ \u2654e7 21.\u2654xg2 \u2656hg8+ there is no more in it for Black than a perpetual check. (Anand)\n\n**16... \u2656g8 17.g3!**\n\nNot 17.\u2656xd4 \u2656xg2+ 18.\u2654xg2 \u2655xd4\u2013+.\n\n**17... \u2656g4! 18.\u2657f4**\n\nIn this complicated situation a whole series of other moves came into consideration, of which the best was 18.\u2658d2!? with the threats of \u2655xg4 and \u2658c4. According to Anand it is followed by 18...\u2654e7!! with unclear complications.\n\n**18... \u2657xf4**\n\n18...\u2656xf4 is probably too speculative. After 19.gxf4 \u2654e7 20.a4 \u2656g8+ Black has an attack, but White can defend against it: 21.\u2654f1 \u2657d5 22.\u2656ac1 etc.\n\n**19. \u2658xd4!?**\n\nThe critical line is 19.\u2656xd4!? \u2654f8! 20.\u2657xd7 \u2656d8 21.\u2656ad1 \u2656xd7 22.\u2656xd7 \u2657xg3 23.hxg3 \u2656xg3+ 24.\u2654h2 \u2657xf3 25.\u2655e3 \u2656g2+ 26.\u2654h3 \u2655xe3 27.fxe3 \u2656xb2 28.\u26561d2 \u2656b1 29.a4 \u2657d5 with roughly level chances in the endgame.\n\n**19...h5**\n\n19...\u2656g6!? was worth considering. Maybe White's best continuation now is 20.a4!?.\n\n**20. \u2658xe6**\n\n20.\u2657xd7+ \u2654xd7 21.\u2658xe6+ is followed by 21...\u2657d6 22.\u2656xd6+ \u2655xd6 23.\u2656d1 fxe6 24.\u2656xd6+ \u2654xd6 25.\u2655d2+ \u2657d5\u2013+.\n\n**20...fxe6 21. \u2656xd7 \u2654f8 22.\u2655d3**\n\nIntending 23.\u2655h7 and mate on f7.\n\n**22... \u2656g7!?**\n\nInstead 22...\u2657xg3!? leads to a forced draw: 23.hxg3 h4! 24.\u2656d6 \u2655c5 25.b4 \u2655e5 26.\u2656d8+ \u2656xd8 27.\u2655xd8+ \u2654g7 28.\u2655e7+ \u2654h6 29.\u2655f8+ \u2656g7 30.\u2655h8+ \u2656h7 31.\u2655f8+ with perpetual check. (Anand)\n\n**23. \u2656xg7 \u2654xg7 24.gxf4 \u2656d8! 25.\u2655e2 \u2654h6 26.\u2654f1 \u2656g8 27.a4!**\n\n27.f5? is followed by 27...\u2657g2+! 28.\u2654e1 \u2657c6! with advantage to Black, since the white king is insecure and the white rook is in acute danger.\n\n**27... \u2657g2+ 28.\u2654e1 \u2657h3! 29.\u2656a3?**\n\nCorrect was 29.\u2656d1! and White holds on, for example: 29...\u2657f5!? and White must find 30.h3 or 30.\u2655f1 here, after which the chances are level. After 30.\u2655e3? \u2656g1+ 31.\u2657f1 \u2655a6! Black would win.\n\n**29... \u2656g1+ 30.\u2654d2 \u2655d4+ 31.\u2654c2 \u2657g4?**\n\nIn spite of having a lead of 75 minutes on his clock, Anand, intending to provoke 32.f3, commits an inaccuracy. The correct move was 31...\u2657f5+! and now the best is 32.\u2656d3!. But after 32...\u2656g4!? or 32...\u2656g2 Black retains winning chances. (Anand)\n\n**32.f3?**\n\nReturning the compliment. Kramnik misses 32.\u2656d3! after which Black has nothing better than 32...\u2657f5 33.\u2654b3 \u2657xd3 34.\u2655xd3 \u2655xf2 and 35.\u2655d8! secures the draw. (Anand)\n\n**32... \u2657f5+ 33.\u2657d3 \u2657h3**\n\nThere was an immediate win with 33... \u2657xd3+! 34.\u2656xd3 (34.\u2655xd3 \u2656g2+ 35.\u2654c1 \u2655xb2+) 34...\u2655c4+ 35.\u2654d2 \u2655c1#.\n\n**34.a5**\n\nKramnik throws the a-pawn forward, so as to compensate for the subsequent loss of material. Other moves also lose.\n\n**34... \u2656g2 35.a6 \u2656xe2+ 36.\u2657xe2 \u2657f5+ 37.\u2654b3**\n\n37.\u2657d3 \u2657xd3+ 38.\u2656xd3 \u2655c4+ \u2013+, 37.\u2654c1 \u2655xf4+ 38.\u2654d1 \u2655d4+ 39.\u2654c1 \u2655e5!\u2013+.\n\n**37... \u2655e3+ 38.\u2654a2 \u2655xe2 39.a7 \u2655c4+ 40.\u2654a1 \u2655f1+ 41.\u2654a2 \u2657b1+**\n\nResigned on account of 42.\u2654b3 \u2655xf3+.\n**43. Blackout in Sofia**\n\n**The World Championship 2010:  \n_Viswanathan Anand against Veselin Topalov_**\n\nVeselin Topalov had actually been eliminated from the World Championship cycle as a result of his defeat at the hands of Kramnik and should have had to re-qualify. With the threat of suing FIDE for millions, his manager Silvio Danailov obtained that Topalov \u2013 in contravention of the rules in force \u2013 should be seeded into the candidates' final.\n\nThe original plan had been that the winner of the World Cup, organised by FIDE in the k.-o. system, would be the challenger to the World Champion. The winner of the 2007 World Cup was Gata Kamsky, who to his surprise now first had to play a candidates' match against Topalov. Kamsky, born in 1974 in Novokusnetsk (Tatarstan, USSR) had been part of the world elite for 20 years. He emigrated to the USA together with his father in 1989. In 1994 he took part in the WCh cycles of the PCA and FIDE and played in 1996 a FIDE WCh match against Karpov, which he lost. After that he withdrew from chess and studied law. In 1999 he surprisingly took part in the FIDE k.-o. World Championship in Las Vegas, and then disappeared once more from professional chess. In 2004, however, he returned and since then has again been regularly playing in tournaments and championships.\n\nThe Bulgarian Chess Federation wanted to hold the candidates' final in Bulgaria, but Kamsky had no interest in playing in his opponent's home country. Kamsky's manager Alexander Chernenko, a diamond dealer, finally in May 2008 made FIDE an offer of 750 000 dollars in prize money, thus clearly outbidding the Bulgarian federation by 150000 dollars, and was awarded the match, which was to be played in Lviv (Ukraine), therefore on neutral territory. In June 2008 FIDE announced that Ilyumzhinov would personally guarantee the playing of this match in Lviv and the prize money of 750000 dollars.\n\nChernenko stalled the FIDE for a long time with promises, but what he could not come up with before the start of November 2008 was the bank guarantee demanded by FIDE for the promised amount of prize money, so that on the 8th November FIDE, contrary to their declaration of June 2008, again offered the match to the Bulgarian Chess Federation, with a prize fund of 250000 dollars plus 50000 dollars contribution to FIDE. The Bulgarian federation accepted, but Kamsky continued to refuse to play in Bulgaria, since Topalov would have a clear home advantage there. After FIDE threatened Kamsky that if necessary they would replace him with the defeated World Cup finalist Alexei Shirov, Kamsky accepted the conditions. The contracts were finally signed during the Chess Olympiad in Dresden 2008.\n\nThe candidates' final then took place from 16th till 28th February 2009 in room six of the National Cultural Palace in Sofia under the title 'World Chess Challenge' and was for eight games. After five games Topalov was leading 2:1 in wins. In the seventh game Kamsky blundered in a clearly winning position and lost. So instead of equality this became the final score of 4\u00bd:2\u00bd. This fixed Topalov as the challenger to Anand.\n\nBecause of the delays leading up to the candidates' final, the World Championship match was also postponed. According to the original plan it should have been played in the first half of 2009. It finally took place from 24th April till 13th May 2010.\n\nOfficially the Bulgarian Chess Federation, the Turkish Chess Federation and the Chess Federation of Singapore bid to stage the match. But apparently the bids from Turkey via the president of the Turkish federation Ali Nihat Yazici and from Singapore via Ignatius Leong were merely sham offers, which were later withdrawn and the only purpose of which was to frighten off other bidders and thus to clear the way for the Bulgarian bid.\n\nThe venue for the match between Anand and Topalov was the Central Military Club in Sofia. The match took place in the building's theatre. The price for entry was five dollars \u2013 obviously too much for Bulgarian chess lovers who for the greater part stayed away from the match. The prize fund was set at 1.2 million dollars for the winner and 800000 dollars for the loser. FIDE received a further 400000 dollars. The event was sponsored by the Bulgarian state and the telecommunications firm Spectrum Net.\n\nThe time control was the classical tournament thinking time of two hours for 40 moves, one hour for the next 20 moves and 15 minutes for the remainder of the game with an increment of 30 seconds per move. The match was for twelve games. In the event of a draw there was to be a playoff of four rapid games, followed by a maximum of five rounds of two blitz games per round. If there were still no decision, a 'sudden death' game would be played. Topalov's manager Danailov wanted to use the so-called 'Sofia rules' for the WCh match, according to which the players were not allowed to end a game by agreeing a draw. Since FIDE refused to agree to this until then unusual rule in WCh matches, Danailov announced that Topalov would for his part play according to that rule.\n\nAt Anand's request the stage was separated from the spectators' area by a semi-transparent curtain to prevent the possibility of signals being given to the players from the spectators' area. What was not made public was that there was always someone from Anand's team 'accompanying' Danailov during the games so that the latter was never unobserved. This measure also served as anti-cheating prophylaxis.\n\nIn his game preparation in Sofia Anand was supported by his team from Bonn: Peter Heine Nielsen, Rustam Kasimdzhanov, Surya Shekhar Ganguly and Radoslaw Wojtaszek. In advance of the match Anand had in addition received help from Garry Kasparov, Magnus Carlsen and Anish Giri. He had played training matches with Carlsen and Giri.\n\nKasparov had offered his help for reasons of chess politics. The ex-World Champion was supporting Anatoly Karpov in his candidacy for FIDE president and moreover had persuaded the president of the German Chess Federation Robert von Weizs\u00e4cker to stand in the election for president of the European Chess Union (ECU). The opposing candidates in the ECU elections included for example Silvio Danailov. Kasparov feared that if Topalov were to win Danailov's influence would grow too much and thus he supported Anand. In addition, he hoped that Anand would help him bring the Indian Chess Federation on to his side for the pending FIDE presidential election.\n\nDuring the match Vladimir Kramnik also got in touch with Anand and also offered his help. It had occurred to Kramnik that in his choice of openings Anand was being guided by his own choices in his Elista match against Topalov and he informed the Anand team about some critical variations. In addition he also had some new ideas. Rarely in any previous WCh match had there been such wide support for one player, in this case support against one player \u2013 against Topalov, but probably more against the latter's manager Danailov. Danailov had polarised opinion too greatly in the past.\n\nTopalov had as seconds in addition to his constant collaborator Ivan Cheparinov the Dutch players Erwin l'Ami and Jan Smeets, in addition the computer specialist Jiri Dufek, who had prepared the opening book for the author of the chess engine _Rybka_ , Vasik Rajlich. Before and during the match Topalov used a beta version of the Rybka4 engine on a mainframe IBM Blue Gene\/P with 8192 processors, to which he had received access with the help of Bulgaria's Premier Boyko Borissov. The Topalov team is reputed to have spent a great deal of money to delay the public launch of the commercial Rybka4 engine till after the match.\n\nWhen Anand learned about this before the match, he and his team were somewhat concerned about the calculating power available to the opposing side. Anand there-upon rented a computer cluster with the _Hiarcs_ engine by the English programmer Mark Uniacke and moreover put his trust in his 'human cluster' \u2013 his seconds.\n\nIn Sofia, Anand stayed in the Hilton Hotel, where he rented a whole section of the 9th floor of the hotel including the presidential suite for himself and his team. The rooms were partly modified according to Anand's wishes. Access to the area of the hotel rented by Anand was off-limits for all other guests. The Anand team received two dedicated 'secure' internet connections. Moreover, security experts had swept the hotel rooms for bugs.\n\nThe title defender had planned to arrive in Sofia with his team on the 16th April. They met on that same day in Frankfurt and wanted to fly together from there to the Bulgarian capital. Then the World Champion and his seconds were caught out by the eruption of the Icelandic volcano Eyjafjallaj\u00f6kull. This had spread a cloud of ash over Europe, whereupon for safety reasons all European air traffic was completely cancelled for several days after the 16th April. Even the attempt to hire the private jet of Wolfgang Grenke, sponsor of OSG Baden-Baden, for whom Anand played in the German Chess Bundesliga, failed since small planes were also subject to the flight ban. Anand was stuck in Frankfurt with his seconds. In order to even get to Sofia, Anand rented with the help of Erik van Reem in the Netherlands a minibus and driver, which took him and his team on a 40 hour journey of over 2000 kilometres from Frankfurt to the Bulgarian capital.\n\nSince the shortness of time meant that visas could not be arranged for all the team members for all the countries along the way, the bus had to take a slightly longer route via Romania instead of Serbia. Nevertheless they still managed to find time along the way to visit Peter Leko and Judit Polgar in Hungary. During the trip they watched the films of the _Lord of the Rings_ trilogy, listened to music by Rammstein and discussed some critical variations of their opening preparation. One special problem before their departure had been getting all their luggage, including the computers of the World Champion and his seconds, removed from the Lufthansa flight they had booked, something which is normally no easy matter. Even this was achieved with the help of Erik van Reem, whose main job is with Lufthansa. In the meantime Anand had requested a three-day postponement of the match, which was turned down by the organisers. FIDE vice-president Georgios Makropoulos decided as a compromise solution that the match could be postponed for one day.\n\nThe match was therefore started on the 24th April by Bulgarian prime minister Boyko Borissov, the patron of the event, with the symbolic first move. Anand's head of delegation Hans-Walter Schmitt had not been affected by the travel problems as he had arrived in Sofia on the 15th April with more or less the last plane to fly, so as to prepare everything for the arrival of the Anand team.\n\nThe organisers offered internet spectators a live broadcast of the games on its own match website and also allowed various media outlets to have the moves. However, the firm ChessBase was the only one which was not allowed to have the moves and after ChessBase nevertheless broadcast and commented on the games 'live' (that is with a slight delay) on its news pages and chess server, Danailov filed via the Bulgarian federation a complaint, which was dealt with on the 29th March 2011 by the Berlin Regional Court and thrown out on all points.\n\nOn the scene the commentators were Zurab Azmaiparashvili and the Bulgarian women's ex-World Champion Antoaneta Stefanova. The players were fetched from their hotels by cars and then taken to the military club with flashing blue lights and a police escort. Anand always arrived first. Topalov's car always came three minutes later. As they entered the building both players were checked for electronic devices by a detector. Cameras had been installed behind the stage to supervise all areas of the building there, with the exception of the toilets. The players were accompanied to the toilet by the second arbiter Werner Stubenvoll. The chief arbiter was Dimitrios Nikolaidis. Even the spectators had to be screened by a detector as they entered the building. Taking mobile telephones into the tournament hall was strictly forbidden.\n\nTopalov won the first game straight out of the opening in a sharp Gr\u00fcnfeld Indian variation, after Anand had forgotten his preparation. Anand, however, managed to equalise the score immediately in the second game. This time his preparation in the Catalan Opening was better than that of Topalov. After a draw in game three Anand struck again in the fourth game and again won with the white pieces and the Catalan Opening. There followed three draws, then Topalov was successful against Anand's Slav Defence and equalised the score.\n\nThere was an unusual incident during the fifth game. Just as Anand was pondering his 17th move, there was a power cut all over Sofia, including the military club, where the lights went out. Players and spectators sat in darkness. The arbiter stopped the clock and interrupted the game. After just 40 minutes the electricity came back on and the game could be continued. At the behest of prime minister Boyko Borissov the energy minister and energy firm CEZ had to send a letter of apology to the players.\n\nThe ninth game was played on the 6th May 2010, St George's day, the day of the Bulgarian spring festival. For that reason there were several noisy parades outside of the military club, the noise penetrated the building and disturbed the players. Topalov was particularly annoyed by the noise. The game turned into the most exciting one of the whole match. Topalov, with the black pieces, had unnecessarily exchanged his two rooks for the queen and then had to fight for a long time for the draw. In a complicated game Anand missed the win several times. At the end the point was shared.\n\nThe twelfth game was played on a score of 2:2 in wins; in it Topalov with white proceeded really aggressively against Anand's Lasker Variation of the Queen's Gambit and then ran into the counterplay of the World Champion. Journalists and chess lovers asked why Topalov apparently went for all or nothing in the final game. Obviously the challenger considered himself hopelessly inferior in a playoff and wanted to avoid this under all circumstances. Thus Anand defended his title successfully for the second time with a score of 6\u00bd:5\u00bd.\n\n **Topalov \u2013 Anand**\n\nSofia, 12th game  \n11th May 2010  \nQueen's Gambit Declined (D56)\n\n**1.d4 d5 2.c4 e6**\n\nAfter Anand had first tried the Gr\u00fcnfeld Indian Defence in this match, then switched to the Slav Defence to the Queen's Gambit, in the final game of the match he chose the Queen's Gambit Declined.\n\n**3. \u2658f3 \u2658f6 4.\u2658c3 \u2657e7 5.\u2657g5**\n\nThe alternative is 5.\u2657f4.\n\n**5...h6 6. \u2657h4 0-0 7.e3 \u2658e4**\n\nThis move starts the Lasker Variation.\n\n**8. \u2657xe7 \u2655xe7 9.\u2656c1**\n\nRecently, 9.\u2655c2 was not so popular: 9...\u2658xc3 10.\u2655xc3 c6 11.\u2657d3 dxc4 12.\u2657xc4 \u2658d7 or 12...b6.\n\n**9...c6 10. \u2657e2 \u2658xc3 11.\u2656xc3 dxc4**\n\n11...\u2658d7 12.0-0 dxc4 is a transposition of moves.\n\n**12. \u2657xc4 \u2658d7 13.0-0 b6**\n\nThe alternative is 13...e5.\n\n**14. \u2657d3 c5 15.\u2657e4 \u2656b8 16.\u2655c2**\n\n16.\u2655a4 is also a favourite move here.\n\n**16... \u2658f6**\n\nAn idea of the Polish player Miroslaw Grabarczyk, which Topalov perhaps did not know. Previously 16...a5 was played.\n\n**17.dxc5**\n\n17.\u2657c6 allows the reply 17...cxd4 18.\u2658xd4 e5!. (Giri)\n\n**17... \u2658xe4 18.\u2655xe4 bxc5**\n\nBlack's active light-squared bishop compensates easily for the weak pawns on a7 and c5.\n\n**19. \u2655c2 \u2657b7 20.\u2658d2**\n\n20.\u2656xc5?! allows the trick 20...\u2657xf3 21.gxf3 \u2656xb2 and after 20.e4 there is the interesting option 20...f5!?.\n\n**20... \u2656fd8 21.f3**\n\nTo limit the effectiveness of the \u2657b7. 21.\u2656xc5? \u2656xd2!\u2013+.\n\n**21... \u2657a6**\n\nSo the bishop changes diagonals.\n\n**22. \u2656f2?!**\n\nFrom here the rook protects the \u2658d2. Nevertheless the rook is awkwardly placed on f2. For the moment it is completely dominated by the \u2657a6 and has no square to go to. 22.\u2656c1 \u2655d7 23.\u2658b3 c4!?= (23...\u2657d3 24.\u2658xc5 \u2657xc2 25.\u2658xd7 \u2656xb2 26.\u26561xc2 leads to a draw).\n\n**22... \u2656d7 23.g3 \u2656bd8 24.\u2654g2 \u2657d3**\n\nAnand, whose position is freer, first makes another waiting move. Black could instead become active here with 24...h5!? or 24...e5!? 25.e4 h5.\n\n**25. \u2655c1**\n\nIn the event of 25.\u2655a4 there follows 25...\u2655g5 and after 26.e4?! (26.\u2658e4! =) 26...\u2655e3 27.\u2655a5 \u2655e1 28.\u2655xc5 \u2657e2! 29.\u2658b3 \u2656d1 30.\u2654h3 \u26568d3 Black has taken control of the action.\n\n**25... \u2657a6! 26.\u2656a3**\n\n26.\u2655c2 \u2657d3 would be a repetition of the position, but here Black can also continue with 26...h5 or 26...e5.\n\n**26... \u2657b7 27.\u2658b3**\n\n27.e4 is met with 27...f5! 28.\u2655c2 g5!. The threat is...g5-g4. It would be wrong to play 27.\u2656xa7? \u2657xf3+.\n\n**27... \u2656c7 28.\u2658a5 \u2657a8 29.\u2658c4**\n\n29.e4 can now also be followed by 29...g5!, for example: 30.\u2656e3 \u2656d4 31.h3 h5! 32.g4? hxg4 33.hxg4 f5! 34.gxf5 g4! and Black wins. (Giri) If 29.\u2656c3 then 29...g5 30.\u2658b3 g4 31.e4 gxf3+ 32.\u2654xf3 f5 33.\u2658xc5 \u2655g7!? with the threat 34...\u2656xc5 35.\u2656xc5 \u2655g4+ 36.\u2654g2 \u2657xe4+ 37.\u2654g1 \u2656d1+ \u2013+.\n\n**29...e5**\n\nIt was also worth considering the double-edged 29...g5!?.\n\n**30.e4**\n\nOr else Black plays...e5-e4.\n\n**30...f5! 31.exf5?**\n\nA mistake which is hard to comprehend. The correct move was 31.\u2658d2.\n\n**31...e4!**\n\n**32.fxe4??**\n\n'Equivalent to resignation.' (Marin) For example 32.\u2656e3 was better, but after 32...exf3+ 33.\u2654g1 \u2655g5 Black has all the trumps.\n\n**32... \u2655xe4+ 33.\u2654h3 \u2656d4**\n\nThe threat is mate on g4.\n\n**34. \u2658e3 \u2655e8!**\n\nThe key move of Black's attack, overlooked by Topalov in his calculations, as he admitted after the game. The threat is mate on h5.\n\n**35.g4 h5 36. \u2654h4**\n\n**36...g5+**\n\nThere was also a win after 36...\u2655d8+ 37.f6 hxg4, for example 38.\u2658xg4 gxf6 39.\u2656f5 \u2656h7+ 40.\u2654g3 \u2655d6+ 41.\u2656f4 \u2656xf4 42.\u2655xf4 \u2656h3+ \u2013+, whilst 36... hxg4? surprisingly gives away the advantage: 37.\u2658xg4 g5+ 38.fxg6 \u2655xg6 39.\u2656f4 \u2656h7+ 40.\u2654g3 and White holds on.\n\n**37.fxg6 \u2655xg6 38.\u2655f1 \u2656xg4+ 39.\u2654h3 \u2656e7**\n\nThe threat is...\u2656xe3+,...\u2656h4+ and...\u2655g4#.\n\n**40. \u2656f8+ \u2654g7**\n\nAfter 40...\u2654h7 White can still muddy the waters with 41.\u2656h8+ (41.\u2656xa8 \u2656xe3+ 42.\u2656xe3 \u2656h4+ 43.\u2654xh4 \u2655g4#) 41...\u2654xh8 42.\u2655f8+ \u2655g8 43.\u2655h6+ (43.\u2655xe7 \u2655c8!\u2013+) 43...\u2656h7 44.\u2655f6+ \u2656hg7 45.\u2655h6+ \u2655h7 46.\u2655xh7+ \u2654xh7 47.\u2658xg4 hxg4+ and wins.\n\n**41. \u2658f5+**\n\n41.\u2656xa8 \u2656xe3+! 42.\u2656xe3 \u2656h4+!! 43.\u2654xh4 \u2655g4#.\n\n**41... \u2654h7!**\n\nBut not 41...\u2654xf8 42.\u2658xe7+ \u2654xe7 43.\u2656xa7+ \u2654d6 44.\u2655f8+ \u2654e5 45.\u2655xc5+ \u2657d5 46.\u2656e7+ +\u2013 and Black must give up the queen.\n\n**42. \u2656g3 \u2656xg3+ 43.hxg3 \u2655g4+ 44.\u2654h2 \u2656e2+ 45.\u2654g1 \u2656g2+ 46.\u2655xg2 \u2657xg2 47.\u2654xg2**\n\n47.\u2656f7+ \u2654g6! 48.\u2656g7+ \u2654xf5 49.\u2656xg4 hxg4! 50.\u2654xg2 \u2654e4 51.\u2654f2 \u2654d3 leads to a won pawn ending.\n\n**47... \u2655e2+ 48.\u2654h3 c4**\n\nWhite is in zugzwang.\n\n**49.a4 a5 50. \u2656f6 \u2654g8! 51.\u2658h6+ \u2654g7 52.\u2656b6 \u2655e4**\n\nThreatening mate on h1. 52...\u2655f3 53.\u2654h4 \u2655e4+! 54.\u2654xh5 \u2655d5+ \u2013+.\n\n**53. \u2654h2 \u2654h7!**\n\nRenewing the zugzwang motif.\n\n**54. \u2656d6 \u2655e5 55.\u2658f7**\n\n55.\u2656b6 h4\u2013+.\n\n**55... \u2655xb2+ 56.\u2654h3 \u2655g7**\n\nWhite resigned due to 57.\u2658h6 c3\u2013+.\n**44. Chess is art**\n\n**The World Championship 2012:  \n_Viswanathan Anand against Boris Gelfand_**\n\nAfter Kirsan Ilyumzhinov and FIDE with the help of Vladimir Kramnik had succeeded in reuniting the two World Championships, the world federation subsequently showed itself incapable of coming up with even a medium term plan for a WCh cycle with a transparent sporting qualification system and in doing so gathering together all the threads of their previous mistakes in planning. At the presidential election of 2006, which was run parallel to the Chess Olympiad in Turin at the end of the FIDE congress, the current office bearer Kirsan Ilyumzhinov and his team saw themselves challenged by Bessel Kok.\n\nKok, an enthusiastic chess fan, had in the 1980s as the managing director of a communications company for banks, SWIFT, sponsored a series of chess tournaments. In 1985 he founded along with Kasparov the Grandmaster Association (GMA), a first attempt of the then World Champion to side-line FIDE with a competing organisation. The GMA successfully set up a series of tournaments under the name Grand Prix. In 1991, however, it was dissolved after a dispute between Kasparov and Kok. In 2002 Kok, in the meantime manager of the Czech company Eurotel, played a decisive part in bringing about the Prague meeting, which finally, though not until 2006, led to the reunification of the World Championships. Many top players who were extremely dissatisfied with FIDE's policy of recent years, supported Kok's campaign 'The Right Move'. The western federations were also behind him.\n\nNevertheless, the Dutchman had no chance of winning the election against the current office bearer. Ilyumzhinov and his supporters had no great trouble getting a clear two thirds majority (54:96 delegates) with the votes of the federations in Eastern Europe, in Central and South America and in the developing chess countries of Asia and Africa. According to the statutes of the world chess federation, every member country had exactly one vote no matter whether the federation has, like Germany for example, 90000 members, or in many chess developing countries perhaps only four or five chess enthusiasts.\n\nOrganising the votes of such mini-federations is easy, since just the promise of some equipment or other concessions leads to the acquisition of the vote and there is hardly anybody back home to call the presidents of these mini-federations to account. Observers, who were at the congress, reported that the existing committee was in no way squeamish about procuring the votes. Many delegates were simply paid for their votes in the corridors of the congress centre, or so it was claimed.\n\nKirsan Ilyumzhinov's policy of many years has been to come to an understanding with critics and the forces of the opposition. So he went to Bessel Kok too and founded with him in December 2006 the firm 'Global Chess', which was according to FIDE the 'commercial arm' of the world chess federation, the aim of which was a better marketing of international professional chess. 'Global Chess', with a share capital of 4.5 million euros, mainly raised by Kirsan Ilyumzhinov, set up its headquarters in Amsterdam and announced that in the near future it would raise the share capital to ten million.\n\nThe description 'commercial arm' of FIDE was, however, somewhat misleading, a point Bessel Kok himself referred to in an interview since it was a private undertaking which was not legally linked with FIDE. Finally all that was realised was the idea of a Grand Prix series of tournaments, essentially no different from the series of the same name which Kok had already instituted in the 1980s as the president of the GMA.\n\nWhen FIDE finally gave up the k.-o. format for the World Championships after the 2004 WCh which had been held in scandalous circumstances in Tripoli, this 128 player tournament was then continued as the 'World Cup' and at first boosted in value because the winner of the World Cup was to become at the same time the challenger of the World Champion. FIDE, moreover, found a financially strong organiser and sponsor in the West Siberian city of Khanty-Mansiysk, which is situated among the Siberian oilfields. As well as chess tournaments they also regularly stage international biathlon competitions.\n\nAfter the founding of Global Chess and the setting up of the Grand Prix series FIDE changed its plans and in June 2007 published a new plan for the WCh cycle for the next five years, in which the new Grand Prix series was included. Now the winner of the World Cup was to play a candidates' final against the victor of the Grand Prix series. The winner of that would be the challenger of the World Champion.\n\nAt their 79th congress in Dresden 2008 FIDE threw out their five year plan of the previous year in view of organisational difficulties with the Grand Prix series. Some tournament venues for the Grand Prix series such as Karlovy Vary, Montreux or Doha were obviously announced much too soon by Global Chess and then withdrew. Magnus Carlsen, one of the favourites to win the Grand Prix series, withdrew from the WCh cycle after the supposed venues dropped out. At the congress in Dresden FIDE now decreed that the challenger of the World Champion should be decided in a candidates' tournament. The participants were to be the finalists of the World Cup of 2009, the losing player from the candidates' final ('World Chess Challenge'), the losing player from the WCh match of 2009, the two best players from the Grand Prix series, the best player on the Elo list not to be qualified via a different route and one player (over 2700 Elo) to be nominated by the organisers.\n\nAt the congress there was heavy criticism of the work of Global Chess, which in its negotiations with sponsors and organisers had relied on verbal promises instead of signing contracts. The criticism within FIDE did not become any less when it became known that for his efforts Bessel Kok was supposed to have received 30 000 euros per month and his managing director Geoffrey Borg another 15 000 euros per month.\n\nBoris Gelfand was born on the 24th June 1968 to a Jewish family in Minsk (Belarus). At the age of four Boris was given as a present by his father his first chess book: _Die Reise ins Schachk\u00f6nigreich_ by Yuri Averbakh and Mikhail Beilin. From 1980 to 1983 he attended the 'Tigran Petrosian Chess School', getting to know the former World Champion personally and being taught by him. In 1988 Gelfand took the silver medal at the World Junior Championship with the same number of points as the winner and in the same year won together with Alexey Dreev the European junior championship.\n\n**Boris Gelfand (born in 1968)**\n\nIn 1990 Gelfand won along with Ivanchuk the interzonal tournament in Manila, but was eliminated in the subsequent candidates' matches in the quarter-final against Nigel Short. In 1993 Gelfand again won the interzonal tournament, this time in Biel, and in 1994 played in the FIDE candidates' matches. He defeated Michael Adams in the last sixteen, then won the quarter-final against Vladimir Kramnik and was eliminated in the semi-final against Anatoly Karpov. At the candidates' tournament of 1997, organised in k.-o. mode by FIDE in Groningen, Gelfand reached the semi-final and was defeated by Anand. In 1998 Boris Gelfand emigrated from Belarus to Israel. At the k.-o. WCh in Las Vegas 1999 Gelfand was eliminated from the last sixteen by the future winner Alexander Khalifman. At the FIDE k.-o. WCh in New Delhi and Teheran Gelfand was defeated in the last sixteen by Alexei Shirov. At the k.-o. WCh in Moscow 2001 the quarter-final against Peter Svidler was the end for Gelfand.\n\nIn 2004 Gelfand refused to take part in Libya, which the Israeli players were allowed to enter after a lot of toing and froing, but where they obviously received a hostile reception. With his result in the World Cup of 2005 in Khanty-Mansiysk, where Gelfand was amongst the first ten, he qualified for the 2007 candidates' matches in Elista. From there, after victories over Kasimdzhanov and Kamsky, he reached the 2007 WCh tournament in Mexico City. Gelfand shared third place with the same number of points as Kramnik.\n\nIn 2009 Gelfand won the FIDE World Cup and with it the right to take part in the candidates' matches from 3rd to 27th May 2011 in Kazan. These were originally scheduled to take place in Baku, but after one of the favourites, the Armenian Levon Aronian, declared that he did not wish to play in Azerbaijan \u2013 a state of war still existed de jure between Armenia and Azerbaijan \u2013 FIDE's competition was moved to Russia. Since the Azeri Chess Federation was nevertheless still financially involved or had already paid out an advance, Shakhriyar Mamedyarov was allowed to retain his place as the organisers' choice. Magnus Carlsen, who would have qualified because of his Elo rating, did not take part as FIDE was not prepared to assure him contractually of a clear timetable for the whole cycle.\n\nThe candidates' matches turned out to be really boring for the spectators, of whom the great majority followed the games on the internet. 27 of the total of 30 'long' games were drawn. Most ties were not decided until the tiebreaks. The players said that the reason for this was the short distance of only four regular games per match. Gelfand won his matches against Shakhriyar Mamedyarov, Gata Kamsky and Alexander Grischuk, whilst the top favourites Vladimir Kramnik and Levon Aronian were both eliminated, and this surprisingly left him the next challenger to World Champion Anand.\n\nAs far as the venue for the World Championship match was concerned (according to the original schedule it should have been played in 2011, but with delays at the various levels this had gradually stretched into 2012), FIDE had in advance been dealing with various bidders. Thus, Universal Event Promotion and Josef Resch, organisers of the successful World Championship match of 2008, would have liked to organise the candidates' matches of 2010 and the subsequent 2011 WCh match. In a statement in February 2009 the UEP gave notice of their bid and offered a prize fund of 430000 euros net (650000 euros gross) for the players in a candidates' tourna-ments and 1.5 million euros for the WCh match. Together with all other costs in staging the events the UEP offer involved up to 4 million euros.\n\nIn May 2009, however, it was announced that the negotiations between FIDE and UEP had broken down. According to UEP's account, FIDE had made demands which went over and above the bid. In addition it had not been possible to agree who would hold the marketing rights. FIDE said it had other bidders for the WCh \u2013 though at this point it had none.\n\nBut in July 2010 another bidder announced an interest in staging the WCh match \u2013 organiser Malcolm Pein and money man Peter Davies. The two Englishmen had got to know each other when Davies, who was interested in chess, received chess lessons as a birthday present from his brother. Malcolm Pein became his chess teacher. Pein's family originated in the Baltic area. His grandfather had once fought for the revolution at the side of Leo Trotzky in the Ukraine, but had then emigrated to South Africa. Pein emigrated back to Europe, to England. As a player he earned the title of International Master. At the Chess Olympiad in Manila 1992 he began with a chess stand his business success as a chess dealer and publisher. Pein founded with a partner the London chess shop 'Chess & Bridge' and over the years bought up various chess publishers in England and the USA.\n\nThe sponsor Peter Davies is the joint proprietor of the British investment company Lansdowne Partners Limited, which administers shares to a value of 13.3 billion dollars and is the second biggest investor in the Lloyds Banking Group \u2013 after the British government. Davies and Pein together visited the 2008 World Championship in Bonn and Davies was enthused. He wanted to bring a World Chess Championship to London. But Pein advised him to start one level below that. That brought in 2009 the start of the successful tournament the London Chess Classic, which was staged for the sixth time in 2014.\n\nIn July 2010 Pein and Davies made to FIDE under the name of 'Chess Promotion London' an offer to stage the next World Championship match with a prize fund of two million euros for the players and 400 000 Euro for FIDE. It was planned for May 2012 in order to bring the World Championship close in time to the summer Olympic Games of 2012 in London, which would have attracted extra attention to it. Pein required from FIDE an acceptance by January 2011, in order to have enough time to prepare the organisation.\n\nDespite the good offer from London, the negotiations between FIDE and Pein ran very haltingly. There was little enthusiasm in London above all for the demands of FIDE with respect to their representatives and the extra costs linked to this for accommodation, travel, etc. The UEP had already complained about the amplitude of demands of this sort. No agreement could be reached before the appointed date. London withdrew its bid. FIDE gave as a reason for London's cancellation that the organisers there were disappointed that Magnus Carlsen had at the beginning of November withdrawn from the WCh cycle and that the Norwegian would be missed as a spectator magnet.\n\nIn July, Indian media announced that the WCh match between Anand and Gelfand would take place in Chennai. The 'All Indian Chess Federation', supported by the provincial government of Tamil Nadu, had put in a bid of 2.35 million dollars for a prize fund out of a total budget of four million dollars. In the middle of July, Kirsan Ilyumzhinov visited India for talks about the details with the prime minister of Tamil Nadu Jayalalithaa Jayaram.\n\nOn the 9th August, however, FIDE issued a press statement in which it surprisingly named Moscow as the venue of the WCh match between Anand and Gelfand and the Russian Chess Federation as its organisers. The precise venue was first announced as being the 'Skolkovo Innovation Centre'. The offer for the prize fund consisted of 2.55 million dollars. Behind this development there was the patron Andrei Filatov, a businessman and millionaire from the Ukraine who was a personal friend of Boris Gelfand.\n\nFilatov had also played chess in his youth and had even worked the demo board during the WCh match between Kasparov and Karpov in 1985 in Moscow. In 1990 he studied at the sports institute in Minsk \u2013 together with Gelfand and Ilya Smirin. In the third term, as the USSR fell apart, he broke off his studies and went into business. He went on to become the proprietor of N-Trans, the biggest transport firm in Russia. Having been booted out, the Indian Chess Federation received as compensation precedence for the staging of the World Chess Championship of 2013.\n\nOn the 26th November 2011 the players signed the contract. The match venue finally became the Moscow Tretyakov Gallery. The match was for twelve games and scheduled for the 11th to the 30th May. In the event of a draw after twelve games there was to be a playoff, like in previous WCh matches, thus four rapid chess games, then blitz games if things were still level. It would finish with a 'sudden death' game. The prize fund of 2.55 million dollars would be divided 60% to 40% (1.53 million to 1.02 million dollars) in favour of the winner, insofar as there was one after the 12 regular games. Otherwise the ratio would be 55% to 45% (roughly 1.4 million to 1.15 million dollars).\n\nIn this WCh match Anand also worked with his regular seconds Peter Heine Nielsen, Rustam Kasimdzhanov, Surya Shekhar Ganguly and Radoslaw Wojtaszek. Before the match he had cloistered himself with his team from January till April 2012 in his European base in Bad Soden, and in addition played some training matches against selected opponents.\n\nGelfand's seconds were his trainer of many years Alexander Khuzman, and in addition Maxim Rodshtein and Pavel Eljanov. Moreover he received 'machine' help from Amir Ban and Shay Bushinsky and their chess engine _Junior_. After the match Gelfand revealed that he also worked with Michael Roiz and Evgeny Tomashevsky. At a certain point Garry Kasparov had also offered his support. Gelfand had declined this. He felt it inappropriate to accept help from someone who had previously had insight into his opponent's secrets. Kasparov had helped Anand two years previously in the match between Anand and Topalov.\n\nGelfand withdrew with his seconds to the Swiss Alps for a month and a half and also worked on his physical fitness there. The decision as to which openings he would employ in the WCh match had already been taken by the Israeli a year before the match.\n\nTo accompany the WCh match the organisers had prepared a very attractive programme, with numerous guests of honour including Mikhail Gorbachov, Garry Kasparov and Vladimir Kramnik. The game scores were, as had become usual everywhere, broadcast live on the internet, and in addition there was a video stream with images of the games, also live commentary, either in Russian or in English, and press conferences with the guests of honour and the players after the game.\n\nAnand lived close by with his team in the Baltschug Kempinski Hotel, approximately five minutes on foot from the Tretyakov Gallery. The hotel had been chosen by Aruna Anand and Hans-Walter Schmitt back in September 2011 during the Tal Memorial. After lunch a few days before the start of the match Anand and his seconds ran into a confrontation between demonstrators and police against the re-election of Putin as Russian president and were escorted back into their hotel by members of the special security unit OMON. Although Anand's hotel lay close to the Tretyakov Gallery, the World Champion was fetched every day before the game by a car. Boris Gelfand went on foot to the gallery along with his seconds.\n\nThe chief arbiter for the match was the Armenian Ashot Vardapetian. The Canadian Hal Bond was the second arbiter. Unlike for the match between Anand and Topalov two years previously, there were no special security measures to check the players for technological aids. There was also no screen between the players and spectators. Anand and Gelfand, who had known each other well for years and who were friends, obviously did not mistrust each other.\n\nThe match turned out to be one between two players who were on a par, though before it Anand was considered the favourite. The World Champion began the first game with the white pieces. After six games the order of colours was changed, that is to say that Gelfand had white in the sixth and seventh games. To general surprise, as his reply to 1.d4 Gelfand made use of the Gr\u00fcnfeld Indian Defence, which he had never before employed in his career. Against 1.e4 too, Gelfand had prepared a completely new defence \u2013 the Sicilian Sveshnikov Variation. Anand chose as his main weapon with the black pieces against Gelfand's 1.d4 the Chebanenko Variation in the Slav Defence.\n\nThe title defender was close to victory in the first game against Gelfand's surprise weapon, the Gr\u00fcnfeld, but gave away his advantage and could only draw. In the third game the title defender tried a sharp Anti-Gr\u00fcnfeld Variation, in which Gelfand, however, always kept things on an even keel. Later Anand tested Gelfand's preparation after 1.e4, but could get nothing out of the fifth game against the Sveshnikov Variation, and just as little with his Rossolimo Variation in the tenth and twelfth games.\n\nGelfand always opened with 1.d4. After three tries without success Gelfand finally managed the full point against Anand's Chebanenko Variation in the seventh game, and took the lead. The subsequent eighth game then followed a strange course. Anand again played his Anti-Gr\u00fcnfeld opening. This time Gelfand did not want to repeat his variation from the third game, but changed to the King's Indian Defence. Anand then chose a very rare continuation. Instead of aiming to follow up his win in the previous game by aiming for a draw with the black pieces, the challenger chose some very ambitious moves and fell on move 14 into a trap which cost him the queen and the game. Gelfand had to resign on move 17 and in doing so created the shortest decisive WCh game of all time. Anand had equalised.\n\nIn the ninth game Anand was once more in a critical position, but managed to reach a draw. In the twelfth game there appeared on the board a deeply analysed position, which Anand and his team had previously evaluated as very favourable, but Gelfand solved the problems at the board with the ingenious move 10...c4.\n\nAfter 12 games the score was 6:6. The tiebreaks would have to decide. These took a dramatic course, especially for the challenger. After a draw in the first rapid chess game, Gelfand lost the second game. In the third game Gelfand had a theoretically winning rook ending on the board but he botched it and the game was drawn. The fourth game ended in a draw. Anand was the winner and had once more successfully defended his title.\n\n **Anand \u2013 Gelfand**\n\nMoscow, 8th game  \n21st May 2012  \nKing's Indian Defence (E60)\n\n**1.d4 \u2658f6 2.c4 g6 3.f3 c5**\n\nThis forces a transposition to a variation of the King's Indian Defence with a Benoni structure. In the 3rd game Gelfand chose with 3...d5 the route to the Gr\u00fcnfeld Indian Defence. After 3...\u2657g7 4.e4 d6 5.\u2658c3 0-0 the S\u00e4misch Variation of the King's Indian Defence would have arisen. The King's Indian was Gelfand's main weapon against 1.d4 at the start of his career.\n\n**4.d5 d6 5.e4 \u2657g7 6.\u2658e2**\n\nHere Anand had different marching orders for his knights compared to the usual ones. Normally in this variation they are placed on c3 and e2. Anand here develops the king's knight via e2 to c3 and delays the development of the queen's knight. The usual moves are 6.\u2658c3 0-0 7.\u2657g5 or 7.\u2657e3.\n\n**6...0-0 7. \u2658ec3!? \u2658h5!?**\n\nPlayed by Gelfand after a long period of thought. A usual setup for Black would have been for example 7...e6 8.\u2657e3 exd5 9.cxd5 a6 10.a4 \u2656e8.\n\n**8. \u2657g5**\n\n8.g4!? would be principled and double-edged.\n\n**8... \u2657f6?!**\n\nA surprising decision. Black offers to exchange his 'bad' bishop, which is really not so bad here. Anand suggested 8...h6: 9.\u2657e3 (9.\u2657h4 \u2658d7) 9...f5!? 10.exf5 gxf5 11.\u2655d2 f4 12.\u2657f2 \u2658d7 with counterplay.\n\n**9. \u2657xf6**\n\n9.\u2657e3!? (Leko) would have been a clever move, after which both black minor pieces on the kingside are misplaced.\n\n**9...exf6**\n\n9...\u2658xf6!? was absolutely playable: 10.\u2655d2 e5. (Stohl)\n\n**10. \u2655d2**\n\nOnce again 10.g4 was committal: 10... \u2658f4 11.\u2655d2 g5 12.h4 \u2658d7 13.\u2655h2 h5 14.gxh5 (Anand did not like 14.hxg5 fxg5 15.gxh5 \u2658e5 16.\u2658d2 \u2655f6) 14...\u2658e5 15.\u2658d2 \u2654h8 with an unclear situation or 15...b5!? 16.cxb5 a6 with compensation for the material.\n\n**10...f5 11.exf5**\n\n**11... \u2657xf5**\n\n11...\u2656e8+ 12.\u2654d1 \u2657xf5 13.g4 is a transposition to the continuation in the game.\n\nIt was worth considering 11...\u2655h4+!? 12.\u2654d1 (12.\u2655f2 \u2655e7+ 13.\u2654d1 \u2657xf5 14.g4 \u2657xb1 15.\u2656xb1 \u2658f4 achieves nothing for White) 12...\u2657xf5 (12...\u2658g3? is bad: 13.\u2655f2 \u2658xf5 14.\u2655xh4 \u2658xh4 15.\u2658b5 and White wins a pawn) 13.g4 \u2657xb1 14.\u2656xb1 \u2658g7 (14...\u2658g3? fails to 15.\u2655e1+\u2013) 15.\u2654c2 \u2658d7 16.\u2655e1!? (Shipov) 16...\u2655xe1 17.\u2656xe1 with a slightly better endgame for White.\n\n**12.g4!**\n\n12.\u2658a3 \u2655h4+!? (12...\u2656e8+ 13.\u2654f2 \u2658f6 is also not bad for Black) 13.\u2654d1 \u2658g7 and Black has no problems at all.\n\n**12... \u2656e8+?!**\n\nThis brings the rook into the game with tempo, but also forces the white king on to a better square. 12...\u2657xb1 13.\u2656xb1 \u2658g7 (13...\u2658f6 14.h4 with an attack on the kingside for White) 14.h4 h5 15.\u2654d1 \u2658d7 16.\u2654c2 \u2658e5 17.\u2657e2 was given by Anand as slightly better for White. Once again 12...\u2655h4+!? was possible: 13.\u2654d1 (not 13.\u2655f2? \u2656e8+ 14.\u2657e2 \u2655g5! intending 15.gxf5? \u2655c1+ 16.\u2658d1 \u2658f4\u2013+) 13...\u2657xb1, transposing into the variation beginning with 11...\u2655h4.\n\n**13. \u2654d1**\n\nAfter 13.\u2657e2?! Black would get good counterplay: 13...\u2655h4+ 14.\u2654d1 \u2657xb1 15.\u2656xb1 \u2658f6 (15...\u2658g7!?) 16.\u2658b5 \u2658a6 17.\u2658xd6 \u2656e7 18.a3 \u2656d8 19.\u2658b5 h5! 20.gxh5 (20.g5? \u2658e4! 21.fxe4 \u2655xe4 with advantage to Black) 20...\u2658xh5 with compensation. (Stohl)\n\n**13... \u2657xb1 14.\u2656xb1 \u2655f6?**\n\nSo as to bring the knight to f4 via tactics. However, the tactics do not work. After 14...\u2658f6 15.\u2654c2 (or 15.h4) or 14...\u2658g7 15.h4 in both cases it is White who has the slightly better prospects.\n\n**15.gxh5! \u2655xf3+ 16.\u2654c2 \u2655xh1**\n\nThat was the tactical justification, however...\n\n**17. \u2655f2!**\n\nTrapping the queen, since the black queen has no square to which to retreat. Black resigned. One attempt was 17... \u2658c6 (17...\u2658a6 18.a3!?+\u2013, there is also a win after 18.\u2657d3 \u2658b4+ 19.\u2654d2 \u2658xd3 20.\u2654xd3+\u2013 or 18.\u2657h3+\u2013) 18.dxc6 \u2655xc6, but after 19.\u2657g2 (or 19.\u2657d3 \u2656e5 20.\u2656f1) 19...\u2655c8 (19... \u2655d7 20.\u2658d5+\u2013) 20.\u2656f1+\u2013 White's position is overwhelming. Another good move is 20.hxg6 hxg6 21.\u2656f1+\u2013.\n\nThe fact that the match ended in a draw after the regulation twelve games meant that the prize fund was divided in a ratio of 55% to 45% (instead of 60% to 40%), which meant a difference of 127 500 dollars in Gelfand's favour.\n\nAt the victory ceremony on the 31st May 2012 Anand received as well as the victor's cheque and the World Champion cup from FIDE a 'Tretyakov Cup' donated by Alexander Molshanovsky, as well as the laurel wreath and winner's medal. At the suggestion of the sponsor Alexander Filatov, the painter Yuri Krotov made two paintings of the World Championship match, one of which went to the Tretyakov Gallery and the other to Anand. Even before the victory ceremony Anand and Gelfand had been received by the Russian president Vladimir Putin. On the evening the Indian ambassador in Moscow, Ajai Malhotra, invited the whole Anand delegation to a reception followed by a dinner in the Indian embassy.\n\nThe tournament hall in the Tretyakov Gallery was sold out every day with 400 spectators \u2013 entrance was in any case free. On some days the capacity of the hall was not sufficient for all the chess lovers who wished to see the WCh match. The press centre was just as well filled. Some 400 journalists had taken out accreditation. On several Russian TV channels there were daily reports about the World Chess Championship. On the internet portal for the match a total of more than two million spectators was counted. On individual match days 200000 spectators followed the live broadcast of the games at the same time.\n**45. The high-flyer from Norway**\n\n**The World Championship 2013:  \n_Viswanathan Anand against Magnus Carlsen_**\n\nMagnus Carlsen was born on the 30th November 1990 to Sigrun \u00d6en and Henrik Albert Carlsen, both engineers by profession, in T\u00f6nsberg, Norway. Sigrun and Henrik Carlsen also have three daughters: Ellen, Ingrid and Signe, of whom Ellen is older than Magnus and the other two younger. They lived at times in Espoo and Brussels. In 1998 the Carlsens moved back to Norway and then lived in Lommedalen, a sub-urb of Oslo. Later the family moved to Haslum, another such suburb.\n\n**Magnus Carlsen (born in 1990)**\n\nIn his youth Henrik Carlsen had himself been an active chess player, but he somewhat lost his motivation after joining the Oslo chess club and there running into several very strong players, including Espen and Simen Agdestein. Henrik Carlsen then took more interest in advancement in his profession but remained interested in chess throughout his life and taught his children the game. Magnus, who as a young child already demonstrated a rapid grasp of things and a very good memory, learned to play chess at the age of five, but at the start showed no particular interest in the game.\n\nBut when he was eight he developed the ambition to beat his elder sister Ellen at chess and began to take a closer interest in the game. His first chess books were Bent Larsen's _Finne Planen_ (Find a plan) and Eduard Gufeld's opening book _The complete Dragon_. Carlsen would later read with great enthusiasm for example Garry Kasparov's series _My Great Predecessors_. He gained his endgame knowledge from reading _Fundamental Chess Endings_ by Karsten M\u00fcller and Frank Lamprecht.\n\nAt the age of eight years and seven months Carlsen took part in 1999 in the youngest age grouping of the Norwegian championship and scored 6\u00bd points from eleven rounds. At the tournament in Gausdal Carlsen met for the first time Simen Agdestein, who was a chess teacher at the Norwegian Sport Elite School NTG (Norges Toppidrettsgymnas). Carlsen was then enrolled in it. Agdestein at first organised as a trainer Torbj\u00f6rn Ringdal Hansen, who was doing civilian (instead of military) service at the sports school. Even Agdestein was amazed by Carlsen's phenomenal feats for memory. If Magnus Carlsen was given an opening book to study, then a day later he had already studied all the variations and could play the opening almost at master level.\n\nThe computer too then became an important training partner. Carlsen trained with the ChessBase database, with chess engines like _Fritz_ and _Rybka_ , and on the internet played countless blitz games against chess partners all over the world on the chess server.\n\nCarlsen also learned from Agdestein, who in addition to his career in chess had also played professional football with 'Lyn Oslo' and been selected eight times for the Norwegian national team, the link between the physical sports and good performances in chess. Since in any case Carlsen was enthusiastic about sport, it was not difficult for him to take on board these lessons. Carlsen played in the football club and also played basketball, he is a good skier and has even practised ski jumping. Moreover he takes part in various other sports.\n\nMagnus Carlsen now regularly took part in tournaments, in a tournament against Norway's best juniors in September 2000 he scored 3\u00bd points out of five games, thus keeping up as a ten year old with some of the clearly older and most talented young players in the country, and between 2000 and 2002 he played in no less than 300 tournaments. In 2003 he scored the three IM norms and became an International Master. After the end of primary school Henrik Carlsen temporarily took his son out of school, bought a camper van and from the autumn of 2003 travelled throughout Europe for ten months with the family to enable Magnus to play in as many tournaments as possible. In the same year he took third place in the U13 European championship.\n\nIn January 2004 Carlsen was invited to Wijk aan Zee, where he won the C-group at his first attempt. At a combined blitz and rapid chess tournament in Reykjavik in March 2004 he defeated Anatoly Karpov and brought Garry Kasparov to the brink of defeat in rapid chess, but then lost as a result of his lack of experience. The 13 year old said about his defeat at the time 'I played like a child!' Kasparov was very impressed by the talent of the Norwegian, especially because this had not developed in one of the 'chess factories' in Russia or the Ukraine, but 'in the middle of nowhere'. When in April 2004 at the Dubai Open he made his third GM norm, Carlsen became the youngest grandmaster at that time and the third youngest in the history of chess, after Sergey Karjakin and Parimarjan Negi. Carlsen was then the youngest player to take part in the FIDE k.-o. World Championship in 2004 in Tripoli, but was eliminated in the first round by Levon Aronian, though only in the tie-breaks.\n\nIn the same year Simen Agdestein published his book about the new super talent, entitled _Wonderboy_. In addition he sought for his prot\u00e9g\u00e9 another strong player as a trainer. At first there were contacts with Kasparov, but then at the start of 2005 the Danish grandmaster Peter Heine Nielsen took on the task of further polishing the rough diamond of the chess scene. At this point Nielsen was also active as a second and a trainer for Anand and later arranged for Carlsen to be the future World Champion's sparring partner for various WCh preparations.\n\nIn 2005 Carlsen shared first place in the Norwegian national championship, but Simen Agdestein was still able to defeat him in the tiebreak. His tenth place in the FIDE World Cup already qualified Carlsen for the candidates' matches for the World Championship. In January 2006 he shared first place in the B-tournament in Wijk aan Zee with Alexander Motylev. In the same year he became for the first time Norwegian champion, after beating Simen Agdestein in the tie-breaks. In 2007 he finished second in the top tournament in Linares. At the candidates' matches he was eliminated in the first round against Levon Aronian, though not until an exciting playoff.\n\nAt the Arctic Chess Open in Troms\u00f8, which at that time was no longer a tournament which was appropriate for the young shooting star, Carlsen curiously was drawn against his father Henrik, who was also playing. He was able to defeat his father but he was only able to manage a draw against his former classmate from the sports school Brede Hagen, who had a clearly inferior rating, and at one point he was even losing. At the World Cup in Khanty-Mansiysk 2007 Carlsen reached the semi-final and was beaten by Gata Kamsky. He finished the Corus A-tournament in Wijk aan Zee 2008 on second place according to tie-break scores, behind Levon Aronian though on the same number of points. Shortly after that he also came second in the top tournament of Morelia\/Linares, this time half a point behind Anand. Now at the latest it had become clear that the 17 year old Norwegian had taken his place in the world elite.\n\nIn June 2008 Carlsen won his first major tournament, the Foros Aerosvit tournament in the Crimea. In the Grand Prix series of 2008-2009 Magnus Carlsen was one of 21 players taking part, but after three tournaments he withdrew when at short notice FIDE changed the venues and dates for the tournaments.\n\nCarlsen's successes in 2009 include second place in the M-Tel Masters in Sofia and a majestic first place with eight points out of ten games at the Pearl Spring tournament in Nanjing. His Elo performance of 3002 was one of the 20 best tournament performances in the history of chess and the best performance by any young player. Immediately before the tournament, Carlsen had made it known that since the beginning of the year he had been working with Garry Kasparov as a trainer. In November 2009 in Moscow Carlsen won the World Blitz Championship. After his tournament victory at the end of 2009 in the London Chess Classic, the Norwegian went on 1st January 2010 for the first time to the top of the world ranking list with an Elo rating of 2810 and was the youngest player to have been first in the world ranking list in the history of chess.\n\nHis high Elo rating qualified Carlsen for the candidates' matches, but he soon explained he would not be taking part and gave as a reason for this the poor organisation of the WCh cycle, the fact that it lasted too long and the privileges which the World Champion enjoyed. The current World Championship cycle of 2008 to 2012 was not 'sufficiently modern and fair', was Carlsen's reason for not taking part. It was suspected that there was advice in the background from Kasparov, who, together with Karpov, whom he was supporting as a candidate for the forthcoming presidential elections, found himself in opposition to FIDE president Kirsan Ilyumzhinov, but Carlsen and also Kasparov denied this. The cooperation with Kasparov had been ended by Carlsen in the spring of 2010.\n\nDuring 2010 Carlsen won the Corus tournament, the Kings tournament in Bazna, the second Pearl Spring tournament in Nanjing and, thanks to the three-point rule in use there, also the London Chess Classic. After the Corus tournament in Wijk aan Zee Magnus Carlsen was hired by the Dutch clothing firm G-Star as the model for its advertising campaign. Carlsen's partner was the actress Liv Tyler. In the years which followed, Gemma Arterton and Lily Cole were placed beside him. In September 2010 G-Star organised a fashion party high above New York in the penthouse of the Cooper Square Hotel, but this time chess was the focal point. Magnus Carlsen played a game on the internet against 'The World'. Hikaru Nakamura, Judit Polgar and Maxime Vachier-Lagrave were invited to give advice over the internet. Present as a guest of honour was Garry Kasparov. In the meantime Espen Agdestein had taken over from Henrik Carlsen the duties as manager of Magnus Carlsen.\n\nIn 2011 Carlsen was successful with victories in the Kings tournament in Bazna, the Chess Festival in Biel and the Masters Final in Bilbao after a tie-break against Vassily Ivanchuk. At the Tal Memorial in Moscow he came in second on tiebreak score behind Levon Aronian. After having for a time ceded the lead in the world ranking list, Carlsen won back first place in the FIDE Elo list in July 2011 and pushed Anand into second place. After that he increased his Elo rating from 2821 (July 2011) to the new record of 2882 (May 2014). In January 2013 Carlsen had with 2862 Elo surpassed for the first time Kasparov's old best rating of 2851 from July 1999. When comparing with Kasparov, however, it needs to be borne in mind that since then the general Elo level has risen clearly: in 1999 the Elo average of the top 100 was about 2644. In May 2014 it was 2703 Elo. Thus the Elo level of the best players had increased by 60 points over these years. The distance between Kasparov and the average Elo of the top 100 was therefore clearly greater in 1999.\n\nIn the meantime Carlsen had become known world-wide: _Time magazine_ chose him in April 2013 as one of the 100 most important people in the world.\n\nCarlsen's rise in Elo rating came as the result of numerous tournament successes, which he scored as number one in the world ranking list. They include tournament wins in Biel 2012, the Masters Final 2012 in Bilbao after a playoff against Caruana, the London Chess Classic 2012 and the Tata Steel tournament in Wijk aan Zee with 10 out of 13 and 1\u00bd points of a lead over Levon Aronian.\n\nAfter numerous delays in the staging of the WCh cycle, as a result of which the WCh match planned for 2011 had actually been postponed until 2012, FIDE, or, more accurately, their new investor Andrew Paulson, stepped up the pace with regard to the WCh cycle of 2012\/13. The US American Paulson began his career as a glamour photographer, became an entrepreneur and was then successful in Russia with the publishing house 'Afischa' and the social network 'SUV'. In September 2011 he met quite by chance Kirsan Ilyumzhinov. He discussed with him and other members of the FIDE executive how chess could be marketed professionally. All his ideas and suggestions as to what could be changed met with rejection by the FIDE officials. It was said that all that had already been tried and it had not worked.\n\nFinally Paulson came to the conclusion that he would have to take things in hand himself and made FIDE an offer to buy the marketing rights for the whole WCh cycle. On the suggestion of FIDE he set up a new firm, called Agon, the sole business of which was to be the marketing of the World Championship cycle, and he guaranteed FIDE a fixed sum as well as a share of the hoped for revenue to be brought in above all by advertising. After marathon negotiations of six months an agreement was finally reached. In February 2012 the transaction was made public, with some previous opposition to this business having taken place within FIDE. Especially Ilya Levitov, marketing director of the Russian federation, spoke out against it. Silvio Danailov cast doubt on the legality of the decision to hold the candidates' tournament in London and announced a lawsuit.\n\nAs a down payment for the deal, Paulson gave FIDE 500 000 dollars cash. As Paulson understood things, the whole WCh cycle with the World Cup, Grand Prix tournaments, candidates' tournament and WCh match would be more reliably managed as to its course and scheduling. In addition the big tournaments should take place in the major metropolises of the world and not in provincial cities in Eastern Europe. If enough advertising partners were found, the amount of the total prize fund would gradually be increased. For 2014 a sum of over five million dollars total prize fund for the whole cycle was announced.\n\nWith all the finesse of a bull in a china shop FIDE and Agon scheduled the London candidates' tournament for the end of October 2012 and thus immediately caused difficulties for two top tournaments, the London Chess Classic, which was usually held in November\/December, and the Masters in Bilbao, which was also staged in autumn. The Moscow Tal Memorial, also usually held in the final quarter of the year, had been safely moved to June in good time by the Russian federation. After protests from the organisers of these tournaments and also some players, FIDE took pity and announced new dates for the candidates' tournament: the 13th till the 30th March. The dates for the WCh match were fixed at the start of 2013 as the 6th till 26th November 2013.\n\nThe candidates' tournament in London was spectacular and the excitement during it justified the decision to once more stage these matches in the form of a tournament. Of the eight participants it was Levon Aronian who had the best start. But in the middle of the tournaments he had first been caught by Magnus Carlsen and then overtaken. The best final spurt, however, was that of Vladimir Kramnik. In the third last round Carlsen, who with the highest Elo rating of all time had started as clear favourite, was surprisingly beaten by Vassily Ivanchuk and now had to watch Kramnik, who in the return round of games had enjoyed victories over his fellow countrymen Peter Svidler and Alexander Grischuk as well as over Teimour Radjabov, pass him by. In the last round but one Carlsen urgently needed a win against Radjabov, which he finally managed to get when Radjabov lost the thread of the game in a drawish ending \u2013 at the same time Kramnik was unable to defeat Boris Gelfand.\n\nThus, before the final round Kramnik and Carlsen were level. The tiebreak system meant that with two players on an equal number of points, their face-to-face meeting would decide \u2013 Carlsen and Kramnik had had two draws \u2013 then the greater number of wins, finally their scores according to the Sonneborn-Berger system, in which a player receives the full total points of any player he has defeated and added to this half the points of any player with whom he has drawn. Carlsen and Kramnik both lost their final round games, against Svidler and Ivanchuk respectively, and so Carlsen became the winner of the tournament on account of having one more win. The tiebreak rules used were heavily criticised later. From a sporting point of view, a proper playoff with rapid games would have been fairer for such an important decision. Kasparov pointedly stated in an interview that the player 'who had lost one more game' became the winner of the tournament.\n\nAt the start of April 2013, Indian media made known that Chennai had been awarded the staging of the WCh match between Anand and Carlsen. FIDE thus fulfilled the pledge it had given in 2011 to the government of the Indian state of Tamil Nadu. Back in 2011 Chennai had after all bid to stage the WCh match between Anand and Gelfand and already been accepted by FIDE, when at the final moment it had been outmanoeuvred by an only slightly better offer from Moscow. FIDE had promised the Indian organisers that the next WCh they would take precedence in the allocation of the match.\n\nOn the 19th April Israel Gelfer, representing FIDE, met representatives of the All-Indian Chess Federation and the chess federation of Tamil Nadu and signed with them a 'memorandum of understanding', a sort of pre-contract. The Indian organisers were prepared to put up 2550000 dollars of a prize fund for the players, and in addition 510000 dollars contribution to FIDE (20% of the prize fund) and a further 255000 dollars for the acquisition of the commercial rights and as compensation for the work of FIDE relating to the WCh. For the appeals committee and the arbiters further costs of 42 250 dollars were available. Thus the chairman of the appeals committeee alone received 7000 dollars remuneration for his work.\n\nTwo days later during the Alekhine Memorial, on the 21st April, Magnus Carlsen's father Henrik gave an interview during the live commentary and expressed his dissatisfaction with FIDE's decision to award the WCh match between his son and Anand to Chennai. Henrik Carlsen argued that this decision was not in accordance with FIDE statutes since there had been no previous formal invitation to tender. The interest in the match was very great world-wide, he argued, and an open bidding process would certainly have obtained a higher prize fund than was being made available by the organisers in Chennai. Carlsen had also explained his point of view to FIDE directly by e-mail and in a face-to-face meeting in Athens with Georgios Makropoulos.\n\nWhen FIDE did not react, the president of the Norwegian federation J\u00f6ran Aulin-Jansson directed an official complaint in an open letter to FIDE on the 3rd May 2013 and demanded that FIDE open a bidding process for the WCh match according to its own statutes. On the same day Philippe Mouttou, as the speaker for a group of businessmen, put in an official offer to stage the WCh match. The prize fund on offer was with 2.65 million euros (plus 800 000 euros to FIDE) clearly superior to that from the organisers in Chennai. The offer was supported by the French federation and the city of Paris, the mayor of which, Bertrand Delano\u00eb, gave notice of this in a letter to the FIDE president.\n\nFIDE, however, stuck to its decision. Carlsen reacted to this by describing himself as 'deeply disappointed', but was obviously also prepared to play in India. FIDE statutes lay down that if a player takes part in a WCh match in the home country of his opponent, he receives an additional 100000 euros. So for the challenger there was in any case some financial compensation for his readiness to play in India. On the 5th May FIDE president Kirsan Ilyumzhinov and FIDE general secretary Ignatius Leong met the secretary of the All-Indian Chess Federation Bharat Singh Chauhan in India to sign the contract.\n\nWhen it became clear who his challenger would be, Anand set up his training camp in Bad Soden, as for the previous matches. Out of his former team, however, Rustam Kasimdzhanov and Peter Heine Nielsen had left. Kasimdzhanov had taken on other commitments, Nielsen had in the meantime been re-hired by Carlsen. Out of loyalty to Anand, however, the Danish grandmaster explained that he wanted to remain neutral for the forthcoming match. Surya Shekhar Ganguly had also left, so that only Radoslaw Wojtaszek was left from the old team. To it were added the two Indian grandmasters Krishnan Sasikiran, Sandipan Chanda, and in addition Peter Leko. At the opening press conference Anand named these players as members of his team. During the match it leaked out that Vladimir Kramnik had probably belonged to the title defender's team.\n\nWhen at the first press conference Anand announced his team, or at least a part of it, Carlsen thanked him for his openness but at the same time explained that he would not be returning the compliment and wanted to continue to keep his team members secret. There was a lot of speculation about the names of Carlsen's seconds. It was a quite safe supposition that there were Carlsen's fellow countryman Jon Ludvig Hammer, and in addition Laurent Fressinet, as well as the young Russian grandmaster Ian Nepomniachtchi and perhaps Vladimir Potkin. Even at the final press conference after the tenth game Carlsen divulged only the name of Hammer. The only thing which was certain was that Carlsen had brought a cook and a personal physician from Norway.\n\nWhilst in Bad Soden Anand was not only working on opening variations, but also seeking to improve his fitness with runs through the woods and swimming, losing ten kilos along the way, Carlsen had set up his training camp in the summer in the Norwegian holiday resort of Krager\u00f8. Carlsen enabled his fans to share via his Facebook page his activities, which also consisted of a great deal of sport. Carlsen let himself be photographed bare-chested at gymnastics, diving or on a jet-ski. He was seen playing tennis and football.\n\nThe Norwegian also put a video on the net showing Carlsen playing a blitz game against Laurent Fressinet. An attentive observer could spot lying around a tennis racquet, tennis balls, a half-eaten apple and even a chess book, though it was not a book on openings but an apparently well-thumbed copy of the endgame manual by the Hamburg masters Karsten M\u00fcller and Frank Lamprecht. Was it a bluff? Did the Norwegians want to lull Anand into a false sense of security with this video and make the World Champion believe that Carlsen was not preparing any openings at all but simply doing endgame training?\n\nShortly before the start of the match there appeared a documentary from the Norwegian multimedia newspaper _Verdens Gang_ (abbreviated to VG), which made available a further insight into Carlsen's training in Krager\u00f6. Actually the 22 year old challenger had Jon Ludvig Hammer produce endgame exercises for him and learned by heart classic games from the history of chess. Unusual preparation for a WCh match!\n\nAt the end of August Carlsen visited Chennai for the first time, so as to meet the organisers there and possibly clarify some still undecided details. There the challenger and the Norwegians got to grasp something of the enthusiasm felt for chess, for the forthcoming WCh match, but also greatly for the challenger personally. When Carlsen visited in Chennai the 'M.O.P. Vaishnav Women's College', he was welcomed there like a pop star by some 1000 pupils. During the visit Magnus Carlsen was also received by Mrs Jayaram Jayalalithaa, the prime minister of Tamil Nadu. As the main sponsor of the forthcoming match for the World Championship, the Indian federal state was responsible for the lion's share of the costs.\n\nJ. Jayalalithaa had been a very popular actress in the 1960s and 1970s and had been in over 130 films. She later followed her fellow actor Marudur Gopalamenon Ramachandran on his way into politics and became his successor in the office of 'Chief Minister' of Tamil Nadu. From Ramachandran Jayalalithaa had not only learned the trade of politics, but also the art of self-promotion, and she used the WCh match between the extraordinarily popular Indian sporting idol Viswanathan Anand and his challenger Magnus Carlsen to present herself to the public. For that reason the fairy-tale opening ceremony, with lots of Indian folklore, dances, colourful costumes \u00e0 la 'Lion king' and chess motifs was moved into the 8000 seater Jawaharlal Nehru Indoor Stadium. Chief Minister Jayalalithaa Jayaram conducted the proceedings as the guest of honour. Her likeness was also emblazoned everywhere on the official posters. One year after this World Championship match, however, J. Jayalalithaa and three co-accused were found guilty after a trial which had dragged out over 18 years of the embezzlement of 66.65 crore rupees (approx. 8 million euros) removed from office and sentenced to four years in prison and a fine of 12.6 million euros. After payment of a bail she was released from custody in October 2014.\n\nThe hosts and venue for the WCh match was the Hyatt Regency Hotel, which set up the hotel ballroom for the match. Between the spectators and the slightly raised stage the organisers installed a soundproof glass barrier, which prevented any noise from the spectators' area reaching the players. The entry price for spectators was 24 dollars for one game, 248 dollars for the whole match. The hotel was at the same time the official hotel for the event. The World Champion Anand, who was born in Chennai and lives there, even only a quarter of an hour from the venue, also moved into the hotel for the period of the match and occupied the presidential suite on the tenth floor. Magnus Carlsen, accompanied amongst others by his manager Espen Agdestein, his father Henrik and his sister Ingrid, also stayed in the hotel, as did numerous international journalists who were reporting on the match.\n\nThe organising body for the match was the All Indian Chess Federation (AICF). The main organiser was DV Sundar, honorary president of the AICF and moreover one of the vice-presidents of the world chess federation FIDE. He was supported by the secretary of the AICF, Bharat Singh, and by the general secretary of the chess federation of Tamil Nadu, V. Hariharan.\n\nThe match was to be over twelve games. The drawing of lots saw Carlsen awarded the white pieces for the first game. A change of colours was foreseen after six games. According to it, Anand not only had the white pieces in the sixth game, but also started the second half of the match with them. The time control was 120 minutes for 40 moves, then 60 minutes for the next 20 moves, and finally 15 minutes for the remainder of the game with an increment of 30 seconds per move from move 61. In the event of a drawn match after twelve games, there was to be a playoff over four rapid chess games with 25 minutes plus ten seconds increment per move. There was to be a fresh drawing of lots for colours for the playoff. If things were still level, then up to five two-game blitz matches were to be played with a time control of five minutes plus three seconds increment. Lastly, if required, a final blitz game, the 'sudden death' game. The players were not allowed to agree a draw before move 30.\n\nThe prize fund was around 1.89 million euros, of which 60%, thus approx. 1.13 million, would go to the winner. The players were obliged to appear in the tournament arena ten minutes before the start of the game and had to submit to a security check. Mobile phones and other technology were not to be brought in the tournament arena by the players.\n\nWhilst Anand went into the World Championship matches of 2008, 2010 and 2012 as favourite, this time he was only the outsider. After 2008 it had become harder and harder for the Indian to defend his title. In January 2010 Carlsen had already for the first time reached the top of the world ranking list. In November 2010 Carlsen fell back to second place and Anand was just ahead of the Norwegian.\n\nDuring 2011 Anand and Carlsen continued to change places as first in the world ranking list, but at the end of 2011 Carlsen started to extend his lead and at the time of the WCh match in Chennai his lead over the second placed player, now Levon Aronian, was 69 points. The Elo difference was as great as that between the number two and the number twenty in the world ranking list. Anand had in the meantime slipped back to eighth place with Elo 2775. So the gap between Carlsen with his 2870 Elo and Anand consisted of almost 100 Elo points. From the purely mathematical point of view, a victory by Carlsen with a lead of three points was absolutely in the realms of the possible. Added to that, in the final tournaments before the match Anand had lost some games as a result of blunders. In 2012 and 2013, moreover, Anand had had to suffer painful defeats in two games against Carlsen.\n\nWhereas the match for the World Championship in 2012 between Anand and Boris Gelfand had attracted hardly any attention outside of the chess scene, Magnus Carlsen's attack on the throne caused a powerful press storm. Naturally, the match attracted the greatest attention in Norway and India, the home countries of the two players. In advance of the match previews were published more and more frequently in the major Indian newspapers; then there were daily articles and reports on the games. In Norway too, the press got involved early on. Even the Norwegian tabloids, headed by the multimedia magazine _Verdens Gang_ (VG), provided Carlsen fans with articles, photos or video. Carlsen himself, or his management, did not stand by passively and provided photos on Carlsen's Facebook page and comments on his Twitter account.\n\nBut in other countries too the interest among the mass media was also enormous and it grew when the editorial teams noticed by the hits on their web pages just how great the interest was, and not only among chess fans. As the match unfolded, more and more people were gripped by the chess virus. At the end the interest in the chess happenings in Chennai even surpassed by far the attention which Robert Fischer had once enjoyed in his match against Spassky in 1972 in Reykjavik. In Germany all the major newspapers and weekly magazines finally reported daily on their web pages on the preparations and the games. During the games several websites had live tickers with the moves and experts explaining them to the public. After the games _Der Spiegel_ published video analyses with ChessBase technology and recorded over 50000 hits for every video.\n\nThe World Championship was the absolute top topic everywhere in the world. It was Magnus Carlsen above all who provided the material for the stories. The new 'super-brain' was only 22 years old and in addition to his commitments as a chess professional was also available as a model for the Dutch fashion label G-Star. The media portrayed the encounter as a duel between the model and the tiger. The Madras Tiger was Anand's nickname in the press.\n\nFIDE's official match site and the local organisers provided for every game live video-stream with the players and had the game commentated on in parallel by experts. The games were explained for the spectators in turn by Susan Polgar and the Indian grandmaster Ramesh or the English IM Lawrence Trent and the Indian women's grandmaster Tania Sachdev. Reports on the World Championship were followed on Indian sports television by 80 million viewers. Norwegian television broadcast all the match games live, some lasting over six hours, and in doing so got a market share of 40%. After the match it was learned in interviews, carried out by the Norwegian media in the streets, that chess sets had practically been sold out in Norway.\n\nThe first game was played on the 9th November. There was a rest day between every two games. Carlsen got nothing out of the opening in the first game and after a few moves had to content himself with a draw by a repetition. The second game also quickly ended as a draw. In the third game Carlsen was at a disadvantage and only just avoided a possible defeat. The fourth game was later described by Carlsen as the best in the match. Anand had worked out an interesting attacking idea against Carlsen's Berlin Defence. But Carlsen defended successfully and Anand then had to fight for a draw, which the title defender managed to achieve. Carlsen finally won the fifth game after a mistake by Anand in the endgame. The sixth game was also won by Carlsen in the endgame.\n\nAnand was unable to recover from this double blow. His play in the seventh and eighth games was listless. It was only in the ninth game that he again attempted to win and obtained a promising position after the opening, but he could not find a decisive attacking idea. Carlsen defended with precision. A draw was probable, but Anand had to resign this game too after an oversight. After a draw in the tenth game Carlsen was confirmed as the new World Champion on the 22nd November 2013. That made him the 16th World Champion in the tradition begun by Wilhelm Steinitz.\n\n **Anand \u2013 Carlsen**\n\nChennai, 9th game  \n21st November 2013  \nNimzo-Indian Defence (E25)\n\n**1.d4**\n\nIn the previous games in the match Anand had played exclusively 1.e4 but achieved nothing with it.\n\n**1... \u2658f6 2.c4 e6 3.\u2658c3**\n\nWith 3.g3 Anand could transpose to the Catalan Opening, with which he was successful in the WCh match of 2010 against Topalov. Another popular move is 3.\u2658f3.\n\n**3... \u2657b4 4.f3**\n\nAnand had never previously played this aggressive move, whereas Carlsen had already employed this variation himself as White.\n\n**4...d5 5.a3 \u2657xc3+ 6.bxc3 c5 7.cxd5 exd5**\n\nThe main line arises after 7...\u2658xd5.\n\n**8.e3 c4**\n\nPlayed quickly by Carlsen. The structure is reminiscent of the famous Botvinnik-Capablanca game, AVRO 1938. There Black won a pawn on the queenside, but was then outplayed on the kingside. After 8...0-0 9.\u2657d3 White has so far had good practical results.\n\n**9. \u2658e2 \u2658c6 10.g4 0-0 11.\u2657g2 \u2658a5 12.0-0 \u2658b3 13.\u2656a2**\n\nThe players are following a game by Kasparov against Judit Polgar from 1997.\n\n**13...b5**\n\nHere Judit Polgar played the less accurate 13...h6?!. After 14.\u2658g3 \u2657d7 15.\u2655e1 \u2656e8 16.e4 dxe4 17.fxe4 \u2658xg4 18.\u2657f4 Kasparov obtained a strong initiative and won the game after 32 moves, Kasparov-Polgar, Tilburg 1997.\n\n**14. \u2658g3**\n\nUp till here Anand had used 15 minutes, Carlsen already roughly 30 minutes.\n\n**14...a5**\n\nA new move by Carlsen. Previously there had been two examples with this position. In the previous games 14...\u2657b7 and 14...\u2656e8 had been played.\n\n**15.g5 \u2658e8 16.e4 \u2658xc1**\n\nBefore the bishop can make it into the open, it is eliminated. Vachier-Lagrave suggested 16...\u2658d6!? as an alternative.\n\n**17. \u2655xc1 \u2656a6**\n\nPreparing the...b5-b4 advance. After the exchange of pawns the \u2656a6 is protected by the bishop.\n\n**18.e5**\n\nA dynamic position has arisen. White has a space advantage on the kingside, Black on the queenside. For Anand, who absolutely had to win the game, the opening has at least been successful in the sense that the position which has arisen is complicated and allows a lot of possible options. During the game Kasparov suggested 18.\u2656b2!?, so as to transfer the rook to the kingside.\n\n**18... \u2658c7**\n\nThe immediate 18...b4!? was also worth considering: 19.axb4 axb4 20.\u2656xa6 \u2657xa6 21.cxb4 \u2658c7 22.f4 \u2655e7 23.\u2655a3 (23.\u2658f5 \u2655xb4) 23...g6 24.\u2656b1 \u2656a8=.\n\n**19.f4 b4**\n\nCarlsen is the first to get in a move on the queenside. In the long run a protected passed pawn will come about on c4 or b3.\n\n**20.axb4 axb4 21. \u2656xa6 \u2658xa6 22.f5!?**\n\nAnand sharpens up the game. He could also play 22.cxb4 \u2658xb4 23.f5. Unlike in the game, the black passed pawn here is on c4 and not so close to the promotion square; and d4 is weak.\n\n**22...b3**\n\nAnd Carlsen also chooses the most principled continuation. Here Anand still had more than an hour on his clock, Carlsen roughly half an hour. But Anand then invested more than half of his time thinking out a plan of attack. The white position may look very promising, but the title defender was not able to find a forced win.\n\n**23. \u2655f4**\n\nAlternatives were:\n\nA) 23.f6 g6, and now:\n\na) 24.h4 \u2658c7 25.h5 \u2657g4 26.hxg6 hxg6 27.\u2655f4 \u2655d7 and White cannot make any further progress;\n\nb) 24.\u2655f4 (so as to deliver mate with \u2655h4-h6-g7) 24...\u2654h8 25.\u2655h4 b2 26.\u2655h6 \u2656g8 27.\u2656f4 b1\u2655\\+ 28.\u2657f1 \u2655d1 29.\u2656h4 \u2655h5 30.\u2658xh5 gxh5 31.\u2655xh5 \u2657f5 32.\u2655xf7 \u2658c7 with an unclear situation;\n\nc) 24.\u2656f4 (this typical mating plan only works with the cooperation of the opponent, but it is somewhat dangerous) 24...\u2657e6 25.\u2656h4 \u2655b6 26.\u2655f4:\n\nNow it would be wrong to play:  \nc1) 26...b2?? 27.\u2656xh7 \u2654xh7 28.\u2655h4+ \u2654g8 29.\u2655h6 b1\u2655\\+ 30.\u2658f1 followed by mate on g7.\n\nA better way is:  \nc2) 26...h5 27.\u2658xh5 gxh5 28.\u2656xh5 b2 29.\u2655h4 b1\u2655\\+ 30.\u2657f1 \u2655xf1+ 31.\u2654xf1 \u2655b1+ 32.\u2654f2 \u2655c2+ 33.\u2654g1 \u2655c1+ 34.\u2654f2 \u2655c2+ 35.\u2654g1 and Black must give perpetual check in order not to be mated.\n\nB) 23.h4 \u2658c7 24.h5 (24.\u2655a3!? Kasparov), with the possibilities:\n\na) after 24...\u2658b5? 25.f6 White obtains a strong attack: 25...\u2657e6 (25...g6 26.hxg6 fxg6 (26...hxg6 27.\u2655f4+\u2013) 27.\u2658e2 \u2657e6 28.\u2658f4 \u2658c7 29.\u2655a3+\u2013 with the threat \u2658xe6 and \u2655a6) 26.g6 fxg6 27.hxg6 hxg6 28.\u2655g5 \u2658xc3 29.\u2655xg6 with the threat of \u2654h2, followed by \u2658h5.\n\nA better way is:\n\nb) 24...\u2657d7!? 25.f6 g6 26.hxg6 hxg6 27.\u2655f4 \u2655b8 28.\u2655h4 b2 and Black holds.\n\n**23... \u2658c7 24.f6**\n\nKasparov preferred 24.\u2655h4, in order to maintain the tension. 'Vishy probably thought that 24.f6 wins by force.' (Kasparov)\n\n**24...g6 25. \u2655h4 \u2658e8**\n\nA remarkable situation: all the black pieces are on the back rank, and yet Black's position offers all the pre-conditions for successful resistance. Instead of the move in the game it would have been wrong to play 25...\u2658e6? in view of 26.\u2655h6 b2 27.\u2656f4 b1\u2655\\+ 28.\u2657f1 \u2655c1 29.\u2656h4 \u2655e3+ 30.\u2654g2 \u2658f4+ 31.\u2656xf4 \u2655xf4 32.\u2655g7#.\n\n**26. \u2655h6!?**\n\nThe threat of \u2656f4 and \u2656h4 looks deadly. But Black still has an antidote. The alternative to the text move was 26.\u2658e2, with the possible continuation 26...\u2657e6 27.\u2658f4 \u2655a8 (after 27...b2? 28.\u2655f2 \u2655b6 29.\u2656b1 +\u2013 the pawn would be lost) 28.\u2657h3 \u2657xh3 29.\u2655xh3 b2 30.e6 \u2658d6 31.\u2658xg6 fxg6 32.e7 \u2656f7 33.\u2655d7 \u2658e8 with an unclear position.\n\n**26...b2!**\n\nThe only defence.\n\n**27. \u2656f4**\n\n27.\u2658e2!? (Kasparov) 27...\u2655a5 28.\u2658f4 \u2657e6 29.\u2658xe6 fxe6 30.\u2657h3 ('The black position looks o.k.') 30...\u2655a6=.\n\n**27...b1 \u2655\\+ 28.\u2658f1??**\n\nA decisive mistake in a critical position. Anand had planned \u2657f1 and then decided at the last moment to go for a supposedly better move. 28.\u2657f1 probably leads to a draw. The variation 28...\u2655d1 29.\u2656h4 \u2655h5 30.\u2658xh5 gxh5 31.\u2656xh5 \u2657f5 32.\u2657h3 (or 32.g6 \u2657xg6 33.\u2656g5 \u2655b6 34.h4 \u2658xf6 35.exf6 \u2655xf6 36.h5 \u2655f3 37.hxg6 fxg6 38.\u2657h3 \u2655e3+ with perpetual check) 32... \u2657g6 33.e6 \u2658xf6 (33...fxe6 34.\u2657xe6+ \u2656f7 35.\u2656h3+\u2013) 34.gxf6 \u2655xf6 was shown by Carlsen at the press conference. 35.e7 \u2655xe7 36.\u2656e5 and Black is in no way worse.\n\n**28... \u2655e1!**\n\nWhite resigned.\n\nKasparov commented on Twitter: 'Has any player ever before won a WCh game without moving his queen?' Since the knight is no longer on g3, 28...\u2655e1 does work here on account of 29.\u2656h4 \u2655xh4 30.\u2655xh4 \u2655a5 and Black wins. Anand had calculated: 28...\u2655d1 29.\u2656h4 \u2655h5 30.\u2656xh5 gxh5 31.\u2658e3 \u2657e6 32.\u2657xd5 \u2657xd5 33.\u2658f5 and the knight comes to e7. White wins in this variation.\n\nTo the delight of the representatives of the media who were there, after winning the title of World Champion, Carlsen, in high spirits, leapt fully-clothed into the hotel pool. In the period after winning the World Championship the new young World Chess Champion from Norway was passed around various publicity events. Immediately after the match, football fan Carlsen was invited by Real Madrid to give the kick-off to their game against Valladolid (4:0 for Real) in the Bernabeu Stadium. In the middle of December at a schools chess tournament he met the Norwegian prime minister Erna Solberg. In January 2014 he played a game of chess with Facebook inventor Mark Zuckerberg. A few days later Carlsen appeared as a guest in the talk show of Fredrik Skavlan and in front of the television public he played a game against Microsoft chief Bill Gates.\n**46. Chess blindness in Sochi**\n\n**The World Championship 2014:  \n_Magnus Carlsen against Viswanathan Anand_**\n\nAfter the World Championship matches of 2012, Anand against Gelfand, and 2013, Anand against Carlsen, FIDE had also scheduled a World Championship match for 2014. That meant that World Championship matches were taking place in three successive years, without FIDE ever explaining why they had deviated from the two year cycle fixed by the regulations. When asked, FIDE president Kirsan Ilyumzhinov in an interview during the Chess Olympiad of Troms\u00f8 even again explained explicitly that the World Championships took place in a two year cycle, without going into the obvious contradiction.\n\nThe challenger was decided in a double-round candidates' tournament, which was played in the second half of May 2014 with eight players in Khanty-Mansiysk. Levon Aronian and Vladimir Kramnik went into the tournament as favourites. Sergey Karjakin was also held to have a chance. But then the majestic victor of the tournament was Viswanathan Anand, who had qualified for the candidates' tournament as the losing player in the World Championship match of 2013. Anand won three games \u2013 against Mamedyarov, Aronian and Topalov \u2013 and did not lose a single one. All the other players lost at least two games. Sergey Karjakin came in second behind Anand. Kramnik was third, Aronian only sixth.\n\nThe victory of Anand constituted for FIDE, or for the marketing company Agon, the problem of finding an organiser for the World Championship match. Naturally the most interest in this new WCh match between Anand and Carlsen was to be found in the home countries of the two players. But the WCh match of 2013 had already been organised in India.\n\nThis pushed Norway into first place as a potential organiser. But the Norwegian federation was already heavily committed financially in 2014 by its staging of the Chess Olympiad in Troms\u00f8. For the organisation of the Chess Olympiad the Norwegian Chess Federation and the organisers in the chess friendly city above the Arctic Circle had received a subsidy of 75 million Norwegian kroner (i.e. 8.9 million euros) from the Norwegian government. When calculating the costs, however, the Norwegians had not taken into account the organising of the FIDE World Cup, which had also been confided to them by FIDE. After some supplementary negotiations with the Norwegian government the organisers received a further grant of 12 million kroner (approx. 1.4 million euros). After that the subject of chess was pretty well exhausted for potential sponsors in Norway.\n\nOther possible organisers and sponsors, for example in Azerbaijan or Armenia, showed no interest because the representatives of their countries, respectively Mamedyarov and Aronian, had failed at the candidates' tournament. On the 10th January FIDE opened the bidding process and set the 10th March as a deadline. At the end of the period, however, there were no bids. FIDE therefore extended the deadline to the 20th April 2014, though also without success.\n\nThe lack of an organiser for the World Championship match between Magnus Carlsen and Viswanathan Anand offered Garry Kasparov a welcome target in the campaign for his bid for the office of FIDE president. After Kasparov had already supported Anatoly Karpov as Ilyumzhinov's challenger for the FIDE presidency in 2010 in Khanty-Mansiysk, the ex-World Champion now stood in person against Ilyumzhinov.\n\nKasparov's two year long election campaign was paid for, as had been the previous joint campaign of Karpov and Kasparov four years previously, by the US American multi-millionaire Rex Sinquefield, who had already set up a major chess centre in his home city of St. Louis and organised several important tournaments there. Kasparov and his supporters and the incumbent FIDE executive engaged in a distinctly dirty campaign with mutual accusations of corruption continuing even through the Chess Olympiad in Troms\u00f8. If he were elected, according to Kasparov, he would immediately find a sponsor for the World Championship match and name an organiser.\n\nIt did not require much imagination to suppose that Rex Sinquefield and the chess centre in St. Louis were what he meant. Kirsan Ilyumzhinov saw himself, as it were, in zugzwang. During the Russian junior chess tournament 'Belaya Ladya' (White rook) in Dagomys in June 2014, Ilyumzhinov met Vladimir Putin, described his problems to him and managed to convince the Russian president that Russia and the Russian Chess Federation had to come on board as organisers and sponsors of the World Championship match between Carlsen and Anand. The FIDE president was pushing at an open door, since the Russian government was in any case planning to use the Olympic park left abandoned after the Winter Olympics for staging international sporting events. Thus in October 2014 the first Formula 1 race on Russian soil was run there.\n\nOn the 11th June, FIDE announced in a press conference with the Interfax agency that the World Championship match would take place in Sochi, or more precisely in a media centre for the past Winter Olympics some 40 kilometres from the city. The budget of the forthcoming WCh match was to be three million US dollars, of which 1.5 million dollars made up the prize fund for the players. After this news was made public, the president of the Norwegian Chess Federation J\u00f6ran Aulin-Jansson commented on it with the sybilline sentence that he was not sure if the match would actually be played in Sochi.\n\nFor the imminent presidential elections of the European Chess Union, which, like the FIDE elections, would be held during the Chess Olympiad in Troms\u00f8, Aulin-Jansson was a candidate on the 'ticket' of the incumbent ECU president Silvio Danailov. The cooperation between the Norwegian and Bulgarian federations was the result of an arrangement about the Norwegian bid for the Olympiad. The Bulgarian Chess Federation had also offered to host it in the city of Albena, but withdrew in favour of Troms\u00f8 after the Norwegians had promised to support Danailov's candidacy for ECU president. Danailov was also allied with Kasparov, therefore meaning that Kasparov and Aulin-Jansson were in the same boat in common opposition to the incumbent FIDE leadership around Kirsan Ilyumzhinov.\n\nThis had already created problems before the first World Championship match between Anand and Carlsen, when the Norwegian federation protested against the awarding of the WCh match to Chennai and in the lead-up to the Chess Olympiad it also led to some tension and conflicts between the Norwegian federation and FIDE. Thus the Norwegians, at short notice and without consulting FIDE, removed the Russian women's team, actually the title defenders, from participation in the Chess Olympiad after the Russians had missed the deadline for enabling Katerina Lagno to make her change from the Ukrainian to the Russian Chess Federation. Under pressure from the FIDE executive, the Norwegians had to rescind their decision.\n\nAfter the finish of the Chess Olympiad, all the rounds of which had been broadcast live by the Norwegian television company NRK in real time, some mutterings could be heard in the Carlsen camp about FIDE's decision to stage the match against Anand in Sochi. Carlsen's manager Espen Agdestein complained moreover about the reduction in the prize fund compared to the first match and turned to the FIDE president with the request to postpone and to move the WCh match, which Ilyumzhinov, however, rejected with the hint that if Carlsen did not play he would lose the title of World Champion without a contest.\n\nDuring the Rex Sinquefield Cup in St. Louis there was another little war of nerves between the Carlsen team and FIDE, since Carlsen let pass the deadline for the signing of the contract for the World Championship and explained that he wanted to finish the tournament in peace and only then worry about the details of the contract. This was reported by the Norwegian and to some extent also by the international press as a serious crisis. In sections of the press it was even suggested that Carlsen did not want to play in Russia and Sochi on account of the annexation of the Crimea by Russia in February 2014 and the subsequent EU sanctions. But FIDE extended the deadline for signing until the 8th September 2014 and Carlsen signed the contract in time, a few hours before the expiry of the deadline. Neither of the two sides had any serious interest in seeing the breakdown of the WCh match, since in that case there would only have been losers.\n\nAnand again prepared himself for the match in his European domicile of Bad Soden. In his team there were Krishnan Sasikiran, Radoslaw Wojtaszek and Grzegorz Gajewski, three players, moreover, who all prefer as White's opening move 1.d4. In addition, during his preparation in Bad Soden he also trained with Alexander Grischuk. After his victory in the candidates' tournament Anand had also won the 'Chess Masters Final' in Bilbao in September, so he could point to a rising curve in his form.\n\nCarlsen, on the other hand, had experienced a few setbacks. At the Chess Olympiad he lost two games against nominally weaker opponents and at the Rex Sinquefield Cup he had to take second place to the outstanding play of Fabiano Caruana. Nevertheless, his results in the two tournaments were still excellent and so the Elo difference in November 2014 between Carlsen (2863) and Anand (2792) was still over 70 points. Carlsen therefore went into the match as the clear favourite. However, the chess world expected that Anand would have drawn his conclusions from his earlier clear defeat.\n\nAt the opening press conference in Sochi, Magnus Carlsen named to the press Peter Heine Nielsen and Jon Ludvig Hammer as his seconds. After the match it turned out that Carlsen had also been receiving help from Laurent Fressinet and Michael Adams. In addition, Garry Kasparov had spoken with Peter Heine Nielsen during the match and given advice over the telephone.\n\nBoth teams, like the organisers and most of the journalists, were lodged in the hotel complex of the Radisson Blue Paradise Resort. The players and their companions occupied sections of the two wings on the 8th floor of the building. For their safety the two players were always accompanied by a bodyguard, who was hired by the organisers.\n\nFIDE had fixed the match for the 7th to 28th November. The distance of the match was once more a maximum of twelve games with exactly the same rules concerning time controls and possible playoffs as in the previous year. The chief arbiter for the match was Andrzej Filipowicz, his assistants Husan Turdaliev and Anatoly Bykhovsky. The appeals committee comprised Georgios Makropoulos, Jorge Vega and Vanik Zakarian. The chief press officer was, as on many FIDE occasions previously, Anastasia Karlovich. Jana Bellin was responsible for medical matters.\n\nDuring the match it became known that Andrew Paulson had sold the firm Agon before the start of the WCh match for one (!) pound to the Russian chess organiser Ilya Merenzon. The latter had previously been the link between Paulson and Kirsan Ilyumzhinov. After Paulson had realised that he would not be able to put into effect his plan for the marketing of the World Chess Championship because he could not find any sponsors for it, he lost interest in Agon. In the election campaign for the FIDE presidency between Kirsan Ilyumzhinov and Garry Kasparov, Andrew Paulson, who saw himself as a businessman and not as a politician, found himself caught between the lines and became the target of attacks emanating from the Kasparov camp. The FIDE election finished as 110:61 in favour of Ilyumzhinov. And Kasparov's ally, ECU president Silvio Danailov, was voted out of his office with a clear majority of 18:33 votes. He was replaced in the office of president of the European Chess Federation by Zurab Azmaiparashvili.\n\nThe new owner of Agon Ilya Merenzon and Nikita Kim were the main organisers of the WCh matches in Sochi. They were supported locally by the Russian Chess Federation and its president Andrei Filatov, who stayed there for the whole period of the match. The conditions in the spacious media centre of the 2014 Winter Olympics park were excellent. Good conditions had also been provided for the visitors, though only very few chess lovers found their way to the somewhat isolated resort on the Black Sea. For some games only 16 spectators were present. Things were different on the internet, which has become more and more the actual spectators' area for the major chess tournaments. Live video-stream allows chess lovers all over the world to follow the games and events as they happen. Peter Svidler and Sopiko Guramishvili provided the majority of the English language commentary.\n\nThe drawing of lots for colours took place at the opening ceremony. Carlsen drew the black pieces for the first game. Since the order of colours was changed half-way through the match \u2013 the player who started the first half with the white pieces, started the second half with the black ones \u2013 Carlsen would therefore in the middle of the match, in the sixth and seventh games, have White twice in succession, which could be a certain advantage.\n\nIn the very first game of the match, with the black pieces, Carlsen achieved an advantage in the late middlegame, but missed his best chance and had to content himself with a draw. In his first game with white, Carlsen was able to score when he outplayed Anand in a variation of the Ruy Lopez which was considered to be harmless. But Anand struck back immediately in the 3rd game, when Carlsen ran into a deeply prepared variation, and equalised the score. The fourth and fifth games were drawn. In the sixth game Anand's preparation in the Paulsen Variation of the Sicilian Defence was revealed to be unfavourable and the challenger got into a passive position. After a major inaccuracy by Carlsen, Anand missed a combination which might have brought him a winning advantage and lost the game instead.\n\n **Carlsen \u2013 Anand**\n\nSochi, 6th game  \n15th November 2014  \nSicilian Defence (B41)\n\n**1.e4 c5 2. \u2658f3 e6 3.d4 cxd4 4.\u2658xd4 a6**\n\nUntil then Anand had hardly ever played the Paulsen Variation. The related Taimanov Variation with 4...\u2658c6 has belonged for a long time to his regular repertoire.\n\n**5.c4**\n\nAt the start of the year in the tournament in Zurich Carlsen chose 5.\u2657d3 \u2657c5 6.c3 d6 7.\u2658d2 \u2658f6 8.0-0 0-0 9.a4 e5 10.\u26584b3 \u2657a7 11.\u2658c4 \u2657e6 12.\u2655e2=, Carlsen-Caruana, Zurich (Blitz) 2014 (0-1\/35).\n\n**5... \u2658f6 6.\u2658c3 \u2657b4**\n\nAlso popular are 6...\u2655c7 and 6...d6.\n\n**7. \u2655d3**\n\n7.e5 \u2657xc3+ 8.bxc3 \u2655a5 is considered very playable for Black. In the main line after 7.\u2657d3 \u2658c6 8.\u2658xc6 Black has hardly any problems either. Another interesting and more recent try by White is 7.\u2655f3.\n\n**7... \u2658c6 8.\u2658xc6 dxc6 9.\u2655xd8+**\n\nAfter 9.e5 Black can avoid the exchange of queens with 9...\u2658d7.\n\n**9... \u2654xd8 10.e5 \u2658d7**\n\nIn two earlier games 10...\u2658e4 was played.\n\n**11. \u2657f4**\n\nThe move in the game is much better than the obvious 11.f4, which limits the effectiveness of the dark-squared bishop. It is an open question whether it was a good idea for Anand to choose a variation with an early exchange of queens, since Carlsen is particularly strong in slightly superior endgames.\n\n**11... \u2657xc3+**\n\nAnand gives up the bishop pair, but degrades the white pawn structure.\n\n**12.bxc3 \u2654c7 13.h4 b6 14.h5**\n\nWhite is slightly better. Thanks to the advantage in space it is easier for him to manoeuvre. Moreover the d6-square is a lovely outpost. Black may have a really solid position but is extremely passive. By advancing the h-pawn Carlsen prepares the attack on the weak pawns on the black kingside.\n\n**14...h6 15.0-0-0 \u2657b7 16.\u2656d3**\n\nThe rook is transferred to the kingside. Here both sides had used 30 minutes.\n\n**16...c5 17. \u2656g3 \u2656ag8**\n\nBlack is planning...\u2658f8 and...g7-g6. The black position appears very passive. But Peter Svidler suggested in his live commentary that this position had been prepared by Anand.\n\n**18. \u2657d3 \u2658f8 19.\u2657e3**\n\nWith the move in the game Carlsen sidesteps possible complications, for example after 19.\u2656g4 g5 20.hxg6 \u2658xg6 21.\u2657e3 \u2658xe5 22.\u2657f4 f6 (not 22...\u2656xg4? 23.\u2657xe5+ +\u2013). The game completely peters out after 19.\u2656h4 g5 20.hxg6 \u2658xg6 21.\u2657xg6 \u2656xg6 22.\u2656xg6 fxg6 23.\u2656xh6 \u2656xh6 24.\u2657xh6 \u2657xg2=.\n\n**19...g6 20.hxg6 \u2658xg6 21.\u2656h5**\n\n21.f4? loses: 21...\u2658xe5! 22.\u2656xg8 \u2658xd3+ 23.\u2654d2 \u2656xg8 24.\u2654xd3 \u2657xg2 and White is two pawns down. Complete equality follows 21.\u2656xh6 \u2658xe5 or even 21.\u2657xg6 \u2656xg6 22.\u2656xg6 fxg6 23.f3 h5.\n\n**21... \u2657c6**\n\nIf f7 were to become weak, Black then has...\u2657e8 available. But from c6 the bishop can also get to a4. 21...\u2658e7 22.\u2656xg8 \u2656xg8 23.g3 with advantage to White.\n\n**22. \u2657c2**\n\nThis removes the bishop from any possible knight fork on e5. But 22.\u2657xh6 achieves nothing: 22.\u2657xh6 \u2658xe5 23.\u2657f4 \u2656xg3 (oder 23...\u2656xh5 24.\u2656xg8 \u2654d7=) 24.\u2656xh8 \u2656xd3 25.\u2657xe5+ \u2654d7 26.\u2657f6 \u2657e4 27.f3 \u2657g6=.\n\n**22... \u2654b7**\n\nThe king leaves the f4-b8 diagonal so as to remove from the position any possible pin after...\u2658xe5, \u2657f4.\n\n**23. \u2656g4**\n\nSo as to perhaps come to f4 with the rook. However, the rook is unprotected on g4 and offers a target for tactical attacks.\n\nBut there were hardly any promising alternatives: 23.\u2656xh6? \u2658xe5 24.\u2656xh8 \u2656xh8 favours Black. Or: 23.\u2657xh6 \u2658e7 24.\u2656hg5 (24.\u2656xg8? \u2658xg8\u2013+) 24...\u2656xg5 25.\u2657xg5 \u2658g6 26.\u2657xg6 fxg6 27.f3 with an extra pawn for White, but bishops of opposite colours. Anish Giri recommended 23.\u2654d2. Carlsen here had something of an advantage in time: 1 hour 7 minutes remaining, Anand 53 minutes left.\n\n**23...a5**\n\nAnand's idea, which was possibly part of his preparation, consisted of advancing the pawn to a3, so as to then attack the weak pawn on a2, perhaps even with a bishop which went via a4 to b3 to offer itself as a sacrifice and thus clear the way for the black a-pawn. Another idea here was 23...\u2658e7!? 24.\u2656xg8 \u2656xg8 25.g3 \u2658g6 26.\u2657xh6 \u2657f3 27.\u2656g5 \u2657e2 28.\u2657b3 \u2657d3 with counterplay.\n\n**24. \u2657d1**\n\nThe h6-pawn may be very weak, but for the moment there is no way to take it under favourable circumstances. So Carlsen protects his rook and prevents...\u2657f3, so as to enable the advance of the g-pawn at some point.\n\n**24... \u2656d8**\n\n24...\u2658e7 25.\u2656xg8 \u2656xg8 26.g4 with advantage to White.\n\n**25. \u2657c2**\n\nHe retreats with the bishop, which leads us to conclude that Carlsen had not yet found a specific plan for the improvement of his position. 25.\u2657xh6?! \u2658xe5 26.\u2656xe5 \u2656xh6 is benefiting Black, who has got rid of his weakness on h6.\n\n**25... \u2656dg8**\n\nCarlsen now decided to bring the king over to the kingside in order to strengthen his attack.\n\n**26. \u2654d2??**\n\nA major blunder. Carlsen noticed his mistake the very moment he had moved his king. 'At that moment I was overwhelmed by a feeling of panic! White does not have a direct win here, but a lasting advantage and there are hardly any moves with which White can spoil his position. \u2654d2 is actually the only one', was what Carlsen had to say about his mistake at the press conference. After 26.\u2654d1 the king and the two rooks are on the same diagonal. As soon as the g-pawn moves, Black would play...\u2657f3, causing material losses. For that reason Carlsen rejected this move probably on grounds of principle. 26.f3 was instead a plausible move. Giri also suggested here 26.\u2656g3.\n\n**26...a4?**\n\nPlayed \u00e0 tempo. Anand absolutely did not reckon on the possibility that Carlsen would offer him such a present and therefore did not spend any time checking the position. Otherwise he would immediately have seen the possible tactical trick. With the king on d2 and on account of the unprotected \u2656g4 the following combination was possible instead: 26...\u2658xe5 27.\u2656xg8 (27.\u2656f4 f5\u2013+) 27...\u2658xc4+ 28.\u2654d3 \u2658b2+ 29.\u2654e2 \u2656xg8 30.g3 \u2657b5+ 31.\u2654f3 \u2658d3. Black has won two pawns and is winning.\n\n**27. \u2654e2**\n\nNow the chance has gone since the knight can no longer take on c4 with check. At this point Anand then realised what a chance he had passed up on.\n\n**27...a3 28.f3 \u2656d8 29.\u2654e1**\n\nCarlsen does not play 29.\u2657xg6!? fxg6 30.\u2656xg6 \u2657e8 31.\u2656g7+ \u2656d7 32.\u2656xd7+ \u2657xd7 33.\u2656xh6 \u2656xh6 34.\u2657xh6. Perhaps the endgame advantage of two pawns was not big enough for him because of the bishops of opposite colours.\n\n**29... \u2656d7 30.\u2657c1 \u2656a8**\n\nIt was worth considering 30... \u2656hd8. After 31.\u2657xh6 (31.\u2657xa3? is followed by 31...\u2656a8. Instead, 31.\u2654e2 seems better but Giri saw counterplay for Black after 31...b5) 31...\u2656h8 32.\u2657xg6 fxg6 33.\u2656xg6 Black could profit from his advanced a-pawn: 33...\u2657a4 34.g4 \u2657b3!.\n\n**31. \u2654e2 \u2657a4 32.\u2657e4+**\n\nAn important intermediate check. It would be a mistake to play 32.\u2657xg6? fxg6 33.\u2656xg6 since after 33...\u2656d1 34.\u2657xa3 (34.\u2657xh6 \u2656a1 35.\u2654e3 \u2656xa2\u2013+) 34...\u2656a1 the \u2657a3 is lost.\n\n**32... \u2657c6**\n\nNow Black's position rapidly falls apart. He had better chances of a defence after the exchange sacrifice 32...\u2654a7 33.\u2657xa8 \u2654xa8, for example: 34.\u2656xh6 \u2656d1 35.\u2657xa3 (35.\u2657d2 \u2656a1 36.\u2656h7 \u2658xe5 with counterplay) 35...\u2656a1 36.\u2654e3 \u2658xe5 37.\u2656g7 \u2658xc4+ 38.\u2654f4 \u2658d6 (not 38... \u2658xa3? 39.\u2656xf7 with unavoidable mate after \u2656h8) 39.\u2654e5 \u2656d1 and White still has a few practical difficulties to overcome on his way to victory.\n\n**33. \u2657xg6 fxg6 34.\u2656xg6 \u2657a4**\n\nBlack's counterplay is too late.\n\n**35. \u2656xe6**\n\nWhite is now winning.\n\n**35... \u2656d1**\n\n35...\u2657c2 36.\u2656hxh6 \u2656a6+\u2013.\n\n**36. \u2657xa3 \u2656a1 37.\u2654e3**\n\nAnother possibility was 37.\u2656e7+ \u2654a6 38.\u2657xc5 bxc5 39.\u2656xh6+ \u2654a5 40.\u2656c7 e.g.: 40...\u2656xa2+ 41.\u2654e3 \u2656xg2 42.\u2656xc5+ \u2657b5 43.\u2656xb5+ \u2654a4 44.\u2656h1+\u2013.\n\n**37... \u2657c2**\n\n37...\u2656xa2 38.\u2657c1+\u2013.\n\n**38. \u2656e7+1-0**\n\n38...\u2654a6 39.\u2656xh6 \u2654a5 (39...\u2656xa2 40.\u2657xc5+\u2013) 40.\u2656c7 \u2656xa2 41.\u2657c1+\u2013.\n\nIn the press conference after the game Carlsen spoke about his feelings: 'I am relieved, simply only relieved. Usually one is happy about a win, but here all I can feel is relief. Up to a certain point it was a good game.' In an interview after the match Carlsen thought that he had never felt that he might lose the match. If he had lost the 6th game as the result of a mistake, the match might have taken another course but in spite of that he would have won it \u2013 Carlsen was sure of that.\n\nAfter this defeat Anand had to defend with a material deficit in the seventh game and he finally reached a draw, but only on move 122. In the ninth and tenth games too, Anand did not manage to draw level in the match. In the 11th game, in which he had again chosen the Berlin Defence to the Ruy Lopez, Anand tried to achieve success with an exchange sacrifice which had not been well enough conceived, but Carlsen found the counter-moves and Anand lost this game too, bringing the match to a premature end with a score of 6\u00bd:4\u00bd.\n\nRussia's head of state Vladimir Putin appeared for the closing ceremony and the awarding of the prizes. The closing ceremony was originally scheduled for the day after the final game. It was postponed for a day on account of Putin's visit. The Russian head of state was then on the stage as the guest of honour at the presentation of the prizes. It was the first time in the history of the World Chess Championships that the head of state of the host country had attended the closing ceremony of a World Championship.\n\nInternational interest in the WCh match was immense. According to the data from Agon the final day's play saw 2.5 million visitors to the tournament website. Most of those visitors, 18%, were from India. German chess lovers, 12.8%, made up the second largest group of visitors, followed by the USA with 12.5%. Great Britain was the fourth largest group with 4.5%. All the other countries were below 4%. There was once more special media interest in Norway. State-owned television NRK and the media concern Verdens Gang broadcast the games live and were on site with camera teams. Compared to that, the Indian media were hardly present in Sochi at all. A camera team from India did not get there until the closing ceremony.\n**Footnotes**\n\nValue measured at Measuring Worth, <http:\/\/www.measuringworth.com\/calculators\/ppoweruk\/>\n\nJimmy Adams: The Zukertort Legend, in: Jimmy Adams: Johannes Zukertort, p. 24 ff.\n\nDomanski, Lissowski: Der Gro\u00dfmeister aus Lublin, p. 123\n\nQuotation from Domanski, Lissowski: Der Gro\u00dfmeister aus Lublin, p. 141\n\nAssiac: Noch ein vergn\u00fcgliches Schachbuch, Hamburg 1974, p. 55\n\nWinter: Chess Notes: Isidor Gunsberg, <http:\/\/www.chesshistory.com\/winter\/winter37.html#5129._Isidor_Gunsberg_C.N._5113_>\n\nGeza von Cziffra, Der Kuh im Kaffeehaus, p. 31ff\n\nChess Archeology, <http:\/\/www.chessarch.com\/archive\/000A_lask_stei\/ca0019a.shtml>\n\nNewark Sunday Call, July 7th 1885, quoted in Hannak: Emanuel Lasker, p. 52\n\nHannak: Emanuel Lasker, p. 150\n\nKmoch: Grandmasters I have known, on: www.chesscafe.com\n\nVidmar, Goldene Schachzeiten, p. 31\n\nTarrasch: Der Schachwettkampf Marshall-Tarrasch im Herbste 1905, p. 62\n\nLasker's Chess Magazine, 1906\/1907, p. 125 ff.\n\nHebeker, 'Hoffen auf das H\u00f6henklima in M\u00fcnchen...' Zum Weltmeisterschaftskampf 1908 zwischen Lasker und Tarrasch, Karl 4\/2008, p. 41\n\nKamm: Siegbert Tarrasch, Leben und Werk, p. 337\n\nKamm: Siegbert Tarrasch, Leben und Werk, p. 437\n\nH\u00fcbner: Der Wettkampf Lasker gegen Schlechter im Jahre 1910, Schach 6\/1999, p. 58\n\nDeutsche Schachzeitung, 1910, p. 30-31, quoted according to H\u00fcbner: H\u00fcbner, Robert: Der Wettkampf Lasker gegen Schlechter im Jahre 1910, Schach, 6\/1999, p. 58\n\nMark Weeks, <http:\/\/chessforallages.blogspot.de\/2007\/01\/most-infamous-world-championship-game.html>\n\nLasker's Chess Magazine, 1906\/1907, p. 125 ff.\n\nVidmar: Goldene Schachzeiten, p. 47\n\nKamm: Siegbert Tarrasch, Leben und Werk, p. 374\n\nWinter: Chess Notes, <http:\/\/www.chesshistory.com\/winter\/extra\/nardus.html>\n\nDeutsches Wochenschach, 1909, p. 178, quoted from Ackermann: Vabanque, Dawid Janowsky\n\nAckermann: Vabanque, Dawid Janowsky, p. 500\n\nWiener Schachzeitung 1911, p. 32, quoted from Ackermann: Vabanque, Dawid Janowsky.\n\nWinter, Chess Notes, <http:\/\/www.chesshistory.com\/winter\/extra\/capablancalasker.html>\n\nV\u00f6lker: Fritz Kortner, p. 75, quoted from Forster, Richard, Stefan Hansen und Michael Negele: Emanuel Lasker. Denker, Weltenb\u00fcrger, Schachweltmeister, Exzelsior Verlag, Berlin 2009, p. 201\n\nAm Grab von Emanuel Lasker: eine Spurensuche: <http:\/\/de.chessbase.com\/post\/am-grab-von-emanuel-lasker-eine-spurensuche>\n\nM\u00fcller: Schachgenie Aljechin, pp. 44\/45\n\nWinter: Chess Notes, <http:\/\/www.chesshistory.com\/winter\/extra\/london.html>\n\nWinter: Chess Notes, <http:\/\/www.chesshistory.com\/winter\/extra\/capablancaalekhine1927.html>\n\nKoblenz, Schach lebensl\u00e4nglich, p. 57\n\nKasparov: My great Predecessors Vol. 1, p. 405\n\nQuoted from: Kasparov, My great Predecessors Vol. 1, p. 413\n\nKasparov, My great Predecessors, Vol. 1, p. 413\n\nDas G\u00e4stebuch der Schachweltmeisterschaft 1934 in Deutschland, M\u00fcnster 2014, p. 29\n\nRainer Buland, in: Das G\u00e4stebuch der Schachweltmeisterschaft 1934 in Deutschland, M\u00fcnster 2014, p. 28\n\nFine: Die gr\u00f6\u00dften Schachpartien der Welt, p. 139\n\nKmoch: Grandmasters I have known\n\nHans Kmoch: Chess under the Nazi Jack-Boot, in: Chess, Oktober 1945, p. 9, quoted from Wikipedia: Salo Landau: <http:\/\/de.wikipedia.org\/wiki\/Salo_Landau>\n\nM\u00fcnninghoff: Euwe\n\nChessGames, <http:\/\/www.chessgames.com\/perl\/chess.pl?tid=54135>\n\nAssiac: Noch ein vergn\u00fcgliches Schachbuch, p. 128\n\nSosonko: The Reliable Past\n\nSpanier: Total Chess, p. 25\n\nHannak: The Lead-up to the great Tournament, in: Euwe: The Hague-Moscow 1948, p. 11\n\nWall: Alekhine and the Nazis, <http:\/\/www.geocities.com\/SiliconValley\/Lab\/7378\/nazi.htm>\n\nWikipedia, <http:\/\/en.wikipedia.org\/wiki\/Grace_Alekhine>\n\nM\u00fcller: Schachgenie Aljechin, p. 34\n\nKoblenz, Schach lebensl\u00e4nglich, p. 28\n\nRee: Preface to Euwe: The Hague-Moscow 1948, p. 5\n\nRee: Preface to Max Euwe: The Hague-Moscow 1948, p. 8\n\nKingston: The Keres-Botvinnik Case Revisited: A Further Survey of the Evidence, in ChessCafe.com, <http:\/\/www.chesscafe.com\/text\/skittles165.pdf>\n\nSoltis: Mikhail Botvinnik, p. 11f.\n\nBotvinnik: Achieving the Aim, p. 8\n\nSoltis: Mikhail Botvinnik, p. 20\n\nSoltis: Mikhail Botvinnik, p. 87\n\nSoltis: Mikhail Botvinnik, p. 106 f.\n\nBotvinnik: Achieving the Aim, p. 71\n\nSoltis: Mikhail Botvinnik, p. 189\n\nSoltis: Mikhail Botvinnik, p. 192\n\nSoltis: Mikhail Botvinnik, p. 199\n\nQuoted from Stolze: Umk\u00e4mpfte Krone\n\nAndrew Soltis in New York Post, 4th April 2010, <http:\/\/nypost.com\/2010\/04\/04\/champs-tragedy\/>\n\nMachhatschek: Der erste Revanchekampf um die Schachweltmeisterschaft, Botwinnik-Smyslow 1958, p. 10.\n\nSosonko: Russian Silhouettes, p. 24\n\nQuoted from Stolze: Umk\u00e4mpfte Krone, p. 123\n\nTal: Life and Games of Mikhail Tal, p. 170\n\nTal: The Life and Games of Mikhail Tal, p. 174\n\nKarpov on Karpov, p. 59\n\nSoltis: Soviet Chess 1917-1991, p. 304\n\nSpanier: Total Chess, p. 24\n\n<http:\/\/de.chessbase.com\/post\/interview-mit-anatoly-karpov-iii>\n\nKarpov on Karpov, p. 42\n\nSosonko: The Reliable Past, p. 115\n\nSoltis, Soviet Chess 1917-1991, p. 337\n\nDer Spiegel, 51\/1977, <http:\/\/www.spiegel.de\/spiegel\/print\/d-40693952.html>\n\nKortschnoi: Mein Leben f\u00fcr das Schach, p. 136 f.\n\nKortschnoi: Mein Leben f\u00fcr das Schach, p. 142\n\nKasparov on Kasparov, Part I, 1973-1985\n\nLawson, Keene: Kasparov versus Korchnoi, The London Contest\n\nKeene, Lawson: Kasparov versus Korchnoi, The London Contest\n\nDer Spiegel, 40\/1987, <http:\/\/www.spiegel.de\/spiegel\/print\/d-13526096.html>\n\nKasparow, Politische Partie, p. 192\n\nThe Guardian, Karpow v Kasparow: the Guardian's coverage of an epic world chess championship match, <http:\/\/www.guardian.co.uk\/news\/blog\/2009\/sep\/21\/Kasparow-Karpow-chess-rematch>\n\nChessBase News, <http:\/\/www.chessbase.de\/nachrichten.asp?newsid=13063>\n\nBudde, Nikolaiczuk: Schachweltmeisterschaft 84-85, p. 335\n\nBudde, Nikolaiczuk: Schachweltmeisterschaft 84-85, p. 335\n\nDer Spiegel, 51\/1988, <http:\/\/www.spiegel.de\/spiegel\/print\/d-13532173.html>\n\nKasparov on modern chess, Part Three, Kasparov vs Karpov, 1986-87\n\nDer Spiegel, 26\/1988, <http:\/\/www.spiegel.de\/spiegel\/print\/d-28042616.html>\n\nGarry Kasparov on Garry Kasparov, Part II 1985-1993, p. 490\n\nGarry Kasparov on Garry Kasparov Part III: 1993-2005, p. 101\n\nCrowther: The Week in Chess 200, September 1998, <http:\/\/www.chesscenter.com\/twic\/twic200.html>\n\nArticle from Wikipedia on Raymond Keene, <http:\/\/en.wikipedia.org\/wiki\/Raymond_Keene>\n\nCounterpoint: Levy vs Keene, The Chess Journalist, Vol. 32, No. 4, December 2003, p. 5\n\nBrain Games: The Full Truth by David Levy, Chess Caf\u00e9, <http:\/\/www.chesscafe.com\/text\/skittles161.pdf>\n\nCrowther: The Week in Chess 281, <http:\/\/www.chesscenter.com\/twic\/twic281.html>\n\nBareev, Levitov: From London to Elista, p. 85\n\nCrowther, The Week in Chess 313, <http:\/\/www.chess.co.uk\/twic\/twic313.html>\n\nBareev, Levitov: From London to Elista, p. 306\n\nOpen letter from Lela Javakhishvili and Ana Matnadze, <http:\/\/www.chessbase.com\/newsdetail.asp?newsid=1702>\n\n<http:\/\/whychess.com\/node\/1605>\n\nWhychess: Kramnik on chess, Anand, Topalov and his future, <http:\/\/whychess.com\/node\/1605>\n\nFIDE President guarantees Topalov-Kamsky Match <http:\/\/www.fide.com\/component\/content\/article\/1-fide-news\/2970-fide-president-guarantees-topalov-kamsky-match>\n\nChessBase News: Anand in Playchess \u2013 the helpers in Sofia, <http:\/\/www.chessbase.com\/newsdetail.asp?newsid=6346>\n\nFIDE President and Bessel Kok form Global Chess Company, <http:\/\/www.fide.com\/component\/content\/article\/3-news\/1898-389-fide-president-and-bessel-kok-form-global-chess-company>\n\nChessBase News: Ein f\u00fcnf Mio. Dollar Match. Interview with Andrei Filatow, <http:\/\/www.chessbase.de\/nachrichten.asp?newsid=12999>\n\nChessvibes: Interview with Boris Gelfand, <http:\/\/www.chessvibes.com\/reports\/boris-gelfand-i-was-by-no-means-inferior-in-this-match-interview-part-1-of-2>\n\nMichaltschischin\/Stetsko: K\u00e4mpfen und Siegen mit Magnus Carlsen, p. 10.\n\nKasparov at the inauguration of the ZMD campus, March 2004 in Dresden.\n\nSimen Agdestein: Wonderboy, New in Chess, 2004\n\nChessBase News: <http:\/\/en.chessbase.com\/post\/facts-and-figures-magnus-carlsen-s-performance-in-nanjing>\n\nChessBase News: <http:\/\/en.chessbase.com\/post\/magnus-carlsen-drops-out-of-world-championship-cycle>\n\nChessBase News, Paulson: 'World cities will glorify chess'. <http:\/\/www.chessbase.com\/newsdetail.asp?newsid=7985>\n**All (classical) World Champions**\n\n1st World Champion | 1886-1894 | Wilhelm Steinitz\n\n---|---|---\n\n2nd World Champion | 1894-1921 | Emanuel Lasker\n\n3th World Champion | 1921-1927 | Jos\u00e9 Raul Capablanca\n\n4th World Champion | 1927-1935, 1937-1946 | Alexander Alekhine\n\n5th World Champion | 1935-1937 | Max Euwe\n\n6th World Champion | 1948-1957, 1958-1960, 1961-1963 | Mikhail Botvinnik\n\n7th World Champion | 1957-1958 | Vassily Smyslov\n\n8th World Champion | 1960-1961 | Mikhail Tal\n\n9th World Champion | 1963-1969 | Tigran Petrosian\n\n10th World Champion | 1969-1972 | Boris Spassky\n\n11th World Champion | 1972-1975 | Robert Fischer\n\n12th World Champion | 1975-1985 | Anatoly Karpov\n\n13th World Champion | 1985-2000 | Garry Kasparov\n\n14th World Champion | 2000-2007 | Vladimir Kramnik\n\n15th World Champion | 2007-2013 | Viswanathan Anand\n\n16th World Champion | 2013-? | Magnus Carlsen\n\n**Ranking list by years**\n\n1st Emanuel Lasker: 27 years World Champion\n\n2nd Alexander Alekhine: 17 years\n\n3th Garry Kasparov: 15 years\n\n4th Mikhail Botvinnik: 13 years\n\n5th Anatoly Karpov: 10 years\n\n6th Wilhelm Steinitz: 8 years\n\n7th Vladimir Kramnik: 7 years\n\n8th Jos\u00e9 Raul Capablanca, Tigran Petrosian, Viswanathan Anand: 6 years\n\n11th Boris Spassky, Robert Fischer: 3 years\n\n13th Max Euwe: 2 years\n\n14th Vassily Smyslov, Mikhail Tal: 1 year\n\n**Ranking list by WCh matches or tournaments won**\n\n(including title defences after a drawn match)\n\n6 World Championship matches won: Lasker, Kasparov\n\n5 World Championship matches won: Botvinnik\n\n4 World Championship matches won: Steinitz, Alekhine, Anand\n\n3 World Championship matches won: Kramnik\n\n2 World Championship matches won: Petrosian, Karpov (plus one walk-over), Carlsen\n\n1 World Championship match won: Capablanca, Euwe, Smyslov, Tal, Spassky, Fischer\n**All (classical) World Championships**\n\n**1 st World Championship 1886**\n\nWilhelm Steinitz against Johannes Zukertort; New York, St. Louis, New Orleans (USA)\n\nTo 10 wins\n\n10:5 with 5 draws. Wilhelm Steinitz becomes World Champion\n\n**2 nd World Championship 1889**\n\nWilhelm Steinitz against Mikhail Chigorin; Havana (Cuba)\n\nFor 20 games\n\n10:6 with 1 draw. Wilhelm Steinitz remains World Champion\n\n**3 th World Championship 1890**\n\nWilhelm Steinitz against Isidor Gunsberg; New York (USA)\n\nFor 20 games\n\n6:4 with 9 draws. Wilhelm Steinitz remains World Champion\n\n**4 th World Championship 1892**\n\nWilhelm Steinitz against Mikhail Chigorin; Havana (Cuba)\n\nFor 20 games or 10 wins, if drawn after 20 games additional games. As the match was tied after 20 games, it was continued until the 10th win of a player\n\n10:8 with 5 draws. Wilhelm Steinitz remains World Champion\n\n**5 th World Championship 1894**\n\nWilhelm Steinitz against Emanuel Lasker; New York, Philadelphia (USA), Montr\u00e9al (Canada)\n\nTo 10 wins\n\n10:5 with 4 draws. Emanuel Lasker becomes World Champion\n\n**6 th World Championship 1896\/97**\n\nEmanuel Lasker against Wilhelm Steinitz; Moscow (Russia)\n\nTo 10 wins. 10:2 with 5 draws. Emanuel Lasker remains World Champion\n\n**7 th World Championship 1907**\n\nEmanuel Lasker against Frank Marshall; New York, Philadelphia, Washington, Baltimore, Chicago, Memphis (USA)\n\nTo 8 wins. 8:0 with 7 draws. Emanuel Lasker remains World Champion\n\n**8 th World Championship 1908**\n\nEmanuel Lasker against Siegbert Tarrasch; D\u00fcsseldorf, Munich (German Empire)\n\nTo 8 wins. 8:3 with 5 draws. Emanuel Lasker remains World Champion\n\n**9 th World Championship 1910**\n\nEmanuel Lasker against Carl Schlechter; Vienna (Austria-Hungary), Berlin (German Empire)\n\nFor 10 games, if drawn the World Champion retains his title\n\n1:1 with 8 draws. Emanuel Lasker remains World Champion\n\n**10 th World Championship 1910**\n\nEmanuel Lasker against Dawid Janowski; Berlin (German Empire)\n\nTo 8 wins. 8:0 with 3 draws. Emanuel Lasker remains World Champion\n\n**11 th World Championship 1921**\n\nEmanuel Lasker against Jos\u00e9 Raul Capablanca; Havana (Cuba)\n\nTo 8 wins, maximum 24 games. If drawn the World Champion retains his title. Lasker resigned after 14 games\n\n4:0 with 10 draws. Jos\u00e9 Raul Capablanca becomes World Champion\n\n**12 th World Championship 1927**\n\nJos\u00e9 Raul Capablanca against Alexander Alekhine; Buenos Aires (Argentina)\n\nTo 6 wins\n\n3:6 with 25 draws. Alexander Alekhine becomes World Champion\n\n**13 th World Championship 1929**\n\nAlexander Alekhine against Efim Bogoljubow; Wiesbaden, Heidelberg, Berlin (German Empire), The Hague (Netherlands)\n\nFor 30 games, if drawn the World Champion retains his title\n\n11:5 with 9 draws. Alexander Alekhine remains World Champion\n\n**14 th World Championship 1934**\n\nAlexander Alekhine against Efim Bogoljubow; Baden-Baden, Villingen, Freiburg, Pforzheim, Stuttgart, Munich, Bayreuth, Bad Kissingen, Mannheim, Berlin (German Empire)\n\nFor 30 games, if drawn the World Champion retains his title\n\n8:3 with 15 draws. Alexander Alekhine remains World Champion\n\n**15 th World Championship 1935**\n\nAlexander Alekhine against Max Euwe; Amsterdam, Delft, Rotterdam, Utrecht, Gouda, The Hague, Groningen, Baarn, 's-Hertogenbosch, Eindhoven, Zeist, Ermelo, Zandvoort (Netherlands)\n\nFor 30 games, if drawn the World Champion retains his title\n\n8:9 with 13 draws. Max Euwe becomes World Champion\n\n**16 th World Championship 1937**\n\nMax Euwe against Alexander Alekhine; The Hague, Rotterdam, Amsterdam, Haarlem, Leiden, Groningen, Delft (Netherlands)\n\nFor 30 games, if drawn the World Champion retains his title\n\n4:10 with 11 draws. Alexander Alekhine becomes World Champion again\n\n**17 th World Championship 1948**\n\nTournament with 5 players and 5 rounds; The Hague (Netherlands), Moscow (USSR) Botvinnik wins with 10 wins, 2 defeats and 8 draws. Mikhail Botvinnik becomes World Champion\n\n**18 th World Championship 1951**\n\nMikhail Botvinnik against David Bronstein; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n5:5 with 14 draws. Mikhail Botvinnik remains World Champion\n\n**19 th World Championship 1954**\n\nMikhail Botvinnik against Vassily Smyslov; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n7:7 with 10 draws. Mikhail Botvinnik remains World Champion\n\n**20 th World Championship 1957**\n\nMikhail Botvinnik against Vassily Smyslov; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n3:6 with 13 draws. Vassily Smyslov becomes World Champion\n\n**21 st World Championship 1958**\n\nVassily Smyslov against Mikhail Botvinnik; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n5:7 with 11 draws. Mikhail Botvinnik becomes World Champion again\n\n**22 nd World Championship 1960**\n\nMikhail Botvinnik against Mikhail Tal; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n2:6 with 13 draws. Mikhail Tal becomes World Champion\n\n**23 rd World Championship 1961**\n\nMikhail Tal against Mikhail Botvinnik; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n5:10 with 6 draws. Mikhail Botvinnik becomes World Champion again\n\n**24 th World Championship 1963**\n\nMikhail Botvinnik against Tigran Petrosian; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n2:5 with 15 draws. Tigran Petrosian becomes World Champion\n\n**25 th World Championship 1966**\n\nTigran Petrosian against Boris Spassky; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n4:3 with 17 draws. Tigran Petrosian remains World Champion\n\n**26 th World Championship 1969**\n\nTigran Petrosian against Boris Spassky; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n4:6 with 13 draws. Boris Spassky becomes World Champion\n\n**27 th World Championship 1972**\n\nBoris Spassky against Robert Fischer; Reykjavik (Iceland)\n\nFor 24 games, if drawn the World Champion retains his title\n\n3:7 with 11 draws. Robert Fischer becomes World Champion\n\n**28 th World Championship 1975**\n\nRobert Fischer against Anatoly Karpov; Manila (Philippines)\n\nTo 10 wins. Robert Fischer did not appear. Anatoly Karpov becomes World Champion without a match\n\n**29 th World Championship 1978**\n\nAnatoly Karpov against Viktor Kortchnoi; Baguio City (Philippines)\n\nTo 6 wins. 6:5 with 21 draws. Anatoly Karpov remains World Champion\n\n**30 th World Championship 1981**\n\nAnatoly Karpov against Viktor Kortchnoi; Merano (Italy)\n\nTo 6 wins. 6:2 with 10 draws. Anatoly Karpov remains World Champion\n\n**31 st World Championship 1984\/85**\n\nAnatoly Karpov against Garry Kasparov; Moscow (USSR)\n\nTo 6 wins. 5:3 with 40 draws. The match was abandoned without a winner\n\n**32 nd World Championship 1985**\n\nAnatoly Karpov against Garry Kasparov; Moscow (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n3:5 with 16 draws. Garry Kasparov becomes World Champion\n\n**33 rd World Championship 1986**\n\nGarry Kasparov against Anatoly Karpov; London (Great Britain), Leningrad (USSR)\n\nFor 24 games, if drawn the World Champion retains his title\n\n5:4 with 15 draws. Garry Kasparov remains World Champion\n\n**34 th World Championship 1987**\n\nGarry Kasparov against Anatoly Karpov; Sevilla (Spain)\n\nFor 24 games, if drawn the World Champion retains his title\n\n4:4 with 16 draws. Garry Kasparov remains World Champion\n\n**35 th World Championship 1990**\n\nGarry Kasparov against Anatoly Karpov; New York (USA), Lyon (France)\n\nFor 24 games, if drawn the World Champion retains his title\n\n4:3 with 17 draws. Garry Kasparov remains World Champion\n\n**36 th World Championship 1993**\n\nGarry Kasparov against Nigel Short; London (Great Britain)\n\nFor 24 games, if drawn the World Champion retains his title\n\n6:1 with 13 draws. Garry Kasparov remains World Champion\n\n**37 th World Championship 1995**\n\nGarry Kasparov against Viswanathan Anand; New York (USA)\n\nFor 20 games, if drawn the World Champion retains his title\n\n4:1 with 13 draws. Garry Kasparov remains World Champion\n\n**38 th World Championship 2000**\n\nGarry Kasparov against Vladimir Kramnik; London (Great Britain)\n\nFor 16 games, if drawn the World Champion retains his title\n\n0:2 with 13 draws.\n\nVladimir Kramnik becomes World Champion\n\n**39 th World Championship 2004**\n\nVladimir Kramnik against Peter Leko; Brissago (Switzerland)\n\nFor 16 games, if drawn the World Champion retains his title\n\n2:2 with 10 draws. Vladimir Kramnik remains World Champion\n\n**40 th World Championship 2006**\n\nVladimir Kramnik against Veselin Topalov; Elista (Russia)\n\nFor 12 games. If drawn, additional games with reduced time limit\n\n3:3 with 6 draws. Kramnik wins the playoff 2:1 with 1 draw. Vladimir Kramnik remains World Champion\n\n**41 st World Championship 2007**\n\nDouble round-robin tournament with 8 players; Mexico City (Mexico)\n\nAnand wins with 4 wins, 10 draws and no losses. Viswanathan Anand becomes World Champion\n\n**42 nd World Championship 2008**\n\nViswanathan Anand against Vladimir Kramnik; Bonn (Germany)\n\nFor 12 games. If drawn, additional games with reduced time limit\n\n3:1 with 7 draws. Viswanathan Anand remains World Champion\n\n**43 rd World Championship 2010**\n\nViswanathan Anand against Veselin Topalov; Sofia (Bulgaria)\n\nFor 12 games. If drawn, additional games with reduced time limit\n\n3:2 with 7 draws. Viswanathan Anand remains World Champion\n\n**44 th World Championship 2012**\n\nViswanathan Anand against Boris Gelfand; Moscow (Russia)\n\nFor 12 games. If drawn, additional games with reduced time limit\n\n1:1 with 10 draws. Anand wins the playoff 1:0 with 3 draws. Viswanathan Anand remains World Champion\n\n**45 th World Championship 2013**\n\nViswanathan Anand against Magnus Carlsen; Chennai (India)\n\nFor 12 games. If drawn, additional games with reduced time limit\n\n0:3 with 7 draws. Magnus Carlsen becomes World Champion\n\n**46 th World Championship 2014**\n\nMagnus Carlsen against Viswanthan Anand; Sochi (Russia)\n\nFor 12 games. If drawn, additional games with reduced time limit\n\n3:1 with 7 draws. Magnus Carlsen remains World Champion\n\n**Venues for classical World Championships**\n\n**Argentina**\n\n---\n\nBuenos Aires | WCh 1927 Capablanca\u2013Alekhine\n\n**Austria**\n\nVienna | WCh 1910 Lasker\u2013Schlechter\n\n**Bulgaria**\n\nSofia | WCh 2010 Anand\u2013Topalov\n\n**Canada**\n\nMontr\u00e9al | WCh 1894 Steinitz\u2013Lasker\n\n**Cuba**\n\nHavana | WCh 1889 Steinitz\u2013Chigorin, WCh 1892 Steinitz\u2013Chigorin, WCh 1921 Lasker\u2013Capablanca\n\n**France**\n\nLyon | WCh 1990 Kasparov\u2013Karpov, games 13-24\n\n**Germany**\n\nBad Kissingen | WCh 1934 Alekhine\u2013Bogoljubow\n\nBaden-Baden | WCh 1934 Alekhine\u2013Bogoljubow\n\nBayreuth | WCh 1934 Alekhine\u2013Bogoljubow\n\nBerlin | WCh 1910 Lasker\u2013Schlechter, WCh 1910 Lasker\u2013Janowski, WCh 1929 Alekhine\u2013Bogoljubow, WCh 1934 Alekhine\u2013Bogoljubow\n\nBonn | WCh 2008 Anand\u2013Kramnik\n\nD\u00fcsseldorf | WCh 1908 Lasker\u2013Tarrasch\n\nFreiburg | WCh 1934 Alekhine\u2013Bogoljubow\n\nHeidelberg | WCh 1929 Alekhine\u2013Bogoljubow\n\nKarlsruhe | WCh 1934 Alekhine\u2013Bogoljubow\n\nMannheim | WCh 1934 Alekhine\u2013Bogoljubow\n\nMunich | WCh 1908 Lasker\u2013Tarrasch, WCh 1934 Alekhine\u2013Bogoljubow\n\nNuremberg | WCh 1934 Alekhine\u2013Bogoljubow\n\nPforzheim | WCh 1934 Alekhine\u2013Bogoljubow\n\nStuttgart | WCh 1934 Alekhine\u2013Bogoljubow\n\nVillingen | WCh 1934 Alekhine\u2013Bogoljubow\n\nWiesbaden | WCh 1929 Alekhine\u2013Bogoljubow\n\n**Great Britain**\n\nLondon | WCh 1986 Kasparov-Karpov, games 1-12, WCh 1993 Kasparov\u2013Short, WCh 2000 Kasparov-Kramnik\n\n**Iceland**\n\nReykjavik | WCh 1972 Spassky\u2013Fischer\n\n**India**\n\nChennai | WCh 2013 Anand\u2013Carlsen\n\n**Italy**\n\nMerano | WCh 1981 Karpov\u2013Kortchnoi\n\n**Mexico**\n\nMexico City | WCh tournament 2007\n\n**Netherlands**\n\nAmsterdam | WCh 1935 Alekhine\u2013Euwe, WCh 1937 Euwe\u2013Alekhine\n\nBaarn | WCh 1935 Alekhine\u2013Euwe\n\nDelft | WCh 1935 Alekhine\u2013Euwe, WCh 1937 Euwe\u2013Alekhine\n\nEindhoven | WCh 1935 Alekhine\u2013Euwe\n\nErmelo | WCh 1935 Alekhine\u2013Euwe\n\nGouda | WCh 1935 Alekhine\u2013Euwe\n\nGroningen | WCh 1935 Alekhine\u2013Euwe, WCh 1937 Euwe\u2013Alekhine\n\nHaarlem | WCh 1937 Euwe\u2013Alekhine\n\nLeiden | WCh 1937 Euwe\u2013Alekhine\n\nRotterdam | WCh 1935 Alekhine\u2013Euwe, WCh 1937 Euwe\u2013Alekhine\n\n's-Hertogenbosch | WCh 1935 Alekhine\u2013Euwe\n\nThe Hague | WCh 1935 Alekhine\u2013Euwe, WCh 1937 Euwe\u2013Alekhine, WCh tourn. 1948\n\nUtrecht | WCh 1935 Alekhine\u2013Euwe\n\nZandvoort | WCh 1935 Alekhine\u2013Euwe\n\nZeist | WCh 1935 Alekhine\u2013Euwe\n\n**Philippines**\n\nBaguio City | WCh 1978 Karpov\u2013Kortchnoi\n\n**Russia\/Soviet Union**\n\nElista | WCh 2006 Kramnik\u2013Topalov\n\nLeningrad | WCh 1986 Kasparov-Karpov, games 13-24\n\nMoscow | WCh 1896 Lasker\u2013Steinitz, WCh tournament 1948, WCh 1951 Botvinnik\u2013Bronstein, WCh 1954 Botvinnik\u2013Smyslov, WCh 1957 Botvinnik\u2013Smyslov, WCh 1958 Smyslov\u2013Botvinnik, WCh 1960 Botvinnik\u2013Tal, WCh 1961 Tal\u2013Botvinnik, WCh 1963 Botvinnik\u2013Petrosian, WCh 1966 Petrosian\u2013Spassky, WCh 1969 Petrosian\u2013Spassky, WCh 2012 Anand\u2013Gelfand\n\nSochi | WCh 2014 Carlsen\u2013Anand\n\n**Spain**\n\nSevilla | WCh 1987 Kasparov\u2013Karpov\n\n**Switzerland**\n\nBrissago | WCh 2004 Kramnik\u2013Leko\n\n**USA**\n\nBaltimore | WCh 1907 Lasker\u2013Marshall\n\nChicago | WCh 1907 Lasker\u2013Marshall\n\nMemphis | WCh 1907 Lasker\u2013Marshall\n\nNew Orleans | WCh 1886 Steinitz\u2013Zukertort, games 10-20\n\nNew York | WCh 1886 Steinitz\u2013Zukertort, games 1-5, WCh 1890 Steinitz\u2013Gunsberg, WCh 1894 Steinitz\u2013Lasker, WCh 1907 Lasker\u2013Marshall WCh 1990 Kasparov\u2013Karpov, games 1-12, WCh 1995 Kasparov\u2013Anand\n\nPhiladelphia | WCh 1894 Steinitz\u2013Lasker, WCh 1907 Lasker\u2013Marshall\n\nSt. Louis | WCh 1886 Steinitz\u2013Zukertort, games 6-9\n\nWashington | WCh 1907 Lasker\u2013Marshall\n**Glossary**\n\n_adjourned game_\n\nGame which is interrupted after a fixed number of moves, usually after 40 moves, and play is resumed on another day. No longer usual nowadays.\n\n_all-play-all_\n\nTournament with more than two players, in which each player meets every other player.\n\n_analysis_\n\nThe more accurate examination of a game including the evaluation of moves and suggestions of alternative options and reasons for error. Common analysis between the players after a game is also called a 'post mortem'.\n\n_appeals committee_\n\nUsually made up of three officials who check the decisions of the arbiter in the event of disputes.\n\n_armageddon_\n\nSee: _sudden death game_.\n\n_chess clock_\n\nThe chess clock measures the thinking time use and consists of two linked clocks. By pressing a button or a lever one or the other clock is set in motion and measures the time taken by one or other of the players.\n\n_demo board_\n\nLarge chess board in the form of a chess diagram, on which the moves of a game are shown to spectators during chess tournaments or matches.\n\n_draw_\n\nA game which ends without a winner. This can arise by stalemate, the lack of enough material to force mate (technical draw), threefold repetition of a position, the passing of 50 moves without a pawn having been moved or any piece being taken, or agreement between the players.\n\n_Elo rating_\n\nNumber of a player, which informs us about his or her playing strength according to a formula developed by the Hungarian mathematician, Arpad Elo. Beginners have about 800 Elo, club players on average 1600 Elo, tournament players 2200 Elo and more, grandmasters 2500 Elo and more and the top professionals 2700 Elo and above. The highest Elo rating so far was reached in 2014 by Magnus Carlsen with 2882.\n\n_FIDE_\n\nThe world chess federation. Founded as the 'F\u00e9d\u00e9ration Internationale des \u00c9checs' on the 20th July 1924 in Paris. Motto: 'Gens una sumus' (We are one family). Since 1927 FIDE has been organising the team World Championship _Chess Olympiads_ , since 1948 also the organisation of the individual World Championship. Over and above this FIDE organises many other tournaments and manages the Elo system.\n\n_grandmaster title_\n\nOfficially: international grandmaster (IGM). Honorary title in chess, which is awarded by FIDE according to specific rules. Whilst until the 1960s only the best players were named grandmaster (1957: 50 grandmasters in the whole world), nowadays very many professionals have the title of grandmaster (2014: 1440 grandmasters).\n\n_k.-o.-tournament_\n\nTournament in knock-out mode. The player who loses a game or a round is eliminated. The winner qualifies for the next round.\n\n_kiebitzer_\n\nSpectator in chess.\n\n_live-stream_\n\nVideo broadcast of chess tournament or match on the internet.\n\n_match_\n\nTwo players meet for a fixed number of games. The match is the classic format for World Chess Championships.\n\n_mate_\n\nThe end of a game of chess. A king which is being attacked can neither avoid the attack nor eliminate the attacking piece.\n\n_notation_\n\nTranscript of a game of chess.\n\n_opening_\n\nThe starting phase of a game in which the pieces are brought into play from their starting squares.\n\n_playoff_\n\nWhen two or more players are equal on points at the end of a tournament or match, a playoff is sometimes used to come to a decision, usually carried out in the form of rapid or blitz chess games.\n\n_sealed move_\n\nOn the   _adjournment_ the player whose move it is writes down the move which he will play on the resumption of the game and gives it in a sealed envelope to the arbiter.\n\n_second_\n\nA player who supports another player in a tournament or match with preparatory work and analyses.\n\n_simultaneous display_\n\nIn a simultaneous display a master or grandmaster plays against several opponents at the same time.\n\n_stalemate_\n\nOne player finds it impossible to move a single piece without putting his own king in check. This ends the game as a draw.\n\n_sudden death game_\n\nThe decisive game after an undecided series of tie-break games. Black obtains one minute less than White, usually it is four minutes against five minutes, but he only needs a draw to 'win' the game.\n\n_thinking time or time control (limit)_\n\nThe time a player has available to make a specified number of moves or to play the whole game. A usual thinking time for tournaments is, for example, two hours for 40 moves, one hour for 20 moves and 30 minutes for the rest of the game. In _rapid chess tournaments_ between 15 and 25 minutes thinking time per game is allocated, in _blitz tournaments_ 5 minutes. Nowadays an increment is frequently used. For every move he has made the player receives some additional seconds as thinking time.\n\n_time trouble_\n\nTime trouble occurs when a player has used almost all of his thinking time and has to make his moves quickly to avoid losing by over-stepping the time control.\n\n_tournament_\n\nCompetition with more than two players.\n\n_trainer_\n\nTeacher or personal adviser of chess students or also of a professional chess player.\n\n_WCh cycle_\n\nThe time after one World Championship match during which qualification tournaments decide on the challenger to the World Champion, including the next WCh match.\n\n_zugzwang_\n\nA situation in which any move a player makes decisively harms his or her own position; the player would like to be able not to make a move. Zugzwang motifs play a major role particularly in elementary endgames.\n**Bibliography**\n\nThe only sources which are listed in the bibliography are those actually used as such in the composition of the text. Various transcriptions of names in the Cyrillic alphabet have been used over the years, depending on the language in which a book was published. In this bibliography we have given them as originally published.\n\nAckermann, D.: _Vabanque, Dawid Janowsky (1868-1927)_ , Das Schach-Archiv, Ludwigshafen 2005\n\nAdams, J.: _Johannes Zukertort_ , New in Chess, Alkmaar 2014\n\nAlexander, C.H.O'D.: _Spassky_ \u2013 _Fischer, das gr\u00f6\u00dfte Schach-Duell der Geschichte_ , Heyne, M\u00fcnchen 1972\n\nAnand, V.\/Nunn, J.: _Vishy Anand: World Chess Champion_ , Gambit, London 2012\n\nAnderberg, P.: _Die WM-K\u00e4mpfe Aljechin_ \u2013 _Bogoljubov, 1929 &1934_, in: Karl, 4\/2008, S. 45ff.\n\nAssiac, alias Heinrich Fraenkel: _Noch ein vergn\u00fcgliches Schachbuch_ , Hamburg 1974\n\nBall\u00f3, H.: _Siegbert Tarrasch: Schachspieler, Arzt, Jude_ , <http:\/\/www.ballo.de\/tarrasch.htm>\n\nBareev, E.\/Levitov, I.: _From London to Elista_ , New in Chess, Alkmaar 2007\n\nBaturinski, V.: _Schachgenie Karpow_ , Sportverlag, Berlin 1991\n\nBijl, C.\/Kortschnoi, V.\/Lind\u00f6rfer, K. (Hrsg.): _Die Schachk\u00e4mpfe um die Weltmeisterschaft zwischen Aljechin und Bogoljubow 1929 und 1934_ , Edition Olms, Z\u00fcrich 1983\n\nBotvinnik, M.: _Match for the World Chess Championship Botvinnik_ \u2013 _Bronstein_ , Moscow 1951\n\nBotvinnik, M.: _Revanchewettkampf um die Schachweltmeisterschaft Botwinnik_ \u2013 _Tal_ , Moskau 1961\n\nBotvinnik, M.: _Achieving the Aim_ , Pergamon Press, Oxford 1981\n\nBotwinnik, M.: _15 Schachpartien und ihre Geschichte_ , Franck'sche Verlagshandlung, Stuttgart 1981\n\nBotwinnik, M.: _Wettk\u00e4mpfe um die Schachweltmeisterschaft Botwinnik_ \u2013 _Smyslow 1954-1957-1958_ , published by Igor Botwinnik, Techalbo, K\u00f6ln 2007\n\nBotwinnik, M.: _Wettkampf um die Schachweltmeisterschaft Botwinnik_ \u2013 _Petrosjan_ , published by Igor Botwinnik, Techalbo, K\u00f6ln 2006\n\nBreutigam, M. et al.: _World Chess Championship 2004, Centro Dannemann, Kramnik vs Leko_ , Chessgate, Nettetal 2004\n\nBuland, R.\/Edtmaier B.\/Schweiger G.: _Das G\u00e4stebuch der Schachweltmeisterschaft 1934 in Deutschland_ , Lit-Verlag, M\u00fcnster 2014\n\nBudde, V.\/Nikolaiczuk, L.: _Schachweltmeisterschaft 84-85 Karpow_ \u2013 _Kasparow_ , Beyer Verlag, Hollfeld 1985\n\nByrne, R.\/Nei I.: _Both Sides of the Chessboard_ , Batsford, London 1974\n\nDomanski, C.\/Lissowski T.: _Der Gro\u00dfmeister aus Lublin_ , Exzelsior Verlag, Berlin, 2005\n\nEdmonds, D.\/Eidinow, J.: _Bobby Fischer Goes to War_ , Harper Collins, New York 2004\n\nEhn, M.: _Der junge Steinitz \u2013 Legenden und Wirklichkeiten_ , in: Karl, 1\/2012, S. 10-18\n\nEuwe, M.\/Prins, L.: _Capablanca_ , Verlag Das Schacharchiv, Hamburg-Bergedorf 1979\n\nEuwe, M.: _The Hague-Moscow 1948, Match\/Tournament for the World Championship 1948_ , Russell Enterprises, Milford 2013\n\nFine, R.: _Die gr\u00f6\u00dften Schachpartien der Welt_ , Heyne Verlag, M\u00fcnchen 1976\n\nForster, R.\/Hansen S.\/Negele, M.: _Emanuel Lasker. Denker, Weltenb\u00fcrger, Schachweltmeister_ , Exzelsior Verlag, Berlin 2009\n\nGligoric, S.: _Fischer \u2013 Spasskij_ , Droemer Knaur, Z\u00fcrich 1972\n\nGutman, L.\/Treppner, G.: _Schach-WM '87 Garri Kasparow \u2013 Anatoli Karpow_ , Beyer-Verlag, Hollfeld 1987\n\nHannak, J.: _Emanuel Lasker_ , Siegfried Engelhardt Verlag, Berlin 1962\n\nHarenberg, W.: _Schachweltmeister_ , Spiegelbuch, Rowohlt, Reinbek 1981\n\nHebeker, F.-K.: _'Hoffen auf das H\u00f6henklima in M\u00fcnchen...'. Zum Weltmeisterschaftskampf 1908 zwischen Lasker und Tarrasch_ , in: _Karl_ 4\/2008, p. 38 ff.\n\nHecht, H.-J.\/Treppner, G.: _Schach-WM Revanche-Kampf 1986_ , Beyer-Verlag, Hollfeld 1986\n\nH\u00fcbner, R.: _Der Wettkampf Lasker gegen Schlechter im Jahre 1910_ (in five parts), in: _Schach_ 1999, issues five, six, eight, eleven and twelve, Exelsior Verlag, Berlin 1999\n\nKamm, W.: _Siegbert Tarrasch, Leben und Werk_ , Manuel Fruth Verlag, Unterhaching 2004\n\nKarpov, A.: _Karpov on Karpov_ , Macmillan Publishing Company, New York 1990\n\nKasparow, G.\/Trelford, D.: _Politische Partie_ , Droemersche Verlagsanstalt, M\u00fcnchen 1987\n\nKasparov, G.: _Garry Kasparov on My Great Predecessors_ (book series in five volumes), Everyman Chess, London 2003-2006\n\nKasparov, G.: _Garry Kasparov on Modern Chess Part Three: Kasparow vs Karpow 1986-87_ , Everyman Chess, London 2009\n\nKasparov, G.: _Garry Kasparov on Modern Chess Part Four: Kasparow vs Karpow 1988-2009_ , Everyman Chess, London 2010\n\nKasparov, G.: _Garry Kasparov on Garry Kasparov_ (book series in three volumes), Everyman Chess, London 2011-2014\n\nKeene, R.\/Lawson, D.: _Kasparov Versus Korchnoi_ , Batsford, London 1983\n\nKingston, T.: _The Keres-Botvinnik Case Revisited: A Further Survey of the Evidence_ , <http:\/\/www.chesscafe.com\/text\/skittles165.pdf>\n\nKoblenz, A.: _Schach lebensl\u00e4nglich_ , Beyer-Verlag, Hollfeld 1997\n\nKortschnoi, V.\/Pachman, L.: _Schach WM '78 Kortschnoi \u2013 Karpow_ , Walter Rau Verlag, D\u00fcsseldorf 1979\n\nLandsberger, K.: _William Steinitz, Chess Champion_ , McFarland & Co, Jefferson\/London 1995\n\nLawson, D.: _The Inner Game_ , Macmillan, London 1993\n\nLind\u00f6rfer, K.: _Das rororo-Schachbuch von A-Z_ , Rowohlt-Verlag, Reinbek bei Hamburg 1984\n\nMachhatschek, H.: _Der erste Revanchekampf um die Schachweltmeisterschaft, Botwinnik-Smyslow 1958_ , Verlag Das Schach-Archiv, Rattmann, Hamburg 1962\n\nMichaltschischin, A.: _Das erste K.u.K.-Match in Moskau 1984\/85_ in: Schach 11\/2009, p. 48 ff.\n\nMichaltschischin, A.\/Stetsko, O.: _K\u00e4mpfen und Siegen mit Magnus Carlsen_ , Edition Olms, Z\u00fcrich 2012\n\nMinckwitz, J.: _Der Entscheidungskampf zwischen W. Steinitz und J. H. Zukertort um die Meisterschaft der Welt_ , Adolf-Roegner Schachverlag, Leipzig 1886. Reprint Jens-Eric Rudolph Verlag, Hamburg 2010\n\nM\u00fcller, H.\/Pawelczak, A.: _Schachgenie Aljechin, Mensch und Werk_ , Siegfried Engelhardt Verlag, Berlin 1962\n\nM\u00fcnninghoff, A.: _Max Euwe, The Biography_ , New in Chess, Alkmaar 2001\n\nNikitin, A.: _Mit Kasparow zum Schachgipfel_ , Sportverlag, Berlin 1991\n\nNilolaiczuk, L.: _Schachweltmeisterschaft 87_ , Karpow-Kasparow, Beyer-Verlag, Hollfeld 1988\n\nRoschal, A.\/Karpow, A.: _Schach mit Karpow_ , Mosaik-Verlag, M\u00fcnchen 1977\n\nPfleger, H.\/Borik O.: _Schach-WM '81 Karpow-Kortschnoi_ , Falken-Verlag, Niederhausen 1981\n\nPfleger, H.\/Borik O.\/Kipp-Thomas, M.: _Schach-WM '87: Kasparow\/Karpow_ , Falken-Verlag, Niederhausen\/Ts 1987\n\nSchonberg, H.: _Die Gro\u00dfmeister des Schachs_ , Moewig-Verlag, M\u00fcnchen 1982\n\nSchultz, D.: _Chessdon_ , Chessdon Publishing, Boca Raton 1999\n\nSeirawan, Y.: _Chess Duels_ , My Games with the World Champions, Everyman, London 2010\n\nSosonko, G.: _Russian Silhouettes_ , New in Chess, Alkmaar 2001\n\nSosonko, G.: _The Reliable Past_ , New in Chess, Alkmaar 2003\n\nSosonko, G.: _The World Champions I Knew_ , New in Chess, Alkmaar 2013\n\nSoltis, A.: _Frank Marshall, United States Chess Champion_ , McFarland & Co, Jefferson\/London 1994\n\nSoltis, A.: _Soviet Chess 1917-1991_ , McFarland & Co, Jefferson\/London 2000\n\nSoltis, A.: _Mikhail Botvinnik_ , McFarland & Co., Jefferson\/London 2014\n\nStolze, R.: _Umk\u00e4mpfte Krone \u2013 Die Duelle der Schachweltmeister von Steinitz bis Kasparow_ , Sportverlag, Berlin 1992\n\nSuetin, A.: _Stunde der Sekundanten_ , Verlag Bock & K\u00fcbler, Berlin 1995\n\nSpanier, D.: _Total Chess_ , Sphere Books, London 1984\n\nTal, M.: _The Life and Games of Mikhail Tal_ , Cadogan Chess, London 1979\n\nTopalov, V.\/Ginchev, Z.: _Topalov-Kramnik, 2006 World Chess Championship, On the Edge in Elista_ , Russell Enterprises, Inc., Miltford 2007\n\nvan Reek, J.: _Grand Strategy, 60 Games by Boris Spassky_ , Margraten 2000\n\nVasiliev, V. L.: _Tigran Petrosjan, His Life and Games_ , Batsford, London 1974\n\nVidmar, M.: _Goldene Schachzeiten_ , Walter de Gruyter & Co., Berlin 1961\n\nvon Cziffra, G.: _Der Kuh im Kaffeehaus_ , Ullstein, Frankfurt\/M. 1993\n\nWolff, P.: _Kasparow versus Anand, The inside Story of the 1995 World Chess Championship Match_ , H3 Publications, Cambridge, Massachusetts 1996\n\n**Magazines**\n\n_New in Chess_ , Alkmaar\n\n_Schach_ , Excelsior-Verlag, Berlin\n\n_Karl \u2013 das kulturelle Schachmagazin_ , Thematical issues on World Championships (4\/2008, 2\/2010, 2\/2012)\n\n**Digital sources**\n\n_ChessBase Mega Database 2015_\n\n**Web publications**\n\nBill Wall, _<http:\/\/www.geocities.com\/SiliconValley\/Lab\/7378\/alllinks.htm>_\n\nHarald Ball\u00f3: _www.ballo.de_\n\nCalifornia Chess Articles: _<http:\/\/www.chessdryad.com\/articles\/index.htm>_\n\nChess Archeology, _www.chessarch.com_\n\nChessBase: _www.chessbase.com_\n\nChess Caf\u00e9: _www.chesscafe.com_\n\nChessgames: _www.chessgames.com_\n\nChessVibes: _www.chessvibes.com_\n\nEdward Winter: Chessnotes, _<http:\/\/www.chesshistory.com\/winter\/>_\n\nFIDE: _www.fide.com_\n\nMark Week: _<http:\/\/chessforallages.blogspot.de\/>_\n\nThe Week in Chess: _www.theweekinchess.com\/_\n\nWikipedia: _www.wikipedia.org_\n\nWhyChess: _<http:\/\/whychess.com\/>_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Al Haines\n\n\n\n\n\n\n\n\n\n\nENGLISH LANDS LETTERS AND KINGS\n\n\n\n\nQueen Anne and the Georges\n\n\nBY\n\nDONALD G. MITCHELL\n\n\n\n\nNEW YORK\n\nCharles Scribner's Sons\n\nMDCCCXCVII\n\n\n\n\nCopyright, 1895, by\n\nCHARLES SCRIBNER'S SONS\n\n\n\nTROW DIRECTORY\n\nPRINTING AND BOOKBINDING COMPANY\n\nNEW YORK\n\n\n\n\nENGLISH LANDS LETTERS AND KINGS\n\n_By Donald G. Mitchell_\n\n\nI. from Celt to Tudor\n\nII. From Elizabeth to Anne\n\nIII. Queen Anne and the Georges\n\nIV. The Later Georges to Victoria\n\n_Each 1 vol., 12mo, cloth, gilt top, $1.50_\n\n\nAMERICAN LANDS AND LETTERS\n\nFrom the Mayflower to Rip Van Winkel\n\n_1 vol., square 12mo, Illustrated, $2.50_\n\n\n\n\n_LETTER OF DEDICATION_\n\n[To Mrs. Grover Cleveland.]\n\nMY DEAR MADAM:\n\n_Many bookmakers of that early Georgian period covered by this little\nvolume eagerly sought to dignify their opening pages with the name and\ntitles of some high-placed patron or patroness.  It is not, my dear\nMadam, to revive this practice that I have asked permission to inscribe\nthis little book to so worthy an occupant of the Presidential Mansion;\nbut, rather, I have had in mind the courteous reception which--while\nyet an inmate of a college on the beautiful banks of Cayuga Lake--you\nonce gave to some portions of the literary talk embodied in these\npages; and remembering, furthermore, the unswerving dignity, and the\nunabating womanly gentleness by which you have conquered and adorned\nthe trying conditions of a high career, I have wished to add my\napplause (as I do now and here) for the grace and kindliness which have\nennobled your life, and made us all proud of such an example of\nAmerican womanhood._\n\n_Very respectfully yours,_\n  Don^d. G. Mitchell.\n\n_Edgewood, June,_ 1895.\n\n\n\n\n{vii}\n\n_CONTENTS._\n\n\nCHAPTER I.\n\n                                                             PAGE\n\n  An Irish Bishop, . . . . . . . . . . . . . . . . . . . . .    3\n  A Scholar, . . . . . . . . . . . . . . . . . . . . . . . .    9\n  Two Doctors, . . . . . . . . . . . . . . . . . . . . . . .   12\n  Lady Wortley Montagu,  . . . . . . . . . . . . . . . . . .   21\n  Alexander Pope,  . . . . . . . . . . . . . . . . . . . . .   30\n  His Poetic Methods,  . . . . . . . . . . . . . . . . . . .   35\n  The Rape of the Lock,  . . . . . . . . . . . . . . . . . .   39\n  Pope's Homer, and Life at Twickenham,  . . . . . . . . . .   43\n  His Last Days, . . . . . . . . . . . . . . . . . . . . . .   48\n\n\nCHAPTER II.\n\n  From Stuart to Brunswick,  . . . . . . . . . . . . . . . .   53\n  Samuel, Richardson,  . . . . . . . . . . . . . . . . . . .   62\n  Harry Fielding,  . . . . . . . . . . . . . . . . . . . . .   67\n  Poet of the Seasons, . . . . . . . . . . . . . . . . . . .   73\n  Thomas Gray, . . . . . . . . . . . . . . . . . . . . . . .   79\n  A Courtier,  . . . . . . . . . . . . . . . . . . . . . . .   83\n  Young Mr. Johnson, . . . . . . . . . . . . . . . . . . . .   88\n\n\n{viii}\n\nCHAPTER III.\n\n  Johnson and Rasselas,  . . . . . . . . . . . . . . . . . .  104\n  The Painter and the Club,  . . . . . . . . . . . . . . . .  108\n  Some Old Club-Men, . . . . . . . . . . . . . . . . . . . .  113\n  Mr. Boswell, . . . . . . . . . . . . . . . . . . . . . . .  118\n  Gibbon,  . . . . . . . . . . . . . . . . . . . . . . . . .  122\n  Oliver Goldsmith,  . . . . . . . . . . . . . . . . . . . .  130\n  The Thrales and the End, . . . . . . . . . . . . . . . . .  135\n\n\nCHAPTER IV.\n\n  A Scottish Historian,  . . . . . . . . . . . . . . . . . .  145\n  A Pair of Poets, . . . . . . . . . . . . . . . . . . . . .  157\n  Miss Burney, . . . . . . . . . . . . . . . . . . . . . . .  164\n  Hannah More, . . . . . . . . . . . . . . . . . . . . . . .  171\n\n\nCHAPTER V.\n\n  King George III.,  . . . . . . . . . . . . . . . . . . . .  181\n  Two Orators, . . . . . . . . . . . . . . . . . . . . . . .  188\n  An Orator and Playwright,  . . . . . . . . . . . . . . . .  195\n  The Boy Chatterton,  . . . . . . . . . . . . . . . . . . .  202\n  Laurence Sterne, . . . . . . . . . . . . . . . . . . . . .  211\n\n\nCHAPTER VI.\n\n  Macpherson and other Scots,  . . . . . . . . . . . . . . .  221\n  George Crabbe, . . . . . . . . . . . . . . . . . . . . . .  231\n  William Cowper,  . . . . . . . . . . . . . . . . . . . . .  239\n  His Later Life,  . . . . . . . . . . . . . . . . . . . . .  249\n\n\n{ix}\n\nCHAPTER VII.\n\n  Parson White,  . . . . . . . . . . . . . . . . . . . . . .  259\n  A Hampshire Novelist,  . . . . . . . . . . . . . . . . . .  265\n  Old Juvenilia, . . . . . . . . . . . . . . . . . . . . . .  271\n  Miss Edgeworth,  . . . . . . . . . . . . . . . . . . . . .  277\n  Some Early Romanticism,  . . . . . . . . . . . . . . . . .  281\n  Vathek,  . . . . . . . . . . . . . . . . . . . . . . . . .  285\n  Robert Burns,  . . . . . . . . . . . . . . . . . . . . . .  291\n\n\nCHAPTER VIII.\n\n  A Banker Poet, . . . . . . . . . . . . . . . . . . . . . .  301\n  Coleridge, . . . . . . . . . . . . . . . . . . . . . . . .  309\n  Charles Lamb,  . . . . . . . . . . . . . . . . . . . . . .  319\n  Wordsworth,  . . . . . . . . . . . . . . . . . . . . . . .  327\n  His Poems, . . . . . . . . . . . . . . . . . . . . . . . .  330\n  Personal History,  . . . . . . . . . . . . . . . . . . . .  337\n\n\n\n{1}\n\n_ENGLISH LANDS, LETTERS, & KINGS._\n\n\n\nCHAPTER I.\n\nWe open in this book upon times--belonging to the earlier quarter of\nthe eighteenth century--when, upon the Continent of Europe, Peter the\nGreat was stamping out sites for cities in the bogs by the Finland\ngulf--when that mad-cap Swedish King Charles XII. was cutting his\nbloody swathe through Poland--when Louis XIV., tired at last of wars,\nand more tired of Marlborough, was nearing the end of his magnificent\ncareer, and when King Mammon was making ready his huge bloat of the\nMississippi Bubble for France and of the South Sea Company for England.\n\n{2}\n\nQueen Anne, that great lady of the abounding ringlets--so kindly and so\nweak--was now free from the clutch of Sara of \"Blenheim\"; and veering\nsometimes, under Harleyan influences, toward her half-brother the\n\"Pretender;\" and other times under persuasion of such as Somers,\nfavoring her cousins of Hanover.\n\nThe visitor to London in those times could have taken the \"Silent way\"\nalong the river--a shilling for two oarsmen and sixpence for a\n\"scull\"--from the Bridge to Limehouse; or he might encounter, along the\nStrand, sooty chimney sweepers and noisy venders of eggs and butter,\nwith high-piled baskets upon their heads.  Sir Roger de Coverley coming\nto town--if we may believe Addison--cannot sleep the first week by\nreason of the street cries; while Will Honeycomb, on the other hand,\nlikens these cries to songs of nightingales: always and everywhere this\ndifference of ear, between those who love the country and those who\nlove the towns!\n\nThere were lumbering hackney cabs in London streets to be hired at ten\nshillings a day (of twelve hours) for those who preferred this to the\n\"Silent {3} way\"; and there were grand coaches for those who could pay\nfor such display; evidences of wealth were growing year by year.  The\nVenetian Republic, now in its last days of power, made a brave if false\nshow upon London streets in those times.  Luttrel[1] says, under date\nof May, 1707:--\n\n\n\"Yesterday the V^n ambassadors made their public entry thro' the city\nto Somerset House in great state and splendor; their coach of state\nembroidered with gold, and the richest that ever was seen in England:\nThey had two with 8 horses, and eight with 6 horses, trimmed very fine\nwith ribbons; 48 footmen in blue velvet covered with gold lace; 24\ngentlemen and pages on horseback with feathers in their hats, etc.\"\n\n\nDr. Swift, four years after, writes to Stella--\"The Venetian coach is\nthe most monstrous, huge, fine, rich, gilt thing I ever saw.\"\n\n\n_An Irish Bishop._\n\nIt could not have been more than two or three years after this sight of\nthe Venetian Coach that Dean Swift introduced to his friend Miss {4}\nVanhomrigh (Vanessa) a young protege of his, whom he had known at\nDublin, and who had made a great reputation there among thinkers, by an\ningenious _Theory of Vision_, and by his eloquent advocacy of an\nIdealism, which he believed would cut away all standing ground for the\nmaterialism that threatened Christian Faith.\n\n[Sidenote: Bishop Berkeley.]\n\nThis protege was George Berkeley[2]--afterward Dean and Bishop--a most\nengaging and winning person then and always.  Addison befriended this\nyoung philosopher, who wrote half a dozen papers for Steele's\n_Guardian_, with much of Steele's grace in them, and more than Steele's\nChristian earnestness.  He went over to the Continent in the wake of a\nBritish Ambassador--was four or five years there, variously employed,\nequipping himself in worldly knowledge, and came back to warn[3]\nEnglishmen against that extravagance and {5} greed for money, which had\nmade possible the South-Sea disaster.  New Yorkers might read the\nwarning with profit now.  For himself, he comes presently to the\nDeanship of Derry, and to a considerable legacy from that Miss\nVanhomrigh--the acquaintance of an hour--so impressed had she been by\nBerkeley's promise of good.  Nor was the promise ever belied.\n\nWith an altruism unusual then, and unusual now, he braved the loss of\nhis Deanship, and current friendships in England, and set his heart,\nhis energies, and his fortune upon a scheme for building up the English\ncolonies in America in ways of Christian living, and of learning.  Long\nbefore, the devout George Herbert had said that Religion was \"ready to\npass to the American Strand;\" and now Berkeley, fresh from the sight of\ndearth and decay in Europe, was earnest in the belief that Christian\ncivilization was to win its greatest coming conquests \"over seas.\"  His\nenthusiasms had, for once, carried him into verse, of which a prophetic\nrefrain has tingled in many an American ear:--\n\n  Westward the course of Empire takes its way!\n\n{6}\n\nThe nidus of the good Dean's hopes and schemes lay in a great college\nwhich was to be built up in the Summer Islands (Bermuda) where the air\n\"is perpetually fanned and kept cool by sea-breezes.\"  But his\nstepping-stone on the way thither was Rhode Island; and for the harbor\nof Newport he sailed, with a few friends, and a newly married wife in\nthe year 1728, after long and weary waiting for a grant, which at last\nis made good on parchment, but never made good in money.\n\n[Sidenote: Berkeley at Newport.]\n\nYet he has faith; and for nearly three years lingers there at his farm\nof Whitehall (the old house still standing), within sound of the surf\nthat breaks upon the ribbed and glistening sands of Newport beaches.\nThe winter is not so mild as in England, but he \"has seen colder ones\nin Italy.\"  Possibly it may be well to set up the college in Newport\nrather than the Summer Islands--when the grant comes: but the grant\ndoes _not_ come.  He makes friends of the farmers about him--of the\nQuakers, the Methodists; sometimes he preaches at Trinity Church (still\nthere), and his sermons are unctuous with the broadest and most liberal\nChurchism: \"Sad,\" he says in one, \"that {7} Religion, which requires us\nto love, should become the cause of our hating one another.\"  He\ncorresponds with Samuel Johnson, of Stratford, Ct.;[4] also, possibly,\nwith Mr. Jonathan Edwards, not as yet driven away into the wilds of\nWestern Massachusetts, by theologic contumacies, from his pleasant\nNorthampton home.  In the hearing of the pleasant lapse of the waters\nupon the beaches--while he waits--the Dean sets himself to that\npleasant, curious writing of _The Minute Philosopher_ in which he\nadroitly parries thrusts with the whole tribe of Free Thinkers, and\nsublimates anew his old and cherished theory--that the spiritual\napprehension of material things is the only condition (or cause) of\ntheir being.\n\nChildren are born to him--and death winnows his small flock--while he\nwaits.  John Smibert, who was fellow-voyager with him, painted that\nlittle family of the Dean, and the picture is now in possession of Yale\nCollege.  At last, in despair of receiving the royal grant, he goes\nback with his {8} family to England (1731).  Many of his books,[5] and\neventually his Whitehall farm, were bestowed upon Yale; and in that\nlively institution year after year, there be earnest students who\ncontend still for Berkeley scholarships and Berkeley prizes; while the\nname of the good Dean is still further kept in American remembrance, by\nthat noble site of a Great Pacific University, which on the Californian\nshores, looks through a Golden Gate to a pathway still bearing\n\"Westward.\"\n\nWe may well believe that the Dean was disheartened by the breaking\ndown--through no fault of his own--of the great scheme and hope of his\nlife.  But he found friendly hands and hearts upon his return to\nEngland.  Through the influences of Queen Caroline (consort of George\nII.) he was given the bishopric of Cloyne--seated among the heathery\nhills which lie northward of the harbor of Queenstown.  All the poor\npeople of that region loved him: and who did not?\n\n{9}\n\nHe was never so profound a thinker, as he was ingenious, subtle, and\nacute.  Though his philosophies all were over-topped by his sweet\nhumanities,[6] yet American students may well cherish his memory, and\nkeep his _Alciphron_--if not his _Hylas and Philonous_--upon their\nbook-rolls.\n\n\n_A Scholar._\n\n[Sidenote: Richard Bentley]\n\nIt is certain that in your forays into the literature of these\ntimes--if made with any earnestness--you will come upon the name of Dr.\nBentley;[7] if nowhere else, then attached to critical footnotes at the\nbottom of books.\n\nHis demolition of the claims, long maintained by an older generation of\nscholars, respecting certain _Epistles of Phalaris_, commanded\nattention {10} at an early stage of his career, and showed ability to\ncross swords, in a scholastic and bitter way, with such men as\nAtterbury and Boyle; and--if need were--with such others as Sir William\nTemple and Dr. Swift.\n\nAs early as 1700 he had come to the mastership of Trinity College,\nCambridge (where a portrait of him by Thornhill now hangs in the\nMaster's Lodge), a proud position--made prouder by his large\nhospitalities.  He had a sensible wife, courteous \"for two\"--as many\nscholars' wives have need to be--and two daughters; one of whom\ninheriting the father's sharp tongue, made a good many young fellows of\nthe college sing; and made some of them sigh too--marrying at last a\ncertain young Cumberland, who became the father of Richard Cumberland,\nthe poet and dramatist.[8]\n\nSome small chronicler tells us of his preference for port over claret;\nindeed he loved all intense things, rather than things diluted, and was\ninaccessible to those finer, milder, delicater {11} graces--whether of\nwine or poetry--which ripen under long reposeful workings.  I spoke of\na portrait of him in the Master's Lodge; there was another in Pope's\n_Dunciad_--not so flattering:\n\n  \"The mighty scholiast, whose unwearied pains\n  Made Horace dull, and humbled Milton's strains;\n  Turn what they will to verse, their toil is vain,\n  Critics like me shall make it prose again.\"\n                                  --Lib. iv., 211 _et seq._\n\n\n[Sidenote: Bentley's scholarship]\n\nHe left no great work; yet what he did in lines of classical criticism\ncould not by any possibility have been better done by others.  He\nsupplied interpretations--where the world had blundered and\nstumbled--which blazed their way to unquestioned acceptance.  He\nmastered all the difficulties of language, and wore the mastership with\na proud and insolent self-assertion--a very Goliath of learning, with\nspear like a weaver's beam, and no son of Jesse to lay him low.  One\nwishing to see his slap-dash manner and his amazing command of\nauthorities should read the _Dissertation on Phalaris_; not a lovable\nman surely, but prince of all schoolmastery lore: and how rarely we\nlove the schoolmaster!  When you meet with that name of {12} Bentley\nyou may safely give it great weight in all scholarly matters, and not\nso much in matters of taste.  Trust him in foot-notes to Aristophanes\n(a good mate for him!) or to Terence; trust him less in foot-notes to\nMilton,[9] or even Horace (when he leaves prosody to talk of rhythmic\n_susurrus_).  You will think furthermore of this Dr. Bentley as living\nthrough all his fierce battles of criticisms and of college mastership\nto an extreme old age, and into days when Swift and Pope and Steele and\nAddison were all gone--a gray, rugged, persistent, captious old man,\nwith a great, full eye that looked one through and through, and with a\nshort nose, turned up--as if he always scented a false quantity in the\nair.\n\n\n_Two Doctors._\n\nWe approach a doctor now as mild and gentle as Bentley was irritable\nand pugnacious; a man not {13} often enrolled among literary veterans;\ntreated with scorn, maybe, by the professional critics; and yet this\nname now brought to your attention is I think, tenderly associated with\nNew Englanders' earliest recollections of rhyme or verse; and it is\nspecially these literary firstlings of the memory that it is well for\nus to trace and hold in hand.  Let us listen for a moment to that old\ncradle hymn:\n\n  \"Hush, my dear, lie still and slumber,\n    Holy angels guard thy bed;\n  Heavenly blessings without number\n    Gently falling on thy head.\"\n\nHow the quaint, simple melody lingers yet, coming from far-away times,\nwhen it drifted over hundreds of New England homes, which as yet knew\nnot _Pinafore_ nor Mr. Sankey!\n\n[Sidenote: Isaac Watts]\n\nIt is of Dr. Watts's[10] familiar name that I speak: he was the son of\na lodging-house keeper in Southampton--in which city a Watts memorial\nHall was dedicated as late as 1875.  Being a {14} dissenter, he was\ndebarred the advantages of a university education, but he taught\ndissenters how to put grace into their hymns and sermons; and without\nbeing a strong logician, he put such clearness into his _Treatise upon\nLogic_ as to carry it for a time into the curriculum of Oxford.\n\nOur American poet, Bryant, had great admiration for the familiar\nWatts's version of the 100th Psalm:--\n\n  We'll crowd thy gates with thankful songs,\n    High as the heavens our voices raise;\n  And earth, with her ten thousand tongues,\n    Shall fill thy courts with sounding praise.\n\nAnd what pious tremors shook the air, when the country choirs in New\nEngland meeting-houses lifted up their voices to the old hymn,\ncommencing:--\n\n  There is a land of pure delight!\n\n\nI don't know but these bits of moral music may have been hustled out\nfrom modern church primers for something more aesthetic; but I am sure\nthat a good many white-haired people--of whom I hope to count some\namong my readers--are carried back pleasantly by the rhythmic jingle of\nthe good Doctor to those child days when hopes were {15} fresh, and\nholidays a joy, and summers long; and when flowery paths stretched out\nbefore us, over which we have gone toiling since--to quite other music\nthan that of Dr. Isaac Watts.  And if his songs are gone out of our\nfine books, and have fallen below the mention of the dilettanti\ncritics, I am the more glad to rescue his name, as that of an honest,\ndevout, hard-working, cultivated man who has woven an immeasurable deal\nof moral fibre into the web and woof of many generations of men and\nwomen.\n\nBy the generosity of a friend he was endowed with all the privileges of\na beautiful baronial home (Abney Park) where he lived for thirty odd\nyears--reaching almost four score--never forgetting his simplicities,\nhis humilities, his faith, his sweet humanities, and never having done\nharm, or wished harm, to any of God's creatures; and this cannot be\nsaid of many who preach, and of many of whom we are to talk.\n\n[Sidenote: Edward Young.]\n\nThere was another clerical poet of less private worth, who had a very\ngreat reputation early in the eighteenth century.  Fragments of his\nsombre- and magniloquent _Night Thoughts_ {16} are still\nfrequently encountered in Commonplace Books of Poetry; while some of\nhis picturesque or full-freighted lines, or half lines, have passed\ninto common speech; such as--\n\n    \"The undevout astronomer is mad;\"\n\n  \"Tired Nature's sweet restorer, balmy sleep;\"\n\n    \"Procrastination is the thief of time.\"\n\n\n[Sidenote: Doctor Young.]\n\nYou will recognize these as old acquaintances; and you are to credit\nthem to Dr. Edward Young,[11] who was born about two hundred years ago\ndown in Hampshire, son of a father who had been Chaplain to King\nWilliam III.  He was an Oxford man, lived a wild life there--attaching\nhimself to a fast young Duke of Wharton, who led him into many awkward\nscrapes--and developing an early love, which clung by him through life,\nfor attaching himself to great people.  He wrote plays which were not\ngood, and odes which were worse than the plays, but touched off with\nlittle jets of terrific adulation:--\n\n  \"To poets, sacred is a Dorset's name,\n  Their wonted passport thro' the gates of fame;\n\n{17}\n\n  It bribes the partial reader into praise\n  And throws a glory round the sheltered lays.\"\n\nAnd so on--to a Compton, a Lady Germaine, a Duke, in nauseous\nsuccession.  In fact, he seemed incapable of using any colors but gaudy\nor resplendent ones, and is nothing if not exaggerated, and using heaps\nof words.  Would you hear how he puts Jonah into the whale's mouth?--\n\n  \"As yawns an earthquake, when imprisoned air\n  Struggles for vent, and lays the centre bare,\n  The whale expands his jaws' enormous size.\n  The prophet views the cavern with surprise,\n  Measures his monstrous teeth, afar descried,\n  And rolls his wondering eyes from side to side,\n  Then takes possession of the spacious seat\n  And sails secure within the dark retreat.\"\n\n\nThis is from his poem of the _Last Day_, which has some of his best\nwork in it.  He wrote flattering words of Addison, which Addison could\nnot return in the same measure.  He had acquaintance with Pope, with\nSwift, with Lady Mary Montagu, and others whom he counted worth\nknowing.  He made a vain run for Parliament, and ended by taking church\norders somewhat late in life--staying {18} one of his plays,[12] which\nwas just then in rehearsal, as inconsistent with his new duties.  He\nmarried the elegant widowed daughter of an earl, who died not many\nyears thereafter; and from this affliction, and his brooding over it,\ncame his best-known poem of _Night Thoughts_.  It had great currency in\nEngland, and was admired, and translated, and read largely upon the\nContinent.  For many a year, a copy of Young's mournful, magniloquent\npoem, bound in morocco and gilt-edged, was reckoned one of the most\nacceptable and worthy gifts to a person in affliction.\n\n[Sidenote: Young's Night Thoughts.]\n\nBut of a surety it has not the same hold upon people in this century\nthat it had in the last.  There are eloquent passages in it--passages\nalmost rising to sublimity.  His love of superlatives and of wordy\nexaggerations served him in good stead when he came to talk of the\nshortness of time, and the length of eternity, and the depth of the\ngrave, and the shadows of death.  Amidst these topics he moved on the\ngreat sable pinions of his muse with {19} a sweep of wing, and a\nsteadiness of poise, that drew a great many sorrowing and pious souls\nafter him.\n\nThis is his Apostrophe to Night:\n\n          \"O majestic Night!\n  Nature's great ancestor!  Day's elder born!\n  And fated to survive the transient sun!\n  By mortals and immortals seen with awe!\n  A starry crown thy raven brow adorns,\n  An azure zone thy waist; clouds in Heaven's loom\n  Wrought through varieties of drapery divine\n  Thy flowing mantle form, and heaven throughout\n  Voluminously pour thy pompous train.\"\n\n\nThere is no well-considered scheme or method in his poems; but his\naugust sorrowing and devout meditations, clothed in a great pomp of\nlanguage, chase each other over his mind, as vagrant high-sweeping\nclouds chase over the sky.  You may watch and follow them in dreamy\nhours, with a languid pleasure; but a real sorrow, or a real task do\nnot, I think, find much help in them.\n\nDr. Young believed, in the moodiness of his grief, that he was going to\nbid adieu to the world; but he did not; we find him back at court long\nafter the funeral bells had sounded in his verse:--back {20} there too,\nin search of offices of some sort; bowing obsequiously to those who had\ngifts in their hands.\n\nGood Mrs. Hannah More tells us that being on one occasion at a\nParliamentary party, where some volumes of original letters were shown,\nshe was specially anxious to see one of her dear Dr. Young, for whose\n_Night Thoughts_ she expressed enthusiastic admiration.  Her anxiety\nwas gratified, and she adds that she had\n\n\n\"the mortification to read the most fawning, servile, mendicant letter\nthat was perhaps ever penned by a clergyman, imploring the mistress of\nGeorge II. to exert her interest for his preferment.\"\n\n\nI do not like to tell such things to those who admire the poet; but we\nare after the truth--first of all.  A curious mixture he was, of\nfrugality and piety--of love for reputation and emotional religion.  He\nessayed the writing of some of his tragic episodes in a dark room,\n\"with a candle stuck in a skull;\" and such love of claptrap abode with\nhim and qualified most of his work.\n\n_Night Thoughts_ has some unforgetable things in it: there is a lurid\nsplendor in many of the {21} lines, and great imaginative range.  But\nhis was an imagination not chastened by a severe taste or held in check\nby the discretions of an elevated and cultured judgment.  Upon the\nwhole, I have more respect for the memory of Dr. Watts, than for the\nmemory of Dr. Young.\n\n\n_Lady Wortley Montagu._\n\n[Sidenote: Mary Wortley Montagu.]\n\nIt is a lady that I next introduce; a very much admired lady in her\nday; and much admired by many even now.  She was correspondent at one\ntime of Dr. Young, as well as of Pope, Steele, and Swift (who was one\nof the few men she feared).  She knew and greatly admired Congreve, had\nfree entree to the palace in time of George I., could and did translate\nEpictetus before she was turned of twenty, and wrote letters to her\ndaughter, Lady Bute, that were long held up to young ladies as patterns\nof epistolary work: of course it is Lady Mary Montagu,[13] of whom I\nspeak.\n\n{22}\n\n[Sidenote: Lady Mary Montagu.]\n\nShe was born at Thoresby Park, a little northward of Sherwood Forest in\nNottingham; was the petted daughter of the Earl of Kingston, and he\nintroduced her (as the story runs) when only eight years old to that\nfamous Kit-Kat Club, which held its summer sessions out by Hampstead\nHeath; and the applause that greeted her beauty and sprightliness\nthere, very likely fastened upon her that greed for public triumphs\nwhich clung to her all her life.  She presided at her father's table,\nwas taught in Greek, Latin, French, Italian; was full of\naccomplishments, and at twenty-one fell in with Mr. Montagu, similarly\naccomplished, whom she had a half mind to marry.  Her father, however,\nhad other views, against which the self-willed young lady rebelled; she\nhad, however, her hesitations--sometimes flinging a new bait to Mr.\nMontagu and then showing a coquettish coolness.  Finally, between two\ndays, she decides; orders Mr. Montagu to have his chaise and four in\nreadiness and makes a runaway match of it.\n\nTheir life for some time is in a suburb of London; where the Lady Mary\nchafes at the retirement, {23} in a way which is not very agreeable to\nMr. Montagu and nettles him; and the nettles creep into their future\ncorrespondence.  But her husband being appointed (1716) ambassador to\nConstantinople, her Ladyship sets off delightedly with a retinue of\nattendants to the shores of the Bosphorus; and writes thence and on her\nway thither, letters full of piquancy and charm.\n\nTo the distinguished Mr. Pope, who has addressed her in almost a\nlover's strain, she says:\n\n\n\"'Tis certain that I may, if I please, take the fine things you say to\nme for wit and raillery; and, it may be, it would be taking them right.\nBut I never in my life was half so well disposed to believe you in\nearnest as I am at present.\"\n\n\nAnd thereupon she goes on to describe a Sunday at the opera in the\ngarden of the Favorita at Vienna.\n\nFirst of all Englishwomen, she had her son inoculated for the\nsmall-pox; this method of prevention being practised at that time in\nportions of Turkey.  Succeeding in this, she brought the method, and\nstrong advocacy of it, back to England with her.  It was a bold thing\nto do, and she {24} always loved boldnesses.  It was a humane thing to\ndo, and her humanities were always active.  The medical professors\nlooked doubtingly upon it; even the clergy preached against it as\ncontravening the intentions of Providence--just as some zealots, fifty\nyears ago, declared against the employment of chloroform and other\nanaesthetics.  But Lady Mary succeeded in her endeavors, and inoculation\nbecame shortly after an approved and adopted practice.\n\nOn the return from the Turkish embassy Mr. Montagu, perhaps at the\ninstance of Pope, bought a home for her at Twickenham, a delightful\nsuburb of London, where the poet was then residing, and at the zenith\nof his fame.  His poetic worship at her shrine was renewed with all the\nold ardor.  He gave Sir Godfrey Kneller a commission to paint her\nportrait in Turkish dress, with which she had done great execution at\ncourt balls.\n\n\"The picture,\" says Pope, in a letter to her, \"dwells really at my\nheart, and I have made a perfect passion of preferring your present\nface to your past.\"\n\n{25}\n\nWhat the past had been we may infer from this bit of verse, written\nwhile she was in the East:\n\n  \"In vain my structures rise, my gardens grow,\n  In vain fair Thames reflects the double scenes\n  Of hanging mountains and of sloping greens.\n  Joy dwells not there; to happier seats it flies,\n  And only dwells where Wortley casts her eyes.\n  What are the gay parterre and checkered shade,\n  The morning bower, the evening colonnade,\n  But soft recesses of uneasy minds\n  To sigh unheard into the passing winds;\n  So the struck deer in some sequestered part\n  Lies down to die, the arrow at his heart;\n  There, stretched unseen, in coverts hid from day\n  Bleeds drop by drop and pants his life away.\"\n\n\nBut this worship is not for very long; there comes a quarrel, which is\nso sharp and bitter, and with such echoes in ode or satire, as to\nbecome the scandal of the neighborhood.\n\nWhat brought it about cannot be so distinctly told.  Lady Mary\npersisted in saying that the crippled sensitive poet had forgotten\nhimself to so impudent an avowal of love that she had repelled him with\na shout of laughter, and so turned his heart into gall.\n\nThat his heart was all gall toward her {26} thereafter there needed no\nproof beyond his stinging couplets; and though he denied her tale with\nunction, he never told a story of his own in respect to this affair\nwhich made _her_ character seem the worse, or _his_ the better.\n\nIn an evil hour her ladyship (who had written verse already, which for\nher fame's sake it were better she had never written), undertook, with\nthe aid of her friend Lord Hervey, to reply to the lampoons of Pope.\nThereupon the shrinking, keen-smarting poet made other burning verses,\nby which the Hervey and the Montagu were both put to the torture.  It\nmust have been uncomfortable weather for her ladyship at Twickenham in\nthose days.  True, Hervey, Peterborough, Bolingbroke, and many of the\ncourtiers were at her service; and she was a favorite of George I.--so\nfar as any respectable woman could be called a favorite of that gross\ncreature; but Pope's shafts of ridicule had a feather of grace about\nthem that carried them straight and far.  Mr. Montagu himself was a\nhusband who loved London and his coal-fields without her ladyship,\nrather better than Twickenham gardens _with_ her ladyship.\n\n{27}\n\nTwenty years of gay \"outing\" she lives, between London and its suburbs;\nhappy, yet not happy; courted and not courted.  She writes to her\nsister Lady Mar[14] in these times:\n\n\n\"Don't you remember how miserable we were in the little parlor at\nThoresby?  We then thought marrying would put us at once in possession\nof all we wanted....  One should pluck up a spirit and live upon\ncordials, when one can have no other nourishment.  These are my present\nendeavors, and I run about though I have five thousand pins and needles\nrunning into my heart.  I try to console myself with a small damsel\n[her daughter, afterward Lady Bute] who is at present everything I\nlike; but, alas, she is yet in a white frock.  At fourteen she may run\naway with the butler.\"\n\n\nAnd when this maiden in white had married (better than the mother dared\nhope), and her son, a vagrant, had gone out into the world and the\nnight, Lady Mary--believing in \"cordials\"--gathered her robes about\nher, and took her fading face into the blaze of the Continental cities.\n\nHer reputation for wit, and daring, and beauty has gone before her, and\nshe writes piquantly and with great complacency of the attentions and\n{28} greetings that meet her in Venice, Florence, and Milan.  The\nappetite for this life grows with feeding; so it becomes virtually a\nseparation from her husband, though cool, business-like letters\nregularly pass between them.  Her son, though grown up into an\n\"accomplished\" man, is a scoundrel--drifting about Europe; and when\nthey encounter the mother insists that he shall drop his name, and deny\nrelationship.\n\nTwenty-two years she lives in that Continental exile, writing all the\nwhile letters to her daughter, which she loved to compare with the\nletters of Madame de Sevigne.  They are witty and sparkling and have\npassed into a certain place in English literature, but they are not\nSevigne letters.  Toward the last of her residence abroad she bought an\nold ruinous palace in Lombardy, not far from Lago di Guarda, equipped\nthree or four of its rooms, and with a little bevy of servants, lived\nin retirement--busied with reading, with her ducks, her pigeons, and\nher garden.\n\nShe writes her daughter:\n\n\n\"The active scenes are over at my age; I indulge, with all the art I\ncan, my taste for reading.  If I _could_ confine it to {29} valuable\nbooks; they are almost as scarce as valuable men....  As I approach a\nsecond childhood I endeavor to enter into the pleasures of it....  I am\nreading an idle tale, not expecting wit or truth in it; and am very\nglad it is not metaphysics to puzzle my judgment, or history to mislead\nmy opinion.\"\n\n\nShe is well past sixty and has lost all her old graces when she falls\ninto this misanthropic spirit; has grown strangely neglectful of her\nperson too; she says that for eleven years now she has not looked in a\nmirror.[15]\n\nBut presently Mr. Montagu dies leaving an immense fortune; there are\nbusiness reasons demanding her return; so she brings back that\nshrunken, unseemly face, and figure of hers to London; takes a house\nthere and fills it with servants.  A cousin, speaking of a call upon\nher, says:\n\n\n\"It is like the Tower of Babel; a Hungarian servant takes your name at\nthe door, he gives it to an Italian, who delivers it to a Frenchman.\nThe Frenchman to a Swiss, and the Swiss to a Polander; so that by the\ntime you get to her ladyship's presence you have changed your name five\ntimes, without the expense of an Act of Parliament.\"\n\n{30}\n\nHorace Walpole pays her a visit, and says, \"she was old, dirty, tawdry,\nand painted.\"  But he did not like her: I do not think she liked him.\n\nCould it be that this old lady--past seventy--with her fine house and\nher polyglot of service and her flush purse, thought to call back the\nold trail of flatterers?  I do not know.  I know very well she did not,\nand that within a twelvemonth she died.\n\nThere is in Lichfield Cathedral a cenotaph representing Beauty weeping\nthe loss of her Preserver; it was placed there by some grateful person\nto perpetuate the memory of the Lady Mary's benevolence in introducing\ninoculation; and I think it is the only eulogy to be found on any\nmemorial tablet of this strange, witty, beautiful, indiscreet,\nstudious, unhappy, disappointed woman.\n\n\n_Alexander Pope._\n\n[Sidenote: Alexander Pope.]\n\nWe close our chapter with some mention of that proud, shy, infirm poet\nof whom we have caught shadowy glimpses in the story of Wortley {31}\nMontagu.  There are scores of little crackling couplets floating about\non the lips of people well known as Pope's.[16]\n\n  \"A wit's a feather and a chief's a rod,\n  An honest man's the noblest work of God.\"\n\n  \"Know then, this truth, eno' for man to know,\n  Virtue alone is happiness below.\"\n\n  \"Honor and shame from no condition rise,\n  Act well your part; _there_ all the honor lies!\"\n\n\nThese must be familiar; and your school must differ from most schools,\nif some of these or other such, from the same author, have not one time\ndone service as snappers at the end of a composition, or as a bit of\ndecoration in the middle of it.\n\nAll know, too, in a general way, that Pope was an infirm man, without\nperhaps a clear idea of what his infirmity may have been; some of those\n{32} fierce lampoons already alluded to, which went flying back and\nforth around the shades of Twickenham, speak of the poet as an ape, a\nhunchback, a monster.  The truth is that he inherited from his father a\nfeeble and crooked frame with some spinal weakness which did give a\nmeasure of excuse to the coarse and brutal satirists of those days.\nHis height was much below that of ordinary men, so that cushions or a\nhigher chair were always necessary at table to bring him to the level\nof his friends; his legs were thin and shrunken and he walked feebly;\nhis countenance was drawn and pinched; yet he had good features, with\nthe delicate complexion of a woman, and a great blue eye, full of\nexpression.  His toilette was always a serious affair for\nhim--specially when he went abroad or would appear at his best (as he\nalways wished to do)--involving the assistance of one or two attendants\nto adjust his paddings, his stays, his canvas jackets, and his twice\ndoubled hose.\n\nI have dwelt with more particularity upon his personal aspect, because\nit serves to explain, or at least largely to qualify, a great many\napparent mysteries in his social career.\n\n{33}\n\nHe was a London boy, born of Romish parents; his father being a small\ntrader in the city, but retiring, about the time of this weakly boy's\nbirth, to a home at Binfield--a country parish lying between Windsor\nand Reading, where they show now a grove of beaches which was a\nfavorite haunt of the boy poet.  He caught schooling in a hap-hazard\nway, as Romanists needed to do in those times; but had a quick, big\nbrain, that made up for many shortcomings in teachers.  Before twelve\nhe had his Latin with some Greek, and had written verse; and after that\nage was his own master--sucking literary sweets where he could find\nthem.\n\nBefore twelve, too, he had made many London visitations--partly to\nstudy French there and partly to find his way to Will's coffee-house,\nand catch sight of old John Dryden, then drawing near to the end of his\nworldly honors.  And this thin, white-faced, crippled boy looking\nstealthily up at the master, even then had wild ambitious dreams of the\nday when he too should have his dignities and lay down the law for\nEnglish letters.\n\nOut by Binfield he happened upon good friends.  {34} Among others a\nBlount family to which belonged two daughters Blount--sympathetic\ncompanions to him then and long afterward; scores of letters, too,\nthere were, to which now Teresa Blount and now Miss Patty Blount were\nparties: He seeming in those romantic days (upon the edge of Windsor\nForest) sometimes in love with one and sometimes the other; and they,\nin this mixing of letters getting probably as confused as he, and a\ngreat deal more vexed; and so came coldness and short-lived\nquarrelling, making one thing pretty sure--that when a young man or\nwoman begins to play with the different tenses of the verb \"I love,\" a\nsingle correspondent is much better than two.  However, his friendship\nwith Miss Patty Blount lasted his life out.\n\nAn old baronet of the neighborhood, who had been diplomat in James I.'s\nday, took a fancy to this keen-thoughted lad and made a companion of\nhim.  He came to know old Wycherly too, and scores of men about town;\neven Jacob Tonson, the famous publisher of those times, had written to\nPope before he was twenty, asking the privilege of printing certain\npastorals of his writing, which {35} had been handed about in the\nclubs; and thought them--what they really were--astonishing for their\nliterary finish.\n\n\n_His Poetic Methods._\n\n[Sidenote: Poetry of Pope.]\n\nBut young Mr. Pope does not think much of the pastorals, save as\nstepping-stones; they paved his way to a large acquaintance with the\nLondon wits; and it would seem that at one time he thought of living at\nthe dreadful pace of these gentlemen--in bottles and midnight routs;\nperhaps he tried it for a while; but his feeble frame could stand no\nsuch neck-breaking gallop.  He can, however, put more of wearisome\nelaboration and pains-taking skill to his rhymes than any of the\nverse-makers of his time.  He has by nature a mincing step of his\nown--different as possible from the long, easy lope of Dryden--and that\nstep he perfects by unwearied practice, and word-mongering, until it\ncomes to the wondrous ten-syllabled movement, which for polish, and\nrhythmic tric-trac is unmatchable.\n\nThe _Essay on Criticism_, _Windsor Forest_, and {36} the _Rape of the\nLock_, all belonged to those early years at Binfield, and I give a test\nof each; first, from the _Essay_:--\n\n  \"Where'er you find 'the cooling Western breeze,'\n  In the next line, it 'whispers through the trees:'\n  If crystal streams 'with pleasing murmurs creep,'\n  The reader's threatened (not in vain) with 'sleep;'\n  Then, at the last and only couplet fraught\n  With some unmeaning thing they call a thought,\n  A needless Alexandrine ends the song,\n  That, like a wounded snake, drags its slow length along.\"\n\n\nNext this bustling bit, from _Windsor Forest_:--\n\n  \"See, from the brake the whirring pheasant springs\n  And mounts exulting on triumphant wings.\n      *      *      *      *      *\n  Ah! what avail his glossy, varying dyes,\n  His purple crest, and scarlet-circled eyes,\n  The vivid green his shining plumes unfold,\n  His painted wings, and breast that flames with gold.\"\n\n\nAnd again, this, from the _Rape of the Lock_:--\n\n  \"Just then, Clarissa drew with tempting grace\n  A two-edged weapon from her shining case;\n  So ladies in romance assist their knight,\n  Present the spear, and arm him for the fight,\n  He takes the gift with reverence, and extends\n  The little engine on his fingers' ends;\n\n{37}\n\n  This just behind Belinda's neck he spread,\n  As o'er the fragrant steams she bends her head.\n  Swift to the lock a thousand sprites repair,\n  A thousand wings, by turns, throw back the hair;\n  And thrice they twitched the diamond in her ear,\n  Thrice she looked back, and thrice the foe drew near.\"\n\n\nAnd yet again--this worthier excerpt from the same dainty poem:--\n\n  \"Fair nymphs, and well-drest youths around her shone\n  But every eye was fixed on her alone.\n  On her white breast a sparkling cross she wore,\n  Which Jews might kiss, and infidels adore.\n  Her lively looks a sprightly mind disclose,\n  Quick as her eyes, and as unfixed as those;\n  Favors to none, to all she smiles extends;\n  Oft she rejects, but never once offends.\"\n\n\nTen pages of extracts would not show better his amazing attention to\ndetails--his quick eye--his gifts in word-craft, and his musical\nexploitation of his themes.  I know that this poet works in harness,\nand has not the free movement of one who gallops under a loose rein;\nthe couplets fetter him; may be they cramp him; but there is a blithe,\nstrong resonance of true metal, in the clinking chains that bind him.\nNo, I do not think that Pope is to be laughed out of court, in {38} our\nday, or in any day, because he labored at form and polish, or because\nhe loved so much the tingle of a rhyme; I think there was something\nelse that tingled in a good deal that he wrote and will continue to\ntingle so long as Wit is known by its own name.\n\nThe good word spoken for him in the _Spectator_--the great printed\nauthority in literary matters--brought him into more intimate\nassociation with the Literary Guild of that paper; he wrote for the\n_Spectator_ on several occasions.  An early contribution is that of\n1712 (November 10th), where he calls attention to the famous verses\nwhich the Emperor Adrian spoke on his death-bed; he says:--\n\n\n\"I was in company the other day with five or six men of learning, who\nagreed that they showed a gayety unworthy that prince in those\ncircumstances;\" and he quotes the lines:\n\n  Animula vagula, blandula\n  Hospes Comes que Corporis\n  Pallidula, rigida, nudula, etc.\n\n\"But,\" he says, \"methinks it was by no means a gay, but a very serious\nsoliloquy to his soul at the point of his departure.\"\n\n\n{39}\n\nAnd out of this comment and thought of Pope's, contributed casually (if\nPope ever did anything _casually_) to the _Spectator_, came by and by\nfrom the poet's anvil, that immortal hymn we all know,--\n\n  \"Vital spark of heavenly flame,\n  Quit, oh quit, this mortal frame;\n  Trembling, hoping, lingering, flying,\n  Oh the pain, the bliss of dying!\"\n\n\n_The Rape of the Lock._\n\n[Sidenote: Rape of the Lock]\n\nI cited two significant fragments from the _Rape of the Lock_, a poem\nbelonging to Pope's early period, and which is reckoned by most poets\nand critics,[17] as well as biographers, his masterpiece, and a\nbeautiful work of the highest literary art.  I recognize the superior\nauthority, but cannot share the exalted admiration; at least, it does\nnot beget such loving approval as brings one back again and again to\nits perusal.  It does not seem to me to furnish very inspiring reading.\n\n{40}\n\nThe setting of this little poem is not large; the story is of a stolen\nlock of hair, and of the resentments that follow; and if one might\nventure upon a synopsis of so delicate a feat of workmanship, it might\nrun in this way:--Belinda, the despoiled heroine, sleeps; sprites put\ndreams in her head and give warning of impending woe.  \"Shock\" (her\ndog) barks and wakes her; she betakes herself to her toilet--the\nfairy-fingered sylphs assisting:\n\n  Some thrid the mazy ringlets of her hair;\n  Some hang upon the pendants of her ear,\n\n--all pictured like carving on a cherry-stone.  At last, fully\nequipped, she goes to a fete upon the Thames; pretty glimpses of the\nriver scenes follow; a crazy baron covets a lock of Belinda's hair.\nThe zephyrs play; day fades; cards come; crowding sprites pile into the\ngame, and twist all into a fairy cable.  The covetous baron snips off a\nlock of Belinda's hair, while she bends over the tea-pot.  The nimble\nsylphs bring from the \"Cave of Spleen\" a stock of shrieks, and tears,\nand megrims.  Sir Plume (\"of amber snuff-box justly vain\") champions\nBelinda, and demands satisfaction of the {41} ravisher--which he does\nnot win; so the battle rages--\"Fans clap, silks rustle, and tough\nwhalebones crack,\" and in the hurly-burly the stolen lock gets wafted\ninto \"lunar spheres,\" and comet-like, closes the shining tale:\n\n  \"This lock the muse [thus] consecrates to Fame\n  And midst the stars inscribes Belinda's name.\"\n\n\nYet Belinda's sovereignty is of an ignoble sort; her tiara made up of\npins and pomades; indeed the women all are as small as the sylphs; toy\ncreatures, and creatures of toys; no nobility, in or about them; and\nvery much to make an honest, self-respecting woman of our time fling\ndown the silvery poem with a wearisome distaste.\n\nAll this is said with a thorough recognition of its art--its amazing\ndexterities of verse--its playful leaps of fancy--its bright shimmer of\nover-nature; and yet those gossamer gnomes seem to me like an\nintrusion; I cannot forget that they were an afterthought of Pope\nhimself; I cannot bring myself to think of the charming fairy-folk of\nFletcher, or of Drayton's _Nymphidia_, or of the _Midsummer Night's\nDream_ wallowing in pomades, {42} and straining at whalebone stays!\nThese live through an eternal frolic in the air; those--of the _Rape of\nthe Lock_--lie in a literary show-case, like a taxidermist's trophies.\n\nIn the sobered time of life, when the iris hues have only fitful play,\nI think a man goes away from these earlier poems of Pope (if he reads\nthem) with new zest, to those wonderful metric condensations of old\ntruths, which flash and burn along the lines of his moral essays.\nThere could be few more helpful rhetorical lessons, for boy or girl,\nthan the effort to pack some of Pope's stinging couplets, or decades of\nlines, into an equal number of lines in prose; the difficulties would\nbe great indeed and would vitalize the lesson; and the lesson, I think,\nwould be far fuller of profitable ends, than the old \"parsing\"\nexercise, and syntactic analysis and description of sentences according\nto the nomenclature of Mr. Lindley Murray or of Mr. Somebody-else.\n\n\n{43}\n\n_Pope's Homer, and Life at Twickenham_.\n\n[Sidenote: Homer of Pope.]\n\nNotwithstanding his much writing, Pope in those early days under the\nbeeches of Windsor forest, was not winning such financial rewards as\nhis friends thought he deserved.  The _Spectator_ did not pay much\nmoney for little poetic trifles--such as the _Messiah_; and Jacob\nTonson was the screw which some publishers are.  There can be no doubt\nthat the poet, with his fine tastes, felt the restraints of a limited\nincome; his old father, who perhaps did not carry sharp business habits\ninto his retirement, had been compelled to leave the country house of\nBinfield, and had gone over to a suburban street dwelling near to\nChiswick.  In this emergency, (if emergency it were,) was it not the\noddest thing in the world that his friends should have advised a\ntranslation of Homer?\n\nYet they did; and so this dauntless young fellow, not over-critical in\nhis Greek knowledge, but with an abounding sense of the marvellous\nbeauties that lay in the old Homeric hexameters, {44} sets about his\ntask; and after five years' toil accomplishes it in such a way as makes\nit probable that there can never be an English Homer that will quite\nmatch it.  There are juster ones; there are faithfuller ones; but not\none that has been so enduringly popular.  Steeping himself in the\nmythologies and the Trojan traditions, he has grafted thereupon his\nstock of British word-craft: Ajax, Achilles, and the rest range to\ntheir places in the martial clank of his couplets, with a life and\ncharm which, if not imbued with Homeric limpidities and melodies,\npossess an engaging picturesqueness that belongs to few long English\nepics.\n\nAnd the poem took: that trenchant Dean Swift strode into the ante-rooms\nof the great men of Court, and swore that he must have a hundred or a\nthousand pounds subscribed for the new Homer of Mr. Pope; and he got\nit; Mr. Pope was the fashion.\n\nUp to that time in the whole history of English literature there had\nbeen no such payment for literary wares as accrued to the author of the\nnew Homer--the sum reaching, for both Iliad and {45} Odyssey, some\nL9,000; with which the shrewd poet bought an annuity (cheaper then than\nnow) of some L500, and a long lease of the Twickenham house and\ngardens; where, thereafter, amidst his willows and his grottos, he\nlived until his death.\n\nThe house[18]--if indeed any part be now the same--has been built over\nand enlarged, and has a jaunty suburban villa pretension that does not\nlook Homeric; but the grotto, or tunnel, which he cut under the high\nroad running parallel with the Thames, and through which he might pass\nunobserved from garden to garden and from his house to the river, is\nstill to be seen there; and trees of his planting still hang their\nlimbs over the pretty greensward that goes down in gentle <DW72> to the\nThames banks.  He put the same polish upon his grounds he did upon his\nverse: his grotto flashed with curious spars, glass jewels, and\nprismatic tinted shells; his walks were decorously {46} paved and\nrolled and his turf shorn to a nicety.  He entertained there in his\nthrifty way, watching his butler very sharply, and by reason of his\ninfirmities, was very measured in his wine-drinking.  Swift, who used\nto come and pass days with him, may have made the glasses jingle: and\nthere were other worthy friends who, when they came for a dinner, kept\nthe poet in a tremor of unrest.  The Prince of Wales, after the Georges\nof Hanover had come in, used sometimes to honor the poet with a visit;\nand the rich and powerful Bolingbroke--what time he lived at\nBattersea--used to come up in his barge, landing at the garden\nentrance--as most great visitors did--and discuss with him those\nfaiths, dogmas, truisms, and splendid generalities which afterward took\nform in the famous _Essay on Man_.\n\nThough the Twickenham home was on a great high road from London to\nTeddington and Hampton Court, and the greater high road of the river,\nit had, like all English suburban places now, its high enclosing walls\nthat gave privacy; and the river shores had their skirting of\nrhododendrons and willows and great beds of laurestina, so that {47}\nthe weak, misshapen poet might take his walks unobserved.  He had his\nvanities, but he did not love to be pointed at.  He carried a mind of\nextreme sensitiveness under that dwarfed figure; and is mad--maybe,\nsometimes, with destiny, that has crippled him so; and bites that thin\nlip of his till the blood starts.  But he does not waste force or pride\non repinings; he feels an altitude in that supple mind of his which\nlifts him above the bad lines of portraits or figures.  He knows that\nthe ready hand and brain, and the faculty of verse which comes tripping\nto his tongue, and the wit which flashes through and through his\nutterance, will make for him--has made for him--a path through whatever\nbeleaguerments of sense, straight up and on to the gates of the Temple\nof Fame.\n\n[Sidenote: Pope's vanities.]\n\nWe have had many vain men to encounter in these talks of ours--men\nassured of their own judgment and taste; but not one, I think, as yet,\nso thoroughly and highly conscious that his cleverness and scholarship\nand deftness and wit were as sure of their reward as the sun was sure\nto shine.\n\nI can fancy him pausing after having wrought {48} some splendid score\nof Homeric lines, which blaze and palpitate with new Greek fire: I can\nfancy him humming them over to himself--growing heated with the flames\nthat flash and play in them--his slight, frail figure trembling with\nthe rhythmic outburst, and he smiling serenely at a mastery which his\nwill and wit have brought to such supreme pitch of excellence that no\nhandling of English will go beyond it.\n\n\n_His Last Days._\n\n[Sidenote: Last days of Pope.]\n\nI have spoken of one face--I mean Lady Mary Montagu's--which used\nsometimes to light up the grotto of Mr. Pope, and have told you how\nthat badly managed friendship went out in a great muddle of sootiness\nand rage; nor were the mud and the filth, which he used in that\ndirection with such cruel vigor, weapons which he was unused to\nhandling: poor John Dennis, a poet and critic of that day, had been put\nin a rage over and over.  Lord Hervey had been scarified.  Blackmore\nand Phillips and Bentley had caught his stiletto thrusts; even Daniel\nDefoe had been subject of his sneers; and {49} so had the bland,\ncourteous Addison.  This sensitive, weak-limbed man saw offence where\nother men saw none; and straightway drew out that flashing sword of his\nand made the blood spurt.  Of course there were counter-thrusts, and\nheavy ones, that caused that poor decrepid figure of his to writhe\nagain--all the more because he pretended a stoicism that felt no such\nattack.  To say that he often made his thrusts without reason, and that\nmuch of his satire was dastardly, is saying what all the world knows,\nand what every admirer of his fine powers must lament.  But he had his\nsteady friendships, too, and his tendernesses.  Nothing could exceed\nthe kindly consideration and affectionate watchfulness which belonged\nto his protection and shelter of his old mother, lingering in that\npoet's faery home of Twickenham till over ninety.  A strange, close\nfriendship knit him to Dean Swift, who had seemed incapable of rallying\nthis sensitive man's--or, indeed, any man's--affections.  Pope, and\nBolingbroke--the brilliant and the courted--were long bound together in\nvery close and friendly communion; the tears of this latter were among\nthe honestest which {50} fell when the poet died.  Bishop Warburton,\ntoo, was most kindly treated by Pope in all his later years, and to\nthis gentleman most of his books were left.  There can be no doubt,\nalso, that the poet felt the tenderest regard for that neighbor of his,\nMiss Patty Blount, who had grown old beside him, and who used at times\nto bring her quiet face into the parlors of Twickenham.  Pope in his\nlast days would, I think, have seen her oftener--did covertly wish for\na sight of that kindly smile, which he had known so long and perhaps\nhad valued more than he had dared to confess.  But in those final days\nshe had gone her ways; maybe was grown tired of waiting upon the\npeevish humors of the poet; certainly was not seen by him more often\nthan a fair neighborly regard would dictate.  Yet he left her all his\nrights there at Twickenham, and much money beside.\n\n[Sidenote: Death of Pope.]\n\nThey say that at the last he complained of seeing things dimly--seeing\nthings, too, which others did not see (as the bystanders told him).\n\"Then, 'twas a vision,\" he said.  Two days thereafter he entered very\nquietly upon the visions all men see after death; leaving that poor,\nscathed, {51} misshapen body--I should think gladly--leaving the\npleasant home shaded by the willows he had planted; and leaving a few\nwonderful poems which I am sure will live in literature as long as\nbooks are printed.\n\n\n\n[1] Narcisse Luttrel: _A brief historical Relation of State affairs_\nfrom September, 1678, to April, 1714.\n\n[2] George Berkeley, b. 1685; d. 1753.  His works (3 vols.) and Life\nand Letters (1 vol.); edited by Fraser, in 1871.  See also very\ninteresting monograph on Berkeley, in Professor Tyler's _Three Men of\nLetters_, Putnam, 1895.\n\n[3] _An essay toward preventing the Ruin of Great Britain_, 1721.\n\n[4] Dr. Samuel Johnson, afterward, 1754, first President of King's (now\nColumbia) College, New York; he was a graduate of Yale; life by Dr.\nBeardsley.\n\n[5] In 1730, he writes to Samuel Johnson, of Stratford, Ct.: \"Pray let\nme know whether they [the college authorities] would admit the writings\nof Hooker and Chillingworth to the Library of the College of New Haven?\"\n\n[6] One of his last publications was, \"_Siris: a chain of Philosophical\nReflections and inquiries concerning the Virtues of Tar-water._\"  And\nit is remarkable that its arguments and teeming illustrations have not\nbeen laid hold of by our modern venders of Tar-soap.\n\n[7] Richard Bentley, b. 1662; d. 1742.  Native of Oulton, Yorkshire.\nWas first Boyle Lecturer, 1692; Master of Trinity, 1700; _Works_,\nedited by Dyce, London, 1836 (only 3 vols. issued of a proposed 8 vol.\nedition).  _Life_, by Jacob Maehly, Leipsic, 1868.\n\n[8] B. 1732; d. 1811.  Best known by his _Memoirs_, 1806; among his\nplays is _False Impressions_, in which appears Scud, the forerunner of\nDickens's _Alfred Jingle_.\n\n[9] All along the foot-notes in a great Quarto of the _Paradise Lost_\n(London, 1732) Bentley's critical pyrotechnics flame, and flare; and he\ncloses a bristling preface with this droll caveat;--\"I made [these]\nnotes _extempore_, and put them to the Press as soon as made; without\nany Apprehension of growing leaner by Censures, or plumper by\nCommendations.\"\n\n[10] Isaac Watts, b. 1674; d. 1748.  _Horae Lyricae_: Memoir by Southey\n(vol. ix., _Sacred Classics_: London, 1834).  Lowndes (_Bib. Manual_)\nsays, that up to 1864, there were sold annually 50,000 copies of\nWatts's Hymns.\n\n[11] B. 1681; d. 1765.  Works, with memoir, by J. Mitford.  2 vols.,\n12mo.  London, 1834.\n\n[12] Only _staying_; since the play (of _The Brothers_) was brought out\nin 1753, some twenty years after his establishment in the rectory of\nWelwyn.\n\n[13] Lady Mary Wortley Montagu, b. 1690 (or 1689?); d. 1762.  Works (3\nvols.), edited by her great grandson, Lord Wharncliffe: Later edition\n(1861), with life by Moy Thomas.\n\n[14] Wife of Lord Mar, who was exiled for his engagement in the\nabortive rebellion of 1715.\n\n[15] Dilke; _Papers, etc._, vol. ii. pp. 354-5.\n\n[16] Alexander Pope, b. 1688; d. 1744.  Editions of his works are\nnumerous.  I name those by Bowles and Roscoe, with that of Elwin and\nCourthope; see also Dilke's _Papers of a Critic_, Leslie Stephen's\n_Life_, and notices by Lowell, Minto, and Mrs. Oliphant.\n\n[17] Lowell, Professor Minto, De Quincey, Hazlitt, Covington, etc.  De\nQuincey says, \"It is the most exquisite monument of playful fancy that\nuniversal literature offers.\"\n\n[18] The identity of the house of Pope was destroyed by a lady owner\n(widow of Dr. Phipps, the Court oculist) in or about 1807.  Pope loved\nlandscape gardening and was aided by Kent and Bridgeman.  Warburton\nspeaks extravagantly of the poetic graces which he lavished upon his\ngrotto.\n\n\n\n\n{52}\n\nCHAPTER II\n\nThe name of Dean Berkeley--an acute and kindly philosopher--engaged our\nattention in the last chapter.  So did that ripe scholar and master of\nTrinity, Richard Bentley;[1] then came that more saintly Doctor--Isaac\nWatts, whose Doxologies will long waken the echoes in country churches;\nwe had a glimpse of the gloomy and lurid draperies, with which the muse\nof Dr. Edward Young sailed over earth and sky; sadly draggled, too, we\nsometimes found that muse with the stains of earth.  We spoke of a\nLady--Wortley Montagu--conspicuous for her beauty, for her\nacquirements, for her vivacity of mind, for her {53} boldness, for her\ncontempt of the convenances of society, and at last, I think, a\ncontempt for the whole male portion of the human race.\n\nThen came that keen, discerning, accomplished poet, Alexander Pope,\nwith a brain as strong and elastic as his body was weak and shaky; and\nwho, of all the poets we have encountered since Elizabeth's day, knew\nbest how to give to words their full forces, and how to make them\njingle and shine.\n\nBut the lives of these I have now named, and of those previously\nbrought to your notice[2] overreached the reign of Queen Anne, and\ndropped off--some in the time of George I., some under his son George\nII., and others in an early part of the long reign of George III.\n\n\n_From Stuart to Brunswick._\n\nBut how came the Georges of Hanover and Brunswick to succeed Anne\nStuart?  Yes, there was a son of the deposed and exiled James II.\n(whose {54} mother was an Italian princess--making him half-brother to\nQueen Anne) known, sometimes as James Edward, and sometimes as The\nPretender.  He had favorers about the Court of Anne; and if the Queen\nhad lingered somewhat longer, or if the Jacobite or Tory political\nmachine had been a little better oiled and in better play, this\nPretender might have come to the throne instead of Hanover George.\nPoet and Ambassador Prior, who was suspected of favoring this, was one\nof those who went to the Tower, and came near losing his head in the\nearly days of King George; and Bolingbroke, the friend of Pope, a known\nplotter for the Stuarts, took himself off hastily to France for safety.\n\nJames Edward, however, did not give the matter up, but made a landing\nin Scotland in 1715 and led that dreary rebellion, in which the poor\nEarl of Mar went astray, and in which Argyle figured; a rebellion which\ngives its small scenes of battle and its network of conspiracies to\nScott's story of _Rob Roy_.  The Pretender escaped with difficulty to\nFrance, made no succeeding attempt, lived in comparative obscurity, and\ndied in Rome fifty years {55} later.  He was, according to best\naccounts, a poor, weak creature, of dissipated habits--of melancholy\naspect--dubbed King of England[3] by the Pope--given a stipend by the\nover-gracious Holy Father--and at last a costly tomb in St. Peter's,\nwhich is dignified by some good sculptural work.  Travelling\nsentimentalists may meditate over its grandiose inscription of James\nIII., King of England!\n\nJames Edward had married, however, a Princess Sobieski of the Polish\nfamily, by whom he had two sons, Charles Edward and Henry.  The elder,\nCharles Edward, an ambitious, handsome, gentlemanly, and amiable\nman--known as the Young Pretender--did, by favor of French aid, and\nstimulated by larger French promises, make a landing in Scotland in\n1745, which was successful at first, but ended with that defeat on\nCulloden Moor, which--with pretty romantic broidery--gives a gloomy\nsetting to Scott's first novel of _Waverley_.\n\n{56}\n\nA second plotting of some friends of the Young Pretender, somewhere\nabout 1751-1752 (dimly foreshadowed in the story of _Redgauntlet_),\nproved abortive.  Thenceforward he appears no more in English history.\nWe know only that this bright, clever, brave Chevalier, who bewitched\nmany a Highland maiden, lived a corrupt life, made a dreary and\nunfortunate marriage (1772), and, bloated with drink and blighted in\nhopes, died at Rome in 1788.\n\nHis brother Henry was a priest, and was made a cardinal.  He spent all\nhis money in pompous living, became miserably poor, and died in Venice\nearly in the present century--the last of his family.  There is in St.\nPeter's Church at Rome, in the Chapel of the Presentation, a great\ntomb, showy with the sculptures of Canova, which commemorates all these\nStuarts, and--so far as Latin inscriptions can do it--makes kings and\nprinces of these unfortunate representatives of the family of King\nJames II.\n\nStill we are without an answer to our question: How and why did the\nGeorges of Hanover come to the British throne?\n\n{57}\n\nThose who recall my mention[4] of that slip-shod pedantic king, James\nI., who came from Scotland, and who brought the Stuart name with him,\nwill remember an allusion to an ambitious daughter of his, Elizabeth\nStuart, who married a certain Frederic of the Palatinate, and possessor\nof the famous chateau whose beautiful ruins are still to be seen on the\nhill above Heidelberg.  You will remember my mention of that\nextravagant ambition which brought her husband to grief and to an early\ndeath.  Well, she had many children; and among them one named Sophia,\nwho married, in 1658, Ernest Augustus, Duke of Brunswick\nand--afterward--Elector of Hanover.  She was a good woman, a fairly\npronounced Protestant--unlike some sisters she had; so that in casting\nabout for a Protestant successor to William III. and to Anne, the\northodox wise ones of England fixed upon this Sophia, the\ngrand-daughter of old James I.  She died, however, before Anne died and\nin the same year; so that the succession fell to her son George Louis,\nwho became George I. of Great Britain.\n\n{58}\n\nHe was well toward sixty when he came to England--did not care overmuch\nto come; loved his ease; loved his indulgences, of which he had a good\nmany, and a good many bad ones; was a German all over; not speaking\nEnglish even, nor ever learning to speak it; had been a good soldier\nand fought hard in his day, but did not care for more fighting, or\nfatigue of any sort; had little culture, and minded the welcoming odes\nwhich English poets sang to him less than he would mind the gurgling of\ngood \"trink\" from a beer-bottle.  Yet withal, he was fairly\nwell-intentioned, not a meddler, never wantonly unjust, willing to do\nkindnesses, if not fatiguing; a heavy, good-natured, heathenish,\nsottish lout of a king.\n\nYet, as I have said,[5] Addison could not find words noble enough to\ntell this man how Anne was dead and he was king; if Addison had made\nhis letter as noble as the drama of _Cato_, George I. would have yawned\nand lighted his pipe with it.\n\nThis George I. had married in early life a beautiful cousin, and a rich\none, but without much {59} character; perhaps he treated her brutally\n(it was certainly a Georgian fashion); and she, who was no saint, would\nhave run away from that Hanover home--had plotted it all, and the night\ncame, when suddenly her lover and the would-be attendant of her flight\nwas savagely slain; and she, separated from her two children and\nspeaking no word more to her grim husband, was consigned a prisoner to\na gloomy fortress in the Aller valley, where she dragged out an\nembittered and disappointed life for thirty odd years; then, Death\nopened the gates and set the poor soul free.\n\nThis was the wife of George I., and the mother of George II.; this\nlatter being over thirty at the time of his father's coming to England,\nand not getting on over-well with the king--the son, perhaps, resenting\nthat confinement of his mother in the Ahlden fortress.\n\nThis Prince of Wales had no more love for letters than his father\nGeorge I.; would have liked a jolly German drinking song better than\nanything Pope could do; was short, irascible, as good a fighter as the\nfather, swore easily and often; had a good, honest wife though, who\nclung to him {60} through all his badnesses.  He had a city home in\nLeicester Square and a lodge in Richmond Park, whence he used to ride,\nat a hard gait, with hunting parties (Pope speaks of meeting him with\nsuch an one) and come home to long dinners and heavy ones.\n\nIt was at this lodge in Richmond Park (which is now less changed than\nalmost any park about London and so one of the best worth seeing) that\na messenger came galloping in jack-boots one evening, thirteen years\nafter George I. had come to the throne, to tell the Prince that old\nGeorge was dead (over in Osnaburg, where he had gone on a visit) and\nthat he, the Prince, was now King George II.[6]\n\n{61}\n\n\"Dat is one big lie\"--said the new and incredulous King with an oath.\nBut it was not a lie; the King was wrathy at being waked too early, and\nwanted to swear at something or somebody.  But having rubbed his eyes\nand considered the matter, he began then and there those thirty-three\nyears of reign, which, without much credit to George II. personally,\nwere, as the careful Mr. Hallam says in his history, the most\nprosperous years which England had ever known.\n\nRemember please, then, that George I., who succeeded Anne, reigned some\nthirteen years; and after him came this short, sharp-spoken George II.,\nwho reigned thirty-three years--thus bringing us down to 1760.  I have\ndwelt upon the personalities of these two monarchs, not because they\nare worthy of special regard, but rather that they may serve more\neffectively as finger-posts or clumsy mile-stones (with wigs upon\nthem)--to show us just how far we are moving along upon the big\nhigh-road of English history.\n\n\n{62}\n\n_Samuel Richardson._\n\nQuite early in that century into which these royal people found their\nway, there lived over beyond Temple Bar, near to St. Bride's Church, in\nthe City of London, a mild-mannered, round-faced, prim little man who\nwas printer and bookseller--in both which callings he showed great\nsagacity and prudence.  He was moreover very companionable, especially\nwith bookish ladies, who often dropped in upon him--he loving to talk;\nand to talk much about himself, and his doings, and the characters he\nput in his books.  For this was Samuel Richardson[7]--the very great\nman as many people thought him--who had written _Pamela_, _Clarissa\nHarlowe_, and _Sir Charles Grandison_.  It is doubtful if he knew Pope\nor Swift or Berkeley; he was never of the \"Spectator set.\"  Pope we\nknow read his {63} _Pamela_ and said there was as much good in it as in\ntwenty sermons: yet I do not think he meant to compliment it--or the\nsermons.  Neither did Bookseller Richardson know people in high\nposition, except Hon. Mr. Onslow the Speaker, who gave him some of the\npublic printing to do and put him in way of business by which he grew\nrich for these times and had a fine large house out by Hammersmith,\nwhere he kept a little court of his own in summer weather; the\ncourtiers being worthy women, to whom he would read his books, or\ncorrespondence relating to them, by the hour.  Possibly you have not\nread his novels; but I am sure your grandmothers or great-grandmothers\nhave read some of them, and wept over them.  He was not learned; was\nthe son of a country carpenter, and in his early days was known for an\neasy letter-writing faculty he had; and he used to be set upon by\nsighing maidens--who were suffering under a prevalent contagious\naffection of young years--to write their love-letters for them; and so\nat last, in busy London, when his head was streaked with gray, he began\nto put together books of letters--written as if {64} some suffering or\nwishful one had whispered them all in his ear.  There was no machinery,\nno plot, no classicism, no style--but sentiment in abundance and vast\nprolixity, and ever-recurring villanies, and \"pillows bedew'd with\ntears.\"  The particularity and fulness of his descriptions were\nsomething wonderful; every button on a coat, every ring on the fingers,\nevery tint of a ribbon, every ruffle on a cap, every ruffle of emotion,\nevery dimple in a cheek is pictured, and then--the \"pillows bedew'd\nwith tears.\"\n\nThere's a great budget of Richardson correspondence that shows us how\nthe leaven of such stories worked; letters from Miss Suffern and Miss\nWestcomb, and Mr. Dunallan, and a dozen others, all interlaced with his\nown; for it does not appear that the old gentleman ever refused the\nchallenge of a letter, or grew tired of defending and illustrating his\ntheories of literary art and of morals, which in his view were closely\njoined.  The stories were published by himself--volume by volume, so\nthat his correspondents had good chance to fire upon him--on the wing\nas it were: \"Poor Clarissa,\" they say; \"my heart bleeds for her, and\nwhat, {65} pray, is to become of her; and why don't you reform\nLovelace, and sha'n't he marry Clarissa?  And I do not believe there\nwas ever such a man as Sir Charles in the world.\"  The old gentleman\nenjoys this and writes back by the ream; has his own little sentiment\nof a sort too, even in the correspondence.  Mme. Belfour wants to see\nhim--\"the delightful man\"--without herself being observed; so entreats\nhim to walk some day in the Park (St. James') at a given hour; and\nRichardson complies, giving these data for his picture:--\n\n\n\"I go through the Park, once or twice a week to my little retirement;\nbut I will for a week together, be in it, every day three or four\nhours, till you tell me you have seen a person who answers to this\ndescription, namely, short--rather plump--fair wig, lightish cloth\ncoat, all black besides; one hand generally in his bosom, the other a\ncane in it, which he leans upon under the skirts of his coat ...\nlooking directly fore-right as passers-by would imagine, but observing\nall that stirs on either hand of him; hardly ever turning back, of a\nlight brown complexion, smoothish faced and ruddy cheeked--looking\nabout sixty-five, a regular even pace, a gray eye sometimes\nlively--very lively if he have hope of seeing a lady whom he loves and\nhonors.\"\n\n\n{66}\n\nThen he writes to Miss Westbrook--an adopted daughter as he calls her:--\n\n\n\"You rally me on my fears for your safety, and yet I know you to be\nnear a forest where lies a great wild bear: I am accused for these\nfears--I am accused for playing off a sheet-full of witticisms, which\nyou, poor girl, can't tell what to do with.  Witticism! Miss W.  Very\nwell, Miss W---- But I did not expect--but no matter;--what have I done\nwith my handkerchief--I--I--I did not really expect; but no matter,\nMiss W----\"\n\n\nA man who can put tears so easily, and for so little cause, into a\nletter, can put them by the barrelful in his books: and so he did, and\nmade Europe weep.  Rousseau and Diderot from over in France,\nphilosophers as they professed to be, blubbered their admiring thanks\nfor _Clarissa Harlowe_.\n\nI have spoken of him not because he is to be counted a great classic\n(though Dr. Johnson affirmed it); not because I advise your wading\nthrough six or seven volumes of the darling _Sir Charles Grandison_--as\nsome of our grandames did; but because he was, in a sense, the father\nof the modern novel; coming before Fielding; in fact, spurring the\nlatter, by _Pamela_, to his great, {67} coarse, and more wonderful\naccomplishment.  And although what I have said of Richardson may give\nthe impression of something paltry in the man and in his works, yet he\nwas an honest gentleman, with good moral inclinations, great art in the\ndissection of emotional natures, and did give a fingering to the\nheart-strings which made them twang egregiously.\n\n\n_Harry Fielding._\n\nThe British Guild of Critics is, I think, a little more disposed to\n_admit_ Richardson's claims to distinction than to be proud of them: it\nis not so, however, with Fielding;[8] if Richardson was \"womanish,\"\nFielding was masculine with a vengeance; gross, too, in a way, which\nalways will, and always should, keep his books outside the pale of\ndecent family reading.  Filth is filth, and always deserves to be\nscored by its name--whatever blazon of genius may compass it about.  I\nhave no {68} argument here with the artists who, for art's sake, want\nto strip away all the protective kirtles which the Greek Dianas wore:\nbut when it comes to the bare bestialities of such tavern-bagnios as\npoor Fielding knew too well,[9] there seems room for reasonable\nobjection, and for a strewing of some of the fig-leaves of decency.\nAnd yet this stalwart West-of-England man, \"raised\" in the fat meadows\nof Somersetshire, and who had read _Pamela_ as a stepping-stone for his\nfirst lift into the realms of romance, was a jovial, kind-hearted,\nrollicking, dare-devil of a man, with no great guile in him, and no\nhypocrisies and no snivelling laxities.  He had a great lineage,\ntracing back to that Landgrave of Alsace, from whom are descended the\nkings and emperors of the House of Hapsburg: and what a warrant for\nimmortality does this novelist carry in those words of Gibbon!--\n\n\n\"The successors of Charles V. may disdain their [Somersetshire]\nbrethren of England; but the romance of _Tom {69} Jones_--that\nexquisite picture of humor and manners--will outlive the Palace of the\nEscurial and the imperial eagle of Austria.\"\n\n\nIt was at home or near by that Henry Fielding found his first\nschooling; at the hand--a tradition runs--of that master who served as\nthe original for his picture of Parson Trulliber: if this indeed be so,\nnever were school-master severities so permanently punished.  After\nthis came Eton, where he was fellow of Lord Lyttleton, who befriended\nhim later, and of William Pitt (the elder), and of Fox--the\nrattle-brain father of Charles James.  Then came two or more years of\nstay at the University of Leyden, from which he laid his course\nstraight for the dramatic world of London; for his father, General\nFielding, had a good many spendthrift habits, with which he had\ninoculated the son.  There was need for that son to work his own way;\nand the way he favored was by the green-room, where the sparkle of such\nlively elderly ladies as Mrs. Oldcastle and Mrs. Bracegirdle had not\nyet wholly gone out.\n\nHe wrote play upon play with nervous English, and pretty surprises in\nthem; but not notable for {70} any results, whether of money-making or\nof moral-mending.  He also had his experiences as stage manager; and\nbetween two of his plays (1735 or thereabout) married a pretty girl\ndown in Salisbury; and with her dot, and a small country place\ninherited from his mother, set up as country gentleman, on the north\nborder of Dorsetshire, determined to cut a new and larger figure in\nlife--free from the mephitic airs of Drury Lane.  There were\nstories--very likely apocryphal--that he ordered extravagant liveries;\nit is more certain that he gave himself freely, for a time, to hounds,\nhorses, and friends.  Of course such a country symposium devoured both\nhis own and his wife's capital; and we find him very shortly back in\nLondon, buckling down to law study; very probably showing there or\nthereabout the \"inked ruffles and the wet towel round his head,\" which\nappear in the charming retrospective glasses of Thackeray.[10]\n\nBut times are hard with him; those fast years of green-room life have\ntold upon him; the \"wet {71} towels\" round the head are in demand; some\nof his later plays are condemned by the Lord Chancellor;[11] in 1742,\nhowever, he makes that lunge at the sentimentalism of Richardson which,\nin the shape of _Joseph Andrews_, gives him a trumpeting success.  It\nencourages him to print two or three volumes of miscellanies.  But\nshadows follow him; a year later, his wife dies in his arms; Lady\nWortley Montagu (who was a cousin) tells us this; and tells us how\nother cousins were scandalized because, a few years afterward, the\nnovelist, with an effusive generosity that was characteristic of him,\nmarried his maid, who had lamented her mistress so sincerely, and was\ntenderly attached to his children.  At about the same period he\naccepted office as Justice of the Peace--thereby still further\ndisgruntling his aristocratic Denbigh cousins.  But the quick-coming\nvolumes of _Tom Jones_ and their wonderful acclaim cleared the space\naround him; he had room to breathe and {72} to play the magistrate; it\nis Henry Fielding, Esq., now,--of Bow Street, Covent Garden.  _Amelia_\nfollowed, for which he received L1,000; and we hear of a new home out\nin the pleasant country, by Baling, north of Brentford, and the Kew\nGardens.\n\nFinally on a June day of 1754 we see him leaving this home; \"at twelve\nprecisely,\" he says in his last Journal, \"my coach was at the door,\nwhich I was no sooner told than I kissed my children all around, and\nwent into it with some little resolution.\"  There needed resolution;\nfor he was an utterly broken-down man, the pace of his wild, young days\ntelling now fearfully, and he bound away for a voyage to the sunny\nclimate of Portugal--to try if this would stay the end.\n\nBut it does not; in October of the same year he died in Lisbon; and\nthere his body rests in the pretty Cemetery of the Cypresses, where all\nvisitors who love the triumphs of English letters go to see his tomb,\namong the myrtles and the geraniums.  If he had only lived to pluck\naway some of those grosser stains which defile the pages where the\ncharacters of an Allworthy and of a Parson Adams will shine forever!\n\n\n{73}\n\n_Poet of the Seasons._\n\nIt was just about the opening of the second quarter of the eighteenth\ncentury--when Fielding was fresh from Eton, fifteen years before\n_Pamela_ had appeared and while George II. was in waiting for the\nslipping off of Father George at Osnaburg--that a stout Scotch poet\nfound his way to London to try a new style of verses with the public\nwhich was still worshipping at the shrine of Mr. Pope.  This was the\npoet of _The Seasons_,[12] whose boyhood had been passed and enriched\nin that bight of the beautiful Tweed valley which lies between\nColdstream and the tall mass of Kelso's ruin,--with Melrose and\nSmailhome Tower and Ettrickdale not far away, and the Lammermuir hills\nglowering in the north.  He had studied theology in Edinboro', till\nsome iris-hued version of a psalm (which he had wrought) brought the\nwarning from some grim orthodox friend--that {74} a good Dominie should\nrein up his imagination.  So he set his face southward, with the\ncrystal scenery of a winter on Tweed-side sparkling in his thought.  He\nlived humbly in London, for best of reasons, near to Charing Cross; but\nby the aid of Northern friends, brought his _Winter_ to book, in the\nspring of 1726.\n\nIt delighted everybody; the tric-trac of Pope was lacking, and so was\nthe master's arrant polish; but the change brought its own blithe\nwelcome.\n\nWe will try a little touch from this first poem of his which he brought\nin his satchel, on the boy journey to London:--\n\n  \"Thro' the hushed air the whitening shower descends,\n  At first, thin, wavering, till at last the flakes\n  Fall broad and wide and fast, dimming the day\n  With a continual flow....\n\n          Low, the woods\n  Bow their hoar heads; and ere the languid sun\n  Faint from the west emits his evening ray,\n  Earth's universal face, deep hid and chill,\n  Is one wide dazzling waste.\n\n          The fowls of heaven,\n  Tamed by the cruel season, crowd around\n  The winnowing stone....\n\n{75}\n\n          One alone,\n  The red-breast, sacred to the household gods,\n  Wisely regardful of the embroiling sky\n  In joyless fields and thorny thickets, leaves\n  His shivering mates.\n\n          Half afraid, he first\n  Against the window beats; then, brisk, alights\n  On the warm hearth; then hopping o'er the floor\n  Eyes all the smiling family askance\n  And pecks, and starts, and wonders where he is.\"\n\n\nThat robin red-breast has hopped over a great many floors in his time;\nand now after a hundred and sixty years he comes brisk as ever out of\nthat Winter poem of Thomson's.  This Scotch poet is wordy; he draws\nlong breaths; he is sometimes tiresome; but you will catch good honest\nglimpses of the country in his verse without going there--not true to\nour American seasons in detail, but always true to Nature.  The sun\nnever rises in the west in his poems; the jonquils and the daisies are\nnot confounded; the roses never forget to blush as roses should; the\noaks are sturdy; the hazels are lithe; the brooks murmur; the torrents\nroar a song; the winds carry waves across the grain-fields; the clouds\nplant shadows on the mountains.\n\n{76}\n\nThomson was befriended by Pope, who kindly made corrections in the\nfirst draught of some of his poems; and that you may see together the\nwordy ways of these two poets I give a sample of Pope's mending.\n\nThomson wrote--speaking of a gleaning girl:--\n\n  \"Thoughtless of Beauty, she was beauty's self\n  Recluse among the woods; if city dames\n  Will deign their faith; and thus she went, compelled\n  By strong necessity, with as serene\n  And pleased a look as Patience ere put on,\n  To glean Palemon's fields.\"\n\nAnd this is the way in which Pope does the mending:--\n\n  \"Thoughtless of Beauty, she was beauty's self\n  Recluse among the close embowering woods.\n  As in the hollow breast of Apennine,\n  Beneath the shelter of encircling hills,\n  A myrtle rises far from human eyes,\n  And breathes its balmy fragrance o'er the wild;\n  So flourished, blooming, and unseen by all,\n  The sweet Lavinia; till at length compelled\n  By strong necessity's supreme command,\n  With smiling patience in her looks, she went\n  To glean Palemon's fields.\"\n\n\n{77}\n\nThere are more words, but the words gleam!  Pope is the master, yet\nmastered by rules; Thomson less a master, but free from bonds.\n\nHe tried play-writing, in those days when Fielding was just beginning\nin the same line, but it was not a success.  After a year or two of\ntravel upon the Continent, on some tutoring business, he published an\nambitious poem (1734-1736) entitled _Liberty_--never a favorite.  He\nhad made friends, however, about the Court; and he pleasantly contrived\nto possess himself of some of those pensioned places, which fed unduly\nhis natural indolence.  But all will forgive him this vice, who have\nread his fine poem of the _Castle of Indolence_ in Spenserian verse.\nIt was his last work--perhaps his best, and first published in 1748,\nthe year of his death.\n\nOne stanza from it I must quote; and shall never forget my first\nhearing of it, in tremulous utterance, from the lips of the venerable\nJohn Quincy Adams, after he had bid adieu (as he thought) to public\nlife and was addressing[13] a {78} large assemblage in the university\ntown of New Haven:\n\n  \"I care not, Fortune, what you me deny!\n    You cannot rob me of free Nature's grace,\n  You cannot shut the windows of the sky\n    Through which Aurora shows her brightening face;\n    You cannot bar my constant feet to trace\n  The woods and lawns by living streams at eve;\n    Let health my nerves and finer fibres brace\n  And I their toys to the great children leave,\n  Of fancy, reason, virtue, nought can me bereave.\"\n\n\nMost readers will think kindly and well of this poet; and if you love\nthe country, you will think yet more kindly of him; and on summer\nafternoons, when cool breezes blow in at your windows and set all the\nleaves astir over your head, his muse--if you have made her\nacquaintance--will coo to you from among the branches: but you will\nnever and nowhere find in him the precision, the vigor, the point, the\npolish, we found in Pope; and which you may find, too, in the fine\nparcel-work {79} done by Thomas Gray, who was a contemporary of\nThomson's, but younger by some fifteen years.\n\n\n_Thomas Gray._\n\nYou will know of that first poem of his--_Ode to Eton College_; at\nleast you know its terminal lines, which are cited on all the\nhigh-roads:--\n\n  \"Where ignorance is bliss\n  'Tis folly to be wise!\"\n\nAll the world knows, too, his _Elegy_, on which his fame principally\nrests.  Its melancholy music gets somehow stamped on the brain of\nnearly all of us, and lends a poetic halo to every old graveyard that\nhas the shadow of a church tower slanted over it.\n\nGray[14] was, like Milton, a London boy--born on Cornhill under the\nshadow almost of St. Paul's.  The father was a cross-grained man,\nliving apart from Mrs. Gray, who, it is said, by the gains of some\nhaberdashery traffic which she set up in {80} Cornhill, sent her boy to\nEton and to Cambridge.  At Eton he came to know Horace Walpole,\ntravelled with him over Europe, after leaving Cambridge, until they\nquarrelled and each took his own path.  That quarrel, however, was\nmended somewhat later and Walpole became as good a friend to Gray as he\ncould be to anybody--except Mr. Walpole.\n\nThe poet, after his father's death, undertook, in a languid way, the\nstudy of law; but finally landed again in Cambridge, and was a\ndilettanteish student there nearly all his days, being made a Professor\nof History at last; but not getting fairly into harness before the gout\nlaid hold of him and killed him.  Probably no man in English literature\nhas so large a reputation for so little work.  Gibbon regretted that he\nshould not have completed his philosophic poem on education and\ngovernment; Dr. Johnson, who spoke halting praise of his poems, thought\nhe would have made admirable books of travel; Cowper says, \"I once\nthought Swift's letters the best that could be written, but I like\nGray's better.\"\n\nThe truth is, he was a fastidious, scholarly man, whose over-nicety of\ntaste was always in {81} the way of large accomplishment.  He was\ncontent to do nothing, except he did something in the best possible\nway.  He so cherished refinements that refinements choked his impulses.\n\nA great stickler he was, too, for social refinements--distinctions,\npreferments, and clap-trap--wanting his courtesies, of which he was as\nchary as of his poems, to have the last stamp of gentility; thus\nrunning into affectations of decorum, which, one time, made him the\nbutt of practical jokers at his college.  Some lovers of fun there\nsounded an alarm of fire for the sake of seeing the elegant Mr. Gray\n(not then grown famous, to be sure) slipping down a rope-ladder in\nundress, out of his window; which he did do, but presently changed his\ncollege in dudgeon.  He had, moreover, a great deal of Walpole's\naffected contempt for authorship--wanted rather to be counted an\nelegant gentleman who only played with letters.  He writes to his\nfriend that the proprietors of a magazine were about to print his\nElegy, and says:--\n\n\n\"I have but one bad way to escape the honor they would inflict upon me,\nand therefore desire you would make {82} Dodsley print it immediately,\nwithout my name, but on his best paper and type.  _If he would add a\nline to say it came into his hands by accident, I should like it\nbetter_.\"\n\n\nI think he caught this starched folly (if it were folly) from Walpole.\nI have heard of over-elegant people in our day with the same\naffectation; but, as a rule, they do not write poems so good as the\n_Elegy_.\n\nGray died, after that quiet life of his, far down in the days of George\nIII., 1771, leaving little work done, but a very great name.  He was\nburied, as was fitting, beside his mother, in that churchyard at Stoke,\nout of which the Elegy grew.  And if you ever have a half day to spare\nin London, it is worth your while to go out to Slough (twenty miles by\nthe Great Western road), and thence, two miles of delicious walk among\nshady lanes and wanton hedges, to where Stoke-Pogis Church, curiously\nhung over with ivy, rises amongst the graves; and if sentimentally\ndisposed, you may linger there, till the evening shadows fall, and\nrepeat to yourself (or anybody you like)--\n\n  \"The curfew tolls the knell of parting day,\n  The lowing herd winds slowly o'er the lea.\n  The ploughman homeward plods his weary way\n  And leaves the world to darkness and to me.\"\n\n\n{83}\n\n_A Courtier._\n\nI have spoken of the association of Walpole with Gray; it was not an\nintimate one after the two had outgrown their youth-age; indeed\nWalpole's association with nobody was intimate; nor was he a man whose\nliterary reputation ever was, or ever can be great.  He was son[15] of\nthat famous British Minister of State, Sir Robert Walpole, who for many\nlong years held the fate of England in his hand.  But his son Horace\ncared little for politics.  He was unmarried, and kept so always; had\nmoney in plenty (coming largely from Government sinecures) and a fat\nplace at Twickenham--called Strawberry Hill; which by his vagaries in\narchitecture and his enormous collection of bric-a-brac, he made the\nshow place of all that region.  He established a private press at this\ncountry home, and printed, among a {84} multitude of other books, a\ncatalogue of royal and noble authors--not reckoning others so worthy of\nhis regard; indeed, he had a well-bred contempt for ordinary literary\navocations; but he wrote and published (privately at first) a romance\ncalled _The Castle of Otranto_.[16]  It was \"a slight thing,\" he told\nhis friends, which he had dashed off in an idle hour, and which he \"had\nnot put his name to; but which succeeded so well that he did not any\nlonger entirely keep the secret.\"  It is a tale, quite ingenious, of\nmingled mystery and chivalry; there are castles in it, and huge\nhelmets, that only giants could wear; and there are dungeons, and\nforlorn maidens; ghosts, and sighing lovers; mysterious sounds, and\npictures that come out of their frames and walk about in the\nmoonlight--it is a pattern book to read at night in an old country\nhouse which has long corridors and deserted rooms, where the bats fly\nin and out, and the doors clang and clash.\n\nBut this strange creature, Horace Walpole, is {85} known best of all by\nhis letters[17]--nine solid volumes of them, big octavo--covering\nnearly the whole of his life and addressed to a half score or so of men\nand women on all possible topics except any serious one; and all made\nready, with curious care, for publication when his death should come.\nOn that one point he did have serious belief--he believed he should\ndie.  This great budget of his letters is one of the most extraordinary\nproducts--if we may call it so--of literature.  It is hard to say what\nis not touched upon in them; if he is robbed, you hear how a voice out\nof the night said \"stop\"--how he slipped his watch under his\nwaistband--how he gave up his purse with nine guineas in it--how Lady\nBrowne was frightened and gave up _her_ watch; if the king has gout in\nhis toe you hear of that; if he goes to the palace he tells you who was\nin the ante-room and how two fellows were sweeping the floor, dancing\nabout in sabots; how the Duc of Richelieu was pale except his nose,\n\"which is red and wrinkled.\"  Great hoops with brocade dresses come\nsailing into {86} his letters; so do all the scandals about my lady\n_this_, or the duchess _that_; so do the votes in Parliament and\nreports about the last battle, if a war is in progress; and the French\nnews, and new things at Strawberry Hill--over and over.  And he does\nnot think much of Gibbon, and does not think much of Dr. Johnson--who\n\"has no judgment and no taste;\" and why doesn't his friend Mason[18] (a\nthird-rate poet) \"show up the doctor and make an end of him?\"--which is\nmuch like saying that Mr. Wardle's fat boy should make an end of Mr.\nPickwick.\n\nYet do not think there is no art in all this, and that you would not\nlike them: there is art of the highest gossipy kind; and I can readily\nunderstand how his correspondents all relished immensely his letters\nwhenever they came.  There is humor and sparkle, and there are delicate\ntouches; he approaches his lighter topics as a humming-bird approaches\nflowers--a swift dart {87} at them--a sniff, a whirl of wings, and away\nagain.\n\nThen he has that rare literary instinct of knowing just what each\ncorrespondent would like best to hear of--that's the secret of writing\nletters that will be welcome.  You cannot interchange his letters.  He\ntickles Lady Ossory's ear with sheerest gossip, and Lady Suffolk with\ntalk of dress and of the last great Paris ball, and the poet Mason with\nbookish platitudes, and Conway with the leakings of political talk, and\nCole with twaddle on art or science.  You want to turn your back on him\nagain and again for his arrant snobbish pretensions or some weak and\nviolent prejudice; yet you want to listen again and again.  It is such\na pretty, lively, brisk, frolicsome, _petillant_ small-beerish talk,\nthat engages and does not fatigue, and piques appetite yet feeds you\nwith nothings.\n\nHe grew old there in his _gim-crack_ of a palace, cultivating his\nflowers and his complexion; tiptoeing while he could over his waxed\nfloors in lavender suit, with embroidered waistcoat and \"partridge silk\nstockings,\" with _chapeau bas_ held before him--very reverent to any\nvisitor of {88} distinction--and afterward (he lived almost into this\ncentury), when gout seizes him, I seem to see still--as once\nbefore[19]--the fastidious old man shuffling up and down from\ndrawing-room to library--stopping here and there to admire some newly\narrived bit of pottery--pulling out his golden snuff-box and whisking a\ndelicate pinch into his old nostrils--then dusting his affluent\nshirt-frills with the tips of his dainty fingers, with an air of\ngratitude to Providence for having created so fine a gentleman as\nHorace Walpole, and of gratitude to Horace Walpole for having created\nso fine a place as Strawberry Hill.\n\n\n_Young Mr. Johnson._\n\nAnd now what a different man we come upon, living just abreast of him\nin that rich English century and that beautiful English country!  We go\ninto Staffordshire and to the old town of Lichfield, to find the boy\nwho afterward became the great lexicographer[20] and the great talker.\nThe {89} house in which he was born is there upon a corner of the great\nbroadened street, opposite St. Mary's Church.  We get a pleasant\nglimpse of the house on a page of _Our Old Home_, by Hawthorne; and\nanother glimpse of the colossal figure of Dr. Johnson, seated in his\nmarble chair, upon that Lichfield market-place.\n\nHis father was a bookseller; held, too, some small magistracy; was\neminently respectable; loved books as well as sold them, and had a\ncorresponding inaptitude for business.  The son added to indifferent\nschooling, here and there, a habit of large browsing along his father's\nshelves; was a great, ungainly lout of a boy, but marvellously\nquick-witted.  With some help from his father, and some from friends,\nand with a reputation for making verses, and tastes ranging above\nbookstalls, he entered at Oxford when nineteen; but {90} the stings of\npoverty smote him there early; and after three years of irregular\nattendance, he left--only to find his father lapsing into bankruptcy\nand a fatal illness.  On the settlement of the old bookseller's estate,\nL20 only was the portion of the son.  Then follow some dreary years; he\nis hypochondriac and fears madness; he is under-teacher in a school; he\noffers to do job-work for the book-makers; he translates the narrative\nof a Portuguese missionary about Abyssinia; he ponders over a tragedy\nof _Irene_.  Not much good comes of all this, when--on a sudden, our\nhero, who is now twenty-six, marries a widow--who admired his\ntalents--who is twenty years his senior and has L800.  Johnson was not\na person to regard closely such little discrepancies as that difference\nin age--nor she, I suppose.\n\nThe bride is represented as not over-comely, and as one--of good\njudgment in most matters--who resorted to some vulgar appliances for\nmaking the most of her \"good looks.\"  Lord Macaulay[21] uses a very\nrampant rhetoric in his encyclopaedic {91} mention of the paint she put\nupon her cheeks.  With the aid of her L800, Johnson determined to set\nup a boarding-school for young gentlemen; a gaunt country-house three\nmiles out of Lichfield was rented and equipped and advertised; but the\nyoung gentlemen did not come.\n\nHow could they be won that way?  The mistress frowsy, simpering,\nancient, painted, and becurled; and Mr. Johnson, gaunt, clumsy,\nsquinting--one side of his face badly scarred with some early surgical\ncut; one eye involved and drooping, and a twitchy St. Vitus's dance\nmaking all uglier.  What boy would not dread a possible whipping from\nsuch a master, and what mamma would not tremble for her boy?  Yet I do\nnot believe he ever whipped hard, when he had occasion; he was\nkind-hearted; but his scolds at a false syntax must have been terrific\nand have made the floors shiver.\n\nAmong the boys who did venture to that Edial school was one David\nGarrick, whose father had been a friend of the elder Johnson; and when\nthe school broke up--as it did presently--Johnson and David Garrick set\nout together for London, to {92} seek their fortune--carrying letters\nto some booksellers there; and Johnson carrying that half-written\ntragedy of _Irene_ in his pocket.  Garrick's rise began early, and was\nbrilliant, but of this we cannot speak now.  Johnson knocked about\nthose London streets--translating a little, jobbing at books a little,\nstarving and scrimping a great deal.  He fell in early with a certain\nRichard Savage,[22] a wild, clever, disorderly poet, as hard pinched as\nJohnson.  According to his story, he was the son of the Countess\nMacclesfield, but disowned by her--he only coming to knowledge of his\nparentage through accident, when he was grown to manhood.  Johnson\ntells the pathetic tale of how Savage paced up and down, at night, in\nsight of his mother's palatial windows, gazing grief-smitten at them,\nand yearning for the maternal recognition, which the heartless,\ndishonored woman refused.  So, this castaway runs to drink and all\ndeviltries; Johnson staying him much as {93} he can--walking with him\nup and down through London streets till midnight--talking poetry,\nphilosophy, religion; hungry both of them, and many a time with only\nten pence between them.\n\nWell, at last, Savage kills his man in a tavern broil; would have been\nhung--the mother countess (as the story runs) hoping it would be so;\nbut he escapes, largely through the influence of that Queen Caroline,\nto whom Jeanie Deans makes her eloquent plea in Scott's ever-famous\nnovel of _The Heart of Mid-Lothian_.  Savage escapes, but 'tis only to\ngo to other bad ways, and at last he died in a Bristol jail.\n\nAll this offered material for a pathetic story, and Johnson made the\nmost of it in his _Life of Savage_--afterward incorporated in his\n_Lives of the Poets_, but first published in 1744, about seven years\nafter his coming to London.  The book appeared anonymously; but its\nqualities gave it great vogue; and its essential averments formed the\nbasis of all biographic and encyclopaedic[23] notices for nearly a\ncentury thereafter.\n\n{94}\n\nBut was the story true?  There were those who doubted at the time, and\nhad an unpleasant sense that Johnson had been wheedled by an\nadventurer; but demonstration of the imposture of Savage did not come\ntill the middle of the present century.  The investigations of Moy\nThomas[24] would go to show that the Savage friend of Johnson's early\ndays in London was the most arrant of impostors; and that of all the\nshame that rests upon him, he can only justly be relieved of that which\ncounts him a child of such a woman as the Countess of Macclesfield.  I\nhave dwelt upon the Savage episode, not alone because it provoked one\nof Johnson's best pieces of prose work, but because it shows how open\nwere his sympathies to such tales of distress, and how quick he was to\nlift the rod of chastisement upon wrong-doers of whatever degree.\n\nIn _London_, too, that imitative classic poem, there shone in a glitter\nof couplets (which provoked Pope's praises) the same righteous\nindignation, and the stings--pricking through all his big {95}\nStaffordshire bulk--of supperless-days and of shortened means:--\n\n  \"By numbers here from shame or censure free\n  All crimes are safe, but hated Poverty;\n  This, only this, the rigid Law pursues,\n  This, only this, provokes the snarling muse.\n\n  \"The sober trader at a tattered cloak\n  Wakes from his dream, and labors for a joke;\n  With brisker air the silken courtiers gaze\n  And turn the varied taunt a thousand ways.\"\n\n\nBetter than this was that poem (_Vanity of Human Wishes_) in which,\neven now, some of us--admiringly--\n\n  \"In full flown dignity see Wolsey stand,\n  Law in his voice, and fortune in his hand.\"\n\nAnd the couplet leads on through Wolsey's story to the poet's coupleted\nsermon, with its savors of a church-bell--\n\n  \"Still raise for good the supplicating voice,\n  But leave to Heaven the measure and the choice.\n  Safe in his power whose eye discerns afar\n  The secret ambush of a specious prayer;\n  Implore his aid, in his decisions rest,\n  Secure whate'er he gives, he gives the best.\n      *      *      *      *      *\n\n{96}\n\n  Pour forth thy fervors for a healthful mind,\n  Obedient passions, and a will resigned;\n      *      *      *      *      *\n  For faith, that, panting for a happier seat,\n  Counts death kind nature's signal of retreat.\"\n\n\nWe must note also that famous Prologue, spoken at Drury Lane in 1747,\nwhen the theatre came first under control of his old friend, Garrick.\nNever had the stage, before nor since, a nobler summons in worthier\nverse: it closes--\n\n  \"Then prompt no more the follies you decry,\n  As Tyrants doom their tools of guilt to die:\n  'Tis yours, this night, to bid the reign commence\n  Of rescued Nature and reviving Sense:\n  To chase the charms of Sound, the pomp of Show,\n  For useful Mirth and salutary Woe:\n  Bid scenic Virtue form the rising age\n  And Truth diffuse her radiance from the Stage.\"\n\n\nGarrick must have been proud to act under such banner of song as that.\nThe tragedy of _Irene_ came to its first representation a short time\nafterward; and surely it would have been worth one's while to see the\nstout, awkward gerund-grinder of forty, slipping into a side-box, or\neven behind {97} the scenes \"in a scarlet waistcoat, with rich gold\nlace and a gold-laced hat!\"  The play, however, did not prove a great\nsuccess either then or thereafter.  The Dictionary, for which proposals\nhad already been issued, promised better things.  That Dictionary did\nultimately give him a great lift--as it has to a good many, since.  The\nponderous volume furnished very many New England households seventy\nyears ago; and I can remember sitting upon it, in my child-days, to\nbring my head properly above the level of the table.  An immense and\nlong-continued toil went to the Dictionary.  Lord Chesterfield,[25] the\nfinished orator and the elegant man--not unwilling to have so great a\nwork bear his name--called attention to the book and the author, when\nnearly ready; but Johnson was too sore with hope deferred to catch {98}\nthat bait; he writes an indignant letter (not published until 1790) to\nthe elegant Chesterfield:--\n\n\n\"Seven years have now passed, my Lord, since I waited in your\noutward-rooms, or was repulsed from your door--during which time I have\nbeen pushing on my work, thro' difficulties of which it is useless to\ncomplain, and have brought it at last to the verge of publication\nwithout one act of assistance, one word of encouragement, or one smile\nof favor....  The notice which you have been pleased to take of my\nlabors--had it been early--had been kind; but it has been delayed till\nI am indifferent and cannot enjoy it; till I am solitary [his wife dead\nnow] and cannot impart it, till I am known and do not want it.\"\n\n\nThis does not show the stuff which went to the making of such a man as\nWalpole!\n\nThe _Rambler_, too, it must be remembered, is making its periodic\nvisits in those early days of the Dictionary toil.  Heavy it is, like\nthe master; and his prejudices as arrant Churchman and sturdy Tory do\nindeed break through its piled-up pages; but never insidiously: he\nsounds a trumpet before he strikes.  Perhaps a little over-fond of\ntrumpeting; loving so much his long sonorous roll of Ciceronian\nvocables.\n\nBut I have not the same dislike of long {99} Johnsonian periods that a\ngood many people have--provided always there is a Johnson to utter\nthem.  They belong to _him_; they match with their wordy convolutions\nhis great billowy make of mind; and short, sharp sentences would be as\nincongruous as a little spurting _jet d'eau_ where great waves come\nrocking on the beach.\n\nIn fact, I have a large unbelief in much of current pedagogic talk\nabout style, and \"getting a good style,\" and \"reforming style,\" and\n\"Saxon style,\" and so on.  To be thoroughly possessed of one's own\nthought, and then to tell it, in the clearest possible way, is the best\nlaw I know for a good style; and a proper following of it will give to\nevery mind that has any color of its own a style of its own.  To putter\nabout the rhetorics in search of fine phrases to wrap your thoughts in,\nis like going in masquerade; furbish it as you will, people will see\nthe smear of old wear in the tinsel trappings, and smell it too.\n\nIf short, homely Saxon will serve one's purpose best in giving sharp,\nshrewd expression to thought, as most times it will, use Saxon; but if\na Latin derivation will hit the very shade of your thinking {100} more\naptly, do not affect to scorn the Latin.  Even if a French\nword--provided always it be at once and easily comprehensible by all\nwhom you address--shall touch the very eye of your purpose better than\nanother, do not scruple to use it.\n\nBut we must ask pardon for this intrusion of small school-mastery talk,\nwhile the great master of the Dictionary and of the _Rambler_ waits.\nAs yet we have followed him through only half of his career; a stalwart\nman, still in the full prime of his years; and I see grouping about him\nat the Turk's Head many another whom we wish to follow; a Boswell and a\nBurke; Reynolds and Beauclerk and Goldsmith--these all are in waiting.\nBut for a fuller and nearer view of these old club-men of more than a\ncentury ago, we open upon another chapter of these _Lands and Letters_.\n\n\n\n[1] Whoso would take measure, of his scholarly thoroughness, his reach,\nhis pertinacity, and his capacity for striking sharp blows, should\nstruggle through his _Dissertation on Phalaris_.\n\n[2] Swift, Addison, Steele, Gay, _et al._, in preceding volume of\n_Lands, Letters, and Kings_ (\"Elizabeth to Anne\")\n\n[3] He lived for many years in the _Palazzo Muti_ near to the church of\nthe _SS. Apostoli_, in Rome; his disorderly life there made it a _Regio\nPalazzo_!\n\n[4] _Lands and Letters_: \"From Elizabeth to Anne,\" p. 100.\n\n[5] _Lands and Letters_: \"Elizabeth to Anne.\"\n\n[6] This is one contemporary account of it--adopted by Thackeray; but\nWraxall (1st vol., pp. 384-385 American reprint, Lea & Blanchard) says\nthat the Duke of Dorset was commissioned to carry the news; but some\nlittle time being required to make himself ready, the Duchess was sent\nin advance.  She arrived at Kew (where the Prince was staying) just as\nthat Prince had gone to bed, as was his wont, after dinner.  The\nPrincess undertook the announcement--though demurring at the duty, and\nanticipating a brutal reception for one who should disturb his\nafter-dinner nap; he was in a huff and _did_ make the comment, noted in\nthe text; but it was not (says Wraxall) to a messenger in jack-boots,\nbut to the Princess of Wales herself.\n\n[7] Richardson: b. 1689; d. 1761.  Various editions of his works.\nKnown quite generally to buyers of cheap books in our day by an\nabbreviated issue of _Clarissa Harlowe_ (Routledge & Sons).\n\n[8] Henry Fielding: b. 1707; d. 1754.  Editions of his works have been\nedited by Arthur Murphy, William Roscoe, and Leslie Stephen; (10 vols.,\n1882-1883.)  Life by Sir Walter Scott in Ballantyne Library; more\ntrustworthy one is that by Austin Dobson.\n\n[9] It is perhaps to be doubted if the bare-faced coarsenesses of\nFielding (much as they are to be condemned) would provoke pruriency so\nmuch as the sentimental and sensuous languors of Richardson.\n\n[10] _History of Pendennis_, Household Ed., Boston: Chap. xxix.\n\n[11] It was in virtue of some altercations growing out of Fielding's\nplays that British censorship was established in 1737, and (perhaps)\nFielding thereby diverted to the study of Law.\n\n[12] James Thomson, b. 1700; d. 1748.  Various editions of his poems; a\nvery elegant one, illustrated by the Etching Club, published 1842-62.\n\n[13] _The Jubilee of the Constitution_, a discourse delivered by\nrequest of the New York Historical Society, April 30, 1839, and\nrepeated shortly after in the old \"Ludlow\" Church, (now \"Dime\nTheatre\"), in Church Street, New Haven.\n\n[14] Thomas Gray, b. 1716; d. 1771.  See Gosse's recent biography for\ncritical as well as sympathetic account of his life and writings.  See\nalso Mitford's edition of his works, with life, London, 1836.\n\n[15] Horace Walpole, b. 1717; d. 1785.  The enumeration of his books,\npamphlets, and of titles relating thereto fill a dozen columns of\n_Lowndes_.  His letters give best measurement of the man.\n\n[16] It purported to be a translation from the Italian of Onuphrio\nMuralto.\n\n[17] Peter Cunningham Edition.  London, 1857-1859.  See also _Horace\nWalpole and His World_, by L. B. Seeley.  1884.\n\n[18] Rev. William Mason, b. 1725; d. 1797; author of _The English\nGarden_, published at intervals (its successive books) between 1772 and\n1782.  It has little merit--Walpole to the contrary.\n\n[19] _Wet Days at Edgewood_, p. 239.\n\n[20] Samuel Johnson, b. 1709; d. 1784.  Boswell's the standard life of\nhim, and Birkbeck Hill's the best edition of that life.  We miss in it,\nindeed, some of the \"Croker\" notes, which made such inviting quarry for\nthe sharp huntsmanship of Macaulay.  But the editing is done with a\nlove and a tirelessness which are as winning as they are rare.  See,\nalso, Leslie Stephen's sketch--which is the best short life.\n\n[21] _Ency. Britannica_; Art. Johnson.\n\n[22] B. 1698; d. 1743.  Poet and dramatist.  Collected edit. of his\nwritings published in 1775.  His largest claim to distinction is due to\nthe _Life of Richard Savage_, by Samuel Johnson; first published 1744.\n\n[23] _Vide_ old edition of _Ency. Britannica_, also Strahan's\n_Biographical Dictionary_ of 1784; _Biographie Universelle, et al._\n\n[24] See _Notes and Queries_, November and December, 1858.\n\n[25] Philip Dormer Stanhope (Earl of Chesterfield), b. 1694; d. 1773,\nbest known by his _Letters to His Son_, first published in 1774.\nJohnson said they taught \"the morals of a courtesan, and the manners of\na dancing-master.\"  This was perhaps over-severe.  People who do not\nlove to disport in fashionable waters are apt to be severe upon those\nwho spend their faculties upon the coquetries of bathing costume.\n\n\n\n\n{101}\n\nCHAPTER III.\n\nIt was a little after the middle of the last century that our story\nopens again.  George II., whose virtues and vices were clock-like in\ntheir regularities, was on the throne; Queen Caroline, whom he had\nalways abused and always venerated, was in her grave for twelve or more\nyears past.  Outside politics were ripening for that French and English\nwar--in which a Montcalm and a Wolfe figured upon our side the water,\nand which has been put in picturesque array by Francis Parkman; the\ngeraniums and oleanders were blossoming over the Portuguese grave of\nHarry Fielding; Thomson had sung his last notes in his _Castle of\nIndolence_ and was laid to rest--not in Kelso, or Dryburgh, where his\nbody should have mouldered--but in a little Richmond Church, within\ngunshot of the \"Star and Garter.\" {102} Gray was still studying the\nscholarly measures of the _Bard_, in his beloved Cambridge; Horace\nWalpole playing the _elegant_ was fattening on his revenues at\nStrawberry Hill; while Dr. Johnson--notwithstanding the Dictionary and\nthe _Rambler_--had been latterly (1756) in such sore straits as to\nappeal to his friend Richardson for the loan of a few guineas to save\nhim from jail; and Richardson, fresh then in his triumphs from\n_Clarissa Harlowe_ and the great _Grandison_, was not slow to grant the\nrequest,[1] and to enjoy all the more his Kingship among the women, in\nhis great house out at Hammersmith.\n\n[Sidenote: London streets.]\n\nA sharp walk of a quarter of an hour from St. Paul's would, in that\ntime, take one into the green fields that lay in Islington; and beyond,\nupon the Waltham road, were the hedges, pikes, and quiet paddocks,\nthrough which went galloping--at a little later day--that citizen of\n\"credit and renown,\" John Gilpin, instead of the clattering suburbs\nthat now stretch nearly all the way between Cheapside and the \"Bell\" at\nEdmonton.\n\n{103}\n\nOf the many bridges which now span the Thames, only two[2]\nrepresentatives were in existence; the old Westminster was there in its\nfirst freshness, and ferrymen quarrelling with it, because it spoiled\ntheir carrying trade to Vauxhall and parts adjacent; and the old London\nBridge was cumbered by lumbering houses, held up by trusses and\ncross-beams, while its openings were so low and its piers so many as to\nmake, at certain stages of the tide, furious cascades which drove great\nwheels geared to cumbrous pumping machinery, to throw up water for the\nbehoof of London citizens.  The old Fleet Prison was in existence, and\nits smudgy stifling air hung over all that low region above which now\nleap the great arches of the Holborn Viaduct; and round the corner, in\nthe reek and smoke of Fleet Street, half way between the spire of St.\nBride's and the spire of St. Clement's Danes--up a grimy court that is,\nvery likely, just as grimy to-day, lived that Leviathan of a man, Dr.\nSamuel Johnson.\n\n\n{104}\n\n_Johnson and Rasselas._\n\n[Sidenote: Rasselas.]\n\nHe had passed through his green days, and the nights when he strolled\nsupperless about London with that poor wretch of a poet Richard Savage.\nThe school at Edial with its three pupils was well behind him; so was\nthe dining behind the screen at Cave's (the bookseller who presided\nover the _Gentleman's Magazine_, with St. John's Gate on the cover\nthen, and on the cover now): so was his age of sentiment ended.\n\nHis wife Tetty had gone the way of all flesh (1752) and he had mourned\nher truly: in proof of this may be counted the presence under his roof\nof a certain old lady, Miss Williams, who is peevish, who is\ntempestuous, who is blind, who tests the tea with her fingers, who\n_will_ talk, and then again, she _won't_ talk; yet Johnson befriends\nher, pensions her--when he has money,--sends home sweetbreads from the\ntavern for her; and when his friends ask why he tolerates this vixen,\nhe gives the soundest reason that he has--\"she {105} was a friend of\nTetty; she was with poor Tetty when she died!\"\n\nAnd his brain was as big, or bigger, than his heart; it had made itself\nfelt all over England by long, honest work--by brave, loud speech.  He\nhad snubbed the elegant Lord Chesterfield, who would have liked to see\nhis name upon the first page of the great Dictionary.  Not an outcast\nof the neighborhood but had heard of his audacious kindness; not a\nlinkboy but knew him by the chink of his half-pence; not a beggar but\nhad been bettered by his generous dole; not a watchman but knew him by\nhis unwieldy hulk, and his awkward, intrepid walk; and we know him--if\nwe know him at all--not by his _Rambler_ and his _Rasselas_, so much as\nby the story of his life.  Who rates _Rasselas_ among his or her\ncherished books of fiction?\n\nWhat an unlikely, and what a ponderous beginning it has!\n\n\n\"Ye who listen with credulity to the whispers of fancy and pursue with\neagerness the phantoms of hope; who expect that age will perform the\npromises of youth, and that the deficiencies of the present day will be\nsupplied by the morrow, attend to the history of Rasselas, Prince of\nAbyssinia!\"\n\n\n{106}\n\nWhen, in days long past, I have read thus far in this elephantine\nnovelette to my children, they were pretty apt to explode upon me\nwith--\"Please try something else!\"  Yet this elephantine novelette has\na host of excellent and eloquent moral reflections in it, shouldering\nand elbowing themselves out from its flimsy dress of fiction.  Shall I\ngive a hint of the scheme of this old story?  An Abyssinian prince\nliving in the middle of a happy valley, walled in by mountains that are\nbeautiful, and watered by rivers that are musical, in the enjoyment of\nall luxuries, does at last become restless--as so many people do--not\nso much from a want, as from the want of a want.  So he conspires with\nImlac, a poet, to escape from the thraldom of complete ease: a sister\nof the prince and her handmaid steal away with them; and with plenty of\njewels the party enter upon their exploration of the ways of outside\nlife.  They encounter hermits whose solitude does not cure their pains,\nand shepherds whose simplicities do not conquer misfortune, and\nphilosophers whose philosophy does not relieve their anxieties, and\nscholars whose learning does not make them happy.\n\n{107}\n\nImlac, the poet, sums up their findings in saying--\"You will rarely\nmeet one who does not think the lot of his neighbor better than his\nown.\"  This is its whole philosophy.  There are interlarded discourses\nupon learning, and marriage, and death, and riches, which might have\nbeen cut from a _Rambler_ or from a sermon.  They travel through upper\nEgypt, and sojourn in the grand Cairo; but there is no shimmer of the\ndesert, and no flash of crescent or scimitar, and no dreamy\norientalism; its Eastern sages talk as if they might have thundered\ntheir ponderous sentences from the pulpit of St. Bride's.  As a\nfinality--if the tale can be said to have any finality--the princess\nthinks she would like--of all things--Knowledge: the poor handmaid, who\nhas had her little adventure, by being captured by a Bedouin chief,\nthinks she would like best a convent on some oasis in the desert; while\nthe prince would like a miniature kingdom whose rule he might\nadminister with justice as easily as one might wind a watch; but all\nagree that, when the Nile flood favors, they will go contentedly back\nto the happy valley from which they set out upon their {108}\nwanderings.  It is interesting to know that the story was written by\nDr. Johnson on the evenings of a single week; and written--before he\nhad come to his pension[3]--to defray the expenses of his mother's\nfuneral; and it is interesting further to know that the magniloquent\ntale did forge its way into the front rank of readers at a time when\n_Roderick Random_ and _Tom Jones_ were comparatively fresh books, and\nonly five years after Mr. Richardson had issued from his book-shop\nunder the shadows of St. Bride's, hardly a gunshot away from the house\nof Johnson, the voluminous history of _Sir Charles Grandison_.\n\n\n_The Painter and the Club._\n\n[Sidenote: Sir Joshua Reynolds.]\n\nAmong the friends the Doctor made in those days of _Ramblers_ and\n_Idlers_ was one Joshua Reynolds,[4] some fourteen years the junior of\nthe Doctor, but sedate and thoughtful beyond his age; with an eye, too,\nfor the beautiful faces of young {109} English girls which had never\nbeen opened on them before; and doing artist work that is quite\ndifferent in quality and motive from that of the old stand-by Mr.\nHogarth, who not long before this time had been preaching his painted\nsermons of the _Rake's Progress_.\n\nReynolds had made his trip to Italy, and had brought back from Rome, in\naddition to his studies of Raphael--an affection of the ear--caught, as\nhe always said, in the draughty corridors of the Vatican, which obliged\nhim ever after to carry an ear-trumpet; but his courtesy and grace and\nprecision of speech made the awkwardness forgotten.  Looking at the\nexquisite child's face of his little Penelope Boothby, expressing all\nthat was most winning in girlhood for him who was so reverent of\nexterior graces, and looking from this to the leathern, seamy face of\nJohnson, and his unlaundered linen, and snuffy frills (when he wore\nany), and it is hard to understand the intimacy of these two men; but\nthere was a tenderness of soul under the Doctor's slouchy ways which\nthe keen painter recognized; and in the painter there was a resolute\nintellection, which Johnson was not slow to {110} detect, and which\npresently--when the new Royal Academy was founded by George III.--was\nto have expression in the great painter's discourses on Art--discourses\nwhich for their courageous common-sense will, I think, outrank much of\nthe art-writing of to-day.\n\n[Sidenote: Turks-Head Club.]\n\nIn 1760 (the year after _Rasselas_ appeared) Reynolds moved into a fine\nhouse, for that day, in Leicester Square--a quarter now given over\nmostly to French lodgers; but in its neighborhood one may find a marble\nbust of the eminent painter; and the house where he gave great steaming\ndinners--famous for their profusion and disorderly array--is still\nthere, though given over to small artists and sellers of bric-a-brac.\nHis good sister, Miss Fanny, who was his housekeeper, loved painting\nand poetry, and a drive in the painter's chariot, which he set up in\nlater days, better than she loved housewifery.  Over-shrewd ones said\nthat Sir Joshua (the title came to him a few years after with the\npresidency of the Royal Academy) did not marry because he had wholesome\ndread of a wife's extravagance; certain it is that he remained a\nbachelor all his life, and {111} thereby was a fitting person to\ndiscuss with the widowed Johnson the formation of a club.  The Doctor\nwas always clubably disposed; so he caught at the idea of Sir Joshua,\nand thence sprung that society--called \"The Literary Club\" afterward,\nwhich held its sessions, first at the Turk's Head, in Gerrard Street,\nSoho Square--on Monday evenings at the start, and afterward on\nFridays--numbering among its early members Johnson, Burke, Reynolds,\nHawkins, Beauclerk, and Goldsmith.  This famous club, though moving\nfrom place to place in the closing years of the last century, still\npreserved its identity; it took a new lease of life in the first\nquarter of the present century, and it still survives in a very quiet\nold age, holding its fortnightly meetings--rather sparingly attended,\nit is true--at Willis's Rooms, St. James's Street, in the west of\nLondon.  Among recent members may be named Gladstone, Sir Frederick\nLeighton, Lord Salisbury, the Duke of Argyle, Tennyson, and Matthew\nArnold.[5]\n\n\n{112}\n\n_Some Old Club-men._\n\n[Sidenote: Edmund Burke.]\n\nBurke,[6] who was among the original nine members, was very much the\njunior of Johnson; but known to him as a sometime Irish student at law,\nwho had written only a few years before two brilliant treatises; one on\n_Natural Society_, and the other on the _Sublime and Beautiful_.\n\nLater he had done excellent historic work in connection with _Dodsley's\nAnnual Register_; but he had not yet entered upon that sea of political\nturmoil over which he was to sweep in so grand a way and with such\nblaze of triumph.  It is possible indeed that he was indebted to the\nassociations of the club for some of the initiative steps toward that\nwonderful career whose outcome in Parliament, in the courts, and in\npamphleteering, has become a component part of the {113} literature of\nEngland.  Burke, even at that early stage of his progress (his first\nspeech was made in 1766) had all his vast resources at ready command;\nJohnson did not wish to meet him in debate without warning; true he was\nafraid of no mere eloquence; he was used to puncturing bloat of that\nsort; but Burke's most fiery speeches were beaded throughout with\nglobules of thought, which must be grasped and squelched one by one, if\nmastery were sought.  He was impetuous, too, and aggressive, but\nreverent of the superior age and reputation of the Doctor; and I\ndaresay coyly avoided those American questions which later came to the\nfront, and upon which they held views diametrically opposed.  In after\nyears it used to be said that Burke's speeches would empty the benches\nof the Commons--ye philosophized; and when not heated, spoke with a\ndrawling utterance and a touch of Irish brogue flavoring his voice;\nindeed he talked so well he was never tired of talking; his sentences\nso swelled out under the amplitude of his illustrations and allusions\nthat I think he came at last to take a pride in their very longitude,\nand trailed his gorgeous convolutions of {114} speech with the\ndelighted eagerness with which a fine woman trails her sheen of satins\nand velvets.\n\n[Sidenote: Topham Beauclerk.]\n\nDr. Nugent, a physician of culture, father-in-law of Mr. Burke, was\nalso one of the original members of the club--getting the\npreferment--as so many in all times do get preferment--simply because\nson-in-law, father-in-law, or nephew--to somebody else.  Another\nnoticeable member of the club was Topham Beauclerk, not by any means\nthe man a casual observer would have taken for an associate of Johnson.\nHe was courtly and elegant in bearing, a man of fashion, smiled upon by\nsuch as Lord Chesterfield and Horace Walpole, and who traced his\ndescent back through the first Duke of St. Albans to Nell Gwynne and\nCharles the Second.  He inherited by right, therefore, gayety and humor\nand wit, and rare histrionic power, and Satan-ry to match.  Old Dr.\nJohnson fairly languished in his admiration of the way in which Topham\nBeauclerk could tell a story.  \"It costs me fearful pains,\" he was used\nto say; \"but this fellow trips through with an airy grace that costs\nhim nothing.\"\n\nBeauclerk was proud of his membership, and {115} brought his own share\nof wit, of general information, and of cheery _bonhomie_ to the common\nreckoning.  He married a certain well-known and much-admired Lady Diana\nBolingbroke--a divorcee of two days' standing--and treated her\nshamefully; that being the proper thing for a fashionable man to do,\nwho was emulous of the domestic virtues of George II.  At his death,\nwith a large jointure in hand, she had peace; and Burke said, with a\nhumor that was uncommon to him: \"It was really enlivening to behold her\nplaced in that sweet house, released from all her cares: L1,000 per\nannum at her disposal, and her husband dead!  It was pleasant, it was\ndelightful, to see her enjoyment of the situation!\"  Beauclerk was too\nfine a fellow to think well of the domesticities; there was a good deal\nof the blood of Charles the Second in him.  Over and over we come upon\nsuch--men of parts squandered in the small interchanges of fashionable\nlife; perpetually saying slight, good things for a dinner-table;\ntelling a story with rare gusto; the envy of heavy talkers who can\nnever catch butterflies on the wing; looking down upon serious duty\nwhether in art or {116} letters; and so, leaving nothing behind them\nbut a pretty and not always delicate perfume.\n\n[Sidenote: David Garrick.]\n\nAnother of the clubmen was David Garrick--not one of the original nine,\nbut voted in a few years after.  Dr. Johnson does indeed give a\ncharacteristic growl when his name is proposed--\"What do we want of\nplay-actors?\" but his good nature triumphs.  Little Garrick was an old\nscholar of his at Edial; and though he has conquered all theatric arts\nand won all their prizes, he is still for him, \"little Garrick.\"  A\ntaste for splendor and dress had always belonged to him.  In his\nboy-days he had written to his father, who was stationed at Gibraltar,\n\"I hope, Papa, you find velvet cheap there; for some one has given me a\nknee-buckle, and it would go capitally with velvet breeches.  Amen, and\nso be it!\"\n\nThat love for the buckles and the velvet clung to him.  When Edial\nschool broke up, he tramped with Johnson to London--the master with the\npoor tragedy of _Irene_ in his pocket, and the boy with such gewgaws\nand pence as he could rake together.  Perhaps, also, the tragic\nsplendor of Shakespeare's verse shimmering mistily across his {117}\nvisions of the future, making his finger-ends tingle and his pulse beat\nhigh.\n\nBut a legacy of L1,000 comes to the Garrick lad presently, which he\ninvests in a wine business, in company with his staid brother, Peter\nGarrick, who looks after affairs in Lichfield, and who is terribly\ndisturbed when he hears that David is taking to theatric studies;--has\nacted parts even!\n\nAnd Davy writes back relenting, and sorry to grieve them at home; but\nkeeps at his parts.  And Peter writes more and more disconsolately,\nlamenting this great reproach, and David writes pretty letters of\nfence, and the wine business leaks away, and Peter is in despair; and\nDavy sends remittances which are certainly not legitimate business\ndividends, thus propping up the sinking wine venture; and before Peter\nis reconciled, has become the hero of the London boards, with a bank\ncredit that would buy all their ports and clarets twice over.\n\nAnd this wonderful histrionic genius, probably unparalleled on the\nEnglish stage before or since his day, so gay, so brisk--so witty\nbetimes--so capable of a song or a fandango, brought life to the {118}\nclub.  Nor was there lack in him of literary qualities; his prologues\nwere of the best, and he had the charming art of listening\nprovocatively when the great doctor expounded.\n\n\n_Mr. Boswell._\n\n[Sidenote: James Boswell.]\n\nAnother early member of the club, whom I think we should have liked to\nsee making his way with a very assured step into the Turk's Head, of a\nMonday or a Friday, was James Boswell, Esquire.[7]  It is a household\nname now, and will remain so for years to come by reason of the\nextraordinary life which he wrote, of his master and patron, Dr.\nJohnson.  Yet it was only a year or so before the formation of the club\nthat this jaunty Scotch gentleman, son of a laird, and of vast\nassurance--having been a tuft-hunter from his youth--caught his first\nsight of the great Doctor, in the little shop of Davies the bookseller;\nand the great man had given a snubbing, then and there, to the pert,\nbut always obsequious Boswell; the future biographer, however, digested\n{119} excellently well provision of that sort, and I think the Doctor\nhad always a tenderness for those who took his flagellations without\ncomplaint.  Certain it is that there grew up thereafter an intimacy\nbetween the two, which is one of the most curious things in the history\nof English Men of Letters.  I know that hard things are said of Mr.\nBoswell, and that every tyro in criticism loves to have a blow at the\nwell-fed arrogance of the man.  Macaulay has specially given him a\ngrievous black-eye; but Macaulay--particularly in those early review\npapers--was apt to let his exuberant and cumulative rhetoric carry him\nup to a climacteric which the ladder of his facts would scantly reach.\nTo be sure Boswell was a toady; but rather from veneration of those he\nworshipped than desire of personal advancement; he was an arrant\ntuft-hunter, thereby enlarging the sphere of his observations; but he\nwas fairly up in classical studies; had large fund of information; was\nsufficiently well-bred (indeed, in contrast with the Doctor, I think we\nmay say excellently well-bred); he rarely, if ever, said malicious\nthings, though often impertinent ones; {120} his conundrums again and\nagain gave a new turn to dull talk; and he had a way, which some even\nmore stolid people possess in our time, of _baiting_ conversation by\ninterposing irrelevant matter, with an air of innocence that\ncaptivates; then there was the pleasant conceit of the man--full-fed,\nsleek, and shining out all over him--over his face, and his erect but\nsomewhat paunchy figure; all which qualities were contributory to the\nhumor and fulness and charm of that famous biography which we can read\nbackward or forward--in the morning or at night--by the chapter or by\nthe page--with our pipe or without it--with our knitting or without\nit--and always with an amazed and delighted sense that the dear, old,\nclumsy, gray-stockinged, snuff-ridden Doctor has come to life once\nmore, and is toddling along our streets, belching out his wit and\nwrath, and leaning on the arm of the ever-ready and most excellent and\nobsequious James Boswell, Esquire.\n\nSuch a book is not to be sneered at, nor the writer of it; perhaps we\nthink it would be easy for us or anybody to write such another, if we\n{121} would only forget conventionalisms and have the courage of our\nimpressions; but if we made trial, I daresay we should find that to\nforget conventionalisms is just what we can't and do not know how to\ndo; and so our impressions get bundled into the swathings of an\nambitious rhetoric which spoils our chances and vulgarizes effort.  I\ndo not say Boswell was a very high-toned man or a very capable man in\nmost directions; but he did have the art of easy narrative to a most\nuncommon degree; and did clearly perceive and apprehend just those\npoints and qualities which go to make portraiture complete and\nsatisfying.[8]  I do not believe that he stupidly blundered into doing\nhis biographic work well; stupid {122} blundering never did and never\ncould accomplish work that will meet acceptance by the intelligence of\nthe world.\n\n\n_Gibbon._\n\n[Sidenote: Edward Gibbon.]\n\nI come now to speak of a more respectable personage--one of whom you\nhave often heard, and whose resounding periods, full of Roman History\nyou will most surely have read; I mean Edward Gibbon[9]--not an\noriginal member of the club, but elected at an early day.  His life has\ngreat interest.  He was the sole survivor of seven children; his father\nbeing a Member of Parliament--very reputable, but very inefficient.\nThere were fears that his famous son would be a <DW36> for life, so\nweakly was he, and so ill put together; but growing stronger, he went\nto Oxford; was there for only a short time; did not love Oxford {123}\nthen, or ever; inclined to theologic inquiry and became Romanist; which\nso angered his father that he sent him to Lausanne, Switzerland, to be\nre-converted under the Calvinist teachers of that region to\nProtestantism.  This in due time came about; and it was perhaps by a\nsort of compensating mental retaliation for this topsy-turvy condition\nof his youth that he assumed and cultivated the pugnacious indifference\nto religion which so marked all his later years and much of his work.\nHe had his love passages, too, there upon the beautiful borders of Lake\nGeneva; a certain Mademoiselle Curchod, daughter of a Protestant\nclergyman, lived near by; and with her the future historian read\npoetry, read philosophy, read the skies and the mountains, discoursed\nupon the conjugation of verbs, and upon conjugalities of other sorts;\nbut this the English father disapproved as much as he had disapproved\nof Romanism; and by reason of this--as Gibbon tells us, in his\ndelightful autobiography--that \"sweet dream came to an end.\"  It is\ntrue the French biographers[10] {124} put a rather different phase upon\nthe story, and represent that while Mademoiselle respected young Mr.\nGibbon very much, she could not return his ardor.  Two colors, I have\nobserved, are very commonly given to any sudden interruption of such\nfestivities.\n\nMademoiselle, however, did not pine in single blessedness; she had a\ncareer before her.  She became in a few years the distinguished wife of\nNecker, the great finance minister of France in the days immediately\npreceding the Revolution, and the mother of a still more famous\ndaughter--that Mme. de Stael who wrote _Corinne_.\n\nThough Gibbon lived and died a bachelor, he always maintained friendly\nrelations with his old flame Mme. Necker, being frequently a guest at\nher elegant Paris home; and she, on at least one occasion, a guest of\nthe historian in London.  It was in the year 1774--ten years after its\nfoundation, that Gibbon was elected member of the Literary Club; he\nbeing then in his thirty-seventh year and well known for his wide\nlearning and his conversational powers.  He was recognized as an\nauthor, too, of critical acumen, and great range {125} of language;\nsome of his earlier treatises were written in French, which he knew as\nwell as English; German he never knew; but the first volume of the\n_History of the Decline and Fall of the Roman Empire_ did not appear\nuntil the year 1776--a good tag for that great American date!  That\nfirst volume made a prodigious surprise, and immense applause.  Poor\nHume[11] (whose story waits), struggling with the mortal disease which\nwas to carry him off in that year, wrote his praises from Edinburgh.\nHorace Walpole, who had the vanity of professing to know everybody\nworth knowing, says, \"I am astonished; I know the man a little; I could\nnot believe it was in him; I must get to know him better.\"\n\nYet Gibbon was not a modest man in the ordinary sense; never, except\nwhen--very rarely--warmed into a colloquial display of his\nextraordinary learning, did he impress a stranger with any sense of his\npower.  He was short and corpulent; had a waddling walk and puffy\ncheeks and a weak {126} double chin; with very much in his general\naspect and manner to explain the miscarriage of his love-affair, and\nnothing at all to explain the Decline of the Roman Empire.  Withal, he\nwas obsequious, studiously courteous; had ready smiles at command; had\na mincing manner; his wig was always in order, and so was his flowered\nwaistcoat; and he tapped his snuff-box with an easy degage air, that\ngave no warrant for anything more than an agreeable titillation of the\nnerves.  But if an opening came for a thrust of his cumulated learning\nin establishing some historic point in dispute, it poured out with a\ngush, authority upon authority, citation on citation, as full and\nimpetuous and unlooked for as a great spring flood.\n\nHe went over to Paris with his honors fresh upon him; was cordially\nreceived there; the Necker influence, and his familiarity with French,\nstanding him in good stead.  He affected a certain style too.  \"I\nhave,\" he says, \"two footmen in handsome liveries behind my coach, and\nmy apartment is hung with damask.\"  He loved such display, though only\nthe hired luxury of a hotel.  He had never a taste for the simpler\nenjoyments of {127} English country life; never mounted a horse and\nscorned partridge shooting or angling.  In a letter to a friend he\nsays, \"Never pretend to allure me by painting in odious colors the dust\nof London.  I love the dust, and whenever I move into the Weald, it is\nto visit _you_, and not your trees.\"\n\nIt does not appear that he went frequently to the Turk's-Head Club.\nThe brusquerie of Johnson would have grated on him--grated on him in\nmore senses than one, we suspect; and the gruff Doctor would have\nscorned his dilettanteism as much as his scepticism.  Gibbon took\nkindly, though, to Goldsmith; but he hated Boswell honestly, and\nBoswell honestly hated back.[12]\n\nHis letters were never strong or bright, nor were his occasional\nliterary criticisms either acute or profound; all his great powers were\nkept in reserve for his _magnum opus_--the History.  For the quietude\nhe thought necessary to its completion he went again to the home of his\nyouth at Lausanne, and there, in sight of that wondrous {128} panorama\nof lake and mountain, upon a site where now stands the Hotel\nGibbon,[13] and a few acacia trees under which the historian meditated,\nthe great work was brought to completion--a great work then, and a\ngreat work now, measured by what standard we will.  To say that one\napproaches the accuracy of Gibbon is to exhaust praise; to say that one\nsurpasses him in reach of learning is to deal in hyperbole.  Even the\nhistorian, Dr. Freeman, who, I think, did much prefer saying a critical\nthing to saying a pleasant thing, testified that--\"He remains the one\nhistorian of the eighteenth century whom modern research has neither\nset aside, nor threatened to set aside.\"  Modern high critics sneer at\nhis large, ceremonious manner; Ruskin pronounces \"his English the worst\never written by an educated Englishman\"[14] (the same Ruskin who found\na \"mass of errors\" under the sunshine of Claude).  But let {129} us\nremember what burden of knowledge those grandiloquent sentences of\nGibbon had to carry; what reach of empire they had to cover!  Here be\nno pigmies, predicating the outcome of little factions, no discourse\nabout the smallness of word-meanings; but vast populations are arrayed\nunder our eye.  We cannot talk of the stars in their courses as we talk\nof the will-o'-wisps of politicians.  Rome marching to its dissolution,\nwith captive nations in its trail,[15] must put a lofty strain upon the\npage that records her downfall.\n\nThrough all, this corpulent, learned, dainty, keen-eyed, indefatigable\nlittle man, is cool--over cool; he has no enthusiasms but the\nenthusiasm of knowing things.  No wrongs that he records seem to chafe\nhim; his blood has no boiling-point; his love no flame; his indignation\nno scorching power.  A great, imposing, processional array of\nsovereigns, armies, nations--of the wise, the vicious, the savage, the\nlearned, the good; but not a figure in it all, however pure or\ninnocent, which kindles his sympathies into a glow; not one {130} so\nprofligate as to make his anger burn; not one so lofty or so true as to\ngive warmth to his expressions of reverence.\n\nYet notwithstanding, if any of my young readers are projecting the\nwriting of a history, I strongly advise them to avoid the subject of\nthe _Decline and Fall of the Roman Empire_.\n\n\n_Oliver Goldsmith._\n\n[Sidenote: Oliver Goldsmith.]\n\nAnd now we come to another member of our club, who reaped far fewer of\nthe substantial rewards of life.----Who, with any relish for the\nbeatitudes of letters, has not tender reverence for the memory of\nGoldsmith?  He was the youngest member of the club at its start, and\nyet the thirty-four years he then counted had been full of change and\nadventure: he had wandered away early from the beautiful paternal home\nof Lissoy in Ireland; had studied in Scotland and in Leyden; had idled\nin both; had been vagrant over Europe; had tried medicine, tried\nflute-playing, tried school-keeping, tried proof-reading for the old\nshopkeeper, Samuel Richardson, and had finally landed in a court not\n{131} far from Johnson's, where he did work for the booksellers.\nAmongst this work were certain essays which attracted the old Doctor's\nattention by their rare literary qualities; and the old gentleman had\nbefriended the author--all the more when he found him a man who did not\nbefriend himself; and who, if he had only sixpence in his pocket (and\nhe was not apt to have more), would give the half of it to a beggar.  A\nlittle over-love for wine, too--when the chance of a tavern dinner came\nto him--was another weakness which the great Doctor knew how to pardon;\nand so Goldsmith became one of the original clubmen; Reynolds, with all\nhis courtly ceremony, growing to love the man; so did Burke; but\nBoswell was always a little jealous of him, and Goldsmith caught at any\noccasion for giving a good slap to that sleek self-consequence which\nshone out all over Boswell--even to his knee-buckles and his silken\nhose.  I do not suppose that Goldsmith contributed much to the\nweightier debates of the club, and can imagine him sulking somewhat if\nhe found no good opening in the troubled waters in which to feather his\ndainty oar.  Again there was an {132} awkwardness, partly\nself-consciousness, partly organic tremor, which put him at bad odds in\npromiscuous talk; to say nothing of the irascibility which he had not\nlearned to control, and which sometimes put a stammer to the tongue;\nhence, Boswell says, \"poorest of talkers;\" but around in his chambers,\nwith one or two sympathetic listeners only, and perhaps a bottle of\nCanary flanking him, and with a topic started that chimed with the\nemotional nature of the man, and I am sure he would have talked out a\nwhole chapter of a new _Vicar of Wakefield_.\n\nBut whatever the tongue might do, there was no doubt about the pen; we\nfind him even undertaking discourses upon _Animated Nature_, and\nhistory--of Greece or of Rome.  Has he then the plodding faculty, and\nis he a man of research?  No; but he has the aptitude to seize upon the\nplums in the researches of others, and embody them in the amber of his\nlanguage.  He poaches all over the fields of history and science, and\nbags the bright-winged birds which the compilers have never seen, or\nwhich, if seen, they have classed with the gray and the dun of the\n{133} sparrows.  His poetry, when he makes it, may not have so much of\npolished clang and witty jingle as the verse of Pope; it may lack the\ngreat ground-swell of rhythmic cadence which belongs to Johnson;\nbut--somewhere between the lines, and subtly pervading every pause and\nflow--there is a tenderness, a suave, poetic perfume, a caressing touch\nof both mind and heart which we cannot describe--nor forget.\n\nOf the original club-men, Goldsmith[16] died first, in 1774, at\nBrick-court in the Temple; he was forty-five years old, and yielded to\na quick, sharp illness at the last, into which all the worries of a\nmuch worried life seemed to crowd him.  He had been plotting new works,\nand a new life too; a getting away (if it might be) from the smirch\nthat hung about him in the Temple corridors, out to the Edgware\nfarmery, where primroses and hedges grew, and where there was a scent\nupon the air, of that old country home of Lissoy:\n\n  \"I still had hopes my latest hours to crown,\n  Amidst those humble bowers to lay me down;\n\n{134}\n\n  To husband out life's taper at the close,\n  And keep the flame from wasting by repose.\n  I still had hopes, for pride attends me still,\n  Amidst the swains to show my book-learned skill;\n  Around my fire an evening group to draw\n  And tell of all I felt and all I saw.\n  And, as a hare, whom hounds and horns pursue,\n  Pants to the place from which at first he flew,\n  I still had hopes, my long vexations past,\n  Here to return, and die at home at last.\"\n\n\nA stolid physician, called in consultation in those last days, and\nseeing his disordered state, asked, \"If his mind was at ease?\"  Mind at\nease!  Surely a rasping question to put to a man whose pulse is\nthumping toward the hundreds, whose purse is empty, plans broken up,\ncredit gone, debts crowding him at every point, pains racking him, and\nthe grimy Fleet Prison close by, throwing its shadow straight across\nhis path.  No, his mind is _not_ at ease; and the pulse does gallop\nfaster and faster, and harder and harder to the end; when, let us\nhope--ease did come, and--God willing--\"Rest for the weary.\"\n\n\n{135}\n\n_The Thrales and the End._\n\nMeantime Dr. Johnson has been withdrawing somewhat from his old regular\nattendance upon the club.  New men have come in, of whiggish\ntendencies; he hears things he does not like to hear; the Americans are\nat last making a fight of it; he is a heavier walker than once; besides\nwhich his increased revenue has perhaps made him a little more free of\nthe Mitre tavern than of old; then he has made the acquaintance of Mr.\nThrale and of Mrs. Thrale--an every-way memorable acquaintance for him.\nMr. Thrale is a wealthy brewer, one while Member of Parliament--his\nworks standing on the ground in Southwark now held by Barclay &\nPerkins, some of whose dependencies cover the site of that Globe\nTheatre where William Shakespeare was sometime actor and shareholder.\nWithal, Mr. Thrale is a most generous, sound-headed, practical, kindly\nman, without being very acute, or cultured, or any way accomplished.\nMrs. Thrale, however, {136} has her literary qualities; can jingle a\nlittle of not inharmonious verse of her own; reads omnivorously; is apt\nin French or Latin; is full of esprit and liveliness, and is not\nwithout a certain charm of person.  She is small indeed, but with\nstriking features and picturesque; easily gracious at her table; witty,\nheadstrong, arch, proud of association with the great Dr. Johnson;\nreally having strong friendship for him; enduring his rudenesses;\nyielding to him in very much, but not so submissive as to take his\nopinion (or that of any other man) about whether she should or should\nnot marry Signor Piozzi, when afterward she came to be a widow.  In\nfact, she had in fine development the very womanly way--of having her\nown way.\n\nThe Thrales owned a delightful country place at Streatham, a pleasant\ndrive out from the city, down through Southwark and Brixton and on the\nroad to Croydon; and there Johnson went again and again: Mr. Thrale was\nso kind, and Mrs. Thrale so engaging.  At last they put at his service\na complete apartment, where he could, on his blue days, growl to his\nliking.  Who can say {137} what might have been the career of the great\nlexicographer if he had fallen into such downy quarters in his callow\ndays; should we have had the Dictionary?  Surely never the life of\nSavage, with its personal piquancy, and possibly never the Boswelliana.\n\n[Sidenote: Tour to the Hebrides.]\n\nBut Johnson was not wholly idle; neither the luxuries of Streatham, nor\nthe chink of his pension money, could stay the unrest of his mind: he\nwrites dedications for other people--shoals of them; he re-edits twice\nover the great Dictionary; publishes _The False Alarm_; completes his\n_Lives of the Poets_; and in the interim--between visits to Oxford,\nBrighton, and Lichfield--he makes that famous trip, with Boswell, to\nScotland and the Hebrides; and never, I think, was so unimportant a\njourney so known of men.  Every smart boy in every American school,\nknows now what puddings he ate, and about the cudgel that he carried,\nand the boiled mutton that was set before him.  The bare mention of\nthese things brings back a relishy smack of the whole story of the\njourney.  Is it for the literary quality of the book which describes\nit?  Is it for our interest {138} in the great, nettlesome, ponderous\ntraveller; or is it by reason of a sneaking fondness we all have for\nthe perennial stream of Boswell's gossip?  I cannot tell, for one: I do\nnot puzzle with the question; but I enjoy.\n\n[Sidenote: Last days of Johnson.]\n\nIn the year 1779 his old friend Garrick died,--leaving nearly a million\nof dollars, which came to him by that stage following and thrift which\nhad so worried the orthodox and respectable brother Peter of the\nwine-shop.  The interesting Mrs. Garrick came, after a time, to a\nlively widowhood on the Adelphi Terrace--looking out over what is now\nthe London Embankment, and with such friends as Miss Hannah More, and\n\"Evelina\" Burney, and the old wheezing Doctor himself, to cheer her\nloneliness and share her luxurious dinners.  The year after, in 1780,\nTopham Beauclerk died; and so that other bright light in the Turks-Head\nClub is dashed forever.\n\nThese, things may well have put new wrinkles in the old Doctor's\nvisage; but he still keeps good courage; works in his spasmodic\nway;--dines with the printer Strahan; dines at the Mitre; dines at\nStreatham; coquettes, in his lumbering {139} way, with Mrs. Thrale, and\ngoes home to the fogs and grime of Bolt Court.\n\nShall I quote from a letter to the last-named lady, dating in the year\n1780?\n\n\n\"How do you think I live?  On Thursday I dined with Hamilton and went\nthence to Mrs. Ord.  On Friday at the Reynolds'--on Sunday at Dr.\nBurney's with the two sweets [daughters of Mrs. T.] from Kensington; on\nMonday with Reynolds; to-day with Mr. Langton; to-morrow with the\nBishop of St. Asaph.  I not only scour the town from day to day, but\nmany visitors come to me in the morning, so that my work [_Lives of the\nPoets_] makes little progress.\n\n\"You are at all places of high resort, and bring home hearts by dozens,\nwhile I am seeking for something to say of men about whom I know\nnothing but their verses....  Congreve, whom I despatched at the\nBorough, is one of the best of the little lives: but then _I had the\nbenefit of your conversation_.\"\n\n\nThis is very well for a plethoric old gentleman of seventy-one.  The\nnext year, 1781, his friend and patron Mr. Thrale died.  This loss was\na grievous one for Johnson.  He had relished his kindliness and his\nlarge, practical sagacity: indeed I think he had relished in him the\nlack of that literary talk and allusion which so many of {140} his\nacquaintances thought it necessary to throw out as bait for the\nLeviathan.  But was the Doctor to enjoy still the delights of that\nStreatham retreat?  It is certain that a year did not pass before there\nwas much gossip, in neighboring gossiping circles, that associated the\nname of Johnson with the clever and wealthy widow, as a possible\nsuccessor to Mr. Thrale.  I do not think any such gossips of the male\nkind ever ventured within easy reach of the Doctor's oaken cudgel.\nThere is no evidence that any thought of such alliance ever came into\nJohnson's mind; but I _do_ think he had sometimes regaled himself with\nthe hope of a certain kindly protectorate over the luxuries and the\nmistress of Streatham, which would keep all its old charms open to him,\nand permit of a fatherly dalliance with the family there.  It appeared,\nhowever, that the clever lady had other views; and did marry three\nyears after--very much to the disgust of her children--Signor Piozzi, a\nmusician of very fair reputation; did live a happy enough life with\nhim; did publish a book or two full of sparkle and many errors, and\nsome mischievously strong cuts at people she disliked; did live {141}\nthereafter to a great old age, and carried roses in her cheeks amongst\nthe eighties; though I think these roses came from the apothecaries.\nShe was always fond of decoration.\n\nIn 1783 the Doctor had a stroke of paralysis, from which, however, he\nrallied and was himself once more--dining with Dilly, with Reynolds, at\nthe Mitre too, with Boswell; he even projects new work--suggests the\nformation of another club in the city, and more within reach: So\ntenaciously do we cling, and so hopefully do we keep plotting!  Finally\nin June, 1784, he takes his last dinner at the old club; Reynolds and\nBurke and Langton and Boswell are there, with others he does not know\nso well; he is feeble at this sitting and ill at ease; clouds gathering\nover him, from which, however, there flashes out from time to time a\nblaze of his old wit.\n\nThereafter, it is mostly Bolt-court--poor blind Miss Williams gone, by\nthis time, and also the sorry physician who had been long a pensioner\non him, and whose nostrums he had taken out of charity.  Of all the\nfaces that once welcomed him {142} there in their way, only his black\nman Francis left.\n\nLangton comes to see him; and Reynolds comes bringing more cheer,\nthough the ear trumpet is awkward for the sick man; Burke comes and\nshows all the melting tenderness of a woman; Boswell, too--before he\ngoes north--bounces in and out, his conceit and assurance mollified and\ndecently draped by the sorrow that hung over him.  Little Miss Burney\nrushes in to the ante-room and stays there hours, hoping some shortest\nlast interview with the great man who had said kindly things to\nher--never thinking that he could not relish her gossippy prattle about\nthe court, and the royal George, now that a great, swift tide was\nlifting him into the presence of another king.\n\n[Sidenote: Death of Johnson.]\n\nThe old superstitious awe and dread of death, which had belonged to him\nthroughout life, disappeared in these latter days, and the gloom--with\nits teasing vampires--was rarefied into a certain celestial haze that\nhung over him tenderly.  He did not excitedly wrestle with the awful\npossibilities the change might bring, nor work himself {143} into any\ncraze of pious exhilaration to bridge the gap; but was restful as a\nbabe at last, and so was led away tranquilly, by his own child-like\ntrust, over the threshold of the mysteries we must all confront.\n\n\n\n[1] See note, Hill's Boswell, p. 304, vol. i.\n\n[2] Blackfriars was not built until 1769, and the old Westminster in\n1750.\n\n[3] Pension granted, 1762: _Rasselas_ published, 1759.\n\n[4] Joshua Reynolds, b. 1723; d. 1792.  His _Discourses_ published,\n1771.  Life by Leslie, 1867.\n\n[5] It is from this latter gentleman--whom I had the good fortune to\nmeet in the course of his visit to this country--that my information in\nregard to the latter _status_ of the club is derived.\n\n[6] Edmund Burke, b. 1729; d. 1797.  Editions of his works are various.\nBest life of him is by John Morley (1867).\n\n[7] B. 1740; d. 1795.\n\n[8] There is, to be sure, a great deal of what the natural reserve of\nmost men would lead them to withhold.  But if this \"free-telling\" does\nadd some of the finer lights and most artistic touches to his picture,\nand if he perceives this to be so (and have we any right to assume the\ncontrary?) shall we not credit it rightly to his book-making art and\ncommend it accordingly?\n\nThat his gentlemanly reserves are not of a pronounced sort may count\nagainst the delicacy of his nature, but not necessarily against his\ncapacity as a literary artist.\n\n[9] Edward Gibbon, b. 1737; d. 1794.  Dr. Milman's is the standard\nedition of his History.  Bowdler's edition (1825) is noticeable for its\nexpurgations.  The work, through its translations, holds as large a\nplace in the historic _curriculum_ of French, Italian, and German\nstudents, as in that of English-speaking nations.\n\n[10] _Biographie Universale_; Article Necker (Mme. Necker, _nee Susanne\nCurchod_).\n\n[11] Hume's first volume of English History appeared in 1754--just\ntwenty-two years before the _Decline and Fall_.  Hume was about\ntwenty-six years Gibbon's senior.\n\n[12] Boswell says in his Diary (1779): \"Gibbon is an ugly, affected,\ndisgusting fellow, and poisons our literary club to me.\"\n\n[13] The old house has wholly disappeared; the hotel covers a portion\nof Gibbon's garden.\n\n[14] Letter in the _Pall Mall Gazette_ in relation to Sir John\nLubbock's \"List of Hundred Best Books.\"  Reprinted in _Critic_\n(American) of March 20, 1886.\n\n[15] See, for instance, account of Julian's march, and of the taking of\nConstantinople.\n\n[16] Oliver Goldsmith, b. 1728; d. 1774.  Fullest and best _Life_, that\nof John Forster.\n\n\n\n\n{144}\n\nCHAPTER IV.\n\nWe parted company, in our last chapter, with Dr. Johnson, of whose work\nand career every educated person should know; we parted company also,\nwith that more lovable, though less important man, Dr. Goldsmith--of\nwhom it would have been easy and pleasant to talk by the hour; we all\nknow him so well; we all would have wished him so well--if wishes could\nhave counted.  And as we sidle into the Poets' corner of Westminster\nAbbey--on whatever visit we make there--we put a friendly eagerness\ninto our search for the medallion effigy of Goldsmith over the door,\nwhich we do not put into our search for a great many entombed under\nmuch greater show of marble.  But Goldsmith's bones do not lie in the\nAbbey; he was buried somewhere under the wing of the Old Temple\nChurch--the particular locality {145} being subject of much doubt;\nwhile the memorial statue of Johnson--his body lying in\nWestminster--must be sought for, still farther down in the city, under\nthe arches of St. Paul's Cathedral.\n\nGarrick has what we might almost call melodramatic monument among the\nmarbles of the Poets' corner; Reynolds has abiding memorials in the\ndashes of mellowed coloring and in the tender graces of those cherished\nportraits, some of which belong to every considerable gallery of\nEngland; Burke and Gibbon lie in quiet country places--the first near\nto his old home of Beaconsfield; and the historian among those southern\ndowns of Sussex which look upon the Channel waters; his books may never\nhave touched us to tenderness; but he bows his way out of our presence,\nwith the grandest history belonging to the eighteenth century for a\nmemorial.\n\n\n_A Scottish Historian._\n\n[Sidenote: David Hume.]\n\nWe must not forget that hard-headed man who wrote Hume's History of\nEngland, who was born twenty-two years before the historian of Rome,\n{146} and died in the year in which Gibbon was reaping his first\nrewards.  He[1] was a sceptic too of even more aggressive type than\nGibbon--was, like him, somewhat ungainly in person, and though of\nlarger build and of coarser mould, possessed a cheery good humor, and a\nbright colloquial wit which made him sure of good friends and many.\nLike Gibbon he lived and died a bachelor: like him, he leaned toward\ncontinental ways of living, and like him garnered some of his highest\nhonors in France.  Of course you know his History of England--where it\nbegins, where it ends--but we do not press examination on these points.\nIn most editions you will find--(it should be found in all)--among the\nforeleaves, a short autobiographic sketch, written in his most neat,\nperspicuous, and engaging manner, which is well worth the close\nattention of every reader, even if he do not wade through the royal\nextensions of the History.  You will learn there that David Hume was\nborn in that pleasant border land of Scotland which is watered {147} by\nthe Tweed, the Yarrow, and the Teviot--where we found the poet Thomson.\nNorth of his boyish home stretched Lammermoor, and westward within easy\ntramping distance lay Lauderdale; but in that day these names had not\nbeen illuminated by a touch of the magician's wand, nor was his mind\never keenly alive to the beauties of landscape.  Hume's childhood knew\nonly great stretches of brown heather, bounded by bare bluish-gray\nheights, with the waves of the German Sea pounding on the rocky,\ndesolate shores--where stands the ruin of Fast Castle, the original of\n\"Wolf's Crag\" of the Master of Ravenswood.\n\nYou will learn further from that precious bit of autobiography--which\nhe calls with a naive directness, \"My Own Life\"--that he was younger\nson of a good family; that he came to fairish education thereabout, and\nin Edinboro'; that his family would have pushed him to the study of\nlaw; but he--loving philosophy and literature better, and in search of\nsome method of increasing his means for their pursuit--wandered\nsouthward to study business in the city of Bristol.  This was a place\nof much greater commercial importance, {148} relatively, then than now:\nbut Bristol merchandizing presently disgusts him; and husbanding\ncarefully his small moneys, he goes across the Channel--to study\nphilosophy, while practising the economies of French provincial life in\na small town of central France.  A few years thereafter he prints his\nfirst book in London on _Human Nature_; and he says it fell \"dead-born\"\nfrom the press; but he is still sanguine and cheery; writes other\nessays after his return from France--hovering between Edinboro' and his\nold Berwickshire home; studying Greek the while, and for a year\nserving, as secretary, the crazy Marquis of Annandale.  Shortly\nthereafter (1746) began his official connection with the General St.\nClair, involving a new and pleasanter experience of European life.  On\nhis return, after three years, he goes to cover again in his old\nBerwickshire home, where he elaborates the _Political\nDiscourses_--setting forth those broad views of trade and commerce,\nwhich came to larger illustration later, under the pen of his good\nfriend Adam Smith.[2]\n\n{149}\n\n[Sidenote: Hume's England.]\n\nIn 1751 he removed from country to town--the true scene he says \"for a\nman of letters,\" and established himself in a small flat of one of\nthose lofty houses which still look down over the New City and the\nvalley gardens, and lived there comfortably--with his sister for help\nmate--on some L50 a year.  He tried vainly for a professorship in one\nof the Scottish universities, but was counted too unsafe a man.  As\nCustodian of the Advocates' Library of Edinboro', a place which he\nsecured shortly after--largely through the influence of lady\nfriends--he came to that familiar fellowship with books which prompted\nhim to the making of his History of England.  He does not begin at the\nbeginning: he tells of the Stuarts first; then goes back to the Tudors;\nand then back of these to the dull (dull to him and dull to us)\nAnglo-saxon start point: Stubbs and Freeman had not in that {150} day\nmade their explorative forays and set up their scaffoldings.\n\nHume's ambition was high and sensitive: he was intensely disappointed\nwith the reception of the earlier volumes of his history.  \"I was\ndiscouraged,\" he says, \"and had not the war been at that time breaking\nout between France and England, I had certainly retired to some\nprovincial town of the former kingdom, have changed my name, and never\nmore have returned to my native country.\"\n\nBut his writings had qualities which were sure in the end to provoke\nthe reading and discussion of them by thoughtful men and women.  He is\nknown wider than he thinks; his books have been translated; Montesquieu\nhas corresponded with him; so has a certain Mme. de Boufflers--a pet of\nthe Paris salons--who has written gushingly of her admiration; and the\nstolid, good-humored, cool-blooded Hume has responded in his awkward\nmanner; other missives, with growing confidences have passed; she\nalways clever, and witty and full of adulation; and he clumsy and\nclever, and with such tenderness as an elephant might show toward {151}\na gazelle.  And the shining side of life opens bewitchingly upon him\nwhen he goes to Paris in 1763 as an attache to the Embassy of Lord\nHertford.\n\n[Sidenote: Hume in Paris.]\n\nIn place of Scotch kerseys, his square, massive figure is set off with\nthe golden broidery of a diplomat.  His reputation as a philosopher and\nas a historian had been confirmed by all the literary magnates of\nParis; and the queens of society in that gay capital, Mme. de Boufflers\namong them, pounced upon the big Scotch David, to be led away through\nthe pretty martyrdoms of the salon.  And he bore it bravely; he had\nfeared, indeed, that his inaptitudes and inexperience would have made\nsuch a life irksome to one of his quiet habits; but he good-humoredly\nand complacently accepted the sacrifice and came to love the\nintoxicating incense.  Sterne, who happened in Paris in those days,\nsays that Hume was the lion of the city; no assemblage was complete\nwithout his presence.  Yet he did not lose his cool philosophic poise.\nHe carried his good humor everywhere, and an indifference that made him\nengaging; if arrows of Cupid were launched at him, they did not pierce\nthrough the wrappings of his thick Scotch phlegm.\n\n{152}\n\nMme. d'Epinay tells a good story of these times about his taking part\nin some tableau where he was to personate an Eastern sovereign, seated\nbetween two beautiful Circassian damsels, to whom he was expected to\nshow devotional assiduities of speech.  But the frigid philosopher,\nbanked in between those feminine piles of silk and jewels, only rubs\nhis hands, slaps his knees, purses up his mouth, and says over and\nover, in his inconsequent French,--\"Eh bien, Mesdemoiselles, vous\nvoila! vous voila donc!  Eh bien, nous voici!\"  Whereat we may be sure\nthat his pretty companions let fall slily a disparaging \"_qu'il est\nbete!_\"--As if the man who had traced to their ultimate issues the\nsubtleties of the _Principles of Morals_ could parry and thrust with\nthe pretty conversational foils of a Pompadour!\n\n[Sidenote: David Hume.]\n\nIt chanced that by the unexpected withdrawal of Lord Hertford, Hume was\nfor a time chief of the Embassy, and for the first and last time (in so\nfull a sense) did a historian of England thus become British Ambassador\nto the Court of France.  But Hume does not love the English or England;\nhe resents their neglect of him; he never {153} forgets that he is a\nScotsman; it twangs in his speech; it twangs louder in his heart; he\nwould like to live in that pleasant country of France:--\"They are all\nkind to me here,\" he says; \"but not one of a thousand in all England\nwould care a penny's worth if I broke my neck to-morrow.\"  And though\nhis reputation is now largely upon the growth at home, still he is not\npleasantly _lie_ with the masters.  Somewhat later, when by another\nunexpected good turn he is made Under-Secretary of State and has\nofficial position in London, he writes to Dr. Blair, of Edinboro', who\nhas offered to give him a letter to Bishop Percy--\"I thank you, but it\nwould be impracticable for me to cultivate his friendship, as men of\nletters have here no place of rendezvous; and are indeed sunk and\nforgot in the general torrent of the world.\"  And yet this was at a\ndate (1763), when the Turk's Head gathering was all alive, when Sterne\nhad recently published the last volume of his _Tristram_; when poor\nSmollett[3] (of {154} _Roderick Random_ fame) has won success by a\nflimsy, but popular continuation of Hume's History; when the _Vicar of\nWakefield_ was fresh (though as yet unprinted); when Mason and Gray and\nWarburton and Johnson were all sounding their trumpets.  With such\nfeelings of alienation it is not strange that Hume did not nestle into\nthe hearts of great Londoners as he had nestled into the good-will of\nParisians.\n\nUnder the influences of Mme. de Boufflers he tried to make a home in\nEngland for that strange creature, Rousseau, who had become an exile,\nand who brought with him--to the torment of Hume--all his\neccentricities, his peevishness, his inhuman vanities, his abnormal\nsensitiveness, his wild jealousies, and his exaltations of genius.\nThese things work a rupture between the two in the end--as they should\nand ought to do,--and the next good sight we have of the Scotch\nphilosopher is in a new home of his own (1772), which he has built in\nthe new part of Edinboro'.  Twenty odd years before he had lived in the\nold city on an income of L50 a year; and now he lives in the new with\nan income of L1,000 a year.  In {155} the old times he had hardly\nsecured place as Custodian of the mouldy Library of the Advocates; now\nhe is the marked potentate in the literary world of Scotland.  Stanch\nPresbyterians do indeed look at him askance, and shake their heads at\nhis uncanny beliefs, or rather lack of beliefs.  Old nurses put\nhobgoblin wings upon him to frighten good children; but he has stanch,\nloving friends among the best and the clearest sighted.  Dr. Blair is\nhis friend; excellent Dr. Robertson is his friend; his good nature, his\nkindness of heart, his rectitude of life, his intellectual charities,\nwon even those who shuddered at his disbeliefs; that sceptical\nmiasma--born in his blood--and harmful to many (as it was to himself),\nseemed to lose its malarious taint in the large, free, intellectual\natmosphere in which the philosopher lived.  Honest doubts were then,\nand always will be, better than dishonest beliefs; just as honest\nbeliefs are a thousand fold better than dishonest doubts.\n\n[Sidenote: Death of Hume.]\n\nIt was in our year of 1776--when his reputation was brightening and\nwidening month by month, that David Hume, the author of the first\nscholarly History of England, died, and was buried {156} on a shoulder\nof the Calton Hill, from which one may look eastward across the valley\n(where lies Holyrood Palace) to the Salisbury Crags on the left and to\nthe Castle Rock on the right.\n\nIt is probable that his History will long hold place on our library\nshelves; its style might almost be counted a model historic style--if\nwe were to have models (of which the wisdom is doubtful).  It is clear,\nit is precise, it is perspicuous, it is neat to a fault.  It might\nalmost be called a reticent style, in its neglect of those wrappings of\nwordy illustration and amplification which so many historians employ.\nHe makes us see his meaning as if we looked through crystal; and if the\ncrystal is toned by his prejudices--as it is and very largely--it is\naltogether free from the impertinent decorative arabesques of the\nrhetorician.  Many of the periods of which he gives the record, have\nhad new light thrown upon them by the searching inquiries of late days.\nOld reputations with which he dealt reverently have suffered collapse;\npolitical horizons which were limited and gave smallness consequence,\nhave widened; but for good, straightforward, lucid, {157} logical\nsetting forth of the main facts of which he undertook the record, Hume\nwill long remain the reference book.  There will be never a time when\nlovers of good literature will not be attracted by his pathetic picture\nof the career of Charles I.; and never a time when the judicious reader\nwill accept it as altogether worthy of trust.\n\nThe life of the historian--by Dr. Huxley--is rather a history of his\nphilosophy than of his life; in which the eminent scientist--with due\napology for intrusion upon literary ground--sets his logic to an easy\ncanter all around the soberer paces of the great Scotch\ncharger--showing nice agreement in the paces of the two, and commending\nand illustrating the metaphysics of the Historian, with a pretty\nfanfaronade of Exposition and Applause.\n\n\n_A Pair of Poets._\n\n[Sidenote: Two poets.]\n\nWere it only to change the current of our talk, I bring now a brace of\npoets to your notice; not well paired indeed, as you will find: but\neach one in his own way giving us music that strongly {158} contrasts\nwith _The Deserted Village_, and the ponderous Satires of Johnson.\n\n[Sidenote: Shenstone.]\n\nShenstone[4] is a name not very much known--not very much worth\nknowing: he was a big, somewhat scholarly, fastidious, indolent,\nrhyme-haunted man, who had studied at Oxford, and who, when the muses\nwere buzzing about his ears, came into possession of a pretty farm in\nthat bit of Shropshire which (by queer English fashion) is planted\nwithin the northern borders of Worcestershire; and it was there that he\nwrote--what is typical of all that he ever wrote, and what has his\ncurrent and favorite sing-song in it:--\n\n  \"Since Phyllis vouchsafed me a look\n    I never once dreamt of my vine.\n  May I lose both my pipe and my crook\n    If I knew of a kid that was mine!\n  I prized every hour that went by\n    Beyond all that had pleased me before;\n  But now they are past, and I sigh;\n    And I grieve that I prized them no more.\"\n\nAnd again--\n\n  \"When forced the fair nymph to forego\n    What anguish I felt at my heart!\n\n{159}\n\n  Yet I thought--but it might not be so--\n    'Twas with pain that she saw me depart.\n  She gazed as I slowly withdrew.\n    My path I could hardly discern;\n  So sweetly she bade me Adieu\n    I thought that she bade me return!\"\n\nWhat should we think of that if we encountered it fresh in a corner of\none of our Sunday newspapers?  We should hardly reckon its author among\nour boasted treasures; yet Burns says \"his elegies do honor to our\nlanguage,\" and a great deal of the same guileless tintinnabulum did\nhave its admirers all over England a century ago; and some of\nShenstone's pretty wares have come drifting down on the wings of albums\nand anthologies fairly into our day.\n\nYet I should rather have encountered him in his fields, than in his\ngarret; for he made those fields very beautiful.  He was a bad farmer,\nto be sure; and sacrificed turnips to marigolds; and wheat to primroses\nand daisies, fast as the season went round; but his home at Leasowes\nwas a place worth visiting for its charming graces of every rural sort;\neven our staid John Adams, when he was in England in those days,\nlooking after American {160} colonial interests--must needs coach it in\ncompany with Jefferson from Cirencester to Leasowes, for a sight of\nthis charming homestead.  Goldsmith too gave its beauties the\nembalmment of his language; and Dr. Johnson sat down upon it, with the\nweight of his ponderous sentences.  One echo more we will have of him,\nas it comes fresh from his pet paradise of that corner of\nShropshire--and certainly carrying a honeyed rhythmic flow:--\n\n  \"My banks they are furnished with bees,\n    Whose murmur invites one to sleep;\n  My grottoes are shaded with trees,\n    And my hills are white over with sheep.\n  I seldom have met with a loss,\n    Such health do my fountains bestow;\n  My fountains all bordered with moss\n    Where the hare-bells and violets grow.\"\n\n\n[Sidenote: William Collins.]\n\nWilliam Collins[5] was a man of a totally different stamp--better worth\nyour knowing--yet maybe with the general public not so well known.\n{161} There is the chink of true and rare poetic metal in his verse,\nand it is fused by an imagination capable of intense heat and wonderful\nflame.  He was only a hatter's boy from Chichester, in the South of\nEngland; was at Oxford for a while, and left there in a huff--though\nsecuring a degree, 1743; afterward went to London; wrote and printed\nsome odes, which he knew were better than most current poetry, but\nwhich nobody bought or read.  He sulked under that neglect, and his\nrage ran--sometimes to verse--sometimes to drink; he had known Thomson\nand Johnson, and both befriended him; but the world did not; indeed he\nnever met the world half way; the poetic phrenzy in him so fined his\nsensibilities that he could not and would not put out a feeling hand\nfor promiscuous greetings.  Poverty, too, came in the wake of his\npoetic cultures, to aggravate his mental inaptitudes and his moral\ndistractions--all ending at last in a mad-house.  He was not, to be\nsure, continuously under restraint--such terrific restraints as\nbelonged to treatment of the insane in that day; but for a half dozen\nor more years of the latter part of his life--wandering all\nawry--saying {162} weak and pointless things, in place of the odes\nwhich had coruscated under his fine fancy; lingering about his\nchildhood's home; stealing under the cathedral vaults of Chichester\n(where his body rests now), and lifting up a vacant and wild treble of\nsound in dreary sing-song to mingle with the music from the choir.\n\nThere are accomplished critics who insist that the odes of Collins\ncarry in them the finest and the loftiest strains which go to marry the\nmusic of the nineteenth century poets to the music of the days of\nElizabeth.  Certain it is, that he loomed far above the ding-dong of\nsuch as Shenstone--that he scorned the classic trammels of the empire\nof Pope--certain that there were fires in him which were lighted by\npoets who lived before the time of the Stuarts, and which gave\nforetaste and promise of the freedom and the graces that shine\nto-day.[6]\n\nI cannot quote better to show his quality than {163} from that \"Ode to\nEvening\" which is so often cited:--\n\n  \"For when thy folding star arising, shows\n  His paly circlet at his warning lamp,\n    The fragrant hours and elves\n    Who slept in flowers the day,\n\n  \"And many a nymph who wreathes her brows with\n  And sheds the freshning dew, and, lovelier still,\n    The pensive pleasures sweet,\n    Prepare thy shadowy car.\n\n  \"Then lead, calm votress, where some sheety lake\n  Cheers the lone heath, or some time hallowed pile,\n    Or upland fallows gray\n    Reflect its last cool gleam.\n\n  \"But when chill, blustering winds or driving rain\n  Forbid my willing feet, be mine the hut\n    That from the mountain's side,\n    Views wilds and swelling floods,\n\n  \"And hamlets brown and dim discovered spires,\n  And hears their simple bell, and marks o'er all\n    Thy dewy fingers draw\n    The gradual dusky veil.\"\n\n\nThis is poetry that goes without help of rhyme; even its halts are big\nwith invitations to the \"upland fallows gray,\" and to the \"pensive\npleasures sweet.\"  Swinburne says, with piquancy and truth, {164}\n\"Corot, on canvas, might have signed the 'Ode to Evening.'\"\n\nDr. Johnson, who was a strong friend of Collins, tells us, in his\n_Lives of the Poets_, that he died in 1756; and that story is repeated\nby most early biographies; the truth is, however, that after that date\nhe was living--only a sort of death in life, under the care of his\nsister at Chichester; and it was not until 1759, when--his moral and\nphysical wreck complete--the end came.\n\n\n_Miss Burney._\n\n[Sidenote: Miss Burney.]\n\nWe have next to bring to your notice, a clever, somewhat frisky,\n_debonnaire_ young person of the other sex, whom you should know--whom\nperhaps you do know; I mean Miss Frances Burney.[7]  You will remember\nthat we have encountered her once before pushing her kindly way into\nold Dr. Johnson's ante-room when he was near to death.  The old\ngentleman had known intimately her {165} father, Dr. Burney, and had\nalways shown for her a strong attachment; so did a great many of Dr.\nBurney's acquaintances, Garrick among them and Burke; and it was\nprobably from such men and their talk that she caught the literary bee\nin her bonnet and wrote her famous story of _Evelina_.  You should read\nthat story--whatever you may do with _Cecilia_ and other later ones--if\nonly to see how good and cleanly a piece of work in the way of a\nsociety novel can come out of those broiling times, when _Humphrey\nClinker_ and _Tom Jones_ and the prurient and sentimental languors of\nRichardson were on the toilette tables of the clever and the honest.\n\nThe book of _Evelina_ is, all over, Miss Burney; that gives it the\nrarest and best sort of realism.  Through all her work indeed, we have\nthis over-jubilant and gushing, yet timid and diffident young lady,\nwriting her stories--with all her timidities and large, unspoken hopes,\ntumbling and twittering in the bosoms of her heroines: if my lady has\nthe fidgets, the fidgets come into her books; and you can always chase\nback the tremors that smite from time to time the fair Evelina, to\n{166} the kindred tremors that afflict the clever and sensitive\ndaughter of old Dr. Burney.\n\nThe book was published anonymously at first, and the secret of\nauthorship tolerably well kept; she says her papa did not know; but\nyoung ladies are apt to put too small a limit to the knowledge of their\npapas!  It is very certain that her self-consciousness, and tremulous,\naffected, simpers of ignorance, were not good to stave away suspicion.\nIt was not long before the world, confounding book and person, came to\ncall her \"Evelina.\"\n\nA pretty picture with this motif comes into her Diary: On a certain\nmorning, our Dr. Franklin--being then in London on colonial\nbusiness--makes a call upon Dr. Burney:--and in absence of her father,\nmeets the daughter: a big, square-shouldered man, very formal, very\nstout, but very kindly, approaches her and says--\"I think I have the\npleasure of speaking with Evelina.\"\n\n\"Oh, no,\" she replies, \"I am Frances Burney,\" and he--\"Ah--indeed!  I\nthought it had been Evelina:\" and there it ends, and we lose sight of\nour broad-shouldered Dr. Franklin, with only this \"Ah!\" upon his lips.\n\n{167}\n\nShe had a modesty that was vain by its excess, and was awkward when\ncaught unexpectedly or with strangers; in great trepidation lest her\nbooks might be talked of--yet with her books and her authorship always\ntormentingly uppermost in her thought.  Her Diary and letters are full\nof them.  Yet she is attractive--strangely so--by her sympathetic\nqualities; so responsive to every shade of sorrow or of joy; winning,\nbecause so tell-tale of heart; and with a tongue that could prattle\ngracefully when at ease; Evelina, in short, without Evelina's beauty or\nexpectations.\n\nI have read the book over again after a gap of many years--with a view\nto this talk of the authoress, and find myself wondering more than\never, how so many of great and commanding intellect should have so\nheartily admired it.  Burke read it with most eager attention and\nlargest praise; old Dr. Johnson delighted in it, and declared it\nsuperior in many points to Richardson (which for him was extravagant\ncommendation).  Even Mme. de Stael, some few years later, gave it her\napplause; and the quick and swift-witted Mrs. Thrale was in raptures\nwith it; {168} and Mrs. Thrale knew a dunce, and detested dunces.\nThere must have been a deftness in her touch of things local,--of\nwhich, I think, she was but half conscious; there was beside a pretty\ndramatic art which found play in many pages of her Diary, and in all\nshe did and all she spoke.  For her third novel of _Camilla_, which\nscarce ever comes off the shelf of old libraries now--where it survives\nin deserved retirement--she received, according to the rumors current\nin those days, the sum of L3,000; such rumors, very likely exaggerated\nthe amount; they are apt to do so--in all times.\n\nHer Diary[8] is of special interest; particularly the portion which\ntakes one into the domestic life of Royalty.  For one of the bitter\nfruits of her celebrity, was her appointment as Lady of the Robes (or\nother such title), to the Queen.  The service indeed did not last many\nyears, but long enough to give us a good sight of the well-disposed,\n{169} fussy, indolent, kind-hearted Queen, and of the inquisitive,\nobstinate, good-natured King.\n\n[Sidenote: Trials of the Queen.]\n\nShe was at the palace, indeed, when one of the earlier attacks of that\nmental ailment which at last slew George III.--fell upon him.  She sees\nthe poor Queen growing wild with dread--disturbed and trembling under\nthose flashes of disorderly talk which smite upon her ear.  She watches\nthe King as he goes out to his drive on a certain fatal day;--hears the\nhushed, muffled steps and the babel of uncertain sounds, as he comes\nback late at night,--waits hour on hour for her usual summons to the\nQueen's presence, which does not come.  At last, midnight being long\npast (and she meantime having hint of some great calamity) goes to the\nQueen's chamber; two other lady attendants were with her, she says; and\nthe Queen, ghostly pale and shuddering--puts her hand kindly upon that\nof the poor little trembling Miss Burney and says \"I am like ice--so\ncold--so cold!\"\n\n\"I tried to speak,\" says Miss Burney in her Diary, \"but burst into\ntears: then the Queen did.\"  And there was cause: for from beyond the\nchamber--along the corridor,--came the idle {170} jabbering of King\nGeorge; and the intellectual power (such as there was of it) \"thro'\nwords and things went sounding on its dim and perilous way.\"\n\nI tell this not to test the reader's capability for sympathy, but to\nfasten poor little Miss Burney, the author of _Evelina_ and _Cecilia_,\nin mind; and to connect her service in the palace of St. James, in the\nyear 1788, with the first threat and the first real attack of the\nKing's insanity.  I am afraid we must set down, as one helping cause to\nthe King's affliction, the American obstinacy in maintaining their\nIndependence.\n\nMiss Burney shortly after, with a pension of L100, retired from the\nroyal duties, which had tried her sadly; and some years later\nencountering and greatly admiring General d'Arblay, who had come over\nan exile from France, in company with other distinguished emigrants, on\nthe outbreak of the French Revolution, she married him (1793), and gave\nhim a home that grew up out of the moneys received from her\n_Camilla_--hence called by old Dr. Burney, \"Camilla Cottage.\"\n\nShe survived her husband and a son (a clergyman {171} of the\nEstablished Church), and lived to so great an age as to find all her\nconquests in fiction over-run at last by the brilliant successes of\nMiss Austen, of Miss Edgeworth, and the more splendid triumphs of\nWalter Scott.  She died almost in our day (1840) and was buried in\nBath; but her best monument you can see without going there; it is her\nbook of _Evelina_ and her Diary.\n\n\n_Hannah More._\n\n[Sidenote: Mistress More.]\n\nOver-fine literary people will, I suppose, hardly recognize Hannah\nMore--or Mistress Hannah More,[9] as I prefer to call her, in virtue of\na good old English, and a good old New Englandish custom, too, which\ngave this title of dignity to matronly women--married or unmarried, of\nmature age, and of worthy lives.\n\nWe must go into the neighborhood of that picturesque old city of\nBristol, in the West of England, to find her.  She was one of the five\ndaughters of a respectable schoolmaster in Gloucestershire.  Hannah,\nthough among the youngest, {172} proved the clever one, and had written\npoems, more than passably good, before she was fifteen; and had\ncompleted a pastoral drama, when only seventeen.  She was, moreover,\ncomely; she was witty and alert of mind, and had so won upon the\naffections of a neighbor landholder, and wealthy gentleman of culture,\nthat a marriage between the two came after a while to be arranged for;\nbut this affair never went beyond the arrangement,--for reasons which\ndo not clearly appear.  It does appear that the parties remained\nfriendly, and that Mistress Hannah More was in receipt of an annual\npension of L200--in the way of _amende_ perhaps--her life through, from\nthe backsliding but friendly groom.  I am sorry to tell this story of\nher (about the L200).  I think so well of her as to wish she had put it\nin an envelope, and returned it with her compliments--year after\nyear--if need were.  However, it went, as did many another hundred\npounds and thousand pounds of her earnings, in the line of those great\ncharities which illustrated and adorned her life.\n\nHer elder sisters as early as 1757 established in Bristol a school for\nyoung ladies, which became {173} one of the most popular and favorite\nschools of the West of England; and when Hannah, some fifteen years\nlater, went up to London, to look after the publication of her _Search\nafter Happiness_, one or two of the sisters accompanied her; and Miss\nHannah, who was \"taken off her feet\" by the acting of Garrick, was met\nmost kindly by the great tragedian--was taken to his house, indeed, and\nbecame thereafter one of the most intimate of the friends of Mrs.\nGarrick.  Dr. Johnson, too, was enchanted by the brisk humor and lively\nrepartee in these clever West-of-England girls; and we have on record a\nbit of his talk to one of them.  He said, in his leviathan way: \"I have\nheard that you are engaged in the useful and honorable employment of\nteaching young ladies.\"  Whereupon they tell the story of it all, in\ntheir bright, full, eager way, and of their successes: and the Doctor,\nsoftened and made jolly and companionable, says, \"What, five women live\nhappily together in the same house!  Bless me!  I never was in\nBristol--but I will come and see you.  I'll come; I love you all five!\"\nOne of the sisters wrote home that {174} she thought--\"perhaps--the big\nDoctor might marry Hannah; for 'twas nothing but--'My love,' and 'My\nlittle Kitten,' between them all the evening.\"\n\nShortly thereafter Mistress More wrote her tragedy of _Percy_; nobody,\nI think, reads it now; but Garrick became sponsor for it--writing both\nprologue and epilogue; and by reason perhaps of his sponsorship it ran\nsome twenty nights successively; the tale of her stage profits running\nup to L600, which would pay for a good many trips from London to\nBristol.  When she came to treat for the publication of a poem which\nshe wrote at that period--she being ignorant of rates,--it was arranged\nwith her publisher that she should receive the sum, whatever it might\nhave been, which was paid Goldsmith for _The Deserted Village_!\n\nIn those early years she was the lively one, and the gay one, and the\nworldly one of the family; but with the death of Garrick, which came\nupon her like a blow, life and all its colors seemed to change.  She\nhaunted London and the theatres no more; she went up to weep indeed at\nher home {175} on the Adelphi Terrace[10] with the disconsolate Mrs.\nGarrick; but all phases of life have now, for Miss More, taken on a\nsoberer hue; she teaches; she founds schools for the poor; she founds\nchapels; she writes tracts; her forward and sturdy evangelical\nproclivities involve her indeed in difficulties with the local church\nauthorities; for her charities go vaulting over their canons; whereupon\nshe relents and abases herself--and then sins in the same holy and\nbeneficent ways of charity again--canons or no canons.\n\nAs a worker she is indefatigable; she drives, rides, and walks over her\nmissionary ground near to Bristol, with the zeal of a gold-hunter.\nThere were those who questioned her wisdom and who questioned the\nquality of her wit, but never one, I think, anywhere, who questioned\nher goodness.  She wrote a novel called _Coelebs in Search of a Wife_.\nDo you happen to have read it?  I hardly know whether to advise it, or\nnot; there is so much to read!  But if you do, you will find most\nexcellent English in it, and a great deal of {176} very good preaching;\nand many hints about the social habits of that time--trustworthy even\nto the dinner hour and the lunch hour; and maxims good enough for a\ncopy book, or a calendar; and you will find--what you will not find in\nall stories nowadays--a definite beginning and a definite end.  I know\nwhat you may say, if you do read it.  You would say that the sermons\nare too long, and that the hero is a prig; and that you would never\nmarry him if he were worth twice his fortune, and were to offer himself\nten times over.  Well--perhaps not; but he had a deal of money.  And\nthat book of _Coelebs_--whatever you may choose to say of it, had a\ntremendous success; it ran over Europe like wildfire; was translated\ninto French, into German, into Dutch, into Polish, and I know not what\nlanguage besides; and across the Atlantic--in those colonial days, when\nbook-shops were not, as now, at every corner--over thirty thousand\ncopies were sold.  Those of us who can remember forty and fifty years\nback, and who knew anything of the inner side of an old-fashioned New\nEngland homestead, must recall the saintship that invested good\nMistress Hannah More!  What unfailing {177} Sunday books her books did\nmake! and with what child-like awe we looked upon her good, kind, old,\npeaked face as it looked out from the frontispiece--with soberly\nfrilled hair all about the forehead, and over this a muslin cap with\nhuge ruffles hemming in the face, and above this circumambient ruffle\nand in the lee of the great puff of muslin--which gave place, I\nsuppose, to the old lady's comb--a portentous bow, constructed of an\nawful quantity of ribbon and crowning that saintly, kindly, homely face\nof Hannah More.\n\nDo you remember--I wonder--that in the early pages of \"The\nNewcomes\"--the Colonel tells Olive Newcome, how he used in his boy days\nto steal the reading of some of Fielding's famous novels; and how\n_Joseph Andrews_, in that forbidden series, had a very sober binding;\nso that his mamma, Mrs. Newcome, when she observed the boy reading it,\nthought--deceived by that grave binding--that the boy might be regaling\nhimself with some work of Mistress Hannah More's; and how, under this\nbelief, she took up the book when he had laid it by; and read and read,\nand flung it down all on a sudden with such a killing, scornful {178}\nlook at the young Colonel, as he never, never forgot in all his life.\n\nIt was unfair of Thackeray to poke fun in this way at good Mistress\nHannah More!  We may smile at her quaintness--her primness--her starch;\nbut there is that in her industry, her courage, her mental range, her\nwide Christian beneficence which we must always venerate.\n\nWe have run on so far, that we have no words to-day for the sturdy old\nKing George.  We turn him over to another chapter, when we will speak\ntoo of Sterne--whom we had almost forgotten--and of Chatterton and of\nsome writing men who sometimes lifted up their voices in the British\nParliament.\n\n\n\n[1] David Hume, b. 1711; d. 1776.  Best edition of his works edited by\nGreen and Grose, 4 vols., 1874.  For life, see Burton and Huxley.\n\n[2] Adam Smith, b. 1723; d. 1790.  A Fifeshire man, and author of that\nfamous book--_The Wealth of Nations_; a good book to read in these\ntimes, or in any times.  He may indeed say rash things about \"that\ncrafty animal called a Politician,\" and the mean rapacity of\ncapitalists; but he is full of sympathy for the poor, and for those who\nlabor; and is everywhere large in his thought and healthy and generous.\nI am glad to pay this tribute, though only in a note.\n\n[3] George Tobias Smollett, b. 1721; d. 1771.  A Scottish physician,\nauthor of various popular novels, of which _The Expedition of Humphrey\nClinker_ is, by many, counted the best.\n\n[4] William Shenstone, b. 1714; d. 1763.  His works (verse and prose)\nwere published in 1764-69.\n\n[5] William Collins, b. 1731; d. 1759.  Interesting memoir by Moy\nThomas, published in 1858.\n\n[6] Swinburne says, with something more than his usual\nexaggeration--\"the only man of his time who had in him a note of pure\nlyric song\";--excluding Gray, and both the Wesleys!\n\n[7] Frances Burney, b. 1752; d. 1840.  She is perhaps better known as\nMme. D'Arblay; though she married somewhat late in life, and after her\nreputation had been won.\n\n[8] The newest and most faithful copy of her _Diary and Letters_ has\nbeen published by George Bell & Sons, London, 1889, 2 vols., 8vo.\n\n[9] Hannah More, b. 1745; d. 1833.\n\n[10] Near present London \"Embankment\"; John Adams was in that day\nstopping at a tavern near by.\n\n\n\n\n{179}\n\nCHAPTER V.\n\nI have spoken within the last few pages of David Hume--philosopher and\nhistorian; he was kindly natured, witty, serene, with a capacity for\nlarge and enduring friendships; yet with not much beguiling warmth in\nhim; leaving a much accredited history, and philosophical writings\neminent for their ingenuity, acuteness, and subtlety.  Under our larger\nand freer range of thinking to-day, it is hard to understand how he\nbecame such a bugbear to so many, and was so unwisely set upon with\npersonal scourgings; even if a man's religious conclusions be all awry,\nwe shall make them no better, nor undo them, by tying a noisy kettle of\nmaledictions at his heels, and goading him into a yelping and maddened\ngallop all down the high ways.  He died unmarried in 1776; his elder\nbrother John, for some reasons of {180} property--which he counted\nlarger than the historian's large repute--changed his name to Home; so\nthat there is not now in Scotland any representative of the immediate\nfamily of this Scotch metaphysician, who bears his name.  I spoke of\nShenstone and gave some specimens of his rhythmic and tender graces;\nbut he never struck deeply into the poetic mine, whether of passion or\nof mystery.  William Collins, however, did; he was not among the very\nforemost poets certainly, but he gave to us tingling and sonorous\nechoes of the great utterances of olden times, and piquant foretaste of\nnobler utterances that were to come.  We had our little social brush\nwith the lively and chatty \"Evelina\" Burney; we paid our worship at the\nshrine of Mistress Hannah More--and I tried hard to fix her quaint,\nhomely, kindly figure in your gallery of literary portraits.\n\nShe lived, like Mme. d'Arblay, to a very great age--eighty-eight, I\nthink, and was (with the exception of the last-named lady) the latest\nsurvivor of all those whose lives and works we have thus far made\nsubject of comment in the present volume.  And the life and works of\nthese people {181} about whom we have latterly spoken, have had steady\nparallelism--longer or shorter--with the life and reign of George III.\n\n\n_King George III._\n\n[Sidenote: George III.]\n\nWe ought to know something of the personality of this king who came to\nthe head of the British household while all these keen brains were\nastir in it, and within the limits of whose rule the American\nRevolution began, and ended in the establishment of a new nationality;\nwhile the French Revolution too gathered its seething forces, and shot\nup its lurid flame and fell away into the fiery mastership of Napoleon.\n\nYou will remember that George II. was son of George I., who inherited\nthrough his mother, Sophia (of Brunswick), who was granddaughter of old\nKing James I. of Scotland and England.  George III. was not the\nson--but a grandson--of George II.  His father, Prince Frederic, who\nlived to mature years, who wrote some poor poetry--who was generous,\nwayward, incompetent, always at issue with father and mother {182}\nboth--was a man nobody much respected and nobody greatly mourned for.\nIt was of him that a squib-like epitaph was written, which I suppose\nexpressed pretty justly popular indifference respecting him and others\nof his family:--\n\n  Here lies Fred,\n  Who was alive and is dead.\n  Had it been his father\n  I had much rather;\n  Had it been his sister,\n  No one would have missed her.\n  But since 'tis only Fred\n  Who was alive and is dead,\n  There's no more to be said.\n\n\nGeorge III. was severely brought up by his mother and by old Lord Bute;\ntaught to be every inch a king; and he was royally stiff and obstinate\nto the last.  Two romantic episodes attaching to his young days belong\nto the royal traditions--in which a pretty Quakeress, and that\nbeautiful Sarah Lennox--whose portrait by Reynolds now hangs in Holland\nHouse--both figure; but these episodes are of vague and shadowy\noutline, almost mythical, with issues only of the Maud Muller\nsort--they sighing \"it might have been,\" and he--not {183} sighing at\nall.  It is certain that he accepted complacently and contentedly the\nbride Charlotte, who came over to him from Germany; and alone of all\nthe quartette of Georges, made a devoted and constant husband as long\nas he reigned.  But if he did not give his queen heart-aches in the\nusual Georgian fashion, I have no doubt that he gave her many a\nheart-ache of other sorts; for he was bigoted, unyielding, austere,\nand, like most men, selfish.  He had his notions about meal-times and\nprayer-time, and getting-up time, and about what meals should be eaten\nand what not eaten; under this discipline wife and children grew\nup--until the boys made their escape, which they did actively.  Yet\nthis old gentleman of the crown is considerate too--more perhaps\noutside his palace than within: he purposes no unkindness; he indulges\nin pleasant chit-chat with his humble neighbors at Windsor; has\nsometimes half-crowns by him for poor favorites; cherishes homely\ntastes; knows a good pig when he sees it, and can test the fat upon a\nbullock with a punch of his staff.  He professed a certain art\nknowledge, too--but always loved the spectacular, melodramatic works of\nour {184} Benjamin West (in which, art-heresy of the time he had\nexcellent company), better than the rare sweet faces of Reynolds, or\nthe picturesqueness of Gainsborough.\n\nHe was English in his speech (though familiar with French and German);\nEnglish, too, in his contempt for the mere graces of oratory; loving\nbetter point-blank talk, fired with interrogation points and\ninterjections.  Mme. d'Arblay, whose acquaintance we made, makes us a\nparty to some of this talk:--\"And so you wrote 'Evelina,' eh? and they\ndidn't know; what--what?  You didn't tell? eh?  And you mean to write\nanother--eh--what?\"\n\nYet withal, Dr. Franklin--whose name is entered in the London Directory\nof 1770, as \"Agent for Pennsylvania,\" Craven Street, Strand--says of\nthe king: \"I can scarcely conceive a man of better disposition, of more\nexemplary virtues, or more truly desirous of promoting the welfare of\nhis subjects.\"  Ten years later, I think Dr. Franklin would have\nqualified the speech.\n\nBut he never could have gainsaid the exemplary virtues of that quiet\nhousehold--where king and {185} queen lived like Darby and Joan--going\nbefore light through the chilly corridors to morning prayers; with\nearly dinners, no suppers, no gambling, no painted women coming between\nthem.  Yet the king, as he grew old, loved plays and farces, and used\nto laugh obstreperously at them, till Charlotte would tap him with her\nfan and pray his majesty to be \"less noisy.\"\n\nHe knew genealogies and geography; he could talk with courtiers about\ntheir aunts and cousins, and stepfathers and mothers-in-law--which is a\ngreat lift to conversation for some minds.  He knew all parts of his\nestablishment--who cleaned the silver and the brass; and what both\ncost.  Like all such meddling, fussy masters of households, he believed\nhimself always right; prayed himself into accessions of that belief:\nand on that belief went on pounding and pumelling the American branch\nof his family into a state that proved explosive.  In short he was one\nof those methodic, obstinate, sober, stiffly religious, conventional,\nstraight backed, economic, terrific, excellent men whom we all like to\nlook at, and read about, rather than to live with.\n\n{186}\n\nAs a school-master he would have set the old lessons in Cocker (if it\nwere Cocker) and recognized nothing better; and if the sums were not\ndone, you would hear of it.  \"What, what? not done? sums not done!\" and\nthen the old red ruler, and the hand put out, and a spat, and another\nspat.  This was George III.  \"Those colonists not going to pay taxes,\neh? and throwing tea into Boston harbor?  What--what?  Zounds--punish\nthe rebels.  Punish 'em well!  I'll teach 'em.  Flinging tea\noverboard--what--eh?\"\n\nAnd so the war crept on; and all through it the great old stiff\nschool-master brandishing his red ruler and making cuts with it over\nseas.  But the time came when he couldn't reach his rebels; and then\nthe long ruler, which was the national power, got broken in half, and\nit has stayed broken in half ever since.\n\nThere is interesting record of the first approaches of that insanity\nwhich ultimately beset the king, in Mme. d'Arblay's Diary, which we\nhave already mentioned; but he made what seemed an entire recovery from\nthe early stroke of 1788; and was king, in all his headstrong and\nkingly ways, once {187} more.  It was in 1785 when John Adams was\npresented to him as Envoy of the United States of America--not a\npresentation, it would seem, that would have any soothing aspect.\n\nYet the old king received Mr. Adams courteously; and under the pretty\nfustian of conventional speech the one covered his regrets and the\nother covered his exultation.  But it was not many years before the\ndistraught brain--after renewed threats--waylaid the monarch\nagain--this time with a surer grip; his speech, his sight, his hearing,\nall lost their fineness of quality and went down in the general wreck;\nin 1810, that mad-cap, that posture-master, that over-fine\ngentleman--so far as dress and carriage and polite accomplishment could\nmake George IV. a gentleman--took rule; but for years thereafter, his\nlunatic father, in white hair and long white beard, might be seen\nstalking along the terrace at Windsor, babbling weak drivel, and\nhumming broken tunes, leading no whither.\n\n\n{188}\n\n_Two Orators._\n\n[Sidenote: Charles James Fox.]\n\nAmong the younger members of the famous Literary Club, some ten years\nafter its foundation, was a muscular, swarthy young fellow[1]--full of\nwit and humor, a great friend of Burke's until the bitterness of\npolitics parted them; shy of approaches to Dr. Johnson, with whom he\ndiffered on almost all points; a man known now in literary ways only by\nthe fragment of British History which he wrote, but known in his own\ntimes as the most brilliant of debaters, most liberal in his politics,\nand always an ardent friend of America.  This was Charles James Fox,\nwho could trace back his descent--if he had chosen--through a Duchess\nof Gordon, to Charles II., and who was a younger son of a very rich\nLord Holland, owner and {189} occupant of that famous Holland House,\nwhich with its remnant of evergreen garden (in whose alleys we found\nAddison walking) still makes a venerable breakwater against the waves\nof brick and mortar which are piling around it.\n\nLord Holland was over-indulgent to this son of his, allowing him, when\na boy on his first visit to Bath, five guineas a night to \"risk\" at\ncards; and the boy took with great kindness to that order of training,\nsending home to his father, when he came to travel (after a brief\ncareer at Oxford) vouchers, and honest vouchers too, for gaming debts\nof one hundred thousand dollars from the city of Naples alone.  And he\nmatched these losses, and larger ones, at Brooks's in London.  Old\nstagers said that he was so sagacious and brilliant at whist, that he\ncould easily have won his five thousand a year; but he took to hazards\nat dice that brought him losses--on one occasion at least--of four\ntimes as much in a night.  It is a wonder he ever became the man in\nParliament that he was, after such dandling as befell him in the lap of\nluxury.  Yet he was an accomplished Greek scholar; loving the finesse\nof the language, and loving more {190} the exquisite tenderness of such\nlamentations as that of Alcestis; his sympathies all alive indeed, in\nyouth and manhood, to humane instincts--the pains and pleasures of the\nrace touching his heart-strings, as winds touch an Eolian harp.  Study\nof exact sciences put him to sleep; he loved the game of Probabilities\nbetter than the certainties of mathematics--gambling away great\nestates, and put to keenest endeavor by the tears of a woman; speaking\nwith his heart on his tongue--too much there indeed--carrying the\ncomradery of the clubs into public life; sharp as a knife to those who\nhad done him, or his, injury; but unbosoming himself with reckless\nfreedom to those who had befriended him; never un-ready in debate;\nwarming easily into an eloquence that charmed men.  But there must have\nbeen much in the voice and eye to explain the force of speeches which\nnow seem almost dull;[2] the best elocutionist cannot read the\nmagnetism into them which electrified the Commons, and which made {191}\nBurke declare him the \"most brilliant debater the world ever saw.\"\n\nIndeed we can only account for his great successes as an orator, his\namazing repute, and his exceptional popularity, when we sum up a half\nscore of contributory causes, which lie outside of the cold print of\nthe Parliamentary record; among these, we count--his Holland wealth and\ntraining, his environments of rank and luxury, his picturesque bearing,\nhis _bonhomie_, his scorn of the rank he held, his accessibility to\nall, his outspoken, democratic sympathies, that warmed him into\noutbursts of generous passion, his fearlessness, his bearding of the\nking, his earnestness whenever afoot, his very shortcomings too, and\nthe crowding disabilities that grew out of his trust--his\nsimplicities--his lack of forethought, his want of moneyed prudence,\nhis free-handedness, his little, unfailing, every-day kindnesses--these\nall backed his speeches and put a tender under-tone, and a glow, and a\ndrawing power in them, which we look for vainly in the rhetoric or the\nargumentation.  He was often in Parliament--sometimes in the Ministry;\nbut his disorderly and reckless life (gaming was not his {192} worst\nvice) made his fellow-politicians wary, and put a bar to any easy\nconfidences between himself and the old-fashioned, sober-sided, orderly\nGeorge III.  We must think of him as an accomplished, generous-hearted,\nimpulsive, dissolute wreck of a man.\n\n[Sidenote: William Pitt.]\n\nIf I mention Pitt,[3] it is only because you will find in your\nhistorical reading, his name always coupled with that of Fox; but he\nnever went to our Literary Club; had little companionship with literary\nmen; yet he had keen scholarship--within a somewhat limited range--and\nan insatiate ambition.  He was tall, spare, pale-faced, haughty, with a\ncontempt for sentiment, and a contempt for money; and of intellect--all\ncompact.  At an age when many are still at college, he had made amazing\nspeeches in Parliament; not profuse, not swollen with words, not\nrhetorical--but clear, sharp, polished, strenuous, with now and then\nthe glitter of some apt and resonant line from his classics.[4]\n\n{193}\n\nHis perspicuous and never-failing flow of language was due, not a\nlittle, to an early habit of translating at sight, from Greek and Latin\norators, under direction of his father the Earl of Chatham--not taught\nby this great master to give slavish word for word translation; but as\napt and polished and vigorous a rendering as he could accomplish,\nwithout any surrender, or mal-presentment of the leading thoughts.  Nor\ndo I know any class-room exercise, nowadays, which would so test and\namplify a young student's vocabulary, or teach him better the easy and\nforcible use of his own language.  But, to have its full disciplinary\npower, it should be a loud, _ore rotunda_ rendering--not a {194} mere\nlip-service; a launch, straight out from shores, into whatever waters\nor wilds the heathen orators may be sailing upon, and a full showing of\ntheir changing drift--whether in the eddies of a playful irony, or\nunder the driving sweep of their storms of denunciation.\n\nSingularly apart from literary men, and most literary influences,\nMacaulay has objected (perhaps with some reason) to Pitt's cruel\ndisregard of Dr. Johnson's needs and longings in his latter years; it\nwould have been a charming thing, for instance, for the son of Chatham\nto put a Government ship at the service of the invalided philosopher,\nfor a voyage under Italian skies; but with Pitt, the large political\nends which were taking shape in his mind, and in process of evolution,\nblinded him to lesser and personal or kindly interests.  A nod of the\nobstinate old king would have counted for more than a tragedy of\n_Irene_.  All his classicism was but a weapon to smite with, or from\nwhich to forge the links of those shining parentheses by which he\nstrangled an opponent.  Nothing beyond or below the cool, considerate\nhumanities of the cultured, self-poised gentleman {195} (unless we\nexcept some rare outbreak of petulance) belongs to this great orator,\nwho could thrust one through with a rapier held by the best rules of\nfence; and who never did or could say a word so warm as to touch a\nfriend or make an enemy forget his courtliness.  Guiding the political\nfate of England through a period of such strain, as demanded more nerve\nand more discretion than any period of a century before, or of a\ncentury thereafter--admired by all, and loved by very few, Pitt died\nquite alone, in a little cottage on Wimbledon Common[5]--even his\nservants had left;--died too of old age; an old age that grew out of\nhis tormenting labors and ambitions--before he was fifty.\n\n\n_An Orator and Playwright._\n\n[Sidenote: Sheridan.]\n\nSheridan is another name about which you have a better right to hear,\nsince he was a favorite member of the Turk's Head coterie, and is a\ndistinct literary survivor of that epoch.[6]  He was {196} son of\nThomas Sheridan, author of a life of Swift and of a now rarely cited\nEnglish Dictionary.  The son Richard, after studying at Harrow, and\nafterward with his father, made a runaway match with a beautiful Miss\nLinley; and he continued doing runaway things all his life.  A duel\nwhich his sharp marriage provoked, gave him material for his early play\nof _The Rivals_,--a play which has come to renewed popularity in our\nday, and country, under the pleasant humor of Jefferson.  _The School\nfor Scandal_ is another of his comedies which makes its appearance from\nyear to year: and Charles Surface and Lady Teazle--no less than Mrs.\nMalaprop, and Lydia Languish, are people who hang by, very\npersistently, and with whom we are pretty sure to make acquaintance at\nsome time in our lives.\n\nMrs. Sheridan proved a much better wife than the conditions of the\nmarriage promised; and I suppose that she was, in a way, contented with\nthe ribbons and fine gowns, and equipages he {197} provided for her\n(when he could); and with his unctuous, tender speeches, and his fame,\nand an occasional tap under the chin,--and with his forgetfulness of\nher when he went to the clubs, or the green-room, or the tavern--as he\ndid very often, and stayed very late.  Indeed \"staying late,\" was the\nruin of him.  But this language into which I have fallen--not without\nwarrant--should not convey the idea that this man was a commonplace,\ndissolute spendthrift; far from it.  His spendings were sublimated by a\ncrazy splendor of ungovernable and ill-regulated generosities, in which\nhis Irish nature bubbled over; and his dissipation wore always the\nblazon of high social cheer; his excesses not sordid or grovelling, but\nthey carried a quasi air of distinction, and were illuminated by the\nglow of his easy talk and the flashes of his wit.\n\nHis wildest spendings were always made without shamefacedness; but, on\nthe contrary, with a bold alacrity, that gave assurance of riches as\nheaped up as those of an Arabian Night's tale.  That wife of his,\ntoo--with her peachy tint, her faery grace, and her syren voice--seemed\n{198} altogether a fit portion and adornment of the oriental profusion\nhe always coveted and always owed for.  His longings and ambitions were\npitched upon a high key--a key to which his social aptitudes were\ncharmingly attuned; and there was a time early in his career when it\nwas a distinction to have the privilege of entree at his beautiful home\nin Orchard Street, Portman Square, to share his sybaritic tastes, and\nto listen to the siren who warbled there.\n\nAt twenty-four this favorite of fortune had written that play which\ndrew all London to see Captain Absolute; at twenty-five he had become\nhalf owner of that great theatre of Drury Lane, from whose till the\nhands of Garrick had drawn out a great fortune, and from which Richard\nSheridan was to draw, often--more than was fairly in it.  Meantime he\nhad inspired, and, in connection with his father-in-law, had composed,\nthe comic opera of the _Duenna_, whose success was enormous, and whose\nbouncing bits of lyrical jingle have come quivering through all the\n_couloirs_ of intervening days, to ours: instance,--\n\n{199}\n\n  \"I ne'er could any lustre see\n  In eyes that would not look on me.\n  Is her hand so soft and pure?\n  I must press it to be sure.\"\n\n\nThen comes the _School for Scandal_, and--two years later--the\n_Critic_; and always the steaming suppers and the singing of many\nsirens, and deeper thrusts into the till of Old Drury; stockholders may\nwince and creditors too; but who shall gainsay or doubt the imperial\ngenius who is winged with victory?  Garrick, whose days of conquest are\nnearly over--is his friend; so is Burke, won by his wit, and by his\nrolling Irish r's; Goldsmith acknowledges his sovereignty: Dr. Johnson\nveils dislike of his radicalism and of his tirades against taxation, as\nhe welcomes him to the Club.\n\nIn 1780, while still under thirty, he entered Parliament (for Stafford)\nand posed there for new conquests.  There came frequent occasions for\nthe interjection of his witty collocation of apothegms, lighted by his\nbrilliant elocution; but there was not much in his parliamentary career\nto attract national attention until the debates opened with reference\nto the Warren Hastings impeachment.  {200} These offered topics which\nappealed to his emotional nature, and under the indoctrination and the\ncoaching of Burke, he made such appeal for the far-off, down-trodden\nprincesses of India as electrified the nation.  \"Whatever the acuteness\nof the bar, the dignity of the Senate, or the morality of the pulpit\ncould furnish [in eloquence] had not been equal to it.\"  This was the\nverdict of so good a judge as Burke.  Yet, reading this speech--or so\nmuch of it as the records show--or those others which followed,[7] when\nthe great trial had opened in Westminster Hall, we find it hard to\nunderstand the enthusiasm of the old plaudits.  There is wit, indeed,\nin whatever work warms him to a glow; old truisms get a setting in his\noratorio reaches which make them gleam like diamonds; but there is none\nof that logical method {201} which wraps one around with convictions;\nbut in place of it a beautiful mass of rhetorical spray, that delights\nand refreshes and passes--like a summer cloud.\n\nMeanwhile the suppers abound, and so do the debts: that siren wife, who\nhad kept his accounts, and made extracts and filled his note-books (and\nhis flasks), passes away.  It is a shock that does not rally his\nforces, but rather disperses them.  He is _lie_ in these times with the\nPrince of Wales; dines with him; wines with him.  Who shall say he does\nnot troll with him some of the piquant snatches of his own verse?  As\nthis:\n\n  \"A bumper of good liquor\n  Will end a contest quicker\n  Than justice, judge, or vicar;\n  So fill a cheerful glass\n  And let good humor pass.\n  But if more deep the quarrel,\n  Why, sooner drain the barrel\n  Than be the hateful fellow\n  That's crabbed when he's mellow.\"\n\nHe _did_ drain the barrel; he did fall from all his dizzy eminence; he\ndid die a drunkard of the grosser sort; without money, almost without\n{202} friends.[8]  There was a great rally of coronets at his funeral,\nand a pompous procession of those who went to bury him at Westminster.\nYou will find his name there, in the Poets' Corner of the Abbey, and\nwill give to his memory your wonder and your pity; but not, I think,\nmuch veneration.\n\n\n_The Boy Chatterton._\n\n[Sidenote: Chatterton.]\n\nWe shift the scenes now for a new episode in our little story of\nletters, although we are under the same murky sky of London.  George\nthe Third is just finishing the first decade of his long reign; most of\nthe clubmen of whom we have spoken are still alive, and go up, with\nmore or less of regularity, to pay their court to Dr. Johnson; but we\nhave our eye specially upon a pale, handsome-faced, long-haired lad,\nnot beyond the schooling age, who knows nothing of courts or clubs, who\nhas stolen away from the thraldom of a small {203} attorney's office in\nBristol, in the West of England, to come up to London and face the\nworld there, and try to conquer it.  He does not know the task he has\nundertaken.  His brain, indeed, is full of fine fancies; he has the\npoetic fervor in full flow upon him.  He has left a mother and a\nsister--whom he loves dearly--his only near relatives; and he writes to\nthat mother under date of May, 1770:\n\n\n\"I am settled, and in such a settlement as I would desire.  I get four\nguineas a month by one magazine; shall engage to write a History of\nEngland, or other pieces, which will more than double that sum.\nOccasional essays for the daily papers would more than support me.\nWhat a glorious prospect!\"\n\n\nAnd, again, a few weeks later to his sister:\n\n\n\"I employ my money now in fitting myself fashionably, as my profession\n(of letters) obliges me to frequent the places of best resort.  But I\nhave engaged to live with a gentleman, the brother of a lord, who is\ngoing to advance pretty deeply into the bookselling branches.  I shall\nhave lodging and boarding, genteel and elegant, gratis.  I shall have\nlikewise no inconsiderable premium.  I will send you two silks this\nsummer, and expect, in answer to this, what colors you prefer....\nEssay writing has this advantage: you are sure of constant pay; and\nwhen you have once wrote a piece which makes the author inquired after,\nyou may bring the booksellers to your own terms.\"\n\n\n{204}\n\nAh, how young he was!  If only those first literary dreams and hopes\ncould be realized, which nestle in the brains of so many--what\nsilks--what houses--what gold--what fame!  Yet this stripling not yet\neighteen could write.  I will give you a taste of his quality--in\nverses shorn of some of the old words he put in them for sake of\ndisguise:--\n\n  \"The budding floweret blushes at the light,\n    The meads are sprinkled with the yellow hue,\n  In daisied mantles is the mountain dight,\n    The nesh young cowslip bendeth with the dew;\n  The trees enleafed, into heaven straught,\n    When gentle winds do blow, to whistling din are brought.\n\n  The evening comes and brings the dew along,\n    The ruddy welkin sheeneth to the eyne;\n  Around the ale stake minstrels sing their song;\n    Young ivy round the door-post doth entwine.\n  I lay me on the grass; yet, to my will\n    Albeit all is fair, there lacketh something still.\"[9]\n\n\nAnd again, of a different order, this--from the same long poem:--\n\n  \"O sing unto my Roundelay,\n    O drop the briny tear with me,\n  Dance no more at Holy day\n    Like a running river be.\n\n{205}\n\n      My Love is dead,\n      Gone to his death bed,\n  All under the willow tree.\n\n  Come with acorn cup and thorn\n    Drain my hearte's blood away;\n  Life and all its good I scorn,\n    Dance by night or feast by day.\n  My love is dead\n    Gone to his death-bed\n  All under the willow tree.\"\n\n\nWell, this is the poetry of the marvellous boy\nChatterton[10]--fragments of which you will find in all the\nanthologies.  That last tender letter to his sister, which I set before\nyou--so gleeful, with promise of silks and of brilliant essays, was\nwritten on the last day of the month of May, 1770; and on the 24th of\nAugust--not three months later--after three days of starvation enforced\nby a poverty of which his pride would not let him tell, he took poison\nand made an end of his career.\n\nFew knew of this; few knew that there had {206} been any such\nadventurer in London; fewer yet, knew what poems--brimming, many of\nthem, with fine fancies--he had left behind him.\n\nA few months after, at the first annual dinner of the newly founded\nRoyal Academy of Art, Goldsmith,[11] being present, talked at table of\na certain extraordinary lad who had come up the year before from\nBristol--and had died the summer past--literally of starvation--leaving\nbehind him certain wonderful poems, which in their phrases, he said,\nhad an air of great antiquity.  And Horace Walpole being also\npresent--he never omitted being present at a Royal Society dinner, when\nit was possible for him to go--overhearing the talk and the name, said\n(we may fancy), \"Bless me, young Chatterton, to be sure!--I had some\ncorrespondence with the young man; nice poems--but apocryphal--poor\nfellow; dead is he--starved, eh? dear me?--shocking--quite so!\" and I\nsuppose that he took snuff and dusted his ruffles thereafter, and then\ntoyed with his delicate glass of fine old Sercial Madeira.  This was\nlike {207} Walpole--wantonly like him.  There had been a\ncorrespondence, as he condescendingly admitted, that I will tell you of.\n\nThis Bristol boy, growing up in sight of Durdham downs, and the gorge\nof the Avon and blue hills of Wales--with poetic visions haunting\nhim--had somehow come upon old parchments--perhaps out of the muniment\nrooms of St. Mary's Redcliffe church, where his father had been sexton;\nhe had been captivated by the quaint lettering, and awed by the odor of\nsanctity; and straightway imaged to himself an old mediaeval priest, to\nbe clothed upon with his own poetic sensibilities, and in the rusty\nphrases of the fourteenth century, to unfold to the world the poetic\nyearnings and aspirations that were seething in the brain of this\nwonderful boy.  The ancient Dictionaries and old copies of Chaucer\nsupplied the language; the antique parchments gave local allusions and\nthe nomenclature; and for inspiration and motive--the winds that blew\nfrom over Chepstow and Tintern Abbey, and Caerleon, and whistled round\nthe buttresses of St. Mary's Redcliffe--supplied more than enough.  So\nbegan {208} the modern antique poems of Thomas Rowley; not a new device\nin the literary world; for Macpherson, whom we shall encounter\npresently, only a few years before had launched some of the \"Ossian\"\npoems, to the great wonderment and puzzle of the literary world; and\nWalpole, still earlier, had claimed a false antiquity and Neapolitan\norigin for his _Castle of Otranto_.  To Walpole, therefore, the eager\nboy sent some fragments of his Rowley poems, which Walpole courteously\nacknowledged, and asked for a continuance of such favors.  Poor\nChatterton, presuming on this courtesy wrote again, declaring his\ndependent condition--apprenticed to a scrivener, and with mother and\nsister dependent on him--but believing that with God's help, and the\nencouragement of his distinguished patron, he might find the way to\nother and better Rowley poems.\n\nMeantime Walpole, through his scholarly friend the poet Gray, had come\nto doubt the antiquity of Rowley's verse; and the plebeianism of this\ncorrespondent has shocked his gentility.  He replies coolly, therefore;\nexpresses doubts of the Rowley authorship, and advises poor Chatterton\nto keep {209} by his apprenticeship at the scriveners.  This sets the\nyoung poet's blood on fire; he will go to London; he will win his way;\nhe will smite the Philistines hip and thigh.  And--as I have told\nyou--he did go; did work; did struggle.  But it is a great self-seeking\nworld he has to face, full throughout of thwarting circumstance.  Yet\ncourage and pride hold him up--hold him up for months against terrific\nodds; at least he will tell nothing of his griefs.  Thus his last\npennies, which should have gone for bread, go to carry little\nlove-tokens to the dear ones he has left.  So lost is he in his little\nHolborn chamber, in that great seething, turbulent whirl of London,\nthat he thinks--even as he mixes his death potion--they will never\nknow; they will never hear: \"Gone\"--that is all!  But they do know: and\nfor them it is to chant broken-hearted the refrain of his own roundelay,\n\n    My love is dead,\n    Gone to his death bed\n  All under the willow tree.\n\n\nIt is not alone for reason of the romantic aspects of the story that I\nhave given you this glimpse of the boy Chatterton, but because there\nwas really {210} much literary merit and great promise in his work; in\nsome respects, he reminds us of our American Poe--the same disposition\nto deal with mysteries, the same uncontrolled ardors, the same haughty\npride; and although Chatterton's range in all rhythmic art was far\nbelow that of Poe, and although he did not carry so bold and venturous\na step as the American into the region of _diableries_, he had perhaps\nmore varied fancies and more homely tendernesses.  The antique gloss\nwhich he put upon his work was unworthy his genius; helping no way save\nto stimulate curiosity, and done with a crudeness which, under the\nlight of modern philologic study, would have deceived no one.  But\nunder this varnish of archaeologic fustian and mould, there is show of\nan imaginative power and of a high poetic instinct, which will hold\ncritical respect[12] and regard as long as English poetry shall be read.\n\n\n{211}\n\n_Laurence Sterne._\n\n[Sidenote: A sentimentalist.]\n\nJust two years before Chatterton died in Holborn, another noted\nliterary character--Laurence Sterne[13]--died in Old Bond Street, at\nwhat were fashionable lodgings then, and what is now a fashionable\ntailor's shop; died there almost alone; for he was not a man who wins\nsuch friendships as hold through all weathers.  A well known friend of\nthe sick man--Mr. Crawford--was giving a dinner that day a few doors\noff; and Garrick was a guest at his table; so was David Hume, the\nhistorian; half through the dinner, the host told his footman to go\nover and ask after the sick man; and this is the report the footman\ngave to outsiders: \"I went to the gentleman's lodgings, and the\nmistress opened the door.  Says I--'How is Mr. Sterne to-day?'  She\ntold me to go up to the nurse; so I went, and he was just a-dying; I\nwaited a while; but in {212} five minutes he said, 'Now it's come.'\nThen he put up his hand, as if to stop a blow, and died in a minute.\nThe gentlemen were all very sorry.\"  And all the sorrow anywhere--save\nin the heart of his poor daughter Lydia--was, I suspect, of the same\nstamp.  His wife certainly would get on very well without him: she had\nfor a good many years already.\n\n[Sidenote: Laurence Sterne.]\n\nYou know the name of Mr. Sterne, I daresay, a great deal better than\nhis works; and it is well enough that you should.  A good many\nfragments drift about in books of miscellany which you are very likely\nto know and to admire; for some of them are surely of most exquisite\nquality.  Take for instance that talk of Corporal Trim with Uncle Toby\nabout the poor lieutenant, and of his ways and times of saying his\nprayers:--\n\n\n\"When the Lieutenant had taken his glass of sack and toast he felt\nhimself a little revived, and sent down into the kitchen to let me know\nthat in about ten minutes he would be glad if I would step upstairs.\n'I believe,' said the landlord, 'he is going to say his prayers, for\nthere was a book laid on the chair by the bedside, and as I shut the\ndoor I saw him take up a cushion.'\n\n\"'I thought,' said the curate, 'that you gentlemen of the army, Mr.\nTrim, never said your prayers at all.\"\n\n{213}\n\n\"'A soldier, an' please your Reverence,' said I, 'prays as often as a\nparson; and when he is fighting for his king and for his own life, and\nfor his honor too, he has the most reason to pray to God of any one in\nthe whole world.'\n\n\"''Twas _well_ said of thee, Trim!' said my Uncle Toby.\n\n\"'But when a soldier,' said I, 'an' please your Reverence, has been\nstanding for twelve hours together in the trenches, up to his knees in\ncold water, or engaged for months together in long and dangerous\nmarches--detached here--countermanded there; benumbed in his\njoints;--perhaps without straw in his tent to kneel on, he must say his\nprayers how and when he can.'  'I believe', said I, for I was piqued,\nquoth the Corporal, 'for the reputation of the army--I believe, an't\nplease your Reverence--that when a soldier gets time to pray he prays\nas heartily as a Parson--though not with all his fuss and hypocrisy.\"\n\n\"'Thou should'st not have said _that_, Trim,' said my uncle Toby; 'for\nGod only knows who is a hypocrite and who is not.  At the great and\ngeneral review of us all, Corporal, at the day of judgment (and not\ntill then) it will be seen who have done their duties in this world and\nwho have not, and we shall be advanced, Trim, accordingly.'\n\n\"'I hope we shall,' said Trim.\n\n\"'It is the Scripture,' said my uncle Toby, 'and I will show it thee in\nthe morning.'\"\n\n\nNow this beautiful naturalness, this delightful, artistic abstention\nfrom all rant or extravagance, makes us wish overmuch that the whole\nguileless character of my uncle Toby had been as {214} charmingly and\nas decently set in the text; but unfortunately, there is a continuous\nembroidery of it all with ribald blotches, and far-fetched foulness of\nspeech; nor is his coarseness--like that of Fielding--half excused by\nthe coarseness of the age; it is inherent and vital: Fielding, indeed,\nis vulgar and coarse, and obstreperous--with the scent of bad spirits\nand bad company on him;[14] but this other, though a parson, and\nperfumed, and wearing may-be, satin small-clothes, has vile and\ngrovelling tastes that overflow in double-meanings of lewdness: even\nGoldsmith, who was not squeamish, calls him \"the blackguard parson.\"\nIt is not probable that Goldsmith ever encountered him; nor did Dr.\nJohnson.  Beauclerk, Garrick, and Walpole would have been more in his\nline; for he loved the glint, and the capital letters, and the showy\ntag-rags of fashion.  And on the strength of his literary reputation,\nwhich had sudden and brilliant burst, and of his good family--since a\nnot far-off ancestor had been Archbishop of York--he {215} conquered\nand enjoyed, for his little day, all that London fashion had to offer.\nI suspect he took a solid comfort in dying in so respectable a quarter\nas Old Bond Street.  He was buried over Bayswater way, not far from the\nMarble Arch, in the graveyard then pertaining to St. George's (Hanover\nSquare) church.  And there was a story, supported by a good deal of\ncircumstantial evidence, that his body was spirited away and recognized\na few days afterward by a medical student among the spoils of a\ndissecting-room.  This story would horrify more than it did, had it\nattached to an author whose humor had kindled love;--as if this man did\nsomehow deserve a more effective \"cutting-up\" after death than he ever\nreceived before it.\n\nThe Rev. Laurence Sterne had--I should have told you--a church-living\ndown in Yorkshire, to which was afterward added, by adroit diplomacy of\nhis friends, an official position in connection with York Cathedral.  I\ndo not think the people of his parish missed him much when he was away;\nand I am very sure they missed him a good deal, whenever he\nwas--nominally--there: painting, {216} fiddling, shooting, and\ndining-out, took very much of his parochial time; and _Tristram Shandy_\nand its success, literary and pecuniary, introduced him to a career in\nLondon, and in Paris afterward--for he was always an immense favorite\nwith the French (instance Tony Johannot's illustrations)--to which he\nyielded himself with a graceful acquiescence that, I am afraid, put his\nparishioners more out of mind than the fiddling and the shooting had\ndone.\n\nI believe that he loved his daughter Lydia with an honest love; with\nrespect to his wife, one cannot be so sure; some of the most tender\nletters he left, are addressed to a Mrs. Draper, who was his \"dear\nEliza\"--through a great many quires of paper.  He was a Cambridge man\nand well taught;--of abundant reading, which he made to serve his turn\nin various ways, and conspicuously by his stealings; he stole from\nRabelais; he stole from Shakespeare; he stole from Fuller;[15] he stole\n{217} from Burton's _Anatomy of Melancholy_; not a stealing of ideas\nonly, but of words and sentences and half-pages together, without a\nsign of obligation; and yet he did so wrap about these thefts with the\nstrings and lappets of his own abounding humor and drollery, as to give\nto the whole--thieving and Shandyism combined--a stamp of\nindividuality.  Ten to one that these old authors who had suffered the\npilfering, would have lost cognizance of their expressions, in the new\nsurroundings of the Yorkshire parson; and joined in the common grin of\napplause with which the world welcomed and forgave them.\n\nBut I linger longer on this name than the man deserves.  Pathos there\nis in his stories, to be sure, that makes you wilt in spite of\nyourself; but a mile away from those Bond Street chambers where this\npale, thin, silk-stockinged clergyman lives, and has his dinner\ninvitations ten deep, is that old scar-faced Dr. Johnson about whom the\nbeggars crowd; who can put no such pathos into his {218} cumbrous\nsentences indeed; but the presence of that old, blind, petulant woman\nin his house--who had waited on his lost wife--is itself a bit of\npathos that I think will outlast the story of _Maria_--and that should\ndo so forty times over.  I wish I could blot out the silk stockings,\nthe rustling cassock, the simper, the pestilent love letters, the\npretences, the artificialities of the man; they are oppressive; they\nrob his words of weight.  Wit--to be sure, and humor--truculent,\nsparkling--more than enough; for the rest, there is hypocrisy,\npretension--beastliness--untruth--all pinned under a satinquilted cloak\nof vague and unreal piety.\n\n\n\n[1] Charles James Fox, b. 1749; d. 1806.  Elected to club membership in\n1774.  His great great-grandmother was the Duchess of Portsmouth; and\nthe Lord Holland so well known for his entertainments at Holland House,\nearly in this century, was a nephew of Charles James Fox.  Life by\nGeorge Otto Trevelyan.\n\n[2] Instance, speech on French affairs and the question of making peace\nwith Napoleon--just then elected First Consul.  Date of February, 1800.\n\n[3] William Pitt, b. 1759; d. 1806.  Younger son of the Earl of\nChatham.  He entered Pembroke Hall, Cambridge, 1773.\n\n[4] Wraxall in his Memoirs (p. 344) cites special instance in the\nspeech, where he deprecates new alliance between North and\nFox--alluding to personal results to himself:--\n\n  \"Fortuna saevo laeta negotio et----\"\n\n(leaving out the _mea virtute_) then pounding on the table, and adding\nwith oratorical vim\n\n      \"----probamque\n  Pauperiem sine dole quaero.\"\n\n\nHere (says Wraxall, who was an auditor) he cast his eyes down--passing\nhis handkerchief across his lips--to recover breath only.  Certainly he\nwas grandly clear of anything like avarice; no great statesman of\nEngland (unless Gladstone) ever thought so little of money.\n\n[5] See Francis Horner article in _Edinburgh Review_, October, 1843.\n\n[6] Richard Brinsley Sheridan, b. 1751; d. 1816.  Moore's Biography,\ninteresting but not authoritative.  Mrs. Oliphant's sketch in the\nMorley _Lives_, is one of that lady's most charming books.\n\n[7] It was on February 7, 1787, that Sheridan made his first notable\nspeech on the Begum charge in the House of Commons; the second, in the\nimpeachment trial in Westminster Hall, in June, 1788.  Others followed\nof less interest toward the close of the trial in 1794.  The best\nreports are of the speeches made in 1788, published at the instigation\nof Sir Cornewall Lewis, in 1859.  See _Wilkes, Sheridan, and Fox_, by\nW. Fraser Rae.  1874.\n\n[8] A fearful account of Sheridan's condition in his last days is to be\nfound in the _Croker Papers_ (1884), chap. x.  It is embodied in what\npurports to be a literal transcript of a conversational narrative by\nGeorge IV., J. Wilson Croker being interlocutor and listener.\n\n[9] [OE]lia (Humphry Ward's version).\n\n[10] Thomas Chatterton, b. 1752; d. 1770.  Tyrwhitt's edition, \"Poems\nsupposed to have been written by Thomas Rowley,\" etc., dates from 1777.\n\n[11] Foster's Goldsmith, vol. ii., p. 248.\n\n[12] Dr. Skeat--as a philologist--is naturally severe upon a thief of\narchaisms, whose robberies and arrogance did puzzle for a while even\nthe archaeologists.\n\n_Per contra_--there is a disposition among many recent critics to rank\nhim high among the pioneers of the \"New Romantic\" movement in England;\n_Vid._ Rodin Noel--_Essays on the Poets_; also, _Athenaeum_, No. 3073.\n\n[13] Sterne: b. 1713; d. 1768.  _Life_, by H. D. Traill; a fuller one\nby Percy Fitzgerald.\n\n[14] Notwithstanding there was almost always evidence of gentlemanly\ninstincts at bottom; and under the scoriae of a dissipated life and\nhabits the sparkling of a soul of honor.\n\n[15] In a sermon read by Corporal Trim (p. 209, _Tristram Shandy_, vol.\ni., London, 1790) are a good many strong points taken, without\nacknowledgment, from one of Richard Bentley's sermons, preached at\nCambridge against Popery, on November 5th--shortly after the first\nattempt of \"the Pretender.\"  This strange similitude is not noticed in\nDr. Ferrier's summing up of Sterne's sinning in this line.\n\n\n\n\n{219}\n\nCHAPTER VI.\n\nWe had sight of George III. in our last chapter, and we shall catch\nsight of him again from time to time; for he was a persistent lingerer,\nand a most obstinate liver.  We had glimpses, too, of that cheery,\nsunny-faced, eloquent, ill-balanced man, Charles James Fox, whom we\nought to remember as a true friend to America, in those critical days\nwhen taxation was swelling into tyranny.  William Pitt, whom we also\nsaw, and to whom we would have been delighted to listen, would never\nhave won greatly upon American sympathies; too cold, too austere, too\nclassic, too fine.  Sheridan, on the other hand, would, and did,\nconquer hearts everywhere; but unfortunately spending his forces in\ngreat paroxysms of effort; one while the greatest comedist, and again\nthe greatest orator, always the greatest spendthrift; {220} and anon\nthe greatest debtor, who only pays his debts by dying.\n\nSterne covered better his deficiencies of money and of soul.  Who could\nhave put more or truer feeling into the story of the poor ill\nlieutenant of the inn, whom Corporal Trim (at Uncle Toby's instance)\nhad gone to see, and of whom he makes report?  And uncle Toby says he\nwill fetch him home and set him afoot in his regiment.\n\n\"Never,\" says Trim, \"can he march.\"\n\n\"But he _shall_ march,\" says uncle Toby.\n\n\"He will die in his tracks,\" says Trim.\n\n\"He shall _not_ die,\" says Toby, with an oath--which oath, says Sterne,\nthe recording angel washed away, so soon as it was uttered.  The Rev.\nLaurence Sterne, it is to be feared, counted too largely upon the swash\nof such tender recording angels.  Only a host of them, with best\nlachrymal equipment, could float away poor Sterne's misdeeds!\n\nWe touched upon the sad life and fate of the marvellous boy,\nChatterton--not a great poet, but with an exuberant poetic glow within\nhim which gave new brightness to old Romanticism, and {221} which\nkindled in after days many a fancy into flame--up and down the pages of\nlater and bolder poets.  Were his forgeries perhaps instigated by the\nOssianic mystification?\n\n\n_Macpherson and other Scots._\n\n[Sidenote: James Macpherson.]\n\nI do not know if you have ever encountered the poems of Ossian.  They\nare out of fashion now; I doubt if fragments even get into the\nschool-books; but some of my readers may remember in a corner of the\nart-gallery of Yale University a painting, with two life-size figures\nin it, by Colonel John Trumbull--a limp and bleeding, and somewhat\ndainty warrior, leaning upon the shoulder of a flax-haired maiden; with\na little strain from Ossian's Fingal, in the placard below, to tell the\nstory.  The mighty Lamderg (who is the warrior) died: and Gelchossa\n(the flax-haired young woman) \"mourned three days beside her love.  The\nhunters found her dead.\"  The picture is, I suspect, almost the only\npermanent mark in America of the amazing popularity which once belonged\nto the strange, weird, monotonous, gloomy, {222} thin poems of Ossian.\nThere are descriptions of mountain crags in them, and of splintered\npines, of thunder blasts and of ocean hoar; and there are crags again,\nand bleeding warriors, and flax-haired women; harps, moonlight, broken\nclouds, and crags again: I cite a few fragments:\n\n\n\"The oaks of the mountains fall; the ocean shrinks and grows again; the\nmoon herself is lost in heaven; but thou art forever the same,\nrejoicing in the brightness of thy course.  When the world is dark with\ntempests, when thunder rolls and lightning flies, thou lookest in thy\nbeauty from the clouds and laughest at the storm.\n\n... \"Rise, moon, from behind thy clouds!  Stars of the night arise,\nlead me, some light, to the place where my love rests from the chase\nalone--his bow near him unstrung; his dogs panting around him.  The\nstream and the wind roar aloud, I hear not the voice of my love.\"\n\n\n[Sidenote: Poems of Ossian.]\n\nAnd yet these poetic flights, which, it would seem might be made up\nfrom collective but injudicious use of the Songs of Solomon and the\nmental exaltations which come from over-indulgence in tea drinking, or\nother strong waters, were borne, on a swift gale of plaudits in the\nlatter half of the last century all over Europe.  Professors of\nLiterature (such as Dr. Hugh Blair) wrote {223} treatises upon their\nfire and grace; such men as Goethe and Schiller were fast admirers;\nNapoleon was said to be bewildered by their beauty.  Of course they had\nFrench translation; and there were versions in German, Greek, Dutch,\nSpanish, and Latin.  The Abbe Cesarotti, besides writing a dissertation\nin favor of the authenticity of the Gaelic poems, gave an Italian\nversion (the favorite one of Napoleon) which in parts has a rounded\nplay of vocables that makes one forget all poverty of invention.  Thus\nwhen Ossian says,\n\n\"Thy side that is white as the foam of the troubled sea, when dark\nwinds pour it on rocky Cuthon----\" it is rounded by the Italian into\nthis pretty bit of mellifluence:--\n\n  \"Il tuo fianco ch' e candido come la spuma del turbato mare,\n  Quando gli oscuri venti lo spingono contro la mormorante\n  Roccia di Cutone----\"[1]\n\n\n{224}\n\nAnd who, pray, was this Macpherson[2] of the Ossian poems, and what was\nhis claim?  He was a Scotch school-master; born somewhere in the upper\nvalley of the Spey, beyond the Grampians and in the heart of the\nHighlands.  He had been at Aberdeen University awhile, and again at\nEdinboro'; but took no degree at either.  He wrote and printed some\npoor verse when twenty; followed it up with some fragments of old\nGaelic song, which commanded wide attention; and in 1762 published that\npoem of Fingal--professedly by Ossian, an old Gaelic bard; and this\nmade him famous.  The measure and range were new, and there was a\ntorrent of flame and thunder and love and fury running through it which\ncaptivated: he went up to London--was appointed to go with Governor\nJohnston to Florida,[3] in America; remained there at Pensacola, a year\nor more; but quarrelled with his chief (he had rare aptitude for\nquarrelling) and came back in 1766.  Some English {225} historical work\nfollowed; but with little success or profit.  Yet he was a canny\nScotchman, and so laid his plans that he became agent for some rich\nnabob of India (from those pickings winning a great fortune\neventually); entered Parliament in 1780; had a country house at Putney,\nwhere he entertained lavishly; and at last built a great show place in\nthe Highlands, near to his birth-place--which one may see to-day--with\nan obelisk to his memory, looking down on the valley of the Spey; and\nnot so far away from the old coach-road, that passes through\nKilliecrankie, from Blair Athol to Inverness, but the coachman can show\nit--as he did to me--with his whip.\n\nThere were those who questioned from the beginning whether the Ossianic\npoems did really come from the Gaelic;--Dr. Johnson among them, whose\ncontemptuous doubts infuriated the Macpherson to such a degree that he\nchallenged the doughty Doctor.  Johnson replied in what may be called\nforcible speech:--\n\n\n\"Mr. James Macpherson, I received your foolish and impudent letter.  I\nhope I shall never be deterred from detecting what I think a cheat, by\nthe menaces of a ruffian.  What {226} would you have me retract?  I\nthought your book an imposture; I think it an imposture still.  Your\nrage I defy.  Your abilities, since your Homer,[4] are not so\nformidable: and what I hear of your morals inclines me to pay\nregard--not to what you shall _say_, but to what you shall _prove_.\nYou may print this if you will.\"\n\n\nDr. Johnson carried a big oaken cudgel with him, when he travelled in\nScotland.  Hume, on the other hand, was, with Scotch patriotism,\ninclined to accept at first, Macpherson's story of authenticity:[5] but\neven he says of this author, with whom he came into altercation--\"I\nhave scarce ever known a man more perverse and unamiable.\"  The\nHighland Society investigated the matter, and reported that while there\nwas no trace of a complete poem in Gaelic corresponding {227} to\nMacpherson's verse, there were snatches of Highland song and ballads\nwhich supported his allegations.  The question is not even yet fully\nsettled, and is hardly worth the settlement.  Macpherson's own\nobstinacies and petulancies put unnumbered difficulties in the way; he\nresented any denial of Gaelic origin for his verse; he resented any\ndenial of his capacity to sing better than the Gael; he promised to\nshow Highland originals, and always made occasions for delay; withal he\nwas as touchy as a bad child, and as virulent as a fish-woman.  Nothing\nsatisfied him; one of those men whose steak is always too much done--or\ntoo little;--the sermon always too short or too long.  He might have\nbeen the \"Stout Gentleman\" of _Bracebridge Hall_: for he was a big man,\nand always wore wax-topped boots.  Old Mrs. Grant too--who must have\nbeen a neighbor of his, when she lived at Laggan--says that he had\nhabits (with theories about social proprieties) which \"excluded him\nfrom decent society.\"  Mrs. Grant was, however \"verra\" correct, and a\nstickler for the minor, as well as the major virtues.\n\nMacpherson left inheritors of his name, and of {228} his estates in\nthat upper valley of the Spey; and a daughter of his became the wife of\nSir David Brewster, the eminent scientist.  He was buried \"by special\nrequest\" in Westminster Abbey; he had been always covetous of such\npublic testimonials to his consequence.  Yet if his book of Ossianic\npoems was ten-fold better than it is, it would hardly give an enduring,\nor a brilliant gloss to the memory of James Macpherson.\n\nBut whatever may be said for the Gaelic, it is certain that Scotticisms\nwere in those days winning their place in song and in tale.  Since the\nday, in the first quarter of the century (1725), when Allan Ramsay had\nsent out from his book shop in Edinboro', his rustic eclogue of the\n_Gentle Shepherd_, a love had been ripening and growing for those\nScottish strains which were to find their last and unsurpassable\nexpression by and by, in the glow and passion of Burns.\n\nMeantime there were hundreds along the Teviot, and the Esk, and by\nEttrickdale, who rolled under their tongues delightedly the Scottish\nbubbles of song, which broke--now from a bookseller, now from a\nschoolmaster, now from a Jacobite, {229} and now from a \"stickit\"\nminister.[6]  I will give you one taste of this Scotticism of the\nborders, were it only to clear your thought of the gloom and crags of\nOssian.  It is usually attributed to Halket, a Jacobite school-master,\nnot so well known as Ramsay or Robert Ferguson:--\n\n[Sidenote: Logie O'Buchan.]\n\n  \"O Logie o' Buchan, O Logie the laird,\n  They ha'e ta'en awa' Jamie, that delved in the yard,\n  Wha played on the pipe, and the viol sae sma',\n  They ha'e ta'en awa' Jamie, the flower o' them a'.\n\n  \"Tho' Sandy has ousen, has gear and has kye,\n  A house and a hadden, and siller forbye;\n  Yet I'd tak' my ain lad, wi' his staff in his hand,\n  Before I'd ha'e him, wi' the houses and land.\n\n  \"My Daddie looks sulky, my Minnie looks sour,\n  They frown upon Jamie because he is poor;\n  Tho' I lo'e them as weel as a daughter should do\n  They're nae half sa dear to me, Jamie, as you.\n\n  \"I sit on my creepie, I spin at my wheel\n  And think on the laddie that lo'ed me sae weel,\n  He had but ae saxpence, he brak it in twa\n  And gied me the hauf o't, when he ga'd awa'.\"\n\n\n{230}\n\nYet the poet, from whom we quote, died only three years before Burns\nwas born; but I think we can see from the graces of this modest\nschoolmaster singer, that taste and accomplishment were both ripening\nin those north latitudes for the times and the man, in which and in\nwhom, such poetry as that of Burns should be possible.\n\nThere was, too, another growth in those days in that northern capital\nof Great Britain; Dr. Robertson had written his History of America and\nhis History of Charles V.  Adam Smith (the friend of Hume) was busy on\nhis _Wealth of Nations_ (published during the year in which Hume died).\nHugh Blair, the eloquent doctor, was delivering his lectures on\nrhetoric.  Henry Mackenzie, the amiable Dean of the Edinboro' literati,\nwas writing his _Man of Feeling_ and his _Julia de Roubigne_,--books of\ngreat reputation in the early part of this century, but with graces in\nthem that were only imitative, and sentiment that was dismally affected\nand over-wrought; and there was Lord Kames, the _Gentleman Farmer_,\nwith a fine great house in the Canongate, who wrote on criticism, with\nacuteness and taste.  You will not read any of the {231} books of these\nlast-named people; 't were unfair to ask you to do so; but they were\npreparing the way for that literary development which will find\nexpression before many years in the columns of the _Edinburgh Review_\n(established at the beginning of this century), and in the border\nminstrelsy of Scott.\n\n\n_George Crabbe._\n\n[Sidenote: Crabbe.]\n\nWe step back into England now, to find two poets whose principal work\nbelonged to the closing years of the last century; and with echoes,\nfresh and strong, trailing over into the beginning of this.  Neither\ntheir work nor their lives belonged to the noises or to the atmosphere\nof London.  City sounds do not press into their verse; but instead are\nthe sounds of sea-waves or of winds on woods, or of church bells, or of\nthe clink and murmur of the lives of cottagers.  The first I name to\nyou of these two is George Crabbe[7]--a name that {232} may sound\nstrangely, being almost unknown and unconsidered now; yet fifty years\nago there was not a reading-book in any of the schools, nor an album\nfull of elegant selections, which was not open for the story of\nPh[oe]be Dawson, or a glimpse of the noble peasant, Isaac Ashford.  But\nall that is gone:[8]\n\n  \"I see no more those white locks thinly spread\n  Round the bald polish of that honored head;\n  No more that awful glance on playful wight\n  Compelled to knee and tremble at the sight,\n  To fold his fingers all in dread the while\n  Till Mister Ashford softened to a smile.\"\n\n\nThis gives the manner and strain of Crabbe; it is Pope, but it is Pope\nmuddied and rusticated; {233} Pope in cow-skin shoes, instead of Pope\nin prunella.\n\nCrabbe was born in the little shore town of Aldborough--looking\nstraight out upon the North Sea; and the rhythmic beat of those waves\nso stamped itself on his boyish brain, that it came out afterward--when\nhe could manage language, in which he had great gift--very clear and\nvery real; there's nothing better, all up and down his rural tales,\nthan his fashion of putting waves into his rhythmic measures--as you\nshall see:--\n\n  \"Upon the billows rising--all the deep\n  Is restless change: the waves so swelled and steep,\n  Breaking and sinking, and the sunken swells\n  Nor one, one moment, in its station dwells.\n      *      *      *      *      *\n  Curled as they come, they strike with furious force,\n  And then, renewing, take their grating course,\n  Raking the rounded flints, which ages past\n  Rolled by their rage, and shall to ages last.\n      *      *      *      *      *\n  In shore, their passage tribes of sea-gulls urge,\n  And drop for prey within the sweeping surge,\n  Oft in the rough opposing blast they fly\n  Far back, then turn and all their force apply,\n  While to the storm they give their weak complaining cry,\n  Or clap the sleek white pinion on the breast,\n  And in the restless ocean dip for rest.\"\n\n\n{234}\n\nFashions of poets and of poetry may go by, but such scenes on those\nNorth Sea shores will never go by.  Crabbe was son of a customs' man,\nof small, turbulent character, and the boy had starveling education; he\npicked up so much as qualified him at length for surgeon (or doctor, as\nwe say) in that small shore town, but gained little: so, threw all\nbehind him--a girl he loved, and a town he did not love--and with three\nguineas in his pocket, and a few manuscript poems, set off for London.\nHe was there, indeed, in the very times we have talked of; when wits\nmet at the Turk's Head, when Fox thundered in Parliament, when Sterne\nwas just dead; but who should care for this stout young fellow from the\nshore?  One man--one only, does care; it is the warm Irish-hearted\nEdmund Burke, who being appealed to and having read the verses which\nthe adventurer brought to his notice, befriends him, takes him to his\nhouse, makes him know old Dr. Johnson; and his first book is launched\nand talked of under their patronage.  Then this great friend Burke\nconspires religiously with the Bishop of Norwich to plant the poet in\nthe Church.  Why not?  He has some Latin; he {235} means well, and can\nwrite a sermon.  So we find him returned to that wild North Sea shore\nwith a little church to feed, and the church people, in their turn, to\nfeed him.  But the arrangement does not run smoothly; those verses of\nhis, unlike most rural verse, have shown all the darker colors of\npeasant life; if full of sympathy, they were full of bitter, homely\ntruth.  The muck, the mire, the griefs, the crimes, the unthrift, the\ndesolation, have given sombre tint to his village pictures; perhaps\nthose shore people resent it; perhaps he is incapable of the cheeriness\nwhich should brighten charity; at any rate he goes away under private\npreferment to a private chaplaincy at Belvoir Castle, the seat of the\nDuke of Rutland.\n\nThere is not a more princely house among the baronial homes of England.\nIt sits among wooded hills--which to the eye of a Suffolk man would be\nmountains--where Lincolnshire and Leicestershire join: the towers of\nLincoln Cathedral are in sight at the north, and Nottingham Castle in\nthe west: and there is a glitter in some near valley of an affluent of\nthe Trent, shining amid billows of foliage; while within the stately\n{236} home,[9] the Suffolk doctor could have regaled himself with\nexamples of Rubens and of Murillo, of Teniers, Poussin, and Vandyke.\n\nThe Duke of Rutland was a kindly man, a sentimental lover of\nliterature, enjoying the verse of Crabbe, and proud of patronizing him,\nbut lacking the supreme art of putting him at ease among his titled\nvisitors; perhaps enjoying from his high poise, the disturbing\nembarrassments with which the good-natured poet was beset under the\nbewildering attentions of some butler, who outshone the host in his\ntrappings, and in his lordly condescension to the level of an\napothecary's apprentice.\n\nIt was not altogether pleasant for Crabbe; and when afterward he had\nmarried his old flame of Aldborough, and by invitation of the Duke (who\nwas absent in Ireland) was allowed to partake of the hospitalities of\nthe castle, the ironical obsequience of the flunkeys all, drove him\naway from the baronial roof.  Through the influence of friends he\nsecures livings,--first in Dorset, and afterward {237} in\nLeicestershire (1789), almost within sight of Belvoir towers.\nHereabout, or in near counties, where he has parochial duties, he\nvegetates slumberously, for twenty years or more.  He preaches,\npractises his old apothecary craft, drives (his wife holding the\nreins), idles, writes books and burns them, grows old, has children,\nloves flowers, and on one occasion, mounts his horse and gallops sixty\nmiles for a scent of the salt air which he had snuffed as a boy.\nMeanwhile the old haunts in London, which he knew for so brief a day,\nknow him no more; his old friends are dead, his hair is snowy, his\npurposes wavering.\n\nBut his children are of an age now to spur him to further literary\neffort; with the opening of the present century he rallies his power\nfor new songs; and thereafter the slowly succeeding issues of the\n_Parish Register_, _The Borough_, and the _Tales of the Hall_, pave a\nnew way for him into the courts of Fame.  He secures another and more\nvaluable living in the South of England (Wiltshire), where the incense\nof London praises can reach him more directly.  One day in 1819 he goes\naway from his publishers with bills for {238} L3,000[10] in his pocket;\nmust take them home to show them to his boy, John; he loves that boy\nand other children over much--more, it is to be feared, than he had\never done that mother, the old flame of Aldborough, in respect of whom\nthere had been intimations of incompatibility; hence, perhaps, the\ninterjection of that sixty-mile ride for a snuff of the freedom of the\nwaves.  He died at last down in Trowbridge (his new living), a little\nway southward of Bradford in Wiltshire; and his remains lie in the\nchancel of the pretty church there.\n\nWe must think of him, I believe, as a good, honest-minded, well-meaning\nman; dull, I dare say as a preacher; diffuse, meandering, homely and\nlumbering as a poet; yet touching with raw and lively colors the griefs\nof England's country-poor; and with a realism that is hard to match,\npainting the flight of petrels and of the curlew, {239} and the great\nsea waves that gather and roll and break along his lines.\n\n\nWilliam Cowper.\n\n[Sidenote: Cowper.]\n\nThe other poet, to whom allusion has been made, living beside him, in\nthat country of England, yet not near him nor known to him, was William\nCowper.  You know him better: you ought to know him better.  Yet he\nwould have managed a church--if a parish had been his--worse than\nCrabbe did.  I fear he would not have managed children so prudently;\nand if he had ever married, I feel quite sure that his wife would have\nmanaged him.\n\nCowper was of an excellent family, being the son[11] of a church\nrector, and was born at the rectory (now destroyed), which once stood\nunder the wing of the pretty church that, with its new decorations,\nstill dominates the picturesque valley {240} town of Great-Berkhamsted,\non the line of the London and Birmingham Railway.  He studied at\nWestminster--being school-fellow with Churchill, the poet, and with\nWarren Hastings--of whose Trial we have had somewhat to say: afterward\nhe entered a solicitor's office at the Temple, where Thurlow (later,\nLord Chancellor) chanced to be clerk at the same time.  He had fair\namount of money, good prospect of a place under Government--his uncle,\nAshley Cowper, being a man of position and influence.\n\nThis uncle had two daughters, to one of whom this young gentleman said\ntender things;--too tender to be altogether cousinly--in which regard\nshe proved as over-cousinly as he.  But the Papa stamped out that\nlittle fire of love before it had grown into great flame.  There is\nreason, however, to believe that the smouldering of it had its\ninfluences upon Miss Theodora all through her life; and who shall say\nthat it did not touch the great melancholy of the future poet with a\nsting of tenderness?  There was, however, no outspoken lamentation; the\nfeminine nature of the man accepted the decision of the uncle as a\ndecree of fate; {241} there was never any great capacity in him for\nstruggle or for controversy, either with men, or with untoward\ncircumstance.\n\nMeantime, the expected preferment came to young Cowper--a place, or\nplaces of value and of permanence, which he had need only to take with\na bold hand and purpose; but the bold hand was lacking; and his\nhesitancy multiplied difficulties which could only be met by\nexamination for fitness before the Lords; that examination stares him\nawfully in the face; he wilts under the bare prospect; is hedged by\ndoubts; palters with his weakness; falls into a wretched state of\nmelancholy, and--buys laudanum to make an end of it all;--then, he has\nflashes of light, and waves of a redeeming firmness chase over his\nmind; but finally, on the very day on which the examination was to take\nplace, he makes a miserable effort at self-destruction.  Was ever a\nman, before or since, who would commit suicide to avoid lucrative\noffice?  William Cowper, with only an ordinary share of average common\nsense, and unhampered by the trappings of genius that belonged to him,\nwould have \"gone on\" for this place; secured it; {242} made his easy\nfortune; lived a good humdrum life; died lamented, and never heard of.\nThe poet's fine brain, however--which had been exercised already in\nmusical verse--built up mountains of difficulty; he told in after\nyears, with a curious sincerity, all the details of his struggle--how\nhe held the phial of laudanum to his lips and how he flung it away; how\nhe held a knife at his heart; and finally, how he threw his garter,\nwhich served for a gallows-rope, over the chamber door, and hung \"till\nthe bitterness of temporal death was past.\"  Righteously enough, after\nall these weakly resolves, which a man of energy would have made\nstrong, he falls into utter distraction; religious doubts and fears\nracking him, and lunacy throttling him; and so this young Templar of\nthe bright prospects goes away to the care of a mad-doctor.  But long\ncurative processes are needful; and he emerges at last--the blush of\nhis youth all gone, and he lighted up and a-flame with tempestuous\nreligious exhilaration.  He would go into orders, but he can never face\na congregation; so he plants himself, by the advice of friends, who\nprop up his waning income, in the flats of Huntingdon, {243} where the\nriver Ouse winds round and round amid the low lands, and sighs among\nits sedges.  He seems like a castaway; what he has written has been\nlittle--a boy's pastime; what he has purposed has been weak; and I\ndaresay that his uncle Ashley Cowper, and his cousin Theodora, and his\nfellow-clerk Thurlow, thought they would never hear of him more, until,\non some far-off day, a funeral invitation might come.\n\nBut Cowper was presently domesticated in the home of a Rev. Mr.\nUnwin--an old gentleman, who has a youngish wife (though eight or ten\nyears Cowper's senior) and a son, who is also a preacher.  These take\nkindly to the invalid; they relish his religious exuberance; they pity\nhis frailties; and then and there begins an intimate friendship between\nMrs. Unwin and our poet, which for its purity, its strength, its\nconstancy, is without a parallel, I think, in English literary annals.\nIt was about the year 1765 that he first fell in with Mrs. Unwin, and\nhe was never thereafter separated from her--for any considerable time,\ncounting by days--up to the year of her death in 1796.\n\nFor the first sixteen years of this exile upon {244} the flats along\nthe Ouse--whether at Huntingdon or at Olney (where they removed after\nthe death of the elder Unwin) there must have been, what most\nmen--whether poets or not--would count a weary and monotonous\nsuccession of weeks and days and months.  There were few neighbors of\nculture; no village growth or stir; lands all tamely level; streams all\nsluggish; industries of the smallest; no shooting--no fishing--no\ncards--no visitors--no driving; sermon reading in the morning; sermon\nreading in the evening; walks in the garden; digging in the garden\n(mild insanity intervening); petting the tame hares; feeding the doves;\nreading Mistress More; singing hymns; drinking tea; listening for the\nlarks; listening to Mrs. Unwin; drowsing--sleeping--dreaming!  Only\ncontrast that dreary trail of days with those passed by Goldsmith, or\nby Johnson, or by Hume!\n\nBut good Mrs. Unwin, who is not only kindly, but has some dormant\nliterary tastes, does rouse him to some poetic labors; she does have\nfaith in his talent; and it was in 1782, I think, that his book\ncontaining _Table-talk_ and other {245} verse, first appeared, and by\nits quiet graces and naturalness provoked inquiry in London, and\namongst cultured readers everywhere--as to who this \"William Cowper of\nthe Inner Temple\" might possibly be?  The Rev. John Newton of Olney\nknew, for the poet had assisted him in the preparation of a certain\n_Olney Hymn Book_, published not long before: and then and thereafter\nthis John Newton---a good-hearted, well-meaning divine of an\nold-fashioned stamp, was pounding, as occasion served, with the hard\nhammer of his unblinking Calvinism upon the quivering sensibilities of\nthe distraught poet.\n\nBut on the breezes of this new reputation which Cowper had wrought came\nin these times (1782) a fresh bird, in fine feathers, floating into the\ndomestic aviary of Olney.  This was Lady Austen, the widow of a\nbaronet--who planted herself there--not without due graces of previous\nintroduction (1781)--between the Unwin and the Cowper for three years,\ngiving a new stir to the poet's brain.  Out of that quickening came,\nafter a night of travail, that ever-fresh ballad of _John Gilpin's\nRide_; it was popular from the first; and {246} some two years\nlater--it was publicly recited by Henderson--a famous Falstaffian actor\nof that epoch, it ran like wild-fire through the journals of the day,\nwhile the shops along Fleet Street showed in their windows a great\njolly picture of Gilpin and his intractable nag cantering past the Bell\nat Edmonton.\n\nThe shy poet, however, did not go to London to reap any honors which\nmight have accrued; he stayed at Olney, working at a new _Task_, toward\nthe conception and accomplishment of which he was led by the witty\nsallies and engaging devices of the new favorite--Lady Austen.  This\npiquant woman, with her charming vivacities, her alluring airs, her\ndazzling chat, had wrapped the quiet, melancholy poet all around with a\nwitchery to which he was unused and which tempted him to his best\npowers of song.  He was proud of his fresh successes, and grateful to\nthat new and fascinating member of their little household who had\nprovoked and prompted them.  What should disturb this cheery party of\nthree--save the ever-lasting unfitness of the odd number?  Perhaps the\nthought of this came first through some {247} tender reproachful look\nof good Mrs. Unwin; perhaps the poet, stirred to some new wrestle with\nhis withered heart, found out its emptiness; perhaps the gay,\nenchanting new-comer grew weary of the song she had provoked--or weary\nof a welcome that stayed so calm.  At any rate she took wing;[12] there\nwas a little flurry of correspondence to mark the parting, which, I\ndare say, both may have wished should be forgotten.\n\nMeanwhile the new, and much-loved poem which had grown out of this\nintimacy did worthily, and very largely, extend Cowper's fame.  Miss\nHannah More was enchanted by it; \"such an original and philosophic\nthinker,\" she says; \"such genuine Christianity, and such a divine\nsimplicity!\"  Even Corsica Boswell calls him \"a genius;\" and Lord\nThurlow (whose favors to the poet never went beyond words) says of his\nold chum, \"If there is a good man on earth, it is William Cowper!\"\n\nBut the waves of applause break only with a {248} low dolorous murmur\nupon the threshold of that Olney home.  A cruel sense of his own\nundeservings weighs upon his spirits; he cannot ask a blessing at his\nmeals, for who would listen? he cannot pray, for it would be mockery;\nand he consoles himself with the poor satisfaction of not being a\nmocker.  He discusses village and public affairs with his barber,\nWilson (who had conscientiously refused to dress Lady Austen's hair\nupon a Sunday).  Alluding to American affairs, in that crisis when a\ntreaty of peace was discussed at Versailles (1783) between France and\nAmerica, he speaks of the \"thirteen pitiful colonies which the king of\nEngland chose to keep and the king of France to obtain--if he could.\"\nA little later, at the same crisis, he says:\n\n\n\"I may be prejudiced against these [Americans], but I do not think them\nequal to the task of establishing an empire....  You will suppose me a\npolitician; but, in truth, I am nothing less.  These are the thoughts\nthat occur to me while I read the newspaper; and when I have laid it\ndown, I feel myself more interested in the success of my 'early\ncucumbers' than in any part of this important subject.\"[13]\n\n\n{249}\n\n_His Later Life._\n\nIt was only in the latter part of his career that the poet made the\nacquaintance of William Hayley,[14] his future biographer, who had been\ndrawn toward Cowper by the charms of his verse and who came to visit\nhim: this friend, through his wide familiarity with the outer world,\nhad suborned bishops and clergy and public men to write to this\nmelancholy exile of Olney and cheer him with their praises--all which\npraises fell like hail upon Cowper's window pane.  And there had been a\nlittle trip devised, to divert that weakened and fatigued mind, down to\nEartham in Sussex, where his friend Hayley has a beautiful place, and\nwhere he brings the artist Romney, to paint the well-known portrait;\nbut there is no long stay away from the old covert on the flats of\nBuckinghamshire; indeed this covert had taken new life within a few\nyears by the advent of a cousin, the Lady {250} Hesketh, the widowed\nsister of his old lost Theodora; she had come with her carriage and\ntrappings, and taken a fine house, and sought to revive pleasantly all\nthe mundane influences of Lady Austen.\n\nFrom Olney there had come about in those times--at the wise suggestion\nof Lady Hesketh--a move over to the near village of Weston, which\nthereafter became the poet's home.  [On an April day many years\nago--moved by an old New England cleaving to the poems and the poet--I\nstrolled down from Newport Pagnell--to which place I had taken coach\nfrom Northampton--following all the windings of the sluggish Ouse, to\nWeston; stopping at the \"Cowper's Oak\" inn, I found next door his old\nhome--its front overgrown with roses--and strolled into his old garden;\nand thence, by a door the gardener unlocked, into the \"Wilderness;\" the\nusher regaling me with stories of the crazy poet whom he had seen in\nhis boyhood, and who loved the birds, and who wore a white tasselled\nnight-cap as he wandered in the garden alleys at noon.]\n\nIt was at Weston, I think, that the translation of Homer was--if not\nundertaken--most largely {251} wrought upon.  The regular occupation\ninvolved counted largely in the dispersion of those despondent mists\nthat were gathering round him.  He brought scholarly tastes and a quick\nconscience to the work; a boy would be helped more to the thieving of\nthe proper English by Cowper's Homer, than by Pope's; but there was not\n\"gallop\" enough in his nature for a live rendering; and he was too far\nin-shore for the rhythmic beat of the multitudinous waves and too far\nfrom the \"hollow\" ships.\n\nIn the intervals of this important labor--which was only fairly\nsuccessful, and gave him no such clutch upon the publisher's guineas as\nCrabbe gained at a later day--only chance things were written.  But\nsome of these chances were brimful of suggestion and of most beautiful\nissues.  That relating to his mother's picture--sent to him by some\ncousinly hand--a flashing from the embers of his life, as it were, the\nreader must know; who knows it too well?\n\n  \"Oh that those lips had language!  Life has passed\n  With me but roughly since I heard thee last.\n  Those lips are thine--thy own sweet smile I see,\n\n{252}\n\n  The same that oft in childhood solaced me.\n  Voice only fails, else how distinct they say,\n  Grieve not, my child, chase all thy fears away!\"\n\n\nBut it is a poem from which quotation will no way serve.  After the\ndeath of Warton, poet Laureate (1790), Lady Hesketh, and other friends\nwere anxious that the Olney poet should succeed to that honor; Southey\nsays, he might have secured it; but Cowper can never, never go up to\ncourt for a kissing of the king's hand.\n\nAnd now there are coming fast drearier days and months to these good\npeople of the Weston home.  The poet's mind, staggered perhaps by those\nlater Homeric labors, but more likely by the grievous religious doubts\nwhich overhang him, loses from time to time its poise; and he goes\nmaundering, or silent, and with no smile for days, into the deserts of\nmelancholy.\n\n[Sidenote: Death of Cowper.]\n\nMrs. Unwin, worn down by long fatigues, is at last smitten by\nparalysis; and she whose life has been spent in serving must herself be\nserved; the poor poet bringing to that service all the instincts of\naffection, and the wavering purpose of a shattered mind.  Yet out of\nthis new gloom and {253} these terrors of the home comes that faultless\nlittle poem inscribed to \"My Mary.\"\n\n  \"Thy silver locks, once auburn bright,\n  Are still more lovely in my sight\n  Than golden beams of Orient light,\n              My Mary.\n\n  \"For could I view--nor them--nor thee\n  What sight worth seeing could I see?\n  The sun would rise in vain for me,\n              My Mary.\n\n  \"Partakers of thy sad decline\n  Thy hands their little force resign,\n  Yet gently prest, press gently mine,\n              My Mary.\"\n\n\nBut here, as before, quotation counts for nothing; it cannot bring to\nmind the mellowness and the tenderness which lurk in so many of the\nlines and in all the flowing measure of the little poem.  Mrs. Unwin\nhas embalmment in it that will keep her memory alive, longer than would\nany tomb in Westminster.\n\nWell, Mrs. Unwin dies at last in the town of East Dereham, Norfolk,\nwhere they had taken her for \"diversion\"; and the poor poet died there\nthree years later and was buried beside her.  {254} They were three\ndreary years--which followed upon her death--for him and for those\nabout him.  From time to time he touched a little bit of old work, but\nput no joy in it; distraught--weary--smileless--only waiting.\n\n[Sidenote: Cowper's poetry.]\n\nCritics are agreed that we shall not rank him among the great poets;\nbut he comes nearer to their rank than anybody in his day believed\npossible.  He is so true; he is so tender; he is so natural.  If in his\nlonger poems there is sometimes a lack of last finish, and an overplus\nof language--there is a frankness of utterance and a billowy undulation\nof movement that have compensating charms.  He loves Nature as a boy\nloves his play; his humanities are wakened by all her voices.  He not\nonly seizes upon exterior effects with a painter's eye and hand, but he\nhas a touch which steals deeper meanings and influences and transfers\nthem into verse that flows softly and quietly as summer brooks.  He\ncannot speak or rhyme but the odors of the country cling to his words.\nThere is no crazy whirl of expletives which would apply to a hundred\nscenes, but clear, forceful epithet, full of singleness of story:--\n\n{255}\n\nFar spires lifting over stretches of yellow grass-grown plain; marsh\nbirds trailing their flight by sluggish rivers; boats dragged\nslumberously at noon-tide with seething bubbles in their wake; great\nbanks of woodland, wading through snows, or throwing shadows by\nmorning, and counter-shadows at evening, over the flanks of low hills\non which they stand in leafy platoons.  And for sounds--far off\nchurch-bells waking solitudes with their tremulous beat and jangle;\nbirds chasing the echoes of their own songs; bees murmurous over banks\nof thyme; cattle lowing in the meadows; or the bay of some\nhound--breaking full and clear, and lost again--as he follows, far off,\nsome cold trail amongst the hills.\n\nAbove all--he is English; the household has for him the sanctity of an\naltar; firesides are lighted and glow with a sacred warmth; home\ninterests are always golden.  Prone to idleness he is perhaps--mental\nand physical; much femininity in him; his thought wavering and riding\non his rhyme.  But he is good, kind; crudest to himself--sticking the\nJohn Newton darts of Calvinism into his conscience, and loving the pain\nof them.\n\n{256}\n\nI think we must always respect the name and the work of William Cowper.\nIn our next chapter we shall listen to the music of a different singer,\nand to the story of a jollier, and yet of a far sadder life.\n\n\n\n[1] As a matter of curiosity I give what appears to be the\ncorresponding Gaelic in a couplet of lines, from the version in Rev.\nArchibald Clerk's Ossian:--\n\n  \"A's gile na 'n cobhar,' tha sgavilte\n  Air muir o ghaillinn nan sian.\"\n                    l. 75, Duan 1, Fionnghal.\n\n[2] James Macpherson: b. 1736; d. 1796.\n\n[3] Mr. Mackenzie (_Diss. lxxxvii., Edit. Highland Soc._, London, 1807)\nsays that he (Macpherson) took some of his Gaelic MSS. to Florida with\nhim and many were lost there.\n\n[4] Macpherson had translated and published the Iliad in 1773.  It will\ninterest my readers to know that a copy of this letter in Johnson's\nhand-writing, was sold in 1875 for L50--five times the sum which he\nreceived for the tale of Rasselas!\n\n[5] Sir John Sinclair, a voluminous agricultural writer of Scotland,\nwas strenuous supporter of Macpherson's claims--respecting Ossianic\norigin, etc.  The best exhibit, however, of the Gaelic side of the\nquestion may be found in the prefatory _Dissertation_ by Rev. Archibald\nClerk, to the beautiful edition of Ossian published by Blackwood & Sons\nin 1870.\n\n[6] George Halket, a Jacobite schoolmaster, d. 1756; Alexander Ross,\nminister, b. 1699; d. 1784; John Skinner, Episcopal clergyman, b. 1721;\nd. 1807.\n\n[7] George Crabbe: b. 1754; d. 1832.  _The Village_, _The Borough_, and\n_Tales of the Hall_, are his best-known works.  _Life_, by his son\n(1834), is a very full and filially devout book of interesting reading.\n\n[8] So late as 1808, the Edinburgh Review, after speaking of\nWordsworth, Coleridge, Southey, etc., continues in language which I\nsuppose is Jeffery's own:--\n\n\"From these childish and absurd affectations we turn with pleasure to\nthe manly sense and correct picturing of Mr. Crabbe; and after being\ndazzled and made giddy with the elaborate raptures and obscure\noriginalities of these new artists, it is refreshing to meet again with\nthe spirit and nature of our old masters in the nervous pages of the\nauthor [Crabbe] now before us.\"  Vol. xii., p. 131, Edinboro' Edition.\n\n[9] The old castle was burned in 1816, but has been rebuilt with more\nthan its old splendor.\n\n[10] Smiles, in his _Memoirs of John Murray_--the publisher in\nquestion--intimates, however, that the sum was far too large, and\nMurray a loser by the bargain.  Chap. xxii., p. 72, vol. ii.  See also\nMurray's own statement to that effect, p. 385, vol. ii.\n\n[11] William Cowper, b. 1731; d. 1800.  Life by Hayley, 1804; another,\nby Southey (regarded as standard), published with edition of his works\nin 1833-37.  A recent life by Thomas Wright, chiefly valuable for its\nlocal details.\n\n[12] Lady Austen married some years later a French gentleman, M. de\nTardif, and died in Paris in 1802.  She may be counted almost\njoint-author (with Cowper) of _The Task_.\n\n[13] P. 325, Life, etc., by Thomas Wright, London, 1892.\n\n[14] William Hayley, b. 1745; d. 1820.  Life of Cowper, 1803.\n\n\n\n\n{257}\n\nCHAPTER VII\n\nBeyond Dunkeld--which is the southern gateway of the Scottish\nHighlands--there stretches a great wood, within the domain of the Duke\nof Athole, where one can wander for miles; the path sometimes mossy,\nalways inviting; now threading dark glens, and again winding under\nhoary forest trees that grow on uplands; now giving glimpses of brook\nor pool, and now of grassy glade on which some group of century-old\nlarches slant their shadows; one may hear noises of chattering\nsquirrels, of whirring pheasants, of roaring wood-streams, of pines\nsoughing in the wind; and at last, going up a side-path, the visitor\nwill come to the door of a Hermitage, bedded in densest mass of\nfoliage.  Fifty years ago--to a month--the guide opened that door for\nme, entered with me, and closed it behind us.  I then {258} observed\nthat the whole inner surface of the door was one great mirror, and that\nthere were other mirrors around; while directly opposite was a\nlife-size painting of Ossian fingering his harp; and as I was scanning\nthe details of this picture, the guide touched some hidden spring;\nOssian straightway disappeared, sliding into the wall, and through the\nchasm one looked out upon clouds of spray, behind which an Alpine\nwater-fall with roar and foam plunged down sheer forty feet into a\nseething pool below.  The water-fall through an artful collocation of\nmirrors seemed to pour down behind you as well; and from the ceiling to\npour down above you, and to gird you all about with its din and splash\nand spray.  With the cliffs and the pine boughs it made a pretty\ngrouping of Ossianic charms; and I am sorry to hear that since 1869 or\nthereabout, the Hermitage, by reason of some vandal outrage, has wholly\ndisappeared.\n\nThe only memorial the traveller will find now in that region of the\nOssianic harping, of which we spoke in the last chapter, is the\nMacpherson Stone, which some twenty-five miles farther northward, {259}\non the Highland trail, peers out from green copses in the upper valley\nof the Spey.\n\nI spoke also in our last talk of the literary ferment that had declared\nitself, and was in active progress along the Scottish border, and in\nEdinboro'.  We had somewhat to say of the poet Crabbe, and of his long\nand successful poems--now little read; and of those other poems by\nCowper, some of which will be always read, and which, when their art\nshall grow old-fashioned and out of date, will show a tender humanity\nand a kindly purpose, which I trust will never go out of date.\n\n\n_Parson White._\n\n[Sidenote: White of Selborne.]\n\nYou will remember that we found both of the last-named poets in the\ncountry; and that their work concerned itself largely with country life\nand with country scenes.  And now we sidle into the country again, for\nour first studies to-day;--into the county of Hampshire, where lived,\ntoward the close of the last century, two personages--not far apart in\nthat pleasant region of rolling downs; unknown to each other; their\nages, indeed, {260} differing by more than a score of years; but both\nleaving books you ought to know something about.\n\nThe first of these personages was a quiet clergyman[1] of very simple\ntastes and simple habits, who lived in a beautiful parsonage--still\nstanding, and still overgrown with ivies and banked about with great\nwaving heaps of foliage--where he wrote _The Natural History of\nSelborne_.  It is not a formal book or an ambitious book; it is simply\na bundle of short letters extending over dates that cover twenty years\nin their stretch; and yet the book is so small you could carry it in\nyour pocket.  Its title describes the book; it tells what this quiet\nold gentleman saw and learned through twenty odd years of observation,\nabout the birds, the beasts, the fishes, the trees, the flowers, the\nstorms, the sunshine and the clouds of that little country parish of\nSelborne.  And yet that simple story is told with such easy frankness,\nsuch delicacy, such simplicity, such truthfulness, such tender feeling\nfor all God's creatures, whether beast or bird, that the little book\nhas become almost as much a classic {261} as _Walton's Complete\nAngler_; and the name of Gilbert White, which scarce a hundred\nLondoners knew when he died, is now known to every well-equipped\nEnglish library everywhere.  I have compared it with _Walton's Complete\nAngler_, though it has not the old fisherman's dalliance with the\nmuses; nor has it much literary suggestiveness.  There are no milkmaids\ncourtesying to its periods, nor any songs, except those of the birds.\nGood old Parson White is simpler (if maybe); he is more homely; he is\nmore direct; and by his tender particularity of detail he has given to\nthe winged and creeping creatures of his pleasant Hampshire downs the\nfreedom of all lands.\n\nIt is true, indeed--as I have said in another connection--that we\nAmericans do not altogether recognize his chaffinches and his titlarks;\nhis daws and his fern-owl are strange to us; and his robin\nred-breast--though undoubtedly the same which in our nursery days\nflitted around the dead \"Children in the Wood\" (while tears stood in\nour eyes) and\n\n          \"Painfully\n  Did cover them with leaves,\"\n\n{262} is by no means our American red-breast.  For one, I wish it were\notherwise; I wish with all my heart that I could identify the old\npitying, feathered mourners in the British wood, with the rollicking,\njoyous singer who perches every sunrise, through all the spring, upon\nsome near tree, within stone's throw of my window, and stirs the dewy\nair with his loud _bravura_.\n\nAnother noticeable thing about this old country parson is his freedom\nfrom all the artifices and buckram and abbreviations of learning, so\nthat he is delightfully comprehensible by everybody.  If only we could\nhave an edition of Gray's Botany--for instance--with some ten lines of\nParson White's homely descriptive English about the height and bigness,\nand color and habit of the flowers, instead of symbols and Latin\ngenealogies and scholastic reticence--what a God-send it would be to\nthe average country gentleman or country woman!\n\nI want specially to call the attention of those young people in whose\ninterest I am supposed to talk--to that homely truthfulness, and\nunabating care of this old gentleman, as giving value to a {263} book\nor to any literary work whatever.  They are not qualities, to be sure,\nwhich of themselves carry performance to a high poetic level; but they\nare qualities which give to it practical and picturesque values, and\nwhich--well laid in--will make work survive.\n\nIf I were to undertake on any occasion the direction of the\ncomposition-writing of young people, I should surely counsel\npainstaking and minute description of homely natural objects.  Nature\nis better than millinery.  Yet out of ten young ladies of average\nculture you shall be able to pick nine who shall tell a listener\nflowingly of the last new dress she has seen, and the stuff, and the\ntrain, and the lace, and the sleeves, and the trimmings, and all the\nmysteries of its fit--to one who shall give a simple, clear-drawn, and\nintelligible account of a new flower, or new tree, or a strange bird.\nThus you will perceive that I have made of this old gentleman--whom I\ngreatly respect--a stalking horse, to fire a sermon at my readers; and\nI am strongly of opinion that there are a great many country clergymen\nof our time and day, who, if they would bring old Parson White's zeal\nto the {264} encouragement of a love and a study of natural objects,\nwould do as much thereby to humanize and Christianize the younger\nmembers of their flocks as they can possibly do by Vanity Fairs or\nparochial oyster suppers.\n\nThe modest house of Gilbert White[2] was occupied very many years by\nthe venerable Professor Bell, late president of the Linnean Society,\nwho died in 1880.  The study of the old naturalist remained long as the\nmaster left it; his oaken book-case was still there; so was the\nthermometer attached to the shelves by which he made his observations;\nhis dial by which he counted the hours stands at the foot of the\ngarden; and in the churchyard near by is his grave; while within the\nquaint old church, to the right of the altar, is a tablet in his honor;\nand in his honor, too, all the birds of Selborne will sing night and\nmorning year after year.\n\n\n{265}\n\n_A Hampshire Novelist._\n\n[Sidenote: Jane Austen.]\n\nAnd now for that other Hampshire personage, of whom I gave you a hint,\nas being also guiltless of London life and almost of London\nacquaintances; it is a lady now of whom I have to speak,[3] and one who\ndeserves to be well known.  She lived, when her books were published,\nonly three or more miles away from Selborne, across the hills\nnorthward--at the village of Chawton, which lies upon the old coach\nroad from Farnham to Winchester.  Miss Austen was much younger--as I\nhave said--than our old friend the parson; indeed she was only\nbeginning to try her pen when Gilbert White was ready to lay his down.\nShe had all his simplicities of treatment and all his acuteness of\nobservation--to which she added a charming humor and large dramatic\npower; but her subjects were men and women, and not {266} birds.  She\nwrote many good old-fashioned novels which people read now for their\nlight and delicate touches, their happy characterizations, their\ncharming play of humor, and their lack of exaggeration.  She makes you\nslip into easy acquaintance with the people of her books as if they\nlived next door, and would be pulling at your bell to-morrow, or\nto-night.  And you never confound them; by the mere sound of their\nvoices you know which is Ellinor, and which is Marianne; and as for the\ndisagreeable people in her stories, they are just as honestly and\nnaturally disagreeable as any neighbor you could name--whether by\ntalking too much, or making puns, or prying into your private affairs.\n\nWalter Scott, who read her books over and over, says, \"That young lady\nhad a talent for describing the involvements and feelings and\ncharacters of ordinary life, which is to me the most wonderful I ever\nmet with.\"  Macaulay, too, admired her intensely; ventured even to\nspeak of her amazing, effective naturalness--in the same paragraph with\nShakespeare.  Miss Mitford confided to a young niece of the authoress,\nthat \"she would {267} give her hand,\" if she could write a story like\nMiss Austen.  We may not and must not doubt her quality and her genius,\nwhatever old-time stiffness we may find in her conversations.  One book\nof hers at least you should read, if only to learn her manner; and as\nyou read it remember that it was written by a young woman who had\npassed nearly her whole life in Hampshire--who knew scarce any of the\nliterary people of the day; who had only made chance visits to London,\nand a stay of some four years in the lively city of Bath.  She was very\nwinning and beautiful--if her portrait[4] is to be relied upon--with a\npiquant, mischievous expression--looking very capable of making a great\nmany hearts ache, beside those which ache in her books.\n\nIt would be impossible to cite fragments from her stories that would\ngive any adequate notion of her manner and accomplishment; it would be\nvery like showing the feather of a bird, to give an {268} idea of its\nswoop of wing.  Perhaps _Pride and Prejudice_, though her first written\nwork, is the one most characteristic.  You do not get lost in its\nsentimental strains; you do not find surfeit of immaculate conduct.\nThere are fine woods and walks; but there is plenty of mud, and\nbad-going.  The very heroines you often want to clutch away from their\nuncomely surroundings; and as for the elderly Mrs. Bennett, whose\ntongue is forever at its \"click-clack,\" you cannot help wishing that\nshe might--innocently--get choked off the scene, and appear no more.\nBut that is not the deft Miss Austen's way; that gossiping, silly,\nirritating _mater familias_, goes on to the very end--as such people do\nin life--making your bile rise; and when the rainbows of felicity come\nat last to arch over the scenes of _Pride and Prejudice_, Mrs.\nBennett's clacking tongue is still strident, and still reminds you in\nthe strongest possible way, that Miss Austen has been busy with the\nveriest actualities of life, and not with its pretty, shimmering vapors.\n\n_Persuasion_ is a less interesting book, and less complete than _Pride\nand Prejudice_; its heroine, {269} Anne Eliot, is not possessed of very\nsalient qualities; hardly gaining or holding very earnest attention;\nyet with a quiet sense of duty, and such every-day fulfilment of it, as\nmakes her righteously draw toward her all the triumphs of the little\ndrama; a lost love is reclaimed by these quiet forces, and victory\ncomes to crown her easy gentleness.  _Northanger Abbey_ is weaker, but\nwith bold, striking naturalism in it; all the littlenesses and\nplottings and vain speech of the Bath Pump-Room seem to come to life in\nits pages; to just such life as we may find about our Cape Mays, and\nPequod, and Ocean houses, every blessed summer's day!  Miss Austen's\nearlier novels, which made her reputation, were written before she was\ntwenty-five, and published later, and under many\ndifficulties--anonymously; so she had none of that public incense\nregaling her, which was set ablaze for the less capable Miss Burney;\nand it was almost as an unknown, strange, quiet gentlewoman that she\nwent down, in the later years of her life, to the shores of the\nbeautiful Southampton Waters--seeking health there; and again, on the\nsame search to the higher lands of the {270} Hampshire downs--where she\ndied, only forty-two, and lies buried under a black marble slab, which\nyou may find under the vaults of the interesting old Cathedral of\nWinchester.\n\nThe recognition of her high qualities was not so extended in her\nlife-time, as it is now; and thirty years after her death, a visitor to\nthe great Hampshire Cathedral was asked by the respectable verger:\n\"What there was _particular_ about Miss Austen, that so many people\nshould want to see her grave?\"  Even the most wooden of vergers would\nhardly ask the question now; her extraordinary quickness and justness\nof observation astonish every intelligent reader.  All the more, since\nher life was lived within narrow lines; but what she saw, she saw true,\nand she remembered.  That wonderful masterly Shakespearian alertness of\nmind in seizing upon traits and retaining their relations and colors,\nis what distinguishes her, as it distinguishes every kindred genius.  I\ncan understand how many people cannot overmuch relish the stories of\nMiss Austen--because they do not relish the people to whom she\nintroduces us; but I cannot understand how any reader can fail to be\n{271} impressed and electrified by her marvellous photographic\nreproduction of social shades of conduct.  How delightful is that\nindignation of Sir John Middleton, when he learns of the villainy and\nfalsity of Willoughby.  \"To think of it! and he had offered the\nscoundrel one of Follies' puppies!\"  And then--reflectively--\"A pretty\nman he was too, and owner of one of the finest pointer bitches in\nEngland!  The devil take him!\"  What a synopsis of the man's qualities,\nand of Sir John's measurement of them!\n\n\n_Old Juvenilia._\n\n[Sidenote: Sandford & Merton.]\n\nI cannot pass from this epoch, without saying somewhat concerning that\ntide of literature for young people which set in strongly about those\ntimes.  There was _Sandford and Merton_, for instance; can it be that\nthe moderns are growing up to maturity without a knowledge of the wise\ninculcations of that eminently respectable work?  Sixty years ago it\nwas a stunning book for all good boys, and for the good sisters of good\nboys.  Whoever was at the head of his class was pretty apt to {272} get\n_Sandford and Merton_; whoever had a birthday present was very likely\nto get _Sandford and Merton_; if a good aunt was in search of a proper\nNew Year's gift for a lad the bookseller was almost sure to recommend\n_Sandford and Merton_; and when a boy went away to school, some\nconsiderate friend was very certain to pop a copy of _Sandford and\nMerton_ into his satchel.\n\nIt is in the guise of a great lumbering narrative--supposed to be\ntrue--into which are whipped a score or more of little stories, each\none capped with a bouncing moral.  Thus, there is an ill-natured boy\ngoing out for a day's scrimmage, and playing his tricks--on a poor\ngirl, and a blind beggar, and a lame beggar, and a farmer, and a\ndonkey.  This goes on very well for awhile; but at last the tables are\nturned, and he gets bitten by the blind beggar, and beaten by the lame\nbeggar, and thrashed by the farmer, and is thrown by the donkey, and a\nlarge dog seizes him by the leg; this latter is printed in capitals,\nand there is a picture of it.  At last, in bed, and with watery eyes,\nthe boy reflects--that \"no one can long hurt others with impunity;\" so\nhe determined to \"behave {273} better for the future.\"  Is it any\nwonder that those who had access to such instructive tales a half a\ncentury ago should have grown up to be excellent men!\n\nThis book of _Sandford and Merton_ was written by Thomas Day,[5] an\neccentric rich man (the world of to-day would have called him a crank),\nwho had a fine place near to Putney on the Thames, who sympathized\nstrongly with Americans in Revolutionary times; who was also a disciple\nof Rousseau, and undertook to educate a young girl--two of them in\nfact, one being a foundling--so that he might have a wife of his own\ntraining, after the Rousseau standard; but the young persons did not\ntrain as he wished; so he found his mate otherwheres.\n\nAnother comfit of a book for young people, but with fewer plums of\nromance in it, was _Evenings at Home_ by Dr. Aikin and Mrs. Barbauld.\nI am sure the very name must bring up tender memories to a great many;\nfor it was a current book down to a time when respectable, and even\nmirth-loving {274} people, _did_ pass their evenings at home, and\nenjoyed doing so.  The book commands even now, in some old-fashioned\nhouseholds, about the same sort of consecration which is given to an\nantique blue and white china tea-pot--not nearly so fine as the newer\nFrench ones--but which by the aid of a little imagination can be put to\nvery pretty simmer of old times and tunes.\n\n[Sidenote: Mrs. Barbauld.]\n\nMrs. Barbauld[6] was worthier than this book; she was a sister of Dr.\nAikin--had distinction for great beauty in her youth; married a French\nclergyman of small parts and weak mind, whose intellect, in his later\nyears, went wholly awry and made her home a martyrdom for her, against\nwhich she struggled bravely.  That home was for a time out at\nHampstead, only a half hour's drive from London, and she knew people\nworth knowing there; Fox and Johnson among the rest--though {275}\nJohnson did give her a big slap for marrying as she did and for\nteaching an infant school.[7]  She wrote poetry too, one verse at least\nwhich Wordsworth greatly admired, and with condescension declared that\nhe would have liked to be the author of such a verse himself.  I cite\nthe verse (with some of the context), which is from an apostrophe to\n_Life_; doubtless suggested by the\n\n  \"Animula, vagula, blandula\"\n\nof Adrian, to which allusion has been made in a previous chapter; but\nthe good woman's evolution of the thought is curiously different from\nthat of Pope:--\n\n  \"Life!  I know not what thou art.\n  But know that thou and I must part;\n  And when, or how, or where we met,\n  I own to me's a secret yet.\n  But this I know, when thou art fled\n  Where'er they lay these limbs, this head,\n  No clod so valueless shall be\n  As all that then remains of me.\n    O whither, whither dost thou fly,\n  Where bend unseen thy trackless course,\n\n{276}\n\n  And in this strange divorce,\n    Ah, tell where I must seek this compound I?\n      *      *      *      *      *\n  Life! we've been long together,\n  Through pleasant and through cloudy weather;\n  'Tis hard to part when friends are dear;\n  Perhaps 'twill cost a sigh, a tear;\n    Then--steal away, give little warning,\n  Choose thine own time;\n  Say not--Good night; but in some brighter clime\n    Bid me--Good morning.\"\n\n\nI cannot part from this excellent old friend of British boys, without\ncalling to mind her ardent Whiggism, and her very pronounced advocacy\nof the American cause, in her last poem of _Eighteen hundred and\neleven_; the republican sympathies alienated a good many of her Tory\nfriends, and brought to her temporary disrepute.  Wherefor, I think,\npatriotic American boys may, on some coming fourth of July, fling their\ncaps into the air for the kindly, brave-speaking Mrs. Barbauld, and for\nher _Evenings at Home_!\n\n\n{277}\n\n_Miss Edgeworth._\n\n[Sidenote: An Irish story-teller.]\n\nYou may be sure that I have not forgotten Miss Edgeworth, who was a\ngood friend of Mrs. Barbauld, and who scored Dr. Johnson and Boswell\ntoo, for the printing of their slurs upon Miss Aikin.[8]\n\nI suspect it would not be an easy task to bring young people, nowadays,\nto much enthusiasm about Miss Edgeworth[9] and her books; and yet if I\nwere to tell all that \"we fellows\" used to think about her when her\n_Popular Tales_, and her delightful _Parent's Assistant_, with its\nstories exactly of the right length--about Lazy Lawrence, and Simple\nSusan, and the False Key, and Tarlton--were in vogue, I am afraid you\nwould give me very little credit for critical sagacity.  A most proper\nand interesting old lady we reckoned her, and do still.  I for one\nnever counted on her being {278} young; it seemed to me that she must\nhave been born straight into the severities of middle age and of\nstory-telling.  I could never imagine her at a game of romps, or buying\ncandies on the sly.  Though I had never seen her portrait--and no one\nelse, for that matter--yet I knew the face--as well as that of my own\ngrandmother; and what a good, kind, serene, motherly face it was!\nThere was dignity in it, however; no boy would have thought of\napproaching her without a study of his deportment; he would see to it\nthat his shoe-lacings were tied and his waistcoat buttons all in\nplace--else, a shake of the head that would have made the cap-strings,\nand the frisette, and the starched ruffles shiver.  But we must not\nspeak lightly of the authoress, to whom thousands of elderly people owe\nso much of instruction and of entertainment.\n\n[Sidenote: Miss Edgeworth.]\n\nShe was the daughter of an Irish gentleman who made a runaway match at\nGretna Green, Maria Edgeworth being a child of that irregular marriage;\nand her father being widowed shortly after, married three other\nwives[10] successively, whose {279} children filled the great house at\nEdgeworthtown in Ireland, where the authoress grew up (though born in\nEngland), and where she came to that knowledge of Irish character and\nhabit which gives distinction and the greatest charm to her books.\n\nScott read them gleefully and admiringly, and as he himself confesses,\ntook a hint from them, to put Scottish character into story, as this\nEnglish-Irish lady had put Irish character into hers; and he says in\nhis first outspoken preface to the _Waverley_ series--that Miss\nEdgeworth in \"making the English familiar with the character of their\ngay and kind-hearted neighbors may truly be said to have done more\ntoward completing the union than perhaps all the legislative enactments\nby which it has been followed up.\"  Such laurels were enough for her\nfame--did not braver ones grow out of the thumb-worn edges of her\nbooks.  I think it would be safe to distrust the honor and directness\nof purpose of any boy or man who, after reading--has either scorn or\ndread of Maria Edgeworth.\n\nOne will not find startling things in her {280} writing; nor will you\nfind great brilliancy of execution--nor the pretty banter and delicate\nEnglish humor, and finer touches which belong to Miss Austen: but you\nwill find orderly progress and a good orderly story--illuminated by\nflashes of Irish wit, and glowing through and through with the kindness\nof a heart which never saw suffering without sympathy, and never any\njoys of even the most vulgar, without a tender satisfaction.  Add to\nthis a shrewd common sense--which never lost its way in romantic\npitfalls, and an unblinking honesty, and charity of purpose--always\nmaking itself felt, and always driving a nail--and you have an array of\nqualities which will, I think, keep good Miss Edgeworth's name alive\nfor a long period to come.  Few people will have the courage to invest\nin the whole of her score of volumes octavo.  It is hardly to be\nadvised; but you may wisely choose a sprinkling of them; her _Frank_,\nfor instance--her _Rackrent_--her _Ormond_, and a volume or two of her\nshorter tales, which will bravely hold their own amongst all the goody\nbooks of a later generation.\n\nTwo specimens of that Irish humor, which she {281} is so apt at\nreporting, and which shine by their pretty flicker of unconsciousness,\nI must cite: the first is that of the politician--a charming type of\nour municipal Milesians--who resented highly his non-appointment to\nsome fat place, after unwearied support of the government, \"against his\nconscience, in a most honorable manner.\"  The second is that of the\nhopeful old Irish dame, who trusted she might die upon a fete day, when\nthe gates of Heaven were opened wide, and a poor \"body might slip in\nunbeknownst.\"\n\nFor our good friend, Miss Edgeworth, we believe that those gates were\nwide open, on every day of the year.\n\n\n_Some Early Romanticism._\n\n[Sidenote: Early Romanticism.]\n\nWhile that clever and attractive Miss Jane Austen was engaged upon her\nstories in her quiet study in Steventon, Hampshire, there was opened\nupon England, by certain other ladies, a new sluice of literature--from\nwhich some phosphorescent sparkles are still distinguishable in our\n{282} time--in brilliant red and yellow covers.  I allude to the\n_Children of the Abbey_, by Miss Roche[11] (an Irish-French lady, who\nlived in Waterford, Ireland), to _Thaddeus of Warsaw_ and the _Scottish\nChiefs_ by Miss Jane Porter, and the _Mysteries of Udolpho_ by Mrs.\nRadcliffe, of London.[12]\n\nVery few middle-aged readers have passed their lives without hearing of\nthese books; the chances are strong that most of such readers have\ndipped into them; and if people dipped at all, before the age of\nfourteen, they were pretty apt to undergo complete submergence.\n\nFrom ten to twelve was--as nearly as I now recollect--about the\nsusceptible age for the _Children of the Abbey_; and if the book came\ninto the {283} hands of one of a bevy of boys or girls, in such tender\nyears, it was pretty apt to run through them all, eruptively--like\nmeasles.\n\nIt was a book that even young people had an inclination to put under\ncover, if detected or liable to be detected in the reading of it; and\nelderly people so caught were understood to be only \"glancing at it;\"\nthe sentiment is so very profuse and gushing.  None of us like to make\na show of our allegiance to Master Cupid.  Miss Roche wrote other\nbooks--but none beside the _Children of the Abbey_ have come down to us\nin the yellow and red of sixpenny form; for which we ought to be\nthankful.\n\n_Thaddeus of Warsaw_ had more excuse in the expression of tender\nsympathies for Poland and all Polish people, at a crisis in the history\nof that unfortunate kingdom.  The success of the book was immense.\nKosciusko sent his portrait and a medal to the author; she was made\nmember of foreign societies, received gold crosses of honor; and oddly\nenough, even from America there came, under the guiding providence of\nMr. John {284} Harper,[13] then I believe Mayor of the City of New\nYork, an elegant carved armchair, trimmed with crimson plush, to\ntestify \"the admiring gratitude of the American people\" to the author\nof _Thaddeus of Warsaw_.  The book, by its amazing popularity, and by\nthe entertaining way in which it marshals its romantic effulgencies in\nfavor of a great cause, may very naturally suggest that other, later\nand larger enlistment of all the forces of good story-telling,\nwhich--fifty years thereafter--in the hands of an American lady (Mrs.\nStowe) contributed to a larger cause, and with more abiding results.\n\n_The Scottish Chiefs_ has less of gusto than the Polish novel--and as I\ntook occasion to say when we were at that date of Scottish history--is\nfull of bad anachronisms, and of historical untruths.  Yet there is a\ngood bracing air of the Highlands in parts of it, and an ebullient\nmartial din of broadswords and of gathering clans which go far to\nredeem its maudlin sentiment.  Mrs. Radcliffe's _Mysteries of Udolpho_\nhad more of the {285} conventionally artistic qualities than either of\nthose last named, though never so infectiously popular.  There are\ngloomy Italian chieftains in it, splendid dark fellows with swords and\npistols and plumes to match; and there are purple sunsets and massive\ncastles with secret passages and stairs; and marks of bloody fingers,\nand papers that are to be signed--or not signed; and one ineffable\nyoung lady--Emily, I think, is her name--who by her spiritual presence\nand lovely features serves to light up all the gloom and the mystery\nand makes the castle, and the dark woods, and the reeking vaults, and\nthe secret paths all blossom like a rose.  I cannot advise the reading\nof the book.\n\n\n_Vathek._\n\n[Sidenote: Wm. Beckford.]\n\nWhen poor Chatterton--of whom we had speech not far back--was near to\nstarving in London, he made one desperate effort to secure the favor\nand patronage of the Lord Mayor of the city, who was a very rich West\nIndia merchant, by the name of Beckford.  Chatterton did gain an\ninterview; did get promise of aid, and win strongly upon the {286} good\nwill of the Lord Mayor; but unfortunately his honor died only a few\ndays thereafter.  Had he lived, the young poet might have had a totally\ndifferent career; and had he lived, the only son and heir of this\nbenevolent Mayor,--William Beckford,[14] then a boy of ten,--would have\nhad a different bringing up.  At twenty, this youth printed--though he\ndid not publish--some journals of continental travel which he had\nconducted in the spirit and with the large accompaniments of a young\nman who loves the splendor of life, and who had at command an annual\nrevenue of six hundred thousand dollars, at that day said to be the\nlargest moneyed income in England.  What a little fragment of this sum\nwhich was squandered upon that splendid trail of travel through Europe\nwould have made poor Chatterton happy!  But young Beckford was by no\nmeans a brainless spendthrift; he had strong intellectual aptitudes;\nwas a scholar in a certain limited yet true sense; and when twenty-two\nonly, had written (in French) {287} that strange, weird romance of\n_Vathek_; well worth your reading on a spare day, and which in its\nEnglish version has made his fame, and keeps his name alive, now that\nhis great houses and moneys are known and reverenced no more.\n\nIt is an Eastern story, with all the glow, color, and splendors of the\ndays of the good _Haroun al Raschid_ in it.  There are crime and love\nin it too; and phantoms and beautiful women, and terrific punishment of\nthe wicked.  Vathek, the hero, who might be Beckford himself, wanders\nthrough a world of delights, where evil phantoms and genii assail him,\nand fascinating maidens allure him; and after adventures full of\nescapes and dangers and feastings, in which he listens to the melody of\nlutes and quaffs the delicious wine of Schiraz, he reaches at last, in\ncompany with the lovely Mironihar, the great hall of Eblis; here we\ncome to something horrific and Dantesque--something which I am sure had\nits abiding influence upon the work of Edgar Poe.\n\n\n\"The place, though roofed with a vaulted ceiling, was so spacious and\nlofty that at first they took it for an immeasurable plain.  But their\neyes at length growing familiar with {288} the grandeur of surrounding\nobjects, they extended their view to those at a distance, and\ndiscovered rows of columns and arcades which gradually diminished till\nthey terminated in a point radiant as the sun when he darts his last\nbeams athwart the ocean....  The pavement, which was strewed over with\ngold-dust and saffron, exhaled so subtle an odor as almost overpowered\nthem....  In the midst of this immense hall a vast multitude was\nincessantly passing, who severally kept their right hands on their\nhearts, without once regarding anything around them.  They had all the\nlivid paleness of death.  Their eyes deep sunk in their sockets,\nresembled those phosphoric meteors that glimmer by night in places of\ninterment.\"\n\n\nAnd afterward, when a royal sufferer, who from livid lips had made\nwarning exhortation to these wanderers, lifts his right hand in\nsupplication, Vathek sees--through his bosom which was \"transparent as\ncrystal\"--his heart enveloped in flames.  Perhaps Hawthorne, in certain\npassages of the _Scarlet Letter_, may have had these red, burning\nhearts of this famous Hall of Eblis in mind.\n\nBeckford wrote also a very interesting account of certain religious\nhouses in Portugal which were the wonder of old days and are a wonder\nnow.  At Cintra, the picturesque suburb of Lisbon, he {289} established\na great Moorish country house within sight of the sea.  Byron gives a\nglimpse of this in _Childe Harold_:--\n\n  \"Here didst thou dwell, here schemes of pleasure plan,\n    Beneath yon mountain's ever beauteous brow;\n  But now, as if a thing unblest by man,\n    Thy fairy dwelling is as lone as thou!\n    Here giant weeds a passage scarce allow\n  To halls deserted, portals gaping wide.\n    Fresh lessons to the thinking bosom, how\n  Vain are the pleasaunces on earth supplied,\n  Swept into wrecks anon by Time's ungentle tide.\"\n\n\nByron would now have to mend his description, since the estate is at\npresent owned by a London merchant, who has bought a title from the\nweak king-folk of Portugal, and keeps the great house in Pimlico order.\nIt is one of the show places of Cintra; and if Moorish domes, and\nmarble halls, and sculpture delicate as that of the Alhambra, and\nfountains, and palms, and oranges, and bowers of roses, and century-old\noaks, and cliffs, and wooded dells, and far-off sight of sails from the\nBay of Biscay are deserving of show, surely this old palace of the rich\nEnglishman is.\n\nAnother palace--for Beckford had an {290} architectural mania--was\nbuilt at Fonthill, the place of his birth, not far east of Salisbury.\nHere was a great ancestral estate, around which he caused to be erected\na huge wall of masonry, some ten or twelve miles in length, to secure\nprivacy and protect his birds.  Within he built courts, towers, and\nhalls--some six hundred men often working together night and day on\nthese constructions--which he equipped with the rare and munificent\nspoils brought back from his travel.  To this Fairy land, however,\nByron's lament would better apply; the walls are down and the towers\nhave fallen; the property is divided; only here and there and blended\nwith new structures and new offices can you see traces of the old\narchitectural extravagance.  The spoiled plantations of Jamaica--whence\nthe Vathek revenue mostly came--brought the change; enough, however,\nremained for the erection of a costly home in Bath, portions of which\nmay still be seen.\n\nA daughter of Beckford's became Duchess of Hamilton; another daughter,\nwho declined Ducal overtures which the father favored, was treated\ntherefor with severities that would have become an {291} Eastern\ncaliph--for which, maybe, he now, like the poor creatures of Eblis\nHall, is holding his right hand over \"a burning heart.\"\n\n\n_Robert Burns._\n\n[Sidenote: Burns.]\n\nWe go now out of England, northward of the Solway, to find that peasant\npoet[15] at whose career I hinted in the last chapter, and whose burst\nof Scotch song was a new wakening for that kingdom of the highlands and\nthe moors.  I dare not, and will not speak critically of his verses;\nthere they are--in their little budget of gilt-bound, or paper-bound\nleaves; rhythmic, tender, coarse, glowing, burning, with a grip in many\nof them at our heart-strings which we may not and cannot shake off.  To\ntell you about these poems and of their special melodies would be like\ntaking you to the sea and telling you how the waves gather and\nroll--with murmurs that you know--along all the shore.\n\n{292}\n\nNor can I hope to tell any more of what will be new to you about his\nlife and fate.  We all know that white-washed, low, roadside cottage--a\nlittle drive out from the old Scotch town of Ayr--where he was born; we\nhave been there perhaps; we have seen other Scottish peasants boozing\nthere over their ale; and have noted the names scribbled over tables\nand cupboards and walls to testify to the world's yearnings and to its\npilgrimages thither.  We know, too, that other low cottage of Mossgiel,\nwhere his poor father--a gospel abiding man--made his last struggle\nagainst the fates--and who of a Saturday night--\n\n  \"Collects his spades, his mattocks, and his hoes,\n  Hoping the morn in ease and rest to spend,\n  And weary, o'er the moor, his course does homeward bend.\"\n\n\nWe all know what a brave fight the two Burns boys, Gilbert and Rob,\nmade of it when Death, \"the poor man's dearest friend,\" took off the\nfather; Gilbert the elder; but Robert the brighter and keener--making\nverselets in the fields which the elder brother approves, and says\nwould \"bear {293} to be printed;\" and so presently after, the first\npoor, thin, dingy volume finds it way to the light, and gives to\nfar-away Edinboro' people their earliest hint of this strange, fine,\nnew, human plant which has begun to blossom under the damps of\nMossgiel.  But the farm life is hard; the poet is wayward; his jolly\nfriends near by who chant his songs are not helpful; his love affairs,\nof which he has overstock in his young wildness, run to confusion;\nquarrels threaten; so he books himself with what moneys the thin, dingy\nvolume of poems have brought him, for America.\n\nWhat if he had come!\n\nBut no; one low, wee encouraging voice--the piping answer to those\npoems--reaches him from Edinboro', and the poet goes thither in his\nbest gear; Dugald Stewart, and Dr. Blacklock the blind poet, and\nMackenzie, of whom I have already made mention, all befriend him.  The\ngentlewomen of Edinboro' entertain him, and admire him, and flatter\nhim; and he, in best blue and buff, with his dark, rolling eyes, and\nlips that command all shapes of language, holds his dignity with these\nfine ladies of the Northern capital; {294} gives compliments that make\nthem tremble; prints other and fuller edition of his poems; goes\nnorthward amongst the highlands--dropping jewels of verse as he\ngoes--to beautiful women, to waterfalls, to noble patrons.  The next\nseason in Edinboro', however, is no longer the same; that brilliant\nseries of fetes and of conquests has gone by; the new lion is too\naudacious; he shakes his fetters with a bold rage that intimidates.  So\nwe find him with some three hundred pounds only, saved out of the new\nbook and the junketings of the Capital, going off to lease quietly the\nfarm of Ellisland, near to Dumfries, and turn ploughman once more.\n\nIt is a poor place, but very beautiful; it is in Nithsdale, and the\nmurmur of the river through its wooded banks makes the poet forget the\ncrop of pebbles which every ploughing turns to the top.  He is\npresently in the Excise too (1789): so gets some added pence by the\ngauging of beer-barrels and looking after frauds upon the revenue;\nmarried too--having out of all the loose love-strings, which held him\nmore or less weakly, at last knotted one, which ties the quiet, pretty,\nwomanly, much injured Jean Armour to his hearth and {295} home,\nforever.  And he begins that Ellisland life bravely well; has prayers\nat night; teaches the \"toddlin' wee things\" their catechism; has hope\nand faith, and sings--and sings; and this, amongst other things, was\nwhat he sang--\n\n  \"O, Willie brewed a peck o' maut,\n    And Bob and Allen cam to see;\n  Three blither hearts that lee-lang night\n    Ye wad na find in Christendie.\n  We are na fou, we're na that fou,\n    But just a drappie in our e'e;\n  The cock may craw, the day may daw,\n    And ay we'll taste the barley bree.\n  It is the moon, I ken her horn,\n    That's blinkin in the lift sae hie;\n  She shines sae bright, to wyle us hame;\n    But, by my sooth, she'll wait a wee.\n  The cock may craw, the day may daw,\n    And aye we'll taste the barley bree.\"\n\n\nNo wonder the pebbles began to show more and more in the plough-land;\nno wonder the jolly fellows of Dumfries came oftener and oftener; the\nlong bouts too amongst the hills chilled him; the crops grew smaller\nand smaller; the \"barley bree\" better and better; he has no tact at\nbargaining; a stanza of Tam O'Shanter is worth more than ten {296}\nplough-days, yet he makes gifts of his best songs.  Household affairs\ngo all awry, let poor Jeanie Armour struggle as she may; the cottage\npalings are down; debts accumulate; and so do those rollicking nights\nat the Globe, or in a shieling amongst the hills.  Yet from out all the\nimpending want, and the gloom, and the desperation, come such sweet\nnotes as these, reaching the ear of humanity everywhere:--\n\n  \"John Anderson, my jo, John,\n    We clamb the hill thegither;\n  And mony a canty day, John,\n    W've had wi' ane anither:\n  Now we maun totter down, John,\n    But hand in hand we'll go,\n  And sleep thegither at the foot,\n    John Anderson, my jo.\"\n\n\nAt last Ellisland must be given up--crops, beasties and all; and never\nmore the wooded banks of Nithsdale shall feel his tread, or hear his\nchant mingling with the river murmurs.  He, and they all--five souls\nnow--just of an age to relish most the woods, the range, the fields,\nthe daisies of Ellisland, must go to one of the foulest and least\nattractive streets of Dumfries, and to a {297} home as little\nattractive as the street.  Fifty years thereafter I went over that\nhouse and found it small, pinched, and pitifully meagre in all its\nappointments; twenty years later, Hawthorne speaks of both house and\nstreet as _filthy_.  What could or should supply the place now--to the\npeasant poet--of the fields, the open sky, the gentle fret and murmurs\nof the streams of Nithsdale?\n\nThe foul fiends who taunted him in the woods now lay hold upon him in\nearnest; every day his fame is flying over straits and seas; every day\nhis poems, old and new, are planting themselves in fresh hearts and\nbrains; every day his wild passions are dealing him back-handed blows.\nOld neighbors have to pass him by; modest women look away; he has\nforfeited social position; and I suspect, welcomed in those days of\nJuly, 1796, the approaches of the disease which he knew was sapping his\nlife:--\n\n  \"Oh, Martinmas wind! when wilt thou blaw\n    And shake the dead leaves frae the tree?\n  Oh gentle death! when wilt thou come\n    And tak a life that wearies me?\"\n\n\n{298}\n\nAnd it comes, in that dismal, miserable upper chamber that you can see\nwhen you go there;--his wife ill; his little children wandering\naimlessly about; it comes sharply; he is on his back--\"uneasy\" the\nnurse said, and \"chafing\"; when suddenly by a great effort--as if at\nlast he would shake off all the beleaguerments of sense, and the\nhaunting phantoms swarming about him--he rallied all his powers--rose\nto his full height from the bed--tottered for a moment, then fell prone\nforward a dead man.\n\nThis was in the month of July, 1796; Burns being then only\nthirty-seven.  Walter Scott, a young fellow of twenty-five, living in\nEdinboro', had just printed his translation of _Leonora_.\nWordsworth--unknown save for a thin booklet of indifferent verse--was\nliving down in Dorsetshire, enjoying the \"winding wood-walks green,\"\nwith that sister Dorothy, who \"added sunshine to his daylight.\"  These\ntwo had not as yet made the acquaintance of that coming man, S. T.\nColeridge, who is living at Clevedon, over by Bristol Channel, with\nthat newly married wife, who has decoyed him from his schemes of\nAmerican migration; {299} and the poet of the Ancient Mariner (as yet\nunwritten) has published his little booklet with Mr. Cottle, of\nBristol, in which are some modest verses signed C. L.  And Charles Lamb\n(for whom those initials stand) is just now in his twenty-first year,\nand is living in humble lodgings in Little Queen Street, London, from\nwhich he writes to Coleridge, saying that \"Burns was a God of my\nidolatry.\"  And in that very year (1796) the dismalest of tragedies is\nto overshadow those humble lodgings of Little Queen Street.  Of this\nand of Coleridge and of Wordsworth, we shall have somewhat to say in\nthe chapter we open upon next.\n\n\n\n[1] Gilbert White, b. 1720; d. 1793.  Oxford man; Fellow in 1744;\ncurate of Faringdon 1758; after 1784, at Selborne.\n\n[2] A charmingly illustrated edition of _The Natural History of\nSelborne_--showing his ivy-covered home and other objects of interest,\nwas published by Macmillan & Co. in 1875 (edited by Frank Buckland).  I\nam indebted for a copy to my friend, Wm. Robinson, of the London\n_Garden_.\n\n[3] Jane Austen, b. 1775; d. 1817.  _Sense and Sensibility_, published\n1811.  Life was written by her nephew J. Austen-Leigh.  Her _Letters_,\nedited by Lord Brabourne, 1884.\n\n[4] Not the dreadful, seamy, photographic reproduction of an old oil\npainting that Lord Brabourne gives, which must be wholly unfair to her;\nbut the earlier engravings.\n\n[5] Thomas Day, b. 1748; d. 1789.  Oxford man; married, 1778; _Sandford\nand Merton_ published 1783.\n\n[6] Mrs. Barbauld (Anna Letitia Aikin), b. 1743; d. 1825.  There is a\npleasant sketch of Mrs. Barbauld and (for a wonder) an approving and\ncommendatory notice of her in Miss Martineau's _Autobiography_, vol.\ni., pp. 228-39.\n\nMiss Martineau's father, it appears, had been a pupil of Mrs. Barbauld.\n\n[7] Boswell's Johnson, vol. vi., p. 28.\n\n[8] The circumstances are given in _Crabb Robinson's Diary_.\n\n[9] Maria Edgeworth, b. 1767; d. 1849.  First volume of _Parent's\nAssistant_ was published, 1796; _Castle Rackrent_, 1800; _Popular\nTales_, 1804.\n\n[10] Miss Honora Sneyd among them, in 1773.\n\n[11] Maria Regina Roche, b. 1766; d. 1845.  The _Grand Dict. Universal\ndu XIX. Siecle_ enumerates no less than thirteen other romances by\nher--in forty odd volumes, all translated, and now utterly forgotten!\n\n[12] Mrs. Radcliffe (Ann Ward), b. 1764; d. 1823; _Romance of the\nForest_, 1791; _Mysteries of Udolpho_, 1794.\n\nMiss Jane Porter, b. 1776; d. 1850; _Thaddeus of Warsaw_, published\n1803; _Scottish Chiefs_, 1810; _Sir Edward Seaward's Narrative_ (in\nconcert with her sister Anna Maria Porter), published in 1826.\n\n[13] Senior member of the old firm of J. & J. Harper, 82 Cliff Street.\n\n[14] William Beckford, b. 1759; d. 1844.  _Vathek_, published (in\nFrench), 1787; better known by an unauthentic English translation,\npublished 1784.\n\n[15] Robert Burns, b. 1759; d. 1796.  Poems published 1786.  First\ncollected edition, 1800; Cunningham edition, with life, in 1834, 4 vols.\n\n\n\n\n{300}\n\nCHAPTER VIII\n\nWe have still in our mind's eye, and very pleasantly, that quaint old\nclergyman of Hampshire, who wrote about the daws, and the swallows, and\nthe fern-owl, in a way that has kept the name of Gilbert White alive,\nfor a great many years.  And who that has read them can ever forget the\nstories of that winning Hampshire lady, whose fame takes on new\ngreenness with every spring-time?  Following upon our talk of this\ncharming authoress, we had a little discursive mention, in our last\nchapter, of certain books which at the close of the last century, or\nearly in this, were written for boys and girls; chiefest among these we\nnoted those written by that excellent woman, Miss Edgeworth.  We spoke\nof Miss Roche, who gushed over in the loves of Amanda and\nMortimer--those fond and sentimental _Children {301} of the Abbey_; and\nof Miss Porter, with her gorgeous heroics about Poland and Scotland,\nand of Mrs. Radcliffe's stunning _Mysteries of Udolpho_.  We had a\nglimpse of the strange work and life of William Beckford--son of the\nrich Lord Mayor Beckford; and we closed our chapter over the grave of\nthat brilliant poet and wrecked man Robert Burns.\n\nThat grewsome death of the great Scotch singer occurred in a miserable\nhouse of a disorderly street in Dumfries, within four years of the\nclose of the last century; his children--without any mastership to\ncontrol, and the love that should have guided dumb--wandering in and\nout; no home comforts about them; the very necessities of life\nuncertain and precarious; all hopes narrowed for them, and all memories\nof theirs full of wildest alternations of joyousness and fright.\n\n\n_A Banker Poet._\n\n[Sidenote: Samuel Rogers.]\n\nYou have perhaps read and enjoyed a poem called _The Pleasures of\nMemory_.  It has tender passages in it; it has an easy, melodious\nswing:--\n\n{302}\n\n  \"Twilight's soft dews steal o'er the village green,\n  With magic tints to harmonize the scene;\n  Stilled is the hum that thro' the hamlet broke,\n  When round the ruins of their ancient oak\n  The peasants flocked to hear the minstrel play\n  And games and carols closed the busy day.\n      *      *      *      *      *\n  Up springs, at every step--to claim a tear,\n  Some little friendship formed and cherished here;\n  And not the lightest leaf, but trembling teems\n  With golden visions and romantic dreams.\n\n\nThis poem, with echoes of Goldsmith in it, with echoes of Dryden, with\nechoes of Cowper--all caught together by a hand that was most deft, and\nby a taste that was most fastidious--was written and published in\nLondon, four years before Burns died, by the poet-banker Samuel\nRogers.[1]  It is not a name that I feel inclined to glorify very much,\nor that should be honored with any large reverence; but it is brought\nspecially to the reader's notice here, because the life, career, and\naccomplishment of the man offers so striking a contrast to that of the\nScottish poet who was his contemporary.  They were born within four\nyears {303} of each other.  One under the bare roof of an Ayrshire\ncottage, the other amid the luxuries of a banker's home in London; one\ncaught inspiration amongst the hills and the woods; the other was\ntaught melody in the drawing-rooms and libraries of London; one wrested\nhis conquests in the kingdom of song, single-handed; and the other, his\nlesser and feebler ones, bolstered with all the appliances that wealth\ncould give, or long culture suggest.  The poetry of the one is rich,\nindividual, and spirited, with sources in nature and in the passions of\nthe man; the poetry of the other has only those congruous and tamer\nharmonies, whose sources lie in the utterance of deeper and stronger\nsingers before him.  Yet the life of that Ayrshire poet was a miserable\nfailure; and the life of this other, Samuel Rogers, was--as the world\ncounts things--a complete success.  No half-starved children pulled at\nhis skirts for bread.  All luxuries were about him, and from the\nbeginning life flowed with him as calmly as a river.\n\nOf his early history there is not much to be said.  We know that he was\nborn at Newington Green--an old suburb lying directly north of the\ncity, {304} toward Stamford Hill--and now engulfed by the tide of\nLondon houses; we know he studied at good schools there, and under\ncareful teachers at home; we know that he used to read and love Dr.\nBeattie's minstrel; we know that once, in boyhood (he tells the story\nhimself), craving a sight of the great Dr. Johnson, he went to his\ndoor, but scared by the first tap of the knocker, sidled away, and so\nnever saw that literary magnate.  It was a timidity that did not cling\nto Mr. Rogers; in all his later years no man in London was less afraid\nof the pounding of a knocker.\n\nHis first volume was printed in the very year on which the poor thin\nbook of Burns's first poems saw the light at Kilmarnock.  This,\nhowever, did not make his reputation; _that_ came six years later with\nthe _Pleasures of Memory_, of which I cited a fragment; and thereafter,\nall down through the earlier half of the present century, there was\nhardly a better known man in London than Samuel Rogers, banker and\npoet.  He voyaged widely and brought back many spoils of travel; he had\nluxurious tastes and fed them with the utmost discretion.  He had\nsocial ambition, and rare sagacity in {305} selecting his companions,\nand in timing his courtesies; he flattered critics, and was obsequious\nto men with titles.\n\nHis house in St. James's--with its broad upper double window, looking\nout upon the Green Park--was known of all men.  Before yet the days of\nbric-a-brac had come, it was filled with beautiful things and with\ntrophies of art.  It was not large nor pretentious; but on its walls\nwere paintings, or sketches by Raphael, by Rubens, by Titian, by\nGainsborough, by Rembrandt, and by Reynolds; and in its ante-rooms,\nmarbles by Thorwaldsen and Canova.  There were no children of the\nhouse, nor was there ever a wife there to aid, or to lord the master.\nYet many a lady, ranking by title, or by cleverness, has enjoyed the\ndinners and the breakfasts for which the house was famous.  The cooking\nwas always of the best; the wines the rarest; the meats and fruits the\nchoicest, and the porcelain superb.  Like most who have richly equipped\nhouses, he loved to have his fine things admired; and he loved to have\nhis fine words echoed.  Few foreigners of any literary distinction\nvisited London from 1815 to 1850, without coming to a taste {306} of\nthe poet's hospitality, and to a taste too, very likely, of his pretty\nsatire.  His wit flashed more sharply in his talk than in his verse;\nand his dinner stories were fabulous in number, in piquancy, and in\nsting.  Like all accomplished _raconteurs_, he must needs tell his good\nstories over and over, so that Rogers's butler, it was wittily said,\nwas next best to Rogers.\n\nHe could hardly have been called a good-natured man, and was always, I\nthink, keener for a good thing to say, than for a good thing to do.  He\ngave, it is true, largely in charities; but in orderly, business-like\nways and with none of the unction and kindly indirectness[2] which\ndoubles the {307} warmth of the best giving.  All London knew him as a\ndiner out, as a connoisseur, as an opera-goer, as a patron of clever\npeople, as a friend to those in place, as a _flaneur_ along Piccadilly.\nHe was cool, unimpassioned, blase in look, never doing openly\ndiscreditable things; and he carried his reputation for unmitigated\nrespectability, for wealth, for sharp speeches, for cleverness, for\nsagacious charities, down to extreme age; dying as late as 1855,\nninety-three years old.\n\n[Sidenote: Rogers' poems.]\n\nThough the poem entitled _The Pleasures of Memory_ made his fame, a\nlater descriptive poem, embodying the gleanings from a trip in Italy,\nis perhaps better known; and it enjoys the distinction of having been\nillustrated and printed at a cost of $70,000 of the banker's money.\nFragments of that poem you must know; the story of Ginevra, perhaps,\nbest of all; so daintily told that it is likely to live and be\ncherished as long as any of the bric-a-brac which the banker poet\ngathered in his travels.  'Tis a story of a picture that he saw--a\n\"lady in her earliest youth.\"\n\n{308}\n\n  \"She sits inclining forward as to speak,\n  Her lips half open, and her finger up\n  As though she said--Beware!  Her vest of gold\n  Broidered with flowers, and clasped from head to foot,\n  An emerald stone in every golden clasp,\n  And on her brow, fairer than alabaster,\n  A coronet of pearls....  Alone it hangs\n  Over a mouldering heirloom, its companion,\n  An oaken chest, half eaten by the worms.\n      *      *      *      *      *\n  Just as she looks there in her bridal dress\n  She was all gentleness and gaiety.\n      *      *      *      *      *\n  And in the lustre of her youth, she gave\n  Her hand with her heart in it, to Francesco.\n  Great was the joy; but at the bridal feast\n  When all sat down, the bride was wanting there,\n  Nor was she to be found!  Her father cried\n  \"'Tis but to make a trial of our love!\"\n  And filled his glass to all; _but his hand shook_,\n  And soon from guest to guest the panic spread.\n  'Twas but that instant she had left Francesco\n  Laughing, and looking back and flying still,\n  Her ivory tooth imprinted on his finger.\n  But now, alas, she was not to be found;\n  Nor from that hour could anything be guessed\n  But that she was not!  Weary of his life\n  Francesco flew to Venice, and forthwith\n  Flung it away in battle with the Turk.\n  Orsini lived; and long mightest thou have seen\n  An old man wandering as in quest of something--\n  Something he could not find--he knew not what\n\n{309}\n\n  When he was gone, the house remained awhile\n  Silent and tenantless; then, went to strangers.\n\n    Full fifty years were past, and all forgot,\n  When on an idle day--a day of search\n  Mid the old lumber in the gallery--\n  That mouldering chest was noticed; and 'twas said\n  By one as young, as thoughtless, as Ginevra,\n  Why not remove it from its lurking-place?\n  'Twas done, as soon as said; but on the way\n  It burst--it fell; and lo, a skeleton,\n  With here and there a pearl, an emerald stone,\n  A golden clasp--clasping a shred of gold.\n  All else had perished, save a nuptial ring\n  And a small seal--her mother's legacy,\n  Engraven with a name--the name of both--\n  \"Ginevra.\"\n\n\nA pretty delicacy certainly goes to the telling of that story; but in\nthe tale of Christabel and of the Ancient Mariner there is something\nmore than delicacy--more of brain and passion and far-reaching poetic\ninsight in the poet Coleridge, than in ten such men as Samuel Rogers.\n\n\n_Coleridge._\n\n[Sidenote: Coleridge.]\n\nYet what a sad life we have to tell you of now!  A life without any\nrepose in it;--a life haunted and goaded by its own ambitions--a life\nput to {310} wreck by lack of resolute governance--a life going out at\nlast under the shadows of great clouds.\n\nColeridge[3] was the son of a humble, quiet, self-forgetting, earnest\nclergyman in the West of England; and the boy, having no other\nopportunity, came to be billeted upon that famous Christ Hospital\nschool in London--whose boys in their ancient uniform of yellow\nstockings and blue coats, and bare heads, still provoke the curiosity\nof those western travellers who wander down Newgate Street, and gaze\nthrough the iron grill upon the paved approach-way.\n\nHe knew Lamb there--Charles Lamb, who in the Essays of Elia addresses\nto him that famous apostrophe: \"Come back into memory, like as thou\nwert in the dayspring of thy fancies, with hope like a fiery column\nbefore thee--the dark pillar not yet turned--Samuel Taylor\nColeridge--Logician, Metaphysician, Bard!\"  Yet this pale-faced\nmetaphysician and friend of Lamb gets {311} severe beatings at the\nhands of the Greek master, though his sweet intonations make the\ncorridors resound with the verse of Homer.  At Cambridge, where he goes\nafterward for a time, he is cheated and bullied; his far-off and dreamy\nlook upon the symphonies of a poetic world not qualifying him for the\nevery-day contests of the cloisters; in the haze in which he lives, he\nloses scent of the honors he had hoped to win; there is no prospective\nfellowship and no establishment for him there.  Disappointed and\ndespairing he goes up to London and enlists as private in the dragoons\nunder a feigned name; but friends detect and prevent the military\nsacrifice.\n\nA little later, we find him in his own West of England again, at\nBristol--whither we have wandered so often in search of poets--and he\nencounters Southey thereabout, whom he had met for the first time on a\nvisit to Oxford in 1794; this brother poet being as hazy, and dreamy,\nand theosophic, and hopeful in those days as Coleridge himself.  The\ntwo form a sort of garret partnership--lecture to the savages of\nsurrounding towns--are inoculated both with the {312} \"fraternity and\nequality\" fever which had grown out of the French Revolution--they\nbelieving that this French car of Juggernaut is to be dragged with its\nbloody wheels over the whole brotherhood of nations.  In this faith\nthey plot a settlement, in the new region--of which they know nothing,\nbut the sweetly sounding name of Wyoming--upon the banks of the\nSusquehanna.  There they would dig, and build cottages, and\nphilosophize, and found Arcadia.  With kindred poetic foresight,\nColeridge marries in these days a bride as inexperienced and as poor as\nhimself; and for a little time there is a one-volumed Arcadia on the\nbanks of the Bristol Channel, with a lovely and pensive Sara for its\npresiding nymph.  Only for those few early years does this nymph enter\nfor much into the career of Coleridge.  Domesticity[4] was never a\n{313} shining virtue in him; and wife, and cottage, and Arcadia somehow\nfade out from the story of his life--as pointless, unsaving, and\nineffective for him, all these, as the blurred lines with which we\nbegin a story, and cross them out.  Southey, with a practical old aunt\nto look sharply after his youngness, is quickly driven from his\nArcadian feeding ground and for the present disappears.\n\nBut Coleridge is still in the wallow of his wild vain hopes and wild\ndiscourse, when he encounters another poet--his elder by a few years\nand of a cooler temperament--William Wordsworth; who about that time\nhad established himself, with his sister Dorothy, upon the borders of\nSomersetshire.  These two men, so unlike, cleave together from the\nbeginning; there is a flagging now in the Unitarian discourses of\nColeridge in country chapels; and instead, wanderings with the brother\npoets over the fair country ways that border upon the Bristol\nstraits--looking off upon the green flats of Somerset, the tufted banks\nof the Avon, the shining of the sea, with trafficking ships, to the\nwest.  Out of these, and of their meditations grow the first book--a\njoint one--of {314} _Lyrical Ballads_; its issue not making a ripple on\nthe tide where Crabbe and Cowper were then afloat; and yet creating an\nepoch in the history of British verse.  For in it was the story of the\nAncient Mariner, and words therein that will never grow old:\n\n  \"Farewell, farewell! but this I tell\n    To thee, thou wedding guest!\n  He prayeth well, who loveth well\n    Both man and bird and beast;\n  He prayeth best, who loveth best\n    All things both great and small;\n  For the dear God who loveth us\n    He made and loveth all!\"\n\n\nYet the poet does still--from time to time wandering into country\nchapels--hammer at strange, irregular sermons, with a mixed metaphysics\nand poetry; and theologies of a dim vague sort which beat on ear and\nhearts, like sleet on slated roofs, and bring never a beam of that\nwarming sunshine which lies in the lines I have quoted from the Mariner.\n\nOne wonders how he lived in those times; with no moneys coming from\nbooks; only driblets from his preachments; and with not enough of {315}\ncommercial aptitude in him to audit a grocer's bill.  The\nWedgewoods--so well known by their pottery--who have a quick eye for\nfine wares of all sorts--recognize his rare brain, and send him over to\nGermany, bestowing upon him an annuity, which enables him to forego his\ntravelling priesthood, and gives him the means of visiting various\ncities of the continent.\n\nThe Wordsworths make the trip with him; and after a stay of a\ntwelve-month--mostly in Gottingen--Coleridge returns, with his\ntranslation of Wallenstein; but this counts for little.  A year later,\nhe finds his way to Keswick--to a beautiful, wooded bay, where Southey\nultimately established his anchorage for life;[5] the Wordsworths were\nnot far off, at Grasmere; and Coleridge plans that weekly paper--_The\nFriend_ (finding issue some years later) with wonderful things in it,\nwhich few people read then; and so fine-drawn, that few read them now.\nThe damps of Keswick give him {316} rheumatic pains, for which he uses\nprotective stimulants; good Dorothy Wordsworth has fears thereanent,\nand regards hopefully his appointment to some civil station at Malta.\nBut his impracticabilities lose him the place after a very short\nincumbency; he crosses to Italy; sees Naples, Amain, and Vesuvius;\nsees, and knows well at Rome, our American painter, Washington Allston.\nThere are bonds of sympathy we might have looked for between the author\nof _Monaldi_ and the author of _Christabel_.\n\nIn England again, the fogs bring back old rheumatic pains; the\nalienation from his wife is declaring itself in more unmistakable ways;\nand then, or thereabout,[6] begins that terrible slavery to opium,\nwhose chains he wore thenceforth, some twenty years, and was not\nentirely free until death broke his bonds.  There is a dreary, yet\ntouching pathos in this confession of his--\"Alas, it is with a bitter\nsmile, a laugh of gall and bitterness, that I recall that period of\nunsuspecting delusion, and how I first became aware of the {317}\nmaelstrom, the fatal whirlpool to which I was drawing, just when the\ncurrent was already beyond my strength to stem.\"\n\nBut against the circling terrors of that maelstrom he does make now and\nthen gallant struggle--goes to the house of that kindly surgeon,\nGillman, at Highgate, who is charged to guard him--does guard him with\nexceeding kindness; the servants have orders to watch him--to follow\nhim in the street on his lecture days.  But the cunning of a man crazed\nby his insatiate appetite outwits them; and over and over the turbid\nroll of his speech--with flashing splendors in it, that give no\nlight--betrays him.  And yet it was in those very days of alternate\nheroic struggle and of devilish yielding that he re-vamps and extends\nand retouches that sweet, serene poem of Christabel, with the pure,\ninnocent, loving, trustful, winning, blue-eyed daughter of Sir Leoline\npraying under the oaks, and contrasted with her that graceful, mocking,\nradiant Geraldine--with smiles that enchant, and alabaster front, and\nundying graces, and wiles of the serpent, and the damps of the pit in\nher breath--as if the demon that pursued and {318} pushed him to the\nwall had foreshadowed himself in that mocking and most beautiful\nGeraldine.\n\nIn those days, too, it was that the young Carlyle used to come to\nHighgate and watch those bulging eyes--pressed out with excess of brain\nsubstance behind them--and listen to his poetic convolutions of speech.\n\"The eyes,\" he says, \"were as full of sorrow as of inspiration.  I have\nheard him talk with eager, musical energy two stricken hours, his face\nradiant and moist, and communicate no meaning whatsoever to any\nindividual of his hearers, certain of whom--I for one--still kept\neagerly listening in hope.\"\n\nThe very children of the neighborhood stood in awe of this wildish\nman--who seemed talking to the trees at times; and yet their awe was\nbroken by fits of mocking courage, and they made faces at him across\nthe high road.  He died there at last--1834 was the year; within sight\nof the smoke of London and the dome of St. Paul's, toward which from\nHighgate there stretched in that day a long line of suburban houses,\nwith scattered open fields, hedges, trees, flowers, and the hum of bees.\n\n\n{319}\n\n_Charles Lamb._\n\n[Sidenote: Essays of Elia.]\n\nAmong those who used to come somewhiles to follow that fine, confused\nstream of poetic talk which poured from Coleridge's lips, was Charles\nLamb,[7] his old school-fellow and friend in the blue-coat days of\nChrist's Hospital.  And what a strange, odd friendship it seems when we\ncontrast the tender and delicious quietude of the Essays of Elia with\nthe portentous flow of Coleridge's speech!  A quiet little stream,\npurling with gentle bendings and doublings along its own meadows--mated\nagainst a river that whirls in mad career, flinging foam high into\ntrees that border it, and only losing its turbidness when it is tided\naway into the sea, where both brook and river end.\n\n[Sidenote: Charles Lamb.]\n\nI love Charles Lamb and his writings so much, that I think everybody\nelse ought to love them.  There is not great weight in those essays of\nhis; you cannot learn from them what the capital of Hindostan is, or\nwhat Buddhism is, nor the date of the capture of Constantinople.\nMeasured by {320} the Dry-as-Dust standard, and there is scarce more in\nthem than in a field of daisies, over which the sunshine and the summer\nbreezes are at play.  But what delicacy there is! what a tender humor;\nwhat gentle and regaling lapses of quaint thought that beguiles and\ninvites and is soothing and never wearies.\n\nLamb's poems are not of the best; they have a haltingness--like that in\nhis speech,--with none of Rogers's glibness and currency, and none of\nhis shallowness either.  Constraint of rhyme sat on Elia no easier than\na dress-coat.  But in prose he was all at home; it purled from his pen\nlike a river.  It was quaint, kindly, utterly true--with little yaws of\nhumor in it, filling his sails of a sudden, and stirring you to smiling\noutbreak--then falling away and leaving him to a gently undulating\nforward movement which charms by its quietude, serenities, and\ncheerfulness.\n\nThere was not much in his life to tell you of; no cannon firing, no\ndrum beats, no moving splendors.  A thin, kindly face he had, and thin\nfigure too; in dark or grayish clothes ordinarily, that a clerk might\nwear; threadbare perhaps at the {321} elbows; not a presentable man\namongst swell people; never aspiring to be;--as distinct indeed as a\nbrown hermit-thrush amongst chattering parrots.  He has a stammer, too,\nas I have hinted, in his voice, which may annoy but never makes this\nquiet man ashamed; in fact, he deploys that stammering habit so as to\nallow of coy advance, and opportunity for pouncing with tremulous\niteration upon his little jokelets, in a way to double their execution;\nhe put it to service, too, in some of his tenderer stories, so as to\nmake, by his very hesitancies, an added and most touching pathos.\n\nHe was of humble origin, his father a servitor about Temple\nCourts--only long gunshot away from Newgate Street; and when the\nson--through with his Christ Hospital schooling--came to have a small\nstipend (first, from the South Sea House and later from the East India\nCompany), he had his little family--the only one that ever belonged to\nCharles Lamb--all about him in his lodgings in Little Queen Street.\nThere was Mary, his sister, ten years older; his poor, bedridden\nmother, and his father, lapsing into dotage and only happy with\ncribbage-board at his elbow, and {322} Charles or other good friend to\nmake count.  It was this quiet household on which a thunderbolt fell\none day.  This is Lamb's mention of it in a letter to Coleridge:--\n\n\n\"My poor dear, dearest sister, in a fit of insanity, has been the death\nof her own mother.  I was at hand only time enough to snatch the knife\nout of her grasp.  She is at present in a mad-house, from whence I fear\nshe must be moved to a hospital.  My poor father was slightly wounded.\nGod has preserved to me my senses.  Write as religious a letter as\npossible, but no mention of what is gone and done with.\"\n\n\nAnd only a day after this, the weak old father, with his plastered\nhead, is playing cribbage; and again, on another day, friends having\ncome in--very many for those small rooms--and the last ceremonies not\nyet over, and they all sitting down at some special repast--Lamb\nbethinks himself of all that has happened, of what lies in the next\nroom (he tells this in a letter to Coleridge), and rushes thither to\nkiss once more the cold face and to pray forgiveness that he has\nforgotten her so soon.\n\nPoor Mary recovers; she lives for years with her brother; the horror of\nthe past staying like a {323} black dream in their thought--of which\nthey dare not speak.  And when new visitations of estrangement\nthreaten, they two, brother and sister, walk away out from the\nstreets--on to Edmonton, through green fields, by hedges, under trees\nwhich they much enjoy, to the doctor's strong guardianship and ward,\nuntil repose comes again and a return.  Lamb at last goes to live at\nEnfield, which is close by Edmonton, north of London, that he may be\nnear her prison-house at all times and seasons.\n\nYet in all these days when the pains and fears of that distracting life\nare resting on him, he is putting those tender and playful touches into\nthe pleasant essays we know so well; conjuring for himself and for\nthousands everywhere a world of sunshine that shall overlap the dreary\none in which he lives, and spend its graces and cheeriness upon the\nmind of the poor forlorn one, who with sisterly affection cleaves there\nand journeys meekly and obediently and sadly beside him.\n\nI do not know how to trust myself to make a citation from those essays\nwhich shall carry to those not over-familiar some good hint of their\n{324} qualities; but I venture upon a bit from his _Dream-Children_:--\n\n\n\"Then I told how for seven long years, in hope sometimes, sometimes in\ndespair, yet persisting ever, I courted the fair Alice W----; and as\nmuch as children could understand, I explained to them what coyness,\nand difficulty, and denial, meant in maidens--when suddenly, turning to\nlittle Alice, the soul of the first Alice looked out at her eyes with\nsuch a reality of re-presentment that I became in doubt which of them\nstood there before me, or whose that bright hair was; and while I stood\ngazing, both the children gradually grew fainter to my view, receding,\nand still receding, till nothing at last but two mournful features were\nseen in the uttermost distance, which--without speech--strangely\nimpressed upon me the effects of speech:--We are not of Alice, not of\nthee, nor are we children at all.  The children of Alice call Bartram\nfather, we are nothing--less than nothing and dreams.  We are only what\nmight have been, and must wait upon the tedious shores of Lethe,\nmillions of ages, before we have existence and a name; and immediately\nawaking, I found myself quietly seated in my arm-chair--where I had\nfallen asleep, with the faithful Bridget unchanged by my side.\"\n\n\nLamb was not deep-thoughted; he would have lost the trail in those\nmeditations and searchings to which Coleridge in his cooler and clearer\nmoments invited and led the way; but there was about him an\nindividuality, a delicacy of thought, {325} a quaint play of airy\nfancies, a beguiling inconsequence, that have made his path in letters\na delightful one for thousands to follow.\n\nI cannot leave his name without calling attention to the charming\nlittle stories of Mrs. Leicester's School--written by Charles Lamb and\nhis sister jointly.  They are--or profess to be--the tales told by\nschool children themselves of their memories--whether sorrows or joys;\nand are so artless in their narrative, so pathetic often, that you\ncannot help but follow the trend of their simple language as you would\nfollow a story which an older sister might tell you about your own\nhomes and your own father and mother.\n\nThose essays of Lamb may sometimes show a liking for things we cannot\nlike; in his dealings with the old dramatists he may pour chirrupy\npraises where we cannot follow with ours.  We may not be won over,\nthough we see Marston through those pitiful eyes and the lens of that\nalways tender heart.  And why should we?  That criticism is not the\nbest which serves to put us in agreeing herds, and to leash us in a\nbundled cohesion of opinion; but it is better worth if it {326}\nstimulate us by putting beside our individuality of outlook the warming\nor the chafing or the contesting individuality of another mind.  There\nis never a time when Lamb's generous, kindly, witty opinions--whether\nabout men or books, or every-day topics--will not find a great company\nof delighted readers, if not of ardent sponsors.  Then, for style--what\nis to be said, except that it is so gracious, so winning, we are\ndelighted with its flow, its cadences, its surprises, its charming\nlapses--like waves on summer beaches--or like an August brook,\nprattling, babbling, and finding spread and pause in some pellucid,\novershadowed pool--where we rest in fulness of summery content.\n\nHe was never a strong man physically, and his poor thin form vanished\nfrom the sight of men in 1834, six months after Coleridge died; and the\npoor sister--unaware what helplessness and loneliness had fallen on\nher, lingered for years in blessed ignorance; she then died; and so we\nturn over that page of English letters on which are scored _Elia_ and\nthe _Tales of Shakespeare_ and pass to others.\n\n\n{327}\n\n_Wordsworth._\n\n[Sidenote: A lake poet.]\n\nOn the 29th day of June, just half a century ago, upon a beautiful\nsunny afternoon--most rare in the Lake Counties of England--I had one\nof the outside places upon an English coach, which was making its daily\ntrip from Kendal, along the borders of Lake Windermere, and on by\nGrasmere and under the flank of Helvellyn, to Derwent-Water and\nKeswick.  I stopped halfway at the good inn of the \"Salutation\" in\nAmbleside, with the blue of Windermere stretching before me; and in the\ntwilight took a row upon the lake--the surface being scarce ruffled,\nand the shores, with their copses of wood, and their <DW72>s of green\nlawn, as beautiful as a dream.\n\n  \"I dipped my oars into the silent lake,\n  And as I rose upon the stroke, my boat\n  Went heaving thro' the water like a swan.\"\n\n\n[Sidenote: Wordsworth.]\n\nThe words were Wordsworth's[8] own; and this was his country; and he\nwho was counted the {328} King-poet in those College Days which were\nnot then long behind me, was living only a little way off.  From\ndifferent points in the embowered roads I could catch a glimpse of the\nlight in his window, at Rydal Mount.  Stratford had been seen indeed,\nbut there were only memories there; and Abbotsford, but Scott and the\nlast of his family were gone; and Olney, but Cowper had been silent a\nmatter of forty years; and here, at last, I was to come into near\npresence of one of the living magicians of English verse--in his own\nlair, with his mountains and his lakes around him.  But I did not\ninterview him: no thought of such audacity came nigh me: there was more\nmodesty in those days than now.  Yet it has occurred to me since--with\nsome relentings--that I might have won a look of benediction from the\nold man of seventy-five, if I had sought his door, and told him--as I\nmight truthfully have {329} done--that within a twelvemonth of their\nissue his beautiful sextette of \"Moxon\" volumes were lying, thumb-worn,\non my desk, in a far-off New England college-room; and that within the\nmonth I had wandered up the Valley of the Wye, with his _Tintern Abbey_\npulsing in my thought more stirringly than the ivy-leaves that wrapped\nthe ruin; and that only the week before I had followed lovingly his\nWhite Doe of Rylstone along the picturesque borders of Wharfdale, and\nacross the grassy glades of Bolton Priory and among the splintered\nledges\n\n  \"Where Rylstone Brook with Wharf is blended.\"\n\nPoets love to know that they have laid such trail for even the youngest\nof followers; and though the personal benedictions were missed, I did\ngo around next morning--being Sunday--to the little chapel on the\nheights of Rydal, where he was to worship; and from my seat saw him\nenter; knowing him on the instant; tall (to my seeming), erect, yet\nwith step somewhat shaky; his coat closely buttoned; his air serious,\nand self-possessed; his features large, mouth almost coarse; {330} hair\nwhite as the driven snow, fringing a dome of baldness; an eye with a\ndreamy expression in it, and seeming to look--beyond, and still beyond.\nHe carried, too, his serious air into his share of the service, and\nmade his successive responses of \"Good Lord deliver us!\" and \"Amen!\"\nwith an emphasis that rung throughout the little chapel.\n\nI trust the reader will excuse these personal reminiscences, which I\nwrite down to fix in mind more distinctly the poet, whose work and life\nwe have only space to glance at now, and whose name will close the roll\nof poets for the present volume.\n\n\n_His Poems._\n\nThere is, and always has been, on the part of too many admirers of\nWordsworth a disposition to resent any depreciation or expression of\ndissent from fullest praise, which has counted against his reputation.\nWe do not like--any of us--to be forced into our admiration of this or\nthat poet, and will not be, for long whiles together.  There is no\n{331} bolstering of bad work that will make it permanently sound; so,\ntoo, what good things are done--whatever opposing sneers or silence may\ndo--will surely, some day or other, be found out.  A book or a poem\nthat needs careful and insistent pilotage by critics, into the harbor\nof a great Fame, will not be so sure of safe anchorage and good\nholding-ground as one that drifts thither under stress of the unbroken,\nquiet, resistless tide of a cultivated popular judgment.  Wordsworth's\nplace is a very high one; some things he has done are incomparable;\nsome altitudes of thought he has reached range among the Miltonic\nheights.  But he has printed--as so many people have--too much.  His\nvanities--which were excellently well developed--seem to have made him\ninsensible to any demerits in his own work and incapable of believing\nthat hand or brain of his could do aught that was not so far above\ncommon level as to warrant its acceptance by the world.  I think he was\nconscientious in this; I do not believe that, like many an author, he\nput before us what he knew or suspected to be inferior, simply because\nhe knew it would be devoured.  There was {332} none of that dishonesty\nin Wordsworth.  He religiously believed that even \"Peter Bell\" and the\ndreariest lines of the \"Idiot Boy\" had a mission.\n\nIf Wordsworth had possessed Browning's sense of humor, he would have\nwithdrawn an eighth of his published works; if he had possessed Hood's\nsense of humor, I think he would have withdrawn a third.  Humor is a\ngreat and good shortener.  Humor seeks to provoke mirth and ripples of\ncheery satisfaction, so it shuns length and prosiness.  Humor is a\ncharming quality in either preacher or poet; and brevity is one of the\nbest parts of humor; indeed brevity and humor always lock hands.\nUnfortunately, Wordsworth had no humor.  Again, that too free and lax\nplay of language in Wordsworth--that told nothing vital, but only\nserved to tie together, by loose and swaying looplets, the flashing\njewels wherein his real genius coruscated and crystallized--not only\nfatigued us who followed and wanted to follow, but it filled the\nmaster's time and books and thought to the neglect of that large\nentertainment of some systematized purpose--some great, balanced, and\nconcreted scheme of poetic story, {333} which he always hinted at, but\nnever made good.  Take that budget of verse which went toward the\nmaking of the \"Recluse\"--how incomplete; how unfinished even in detail;\nyet splashed up and down with brilliancies of thought and fancy; with\nhere and there noble, statuesque, single figures; like a great\nantechamber, detaining us with its diverting objects, with interposed,\nwearisome, official talk--we all the while hoping to fare through to\nsome point where we shall see the grandeur of the house and take in\nreverently its great proportions, and pay homage to the master.  But we\nnever come to those Arcana; we end in waiting; great, fine bursts of\nsong, and of glowing narrative--sun, mountains, and clouds giving us\naugust attendance--but no mapping of a whole, whose scheme is fitly\nbalanced, and whose foundations bear up a completed body and dome, with\ncross and crown.  But though his languors of language, his prosiness,\nhis self-satisfaction do madden one to damnatory speech, yet when his\nsong breaks out at its best--seeming to tie the upper mysterious world\nto this mundane level--to make steps of melody and of heavenly lift to\n{334} invite and charm as toward the Infinite, we are ashamed of our\ntoo easy discomfiture:--\n\n    \"Not in entire forgetfulness,\n    And not in utter nakedness,\n  But trailing clouds of glory do we come\n    From God, who is our home:\n\n  \"O joy! that in our embers\n    Is something that doth live,\n  That Nature yet remembers\n    What was so fugitive!\n  The thought of our past years in me doth breed\n  Perpetual benediction: nor indeed\n  For that which is most worthy to be blest;\n\n    \"But for those first affections,\n    Those shadowy recollections,\n    Which be they what they may,\n    Are yet a fountain light of all our day,\n  Are yet a master light of all our seeing;\n    Uphold us, cherish, and have power to make\n  Our noisy years seem moments in the being\n    Of the Eternal silence: truths that wake,\n          To perish never;\n  Which neither listlessness, nor mad endeavor,\n          Nor man, nor boy,\n  Nor all that is at enmity with joy.\n  Can utterly abolish or destroy!\n    Hence in a season of calm weather\n    Though inland far we be,\n  Our souls have sight of that immortal sea\n\n{335}\n\n    Which brought us hither,\n    Can, in a moment travel thither,\n  And see the children sport upon the shore,\n  And hear the mighty waters rolling evermore.\"\n\n\nThese verses belong to an ode that should never be forgotten when we\nreckon up the higher reaches of the poetic tides of this generation.\n\nI am disposed to think that all of us, as we grow older, come into\nlarger and fuller appreciation of the wonderful intuitions of this poet\nand of his marvellous grasp of all the subtler meanings in Nature's\naspects.  Certainly those lines composed above _Tintern Abbey_, do not\noffer food for babes.  Only older ones know that--\n\n              \"Nature never did betray\n  The heart that loves her; 'tis her privilege,\n  Through all the years of this our life, to lead\n  From joy to joy; for she can so inform\n  The mind that is within us, so impress\n  With quietness and beauty, and so feed\n  With lofty thoughts, that neither evil tongues,\n  Rash judgments, nor the sneers of selfish men,\n  Nor greetings where no kindness is, nor all\n  The dreary intercourse of daily life,\n  Shall e'er prevail against us, or disturb\n  Our cheerful faith, that all which we behold\n  Is full of blessings.\"\n\n\n{336}\n\nSo, too, in the Excursion, whose mention we perhaps dwelt upon too\nlightly--that grand Wordsworthian mating of man with Nature is always\nshining through the poet's purpose, and gleaming along his lines: a\ndeep and radical purpose it is; all else sways to it; all else is\ndwarfed and made small in the comparison.  Hence, poor Mary Lamb is\nhalf-justified in her outcry--that under its dominance a poor dweller\nin town has hardly \"a soul to be saved.\"[9]  Grand, surely, are many of\nhis utterances, morally and intellectually, and carrying richest\nadornments of poesy to their livery; immortal--yes; yet not favorites\nfor these many generations: too encumbered; sheathed about with tamer\nthings, that will not let the sword of his intent gleam with a vital\nkeenness and poignancy.  Always the great lesson which the stars and\nthe mountains and rolling rivers sing--sing in his lines; but\nbuttressed with over-much building up of supporting and flanking words.\nAlways the grand appeal to man's moral nature and instincts is\nimminent; always the verse radiant with the {337} beguiling lights\nwhich he has set to burn upon the hills and in the skies; but, too\noften, even the sunset glories pall, and weary with their over-painting\nand golden suffusions of language.\n\nIf one is tempted to go back to the contemporary criticism of the\n_Excursion_, he should temper the matter-of-fact admeasurement and\nantipathies of Jeffrey in the _Edinburgh_, with the kindlier and more\nfeeling discourse of Charles Lamb in the _Quarterly_ (1814).  And of\nthis latter, it is to be remembered that its warm unction and\nearnestness were very much abated by editorial jugglery.  Lamb never\nforgave Gifford for putting \"his d----d shoemaker phraseology instead\nof mine;\" and in an explanatory letter to Wordsworth he tells him that\nmany passages are cut out altogether, and \"what is left is of course\nthe worse for their having been there,\" and in a wonderful figure\ncontinues,--\"the eyes are pulled out, and the bleeding sockets are\nleft.\"\n\n\n_Personal History._\n\nWordsworth was a Cumberland man by birth, and from the very first\nopened his young eyes {338} upon such scenes as lay along the Derwent.\nHis father was an attorney-at-law and agent for the Lonsdale estates;\nnor does the poet fail to assure us in his autobiographic notes--with a\npride that is only half veiled--of the gentle blood that flowed in his\nmother's veins.  But the family purse was not plethoric; and--his\nfather dying, when Wordsworth was only fourteen--it was through the\nkindness of his uncles that he had his \"innings\" at Trinity College,\nCambridge, and felt his poetic pulses stirred by the memory of such old\nCambridge men as Milton, and Waller, and Gray.  The flat meadows\nbordering the Cam were doubtless tame to his Cumberland eyes, nor do\nUniversity memories count for much in irradiating his future work;\nperhaps the brightest gleam that comes from those cloistered sources\nupon his verse is that which is reflected from the wondrous vaulted\nceiling of King's College Chapel:--\n\n              \"That branching roof\n  Self-poised, and scooped into ten thousand cells\n  Where light and shade repose, where music dwells,\n  Lingering and wandering on as loth to die.\"\n\n{339}\n\nA vacation passed in the mountains of Switzerland sharpened an appetite\nfor travel upon the Continent; and thither he went shortly after taking\nhis degree (1791); was in Orleans and in Paris the succeeding year;\ncaught the fever of those revolutionary times, and for a while\nseriously entertained the purpose of throwing himself into the swirl of\nthat tide of Girondism which was to fall away so shortly after, leaving\ntracks of blood.\n\nThere was a short stay in London on his return--counting for very\nlittle in the story of his life.  _Westminster Bridge_ and _A Farmer of\nTilsbury Vale_ are all that bring a glimmer of remembrance to the lover\nof his books, out of the tumult and roar of \"Lothbury\" and Cheapside.\nThereafter came the quiet life in Dorsetshire with his good sister\nDora--where his poetic moods first came to print--and where Coleridge\nfound him (1796) and cemented that friendship which drew him next year\ninto Somersetshire--a friendship, which, with one brief interruption,\nthat promised a bitter quarrel--lasted throughout their lives.\nThere--at Alfoxden, in Somersetshire--was forged that little book of\n_Lyrical Ballads_, containing the _Ancient {340} Mariner_ and _Tintern\nAbbey_--the best possible types of the respective powers of the two\npoets.\n\nIn 1799 Wordsworth established himself at Grasmere, in Westmoreland,\nhis sister remaining--as she always did--a beloved inmate of his home.\nIn 1802 he married, most fortunately, a woman who was always\nsympathetic and kindly, as well as an excellent and devoted mother of\nthe children born to them;[10] moreover, she was exceptionally endowed\nto stimulate and give range to his poetic ambitions.  Between Grasmere\nor its neighborhood, and the better-known home of Rydal Mount, the poet\npassed the remainder of his life.  There were, indeed, frequent\ninterludes of travel--to Scotland, to Leicestershire, to Southern\nEngland, to Ireland, and the Continent--from all which places he came\nback with an unabated love for the lakes and mountains which bounded\nhis home.  Never did there live a more exalted lover of {341} Nature;\nand specially for those scenes of Nature which cradled him in infancy\nand which cheered his manhood.  Without being largely experienced in\nthe devices of gardening craft, he yet gave frequent and profitable\nadvice to those among his friends who were building up homes in the\nsurrounding lake district; and the Beaumont family of Leicestershire\nshow with pride a winter garden at Coleorton, which is an evergreen\nremembrancer of the poet's skill and taste.  He resented all undue\ninterference with natural surfaces; his art was the larger art of\nwinning one to the reasonableness and beauty of nature's own purposes.\n\nNot a resident in the neighborhood of Ambleside but knew his gaunt\nfigure stalking up and down the hills; yet not counted over-affable;\nthe villagers report him--\"distant, vera distant.  As for his habits he\nhad none--niver knew him wi' a pot i' his hand or a pipe i' his mouth.\"\nAnd another says--\"As for fishing, he hadn't a bit of fish in him,\nhadn't Wordsworth--not a bit o' fish in him!\"[11]  This sounds\nstrangely to one familiar {342} with _Lines to gold and silver fish in\na glass globe_.\n\nCertainly he did not love babble nor little persiflage; he had neither\nthe art to coin it nor the humor to redeem it.  But he was capable of\nsensible, heavily-charged talk, even upon practical themes, showing a\ncapacity for, and a habit of, consecutive and logical thinking.  Often\nreading and discoursing on poets and their work, but chary of any\nexuberance of praise; if ever cynical, tending that way under such\nprovocation.  Not indisposed--for small cause--to recite from\nWordsworth (as Emerson tells us in the story of his first visit to\nRydal Mount); but reciting well, and putting large, dashing movement\ninto the verse--as of faraway rebounding water-falls.  His egotism,\nthough not easily kept under, was not riotously exacting or audacious;\none could see at the bottom of it--not the little vanities of a\nflibbertigibbet, but respect and reverence for his inborn seership and\nfor his long priesthood at the altar of the Muses.\n\nHe had no musical ear, no power of distinguishing tunes, yet was rapt\ninto ecstatic fervor by the {343} near and sweet warbling of a bird.\nBooks he loved only for their uses; he favored no finical \"keeping\" of\nthem, but plunged into an uncut volume with a smeared fruit-knife--if\nneed were.  Southey dreaded his visits to his Keswick library, saying\nhe was \"like a bear in a tulip garden.\"  He was parsimonious too;\ngenerosity in praise, or in purse, was unknown to him; and he had stiff\nschool-mastery ways with youngish men--craving oblation and large\ntokens of respect.  De Quincey said he never offered to carry a lady's\nshawl; hardly offered a hand to help her over a stile.  He was not\nmobile, not adaptive, not gossipy; last of men for a picnic or a\ntea-party.  His shaking of hands was \"feckless;\" which to a Scottish\near means a hand-shake not to be run after and with no heartiness in\nits grip.  That home of Rydal Mount was a modest and charming one;\nwithin--severely simple; in abstemiousness the poet was almost an\nanchorite: without--a terrace walk, a velvety stretch of turf, mossy\nvases, a dial, a few patches of flowers, grayish house-walls on which\nthe clambering vines took hold, quaint stone chimney-tops on which the\n{344} lichens clung and around which the swallows played, views of\nRydal Water, glimpses of Windermere, of Nab-scar, and of nearer heights\ncrowned with foliage.\n\nWordsworth was never a man of large means; his poems gave only small\nmoneyed returns; nor did he care overmuch for expensive indulgences;\ntravelling was his greatest and most coveted luxury.  All new scenes in\nnature came to his eye as so many new phases of his oldest and\ntenderest friend.\n\nFor a considerable period he was in receipt of a small revenue from a\nlocal Commissionership of Stamps, and during the last eight years of\nhis life received a pension of L300 from the Government.  A year after\nthe grant, upon the death of Dr. Southey, he was, through the urgence\nof friends, and at the solicitation of Sir Robert Peel, induced to\naccept the post of Poet Laureate--going up to London, at the age of\nseventy-three, to kiss the hand of the young Queen, in recognition of\nthat honor.  This young Queen, then in her twenty-fourth year, was her\npresent gracious lady, Victoria, who had succeeded to her bluff\nsailor-uncle, {345} William IV., in 1837, and to her sorrier uncle,\nGeorge IV., who had died in 1830.\n\nWordsworth was among those stately country gentlemen who believed that\nwith the passage of the great Reform Bill of 1832, England was about to\nenter upon her decadence.  Like many another poet, he had faith in\nestablished privileges, and faith in grand traditions.  He bestirred\nhimself, too, in the latter years of his life, to defeat--if it might\nbe--the scheme for pushing railways across his quiet and beautiful\nregion among the lakes of Westmoreland and Cumberland.  Happily he did\nnot live to see the desecration of his charming solitudes; it would\nhave made him wroth to watch the wreaths of vapor from the engines\nfloating around the chimney-tops of Rydal Mount.[12]  The lines he\nwrote fifty years before his death, he lived by to the last:--\n\n  \"To her fair works did Nature link\n  The human soul that thro' me ran;\n\n{346}\n\n  And much it grieved my heart to think\n  What man has made of man.\n  The budding twigs spread out their fan\n  To catch the breezy air;\n  And I must think, do all I can,\n  That there was pleasure there.\"\n\n\nHe had not only a poet's, but a Briton's love for that old England--of\nmossy roofs, and park lands, and smoking chimneys, and great old\nhouses, and gnarled oaks, and way-side cottages.  He cherished all\nRaskin's antipathy to huge manufacturing centres, and the din of\nmachinery and trip-hammers; he would have no pounding to fright the\ncuckoos, and no reservoirs among the hills to choke the rills; but\neverywhere the brooks purling their own murmurous ways through leafy\nsolitudes and sweet, open valleys.\n\nWell, those are the sights that win most, I think, toward the celestial\nvisions which the good poet always cherished, and which symbolized best\nthe \"dear Jerusalem,\"--\n\n  \"Along whose streets, with pleasing sound,\n    The living waters flow,\n  And on the banks, on either side.\n    The trees of life do grow.\"\n\n\n{347}\n\nOnly the name--William Wordsworth--is graven upon the simple stone\nwhich marks the poet's grave, in a corner of the church-yard at\nGrasmere; and the bodies of wife and children lie grouped there beside\nhim.\n\n\n\n[1] Samuel Rogers, b. 1768; d. 1855.  His _Pleasures of Memory_,\npublished 1792; _Italy_, 1822-28.\n\n[2] Crabb Robinson, chap. ix., 1881, p. 165, vol. ii., says he \"was\nnoted for his generosity toward poor artists.\"  The story he tells in\nconfirmation is, that Sir Thomas Lawrence appeared at his door and\nbegged him to save the president of the Royal Academy from disgrace,\nwhich must follow except a few thousands were raised next day; he (Sir\nThomas) offering his paintings, drawings, etc., in guarantee.  Crabb\nRobinson continues that \"Rogers saw Lord Ward [a nobleman of great\nwealth] next day and arranged for the advance by him;\" an advance that\nnever brought loss to either Ward or Rogers.  The latter's\n\"generosities\" were a good many of them of this color; _i.e._, securing\nadvances which were pretty sure to be repaid.\n\n[3] S. T. Coleridge, b. 1772; d. 1834.  Many of his works edited by H.\nN. Coleridge, husband of his only daughter Sara.  Special mention\nshould be made of the Coleridgean labors of that indefatigable worker,\nthe late J. <DW18>s Campbell.\n\n[4] He had a son Hartley, whom Crabb Robinson describes in 1816 as \"one\nof the strangest boys I ever saw.  He has the features of a foreign\nJew, with starched and affected manners.\"  He also speaks of the other\nson, Derwent, as a \"hearty boy, with a good-natured expression.\"  The\ndaughter--afterward Mrs. Henry Nelson Coleridge, editress of many of\nher father's works (continues Robinson), \"has a face of great\nsweetness.\"\n\n[5] Southey did not go to Keswick to reside until 1803-4.  Coleridge,\nhowever, was there as an occupant of a portion of the future Southey\nhome in 1800.  Southey paid him a visit in the summer of 1801.  See\nTraill, chap. v.  See also _Memorials of Coleorton_, passim.\n\n[6] Probably some time between 1803 and 1806.\n\n[7] Charles Lamb, b. 1775; d. 1834.\n\n[8] William Wordsworth, b. 1770; d. 1850.  _Evening Walk_ published\n1793; _Lyrical Ballads_ (in conjunction with Coleridge), 1798;\n_Excursion_, 1814; _White Doe of Rylstone_, 1815; first collected\nedition of poems, 1836-37; _Life_ by W. H. Myers; a much fuller, but\nsomewhat muddled one, by William Knight, 3 vols,, 8vo, 1889.  Dowden's\nedition of Wordsworth's poems (Aldine Series) is latest and best.\n\n[9] See Lamb's Letters, cited in Knight, vol. ii., p. 235.\n\n[10] His wife was Mary Hutchinson, of Penrith.  Their children were\nJohn, b. 1803; Dorothy, b. 1804 (became Mrs. Quinlan and died before\nher father); Thomas, b. 1806; Catharine, b. 1808; and William, b.\n1810--the last being the only one who survived the poet.\n\n[11] This based on \"Mr. Rawnsley's Gleanings amongst the Villagers.\"\nSee _Athenaeum_, February 23, 1889.\n\n[12] There is a very interesting account of Wordsworth's home life,\netc., in Miss Martineau's _Autobiography_, vol. i., p. 504 _et\nseq._--but very much , as all her pictures are, by her own\nmegrims and disposition to sneer at all the world--except Miss\nMartineau.\n\n\n\n\n{349}\n\nINDEX.\n\n\nAdams, John, 187.\n\nAddison, Joseph, 4.\n\nAikin, Dr., 273-276.\n\nAllston, Washington, 316.\n\nAnne, Queen, the times of, 1-3.\n\nAusten, Jane, her life and personality, 265-267; opinions of Walter\nScott, Macaulay, and Miss Mitford concerning, 266, 267; her _Pride and\nPrejudice_, 268; _Persuasion_ and _Northanger Abbey_, 268, 269; her\nqualities, 270, 271; burial-place, 270.\n\nAusten, Lady, and William Cowper, 246, 247.\n\n\nBarbauld, Mrs., 273-276.\n\nBeauclerk, Topham, 114-116.\n\nBeckford, William, and his _Vathek_, 285-291.\n\nBentley, Richard, his _Siris: A chain of Philosophical Reflections and\nInquiries concerning the Virtues of Tarwater_, 9; writes on the\n_Epistles of Phalaris_, 9-11; his family, 10; portrait of, 10, 11; as a\nwriter and as a man, 11, 12.\n\nBerkeley, George, his _Theory of Vision_, 4; his career, 4-9; his\nverse, 5; his sermons, 6; _The Minute Philosopher_, 7; his family, 7;\nhis philosophy, 9.\n\nBlair, Hugh, 230.\n\nBlounts, Alexander Pope and the, 34.\n\nBoswell, James, and his _Life of Dr. Johnson_, 118-122.\n\nBoufflers, Madame de, and David Hume, 150.\n\nBurke, Edmund, 112, 113; his words concerning Beauclerk's widow, 115;\nhis burial-place, 145.\n\nBurney, Frances, and Dr. Johnson, 138, 142, 164, 165; her stories, 165;\n_Evelina_, 165-168; _Camilla_, 168; her Diary, 168-169; last years,\n170, 171.\n\nBurns, Robert, his poetry, 291; his career, 292-297; his death, 298,\n301; compared with Samuel Rogers, 302, 303.\n\n\n_Camilla_, Miss Burney's, 170.\n\nCarlyle, Thomas, his words concerning Coleridge, 318.\n\n_Castle of Otranto_, The, Walpole's, 84.\n\nChatterton, Thomas, the young poet, 202-205; his end, 205, 206, 209;\nand Horace Walpole, 206-209; the Rowley Poems, 207, 208; compared with\nPoe, 210.\n\nChesterfield, Lord, and Dr. Johnson, 97, 98.\n\n_Children of the Abbey_, Miss Roche's, 282, 283.\n\n_Christabel_, Coleridge's, 317, 318.\n\nCoach, the Venetian, 3.\n\n_C[oe]lebs_, Hannah More's, 175, 176.\n\nColeridge, S. T., 298, 299; his life, 309-317; Lamb's apostrophe to,\n310; and Southey, 311, 312; and Wordsworth, 313; his _Ancient Mariner_,\n314, and Washington Allston, 316; his opium habit, 316, 317; his\n_Christabel_, 317; Carlyle's words concerning, 318; his death, 318.\n\nCollins, William, 100-163; his _Ode to Evening_, 163, 180.\n\nCoverley, Sir Roger de, 2.\n\nCowper, William, his family and education, 239, 240; his love affair,\n240; mental trouble, 241, 242; and Mrs. Unwin, 243-245, and Rev. John\nNewton, 245; _John Gilpin's Ride_, 245, 246, and Lady Austen, 246; _The\nTask_, 240, 247; on American affairs, 248; later life, 249-253; his\nHomer, 250, 251; his place as a poet, 254-256.\n\nCrabbe, George, compared with Pope, 232, 233; his birth and early work,\n233-235; private chaplain to the Duke of Rutland, 235, 236; his life\nand character, 237, 238.\n\nCurchod, Mademoiselle, afterward Madame Necker, 123.\n\n\nDay, Thomas, and _Sandford and Merton_, 271-273.\n\n_Decline and Fall of the Roman Empire_, Gibbons's History of, 125, 127,\n130.\n\n\nEdgeworth, Maria, 277-281.\n\nErnest, Augustus, Duke of Brunswick, 57.\n\n_Evelina_, Miss Burney's, 165-168.\n\n_Evenings at Home_, by Dr. Aikin and Mrs. Barbauld, 273-276.\n\n\nFerguson, Robert, 229.\n\nFielding, Henry, his coarseness, 67, 68; his character and ancestry,\n68; his schooling, 69; his dramatic work, 69, 70; his _Joseph Andrews_,\n_Amelia_, and _Tom Jones_, 71, 72; his marriage, 70, 71; his death, 72.\n\nFox, Charles James, 188-192.\n\nFranklin, Benjamin, and Miss Burney, 166; his words concerning George\nIII., 184.\n\nFreeman, Edward, his words concerning Gibbon, 128.\n\n\nGarrick, David, at Dr. Johnson's school, 91, 92; as a boy, 116; a\nmember of the \"Literary Club, \" 116; as an actor, 117, 118; his death,\n138; Hannah More and, 173, 174.\n\nGeorge I., ancestry, 57; comes to England, 58; his character, 58; his\nwife, 58, 59.\n\nGeorge II., 59-61; his reign, 61.\n\nGeorge III., character and personality of, 181-187.\n\nGibbon, Edward, birth, parentage, and education, 122; his love for\nMlle. Curchod, afterward Madame Necker, 133, 124; a member of the\n\"Literary Club, \" 124, 127; as an author, 124, 125; his _Decline and\nfall of the Roman Empire_, 125, 127-130; as a man, 125, 126; in Paris,\n126; his burial-place, 145.\n\nGoldsmith, Oliver, a member of the \"Literary Club, \" 130, 131; as a\nwriter, 132, 133; his death, 133, 134; his burial-place, 144, 145.\n\nGray, Thomas, birth, parentage, and education, 79, 80; opinions of his\nwork, 80; his fastidious refinement, 80-82; the _Elegy_ churchyard, 82;\nand the Rowley Poems, 208.\n\n\nHalket, George, 229.\n\nHayley, William, a friend of Cowper's, 249.\n\nHesketh, Lady, her interest in Cowper, 250, 252.\n\nHomer, Pope's translation of, 43-45; Cowper's translation, 250, 251.\n\nHoneycomb, Will, 2.\n\nHume, David, compared with Gibbon, 145, 146; his birth and early years,\n146-148; his _Political Discourses_, 148; his _History of England_,\n146, 149, 150, 156, 157; and Madame de Boufflers, 150; in Paris,\n151-154; ambassador to the Court of France, 152; did not love England,\n152, 153; his home in Edinboro', 154, 155; his death, 155, 156, 179;\nhis words concerning James Macpherson, 226.\n\n\n_John Gilpin's Ride_, Cowper's, 245, 246.\n\nJohnson, Samuel, his birth, parentage, and early career, 88-90; his\nmarriage, 90, 91; his boarding-school, 91; his personal appearance, 91;\ngoes to London, 91, 92; his _Irene_, 90, 92, 96, 97; and Richard\nSavage, 92-94; his _London_, 94, 95; his _Vanity of Human Wishes_, 95,\n96; his Prologue spoken at Drury Lane, 96; his Dictionary, 97, 98; his\nletter to Lord Chesterfield, 98; in poverty, 102; death of his wife,\n104; and Miss Williams, 104, 105; his power felt, 105; his _Rasselas_,\n105-108; his friendship with Sir Joshua Reynolds, 108, 109; Boswell's\nLife of, 118-122; and the Thrales, 135-137, 139, 140; his journey to\nthe Hebrides, 137, 138; his last years, 137-143; his burial-place, 145;\nHannah More and, 173; his reply to James Macpherson, 225, 226.\n\n_Joseph Andrews_, Fielding's, 177.\n\n\nKames, Lord, 230.\n\n\nLamb, Charles, his words on Burns, 299; his apostrophe to Coleridge,\n310; his writings, 319, 320, 323-326; his personality, 320, 321; his\nfamily afflictions, 321-323; his death, 326.\n\nLamb, Mary, 321-323, 326.\n\n\"Literary Club, \" the, 111.\n\nLondon Bridge, 103.\n\n\nMacaulay, T. B., on Boswell, 119; his opinion of Jane Austen, 266.\n\nMackenzie, Henry, 230.\n\nMacpherson, James, and the Ossian poems, 221-227; his life, 224, 225;\nhis habits and disposition, 226, 227.\n\nMitford, Miss, her words concerning Jane Austen, 266, 267.\n\nMontagu, Lady Mary Wortley, her birth, parentage, and early life, 21,\n22; her marriage, 22; her letters, 21, 23, 28; has her son inoculated\nfor smallpox, 23, 24; Pope's admiration for, 23-25; quarrels with Pope,\n25, 26; a favorite of George I., 26; her later life, 27-30; Horace\nWalpole's words concerning, 30, 52, 53.\n\nMore, Hannah, her words concerning Dr. Edward Young, 20; her youth,\n171, 172; her pension, 172; acquaintance with Garrick and Johnson, 173,\n174; her tragedy of _Percy_, 174; as a worker, 175; her _C[oe]lebs_,\n175, 176; her goodness, 175-178; Thackeray's reference to, in _The\nNewcomes_, 177, 178; her age, 180.\n\n_Mysteries of Udolpho_, Radcliffe's, 284, 285.\n\n\nNecker, Madame, 123, 124.\n\nNewton, Rev. John, of Olney, and William Cowper, 245.\n\n_Night Thoughts_, Young's, 15, 16, 18-30.\n\n_Northanger Abbey_, Jane Austen's, 269.\n\nNugent, Dr., 114.\n\n\n_Ode to Evening_, Collins's, 163.\n\nOssian's Poems, 221-227; the Ossianic Hermitage, 257, 258.\n\n\n_Percy_, Hannah More's tragedy, 174.\n\n_Persuasion_, Jane Austen's, 268, 269.\n\nPitt, William, 192-195.\n\nPope, Alexander, his admiration for Lady Mary Wortley Montagu, 23-25;\nfamiliar couplets of, 31; his infirmity and personal appearance, 31,\n32; his birth and early years, 33, 34; and the Blounts, 34; his poetic\nmethods, 35-39: his _Essay on Criticism_, 36; his _Windsor Forest_, 36;\nhis _Rape of the Lock_, 36, 39-42; writes for the _Spectator_, 38, 39;\nhis translation of Homer, 43-45; his house and friends at Twickenham,\n45-50; his last days, 48-51, 53.\n\nPorter, Jane, her _Thaddeus of Warsaw_, 283, 284; her _Scottish\nChiefs_, 284.\n\n_Pride and Prejudice_, Jane Austen's, 268.\n\n\nRadcliffe, Ann Ward, her _Mysteries of Udolpho_, 284, 285.\n\n_Rambler, The_, 98.\n\nRamsay, Allan, 228.\n\n_Rape of the Lock_, Pope's, 36, 39-42.\n\n_Rasselas_, Dr. Johnson's, 105-108.\n\nReynolds, Sir Joshua, 108-111.\n\nRichardson, Samuel, a printer and book-seller, 62; his friends, 63, 64;\nas a writer of letters, 63-66; the father of the novel, 66, 67; assists\nDr. Johnson, 102.\n\nRobertson, Dr., 230.\n\nRoche, Maria Regina, her _Children of the Abbey_, 282, 283.\n\nRogers, Samuel, his _Pleasures of Memory_, 301, 302, 307-309; compared\nwith Burns, 302, 303; his career and character, 303-307.\n\nRousseau, J. J., 154.\n\nRowley Poems, The, 208.\n\nRuskin, John, on Gibbon's style, 128.\n\n\n_Sandford and Merton_, Day's, 271-273.\n\nSavage, Richard, and Dr. Johnson, 92, 94.\n\nScott, Walter, his opinion of Jane Austen, 266; his translation of\n_Leonora_, 298.\n\n_Scottish Chiefs_, Jane Porter's, 284.\n\n_Selborne, Natural History of_, White's, 260-262.\n\nShenstone, William, 158-160, 180.\n\nSheridan, Thomas Brinsley, 195-202; as an orator, 199, 200; his end,\n201, 202, 219.\n\nSmibert, John, his painting of Berkeley and family, 7.\n\nSmith, Adam, 230.\n\nSophia, grand-daughter of James I. and mother of George I., 57.\n\nSouthey, Robert, and Coleridge, 311, 312.\n\nSterne, Laurence, his death, 211, 212; his style, 212-214; his\nburial-place, 215; his character and habit, 215, 216; his literary\npilferings, 216, 217; pathos of his life, 217, 218, 220.\n\nStoke-Pogis Churchyard and Gray's _Elegy_, 82.\n\nStuart, Charles Edward, the Young Pretender, 55, 56.\n\nStuart, Elizabeth, daughter of James I., 57.\n\nStuart, Henry, 56.\n\nStuart, James Edward, the Pretender, 53-55.\n\nSwift, Dean, and Pope's Homer, 44.\n\n\nThackeray, W. M., and Hannah More, 177, 178.\n\n_Thaddeus of Warsaw_, Jane Porter's, 283, 284.\n\nThomson, James, his boyhood, 73; brings his poetry to London, 73, 74;\nhis _Winter_, 74, 75; befriended by Pope, 76; his _Liberty_ and _Castle\nof Indolence_, 77, 78; his burial-place, 101.\n\nThrales, The, and Dr. Johnson, 135-137, 139, 140.\n\nTurk's Head Club, The, 111 _et seq._\n\n\nUnwin, Mrs., and William Cowper, 243-245, 252, 253.\n\n\nVanhomrigh, Miss, 4, 5.\n\n_Vathek_, Beckford's, 285-288.\n\n\nWalpole, Horace, his words concerning Lady Mary Wortley Montagu, 30;\nhis parentage and life at Twickenham, 83, 84, 87, 88; his _Castle of\nOtranto_, 84; his letters, 85-87; his words concerning Gibbon, 125; and\nthe poet Chatterton, 206-209.\n\nWatts, Isaac, associations of the name, 12, 13; birth, parentage, and\neducation, 13, 14; Bryant's admiration for, 14; his hymns, 14, 15;\nendowed with a home, 15.\n\nWestminster Bridge, 103.\n\nWhite, Gilbert, and the _Natural History of Selborne_, 259-264; his\nhouse, 264.\n\nWilliams, Miss, and Dr. Johnson, 104, 105.\n\nWordsworth, William, 298; and Coleridge, 313; the author's personal\nreminiscence of, 327-330; his poetry, 330-337; his parentage and early\nyears, 337-340; his marriage, 340; his love of Nature, 340, 341;\npersonal traits, 341-343; his home at Rydal Mount, 343, 344; his\npension, 344; made Poet Laureate, 344; opposed to railways and\nmanufactures, 345, 346; his burial-place, 347.\n\n\nYoung, Dr. Edward, his _Night Thoughts_, 15, 16, 18-20; his birth,\nparentage, and early work, 16; his _Last Day_, 17; his marriage, 18;\nback at court, 19, 20; Hannah More's words concerning, 20.\n\n\n\n\n[Transcriber's note: the source book's odd-numbered pages had varying\nheaders.  In this etext, they have been converted to sidenotes and\nplaced where appropriate.]\n\n\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of English Lands Letters and Kings, by\nDonald G. Mitchell\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nHonorary   \nWhite\n\nE. R. Braithwaite\n\n# Contents\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nChapter Ten\n\nChapter Eleven\n\nChapter Twelve\n\nAbout the Author\n\n# Chapter  \nOne\n\nMY VISIT TO SOUTH Africa really began the moment the huge Boeing 747 lifted off the tarmac at London's Heathrow airport and shuddered its way into the darkening sky. From that moment, my section of the lower rear cabin was transformed into a separate little world peopled by an impromptu, noisy mix of British, Amer\u00adicans, Swiss, Germans, South Africans, a few French, and me.\n\nI sat watching and listening, especially to the emigrants. They were all British, suspended between the certainty of having finally discarded a familiar way of life and the uncertainties toward which they were being inexorably propelled. They were all young, aged between eighteen and thirty, and seemingly ill at ease as they recited the litany of troubles which had precipitated their decisions to leave Britain\u2014the skyrocketing cost of living, restrictions on heating, lighting, and gasoline, the excessive cost of mortgages, strikes, competition with Blacks for jobs, and the inclement weather.\n\n\"We're doing it mainly for the children\" was the excuse most frequently exchanged between them. I wondered why they found it necessary to make excuses for the decision to make a change, especially one so exciting and adventurous. The children, for their part, were soon running up and down the aisles and in and out of the toilets, happily unaware of the role they played in their parents' momentous decision.\n\nAt Nairobi, our first stop on African soil, some of the passengers departed and were replaced by British and South Africans linking up with us from other routes. All white. Not another black face in sight.\n\nThe other travelers had something else in common. They were all, in varying degrees, pleased and excited to be going or returning to South Africa. I was odd man out, wrapped in layers of uncertainty and apprehension, wondering whether and for how long I would be able to stay in an environment which would deliberately seek to humiliate and degrade me.\n\nMy fellow travelers knew the good life or anticipated a better one. Would any of them be able to understand my decision to expose myself voluntarily to a social order which would not only deny my humanity, but claim divine guidance and support for doing so? Ever since leaving the London airport, I had not exchanged a single word with anyone but the stewards, and then only in response to queries about meals. Was this a foretaste of what lay ahead?\n\nSince boyhood in Guyana I had heard stories about horrors of life for Blacks in the gold, diamond, and coal mines of South Africa, and the cruel oppression they suffered at the hands of their European conquerors. I remember hearing about Blacks working deep in the bowels of the earth, day after day, ill-fed and poorly paid, completely at the mercy of the Whites who tyrannized and bullied them. Floods and cave-ins had trapped hundreds of these Blacks, and only token rescue efforts were made; their fate was of little consequence because they could so easily be replaced.\n\nWe talked of these things, my boyhood friends and I, happily ignorant of the grimmer realities, safe in our freedom to move and speak, to see and learn, our discussions of the plight of our faraway black brethren hardly more than an academic exercise. In Guyana, the men who worked the gold, diamond, and bauxite deposits were called miners, but they all worked above ground, not like moles burrowing deep out of sight. The gold and diamond miners usually worked their own \"claims,\" each hoping for the one big \"strike\" which would lift him overnight from penury to riches, meanwhile scratching a bare living from the reluctant earth. The bauxite miners worked for the Bauxite Company, balancing precariously between negotiations for better conditions and threats of a strike.\n\nIt was hard for my friends and me to take in the horror stories of long lines of ragged black men led docilely to and from the deep pits each day, under the cruelly watchful eyes of armed white guards. Why did they not turn on their oppressors the same way the Guyana sugar plantation workers sometimes did when the burden of long laborious hours with poor pay became unbearable? Blissfully young and arrogantly uninformed, we blamed the South African Blacks for being too timid and boasted among ourselves of what we would have done in similar circumstances.\n\nAs I grew up, it seemed that each successive South African Government instituted new and more oppressive laws against the black population, who seemed more and more resigned to their fate, or more and more helpless to change it. I met African Blacks for the first time during my student days in England. Though none of them was from South Africa they seemed well informed about life in the Republic and excited my imagination with horror tales of white-settler inhumanity to the native Blacks. As they told it, the whole sorry business began with the establishment of a Dutch East India Company Trading Station at Table Bay and some mutinous personnel who later settled there as free farmers. Slaves from other parts of Africa were shipped there to help in the development of the settlement but were rigidly segregated, being denied even the right to wear shoes. As the settlement developed, the settlers or trekboers pushed into the interior, seizing the wide grazing grounds of the pastoral Hottentots, stealing their cattle, and killing the virtually defenseless Blacks.\n\n\"Those trekboers were all Calvinists and believed that God made the white man to rule over Blacks,\" one friend reminded me. \"In fact they conceived of Blacks as being little more than animals. They would hold shooting competitions, with prizes going to the men who, in a stipulated time, killed the largest number of Blacks. Proof of a 'kill' was the severed penis of a male or two breasts of a female. Nothing new. The British did the same thing when they colonized Australia and Tasmania.\n\n\"Nothing has changed. Nothing,\" he said. \"Sure, Blacks in South Africa wear shoes nowadays and are not arrested for smoking in the street. They're even allowed to work in offices or on building sites beside white men. But that means nothing. Whites still think the Blacks are animals. True, they're not shot for bounty like in the old days. Nowadays Blacks who step out of line are quickly arrested. The lucky ones are sent to jail; the others simply disappear.\"\n\n\"For what crime?\" I asked. My friend evidently had a flair for dramatic overstatement.\n\n\"Crime? If you're black in South Africa that's the crime. Everything else is merely supporting evidence. To look a white man in the eye is a crime. To object when he abuses you is a crime. Everything you do is a crime. The State says so and the Church agrees.\"\n\nWhen job-hunting in London in the summer of 1947, I met a black escapee from South Africa. At that time I was in the throes of despondency and disillusionment at the prejudice and discrimination I had encountered in Britain. It was in a coffee shop off Piccadilly Circus where I occasionally had a mug of hot tea and a thick cheese sandwich for my midday meal. Cheap and satisfying. He was there when I arrived and I deliberately took a seat near him. He smiled as I sat down.\n\n\"Hello, there,\" he greeted me, speaking with difficulty through a mouthful of fish and chips.\n\nI was pleased at the prospect of a friendly interlude with another human being, particularly a black one. We'd understand each other.\n\n\"Where you from?\" he wanted to know.\n\n\"Guyana.\"\n\n\"Been over here long?\"\n\nI mentioned that I'd come over eight years earlier and taken a degree at Cambridge and gone on to post-graduate studies. Then I had served in the R.A.F. Now I was job-hunting.\n\n\"You'll do all right,\" he assured me.\n\n\"Like hell I will. I've already been hunting nine months for a job. Any job.\"\n\n\"You've not been looking in the right places,\" he said. \"I've been here two years and I've got a job. Got it six weeks after I arrived. Good job. No complaints.\" Smiling a white-toothed, superior, employed smile.\n\n\"Doing what?\" I asked.\n\n\"Hospital orderly at Lewisham General.\"\n\n\"You like it there?\"\n\n\"Sure. The pay's good. Lots of time off, and the work's not hard. You should try it.\"\n\n\"No thanks.\" The idea of working in a sterile hospital ward ministering to ailing, irritable people did not appeal to me.\n\n\"Then what are you complaining about? You want a job, don't you?\"\n\n\"I'm looking for a job I'm qualified to do.\"\n\n\"Man, you're lucky you can pick and choose. Where I come from you take what they give you.\"\n\n\"Where's that?\"\n\n\"Cape Town. South Africa. I finished high school, but the only job I could get was messenger boy in a grocery store. Sweep out the store and carry groceries for the Whites. One Rand a week.\"\n\n\"How much is that in English money?\"\n\n\"About eighteen shillings.\"\n\n\"So you decided to come over here,\" I said.\n\n\"You make it sound so easy, man. As if I walked up to the steamship office and said 'I'm going to England, here's the money for my fare.' Hell, I walked to Port Elizabeth, then stowed away on a British ship for Southampton.\"\n\n\"Just like that?\"\n\n\"Oh, they found me when we were halfway across. Made me wash decks and help in the galley. But here I am.\"\n\n\"And here you stay,\" I added.\n\n\"Hell, yes. I'd rather die than go back. In South Africa a black man is treated like an animal. Your degrees wouldn't be worth a damn. Anyway, you couldn't get into the universities, not the white ones.\"\n\nIn 1960 I received an appointment as Human Rights Officer for the World Veterans Federation at the Federation's headquarters in Paris and met many South Africans who lived there, most of them painters, writers, or musicians. They all saw themselves as voluntary refugees from a repressive police state. Sometimes over coffee at an outdoor Left Bank restaurant I'd join in lively discussion with them and a few expatriate Americans, black and white. I learned that, even with lynch law at its worst, life in the southern United States was far better for Blacks than in South Africa. At that time Blacks in the American South had limited voting rights while Blacks in South Africa were not even included in the national census, because they were not considered human. In the United States Blacks could seek redress for injustice in the Federal courts, either directly or through legal representation; in South Africa the courts themselves enforced discriminatory practices. Apart from me, all those who took part in the conversations had firsthand experience of the oppression they described, and the picture they drew of the life of Blacks in South Africa was frightening\u2014a society where the owner of a black skin was helplessly subjected to exploitation, ill-treatment, and the death penalty.\n\nEarly in 1965 I was appointed Guyana's Ambassador to the United Nations and met more South Africans in New York, white and black. Some of the Whites, churchmen and others of liberal persuasion, were petitioners against the racist regime; but the majority were businessmen, employees of the South African Government, and tourists. Always acutely aware of the overwhelming antipathy to their Government, they were continually on the defensive, insisting that outsiders could not appreciate the peculiar conditions of South Africa. Without exception, the South African Blacks I met were petitioners, escapees, and permanent exiles from their country, and committed to persuading United Nations member states to deny any aid or support to South Africa.\n\nIn 1966 the United Nations General Assembly voted overwhelmingly to terminate South Africa's mandate to supervise and control South West Africa, now Namibia, and itself assumed full responsibility for the territory. I was very active in the special committee which designed the resolution and lobbied for its passage until finally the United Nations Council for South West Africa was set up. I was made President of that Council.\n\nI plunged enthusiastically into the struggle to free Namibia from the defacto control of South Africa. But although I then became directly involved with the fate of the Blacks of Southern Africa, my prime interest at the time lay in showing that I was fully professional, so as to demonstrate that representatives from the smallest, least powerful, and poorest of member nations could effectively conduct the business of the United Nations.\n\nSome of the black petitioners were impatient with the careful way in which I approached their complaints. They clearly expected of me a more immediate identification with their situation and were not the least bit impressed by my posture of scrupulous impartiality. During a session with a group of exiles from Namibia one of them impatiently asked,\n\n\"Mr. Ambassador, whose side are you on?\"\n\nMany a petitioner would say \"You ought to go and see for yourself,\" as I failed to comprehend the horrors they tried to convey: police brutality; arrest and imprisonment without legal defense; trials which mocked justice; Blacks routinely sentenced to banishment and death; Blacks barred by law from voting; Blacks forbidden to organize labor unions, banned from all but the lowest work categories. It was all new to me. The British who had governed Guyana were no strangers to prejudice and discrimination, but nothing in their treatment of that predominantly black population in any way compared with the terrible stories I heard from the South African petitioners. It was not painful for me to reflect on British Guiana's civil service, schools, colleges, police, communications, utilities, and courts\u2014all managed and operated by Blacks under white supervision which was more ritual than functional. Then, as now, the busiest places in the country were the courts, forever crowded with litigants and their representatives, as if everyone was determined to prove his access to equal justice.\n\nThe Government of South Africa pointedly ignored the new Council. Several requests for permission to visit Namibia were either unanswered or peremptorily refused. On one occasion, I made a personal appeal for a meeting of the U.N. Security Council on behalf of a group of Namibian Blacks who had been abducted within their country by South African Security Police and taken to South Africa where they were charged with treason, tried, and pronounced guilty. The members of the Security Council individually expressed their distaste for South Africa's action and, in carefully guarded language, condemned her intrusion into the territory over which she no longer had legal control. However, no resolution emerged from the meeting, nor was there any collective statement of condemnation of South Africa. The Blacks were hanged.\n\n\"Go and see for yourself,\" the petitioners urged, though they knew that the Council and I would not be permitted to enter South Africa or Namibia. And while I sometimes heard myself echoing the same refrain, I knew that I would not willingly have exposed myself to life in South Africa. So I played with the thought of going there, secure in the knowledge that I would not be allowed in. But the thought often haunted me: Just suppose the South African Government suddenly relented?\n\nJust supposing one day there came a letter of clearance for the Council to visit Namibia? My colleagues and I, black and white, would have to go through South Africa. Would we be allowed to travel together, eat together, use the same hotel? Or would we be segregated according to South Africa's racial policies? Would I accept such segregation as secondary to the main issue of fact-finding? Perhaps, as a native of another sovereign country and under the protection of the United Nations I would not be subject to South Africa's segregation laws. But, if insulated from them, how could I truly appreciate their effect?\n\nIt would be a harsh irony for me, a black man, to visit a country like Namibia or South Africa and be isolated from the cruelties to which other black men were continually subjected. How would the Blacks themselves react to me, a \"protected\" person? Would they respond to me as a black \"brother\" or merely as a representative of the United Nations who happened to be black but was unlikely to be concerned for their plight? I was not an African, had no knowledge of their languages and no real understanding of their traditions, so I would be as much an outsider as anyone else.\n\nEarly in 1973, long after I had left the U.N., a friend in Guyana sent me a clipping from the South African Official Gazette. The clipping stated that, as of that date, the ban on all books by E. R. Braithwaite was lifted. I was surprised and, on impulse, telephoned the South African Consul General in New York. I said that I had just learned of the lifting of the ban from my books, and even as I thought of it, asked whether the ban was lifted from the author as well.\n\nThe Consul General was friendly and charming and completely and happily unaware of author, books, or ban. We chatted awhile and he suggested that the best way of checking any ban on myself was by applying for a visa. All visa applications are processed in Pretoria and a successful application would mean there was no ban. He invited me to visit his office at my convenience and make the application.\n\nHe was as friendly, courteous and urbane as his voice had promised. With the utmost civility we talked about his country and its policies and he said that he sincerely hoped my visa application would be favorably considered and that I might at close quarters come to an appreciation of why such policies and practices were necessary in the prevailing circumstances. I told him that everything I had heard about his country had already prejudiced me against that likelihood.\n\n\"Be patient. Go and see for yourself,\" he said, unwittingly echoing the others. He sounded as if he expected the visa to be granted. Would he, as Consul General, advise that I be given one? I suddenly felt cornered. Supposing, just supposing the visa was granted. What then? I was not attracted to the idea of spending any time in a racist society. So why bother to apply for a visa? If it was granted and I refused it, that would be the end of me as a critic of South Africa.\n\nBut there seemed little chance that the visa would actually be granted. Surely it would suffer the same fate as those earlier Council appeals. So security-conscious a state as South Africa would certainly investigate my background, especially my anti-South Africa position at the United Nations. Their conclusions would certainly be negative. In any case, I could tell myself that I had tried to visit South Africa. I'd be able to put the \"Go see for yourself\" thing to rest once and for all.\n\nFive months passed with no word from Pretoria and I had convinced myself that my application had been ignored, when there was a telephone call from the Consul General. My visa had been granted. My immediate reaction was one of acute distress. Now that my way was clear, the thought of actually going to South Africa was abhorrent. For his part, the Consul General believed I sincerely wished to visit his country and invited me to come to the Consulate again and meet other members of his staff who would provide me with a useful overview. I accepted.\n\nThey seemed to be calling my bluff. For years now I had been so safe in my posture of justified condemnation of South Africa's racial policies, isolated from whatever the grim realities might be. Everyone knew that South Africa was closed to critical inspection, especially by Blacks, so I was safe in my hawk's nest. Until now. What would I say to the Consul General? What excuse could I fabricate to explain my rejection of the visa?\n\nBut, on the other hand, why reject it? So far, I had given full credence to South Africa's critics and had readily allied myself with them. Well, why not see for myself, as so many of them had advised? No matter how trying the circumstances, I had the right, as a visitor, to leave whenever I chose. Yet, at my age, and accustomed to freedom of movement, speech and association, could I tolerate even for a short time the contempt, restriction, and discourtesy which were inescapable if I entered South Africa?\n\nWould I be willing to obey the \"Whites Only\" signs, ignore the \"kaffir\" epithet, and give way to Whites? I doubted that I could. Yet how could I ever meet and talk with South African Blacks on their own earth except by going there? Out of the blue was handed me the opportunity to \"see for myself.\" Didn't I have a duty to seize it? Should I reject it on the flimsy excuse of safety or sensitivity to anti-Black attitudes? My doubts and dread nagged me like a toothache, but I knew I was going to go.\n\nThe stewardess announced that we were making our descent to Johannesburg and I began mentally preparing myself for what I felt sure would be my first test. What would I do when confronted with the \"Whites Only\" signs? Would I have to undergo a separate passport and customs check? Would the humiliations I had heard about begin then or later?\n\nPreoccupied with these speculations, I hardly noticed that the huge plane had landed, was taxiing to its gate. There followed the gathering up of personal belongings and the long line through the narrow exit to the shock of warm sunshine on the short walk to the cavernous customs hall.\n\nTry as I might, the only signs I could locate were those over the narrow gateways to the passport control desks distinguishing between South African nationals and others. In my turn I was shown the same courteous treatment as anyone else and moved into the baggage claim area where I grabbed a metal pushcart just ahead of someone else. With nothing to declare I pushed my bags through customs and outside into whatever the next several weeks would disclose.\n\n# Chapter  \nTwo\n\nMY HOTEL WAS A new one on the edge of the business district, pompously dominating a busy crossroads and overlooking a block-square park, a green oasis amid the steel, glass, and concrete. My car had barely stopped when the door was yanked open by a Black, the doorman\u2014tall, muscular, and resplendent in gray top hat, matching pearl gray tail suit, black tie, and gleaming black shoes. He helped me out of the car, smiled broadly, and greeted me in what sounded like Afrikaans but changed quickly to English when he noticed my failure to respond. He seemed surprised that only I, and not my white driver as well, would be staying at the hotel.\n\nInside, the hotel was even more imposing. The lobby was spacious, with leather divans spotted like islands on a placid sea. Artfully carved and paneled woodwork on the walls highlighted a wide wooden staircase leading upward to the mezzanine floor. The doorman led me to the reception desk and presented me with something of a flourish. The white reception staff were all aplomb and courtesy as if well prepared for my coming. Gray-suited porters everywhere. \"Good morning, sir.\" One of them introduced himself to me as the manager. \"I trust you had a comfortable flight.\" I thanked him and said I had. \"We are very happy to have you staying with us,\" he went on, motioning me to a table to which he brought a pen and registration cards and showed me where he needed my signature. My name and flight particulars had already been entered on the card, which surprised me until I remembered that in New York I had been advised to make my travel plans through the Grosvenor Tours Company and they had chosen this hotel for me.\n\nThe formalities completed, I was shown to my suite, large and comfortably cool. Two porters followed with my luggage. They were black, gray suited, and, I noticed, were scrutinizing me carefully. When the manager left, one of the porters addressed me in what I presumed was an African dialect.\n\n\"I don't understand,\" I told him.\n\n\"You're not African?\" he asked, in English.\n\n\"No.\"\n\n\"Where are you from?\" meanwhile busying themselves with my luggage.\n\n\"Guyana.\" The look on their faces told me the name meant nothing to them.\n\n\"Where's that?\"\n\n\"South America.\"\n\n\"America. That where Mr. Bob Foster is from. Do you know Mr. Bob Foster, sir.\"\n\n\"No. Who is he?\"\n\n\"A boxing champion.\" Proudly. \"He stayed here in this hotel.\" Looking at me as if that bit of information was important and should be received respectfully. I nodded, accepting.\n\n\"You a boxer?\" he asked.\n\n\"No. I write books.\"\n\nHe left me with the feeling that as a non-boxer, I held no further interest for him. Later I learned that Bob Foster, the boxer, had not only stayed here but had been the guest of honor, cutting the ribbon which officially opened the hotel for public business. I also learned that it was no accident which brought me here.\n\nAccording to South African law, a hotel can accept non-white guests only if it obtains a special permit or license to do so, and very few such permits are issued. Non-Whites are Blacks, Asians, and those of mixed blood (Coloreds). Ironically, only the best, the five-star hotels, are licensed to accommodate Non-Whites. Native Non-Whites, of course, rarely have either the means or the temerity to use these hotels. To complicate the situation further, visiting Non-Whites are designated \"Honorary White\" to insure, it is claimed, their insulation and exemption from the many embarrassments which would otherwise attend them. I discovered that this title was first conceived to meet the special circumstances of Japanese businessmen who came to establish footholds for their companies in the South African market. They could not, like indigenous Non-Whites, be contemptuously restricted and segregated, so it was decided to \"whiten\" them for as long as they lived and worked in South Africa. Eventually, all non-white visitors were called \"Honorary White.\"\n\nOutside, it was sunny and uncomfortably hot; inside it was refreshingly cool from air conditioning and the fine mesh curtains drawn across the large windows which overlooked the street. I prowled around to familiarize myself with what would be my point of departure for the next six weeks. The vestibule was equipped with a washroom and cloak room for visitors and led into the spacious, attractive dining area. This contained a large wooden table, polished to a dazzling shine, and six matching chairs. The nearby wall was really a cupboard artfully contrived to hide a small refrigerator and shelves for pots and pans, cutlery and glassware. A room divider of simulated bamboo partly separated this from the lounge, large and luxurious and painfully overdone in green\u2014olive green carpet, paler green walls, a glass-topped center table which held a large basket of fruit, lime green upholstered furniture, pictures in contrasting shades of green, and, scattered about the room, an abundance of artificial plants.\n\nLuckily, the bedroom door could be closed to shut out the green menace from the more somber but equally lush comfort of the large, canopied bed in polished dark wood, matching side tables, highboy, and chest of drawers. Near the window was a wide writing table and two chairs with elephant hide seats. One entire wall seemed to glide away at a touch to reveal ample closet space for clothing and luggage.\n\nThe bathroom was nearly as large as the bedroom and completely lined in glistening brown tile. Twin washbowls and mirrors, a large deep bath, bidet, separate shower stall, a telephone, and piped music. Many towels were piled beside the washbowls and hung from racks near the bath and shower stall. The radio and piped music could be controlled from several points throughout the suite.\n\nSo this was the five-star treatment. It was not what I would have chosen if the choice had been mine. Whites could choose according to the dictates of their pocketbooks; visiting Blacks must pay the top price.\n\nI dialed room service for a cold drink. The young, black attendant seemed very surprised yet pleased to discover that I was black, and said something to me in a language I could not understand.\n\n\"I'm afraid I don't speak your language,\" I replied.\n\n\"You're not Zulu?\" he asked.\n\n\"No.\" I was secretly flattered at his mistake.\n\n\"Where you from?\" he wanted to know.\n\n\"South America,\" I said.\n\n\"You know Mr. Bob Foster, sir?\"\n\n\"No.\"\n\n\"He's from your country.\"\n\n\"No. He's from the United States.\" Realizing, from his expression, that the small geographical difference did not impress him.\n\n\"He lived in this hotel,\" he said. Then, smiling, \"He's a great boxer. A big champion. He beat the white man. He beat the South African.\" The smile was wide. I paid and he left. Evidently, Mr. Bob Foster had made a deep impression here.\n\nSipping my drink, I opened the curtains and looked out onto the small park which occupied the block directly opposite. It contained neatly trimmed lawns, flowering shrubs, a central fountain of concrete slabs arranged to simulate a miniature waterfall, and shade trees casually spaced around its perimeter. A tiled walkway neatly bisected this handsome park, and an iron fence enclosed it on all sides, broken only by the wide gates at each end of the walkway. Benches were scattered under the trees, and these were all occupied by young black men and women chatting together or merely dozing in the sun. Sprawled on the grass near one flowering bush were three men, two of them white and all of them unkempt, who lazily passed a bottle from one to another. Here and there were forms face down on the lawn, seemingly asleep. White men and women hurried through the park, intent on whatever their business might be; the unemployed sat in the sun, in their idleness and, perhaps, in their dreams.\n\nI took the lift downstairs and crossed the street into the park. This was as good a place to begin as any; I might as well plunge in. I walked across the lawn to a group sitting under a tree. Two men and a woman, all black, watched my approach in silence.\n\n\"Good afternoon,\" I greeted them. No sign of welcome on any face. Then one of the men responded with a slight nod and a barely audible growl. Not to be put off, I persisted.\n\n\"I'm a stranger visiting your country.\" This seemed to stir some small interest. Press on, I told myself.\n\n\"If I wasn't sure that I'd made a long trip to be here, I could easily imagine this was England. Same lawns, same trees, and same green benches.\" I waited to see some faint hint of interest.\n\n\"You from England?\" the woman asked, making it sound like an accusation, not believing it.\n\n\"I took the plane in London,\" I replied. \"Actually, I now live in the United States, but I once lived in London for many years.\"\n\n\"Yes, but where are you from?\" the woman persisted.\n\n\"South America. Guyana. That's where I was born.\"\n\n\"Bob Foster is from America,\" one man said, smiling not at me, but to the happy memory of whatever images the name Foster conjured up for him. \"You know him?\"\n\n\"No,\" I said. \"How are things with you?\" I felt somewhat intrusive but needed to establish some basis for conversation. They exchanged glances and one of the men, bald and sparsely bearded, said something in what I guessed was an African language or dialect. Not knowing what he'd said, I said nothing.\n\n\"You from Lesotho?\" the bald one asked. That surprised me because I'd already told him where I came from and I was sure he'd heard enough to know that I was not indigenous African. Maybe they were playing a little game with me.\n\n\"No. I'm from America.\" North or south was not really important at this point. The woman said something quite unintelligible, and the bald one said, \"No work,\" spreading his long-fingered hands in a wide gesture to include his companions. They were all neatly dressed, the men in dark suits, white open-necked shirts, and shoes thinly filmed with red dust as if they had done much walking. The woman, young, round-faced, and sturdily built, wore a simple cotton frock in a bright print, her stockingless feet brown and shapely in white sandals.\n\n\"You working?\" she asked.\n\n\"Yes.\"\n\n\"But now? Here?\"\n\n\"Here I'm on holiday. Just visiting,\" I replied. \"Do you live nearby?\" They looked at each other and laughed in that sharp humorless way which is both bitter and contemptuous, as if I had committed some small stupidity.\n\n\"Live nearby?\" they mimicked. \"No, we live in Soweto.\" Then, waving an arm to include the whole park, the bald man added, \"We all live in Soweto. We come in each morning looking for work and we go back each night. We don't live here.\"\n\nAbruptly he turned away and talked rapidly with his companions, their unfamiliar language shutting me out completely.\n\nThey seemed to have no further interest in me, offering no response when I said goodbye and left them to wander around the park and out into the bustling streets amid the noise of traffic and construction.\n\nAlong the narrow pavements, Whites hurrying to and fro, purposefully. Blacks moving with the stream, many of them in the uniforms of servitude\u2014messengers, maids, porters; on their faces the patient dignity etched deeply by centuries of survival. I wondered what went on behind these smooth black masks of people forced by law into the most menial of work and always under the watchful eye of police who were everywhere in view: large powerful men red-faced from the heat, projecting a certain surly contempt for everyone in general and Blacks in particular. Jackbooted, helmeted, and sometimes armed, they seemed hand-picked for the role of controlling others through fear.\n\nI returned to my hotel to make some telephone calls, contacts with friends of friends, people who might be able to tell me about various aspects of life in South Africa, and was deeply encouraged by their friendliness.\n\nI switched on the radio in my bedroom. After a few moments of music, a program was announced entitled \"Annie, Get Your Gun.\" I was about to change the station, thinking it was the old musical production, when the announcer explained that it was that week's installment of a program on guns for housewives. Fascinated, I listened to the advice on the purchase, handling, and maintenance of firearms and ammunition of various types.\n\nThe implication was inescapable. The enemy against whom the radio audience was warned, the \"they\" against whom Annie was being taught to point her gun, aim, and slowly squeeze the trigger was the Blacks, the same who cooked Annie's meals, cared for her children, cleaned her house, washed and ironed her clothes, trimmed her lawns, ran errands for her husband and provided the basic foundation from which she enjoyed a comfortable living with enough left over for guns and bullets.\n\nThat evening I made my first social call in South Africa on Helen Suzman, to whom I had been introduced through letters by a mutual friend in New York. A Progressive Party member of the South African Parliament, Helen Suzman was internationally known as an outspoken critic of apartheid. She had invited me to dine with her family and a small group of personal friends. At her suggestion, I arrived early to give us an opportunity to talk before the other guests arrived.\n\nShe met me at the door and led me through the house to a rear patio which overlooked a spacious tree-shaded lawn.\n\n\"I'm baby-sitting, so I hope you don't mind if we sit out here. I can keep an eye on my grandchildren,\" she said, pointing to two small, chubby children playing in a corner of the lawn. Tall and suntanned, she moved with an easy grace, as if completely confident of herself.\n\n\"My son and his wife are visiting from England, and one of my daughters is home from the United States. Those two are my son's children. Wonderful to have them around. Keeps me young,\" she said, smiling. \"I'm Helen. What do I call you?\"\n\n\"Ted.\"\n\n\"Well, Ted, welcome to South Africa, and I hope you see and hear enough to make the trip worthwhile.\"\n\n\"Thank you.\"\n\n\"I know a little about you. When Lillian Poses wrote me that you were coming I checked you out. Your books, I mean. From the library. The film of To Sir, with Love was very popular here. Especially the private showings, you know, the uncut version.\"\n\n\"Can't think what anyone could find necessary to cut in that film,\" I said.\n\n\"This is South Africa, my friend,\" she said. \"Can't publicize the idea of a black and a white teacher getting too chummy. Especially if one of them is a woman. Worse yet, teenage, white, girl students having a crush on a black teacher! Tut, tut.\" The smile breaking through to undermine the mock severity of her tone.\n\n\"Couldn't have been much left of the film if they cut all that out,\" I suggested.\n\n\"I wouldn't know, as I didn't see the cut version. But I do know it was very popular. People crying buckets into their hankies. It was banned for a while, you know.\"\n\n\"Yes, I heard. I even met one of the MPs who sat on the committee which imposed the ban. A Mr. Englebrecht. But that same committee later rescinded the ban and Englebrecht admitted that he and his family enjoyed the film.\"\n\n\"That's part of our problem, going around in circles where Blacks are concerned. On the one hand we promote the myth of the inferior Black while on the other we refuse to look at him for fear of discovering his equal humanity. I hear that you plan to spend some time in South Africa. How long?\"\n\n\"As long as I can bear it,\" I replied.\n\n\"Oh, you look fit enough,\" she laughed.\n\n\"I was thinking of my spirit,\" I said.\n\n\"So was I. How do you plan to move about and where do you intend to go?\"\n\n\"I've arranged for a car and driver for trips outside Johannesburg. In the city, I intend to use whatever public transport is available. I'd like to visit as much of the country as possible, particularly the Bantustans.\"\n\n\"The new name for them is 'Homelands,'\" she smiled, as if the name conjured up for her some particular irony. \"One word of advice. This is not London or New York. You can't get on any bus or hail any taxi you see. If you have a car at your disposal, use it. Understand?\"\n\n\"Understand.\"\n\n\"No point in exposing yourself to unnecessary embarrassment.\" She excused herself to step into the garden and adjudicate a minor argument between the children, returning within a few minutes. Her movements were quick and controlled.\n\n\"That's part of my dilemma,\" I continued when she returned. \"I want to avoid embarrassment to myself, but I also want to have a clear idea of what life is like for a Black in this society. I'm sure I'll learn nothing if I'm preoccupied with my own comfort and sensitivity.\"\n\n\"Being insulted and abused won't help either. If you want to know what it's like for Blacks in this society, talk to them. Ask them. They might not tell you, but ask them just the same.\"\n\n\"Why wouldn't they tell me?\"\n\n\"They might not trust you.\"\n\n\"I'll take that chance. Could you introduce me to some of them?\"\n\n\"I don't know that that will help you. Some of them talk with me, but I'm not sure that they trust me. Don't blame them. In their position I might not trust me either.\"\n\n\"In New York, I was told that you are perhaps the only White in Parliament who speaks on behalf of Blacks.\"\n\n\"You were told wrong. I speak up against repressive governmental policies. I speak against the arbitrary way in which those policies are imposed on our citizens, black and white. I speak against house arrest, banning and jail sentences for those who criticize the Government. I speak against disenfranchisement of all Blacks. Actually, I think it would be truer to say that I speak against the inequities in our society rather than for any particular group.\"\n\n\"But I heard that Blacks are more favorably disposed to you than to other Whites.\"\n\n\"You're very kind.\" Again that quick, lively smile. \"Although, come the crunch, I don't know that that would save me. Anyway, I don't think I can be much help with introductions. At this time, most politicians are busy in their constituencies getting themselves ready for the opening of Parliament next week in Cape Town.\"\n\n\"I plan to visit Cape Town. Mr. Englebrecht promised to arrange meetings with the Foreign Minister, the Minister for Bantu Affairs, and, if possible, the Prime Minister.\"\n\n\"Fine, then you'll be well taken care of. Anyway, phone me when you're there and we'll have lunch together or something.\"\n\nWe were joined by Helen's daughter, son and daughter-in-law, and Helen's husband. The children ran in from their play to be fussed over and conversation became general. I learned that the son and daughter were both living and practicing their separate professions overseas because they preferred the freer societies of Britain and the United States. Dr. Suzman, a slight, graying man, said little, yet there was an aura of strength about him. Perhaps he supplied the anchorage which secured and sustained Helen.\n\nIn time, the other guests arrived and we were introduced. Most of them were Afrikaners, members of the dominant white group, supporters of the Nationalists, the political party in power. I had no idea whether Helen had told them much or anything about me to prepare them for the encounter, but I immediately sensed their effort to appear cosmopolitan, able to consort easily with anyone. The handshakes pumped a bit too hard, the greetings a shade too hearty. The few other guests were British, that is, they were of British rather than Boer extraction and proudly English-speaking. I'd heard that there existed a wide philosophical gulf between these people, their common whiteness notwithstanding. Perhaps there is a real difference, but apart from the somewhat heavily accented English of the Afrikaners, to which my ear quickly became attuned, they appeared the same to me. White.\n\n\"Tell me, Mr. Braithwaite,\" I was asked, \"what's your impression of our country?\" A stocky, florid man in, I guessed, his early fifties, well-groomed, well-rounded, exuding an air of substance. He had been introduced as a banker, and looked the part, although his grip as we shook hands was strong and forceful and suggested he spent as much time outdoors using his muscles as indoors using his banking skills. His round, pleasant face seemed accustomed to smiling easily as if his course through life avoided the rocks and shoals which battered the less fortunate.\n\n\"I've been here only a day,\" I replied, \"hardly enough time to form an impression.\"\n\n\"But surely you have some feel of the place,\" he countered, smiling. \"You writers are supposed to possess a special sensitivity to atmosphere. You have the advantage of viewing things with both an inner and outer eye, which suggests that you see more and in a shorter time than the rest of us.\"\n\nI wasn't sure about him. The bonhomie came so easily. All I'd heard about South Africans in general and Afrikaners in particular had warned me to be wary of them. Was this one being complimentary or mocking? I thought I'd play it safe.\n\n\"I don't consider myself specially equipped to view you or anyone else, so I prefer to take time in looking.\" The rest of them were looking and listening to us.\n\n\"May I ask the same question, but in another way?\" another guest interposed. Voice, casual manner, all of a piece, proclaiming the Britisher. Perhaps deliberately so to emphasize some difference from the Afrikaners. This gentleman was tall, lanky in his baggy but well-cut clothes. Thin-faced and sad-eyed. I wondered whether he was an immigrant or a native. So difficult to tell with the British. They can remain considerably aloof from a community even if they were born in it, as if geographical locations were merely accidents of fortune with no formative influence on their ancestral character. He went on.\n\n\"Did you have some personal view of South Africa in advance of your decision to visit us?\" Even if he wasn't a native, he certainly seemed to feel at home. \"Us,\" he'd said.\n\n\"Certainly.\"\n\n\"Would you like to tell us about it?\"\n\n\"Why not?\" I decided to lay it on their collective plate and watch the reaction.\n\n\"Simply stated, it was a negative view. Some of it derived from those white South Africans, officials and others, who tried to defend your policies and were obviously uncomfortable about it; some of it from other white South Africans, mainly churchmen, whose conscience made them resist those policies and who suffered house arrest, banning, and sometimes imprisonment. But most of it came from Blacks of both South Africa and Namibia who were victimized by those policies and were lucky enough to escape. I found their stories most persuasive.\"\n\n\"And would you, as a writer, be content with that?\"\n\n\"Surely my presence here is answer enough. However, while at the United Nations I noticed that even those countries which seemed most friendly to South Africa never publicly defended her policies. Still, I am here and will try to be as objective as the situation will let me.\"\n\nHell, I didn't need to sugarcoat anything for them.\n\n\"Perhaps, while you're here, we can change your view,\" the banker said. \"Providing you are willing to subdue your prejudices. Many people from outside our country are deeply prejudiced against us without knowing anything about how we came to be what we are, how we function as a people, and the real nature of the relationship between us and the Bantu nation.\"\n\n\"Nation?\" I asked. \"I thought that, Black and White, you were all one nation.\"\n\n\"That's a common misconception.\" He smiled, assured that he spoke for all of them. \"The Blacks are a separate people, several nations, in fact. Language, customs, religions. They're not the collective group outsiders imagine them to be. I know. I grew up with them and speak several of their languages. Among themselves they are as different from each other as they are from us. Our policy, simply stated, is to respect those differences, and as circumstances dictate, preserve them.\"\n\n\"Have you decided all this for them or with them?\"\n\n\"Come now, let's be quite frank with each other,\" he admonished, still smiling. \"Our predecessors fought and conquered the Bantu and, like conquered people everywhere, they became subject people. Subject people are never treated as equals, at least not until prevailing political and economic conditions dictate such a step. The Bantu outnumber us ten to one, at least, and we cannot now or in the foreseeable future allow them any conditions or circumstances which could precipitate armed conflict with us. We must protect ourselves against them. Outsiders don't understand this. Actually, we live in fear of them.\"\n\n\"You, in fear of them? In the few hours I've been here, I would guess that the shoe is on the other foot,\" I said.\n\n\"He's right, but for all the wrong reasons,\" said Helen's daughter. Her dark eyes flashed under a short crop of brown hair, everything about her explosively vital, in marked contrast to her calm, unflappable mother. \"Of course we're afraid, but we deliberately create and maintain the awful conditions under which the Blacks live, then we watch them for signs of revolt. If there's no sign, we pressure them a little more. So it goes on. We're afraid of their numbers, but, in our fear we seem to want to woo the very danger which threatens us. It's a vicious cycle. I couldn't function in such an atmosphere, so I cleared off.\"\n\n\"We can't all exercise such a happy choice,\" the Englishman said. \"Some of us must accept the responsibility for finding a formula which would allow\u2014\"\n\n\"What formula?\" I interrupted. \"For more than a century, the Blacks have been completely disarmed, tribally dislocated, disenfranchised, and displaced. Given your economic power, your command of military personnel and weapons, the fear of them which you express seems to me at best dubious.\"\n\n\"It's not as easy as that,\" the banker interposed. \"I'm sure you appreciate that even the most sophisticated arms in the hands of a few cannot always resist the resolute pressure of an unarmed or primitively armed mob.\" The smile was there, as if he already rejected the image his words conjured up. \"However, we hope it will never come to such a bloody test. In spite of what you have certainly heard to the contrary, we are not completely against change. We welcome change, providing it is orderly. We welcome evolution, with everyone developing in his own way, at his own pace, with his own kind. It is revolution that we oppose.\"\n\nSeveral others intervened now, as if triggered by the word revolution. One elderly gray-haired couple kept determinedly out of it. From the few words they spoke, I guessed that they were Afrikaners. They seemed ill at ease and I wondered why they were there. Maybe Helen had her reasons. Maybe they simply weren't used to meeting Blacks, even one unarmed Black from overseas.\n\n\"How can you claim to favor the development of the Blacks in their own way, at their own pace and with their own kind if you reserve to yourself the right to control that way and that pace?\"\n\n\"For the time being, my friend,\" the banker insisted, \"only for the time being. Our Bantu people are not like you, educated and sophisticated... \"\n\n\"I met some in New York, petitioners against your policies. They seemed sophisticated enough for me to believe them highly educated,\" I told him. \"Some of them are products of your university system.\"\n\n\"Any glib dissident could sound off at the United Nations,\" returned the banker, with a trace of heat. \"Anyone who is against South Africa is sure of a hearing there. Our Bantu people need to be educated into the responsibilities of government. We have designed an educational system which will provide them with the necessary skills.\"\n\n\"Wouldn't they have acquired those skills in your established universities?\"\n\n\"We do not wish the black man to be a carbon copy of ourselves. Anyway, it is easy to see you have been told a great deal about us, all of it to our disadvantage.\" They were all watching me, Helen aloof from it as if she had provided the stage for this encounter and was letting it take its course.\n\n\"I told you so earlier this evening,\" I reminded him. \"It seems to me that if your claims of goodwill are genuine, you should be having this kind of dialogue with South African Blacks. Do you? I'll be here today and gone tomorrow. Why not give them an opportunity of testing your goodwill?\"\n\n\"It's not easy,\" someone else interrupted. \"Few of us know the Bantu except as employees, and fewer of us want to know him in any other relationship.\"\n\n\"That's only half of it,\" from another. \"Overtures of friendship from us are likely to be met with hostility and suspicion.\"\n\n\"Only a moment ago, you claimed the right to determine the pace and scope of their destiny, now you say you don't know them,\" I said. \"Anyway, shouldn't you take a risk with them, just as you did with me?\"\n\n\"It was no risk at all,\" the Englishman intervened. \"We know your books and they gave us a rather good idea of who you are.\"\n\n\"That may be,\" I retorted, not wishing to let him off, \"but I am no less opposed to your policies than is any local black man you can think of. I hope my books made that clear.\"\n\n\"Perhaps, perhaps,\" replied the banker, \"but for the moment you are here with us, so let's talk with you. You represent the world view of us, and we are not insensitive to that view. What many outsiders do not know is that, in our own way, we are striving to redress some of the inequities in our society. Compared to what is happening in some other places, you might notice nothing or very little, but change is occurring nonetheless. In sport, for example.\"\n\n\"Are you referring to the fact that a black American fought your white champion here recently? And defeated him?\" I deliberately added the last bit for good measure, trying to put a small dent in the armor of their secure rightness. He ignored it.\n\n\"That, and other things. Arthur Ashe was here competing in our tennis tournament. Insignificant it may seem to you, but for us it is an important beginning. Let me tell you of another advance. I have for years been a member of an exclusive club here in Johannesburg whose members were only Afrikaner Whites. English-speaking Whites were not admitted. Not long ago, on my initiative, an English-speaking person was elected, so, quietly but effectively, the old order has changed.\"\n\n\"The next move should be to offer membership to a Black.\" This was deliberate provocation, but nobody bit.\n\n\"No, the next move is to invite you to come and meet us. Come and talk with the members of my club, all of us dyed-in-the-wool, intransigent conservative Boers, as you see us. Come and do us the courtesy of hearing our point of view. Doesn't mean you have to agree with it, but at least hear us.\"\n\nThe others seemed as surprised as I was by his invitation. Shocked even. I watched the man, his eyes mischievously glistening in the smooth, smiling face. Was he playing with me, knowing I would refuse his offer? Was this a way of finally stifling my criticism? No doubt I'd been coming on a bit strong.\n\n\"I can't imagine that any useful purpose could be served by my visiting your club,\" I temporized.\n\n\"How do you know?\" he replied. \"What's your private formula for change? How do you decide what might precipitate some small change, or even some big change? Come and talk with us. I promise you we will listen courteously to your criticism of us and I hope you in turn will hear us out.\"\n\n\"Why don't you?\" Helen urged me.\n\nThe elderly, graying couple were frowning, as if they were not very happy with this turn of events.\n\n\"Perhaps our visitor prefers to condemn from the outside,\" the Englishman offered. \"It's simpler that way.\"\n\n\"That's not my concern,\" I said to him, to all of them. \"I'm asking myself why I should be the one when it would be even easier for you to invite a local Black. From what I've already heard, they speak both English and Afrikaans. If some challenge is intended, wouldn't it be better directed to one of them?\"\n\n\"Ah, a challenge is intended,\" the banker seized on it. \"You are here, you have been critical of us and I'm saying to you, 'Come and meet us in one of our strongholds.' Yes, it is a challenge, Mr. Braithwaite. From outside, you castigate us without really knowing anything about us. Now, in fact, I'm going out on a limb by inviting you, a black man, to come and justify the invitation to my brother members. You might not think so, but this in itself is a giant step.\"\n\nA friend in New York, on hearing of my intention to visit South Africa, had warned that the granting of a visa to me was no innocent act; it meant that there was a plan to use me somehow to South Africa's advantage. Now I asked myself, was this part of the plan? Had I been deliberately inveigled into something? I couldn't believe that. The invitation, challenge, had emerged far too naturally, and besides, Helen would never lend herself to that kind of sleazy plotting. I must be careful to avoid reading evil intent into everything.\n\nThe challenge was more like a dare, and he seemed to be daring himself rather than me. Daring himself to carry off another first? An Englishman, and now a Black? Perhaps I did him an injustice, perhaps he was completely sincere in all he said. But how would I feel meeting with a group of men psychologically, philosophically, and spiritually conditioned to see Blacks, myself included, as barely human and undeserving of ordinary human treatment?\n\n\"Well?\" he prodded. They all seemed to be waiting for my answer. I wondered if I was making far too much of the situation. In my work and travel in the United States, I'd eaten and talked with men who, upon examination, were no less bigoted than South Africans are reputed to be. If this man, himself a product of the environment and conditioning which nurtured the hates and fears within this society, was willing to make a gesture, should I reject it? Call it dare or challenge, what the hell? From the far distance of New York, I had cried for dialogue. Well, here it was offered, in the very heartland of racism. He'd said that native Blacks would very likely treat an overture to friendliness with suspicion and distrust. He didn't say how they would treat a dare, a challenge. I'd come this far to see and hear for myself, from anyone who would show me and tell me. Would I reject such an invitation if it came from a liberal or a Black?\n\n\"Okay,\" I said to him.\n\n\"Then you'll come?\"\n\n\"Yes. I'll come.\"\n\nSoon after we sat down to dine. The banker excused himself and left. Now the conversation shifted to other things, gas shortage, the state of the economy, the imminent elections, etc. Like any other dinner party anywhere. Relaxed with them, I enjoyed the food, the company, the talk, but from time to time would pull myself up wondering if it was all a special exercise in good manners for my benefit. Impossible. They were talking among themselves in a familiar, ordinary way. My being there imposed no strain upon them. I'd have to watch myself and not let my blackness become my own handicap.\n\n# Chapter  \nThree\n\nABOUT NINE O'CLOCK THERE was a call for me from a friend of a friend in England. I'd spoken to him earlier and mentioned that I would be dining at Helen's. Now he phoned to let me know that he'd arranged for me to meet a group of people and, if I wished, he'd come to the Suzman home to fetch me. I agreed.\n\nIn London and New York, friends had said, \"While you're in South Africa, you must meet so-and-so, a really fine person who could be very helpful to you over there. I'll give you his (her) phone number and drop him (her) a line to say you'll be in touch.\" I've never been very enthusiastic about that sort of introduction, and was even less so in the case of South Africa. My friends in New York and London were white and the introductions were to white South Africans. Were they really liberal or would they put on an act of liberalism for the sake of their distant friends? So far, Helen was everything that had been claimed for her. I'd soon find out about John.\n\nHe arrived about an hour later and I left with him. On the way to his home, he told me that the people he'd invited to meet me were all involved in the arts\u2014in an amateur way, because there was little opportunity for them professionally. The arts were still struggling in South Africa, and though there were rich veins of talent running throughout the society, too little attempt was made to tap and promote them. As we drove, he pointed out places of local interest, mostly new multistoried buildings, and commented on the elaborate highway system under development.\n\nAt his home, I met his wife, two teenaged sons, and his other guests, all men, four black and one white. I was quite surprised, perhaps foolishly so, because he'd given no hint of their color, but spoke merely of artists. At Helen's, the only other Blacks in sight were servants. Judging by the large living room, the house was commodious, but the shabbily comfortable furniture suggested a modest budget. I wondered what John did for a living; his strong handclasp in greeting had been made with a calloused hand, hinting at outdoor work. What was his relationship to these Blacks and how was he able to bring them together at such short notice?\n\nThe family and guests welcomed me. I wondered how I should approach the black men, my memory of my encounter in the park bidding caution, so I expressed my pleasure at being in their country and hoped they would take it from there.\n\n\"My name's Obie and this is James and Kebo and Molefe. We've all read at least one of your books, To Sir, with Love. When we heard from John here that you were here, we agreed to take a chance and stay in town a little later to meet you.\"\n\n\"Take a chance?\" I asked. \"What do you mean?\"\n\n\"Didn't you know? Blacks are not allowed in the city at night,\" he replied, his easy smile sugarcoating the words.\n\n\"So where do you live?\"\n\n\"We all live outside the city in a place they created for us called Soweto. But tell us, what are you doing in this place? John wouldn't give us any details.\"\n\n\"I'm just visiting,\" I replied.\n\n\"Did you know that your books were banned here?\" the one named Molefe asked. He was short, stocky, with a shiny, hairless head and tiny curls on his upper lip and chin. The total effect was a little startling, as if he would be more at home on a pirate ship with a cutlass held between his large, white teeth.\n\n\"Yes. I knew.\"\n\n\"What we've been wondering,\" Molefe continued, \"is why they issued you a visa after banning your books.\"\n\n\"The ban was lifted,\" I replied. \"There was an announcement in the Official Gazette.\" They were making me feel uncomfortable; the warmth of their greeting had quickly evaporated. I looked at John, wondering whether he had deliberately set this up. Hell, I'd never met him before tonight and he had knowingly brought me into this. Since entering his house, he'd retreated into the background, saying nothing, leaving it to the others to grill me. His wife sat on a divan with her sons, silent, observing.\n\n\"Makes no difference, man,\" Obie interposed. \"The things you say in your books remain. I read Reluctant Neighbors. No white man is going to love you for that one and no South African White would forgive you for even thinking such things. So we want to know if the ban was lifted after you applied for a visa.\"\n\n\"No. I saw the notice of the lifting of the ban and then inquired about the visa.\"\n\n\"Did you have any trouble getting it?\"\n\n\"I wouldn't call it trouble. When I learned that the ban was lifted I spoke with the South African Consul General in New York. He suggested that I formally apply for a visa to test the lifting of the ban on my books. I waited about five months after applying, then I learned that the visa was granted.\"\n\n\"As easy as that? No restrictions, no limitations?\" Obie was slim, in a neat dark suit, and soft spoken with an easy, intellectual air. He chose his words carefully, as if weighing each one to insure its fullest impact. His heavy-lidded eyes always seemed half closed.\n\n\"When the visa came through, I arranged to see the Consul General and asked whether, if I visited his country, I would be allowed to move about freely and talk with persons black and white. He assured me that I'd have no difficulty in either respect. After giving the matter considerable thought, I decided to make the visit and here I was.\"\n\n\"As easy as that?\" from Obie again.\n\n\"As easy as that,\" I answered.\n\n\"Doesn't it tell you anything, man?\" Molefe asked. \"Your books were banned. The film of the first one was first banned, then released but restricted to White-only bijous**. They ban the works of people they consider dangerous, or they consider the works dangerous, whatever way you look at it. They go to all that trouble against you and then hand you a visa. Doesn't it tell you anything?\"\n\n\"They're using you, man,\" the deep, resonant voice of Kebo intervened. He was big, his bulk further emphasized by the bulge of his belly under the loose-fitting, short-sleeved caftan. A tiny golden earring fitted snugly into the lobe of his left ear. A large, handsome man, I could imagine him a fierce Othello.\n\n\"I don't agree.\" They were getting to me, stirring up resentments I didn't imagine I'd feel against fellow Blacks. Did they think I was some kind of cretin? In New York, I'd asked myself all these questions and more. Not only about the banned books, but about my United Nations speeches and statements as well. These men were suggesting that the lifting of the ban on my books was a deliberate ploy to entice me into visiting South Africa! By implication they were crediting the South African Government not only with the highest intelligence, but with prescience as well. I didn't buy that.\n\n\"Think about it,\" he went on. \"They ban your books and your film. Okay. Now they lift the ban and give you a visa. No restrictions. Ergo, South Africa is pursuing more liberal policies, see? They let you in, a black man with an international reputation as a critic of racist and discriminatory policies. That means something, man. It's like Arthur Ashe playing in their tennis tournament and Bob Foster fighting their lily-white champion. Liberal South Africa.\" He made a brushing gesture with his large hand as if to erase an unwelcome vision.\n\n\"I'm here because I wanted to come here,\" I said. \"If your Government so cleverly anticipated my moves, so what? I'm still in charge of my own eyes and ears. I'm still in control of my own mind.\"\n\n\"Happy to hear it,\" said Obie.\n\n\"Famous last words,\" from Molefe, a sly grin pulling down one side of his mouth. Okay, if these bastards were playing some game, I'd had enough of it. I looked at John, but he refused to meet my eye.\n\n\"Where are you living?\" James asked. He had been sitting all the while in a large overstuffed chair which seemed to hold his tiny body captive. In the deepest chair, in the darkest corner of the room, he'd become inconspicuous and now I could see little more than the narrow face. However, I was grateful for his intervention.\n\n\"I'm staying at the Landdrost.\"\n\n\"That's the new hotel on Plein Street opposite the park,\" John said.\n\n\"Bob Foster stayed there,\" Molefe said.\n\n\"Do you plan to write a book about South Africa, after your visit?\" Obie asked, smiling his soft smile.\n\n\"It's very possible.\"\n\n\"How do you plan to see the country and the people? Will you just wander around by yourself or will you be shown, officially?\"\n\n\"Whichever way will help me see what I want to see,\" I said. \"I was told in New York that the Information Office would give me any help I need. I'd be very grateful if any of you can give me any leads.\" Maybe I was missing something here. Could be that these fellows were trying to be helpful, in their own way. What was it Helen Suzman had said about them not trusting me? Maybe that was it. Perhaps they'd learned to be damned careful, even with other Blacks. Well, they had a perfect right to question my motives, but I wished it could have been done in a more friendly manner.\n\n\"The Information Office!\" he exclaimed. \"So you'll be given the conducted tour and shown only what they want you to see. The white tour. Then you'll go back to where you come from and say South Africa's a lovely place.\"\n\n\"Look, I'm a stranger here. I don't know my way around, so I'll have to depend on someone to tell me things. If you don't trust the official line, why don't you help? Why don't you show me what you think I ought to see?\"\n\n\"The Landdrost is a far cry from the way Blacks live in this country,\" Kebo chimed in.\n\n\"I have no choice but to stay at the Landdrost.\"\n\n\"Why don't we cut the shit and tell our brother what it's really like to be black in this place. If he's willing to listen. After all, he's come to see us, so let's tell him what it's like to be treated like shit in the land of his forefathers.\" Kebo stood up, looking large and threatening as the light caught the shiny smoothness of his massive forearms. \"I read your book, my brother. It hurt you when you couldn't get the job you wanted, because of your black skin. You think that's something? Here you won't even be allowed to apply. Here, no Black would dare raise his ambitions that high. Any job higher than shit carrier is reserved for the white man. By law.\"\n\nReaching under his caftan into a pocket of his trousers he produced a flat, worn little book and flicked it open before my face.\n\n\"This is what every black man and woman is reduced to in this place. This thing. It governs our lives. Because of it, you're nothing. Without it, you're less than nothing. Man, you could leave your country thousands of miles away and come here, just because you wanted to see how we live! All you needed was a visa. We can't move a single step without this thing, day or night.\"\n\nI wanted to take a closer look at the thing he held under my nose, but thought it unwise to interrupt him. His anger was all the more powerful because it was so controlled.\n\n\"Listen, brother,\" he said, \"John got in touch with us today and told us you were in town. We wanted to meet you, to meet a black brother who can come and go as he pleases, write as he pleases, think as he pleases. But when we meet you, we realize how it is possible to live differently from the way we are. We give you some shit because we are angry at the difference between you and us. Christ, even to meet you we have to creep about in the dark like criminals. We are here in John's house. When it's time to go, do you think we can just walk out the door? No. However late it is, John will have to take us out of the city to Soweto where we live.\"\n\n\"Why?\" I asked, directing the question to John, wanting to hear from him about it. Before he could reply, Molefe said, \"Any Black found in the city after eleven o'clock at night is in trouble. The police cruise around in vans looking for Blacks. Only a few house servants or restaurant employees or watchmen are allowed here at night; they have special permission obtained by their employers. For any other, it's into the van and off to the police station.\"\n\n\"Even if you have one of those books?\" I pointed to the one which Kebo still held in his hand.\n\n\"Yes. Even if you have the Book. That only permits you to be in the city by daylight. Not at night. It's called the Book of Life. Here, take a look at mine.\"\n\nHe handed it to me. It was a thin group of printed forms stapled together inside a black leatherette cover and arranged alphabetically as follows:\n\nPage A. Residential address:\n\nPermit to be employed in Johannesburg daily from 8 A.M. to 11 P.M.\n\nPage B. Reserved for monthly signature of employer.\n\nPage C. Poll tax stamps.\n\n(Poll tax must be paid by June of each year.)\n\nPage D. Homeland tax stamps.\n\n(Rated according to individual's earnings.)\n\nPage E. Bantu Labour Health Regulations.\n\nPage F. Driver's license.\n\nPage G. Reserved for Arms License.\n\n(This is a mockery as no arms licenses are issued to Blacks. Any Black found in possession of even a penknife is liable to arrest and prosecution.)\n\nPage H. Personal particulars, including those of wife and children, if any.\n\nPage I. Reserved for photograph of individual.\n\n\"Think of it, my friend,\" Molefe continued. \"You walk free. Every one of us Blacks, from the age of sixteen, must carry one of these at all times. Without it you have no identity, no life, so you spend your life safeguarding it.\"\n\n\"Do you know what's one of the most humiliating acts of my life?\" Obie said. \"Getting my Book signed each month. It says that it should be signed by the employer, but that really means that any white man can sign it, and the signing is usually assigned to the most junior white in the job. Boy, and do they shit on you! They're so happy to have someone below them, they make you crawl for that signature. And I crawl, my brother. Me, who would love to take them by the throat, I crawl for that signature. In your book, Reluctant Neighbors, you talk of pride as if it is every man's birthright. Here the black man has no birthright, not even the pride in being a man.\"\n\n\"You asked me earlier how I planned to move around and learn about conditions. I'll see what the officials want to show me, but I'd like to hear from you and other Blacks, too, about conditions as you see them, if you'd be willing to talk to me and show me.\"\n\n\"Only if you're prepared to come and see us where we live,\" Molefe replied. \"We can't come often to John's house. Too risky for him and his family. Before you know it the Security Police will be on to him for consorting with Blacks. They'll think we're plotting something. In any case, it would be better to show you how we live than merely tell you about it. Think your stomach can stand it?\"\n\n\"Certainly. If yours can.\"\n\nJohn's wife and children served some cool drinks and the discussion switched to other things. Mainly writing. Poetry. Now I discovered that the white guest, Brian, was involved in publishing and promoting the works of black authors and poets in South Africa. Without a white person to help, Blacks had no access to the publishing houses. Obie, whose recently published book of poems had been very well received, was particularly bitter about this.\n\n\"In every way, at every turn, we're made and kept dependent on the white man. Brian here's okay, but why should we have to need even him? Whites come along and claim to be interested in our poetry, novels and plays and promise to act on our behalf. Then they promote our work for their own benefit, they complete the negotiation without a word to us, they give us what they choose. They know we can't fight them in the courts. God, let our day come!\"\n\nThe Whites seemed quite unmoved by these outbursts as if they'd either heard them all before or were confident that they occupied a separate and different place in the Blacks' regard. Perhaps people like John, his family, and Brian represented a bridgehead of interracial trust and understanding. Maybe there were others like him. John's surname suggested that he was of Boer stock. His hands were rough and calloused by hard work. How did he manage to win the trust of Blacks like these bright, intelligent men? How deep and real was his liberal stance? His children seemed comfortable in the company of Blacks, and children of that age are usually an excellent barometer of a family's racial attitudes. Christ, there was so much to be learned and so little time. I'd planned to stay six weeks and already that seemed too short.\n\nOn my way to the hotel that night, driven by John's wife, I saw two of the police vans making their rounds; mobile gray boxes already heavy with the nightly haul. There, I thought, but for the grace of God...\n\nNext morning, I was up early\u2014the noise of traffic from the street below nudged me awake. I showered, breakfasted, and I decided to take a stroll, perhaps to window-shop for some small memento of my visit to this city. At a street intersection near the hotel, a police minicar stood by the curb. I crossed the road, giving no more than a fleeting glance to the burly, red-faced officer at the wheel, and made my way slowly down the block, pausing outside a fruitstand to admire the racked display boxes laden with luscious fruit. Yellow mangoes, dark purple plums, large tight bunches of black grapes, red flecked yellow peaches nestling in pockets of soft paper, pears, grapefruit, bananas, nectarines, all looking so fresh and delectable. Two dark-skinned men, Indians, waited courteously on the customers. I wondered if the Indians owned the shop or merely worked there, and promised myself to buy some of the fruit on my way back.\n\nA little farther along, I stopped to watch a construction site across the street. Blacks and Whites pulling and carrying, hammering and drilling amid the noise and bustle, the towering naked girders and the swinging crane cables. From where I stood, there seemed to be harmony on that job, the natural, active interdependence I'd observed on building sites in New York, London, or Paris. I'd inquire about it. I noticed that the policeman was approaching on foot from the right, jackbooted and helmeted, the leather thongs from a thick club dangling beside his right leg, his face anonymous behind large dark goggles.\n\nHuge, powerful, and casual, he seemed to be walking directly toward where I stood. I wondered if I should move out of his way, but quickly rejected the idea. Hell, the sidewalk was wide enough for both of us and more. In New York, black petitioners had told of brutal treatment at the hands of the South African police. One had said to me, \"The police come along and toss you into their car and take you to the police station. They ask you questions and you must remember to say 'Baas,' each time you answer. If you don't, they beat you across the mouth with their short clubs to teach you how to speak to a white man.\" Blacks aren't allowed to say Yes and No. They must always show proper humility to the white Baas. Fascinated, I watched him approach. As he came nearer and nearer, I felt nervous, fearful flutterings in my gut and sudden perspiration ran coldly down my armpits. Good sense told me to move, step back or forward, but stubbornness made me stay where I was.\n\nI felt afraid, awed by the towering faceless size of him, until it was too late to move; I braced myself for the inevitable crunching contact\u2014but suddenly, nimbly, he sidestepped away from me and continued his slow, deliberate way until he was out of sight.\n\nLater, I mentioned the near-incident in a telephone conversation with a friend who advised me to do the safe thing in the future and give way, unless I wanted a brutal beating with no redress. He reminded me that the policeman would have no way of knowing that I was an overseas visitor and would merely consider my behavior another instance of \"kaffir cheek\" which deserved whatever it got, and no one, Black or White, would intervene on my behalf. I could easily be hauled off to the nearest police station and humiliated before any attempt was made to identify me, my unfamiliar accent notwithstanding.\n\nI paid a courtesy call at the Office of Information to let the officials know I had arrived in Johannesburg and to find out what formalities I would have to go through to achieve my purpose, learning about the lives of Blacks. I was courteously received and told that the information offices in every city would be happy to facilitate my inquiries in every way. No formalities were necessary except when I wished to visit the black enclaves in or near the major cities. For entry into these a special police pass was required. This pass, I was told, was intended for my own protection; the crime rate in those enclaves was very high, and the authorities were concerned for my safety. I thanked the Information Officer and expressed the wish then and there to visit Soweto, the largest black township within the immediate environs of Johannesburg. I was promptly given permission, together with a guide, an employee of the Information Office, and promised a comprehensive look at every facet of life in the black community. I was told that my guide, a blonde young woman, was very knowledgeable about Soweto and would be able to answer any question put to her about the township and its inhabitants. A white guide to inform me about the living conditions of blacks in a Black enclave in which not even a single White lived or was allowed to live? What would she really know? I'd wait and see.\n\nSoweto is the largest of several black townships within the jurisdiction of Johannesburg, about fifteen miles outside the city and far enough away for the Whites not to be offended by its ugliness or threatened by the violence which frequently erupts there. It is situated in a natural hollow, the inhabitants restricted to an area of approximately thirty-four square miles. There is only one road in or out, wide and hard-surfaced to the edge of the township, and readily blocked off and controlled if necessary.\n\n\"What's the population of Soweto?\" I asked my guide as we stopped briefly on a rise overlooking the township, the low, tightly packed, box-like white houses glimmering in the sunny heat, and reminiscent of the packed graveyards between Manhattan and La Guardia Airport.\n\n\"About six hundred thousand.\"\n\nShe had the figure pat and ready for me. Hadn't I heard somewhere that Blacks in South Africa did not vote and no official census was taken of them? As if it had served its intended purpose, the hard-surfaced road ended abruptly at the entrance to Soweto, opposite the huge General Hospital. From there on into the township, the road was pitted and worn, with deep ruts holding water from some recent rain. Now we came upon row upon row of prefabricated four-roomed concrete houses, built closely together and separated from each other by a narrow grassy alleyway into which grew a few trees or shaggy shrubs. Some of these houses were roofed in concrete while others had corrugated galvanized metal roofs which caught and reflected the sunlight. Each house was fitted with four small windows and a door. I guessed that there was a window for each room, but those gleaming roofs worried me.\n\n\"What's it like inside?\" I asked my guide. \"In this heat it must be really awful.\"\n\n\"Not really,\" she replied. \"They're designed to stay cool in summer and warm in winter.\"\n\n\"How?\"\n\n\"Something to do with the way they're built,\" she replied lamely.\n\n\"Have you ever been in one?\"\n\n\"No, but I've talked with some builders on the project.\"\n\nWe stopped the car so that I could take a closer look at one of the houses. I saw no sign of electric cables or the familiar exhaust outlet which indicated internal sewerage.\n\n\"What about electric lighting?\" I asked her.\n\n\"Most of the houses are fitted with electricity,\" she replied. \"Some of these older ones are without, but the plan is to extend it to all of them. The houses in this section are among the first built in Soweto. You can tell that because many of them have the old concrete roofs. The locals call them sleeping elephants. The newer ones are a real improvement. They have electricity and running water. The older ones have outdoor water taps.\"\n\nTen or fifteen years earlier, she went on, Soweto was a terrible slum and the Government, in its concern for the welfare of the inhabitants, had embarked on a comprehensive rehousing scheme. From time to time, the scheme was revised and improvements included. But inevitably housing needs outstripped the pace of construction. As new houses were built, the slums were bulldozed out of existence and their occupants relocated. The houses could be bought or rented. When they were bought, the purchase related only to the building, not to the land. In Soweto and similar black townships, the house may be bought but the land on which it stands can only be leased. The length of such a lease is usually less but never more than thirty-five years, and this period may be extended or not, at the discretion of the authorities. If an extension of his lease is denied, the black lessee has no hope of appeal. My guide supplied these interesting pieces of information matter-of-factly. There was nothing bitter or vengeful about her statements or observation; she was merely providing information on a state of affairs which exists, and she was in no way personally involved.\n\nTenants, she informed me, fared no better. A house is rented to an individual who occupies it with his family, usually a wife and three or four small children. That is the \"official\" family. Because there are not enough houses to accommodate in comfort even sixty percent of those needing shelter, subletting is encouraged and practiced. The tenant benefits little from this, however, for subtenants must pay their rent, not to him, the \"official\" tenant, but to the landlord, the city of Johannesburg, through its local agent. It can be assumed that there is hardly a house in Soweto without its quota of subtenants, so considerable revenues must accrue to the city over and above the basic rents anticipated for the scheme.\n\nI asked her if she could arrange for me to look inside one of the houses, but she merely smiled at that. Actually, I could not see her approaching any of those black residents. Many of the women interrupted their chores long enough to stare at us, my blonde guide and me. I wondered what they thought of us.\n\nAs the tour progressed, there was no escaping the drab sameness of the houses, the garbage-littered streets, or the few shoddy shops. Groups of youths sat outside the shops or wandered about aimlessly. My guide explained that the schools were still out for the long Christmas vacation. She pointed out what she called some of the special advantages of Soweto. Picnic grounds, a pleasantly green though unkempt oasis; a large football stadium where all the main outdoor social and athletic events, such as boxing matches, were held; a nursery school for children of working mothers; the empty Soweto High School. We pulled into the high school yard and I peeped into a classroom through a broken window. Row upon row of dusty wooden desks, the walls unrelieved by even a map. Gloomy.\n\nOur tour continued along roads now generally tar-surfaced and comfortably passable and we stopped at the only vocational school in Soweto. About two hundred youths annually, as many as the school can now accommodate, are selected out of more than a thousand who have passed a qualifying examination, and are taught the rudiments of electrical wiring, plumbing, bricklaying and masonry, and carpentry.\n\nThe school's principal was an Englishman long resident in South Africa, and, like most school principals, complained of the acute shortage of basic equipment, materials, and textbooks, in spite of which the youths were making extraordinary progress. I saw some of the models made by the students and some of their drawings, and they compared very favorably with work I'd seen by design students in well-equipped classrooms in London and New York. One student's work was so outstanding that a visiting Swiss diplomat had given him a very expensive watch in encouragement.\n\nThe Principal said that, given the opportunity and further training, the black students could excel in the building and other industries which are clamoring for skilled labor. Unfortunately, they are victimized by South Africa's \"job reservation\" laws, by which all skilled and some semi-skilled jobs are reserved to Whites. A bricklayer, plasterer, or electrician must be white. The young black students, ambitious and enthusiastic while training, face a very frustrating future. They are likely to be employed as low-paid helpers to Whites less skilled than themselves and might even do the work without receiving the pay.\n\nThe Principal told me that present building needs have forced some builders to let Blacks do skilled work, even at the risk of prosecution. Reflecting a booming economy, contractors are enjoying their busiest times and there is an acute shortage of skilled white labor. There are many Blacks on their payrolls fully capable of skilled work without supervision. To meet their pressing deadlines, the contractors put the Blacks on skilled jobs and keep legal representation readily available to deal with such prosecution and fines as are incurred. Legal fees and fines are prorated into each building estimate. The Principal hoped that, eventually, the job reservation laws would crumble under pressure from public need for housing and the industry's need to expand.\n\nWe now drove through the so-called elite section of the town. Most of the homes here were attractive bungalows surrounded by neatly trimmed lawns, with flowering shrubs and fruit trees. These were the homes of Soweto's tiny \"black bourgeois\" community, the local doctor, dentist, grocer, gas-station operator, etc., all of whom had struggled and saved to rise above the depressing sameness. Each of them had begun by buying the government-built four-room square structure and added rooms to it as they could. They had had to install at their own expense running water, plumbing facilities, electricity, and whatever other household devices they could afford. All this on a flimsy lease which could be rescinded at the Government's whim.\n\nIronically, my guide spoke of the bungalows and their owners proudly, as if those people had been specially \"allowed\" to achieve that much, her voice crisp and objective as if she were speaking of cold, inanimate things, not insecure human beings who were forced to live in fear that one fine day the dreams they'd earned would be snatched away from them. I thought of myself, my own pride in ownership of a home thousands of miles away, my security in the knowledge that I had the right to defend it against all comers, supported by the full weight of the law.\n\nMy guide now promised a big surprise and we drove to the Bantu Council Building. It was much more than a surprise, the sight of that modern red-brick building, graceful in its simple lines against a dramatic background of neatly trimmed lawns and darkening sky. A macadam driveway circled in front of the building before coming to rest at the base of a wide wooden stairway which led upward to carved wooden doors. A uniformed doorman led us inside and then hurried away to find the Secretary of the Council. My guide proudly showed me the large Council Chamber, paneled in wood and thickly carpeted, and the smaller offices of the President and Secretary of the Council.\n\nWe found the Secretary in his office in conversation with someone, so we waited for him in the Council Chamber. My guide told me that members of the Council are mainly drawn from the small business community of Soweto. The Council is supposed to oversee Soweto's health and educational and social welfare, and make recommendations to the white Johannesburg Council which has the final decision as to which, if any of them, are expedited.\n\nWhen the Council Secretary finally joined us, he proceeded to give me a very careful review of the Council and its work. He seemed primarily concerned with impressing my guide whom he knew to be a Government official. Smiling broadly, he invited me to ask him questions.\n\n\"How much freedom can your Council exercise in the management of Soweto's affairs?\" I asked.\n\n\"Well,\" glancing nervously at my guide, \"we have a pretty free hand. We're on the spot, we know what the township needs, and our recommendations are generally honored.\" Nodding his head affirmatively all the while.\n\n\"Does the Council collect the rents on the houses?\"\n\n\"Yes.\"\n\n\"Would you tell me about subletting and how it works here?\"\n\n\"Well, I couldn't go into that. That's the Council's business. I can't discuss that.\"\n\n\"I understand you maintain supervision of the schools.\"\n\n\"Yes.\"\n\n\"I've just been looking at your high school. From the outside\u2014\"\n\n\"The schools are still out for the Christmas holidays,\" he interrupted.\n\n\"I know. But could you\u2014\"\n\n\"The Chairman of the Schools Committee would be the best person to talk to you about the schools.\" Again he interrupted me, anticipating my question, meanwhile looking at my guide as if to assure her that he would say or do nothing contrary to official policy.\n\nUnexpectedly we were joined by a little man, hardly over five feet tall, shiningly bald, and spry. On being introduced to me, he seemed surprised.\n\n\"I thought you were a plainclothes policeman,\" he said. \"I was planning to ask you to help me get a new pass.\" Grinning meanwhile.\n\n\"At your age, why would you need a pass?\" I asked him.\n\n\"Every black man needs a pass,\" he replied, the smile vanishing. \"I am a member of this Council. I live and work here in Soweto. Been here nearly all my life. I'm seventy and still I need a pass.\" His watery eyes staring balefully at my guide, he continued, \"Blacks are not human, so they need passes to move among the humans. What about you?\"\n\n\"I'm a visitor from overseas. This is my first visit to your country and this lady is guiding me around Soweto.\"\n\n\"Before people try to guide others, they should try guiding themselves,\" he replied, looking at me. \"How can you guide when you don't know Soweto yourself? Blacks live in Soweto. Only Blacks. They're forced to live in Soweto. They know what is Soweto. The white man comes here and says to us, Come. Go. Fetch. Carry. Live. Die. Show your permit. Show your pass. That's all the white man knows about Soweto. Busloads of white tourists drive through the township with somebody in the bus showing them how the Bantu live. Somebody white, from the Information Office. They say, 'Look at the Bantu, how happy they are in Soweto. Look at them smile. Look at the happy children playing football. Look at the happy old men drinking Bantu beer.' Guides!\"\n\nHe made the last word sound like an insult, speaking his mind, careless of any effect it might have on the now pale white woman. The Council Secretary nervously wet his lips from time to time as if preparing to intervene, but the old one seemed beyond caring, beyond fear. Perhaps, I thought, he has finally come to terms with himself, his life and his dignity, and has decided to make his stand.\n\n\"You want to see Soweto, come to us,\" he told me. \"Come as a brother.\"\n\nI apologized for the impromptu visit, saying that my stay in Johannesburg was short and I'd taken advantage of the opportunity provided me to see his township. But he would not be pacified.\n\n\"If you want to know about us, make time. Don't tell me you have too little time. You're one of us, black like us. You do not need any White to tell you about us or show you how we live. We'll make time to see you, talk with you. Let us know when you can come, but come. We need to meet our brothers from far away. You've come this far, don't tell me you have no time.\"\n\nI felt humbled and promised that I'd make the time to be with them. Somehow. He was good for me. I felt elated, and at the same time, reminded of my priorities.\n\nThat was the end of my guided tour. On the way back to Johannesburg, my guide and I talked, but desultorily. She seemed to have lost much of her enthusiasm. At my hotel, there were telephone calls for me from a local newspaper, the Johannesburg Times, seeking an interview, and from a black poet I'd met. I returned the Times call and agreed to be interviewed, then called the poet and, in passing, mentioned that I'd just made a guided tour of Soweto. He laughed at the idea of the white guide and suggested that it was a deliberate ploy on the part of the Office of Information to keep me away from the inhabitants of Soweto. He himself offered to take me there or anywhere else so that I could really meet the people. I told him that I'd been warned not to go into a township without a permit, but he brushed that aside, asking who the hell would know the difference. I'd be a black man in a black township. \"They say we all look alike, don't they?\" he laughed. I agreed to take the risk and go with him.\n\n* Cinemas.\n\n# Chapter  \nFour\n\nON THE APPOINTED DAY, we met in front of the hotel and drove to Alexandra, six miles outside of the city in the opposite direction from Soweto. We drove through lovely suburbs of wide, clean streets and charming villas surrounded by neat lawns and carefully nurtured hedges and the ubiquitous blue-tinted swimming pool. All along the route were the separate bus stops for Blacks and Whites.\n\nMy first impression of Alexandra was of a garbage dump. Everywhere the garbage was piled as if the inhabitants had long given up the struggle to remove it and just let it accumulate. Where Soweto had roads and drearily similar box-like houses, Alexandra had a jumble of narrow, garbage-clogged foot paths worn out of the naked earth by decades of footsteps, intersecting with shallow gullies which wound their way erratically here and there until they were lost in sudden overgrowths of weeds. What had once long ago been neat houses had deteriorated into dilapidated wrecks patched with tin, cardboard, or even strips of plastic, their squalor emphasized by the uglier little tin outhouses scattered around them. In the middle of all this, two buildings rose ten or twelve stories into the air, straight sided, red-bricked, and looking clinically functional, as if contemptuous of the squalor though firmly anchored in it.\n\nThese were the hostels, one for men and one for women, built to house cheap black labor necessary for the numerous manpower-hungry industrial projects which are mushrooming around Johannesburg. They were designed to accommodate the largest number of workers in the least possible space, and are a honeycomb of tiny, cell-like rooms. Cold running water and toilets are provided at one central location in each building.\n\nMost of the black workers in Johannesburg and its environs are young men born in ghettos like Soweto and Alexandra; others are migrant workers from the Bantustans of the Transvaal, Transkei\u00ad, Zululand, and other outlying territories, or immigrants from Rhodesia, Botswana, Lesotho, Swaziland, and even Mozambique. These migrant and immigrant workers are not permitted to travel with their wives, and see them for only a short time each year, at Christmas, when they are given leave to return home.\n\n\"Well, what do you think of all this?\" my companion asked.\n\n\"It stinks,\" I replied, meaning the garbage.\n\n\"What about those?\" pointing to the hostels.\n\n\"At least they're an improvement on the tin shacks around them.\"\n\n\"You think so? Talk to some of the fellows who live in them. Best that can be said of them, they have electricity and running water. They'd never house white men in places like that.\"\n\n\"Could I take a look?\"\n\n\"Doubt it. They don't encourage outsiders. We could peep in if you like. Many of the ground floor rooms have broken windows and nobody seems to be in any hurry to repair them.\"\n\nI declined, not wanting to intrude on the hostel residents, but my friend led the way through the weeds and stunted trees to the base of one of the hostels.\n\n\"This is the women's unit,\" he said. \"All these lower rooms are empty. Peeping Toms and things like that, you know. And besides, most of the women stay in the hostel only for a short while then try to find jobs as domestics with the chance of living in their employers' houses.\"\n\nI looked through one of the broken windows. The narrow room contained a small iron cot with a thin, plastic-covered mattress. A rickety wooden table completed the furniture.\n\n\"The men's hostel is always filled to capacity, with a waiting list of others wanting to get in,\" he said.\n\n\"Where do they live while they're waiting?\"\n\n\"You really want to see?\"\n\nHe led me back toward the shacks and we had to pick our way over piles of garbage and around a partly enclosed but uncovered hole which evidently served as the communal lavatory. There was a water spigot a few yards away. We entered one of the ramshackle houses and he knocked on an inner door, then pushed it open to let us into a pitiful room about eight foot square. Although we had come in from bright sunlight the room was in near total darkness; the only window was tightly sealed with burlap. A young man crouched by a single lighted candle, eating something with his fingers from a metal pot. My companion made the introduction, but, in the circumstances, no attempt was made to shake hands. I could not see the young man's face clearly in the prevailing gloom and I knew he could not see mine. He sat on the edge of a cot, one of three which ringed the room. He told us that six of them, five other men and himself, lived there, sleeping two to a cot. I tried to imagine what it was like.\n\nThe young man finished his meal, wiped his fingers with a piece of paper and stood up. I saw that he was neatly dressed in short, sharply creased slacks and shiny shoes. He said he was employed as a clerk with a local engineering firm, having graduated from the Orlando High School in Soweto. He suggested that we leave as he had arranged to meet some of his friends nearby. As we were leaving, I noticed that another of the cots was occupied, someone making the most of having the entire cot to himself. No electricity, no running water, no sewage facilities, no privacy, no sunlight, no air.\n\n\"Christ!\" the word slipped out.\n\n\"Getting to you, eh?\" my friend said. \"I sometimes read about how you black Americans riot because of your living conditions. We'd trade with you any day, and think ourselves lucky. How long do you think you'd remain human in a room like that? You can hardly close your eyes before there's someone wanting to stick it up your ass. Hell, no women available, so what can you expect. The women who live here stay indoors at night. Rape around here is less a crime than a daily hazard.\"\n\n\"Don't these fellows ever take any action?\"\n\n\"Action? What action? In this country you work or you starve. If you have a job you hang onto it because you know that there are several others just waiting for you to slip. Action? You mean like striking? Shit, they'd throw you in jail so fast! Don't forget you're talking about Blacks.\"\n\nGod, no wonder the white guide had kept far away from this place! These black men and women actually had to pay to stay in stinkholes like this. Somebody was making a fortune out of all this misery and it wasn't the Blacks. They could not own property, thus could not be landlords. So it had to be either the Government, through its Bantu Councils, or private industry, growing fat on Government contracts.\n\n\"We welcome evolution but we are opposed to revolution.\" The banker had repeatedly chanted the Government's slogan. So had the MP, Englebrecht. Didn't they realize that it was in places like this that revolutions were born and bred? Maybe they'd never seen sights like these, even though they festered right under their very eyes. In other places, others had been similarly blind and uncaring until someone had rubbed their insensitive noses in the shit.\n\nMy friend led me on through the darkening township. I felt that he was slyly pleased at the way I had been affected by the hostel visit, and how carefully I was stepping around the mounds of garbage in our path. People sat on the stoops of the shacks chatting with each other, seeming unmindful of the ugly chaos around them.\n\n\"How would you like to spend a week or two here?\" He was smiling. Laughing at me.\n\n\"Not for anything,\" I answered. I wanted to get away from there, away from the stench, the dilapidation, the all-pervading air of decay. It was getting me down. I couldn't understand how he could be so at ease, so comfortable. Then I remembered this was what he wrote about in his poems.\n\n\"A bit different from your guided tour of Soweto, isn't it?\" he grinned. \"The Information Office never brings tourists down here. No smooth roads for the cars, no fancy playgrounds for happy, smiling black children. No Government show pieces. All we have is what you see. Decay and death, and we're forced to live in it. Nowhere to move to, and even if we found somewhere better, how the hell would we get permission to move? You saw what the inside of that hostel unit looked like? Some rooms in these other places are worse. Much worse. And people live here and rear their children. Right here in these miserable holes. Christ Almighty, it's inhuman!\"\n\n\"I agree,\" I told him.\n\n\"You agree!\" He suddenly turned on me, the thin face tight with anger, a trickle of spittle escaping his mouth. \"You agree! That's mighty big of you, my friend. But in a few minutes you'll walk away from it, back to your fancy hotel. I suppose you'll take a nice hot bath and wash away every memory of this stinking slum. You agree! That's nice. That's very nice. We agree too, but we still have to live in this shit. And pay for the privilege. Do you realize that? We pay rent to live in these stinking, rotten holes. Come with me, man. There's something else you should see!\"\n\nWith that he started off down a narrow alleyway between some shacks, not even waiting or looking back to see if I was following him. I hurried after, not daring to risk losing him. He led me beyond the shacks, across an open piece of uneven ground where some kind of dwelling had been bulldozed away, and on to another group of run-down houses, rotting and ready to cave in on themselves. Outside one of these, a very old woman sat in the middle of some cardboard cartons and paper bags filled to bursting with rags. Here my friend stopped and pointed a thin arm.\n\n\"Look at her, my friend. She's too old to clean the white man's house or mind his children, so she's discarded, useless as the stinking stuff around her. She can't pay the miserable rent for that shack, so they've thrown her out of it. Look there.\"\n\nHe pointed to the heavy padlock on the rickety door. Unimpressed, uncaring, the aged one sat, staring at nothing in particular, her eyes red and rheumy, her lined face set in final resignation, showing neither pain nor anxiety nor interest in whatever the next unhappy step might be.\n\n\"What will happen to her?\" I asked.\n\n\"If she's lucky she'll die soon,\" he replied, bitterly. \"Maybe someone will take her in for the night. There are lots like her, the white man's garbage. I can show you some more, if you like.\"\n\n\"No, thank you.\" I was becoming thoroughly irritated with his sneering and his jibes. I'd not created these ugly situations. He'd invited me to come and see, and now he was treating me as if all this was my responsibility.\n\n\"Why don't you take her in?\" I asked him, striking back.\n\n\"Me? Take her where? All I have right now is bed space, and I was damned lucky to find that.\"\n\n\"So we'll both walk away from her, won't we?\"\n\n\"Yes, my friend, we'll both walk away. But I won't walk far. I can't walk far. I'll always be near enough to see it and hear it and smell it. Every minute of the day it is with me. So I write about it. Me and others like me. We write about the things that hurt us and degrade us, but unlike you, we have no outlets for the things we write. Shit, man, even there we need the white man, and how he exploits our need! But, let's get the hell out of here, if you've had enough.\" Again he was smiling.\n\n\"I've had enough.\" In silence we returned to Johannesburg.\n\nThe next afternoon I went to Dorkay House, a center for the arts in downtown Johannesburg, where I had been invited to hear some black musicians give a private performance for a visiting white American impresario. I was there early and wearily walked up six flights of stairs to a narrow room, with a raised platform at one end, in front of which were rows of metal chairs. The small audience, most of whom were already seated when I arrived, was mostly African with a sprinkling of Whites and Indians who all seemed to know one another. Before the performance began they called to each other in familiar terms, the way artists do everywhere asking about mutual friends, their whereabouts, and whether or not they were working.\n\nThe first group to perform, the Batsumi or Hunters, consisted of a flautist, a saxophonist, a pianist, a guitarist, a vocalist who doubled on a huge bongo drum, and two drummers, one who sat enthroned among a glittering assortment of drums and another who beat dexterously on twin, supported kettle drums with padded drumsticks. Two of this group, the kettle drummer and the guitarist, were blind.\n\nFrom the moment the performance began, it became evident that this was no ordinary group of men. They seemed to enter an immediate dialogue with each other, the pianist provoking the conversational pattern which the others took up, shaped and shaded as their impulses and instruments dictated. With any group, conversational patterns form and re-form. So it was here. They played, or rather they spoke and sang with and through their instruments, many instruments and sounds completely integrated, blending with each other. At times, the flute and saxophone were in private communion, whispering to each other yet providing a variable obbligato to the insistent piano and plaintive guitar. From time to time a musician would break into song, his wordless sounds giving the music an additional, strange dimension.\n\nThe flute seemed to be made in two sections, that which contained the mouthpiece being several inches shorter than the shank which held the spaced apertures. Caught in an occasional frenzy of expression, the flautist would pull one part from the other, and, using the palm of one hand as a mute, produce from the truncated instrument extraordinary sounds. At other times, he would blow through the lower part as if it were a trumpet, muttering into it at the same time, this creating a hoarse sound, simultaneously strange and familiar.\n\nFrom a young woman sitting near me, I learned that they were all from Soweto and were only spare-time musicians; they earned their living as messengers, garage helpers, gardeners, and watchmen. Occasionally they made a recording, but they never received enough from such a venture to make any real difference to their impoverished state.\n\nThe second group, the Alan Kinela Quartet, used the same drummers to provide background for a tenor saxophonist and an electric guitarist. Their music was less introspective, more in the familiar, traditional jazz idiom. After a few numbers, this group joined with the first and played Stumeleng (\"Be Happy\"), a lively, exuberant piece from their common tribal heritage.\n\nAt the end of the concert, I stayed to meet the musicians. I expressed my delight at the power and joy of their music, all the more impressive in the face of the white South Africans' determined attempts to humiliate and degrade the black man. The vocalist-bongo drummer spoke for the others.\n\n\"How is it where you come from?\" he asked me. \"Where is this place Guyana? In Africa? Where?\" His face dripped perspiration which he occasionally scooped away with a forefinger. A handsome young man of medium height, filled with energy which seemed ready to erupt out of him. I told him where Guyana is, pinpointing it on that other continent.\n\n\"Tell me about the people,\" he said. \"Are they all black, like you? Tell me about them.\" Looking me up and down as if to discover any difference between him and me, them and me.\n\nI told him briefly something of my people. He then asked how long I'd been in Johannesburg, how much I'd seen of the city and the black people and what was my general impression of their condition. Before I could reply, he held up a broad, thick-fingered hand and warned me:\n\n\"When you talk about my country, don't pity me. Look at us.\" Here he moved closer to me as if to emphasize that we were of approximately the same height.\n\n\"Talk to us as one of us. I will tell you how I live here and you will tell me about life in your country. I will tell you that I am deeply dissatisfied with the conditions of my life here and perhaps you will tell me that you are dissatisfied with conditions in your own country. We black men have been here for thousands of years. We have learned how to survive the heat and the floods and the drought, the hunger and the times of plenty. Now we must learn to live through slavery, right here in our homeland. We will live through this present experience. Our music is an expression of the spirit, just as survival springs from the spirit, just as hope, love, and strength are things of the spirit. Come, my brothers,\" he beckoned to the others to come nearer around us. \"Come and tell our friend here how we can live in shit and still make music.\"\n\nHis voice had acquired a sharp edge, cutting into me. His round face was grave, the eyes hard, glittering. I guessed his age at twenty-eight or thirty. The voice which had given such poignancy and power to his songs was now low and sonorous, the words tinged with bitterness.\n\n\"Tell our friend here that we are of Africa as the dust of the veld and the wind which blows it and as the rivers which are its blood. We are permanently of Africa, as the dust of our fathers is mixed with the dust of the veld. Now we are humiliated here and must bend in the dust. But we will be established again in our rightful place when we learn to pay more attention to things of the spirit. Do you hear me?\"\n\nI nodded. I was hearing him.\n\n\"I do not speak of your church. I speak of the spirit of man. When we learn, as our fathers did, to pay more attention to things of the spirit, we will know how to work together and suffer together and, once again, be established together in our fatherland.\"\n\nAbruptly he walked away, the others breaking up to follow him.\n\nTwo of his companions, Jim and Joe, remained with me and told me more of their daily lives. They seemed to take it as a matter of course that they would encounter great difficulty in everything.\n\n\"Wish we could invite you somewhere for a drink or a cup of coffee,\" Joe said, \"but there is nowhere in this damned city where a black man can buy something even as simple as that. Sorry.\"\n\n\"Don't worry,\" I told him. \"It's enough for me to be able to talk with you. How much do you practice?\"\n\n\"Oh, we manage about five or six hours a week. We all have jobs here in the city, but we are arrogant enough to believe that we can do better. Much better. Don't you agree?\"\n\n\"Readily. Listening to you play, now hearing you talk, I've no doubt whatever of your abilities.\"\n\n\"You flatter me, my friend. Say, why don't we get the hell out of this place?\" Nodding to Jim and me to follow him out. Downstairs he said, \"Wouldn't do much good to talk too much up there. Too many ears listening to everything.\"\n\n\"I'm staying at the Landdrost Hotel not far from here,\" I told them. \"Why don't we go there and have a drink or whatever?\"\n\n\"That the new big hotel that Bob Foster stayed at?\" Jim asked.\n\n\"Yes.\"\n\n\"Okay! I'd like to see inside one of those places.\"\n\n\"A hotel is a hotel,\" I said.\n\n\"So speaks the rich American visitor. You're beginning to sound like Bob Foster. You know what that son-of-a-bitch said when he was here? He said that South African Blacks were well off. Said he wouldn't mind buying a house and living here. You know what's funny about that? If he wanted to, he could buy a house here. As an American he would be treated differently. Even if we had the money we are prevented by law from owning land. But they'd let that loud-mouthed bastard buy land if he wanted it. Hell, he's a big-shot boxing champion and he's American.\"\n\nAt the hotel, I noticed their nervousness as they followed me to the desk to collect my key, then up the elevator to my room. We ordered drinks and sat down.\n\n\"Christ, just look at me,\" Joe suddenly exclaimed. \"I'm as nervous as a kitten, just coming into this place. A grown man, but the Whites have got me so that I'm scared of my own shadow. Scared to be in their big, shiny hotel even as the guest of a black man like myself. Isn't that too funny, for Christ's sake? What am I? Jesus Christ! What am I?\"\n\n\"Take it easy, Joe,\" from Jim.\n\n\"Shit with take it easy. He's black like us, so he should understand. Look at me, damn it, look at me and tell me what I am. Our friend here can come into our country, move about as he pleases, live in a hotel like this. In short, he can live as a man. We heard he writes books. He can afford to come all the way here to look at our country and us.\" Then to me. \"Tell me, Mr. Braithwaite, author and VIP, in what way are you different from me?\"\n\n\"In no way that I can think of,\" I replied.\n\n\"Thank you, my friend, for nothing. Let me tell you a little about me. I have a university degree. From Fort Hare, the black university. That means it is not as good a degree as if I'd had it from a white university. Anyway, that degree suggests that at some time in my life I was ambitious, imaginative, and hopeful. At Fort Hare I used to talk with others like myself, mostly about our hopes and plans for the future. Look at me now. Each day from eight to five, I stack goods in the carrier of a bicycle and deliver them to contemptuous white housewives who never see me, never address me directly. They just point to where they want the stuff put. I've been doing that job for three years and I bet not one of them knows my name.\"\n\n\"Don't think about it,\" suggested Jim. \"We're all in the same boat.\"\n\n\"Think, hell! I don't think about it. To think is to see myself, to recognize the thing I have become in three short years. Sometimes I wish I could kill myself, but what then would become of my wife and two small children? I once read somewhere that prisoners held in solitary confinement spend hours watching ants. I tried that. Do you know what happened? The ant walked away. It had somewhere to go. I have nowhere to go. I no longer think. I am one of the living dead of Soweto.\"\n\nListening to him I was hearing myself again. I, too, had thought my situation hopeless.\n\n\"What about your music?\" I asked, anxious to disperse the painful reflections.\n\n\"What about it? Did you enjoy it?\"\n\n\"Immensely.\"\n\n\"Listen to the man, Joe,\" Jim said, sarcastically. \"He says he enjoyed our music immensely. What do you think of that? He is able to enjoy, which is a luxury we can't afford. We don't enjoy our music, man. We need it, like an addict needs his dagga\u2020. It's our survival kit. And there's the other thing. Somebody might like our music well enough to want to do something about it. Cut a record! Arrange a tour! You never know.\"\n\nSmiling his sad smile as if accustomed to watching his dreams fade and die.\n\n\"Shit, man, you never know,\" said Joe. \"Like today. The white men there at Dorkay House. That one in the blue blazer. The bald one. American, I think he is. The other one is local and he told his American friend about us, so he came to listen. Not for nothing. We're tired of performing for them for nothing. He, the American, paid for the session. Anyway, there's always the chance they might like what they hear\u2014\"\n\n\"Sure, sure,\" Jim interrupted. \"How many times have we gone through that exercise. You know, I've come to the conclusion that I'm mad. Totally mad. Why else would I be sitting here in this fucking hotel where I am not welcome, drinking this man's booze, which is against the law for Blacks, and being what I am not. I could not buy him a stinking cup of coffee, so he brings me here, in this white man's luxury pit, and orders whisky, served to us obsequiously by one like us. It's all madness. I am a human being but I'm restricted to performing menial functions for a miserable pittance. You say, what about my music? I would like to play it and enjoy it, but I play it to relieve myself, not to enjoy. Like the job I do. I don't enjoy it. I just do it and try not to think of it. If I let myself think of it, I'd hate myself and choose to die. I'm dislocated from life. I'm quite mad.\"\n\n\"Don't listen to him,\" Jim cautioned. \"He's as sane as you are, but he gets that way sometimes. A drink or two and he'll snap out of it.\"\n\nI nodded, but I understood what Joe was saying, because I'd already seen enough to convince me that, forced to live as they were in the ghettos of Soweto and Alexandra, I would surely go mad. Christ, how could anyone feel ambition and hope while restricted to such a stinking environment? Most of the Blacks I'd seen in the city were neatly dressed. How did they manage it? What very special kind of fortitude was required each day and every day in this place to survive an hour, let alone a day? Maybe Joe was right. Maybe madness was the answer.\n\nWe spent another hour talking. They were like any black men I'd known in London, New York, Paris, or Jamaica\u2014anywhere. Intelligent, sensitive, and smarting under racial pressure. We had more than enough in common to draw us close together.\n\nI wanted them to be comfortable with me in the only place where we could be together in Johannesburg, but that very comfort was a continuing irritant. They'd come willingly with me, but, once in the room, seemed angry with themselves for being there and angry with me for causing them to be there. They drank, but without conviviality. We'd come up to talk about their music and their lives, but their only comments were on the \"white man's luxury\" of the hotel and my freedom to enjoy it. I wanted to remind them that I was paying for that suite but decided to keep my mouth shut. Perhaps they were aware of the \"Honorary White\" thing and were making sly digs at me. What did I have to do to prove that I was with them, sharing their identity? But was I? And what identity? Would I wish to live where they were forced to live, share their lives, suffer the same daily prohibitions and restrictions?\n\nWould I live in Alexandra even for a day? What was my feeling of identity worth if I would not voluntarily share with them? At this level, fine. But what about the levels on which their lives were lived? Could I fetch and carry for the white man and call him Baas? The very thought of it sent cold shivers through me. Everything about my life had always pointed in the opposite direction. From childhood.\n\nInto my mind flashed the memory of the Indian cane-cutter named Mungal Sirgh. A white manager on the Berbice sugar estate in Guyana where Mungal Sirgh worked had become impatient with the \"lazy coolie bastard\" and kicked him. Mungal Sirgh had replied by swinging his machete at the offending leg, slicing through the thick leather and opening the limb neatly from knee to ankle. Taken into custody, his repeated comment was, \"He kick me. Kick is for dog. Mungal Sirgh not dog.\" Christ, why did that come to mind after more than forty years? The \"Honorary White\" thing was no better than a kick in the ass. The intention was the same. To humiliate the black visitor; to deny him the dignity of his blackness; to remind him that in that society he had no identity except that which they, the Whites, chose to let him have. As a Black I was invisible, not there, not to them. To be seen and heard, I needed an overlay on my invisibility.\n\nIf that's how the Whites felt, to hell with them. But what about these men who called me brother? Why was there this gap between us? Perhaps they were saying something to me. Maybe they knew of the Honorary White label and resented it, for my sake. Or maybe they resented me for allowing myself into the situation\u2014a black man labeled white and seeming to enjoy it. I was relieved when they left.\n\nThat evening, I asked the hotel doorman to call a taxi to take me to Parktown, a residential suburb of Johannesburg, where I'd accepted an invitation to dine. When the taxi arrived, the doorman opened the door for me and I gave the white driver the address. I could see him eyeing me speculatively in the driving mirror as we got under way. Eventually, he opened up.\n\n\"Are you from Botswana?\"\n\n\"No,\" I replied.\n\n\"Swaziland?\"\n\n\"No. I'm not African.\"\n\n\"Oh, you're a VIP from overseas.\" Sounding pleased with himself as if he'd happened on the answer to the sixty-four-thousand-dollar question.\n\nWhen I paid him, the taxi driver gave me a card with the name and telephone number of his taxi company. At the end of the evening, I telephoned the taxi company and requested a cab to pick me up. I said goodbye to my hosts and waited outside for the cab which soon arrived.\n\nI was about to enter it when the driver called to me.\n\n\"Hey you, wait a minute. This is not for you.\"\n\n\"Why not?\" I asked.\n\n\"This is not for Non-Whites. I'm not allowed to take Non-Whites in this taxi.\" Meanwhile reaching backward in an attempt to shut the door which I held open.\n\nSo, it had happened. After all the fancy official footwork it had happened. Here I was, miles away from the city and without other means of reaching it. I felt suddenly angry at the thought that the taxi would drive away, leaving me there, helpless in an unfamiliar place. On impulse, I climbed in.\n\n\"I can't take you,\" the driver insisted.\n\n\"Then we'll damned well both stay here.\" My anger spilling out. \"I telephoned you from this address and you were sent here to collect a passenger and take him to the Landdrost Hotel, weren't you? Well, I'm that passenger and I'll be damned if I'll get out of this taxi.\" Without another word he turned the vehicle around and headed toward the city.\n\n\"If a policeman stops us, I could lose my license,\" he complained.\n\n\"If a policeman stops us, tell him to talk to me!\" I responded.\n\n\"The bloody dispatcher didn't tell me you were non-white,\" he went on. \"If he'd told me, I'd have known.\"\n\n\"How would he know from the sound of my voice?\" I asked.\n\n\"Well, non-white VIPs stay at the President or the Landdrost. It's not that I don't want to take Non-Whites in this taxi. It's not me. It's the law. If a policeman stops me with a Non-White in my taxi I could lose my license, and my job. But I suppose it's okay, if you're a VIP.\"\n\nJust listen to him! This same bastard would have left me stranded back there just because of my black skin.\n\n\"Do they ever tell you if a fare is black or white?\" I asked.\n\n\"Well, no, because we don't normally pick up Non-Whites.\"\n\n\"One of your taxis took me to Parktown from the hotel. In broad daylight, so he knew I was black. He didn't tell me he couldn't carry Blacks. So, if he could carry me there, why all this fuss about taking me back to my hotel? Does the policy of your company change with the drivers or from daylight to night?\"\n\n\"It's the same policy, but\u2014\"\n\n\"But you don't want to carry Blacks.\" I interrupted whatever excuse he was about to give.\n\n\"Look, I don't have anything against you\u2014\"\n\n\"Like hell you don't.\" In spite of myself the violence was spilling over. \"They sent you to pick me up, but one look at this black face and you were ready to fly off and leave me back there in the dark.\"\n\n\"I was only doing my job,\" I heard him say.\n\n\"Hell, no. This is your bloody job. Carrying passengers who call you is your job.\"\n\nI leaned back, swallowing the rest I wanted to say. What the hell was the use? A bastard like this would do the same thing again five minutes from now. He made some comment, but I didn't hear it. I lost interest in him and anything else he had to say. Just get me to that bloody hotel, I thought. Just get me there.\n\nA few weeks later I learned from the doorman at the Landdrost that some men from the Security Police had been making inquiries about me. They'd questioned the doorman and referred to a comment I'd made to the press about a white taxi driver who'd refused to take me in his cab and had only complied when I'd climbed in over his objections. Apparently they wanted to question the driver and needed some identification from me. They said they would be returning to see me. To hell with them. As far as I was concerned the matter was closed.\n\nThe following week, I went to visit a young Indian, living a few blocks from my hotel in a small area temporarily designated \"Indian,\" who recently had been released from the political prison on Robben Island, the same prison in which Chief Nelson Mandela has been held for years. About seven miles offshore from Cape Town, it houses several hundred political dissidents, all black and serving sentences which range from one to twenty or more years. I was eager to hear about conditions there at first hand.\n\nThe young Indian had been active as a publisher and distributor of newsletters attacking the Government's racist policies. He was caught, tried under the Suppression of Communism Act, and jailed for ten years without right of appeal. Now, even though he had been released, this young man was under a restriction order prohibiting him from having visitors. On entering his house it was agreed that, in the event of a visit from the police or security agents, I was to say that I was visiting his brother who lives in the same house.\n\nHe was full-bearded, thin, and hollow-cheeked as if recently recovered from a long illness, but his handshake was firm and he greeted me enthusiastically, mainly because I was from the same country as Dr. Cheddi Jagan whom he admired tremendously for his resolute position against the British during Guyana's struggle for independence. He had heard that I was in Johannesburg and wished to talk with me, to \"set me straight,\" as he put it. He made reference to the recent visits of Arthur Ashe and Bob Foster, both of whom, he claimed, played into the hands of the racist South African Government which sought to use such visits to divert international pressure from their policies of segregated sport. He seemed to believe that any Black from outside who visited South Africa was, by implication, accepting the prevailing policies as valid. He wanted to know how I had managed to acquire a visa in the first place and how was it that the author of a book like Reluctant Neighbors could persuade the South African Government to let him in. He fired off these and other questions without waiting for answers. He insisted that the Government was deliberately inviting well-known overseas Blacks, particularly Americans, to South Africa and showing them certain isolated aspects of the lives of Blacks in the Republic, so as to brainwash them into supporting the Government's racist philosophy. Bob Foster, he said, was a case in point.\n\n\"That black American went so far as to state that he liked this country so much he was seriously considering building a house here,\" he sneered. \"The idiot doesn't realize that if he lived here, he, too, would soon be compelled and condemned to live in a black township like Soweto or Alexandra, instead of a fancy suite at the Landdrost Hotel where you, too, are staying.\" Looking at me as if I shared Foster's guilt.\n\n\"I had the choice of three hotels here which are allowed to take Blacks,\" I told him.\n\n\"Yes, I know,\" he interrupted.\n\n\"Let me explain. I was told this at the airport the moment I arrived. I was told that there were no other hotels I could go to, none owned by Blacks or Indians or Coloreds or anyone else other than Whites. I make no apology for staying there.\"\n\n\"Okay. Okay. I accept that you had no choice, but people like you and Foster and Ashe are setting back the black struggle ten years. By coming here. By letting South Africa use the fact of your coming to counter our accusations of discrimination.\"\n\nIt finally got through to me that he had invited me to see him, not really to tell me about Robben Island, though he answered my questions, but to protest my visit to his country. He'd mentioned that he'd tried to reach both Foster and Ashe without any success.\n\n\"How do you imagine anyone outside your country would know anything about conditions here if no one made any attempt to learn at first hand?\" I asked.\n\n\"You could learn without coming here. Especially you. You were at the United Nations. Didn't you meet any of our brothers who went there to petition? Some of our brothers from here and South West Africa made it over to the States. Didn't any of them see you?\"\n\n\"Yes. I met some of them.\"\n\n\"Didn't you believe what they told you?\"\n\n\"I was persuaded by what they told me.\"\n\n\"Don't give me all that diplomatic shit, man. Either you believed them or you didn't.\"\n\n\"I was generally persuaded by them, but I welcomed the opportunity to see the situation for myself. This is it.\"\n\n\"Do you dash off to every country to check everything for yourself?\"\n\n\"No.\"\n\n\"Then why this? Did you have any difficulty getting a visa from this government?\"\n\n\"No.\"\n\n\"Shit, man, doesn't that tell you anything? Your books were banned in this country. Even today Blacks can't see your film in the public bijou, and that, too, was banned to Whites for some time. In spite of all that these Afrikaners gave you a visa to come here. Think, man! Can't you see they're planning to use you?\"\n\n\"Look, they can plan what the hell they like, that has nothing to do with me. I was issued a visa. Fine. But nobody can control how I think about what I see and hear and feel.\" And, on impulse added, \"Not even you.\"\n\nHe laughed, reaching forward to touch me.\n\n\"You think so? You really think so? By the time these sons-of-bitches are through with you, you'll be singing their tune without realizing it. You'll go back to the States and tell people all about how freely you were allowed to move about. No supervision, therefore, no police state. Which makes a liar out of all of us. Right? They'll wine you and dine you and prove that educated Blacks can make it anywhere. Only the lazy Bantu has to be kicked in the ass and locked in a ghetto to make him stir himself. They'll forget to tell you that he is disenfranchised, denied a reasonable education and the right to bargain for his labor and compete for the job he wants to do. Yes, friend, they'll tell you you're different and, you know something, you'll end up believing it.\"\n\n\"Think what you like,\" I said.\n\n\"Eh?\"\n\n\"Think what the hell you like,\" I repeated and stood up to leave. \"Look, you invited me and I came to talk with you. I thought you'd tell me about what the life is like for you and others. I came because I wanted to learn the truth, to hear it for myself so I can write about it. I expected that you, black like me, would lay it on me, without all this bullshit. You think I was born yesterday? I've lived most of my adult life among Whites. London, Paris, New York, Rome. I've no illusions about them, but I don't see them as bloody supermen either. They can't control how I think and what I'll write.\" He had needled me to this point. He and the others. Who the hell did they think they were? Pouring their suspicions over me. Here they were locked tight in the rotten ghetto and wanting the outside world to know of their plight. Okay. I'd come in. Of my own free will. So tell me and I'll write it. That's what I was saying to them, but all I was getting was their suspicion and scorn.\n\n\"Hey, cool it, man.\" He reached forward and pushed me back into the chair. \"Don't get excited. We're talking. Relax.\"\n\n\"You relax. You call that talking, making me seem like some half-assed idiot just because I've visited your country? White newsmen and writers fly here regularly, write their pieces and fly out again. Do you warn them that they're being used?\"\n\n\"Fuck them.\"\n\n\"And fuck you, too, mate. What gives you the right to be so high and mighty? Your years on Robben Island? Okay, I sympathize.\"\n\n\"Stuff your sympathy. Hell, man, you're beginning to sound like Whitey. Cool down. I'm only trying to help you. And don't hand me that shit about Paris and London. Over there they might hate your guts, but the law limits what they can do to you. Here Whitey is the law. Blacks can't command the law because it was not intended for them. They can't demand justice, because it was not intended for them. Justice and the law are concepts which apply to men. To humans. In this society Blacks are not considered human so they are not sheltered by those concepts. Did you know that, in this society we have no vote? We're not even on the official census. Shit man, we're not here. Don't talk to me about Whites in Europe or America. These here are different. They're fascists of the worst kind.\n\n\"Look,\" he was leaning forward, tapping on my knee with a long finger. \"All I'm begging you to do is think. I'm black. You're black. I published a few newsletters which nobody outside this town ever heard of and they threw me into jail. You've written books which have been read by millions. Attacking the very policies they live by. Okay, they try to keep those books out, but they're brought in anyway and read, so to save their own fucking face, they lift the ban. That makes this a liberal society. Right? And to cap it all, they let you in. Man, they used you before you stepped into that airplane.\"\n\nThe logic of it hit me hard, killing my anger and stirring up the fears I'd earlier had about making the visit. The visa was five months in coming. Perhaps all that time was necessary while the design was worked out. Christ, I was beginning to think like him.\n\n\"Okay, you made your point. Now I must be running along. I've a few things to do.\" I wanted to be out of this.\n\n\"Like a dinner engagement, maybe? With some of your white friends?\" Grinning.\n\n\"Perhaps.\" He had the knack of finding the nerve.\n\n\"Don't worry. They've enough black slaves to keep it hot for you. Okay, man. Like you say, you can see and hear and think for yourself, but I tell you they'll use you. They do it all the time. Among us. Even out there on the Island. Can you imagine that? Even out there where you'd think we were all brothers, all there for the same reason, all united against the fascist bastards. Even there they managed to use some of us against others. And for what? Some fucking little privilege we'd already learned to live without. After all we'd been through, to sell one's soul for shit like that! So you see man, telling me that you can see and hear and think for yourself doesn't mean a damn thing. Anyway, while you're thinking for yourself, think about us and remember that in the eyes of these fascists you're no better than the rest of us.\"\n\n\"I'll remember,\" I said. I'd come to this house with a gutful of goodwill toward this man. Now all I could feel was a nagging suspicion that somehow I'd been trapped into betraying him and others like him. Just by being in their country.\n\n\"In prison the payoff was some worthless little privilege,\" he was still with it. \"What are they giving you? The 'Honorary White' bit, so you can believe yourself different from the rest of us? Fancy hotel, your face in the white newspaper, moving around freely? Same thing, man. Privileges bought\u2014\"\n\n\"Nobody's bought me,\" I said, lamely.\n\n\"\u2014And paid for, man. And when you think you're moving about more freely than the rest of us, just look over your shoulder. If you're quick enough you might learn something.\"\n\nEverything he said struck home. Sure, I had been telling myself that nobody was restricting or supervising my movements. I'd been in and out of Soweto and Alexandra, hadn't I? My only problem had been my inability to make contact with the so-called black representatives. Buthelezi. Matanzima. The Information Office had promised me meetings with them but had only come up with excuses. Always at the last moment. But I must not let the things this man was saying color everything that happened. If I couldn't reach the big Blacks, there would be others.\n\n\"Are you concerned for me or just sorry for yourself?\" I asked, trying to throw him on the defensive, and free myself from the suffocation of his penetrating insight.\n\n\"I'm not sorry for me, man. I'll live. I lost ten years of my life out there on the island. Doing shit, man. Breaking rocks for the sake of breaking rocks. You're sitting on a pile of rocks today with a hammer in your hand and sometime next week or the week after it's a pile of pebbles and you can't remember how it happened. You've used two weeks of your life watching rocks turn to dust. And the next week you're sitting on another pile of rocks. Or is it the same one? You know what they did with the pebbles, man? They just left them there to remind us that we were just shit. You know what our ambition was? To stay alive. Staying alive, that's all. Living for news from outside. Do you know what was the most important thing to us in there? Not money, man. Not pussy. A newspaper. Any old newspaper. We read every word. Everything. And we talked. Can you understand? Those fucking Afrikaner guards watched us to prevent us from talking. Threatened us. Punished us. But we talked. Even with our mouths shut like, what you call them, ventriloquists, man. Whoever found a piece of newspaper read it, then passed it on and told everyone what he'd read. After a while we were reading more closely, more perceptively than when we were free. We shared our points of view. We talked. Especially about the political situation.\" Here he laughed again, scratching his head, remembering.\n\n\"Once a priest came into the prison carrying a briefcase with a newspaper, the Times, stuck under the flap. Like lightning, it disappeared. He never made a fuss about it. That Sunday we had a whole newspaper to read. After that, whenever that priest came to see us, he brought a newspaper and it always disappeared from his briefcase. Survival, man, that's the word. Nelson Mandela is up there. Living it out from hour to hour. That's where you learn about hope, man. Without it you're dead.\"\n\nHe came and placed a hand on my arm, a conciliatory gesture.\n\n\"Will you come and see me again, friend? I promise to be nice.\"\n\n\"Don't strain yourself on my account.\"\n\n\"That's not a strain. Living like this is a strain. Shit, I can't even see you to the door. Never know who might be checking on me from outside. If I'm seen talking to you, they could come and take me away. Fucking lovely way to live, isn't it? I'm jealous, man. You, a stranger, can move about as you wish. Right? Me, a native son, I'm denied the right to step outside. Goodnight, man.\"\n\nI left him, his words continuing their disturbing refrain in my ear. I'd gone to his house to talk with him about his time in prison. He'd talked about my visit to his country, sowing in my mind a very sizable seed of doubt about my own motives, and my possible malleability by the South African authorities. Walking away from the Indian's irritating sneers, I wondered if he was right.\n\nHe'd questioned my coming to South Africa but he'd either forgotten or ignored the fact that my coming made it possible for me to see him and hear his cynical censure. In his position, I'd be just as embittered, seeing strangers move about with ease while I was restricted to my own house. But what the hell did he want of me?\n\n\u2020 Marijuana.\n\n# Chapter   \nFive\n\nON MY WAY BACK to the hotel, I passed a restaurant, brightly lit and attractive, and suddenly realizing I was hungry, I decided to go in. I pushed the door but got no further than a step inside, where I was confronted by a waiter, dead-faced and stony-eyed, who placed himself in front of me. He said something to me which I supposed was in Afrikaans.\n\n\"What did you say?\" I asked.\n\n\"You do not come in.\" This was stupid. I was already in and thinking out my next move. Now I fully realized why the hotel people had repeatedly suggested that I let them know whenever I wanted to dine out and they would make the arrangements for me, claiming that they knew where all the best eating places were located. This waiter looked as if he would have welcomed a fracas, eyes pale, pugnacious jaw thrust forward. I was turning to leave when another man approached and asked him something in Afrikaans. The waiter replied, and the newcomer then addressed me.\n\n\"I don't speak your language,\" I said.\n\n\"You're not African?\"\n\n\"No, I'm a visitor.\" At which he spoke again to the pale-eyed waiter, this time impatiently, but I walked out, wishing them both to whatever hell was reserved for Afrikaners.\n\nIn my room, the things the Indian had said teased and tormented me, throwing into sharp relief what had happened at the restaurant. The waiter's contempt for Blacks was ready and waiting for expression. A waiter! His awkward English indicated that he may well have been a foreigner, an immigrant. How quickly people took on the local social coloration. Like chameleons. Come to think of it I hadn't seen a restaurant in Soweto or Alexandra. Maybe I passed them and didn't notice. What were they like? Could I eat a meal in one of them? Christ!\n\nMy reflections were interrupted by a telephone call from a young black newsman I'd met a few days earlier.\n\n\"How are you doing?\"\n\n\"Fine,\" I lied.\n\n\"How would you like to come out here and see how some of us live?\"\n\n\"Where's 'out here'?\"\n\n\"Soweto.\"\n\n\"I've been there.\"\n\n\"Soweto's a big place. I don't think you'd have come to this part. I heard you'd visited with the big boys here. Come and see how the little people live.\"\n\nSafely indoors, I wasn't keen to go out again. Besides, I'd had enough of social exposure for one night. A quick tray from room service seemed a more attractive alternative.\n\n\"How could I get there now,\" I temporized. \"It's nearly eight o'clock.\"\n\n\"By taxi. Black taxi. No white taxi will bring you out here. Get a black taxi from the taxi stand near the black bus stop. I'll meet you at this end. It will do you good to travel the way the rest of us do.\" I was still far from enthusiastic.\n\n\"How will I get back here?\" I asked, thinking of the special permit required of Blacks in the city at night.\n\n\"I'll see to it, don't worry about that.\"\n\n\"Okay,\" I surrendered and went out to the black bus stop across the park. I was directed to a taxi, empty while the driver stood around joking with friends. Loud laughter punctuated each sally. He waved me to sit inside. Soon I was joined by another passenger who sat beside me without saying a word. Then another and another, followed by two more, these sitting in the front. Not a word from anyone. Another person pushed in the back and we were all forced to sit diagonally pressed together. Another passenger slipped into the front. A woman. At first I thought she'd be driving because she sat at the wheel, but now the driver got in, pushing against the woman until he could take hold of the wheel even though his body was only halfway under it. Somehow he started the vehicle and we were off.\n\nIt was the most uncomfortable taxi ride I'd ever taken. Eight adults cramped uncomfortably into space designed for five, the driver miraculously shifting gears and steering from his sideways position. We passed several taxis similarly overloaded, always with Blacks.\n\nIn Soweto, my acquaintance was waiting as promised, standing beside his car. He said he could have fetched me, but thought the experience of riding as he did twice each day would help me to understand better what was normal for a Black. He kept his car for after-work use.\n\nWe drove to his home, one of the square concrete boxlike structures in the northeast part of Soweto. Instead of the corrugated metal roof I'd seen on some of the other houses, this one and its neighbors wore bulky concrete tops, making them seem humpbacked in the nighttime gloom, very much, in fact, like huge sleeping elephants. Inside it was hot, even with the few windows open. Several candles were strategically placed about the room for light. Indoors he turned to me and said, \"Welcome to the real Soweto.\"\n\nThe house was sparsely furnished. The main room in which I stood contained a wooden table with three wooden chairs around it, a rough chest of drawers reaching nearly to the low ceiling, and a narrow wooden cot. In a corner another table, roughly made but sturdy, supported some cooking utensils and a Primus stove. No electricity. No signs of running water.\n\n\"Six of us live here, in four little rooms,\" he said, his eyes brightly on me as if to note my slightest reaction. He led me into another room which was furnished in nearly the same way, except that there was no cooking equipment. A central wooden table, two low cots opposite each other and two wooden cupboards. Near one of the cots was a small upended box, centrally divided, which contained several books. Crowning the box was a half-worn candle stuck in a Coke bottle.\n\n\"My brother is a medical student, one of the very few. That's where he studies. He leaves here at five o'clock each morning to make his way into town and compete with white boys who read by electric light, sleep in comfortable beds and eat a good breakfast.\" Saying it all so matter-of-factly. I looked at him and surprised the pain on his face.\n\n\"Different from your hotel, don't you think?\" he asked.\n\n\"Yes, different.\"\n\n\"Different from those houses they showed you on your officially conducted tour, I'm sure. Then, you saw houses like this, but with electric lighting and a kitchen sink and a water toilet out back. I'm sure they didn't show you these. How would you like to live here for a month? No, a week, or even a day?\"\n\nI was feeling battered, first by the Indian, then that bastard at the restaurant, now this. How much crap was I supposed to take? I wanted to see, at first hand, the conditions under which my fellow Blacks lived, but why should they think they had a right to cram it down my throat?\n\n\"I wouldn't wish to live here,\" I replied. Leaving it at that.\n\n\"How about a drink?\" he asked, not waiting for my response, but reaching into a cupboard for two china mugs and a tin of powdered coffee. He poured water from a container into a tin kettle, then set to pumping away at the Primus stove, pricking at the jet from time to time and cursing under his breath as it defied his efforts.\n\n\"Not to worry,\" I said, actually relieved that the little stove was defiant. After all he'd said I wasn't too anxious to risk the water, even though it would of course be boiled. Wondering if he had to go through this same exercise early every morning to have hot water for shaving. What happened in the winter? Apart from the bare necessities there was nothing. No curtains, no posters, no pictures.\n\nThe failure of the coffee project seemed to cause us both some embarrassment, but he saved the situation by inviting me to take a walk around the neighborhood. Outside the night was star-\u00adstudded and pleasantly warm, the night shadows smudging the outlines of bush, tree, and house, giving the whole place a romantic softness. No street lamp in this part of town, only the candleshine from open doorways and the starlight from above.\n\n\"It's okay if we walk around here together at night,\" he told me, \"but any one of us alone would be asking for trouble. You call it mugging in the United States. Bands of young boys roam the streets at night, preying on men who've been drinking in the local beer gardens or in the shebeens. Beat them up and rob them. Sometimes kill them. Know how they do it? They push a piece of sharpened wire, something like a short knitting needle, into the neck at the base of the skull. Paralyzes those who survive. Many of the paraplegics in the local hospital are victims of the Tsotsis.\"\n\n\"Why do you call them that? What does it mean?\"\n\n\"Not too sure. Something to do with the Zoot Suit gangs of the United States, I've heard. Anyway, so the story goes. Most of them are boys without parents or even relatives. School dropouts. Or maybe they couldn't get into school. Couldn't afford the fees, or clothes, or books. So they don't go. After all, schooling is compulsory only for Whites. Optional for Blacks.\"\n\nWe walked around, listening to the night sounds, people talking to each other, snatches of conversation floating out from the houses, all in an African dialect. Music. Edmundo Ros swinging his inimitable way through a rumba rebroadcast from London. The sudden scream of a night bird in the near distance. We could have been light years away from the neat, trim suburbs designated \"White.\" In order to reach this place from my hotel the route had been through suburb after suburb of affluence and comfort. My companion and hundreds of thousands like him made the same trip to and fro each day, seeing the affluence, envying the comfort. Inevitably hating.\n\n\"I think I should be heading back to town.\" I said.\n\n\"Had enough?\"\n\nI told him it wasn't that. I was anxious about being stopped by the police. If that happened and they found out that I was a visitor they'd also discover that I had no permit to be in Soweto.\n\n\"Nobody's going to stop us. Not unless there's a police raid to find people illegally living here.\"\n\n\"Do you know when a raid is likely to happen?\"\n\n\"No.\"\n\n\"Then I'd better get back.\"\n\nHoping he'd see the point and agree. Realizing, belatedly, that I was completely dependent on him now for leaving that place. There were no telephone booths to be seen, no taxi stands, no bus stops. If there were any such places, only he knew where they were.\n\nNow and then a car passed us, always hurrying. No pedestrians. Perhaps the Tsotsis keep nighttime walking to a minimum. Abruptly he turned and we retraced our steps to his house where we got into his car and headed out of Soweto.\n\n\"The trick is not to have a breakdown at night,\" he said. \"No help for the black motorist, not even from the police. If you have a breakdown and a policeman approaches you, the first thing he asks is not what's wrong with the car, no, he wants to see your pass.\"\n\nWe made it without incident to my hotel where he left me and hurried off home. I wished him a safe journey, asking myself if the trips, his and mine, were worth the risk and anxiety. I could so easily have jeopardized the rest of my stay. I promised myself I'd not do it again.\n\nI sat and thought over the events of the day which had been painful and very irritating. Here I was a black stranger in this country and it was becoming more and more difficult to meet and consort with Blacks without being subject to suspicious inquiry. Inside me, I felt deep identification with them in their unhappy state. Everything I'd seen and heard since entering the country merely strengthened that feeling, because I knew that the only thing which saved me from the same fate was the fact that I was a national of another sovereign state. I wouldn't want to live as they lived, but neither did they. I was prepared to be with them whenever they wished, to learn from them, about them. I didn't wait for them to seek me out. I sought them. But evidently that was not enough. Okay, so they thought I'd be used by their Government. I believed them to be wrong. So why couldn't they give me the benefit of the doubt? I was already in the country. If I looked and listened and heard and then went off and wrote laudatory pieces in spite of all the evidence to the contrary, then they could call me traitor.\n\nWhite journalists I'd met in London and New York had given me the impression that they'd been able to talk with Blacks in South Africa without much difficulty. If that were true why were these Blacks making things so difficult for me? So different from my visits to other parts of Africa, where I'd been made to feel welcome. Immediately. My black skin was my ticket to enter. Here, it was the reverse; my very blackness was the barrier. Well, perhaps I should be patient. After all, the conditions I'd already seen were worse than I'd dreamed possible and those very conditions might be the reasons for my black friends' suspicions and reservations.\n\nThe hotel's public relations officer telephoned and said she hoped that all was well with me.\n\n\"If there's any way in which I can be of help to you, please don't hesitate to ask,\" she said.\n\n\"Right now, all I ask of life is a hot bath, a cold drink and a funny movie,\" I replied, lightheartedly.\n\n\"The bath and drink are no problem,\" she told me, \"but the movie is another matter. If you're really keen to go to the bijou, though, and decide what you want to see, I could telephone the management and I'm sure it could be arranged.\"\n\n\"Why telephone? I don't understand.\"\n\n\"I think I'd better come up to your suite and explain,\" she said. A few minutes later, she arrived, blonde and well-groomed, with that quiet confidence which seems to be the stock-in-trade of the public relations fraternity.\n\nSeated, she said, \"I think I must explain the bijou situation here in South Africa. Most cinemas are operated for Whites only. We call them bijous here. There are a few for Coloreds and Blacks in their own areas. Indians have their own. In any case, those in Johannesburg are for Whites only. As an important visitor to our country you are allowed, shall we say, special status. I feel sure that if you decide on the film you want to see, I can telephone the management and there will be no difficulty.\"\n\n\"You mean I couldn't just go to the box office and buy a ticket?\"\n\n\"Not unless they're expecting you. It's the law, I'm afraid.\" It suddenly struck me that in this country I could not, for an hour or two, lose myself in the temporary anonymity of a darkened cinema the way I'd done in every other country in which I'd lived or visited. It had been a favorite way of slipping away from pressing reality, a painless, absorbing way of insuring the quick passage of time. A thought occurred to me.\n\n\"What happened when the film To Sir, with Love was shown here?\"\n\n\"Same thing.\"\n\n\"No Blacks allowed to see the black actor?\"\n\n\"Not here in Johannesburg. Blacks and Whites are prohibited by law from congregating in the same place. Anyway, they wouldn't have missed much, because the film was so badly censored it was difficult to follow the sequence of events. Anything between the black teacher and the white one, Gillian, was cut out. I know, because I saw it first at the bijou and then I saw the whole movie at a private showing in a friend's house. It's possible to rent an uncut film from one of the rental agencies. Anybody can rent them, Black or White, as long as they have the money.\"\n\n\"So if I wanted to see a movie, I'd have to pay for my ticket then beg permission to get in, right?\"\n\n\"Well, I wouldn't put it quite like that. We at this hotel would do whatever we could to avoid any embarrassment to you.\"\n\nI thanked her and, soon after, she left. I had no quarrel with her. I thought of walking off my irritation in the streets, but then remembered my near encounter with the policeman. The only alternative was a bath, a cold beer, and settling down with pencil and paper to review the days already passed in this beautiful city, this very uncomfortable society.\n\nSome days later, early in the morning, a group of black welfare officers came to the hotel asking to see me. I invited them up to my room, wondering how the word of my presence had got around.\n\nThere were five of them, two women and three men, all of them employed to service one or other area of welfare for Blacks. They complained of the inadequacies of the service and reminded me that I had been involved in nearly similar situations as recorded in my book Paid Servant, which they had read. They could not begin to cope with even a tiny part of the problems people brought to them, and they wondered if I could offer them any advice. Listening, I learned that ten of them were expected to service Soweto and Alexandra, a total population of well over a million people. Their office was in white Johannesburg, and both the limited programs undertaken and the minimal funds available were subject to white control. The biggest problem was the many dislocated, neglected children who roamed the streets of the townships living on what they could beg or steal, homeless, without parents or relatives. According to the supervising authorities these children were not legally of the townships and therefore did not qualify for any official assistance. They could not go to school even if they wanted to because a new law required all legitimate parent-residents to have a pink school card for their child. Without it no child was accepted as a pupil.\n\nWhite children, if neglected or abandoned, are cared for at Government expense. They would in no circumstances be allowed to wander about like homeless dogs eating garbage. I told them that I had seen an elderly white woman throwing chunks of bread to the pigeons and how, as soon as she'd wandered away, the children drove the pigeons off and collected the bread, eating it hungrily. Oddly I saw none of them begging.\n\nThe white welfare officers were the bosses and called the tune. Colored officers dealt with colored clients, Asian officers with Asian clients, black officers with Blacks. Levels of payment differed for each officer group. It became apparent that they were their own most needy clients, their own welfare their most pressing priority.\n\nWe talked for most of the morning, taking a short break for coffee which I ordered. There was something a little bizarre about us. In the park outside were living reminders of the urgency of their work, yet they sounded so much like welfare workers I've known in England, France, and the United States. The same pompous preoccupation with the jargon of their profession, the same insistence on separating themselves in every way from their \"clients.\" They emphasized that they were trained and qualified as sociologists, forgetting their earlier complaint that Blacks are denied access to anything more than the median levels of qualification.\n\nFrom time to time, the door to my suite would be opened and one of the floor supervisors, white, would peer at us from the vestibule and quickly retreat with a \"Sorry, just checking to see that everything's okay.\" I assumed that it was just the normal hotel practice. It has happened to me in many parts of the world. My visitors, however, felt quite differently. They believed that we were being watched and that the interruptions would continue as long as they remained with me.\n\n\"They think we're plotting something up here,\" one of the women said.\n\n\"Blacks talking together are always supposed to be plotting. To kill them or steal from them. That's why that one comes in without knocking.\"\n\n\"What's surprising about that?\" from another. \"We're watched, and I'm sure our American friend is being watched. Wouldn't surprise me if these rooms are bugged.\" Suddenly getting up to peer among the artificial plants, behind the sofas, under the tables. Everywhere.\n\n\"Why worry?\" I asked, smiling. \"We've not been plotting, so who cares if someone's listening to us? Surely it should come as no surprise to anyone to hear that black children are homeless and starving.\"\n\n\"Survival, friend, survival. Mustn't make anything too easy for them. You probably think we've been reading too many spy novels. In this place you say something out of line and they have you hanging, like a fish. You're lucky you can afford to be amused at us.\"\n\n\"I'm not amused at you.\" Christ, would I never escape having to defend myself.\n\n\"Stay here six months, or three months, and you'll understand what we're talking about.\"\n\nOn this subdued note, they left me.\n\nThe meeting with the Bantu Council had been arranged for eight o'clock that evening in the home of one of the members. I would have preferred to meet them in the Council building where I could see them against their working background but naturally I was obliged to follow their arrangements.\n\nOne of them called for me promptly at seven o'clock. On the way to Soweto, I told him of my earlier visit with the white guide and the old man's outburst which had led to this meeting. He seemed preoccupied, glancing in the rear-view mirror more often than I thought necessary, and I was surprised when he suddenly asked, \"Is anyone following you?\"\n\nI didn't know how to respond to that. \"Why?\" I asked.\n\n\"Just trying to make sure,\" he replied, and said no more about it. On arrival at his home, his wife greeted us with the news that two members of the Security Police, one black and one white, had been there asking about me and the meeting, claiming that they wanted to make sure I would be quite safe. She replied that I would be under their roof, as safe as they were, as protected as they were, if any protection was necessary. They replied that it was their duty to prevent an international incident and merely wanted assurance that all would be well.\n\n\"I gave them short shrift,\" she said, smiling.\n\nThis was my first experience of the Security Police actually monitoring my movements. It was no longer a joke, an offshoot of my friend's paranoia, but undeniable proof of the Big Brother interest in my movements. My hosts seemed to take it all in stride. They told me that police spying was merely another fact of daily life; it pervaded every area of living to the point where no one fully trusted his neighbor or associates or friends. This was equally true of the Council: although they were all black, each one was afraid that another might report something said or done in the hope of receiving some minuscule concession from the Security Police.\n\n\"That way they keep us distrustful of each other, suspicious, so we're unwilling to come together in any real way to help each other. If I have a new idea, I don't know where to start. I can discuss it with my wife, but who else? Sometimes people come to me with ideas. I've got to listen very carefully. If their ideas have the slightest hint of opposition to Government policy, my first reaction is that they're trying to trap me. Oh, yes, that's part of the technique. They come to you with an idea and the next thing you know they claim it was your idea in the first place and you have the Security Police on your back. All the time the police hold over your head the threat of sending you off to one of the Homelands. They could come here tomorrow and claim I'd been instigating something and deport me out of here. They'd tell me to remove my house from the Government's land. How do I go about picking up a house like this? They've got us Blacks in a vise. As a result we don't trust each other. We talk, sure, we talk, but we watch what we say. So, now you know. We'll meet here tonight with you and we'll tell you some things you could hear from anyone else. That's safe. But wait till you ask us questions that require us to express our deep feelings, questions that get to the bone. Then see what happens. We begin to look at each other. My friend, the Whites have got us so that each one of us has become the other's policeman.\"\n\n\"Then this exercise tonight is likely to be a waste of time,\" I suggested.\n\n\"Not altogether. Wait and see for yourself.\" But I was right.\n\nBy eight o'clock no one else had arrived. At eight thirty-five one showed up bringing his wife and a friend as if it were a party. Ten minutes later another arrived, with his wife and a local schoolmaster. No sign of the aggressive little councillor whose challenge to me had precipitated the whole thing. I asked my host about him and learned that he had been informed of the meeting and had promised to attend. By nine fifteen, he had still not come and someone left to fetch him but soon came back saying that he was not at home.\n\n\"After his little performance at the Council Office in front of the white woman guide, someone's had a word with him, I suppose,\" one of the men suggested.\n\nI was surprised to learn that everyone knew of the incident, insignificant though it had seemed to me.\n\nThree young men showed up, none of them councillors, one of them a newsman working for a city newspaper. Conversation settled on the safe topic of the schools. The schoolmaster was praised on all sides for the wonderful job he was doing, even though it emerged that the children's big successes were in their competitive singing, rather than in their academic work. Of the six to seven hours of the school day, at least two hours were spent rehearsing songs, mostly European songs.\n\nDiscreet questioning disclosed that the schools in Soweto are poorly equipped, the teachers poorly trained, the pupils ill-\u00adprepared to compete in the harshly competitive society; and here were these black men congratulating each other. When I probed further on the schools, on teacher and pupil performance, they readily resorted to a lengthy litany of woes, all of which were blamed on the Government and so outside their control.\n\nI was soon bored with it all. I had been led to believe that they were ready and able to talk freely with me about their community, but all that had taken place were moans, evasions, and backslapping. What the hell had they to be proud of? The few schools they had were overcrowded, understaffed, and ill-equipped. Large numbers of children were roaming the streets instead of being in school, and the devilish \"pink card\" system kept it so. More and more of these children were pressured into Tsotsi gangs, and these men, each secure in his own circumscribed job, did nothing to change the situation.\n\nI stood up to leave, just as the hostess brought a tray of drinks on which the other guests avidly fell as if that were the real reason for the gathering. If real change would come to places like Soweto, it would not be through the efforts of men such as these, I realized.\n\nOn the way to my hotel, my host and I said little, each wrapped in his own assessment of the abortive meeting. Now and then he slowed down to allow another car to pass us, and I realized he was still afraid that I was being followed. Or perhaps he was concerned for himself. So easily one could be caught in the grip of paranoia.\n\nLying on my bed, reviewing the day's events, I was disturbed by the non-appearance of the little Soweto councillor. I had been told that he knew I would be there to see him. After his spirited effort in front of my white guide, he would want to be there, as face-saving is very important among people. I wondered if she had complained to her superiors about his outburst and a decision had been made to silence him, at least for the duration of my stay. With all that I heard about the police and their tactics, he had invited a pack of trouble for himself. But perhaps it was worth it, to him. Perhaps he had reviewed his life and had seized the opportunity to make a gesture, to himself. In the prevailing circumstances, that small gesture assumed heroic proportions. No other Black had said or done as much. Not publicly. Not in the presence of a member of white officialdom.\n\n# Chapter  \nSix\n\nTHE NEXT DAY WAS the day of the promised lunch at the Afrikaner businessmen's club arranged by the banker I had met at Helen Suzman's. On the way there he explained the growth and development of the white community.\n\nA man eminently knowledgeable about money, its power and influence, he spoke easily of his plans for the future. He spoke of the club to which we were going, its founding and the type of people who were its members. He warned that I might find them inflexible in their social attitudes, but hoped I'd be patient and remember that they were the products of a grim period in South Africa's history when men and women needed to fight for the land on which to settle and establish communities. For my benefit, he recounted the Afrikaner version of those wars of conquest, stressing the courage and fortitude of the voortrekkers and their womenfolk. He spoke of the bloody conflicts in which Afrikaners of earlier generations had frequently been involved and made it seem that they had invariably been on the defensive against a persistent, devious, and intractable enemy. Memory dies hard and I got the impression that Afrikaner hatred of Blacks is deliberately kept alive today, primarily for tactical political purposes. I reminded him that the wars of which he spoke were several generations old. Since then the whole world had been torn by wars far graver than those the voortrekkers fought and yet had shown a willingness to rise above the hates and fears which had given rise to those crises. What was there so special about South Africa that it needed to \"feed fat its ancient grudge\"?\n\nIt is both distressing and fascinating to hear people defend their contempt and hatred of Blacks, especially to me, a man as black or blacker than their enemy. I asked him if he and his kind had no concern for the inevitable bloody results if they persisted in their despotic pressure of the Blacks.\n\nAt this, his tune changed. He denied contempt, citing his own friendly relationship with the Blacks he employed on his farm.\n\n\"I'm willing to admit that changes must come,\" he said. \"They will come. But we must not expect them overnight. We can't have revolution here. Evolution yes, but not revolution.\" The words flowing so easily from him, cushioned in comfort as he was by the blood, sweat, and toil of the Blacks whom he despised. He could talk of evolution, secure behind the vast stockpiles of armaments and the military manpower deployed strategically all over the country\u2014I pulled myself up short. I was on my way to hear from him and others like him, so the thing to do was wait and listen to them.\n\nThe Clubhouse was much as I expected, an attractive red-brick building set against a pleasant background of carefully nurtured trees and trimmed lawns and flower beds. Beginning at the doorway, uniformed servants everywhere, all black, ready to dart off at the master's bidding, all eyeing me with surprise and speculation, the first black person ever to set foot in that building in other than a menial capacity.\n\nSettled in the well-appointed lounge with my host to await the other guests, I drank a glass of sherry with him, amused that we were in fact breaking the law which forbade Blacks and Whites to drink alcoholic beverages together. Unable to keep the thought to myself, I shared it with him.\n\n\"Let's put it this way,\" he told me. \"You're an overseas visitor, a world famous author, VIP. During your stay in South Africa you have the honorary status of a white man.\"\n\nThat spoiled it for me, my mood of friendly ease evaporating completely, giving way to a rage which I fought to control. I put my glass down and looked at him, hating the arrogance which led him to assume that he, they, could change the color of my skin to suit their whim. No, not change it. Just overlook it, ignore it to the point where it did not exist for them and they could superimpose their choice upon it. But no, just looking at them convinced me that my blackness was there before them, large and unavoidable; it was plain from the way they behaved when we were introduced\u2014the hurried pleased-to-meet-you, the words rushed out to belie their meaning; the quick retreat from my deliberately firm handshake.\n\nI had half expected to meet a group of highly intelligent, urbane men, as conversant with world affairs as they were knowledgeable about their particular interests, articulate and arrogantly relaxed in the assurance of their power and prestige. Instead, I found myself in a group of rather ordinary people, most of them painfully hesitant on matters outside their parochial concerns and generally uncomfortable in the unfamiliar company of a black man who did not treat them as his betters.\n\nTwo of them, an elderly economist and a physicist, seemed more relaxed than the rest, and keen to discuss South Africa's international image, even though they took a lofty view of the criticism directed at her. They argued that the continuing international economic crises were working to South Africa's advantage, and would eventually have the effect of forcing some accommodation to her domestic policies. In support of this, they pointed to their country's considerable gold reserves, the rapid rise in the price of gold, and the new political leverage which, they claimed, South Africa could now exercise.\n\n\"In this world, money talks,\" the economist said, \"and the loudest, most persuasive voice is that of gold. Even some African countries which publicly criticize us because of our domestic policies are willing to make private economic agreements with us. Out of such agreements political accommodations are born.\"\n\nSo we drank and talked, watched covertly by the black serving team which quietly and efficiently attended us. I wondered how they viewed my presence among the white men. Did they too assume that I was being used by the Whites? I could read nothing behind their unsmiling faces and courteous manner.\n\nThen lunch was ready and we were seated, and I realized that this was a first for all or nearly all of them. They were sharing the same board on equal terms with a black man, and no matter how they might rationalize it to themselves that simple fact was incontrovertible. There was the usual friendly chitchat as each tried to settle down. I wondered how each would report this meeting to wife, children, and business associates. And what would they tell the Blacks who serve them at home and with whom they claimed to have good personal relationships?\n\nI remembered chatting in a park a few days before with a maid who was supervising a small white child and a dog. I tried to question her about social conditions in South Africa, but, inevitably, she brought up Bob Foster.\n\n\"How he beat that white man! It was so good.\"\n\nHer whole body glowed with the sharing in that small victory, this woman whose life was destined to be spent in lowly service, nurturing the children who would one day grow up to treat her with casual contempt, whether it was personal or public. She would be used, underpaid, kept in her place...\n\nNow here were these men, most of whom had in their time been bathed and comforted by black women, casually defending their inhuman policies with the spurious claim of \"good relationships.\" Spurious? They were completely sincere and convinced of their righteousness.\n\nAfter lunch, I was formally introduced to the group and invited to address them. On the spur of the moment I decided to talk on the economics of waste, deliberately choosing that neutral approach to tease them out of their shells, to let them feel comfortable with the Honorary White and open up so that I might learn about them. I said that I had been impressed by Johannesburg and its flourishing suburbs, but sickened by the wide evidence of the exclusion of Blacks from the essential life of the community. Blacks were everywhere, cleaning, serving, providing an inescapably solid base to the community's economic life, but resentfully, unwillingly, because they were denied the right to exercise their imaginative potential. I asked them to explain how a community could ever reach its full growth if the greater part of its people were restricted to minimal contribution. As I saw it the result was waste on an unbelievable scale, shrouded behind the absurdities of discrimination.\n\nThey listened in silence but when I sat down they defended themselves vociferously. They insisted that the Blacks of South Africa are better off economically than Blacks in any other part of Africa. They told me that though there was job reservation which favored Whites, the law required every man to be paid the rate for the job, and that those employers guilty of evading that law were invariably foreign firms, particularly American.\n\nThey insisted that I had not been in the country long enough to see and understand the complexities of the labor structure in general, nor the conditions affecting the black role in particular. Very few Blacks, they claimed, were capable of other than menial employment. South African Blacks had changed little from their original primitive state and were, for the most part, still happier living in the rural Homelands in their traditional way. On the other hand, the grim conditions in which the black workers lived were not really intolerable to them, being an improvement on what they knew in their familiar rural living. It was Communists and outside agitators who stirred them up and tried to make them dissatisfied with their lot. Rural Blacks were discouraged from taking their families with them to the urban centers only because that would have meant too great a dislocation, in addition to the aggravated problems of housing, feeding, and educating their children. On and on. The old familiar clich\u00e9s, but trotted out with the utmost sincerity. As I listened it was difficult for me to keep my mounting irritation under control. I am as black as the men and women they were talking about.\n\nBut I was a stranger. I would be here today and gone tomorrow. I needed nothing from them, so they could afford to be generous with their time and their rhetoric. Perhaps they expected me to be flattered by being among them, treated as an equal by them. Nudged by the irritation which would not subside I said, \"I understand you've broken the 'Afrikaner Only' rule in this club and admitted Englishmen. That tells me you're getting around to forgiving and forgetting what Kitchener and his redcoats did during the Boer War.\" There was silence for a few moments, not even the tinkle of ice in a glass. Then someone said:\n\n\"We've come a long way since those days. Language aside, we're all South Africans here.\"\n\n\"That's what I was thinking,\" I said. \"Maybe the same spirit will foster a similarly reasonable attitude to the Zulu Wars and the descendants of those who fought in them.\"\n\nSilence.\n\n\"Here am I,\" I went on, \"sitting with you. I have no way of knowing where my ancestors came from. History suggests that nowhere in Africa was secure from the slaver's nets.\"\n\nThey were watching me, most faces wearing that pained half-smile which was as much as courtesy demanded.\n\n\"What's your point, Mr. Braithwaite?\" one asked.\n\n\"I'm anticipating the day when Blacks might be admitted to membership of your club. After all, one ex-enemy is as good as another. You could always designate them Honorary White.\" My little quip fell flat. Even the half-smiles had vanished.\n\n\"The designation Honorary White is merely a convenience reserved for overseas visitors,\" one said. \"We do not wish to embarrass them by any regulations designed specifically to deal with domestic circumstances.\"\n\n\"Yes. I know,\" I replied, turning the needle. \"Yesterday, some men I met in a park near my hotel mistook me for a Zulu, so I must look like one. I'm merely considering the possibility that I might be descended from one.\"\n\n# Chapter  \nSeven\n\nMY NEXT PLAN WAS to visit the Transkei, one of the Government-designated \"Homeland\" areas. It was an hour's plane ride from Johannesburg to Durban, the nearest airport to the Transkei. A car with a driver awaited me at the airport and we immediately took off on the three-hundred-fifty-mile road journey to Umtata, the capital town of the Transkei. I had expected that here, in a predominantly black enclave which was supposedly preparing itself for independence, I would find Blacks in control in all departments and at all levels of political, social, and economic life. My eyes were soon opened. At the Information Office, my first stop, the staff were all Afrikaners, officials of the central Government. The Information Officer welcomed me and promised to arrange for me to tour the Transkei. He would be in touch with me later that morning. I decided to use the time to look around Umtata.\n\nThe Transkei capital looked thriving and prosperous. Every kind of business enterprise was represented, including automobile and farm machinery showrooms, supermarkets, banks, filling stations, and several hotels. All of them White-owned. No signs that Blacks had any kind of economic foothold in this, their own community. I passed the neat new police station, the white policeman leaning lazily against the door, looking toward the new multi\u00adstoried Government buildings. Truly a thriving town, showing off its potential for growth and development. Blacks everywhere, but not in command, not in authority. About half a mile from the hotel I saw a charming single-storied building, evidently a school, attractive in its simplicity of design, the large windows promising excellent natural lighting for the rooms. A well-kept grassy playground occupied the adjoining lot. On inquiring about it from a passerby I learned that it was the white school\u2014a school for the children of white administrators and businessmen. Here in the heart of a black enclave, the White-only restrictions still applied. The charming bungalows, offices, shops, everything carried the invisible but unmistakable label, \"White.\"\n\nAfter lunch I set out, with the Information Officer, for a tour of some parts of the Transkei. It could not be accidental that this so-called black Homeland was, for the most part, rocky, infertile land which can barely support the local herdsmen's scrawny cattle and goats. Adjacent to the township were many neat, small bungalows, silent evidence of the social changes which have overtaken the region, as the men are lured away from the small farm holdings to the unskilled jobs in the township. The horse is less in evidence than the car. Beyond the township the bungalows gradually gave way to the traditional circular Zulu huts of thatch and clay, each with its small patch of maize; women working among the long rows of green stalks, men tending their cows on sparsely covered hillsides. Even here, in their supposed \"Homeland,\" Blacks were literally restricted to the outer limits of the township, out of sight of progress, needed only to grease its wheels.\n\nThe more I saw of the Transkei the more I sympathized with those urban blacks who were so determined to avoid being relocated to the Homelands. The Government's stated policy foresaw eventual independence for regions such as the Transkei. On what kind of economic base could such independence be founded? The businesses in Umtata were all White-owned, their profits surely siphoned out of the black community. I asked the Information Officer about this. He told me that the overall plan envisaged a gradual takeover of all businesses by Blacks. White businessmen were encouraged to employ Blacks and train them into the techniques of management. When a trainee showed himself capable of taking over, the Government could purchase the business from the owner at current market prices and resell it to the trainee-manager on extended terms. I remarked that I saw no sign of any Blacks being trained. The scheme was new, but was slowly getting under way, he claimed. I said that the places I'd visited all showed clear evidence of prosperity, and it seemed unlikely their owners would easily relinquish them. Umtata is the largest and busiest of the Transkei towns. I could not see the businessmen walking away from such a gold mine. He had no answer.\n\nEverywhere we drove the situation was the same. Blacks following their \"traditional lifestyle\" on land which grudgingly and barely supported them. The more I saw the more absurd became the Government's claim that it was nurturing these Homelands toward independence. I reflected on the recent turbulent history of my own homeland, Guyana, and the years of preparation in the management of government and services. How could powerless people learn to exercise power wisely except through experience?\n\nBack in Umtata, I took a stroll to a place which seemed to fulfill the joint purposes of bus stop, taxi stand, and open-air market. Only Blacks in sight. Overlooking this crossroads was an imposing new hotel. A fruit vendor told me that it was a new hotel for Blacks only, as they were not welcome at the other hotels. I didn't tell him where I was staying but, in reply to his question, admitted merely that I was an overseas visitor passing through the town. Inquisitively, two or three others strolled over to listen in on our conversation. I asked about the fruit on sale, tiny bananas, some hard peaches, and mangoes, and learned that they were grown on the patches of land tended by the vendors themselves. It was too early in the season for anything except bananas and peaches. They were surprised to discover that I knew about mangoes and could tell them about varieties familiar in the West Indies but which they'd never heard of.\n\nGradually, carefully, I steered the talk to independence, saying I'd heard in Johannesburg that the Transkei would become independent, and adding that perhaps some of them might be in the Government. This amused them.\n\n\"Who's been telling you those stories? Buthelezi?\" one asked.\n\n\"I read it in the newspapers,\" I replied.\n\n\"The newspapers are not for African people. They say what the white man wants to hear.\"\n\n\"I read that this homeland will become independent as a separate state, like Botswana or Lesotho.\"\n\n\"I'll tell that to my grandchildren,\" one young man said, \"and even then they will not believe it.\" He was about twenty years old. I suddenly realized that we were conversing easily in English. These rural Blacks were not educated men but they were able to converse with me in my language, and, most likely, they were as comfortable with Afrikaans. Now and then, they would revert to their tribal languages as if to underscore their linguistic range.\n\n\"What about you?\" I asked. \"Any of you preparing to be leaders?\" saying it with a smile, making my inquiry sound casual and unimportant. Immediately there was that exchange of glances I'd come to recognize, and with it the withdrawal. Two of the young men walked away.\n\n\"Did you say you are from overseas?\" the vendor asked.\n\n\"Yes. Why?\"\n\n\"Sometimes strangers come here asking questions.\" Then turned to one of his companions and spoke in an African language which ignored and dismissed me. Even so tentative an inquiry about political activity had been enough to excite suspicion and distrust. People thinking of independence would be preparing for it, somehow, and there must be some evidence of that preparation. Perhaps, as the man said, it was all white newspaper talk.\n\nBeyond the Transkei borders and into Natal, the countryside changed dramatically. The land was predominantly flat or rolling, perfect for farming on a vast scale. The road wound itself through lovely rural areas with attractive townships spaced between the wide expanses of farmland, mile upon mile of the lush green of wheat or maize, with here and there orchards heavy with oranges, peaches, or mangoes. The glow of prosperity lay over the neat, freshly painted bungalows with smooth, trimmed hedges and lawns. The modernistic spires of the calvinist kirks were a particularly dominant feature of each township. Wealth, comfort, and prosperity everywhere, the well-fed burghers chatting outside their houses, the ubiquitous black servants carefully sweeping, clipping, and tending.\n\nBlacks everywhere in each town, manning the filling stations and delivery trucks, always in the servant roles, the local burghers slow-moving in their untroubled security, seeming hardly to see the Blacks who fetched and carried for them.\n\nOutside Pietermaritzburg we needed directions for the shorter route to Durban and sought them at a police station. Two entrances to the same office, one for Blacks (all Non-Whites) and one for Whites. Three policemen standing outside, two Indians and one White. I approached the white one and asked directions to Durban. He merely stared past me, his pale eyes seeking some distant point beyond my shoulder. After a few moments I left him and returned to the car. My driver sought and received the information from the Indian policemen, the white one looking on. Perhaps the bastard thought himself too important even to speak to a black man. I wondered what kind of relationships obtained in that police station. We drove away.\n\nMy driver complained that I should not have spoken to the policeman, and said that, in his view, it would help me if I observed the \"Black\" and \"White\" signs where they appeared. He claimed that he did not support the Government's racial policies, arguing that he was of British stock. Yet he was obviously irritated with me for not falling in line. He predicted that, with my attitude, I'd have a rough time in Cape Town. I told him I'd be happy if he did the driving and left my behavior to me.\n\nReturning to Durban, I telephoned several people, friends of Johannesburg friends, hoping to arrange meetings. They were all Indian, which was not surprising as most of the Indians in the country are located in Natal Province of which Durban is the capital. Indians were originally brought to South Africa as indentured laborers for the sugar plantations in much the same way as they were first taken to Guyana. In both countries they had prospered, emerging mainly as truck farmers, sugar cane planters, and small businessmen. Under the Nationalist Government their fortunes had altered dramatically and, though they still enjoyed a few privileges denied the black African, they were subject to many restrictions. Like other Non-Whites, they are consigned to enclaves and though unlike black Africans they are allowed to purchase land on which to build homes, they may at short notice be moved to some other location if the authorities decide that the one they occupy is more suitable or desirable for Whites. One of these Indian friends, a doctor, accepted my invitation to come and share some tea within the hour.\n\nPunctually on the hour, my telephone rang. The doctor was calling from the lobby. She had arrived, but on entering the lobby, had been stopped and told that Non-Whites were not allowed in the hotel. She explained that she was calling on me and was eventually allowed to telephone my room. Angered, I went down to the lobby and without a word to anyone, escorted her up to my room. She seemed quite unperturbed by the experience, and wryly amused at my anger.\n\n\"I quite expected that they'd stop me,\" she said.\n\n\"Even when you said you were here to see me?\"\n\n\"Sure. They have to remind us that the presence of a black visitor in the hotel really makes no difference. We are still not welcome. Anyway, welcome to Durban.\"\n\n\"Thank you. Shall we go down and have some tea?\"\n\n\"I think it would be better if you had it sent up. I've had enough of white contempt for one day. Besides, we can talk more freely up here.\"\n\nWhile waiting for our tea she told me of her practice among her people, many of whom were able to make a good living, in spite of the increasing restrictions placed upon them. She, like so many of her friends and clients, had been born and raised in Durban where a thriving Indian community had developed. They had built a mosque and several good schools, cinemas, a community center. About seven years ago, the Government had rezoned that part of Durban where they lived and redesignated it a white area. Except for some businesses, the Indians were to be relocated some miles out of Durban. Their homes would have to be sold, either privately to Whites, or through compulsory purchase by the Government. In either event, the purchase price was frozen at the price obtaining when the order was first announced. The Indians protested the order but they had no political power base from which to make their protest effective. She said that as a result of the order, many families had become dislocated, and the community demoralized.\n\n\"If you'd like to come with me, I'll show you,\" she offered. The tea arrived, we drank it and set off. We drove about the Indian section of Durban while she pointed out a house here, a bungalow or office there, all evacuated and desolate awaiting either new white occupancy or the bulldozer which would level the lot for a new park or shopping center. We visited an arcade where I was introduced to some Indian traders, nearly inarticulate in their bewilderment and frustration.\n\n\"Is there no way of protesting these orders?\" I asked.\n\n\"It is dangerous for Blacks to protest,\" I was told. \"All it could bring you would be a cell in jail, or, at best, a cracked head.\"\n\nI was struck by this Indian doctor's inclusive use of the word \"Blacks,\" especially in a country where shades of color were so important in determining where one lived, or worked. Perhaps it was her way of responding to my sympathetic interest.\n\n\"Have you heard?\" someone interposed, excitedly. \"They're rioting out at New Germany.\"\n\n\"Who's rioting?\" I inquired.\n\n\"The Blacks at Frames.\"\n\n\"What's Frames?\" I asked the doctor.\n\n\"It's a textile factory complex, with plants scattered around Durban. Cheap black labor, Indian and African, at starvation wages. Seems the workers have gone on strike. Something must have happened out there to make the workers risk a confrontation with the police and their guns and their dogs.\"\n\n\"Where's this New Germany?\"\n\n\"In the suburbs. Not far away. Would you like to go out there?\"\n\n\"Could we?\"\n\n\"Oh, we could. The important thing would be to keep well away from any trouble areas, but it would be an eye-opener for you if your stomach can take it. The police can be quite brutal, you know.\"\n\nI assured her that my stomach would be fine and we drove off toward New Germany and the Seltex factory. My friend told me that nearly ten thousand workers were employed at ten factories around Durban, five of them owned by the Frames company, reportedly notorious for their bad working conditions.\n\nEven before I caught sight of the factory, we could hear the sound of human voices, a loud, unintelligible rumble punctuated by shouts. Around a bend in the road, we saw a large number of Blacks milling about in front of the main office building, and my friend pulled the car onto the grass verge, well away from the action. We got out and walked to a point of vantage some distance away from the crowds but with a clear view of what was taking place. Several police vans were parked in an orderly row, and near them stood several groups of policemen, both black and white, some holding large dogs on the leash, others carrying walkie-\u00adtalkies or armed with riot helmets, nightsticks, and rifles or Sten guns. A few dogs were straining at their leashes, but most of them sat obediently at the heels of their handlers.\n\nWe moved to the outer edge of the crowd to find out what had happened. My friend beckoned two Indian workers who came over to speak with us. They said that the situation was very explosive. The strike had been in progress since early morning and, so far, nearly four hundred workers had been arrested and carried off. Similar strikes were in progress at factories in Pinetown and Hammarsdale, all located within the environs of Durban.\n\nFrom some of the workers I was able to get the story of the circumstances which led to the strike. They had long been promised a pay raise, ranging from ten cents to two Rand per week, depending on length of service. When they received their latest pay envelopes it was discovered that management had reneged on the promise. Such conduct by management was familiar, but the decision to strike was spontaneous and unexpected.\n\nThe law of South Africa expressly forbids strikes by Blacks, and only very grudgingly nowadays allows them limited negotiating action. In striking the men were risking prosecution, especially under the vague but comprehensively punitive Suppression of Communism Act. I noticed, here and there among the crowd, white men in civilian clothes equipped with cameras and recording devices. I was told that they were members of the Security Police, collecting information for use in later prosecution of individuals.\n\nThe workers had reported for work as usual at 6:00 A.M. but had refused to go into the factory. Management had promptly called the police who arrived in their vans within minutes and stationed themselves in full, threatening view of the workers, waiting for any provocation. I noticed two Whites, a man and a woman, both civilians, moving among the Blacks, talking and quietly exhorting small groups. I was told that they were labor union organizers, advising the Blacks to continue to strike until better pay and working conditions were negotiated. In so doing, they were breaking the law, on several counts, and my informants assured me that, eventually, they would be severely punished.\n\nWhy eventually, I wanted to know.\n\n\"Probably any attempt to arrest them here would ignite the already volatile situation, beyond any power of control by the police. However, the police will get those two\u2014later.\"\n\nSuddenly the police were moving forward, in a spaced line, the dog handlers in the vanguard. The black officers seemed as ruthless and brutal as their white comrades. The crowd fell back before them. From the front ranks of the crowd a young black man broke away, hat in hand, moving toward the edge of the crowd. In an instant a police dog was loosed, swiftly and silently overtaking him. As the crowd roared in protest, he looked backward, stumbled and fell. The dog seized him by the bare forearm, worrying it from side to side as the young man screamed. The dog's handler reached the young man, roughly dragged him upright before snapping the leash onto the snarling animal, which reluctantly let go of his arm.\n\nThe young man had in no way threatened anyone. He was now roughly handled toward one of the police vans and shoved inside. The crowd kept up its noisy protests as the police continued their steady advance. Suddenly, without a sound, one of them clutched his face in both hands and fell forward. Someone whispered that he had been struck by a stone. The advancing police stopped and two of them rushed to help their brother officer. To my amazement, several black workers left the crowd and knelt to help the stricken policeman, gently cradling his head on their knees. Two policemen quickly brought a stretcher and placed their comrade on it, brushing aside the helpful Blacks.\n\nThe injured policeman safely away in an ambulance, the police continued their advance on the crowd until it was encircled on three sides.\n\n\"Start thinking up a story to explain why you're here,\" the doctor whispered to me. \"Just in case they come this way.\" A young black worker came over to us.\n\n\"You reporters?\" he asked.\n\n\"No. Just strangers, looking on,\" I said.\n\n\"Strangers from where? Durban?\"\n\n\"No, from overseas. I heard about the strike and came along to see what was happening.\"\n\n\"This isn't a strike,\" he said. \"It's a set up. The big boss Frame has been promising a raise for weeks, then they said we'd have it last Friday. So we get our pay and it's the same as always. No raise. So we walk out and they bring in the police. Always the same. They're saying that the news from the other factories is the same. I just heard that the Minister for Community Affairs of the Kwa-Zulu Homeland is on his way here for a meeting with management representatives.\"\n\n\"Is he white?\"\n\n\"No, he's black.\"\n\n\"Don't you have any local representatives to negotiate with management?\" I asked him.\n\n\"Representatives? The police look to see who's doing the talking and they pick him up. Sometimes the police send in their spies among us to start us talking, then they come along and pick us up. What about you? Suppose they find you here?\" he asked me.\n\n\"Suppose who finds me here?\"\n\n\"The Security Police.\"\n\n\"Well, I'm not interfering. Just looking on. The most they can do is ask me to leave.\"\n\n\"Don't be too sure. You're black. They could rough you up first and ask who you are later. In any case, my friend, when you move, walk. Never run. They're always looking out for black men running.\" With that cryptic remark, he wandered off.\n\nIt was frightening. All of it. The bewildered Blacks temporarily courageous in the rightness of their cause; invisible management exploiting their weakness; and the big brutal police, ready, willing, and able to wield their power. Helen Suzman's daughter was right\u2014it seemed as if the Blacks were being deliberately harried toward breaking point. The time must soon come, I told myself, when no show of police strength would be enough to contain the workers' explosive anger and destructive rage.\n\nEarly next morning, I left by plane for Cape Town, after promising to route myself once more through Durban on my return journey to Johannesburg. Arrived at Cape Town I booked into the President Hotel at Sea Point. I had hardly settled into my room when the Assistant Manager came to see me, apologizing for what he had to tell me. It appeared that when my reservation had been requested by telephone from Johannesburg, he, knowing who I was, had applied for and received a permit allowing the reservation. However, the permit was granted on condition that I, a black man, did not use the bar at the same time as white patrons. He said he way sorry to inform me of that proscription, but he had no choice in the matter.\n\nI was furious and told him so. I was paying for my accommodation at the same rate as anyone else and warned him that if I wanted to take a drink in the bar at any time, I would do so. Either he accepted my reservation or he did not, but if he did I demanded to be treated exactly the same as any other guest. He showed me a copy of the letter and permit issued by the Interior Ministry. I asked for copies of both so that I could study them and he promised to let me have them, but never did.\n\nNext day, Mr. Englebrecht MP, whom I'd met a year ago in New York, called for me and, with his wife, took me on a leisurely tour along the shoreline of Cape Town. As we talked I tried to measure his political acumen and sophistication. An Afrikaner, he had been a schoolmaster before entering politics as a supporter of the Nationalist Party and seemed to be in complete agreement with the Government's racial policies. To my critical remarks on the education, housing, and employment of Blacks he replied that the Bantu Act and similar laws were designed specifically for the benefit of Blacks. It became clear that, to him, Blacks are intellectually and psychologically inferior to Whites.\n\nHe spoke of how Blacks lived in the rural Homelands, the women in servitude, tending the fields while rearing children, the men content in continual idleness except for the little effort required to tend their cattle, sheep or goats. I asked how, if that were the general rule, the mines and other industries were able to recruit all the black workers they needed. Were the Blacks impressed into service like the sailors of old?\n\nHe complained that outsiders were far too eager to criticize South Africa without fully appreciating the prevailing situation. Few of the critics had even visited the country as I was doing for a firsthand view. The Blacks, he believed, were, for the most part, content with their circumstances, and would be quite responsive to gradual improvements, were they let alone and not incited to strikes and other anti-social actions by activists, often from outside. (An outsider from where? The only outsiders with easy access to South Africa are the British. Perhaps that's what he meant.) The Government was aware of such activities and had the means to stop them.\n\nEvolution, not revolution, he reminded me, and referred pityingly to the parlous state of the British economy which was crippled perhaps beyond repair by strikes which were all, he believed, Communist inspired. That would never happen in South Africa. In any case, he believed that many people in Britain were disillusioned with conditions there and were thinking of emigrating to South Africa. I argued that though many young people were leaving Britain, South Africa was lowest among their choices of a new home, and suggested that the near inevitability of bloody confrontation between Black and White might be responsible. He said that such a confrontation was most unlikely, because it would not be allowed to happen. His haughty assurance was disturbing. How could anyone with any claim to sensitivity be so determined to ignore all the ominous signs? Or was he only saying those things to me? In the time I'd already spent in the country, I'd already heard and seen enough to convince me that events were moving, however slowly, to some awful, cataclysmic d\u00e9nouement. Perhaps this man and his wife, so comfortable under the prevailing system, were really reassuring themselves that it would continue. At least for their lifetime. I then remembered that, when I had first met Englebrecht at a cocktail party in New York, he had told me that he was one of the group of censors in South Africa who had voted to ban the showing of the film of my autobiography, To Sir, with Love. When I'd asked him the reason for the ban, he'd replied,\n\n\"As we watched that film at a private showing, we were all irritated by the sight of that black teacher being so right all the time.\"\n\n\"What do you mean by being right?\" I'd asked.\n\n\"Well, being so knowledgeable, so bright, setting the example for white students.\"\n\n\"But the film was eventually released to the public.\"\n\n\"Yes, with a change here and there.\"\n\nI wondered if he was remembering that I was that same teacher in person, unchanged, still willing to make my small challenge to social absurdities.\n\nWe stopped for tea at a teashop and sat at one of the outdoor tables. When the black waiter served our tea he asked me if I was the same person whose photograph was in the morning newspaper. I said I was. He told me he'd read To Sir and would like me to autograph his personal copy of the paperback edition. I wondered if my companions appreciated the implication. As long as people could think and read, there was no knowing what they might do, for themselves. Perhaps the small miracle needed to change the course of events in South Africa was already on its way.\n\n# Chapter  \nEight\n\nLATER THAT WEEK I met with the Deputy Minister for Bantu Education. Formerly a dominie in the Reformed Dutch Church, he spoke of Blacks as if they were a lesser breed of man whose collective course must be carefully plotted and controlled by those whom God had elected their masters, the Whites. Painstakingly he detailed the many and varied educational programs he claimed were designed to help Blacks function in the South African society, and made it seem that huge sums of money were spent annually to that purpose. Courteously he brushed aside my suggestion that if the proscriptions against multi- and interracial education were removed, education for all might be less costly.\n\n\"Our Blacks are incapable of learning at the same pace as Whites,\" he told me, smiling his pontifical smile as if he knew he was quoting Holy Writ. \"They need to be helped, slowly and carefully.\"\n\n\"Toward what level?\" I asked.\n\n\"Toward their own level. It would be foolish to place them in any educational situation beyond their limited potential.\"\n\n\"I've discovered that under previous governments, Blacks attended your universities and gave an excellent account of themselves. In those days their potential was never in question.\"\n\n\"You were misinformed,\" he disagreed, still smiling. \"In those days there were those who experimented with the educational process and were anxious to prove themselves right. Many of our Blacks were frustrated by a university experience specifically designed for Whites. This Government has learned from those mistakes.\"\n\n\"I'm thinking of my own university education,\" I told him. \"At no time did I feel frustrated by it or incompetent to deal with it.\"\n\n\"You cannot compare your background with that of our Bantu.\" Nothing seemed likely to disturb his composure. \"I've dealt with them all my life. As you move about through our country and see them, I'm sure you will appreciate the difference between them and yourself.\" He stood up in dismissal.\n\nNext was a luncheon with Mr. Englebrecht and some of his fellow MPs, who were gathered and waiting for me in a small, private dining room in the House. When the introductions were made, I realized that they were all of the ruling Nationalist Party, and as we settled into conversation, I noticed how eager each seemed to prove his party loyalty and political savvy. One of them had recently \"crossed over\" from the opposition United Party.\n\n\"Would that be a case of political defection?\" I needled him.\n\n\"Not at all. For as long as I shared the views and objectives of the United Party members, I worked in pursuit of those objectives. But political attitudes should be responsive to the people's interests, and the United Party is philosophically behind the times. I believe in this country and its destiny so I joined the party whose objectives and vision matched mine.\"\n\nHe was going to some lengths to convince me that his move was not dictated by political opportunism. I wondered about his constituents. English parliamentarians seemed generally flexible in the face of political realities. These South Africans, without exception, demonstrated an unfailing inflexibility, making me wonder if it was because they dared not drift from the rigid philosophical center, for fear of being labeled liberal, the political kiss of death. They spoke of their country and its people, but left me in no doubt that they meant white people and were completely insensitive to the plight of anyone else. They seemed to believe that their political supremacy would last forever and that no effort by the Blacks inside or anyone else outside could reverse the situation.\n\nThere were no political signs to contradict them. Their main opposition, the United Party, was torn by internal strife and, in fact, offered no realistic alternative platforms. Differing only in degree, the United Party's attitude to Blacks was paternalistic and repressive, as they were careful to avoid any progressive recommendations beyond the minimal sops to overseas criticisms. The brand new Nationalist was in his element, an enthusiastic convert among those diehards. He suggested that South Africa's policies were hardly different from those practiced in some parts of the United States.\n\nI countered this by saying that I was currently on the faculty of a Southern university and had discovered that some of the most adventurously progressive changes in American social and educational attitudes were taking place in the South. Whereas a few years ago Southern American Blacks were denied even the right to vote, today they were competing for and sometimes being elected to high office, including that of mayor and could soon be expected to aim higher, much higher.\n\nLest there be any misunderstanding of where I stood, I reminded them that I was a native of a country of Blacks, governed by Blacks, descendants of slaves and indentured laborers. As I walked and drove around their country, I could discover no genetic difference between the black South Africans and myself, and felt confirmed in my confidence that I was the equal of anyone, black or white.\n\nThey remained courteously adamant, refusing even to consider the existence of any parallels. They believed they were in the right, the God-given right of Afrikaners who understood their place in Africa and were not interested in references to other, distant situations. They made no attempt to support their positions by argument. Each one would make a statement, either well-rehearsed or familiar through frequent use, and the others would express complete agreement. The wonder of it all was that they were saying it to me.\n\nI examined my own reactions to them. They were bigots, just like others I had met in other countries. Their bigotry was no less offensive to me because they were South African. On the contrary. How could they imagine that I, in my black skin, was different from the Blacks all around them? If they knew that the native Blacks feared and hated them and would seize the opportunity to be revenged on them, why did they assume that I would feel differently?\n\nMaybe they really didn't give a damn about what I thought or how I felt. Maybe their sitting, eating, and talking with me was the real measure of their contempt. I was black. In these near-intimate circumstances they could tell me and show me what they felt about the millions of Blacks like me whom they ruled with cruelty and contempt, enforced by banning, restriction, imprisonment, and death.\n\nI asked them about their political opposition in general and Mrs. Helen Suzman in particular. Their responses demonstrated their complete confidence in the power of their party and in the prevailing national indifference. They showed little respect for Mrs. Suzman's politics but were eager to claim her as proof of the Government's ability to accommodate criticism. Could these men ever conceive of relinquishing this enormous power or sharing it with Blacks?\n\nOn returning to my hotel, I received a telephone call from someone who introduced himself as Mr. Welcome Msomi. He told me that he knew of me, had heard and read of my presence in South Africa, was an ardent fan especially after reading Reluctant Neighbors with which he fully identified, and wished to invite me to a performance of the Zulu Theatre Company's version of Macbeth, or, in Zulu, Umabatha, written in Zulu by Mr. Msomi himself, who also played the lead. I promptly accepted the invitation.\n\nUmabatha was staged at the Maynardville Open Air Theatre. The aged, gnarled trees which ringed the grassy stage were an ideal background for the grim events of Shakespeare's bloody tragedy. From the opening moments when the three witches emerged leaping and shrieking from the cavernous dark between the trees, through the spectacular tribal dances, the plotting to kill Duncan (Dungane) and Banquo (Bhangane) and their deaths, and the final defeat and death of Macbeth (Babatha), the audience was held enthralled.\n\nSuch was the vigor of the play that, though I understood not a single word of Zulu, I was irresistibly carried along with its flow, its pace, its power, and above all the natural way in which it blended into its element, the starlit, African night. Whether crossing the stage in military elegance, spears closely aligned to present a colorful phalanx, bare feet pounding rhythms from the green turf, or leaping at each other in angry confrontation, the actors filled the stage with movement, now vibrant, now attenuated as death itself. Towering above all was Umabatha, power-hungry yet fearful, a willing pawn in the hands of his ambitious, resourceful wife, magnificently played by Daisy Dumakudi, who drew repeated cheers from the audience.\n\nAt the end of the performance, I made a surprising discovery. Beside me was a black couple, and we introduced ourselves and chatted for a few moments before the curtain went up. On the other side of me and in front were Whites, and I had felt pleased that there was no discrimination in this theatre. But looking around at the end, when the lights came up, I saw that behind me for a few rows were a group of seats occupied by Blacks, so, together we were a tiny enclave in the white audience, altogether little more than a score of black faces, but grouped together. Was it purely accidental that the only seat available was in exactly that spot? I'll never know, but it is possible to stretch the long arm of coincidence too far. I told myself to think only of the joy of the evening.\n\nAfter the show, I was introduced to the organizers, Mrs. Ren\u00e9e Ahrenson and Mrs. Cecelia Sonnenberg, and the black cast. In the men's dressing room they gathered about me, enveloping me with their enthusiastic welcome, telling me I was Zulu, my face, build, everything. They'd read and talked about my books, especially Reluctant Neighbors, because they knew from immediate and painful experience about white contempt for the black man. They appreciated the irony of this very evening, cheered so enthusiastically tonight by Whites who would not look at them in the street tomorrow.\n\n\"We are like court jesters,\" one said, \"entertaining them at their command, compensated by the scraps from their table.\"\n\nEven among the gaiety of our meeting the pervasive bitterness was there, yet it could not dampen their spirit or inhibit their determination to pursue their profession. They talked of other plays by Shakespeare and other playwrights which they planned to examine for their translatability into the fluid, powerful Zulu. One young actor asked me about my stay in Cape Town, about my movements, and whether I was being followed. He hinted that he would like to come to my hotel to talk with me privately. I told him I had no way of knowing if I was followed, but, if he was willing to take the risk, I'd be pleased to talk with him at my hotel.\n\n\"Ride back with me to the hotel,\" I invited.\n\n\"No,\" he replied. \"I must go back to town with the others, but I could come to see you tomorrow morning.\"\n\n\"Come and have breakfast with me.\"\n\n\"Okay. Around nine?\"\n\n\"Fine.\"\n\nHe arrived a little before nine next morning, with three others from the Umabatha cast. We ordered breakfast and I told them how deeply impressed I had been by their performance the night before. Tremendous.\n\n\"Sure. For the Whites,\" one said.\n\n\"I enjoyed it,\" I reminded them.\n\n\"Yes. We understand that. But who do you think benefits from it? The black actors and actresses? Never. Did you hear what happened to them in London? Same rave reviews. Same sold-out performances. Yet there wasn't enough money to pay the boys' hotel bills. Did you know that? Welcome might be doing all right, but what about the others? If it had been a white company with white actors on such a successful run, everybody would have been doing fine, Umabatha's a black company, of black actors. But the management is white. The Whites are doing nicely with the production. Only the Blacks are having a rough time.\"\n\nI needed to know where all this was leading, so I said nothing.\n\n\"We hear you've talked with some of the boys, some poets you met here in town, and that you might be writing about it. So we'd like you to get the story straight.\"\n\n\"What story?\"\n\n\"The story about how helpless the black man is in this country. You'll hear about the black poet and the black musician and the black writer and the black actor. We're all slaves, my friend. You know how ants keep aphids to milk them. It's that way with the black artist. If he has any talent he's milked by some white bastard until he's dry. That's how this country is. A black man cannot talk with publishers or promoters or people like that, so he has to have a white man to do it for him. So the white man takes over. He doesn't really represent the black artist, doesn't work for him. He becomes the boss. He calls the tune. He is the employer instead of the servant. He pays what suits him. The Black sweats his ass out. He creates. He writes. He directs. But at the end of it all, he's never mentioned in the program notes. Whitey takes Blackey to London, Tokyo, Perth. He fills his bank account with loot and the black man returns home with nothing.\"\n\n\"Wait a minute,\" I said, and picked up the Umabatha program. \"Look here. Welcome's mentioned quite prominently as originator, writer, and leading man.\"\n\n\"Shit, man. Have you talked with him? Sure his name is in the book, but who do you think reaps the harvest? Welcome thinks he's doing okay, but what about all the others in the cast? He couldn't do it alone.\"\n\nI wondered about that, trying to relate what these men were saying to the exuberance and enthusiasm I'd just witnessed. Was it possible for unhappy, dissatisfied players to perform with such verve? What was that line so familiar in the theater? The play's the thing. Perhaps it was true here in Africa, here among Blacks who considered themselves as professional as any Shakespearean specialist.\n\n\"Like all men, the black artist has to eat, my friend,\" one said. \"He has to have a family, he needs a decent roof, needs to buy material for his craft and all that. The white man knows all this, selects his talented lamb and draws him to the slaughterhouse. The throat is cut and the blood is drawn until the carcass is useless.\"\n\n\"Wait a moment,\" I interrupted. \"How do you know all this?\"\n\nThey exchanged glances with each other, and a few words in their own language. Then one said:\n\n\"We're artists. We're still students as every artist is always a student. We do different things, but we're all caught in the white man's snare. Without him we're helpless, we don't eat. With him we're meat which he slices as he wishes. The system protects him as he ravages our flesh. He smells out our talents before we're aware of it, then he sells our talent for his benefit. He becomes the agent. In the case of a painter or sculptor, he offers the work to galleries in the cities where we couldn't get into the door without his help, then he prices the work and takes the biggest slice. In the case of a musician, there is the recording studio. The white agent negotiates and often winds up owning the copyright. Sometimes his name appears on the record as composer. Same thing with live performances. I've yet to hear of a black musician earning thirty Rand a performance in spite of a full house.\"\n\nThe faces around me were now grim, bitter masks, the suppressed hatred of years spilling out with their words.\n\n\"You write books, my friend, and you get the credit. They make movies of your work and you get the credit and the money. You can come here and live in a hotel like this, a big room like this. In your country you can be a real artist. Here we are shit.\"\n\nA waiter wheeled our breakfast into the room, looking with some surprise at me and my guests. I told him we'd help ourselves, tipped him, and he left.\n\n\"You have an agent?\" one asked. \"What's he like? Black? White?\"\n\n\"White and she's a woman.\"\n\n\"I hear that over there you pay the agent a percentage. Right?\"\n\n\"Right.\"\n\n\"Here the man who represents you pays you something\u2014or nothing. What we tell ourselves all the time is 'Come together. Close ranks. End the squabbles among ourselves. Stop underselling each other. Let's find our direction, present what we want, where we want it and how we want it.' How will we achieve this? By striving for good standards like the white man did. By working at perfecting our art, not for any exotic superficialities, but for quality. Shit, the white man was not born with his high standards. He worked for them. We can work for them too. Not his, but our own.\"\n\nI liked what they were saying and felt my reservations evaporating rapidly.\n\n\"How will you achieve all this?\" I asked.\n\n\"By working together, supporting each other. By accepting hunger and pain until we can speak for ourselves, negotiate for ourselves. Some of our brothers have formed an organization. Mdali, in Johannesburg. Its purpose is to get all our people involved in discovering our talents and our arts. Up to now we've been preoccupied with showing off our talents and our arts to the white man, selling ourselves to him. Now we must forget that shit. We must take our art and talent to our people in the ghettos. If and when our art is good enough, the white man will come to us. In the ghettos. He will watch us perform among our people and will respect our people because he'll have to respect all of it. The art and the people.\"\n\n\"Listen, friend,\" one said. \"When we talk of supporting our own, we are talking more to ourselves than to you. The habits of eating, of warm clothing, of sleeping under a roof with a woman make slaves of us all. I am also a sculptor and after years of work, what do I have to show? Nothing. No house, no money, no clothes, no wife, no studio, no kiln!\" The last words said in a sudden shout as he jumped up from his seat and walked over to stare outside through the sunlit window. The others looked over at him but made no move to go to him.\n\n\"Friend,\" one said. \"We came to talk with you as a fellow artist, not to burden you with our pain. We really came to talk to you about acting and writing prose and poetry, and painting pictures and carving wood. But here we are sharing our pain with you. We look at you, at how you live here, the confidence with which you speak and we envy you. You are an artist, independent. That's why this government lets you come into this country, into this hotel. We want to be independent as artists. Independent of the white man, employing him only as we need him. We envy you. Look at me. Until coming into this room I've told myself I needed the white man, because he stood between me and the door. Every door. Now I feel I can promise the white man that I don't really need him, have never needed him. I should have known it all the time.\"\n\n\"Brother,\" another said. \"You see, we're now talking for ourselves. Each one for his work. We're actors because we need to eat. Each of us is something else. Something individual. Ben's a sculptor and gets very little for the work he does, then later sees pictures of his pieces in glossy magazines. Not even his name to them. Same thing with Biki's painting. Tom and me, we write. Some poetry, some prose. I would have liked to go on to the university to study literature, but that's a dream. Me, when the frustration gets me, I drown myself in brandy. If I can afford it. Like my brothers here, I want a chance to exhibit, to expose myself to the world, to compete. But what can I do here. Vokol. Nothing.\"\n\n\"Anything I can do?\" I asked.\n\n\"You're doing it, brother,\" one said. \"You invited us to come and talk with you. You listen. We heard about you but we needed to see and hear you for ourselves. We've seen Blacks from your country come here before. Bob Foster for one. He stayed here, hidden from us behind his managers and secretaries. We weren't sure about you. Come see us in Jo'burg. Come to Mdali and meet some more of the brothers. Will you?\"\n\nI promised I would. I remembered my early suspicion and distrust. Now I felt humbled. I could think of nothing to say to reach and touch them, to convey my oneness with them and their plight, for anything I said would merely emphasize my own fortunate position.\n\nI thought of suggesting that they flee the country and try their fortune in some more sympathetic society, but swallowed the words before they could leak out. These men were talking of developing themselves, proving themselves among their own people in their own land.\n\nI poured coffee and drank to them. To the cast of Umabatha, to those isolated on Robben Island, to Mdali, to Blacks everywhere involved in the struggle for freedom and dignity.\n\n\"And you with us, brother,\" they said.\n\n# Chapter  \nNine\n\nON MY RETURN TO Johannesburg, I accepted an invitation to address the members of the Executive of the Young Women's Christian Association in Soweto. Two members of the Executive, Mrs. Meteni and Mrs. Iowele, called for me at the hotel and visited for a few minutes. Both housewives, both married to men who worked in Johannesburg, they told me that their YWCA branch was all black, in keeping with the Government's segregationist policies. Both had attended YWCA conferences outside South Africa and sat in the same room with delegates from the white South African branch.\n\n\"Did that cause you any difficulty?\" I asked.\n\n\"Not us. We are not the ones who hold ourselves separate. Outside this country we meet as equals.\"\n\n\"Do overseas organizations know that here in South Africa you are kept separate?\"\n\n\"Of course they do, but they cannot interfere.\"\n\nOn the way to their meeting room they took me through a part of Soweto I had not previously seen to show me, they said, the reasons why the YWCA served a useful purpose in the community. Women everywhere, many of them with very young children.\n\n\"Each day, except Sundays, Soweto becomes a community of women. The men are away at work, so most of us are left to clean house, tend the children, and watch the days slip away with little or nothing accomplished. We don't get any satisfaction from recounting our common miseries. Whatever improvement is needed here must come from us. The Bantu Council meets and talks but nothing happens, because all money decisions must be made by Whites in Johannesburg. So we've begun. Out of money we raised ourselves, through dances and picnics and collections, we've built one meeting hall. It's a beginning.\"\n\nWhen we reached the meeting hall, I understood what she meant. It was a squat, L-shaped, solidly built red-brick structure, set on a small rise, the interstices between the bricks highlighted in white. Larger than any of its neighbors, it exuded an aura of elegance and permanence. The grass around it was neatly trimmed, and here and there recently transplanted trees stood like symbols of growth and hope.\n\n\"What do you think?\" Mrs. Iowele asked, proudly.\n\n\"Very impressive,\" I replied.\n\n\"It's been eleven years in getting to this stage,\" Mrs. Meteni said. \"When we can, we hope to add wings on either side of the main hall. Might take another eleven years, but we'll do it.\"\n\nWe went in and I was introduced to the other members. We sat down and they quickly and professionally attended to their business, reviewing their work among the young people, housewives and pre-school children of Soweto. From time to time their comments clearly indicated their separateness from the Whites, especially when they referred to invitations received to attend overseas conferences. They spoke of several fund-raising projects planned for the months ahead.\n\nTheir agenda completed, they invited me to address them. At this moment, two other women joined us, one of them white. Someone sitting next to me explained that she was a graduate student from an American university who was examining the social conditions in Soweto as a basis for a doctoral thesis. I spoke briefly, of the reason for my visit to South Africa and what I had so far seen as I traveled about the country.\n\nAfterward, they spoke about Soweto and about the very few options open to black women. A few were teachers, some were nurses, or helpers at the local nursery schools, and many were domestics. All were frustrated by the narrowness of their lives.\n\n\"We read the YWCA publications sent to us from overseas,\" one said. \"We read of the things women do, are allowed to do, in other countries, in other parts of Africa and here we are, forced to confine our interests and ambitions to Soweto, and, even so, what we do here depends on what the white man lets us do.\"\n\nSuddenly one of the group, a graying, buxom woman stood up, interrupting the others with, \"Sisters, we are wasting time, our own and that of our visitor. Let's not talk about the stupid irritations like where we can or cannot go. He didn't come all the way from America to hear that. Let's tell him about what's really important to us. Let's tell him about the thing which frightens us and keeps us helpless. Let's tell him about our Fear.\"\n\nShe seemed to capitalize the word, giving it a dimension of immediate threat, bringing it into the room with us. Looking directly at me she said, \"My brother, we women live in Fear, every moment of every day of our lives. So do our men, but they, most of them, go off to work each morning and can temporarily forget their Fear in doing their jobs. But we think of them at those jobs and we fear for them. I have a man, sir, a lovely man, a good man who looks after me and our children. He's a very intelligent man. I know, because I live with him. I love to hear him talk with me and the children or with our friends when they come to visit. I'm very proud of him.\" She paused and looked around at the others, all with their eyes focused on her.\n\n\"My man works as an ordinary clerk in the city, a job any boy could do, but he has no way of seeking or getting promotion. Young white boys order him about, calling him 'boy' sometimes. My man. My proud man. It's eating away at him inside.\"\n\nOne of the women who had come to my hotel looked questioningly at me, as if wondering whether I wanted to hear what her colleague was saying. I nodded affirmatively.\n\n\"I watch my man go off at five each morning, and I wonder whether today will be the day when the dam will break, when somebody will say the final word to him, when some Whitey will heap on his head the final indignity and my man will lose control.\" Some of the others were nodding their heads as she spoke, their faces grave, living with her in her pain.\n\n\"It will happen one day,\" she went on. \"I can feel it in my bones, because I know my man. One day they will say something or do something to him and he will blow up. Do you know what will happen then to him, to us? They will call the police and take my man to jail, and I won't know anything about it. I will wait for him and when night comes and he is not here, I'll know. And the children will know. And tomorrow I must go search for him.\n\n\"Do you think the police will come to tell me my man is in jail? Never. I must search for him. One by one I must visit the police stations and ask for him. Always I must wait. I must swallow my anxiety and wait while they look at me, hating me because my husband is a man, laughing at me because I am black and helpless. I must wait until they check their lists. Sometimes they spell his name or pronounce his name their way and do not recognize the name I give them, so they say 'Go. We do not have your man.'\"\n\nThe room was quiet yet vibrant with the spell of the woman's pain. It seemed as if it was there with us, happening before our eyes.\n\n\"So, day after day I must search, living with my Fear, living with my children who will draw upon my Fear and be frightened. Maybe in three or four days I'll find him, thrown into a stinking cell with many others, stinking with his shit and his Fear, as frightened as I am. The police will say 'Bring forty Rand to pay your husband's fine.' Forty Rand! It's as easy to tell me go steal a star from the sky. Without forty Rand my husband may be deported somewhere up North. Somehow I must get that money. So it is, my brother. We busy ourselves with this place, and whatever we can do here, to distract us from our Fear, for our men, for our children and for ourselves.\"\n\nHer voice broke but her eyes were dry, though I knew she was weeping behind her eyes. The looks on the other faces told me they were with her on every frightening step she took.\n\n\"There's something else,\" she went on. \"We also live in fear of each other. Look at us. All black. All poor. And yet, even among us, sisters you might say, there may be one or more who will later report to the Security Police about what has been said here, by you and by us, but especially by us. So, while we fear for each other, we go in fear of each other, selling each other for the puny privilege of a pass, a permit for a relative, or, worst of all, a few Rand. I read some of your books, my brother. You are a teacher. Tell us how to trust each other. That's what we need to know. Talk to us about trusting each other, because when we can trust each other, we can together be strong against the white man's tyranny.\"\n\nI was about to reply, thinking that she was finished, but she lifted a hand to bid me wait.\n\n\"The white man wants to keep us afraid. Do you know how he does it? Ask anyone here. The Security Police raid our homes. Everyone's homes. It's to see if anyone's living there illegally, without a pass. They always come late at night or very early in the morning when we are dazed by sleep. They pound on our doors to frighten us, and if we're not quick to open them, they break them in. They love to see us huddled in our beds, cowering against their flashlights and their guns. And their dogs. They love to pull the bedclothes off us and look at our nakedness. So we live, my brother. In spite of all that we've come together and built this place. Now tell us how we can build ourselves, that we might be stronger than our fear.\"\n\nAbruptly she sat down, leaving me weakly unequal to the challenge of responding to her, to them.\n\n\"My sisters,\" I said, letting the moment and the feeling dictate whatever I'd say to them, \"you've opened doors to a world I'd never known existed. I've lived in countries where Blacks must fight for everything they get and have, and in my own struggle I had imagined myself confronted with formidable difficulties. In the face of what I've heard today, I feel humble. I believe that people who suffer as you do and survive as you do, can discover in yourselves reasons to trust each other. It must have taken great courage for our sister to speak the way she did, and I would like to believe that we all respect that courage. Perhaps, at times, our personal, private needs seem greater than the collective good and I can only be sympathetic with those who must make choices under these terrible conditions. Be patient with each other.\"\n\nBefore leaving I talked informally with them, especially the matron who had spoken so eloquently and moved me so deeply. I loved her, the dignity and majesty limned in her smooth black face and I knew that the spirit of freedom glowed strongly in her and could ignite the feebler ones. I felt encouraged and strengthened.\n\nBack at the hotel I was overcome by restlessness. I missed the casual ease of consorting with my friends in the U.S.A. and the challenge of my work. Here and now I was surrounded by hate and anger and menace and people engulfed in suffering. From my window I could see groups of young men scattered about the park and suddenly decided to go down among them.\n\nI chose a group at random and sat on the grass nearby, within easy earshot, looking at them, evidently taking an interest. They were conversing in an African language, very animatedly, waving arms, sometimes jumping up the better to emphasize a point. I was fascinated by it all, and by the realization of being so much the outsider, not understanding a word of what went on.\n\nSuddenly one of them noticed me and said something. I shook my head, smiling, and explained that I was a visitor from overseas, staying across the street at the Landdrost Hotel. They looked at me in surprise, then at the hotel and suddenly gathered around me, bombarding me with questions, in English. Suddenly one of them remembered reading something about me in one of the newspapers and asked if I was the author of Reluctant Neighbors. He had not read it but knew someone who had, and remembered some of the things discussed in it. For a while, we talked about me and my books, but gradually they got back to what they had been talking about before. It was the letter-bomb murder of Abraham Tiro in Gabarone, Botswana.\n\nMr. Tiro, self-exiled in Botswana, had formerly been the leader of one student organization, and was, at the time of his death, president of another. In a recent letter Mr. Tiro had affirmed their joint solidarity and had concluded with the enigmatic line, \"No struggle can come to an end without casualties.\"\n\nHardly noticing me, they continued the discussion in English, arguing that the assassination was engineered by agents of the South African Security Forces, because Mr. Tiro had cleverly eluded them and fled to Botswana. The letter-bomb, according to them, was a highly sophisticated Western device, far removed from the kind of thing any African would ever think of using against another African. Furthermore, it was another attempt by the South African Security Forces to intimidate Africans by showing how capable they were of reaching their quarry, no matter where they hid. Suddenly, one of them pounded his fists on the ground, his face distorted with an anger he could barely contain.\n\n\"Look at us,\" he cried, \"just look at us. Do you know what we're doing? We're mourning a brother. The white man has murdered our brother in Botswana and we are mourning him. But even that we must do here, in the open, away from prying eyes and ears. Why do we live like this? Our brother is dead. He killed no one, harmed no one. He merely spoke of the imperative of human dignity. He only spoke out against white brutality, and for that he was hunted and hounded from his school, his home, his country, and now his life. They watch us, his mourners, listening for some sign of protest. They let us know they're watching us, to make us fearful. And yet they say they fear us. They say we outnumber them and they fear us. From outside you are likely to believe them. Think of it. Twenty million Blacks against about four million Whites. From outside you will think of those numbers and their imbalance and you will believe that the white man really lives in fear of us. From outside you cannot see the way in which we are dispersed and weakened. Effective resistance is not established overnight. It needs to be planned, nurtured, and led. We need to see leaders, to give to any movement a point of reference. We need to know a man, with a name that we can talk about to our wives, our children, ourselves. We need to see our pride, clearly, in the flesh, to be reminded who we were and could be.\n\n\"We are ashamed, my friend, to sit here like women and weep for those whom they ban and imprison and kill. We, the sons of sons of Zulu warriors sit and cry because we are powerless in our fear. We should rise up against them. Perhaps you, too, sit there thinking that we should forget our fear and rise against those who make us afraid.\"\n\n\"You were at Dorkay House,\" one said, pointing at me. \"The night some white men came to listen to the bands from Soweto.\"\n\nI admitted that I had been.\n\n\"Somebody said you write books,\" he said. I agreed.\n\n\"Will you write about us?\"\n\n\"I cannot be certain. After all I do not know your names. I will write about the conditions under which you live and perhaps, if you ever read the book, you might be able to identify yourselves.\" Saying it to have some real part in the conversation, not really knowing if I would write about them.\n\n\"That's good, friend. No names. Please, no names. But if you write about conditions here we shall never be able to read your book. Not here. Don't you know that books critical of the Government are banned?\"\n\n\"I know. My books were banned for a time.\"\n\n\"You heard about that Afrikaner writer, Brink? He's written a book about a White sleeping with a black woman. Hell, they're after him. The politicians, the church, everybody. Shit, everybody knows it happens. That's why they've made laws against it. But the laws don't stop people from doing what they want, it only makes them cautious. Where the hell do they think all the half-breeds, the Coloreds, come from?\n\n\"Shit, brother. Can you imagine what would happen if a Black wrote a book like that? Nobody would even publish it. Look at us. We write simply stories and songs about ourselves and nobody wants to touch them. Naturally we write about our pain and our problems, not about sleeping with some white woman. You ever heard of Benedict Vilakaizi?\"\n\n\"No. Who is he?\"\n\n\"He was one of the few Blacks to teach at Witwatersrand University, years ago, before this Government changed all that. He was also a poet. He wrote many poems. Go to a bookstore and ask for Zulu Horizons. That's the name of his book of poems. Listen to this:\n\n> 'Yes, when a siren screeched one day,\n> \n> A poor black dassie\u2021 heard its call\n> \n> And, answering its summons in confusion\n> \n> Was trapped.\n> \n> And then\n> \n> Transformed into a mole,\n> \n> Was forced to burrow deep and search for gold.'\"\n\n\"That's very good,\" I said.\n\n\"Shit, brother. It's more than good. It's beautiful. And it's real.\"\n\n\"I'd like to meet the poet,\" I said.\n\n\"You will, one day.\" Laughter. \"For now, he's dead. Died about thirty years ago. But he was a real Zulu. He spoke to the heart and soul of his people, to their pain and their pride. He was a father to us, not like Buthelezi.\"\n\n\"What about other black writers?\" I asked, wishing to keep him off Buthelezi.\n\n\"Oh, there are many. Many. But nobody wants to publish them. When a few of them get published, there's nothing much in it for them. But they, too, speak for us. Like Stanley Wotzuwadi. Listen to this:\n\n> 'I get my cue\n> \n> From the glint in the cop's eye.\n> \n> I have seen it before.\n> \n> So I have to find it.\n> \n> I pull away from Mons and hug myself in desperation.\n> \n> Up, down, back, front, sides, like a\n> \n> Crazed tribal dancer\n> \n> I have to find it.\n> \n> Without it I'm lost.\n> \n> With it I'm lost, a cipher in Albert Street.\n> \n> I hate it. I treasure it.\n> \n> My pass. My everything.'\n\n\"You understand. He's talking to me. About me. About us. No white man can understand those words.\"\n\n\"You know about the Book of Life? Our pass book?\" one asked.\n\n\"Yes. I've seen one,\" I replied. \"Where can I buy some books by these black writers?\"\n\nThey told me of a bookshop on Commissioner Street, a few blocks away, which was known to carry books by Blacks, native and foreign.\n\nAfter some further conversation they invited me to have a drink with them later and I agreed to meet them outside the railroad station a short distance from my hotel. I left them, to search for the bookshop and purchase copies of books they'd mentioned, poetry and prose by Blacks, the few who had managed to get some of their work published. As they'd directed me I took the route through Eloff Street toward Commissioner Street.\n\n\u2021 Black peasant.\n\n# Chapter  \nTen\n\nHALF A BLOCK AWAY from Commissioner Street I saw a black man running, just running along Eloff Street toward me, on the roadway near the sidewalk to avoid the traffic. There seemed to be no one pursuing him and I was not alerted to any trouble until two white men stepped in his way and grabbed him, one of them holding him tightly by the collar of his shirt. They were shouting at him in Afrikaans, so I could understand nothing except the repeated, \"No, no, Baas\" from him. They cuffed and kicked him, shaking him meanwhile.\n\nA crowd quickly gathered, some of them asking what the man had done. Some of these spoke English but all I could make out was that the man had been running. The men who held him had not been chasing him, nor did they know why he'd been running. Someone called the police, who quickly arrived and took over from the two civilians, handling the man just as roughly. One policeman, a big, red-faced, crew-cut fellow slapped the man on the side of the head with a huge meaty hand, snarling at him in Afrikaans. Someone near me called out to the policeman in English to stop hitting the man. The policeman told the bystander to shut up or he'd take him down to John Vorster Square for interfering in the due process of justice.\n\n\"Christ, is that justice?\" I heard myself ask. The big policeman glared at me. I felt pitifully helpless, a stranger, unable to do anything but look on, as they dragged the man to where they'd parked their car.\n\nThe bystanders began chattering among themselves, some expressing outrage at the two who had stopped the running man, others angrily defending their action. A crazy mixture of English and Afrikaans. Whites shouting at each other. A few Blacks standing well away from it, watching but saying nothing. I was too near the center of it for comfort, dazed by the suddenness, the sheer brutality of it. I felt myself shaking and knew that I was frightened. Suddenly, unexpectedly frightened. Christ, a minute ago the black man had been running here, the slap, slap of his feet still a faint echo in my ear with the memory of his loose shirttails fluttering behind him; now, he had vanished, in the company of brutal men, probably to be beaten while screaming for mercy, no one asking questions, no one interested in explanations. If it had been nighttime, he might have been killed on the spot. For what? For being in a hurry?\n\nAt long last I realized what so many people had been trying to tell me. It finally sank in. If you're Black in South Africa, there's nothing between you and sudden violence, nothing to protect you from the hate of centuries. On the street, in your home, anywhere. Not the laws, not the courts, not the police. \"Walk,\" they said to me, \"don't run. Don't ever run.\" I felt exposed, brought up hard against the fact that here I could not take for granted even the simple protection of personal space that strangers in all other places respect between each other.\n\nSlowly I walked away from there to the bookshop. The bookshop attendant, a young, dark-haired woman, not only had the books in stock, but had read them and spoke knowledgeably and enthusiastically about them. We went downstairs to the lower floor of the shop. While I waited for the books, there would be footsteps from time to time on the stairs which were near the cashier's desk. Each time the young woman would quickly, furtively glance behind her.\n\nIt happened again and I asked, \"Why are you so nervous?\"\n\n\"Nervous?\"\n\n\"Yes. Every time there are footsteps on those stairs you look frightened.\"\n\nShe blushed. \"I guess you're right. It's become a sort of occupational disease. But I can't help it. Each time it happens I promise myself that the next time I'll just ignore it, but I can't.\"\n\n\"But why? They're only customers, like myself, coming to look and buy.\"\n\n\"Not always. Regularly the Security Police drop in. It's got so we can nearly recognize their feet. They come in and flip through the books on display, deciding in their own strange way what must be withdrawn, what may be sold, what jackets must be covered. It's a kind of harassment, and there's nothing we can do about it. They're always looking for Communist literature and any title that's the least bit dubious, to them, gets the book banned.\"\n\n\"What do you mean when you speak of covering jackets?\"\n\n\"I mean just that. I'll give you an example.\"\n\nWith that she went to a rack and returned with a paperback copy of The Great Gatsby, which was published in England. The cover carried palely colored drawings of three people, two women and a man, dressed in the style of the twenties, the women's clothing little more than a frothy film over the thin pencil lines of their bodies. There was nothing even vaguely suggestive about the drawings; one could differentiate between the sexes, that was all.\n\n\"They've ordered us to place covers over these jackets,\" she said. \"The book has recently been prescribed for Secondary Schools, and these men claim that the cover is too suggestive for exposure to the students.\"\n\nI was tempted to disbelieve her but for the seriousness in her clear gray eyes.\n\n\"Jesus Christ,\" I said.\n\n\"Oh, yes,\" she replied, \"They claim they are doing it in his name and in the interests of morality. Several times each week, without any advance notice, they come in, paw their way through our books and take some away, ostensibly to read and inspect them for questionable ideas and philosophies.\"\n\n\"Mine were banned,\" I said.\n\n\"Yours?\" Her eyebrows raised in surprise. \"Are you a writer, too? What's your name? You're not South African, are you? Your accent. Sounds English to me.\"\n\n\"Yes, I write. The name's Braithwaite and I'm not South African.\"\n\n\"Braithwaite! Of course. To Sir, with Love. I read in the Mail that you were in Jo'burg. Yes. Of course. For a while we were prevented from selling your books, even though To Sir was prescribed for the Training College. Can you make any sense out of that? You know they banned Black Beauty, the famous children's book, some years ago? That's the kind of mentality we have to deal with.\"\n\n\"Now I understand your nervousness. I'm sure I'd behave in the same way under similar pressures.\"\n\n\"Oh, in my case it's more than that. I was in jail for several years. For opposition to the Government. They jailed me under the Suppression of Communism Act, though they knew I'm no Communist. But that's a sort of catch-all. After I came out of prison I was restricted to my home for a long time. Couldn't travel or entertain friends or anything. It's like if you stopped living but went on breathing. I thought I'd get over my fear of them, but it happens in spite of myself. I suppose it's because I can't bear the thought of being imprisoned again.\"\n\nSomehow her words made a far deeper impression on me than did those of the Indian who had been held in Robben Island. He was still full of pep and vinegar and, given the chance, would again be inciting resistance. But this young woman had been hurt deeply as only the insider can be hurt when visited by the wrath of his own kind.\n\nI paid for my books and left her. At every turn the underlying ugliness in the society was breaking through the upper crust of comfort and prosperity. To keep the majority in a state of fear some of the minority must themselves become victims, spreading the virus of fear within as well as without. With my small burden of books, I walked the short distance to the railway station. The five young men were waiting for me.\n\n\"Got your books?\" one asked.\n\n\"Yes.\" I decided to say nothing of the incident in Commissioner Street. \"Where are we going for our drink?\"\n\n\"Come to where we live. In Soweto.\"\n\nMy immediate impulse was to refuse. I'd seen enough of that place.\n\n\"I've been there. I was there today, visiting with the YWCA.\"\n\n\"Women!\" one said, scornfully.\n\n\"We're inviting you to come and have a drink with us, brother,\" another said, pointedly. I realize I should have anticipated this and invited them to my hotel. They watched me, waiting.\n\n\"Okay,\" I said, hating myself for my weakness. What the hell was I trying to prove?\n\nOver my objections one of them said he'd get the tickets and we walked on to the station platform. The train was there, its engine hissing impatiently, perhaps at the Blacks who were shoving and pushing each other to get on. My friends pulled me into the surging mass.\n\n\"Why do we have to get into this one?\" I protested. \"I can see other carriages further along.\"\n\n\"Those are for Whites,\" was the reply, by which time I was caught in the surge and propelled along into the carriage which was already packed beyond capacity. All black people, those who were lucky enough to be seated, uncomfortable under the leaning weight of the rest of us. The place stank from the limited ventilation and the crush of human bodies. I wished I'd been strong enough to refuse this invitation.\n\nSoon I was perspiring, the cool rivulets trickling their way down my armpits and my back. My arms were pinned to my sides, the package of books a rough discomfort against my ribcage. The knowledge that the man braced against me was feeling the same discomfort helped.\n\n\"You could have been riding comfortably with the Whites,\" one of my friends whispered hoarsely in my ear. \"For as long as you're visiting here you're regarded as White, did you know? Honorary White.\" He made the two words sound like a curse.\n\n\"Now you tell me,\" I whispered back, letting him gloat over his small imagined victory.\n\nThrough the crush, the white conductor forced his way, grabbing tickets from outstretched hands, quickly examining them, shouting at those whose tickets were not in order, abusing them, beefy, red-faced, belligerent. We swayed against his rough, onward passage. I felt helpless, with one foot barely touching the ground, dependent entirely on the pressures which kept me erect. As he elbowed his way past me, I saw the anger in his pale gray eyes, anger and contempt for the black mass through which he must claw his way.\n\n\"He's got to get the tickets before we reach Soweto,\" was whispered to me. \"Once we got there they'd walk all over him.\"\n\nWith a squeal of brakes, the train shuddered to a stop. Gratefully I waited for some easing of the pressure. By twisting my head around I caught the eyes of one of my friends.\n\n\"This is a white stop,\" he explained. \"No Blacks getting off here. We've got more than an hour of this.\"\n\nThrough a window I saw the legend, Braamfontein in bold lettering on the fence along the station, and beyond the red gleam of a bungalow's roof nestling under trees. Evidently a white suburb. Soon we were on our way, making more stops for the departing Whites\u2014Langlaate, Croesus, Canada\u2014depositing those who had been riding in cushioned ease.\n\nI could have been riding comfortably too, as an Honorary White. Thinking of it I felt the full impact of its debasement. For all these years I'd been living proudly in my black skin, doing very satisfying things in it. In this same skin I'd spent a happy boyhood in Guyana, learning about ambition and pride and the pleasure of competitive effort from parents and teachers and others, most of them in black skins like mine\u2014some white, but treating our black skins with respect. In this skin I'd sat with other undergraduates in an English university, pitting my intellect against theirs, confident in my abilities. In this skin I'd flown a fighter aircraft during the war, had known love, anger, despair, and success. In this same skin I'd written my books, taught Whites, and represented my country as a diplomat.\n\nThis skin had always been good enough for me. Men had admired my prowess in it. Women of many colors had found it beautiful. Never before had anyone, anywhere, attempted to change it. Yet now my color was far more important than anything I might be or do. Piss on their Honorary White! I'll ride Black.\n\nAt Canada Junction some of the crowd dismounted to find their branch lines to various parts of Soweto. We continued on to Orlando, from which we had a long walk to where they lived. We walked along a dusty, rutted road, through rough weed-grown land cluttered with stones and here and there a pile of rubble, all that remained of former homes that had been bulldozed to the ground. In the near distance the houses were little square boxes softened by the shrubs and small trees which grew around them.\n\nAt a crossroads a group of men stood in excited conversation near a car which had run off the road and now was tilted lopsidedly on the grass verge. Along both sides of the roadway people were at their doors or windows looking toward the group of men. We stopped and one of my acquaintances spoke to them in their language. There followed an outburst of sound accompanied by much arm waving.\n\nThis car and another had been involved in an accident. An argument had ensued and the police had arrived. One driver had promptly taken to his heels. The police had shouted to him to stop, and without waiting for him to comply, had shot him, there at the corner, the bloodstained grass providing mute evidence. The wounded man had been removed in the police car to hospital, but was not expected to live. A crowd had formed and the police had ordered them to disperse. The people had merely retreated to stand outside their homes, in an ugly mood, but helpless and defeated.\n\nWho cared whether another Black was shot and killed! Perhaps, at the hospital it would be discovered that he did not have a residence permit or Book of Life, and was therefore in Soweto illegally. After all, once a black man was in the hands of the police, identity material could easily disappear. No local resident would dare complain about the actions of the police. In any case, who could they complain to? The police themselves? The white courts? No. Fear of the white man dominated their lives. Fear of sudden violence, arrest, deportation to some remote rural area.\n\nAlong the dusty street, small groups of black men and women whispered together, their voices subdued even though the police were long gone. I couldn't imagine this fearful numbness among Blacks in New York or Chicago or London or Birmingham, in Jamaica or anywhere else where large groups of Blacks lived. Perhaps, in spite of my acquaintances' optimistic talk, they were already demoralized past any resistance.\n\nI said as much, but was told that no one trusted his neighbor enough to band together. Resisting vicious police tactics in broad daylight was one thing, but where would there be any help if the police called at night, with their guns and their dogs, carefully selecting the houses of those who'd spoken up against them? Who'd lend a hand when his neighbor was dragged out and away, no longer militant, but abjectly groveling and begging for mercy?\n\nThis they lived with, these young men, desperately trying to blow some faint sparks from their despondent spirits, willing themselves to cling to faint dreams of a freedom which grew fainter and more distant each hour. They asked me about Blacks in Harlem and the deep South of the United States, hoping to hear from me accounts of white brutality to Blacks which might offer them some small consolation in their own desperate situation. But I told them that though the American police, given the opportunity, could be just as racist and brutal, Blacks in the United States were militantly aggressive in their own defense. Somehow, these young Africans had been fed the idea that their condition was in no way different from that of American Blacks. They quoted stories of Lester Maddox of Georgia and his ax-handles and of George Wallace of Alabama defying the Court's orders, but did not know that history had already overtaken these men. They liked to tell themselves that they would one day rise up against their oppressors; they even imagined themselves engaged in covert activities against the Whites. But it was all bravado. Empty. Mere posturing.\n\nThey angered me, these young men. I thought of people I knew in Europe and the United States, black and white, who had talked with me in the fond hope that the black South African would eventually rid himself of the incubus of oppression, by the bloodiest means, if necessary. I thought of the young Blacks in my classes at New York University who'd believed that the militant projection of their blackness was a part, perhaps the most important part of their African identity. Were they identifying with this weakness, this demoralized hopelessness? These men had been bleating about the death of a brother, but in fact Tiro's death had kindled no fire, had engendered no rage. A few minutes of huddling in grief, or beating the unresponsive earth, that was all. The realities had to be faced, clamored to be faced. Tiro was dead and had already faded into the pitiful legends of yesterday. Today was now, the job in Johannesburg, the Book of Life, the quarter room in Soweto, the paralyzing fear of Whites.\n\nAs we continued on our way to the houses we passed the elementary school. From the outside it was a large solid enough structure of reddish brick built to form a hollow square, single-storied and squat, the rough mortar between the bricks suggesting haste in construction. The ground around it was red clay nearly covered by a ragged growth of weeds, kept somewhat in check by human feet, because it provided the playground area for the school. The whole was enclosed by a wire fence torn in several places.\n\n\"I went to this school years ago,\" one said.\n\n\"I'd like to take a look without interrupting anything,\" I said. \"I'd like to see them without anyone putting on a show for me.\"\n\n\"Then just look inside. They're accustomed to people looking in. The teacher won't mind. Nobody will put on any show for you.\"\n\n\"How do you know?\"\n\n\"I'm always poking my nose in. We need to know what's going on in the schools, so we check from time to time. Go ahead, if you like.\" We stopped beside a classroom. He'd said \"we\" as if he were speaking for some absent group or organization.\n\nFrom outside the only sound we could hear was the low voice of one person, perhaps the teacher, rising and falling softly. The door was slightly ajar, and, urged by my companions, I opened it further and went in, followed by them.\n\nThe room was small and very overcrowded. The benches and few desks supported twice as many children as they were designed for, squashed and perched wherever they could squeeze their thin bodies. Those who missed out on the benches and desks were either standing at the back or squatting on the bare concrete floor along the sides of the room. In a room probably built to accommodate fifteen or twenty children in comfort, there were eighty or ninety. Most of them had plastic sacks in which they carried and zealously guarded their books, rulers, pencils, and other supplies.\n\nConsidering the external surroundings, the children were tidily dressed, the boys, for the most part in gray shorts and gray or white shirts, the girls in dark blue uniform dress with white bodices. They hardly noticed our entrance. Without exception their attention was raptly fixed on the teacher, absorbing his every word.\n\nI leaned against a wall, watching what was, to me, a miracle. I'd forgotten that, in many parts of the world there still existed a ravenous hunger for learning and knowing. If had been so in my boyhood days in Guyana, even though we were never cramped like this, never oppressed in this way. But in the long intervening years of watching students and being a teacher, I'd become accustomed to other conditions, in which students needed to be inveigled, coerced, bribed, or flattered into making the smallest intellectual effort.\n\nThese youngsters were eager, their faces and eyes bright with either the enthusiasm of discovery or competition, perhaps already aware that a great deal depended on them, and knowing well that outside there were others who would gladly take their places.\n\nThe teacher nodded in our direction but continued with his lesson, apparently unperturbed by the interruption. The lesson was conducted in Afrikaans. He would ask questions and then have to select an answer from among the forest of waving hands which clamored for the chance to reply. None of the children seemed to notice the heat or the overcrowding. They were in an intensely competitive situation and were fully responsive to it. This lesson would be followed by one in English. Until the age of eight or, in some cases, ten years, the child reads, writes and does his counting in an African language, then is abruptly switched to studies conducted in Afrikaans or English or both. Blacks view this as a deliberate plan to inhibit their progress in a society which uses Afrikaans and English exclusively and interchangeably.\n\nObviously, there never would be any problem of discipline here, because there was no boredom. The children seemed to be soaking in every tidbit of information through eyes and ears, through their very skin. But what of tomorrow when even the minimal haven of this school would be denied them? All this youthful energy and thrust must inevitably collide with the white man's blockades and become poisoned with frustration, anger, and hate. I could almost feel it, a near tangible force, the accelerating buildup of energy as each graduating group was forced out into its own confrontation with the cold, closed world. How long would they be denied? Eventually, clubs, police dogs and even guns would not be able to subdue them, and that was exactly what the Whites feared.\n\nOutside, my companions decided to stay in Soweto and offered to accompany me to the railway station. They seemed to have forgotten all about the drink they'd promised me. I told them I'd had enough of that and would rather suffer a taxi, if they'd help me find one.\n\n\"Can't take it more than one way?\" one asked, grinning. \"We make it both ways, five times a week. Anyway, you were lucky today. Nobody picked your pocket.\"\n\nInvoluntarily I checked. He was right.\n\n\"Would you show me where I'd find a taxi?\" I asked.\n\n\"Not easy at this hour,\" one answered. \"But don't worry, I'll run you into Jo'burg.\" He led the way to his home, next to which a shiny, near-new car was parked.\n\n\"I never use it for work,\" he said, touching it fondly. \"Can't afford to run it to the city every day. Then the cost of parking it. So I use it mainly on weekends.\"\n\n\"As many a late virgin will certify,\" another added, and all joined in the ribald laughter. Now that the matter of my transportation was settled we stood around outside his house, chatting in lighter vein with each other. Eventually, over my broad hints, the owner of the car started it up and we climbed in, the others riding only as far as their homes. On the way to the city, he was more relaxed with me, talking about his job as a warehouseman\u2014a dead-end, but it kept him alive. In contrast to the power-cut gloom of London which I'd recently left, Johannesburg's night was lit like a fairyland, its power stations all fueled by coal. A few blocks from my hotel we stopped at a traffic light and I noticed three smartly dressed young women, black, chatting together on the pavement, their lush bodies, bright, lipsticked mouths, and bold postures seeming out of place at that hour in Johannesburg.\n\n\"If Blacks are not allowed to live in this city, where would they find clients?\" I asked my companion.\n\n\"If you wait long enough you'll see,\" he replied. \"They're waiting for Whitey. When it's dark and he thinks nobody's seeing him he leaves his wife and goes looking for black pussy.\"\n\n\"But what about the police? Don't they pick them up?\"\n\n\"Only those who don't pay.\"\n\nSo much for apartheid.\n\n\"You want to hear a famous saying?\" he asked me, smiling wickedly. \"A real proverb? A Soweto proverb?\"\n\n\"Go ahead,\" I said.\n\n\"The final destiny of the white man lies between the black woman's legs. Work that out, my friend.\"\n\n# Chapter  \nEleven\n\nTWO DAYS BEFORE I was scheduled to leave South Africa, some students from Witwatersrand University telephoned. They said they wished to visit and talk with me, and I agreed to have dinner with them at my hotel; they wanted to take me to a local restaurant, but one near-experience of that was enough for me.\n\nI suddenly had a feeling of confusion. Talking with one of the students on the telephone had brought the old, familiar feeling of excitement which always comes to me at the prospect of meeting young people, challenging their intellects and having them challenge mine, learning from them and hoping to teach them. Life had, so far, favored me with a wide variety of experiences which lent themselves to excellent illustrations whenever I needed to enliven an academic topic. These students had invited me to meet and talk with them. They were White. For weeks now I had been bombarded by the ugliness of white bigotry toward Blacks. I'd seen a young man beaten and humiliated for no reason. He'd been running, that's all. I'd heard lovely black women talk of the fear which was a major ingredient of their daily lives. I'd traveled, cooped up with other Blacks like cattle in a truck, while Whites rode in comfort on the same train. Now here I was, reacting with pleasure to an invitation to meet and consort with Whites. Did the fact that they were students make the difference?\n\nI wondered what the young men I had so recently visited in Soweto would think of me, if they knew I was entertaining a group of Whites. Would they consider me insensitive to their plight? But why worry about what they would think? What did I think? In the face of all the injustice I saw all around me, how could I justify to myself the feeling of pleasure at meeting the students? Perhaps, I thought, they were denied the opportunity to meet and talk with Blacks. Perhaps meeting and talking with me might sow some tiny seed of tolerance and respect which might take root. Or would it? Hell, I was not the first Black any of them had met or could meet if they wished. Or wasn't I? Maybe they'd never met another Black who'd had the opportunities to do what I had done. Perhaps, in their eyes, I was different. But, wait, wasn't that exactly what the Indian ex-Robben Island prisoner had predicted would happen? That the Whites would get to me and seduce me into believing myself different from local Blacks?\n\nI was feeling quite low when the students arrived, but tried to hide it in welcoming them and making them comfortable. Eight of them, five men and three women, young and, at first, somewhat ill at ease. One dark-haired woman who seemed to be the leader of the group apologized for encroaching on my time, particularly as they knew from the newspapers how busy I was.\n\n\"We just had to take the chance, sir,\" she said. \"We've read your books, we know you've lived in England, France, and the United States and we'd like to talk to you about things we'd never be able to discuss with anyone here. We just had to take the chance that you'd see us.\"\n\nIn the face of her plea my misgivings subsided. Hell, these young people looked no different from other groups of young Whites I'd taught in London or Denmark, New York or Florida. Perhaps, in some small way I might be useful. Wasn't this what I had always tried to do as a teacher?\n\nWe talked. At first about my books, my teaching, my travels and my diplomatic service, gradually moving on to themselves as members of their university and citizens of their country.\n\n\"All our philosophy courses teach us to examine the human condition continually and try to improve it,\" said one young woman, whose two thick braids emphasized the youthfulness of her serious face. \"We read about social structures, historical and modern, and it is inevitable that we compare them with our own. We talk about the anomalies among ourselves. That's fine. But then we try to discuss them in class and that's where the trouble starts. How can we talk about the human condition without referring to the Blacks in our society? As soon as you mention Blacks, professors get uptight.\"\n\n\"Unless you refer to them only as statistics,\" another said.\n\n\"In High School everyone was eager to get to the university,\" a young man said. \"We came, believing that we should develop as thoughtfully intelligent people, prepared to assume future responsibilities. And we are encouraged, as long as our inquiries and interests are not directed toward real social change!\"\n\n\"I had this thing with my philosophy professor,\" one said. \"We were discussing social change and after a while it struck me that our entire discussion was limited to intellectual speculation. No one had tried to draw any parallels between what we were philosophizing about and the social realities around us. No one had made any reference to Blacks; no one had commented on apartheid. Of course we talked about injustice, but not as if any of us was even tangentially involved in it. We even reviewed research that had been done, but it was as if we were discussing the behavior of caribou in Canada. So I finally spoke up and said, 'Why don't we, as students, examine our own attitudes to Blacks?' In as many words I was told to forget it.\"\n\n\"Why?\" I asked.\n\n\"It's dangerous to display a social conscience,\" a young man said. \"If you have a social conscience you will inevitably get around to examining Government policies and practices. So you raise a question involving the slightest criticism of that policy and the trouble starts! Some of our professors are members of the Nationalist Party and ardent defenders of Government policy. Before you know it you're under some kind of investigation.\"\n\n\"From your professors?\"\n\n\"Worse. Much worse. From the Security Police. It's a grim situation and you find yourself spinning in circles. We study logic, so Plato's Republic is part of our reading. We read it and we look around to test the validity of the things we read which seem sound against the reality around us. When faced with conflicting concepts, we naturally expect to be able to talk with our professors about them. We read Mann and Thoreau and Steinbeck. French, English, German, Italian, Russian\u2014we try to understand the views and opinions of those considered the world's foremost thinkers. Some of us have been reading Solzhenitsyn. Isn't it to be expected that we will look at our own social structure, if only to reassure ourselves that it is a good one?\"\n\n\"In this society,\" another said, \"if you entertain liberal views, you are soon forced by circumstances into testing the strength and honesty of your liberalism.\"\n\n\"Strength?\" a young woman asked. \"How can you use that word? The fact is that we assume postures. For a short while. When the pressures begin we cave in. We don't necessarily change our views but we do the treacherous thing, the humiliating thing. We cave in to the pressures applied by the university and the Security Police. We learn that there is no such thing as intellectual independence. We don't see it in our professors. We don't see it in our parents. We don't see it. Period.\"\n\n\"You spoke of having liberal views,\" I said to them. \"What views?\"\n\n\"That's what we wanted to talk to you about,\" one young man replied. Red-haired, with a full, neatly trimmed beard, he'd sat quietly since they'd arrived. \"We read an interview of you in the paper the other day, and some of us have been talking about what you said. You spoke about 'social conscience.' Okay, in this society the moment the words 'social conscience' are used, we're talking about our racial situation. About Blacks. As soon as we look around to assess our social or economic situation we see Blacks. Everywhere. And we see what we do to them.\"\n\n\"Well, what's your attitude to them? I mean you, individually, how do you react to them?\"\n\n\"I'm not sure. I try to be\u2014\"\n\n\"We're supposed to be afraid of them,\" the brunette interrupted him. \"Everyone warns that they'll kill us in our beds one day. We're advised to be watchful of them, to keep ourselves armed always.\"\n\n\"Are you afraid of them?\" I pointed the question directly at her.\n\n\"Not ordinarily. Not in the streets, if I see them walking. Not in the shops or offices,\" she stammered, then recovered. \"Well, not those in our home. I mean, not when you can see them. But it's different when you think of them away from you. Where they live. What do they think? What do they talk about? My father worked many years in the Ciskei. He says they never forgive, they can go years biding their time.\"\n\n\"Do you think that's because they're Black? People everywhere resent injustice. If you mistreat people they are likely to turn on you. Whites do. Why should Blacks be any different?\"\n\n\"It's not that they're different.\" From the bearded young man. \"The real fact is that we don't really know them. They're all around us and we don't know them.\" I was glad he'd cut in. He seemed to have given the matter a great deal of thought.\n\n\"What's stopping you from knowing them?\" I wanted to stir something up, get under their skins. \"They're in your homes all day long. You can always begin there.\"\n\n\"My father said that when he was at Wits it was multiracial,\" a blonde girl intervened, but before she could divert him the bearded young man said, \"Last summer I worked on a job with two Blacks. Okay, it was only a summer job, not much for me to do. They did all the work. One thing I had to do was sign their work permits. I suppose you've heard that a Black has to have his work permit signed by a White. Each month. Sometimes the most junior White is assigned that job. Even a girl typist might be the one to sign the permits of men old enough to be her father. I think the idea is to keep them in line, you know, humiliate them, remind them of their dependence.\" He licked his lips, looking around at his peers.\n\n\"Anyway,\" he went on, \"there wasn't much to do so I'd get talking with those two. We never talked about politics and they never talked about themselves, I mean about where they lived, or their families, anything like that. We talked about sports or books or about the American and Russian moon trips. Things like that. It was nice talking with them, they seemed to know a hell of a lot, you didn't get the feeling they were inferior or anything like that. Anyway, they got laid off, for some reason or another. I can't remember. Well, if a Black is unemployed he cannot get a monthly signature and is likely to be picked up and jailed or deported. If he goes looking for a job people always suspect the worst as the reason he was sacked. See the dilemma? Anyway, these two came to see me, met me outside one day and asked me to help them out with the signature until they could find another job. They were having a tough time, but they didn't ask for money, only the signature. Well, I knew nobody would check it, so I signed their permits. It was a funny feeling. Twenty-two years old and I held the destiny of two men at the tip of my pen. If I'd refused to sign they would have been lost. You know how it feels to have people beg you for their lives? I did it, but I never felt good or proud. Each month they'd meet me in a park and I'd sign for them. Nearly eight months. Now they've got jobs in a warehouse. I was glad when it ended, my signing, I mean. I think they even hated me a bit. They never came to tell me they'd got the new jobs; I heard it by accident. When they came for the signatures I'd sign and they'd go. No more talking together, so we never really knew each other. I don't even know what they really thought about me. Could be they hated me for having become so dependent on me. Thing is, I never felt good about helping them.\"\n\n\"If it's so difficult for you, it's a hundred times worse for a woman,\" the blonde said. \"We cannot be seen talking to a black woman in a public place, let alone a black man.\"\n\n\"We're getting away from the point,\" a young man intervened, then to me.\n\n\"We're here because of the things you said in your interview. We think you probably believe that we are either unwilling or unable to protest Government policies.\"\n\n\"Critical examination and challenge mean nothing in the abstract. The social, political, and economic realities all around you are crying out to be challenged,\" I insisted.\n\n\"You may not know it,\" he replied, a bit testily, \"but Wits does have a reputation for criticism and protest, in spite of administration pressure and police harassment. Have you heard anything of NUSAS, the National Union of South African Students? Let me tell you. It was founded by an Afrikaner student in the old days, to encourage and support the interests of white students. It had its beginning at an Afrikaans university, and in spite of the prevailing policies of discrimination and bigotry, opened its membership to black students. That hasn't changed. If you charge that the society at large is becoming more and more polarized you may be correct, but if you looked a little closer you would soon discover that NUSAS members, black and white, are in the vanguard of action for social and political change. Many former members of NUSAS were imprisoned, sometimes in solitary confinement, for pressing for social reform. My father was one of them. I was in prep school in 1964 when the Security Police began mopping up anyone who was an activist, faculty or student. Wits. Cape Town. Rhodes and Natal. They raided all the places with NUSAS affiliation. The economics lecturer at Rhodes, Norman Bromberger, was picked up and held in solitary for a hundred and sixty-eight days. Did you know that?\"\n\n\"No.\"\n\n\"Nowadays students are not openly activist, but they're active nonetheless. No point in getting beaten up or jailed if you can avoid it. Long before black students had formed unions, the Whites were bearing the heat of protest. Did you know that until this Government came to power there were black students at the English language universities? Did you know there were even integrated campus dances? The Government banned them in 1965. Ever heard of Sir Richard Luyt?\"\n\n\"Yes. He was Governor of British Guyana just before it became independent.\"\n\n\"Right. He's now Principal of Cape Town U. After his appointment they resumed integrated dances there. He even offered a lectureship to a black anthropologist trained at Oxford, Archie Mafeje, but the Government forced the U. to withdraw the offer. Nearly a thousand students and faculty sat-in to protest. Dr. Mafeje was not appointed, even though the sit-in lasted nearly two weeks. Would you say that the students have been sitting on their hands?\"\n\n\"No, I wouldn't.\" These were the things I'd been hoping to hear, not the nice, self-deprecating noises which had been made so far. But still, he was telling me of the old times. I wanted to hear about now. About here. About them.\n\n\"Meanwhile the police have become rougher in their tactics and more sophisticated in their methods. Their chief weapon was intimidation. They could easily get Afrikaner students to infiltrate student organizations. Anyone who expressed anti-Government sentiments was a target, no matter who he or she might be. Did you ever hear of Philip Golding or John Schlapobersky? They were both at Wits and were detained in 1969. A favorite police ploy was to confiscate the passport of any student or faculty member suspected of anti-Government sentiments. An American student named Rex Heinke was deported for the same reason. Don't think that all these activities were by Whites pursuing their own selfish interests.\"\n\n\"Weren't they?\"\n\n\"No. Even after Blacks were banned from attending white universities the white activists would meet with Blacks, surreptitiously, even in the black townships. Like everything else, the informers soon got wind of it. Some white students were arrested in Soweto, for being there without permits. That was really the year of the raids. The Security Police were everywhere, picking people up and detaining them on the flimsiest pretext. One student named Ahmed Timol died in detention it is said because of a severe beating by the police.\"\n\nHe seemed unable to stop himself. The stuff was flowing from him, rushing out of him, while the rest of us listened.\n\n\"Did you see in the newspaper that Abraham Tiro has been killed by a parcel bomb? They've been after him a long time. In 1972, soon after I came up to Wits, he made a graduation speech at Turfloop U., the black university in Natal. In it he attacked the racist philosophy of the Bantu Education Act. For this he was expelled. As a result all the Turfloop students staged a sit-in and all were expelled, the police helping with the expulsions. Naturally. This set off a chain reaction, as students from black universities at Fort Hare, Westville, Bellville, Zululand, everywhere, joined in spontaneous protest. The high point was a peaceful demonstration by more than ten thousand Whites on the steps of St. George's Cathedral at Cape Town. As you would expect, the police charged at them, with batons and guns, and even followed them into the Cathedral to beat them up and arrest them. Can you imagine that? Inside the Cathedral. And the Prime Minister defended them.\n\n\"I could tell you more. Do you know why? I plan to document it all; maybe I'll use it as the basis for my thesis. I was in Cape Town for that protest demonstration. I saw what happened with my own eyes. I stayed out of the way. I don't think I could take that kind of brutal beating. The police don't care whom they hit, man or woman. Or where.\"\n\n\"What about the black students?\" I asked.\n\n\"They're harassed all the time. Even before they open their mouths they're banned. I guess the informers are even more active among them. Black informers. Every day you read of some more being banned, restricted to their own homes from dusk to dawn. However, you seem to take their protest for granted. What you wanted to know about was our involvement. If you see little evidence of it, that's because we're copying the tactics of the Blacks. We've been operating underground since 1972, when the editor and the cartoonist of the \"Wits Student,\" our university publication, were suspended from the U. for publishing anti-government material. They were subsequently charged with offenses under the Publications and Entertainments Act and convicted. The same thing has happened at other universities, black and white. The Government has strangled protest in any overt form. I don't believe they will succeed in silencing protest. We're being forced to operate differently, that's all.\"\n\n\"How has the murder of Tiro affected your underground activities?\" I asked. \"Seems to me the Security Police have demonstrated how very easily they can reach anyone, anywhere.\"\n\n\"Perhaps,\" a young woman said, \"but not necessarily. The police knew that Tiro depended on help from friends. He'd fled to Botswana, but he needed help, and the police found a way to get to him. Here we are learning to be very careful. We hadn't agreed to tell you all these things because we couldn't take the risk of your mentioning your source, if you decided to write about it.\"\n\n\"I won't mention names,\" I promised, \"nor will I ask which of you are involved in what movement. But I'd like to know whether it really is a movement or just the posturings of a few?\"\n\n\"We're more than a few. Much more. Even some faculty members are with us. The same is true at other universities. Unfortunately, we can't expect any public support, not even among our parents or relatives. Everybody is afraid. Even the priests. Most of them, the English-speaking ones, are sympathetic, but they can't all be martyrs. Did you know that there's a violent right-wing group operating with the knowledge and consent of the Government and Security Police? They call themselves Scorpio. They're terrorists, but because they're White there's no mention of them. In '72 they fire-bombed the home of the student president of Cape Town U., Geoff Budlender. Naturally, no arrests, no prosecutions. So, you see, we're like lambs surrounded by predators. From the distance of the U.S.A. you might imagine that this is a struggle between Blacks and Whites. It's much more than that. Under this government it's also become a struggle by some Whites for freedom to live in peace with justice for everyone, Black and White. We do not plan to turn this country over to the Blacks, but we'd prefer to see it governed democratically, justly, and without recourse to fear and intimidation.\"\n\nWe were interrupted by a call from the hotel lobby. Dr. Bozzoli, President of Witwatersrand University, wished to speak to me. He came on the line and reminded me that I'd promised to dine with him at his house that evening. Covering the mouthpiece, I told the students that the President of the U. was at the hotel and I thought I'd better meet him downstairs.\n\n\"On account of us?\" one asked.\n\n\"Well... \"\n\n\"I think you should ask him up,\" one said. The others nodded.\n\nI asked him to come up, checked my diary and found that it was true. I'd made the entry, but on another page. I told the students of my dilemma, adding that I was sure Dr. Bozzoli had heard at the desk that I was entertaining a group of his students. The front desk seemed to know everything I did.\n\n\"They wouldn't know,\" I was told. \"We drove into the basement garage and took the lift right up here. We knew the number of your suite from when we'd phoned earlier.\"\n\n\"You'll have to go with him,\" one said.\n\n\"I suppose so, but I was really enjoying this talk with you. And I've invited you to be my guests for dinner. What shall we do about it?\"\n\n\"Whatever you say,\" they told me. I was unhappy to have to leave them, because we'd finally broken through the earlier resistance and were reaching each other.\n\n\"Here's my suggestion,\" I offered. \"I'll go with Dr. Bozzoli for two hours. Meanwhile you order your meal here, whatever you like, and take your time about it. I'll be back before you're through and we can have coffee together and continue our talk.\"\n\nThey agreed. The President arrived and seemed quite surprised to see me with a roomful of young people. I made no attempt at individual introductions, bearing in mind some of the things we'd been discussing. They seemed a little jittery at having been discovered by him.\n\nOn the way to Dr. Bozzoli's home, I told him of my predicament and explained that the students had offered to wait for me while I kept my dinner engagement with him.\n\nWith the Bozzolis were a few members of the university community, one or two businessmen and their wives and the author Nadine Gordimer with her husband. Before dinner we sat outdoors and made small talk. For much of this pleasant interlude I listened, observing these charming, urbane, gracious people who seemed untouched and untroubled by the sinister air which foamed and rumbled about them. They talked knowledgeably about their country's economy and the implications of the extraordinary fluctuations of the gold prices. They commented on the equally dramatic changes taking place in diplomatic procedures, largely due to what they called American \"instant\" statesmanship as practiced by the very peripatetic American Secretary of State. They discussed the international effect of oil shortages, but assured themselves that South Africa would suffer less than most because her economy depended more on her massive coal reserves than on oil.\n\nFrom where we sat, the women gowned and coiffed, the men elegant and worldly, national disaster seemed light years away. Part of me wanted only to enjoy this short respite from the hustle of six extremely busy weeks, but another part of me was watchful and listening, remembering that under the selfsame stars that glistened overhead, and within a short distance from where we sat, the pain of exclusion was being acutely felt and deeply resented by others, Blacks like myself. Within earshot of our sophisticated banalities, the fuse was already set for a tragic explosion.\n\nThings changed when we went in to dinner. Mrs. Bozzoli, who had said so little outdoors, being content to supervise the introduction of newcomers and settle them down with drinks, now emerged as a highly articulate and well-informed hostess, displaying a surprising independence of spirit which defied compromise of her personal principles. As if under her stimulating influence, the conversation became more sober and careful, and was directed to the fundamental issues of South Africa's domestic and international situation. Inevitably, I was asked to state my impressions of the society even though I protested that I had seen very little and learned even less in the six weeks of my visit. However I said that it seemed to me the Government was deliberately trying to goad the black people to the point of revolt.\n\nEven though their expressed attitudes vary in form, all Whites benefit from the cruel exploitation of the Blacks and are disinclined to any change likely to threaten those benefits in any way. Some of those sitting and talking with me took a distant, intellectually objective view of the racial situation, assuring me that in spite of what I might see many changes for the better were in progress. They could recall the conditions and circumstances of ten or fifteen years earlier, and were themselves impressed by the dramatic way in which changes had occurred since. They drew my attention to the recently publicized decisions to abandon some of the \"petty apartheid,\" the segregation signs so familiar on park benches, buses, and public buildings.\n\nImpatiently, I applauded their objectivity but insisted that I could not share it. They could afford their distance from the Blacks, because at every level that distance was maintained and encouraged. They all had black servants who were denied the right to bargain for their labor and could hardly protest their treatment. I could not be \"objective.\" I was black and could not, would not wish to avoid identity with those of whom they spoke so impersonally, so unfeelingly. I knew that I was sitting there with them only because I was an overseas writer whose work they admired. Did they care about the authors and poets of equal or greater potential vegetating among them?\n\n\"Let's get to the heart of the matter,\" one woman said, her face set in a mold of aggressive determination. \"I'm a sociologist. The very nature of that discipline requires that we regularly examine our society for strengths and weaknesses. The moment we begin we're confronted with the inequities imposed on Blacks. Okay. But consider for a moment what would happen if those inequities were suddenly removed. Our elders remind us of what they endured at the hands of Blacks when this country was settled. The disparity in numbers remains, perhaps it has even increased. Just imagine the Blacks in power. Given the present conditions, what could we do to reduce what you call polarization without tipping the balance of power in favor of the Blacks?\"\n\nThere was a sudden stirring among the group. Clearly, she had posed the question of general concern to them all. She caught me unprepared. I had not, so far, been thinking along those lines.\n\n\"I would prefer to speculate on the sharing of power rather than on a reversal of roles,\" I said. \"Think what a willing and conscientious black population could contribute to the society. Not as near-slaves but as citizens proud of their rights. In many other societies, given the opportunity, Blacks have proved themselves as capable as anyone else of setting national needs at the top of their priorities.\"\n\n\"It would be unreasonable not to expect them to want to revenge themselves on us for past injustices,\" another suggested.\n\n\"If Germany and Israel can find bases for mutual cooperation, I imagine it is quite possible for anyone else.\"\n\n\"In this society, the individual is expected to conform politically and socially,\" a bald man said. \"The attitude to Blacks is both social and political. If I, as an individual, wished to adopt a humanitarian attitude to Blacks whom I meet, work with, or employ, I would automatically be assuming a posture politically at variance with the prevailing governmental policies.\"\n\n\"If, as an individual, in spite of the attitudes of others, you can recognize and respect the humanity of Blacks, I cannot see how that would force you into any political posture. I am Black. You can sit here and converse courteously with me; that does not suggest a political posture. You say you work with Blacks. I cannot see that, should you treat them with courtesy and respect you are assuming a political posture. If you employ Blacks as domestic servants and decide to pay them a wage you can afford and they are worth, I do not consider that a political posture, unless you wished to make it so.\"\n\n\"I don't think your reference to yourself is relevant,\" someone intervened. \"You are a famous author and a stranger, so immediately our attitude to you is one of respect.\"\n\n\"But I am black and my presence among you should help you to appreciate the stupidity of the assumptions that Blacks are less capable, less intelligent and less human than Whites. The real difference between your black countrymen and myself lies in access to opportunity. The question you'll have to face is, how much opportunity would you wish to see granted to Blacks, the opportunity to vote, to negotiate the sale of their labor, to own land on which to build their homes, to compete according to their abilities?\"\n\n\"What about the risks?\" from another. \"The risks of revenge on the part of the Blacks? You seem to expect us to take an objective view of what is primarily an emotional matter.\"\n\n\"Just as you expect me to respond objectively to your questions about Blacks,\" I replied.\n\n\"The point is well taken,\" interposed President Bozzoli. \"Now, Mr. Braithwaite, in the light of what you've been saying to us, would you consider returning to spend some time here on campus? Say, three to six months as guest professor? My feeling is that it would be of tremendous benefit to this university.\"\n\nHearing the words coming from him in soft, measured tone I was immediately flattered. Then in a flash came the image of what would happen to me.\n\n\"Good God,\" I exclaimed.\n\n\"What's the matter?\" he asked. \"Have I taken you by surprise?\"\n\n\"You have, but I've just realized what that would mean. Your Government could require me to live in one of the black townships. Soweto, possibly.\"\n\n\"What's the objection?\" someone asked.\n\n\"Objection! Have you been to Soweto? No, thank you. I like the idea of living where I can afford and moving about with freedom. I won't be herded.\" That train ride appeared in my mind, a horrifying apparition.\n\n\"That could easily be taken care of,\" Mrs. Bozzoli intervened. \"You could live in our house, be part of our household for the time you are here.\" Said so naturally, not a moment of hesitation.\n\n\"But there's the matter of movement.\" I was frantically searching for excuses, knocked off base by my own unpreparedness for this. Who would have imagined an invitation to stay on this White-only campus? \"I'd be literally restricted to the campus. I couldn't walk into a cinema in town, or a restaurant or anything. I don't think I could stand that.\"\n\n\"Seems like a small sacrifice, in view of the arguments you've recently raised,\" some remarked. Belatedly I remembered Mrs. Bozzoli's invitation to share their home. How would I get out of that.\n\n\"I could not accept your invitation if I agreed to come back,\" I said to her. \"It would be quite unworkable. I would need a separate place where I could meet and talk with anyone at any time without having the comings and goings imposed upon you.\"\n\n\"Then we'd arrange for you to have a faculty apartment.\" She had the reply in a moment. I felt cornered.\n\n\"Your being here would be wonderful for us, for our students and for our community,\" Dr. Bozzoli said. \"The response you've already had should be some indication of how deeply we all appreciate what you've been saying to us. I'm sure we would be taking some small risk in having you disturb our complacency, but we are willing to take it. What about you? If we are to work together for realistic social change, it will take more than a short lecture or two. Our staff and students would need to talk with you, get a feel of your mind and spirit at close quarters. They would need to touch you, intellectually and spiritually. It would take a little time for us to learn about you and from you, as it would take a little time for you to learn about us.\"\n\n\"Perhaps Mr. Braithwaite's comfort is more important than his political posture,\" someone suggested softly, coating the barb with a light layer of laughter. I heard it and it disturbed me. Was that how they saw me?\n\n\"Dr. Bozzoli, I was completely unprepared for your invitation and would like to think about it. I must confess that my very immediate reaction to it is negative, because of how this society treats its Blacks. However, please let me have some time to think about it.\"\n\n\"That's very reasonable,\" he replied. \"Sorry I sprang it on you like that, but the thought came to me just as unexpectedly. I'd like to see this university as an institution that's able to meet and accommodate a challenge. Your presence here would be very timely. If we all really want positive change to take place, some of us critics must step off the sidelines and jump into the game. I suppose I'm inviting you to take risks, but the decision must be yours. We would welcome your presence at this university.\"\n\nI thought of all the Blacks who had warned me that, somehow, I would be used by the Whites. Could this invitation be regarded as a way of using me? The Principal had no doubt about my uncompromising attitude to his Government's racial policies. He was not, in any way, asking me to dilute or abandon those views. It was an invitation, pure and simple. Why should I be looking over my shoulder for those Blacks who might censure me. Hadn't I always taken pride in my personal freedom of spirit? If one of the black colleges had extended such an invitation, would I have hesitated? Was my personal comfort at issue here, as had been hinted?\n\nI didn't know. I really needed time to think.\n\n\"Don't worry for a moment about your accommodation, or anything else,\" Mrs. Bozzoli insisted. \"Just come, we need you.\"\n\n\"Very few outsiders find their way to South Africa's academic centers,\" a professor added. \"Our students and teachers are facing the grim prospect of intellectual ingrowth. Challenge from a source uncommitted to this country could be a very important catalyst, at this time when we are all searching for answers. At least, give us an opportunity to see ourselves as we are seen, from the outside.\"\n\nI wanted to shout at them, \"I'll think about it!\" Couldn't they imagine how depressing, how nerve-racking it was for me in their city? Would it be enough for me to be with them and the students? Was there no way in which I might, at the same time, be of service to the Blacks, for my soul's sake?\n\nDr. Bozzoli interrupted to explain that I had been dragged away from some students who were still waiting for me at my hotel room, thus making it easy for me to say my adieux and leave.\n\nThe students had finished their meal and were dallying over coffee and cigarettes, looking relaxed and comfortable. I wondered again what my black friends in Soweto would think of this scene if they could see it. And the young, aggressive Indian of Robben Island: wouldn't he see this as definitive evidence that I had sold out?\n\nBefore I'd left the students, there'd been some talk of underground action. I wished to learn more about that and asked them. Perhaps, in my absence they had decided they'd said enough; I made several attempts to steer the conversation back to the topics we had been discussing, but soon became aware of their resistance.\n\n\"I think we were sort of blowing off steam,\" one said. \"When one is talking with someone like you, one is apt to overreach oneself. If you have a social conscience you cannot avoid awareness of the inequities around you. You want to do something about them, and that brings you up hard against authority. In this society authority is inflexible. So you have a choice. Either you follow the lead of your social conscience and take the risks or you shut up. If you're wealthy, you can get away with an occasional gesture, like taking some token interest in a nursery school for black children, or inviting a black writer or politician into your home. If you're not wealthy or influential, you compromise. They have ways of making you compromise, or they break you. By fear and intimidation. You know, we've been wondering something while you were away. If you were South African, how long do you think you'd last?\"\n\n\"The point is,\" I replied, \"I'm not South African. If I had been I might never have become the man I am. All the odds are against it.\"\n\nFor a while we continued to chat about other things, the U.'s drama program, films, sports.\n\nThen they left. I wondered whether Dr. Bozzoli's appearance had affected them. I had introduced them collectively, so it was very unlikely that he had taken particular notice of any individual. In any case, I felt sure he was not more than pleasantly surprised to find me surrounded by a group of students, none of whom was personally known to him.\n\nYet it was clear that they had been turned off. For some reason fear had got into them, so there it was again. From every side I was hearing about it. Everywhere I saw evidence of it. From Blacks. From Whites. From black housewives and workers, and now from white students.\n\nAll of them, black and white, seemed powerless against the forces which intimidated and frightened them, yet there were undercurrents of rebellion; the earlier outburst of the students, the blacks' smoldering rage. With the entrenched Government resisting every effort toward positive change, the future offered nothing but violence and bloodshed. The Government was ready for a confrontation. Their huge stockpiles of armaments and their aggressive intransigence suggested that they would even welcome it. Everything pointed to a collision course.\n\n# Chapter  \nTwelve\n\nTHE FOLLOWING DAY WAS spent in preparation for my departure from South Africa. The officials at the Tourist Bureau who advised me were courteous and helpful, clearly concerned with projecting the right image. While waiting there, I picked up a copy of the day's Rand Daily Mail. It carried a story on the Government's intention to impose new curbs on \"some groups.\" The State President, Mr. Fouche, warned that a number of pressure groups were \"trying to bring about unconstitutional political, social, and economic change in South Africa,\" and claimed that \"implicit in their call for change was the threat of internal violence. These groups do not have in mind normal evolutionary change,\" Mr. Fouche was quoted as saying. \"They are bent upon radical, even revolutionary political activities.\" He claimed that they were financed from abroad and expressed the Government's determination to curb them. Oblique reference was made to NUSAS and the black South African Students' Organization, most of whose leaders were already banned or restricted.\n\nIn the opposition's response to Mr. Fouche, Mrs. Helen Suzman, the most vocal, spoke in defense of the predominantly white NUSAS, saying, \"I do not always agree with everything NUSAS does, but I hope there are enough who will see that the Government does not financially starve NUSAS into submission. I believe it is important that NUSAS be allowed to operate. This is an organization of young people who care about racial injustices... \"\n\nNot a single word in defense of SASO, the black organization.\n\nA shocking story caught my attention. An eleven-year-old black child, Godfrey Lambert, had been picking up pieces of coal at the Beaufort West railway yard and was caught by three white railway workers who undressed him, smeared his body with grease and held him in front of the fire door of a blazing locomotive engine. The child had been literally roasted and was horribly scarred, physically and psychologically.\n\n\"At night he wakes up screaming,\" his mother said, adding that she feared her son's mind was permanently damaged.\n\nThe white railway workers were each sentenced to six lashes and a year's imprisonment, suspended for three years.\n\nI tried to imagine a parallel situation in which three Blacks brutalized a white child. In a South African court, the death penalty would have seemed lenient punishment indeed!\n\nGod, could I in my right mind consider returning to this country to work and live among people who could condone such atrocities? To suspend sentence on three such savages was nothing short of condoning their act. Reading about it stirred the rage inside me. In fact, reading the newspaper altogether was a frustrating exercise. The entertainment page advertised theatre, cinema, concerts, dogracing, etc., but I knew those ads were not directed to me and other Blacks. No need to include the \"Whites Only\" tag. But why the hell was I even bothering to think about it? Had I indeed been seduced into imagining myself exempt? Christ! Chasing around the country as a tourist with hardly a moment to spare was one thing. Wherever I went the Information Office would get the word there ahead of me and dictate that courtesy be shown me. Living in Johannesburg for three months within sight and sound of familiar amenities would be quite another thing.\n\nCould I cope with it? What did it matter if my color barred me from cinemas and restaurants and bars? Was my own comfort so overriding a priority? The native Blacks had each known a lifetime of exclusion from all these things. Wouldn't the work with students and faculty be enough? I could always seek out my fellow Blacks in the black townships. If my feeling of identification was real, why did that prospect bother me?\n\nAnd that other thing, the continued probing and questioning from the Blacks themselves. How much more of that could I take? Their sly, oblique quizzing always made me feel guilty of the comfort I enjoyed, the privileges I could exercise. But had I not earned each privilege, each area of comfort? Was I not autonomous, free to design my own life, answerable only to myself? Outside South Africa any of these Blacks could live as I live, be free to fulfill himself. Why should I be on the defensive? Why should I allow myself to be forced into playing someone else's role?\n\nIf I returned to South Africa, Dr. Bozzoli promised complete access to students and faculty, and complete freedom to be myself, thinking as I thought, believing as I believed, making no secret of my distaste for their racial policies. How long would they and the Government accommodate me and my views? Dr. Bozzoli probably thought he was making me a simple proposition. Hell, about as simple as the maze of a man's life.\n\nEarly next morning I packed my bags. All that remained was to pay my bill and get out to the airport. In the park across the street the scrawny black boys were already at their interminable games, dodging between the black workers who hurried past them. Hungry, uncared-for, this was their youth. What of their tomorrow and adulthood? Did they have any hopes, any aspirations? Did they know anything about love of country, these children who had been denied even love of family? It is said that the future of a country is invested in its youth. What part of South Africa's future was invested in these boys?\n\nThe hotel receptionist had proudly mentioned the national motto, \"Unity is Strength.\" Where was the unity? Was the strength that of arms and armaments? Was it understood that the words completely excluded any participation by Blacks? From my window the view offered a sense of ordered progress, of controlled growth, with no sign of the suppurating evils and the frustration and hatred they bred.\n\nThose hurrying black figures seemed concerned only with getting to their jobs on time, neatly dressed, faces revealing nothing of their thoughts. Where did I hear that the black man was content with his lot? Those hurrying men and women were no different from myself except in experience of living in freedom.\n\nAt the hotel entrance, the car was waiting to take me to the airport.\n\n# About the Author\n\nE. R. Braithwaite was born in British Guiana (now Guyana) in 1912. Educated at the City College of New York and the University of Cambridge, he served in the Royal Air Force during World War II. Braithwaite spent 1950 to 1960 in London, first as a schoolteacher and then as a welfare worker\u2014experiences he describes in To Sir, With Love and Paid Servant, respectively. In 1966 he was appointed Guyana's ambassador and permanent representative to the United Nations. He has also held positions at the World Veterans\u00ad Federation and UNESCO, was a professor of English at New York Univer\u00adsity's\u00ad Institute for Afro-American Affairs, and taught creative writing at Howard University. The author of five non\u00adfiction books and two novels, he currently lives in Washington, DC.\nAll rights reserved, including without limitation the right to reproduce this ebook or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.\n\nCopyright \u00a9 1975 by E. R. Braithwaite\n\nCover design by Mauricio D\u00edaz\n\nISBN 978-1-4804-5741-6\n\nThis edition published in 2014 by Open Road Integrated Media, Inc.  \n345 Hudson Street  \nNew York, NY 10014  \nwww.openroadmedia.com\n\n# **EBOOKS BY E. R. BRAITHWAITE**\n\n**FROM OPEN ROAD MEDIA**\n\n**Available wherever ebooks are sold**\n\n**Open Road Integrated Media** **is a digital publisher and multimedia content company. Open Road creates connections between authors and their audiences by marketing its ebooks through a new proprietary online platform, which uses premium video content and social media.**\n\n**Videos, Archival Documents, and New Releases**\n\n**Sign up for the Open Road Media newsletter and get news delivered straight to your inbox.**\n\n**Sign up now at**\n\n**www.openroadmedia.com\/newsletters**\n\n**FIND OUT MORE AT**\n\n**WWW.OPENROADMEDIA.COM**\n\n**FOLLOW US:**\n\n**@openroadmedia** **and**\n\n**Facebook.com\/OpenRoadMedia**\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n## By The Same Author\n\nWhite Doves at Morning\n\nJolie Blon's Bounce\n\nBitterroot\n\nPurple Cane Road\n\nHeartwood\n\nLay Down My Sword and Shield\n\nSunset Limited\n\nHalf of Paradise\n\nCimarron Rose\n\nCadillac Jukebox\n\nHeaven's Prisoners\n\nBurning Angel\n\nThe Lost Get-Back Boogie\n\nThe Convict\n\nDixie City Jam\n\nIn the Electric Mist with Confederate Dead\n\nA Stained White Radiance\n\nThe Neon Rain\n\nA Morning for Flamingos\n\nBlack Cherry Blues\n\nTwo for Texas\n\nTo the Bright and Shining Sun\n\nSIMON & SCHUSTER  \nRockefeller Center  \n1230 Avenue of the Americas  \nNew York, NY 10020\n\nThis book is a work of fiction. Names, characters, places, and incidents either are products of the author's imagination or are used fictitiously. Any resemblance to actual events or locales or persons, living or dead, is entirely coincidental.\n\nCopyright \u00a9 2003 by James Lee Burke  \nAll rights reserved, including the right of reproduction in whole or in part in any form.\n\nSimon & Schuster and colophon are registered trademarks of Simon & Schuster, Inc.\n\nLibrary of Congress Cataloging-in-Publication Data is available.\n\nISBN-13: 978-0-7432-6097-8  \nISBN-10: 0-7432-6097-X\n\nVisit us on the World Wide Web:  \n<http:\/\/www.SimonandSchuster.com>\n\n## Acknowledgments\n\nI would like to thank Leslie Blanchard at the Iberia Parish Library and Vaun Stevens and Don Spritzer at the Missoula Public Library for their friendship and generous help over the years.\nTo my wife, Pearl, and my children,\n\nJim, Jr., Andree, Pamala, and Alafair\n\n##  \n## Chapter 1\n\nThe first week after Labor Day, after a summer of hot wind and drought that left the cane fields dust blown and spiderwebbed with cracks, rain showers once more danced across the wetlands, the temperature dropped twenty degrees, and the sky turned the hard flawless blue of an inverted ceramic bowl. In the evenings I sat on the back steps of a rented shotgun house on Bayou Teche and watched the boats passing in the twilight and listened to the Sunset Limited blowing down the line. Just as the light went out of the sky the moon would rise like an orange planet above the oaks that covered my rented backyard, then I would go inside and fix supper for myself and eat alone at the kitchen table.\n\nBut in my heart the autumnal odor of gas on the wind, the gold and dark green of the trees, and the flame-lit edges of the leaves were less a sign of Indian summer than a prelude to winter rains and the short, gray days of December and January, when smoke would plume from stubble fires in the cane fields and the sun would be only a yellow vapor in the west.\n\nYears ago, in both New Orleans and New Iberia, the tannic hint of winter and the amber cast of the shrinking days gave me the raison d'etre I needed to drink in any saloon that would allow me inside its doors. I was not one of those valiant, alcoholic souls who tries to drink with a self-imposed discipline and a modicum of dignity, either. I went at it full-bore, knocking back Beam or Black Jack straight-up in sawdust bars where I didn't have to make comparisons, with a long-necked Jax or Regal on the side that would take away the aftertaste and fill my mouth with golden needles. Each time I tilted the shotglass to my lips I saw in my mind's eye a simian figure feeding a fire inside a primeval cave and I felt no regret that I shared his enterprise.\n\nNow I went to meetings and didn't drink anymore, but I had a way of putting myself inside bars, usually ones that took me back to the Louisiana in which I had grown up. One of my favorites of years past was Goldie Bierbaum's place on Magazine in New Orleans. A green colonnade extended over the sidewalk, and the rusted screen doors still had painted on them the vague images and lettering of Depression-era coffee and bread advertisements. The lighting was bad, the wood floor scrubbed colorless with bleach, the railed bar interspersed with jars of pickles and hard-boiled eggs above and cuspidors down below. And Goldie himself was a jewel out of the past, a seventy-year-old flat-chested ex-prizefighter who had fought Cleveland Williams and Eddie Machen.\n\nIt was night and raining hard on the colonnade and tin roof of the building. I sat at the far end of the bar, away from the door, with a demitasse of coffee and a saucer and tiny spoon in front of me. Through the front window I could see Clete Purcel parked in his lavender Cadillac convertible, a fedora shadowing his face in the glow of the streetlight. A man came in and removed his raincoat and sat down on the other end of the bar. He was young, built like a weight-lifter whose physique was earned rather than created with steroids. He wore his brown hair shaved on the sides, with curls hanging down the back of his neck. His eyebrows were half-moons, his face impish, cartoonlike, as though it were drawn with a charcoal pencil.\n\nGoldie poured him a shot and a draft chaser, then set the whiskey bottle back on the counter against the wall and pretended to read the newspaper. The man finished his drink and walked the length of the bar to the men's room in back. His eyes looked straight ahead and showed no interest in me as he passed.\n\n\"That's the guy,\" Goldie said, leaning close to me.\n\n\"You're sure? No mistake?\" I said.\n\n\"He comes in three nights a week for a shot and a beer, sometimes a catfish po'boy. I heard him talking about it on the payphone back there. Maybe he's not the guy who hurt your friend, but how many guys in New Orleans are gonna be talking about breaking the spokes on a Catholic priest?\"\n\nI heard the men's room door open again and footsteps walk past me to the opposite end of the bar. Goldie's eyes became veiled, impossible to read. The top of his head looked like an alabaster bowling ball with blue lines in it.\n\n\"I'm sorry about your wife. It was last year?\" he said.\n\nI nodded.\n\n\"It was lupus?\" he said.\n\n\"Yeah, that's right,\" I replied.\n\n\"You doin' all right?\"\n\n\"Sure,\" I said, avoiding his eyes.\n\n\"Don't get in no trouble, like we used to do in the old days.\"\n\n\"Not a chance,\" I said.\n\n\"Hey, my po'boy ready?\" the man at the end of the bar asked.\n\nThe man made a call on the payphone, then ate his sandwich and bounced pool balls off the rails on the pool table. The mirror behind the bar was oxidized an oily green and yellow, like the color of lubricant floating in water, and between the liquor bottles lined along the mirror I could see the man looking at the back of my head.\n\nI turned on the bar stool and grinned at him. He waited for me to speak. But I didn't.\n\n\"I know you?\" he said.\n\n\"Maybe. I used to live in New Orleans. I don't anymore,\" I said.\n\nHe spun the cue ball down the rail into the pocket, his eyes lowered. \"So you want to shoot some nine ball?\" he said.\n\n\"I'd be poor competition.\"\n\nHe didn't raise his eyes or look at me again. He finished his beer and sandwich at the bar, then put on his coat and stood at the screen door, looking at the mist blowing under the colonnade and at the cars passing in the neon-streaked wetness in front of Goldie's bar. Clete Purcel fired up his Cadillac and rattled down the street, turning at the end of the block.\n\nThe man with the impish face and curls that hung on the back of his neck stepped outside and breathed the air like a man out for a walk, then got into a Honda and drove up Magazine toward the Garden District. A moment later Clete Purcel pulled around the block and picked me up.\n\n\"Can you catch him?\" I asked.\n\n\"I don't have to. That's Gunner Ardoin. He lives in a dump off Tchoupitoulas,\" he said.\n\n\"Gunner? He's a button man?\"\n\n\"No, he's been in two or three of Fat Sammy Figorelli's porn films. He mules crystal in the projects, too.\"\n\n\"Would he bust up a priest?\" I asked.\n\nClete looked massive behind the steering wheel, his upper arms like big, cured hams inside his tropical shirt. His hair was sandy, cut short like a little boy's. A diagonal scar ran through his left eyebrow.\n\n\"Gunner?\" he said. \"It doesn't sound like him. But a guy who performs oral sex for a hometown audience? Who knows?\"\n\nWe caught up with the Honda at Napoleon Avenue, then followed it through a dilapidated neighborhood of narrow streets and shotgun houses to Tchoupitoulas. The driver turned on a side street and parked under a live oak in front of a darkened cottage. He walked up a shell driveway and entered the back door with a key and turned on a light inside.\n\nClete circled the block, then parked four houses up the street from Gunner Ardoin's place and cut the engine. He studied my face.\n\n\"You look a little wired,\" he said.\n\n\"Not me,\" I said.\n\nThe rain on the windshield made rippling shadows on his face and arms. \"I made my peace with N.O.P.D.,\" he said.\n\n\"Really?\"\n\n\"Most of the guys who did us dirt are gone. I let it be known I'm not in the O.K. Corral business anymore. It makes life a lot easier,\" he said.\n\nThrough the overhang of the trees I could see the Mississippi levee at the foot of the street and fog billowing up from the other side. Boat lights were shining inside the fog so that the fog looked like electrified steam rising off the water.\n\n\"Are you coming?\" I asked.\n\nHe pulled an unlit cigarette from his mouth and threw it out the window. \"Why not?\" he said.\n\nWe walked up Gunner Ardoin's driveway, past a garbage can overflowing with shrimp husks. Banana trees grew against the side of the house and the leaves were slick and green and denting in the rainwater that slid off the roof. I jerked the back screen off the latch and went into Gunner Ardoin's kitchen.\n\n\"You beat up Catholic priests, do you?\" I said.\n\n\"What?\" he said, turning from the sink with a metal coffeepot in his hand. He wore draw-string, tin-colored workout pants and a ribbed undershirt. His skin was white, clean of jailhouse art, his underarms shaved. A weight set rested on the floor behind him.\n\n\"Lose the innocent monkey face, Gunner. You used a steel pipe on a priest name of Jimmie Dolan,\" Clete said.\n\nGunner set the coffeepot down on the counter. He studied both of us briefly, then lowered his eyes and folded his arms on his chest, his back resting against the sink. His nipples looked like small brown dimes through the fabric of his undershirt. \"Do what you have to do,\" he said.\n\n\"Better rethink that statement,\" Clete said.\n\nBut Gunner only stared at the floor, his elbows cupped in his palms. Clete looked at me and raised his eyebrows.\n\n\"My name's Dave Robicheaux. I'm a homicide detective with the Iberia Parish Sheriff's Department,\" I said, opening my badge holder. \"But my visit here is personal.\"\n\n\"I didn't beat up a priest. You think I did, then I'm probably in the shitter. I can't change that.\" He began picking at the calluses on his palm.\n\n\"You get that at a twelve-step session up at Angola?\" Clete said.\n\nGunner Ardoin looked at nothing and suppressed a yawn.\n\n\"You raised Catholic?\" I said.\n\nHe nodded, without lifting his eyes.\n\n\"You're not bothered by somebody hospitalizing a priest, breaking his bones, a decent man who never harmed anyone?\" I said.\n\n\"I don't know him. You say he's a good guy, maybe he is. There's a lot of priests out there are good guys, right?\" he said.\n\nThen, like all career recidivists and fulltime smart-asses, he couldn't resist the temptation to show his contempt for the world of normal people. He turned his face away from me, but I saw one eye glimmer with mirth, a grin tug slightly at the corner of his mouth. \"Maybe they kept the altar boys away from him,\" he said.\n\nI stepped closer to him, my right hand balling. But Clete pushed me aside. He picked up the metal coffeepot from the counter and smashed it almost flat against the side of Gunner Ardoin's head, then threw him in a chair. Gunner folded his arms across his chest, a torn grin on his mouth, blood trickling from his scalp.\n\n\"Have at it, fellows. I made both y'all back on Napoleon. I dialed 911 soon as I came in. My lawyer loves guys like you,\" he said.\n\nThrough the front window I saw the emergency flasher on an N.O.P.D. cruiser pull to the curb under the live oak tree that grew in Gunner Ardoin's front yard. A lone black female officer slipped her baton into the ring on her belt and walked uncertainly toward the gallery, her radio squawking incoherently in the rain.\n\nI slept that night on Clete's couch in his small apartment above his P.I. office on St. Ann. The sky was clear and pink at sunrise, the streets in the Quarter puddled with water, the bougainvillea on Clete's balcony as bright as drops of blood. I shaved and dressed while Clete was still asleep and walked past St. Louis Cathedral and through Jackson Square to the Cafe du Monde, where I met Father Jimmie Dolan at a table under the pavilion.\n\nAlthough we had been friends and had bass fished together for two decades, he remained in many ways a mysterious man, at least to me. Some said he was a closet drunk who had done time in a juvenile reformatory; others said he was gay and well known among the homosexual community in New Orleans, although women were obviously drawn to him. He had crewcut, blond good looks and the wide shoulders and tall, trim physique of the wide-end receiver he had been at a Winchester, Kentucky, high school. He didn't talk politics but he got into trouble regularly with authority on almost all levels, including six months in a federal prison for trespassing on the School of the Americas property at Ft. Benning, Georgia.\n\nIt had been three months since he had been waylaid in an alley behind his church rectory and methodically beaten from his neck to the soles of his feet by someone wielding a pipe with an iron bonnet screwed down on the business end.\n\n\"Clete Purcel and I rousted a guy named Gunner Ardoin last night. I think maybe he's the guy who attacked you,\" I said.\n\nFather Jimmie had just bitten into a beignet and his mouth was smeared with powdered sugar. He wore a tiny sapphire in his left earlobe. His eyes were a deep green, thoughtful, his skin tan. He shook his head.\n\n\"That's Phil Ardoin. Wrong guy,\" he said.\n\n\"He said he didn't know you.\"\n\n\"I coached his high school basketball team.\"\n\n\"Why would he lie?\"\n\n\"With Phil it's a way of life.\"\n\nAn N.O.P.D. cruiser pulled to the curb out on Decatur and a black female officer got out and fixed her cap on her head. She looked like she was constructed of twigs, her sky blue shirt too large on her frame, her pursed lips layered with red lipstick. Last night Clete had said she reminded him of a black swizzle stick with a cherry stuck on the end.\n\nShe threaded her way through the tables until she was abreast of ours. The brass name tag on her shirt said C. ARCENEAUX.\n\n\"I thought I should give you a heads-up,\" she said.\n\n\"How's that?\" I asked.\n\nShe looked off abstractly at the traffic on the street and at the artists setting up their easels under the trees in Jackson Square. \"Take a walk with me,\" she said.\n\nI followed her down to a shady spot at the foot of the Mississippi levee. \"I tried to talk to the other man, what's his name, Purcel, but he seemed more interested in riding his exercise bike,\" she said.\n\n\"He has blood pressure problems,\" I said.\n\n\"Maybe more like a thinking problem,\" she replied, looking idly down the street.\n\n\"Can I help you with something?\" I asked.\n\n\"Gunner Ardoin is filing an assault charge against you and your friend. I think maybe he's got a civil suit in mind. If I was you, I'd take care of it.\"\n\n\"Take care of it?\" I said.\n\nHer eyes squinted into the distance, as though the subject at hand had already slipped out of her frame of reference. Her hair was black and thick and cut short on her neck, her eyes a liquid brown.\n\n\"Why are you doing this?\" I asked.\n\n\"Don't like people who mule crystal into the projects.\"\n\n\"You work both the night and the morning watch?\"\n\n\"I'm just up from meter maid. Low in standing, know what I mean, but somebody got to do it. Tell the priest to spend more time with his prayers,\" she said, and started to walk back to her cruiser.\n\n\"What's your first name?\" I asked.\n\n\"Clotile,\" she said.\n\nBack at the table I watched her drive away into the traffic, the lacquered brim of her cap low on her forehead. Meter maid, my ass, I thought.\n\n\"Ever hear of Junior Crudup?\" Father Jimmie asked.\n\n\"The blues man? Sure,\" I said.\n\n\"What do you know about him?\"\n\n\"He died in Angola,\" I said.\n\n\"No, he disappeared in Angola. Went in and never came out. No record at all of what happened to him,\" Father Jimmie said. \"I'd like for you to meet his family.\"\n\n\"Got to get back to New Iberia.\"\n\n\"It's Saturday,\" he said.\n\n\"Nope,\" I said.\n\n\"Junior's granddaughter owns a twelve-string guitar she thinks might have belonged to Leadbelly. Maybe you could take a look at it. Unless you just really don't have the time?\" he said.\n\nI followed Father Jimmie in my pickup truck into St. James Parish, which lies on a ninety-mile corridor between Baton Rouge and New Orleans that environmentalists have named Toxic Alley. We drove down a state road south of the Mississippi levee through miles of sugarcane and on through a community of narrow, elongated shacks that had been built in the late nineteenth century. At the crossroads, or what in south Louisiana is called a four-corners, was a ramshackle nightclub, an abandoned company store with a high, tin-roofed gallery, a drive-by daiquiri stand, and a solitary oil storage tank that was streaked with corrosion at the seams, next to which someone had planted a tomato garden.\n\nMost of the people who lived at the four-corners were black. The rain ditches and the weeds along the roadside were layered with bottles of beer and pop cans and trash from fast-food restaurants. The people who sat on the galleries of the shacks were either old or infirm or children. I watched a car filled with teenagers run a stop sign and fling a quart beer bottle on the side of the road, ten feet from where an elderly woman was picking up litter from her lawn and placing it in a vinyl bag.\n\nThen we were out in the countryside again and the sky was as blue as a robin's egg, the sugarcane bending in the wind as far as the eye could see, egrets perched like white sculptures on the backs of cattle in a roadside pasture. But inside the loveliness of the day was another element, discordant and invasive, the metallic reek of natural gas, perhaps from a wellhead or a leaking connection at a pump station. Then the wind shifted and it was gone and the sky was speckled with birds rising from a pecan orchard and from the south I could smell the brassy odor of a storm that was building over the Gulf.\n\nI looked at my watch. No more than one hour with Father Jimmie's friends, I told myself. I wanted to get back to New Iberia and forget about the previous night and the trouble with Gunner Ardoin. Maybe it was time to let Father Jimmie take care of his own problems, I thought. Some people loved adversity, got high on it daily, and secretly despised those who would take it from them. That trait didn't necessarily go away because of a Roman collar.\n\nThe state road made a bend and suddenly the endless rows of sugarcane ended. The fields were uncultivated now, empty of livestock, dotted with what looked like settling ponds. The Crudup family lived down a dirt lane in a white frame house with a wraparound veranda hung with baskets of flowers. Three hundred yards behind the house was a woods bordered with trees that were gray with dead leaves and the scales of air vines, as though the treeline had been matted with premature winterkill.\n\nFather Jimmie had set the hook when he had mentioned Leadbelly's name, but I knew as we drove down the road toward the neat white house backdropped by a poisoned woods that this trip was not about the recidivist convict who wrote \"Goodnight Irene\" and \"The Midnight Special\" and who today is almost forgotten.\n\nIn fact, I wondered if I, like Father Jimmie, could not wait to fill my day with adversity in the way I had once filled it with Jim Beam and a glass of Jax with strings of foam running down the sides.\n\nWhen I cut my engine in front of the house, I took a Dr Pepper from the cooler on the seat and raked the ice off the can and drank it empty before stepping out onto the yard.\n\n## Chapter 2\n\nJunior Crudup's granddaughter had a face like a goldfish, light skin that was dusted with freckles, and glasses that turned her eyes into watery brown orbs. She sat in a stuffed chair, fanning herself with a magazine, her rings of fat bulging against her dress, waiting for me to finish examining the Stella guitar that had lain propped in a corner of her attic for thirty years. The strings were gone, the tuning keys stiff with rust, the sound hole coated with cobweb. I turned the guitar on its belly and looked at three words that were scratched into the back of the neck: Huddie Love Sarie.\n\n\"Leadbelly's real name was Huddie Ledbetter. His wife was named Sarie,\" I said.\n\nJunior Crudup's granddaughter looked through a side window at two children playing on a rope swing that was suspended from a pecan tree. Her name was Doris. She kept straightening her shoulders, as though a great weight were pressing on her lungs. \"How much it wort'?\" she asked.\n\n\"I couldn't say,\" I replied.\n\n\"Four or five songs were in the bottom of the guitar case, each with Junior's signature,\" Father Jimmie said.\n\n\"Yeah, what they wort'?\" Doris Crudup asked.\n\n\"You'd have to ask somebody else,\" I said.\n\nShe gave Father Jimmie a look, then got up from her chair and took my coffee cup into the kitchen, although I had not finished drinking the coffee in it.\n\n\"Her husband died three years ago. Last month the social worker cut off her welfare,\" Father Jimmie said.\n\n\"Why?\"\n\n\"The social worker felt like it. That's the way it works. Take a walk with me,\" he said.\n\n\"I need to get back home.\"\n\n\"You have time for this,\" he said.\n\nWe went outside, into the sunlit, rain-washed loveliness of the fall afternoon. The pecan tree in the side yard puffed with wind and a yellow dog rolled on its back in the dirt while the children swung back and forth above it on their rope swing. But as I followed Father Jimmie down an incline toward the woods in back I could feel the topography changing under my feet, as though I were walking on a sponge.\n\n\"What's that smell?\" I said.\n\n\"You tell me.\" He tore a handful of grass from the soil and held the roots up to my nose. \"They truck it in from all over the South. Doris's lungs are as much good to her as rotted cork. People around here carry buckets in their cars because of their children's constant diarrhea.\"\n\nI held onto the trunk of a withered persimmon tree and looked at the soles of my shoes. They were slick with a black-green substance, as though I had walked across a factory floor. We crossed a board plank spanning a rain ditch. The water was covered with an iridescent sheen that seemed to be rising in chains of bubbles from the bottom of the ditch. Perhaps twenty settling ponds, layered over with loose dirt, were strung along the edge of the woods, each of them crusted with a dried viscous material that looked like an orange scab.\n\n\"Is this Doris's property?\" I said.\n\n\"It belonged to her grandfather. But twenty years ago Doris's cousin made his 'X' on a bill of sale that had Junior's name typed on it. The cousin and the waste management company that bought the land both claim he's the Junior Crudup of record and Doris is out of luck.\"\n\n\"I'm not following you.\"\n\n\"No one knows what happened to the real Junior Crudup. He went into Angola and never came out. There's no documentation on his death or of his release. Figure that one out.\"\n\n\"I don't want to.\"\n\nFather Jimmie studied my face. \"These people here don't have many friends,\" he said.\n\nI slipped the flats of my hands in my back pockets and scuffed at the ground with one shoe, like a third-base coach who had run out of signals.\n\n\"Think I'll pass,\" I said.\n\n\"Suit yourself.\"\n\nFather Jimmie picked up a small stone and side-armed it into the woods. I heard it clatter among the tree trunks. Birds should have risen from the canopy into the sky, but there was no movement inside the tree limbs.\n\n\"Who owns this waste management company?\" I asked.\n\n\"A guy named Merchie Flannigan.\"\n\n\"Jumpin' Merchie Flannigan? From New Iberia?\" I said.\n\n\"One and the same. How'd he get that name, anyway?\" Father Jimmie said.\n\n\"Think of rooftops,\" I said.\n\nAs I drove back to New Iberia, through Morgan City, and down East Main to my rented house on Bayou Teche, I tried not to think anymore of Father Jimmie and the black people in St. James Parish whose community had become a petro-chemical dumping ground. As sad as their story was, in the state of Louisiana it wasn't exceptional. In fact, on television, the current governor had threatened to investigate the tax status of some young Tulane lawyers who had filed suit against several waste management companies on the basis of environmental racism. The old plantation oligarchy was gone. But its successors did business in the same fashion\u2014with baseball bats.\n\nI fixed an early supper and ate it on an ancient green picnic table in the backyard. Across the bayou kids were playing tag football in City Park and smoke from meat fires hung in the trees. In the deepening shadows I thought I could hear voices inside my head: my adopted daughter, Alafair, away at Reed College; my deceased wife Bootsie; and a black man named Batist, to whom I had sold my bait and boatrental business south of town. I didn't do well on Saturday afternoons. In fact, I wasn't doing well on any afternoon.\n\nOn some weekends I drove out to the dock and bait shop to see him. We'd fish the swamp for bass and sac-a-lait, then head home at sunset, the cypress trees riffling like green lace in the wind, the water back in the coves bloodred in the sun's afterglow. But across the road and up the incline from the dock were the burned remains of the house my father had built out of notched and pegged timbers during the Depression, the home where I had lived with my wife and daughter, and I had a hard time looking at it without feeling an indescribable sense of loss and anger.\n\nThe inspector from the fire department called it \"electrical failure.\" I wished I could accept the loss in terms as clinical as those. But the truth was I had trusted the electrical rewiring on my home to a fellow A.A. member, one who had stopped attending meetings. He filled the walls with cheap switches that he did not screw-wrap and inserted fourteen-gauge wire into twelve-gauge receptacles. The fire started inside the bedroom wall and burned the house to the ground in less than an hour.\n\nI went into the house and looked up Merchie Flannigan's name in the directory. I had known his parents in both New Orleans and New Iberia, but I'd never had reason to take official notice of Merchie until I was a patrolman near the Iberville Welfare Project off Basin Street, back in the days when cops still rang their batons off street curbs to signal one another and white kids would take your head off with water-filled garbage cans dropped from a five-story rooftop.\n\nLong before Hispanic and black caricatures acted out self-created roles as gangsters on MTV, white street gangs in New Orleans fought with chains, steel pipes, and zip guns over urban territory that a self-respecting Bedouin wouldn't live in. During the 1950s, the territorial war was between the Cats and the Frats. Frats lived uptown, in the Garden District and along St. Charles Avenue. Cats lived in the Irish Channel, or downtown or in the projects or out by the Industrial Canal. Cats were usually Irish or Italian or a mixture of both, parochial school bust-outs who rolled drunks and homosexuals and group-stomped their adversaries, giving no quarter and asking for none in return.\n\nIn a back-alley, chain-swinging rumble, their ferocity and raw physical courage could probably be compared only to that of their historical cousins in Southie, the Five Points, and Hell's Kitchen. Along Bourbon Street, after twelve on Saturday nights, the Dixieland bands would pack up their instruments and be replaced by rock 'n' roll groups that played until sunrise. The kids spilling out the front doors of Sharkey Bonnano's Dream Room, drinking papercup beer and smoking cigarettes on the sidewalks, their motorcycle caps and leather jackets rippling with neon, made most tourists wet their pants.\n\nBut Jumpin' Merchie Flannigan could not be easily categorized as a blue-collar street kid who had made good in the larger world. In fact, I always had suspicions that Jumpin' Merchie joined a gang for reasons very different from his friends in the Iberville. Unlike most of them, he was not only streetwise but good in school and naturally intelligent. Merchie's problem really wasn't Merchie. It was his parents.\n\nIn New Iberia Merchie's father was thought of as a decent but weak and ineffectual man whose rundown religious store was almost an extension of its owner's personality. Many nights a sympathetic police officer would take Mr. Flannigan out the back door of the Frederic Hotel bar and drive him to his house by the railroad tracks. Merchie's mother tried to compensate for the father's failure by constantly treating Merchie as a vulnerable child, protecting him, making him wear short pants at school until he was in the fifth grade, denying him entry into a world that to her was as unloving as her marriage. But I always felt her protectiveness was of a selfish kind, and in reality she was not only sentimental rather than loving, she could also be terribly cruel.\n\nAfter the family moved to New Orleans and took up life in the Iberville, Merchie became known as a mama's boy who was anybody's punching bag or hard-up pump. But at age fifteen, he threw a black kid from the Gird Town Deuces off a fire escape onto the cab of a passing produce truck, then outraced a half dozen cops across a series of rooftops, finally leaping out into space, plummeting two stories through the ceiling of a massage parlor.\n\nHis newly acquired nickname cost him a broken leg and a one-bit in the Louisiana reformatory, but Jumpin' Merchie Flannigan came back to Canal Street and the Iberville Project with magic painted on him.\n\nWhen I called him at home he was gregarious and ingratiating, and said he wanted to see me. In fact, he said it with such sincerity that I believed him.\n\nHis home, of which he was very proud, was a gray architectural monstrosity designed to look like a medieval castle, inside acres of pecan and live oak trees, all of it in an unzoned area that mixed pipeyards and welding shops with thoroughbred horse barns and red-clay tennis courts.\n\nHe greeted me in the front yard, athletic, trim, wearing pleated tan slacks, half-top, slip-on boots, and a polo shirt, his long hair so blond it was almost white, a V-shaped receded area at the part the only sign of age I could see in him. The yard was covered in shadow now, the chrysanthemums denting in the wind, the sky veined with electricity. In the midst of it all Merchie seemed to glow not so much with health and prosperity as confidence that God was truly in His heaven and there was justice in the world for a kid from the Iberville.\n\nHe meshed his fingers, as though making a tent, then pointed the tips at me.\n\n\"You were out at the Crudup farm in St. James Parish today,\" he said.\n\n\"Who told you?\" I asked.\n\n\"I'm trying to clean up the place,\" he replied.\n\n\"Think it might take a hydrogen bomb?\"\n\n\"So give me the gen on it,\" he said.\n\n\"The Crudup woman says she was cheated out of the title.\"\n\n\"Look, Dave, I bought the property three years ago at a bankruptcy sale. I'll check into it. How about some trust here?\"\n\nIt was hard to stay mad at Merchie. I knew people in the oil business who were openly ecstatic at the prospect of Mideastern wars or subzero winters in the northern United States, but Merchie had never been one of them.\n\n\"Been out of town?\" I said.\n\n\"Yeah, Afghanistan. You believe it?\"\n\n\"Shooting at the Taliban?\"\n\nHe smiled with his eyes but didn't reply.\n\n\"The woman in St. James Parish? Her grandfather was Junior Crudup,\" I said.\n\n\"An R&B guy?\"\n\n\"Yeah, one of the early ones. He did time with Leadbelly. He played with Jackie Brenston and Ike Turner,\" I said. But I could see him losing interest in the subject. \"I'd better go. Your place looks nice. Give me some feedback later on the Crudup situation, will you?\" I said.\n\n\"My favorite police officer,\" I heard a woman say.\n\nThe voice of Theodosha Flannigan was like a melancholy recording out of the past, the kind that carries fond memories but also some that are better forgotten. She was a member of the LeJeune family in Franklin, down the Teche, people whose wealth and lawn parties were legendary in southwest Louisiana, and she still used their name rather than Merchie's. She was tall, darkly beautiful, with hollow cheeks and long legs like a model's, her southern accent exaggerated, her jeans and tied-up black hair and convertible automobiles an affectation that belied the conservative and oligarchical roots she came from.\n\nBut in spite of her corn bread accent and the pleasure she seemed to take in portraying herself as an irreverent and neurotic southern woman, she had another side, one she never engaged in conversation about. She had written two successful screenplays and a trilogy of crime novels containing elements that were undeniably lyrical. Although her novels had never won an Edgar award, her talent was arguably enormous.\n\n\"How you doin', Theo?\" I said.\n\n\"Stay for coffee or a cold drink?\" she said.\n\n\"You know me, always on the run,\" I said.\n\nShe curled her fingers around the limb of a mimosa tree and propped one moccasin-clad foot against the trunk. Her breasts rose and fell against her blouse.\n\n\"How about diet Dr Pepper on the rocks, with cherries in it?\" she said.\n\nDon't hang around. Get away now, I heard a voice inside me say.\n\n\"I'm just about to fix some sherbet with strawberries. We'd love to have you join us, Dave,\" Merchie said.\n\n\"Sounds swell,\" I said, and dropped my eyes, wondering at the price I was willing to pay in order not to be alone.\n\nOn the way into the backyard Theodosha touched my arm. \"I'm sorry about your loss. I hope you're doing all right these days,\" she said.\n\nBut I had no memory of her sending a sympathy card when Bootsie died.\n\nI went to an early Mass the next morning, then bought a copy of the Times-Picayune and drank coffee at the picnic table in the backyard and read the newspaper. I read three paragraphs into an article about an errant bomb falling into a community of mudbrick huts in Afghanistan, then closed the paper and watched a group of children throwing a red Frisbee back and forth under the oak trees in the park. A speedboat full of teenagers roared down the bayou, swirling a trough back and forth between both banks, splintering the air with a deafening sound. I heard my portable phone tinkle softly by my thigh.\n\nThe operator asked if I would accept a collect call from Clete Purcel.\n\n\"Yes,\" I said.\n\n\"Streak, I'm in the zoo,\" Clete shouted.\n\nIn the background I could hear voices echoing down stone corridors or inside cavernous rooms.\n\n\"What did you say?\"\n\n\"I'm in Central Lock-Up. They busted me for assaulting Gunner Ardoin. I feel like I've been arrested for spraying Lysol on a toilet bowl.\"\n\n\"Why haven't you bonded out?\" I asked.\n\n\"Nig and Willie aren't answering my calls.\"\n\nI tried to make sense out of what he was saying. For years Clete had chased down bail skips for Nig Rosewater and Wee Willie Bimstine. He should have been out of jail with a signature.\n\nI started to speak, but he cut me off. \"Gunner is a grunt for Fat Sammy Fig, and Fat Sammy is connected up with every major league piece of shit in Louisiana. I think Nig and Willie don't want trouble with the wrong people. Arraignment isn't until Tuesday morning. Been down to Central Lock-Up lately?\"\n\nI took the four-lane through Morgan City into New Orleans. But I didn't go directly to the jail. Instead, I drove up St. Charles Avenue, then over toward Tchoupitoulas and parked in front of Gunner Ardoin's cottage. His Honda was in the driveway. I walked down to a corner store and bought a quart of chocolate milk and a prepackaged ham sandwich and sat down on Gunner's front steps and began eating the sandwich while children roller-skated past me under the trees.\n\nI heard someone open the door behind me.\n\n\"What the fuck you think you're doin'?\" Gunner's voice said.\n\n\"Oh, hi. I was about to ask you the same thing,\" I said.\n\n\"What?\" he said. He was bare chested and barefoot, and wore only a pair of pajama bottoms string-tied under his navel. The breeze blew from the back of the cottage through the open door. \"What?\" he repeated.\n\n\"Toking up kind of early today?\"\n\n\"So call the DEA.\"\n\n\"Father Jimmie Dolan was your basketball coach. Why did you say you didn't know him?\"\n\n\"'Cause I can't remember every guy I knew in high school with a whistle hanging out of his mouth.\"\n\n\"Father Jimmie says it wasn't you who attacked him, Gunner. But I think somebody told you to bust him up, and you pieced off the job to somebody else. Probably because you still have qualms.\"\n\n\"Is this because I filed on your friend?\"\n\n\"No, it's because you're a shitbag and you're going to drop those charges or I'll be back here tonight and jam a chainsaw up your ass.\"\n\n\"Look, man\u2014\" he began.\n\n\"No, you look,\" I said, rising to my feet, shoving him backward through the door into the living room. \"Fat Sammy is behind the job on Father Jimmie?\"\n\n\"No,\" he said.\n\nI shoved him again. He tripped over a footstool and fell backward on the floor. I pulled back my sports coat and removed my .45 from its clip-on holster and squatted next to him. I pulled back the slide and chambered a round, then pointed the muzzle at his face.\n\n\"Look at my eyes and tell me I won't do it,\" I said.\n\nI saw the breath seize in his throat and the blood go out of his cheeks. He stretched his head back, turning his face sideways, away from the .45.\n\n\"Don't do this,\" he said. \"Please.\"\n\nI waited a long time, then touched his forehead with the gun's muzzle and winked at him.\n\n\"I won't. I'd think about my request on those charges, though,\" I said.\n\nJust as I eased the hammer back down, his bladder gave way and he shut his eyes in shame and embarrassment. When I looked up I saw a little girl, no older than six or seven, staring at us, horrified, from the kitchen doorway.\n\n\"That's my daughter. I get her one day a week. I've known some cruel guys with a badge, but you take the cake,\" Gunner said.\n\nThe charges against Clete were dropped by three that afternoon. I drove him from Central Lock-Up to his apartment on St. Ann, where he fell asleep on the couch in front of a televised football game. Fat Sammy Figorelli's home was only three blocks away, over on Ursulines. The temptation was too much.\n\nFat Sammy had grown up in the French Quarter, and although he owned homes in Florida and on Lake Pontchartrain, he spent most of his time inside the half city block where the Figorelli family had lived since the 1890s. It seemed Sammy had been elephantine all his life. As a child the balloon tires of his bicycle burst under his weight. His rump wouldn't fit in the desk at the school run by the Ursuline nuns. In high school he got stuck inside his tuba while performing with the marching band at an LSU football game. The paramedics had to scissor off his jacket, smear him with Vaseline, and pry him loose in front of ninety thousand people. In his senior year he mustered up the courage to invite a girl to the Prytania Theater. A gang of Irish kids in the balcony rained down a barrage of water-filled condoms on their heads.\n\nAs an adult he filled his body with laxatives, tried every diet program imaginable, trained at fat farms, sweated to the oldies with Richard Simmons, attended a fire-walker's school run by a celebrity con man in California, almost died from liposuction, and finally had a gastric bypass. The consequence of the latter was a weight loss of 170 pounds in a year's time.\n\nAll of the wrong kind.\n\nHe lost the blubber, but under the blubber was a support system of sinew that hung on his frame like curtains of partially hardened cement. If this was not enough of a problem, Fat Sammy had another one that was equally egregious and beyond the scope of medicine. His head was shaped like a football, his few strands of gold hair brushed like oily wire into his scalp.\n\nI twisted an iron bell on the grilled door that gave onto a domed archway leading into Fat Sammy's courtyard.\n\n\"Who is it?\" a voice said from a speaker inside the gate.\n\n\"It's Dave Robicheaux. I've got a problem,\" I said.\n\n\"Not with me, you don't.\"\n\n\"It's about Gunner Ardoin. Open the door.\"\n\n\"Never heard of him. Come back another time. I'm taking a nap.\"\n\n\"There're some movie people in New Iberia. They want to work with some local guys who know their way around,\" I said.\n\nThe speaker box went dead and the gate buzzed open.\n\nThe courtyard was surfaced with soft brick, the flower beds blooming with yellow and purple roses, irises and hibiscus and Hong Kong orchids. Banana and umbrella trees and windmill palms grew along the walls, and the balconies dripped with bougainvillea and passion vine. Fat Sammy lay in a hammock like a beached whale, a Hawaiian shirt unbuttoned on his chest, his skin glazed with suntan lotion. A portable stereo and a mirror and a hairbrush sat on a glass-topped table next to him. The stereo was playing \"Clair de Lune.\"\n\n\"Who are these movie people?\" he asked.\n\n\"Germans. They're making a documentary. I think you're the man to show them around,\" I said.\n\nI pulled up a deep-backed wicker chair and sat down without being asked. He sat up in the hammock and turned down the volume on the stereo, his scalp glistening in the sunshine. He wiped his head with a towel, his eyes neutral, his mouth down-turned at the corners. \"Documentary on what?\" he asked.\n\n\"Let me clear the decks about something else first. Somebody beat up a priest named Father Jimmie Dolan. It's a lousy thing to happen, Sammy, something no respectable man would be involved in. I thought you'd want to know about it.\"\n\n\"No, I don't.\"\n\n\"In the old days elderly people in New Orleans didn't get jack-rolled and their houses didn't get creeped and nobody murdered a child or abused Catholic clergy. If N.O.P.D. couldn't take care of it, we let you guys do it for us.\"\n\nHis eyes were hooded, like a frog's. \"You were kicked off the force, Robicheaux. You don't speak for nobody, at least not around here.\" He paused, as though reconsidering the tenor of his rhetoric. \"Look, this used to be a good city. It ain't no more.\"\n\nWhen I didn't speak he took a breath and started over. \"This is the way it is. I make movies. I build houses. I'm developing shopping centers in Mississippi and Texas. You want to know who's running New Orleans? Flip over a rock. Welfare pukes hustling bazooka and blacks and South American spics and bikers muleing brown skag out of Florida. Nothing against the blacks or the spics. They're making it just like we did. But I wouldn't be in a room with none of them people unless I was encased in a full-body condom.\"\n\n\"Who did the job on Father Dolan?\"\n\nHis eyes were pale blue, almost without color, his expression like that of a man who had never learned to smile. \"Somebody saying it's on me? This guy Ardoin you mentioned?\"\n\nI looked at a strip of pink cloud above the courtyard. \"You're the man in New Orleans,\" I said.\n\n\"Yeah, every whore in the city tells me the same thing. I wonder why. I ever jerk you around, Robicheaux?\" he said.\n\n\"Not to my knowledge.\"\n\n\"Then I ain't going to now. That means I didn't have nothing to do with hurting a priest, and what I might know about it is my own business.\"\n\n\"I'm a little disappointed, Sammy. Within certain parameters you were always straight up,\" I said. I got up to go.\n\nHe brushed at his nose, his pale blue eyes burrowing into my face. \"You lied your way in here? About them movie people?\" he said.\n\n\"That was on the square.\" I handed him a business card that had been given to me by a member of a visiting German television crew the previous week. \"These guys are doing a story on the New Orleans connection to the assassination of President Kennedy. They believe it got set up here and in Miami.\"\n\n\"You saying I\u2014\" His voice broke in his throat. \"I voted for John Kennedy.\"\n\n\"I'm saying nothing had better happen to Father Dolan again.\"\n\nFat Sammy rose from the hammock, wheezing in his chest, like an angry behemoth that couldn't find its legs. I had forgotten how tall he was. He picked up a glass of iced tea from the table, gargled with it, and spit it in the flower bed.\n\n\"You own your soul?\" he asked.\n\n\"What?\"\n\n\"If so, count yourself a lucky man. Now get the fuck out of here,\" he said.\n\nI ate dinner with Clete at a small restaurant up the street from the French Market, then shook hands with him and told him I had better head back for New Iberia. I watched him walk across Jackson Square and pass the cathedral, pigeons flapping in the shadows around his feet, and disappear down Pirates Alley. I started to get into my truck, but instead, for reasons I couldn't explain, I sat down on one of the iron benches by Andrew Jackson's equestrian statue, and listened to a black man playing a bottleneck guitar.\n\nIt was the burnt-out end of a long day and a longer weekend. The wind was cold off the river, the light cold and mauve colored between the buildings that framed the square, the air tinged with the smell of gas from the trees and flower beds. The black man worked the glass bottleneck up and down the frets of his guitar and sang, \"Oh Lord, my time ain't long. Rubber-tired hack coming down the road, burial-ground bound.\"\n\nAn N.O.P.D. cruiser pulled to the curb on Decatur. A black woman in uniform got out and fixed her cap, adjusted the baton on her belt, and walked toward me. She positioned herself between me and the sun, like an exclamation point against a fiery crack in the sky. I picked at my nails and didn't return her stare.\n\n\"Can't stay out of town?\" she said.\n\n\"I have an addictive personality,\" I replied.\n\nShe sat down on the corner of the bench. \"You got a bad jacket for a cop, Robicheaux.\"\n\n\"Who the hell are you?\" I said.\n\n\"Clotile Arceneaux. See,\" she said, lifting her brass name tag with her thumb. \"Your friend, Father Dolan? He's an amateur, and they're going to take his legs off\u2014yours, too, you keep messing in what you're not supposed to be messing in.\"\n\n\"I'm not big on telling other people what to do. I ask they show me the same courtesy,\" I said.\n\nThe baton on her hip kept banging against the bench. She slid it out of the ring that held it and bounced it between her legs on the cement. Her pursed lips looked like a tiny red rose in the gloom. I thought she would speak again, but she didn't. The sun went down behind the buildings in the square and the wind gusted off the levee, smelling of rain and fish-kill in the swamps.\n\n\"Can I buy you coffee, officer?\" I said.\n\n\"Your friend is off the hook on the assault beef. Time for you to go home, Robicheaux,\" she said.\n\nHome, I thought, and looked at her curiously, as though the word would not register in my mind.\n\n## Chapter 3\n\nOn Monday I left the department at mid-morning and checked out a history of Louisiana blues music and swamp pop from the city library and began reading it in my office. It was raining outside, and through my window I could see a freight train, the boxcars shiny with water, wobbling down the old Southern Pacific tracks through the black section of town. The longtime sheriff, an ex-marine who had marched out of the Chosin Reservoir, had retired and been replaced by my old partner, Helen Soileau.\n\nI saw her stop in the corridor outside my office and bite her lip, her hands on her hips. She tapped on the door, then opened it without waiting for me to tell her to come in.\n\n\"Got a minute?\" she asked.\n\n\"Sure.\"\n\n\"A couple of N.O.P.D. plainclothes picked up a prisoner this morning. They said you and Clete bent a pornographic actor out of shape. They thought it was funny.\"\n\n\"Pornographic actor?\" I said vaguely.\n\n\"Ardoin was his name.\"\n\n\"Clete flattened a coffeepot against the side of the guy's head, but it wasn't a big deal,\" I said.\n\nShe had the muscular build of a man and blond hair that she cut short, tapering it on the sides and neck so that it looked like the freshly cropped mane on a pony. She wore slacks and a white, short-sleeve shirt, a badge holder hooked on her belt. She sucked in her cheeks and watched a raindrop run down the window glass above my head.\n\n\"Not a big deal? Interrogating people outside your jurisdiction, banging them in the head with a coffeepot? Dave, I never thought I'd be in this situation,\" she said.\n\n\"Which one is that?\"\n\nShe leaned on the windowsill and looked at the lights of the freight caboose disappearing between a green jungle on each side of the tracks.\n\n\"You and Cletus work it out, but I don't want anybody, that means anybody, dragging N.O.P.D.'s dogshit into this department. I don't want to be the dartboard for those wise-asses, either. We straight on this?\" she said.\n\n\"I hear you.\"\n\n\"Good.\"\n\n\"Remember an R&B guitarist named Junior Crudup?\" I asked.\n\n\"No.\"\n\n\"He went into Angola and never came out. I think his granddaughter got swindled out of her land over in St. James Parish. I think Merchie Flannigan is mixed up in it.\"\n\nShe straightened her back, then looked at me for a long moment. But whatever she had planned to say seemed to go out of her eyes. She grinned, shaking her head, and walked out into the corridor.\n\nI followed her outside.\n\n\"What was that about?\" I asked.\n\n\"Nothing. Absolutely nothing,\" she said. \"Streak, you're just too much. God protect me from my own sins.\"\n\nThen she laughed out loud and walked away.\n\nMonday night I listened to two ancient .78 recordings made by Junior Crudup in the 1940s. As with Leadbelly, the double-strung bass strings on his guitar were tuned an octave apart, but you could hear Blind Lemon and Robert Johnson in his style as well. His voice was haunting. No, that's not the right word. It drifted above the notes like a moan.\n\nThere are some stories that are just too awful to hear, the kind that people press on you after A.A. meetings or in late-hour bars, and later you cannot rid yourself of. This is one of them.\n\nOldtime recidivists always maintained that the worst joints in the country were in Arkansas. Places like Huntsville and Eastham in the Texas penal system came in a close second, primarily because of the furious pace at which the convicts were worked and the punishment barrels they were forced to stand on throughout the night, dirty and unfed, if a gunhack decided they were dogging it in the cotton field.\n\nBut Angola Pen could lay claims that few other penitentiaries could match. During Reconstruction Angola became the model for the rental convict system, one emulated throughout the postbellum South, not only as a replacement for slave labor but as a far more cost-efficient and profitable successor to it. Literally thousands of Louisiana convicts died of exposure, malnutrition, and beatings with the black Betty. Each of the camps made use of wood stocks that were right out of medieval Europe. The scandals at Angola received national notoriety in the 1950s when convicts began slashing the tendons in their ankles rather than stack time on what was called the Red Hat Gang.\n\nI drove up Bayou Teche to Loreauville, where the black man to whom I had sold my boat and bait business now lived with his daughter on a small plot of land not far from town. His house was set back in the shadows, on the bayou's edge, the tin roof almost entirely covered by the overhang of pecan and oak trees. I parked my pickup truck in the trees and walked up to the gallery, where he sat in a wood rocker, a jelly glass filled with iced coffee in his massive hand.\n\nHis name was Batist, and he was both older than he would concede and yet indifferent to what the world thought of him. He had worked most of his life as a farmer, a muskrat trapper and commercial fisherman with my father, and as a packer in several canneries. He could not read or write, but he was nonetheless one of the most insightful people I had ever known.\n\nA fat, three-footed raccoon named Tripod was eating out of a pet bowl on the steps.\n\n\"What's the haps, 'Pod?\" I said to the raccoon, scooping him up in my arms.\n\nBatist's whiskers were white against his cheeks. He removed a cigar from the pocket of his denim shirt and slipped it into his jaw but didn't light it.\n\n\"You ain't come to see me this weekend,\" he said.\n\n\"I had to take care of some business in New Orleans,\" I said. \"Years ago, you knew Junior Crudup, didn't you?\"\n\nHe raised his eyebrows. \"Oh yeah, ain't no doubt about that,\" he replied.\n\n\"What happened to him?\"\n\n\"What always happened to his kind back then. Trouble wherever he went.\"\n\n\"Want to be a little more specific?\"\n\n\"Back in them days there was fo' kinds of black folks. There was people of color, there was Negroes, and there was colored people. Under all them others was niggers.\"\n\n\"Crudup was in the last category?\"\n\n\"Wrong about that. Junior Crudup was a man of color. Called his-self a Creole. He wore an ox-blood Stetson, two-tone shoes, and a shirt and suit that was always pressed. Used to have a cherry red electric guitar he'd carry to all the dances. If a man could be pretty, that was Junior.\"\n\n\"How'd he end up in Angola?\"\n\n\"Didn't fit. Not in white people's world, not in black people's world. Junior had his own way. Didn't take his hat off to nobody. He'd walk five miles befo' he'd sit in the back of the bus. Back in them days, a black man like that wasn't gonna have a long run.\"\n\nTripod was struggling in my arms and kicking at me with his feet. I set him down and looked at the fireflies lighting in the trees. The air was cool and breathless, the surface of the bayou layered with steam. An electrically powered boat hung with lanterns was passing through the corridor of oaks that lined the banks. Batist's attitudes on race were not conventional ones. He never saw himself as a victim, nor did he ever act as the apologist for black men who were forced into lives of crime, but by the same token he never told less than the truth about the world in which he'd grown up. So far I could not determine where he stood on Junior Crudup.\n\n\"It started at a dance at the beginning of the Depression,\" he said. \"Junior was about t'irteen or fo'teen years old, working in a band for a black man had the most beautiful voice you ever heard. They was playing in a white juke by Ville Platte, on a real hot night, the place burning up inside. The singer, the man wit' the beautiful voice, he was playing the piano and singing at the same time, sweat pouring down his face. A white woman come off the dance flo' and patted her handkerchief on his brow. That's all she done. That's all she had to do.\n\n\"After the juke closed up, five white men drunk on moonshine caught the singer out on the road and beat him till he couldn't get off the ground. But that wasn't enough for them, no. They was in an old Ford, one wit' them narrow tires, and they run the tire right acrost his t'roat and busted his windpipe. Man never sung again and died in the asylum. Junior seen it all, right there on the side of the road, and couldn't do nothing about it. I don't t'ink there was a person in the whole round world he trusted after that.\"\n\n\"Why'd he go to the joint, Batist?\"\n\n\"Got caught sleeping wit' a white man's wife. That was 1934 or '35. But you want to know what happened in there, we got to talk to Hogman.\"\n\n\"Batist, I'd really like to keep this simple.\"\n\n\"They put Junior Crudup on the Red Hat Gang. Every nigger in Lou'sana feared that name, Dave. The ones come off it wasn't never the same.\"\n\nHogman Patin was a big, powerful man, an ex-con musician who had done time at the old camps in Angola with Robert Pete Williams, Matthew Maxey, and Guitar Git-and-Go Welch. His arms were coal black and laced with pink scars from a half dozen knife beefs inside the prison system. Now he ran a cafe in St. Martinville, appeared once a year at the International Music Festival in Lafayette, and sold scenic postcards with his signature on them for a dollar a piece. Batist and I sat with him in his side yard, a mile up the bayou, while he threw scrap wood on a fire and told us about Junior Crudup and the Red Hat Gang.\n\n\"See, Junior run the first year he was on the farm. Gunbull put a half cup of birdshot in his back, but he whipped a mule into the water and held onto its tail till it swum him all the way acrost the Miss'sippi,\" Hogman said, flinging a board into the fire, the sparks fanning across the bayou's surface. \"A young white doctor on the other side picked the shot out of his back and tole Junior he had a choice\u2014he'd give Junior ten dollars and forget he was there or the doctor would carry him on back to the penitentiary.\n\n\"Junior said, 'They'll whup me with the black Betty if I go back.'\n\n\"The doctor say, 'No, they ain't. I'm gonna make sure they ain't.'\n\n\"The doctor carried him on back to the farm and tole the warden he was gonna come see Junior every mont', and if Junior was whupped, the doctor was gonna have the warden's job.\n\n\"When Junior come out of the infirmary, they sent him to the Red Hat Gang. There was two captains running the Red Hat Gang then, the Latiolais brothers. First day they tole Junior they knowed they couldn't whup him, but by God they was gonna kill him.\n\n\"See, there was several t'ings special about the Red Hat Gang. Everybody wore black-and-white stripes and straw hats that was painted red. But didn't nobody walk. From cain't-see to cain't-see, it was double-time, hit-it-and-git-it, roll, nigger, roll.\n\n\"The Latiolais brothers was both drunkards. One of them might drink corn liquor under a tree and take a nap, then wake up and point his finger at a man and say, 'Take off, boy.' The next t'ing you'd hear was that shotgun popping.\n\n\"If a man fell out under the sun, he'd get put on an anthill. If a man was dogging it on the wheelbarrow, the captain would say, 'I need me a big wet rock.' There was a mess of rocks piled up down in the shallows, see. A convict would have to find a big one, a twenty-five pounder maybe, wet it down, and run it back up the slope to the captain befo' it was dry. Course, the faster the convict run, the quicker the rock got dried.\n\n\"So one day the captain tole Junior he was dogging it and he better get his ass down on the river and bring the captain the biggest wet rock he could find. Now, them rocks was a good half mile away and the captain knowed Junior was gonna be one wore-out nigger by the end of the day.\n\n\"Except Junior toted the rock on up the slope, then when the captain wasn't looking, he ducked behind some gum trees and pissed all over it. Then he holds up the rock to the captain and says, 'This wet enough for you, boss?'\n\n\"The captain touches the rock and looks at his hand and smells it. He cain't believe what Junior just done. Everybody on the Red Hat Gang started laughing. They was trying to hide it, looking at the ground and each other, but they just couldn't hold it inside. It was so funny they thought for a minute even the captain would laugh. They was sure wrong about that.\"\n\n\"What happened?\" I asked.\n\nHogman wore a strap undershirt that hung like rags on his body. His eyes took on a melancholy cast.\n\n\"The captain took Junior to the sweatbox on Camp A. It was an iron box no bigger than a coffin, standing straight up on a concrete pad. They kept that boy in there seven days, in the middle of summer, no way to go to the bat'room except a bucket between his legs,\" he said.\n\n\"What became of Junior?\" I asked.\n\n\"Don't know. He was in and out of 'Gola a couple of times. Maybe they buried him in the levee. I reckon there's hundreds in that levee. I don't study on it no mo',\" he said.\n\nHis eyes seemed to focus on nothing, his forehead glistening in the firelight.\n\nEarly the next morning I picked up my mail in my pigeon hole at the department and sorted through it at my desk. In it was an invitation, written in a beautiful hand on silver-embossed stationery.\n\nDear Dave,\n\nCan you come to Fox Run Saturday afternoon? It's lawn tennis and drinks and probably a few self-satisfied people talking about their money. In fact, it's probably going to be a drag. But that's life on the bayou, right? Merchie and I do want to see you.\n\nCall me. Please. It's been a long time.\n\nUntil then, Theodosha\n\nA long time since what? I thought.\n\nBut I knew the answer, and the memory was one I tried to push out of my mind. I dropped the invitation into a drawer and glanced out the window at a car with two men in it, pulling to the curb in front of the courthouse. The driver wore a black suit and a Roman collar. His passenger twisted his head about, his face bloodless, like someone on his way to the scaffold.\n\nTwo minutes later the pair of them were at my door.\n\n\"Phil came to the church and made his reconciliation,\" Father Jimmie said, closing the door behind him. \"If you don't mind, he'd like to talk over some things with you. Maybe in private.\"\n\nGunner Ardoin, whom Father Jimmie referred to as Phil, looked at me briefly, then out the window at a trusty mowing the grass.\n\n\"You want to tell me something, Gunner?\" I asked.\n\n\"Yeah, sure,\" he replied.\n\nFather Jimmie nodded and left the room. I told Gunner to take a seat in front of my desk. He breathed through his mouth, as though he were inside a walk-in freezer.\n\n\"I'm doing this for Father Dolan,\" he said.\n\n\"You're doing it to save your ass,\" I said.\n\nHis eyes didn't look at me but his face hardened.\n\n\"You went to confession?\" I said.\n\n\"They call it reconciliation now. But, yeah, I went,\" he said.\n\n\"So who put the contract on Father Jimmie?\"\n\n\"I got a phone call. From a guy named Ray. He don't have another name. He just said I was supposed to take care of Father Dolan. When I got a delivery to make, Ray is the guy who calls me. I told Ray I didn't do stuff like that. He says I do it or I find a new source of income. So I called up a guy. He rolls queers in the Quarter and at some sleaze joints on Airline. For a hundred bucks he does other kinds of work, too.\"\n\n\"Do you have any idea what you did to a decent and fine man?\"\n\n\"You want the guy's name?\"\n\n\"No, I want Ray's last name and I want the guy Ray works for.\"\n\n\"Man, you don't understand. Father Dolan's got enemies all over New Orleans. He's trying to shut down drive-by daiquiri windows and trash incinerators and these guys who been dumping sludge out in the river parishes. He told the Times-Picayune these right-to-life people were committing a sin by putting these women's pictures and names on the Internet.\"\n\n\"What are you talking about?\"\n\n\"These anti-abortion nutcases. They take pictures of women going into abortion clinics, then put the pictures and the women's names and addresses on the Internet. Father Dolan spoke up about it, a Catholic priest. How many enemies does one guy need?\"\n\n\"Our time is about up, Gunner,\" I said.\n\n\"The queer-bait from the Quarter was supposed to scare Father Dolan, not go apeshit with a pipe. Hey, are you listening? It's on the street I snitched off Sammy Fig. You must have given up my name to Fat Sammy.\"\n\n\"Sammy says he never heard of you. You shouldn't have anything to worry about.\"\n\n\"I knew it.\" His face turned gray. He wiped his mouth and looked at the trusty gardener clipping a hedge outside the window. \"Why you staring at me like that?\" he said.\n\n\"I think you're using the seal of the confessional to keep Father Dolan from testifying against you.\"\n\n\"Maybe that was true at first. But I'm still sorry for what I done. He's a good guy. He didn't deserve what happened to him.\"\n\nI glanced at my watch. \"We're done here. So long, Gunner,\" I said.\n\nHe rose from his chair and walked to the door, then stopped, his shoulders slightly stooped, his impish features waiting in anticipation, as though an act of mercy might still be extended to him.\n\n\"What is it?\" I said.\n\n\"Call Sammy Fig. Tell him I didn't rat him out.\"\n\n\"What's Ray's last name?\" I asked.\n\n\"I don't know.\"\n\n\"Adios,\" I said.\n\nI went back to reading my morning mail. When I looked up again, he was gone. A moment later Father Jimmie stuck his head in the door, his disappointment obvious.\n\n\"You couldn't help Phil out?\" he asked.\n\nThe next day I called the warden's office at Angola Penitentiary and asked an administrative assistant to do a records search under the name of Clarence \"Junior\" Crudup.\n\n\"When was he here?\" the assistant asked.\n\n\"In the forties or fifties.\"\n\n\"Our records don't go back that far. You'll have to go through Baton Rouge for that.\"\n\n\"This guy went in but didn't come out.\"\n\n\"Say again?\"\n\n\"He was never released. No one knows what happened to him.\"\n\n\"Try Point Lookout.\"\n\n\"The cemetery?\"\n\n\"Nobody gets lost in here. They either go out through the front gate or they get planted in the gum trees.\"\n\n\"How about under the levee?\"\n\nHe hung up on me.\n\nAt noon I walked past the whitewashed and crumbling brick crypts in St. Peter's Cemetery to Main Street and ate lunch at Victor's Cafeteria, then returned to the office just as the sun went behind a bank of thunderheads and the wind came up hard in the south and began blowing the trees along the train tracks. There were two telephone messages from Theodosha Flannigan in my mailbox. I dropped them both in the dispatcher's wastebasket.\n\nAt 4:00 P.M., in the middle of a downpour, I saw her black Lexus pull to the curb in front of the courthouse. She popped open an umbrella and raced for the front of the building, water splashing on her calves and the bottom of her pink skirt.\n\nI went out into the corridor to meet her, feigning a confidence that masked my desire to avoid seeing her again.\n\n\"Did you get my invitation?\" she said, her face and hair bright with rain.\n\n\"Yes, thanks for sending it,\" I replied.\n\n\"I called earlier. A couple of times.\"\n\nTwo deputies at the water cooler were looking at us, their eyes traveling the length of her figure.\n\n\"Come on in the office, Theo. It's been a little busy today,\" I said.\n\nI closed the door behind us. \"If you can't come Saturday, I understand. I need to talk to you about something else, though,\" she said.\n\n\"Oh?\"\n\n\"I've got a problem. It comes in bottles. Not just booze. Six months ago I started using again. My psychiatrist gave me the keys to the candy store,\" she said.\n\nHer voice was wired, the whites of her eyes threaded with tiny veins. She let out a breath in a ragged sigh. Her breath smelled like whiskey and mint leaves, and not from the previous night. \"Can I sit down?\" she asked.\n\n\"Yes, I'm sorry. Please,\" I said, and looked over my shoulder at Helen Soileau passing in the corridor.\n\n\"Dave, I have little men with drills and saws working in my head all day. Sometimes in the middle of the night, too,\" Theodosha said.\n\n\"There's a meeting tonight at Solomon House, across from old New Iberia High,\" I said.\n\n\"I've been in treatment twice. I was in analysis for seven years. I get a year of sobriety, then things start happening in my head again. My most recent psychiatrist shot himself last week. In Lafayette, in Girard Park, while his kids were playing on the swings. I keep thinking I had something to do with it.\"\n\n\"Where's Merchie in all this?\"\n\n\"He makes excuses for me. He doesn't complain. I couldn't ask for more. You know, he's not entirely normal himself.\" She took a handkerchief from her purse and blotted the moisture from her eyes. \"I don't know what I'm doing here. Merchie's bothered because you think he's dumping oil waste around poor people's homes. He looks up to you. Can't you come out to Fox Run Saturday?\"\n\n\"I'm kind of jammed up these days.\"\n\n\"How long were you drunk?\"\n\n\"Fifteen years, more or less.\"\n\n\"You didn't want to drink when your wife died?\"\n\n\"No,\" I said, my eyes leaving hers.\n\n\"I don't know how anybody stays sober. I feel dirty all over.\"\n\n\"Why?\"\n\n\"Who cares? Some people are born messed up,\" she said. \"I'm sorry for coming in here like this. I'm going to find a dark, hermetically sealed, air-conditioned lounge and dissolve myself inside a vodka collins.\"\n\n\"Some people just ride out the hangover. Today can be the first inning in a new ballgame.\"\n\n\"Good try,\" she said, rising from her chair.\n\nI thought she was about to leave. Instead, she fixed her gaze on me, waiting. Her hair had the black-purplish sheen of silk, the tips damp and curled around her throat.\n\n\"Is there something else?\" I asked.\n\n\"What about Saturday?\" Her face softened as she waited for an answer.\n\n## Chapter 4\n\nThat evening, at twilight, a Buick carrying three teenage girls roared around a curve on Loreauville Road, passed a truck, caromed off a roadside mailbox, then righted itself and slowed behind a school bus as someone in the backseat flung a box of fast-food trash and plastic cups and straws out the window. The truck driver, a religious man who kept a holy medal suspended from a tiny chain on his rearview mirror, would say later he thought the girls had settled down and would probably follow the church bus at a reasonable speed into Loreauville, five miles up Bayou Teche.\n\nInstead, the driver crossed the double-yellow stripe again, into on-coming traffic, then tried to cut in front of the church bus when she realized safe harbor would never again be hers.\n\nHelen Soileau, four uniformed deputies, two ambulances, and a firetruck were already at the accident scene when I arrived. The girls were still inside the Buick. The telephone pole they had hit was cut in half at the base and the downed wires were hanging in an oak tree. The Buick had slid on its roof farther down the embankment, splintering a white fence before coming to rest by the side of a fish pond, where the gas tank had exploded and burned with heat so intense the water in the pond boiled.\n\n\"You run the tag yet?\" I said.\n\n\"It's registered to a physician in Loreauville. The baby-sitter says he and his wife are playing golf. I left a message at the country club,\" Helen said.\n\nShe wore her shield on a black cord around her neck. The wind shifted, blowing across the barns and pastures of the horse farm where the Buick had burned. But the odor the wind carried was not of horses and alfalfa. Helen held a wadded-up piece of Kleenex to her nose, snuffing, as though she had a cold. Two firemen used the jaws-of-life to pry apart the window on the driver's side of the Buick, then began pulling the remains of the driver out on the grass.\n\n\"The bus driver says the Buick was swinging all over the road?\" I asked.\n\n\"Yep, they were having a grand time of it. Life on the bayou in 2002,\" Helen said.\n\nThe water oaks along the Teche had already lost their leaves and their branches looked skeletal against the flattened, red glow of the sun on the western horizon. A spruce green Lincoln with two people in the front seat approached us from the direction of Loreauville, slowing in the dusk, pulling onto the shoulder. The driver got out, looking over the top of his automobile at the scene taking place by the fish pond, his face stenciled with a sadness that no cop, at least no decent one, ever wishes to deal with.\n\nI reached through the open window of Helen's cruiser and picked up a pair of polyethylene gloves and a vinyl garbage bag.\n\n\"Where you going?\" she said.\n\n\"Litter patrol,\" I replied.\n\nI walked back along the road for two hundred yards or so, past a line of cedar trees that bordered another horse farm, then crossed the road to the opposite embankment where a spray of freshly thrown trash bloomed in the grass. I picked up chicken bones, half-eaten dinner rolls, soiled paper napkins, a splattered container of mashed potatoes and gravy, three blue plastic cups, three lids and straws, and broken pieces of a plastic wrap that had been used to seal the lids on the cups.\n\nThere were still grains of ice in the cups, along with the unmistakable smell of sugar, lemon juice, and rum. I found a paper sack and placed the cups and lids in it, then deposited the sack in the garbage bag. When I got back to the accident scene, Helen was talking to the father and mother of the girl who had driven the Buick. The father's face was dilated with rage as he pointed his finger at the drivers of both the truck and the church bus, both of whom had said his daughter was speeding and crossing the double-yellow stripe.\n\n\"Maybe you boxed her off, too. Why would she go off the lefthand embankment unless you wouldn't let her back in line? Answer me that, goddamnit,\" he said.\n\nAn ambulance containing the bodies of the three girls was working its way around the other emergency vehicles, its flashers beating silently against the dusk.\n\nI dropped the evidence bag in Helen's cruiser and drove home, passing a rural black slum at the four corners, where several cars and pickup trucks were lined up at the service window of a drive-by daiquiri store.\n\nEarly the next morning, when the streets were still empty and the light was gray and streaked with mist in the backyards along the bayou, Fat Sammy Figorelli parked his Cadillac in front of my house, puffed on a cigarette while studying the live oaks and antebellum homes that lined East Main, then walked up on my gallery and began knocking so hard the walls shook.\n\n\"You mad at my door?\" I said.\n\n\"I need to straighten you out about a certain issue,\" he said.\n\nI stepped outside, barefoot, still unshaved, dressed only in a T-shirt and khakis. He wore a rust-colored shirt and brown knit necktie and knife-creased slacks. He stood a half-head taller than I, his porcine face shiny with cologne.\n\n\"A little early, isn't it?\" I said.\n\n\"I get up at four every morning. I think sleep sucks,\" he said.\n\n\"I see. Then you wake up other people. Makes sense.\"\n\n\"What?\" he said.\n\n\"Why are you here, Sammy?\"\n\n\"I got this punk Gunner Ardoin calling me up, telling me he didn't rat me out, that he's got a little girl, that he can't afford to lose work 'cause he's in the hospital.\"\n\n\"Why tell me about it?\" I asked.\n\n\"Thanks to you and that animal Purcel, my name is getting drug into all this.\"\n\n\"Into what?\"\n\n\"Stories about a priest getting bashed. I don't want to hear my name coming up no more in regard to Father Jimmie Dolan. This guy is a world-class pain in the ass and I got nothing to do with him. What kind of priest punches out the owner of a health salon, anyway?\"\n\n\"I hadn't heard that one.\"\n\n\"He probably left it out of his homily.\"\n\n\"I'll try to remember all this. Thanks for dropping by,\" I said.\n\nSammy looked at me for a long time, his nostrils swelling with air, his small mouth a tight seam, as though he had been talking futilely to either a deaf or stupid man. A delivery truck smelling of donuts or freshly baked bread passed on the street. Fat Sammy watched the truck turn the corner by a huge, redbrick, tree-shaded antebellum home called the Shadows and disappear down a side street.\n\n\"This is a nice town,\" he said.\n\nI realized that whatever was really bothering him was probably not within his ability to explain. He watched a blue jay lighting on a bird feeder that hung from an oak limb in the yard. Then, like every mainstream American gangster I had ever known, almost all of whom struggle to hold onto some vestige of respectability, he unknowingly opened a tiny window into a childlike area of his soul.\n\n\"I talked with them German film people who's doing a documentary. They say you told them I used to be on a first-name basis with a Miami guy who helped kill President Kennedy. It's true, you told them people that?\" he said.\n\n\"You know the same stories I do, Sammy. They just sound better coming from you. You were born for the screen, partner,\" I said.\n\nHe seemed to think about my explanation, but showed no indication of wanting to leave my gallery.\n\n\"You care to come inside and have some coffee?\" I said.\n\n\"Got any donuts?\" he said.\n\nI opened the door for him and watched his enormous bulk move past me into my house. I could smell an odor like testosterone ironed into his clothes.\n\nThat morning I drove to the high school that the three dead girls had attended up the bayou in the little town of Loreauville. The registrar gave me a copy of the yearbook from the previous year and I found the three girls' photographs among members of the junior class. All three had been either class officers, prom queens, members of the drama club and speech team, or participants in Madrigals. They had been scheduled to graduate in the spring.\n\nBut one of the girls had a different kind of distinction. The driver, Lori Parks, had been on probation for possession of Ecstasy and had been driving with a restricted license for a previous DWI. By late afternoon the forensic chemist at our crime lab had matched a latent print from one of the plastic cups I had picked up two hundred yards from the crash site. The latent belonged to Lori Parks.\n\nThere is no open-container law in the State of Louisiana. It is supposedly illegal to drink and drive in the state, but a vendor can sell mixed drinks at drive-by windows to people in automobiles, provided the container is sealed. Wrapping a piece of plastic around the lid of a daiquiri cup satisfies the statute, and the passengers in the automobile are allowed to open the cups and consume any amount of alcohol they wish as long as they do not give alcohol to the driver.\n\nIf the driver is drinking and sees a state trooper or sheriff's deputy hit his flasher, he only needs to hand his cup to a passenger and instantly he comes into compliance with the law.\n\nThe only person legally liable for any violation of the statutes governing the drive-by window sale of mixed drinks is the clerk who actually makes the sale, never the owner. Sometimes the clerk, who is usually paid no more than minimum wage, is fined or jailed or both for selling to underage customers. But the daiquiri windows remain open seven days and nights a week, positioned on each end of town, thriving on weekends and on all paydays.\n\nJust before I started to drive out to the daiquiri store at the four corners on Loreauville Road, the phone rang on my desk. It was the administrative assistant in the warden's office at Angola Penitentiary, the same man who had hung up on me when I had mentioned the possibility of Junior Crudup being buried under the levee along the Mississippi River.\n\n\"I did some digging around,\" he said.\n\nI laughed into the receiver.\n\n\"You think this is funny?\" he said.\n\n\"No, sir. I'm sorry.\"\n\n\"Ever know an old gunhack by the name of Buttermilk Strunk?\"\n\n\"Cain't-See to Cain't-See Double-Time Strunk?\" I said.\n\n\"That's the man. He was working levee gangs from Camp A in 1951. He says Crudup was a big stripe back then and on the shit list of a couple other gunbulls who wanted to make a Christian out of him, get my meaning?\"\n\n\"I think so,\" I said.\n\n\"They worked him over pretty bad. Strunk says that's about the time a man came to the penitentiary and made recordings of some of the convicts. According to Strunk, this man probably saved Crudup's life.\"\n\n\"You mean John or Allen Lomax, the folk music collectors?\"\n\n\"No, this guy lives in Franklin. You ought to know him. He only owns about half the goddamn state.\"\n\n\"Who are we talking about?\" I said, my impatience growing.\n\n\"Castille LeJeune. Strunk says LeJeune came to Angola with a man from a record company and got Crudup pulled off the levee gang. He doesn't know what happened to him after that.... You still there?\"\n\n\"Castille LeJeune saved the life of a black convict? I'm having a hard time putting this together.\"\n\n\"Why's that?\"\n\n\"He's supposed to be a sonofabitch.\"\n\n\"Remind me not to waste my time on bullshit like this again,\" the administrative assistant said.\n\nThat night my old enemy was back. According to his friends, Audie Murphy fashioned a bedroom out of his garage in the hills overlooking Los Angeles and slept separately from his wife, a loaded army-issue .45 under his pillow. After World War II he had become convinced that, before he could sleep a full night again, he would have to spend five days in peacetime for every day he had spent on the firing line. For him that meant twenty years of sleeplessness.\n\nI couldn't offer my limited experience in Vietnam as the raison d'etre for my insomnia. I drank before I went there and I drank more when I came back. Now I did not drink at all and my nocturnal hours were still filled with the same visitors and feelings; they simply took on different shapes and faces.\n\nThe night seemed alive with sound\u2014the clatter of red squirrels on the roof, a dredge boat out on the bayou, a brief rain shower that swept across the trees in the yard. When I finally fell asleep I dreamed of my dead wife Bootsie and Father Jimmie Dolan and the three girls who had died in a burning automobile and of a Negro convict who had been ground up in a system that loathed courage in a black man.\n\nWhat were the dreams really about? An imperfect world, I suspect, one over which death and injustice often seemed to hold dominion. But what kind of fool would surrender his sleep over a condition he could not change?\n\nSleeping with a .45 did not bring Audie Murphy peace of mind, nor did gambling away millions in Las Vegas. I had slept with firearms, too, and invested substantial sums of money in the parimutuel industry at racetracks all over the country, but I was no more successful in my attempt at redress with the world than he was. That said, I did have an answer for insomnia, one that was surefire and one that Murphy evidently did not try. But just the thought of its coming back into my life made sweat pop on my forehead.\n\nWhen I went to the office in the morning a faxed message was waiting for me from the Department of Public Safety and Corrections in Baton Rouge. Since there was no record of Junior Crudup's discharge from Angola or his death while on the farm, it was the department's contention he had served his full sentence and gone out \"max time,\" which meant he would have been released without parole stipulations or supervision sometime in 1958.\n\nIt was pure blather.\n\nI called Father Jimmie Dolan at his rectory in New Orleans and was told he was working in the garden. Fat Sammy had said Father Jimmie was a global-size pain in the ass. The archdiocese must have felt the same. He had been assigned to an ancient, downtown church in a dirty, dilapidated neighborhood off Canal, where Mass was still said in Latin, women in the pews covered their heads, and communicants knelt at the altar rail when they received the Eucharist, as though the 1960s reforms of Vatican II had never taken place.\n\nLast year, when I remarked to Father Jimmie on the obvious bad judgment if not punitive intention on the part of the diocese in placing a minister such as himself in a parish with an anachronistic mindset, he replied, \"Some people can't accept change. So the church lets a few wall themselves up in a mausoleum and pretend the past is still alive. Know anybody else who has that kind of problem?\"\n\n\"Excuse me?\" I said.\n\n\"They're not bad guys,\" he said, grinning from ear to ear.\n\nMy mind came back to the present and I heard Father Jimmie scrape the phone receiver off a hard surface.\n\n\"Fat Sammy Figorelli says you punched out the owner of a health salon,\" I said.\n\n\"Not exactly.\"\n\n\"How 'not exactly'?\"\n\n\"The guy we're talking about runs a massage parlor and escort service. He forced a seventeen-year-old Vietnamese girl from our parish to give a blowjob to one of his customers. Is this why you called?\"\n\n\"The Department of Corrections says Junior Crudup's last sentence was up in 1958. They say he wasn't paroled and he didn't die inside the prison, so he must have gone out max time in '58.\"\n\n\"He was probably killed and buried on the farm. But I doubt if we'll ever know.\"\n\n\"There's more. An oldtime gunbull says a man by the name of Castille LeJeune got Junior off the levee gang around 1951. But that's where the trail ends.\"\n\n\"Castille LeJeune, in Franklin? That's Theodosha Flannigan's father. She's married to Merchie Flannigan.\"\n\n\"How'd you know that?\" I said.\n\n\"She used to live in New Orleans. She was one of our parishioners. Can we have a talk with Mr. LeJeune?\"\n\n\"I don't like to get too close to Theodosha.\"\n\nThere was a beat, then he said, \"Oh, I see.\"\n\nWay to go, Robicheaux, I thought.\n\nThat afternoon I went to each drive-by daiquiri store in New Iberia. Each of the stores used the same type of blue plastic cups that I had picked up near the accident scene, the same type lids, the same type sealing wrap. I showed each of the clerks working the window the yearbook photographs of the three girls killed on Loreauville Road. Each clerk looked at them blankly and shook his head. At the first three stores I believed the denials given me by the clerks. At the fourth my experience was different.\n\nThe store was a boxlike, plywood structure, painted white, located inside an oak grove just outside the city limits. I parked my cruiser in the trees and waited in the shade while the clerk, a kid probably not much over legal age himself, serviced three drive-by customers. Then I walked to the window, which had a flap on it propped up by a stick. I opened my badge on him.\n\n\"What's your name?\" I said.\n\n\"Josh Comeaux.\"\n\n\"You work here every evening, Josh?\"\n\n\"Yes, sir. Unless I have a basketball game. Then Mr. Hebert lets me off,\" he answered.\n\nI flipped the high school yearbook open to a marked page and showed him pictures of two of the dead girls.\n\n\"You know either one of these girls?\" I asked.\n\n\"No, sir, I can't say I do,\" he said. He wore khakis and a starched, print shirt, the short sleeves folded in neat cuffs on his upper arms. His hair was black, combed back with gel, boxed on the neck, his skin tanned.\n\n\"Can't or won't?\" I said, and smiled at him.\n\n\"Sir?\" he said, confused.\n\nI turned to another marked page in the yearbook and showed him a picture of Lori Parks.\n\n\"How about this girl?\" I said.\n\nHe shook his head, his eyes flat. \"No, sir. Don't know her. I guess I'm not much help on this. These girls do something wrong?\"\n\n\"You seem out of breath. You all right?\" I said.\n\n\"I'm fine,\" he said, and tried to smile.\n\n\"What time did you serve her?\" I asked.\n\n\"Serve who?\"\n\n\"Lori Parks,\" I said, tapping the picture of the driver.\n\n\"I haven't said I did that. I haven't said no such thing. No, sir.\"\n\n\"The autopsy on this girl indicates she was alive when the gasoline tank on her car exploded. She was seventeen years old. I think you're in a world of shit, partner.\"\n\nHe swallowed and looked at the smoke hanging in the trees from a barbecue joint. He opened his mouth to speak, but a middle-aged, balding man who wore a cowboy vest and a string tie and hillbilly sideburns that looked like grease pencil cupped his hand on the boy's shoulder and glared at me through the service window.\n\n\"You saying we served somebody under age?\" he asked.\n\n\"I know you did,\" I said.\n\n\"Every young person who comes by this window has to show ID. That's the rule. No exceptions,\" he said.\n\n\"You the owner?\" I said.\n\nHe ignored my question and addressed his clerk. \"You serve anybody who looked like a minor yesterday?\"\n\n\"No, sir, not me. I checked everybody,\" the clerk said.\n\n\"That's what I thought,\" the man in the vest said. \"We're closed.\"\n\n\"How did you know the problem sale was yesterday?\" I asked.\n\nHe pulled out the support stick from under the window flap and let it slam shut in my face.\n\nWhile I had spent the afternoon questioning the employees of New Iberia's drive-by daiquiri stores, an unusual man was completing his journey on the Sunset Limited from Miami into New Orleans. He had small ears that were tight against his scalp, narrow shoulders, white skin, lips that were the color of raw liver, and emerald green eyes that possessed the rare quality of seeming infinitely interested in what other people were saying. He sat in the lounge car, wearing a seersucker suit and pink dress shirt with a plum-colored tie and ruby stick pin, sipping from a glass of soda and ice and lime slices while the countryside rolled by. An elderly Catholic nun in a black habit sat down next to him and opened a book and began reading from it. She soon became conscious that the man was watching her.\n\n\"Could I help you with something?\" she asked.\n\n\"You're reading The Catholic Imagination by Father Andrew Greeley. A fine book it is,\" the man said.\n\n\"I just started it. But, yes, it seems to be. Are you from Ireland?\"\n\nHe considered his reply. \"Umm, not anymore,\" he said. \"Are you going to New Orleans, Sister?\"\n\n\"Yes, I live there. But my parents came from Waterford, in the south of Ireland.\"\n\nBut he didn't seem to take note of her parents' origins. His eyes were so green, his stare so invasive, she found herself averting his gaze.\n\n\"Would you be knowing a Father James Dolan in New Orleans?\" the man asked.\n\n\"Why, yes, he's a friend of mine.\"\n\n\"I understand he's a lovely man. Works in a parish where they still say a traditional Mass, does he?\"\n\n\"Yes, but he's\u2014\"\n\n\"He's what?\"\n\n\"He's not a traditional man. Excuse me, but you're staring at me.\"\n\n\"I am? Oh, I beg your pardon, Sister. But you remind me of a mother superior who ran the orphanage where I once lived. What a darlin' sack of potatoes she was. She used to make me fold my hands like I was about to pray, then whack the shite out of me with a ruler. She was good at pulling hair and giving us the Indian burn, too. Have you done the same to a few tykes?\"\n\nHe drank from his glass of ice and soda and lime, an innocuous light in his eyes. \"Not running off, are you? You forgot your book. Here, I'll bring it to you,\" he said.\n\nBut she rushed through the vestibule into the next car, the big wood beads of her fifteen-decade rosary clattering on her hip, the whoosh of the doors like wind howling in a tunnel.\n\nOn Saturday afternoon Father Jimmie and I went together to the Flannigan lawn party at Fox Run, down Bayou Teche, in St. Mary Parish. The home had been constructed during the early Victorian era to resemble a steamboat, with porches shaped like the fantail and captain's bridge on a ship and cupolas and balconies on the upper stories that gave a spectacular view of the grounds, the antebellum homes on the opposite side of the bayou, and the sugarcane fields that seemed to recede over the rim of the earth.\n\nLive oaks draped with moss arched over the roof of the house, and palm trees grew in their shade to the second-story windows. A visitor to the lawn party could ride either western or English saddle around a white-fenced track by the horse stables, or play tennis on either a grass or red-clay court. The buffet tables groaned with food that had been prepared at Galatoire's and Antoine's in New Orleans. The drink table was a drunkard's dream.\n\nThe guests included the state insurance commissioner, who was under a federal grand jury indictment and would later become the third state insurance commissioner in a row to go to prison; petro-chemical executives from Oklahoma and Texas whose wives' voices rose above all others; two New York book editors and a film director from Home Box Office; an ex-player from the National Football League who rented himself out as a professional celebrity; career military officers and their wives who had retired to the Sunbelt; the former governor's mistress whose evening gown looked like pink champagne poured on her skin; and state legislators who had once been barbers and plumbers and who genuinely believed they shared a common bond with their host and his friends.\n\nFather Jimmie had worn his Roman collar, and the consequence was that he and I stood like an island in the middle of the lawn party while people swirled around us, deferential and polite, touching us affectionately if need be but avoiding the eye contact that would take them away from all the rewards a gathering at Castille LeJeune's could offer.\n\nAfter a half hour I wished I had not come. I went inside the house to use the bathroom, but someone was already inside. A black drink waiter in the kitchen directed me to another bathroom, deeper in the house, one I had to find by cutting through a small library and den filled with fine guns and Korean War\u2013era memorabilia.\n\nA steel airplane propeller was mounted on the wall, and under it was a framed color photograph of Castille LeJeune and a famous American baseball player, both of them dressed in Marine Corps tropicals, standing in front of two vintage Grumman Hellcat fighter planes parked on a runway flanged by Quonset huts and palm trees. In another photo LeJeune stood at attention in his dress uniform while President Harry Truman pinned the Distinguished Flying Cross on his coat.\n\nBut the photos that caught my eye were not those of Castille LeJeune's career as a Marine Corps pilot. A picture taken at his wedding showed him and his young wife, in her bridal gown, standing on the steps of a church. She was tall, dark featured, and absolutely beautiful. She also looked like the twin of her daughter, Theodosha.\n\nWhen I went back outside the sun was setting beyond the trees on the bayou, the sugarcane fields purple in the dusk, the air cool and damp, thick with cigarette smoke and smelling of alcohol that had soaked into tablecloths or had been spilled by the guests on their clothes.\n\nIn my absence Father Jimmie had cornered both Castille LeJeune and Merchie Flannigan and was talking heatedly with them, his coat separating on his chest when he raised his arms to make a point, one foot at a slight angle behind the other, in the classic position of a martial artist.\n\n\"Let me finish if you would,\" he said when Merchie Flannigan tried to speak. \"You say you're cleaning up the Crudups' property? The place is floating in sludge.\"\n\n\"I'm sure Merchie is doing his best. Why don't you help yourself to the food, Father?\" Castille LeJeune said.\n\nHe was a trim, nice-looking man, with a lean face and steel gray hair that he combed straight back. He wore a white sports coat and dark blue shirt and a gold and onyx Mason's ring on his marriage finger.\n\n\"No, thanks,\" Father Jimmie said, wagging two fingers as though brushing Castille LeJeune's words from the air. \"So let me see if I understand correctly. In 1951 you took a friend to Angola Prison to record Junior Crudup, but you have no idea what happened to Junior later?\"\n\n\"I was doing a favor for my wife. She was fond of folk music. That was a long time ago,\" LeJeune answered, his eyes crinkling at the corners, his gaze wandering among his guests.\n\n\"But a retired guard, a man named Strunk, says you got Junior pulled off the levee gang.\"\n\n\"I don't remember that. I wouldn't have had that kind of influence,\" LeJeune said.\n\n\"Really?\" Father Jimmie said. \"You wouldn't throw a fellow a slider, would you?\"\n\nThe insult to a man of his age and position seemed not to register in LeJeune's face. Instead, his eyes crinkled again. \"Have a good time,\" he said. He placed his hand warmly on Father Jimmie's arm and walked away.\n\n\"Let me get you a beer, Father,\" Merchie Flannigan said.\n\n\"Shame on you for what you're doing to those black people in St. James Parish,\" Father Jimmie said.\n\nFather Jimmie's heart might have been in the right place, but it was embarrassing to listen to him berate Merchie Flannigan in front of others and I didn't wait to hear Merchie's reply. I walked out of the backyard and into the oak trees, then witnessed one of those moments when you realize that each human being's story is much more complex than you could have ever guessed.\n\nBetween the horse stables and the bayou was a white-railed, sloping green pasture containing a fish pond and a small dock. A gas lamp mounted on a brass pole burned above the dock, and I could see moths flying into the flame, then dropping like pieces of ash into the water. As I stood among the trees I saw Theodosha watching the same scene, one hand on the fence railing. The electric lights were on in the stables and I could see her face clearly in the illumination, her brow knitted, the muscles in her throat taut, her hand gripped tightly on the rail.\n\nI walked toward her but her attention had been distracted by the strange red reflection of the sun's afterglow on the bayou. A little boy and girl, not older than four or five, climbed through the fence on the opposite side of the fish pond and ran giggling toward the dock. I had no way of knowing the depth of the pond, but a spring board was attached to the end of the dock, which meant the depth was certainly over a child's head.\n\nTheo looked back from the sunset at the pond and saw the children almost the same time as I. She bit her lip and raised her hand as though to warn them off, but she remained outside the fence, frozen, as though an invisible shield prevented her from entering the pasture. The children thumped onto the dock and danced up and down, then bent over the edge of the dock and peered at the fish feeding on the moths dropping from the flame in the gas lamp.\n\nTheodosha heard me walk up behind her. She turned abruptly, startled, her expression one of both fear and shame.\n\n\"That water is fairly deep, isn't it?\" I said.\n\n\"Yes,\" she said, turning back toward the pond. \"Yes, those children shouldn't be out there. Where are their parents?\"\n\nI started to climb through the fence.\n\n\"No, I'll do that. I'm sorry. I'm\u2014\" She didn't finish whatever she was going to say. She ducked under the top rail of the fence and ran awkwardly onto the dock, then returned, clasping each of the children by the hand.\n\nThe children's faces were hot, angry, a bit frightened, their cheeks pooled with color.\n\n\"We didn't know we did anything wrong, Miss Theo,\" the little boy said.\n\n\"You shouldn't go near a lake or pond or the bayou without your mother or father. Don't you ever do this again,\" Theo said, and shook him.\n\nBoth of the children began to cry.\n\n\"Hey, you guys, let's get a soft drink,\" I said.\n\nI took them by the hand and walked them to the drink table and asked the waiter to give each of them a Coca-Cola. Through the trees I saw Theodosha walking rapidly toward the back of her house, her arms clinched across her chest, as though the temperature had dropped thirty degrees.\n\nI decided I'd had enough of the LeJeune family for one evening. I told Father Jimmie I'd say good night to our hosts for both of us and went to find Theodosha inside the house. I didn't have to look far. She was in the den with her father, sitting on a stuffed leather footstool beneath the mounted airplane propeller, her face in her hands. Castille LeJeune stood above her, stroking her hair, his eyes filled with pity.\n\nNeither one of them saw me. I backed out of the doorway and joined Father Jimmie outside.\n\n\"Do you know where Merchie is?\" I asked.\n\n\"He and another man went to the stables. The other guy seems to have his own Zip code,\" he said.\n\n\"Let's go, Father.\"\n\n\"I was too hard on Flannigan?\"\n\n\"What do I know?\" I said.\n\nWe got in my pickup truck and headed down the long driveway toward the state road. I thought the bizarre nature of my visit to the plantation home of Castille LeJeune was over. It wasn't. In the glare of floodlamps, by a long white, peaked stable, Merchie Flannigan was perched on top of a fence, drinking from a bottle of Cold Duck, while a tall, gray-headed, crew-cropped, angular man in cowboy boots and western-cut slacks was lighting strings of Chinese firecrackers and throwing them in the air while a group of children screamed in delight. In the background, a half-dozen thoroughbred horses raced back and forth across a fenced pasture.\n\nMerchie flagged me down and walked toward my truck, slightly off balance.\n\n\"Not leaving, are you?\" he said.\n\n\"Looks like it. Thanks for having us out,\" I said.\n\nMerchie bent down to window level to see across me. \"I'm a bum Catholic, Father. But I try,\" he said.\n\n\"You were in the reformatory?\" Father Jimmie asked.\n\nMerchie's face reddened. \"Yeah, I guess I was.\"\n\n\"We'll compare stories sometime,\" Father Jimmie said.\n\nThe tall, crew-cropped man lit another string of firecrackers and threw it popping into the air. One of the thoroughbreds struck the fence and knocked a slat onto the grass.\n\n\"Why are you letting that guy panic those horses like that?\" I said.\n\n\"That's Will Guillot. Those are his kids,\" Merchie replied, then seemed to look into space at the vacuity of his words. \"Will does things for my father-in-law. You don't know him?\"\n\n\"No.\"\n\n\"You should,\" he said.\n\n\"Why?\"\n\n\"You're a police officer,\" he said. He leaned on his arms against the side of my truck, his eyes slightly out of focus, his breath like a wine vat.\n\n## Chapter 5\n\nThe telephone call to Father Jimmie came on Sunday afternoon, while he was watching a pro football game on television at the rectory. It was raining, and through the window he could see the rainwater cascading off the roof, pounding the small garden he tended in the green space between the gray, back wall of the church and the alley where the sanitation service picked up the garbage.\n\n\"I need to go to confession, Father,\" the voice said.\n\n\"Reconciliation is scheduled every afternoon at four, except Sundays,\" he said.\n\n\"I need to go now.\"\n\nFather Jimmie looked over his shoulder at a quarterback completing a thirty-yard pass on the television screen.\n\n\"Can it wait?\" he asked.\n\n\"I have to get something of a serious nature off my conscience.\"\n\nIn the silence Father Jimmie could hear the man breathing into the receiver. \"I'll be in the confessional at four o'clock,\" he said.\n\nHe finished his sandwich in front of the television, and a half hour later walked down the center aisle of the church toward the three confessionals that were inset in a side wall at the rear of the building. The inside of the church was magnificent. Twin balconies draped with brilliant red tapestries extended all the way from the choir to the altar area. The pulpit was hand-carved from teak wood and had been constructed high above the laity, in a time when there were no microphones to magnify the minister's voice. Whenever the sunlight struck the stained-glass windows, the effect inside the church was stunning. The celestial scenes on the ceiling and the paintings depicting Christ's passion in the Garden of Gethsemane and his ordeal by scourge and mockery and spittle and finally crucifixion made the viewer swallow in both reverence and trepidation.\n\nThe front doors of the church were open, and Father Jimmie could see the grayness of the afternoon out on the street and the drabness of the neighborhood and the rainwater welling up from the storm sewers. Perhaps a dozen people were in the pews, all of them old, their clothes shabby, their rosary beads wrapped around their hands. Some nodded at him and smiled as he passed. Their faith was genuine, he thought, their level of devotion long since proven by the lives they had led, but if they did not have this place to visit, where they could say their beads and confess sins that were either imaginary or inconsequential, he knew they would have no lives at all.\n\nA homeless man slept in a back pew, curled up in a fetal position, his odor rising from his clothes like a living presence. A bottle of fortified wine had fallen from his coat pocket and was precariously balanced on the edge of the pew.\n\nFather Jimmie picked it up, tightened the cap, and placed it on the floor, within arm's reach of the sleeping man.\n\nThen, on the far side of the church, he saw a man he had never seen before. The man wore a tight-fitting tan raincoat buttoned to his neck, like a prison on his body. His face was beaded with water, his ears like small cauliflowers, his hair cut short, combed neatly, reddish in color. He was sitting rather than kneeling, his hand resting on a domed, black lunch box. His eyes never made contact with Father Jimmie's.\n\nFather Jimmie went into the vestibule of the church and smelled the wind and rain and leaves blowing in the street. He wished he had not answered the phone in the rectory. It was a gray, wet day, with a touch of winter in the air, but it reminded him of Kentucky in the late fall, just before Advent, when a great dampness would settle on the Cumberland Mountains and the color would drain out of the sky and the fields and the leaves of the hardwoods would turn to flame in the hollows. It should have been a day to watch football and eat soup and hot bread and perhaps jog in Audubon Park. But he could not refuse a request for reconciliation, no matter how neurotic, self-absorbed, or irritating the source was.\n\nHe opened the door to a side corridor that led to the back entrance of a confessional, placed his stole around his neck, and sat down inside. He heard someone open the door to the adjoining box and the person's weight depress the kneeler that was attached to the partition separating the penitent from the confessor. Father Jimmie pushed back the wood slide that covered the small, grilled, gauze-covered window through which the penitent, in this case a man who smelled of street damp and hair tonic, would make his confession.\n\nBut the man did not speak.\n\n\"Are you the gentleman who called the rectory?\" Father Jimmie asked.\n\n\"That I am, Father.\"\n\n\"What is it you'd like to tell me?\"\n\nFather Jimmie could see the outline of the man's head. The ears looked like they had been carved around the edges with a paring knife. He heard the man snuff down in his nose and shift his weight on the kneeler.\n\n\"Been a while since I've visited one of these,\" the man said.\n\n\"Yes?\"\n\n\"I'm a bit flummoxed. Hold on a bit, Father, while I organize my thoughts.\"\n\nFather Jimmie heard what he thought was the man's lunch box clattering open inside the confessional. \"What are you doing in there?\" he asked.\n\n\"Nothing.\" The man was breathing hard now. \"I met a Catholic sister on the train. I was rude to her. She's a friend of yours. So I apologize for that.\"\n\n\"Oh, you're the fellow. Well, she already called me. I'll pass on your apologies. Is that it?\"\n\n\"I scared the shite out of her. She tell you that?\"\n\n\"Don't do it anymore and it won't be a problem. Is that all you have to tell me. Because if it is\u2014\"\n\n\"No, it is fucking not, sir.\"\n\n\"What did you say?\"\n\nThe man was breathing hard through his nose now, a ray of light from outside the confessional glimmering on the planed surfaces of his face.\n\n\"I said give me a fucking minute, if you please,\" he said.\n\n\"Are you drunk?\"\n\nThe man did not reply. He seemed to burn with energies he couldn't express. He rocked on the kneeler and twisted his head from side to side, then made a grinding noise in his throat. The lunch box clattered with sound again, as though the man had dropped a heavy object in it and snapped the latch on the lid.\n\n\"Tell the nun she's a splendid woman and I hope she lives long enough to have a bishop for a son. Send up a thanks to your patron saint, Father. Maybe buy a Powerball ticket while you're at it,\" the man said.\n\nHe flung open the door of the confessional and stalked through the vestibule and out the front of the church. Father Jimmie followed him as far as the front steps and watched him walk toward Canal, a golfer's cap pulled down on his head, his narrow shoulders hunched forward in the rain, his lunch box glistening with moisture. The man looked back over his shoulder at Father Jimmie, his face contorted, as though he had just fled a burning building.\n\nIt had rained through the night in New Iberia, and in the morning the sun rose like a pink wafer out of a blanket of fog that covered the cane fields. When I got to the office the parents of Lori Parks were waiting for me. Sometimes the survivors of family members who meet violent deaths have no place to direct their anger and loss other than at the police officer who is assigned to help them. Their rage is understandable, particularly when a cop is straight up and informs them the percentages are not in favor of justice being done. But sometimes the anger of the survivors has more to do with guilt than grief.\n\nThe father was sandy haired and tall, with an aquiline nose, the tops of his forearms sun freckled, his hands long and tapered. The wife was built like a stump, a ring of fat under her chin, her hair dyed dark red, her perfume a chemical fog.\n\n\"I hear you're questioning the employees of the daiquiri shops in town,\" the father said.\n\n\"Yes, sir, that's correct,\" I said.\n\nHe and his wife had not taken a seat when I offered them one. They looked down at me, from across my desk, stolid, angry, their defenses and denial rooted in concrete.\n\n\"Are you saying our daughter was DWI?\" he asked.\n\n\"That's the conclusion of our lab.\"\n\nHe nodded silently, the color in his eyes deepening, the skin around the rim of his nostrils whitening.\n\n\"So the truck and bus drivers are off the hook?\" he said.\n\n\"I don't think they're players in this,\" I said.\n\n\"Excuse me?\" the wife said.\n\n\"I think your daughter and her friends were served alcohol illegally. I'd like to put the people in jail who empowered them to drink and drive. But to be truthful I don't think that's going to happen.\"\n\n\"Our daughter is responsible for her own death? Is that it? A seventeen-year-old girl burns to death and it's her goddamn fault?\" the father said.\n\nI leaned forward on my desk and picked up a paper clip from the ink blotter, then dropped it. \"Dr. Parks, I'm sorry for your loss. Your daughter had a history. It's one a lot of kids have today. But the fact won't go away that she'd had her license suspended previously and she was on probation for possession of Ecstacy. Was she ever in any kind of treatment program?\"\n\n\"How dare you?\" the wife said.\n\n\"How about it, sir?\" I said to her husband.\n\n\"You're scapegoating my daughter, you sonofabitch,\" he said.\n\n\"We're done here,\" I said. I folded my hands on my desk blotter and avoided eye contact with them.\n\n\"We'll be back,\" the father said.\n\n\"I have no doubt about that,\" I replied.\n\nAt mid-morning I walked down the street, across the railroad tracks, and had coffee and a piece of pastry at Lagniappe Too on Main. When I got back to the department a black woman in blue slacks, a beige shirt, and polished black shoes was waiting for me by the dispatcher's cage. She carried a zippered satchel under her arm.\n\nWhat was her name? Andrepont? No, Arceneaux. Clotile Arceneaux. Clete had said she looked like a black swizzle stick with a cherry stuck on the end. He should have been a writer rather than a chaser of bail skips, I thought.\n\n\"Got a minute?\" she said.\n\n\"For you, anytime,\" I said.\n\nShe walked with me to my office. I closed the door behind her. \"N.O.P.D. hasn't busted you back to meter maid, have they?\" I said.\n\n\"Thought I might show you some photos of an interesting guy who just got to town,\" she said.\n\n\"You want to tell me who you are?\"\n\nShe smiled at me with her eyes and removed a manilla folder from her satchel. \"You ever see this guy before?\" she asked.\n\nThere were four black-and-white photographs inside the folder, three taken with a zoom lens, one taken in the garish light of a Toronto booking room. The man in the photographs made me think of a ring attendant at a boxing gym or a horse groom at the track. \"Nope, I don't know him,\" I said.\n\n\"His name is Max Coll. He's been questioned or been a suspect in thirty-two homicides. Not one conviction. Interpol thinks he worked for the IRA but they're not sure. Miami P.D. says he's freelance and jobs out for the Mob. We had a tail on him yesterday, but he shook it. We think he showed up at your friend Father Dolan's.\"\n\n\"Think?\" I said.\n\n\"A detective talked to Father Dolan. Seems like Father Dolan has got us mixed up with the bad guys,\" she said.\n\n\"Why you showing me this stuff?\"\n\n\"Hate to see your friend get clipped 'cause he's a poor listener. That goes for you, too, handsome.\"\n\n\"You're with the G?\"\n\n\"We think the priest was lucky yesterday. What we can't figure is why. Max Coll is a lot of things but fuck-up isn't one of them,\" she said.\n\n\"You're DEA?\"\n\nShe looked up into my face, her head tilted at an angle, her teeth white behind her grin. \"I heard you had a cinder block for a head,\" she said.\n\n\"Have you had lunch yet?\" I said.\n\n\"Some people are all work and no play. That's me, Robicheaux. Max Coll uses a silencer, sometimes an ice pick. You heard it first from your ex\u2013meter maid friend at N.O.P.D.\"\n\n\"Right,\" I said.\n\nShe stuck a business card in my shirt pocket and hit me on the hip with her satchel. \"See you around, darlin',\" she said.\n\nI walked with her to the front door of the building and watched her get in her automobile and drive away. Helen Soileau was standing behind me.\n\n\"What's with Miss Hip-Slick?\" she said.\n\n\"She's with N.O.P.D.,\" I said.\n\n\"The hell she is. She's a state trooper. She used to work undercover narcotics in Shreveport. She got into a firefight with some dealers about ten years ago and shot all five of them.\"\n\nLater, while I was out of the office, Clete Purcel left a message that he had checked into the old motor court on East Main, one that had long served as his field office in southwest Louisiana and his home away from home. The motor court was located inside a massive bower of live oak trees and slash pines on the bayou, and when I drove through the entrance that evening I saw Clete in front of the last cottage, barechested, wearing shorts with dancing elephants on them, flip-flops, and a Marine Corps utility cap, drinking from a bottle of Dixie while he flipped a steak on a flaming grill.\n\n\"Running down bail skips?\" I said.\n\n\"No, I just had to get out of the Big Sleazy for a while. Gunner Ardoin is driving me nuts,\" he said.\n\n\"What's happening with Gunner?\"\n\n\"He thinks somebody's going to clip him. Maybe he's right. So I...\"\n\n\"So you what?\"\n\n\"Gave him my apartment.\"\n\n\"Your apartment? To Gunner Ardoin?\"\n\n\"His wife skipped town and left his little girl with him. What was I supposed to do? Quit looking at me like that,\" he said. He picked up a can of diet Dr Pepper from an ice chest and tossed it at me.\n\nI sat down in a canvas chair, out of the smoke from the grill. Through the trees the sunlight looked like gold foil on the bayou. A tugboat passed, its wake slapping against the bank.\n\n\"Ever hear of a button man by the name of Max Coll?\" I said.\n\n\"A freelance guy out of Miami?\"\n\n\"That's the one.\"\n\n\"What about him?\"\n\n\"That black patrolwoman who answered the complaint in Ardoin's kitchen, Clotile Arceneaux? She's an undercover state trooper. She told me this guy Coll tried to kill Father Dolan yesterday,\" I said.\n\n\"Dolan thinks he walks on water. You might tell him the saints died early deaths.\"\n\n\"He's not a listener,\" I said.\n\n\"Yeah, like somebody else I know,\" Clete said.\n\nI walked down in the trees and watched the boats pass on the bayou while Clete finished grilling his steak. On the opposite bank two black laborers were trenching a waterline while a white man in a straw hat supervised them. When I walked back out of the trees Clete was laying out two plates, paper napkins, and knives and forks on a picnic table.\n\n\"I don't want to steal your supper,\" I said.\n\n\"Don't worry about it. My doctor says when I die I'll need a piano crate just to put my cholesterol in,\" he said.\n\n\"I'm trying to find out what happened to a convict in Angola back in the fifties. A guy named Junior Crudup. He went in and never came out,\" I said.\n\n\"Yeah?\" Clete said, dividing up his steak, looking at a woman in a bathing suit on the bow of a speedboat.\n\n\"Father Jimmie and I were at the house of Castille LeJeune Saturday evening. LeJeune got Crudup off the levee gang back in 1951. But he said he has no memory of it,\" I said.\n\n\"You're talking about stuff that happened a half century ago?\" Clete said.\n\n\"Crudup's family got swindled out of their property.\"\n\nClete plopped a foil-wrapped potato on my plate and sat down. He looked at me for a long time. \"So you think this character LeJeune is lying?\" he said.\n\n\"I couldn't tell.\"\n\n\"Wake up, big mon. Rich guys don't care whether the rest of us believe them or not. That's why they're great liars.\"\n\n\"His daughter saw two kids about to fall into a fish pond. But she was afraid to climb inside a fence and get them,\" I said.\n\n\"Is Father Dolan part of this?\"\n\n\"He took me out to the Crudup place in St. James Parish.\"\n\n\"This guy is playing you, Dave. He knows you don't like authority or rich people and you're a real sucker for a sob story. How about letting Dolan and the Throw-ups or whatever clean up their own shit?\"\n\n\"I'm getting played? You just gave a pornographic actor your apartment. The same guy you hit in the head with a coffeepot. You go from one train wreck to the next.\"\n\n\"That's why I never listen to my own advice.\"\n\nHe drank from his bottle of Dixie beer, his green eyes filled with an innocent self-satisfaction, his jaw packed with steak.\n\nThe next morning I drove to the house of Josh Comeaux, the clerk who I believed had sold daiquiris to Lori Parks and her friends the afternoon they burned to death. He lived with his mother in a small, weathered frame house not far from the Southern Pacific railway tracks. In the front yard was a post with hooks on it, from which vinyl bags of garbage hung so they would not be torn apart by dogs before the trash pickup.\n\nJosh pushed open the screen door and stepped out on the gallery. He was barefoot and wore recycled jeans without a belt and a black T-shirt with the sleeves cut off. A heart with a circle of thorns twisted around it was tattooed high up on his right arm. Through the screen I could see a fat woman in a print dress watching a television program.\n\n\"You come to arrest me?\" he said.\n\n\"Not yet. Who bruised up your face?\"\n\nHe touched the yellow-and-purple discoloration below one eye.\n\n\"Dr. Parks did. Last night. After I got off from work.\"\n\n\"Lori's father?\" I said.\n\n\"Yes, sir. That's why I figured you were here.\"\n\n\"He knocked you around?\"\n\n\"I went in for gas at the all-night station. He walked me out in the shadows and hit me. He was pretty mad.\"\n\n\"Are you telling me you confessed something to Dr. Parks?\"\n\n\"Yeah. I mean yes, sir. I told him what I did.\"\n\n\"Before you go any farther, I need to advise you of certain rights you have, the most important of which is your right to have an attorney.\"\n\n\"Who is that?\" the fat woman in the chair yelled through the screen.\n\n\"Just a guy, Mom,\" Josh said, and walked out into the yard, out of earshot from his mother. \"I told Dr. Parks I sold daiquiris to Lori and her friends. They were there three times that afternoon. It's not the only time I've sold to underage kids, either. Mr. Hebert tells us not to hold up the line 'cause somebody can't find their driver's license. But what he means is on weekend nights don't pass up any business.\"\n\n\"Mr. Hebert is your employer?\"\n\n\"Yes, sir. At least till this morning. He fired me when I told him I'd served Lori and the other girls.\"\n\n\"Did Lori give you an ID of any kind?\"\n\nHe shook his head. \"When Lori Parks wanted something, you gave it to her. She was the prettiest girl in Loreauville.\"\n\n\"Josh, I'm placing you under arrest. Turn around while I hook you up.\"\n\n\"Am I going to prison?\"\n\n\"That's up to other people, partner,\" I said, and put him in the backseat of the cruiser, my hand on top of his head.\n\nAs we drove away I saw his mother walk out on the gallery and look in both directions, wondering where her son had gone.\n\nThat afternoon I called Lori Parks's father at his office. His receptionist told me he was not expected in that day.\n\n\"Is the funeral today?\" I asked.\n\n\"It was yesterday,\" she replied.\n\n\"Would you give me his home number, please?\"\n\n\"I'm not supposed to do that.\"\n\n\"We can send a cruiser out there and bring him in, if you like,\" I said.\n\nWhen I called his home no one answered and the message machine, if he had one, was turned off. I checked out a cruiser and drove to Loreauville, nine miles up the Teche, and found his house in a wooded, hilly area on the bayou, just outside of town.\n\nThe one-story house was long and flat and constructed of what is called South Carolina brick, torn down from nineteenth-century buildings and shipped to Louisiana for use in custom-built homes. Apple green wood shutters that were ornamental rather than operational were affixed to the walls on each side of the windows and looked as if they had been painted on the brick. The porch ran the width of the house and was intersected with a series of miniature fluted columns. With its flat roof and squeezed windows, the house looked like a constipated man crouched back in the trees. It had probably cost a half million dollars to build.\n\nDr. Parks stood on a shady knoll overlooking the bayou, slashing golf balls across the water into a grove of persimmon trees. When I walked up behind him, leaves crackling under the soles of my shoes, he glanced at me for only a moment, then whacked another ball into the persimmons.\n\n\"I arrested Josh Comeaux this morning,\" I said.\n\n\"Glad to hear it,\" he said. His face was heated, freshly shaved, even though it was late in the day. He picked another ball out of a bucket and set it on a tee.\n\n\"He says you knocked him around.\"\n\n\"What's your business here, Detective?\" He rested his driver by his foot. He wore doeskin gloves that had no fingers and a long-sleeve maroon polo shirt and casual slacks that accentuated the flatness of his stomach and the graceful line of his hips.\n\n\"I'd like to see the owners of these drive-by daiquiri stores run through a tree shredder. But you're taking out your anger on the wrong person, Dr. Parks,\" I said.\n\n\"I moved my family here from Memphis. We thought small-town America wouldn't have drugs and political officials on the take and bastards who sell children booze to kill themselves with. I've been a stupid man.\"\n\nHe took his position on the tee, lifted his golf club with perfect form, and whipped it viciously into the ball.\n\n\"Don't add to your grief, sir,\" I said.\n\nHe turned and faced me. \"You have any idea of what it might have been like inside that car?\" he said.\n\n\"The tox screen showed traces of marijuana in Lori's blood,\" I said.\n\n\"So what?\"\n\n\"Maybe Josh Comeaux is a victim, too.\"\n\n\"I must have done something wrong in a former life,\" he said.\n\n\"Pardon?\"\n\n\"My daughter was burned alive and the cop who should be kicking somebody's ass is a goddamn titty-sucking liberal. You need to leave my property.\"\n\nI took my sunglasses out of their case, then replaced them and stuck the case back in my shirt pocket. The wind was cold blowing out of the trees and I could smell the heavy odor of the bayou in the shadows. The skin under Dr. Parks's right eye seemed to twitch uncontrollably.\n\n\"Are you hard of hearing?\" he asked.\n\n\"The judge will probably go light on Josh's bond. That means he'll probably be back home in a day or so. Are we clear on the implication, sir?\" I said.\n\n\"That I'd better not hurt him?\"\n\nHe waited for an answer but I didn't give him one. I fitted on my sunglasses and walked back to the cruiser, my shoes crunching through the leaves the doctor had raked into piles, only to see them blown apart by the wind. The doctor's wife emerged from the front door, wearing a house robe and slippers, a drink in her hand, the makeup on her face like a theatrical mask.\n\n\"You think I care about that boy? You think that's what this is about? Where are your brains, man?\" the doctor shouted after me.\n\nThe following evening I ate supper in the backyard, then went to the old cemetery by the drawbridge in St. Martinville where Bootsie was buried. The air was cold and smelled of distant rain, the sky yellow with dust blown from the fields. Several of the houses bordering the cemetery had signs on the galleries announcing TOMBPAINT FOR SALE. In south Louisiana we bury the dead on top of the ground and it's a tradition to whitewash the crypts of family members on All Saints Day. But it wasn't November yet. Or was it? I had to look at the calendar window on my watch to assure myself the month was still October.\n\nBootsie's crypt was located by the bayou, and standing next to it I could look downstream and see on the opposite bank the ancient French church and the Evangeline Oak where she and I had first kissed as teenagers and the stars overhead had swirled like diamonds inside a barrel of black water.\n\nI removed the three roses I had placed in a vase two nights previous and washed and refilled the vase under a tap by the gravel path that led through the cemetery. Then I put three fresh roses in the vase and set it in front of the marble marker that was cemented into the front of Bootsie's crypt. The roses were yellow, the petals edged with pink, the stems wrapped in green tissue paper by a young clerk at the Winn-Dixie store in New Iberia. When he handed me the roses I was struck by the bloom of youth on his face, the clarity of purpose in his eyes. \"I bet these are for a special lady,\" he had said.\n\nI sat on a metal bench with a ventilated backrest for a long time and drank a bottle of carbonated water I had brought from home. Then the wind came up and scattered the leaves from a swamp maple on the bayou's surface, and inside the sound of the wind I thought I heard a loon calling.\n\nI finished the bottle of carbonated water, screwed the top back on, and pitched the bottle at a trash barrel. But the bottle bounced on the rim of the barrel and fell on the gravel path. Rather than get up from the bench and retrieve it, I looked at it dumbly, all my energies dissipated for reasons that made no sense, the light as cold and brittle as if the sun were layered with ice.\n\nI heard footsteps behind me.\n\n\"I wasn't going to disturb you but I have to get back home,\" Theodosha Flannigan said.\n\n\"Pardon?\" I said.\n\n\"Your neighbor told me you'd be here if you weren't at home,\" she said. \"I was parked in my car, waiting for you to come out. Merchie doesn't know where I am. He ducks bullets in Afghanistan, then gets strung out if he breaks a shoelace. It's because of his mother. I think she was lobotomized. That's not a joke.\"\n\nI couldn't follow what she was saying. I started to get up, but she put her hand on my shoulder and sat down beside me.\n\n\"It's about Saturday night. Those two children were in danger of falling in the pond and I just stood there and watched it happening. I feel like shit,\" she said.\n\n\"'Bravery' and 'fear' are relative terms. What counts is you went after them,\" I said.\n\n\"I have some bad memories about that pond,\" she said. She bit on a hangnail and stared into space. \"I never go inside that fence. You must think I'm an awful person.\"\n\nBut the truth was I didn't want to talk about Theo's personal problems. I stood and picked up the plastic bottle that had bounced off the trash can and dropped it inside. When I sat back down I felt the blood rush from my head.\n\n\"Are you okay?\" she said.\n\n\"I still have bouts with malaria sometimes,\" I said.\n\nShe wore a scarf tied under her chin, the points of her hair pressed flat against her cheeks. \"Something else is bothering me, too, Dave. I think I make you uncomfortable,\" she said.\n\n\"No, that's not true. Not at all,\" I said, focusing my eyes on the bayou.\n\n\"That night we had the little fling? We'd both been drinking our heads off. Neither one of us was married at the time. I admit I thought you might come back around, but you didn't. So I wrote it off. It's no big deal.\"\n\n\"You're right, it's no big deal. I didn't say it was a big deal,\" I replied.\n\n\"Then why are you so\u2014\"\n\n\"It's not a problem. That's really important to understand here,\" I said.\n\n\"I'm afraid I've intruded upon you.\"\n\n\"No, you haven't. Everything is fine. Give Merchie my best.\"\n\n\"Will you come out to dinner?\"\n\nI pinched my temples and looked down the bayou at the Evangeline Oak looming over the water and at the spire of the old French church, a sliver of moon rising behind the steeple.\n\n\"Maybe we can talk about it later,\" I said.\n\n\"Sure. I'm sorry for being here like this. Since my psychiatrist died...No, that's the wrong word. Since he shot himself I feel this terrible sense of guilt. I've got two days' sobriety now. That's pitiful, isn't it? I mean, taking pride in staying off the hooch for two days, like I invented the wheel?\"\n\n\"I'll see you, Theo.\"\n\nShe exhaled her breath and I felt it touch my skin. She raised her eyebrows, staring inquisitively into my face, as though I needed to supply the endings to all her unfinished thoughts. Then she seemed to give it up and kissed the tips of two fingers, pressed them against my cheek, and walked out of the cemetery, a solitary firefly lighting in a tree above her head.\n\nIn the morning I called a homicide detective at the Lafayette City Police Department by the name of Joe Dupree. He had been in the 173 Airborne Brigade in Vietnam, but never spoke of the war and ate aspirin constantly for the pain he'd carried in his knees for thirty-five years. He was also one of the most thorough investigators I had ever known.\n\n\"What do you have on this psychiatrist who shot himself in Girard Park?\" I said.\n\n\"Dr. Bernstine? It's going down as a suicide. Why do you ask?\"\n\n\"A woman named Theodosha Flannigan has brought it up a couple of times.\"\n\n\"Merchie Flannigan's wife?\"\n\n\"Yeah, how'd you know?\"\n\n\"Her name was in Bernstine's appointment book,\" he replied.\n\n\"You don't buy the suicide?\"\n\n\"He took two .25 caliber rounds in the right side of the head. The muzzle burns were an inch apart, just above the ear. If the second round was discharged as a spastic reaction, why were the entry wounds almost identical?\"\n\n\"Any witnesses?\" I asked.\n\n\"None who could give a visual. But a kid said he heard two pops. At first he said they were a few seconds apart. Then he said they were together. Finally he said he couldn't be sure what he heard. Anyway, Bernstine had powder residue on his right hand. I'd like to say he was left-handed so my suspicions would have more basis. But he was ambidextrous.\"\n\n\"What's bothering you, besides the kid originally saying there was a time lag between the shots?\"\n\n\"Bernstine died on a Saturday. The Flannigan woman was scheduled to see him the following Tuesday. But there was no case record on her in his files.\"\n\n\"Maybe he had just started seeing her.\"\n\n\"No, I called Ms. Flannigan. She said she'd been going to Bernstine for six months. Anyway, Bernstine's wife calls me every day and tells me no way in hell he shot himself. Maybe not. But he'd lost his butt in the stock market and rumor has it he was messing around on his wife. So it's going down as a suicide.\"\n\n\"Thanks for your time, Joe.\"\n\n\"You haven't told me what Ms. Flannigan said to you.\"\n\n\"For some reason she feels guilty about Bernstine's death,\" I said.\n\n\"Think she was in the sack with him?\"\n\n\"If she had been, she would have told you about it. She's a little neurotic,\" I said.\n\n\"I'm shocked you'd know anybody like that, Dave.\"\n\nThe following Monday Father Jimmie Dolan had just returned to the rectory after saying a 7:00 A.M. Mass when the phone rang in his office.\n\n\"Hello?\" he said.\n\nThere was no reply. He heard a streetcar bell clanging in the background.\n\n\"Hello?\" he repeated.\n\n\"Oh hello, Father. Sorry. I couldn't get the bloody door closed on the booth,\" a voice said.\n\n\"It's you again, is it?\"\n\n\"Father, you've put me seriously in the shitter.\"\n\n\"I think you need counseling, my friend.\"\n\n\"Sir, you're a prelate and hence I believe a man of honor. Can you give me your word you won't continue to interfere in certain enterprises that are fully legitimate and doing little if any harm to anyone?\"\n\nFather Jimmie shuffled some papers around on his desk, then picked up a page torn off a note pad. \"Your name is Max Coll?\" he said.\n\n\"The coppers must have paid you a visit.\"\n\n\"Are you on Canal or St. Charles?\"\n\nThere was a pause, then Max Coll said, \"Now, how would you be knowing where I am?\"\n\n\"There's only one streetcar line in operation today. It runs only on those two streets. So that means you're not too far away from me.\"\n\n\"You're a mighty intelligent man. But I need to\u2014\"\n\n\"You stay out of my church.\"\n\n\"Sir?\"\n\n\"You heard me. If you ever bring a weapon into my confessional again, I'll tear you apart.\"\n\n\"Excuse me for saying this, Father, but that is a fucking mean-spirited statement for a Christian minister to make.\"\n\n\"Be thankful I don't have my hands on you,\" Father Jimmie said, and hung up.\n\nThen he stood motionlessly by his desk, his heart hammering against his chest.\n\n## Chapter 6\n\nThat same evening, Leon Hebert, the daiquiri-store operator who had fired Josh Comeaux, had to handle the window by himself because Josh's replacement had called in sick. Hebert didn't like to work alone, at least not at night. He was a cautious man, both with money and people, and had made his living over the years on the soiled edges of society wherever he had gone. If there was any group of people he understood in this world, it was his clientele.\n\nAfter he was discharged from the United States Navy, he had owned a liquor store on South Central Avenue in Los Angeles. The profits were huge and, except for the insurance, the overhead minimal. He accepted food stamps, welfare grocery orders, and even Bureau of Public Assistance bus tokens in place of money. After 2:00 A.M. he and a hired man would drive a panel truck down to East Fifth Street and sell eighty-nine-cent bottles of fortified wine, called short dogs, for two dollars apiece to the desperate souls who could not wait for the bars to open at 6:00 A.M.\n\nBut Leon Hebert learned there was a downside to running a business in a ghetto. On a warm summer night a white L.A. patrolman tried to hook up a drunk driver and force him into the back of a cruiser. In five minutes bricks, bottles, and chunks of curb stone were being flung into the traffic on Century Boulevard. This was in the era before the Crips and Bloods, but their predecessors\u2014the Gladiators, Choppers, Eastside Purple Hearts, Clanton 14, and the Aranas\u2014rose to the occasion and strung fires all over the south and east sides of Los Angeles.\n\nA Molotov cocktail crashed through the window bars and front glass of Leon Hebert's store. The inventory went up like gasoline.\n\nIn the riot only two groups of white-owned businesses were spared: funeral homes and the offices of bail bondsmen. The lesson was not lost on Leon. When he got back to New Iberia, his birthplace, he sold burial insurance to people of color, collecting their half-dollar and seventy-five-cent premiums weekly, wending his way without fear through every back-of-town slum in south Louisiana.\n\nThen he discovered the fast lane to prosperity was still available. He didn't have to go into the ghetto to sell his wares, either. The ghetto dwellers came to him, inside a shady grove on the four-lane, their gas-guzzlers smoking at his drive-by window, his ice-packed daiquiris, sweet and cold, ready to go at five bucks a pop.\n\nHe should have felt good about his situation, he told himself. He'd saved every cent he'd made peddling burial insurance and put it in a surefire franchise that gave him 60 percent of the profits. He made people happy, didn't he? Why did these damn kids from Loreauville get themselves killed with his cups in their car? And how about Josh Comeaux telling the physician, what was his name, Dr. Parks, teenagers were always served at Leon's drive-by?\n\nMondays were slow and Leon thought about closing up early. What was it that was bothering him? The doctor? The sheriff's detective who got in his face? He looked out the service window into the dusk and saw blue-collar families leaving a barbecue-and-po'boy place on the corner of the short span of asphalt that joined the eastbound and westbound highways, between which he operated his store. The evening was warm and fireflies floated in the oak trees. He watched the people from the barbecue place getting into their cars and pickups, their children bouncing up and down on the seats. For just a moment he wanted to join them and free himself from whatever presence it was that seemed to cling to his skin like road film.\n\nThree spoiled brats from Loreauville run themselves off the road into a telephone pole and he's in the toilet. There was no justice, he told himself.\n\nA junker car filled with black men pulled to the window and Leon removed six plastic-sealed daiquiris from the ice compartment of his giant refrigerator and handed them one by one through the driver's window.\n\nLeon waited for the car to pull away, but it didn't. The driver continued to stare into Leon's face, a toothpick in the corner of his mouth. A passenger in the backseat was smoking a cigarette, the ash glowing in the darkness. The passenger by the right front window held a metallic object in his lap, one that glinted dully in the light from the dashboard.\n\n\"I'm not alone. I got a man here with me,\" Leon said, his pulse quickening.\n\n\"What you talking 'bout, man?\" the driver said.\n\nThe passenger by the right front window lifted a Zippo from his lap and lit a cigarette with it.\n\nLeon let out his breath. \"Y'all want something else?\" he said.\n\n\"Yeah, you ain't give me my change,\" the driver said.\n\nThree hours later Leon Hebert put his money bag in the floor safe, locked the doors, and turned off the lights. It was a beautiful night. The wind rustled in the trees overhead and the constellations were stenciled across the sky. An eighteen-wheeler passed on the highway, then an ambulance with its flashers on. The ambulance continued on past the hospital and turned onto the drawbridge and the state highway that fed into Loreauville Road, where the three girls had been trapped inside their burning Buick.\n\nWhy did he have to think in images like that? He didn't do it, he told himself. That kid who worked for him, Josh Comeaux, had a boner in his pants for the Parks girl and would have let her slam it in the car door if she'd wanted to. Why didn't they put that in the paper? he said to himself.\n\nNo justice, he thought.\n\nSomeone started a pickup truck in front of the barbecue place and backed out of the parking lot, then headed slowly down the asphalt strip toward Leon Hebert's store, pieces of gravel clicking under the tires.\n\nLeon fished his car keys out of his pocket, then dropped them in the darkness. When he bent over to retrieve them, the driver of the pickup turned into the oyster-shell loop that curved past Leon's drive-by service window.\n\n\"We're closed,\" Leon said, the high beams of the pickup burning red circles into his eyes.\n\nBut there was no response from inside the truck.\n\n\"Who is that?\" he said, trying to smile.\n\nA figure opened the truck door and stepped out on the oyster shells. Leon raised his hand to shield his eyes and squinted into the brilliance of the headlights. \"My clerk already went to the bank. There's nothing here for you,\" he said.\n\nThe first pistol shot hit him high up on the chest with an impact like an anvil iron, knocking him backward, the hard-packed oyster shells slamming into the back of his head. The shooter had cut the lights on the pickup and was walking toward him, stooping for just a second to pick an object off the ground. The shooter stared down at Leon, perhaps realizing a mistake had been made, that the wrong person had been shot, that Leon Hebert should not have had a fate like this imposed upon him.\n\nThe figure leaned over him, blotting out the sky. Leon tried to speak, but the only sound that issued from his body was the air wheezing through the hole in his lung.\n\nThen his mouth was pried apart and something that was stiff and bittersweet and crusted with dirt was shoved between his teeth and forced deep into his throat. Leon's right hand tried to clasp the shoe of the figure bending over him, to somehow telegraph the plea for mercy that his lungs and throat could not make. At that moment he looked into the face of his tormentor and knew what his final moments on earth would be like. He twisted his head sideways and looked desperately out at the highway, wondering how the world of normal people and normal events could be only a heartbeat away.\n\nNo one reported the shooting until just before sunrise, when a tramp who had been sleeping in the weeds by the railway track crossed the road and tripped over the body. Helen Soileau picked me up at my house in a cruiser and handed me a thermos filled with coffee and hot milk. She hit the flasher and we rolled through town to the crime scene.\n\n\"You're the skipper now. You don't have to do this early A.M. stuff anymore,\" I said.\n\n\"Somebody has to keep you guys on your leash,\" she replied.\n\nHer eyes looked straight ahead, her expression flat. We passed a long row of shacks, the reflection of the flasher rippling across the house fronts.\n\n\"This isn't a robbery-homicide, is it?\" I said.\n\nCane trucks packed to the top were already on the road when we got to the crime scene, snarling traffic at the intersection by the drawbridge. The early sun was red through the trees and mist was rising off the bayou behind the hospital. Leon Hebert lay on the oysters shells a few feet from his drive-by window, a bullet wound in his chest, a second one puckered in the center of his forehead, a third through the eye. A blue daiquiri cup had been compressed into a cone and stuck in his mouth.\n\nAn ambulance and three sheriff's department vehicles were parked outside the yellow crime-scene tape that had been strung through the oak trees. The coroner had not arrived but our forensic chemist, Mack Bertrand, was kneeling beside the body, slipping plastic bags over the hands of the dead man. A small man in tattered clothes and tennis shoes without socks sat outside the tape, his back against a tree trunk, his knees drawn up before him.\n\n\"How do you read it, Mack?\" I said.\n\n\"The shooter used a revolver or he picked up his brass. I'd say the wounds were made with either a .38 or a nine-millimeter,\" he replied. He had ascetic features and wore a clip-on bow tie, suspenders, a crinkling white shirt, and a briar pipe in a little leather holster on his belt.\n\nHe lifted the dead man's right wrist. \"It looks like there's shoe polish under his fingernails,\" he said. \"My guess is the first round was fired from a distance and hit him in the chest. Then the shooter walked up close and put the next two in him point-blank. The victim probably looked up into the shooter's face and grabbed his shoe before he died.\"\n\n\"Why would he do that?\" Helen asked.\n\nMack shook his head. He popped open another plastic bag and with a pair of tweezers lifted the coned plastic cup from the dead man's mouth, then dropped it inside the bag. \"Take a look at this,\" he said, getting to his feet. \"There's blood on the bottom of the cup. That means the victim's heart was probably still pumping when the cup was shoved into his mouth.\"\n\n\"Meaning?\" Helen said.\n\n\"Who knows?\" he said.\n\n\"No forced entry on the building?\" I said.\n\n\"None that I could see,\" he said.\n\n\"How about tire impressions?\" Helen asked.\n\n\"Probably every kind of tire made in the western world has been through here. Did y'all know this guy?\" Mack said.\n\n\"He moved back here from L.A. He used to sell burial insurance,\" Helen said.\n\nI looked at the small man in tattered clothes sitting against the tree trunk outside the tape. \"Is that the guy who found the victim?\"\n\n\"Yeah, good luck. I get the impression he's a traveling wine connoisseur,\" Mack said.\n\nI stepped outside the crime-scene tape and squatted down eye-level with the man in tattered clothes. His skin was grimed with dirt and he wore a greasy cap crimped down on his head. Like all men of his kind, his origins, the people who had conceived him, the place or home where he grew up had probably long ago ceased being of any importance to him.\n\n\"You were sleeping by the tracks?\" I said.\n\n\"I fell off the train. I was pretty much knocked out,\" he said.\n\n\"Did you see or hear anything that might be helpful to us?\" I asked.\n\n\"I told it to that other guy.\" He nodded toward Mack Bertrand.\n\n\"Nothing bad is going to happen to you, podna. You're not going to jail. We're not holding you as a material witness. All those things are off the table. Just tell me what you saw.\"\n\nHe wiped his nose with his wrist. \"Late last night I heard something go 'pop.' Then I heard it again. Maybe twice. Then a pickup truck drove off.\"\n\n\"Did you see the driver?\"\n\n\"No.\"\n\n\"What did the pickup truck look like?\"\n\n\"Just a truck. It was going down toward the bridge there.\"\n\n\"Why'd you come across the road this morning?\"\n\n\"They got free coffee at the hospital,\" he replied.\n\nMy knees ached when I stood up. I took two dollars from my wallet and gave it to him. \"There's a donut shop back toward town. Why don't you get yourself something to eat?\" I said. I started to walk away from him.\n\n\"I seen something go flying out the truck window. Under the streetlight. Down toward the drawbridge. I don't know if that's any help to you or not,\" he said.\n\nA few minutes later the coroner arrived. Later, the paramedics unzipped a black body bag and placed the remains of Leon Hebert inside it and lifted it onto a gurney. Mack Bertrand fiddled with his pipe and put it between his teeth, upside down. He was a family man, a Little League coach and regular church goer, and usually not given to a public expression of sentiment.\n\n\"You asked why the victim grabbed the shooter's shoe,\" he said. \"He was asking for mercy.\"\n\nI waited for him to continue. But he didn't.\n\n\"Go on, Mack,\" Helen said.\n\n\"That's all. He had a sucking chest wound and couldn't speak. It was probably like drowning while somebody watched. So he tried to beg with his hand. He must have been a bad judge of character.\"\n\n\"How's that?\" I asked.\n\n\"Whoever did this poor bastard wanted him to go out as hard as possible,\" Mack said.\n\nHelen and I and a uniformed deputy searched along the edges of the road by the drawbridge, looking for the object the hobo said he had seen thrown from the fleeing pickup truck. But we found nothing of consequence. Helen dropped me at my house and I shaved and showered and drove to the office. At 9:15 A.M. I called the office of Dr. Parks. The receptionist said he would not be in. I called his home.\n\n\"What do you want, Mr. Robicheaux?\" he said.\n\n\"How did you know it was\u2014\"\n\n\"Caller ID. What's the problem now?\"\n\n\"I'd like to come out to your house a few minutes.\"\n\n\"You're not welcome at my house.\"\n\n\"Sorry to hear you say that,\" I replied.\n\nI drove up Loreauville Road, through horse-farm country and fields bursting with mature sugarcane, under a hard blue sky you could have scratched with a nail. The air was cool and sweet smelling, like cinnamon burned on a woodstove, and through the cypress and oak trees that lined the Teche the sunlight glittered like goldleaf on the water's surface.\n\nBut when I turned into Dr. Parks's driveway I seemed to enter a separate reality. His house was covered with shadow, the air cold, the birdbaths and empty fishpond and flagstone walkways moss stained and smelling of night-damp. The back end of a battered beige pickup truck stuck out of a shed in the rear of the house. Next to it was a stack of hay bales with a plastic bull's-eye pinned to them and a dozen arrows embedded in the straw. I had to ring the bell twice before he answered the door.\n\nHe was unshaved, the whites of his eyes shiny with a yellow cast, as though he had jaundice, a sour odor emanating from his clothes.\n\n\"Say it,\" he said.\n\n\"May I come in?\" I asked.\n\n\"Suit yourself,\" he said, and walked deeper into the house.\n\nWe entered a large, cheerless room with an unlit gas log fireplace and dark paneling on the walls and windows covered by thick velvet curtains. Track lights on the ceiling were focused on a huge gun case that was filled with both modern and antique firearms.\n\n\"That's quite a collection,\" I said.\n\n\"Get to it, Detective,\" he said.\n\n\"Somebody waxed Leon Hebert last night. Somebody who really had it in for him.\"\n\n\"That breaks me up.\"\n\n\"You own a .38 or a nine-Mike?\"\n\n\"A what?\"\n\n\"A nine-millimeter.\"\n\n\"Yeah, a half dozen of them.\"\n\n\"You drive your pickup truck last night?\"\n\n\"No.\"\n\n\"Where were you last night?\"\n\n\"Home, with Mrs. Parks. And that's the last question I'm answering without my attorney being present.\"\n\nWe were standing no more than one foot apart. I could see the fatigue in his face, the sag in his skin, the manic shine of grief and anger in his eyes.\n\n\"My second wife died at the hands of violent men, Dr. Parks. The sonsofbitches who did it are all dead and I'm glad. But their deaths never brought me peace,\" I said.\n\n\"Is that your evangelical moment for the day?\"\n\n\"I recommend you not leave town.\"\n\n\"One question?\" he said.\n\n\"Go ahead.\"\n\n\"Did Hebert see it coming? Because I hope that motherfucker suffered just the way my daughter did before he caught the bus.\"\n\nI left his house without answering his question. There are times as a law officer when you wish you did not have to look into the soul of another, even a grieving victim's.\n\nThat afternoon a seventeen-year-old black kid by the name of Pete Delahoussaye came into my office. Pete was over six feet and walked like he was made from coat hanger wire, but he had a fast ball that came down the chute like a B.B. and LSU and the University of Texas had both offered him athletic scholarships. Seven days a week, at 5:00 A.M., Pete and his widowed mother delivered the Baton Rouge Morning Advocate from one end of town to the other.\n\nHe stood in front of my desk, a paper sack hanging from his left hand.\n\n\"What's happenin', Pete?\" I said.\n\n\"Found something early this morning. Thought maybe I should bring it in,\" he said.\n\n\"Oh?\"\n\n\"Yeah,\" he said, sticking his hand in the bag. \"I was passing Iberia General, going toward Jeanerette, when something come sailing out of a pickup.\"\n\n\"Whoa,\" I said, rising from my chair, just as he lifted a blue-black, pearl-handled revolver from the paper sack. I could see the leaded ends of bullets inside the cylinder. I stepped away from the muzzle and took the gun from him.\n\n\"How much have you handled this, partner?\" I asked.\n\n\"A little bit,\" he replied, his eyes leaving mine.\n\n\"Did anyone else handle it?\"\n\n\"No, suh.\"\n\n\"Did you see the person inside the truck?\"\n\n\"No, suh, I ain't.\"\n\n\"What kind of pickup was it?\"\n\n\"Just a beat-up old truck. Brown, I think. I would have brought the gun in this morning, but I had to go to school.\"\n\n\"You did fine.\"\n\n\"Mr. Dave?\"\n\n\"Yeah?\"\n\n\"I didn't know about the man getting killed at the daiquiri drive-by till this afternoon. My mother thinks I'm in trouble.\"\n\n\"You're not. You're a good guy, Pete. Mind if we fingerprint you?\"\n\n\"So you won't get my prints mixed up with somebody else's?\"\n\n\"You got it.\"\n\n\"That's it?\"\n\n\"That's it.\"\n\nI watched him walk down the hall, grinning, his day back in place. Keep playing baseball, kid, and don't ever grow up, I thought.\n\nMack Bertrand, our forensic chemist, called me from the lab the next afternoon. \"We've got a ballistics match on the .38,\" he said.\n\n\"How about latents?\" I asked.\n\n\"They all belong to Pete Delahoussaye,\" he said.\n\n\"None on the rounds in the cylinder?\" I asked.\n\n\"Absolutely clean. I think that gun was oiled and wiped down before it was fired.\"\n\n\"What did you get off the plastic cup?\"\n\n\"Smudges that had dried dirt on top of them. I'm sure they were there long before our shooter arrived.\"\n\n\"Anything else?\"\n\n\"The victim had shoe polish and grains of leather under the nails of his right hand. But we knew that at the crime scene. Except for the discarded weapon, I'd say our perp was a professional.\"\n\n\"Thanks, Mack. By the way, what would you say the value of the gun is?\"\n\n\"It's a single-action army Colt, fairly rare. A lot of collectors have them. Maybe fifteen hundred dollars.\"\n\nI walked down to Helen's office and opened the door. She was just getting off the phone. \"I'd like to get a warrant on Dr. Parks's house,\" I said.\n\n\"Looking for what?\" she asked.\n\n\"Mack Bertrand says there were leather scrapings under the victim's nails.\"\n\n\"Think Parks is our man?\"\n\n\"He had both motivation and opportunity.\"\n\nHer eyes searched my face. \"That isn't what I asked,\" she said.\n\n\"I went out to his house yesterday. He didn't attempt to hide his hatred of the victim. He even wanted to know if Hebert suffered. Later I wondered if it was an act.\"\n\n\"Like he's trying to brass it out?\"\n\n\"Maybe. What doesn't make sense is the shooter throwing the gun out his truck window right by the drawbridge. Unless he wanted us to find it.\"\n\n\"Why do perps do anything?\" She glanced down at the legal pad by her telephone. \"We ran the serial number on the gun. It's registered to a William Raymond Guillot. He lives in Franklin.\"\n\n\"Guillot?\" In my mind's eye I saw a tall, gray-headed, crew-cropped man by a slat fence, lighting a string of firecrackers, pitching it into the air, while behind him a half-dozen thoroughbreds thundered back and forth across a pasture.\n\n\"You know him?\" Helen said.\n\n\"If it's the same guy, I saw him with Merchie Flannigan at Castille LeJeune's place.\"\n\nShe bit down on the corner of her lip. \"I think the ante just got raised on us,\" she said.\n\n\"Say again?\"\n\n\"I checked out Hebert's liquor license with the state board. He didn't own the daiquiri shop. It's part of a corporation called Sunbelt Construction. Guess who's listed as the CEO?\"\n\nBefore I could answer, she said, \"You got it, bwana. Castille LeJeune. Hope you enjoy charging howitzers with a popgun.\"\n\n## Chapter 7\n\nMax Coll could not believe his bad luck. Not only had he blown the job on the priest in the confessional, his efforts at researching the priest's schedule for another run at the situation had been blessed with an electric storm from hell. By late Wednesday afternoon the streets of New Orleans were flooded and lightning had crashed into an oak tree on St. Charles, dropping most of the canopy into the center of the avenue. The consequence was a traffic jam from Canal all the way uptown to Carrollton Avenue. Max could not even get a taxi from the edge of the Quarter to Father Dolan's church and had to walk ten blocks in a driving rain, a scoped and silenced .223 carbine banging against his rib cage.\n\nHe looked like a drowned rat when he entered the church. Water poured out of his shoes and each time he coughed he experienced a sensation like a sawblade splitting his sternum. He began sneezing and couldn't stop. He honked his nose into a wad of paper towels until he was light-headed, then was almost run down by a beggar woman pushing her way out of the vestibule with a shopping cart.\n\nWhy had he taken this job? It was jinxed from the start. New Orleans wasn't a city. It was an outdoor mental asylum located on top of a giant sponge.\n\nGet a hold of yourself, he thought. Take care of business, do a proper job of it, and never come back here again. It was almost 6:00 P.M. and the sky outside was absolutely black. The priest had finished his afternoon stint in the confessional and was no doubt having his supper, Max told himself. If the priest was true to his schedule, he would be saying his evening prayers in a front pew soon, his wide back presenting itself in lovely fashion to Max's crosshairs up in the choir. It was all going to be neat and tidy, nothing personal involved, no unnecessary pain. We all got to earn our keep, Father, he said to himself.\n\nMax waited until the vestibule was empty, then darted up the side stairs into the choir area. Ah, that was easy enough, he thought, looking down on the half dozen or so old people praying in the pews. Through a side window he saw lightning leap above the adjacent rooftops, illuminating the fire escape and the alleyway down below. Max did not like lightning. It brought back memories and catechism lessons he saw no point in reliving. He blew his nose softly, unbuttoned his raincoat, and unsnapped the carbine from the sling under his armpit. When he sat down in a chair among a pile of hymnals in the corner he unconsciously glanced upward at the celestial paintings on the ceiling, then quickly shifted his attention back to the nave of the church before he got lost in troubling thoughts that would be of no help in concluding the business at hand.\n\nHe surveyed the marble pillars, the tapestry-draped banisters on the balconies, the apse over the altar, the hand-carved pulpit. The place looked like it had been transported from the Middle Ages and dropped from a hundred-thousand feet into the middle of a slum, he thought. Even the parishioners could have been street beggars out of the fifteenth century. All the place needed was Quasimodo swinging on the bells. What was the matter with these people? Hadn't they heard of modern times? And how about this Father Dolan, threatening him with physical violence over the telephone? Now, that was a sad state of affairs, an Irish-American priest berating a man who had worked in the service of the IRA. Pitiful, Max thought.\n\n\"What are you doing, Mister?\" a little boy's voice said.\n\nOh shite, he thought.\n\n\"Are you here for choir practice?\" the child said. He was not over nine or ten and wore long pants and a white shirt with a tie. His hair was wet and freshly combed, his nails pink and trimmed.\n\nMax closed his raincoat, covering his carbine. \"Choir practice? Not exactly,\" he said.\n\n\"Then what are you doing?\"\n\n\"Examining the roof for rain leaks. I work for the bishop.\"\n\n\"How come you're all wet?\"\n\n\"I told you. Now get lost.\"\n\n\"I'm here with my mother for Father Jimmie's choir practice. I don't have to do what you say.\"\n\n\"Now, you listen, you malignant pygmy\u2014\" Max said.\n\n\"Screw you,\" the little boy said.\n\nMax coughed violently into his palm. His head was splitting, his nose running. \"Here's five dollars. Go buy yourself a hot chocolate,\" he said.\n\n\"Screw you twice,\" the little boy said.\n\n\"How would you like your dork stuffed in a light socket?\" Max said.\n\n\"Make it ten bucks,\" the little boy said.\n\n\"What?\"\n\nThe little boy peered over the balcony. \"Here comes Father Dolan now. Ten bucks or I start screaming,\" he said.\n\nMax shoved the money in the boy's hands and watched him run down the stairs. The little bastard, he thought. I hope the vendor pours Liquid Drano in his hot chocolate.\n\nThen Max heard footsteps, many of them, clopping up the wooden stairs. Either this is not happening or I'm being fucked with a garden rake, he thought.\n\nHe jerked open the window on the fire escape and climbed outside into rain that was now mixed with hail, closing the window halfway behind him. The icy pellets pounded his head, scalded his face, and slid down his coat collar inside his clothes. And if that wasn't enough, a bolt of lightning crashed into the alley, filling the air with the stench of sulphur and scorched electrical wiring. Jesus God, why was this being visited upon him? Then he looked down below and realized there were no steps below the fire escape, only rusted fastenings in the stone wall where a steel extension had once been in place. He was trapped like a rain-soaked parrot on a perch in an electrical storm, while inside the church Father Dolan's parishioners were dry and warm, passing out hymnals to one another.\n\nWell, maybe it was time to spread the discomfort around a little bit, forget neat and tidy and simply splatter the good father's porridge and be on his way, Max thought. Why not? Click off the safety, burn the whole magazine if need be, then haul ass right through the choir and on downstairs into the street. Father Dolan's singing parishioners would be too busy climbing under the furniture or shaking the crab cakes out of their drawers to worry about describing Max Coll to the authorities.\n\nHe knelt down in a shooting position on the fire escape, squinted into the carbine's scope, and saw the priest's magnified face swim into the crosshairs. In fact, the magnification of the priest's head was so great Max could not make out detail but see only hair and skin and perhaps just a touch of beard stubble. The hail clattered and danced like mothballs on the steel mesh of the fire escape, stinging the backs of Max's hands, drumming softly on his cap.\n\nThe carbine was loaded with soft-nose rounds and two of them impacting inside the priest's face would undoubtedly blow the back of his head into the wall like pieces from a broken watermelon. Max ground his molars, breathed hard through his nose, and felt his finger tighten inside the trigger guard. Squeeze it off, he told himself. Do it, do it, do it.\n\nBut he froze again, his hands trembling, just as they had trembled inside the confessional.\n\nHe was disgusted with himself. As he started to get to his feet, the silencer on the muzzle of the carbine scraped against the window glass. Suddenly he was not only looking straight into the priest's face, the priest was actually charging toward him.\n\nThere was no place to run. The priest jerked the window open, ripped the carbine from Max's hands, then gripped the stock with both hands and drove the steel-plated butt into Max's mouth. Max felt his lip burst like a grape against his teeth, then the guardrail behind him peeled from its fastenings. In the wink of an eye he was plunging backward through space, his arms outspread, preparing himself for the impact on the brick-paved alley below.\n\nInstead, he crashed into the middle of an opened Dumpster loaded to the gunnels with rotten produce and the leftovers from a parish shrimp boil. He stared upward from the garbage like a crucified man, right into the angry face of Father Dolan, who peered down at him from the edge of the broken fire escape. Max extracted himself from the softness of garbage that seemed to be sucking him into its maw and began pulling himself over the side of the Dumpster.\n\n\"Don't forget this,\" he heard Father Dolan call.\n\nMax looked up in time to see his carbine plummeting through the rain and hail, just before it bounced off his uplifted face.\n\nOn Thursday morning I took the four-lane into Franklin, then checked in with the St. Mary Parish Sheriff's Department and was given directions to the home of William Guillot. It was a lovely old Victorian house, located in a tree-covered, residential neighborhood, one of deep green lawns and hydrangeas and impatiens blooming in the shade and wide galleries hung with porch swings. But the gardener told me Guillot wasn't there and I could probably find him at the subdivision he was building not far from the four-lane.\n\nIt wasn't hard to find. Five hundred yards from the road, where two tin-roofed farmhouses had once stood amidst cedars and poplar trees, bulldozers had scoured a thirty-acre wound in the earth for the construction of houses that looked as if they had been designed by a man with delirium tremens. At the entrance to the subdivision-in-progress a workman was spreading kerosene on a huge pile of oaks and slash pines that had been recently lopped into segments with chainsaws.\n\nI parked my cruiser in a cul-du-sac flanked by three framed structures that several electricians were wiring. The man I had seen throwing firecrackers in the air by Castille LeJeune's horse barns was talking with a truncated, moon-faced workman in a yellow hard hat.\n\nWhen the workman saw me, he turned his face away, mounted the steps of a framed structure, and busied himself with a nest of wiring hanging from the back of a breaker box.\n\nWilliam Guillot wore shined cowboy boots and dark blue western slacks with high pockets and a gray snap-button shirt. He seemed to be one of those men to whom age was an asset and maturity a source of power and confidence. His skin was grainy, his profile rugged; in fact, he had all the handsome characteristics of the archetypical western horseman, except for a purple birthmark that was like dye that had leaked from his hairline into the corner of his left eye.\n\n\"Help you?\" he said.\n\n\"My name's Dave Robicheaux. I'm a detective with the Iberia Sheriff's Department. Are you William R. Guillot?\" I said, my gaze wandering from him to the electrician in the yellow hard hat.\n\n\"Call me Will. What can I do for you?\" he said.\n\n\"Where were you Monday night, Mr. Guillot?\"\n\n\"At my fish camp. Down at Pecan Island.\"\n\n\"Anybody with you?\"\n\n\"Maybe. What is this?\"\n\n\"We're in possession of a revolver that's registered in your name. It's a single-action Colt .38. You own a weapon like that, sir?\"\n\nHis hazel eyes fixed on mine and never blinked. \"Say that again.\"\n\nI repeated my statement.\n\n\"Yes, I do own one. But it's at my house,\" he said.\n\n\"Not anymore.\"\n\n\"Bullshit,\" he said, half smiling.\n\n\"I think we'd better take a ride to your house and check it out.\"\n\n\"If you haven't noticed, I'm building a subdivision.\"\n\n\"You an architect?\"\n\n\"No.\"\n\n\"The revolver registered in your name is part of a homicide investigation, Mr. Guillot. If I were you, I'd get my priorities straight.\"\n\n\"Homicide?\" he said, genuinely surprised.\n\n\"You own a brown pickup truck?\"\n\n\"I don't. The company does. What about it?\"\n\nBut I was looking at the back of the electrician who had walked away, and was not listening to William Guillot anymore.\n\n\"Did you hear me? What the hell is going on? Why are you staring at my electrician like that?\"\n\n\"Is he your subcontractor?\"\n\n\"What about it?\"\n\n\"He installed defective wiring in the walls of my house. It burned to the ground,\" I said.\n\nGuillot's eyes narrowed and dropped briefly to my person, as though he were filing away my inventory in a private compartment. \"Follow me to my house,\" he said.\n\nTwenty minutes later I stood in his home office, the sunlight breaking through a pecan tree by the side window, while he searched his desk, a wall safe, and the drawers of a gun cabinet. \"It's gone,\" he said.\n\n\"You have a break-in recently?\"\n\n\"Six or seven months ago.\"\n\n\"You reported it?\"\n\n\"Yeah, but I didn't miss the .38. Why would somebody steal only the .38 and none of my other guns?\"\n\n\"Write down the names of the person or persons you were with Monday night.\"\n\n\"Maybe I don't want to do that.\"\n\n\"I see. Maybe you can work through that problem in a jail cell.\"\n\nHe wrote a woman's name and address and telephone number on the top page of a scratch pad and handed it to me. \"My wife and I are separated. Her lawyer is trying to clean my clock. This isn't information that will help my situation,\" he said.\n\n\"It's not our intention to compromise your privacy,\" I said.\n\nBut his eyes grew heated, as though he were remembering an unfinished, angry thought. \"Back there at the house site, you made a serious accusation about my electrician. Did you file charges against him?\" he said.\n\n\"In New Iberia we have no inspection system outside the city limits. Also, in Louisiana an electrical contractor has no liability one year after the work is done. You like building homes in Louisiana, Mr. Guillot?\"\n\n\"I think you've got an ax to grind, Mr. Robicheaux. Let me say this up front. When I get pushed, I push back.\"\n\n\"Really?\"\n\n\"Yeah, really,\" he said.\n\nI tossed my business card on his desk. \"Give me a call when I can be of service,\" I said.\n\nThat same afternoon the phone rang on the desk in Father Jimmie Dolan's office. He stared at the phone as it rang four times, then listened to the voice that came through the speaker on the message machine.\n\n\"Are you there, Father? Excuse me if I sound strange, but I have a broken nose, a mouth that looks like a smashed plum, and a tooth knocked out of my head. All done by a Catholic priest,\" the voice said.\n\nIn the background Father Jimmie could hear piano music and the sounds of street traffic.\n\n\"I know you're listening, Father. Would you please have the courtesy to pick up the fucking phone,\" the voice said.\n\n\"What is it this time?\" Father Jimmie said.\n\n\"Because of you I'm up to my bottom lip in Shite's Creek and the motorboat is about to go roaring by.\"\n\n\"Could you do something about your language, please?\"\n\n\"My language?\" Coll said, his voice like a nail being pried out of dry wood. \"I took ten thousand dollars up front for the whack on you. Now I have to pay it back or prepare to go through life with no thumbs.\"\n\n\"Then return it.\"\n\n\"I lost it at the dog track.\"\n\n\"Change your way, Coll.\"\n\n\"Sir, please don't be talking to me like that. I'm miserable enough.\"\n\n\"I called the police on you yesterday. If you won't worry about your soul, you might give some thought to what New Orleans' finest will do to you.\"\n\n\"If there's a trace on your line, it won't help. I'm on a cell.\"\n\n\"You're close by the little alcove in the French Market. I know the pianist who plays there. She's playing her theme song, 'Down Yonder,' right now.\"\n\n\"You leave a man no dignity. Can you help with the ten thousand? Maybe I could borrow it from one of your charities?\"\n\n\"I'm hanging up now. I don't want you to contact me again.\"\n\n\"Oh, sir, don't do this to me. Don't fucking do this to a man who\u2014\"\n\n\"Who what?\"\n\n\"Maybe wants to remember who he used to be.\"\n\nFather Jimmie replaced the receiver in the phone cradle, the plastic surface as warm as human tissue against his palm, his hand trembling for reasons he couldn't readily explain.\n\nEarly the next morning I drove to Abbeville and interviewed Gretchen Peltier, the woman whose name had been given to me by Will Guillot as his alibi witness. She was middle-aged, slightly overweight, her hair dyed a deep black to hide the white roots. She worked as a secretary for an insurance agency and her hands trembled on the desktop when I asked her about her whereabouts Monday night. Her employer was inside a glass-windowed office, his door closed.\n\n\"Can't we do this somewhere else?\" she said.\n\n\"Sorry,\" I replied.\n\n\"I was with Mr. Will. At his camp. We're friends.\"\n\n\"What hours were you with him?\"\n\n\"I left his camp at dawn. The next day. Does that satisfy you?\" Her eyes were filmed with embarrassment.\n\nLater the same morning, Helen Soileau and I and another plainclothes served the search warrant on Dr. Parks at his home. His face looked sleepless; he had just finished shaving and a piece of bloody tissue paper was stuck to a cut on his chin. He stared at the warrant incredulously. \"Search for what?\" he said.\n\n\"Let's start with your shoes. Take them off, please,\" I said.\n\nHe stared long and hard at me, then the resolution seemed to go out of his eyes. He sat on a footstool in the living room and unlaced each of his black dress shoes and handed them to me. The shoes were new and the leather on them was buffed and smooth and bright as mirrors. \"Let's take a look in your closet, Doctor,\" I said.\n\nWe went inside the master bedroom. The curtains were closed, the air oppressive. I felt almost claustrophobic inside the room. \"Could you open the curtains, please?\" I said.\n\nHe started to turn on the overhead lighting.\n\n\"No, sir. Open the curtains,\" I said.\n\n\"Why?\" he said.\n\n\"Because I see better with natural light,\" I said.\n\nWhen he pulled back the curtains the room was immediately flooded with sunshine. The window gave onto a patio and a beautiful view of the bayou and the live oaks in the side yard. But the potted plants on the patio were dead, the glass-topped table marbled with dirt and the dried rings of evaporated rainwater. Helen and I pulled all the shoes out of the closet and bagged two pairs of black ones.\n\nDr. Parks sat on the side of the bed, his shoulders rounded. His wife opened the bathroom door, looked at us briefly, then closed it again. \"Look, you've got your job to do. I accept that. But I heard...\" he said.\n\n\"Heard what?\" I said.\n\n\"You people found the gun that killed the daiquiri-shop operator,\" he said.\n\n\"The man who owns the weapon makes a convincing case it was stolen,\" I said.\n\n\"You think I go around stealing guns from people?\"\n\n\"You attend gun shows, Dr. Parks?\" Helen asked.\n\n\"Sure. All over the country.\"\n\n\"Ever buy a firearm at a tailgate sale?\" she asked.\n\nHe rubbed his brow. \"It's hopeless, isn't it?\" he said.\n\n\"What do you mean?\" I said.\n\n\"I've heard about stuff like this. You can't make your case and you zero in on the survivors of the victim,\" he said.\n\nThere were many rejoinders either Helen or I could have made. But you don't break off the barb of a harpoon in a man who has already been ripped from his liver to his lights.\n\nWe got back in the cruiser and crossed the drawbridge in Loreauville, then headed up the state highway toward New Iberia. We passed cane trucks and the old Negro quarters left over from plantation days and an emerald green horse farm with big red barns and pecan trees next to a white house.\n\n\"Why'd you want the curtains open back there?\" Helen asked, watching the road.\n\n\"Their bedroom was like a grave. I couldn't breathe.\"\n\nShe glanced sideways at me.\n\n\"You didn't feel it?\" I asked.\n\n\"You worry me, bwana,\" she said.\n\n## Chapter 8\n\nOn Saturday morning I drove with Clete to New Orleans to check out his apartment, which he had loaned to Gunner Ardoin and his little girl. We crossed the Atchafalaya on the arched steel bridge at Morgan City, the docked shrimp boats and old brick buildings and tile roofs and palm-dotted streets of the town spread out below us in the sunshine. Then we drove into rain that seemed to blow out of the cane fields like purple smoke, and by the time we approached the giant bridge spanning the Mississippi, Clete's Cadillac was shaking in the wind, the fabric in the top denting with hailstones.\n\nWe drove into the French Quarter and parked in front of his apartment on St. Ann. He ran through the rain and went upstairs into his apartment. A few minutes later he was back in the car, his brow knitted.\n\n\"Gunner taking care of the place?\" I said.\n\n\"Yeah, everything looks fine,\" he said.\n\n\"What's wrong?\" I said.\n\n\"He left a message on the machine. He said an Irish guy was asking around in the neighborhood a couple of days ago. A weird-looking dude with little ears. Gunner thought maybe this guy had business with me.\"\n\n\"Max Coll?\" I said.\n\n\"Yeah. I think Gunner's got it wrong, though. Coll doesn't have any reason to be interested in me. Gunner might get himself popped.\"\n\n\"Where's Gunner now?\"\n\n\"He didn't say. How do I get involved in crap like this?\"\n\n\"Let's have a talk with Fat Sammy.\"\n\n\"I can't stand that guy. He looks like a blimp after all the air has gone out of it.\"\n\n\"There're worse guys in the life.\"\n\n\"Oh, I forgot, he gives discounts to the meth whores who work in his porn films,\" he said.\n\nHe fired up the Caddy, the rust-eaten muffler roaring against the asphalt, and we drove in the rain to Fat Sammy's house on Ursulines.\n\nI rang the iron bell at the entrance.\n\n\"Who is it?\" Sammy's voice said from the speaker inside the archway.\n\n\"Dave Robicheaux,\" I replied.\n\nHe buzzed open the gate and we walked through the flooded courtyard to the door of his house, which he had already unbolted and left ajar. I had not told Sammy that Clete was with me. When we stepped inside the living room he was lying on the floor, dressed in purple gym trunks and a strap undershirt, watching an opera on cable TV while he curled dumbbells into his chest. His massive legs were as white and hairless as a baby's, his pale blue eyes looking at us upside down.\n\n\"What's the haps, Sammy?\" Clete said.\n\n\"Who said you could come in here, Purcel?\" Fat Sammy asked.\n\nClete looked at me. \"I'll wait in the car,\" he said.\n\n\"Clete's my friend, Sammy.\"\n\nSammy set down the dumbbells and got to his feet, his lungs wheezing. The living room was dark, the windows covered with thick velvet curtains. Through a side door I could see two men, neither of whom I recognized, shooting pool. Sammy looked down from his great height at both me and Clete.\n\n\"So you want to watch some opera?\" he asked. He spread his feet and began touching his toes.\n\n\"You know a guy named Max Coll?\" I said.\n\n\"Do I know him? No. Do I know who he is? Yeah, he works out of Miami 'cause it's suppose to be an open city there. Here's the short version. You want somebody clipped, there's guys in Little Havana who work for a service. You want it done right, you ask for this Irish character. Except some people say he's a wacko.\"\n\nOut of the corner of my eye I saw Clete staring intently through the side doorway at the two men shooting pool.\n\n\"Wacko how?\" I said.\n\n\"I don't know, 'cause I don't keep company with them kind of people,\" Sammy said. \"Look, what I hear is the wacko screwed up a job in New Orleans and stiffed the wrong people. That means if he goes back to Miami he might float up in a barrel. Now, we done with this?\"\n\n\"The guy in there with the patent-leather hair? Is that Frank Dellacroce?\" Clete asked.\n\n\"What about it?\" Sammy said.\n\n\"Nothing. I thought he was down on a murder beef in Texas. Maybe George W. slipped up during his days as chief needle injector,\" Clete said.\n\nSammy's eyes looked at nothing while he scratched at his cheek with three fingers. \"Come back another time, Robicheaux,\" he said.\n\nOutside, rain was sluicing off the rooftops while Clete and I ran for his Cadillac. We got inside and slammed the doors. \"Why do you always have to start up the garbage grinder?\" I said.\n\n\"That greaseball shooting pool put his infant daughter in the refrigerator and held a gun to his wife's head while he did it. You think Sammy is on the square? I think he's a fat douche bag who should have been blown out of his socks years ago.\"\n\n\"You don't listen, Clete. It's hopeless. You'll never change.\"\n\n\"Neither will you, Dave. You'd like to splatter every one of these shitheads, but you won't admit it. Bootsie's death is eating your lunch. You talk about getting honest at meets? Why don't you stop stoking up your own fires?\"\n\nWe drove over to Decatur in silence, wrapped in anger, with no destination, the sky as gray as dirty wash. Rainwater was spouting from the sewer grates, the guttural roar of the ruptured muffler vibrating through the Cadillac's frame.\n\n\"If you want to attack me, Clete, do it. But don't drag my wife's death into it,\" I said.\n\n\"I'm finished talking about it. Live your own life,\" he replied.\n\nAt the traffic light in front of the Cafe du Monde I got out of the car, slammed the door behind me, and ran through the rain to the pavilion. When I looked back over my shoulder Clete was gone and Jackson Square looked as cold and stark as a black-and-white photograph taken in the dead of winter.\n\nI ordered coffee and hot milk and a plate of beignets, but couldn't eat. I walked the streets in the rain, keeping under the balconies, threading through the tourists carrying street-sale ten-dollar umbrellas. I looked through steam-coated windows of cafes and bars where people were watching Saturday-afternoon football on television. On Dauphine I went into a bar that was packed with gay men, all of them shouting in unison to punctuate the gyrations of a famous transvestite dancing on the stage. The bartender wore a pencil-line mustache and earrings and a black leather cap and leather vest without a shirt. He stared at me across the bar.\n\n\"You have coffee?\" I asked.\n\n\"This look like a Starbucks?\" he replied in a New England accent.\n\n\"Give me a soda with a lime twist,\" I said.\n\nHe fixed my drink and set it on the bar. He smiled to himself, but not offensively.\n\n\"On the job?\" he said.\n\n\"No, not on the job,\" I said.\n\n\"No problem, sir,\" he said.\n\nI closed my eyes as I drank down the soda and lime in the glass. I could have sworn I tasted the traces of bourbon in the ice. I used the rest room and walked back out on the street, my skin and clothes reeking of cigarette smoke, my head buzzing with sounds like an electric wire popping in a rain puddle.\n\nI lost track of time. It stopped raining toward evening and a wet fog settled on the French Quarter and drifted like colored smoke off the neon lights over the clubs. Bourbon Street, which was closed at night to automobile traffic, became filled with college boys drinking beer out of plastic cups, conventioneers and tourists strung with cameras peering into strip joints that featured both topless and bottomless performances, and black kids tap dancing like minstrel caricatures or running a shuck that begins, \"Bet you five dollars I can tell you where you got your shoes at.\"\n\nI walked along the river where bums sat on stone benches with sack-wrapped bottles of fortified wine between their thighs. I turned up Esplanade and walked all the way to the ragged edge of the Quarter at Rampart, past a hallelujah mission with a neon cross above its door, past Louis Armstrong Park, a place no white person in his right mind enters either day or night, over to Basin Street and the long white wall that fronted St. Louis Cemetery. Through the gates I could see row upon row of whitewashed crypts and stone crosses, framed against the sodium lamps of the Iberville Project that burned in the fog with the incandescence of pistol flares.\n\nI sat down on a bus bench next to a huge man with a wild beard and head of black hair. He wore a suit that looked like it had been pulled from a garbage can, a tie knotted like a garrote in the collar of his flannel shirt. His skin was so grimed with dirt it was hard to tell his race. His eyes made me think of the renegade Russian priest Rasputin.\n\n\"You got any money?\" he said.\n\n\"What do you need it for?\" I answered.\n\n\"Something to eat. Maybe a drink or two.\"\n\nI found four dollars and seventy-three cents in my pocket and gave it to him. He clenched it in his hand but remained seated on the bench. \"I got me a dry place in one of the tombs. The mission is all full on Saturday nights,\" he said.\n\nI nodded. A group of tourists were walking by, talking among themselves about either A Streetcar Named Desire, the play by Tennessee Williams, or the original streetcar itself, which today sits like an immobile and disconnected anachronism on a cement pad down by the river.\n\nThe disheveled man stood up and began waving his arms at them. \"That streetcar didn't go out to Desire,\" he yelled. \"It went out to Elysian Fields. It was the last car that still run out to Elysian Fields. All these streets here was Storyville. It was full of colored whorehouses and women who killed themselves with morphine. Hey, don't you go in them crypts! The kids from the Iberville Project climb over the wall and bust people like you in the head. Are you listening to me? This ain't New Orleans. You're standing in the city of the dead. You just don't know it yet.\"\n\nThe tourists walked quickly up the street toward Canal, their faces ashen.\n\nA minute later Clete Purcel's Cadillac came around the corner, oil smoke leaking from under the frame, a hubcap rolling loose across the asphalt, like a paean to the disorder in his life. He pushed open the passenger door.\n\n\"Want to go back to New Iberia?\" he said.\n\n\"Why not?\" I said, and got inside. I looked through the back window at the silhouette of the disheveled man receding behind us.\n\n\"Sorry I got on your case. But I think Fat Sammy has been putting the slide on you,\" Clete said.\n\n\"Maybe he has.\"\n\n\"No maybe about it, Streak. Every ounce of meth going into the projects has Sammy's greasy prints all over it. He makes me think of a giant snail trailing slime all over the city.\"\n\n\"You're one in a million, Cletus.\"\n\nHe looked at me uncertainly, a pocket of air in one cheek, then roared up the ramp onto I-10. We poured it on all the way back to New Iberia, like two over-the-hill low riders who no longer look at calendars or watch the faces of clocks.\n\nOn Monday morning Mack Bertrand called me from the lab and said the shoes we had removed from Dr. Parks's house were not the source of the leather scrapings found under the fingernails of the dead daiquiri vendor, Leon Hebert. A few minutes later Helen came into my office and I told her of the lab's findings.\n\n\"So where does that leave us?\" she said.\n\n\"A revenge killing of some kind. The daiquiri cup stuffed down the victim's throat indicates a high level of rage. Dr. Parks had motivation.\"\n\n\"You don't sound convinced,\" she said.\n\n\"Parks has so much anger I doubt he'd deny killing the man if he did it.\"\n\n\"How about this guy Guillot?\"\n\n\"He's a poster child for obnoxiousness. But why would he shoot someone and throw the weapon, registered in his name, on the side of the road?\"\n\n\"We're talking about middle-class people, Streak. Career perps are predictable. Dagwood and Blondie aren't.\"\n\nBeautiful.\n\nBut I believed there were other factors at work in this case that were more complex than a simple act of vengeance. It was too much for coincidence that Castille LeJeune's corporation owned the daiquiri store where Leon Hebert had been murdered and that the murder weapon belonged to Will Guillot, one of his employees.\n\nBut Helen was right. We were dealing with middle-class people who didn't have the proclivities and personal associations of career criminals, most of whom were basket cases who left a paper trail through the system from birth to the grave.\n\nWhy had Theodosha Flannigan been afraid to climb through the fence surrounding the fish pond on her father's property? Why did Castille LeJeune say he had no memory of using his influence to get Junior Crudup off the levee gang at Angola? People denied evil deeds, not good ones.\n\nAnd how about the suicide of Theodosha's psychiatrist? If she was his regular patient, why wasn't her case file in his records?\n\nI long ago became convinced that the most reliable source for arcane and obscure and seemingly unobtainable information does not lie with government or law enforcement agencies. Apparently neither the CIA nor the military intelligence apparatus inside the Pentagon had even a slight inkling of the Soviet Union's impending collapse, right up to the moment the Kremlin's leaders were trying to cut deals for their memoirs with New York publishers. Or if a person really wishes a lesson in the subjective nature of official information, he can always call the IRS and ask for help with his tax forms, then call back a half hour later and ask the same questions to a different representative.\n\nSo where do you go to find a researcher who is intelligent, imaginative, skilled in the use of computers, devoted to discovering the truth, and knowledgeable about science, technology, history, and literature, and who usually works for dirt and gets credit for nothing?\n\nAfter lunch I drove to the city library on Main and asked the reference librarian to find what she could on Junior Crudup.\n\nShe looked thoughtfully into space. She had a round face and wore glasses with pink frames and parted her hair down the middle. \"I have a history of blues and swamp pop here. That might be helpful,\" she said.\n\n\"I've already used that. This guy disappeared from Angola about1951. There's no record anywhere of what happened to him.\"\n\n\"Wait here a minute,\" she said.\n\nI watched her moving around in the stacks, sliding a book off a shelf here and there, then clicking on a computer keyboard. A few minutes later she waved for me to join her at a back table, where she had spread open several books that contained mention of Junior Crudup.\n\n\"I looked at those already, I'm afraid,\" I said.\n\n\"Well, there's a photographic collection in Washington, D.C., that might be worth looking at,\" she said.\n\n\"Pardon?\"\n\n\"In the forties and fifties a photographer who once worked with Walker Evans photographed convicts all over the South. He had a penchant for black musicians. He tracked some of their careers for decades. There are hundreds of photographs in his collection.\"\n\n\"Is he still alive?\"\n\n\"No, he died twenty years or so ago.\"\n\n\"How do we get a hold of the collection?\"\n\n\"All the ones he took of Crudup or of Louisiana prisons are downloading and printing right now. You need anything else?\"\n\nThe photographs were stunning, shot with grainy black-and-white film in Jim Crow jails and work camps, when the convicts still wore stripes and the hacks carried lead-weighted walking canes and made no attempt to hide the spiritual cancer that lived in their faces. Nor was there any attempt to hide the level of severity and privation that characterized the lives of the prisoners in the photographs. In each photo the camera caught an image or a detail that left no doubt in the viewer's mind about what he was seeing: a wheeled cage tiered with bunks parked inside a swamp; a convict sitting in the bottom of a wood sweatbox, a forced grin on his face, a waste bucket by his foot; a work gang assembled at morning-bell count, while in the background two men tried to balance themselves barefoot atop a case of empty pop bottles; a mounted gunbull in a cowboy hat framed against a boiling sun, his arm pointed, yelling a command at a convict pulling a fourteen-foot cotton sack behind him.\n\nIt was called stacking time on the hard road.\n\nBut in each of the photographs the reference librarian had downloaded, Junior Crudup was obviously the odd piece in the puzzle box, regardless of his surroundings. In a ditch with a dozen other convicts, he was the only light-skinned man, the only one with an etched mustache, and the only one to look directly into the camera. His eyes were clear, his face marked by neither resentment nor grandiosity. I suspected he was one of those for whom the gunbulls did not have a category, which would not have been good news for Junior Crudup.\n\nBut some of the photographs were taken outside of prison. One showed him with Leadbelly, the two of them laughing at a joke in front of what appeared to be a practice session of Cab Calloway's orchestra. Another showed him at a crowded table in a supper club, a beautiful black woman in a pillbox hat and polka-dot organdy dress, with an orchid pinned to her shoulder, seated next to him. Everyone in the picture was grinning at the camera, except Junior Crudup. He was dressed in a tuxedo, his tie pulled loose, a cigarette trailing a line of smoke from between two fingers. There was a half grin on his mouth, his eyes focused on a neutral spot, as though he were not entirely connected to the environment around him.\n\nI got a manilla envelope from the reference librarian and began slipping the printouts of the photos inside it. Then a detail in the last photo caught my eye and made me pull it back out. The photo was far less dramatic than the others and showed eight or nine convicts in denims, not stripes, plowing under cane stubble with mules in a sugarcane field that sloped down to a bayou.\n\nAn obese white man in a straw hat, with a doughlike face and a shotgun propped on his thigh, was watching them from atop his horse. Junior was staring up at the gunbull, a hoe at an odd angle over his shoulder, his face puzzled, as though he had just been told something that made no sense. It was wintertime and the bayou was low, the roots of the cypress trees exposed along the banks. A stump fire was burning on the edge of the field, the smoke drifting like a dirty smudge across the sun. Across the bayou, on the edge of the picture, was the back of a Victorian home that had obviously been built to resemble a steamboat.\n\nThe home of Castille LeJeune.\n\nA half hour later I rang the bell on his front porch, without having called or gone through his corporate office in Lafayette. \"I thought you might be interested in this photo. According to the cutline on it, it was taken in 1953,\" I said when he opened the door.\n\nHis eyes dropped to the photo briefly but he did not take it from my hand. \"Mr. Robicheaux, how nice of you to drop by,\" he said.\n\n\"That's Junior Crudup in the picture, Mr. LeJeune. That's your house in the background.\"\n\nHe wore slacks and a tie and a blue sweater with buttons on it. His eyes fixed on mine, twinkling. \"I'm sure what you say is true. But the burning issue here seems to escape me.\"\n\n\"You said you had no memory of getting Crudup off the levee gang. But here he is, harrowing your sugarcane field across the bayou from your house.\"\n\nHe tried to suppress a laugh. \"Let's see if I understand. You've driven out here to talk to me about a photo taken of convicts almost fifty years ago?\"\n\n\"Did you rent convict labor back then, Mr. LeJeune?\"\n\n\"The people who ran my family's agricultural interests might have. I don't remember.\" He looked at his wristwatch and raised his eyebrows. \"Oh heavens, I'm supposed to leave for New Orleans shortly.\"\n\nHis patrician insouciance, his disingenuousness and contempt for the truth were part of a lifelong attitude on which there were no handles. I could feel words breaking loose in my throat that I didn't want to say. \"You received the Distinguished Flying Cross from Harry Truman, did you?\"\n\n\"Do you wish me to confirm what you already know, or do you wish to ask me a meaningful question?\" he said, his eyes gazing benignly out on the flowers and palm and oak trees in his yard.\n\nI could feel my left hand opening and closing against my thigh, the veins tightening in the side of my head. Don't get into this, I heard a voice say in the back of my mind. \"I met Audie Murphy once. It was a great honor,\" I said.\n\n\"I'm happy to hear that,\" he said.\n\n\"Thank you for your time, Mr. LeJeune,\" I said.\n\nHe made no reply. Even though I had managed to control my anger I felt like a fool, one of that great army of salaried public servants who were treated by the very rich as doormen and security guards. I got in my cruiser and began backing down the long, shaded driveway to the state road, the sun flashing through the canopy like the reflection off a heliograph. When I reached the entrance to the state road I had to wait for a long line of cane trucks to pass, the wagon beds swaying heavily with the enormous loads they carried. In the meantime Castille LeJeune had gotten into his Oldsmobile and was driving toward me.\n\nI got out of the cruiser and walked to his car, then waited for him to roll down the window. \"I'm sorry, I forgot to leave you a business card,\" I said, and placed it on his dashboard. \"I think something real bad happened to Junior Crudup. Please be advised there's no statute of limitation on murder in the state of Louisiana, Mr. LeJeune. By the way, it was an honor to meet Audie Murphy because he seemed to be both a patriot and a straight-up guy who didn't try to get by on bullshit.\"\n\nOn Tuesday morning Helen called me into her office. \"I just got off the phone with Castille LeJeune's attorney. He says you made a nasty accusation yesterday at LeJeune's house,\" she said.\n\n\"News to me.\"\n\n\"You think you can jam a guy like Castille LeJeune?\"\n\n\"He's lying about Junior Crudup.\"\n\n\"The R&B convict again?\"\n\n\"Right.\"\n\n\"How about we concentrate on crimes in this century? Starting with the homicide at the daiquiri store.\"\n\n\"No matter what avenue we take, I think it's going to lead back to LeJeune.\"\n\n\"Maybe because you want it to.\"\n\n\"Say again?\"\n\n\"You hate rich people, Dave. You can't wait to get into it with them.\"\n\n\"No, I just don't like liars.\"\n\n\"Can you do me a favor?\"\n\n\"What?\"\n\n\"Go somewhere else. Now.\"\n\nThat afternoon Father Jimmie Dolan was at a basketball practice in a Catholic high school gymnasium not far from his church, when his cell phone rang inside his gym bag. \"Father Dolan,\" he said into the receiver.\n\n\"I need only a quick word. Don't be hanging up on me now,\" the caller said.\n\n\"How did you get this number?\"\n\n\"Told the secretary at the rectory I was your grandfather. I need something from you.\"\n\n\"What could I possibly have that you want?\"\n\n\"I was paid to take out this fellow Ardoin. But I'm not going to do it.\"\n\n\"You didn't answer my question. What is it you want?\"\n\n\"There's an open contract on me, Father. That means I'm anybody's fuck. But they messed with the wrong fellow, you get my drift?\"\n\n\"No, and I don't want to.\"\n\n\"I'm going to loosen some people's earthly ties.\"\n\nFather Jimmie stared listlessly across the gym at the boys who were taking turns laying up shots under the basket. He had a sore throat and fever and wanted nothing else in life at that moment except a glass of whiskey and a warm bed to lie down in.\n\n\"You know what I'm asking from you, don't you?\" Max Coll said.\n\n\"I think you want absolution for your sins, Max. But you can't have it. Not over the phone, certainly. And perhaps never, not unless you give up your violent ways.\"\n\nThe cell phone was silent.\n\n\"Did you hear me?\" Father Jimmie said.\n\n\"I think I've misjudged you. Under it all you're a hard-nosed bastard of a kind I remember only too well, one whose cassock and collar come before his humanity. Shite if you're not a disappointment to me.\"\n\nThe transmission went dead. Father Jimmie's cheek stung as though it had been slapped.\n\n## Chapter 9\n\nThat evening I fixed a bowl of milk for a stray cat and watched him drink it on the gallery. He was a hard-bodied, short-haired, unneutered white cat with chewed ears and pink claw scars inside his coat. His tail was as thick as a broom handle. When I petted him he looked at me blankly, then went back to his milk.\n\nTheodosha Flannigan pulled her Lexus into the driveway and parked under the pecan tree by the side of the house. A guitar in an expensive case was propped up in the backseat. She wore loafers and a blue terry cloth blouse and jeans low on her hips so they exposed her stomach. The wind gusted and leaves swirled around her, and a single band of dusky sunlight cut across her face.\n\n\"What's the name of your little friend?\" she asked, sitting down on a step next to the cat.\n\n\"He didn't say,\" I replied.\n\nShe picked the cat up in her arms and kissed him on top of his head. Then she flipped him on his back and set him in the crevice formed by her thighs and straightened his body by pulling his tail as though it were a strap on a piece of luggage. She scratched him between his ears and under his chin. \"We're going to call him Mr. Adorable. No, we're going to call him Snuggs,\" she said.\n\n\"What's happenin', Theo?\" I said.\n\n\"I heard about your visit to my father's house.\"\n\n\"Your father has a problem with the truth. He doesn't think he needs to tell it.\"\n\n\"He says you talked to him as though he were a criminal.\"\n\n\"I talked to him as though he were an ordinary citizen. He didn't like it. Then, rather than confront me about it, he used his attorney to report me to the sheriff.\"\n\n\"He comes from a different generation, Dave. Why don't you have a little compassion?\"\n\nTime to disengage, I said to myself. The streetlights were coming on under the oak trees, and the air was cool and damp and I could smell an odor like scorched brown sugar from the mills. Theo set down the cat and stroked his back, then stood up. \"You want to see my new guitar?\" she asked.\n\n\"Sure. I didn't know you played,\" I said.\n\nShe came back from the car with her guitar and unsnapped the case. \"I'm not very good. My mother was, though. I have some old tapes of her singing some of Bessie Smith's songs. She could have been a professional. The only person I've ever heard like her is Joan Baez,\" she said.\n\nTheo removed the guitar from its case and sat down again on the steps. She made a chord on the neck and brushed her thumb across the strings, then began singing \"Corina, Corina\" in Cajun French. She had been much too humble about her ability. Her voice was lovely, her accompaniment with herself perfect as she ran each chord into the next. In fact, like all real artists, she seemed to disappear inside the thing she created, as though the identity by which others knew her had nothing to do with the inner realities of her life.\n\nShe smiled at me when she finished, almost like a woman delivering a kiss after she has made love.\n\n\"Gee, you're great, Theo,\" I heard myself saying.\n\n\"My mother used to sing that. I don't remember her well, but I remember her singing that song to me before I went to sleep,\" she said. She began putting away her guitar.\n\nThe cat she had named Snuggs nuzzled his head against her knee. The wind riffled through the oak and pecan trees overhead, and a group of children on their way to the library rode by on bicycles, laughing, the streetlights glowing in the dampness like the oil lamps in a Van Gogh painting. There was not a mechanized sound on the street, only the easy sweep of wind and the scratching of leaves on the sidewalk. I didn't want the moment to end.\n\nBut like the canker in the rose or the serpent uncoiling itself out of an apple tree, there had been an element in Theo's song that disturbed me in a way I couldn't let go of.\n\n\"The melody for 'Corina, Corina' is the same as 'The Midnight Special,'\" I said.\n\n\"Un-huh,\" she said vaguely.\n\n\"That was Leadbelly's song. The Midnight Special was a train he rode into the Texas State Penitentiary at Huntsville. According to the prison legend, the convict who saw the headlight on the locomotive shining at him in his sleep was going to be released in the coming year.\"\n\nBut I saw she had still not made the connection.\n\n\"Your father didn't want to answer questions about Junior Crudup, Theo,\" I said. \"Crudup was Leadbelly's friend inside Angola. They probably composed songs together. I think Crudup was a convict laborer on your father's plantation.\"\n\nShe continued snapping her guitar case shut and never looked at me while I spoke. But I could see what I thought was a great sadness in her eyes. She reached over and petted the cat good-bye, then turned toward me. \"You have an enormous reservoir of anger inside you, Dave. I guess I feel sorry for you,\" she said.\n\nThe next morning events kicked into overdrive, beginning with a phone call from Clotile Arceneaux, the black patrolwoman who Helen said was an undercover state trooper.\n\n\"We've got Father Jimmie Dolan in custody,\" she said.\n\n\"Are you serious?\" I said.\n\n\"As a material witness. He won't give up Max Coll's whereabouts.\"\n\n\"Which administrative moron is behind this?\" I said.\n\nShe paused before she spoke again. \"Coll tried to kill the priest but he won't press charges. So a couple of detectives figured Father Jimmie is not a friend of N.O.P.D. and decided to put the squeeze on him. Look, the word on the street is there's an open contract on Max Coll. We need this guy out of town or in lock-up. We also don't need trouble from Catholic priests.\"\n\n\"Can't help you,\" I said, and hung up the phone.\n\nShe called back three hours later. \"Guess who?\" she said.\n\n\"Same answer as before,\" I said.\n\n\"Try this. We just heard from Miami-Dade P.D. Max Coll flew into Ft. Lauderdale, whacked two greaseballs who were getting laid on a yacht, then caught the last flight back to New Orleans. At least that's what they think. Get Dolan out of Central Lock-Up. Better yet, get him out of the state,\" she said.\n\nBut I didn't have to spring Father Jimmie. The bishop and Father Jimmie's conservative colleagues at his church came through for him, evidently making trouble from the mayor's office on down through the chain of command at N.O.P.D.\n\nFather Jimmie called me at home that evening. \"You know the story of Typhoid Mary?\" he said.\n\n\"A nineteenth-century cook or kitchen helper who caused problems everywhere she went?\" I replied.\n\n\"The bishop is recommending I travel somewhere that's quiet and rustic. Maybe do a little bass fishing. I think anywhere outside of New Orleans would be fine with him,\" he said.\n\nI shut my eyes and tried not to think about what he was obviously suggesting. \"Straight up, Jimmie. Do you know where Max Coll is hiding?\"\n\n\"Absolutely not,\" he said.\n\n\"Why didn't you file charges against him?\"\n\n\"The cops need a Catholic minister to tell them Coll's a killer?\"\n\nI rubbed the back of my neck. \"Want to entertain the bass?\" I asked.\n\nFather Jimmie moved into a back room of my house and the weekend passed uneventfully. On Monday Clete called the department and asked me to meet him for lunch at Victor's Cafeteria.\n\nIt was crowded with noontime customers, the wood-bladed fans turning high above us on the stamped-tin ceiling, the steam tables arrayed with Friday specials featuring shrimp or catfish or \u00e9touf\u00e9e. Clete's plate was piled with dirty rice and brown gravy, kidney beans, and two deep-fried pork chops. He wore an electric blue shirt and white sports coat, his face red with sunburn from a tarpon-fishing trip out on the salt. \"Dolan's at your place, huh?\" he said.\n\nI nodded, waiting for him to begin one of his lectures. But he surprised me.\n\n\"There's an N.O.P.D. snitch I pay a few bucks to. He called me this morning about a bail skip who's hid out in Morgan City. Then he mentions this guy Max Coll. He says Coll capped two high-level Miami greaseballs and there's a fifty thou open whack on him. Which means every street rat in New Orleans is crawling out of the sewer grates.\"\n\n\"Yeah, I heard about it.\"\n\n\"Right,\" Clete said, feeding a half piece of bread into his mouth. \"Well, tell me if you've heard this. At seven this morning either Frank Dellacroce or his clone was in the donut shop by the railway tracks.\"\n\n\"Here, in New Iberia? The guy you saw shooting pool in Fat Sammy's house?\"\n\n\"He came out of the donut shop just when I was going in. At first he couldn't believe his bad luck. Then he puts on a wise-ass grin and says, 'You fish for green trout over here, Purcel?' I go, 'No, I'm looking for a needle dick who puts his own child in a refrigerator. Know anybody like that, Frank?'\n\n\"He goes, 'That story is a lie my wife's lawyer spread about me during our divorce. So why don't you either pull your head out of your ass or mind your own fucking business?'\"\n\nPeople around us were quietly picking up their plates and trays and moving to tables farther away from us.\n\n\"Just then two more greaseballs come out of the donut shop. One used to be a shooter for the Giacanos. The other one I don't know.\"\n\n\"How do you read it?\" I asked.\n\n\"They think Dolan knows where Coll is hiding. Any way you cut it, big mon, you've let Dolan piss in your shoe.\"\n\n\"Can we take our food to the park?\" I said.\n\n\"What's the problem?\"\n\n\"I think we're about to get thrown out of here.\"\n\n\"What for?\" he said, still chewing, his face filled with puzzlement.\n\nAfter I returned from lunch I went into Helen's office. She was talking on the phone, standing up, a pair of handcuffs pulled through the back of her belt. Before she hung up I heard her say, \"You don't have to tell me.\" Then she looked at me blankly. \"What is it?\" she said.\n\n\"Clete says three New Orleans wiseguys are in town. They're after a rogue button man by the name of Max Coll,\" I said.\n\n\"They're staying at the Holiday,\" she said.\n\n\"How do you know that?\"\n\n\"The manager called earlier. The greaseballs have hookers in their rooms and are scaring the shit out of the staff. I was about to tell you about it but I got a call from a guy at the chamber of commerce. He says you and Clete Purcel had a conversation in Victor's Cafeteria that made a third of the room move their tables.\"\n\n\"I'm sorry.\"\n\n\"Dave, I've told you before, we have enough problems of our own. What does it take to make you understand that?\"\n\nThe room was silent. I heard a warning bell clanging at the railroad crossing and a freight train clattering down the tracks. \"You want the wiseguys out of town?\" I said.\n\n\"I hate to tell you what I want,\" she said.\n\n\"Just say it, Helen.\"\n\nShe spit a hangnail off her tongue. \"Meet you outside,\" she said.\n\nWe arrived in four cruisers at the Holiday Inn out by the four-lane. My experience with the Mob or its members had never been one that possessed any degree of romance. In fact, my encounters with them always made me feel as though I had walked inside the drabness and urban desperation of an Edward Hopper painting. Although it was Monday and the motel was almost empty, Frank Dellacroce and his two friends had taken a row of rooms in back, facing the highway, where road noise echoed off the windows and doors of their building. Their cars were brand new, waxed and shining, but were parked by an overflowing Dumpster, out of which trash feathered in the wind and scudded across the asphalt. The sun was barely distinguishable in the sky, the air close with an odor like fish roe that has dried on a beach; the only sign of life in the scene was a palm tree whose yellowed fronds rattled dryly in the wind.\n\nHelen got out of her cruiser, her arms pumped, her shield hanging from a black cord around her neck. A cleaning woman was passing on the walkway, a plastic bucket filled with detergent bottles on her arm. \"You smell marijuana coming from that room?\" Helen asked.\n\n\"Ma'am?\" the cleaning woman said.\n\n\"That's what I thought,\" Helen said. She banged her left fist on the door of the room registered to Frank Dellacroce, her right hand resting on the butt of her holstered nine-millimeter. \"Iberia Parish Sheriff's Department! Open the door!\" she shouted.\n\nWith few exceptions, television and motion pictures portray members of the Mafia or the Mob or the Outfit as dapperly dressed, Plotinian emanations from an ancient ethnic mythos. They are not only charismatic\u2014they take on the proportions of protagonists in Elizabethan tragedy, with accents from Hell's Kitchen.\n\nThe truth is most of them are stupid and at best capable of holding only menial jobs. They use dog-pack intimidation to get what they want, whether it involves preferential seating in a restaurant or taking over a labor union. On a personal level their sexual habits are adolescent or misogynistic, their social behavior inept and laughable.\n\nIn terms of health, they're walking nightmares. Listen to any surveillance tape: After age fifty, they complain constantly about clap, AIDS, obesity, impotence, emphysema, clogged arteries, ulcers, psoriasis, swollen prostates, the big C, and incontinence.\n\nThe room door opened and a man with black, freshly barbered hair and pale features and dark eyes stepped outside. He was barefoot and wore slacks without a shirt. His chest was triangular in shape and covered with a fine patina of hair, his upper arms well developed. He started to pull the door shut behind him.\n\nHelen pushed the door back on the hinges. \"Your name Dellacroce?\" she said.\n\n\"Frank Dellacroce, yeah. Why the roust?\" he said.\n\n\"We have a complaint you're soliciting prostitution and using narcotics in the motel. Place both your hands against the building and spread your legs, please,\" she said. She crooked a finger at a figure inside the room. \"You need to come out here, Miss. Bring your purse with you.\"\n\nThe girl who emerged from the room was probably not over nineteen, dressed in sandals; skintight, cut-off jeans; and a Donald Duck T-shirt that hung on the points of her breasts. She wore no makeup and her hair was bunched on the back of her head with a rubber band. \"I didn't do anything,\" she said.\n\n\"Get out your ID,\" Helen said.\n\nThe girl's hands were shaking as she removed her driver's license from her billfold and handed it to Helen.\n\nHelen looked at the photo and the birth date on the card, then gave it back to her. \"Beat it.\"\n\n\"Ma'am?\"\n\n\"Your trick is a guy who put his infant child inside a refrigerator. You want a fuckhead like that in your life?\" Helen said.\n\nThe girl walked hurriedly across the parking lot toward the street. The uniformed deputies had pulled Dellacroce's two friends out of the adjoining rooms and were shaking them down against a cruiser. But they found no weapons or dope on them and none in their rooms.\n\nDellacroce was still leaning against the wall, his feet spread. \"We done with this?\" he said.\n\nHelen didn't answer. I could see the frustration building in her face.\n\n\"Hey, we're here for the tarpon rodeo. We ain't broke any laws. You get off squeezing my sack, fine. But I want a lawyer,\" Dellacroce said.\n\n\"Better shut up,\" I said.\n\n\"I'd show you where to bite me, but I'm holding up the building here,\" he said.\n\n\"Helen, could I have a word with Mr. Dellacroce?\" I said.\n\n\"Please do,\" she replied.\n\nDellacroce took his hands off the wall and watched her and the deputies get back in their cruisers. I told Dellacroce's two friends to go inside their rooms and to keep their doors shut. Dellacroce stared at me, a cautious light in his eyes.\n\n\"My house is off-limits to you, Frank. So is Father Jimmie Dolan,\" I said.\n\nHis slacks hung just below his navel. He traced the tips of his fingers up and down the smooth taper of his stomach, almost as though he were caressing a woman's skin. \"You were Purcel's partner in the First District?\" he said.\n\n\"At one time.\"\n\n\"Mind if I get my shirt?\" he said.\n\n\"No, I don't mind,\" I said.\n\nHe reached inside the door and picked up a long-sleeve pink shirt and began drawing a sleeve up his arm. His hair was tapered, lightly oiled, iridescent on the tips. \"Purcel was on a pad for us,\" he said.\n\n\"Yeah?\" I said.\n\n\"That's all. He made himself a little change.\"\n\n\"What are you saying, Frank?\"\n\n\"Nothing. Just talking about the history of your friend.\"\n\n\"Tell me, is that story about your infant child true?\"\n\n\"No,\" he said. His eyes held on mine, devoid of any sentiment or moral consideration I could see, indifferent to the lie they either contained or did not contain. His mouth was slightly parted and his teeth were wet with his saliva. I could feel his breath puff against my skin like a presence released from a poisonous flower. Involuntarily I stepped back from him.\n\n\"Word of caution, Frank. Max Coll was a shooter for the IRA,\" I said.\n\n\"The what?\"\n\n\"I hope you find Coll. I really do. Have a nice day,\" I said, and grinned at him.\n\nThe sun came out late in the afternoon, the wind died, and the sky was marbled with crimson clouds. When I got home from work Father Jimmie was raking leaves in the backyard.\n\n\"Clete and I are going to throw a line in. How about joining us?\" I said.\n\n\"Not today,\" he said. He picked up a huge sheaf of blackened pecan and oak leaves and dropped them on a fire burning inside a rusted oil barrel. The smoke rose in thick curds and twisted through the canopy like a yellow handkerchief.\n\n\"Never knew you to pass up a fishing trip,\" I said.\n\n\"I saw Max Coll,\" he said.\n\n\"Don't say that.\"\n\n\"I was coming out of Winn-Dixie. He was standing across the street.\"\n\n\"Maybe you're imagining things.\"\n\n\"No, I saw him, Dave.\"\n\n\"Then he'd better not come around here.\"\n\n\"He's a sick man. He needs help.\"\n\n\"I'm not buying into this discussion,\" I said, and walked away.\n\nWhen I looked back out the kitchen window Father Jimmie was heaving more leaves onto the fire, his clothes and skin auraed with smoke and dust in the shafts of sunlight breaking through the trees.\n\nGod protect me from martyrs and saints, I thought.\n\nClete and I hitched up my boat trailer to the back of my pickup and a half hour later slid the boat into the water at Bayou Benoit in St. Martin Parish. The surrounding water shed looked both enormous and desolate in a strange, autumnal way. There wasn't a sound from the bays or the inlets, not even the flopping of a bass or a gator back in a cove. A painter would have called it a beautiful evening. The western sky was still pale blue, the clouds like strips of fire, the leaves of the cypress and willow trees golden and motionless in the dead air. But the closed shutters on the houseboats and the lines of ducks and geese transecting the sun made something sink in my heart, as though I were the last man standing on earth.\n\nAs we headed across a long bay into a flooded woods, Clete sat in the bow, humped over, his back to me, the collar of his denim coat pulled up, his Marine Corps utility cap snugged down on his head. He ripped the tab off a can of beer and drank it, then began eating a Vienna sausage sandwich. I cut the engine and let the boat drift on its wake into the trees. Clete reached into the ice chest and tried to hand me a diet Dr Pepper.\n\n\"No, thanks,\" I said.\n\nHe clipped a Mepps spinner on his monofilament and cast it deep into the cove. \"Something happen today?\" he asked.\n\nI told him about my encounter with Frank Dellacroce at the motel, about his attempt to put me on a pad, about his mention that Clete had once taken juice from the Mob. Clete retrieved his lure, his face never changing expression.\n\n\"So what's the point?\" he said.\n\n\"I don't like a degenerate bad-mouthing my friends. I don't like being offered a bribe,\" I replied.\n\nHe waited a long time before he spoke again. \"I don't think that's the problem, noble mon,\" he said.\n\n\"Oh?\"\n\n\"You think all this belongs in a time capsule,\" he said, making a circle in the air with his hand. \"Outsiders aren't supposed to come here, particularly greaseballs and Wal-Mart and these cocksuckers grinding up the trees with bulldozers. It's always supposed to be1950.\"\n\n\"I see.\"\n\n\"The truth is you wish you had all these bastards locked in your sights inside a free-fire zone.\"\n\n\"Glad you've figured it all out.\"\n\n\"At least I don't sleep with a nine-millimeter anymore.\"\n\n\"Don't be offended when I say this, but, Clete, you can really piss me off sometimes.\"\n\n\"You worry me, mon. I think you're going into a place inside yourself that people don't come out of.\"\n\nI saw a bass roll among the flooded trees, like a green-gold pillow of air violating the symmetry of the surface. I cast my Rapala above the place he had broken the water, hoping to retrieve it across his feeding area. Instead, the balsa wood lure clacked against the trunk of a willow and the treble hooks went deep into the bark.\n\n\"I'll row us over there,\" Clete said.\n\n\"Not on my account,\" I said. I jerked the monofilament with my hand and snapped it off. The sun disappeared on the horizon like a flame dying on a wet match.\n\nWay leads on to way.\n\nI tried to go to bed early that night but I couldn't sleep. Rain began to click on the trees, then on the tin roof of my house, and I dressed and drove up the bayou road in the rain to St. Martinville. On the edge of the black district I went into a brightly lit cafe and ordered a cup of coffee and a small bowl of gumbo at the counter. A door with a beaded curtain was cut in one wall, and in the adjoining room a man was playing an accordion, while another man, with thimbles on his fingers, accompanied him on an aluminum rub board that had been molded to fit the contours of his chest.\n\nThe people in the other room were all light-skinned people of color, often called Creoles, although originally the term Creole had denoted a person of French or Spanish ancestry who had been born in the New World. The people in the next room were blue-collar mulattos whose race was hard to determine. They drifted back and forth across the color line, married into both white and black families, still spoke French among themselves, and tended to be conscious of manners and family traditions.\n\nSeated in one corner by himself was Frank Dellacroce, a shot and a glass of beer by his hand, his legs crossed, his silk shirt unbuttoned in order to expose his chest hair and the gold chain and medallion that rested on it. He tossed back the whiskey and flexed his mouth as though he had just performed a manly act. Then he tilted back his head, the small of his back against the seat of the chair, and seemed to resume his concentration on the music. The song the accordionist was playing was \"Jolie Blon,\" the most haunting and unforgettable lament I have ever heard. Then I realized that the object of Frank Dellacroce's attention had nothing to do with music, or a song about unrequited love and the loss of the Cajun way of life: Frank Dellacroce's attention was fixed on the shapely form of a young Creole woman dancing by herself.\n\nHer name was Sugar Bee Quibodeaux. Her eyes were turquoise, her hair the color of mahogany, fastened in back with a silver comb, her gold skin dusted with sun freckles. She also had the mind of a seven year old. She had conceived her first child when she was twelve and at age fifteen was taken to a state hospital by her grandparents and sterilized. Sometimes a local cop or a kind neighbor or business person tried to protect her from herself, but ultimately no one could restrain Sugar Bee's love of boys and men and the excitement and joy her own body gave her.\n\nI finished eating and paid my check at the register. Through the beaded curtain I could see Sugar Bee sitting at Frank Dellacroce's table, a bottle of beer and a glass in front of her. She was leaning forward, listening to something he was saying. He leaned forward, too, his hand deep under the table. Then the two of them stood up and she picked up her purse, one with white sequins and tassels on it, and hung it by a string from her shoulder. They walked through the beaded curtain toward the front door.\n\n\"That's far enough, Frank,\" I said.\n\nHe turned around, half smiling. \"You following me?\" he said.\n\n\"Nope.\"\n\n\"Then we got no problem here. Right?\"\n\n\"Yeah, I think we do,\" I said.\n\n\"No, no, man,\" he said, wagging his finger. \"I ain't done nothing wrong.\"\n\n\"That's a matter of definition, Frank,\" I said.\n\n\"We talking about a racial issue here?\"\n\n\"You're going back to your motel, Frank. You're going back alone. Got the drift?\"\n\n\"I checked you out, Robicheaux. You're an A.A. rum-dum people around here feel sorry for. But that don't mean you get to beat up on guys like me 'cause I'm Italian or from New Orleans or whatever the fuck it is about me that bothers you.\"\n\nI looked at my watch. \"Your coach is about to turn into a pumpkin,\" I said.\n\nHe stepped toward me. \"This is a free country. You don't like what me and the lady are doing, I say suck my dick. Now, you get out of my face and out of my space 'cause I really fucking don't like you, man.\"\n\n\"At this point I'm placing you under arrest. Put your hands behind you and turn around, please,\" I said.\n\n\"Arrest? For what?\" he said, his face incredulous.\n\n\"Disturbing the peace, creating a public nuisance, using profanity in public, that sort of thing. I'll think of some more charges on the way down to the jail,\" I said.\n\n\"This ain't even your jurisdiction,\" he said.\n\nBut I wasn't listening now. I turned him toward the wall and hooked him up, then pushed him out the door into the parking lot. It had stopped raining, and the air was cold and wet, and fog was rolling out of the trees across the road. Sugar Bee and several other patrons of the cafe and bar had walked outside and were watching us.\n\n\"You armed, Frank?\" I said.\n\n\"Want to search my crotch? Be my guest,\" he replied.\n\nI fitted my hand under his arm and moved him toward the hood of my truck. That's when he hawked phlegm out of his throat and spat it in my face.\n\nI felt it in my eyelashes, on my mouth, in my hair, like a skein of obscene thread clinging to my person. I picked him up by his belt and slammed him into the fender of the truck, then drove his head down on the hood. But Frank Dellacroce was not one to give up easily; though his wrists were cuffed behind him, he brought one hand up and clenched it into my scrotum.\n\nI smashed his head into the hood again, then got my handcuff key out of my pocket and unhooked him. I spun him around and drove my fist into his mouth, throwing all my weight into the blow, snapping his head back as though it were on a spring. I saw his lip burst against his teeth, and I hooked him in the eye with a left, caught him on the jaw and in the throat and on the nose as he went down.\n\nHe was whipped, but I couldn't stop. I picked him up by his shirt and hit him again, rolled him off a car fender and drove my fist repeatedly into his kidneys. He collapsed in a mud puddle and tried to drag himself away from me. But I knelt beside him and twisted his shirt in my left hand and drew back my fist to hit him again. He tried to speak, his ruined face pleading. I heard people screaming and felt Sugar Bee slapping at my head with a shoe, her voice keening in the damp air.\n\nA light on a pole burned overhead. I stared at the circle of faces around me, like a drunkard coming out of a blackout. Their eyes were filled with fear and pity, as though they were watching a wild animal tear his prey apart inside a cage. But there was one man in the crowd who did not belong there. He was white and had narrow shoulders and wore a seersucker suit with a pink tie. His ears were small, convoluted, hardly more than stubs on the sides of his head. His face and expression made me think of the bleached hide on a baseball.\n\nAs I looked up into his eyes I had no doubt in the world who he was, no more than you can doubt the presence of death when it suddenly steps into your path. I got to my feet and helped Frank Dellacroce up, then propped him against the grill of an ancient gas guzzler, no more than five feet from the man in the seersucker suit.\n\n\"Frank, meet a guy you've probably been looking for all your life,\" I said.\n\nThen I walked off balance to my truck and drove away.\n\n## Chapter 10\n\nEarly the next morning I soaked my hands until the swelling had gone out of my fingers, then I put Mercurochrome on the cuts in my knuckles and tried to cover them unobtrusively with flesh-colored Band-Aids. I picked up the morning paper off the gallery and went through it page by page, just as I had done for years when I was coming off a drunk, wondering what kind of carnage I may have left in an alley or on a rain-swept highway.\n\nBut this morning the paper seemed filled with cartoons and sports and wire-service and local feature stories that had nothing to do with events in front of a cafe-and-bar on the St. Martin Parish line. Snuggs, my newly adopted cat, followed me back inside and I opened a can of food for him and put it in his bowl and sat with him on the back porch while he ate. The wind was cool and damp and sweet smelling through the trees, but each time I closed my eyes I saw the terrified, blood-streaked face of Frank Dellacroce and wondered who lived inside my skin.\n\nFather Jimmie was still asleep, so I drove over to Clete's cottage at the motor court and took him for breakfast at the McDonald's on Main Street. Then I cleared my throat and told him about the previous night\u2014at least most of it.\n\n\"Wait a minute,\" he said, raising his hands from his food. \"You had your piece and your cuffs with you?\"\n\n\"Right,\" I said.\n\n\"Why?\" he said.\n\nI shrugged.\n\n\"Maybe because you were looking for trouble when you left home?\" he said.\n\nI looked at an oak tree out on the street, one that was strung with moss and lighted by the pinkness of the early sun. \"I saw Max Coll there,\" I said.\n\n\"You did what?\"\n\n\"In the crowd. I've seen pictures of him. It had to be Coll. His head looks like a used Q-tip,\" I said.\n\nClete's eyes studied my face. They seemed to contain a level of sorrow that I could not associate with the man I knew. \"What are you doing to yourself, Streak?\" he said.\n\nAt 11:30 A.M. Helen leaned her head in my door. \"Pick up line two. See how much this has to do with us. If it doesn't, don't let it get on our plate,\" she said.\n\nThe man on the other end of the line was a St. Martin Parish plainclothes named Dominic Romaine. He was a big, fat, sweaty man, known for his rumpled suits, horse-track neckties, and general irreverence toward everything. He had emphysema and his voice wheezed into the phone when he spoke.\n\n\"That guy you beat the shit out of last night, Frank Dellacroce?\" he said.\n\n\"Uh, there's a bad connection, Romie. Say again.\"\n\n\"Pull on your own joint, Robicheaux. I don't know why you busted this guy up, but it don't matter. In other words, you're not gonna be up on an IA beef.\"\n\n\"Sorry, I'm just not reading you, partner.\"\n\nI heard him take a deep breath, the air in his lungs whistling like wind in a chimney. \"After you got finished with Dellacroce, he drove to a cabin by Whiskey Bay. It's actually a fuck pad a bunch of greaseballs out of Houston use. Get this\"\u2014he broke off and started laughing, then fought to catch his breath again\u2014\"he was behind the wheel of his car, sucking on a bottle of tequila, while this mulatto broad was giving him a blowjob, when a guy comes out of the dark and parks a big one in the back of his head. I mean a big one, too, like a .44 mag. His brains were still running out his nose when we got there.\"\n\nDominic Romaine started laughing again. I felt my vision go in and out of focus. Outside, an ambulance passed the courthouse, its siren screaming. \"You still there?\" he said.\n\n\"Who was the shooter?\"\n\n\"No idea. No description, either. The mulatto handing out the blowjob is retarded or something. Dave, there's a question that needs to go into my report.\"\n\n\"I didn't see Dellacroce after my encounter with him,\" I said.\n\n\"Got any speculations on the shooter?\"\n\nMy head was pounding, my stomach churning. \"Check with N.O.P.D. Dellacroce was a hitman and fulltime wise-ass. I think he was a grunt for Fat Sammy Figorelli.\"\n\n\"It sounds like his passing will go down as a great tragedy. Hey, Dave? You know that song by Louie Prima? 'I'll be standing on the corner plastered when they bring your coffin by'? I love that song. Hey, Dave?\"\n\n\"What?\"\n\n\"Next time you go looking for a punching bag, make sure it ain't in St. Martin Parish,\" he said.\n\nI barely got through the day. I tried to convince myself the man I had seen in the crowd the previous night was not Max Coll. I had seen only photos of him, taken through a zoom lens or in a late-night booking room. The man in the crowd could have been a tourist, or someone who had walked over from the convenience store next door, I told myself. And even had it been Max Coll, was I my brother's keeper, particularly if my \"brother\" was a dirtbag like Frank Dellacroce?\n\nBut I knew in my heart my thought processes were self-serving and futile and that I had helped set up a man's death. I worked late at the office, past sunset, then turned out the light on my desk and drove home, just as it began to rain.\n\nI pulled into my drive, expecting to see Father Jimmie's car under the porte cochere. Instead, I saw Theodosha Flannigan's Lexus parked in the shadows and a light burning in the kitchen. The trees in the yard and the bamboo along the edge of the driveway were shrouded in mist, and yellow leaves floated in the rain puddles. The front door and the windows of the house were open, and I thought I could smell the odor of freshly baked bread. In fact, the entire scene, the dark cypress planks in the walls of the cottage, the rusted tin roof, the black-green overhang of the oaks and pecan trees, and the warm radiance emanating from the kitchen windows, all made me think of the house where I had lived many years ago with my father and mother.\n\nAs soon as I stepped into the house I saw Snuggs resting on the arm of the couch, his eyes shut, his paws tucked under his chest, a red satin bow tied around his neck. I walked into the brightness of the kitchen and stared woodenly at Theodosha, who was lifting a loaf of buttered French bread out of the oven. Behind her, steam curled off a pot of gumbo. Her mouth parted slightly when she saw me, as though I had dragged her away from a troubling thought.\n\n\"I fixed you some supper. Hope you don't mind,\" she said.\n\n\"Where's Father Jimmie?\" I asked.\n\n\"He went to Lafayette. He said he's probably going to stay over.\"\n\n\"Is Merchie here?\" I said.\n\n\"I'm not sure where he is. He's just out being Merchie. Do you want me to go?\"\n\n\"No, I didn't mean that. I'm just a little disconnected today.\"\n\nShe began setting the table as though I were not there. Her hair looked like it had just been cut and shampooed. She wore Mexican sandals and khakis with big pockets and a denim shirt embroidered with roses and stovepipe cactus. In fact, as I looked at her moving about the room, I realized what it was that drew men to her. She was one of those women whose intelligence and \u00e9lan and indifference to public opinion allowed her to give symmetry and order to what would have been considered chaos in the life of a lesser person.\n\n\"Theo, I'd feel a lot better if we could ask Merchie over,\" I said.\n\n\"I knew you'd say something like that.\"\n\nShe set a gumbo bowl on the table and stared at it emptily. She removed a strand of hair from the corner of her mouth and walked to within a foot of me. She started to touch me, then folded her arms in front of her, as though she had no place to put her hands. Her breath was cold and smelled of bourbon and orange slices.\n\n\"I was going to a meeting today. I had no plans to drink. I swear. I drove twice around the block, then went into a bar and drank for two hours.\" She looked up at me desperately. \"Dave, I'm seriously fucked up. Nothing I do works.\"\n\nShe lowered her head and inverted her palms and clasped them around my wrists. She stood on my shoes with her sandals and her stomach touched my loins. I could smell the shampoo in her hair and the perfume behind her ears. She pulled my hands to her sides and held them there. I could feel a thickness growing in me, a dryness like confetti in my mouth. She slipped her arms around my waist and pressed her face sideways against my chest.\n\n\"Dave, why didn't you ask me to marry you?\" she said.\n\n\"This is no good, Theo.\"\n\n\"We had fun together. Why did you go away?\"\n\n\"I was a drunk. I would have made any woman unhappy.\"\n\nHer eyes were wet against my shirt. I patted her on the back and tried to step away from her. Then she turned up her face to be kissed.\n\n\"I'll see you,\" I said.\n\n\"What?\"\n\n\"I have to go back to the office,\" I lied. \"I just came home to get something. I don't even remember what it was.\"\n\nThen I left my own house, feeling stupid and inadequate, which was perhaps an honest assessment.\n\nWhen I returned to the house two hours later she was gone. The kitchen was immaculate, the food she had fixed carted away. I didn't fall asleep until after midnight. Then I woke at three in the morning and sat on the edge of the mattress, my skin filmed with sweat, my loins like concrete, the darkness creaking with sound. I put my loaded .45 under the pillow and when the sun came up the hardness of the steel frame was cupped in my palm.\n\nLater, I ate a bowl of Grape-Nuts and milk and sliced bananas on the kitchen table, then heard Snuggs at the back screen. I opened the door for him and he walked to his pet bowl under the kitchen sink and waited for me to fill it with the box of dry food I kept on top of the icebox. The red silk bow Theodosha had tied around his neck was coated with mud. I took a pair of scissors from the dresser in the hallway and snipped the bow loose from his fur. \"It looks like Theo's concern for you was limited, Snuggs,\" I said.\n\nSomehow that thought made me feel more comfortable about leaving her and the meal she had prepared for me the previous night. I returned the scissors to the dresser drawer. But before I shut it I glanced down at the box where I kept all the sympathy cards that had been sent to me when Bootsie died. A corner of an envelope stuck out of the pile and the return address on it made me wince inside. On my visit to Theo and Merchie's house several weeks ago she had expressed her sympathies about Bootsie's death, but I'd had no memory of her sending a card and had concluded her sentiments were manufactured.\n\nBut her card was in the pile and the statements on it were obviously heartfelt. I picked up Snuggs and set him on the countertop and patted his head. \"How can one guy's thought processes be this screwed up?\" I asked.\n\nSnuggs rubbed against me, brushing his stiffened tail past my nose, and made no comment.\n\nThe phone on the counter rang. I started to pick it up, then hesitated and stared at it, my heart quickening, because I knew who it was, who it would have to be, if he was the obsessed and driven man I thought he was.\n\n\"Hello?\" I said.\n\n\"Is the good father there?\" the voice asked.\n\n\"No, he's not.\"\n\n\"Would you be knowing his whereabouts?\"\n\n\"No, I don't. But I recommend you not call here again.\"\n\n\"Oh, do you now?\"\n\n\"Mr. Coll, I'm a lot less tolerant about you than Father Dolan. You drag your sickness into my life and I'm going to put a can of roach spray down your throat.\"\n\n\"I'm the sick one? Two nights ago you kicked the bejesus out of that poor fuck in front of the bar. I'd say you're a piece of work, Mr. Robicheaux.\"\n\nUse the cell phone to call the office and get the line open, I told myself. But Max Coll was ahead of me. \"I'm not on a ground line, sir. You needn't fiddle around with technologies that will serve no purpose. Tell Father Dolan he and I share a common destiny.\"\n\n\"Are you insane? You're talking about a Catholic priest.\"\n\n\"That's the point. It's the likes of me who keep him in business. Thanks for your time, Mr. Robicheaux. I hope to meet you formally. I think you might be my kind of fellow.\"\n\nHe hung up.\n\n\"So the guy's a nutcase,\" Clete said at lunch.\n\nI pushed my food away. We were in a place called Bon Creole, a small family-owned cafe that specialized in po'boy sandwiches. It was two in the afternoon and the other tables were empty. \"I've got another problem, Clete,\" I said.\n\n\"No kidding?\"\n\n\"It's not funny.\"\n\n\"Look, big mon, Frank Dellacroce's mother was probably knocked up by leakage from a colostomy bag. He got what he deserved. Stop thinking about it.\"\n\n\"I'm not talking about Dellacroce.\"\n\n\"Then maybe you should take whatever it is to Father Dolan. I don't know what else to say.\"\n\nHe waited for me to reply. When I didn't, he widened his eyes and opened his hands, as if to say, What?\n\nI want a drink. Worse than I've ever wanted one in my life, I heard a voice say.\n\nClete's next remark did not help. \"I'm a bad guy to ask for advice. I always handled my problems with a pint of Beam and a six-pack of Dixie, then I wake up the next morning with a Bourbon Street stripper whose idea of world news is the weather channel.\" He read the expression on my face and grimaced. \"Sorry, Streak. Sometimes I don't know when to shut up,\" he said.\n\nWhen I got back to the office, Wally, our three-hundred-pound, hypertensive dispatcher, gestured at me from the cage. Long ago every plainclothes in the department had become inured to Wally's sardonic sense of humor and his comments about our bumbling ways and collective lack of intelligence. But this afternoon he was different. His eyes were evasive, his smile like an incision in clay. \"Been to lunch, huh?\" he said.\n\n\"Yeah. What's up?\"\n\n\"That fellow Flannigan was in here.\"\n\n\"Merchie Flannigan?\"\n\n\"He was pacing up and down for an hour, like he was about to piss his pants. When he got ready to go I axed him if he wanted to leave a message.\" Wally shifted in his chair, arching his eyebrows.\n\n\"Would you just spit it out?\" I said.\n\n\"He said tell Dave not to be running his pipeline under the wrong man's fence. The district attorney and some Chamber of Commerce people was in the waiting room. So was Helen.\"\n\nA woman passed us and looked back at me briefly. \"Okay, Wally, I appreciate it,\" I said, and started to walk away.\n\n\"Hey, Dave?\" he said.\n\n\"Yeah?\"\n\n\"I never liked that guy. He's a bum. Put a cork in his mout'.\"\n\nI walked back to the cage. \"What are you telling me?\" I said.\n\nWally picked up a pencil and went back to his paperwork. \"Nothing. I didn't mean to mix in nobody's bidness,\" he replied.\n\nI went into my office and stood at the window, tapping my fingers on the sill. I had no doubt Merchie wanted trouble. Otherwise he would not have brought his complaint into the place where I worked. Well, sometimes the best way to deal with the lion is to spit in the lion's mouth, I told myself. At 5:00 P.M. I drove to Merchie and Theo's home on the edge of town.\n\nEven though I had passed the house a thousand times, I still could not get over the juxtaposition of imitation thirteenth-century battlements with a boiler works across the highway. But perhaps the conjuncture of nouveau riche vulgarity with pecan orchards and horse barns and the softly lit ambiance of Bayou Teche was the perfect stageset for a man like Merchie Flannigan. Strip away the guise of the reformed street hood and self-made egalitarian success story, and there was little difference between Merchie and his father-in-law, Castille LeJeune. They didn't go after their enemies head-on; they poisoned the environment where they worked.\n\nI saw Theo look out the living room window as I parked my truck.\n\n\"What's wrong, Dave?\" she said, opening the front door.\n\n\"Merchie was looking for me at the department. He seems to think I'm causing a problem in his marriage,\" I said.\n\n\"Come in.\"\n\n\"Where is he?\"\n\n\"At my father's. Wait, don't leave like this.\"\n\n\"Straighten him out on this, Theo,\" I said.\n\nHer face slipped by the driver's window as I turned around and headed back out the driveway.\n\nA half hour later, I pulled up to the front of Fox Run, Castille LeJeune's home outside of Franklin. I rang the front doorbell, but no one answered. The wind was balmy out of the south, smelling of brine and schooled-up speckled trout at Cote Blanche Bay, the setting so tranquil that my anger at Merchie, which I had fed all the way down the road, made me feel like a spiritually unclean visitor inside a church. The house itself was deep in shadow, the oak trees creaking overhead, but the surrounding fields and horse pasture were still lit by the last rays of the sun, and in the distance I thought I saw Merchie walk from behind a row of abandoned cabins to a promontory that overlooked the bayou.\n\nI went around the fenced pond that Theo feared for reasons she did not share and walked past a row of shotgun cabins that were probably built in the 1890s for the black people who planted and harvested the LeJeune family's sugarcane and drove it to the grinding mill in mule-drawn wagons without a member of the LeJeune family ever putting a hand on it. The cabin doors were gone, the tin roofs buckled loose from the joists, the plank floors blown with grit and scoured by the hooves of livestock. The privies were still standing, the eaves clotted with the nests of yellow jackets and mud-dobbers; the wood seats, once streaked with urine, now dry and smooth as old bone; the grass around the walls a bright green.\n\nI wondered if Junior Crudup had once slept in these cabins or used these privies, coming in hot and dirty from the fields, perhaps in leg irons, his evening meal a jelly glass of Kool-Aid and a tin plate of greens, fried ham fat, corn bread and molasses. I wondered how many lyrics in his songs had their inception right here, among these desiccated shacks that perhaps told more of a people's history than anyone wished to remember.\n\nI had left work ready to bend Merchie's day out of shape and now I had managed to link him in my mind with his father-in-law and the cruelties and racial injustices of Louisiana's past. What was my motivation? Easy answer. I didn't have to think about the fact I had deliberately put Frank Dellacroce in Max Coll's gunsights.\n\nMerchie was standing on a grassy knoll, his back turned to me, and did not hear me walk up behind him. A solitary white crypt, closed in front by a black marble slab that was chiseled around the edges with strings of flowers and clusters of angels, rested at a slight angle in the softness of the ground. Merchie squatted down with an orchid he inserted in a green water vase. The name on the slab was Viola Hortense Flannigan, Merchie's mother, the strange, neurotic, possessive woman who used to wash out his mouth with soap and whip his bare legs with a switch until he danced.\n\nEarlier I had been ready to tear him apart. Now I felt my anger lift like ash from a dead fire.\n\n\"I apologize for intruding on you,\" I said.\n\n\"You're not,\" he said, rising from his crouched position, a bit like a man waking from sleep.\n\n\"You were looking for me at the department?\"\n\nHe scratched the top of his arm idly and looked at the wind blowing in the grass. \"I get hot under the collar sometimes. Things aren't always right with me and Theo. So I take it out on the wrong people,\" he said.\n\n\"No harm done,\" I said.\n\nHe combed his hair and put his comb away, then watched a flock of black geese freckle the sun. \"My mother always wanted to be a southern lady. She told people she grew up in the Garden District in New Orleans. The truth was her old man ran a produce stand in the Irish Channel. So I bought this little piece of land from my father-in-law and buried her in it.\"\n\nI nodded, my eyes averted. In the distance I could see the railed fish pond that caused Theo such fear she had almost let two children drown rather than climb through a fence and approach the water.\n\n\"What happened at that pond, Merchie?\" I asked.\n\nHe opened and closed his hands, the veins in his forearms filling with blood. \"This place is a living curse. I'd like to set fire to it and plow its earth with salt. Outside of that, I don't have much to say about it,\" he said. Then he walked away, accidently kicking over the vase into which he had placed an orchid for his mother.\n\n## Chapter 11\n\nSome people seem to be born under a bad sign.\n\nAt 8:30 A.M. the following day an arson inspector called me at the office. In the early hours of the morning a fire had broken out in Dr. Parks's game room and had quickly spread through the roof, destroying the back third of his house. \"I know the guy just lost his daughter, but he's hard to take. How about coming out here, Dave?\" the inspector said.\n\n\"What's the deal?\" I said.\n\n\"Parks is convinced somebody tried to burn him out.\"\n\n\"My relationship with Dr. Parks isn't a very good one.\"\n\n\"You could fool me. He seems to think you're the only guy around here with a brain.\"\n\nI drove up to Loreauville and crossed the drawbridge there and followed the state road to the shady knoll where Dr. Parks's home sat among the trees like a man with an angry frown. A solitary firetruck was still there and two firemen were ripping blackened wood out of a back wall with axes. Dr. Parks approached me as though somehow I were the source of all the problems and missing solutions in his life. \"I want an arson investigation initiated right now,\" he said.\n\n\"That's a possibility, but so far there doesn't seem to be enough evidence to warrant one.\" I raised my hand as he started to interrupt. \"No one is saying your suspicions don't have merit. These guys just haven't found an accelerant or a\u2014\"\n\n\"It's connected to my daughter's death.\"\n\n\"No, it's not, sir.\" I fixed my eyes on the blackened back of his house and the roof that had caved in on the kitchen and master bedroom. It was so quiet I could hear my watch ticking on my wrist.\n\n\"Look here, Mr. Robicheaux, I asked that you come out because I know about some of the losses in your own life. I thought you would understand what's going on here,\" he said.\n\nI tried to ignore the personal nature of his statement. \"These firemen are good guys. You can trust what they tell you. I think you've just had a lot of bad luck,\" I said.\n\n\"There's no such thing as luck,\" he replied.\n\nJust then an unshaved, mustached fireman in rubber pants and suspenders and a big hat walked from behind the house with a clutch of fried electrical wiring in his hand. \"We got an ignition point,\" he said.\n\n\"What?\" Dr. Parks said.\n\nThe fireman spread the wires across his palm and cracked open the insulation on them. \"These were in the wall of your game room. See, they're burned from the inside out,\" he said.\n\n\"That's impossible. I just had that game room added on two years ago,\" Dr. Parks said.\n\n\"It's not impossible if somebody installed oversized breakers in your breaker box,\" the fireman said.\n\n\"Who did the work on your house, Doctor?\" I said.\n\n\"Sunbelt Construction,\" he said.\n\nI tried to walk away from him, as though I were preoccupied with the destruction at the back of his home. But he grabbed my arm roughly. \"What do you know about Sunbelt Construction?\" he asked.\n\n\"It's owned by Castille LeJeune,\" I replied.\n\n\"Who the hell is Castille LeJeune?\"\n\n\"His company owns the daiquiri store where your daughter and her friends bought their drinks on the day they died,\" I said.\n\nHad I just set up another man, in this case Castille LeJeune? I asked myself on the way back to the department.\n\nNo, I had simply told the truth.\n\nBut that did not change the fact I had let Frank Dellacroce take the big exit at the hands of Max Coll.\n\nLater I went home for lunch and found Father Jimmie on a ladder, screwing a basketball hoop to the back of the porte cochere.\n\n\"You do open-air reconciliations?\" I said.\n\n\"Yeah, hold the ladder for me. What's the problem?\" he replied, still concentrated on his work.\n\n\"It's not overdue library books,\" I said.\n\nHe looked down at me.\n\n\"I think Max Coll capped a wiseguy out at Whiskey Bay. I probably could have prevented it,\" I said.\n\nHe climbed down from the ladder and replaced his tools in a metal box and clicked it shut. \"Run that by me again,\" he said.\n\nWe walked toward the bayou while I told him what had happened\u2014the abiding anger that had made me seek out a violent situation, the savage beating I had given Frank Dellacroce, my recognizing Coll among the crowd in front of the cafe, and, most serious of all, my releasing Dellacroce from custody when I knew, with a fair degree of certainty, I was turning him over to his executioner.\n\nFather Jimmie picked up a pine cone and tossed it into the middle of the bayou. \"Dave, if you share responsibility for this man's death, then so do I,\" he said.\n\n\"How?\"\n\n\"I was uncooperative with N.O.P.D. I could have worked with them and helped bust Coll. He would have been past history now.\"\n\nI sat down on a stone bench by the edge of the bayou. Its surfaces felt cold and hard through my trousers. The wind gusted and red and yellow leaves tumbled out of the trees into the water. \"You going to give me absolution?\" I asked.\n\n\"You were forgiven as soon as you were sorry for what you did. But you need to tell this to someone else or you'll have no peace of mind.\"\n\n\"Sir?\"\n\n\"What's the new sheriff's name? The woman who used to be your partner? Let me know how it comes out,\" he said.\n\nHe walked back up the slope and removed a basketball from a cardboard box and swished it through the hoop. You got no free lunch from Father Jimmie Dolan.\n\nHelen listened quietly while I told her about the events of the night I beat Frank Dellacroce within an inch of his life. Her elbows rested on the ink blotter, her chin resting on her thumbs, her fingers knitted together. \"This guy Coll is wanted in Florida on two murders?\" she said.\n\n\"For questioning, at the least.\"\n\n\"What do you think he's doing around here?\"\n\n\"That's open to debate,\" I said.\n\n\"Meaning what?\"\n\n\"He has an obsession with the priest who's staying at my house. He's obviously hunting down the people who are trying to take him out. His brains were probably in the blender too long. Take your choice.\"\n\nShe stood up from her chair and stared out the window, her fingers opening and closing against the heel of her palm. \"So far there's no evidence it was Coll who shot Dellacroce?\" she asked.\n\n\"No.\"\n\n\"And you never saw Coll in person?\"\n\n\"Only in photographs.\"\n\n\"I think you're under a lot of strain. And that's where we're going to leave it for now.\"\n\nShe had given me a temporary free pass, a complicitous wink of the eye; all I had to do was acknowledge it. \"My perceptions aren't the issue here. Coll called me at my house. He told me he was in the crowd the night I busted up Dellacroce.\"\n\n\"Coll called you?\"\n\n\"That's right.\"\n\n\"This isn't police work. It's a soap opera. Are you drinking?\"\n\n\"No.\"\n\n\"Dave, you either get your act together or we seek other alternatives. None of them good.\"\n\n\"You want my shield?\"\n\n\"I won't be a party to what you're doing,\" she said.\n\n\"Doing what?\"\n\n\"Ripping yourself apart so you can get back on the bottle. You don't think other people read you? Give yourself a wake-up call.\" She wadded up a piece of paper and tossed it angrily at the wastebasket.\n\nThat evening I went to an A.A. meeting in a tan-colored, tile-roofed Methodist church, not far from the railway tracks. From the second-story window I could see the palm trees in the churchyard, the old brick surfacing in the street, the green colonnade of an ancient firehouse, the oaks whose roots had wedged up the sidewalks, and the strange purple light the sun gave off in its setting.\n\nAcross the railway tracks was another world, one that used to be New Iberia's old redlight district, whose history went back to the War Between the States. But today the three-dollar black prostitutes and five-dollar white ones were gone and the cribs on Railroad and Hopkins shut down. Instead, white crack whores, called rock queens, and their black pimps worked the street corners. The dealers, with baseball caps reversed or black silk bandannas tied down skintight on their scalps, appeared in the yards of burned-out houses or in the parking lots of small grocery stores as soon as school let out. After sunset, unless it was raining, their presence multiplied exponentially.\n\nThey offered the same street menu as dealers in New Orleans and Houston: weed, brown skag, rock, crystal meth, acid, reds, leapers, Ecstasy, and, for the purists, perhaps a taste of China white the customer could cook and inject with a clean needle in a shooting gallery only four blocks from downtown.\n\nDown the hall, on the second floor of the Methodist church, was a Narcotics Anonymous meeting. Most of the attenders there had been sentenced by the court. Few were people you would normally associate with criminality. Almost all of them, in another era, would have been considered run-of-the-mill blue-collar people whose lives had nothing to do with the trade on Hopkins and Railroad avenues.\n\nBut on that particular evening I was not thinking about the ravages of the drug trade. Instead, I was wondering how long it would be before I walked into a saloon and ordered four inches of Black Jack or Beam's Choice with a long-neck Dixie on the side.\n\nThen I looked across the room and saw a man who was geographically and psychologically out of place. He saw me staring at him and raised one meaty paw in recognition. His eyes were like merry slits, his jowls glowing with a fresh shave, his sparse gold hair oiled and flattened into his pate. I crossed the space between us and sat in the chair next to him.\n\n\"This is a closed meeting of A.A. What are you doing here?\" I said.\n\n\"I checked it out. It's an open meet. Besides, I belong to Overeaters Anonymous, which means I probably got trans-addictive issues. That means I can go to any fucking meeting I choose,\" Fat Sammy Figorelli replied.\n\n\"That's the worst bullshit I ever heard. Get out of here,\" I said.\n\n\"Fuck you,\" he said.\n\n\"Is there a problem over there?\" the group leader said.\n\nSammy didn't speak during the meeting. But afterward he helped stack chairs and wash coffee cups and put away all the A.A. literature in a locker. \"I like this place,\" he said.\n\n\"You're about to have some major trouble,\" I said.\n\n\"I'm gonna have trouble? You're beautiful, Robicheaux. Take a walk with me,\" he said.\n\nI followed him down the stairs, into the darkness outside and the odor of sewer gas and wet leaves burning. \"If you're using A.A. to\u2014\" I began.\n\n\"You drunks think you're the only people who got a problem. How would you like food to be your enemy? Anybody can stay off booze a hunnerd percent. Try staying off something just part way and see how you feel,\" he said.\n\n\"What's your point?\"\n\n\"My sponsor says I got to own up to a couple of things or I'm gonna go on another chocolate binge, which don't do my diabetes a lot of good. Max Coll not only cowboyed a couple of high-up guys in Miami, he stiffed the sports book they owned for a hundred large. The word is he's gonna be hung by his colon on a meat hook. Last point, there's a guy around here you don't want to mess with.\"\n\nHe stopped and lit a cigarette. The cigarette looked tiny and innocuous in his huge hand. He watched a car full of black teenagers pass, their stereo thundering with rap music, his face clouding with disapproval.\n\n\"Which guy?\" I said.\n\n\"A guy who hurts people when he don't have to. You want to find him, follow the cooze. In the meantime, don't say I ain't warned you.\"\n\nThen he labored down the sidewalk toward his Cadillac, his football-shaped head twisted back at the sunset.\n\n\"Come back here,\" I said.\n\nHe shot me the finger over his shoulder.\n\nI thought I was finished with Sammy Fig for awhile. Wrong. The phone rang at 2:14 in the A.M. \"There's something I didn't tell you,\" he said.\n\nI sat on the edge of the bed, the receiver cold against my ear. Outside, the moon was bright and glowing with a rain ring behind the sculpted limbs of a pecan tree. \"Time to desist, Sammy. That means join Weight Watchers or go to the fat farm, but stay out of my life,\" I said.\n\n\"Frankie Dellacroce's family is in Fort Lauderdale. A couple of them are on their way here.\"\n\n\"So long,\" I said, and started to lower the receiver from my ear.\n\n\"They got you made for the pop on Frankie.\"\n\n\"Me?\"\n\n\"You broke his sticks in front of a bunch of colored people earlier in the night. Later the same night he catches a .44 mag in the head. You're a cop. Who would you put it on?\" he said.\n\nI could hear my breath against the receiver. \"This is crazy,\" I said.\n\n\"I got to get some sleep. You're lucky you ain't got insomnia,\" he said, and hung up.\n\nIn the morning I confronted Father Jimmie at the breakfast table. \"Sammy Figorelli says a couple of Frank Dellacroce's relatives might be coming around,\" I said.\n\n\"What for?\" he said.\n\n\"They think I killed him.\"\n\n\"Not too good, huh?\"\n\n\"Where can I find Max Coll, Jimmie?\"\n\n\"If I knew, I'd tell you,\" he replied.\n\n\"I'd like to believe that. But I'm starting to have my doubts.\"\n\n\"Want to repeat that?\" he said, chewing his food slowly.\n\n\"He's going to call again. When he does, I'd like for you to set up a meet with him.\"\n\nI saw his brow furrow. \"I can't do that,\" he said.\n\n\"You sentimental about this guy?\"\n\n\"He's a tormented man,\" he said.\n\n\"Tell yourself that the next time he empties somebody's brainpan.\" I picked up my cup of coffee and took it with me to work.\n\nExcept I did not go to work. I turned around in the parking lot and drove to the cemetery in St. Martinville, where Bootsie was buried in a crypt right up the bayou from the Evangeline Oak. I sat on the ventilated metal bench in front of the crypt and said the first two decades of my rosary, then lost my concentration and stared woodenly at the bayou and the leaves swirling in the current and the ducks wimpling the water around lily pads that had already turned brown from early frost. My skin felt chafed, as dry as paper, my palms stiff and hard to close. I replaced my rosary in my coat pocket and put my face in my hands. The sun went behind a cloud and the wind was like ice water on my scalp.\n\nWhy did you go and die on us, Boots? I heard myself say, then felt ashamed at the selfish nature of my thoughts.\n\nAn hour later I walked into the department, washed my face in the men's room, then undertook all the functions of the working day that give the illusion of both normalcy and productivity. Clete Purcel dropped by, irreverent as always, telling outrageous jokes, throwing paper airplanes at my wastebasket. He even used my telephone to place an offtrack bet. By noon the day seemed brighter, the trees outside a darker green against blue skies.\n\nBut I could not concentrate on either the growing loveliness of the day or the endless paperwork that I was sure no one ever looked at after it was completed.\n\nWe had no one in custody for the shooting of the drive-by daiquiri store operator, even though we had a suspect with motivation in the form of Dr. Parks, and a connection, through the murder weapon, to an employee of Castille LeJeune. In the meantime a Celtic killing machine like Max Coll was running loose in our area; I had been made by the family of Frank Dellacroce for the murder of their relative; and Theo and Merchie Flannigan continued to hover on the edge of my vision, chimeric, protean, like the memory of a college prom that, along with youth, belongs in the past.\n\nIt was the kind of criminal investigation in which thinking served no purpose. The motivation in most crimes was not complex. Usually people steal and cheat because they're either greedy or lazy or both. People kill for reasons of money, sex, and power. Even revenge killings indicate a sense of powerlessness in the perpetrator.\n\nAt least that was the conventional wisdom of duffer cops who think psychological profiling works best in films or TV shows that have little to do with reality.\n\nBut where did Junior Crudup fit into this? Or did he? Maybe Helen was right, I just wanted to nail the Daddy Warbucks of St. Mary Parish, Castille LeJeune, to a tree.\n\nI spread the photos of Junior Crudup given me by our reference librarian on my desk blotter. Did you dream at night of the black Betty slicing across your back? I wanted to ask him. Didn't you learn you can't beat the Man at his own game? What happened to you, partner?\n\nI picked up the last photo in the series and looked again at the image of Junior staring up at a mounted gunbull, across the bayou from Castille LeJeune's home, his hoe at an odd angle on his shoulder, his face puzzled by a world whose rules ensured he would never have a place in it. But the focus of my attention was not Junior. In the wintry background, guiding a single-tree plow through the cane stubble, was a muscular, coal black convict, with the clear detail of welted scars on his forearms, the kind a convict might earn in a half dozen knife beefs.\n\nI held a magnifying glass to the grainy black-and-white image. I was almost sure the face was that of a youthful Hogman Patin, the longtime recidivist who had been on the Red Hat Gang with Junior but had said he did not know Junior's fate.\n\nI picked up the telephone and called my house.\n\n\"Hello?\" Father Jimmie said.\n\n\"Want to check out some Louisiana history you don't find in school books?\"\n\n\"Why not?\" he said.\n\n## Chapter 12\n\nWherever Hogman lived, he created a bottle tree, for reasons he never explained. During winter, when the limbs were bare, he would insert the points of the branches into the mouths of colored glass bottles until the whole tree shimmered with light and tinkled with sound.\n\nFather Jimmie and I parked in the front yard of his house on the bayou and walked around to the back, where Hogman was hoeing weeds out of a garden next to his bottle tree. He stopped his work and smiled, then saw my expression.\n\n\"Why'd you jump me over the hurdles?\" I said.\n\n\"You mean about Junior?\" he said.\n\n\"You got it,\" I replied.\n\n\"Junior punched his own ticket. You might t'ink he was a hero, but back in them days, if a nigger got mixed up wit' a white woman, all of us had to suffer for it.\"\n\n\"How about spelling that out for me?\" I said.\n\nThe year was 1951. Hank Williams and Lefty Frizzell played on every jukebox in the South, and across the ocean GIs packed snow on the barrels of .30 caliber machine guns to keep them from melting while they mowed down wave after wave of Chinese troops pouring into North Korea.\n\nBut in central Louisiana, a group of black convicts who knew little or nothing of the larger world suddenly found themselves transferred from Angola Penitentiary to a work camp for nonviolent offenders deep down in bayou country. The camp had been created out of the remnants of what had been called the quarters on Fox Run Plantation. None of the convicts knew what to expect. The first morning they found out.\n\nThey were given clean denims, soap, toothpaste, good work shoes, and were told to burn their striped pants and jumpers in a trash barrel behind the camp. The beatings with the black Betty, the sweatboxes and anthill-treatment, the fecal-smelling lockdown units, the killings by guards on the Red Hat Gang, became only a memory at Fox Run. Sometimes a truculent inmate was forced to wear leg irons or stand all night on an upended bucket, and the food they ate\u2014the greens, fatback, beans, corn bread, and molasses\u2014was the same fare served at Angola; but the guards were not allowed to abuse them, and at night the inmates slept in cabins with mosquito screens on the windows, boiled coffee in the fireplace, played cards and listened to radios, and on holidays had preserves and cookies to eat.\n\nThe humane treatment they received was due solely to one person: Miss Andrea, as they called her, the wife of Castille LeJeune.\n\nThe other inmates had been in the camp six months when Junior transferred in from the Red Hat Gang. The first time he saw her he was in the bottom of an irrigation canal with Hogman Patin, raking mounds of yellowed weeds out of the water and flinging them up on the embankment. She was riding English saddle on a black gelding, her long hair tied behind her head, her white riding pants skintight across her rump and thighs. Her small hand was cupped around a braided quirt.\n\n\"That's her, huh?\" Junior said.\n\n\"Who?\" Hogman said.\n\n\"Miz LeJeune,\" Junior said.\n\n\"What you care who she is?\"\n\n\"She wrote me a letter.\"\n\n\"Shit.\"\n\n\"That's right. Up at the joint. Tole me how much she liked my music. She's a fine-looking woman.\"\n\n\"You get them t'oughts out of your head, nigger,\" Hogman said.\n\n\"You boys eye-balling down there?\" the guard said from horseback.\n\nAmong Junior's few possessions was a guitar, a twelve-string Stella he had bought in a New Orleans pawnshop. He tuned the double-strung E, A, and D strings an octave apart so that the chords reverberating out of the sound hole gave the impression of two guitars being played simultaneously. Each evening, after supper, he played on the front steps of his cabin, his steel finger picks glinting in the setting of the sun, his voice rising into a sky filled with clouds that looked like colored smoke.\n\nThen, one spring night, while he played on the steps, he saw her car stop on the road. It was a purple 1948 Ford convertible, with an immaculate white, buttoned-down top. She was smoking a cigarette behind the wheel, her skin softly lit by the green illumination of the dashboard. She listened to him play until she had finished her cigarette, then she dropped it outside the window, restarted her engine, and drove away.\n\nIn July, on a languid Saturday morning, a guard by the name of Jackson Posey told Junior to put on a fresh change of state blues, to brush his shoes, comb his hair, bring his guitar, and get in the guard's pickup truck. As the two of them drove toward the big house, Junior could feel the guard's irritation like a palpable presence inside the cab.\n\n\"What's going on, boss?\" Junior asked.\n\nJackson Posey did not reply. Although he was often called boss, he held the rank of captain, one he had earned by shepherding convicts under the gun for two decades, pulling almost the same kind of time as his charges. But the fact he was a captain was a matter of great pride to him, because it meant he was literate and had administrative duties within the penal system. His forearms were pocked with early indications of skin cancer, the top of his forehead half-mooned like a sliver of melon rind where he normally wore a hat. He put three fingers into a pouch of Red Man and inserted the string tobacco into his jaw, then drove around to the back of the big house and parked under a mulberry tree.\n\nJunior could see Andrea Castille seated on the patio, a pitcher of lemonade on a glass table beside her. A recording machine, the kind that made use of wire spools, rested on the brickwork by her foot, an extension cord running back through the French doors into the house. Inside the living room a little girl, a miniature of her mother, played on the rug with wood blocks.\n\n\"I always treated you fair, ain't I?\" the guard said.\n\n\"Yes, suh,\" Junior said.\n\n\"Then it don't hurt to tell Miss Andrea that, does it?\"\n\n\"No, suh.\"\n\n\"You stay where I can see you,\" he said.\n\n\"Wouldn't have it no other way, boss.\"\n\nJackson Posey narrowed one watery blue eye, as though squinting down a rifle barrel. \"You sassing me?\" he said.\n\nJunior shut the truck door behind him and approached Andrea Castille with his guitar cradled under his right arm. She wore a pink sundress and dark glasses and a gold cross on a chain around her throat. \"Can you play 'Goodnight Irene'?\" she said.\n\n\"Yes, ma'am, I learned it from the man who wrote it,\" he replied.\n\n\"I'd like to record you while you do it. That is, if you don't mind.\"\n\n\"No, ma'am, I'm glad to.\"\n\n\"Would you like to sit down?\"\n\n\"Standing is just fine, ma'am.\"\n\nHe slipped the cloth strap of the guitar around his neck and sang for her, feeling foolish at the contrived nature of the situation, wondering if the guard's eyes were burrowing into his back or if Andrea Castille's husband was watching him from an upstairs window.\n\n\"You have a wonderful voice,\" she said. \"Sit down. Please, it's all right.\"\n\n\"Ma'am, I'm a convict.\" Involuntarily his eyes swept across the back windows of the house.\n\nShe seemed to resign herself to his recalcitrance. \"Would you sing another song?\" she said.\n\nHe sang one of his own compositions. The breeze had dropped and his shirt was damp against his skin. He could not see her eyes behind her dark glasses, but he believed they were invading his person. His fingers were moist and clumsy on the frets, his voice uncertain. A muscle spasm sliced across his back from the odd angle in which he was holding the Stella.\n\nHe stopped and blotted his face on his sleeve, his heart beating. Why was he behaving like this?\n\nBut he already knew the answer. He wanted her approval\u2014just like an organ grinder's monkey.\n\n\"I hurt my back in the field yesterday. Just ain't myself,\" he said.\n\n\"Maybe you can come another time, when you're feeling better,\" she said.\n\nHe shook his head negatively, his eyes lowered, his frustration and anger at himself rising. But she didn't give him time to speak. \"I have something for you. I'll be just a minute,\" she said.\n\nHe waited patiently in the dappled sunlight, the heat rising from the bricks around him. What was she up to? He had known white women like her in the North, he told himself. They liked to stick their hand in the tiger cage. Sometimes they even brought the tiger into their bed. Well, if that was what she wanted, maybe she might just find out who sticks what in who, he said to himself.\n\nShe emerged from the French doors with a narrow, blue-felt, brass-hinged box in one hand. She removed her dark glasses and handed him the box. For the first time he saw her eyes. They were the color of violets, like none he had ever seen, and there was a kindness and honesty in them that caused a thickening sensation in his throat.\n\n\"I've heard you play these on your records. I didn't know if you had one now or not,\" she said.\n\nHe pried the lid back stiffly and looked down at a chrome-plated harmonica cushioned inside the white satin interior of the box.\n\n\"It's an E-major Marine Band,\" she said.\n\n\"Yes, ma'am. I know. This is a fine instrument, Miss Andrea.\"\n\n\"Well, thank you for coming to my house,\" she said. Then she shook his hand, something no southern white woman had ever done.\n\nOn the way back to the camp, the guard, Jackson Posey, kept turning and staring at the side of Junior's face. Junior looked straight ahead, the harmonica case gripped in his palm. Just before they drove past the wire into the cluster of cabins that made up the improvised work camp, Posey braked the truck and shifted the floor stick into neutral. A cloud of dust floated by his open window.\n\n\"You got no control over what that woman does, so I ain't holding it against you,\" he said.\n\n\"Suh?\" Junior replied.\n\n\"You know what I'm talking about. Her husband's coming home from the arm service next week,\" the guard said.\n\n\"Yes, suh,\" Junior said, still uncertain about the direction of the conversation.\n\n\"I ain't gonna lose my job 'cause I let his wife shake hands with a nigger convict. You hearing me, Junior?\"\n\nJunior could feel the softness of the felt box in his fingers. \"You don't like what she done, lock me down, bossman,\" Junior replied.\n\n\"You just earned yourself a night on the bucket. Sass me again, and Miss Andrea or no Miss Andrea, you're gonna be the sorest nigger in the state of Lou'sana.\"\n\nTwo weeks later, while Junior and Hogman were pulling stumps on the far side of the bayou, he saw Andrea LeJeune and her husband cantering their horses through a field of buttercups. They clopped across a wood bridge that spanned a coulee, disappearing into a grove of live oaks. A few minutes later she emerged by herself, her face pinched with anger, and slashed her quirt across her horse's flank. She galloped past Junior toward the drawbridge, her thighs crimped tightly into the horse's sides, dirt clods flying off her horse's hooves. She was so close Junior could have reached out and touched her leg.\n\nBut if she saw him, she showed no recognition in her face.\n\nThat night another convict in Junior's cabin was looking at the pages of a newspaper that had blown from the road into the camp's wire fence. A photograph on the front page showed Castille LeJeune in a dress Marine Corps uniform with a medal hanging on a ribbon from his neck. \"That's the man own Fox Run, ain't it?\" the convict said. His name was Woodrow Reed. He wore a goatee that looked like a cluster of black wire on his chin, and the other inmates believed he could tell fortunes with a greasy pack of cards he carried in his shirt pocket.\n\n\"That's the man,\" Junior replied.\n\n\"What it say about him?\" Woodrow asked.\n\n\"He saved a bunch of lives, then he shot down a Nort' Korean name of Bed Check Charley.\"\n\n\"Bed Check who?\"\n\n\"That's a guy used to fly over the Americans in a Piper Cub and drop hand grenades on them. The F-80s couldn't nail him 'cause they was too fast. But Mr. LeJeune went after him in a World War II plane that was a lot slower and blew his ass out of the sky.\"\n\n\"How come you know all this?\" Woodrow asked.\n\n\"Read about it in a magazine.\"\n\n\"You somet'ing else, Junior,\" Woodrow said.\n\nBut secretly Junior did not feel he was something else. One out of three of his adult years had been spent in prison. He had made race records in Memphis, been interviewed in Downbeat magazine, and performed with Cab Calloway's orchestra in New York City, all before he was thirty years old. But what had he done with his success? Rather than build upon it, he had gotten into trouble every place he went. Now he was the man with one eye in the country of the blind, sassing redneck prison guards, a hero to hapless, illiterate, and superstitious men because he could read a magazine.\n\nOne month later, on a Saturday afternoon, Andrea LeJeune had him brought to the big house again. This time her husband was with her on the patio, seated under an umbrella, a tropical drink in his hand. Their daughter, who must have been around three or four years of age, was throwing a ball back and forth on the lawn with a black maid.\n\n\"This is my husband, Junior. He'd love to hear you sing 'Goodnight, Irene,'\" she said.\n\nLeJeune's legs were crossed. He wore socks with his sandals and seemed to be studying the points of his toes.\n\n\"Huddie Ledbetter done it a lot better than I can,\" Junior replied. He shifted his weight and felt the belly of the guitar scrape hollowly against his belt buckle.\n\n\"Then play something of your own choosing,\" Castille LeJeune said, his gaze still fixed on the end of his foot.\n\n\"Suh, I ain't all that good,\" Junior said. His eyes met LeJeune's briefly, then slipped away.\n\n\"You uncomfortable for some reason?\" LeJeune asked.\n\n\"No, suh.\"\n\n\"Then play. Please,\" LeJeune said.\n\nHe sang \"Dig My Grave with a Silver Spade,\" running quickly through the verses, leaving out the treble string improvisations he usually ran high up on the guitar's neck. When he finished he looked at nothing, the guitar strap biting into the back of his neck. He could smell the exhaled smoke from LeJeune's cigarette drifting into his face.\n\n\"You seem to be a man of considerable accomplishment. How is it you spent so many years in jail?\" LeJeune said.\n\n\"Don't rightly know, suh. Guess some niggers just ain't that smart,\" Junior replied.\n\nHe heard the guard's shoes crunch on the gravel drive, as though the guard were experiencing a tension he had to run through the bottoms of his feet into the ground. But LeJeune seemed to take no notice of any sardonic content in Junior's remark.\n\n\"Maybe you should have joined the military and found a career for yourself that didn't get you into trouble,\" LeJeune said.\n\n\"I served in the United States Navy, suh. Under another name, but in the navy just the same.\"\n\n\"You were a stewart?\"\n\n\"No, suh. I was a munitions loader. I loaded munitions right next to Harry Belafonte.\"\n\n\"Who?\"\n\n\"He's a singer, suh.\"\n\n\"Obviously my knowledge of pop'lar music isn't very extensive,\" LeJeune said, and smiled self-indulgently at his wife.\n\nWhy had Junior just told LeJeune of his military record or the fact he had known Harry Belafonte? It was like flipping a piece of gold through a sewer grate. At that moment he hated LeJeune more than any human being he'd ever met.\n\n\"Would you like something to eat before you go?\" LeJeune said. He held up a crystal plate on which a thick ring of crushed ice was embedded with peeled shrimp.\n\n\"No, thank you, suh.\"\n\n\"I insist,\" LeJeune said. He used a fork to scrape a pile of shrimp and ice on a paper plate, then inserted a toothpick in a shrimp and handed the plate to Junior. \"Go back yonder and sit in the shade and eat these.\"\n\nJunior looked at the yard, the absence of chairs or scrolled-iron benches on the grass or even a glider hanging from an oak limb. \"Where, suh?\" he said.\n\n\"Behind the carriage house. There's a box you can sit on. Enjoy your snack and then Mr. Posey will take you back to the camp,\" LeJeune said.\n\n\"You sit right here at the table. I'm going to get you some gumbo and a Coca-Cola from the house,\" Miss Andrea said. \"Did you hear me? Put your guitar in the chair and sit down.\"\n\n\"I think Mr. Crudup knows where he should eat,\" her husband said.\n\n\"Castille, if you weren't so miserably stupid and insensitive, I think I'd shoot you,\" she replied. Then she added \"God!\" and went inside the house.\n\nLeJeune got up from his chair and walked to the driveway, where he talked quietly with the guard, Jackson Posey. Junior Crudup felt as though he were sliding to the bottom of a dark well from which he would never emerge.\n\nJackson Posey did not drive the pickup truck directly back to the work camp. Instead, he crossed the bayou on the drawbridge and parked between a sugarcane field and a persimmon grove, out of sight of either the LeJeune home or the camp. He breathed hard through his nose, his mouth a tightly crimped line.\n\n\"Get out of the truck,\" he said.\n\n\"I ain't did nothing, boss.\"\n\n\"You got that sonofabitch on my ass. You call that nothing?\"\n\n\"Not my fault, boss.\"\n\nThey were both standing outside the truck now. The sky was hot and bright and wind was blowing dust out of the cane field and birds were clattering in the persimmon trees. Jackson Posey reached behind the driver's seat. Junior heard something hard clank against metal.\n\n\"Drink it,\" Posey said.\n\nBut Junior shook his head.\n\n\"Good 'cause now I can send your skinny black ass right back up to 'Gola.\"\n\n\"Ain't nobody in the camp supposed to get the Mussolini treatment. Miss Andrea don't allow it.\"\n\n\"Miz LeJeune don't write the rules now. What's it gonna be? Don't matter to me one way or another.\" Posey shook a cigarette loose from a package of Camels and inserted it in his mouth.\n\nJunior took the bottle of castor oil from the guard's hand and unscrewed the cap. The bottle was brown and heavy, the oil as viscous as syrup. He began to drink, then gagged and started again. The guard looked at his watch.\n\n\"All of it,\" Posey said.\n\n\"Ain't right, boss.\"\n\n\"You messed up the man's pussy. What do you expect him to do? Like my daddy used to say, life's a bitch, then you die. Chug it down, boy.\"\n\nPosey watched while Junior finished the bottle, then fingered a reddish purple scab on his arm, one that had not been there only two days ago. He drew in heavily on his cigarette, his eyes draining, as though he were purging himself of any intimations of his own mortality.\n\n\"It ain't nothing personal, Junior,\" he said.\n\n\"It's real personal, boss.\"\n\nThe guard stared emptily at the heat waves bouncing off the bayou and flicked his cigarette into the wind.\n\nBy the time Junior got back to the camp his bowels were collapsing inside him.\n\nHogman stopped his account and picked up a bottle that had fallen from his bottle tree. He wedged it in the fork of the tree and seemed to lose interest in both Father Jimmie and me and the story he had been telling.\n\n\"Go on, Hogman,\" I said.\n\n\"Junior started believing he was gonna have a life besides jailing and road-ganging. Gonna get a pardon from the governor and be a big star up Nort'. Just like Leadbelly.\"\n\n\"Andrea LeJeune was going to work a pardon for him?\"\n\n\"That's what he t'ought. She made Jackson Posey keep taking Junior up to the house when Mr. LeJeune was gone. Junior talked about her all the time, how pretty she was, what she smelled like, how she had all these fine manners, how she knew everyt'ing about his music. A whole bunch of people come up from New Orleans to hear him sing and play his twelve-string in the backyard.\"\n\n\"What happened to him, Hogman?\"\n\n\"Don't know. I got paroled. Last time I seed Junior he was playing 'Goodnight Irene' on the steps of his cabin, waiting to see if Miss Andrea was gonna drive by in her li'l convertible.\"\n\n\"I think you're holding out on me, partner.\"\n\n\"Miss Andrea got killed in a car wreck two or t'ree years after I left the camp. Mr. LeJeune lived up in that big house wit' just himself and his li'l girl. Junior disappeared. Ain't nothing left of him but a voice on scratchy old records. Nobody cared what happened back then. Nobody care now. You axed for the troot'. I just give it to you.\"\n\nHogman walked inside the back of his house and let the screen door slam behind him.\n\n## Chapter 13\n\nOrdinary people sometimes do bad things. A wrong-headed business decision, a romantic encounter in a late-night bar, a rivalry with a neighbor over the placement of a fence, any of these seemingly insignificant moments can initiate a series of events that, like a rusty nail in the sole of the foot, can systemically poison a normal, law-abiding person's life and propel him into a world he thought existed only in the perverse imaginings of pulp novelists.\n\nAt sunrise on Saturday morning the sky was pink and blue, the trees in my yard dripping from a thunder shower during the night, and I took a cup of coffee and hot milk and a bowl of Grape-Nuts out on the gallery and read the morning paper while I ate. When I was halfway into the editorial page Dr. Parks pulled his battered, beige pickup to the curb and got out. His jaws were heavy with beard stubble, one eye clotted with blood; he wore no socks and jeans that were grass-stained at the knees.\n\n\"I need help,\" he said.\n\n\"In what way?\"\n\nHe sat down on a step, a few inches from me. His long, tapered hands rested between his legs and his body gave off an odor like sour milk. His mouth began to form words, but nothing came out.\n\n\"Take it easy, Doctor. This stuff will pass with time. A guy just needs to put one foot in front of the other for a while,\" I said.\n\n\"There's no justice. Not for anything,\" he said.\n\n\"Pardon?\"\n\n\"My daughter's death. The electrical fire at my house. I bought a home warranty policy from Sunbelt Construction. The policy is underwritten by a bunch of criminals in Aurora, Colorado. I tried to talk to the Louisiana insurance commissioner about it and was told he's on his way to the federal pen.\"\n\nLike most people whose lives have been left in disarray by events so large he couldn't even describe them to himself, his rage against the universe had now reduced itself to the level of a petty financial quarrel with a fraudulent home warranty company.\n\n\"There might be a state senator or two we can call on Monday. How about a cup of coffee?\" I said. I rested my hand on his shoulder and tried to smile, then I saw the green cast in the skin under his eyes and the detached stare that made me think of soldiers I had known many years ago.\n\n\"I was on a medevac at Khe Sanh. I was in two crashes and one shoot-down. I put my best friends in body bags. It was all for nothing. This goddamn country is going down the sewer,\" he said.\n\n\"I was over there, too, Doc. We can always be proud of what we did and let the devil take the rest of it. Sometimes you've got to throw the bad times over the gunnels and do the short version of the Serenity Prayer. Sometimes you just say full throttle and fuck it.\"\n\nBut my words were of no value. He got to his feet like a man walking in his sleep, then turned and extended his hand. \"I insulted you at my home and in your office. I didn't mean what I said. My wife and I are better people than we seem,\" he said.\n\nHe pressed the fingers of one hand against the side of his head, like a man experiencing a pressure band or a level of cerebral pain that gave him no relief. He pulled open the door of his pickup and got inside, holding the steering wheel to steady himself. I walked to the passenger window.\n\n\"Where you headed?\" I asked.\n\n\"To confront the people who cheated me, the ones who put defective wiring in my house, the ones who shouldn't be on the goddamn planet.\"\n\n\"I don't think that's a good idea, Doc.\"\n\n\"Stand away from the truck,\" he replied. He ground the transmission into gear and swung the truck into traffic, almost hitting an automobile packed with Catholic nuns.\n\nI went back inside and called the dispatcher. Wally happened to be on duty. \"You want us to pick up this guy, Dave?\" he asked.\n\nI thought about it. Roust Dr. Parks now, in his present state of mind, and we would probably only add to his grief and anger. With luck he would eventually go home or at worst get drunk somewhere, I told myself. \"Let it go,\" I said.\n\nHelen Soileau called me just after lunch. \"How busy are you?\" she said.\n\n\"What's up?\"\n\n\"It's Dr. Parks. Wally said you called in on him earlier.\"\n\n\"What about him?\"\n\n\"Evidently he went looking for Castille LeJeune. He didn't find him, so he went after this guy Will Guillot.\"\n\n\"What do you mean 'he went after him'?\"\n\n\"With a cut-down double-barrel twelve-gauge.\"\n\n\"He shot Guillot?\" I said.\n\n\"You got it backwards. Parks is dead. Say good-bye to our prime suspect in the drive-by daiquiri shooting.\"\n\n\"Wait a minute. I can't get this straight. Parks is dead?\"\n\n\"At least he was five minutes ago. Get pictures if you can,\" she said.\n\nWhen I got to the home of Will Guillot emergency vehicles were still parked along the street and barricades set up to prevent the curious and the voyeuristic from driving past the house. The incongruity of the images there would not fit in time and place. In a tree-covered neighborhood of nineteenth-century homes and thick St. Augustine lawns, where the hydrangeas and impatiens and Confederate roses were softly dented by the breeze, and blue jays and robins sailed in and out of the live oaks, Dr. Parks lay on his side in the driveway, his mouth and eyes locked open, one cheek pressed flat against the cement, a pool of dried blood issuing from a ragged hole in his throat into the sunlight. Six inches from his outstretched hand lay a cut-down twelve-gauge, the stock wood-rasped and sanded into a pistol grip.\n\nThe crime-scene investigator was a nervous, tightly wrapped man with a strong cigarette odor by the name of Dale Louviere. When I ducked under the crime-scene tape he glared into my face, as though challenged, nests of green veins pulsing in his temples. Before he had entered law enforcement he had been a gofer and point man for a notorious casino operator in Lake Charles.\n\n\"What do you want, Robicheaux?\" he said.\n\n\"Dr. Parks was part of an Iberia Parish homicide investigation. Where's the coroner?\" I said.\n\n\"Him and the sheriff fish together on Saturday. We're still waiting on them,\" Louviere replied.\n\n\"Are there any witnesses?\"\n\n\"Yeah, the shooter, Will Guillot. He's in the kitchen.\"\n\n\"How do you read it?\" I asked.\n\n\"Open and shut. The vic went nuts about a house fire or a home warranty policy or something. He came here to wax Guillot and instead caught a .45 in the throat. The round hit the oak tree in front.\"\n\nI leaned over to look more closely at the cut-down twelve-gauge. I couldn't see a brand name on it, but the steel around the magazine was incised with delicately engraved images of ducks and geese in flight. \"Handsome gun to chop down with a hacksaw,\" I said.\n\n\"Get some mud in the barrel and that's what people do, Robicheaux,\" Louviere replied.\n\n\"Except this guy was a collector. How many collectors spend their time converting their firearms into illegal weapons?\"\n\n\"The next time I investigate a homicide, I'll have the crime scene shipped to Iberia Parish so you can supervise it,\" he said.\n\nI walked through the porte cochere to a back door and entered the kitchen without knocking. Will Guillot was at the counter, gazing out the back window into the yard, while he ate a ham-and-lettuce sandwich. A tall, half-empty glass of milk rested by his sandwich plate. He turned and looked at me quizzically, the birthmark that drained like purple dye from his hairline to the corner of his eye almost obscured by shadow, so that one side of his face looked like the marred half of a large coin.\n\n\"You were in fear for your life, were you, Mr. Guillot?\" I said.\n\n\"Yeah, I guess that describes it,\" he answered, one cheek stiff with a piece of bread. \"You have jurisdiction here?\"\n\n\"You don't have to talk to me if you don't want to.\"\n\n\"I don't want to.\"\n\n\"Fair enough. On an unrelated subject, are you a hunter or a gun collector?\"\n\n\"I hunt. Why?\"\n\n\"No reason. Were you in 'Nam?\"\n\n\"No. What's that have to do with anything?\"\n\n\"Dr. Parks was on a medevac. He had his problems, but I don't think he was a violent man. I don't think that cut-down twelve on the driveway was his, either.\"\n\n\"This conversation is over, Mr. Robicheaux, and you can get out of my house.\"\n\n\"Does it bother you?\" I said.\n\n\"Bother me? That I defended myself against a lunatic?\"\n\n\"His daughter was burned alive after buying liquor illegally at one of Castille LeJeune's daiquiri shops. His house burned after you put bad wiring in it, and you shot him to death after he came here to complain about a fraudulent home warranty policy you sold him. It's hard to believe one guy can have that much bad luck, isn't it? Enjoy your sandwich, Mr. Guillot. I'll be in touch,\" I said.\n\n\"Kiss my ass,\" he said.\n\nSunday Father Jimmie had gone to Lafayette to collect signatures on a petition to shut down drive-by daiquiri windows and had stayed the night at a retreat house in Grand Coteau. I ate a plate of clam spaghetti at a cafe in Jeanerette, then went to sleep reading T.E. Lawrence's Seven Pillars of Wisdom, with Snuggs on the foot of the bed. My windows were open and in my sleep I heard the wind in the trees, a solitary pecan husk rattle on the tin roof, a workboat chugging heavily on the bayou. The air was cool and clean smelling with ground fog, rainwater ticking in the trees, and I felt Snuggs walk across my back so he could sniff the breeze blowing through the screen. Just after midnight, my bowels constricted as though I had swallowed a piece of broken glass. I went into the bathroom and sat on the toilet, my thighs trembling with nausea.\n\nThen I heard someone wedge a tool between the back door and the jamb, splinter the deadbolt, and enter the house. Whoever it was moved quickly toward the band of light at the bottom of the bathroom door, opened it slightly, and looked in at me.\n\n\"I wasn't planning to meet you like this, but I couldn't resist the opportunity. Can I be getting you anything? You don't look too well,\" the figure said.\n\n\"Coll?\"\n\n\"Right you are. No, don't get up. Take care of business while I have my say, then I'll be off.\" His hand came through the opening and removed the key from the lock. He shut the door and locked it from the outside.\n\n\"What do you think you're doing?\" I said.\n\nI heard him go into the bedroom, then scrape a chair into position. \"This is a fine cat you have here. Been in a few fights, has he?\"\n\n\"Listen, Coll\u2014\"\n\n\"He's got a real pair of bandoliers back there.\"\n\nMy face was cold with sweat, a bilious fluid rising from my stomach. Gray spots danced before my eyes.\n\n\"Father Dolan and I have nothing to do with your life,\" I said.\n\n\"Oh, but you do. Two rather nasty cretins just arrived in town, Mr. Robicheaux, the cousins of Frank Dellacroce. Stone killers, they are, sir, with no parameters and no charitable impulses. Evidently a few of the greaseballs think you blew poor Frank's head off. Would you like to hear what they did to a friend of mine?\"\n\n\"No.\"\n\n\"Took a blowtorch to him. What's the name of your cat?\"\n\n\"Snuggs.\"\n\n\"What a fine little fellow. Built like a fucking fire hydrant. It's a shame the innocent suffer. But maybe that's the only thing that causes us to take action.\"\n\nI could feel my heart quicken. \"What are you saying?\"\n\n\"I didn't make the world. I just live in it as best I can. I'll be going now.\"\n\n\"You leave that cat here.\"\n\nBut he didn't reply. I heard his chair scrape but did not hear him set Snuggs down. \"Coll? Did you hear me?\" I yelled.\n\nI heard him banging about in the kitchen, then a hard, clunking sound and his footsteps going heavily through the house and out the front door. By the time I was able to climb out the bathroom window, the yard and the street were empty, the ground puffing with fog, the moon as bright as a white flame behind the skeletal outline of a water oak.\n\nI went around back and entered the house through the kitchen door. A pitcher of milk rested on the drainboard and Snuggs was lapping from a bowl next to it, one Max Coll had filled with both milk and dry cat food.\n\nI started to dial 911, then gave it up, propped a chair against the kitchen door, and went back to sleep, my .45 under my pillow.\n\nAt 8:05 Monday morning Clotile Arceneaux walked into my office. She wore a pair of navy blue slacks, a blouse printed with tropical flowers, and a polished black gunbelt with her badge holder hung from the front and her cuffs pushed through the back. She had the blackest hair and wore the brightest lipstick I had ever seen.\n\n\"How's life in the Big Sleazy?\" I said.\n\nShe grinned broadly, then sat down without being asked. \"You're a magnet, Robicheaux,\" she said.\n\n\"For what?\"\n\n\"Trouble. We keep a few people at the New Orleans airport, watching to see who comes and goes, know what I mean? Three days ago a couple of greaseballs from Ft. Lauderdale got into town, spent the night with some hookers, then caught a flight to Lafayette. Guess what their last names are?\"\n\n\"Dellacroce?\"\n\n\"How'd you know?\"\n\n\"Max Coll was at my house last night.\"\n\n\"Say again?\"\n\n\"He was walking around inside my house. He talked to me through the bathroom door.\"\n\nShe looked up at one corner of the ceiling, her eyelids fluttering. Then she scratched her neck and looked at me. \"I brought mug shots of the Dellacroces. They're brothers, Tito and Caesar. Tito's friends call him the Heap, 'cause he looks like a haystack with eyes. But the mean one is Caesar. He's short and not very bright.\"\n\n\"He uses a blowtorch on people?\"\n\n\"You do know about these guys.\"\n\n\"Max Coll is tops when it comes to intel.\"\n\n\"I've got to get a job over here. New Orleans just doesn't cut it.\"\n\n\"Want to go to lunch later?\"\n\n\"Like to, slick, but the Big Sleazy calls. I've got a little more here on your man Coll.\"\n\n\"He's not my man. He's a meltdown you guys shipped to New Iberia.\"\n\nShe raised her eyebrows and made an innocent face as she opened a manilla folder in her lap. \"The Coll family was hooked up with the IRA for generations,\" she said. \"Some of them may have been behind the bombing of a pub in Belfast. Some Protestant militants decided to get even and took Max's whole family out, including an older brother who was a Catholic priest.\"\n\n\"That's how he ended up in the orphanage,\" I said, more to myself than to her.\n\nShe looked down again at the open folder in her lap. \"Yeah, that's right. He was there until he was fifteen,\" she said.\n\n\"Go to lunch with me,\" I said.\n\nShe thought about it. \"Make it a beignet and a cup of coffee,\" she said. She studied me with one eye half closed.\n\nThat afternoon I looked down at the booking photos of Tito and Caesar Dellacroce she had left on my desk. Tito, known as the Heap by his peers, stared back at me with eyes that were like cups of black grease. His brother made me think of a ferret in need of a haircut. Both Max Coll and Fat Sammy Figorelli had indicated Frank Dellacroce's relatives had put his death on me. Maybe. But I believed their real target was still Max Coll, and Max was in New Iberia for reasons other than a religious obsession with Father Jimmie. I believed Max had intimations about where the hit on Father Jimmie had come from, and Max blamed that same person for putting a contract on him and had come to our area to wipe the slate clean.\n\nOr perhaps he was simply crazy.\n\nRegardless, it was time to dial up Max's head and see how he liked having things turned around on him. I called the Daily Iberian and scheduled an ad to run in the next day's personal notices.\n\n\"Let me read this back to you,\" the clerk said. \"'Max, you owe me $57.48 for the damage you did to my back door. Why don't you pay your debts instead of acting like a window-licking voyeur who breaks into people's houses and molests their pets? Tito and Caesar just blew into town and seem upset because you canceled their cousin's ticket. Have a nice day\u2014Dave.'\"\n\n\"Perfect,\" I said.\n\n\"Mr. Robicheaux, this ad doesn't make much sense.\"\n\n\"It does if you're morally insane,\" I replied.\n\nDid you ever have a song in your mind you couldn't get rid of? For me, at least on that Monday afternoon, it was \"Goodnight Irene.\" I kept thinking of Junior Crudup sitting on the steps of his cabin in the work camp, playing his twelve-string guitar, singing the words to Leadbelly's most famous composition, while he waited to catch a glimpse of Andrea LeJeune's purple Ford convertible passing on the dirt road. Did she arrange for him to return to the house again? Did the guard, Jackson Posey, continue to torment him because of the hatred Posey felt for himself and the lot the world had dealt him?\n\nIf God in that moment looked down upon His creations, I wondered if He wasn't terribly saddened by the level of madness that had become the province of His children.\n\nThe song was still in my head when I went that afternoon to Baron's, the health club where I worked out, and saw Castille LeJeune seated on a hardwood bench in the dressing room, his face bright with sweat from his racquet ball game, a towel wrapped around his neck. He was jovial and expansive, sipping from a glass of icewater while he talked with a group of businessmen, although a sign on the wall stated no glass containers were allowed in the room. It was 5 P.M. and both black and white workers from the salt mines out in the wetlands and the sugar mills that ringed the town burst loudly into the dressing room. Instead of being intimidated by LeJeune's presence, they treated him as they would a celebrity, greeting him as \"Mr. Castille.\" Somehow he was one of them, at least for the moment, a patrician who knew them by their first names and spoke both demotic French and English without being patronizing.\n\nThere were great differences in the room, but not between the races. The black and white working men spoke the same regional dialect and shared the same political attitudes, all of which had been taught them by others. They denigrated liberals, unions, and the media, considered the local Wal-Mart store a blessing, and regularly gave their money to the Powerball lottery and casinos that had the architectural charm of a sewer works. They were frightened by the larger world and found comfort in the rhetoric of politicians who assured them the problem was the world's, not theirs. And most heartening of all was the affirmation lent them by a genteel person like Castille LeJeune, a Distinguished Flying Cross recipient who, unlike many members of his class, showed no fear or lack of confidence in their midst, which told them of his respect for their humanity.\n\nI dressed in a corner of the room, my back turned to LeJeune and the cluster of men around him. Maybe I was wrong about him, I thought. Maybe Helen and Theodosha were justified in their criticism of my attitudes. I was born in the late Depression and bore an ingrained resentment toward the wealthy and the powerful. All drunks fear and desire both power and control, and sometimes even years of sobriety inside A.A. don't rid alcoholics of that basic contradiction in their personalities. Why should I be any different?\n\nWhen I had almost thought my way into a charitable attitude toward Castille LeJeune, I felt a hand touch my shoulder. \"Would you like to play a round of racquet ball, Mr. Robicheaux?\" he said.\n\n\"Never learned how,\" I said.\n\n\"Do you have any idea why this deranged physician, what's-his-name, Parks, would have come to my home, then to my foreman's?\"\n\nSo you're a showboat as well as a hypocrite, I thought. \"His daughter was served illegally at your daiquiri shop before she died in a car crash. Your company defrauded him on the house-remodeling job it did at his home. He also said you sold him a bogus warranty on his house. Maybe that might have something to do with it,\" I replied.\n\n\"I'd like to say your reputation precedes you, Mr. Robicheaux. But your potential seems to have no limits,\" he said.\n\n\"Your deceased wife brought a black convict to your house out of respect for his musical talent, an event evidently you couldn't abide. That same convict, Junior Crudup, disappeared from the face of the earth. I suspect, on the day of your death, his specter will be standing by your bed.\"\n\nThe only sound in the room was the hum of the overhead fans.\n\n\"How dare you?\" he said.\n\nI'm going to get you, you sorry sack of shit, I said to myself, my eyes fixed six inches from his.\n\nThe days were growing shorter, and by 6 P.M. the sun had set, the sky was black and veined with lightning, and Bayou Teche was high and yellow and chained with rain rings in the glow of the lamps along the banks of City Park. Father Jimmie walked about in the backyard, his hands in his pockets, examining the sky, the wind swirling leaves around his ankles. He came back in the house smelling of trees and humus, his eyes purposeful.\n\n\"I need to work things out with Max Coll,\" he said.\n\n\"You have to do what?\" I said.\n\n\"He's in New Iberia because I'm here. Now, these other criminals are showing up because he's here. Where does it end? One man is already dead.\"\n\n\"Frank Dellacroce sexually exploited a retarded girl. I think he got off easy.\"\n\n\"I had to own up to some things at the retreat, the big one being pride.\"\n\n\"In what?\"\n\n\"My feeling of virtuous superiority to others,\" he said.\n\n\"You don't call self-flagellation a form of pride?\"\n\n\"You're a hard sell, Dave.\"\n\nThe phone rang like a providential respite. Or at least that's what I thought until I realized who was on the other end of the line.\n\n\"Where do you get off embarrassing my father in a public place?\" a woman's voice said.\n\n\"Your father is neither a victim nor a martyr. Cut the crap, Theo,\" I said.\n\n\"Your anger taints everything in your life. You disappoint me in ways I can't describe.\"\n\nI heard a sheet of rain clatter across the tin roof. I wanted to pretend I was impervious to her words, but the element of truth in them was like a thorn pressed into the scalp. \"Where are you?\" I said.\n\n\"In a bar.\" She gave the name, a box of a place squeezed between shacks in New Iberia's worst neighborhood.\n\n\"How much have you had?\" I asked.\n\n\"I'm drinking a soda and lime, believe it or not. But I'm about to change that. Why, you want to get loaded?\"\n\n\"You wait there,\" I said.\n\nAs I backed out of the driveway, the canopy of oaks over the street stood out in lacy, black-green relief against the lightning rippling across the sky. I did not pay particular attention to the car that rounded the corner and followed me past the Shadows.\n\nInside the house Father Jimmie tore the wrapper off his hangered dry cleaning and discovered his black suit was missing. He would have sworn it had been with his other things when he had brought them from the laundry three days ago. He searched the rack, then checked the top drawer where he kept his Roman collar and rabat, the backless garment that serves as a priest's vest.\n\nBoth collar and rabat were gone.\n\n## Chapter 14\n\nI drove to the bar Theodosha had called from and parked on the street. The bar was a gray, dismal place, ensconced like a broken matchbox under a dying oak tree, its only indication of gaiety a neon beer sign that flickered in one window. She was at a table in back, the glow of the jukebox lighting her face and the deep blackness of her hair. She tipped a collins glass to her mouth, her eyes locked on mine.\n\n\"Let me take you home,\" I said.\n\n\"No, thanks,\" she replied.\n\n\"Getting swacked?\"\n\n\"Merchie and I had another fight. He says he can't take my pretensions anymore. I love the word 'pretensions.'\"\n\n\"That doesn't mean you have to get drunk,\" I said.\n\n\"You're right. I can get drunk for any reason I choose,\" she replied, and took another hit from the glass. Then she added incongruously, \"You once asked Merchie what he was doing in Afghanistan. The answer is he wasn't in Afghanistan. He was in one of those other God-forsaken Stone Age countries to the north, helping build American airbases to protect American oil interests. Merchie says they're going to make a fortune. All for the red, white, and blue.\"\n\n\"Who is they?\"\n\nBut her eyes were empty now, her concentration and anger temporarily spent.\n\nI glanced at the surroundings, the dour men sitting at the bar, a black woman sleeping with her head on a table, a parolee putting moves on a twenty-year-old junkie and mother of two children who was waiting for her connection. These were the people we cycled in and out of the system for decades, without beneficial influence or purpose of any kind that was detectable.\n\n\"Let's clear up one thing. Your old man came looking for trouble at the club today. I didn't start it,\" I said.\n\n\"Go to a meeting, Dave. You're a drag,\" she said.\n\n\"Give your guff to Merchie,\" I said, and got up to leave.\n\n\"I would. Except he's probably banging his newest flop in the hay. And the saddest thing is I can't blame him.\"\n\n\"I think I'm going to ease on out of this. Take care of yourself, kiddo,\" I said.\n\n\"Fuck that 'kiddo' stuff. I loved you and you were too stupid to know it.\"\n\nI walked back outside into a misting rain and the clean smell of the night. I walked past a house where people were fighting behind the shades. I heard doors slamming, the sound of either a car backfiring or gunshots on another street, a siren wailing in the distance. On the corner I saw an expensive automobile pull to the curb and a black kid emerge from the darkness, wearing a skintight bandanna on his head. The driver of the car, a white man, exchanged money for something in the black kid's hand.\n\nWelcome to the twenty-first century, I thought.\n\nI opened my truck door, then noticed the sag on the frame and glanced at the right rear tire. It was totally flat, the steel rim buried deep in the folds of collapsed rubber. I dropped the tailgate, pulled the jack and lug wrench out of the toolbox that was arc-welded to the bed of the truck, and fitted the jack under the frame. Just as I had pumped the flat tire clear of the puddle it rested in, I heard footsteps crunch on the gravel behind me.\n\nOut of the corner of my eye, I saw a short, thick billy club whip through the air. Just before it exploded across the side of my head, my eyes seemed to close like a camera lens on a haystack that smelled of damp-rot and unwashed hair and old shoes. I was sure as I slipped into unconsciousness that I was inside an ephemeral dream from which I would soon awake.\n\nI knew it was sunlight when I awoke. I could feel its warmth on my skin, see its red-edged radiance at the corners of the tape that covered my eyes. Along with the chemical odor, perhaps ether or chloroform, that still clung to my face I could smell dead fish and ponded water that had gone stagnant inside shade and blackened leaves freshly broken by someone's shoes. I was seated in a chair, my wrists cuffed behind me with a plastic band. I turned my head into a breeze blowing from a window or door, like a blind man entering his first day without sight, vainly hoping the world around him was not filled with adversaries.\n\nA motorboat passed a short distance away. When the sound of the wake sliding through flooded trees died, I heard two men talking about a football game in another room. I tried to rise from the chair, then realized both my ankles were strapped to the legs. \"Asshole is awake,\" I heard one of the men say.\n\nA door opened and I felt the planks under my feet become depressed by the weight of the men entering the room. \"How you feel?\" one of them said.\n\n\"You're kidnapping a police officer,\" I said.\n\n\"I asked you how you felt.\"\n\n\"All right. I feel all right,\" I replied.\n\n\"Hear that? He's all right,\" the second man said. \"Frank Dellacroce is not all right. Somebody blew most of his head off.\"\n\n\"It wasn't me,\" I said.\n\n\"It wasn't him,\" the second man said. \"That's good to hear, 'cause people say you kicked the shit out of him the night he was killed. While he was in handcuffs.\"\n\n\"You got it wrong,\" I replied.\n\n\"He says we got it wrong. That's good, 'cause what we hear about you ain't so good. We hear you got a hard-on 'cause you can't drink, that you like to beat up people, that you got some kind of problem with Italians in general,\" the same voice said.\n\n\"I haven't seen you. I don't know who you are. I think what we have here is a misunderstanding. I'm ready to let it go at that,\" I said.\n\n\"He's ready to let it go. I like that. We're talking about a generous man here,\" the same voice said. \"You want a beer?\"\n\n\"No.\"\n\n\"Yeah, you do.\"\n\nI turned my face toward the voice. \"Why pull a federal beef down on yourselves? Use your head,\" I said.\n\n\"Oh, we'll use our head, all right. You bet your life,\" the same man said.\n\nI heard a tab being torn loose from a pressurized can, then smelled beer and heard foam splattering on the floor. I could hear someone drinking from the can, swallowing thirstily. He pressed the can against my mouth, clicking it against my teeth, then forcing the aluminum rim between my lips.\n\n\"Don't do that,\" the first voice said.\n\n\"He wants it. He just don't know it yet,\" the second man said.\n\nSomeone, I think the second man, pulled loose my belt, then inserted his fingers between my trousers and stomach and poured the remnants of the can into my underwear. \"You already pissed your pants while you were asleep, so I'm just cleaning you up,\" he said.\n\nI felt the beer run down my thighs and calves. The wind blew through the windows and puffed the room and tin roof with air that smelled of brine and ozone and thunderheads out on the Gulf. Try to think clearly, I told myself. If they simply wanted to kill you, you would already be dead. They're not using their first names with one another and your eyes are taped because eventually they're going to release you. Don't change their agenda, I thought.\n\n\"Where's this guy Max Coll?\" the first man said.\n\n\"If you find out, I'd like to know. He creeped my house,\" I replied.\n\n\"He creeped a cop's house?\" the same man said.\n\n\"He's not your ordinary button man,\" I said.\n\n\"What's he look like?\"\n\n\"I never saw him,\" I lied.\n\n\"He's a Mick, though, right?\" the same voice said.\n\n\"We know he's in the area. We think he popped Frank Dellacroce. But we don't have much information on him.\"\n\nSuddenly a steel instrument bit into my left thumb and mashed the tissue and veins into the joint. I tried to clench my jaws on the scream that came out of my throat.\n\n\"That's what happens when you try to jerk us around,\" the second man said. He was behind me now, his breath touching the back of my ear. \"Guess where these pliers are going next?\"\n\n\"No more of that,\" the first man said.\n\n\"He killed Frank,\" the second man said.\n\n\"Maybe. But we wait on the man and see what he wants. Get out of the way,\" the first voice said.\n\nI smelled his presence in front of me, like hair with sweat dried in it and clothes with soap still in the fabric. Then his huge hand molded a chemical-soaked towel over my face and I felt myself floating to the bottom of a dank well where laughing faces stared down at me from a circle of blue overhead.\n\nI lay sideways on a floor through most of the afternoon, my eyes still taped, my knees and ankles now wrapped tightly with tape as well. In my mind's eye I tried to see the faces of all the people who had been important in my life. I thought of my mother and father, illiterate Cajuns who had done the best they could with what little they owned and who struggled through the Depression and the war years to create a decent home for themselves and their only son. I thought of the two Catholic nuns who had been my first-and second-grade teachers and the time when I accidently walked into a room where they were jitter-bugging to a phonograph, their beads and habits flying. The other clergy I had known in my early years had disappeared from memory, but those two remained with me, as though framed inside a secular holy card.\n\nI thought about the members of my platoon, deep in Indian country, blade faced, stinking of funk and rotted socks and mosquito repellant, their skin twitching as they worked their way down a night trail strung with toe-poppers and booby-trapped 105 duds. I thought about my dead wives, Annie and Bootsie, who were always my steadfast friends as well as spouse and lover, and I thought about Alafair, my adopted daughter, studying at Reed College in Portland, and wondered if I would ever see her again.\n\nI thought about the country in which I had grown up and which I had served as a soldier and police officer. It was the best country on earth, the most noble, egalitarian, democratic experiment in human history. It was a grand and wonderful place to live, well worth the fighting for, as Ernest Hemingway would say. Thomas Jefferson knew that, and so did Woody Guthrie, Dorothy Day, Joe Hill, Molly Brown, and the IWW.\n\nTo hell with the likes of my warders, who I was sure were Tito and Caesar Dellacroce. Let them do their worst, I told myself. And to hell with all the politicians on the take and the princes of industry who lionized Third World bedbugs in order to carry out their agenda of inculcating fear in the electorate at home. America was still America, the country everyone in the world wanted to emulate, where rock 'n' roll and the Beat lyrics of Jack Kerouac would outlive all the venal interests that threatened her.\n\nDying wasn't so bad, not if you faced it bravely, with a clear conscience and your principles still intact. But maybe it wouldn't come to that, I told myself. The tape was still on my eyes, my tormentors ostensibly still unidentifiable.\n\nAt least that is what I told myself.\n\nThen I heard movement in a room beyond the door of the room in which I lay, and the muffled voices of at least three men talking, and I felt my sense of personal resolve begin to drain like water from the bottom of a gunnysack.\n\nThe door opened and two sets of hands lifted me into a chair. The room was silent, the tin roof creaking from the cooling of the day. Someone wrapped tape both around my waist and the back of the chair.\n\n\"I don't know where Max Coll is. What purpose would I have in concealing his whereabouts?\" I said, although no one had spoken to me.\n\n\"See, he knows what we want. He don't even wait to be asked the question. That shows us he's a smart guy who can look into the minds of other people. That shows us he's smart and we're dumb,\" said the voice of the man who had applied a pair of pliers to my thumb.\n\n\"How you want this to play out, 'cause we got a flight to catch?\" said the voice of the other man, who I now believed to be Tito Dellacroce, also known as the Heap. But he was speaking to someone else, and not to his brother, either.\n\nWhoever he asked the question of did not respond. Instead, I heard the soft sound of a clothing zipper sliding on its track, followed by a pause, just before a warm stream of urine splashed in my face and ran down inside the tape that bound my eyes. I twisted my head from side to side, but the person urinating on me painted my mouth, hair, and neck and drenched my shirt before he zipped up his fly again.\n\n\"We're naming this place Yellow Springs, Louisiana, in your honor, Robicheaux,\" said the voice of the man with the pliers.\n\nThey left the room and closed the door behind them. I leaned forward and spit, then sucked saliva out of my jaws and spit again. I heard a car door slam and the car drive away. Two men reentered the room and one of them grabbed a corner of the tape and ripped it loose from my eyes and the back of my head.\n\n\"You're shit out of luck,\" said the man with the tape hanging from his fingers. He was short, with a pointed face, and small, energized, deep-set eyes, his hair scalped above his ears like bowl-cut animal fur.\n\nNext to him was his brother, Tito the Heap. His hair was braided in dreadlocks that hung to his shoulders, which sloped away from his thick neck like the sides on a tent. One jawbone kept flexing like a roll of pennies.\n\nThe room was bare, except for a table on which a tool box and a camcorder rested. The walls and floor were constructed of rough planks, and through the screen window I could see a woods strung with air vines and dotted with palmettos and beyond the tree trunks a bay and the red sun low on the horizon. In the distance somebody was firing a shotgun, perhaps popping skeet over the water.\n\n\"Are you listening, asshole? The man says the whack goes down an inch at a time. You get to be in your own movie,\" said the short man, whom I recognized from his mug shot as Caesar Dellacroce.\n\n\"Get it over with,\" I said.\n\n\"I think if you knew what was coming, you wouldn't say that,\" Caesar said.\n\nI looked into space, my eyes slightly out of focus with fatigue and hopelessness and now resignation.\n\n\"I'm talking to you,\" Caesar said. He popped my cheek with his hand.\n\n\"I figure I'm done, so what I'm about to tell you is the truth. I didn't smoke Frank Dellacroce, but I wish I had. He was a punk and a bully and somebody should have put the electrodes on him and blown out his grits a long time ago. When you get finished with me, Clete Purcel is going to turn over every rock in New Orleans and Fort Lauderdale until he finds you, then make you wish your mother had flushed you down the toilet with the afterbirth.\"\n\nCaesar stared at me, his mouth parted slightly, his jaws slack. \"Say that again?\"\n\n\"Go fuck yourself,\" I said.\n\n\"You believe this guy?\" Caesar said to his brother. But he was clearly distracted now, not quite in charge anymore.\n\n\"We wasted too much time on this,\" Tito said reflectively. His eyes, like his brother's, were inset deeply in the skull, his nostrils flaring when he breathed, as though the plates of muscle on his chest and shoulders were squeezing the air from his lungs. \"Here's what it is, ace. You rolled the dice with the wrong guy and lost. We ain't responsible for this. So take your medicine like a man. I'll make it short and sweet as possible. You want to say anything?\"\n\n\"No,\" I replied, and fixed my gaze out the window on a watery, red sunset barely showing behind the thin trunks of trees that had already turned dark with the gloaming of the day. Tito Dellacroce pushed a sponge into my mouth with the heel of his hand, then began winding tape around my head.\n\n\"Hang on,\" Caesar said, staring out the same window but at a different angle.\n\n\"What?\" Tito said.\n\n\"There's a priest out there,\" Caesar said.\n\n\"Where?\"\n\n\"Walking down off the levee. He's carrying a briefcase. Look for yourself. He's got a bandage around his throat,\" Caesar said.\n\nTito went to the window, then pulled a curtain across it. \"You ever seen a priest around here?\" he asked.\n\n\"Yeah, lots of priests hung out at Frank's old fuck pad.\"\n\n\"His fuck pad was up the road. Our father used to take us fishing here. It ain't a fuck pad,\" Tito said.\n\n\"Enough, already. It's a priest carrying a pro-life petition around or something. It ain't a big deal,\" Caesar said.\n\n\"Get outside.\"\n\n\"Do it yourself. The mosquitoes out there eat cows for lunch.\" Caesar peeked through the side of the curtain. \"See, he's gone.\"\n\nJust as he dropped the curtain back in place someone in heavy shoes walked up on the porch and banged hard on the door. Tito and Caesar looked at each other. Then the visitor on the porch banged even harder, shaking the entire cabin. \"I'll get rid of him. Stay with asshole,\" Caesar said.\n\nHe removed a .25 caliber automatic from his side pocket, snicked a round into the chamber, set the safety, and replaced the gun in his pocket. He opened the door and stepped into the front room. Tito Dellacroce stood behind me, one huge hand resting on my shoulder, the lower portion of his stomach touching the back of the chair. I could hear him breathing and smell the food he had eaten for supper on his skin. Caesar had left the door between the rooms ajar so Tito could listen.\n\n\"What can I do for you, Father?\" I heard Caesar say.\n\nThe reply was muffled, a wheezing sound, like a man speaking through a rusty clot in his windpipe.\n\n\"What's that?\" Caesar said.\n\nThe priest tried again, his voice barely a whisper.\n\n\"You're signing up people for a retreat?\" Caesar said. \"No, we belong to a church in Florida. We're just doing some fishing. Here's five bucks for your missions or whatever. No, I don't need no holy card.\"\n\nThe priest spoke again.\n\n\"We ain't got a bathroom. Just a privy out back no white person would want to slap his keester on. Try the filling station up on the state road. Okay, vaya con dios. That's Latin for 'see you around,' right?\"\n\nA moment later Caesar came back through the door that separated the two rooms of the cabin.\n\n\"So?\" Tito said.\n\n\"So nothing. The guy had a tracheotomy or something. He sounded like all his gas was coming out the wrong end,\" Caesar said.\n\n\"Check.\"\n\n\"On what?\"\n\n\"On where he is. I got to draw a picture on your forehead?\"\n\n\"You worry too much,\" Caesar said irritably, and jerked the window curtain aside again. Then he froze. \"I told him not to go back there.\"\n\n\"Go back where?\" Tito said.\n\n\"To our privy. I told him not to do that.\"\n\n\"Give me your piece. Get away from the window,\" Tito said.\n\nThe wind gusted off the water, stressing the tin roof against the joists. Then someone stepped onto the back porch. Tito jerked the .25 caliber automatic from his brother's hand and clicked the safety off with his thumb. \"Is that you, Father? 'Cause if it is this is getting to be a headache we don't need\u2014\"\n\nThe door burst open and, framed against the light, dressed in a black suit and Roman collar and black rabat, was a compact, well-groomed man with a 1911 U.S. Army model .45 automatic in each hand.\n\n\"Oh, it's a darling pair we have here. Suck on this,\" he said. He began firing with both guns, shooting Tito in the mouth and through the throat, hitting his brother Caesar Dellacroce in the sternum and thigh.\n\nTito crashed into a wall and collapsed on his spine, his legs spread, his jaw torn loose from his head. Caesar tried to crawl away from the rounds that blew the sole of his shoe off his foot, tore through a buttock, and splattered blood off his shoulder in a horsetail on the floor.\n\nThe room was littered with ejected shell casings when Max Coll finally stopped firing. He nudged Tito in the chest with his polished shoe, satisfying himself that Tito was dead, then leaned down and studied Caesar's face. \"Oops, looks like you're still on board, little fellow,\" he said, and fired a round into the side of Caesar's head, stepping back to avoid the splatter.\n\nHe stood erect and took my measure, his cheeks rosy, a cleft in his chin slick with sweat. He pulled the sponge from my mouth. \"You all right, Mr. Robicheaux?\" he asked.\n\nMy heart was pounding, my ears almost deaf. \"Cut me loose,\" I said.\n\n\"Can't do that, sir. You're a copper through and through. You'd figure out a way to have me in cuffs for sure. Give my best to Father Dolan. He's a bit hard-headed, but under it all I think he's a fine man of the cloth. His kind make me proud I'm a Catholic,\" he said.\n\nAnd with that he was gone.\n\nFifteen minutes later three cruisers from the St. Martin Parish Sheriff's Department arrived at the fish camp, having been notified of my situation from a payphone by Max Coll.\n\n## Chapter 15\n\nOn Wednesday afternoon, after sleeping almost fifteen hours, I drove with Clete Purcel in his Caddy to City Park and sat under a barbecue pavilion in the rain on the banks of Bayou Teche.\n\n\"A guy pissed in your face?\" he said.\n\n\"No, first he pissed in my face. Then he pissed all over me,\" I replied.\n\nHe lit a Lucky Strike and spit a piece of matter off his tongue. A moment later he flipped the cigarette into the bayou and watched it float away. \"Don't let me light one of these again,\" he said.\n\n\"I won't.\"\n\n\"The Flannigan broad set you up,\" he said.\n\n\"I don't believe that.\"\n\n\"She got you out of your house and into a bar. What's that, working the Steps one drink at a time?\"\n\n\"It was my idea to go over there.\"\n\n\"Why? You got some big obligation to keep other people from drinking if they want to?\"\n\nI didn't answer. I tried to avoid his eyes. \"Are we talking about boom-boom out of times past?\" he said.\n\n\"Why don't you give some thought to the way you talk to other people, Clete?\"\n\n\"Did you ever get it on with her or not?\" he asked.\n\n\"Maybe.\"\n\n\"Maybe?\" He nodded profoundly. \"So after you made your expunch's father look like a vindictive prick in front of his friends, you don't think she would lure you to a slop chute in hopes you'd either get killed or drunk again? Perish the thought.\"\n\nI stared at the rain dimpling the surface of Bayou Teche. \"Theo isn't connected with people like Tito and Caesar Dellacroce,\" I said.\n\n\"Merchie worked for the Teamsters in Baton Rouge. They'd force guys to buy a union book, then get them fired after a month so they could crank up their membership numbers. That's how he got into the pipeline business.\"\n\n\"That doesn't mean he's mobbed up today.\"\n\n\"A guy who trucks oil waste into black neighborhoods? Not a chance. When I was a kid we had a rumble with the Ibervilles. It was supposed to be fist, feet, and elbows, no shanks, no chains. Merchie opened a switchblade and busted it off in my cousin's arm. In my opinion he's still a project street rat as well as full-time punk and gash hound. Quit defending these assholes.\"\n\n\"Gash hound?\" I said.\n\n\"Forget it, big mon. I don't want to talk about it anymore. Your head is encased in cement.\"\n\nI had long ago learned there was no point in arguing with Clete or expecting him to understand that the people he resented most were those who came from the same background he did. He pushed his porkpie hat down on his brow and stared disgustedly at the rain. \"I'm going to cripple the motherfuckers behind this, Dave. I mean that literally,\" he said.\n\nHe walked away under a dripping live oak toward his Caddy, his sports coat stretched to splitting on his huge shoulders.\n\nHe dropped me off at the house and I went inside and lay down on the bed in the back room. Earlier I said I had slept for fifteen hours. The truth is a little different. I could not rid myself of the sense of violation I had experienced at the hands of Tito and Caesar Dellacroce and the man who had urinated on me. I felt that soap could not cleanse my skin or my hair. When I closed my eyes and began to drift into sleep, I didn't dream of the Dellacroces but instead of a war few people are interested in today. I heard automatic weapons fire, the thropping of helicopter blades, and I saw strings of white light fountaining inside jungle foliage from the explosion of a phosphorus round. I felt a medic from Staten Island tying my wrists so I would not tear at the compress on my side. I smelled the odor of blood and feces in the uniforms of both the living and the dead being piled around me on the floor of an overloaded slick piloted by a nineteen-year-old warrant officer who had taken a steel splinter in one eye.\n\nSleep occurred in ten-minute intervals, and each time I awoke I wanted four inches of Black Jack straight up, vodka that had been at least twelve hours in a freezer, beer that hit the back of the throat like a spray of golden needles, yellow mescal with a thick green worm in the bottom of the bottle.\n\nAn hour after Clete had dropped me off I sat on the side of the bed with a head full of cobwebs, my mouth dry and tasting like bitters. Helen had told me not to come back to work until the following Monday. But memory was the enemy, and solitude and inactivity gave me no respite from it. I called N.O.P.D. and left a message for Clotile Arceneaux. A half hour later she called me back. \"What's happenin', baby cakes?\" she said.\n\n\"Baby cakes?\"\n\nI heard her laugh. \"What can I help you with?\" she said.\n\n\"What have you got on Merchie Flannigan?\"\n\n\"A pipeline or oil guy, grew up in the projects, did some time when he was a kid?\"\n\n\"That's the one.\"\n\n\"I'll check but I think he's pretty inactive.\"\n\n\"Clete thinks maybe Merchie and his wife might have been mixed up with the Dellacroce brothers.\"\n\n\"What about the Dellacroces?\"\n\n\"They're dead. Max Coll smoked them both.\"\n\n\"So much for inner-department communications. Coll killed them?\"\n\n\"He's posing as a priest and carrying a couple of .45 autos in a briefcase. Tito and Caesar Dellacroce abducted me. They took me to a fish camp not far from where Coll killed their cousin.\" It sounded foolish when I said it.\n\nShe paused a moment. \"What did they do to you at this fish camp?\" she asked.\n\n\"Nothing. Coll capped them.\"\n\nShe paused again and I could tell she didn't believe me. \"Let me give you a tip. Screw Max Coll and screw the Dellacroces. The issue is porn and crystal meth. Everything else is secondary. New Orleans was made for it. You with me on this?\"\n\n\"No.\"\n\n\"That's what I thought.\"\n\n\"Sorry to bother you,\" I said.\n\n\"Don't give me any of your guff, Robicheaux. You doin' okay over there?\"\n\n\"Why?\"\n\n\"'Cause you don't sound like it,\" she said.\n\nSo that's why she was undercover at N.O.P.D., I thought after I hung up. Some cops were probably on a meth pad and maybe the pornographers had gotten to a few of them, too. Porn had always been there, in one form or another, and sex and the economics of New Orleans tourism were longtime business companions. The Mob maintained they didn't traffic in porn, just as they claimed they didn't deal in narcotics. But they lied. They were involved in every pernicious enterprise in the United States, and decades ago had branched into shipping, the meat industry, and coal mining. The numbers racket used to be the lubricant that fueled and greased all their other machinery, but since state lotteries and legalized gambling had replaced numbers as their chief source of money, the progeny of Lucky Luciano and Benny Siegel had shifted gears to keep up with the times.\n\nNot only had the Internet provided huge new markets for porn producers, their businesses had a built-in edge on dope trafficking. They had the First Amendment to hide behind, and most zoning boards had no problem in allowing them to open their businesses in neighborhoods where the residents, usually the poor and elderly, had no power.\n\nThe overhead was low. Junkies, demented sluts, and perverts of every stripe couldn't wait to take off their clothes in front of the camera, convinced their acting careers were just beginning.\n\nThe subject of pornography brought to mind Fat Sammy Figorelli again. He had warned me about a man he said hurt people without cause, although Sammy, in his self-serving fashion, managed not to mention the man's name. Clete was right. I had given Sammy a free pass too long. I called Clotile Arceneaux again.\n\n\"I need a favor,\" I said.\n\n\"What kind?\"\n\n\"While my eyes were taped shut a guy urinated in my face. I think Fat Sammy Figorelli knows who he is.\"\n\n\"Say all that again?\"\n\nI did, this time in detail. She was quiet a long time. \"What do you want from me?\" she said.\n\n\"Help me jam up Sammy Fig.\"\n\n\"Can't do it.\"\n\n\"Why not?\"\n\n\"We think Fat Sammy might be talking to us soon.\"\n\n\"As an informant?\"\n\n\"Think FBI and Witness Protection.\"\n\n\"These guys were going to burn my kite, on film, one frame at a time. I'm not too interested in hearing about federal needs right now.\"\n\n\"Too bad. Stay in New Iberia, Robicheaux. That's not just a cautionary statement, either,\" she said.\n\nThat evening I took Clete to dinner at the Patio in Loreauville. After we ate we walked to the iron bridge over Bayou Teche and stared down at the water. The sky was crimson, full of birds, the air heavy with the smell of the sugar mills grinding cane. In the distance I heard a boat horn blowing on the water.\n\n\"I'm worried about you, noble mon,\" Clete said.\n\n\"You shouldn't.\"\n\n\"You fool lots of people. But you never fool your old podjo. Tell me I'm wrong.\"\n\nI couldn't, so I changed the subject. \"Fat Sammy knows who put the hit on me,\" I said.\n\n\"I told you he was a grease bag.\"\n\n\"I need to put the squeeze on him. N.O.P.D. was no help.\"\n\n\"You mean the black broad, what's-her-name, Clotile Whatever?\"\n\n\"She's got her own problems.\"\n\n\"Save the St. Francis of Assisi routine for another time. What's today?\"\n\n\"Wednesday,\" I said.\n\nClete put a stick of gum in his mouth and looked at the shadows the trees made on the bayou's surface. \"You really want to put a freight train up Sammy's cheeks?\"\n\n\"I couldn't have said it better.\"\n\n\"Remember Janet Gish? Used to be a dancer out on Airline?\" he said.\n\n\"What about her?\"\n\n\"She was Gunner Ardoin's costar in one of Fat Sammy's films. You like Italian opera?\"\n\nDuring the next two days Clete made several phone calls to New Orleans and was mysterious about all of them. But taciturnity in Clete, at least with me, usually meant he was working on a scheme that was so outrageous no sane person would involve himself in it. No one who reviewed Clete's record could doubt his creativeness when it came to spreading mayhem and chaos wherever he went. He not only shot a federal witness to death in a hog lot, he filled a New Orleans' gangster's vintage convertible with cement, destroyed a half-million-dollar home out on Lake Pontchartrain with an earth grader, pinned a hitman on the floor of the Greyhound depot's men's room and poured the contents of a liquid soap container down his throat, dropped a Teamster steward off a fourth-floor hotel balcony into a dry swimming pool, handcuffed a U.S. congressman to a fire hydrant on St. Charles, cuffed a dirty cop to the conveyor chain in a car-wash and hot-wax machine, and was believed to have put sand in the fuel tank of an airplane that crashed and exploded in the mountains of western Montana, stringing the spruce trees with the remains of several Galveston and Las Vegas mobsters.\n\nHe considered his own behavior perfectly reasonable and did many of the above deeds and others that were worse with a lopsided grin on his face, thinking them hardly worthy of mention.\n\nHis best friends were drunks, grifters, and brain-fried street people, his girlfriends strippers and junkies. Gangbangers, pushers, strong-arm robbers, and dirty cops crossed the street when they saw him coming. He swallowed his blood and ate his pain and never flinched in a fight, no matter what his adversaries did to him. He was the bravest and most loyal man I ever knew, and also the most irreverent, reckless, irresponsible, and self-destructive.\n\nI tried not to think of how Janet Gish could be a player in Clete's plan to jam up Fat Sammy Figorelli. Friday evening I found out.\n\nHe told me to meet him in Metairie, in front of a rented hall on the edge of a middle-class neighborhood. Metairie had become a white-flight refuge during the mass exodus from New Orleans in the 1970s, known for its strict law-and-order attitudes and the distinction of having given David Duke his start in the state legislature.\n\nI waited for Clete in the parking lot, the sky ribbed with strips of pink cloud, the trees ruffling in the yards of the modest homes beyond a shopping mall, the rental hall filling with families dressed as though they were going to church. The scene made me think of Levittown but not in a bad way. The rental hall, with its gravel roof and artificial brick shell, seemed to transcend its own cheapness, like an excursion back into an earlier era when American neighborhoods had sidewalks and were defined by their sense of community and generational continuity.\n\nI looked again at my watch. Where was Clete? The light was fading, the air growing cold. From inside the hall I could hear someone adjusting the volume on a microphone. Then I saw Clete's lavender Cadillac coming hard down the street, the front and back seats packed with people, slowing down for a stop sign just before he bounced into the parking lot, dust and exhaust fumes rising like a dirty halo from the car frame. When he cut the engine the entire car body seemed to gasp and shrivel like an animal that had been mortally wounded. The windows were open and I could smell a heady, thick odor, like burning leaves, drifting out on the wind, then someone flicked a marijuana roach sparking onto the pavement.\n\nClete got out of the car and closed the door behind him, then leaned down to the window. \"Crack open another six-pack and go easy on the stash. I'll be right back,\" he said.\n\n\"Where's the fucking opera? You said we were gonna see an opera,\" a woman in back said.\n\n\"I've got reserved seats. Trust me. Just be cool. Everything's copacetic,\" he replied.\n\nHe walked past me, so I would have to follow him, out of earshot of the people in the car. He lifted his shirt off his chest and sniffed at it. \"Do I smell like a whorehouse?\" he asked.\n\n\"What's going on?\" I said.\n\n\"Fat Sammy belongs to this group of amateur opera singers. They perform once a month at the hall. It's Ozzie and Harriet night by way of Palermo. The archbishop is a big fan and sits up on the front row. Starting to get the picture?\"\n\n\"No.\"\n\n\"You want to squeeze Fat Sammy, forget conventional methods. Sammy's a geek and closet pervert who always wanted people to like him. So he comes out here and pretends he's a normal member of the human race. That's about to end.\"\n\n\"Who's in the car?\"\n\n\"Janet Gish and Big Tit Judy Lavelle and four others who got bonds with Willie Bimstine and Nig Rosewater. Either Sammy gives up the guy who put the whack on you or I'm marching all of them right up the front aisle and turning them loose.\"\n\n\"This doesn't sound too good, Clete.\"\n\n\"Oh, Sammy Fig as victim, I forgot. Every one of those broads has worked in either his porn films or his massage parlor. Ask them how they like giving twenty-dollar blowjobs to conventioneers from Birmingham.\"\n\nI walked back to the Cadillac and looked inside. \"How y'all doin'?\" I said.\n\n\"Hey, Robicheaux, Clete say you taking us to supper at Galatoire's,\" a black woman in shades said. She called herself Cody Wyoming, although she had grown up on Prytania Street in New Orleans, not far from where Lillian Hellman was born.\n\n\"He hasn't filled me in on that yet,\" I replied.\n\n\"You might be getting old, Streak, but I bet you still got the thrust under the hood,\" she said. Everybody in the car roared.\n\nI walked back to Clete. \"Galatoire's?\" I said.\n\n\"Nig and Willie owe me a thousand for running down a skip in Mobile. Except they say they don't owe me anything because I told Willie to write the bond on this guy when I knew he was mainlining six balloons a day. So I told them they pay for the dinner at Galatoire's, I tell the girls it's on Willie and Nig, which means they'll tell all the other hookers in New Orleans Willie and Nig are great guys, and we call it even.\"\n\n\"I don't think this is going to work.\"\n\n\"It'll work. Ever hear that story about Sammy taking a girl to the Prytania and a bunch of kids in the balcony hitting them with water bombs made from condoms? I was one of the kids in the balcony. I guess I'm sorry for what we did, but that's the way it was back then. Come on, Streak, this is the life we chose.\"\n\nOn that note I walked through the double doors of the hall into the heart of middle America, cloistered, far from the inner city, passenger jets decelerating overhead as they approached the airport, a bustling shopping mall close by, and a freeway streaming with headlights to reassure everyone God was in His heaven and all was right with the world.\n\nClete had not lied to Janet Gish and her friends about reserved seating. Eight folding metal chairs in the front row remained empty, a program resting on the seat of each one. Otherwise the house was packed. Sammy Figorelli stood resplendent on the stage with his fellow singers, beaming, stuffed inside a summer tux, the footlights surrounded by bouquets of plastic flowers. Clete took out his cell phone and pushed a button on the speed dial.\n\n\"I'm down in front, Janet. I'll wave to you when I'm sure we've got the right seats. Yeah, wait for me to wave. It's mass confusion here,\" he said, and clicked off his phone.\n\nBy now Sammy had seen us and was watching us out of the corner of his eye while he tried to hold a conversation with the other singers. Clete mounted the wood steps that led onto the stage as though he were part of the production, stepping carefully over the plastic flowers clumped around the footlights. \"Got a second, Mr. Figorelli?\" he said.\n\nFat Sammy walked toward him, his eyes like hot BBs. \"What do you think you're doing, Purcel?\" he asked.\n\n\"Check out the ladies in the doorway at the back of the hall. They've been doing a little weed, so I hope they don't get too giggly,\" Clete replied.\n\nSammy stared at the back of the hall like a man witnessing the erection of his own gallows. His cheeks bladed with color and pinpoints of sweat popped on his forehead. He labored down the steps, forcing Clete to follow him. \"You get rid of them people,\" he said hoarsely.\n\n\"And miss the reception afterwards? You kidding? Can we get introductions to the archbishop?\" Clete said.\n\n\"What are you after?\" Sammy said, his breath coated with funk.\n\n\"Give us the name of the guy who sicced the Dellacroces on Dave.\"\n\nSammy's face was shiny with a greasy film now, his boutonniere like a red wound on his jacket. \"You got no right to do this to me, Purcel,\" he said.\n\n\"I'm counting to three, then waving Janet Gish into action.\"\n\n\"The guy's out there now, you dumb Mick.\"\n\n\"Where?\" Clete said, twisting his head to survey the crowd.\n\n\"Don't do that. You're gonna get me clipped,\" Sammy said.\n\n\"I don't see anybody out there I know. Do you, Dave?\"\n\n\"We're done here,\" I said.\n\n\"No, no. Sammy's going to give us a name,\" Clete replied, waving a finger.\n\n\"Sammy's going down with the ship. Right, Sammy?\" I said.\n\nBut Sammy Fig's embarrassment was such he could no longer speak. In fact, I thought he was on the edge of having a coronary attack. The fatty layer under his chin trembled, his chest heaved, and sweat ran like hair oil into his shirt collar. I was convinced, at that moment, that inside every adult human being the child was still present, in this case an obese little boy struggling to free himself from the metal coils of a tuba while a packed football stadium laughed at his discomfort.\n\n\"We're going to boogie. Tell the guy who pissed on me I'll be looking him up,\" I said.\n\n\"You already burned me. Y'all don't know what you've done,\" Sammy said.\n\n\"That's the breaks. Anything else happens to Dave, I'm going to see you first. That means you're going to be the deadest douche bag in New Orleans,\" Clete said, jabbing Sammy in the chest with his finger.\n\nWe left Sammy standing numb and shaken in front of his audience and rounded up Janet Gish and her friends and headed for Galatoire's on Bourbon Street.\n\nOn the way out of the rental hall I searched the crowd for a familiar face, one that might belong to the man who had crisscrossed me from head to foot with his urine. But if he was there, I did not see him.\n\n\"You blew it, Dave. Fat Sammy would have cracked,\" Clete said later.\n\n\"What did Sammy do when you and your friends threw waterbomb condoms at him and his girlfriend?\" I said.\n\nWe were coming out of Galatoire's, into the pre-Christmas holiday atmosphere of late-night Bourbon Street. The street was loud with music, the neon like purple and pink angel hair inside the fog blowing off the river. \"He cried and came at us with both fists,\" Clete said.\n\n\"He's still the same kid.\"\n\n\"All of us are. Except Fat Sammy became a pimp and dope pusher. It's only rock 'n' roll, Dave. Everybody dies. Go with the flow and try to have a few laughs,\" Clete said. He propped his shoe on a fire hydrant and buffed the tip with a cloth napkin he had taken from the restaurant.\n\n## Chapter 16\n\nI went back to work Monday morning. I took a legal pad from my desk drawer and wrote Junior Crudup's name at the top of it, then drew a circle around it. This is where it had all started, I thought, both for me and the LeJeune family. Under Junior's name I wrote the names of Castille LeJeune, Theodosha, Merchie, and Theodosha's psychiatrist in Lafayette, the man who supposedly committed suicide.\n\nThen I angled a line from Castille LeJeune's name to the names of Will Guillot and the dead daiquiri shop operator and Dr. Parks, who had died in Will Guillot's driveway.\n\nTo one side I placed the names of the New Orleans players\u2014Father Jimmie Dolan, Max Coll, the Dellacroce family, and Gunner Ardoin, the part-time porn actor.\n\nThe connections between the names and the deeds associated with them seemed byzantine on the surface, but for me the answers in the investigation lay in the past and the key was still the first name on the page, Junior Crudup.\n\nHelen opened my office door. \"The Lafayette Sheriff's Department just called. Get this,\" she said. \"The archdiocese is having a clerical conference of some kind. One of the out-of-towners happened to be an Irish priest. His jokes were a big hit. Then a pistol fell out of his shoulder bag in the lobby of the Holiday Inn.\"\n\n\"Our man Max?\"\n\n\"What's with this guy?\"\n\n\"He's nuts.\"\n\n\"That's the best you can do?\"\n\n\"Got a better explanation? Where'd he go?\"\n\n\"They don't know. They think he was driving a rental.\"\n\n\"He'll be back.\"\n\n\"You sound almost happy.\"\n\n\"He saved my life. Maybe he has redeeming qualities,\" I said, grinning at her.\n\n\"The guy who said 'suck on this' and blew away two people?\"\n\n\"It's only rock 'n' roll,\" I said.\n\n\"Fire your psychiatrist,\" she said, and closed the door.\n\nI studied the names and lines on my notepad. Years ago, after the murder of my wife Annie, I went twice a week to sessions with an analytically oriented therapist in Lafayette. He was one of those who believed most aberrations in behavior and personality development were caused by fairly obvious dysfunctions in the patient's environment. The problem in treating them, he maintained, was that they were so obvious the patient usually would not buy the connection between the cause and the problem.\n\nTheodosha had told me her husband, Merchie, was having what she called another flop in the hay and that she couldn't blame him for it. I took that to mean she had a sexual problem of her own, one that had sent her husband elsewhere. But I also remembered a remark our dispatcher Wally had made about Merchie Flannigan, as well as one made by Clete Purcel.\n\nI walked up front and leaned on the half-door that enclosed Wally in the dispatcher's cage. He was writing on a clipboard, the top of his head and his neatly parted, little-boy haircut bent down. His shirt pocket was stuffed with cellophane-wrapped cigars. \"What chu want, Dave?\" he asked without looking up.\n\n\"You told me Merchie Flannigan was a bum, that he was a guy you never liked. Let's clear that up,\" I said.\n\n\"So I got a big mout',\" he replied.\n\n\"This is part of a murder investigation, Wally. I'm not going to ask you again.\"\n\n\"He's got a wife, but he messes around on the side.\"\n\n\"A lot of men do.\"\n\n\"He was driving my wife's niece home. She was working at his office in Lafayette. She was seventeen years old at the time. He axed her if she wanted to go swimming at his club. It was late and the club was closed, but he said it didn't matter 'cause he had a key and the owner and him was golf buddies. She didn't have a suit, but he said that wasn't no problem 'cause they'd get one from behind the counter and put it on his tab.\n\n\"There wasn't no lights on in the pool when she came out of the dressing room. She started swimming back and fort' across the shallow end, then he come up to her and axed her if she could swim on her back. She said she always got water up her nose, and he says just turn over and rest on my hands and I'll show you how to do it.\"\n\nI waited for him to go on but he didn't.\n\n\"What happened?\" I said.\n\n\"He tole her how pretty she was, how she had to be careful about young boys only got one thing on their mind. She tole him she was cold and she better go back inside and get dressed. He said it was okay, they'd come back another time, that she was the prettiest girl he'd ever seen.\"\n\nHe stopped again, ticking his pencil on the clipboard, looking at nothing.\n\n\"That was it?\" I said.\n\n\"It was enough for her daddy. He was gonna go over to Flannigan's house and break his jaw but his wife hid the car keys. So the next morning he walked into Flannigan's office and made sure the door was open so everybody could hear it and tole him his daughter wouldn't be coming back to work no more.\"\n\n\"Thanks, Wally.\"\n\n\"What do I know?\" he said.\n\nA lot, I thought.\n\nI went back to my office and started in on the paperwork that had built up during the days I was off. The phone on my desk rang.\n\n\"Tell me what I'm hearing isn't true,\" the voice of Clotile Arceneaux said.\n\n\"I'm not too keen on rumors.\"\n\n\"Did you and your buddy Purcel brace Sammy Fig out in Metairie Friday night?\"\n\n\"Maybe.\"\n\n\"Some federal agents are seriously pissed off about this, as well as somebody else, meaning myself. What gives you the right to go into another jurisdiction and intimidate other people's witnesses?\"\n\n\"I don't read it that way.\"\n\n\"Well, read this. Sammy Fig thinks either I or federal agents gave you information that sent you over to Metairie. He says he'll no longer be cooperating with us and we can shove Witness Protection up our ass.\"\n\n\"That's the way it flushes sometimes.\"\n\n\"I love your metaphors. I even like you. But right now I'd like to push you off a tall building.\"\n\n\"Where's Sammy now?\"\n\n\"I left that part out, did I? We have no idea. Gone. My guess is he's gonna try to take it to them before they get to him first.\"\n\n\"Take it to whom?\"\n\n\"To whom? I love talking to cops who need to show me how educated they are. How would we know, since eighteen months of casework just got dumped in the toilet? You're something else, Robicheaux. I hope you come out of this all right, but remind me to be on vacation the next time I catch a case you're involved with. Did you and Purcel really take a bunch of hookers to Galatoire's?\"\n\n\"I think we've got a bad connection. Let me call you back later.\"\n\n\"Not necessary. I've had all the horse shit I can take in one day,\" she said.\n\nTop that.\n\nAt noon I signed out of the office and drove up the bayou to Hogman Patin's house. He was building a chicken coop under a pecan tree in his side yard and pretended not to see me when I turned into the drive. He slipped his hammer through a hole in a leather pouch on his belt, looking intently at his creation, then walked around the back of his house, out of sight.\n\nI left my truck on top of the oyster-shell drive, the engine ticking with heat, and followed him. He was sitting on his steps, his big hands cupped on his knees, the knife scars on his arms like the backs of worms that had burrowed under the skin. The sun's reflection wobbled brightly on the bayou's surface, but he stared at it without blinking. \"Ain't goin' to let the past alone, are you?\" he said.\n\n\"You have to confront it to get rid of it, Hogman,\" I replied.\n\n\"I done tole you almost all I know. Why don't you let it be?\"\n\n\"What happened to Jackson Posey, the guard who had to keep taking Junior up to Miss Andrea's house?\"\n\n\"Cancer eat him up. Heard he died at Charity Hospital in Lafayette. Died hard, too.\"\n\nI picked up a handful of moldy pecans from a shady, damp area and began chunking them into the bayou. \"You've never told anybody why you made a bottle tree in your backyard, have you?\" I said.\n\n\"Ain't nobody else's bidness.\"\n\n\"You're a religious man, Hogman. Each one of those bottles represents a different prayer. Every time the wind makes the glass sing in the branches, a prayer goes up from each of those bottles, doesn't it?\"\n\nHe lowered his eyes and pared one of his fingernails with a toothpick. \"What a man do in his home is what he do in his home,\" he said.\n\n\"You helping cover up a murder, Hogman.\"\n\n\"Ain't right you talk to me like that, Dave. No, suh.\"\n\n\"Maybe not. But why do you want to protect the LeJeune family?\"\n\n\"I ain't seen what happened after I left the camp. Cain't tell you about what I ain't seen. Don't want to tell you about what I ain't seen, either.\"\n\n\"Somebody saw. Somebody knows.\"\n\nHe breathed hard through his nose, his nostrils flaring in his frustration with me and his own conscience. The wind was cool and wrinkled the bayou's surface, and Hogman's bottle tree rang like spoons clinking on crystal. \"There's a man down at Pecan Island stacked time in the same camps as me and Junior. He was a check writer and used to carry the water can when we road-ganged. Him and his gran'daughter sell crabs and vegetables off a truck out on the state road. His name is Woodrow Reed.\"\n\n\"How does he feel about talking to a white man?\"\n\n\"He don't care what color you are. He climbed up on a power pole to get a cat down and got 'lectrocuted. His eyes cooked in his head. You'll t'ink he's looking at you but don't no light go t'rew his eyes. His eyes scare people. Maybe that's why ain't nobody ever been around axing Woodrow questions about what he seen.\"\n\nI drove back to New Iberia and on south of Abbeville, where sugarcane acreage gave way to sawgrass and clumps of gum trees and the miles of wetlands that bled into the Gulf of Mexico, forming the watery, ill-defined coastline of southwest Louisiana. I crossed a bridge onto one of the few remaining barrier islands left in Louisiana, a reef composed of hard-packed shell ground up by the tides, the crest topped with alluvial soil that is among the richest in the western hemisphere. The adjacent islands had been dredged and scooped out of the surf and hauled away on barges decades ago for highway-construction material, but portions of Pecan Island, preserved largely by an oil corporation as a recreational area for its CEOs, contains wooded acreage where the canopy of live oaks rises perhaps two hundred feet into the sky and the sunlight breaking through the moss and branches and air vines is the same color as light filtering through green water in the Florida Keys.\n\nIn the midst of duck-hunting camps with wide, screened-in porches and adjacent boathouses was the tiny vegetable farm and blue-point crab business of Woodrow Reed. Stacks upon stacks of collapsible wire crab traps, webbed with dried river trash, stood by the side of his small, paintless house. A middle-aged black woman was chopping up nutria parts on a butcher block a short distance away, the rubber gloves on her hands spotted with brown matter.\n\nWoodrow Reed's eyes were large, round and flat, unblinking, like painted facsimiles that had been cut out of paper and pasted on the face of a mannikin. They stared at me intently, the pupils dilated and black, although it was obvious Woodrow Reed was sightless.\n\n\"I'm Dave Robicheaux, with the Iberia Parish Sheriff's Department,\" I said. I opened my badge holder and held it aloft so the middle-aged woman in the side yard could see it.\n\n\"I knowed you was coming,\" he said, rising from where he sat on the front steps.\n\n\"Hogman called you?\" I said.\n\n\"Yeah, but he didn't have to. I knowed somebody was coming one day. Want to come in, suh?\" He opened the rusted screen door to his front porch and waited for me to enter.\n\nHe could not have been over five feet. His skin was the color of a razor strop that has yellowed with wear, his body compressed and hard looking, his cheeks and chin scrolled with gray whiskers. But I could not get over his eyes. I had seen eyes like his only once before, in the body of a man who had been exhumed from a grave in northern Montana where he had lain for decades under frozen ground.\n\n\"How'd you come by your farm, Mr. Reed?\" I asked.\n\n\"You already know the answer to that.\"\n\n\"Can you tell me how Junior Crudup died?\" I asked.\n\nWoodrow Reed was sitting on what looked like a motion-picture theater seat mounted on a wood block, his palms propped on his thighs. His denim pants were neatly pressed, the cuffs and pockets buttoned on his long-sleeve work shirt.\n\n\"The doctor give me another year. I already put my farm in my daughter's name. Ain't a whole lot can touch me no more. I got cancer, just like Jackson Posey, although I never smoked like he did or had no problems with my skin,\" he said.\n\n\"Tell me about Junior, sir.\"\n\n\"Junior was gonna be Junior. He didn't wear no other man's hat. That was Junior,\" he said. For the first time he smiled.\n\nIn the waning days of summer, when the amber light at evening turned the countryside into a yellowing antique photograph, Junior Crudup took his twelve-string Stella guitar out on the steps of the cabin in the work camp and began composing a song whose lyrics he penciled on a paper bag flattened down on the board plank beside him.\n\n\"What you calling your song?\" Woodrow asked, sitting down next to him in the dusk.\n\n\"'The Angel of Work Camp Number Nine,'\" Junior replied.\n\nWoodrow rubbed the whiskers that grew like black wire on his chin. \"T'ink that's a good idea, Junior?\" he asked.\n\n\"Gonna record it up in Memphis one day. You gonna see,\" Junior replied.\n\n\"I seen her car out here last night. Parked right there on the road. She was smoking a cigarette behind the wheel and playing the radio in the dark.\"\n\n\"You better not be fooling with me, Woodrow.\"\n\n\"It was her. Cap'n Posey walked up to her window and axed if anyt'ing was wrong. She said she was just taking a drive. Then she drove on down the road toward the li'l sto' by the bridge. A li'l while later I seen her drive on back to the big house. She was drinking a bottle of beer, tilting her chin up each time she took a sip.\"\n\n\"Why didn't you come get me?\"\n\n\"You spent too much time up Nort', Junior. You're having t'oughts ain't no nigger in Lou'sana ought to be having.\"\n\n\"Maybe it was that way at first. But not now. You know what she got that make her special?\"\n\n\"Her tits ain't bad.\"\n\n\"Don't be talking that way, Woodrow. She's special 'cause she got respect for other people.\"\n\nJunior adjusted the belly of his guitar on his thigh and slipped his three steel finger picks on his right hand, then corded the neck of the guitar and began singing:\n\nAt Camp Number Nine it's \"Roll, nigger, roll,\n\nNo heaven for you, boy, the state own your soul.\"\n\nThey took my home and family,\n\nGive me chains, fatside, and beans,\n\nBossman making me a Christian,\n\nGod Almighty, hear that Betty scream.\n\n\"You risking your ass for somebody don't know you alive,\" Woodrow said.\n\n\"Rich ladies like that got all kinds of things they got to do, places they got to travel to, Woodrow. She cain't be coming down here all the time.\"\n\n\"Don't let Boss Posey hear that song.\"\n\n\"When she invites me back up to the house?\" Junior said.\n\n\"Yeah?\"\n\n\"That's the first song I'm gonna play.\"\n\nThere was drought in the fall and the fields hardened and cracked under a merciless sun and an empty sky that by noon was like white glass. The leaves of the cane baked in the wind and frayed into thread on the ends and rattled dryly on the stalks, and by evening the sky was cinnamon colored with dust and the convicts filling mule-drawn water tanks with buckets they flung into the bayou on ropes had to tie wet handkerchiefs across their nostrils and mouths. To conserve water the convicts bathed in the bayou, then sat listlessly on the porches of their cabins until lock-up. Every third or fourth evening, while the cicadas sang in a grove of cedar trees near the camp, Junior worked on the song he was composing in tribute to Andrea LeJeune, waiting for the invitation to play on her lawn again, telling himself she was contacting the governor and that any day a parole order for his release would be delivered at the camp's front gate.\n\nAt bell count on a September morning Jackson Posey saw the folded brown paper sack covered with penciled lyrics sticking from Junior's back pocket.\n\n\"What you got there, Junior?\" he asked.\n\nThe early sun was already a dull red inside the dust blowing out of the fields. At the bottom of the slope that led down to the bayou, the water was low and swarming with gnats, algae-webbed snags protruding from the surface, all of it smelling of dead fish that lay bloated and fly-specked on the banks.\n\n\"Just li'l notes I keep for myself, boss,\" Junior replied.\n\n\"Let's see it,\" Jackson Posey said, fitting a pair of glasses on his nose. He took the bag from Junior's fingers and studied the words on it, his lips moving slightly as he read. The sores on his arms seemed deeper, more black than purple now. His eyes fixed on Junior's. \"You got Camp Number Nine in here?\" he said.\n\n\"Yes, suh.\"\n\n\"Camp Number Nine is us.\"\n\n\"It is and it ain't, boss.\"\n\nThe guard read both sides of the paper bag, then shook a Camel loose from his cigarette pack and slipped it into his mouth. He laughed to himself and handed the song lyrics back to Junior. \"I ain't a big judge of poetry, but I'd say keep this one.\"\n\n\"Thank you, suh.\"\n\n\"To wipe yourself with. You never cease to entertain me, Junior,\" Posey added.\n\nAt morning bell count two days later Andrea LeJeune got out of her Ford convertible at the camp's front gate, wearing a polka-dot sun dress and dark glasses and a blue bandanna tied tightly on her head, the wind whipping her dress around her legs.\n\n\"We're taking Junior to a recording studio in Crowley, Mr. Posey. Make sure he brings his guitar and his harmonica and a sack lunch. Y'all will follow me in your truck,\" she said.\n\nJackson Posey involuntarily looked toward the big house. \"Mr. LeJeune at home, ma'am?\" he asked.\n\n\"No, he's not, and I resent your asking,\" she replied.\n\nJunior wrapped his Stella in a blanket, tied string around the belly and the neck, and slipped his E-major Marine Band harmonica in his shirt pocket. Before they left the camp, Posey put chains on Junior's ankles and handcuffs on his wrists, and set the guitar in the bed of the truck. As they drove away Junior looked out the back window at his friend Woodrow flinging a bucket into the bayou on a rope under the gaze of a mounted gunbull.\n\nThen Junior and Jackson Posey were on the highway, driving through a long tunnel of oak trees behind Andrea LeJeune's purple convertible, the broken sunlight flicking by overhead, the wind cool in their faces.\n\n\"You gonna make the big time, huh?\" Posey said.\n\n\"Don't know about that, suh.\"\n\n\"Think it's coincidence she's taking you to Crowley?\"\n\n\"I ain't following you, boss.\"\n\n\"That's where she meets a man I wouldn't take time to spit on. Castille LeJeune should have invested some of his money in a chastity belt. Know the difference between rich people and us?\" Posey said.\n\n\"No, suh,\" Junior answered.\n\n\"They don't get caught.\"\n\nWhen they pulled into the Crowley town square Andrea LeJeune parked her car next to one of the old elevated sidewalks and went inside the dime store, one with a popcorn machine in front, to use the pay telephone. Then they drove out into the countryside again, through rice fields that were separated by hedgerows, to a white-painted, flat-top building constructed entirely of cinder blocks that was located inside a grove of cedar and pine trees like a machine-gun bunker.\n\nThis was the same primitive studio where a few years later Warren Storm and Lazy Lester would record and Phil Phillips would cut the master for \"Sea of Love,\" which would sell over one million copies. The equipment was prewar junk, the resonator for Junior's acoustic Stella a chunk of storm sewer pipe with a microphone on the other end. But each person working in the studio knew who Junior Crudup was, and his identity as both a black man and a convict seemed to melt away as the session progressed.\n\nHe recorded eight pieces, the last of which was \"The Angel of Work Camp Number Nine.\" As he sang the lyrics he looked through a greasy side window and saw her by the front fender of her convertible, talking to a tall white man who had just gotten out of an Oldsmobile with grillwork that resembled chromium teeth. The white man was thin, dark haired, his crisp shirt tucked tightly inside his seersucker slacks. He rested one foot on the bumper of his car and removed a blade of grass from the tip of his two-tone shoe, then took his car keys from his pocket and inserted his finger through the ring and spun them in the air.\n\nHe drove away toward town in his Oldsmobile and Andrea LeJeune followed him. Junior's voice broke in the middle of his song and he had to start again.\n\nLater, Junior and Jackson Posey rode back through the town square of Crowley, past the colonnaded storefronts and tree-shaded elevated sidewalks inset with iron tethering rings, past the dime store with a popcorn machine in front from which Andrea had made a phone call.\n\nJunior was hunched forward on the seat, his wrists cuffed, the chain between his ankles vibrating with the motion of the truck, his expression concealed from Jackson Posey.\n\n\"I'll show you something,\" Posey said, and cut down a side street and out onto a state road, past a shady motor court that featured a swimming pool in back and a supper club in front. Posey slowed the truck so he and Junior could have a clear view of the stucco cottages inside the trellised entrance.\n\n\"Don't need to be seeing none of this, boss,\" Junior said.\n\n\"There's his Oldsmobile. There's her little Ford. What do you reckon he's doing to her right now?\"\n\nJunior stared at the tops of his cuffed hands and did not speak again until they were back at the camp.\n\nBut his day was not over. Just after supper Jackson Posey came for him again. \"She wants to see you,\" he said.\n\n\"Wore out, boss.\"\n\nHe was alone, sitting on an upended Coca-Cola box in the corner of the dirt yard, next to the fence topped by five strands of barbed wire tilted back at an inward angle, his guitar still wrapped with a blanket and tied with string on top of his bunk inside. The sun was only a smudge on the western horizon and the lilac-colored sky throbbed with the droning of cicadas.\n\n\"Get your skinny ass up before I kick it up between your shoulder blades,\" Posey said. \"One other thing?\"\n\n\"What's that, boss?\"\n\n\"You tell her I drove you past that motor court today, I'm gonna take you out to a stump, nail your balls to it, and leave you there with a knife. Ain't storying to you, Junior. I seen my daddy do it when I was a boy,\" Posey said.\n\nBut Junior did not get up from the Coca-Cola box. \"I ain't playing no more today,\" he said.\n\nPosey raised his fist and knocked him to the ground.\n\n\"Whup me or put me on the bucket. I ain't going to play no more,\" Junior said.\n\n\"I don't have to whup you. I'm gonna do it to Woodrow Reed instead,\" Posey said.\n\nOn the way to the house of Castille and Andrea LeJeune, Junior wondered what he had done in this world to earn the grief that seemed to be his daily lot.\n\nHe waited on the patio with his guitar and harmonica for Andrea LeJeune to come downstairs and through the French doors. When she emerged she was still wearing the polka-dot dress she had worn earlier. Her face looked haggard, somehow thinner in the evening light.\n\n\"I wanted you to know the producer at the studio called to say how thrilled he was. I'm just sorry I didn't get to hear you perform,\" she said.\n\n\"I understand, ma'am,\" he replied.\n\n\"I have to go away, Junior. But I'm going to do everything I can to see you released from prison. What happened to your head?\"\n\n\"Fell down the steps,\" he replied, his face empty.\n\nShe gave a long, hard look at Jackson Posey standing by the pickup truck in the driveway. \"Come in the house,\" she said.\n\n\"That ain't a good idea, Miss Andrea,\" Junior said.\n\nShe walked to the edge of the drive. \"Mr. Posey, Junior is coming into the living room for a few minutes. We're not to be disturbed,\" she said.\n\n\"I cain't allow that, ma'am.\"\n\n\"You can't what?\" she said.\n\nShe stared him down, then turned on her heel and marched inside her house, curling one finger for Junior to follow her.\n\n\"Sit down,\" she said.\n\n\"Miss Andrea, Boss Posey ain't an ordinary man,\" Junior said.\n\n\"I'm going to call every week and have someone check on you. You have nothing to be afraid of.\"\n\n\"It don't work like that.\"\n\nShe sat down in an antique chair with an egg-shaped crimson pad inset in the back and folded her hands in her lap. \"The producer said you wrote a song called 'The Angel of Camp Number Nine.' Is that about me?\"\n\nHe hesitated, then said, \"Yes, ma'am, I reckon it is.\"\n\n\"That's one of the most touching compliments I've ever received. I'd appreciate it very much if you'd play it.\"\n\nHe slipped the guitar over his neck and began to sing:\n\nWhite coke and a red moon sent me down,\n\nJudge say ninety-nine years, son, you Angola bound,\n\nIt's the Red Hat Gang from cain't-see to cain't-see,\n\nThe gunbulls say there the graveyard, boy,\n\nIf you wants to be free.\n\nLady with roses in her hair come to Camp Number Nine,\n\nSay you ain't got to stack no mo' Lou'sana time,\n\nGonna carry you up to Memphis in a rubber-tired hack,\n\nBuy you whiskey, cigars, and an oxblood Stetson hat.\n\nMiss Andrea is an angel drive a li'l purple car,\n\nLive on cigarettes, radio, and a blues man's guitar\u2014\n\nEven before he looked through the front window and saw the automobile of Castille LeJeune approaching the house, he knew there was something terribly wrong. Andrea LeJeune's face seemed repelled, as though someone had touched it with a soiled hand.\n\n\"You don't need to sing anymore,\" she said.\n\n\"Ma'am?\"\n\n\"What you've done is very nice, but I don't think this song needs to be recorded.\"\n\n\"I don't rightly understand,\" he said.\n\n\"This particular composition would probably be better deleted from your recording session. I think that's clear enough, isn't it?\"\n\nHe felt his mouth pucker as though a nerve ending had been cut in his face. From outside he heard a car door slam, then footsteps on the gallery. He lowered his eyes. \"Why ain't it supposed to be recorded?\" he asked.\n\n\"I don't think I should have to explain that to you,\" she replied.\n\nHis throat felt as though he had swallowed a handful of needles. \"I'm ready for Boss Posey to take me back now,\" he said. He pulled the Marine Band harmonica from his shirt pocket and set it on a flower-patterned couch by the French doors.\n\n\"I'm not in the habit of having people return gifts to me,\" she said.\n\n\"I'd really appreciate it, ma'am, I mean appreciate more than anything else in the world, if you could just yell at Boss Posey for me, tell him I'se on my way,\" Junior said.\n\nJust then Castille LeJeune opened the front door and walked into the living room, a Panama hat hanging from his fingertips, his mouth twisted in an incredulous smile.\n\n\"Please explain it to me, or I'll have to conclude I've either lost my mind or walked into the wrong house,\" he said.\n\nI heard the cell phone ring on the front seat of my truck. I went outside and picked it up.\n\n\"Where are you?\" Helen Soileau's voice said.\n\n\"Pecan Island.\"\n\n\"What are you doing at Pecan Island?\"\n\n\"Interviewing a man who did time with Junior Crudup.\"\n\nShe exhaled her breath into the phone. \"We've got a submerged car in West Cote Blanche Bay. The driver's still in there. A witness says he heard firecrackers going off before the car went into the water. Then the car drove off a pier.\"\n\n\"How about sending someone else?\"\n\n\"Dave, your separate itinerary ends right now. Get your butt over there.\"\n\n\"Soon as I can,\" I said.\n\n\"Not good enough.\"\n\nI turned off the ringer on the cell and went back inside to finish my interview with Woodrow Reed.\n\n## Chapter 17\n\nMr. LeJeune and Miss Andrea had a big fight that night,\" Woodrow said.\n\n\"How do you know?\"\n\n\"My cousin was the maid. She tole me later, that was after I was out of the joints, she tole me Mr. LeJeune went crazy that night. He picked up Miss Andrea's clothes off the flo' and smelled them.\"\n\n\"He did what?\"\n\n\"He smelled her clothes and knowed she was messing around on him. He was yelling all over the house, saying his wife went to bed wit' a nigger. My cousin was so scared she run out the do' and hid in the trees down by the bayou. She said Mr. Castille come crashing out of the house and drove his car down to the work camp.\"\n\n\"Looking for Junior?\"\n\n\"No, suh. He was after Boss Posey. A man like Castille LeJeune don't go after a nigger convict. It was Boss Posey he took it out on.\"\n\n\"I don't understand. Jackson Posey knew Junior was innocent, that Andrea LeJeune was having an affair with a man in Crowley.\"\n\n\"What was Boss Posey gonna say? 'Your wife been sleeping wit' another white man and I knowed about it and I ain't said nothing'? Boss Posey was caught, just like Junior. Boss Posey was gonna save his job and his ass only way he knew how.\"\n\nWoodrow Reed stopped his account, his hands fixed rigidly on his thighs, staring at me with his flat, sightless eyes. The pupils were overly large, like black dimes, as though they contained thoughts and remembered images that were bursting inside his head.\n\n\"Save his ass how, Woodrow?\" I said.\n\n\"I got great shame about this, Mr. Robicheaux. The story of Judas ain't only in the Bible. Thirty pieces of silver can come to you in lots of ways.\"\n\nHe looked at me a long time while fireflies sparked in the darkness outside and moths thudded softly against the screens, then he told me the rest of it.\n\nTwo weeks passed at the camp, and still there was no rain, only heat and dust blowing from the fields and dry lightning at night and the rumble of distant thunder over the Gulf. Cigarettes thrown from automobiles and pickup trucks started roadside grassfires that spread into the cane, and after sunset Woodrow and Junior sat on the front steps of their cabin and watched the dull red glow inside the clouds of brown smoke on the horizon.\n\nJunior no longer played his guitar or sat in on bouree games or sassed the guards. Until lock-up he loitered in the corners of the yard, or sat on his upended Coca-Cola box, which everyone now called \"Junior's box,\" or sat on the steps with Woodrow, staring at the empty dirt road that led down to a small general store by the drawbridge.\n\n\"You tearing yourself up over somet'ing that was never real,\" Woodrow said. \"Miss Andrea is a nice white woman. But that's all she is. She ain't sent down by God to take care of Junior Crudup.\"\n\n\"Shut up, Woodrow,\" Junior replied.\n\n\"Sure, I can do that. Then you can talk to yourself 'cause everybody else around here t'inks you done lost your mind.\"\n\nWoodrow took a worn pack of playing cards out of his shirt pocket, shuffled them, then cupped and squared them in his palm. \"Here, I'm gonna give you one of my readings. Won't cost you a cent,\" he said.\n\n\"Don't be giving me none of your truck,\" Junior said.\n\nBut Woodrow went ahead and turned the cards over one at a time, placing them in a circle in the space between him and Junior. \"See, there's you, the one-eyed Jack. Slick, wit' a li'l thin mustache, got the mojo going on the rest of the world. Up top there is the queen of hearts. Guess who that is. Over here is the king of diamonds. Guess who that is. Notice the king and the queen ain't interested in whether the one-eyed Jack is playing pocket pool wit' himself or not. What that mean, Junior, is that rich white people don't care about what goes on down here in this camp.\"\n\n\"Ain't got time for this, Woodrow.\"\n\nWoodrow peeled three more cards off the deck and snapped them down in a vertical line traversing the circle. \"See, there's the joker, right over the head of the one-eyed Jack. That means our man, the one-eyed Jack, is a full-time fool. Sure you don't want to rename your song 'The Dumbest Nigger in Camp Number Nine'?\"\n\nBut Junior only stared at the fires and brown clouds of smoke on the horizon and the buzzards that were slowly descending in a vortex toward a woods on the far side of the bayou.\n\nWoodrow put three cards down on the step in a horizontal line, completing a cross inside the circle. Junior expected another ridiculing remark but instead there was only silence. He glanced sideways at Woodrow. \"Why you got that look on your face?\" he said.\n\nWoodrow started to scoop the cards up. But Junior held his wrist. \"Answer me, Woodrow,\" he said.\n\n\"It's just a card trick. Been playing it on people for years. Don't none of it mean anyt'ing,\" he replied.\n\nJunior peeled loose a card that was cupped inside Woodrow's palm. \"How come you trying to hide the Jack of spades?\" he asked.\n\nWoodrow rubbed one eye with the heel of his hand and stared sadly at the bayou. \"It's Boss Posey, Woodrow. Lawd Gawd, it's Boss Posey. Why you gone and done this to yourself?\" he said.\n\nThen he rushed away to be by himself, leaving his deck of cards scattered on the steps.\n\nThe next day Junior received a contract in the mail from the recording studio. He sat on the edge of his bunk and read the letter that accompanied it, then walked to the fireplace and held a match to the letter, the contract, and the envelope they came in and watched the pages blacken and curl into ash on the hearth. The next morning at bell count Junior stood unshaved and dirty in the front row of men who were about to go into the fields to trench firelines around unburned cane and shovel dirt over stubble that was still smoldering. Jackson Posey looked at the puffiness around his eyes and sniffed at his breath. \"Where'd you get the julep?\" he said.\n\n\"Don't remember, boss,\" he replied.\n\n\"Woodrow, run back to the shed and bring me a case of them empty pop bottles,\" Posey said.\n\nWoodrow started toward the rear of the camp.\n\n\"I said run, boy.\"\n\n\"Yow, boss,\" Woodrow said.\n\nHe ran to the shed and lifted a wood case of Royal Crown Cola bottles by the handles and closed the door behind him with his foot, the bottles clinking between his hands. Then, as though a choice lay before him that would forever define who he was and the place he would inhabit in the world, he hesitated. On the perimeters of his vision he could see the LeJeune home high up on the slope, built to resemble a steamboat, surrounded by live oaks and palm trees; he could see a bulldozer and scooped out hole between the camp and the house where a damaged gas storage tank had just been removed; he could see the soot and brown smoke blowing out of the fields, the buzzards circling in the sky, the barbed wire that surrounded the camp, the tin roofs of the cabins already expanding against the joists with the heat of the day, the hard-packed clay smoothness of the yard, the gunbulls and trusty guards already mounted on their horses, most of them armed with double-barrel, cut-down shotguns whose steel was the color of a worn five-cent piece, and in the midst of it all, Woodrow's best friend, Junior Crudup, drunk on julep made from yeast, raisins, and cracked corn boiled in a syrup can, about to be destroyed by his own pride.\n\nDrop the bottle case on the ground, he told himself. Let them ship you back to 'Gola. Do cain't-see to cain't-see on the Red Hat Gang, take the sweatbox treatment on Camp A, but don't hep them to hurt Junior. Please, Lawd, make me be strong when I am weak, he prayed.\n\n\"Goddamn it, boy, move your ass!\" Jackson Posey shouted.\n\n\"I'm coming, boss!\" Woodrow said, running, the empty pop bottles rattling inside their wooden slots.\n\nJunior sat down on the ground, pulled off his shoes and socks, and mounted the pop bottles, extending his arms out sideways for balance. The other men marched out the front gate, their eyes straight ahead, and began climbing into the trucks that waited for them. When the trucks drove away in the dust, Woodrow looked through the slats in the tailgate and saw his friend quivering like Jell-O atop the rows of R.C. Cola bottles, his pain sealed inside his closed eyelids.\n\nJunior was still there when the trucks returned in the evening. Except he didn't look like Junior anymore. There were skinned places on his face and knots on his head; one eye was swollen shut and his denims were dark with his own urine.\n\nAt sunset Junior was allowed to come off the box and sit in one corner of the yard. As the other men passed on their way to the mess shack, they saw the bottoms of Junior's feet and had to look away. But Junior's trial by ordeal was not over. Jackson Posey stood over him, thinking private thoughts, touching at the corner of his mouth with one finger. Posey looked up the slope toward the gouged hole in the landscape where a gas storage tank had been pried out of the ground.\n\n\"Get your shoes on, Junior. Woodrow, bring a spade from the shed and get my lunch bucket and a chair from my office,\" Posey said.\n\nThe three of them walked together up the slope in the twilight, Junior limping like he had glass in his shoes, while purple martins darted through the haze of smoke in the air. A fat, thumb-buster .45 revolver creaked in a holster on Boss Posey's hip. Woodrow set down the chair for Boss Posey to sit in and speared the spade into a huge mound of wet clay by the hole, then set down Posey's lunch bucket on the ground by the chair. For just a moment he thought he smelled rain inside the wind.\n\n\"You don't need me no more, huh, boss?\" he said.\n\n\"Hunker down on the dirt pile and keep me company,\" Posey replied, opening his lunch bucket and removing a pint of whiskey.\n\nHe wants you to attack him, Junior. Then he's gonna kill you. He brung me to be a witness and cover his ass, Woodrow said to himself. Look at me, Junior. Can you hear the words I'm t'inking?\n\n\"Dozer man run out of gas today, Junior. So you got to fill up that hole for me. Better get on it,\" Posey said.\n\n\"Stood all day on the bottles, boss. Ain't got nothing left,\" Junior said.\n\n\"You done this to yourself, boy.\" Posey unscrewed the cap on his whiskey bottle and took a sip, rolling it in the corners of his mouth before he swallowed. Then he seemed to think a long time before he spoke again. \"You believe you're better than me, don't you?\"\n\n\"No, suh,\" Junior replied.\n\n\"Smarter, been more places, slept with better-looking white women than I have. Been wrote up in northern magazines. A man like me don't get his name in the paper lessen it's in the obituary.\"\n\nJunior pulled the spade out of the clay mound and began shoveling into the hole, keeping his bruised feet stationary, swiveling his back to throw each spadeful. Boss Posey drank from the bottle again, then removed a piece of waxpaper-wrapped chocolate cake and a slapjack from his lunch bucket. The slapjack was perhaps eight inches long, thin, mounted on a spring, lead-weighted and swollen at the tip, like the head on a snake. He rested it on his thigh and ate part of the cake, then put both the slapjack and the remnant of the cake back in the lunch bucket.\n\nThe sun dipped over the rim of the earth and the fields went dark and nightbirds began calling to one another in the woods across the bayou. At first Woodrow tried to close his eyes and sleep on his feet. Then, without asking permission, he sat down on the back side of the pile Junior was spading into the hole. But Boss Posey didn't seem to mind. He was drinking steadily from the bottle now, bent slightly forward in the chair, the cancer on his arms like small poisoned roses buried in his skin.\n\nOff in the distance Woodrow heard the dry rumble of thunder and saw a tree of lightning splinter across the sky. Junior's movements with the shovel became slower and slower, then it slipped out of his hands and clattered down into the darkness.\n\n\"I had it, boss. You gonna shoot me, go 'head on and do it,\" he said. He stood erect, his face slick with sweat, his body glowing with stink, one eye swollen into a knot with a slit in it.\n\n\"I'm about to lose my job 'cause of you. My pension goes out the window with it. That's what you done, you black sonofabitch. Now, you fill that goddamn hole.\"\n\n\"Know what the problem is, boss?\" Junior asked. \"It ain't Miss Andrea. It ain't Mr. LeJeune, either. It's 'cause you ain't no different from us. You eat the same food, stack the same time, kiss the same pink ass the niggers do. Maybe it's time you wise up.\"\n\nThe first blow with the slapjack caught Junior across the temple, splitting the skin to the bone. Then Jackson Posey whipped him to the ground, just as though he were chopping on a piece of wood.\n\nBut Woodrow believed it was the first blow that killed Junior and that the others were visited upon the body of a dead man, because Junior made no sound as the slapjack whistled down on his head and neck and back, thudding to the ground on his knees, his eyes already rolled upward in his head.\n\nAnd while his friend died Woodrow stood by impotently, his fists balled in front of him, a cry coming from his throat that sounded like a child's and not his own.\n\nJackson Posey's chest was heaving when he looked down at his work. He flung the slapjack aside. \"Damn!\" he said. He paced up and down, staring back at the camp, then at the lights burning in the LeJeune house. Woodrow was so frightened his teeth knocked together in the back of his mouth.\n\nPosey steadied his foot against Junior's shoulder and tried to shove his body over the edge of the hole. But Junior's body fell sideways and Boss Posey couldn't move it with his foot. In fact, Woodrow could not believe how weak Posey was.\n\n\"Get a holt of his feet,\" Posey said.\n\n\"Suh?\"\n\n\"Pick up his feet or join him. Which way you want it?\"\n\nWoodrow gathered up Junior's ankles while Boss Posey lifted his arms, and the two of them flung Woodrow's friend over the rim of the hole. The thump it made when it hit the bottom was a sound Woodrow would hear in his sleep the rest of his life.\n\n\"Go over there and set on the ground,\" Posey said.\n\nPosey mounted the bulldozer and started the engine. With the lights off he lowered the blade and pushed the huge pile of clay into the hole, backing off it, packing it down, scraping it flat, until the hole was only a dimple in the landscape. When he cut the engine Woodrow could hear the first drops of rain pinging on the steel roof over the driver's seat.\n\n\"Junior transferred out of here tonight. Ain't none of this happened. That's right, ain't it, Woodrow?\"\n\n\"If you say so, boss.\"\n\n\"There's a half inch of whiskey left in that bottle. You want it?\"\n\n\"No, suh.\"\n\n\"Have a Camel,\" Posey said, and shook two loose from his pack. \"Go ahead and take it. It's a new day tomorrow. Don't never forget that. Sun gonna be breakin' and a new day shakin'. That's what my daddy always used to say.\"\n\nHow'd you come by this little farm here?\" I asked Woodrow.\n\n\"Mr. LeJeune sold it to me. Give me a good price wit'out no interest,\" he replied.\n\n\"To shut you up?\"\n\n\"He sent a black man to me wit' the offer. Never saw Mr. LeJeune.\" Woodrow stared at me with his flat, sightless eyes that could have been large painted buttons sewn on his face. Lightning jumped in the clouds over the Gulf.\n\nI slipped my business card between his fingers. \"Let me know if I can do anything for you,\" I said.\n\nHis hand folded around the card. \"Whatever happened to Mr. LeJeune's li'l girl, the one named T'eo?\" he asked.\n\n\"Theodosha? She's around.\"\n\n\"My cousin, the maid for Mr. and Miz LeJeune? She always worried about that li'l girl. She said t'ings wasn't right in that house.\"\n\nI asked him what he meant but he refused to explain.\n\n\"How long were you inside?\" I said as I was leaving.\n\n\"Five years.\"\n\n\"What'd you go down for?\"\n\n\"Fifty-t'ree-dol'ar bad check,\" he replied.\n\n## Chapter 18\n\nAs I drove back toward New Iberia a thunderstorm blew in from the Gulf and marched across the southern tip of Vermilion Parish, thrashing the sugarcane in the fields, the rain twisting in my headlights. I could not shake the tale told me by Woodrow Reed, nor the sense of needless death and cruelty and loss that it instilled in the listener. I turned on my radio and tried to find a station that was playing music, but my radio went dead, although it had been working fine earlier.\n\nI tried to get Helen again on my cell phone, but I couldn't raise the wireless service and gave it up and tossed the cell phone on the seat. I passed flooded rice fields wrinkled with wind and lighted farmhouses that looked like snug islands inside the storm. Then I passed a billboard on a curve and my lights flashed across a woman standing by the side of the road.\n\nShe wore blue jeans and an unbuttoned tan raincoat that whipped back in the wind. Her hair was honey colored, tapered on her neck, her skin almost luminous in the glare of headlights. Hey, G.I., give a girl a ride? I thought I heard a voice say.\n\nI braked the truck to the side of the road, my heart beating, and looked through the back window. The woman stood on the shoulder of the road, silhouetted against a light that shone on the face of the billboard. Don't buy into this, I told myself. It's not her. Your wife is dead and all the delusions and misery you inject into your life will not change that inalterable fact.\n\nThen I put the truck in reverse and began backing toward the figure on the side of the road.\n\nShe glanced back over her shoulder once and began running. I accelerated faster, swerving on and off the pavement, until I was abreast of her. Through the rain-streaked glass her face stared at me, beaded with water, eyeshadow running down her cheeks, her mouth glossy with lipstick. I closed and opened my eyes, like a man coming out of darkness into light, her face forming and reforming in the rain.\n\nI shoved open the passenger door and held up my badge holder. \"Get in,\" I said.\n\nShe hesitated a moment, then sat down in the passenger seat and slammed the door behind her. She gave me a hard look in the glow of the dash panel. Her cheeks were pitted and heavily made up, her clothes reeking of cigarette smoke and booze. \"Thanks for the ride. My old man threw me out,\" she said.\n\n\"Where do you want to go?\" I asked.\n\n\"First bar we pass,\" she said. \"For a minute you scared me. I had trouble with a couple of black guys last night. You stopped just 'cause you saw me in the rain?\"\n\n\"I thought you were somebody else,\" I said.\n\nShe gave me a look. \"There's a bar past the curve. Right by the motel,\" she said.\n\nI put on my turn indicator and began to slow the truck. I knew the bar. It was a ramshackle, sullen place owned by a man who ran dog fights.\n\n\"I left my purse at the house. The sonofabitch I live with has probably drunk it up by now,\" she said.\n\nI stopped in the parking lot and waited. She took a cigarette from her shirt pocket and lit it with a plastic butane lighter. She continued rubbing the striker wheel under her thumb. \"Look, I can't drink in there for free. You want some action or not?\" she said.\n\n\"Get out,\" I said.\n\n\"I can really pick them,\" she said. She stepped out into the storm and slammed the truck door as hard as she could.\n\nLesson? Chasing a nighttime mirage on a rain-swept highway has no happy ending for either the quick or the dead.\n\nThe one-car fatality at West Cote Blanche Bay seemed to lack any plausible explanation. The witness, an elderly Cajun hired to pick litter out of the ditches along the roadside, had seen an expensive, large car parked next to a compact in a grove of pine trees. Children had been lighting fireworks all evening, shooting Roman candles and rockets over the bay. Then he had heard firecrackers in the trees, just before the compact had driven away. When he looked again at the grove of pines, the large car started up and drove out onto a pier, snapping the supports on the guardrail into sticks, finally plunging off the end of the pier into the water.\n\nHelen Soileau had arrived at the bay only a few minutes before me. She walked with me up a shell ramp and introduced me to the witness. As with most elderly Cajun men, his handshake was as light as air. \"How many firecrackers did you hear?\" I asked him.\n\n\"Two, maybe t'ree,\" he replied.\n\nHe was a tiny man, dressed in neat khakis, with cataracts and a supple face that resembled brown tallow. He seemed nervous and kept glancing over his shoulder at the bay and at the splintered guardrail on the pier and at the wrecker that so far had not been able to pull the sunken car off a submerged pipeline, all of it lit in the glare of searchlights mounted on a firetruck.\n\n\"Is anything wrong?\" I asked.\n\n\"I seen a big man behind the wheel. Seen him go crashing right off the end of the pier there. I cain't swim, me. I keep t'inking maybe there was air inside the car. Maybe if I'd brung hep sooner\u2014\"\n\n\"You have no reason to feel bad about anything, sir. Who was in the compact?\"\n\n\"Just somebody driving a li'l car. It was an old one. I ain't sure what kind.\"\n\n\"Was a man or woman driving?\"\n\nHis shook his head, his face blank.\n\n\"What color was the car?\" I asked.\n\n\"I just ain't paid it much mind, no.\"\n\n\"You see a license tag?\" I asked.\n\n\"No, suh.\"\n\n\"The firecrackers you heard, those were in the pine trees? You're sure about that?\" I said.\n\n\"No, suh, I ain't sure about none of it no more.\"\n\nI patted him on the shoulder and walked down to the water's edge. The bay was black, dimpled with rain rings, and the tide was pushing small waves that glistened with gasoline up on the sand. Two scuba divers, both of them sheriff's deputies, had already been down on the wrecked car. They were sitting on the running board of the firetruck in their wetsuits, sharing a thermos of coffee.\n\n\"What's it look like down there?\" I asked.\n\n\"The vehicle landed on its side. Driver's face is down in the silt. The ignition is on and the gearshift in 'Drive,' \" one of them said. His name was Darbonne. He was unshaved and had curly black hair, his throat prickled with cold.\n\n\"Any chance air was trapped in there?\" I asked.\n\n\"The front windows were down. The driver's arm is tangled up in the seat belt, like he couldn't find the release button. All that water probably hit him like a hammer,\" Darbonne said.\n\n\"The witness blames himself for not getting back with help sooner. Tell him about the air situation, will you?\" I said.\n\nDarbonne nodded and yawned. \"When they drive off bridges or piers, they're drunks, nutcases, or suicides,\" he said. \"If a guy in a Caddy ices himself, he should have the courtesy to do it without inconveniencing people who make twenty-five grand a year.\"\n\n\"Say again?\"\n\n\"The whale who just offed himself. I wish he'd gone to a heated, indoor pool to do it,\" the driver said, then looked at my expression. \"What, I just spit on the floor in church?\"\n\nA few minutes later the divers went down again to reset the hook on the Cadillac's frame so the car could be flipped over on its top and slid off the pipeline it partially rested on. Helen and I stood by the water's edge and watched. The moon had broken through a slit in the clouds, and far out on the horizon there were whitecaps that looked like tiny bird's wings.\n\n\"Castille LeJeune's lawyer called again. He's talking about a harassment suit against the department,\" she said.\n\n\"He'd like my job?\"\n\n\"What did you find out down at Pecan Island?\" she said, ignoring my question.\n\n\"Castille LeJeune had Junior Crudup killed. He was beaten to death by a prison guard, a guy named Jackson Posey,\" I replied.\n\nShe looked at the black surface of the bay and at the slickness of the wrecker cable as it extracted the submerged car from the water. Her face did not change expression. She wiped away a raindrop that had caught in her eyelash. \"Where's Crudup's body?\" she asked.\n\n\"Probably still buried on the LeJeune's property,\" I said.\n\n\"Get a search warrant,\" she said.\n\nThe wrecker man winched the Cadillac upside-down out of the shallows and slid it up on the bank, the front windows gushing with water and oil-blackened silt. The body of a huge man hung against the safety strap, his shoulders and neck pressed against the roof, his face twisted toward the open window so he appeared to be staring at a bizarre event taking place outside his automobile.\n\nI squatted down to eye-level with him and shone a flashlight on his face and inside the rest of the car. There was a small entry hole in his neck, his cheek, and the side of his head. The wounds had bled out and had washed clean in the water and had started to pucker around the edges.\n\n\"Ever think anybody could sucker-drop Fat Sammy Figorelli?\" Helen said behind me.\n\n\"No,\" I said. I reached inside the car and closed Sammy's eyes. The inverted weight of his massive buttocks and thighs had curved his spine so that his back and neck were compressed like a gargoyle's.\n\n\"Don't waste your sympathies, Streak. He was a pimp and a pusher and the world's a better place every time one of these shitbags gets stuffed into a hole,\" Helen said.\n\n\"I guess you're right,\" I said. But I could not help remembering the stories of a French Quarter fat kid who had spent years being the butt of people's jokes.\n\nHelen stood up from the spot where she had crouched behind me. \"Wrap it up here. At oh-eight-hundred tomorrow go to work on the warrant. It's time Castille LeJeune learned this is the United States,\" she said.\n\n\"You got it, Top,\" I said, referring to her old rank in the U.S. Army.\n\n\"Call me that again and I'll tear off your head and spit in it,\" she replied.\n\nI think even Fat Sammy would have enjoyed that one.\n\nWe had the warrant by late Tuesday afternoon. Without announcement and with a balmy breeze at our backs and a sky the color of a ripe peach, two cruisers from the Iberia Sheriff's Department, three from St. Mary Parish, a front-end loader, and a bulldozer chain-boomed on a flatbed tractor-trailer rig all came down Castille Le Jeune's front drive, raking through the lone tunnel of oaks, right into the middle of an outdoor dinner party LeJeune was holding on his terrace.\n\nHelen and I and a plainclothes from the St. Mary sheriff's office served the warrant on him in front of his guests, who included, among at least a dozen others, Theo and Merchie Flannigan. LeJeune tried to feign an amused dismay and the good cheer of the professional bon vivant, but Theo imposed no such restraints on herself.\n\nShe wore a low-cut white evening dress and a necklace of red stones around her throat. Her skin was flushed with either the challenge of the moment or the glass of bourbon and crushed ice with a sprig of mint she had been drinking. She placed her small fists on her hips, as a drill instructor might, and turned her face up into mine. \"You're an idiot,\" she said.\n\n\"Excuse me, madam, but you need to sit down and stay out of this,\" Helen said.\n\n\"And you need to work on your sexual-identity problems before you lecture other people in their homes,\" Theo said.\n\nHelen gazed through the trees at the bayou and the deserted shacks that had once housed prison inmates, her breasts hard-looking as softballs against her shirt. She reread the warrant to herself, seemingly indifferent to Theo's insult. Then she lifted her eyes into Theo's. \"Repeat what you just said.\"\n\n\"You have no business here,\" Theo said.\n\n\"Where do you think the burial site is?\" Helen said to me, ignoring Theo.\n\n\"On a line between here and what would have been the front gate of the prison camp. I'd put it pretty close to that pond inside the fenced area,\" I said.\n\nLeJeune raised his hands. \"Listen to me,\" he said. \"I don't know anything about this man Junior Crudup or whatever his name is. My wife befriended the convicts who worked out their sentences on our farm. She was a kind, gentle, decent person. How in God's name can you accuse us of hiding the remains of a murdered man on our property?\"\n\nHelen walked out into the yard. \"Take out that fence and start in a circle. Drain the pond if you have to,\" she said to the two heavy-equipment operators.\n\nHelen went back to her cruiser and I began walking down the slope toward the old work camp. Inside the evening shade of the trees I could hear the conversation and tinkle of glasses resume among LeJeune's guests on the patio.\n\n\"Dave, stop,\" Theo said, catching my arm.\n\nShe'd just had her hair cut and it was thick and even and shiny on the whiteness of her shoulders. The bourbon and smell of ice and mint on her breath touched my face like the tracings of a kiss.\n\n\"Your father commissioned a murder,\" I said.\n\n\"You have it all backwards,\" she said.\n\n\"Then why are you afraid to go down to the pond?\"\n\n\"For reasons you don't understand.\"\n\n\"You can tell the jury that at your father's trial.\"\n\n\"Why do you hate him so much?\"\n\n\"Because he's a sonofabitch.\"\n\n\"I'll remember you said that to the day I die.\"\n\n\"Go back home, Theo. Your guests are waiting.\"\n\n\"I can't believe I slept with you. I want to peel my skin off.\"\n\nPerhaps her response was justified, but at that moment I didn't care one way or another. Down below, the bulldozer and front-end loader were tearing apart a white-rail fence and a sloping green pasture, looking for the bones of a man who had been beaten to death so a cancer-ridden prison guard could keep his pension and a cuckolded husband his pride.\n\nThe heavy-equipment operators worked by gasoline-powered light until midnight, blading away the grass and topsoil, pushing it into water-beaded, black-green mounds. They came back at sunrise and started in again, scooping huge amounts of wet clay and feeder roots from the oak trees onto LeJeune's lawn, trenching a drainage into his fish pond, smashing his dock into kindling. By noon the entire landscape between the trees in his backyard and the cluster of cabins by the bayou was an ecological disaster, water oozing from the substrata, perch and catfish fighting for survival in small pools, a cow's ribs arching out of the clay like a woman's comb.\n\nA half dozen uniformed deputies in rubber boots raked and probed for hours but found no sign of a human burial. By Wednesday afternoon the excavation area had become a giant, water-filled pit. Since the previous day I had slept three hours. My eyes stung, my jaws were like sandpaper, and a stale, clammy odor rose from my clothes. The heavy-equipment operators shut down their machines and waited. Helen shook her head and the operators climbed down and began packing up.\n\n\"We're in the Dumpster, bwana,\" Helen said.\n\n\"That body was here. He moved it,\" I said.\n\n\"Ride back with me. You look like a car wreck,\" she said.\n\n\"He's not going to get away with it. I'm going to fry that bastard.\"\n\n\"You probably will. Even if you have to take everybody down with you. You might give that some thought,\" she said.\n\nI opened and closed my mouth and felt my ears popping, the horizon tilting slightly, a buzzing sound inside my head, as though my old companion the malarial mosquito was having its way with me again.\n\nHelen cupped her hand around my upper arm and kneaded the muscles in it. \"Come on, Loot, give a girl a lift,\" she said.\n\n\"What? What did you say?\" I said.\n\nShe looked at me strangely, her eyes filled with a mixture of pity and sadness.\n\nNot far away, just outside the little town of Jeanerette, Clete Purcel drove down a back road past three antebellum homes that were so stunning in appearance, the tree-shaded lots they sat on so perfect in arboreal and floral arrangement, they looked like Hollywood movie fabrications rather than homes that people of enormous wealth actually lived in. He turned at the green, embanked property corner of the last house in the row, crossed a steel bridge over the Teche, and passed, within fifty yards of the last antebellum home, a rural slum composed of rusted trailers, desiccated sheds, and junker cars that could have been replicated from a photograph taken in Bangladesh.\n\nHe removed a pair of binoculars from his glovebox and went inside a cafe from which his line of sight allowed him to see the trailer slum that spilled haphazardly to the edge of the bayou. It had not been a good day for Clete. Early that morning he had picked up a bail skip for Wee Willie Bimstine in Opelousas and was about to transport him back to New Orleans, when the skip began jerking against the D-ring anchored on the floor of the Caddy, his face twisted with visceral pain, threatening to soil himself and the convertible if he wasn't allowed to use the bathroom. Clete cuffed him to a pipe next to the toilet in a filling station and waited outside. In less than two minutes the skip managed to put seventy-five cents in a sexual-enhancement dispenser, smear his wrist with a desensitizing lubricant, slip the cuff, and escape out a window.\n\nScore one for the meltdowns, Clete thought.\n\nA half hour later a woman did a hit-and-run on his convertible in a church parking lot; the investigating traffic officer gave him a citation for an expired inspection sticker; and while Clete argued the situation a flock of robins lit in a tree above his car, the top of which was down, and defecated all over the seats and upholstery.\n\nHe drank coffee and focused his binoculars on a trailer that was broken in the center and had vinyl garbage bags taped across the windows. It was the home of the skip's one-time fall partner, an Angola parolee who had been down twice for sexual battery against children. There was no movement inside the trailer, but next door a woman in faded jeans, tennis shoes without socks, and peroxided hair that was waved on only one side walked down to the school-bus stop and waited for her child. Then she escorted the child, a boy of about eight, back home and closed the door behind her.\n\nA moment later she reemerged with a tall, equestrian-looking man who had a hard, flat stomach and a purple birthmark that seeped from his hairline to the corner of his eye. They kissed on the mouth and the man put on a yellow hard hat and got into a waiting car driven by another man wearing an identical hard hat. The two men parked in front of the cafe and came in and sat down in the booth next to Clete's.\n\n\"Kid come home a little soon?\" the driver of the car, a truncated, moon-faced man, said.\n\nThe man with the birthmark didn't reply but instead snapped his fingers repeatedly for the waitress's attention. After she took the order and went away, he said, \"This guy Robicheaux is a walking hemorrhoid. You should see the old man's property. It looks like a bombing zone.\"\n\n\"Tell me about it,\" the other man replied. His blond hair was combed straight back from a receding hairline, and he kept leaning forward, reverentially, each time the other man spoke. But the man with the birthmark was silent now, not interested in whatever the blond man had intended to say. The blond man, who wore a pair of electrician's wire snips in a leather case on his belt, tried again. \"His house was so dried out a popcorn fart could have set it on fire but he blames me for it. He tried to screw me with the Better Business Bureau and get my license pulled.\"\n\nBut the man with the birthmark, whom Clete had now connected with the name Will Guillot, only sipped his coffee and looked out the window at the bayou and the antebellum home on the far side of the steel bridge.\n\n\"You think he sent that doctor to your house?\" the moon-faced man said.\n\n\"Probably.\"\n\n\"You're a mean machine, Will.\"\n\n\"Nope.\"\n\n\"The guy came at you with a sawed-off shotgun?\"\n\n\"He thought he could go into a man's house and kick ass. He lost. End of story,\" Will Guillot said.\n\n\"Pow!\" his friend said.\n\nBoth men became silent, eating slices of apple pie, drinking their coffee, picking their teeth. Clete went to the rest room, then waited for his check. The men in the other booth were talking about football now. Go home, he thought. You don't need any more bad luck today.\n\nHe looked out the window and saw the child of Will Guillot's girlfriend playing on a swing set, a cheap one that was probably bought at Wal-Mart. The skip's fall partner, the sex predator, pulled up next door, talked to the boy briefly, tousling his hair, then went inside his trailer.\n\nClete paid his check and started toward the door. He paused, thinking to himself, then reset his porkpie hat and walked back to Will Guillot's table. He grinned without speaking, his Hawaiian shirt partially unbuttoned on his chest, his eyes flicking sideways as though he did not know how to introduce himself.\n\n\"Help you?\" Guillot said.\n\n\"You guys were in the Crotch?\" Clete said.\n\n\"The what?\" the blond man said.\n\n\"I heard you say something about a 'mean machine,' so I thought you were talking about Mother Green's Mean Machine. See, jar-heads call\u2014\"\n\n\"Yeah, I know all about that. What can I do for you?\" Guillot said.\n\nClete cleaned an ear with one finger, looking sideways again as he did it, his face filling with thought. \"I think I know who you are,\" he said.\n\n\"You do?\" Guillot said.\n\n\"You popped a doctor from Loreauville in your driveway. Guy was some kind of weirded-out Vietnam vet, right? That's some kind of irony, huh? Guy probably had a thousand AK rounds shot at him, then loses his Kool-Aid and gets smoked in the suburbs.\"\n\nGuillot looked across the table at his friend and tapped his fingernail on the cover of his wristwatch. The two men started to get up.\n\n\"Whoa,\" Clete said.\n\n\"Whoa, what?\" Guillot said.\n\n\"The lady up there in the trailer, the one you're banging? She's got a little boy. The guy next door happens to be a sex predator. So while you're getting your twanger taken care of, the freak who was just patting her kid on the head is figuring out ways to sodomize him. My suggestion is you take your mind off your dick long enough to move the lady and her son out of that shithole before the kid's life is ruined. Can you relate to that?\"\n\n\"You've got some fucking nerve,\" Guillot said.\n\nThe owner of the cafe had come from behind the counter and was standing behind Clete now, resolute, his feet planted, his thumb raised in the air.\n\n\"Out,\" he said.\n\n\"No problem,\" Clete said. He pulled two one-dollar bills from a brass money clip and dropped them on his table.\n\nBut outside Clete could not give it up, standing by his car door, flipping his keys back and forth, his face growing darker. He watched Will Guillot and the electrical subcontractor with him get in their car. \"Hold on a minute,\" he said.\n\n\"Get a life, queer bait,\" Will Guillot said from the passenger window as his car rolled past Clete.\n\nClete watched the two men cross the steel bridge over the Teche and turn down the tree-shaded back road that led past the row of antebellum homes. In his mind's eye he saw himself running them off the road, strolling back to their car, his blackjack in his side pocket, moving the situation on up to the full-tilt boogie. Why not? he thought. The day couldn't get any worse than it was already.\n\nHe got into his Caddy, slammed the door, and turned the ignition. He heard a dry, clicking sound, then nothing. The battery was as dead as a butcher block.\n\nIt took an hour for a filling station a half block away to send a truck that gave him a quick-start. He sat behind the wheel, revving the engine to charge the battery, oil smoke pouring from under the frame, bird-shit smears on his clothes, all immediate hope of squaring the beef with Will Guillot gone.\n\nHe looked through the windshield at the trailer slum by the bayou and the parolee who was now drinking a can of beer on his steps and talking to the little boy from next door.\n\nClete retrieved a pair of leather work gloves from under the seat and put them in his pocket, then dropped the Caddy into low gear and rolled into the trailer slum, gravel and oyster shells ticking softly under his tires.\n\n\"You Bobby Joe Fontenot?\" he said.\n\nThe man on the steps was relaxed, smoking a cigarette with his beer, barefoot in the sunshine, his arms flecked with blue tattoos done by an needle improvised from the guts of a ballpoint pen. He wore imitation black leather pants and a tie-dyed strap undershirt, his black hair scalped on the sides and braided into a matador's pigtail in back.\n\n\"I'm gonna take a guess. Casting director from, what's that TV show called, Survivor?\" he said, squinting against the sunlight.\n\nClete grinned and got out of the Caddy, opening his badge holder briefly. \"Looking for your friend who jumped his bond with Wee Willie Bimstine and Nig Rosewater,\" he said. \"Slipped his cuffs this morning and left me with shit on my nose.\"\n\n\"Haven't seen him.\"\n\n\"Mind if I look inside?\"\n\n\"Get yourself a beer. It's in the icebox.\"\n\n\"Thanks,\" Clete said, and gave him the thumbs-up sign.\n\nClete stepped inside. The garbage can in the small kitchen was overflowing, the counters covered with pizza and fried-chicken cartons. A television set was playing without sound, the VCR under it lighted, a cassette pushed halfway into the loading slot. Clete shoved the cassette all the way into the unit with his thumb and waited for the video image to transfer to the screen. Then he clicked off the set and the figures on the screen shrank to a small dot. He slipped on his work gloves and called through the screen door: \"Did you know you have a gas leak in your stove?\"\n\nBobby Joe stepped inside the trailer, sniffing at the air. Clete drove his fist into Fontenot's stomach, burying it to the wrist, so deep he actually felt bone. Then he kicked the wood door shut, flung him headlong into a wall, and pulled a shelf filled with carnival midway ceramics down on top of him. He ripped the cassette from the VCR and bounced it off Bobby Joe's face, then rooted in the refrigerator's freezer compartment and pulled out a box of Popsicles and threw them in Bobby Joe's face, too.\n\n\"You get the kids in here with cartoons and ice cream?\" he said.\n\nBobby Joe tried to raise himself up against the wall, spittle running from the corner of his mouth. \"I'm in treatment. Ask my P.O.,\" he said hoarsely.\n\nClete opened and closed his huge hands, breathing hard, his cheeks pooled with color. He lifted Bobby Joe by his shirt and belt and threw him into the narrow bathroom at the back of the trailer. Bobby Joe grabbed the side of the lavatory and tried to raise himself up again, his face bewildered.\n\n\"What did your P.O. tell you about putting your hands on little kids, asshole?\" Clete said.\n\n\"I ain't put\u2014\"\n\nClete locked one hand on the back of Bobby Joe's neck and drove his head down on the toilet bowl, smashing his mouth against the rim, plunging his head into the water, scouring the bottom of the bowl with his face. It should have been enough but he was beyond controlling it now or even trying. He slammed the toilet seat down on Bobby Joe's neck and head, then grabbed the top of the shower stall and mounted the toilet, crushing the seat down on Bobby Joe's head, tap dancing on it like an elephant on hallucinogens while Bobby Joe's legs thrashed on the linoleum.\n\nOutside he heard children playing and through the top of the window he saw a little girl chasing after a Frisbee that sailed above her head, and like a man descending from an electrical storm high up on a mountain he stepped back down on the floor and pulled Bobby Joe from the toilet bowl, dripping with water and blood.\n\nHe tossed a towel in Bobby Joe's face and leaned back against the wall, out of breath, his fists still knotting. \"I'm going to make regular checks on the kid next door,\" he said. \"If I find out you've been near him, you'll wish you were a bar of soap back in 'Gola. The same goes if you dime me. Maybe you think you got a bad deal here today, but pervs don't get slack. You hearing me on this?\"\n\n\"You fat fuck,\" Bobby Joe said, pressing the towel to the blood that ran off his chin, looking at it in disbelief, his words muffled, his mouth still trembling. \"You like family values? That kid's mother used to be an army whore over by Folk Polk. I'm gonna find out your name. If I ever offend with a kid again, I'm gonna say it each time I poke him. How's that, asshole?\"\n\nWhen Clete got back to the motor court, he stayed under the shower until the hot water tank went empty, burned his clothes in a barbecue pit, drank a quart of whiskey-laced eggnog, and still could not feel clean.\n\n## Chapter 19\n\nFather Jimmie Dolan had done six months federal time for demonstrating at the School of the Americas and probably considered himself jailwise. But in reality, like all people who are intrinsically decent, he was incapable of the cynicism that passes for prison-acquired wisdom.\n\nOn Thursday morning he was in Franklin, in black suit and Roman collar, collecting signatures on his petition to ban the sale of mixed drinks from drive-by windows. During three hours of approaching people in front of strip malls and grocery stores, he had amassed a total of six signatures, one from a retarded man, and two from people who signed their names with an X.\n\nHe bought a take-out lunch from a McDonald's and ate it in his car under the trees in a small park, then fell asleep. The day was unseasonably warm, the live oaks flickering with wind, but he dreamed of snowmelt in the Cumberland Mountains, the bright air of early spring, tea-colored streams that leached out of limestone cliffs, dogwood blooming purple and white on a hillside. When he awoke, children were running by the front of his car, kicking a soccer ball in the leaves, the spangled sunlight racing across their bodies, but somehow there was a continuity between the beauty of the Appalachian spring in Jimmie's dream and the joy of the children at play.\n\nHe got out of his car and began walking toward the public rest room. He had no reason to pay attention to a nervous, agitated plainclothes detective by the name of Dale Louviere, who was parked in a Ford by the swing sets, the same detective who had investigated the killing of Dr. Parks by Will Guillot and called it an open-and-shut case of self-defense.\n\nNor did Father Jimmie pay attention to a man known as Cash Money Mouton standing by the lavatory inside the rest room.\n\nCash Money's last name was French but he was actually a pecker-wood product of north Louisiana. He used to sell fire and accident and term life insurance from door to door in black and poor-white neighborhoods, and was infamous for both his sweaty enthusiasm and his carnival sales rhetoric. He would pull clutches of papers and brochures from a vinyl briefcase, his face bursting with sincerity, tapping his seated listener, usually the man of the house, on the kneecap, saying, \"You run your lawnmower over your foot and chop your toes off, I'll give you twelve-hunnerd dollars, cash money, boy. You stick your hand in your skill saw, I'll pay you five-hunnerd dollars, that's cash money, for every finger you cut off. Splash muriatic acid in your eyes and go blind, I'm talking five-thousand bananas, cash money, boy.\"\n\nThen Cash Money Mouton's uncle became police chief and Cash Money began a new career.\n\nFather Jimmie stood at the urinal and relieved himself. He could feel the man at the lavatory staring at the side of his face. He started to look at him, then thought better of it and kept his eyes straight ahead. But when he tried to get to the lavatory the man known as Cash Money stood in his way.\n\n\"Excuse me,\" Father Jimmie said.\n\nBut Cash Money did not move. He wore sideburns, a Tabasco tie, an American flag in his coat lapel. He smelled of deodorant, hair tonic, and fear. There was almost an iridescent shine on his skin.\n\n\"Is there some difficulty here that I don't quite grasp?\" Father Jimmie asked.\n\n\"Repeat that?\" Cash Money said.\n\n\"Could I be of some assistance to you?\"\n\n\"That's it,\" Cash Money said.\n\nHe stepped into the rest room doorway and waved at the man in the Ford automobile. Father Jimmie rinsed his hands, shook them off, and tried to walk around him.\n\n\"You're not going anywhere, buddy boy,\" Cash Money said.\n\n\"Push me again and we're both going to regret the next couple of minutes,\" Father Jimmie said.\n\nBut Cash Money was looking over his shoulder now and not at Father Jimmie. \"He just threatened me,\" he said to the man approaching the rest room.\n\n\"What else did he do?\" the plainclothes detective named Dale Louviere said. Even in the open air a gray fog of nicotine and ash seemed to enclose his body. Clusters of veins, like tiny pieces of green string, pulsed in his temples.\n\n\"He said he wanted to help me. He was fooling with his fly when he said it,\" Cash Money said.\n\n\"You're a liar,\" Father Jimmie said.\n\n\"We saw you watching those kids, Father,\" Louviere said.\n\n\"How would you like to have your teeth knocked down your throat?\" Father Jimmie said.\n\n\"Hook him up,\" Louviere said.\n\n\"I ain't putting my hands on him,\" Cash Money said. His eyes jumped sideways when Father Jimmie looked him directly in the face.\n\nAt the police station Father Jimmie was charged with sexual solicitation and threatening a police officer and locked in an empty holding cell that was in full view of anyone, male or female, in the booking area. He made a pillow out of his coat, pulled off his collar, and lay down on a wood bench. He stared up at the graffiti and scratched drawings of genitalia that covered almost all the painted surfaces in the cell, and remembered the admonition of the blues singer Lazy Lester: \"Don't ever write yo' name on the jailhouse wall.\"\n\nHe could see Louviere punching in numbers on a phone, calling up first the local newspaper, then a television station in Lafayette and one in Baton Rouge, the Associated Press in New Orleans, and finally the diocese.\n\nLouviere walked to the cell door. \"Want your phone call now?\" he asked.\n\n\"I'd like to ask you a question first,\" Father Jimmie replied.\n\nLouviere unlocked the door and pulled it open. \"If you're wondering whether I'm a Catholic, yeah, I am. And it's perverts like you who give the church a bad name,\" he said.\n\n\"Call yourself whatever you wish, but you're not a Catholic. The real issue is whose pad are you on. Who's paying you to do this to other people, sir? What price have you gotten for your soul?\" Father Jimmie said.\n\nI was at the office when Father Jimmie called.\n\n\"How much is your bond?\" I said.\n\n\"I haven't been arraigned yet,\" he replied.\n\n\"Why'd you have to take your petition into St. Mary Parish?\"\n\n\"What's wrong with St. Mary Parish?\"\n\n\"It's a fiefdom. They think it's the year 1300 down there.\"\n\nI heard him laugh. \"A fiefdom? With serfs in iron collars and that sort of thing? That's an interesting observation. I see,\" he said.\n\nNo, you don't, Jimmie, I thought. But martyrs and saints fly low with the angels, colliding with telephone poles and the sides of buildings, and consider harm's way their natural environment. Who was I to contend with them?\n\nMax Coll didn't like gambling; he loved it and all the adrenaline rush and glittering ambiance that went with it, as passionately as a man could love a woman or a religion. All men had a vice, his father used to say. It was recognition of our moral frailty that allowed us to retain our humanity, he said. The man who wasn't tempted by drink or women or betting the ponies could easily set himself on a level above Christ, and hence become guilty of the most pernicious of the seven deadly sins, namely arrogance and pride.\n\nMax had always remembered his father's words. Drink robbed a man of his intelligence and his organs; women gave a man satiation for only a little while, and memory of it immediately rekindled lust for and dependence on more of the same.\n\nBut gambling gave a man control, allowed him to choose his battleground and make use of his knowledge about both people and mathematics. The losses were only monetary ones, and since gambling was never about money, what difference did the loss make, particularly for a single fellow whose occupation was a bloody affair that should allow for a sybaritic excursion once in a while?\n\nHe was discriminating in the games he played. The slots, video poker, and electronic keno were created for natural-born losers. Jai alai was fun and fast, but what reasonable person would bet on players who all came from the same part of Spain and were related to one another? With the ponies you could dope out the morning line, study the track conditions and the animals in the paddock, and have a fair chance at the windows. Craps was for showboats, roulette for C\u00f4te d'Azur faggots, and dog tracks everywhere strictly for the dogs.\n\nNot to say he didn't bet ball games, boxing matches, and national elections. In fact, Max once bet a window washer on the thirty-first floor of a Chicago hotel that he could climb out on the sill and clean the window faster than the professional washer. He not only won the wager, he enjoyed the experience so much he washed four more windows out of goodwill.\n\nBut the game that got Max in trouble was blackjack, the one game that gave the casino player a running chance at beating the house. Max's memory bank was almost like a computer's, and even when going up against a houseman dealing out of a five-deck shoe, Max's ability to count cards and to successfully stay put or risk another hit was uncanny.\n\nMax's weakness at the blackjack table was his inability to put principles ahead of personalities. He didn't resent losing to a machine or to corrupt jai alai players wanting to keep their family members out of the tomato patch. Max did not like to lose to individuals, particularly stolid and dispassionate people who were paid by the hour and could not wait to get off work. To count cards until his brain was bleeding, then have a joyless clod turn up a blackjack on him out of sheer luck made the scalp recede on his skull.\n\nHe would retaliate by playing multiple hands, progressively increasing his bets, doubling up on splits, until he was broke, exhausted, and depressed, staring out the window at the ragged edges of dawn in Vegas or Reno or Atlantic City, wondering if he could get the casino manager to open a credit line for him.\n\nMax depressed was Max out of control. He would telephone sports books all over the country and lay down fifty thousand dollars in bets without blinking an eye. Then he would dress in a pair of pressed pink pajamas and lie spread-eagled on his back in the center of his hotel bed, the world spinning around him, his heartbeat decreasing, a strange serenity washing through him, as though he had descended to the bottom of a vortex and was no longer at its mercy or required to control it.\n\nUsually his sports-book binges were harmless and his wins canceled out his losses. But contrary to all his wisdom he went in heavy on an insider tip at the jai alai fronton in Dania and took a bath for a hundred large he couldn't pay. Not only was the sports book in Miami unsympathetic with Max's financial situation, they sold his debt to shylocks who informed him the vig was four thousand a week, none of which applied to the principle.\n\nOr he could take out a Catholic priest.\n\nSo he had come to Louisiana on a gray, rain-swept, cold day, trudging through flooded streets floating with garbage, himself no different in aspect than a poor sod on his way to work in the peat bogs. But there had been an upside to it all. He'd found out he didn't have it in him to shoot a priest, which meant perhaps part of his soul was still intact. Secondly, he had discovered a new identity and gambling ambiance.\n\nWearing Father Dolan's black suit and rabat and collar, he had entered a bingo parlor on an Indian reservation in south-central Louisiana and had suddenly found himself a celebrity. People smiled at him, shook his hand, offered him their chairs at the tables, patted him affectionately, brought him beer and sandwiches from the cafe. He began to feel like a mascot being trundled from hand to hand by five hundred people. In fact, he was pinched and pulled and squeezed so many times and places he couldn't concentrate on his bingo board and finally gave it up.\n\nThen he was asked to stand on the stage and call out the bingo numbers. Why not? he thought. It was a grand evening. The weather had turned balmy again; palm trees strung with colored lights were rustling in the breeze outside the windows; the faces of the people around him were warm and filled with goodwill. Maybe his clerical role was a bit cosmetic, but it was still a fine way to be.\n\nThen at 10:00 P.M. he went into the bar and ordered a cup of coffee and sat down to watch the nightly news.\n\nThe lead story was the arrest of one Father James Dolan, charged with sexual solicitation in a public rest room that was located close by a children's playground.\n\nThe arresting officer, Dale Louviere, was interviewed on camera. \"We had this area under surveillance because of previous complaints,\" he said.\n\n\"Regarding the children?\" the reporter asked.\n\n\"Yes, that's exactly correct,\" Louviere replied.\n\n\"Regarding this particular suspect?\" the reporter asked.\n\n\"I'm not at liberty to say that. We're currently involved in a deep background investigation,\" Louviere replied.\n\nIf I ever saw a bull carrying around its own china shop, Max thought. Oh well, it was the good father's cross to bear, not Max's. Maybe Father Dolan would have a little more empathy for professional criminals now that he'd gotten himself jammed up by coppers on a pad, Max told himself.\n\nHe finished his coffee and went back to the bingo game. But the fun was gone and the clothes on his body suddenly felt foreign on his skin, superheated, sticky, smelling of the priest.\n\nHe found himself biting his knuckle, oblivious to the stares around him. What was it that bothered him? The priest was a hardhead, determined to see himself buggered with a posthole digger. Max had nothing to do with it, no obligations to him.\n\nWrong, he thought, lowering his eyes, staring into his lap.\n\nHe had set out to murder an innocent, decent man, something he had prided himself on never having done. In addition, the priest had bested him at every turn; that thought didn't go down well, either. In fact, all of Max's thoughts were like thongs on a flagrum whipping down on his head.\n\nIt was depressing.\n\nHe walked outside into the wind and the sweep of stars overhead and the glow of Christmas lights strung around palm trees and started up his rental Honda. He removed a .45 automatic wrapped in an oily cloth from under the seat and set it beside him. As he drove down the two-lane road toward the interstate, he rested his right hand on the .45 and felt his heart rate decrease and his breath grow quiet in his chest.\n\nThen he looked up through the windshield at the stars and for the first time in years found himself addressing an ancient deity with whom he had once had a relationship.\n\nSir, if you're going to drop problems of conscience on a man like me at this time in his career, he prayed, would you mind doing so in a gentler manner so I don't have to feel I'm being crunched inside the iron maiden? I would very much appreciate it. Thank you. Amen.\n\nIt rained the next morning and Jimmie Dolan was still in jail, waiting to be arraigned at 11:00 A.M. I had just sat down at my desk when I saw an unmarked vehicle of the kind used by N.O.P.D. pull to the curb and Clotile Arceneaux, wearing Levis, a knit sweater, and blue-jean jacket, get out and run through the rain to the courthouse entrance, her hand raised in front of her brow.\n\nShe came into my office, out of breath, her denim jacket streaked with rainwater. She sat down without being invited and said, \"Wow! You're a hard man to catch!\"\n\n\"I don't follow you,\" I said.\n\n\"I left three messages yesterday afternoon,\" she said.\n\n\"I was in Franklin. Father Dolan is in jail,\" I replied.\n\n\"Yeah, I know all about it. Guy really walks into wrecking balls, doesn't he? Look, what have you got on the death of Sammy Figorelli?\"\n\n\"Nothing.\"\n\n\"Nothing?\"\n\n\"He was killed with a .22. He probably knew the shooter. That's about it,\" I said.\n\nI could see her anger at losing months of work rekindling itself in her face. She bit a thumbnail and looked at the rain hitting on the window, then looked back at me. \"I came here for another reason as well. In fact, I'm off work today,\" she said.\n\n\"What is it?\"\n\n\"You already have breakfast?\" she said.\n\n\"No,\" I lied.\n\n\"It's on me,\" she said.\n\n\"Your accent seems to come and go.\"\n\n\"See, I knew you were a smart man.\" She smiled, her mouth pressed into a small flower.\n\nWe got a take-out order at Victor's Cafeteria on Main and drove across the bayou to a giant crab-boil pavilion next to an exhibition hall where, believe it or not, Harry James, Buddy Rich, Willie Smith, and Duke Ellington's arranger, Juan Tizol, performed during the 1950s. The camellias along the bayou were in bloom and looked like red paper flowers inside the grayness of the day, and a tug was moving a huge iron barge loaded with dredged mud through the drawbridge up by Burke Street.\n\n\"So what's the haps?\" I said.\n\n\"I came down on you pretty hard when you and Purcel scared Fat Sammy out of town,\" she replied.\n\n\"Your feelings were understandable.\"\n\nThere was a fried-egg-and-ham sandwich on French bread in her Styrofoam container but she hadn't touched it. \"I talked to Purcel. He told me about your wife's death,\" she said.\n\nI raised my chin to straighten my collar and looked at the tug moving the barge down the bayou.\n\n\"So what I'm saying is\u2014\"\n\n\"Got it. You don't need to explain.\"\n\n\"How about shutting up a minute? My husband was killed in Iraq in '91. He was in a tank. The army said he died instantly but I don't believe them,\" she said.\n\n\"I'm sorry.\"\n\n\"For a long time I thought I saw him at a football game or in a bar or in a crowd at a department store. That ever happen to you?\"\n\n\"No.\"\n\n\"You're lucky. What I'm saying, Robicheaux, is I think you're a good cop and you don't need another cop yelling at you.\" She picked up her sandwich and took a bite out of it. I heard the tug blowing its whistle at the next drawbridge.\n\n\"It's Friday. You want to hang around town, maybe catch dinner and go to a movie?\" I said.\n\n\"What's playing?\" she asked.\n\n\"Really hadn't checked it out.\"\n\n\"Father Dolan's being arraigned at eleven,\" she said.\n\n\"I thought you'd cut loose of Father Jimmie's problems.\"\n\n\"A girl's got to do something for kicks,\" she said, and watched me over the top of her cup while she drank her coffee.\n\nDale Louviere liked being a city police officer, especially since he had been promoted to plainclothes and given his own office, a travel account, and membership in two civic clubs. The pay was nothing to brag about, but good things happened if a man did his job and accorded people respect and made sure he was available to serve in whatever capacity he was needed.\n\nAnything wrong with that? he asked himself.\n\nHe lived a bachelor's life in a freshly painted bungalow out in the country, enclosed by sugarcane fields, cedar trees, flower beds and vegetable gardens tended by a trusty from the parish stockade. The radical priest's accusation that he was on a pad still rankled him. Dale Louviere never accepted a bribe from anyone; he didn't have to. He took care of his own side of the street and the other things took care of themselves. A mortgage or car loan was approved upon application; his drinks were put on a tab at local bars but he was not expected to ever pay the tab; a land developer gave him forty-yard-line tickets to LSU's home games whenever he wanted them; and at Christmas-time cellophane-wrapped baskets of candy, fruit, and wine were delivered to his door.\n\nThe people who owned the sugar mills, drilled the oil wells, and governed the parish's affairs paid most of the taxes, didn't they? They gave other people jobs. The parish would be a giant rural slum without them. So a civil servant had to pay attention to the needs of rich people who could locate elsewhere anytime they chose.\n\nAnything wrong with that?\n\nEarly the same morning Father Jimmie Dolan was to be arraigned, Dale Louviere rose at first light, put on his warm-up suit, and drank coffee and smoked a cigarette at the kitchen table, waiting for the chill to go out of the room. Through the front window he saw a Honda pass on the state road, then return, going in the opposite direction.\n\nHe washed his cup and saucer in the sink, put his spare set of house and car keys around his neck on a braided lanyard, and began his early-morning aerobic walk down the state road. Two hundred yards from his bungalow he crossed a wood bridge over a coulee and entered a long, cleared slash between two unharvested cane fields. The rain had quit temporarily, but fog hung like smoke in the cane and the thatch under his feet was sodden and mud coated, squishing each time he took a step, soaking the bottoms of his sweat pants.\n\nOne of his shoes went down ankle-deep in water. Bad day for aerobics, he thought.\n\nHe heard a car stop on the road. When he looked behind him he saw the Honda again, and a priest with a map spread across his steering wheel, rolling down his window now, his face expressing his obvious need for directions. But secretly Dale Louviere neither liked nor trusted the clergy, and off the clock he gave them no time. He pretended to tie his shoe until he heard the sound of the Honda's engine thinning in the distance.\n\nIt started to sprinkle again and Dale Louviere headed home, walking fast along the edge of the road, through ground fog that welled out of the ditches, his arms pumping the way he had learned in an aerobics class. He wondered if he would ever successfully quit smoking. He had tried many times, but within three days he would be so irritable and agitated his colleagues would toss cigarettes on his desk blotter by way of suggestion. Now the best he could do was pump the smoke out of his lungs and the nicotine out of his blood with a hard, early-morning walk that left his head spinning and his nervous system screaming for another cigarette.\n\nFortunately he had stuffed a pack in his jacket pocket. Just as he fished one out he saw the Honda coming in his direction again. The driver pulled alongside Dale Louviere and rolled down the window with the electric motor. He wore a golfer's cap pulled down on one eye, and had a tight face and small ears, like a fighter who had spent too many years in the ring. A road map was crumpled on the dashboard. His black suit and rabat were dry, his shoulders narrow, his hands round and pink on the steering wheel.\n\n\"Could you be directing me back to Highway 90, sir?\" the priest said.\n\n\"Go to the four corners and turn left,\" Louviere said.\n\nThe priest screwed his head about, his eyebrows raised into half-moons. \"That simple? I must have made a complete circle. I think the bishop served too much of the grog last night.\"\n\nBut instead of driving away he started fiddling with his map, running his finger along a line that marked Highway 90, peering down the road, then through the back window again. Dale Louviere thought he heard a knocking sound in the trunk.\n\n\"What's that?\" he asked.\n\nThe priest clucked his tongue. \"I'm afraid I ran over a dog. I'm taking him to a veterinary if I can find one,\" he said. \"Turn at the crossroads, you say?\"\n\n\"Correct. You can't get lost. Got it now?\" Louviere said impatiently. He lit a cigarette and drew the smoke lovingly into his lungs.\n\n\"I don't see that on this map,\" the priest said.\n\n\"Look, it's not that hard. You see the state road here\u2014\" He held his cigarette to one side and leaned in the window.\n\nThat was as far as he got. The priest grabbed the lanyard around Louviere's neck and rolled up the window on his throat, trapping his head at the top of the glass like a man caught in an inverted guillotine.\n\nHe pressed down on the accelerator and drove his car down the road and into Louviere's driveway, while Louviere held onto the door handle and tried to extend his body like a crane's to keep from being decapitated.\n\n\"Be a good fellow and toggle along as best you can. We'll have you safe and snug in your digs before you know it,\" the priest said. \"Oops, a little bump there. Hang on.\"\n\nDale Louviere felt his head being torn loose from his torso as he tripped over his feet, fighting to find purchase. The Honda moved past the side of his house, his gardens and flower beds and across the thin, wintergreen stretch of grass that comprised his backyard, into a paintless cypress barn left over from an earlier time.\n\nThe priest lowered the window glass and Dale Louviere fell backward into a smell of rotted straw, hard-packed, damp earth, and horse manure that powdered into dust. The priest cut the engine on the car and got out, a .45 automatic hanging from his right hand. \"I have nothing against coppers. Except those who are no better than me and pretend otherwise. On which side of the line would a fellow like you fall, sir?\" he said.\n\nAgain Dale Louviere heard a kicking sound in the Honda's trunk but could not think of anything except the violent pounding in his own chest.\n\nAt 10:55 A.M., while Father Jimmie Dolan sat in a St. Mary Parish courtroom, cuffed to a wrist chain with a collection of drunks, pipeheads, prostitutes, and wife batterers, the prosecutor's office received a call from Dale Louviere. He indicated he was resigning his job and, for personal reasons, moving to an undisclosed city out of state. He also said there was no substance to the charges against Father James Dolan and that his colleague, Cash Money Mouton, who had made the arrest in the public rest room, would confirm the same, provided he could be found.\n\nClotile Arceneaux, Father Jimmie, and I walked out the front door of the courthouse together. The rain had stopped and the town looked washed and clean, the trees green against the grayness of the day, the ebb and flow of the traffic on a wet street somehow an indicator of the world's normalcy.\n\n\"What happened in there?\" Father Jimmie said.\n\n\"I wouldn't worry about it,\" I said.\n\n\"Max Coll is behind this, isn't he?\" he said.\n\n\"Who cares? Those guys deserve anything that happens to them,\" I said.\n\n\"I thought New Orleans was tough. Y'all have death squads over here?\" Clotile said.\n\nI started to make a flippant reply, but saw the troubled expression on Father Jimmie's face. \"I have to get my car from the pound,\" he said.\n\n\"We'll see you at the house. Let it slide, Jimmie,\" I said.\n\n\"One of those men may be dead,\" he replied.\n\nHe walked down the street, his black suit rumpled and stained from sleeping overnight on a cement jailhouse floor.\n\n\"Your friend isn't easily consoled, is he?\" Clotile said.\n\n\"Ever hear about the Jewish legend of the thirteen just men who suffer for the rest of us?\"\n\n\"No. What's the point?\"\n\n\"Some people have to do life in the Garden of Gethsemane,\" I said.\n\nShe picked up my left hand and looked at it, her fingers cool on my skin. \"This is where those greaseballs put the pliers to you?\" she said.\n\n\"Yes.\"\n\nShe patted the top of my hand and released it. \"Take care of your own ass for a change,\" she said.\n\n## Chapter 20\n\nFather Jimmie had not been back at my house ten minutes when the phone rang in the kitchen. He picked it up but did not speak, his breath audible in the silence.\n\n\"Ah, you're a clairvoyant as well as a spiritual man,\" the voice on the other end said.\n\n\"Leave me alone. Please,\" Father Jimmie said.\n\n\"I got you, didn't I?\"\n\n\"What do you mean?\"\n\n\"Oh, you know what I mean, sir. It took a bastard like me with blood on his hands to get you out of the slams. Now it's you who owe me.\"\n\n\"What did you do with those men?\"\n\n\"They're both alive and probably enjoying a cool drink in a warm climate by now. I think one of them mentioned Ecuador. Have to say, though, I was tempted to release them from their earthly bonds.\"\n\nFather Jimmie sat down in a chair and tried to think. \"Perhaps you mean well, but you cannot use violence to solve either your problems or mine,\" he said.\n\n\"What do you know of violence, sir? What do you fucking know of it?\"\n\n\"You're full of hatred, Max. Get it out of your life. You injure yourself with it more than others.\"\n\n\"If I came into your confessional, would you give me absolution?\"\n\n\"Yes.\"\n\n\"There are a couple more house calls I'd like to make.\"\n\n\"You don't negotiate the terms of forgiveness.... Max? Did you year me?\"\n\nBut Max Coll had hung up. Father Jimmie leaned his head down on his hand, the stink of the jail still on his clothes, Snuggs the cat pacing back and forth on the table, his tail dragging across Father Jimmie's face. He felt more tired than he had ever been in his life, vain and used up, now sullied by the accusation of molester, even though it was a lie.\n\nHe knew the rumor would always follow him, regardless of where he went or what he did. A wave of revulsion and anger washed through him and made him clench his fists. Is this what all the years in the seminary, the struggles with celibacy and bigots and dictatorial and obtuse superiors had been about? To end up with his name and life's work soiled by an accusation that made his skin crawl?\n\nWhy didn't he quit running a game on himself? He posed as the altruist, but other people constantly had to get him out of trouble. If he had wanted to be a true missionary and take real risks, why hadn't he joined the Maryknolls? He disdained the role of the traditional priest, but in his self-imposed piety he had become little more than a noisy gadfly dedicated to causes Carrie Nation might have supported.\n\nHe had just lectured a tormented man on his violence, although he, Jimmie Dolan, had just profited from it, and if truth be known he was glad he was on the street and perhaps secretly glad his false accusers had gotten their just deserts.\n\nBetter to marry than to burn, St. Paul had said. Better to be a bourbon priest or a diocesan sycophant than a self-canonized fool, Father Jimmie thought.\n\n\"What do you think, Snuggs?\" he said.\n\nSnuggs answered by nudging his head into Father Jimmie's chin.\n\nFather Jimmie went into the bedroom, flung his clothes in the corner, and got under the shower. The water coughed in the pipes, then seemed to whisper the word hypocrite in his ear.\n\nThe South has changed dramatically since the civil rights legislation of the 1960s. Anyone who says otherwise has either not been there or wishes to keep old wounds green and tender as part of a personal agenda. And nowhere has the change been more visible than in the once recalcitrant states of the Deep South.\n\nBut that evening, when I took Clotile Arceneaux to supper on East Main, I tried to convince myself otherwise. I told myself the furtive glances at our table, the awkwardness of friends who felt they should stop by and say hello, were expressions of narrowness and latent racism to be expected in our culture.\n\nThe truth was no one took exception to Clotile's race. But they did take exception to my being out with another woman in less than a year of Bootsie's death.\n\nIt had turned cold again when we left the restaurant. Stars were spread across the sky, the horizon flaring with stubble fires, smoke boiling out of the electric lights at the sugar mills.\n\n\"You a little uncomfortable in there about something?\" Clotile asked.\n\n\"Not me,\" I replied.\n\nShe opened the door to my pickup by herself and got in and closed it behind her, although I had tried to help her in. \"You're really out of the past, aren't you?\" she said.\n\n\"Probably,\" I said.\n\nShe smiled and didn't say anything. We drove toward the drawbridge and the theater complex on the other side of Bayou Teche. She had checked in to a motel out by the four-lane that afternoon.\n\nWe crossed the bayou and turned in to the theater parking lot. It was filled with teenagers, long lines of them extending out from the ticket windows.\n\n\"Friday night is a bad night for the movies here,\" I said.\n\n\"We don't need to go,\" she said, looking straight ahead.\n\nI turned around in the parking lot, recrossed the bayou, and drove up East Main, without destination. The street seemed strangely empty, the stars shut out by the canopy of oaks overhead, my rented shotgun house dark and blown with unraked leaves. I hesitated, then pulled into my driveway and cut the engine. The ground fog in the trees and bamboo glistened in the lights from City Park across the bayou.\n\n\"Where's Father Dolan?\" she asked.\n\n\"Staying with friends in Lafayette.\"\n\n\"You have a lot of regrets in your life, Robicheaux?\" she said.\n\n\"All drunks do,\" I replied.\n\n\"How do you deal with them?\"\n\n\"I don't labor over them anymore.\"\n\nShe still looked straight ahead. \"I don't want to be a regret in somebody's life,\" she said.\n\n\"Want to meet my cat?\" I said.\n\nAnd that's what we did. I introduced her to Snuggs; then we ate ice cream in the kitchen and I drove her to her motel.\n\nAfterward I went to the cemetery in St. Martinville and sat on the steel bench by Bootsie's tomb and watched the moon rise over the old French church on the bayou.\n\nThat night I dreamed I was in New Orleans in an earlier era, riding on a streetcar out to Elysian Fields. The streets were dark, the palm fronds on the neutral ground yellow with blight. No one else was on the car except the motorman. When he turned and looked back at me his eyes were empty sockets, the skin on his face dried and shrunken into little more than gauze on his skull.\n\nOftentimes police cases are not solved. They simply unravel, by chance and accident. With good luck there will even be an appreciable degree of justice involved, although it often originates from an expected source.\n\nEarly the next morning, Saturday, my lawn was white with frost and the bamboo on the side of the house was stiff and hard and rattled like broomsticks in the wind. I put on my sweat suit, ran three miles through City Park, then showered and drove down to Clete's cottage in the motor court.\n\nHe sat on the side of his bed in the coldness of the room, sleepy, shivering slightly, wearing only a strap undershirt and pajama bottoms. The wastebasket in his kitchen was stuffed with fast-food containers and beer cans.\n\n\"You want to do what?\" he said.\n\n\"Eat breakfast at McDonald's, then maybe knock down some ducks at Pecan Island,\" I said.\n\n\"I'm busy today,\" he replied.\n\n\"I see.\"\n\nIt was quiet in the room. His eyes lingered on mine. \"What's bothering you, big mon?\" he said.\n\nI told him about the dream, the motorman with the skeletal face, the darkness outside the streetcar, the yellowed palm fronds that clattered like bone. \"You ever have a dream like that?\" I said.\n\n\"I used to dream I was on a Jolly Green that was going down. But that was in the hospital in Saigon. It doesn't mean anything. It's just a dream.\"\n\n\"I can't shake it,\" I said.\n\nHe got up from the bed and began dressing. \"Turn on the heat, will you? It feels like it's thirty below in here,\" he said.\n\nWe ate at the McDonald's on East Main. Outside, the sky was blue, the leaves of the live oak in the adjacent lot flickering in the sunlight. \"Can't tempt you into a duck-hunting trip?\" I said.\n\nHe wiped his mouth with a crumpled napkin and dropped it onto his plate. \"That perv I told you about, Bobby Joe Fontenot, the one in the trailer court? I couldn't stop thinking about what he said to me.\"\n\n\"Said what?\"\n\n\"That if he reoffended, he was going to use my name every time he stuck it to a little kid. So I called the perv's P.O. Guess what? The P.O. is on vacation. So I told the guy handling his case file about the little boy in the trailer next door. He did everything except yawn in my ear.\"\n\n\"Call Social Services,\" I said.\n\n\"I already did. I think that kid is shark meat.\"\n\nHe gathered up the trash from both our meals and stuffed them angrily into a bin.\n\n\"Take it easy, Cletus,\" I said.\n\n\"Screw the ducks. Time to spit in the punch bowl,\" he said.\n\nThe mother of the little boy in the trailer court was named Katie Goltz. She sat with us in her tiny living room, still not connecting the reasons we were there, even though Clete mentioned he had been chasing down a bail skip who was the fall partner of Bobby Joe Fontenot, a convicted sex predator living next door.\n\nShe wore no lipstick, old jeans, Indian moccasins, and a colorless pullover. Her hair was cut short, and had probably been brown before it was peroxided and waved on one side to resemble a 1940s leading lady's.\n\n\"Where's your son?\" Clete said.\n\n\"At the strip mall,\" she replied.\n\nClete nodded. \"He went with some friends?\" he asked.\n\n\"Bobby Joe took him. To buy him a comic book for helping clean his trailer,\" she said.\n\nClete leaned forward in his chair. \"Ma'am, we have a Meagan's Law in Louisiana. You must have been notified about Bobby Joe Fontenot's record,\" he said.\n\n\"People change,\" she said.\n\n\"You listen to me. That guy is a degenerate. You keep your son away from him,\" Clete said.\n\nShe focused her eyes on a neutral space, her hands folded in her lap. Her arms were muscular, as though she had grown up doing physical work, her complexion clear. Behind her, framed on the wall, was a black-and-white photograph of her and a man who looked like a power lifter. His hair was shaved on the sides, curly in back, his face impish, like a cartoon drawing of a monkey's.\n\nI stood up and looked closer at the picture. It was inscribed \"To Katie Gee, the girl who made my own screen role a real pleasure, Your pal, Phil.\"\n\n\"That's Gunner Ardoin,\" I said.\n\n\"'Gunner' is his nickname. Phil is his real name. You know him?\" she said.\n\n\"He was involved with the beating of a priest in New Orleans. You made a film with him?\" I said.\n\nShe frowned, unable to process all that she just heard. \"I made just one film. My screen name is Katie Gee. The producer said 'Gee' looks better than 'Goltz' on the credits. Phil was my costar. What was that about a priest?\" she said.\n\n\"You were in one of Fat Sammy Figorelli's porn films?\" Clete said.\n\n\"They're art films. They're shown in art theaters. Listen, nobody has hurt my little boy. I wouldn't let that happen. I have to go to the washateria now,\" she said.\n\nThere seemed nothing left to say. Her mindset, formed out of either desperation, ignorance, or just plain stupidity and selfishness, was armor-plated, and in all probability no amount of attrition in her life or her son's would ever change it.\n\nBobby Joe Fontenot pulled up outside, wearing a foam-rubber collar, his face marbled with bruises. When the little boy got out of his car, Bobby Joe cocked his index finger at him, as though he were pointing a gun, and said, \"Come over and watch some TV tonight. I got some Popsicles.\"\n\nClete and I got up to go, our mission by and large a failure. Her son rushed past us into his bedroom, a new comic book rolled tightly in his hand. Clete twisted the handle on the front door, then stopped and turned around. \"It's not coincidence you let that geek be alone with your kid. There's a financial motive here, isn't there?\" he said.\n\n\"Coincidence?\" she said.\n\n\"You've got more than a neighborly relationship with that asshole next door. He knows you were working the trade around Folk Polk,\" Clete said, tapping the air with one finger. \"Fontenot's in porn films, too, isn't he?\"\n\n\"I'm not saying any more. I have to go to the washateria and fix lunch and do all kinds of things I don't get no help with. Why don't y'all just leave now? I didn't do anything to cause this, and you can't say I did,\" she said.\n\nShe stared at us indignantly, her arms folded across her breasts, as though the irrefutability of her logic should have been obvious to anyone.\n\nClete and I crossed the Teche on the drawbridge behind the trailer court and headed toward New Iberia on the back road, past the row of oak-shaded antebellum homes that belonged on a movie set. Then he mashed on the gas, one hand on top of the steering wheel, the sugarcane fields racing past us, a crazy light in his eyes.\n\n\"What are you thinking about, Clete?\"\n\n\"Nothing. I'll drop you off,\" he said.\n\n\"Clete?\"\n\n\"Everything is copacetic. Just hang loose. I'll check in with you later,\" he said. He whistled an aimless tune under his breath.\n\n## Chapter 21\n\nAt 10:15 Monday morning I received a call from Clotile Arceneaux. \"Did you hear from the FBI yet?\" she asked.\n\n\"No,\" I said.\n\n\"You will. They just left here. They want to put a net over Max Coll real bad,\" she said.\n\n\"A guy crossing state lines to commit a homicide? I guess they would.\"\n\n\"No, you've got it wrong. It's face-saving time. Because he's IRA, he's on a terrorist watch list. In fact, he's been on one for three years. Except he's been going back and forth across the Canadian border like a yo-yo, making a lot of people look like shit.\"\n\n\"That's their problem,\" I said.\n\n\"You're not hearing me. The Feds believe Coll is...\" She paused and I heard her shuffling papers around. \"They say he's a nonpathological compulsive-obsessive with paranoid and antisocial tendencies.\"\n\n\"Antisocial tendencies? This is the kind of crap that comes out of Quantico. Don't buy into it.\"\n\n\"Will you shut up? They're saying Coll kills people because he feels he has a right to. He's not a psychopath or a schizophrenic or anything like that. He's just a very angry man. Have I got your attention?\"\n\n\"Yes,\" I said.\n\n\"He had a wife and son in Belfast nobody in law enforcement knew about. They used a different name so Coll's enemies wouldn't find them. But about five years ago a Protestant death squad of some kind put a bomb under their car and killed both of them. They were on their way to Mass.\"\n\nThe subject wasn't funny anymore.\n\n\"Is there a tap on my home phone?\" I asked.\n\n\"We're in the George W. Bush era. I'd keep that in mind,\" she said.\n\nFifteen minutes later Helen came into my office, a clutch of fax sheets in her hand. \"Did you hear anything about an explosion on the drawbridge in Jeanerette?\" she asked.\n\n\"No,\" I said.\n\nShe sat on the corner of my desk and studied the fax sheets in her hand. \"This is from the St. Mary's sheriff's office. See what you think,\" she said. Her jawbone flexed against her cheek.\n\nI took the sheets from her hand and read them, trying not to show any expression. The details of the investigator's report were incredible. In the early A.M. someone had evidently slim-jimmed a wrecker that was parked in a filling station located a half block from the trailer court by the Jeanerette drawbridge. After hot-wiring the ignition, the perpetrator drove the wrecker down to the trailer court, hooked up the winch to a trailer owned by one Bobby Joe Fontenot, and ripped it off its cinder blocks, tearing loose all the plumbing, electrical, phone and cable connections.\n\nAccording to witnesses, the owner tried to exit the trailer but discovered the door had been sealed shut with a bonding adhesive used to repair the bodies of wrecked automobiles. The perpetrator skidded the trailer out of the court onto the surfaced road, bouncing it across a drainage ditch, smashing mailboxes and parked cars. When the trailer toppled on its side, witnesses thought they saw the owner trying to climb out of an exposed window. But the driver of the wrecker accelerated, knocking Fontenot, the owner, back inside. The driver then dragged the trailer across the steel grid of the drawbridge, geysering rooster-tails of sparks in the darkness.\n\nA liquid blue flame enveloped one of the butane tanks on the rear of the trailer. The explosion that ensued blew burning paper, fabric, and particleboard all over the bayou. The owner, who by this time had broken out a window and cleaned the glass from the frame with a hammer, barely escaped with his life.\n\nThe perpetrator abandoned the wrecker and burning trailer, which was tightly wedged between the steel side beams on the bridge, and disappeared into the darkness on the far side of the bayou. A moment later an ancient Cadillac convertible was seen speeding down the road toward New Iberia, the engine misfiring, leaking oil smoke, the driver wearing a small, short-brim hat perched on the front of his head.\n\n\"Wow, that's something, isn't it?\" I said, handing the fax sheets back to Helen.\n\n\"Any idea who could pull a stunt like that?\" she said.\n\n\"There're a lot of old gas guzzlers like that around,\" I replied, my eyes drifting around the room.\n\n\"Right,\" she said.\n\n\"No mention of the Cadillac's color?\"\n\n\"Nope,\" she said.\n\n\"It's not in our jurisdiction, anyway. Let St. Mary Parish do some work for a change.\"\n\n\"You get Clete Purcel in here right now,\" she said.\n\nBut Clete did not answer his phone, and when I drove by the motor court, the manager told me he had not seen Clete's car in the last day or two. I called Clete's office in New Orleans. The temporary secretary he sometimes used was an ex-nun by the name of Alice Werenhaus who put the fear of God in some of Clete's clients.\n\n\"You are Mr. Robicheaux?\" she said.\n\n\"I was when I got up this morning,\" I replied, then quickly regretted my mistake in attempting humor with Alice Werenhaus.\n\n\"Oh, it is you, isn't it? I should have immediately recognized the quick wit at work in your rhetoric,\" she said. \"Mr. Purcel left a message for you. Would you like me to read it to you?\"\n\n\"Yes, that would be very nice, Ms. Werenhaus,\" I replied.\n\n\"It says, 'Give Alice a pay phone number and a time. Fart, Barf, and Itch probably have you tapped.' \"\n\n\"What's going on?\" I said.\n\n\"I suspect that's why he'd like to talk with you, Mr. Robicheaux. To explain everything to you. I'm sure by this time you're rather used to that,\" she said.\n\nI walked downtown and got the number off a public telephone and called it back to Alice Werenhaus. \"I'll be at this number at one P.M.,\" I said.\n\nI expected another rejoinder at my expense. But she surprised me. \"Mr. Robicheaux, be careful. Watch after Mr. Purcel, too. Under all his bluster he's a vulnerable man,\" she said.\n\nAt 1:04 P.M. the payphone across from Victor's Cafeteria on Main Street rang. I picked it up and didn't wait for Clete to speak. \"Have you lost your mind?\" I said.\n\n\"About what?\" he said.\n\n\"You stole a tow truck out of a filling station. You almost burned Bobby Joe Fontenot to death in his trailer. The drawbridge in Jeanerette is still closed with the melted wreckage you left on top of it. Boat traffic is backed up ten miles.\"\n\n\"Oh, yeah, that,\" he replied. \"Things got a little out of hand. Look, big mon\u2014\"\n\n\"No, you look, Clete. Helen wants to feed you into an airplane propeller.\"\n\n\"She's emotional sometimes. I talked with Clotile Arceneaux. She says your phone is tapped.\"\n\n\"I already got that. Listen to me\u2014\"\n\n\"You think the Feds are tapping a cop's phone because they're worried about an Irish button man whacking out a couple of greaseballs? These guys still haven't found Jimmy Hoffa. It's Merchie Flannigan and his wife they're worried about.\"\n\n\"You're making no sense.\"\n\n\"That broad's been giving you a hand job. I did some checking on Merchie's company. He's in line for some big drilling contracts in Iraq after Shrub turns it into an American colony. That means his father-in-law, what's-his-face, Castille LeJeune, is probably mixed up in it, too. The Feds are after Coll because he's about to pop somebody with a lot of juice, not because they're worried about Coll trying to kill a Catholic priest or smoking the Dellacroce brothers.\"\n\nIt was pointless to argue with Clete. He was the best investigative cop I ever knew, his blue-collar instincts for deception and hypocrisy and flimflam always on target. But his antipathy toward Federal law enforcement agencies, particularly the FBI, was unrelenting, and at best he considered them bumbling and inept and at worst lazy and arrogant.\n\n\"Why'd you say Theodosha Flannigan was giving me a hand job?\" I asked.\n\n\"She and her husband are business partners. She set you up to either get drunk or clipped, she didn't care which. Rich broads look after their money first and think about the size of your Johnson second. You think she's going to let a guy like you screw up her family's finances?\"\n\n\"You really know how to say it, Cletus.\"\n\n\"You want to be a dildo for this broad, that's your choice. She's dirty, Streak, just like her husband and her old man.\"\n\n\"What are you up to?\"\n\n\"I told you before, I'm going to make cripples out of the shitheads who hurt you. Get this. I saw a guy in Franklin who looks just like your description of Max Coll.\"\n\n\"Stay away from him, Clete.\"\n\n\"Lose a resource like that? By the way, what's the name of that electrician who burned down your house?\"\n\nI started to give him the name, then refused.\n\n\"That's all right. I already had a talk with him. He might be contacting your department, but don't believe anything he says.\"\n\nLater, I went into Helen's office. She was on the phone, nodding, while someone on the other end talked, her eyes on mine. \"All right, we'll take care of it.... I agree with you. Absolutely.... This isn't the Wild West. You got it,\" she said, and hung up.\n\nHer face looked scorched.\n\n\"Who was that?\" I asked.\n\n\"The Lafayette sheriff. An electrical contractor by the name of Herbert Vidrine was pulled out of his house at around six-thirty this morning and worked over in his backyard,\" she said.\n\nShe looked down at the yellow legal pad on her desk, widening her eyes, as though she could not quite assimilate what she had just heard and written down. \"By 'pulled out,' I mean just that. His attacker was wearing work gloves of some kind and grabbed Vidrine by the mouth like he was picking up a bowling ball,\" she said. \"He swung him around in a circle and threw him into the side of a garbage truck. Vidrine is in Our Lady of Lourdes now. A neighbor got the tag number of the attacker's car. A lavender Cadillac convertible. Guess who it belongs to?\"\n\n\"I just talked to Clete on the phone. He's not coming in,\" I said.\n\n\"The electrical contractor is too scared to file charges. But Clete's not going to use Iberia Parish as his safe house while he goes around kicking people's asses.\"\n\nI nodded.\n\nThe heat went out of her face. \"What's the score on this electrical contractor?\" she said.\n\n\"He's the guy who installed bad wiring in my house. He works for Will Guillot.\"\n\n\"I'm fed up with the stuff, Dave. Clean it up or you and Clete can start making your own plans,\" she said.\n\nI took the old highway through Broussard into Lafayette and hit a rainstorm just outside of town. By the time I got to Our Lady of Lourdes Hospital the streets were flooding. I ran past a row of blooming camellia bushes into the side entrance of the hospital and asked at the nurse's station on the second floor for directions to Herbert Vidrine's room.\n\n\"Three rooms past the elevator, on your left,\" the nurse said.\n\nI thanked her and started down the hall. Then I stopped and went back to the station. I opened my badge holder. \"How's Mr. Vidrine doing?\" I asked.\n\n\"A concussion and a broken arm. But he's doing all right,\" the nurse replied. She was young and had clean features and brown hair that was clipped on her neck.\n\n\"Has anyone else been in to see him?\"\n\n\"Not since I've been here. I came on at eight A.M.,\" she said.\n\n\"Could I use your typewriter?\" I said.\n\nI had taken a fiction-writing course when I was an English education major at Southwestern Louisiana Institute. I hoped my old prof, Lyle Williams, would be proud of the letter I was now composing. I typed rather than signed a name at the bottom, folded and put the letter in an envelope the nurse gave me, then printed Herbert Vidrine's name on the outside.\n\n\"Would you wait ten minutes, then deliver this to Mr. Vidrine's room?\" I said.\n\n\"I don't know if I should get involved in this,\" she replied.\n\nI placed the envelope on her desk. \"You'd be helping out the good guys,\" I said.\n\nVidrine was sitting up in bed when I entered his room, one arm in a cast, easing a teaspoon of Jell-O past a severely swollen bottom lip.\n\n\"How are you, Herbert?\" I said.\n\nHe put his spoon back in a bowl that rested on his bed tray. \"You're Iberia Parish. What are you doing here?\" he said.\n\n\"We're looking for the guy who hurt you but on different charges,\" I said, laying my raincoat and hat on a chair.\n\n\"Maybe you're here to rub salt in a wound, too,\" he said.\n\n\"You burned my house down, partner. But I'm like you, I'm a drunk. I can't carry resentments. Did you ever go back to meetings?\"\n\nHis eyes left mine. Even though he was a hard-bodied man, he looked small in the bed, his spoon clutched in a childlike fashion. \"I never had that big a drinking problem. It was just when I was married,\" he said.\n\n\"The man who attacked you didn't have the right to do what he did,\" I said.\n\nHe frowned and ran his tongue over the swelling in his bottom lip. \"Just leave me alone,\" he said.\n\n\"One day you're going to have to do a Fifth Step on the injury you caused me and my family. My father built that house in the Depression with his own hands. My second wife was murdered in it. Her blood was in the wood,\" I said.\n\n\"I'm sorry,\" he said.\n\n\"Maybe you are,\" I said. I put my business card on his nightstand. \"I think you have a lot of information about the dealings of some bad people, Herbert. Why take their bounce?\"\n\n\"I haven't done anything wrong,\" he replied.\n\nI drummed my fingers on top of the chair where my raincoat rested and looked out the window at an oak tree whipping in the wind, its leaves shredding high in the air. Then I picked up my raincoat and left, just as the nurse entered with the letter I had typed at the nurse's station.\n\n\"This was left for you, Mr. Vidrine,\" I heard her say behind me.\n\nI waited five minutes, then reentered Vidrine's room. \"I forgot my hat,\" I said, picking it up from the chair.\n\nThe letter I had written lay unfolded on top of his bed tray. He was staring into space, his expression disjointed, like a man at a bus stop who has watched the bus's doors close in his face and the bus drive away without him.\n\nThe letter I had typed at the nurse's station read as follows:\n\nHerbert,\n\nSorry you got your ass stomped by that queer bait we had trouble with at the cafe in Jeanerette. But if you can't deal with a fat shit like that, I don't need you on the job. Take this as your official notice of termination. Also be advised you are forfeiting all fees due on uncompleted work.\n\nWill Guillot\n\n\"Something wrong?\" I asked.\n\n\"Yeah, there is. You want to know about Sunbelt Construction?\"\n\n\"Yeah, what's up with these guys?\"\n\n\"They got connections with gangsters in New Orleans.\"\n\n\"That's not real specific.\"\n\n\"Maybe they're selling dope. I'm not sure. But Will Guillot is going to take over the company. He's got something on the old man.\"\n\n\"Castille LeJeune?\"\n\n\"Yeah, him. The war hero.\"\n\n\"What does Guillot have on him?\"\n\n\"I don't know. I asked him once and all he said was, 'I finally got the goods on both him and that cunt.' I asked him which cunt he meant. He told me it wasn't my business.\"\n\n\"Ever hear the name of Junior Crudup?\"\n\n\"No,\" he said.\n\nIt had stopped raining outside. The sky was gray, the sun buried in a cloud like a wet flame, the hospital lawn blown with camellia petals. \"That's all you got for me, Herbert? It's not too much,\" I said.\n\n\"I'm an electrician. People don't confess their sins to me.\"\n\n\"See you around,\" I said.\n\n\"One time I told Will Fox Run was a beautiful place. He said, 'Don't let it fool you. All these places got a nigger in the woodpile.' I wasn't sure what he meant, though.\" He tilted his head inquisitively, waiting for me to speak, as if somehow we were old friends.\n\nSo Vidrine repeated a racist remark that confirms what you already knew,\" Helen said in her office an hour later. \"Maybe a convict was killed on the LeJeune plantation fifty years ago. Or maybe not. We didn't find a body, bwana.\"\n\n\"That's the point,\" I said. \"How could Will Guillot be blackmailing Castille LeJeune about the death of Junior Crudup? Guillot has something else on him.\"\n\n\"I'm glad we cleared that up. Now get out of here,\" she replied.\n\nI couldn't blame Helen for her feelings. The real issues were the murders of the daiquiri-store operator and Fat Sammy Figorelli, and in both instances we had no viable suspects. In the meantime I had gotten myself abducted, gotten deeply involved in a murder case from a half century ago, and had helped bring Max Coll to our community.\n\nAs a member of Alcoholics Anonymous, the axiom \"keep it simple\" was supposed to guide my daily life.\n\nWhat a joke.\n\nBut Helen herself had said the real problem lay in the fact we were dealing with Dagwood and Blondie. Amateurs hide in plain sight. They also do not feel guilty about the misdeeds they commit. They attend church, Kiwanis meetings, belong to the Better Business Bureau, support every self-righteous moral cause imaginable, and float like helium balloons right over whole armies of cops looking for miscreants in off-track betting parlors, triple-X motels, and crack houses.\n\nThe word criminal is more an emotional than legal term. Go to any U.S. post office and view the faces on the wanted posters. Like Dick Tracy caricatures, they stare out of the black-and-white photographs often taken in late-night booking rooms\u2014unshaved, pig snouted, rodent eyed, hare lipped, reassuring us that human evil is always recognizable and that consequently we will never be its victim.\n\nBut every longtime cop will tell you that the criminals who scared him most were the ones who looked and talked like the rest of us and committed deeds that no one, absolutely no one, ever wants to have knowledge of.\n\nFive or six years ago Helen and I had to fly to Deer Lodge, Montana, and question a kid whose execution was scheduled in three days. We were not prepared for what we saw when he was brought into the interview room in a short-sleeve, orange jumpsuit and leg and waist chains. His first name was Kerry, and the softness in his name was like both his features and his North Carolinian accent. He had no cigarette odor, no tattoos, no needle tracks. His auburn hair was shampooed, clipped on the ends, and kept falling across his glasses, so that he constantly twitched his head to shake a loose strand out of his vision.\n\nWhile we questioned him about a murder in Iberia Parish, his large glasses wobbled with reflected light and a strange, almost self-effacing smile never left his mouth. If he bore anger or resentment toward anyone, I could not detect it.\n\nHe had been sentenced to death for tying a rancher and his wife to chairs in their kitchen and butchering them alive. While on Death Row he helped organize a riot that resulted in the convict takeover of the entire maximum-security area. Kerry also was a chief participant in the fate of five snitches who were pulled out of protection cells, tortured, and lynched with wire loops from the second tier of a lockdown section.\n\nHe said he knew nothing of the homicide in Iberia Parish.\n\n\"Your fingerprints at the murder scene indicate otherwise. Maybe the victim had it coming. Why not get your interpretation of events on the books?\" I said.\n\nHe flipped his head to clear a strand of hair from his glasses and smiled at a joke that only he seemed to understand.\n\nWe gave it up. But before we left the interview room I had to ask him another question. \"What do you think lies on the other side, Kerry?\" I said.\n\nHe had a slight cold and couldn't wipe his nose because his hands were manacled at the waist, so he huffed air out of his nostrils before he answered. \"You just move on to another plane of existence,\" he said.\n\nThe afternoon of his injection he had to be awakened from a sound sleep. Minutes later the death warrant was read and he was videotaped by a member of the medical examiner's office on the way to the execution chamber. He grinned at the camera and said, \"Hi, Mom,\" and jiggled all over with laughter.\n\n## Chapter 22\n\nI went to bed early that night and listened to the rain hitting the tin roof of my rented house. The fog was white in the trees, a lighted tugboat out on the Teche, its gunnels hung with rubber tires, glistening inside the rain. I slept the sleep of the dead.\n\nThe time on my alarm clock was 4:16 A.M. when I heard the unmistakable sound of Clete's automobile engine dying in my driveway. A moment later he tapped softly on the front door. He was wearing gloves and a beat-up leather bomber jacket. The jacket was unzipped, and I could see his nylon shoulder holster and his blue-black, pearl-handled .38 revolver inside it.\n\n\"Where have you been?\" I said.\n\n\"At a fish camp on Lake Fausse Pointe. Get dressed. I know where Max Coll is,\" he said.\n\n\"No more cowboy stuff, Clete.\"\n\n\"Me?\" he said.\n\n\"Where is he?\" I said.\n\nClete stepped inside the living room and started to explain, looking back over his shoulder at the street, then got vexed at being conciliatory. \"You want in on this or not?\" he said.\n\nI left a note on the kitchen counter for Father Jimmie, then Clete and I headed out in the predawn wetness for New Orleans, a thermos of coffee and a box of beignets on the seat between us. The old homes along East Main were still dark, the live oaks dripping on the sidewalks. I was still not quite awake.\n\n\"Run it by me again,\" I said.\n\n\"Janet Gish is trying to get off the nose candy without a program, so she spends most of the night at Harrah's. She says a guy with a Mick accent was in the casino until early Saturday morning, then he left just before seven. He came back at eight-thirty, ate a plate of steak and eggs, played some more blackjack, and drove off in a Honda.\"\n\n\"Why was she paying so much attention to a guy with an accent?\" I asked.\n\n\"One, I'd already described Coll to her, and, two, she still hooks a little on the side and thought he'd be an easy trick. Here's the rest of it. He had on black dress pants, like a priest might wear.\"\n\nIt was raining and still dark when we crossed the high bridge over the Atchafalaya at Morgan City. Down below I could see shrimp boats in their berths, the red-tiled roofs of the town, and the great, cypress-dotted expanse of the wetlands in the south, all of which were being eaten away by saltwater intrusion at a rate of hundreds of square miles a year.\n\n\"Doesn't your heater work?\" I said.\n\n\"It's full blast, mon.\"\n\nClete's cell phone rang. He answered it, listened, then said thanks to someone and clicked it shut again. \"That's Janet. The guy who looks like Coll is still there. By the way, she's got a porn lead for us, too,\" he said.\n\nWe crossed the wide sweep of the Mississippi just as the first cold band of light, like the blunt edge of a sword, appeared on the eastern horizon. Then we were rolling down I-10 past the southern shore of Lake Pontchartrain, into the heart of the city, the welfare projects, the cemeteries where the dead were entombed in white brick, the homeless and the hopelessly addicted gathered around fires next to the cement pillars that supported the elevated highway.\n\nAt the head of Canal Street stood the casino, the royal palms at the entrance beaded with rainwater in the graying of the dawn. The gamblers inside were not a group that took note of changes in either weather or clocks. The rain might beat against the windows and lightning flicker on the streets outside, but the blacks and Hispanics and blue-collar whites who crowded the tables or fed the endless banks of slot machines were committed to their own form of solipsism, one in which the amounts that were lost or gained were far less important than the gamblers' desire to stay in the game, to be a part of the action, at the table or in front of the machine, until they were physically and emotionally sated in a way no sexual or narcotic experience can equal.\n\nJanet Gish was at the bar, a scotch and milk in front of her. Her hair was currently orange, stiff with spray, the tops of both breasts tattooed with a blood red star, her skin rough grained, freckled, layered with makeup. But in spite of all the cosmetics and chemicals she used on herself, she had one natural gift that was unimpaired by the life she lived. Her eyes were like a doll's, with weighted lids that clicked open suddenly, so that she always seemed surprised, somehow still vulnerable.\n\nShe turned on the stool, drew in on her cigarette, and looked at us without expression. \"Lend me twenty bucks, Streak?\" she said.\n\nI took out my wallet and found fifteen. She took it and slipped it under her glass. \"I got to get out of this shit. I just dropped three hundred in a half hour. How about lunch at Galatoire's? God, I hate this place,\" she said, although I had no idea which place she meant.\n\n\"On the clock today. You know how it is,\" I said.\n\nShe was obviously stoned or drunk or both, staying off coke with booze and baccarat, paying the rent with fifty-dollar tricks, starting her daily routine at 4 P.M. with eyewash, thirty-minute hot showers, and white speed on the half shell. Anyone who thinks prostitution is a victimless crime needs his head drilled with a brace and bit.\n\n\"Where's our Irish friend?\" I asked.\n\n\"Just went out the door. Like voom,\" she replied.\n\nClete's face reddened with exasperation. \"Why didn't you call?\" he said.\n\n\"It's been a long night. I don't need criticism right now. I just don't need that kind of unjustified negativity in my morning,\" she said, a thin wire quivering in her throat.\n\n\"Right,\" he said, glancing up and down the bar.\n\n\"Because if that's why you two are here, I'll just go back to the tables,\" she said. She gestured at the bartender. \"This milk is curdled. Give me a tequila sunrise.\"\n\n\"We appreciate everything you've done for us, Janet. How long has our man been gone?\" I said.\n\n\"Ten minutes,\" she said.\n\n\"You saw him drive away?\" I asked.\n\n\"No, he was walking. Right up Canal. Like he was in a hurry,\" she said.\n\n\"When he left Saturday morning for an hour or so, did he walk or drive?\" I asked.\n\nShe thought about it. \"He walked down Canal. Just like this morning,\" she replied.\n\n\"Stay here, Cletus,\" I said.\n\n\"Oh, I got it. I just drive people around, then turn into an ashtray. I'm glad I'm your friend, Dave, because otherwise I don't think you'd have any,\" he said, screwing an unlit Lucky into his mouth.\n\nI didn't try to explain. I hurried down Canal, past smoking sewer grates and gutters dark with rainwater, to the side street that led into the dilapidated downtown area where Father Jimmie Dolan's church was located, like a fifteenth-century fortress inside which its inhabitants refused to accept a tidal wave of ecclesiastical change.\n\nThe early-morning Latin Mass had already begun when I entered the vestibule and dipped my hand in the holy water fount. In a back pew, hard by a marble pillar, I saw the diminutive form of Max Coll, next to a group of elderly, head-covered women, all of whom had rosary beads threaded through their fingers. He wore black trousers and a puffy, tan down jacket that was zipped halfway up his chest.\n\nMy cell phone was in my pocket, my .45 automatic in a clip-on holster attached to my belt. I started to punch in a 911 call on the phone, then thought better of it and instead genuflected at the end of the pew and knelt down next to Max Coll.\n\n\"Walk out of here with me,\" I whispered.\n\nHe glanced at me and showed no sign of either recognition or alarm. \"Bugger off,\" he said.\n\n\"No one needs to get hurt here,\" I said.\n\nHe ignored me and concentrated on the missal in his hands.\n\n\"I know some evil men killed not only your natal family but your wife and son as well,\" I said. \"Both my mother and my second wife died at the hands of murderers. I can understand the feelings you've had to deal with over the years. I think many of the people you killed were bastards and deserved what they got. But it's time to give it up. Take a walk with me, Max. You know it's the right thing to do.\"\n\nOther people were beginning to look at us. \"You're disturbing the Mass, Mr. Robicheaux. Now show some respect and shut your 'ole,\" he replied.\n\nParishioners who had come in late, one of them weighing at least three hundred pounds, began bottling up the open end of the pew. I was trapped with Max Coll. I thought I might have a chance at him during communion, but as soon as the communicants began filing toward the front of the church, Max helped an elderly woman into a wheelchair and pushed her to the altar.\n\nI stayed right behind them, received the Host myself, which he did not, and followed them back into the pew. Through the concluding prayers he kept his eyes straight ahead, one thumb hooked inside his half-zipped jacket. Just as the priest gave the final blessing to the congregation, Max turned to me and calmly whispered, \"Got a Beretta nine-millimeter, fourteen rounds in the mag, all tucked nicely under my armpit. Try to take me and, House of the Lord or not, I'll leave hair on the walls.\"\n\nWith that, he wheeled the elderly woman down the center of the aisle and through a crowd in the vestibule, like a mummy wrapped in black cloth being trundled along a cobbled street. He and two other men lifted her down the steps and fitted her chair into a waiting van, then suddenly Max Coll leaped into the traffic.\n\nI went after him, my shield held up above my head, a wall of water from a passing truck striking me full in the face, horns blowing, a taxi missing me by inches. Somewhere on the edge of my vision two vehicles crashed into each other. Max was now somewhere on the opposite side of the traffic, hidden behind a city bus or a Mayflower van or a refrigerator truck, all of which were moving through the intersection.\n\nI reached the opposite sidewalk and looked in both directions.\n\nNo Max Coll.\n\nI saw the bus stop briefly on the next block, then it turned a corner and headed in the direction of Lee Circle. I started running, threading my way through pedestrians, truck drivers off-loading food for restaurants, winos sitting in doorways with their legs outstretched on the sidewalk. I turned the corner and saw the bus at the curb in the middle of the block, the door opened to allow a passenger to exit.\n\nI ran toward it, breathless, waving my arms at the driver. As the bus pulled away from the curb I struck the side with my fists. Behind the elongated glass windows in the back door I saw Max Coll standing in the aisle, holding a support strap with one hand. He grinned, unzipped his jacket, and pulled out the sides to show me he had no weapon on his person.\n\nThe bus sped through the next intersection and disappeared down the street. I reached for my cell phone to punch in a 911 call, then remembered hearing it clatter across the sidewalk two blocks behind me.\n\nI stopped in the men's room at the casino and tried to dry off with paper towels before I went in search of Janet Gish and Clete Purcel. A few minutes later, my clothes glued to my skin, I found the two of them eating breakfast in the restaurant, Janet looking half revived by food and coffee. Clete chewed his food thoughtfully, his eyes traveling up and down my person. \"I'm not even going to ask,\" he said.\n\n\"He was at Mass. He got away,\" I said.\n\n\"At Mass? A stone killer?\"\n\n\"I just told you.\"\n\n\"So instead of calling the locals, you decided to talk him in?\" he said.\n\n\"Something like that,\" I replied.\n\n\"Couldn't have used any backup from me, of course?\"\n\n\"Lay off it, Clete,\" I said.\n\nHe took a coffee cup and saucer that was set up on an empty table, poured the cup full, and pushed it toward me. \"Sit down, big mon, and let Janet tell you how Fat Sammy was shipping porn out and crystal in,\" he said.\n\n\"It all had to do with those Mideastern degenerates,\" she said.\n\n\"Those what?\" I said.\n\n\"Those Muslim lamebrains or whatever who crashed the planes into the towers. Sammy Fig said he was going to round them up for the FBI,\" Janet said.\n\nI gave Clete a look.\n\n\"You're going to love this, Streak. Sammy straightening out Fart, Barf, and Itch,\" he said.\n\nIt seemed a grandiose and bizarre tale, but in truth no more peculiar than many in New Orleans' long history of political intrigue, from William Walker's military adventurism into Nicaragua during the 1850s to Lee Harvey Oswald's involvement in the city with the Fair Play for Cuba Committee.\n\nAccording to Janet Gish, Fat Sammy felt tainted by a past association with a mobster who had been an enforcer in Brooklyn and later one of the Watergate Plumbers. The mobster was part of a blackmail sting involving Cuban prostitutes in Miami, and just before Kennedy's visit to Dallas on November 22, 1963, the mobster showed up in New Orleans with a hooker and stayed at a motel owned by Sammy's uncle. As soon as Sammy heard John Kennedy had been shot, he was convinced New Orleans had been the staging area for the assassination.\n\nFrom that time on, Fat Sammy did everything in his power to demonstrate his patriotism and disassociate himself from the people who he believed had murdered the president.\n\n\"The night before the planes crashed into the towers, these Mideastern guys were in Sammy's club by the airport. They told one of the girls they were pilots,\" Janet said.\n\n\"Maybe they were,\" I said.\n\n\"Except they were sweating so bad the janitor had to scrape the B.O. off the furniture. They had another problem, too. Like keeping napkins over their boners.\"\n\n\"Sorry, I'm just not following all this,\" I said.\n\n\"Sammy calls the FBI. They send some guys out and Sammy looks at all these photos and says that's not the guys who were in the club. One of the FBI guys says, 'Well, these are the hijackers who died in the planes.'\n\n\"Sammy says, 'Yeah, but there must have been other hijackers whose planes got grounded. The guys in my club are the ones who probably never got off the tarmac.' Even while he's talking you can already hear the toilet flushing.\n\n\"Two weeks go by and Sammy calls the FBI in Washington. He tells some agent there they're looking in the wrong place for terrorists. He says these guys are not Muslim revolutionaries, they're degenerates and losers, just like the other jack-offs who come into the club. Sammy says to the FBI agent, 'Use your fucking head. These guys weren't hanging in mosques or living in Nebraska. They were holed-up in Miami and Vegas and hanging in dumps like mine 'cause they want to get laid. You want to nail 'em, float some cooze out on the breeze and see what happens.' \"\n\nPeople at other tables were turning to stare.\n\n\"Maybe we should move to a quieter spot,\" I said.\n\n\"Well, excuse me. Here's the briefer version so I don't offend anybody,\" she said, her eyelids fluttering. \"The FBI agent blew Sammy off, so he set up an Internet site out in Arizona to sell his movies. He was using a P.I. to run the credit card numbers of anybody with a Mideastern name who bought from the site.\"\n\n\"Who were his partners?\" I said.\n\n\"You met a couple of them,\" she replied.\n\n\"The Dellacroces?\" I said.\n\nShe raised her eyebrows innocuously.\n\n\"Tell him the rest of it, Janet,\" Clete said.\n\n\"Sammy got paid in crystal. It's cooked across the border and comes through Tucson,\" she said. Then she looked at nothing, the whites of her eyes veined, her facial skin like flesh-colored clay that had been molded on bone. \"Sammy wasn't a bad guy. He took us all to Disney World once. He wore a Mouseketeer hat on the plane all the way back home.\"\n\n\"Who popped him, Janet?\" I said.\n\n\"I don't know. Sammy always said it was the normals you got to watch out for, 'cause they never learn who they really are.\"\n\nShe stared through the front windows at the palm trees beating in the wind and the rain slashing on the glass.\n\n## Chapter 23\n\nIt was afternoon when Clete dropped me off at the house. The sky was a cold blue, dense and flawless in texture and color, the lawns along the street ridged with serpentine lines of leaves where the rainwater had receded into the streets. I shaved, showered, changed clothes, and went to the office.\n\nHelen listened quietly while I told her of what had happened in New Orleans, her gaze fixed out the window on the crypts in the old cemetery.\n\n\"You called N.O.P.D. about Coll?\"\n\n\"Yes.\"\n\n\"When?\" she said.\n\n\"When we left town.\"\n\n\"I don't think you wanted to arrest him.\"\n\n\"Then why would I have chased him across town?\"\n\n\"You should have called N.O.P.D. as soon as you saw him inside the church.\"\n\n\"Picture this scene, Helen. A couple of hot dogs coming through the vestibule with M-16s and 12-gauge pumps and Max Coll with a nine-millimeter,\" I said.\n\n\"Coll saved your life. You think you owe him.\"\n\nI started to speak but she raised her hand for me to be quiet. \"The state attorney's office put us on notice this morning. We're going to be investigated for harassment of Castille LeJeune, destruction of his property, and for deliberately damaging his reputation. What do you think of that?\" she said.\n\n\"You warned me,\" I replied.\n\n\"You never understand what I'm saying, Dave. You were right about the murder of Junior Crudup. LeJeune was behind it. He thinks we've got information that in reality we don't. Find out what it is. You're a handful, bwana.\"\n\nShe folded her arms on her chest, shaking her head, a smile tugging at the corner of her mouth.\n\nAt quitting time I drove to the home of Merchie and Theodosha Flannigan. It was almost the winter solstice now, and the sepia-tinted light in the trees and on the bayou seemed to emanate from the earth rather than the sky. Merchie greeted me at the door, wearing glasses, a book in his hand, his long hair like white gold against the soft glow of a living room floor lamp. \"She's not here,\" he said.\n\n\"It's you I want to talk to,\" I replied.\n\n\"Why is it you keep finding reasons to put yourself in my wife's path? Just doing your job?\"\n\n\"You're out of line, Merchie.\"\n\n\"Could be. Could also be you'd like to get into Theo's pants. If that's the case, good luck, because she's out drunk somewhere.\"\n\nI cleared my throat and shifted my eyes off his face. His thoroughbreds were nickering inside a pecan orchard beyond a white fence, their bodies barely distinguishable in the shadows. \"The murder of Junior Crudup isn't going away. His remains were moved, but eventually we'll find out what happened to them. If I have anything to do with it, your father-in-law is going to have an opportunity for on-the-job training in soybean farming,\" I said.\n\n\"So why tell me about it?\"\n\n\"Because I think you wouldn't mind seeing that happen.\"\n\n\"You want to dip your wick, go do it. But leave us out of your personal problems.\"\n\n\"I think Theodosha knows what happened to Junior Crudup's body.\"\n\n\"My wife is a sick person. That's why she's spent a hundred thousand dollars on psychiatrists and clinics. But I think you like stirring her up. I think you like feeding on our troubles.\"\n\nHe started to close the door but I held it open with one hand. \"Your wife's frigid, isn't she?\" I said.\n\nHe released the tension on the door, slipped off his glasses, and dropped them in his shirt pocket. \"If you weren't already an object of pity and public ridicule, I'd splatter your nose all over your face. Now go home,\" he said.\n\nThe door clicked shut. I stared at it stupidly, my ears ringing in the silence.\n\nEarly the next morning Clete picked me up for breakfast, cheerful, wearing his utility cap low on his brow, a Hawaiian shirt under his bomber jacket, driving with one hand down East Main toward Victor's Cafeteria.\n\n\"You moved back into the motor court?\" I said.\n\n\"Yeah, why not?\"\n\n\"You burned a guy's trailer. You assaulted a man in Lafayette.\"\n\n\"They're not filing charges. Not if they want to stay on the planet. So I don't see the big deal. Things get out of control sometimes. I'm cool with it,\" he said, fiddling with the radio.\n\nClete was Clete, a human moving violation, out of sync with both lawful and criminal society, no more capable of changing his course than a steel wrecking ball can alter its direction after it's been set in motion. Why did I constantly contend with him? I asked myself.\n\nBut I knew the answer and it wasn't a comforting one: We were opposite sides of the same coin.\n\nI told him about my visit to Merchie Flannigan's house.\n\n\"That punk said that to you?\" he asked.\n\n\"I got a little personal about his wife,\" I replied.\n\n\"That's another question I have. You actually asked him if his wife wouldn't come across?\"\n\n\"I guess that sums it up.\"\n\n\"I can see that might piss him off. Particularly when he knows you bopped her.\"\n\n\"Can't you show some subtlety, just a little, once in a while?\"\n\n\"You bump uglies with a guy's wife, then tell him she's an ice cube, but it's me who's got a problem with language?\"\n\n\"She was drunk. We both were. Stop harping on it.\"\n\nHe looked at me, then turned into the parking lot across from Victor's. The old convent across the bayou was still in shadow, the live oaks speckled with frost. \"Why get into Flannigan's face about his wife's sex life?\" he said.\n\n\"A psychiatrist would probably say she has trouble with intimacy. So she gets it on when she's drunk, usually with strangers or people she doesn't care about. It's characteristic of women who were molested as children,\" I replied.\n\n\"You're really going to hang LeJeune's cojones over a fire, aren't you?\"\n\n\"You better believe it,\" I said.\n\nLater I signed out a cruiser and drove to the Lafayette Police Department to see my old friend Joe Dupree, the homicide cop and airborne veteran who had investigated the gunshot death of Theo Flannigan's psychiatrist. While I talked he sat behind his desk, picking one aspirin, then another, then a third out of a tin container, swallowing them with water he drank from a cone-shaped paper cup. His tie was configured to the shape of his pot stomach, his hair combed like strands of wire across the bald spot on top of his head.\n\n\"So you think this guy Will Guillot is blackmailing Castille LeJeune and it has something to do with LeJeune's daughter?\" he said.\n\n\"Right.\"\n\n\"About what?\"\n\n\"Molestation.\"\n\nJoe leaned back in his chair and rubbed his mouth. Through the window I could see a chained-up line of black men in orange jumpsuits being placed in a jail van. \"Well, Ms. Flannigan's file was missing from Dr. Bernstine's office. But I found out several other files were missing, too. Maybe Bernstine took them home and they got lost somehow. Or somebody could have stolen several files to throw off the investigation. Anyway, it's been a dead-end case,\" he said.\n\n\"You checked out the secretaries, any reports of forced entry?\" I said.\n\n\"If Bernstine was burglarized, he didn't report it. The alarm company never had to do a 911, either. The secretary is a church-going, family woman, with no reason to steal files from her employer.\"\n\n\"How long was she there?\"\n\nHe looked down at the torn notebook pages that were clipped inside a case folder. \"Seven months,\" he said.\n\n\"Who was the secretary before this one?\" I asked.\n\nHe looked again at his notes. \"A woman named Gretchen Peltier. But she quit before Ms. Flannigan starting seeing Bernstine.\"\n\n\"What was that name again?\"\n\nI drove to the alarm company that had serviced Dr. Bernstine's office. Like most alarm companies, it was an electronic shell that didn't provide security but instead relayed distress signals to the fire department or a law enforcement agency. In other words, the chief expense of home security was passed on to the taxpayers and the alarm company was able to maintain its entire system, which monitored several parishes, with no more than a half dozen technicians and sales and clerical employees.\n\nBut the assistant director of the company, a black woman named Dauterive who had been an elementary school teacher, did her best to help me. A computer record of all electronic warning signals originating during the last year at Dr. Bernstine's office was laid out on the desk. \"See, there were a number of power failures. Those were either during an electrical storm or when a power line was knocked down. These other dates are the times the customer didn't disarm the system fast enough. The dispatcher had to call and get the password.\"\n\nShe was heavyset and wore glasses and a pink suit with a small corsage on the lapel. She glanced at her watch.\n\n\"Am I taking up too much time?\" I asked.\n\n\"Oh, no. It's my anniversary. My husband's meeting me for lunch,\" she replied.\n\n\"Who's the dispatcher?\"\n\n\"We use the Acadiana Ambulance Service. When they receive an emergency signal, they call the residence or the business and clear it up, or they notify the appropriate response service,\" she replied.\n\n\"When was the last time you received an alarm that could have indicated an unauthorized entry?\" I asked.\n\n\"Here,\" she said, and tapped her finger on the computer printout. The date was one day after the gunshot death of the psychiatrist, Dr. Bernstine. \"But the dispatcher called and got the password.\"\n\nI ran my finger up the column on the printout to a billing notation for July and a description of services that amounted to two thousand dollars. \"What's this?\" I asked.\n\n\"It looks like the customer changed out the system. If I remember correctly, a power surge fried the main panel and the customer decided to use the opportunity to upgrade.\"\n\nI was getting nowhere. \"Let me think about this stuff and come back,\" I said.\n\n\"I don't know if this is of any help to you, but the customer changed his keypad code when he got his new system. See?\" she said, and tapped the notation again.\n\n\"Yes?\"\n\n\"He didn't change his password. Sometimes people don't like to change the password, particularly if it's a pet name or part of a private joke in the family,\" she said.\n\nShe looked me flatly in the face.\n\n\"That's a hole in the dike, isn't it?\" I said.\n\n\"You might say that,\" she replied.\n\n\"Did you say today was your anniversary?\" I asked.\n\n\"That's correct. Our twenty-seventh.\"\n\n\"Have a great anniversary, Ms. Dauterive.\"\n\nI headed straight for Abbeville, twenty miles south on the Vermilion River, and the insurance company that employed Gretchen Peltier, the woman who had given Will Guillot his alibi for the night the drive-by daiquiri shop operator was murdered and who had also turned out to be a former employee of the slain psychiatrist.\n\nShe was terrified. Like most people who lead ordinary lives and stray across a line, usually in concert with someone far more devious than themselves, she could neither defend herself nor lie convincingly. Instead, she began to perspire and swallow like someone in an elevator hearing steel cables snap a strand at a time.\n\n\"I don't think you're a bad person, Ms. Peltier. But you're taking the weight for a bad guy,\" I said.\n\n\"Taking the weight?\" she said, more confused and frightened than ever now, her eyes flicking to the open door of her employer's office.\n\n\"You're about to take Will Guillot's fall. That means you'll go to prison. You'll live behind razor wire and cell with murderers and sexual deviates of every stripe. Snitch one of them off and you get glass put in your food. That's where Will Guillot has taken you.\"\n\nMy rhetoric was cruel. She was a sad woman, her eyes etched with mascara, her clothes obviously bought at a discount store. I could only guess at the means of seduction Will Guillot had used to entice her into cooperating with the systematic destruction of her own life.\n\n\"I knew the code numbers to the alarm system in Dr. Bernstine's office,\" she said. \"Dr. Bernstine had shot himself in the park. I gave the numbers to Will because he said his wife, the one he's divorcing, told Dr. Bernstine a lot of lies that were going to be used in court against him. I gave him the password, too.\"\n\n\"How did he get into the building?\" I said.\n\n\"A man who works for him, an electrician, opened the door. But the numbers on the keypad had been changed. The alarm went off. If Will hadn't had the password, the cops would have come out.\"\n\nHer eyes were wet. She rested her forehead on the heel of her hand.\n\n\"You told me Guillot was with you the night the daiquiri store operator was killed. Was that a lie?\"\n\n\"No.\"\n\n\"You sure?\" I said, looking down into her face.\n\n\"I thought I was helping Will. Why have you done this to me?\" she replied. She found a handkerchief in her purse and pressed it against her eyes.\n\n\"What's going on out here?\" her employer said, standing in the doorway of his office, his tie printed with hundreds of tiny blue stars against a red background, a small American flag pinned on the lapel of his suit.\n\nI walked to my cruiser, which was parked on Abbeville's town square. The sun was already deep in the west, the light thin and brittle on the old brick cathedral in the square and the cemetery behind it, where the bodies of Confederate dead from Shiloh and Port Hudson lay in crypts stained with lichen and split with fissures, as though the earth were determined to absorb them and their contents back into itself. I could hear traffic crossing the steel bridge over the Vermilion River and smell the odors of diesel oil and water and shrimp husks piled behind a restaurant, and as I looked at the bare limbs of the willows along the river I was suddenly filled with the sense the sun was not simply completing part of its cycle across the sky, it was about to descend over the rim of the earth for the last time.\n\nIn psychoanalysis it's called a world destruction fantasy.\n\nWere my irrational feelings connected to the fact I had just helped dismantle a woman's life? Or were the rats' nests of rags and bones in those crypts reminders that Shiloh was not a grand moment in history, but a three-day meat-cutter that soaked the hills with the blood of farmboys, most of whom never owned a slave or knew anything about the economics of northern textile mills? Or was the sum total of my own life finally being made apparent to me?\n\nThe streets were almost empty, swirling with dust and pieces of newspaper, the water oaks bare of leaves, many of the old stores permanently closed. The world in which I had grown up was gone. I wanted to pretend otherwise, to find excuses for the decay, the strip malls, the trash strewn along the roadways, the century-old live oaks that developers lopped into stumps with almost patriotic pride. In my vanity I wanted to believe that I and others could turn it around. But it was not going to happen, not in my lifetime nor in my child's.\n\nIt was 4:45 when I got back to the department and rain had begun falling in big fat drops on the sidewalk that led into the courthouse. I pulled my mail out of my pigeon hole and went into my office. A few minutes later Helen came in.\n\n\"So what happened today?\" she asked.\n\nI told her.\n\n\"Will Guillot creeped the psychiatrist's office and stole Theo Flannigan's file so he could blackmail Castille LeJeune?\" she said.\n\n\"It's more serious than that. I think he murdered the psychiatrist on orders from Castille LeJeune. He was probably supposed to deliver the file back to LeJeune, but he either didn't do that or he xeroxed it and is using it to take over the old man's business.\"\n\nThrough the window I saw a hearse pass on its way to the funeral home on St. Peter Street. I got up from my desk and let down the venetian blinds. My office suddenly seemed hermetically sealed, artificially lit, shut off from the rest of the world.\n\n\"You unhappy about something?\" Helen said.\n\n\"No. Everything is fine.\"\n\nShe looked somberly at my face. \"Have dinner with me, Pops,\" she said.\n\n\"Why not?\" I said.\n\n## Chapter 24\n\nThat evening I walked into the kitchen while Father Jimmie was on the phone. Unconsciously he turned his back to me, rounding his shoulders, as though somehow creating a shell around his conversation.\n\n\"I believe you, but we'll do this on my terms. No, you have my word. I'll be there. Now, good-bye,\" he said. After he hung up he turned around and grinned sheepishly. \"I get calls from a neurotic parishioner once in a while,\" he said.\n\n\"Was that one of them?\" I asked.\n\n\"Let's don't clutter up the evening, Dave.\"\n\n\"You're meeting Max Coll?\"\n\n\"He's ready to change his way. I can't deny him reconciliation or communion.\"\n\n\"Coll is planning to kill somebody. But you're supposed to repair his soul so he can sneak into heaven through a side window?\"\n\n\"That last sentence describes two thirds of my constituency,\" he said.\n\nHe picked up Snuggs and a box of cat food and went out on the back steps to feed him.\n\n\"I already fed him,\" I said.\n\n\"He's a warrior. He needs extra rations,\" Father Jimmie replied.\n\nThere was no moon that night. Screech owls were screaming in the trees and the humidity was so thick I could hear moisture ticking in the leaves on the ground. Father Jimmie had gone out, although I had no idea where. I went into the small office I had created in my rented house and sat at the desk and began writing a letter to Alafair.\n\nDear Alf,\n\nWe're going to have a swell time at Christmas. Clete's in town and is anxious to see you, as of course am I. How is your novel going? I bet it's going to be a fine one. Hope you're through exams by now. Don't be too worried about grades. You always did well in school and college is not going to be any different. Would you like to take a ride out on the salt if the weather permits? Batist says he's found a new spot for redfish by Southwest Pass.\n\nThe images out of the past, created by my own words, made my eyes film. I saw Bootsie, Alafair, and me in the stern of our boat, with Batist at the wheel, the throttle full out, slapping across West Cote Blanche Bay at sunrise, the salt spray like a wet kiss on a spring morning.\n\nI put aside the letter and stared at the guns mounted on the gunrack I had screwed into the wall: an AR-15, a sporterized '03 Springfield, and my old Remington twelve-gauge, the barrel sawed off even with the pump, the sportsman's plug long ago removed from the magazine.\n\nI knew what had been on my mind all afternoon and evening. Since I had interviewed Gretchen Peltier at the insurance office in Abbeville I'd had little doubt about Will Guillot's involvement in the burglary of Dr. Bernstine's office and Bernstine's death by gunshot in Lafayette's Girard Park. I also had no doubt he was mixed up in pornography and narcotics and the blackmail of Castille LeJeune. The problem was his crimes had all been committed in other parishes, and there was no way to hang the killing of either Sammy Figorelli or the drive-by daiquiri store operator in New Iberia on him.\n\nIn order to get at him and subsequently Castille LeJeune, I would have to work with at least three other law-enforcement agencies. Then the legal processes of indictment and prosecution would be turned entirely over to others, perhaps in a parish Castille LeJeune controlled.\n\nI turned off the light and sat in the darkness with the twelve-gauge across my lap. The steel and the wood of the stock felt cool against my palms. I opened the breech and smelled the odor of the machine oil I had used to clean the chamber and the magazine, then set the stock butt-down between my legs, moving my thumb along the edges of the barrel where I had sawed it off and sanded it smooth with emery paper. I thought about my dead wife Bootsie and the systemic corruption of the place I loved and the inhumanity and cruelty that had been visited upon a great blues artist like Junior Crudup.\n\nI removed a box of double-ought buckshot from my closet shelf and began pressing a handful of shells one at a time into the magazine of my Remington. I sat in the darkness a long time, the gun resting on my knees, my mind free of all thought, a strange numbness in my body. Then I ejected the shells and replaced them one by one in their box, set the shotgun back in the rack, and took a walk down by the drawbridge. A lighted tug was waiting for the bridge tender to raise the bridge. I waved at him in the pilot house and he waved back at me, then I walked back home and went to bed, with Snuggs sleeping at the foot.\n\nThe next day, Friday, I contacted Joe Dupree in Lafayette, and we went to work on getting a search warrant on Will Guillot's home and place of business. But it was going to be a long haul. The warrant request was based on statements made by Gretchen Peltier, the psychiatrist's former secretary, about a break-in committed in Lafayette by a man who lived in Franklin. Also, Will Guillot was probably many things, but stupid wasn't one of them. It was highly unlikely he would keep the stolen case file, which he was using to blackmail Castille LeJeune, in either his home or office.\n\nThere are days in law enforcement, just like those at the craps table, when you think the dice have no combinations on them except treys and boxcars. Then suddenly they magically bounce off the backboard, all elevens and sevens.\n\nJust before quitting time Helen opened my door and leaned inside. \"The sheriff in St. Mary just called. Will Guillot made a prowler report last night. The city cops who responded told him there'd been a peeping Tom in the neighborhood, but Guillot seemed to think it was someone else.\"\n\n\"Who?\"\n\n\"He was walking around in the yard with a gun and not saying.\"\n\n\"Thanks for passing it on,\" I said.\n\nI continued with the paperwork I was doing, my expression flat. I thought she was about to close the door and go back to her office but instead she approached my desk, her eyes on mine.\n\n\"My words don't have much influence on you. But be careful, Dave. Don't give power to a guy like Castille LeJeune,\" she said.\n\n\"I hear you,\" I said.\n\n\"Yeah,\" she said.\n\nAt 5 P.M. I went home, reloaded my cut-down twelve-gauge, locked it in the steel box that was welded to the bed of my pickup truck, and drove to Clete's cottage at the motor court.\n\nHe was outside, grilling a chicken, drinking from a quart bottle of beer, his eyes watering in the smoke, the collar of his jacket pulled up around his neck, his utility cap cocked sideways.\n\n\"What's shakin', big mon?\" he said.\n\n\"Think the Bobbsey Twins from Homicide should make a house call down in Franklin?\" I said.\n\n\"Oh my, yes indeedy,\" he replied, as though the statement were one word.\n\nThe shrubs and gazebo and wide gallery of Will Guillot's house were threaded with Christmas lights, and sequined cutouts of reindeer, with tinted floodlamps aimed at them, were spiked into the lawn. We pulled into the driveway and parked just inches from where Dr. Parks had bled to death on the cement. I unlocked the steel box in the truck bed, removed my cut-down twelve-gauge, and tossed it to Clete. He went into the shrubbery with it, deliberately silhouetting against the Christmas lights and tinted floodlamps, the barrel held at an upward angle. As I walked up on the gallery I saw Will Guillot pull aside a curtain on a tall window and look outside. I hung my badge holder on the breast pocket of my sports coat and banged hard on the door with the flat of my fist.\n\nEverything I did in the next few minutes would be based on my belief that Gretchen Peltier had truly been sickened by her experience with Will Guillot and had not gone back to him or confessed she had given him up.\n\nHe jerked open the door and stared into my face. He wore a burgundy corduroy shirt and gray slacks and loafers, and in the dim light the birthmark on his face looked like a scar from a hot iron. Behind him I saw a woman get up from the couch and go into the back of the house. \"Do I need to call the cops?\" he said.\n\n\"I'm the least of your troubles, Mr. Guillot. I think your electrician wants to park one in your brainpan,\" I said.\n\n\"What?\" he said, his eyes shifting from me to Clete, who had just walked out of the yard, stepping up on the gallery with the twelve-gauge resting in the crook of his arm.\n\n\"It's clear,\" Clete said to me.\n\n\"What's clear? Why are you walking around in my yard with that shotgun?\" Guillot said.\n\n\"Your electrician, Herbert Vidrine, gave you up. But I guess that wasn't enough for him. Evidently he hates your guts. What'd you do to the poor guy?\" I said.\n\n\"I already found out about that letter you or somebody else sent him with my name on it. It didn't work,\" Guillot said, his eyes flicking from me to Clete and the shotgun again.\n\n\"Try this. You got Herbert Vidrine to help you break into Dr. Samuel Bernstine's office the same weekend Bernstine took two .25 caliber rounds in the head. You set off the alarm, then found out you had the wrong code numbers for the keypad. But fortunately for you somebody had given you the password and you were able to give it to the alarm service when they called.\"\n\nGuillot tried to let my words slide off his face, biting down with his back molars so his jaw didn't sag. \"Then arrest me so I can sue you into the next dimension,\" he said.\n\n\"You think this is about some pissant B&E?\" Clete said.\n\n\"Who is this guy?\" Guillot said to me.\n\n\"There's my buzzer,\" Clete said, opening his P.I. badge, then flipping it closed again before Guillot could look at it carefully. \"The G doesn't spend its time on nickel-and-dime farts who make dirty movies. But unlucky for you, a guy we do care about, a psychopath named Max Coll, is in the neighborhood, and it's got something to do with you and the cocksucker you work for.\"\n\nGuillot looked behind him, as though he did not want our words heard by the woman who had gone into the back of the house. If he had closed the door in our faces and called his attorney, it would have been over. But Clete had set the hook and Guillot couldn't pull it out. He stepped out on the porch with us and pulled the door shut behind him, shivering slightly in the cold.\n\n\"What's the deal on this guy you mentioned, what's his name, this guy Coll?\" he said.\n\n\"He blows heads for the IRA or the Mob or just because he can't get it up in the morning,\" Clete said.\n\n\"He's here, in Franklin?\" Guillot said to Clete.\n\n\"You tell us,\" Clete said.\n\nGuillot looked out into the darkness, as though trying to see beyond the Christmas lights that partly illuminated his yard. \"None of this has anything to do with me,\" he said.\n\n\"Let me ask you this question: When the warrants are cut, or if Max Coll is in town, looking for the people who put the whack on him, whose grits are going into the fire, yours or Castille LeJeune's?\" I said.\n\nClete pumped a shell out of the shotgun's chamber and dropped it into Guillot's shirt pocket. \"Twelve-gauge double-ought bucks. Load up your bird buster and stick it under your bed. Better than a warm glass of milk. You'll sleep like a baby. I guarantee it,\" he said, and gave Guillot the thumbs-up sign.\n\nTen minutes later we turned into Fox Run and drove down the long, oak-lined driveway to Castille LeJeune's front entrance. Almost the entire house was scrolled with white Christmas lights, so that the house glowed like a nineteenth-century paddle-wheeler inside a fog bank on the Mississippi. My guess was that Will Guillot had called LeJeune as soon as we had left his house, and I hoped, in an undeniably mean-spirited fashion, that for the first time in his life Castille LeJeune was genuinely afraid.\n\nI parked at the end of the drive and cut the headlights on my truck. A solitary shadow moved across the windows in the living room. I started to get out, but Clete hadn't moved, the shotgun propped at an angle between his legs, the chamber open.\n\n\"Dave, Guillot's a sex freak and a lowlife and dirty up to his elbows. I'm not so sure about the guy in that house,\" he said.\n\nI looked at him.\n\n\"All this crap isn't adding up for me,\" he said. \"The war hero didn't pop the drive-by daiquiri guy and neither did Guillot, not if you buy his alibi. But for one reason or another we keep looking at the war hero. No matter what happens, it's always the war hero. Meanwhile Merchie Flannigan's old lady gets a free pass, the same broad who got you kidnapped.\"\n\n\"Theodosha is south Louisiana's answer to Bonnie Parker?\" I said.\n\n\"Be a wise-ass if you want. You hate the guy in that house and the class of people he comes from.\"\n\n\"I do? You've been at war with these people all your life.\"\n\nHe took off his utility cap, looked at it as though he had never seen it before, then refitted it on his head. \"He really bagged Bed Check Charley?\" he asked.\n\n\"That's the story.\"\n\n\"I'd like to get his autograph. Hey, I'm serious,\" he said.\n\nHe got out of the truck, trying to suppress his grin, and followed me onto the porch. A white-jacketed black houseman answered door, a broom and dustpan in his hands.\n\n\"Is Mr. LeJeune home?\" I said.\n\n\"Took his guests to the country club a half hour ago. I'm still cleaning up,\" the houseman said.\n\nI opened my badge. \"Did you receive a phone call in the last ten minutes?\" I said.\n\n\"Yes, suh, I sure did,\" he answered.\n\n\"From whom?\"\n\n\"My wife. She tole me to bring home a loaf of bread.\"\n\nOn the way to the country club Clete was still grinning.\n\n\"Why is all this funny?\" I said.\n\n\"I miss the Mob. Shaking up a bunch of Kiwanians just doesn't cut it.\"\n\n\"You're too much, Cletus.\"\n\nIn that mood we pulled into the tree-bowered entrance of a small tennis and golf club outside the city limits. It wasn't hard to find Castille LeJeune. He and his friends were having drinks under a pavilion and driving golf balls on a lighted practice range dotted in the distance with moss-hung live oaks that smoked in the mist. The range looked hand clipped, immaculate, with neither a leaf nor wind-blown scrap of paper on it.\n\nThe pavilion seemed as isolated and disconnected from the outside world as the golf range was from the trash-strewn roads beyond the hedges that bordered the club. Deferential black waiters brought LeJeune and his friends their buttered rum drinks on silver trays; a Wurlitzer jukebox next to the bar played Glen Miller and Tommy Dorsey recordings; a rotund, cherry-cheeked man was speaking affectionately about \"an old nigger\" who had worked for his family, as though the waiters would take no offense at his language.\n\nWe locked the truck, with the twelve-gauge inside, and walked past the clay tennis courts, all of them deserted, the wind screens rattling in the breeze, just as Castille LeJeune whacked a ball off a tee and sent it downrange in a high, beautiful arc. The people at the tables or teeing up from wire buckets filled with golf balls showed no recognition of our presence. LeJeune positioned himself, swung his driver back, and once again lifted the ball surgically off the tee, high into the darkness, a testimony to his health, the power in his wrists and shoulders, and the maturity and skill he brought to his game.\n\nClete used a toothpick to spear a peeled shrimp from a large bowl of crushed ice on the bar, dipping it in hot sauce, inserting it in his mouth. His badge holder was stuck in his belt, mine in the breast pocket of my sports coat. But still no one looked at us.\n\n\"Give me a Jack straight up with a beer back,\" he said to the bartender.\n\n\"Right away, suh,\" the bartender replied.\n\n\"That's a joke,\" Clete said.\n\nLeJeune's friends were not people who had to contend with the world. They may not have owned it, nor would they take any part of it through the grave, but while they were alive they could lay rental claims on a very large portion of it.\n\n\"Mr. LeJeune, we'd like for you to come with us to the Iberia Parish Sheriff's Department,\" I said.\n\n\"Why should I do that, Mr. Robicheaux?\" he replied, addressing the ball on his tee, his feet spread, his thighs flexed tightly.\n\n\"We need you to answer some questions about the murder of Dr. Samuel Bernstine and the fact Will Guillot has been blackmailing you about your molestation of your daughter when she was a child,\" I said.\n\nIn the silence I could hear leaves scraping across the surface of the tennis court. LeJeune seemed to gaze at an isolated thought in the center of his mind, then he sighted downrange and smacked the ball in a straight line, like a rifle shot, so that it did not strike earth again until it was almost to the oak trees smoking in the electric lights.\n\n\"You need to talk to my attorney, Mr. Robicheaux, not to me,\" he said.\n\n\"Did you hear what I said? We're investigating a homicide, the second one that happens to be connected with your name. We don't call attorneys to make appointments,\" I said.\n\nHe turned and dropped his driver in an upended leather golf bag. He wore a silk scarf around his neck, as an aviator might, the ends tucked inside a sweater with small brown buttons on it. In the corner of my eye I saw two security guards walking from the club's main building and a man at the bar punching in numbers on a cell phone.\n\nLeJeune began chatting with a woman seated at a table as though I were not there. Then I started to lose it.\n\n\"You had Junior Crudup beaten to death,\" I said. \"You turned your daughter's childhood into a sexual nightmare. You sell liquor to drunk drivers and probably dope and porn in New Orleans. You think you're going to walk away from all this?\"\n\n\"Mr. Robicheaux, I don't know if you're a vindictive man, or simply well-meaning and incompetent. The truth is probably somewhere in between. But you need to leave, sir, to let this thing go and give yourself some peace,\" he replied.\n\nHis detachment and his pose as a chivalric and charitable patriarch were magnificent. As Clete had always said, some people have no handles on them. Castille LeJeune was obviously one of them, and I felt like a fool.\n\nThen Clete, who all night had been the advocate of reason and restraint, stepped forward, his thick arm and shoulder knocking against mine. \"You were a fighter pilot in the Crotch?\" he said.\n\n\"In the what?\" LeJeune said.\n\n\"I was in the Corps, too. Sunny 'Nam, class of '69, smokin' grass and stompin' ass with Mother Green's Mean Machine. See?\" He removed his utility cap and pointed to the globe and anchor emblem inked on the cloth. \"We used to have a Bed Check Charley, but he was a guy who'd start lobbing blooker rounds in on us at about oh-two-hundred so nobody could get any sleep. Do you have any autographed photos? No shit, it'd mean a lot.\"\n\n\"Sir, I don't ask this for myself, but there're ladies present. Let's don't have this kind of scene here,\" LeJeune said.\n\n\"I can dig it,\" Clete said, putting his cap back on, his eyes cocked up in his head as though he were meditating upon a metaphysical consideration. \"The problem is some greaseballs kidnapped and tortured a police officer and pissed all over his face while he was blindfolded. So how about taking the corn bread out of your mouth? It's getting to be a real drag.\"\n\n\"I apologize for any offense I may have given you,\" LeJeune said. \"Tell me something, that badge you have hanging from your belt? I have the feeling you're not a police officer.\"\n\nI could see the heat climbing into Clete's face. \"Dave, hook up this prick. Work out the legal stuff later,\" he said.\n\nThe situation was deteriorating rapidly now. Two security guards had just walked into the pavilion and were standing behind us, awkward, unsure what they should do next. I turned so they could see my badge. \"It's all right. Iberia Parish Sheriff's Department,\" I said.\n\nThey tried to be polite, their eyes avoiding mine. I felt sorry for them. They made little more than minimum wage, paid for their own uniforms, and possessed no legal powers. They waited for Castille LeJeune to tell them what to do.\n\nBut I raised my finger before he could speak. \"We're leaving,\" I said.\n\n\"Screw that,\" Clete said.\n\nTwo cruisers from the St. Mary Parish Sheriff's office had pulled into the parking lot and three uniformed deputies, one black, two white, were walking toward us, their faces filled with purpose. I slipped my hand around the thickness of Clete's arm and tightened my grip. \"We're done here,\" I said.\n\nBut it was too late. The three deputies went straight for Clete, with the collective instinct of pack hounds who had just gotten a sniff of a feral hog. At first he didn't resist. When they walked him toward a cruiser, he was seemingly in control of himself again, grinning, full of fun, back in his familiar role of irreverent trickster, ready to let it all play out.\n\nMaybe I should have stayed out of it. But I didn't.\n\n\"Let's slow it down a little bit,\" I said to the black deputy, a towering man with lieutenant's bars on his collar.\n\n\"Best let us do our job, Robicheaux,\" he replied.\n\n\"What's the beef?\" I said.\n\n\"Impersonating a police officer,\" he replied.\n\n\"That's bogus. He never claimed to be a police officer.\"\n\n\"Work it out at the jail. We just deliver the freight,\" he said.\n\nIt should have all ended there, a routine roust to appease a rich man, a discussion down at the sheriff's department, maybe a few hours in a holding cell, at worst an appearance in morning court where the charge would be kicked.\n\nBut one of the white deputies, an angry man with corded veins in his neck who had been fired in another parish for abusing a prisoner, had pushed Clete into a search position against the hood of the cruiser and was running his hands down Clete's left leg.\n\n\"Ease up, my man,\" Clete said.\n\n\"Close your mouth,\" the deputy said.\n\n\"That's a slapjack in my right hand pocket. I'm not carrying,\" Clete said, twisting around.\n\n\"I told you to shut up,\" the deputy said, and slapped Clete's utility cap off his head.\n\nClete ripped his elbow into the deputy's face, breaking his nose, then caught him in the jaw with a right hook that lifted him off the ground and knocked him the full length of the cruiser.\n\n\"Ouch,\" he said, trying to shake the pain out of his hand, trying to step back from his own misdeed.\n\nThen they were on him.\n\n## Chapter 25\n\nIt rained at sunrise and kept raining through the morning. Clete was in jail and Father Jimmie had not returned to the house. Because it was Saturday Helen was at home. I called her and told her how it had gone south at Castille LeJeune's golf and tennis club.\n\n\"What did you plan to accomplish over there?\" she said.\n\n\"Not sure.\"\n\n\"I am. You wanted to provoke a confrontation and blow pieces of Castille LeJeune all over the golf tee.\"\n\n\"That's a little strong.\"\n\nI thought she was going to give it to me but she didn't. \"As far as you know, Guillot didn't try to call LeJeune after you went to Guillot's house?\" she said.\n\n\"When we went to LeJeune's house, the man cleaning up said nobody had called except his wife. She wanted him to pick up a loaf of bread.\"\n\n\"Maybe LeJeune is not the guy we should be after.\"\n\n\"He's the guy.\"\n\n\"I think I'm going to do something more rewarding today, like have a conversation with a pile of bricks,\" she said.\n\n\"Did you just hear something on the line?\"\n\n\"Hear what?\"\n\n\"A friend in New Orleans said I probably have a federal tap on my phone.\"\n\n\"Have a nice weekend, Dave.\"\n\nClete was in serious trouble and would not be able to bond out of jail until he was arraigned Monday morning. The impersonation beef was a gray area. A person does not have to specifically claim to be a police officer in order to be guilty of impersonating one. He simply has to give the impression of being one. But Clete had licensed P.I. status and ironically, as an employee of a bail bond service, possessed legal powers that no law officer did, namely, he could cross state lines and even break into residences without a warrant to arrest a bail skip who was a fugitive from a court proceeding.\n\nThe assault-and-battery beef was another matter. With luck and some finesse, an expensive, politically connected lawyer could probably get the charge kicked down to resisting. But it wasn't going to be easy. Clete's reputation for violence, destruction of property, and general anarchy was scorched into the landscape all the way across southern Louisiana. His enemies had longed for the day he would load the gun for them. Now I had helped him do it.\n\nI went to Baron's Health Club, worked out with free weights, then sat for a half hour in the steam room. When I came back outside it was still raining, harder than before, litter floating in the ditches that bordered the streets. I went to an afternoon A.A. meeting above the Methodist church by the railroad tracks and listened to a man talk about nightmares he still had from the Vietnam War. His face was seamed, unshaved, his body flaccid, his clothes mismatched. He had been eighty-sixed out of every bar in the parish and he had been put out of two V.A. alcoholic treatment programs. He began to talk about a massacre of innocent persons inside a free-fire zone.\n\nI couldn't listen to it. I left the meeting and drove home. When I pulled into the driveway my yard was flooded halfway to the gallery and Theodosha Flannigan was waiting for me by the door, a rain-spotted scarf tied on her head, her face filled with consternation. Snuggs was turning in circles around her ankles.\n\n\"I know all about last night,\" she said.\n\n\"Not a good day for it, Theo,\" I said, unlocking the door.\n\nI went in the house without inviting her inside, but she followed me anyway, Snuggs racing past us toward the food bowl in the kitchen.\n\n\"My father didn't molest me. It was a black man. That's why I was seeing Dr. Bernstine,\" she said.\n\n\"Don't do this, Theo.\"\n\n\"When I was a little girl a black convict got in our house and hurt me. He was killed running down toward the bayou.\"\n\n\"Killed by whom?\"\n\n\"A prison guard. He worked at the labor camp. He and the other guards buried him in back. I saw the bones when the fish pond was dug. They were sticking out of the dirt in a front-end loader.\"\n\n\"You've been fed a lie.\"\n\n\"It's the truth. I went over every detail of it with my father.\"\n\n\"Bernstine told you your father raped or molested you, didn't he?\"\n\n\"It doesn't matter. I know what happened.\"\n\n\"When you first told me about Bernstine's death, you said you thought you had something to do with it.\"\n\n\"I was confused. I know the truth now.\"\n\nI gave up. Through the kitchen window I could see steam rising off the bayou in the rain. Theodosha picked up Snuggs, set him on the counter, and rubbed her hand down his back. \"Merchie is leaving me,\" she said.\n\n\"That's too bad.\"\n\n\"We're not good for each other. We never were. I'm too messed up and he's too ambitious.\"\n\n\"I have some things to do today, Theo.\"\n\nI could hear an oak branch slapping against the side of the house, water rushing out of a gutter into the drive.\n\n\"We had fun together, didn't we?\" she said.\n\n\"Yeah, sure,\" I replied.\n\n\"Know why we're alike?\"\n\n\"No.\"\n\n\"We both live in the cities of the dead. We don't belong with other people.\"\n\n\"That's not true. Why did you use that term?\" I said, my heart quickening.\n\nBut she didn't answer. She lifted up Snuggs and set him back down on the floor, then touched me on both cheeks and kissed me on the mouth. \"So long, baby. I never told you this, but you're the only man I ever slept with and dreamed about later,\" she said.\n\nShe went out the front door, letting the screen slam behind her, then ran for her car. I had to force myself not to go after her.\n\nI lay down on my bedspread, with my arm across my eyes, and listened to the rain on the roof. I drifted off to sleep and suddenly saw an image out of my past, one that had no catalyst other than perhaps the story told by the war veteran at the noon A.A. meeting. I saw the members of my platoon marching at night through a rain forest that had been denuded by napalm. Their faces and uniforms and steel pots, even the green sweat towels draped over their heads like monk's cowls, were gray with ash. They cast no shadows and made no sound as they marched and their eyes were all possessed by the strange non-human look that soldiers call the thousand-yard stare.\n\nI sat straight up in my bed, my throat choking.\n\nThe phone was ringing in the kitchen. I went to the counter and picked it up, the dream still more real than the world around me. \"Hello?\" I said.\n\n\"Is Father Dolan there?\"\n\n\"Coll?\"\n\n\"Sorry to be a nuisance, Mr. Robicheaux. I just wanted to pass on something to Father Dolan.\"\n\nMy mind began to race. Castille LeJeune had remained untouchable and was about to skate. Will Guillot could probably not be charged with any crime more serious than breaking and entering, and the evidence against him was problematic and subject to easy dissection by a defense attorney.\n\n\"I owe you one, Max. That means I don't want to see you taken off the board by a couple of local scum wads,\" I said.\n\n\"Could you be speaking a little more plainly, sir?\" he replied.\n\nMy pulse was beating in my wrists, the veins dilating in my scalp. \"I think the clip on you came down from a couple of homegrown characters in the porn and meth trade. Maybe you should stay out of Franklin, Louisiana, and spend more time at Biscayne Dog Track,\" I said.\n\n\"A couple of local fellows, you say? Now, that's interesting, because I'd come to a very different conclusion. I thought the porn connection was the woman, the screenwriter, Ms. Flannigan. She's the brains in the family, not her father. The colored people hereabouts say he may have had his way with her when she was a child. This fellow Guillot is trying to take over the business, so Ms. Flannigan does the daiquiri fellow, draws a lot of attention to her father's selling grog to teenagers and drunk drivers, and uses Guillot's gun to do it. Perfect way to screw both her daddy and her business rival.\"\n\n\"Why would Theo Flannigan be the porn connection?\"\n\n\"I'm ashamed to say I'm well acquainted with a number of lowlifes in the underworld who say Sammy Figorelli's films were successful because they were written by a famous woman author. It's not a big reach to figure out who that might be.... Hello? Are you there?\"\n\n\"Yes,\" I said weakly.\n\n\"I've never harmed a woman, sir, so I let the matter go. But I'll be reamed up the bung hole with a spiked telephone pole if you haven't made me reconsider the LeJeune and Guillot fellows.\"\n\n\"Hold on, Coll.\"\n\n\"No, you've done me a favor. I've got to cancel my flight reservations and give it all a good think. Tell Father Dolan thanks for his help. A tip of the hat to yourself as well.\"\n\nThe line went dead. I replaced the receiver and wiped my face with a dish towel. I tried to sort through the conversation I had just had with Max Coll. My head was a basket of snakes, my mouth dry, my thoughts suddenly centered on a jigger of Beam poured into an iced mug of draft beer inside a Saturday-afternoon bar that was only two blocks up the street.\n\nFather Jimmie Dolan's car pulled into the driveway, pushing a wave of water under the house. When he entered the front door he was smiling, his tan, wide-brim hat dripping. \"Any calls for me?\" he asked.\n\nI drove downtown to the restaurant that used to be Provost's Pool Room. It was warm and cheerful and crowded inside, and I sat at the hand-carved mahogany bar and looked out the window at the wetness of the day and the traffic passing in the street. As a boy I used to come to the pool room on Saturday afternoons with my father, Big Aldous, in a era when the plank floors were strewn with football betting cards and green sawdust and the owner served free robin gumbo out of big pots that he set on an oilcloth-covered pool table. The stamped tin ceilings and mahogany bar and old brick walls still remained, but the building was an upscale restaurant now that catered to tourists who came to see a world that no longer existed.\n\nThe bartender wore his hair slicked back and black pants and a white jacket and black tie. \"You just gonna have coffee, sir?\" he asked.\n\n\"How about I buy you a drink?\" I said.\n\n\"Sir?\"\n\n\"It's not a complicated question.\" It sounded bad but I grinned when I said it.\n\nHe shrugged. \"I get off in a hour,\" he said.\n\nI put several one-dollar bills on the bar. \"Make sure it's Beam or Jack,\" I said.\n\n\"You got it,\" he said, scooping up the bills.\n\nThen I drove back home and went into the kitchen, where Father Jimmie was reading the newspaper. He lowered the paper, then looked curiously at my face. \"It can't be that bad, can it?\" he said.\n\nSo I told him how bad it was, or at least how bad I thought it was; but I was to learn my education about my own obtrusiveness was ongoing. After I finished he sat for a long time without speaking, his gaze turned inward, unable to conceal his disappointment at either me or his own missionary failure, or the world as it really is. I suspect I wanted absolution, like a child going to confession on a Saturday afternoon, leaving behind his imaginary sins, bounding down the street as though a stricken world has just been made whole again. But that wasn't to be.\n\nFather Jimmie had a look of sadness in his eyes that I cannot adequately describe. \"You don't know what you've done,\" he said.\n\n\"Maybe I have at least a fair idea,\" I replied.\n\n\"Max met with me outside Franklin. He expressed what I think was genuine remorse for the evil he's done in his life. I gave him absolution. But you hung the bait in his face and energized him. My God, man, we're talking about his soul.\"\n\nI felt light-headed, as though I were coming down with the flu. When I tried to speak I couldn't clear the obstruction in my throat. Father Jimmie filled a glass with water but did not hand it to me.\n\n\"Listen, Coll changed his direction because he didn't want to kill a woman,\" I said.\n\n\"It makes no difference.\"\n\n\"It does. I never thought about Theo being involved. Even though Clete kept warning me, I never thought about Theo.\"\n\nFather Jimmie realized I had already moved on from my own irresponsibility and was now concentrating on another matter, one that showed a degree of obsession beyond his grasp. He set down the glass and turned on me. I saw his right hand close. His next words were spoken through his teeth: \"Don't deceive yourself. You're a violent and driven man, Dave, just like Max Coll.\"\n\nHis eyelids were stitched to his brows, his throat bladed with anger and rebuke.\n\nThat evening the sky was as dark as I had ever seen it. Lightning rippled like quicksilver across the thunderheads in the south, and the sugarcane in the fields along the road to St. Martinville thrashed and flickered in the wind and rain, the oak canopy blowing leaves that stuck like leeches on my windshield. I went to Mass in the old French church on the square in St. Martinville, then when the church was empty put five dollars in the poor box and removed an unlit votive candle in a red glass receptacle and took it with me down to the cemetery on the bayou.\n\nIt was a foolish thing to do, I suspect, but I had long ago come to view the world as an unreasonable place, not to be contended with, better left to pragmatists and the mercantile who view the imagination and the unseen as their enemy. I parked under the streetlight, opened an umbrella, and walked between the crypts toward Bootsie's tomb. A generic compact car passed behind me, turned at the corner, and disappeared down a side street.\n\nThe bayou was high, dented with rain rings, yellow in the lights from the drawbridge. I placed the votive candle next to the marble tablet on Bootsie's tomb, wedged the umbrella so that it sheltered the candle from the rain and wind, then lit the wick.\n\nThe same compact car came out of the square and crossed the drawbridge, but I paid little attention to it. An event I had never seen in my life was taking place in front of me. Two huge brown pelicans drifted out from under the bridge, floating south on the tidal current, their wings folded tightly against the wind, their long yellow bills tucked down on their chests. I had never seen pelicans this far inland and had no explanation for their presence. Then I did something that made me wonder about my level of sanity.\n\nI rose from the steel bench I was sitting on, pointing at the two birds, and said, \"Take a look, Boots. These guys were almost extinct a few years ago. They're beautiful.\"\n\nThen I sat down and folded my arms on my chest, the rain clicking on my coat.\n\nThat's when I saw the compact in plain relief against the streetlight at the corner. It was pulled into a careless position at the curb, steam rising from the hood, the driver moving around in silhouette, as though he were having trouble with his safety belt.\n\nDave! a voice said, as audibly as a voice speaking to you on the edge of sleep, as defined as a stick snapping inside the eardrum.\n\nI rose from the bench just as the streetlight glinted on the lens of a telescopic sight and the muzzle flash of a rifle splintered from the passenger window of the compact car. The bullet whanged off the steel bench and blew pieces off a statue of Jesus's mother.\n\nI ducked down between the crypts and pulled my .45 from my belt holster and sighted with two hands on the compact. But there were houses on the far side of the street and I couldn't fire. I started running toward the compact, the .45 held at an upward angle, zigzagging between the crypts, my eyes locked on the driver, who was fighting to straighten the car's wheels so he would not hit the curb.\n\nHe pulled around a parked pickup truck and floored the compact down the street. In seconds he would be beyond any safe angle of fire that I would have. I left the sidewalk and ran toward the corner of the cemetery, jumped on top of a crypt, and went over the chainlink fence into the street. The compact was twenty-five to thirty yards away, headed down the bayou in the direction of the church, the license plated patinaed with mud. I stood in the center of the street, both arms extended, and aimed low on the trunk.\n\nI squeezed off three rounds, the recoil knocking my forearms upward, the muzzle throwing sparks into the darkness, the spent shells tinkling on the pavement. I don't know what I hit inside the compact, but I heard the hard slap of all three hollow-point rounds bite into metal.\n\nThe compact swerved around a corner and disappeared down a tree-lined side street that looked like an illustration clipped from a 1940 issue of The Saturday Evening Post.\n\nI went back to my truck and used my cell phone to punch in a 911 on the compact, then walked to Bootsie's tomb, my ears still ringing from the explosions of the .45. The umbrella had not been disturbed by the wind and the candle was burning brightly inside its red receptacle, but the pelicans had flown or drifted southward on the current.\n\nI heard your voice, I said.\n\nBut there was no reply.\n\nI don't care who else knows it, either. That was your voice, Boots, I said.\n\nThen I said a prayer for her and one for me and headed back for the truck, wishing the pelicans had not gone.\n\nDon't worry, they'll be back. One of these days when you least expect it, you'll see them on Bayou Teche, she said.\n\nI turned around, my jaw hanging, the clouds blooming with electricity that made no sound.\n\n## Chapter 26\n\nI rose before dawn Sunday morning and ate a breakfast of Grape-Nuts and coffee and hot milk in the kitchen. When I opened the front door to leave I saw an envelope on the porch with a footprint stenciled across it and realized it must have fallen out of the door-jamb the previous night and been stepped on by either me or Father Jimmie.\n\nThe letter inside was handwritten and read:\n\nDear Mr. Robicheaux,\n\nI must talk to you. I don't know why all this is happening. We moved here to live in a decent environment and look what everyone has done to us. I also do not understand this new development. Nobody will answer my questions. I think all of you people suck. Call me at home. Do it right now.\n\nSincerely,\n\nDonna Parks\n\nIn my memory I saw a stump of a woman, with dyed red hair and perfume that was like a chemical assault on the senses, a ring of fat under her chin. She was the mother of Lori Parks, the teenage girl who had died with two others inside their burning automobile on Loreauville Road. I did not look forward to seeing Mrs. Parks again.\n\nI put away her note and drove to Franklin. The parking compound for Sunbelt Construction was located behind a house trailer that served as a company office. In the lot were trucks of every kind, front-end loaders, bulldozers, and grading machines but no compact car that resembled the shooter's.\n\nI drove back to New Iberia and parked in Merchie and Theodosha Flannigan's driveway. Their faux medieval home was shrouded in fog puffing off the bayou, their horses nickering and blowing inside the pecan orchard. The morning newspaper was still inside the metal cylinder at the foot of the drive, but woodsmoke was rising from a living room fireplace. There was no compact car anywhere in sight, but I did not expect to see one. In fact, I did not know why I had come to the Flannigans' home. Perhaps it was to prove somehow that Theo was not involved with a criminal enterprise, that she was a victim herself and not capable of setting me up to be kidnapped and tortured by the Dellacroce brothers. Maybe I just wanted to believe the world was a more innocent place than it is.\n\nI got out of the truck and rested my hands on the top rail of the white fence that bordered the pecan orchard and watched the Flannigans' thoroughbreds moving about in the fog. I could hear their hooves thudding on the soft earth, smell the fecund odor of the bayou, like the smell of humus and fish roe, and the pecan husks and blackened leaves that had been trodden into pulp in the trees, and I wondered how it was that a place this beautiful would not be enough for anyone, why each morning would not come to the owner like a blessing extended by a divine hand.\n\nTheodosha opened the front door and walked down the drive in her bathrobe and slippers, her hair black and shiny in the grayness of the morning. \"What are you doing out here?\" she asked.\n\n\"How bad would you be willing to screw an old friend?\" I said.\n\n\"It's pretty early in the morning for your craziness, Dave.\"\n\n\"Your novels were nominated twice for Edgars but they didn't win. If your script-writing career was on track, I think you'd be out in the Hollywood Hills, not on the bayou. Maybe Fat Sammy Figorelli's skin films were a shortcut to being back on the big screen.\"\n\n\"You're sickening,\" she said.\n\n\"Somebody shot at me last night.\"\n\n\"I can't imagine why.\"\n\n\"Did you set me up with the Dellacroces?\"\n\nShe walked past me and pulled the morning paper from the metal delivery receptacle, then started back up the drive toward her house. \"Too bad it's Sunday,\" she said.\n\n\"Why's that?\"\n\n\"The state mental hygiene unit in Lafayette is closed. But if I were you, I'd jump right on it first thing in the morning,\" she said, opening the paper, not bothering to even glance at me as she spoke.\n\nWhen I got back home, Father Jimmie was gone, his closet empty. He had left a recording for me on my message machine, its brevity like a shard of glass: \"So long, Dave. Thanks for your hospitality. I hope everything works out for you.\"\n\nThere was also a voice message from Donna Parks: \"Why don't you answer my goddamn letter, you callous fuck?\"\n\nIt was going to be a long day.\n\nI tried to eat lunch but had no appetite. As I washed my dishes and put away my uneaten food, I looked through a window and saw Helen Soileau pull into the driveway. She got out of the cruiser and walked to the gallery, wearing faded jeans, boots, and a mackinaw, her jaw set. I opened the door before she could knock.\n\n\"I was out of town, so I just got the report on the car sniper,\" she said, walking past me into the warmth of the living room. \"Go over it for me.\"\n\nI went over each detail with her and also told her I had been to Franklin that morning to look for the compact car I had put three rounds in.\n\n\"Anybody from St. Mary Parish contact you?\" she asked.\n\n\"No,\" I said.\n\n\"Yesterday somebody got past the alarm system at both Castille LeJeune's and Will Guillot's house. In the middle of the afternoon. A real pro. Know who it might be?\"\n\n\"Max Coll,\" I said.\n\n\"What was he looking for?\"\n\n\"Evidence they put a hit on him.\"\n\n\"I hate to even ask this question. How would you know this?\"\n\n\"He called here yesterday. I more or less told him there were two local guys behind the contract on him and they lived in Franklin.\"\n\nShe stood at the ceiling-high living room window and stared out at the street and at the rain dripping through the canopy of live oaks that arched over it, her fists propped on her hips. \"Want to tell me your motivation for doing that?\" she said.\n\n\"I owed him one.\"\n\n\"We don't owe criminals. We break their wheels and put them out of business. We don't make individual judgments on the people we need to arrest.\"\n\n\"I don't see it that way.\"\n\n\"There are a lot of things you don't see,\" she replied, turning to look directly at me. \"I'm pulling your shield, bwana.\"\n\nI nodded, my expression flat. \"It's been that kind of day,\" I said. I slipped my badge holder out of my pocket and handed it to her. \"Coll thinks Theo Flannigan may have been the porn connection to Sammy Figorelli. Maybe she was the shooter in the daiquiri drive-by. In case you want to follow that up.\"\n\nHelen flipped my badge holder back and forth in her hand while she listened, then she tucked it into her pocket. \"Sometimes you break my heart,\" she said.\n\nI had been suspended before, put on a desk, investigated by Internal Affairs, locked up on at least three occasions, and years ago fired by N.O.P.D. But this time was different. The suspension came not from a career administrator but from my old partner, a woman who had been excoriated as a lesbian and who had never allowed the taunts and odium heaped upon her to diminish either her integrity or the dignity and courage that obviously governed her life.\n\nThe fact that it was she who had pulled my plug made me wonder if indeed I hadn't gone way beyond the envelope and become one of those jaundiced and embittered law officers whose careers do not end but flame out in a curlicue of dirty smoke that forever obscures the clarity of their moral vision.\n\nBut that kind of thinking is what we call in A.A. the paralysis of analysis. In terms of worth it shares commonalities with masturbation, asking a rage-a-holic for advice on spiritual serenity, or listening to your own thoughts while trapped by yourself between floors in a stalled elevator.\n\nI went into the kitchen and called Donna Parks at her home. There was no answer. I left a message on her machine and drove to Franklin to visit Clete Purcel in jail.\n\nA turnkey walked me down a corridor to an isolation cell, one with horizonal bars, flat cross-plates, and an iron food slit in the door, but with nothing inside except a stainless steel toilet and a metal bench bolted to the floor. Clete was sitting on the bench, still in his street clothes, his wrists locked to his hips with a waist chain, another chain locked between his ankles. His right eye was swollen into a puffed knot, his forehead and chin scraped raw. The cement floor outside the cell door was splattered with red beans, rice, two pieces of white bread, and coffee from a broken Styrofoam cup.\n\n\"Who did that to his face?\" I said.\n\n\"He come in like that,\" the turnkey said.\n\n\"That's a lie,\" I said.\n\n\"He wouldn't put on his jumpsuit. He threw his tray at a deputy. You got issues with it, talk to the boss. I just clean up the mess,\" the turnkey said, and walked away.\n\nI hung my hands through the bars. \"How you doin', Cletus?\" I said.\n\nHe stood up from the bench and shuffled toward me, his chains clinking on the cement. \"I'm going to look up a couple of these guys when I get out of here,\" he said.\n\n\"Why do you have to provoke them?\"\n\n\"It's fun.\"\n\n\"I'm suspended. I don't have any clout to help you.\"\n\n\"What'd you do?\"\n\n\"Fired up Max Coll and pointed him at LeJeune and Guillot. I figured my line was tapped and I might get the Feds in here.\"\n\n\"I keep telling you, it's the broad.\"\n\n\"Maybe it is.\"\n\nThen his eyes went away from mine and looked into space. \"Nig and Wee Willie won't go bail,\" he said.\n\n\"Why not?\"\n\n\"They're pissed because of that dinner I charged on their card at Galatoire's. Plus two of the girls skipped their court appearances and Nig's putting it on me.\"\n\n\"What kind of bail are we talking about?\" I asked.\n\n\"A screw tried to do an anal search on me. He's going to need some dental work. So I've got two separate A&B's on a law officer.\"\n\nI touched my forehead against the bars and closed my eyes. Clete kicked the door with the point of his shoe, rattling it in the jamb. \"Listen up, Dave. We're the good guys. The problem is nobody else knows it. But that's their problem, not ours,\" he said.\n\nI left the jail and parked my truck on an oyster-shell road down by Bayou Teche, just outside the Franklin city limits. Rain was falling on the trees around my truck, and across the bayou were a cow pasture, a collapsed red barn, and a solitary black man in a straw hat, sitting on an inverted bucket, cane-pole fishing under a live oak. I got out of the truck, tossed a pine cone into the current, and watched it float southward toward the Gulf.\n\nClete had made a point, one which I don't think was either vituperative or vain. Legal definitions had little to do with morality. It was legal to systemically poison the earth and sell arms to Third World lunatics. Politicians who themselves had avoided active service and never had listened to the sounds a flame thrower extracted from its victims, or zipped up body bags on the faces of their best friends, clamored for war and stood proudly in front of the flag while they sent others off to fight it.\n\nThe polluters and the war advocates are always legal men, as the Prince of Darkness is always a gentleman.\n\nThe John Gottis of the world make good entertainment. The polluters and the war advocates can be seen at prayer, on camera, in the National Cathedral. Unlike John Gotti, they're not very interesting, but they cause infinitely more damage.\n\nThe chances were I would never take down Castille LeJeune for the murder of Junior Crudup. Nor did it look like I would solve the shooting of the drive-by daiquiri store operator or Fat Sammy Figorelli. The people who had committed these crimes did not have patterns and to one degree or another operated with public sanction. They might go down for an ancillary offense, but at worst they would do minimum time, if not get probation.\n\nBut regardless of what occurred in the lives of others, I was going to clear my conscience of a problem I had created because of my desire to control a situation in which I had failed.\n\nI drove through the wet streets of Franklin, out to Fox Run, and lifted the false knocker on the front door that activated the chimes deep inside the house. A moment later Castille LeJeune answered the door, dressed in sweat clothes, a towel twisted around his throat, surprisingly pleasant, his face ruddy from riding an exercise bike by the sun room that gave onto the back patio, the same patio where Junior Crudup had entertained him and his wife fifty years ago.\n\n\"Come in, sir,\" he said, opening the door wider.\n\n\"I don't know if you'll want me in your house after you hear what I have to say,\" I said.\n\nHe laughed and closed the door behind me. \"Go ahead. I know a determined man when I see one. But excuse me just a minute. I have to use the bathroom,\" he said.\n\nHe went into a hallway and closed a door behind him, then I heard him urinating into a toilet bowl. Through the French doors I could see the long slope of his backyard tapering down to the bayou, a yellow bulldozer parked by the area we had excavated during our search for Junior Crudup's remains. Much of the dirt had been filled in, smoothed and tamped down, so that the lawn was now a mottled brown and green, in patterns like camouflage.\n\nI heard LeJeune washing his hands, then he came back into the living room.\n\n\"I couldn't stick you with Junior Crudup's death, so I tried to sic a psychological nightmare by the name of Max Coll on you,\" I said.\n\n\"Ah, a mea culpa because you've put me at risk. Let me clarify something for you\u2014\"\n\n\"If I can finish, please. Using Coll was a gutless act on my part. If I had wanted you smoked, I should have done it myself instead of exploiting a headcase.\"\n\n\"I admire your candor, Mr. Robicheaux. But I'm not bothered by Coll's presence in the community. I walked in on him and he fled. If this fellow is indeed a soldier for the IRA, which is what I've been told, then I understand why the British are still in control of northern Ireland.\"\n\n\"Wait a minute. You saw Coll?\"\n\n\"I just told you that.\" He stared at me, his eyes probing mine.\n\n\"Was he armed?\"\n\n\"He might have been. It's hard to say. I didn't bother to ask.\"\n\n\"Where did he go?\"\n\n\"Out the back door. I've reported all this.\"\n\n\"You might drop by the church today and light a candle, maybe offer a prayer of thanks that a guy like Father Jimmie Dolan is a minister in the Catholic Church,\" I said.\n\n\"As always with you, Mr. Robicheaux, I have no idea what you're talking about. But if this man Coll comes back around, he'll rue the day he left his little shanty back in the peat bogs or wherever he comes from.... Am I losing your attention?\"\n\n\"Hubris has always been my undoing, Mr. LeJeune. Maybe it will be different with you. Anyway, my badge has been pulled and I'm done. Run your happy warrior act on somebody else,\" I said.\n\nWhen I got back home I put on sweat pants and a hooded jersey, tied on my running shoes, and jogged down East Main, past the Shadows and the plantation caretaker's house across the street, which now served as a bed-and-breakfast, and crossed the drawbridge into City Park. I ran along the winding paved road through the live oak trees, my clothes soggy with mist, then cut across the closely clipped grass and ran along the edge of the bayou. In our area the sugar mills are fired up twenty-four hours a day during the cane-grinding season, and in the distance I could see a huge red glow on the horizon, like fire trapped inside a thunderhead, and I could hear the heavy thumping sound of the machines, like the reverberation of giant feet stamping upon the earth. There was not another soul inside the park, and for just a moment my heart quickened and I felt more alone than I had ever felt in my life.\n\nI sat down on a bench, my palms propped on my thighs, my breath coming hard in my throat. What was it Theodosha had said? We were alike because we both lived in the cities of the dead? I wiped the sweat off my face with my jersey and fought to get my breath back, widening my eyes, concentrating on the details around me, as though my ability to remain among the quick depended on my perception of them.\n\nIs this the way it comes? I thought\u2014not with a clicking sound and a brilliant flash of light on a night trail in Vietnam, or with a high-powered round fired by a sniper in a compact automobile, but instead with a racing of the heart and a shortening of the breath in a black-green deserted park smudged by mist and threaded by a tidal stream.\n\nMy head hammered with sound that was like helicopter blades thropping overhead, and for just a moment I was back on a slick piled with wounded and dying grunts, AK-47 rounds vectoring out of the jungle canopy down below, the inside of the airframe crawling with smoke.\n\nI put my head down between my knees, my hands on the pavement, the world spinning around me.\n\nI looked up and saw from out of the mist a pink Cadillac convertible headed toward me, one with wire wheels, tail fins, Frenched headlights, and grillwork that was like a chromium smile, the radio blaring with 1950s Jerry Lee Lewis rock 'n' roll.\n\nThe Cadillac passed me and behind the wheel I saw a man with an impish face, the features cartoonlike, as though they had been sketched with a charcoal pencil, the hair shaved on the sides and left long and curly on the neck.\n\n\"Gunner?\" I said out loud.\n\nBut the driver did not hear me, and the Cadillac wound its way out of the park, the only piece of bright color inside the failing light.\n\nGunner Ardoin in New Iberia? I asked myself. No, I had let my imagination run away with itself. The year was 2002, not 1957, and the rock 'n' roll days of pink Cadillacs, drive-in movies, Jerry Lee Lewis, and American innocence were over.\n\nAt 10:00 P.M. I turned on the local news. The lead story involved a homicide inside a Franklin residence. The television camera panned on a tree-lined street and a Victorian home where paramedics were exiting a side door with a gurney on which a figure inside a body bag was strapped down. The reporter at the scene said the victim had been shot once in the temple and once in the mouth and, according to the coroner, had been dead approximately twelve hours. The victim's name was William Raymond Guillot.\n\n## Chapter 27\n\nIt was still raining Monday morning, the air cold, the fog heavy among the crypts in St. Peter's Cemetery as I pulled into the parking lot at the courthouse.\n\nWally, our leviathan dispatcher, made a face when he saw me come through the front door. \"Dave, you ain't suppose to be here,\" he said.\n\n\"Pretend I'm not,\" I said.\n\n\"Don't jam me up here. I'm your friend, remember?\"\n\n\"Is anybody working the Guillot homicide?\" I said.\n\n\"I didn't even hear you say that. I'm deaf and dumb here. Go home,\" he replied.\n\nHelen's door was ajar. I went inside without knocking. \"What's happening in Franklin with the Guillot shooting?\" I said.\n\n\"None of your business,\" she said.\n\n\"They made Max Coll for the hit?\"\n\n\"One in the temple, one down the throat. The signature of a pro,\" she said.\n\n\"I don't buy it.\"\n\n\"What you need to buy is a hearing aid. You were suspended as of yesterday. Now haul your ass out of here.\"\n\n\"I talked with Castille LeJeune late yesterday afternoon. He says he walked in on Coll while Coll was creeping his house. If Coll was going to pop anybody, he would have done it then.\"\n\n\"You went out to LeJeune's, after I pulled your badge?\"\n\n\"I told him I was suspended. It was a personal visit.\"\n\nShe shook her head, nonplussed. \"We have an attorney in lawyer jail right now. I'm about to put you in there with him,\" she said.\n\n\"Coll isn't the shooter.\"\n\n\"Don't be on the premises when I get back.\" She walked down the hall and into the women's restroom, glancing back at me just before she pushed open the door, as though my argument for Coll's innocence had just sunk a hook on the edge of her mouth.\n\nLouisiana is a small state, with a comparatively small population. In the year 2002 over 950 people were killed and 55,000 injured on our state highways. Booze was a major factor in most of the fatalities. Hence, the presence of a drunk person behind the wheel of an automobile in Louisiana is hardly an anomaly. So I had no reason to be surprised when I picked up the phone in my kitchen and heard a woman's voice say, \"Why don't you do something about this goddamn traffic light out here on the four-lane?\"\n\n\"Who is this?\" I asked.\n\n\"Donna Parks, who does it sound like? The man in front of me is driving a shit box that's smoking up the whole town. He won't turn left because there's no arrow on the traffic light and I have to breathe his goddamn exhaust fumes.\"\n\nFor just a moment I had the uncharitable thought that her husband, Dr. Parks, was better off dead.\n\n\"What could I do for you, Ms. Parks?\"\n\n\"I want to file rape charges.\"\n\n\"You've been sexually assaulted?\"\n\n\"Like my deceased husband said, you people are really dumb. I'll come over there and explain it to you. Where are you?\"\n\n\"Since you dialed me at my home number, I think we should both conclude I'm at home.\"\n\nShe belched softly, then I heard what was probably her car horn blowing just before the line went dead.\n\nWith luck she would have an accident before she got to my house, I thought.\n\nI looked at my watch. Clete's arraignment was at 11:00 A.M. I wrote a note for Donna Parks, included my cell phone number on it, and stuck the note inside the grill on the front screen. Eventually I would have to deal with her, but it would be easier to do by phone than in person. I put Snuggs on the back porch, slipped my checkbook in my pocket, and started out the door, just as Merchie Flannigan pulled into the driveway, blocking my truck. He worked his way around the puddles in the yard and stepped up on the gallery, raking back his long, white-gold hair with his fingers.\n\n\"Hang on, old buddy. Need to clear up my remarks to you when you came by the house,\" he said.\n\n\"I'm in a hurry, Merchie,\" I said.\n\n\"Let's face it. I was jealous. Theo and I haven't had the best marriage. You said I was out of line. You were right.\" He extended his hand, his jaw square, like an imitation of an athletic, educated, country club millionaire, one he had probably seen on a movie screen as a child and had spent a lifetime trying to become.\n\nI didn't take his hand. \"I think you're here covering your wife's ass. Will Guillot got popped and the cops are going to be taking a hard look at his enterprises. I believe Theo is part of a porn operation in New Orleans,\" I said.\n\nThe smile died on his face. \"You're actually serious? You believe Theo is involved with pornography?\" he said.\n\n\"The word is she wrote scripts for Fat Sammy Figorelli. Where was she the night the daiquiri store operator got shot?\"\n\nHe slipped his hands into his pockets and looked at the rain falling through the live oaks onto the street, as though any conversation with me was useless and the problem was mine, not his. \"Theo and I are taking a cruise to the Islands. I came by here to do the right thing. But I can see it was a mistake.\"\n\n\"Where'd it go wrong for you, partner?\"\n\n\"Wrong about what?\" he said.\n\n\"You were Jumpin' Merchie Flannigan, a stand-up kid from the Iberville who did the crime and stacked the time. Why'd you become a hump for a bum like Castille LeJeune?\"\n\nThe skin of his face seemed to crinkle, like a sheet of yellow paper held against a hot light bulb. He raked his hair back over his head again and started to speak, his eyes tangled with thoughts I could only guess at, then stepped off the gallery and walked through a water puddle to his Mercedes.\n\nI headed down the four-lane toward Franklin and five miles outside New Iberia felt a front tire on my pickup go soft and begin to wobble. I pulled to the shoulder and changed the tire in the rain. It was almost 11:30 when I got to the St. Mary Parish Courthouse. Across the street I saw the restored pink Cadillac I had seen in City Park the previous night. A curious black man holding an umbrella was bent down by the driver's window, admiring the interior.\n\n\"Do you know who owns this?\" I asked.\n\n\"A man who got a lot of money,\" he replied.\n\nI went inside the courthouse and peeled off my raincoat in front of a coffee stand run by a blind man. I had no way of knowing the amount of Clete's bail, but obviously it would be high, and the 10 percent bondsman's fee would probably clean out my checking account and part of my savings. Of course, my paying a bondsman's fee was predicated on the assumption a local bondsman would be willing to write a bond on Clete, whose past record included fleeing the United States on a murder warrant.\n\n\"You want a cup of coffee, Dave?\" the blind man behind the counter said.\n\n\"Yeah, sure, Walter,\" I said, distracted by a brown-haired little girl, no older than six or seven, sitting on a bench by the courtroom entrance. A small teddy bear, a red ribbon with a silver bell on it tied around the neck, was perched on her lap. Where had I seen her before? Then I remembered, with a rush of shame. It had been at Gunner Ardoin's house, on the morning I had rousted him last fall, chambering a round in the .45, sticking it in his face, causing him to soil himself while his little girl watched.\n\nI walked up to her, my raincoat slung over my arm. \"Is your daddy here?\" I asked.\n\n\"He's inside the big room,\" she replied.\n\n\"What's he doin' there?\"\n\n\"Helping Clete.\"\n\n\"You remember me?\" I asked.\n\n\"You're the man who pointed a gun at my daddy.\"\n\nI went inside the courtroom just as the morning's proceedings were breaking up. Clete was talking to a local attorney while a deputy put cuffs on him for his trip back to jail. The judge left the room for his chambers, and among the people filing out in the corridor I saw Gunner Ardoin.\n\n\"Clete's going back to lock-up?\" I said.\n\n\"Just till he bonds out,\" Gunner said.\n\n\"How much is his bond?\"\n\n\"Fifty grand,\" he said.\n\n\"How'd he put it up?\"\n\n\"He didn't. I did.\"\n\n\"You went a fifty-thou bond?\"\n\n\"You don't watch the news? I hit the Powerball last week. Three million bucks. I bought him that pink Caddy out front, too.\"\n\nI looked at him, stupefied. He walked past me and took his little girl by the hand. \"Want something to eat? Clete's going to meet us outside in a few minutes,\" he said.\n\n\"Why not?\" I replied.\n\nA half hour later the four of us were eating gumbo at a checker-cloth-covered table inside a cafe one block from the courthouse. The pink Cadillac convertible was parked outside, rainwater standing up in beads as big as marbles on the waxed surface.\n\n\"I appreciate it, Gunner, but I can't accept it,\" Clete said.\n\n\"The title's already in your name, man,\" Gunner said.\n\n\"We'll have to change that,\" Clete said.\n\nGunner looked at a spot on the far wall of the cafe. \"There's something I didn't mention. A couple of guys I was inside with needed a place to crash. Remember Flip Raguzi, used to run a chop shop for the Giacanos over in Algiers? He started a grease fire on the stove. It sort of changed the way your kitchen and the ceiling look.\"\n\n\"You let Flip Raguzi stay in my place? This guy has diseases scientists haven't found names for,\" Clete said.\n\n\"What's he talking about, Daddy?\" the little girl asked.\n\nClete shut his eyes, then opened them. \"Give me the keys,\" he said.\n\nOne of my favorite lines of all time, one excerpted from a 1940s song understood readily by all those who experienced the human and economic realities of the Depression and war years, goes as follows: \"You don't get no bread with one meatball.\"\n\n\"What's funny?\" Gunner said.\n\n\"Nothing,\" I said. \"Take a walk with me, will you?\"\n\nWe went outside and stood under a canvas awning, the mist blowing in our faces.\n\n\"That's a decent thing you did for Clete, Gunner,\" I said.\n\n\"I don't use that name anymore,\" he said.\n\n\"How about Father Jimmie? You do the right thing by him, too?\" I said.\n\n\"Matter of fact, I did. But that's my business.\"\n\n\"I respect that, Phil. But I need your help, too. Know a woman named Theo Flannigan?\"\n\n\"Jumpin' Merchie's old lady? I know who she is, but I don't know her personally.\"\n\n\"Was she writing scripts for Fat Sammy Figorelli?\"\n\nHe shook his head. \"No, but she might as well have. Her books were lying around the set. The director would lift the dialog from the love scenes in her books. So a bunch of degenerates, that includes me, were doing sixty-nines on each other and talking like Shakespeare.\"\n\n\"Why would the director pick her work to steal from?\" I asked.\n\n\"A guy named Ray was involved. His girlfriend was my costar. I never saw him, but I think he was the same guy who'd call me and tell me where to pick up my meth delivery to the projects.\"\n\nRay?\n\nWhy hadn't I seen it? William Ray Guillot, lately of Franklin, Louisiana, now having his blood drained and replaced with formal-dehyde.\n\n\"You're sure Theo wasn't part of Fat Sammy's action?\" I said.\n\n\"Ever see one of Fat Sammy's films?\"\n\n\"No.\"\n\n\"You don't want to,\" he said. \"Let's go inside. Clete needs to drive me and my daughter to the airport in Lafayette. I'm buying a Mexican restaurant in San Antonio. You get to town, have a free dinner on me.\"\n\n\"You're a stand-up guy, Phil.\"\n\n\"I'm out of the life. I'm a millionaire. What's a few bucks to show some gratitude?\"\n\nI started to say something else, but he cut me off.\n\n\"I got your drift. Give it a rest,\" he said.\n\nI drove back to my house on East Main and tried to put the LeJeune family and Junior Crudup out of my mind, but I couldn't rest. I did not believe Max Coll killed Will Guillot, and I couldn't shake the feeling that Castille LeJeune had been unduly happy when I went to his home, as though with a broad sweep of a broom he had gotten rid of a large problem in his life. In fact, I believed Castille LeJeune was about to get away with at least one if not two additional homicides.\n\nAnd I also felt I had a problem of conscience about Theo Flannigan. I had falsely accused her of involvement in the shooting of the daiquiri-store operator and the production of pornographic films.\n\nIn fact, I rued the day I had ever heard of the LeJeunes or Junior Crudup.\n\nOn top of my more elevated level of problems, Batist stopped by the house with another one, namely Tripod, Alafair's three-legged raccoon, whom Batist carried up on the gallery inside Tripod's wood-frame hutch.\n\n\"Cain't keep him at my house no mo',\" he said.\n\n\"Why can't you?\" I asked, looking down at Tripod, who was standing up in the hutch, his claws hooked on the wire screen, his whiskered snout pointed at me.\n\n\"He's old, like me. He went to the bat'room on the kitchen flo',\" Batist said.\n\n\"Thanks, Batist.\"\n\n\"You welcome,\" he replied, and drove off.\n\nI opened the wire door on Tripod's hutch and he stepped out on the floor and looked up at me. \"How's it hangin, 'Pod?\" I said.\n\nHe responded by running into the kitchen and eating Snuggs's food out of the pet bowl.\n\nBut I could not distract myself from my problems with the world of play and innocence represented by animals. I wanted to believe I'd been dealt a bad hand. There was even some truth in my self-serving conclusion. But unfortunately I had dealt the hand to myself, beginning with the day I stepped into the unsolved disappearance of Junior Crudup, a man who had probably sought self-immolation all his life.\n\nI called Theo at her house and apologized for my accusation.\n\n\"Drunks are always sorry. But they do it over and over again,\" she said.\n\n\"Could you define 'it,' please?\"\n\n\"Acting like an asshole.\"\n\n\"I see.\"\n\n\"Have you apologized to my father?\" she asked.\n\n\"Are you serious?\" I said.\n\nShe hung up.\n\nI called Helen Soileau at the department and told her I'd been wrong about Theo.\n\n\"How'd you clear her?\" she asked.\n\n\"A porn actor told me a guy named Ray, as in William Raymond Guillot, was responsible for lifting material from Theo's books for Sammy Figorelli's movies. Theo had nothing to do with it.\"\n\n\"Thanks for telling me.\"\n\n\"Can you get another warrant to search Castille LeJeune's property?\"\n\n\"No.\"\n\n\"I want to resign from the department, Helen. I'll have a formal letter on your desk by tomorrow.\"\n\n\"That's the way you want it?\"\n\n\"Absolutely.\"\n\n\"I love you, bwana, but I don't trust you. And I...\"\n\n\"What?\"\n\n\"Want to kill you sometimes.\"\n\nI got in my truck and backed into East Main. The bamboo and gardens in front of the Shadows breathed with mist that blew into the street, and as I looked at the old, massive brick post office on the corner, where a Creole man sold sno'balls and chunks of sugarcane off a canopy-shaded wagon when I was a kid, and as I watched the traffic turn at the next light onto the drawbridge, just past the Evangeline Theater where my father, mother, and I went to see cowboy movies in the 1940s, I had the feeling, not imagined, not emotional in nature, that I would never see any of these places or things again.\n\n## Chapter 28\n\nAs I approached Fox Run I could see sleet marching across the barren cane fields on the far side of the Teche, the same fields where Junior and Woodrow Reed labored a half century ago under the watchful eyes of Boss Posey and the other mounted gunbulls, all of them, one way or another, controlled by the man who lived across the bayou in the great white house that resembled a Mississippi paddle-wheeler.\n\nI parked by the carriage house. The automobiles were gone and even though the sky was dark, no lights burned inside the main house. I dropped my cell phone in the pocket of my raincoat and walked down the slope toward the bayou, where the yellow bulldozer sat, huge, mud smeared, and clicking with soft white hail.\n\nHelen had said we were looking for Dagwood and Blondie, whose advantage was they did not feel guilty and hence hid in plain sight. But amateur criminals have another kind of problem, one that professionals do not. They're arrogant and they presume. They're psychologically incapable of believing the system was not constructed to benefit them, and consequently they cannot imagine themselves standing in front of a law-and-order judge who can send them away for decades.\n\nThe bulldozer blade was partially raised, the tractor-treads pressed deeply into the earth, fanning back off the rear of the dozer in patterns like horse tails, as though the operator had been involved intensely with one particular area of repair rather than the entire environment. The keys were hanging from the ignition. I turned over the engine, revved the gas once, and clanked the transmission into reverse. As I backed up the dozer, a different kind of topography began to emerge from under the suspended blade\u2014an unevenly filled depression, one that had not been graded and tamped down, so that the surface was spiked with severed tree roots and ground-up divots of grass.\n\nI dropped the blade, shifted into forward gear, and raked off the top layer of the depression, then backed up again so I could see where the blade had cut. The dirt was loose, sinking where there were air pockets, water oozing from the subsoil that had been compressed by the weight of the tractor-treads. I dropped the blade lower, this time cutting much deeper into the hole, trundling up a huge, curled pile of mud, blue clay, and feeder roots that looked like torn cobweb. But this time, when I backed off the hole, I saw something I hoped I would not find.\n\nI cut the engine, pulled loose a shovel that was behind the seat, and walked around the front of the blade to a spot where a human arm, shoulder, and the curved back of a hand protruded from the soil, the hail rolling down the sides of the depression, pooling around them.\n\nI pushed the shovel under the back of the person and wedged the torso and the face free from the soil. The skin had turned a bluish gray, either in the water or because of the clay in the alluvial fan of the bayou, but his eyes were open and still emerald green, his small ears tight against the scalp, his shoulders somehow far too narrow for the violent and dangerous man he had once been.\n\nThere were entrance wounds in his face, under one arm, and in his left temple.\n\nI speared the shovel blade into the clay and reached for the cell phone in my raincoat pocket, just as the cell phone began ringing. I flipped it open and placed it against my ear. \"Dave Robicheaux,\" I said.\n\n\"Are you trying to avoid me?\" a woman's voice said.\n\nThe hail was hitting hard on my hat and the steel frame of the bulldozer and I could hardly hear her. \"Ms. Parks, I'm no longer with the sheriff's department. You need to call\u2014\"\n\n\"I found a diary under Lori's mattress. There were hearts all over the last page and drawings of a man's face. It wasn't some kid's face, either. There was a phone number, too.\" Her voice was starting to crack. \"You know who that number belongs to?\"\n\n\"No, I don't.\"\n\n\"A pipeline company in Lafayette. It's owned by that man who lives in that phony piece of medieval shit across from the junk yard.\"\n\n\"Say his name, Ms. Parks.\"\n\n\"Flannigan. Merchie Flannigan. I'm filing charges for statutory rape.\"\n\n\"Ms. Parks, Lori might have known someone who simply worked at the pipeline company.\"\n\n\"This number goes into Flannigan's office. It's his extension. Why are you covering up for him? I hate you people,\" she said.\n\nShe was obviously still drunk, but I couldn't fault her for her rage. Her daughter had burned to death in an automobile after being sold liquor illegally, and her husband, who had survived a tour as a combat medic, had been killed with impunity by Will Guillot, the investigation written off by a cop on a pad. But family survivors of homicide victims are seldom mentioned in follow-up news stories, even though the grief they carry is like the daily theft of sunlight from their lives.\n\nThe window on my cell phone cleared. Donna Parks was off the line now, but either because of the weather or my location I was losing service as I tried to punch in a 911 call. I heard someone's feet crunch on the hailstones behind me.\n\n\"You must have been a Marine, Mr. Robicheaux. I think you're the most determined man I've ever met.\"\n\nI turned and looked into the face of Castille LeJeune. He wore a silver shooting jacket, one with ammunition loops sewn on the sleeves, a flat-brimmed, pearl-gray Stetson hat, and khaki trousers tucked inside fur-lined, half-topped boots. In his right hand he held a blue-black revolver with walnut grips. But he did not point it at me. Up on the slope, by the carriage house, I could see Merchie Flannigan's Mercedes.\n\n\"You got the jump on me, Mr. LeJeune. You and your son-in-law just pull in?\" I said.\n\n\"The question is what do I do with you, Mr. Robicheaux.\"\n\n\"You didn't just pop ole Max, did you? You executed him.\"\n\n\"Could I see your search warrant?\"\n\n\"Don't happen to have it with me.\"\n\n\"Ah.\"\n\n\"Merchie has been screwing both you and your daughter, Mr. LeJeune. He stole a single-action Army colt from Will Guillot and used it to kill the daiquiri store operator. Then he threw the gun down so we'd put it on Guillot and by extension on you and your enterprises.\"\n\n\"Why would he kill a liquor salesman?\"\n\n\"Merchie was banging a seventeen-year-old girl by the name of Lori Parks. She died in a car wreck after she bought booze from a drive-by store you own.\"\n\nI could see the connections coming together in LeJeune's eyes. Behind him Merchie Flannigan was walking down the slope, his hands in the pockets of his jacket, his shoulders hunched under an Australian flop hat.\n\nLeJeune glanced over his shoulder, then focused on my face again. \"You uncovered evidence in a homicide without a warrant, which destroys the probative value of the discovery,\" he said. \"But you're not a stupid man. Something else is going on here. You quit the sheriff's department, didn't you?\"\n\nI shrugged my shoulders. \"We've got your ass in the bear trap, Mr. LeJeune. How's it feel?\" I said, and actually laughed.\n\nUp on the slope I saw Theodosha Flannigan park her Lexus and walk into the front of the house, carrying a guitar case.\n\n\"Open your coat,\" LeJeune said, raising his pistol toward my chest. \"Use your left hand, unsnap the strap on your sidearm, and drop it on the ground.\"\n\n\"Nope,\" I said.\n\n\"Say again?\"\n\n\"A police officer never surrenders his weapon.\"\n\n\"You're not a police officer anymore.\"\n\n\"Old habits die hard.\"\n\nI'd like to say my behavior was brave, my principles inviolate, but in reality I didn't feel personally threatened by Castille LeJeune. He didn't care enough about me or the social class I represented to hate or fear me, and in all probability he still retained some of the fatalistic views that had allowed him to survive the Korean War as a decorated combat pilot. The system had served him for a lifetime\u2014why should it fail him now?\n\nBut on another level I misjudged him. He could abide a professional enemy such as myself, but treachery inside the castle walls was another matter. He pulled back my coat, removed my .45 from the clip-on holster I wore, and tossed it in the mud.\n\nMerchie Flannigan was standing now on the rim of the depression, his face disjointed as he stared down at the half-exhumed body of Max Coll. \"Who's this dead guy? What's happening here?\" he said.\n\n\"You were having an affair with a seventeen-year-old girl?\" LeJeune said.\n\n\"Hold on, there, Castille,\" Merchie said.\n\n\"I always told Theo you were trash, with your blow-dried hair and Thesaurus vocabulary. You shot my sales person at the drive-by window?\"\n\n\"I think I'm going to boogie and let you and Dave work it out. Maybe y'all can tell each other war stories. But I'd say from the looks of things here, you're genuinely fucked, Castille,\" he said, and began walking back up the slope.\n\nThe temperature had dropped, and the air was bitter, like the taste of copper coins, the tin roofs of the old convict cabins speckled with frost. I could see a lump of cartilage working in LeJeune's jaw. Merchie was halfway up the slope when LeJeune raised the revolver and fired three times, pop, pop, pop.\n\nEither his hand shook from cold or anger or he was simply not a good shot, because he missed with all three rounds, and I heard the bullets break glass in the French doors that gave onto his patio.\n\nMerchie ran past the carriage house and down the drive, hunkered low, the brim of his Australian flop hat angled down over his neck. I walked up behind LeJeune, slipping my hand down his forearm, removing the revolver from his grasp.\n\n\"You killed Will Guillot and were going to put it on Max Coll?\" I said.\n\n\"I have nothing more to say,\" he replied.\n\n\"Guillot killed both Bernstine and Sammy Figorelli and took a shot at me, didn't he?\"\n\n\"Can't help you, sir,\" he replied.\n\nUp at the house there was no sound or any movement behind a window or the French doors. I snicked open the cylinder on LeJeune's revolver, ejected all the shells into my palm, then retrieved my .45 from the mud.\n\n\"My vision isn't very good anymore. You know I tied Ted Williams' gunnery record? Highest ever set by a Marine or Navy aviator. That's God's honest truth,\" he said.\n\n\"I believe you. Better take a walk with me,\" I said, punching in 911 on my cell with my thumb.\n\n\"Of course. We're going up to my house. I'll fix coffee for us. I have no personal feelings about this,\" he said.\n\nHe walked up the slope beside me, his chin lifted, his hands stuck in the pockets of his silver shooting jacket, his nostrils flaring as he breathed in the fresh coldness of the afternoon. I studied the back of the house, but still there was no movement inside. I felt LeJeune's attention suddenly refocus itself on the side of my face.\n\n\"Why are you so somber, Mr. Robicheaux? It should be a red-letter day for you,\" he said.\n\n\"My father taught me to hunt, Mr. LeJeune. He used to say, 'Don't be shooting at nothin' you cain't see on the other side of, no.' He was a simple man, but I always admired his humanity and remembered his words.\"\n\n\"As always, your second meaning eludes me.\"\n\n\"Is that Theo's car in the driveway?\"\n\nHe stared at the rear end of the Lexus that protruded just past the edge of the carriage house. His eyes began to water and he rushed across the patio through a cluster of winter-killed potted plants and tore open the French doors.\n\nTheodosha Flannigan sat in an antique chair, with a crimson pad inset in the back, her guitar perched on her lap, her trimmed fingernails like shavings from seashell, her knees close together, at a slight ladylike angle, her mouth parted in mild surprise, a hole with a tiny trickle running from it in the center of her forehead.\n\nThrough the front window I saw a half dozen emergency vehicles turn off the state road and come roaring up the drive through the long tunnel of live oaks, their flashers beating with light and color, their sirens muted, as though the drivers were afraid they might wake the dead.\n\n## Epilogue\n\nMy daughter Alafair and I flew to Key West for Christmas and hired a charter boat I could scarcely afford and scuba-dived Seven Mile Reef. The water was green, like lime Jell-O, with patches of hot blue floating in it, the reef swarming with baitfish and the barracuda that fed off them. At sundown we set the outriggers and trolled for a stray marlin or wahoo as we headed back into port, the gulls wheeling and squeaking over our wake, the sun bloodred as it descended into the Gulf.\n\nAlafair looked beautiful in her wetsuit, her body as sleek and hard and tapered as a seal's, her Indian-black hair flecked with seaweed. As she stood in the stern, watching our baited hooks skip over our wake, she reminded me of Theo Flannigan and all the innocent victims of violence everywhere, here, in this country, where friends clasped hands and leaped from flaming windows into the bottomless canyons of New York City, or in the Mideast, where a storm of ballistic missiles and guided bombs would rain down upon people little different from you and me.\n\nBut it was the season of Christ's birthday and I did not want to dwell upon all the corporate greed and theological fanaticism that had rooted itself in the modern world. We attended Mass in a church James Audubon had sat in, strolled Duval Street among revelers with New York accents, ate dinner at a Cuban cafe by the water under a ficus tree threaded with Christmas lights, and visited the home of Ernest Hemingway down on Whitehead Street. The sun was gone, the sky full of light, the incoming tide wine dark against the horizon, and bottle rockets fired from Mallory Square were popping in pink fountains high above the waves. How had Hemingway put it? The world was a fine place, and well worth the fighting for.\n\nAs Alafair and I walked back toward the happy throng on Duval Street, the yards around us blooming with flowers, the air touched with salt and the smell of firecrackers, I thought perhaps the world was more than just a fine place, that perhaps it was a domed cathedral and we only had to recognize and accept that simple fact to enjoy all the gifts of both heaven and earth.\n\nCastille LeJeune was sentenced to Angola Prison for manslaughter in the shooting of Max Coll and for first-degree homicide in the death of Will Guillot. Because of his age, he was transferred to an honor farm, where he did clerical work in an office. The correctional officers at the farm admired him for his genteel manners and military bearing and his fastidiousness about his dress. In fact, they came to call him \"Mr. LeJeune\" and often sought his advice about financial matters. But a visiting prison psychologist put an evaluation in his jacket that indicated LeJeune was not only experiencing depression and self-loathing over the death of his daughter but perhaps intense levels of guilt characteristic of a father who has sexually molested his daughter.\n\nAn inmate's jacket is confidential only until the first trusty clerk reads it.\n\nCastille LeJeune became what is known as a short-eyes in the prison population. Other inmates shunned him; the correctional officers became distant and formal in their dealings with him. He was transferred back to Angola after he bit into pieces of broken glass that had been mixed into his food.\n\nIronically, he was placed in a segregated unit within viewing distance of the levee built by the Red Hat Gang on which Junior Crudup had pulled what Leadbelly called his great, long time.\n\nWe couldn't make our homicide case against Merchie Flannigan and he got away with the murder of the daiquiri-store operator. At least legally he did. But Castille LeJeune nailed him from jail by having his lawyers file a wrongful death suit against him, freezing his personal and corporate accounts, then using Donna Parks to bring statutory rape charges against him. Merchie's reputation was ruined and his pipeline business went bust. For a while he ran a welding service, then began hanging out at a bar frequented by Teamsters in Baton Rouge. I ran into him one day by the capitol building, where Huey Long was shot down in 1935.\n\n\"Hey, Dave, no hard feelings, huh?\" he said.\n\n\"Not on my part,\" I replied.\n\nHe smelled of cigarettes and was fat and puffy, sporting a mustache and goatee, driving a junker car that was parked at the curb, a young girl in the passenger seat.\n\n\"That's my niece,\" he said.\n\n\"Right,\" I said.\n\n\"Putting together a drilling deal in Iran, can you believe it?\"\n\n\"That's great, Merch.\"\n\n\"Good seeing you, Dave. I mean that,\" he said, taking my hand, trying as hard as he could to hold my eyes without averting his. The girl tossed a beer can out of the window as they drove away.\n\nFather Jimmie and I and two employees of a funeral home made up the entire retinue at the graveside ritual for Max Coll in a Catholic cemetery outside Franklin. I felt partly responsible for his death, but had he not died, Castille LeJeune would not have gone down, nor would Castille LeJeune have utilized an opportunity to take Will Guillot off the board. Ultimately I came to think of Max Coll in another fashion. In his way he was a brave man who made his own choices, and it was an arrogance on my part and a disservice to him for me to pretend that somehow I was the designer of his fate.\n\nFather Jimmie went back to his conservative parish in New Orleans and worked as a chaplin at Central Lock-Up. After Alafair returned to college in Portland, I invited both him and Clotile Arceneaux to dinner at a Mexican restaurant off the upper end of St. Charles.\n\n\"Pretty slick how you took down LeJeune, searching his property without a warrant,\" she said.\n\n\"It was dumb,\" I said.\n\n\"You never flinched, even though LeJeune had a pistol on you and you thought you didn't have backup,\" she said.\n\n\"Say that again.\"\n\n\"I watched you through a pair of field glasses. An FBI sharpshooter watched LeJeune through a scope on a rifle.\" She put a forkful of food in her mouth and raised her eyebrows at me.\n\n\"Y'all were using me as bait?\"\n\n\"Got a job for you with the state if you want to start putting away bad guys again.\"\n\n\"I'm out.\"\n\nShe placed her foot on mine under the table and squeezed. \"Come see me sometime and we'll talk about it,\" she said.\n\n\"Am I missing something here?\" Father Jimmie said.\n\n\"Dave likes to pretend he can stop being a police officer. Make him go to confession, Father,\" she replied.\n\nClete Purcel spent three months in the St. Mary Parish Prison, paid twenty-thousand dollars in damages to the sheriff's deputies he had busted up with his fists, and upon his discharge moved in with me, saying he was going to start up another P.I. office in New Iberia. We fished for sac-a-lait and bass at Henderson Swamp and Bayou Benoit, and Clete tried to appear light-hearted and unaffected by his time in jail. But I knew better. Clete was a natural-born cop and despised the new breed of criminals and literally washed himself in the shower with peroxide when he got out of the bag.\n\nBut out on Bayou Benoit, with the spring breezes up and the bream spawning back in the bays, the levee sprinkled with buttercups, we didn't talk about the bad times of the past or the present. I had never looked to the skies for great miracles, and, as St. Augustine once indicated, to watch a vineyard soak up the water in a plowed row and produce a grape that could be translated into wine was all the proof we needed of higher realities. But when Clete and I were deep in the swamp, the lacy green branches of the cypress trees shifting back and forth across the sun, I fell prey to a new temptation as well as hope.\n\nI waited to see a pair of pelicans drift down off the windstream, their wings extended and pouched beaks bulging, their improbable presence a harbinger of better times. I waited for them daily and sometimes in the flapping of wings overhead I thought I heard Bootsie's voice, reminding me of her promise about the pelicans, only to discover that a white crane or blue heron had been frightened by our outboard and had flown through the cypress trees back onto open water.\n\nBut I'm sure one fine day, when I least expect it, the pelicans will return to Bayou Teche, and in the meantime I share my thoughts about them with no one, except perhaps Snuggs and Tripod, who, like me, sleep little and wake before first light.\n\n## Contents\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nEpilogue\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by Victorian\/Edwardian Pictorial Magazines, Lesley\nHalamek, Jonathan Ingram and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\n_THE HARMSWORTH MAGAZINE._\n\n\n\n\n[Illustration: A FAIR ANGLO-SAXON.\n\n_From the Painting by A. Seifert._\n\n_By Permission of the Berlin Photographic Co., Bond Street, W._]\n\n\n\n\nTHE\n\nHARMSWORTH\n\nMONTHLY PICTORIAL\n\nMAGAZINE\n\n\nVOLUME I.\n\n1898-9. No. 1.\n\n[Illustration]\n\n\n_PUBLISHED BY_\n\nHARMSWORTH BROS., LIMITED, LONDON, E.C.\n\n\n\n\nINDEX TO VOL. I.\n\nARTICLES.\n\n                                                                 PAGE.\nALBUM, A FAMOUS WIGMAKER'S FAMOUS. By Gavin Macdonald.\n    Illustrated by Facsimiles                                     356\nBALLOON JOURNEY, A GIRL'S, OVER LONDON. By Gertrude Bacon.\n    Illustrated by Photographs                                    400\nBEAUTIES, IRISH. By Ignota.\n    Illustrated by Photographs                                    484\nBLOODHOUNDS, A MAN HUNT WITH. By Alfred Arkas.\n    Illustrated by Photographs                                    383\nCHESHIRE TOWN, IN A DISAPPEARING. By Percy L. Parker.\n    Illustrated by Photographs                                    166\n\"CHRYSANTHEMUMS CURLED HERE.\" A Chat with a Floral Barber.\n    By Alfred Arkas. Illustrated by Photographs                   579\nCRACKERS, COSTLY CHRISTMAS. The Romance of Christmas Presents.\n    Illustrated by Photographs                                    439\nCRICKET AND CRICKETERS. Words by M. Randall Roberts.\n    Pictures by Mr. \"Rip\"                                         212\nCRICKET MATCH, A VERY QUEER. Mr. Dan Leno's Eleven v. Camberwell\n    United C.C. By Gavin Macdonald. Illustrated by Photographs    323\nCYCLIST, THE CLEVEREST AMATEUR, IN THE WORLD. Remarkable Trick\n    Riding by a Military Officer                                  493\nDANGER SIGNALS, NATURE'S. A Study of the Faces of Murderers.\n    By J. Holt Schooling. Illustrated by special Photographs      656\nDARLINGS, LITTLE. By Somers J. Summers.\n    Photographic Illustrations by W. J. Byrne                      99\nDOCUMENTS, INCRIMINATING.\n    With Facsimiles of Fatal Writings                             304\nDOOR-KNOCKERS, FAMOUS LONDON.\n    Illustrated by Photos specially taken.                        216\nDOUBLES IN REAL LIFE, NOTABLE.\n    With Photographic Evidence                                      5\nENGINE MATCH BETWEEN ENGLAND AND AMERICA, AN. By F. A. Talbot.\n    Illustrated by Photographs                                    651\nEXCUSE, OUR, FOR THE ISSUE OF A SIXPENNY MAGAZINE\n    AT THREEPENCE                                                   3\nFIRE BRIGADE HEROES, TRAINING OUR. By Alfred Arkas.\n    Illustrated by Photographs                                    243\nFIRES, SOME SENSATIONAL. By Frederick A. A. Talbot.\n    Illustrated by Photographs                                    529\nFOOTBALL, MAKING A. An Essential Part of a Great Game.\n    Illustrated.                                                  444\nFORTRESS, THE MOST REMARKABLE, IN THE WORLD. By Percy L. Parker.\n    Illustrated by Photographs                                    274\nMAN-OF-WAR, HOME LIFE ON BOARD A.\n    Illustrated by Photographs                                     86\nMAN IS MADE OF WHAT? By T. F. Manning.\n    Illustrated by Photographs                                    339\nMEDICAL DETECTIVE AND HIS WORK, THE. By T. F. Manning.\n    Illustrated by A. Morrow and by Diagrams                      144\nMICE WORTH THEIR WEIGHT IN GOLD. By Gavin Macdonald.\n    Illustrated by Photographs                                    631\nMINIATURE CRAZE, THE MODERN. By H. M. Tindall.\n    Illustrated by Charming Examples                              197\nMONEY, STRANGE KINDS OF. By Robert Machray.\n    Illustrated by Photographs                                    639\nMURDERS, LONDON'S UNDISCOVERED. By Lincoln Springfield.\n    Illustrated by Photographs                                    515\nNEWSPAPER, MAKING A MODERN.\n    By Alfred C. Harmsworth                                        38\n\"PERPETUAL MOTION\" SEEKERS.\n    With Illustrations of Machines recently invented              315\nPHOTOGRAPHIC LIES. With Remarkable Photos, proving\n    the Uselessness of the Camera as a Witness                    259\nPOISON DEVICES.\n    Illustrated                                                   106\nPOSTAGE STAMPS WORTH FORTUNES.\n    Illustrated by Facsimiles of Valuable Stamps                  327\nRAILWAY SMASHES, FAMOUS. By Frederick A. Talbot.\n    Illustrated by Photographs                                    227\nROYALTIES, LITTLE.\n    Illustrated with Photographs by Speaight                      590\nROYALTY, TATTOOED. By R. J. Stephen.\n    Illustrated by Photographs                                    472\nSANDOW, HOW, MADE ME STRONG.\n    Illustrated with Photographs                                   23\nSECRET CHAMBERS, REMARKABLE.\n    Written and illustrated by Allan Fea                          416\nSERMONS WITHOUT WORDS. A Marvellous Performance in Dumb Show.\n    By Alfred Arkas                                                67\nSKELETONS, MODERN FAMILY. By Beatrice Knollys.\n    Illustrated by A. S. Hartrick                                  17\nSLEIGHS FOR CHRISTMAS. By J. E. Whitby.\n    Illustrated by Photographs                                    558\nSMOKER'S MUSEUM, FROM A. By T. C. Hepworth.\n    With Illustrations                                            370\nSPORT, THE MOST CRUEL, IN THE WORLD. By Sidney Gowing.\n    Illustrated by Photographs                                    182\nSTATISTICS GONE MAD. By J. E. Grant.\n    Illustrated by Diagrams                                       609\nTEA, HOME OF FOUR O'CLOCK, THE.\n    Illustrated by Photographs                                    605\nTOY, A \u00a310,000. Complete Working Railway in a Room.\n    By Robert Machray. Illustrated by Photographs                 125\nWEATHER, HOW WE GET OUR. By Gavin Macdonald.\n    Illustrated by Photographs                                     55\nWHISTLER, THE WORLD'S CHAMPION.\n    Illustrated by Photographs and Musical Examples               546\nWHITE \"ZOO,\" A. Lord Alington's Hobby. By Alfred Arkas.\n    Illustrated by Photographs                                    154\nWIVES, AMERICAN, OF ENGLISH HUSBANDS.\n    Illustrated by Portraits                                      289\n1898. Your Everyday Life in the past Twelve Months.\n    By Alfred Arkas                                               455\n3,000 MILES ON RAILWAY SLEEPERS. One Aspect of a Bicycle Tour\n    Round the World. By Edward Lunn. Illustrated by Photographs   619\n\n\nSTORIES.\n\nBABY SANTA CLAUS, A. The Story of a Christmas Reconciliation.\n    By Marion Elliston. Illustrated by Harold Copping             521\nBEHAVIOUR OF WARRINGTON, V.C., THE. By Percy E. Reinganum.\n    Illustrated by W. B. Wollen, R.I.                             236\nCHANCELLOR'S WARD, THE. By Richard Marsh.\n    Illustrated by F. H. Townsend                                  73\nCHOLERA SHIP, THE. By Cutcliffe Hyne.\n    Illustrated by Richard Jack                                   159\nCLEVER MRS. BLADON. By E. Burrowes.\n    Illustrated by Sydney Cowell                                  645\nCOUNT AND I, THE. The Story of a Stolen Letter.\n    By James Barratt. Illustrated by Robert Sauber                447\nCOURTSHIP BY PROXY. By H. A. Therrauld.\n    Illustrated by Fred Pegram                                    461\nCROWDED HOUR, A. By Clarence Rook.\n    Illustrated by B. E. Minns                                    634\nCURSE OF THE CATSEYE, THE. By Alfred Slade.\n    Illustrated by E. Prater                                      623\nDAPHNE. By Walter E. Grogan.\n    Illustrated by Harold Copping                                 361\nDESCENT OF REGINALD HAMPTON, THE. By Halliwell Sutcliffe.\n    Illustrated by W. Rainey, R.I.                                189\nDESPATCHES FOR GIBRALTAR, THE. By Gilbert Heron.\n    Illustrated by D. B. Waters                                   389\nDESTINY, MY. A Wayside Romance. By C. K. Burrow.\n    Illustrated by Fred Pegram                                    347\nEDITOR'S ESCAPADE, THE. By Archibald Eyre.\n    Illustrated by S. H. Vedder                                   405\nFACE AT THE DOOR, THE. By Walter D. Dobell.\n    Illustrated by S. H. Vedder                                   373\nFAIR NEIGHBOUR'S PIANO, MY, AND WHAT CAME OF IT.\n    By Henry Martley. Illustrated by F. H. Townsend               281\n\"FINDER WILL BE REWARDED, THE.\" A Bachelor's Romance.\n    By Gerald Brenan. Illustrated by Sydney Cowell                489\nFIVE HUNDRED POUND PRIZE, THAT. By Richard Marsh.\n    Illustrated by John H. Bacon                                  172\nGASCOYNE'S TERRIBLE REVENGE. A Story of the Indian Mutiny.\n    By J. F. Cornish. Illustrated by Vereker M. Hamilton. R.P.E.  265\nGOLDEN CIRCLET, THE. By Charles Kennett Burrow.\n    Illustrated by Ralph Peacock                                   11\nHER LETTER! By J. Harwood Panting.\n    Illustrated by W. B. Wollen, R.I.                              61\nHIS HIGHNESS THE RAJAH. The Quest of the Yellow Diamond.\n    By Beatrice Heron-Maxwell. Illustrated by E. J. Sullivan      549\nHIS SOVEREIGN REMEDY. By Clarence Rook.\n    Illustrated by B. E. Minns                                     94\nHOW THE BURGLAR HELPED AT CHRISTMAS. By Lucian Sorrel.\n    Illustrated by H. M. Brock    476\nHOW THE MINISTER'S NOTES WERE RECOVERED.\n    By Beatrice Heron-Maxwell. Illustrated by Fred Pegram         250\nIAN'S SACRIFICE. By Alick Munro.\n    Illustrated by Ralph Peacock                                  309\n\"KLONDYKE, OFF TO.\" By George A. Best.\n    Illustrated with Novel Life Photographs                       583\nLONDON'S LATEST LION. By Gilbert Dayle.\n    Illustrated by Fred Pegram                                    595\n\"MAN OVERBOARD!\" An Episode of the Red Sea.\n    By Winston Spencer Churchill. Illustrated by Henry Austin     662\nMISSING Q.C.'s, THE. By John Oxenham.\n    Illustrated by Frank Craig and T. Robinson                    497\nMOTOR-CAR ELOPEMENT, AND HOW IT ENDED, THEIR. By Edgar Jepson.\n    Illustrated by H. R. Millar                                    49\nPRINCESS IN GREEN AND TAN, A. By Arthur Preston.\n    Illustrated by A. Rackham                                     611\nSHORT MEMORY OF MR. JOSEPH SCORER, THE VERY.\n    By John Oxenham. Illustrated by H. M. Brock                   131\nSTIR OUTSIDE THE CAF\u00c9 ROYAL, THE. By Clarence Rook.\n    Illustrated by Hal Hurst, R.B.A.                              319\nSTONE RIDER, THE. By Nellie K. Blissett.\n    Illustrated by Max Cowper                                      30\nTELEGRAPH MYSTERY, A. By W. B. Northrop.\n    Illustrated by H. H. Fl\u00e8re.                                   539\nTRAGEDY OF A THIRD SMOKER, THE. By Cutcliffe Hyne.\n    Illustrated by J. Finnemore. R.B.A.                           297\nTRAVELLING COMPANION, MY. By Catherine Childar.\n    Illustrated by Fred Pegram                                    115\n\nFULL PAGE ILLUSTRATIONS.\n\n\"ANDR\u00c9E, INDEED! I WAS THERE LONG AGO.\"\n    From the Painting by T.C. Hepworth                            669\nBURDEN OF LOVE, A.\n    From the Painting by N. Sichel                                224\nCHARLES I. ON HIS WAY TO EXECUTION.\n    From the Painting by Ernest Crofts, R.A.                      331\nCHRISTMAS, THE FIRST.\n    From the Painting of H. J. Sinkel                             434\nCUBAN BELLE, A.\n    From the Painting by Gabriel Ferrier                          219\nDAUGHTER OF CANADA, A.\n    Photographic Study                                            565\nDECEMBER DAY IN THE OLDEN TIME, A.\n    From the Painting by A. Perez                                 568\nDRAGON AND GEORGE, THE.\n    From the Painting by R. Holyoake                              333\nEMPTY CHAIR, THE.\n    From the Painting by Briton Rivi\u00e8re, R.A.                     336\nEVERYBODY'S FAVOURITE.\n    Photographic Study                                            561\nFAVOURITE, THE.\n    From the Painting by Arthur J. Elsley                         110\nFOR DEAR LIFE.\n    From the Painting by Stanley Berkeley                         329\nGIRL OF THE PERIOD.\n    From the Painting by Heywood Hardy                            668\nGOOD NIGHT!\n    From the Painting by G. Hom                                   112\nGORDONS AND GREYS TO THE FRONT.\n    From the Painting by Stanley Berkeley                         430\nGREEK GIRLS PLAYING BALL.\n    From the Painting by the late Lord Leighton                   577\nGREUZE'S MASTERPIECES, ONE OF.\n    Now in the National Gallery                                   425\nHAPPY AS A KING.\n    Photographic Study                                            671\n\"HUSH!\"\n    From the Painting by Maud Goodman                             109\nIN RUSSIA--THE TERROR OF THE PLAIN.\n    From the Painting by A. Von W. Kowalski                       672\nJOHN BULL FOR EVER--WHAT WE HAVE WE'LL HOLD.\n    From the Painting by Maud Earl                                 56\nJUDITH.\n    From the Painting by N. Sichel                                334\nLAKE WINDERMERE IN THE WINTER OF 1885.\n    From a Photograph                                             564\nLAST ELEVEN AT MAIWAND, THE.\n    From the Painting by Frank Feller                             566\nLAST MINUTE, THE. NOW OR NEVER.\n    From the Painting by T. M. Hemy                               443\nLITTLE DEAR, A.\n    Photographic Study                                            667\nLIVE AND LET LIVE.\n    From the Painting by A. W. Strutt                             332\nMAKING A MARRIAGE IN THE OLDEN TIME.\n    From the Painting by A. T. Vernon                             221\nMANNERS AT TABLE.\n    From the Painting by A. J. Elsley                             330\nMEDITATION.\n    From the Painting by N. Sichel                                111\nMIRIAM THE PROPHETESS.\n    From the Painting by N. Sichel                                574\nMOTHER'S DARLING.\n    Photographic Study                                            569\nNAPOLEON'S FLIGHT AFTER WATERLOO.\n    From the Painting by A. C. Gow, R.A.                          666\nOPPORTUNITY FOR FLATTERY, AN.\n    From the Painting by D. Hernandez                             575\nOVERTAKEN!\n    From the Painting by John A. Lomax                            280\nPETS, SOME COSTLY.\n    From Photographs                                               85\nPRINCE, OUR.\n    From the Painting by A. Stuart Wortley                        567\nPUSHING FAMILY, A.\n    From the Painting by G. A. Holmes                             428\nRUSSIAN BELLE, A.\n    Photographic Study                                            571\nSALMON POACHER, THE.\n    From the Painting by Douglas Adams                            335\nSON AND HEIR, THE.\n    From the Painting by L. Schmutzler                            427\nSPAIN, A FLOWER OF.\n    From the Painting by N. Sichel                                108\nSPAIN, A LITTLE MAID FROM.\n    Photographic Study                                            338\nSPANISH PEACE COMMISSIONER, A.\n    From the Painting by Hal Hurst, R.B.A.                        665\nSUMMER.\n    From the Painting by W. Reynolds Stephens                     220\nSWEET AND TWENTY.\n    From the Painting by G. L. Seymour                              2\nTALLY HO!\n    From the Painting by Heywood Hardy                            572\nTIME TO GET UP.\n    From the Painting by A. J. Elsley                             426\nTURNER'S GREAT WORK--THE FIGHTING TEMERAIRE.\n    Now in the National Gallery                                   429\nVERY OLD, OLD STORY, A.\n    From the Painting by L. Alma Tadema, R.A.                     670\nWAITS, THE.\n    From the Painting by W. H. Trood                              570\nWATER CARRIER, THE.\n    From the Painting by J. W. Godward                            222\nWHICH WINS?\n    From the Painting by Arthur J. Elsley                         223\nWHY NO. I. WAS SO POPULAR.\n    Head, from the Painting by A. Seifert                         563\nWHY THE ANTELOPES STAMPEDED.\n    From the Painting by William Strutt                           226\nWILL HE COME?\n    From the Painting by Marcus Stone, R.A.                       114\nYORKSHIRE LASS, A.\n    Photographic Study                                            573\n\nPOETRY.\n\nBABY BELLE. By Bernard Malcolm Ramsay.\n    Illustrated by Harold Copping                                 482\nBABY, IN PRAISE OF. By Barrington McGregor.\n    Illustrated by C. Robinson                                    661\nGOLDEN HAIR AND CURLYHEAD. By Allan Upward.\n    Illustrated by J. H. Bacon                                    435\nLITTLE MAID.\n    Illustrated by C. Robinson                                    258\nROGUEY MAN, THE.\n    Illustrated by H. H. Fl\u00e8re                                    346\nROSE AT LAST, A. By Clifton Bingham.\n    Illustrated by Harold Nelson                                   22\nSAD FATE OF MISTRESS PRUE, THE.\n    Illustrated by Robert Sauber                                  399\nSHOE, A TINY.\n    Illustrated by Archie Watkins                                 308\nSUNSET, BEYOND THE. By Clifton Bingham.\n    Illustrated by Charles Robinson                               235\nTHREE SCORE AND TEN.\n    Illustrated by T. Walter West                                 388\nTO A BLANK SPACE. By the Rev. J. Hudson, M.A.\n    Illustrated by Robert Wallace                                 576\n[Illustration: SWEET AND TWENTY.\n\n_From the Painting by G. L. Seymour._] [Illustration]\n\n\n\n\nOUR EXCUSE ...\n\nFOR THE ISSUE OF A SIXPENNY MAGAZINE AT THREEPENCE.\n\n... _SOME REASONS WHY_.\n\n\nThe beginning of a new Magazine, once an event, is now so much a\ncommonplace that the ancient excuse of the \"long felt want\" no longer\nserves.\n\nIn the days of the Nabobs, the gentle shaking of the Pagoda tree\nsufficed to bring great stores of wealth, but these be the times of\nthe fallen rupee. Your modern Anglo-Indian toils out his existence for\na bare pittance. And it is so in the making of Magazines. One hundred\nand fifty years ago the mere issue of the \"Gentleman's\" stirred to\ntheir depths the Coffee Houses and the Clubs, not only here in the Old\nCountry, but in our North American Colonies as well.\n\nTimes are changed, alas! \"The Harmsworth Magazine,\" though, indeed, it\nappeals to an English-speaking audience of over one hundred millions,\nwill at best provoke a little favourable comment in the train and\nthe library, for the Magazine field has been vastly exploited, and\nespecially of late. A modern buyer of periodical publications rises as\nwarily to a new lure as a twice-shot-over partridge to the gun.\n\nThe reader of Magazines has of late years been harried by a direct,\nby an enfilading, and a ricochetting fire of new adventures, some\nhonestly and avowedly frivolous, others portentously literary, a few\nloftily artistic. Every imaginable plan has been adopted whereby his\ncapture might be effected. Projectiles calculated to vanquish by size\nand weight of paper have been hurled at him; there have even been\nsurreptitious and spy-like attempts to enter his domestic circle by\nseeking the favour of his wife and daughters by means of \"Women's\nDepartments,\" all frocks, furbelows, and complexion cures; and worse,\nhis very children have been attacked by page on page of \"Nursery Chat\"\nand \"Tiny Tales for Little Listeners.\"\n\nLast straw of all, he has been patronised by the vast army of \"Great\nAuthors\" of the period. And if the chit-chat of the press is to\nbe believed there never were in Rome, in Athens, or in the days of\nElizabeth herself, so many distinguished litterateurs as at present.\nThe unfortunate victim has trembled at the solemn pomp of\n\n    \"The editor of the 'Monster Magazine' has pleasure in\n    announcing he has been so fortunate as to secure the\n    masterpiece of Mr. ----.\"\n\nor,\n\n    \"It is rumoured that Mr. ---- has been induced to enter\n    into an agreement to contribute an important series of short\n    stories to the \"Monster Magazine\" during the Spring of 1905.\n    Mr. ---- is entirely occupied in the fulfilment of various\n    contracts until that time.\"\n\nIt is \"right here,\" as our American kinsmen have it, that \"The\nHarmsworth Magazine\" comes in.\n\nTogether with a great many other people, we came to the conclusion\nlong since that a good deal of the literary wares that are foisted on\nthe public by means of the ordinary advertising methods of personal\nparagraphs and \"interviews\" is mainly rubbish. Frankly and openly\ndo we, therefore, declare that mere \"names\" will never command an\nentrance to the pages of this Magazine. As with our \"Daily Mail\" and\nour other journals, we shall rely on new writers. The public is weary\nof the reiteration of the same contributors to each of the monthly\npublications. He (and she) wants something new. It is our desire, for\nthe sake of the public, for the benefit of young artists and others,\nand for our own profit, to avoid the productions of the professional\n\"ring\" of much advertised mediocrity which most assuredly dominates\nmany of our Magazines to-day, though the work of really representative\nmen and women will always be secured, without regard to its cost.\n\nIn selecting the price at which \"The Harmsworth Magazine\" should be\nissued to the British, Canadian, Australasian, South African, and\nAnglo-Indian public, we choose that of the two most distinguished\njournals in our language, \"The Times\" and \"Punch.\"\n\nCan such a publication as this be sold for 3d.? Provided we reach\na gigantic circulation, we can do it. We are enabled to issue a\nthreepenny Magazine containing more expensive literary matter, more\nnumerous pictures, and more pages than the sixpenny Magazines of a few\nmonths back, at so ridiculous a price, because this Magazine is only a\nsmall incident in an organization controlling four daily journals and\nnearly thirty weekly periodicals; because we already possess and are\nnow building printing machinery of an entirely novel and labour-saving\nnature.\n\nThe Magazine will be cheap as to price only. In every respect, save,\nperhaps, mere bulk, \"The Harmsworth Magazine\" will compete frankly,\nand without reserve, with older friends in the same field.\n\nThe experiment, largely due to a devoted band of workers, headed by my\nbrother Cecil, is at least an interesting one. Will it succeed? Much\ndepends upon the good word of those who read it. If it meets with\nyour approval, if you consider that the enterprise is worthy of\ncommendation, will you make our effort known to your circle?\n\nALFRED C. HARMSWORTH.\n\n[Illustration: WE ARE FIVE.]\n\n\n\n\nNOTABLE DOUBLES IN REAL LIFE.\n\n_With Photographic Evidence._\n\n\n[Illustration: _Elliot & Fry, photo._\n\nTHE DUKE OF NORFOLK.]\n\n[Illustration: _Elliot & Fry, photo._\n\nMR. G. MANVILLE FENN.]\n\nIt is pretty generally believed that the Czars of Russia are in the\nhabit of employing understudies to personate them when some more than\nusually hazardous public appearance has to be made. Whether or not\nthis is true we cannot take upon ourselves to say, but it is very\nclear that if Nicholas II. were in need of a \"double,\" he would not\nrequire to go outside the circle of his own relatives to find an\nalmost exact replica of himself in our Duke of York. The two Princes\nare first cousins, but the facial resemblance existing between them\nis far more remarkable than is ordinarily the case between near\nrelations. It is true, of course, that the Duke of York is a\nbetter-looking man than his cousin, but any make-up artist, by\nthe employment of a few pencilled lines round the eyes, and by\nre-arranging the hair, could transform H.R.H. into an exact likeness\nof the Czar.\n\n[Illustration: _W & D Downey, photo, Ebury Street._\n\nH.I.M. THE CZAR AND H.R.H. THE DUKE OF YORK.]\n\nMore noteworthy still, because of the absence of relationship between\nthem, is the likeness of the present Postmaster-General, the Duke of\nNorfolk, and the veteran novelist, Mr. George Manville Fenn. Looking\nupon the two portraits, it is not easy to believe that Mr. Fenn is\nsixteen years the senior of the head of the great house of Howard.\nAnother curious feature in connection with the two cases before us is\nthe fact that, although the Duke of Norfolk is almost as much like Mr.\nGeorge Manville Fenn as one pea resembles another, his resemblance\nto certain portraits of the great Charles Dickens is rather remote,\nwhereas Mr. Fenn's is very close.\n\nIt should here be mentioned that in the case of most of our doubles\nthe likeness is even more pronounced in actual life than it appears\nfrom the photographs. In many instances the gestures, the walk, and\nthe little mannerisms of the personages here portrayed are practically\nidentical. The writer recalls to mind the example of a gentleman\nwell-known in the West end of London who resembles the present Duke of\nDevonshire as closely as the Duke of York resembles the Czar. The\nDuke of Devonshire's imitator--if he be such--not only wears his hat\npressed down over his eyes in the well-known fashion of the Duke,\nbut assumes almost as inimitably that intensely bored look that has\ndeceived so many people as to the true character of the head of the\nLiberal Unionist party. Mere photographs would inevitably fail to do\njustice to a case of this kind.\n\n[Illustration: _Russell & Sons, photo._\n\nTHE RT. HON. J. CHAMBERLAIN, M.P.]\n\n[Illustration: _London Stereoscopic Co., photo._\n\nMR. AUSTEN CHAMBERLAIN, M.P.]\n\nIn regard to the adjoining portraits of Mr. Austen Chamberlain and\nthat of his scarcely less distinguished father, it is noticeable that\nin addition to the striking facial resemblance, there is the same\ndefect in the sight of the right eye occasioning the use of the\nmonocle. Even if we take it for granted that Mr. Joseph Chamberlain\nhas indulged in the harmless foible of dressing his hair and arranging\nthe cast of his countenance to accentuate his likeness to the member\nfor East Worcestershire, it cannot be gainsaid that the similarity\nbetween the son and the father is real enough to merit illustration in\nthis gallery of \"doubles.\"\n\nJesting apart, those who have studied Mr. Austen Chamberlain in the\nHouse and on the platform, prophesy for him a very remarkable career.\nHe has much of the readiness and all the imperturbability that have\nmade his father the ablest \"parliamentary hand\" since the retirement\nof Mr. Gladstone. It is interesting to note that the disbelief of Mr.\nChamberlain _p\u00e8re_ in exercise, as a means of recruiting the health,\nis not shared by Mr. C. _fils_, who is an enthusiastic cyclist.\n\n[Illustration: _London Stereoscopic Co., photo._\n\nMR. L. ALMA-TADEMA, R.A.]\n\n[Illustration: _Elliott & Fry, photo._\n\nTHE LATE MR. GEO. DU MAURIER.]\n\nThe late Mr. Du Maurier was of French extraction, while Mr.\nAlma-Tadema was born at Dronryp, in Holland, yet they might have been\ntwin brothers, so strangely alike were they. If Mr. Du Maurier had\nworn his hair a little longer and parted it in the middle, the most\nintimate mutual friends of the two distinguished artists must have\nfound it difficult to tell which was which. An amusing story is told\nillustrating this point. Mr. Du Maurier, dining at a friend's house\none evening, was placed next to a lady whom he did not recollect to\nhave met before. A brief dialogue, something to this purpose, ensued:\n\nLady: \"You know, Mr. Alma-Tadema, that you are supposed to resemble\nMr. du Maurier very closely. For my part, I do not see how the most\nsuperficial observer could be deceived in the matter!\"\n\nMr. Du Maurier: \"Pardon me, but I am Mr. Du Maurier!\"\n\nSome people tell the story the other way round--with Mr. Alma-Tadema\nas the second party in the dialogue--with equal effect.\n\n[Illustration: _Elliott & Fry, photo._\n\nPROFESSOR STUART, M.P.]\n\n[Illustration: _Elliott & Fry, photo._\n\nMR. STANLEY J. WEYMAN.]\n\nThese are portraits of Professor Stuart, M.P. for Hackney, and Mr.\nStanley J. Weyman, the novelist. If Mr. Weyman ever becomes a member\nof Parliament it is to be hoped that he will not relinquish his\neyeglass. Were he to do so he would run a great risk of merging\nhis identity in that of the Professor. He might not object to this,\nhowever, nor would Professor Stuart protest very indignantly we may be\nsure, were he to find himself suddenly credited with the authorship\nof Mr. Weyman's fascinating romances. Let us hope that Mr. Weyman will\nnot enter the political arena, bestowing on Westminster the gifts that\nwere meant for mankind.\n\n[Illustration: _Elliott & Fry, photo._\n\nMR. EDWARD GERMAN.]\n\n[Illustration: _Elliott & Fry, photo._\n\nMR. ANTHONY HOPE.]\n\nMost of us have forgotten that Mr. Anthony Hope contested a seat in\nParliament in 1892, but few of us are sorry that the gifted author\nfailed to get in. Mr. Anthony Hope Hawkins, to give him his full\nname, is an excellent speaker, but even that gift is not so useful\nin Parliament as consistent and unquestioning voting-power, and until\nmembers are allowed to read their speeches the gift of authorship\nwill remain at a discount there. A good many of us, perhaps, could cut\ntolerable figures at Westminster, but our Anthony Hopes and Stanley\nWeymans are few and far between, and we would wish to keep them to\ntheir proper work of literature. Mr. Edward German, Mr. Anthony Hope's\ndouble, is a young composer who has done very well already, and may be\nexpected to do better in the future.\n\n[Illustration: _Elliott & Fry, photo._\n\nTHE RT. HON. CECIL J. RHODES.]\n\n[Illustration: _Hills & Saunders, photo._\n\nSIR J. STAINER.]\n\nA close examination of the portraits of the Rt. Hon. Cecil John Rhodes\nand of Sir John Stainer, the Professor of Music at Oxford, should\nwell repay the expert physiognomist. At first blush it seems hardly\nprobable that the man of action, the empire builder, should have much\nin common with the scholarly musician--though indeed Mr. Rhodes has\n\"faced the music\" right manfully more than once in the course of his\nsplendid career. Examine carefully the mouths of our two celebrities,\nand take note of the well-defined lines leading downwards from the\ncorner of the nose. The eyes, too, and the contours of the two faces\nare strangely similar. There is a dimple in Mr. Rhodes' cheeks that\nproves conclusively, if we had no other evidence, that Mr. Rhodes is\na man of humour, nor are similar indications wanting in the adjoined\nportrait of Sir John Stainer. If Sir John had taken himself off to\nSouth Africa in early youth it might have been his fate to add another\nempire to the Queen's dominions; if Mr. Rhodes had stayed on at Oriel\nCollege, Oxford, and devoted his vast abilities to the study of music,\nhe might now be occupying the professional chair in that art at his\nAlma Mater.\n\n[Illustration: _London Stereoscopic Co., photo._\n\nMR. JOHN HARE.]\n\n[Illustration: _London Stereoscopic Co., photo._\n\nMR. ARTHUR ROBERTS.]\n\nThere is a distinct style of theatrical face that we all recognise\ndirectly we see it. For instance, the heavy tragedian with the blue\nchin and luxuriant hair, \u00e0 la Sir Henry Irving, is known wherever he\nis seen, and quite a number of pages of our Magazine might be filled\nwith his doubles. But Mr. John Hare and Mr. Arthur Roberts whose\nportraits we give side by side are comedians (of widely different\nstyles), and are not particularly theatrical in appearance. Off the\nstage Mr. Hare might be taken for an eminent Q.C., while \"Arthur\"\nmight be supposed to move exclusively in turf circles. Mr. Hare, whose\nreal name is Fairs, is, of course, the best \"old man\" actor we have.\nIn connection with this fact he himself tells a rather good story.\nHe was in a carriage on the Underground Railway when he met an old\nschool-fellow. Gradually the conversation turned to theatres. \"Are you\nfond of the stage?\" Mr. Hare was asked by his friend. When the reply\nwas \"Yes,\" he presumed that Mr. Hare had seen a certain play at the\nPrince of Wales's.\n\n\"No,\" said Mr. Hare, \"I can't say I have seen it!\"\n\n\"Then you should go at once,\" said his friend. \"It's a capital play,\nand a devilish clever old man acts in it--a fellow named Hare!\"\n\n[Illustration: _A. Sachs, photo, Bradford._\n\nMR. MARK OLDROYD, M.P.]\n\n[Illustration: _London Stereoscopic Co., photo._\n\nLORD BALFOUR OF BURLEIGH.]\n\n[Illustration: _Elliott & Fry, photo._\n\nSIR THOS. ESMONDE, M.P.]\n\n[Illustration: _Russell & Sons, photo._\n\nSIR E. GREY, M.P.]\n\nLord Balfour of Burleigh, the Secretary for Scotland, and Mr. Mark\nOldroyd, M.P. for Dewsbury, are an interesting pair of political\ndoubles. Lord Balfour (whose title by the way was attainted in 1716\nand only restored to the present peer in 1869) is one of the hard\nworkers in the House of Lords, and knows more about education, water\nsupplies, and Sunday closing, than an omnibus-full of average members\nof the Lower House. When not actively engaged, in his Secretarial\ncapacity, in looking after the interests of the Northern Kingdom, Lord\nBalfour is wont to put in a little light work as chairman of a factory\nor rating committee. Mr. Mark Oldroyd divides his time between his\npolitical duties and his business, as a woollen manufacturer, in\nDewsbury. He has been mayor of the famous Yorkshire town, and is as\nproud of his native place as his townsfolk are proud of him.\n\nTwo youthful baronets and Members of Parliament now claim our\nattention. Sir Edward Grey is almost as distinguished in Parliament\nas he is in the world of athletics--he is once more tennis (not\nlawn-tennis) champion for England. As Under-Secretary for Foreign\nAffairs in the last Government, he was a pronounced success--his\nmanner being voted only less superior than that of the extremely\nsuperior person, the Hon. George Curzon, who ornaments the same office\nat the present time. Sir Thomas Esmonde, born in the same year (1862)\nas Sir Edward Grey, should have a splendid parliamentary future before\nhim, for he is a descendant of no less a celebrity than the great\nHenry Grattan.\n\n[Illustration: _Elliott & Fry, photo._\n\nRT. HON. LORD ROSEBERY.]\n\n[Illustration: _Elliott & Fry, photo._\n\nHON. PHILIP STANHOPE, M.P.]\n\n[Illustration: _London Stereoscopic Co., photo._\n\nRT. HON. ARNOLD MORLEY, M.P.]\n\nLord Rosebery has at least two doubles among public men. This is not\nto be wondered at when one considers how popular a man is the last\nLiberal Prime Minister.\n\nWhen the Duke of Wellington was living, it was the pride of many a\nprivate citizen to be thought like the great Duke; and Disraeli had\nmany doubles, the late Sir James Stansfeld being one of them. In\nGermany, at the present moment, we may meet passable duplicates of\nBismarck in every town. Who does not recollect the perfect army of\nRandolph Churchills that invaded society when that brilliant young\nstatesman's fame was at its greatest? It is surely a harmless conceit\nthat causes an inoffensive private person, if he in any way resembles\na great man of whom everybody is talking, to accentuate the likeness\nby every means in his power.\n\nBut in the case of Lord Rosebery's doubles it is somewhat different.\nBoth Mr. Arnold Morley and Mr. Philip Stanhope are distinguished men\nthemselves, and we may be quite sure that they do not spend much of\ntheir time dressing up to the likeness of their political leader.\nMr. Philip Stanhope is a near relative of Lord Rosebery's, and is of\nexactly the same age. Mr. Arnold Morley is two years younger than Lord\nRosebery (having been born in 1849), was Postmaster-General in the\nlast Liberal Administration, and may some day be Prime Minister.\n\n[Illustration: _Valentine & Sons, photo._\n\nTHE LATE RT. HON. W. E. GLADSTONE.]\n\n[Illustration: _Westfield, photo, Walmer._\n\nMR. H. PAGE, J.P.]\n\nWith doubles of Mr. Gladstone we might easily fill several pages\nof this magazine. Mr. Henry Page, J.P., of Deal, is an almost exact\nreplica of the venerable statesman, and has been the recipient of\nattentions really meant for Mr. Gladstone on more than one occasion.\nIt is a singular fact that Mr. Page's father bore a remarkable\nlikeness to the Duke of Wellington.\n\nThe reader will have noticed already that the greater number of our\ndoubles is to be found in the ranks of the politicians. It is really\nquite astonishing to contemplate how many doubles are to be found in\nthe House of Commons itself.\n\n[Illustration: _Elliott & Fry, photo._\n\nMR. H. O. ARNOLD FORSTER, M.P.]\n\n[Illustration: _Elliott & Fry, photo._\n\nMR. E. F. G. HATCH, M.P.]\n\nMr. H. O. Arnold Forster and Mr. E. F. G. Hatch, M.P. for the Gorton\nDivision of South-West Lancs, for instance, it is said grow more like\none another every day.\n\nThe difficulty experienced by the Speaker in attaching the right name\nto these gentlemen when they rise to \"catch his eye\" must be very\nconsiderable.\n\n[Illustration: _Russell & Sons, photo._\n\nMR. J. ROCHE, M.P.]\n\n[Illustration: _Elliott & Fry, photo._\n\nRT. HON. LORD GEORGE HAMILTON, M.P.]\n\nLord George Hamilton, who, with Mr. J. Roche, M.P., makes up the\nlast pair of our doubles, is an excellent example of the immense\ndisadvantage attaching to a public man whose features do not lend\nthemselves to caricature. Had Lord George overcome his natural\ndeficiencies in this respect by the adoption of an eyeglass, an\norchid, or an eccentric brand of waistcoat, he might ere now have been\nranked among our Prime Ministers, for it is an undoubted fact that\nthese details are better remembered by the public at large than years\nof devoted hard work.\n\nDisraeli's cork-screw curl on the forehead is less likely to be\nforgotten than his splendid services to the Empire, while it may be\nasserted with confidence that Mr. Chamberlain's eyeglass and orchid\nwill linger in the public mind long after his personal sacrifices\nfor the principle of Unionism are familiar to none but the student of\nhistory.\n\nWhen at the General Election of 1868 Lord George captured the seat\nfor the County of Middlesex--then regarded as an impregnable Liberal\nstronghold--a dazzling future was prophesied for him. If these\nprophecies have not been realised to the full extent it is not, as\nwe believe, because Lord George has not lived up to his earlier\nreputation, but simply because Nature has not gifted him with\na remarkable personal appearance, nor art with a satisfactory\nsubstitute. However, a Statesman even of the first rank who has\noccupied with distinction such important offices as First Lord of\nthe Admiralty and Secretary of State for India, has no reason to be\ndissatisfied with himself. No doubt each reader of this article will\nbe able to add considerably to our gallery of \"doubles,\" but we have\ndone enough if we have opened up an amusing and interesting train of\nideas.\n\n\n\n\nTHE GOLDEN CIRCLET.\n\nA COMPLETE STORY BY CHARLES KENNETT BURROW.\n\n_Illustrated by Ralph Peacock._\n\n\n[Illustration: \"HE VENTURED TO GLANCE OUT.\"]\n\nAnnesley walked past the main entrance to the Century Theatre in the\ncurious condition of one who is able partly to regard himself from\nthe outside. The boards were placarded with the announcement of a new\nplay, to be produced that day week, \"The Golden Circlet,\" by Conrad\nHowe. Now Annesley and Conrad Howe were the same person; but it was\ndifficult to convince the former, who had worked so deadly hard and\nfailed so often, that the latter was now within sight of what might\nprove a great success. Annesley saw people stop to look at the\nannouncement and read his other name, with a feeling that he was\nalmost guilty of a serious misdemeanour; he was taking them, as it\nwere, at a disadvantage; he was almost inclined to tap one elderly\ngentleman on the shoulder and assure him that no harm was intended to\nhim or any one else.\n\nThe secret of the authorship of \"The Golden Circlet\" had been well\nkept. Only three people were in the know, and not one of these was\na woman. Annesley therefore felt safe. He had assumed the other name\nbecause his own had brought him no luck; he imagined people shrugging\nshoulders and wagging wise heads; he could hear the murmur,--\"What!\nAnnesley still writing plays? If he hadn't wasted his time over that,\nhe might have had some money left. What a fool the man is!\" Annesley\nhad therefore put down the pen and Conrad Howe had taken it up.\nMoreover, Conrad Howe had actually written a play which seemed to\nhave in it the elements of popularity; hence newspaper paragraphs,\ndiscussions as to identity, and finally the fixing of the first night\nand the appearance of the posters.\n\n\"The Golden Circlet\" represented six months' grinding work. He\nhad practically shut himself away from the world. He had declined\ninvitations, paid no calls, risked everything on a last throw. When\nthe thing was finished it seemed like coming into fresh air again; he\nremembered people whose names he had almost forgotten, and above all a\ngirl whom he had told himself it might be wiser to forget; and, while\nhis passionate working fit was on, he had almost succeeded, seeing\nher only as a possibility at the beginning of success. It is wonderful\nwhat hard work may do for a man, for a time. But when the pause comes\nhuman nature must always have its backward glance, its old heart\nsearchings, its reviving pains.\n\nAnnesley, then, stood watching the entrance to the Century Theatre,\nand, as he stood there, suddenly his heart commenced a wild stampede.\nHe slipped into the doorway of a shop just in time to escape the\neyes of a girl who was walking quickly up the Strand. He waited for a\nmoment; she did not pass. After a time he ventured to glance out; she\nhad left the theatre, and was disappearing in the crowd.\n\nHis first impulse was to overtake her and make a clean breast of\neverything, but a moment's reflection convinced him that, having\nrestrained himself so far, it would be folly to make a doubtful step\nthen. Connie Bolitho had probably no idea that Conrad Howe was a cloak\nfor Herbert Annesley, and he saw an opportunity for a little comedy\nnot to be neglected. Since his position had grown stronger he felt\nfree to indulge his humours; a year before life had seemed all\ntragedy, with a diminishing banking account, and a sheaf of unpaid\nbills. He walked carelessly up to the box-office.\n\n\"Did a lady take seats a moment ago; a lady with a red hat and\nfur-trimmed cloak?\"\n\n\"Pretty?\" asked the clerk.\n\n\"Very pretty,\" said Annesley.\n\n\"Yes,--two stalls.\"\n\n\"Two!\" said Annesley, with an inner question in the word. \"Are the\nnext seats engaged--the ones, I mean, on either side of those two?\"\n\nThe man looked at the plan.\n\n\"No,\" he said.\n\n\"Book them to me, please.\"\n\n[Illustration: \"'YOU ARE FORGIVEN,' SHE SAID, SWEETLY.\"]\n\nThe clerk smiled benignly as he handed the tickets to Annesley; the\nlife in a box-office is dull during business hours.\n\nAnnesley walked away with his tickets, feeling that he had done a\ngood morning's work. He had at any rate made sure of a seat near Miss\nBolitho; if her companion were a man he must brace himself to eclipse\nthat fortunate individual; if a woman, it did not matter. He would\nprefer the woman, for in six months a great deal might have happened.\nMiss Bolitho was not bound to him in any way; they had seemed to\nunderstand each other, but a struggling writer with only debts to his\ncredit, had not dared to lay those debts and a doubtful future at his\nlady's feet.\n\nDuring the next week Annesley's time was fully occupied, but when the\ngreat day came and the final rehearsal was over he had a few hours\nin which to feel that almost unendurable excitement which precedes an\nordeal the result of which is not in our own hands. His part of the\nwork was over, but would the actors rise to theirs? He believed they\nwould, but belief is a poor support when so much depends upon it. His\nexcitement was also doubled by the prospect of watching the effect of\nhis work on Miss Bolitho.\n\nAnnesley reached the theatre five minutes before the curtain rose. The\nhouse was full; the gallery seethed like a hive, people were already\nstanding at the back of the pit. A glance showed him that Miss Bolitho\nwas there, with a man whom he had never seen before at her side. He\nmade his way quickly to his seat and was there before she had observed\nhim.\n\n\"You are as interested in plays as ever?\" he asked.\n\n\"Mr. Annesley!\" she cried. He was sure that the hand she gave him\ntrembled a little.\n\n\"May I ask you to forgive me for the past six months? I've been\nworking terribly hard, almost night and day.\"\n\n\"At a play?\"\n\n\"Yes,--at a play.\"\n\n\"You are forgiven,\" she said sweetly, \"because you are brave and stick\nto your ideals.\"\n\n\"I am rewarded,\" he murmured. A glance at her face assured him that\nher beauty was not less; that, at any rate, had remained unchanged.\n\n\"Do you know who this Mr. Conrad Howe is?\"\n\n\"No one seems to know; his identity has been kept secret most\nsuccessfully.\"\n\n\"Do you suppose it is not his real name?\"\n\n\"I have an idea it isn't; it sounds assumed, doesn't it?\"\n\n\"I'm not sure. What do you think, Tom? Let me introduce you to Mr.\nAnnesley,--my cousin, Captain Bolitho, who is just home from India.\"\nThey bowed severely to each other.\n\n\"We were discussing,\" said Connie, cheerfully, \"whether Conrad Howe\nwas a real or a pen name. What do you think?\"\n\n\"I don't know anything about these writing Johnnies. I don't see why\nthey shouldn't use their own names unless they're ashamed of them.\"\n\n\"Perhaps you don't quite understand, Tom,\" Miss Bolitho suggested.\n\n\"Perhaps I don't!\" said Tom.\n\n\"The climate of India is so trying,\" Miss Bolitho whispered to\nAnnesley.\n\n\"It must be,\" he said, smiling.\n\nThe orchestra glided into a slow movement and the curtain rose. I\nneed not tell you the story of the play; it was simple, but intensely\nhuman, having in it the philosophy learnt in years of struggle, but\nalways with hope and faith in the ultimate good beyond. It presented\nno problem of the gutter raised to drawing-room standard by\nmeretricious gilding; it had the singular distinction of being\nperfectly clean and also entirely dramatic. As Annesley saw his\nwork develop before his eyes, and felt how it was taking hold of a\nbreathless audience, he did not grudge the experience that had gone\nto its making or regret that he had kept his ideals unsoiled. When the\ncurtain fell upon the first act the clamour of applause was the true\nexpression of genuine emotion aroused by legitimate means. Annesley\nfelt weak and almost sick. He realised vividly what it all meant to\nhim; he realised, above all, of what little value it would be if he\nfailed in the greater matter of his love. Connie leaned towards him;\nshe had tears in her eyes.\n\n[Illustration: \"THE MANAGER WAS SIMMERING WITH JOY.\"]\n\n\"This is the kind of thing we've been waiting for,\" she said. \"This is\nquite true and human. Conrad Howe should be a happy man to-night.\"\n\n\"If he is in the house.\"\n\n\"I hope he is; there's sure to be a call.\" Annesley's heart thumped.\n\n\"That must be awfully trying to a man,\" he said.\n\n\"Why don't you write plays of this kind?\"\n\n\"It's rather the sort of thing I've been aiming at.\"\n\n\"Go on aiming at it, then, and you'll succeed.\"\n\n\"With your encouragement I feel I could do anything.\"\n\n\"This isn't a bad play, is it?\" asked Captain Bolitho.\n\n\"It's splendid,\" said Connie.\n\n\"The fellow knows something, too. There's not all that confounded\nfootle that leads you nowhere. The girl's ripping.\"\n\n\"She is,\" said Annesley. As a matter of fact she was a careful study\nof Miss Bolitho; for that reason Miss Bolitho appeared entirely\nunconscious of it.\n\n\"There are only three acts, too,\" said the Captain; \"that's sensible.\nFive acts, with long waits between, are killing. I call it taking your\nmoney on false pretences. You don't come to a theatre to hear the band\nplay.\"\n\nWhen the curtain rose again the house instantly settled into silence,\na sure sign that things were going well. Connie leaned forward with\nsomething of the eagerness of a child; even Captain Bolitho unhinged\nhimself, as it were, and indicated interest by a slightly curved back.\nAnnesley began to feel master of himself again; part of the future,\nat least, was now safe; how much that means to a man who steps from\npoverty to the security of a decent income can only be realised by\nthose who have been in a like case; the mere fact of being able to pay\na debt with promptitude is capable of affording a very exquisite joy.\nBut, now that so much was within his grasp, he longed for all; the\nhorizon of desire, like the horizon of the actual world, always\nrecedes as we advance; since a few months before he had travelled\ninnumerable miles towards success; that being reached, there was still\nan infinite distance beyond.\n\n[Illustration: BEFORE THE CURTAIN.]\n\nIn the second act there was a simple love-scene that appeared to\ntake the audience by surprise; it was direct, touching, convincing.\nAnnesley noticed that no one laughed, a thing almost unprecedented in\na London theatre when sentiment attitudinises upon the boards. This\ngave him a glow of well-earned triumph; he had mentally decided\nbeforehand that that was the crucial point of the play; when it was\npassed he dropped back and closed his eyes.\n\n\"You didn't see all that act,\" Connie said to him in the interval;\n\"are you tired,--were you asleep?\"\n\n\"I'm neither tired nor sleepy, I heard everything.\"\n\n\"Didn't you think the love-scene beautiful?\"\n\n\"Yes,\" he said, blushing at his own candour.\n\n\"I didn't think much of that,\" said Captain Bolitho, \"I suppose\nbecause I can't see myself saying pretty things to a girl. It's not\nin my line, you know. I feel 'em, but can't express 'em. My notion is\nthat the girl should make love to me.\"\n\n\"But you must begin, surely,\" Connie said.\n\n\"That's just the deuce of it,\" said the Captain, \"I can't.\"\n\nAnnesley rose. \"I must go now,\" he said, \"to another part of the\nhouse. When it's over will you remain here till I come? I've an idea\nthat I can find out who this Conrad Howe is. May I bring him to see\nyou if I'm right?\"\n\n\"Do, I'll wait for you.\" He went out into the Strand and lit a\ncigarette. The aspect of the world had changed for him; he even saw\ncabs and busses with different eyes. Every passenger upon the pavement\nseemed a friend, the roar of traffic had new music in it,--the stars\nabove the housetops looked down with kindly eyes. The cool air put\nfresh courage into him, soothed his pulse, made his hope seem real.\nInside the theatre it had been altogether difficult to understand\nsubstantial facts; but out there in the hurry of the street it was\neasy enough. There was no doubt about \"The Golden Circlet,\" or Connie\nBolitho, or about himself; they all existed, they all were of the\nworld. The name of Conrad Howe stared at him from the placards; he\neven touched the letters with his fingers to make quite sure. Ten\nminutes later he re-entered the theatre by the stage door.\n\nHe met the manager in the wings. That gentleman was simmering with\njoy, his congratulations were overwhelming. Annesley bore them with\nresignation.\n\n\"There's sure to be a call for 'Author,'\" said the manager; \"you'll go\nto the front, won't you? It's always better; pleases them, you know.\nDo you feel nervous? Come to my room and have some champagne. This is\na howling success, Mr. Howe--nothing like it for years. Just listen to\nthat applause? You've fetched 'em, no doubt about it. Come along and\nhave that champagne.\" Annesley went readily enough; the atmosphere of\nthe theatre was getting on his nerves again.\n\nWhen the last curtain fell the pit and gallery got upon their feet and\ncheered; the rest of the house was equally decisive if more discreet;\n\"The Golden Circlet\" was a success. And in the midst of the hubbub\nAnnesley found himself before the curtain, bowing, dazzled by the\nfootlights and straining his eyes to see one face. And, as though in\nobedience to his call, it rose before him, flushed, glowing, with eyes\nfrom which the delight and astonishment had hardly died, and with\nlips whose smile seemed tremulous with coming tears. That was the true\nmoment of his triumph.\n\nAs soon as he could escape he found his way into the empty stalls; one\nfigure remained. As he approached Connie raised her head. The colour\nhad died out of her face; she was as pale as Annesley was himself. He\nheld out his hand.\n\n\"I have brought Conrad Howe to see you,\" he said.\n\n\"Why didn't you tell me before? It was cruel of you.\"\n\n\"Perhaps it was because I thought that if I failed I could not bear\nthat you should know it.\"\n\n\"That was not true friendship.\"\n\n\"Did I ever profess friendship for you?\"\n\nShe hesitated, and played with her fan. A little wave of colour flowed\nback into her cheeks.\n\n\"You see,\" he went on, \"I was pretty much alone in the world, and had\nto make my mark in my own way. A few months ago things were very black\nwith me. I shut myself up and worked.\"\n\n\"It must have been hard for you,\" she said, \"to cut yourself off from\neverything like that.\"\n\n\"It was hard, I'm not going to pretend it wasn't. But I had hope--not\nvery bright, perhaps, but still it was enough to keep me from going\nunder.\"\n\n\"You had faith in yourself and in your own work.\"\n\n\"I had more than that. Can you guess what it was?\" Their voices\nsounded curiously hollow in the empty theatre,--the attendants were\nalready putting up and covering the seats.\n\n[Illustration: \"'I WISHED TO WIN YOUR LOVE.'\"]\n\n\"You hoped to get fame and money?\"\n\n\"Yes, but more than either I wished to win your love. Don't kill my\nillusion, don't ring down the curtain on my romance, Connie, and leave\nme in the dark. Everything I did was for you. You inspired whatever\nwas good in 'The Golden Circlet.' The thought of you kept my head\nabove water. I can come to you now without feeling ashamed.\"\n\n\"You might have come before. You need never have been ashamed. I could\nhave helped you, oh, so much!\"\n\n\"But now that the dark days are over, you won't turn your back on me\nand say I don't need your help? I need it more than ever. My love, the\ngolden circlet is yours if you will take it from me.\"\n\nShe, gave him both her hands and lifted her face to his.\n\n\"I am your's always,\" she said, \"but I think, perhaps, I loved you\nbetter when you were quite poor, but you never asked me then to love\nyou. Think of what you've lost!\"\n\nAnnesley took her in his arms in spite of a watchful attendant. \"Never\nmind,\" he said, \"everything's in the future for both of us, never mind\nthe past. They may even damn my play now if they like.\"\n\nAt this point Captain Bolitho's voice was heard in loud protest.\n\n\"I tell you,\" he was saying, \"I left a lady in your confounded\ntheatre, and she hasn't come out. I've had a cab waiting ten minutes.\"\n\n\"It's Tom,\" Connie whispered, \"I forgot all about him. Poor Tom!\"\n\n\"Miss Bolitho's quite safe,\" said Annesley, \"we've just been settling\na little matter of great importance to both of us.\"\n\nCaptain Bolitho peered into the face of each in the uncertain light\nand seemed to understand.\n\n\"The devil you have!\" he murmured under his breath. Then he said\naloud, \"Anyhow, Connie, I can't keep the cab waiting any longer. I\ncongratulate you, Mr. Annesley Howe, on your 'Golden Circlet.' That\nwas a deuced neat little surprise you'd hatched for us. I like your\nplay, and I daresay I shall like you when I know more of you. Dine\nwith me next Thursday, will you? Good-night.\"\n\n\n[Illustration: A MOTHER OF TWO.\n\n_Photo by Landon, Ealing_]\n\n\n\n\nMODERN FAMILY SKELETONS.\n\nBY BEATRICE KNOLLYS.\n\n_Illustrated by A. S. Hartrick._\n\n\nA family ghost is a possession almost as respectable as a patent of\nnobility, and happy is the house reputed, on satisfactory evidence, to\nbe haunted by one. There are still a few hereditary ghosts left, and\na few leasehold and freehold ghosts; but these last are often the\nproperty of retired manufacturers and American millionaires who have\nbought house and lands, pedigrees, portraits, and family ghosts all\ntogether as they stood.\n\nIn this article it is my intention to be the biographer of a few\nancient and well-born ghosts only, as space will not permit me to\ncondescend to mere one-generation ghosts, pedigreeless spirits.\n\n[Illustration: THIS DRUMMER APPEARS WHENEVER A MEMBER OF THE OGILVY\nFAMILY IS GOING TO DIE.]\n\nA. was an Airlie who killed a poor drummer, whose spirit plays a drum\nat Cortachy Castle, Kirriemuir, Scotland, whenever any member of the\nOgilvy family is going to die. The origin of this tradition is that\nthe drummer, for some reason or other, in his lifetime so enraged a\nformer Lord Airlie that he had him thrust into his own drum and flung\nfrom the window of a tower of Cortachy Castle, though the drummer\nthreatened to haunt the family ever after if his life were taken.\n\nHe has seemingly kept his word, for in 1849, before the decease of a\nLord Airlie, and again in 1884, before the death of a Lady Airlie, the\nbeat of the drum was on each occasion distinctly heard by different\nguests of the family. One of these guests was a lady staying in the\ncastle, who was so ignorant of the tradition that, having heard the\nbeating of a drum while dressing for dinner, she innocently asked her\nhost--Lord Airlie--at the table who his drummer was. The question made\nthe peer turn quite white, for the sound had preceded the loss of his\nfirst wife, and it was only a few months after this ominous dinner\nparty that the second wife died.\n\nThe Combermere family have two ghosts in their record. In Combermere\nAbbey there is an old room, once a nursery, and here has been seen the\nspirit-figure of a little girl fourteen years old, dressed in a very\nquaint frock with an odd little ruff round its neck. It appeared to\na niece of the late Lord Cotton as she was dressing for a very late\ndinner one evening in this former nursery, now used as a bedroom. She\nhad just risen from her toilet-glass to get some article of dress when\nshe saw the child standing near her bed--a little iron one which stood\nout in the room away from the wall--and presently the figure began\nrunning round the bed in a wild, distressed way, with a look of\nsuffering in its little face, which the lady could see quite plainly\nas the full light of her candles fell upon it.\n\nOn mentioning this apparition, her widowed aunt, Lady Cotton, called\nto remembrance that the late Lord Cotton had told her of the sudden\ndeath years ago of a favourite little sister of his, with whom he had\nbeen playing, he being also a child then, by running round and round\nthe bed with her, just the night before--indeed, only a few hours\nbefore, her decease.\n\n[Illustration: THE COMBERMERE GHOST--A YOUNG GIRL WHO APPEARS TO\nFORETELL DEATH.]\n\nA stranger story still, and one that has not yet, I believe, appeared\nin print, is that where quite recently a lady took an amateur\nphotograph of the drawing-room of a house once inhabited by the late\nLord Combermere--at Brighton I think it was. The lady in question saw,\nto her horror and astonishment, visible on the plate, the ghost of\nthe old peer--a tall man with rather stout face and a\nmoustache--reproduced sitting in one of the easy chairs of this\ndrawing-room, though not apparent to the naked eye.\n\nThe Drake ghost--the spirit of Sir Francis Drake--might be termed a\nsporting spirit, as it has been frequently seen in different parts of\nDevonshire and Cornwall--notably Plymouth--driving a hearse drawn by\nheadless horses and followed by a pack of headless hounds.\n\nTwo Gordon ghosts live at Fyvie Castle in Scotland. One is a lady\ndressed in a magnificent costume of green brocade, who is seen, candle\nin hand, passing through a tapestried room of the old castle when any\nimportant event is going to happen to the family.\n\nThe other spirit is by profession a trumpeter, who tradition affirms\nhaunts the castle in revenge for having during his lifetime been\nseized by the press-gang at the instigation of the then Gordon of\nFyvie Castle, who wished to get rid of a rival in the affections of a\npretty daughter of his factor or bailiff.\n\nThe girl, however, remained faithful to the trumpeter, the separation\nfrom him making her die of a broken heart; and now, like the drum of\nCortachy Castle, a trumpet is heard whenever misfortune is in store\nfor the unlucky Gordons. Ill-fated they certainly are, as beside\nbeing the hereditary owners of unlucky ghosts, they are also under a\nhereditary curse--the curse of a \"Thomas the Rhymester\"--who, when the\ngates of the castle long years ago were churlishly closed against him\nin the days of wandering minstrelsy, declared that the property should\nnever descend in a direct line till three \"weeping\" stones were found;\nbut up to twenty years ago, when a relative of the writer was staying\nat the castle, only one weeping stone had been discovered.\n\nIn Fyvie Castle there is also a sealed room, which is always kept\nreligiously closed; for the saying is, should the door be ever opened,\nthe master would die and his wife go blind. Faith and fear have\nprevented the saying being proved, as the room has never been opened;\nbut as regards the curse of \"Thomas the Rhymester,\" it is certainly a\nfact that the Gordons have never inherited in a direct line.\n\n[Illustration: ONE OF THE FYVIE CASTLE GHOSTS IS A TRUMPETER BY\nPROFESSION.]\n\n[Illustration: ONE OF GLAMIS CASTLE'S GHOSTLY INHABITANTS--A TALL,\nBEARDED VISITANT IN ARMOUR.]\n\nThere is a perfect spirit vault of ghosts at Glamis Castle, the\nancestral residence of another old and celebrated Scotch family, the\nLyons, the head being the Earl of Strathmore. They also possess a\nsecret chamber, which is supposed to be connected with some\nterrible mystery known only to each owner, the next heir, and the\nhouse-bailiff, of the time being. Even the exact locality of the room\nis never revealed to others than those three, and though more than one\nheir-apparent has promised to tell the secret to his bosom friends as\nsoon as the attainment of his twenty-first year entitled him to learn\nit; yet after he has known it, a solemn silence on the subject has\nbeen maintained, and beyond the fact that a stonemason is supposed to\nbe secretly employed to close the approach to this chamber after each\nvisit, nothing more definite is known. The strangest part of it all\nis the evident necessity that each successive house steward should\nbe made acquainted with this mystery, which looks as if to him\nwas intrusted the duty of providing food for some person or thing\nimprisoned in those walls of fifteen feet thickness. Whether the\nmystery is in any way connected with the apparition of a bearded man,\nwho flits about the castle at night, and hovers over the couches of\nchildren, is not known; perhaps it has something to do with a figure\nwhich appeared at a window to a guest staying at Glamis Castle, and\nsitting up late one moonlight night. The owner of the pale face, lit\nup with great sorrowful eyes, seemed to wish to attract attention, but\nit was suddenly pulled away as if by some superior power. Presently,\nhorrible shrieks rent the night air, and an hour or so later, the\nguest, gazing horror-stricken from the window of the room, saw a dark\nhuddled figure, like that of an old decrepit woman, carrying a bundle,\npass across the waning moonlight outside, and vanish.\n\nPerhaps the most interesting legend attached to this magnificent old\ncastle is the historical tradition that in one of its rooms Duncan was\nmurdered by Macbeth, \"Thane of Glamis,\" and this Duncan is perchance\nthe tall bearded ghost in armour who haunts the old square tower,\nand on one occasion nearly frightened to death a child who, with\nits mother, was on a visit to the castle. The child was asleep in a\ndressing-room off its mother's bedroom. She herself was lying awake,\nwhen a cold blast extinguished her light suddenly, but not the\nnight-light in the dressing-room, from whence, immediately after,\nproceeded a shriek. The mother rushed in and found her child awake,\nand in an agony of fear, because the tall mailed figure she herself\nhad seen pass into the dressing-room had come to the side of the cot\nand leant over the face of the child. As a matter of fact, tradition\nand truth are so mixed up with all the stories connected with\nthis very ancient fortress-palace, that it is difficult, in fact\nimpossible, to know what to believe and what to disbelieve.\n\n[Illustration: \"WHILE SHE PRAYED THE SPIRIT APPEARED AND SAID, 'TAKE\nUP THE CANDLE AND FOLLOW ME.'\"]\n\nA more peaceable spirit is the Townshend ghost of Rainham, in Norfolk,\ncommonly known as the \"Brown Lady.\" She is described as tall and\nstately, dressed in a rich brown brocade, with a sort of coif on her\nhead. The features are clearly defined, but where the eyes should be\nare nothing but hollows. She is seen walking about the old mansion\nevery now and then, though no reason can be discovered to account for\nher restlessness. Lord Charles Townshend, on being asked by a lady if\nhe also believed in the apparition, replied, \"I cannot but believe,\nfor she ushered me into my room last night.\"\n\nThe Lonsdale spirit seems to have been as rowdy in death as it was\nduring life when it inhabited the body of Jemmy Lowther, well known as\nthe \"bad Lord Lonsdale.\" For years after his decease the inhabitants\nof Lowther Hall and the neighbourhood were kept in a constant state\nof excitement by continual disturbances in the house, noises in the\nstables, and the galloping across country of Lord Lonsdale's phantom\n\"coach and six.\"\n\nThe Powys Castle ghost was a much more amiable spirit, and of quite a\nsuperior character to the devil-may-care spirit of Jemmy Lowther. His\nobject was benevolent, and his manners were well-bred and gracious\nwhen he appeared. His last visit was to a poor pious workwoman, who,\nin the absence of the Herberts from Powys Castle, was purposely put by\nthe servants in the haunted bedroom, a handsomely furnished apartment\nwith a boarded floor, a big bedstead in one corner, and two sash\nwindows. A good fire was made up in the room, and a chair and a table\nwith a large lighted candle on it was placed in front of the fire.\nShe had just sat down in the chair to read her Bible, when to her\nastonishment in walked a gentleman. He wore a gold-laced hat and\nwaistcoat, with coat and the rest of his attire to correspond. He went\nover to one of the sash windows, and putting an elbow on the sill,\nrested his face on the palm of his hand. She supposed afterwards\nthat he stood quietly thus to encourage her to speak, but she was too\nfrightened. Then he walked out of the room, and the poor woman, rising\nfrom her chair, fell on her knees and began to pray. Whilst praying,\nthe spirit appeared again, walked round the room, and came close\nbehind her. He again departed, and again appeared behind her as she\nstill knelt. She said, \"Pray, sir, who are you, and what do you want?\"\n\nIt lifted its finger and said--\n\n\"Take up the candle and follow me, and I will tell you.\"\n\nShe did as she was bid, and followed him into a very small room,\nwhere, tearing up a board, he pointed to an iron box underneath,\nand then to a crevice in the wall where lay hidden a key. These he\ncommanded were to be sent to the Earl of Powys, then in London. This\nwas done, though history does not relate what the box contained; but\nit was known that this poor Welsh spinning woman was provided for\nliberally by the Powys family till she died about the beginning of\nthis century.\n\nThough one does not associate ghosts with such a city of excitement,\nlife, and renovation as London, yet it does possess several haunted\nhouses. One belonging to a present-day peer, and situated in Park\nLane, is said to be haunted by fashionable spirits having a dance.\nSome people can only hear the buzz of their voices and the swish\nof dresses and the tap of feet, while others can see the figures\nthemselves talking and dancing.\n\nYes, there are more things in heaven and earth than are dreamt of in\nour philosophy.\n\n[Illustration: LETTING THE FAMILY SKELETONS OUT OF THE CUPBOARD.]\n\n\n\n\nA ROSE AT LAST\n\nBY\n\nCLIFTON BINGHAM\n\n[Illustration]\n\n  It was only a rosetree slender\n    On a dingy window sill,\n  In the heart of the busy City,\n    With its mingled good and ill.\n  And the Angels must have seen it,\n    Unwilling to let it die,\n  For it thrived and bore a rose-bud\n    Under that darksome sky!\n\n  A white face watched it daily\n    With joy in its childish eyes,\n  As she played alone in the garret\n    Under the city skies:\n  It brightened the dingy windows,\n    Each night as she crept to bed,\n  Though hungry and loveless and lonely,\n    \"It will soon be a rose,\" she said.\n\n  There at the window one morning,\n    The bud was a rose so fair,\n  But the garret was still and silent,\n    There was no little white face there!\n  It was smiling in happy slumber,\n    Its pain and loneliness past,\n  For the Angels who loved her were saying,\n    That the bud was a rose at last!\n\n\n\n\nHOW SANDOW MADE ME STRONG\n\nA REMARKABLE PHYSICAL TRANSFORMATION.\n\n\nIt was a question of going to South Africa or running the risk of a\nshort life in England; health dictated the question, and the answer\ndepended on many things. Someone suggested Sandow's School of Physical\nCulture as a compromise; and finally England, backed up by financial\nand other reasoning, carried the day.\n\nI was a puny youth, weak of spirit and frail of frame, when I first\nvisited Sandow's muscle factory in St. James's Street, London,\nand said that I had come to be made into a strong and healthy\nEnglishman--to obtain a fresh lease of life if possible.\n\nSandow fingered my arms and chest as he might a prize ox, and remarked\nthat I should make an admirable subject for his purpose; he liked\npulling folks out of their graves. Whereupon I imagined I should be\npassed into the gymnasium to swing a dumb-bell for an hour or so,\nand be invited to drop in again when I was next that way. But I was\nmistaken. Had my object been to enlist in Her Majesty's forces, the\nexaminations and tests I was subjected to could not have been more\nextensive or peculiar. I was sounded, measured, weighed, pounded and\nquestioned, the results being solemnly entered into a big ledger, as\nthough it might all be used as evidence against me should the need\never arise. Weight 120 lbs., chest measurement 32 in., height 5 ft.\n6-1\/2 in., though the latter is immaterial, as Sandow does not bargain\nto make one grow in that direction when nature considers her duty\ndone.\n\n[Illustration: ON ENTERING SANDOW'S SCHOOL.]\n\n[Illustration: TEN MONTHS AFTER.\n\n(_From Photographs._)]\n\nThough I felt ashamed of the figures myself, they did not seem\nto affect my burly interrogators in any way, and the examination\nproceeded. Had I indigestion, and did I smoke? I confessed to a little\nof either weakness of the flesh. Was there any particular ailment in\nthe family, and would I take a full breath and blow down this tube?\nAs I did so, a little clock-like machine ticked merrily away, till it\nregistered that my pair of lungs--or \"one and a decimal,\" as a blunt\nold doctor had once informed me--could contain at full pressure 185\ncubic inches of air--a poor record, be it said.\n\n[Illustration: BEING SOUNDED.]\n\n[Illustration: \"THE LITTLE MACHINE TICKED MERRILY AWAY.\"]\n\n[Illustration: HEIGHT AND WEIGHT.]\n\nNext came dumb-bell and weight tests, careful note being made of the\nexact number of pounds I could lift with one hand, two hands, hold\nat arm's length, and support above my head. The record ran thus:--One\nhand lift, 65 lbs.; at arm's length, 18 lbs.; raised from shoulders\n(1) 40 lbs., (2) 35 lbs. each. Bar-bell raised above head, 85 lbs. So\nthe examination ended, and when my photograph had been taken as a\nsort of example \"before trying,\" I was free to join the little army of\nhealth-and-muscle seekers whenever I chose.\n\nA very mixed army it was. Stern-visaged men were there going through\nthe exercises as seriously as if life itself depended on them;\nsprightly veterans taking again to regular exercise, so much missed\nsince they joined the half-pays; middle-aged men making up for the\nnegligences of earlier days; clerks and students of all kinds going\ninto strict training in order to be in form for the cricket and\nrunning season; and finally a goodly sprinkling of puny youths working\nhard to attain the weight and chest measurement necessary to give\nthem another chance at Sandhurst or Woolwich, where they had just been\ndeclined \"for physical reasons.\"\n\nThe display was not without its humour. A plump stockbroker is a\ncommon and natural enough sight in the city, but he forms a different\nspectacle as, minus the glossy hat and black coat of his calling, he\nenergetically whirls a pair of dumb-bells in the frantic endeavour to\nexchange his superfluous avoirdupois for sinew and muscle, especially\nwhen his immediate neighbour, a very lean litt\u00e9rateur, is performing\nthe same evolutions with the secret hope of putting on flesh.\n\nIt would require a keen eye, supported by a good imagination, to\ndiscover any outward visible sign of the \"strong man\" about the\nvarious instructors of Sandow's school, dressed as they are in\nordinary attire, to say nothing of fashionable collars and the latest\nthing in neckties. Any one of them might have strolled in from Bond\nStreet, mistaking the place for the club, yet any one of them would\nthink nothing of snatching up a 100 lb. dumb-bell and raising it aloft\nwith the ease with which most people might perform a similar feat with\nan umbrella.\n\nWhen I presented myself at the gymnasium for my first course of\ninstruction I was handed a pair of dumb-bells weighing not more than\n3 lbs. each. I protested that I had been in the habit of using bells\nthree times as heavy. It did not matter, I was informed,--lead pencils\nwould be almost as serviceable, providing I concentrated my whole\nattention on each exercise in turn.\n\nIt must not be supposed, however, that dumb-bells do not play an\nimportant part in Sandow's system. On the contrary, as will be seen\nfrom the photographs herewith, they figure in numerous exercises, but\ntheir weight is practically immaterial. They usually vary according to\nthe physical condition of those using them.\n\nHaving grasped his \"three-pounders,\" the student is made to stand in\nan attitude of ease, the inner side of his arms fronting outwards. His\nvery first step on the road to muscular development is to alternately\nbend each arm at the elbow, bringing the dumb-bell close to the\nshoulder. This has to be repeated some twenty or thirty times, to the\nmeasured \"One, two, three,\" of the instructor.\n\nThe same thing is then gone through with the arms turned the other\nway, so that the knuckles instead of the finger-tips are brought up to\nthe shoulders. Next the arms are extended outwards in a straight\nline, each being bent in turn at the elbow, and the dumb-bell brought\nimmediately above the shoulder. And here comes the student's first\ndifficulty; for in extending the arms each time it is necessary\nto keep them straight and rigid in order that the muscles may be\nbenefited by the strain. It is amusing to watch various pairs of arms\ngradually drooping as this exercise proceeds.\n\nAltogether the dumb-bells are used in about twenty different\npositions, each affecting a different set of muscles. There is the\nlunge, for instance, exercising both arms and legs. First standing\nat ease, the pupil takes a stride forward and strikes out alternately\nwith his left and right, as though an adversary awaited the blow.\nSome twenty-five or thirty such lunges, however, are calculated to\ntransform the most bellicose among Sandow's disciples into members of\nthe Peace Society.\n\n[Illustration: LIFTING 70 LBS. WITH TWO HANDS.]\n\nThe wrists are strengthened in this fashion: once more extending the\narms in a line with the shoulders the pupil now holds the dumb-bells\nby the ends, instead of in the usual way, and with a circular motion\nof the wrists revolves the bells first from right to left, then from\nleft to right.\n\n[Illustration: THE BAR-BELL TEST--SUPPORTING 85 LBS.]\n\nNext comes what the flippant call the \"see-saw\" motion. With the\ninevitable dumb-bell in each hand the student stands erect; the\nsee-saw consists of nothing more remarkable than bending the upper\nportion of the body from side to side, without moving the lower limbs.\nThese are cared for in the next exercise. Lying at full length on the\nground, the pupil actually proceeds to kick his legs in the air! Not\nparticularly graceful, perhaps, but highly beneficial, it is claimed,\nto the \"hinges\" at the knees and hips. What this motion does for the\nlower limbs, the next does for the upper part of the body. Lying at\nfull length on the ground as before, and keeping the legs perfectly\nstiff, the student raises his head and shoulders from the ground, and\nwith a quick movement swings forward until his body is bent almost\ndouble, then returning slowly to the former position. The dumb-bells\nare now forsaken for a time. The lesson to be learned is to support\nthe body on the hands and toes, and to alternately lower and raise it\nby respectively bending the elbows and straightening the arms, taking\ncare not to touch the ground with any part of the body. It looks and\nsounds easy enough; so it is, to do it once, but quite another thing\nto keep it up in quick succession until the instructor sees fit to cry\n\"halt!\" which is timed, it seems to the student, specially to remind\nhim of the penultimate straw and the camel's back.\n\n[Illustration: RAISING 40 LBS. WITH ONE HAND.]\n\nDumb-bells are now resumed, this time attached to stout elastic\nstrands, these in turn being fixed to the wall. Exercises of much the\nsame kind as before are gone through, except that the strain on the\nmuscles is now greater, seeing that almost every movement involves\nstretching the rubber bands to their fullest extent, and allowing them\nto return to their natural state slowly, not with a snap. The\nsame principle is applied to the development of the legs and neck,\ningenious devices in the shape of \"harness\"--forming an interesting\nbranch of the system--being requisitioned for the purpose. In each\ncase the elastics have to be stretched as much as possible, the strain\nbeing in turn centred on sets of muscles that could be reached by no\nother method.\n\n[Illustration: THE LUNGE.]\n\nIf after having gone through all these exercises the pupil should pine\nto develop his knowledge of Physiology as well as his frame, he may\nlearn that this little action affects the latissimus dorsi, that that\ntiny movement seeks out the neglected deltoid, that another bend\nof the body, insignificant though it may seem, means much to the\npectoralis major, and so forth. But the gentle student usually prefers\nnot to burden his brain with these things, and in this respect he is\nperhaps not unlike the gentle reader. So no more shall be inflicted.\n\n[Illustration: THE FIRST STEP.]\n\nEvery pupil has to attend Sandow's School at least twice a week, and\nwhen there to repeat each of the exercises named some twenty\ntimes, though this number is a kind of moveable feast, advancing or\ndecreasing with his condition, reaching as high as sixty and as low\nas ten. Beyond that he is supposed to practise every day at home,\nand regularity in this greatly facilitates the development, just as\nhome-lessons assist a schoolboy's education. There, probably,\nthe simile ends; certainly the majority of Sandow's followers do\nconscientiously work out of school hours.\n\nWhen students have been got into trim generally--this takes about\na month--they are allowed to add weight-lifting, with and without\n\"harness,\" to their regular exercises. To do so before the body was\nin a supple condition might result in serious strains occasionally. A\nstill further stage is practice on the Roman pillar. This consists of\nhanging backwards suspended from the knees, and from that rising to an\nupright position, lifting with the body a bar-bell weighing anything\nbetween 30 lbs. and 120 lbs.\n\n[Illustration: \"WHAT THE FLIPPANT CALL THE 'SEE-SAW' MOTION.\"]\n\nEvery few months examinations are held, the same tests and\nmeasurements as on entering being gone through, and the results put\ndown side by side in the ledger, so that one's weak points can be seen\nat a glance and receive particular attention forthwith.\n\n[Illustration: \"NOT PARTICULARLY GRACEFUL, BUT HIGHLY BENEFICIAL.\"]\n\nPersonally, I had not been in the school a few weeks before I began to\nfeel its benefits. The first signs were the arrival of an appetite and\nthe disappearance of indigestion and insomnia. Gradually I exchanged\nloose flesh for firm muscle; my weight increased; my chest measurement\nadvanced. My weight-lifting crept up by \"fives\" and \"tens,\" till at\nthe end of three months I could raise 70 lbs. with one hand, 350 lbs.\nwith two, and 500 lbs. in \"harness,\" all with comparative ease.\n\n[Illustration: \"UNTIL HIS BODY IS BENT ALMOST DOUBLE.\"]\n\nEvery time I blew into the little lung-testing machine I felt\napprehensive of its breaking or getting out of order under the\nstrain. My course of instruction commenced ten months ago; at the last\nexamination, held recently, my record ran:--One hand lift 130 lbs. (an\nincrease of 65 lbs.). Held at arm's length 35 lbs. (increase 17 lbs.).\nRaised from shoulders, one hand, 90 lbs. (increase 50 lbs.), both\nhands, 160 lbs. (increase 90 lbs.). Raised above head 175 lbs.\n(increase 90 lbs.). Weight, 10 st. 0 lb. (increase 1 st. 6 lb.); chest\nmeasurement, 36 inches (increase 4 inches). Lift with \"harness\" 800\nlbs.; without 550 lbs. Perhaps it should be added that this result\nwas not achieved by irregular attendance at the school or occasional\npractice at home. I worked diligently every day on rising in the\nmorning, and before retiring at night, and I fancy I have no need to\ngo to South Africa now.\n\n[Illustration: FOR THE WRISTS.]\n\nA little about the St. James's School itself. Incredible though it may\nseem, it is not a limited company. Every one connected with the place,\nfrom the manager downwards, has to go through the system. That is\nwhy the door is opened to you by a young Hercules whose clothes are\nbursting over him, and who, rumour says, is afraid to take them off\no' nights lest he should never be able to get into them again; that\nis why, if you call early or late enough, you will see a muscular\ncharwoman scrubbing the front steps to the quick time of \"Sandow's\nMarch,\" for even she is not exempt. There is, by the way, a special\ncourse of training for lady pupils.\n\n[Illustration: NOT SO EASY--]\n\nEvery one connected with the place participates in the profits,\nwhich must be large, from the head-manager down to the two humbler\nindividuals just mentioned. That, doubtless, is why the door is always\nopened to you with commendable alacrity, and may account for the fact\nthat the front steps are the whitest in St. James's Street, and that\nthe brasswork about the establishment positively dazzles the eyes with\nits gleam.\n\n[Illustration: --AS IT LOOKS.]\n\nOf course Sandow has his \"secret.\" It is that he does not believe in\ndeveloping one part of the body at the expense of another. His aim is\nnot to turn out pupils with runners' legs or rowers' arms, but of\ngood physique generally. If a runner enters the school his legs are\nnaturally better developed than the average. They will, therefore,\nrequire less attention than usual, and more will be given to other\nparts of his body. And so forth.\n\n[Illustration: IT IS THE CONSTANT--]\n\nThe exercises are so devised that no set of muscles in the body is\noverlooked. In the ordinary course they are all developed together, at\nmuch the same rate; but this, of course, cannot always be adhered to.\nIt frequently happens that a pupil desires chest expansion above all\nelse, in which case he will devote himself primarily to the exercises\nspecially framed to bring about that result. In several cases a couple\nof inches in the way of chest measurement has stood between pupils at\nSandow's and commissions in Her Majesty's army.\n\nMuch depends, Sandow avers, on mind concentration.\n\n[Illustration: --STRAIN THAT--]\n\n\"It is of little use,\" he says, \"going through the exercises\nmechanically. As each one is performed, it should occupy the whole\nattention. Merely swinging a dumb-bell the regulation number of times\nwill do no good. It should be regarded as serious work, and one's\nheart should be in it. It has not been my aim to produce what are\nknown as strong men; it is a comparatively easy task to pick out a few\nmen exceptionally endowed by nature, and train them until they attain\ngreat proficiency in particular feats of strength and activity. It may\nbe considered somewhat ambitious, but my honest desire is nothing less\nthan to permanently raise the standard of physique in the whole race,\nand to restore, as far as possible, the old types of physical\nstrength and beauty, for the loss of which civilization is so largely\nresponsible.\"\n\nOne naturally asks: What is the age limit at which physical\ndevelopment necessarily ceases? Perhaps Sandow's school-register best\nanswers the question. His pupils range from fourteen to seventy-three.\nThe gentleman of the latter age felt so rejuvenated after one week's\nattendance that he promptly put himself down for a whole year's\ncourse, and has since declared his intention of \"never leaving school\"\nuntil old age compels him.\n\n[Illustration: --DEVELOPS THE MUSCLES.]\n\nIt is interesting to recall how Sandow first came before the public as\nan exponent of strength. Some nine years ago it was the practice of a\n\"strong man\" then performing at a London theatre of varieties to\nissue nightly from the stage a challenge to the world generally to\naccomplish any of his feats, which included the lifting of great\nweights, the snapping of steel chains, and the bending of iron bars.\nOne night, to everyone's surprise, the challenge was accepted by a\nmember of the audience, and a young man stepped upon the stage in\nimmaculate evening dress. When this was removed the customary attire\nof the stage \"strong man\" was revealed. It was Sandow, then unknown.\n\n[Illustration: LIFTING 350 LBS. WITHOUT HARNESS.]\n\nAmid the wildest excitement he performed every one of the wonderful\nfeats. The next day a new \"strong man\" had dawned.\n\n[Illustration: WEIGHT-LIFTING ON THE ROMAN PILLAR.]\n\nIt is Sandow's ambition to start schools of muscular development in\nall the principal cities and towns in the kingdom, and if they\nbecome as popular as those in London, there is hope for the country,\nphysically, yet. The tendency of the Englishman, since he acquired the\nhabit of living in towns, has been to take too little exercise.\nRoast beef and Sandow may do more for the race than the former ever\naccomplished alone.\n\n[Illustration: LIFTING 500 LBS. WITH HARNESS.]\n\nA. E. J.\n\n[Illustration]\n\n\n\n\nTHE STONE RIDER!\n\nA SHORT STORY OF THE WEIRD.\n\nBY NELLIE K. BLISSETT; ILLUSTRATED BY MAX COWPER.\n\n\nIt was a dull day in early spring, and the wind in the pine forest\nbehind the Castle of Salitz was making a melancholy moaning. In one\nof the deep window-seats of the castle I sat, with a book in my hand,\nlooking down at the drowned landscape and the swollen river. I had\ncome to visit that mysterious personage, Count Siebach von Salitz,\nwhose extraordinary powers of thought-reading and prophecy would\nhave brought him in several fortunes had he chosen to use them\nprofessionally. As it was, he was the object of much interest, and\nnot a little awe, in half the capitals of Europe; and it was with some\ncuriosity that I accepted his invitation to his Hungarian estate.\n\nSo far nothing in the least peculiar had occurred to me--a\ndisappointment I was rather inclined to resent.\n\nSiebach's step disturbed my meditations. I turned and saw him coming\ndown the passage--a tall, gaunt man, with a haggard face and evil\neyes. But if Siebach's personal appearance was not prepossessing, his\ncharm of manner was so great that when you knew him well you forgot\nthe small, cruel eyes, the sneering mouth, the curious mixture of\npower and cunning which characterized his countenance. His voice, too,\nwas singularly beautiful, and atoned for many things.\n\nHe smiled as he came up and seated himself beside me.\n\n\"If you admire the view, you shouldn't look so solemn, Bazarac,\" he\nsaid; \"and if you don't, and are bored, shall we go for a ride? Or\nwill you come and look at my study?--you haven't seen it yet, and it\nis worth seeing.\"\n\n[Illustration: \"HAGGARD FACE AND EVIL EYES.\"]\n\n\"Everything here is,\" I answered, as I rose and followed him\ndownstairs.\n\nHe laughed.\n\n\"That is the disadvantage of being born a Siebach of Salitz--there is\nno merit in possessing perfection. It is merely inherited property.\nDon't knock your head against this doorway--it is low. That's right!\"\n\nWe had passed under a low archway into a long room panelled with black\noak. There was a table, littered with papers, near the window, and\nover the hearth hung the portrait of a young man whose countenance,\nparticularly about the mouth, distinctly resembled that of Siebach.\n\n\"How like you that portrait is!\" I exclaimed.\n\nHe looked at it for a moment as though weighing my remark carefully in\nhis mind.\n\n\"Do you think so?\" he said at last. \"It is my poor cousin Franz.\"\n\n\"I didn't know you had one.\"\n\n\"He is dead. He was drowned whilst we were bathing in the river\nbeneath. I was with him at the time, but I could not save him. His\nbody was never recovered--it was an awful affair. He was only seven\nand twenty.\"\n\n\"Younger than you?\"\n\n\"Oh, no--older. He was the heir. Poor Franz!\"\n\nI looked at the portrait with increased interest, and Siebach gazed at\nit too. There was a disagreeable expression on his face.\n\n\"It is a fine portrait,\" I said.\n\n\"Very--an Auberthal. You know Auberthal, of course? A splendid\npainter. Singular, now, I forgot that he will arrive here to-day. He\nhas a long-standing engagement to visit me.\"\n\nI was very glad to hear it, for I had known Auberthal when he was a\nmere boy, studying in Garcia's \"Atelier Espagnol.\" We had seen a great\ndeal of each other, and I had liked him exceedingly. Although Siebach\nwas very entertaining, I did not altogether _trust_ him; a solitude\nonly relieved by his presence did not at the moment appear alluring.\n\nI expressed my pleasure, and began to walk about the study, admiring\nthe family portraits, of which there were a great number. Under one of\nthem I noticed a curtain drawn across the wall, and, supposing it to\nconceal a picture or a cabinet, I very innocently put out my hand as\nif to draw it on one side.\n\n[Illustration: \"TALKING TOGETHER IN THE FOREST.\"]\n\nA sharp exclamation from Siebach stopped me. I dropped the curtain and\nturned to him.\n\n\"What is the matter?\"\n\nHe recovered his self-possession immediately.\n\n\"Nothing. I was cutting a pencil and the knife slipped. Oh, it is only\na scratch!\"\n\n\"What is behind this curtain?\" I asked, returning to my former\noccupation.\n\nHe did not answer at once. Then he laughed, a trifle uneasily.\n\n\"A family superstition--nonsense if you like. You can look.\"\n\nI drew it accordingly. The curtain covered a large recess, and in this\nrecess stood the life-sized statue of a horse in white marble, bearing\na man in armour upon his back. The singular part about this equestrian\ngroup was, that whilst the horse was stone, the trappings and the\nman's armour were real.\n\n\"That is an odd idea,\" I remarked.\n\n\"What, the armour? Oh, it belonged to an ancestor of mine. Of course\nthere is a stone figure underneath to match the horse.\"\n\n\"The vizor of his helmet is down. Why don't you raise it? It would be\nfar more effective.\"\n\nHe laughed again more uneasily than ever.\n\n\"My dear Bazarac, 'let sleeping men lie' is an excellent transposition\nof the old proverb. This gentleman is supposed to 'walk'--or rather\nride. In other words, he is the family apparition. He is supposed to\nride about the castle at night.\"\n\n\"What a very unpleasant idea!\"\n\n\"Do you think so? Well, it is sufficiently ghastly, I admit.\"\n\n\"Have you ever seen him?\"\n\n\"No, but I have often fancied I heard a horse snorting and trampling\nabout the passages. At this time of year he is often heard. The\nservants tell odd stories about him, but I have never encountered him\nmyself.\"\n\n\"It would be an interesting encounter.\"\n\nSiebach shuddered visibly.\n\n\"I think not,\" he said, in an altered tone.\n\nI looked up at him. His face was very pale, and his shifty glance\navoided mine.\n\n\"You are afraid of him,\" I said, laughing.\n\nAn odd light blazed for a second in his eyes. He had a pair of gloves\nin his hand, having just come in from a walk. Suddenly, without any\nwarning, he flung one glove full at the mailed face of the Stone\nRider. The armour rattled, and the glove fell back at Siebach's feet.\nHe picked it up and looked me in the face.\n\n\"You see whether I am afraid,\" he said, haughtily.\n\nI did not understand his manner, but I saw that it would be better to\nchange the subject at once, and avoid it for the future. So I asked\nhim at what time Auberthal would arrive, and we talked of other\nthings.\n\nAuberthal came in time for dinner--a little round man with a face all\nbrown skin and black beard, and extraordinarily bright eyes. I should\nnever have recognized in him the slip of a boy whose genius had\nelectrified the \"Atelier Espagnol,\" but he was as pleasant as ever. We\npassed a very enjoyable evening, and retired in due course to bed.\n\nFrom the moment I had dropped the curtain across the recess in\nthe study, I had never given another thought to the Stone Rider.\nAuberthal's arrival had successfully banished reflection on that\nsomewhat peculiar incident. I undressed, and got into bed, and, as I\nwas not sleepy, began to read. I suppose this was at about half-past\neleven, and I went on reading steadily for over half an hour, at the\nend of which period I laid down my book and prepared to blow out my\ncandles, when a sound arrested my attention, and I paused to listen.\nThe castle had long been silent, and everyone had retired to rest. Yet\nthere was a distinct sound as of someone moving about the corridors\nunder me.\n\nMy room was in the second story of the building, at the head of the\ngrand staircase--an immensely broad and imposing affair of beautifully\ninlaid marble. The corridors, too, were all marble paved, so that the\nslightest sound was noticeable in them. I listened, and distinctly\nheard the noise, whatever its cause, approach the foot of the\nstaircase. Then it paused for a moment, and there followed a curious\nsound of scrambling, as of a large and somewhat unwieldy object coming\nup the stairs.\n\nBy this time my curiosity was thoroughly excited. I got out of bed\nand went to the door. As the room was very long, and the door at the\nfarther end of it, this was a decidedly better post for listening\npurposes. I had not been there a second before I heard the\nunmistakable rattle of armour, and the snuffling sound a horse would\nmake after any unusual exertion. A wild idea flashed across my mind,\nand I pressed closer to the door.\n\nThis was the Stone Rider!\n\nThe sounds came nearer and nearer until they were just outside. Then\ncame another pause, and a heavy sigh--almost a groan--but whether from\nhorse, or rider, I could not decide. Then the horse was turned round,\nand clattered and rattled down the shallow steps of the staircase, and\naway down the corridors, until all was silent once more.\n\nAll this time, though greatly excited, I had not felt the slightest\nsensation of fear; but now that all was still such a feeling of terror\ncame over me that I lay awake for hours scarcely able to breathe,\nlistening for the return of this midnight visitant. But he did not\ncome, and towards morning I fell asleep.\n\n[Illustration: \"IT IS GOING DOWN THE CORRIDOR TOWARDS THE STUDY.\"]\n\nAt breakfast I observed that Auberthal, who had been very lively the\nprevious evening, seemed silent and depressed. Siebach, too, looked\nrather yellower and thinner than usual. I enquired if they had not\nslept well.\n\n\"Oh, yes,\" answered Siebach, hastily, \"I have slept very well indeed,\nthank you.\"\n\nAuberthal said nothing for a moment.\n\n\"You don't look particularly brilliant yourself, Bazarac,\" he remarked\npresently.\n\n\"Somebody was racketing about the staircase last night and disturbed\nme,\" I replied carelessly. \"Didn't you hear it, Auberthal? Your room\nis next mine. I wondered whether the noise would keep you awake.\"\n\nSiebach looked up at me sharply and seemed about to speak. But he\nthought better of it, and returned to his breakfast.\n\n\"Yes,\" said Auberthal, quietly. \"Something certainly kept me awake.\nThat family ghost of yours, Siebach, I expect--the Stone Rider.\"\n\n\"I heard nothing,\" returned the Count, stolidly.\n\nBut Auberthal was not to be silenced.\n\n\"No? That is odd. I heard him distinctly. He stopped outside my door;\nand something groaned. It gave me a peculiar sensation. What makes him\nwalk, Siebach--I suppose there's a legend?\"\n\n\"Oh, there are lots of legends,\" answered Siebach, offhandedly. \"One\nsays that the Ritter von Salitz in the thirteenth century caused\na statue of himself, on his favourite charger, to be set up in the\ncourtyard of the castle, and when he took prisoners of war, he chained\nthem to the Stone Rider and flogged them to death. When he was about\nsixty he married for the second time. His wife was very young and very\nbeautiful, and had been betrothed to his eldest son, whom he hated,\nand banished from the castle. One day he found his son and his wife\ntalking together in the forest. He seized them, had them lashed to the\nstatue, and directed his men to flog them to death, whilst he himself\nstood by and derided them. However, that was the last atrocity he\nperpetrated, for he soon after went mad, and died. And his spirit is\ndoomed to ride the stone horse for ever.\"\n\n\"A sufficiently horrible story, at any rate,\" remarked Auberthal,\ncomposedly. \"Is the horse in your study the original of the\ncourtyard.\"\n\n\"Yes. It has been most carefully preserved, and handed down from\ngeneration to generation.\"\n\n\"No wonder it roams about the castle at night,\" I said.\n\n\"That is mere nonsense,\" returned Siebach, irritably.\n\nI said nothing more; but after breakfast I found an opportunity of\nspeaking to Auberthal alone.\n\n\"I should like to investigate this matter,\" I said. \"Will you help me,\nAuberthal?\"\n\nHe laughed.\n\n\"Certainly; but I don't believe in ghosts, you know, Bazarac. I trust\nyou don't?\"\n\n[Illustration: \"FOR A FEW MOMENTS THE RIDER REMAINED MOTIONLESS.\"]\n\n\"I have seen some very strange things in connection with ghosts; at\nall events, will you keep up to-night, and follow the Stone Rider with\nme?\"\n\n\"If it will afford you any amusement.\"\n\n\"Don't speak to Siebach about it, then. He evidently does not care for\nthe subject,\" and I related to him the incident of the glove.\n\nHe looked rather grave.\n\n\"I am sorry to hear it,\" he said, when I had finished. \"There is\ninsanity in his family, you know--I don't think his brain is what it\nwas. And once he went off his head altogether.\"\n\n\"When?\"\n\n\"Soon after his cousin was drowned. He saw it happen. That was enough\nto drive anyone mad, perhaps. But he was always queer.\"\n\n\"Then, to-night--?\"\n\n\"Yes. When he gets to the bottom of the staircase again we will follow\nhim.\"\n\nThe day passed off very quietly, and nothing more was said about the\nstatue. We went to our rooms at the usual time, and I sat down to\nwait. At a few minutes past twelve I heard the noise beginning. It\ncame up the staircase as it had done before, and paused for a moment\noutside the door. Then I again heard the sigh, or groan, and the\nclattering down the stairs. I opened my door and found Auberthal\nalready on the landing.\n\n\"Make haste,\" he said. \"It is going down the corridor towards the\nstudy.\"\n\n[Illustration: \"HE GRIPPED MY ARM.\"]\n\nWe rushed down, and along the passage, the rattling going in front of\nus. But we were too slow. When we reached the study, the green\nbaize curtain was drawn, and everything was perfectly still. After\na moment's hesitation I pushed back the curtain. There sat the Stone\nRider, immovable as ever, mailed and erect.\n\n\"He looks quite harmless,\" I said, doubtfully.\n\nAuberthal bent down and held the candle closer. On the side of the\nhorse were great dark stains, and the armour glimmered redly in\nthe flame. The painter put his hand on one big patch, and drew back\nquickly.\n\n\"I could swear it was wet,\" he whispered. \"Let us go!\"\n\nWe returned, and I drew him into my room.\n\n\"It's very odd!\"\n\n\"Very!\" He held up his hand. \"Do you see?\"\n\n\"Good Heavens!\" I gasped, \"it's all red!\"\n\n\"With blood,\" he said, solemnly.\n\n       *       *       *       *       *\n\nFor some days neither Auberthal nor I spoke of our adventure with the\nStone Rider. But at last, one evening before dinner he came to me in\nmy room.\n\n\"I shall go down into the study to-night,\" he said, \"and see what\nreally happens. Will you come too?\"\n\n\"Yes. The noise at night still goes on?\"\n\n\"Regularly every night. Bazarac, I mean to get to the bottom of this\nmystery.\"\n\n\"All right. I shall be charmed if you can prove the whole thing a\nhoax, but--\"\n\n\"But what?\"\n\n\"I don't think you will.\"\n\nHe considered for a moment.\n\n\"I don't think I shall either,\" he said, as he left me.\n\nSiebach was unusually brilliant and amusing at dinner. He kept us at\ntable long after our usual hour, and when we at last got away to our\nrooms there was barely time to let the castle become quiet, and get\nback to the study, before twelve o'clock. However, we accomplished the\nfeat, seated ourselves near together, blew out the candle, and waited\nfor the ghost to move.\n\nFor some time everything was silent. Then, all at once, the room\nbecame strangely illuminated. One after another the chairs, and\ntables, and pictures grew out of the gloom, lit up with a pale,\npeculiar light. And at last the curtain was drawn aside--the horse\nshook himself, and snorted--the armour rattled--and the Stone Rider\nrode slowly out into the middle of the room.\n\nThe supernatural radiance streamed from him--it issued from the closed\nbars of his helmet, from the steel breastplate, from the joints of the\nrusted gorget. It seemed to grow brighter every moment, till, almost\ndazzled, I turned my attention to the horse.\n\nI did not at first notice the stain on his side which Auberthal had\nobserved. But as I looked at him, I saw that a dark stream began to\ntrickle down the whiteness of the marble. It dripped from a great dent\nin the breastplate of the Rider--dripped slowly and steadily over the\nhorse's neck, and rolled down to the floor.\n\nFor a few moments the rider remained motionless; then struck his spurs\ninto the marble flanks of his steed, and they moved away. The light\nwent with them through the open door, and Auberthal sprang up and\nrushed after them.\n\nI saw the Stone Rider turn in his saddle and look back as we raced\nafter him; and a flash of flame seemed to shoot out from between the\nhelmet-bars. On they went--clattering, clashing, rattling through the\nstone passages, and we after them. They reached the staircase--the\nRider rose in his stirrups and urged the horse up. The pace was too\nfast--the horse slipped, plunged--and finally recovered himself, just\nas an ordinary horse might do, and halted.\n\nBut the Rider's balance was destroyed. He swayed in the high\nsaddle--his arms went wildly into the air--and he crashed forward,\nand fell, with a horrible rattling sound, at our feet. The clasps that\nfastened the gorget and breastplate burst--the helmet rolled away--and\non the pavement before us lay a skeleton!\n\nFor a time we were too stunned to speak. Then Auberthal uttered an\nexclamation of horror and looked up.\n\nHalf way up the staircase stood Siebach von Salitz. His face was\nghastly white--his eyes were widened with an expression of awful\nterror--his hands were stretched out as though grasping the air. He\nstood motionless for some moments, staring into vacancy; then his\nrigid expression relaxed, his arms dropped to his sides, and he came\ndown the stairs.\n\n\"What has happened?\" he enquired.\n\n\"That!\" said Auberthal, bluntly, pointing to the skeleton.\n\nSiebach bent over it for a moment. Then he kicked it contemptuously\naside.\n\n\"Somebody has been playing a practical joke,\" he remarked.\n\nAuberthal coughed.\n\n\"I have not, nor has Bazarac. Who could have done it?\"\n\n\"Do you suppose I have?\"\n\nSiebach seemed indignant. Auberthal looked at him very quietly.\n\n\"I do not suppose anything,\" he said, \"but there is the skeleton, and\nthere is--\"\n\nHe turned to look for the horse, but it was gone.\n\n\"There was the horse,\" he concluded, \"and to-morrow morning I leave\nfor Paris. Good-night!\"\n\nHe disappeared up the staircase, leaving me face to face with Siebach.\n\n\"What does he mean?\"\n\n[Illustration: \"I RECOGNISED ONE OF SIEBACH'S SERVANTS.\"]\n\n\"I really don't know, Siebach.\"\n\n\"Do you intend to leave for Paris, too?\"\n\n\"I am very sorry,\" I said, \"but my nerves are really not equal to this\nsort of thing. Good-night, Siebach!\"\n\nHe surveyed me with an odd expression; then, suddenly, he gripped my\narm.\n\n\"Do you think--\" he almost gasped in my ear--\"do you think that he\nsuspects anything?\"\n\nI shook him off.\n\n\"Good heavens, Siebach! What should he suspect? Can't you explain this\nhorrible thing?\"\n\nHe recovered his self-command almost immediately, and smiled feebly.\n\n\"No. I can't,\" he said. \"Am I to explain all my family skeletons,\nBazarac?\"\n\n\"Not if you do not wish.\"\n\nAnd I left him standing by the skeleton of the Stone Rider.\n\n       *       *       *       *       *\n\nFor some years I did not come across Count Siebach von\nSalitz--neither, I am afraid, did I wish to do so. Of the Stone\nRider--who had proved to be no stone at all--I often thought, but\nat last I hardly regarded the incident as anything more than the\nrecollection of a bad dream. Auberthal and I met frequently, and\noften discussed our adventure; and I believed that he had suspicions\nconcerning Siebach which I did not care to share. But one evening\nas we sat in the \"Atelier Espagnol\"--Auberthal and myself--someone\nknocked at the door and came hastily in. I recognised one of Siebach's\nservants.\n\n\"What is it?\" I asked.\n\n\"Will M. Bazarac or M. Auberthal come to my master at once? He is very\nill at the H\u00f4tel ----.\"\n\nWe both rose and looked at each other, and Auberthal slipped his arm\nthrough mine.\n\n\"We had better go together.\"\n\nSo we went. The H\u00f4tel ---- was close by. In ten minutes we were in\nSiebach's bedroom.\n\n[Illustration: \"HIS EYES BLAZED WITH FEVERISH LIGHT.\"]\n\nHe lay in bed, looking thinner and more haggard than ever. His eyes\nblazed with feverish light, and he beckoned us eagerly to approach.\n\n\"There is not much time,\" he said, speaking in a weak, strained voice;\n\"I sent for you to tell you--what is that?\"\n\nHis eyes dilated with fear, and he glanced round the room.\n\n\"It is nothing,\" said Auberthal, gently.\n\nHe laughed--a short, bitter laugh.\n\n\"He is not far off--he never is. Don't you hear the horse breathing\noutside the door? I can. I always hear it now. Don't let it come\nin--don't--don't, Auberthal!\"\n\nHis voice rose to a shriek.\n\n\"Nothing shall come in.\"\n\n\"Thank you. I am so foolish to mind! I--I wanted to tell you. I--I\nmurdered him.\"\n\nHe fell back exhausted.\n\n\"Whom?\" asked Auberthal, aghast.\n\n\"My cousin Franz. He was the heir.\"\n\n\"But he was drowned.\"\n\nSiebach struggled up on his elbow.\n\n\"No, I told them that. I shot him; and I knew if they found the body\nthey would accuse me, so I hid it. And when his father died, and I\ngot the castle, I dug him up--and--you know. I could not hide the\nskeleton, so I put it on the horse. Don't you think that was a good\nidea?\"\n\nHe laughed, and Auberthal looked at me with a shudder.\n\n\"The armour hid it,\" went on Siebach, \"and I knew they were all so\nsuperstitious they wouldn't touch it. And then you came--you and\nAuberthal.\"\n\nAt that moment the doctor came in. When he left the room he called me\nout.\n\n\"Count Siebach is mad?\" I questioned,\n\n\"He is not responsible for what he says. Are you a friend of his?\"\n\n\"In a way.\"\n\n\"Then you had better stay with him. Send for me if he gets worse. I\nshall do no good by stopping.\"\n\nI went back to Auberthal. Siebach was obviously too ill to be left. I\nagreed to sit up with him half the night, whilst Auberthal rested.\n\nSiebach was exhausted, and for some hours lay quite still. I think he\nwas insensible. But about 12 o'clock I heard a sound from the bed,\nand went to him. He was sitting up, looking straight before him into\nspace.\n\n\"Don't you hear it?\" he asked.\n\nI listened, to appease him.\n\n\"No.\"\n\n\"Not the horse?\"\n\nI listened more attentively.\n\nYes--the old rattle--the old sound of a horse's hoofs. It was coming\nup the stairs.\n\nSlowly the door opened--slowly the light I had seen before grew in the\ndarkened air--and into the room rode the Stone Rider, rigid, erect,\nwith the unearthly radiance all around him.\n\nHe came up to the foot of the bed, and slowly lifted the vizor of his\nhelmet, disclosing a glistening skull--and, as I looked, the skull\nbecame the face in the portrait over the mantelpiece of the study at\nSalitz. It was too evident that Siebach recognised it. His eyes were\nfixed on the apparition; his thin features were grey, and drawn with\nfear. For a moment he remained motionless, staring at it; then he\nthrew up his arms with an awful cry, and fell back.\n\nSlowly the Stone Rider drew the mailed gauntlet from his right hand.\nFor a moment he poised it deliberately in the air, then flung it full\nin Siebach's face.\n\nA shudder ran through the prostrate figure, but it did not move again;\nand the Stone Rider turned his horse and rode from the room. The light\nfollowed him, and we were again in semi-darkness.\n\nThen I lit a candle and rang for Auberthal and the servants.\n\n       *       *       *       *       *\n\nWhether the story of the murder was correct or not, I cannot say. It\nmay have been the madness of a diseased imagination, or it may have\nbeen the late remorse of a criminal. At any rate, it is not for me to\nthrow suspicion on the name of a dead man. I can only relate what I\nmyself saw and heard. The doctor declared, and maintains to this day,\nthat his patient was insane; and, being a doctor, he very naturally\nhas the world on his side. But, say what he will, there is one thing\nhe can never explain. When I lit the candle that night, and found\nCount Siebach von Salitz lying dead, I found also that on his forehead\nwas the distinct print--purple and bruised--of a clenched fist. The\ndoctor cannot explain this; perhaps I can. For what could it be if it\nwas not left by the gauntlet of the Stone Rider?\n\n[Illustration: \"I ... FOUND COUNT SIEBACH VON SALITZ LYING DEAD.\"]\n\n\n\n\nMAKING A MODERN NEWSPAPER.\n\nSOME SECRETS REVEALED\n\nBY ALFRED C. HARMSWORTH, Editor of the _Daily Mail_.\n\n\n[Illustration: FROM FOREST--]\n\nWhen you casually and carelessly open your newspaper of a morning, how\noften do you realise, even if you are aware, that it is the product\nof a score of busy organisations, with tentacles spread over the whole\nworld, the operation of which involves the best brains and machinery\nof the age; that unlimited capital and thought are devoted to its\ndaily production; that its continual appearance has created a new\nclass of men who work at night and sleep by day; that its distribution\nrequires the use of special trains, and the gathering of its news the\nopening at night of telegraph, cable, and telephone offices; that the\npublic appetite for reading is sweeping away vast Scandinavian and\nAmerican forests for the manufacture of the wood pulp of which the\npaper itself is made; and that the very journal you are reading may\nhave formed part of a growing tree a month ago!\n\nIn the days of wagers, the wool growing on a sheep's back was once\nconverted into a dresscoat by dinner-time--and they dined at four\no'clock then! In the last few years a not dissimilar experiment\nresulted in the conversion of a tree that was growing at dawn into a\nnewspaper by luncheon.\n\nYour daily newspaper is the best bargain you will ever make, and you\nmake it every day. Do you grasp the fact that your newspaper is the\nmost splendid example of co-operation imaginable--that it enables you\nto obtain for a few pence each week that which, if only one copy were\nprinted, would cost you, for telegraphy, for brain work, for machinery\nand building and land, a thousand pounds a day or more? The Duke of\nWestminster or Mr. Astor might buy a better horse, picture, or theatre\nseat than you can, but your newspaper is as good as theirs.\n\nAccording to Mr. Labouchere and some other folk, the mystery of the\npress is the secret of its power. Yet I venture to think that if I\nlift the curtain a little--nay more, if I take the public behind\nthe scenes for a short while--I shall be increasing rather than\nendangering the respect in which the newspaper press is very properly\nheld in this country.\n\nIn the days when many newspapers were small sheets, produced in dark\nalleys, under the charge of disreputable ne'er-do-wells, who veiled\na vast amount of vulgarity under the name of Bohemianism, it was\ndoubtless a wise thing to surround the press with mystery. The less\nthe public knew about a newspaper office the better for the newspaper.\nBut to-day the public press is the concentration of all that is best\nin thought and all that is most modern in mechanism.\n\n[Illustration: --TO FLEET STREET.\n\nA three mile roll of paper.]\n\n[Illustration: HOW THE NEWS COMES--BY CABLE, TELEGRAPH, TELEPHONE,\nETC.]\n\nThe internal construction of a newspaper office is almost as\ncomplicated as that of a battleship--the duties of a modern editor as\nonerous as those of the man in the conning tower.\n\nLet us take a hasty glance at the inside life of a journal.\n\nA newspaper office is one of the few business establishments in which\nthe human machinery is at work the whole twenty-four hours round. The\nbusiness department, which requires the same staff as is needed in an\ninsurance office or bank, starts its operations, as a rule, at nine\nin the morning, when the heads and clerks of the advertising,\ncirculation, and other departments assemble.\n\nWith them arrives the first of the editorial staff. He, in the case of\none newspaper with which I am acquainted, relieves the colleague who\nhas been on duty since the previous midnight. It is his duty to open\nthe editorial letters, to watch the news of the day, to see whether\nthe particular journal on which he is engaged has gained or lost by\ncomparison with its competitors in the collection of news, and to\narrange matters generally for the coming of his co-workers, the\nforeign editor, and others, who assemble at eleven o'clock.\n\nBy this hour many of the reporters are already engaged in their\nmultifarious engagements in various parts of the metropolis. The\npreparation of the next day's paper goes on steadily until five\no'clock, when there is usually a brief conference of the editorial\npowers that be on the policy to be adopted on any particular event,\nand the methods required for obtaining any particular news or\nother features, and then, at six o'clock, the hard work of the day\ncommences.\n\n[Illustration: JUST OUT!]\n\nThe clerks, who have been receiving and checking advertisements all\nday, have sent them to the printing department, where advertisers'\nannouncements are being put into print as rapidly as nimble fingers\ncan operate quick machinery, and then, save for the presence of one\nor two clerks, the advertisement and commercial side of a newspaper\n\"shuts down\" for the day. The sub-editors appear, reporters come in\nwith the results of their day's labours, news arrives by the tape and\nother news machines in a constantly increasing quantity for the next\nnine hours. First comes the news from China or India. The Indian\ncorrespondent puts his telegram on the wire at eight or nine o'clock\nin Bombay, which is equal to four o'clock in the afternoon in London;\nand this difference of time, even allowing a couple of hours for\ntransmission, makes him always first in the field with his news. But,\non the other hand, the American news will not arrive until very late\nindeed, for when it is seven o'clock in the evening at New York it is\nmidnight here.\n\n[Illustration: OLD STYLE.\n\n(Setting type by hand at 10 words per minute.)]\n\n\"How do you manage to find all the little pieces of news to put\ninto your paper?\" is a question that must have been asked of every\njournalist.\n\n[Illustration: NEW STYLE.\n\n(Setting type by machinery at 40 words per minute.)]\n\nThat is not the difficulty. One's heaviest task is the keeping out of\nthe items of news. On an average day it is safe to estimate that twice\nor thrice as much intelligence comes to a newspaper as it can possibly\nuse. At times like, say, the last Jubilee, or at any moment of public\nexcitement, news pours in in a manner appalling to contemplate.\n\nThe wonder is that there are so few mistakes in journals. When it is\nremembered that those who handle and pass the news have often but a\nsecond to decide as to its accuracy, that it often comes from parts\nof the world to which it is impossible to refer speedily by telegram,\nthat it frequently consists of statements made by public men, who\nmay disavow them when put to the test--when it is remembered that\nthe sub-editor has to contend with the errors of shorthand, of the\ntelegraph, the electric cable, and the telephone, I think that British\nnewspapers, and London metropolitan newspapers in particular, are an\nobject lesson to the world in accuracy. Laborious publications like\nthe _Army List_, and the _London Gazette_, which are compiled by a\nleisurely Government staff, contain as many errors in proportion as\nthe hastily produced modern newspaper.\n\nAccuracy, indeed, may be considered to be the feature of English\njournalism. The stress of newspaper competition in New York induces\nthe younger journals to rush anything into type that comes to hand,\nand the American public does not seem to mind it.\n\nBut I pity the English journal which should print one or two items of\nfalse news. The average Briton, who is a plodding, painstaking man,\ntakes his newspaper as seriously as his breakfast, and one or two\nmistakes in his newspaper, or his eggs, would make him change his\ncaterer. He has no sympathy for \"enterprise\" which leads him astray.\nAnd from this fact arises one of the differences between the English\nand the American newspaper. From the American aspect, ours is dull,\nslow, stupid, and behind the times. On the other hand our journals are\ntypical of the painstaking, plodding nature of our people, and, like\nour public buildings, are often much better than they look.\n\n[Illustration: DISTRIBUTING CARTS WAITING FOR THE EVENING PAPER TO\nCOME FROM THE MACHINES.]\n\nTo return to our visit to the newspaper office. All the evening long\nas news arrives it is cut down and measured as to its importance,\ncorrected, given its proper heading, and sent upstairs by pneumatic\nor other lifts to the composing department. Towards eleven o'clock\nat night every brain is concentrated on its task. At one o'clock the\nworst is over. There is time for a cigar or a cigarette. One may be\nwaiting for important news from a war correspondent, or merely keeping\nthe paper open for any news that may arrive between one and three in\nthe morning.\n\n[Illustration: CYCLIST DISTRIBUTORS \"LOADING UP.\"]\n\nThe type is first set into columns by machinery, corrected and\nre-corrected; these columns are then made up into pages, which are\nagain corrected, each page being tightly screwed into an iron frame (I\nam purposely using no technicalities). A papier mach\u00e9 or other mould\nis then taken of each page, and into this mask (or matrix) hot metal\nis poured, and the pages come out in the form of curved plates ready\nfor fixing on the machines. It is a difficult process to explain\nwithout ocular demonstration, and I have been so long accustomed to\nthe work that I have lost all sense of its beauty and ingenuity.\n\nTowards three o'clock in the morning all the curved plates have been\nfixed on the machines; final proof copies--that is to say, first\nimpressions of the paper--have been passed; the machines start, and\nup come complete copies of the paper as you see it at the breakfast\ntable, the club, or in the railway train.\n\nThe first complete copies are carefully scanned by dozens of eager\neyes in the hope of finding some tiny blunder which it is not too late\nto remove.\n\nEach of these modern printing presses depicted here has a nominal\ncapacity of 48,000, or 96,000 copies per hour, according to the size\nof the paper.\n\n[Illustration: THE MACHINE WHICH EATS PAPER AT THE RATE OF 20 MILES AN\nHOUR.]\n\nIt is a speed truly terrific. The carts that are waiting outside the\nnewspaper office in the night seem to be filled almost by magic.\nOne hears the machinery start; a few minutes later the race for the\ndistributing agents and the railway trains begins. Upstairs such of\nthe editorial staff as have not gone home are enjoying the same kind\nof chat at the conclusion of their labours as other men do at their\nclubs. Nor are we newspaper men clubless even at that hour. The Press\nClub, hard by Fleet Street, keeps its doors open for journalists\nuntil five a.m.; and for the printers and others there are special\nhostelries open to them, and to them only, by legal enactment. Railway\ncompanies, too, provide trains for us, though not so many as they\nshould, thus enabling us to get away from the city to the pure air of\nthe suburbs at a time when all the world is sleeping.\n\n[Illustration: HOW THE PAPERS COME UP FROM THE \"INFERNAL REGIONS.\"]\n\nNewspapers are commercial concerns, and their proprietors are as\nanxious to attractively stock their columns as tradesmen their shop\nwindows. We do not say so in our journals, but privately we are\nentirely aware that we are racing each other for attractive news.\nAs to what does or does not sell in a newspaper, always an important\nquestion, opinions differ greatly. I doubt whether any two editors of\nmetropolitan daily journals would agree on that point, the fact being\nthat what pleases one audience does not necessarily interest another.\nSometimes a newspaper will adopt a feature that has proved successful\nin a contemporary with most disappointing results in its own case. Now\nand then a particular feature will spread throughout the whole press.\nAt one time the public is bent upon foreign news, at another time upon\nmatters purely domestic, but I think all are agreed that the average\nmetropolitan reader nowadays turns to his foreign news before he reads\nanything else. Two or three years ago there appeared to be a positive\ncraze for sporting intelligence. To-day mere sporting news seems to\nhave lost much of its attraction. The year before last the amount of\ncricket in the evening journals was a source of amazement. This year I\nventure to think cricket will reach its proper level.\n\n[Illustration: PAPERS BEING TURNED OUT COMPLETE, FOLDED, COUNTED, AND\nREADY FOR THE AGENTS--AT THE RATE OF 48,000 COPIES PER HOUR.]\n\nBut that every section of the public values the quick and accurate\npublication of news is obvious. The desire for speed increases each\nyear, and it is now recognised that the main object of a modern\nnewspaper organisation is the collection of news and the accurate and\nspeedy publication thereof. Incidentally it may be mentioned that of\nthe quickness with which this is performed by the press, the evening\njournals in particular, few of the public have the least appreciation.\nI have known the verdict of a trial, the result of a cricket match,\nor a boat race, published to the world within _ten seconds_ of the\narrival of the news in the newspaper office. The statement seems\nincredible, but the thing can be done in more than one newspaper\noffice in London and the provinces.\n\n[Illustration: AN EDITORIAL CONCLAVE.\n\n(Deciding the policy of the paper.)]\n\nI have asked for and obtained an item of news from New York in\nseven minutes. In this space of time was comprised the writing of my\nquestion in London, its transmission to New York, the writing of the\nnews there, and the telegraphing of it back to London.\n\nThe British evening journals, and more especially those of the\nprovinces, and Scotland, are, in my opinion, ahead of the world in the\nrapidity with which they publish accurate information.\n\nWe newspaper men love to chat among ourselves of great examples of the\npublication of exclusive news, \"beats\" and \"scoops,\" we call them. One\nof the most successful was that achieved by the _Pall Mall Gazette_\nwhen it announced, in the teeth of press and official denials\ninnumerable, the resignation of Mr. Gladstone. I was in the United\nStates at the time, and can truly say that for well-nigh a month the\n_Pall Mall Gazette_ was advertised day after day by a contradictory\ntelegram in every paper in the United States. It is said that \u00a3500\nwas paid for that item of intelligence. It would have been cheap at\n\u00a35,000.\n\nAnother great achievement was the publication by the _New York World_\nof news of the sinking of H.M.S. _Victoria_. It is not pleasant\nfor the British journalist to remember that the full account first\nappeared in a journal published on the other side of the Atlantic,\nand that that account was retransmitted to England. Then among other\nsensational news victories were those of the _Times_ correspondent\nat Pekin, in the recent Far Eastern imbroglio, and of Mr. Archibald\nForbes at the time of the Franco-Prussian war.\n\nThe present generation has almost forgotten a great newspaper\ndevelopment of a generation back. Nearly thirty years ago the whole\nworld was wondering what had become of Dr. Livingstone. Many attempts\nwere made to find him; there were private and semi-official hunts for\nthe missing missionary, but without avail. Then the _Daily Telegraph_\nand the _New York Herald_ despatched Mr. Stanley, who found him at\nUjiji. Next to the splendid war work of Sir W. H. Russell during\nthe Crimea, Stanley's work was the best expeditionary journey of\nthe century. More recently we have seen great feats of newspaper\nenterprise, both in this country and the United States, grow out\nof the Hispano-American war. War news will probably always be a\nnewspaper's greatest luxury.\n\n[Illustration: FLEET STREET BEFORE DAWN.]\n\nThe _Sheffield Daily Telegraph_ did a very big thing in 1867. I\nextract an account of the accomplishment from a recent publication:\n\n\"At that time, although few outsiders suspected it, there existed\nin Sheffield a British Vehmgericht--of which a man named Broadhead,\nsecretary of the Sawgrinders' Union, was president--for the secret\ntrial and punishment of non-unionist workmen. The _Telegraph_, acting\non private and dearly-bought information, attacked this organisation,\nSir William Leng, of course, finding the money, and often personally\nconducting the necessary investigations. It was a delicate as well as\na dangerous task, as he soon found to his cost.\n\n\"One of his reporters was bludgeoned and left for dead in one of the\nprincipal streets of the town, and in broad daylight. The house in\nwhich another lodged was blown up with gunpowder. His own life was\nthreatened day by day, and often many times a day. His leaders were\nwritten with a revolver on his desk and another strapped to his hip,\nand for nearly a year he never went abroad unarmed. At length the\nfamous Royal Commission of 1867 was appointed, with the result that\nthe secret horrors Sir William had so fearlessly denounced were\ndragged into the light of day. All England stood aghast, and the\narch-villain Broadhead, together with Crookes, Hallam, and others of\nhis tools, made full confession in order to save their own miserable\nnecks. The power of the terrible tribunal was broken for ever; but\nthe exposure cost the _Telegraph_, from first to last, some eighteen\nthousand pounds.\"\n\nSir William Leng's daring calls to mind that of Mr. Ross, of _Black\nand White_, who as a young man went through an experience that, while\nit proved a stepping-stone to his fortune (for he made nearly \u00a31,000\nby his exclusive telegrams to the press), thrilled the world for a\nvery long time. The following is an account of the matter given me by\na friend of his:--\n\nIn the memorable winter of 1880, when the snow lay so deep along the\nlines of the North that trains passed through tunnels of ice, and\ntowns were isolated for days, a gruesome incident happened.\n\nThe Earl of Balcarres died at Florence, and the body, having been\nembalmed, was conveyed by tedious stages to Aberdeen, thence to\nbe consigned to the mausoleum which formed part of the magnificent\nmansion at Dunecht, upon which the deceased Earl had spent twenty\nyears of thought and \"tons of money.\"\n\nA hearse, of the lugubrious type one is accustomed to see in country\ntowns, had been sent to await the belated train at Aberdeen, and the\nbody was duly transferred, not without difficulty, for the bulk of the\nsuite of coffins was a little greater than village hearses are made\nto meet. The weary ten mile journey was undertaken in the dark, amid a\ndownfall of snow, over the bleak road that leads from the granite city\nto the village of Skene. Progress was slow, the night grew darker\nand stormier; the snow drifted in wreaths across the road; the horses\nbecame exhausted; the men in charge did their utmost for a time, but\nit seemed as if, in the words of the national poet, \"the De'il had\nbusiness on his hand.\" Hearse and horses became embedded in a bank of\nsnow, and further effort was futile; the body had to be abandoned for\nthe night.\n\nOn the following day the storm abated, assistance arrived, the vehicle\nwas extricated, and the body was conveyed to Dunecht. There the\nfuneral service was conducted in the chapel which is built over the\nfamily vault, and with little ceremony and few attendants the body was\ndeposited on one of the shelves of the underground structure which was\nintended to be the tomb of the family to which its first tenant, the\nnoble Earl, belonged.\n\nThe weird circumstances attending the Lord Balcarres' death and\nfuneral were almost fittingly followed by events of unparalleled\nmystery. Twelve months almost to a day had transpired when a heavy\nodour of spices attracted the attention of the servants moving about\nthe mansion. On examination it was found that the huge slab of stone\nwhich covered the doorway leading into the vault had been disturbed.\nThe stone--seemingly heavy enough to require the strength of a dozen\nmen to move it--had been lifted, the vault had been entered, the\ncoffin \"pinched\" forward till it rested on the floor, the lid had been\ntorn off, the two inner cases had been rent, the body removed, and\nthe floor of the vault was strewn with the red sawdust by which the\nembalming fluid had been absorbed. Here was a mystery indeed.\n\nThe first hint of what had happened appeared in the papers on\nSaturday. The young Earl was telegraphed for, and outposts of police\nwere established round the house, with instructions that no one was to\nbe admitted, and no information was to be vouchsafed. One enterprising\nyoung journalist--Mr. W. D. Ross--who at that time was editing the\nprincipal evening paper in Aberdeen, resolved to break the silence by\nwhich his contemporaries were baffled. He secured the co-operation of\none of the servants on the estate to whom he was known, and, deeming\nboldness best, found his way to the house, and demanded an audience of\nthe Earl. The housekeeper, after some demur, consented. Plain-spoken\ntact was necessary in dealing with so delicate a matter; so when\nthe Earl appeared, the young man explained that he was there as the\nrepresentative of the _Times_ (of which he was then the correspondent)\nto consult the young peer's wishes as to what should be said about\nthis mysterious matter, with a view to obviate malicious and mistaken\nversions.\n\n[Illustration: A CORNER OF MESSRS. W. H. SMITH AND SON'S HEADQUARTERS\nIN LONDON AT 3.30 A.M.]\n\nLord Balcarres wisely accepted this considerate method, and, despite\nthe orders that had been issued, gave special facilities to the\npressman to examine the vault and obtain the facts so far as they\ncould be obtained at the time. The first result was that Mr. Ross\nsecured the monopoly of information, and also the monopoly of the\ntelegraph wires at Aberdeen, and on Monday morning all the papers\nthroughout the country published columns on the Dunecht mystery. It\nwas this publicity that eventuall resulted in the partial elucidation\nof the mystery.\n\n[Illustration: REPORTERS GLEANING \"FULLEST DETAILS OF THE CRIME.\"]\n\nFor days and weeks the telegraph officials at Aberdeen were kept\nbusy transmitting the reams of \"copy\" which, in his capacity of half\ndetective and half reporter, this young man had prepared. Mr. Ross\nprobed the matter minutely, and, apart from his important police work,\nso thoroughly was his newspaper task accomplished, that over thirty\nleading daily papers passed their correspondence into his hands.\nThrough the various phases of the mystery, ample orders and handsome\nrevenue poured into him, since sub-editors put no stint on the\nquantities of matter of vital interest furnished for the public under\nthe heading of \"The Dunecht Outrage.\" The sensation was kept up by\nspeculation, searches by bloodhounds, police investigations, arrests,\nbody-snatching theories, suggestions of black-mail, of malice, and\nevery kind of motive, for twelve months.\n\nDuring this time, the newspaper man, whose detective work was\nconsidered of the greatest value by the police, became an important\nmedium between the parties supposed to be concerned and the detective\nstaff of the city, a position of very considerable personal danger.\n\nThen the interest died away, till in July of 1882, eighteen months\nafter the rifling of the tomb, the body was found buried in the leaf\nmould that lay in the dry bed of a little rivulet that at one time had\nrun through the grounds at Dunecht.\n\nPublic interest was again kept at high tension by the curiosity of\nthe people to account for the motive of the outrage. Then came the\napprehension of suspected persons, afterwards liberated, and finally\nof one named Souter, who was convicted in the High Court at Edinburgh\nand sentenced to penal servitude. The conviction hardly met the\njustice of the case, for it was obvious that there must have been a\ngroup of grave-robbers at work.\n\nOne of the most curious things about the case was that the police\ninformed Mr. Ross that they believed it was the intention of the\nguilty parties to make a confession, and that they had elected to make\nhim the medium of it. It was actually arranged that the parties were\nto travel to Aberdeen by a certain train to reveal the whole mystery,\nbut for reasons that have never transpired this plan was subject to\nsudden eclipse, and to this day the mystery remains as much a mystery\nas ever. The unfortunate man Souter, whose actual guilt was greatly\ndoubted, called upon Mr. Ross the moment he was set at liberty, and\nthrough him communicated to the Press a circumstantial repudiation of\nhis own responsibility, and promised that what he knew about the crime\nand the criminals would ultimately be revealed when considerations of\nhonour which had kept him silent could be removed.\n\nThis is the story of the famous mystery which formed one of the most\nthrilling newspaper sensations of modern times, and which created for\nthe present manager of _Black and White_ a reputation for enterprise\nwhich has lasted till to-day.\n\n[Illustration: IN THE EDITOR'S PRIVATE OFFICE--\"I HAVE AN IMPORTANT\nSECRET TO SELL!\"]\n\nOf a hundred interesting sides of newspaper life I have been unable\nto say anything. The dangers of war correspondents--the humours of the\nsociety column, and the people who want to get into it--the financial\neditor--the lady journalist--the parliamentary staff--the descriptive\nreporter--the newspaper artist--the _\u00a3 s. d._ of journalism--each\nand all of these, and many more, would make a paper of considerable\ninterest; and Mr. Joseph Hatton should write his \"Journalistic London\"\nanew, for the whole newspaper position has changed since his last\nedition.\n\nThe sub-editor and the descriptive reporter appear to me to be the men\nupon whom the chief work of the journalism of the future will fall.\nIn France, where they do many things well, such masters as Zola have\nraised descriptive newspaper writing to the level of an art. Here,\nsave in the case of war correspondence and parliamentary work, we have\nnot specialised much as yet. A descriptive reporter, as one of the\nartists who has illustrated this little chat of mine suggests, may\nbe sent out to describe a murder trial, a fire, an execution, or\ninterview a great novelist!\n\nWe shall improve by-and-by. The old verbatim reporter will always\nremain, but he must give way to the descriptive writer in many\nmatters.\n\nTouching the question of the publishing of great secrets--such as that\nof Mr. Gladstone's retirement already referred to--I claim for the\nnewspaper press of Britain that it refrains from publishing news\ncalculated to needlessly injure or offend. How well do we know the\nfair visitant who comes to us with some great scandal to sell, and\nwho becomes almost indignant when she is politely shown out. Women, I\nfear, are more versed in this matter than men.\n\n[Illustration: SOME DAYS IN THE LIFE OF A NEWS-GATHERER.\n\n  _Out with the River Police._\n  A murder trial.\n  A railway accident.\n  A political meeting.\n  An execution.\n  A colliery disaster.\n  Interviewing a distinguished novelist.\n  A fire.\n\n]\n\n\n\n\nTHEIR MOTOR-CAR ELOPEMENT,\n\nAND HOW IT ENDED.\n\nBY EDGAR JEPSON.\n\n_Illustrated by H. R. Millar._\n\n\n[Illustration]\n\nThe atmosphere of the room was charged almost with storm; there was a\nthrill upon its air, the thrill of pent emotion. Jack stood gazing out\nof the window; Kitty sat by the fire looking at his broad back almost\nhungrily, a craving for the clasp of his arms rending her, her hands\nclenched to the whitening of her finger-nails in the effort to keep\ncontrol of her feelings.\n\n\"What's the use of having fifty thousand a year, if I can't marry the\nman I want!\" she cried, fiercely.\n\nAt her words a sudden spasm of pain caught his breath, and twisted his\naverted face; but he made shift to say in his usual drawl--\n\n\"It does seem rather hard lines, little girl. Who is it?\"\n\n\"Don't call me little girl! I believe you think I'm still a child!\"\nsaid Kitty.\n\n\"Very well, very well--madam. Who is the man? Young Malmesford?\"\n\n\"As if I should tell you!\" cried Kitty.\n\n\"Well, you sent for me. I thought you wanted my advice or help, or\nsomething, don't you know!\" said Jack.\n\n\"I want help badly enough,\" said Kitty; and he turned sharply at her\ntone to see that her face was very pale in the frame of her black\nhair. \"But how could you help me in this? How could anyone help me? I\noughtn't even to talk about it to you!\"\n\n\"Oh, yes; you ought!\" he said, quickly. \"You've always talked about\neverything to me!\" He paused awhile, then added, and he could not keep\nthe sadness out of his voice, \"So you want someone else to talk to\nabout everything? Who is it? I'll deal with him all right.\" The last\nwords came savagely.\n\n\"Oh!\" cried Kitty, \"I believe you'd order him to marry me, and thrash\nhim if he refused!\"\n\n\"I'd see that he did it!\" said Jack, with the same savage earnestness.\n\nA silence fell upon them; Kitty's thoughts seemed to grow more\ndistressful, for now and again she sighed; Jack stared out of the\nwindow, and watched the deepening twilight blacken the park; it seemed\nto him that this confession of Kitty's was so blackening his life; the\nnight was settling down upon it.\n\n\"Jack--do you--do you remember--about two years ago--you stopped\nkissing me. Why--why did you do it?\" said Kitty, softly; she seemed to\nhave wandered from the point. He turned to her; the glow of the fire\nalone lit the room now; and she was sitting full in it. Her face was\nstill pale.\n\n[Illustration: \"CLENCHING HIS FIST AND BANGING IT ON THE TABLE.\"]\n\n\"Oh,\" he said, in discomfort, \"you weren't a child any more. And you\nwere a great heiress--and I was your friend and guardian--and all that\nsort of thing, don't you know!\"\n\n\"Poor Jack! You're very poor, aren't you, Jack?\"\n\n\"No, I'm not! I'm rolling in riches! I've four hundred a year!\" said\nJack, bitterly. \"Besides, there's the Colonial Land Agency; I made\ntwenty pounds out of that last year.\"\n\n\"What's four hundred a year with your tastes?\" queried Kitty.\n\n\"Look here! don't let's talk about me. What about this fellow?\" said\nJack, clenching his fist and banging it on the table.\n\n\"You should never have left Westralia. You kept your horses, you\ngot your sport; you were on the way to becoming the big man of the\ndistrict,\" said Kitty, not to be diverted from her theme. \"Do you\nremember what a swell you were when you first found me, six--no,\nseven--I'm always forgetting that I'm nineteen--years ago, and how\npoor father and I were? Do you know I should never have been anything\nbut a wild bush-girl if you hadn't taken me in hand and looked after\nme? Really you taught me everything! I believe that but for that I\nmight have worn the wrong clothes!\"\n\n\"Oh, nonsense! You were _born_ all right,\" said Jack.\n\n\"Oh, yes, you did,\" said Kitty. \"And when three years ago the gold\nwas found, and father made his million, and died, appointing you my\nguardian, and you thought I ought to come to England and have some\nschooling, I believe you left Westralia just for my sake, to look\nafter me.\"\n\n\"One always comes back to England,\" said Jack, quickly.\n\n\"You wouldn't have come but for that,\" said Kitty.\n\n\"Oh, yes, I should. Of course I should.\"\n\n\"I always thought it strange that father didn't leave you a few\nthousands a year for your trouble in looking after me and my fortune,\"\nsaid Kitty.\n\n\"He knew jolly well I shouldn't have taken it,\" said Jack, hotly.\n\nThere was a pause; and then she said thoughtfully--\n\n\"Do you know I believe father thought you would fall in love with me\nand marry me? Wasn't it a funny idea?\" said Kitty.\n\n\"Oh, v--v--very funny! Very funny!\" said Jack, grinding his teeth\nsoftly.\n\n\"Yes; just think of your age. Why, you'll be twenty-eight on the tenth\nof March,\" said Kitty.\n\n\"Oh! So it's that young fool Malmesford, is it?\" said Jack, viciously.\n\n\"What's that young fool Malmesford?\" asked the innocent Kitty.\n\n\"Look here,\" said Jack, in a quiet, strained voice, \"we're getting\naway from the point. You want to marry a man; and I'm to make him\nmarry you. Who is he?\"\n\n\"Ah,\" said Kitty, plaintively, with a long-drawn breath, \"now I see\nwhy you're so keen about it. You want to get rid of me. You are tired\nof the trouble of looking after my stupid investments. Well, I'm sure\nI don't wonder at it. You want to marry me off, and have done with\nit. I wouldn't have sent for you if I'd known; I've only added to your\ntrouble.\"\n\n\"Well,\" said the goaded Jack, \"thank goodness you'll be of age in two\nyears; and then I sha'n't be plagued like this.\"\n\n[Illustration: \"SHE SET DELIBERATELY TO WORK TO FILE THROUGH THE\nHANDLE.\"]\n\n\"Plagued,\" said Kitty, \"how plagued? I'm so sorry. How was I to know\nyou wanted to be rid of the trouble of me and my fortune? You never\ngrumbled before.\"\n\n\"Oh, your fortune! I tell you I've wished a thousand times that every\ninvestment of yours went to smash, and you lost every penny of it!\nSo there! I'll just leave you for awhile to make up your mind whether\nyou're going to tell me who the man is, or not!\" He flung out of the\nroom in a heat, and banged the door.\n\nKitty laughed a little low laugh of extreme relief; but her eyes\nwere all shining; and she said with a little shiver, \"He loves me--he\ndoes--he does--he does!!!\"\n\nPresently she rose, with a very resolute face, took a hat and coat\nfrom a peg in the hall, went out of the back-door, and down to the\nstables. She went into a coach-house, switched on the electric light\nabove her motor-car, and considered it thoughtfully. It was a big car,\nwith something of the air of a trap, built to hold two. Then she went\nto the box of tools used for its machinery, and selecting a fine file\nstepped into the car, and set deliberately to work to file through the\nhandle of the lever which started and stopped it. Her Australian life\nhad made her a capital work-woman, and she did it neatly; but it was\na long piece of work, and now and again she stopped to test it. She\nwished to file through it, so that she could break it with a jerk.\nAll the while she worked she whistled softly. Something about her task\nseemed to amuse her.\n\nAt last she completed it to her liking, and then sat back in the\ncar, weighing, with a face that grew very serious, the risks of the\ndangerous game she had resolved to play. After a long while she rose\nand said between her teeth, \"I don't care if we are smashed, Jack and\nI, together.\"\n\nShe came back to the house, went to him in the billiard-room, and\nsaid, \"We're going to dine at the Hall to-night. Aunt will go in the\nbrougham, and you and I in the motor-car.\"\n\n\"I hate the beastly thing. I know there will be a smash some day,\" he\nsaid. His temper was still ruffled.\n\n\"Very well,\" said Kitty, gently. \"You go with aunt, and I will go in\nthe car by myself.\"\n\n\"I'll be shot if I do!\" said Jack; then he said, \"I suppose Malmesford\nwill be there?\"\n\n\"I suppose he will,\" said Kitty, very demurely. \"But why do you speak\nso contemptuously of your cousin?\"\n\n\"I didn't choose my cousins, did I?\" said Jack.\n\n\"You're very irritable to-day,\" said Kitty, severely, and she left\nhim.\n\n[Illustration: \"KITTY AND THE MARQUIS WERE PLAINLY GREAT FRIENDS.\"]\n\nLater, as they were settling themselves in the motor-car, Jack, still\ncaptious, said, \"How many more rugs? are we going to the North Pole?\"\n\nKitty's heart jumped: they might be going a good deal further: she\nonly said, \"There are ten degrees of frost already; and it isn't like\na closed carriage.\"\n\nShe handled the lever very gingerly, and brought them to the Hall\nsafely. Jack did not enjoy the dinner. Kitty and the Marquis of\nMalmesford were plainly great friends: she had never, indeed, been so\nnice to him before. Jack tried to regard their friendship with the\neye of an indulgent guardian, hardened, as he believed himself, to\nthe thought of her marrying; he made a very poor hand at it. He had\naccustomed himself, indeed, to looking at her across the great gulf of\nher wealth; but the sight of another man making fortunate love to her\nawoke in him a desperate jealousy.\n\nThey were late leaving the Hall; and it was a bitter black frost. Aunt\nAnne started first in her brougham, and then Kitty, in a long sealskin\njacket and sealskin cape, walked down between Jack and Malmesford to\nthe stables, where the motor-car awaited them. Jack wrapped the rugs\nround her very carefully, and took his seat at her side; she cried a\ncareless \"Good-night!\" to Malmesford, and started the car gently. As\nthey turned into the road at the end of the drive, she moved the lever\nnearly to full speed, and with a sharp jerk of her strong little wrist\nsnapped off the handle.\n\n[Illustration: \"SHE MOVED THE LEVER NEARLY TO FULL SPEED.\"]\n\n\"What's that?\" said Jack.\n\n\"Oh, Jack!\" she cried, with an odd, excited thrill in her voice, \"I've\nsmashed the handle, and we can't stop!\"\n\n\"Good Heavens!\" cried Jack, and threw his arm around her.\n\nThe speed began to quicken.\n\n\"The lever's nearly at full speed,\" said Kitty, quietly. \"What are we\nto do?\"\n\nHis arm tightened round her, and the alternatives raced through his\nmind. \"We must strike the Great North Road at Anderfield, and heaven\nforgive any one who gets in our way!\" he said.\n\n\"Six miles and two turns,\" said Kitty; \"but it's our only chance.\"\n\nThe hedges were flying past. The first turn was two miles away, and\nthey were very soon on it. Kitty put on all the brake she could; and\nthey came round it safely. They came down hill to the second turn:\nfortunately it was not sharp: a long hill fairly steep, and, for all\nthe brake, the machine went quicker and quicker until it seemed almost\nto fly, scarcely touching the ground. The hedge of the other side\nof the Great North Road sprang suddenly up before them: they seemed\nalmost on it; Jack, with his heart in his mouth, lifted Kitty half out\nof her seat as they whizzed round the corner on two wheels: the car\nsettled with a jerk that proved the strength of its springs, and they\nripped down the Great North Road.\n\nKitty laughed a short hysterical laugh.\n\n\"I thought we'd gone to glory together!\" she said: and they both lay\nback panting.\n\n\"How far are we going?\" said Jack.\n\n\"It won't stop for fifty miles,\" said Kitty.\n\n\"Good Lord!\" said Jack. \"Can't I do anything? Let me get at the\nmachinery.\"\n\n\"You can do nothing!\" said Kitty, sharply.\n\nFor a long while neither said a word. The car sped along with a\nquerulous, eerie whirr that rose to a clattering snarl as it hurtled\ndown hill. The cold air stung their faces; the hedges were level,\nblack walls on either side; now and again they flew through a sleeping\nvillage; and the dogs who ran out to bark, turned and fled yelping\nfrom this sinister, rushing monster. Kitty's firm hand steered them\nsteadily, save when the car jerked snarling down hill, out of control;\nnow and again she set the whistle hooting. Jack sat with his mind in\na whirl of fears of what might befall her. Little by little the\noppression of a nightmare began to weigh upon them as a binding spell.\n\nJack broke it by withdrawing his arm from around her, and lighting a\ncigar; he did not slip his arm back.\n\nPresently she said softly, \"Hold me again, Jack, I feel safer\"--his\narm slipped round her--\"I feel--I feel--as if some dreadful beast were\ncarrying us away.\"\n\nShe looked infinitely childlike; and he gripped her closer.\n\n\"Poor aunt Anne, she'll think we've had a smash, as indeed we may,\"\nshe said presently.\n\n\"By Jove, yes; they'll be hunting the neighbourhood for us!\" said\nJack.\n\n\"As for Lord Malmesford, he'll think you've run away with me,\" said\nKitty.\n\n\"Oh, nonsense!\" said Jack, uneasily.\n\n\"He will though. Juliette Halliwell will tell him so. I saw her get\nvery angry at the affectionate way you were looking at me at dinner,\"\nsaid Kitty.\n\n\"I wasn't!\" said Jack.\n\n\"Oh, yes, you were; ever so affectionately. What kind of affection was\nit, Jack--paternal?\"\n\n\"Talk of something else!\" said Jack, in a thick voice; and nestling\nagainst him, she felt him quiver and his heart shake him at each\nthumping beat.\n\nSome miles further on the lights of a town rose suddenly a little way\nahead. Kitty set the whistle hooting, and slowed the car as much as\nshe could, but even then they dashed down the long silent street at a\nvery dangerous pace. It was fortunate that it was empty. They were\na mile beyond it before they breathed easily again, and Kitty said,\n\"What town was that?\"\n\n\"I don't know,\" said Jack. \"We're five-and-twenty miles from home.\"\n\nThe road stretched far away ahead, very white in the moonlight; and\nthe feeling that the car was a malignant living creature came upon\nthem more oppressively than ever, wearing their nerves.\n\n[Illustration: \"THEY RUSHED TOWARDS THE WAGGON.\"]\n\nKitty nestled closer to him--a fear that her desperate freak would\nhave a tragic end invading and filling her heart. They rushed up a\nlong hill--the car seemed to breast it like a strong demon--and at the\ntop saw before them a long steep descent.\n\n\"Now the brute's going to have all its own way,\" said Kitty, between\nher clenched teeth.\n\n\"Never mind, little girl,\" said Jack, cheerily, \"sit tight.\" If she\nhad not been there, he felt that he would have enjoyed the danger; as\nit was, he sat in torture.\n\n\"It is out of control!\" cried Kitty; and, peering ahead:\n\"There's--there's a waggon at the bottom of the hill!\"\n\nThe whistle hooted and hooted; she gave the car the brake; and at each\nleap it jarred every bone in her body. They rushed towards the\nwaggon; if the waggon was not on its right side of the road, they were\nsmashed: they were upon it; Kitty screamed out; there was a snapping\ncrash; then they were rushing along the empty road with the left\nsplash-board torn off. Kitty lay back in a dead faint. Jack caught the\nsteering-gear in his right hand, raised Kitty with his left arm, and\ntwisted into her place, holding her on his knees. The car began to\nslacken and go smoother up the opposite hill; in three minutes it was\nsteady again. Kitty lay heavy and still in his arms, her face very\nwhite in the moonlight; her faint breathing scarce parted her lips.\n\nUphill and downhill, through villages, through another town the car\nfled on. Now and again Kitty murmured a word, now she seemed to sleep.\nThe night was wearing on. At last it seemed to him that the beast was\ntiring; and he scarce dared believe it. But breasting the next long\nhill it slowed and slowed; its moan hushed; it came to a crawl. Thirty\nyards from the top it stopped a moment, moved on again, then stopped\nfor good. For all its danger he sighed that their ride was at an end.\nKitty never stirred; he gave her a little shake; and she sighed too,\nand raised herself. They looked down on a great stretch of country;\nhere and there the dim twinkling showed the lights of a town.\n\n[Illustration: \"KITTY LAY STILL IN HIS ARMS.\"]\n\n\"There are some biscuits and a flask of cherry brandy, if it isn't\nbroken, in the box of your seat,\" said Kitty, slipping into the\nplace at his side. He fished them out unharmed, and they munched the\nbiscuits, and drank from the flask by turns.\n\nHe looked at his watch, and said, \"Ten past three! By Jove, we've had\na narrow squeak!\"\n\n\"Three in the morning, and miles from anywhere. I'm hopelessly\ncompromised,\" said Kitty.\n\nJack knitted his brows, thinking it out; he could not gainsay it. He\nsaid nothing. \"Oh!\" said Kitty, almost in a wail, \"I thought you were\na man of honour, Jack.\"\n\n\"Well?\" said Jack.\n\n\"There is only one course open to you,\" said Kitty.\n\n\"Well, I suppose there is,\" said Jack, a little stiffly. \"Will you\nmarry me?\"\n\n\"Yes: I will--I must--I must,\" said Kitty, with a deep sigh.\n\nPresently she said in a very low voice, \"Have you no sense of what is\nfitting?\" As she spoke she looked into his eyes, swiftly and away.\n\nHe caught her to him, and kissed her; it seemed to him that her lips\nwere responsive.\n\nA sudden jealous pang wrung his heart. \"But--but--the other man: the\nman you want to marry?\" he said.\n\n\"Ah, yes,\" said Kitty, carelessly--\"the other man. It's no use\ntalking about him now. Let us forget him. I will tell you about him\nwhen--when--we are married.\"\n\nShe threw her arms round his neck and whispered, \"Do you think you\nwill learn to love me, Jack?\"\n\nHe pressed her to him and cried passionately, \"For four years I have\nloved you more and more every day. Every day I have cursed your money\nmore!\"\n\n\"Poor Jack!\" said Kitty, and her eyes were full of tears. He lifted\nher out of the car, putting his arm round her, and supporting her;\nand they began to walk down the hill in search of a railway station,\ncareless, in the glow of their happiness, of that bitter cold, and of\nthe inevitable long wait for a train.\n\n\n\n\nHOW WE GET OUR WEATHER.\n\nBY GAVIN MACDONALD.\n\n_With photographs illustrating the queer side of the matter._\n\n\nIn most of the morning papers we are accustomed to the luxury of a\ndetailed weather report and forecast. The majority glance at it with\na sceptical smile. They are of opinion that in order to be on the safe\nside they must invert its message. If fine weather and sunshine\nare predicted, they sagely nod and take down the homely gamp. The\nprediction of a hurricane or stormy showers is the signal for leaving\numbrellas and overcoats at home.\n\nHowever, those who know anything of the gigantic strides meteorology\nhas made within the past few years are aware that in the main its\nprognostications are accurate. In fact, it is a matter for great\nsurprise that its practical uses are not more generally recognised and\ntaken advantage of.\n\nIf you meet your best friend in the street his first six words contain\nsome reference to the weather. The merest stranger looks questioningly\nat the sky when he has made his bow. Two-thirds of the daily\nconversation of the British Isles has to do with this subject; nor is\nthis surprising, for it is a matter of vital importance, affecting all\nclasses alike.\n\n[Illustration: CHURCH LIFTED INTO THE AIR BY A TORNADO AND DROPPED\nROOF-DOWN ON A HOUSE 100 FEET AWAY.]\n\nA wet Bank Holiday may mean thousands of pounds out of a railway\ncompany's pocket, not to mention the disappointment and chagrin of\ncountless thousands of prospective holiday makers. A severe frost may\ndisorganise a whole trade. In 1881, for instance, the whole building\ntrade was at a standstill for a period of nearly three weeks, owing\nto the severity of the frost. And to the farmers, horticulturists, and\nfruit-growers the weather is a matter of financial life or death.\n\nMeteorology is of invaluable assistance in other ways: in warning\nour coasts of coming storms; in deciding the climate and consequent\nhealthfulness of the different parts of the country.\n\nYou can't even build a new town successfully without it, for only by\naccurate meteorological observation can the two most important factors\nof water-supply and sewerage be dealt with. For example, in planning\na new waterworks, the ground subject to the greatest rainfall, and\nhaving the utmost gathering capacity, must be selected; while in\nconstructing the system of sewerage, it is essential for the\nsurveyor to accurately gauge the force and volume of the heaviest\nthunder-shower. If this is miscalculated, pipes of insufficient\ncapacity may be laid with disastrous results to the city and its\ninhabitants.\n\nThese things are only to be learned by a study of meteorology.\n\nFew people have any knowledge of the science beyond that supplied\nthem by the forecasts and charts in the daily papers. Consequently the\ncharts, which are more or less abstruse, are only understood by the\nfew, and the forecasts are indulgently tolerated as a description\nof useless fortune-telling, rendered respectable by scientific\nrecognition.\n\n[Illustration: HOUSES WRECKED BY A TORNADO.]\n\nThe popular idea seems to be that certain scientific men who have\ngiven the subject considerable study, cast a knowing eye on the\nevening sky, and pass on written prognostications for use in the\nmorning papers.\n\nAs a matter of fact the method by which we obtain our weather reports\nand forecasts is very different, and savours even more strongly of\nromance than the clairvoyant system usually identified with the seers\nof the weather office.\n\nTwo institutions look after our weather--the Meteorological Office, a\nGovernment department with a grant of \u00a315,000 per annum, and the Royal\nMeteorological Society, a scientific institution maintained by the\nsubscriptions and donations of its members.\n\nThe Meteorological Office occupies a dull set of rooms in Victoria\nStreet over a shop, and, other than the latest weather chart, hung\nup outside the street door, there is nothing to intimate that the\npresiding wizards of the weather sit upstairs, and that if you\nare particularly anxious to have the latest information in their\npossession you have only to walk up and pay the nominal sum of one\nshilling.\n\nLikewise you may receive the latest information by letter for the same\nfee, or by wiring to \"Weather,\" London, the shilling fee and the cost\nof a telegraphic reply.\n\nFarmers and others to whom the question of weather is a vital one,\nespecially at the hay and harvest seasons, are supplied with harvest\nforecasts for the nominal sum of 2_s._ 6_d._ per quarter, in addition\nto the cost of the telegrams.\n\nIn addition to this, a set of forecasts is daily supplied to the\nnewspapers, and about twenty-eight well-known agriculturists, for\npublic exhibition in their neighbourhoods.\n\nThe system employed in making up the weather is of more than usual\ninterest, and is worthy of some description.\n\nIn connection with the office are some 140 observing stations,\nincluding 17 belonging to the Royal Meteorological Society and 19 to\nthe Scottish Meteorological Society. These stations are divided\ninto classes according to the value and quantity of the observations\nsupplied by them. Excepting the cases of telegraphic stations,\nwhich are subsidised by the central office, the observers are mostly\nvolunteers who are interested in meteorology, and who provide their\nown instruments.\n\nThe office receives sixty telegraphic weather reports each morning,\neighteen every afternoon, and twenty-nine each evening, in addition to\nan enormous mass of data supplied by volunteer and casual observers.\n\nThe forecast we are accustomed to find in our morning paper is\ncompiled from the telegraphic reports of the subsidised stations.\nThere is something peculiarly fascinating in the idea of the clerk of\nthe weather scenting out a big gale and issuing a warning hours before\nits arrival on our coasts. One associates him with a prophet or witch,\nand very naturally wonders how it is done.\n\nAs a matter of fact forecast work is far from romantic, entails\nvery great mental labour, excellent judgment, and great scientific\nknowledge and experience.\n\nThe forecasts are made three times a day--at 11 a.m., 3.30 p.m., and\n8.30 p.m. They are, of course, based on the telegraphic reports\nand observations. The 8.30 p.m. forecast is made for the morning\nnewspapers.\n\nAmong the volunteer observers are representatives of all professions.\nIn one case a deaf and dumb gentleman presides over a station of\nconsiderable importance.\n\nThe stations themselves are mostly situated in the observers' grounds,\nand the surroundings of some of them are very picturesque. The\nstations at Rousdon and Chapel Hill, Torquay, are both beautifully\nsituated. Princetown station is particularly interesting, because of\nits situation in the yard of the great Dartmoor penal establishment!\nWe may be quite sure that its presence in such surroundings has\nnothing to do with the well-being of the convicts themselves, the\ndreary routine of whose lives is little affected by considerations of\nweather. In another case, the meteorological observatory is found\non the tower of a church--that of Boston, Lincolnshire. Among the\ninstruments on the tower is an electrical thermometer connected with\nthe ground by a wire so that it may be read without the necessity\nof ascending. It is impossible to over-estimate the usefulness of\na station such as this, situated as it is in the midst of purely\nagricultural country. The farmers round Boston avail themselves, it\nneed scarcely be said, of the valuable information furnished by the\nmysterious little instruments on their church tower.\n\nMore interesting, perhaps, than any of these is the observatory\nsituated in a London churchyard.\n\nAlthough every day a ceaseless throng of human beings crowd and jostle\nin the streets of the City of London, yet it has always been difficult\nto obtain observations there, for the very good reason that scarcely\nanybody lives within its precincts. The only station of the kind is to\nbe found in the churchyard of St. Luke's, Old Street, one of the few\nrestful spots in this busiest corner of the world.\n\nThe highest station in Great Britain is that on the summit of Ben\nNevis, 4,407 feet above the sea. The northmost station is in the\nShetland Isles.\n\nMany gentlemen among the volunteer observers are leading\nmeteorological experts, and spend much time and money on the equipment\nand maintenance of their stations.\n\n[Illustration: _Messrs. Metcalfe, photo., Richmond, Yorks_ HAILSTONES\n(ACTUAL SIZE) THAT FELL AT YORK, JULY 8, 1893.]\n\nA very fine private observatory is that belonging to Col. Knight, of\nHarestock, Winchester, of which an illustration appears on page 60.\n\nThe scaffolding in the foreground was erected for the purpose of\nlowering an earth thermometer into the ground. This instrument, which\nis constructed to register the temperature seventy feet below the\nsurface, is contained in the wooden chamber standing at an angle to\nthe scaffolding, and was photographed during the sinking process.\n\nBesides the work of preparing weather reports and forecasts, the\noffice fulfils many other functions, such as the study of ocean\nmeteorology, climatology, and so forth. In connection with the former\nwork, the office annually receives some hundreds of reports and\nobservations from officers of ships of the Royal Navy and Mercantile\nMarine.\n\nThe fishermen and sailors round our coasts have much to thank the\noffice for. Besides supplying all the ports with daily weather reports\nand forecasts, it has lent over 200 barometers to fishing villages and\nother places on the coast for the benefit of the seafaring population.\n\nFortunately in this country we suffer comparative immunity from\ntornados, sirrocos, cyclones, and other dangerous natural phenomena.\n\nThat we can produce something more ferocious than an April shower,\nhowever, is amply demonstrated by our illustration of two huge rents\ntorn in a hillside at Langtoft, East Yorkshire, by the bursting of a\nwaterspout.\n\nHailstorms are another great source of destruction. Most people will\nremember the damage caused by a hailstorm in Essex last year, when\nseveral farms and homesteads were utterly wrecked, and great numbers\nof cattle killed.\n\nMany people who have not encountered the big hailstorm regard it with\nthe cheerful scepticism with which they view the sea serpent and\nthe abnormal gooseberry. However, by permission of the Royal\nMeteorological Society, we are enabled to reproduce a photograph of\nsome of the hailstones--actual size--which fell in a great storm at\nYork on July 8th, 1893, together with a section of corrugated iron,\nshowing holes and damage caused by hailstones which fell in a similar\nstorm at Tulcumbah, N.S.W., on Oct. 13th, 1892.\n\n[Illustration: CORRUGATED IRON PIERCED BY HAILSTONES.]\n\nHowever, most people would rather lose a section of corrugated roofing\nthan encounter the flash of lightning that struck the man whose\nclothes appear in the illustration on the next page. As will be seen,\nthe clothes are literally shredded to rags, and the strong leather\nboots are torn as though they were tissue paper.\n\n[Illustration: MIDNIGHT PHOTOGRAPH OF LIGHTNING FLASH IN SHANGHAI\nHARBOUR.]\n\nPhotographs of lightning are no longer novel; but our picture of\na flash taken at midnight in Shanghai Harbour is one of the most\nremarkable ever seen. It is some distance behind the anchored\nsteamer, but the reflection on the water is so vivid as to give it the\nappearance of moonlight.\n\nThe tornado is a phenomenon we can very well do without, and we\nsincerely hope the clerk of the weather will give us ample notice of\nthe very faintest indication that one of these inanimate monsters is\ncoming our way.\n\nThe tornado is soon over, it is true, but hailstorms are to be\npreferred. On May 27th, 1893, a storm of this nature put in an\nappearance at Wellington, Kansas, and practically wrecked the whole\ncity. A horse was picked up, stable and all, and blown some hundreds\nof yards to leeward. The stable was smashed, but curiously enough the\nhorse came down on his feet and escaped unhurt.\n\n[Illustration: CLOTHES OF MAN STRUCK BY LIGHTNING.]\n\nIn the same storm the Lutheran church was lifted bodily from its\nfoundations into the air, and fell, bottom upwards, on top of a new\nresidence 100 feet away, as it appears in the photograph.\n\nIn another photograph are some collapsed houses, the result of a\nsimilar storm in Lawrence, U.S.A.\n\nAlthough our own Meteorological Office and Society have no such\nstartling instances to record, yet they possess much data of equal\ninterest.\n\nFor instance, how many people know that on Dec. 4th, 1879, the\nthermometer registered 23 degrees below zero at a place called Black\nAdder, in Berwickshire? This is the greatest degree of frost ever\nknown in Great Britain. The coldest spot in the world is Verkoianski,\na town in Siberia, where 120 degrees of frost have been registered.\nThe hottest is the Red Sea, where 120 degrees of heat are often\nexperienced.\n\nThe hottest place in Great Britain, curiously enough, is London,\nor rather the Thames Valley. The wettest, Seathwaite, in the Lake\nDistrict, where 8.03 inches of rainfall have been registered in 24\nhours.\n\nTaking 1 inch of rain to represent 101 tons of water per acre, it will\nbe seen that the farmers cannot complain of drought in the Seathwaite\ndistrict.\n\nHowever, the greatest rainfall ever measured in this country occurred\nin Camden Square, London, on June 28, 1878, when 3-1\/4 inches fell in\n1-1\/2 hours.\n\nLately several meteorological experiments have been made with kites\nand balloons, which are expected to enrich the science with many new\ndiscoveries.\n\nMany people have curious ideas of the capabilities and functions of\n\"The clerk of the weather.\" Mr. Robert H. Scott. M.A., F.R.S., the\ngentleman at present occupying this position, in his book on \"Weather\nCharts and Storm Warnings,\" tells some curious stories illustrative of\nthis.\n\nFor instance, in June 1886 he received a letter bearing no less than\nten postmarks. It was addressed \"Weather Office, Strand, London.\" Its\ncontents were--\"Three next days order to be fine.\"\n\nA Boston letter was addressed--Right Hon. Clerk of Weather, 9, Downing\nStreet, London, W.C.\n\nIts contents were--\n\n\"My Lord Clerk,--May it please your lordship you will greatly oblige\nyour humble servant by writing or sending me a telegraph whether it\nwill be fine or no on the 5th of November, 1867.\n\n  \"I have the honour to remain,\n       \"Your lordship's most obedient servant,\n            \"Joseph William ----.\"\n\nSuch letters are by no means rare, though such ignorance seems\nscarcely credible in the nineteenth century.\n\n[Illustration: _Boak & Co., Photo, Bridlington Quay_\n\nHILL TORN BY WATERSPOUT, LANGTOFT, E. YORKS]\n\nFurther, there are many false prophets who prophesy without science,\nand they rarely miss the opportunity of sending along a forecast in\norder to give the constituted clerk of the weather a leg up in his\narduous duties.\n\nThere are also many amateur weather prophets.\n\nOne of these gentlemen issued monthly postcard forecasts for more than\ntwelve months between 1882 and 1883.\n\nIf they are wrong, nobody bothers, but if the S.W. gale predicted from\nthe Meteorological Office fails to put in an appearance, woe to the\nunfortunate clerk of the weather. People forget how many times his\npredictions have been verified.\n\nIf one is interested in meteorological work and is anxious to become\nan observer, the path is by no means difficult. On application at the\noffice a form is sent, which must be filled up. Certain particulars as\nto the observer's fitness are naturally required, and he is invited to\nforward a description of his residence and a plan of the spot on which\nhe would suggest erecting his instruments.\n\nHe must also describe the natural surroundings, so that the office may\ndecide whether they are likely to have any prejudicial effect on the\ninstruments, and therefore affect the accuracy of the records.\n\nEach observer supplies his own instruments, and if his application is\naccepted, a book of instructions on their correct use is sent to him.\n\nOr he may obtain instruction at the London office, or any of its chief\nagencies. In all these places sets of instruments are kept in working\norder for the express purpose of instructing observers in the methods\nof observation.\n\n[Illustration: COLONEL KNIGHT'S 70-FOOT EARTH THERMOMETER,\nWINCHESTER.]\n\nI suppose most of our readers, during a stay at some port or favourite\nwatering place, have observed a curious triangular black object\nsuspended from the pier or jetty signal-mast.\n\nThose who have enquired as to its nature will know that it is the\nsignal of an approaching storm. On receiving telegraphic notice of\nan atmospheric disturbance on or near the British coasts, the\nMeteorological Office telegraphs to all the chief ports and fishing\nstations.\n\nThe telegram is exhibited at the foot of the signal-mast, and the\nwarning signal, a black canvas cone 3 feet high and 3 feet wide at the\nbase, is immediately hoisted.\n\nThe nature and direction of the approaching storm is indicated by the\nposition of the cone.\n\nAt night three lanterns hung on a triangular frame supply its place.\n\nStorm warning telegrams are supplied to some 215 stations, of which\n117 are in England and Wales, 63 in Scotland, 28 in Ireland, 4 in the\nIsle of Man, and 3 in the Channel Islands.\n\nAnother branch of the work, of invaluable service to navigators, is\nthe preparation of monthly current charts of the oceans of the world.\nObservations are constantly being made by captains of ocean-going\nvessels, and the data are forwarded whenever possible to the Weather\nOffice. A strict account of the currents recorded in each month has\nbeen kept for 60 years!\n\n\n\n\nHER LETTER!\n\nONE OF J. HARWOOD PANTING'S INTERESTING NARRATIVES.\n\n_Illustrations by W. B. Wollen, R.I._\n\n\nBrussels--evening--an evening which preceded a still more memorable\nmorn. To be precise, it was the 15th of June, in the year of grace\neighteen hundred and fifteen.\n\nCaptain John Durnford, of the Guards, stood outside the Chapelle du\nSaint Sacrament des Miracles. The air was full of rumours. Napoleon\nhad been striding Europe like a Colossus. No one knew what would be\nhis next move on the strategical chessboard. But it was not of him,\nnor of the events connected with him, that John Durnford was thinking\nas he stood before the Chapelle.\n\nHe had heard of the death of a woman whom he had tenderly loved. Years\nago, before he entered the army, they had been sweethearts. Then they\nhad drifted apart; and now he had discovered, quite accidentally,\nthat she had died but two days ago, homeless, friendless. And yet\nnot entirely that. Her last moments had been tended by Sister Anne, a\n_religieuse_, and it was to see her that Jack was waiting outside the\nChapelle.\n\nPresently, the hour of nine was chimed from the surrounding belfries.\nAlmost simultaneously, the door of the Chapelle was opened, and the\n_religieuse_ came out.\n\n\"Pardon me,\" said Jack, approaching her, hat in hand; \"but am I\nspeaking to Sister Anne?\"\n\n[Illustration: \"'AM I SPEAKING TO SISTER ANNE?'\"]\n\n\"Yes, my son.\"\n\n\"You are the lady, are you not, who so charitably befriended Mdlle.\nDenton?\"\n\n\"I but did my duty, my son.\"\n\n\"Ah, if all the world would but interpret duty in the same way! I am\nan old friend of Mdlle. Denton's, and it was only by chance I heard of\nher death. Could you let me see her before--before----\"\n\nJack's voice faltered. He did not complete the sentence.\n\n\"Before she is buried, you would say? I understand,\" said Sister Anne,\nsympathetically. \"Poor child! I thought she hadn't a friend in the\nworld. It seems I was mistaken. Will you follow me?\"\n\nShe took him through a labyrinth of streets, and paused before a\nramshackle old house which had seen and withstood the storms of more\nthan one revolution.\n\n\"You would like to be alone with the dead?\" asked the Sister.\n\n\"If Madame will grant me that favour.\"\n\nShe rang the bell, whispered to the drowsy old _concierge_, and,\nwith a _Benedicite_, was gone. The _concierge_ conducted him up\nthe staircase, pointed to a door, gave him a lighted candle, and\ndescended.\n\nJack opened the door, and as he did so a gust of wind blew out his\nlight and left him in darkness. He had just time, however, to see\nthe white-shrouded figure stretched on the bed in the corner. He\napproached it reverently, and stood by the side of the shroud, with\nthoughts which choked themselves for utterance.\n\n\"Poor, poor Minnie! This, then, is the finish!\"\n\nWhat was that? His moan, he thought, was echoed by another. He quickly\nput the thought from him.\n\nHe put his hand gently forward to feel the face of the dead woman, and\nin doing so it rested upon something warm, palpable. He could almost\nhave shrieked, the transition of feeling was so great--between the\nice-cold rigour he had anticipated, and the warmth of animate life.\nWhat could it mean?\n\nHe had no time for conjecture, for the hand which he had extended\nto the face of the dead was clasped by another hand--the hand of the\nliving.\n\n\"In Heaven's name, who are you?\" demanded Jack.\n\nThere was no answer; then Jack repeated his question in French. This\ntime there came an answer.\n\n\"One--one who loved her, Monsieur! By what right are you here?\n\n\"By as great a right as yours--as one who loved her, too.\"\n\nJack thought he heard a curse between clenched teeth.\n\n\"Love? _Peste!_ What does a cold-blooded Anglais know of love? You\ncome here as a thief in the night.\"\n\n\"Thief!\" Jack exclaimed. \"I suppose you know the meaning of the words\nyou have used?\"\n\n\"_Parbleu!_ How could I do otherwise, since Monsieur himself has\nprovided me with an illustration? Is it the act of an honest man to\nsteal into a chamber? Is it the act of a gentleman to encroach upon\nanother's grief? No; it is the act of a _vauvien_; for it is insult to\nthe living and profanation to the dead.\"\n\nThe man was evidently distraught with grief; so Jack replied calmly,\n\"You talk of profanation to the dead. It would indeed be profanation\nwere I to imitate your language. I am willing to admit that you excel\nin your nice selection of epithets, but I deny your love for the poor\ndead girl lying here by your use of them.\"\n\n[Illustration: \"BEAUTIFUL SHE LOOKED, EVEN IN DEATH.\"]\n\nJack's calmness of utterance, so strongly in contrast to that of the\nstranger, produced some effect upon his hearer. There was a lengthy\npause. Save for the sharp breathing of the two men confronting each\nother, the chamber might have been given up entirely to the dead.\nIt seemed in that pause as though the still form in the shroud were\nlistening for an answer.\n\nAt length the stranger spoke, his voice now tremulous and pathetic:\n\n\"You doubt my love for her? _Eh, bien!_ I loved her as few men could\nhave loved. I have confronted death once, twice this day to see her\ndear, dead face. I have confronted--still confront--what is worse than\ndeath: disgrace and ignominy. Has Monsieur done as much?\"\n\n\"No,\" said Jack, sententiously, touched yet chagrined by the man's\npassion.\n\n\"Until Monsieur has done as much, has he the presumption to say that\nhe has as great a right to stand here as I?\"\n\n\"Presumption!\" cried Jack. \"By whatever right I stand here, I\ncertainly question your right to use such terms to me. But before we\ndiscuss the point further, would it not be as well to have a light?\"\n\nThere was a hasty movement on the part of the figure opposite.\n\n\"If you stir, you are a dead man.\"\n\nThere was a faint ray of light shining through the window, not\nsufficient for Jack to see the person before him, but sufficient to\nsee the cold gleam of steel. It was a sword. This man was a soldier,\nthen, and an enemy. Jack now understood his allusion to the peril he\nhad run in coming there, and admired his bravery. His love for Minnie\nDenton must indeed have been great.\n\n\"You spoke about ignominy just now,\" said Jack. \"I don't know whether\nyour interpretation of the phrase is the same as mine. But a British\nsoldier--for I, too, am a soldier--considers that there is no greater\nignominy than that of being suspected of cowardice. I should be a\ncoward if I cared for your threats. I'm going to get a light.\"\n\n\"Pardon me. You are a brave man. I did wrong to threaten you.\"\nJack heard the sword return to its scabbard. \"Let me appeal to your\nhonour.\"\n\n\"That is an appeal which has never been made to me in vain.\"\n\n\"My visit here has been a secret. I wish it to remain so. This much\nonly I may tell you--that I am an officer in the French army, enjoying\na position of great responsibility and trust. You see the risk I have\nrun.\"\n\nJack started. This man had indeed risked much to see the last of the\nwoman he loved.\n\n\"You say that your visit here is a secret one; and yet you reproached\nme just now with being a thief in the night. I will not retaliate; for\nI too can respect a brave man. I will only say that your confidence\nwill not be betrayed.\"\n\nJack stretched out his hand. It was again clasped by the stranger.\nThey stood thus for a moment, hand in hand, over the dead.\n\nThen the stranger bent, and Jack could hear him whispering terms of\nendearment to ears that could not hear, and pressing kisses upon lips\nthat could not respond.\n\n\"Now, Monsieur, I am going,\" he said, at length. \"I thank you for your\npatience, and will send up the _concierge_ with a light. You will then\nbe able to read this letter. Oblige me by taking it. From it you will\nsee who is the most entitled to her love. It was the last letter she\never wrote. You say you are a soldier? _Eh, bien_, when next we meet,\nMonsieur, it will be in a different place. As we have learned to\nrespect each other, I hope to show that respect in the best way a\nsoldier can--by crossing swords with you. _Jusqu'au revoir!_\"\n\n[Illustration: \"HE WAS STOPPED AT THE ENTRANCE BY A GENDARME.\"]\n\n\"_Au revoir_, Monsieur!\"\n\nIn a moment or two Jack heard the stranger go out, and the _concierge_\ncame stumbling up with a light. Jack took it from him, and gazed upon\nthe face of Minnie Denton. Beautiful she looked, even in death. The\npain and agony of the last struggle had gone and left the features\nplacid, as one in peaceful sleep.\n\nYears ago he had loved her deeply, tenderly, and she had returned his\nlove. Then they had quarrelled. The breach between them had widened,\nand in a fit of desperation he enlisted. Europe was at the time one\ngreat battlefield, and Jack was immediately sent on active service. So\nhe had altogether lost sight of his old love.\n\nHe had been with Wellington in the Peninsula, and after serving with\na bravery which had gained him the eulogiums of his general, had been\ndrafted with his regiment to Brussels just prior to the time when\nNapoleon escaped from Elba.\n\nThere had not been much time to think of love while these stirring\nevents were transpiring, but the news of his old sweetheart's death,\nin the very city in which he was stationed, had touched a tender\nchord.\n\nJack mused mournfully upon the past as he looked down on the still,\nsilent face. She had been fickle; yet had not he? What would their\nfate have been had they not quarrelled? Would it have been widely\ndifferent? Perchance she would have been a happy mother; he, a happy\nfather; or they might have been utterly miserable.\n\nWhatever Fate might or might not have had in store for them in other\ncircumstances, it was galling to think that her last thoughts had been\nof this stranger--a Frenchman and an enemy.\n\nBut was it true? There could, alas! be little doubt of it, for had\nnot the Frenchman left with him the best--rather he would say, the\nworst--of all testimony: her own letter? What stronger evidence of her\nfickleness could there be than that?\n\nJack turned to the light and looked at the letter which had been\nplaced in his hand.\n\nGood heavens! What was this?\n\n[Illustration: \"THE DUKE AND HE HELD A WHISPERED CONVERSATION.\"]\n\nIt was no love letter, but a document folded in the shape of a letter.\nJack looked at it eagerly, and read it through not once, but twice,\nand thrice.\n\nIt was Napoleon's directions to his generals, signed by the Emperor\nhimself, containing specific instructions respecting the forthcoming\nbattle against the allied forces. The one line that burnt itself into\nJack's brain was that an advance was to be made upon Quatre Bras early\nthe next morning. Wellington had no suspicion that the advance was to\nbe made so soon; for Jack knew that he and many of the officers were\nat a ball given by the Duchess of Richmond in the Grande Place.\n\nThe Frenchman had said that he was an officer, enjoying a position of\ngreat responsibility and trust. Jack saw it all. He had given him this\ndocument instead of, as he supposed, the dead woman's letter. Then\ncame to Jack a question of honour. Had he the right to use this\ninformation?\n\nHe did not pause long to consider the point. The safety of his country\nwas at stake. That was enough. The old maxim, \"All is fair in love and\nwar,\" had now a double signification. So Jack hurried along with all\npossible speed to the Grande Place.\n\nThe ball was at its height. The strains of music, the laughter of the\ndancers, came to Jack as he neared the Duchess's residence.\n\nHe was stopped at the entrance to the hall by a gendarme.\n\n\"Est ce que vous avez votre billet, Monsieur?\"\n\n\"Non.\"\n\n\"Alors je ne puis pas vous admettre.\"\n\nJack explained it was of the utmost importance that he should see the\nDuke of Wellington, and at length he was ushered up the staircase into\nan ante-room, while an attendant went in search of the Duke.\n\nJack had a full view of the ballroom as he waited. As in a\nkaleidoscope he saw the gleam of many uniforms, fair faces, white\nshoulders, slender graceful forms--alternate flashes of scarlet and\nwhite--as couple after couple whirled by in the mazy waltz. Presently\nfrom out the maze came one martial figure which Jack knew well. There\ncould be no mistaking that stern, immobile face, the tightly pressed\nlips, the prominent Roman nose. It was the Iron Duke!\n\n\"Well, sir, you wish to see me?\" was his laconic greeting.\n\n\"Yes, General, on a matter of life and death. Read that.\"\n\nHe handed the Duke the document he had received from the Frenchman.\nHis searching eyes had grasped its contents in a moment; yet he\nbetrayed no excitement or astonishment.\n\n\"Where did you get this?\" he calmly asked.\n\nJack briefly explained the circumstances under which he had obtained\npossession of the document. The Duke turned to his aide-de-camp.\n\n\"Tell General Picton I wish to see him immediately.\"\n\nIn a minute or two the aide-de-camp returned with the General.\n\n[Illustration: \"THE FRENCHMAN WAS A VERY SKILFUL SWORDSMAN, JACK\nEQUALLY SO.\"]\n\n\"Napoleon left Frasne this morning,\" said Wellington. \"The Prussians\nhave fallen back. Ziethen has been beaten. Napoleon is marching now\nupon Quatre Bras. Read that.\"\n\nPicton read the document, and studied the plan. Then the Duke and he\nheld a whispered consultation. The aide-de-camp returned again and\nagain to the ballroom, and Jack saw the officers stealing away one by\none. Then the Duke turned to Jack:\n\n\"You have done well in bringing me this document. I will not forget\nit. Prepare to join your regiment.\"\n\nJack saluted, and passed into the street. As he did so, the bell\nof the Hotel de Ville boomed one. Simultaneously could be heard the\ntramp, tramp of the Highland regiments as they defiled into the Grande\nPlace.\n\nThe British forces were preparing to meet the enemy.\n\nIn the morning the two armies found themselves ranged in battle array\nopposite each other.\n\nThen came the opening struggle at Quatre Bras, followed by the yet\nmore memorable death-wrestle of nations at Waterloo.\n\nNo need to repeat the incidents of that famous day--Picton's bravery,\nNapoleon's strategy, Wellington's tenacity of purpose, the glorious\nstand around the farmhouse of La Haye Sainte.\n\nNapoleon charged again and again the immovable British centre.\nThe destinies of nations hung in the balance, and it was not until\nWellington gave the famous command--\"Up, Guards, and at them!\"--that\nthe balance turned to the side of victory.\n\nJack was foremost in the charge, and as his column swept down the\n<DW72>, he heard a voice cry out to the fleeing Frenchmen:\n\n\"Arr\u00eatez! Arr\u00eatez!\"\n\nHe recognised the voice as that of the man whom he had met at the\nshroud of Minnie Denton. Though the interview had only been brief, he\ncould recall every accent. The voice was one he was never likely to\nforget.\n\nFinding his efforts to check the retreating soldiers unavailing, the\nofficer turned and faced the pursuing column.\n\nJack was the first to reach him. The rest of the column swept on,\nleaving the two face to face, sword to sword.\n\n\"Your prophecy has come true, Monsieur,\" said Jack. \"We have met\nagain--a little sooner probably than you anticipated.\"\n\n\"Ah! it is you,\" said the officer. \"Truly pleased to see you. We are\ndestined, it seems, to be rivals till the last. I beat you in love,\nyou will admit; and I shall do my best to----\"\n\nHe did not finish the sentence. Steel met steel; the sparks flew from\nthe quivering blades. The Frenchman was a very skilful swordsman, Jack\nequally so. Jack at last with an adroit parry sent the sword from his\nadversary's hand.\n\nJack was stooping to pick up the weapon when a stray shot hit the\nFrenchman in the breast. He fell with a groan from his horse. Jack\nquickly dismounted, and knelt by his side.\n\n[Illustration: \"JACK QUICKLY DISMOUNTED, AND KNELT BY HIS SIDE.\"]\n\n\"Ah, Monsieur, it is very good of you,\" he gasped, as Jack raised his\nhead; \"it is very, very good of you; but I am dying. The fortune of\nlove was with me; the fortune of war is with you.\"\n\nJack strove to staunch the blood that was gushing from the wound, but\nin vain. The wound was a mortal one.\n\n\"It is useless,\" gasped the Frenchman. \"Nothing can be done, and I\nwould rather die than be a prisoner. You are my enemy, but you are a\ngentleman. One thing I would ask you. Minnie--Mdlle. Denton--is to be\nburied to-morrow, Bury us in one grave. It is all I ask.\"\n\nJack promised. He felt a great pressure from the hand resting within\nhis; then the head fell back in his arms. A brave soldier had fought\nhis last battle.\n\nSimultaneously there rose on the air a great shout. It was the shout\nof the conquering army announcing that the battle of Waterloo had been\nfought and won.\n\n\n\n\nSERMONS WITHOUT WORDS\n\nA MARVELLOUS PERFORMANCE IN DUMB SHOW.\n\n\nNot a quarter of a mile from the Marble Arch, on the left side of\nOxford Street (No. 419, Oxford Street, as a matter of fact), looking\ntowards the Park, there stands a dull, unpretentious, red brick\nedifice, so unpretentious indeed that in spite of its ecclesiastical\nappearance it is unnoticed by the majority of passers by.\n\nThe bulk of the teeming thousands who pride themselves that they know\ntheir London are ignorant of its whereabouts, nor are the countless\nlegions who daily pass through the busy thoroughfare better informed.\n\nNor is it surprising; for there is little but a tiny cross on the\ncoping stone, and a dingy notice board behind dingier railings,\nto mark one of the most interesting buildings in all London--St.\nSaviour's Church, the cathedral of London's 2,000 deaf and dumb.\n\nHere Sunday by Sunday the silent poor and the silent rich worship\ntogether. Outside, the roll of traffic merges into one long dull roar\nthat may distract the thoughts of worshippers in other churches, but\nto the congregation of St. Saviour's makes no difference. They cannot\nhear it.\n\n[Illustration: JUSTICE (WEIGHING WITH SCALES).]\n\nI had heard much of the Rev. F. W. G. Gilby's wonderful method of\npreaching to his people, how he has become thoroughly conversant not\nonly with the old-fashioned finger spelling familiar to those who have\nwatched the conversation of the deaf and dumb, but can also by means\nof gesture and acting make use of a system of preaching richer in\nsuggestion, wider in range, and infinitely more effective in its scope\nand power of riveting the interest of his flock.\n\nAccordingly, one wet Thursday evening a short time ago, I made one of\nthe congregation at evening service, curious to take part in such a\nservice myself. I am never likely to forget the impression that\nquiet service made on me, nor to relieve my mind of the feeling of\noverwhelming depression at the realisation that this little crowd of\nafflicted people, miserably and unutterably poor in the majority of\ncases, was living, moving, and breathing in our very midst, helpless\nyet happy, willing and intelligent, yet almost entirely dependent\non this one enthusiastic, unselfish man for their comfort--not only\nspiritual, but in many cases, as I discovered, material as well.\n\nI have not the space, nor is it in my province in this short article,\nto describe or appeal on behalf of the needs of this institution, but\nthe interested ones should see for themselves, and if within their\npower, help.\n\nExcepting the chaplain's wife, herself an expert follower of her\nhusband's method, I was the first to arrive. The lights were low,\nand there was nothing about the dim church save the absence of choir\nstalls and pulpit to suggest the unusual nature of its mission.\n\nPresently a distant door opened, a shuffling step dragged along the\naisle. The first member of the congregation took his rags with him\ninto a front seat. He was a shoeblack down on his luck, but nobody\nturned him out. In Mr. Gilby's flock all are equals, all are friends\nin their common adversity. The first seats are for the first comers.\n\nA few moments later and the congregation was nearly complete. Here and\nthere one caught a flash of recognition between two friends, then up\nwent two pairs of hands flashing white in the dim light as an animated\nconversation took place across the church. By the time the church was\nhalf full a whole volley of chatter was playing round; everywhere\nthe darkness was alive with flickering, speaking hands, and faces\nvibrating with expressive gesture. It was an odd scene, weird\nand uncanny to the hearing visitor who sat misunderstood and not\nunderstanding amid the silent throng.\n\n[Illustration: 1. \"DEAR]\n\n[Illustration: 2. DEAF AND DUMB]\n\n[Illustration: 3. FRIENDS]\n\n[Illustration: 4. WE]\n\n[Illustration: 5. WISH YOU]\n\nIn a few moments the chaplain, attired in the usual canonicals,\nappeared, and the service commenced.\n\nThroughout the proceedings there was no sound but the dull roar of\npassing omnibuses and cabs outside. Not a hymn, not a word, only that\nindescribable hush, almost unnerving to one strange to the scene.\nYet throughout the service, in the prayers, in the sermon, not an eye\nstrayed from the slight figure talking in a language of his own at the\nlittle desk on the altar steps.\n\nAt first the ghostly reality of this strange sermon dispelled all\nother thoughts. It did not seem comprehensible that there could be any\nconnection between the chaplain and the attentive congregation, but\nhere and there one could catch a reflection of one of his gestures on\nthe face of an intent watcher.\n\nThen a more than usually familiar passage was signalled, and a broad\nintelligent smile passed swiftly across the faces of the congregation,\nand they nodded and looked towards each other comprehendingly.\n\nThen for the first time one realised that the flying fingers playing\nrapidly above the reading desk, flickering now high and now low,\nlike the figures in a kinetoscope picture, meant something; that the\ngestures, the graceful swaying of the body, the marvellous play of\nthe features, all had their meaning; that each little movement was\nintelligible to the watchers as the word of a spoken sermon, and\ninfinitely more expressive.\n\nAs the utter novelty of the scene became more familiar, I found myself\ntrying to interpret the drift of the sermon, and it was little short\nof marvellous how intelligible a great number of the gestures were,\neven to one untrained and unused to sign language.\n\nThe acting and gestures in many sentences were so obvious, that it was\nalmost as though the words were rather the equivalents of the signs\nthan _vice vers\u00e2_. It was, indeed, an astonishing revelation of the\npossibilities of human expression. When the faculty is combined with\na system of word signs intelligible to the merest child, it will be\nunderstood how much may be done in this way, without recourse to the\nmore tedious method of spelling out each word separately, although\nthis is necessary where the sign imagery is so subtle as only to\nappeal to highly cultivated imaginations.\n\nAs a matter of fact, Mr. Gilby has a marvellous faculty in this\ndirection, that has been fostered and perfected by life-long study. So\nmuch is this the case, indeed, that I doubt if he could be equalled in\nthis direction by any one of our greatest actors.\n\nPresently the service was at an end. There was a little desultory\nsilent conversation, and the congregation dispersed, just as it\ncame, without a sound. Three or four stragglers, clean and\nintelligent-looking, but obviously poor, remained behind, and\npresently made their way up the altar steps, and into the tiny vestry.\n\n[Illustration: 6. JOINED TOGETHER]\n\n[Illustration: 7. IN ONE HEART AND ONE MIND,]\n\n[Illustration: 8. IMITATING]\n\n[Illustration: 9. CHRIST]\n\n[Illustration: 10. UNTIL]\n\n[Illustration: 11. THE CHURCH]\n\n[Illustration: 12. IS COMPLETE.\"]\n\nI followed them, and when each had stated his different wants and\ndifficulties, and received relief and comfort, I persuaded Mr. Gilby\nto assist me in the preparation of this article, illustrative of his\nremarkable work.\n\nIt will be readily granted by those who examine our interesting series\nof photographs, that my demands on his good nature were by no means\nmoderate. Those who object to being photographed almost as much as\nthey dislike the necessary visit to the dentist--and Mr. Gilby is one\nof these--will appreciate Mr. Gilby's feelings when our photographer\ndesired not only one siting, but a dozen. However, Mr. Gilby will\nbe more than compensated if this article is the means of attracting\npublic attention to the afflicted ones that are his especial charge.\n\nThe most important photographs we give are those that illustrate a\nmessage that I have prevailed on Mr. Gilby to issue through these\npages to the deaf and dumb of the British Empire. The message is\nnecessarily brief and short: as it is we are obliged to print twelve\nphotographs in order to do it justice. The exact message is as\nfollows:--\n\n\"Dear deaf and dumb friends, we wish you joined together in one heart\nand one mind, imitating Christ until the Church is complete.\"\n\nThe appropriateness of many of Mr. Gilby's signs becomes immediately\napparent on glancing at the photographs, but some are not as clear as\nothers. \"Deaf and dumb\" is signified by rapidly touching the mouth and\nthe ear; \"friend,\" by shaking hands with oneself; \"we,\" by pointing at\noneself, at the persons addressed, and vaguely to the left to indicate\npeople in general; and \"joined together,\" by opening the hands, and\nthen bringing them together closed. A most interesting sign is that\nrepresenting \"Christ,\" where a finger is pressed into the palm of each\nhand in rapid succession, as if to indicate the piercings of the nails\nof the cross; and scarcely less remarkable is that which denotes the\n\"Church\"--the motions being those of one ringing church bells!\n\nExcellent as these photographs are, they convey but a slight\nimpression of the effect produced by a sermon in Mr. Gilby's\ngesture-language. It must be understood that his is no laborious art.\nDistinct and picturesque as Mr. Gilby's motions are, they succeed\none another with the rapidity of words penned by an expert shorthand\nwriter. On one occasion, indeed, Canon Wilberforce--one of the most\nfiery orators of the day--addressed our deaf and dumb congregation,\nand Mr. Gilby, who stood by the side of the eloquent Canon to\ninterpret the discourse, experienced no difficulty in keeping level\nwith him. It will thus be seen that, as practised by an expert, the\nart of gesture-language leaves little room for improvement. As a\nmatter of fact, the sentence given above would be \"signed\" by Mr.\nGilby, in the course of an ordinary pulpit address, in about three\nseconds.\n\n[Illustration: \"KNOWLEDGE.\"]\n\nNeedless to say those signs that are to be expressive of themselves\nrequire to be of the most suggestive nature in order to be readily\nunderstood, and it is in the invention of these that the teacher\nof the deaf and dumb may find a great field for the exercise of his\ningenuity.\n\nIn a great number of cases there are signs which are universally\naccepted and understood by deaf mutes the world over. On the other\nhand, each school has its own special gestures, equally expressive but\npeculiar to itself, and in the department of versatility of gesture\nMr. Gilby is second to none. In fact, I have seen him express an idea\nin half a dozen ways, and each one of them could have been interpreted\nwith ease by a half wit.\n\nIn the majority of cases the photographs illustrating the gestures\nhave been taken in an entirely novel way.\n\nBy making several exposures on one plate we have sought to illustrate\nthe various movements composing those gestures which are of a\ncomposite description. Where a word or idea is expressed by a\nsingle sign, this is, of course, unnecessary. One photograph is\nall-sufficient.\n\nIt may possibly occur to many that there might be considerable\ndifficulty in conveying a difference of expression in the same idea;\nthat is to say, the difference indicated in spoken language by a mere\nvariation of inflection in the voice. As a matter of fact, the sign\nlanguage is even more expressive in this particular. An excellent\nillustration of this is given in two photographs on page 71. Both\ngestures express the same idea--a parting between two friends. In\nthe first of the two you have the parting in which there is a little\nsadness. The idea of separation is conveyed by the hand leading the\nother away. That it is a matter of regret is shown by the expression\nof the face and the nod of the head.\n\nIn the second photograph of the pair you still have the parting. This\ntime, however, it is a humorous rendering which might be used with\nhappy felicity at the conclusion of a platform speech, where the\nspeaker wished to convey a sort of \"Well, I'm sorry to go, but I\nmust,\" notion. Here the separation is humorously expressed by the\nsuggestion of brute force brought to bear on the speaker's collar.\n\n[Illustration: \"CONVERSATION\" (OPENING AND SHUTTING FINGERS AS THE\nMOUTHS OF BIRDS CHATTERING).]\n\nIn similar fashion many inflections may be given to the same idea, and\nwith the indispensable assistance of facial expression the elements of\nHope, Tragedy, Comedy, Fear, are introduced.\n\nThe extraordinary mobility of Mr. Gilby's features must prove of\nthe utmost service to him. With a scarcely perceptible quiver of the\nfeatures his face expresses alternate Tragedy and Humour. So much so,\nindeed, that one feels that he is throwing his whole nature into each\nand every fleeting gesture. And this is probably the secret of his\nsuccess, for to this pale-faced, highly strung man the cause of the\ndeaf mute is as life itself.\n\nThe education of the deaf and dumb is necessarily limited, though\nthe general impression that they are deficient in mental capacity is\nentirely erroneous. On the contrary, brightness, intelligence, and,\ncuriously enough, content are their chief characteristics. Such\neducational limitations as exist are an unavoidable result of the\ntedious and trying system that must be gone through in order to give a\ndeaf and dumb child even the rudiments of an education.\n\nIf you wish to teach such an one what a cow is and how to spell the\nword, there is only one method, and that is to place a picture of a\ncow before it and write the word on paper till it comprehends that the\nletters C O W represent the name of the animal in question.\n\nIt will be seen, therefore, that only those who have enjoyed very\nexceptional educational advantages are in a position to appreciate\nsome of the deeper abstract ideas of philosophy and the sciences.\n\n[Illustration: \"PARTING.\"]\n\nAbstract ideas are difficult of adequate expression, therefore, not\nbecause they cannot be suggested by the sign language, but by reason\nof the reader's own inability to comprehend their significance. Some\nof the more general ideas of an abstract nature are, however,\ntaught with comparative ease. We give two examples. Both are almost\nself-explanatory. The first (on p. 70) expresses Knowledge, or Wisdom;\nthe second (p. 67) is a sign demonstrative of Justice. Nothing could\nbe clearer, of course. It is simply a mimetic illustration of the\nsymbolical picture of Justice blind, and so impartial, holding the\nscales. The right hand is first placed in the position of holding the\nscales, and is then rapidly brought down on a level with the other,\nthus picturing the scales.\n\n[Illustration: \"STUPIDITY\" (THE ASS).]\n\nTwo other pictures illustrate signs of a peculiarly expressive\nnature. Nobody will want to be told what a deaf and dumb man means\nwho describes you with the sign shown below. The lower picture on the\npreceding page is Mr. Gilby's way of expressing the fact that he has\nbeen holding a conversation with someone. It will be noticed that\nthere are two positions of the forefinger, which are intended to\nindicate that this finger is snapped rapidly against the thumb. It is\nmore or less a humorous way of expressing the idea, and as actually\nillustrated by Mr. Gilby is exceedingly comic. A more sedate way of\nexpressing the idea would be to hold the hands in the same position,\nbut to draw them slowly apart and towards each other.\n\nI feel that no article on this subject would be complete without some\nspecial illustration of the enormous part pure facial expression plays\nin Mr. Gilby's peculiar method. Indeed it is in his case a fine art,\nand must represent an enormous increase in the effectiveness of his\naddresses and lectures, and consequently in the happiness and comfort\nthey give his silent audiences.\n\n[Illustration: \"LET ME THINK.\"]\n\n[Illustration: \"HAD IT ON THE TIP OF MY TONGUE.\"]\n\n[Illustration: \"NOW, WHAT WAS IT!\"]\n\n[Illustration: \"AH! WAIT!\"]\n\n[Illustration: \"NO! I GIVE IT UP.\"]\n\nFive photographs illustrative of a little lapse of memory explain\nbetter than any words what I mean. Without strict attention to\ngrammar, I will call this Forgetfulness, More Forgetfulness, Most\nForgetfulness, Still more Forgetful, Forgotten.\n\nThis series will be an object lesson on the debt we all owe that\nfleeting, intangible thing we call Human Expression.\n\nNo article is complete without its story. Mr. Gilby is full of\nstories, but I have only space for one, and that looks weak on paper\nwhen I remember how inimitably it was acted when he gave it to me.\n\nSome time since he was due to give a short address in the schoolroom\nunder the church. The Vicarage adjoins the sacred edifice, and he\ntherefore decided it was unnecessary to change the light indoor shoes\nhe was wearing in his study. Accordingly he wore them on the platform\ndownstairs and commenced his address.\n\nA few moments later he happened to quote the text in which the\nwords, \"I cast my shoe,\" occur. Now obviously the best sign for the\nexpression of this idea was a gentle kick. Mr. Gilby gave it, but the\ndemonstration proved much more literal than he had intended, for a\nsecond later his shoe flew through the air and dropped in the midst of\nan immensely amused audience.\n\nThe story emphasises Mr. Gilby's belief that humour is, and always\nshould be, a valuable ally in the higher education of the deaf and\ndumb.\n\nIt is an infallible means of securing that closer understanding and\nsympathy between teacher and pupil which raises teaching from the dull\nmechanical level of routine to a fine art.\n\nHumour in his case is a natural gift--perhaps one of his greatest.\nIt peeps out unbidden in his sermons. It renders his lectures and\naddresses delightful to deaf, dumb, and hearing visitors alike, and\none cannot but feel that in all the many branches of his work it turns\nsadness into sunshine and depression into an unfaltering hope for the\nfuture.\n\nThe scope of this article on St. Saviour's Church does not permit of\nour entering upon the hotly-contested methods of educating the deaf,\nwhether by the lips or by manual signs or spelling. Mr. Gilby is one\nof the Government Inspectors of Schools, and, having been born of\ndeaf parents, and brought up amongst the afflicted, may reasonably be\npresumed to have a right judgment in these matters. For himself, he is\nan ardent upholder of the Combined System--often known as the American\nway of instructing the deaf. He differs in toto from any who may think\nthat Missions to the Deaf are unnecessary, for by learning speech they\nare raised to the same level as their more fortunate brethren who can\nhear.\n\nIn conclusion I cannot repay Mr. Gilby's courtesy and kind assistance\nin the preparation of this article better than by repeating the wish I\nfeel to be nearest his heart:--\n\nIf you have an opportunity, help the Deaf and Dumb.\n\nALFRED ARKAS.\n\n\n\n\nTHE CHANCELLOR'S WARD.\n\nPERHAPS MR. RICHARD MARSH'S BEST SHORT STORY.\n\n_Illustrated by F. H. Townsend_.\n\n\nI.\n\nOne really ought to write, She married him, not, He married her.\n\n\"The simple question is, my dear Tommy, are you going to take me or\nleave me?\"\n\nThis was in Hyde Park. They were seated on one of those seats which\nare in front of the police station. Neither of them ought to have been\nthere. Which, of course, was one of the reasons why they were. Mr.\nStanham turned his eyeglass full upon Miss Cullen. Perhaps he thought\nthat that was sufficient answer. Anyhow, she went on--\n\n\"In other words, are you going to marry me, or are you not?\"\n\n[Illustration: \"'ARE YOU GOING TO MARRY ME, OR ARE YOU NOT?'\"]\n\n\"I am; Gad, I should rather hope so. I say, don't be too hard upon a\nfellow, Frank.\"\n\n\"Call me Fanny, don't call me Frank! Don't you know that my name is\nFrances, sir, which has absolutely no connection with Frank!\"\n\n\"That's all right, old man.\"\n\nThat's what Mr. Stanham murmured. Extraordinary how some men do talk\nto women nowadays, even to the women whom they love!\n\n\"Then, if you do intend to marry me, Mr. Thomas Stanham, you'll be so\ngood as to do so on Thursday morning next before noon.\"\n\nMr. Stanham began to scratch the gravel with his stick.\n\n\"And get seven years' penal.\"\n\n\"Stuff! They don't give you penal servitude for marrying wards in\nChancery. It's contempt of court.\"\n\n\"Yes, I know. Have to wash out your cell at Holloway, and stand at\n'attention,' with your hat off, while the governor cuts you dead.\"\n\n\"Then perhaps you will be so good as to tell me what it is that you\ndo propose to do. Do you imagine that you are the sort of person the\ncourt of Chancery will ever allow to marry me?\"\n\n\"Haven't so much imagination, my dear Frank.\"\n\n\"Call me Fanny, not Frank! You are not to call me Frank. Then do you\nsuppose that I'm the sort of girl who's willing to wait, and not\nmarry her sweetheart, until she's twenty-five? Because if you suppose\nanything of that kind, we must be perfect strangers.\"\n\n\"It's very good of you, I'm sure.\"\n\n\"Oh, I daresay. You don't love me that much.\" Miss Cullen flicked her\nparasol. \"Because a horrid old uncle chooses to say that I'm to be a\nward of the court until I'm five and twenty, am I to be a spinster all\nmy life? If you loved me the least little bit, you'd invite the Lord\nChancellor to come and see you marry me in the middle of Hyde Park,\neven if, directly the deed was done, he had your head cut off on Tower\nHill.\"\n\n\"Thanks, dear boy.\"\n\nOf course he married her. On the morning of the specified Thursday she\nwent out for a stroll, and he went out for a stroll, and they met at\nthe registrar's, and, as she put it, the deed was done. And, when the\ndeed was done, she went home to lunch, and he went, not home to lunch,\nbut to a private place, where he could swear. Now here they were, both\nof them, at Tuttenham. They encountered each other on the doorstep.\nShe said, \"How do you do, Mr. Stanham?\" And he said, \"How do you do,\nMiss Cullen?\"\n\n[Illustration: \"THEY ENCOUNTERED EACH OTHER ON THE DOORSTEP.\"]\n\n\"Nice way in which to have to greet your own wife,\" he told himself,\nhaving reached the comparatively safe solitude of his own apartment.\n\nThen the Duke got him into his own particular smoking-room. The Duke\nwas in an armchair. Mr. Stanham stood before the fireplace with his\nhands in his pockets. The talk wandered from Dan to Beersheba. Then,\na good deal _\u00e0 propos des bottes_, the Duke dropped what he evidently\nintended to be taken as a hint.\n\n\"If you take my advice, young man, you'll keep clear of Frances\nCullen. She's here.\"\n\nMr. Stanham winced.\n\n\"Is she? Yes. I know. I met her on the steps.\"\n\n\"Did you!\" The Duke eyed him. He, not improbably, had observed the\nwince. \"Warnings are issued all along that coast. Steer clear.\"\n\n\"What do you think they'd do to a man if he were to marry her?\"\n\n\"Do to him! Tommy! I hope you're not meditating such a crime. She's\nnot an ordinary ward of the court, any more than she's an ordinary\nmillionaire.\"\n\n\"So I suppose.\"\n\n\"You had a little run with her in town. Everybody had their eyes on\nyou, as you're aware. And when the Duchess told me she was coming,\nI'd half a mind to write and put you off--fact! This is not a house\nin which even tacit encouragement can be offered to a dalliance with\ncrime. Not\"--the Duke puffed at his pipe--\"not that she's half a bad\nsort of girl. She's clever. Very pretty. And she's got a way about her\nwhich plays havoc with a man.\"\n\n\"Much obliged to you, I'm sure.\"\n\n\"What do you mean?\"\n\n\"For saying a good word for my wife.\"\n\n\"Your wife?\"\n\n\"Mrs. Thomas Stanham--_n\u00e9e_ Cullen.\"\n\n\"Tommy!--You don't mean it!\"\n\n\"You can bet your pile I do,--and then safely go one better. I've got\na copy of the marriage certificate in my pocket, and I rather fancy\nthat she's got the original document in hers.\"\n\n\"You--young blackguard!\"\n\n\"Sort of cousin of yours, aint I, Datchet? It's all in the family, you\nknow. Blackguard, and all.\"\n\n\"How did you do it?--And when?--And who knows?\"\n\n\"Only you and me, and the lady. That's what's weighing on my mind.\nWhat's the good of having a wife, if she ain't your wife--or, at any\nrate, if you daren't say that she's your wife, for the life of you?\"\n\nThe Duke suddenly rose from his seat. He seemed to be in a state of\nactual agitation.\n\n\"Tommy, do you know that the Chancellor is coming here?\"\n\n\"Who?\"\n\n\"The Lord Chancellor. The carriage went to meet him an hour ago. I\nexpect him every moment.\"\n\nMr. Stanham looked a trifle blank.\n\n\"I didn't know the ministry was formed.\"\n\n\"It's formed, but it's not announced; Triggs is to be the Chancellor.\"\n\n\"And what sort of gentleman may Triggs be, when he's at home?\"\n\n[Illustration: \"'YOU--YOUNG BLACKGUARD!'\"]\n\n\"Sir Tristram? Well!\" The Duke was walking up and down the room. He\nappeared to be reflecting. \"He's rather a queer card, Triggs is. He's\nbeen a bit of a wildish character in his time--and they do say that\nhis time's not long gone. He has a temper of his own--a nasty one.\"\nPausing, the Duke fixedly regarded Mr. Stanham. \"I should say that\nwhen Triggs learns what you have done, he will clap you into gaol, and\nkeep you there, at any rate until Miss Cullen ceases to be a ward of\nthe court.\"\n\nMr. Stanham's countenance wore a look of dire consternation.\n\n\"No! She's to be a ward until she's twenty-five, and she's not yet\ntwenty-two.\"\n\n\"Then, in that case, I should say that, at the very least, you are in\nfor three good years of prison. My advice to you is----\"\n\nThe Duke's advice remained unuttered. Just at that moment the door was\nopened. A servant ushered in a new-comer.\n\n\"Sir Tristram Triggs.\"\n\nThe Duke, striding forward, held out both his hands.\n\n\"Sir Tristram!--And how long is it to be Sir Tristram?\"\n\nThe other shrugged his shoulders.\n\n\"For a few hours, more or less, I suppose. I don't know much about\nthis kind of thing. I daresay I shall know more about it when I've\ndone.\"\n\n\"When you've done? May that not be for many and many a year! Allow me\nto introduce to you a friend of mine,--Mr. Thomas Stanham.\"\n\nSir Tristram turned. For the first time, he appeared to notice Mr.\nStanham.\n\nPhysically, the new, great man was short, and inclined to ponderosity.\nThe entire absence of hair upon his face served to accentuate its\npeculiar characteristics. It was a square face,--and, in particular,\nthe jaw was square. His big eyes looked from under a penthouse formed\nby his over-hanging brows. As one looked at him, one instinctively\nfelt that this was a man whom it would be safer to have as a friend\nthan an enemy. As he turned, a faint smile seemed to be struggling\ninto existence about the corners of his great mouth. But, directly\nhis glance alighted upon Mr. Stanham, that smile vanished into the\n_ewigkeit_. He looked at him very much as a bull-terrier might look\nat a rat. And he said, in a tone of voice which seemed fraught with\ncurious significance--\n\n\"I have had the pleasure of meeting this gentleman before.\"\n\nOn his part, Mr. Stanham regarded Sir Tristram with a supercilious air\nwhich, perhaps unconsciously to himself, was only too frequently seen\nupon his face,--as if Sir Tristram were an inferior thing.\n\n\"I'd no idea that your name was Triggs.\"\n\nThe Duke, standing behind Sir Tristram, clenched his fists, and glared\nat Mr. Stanham as if he would like to have knocked him down.\n\nIt happened, shortly afterwards, that Miss Cullen left her bedroom to\ncome downstairs. As she went along the corridor she met a gentleman\nwho was being conducted by a servant, probably, to his own apartment.\nThe gentleman was Sir Tristram Triggs. When Sir Tristram saw Miss\nCullen, and Miss Cullen saw Sir Tristram, they both of them stopped\nshort. The great man's complexion was, normally, of a ruddy hue. At\nsight of the lady he turned the colour of a beetroot, boiled. She\ndrew herself up to the full capacity of her inches. And she uttered a\nsingle monosyllable.\n\n\"You!\"\n\n[Illustration: \"'YOU!'\"]\n\nThat was all she said--then went sweeping on.\n\n\"That horrid man!--He here!--To think of it!--If I'd only known that\nhe was coming, I do believe, in spite of Tommy, that I'd have stayed\naway.\"\n\nAt the foot of the stairs Miss Cullen encountered Mr. Stanham. That\ngentleman had, as he was wont to have, his hands in his pockets. Also,\nas he was not wont to have, he had a face as long as his arm.\n\n\"I say, Frank, old man, isn't there somewhere where I can have a word\nor two with you on the strict 'Q.T.'?\"\n\n\"Certainly--the library. There's never a soul in there.\"\n\nOne would not like to libel Tuttenham so far as to say, with Miss\nCullen, that the only tenants the library ever had were the books.\nBut, on that occasion, it did chance that the pair had the whole place\nto themselves. Mr. Stanham perched himself on a corner of the table,\nstill with his hands in his pockets.\n\n\"There's going to be a pretty kettle of fish, dear boy.\"\n\nThat was what the gentleman observed.\n\n\"My dear child, what do you mean? What is the matter?\"\n\n\"The Lord Chancellor's here.\"\n\n\"No!--How do you know?\"\n\n\"Datchet just introduced me to him.\"\n\n\"Oh, Tommy, I say, what fun!\"\n\nWith a little laugh, the lady clapped her hands. She appeared to be\ngifted with a keener eye for comedy than Mr. Stanham.\n\n\"I don't know what you call fun. It happens that the new Lord\nChancellor is a man who, I have good reason to believe, would give a\ntidy trifle for a chance of getting his knife into me.\"\n\n\"Whatever for?\"\n\n\"I'll tell you the story. Last year, when I was at Canterstone for the\nshooting, I was placed next to a man whom I had never seen in my life,\nand whom I never wanted to see in my life again. What Charlie asked\nhim for, beats me. I believe, if he knew one end of a gun from the\nother, it was as much as he did know. I doubt if there ever was his\nditto as a shot. I wiped his eye over and over again. I kept on doing\nit. I couldn't help it--I had to. He never hit a bird. But he didn't\nlike it, any the more for that. We had something like a row before the\nday was over. I fancy that I said something about a barber's clerk.\nAnyhow, I know I walked off there and then.\"\n\n\"You nice, agreeable child! It's my opinion that all you men are the\nsame when you are shooting--missing links. And, pray, what has this\npleasant little sidelight on the sweetness of your disposition got to\ndo with the new Lord Chancellor?\"\n\n\"Only this,--the new Lord Chancellor's the man I called a barber's\nclerk.\"\n\n\"Tommy! How horrible!\"\n\n\"It does seem pretty lively. You should have seen how he looked at\nme when Datchet just now introduced us. Unless I am mistaken in the\ngentleman, when this little affair of our's leaks out, and I'm brought\nup in front of him, and he sees who I am, he'll straightway consign me\nto the deepest dungeon, and keep me there, at any rate as long as he's\nLord Chancellor. It's only a cheerful little prophecy of mine. But you\nmark my words, and see.\"\n\n\"My poor, dear boy! Whatever shall we do?\"\n\n\"There's one thing I should like to do, and chance it;--I should like\nto kick Sir Tristram Triggs!\"\n\n\"Kick who? Sir Tristram Triggs! Tommy! Why would you like to kick Sir\nTristram Triggs?\"\n\n\"That's the beggar's name.\"\n\n\"The beggar's name? Can it be that Sir Tristram Triggs is the new\nLord Chancellor?\" She threw out her arms, with a gesture of burlesque\nmelodrama. \"Tommy! Kiss me! Quick. Before I faint!\"\n\n[Illustration: \"'KISS ME! BEFORE I FAINT!'\"]\n\n\"I never saw a chap like you for kissing.\"\n\n\"That's a pretty thing to say! Although we may be married, sir, we\nhave not yet been upon our honeymoon.\"\n\n\"I'll kiss you, if you like.\"\n\n\"Thank you kindly, gentle sir!\" She favoured him with a sweeping\ncurtsey. \"Tommy, even you have no idea of the ramifications and\ncomplications of our peculiar situation.\" Mr. Stanham had removed his\nhands from his pockets. They occupied a more agreeable position round\nthe lady's waist. \"See if I don't snatch you from the lion's jaws.\"\n\n\"Does that mean that you will help me to escape from Holloway?\"\n\n\"It means that you will never get as far as Holloway?\"\n\n\"Am I to die upon the road then?\"\n\n\"Don't talk like that, don't! You don't know what a wife you've got!\nYou don't know how she loves you, worthless creature that you are!\nTommy, do say that you love me, just a little bit! There, you needn't\nsqueeze me quite so tight. I can't explain to you all about it. I will\nsome day! There's going to be a duel, perhaps to the death! between\nthe Lord Chancellor and yours to command; and if that august\npersonage, in the figure anyhow, of Sir Tristram Triggs, is not\nworsted and overthrown, I will give you leave, sir, to say that you do\nnot admire my taste in dress.--Tommy, don't.\"\n\n\nII.\n\nAfter dinner, Miss Cullen, strolling about the great glasshouse, all\nalone, came upon Sir Tristram, also all alone. Although not, probably,\nmore than half an inch taller than the gentleman, she looked, yes,\ndown at him, as if, comparatively, he were but an insect at her feet.\n\n\"Well, Sir Tristram, what amends do you propose to make to me?\"\n\n\"Miss Cullen?\"\n\n\"Sir?\"\n\nShe looked at him; and this famous lawyer who had been more than a\nmatch for the _olla podrida_ of the law courts, and the champions of\nthe political ring, quailed before a young girl's eyes.\n\n\"I fear, Miss Cullen, that I fail to apprehend your meaning.\"\n\n\"Is it possible that you are an habitual desecrator of that law which\nyou have sworn to uphold, and that, therefore, the details of your\ncrimes are apt to escape your memory? More than three months have\nelapsed since you committed your crime. So far as I know, you have\nnot sought as yet to take advantage of any occasion to offer me\natonement.\"\n\nSir Tristram faced round to her with something of the bull-dog look\nwhich had come upon his face when he had found himself in front of Mr.\nStanham.\n\n\"May I inquire, Miss Cullen, why you go out of your way to use\nlanguage of such extravagant exaggeration? It would be gross\nabsurdity, amounting almost to prostitution of language, to call the\noffence of which I was guilty, if it was an offence, a crime.\"\n\n\"Perhaps it is because you are a lawyer that you are unaware that not\nso very long ago a man was sentenced to six months' imprisonment for\nexactly the same thing.\"\n\n[Illustration: \"'THAT FOR THE CHANCELLOR!'\"]\n\nSir Tristram fidgeted. He seemed not to have complete control over his\ntongue.\n\n\"Miss Cullen, I trust that I may never be found lacking in respect\nto a lady. If I have been so unfortunate as to have offended you, I\nproffer you my most sincere apologies, and I humbly entreat for your\nforgiveness.\"\n\nMiss Cullen remained, obviously, wholly unmoved.\n\n\"When a criminal expresses his contrition, is he held, by so doing, to\nhave sufficiently purged himself of his offence?\"\n\n\"What is it that you require of me?\"\n\n\"I am told that you are to be the new Lord Chancellor. I am a ward in\nchancery.\"\n\n\"I learn the fact with the greatest pleasure.\"\n\n\"Do you? Then your pleasure bears a strong resemblance to my pain. I\nam to remain a ward till I am twenty-five.\"\n\n\"Indeed?\"\n\nSir Tristram began to rub his hands.\n\n\"Yes,--indeed! I had an objectionable uncle who was so foolish as to\nsuppose that I could not be a better judge of my own life's happiness\nthan--a number of elderly gentlemen.\"\n\n\"Hem!\" Sir Tristram coughed.\n\n\"If I was willing to overlook your offence--\" Sir Tristram smiled--\"I\nshould require a _quid pro quo_.\"\n\n\"And what, my dear Miss Cullen, would be the nature of the _quid pro\nquo_?\"\n\n[Illustration: \"'WELL, OLD MAN, HAVE YOU HAD IT OUT WITH TRIGGS?'\"]\n\n\"I should want you to consent to my marrying.\"\n\n\"To consent to your marrying?--Ah!--I see!--If the matter is laid\nbefore me in due and proper form--it is possible that you have a\ncertain individual in your mind's eye whom you are willing to make the\nhappiest of men--and I was satisfied that he was a fit, and a proper,\nperson, and every care was taken to safeguard your interests--then, my\ndear Miss Cullen, nothing would give me greater pleasure than to give\nmy consent to your being happily launched on what, I fear, is, too\noften, the troubled sea of marriage.\"\n\n\"That's not the sort of thing I want at all.\"\n\n\"No? Then what is the sort of thing you want, may I inquire?\"\n\nThe young lady tapped her foot against the floor. For the first time,\nshe seemed to be not entirely at her ease.\n\n\"The fact is, I'm married already.\"\n\n\"Married--already?--With the consent of the court?\"\n\n\"Bother the court!\"\n\n\"Young lady!--Are you aware who it is to whom you are speaking?\"\n\n\"I am perfectly aware. I am speaking to the person who kissed me\nagainst my will.\"\n\n\"Miss Cullen!--I'm the Chancellor!\"\n\n\"That for the Chancellor!\"\n\nShe actually snapped her fingers in his face. He seemed to be\nspeechless; though, perhaps, he only seemed so. When he did speak, it\nwas as if he were suffering positive pain.\n\n\"I find myself unable to believe that you are capable of realising\nthe position in which I stand, the position in which you stand,\ntoo. Personal misusage I might endure. But, in this matter, I am\nimpersonal. Take care! I represent, in my poor person, the majesty of\nEnglish law.\"\n\nHe turned as if to go. If he supposed that he had crushed her, he was\nvery much mistaken.\n\n\"Is that your last word, Sir Tristram?\"\n\n\"Miss Cullen, it is my last.\"\n\n\"Then, now, be so good as to listen to my last word. The Duke of\nDatchet is a magistrate. I will go straight to him, and demand from\nhim a warrant for your arrest.\"\n\n\"A warrant, for my arrest? Girl!\"\n\n\"I presume that it is because I am a girl, that you are enough of a\nman, first to assault, and then to bully me.\"\n\nTaking out his handkerchief, Sir Tristram applied it to his brow.\n\n\"Am I mad, or you? Are you utterly impervious to any sort of reason?\"\n\n\"Not more than you are. I have yet to learn that, because you are Lord\nChancellor, you cannot be made to answer for your crimes, exactly like\nany other criminal. Forgive my husband, forgive me, whose only crime\nhas been that we love each other, and who have not offended in the\nsight either of heaven or of earth, and I will forgive you, who have\noffended in the sight of both. Decline to do so, and, unless there\nis one law for the great and another for the small, in which case the\nworld shall hear of it, I promise that you shall learn, from personal\nexperience, what it means to go to gaol.\"\n\nSir Tristram looked about him, as if he wondered why the earth did not\nopen to swallow her. He seemed to gasp for breath.\n\n\"Miss Cullen, I beg that you will not suppose, that, under any\npossible circumstances, I could listen, even for a single instant, to\nwhat, to me, are your hideous insinuations. But one possible solution\nI do see to the painful situation in which you stand. If the person\nwhom you have illicitly and improperly married--\"\n\n\"Not improperly married, how dare you!\"\n\n[Illustration: \"'HER BEHAVIOUR FILLED ME WITH PROFOUND AMAZEMENT.'\"]\n\n\"In the eyes of the court, Miss Cullen, certainly, in the eyes of\nthe court. Hear me out. If this person should prove to be a fit and a\nproper person, of good character, of due position, and so forth, then,\ntaking all the circumstances into consideration, I might be moved to\nleniency. What is the person's name?\"\n\n\"He is of the highest lineage.\"\n\n\"So far, so good.\"\n\n\"He is a gentleman of the noblest character.\"\n\n\"Still better.\"\n\n\"He would be showing honour to any lady in the land if he made of her\nhis wife.\"\n\n\"Hem! Precisely! I asked you for his name.\"\n\n\"Thomas Stanham.\"\n\n\"Thomas Stanham!\" Sir Tristram's countenance went as black as a\nthundercloud. \"Thomas Stanham!\" He turned to her with a look of fury\non his face which took even Miss Cullen by surprise. \"That vagabond!\"\n\n\"How dare you speak so of my husband, sir?\"\n\n\"Your husband? Girl, you are a fool! You, the owner of prospective\nmillions, have thrown them, even before they are in your actual\npossession, into the lap of that pitiful adventurer. You ask me to\nshow him leniency? I will be lenient to you at least. I will protect\nyou from him, in spite of yourself.\"\n\nHe spoke with a degree of dramatic intensity which threw a lurid light\nupon the cause of his success in life. Miss Cullen was silenced after\nall. She stood and watched him as he strode away, with a degree of\ndignity in his bearing which seemed to have suddenly made him taller.\n\n\"Tommy must have wiped his eye!\"\n\nThat was what she said to herself when she was alone.\n\n\"Well, old man, have you had it out with Triggs?\"\n\nTurning, Miss Cullen found that Mr. Stanham had approached from\nbehind. He stood in the doorway--as usual, with his hands in his\npockets.\n\n\"Yes, young man, I've had it out with Triggs.\"\n\nMiss Cullen had a little flush on her cheeks, and an added light\nin her eyes, which superfluities, it might be said, unjustifiably\nheightened her attractions.\n\n\"Softened his adamantine breast?\"\n\n\"Well, hardly. Not what you might call quite. In fact, I should say\nthat, if he remains in his present frame of mind, he will send you,\nfor a certainty, to something much worse than penal servitude for\nlife.\"\n\n\"Is that so? Very kind of you, I'm sure. I knew you'd make a mess of\nit, my love.\"\n\n\"Wait till the play is over. There's always a muddle in the middle.\nThe third act has not begun.\"\n\n\nIII.\n\n\"Triggs, this is the deuce of a nice state of things!\"\n\nThe latest ornament of the woolsack was seated in the privacy of his\nown apartment prior to retiring to rest. But the cares of his position\nhad followed him there. He was working his way through a mass of\npapers when his host appeared at the door.\n\n\"To what state of things does Your Grace refer?\"\n\nThe Duke looked round as if to make sure that they had the room to\nthemselves. He seemed to be in a state of considerable agitation;\nindeed, the abruptness of his entry had in itself suggested agitation.\n\n\"Of--of course you know that I--I'm a magistrate.\"\n\n\"Certainly I know it.\"\n\nSomething in the other's tone seemed to have a soothing influence upon\nthe Duke, possibly because it roused the spirit of mischief that\nwas in him. He sat in an armchair. Crossing his arms upon his chest,\nstretching out his long legs in front of him, he regarded the toes of\nhis evening shoes.\n\n\"Triggs, I have had an application made to me for a warrant for your\narrest.\"\n\nThe Chancellor went a peony hue, as we have seen him do before.\n\n\"Your Grace is joking.\"\n\n\"I wish I were. I found it anything but a joke, and I am afraid that\nyou are not likely to find it one either.\"\n\nSir Tristram removed his glasses. He held them in his hand. His face\nbecame hard and stern.\n\n\"May I ask Your Grace to be more explicit?\"\n\nThe Duke turned. Placing one elbow upon the arm of his chair, he\nlooked at Sir Tristram as he leaned his chin upon his hand.\n\n\"Triggs, Miss Cullen has applied to me to issue a warrant against you\nfor assault.\"\n\n\"Surely such an application was irregular.\"\n\n\"I am not so sure of that, I am not so sure. Anyhow, I told her that\nit was. The only result of which, so far as I can judge, will be that\nshe will make the application, in more regular form, either to me, or\nto someone else, to-morrow. But that is not the point. Triggs, did you\ndo it?\"\n\n\"Is it necessary that Your Grace should ask me?\"\n\n\"You didn't kiss her?\"\n\nSir Tristram took out his handkerchief. He actually gasped for breath.\nIt is to be feared that at that moment the representative of English\nlaw almost told a lie. However, it was only almost; not quite. He\nmerely temporised.\n\n\"The whole affair is a pure absurdity.\"\n\n\"How do you mean? Is the charge unfounded?\"\n\nSir Tristram drew his handkerchief across his brow.\n\n\"Supposing I did kiss her.\"\n\n\"Supposing! Triggs? Good heavens! I remember your leading for a\nwoman who brought exactly such a charge against a man. I remember\nhow clearly you pointed out how, under certain circumstances, such\nan action might be, and was, an offence against good morals. Didn't\nPickum give the man six months?\"\n\nThe lawyer's resemblance to a bull-dog became more and more\npronounced. He all but showed his teeth.\n\n\"I don't know, Duke, if you are enjoying a little amusement at my\nexpense.\"\n\nThe Duke sprang to his feet. His bearing evinced an accession of\ndignity which, in its melodramatic suddenness, almost approached to\nfarce.\n\n[Illustration: \"AN AMAZING RESEMBLANCE TO A SMILE.\"]\n\n\"It is not my habit, Sir Tristram, to regard my magisterial duties as\noffering much scope for amusement. Situated as I am--as you are--as\nwe all are--our party!--in the eyes of the nation, it seems to me that\nthis matter may easily become one of paramount importance. Of such\nimportance that I have come to you as a friend, to-night, to ask\nyou, if there is a chance of Miss Cullen's charge becoming so much as\nwhispered abroad, to seriously consider if it would not be advisable\nfor you to place your resignation in the hands of the Prime Minister\nbefore your appointment to the Chancellorship is publicly announced.\"\n\nSir Tristram's jaw dropped open. His resemblance to a bull-dog\nperceptibly decreased.\n\n\"Duke!\"\n\n\"I am not certain, in coming to-night, that I have not allowed my\nfriendship for you to carry me too far. Still, I have come.\"\n\n\"Your Grace is more than sufficiently severe. If you will allow me to\nexactly explain my position in this matter, I shall have no difficulty\nin making that evident. I fear that Miss Cullen is a dangerous young\nwoman.\"\n\nThe Duke shrugged his shoulders.\n\n\"You, of all men, ought to know that, under certain circumstances,\nwomen are dangerous--and even girls.\"\n\n\"Precisely. That is so. But, I think that, after I have made my\nexplanation, you will allow that Miss Cullen is an even unusually\ndangerous example of a dangerous sex.\" He paused--perhaps for\nreflection. When he continued, it was with a hang-dog air. \"Some short\ntime since I did myself the honour of asking Miss Cullen to become my\nwife. I fear that--eh--circumstances induced me to take her answer too\nmuch for granted. So much so, indeed, that--eh--while I was waiting\nfor her answer, I--eh--I--eh--kissed her. I do not wish to lay stress\nupon the accident that the kiss was but the merest shadow of a kiss.\nBut such, in fact, it was.\"\n\n\"In plain language, Triggs, you kissed her against her will.\"\n\n\"I had no idea that it was against her will, or I should certainly not\nhave done it. Her behaviour after--eh--my action, filled me with the\nmost profound amazement. She jumped up. She addressed me in language\nwhich I can only describe as more pointed than elegant. And--eh--she\nwalked away, leaving me, I do assure Your Grace, dumbfounded.\"\n\n\"Well?\"\n\nThe Duke's back was turned to Sir Tristram, possibly because there was\nsomething on His Grace's face which bore an amazing resemblance to a\nsmile.\n\n\"Well, I heard nothing more of the matter. Indeed, I have heard and\nseen nothing of the lady till I met her here to-day. This evening she\nhas alluded to the matter in a manner and in terms which filled\nme with even more profound amazement than her behaviour on\nthe--eh--original occasion.\"\n\n\"But, man, didn't you apologise?\"\n\n\"I apologised in terms of almost abject humility. But that did\nnot content her. I will be frank with Your Grace. She made me a\nproposition which----\"\n\nThe Duke waved his hands. He cut Sir Tristram short.\n\n[Illustration: \"SHE LOOKED CHARMING.\"]\n\n\"I have heard too much already. Triggs, I have allowed my friendship\nfor you to play havoc with my discretion, let me hear no more. My\nadvice to you is compromise, compromise, at almost any cost. You don't\nwant to have your career ruined by a girl, and for the mere shadow of\na kiss. To consider nothing else, think of the laughter there would\nbe. As you say, the young woman can be dangerous, and, if nothing\nhappens to change her purpose, you may take my word for it that she\nmeans to be.\"\n\nBefore Sir Tristram could reply, the Duke was gone. The newly\nappointed representative of the majesty of English law was left alone\nwith his papers and his reflections. These latter did not seem to\nbe pleasant ones. Words escaped his lips which we should not care to\nprint;--we fear they referred to that undutiful ward of his lordship's\ncourt. Inwardly, and, for the matter of that, outwardly, he cursed her\nwith bell, book, and candle; certainly never was heard a more terrible\ncurse. And, so thoroughly did he enter into the spirit of the thing,\nthat he was still engaged in cursing her when the door opened, and in\nfront of him was Miss Cullen with the handle in her hand.\n\nShe looked charming, and by that we mean even more charming than\nusual. She had changed her dress for a _peignoir_, or a dressing-gown,\nor something of the kind. Beyond question Sir Tristram had no notion\nwhat the thing was called. It suited her to perfection--few men had a\nbetter eye for that sort of thing in a woman than he had. There is no\nfathoming feminine duplicity, but no one ever _looked_ more surprised\nthan did that young woman then. She had thrown the door wide open and\nrushed into the room, and half closed it again behind her before she\nappeared to recognise in whose presence and where she really was.\n\n\"I--I thought--isn't this Mary Waller's room? Oh--h!\"\n\nAs struck with panic she turned as if to flee. But Sir Tristram, who\nwas gifted, before all else, with presence of mind, interposed. He\nrose from his chair.\n\n\"Miss Cullen, may I beg you for moment?\"\n\n\"Sir! Sir Tristram Triggs!\" Miss Cullen's air of dignity was perfect,\nand so bewitching. \"I had something which I wished to say to Lady Mary\nWaller. There has been some misunderstanding as to which was her room.\nI must ask you to accept an apology.\"\n\n\"Unlike you, Miss Cullen, I always accept an apology.\"\n\n\"Indeed. Then my experience in that respect has, I presume, been the\nexception which proves the rule.\"\n\n\"May I ask when you apologised to me,--and for what?\"\n\n\"This evening--,\" the lady looked down; her voice dropped; thrusting\nthe toe of her little shoe from under the hem of her skirt, she tapped\nit against the floor--\"for becoming a wife.\"\n\nThe grim man behind the table regarded her intently. Although he knew\nthat the minx was worsting him with his own weapons, she appealed\nto, at any rate, one side of him so strongly, that he was unable to\nprevent the corners of his mouth from wrinkling themselves into a\nsmile.\n\n\"May I ask, Mrs. Stanham----\"\n\n\"Sir Tristram!\" She threw out her arms towards him with a pretty\nlittle gesture. \"You have set my heart all beating! You have brought\nthe tears right to my eyes! You are the first person who has called me\nby my married name.\"\n\n[Illustration: \"'THEN I'LL KISS YOU.'\"]\n\nHe moved his hand with a little air of deprecation--as if the thing\nwere nothing.\n\n\"May I ask, Mrs. Stanham, if Mr. Thomas Stanham is related to the Duke\nof Datchet?\"\n\n\"Related?--Of course he is!--He's his favourite cousin.\"\n\n\"His _favourite_ cousin?\" We doubt if she was justified in her use of\nthe adjective, but, the simple truth is, she _was_ a dangerous young\nwoman. \"I see. The plot unfolds. May I ask, further, if this little\ncomedy was rehearsed in advance?\"\n\n\"And in my turn, may I ask, Sir Tristram, what it is you mean?\"\n\nThey looked at each other, eye to eye. They understood each other\npretty well by the time Sir Tristram's glance dropped down again to\nthe papers on his table. His tone became, as it were, judicial.\n\n\"Well, Mrs. Stanham, I have been considering the matter of which you\nspoke to me this evening, and, having regard to the whole bearing of\nthe case, to the social position of Mr. Thomas Stanham, and so forth,\nspeaking, of course, _ex parte_, and without prejudice, I may say\nthat, as at present advised, if proper settlements are made,\nthe marriage might be one which would not meet with the active\ndisapprobation of the court.\"\n\nSir Tristram raised his eyes. The lady shook her head--very decidedly.\n\n\"That won't do.\"\n\n\"Won't do?--What do you mean?\"\n\n\"What I say. I'm not going to have Tommy bothered about settlements.\nI'm settlement enough for Tommy. What you have to do is to sit down\nand to simply write this: 'My dear Mrs. Stanham,--Speaking as Lord\nChancellor, it gives me much pleasure in assuring you, as a ward of\nthe court, that your marriage with Mr. Thomas Stanham meets with my\nentire and unreserved approval.--Yours faithfully, Tristram Triggs!'\"\nSir Tristram glowered--he might! But she was undismayed. \"You will\nhave to do it, sooner or later--you're a very clever man, and you know\nyou will!--so why not do it at once?\"\n\nHe did it at once. Actually! Possibly because the whole affair\nappealed keenly to his sense of humour,--one never knows! She read the\npaper, folded it, and then she said--with such a pout! and with such\nmalice in her eyes!--\n\n\"Now you may kiss me again; if you like.\"\n\n\"I am obliged to you; but the costs in the suit have already been too\nheavy.\"\n\n\"Then I'll kiss you!\"\n\nAnd she did--with some want of precision, just over the right eye.\nThen she fled to the door. When she was half-way through it, she\nturned, and waved towards him the hand which held the paper.\n\n\"You are my guardian, you know.\"\n\n[Illustration: THE QUEEN'S BODYGUARD.]\n\n\n\n\nSOME COSTLY PETS.\n\n[Illustration: 1. Mr. S. Woodiwiss's short-haired English tabby,\n\"Champion Zenophon\" (worth \u00a3100). 2. The Hon. Mrs. McLaren Morrison's\nPersian, \"Ameer\" (worth \u00a3100). 3. Mrs. C. Hill's short-haired blue,\n\"Patrick Blue\" (worth \u00a350). 4. Madame Portier's long-haired blue,\n\"Blue Boy\" (worth \u00a3100). 5. Mrs. L. G. Leverson's Siamese, \"Rynda\"\n(worth \u00a330). 6. Miss G. Willoughby's chinchilla long-haired, \"Zaida\"\n(worth \u00a3160). 7. Miss G. Willoughby's Siamese, \"Fulmer Banjo\" (worth\n\u00a350). 8. Mrs. Herring's \"Champion Jemmy,\" English silver tabby (worth\n\u00a3100). 9. Hon. Mrs. McLaren Morrison's long-haired black, \"Satan\"\n(worth \u00a3100).]\n\n[Illustration: Music (BUGLE CALL SUMMONING THE COOKS TO GET THE MEN'S\nGROG FOR THE DAY.)]\n\n\n\n\nHOME LIFE ON BOARD A MAN-OF-WAR.\n\nDESCRIBED BY ONE WHO HAS LIVED THERE.\n\n\nTo the majority of Englishmen the phrase \"Life on board a Man-of-War\"\ncalls up pictures of smart gun-drill, tactical exercises, and other\nmore or less irksome though necessary duties. Few people indeed have\nany cognizance of the way in which our bluejackets live their daily\nlife and how they manage to amuse themselves in the spare time at\ntheir disposal during the three years afloat, which is the usual\nperiod of a seagoing ship's commission.\n\nJack is awakened at 5 a.m. in summer and 6 a.m. in winter by the\nloud blare of a bugle under his hammock, and the hoarse voices of the\nbosun's-mates shouting \"Show a leg there. Arise and shine, 'rise and\nshine. All ha- - - - - -nds lashupandstowhammocks.\" Having lashed his\nbedding in his hammock in the regulation manner, by taking seven turns\nround it with his hammock-lashing, he has his breakfast, for which\nmeal he is allowed half an hour.\n\nHe then works and drills more or less continuously until noon, with\nthe exception of 15 minutes' \"stand easy\" at 8 a.m., when he is\nallowed to smoke, and to go down to his mess and eat and drink if he\nfeels so inclined.\n\nAt noon the ship's company is \"piped to dinner.\"\n\nNoon is the dinner hour of our navy right throughout the world, and\nthough things have greatly changed since the introduction of steam and\nthe torpedo, the navy still retains the \"bosun's pipes\" of the days of\nNelson. No sooner is the shrill pipe sounded than there is an excited\nrush of men to the cook's \"galley,\" whence arises a cloud of odorous\nsteam redolent of baked meats, vegetables, and baked and boiled\n\"duffs\" (so dear to the naval heart of all ages), which are to feed\nthe 600 or 700 odd hungry men just released from work.\n\n[Illustration: _Photo. R. Thiele & Co._\n\nSERVING OUT JACK'S GROG (12.30).]\n\nMen going on watch at noon--as the Marine sentries, for example--are\nallowed to fetch their dinner at \"seven bells\" (11.30), and sometimes\nludicrous mistakes will arise through this privilege. The men take\nturns to prepare the dinner, and the cook of the mess for the day\nusually fetches his mess-mates' dinner from the \"galley.\" On one\noccasion which the writer recalls, the cook was at work on deck\nwhen the bell struck seven, and could not get away. Several of his\nmess-mates (he was a Marine Artillery man) having to go on watch at\nnoon, proceeded to the \"galley\" in quest of their dinner, and \"fisted\"\n(seized) a savoury dish they imagined to be theirs, without first\nexamining the brass mess-number on the side thereof. The dinner was\ndivided and eaten, and the plates were being washed up, when a group\nof excited bluejackets, having questioned every other mess in the\nship, made their way to No. 19 mess and hungrily demanded their\ndinner.\n\n[Illustration: _Photo. R. Thiele & Co._\n\nTEETOTAL SAILORS RECEIVING CASH INSTEAD OF RUM.]\n\nThe Marines had taken the wrong one, but offered their own in\nexchange. Search at the \"galley\" failed to produce the missing meal,\nwhich was eventually discovered stowed away beneath a wash-tub under\nthe Marines' mess-table, uncooked. The absent-minded cook for the day,\nwho was much taken up with a song of his own composition, entitled\n\"A Barrack-room Dinner,\" which he was to sing at a forthcoming\nentertainment, had, in his contemplation of the visionary meal he was\nto sing of, forgotten to take the actual dinner to the galley, and\nthere it lay in the mess in all its uncooked glory.\n\nAfter a somewhat heated discussion, the Marines appeased the\nbluejackets by paying for a dinner of corned beef and pickles from the\ncanteen, and thought they had heard the last of the matter; but the\nsailors had determined to pay the \"Joeys\" in their own coin, and did\nso a few days later, when the ship's company, being at \"collision\nquarters,\" the Marines' messes were emptied of all their inmates. A\nparty of bluejackets was stationed with the diving apparatus on the\nmain-deck near the Marines' messes, and in the party were several who\nhad suffered the loss of their dinner. It was 4 p.m., and noticing a\nlarge \"plum-duff\" on the table, evidently intended as a delicacy for\ntea, they pounced on it to a man. When the Marines came down in hungry\nexpectation, behold! there was but an empty dish.\n\nDinner time lasts an hour and a quarter, and at \"one bell\" (12.30) the\nbugle-call for grog--\"Nancy Dawson,\" as it is nick-named--summons the\ncooks to the grog tub.\n\nThe bugle-call, which is unknown even to army men, is given on the\nprevious page.\n\nEach man above the age of eighteen is allowed half a pint of grog,\nusually mixed in the proportion of one part of rum to three of\nwater, and hence familiarly termed \"three water\"; and the number of\nhalf-pints due to each mess is served out to the cook of that mess\nfor the day. The cooks stand _\u00e0 queue_ in the numerical order of their\nmesses, the mess whose turn it is to pump the grog-water for that\nday (the messes take daily turns at so doing, petty officers' mess\nexcepted) standing first \"on tally,\" and the grog is served out by\na petty officer and the Marine sergeant of the guard, under the\nsupervision of a warrant officer and the ship's steward, who, book in\nhand, checks off the number of pints allotted to each cook.\n\nThe grog-tub is usually decorated with some loyal motto worked in\nbrass, a first favourite being \"The Queen, God bless Her.\"\n\nA large proportion of men, thanks to the praiseworthy exertions of\nthat true friend of Naval mankind, Miss Agnes Weston, are teetotalers;\nand these men, together with the boys under 18, are allowed money\ninstead of rum at the rate of one penny one day and three farthings\nthe next alternately. This is paid them once a quarter (monthly in\nharbour ships) by the paymaster in exactly the same manner in which\nthe entire ship's company receive their ordinary pay.\n\nThe dinner-hour, too, is a convenient time for the sale of dead or\n\"run\" men's effects.\n\nWhen a man has absented himself without leave for seven days he is\nofficially posted a deserter, and any clothes, uniform, &c., he may\nhave left behind him are sold by auction to the highest bidder, the\nproceeds going to the Government.\n\n[Illustration: _Photo. R. Thiele & Co._\n\nAN AUCTION--SELLING A DESERTER'S CLOTHES.]\n\nJack Tar, like a great number of his social superiors, does not\nbelieve in giving a paternal Government any more than he can\nconveniently help; and many a great bargain does he pick up at these\nsales. For instance, a white duck tunic, such as the master-at-arms\nis holding up for inspection in our illustration, and which costs\nJack 4s. to 5s., will start at 3d. and slowly mount up to 6d. or 8d.,\nbeyond which sum the bidding seldom rises.\n\nAt 1.15 p.m., dinner being over, on ordinary week days the bugle\nsounds \"Clean Guns,\" and work recommences; but on Sundays and\nThursdays (known to the bluejacket as \"Spun-yarn Sunday\") the ship's\ncompany are granted an afternoon of rest.\n\nAs soon as dinner time is over the bosun's mate pipes the sufficiently\nobvious pipe \"Hands make and mend clothes\"; and, as Jack makes all\nhis own wearing apparel, he is not slow to take advantage of the time\nallowed him.\n\nIn fine weather the men bring their machines on deck and smoke and\nsew together. Every conceivable kind of needlework does Jack execute\nequally well. And not only the rank and file, but the petty officers\nalso are glad to make their own clothes rather than buy them ready\nmade; and though Jack is generally a self-taught tailor, he turns out\nfar smarter work than the slop-shops. The difference is very obvious\nif one compares a bluejacket wearing uniform \"built\" by his own\ndeft fingers, with one who is wearing a suit bought at some \"Naval\nOutfitter's.\"\n\nThe men have the forward part of the upper deck to themselves, the\npetty officers having the space further aft set apart for them;\nbut this advantage is not without its little drawbacks. Witness an\nincident experienced by the writer.\n\nHe was seated by a ventilator playing chess--a favourite game--with\na comrade. The fleet was about to enter Vigo, and a heavy sea was\nrunning, drenching the fo'c'sle and the other side of the deck, but\nleaving the space where the players were seated dry as a bone. They\nwere just congratulating themselves on their comfortable quarters,\nwhen the ship, suddenly altering course to make the entrance to the\nbay, slewed round to port, and a heavy sea came neatly in and caught\nthem as they sat. Chessmen, board, and players went suddenly floating\nabout the deck in picturesque confusion, to the great amusement of the\nonlookers, who were expecting some diversion. Going below to change\nhis clothes--for he was wet to the skin--the writer had the bad luck\nto stand directly under the same ventilator, and no sooner had he\ndonned dry clothes than another malevolent and illfavoured sea came\ncarefully down the ventilator shaft and rendered him as wet again. He\ntenderly avoided that ventilator during the remainder of the cruise.\n\nThursday afternoon is the recognised time for the opening of the\nmysterious and voracious \"Scran-bag.\"\n\nOn board a man-of-war tidiness is a matter of great importance, and\nwith a view to enforcing it an officer--on Sundays and Thursdays the\npuissant captain himself--makes a tour of the entire ship at certain\nhours. Woe betide the luckless man who has left out of its place the\nsmallest article! For when the decks are being cleared up for the\n\"Rounds\" (as the inspection is termed), here a towel that has been\ninadvertently left on a rack instead of being stowed away in its\nappointed place, the kit-bag--here a book, or a coat, or a pound\nof tobacco, stowed away out of sight behind a scuttle cover, and\ndiscovered by the insinuative, far-reaching hand of the \"Crusher,\" as\nthe ship's corporal is familiarly termed, a hand that has a pleasant\nknack of exploring out-of-the-way nooks and corners--in short, any\narticle that is left about is confiscated, and placed within a huge\ncanvas bag, the \"Scran-bag.\"\n\nEvery Thursday it is opened, and there gathers around it an excited\nknot of men who overhaul its contents thoroughly, a ship's corporal\nstanding by to see that no man claims \"what isn't his'n.\" But before\nthe owner is allowed to take away his article he is mulcted in one\npenny for each article, to be put in the poor-box, or else he has to\nprovide a piece of soap to be used in scrubbing decks.\n\nNearly everyone has seen \"Ship's tobacco\" in some form or other, but\nfew know how the sailor prepares it for use. It is served out to him\nmonthly, at the same time as his soap, in packages of 1 lb., for which\nhe pays 1_s._ 1_d._, being allowed it duty free. It is a dark, rich\nleaf, and the first thing done is to remove the stems. This done, some\nwater is sprinkled on the loose leaves (the old salt will prefer rum,\nto add to its strength and flavour), and the whole is enclosed in\na piece of canvas and tightly bound with twine until it assumes a\ncigar-like shape, pointed at each end. Next some fine line is taken,\none end secured to the tobacco and the other made fast to some strong\nsupport. One or two men now sit astride the line, and the tobacco is\nwound round and round, the weight of the men compressing it to about\nhalf its original bulk. When entirely covered with line it is tightly\nsecured, and in two or three days is ready for use.\n\nAt night the men's time may fairly be considered their own. On certain\nevenings fresh water is served out for the washing of clothes, for\nJack is his own washerwoman as well as tailor. That the marine is no\nless handy than his sailor brother may be gathered from the fact that\nthe ship's cobbler usually belongs to that immensely useful branch of\nthe service so aptly described by Kipling as \"soldier and sailor too.\"\n\n[Illustration: _Photo. R. Thiele & Co._\n\nJACK AS TAILOR--MAKING AND MENDING HIS CLOTHES.]\n\nA number of men who are handy with razor and scissors make a good\naddition to their pay by attending to the tonsorial wants of their\nless gifted brethren, and shave and cut hair in a heavy sea-way with\nthe ship rolling and pitching all over the place as easily as they do\nin harbour with an immovable deck to stand on.\n\n[Illustration: _Photo. R. Thiele & Co._\n\nA SKILFUL MACHINIST.]\n\n\"All work and no play\"--the proverb was made for Jack; and though the\nbluejacket has to make his own amusement he does it as thoroughly as\nhe does all else he puts his hand to. Nearly every ship in the navy\nhas its <DW65> troupe or theatrical party, and some really clever\nperformances are given; the make-up and dress are good, and would be\nno disgrace to a professional company. The fair sex, though absent,\nare hardly conspicuous thereby; few uninitiated eyes would detect\nin the female characters a middle-aged able-seaman or a cheeky young\nordinary.\n\nA more athletic relaxation is boxing, which is--as it should be--a\nfavourite amusement aboard. Many a good man has the Royal Marines or\nthe navy supplied to the professional ring.\n\nWhile the men are amusing themselves in various ways their superiors\nare likewise killing time, and will often indulge in cricket on the\nquarter-deck, which is screened with canvas to avoid losing the ball\noverboard. The game can only be played at sea, for in harbour the\nquarter-deck is required for more serious work. The ball is usually\na soft tennis-ball. The officers don flannels, and many an exciting\ngame, such as Ward-room _v._ Gun-room, is played, and continued at\nevery opportunity till the match is finished.\n\nEvery officer aboard takes an interest in these matches, captain and\ncommander often coming on deck to encourage their juniors with their\naugust presence, and many a match won by the navy ashore has been due\nto the practice aboard. For if a man can play cricket with a sloping\nand mobile deck beneath him he can surely do better on a well-rolled\npitch.\n\nOn Saturday nights, on such ships as carry one, the band discourses\nsweet dance-music for the delectation of the men; and these\nproverbially ardent lovers of Terpsichore are true to their goddess,\neven though the wind is howling great guns, and the ship rolling and\npitching in such a way that none but true sons of Neptune could even\nwalk upright, to say nothing of dancing.\n\nWhen no band is carried, a miscellaneous collection of blue-jacket and\nmarine musical amateurs supply the deficiency; and their music, though\nnot perhaps up to the standard of Mr. Dan Godfrey, amply fulfils its\nrequirements.\n\n[Illustration: _Photo. R. Thiele & Co._\n\nSAILORS CLAIMING THEIR ARTICLES ON THE OPENING OF THE \"SCRAN-BAG.\"]\n\nWhen the ship is in harbour, leave is often allowed, and in connection\nwith shore-going there is an interesting formality not generally\nknown. It is one of the most heinous crimes in the naval decalogue\nto attempt to smuggle any intoxicant aboard; and to obviate such\na possibility every man on returning from shore is searched by the\ncorporal of the gangway, a ship's corporal standing by, book in hand,\nto enter the names of the offenders in the \"black list,\" to be dealt\nwith next day by the commander.\n\n[Illustration: _Photo. R. Thiele & Co._\n\n\"JACK IS HIS OWN WASHERWOMAN.\"]\n\nThe corporal of the gangway is usually an experienced Marine told off\nfor the duty, and under his hands it would be difficult for even the\nmost crafty smuggler to conceal any liquor about his person.\n\nBut to the credit of our navy be it said that cases of smuggling are\nextremely rare.\n\nAs a general rule Jack Tar and Joe the Marine, though certainly\nsometimes labouring under conditions trying to even the most\nlaw-abiding civilian, conform to regulations and discipline with that\nbreezy cheerfulness and brave good-will which makes them, as they\nalways have been and it is to be hoped always will be, the idols of\ntheir countrymen, the proud boast of their nation, and a standing\nmenace to her enemies.\n\n[Illustration: _Photo. R. Thiele & Co._\n\nOFFICERS PLAYING CRICKET ON THE QUARTER DECK.]\n\n\n\n\nCRICKET SKETCHES.\n\n_Two Pages by Mr. \"Rip.\"_\n\n[Illustration: MR. K. J. KEY--A STATELY PROCESSION OF ONE.]\n\n[Illustration: AN UNUSUAL EXPERIENCE FOR THE GREAT ARTHUR SHREWSBURY.]\n\n[Illustration: AN ELEGANT BAT--MR. F. G. J. FORD.]\n\n[Illustration: \"RANJI\" MAKES A BRILLIANT CATCH.]\n\n[Illustration: LITTLE \"BOBBY\" ABEL GETTING AWAY WITH AN AWKWARD ONE.]\n\n[Illustration: FIREWORKS! AN OVER BY MR. JESSOP.]\n\n[Illustration: PHILLIPS, THE INTERNATIONAL UMPIRE.]\n\n[Illustration: BIG TOM RICHARDSON--READY TO KNOCK UP A BRISK DOZEN OR\nSO.]\n\n\n\n\nHIS SOVEREIGN REMEDY.\n\nA COMEDY IN AN OMNIBUS.\n\nBY CLARENCE ROOK; ILLUSTRATED BY B. E. MINNS.\n\n\nThe gloom was gathering. Ten minutes ago the conductor had leaned from\nhis step, taken the lamp from some unseen hand, and stuck it up in\nits place by the door. The bus lurched round the corner into Bishop's\nRoad. It was a Bayswater bus, and the old gentleman who was changing\nhis seat drove his elbow into my hat.\n\n[Illustration: \"DROVE HIS ELBOW INTO MY HAT.\"]\n\n\"Bless me! I'm always doing that. Most extraordinary! I'm sure I beg\nyour pardon.\"\n\nI told him that it was of little consequence, and another swing of\nthe bus seated him suddenly beside the tired-looking girl with a music\nportfolio in her hand. She opened her eyes for a moment, and then\nclosed them again. The woman beyond shifted her baby to the other\narm--the arm furthest removed from the old gentleman--and continued to\nrock it mechanically.\n\nThe old gentleman evinced a restlessness which was not suggested\nby his mild aspect and his white hair, though a closer examination\nrevealed a certain furtive look in his eyes. Four separate times he\nhad shifted his seat since I had taken my place in the corner next\nthe door at Oxford Circus. A slight irritation at his want of repose\ncaused me to shoot a protesting glance at him over the top of\nmy evening paper, for few things annoy me so much as purposeless\nactivity. Old gentlemen should be glad enough to sit still when they\nhave the chance. But I could not find it in my heart to be angry with\nsuch a benevolent-looking old gentleman.\n\nIt was just then, as my eyes were returning to my paper, that the\ndemon of suspicion took tentative hold upon my mind. \"Why,\" I asked\nmyself, \"do nice-looking old gentlemen, with white hair and shifting\neyes, want to change their place in a bus?\"\n\nThe suspicion came--and went, for the kindly and venerable face gave\nno hold for doubt. But I laid down my paper upon my knees and leant\nback in the corner to watch him, speculating whether he would change\nhis place again before we came to Westbourne Grove. The driver's\nwhip-lash sounded on the middle pane opposite to me, and the bus\nslowed down to take up a passenger who, after a glance inside, mounted\nto the roof.\n\nThe conductor shoved his parcel up after him, pulled the string and\nresumed his position against the side of the door, where, upon that\nmysterious block which is kept in a receptacle over the entrance,\nhe was apparently making sketches of the passengers inside. Mentally\ncommending his diligence, I turned my eyes again to the old gentleman,\nwho met my glance for a moment, and seemed to deprecate my displeasure\nby the lifting of his brows and a turn of his head.\n\nAs the bus quickened up again, the tired-looking girl swayed slightly,\nand her head sank upon the shoulder of the old gentleman. The old\ngentleman glanced sideways at the closed eyes of his neighbour, and,\nas a kindly smile stole over his face, his arm slid round the girl's\nwaist. The pair made quite a pretty picture. The conductor at my elbow\nturned slightly, to get a better light upon his sketching block.\n\nAnd then I noticed a curious disturbance--only a momentary rise and\nfall--in the dress of the sleeping girl. No one, so far as I could\ntell, had moved. The girl's hands were lying in her lap, precariously\nclasping her music portfolio. The disturbance occurred on the right\nside of the dress, which was the side furthest from the old gentleman\nin whose kindly embrace the girl lay.\n\nThe explanation came to me in a flash. In so sudden a flash that I\nturned in the same instant to the conductor and found his sidelong\nglance meeting mine.\n\n\"See that?\" he muttered, under the clatter of the bus.\n\n\"I should think I did,\" I said, \"he's picked her pocket.\"\n\n\"I've 'ad a eye on the old josser for the last month,\" he said. \"I'll\nmake it a fair cop this time. You're my witness.\"\n\n\"Well,\" I said, \"I'm not awfully keen on being mixed up----\"\n\n\"Bit of high-spyin' now,\" he said. \"What's the matter with a little\nbit o' high-spyin', eh?\"\n\n[Illustration: \"THE TIRED-LOOKING GIRL SWAYED SLIGHTLY.\"]\n\nThe conductor mounted the steps to the roof. The tired girl, awakened\nsuddenly to her position, straightened herself and peered anxiously\nthrough the window of the bus as though to make certain that she\nhad not been carried to Wormwood Scrubs in her sleep. Reassured, she\ngathered up her portfolio in a firmer grasp with one hand, and with\nthe other searched the back of her head for errant pins.\n\nRound the edge of my paper I watched the old gentleman, whose eyes\nwere now fixed obliquely upon the woman on his left. I distinctly saw\nhis eyes travel down from the woman's face to her black cloth jacket,\nand stop at the outside pocket, from which her omnibus ticket was\npeeping. The pocket was on a level with, and almost touching his\nelbow, and his hand, his left hand, which was resting upon his knee,\nbegan slowly to travel towards the pocket of the tired-looking woman.\n\nThe baby was kicking, grasping at the stuffy air with crinkled\nfingers, and threatened to give voice, and the tired-looking woman,\nrocking more anxiously than before, looked timidly from one neighbour\nto another as though in apology for the wrath to come.\n\nAt that moment my glance was attracted to a point above the old\ngentleman's head, where I met the eyes of the conductor, pressed close\nagainst the window-pane. A little higher was the tip of his nose,\nwhitened by the pressure, and above that his stubby red moustache,\nunderneath which a mouth gaped with inquiry. For a moment or two I\nwas fascinated by the inverted face, which seemed to belong to some\nother-world creature which had tumbled from extra-mundane space and\nstuck fast upon the window of the Bayswater bus.\n\nThe benevolent old gentleman, quite unconscious of the watchful eyes\nbehind his head, was regarding with a bland smile the advertisements\non the window behind me. And as my eyes fell again on the spot where I\nhad last seen his hand, I saw that it was not there. There never was a\nmore unskilful performance. For there sat the old gentleman before my\neyes, looking calmly over my head, with two fingers inserted into the\npocket of the woman who was rocking the baby. As though it knew the\nwrong that was being done, the baby gave vent to the threatened yell,\nand the mother, patting it, and rocking it, and speaking to it in\nunknown tongues, saw nothing and felt nothing else.\n\n[Illustration: \"APPARENTLY MAKING SKETCHES OF THE PASSENGERS.\"]\n\nSuddenly, as I watched, the benevolent old gentleman dropped his eyes\nfrom the advertisements, and mine arrested them as they fell. Never\nwas an old gentleman so vastly perturbed. I almost felt sorry for him;\nfor an aged criminal who has not learned the art of escaping detection\nand is therefore hopelessly incompetent, is a pathetic sight.\n\nThe omnibus stopped with a jerk just as we came within the range\nof the lamps at the corner, and the old gentleman, so evil were his\ndeeds, seemed to shrink from the light. I was not quite certain of the\netiquette with pickpockets. Ought I to leap upon him then and there\nand to denounce him? That would be melodramatic, I reflected; and I\nhate a scene; so I only raised myself from my seat, borrowed support\nfrom the handrail above my head, and waited upon events.\n\n[Illustration: \"TOWARDS THE POCKET OF THE TIRED-LOOKING WOMAN.\"]\n\nThe tired girl bestirred herself and looked round, the woman with the\nbaby changed her burden again from one arm to the other and peered\nanxiously at the door.\n\n\"Royal Oak,\" I said, answering her look of inquiry.\n\nShe sank back in her seat and closed her eyes, and at the same moment\nthe old gentleman jumped up and shambled towards the door, while the\nother passengers carefully drew in their toes.\n\nBy this time I noticed that the conductor's face had detached\nitself from the window. Three people had risen to leave; but the old\ngentleman was first, being clearly in a hurry; and as he found himself\nunable to pass me, half-standing and half-sitting, with my hand on\nthe overhead rail, he looked pleadingly at me, as though imploring my\nsilence. I hesitated a moment. It was none of my business to arrest\ncriminals. But I did not mind giving a passive support to the cause\nof justice, so I stayed where I was. And then the conductor appeared,\nblocking the doorway.\n\n\"No, yer don't,\" he said.\n\n\"My good man,\" began the old gentleman, \"I sincerely trust I have\ngiven no offence. I only----\"\n\n\"I see yer,\" said the conductor, looking over his shoulder towards the\npublic-house, and jerking his head.\n\n\"Then kindly oblige me,\" said the old gentleman, \"by not making a\nfuss. If a sovereign now----\"\n\n\"Oh, stow it,\" said the conductor. \"You've done it once too often,\nthat's what you 'ave. I see yer right enough this time, and you're\ngoing to be give in chawge, that's what you are. Strite.\"\n\nThe old gentleman looked helplessly round him. Impatient passengers\nbegan to remonstrate from the step; others from the kerb.\n\n\"'Old on,\" said the conductor, \"we're all goin' 'ome to tea.\"\n\nA policeman crossed from the opposite corner.\n\n\"'Igher up there!\" he remarked, dispassionately.\n\n\"Look 'ere, constable,\" said the conductor, \"'ere's a job in your\nline.\" Then his tone became official. \"I 'ereby give this man in\nchawge for picking pockets.\"\n\n\"Oh,\" said the policeman, scattering the bunch of people gathered\nround the step.\n\n[Illustration: \"AS I WATCHED.\"]\n\n\"I see him--and this gentleman 'ere see him,\" said the conductor.\n\"'Tain't the first time, neither. Old 'and, he is; that's what _he_\nis.\"\n\nThe doorway was now blocked by the policeman's form.\n\n\"That ain't good enough for me,\" he said. \"Any of you ladies and\ngentlemen lost anything?\"\n\n\"I see 'is 'and in that lydy's pocket,\" said the conductor, pointing\nover the constable's shoulder at the woman with the baby. \"You feel in\nyour pocket, lydy.\"\n\n[Illustration: \"'I GIVE THIS MAN IN CHARGE FOR PICKING POCKETS.'\"]\n\nThen ensued a general searching of pockets, while a rival omnibus\nswept by triumphantly and gathered up such passengers as were too\nimpatient to await the outcome of the situation.\n\nI leaned forward and said in an undertone to the girl with the\nportfolio, who alone of the passengers shewed no interest in the\ncontents of her pocket, \"You had better look in your pocket, I feel\nconvinced it was picked while you were asleep upon his shoulder.\"\n\n\"I wasn't,\" she said, abruptly. Then, reflecting apparently that she\nwas rude as well as tired, she added, \"I've nothing worth stealing,\nthank you all the same.\"\n\nIn a desultory way she began fumbling in the pocket of her dress. The\nold gentleman stood by the policeman. His face had grown very red; his\neyes, wandering from one passenger to another, became suddenly fixed,\nand his face was redder than ever. It was sufficiently obvious that he\nwas very uneasy. Following the direction of his eyes, I saw the baby's\nhead hanging at an alarming angle over the woman's arm. The mother\nwas leaning towards the light and looking at the contents of her free\nhand--a bus ticket, two pennies, a farthing, and a sovereign.\n\n\"Now, then! lost anything, mem?\" asked the conductor.\n\n\"No, _I_ ain't lost nothing,\" she began, slowly.\n\nThe old gentleman nodded to her pleasantly.\n\n\"Though,\" she continued, \"I don't rightly understand why----\"\n\n\"I think this must belong to you, sir,\" said the girl with the\nportfolio, suddenly, holding out a sovereign to the old gentleman.\n\n\"Not at all, my dear; nothing to do with me, nothing whatever,\" he\nsaid, nodding his head at her. \"Old enough to be your grandfather,\ntoo!\"\n\n\"Now then, what's all that?\" asked the policeman.\n\n\"Only this gentleman must have been putting a sovereign into my\npocket, and I insist--oh! I insist----\"\n\n\"Look here, constable,\" said the old gentleman, \"can't you see that\nyou are embarrassing the young lady? Any little transaction between\nher and me is none of your business, or anyone else's either.\"\n\nThe old gentleman stamped impotently upon the floor of the omnibus.\n\n\"He's been giving money away,\" said the policeman over his shoulder to\nthe conductor, \"looks like.\"\n\n\"And why not, why not?\" said the old gentleman. \"What's the good of\nhaving money if you can't make people happy with it?\"\n\nThe constable looked reflectively at him.\n\n\"I dunno,\" he said. \"I'd better take your name and address.\"\n\nThe old gentleman looked apprehensively round. Then he took a card\nfrom his pocket and gave it to the policeman.\n\n\"Please don't read it out,\" he said.\n\nThe policeman looked at the card, put it into his pocket-book, and\nmade a note in pencil. Then he swung himself off the omnibus and\nlooked hard at the old gentleman as he descended slowly.\n\n\"You go home,\" he said. \"You want to be took care of, you do.\"\n\n[Illustration: \"'YOU GO HOME,' HE SAID.\"]\n\nThe conductor stood upon the kerb with his hand on the rail, looking\nafter the old gentleman as he trudged off towards Royal Oak Station.\n\n\"'Urry up there,\" said the constable. \"Wastin' my time,\" he added, as\nhe turned his back.\n\nThe conductor rang the bell and leaned dolefully against the stairway\nas the bus started away from the dispersing crowd.\n\nThe girl with the portfolio was regarding her sovereign thoughtfully,\nholding it between her thumb and forefinger; then she returned it with\nher handkerchief to her pocket, looked doubtfully round and blushed\nslightly.\n\nThe woman with the baby was biting something, which, as she caught my\neye, she hurriedly slipped into her jacket pocket. \"Not that I'd be\nbe'olden to anybody,\" she remarked at large, rocking her baby with\nmuch energy, \"me 'usband earning good money, thanks be. But peliteness\nis peliteness----\"\n\n\"You _may_ think yer know yer way abart,\" said the conductor, looking\nat me and jerking his head up and down, \"but now and then you find\nyou're left--badly left. Now, think o' that! Droppin' sovereigns all\nover the place. Well, I wish I'd a'knowed!\"\n\n[Illustration: _Photo by Landor, Ealing_\n\nTHE HEAVENLY TWINS.]\n\n\n\n\n[Illustration]\n\nLITTLE DARLINGS.\n\nHALF A MILLION PHOTOGRAPHS OF CHILDREN.\n\n_Words by Somers J. Summers. Photographic Illustrations by W. J.\nByrne._\n\n\n[Illustration: SUNSHINE.]\n\nLocked away in the breast of Mr. W. J. Byrne, the children's\nphotographer, is a secret which, when she has read this article, every\nmother of children will want to know. Let it be said at once, however,\nthat her curiosity will have to go unsatisfied; Mr. Byrne has his\nsecret, and wild interviewers cannot drag it from him.\n\nAbility to pose adults gracefully and naturally before the camera is\nan accomplishment admittedly rare; in the case of children, with the\ndifficulties increased tenfold, it must be a gift. It is one thing to\ndump a subject into a chair and obtain a likeness, another to make a\npicture as well. And when a man has taken half a million photographs\nof little sitters, in as many poses, he may be held to be something\nof an authority on the subject. That is Mr. Byrne's record; he is to\nchildren what Rosa Bonheur is to animals, save that he uses neither\npencils nor brush; he is a veritable artist with the camera. Some of\nthe examples of his skill here seen represent, it is not difficult to\nrealise, an infinity of painstaking and experimenting, while others\ntell of patient waiting, followed by considerable alacrity at the\nmoment of a fleeting expression which he desired to preserve. Mr.\nByrne's method is very simple; one half of his secret is soon told.\n\n[Illustration: SHADOW.]\n\n\"Photographing children,\" he says, \"is charming work, but it can never\nbe successful so long as the customary relations between them and the\nphotographer exist. They usually enter a studio with much the same\nsort of feelings as they do a dentist's. They should be made to feel\nat home before the business side of their visit is reached. Instead of\nbeing at once placed in the 'operating' chair, they should be allowed\nto wander about, if old enough, at their own sweet wills and in\nany case become accustomed to their strange surroundings. Wild\ngesticulations, promises of chocolates, stories of 'the little bird,'\nand orders to 'keep like that,' only serve to produce expressions\nof wonder and fear. Personally I let the child amuse itself with new\ntoys, and either pretend to take no notice, or else join in the game.\nThis may go on for half an hour. Meanwhile, an attendant is quietly\nfocussing an almost concealed camera, and when the child begins to\nprattle, I wait for an unconscious and happy expression, then snap\ngoes the shutter, and the thing is done.\n\n[Illustration: \"WHO IS THAT LITTLE FELLOW?\"]\n\n[Illustration: \"I DON'T LIKE HIM!\"]\n\n\"Every child's face is beautiful to at least one pair of eyes. The\nfeatures may not be symmetrical, the eyes may be small and dull, but\nthe charm of childhood does not lie entirely in facial beauty. It is\nthe coy smile and the quaint expression that a parent prizes most. And\nit is these characteristics that a photographer should aim to catch.\nMothers often make the mistake of rehearsing the sitting at home. It\nis even better not to mention the matter in the presence of the little\none; it is usually much more satisfactory if the visit is a surprise\none as far as the child is concerned. It is also unwise to dress the\nyoung sitter in unaccustomed clothes or to warn it to be good. For\ngeneral work, my rule about posing children is, 'Never pose them at\nall.'\"\n\n[Illustration]\n\nMr. Byrne's studios are veritable toy-shops, containing everything\nfrom a jumping frog to a model of an Atlantic liner. Indeed, Mr. Byrne\nhas given a big firm of toy-dealers a standing order to send anything\nnew that comes in the market. Antiquarians will learn with a pang\nthat the dear old Noah's Ark is going the way of all flesh. British\nchildren will have none of it. They refuse to look pleasant for less\nthan a little bicycle with rubber tyres, or a miniature motor-car with\nreal boilers--at least when they go to be photographed.\n\nSo much, then, for how Mr. Byrne's \"happy\" results are produced; what\nabout his \"unhappy\" ones? Both are well represented here. Take the\nfirst pictures, Sunshine and Shadow. What caused the inquisitive\nlittle fellow in the first to find the world all dark and so little of\ninterest in life a few seconds later?\n\nIt must have been something wholly unexpected, for it effected much\nthe same change in his companion's countenance. Was it a pin-point\ngently insinuated between the shoulder blades, or a cold sponge\ndexterously applied to the little spine?\n\n[Illustration: \"QUEEN OF HEARTS.\"]\n\nThat is what mothers would like to know. Mr. Byrne says, with a\nsmile, that it was neither--that the expression was a purely natural\ndevelopment. But will the mothers of England believe him?\n\n[Illustration: UNHAPPY LITTLE PEOPLE.]\n\nThere is, too, the case of the little boy gazing so intently into a\nhand-glass; what did he find hidden in its depths to make him suddenly\ncast it aside, and turn to where the photographer is presumably\nstanding, with such a look of mingled disappointment and disgust? His\ndiscovery was evidently remarkable, for, as will be seen, it had the\nadditional effect of taking his fore-lock out of curl. Again Mr. Byrne\nis appealed to, and again he smiles and vouchsafes the same reply. And\nagain the mothers of England will have their suspicions.\n\n[Illustration: \"RATHER SLOW, THIS!\"]\n\nIn photography, as in many another profession, the path of those who\nwould forsake the beaten track does not lie through acacia groves.\nMany obstacles strew the way. For instance, Mr. Byrne conceived the\nnotion of posing children in a big boot, such as appears in two of the\naccompanying reproductions. It is a simple-looking boot, yet it took\ntwo years to make; that is to say, the day the order was given, and\nthe day it was satisfactorily carried out, were separated by a span\nof such duration. But much happened in between. First of all Mr. Byrne\nmade a rough design of what he wanted, giving the dimensions, etc.,\nand sent it, through a friend, to a local bootmaker. Perhaps, not\nanxious to have his idea noised throughout camera-land, Mr. Byrne's\ninstructions were not as explicit as they might be; it was the time\nof the dynamitard outrages, and the worthy artist in leather grew\nsuspicious. What might his customer want with such a boot? it was most\nunusual; he had never heard of a man with such a large foot; and why\nonly _one_ boot? He didn't like such peculiar orders, but he would do\nwhat he could in the matter; of course it would cost a goodish sum. As\nto whether the poor man had nightly visions of the strange boot being\nfilled with infernal machines and placed under the House of Commons,\nand himself charged by the State with aiding and abetting the plot, no\nreliable information is forthcoming, but certainly, after two months\nhad elapsed, he sent word to say that he found the work more difficult\nthan he had anticipated, and that unless the one-legged individual,\nfor whom the boot was apparently intended, could call and be measured\nin the ordinary way, he must regretfully throw up the job. A carpenter\nwas next tried, but with little more success. The boot actually did\ncome home, after a time, but it was large enough for six children to\nlose themselves in, instead of comfortably accommodating one. At this\nstage it occurred to Mr. Byrne that one of the Drury Lane \"property\"\nmen, used to tailoring for pantomime giants and other unusual\ncreatures, would be able to make a boot a little bit out of the common\nwithout being too inquisitive as to its mission in life. So it finally\narrived, a beautiful creation, fit for any Brobdingnagian dandy, and\nredolent of Day and Martin. But Mr. Byrne wanted a dilapidated boot;\nto save further trouble, however, he proceeded, with the aid of a\npocket knife and an old hatchet, to dilapidate it himself.\n\n[Illustration: \"WHAT DOES ALL THIS MEAN?\"]\n\nIt was much the same with the egg appearing herewith. It was tried in\ncanvas, wood, and papier-mach\u00e9 before the more serviceable aluminium\nproduced a \"lay\" that any pantomime bird might be proud of. Both the\nboot and the egg have done yeoman service since. They have assisted in\nproducing something approaching a thousand photograph-pictures.\n\nWhen Mr. Byrne had shown that the novel could be blended with the\nartistic in child photography, would-be imitators were not slow to\nappreciate the innovation. Photographers wrote from all parts of the\ncountry to inquire where Mr. Byrne obtained his \"properties\"; they\nwould like to add duplicates to their own studios. Guileless Mr. Byrne\nreplied in each case--\"From America.\"\n\nIt would be as difficult for Mr. Byrne to say how he came to make a\nsp\u00e9cialit\u00e9 of child photography as it would to explain what led him to\ntake to the camera at all. He practically drifted into both.\n\n[Illustration: LITTLE MISS PENSIVE.]\n\n\"My life,\" he says, \"might almost be described as one long drift.\nAlthough I was born in Ireland, I entered the Italian army, for some\nreason no one, not even myself, has ever been able to explain, when\nI was fifteen years of age. I fought in the Austro-Franco-Italian\ncampaign of 1859, being one of the only two Britishers engaged in the\nwar. The other was the late Colonel Peard. Like him, I was present\nat the great battles of Magenta and Solferino, and like him I several\ntimes came near ending my career on the plains of Lombardy.\n\n\"When the war was over, and there seemed no prospect of another,\nI drifted away from the colours, back to London, and into the more\npeaceful occupation of portrait making. Photography was in its infancy\nin those days; and I can only presume that I took kindly to it because\nI had always been something of an artist, which was of considerable\nadvantage to me in my new profession. After a while I discovered\nthat I secured happier results with children than was usual, probably\nbecause, being naturally fond of them, I devoted more care and\nattention to them than was customary, for child sitters were rare\nthen, and photographers were apt to regard them as rather bad\nbargains, notwithstanding the higher fees charged.\"\n\n[Illustration: LITTLE DAMES IN FANCY FRAMES.]\n\nIt is not surprising to learn that Mr. Byrne has \"snapped\" nearly all\nthe little English royalties; his studios being located at Richmond in\nSurrey, he has frequently been summoned to the White Lodge and Windsor\nCastle, not to mention more distant royal seats. Regarding juvenile\nprinces and princesses, Mr. Byrne has something interesting to say.\n\n[Illustration]\n\n\"Royal children have charming and simple ways, and it is usually\nan easy matter to establish friendly relations with them. That once\naccomplished, photographing always becomes a pleasure. They are just\nas fond of new toys as other children, just as eager for a romp,\nfrequently more so. Indeed, I have more than once had to 'play\nsoldiers' with some of the Queen's grandchildren while waiting for\nthem to reach a sitting-still mood.\"\n\n[Illustration]\n\nPerhaps much of Mr. Byrne's success is due to the fact that\nphotography is not merely his profession; it is his hobby as well.\nHe is constantly devising new and novel poses, both for his private\nsitters and the child-models he employs. When a mother calls at the\nstudio with her bairn, she is shown numerous bulky volumes, veritable\npicture galleries of children in almost every conceivable attitude,\nand invited to choose which she would prefer for her own little one's\nportrait. Bewildered by such a display, she usually elects to leave\nthe choice to the photographer, and she is invariably wise; for the\npose that shows to perfection the characteristic beauty of one child\nmay be quite unsuitable for another. A trained eye notices these\nthings as quickly as a practised artist can tell whether a certain bit\nof scenery is \"paintable\" or not. One of Mr. Byrne's child-models must\nbe the most photographed little person in the country, for she has\nfigured in no fewer than two thousand studies. To show the variety of\nthese, a page herewith is devoted to a composite reproduction of the\nlittle model in some of her happiest poses. The sweet child, it might\nbe mentioned, is the daughter of Mr. Byrne's laundress.\n\n[Illustration: A CHILD MODEL--SOME OF HER HAPPIEST POSES.]\n\nMr. Byrne has three \"Don'ts\" for mothers who would secure speaking\nlikenesses of their little ones. Don't let the little one know\nbeforehand that it is going to a studio. Don't dress it in any costume\nto which it is unaccustomed. Don't endeavour to arrange its hair; this\nwill look better if allowed to fall naturally.\n\n[Illustration: SOMEBODY'S NEST-EGG.]\n\nIt must not be supposed, however, that the subject of this article is\na photographer of children only. Between them, eighty-nine exhibitions\nin various parts of the world have awarded him something like a\nquarter of a hundredweight of medals for exhibits of all kinds, while\nhe has also photographed nearly every adult member of our royal family\nand innumerable celebrities. In this branch of his work, however, Mr.\nByrne has to content himself with the artistic, and leave the novel\nseverely alone. For the Prince of Wales, considerate sitter though he\nis, would hardly consent to have himself \"caught\" in an old boot, or\nSir Ellis Ashmead Bartlett, though he does much to amuse his fellow\nmembers of Parliament, permit himself to be represented emerging from\nan aluminium egg. So the \"properties\" have to be laid aside at times.\n\n[Illustration: BUILT FOR ONE.]\n\n\"The Queen,\" says Mr. Byrne, \"is one of the best sitters in the\nworld, very rarely moving or spoiling a plate. That abomination of all\nstudios, the head rest, is quite unnecessary in Her Majesty's case.\nThe Prince of Wales takes quite a keen interest in photography\nhimself, and when sitting will go to considerable personal trouble in\norder that the results may be successful. The Prince will get together\na royal group where anyone else would fail. The German Emperor is a\nmost genial sitter; but his pose before the camera is apt to be rather\nstiff, and his expression somewhat stern. Nearly every royal sitter,\nin fact, has some peculiarity which one finds it one's task to\nmoderate without destroying altogether.\"\n\n[Illustration: \"I DON'T LIKE PHOTOGRAPHERS!\"]\n\nBut we are no nearer Mr. Byrne's secret. How were the \"unhappy\"\npictures obtained? Stories are told of fond mothers, waiting\nin anterooms, being horrified to suddenly hear piercing shrieks\nproceeding from the studio. Breathlessly they have dashed in, to find\nMr. Byrne all smiles, baby all tears. When the proofs came home, the\npicture showing baby crying was generally voted wonderfully lifelike,\neven if it was not selected for general distribution.\n\nThese strange rumours are referred to, and once again Mr. Byrne is\nappealed to to withdraw the veil; but yet again he smiles and replies\nas before. Which reply, it is to be feared, the mothers of England\nwill accept with the customary condiment.\n\n[Illustration: POISON\n\n  POISONED RING.\n  POISONED GLOVES.\n  POISONED SWORD.\n  POISONED HOCKEY STICK.\n  POISONED SHIRT.\n  POISONED DAGGER.\n  POISONED BOOTS.\n  POISONED BOWL.\n]\n\n[Illustration: Decorative frame]\n\n\n\n\nDEVICES CONCEIVED BY THE GRIM AND GHASTLY INGENUITY OF OUR\nFOREFATHERS.\n\n\nThe ingenuity of man in devising schemes to perpetuate life has almost\nbeen equalled by his ingenuity in framing devices to abridge it. In\nall ages there have been men who hated convention, even conventional\nmurder. When they desired to remove an enemy secretly, they had\nrecourse to poison; but not to ordinary schemes of poisoning. They\nmade it an art.\n\nAn interesting device, very palpably suggested by the famous shirt\nof Nessus, is that given on the opposite page. The shirt was charged\nwithin with poison of great acerbity, which so acted upon the skin of\nits wearer as to inflame and then blister with incredible malignity,\nuntil the victim died in horrible agonies.\n\nVarious hypotheses have been put forward with respect to the history\nof the shirt we illustrate, which is now in an American museum; but\nthere seems every reason to believe that it is the garment mentioned\nin a German work of the 15th century by Adolph Beckert, as having been\nexhumed with the body of a Greek military captain, with an inscription\nto the effect that, having been guilty of gross insubordination,\namounting to treason, he was ordered to denude himself of his garments\nin the presence of the soldiers, and don the fatal tunic.\n\nOf equal, if not greater antiquity, was the device in vogue among the\nTartar princes.\n\nWhen a prince desired to dispatch an inconvenient subject, he was\ninvited to a game of hockey, participated in by the various nobles,\nofficers and officials of the Court. The Royal Chamberlain always\ntook good care that the technical miscreant should play with a special\nstick, within whose jewelled hilt lurked the deadly poison. But even\nthe drug would have been scarcely sufficient to destroy life if there\nhad not been also concealed in the handle a number of microscopic\nneedle points which, tiny as they were, and almost unobserved at the\nfirst handling, were quite sufficient after a few minutes' play to\npuncture the skin.\n\nIn this connection we may advert to the vast number of swords and\ndaggers extant whose tips were impregnated with poison; these are to\nbe found in many museums in Europe. There are not a few even in our\nown Scotland Yard. Although legitimately poison devices, they betray\nlittle ingenuity.\n\nThe pair of gauntlets shown on the opposite page are said to have been\nthe property of a great foreign political notability. Once well on\nthe hands of the victim, a species of gum, with which the interior\nwas lined, adhered with such tenacity to the flesh that it was next to\nimpossible to remove the gloves, except by cutting them away; and\neven then the gum remained impervious to water, or of solution, until\ninflammation was succeeded by festering, and unless amputation of the\nwhole arm followed, death inevitably ensued.\n\nSeveral deaths from the wearing of poisoned boots have been recorded,\nnotably that of Andre Nolofski, courier to the Russian Empress\nCatherine the Great, who was discovered some nine miles from Moscow,\nlying prone dead on the side of the road, with apparently not a mark\nof disfigurement, until someone removed his boots. Then a surgeon\ndiscovered that tiny poisoned needles had pierced his feet.\n\nPoison rings were not unknown to the Romans. A hollow duct was\ncontained in half the ring, from which poison was supplied to a needle\nof infinitesimal size at the opposite side. The other half of the\ncircle enclosed a very delicate mechanism for operating the needle.\nWhen, with a slight pressure, the hand of the enemy was clasped with\napparent cordiality and good-fellowship, it gently punctured the skin\nwithout exciting suspicion.\n\nAnother device we illustrate was in very common use amongst the\nChinese up to a century ago. It consisted of a bowl heavily coated\nwith a colourless soluble poison on the inside. Upon any hot liquid,\nsuch as tea, being poured into this cup, the poison became dissolved.\nAs a whole service of this ware might come into the possession of one\nfamily without suspicion, it is not difficult to credit the fact that\nin spite of every precaution about food, seven or eight persons were\noften exterminated by this process.\n\n\n\n\nOUR MONTHLY GALLERY OF BEAUTIFUL AND INTERESTING PAINTINGS.\n\n\n[Illustration:\n_By permission of the Berlin Photographic Co., London, W._\n\nA FLOWER OF SPAIN.]\n\n[Illustration:\n_By permission of the Berlin Photographic Co., London, W._\n\n\"HUSH.\"]\n\n[Illustration\n _By permission of the Berlin Photographic Co., London, W._\n\nTHE FAVOURITE.]\n\n[Illustration:\n_By permission of the Berlin Photographic Co., London, W._\n\nMEDITATION.]\n\n[Illustration:\n_By permission of the Berlin Photographic Co., London, W._\n\nGOOD-NIGHT!]\n\n       *       *       *       *       *\n\n\n\n\nTranscriber's Note\n\n\nPage 14: 'busses' corrected to 'buses', though 'busses' may have been\nin use in 1898, or may have been confused with fishing boats (herring\nbusses).\n\n\"... he even saw cabs and buses with different eyes.\"\n\nPage 40: 'our's' corrected to 'ours' (though 'our's' may have been in\nuse in 1898).\n\n\"From the American aspect, ours is dull,...\"\n\nPage 56: 'Meterological' corrected to 'Meteorological'\n\n\"the Meteorological Office, a Government department....\"\n\nPage 78: 'ofence' corrected to 'offence'.\n\n\"... to have sufficiently purged himself of his offence?\"\n\nPages 93-97: 'bluejacket' (noun) appears a number of times;\nblue-jacket (adjective) appears once.\n\n'workwoman' and 'work-woman' also both appear, in different stories.\n\nNumerous occurences of words which are sometimes joined by a hyphen,\nand sometimes separate, also appear (e.g. 'bugle-call' and 'bugle call').\n\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Harmsworth Magazine, v. 1\n1898-1899 - No. 1, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nThank you for buying this ebook, published by NYU Press.\n\nSign up for our e-newsletters to receive information about forthcoming books, special discounts, and more!\n\n**Sign Up!**\n\n### **About NYU Press**\n\nA publisher of original scholarship since its founding in 1916, New York University Press Produces more than 100 new books each year, with a backlist of 3,000 titles in print. Working across the humanities and social sciences, NYU Press has award-winning lists in sociology, law, cultural and American studies, religion, American history, anthropology, politics, criminology, media and communication, literary studies, and psychology.\n**The Queer Renaissance**\n\n# **_The Queer Renaissance_**\n\nContemporary American Literature and the Reinvention of Lesbian and Gay Identities\n\nRobert McRuer\n\nNEW YORK UNIVERSITY PRESS  \nNew York and London\n\n\u00a9 1997 by New York University\n\nAll rights reserved\n\nLibrary of Congress Cataloging-in-Publication Data\n\nMcRuer, Robert, 1966\u2013  \nThe queer renaissance : contemporary American literature and the  \nreinvention of lesbian and gay identities \/ Robert McRuer  \np. cm.  \nIncludes bibliographical references and index.  \nISBN 0\u20138147-5554\u20132 (clothbound : alk. paper).\u2014ISBN  \n0\u20138147-5555\u20130 (paperbound : alk. paper)  \n1. Gays' writings, American\u2014History and criticism.  \n2. Homosexuality and literature\u2014United States\u2014History\u201420th  \ncentury. 3. Literature and society\u2014United States\u201420th century-  \nHistory and criticism. 4. American literature\u201420th century-  \nHistory and criticism. 5. United States\u2014Civilization\u20141970\u20136. Gender identity  \nin literature. 7. Lesbians in literature.  \n8. Gay men in literature. I. Title.  \nPS153.G38M38\u20131997  \n810.9\u2032920664\u2014dc21 96\u201351220  \nCIP\n\nNew York University Press books are printed on  \nacid-free paper, and their binding materials are chosen  \nfor strength and durability.\n\nManufactured in the United States of America\n\n10 9 8 7 6 5 4 3 2 1\n\n## **_Contents_**\n\nPreface\n\nAcknowledgments\n\n[**_Introduction_**  \nReading the Queer Renaissance](intro.html#intro)\n\n[**_Chapter One_**  \nBoys' Own Stories and New Spellings of My Name: Coming Out and Other Myths of Queer Positionality](ch01.html#ch01)\n\n[**_Chapter Two_**  \nQueer Locations\/Queer Transformations](ch02.html#ch02)\n\n[**_Chapter Three_**  \nUnlimited Access? Queer Theory in the Borderlands](ch03.html#ch03)\n\n[**_Chapter Four_**  \nQueer Identities in a Crisis](ch04.html#ch04)\n\n[**_Epilogue_**  \nPost-Queer?](epilogue.html)\n\nNotes\n\nWorks Cited\n\nIndex\n\n## **_Preface_**\n\nEvery year, countless Americans from the forty-eight contiguous states flock to Hawaii, making tourism that state's number one industry Travel agents, hotels, and airlines entice tourists by shaping the Aloha State into the land of beauty, romance, and imagination. A recent _Vacations by Sheraton_ brochure, for instance, announces, \"Hawaii is as much a place of mind as it is a place on the map. A peaceful paradise where golden sunshine warms the spirit along with the body Where worry drifts away on azure waves.\" MTI Vacations adds, \"Hawaii is a honeymoon paradise! You'll be inspired by the breathtaking scenery and swept up in the romance of the islands.\" The industry supports these claims with visual evidence, scattering pictures of men and women in luxurious settings through every brochure: men and women standing beside waterfalls or pools, walking along the beach, or watching the tropical sunset.\n\nGay men and lesbians, however, have disrupted the fantasy of Hawaii as a romantic (heterosexual) paradise. In 1990, three same-sex couples filed a lawsuit claiming the state violated their civil rights by refusing to grant them marriage licenses. The Hawaii Supreme Court agreed and ordered a lower court to review the couples' complaints. Since states have a long history of recognizing marriages legally performed in other states, the events in Hawaii provoked a flurry of legislative activity elsewhere. By the mid-1990s, legislators across the country had introduced bills that would ensure that their states would not have to recognize same-sex marriages performed in Hawaii. Congress even introduced the Defense of Marriage Act (DOMA), and members of both major political parties lined up behind it to show their support for the idea that marriage should be defined only as the union between one man and one woman. DOMA was passed and signed into law by President Bill Clinton in September 1996.\n\nIn this context, the already suspect metaphor of the mainland collapses under the weight of the crisis. Hawaii indeed becomes \"the mainland\" here, given its ability to drive the legislative agenda of the U.S. Congress and of many other states in the Union. The legislators' panic, and that of their constituents, endows Hawaii with a threatening subjectivity that belies characteristic understandings of it as an island paradise passively awaiting heterosexual travelers. No longer simply a beautiful and distant place to escape _to_ , Hawaii\u2014because it broaches the taboo subject of same-sex marriage\u2014becomes a subject to escape _from_.\n\nNonetheless, to many gay men and lesbians (and indeed, to the litigating couples themselves), gay and lesbian marriage hardly constitutes a threat to civilization as we know it. On the contrary: proponents of lesbian and gay marriage openly admit that they don't want to threaten heterosexual marriage; instead, they want to _uphold_ marriage by extending it to lesbians and gay men. Other gay men and lesbians see the issue somewhat differently. While not denying that lesbians and gay men should have equal access to the political and economic benefits marriage bestows on couples in American culture, they argue that the gay and lesbian marriage movement seeks admittance to an institution that gay liberationists have sharply and consistently critiqued.\n\nThe debate over marriage rights in Hawaii might actually extend those critiques, since it again makes visible the crises that circulate around marriage and \"the family\" The legislative panic in particular exposes the ways in which the idea of the family is grounded in a fear of _contiguous states:_ that is, a fear of queer states and eroticisms that are supposed to be safely distant, marginal, and exotic but are, in actuality, always internal to the family, the community, the nation. The debate about marriage rights in Hawaii reveals that those queer states (indeed, like Hawaii) are not \"naturally\" distant, marginal, and exotic; rather, they are distanced, marginalized, and _exoticized_ through the heteronormative processes that construct \"the family,\" \"the mainland,\" and \"civilization as we know it.\" It is crucial that queer activists and theorists continue to draw attention to those heteronormative processes, especially since the lesbian and gay marriage movement in fact repeats them; many advocates for same-sex marriage distance queer identities and communities that don't fit their \"new and improved\" model of family.\n\n_The Queer Renaissance: Contemporary American Literature and the Reinvention of Lesbian and Gay Identities_ examines how queer writers of the past fifteen years have shaped alternative relationships, communities, and identities. The 1980s and 1990s have seen an unprecedented wave of cultural activity by openly queer poets, playwrights, and novelists. _The Queer Renaissance_ not only highlights that phenomenon but also considers how it works in tandem with a \"renaissance\" of radical queer political analysis that reinvents lesbian and gay identities and alliances in order to challenge dominant constructions of sex, sexuality, gender, class, and race.\n\nThe queer alliances and interconnections writers and activists have shaped cannot be contained by the model of a well-behaved same-sex dyad, and _The Queer Renaissance_ thus brings together both women _and_ men from a variety of overlapping communities and locations: women working together as friends and lovers, female and male activists protesting home-lessness and inadequate health care, lesbians of color _hacienda alianzas_ (forging alliances), queers in the streets chanting, \"We _are_ fabulous\u2014get used to it!\" or \"Safe sex\u2014do it!\" Although putting men and women together in one study will undoubtedly strike some readers as monstrously unnatural, such a queer gathering is actually more representative of the ways men and women have indeed worked together in many locations over the past fifteen years. The field of contemporary gay and lesbian literary studies, which generally focuses on writing by _either_ men _or_ women, in this sense lags behind gay and lesbian communities more generally.\n\n_The Queer Renaissance_ brings together Audre Lorde, Edmund White, Randall Kenan, Gloria Anzald\u00faa, Tony Kushner, and Sarah Schulman and considers how or whether these writers breed fluid and disruptive identities and communities. The conversation _The Queer Renaissance_ stages among these writers intends to extend that disruption and fluidity. However, I unavoidably erect other barriers or establish new boundaries that can, and I hope will, be disrupted in turn. Most obviously, I focus here on \"American\" writers. Without question, the cultural phenomenon I examine crosses many different geographical and metaphorical borders, making my American focus in many ways arbitrary. My (partial) rationale for this focus is twofold. First, the field of literary studies continues to be segregated along national lines, and American literary studies provides one venue in which _The Queer Renaissance_ might speak. At the same time, although this study thereby perpetuates the segregation of literary studies, it also (like lesbian and gay studies generally) questions from within many of the academic boundaries we establish. Second, disparate as the writers in this study might be, they are all subject, although in different ways, to various American institutions: religious life, the medical establishment, the U.S. Congress, the border patrol, the marketplace, and so forth. Therefore, many of the disruptions envisioned and detailed by writers in _The Queer Renaissance_ are disruptions of specific American institutions or forces.\n\nStill, the \"unity\" I achieve through this focus is temporary and at many points contested, and the margins of my study threaten to take it elsewhere. Latin America, China, Italy, Australia; Edmonton, Berlin, London, Tel Aviv\u2014these are a few of the places that I tuck away in endnotes or that the writers I examine mention in passing. Other locations\u2014Carria-cou, Mexico, West Africa\u2014already resist marginalization in my study, since they have shaped the consciousness of the writers at its center. These locations begin to reconfigure _The Queer Renaissance_ and to position the writers and texts here in alliance with other people, places, or contiguous states. Such an ongoing reconfiguration, however, is desirable and necessary: although most of the writers I analyze expose the ways normative identities are structured to distance, marginalize, and exoticize others, they do not do so from some imaginary \"uncontaminated\" location; although they refuse to be kept in their place, they simultaneously refuse to establish a pure \"paradise\" elsewhere. The writers at the center of _The Queer Renaissance_ perversely insist that we can and should sustain the messy processes of interrogating the exclusions our identities and communities effect and of imagining our communities and identities otherwise.\n\n## **_Acknowledgments_**\n\nI am grateful to Eric Gardner, Robert Nowatzki, and Michael Thurston for their careful reading of each draft of this book. Our many hours of conversation about this project have been invaluable. I especially thank Michael B\u00e9rub\u00e9 for his ongoing assistance, encouragement, and inspiration, and for his thoughtful criticism of my work at various stages. I am also very grateful to Lisa Duggan, Robert Dale Parker, and Paula Treichler for their insights and encouragement.\n\nElizabeth Davies read and commented on much of the manuscript, and many other colleagues and friends read drafts of individual chapters: Stacy Alaimo, Amanda Anderson, Steve Amarnick, Beth Coleman, Jon D'Errico, Sagri Dhairyam, Amy Farmer, Will Harris, Randall Kenan, Kenneth Kidd, Barb Sebek, Siobhan Senier, Tamise Van Pelt, and Liesl Ward. Linde Brocato and Evan Engwall advised me on the Spanish translations in chapter 3.\n\nI want to thank Jennifer Hammer at New York University Press for soliciting this project, and for her patience and assistance along the way. This book was made possible through fellowships granted to me by the Graduate College and the Department of English at the University of Illinois at Urbana-Champaign. I am grateful to the Department of English and the Unit for Criticism and Interpretive Theory for their support of both my work and the work of lesbian and gay studies more broadly.\n\nFinally, and most important, this project would not have been completed without the love and support of Tom Murray, who saw me through from beginning to end.\n\nAn earlier, shorter version of chapter 1 was published _Genders_ 20 (1994): 260\u201384. Copyright 1994 by New York University Press. Reprinted with permission.\n\n## **_Introduction_  \nReading the Queer Renaissance**\n\nIn one of the most fabulous moments in the introduction to her _Epistemol-ogy of the Closet_ , Eve Sedgwick considers the analysis and recovery work being done in gay and lesbian literary studies on the Harlem Renaissance, the New England Renaissance, and the Renaissance in Italy and England. As Sedgwick sees it, an antihomophobic inquiry into each of these renaissances promises to reconfigure completely the ways in which we conceive of the period in question. Although we \"can't possibly know in advance ... where and how the power in [these renaissances] of gay desires, people, discourses, prohibitions, and energies were manifest,\" Sedgwick asserts that the questions currently being asked by scholars about these movements reveal that such desires and discourses were nonetheless undoubtedly central (58\u201359). She then concludes the section, characteristically, by \"flaunting it\": \"No doubt that's how we will learn to recognize a renaissance when we see one\" (59).\n\nIf the centrality of lesbian and gay desires to a moment of distinctive cultural production is indeed one marker of a renaissance, then we should recognize the past fifteen years as just such a renaissance. Before the 1969 Stonewall Riots ushered in the contemporary gay liberation movement, material representations of same-sex desire in American literature and the arts were few and far between. Even in the 1970s, although openly gay and lesbian literature had begun to appear more regularly, such work was still quite rare. In the 1980s and 1990s, however, all that changed. Literally thousands of novelists, poets, and playwrights published or performed works about lesbian and gay people. Writers' groups and workshops, such as the Violet Quill Club, Other Countries, and Flight of the Mind, helped create and nurture this literary movement; literary magazines and book reviews devoted entirely to gay and lesbian writing propelled the movement forward. Annual Lambda Literary Awards\u2014or \"Lammies\"\u2014even provided institutional recognition of outstanding achievement in lesbian and gay literature. Straight America began to take notice. As journalist David Gates observed, \"At least since Edmund White's best-selling 'A Boy's Own Story' in 1982, major American houses have published more and more of such writing; even the staid Book-of-the-Month Club has announced a gay and lesbian reprint series\" (Gates and Malone, 58). Gates's incredulity about the supposedly staid Book-of-the-Month Club could have been as easily directed inward: his comments about the \"explosion\" of gay writing appeared in a magazine no less mainstream than _Newsweek_. Not surprising, by 1993, when this _Newsweek_ article was published, gay and lesbian writers and publications had been taking note of the cultural phenomenon for quite some time.\n\n_The Queer Renaissance: Contemporary American Literature and the Reinvention of Lesbian and Gay Identities_ puts a name to this unprecedented outpouring of lesbian and gay creative work. By naming and discussing this efflorescence of creative activity as a queer \"event,\" I work in this study both to secure a context for analysis of the period and to promote critical representation of the texts and authors involved. Many gay and lesbian writers have noted this cultural renaissance; my project is among the first to analyze the movement critically.\n\nThrough this analysis, I hope to complicate and illuminate the position of lesbian and gay writers in contemporary literary studies generally. When contemporary gay or lesbian writers are studied or taught individually, they are easily exoticized; in other words, they are inserted into a general academic market for difference that is able to consider and contain queer desire temporarily, before going on with business as usual. \"Business as usual\" in this case entails the reproduction of multicultural narratives of American literary history that celebrate and consume difference, including sexual difference, while simultaneously positing, without acknowledgment, an unmarked (and therefore heterosexual) consumer for whom this \"difference\" is an addendum to the straight narrative. If contemporary queer work is articulated to the idea of a collective movement or renaissance, however, it might be seen as disrupting or transforming the straight narrative and business as usual.\n\nThis disruption or transformation will definitely not be easy or automatic. Certainly, in literary studies there are many institutional pressures and expectations (such as the expectation that a given course will \"survey\" a culture or time period) that are not easily overcome. At the very least, I hope my analysis complicates the tendency to have days or weeks devoted to the study of \"difference.\" By reading contemporary gay and lesbian\n\nwriters as part of a larger contemporary renaissance, I mean to move them from the margins to the center of the stories we tell ourselves about contemporary literary history.\n\nAt the same time, there is more than a little irony involved in considering a \"rebirth\" of any sort during a period when thousands have died of complications related to acquired immune deficiency syndrome (AIDS). For that reason, my study tries to resist\u2014or _queer_ \u2014certain characteristic tendencies of renaissance thinking. This \"renaissance\" is \"queer\" not only because it is by and for lesbians and gay men but also because it is _different_ from other renaissances. Indeed, this is not a renaissance at all, if renaissances are understood\u2014as they traditionally are\u2014as securing a \"great nation's\" cultural position through \"transcendence\" of that nation's historical location. Instead, this is a queer renaissance rooted in particular communities, histories, and struggles. Moreover, this is not simply a cultural renaissance but a rebirth of radical gay and lesbian political analysis, which recalls the formation of the Mattachine Society in the 1950s or the Gay Liberation Front in the 1970s\u2014two other twentieth-century moments when lesbian and gay people came together to reimagine and reshape culture and identity.\n\nSuch a reshaping of _identity_ , like the unprecedented outpouring of gay and lesbian literature that many critics have noted, has not been overlooked by contemporary commentators, who have positioned \"queerness\" in particular as a supplement to understandings of sexuality that posit homosexuality and heterosexuality as fixed, immutable, and\u2014supposedly\u2014equal and opposite identities. Influenced by both poststructuralist theory and the flamboyant queer activism of the late 1980s, these critics have theorized queer identity as an alternative to such rigid understandings of sexuality. Sedgwick, for example, begins _Tendencies_ , the book she wrote concurrently with _Epistemology of the Closet_ , with an extended meditation on the identities being deployed around her:\n\nAt the 1992 gay pride parade in New York City, there was a handsome, intensely muscular man in full leather regalia, sporting on his distended chest a T-shirt that read, KEEP YOUR LAWS OFF OF MY UTERUS.\n\nThe two popular READ MY LIPS T-shirts marketed by ACT UP were also in evidence, and by the thousands. But for the first time it was largely gay men who were wearing the version of the shirt that features two turn-of-the-century-looking women in a passionate clinch. Most of the people wearing the version with the osculating male sailors, on the other hand, were lesbians. FAGGOT and BIG FAG were the T-shirt legends self-applied by many, many women; DYKE and the more topical LICK BUSH by many, many men. .. .\n\nAnd everywhere at the march, on women and on men, there were T-shirts that said simply: QUEER.\n\nIt was a QUEER time. ( _Tendencies_ , xi)\n\nQueer, indeed. Sedgwick's overview of the 1992 parade makes it clear that these new identities were, above all, not fixed in discrete gendered locations; instead, identities were reborn in the interstices between various genders and sexualities. Moreover, this queer identification across difference often sharply (and with a camp flair) critiqued powerful American institutions\u2014most obviously here through the \"Keep Your Laws Off of My Uterus\" and \"Lick Bush\" T-shirts but also through ACT UP's graphics and T-shirts generally, since the AIDS Coalition to Unleash Power (ACT UP) was, after all, formed to address directly and explicitly the government's neglect of people living with AIDS. Indeed, the very slogan \"Read My Lips\" was intended both to ridicule George Bush's vacuous uses of the phrase and to draw attention to the government's overwhelming silence on issues of concern to lesbians and gay men, and to the many different people living with AIDS.\n\nIn this project, I argue that the renaissance of gay and lesbian creative work has not emerged alongside or above the renaissance of queer identities and political analyses; instead, the efflorescence of creative work is contingent on and, in turn, represents and fuels the proliferation of queer identities and political analyses. Both dimensions of the Queer Renaissance provide readers with fluid identities that are shaped and reshaped across differences and that interrogate and disrupt dominant hierarchical understandings of not only sex, gender, and sexuality but also race and class. In the chapters that follow, these identities take different names (Zami, trickster, new mestiza, AIDS activist) but share this queer commitment to fluidity and disruption. Indeed, to paraphrase Sedgwick, it is, no doubt, in such fluidity and disruption that we will learn to recognize a queer renaissance when we see one.\n\nThe remainder of this introduction locates the Queer Renaissance in and around contemporary debates about the uses and abuses of renaissance thinking. In the first section, through a brief survey of critical work on renaissances in literary studies, I consider how the critiques leveled by scholars in various literary fields might be used to read renaissance discourse in American culture more generally (including recent discussions of the supposed cultural renaissance brought on by the AIDS crisis). I then foreground a different kind of renaissance altogether: looking to lesbian and gay history, particularly the history of the 1950s and the 1970s, I argue that the contemporary emergence of radical gay and lesbian political analysis and activism is actually a reemergence, with antecedents in earlier lesbian and gay communities. The cultural production of the past decade and a half can and should be read through this political and historical rebirth, rather than through more traditional, transcendent notions of a renaissance.\n\nReading the Queer Renaissance alongside and against other American renaissances of the twentieth century, particularly the Harlem and Chicano Renaissances, the next section of the introduction considers whether these renaissances might also be understood as antecedents to the Queer Renaissance. The Chicano and Harlem Renaissances at times critique powerful American institutions and construct new, more fluid subjectivities, but despite these similarities, the identities shaped in both are generally less fluid than the identities shaped in the Queer Renaissance. Moreover, in the Harlem Renaissance in particular, the relationship between art and politics is much more tenuous than it is in the Queer Renaissance. Therefore, I argue that contemporary authors reconceive the identities shaped in earlier literary movements such as these.\n\nFinally, I position my project, and the writers who are central to it, within current contested understandings of queerness. Although only a few of the writers in _The Queer Renaissance_ have been associated with the movement overtly labeled New Queer Writing by Dennis Cooper and others, they share (and in many ways extend) the New Queer Writers' emphasis on a critical perversion that continuously forges unexpected alliances and gives voice to identities our heteronormative culture would like to, but cannot, silence.\n\n### **On Renaissances and Revolutions**\n\nRenaissances have long been staples of literary periodization, but in recent decades, scholars influenced by cultural studies and new historicism have tended to scrutinize the concept more closely. In his introduction to _The American Renaissance Reconsidered_ , for example, Donald E. Pease writes:\n\nThe term _American Renaissance_ designates a moment in the nation's history when the \"classics,\" works \"original\" enough to lay claim to an \"authentic\" beginning for America's literary history, appeared. Once designated as the _locus classicus_ for America's literary history, however, the American Renaissance does not remain located within the nation's secular history so much as it marks the occasion of a rebirth from it. Independent of the time kept by secular history, the American Renaissance keeps what we could call global renaissance time\u2014the sacred time a nation claims to renew when it claims its cultural place as a great nation existing within a world of great nations. Providing each nation with the terms for cultural greatness denied to secular history, the \"renaissance\" is an occasion occurring not so much within any specific historical time or place as a moment of cultural achievement that repeatedly provokes rebirth, (vii)\n\nDespite the imperative to be \"reborn\" outside secular history, however, the contributors to the project Pease introduces, as he goes on to explain, met specifically to _resist_ such demands, supposedly characteristic of \"global renaissance\" thinking generally. The ensuing volume consequently purports to reconsider the American Renaissance: \"the demand for rebirth was met, but this time the American Renaissance was reborn not _without_ but within America's secular history\" (vii). The essays that follow Pease's introduction\u2014\"Slavery, Revolution, and the American Renaissance,\" \"The Other American Renaissance,\" and five others\u2014thus consider how American works from this distinctive moment of cultural production neither transcended the contingencies of history and politics nor escaped nineteenth-century controversies about and around race, gender, and class. The volume's contributors, in short, counter the dislocating tendencies of renaissance discourse with an insistence on location and specificity.\n\nSuch an insistence is crucial, given that the dislocating tendencies of renaissance discourse in American culture are in evidence far beyond the literary sphere. Growing up outside of Detroit in the 1970s, for instance, I watched how that urban area was transformed into the \"Renaissance City\" (an epithet I later learned it shared with Pittsburgh). The sign of this supposed transformation was and is the colossal Renaissance Center: a circular, seventy-four-story glass and steel tower flanked by four similar, though somewhat shorter, towers. At first, this literal \"center\" of Detroit's renaissance might be understood as locating that movement, both geographically in the heart of the city and historically in a moment of urban renewal. Yet such an understanding of the Renaissance Center as the located sign and signature of the Renaissance City simultaneously covers over the inevitable dislocations of \"urban renewal.\" _Center_ is, of course, a verb as well as a noun, and we might do well to examine the process of centering that occurred in the construction of, and that continues to occur around, this particular center. Atop the Renaissance Center, what can be seen is decidedly not the poverty, violence, and despair that continue to affect the majority of the city's inhabitants. On the contrary, one is literally above all of that: the centering of vision that takes place at such a height assists in the construction of yet another Great American City filled with glass and steel skyscrapers. Indeed, especially at night with the glittering lights reaching as far as the eye can see, one could as easily be in, say, the Sun Belt as in the Rust Belt; that is, in the region that gained jobs even as the restructuring of labor and industry in the Midwest (a restructuring that the renaming of the Motor City as the Renaissance City covers over) cost jobs.\n\nHence, renaissance discourse in Detroit\u2014centered in a monument that \"transcends\" the material conditions of the city\u2014 _dis_ locates and _de_ specifies. This particular (dis)location ostensibly provides visitors with a plethora of perspectives on the city, but the revolving lounge-and-observation deck at the top of the Renaissance Center, while shifting constantly, actually goes nowhere. Furthermore, the supposed \"heart of the city\" is ironically empty of any long-term life; the \"tallest hotel in the world\" obviously does not house those who actually live in Detroit. Instead, many Detroiters, whose labor has been made marginal or superfluous by corporate and industrial restructuring, have been displaced to ghettos on the city's margins. These margins are precisely what become invisible by the centering of vision that renaissance discourse effects in Detroit.\n\nIn literary studies, the dislocations of renaissance rhetoric are never, perhaps, quite as menacing as those enacted by the idea of a renaissance in Detroit's fantasy of urban renewal. However, there are similarities, and as _The American Renaissance Reconsidered_ confirms, many literary scholars have begun to look carefully at how the concept is deployed. Indeed, in the past few decades an influential critical movement has worked methodically to rename another entire era in literary and historical studies, precisely to shed the negative baggage carried by \"the Renaissance.\" As Leah S. Marcus explains, _\"Early modern_ carries a distinct agenda for historians, who have adopted the name quite consciously as a sign of disaffiliation from what they perceive as the elitism and cultural myopia of an older 'Renaissance' history\" (41\u201342). Like the rhetoric of renewal in Detroit, much traditional\n\nRenaissance scholarship, according to Marcus, centers our vision of that era at the cost of the supposed cultural margins:\n\nThe term _Renaissance_ is optimistic, upbeat\u2014rebirth and renewal are marvelous ideas. One of the reasons many historians have become suspicious of the term is that it buys its optimism at too great a price\u2014the neglect of other cultural currents and forms of cultural production, of a vast sea of human activity and misery that _Renaissance_ either failed to include or included only marginally. The term _Renaissance_ implicitly calls for a perception of historical rupture (in order to be reborn, a culture must previously have died) and, along with that, a subtle hierarchical valuation of disparate cultures. (43)\n\nThe use of the term _early modern_ , in contrast, explicitly resists such hierarchical valuation and refocuses our vision, bringing critics down from the revolving lounge that \"Renaissance\" scholarship is always in danger of becoming.\n\nThe elitism and cultural myopia to which scholars of the early modern period object appear in characterizations of other renaissances, not least in characterizations of the renaissance of contemporary lesbian and gay culture under consideration here. Especially when the topic in question is AIDS, elitist and idealist pronouncements abound. Consciously or unconsciously, commentators often position art as the force that can \"purify\" the supposed stigma of AIDS. For this reason, Douglas Crimp in particular, in his introduction to the influential collection _AIDS: Cultural Analysis\/Cultural Activism_ , sharply chastises those who would deploy the rhetoric of renaissance when discussing the literature and art of AIDS. Writing to counter such \"idealistic platitudes\" as Elizabeth Taylor's assertion that \"art lives on forever\" ( _AIDS_ , 5), Crimp asserts that these clich\u00e9s communicate the message \"that art, because it is timeless and universal, transcends individual lives, which are time-bound and contingent\" ( _AIDS_ , 4). Crimp is critical of any artistic ideology that positions art as outside or above the contingencies of material human existence or\u2014even more perniciously\u2014as that which \"redeems\" human suffering:\n\nRedemption\u2014of course\u2014necessitates a prior sin\u2014the sin of homosexuality, of promiscuity, of drug use\u2014and thus a program such as \"AIDS in the Arts\" [a PBS report against which Crimp is arguing] contributes to the media's distribution of innocence and guilt according to who you are and how you acquired AIDS. Promiscuous gay men and IV drug users are unquestionably guilty in this construction, but so are all people from poor minority populations. ( _AIDS_ , 4n. 4)\n\nIn such an ideological context, in which the guilty offer up art as penance for their misdeeds, Michael Denneny's belief that \"we're on the verge of getting a literature out of this that will be a renaissance\" is insensitive and escapist (qtd. in Crimp, _AIDS_ , 4), and Richard Goldstein's that \"AIDS is good for art\" and \"will produce great works that will outlast and transcend the epidemic\" is politically reactionary (qtd. in Crimp, _AIDS_ ,5). These comments, and others like them, fetishize the artistic work and ignore how certain understandings of art require\u2014and indeed, produce\u2014the abject artist, the sinner whose transfiguration into the saint comes only through her art. Therefore, as Crimp convincingly establishes in his dismissal of such sentiments, \"we don't need a cultural renaissance; we need cultural practices actively participating in the struggle against AIDS. We don't need to transcend the epidemic; we need to end it\" ( _AIDS_ , 7)\n\nAs I have already insisted, however, the term _renaissance_ might be made to signify differently\u2014might, in short, be queered. Crimp himself implicitly opens the door to such a query of the concept, I think, when he faults media reports for failing to make \"any mention of _activist_ responses to AIDS by cultural producers\" ( _AIDS_ ,4). Crimp's point is not that activists are the alternative to artists, as if the two groups were always and everywhere distinct, but rather that activists resignify the \"artistic,\" providing an alternative to the idea of art as somehow redemptive or transcendent. Crimp's own subsequent analysis of _Let the Record Show_. . ., an installation produced by ACT UP in the Broadway window of New York's New Museum of Contemporary Art, presents readers with precisely such an activist\/artistic project: _Let the Record Show_. . ., in contrast to exhibitions sequestered within museums, highlighted to those outside on the street the negligence and bigotry exhibited during the AIDS crisis by the Reagan administration, the medical establishment, and organized religion ( _AIDS_ ,7\u201312). Extending Crimp's linkage of art and politics, I suggest that activism similarly does not preclude us from reading the contemporary queer moment as a renaissance, as if queer politics and renaissances were always and everywhere distinct. On the contrary: we might say that the contemporary moment presents us with a \"queer renaissance\"\u2014not in the sense of a cultural moment for gay men and lesbians that dislocates and despecifies, \"transcending\" history and politics, but in the sense of a cultural moment that engages history and politics and that recalls or cites other such moments. This is not simply a cultural renaissance, \"timeless and universal,\" but rather a specific time of enormous cultural productivity, integrally connected to another kind of renaissance: a renaissance of radical lesbian and gay political analysis. Shaped and reshaped precisely to resist the cultural myopia of those who celebrate art and discount ongoing structural inequities, and of all who maintain such inequities, this renaissance once again focuses our vision on the queer margins that are effaced (or \"redeemed\") by more elitist or escapist notions of cultural production.\n\nTwo moments in twentieth-century lesbian and gay history that might be recalled in a consideration of the contemporary queer period, among the many moments that this period explicitly or implicitly cites, are the early 1950s, when the Mattachine Society was founded by Harry Hay and four other leftists, and the early 1970s, when, after the Stonewall Riots, groups of young people came together to form the Gay Liberation Front (GLF). In his groundbreaking _Sexual Politics, Sexual Communities: The Making of a Homosexual Minority in the United States, 1940\u20131970_ and his subsequent _Making Trouble: Essays on Gay History, Politics, and the University_ , John D'Emilio presents the most comprehensive analysis of these time periods. Throughout his work, D'Emilio makes clear that there has been an ongoing tension in twentieth-century gay and lesbian history between radical liber-ationist and liberal reformist politics.\n\nIn the 1950s, D'Emilio explains, \"Hay and his partners . . . brought to their discussions an interest in uncovering a systemic analysis for social problems. As Marxists they believed that injustice and oppression came not from simple prejudice or misinformation but from relationships deeply embedded in the structure of society\" ( _SP_ , 64). For this reason, the early Mattachine Society set out to formulate a theoretical explanation for the homosexual's inferior status. The Mattachine Society began to argue that this status was not a function of individual deviance but instead resulted from the structure of American society, particularly from the structure of the dominant nuclear, heterosexual family, which \"equates male, masculine, man ONLY with husband and Father . .. and which equates female, feminine, woman ONLY with wife and Mother\" (qtd. in _SP_ , 65). One of the main goals of the early Mattachine Society, then (which was linked to the Marxist goal of distinguishing between a class \"in itself\" and a class \"for itself\"), was to make homosexuals aware of their position as \"a social minority imprisoned within a dominant culture\" (qtd. in _SP_ ,65).\n\nAlthough D'Emilio describes the Mattachine Society's first action outside their discussion group as a \"modest, undramatic one\" ( _MT_ , 30), it is nonetheless significant if we are to read the contemporary Queer Renaissance, which repeatedly stresses coalition across differences, through this earlier period. Police in Los Angeles had a long history of harassing the Chicano community in that city. Not long after the Mattachine Society began to meet in the early 1950s, incidents of harassment of Chicanos received increasing attention in the newspapers. The city government finally held public hearings on the incidents, and the founders of the Mattachine Society attended these hearings to voice their support for disciplinary action against the Los Angeles police officers involved. \"The rationale for their participation,\" D'Emilio explains, \"was their conviction that all socially oppressed minorities had something in common\" ( _MT_ , 30). Hence the early Mattachine Society's radical critique of American society extended beyond an isolated analysis of the homosexual's position within that society to a promotion of intersubjective relations and alliances outside of their own group.\n\nAs the 1950s progressed and the Mattachine Society grew, however, those favoring such a systemic and far-reaching critique of American society came into conflict with those preferring a liberal reformist approach. By mid-1953, leadership had transferred to a group of men intent on redefining the mission of the Mattachine Society. These men sought to replace the structural analysis of Hay and the other founders, with its emphasis on homosexuals as a social minority within a dominant heterosexual order, with a position that argued that, except for their sexual object choice, homosexuals were \"no different from anyone else.\" Assimilation to American society was the goal, and this goal could be met if only \"false ideas\" about homosexuality were corrected. Therefore, the new leadership of the Mattachine Society advocated \"aiding established and recognized scientists, clinics, research organizations and institutions . . . studying sex variation problems\" (qtd. in _SP_ , 81). D'Emilio concludes:\n\nAccommodation to social norms replaced the affirmation of a distinctive gay identity, collective effort gave way to individual action, and confidence in the ability of gay men and lesbians to interpret their own experience yielded to the wisdom of experts. Under its new officers, the Mattachine Society shifted its focus from mobilizing a gay constituency to assisting the work of professionals. ( _SP_ , 81)\n\nAs in American culture generally in the 1950s, so too in the later Mattachine Society: the new atmosphere was no longer congenial to the Marxist insights Hay and his cohorts initially brought to the movement.\n\nA similar \"retreat to respectability\" occurred not long after this within the autonomous lesbian organization the Daughters of Bilitis ( _SP_ , 75). Respectability, indeed, was the very basis for the formation of the Daughters of Bilitis (DOB). When the group first met in 1955, they explicitly emphasized that their organization was to be an alternative to the lesbian bars, which they considered \"vulgar and limited\" (Kennedy and Davis, 68). These lesbian bars, however, although not as numerous as gay male bars, were one of the few arenas where women had the opportunity and space to shape a collective lesbian identity. Elizabeth Lapovsky Kennedy and Madeline D. Davis explain:\n\nTough bar lesbians . . . expanded the presence of lesbians in the world of the 1950s. As they affirmed their right to live as lesbians, they made it easier for other lesbians to find them, and more difficult for the heterosexual community to ignore them. ... In addition, by spending as much time as possible in the bars under difficult conditions, tough bar lesbians created a strong sense of community solidarity and belonging that included women of diverse ethnic and racial groups. (111)\n\nKennedy and Davis's language, especially its emphasis on \"community solidarity,\" diversity, and difference from the heterosexual community, links the incipient analysis occurring in the lesbian bars to the analysis developed by Hay and the early Mattachine Society. The DOB's alternative to such group consciousness can also be linked to the \"alternatives\" emphasized by the leaders of the later Mattachine Society: the founders of the DOB \"saw education\u2014the dispelling of myths, misinformation, and prejudice\u2014as the primary means of improving the status of lesbians and homosexuals\" ( _SP_ , 102). As D'Emilio points out, the lesbian retreat to respectability, with its disavowal of bar culture, was particularly class-bound ( _SP_ , 106). However, the shifts in the Mattachine Society\u2014a demonizing of Marxism, an appeal to middle-class professionals, and a refusal to acknowledge that a group's status is a function of larger societal structures\u2014were also clearly class-bound.\n\nThus the radical beginnings of the lesbian and gay movement were overshadowed by a shift to liberal reformism. Although gay and lesbian organizations continued their activity throughout the 1950s and 1960s, they did so as part of a movement that \"could accurately be described as a reform movement solidly implanted in the American liberal tradition\" (MT, 239). It was not until the formation of the Gay Liberation Front in 1969 that the radical analysis of the early Mattachine Society was decisively \"reborn.\" The time was ripe for such a rebirth. Young lesbians and gay men were already involved in the antiwar movement, the Civil Rights movement, and radical feminism\u2014movements that had spoken and were speaking to American young people generally ( _MT_ , 241). When the police raided the Stonewall Inn Bar on the night of June 27, 1969, the rage and energy generated by the ensuing riots could be easily articulated to communities of young people who were already radicalized by these other movements. In such a context, the GLF was born, and many of the ideas expressed by their early Mattachine forebears were reborn:\n\nGLFers began to construct a rudimentary analysis of gay _oppression_. It was not a matter of simple prejudice, misinformation, or outmoded beliefs. Rather, the oppression of homosexuals was woven into the fabric of sexism. Institutionalized heterosexuality reinforced a patriarchal nuclear family that socialized men and women into narrow roles and placed homosexuality beyond the pale. These gender dichotomies also reinforced other divisions based on race and class, and thus allowed an imperial American capitalism to exploit the population and make war around the globe. ( _MT_ , 242)\n\nAs with the early Mattachine Society, the two main characteristics of the GLF's political vision were a commitment to an ongoing interrogation of the very structures of heterosexual, patriarchal, and capitalist society and a simultaneous recognition that resistant identities needed to be reshaped across differences. For this reason, as Terence Kissack explains, the GLF \"not only acted in defense of gay and lesbian rights, but also participated in antiwar demonstrations, Black Panther rallies, and actions undertaken by radical feminists\" (108). Over the next few years, GLF \"kindred groups\" spread to cities and college campuses all over the country ( _MT_ , 243).\n\nNot long after the formation of the GLF, however, a tension between the radical vision it espoused and more traditional liberal reformist impulses, similar to the tension that shifted the Mattachine Society's goals, emerged. The Gay Activists Alliance (GAA) was formed virtually on the heels of the GLF and was, not surprising, much narrower in its focus: \"Rather than try to destroy the old in order to build something new, they saw themselves as unjustly excluded from full participation in American society and wanted recognition and a place inside\" (MT, 247). The GAA was able to achieve several victories\u2014most notable, the removal of homosexuality from the American Psychiatric Association's list of mental disorders (MT, 249)\u2014but their shift to a single-issue politics eventually trumped the GLF's broader vision. A commercial subculture of restaurants, bars, and bathhouses flourished by the end of the 1970s and can be seen as an indirect consequence of activists' efforts, but this subculture was social and sexual rather than explicitly political ( _MT_ , 250\u201351). Understandably, just as women in many political movements were becoming dissatisfied with the overwhelming focus on male concerns and with ongoing sexist attitudes generally, many lesbians were dissatisfied with these new developments within the gay movement and with groups that focused on sexuality while virtually ignoring gender. This dissatisfaction led to the formation of autonomous lesbian-feminist groups. As the 1970s progressed, these autonomous groups themselves increasingly shifted from a structural critique of society to a more Utopian and separatist outlook ( _MT_ , 252).\n\nD'Emilio's summation of where all these shifts ended up is particularly astute. In stark contrast to the structural change envisioned by the GLF (and the early Mattachine Society), D'Emilio presents what may be the \"quintessential products\" of each wing of what he calls the \"gendered seventies\": \"the elaborate, glitzy, high-tech gay male discos found in many cities, and the self-sufficient, rural communes of lesbian separatists. Here were men, in a public space, spending money, focused on themselves, and searching for sex. And here were women, in a private retreat, financially marginal, focused on group process, and nurturing loving relationships\" ( _MT_ , 258). So much for structural critique and for identities reshaped across differences. Although D'Emilio acknowledges that \"some scrambling of gender characteristics did occur,\" his critical insight is that traditional understandings of sex and gender were, ironically, very much intact in gay and lesbian communities as the 1970s concluded (MT, 258).\n\nDespite what D'Emilio sees as a retrenchment at the end of the 1970s, he is not the only theorist to note that the AIDS crisis and other factors in the 1980s and 1990s have incited another period of political rebirth. Dennis Altman writes, \"It has struck me that AIDS is leading gay men and our organizations towards some of the practices associated with the gay liberation movements of fifteen years ago\" (41). Sarah Schulman broadens this observation, suggesting that feminist _critiques_ of \"practices associated with the gay liberation movements\" have actually reconfigured \"our\" organizations, taking them further than they went in the early 1970s:\n\nThe coming together of feminist political perspectives and organizing experience with gay men's high sense of entitlement and huge resources proved to be a historically transforming event. Necessity was the best motivator for efficiency, and for the most part ACT UP was able to function with a wide coalition and broad divergence of opinion. . . . ACT UP produced the largest grass-roots, democratic, and most effective organizing in the history of both the gay and feminist movements. ( _My American History_ , 11\u201312)\n\nSchulman's account of the early days of ACT UP implies that, if gay men's inability to function with \"a wide coalition and broad divergence of opinion\" led to the fracturing of many gay and lesbian organizations in the early 1970s, a revitalization of lesbian and gay politics occurred in part because some gay men, however tentatively, began to learn the lessons of feminism and to work more effectively with lesbian and straight feminists.\n\nBeyond this, D'Emilio factors in three additional catalysts that provoked a widespread rebirth of radical lesbian and gay political analysis: the rise of the New Right, the \"sex wars\" of the early 1980s, and independent organizing by lesbians and gay men of color ( _MT_ , 258\u201362). As early as 1977, the New Right was specifically targeting gay men and lesbians through Anita Bryant's campaign to encourage voters to repeal gay rights legislation in Dade County, Florida. Bryant's campaign and others brought home to activists both the dangers of complacency and the need for an ongoing analysis of the ways in which gay and lesbian oppression is built into the very structure of society. More positively, the \"collapse of the lesbian-separatist Utopia\" ( _MT_ , 259) and the ensuing sex wars of the early 1980s brought to the forefront the voices of s\/m lesbians and other \"sex radicals.\" These diverse voices called for a more consistent attention to differences within feminist and lesbian communities. The sex wars were fought over issues such as pornography, sadomasochism, sex work, and alliances with men, and they made possible an understanding of sexuality as more fluid and discontinuous than 1970s utopianism had admitted. At the same time, through a series of moves that similarly brought to the forefront voices that had been previously ignored or suppressed, independent organizing by lesbians and gay men of color finally provoked (some) white gay men and lesbians to attend to racial privilege and to the tendency to disavow racial divisions within lesbian and gay communities.\n\nTo all theorists of this period, however, AIDS is the central issue that elicited a sense of political urgency among gay men and lesbians generally.\n\nThe rebirth of radical politics that AIDS sparked eventually led to the formation, in the late 1980s and early 1990s, of other groups not specifically focused on AIDS, such as Queer Nation and the Lesbian Avengers. These later groups, in turn, tended to revitalize ACT UP and AIDS activism.\n\nThe consequences of this political renaissance have been profound. D'Emilio insists that the AIDS epidemic \"has once again given plausibility to understandings of gay and lesbian oppression as systemic; it has exposed the complex ways in which it is tied to a host of other injustices\" ( _MT_ , 268). In short, responses to the AIDS epidemic have brought us back to analytical practices associated with the early Mattachine Society and the GLF. A consideration of the specifically queer identities that emerge from this return, moreover, brings us back to the impudent identities with which I began this introduction: identities refusing fixation in any one sexual or gendered location, reshaped across difference and in opposition to systemic injustice. In response to the challenges of the 1980s and 1990s, queers have collectively _reinvented_ gay and lesbian identities; in unexpected ways and in ever-multiplying locations, they have flaunted new, transgressive identities. And as I have indicated, contemporary queer commentators have been quick to spotlight, and have in turn fueled, this identificatory irreverence. Cherry Smyth writes, \"Queer promises a refusal to apologise or assimilate into invisibility. It provides a way of asserting desires that shatter gender identities and sexualities, in the manner some early Gay Power and lesbian feminist activists once envisaged\" (60). Michael Warner likewise emphasizes:\n\nEvery person who comes to a queer self-understanding knows in one way or another that her stigmatization is intricated with gender, with the family, with notions of individual freedom, the state, public speech, consumption and desire, nature and culture, maturation, reproductive politics, racial and national fantasy, class identity, truth and trust, censorship, intimate life and social display, terror and violence, health care, and deep cultural norms about the bearing of the body. Being queer means fighting about these issues all the time, locally and piecemeal but always with consequences. It means being able, more or less articulately, to challenge the common understanding of what gender difference means, or what the state is for, or what \"health\" entails, or what would define fairness, or what a good relation to the planet's environment would be. Queers do a kind of practical social reflection just in finding ways of being queer. (6)\n\nObviously, I cannot claim to address all the interrelations Warner cites here, though in the pages that follow I do engage many of them. My project is more modest. My study joins this queer parade of commentaries, and\u2014through its reinflection of the insights of Smyth, Warner, Sedgwick, D'Emilio, and many others\u2014aims to couple the renaissance of queer identities and political analyses with the other renaissance with which I began: the unprecedented efflorescence of queer cultural production. Of course, \"couple\" is not exactly the right word, if the concept assumes a prior separation of the parties involved. As I have already implied, neither renaissance was discrete in the first place.\n\n### **Queer America**\n\nThe renaissance under consideration here is \"queer\" not simply because it is by and for lesbians and gay men but because it is different from many other renaissances. As I insisted at the beginning of the last section, the Queer Renaissance resists the discourse of transcendence and historical dislocation that generally accompanies cultural renaissances. In this section of my introduction, I consider more carefully how the Queer Renaissance differs from a few specific twentieth-century renaissances. At the same time, I complicate the sense I gave of the Queer Renaissance's distinction from other renaissance moments. The Queer Renaissance is indeed fabulously distinct from renaissances that strive for transcendence of secular history, but absolute distinction from all other renaissance moments would be impossible, given the permeability of boundaries the Queer Renaissance celebrates. Moreover, many of the central figures of the Queer Renaissance would credit not only gay and lesbian history with shaping their political consciousness but also\u2014or primarily\u2014Chicana\/o, African American, feminist, and working-class history, and hence these earlier literary and political movements are often explicitly or implicitly cited in the Queer Renaissance. The Chicano and Harlem Renaissances, in particular, share some characteristics with the contemporary Queer Renaissance. Nonetheless, vocal factions in both the Chicano and Harlem Renaissances attempted to contain fluidity, and thus writers in the Queer Renaissance who avail themselves of these movements reinvent the identities they proffer.\n\nThe Chicano Renaissance of the late 1960s and early 1970s, for instance, like the Queer Renaissance of the 1980s and 1990s, initially produced a revitalized and politicized identity. \"Chicano\" identity was rooted in _El Movimiento_ and opposed to a hyphenated \"Mexican-American\" identity, which was perceived as more complacent and assimilationist. Systemic critique and an identity reshaped across differences\u2014the two central components I identify with the Queer Renaissance\u2014were also very much components of the Chicano Renaissance, as Rodolfo \"Corky\" Gonzales's epic poem _I Am Joaquin\/Yo Soy Joaquin_ makes clear.\n\nI want to consider a few aspects of _I Am Joaquin_ carefully, since the poem so thoroughly intertwined, in its message and its reception, the political and literary aspects of the Chicano movement. Gonzales's poem is printed in both English and Spanish. The first stanza opens by sharply criticizing _la sociedad gringa_ that is destroying \"Joaquin\":\n\nI am Joaquin,\n\nlost in a world of confusion,\n\ncaught up in the whirl of a\n\ngringo society,\n\nconfused by the rules,\n\nscorned by attitudes,\n\nsuppressed by manipulation,\n\nand destroyed by modern society.\n\nYo soy Joaquin,\n\nperdido en un mundo de confusion\n\nenganchado en el remolino de una\n\nsociedad gringa,\n\nconfundido por las reglas,\n\ndespreciado por las actitudes,\n\nsofocado por manipulaciones,\n\ny destrozado por la sociedad moderna. (6\u20137)\n\nEven as the poem places Joaquin within a desperate situation, however, it begins to shape a Chicano identity that will remake this \"world of confusion.\" The English version of this first stanza appears beneath the picture of a young boy of eight or nine, who holds a sack and is seated atop boxes of El Rancho grapes. The credits identify this boy as a California grape worker (Gonzales, 118). The Spanish version of the first stanza appears on the opposing page, beneath the picture of a young _girl_ of eight or nine, who leans against a wooden shack, also in California (118). The \"I\/Yo\" that is Joaqu\u00edn, then, is not individual but collective: both Spanish-and English-speaking, female and male, child and adult. More than fifty photographs throughout _I'm Joaquin_ \u2014of farmworkers, demonstrators, veterans, children, and so forth\u2014make Chicano identity even more expansive, as does the poem's distribution: thousands of copies were reproduced by Chicano newspapers; _teatro_ groups performed the poem from California to Mexico (Teatro Campesino in particular, transforming it into a film); and\u2014as Gonzales asserts\u2014excerpts from the poem were \"used by almost every Chicano organization in the country\" (2).\n\nThe poem itself represented and refueled the Chicano movement. As Gonzales explains, \"J _Am Joaquin_ was the first work of poetry to be published by Chicanos for Chicanos and is the forerunner of the Chicano cultural renaissance. The poem was written first and foremost for the Chicano Movement\" (3). Hence, far from transcending history or providing an alternative to _El Movimiento_ , the Chicano Renaissance was understood as integrally connected to the political and the historical: \"There is no inspiration without identifiable images, there is no conscience without the sharp knife of truthful exposure, and ultimately, there are no revolutions without poets\" (1). Carlos Mu\u00f1oz, Jr., explains that \"I _Am Joaquin_ did not offer its readers a well-defined radical ideology, but it did provide a critical framework for the developing student movement through its portrayal of the quest for identity and its critique of racism\" (Mu\u00f1oz, 60\u201361). In short, the poem nourished the nascent analysis taking shape within the movement. The sharpened political analysis within the movement, in turn, encouraged the formation of other literary works\u2014such as the plays of Luis Valdez\u2014that sustained a more well-defined critical ideology.\n\nThe poem provided the movement with a historical framework: \"I _Am Joaquin_ filled a vacuum, for most student activists had never read a book about Mexican American history\u2014especially one that linked that history with Mexican history\" (Mu\u00f1oz, 61). To strengthen contemporary Chicano identity, Gonzales drew on a collective Chicano history. The name in the title, for instance, recalls the legend of Joaquin Murrieta, a miner who worked in California from 1849 to 1851. Gonzales explains that Anglo miners raped and killed Murrieta's wife and then drove Murrieta himself away from the claim he had staked. He was later falsely accused of horse stealing and whipped in public for this fictitious crime. Vowing revenge, Murrieta formed a band of outlaws, who commenced a series of \"daring robberies and narrow escapes\": \"He became a legendary figure, _el bandido terrible_ , the 'Robin Hood of El Dorado\/The California government offered a reward of $5,000 for him, dead or alive, and\u2014because his last name was not well known\u2014California became unsafe for anyone whose name happened to be Joaquin\" (Gonzales, 115).\n\nThrough its citation of this legend, Gonzales's poem both valorizes a history of Chicano resistance to Anglo oppression and enacts a reversal in the service of contemporary Chicano identity. In the nineteenth-century context recalled by the legend, the statement \"I am Joaquin\" would have been fraught with danger; if one's name was Joaquin, silence would be the surest way to avoid certain death. Gonzales, in contrast, animates the Chicano movement by championing and disseminating the assertion \"I am Joaquin.\" The dangerous statement is transformed into a byword for all those who join the Chicano movement and thereby becomes a sign that \"Joaquin\" cannot and will not be eliminated. The statement \"I am Joaquin,\" then, is similar to ACT UP's appropriation and resignification of the pink triangle in the 1980s and 1990s. In Nazi Germany, silence about one's homosexuality was the only way to avoid being forced to wear the pink triangle and sent to extermination camps. ACT UP inverted the triangle (in Nazi Germany, it was worn with the point facing down) and appended to it the motto \"Silence = Death.\" Wearing the triangle thus signified both one's membership in a vital, collective movement that refused to be silenced and one's belief that the government's silence on AIDS, in particular, had been murderous.\n\nThe renaissance of African American cultural production in the 1920s, especially if that movement is understood as the \"New Negro Renaissance,\" similarly displayed a dynamic new identity. Although the New Negro Renaissance was certainly much more committed to abstract ideas of beauty than either the Queer or the Chicano Renaissance, all three nonetheless attempt to construct an identity that will speak to the presumed inadequacies of older identities: we are not simply lesbian and gay, but queer; not Mexican American, but Chicano; not Old Negroes, but New Negroes. An \"Old Negro\" identity might be considered politically apathetic and acquiescent in its subordination to whites, but \"New Negro\" identity offered an active, proud, resistant alternative. Hence the New Negro Renaissance, like the Chicano Renaissance forty years later, offered some African Americans of the 1920s a heightened and collective \"race consciousness\" that specifically countered from within the negative representations from without that dominated discussions of Negro life. Many in the New Negro Renaissance also saw art as the force that could bring about change. James Weldon Johnson, for instance, insisted that \"artistic efforts\" would enable the New Negro to overcome racism \"faster than he has ever done through any other method\" (qtd. in Lewis, 193).\n\nHowever, there are important differences between the Queer Renaissance and both the New Negro and the Chicano Renaissance. In the New Negro Renaissance, for example, even though writers such as Johnson and W. E. B. DuBois saw politics and art as compatible and even positioned art as the highest form of social protest, the identity some of these writers favored was much less fluid than the identities favored by writers in the Queer Renaissance. DuBois and others championed the artist's ability to produce \"positive images\" of black people. Representations of the erotic, therefore, were often proscribed. Carla Kaplan notes that \"black publication guidelines warned that nothing liable to add fuel to racist stereotypes of wanton licentiousness and primitivism would be printed\" (122). Kaplan concludes that such proscriptions \"were much more rigorously applied to the work of black women\" (122). Male gatekeepers in the Harlem Renaissance thus scapegoated black women writers as more likely to reflect \"wanton licentiousness\" but simultaneously required them to embody the \"highest moral values.\" Black women such as Bessie Smith did shape more fluid (and sexual) identities, but artists like Smith \"were not often invited to New Negro salons\" (Gates, \"Trope of a New Negro,\" 148).\n\nFurthermore, for some the question of how or whether the \"political\" and \"aesthetic\" functions of art were related remained open. The debate was ongoing and often vigorous. Alain Locke and others resisted DuBois's insistence that literature portray black people in a positive light and advocated, in contrast, unfettered individual self-expression. Locke and his cohorts understood DuBois's desire for \"positive images\" as propagandistic. The debate about art paralleled a more general controversy over New Negro identity. As Henry Louis Gates, Jr., argues, \"At least since its usages after 1895, the name [New Negro] has implied a tension between strictly political concerns and strictly artistic concerns\" (\"Trope of a New Negro,\" 135). Locke's usage of the term to describe a literary movement, according to Gates, celebrated a mythological racial self and diluted the connotations of political activism that New Negro identity carried. Other writers took this separation of the artistic from the political to the extreme: in 1924, Charles Johnson characterized Jean Toomer as \"triumphantly the negro artist, detached from propaganda, sensitive only to beauty\" (qtd. in Lewis, 90). Thus, in spite of its articulation of a collective new identity for some, and in spite of some writers' linkage of the political and the aesthetic, the New Negro Renaissance as often positioned the two realms at odds, and this sharply contrasts to the Queer Renaissance, where they have consistently been mutually constitutive.\n\nFinally, if the New Negro Renaissance is understood, more traditionally, as the Harlem Renaissance, then its limited geographic location and availability contrast with the ubiquity of the Queer Renaissance. The \"here\" of the queer pronouncement \"We're here, we're queer, get used to it\" is as valid, and perhaps more threatening, in Peoria as in New York or San Francisco. I do not mean by this assertion to contradict my argument in favor of location and specificity but rather to affirm that queers in specific locations everywhere have reinvented gay and lesbian identity. We are everywhere, but the terms of our existence shift according to distinct histories and locations. Thus my first chapter is critical of how Edmund White's Ohio in _A Boy's Own Story_ becomes a blank \"Everyplace,\" whereas my second and third chapters examine specific and very different queerings of rural North Carolina and the U.S.-Mexican border region, respectively.\n\nAs with the Harlem Renaissance, there are differences between the Chicano and the Queer Renaissance\u2014in this case, precisely to the extent that the former is, relentlessly, the Chicano Renaissance. As Lorna Dee Cervantes writes, \"You cramp my style, baby \/ when you roll on top of me \/ shouting, 'Viva La Raza' \/ at the top of your prick\" (qtd. in Gutierrez, 48). In other words, the Chicano Renaissance, despite its expansiveness and despite the gender inclusivity suggested by the first few pages of \/ _Am Joaquin_ , often depended for its meaning on the containment of Chicanas. \"Queer,\" too, can and does generate its own exclusions (some of which are considered more fully in my third chapter), but I think that, at its best, the concept is unruly and undermines attempts at fixation or containment. \"To queer,\" in my mind, is precisely \"to bring out the difference that is compelled to pass under the sign of the same.\" As Sedgwick notes, \"The most exciting recent work around 'queer' spins the term outward along dimensions that can't be subsumed under gender and sexuality at all: the ways that race, ethnicity, postcolonial nationality criss-cross with these _and other_ identity-constituting, identity-fracturing discourses, for example\" ( _Tendencies_ , 8\u20139). Isaac Julien, Gloria Anzald\u00faa, and Richard Fung are exemplars of this type of queer theory, according to Sedgwick: \"[These] intellectuals and artists of color . . . are using the leverage of 'queer' to do a new kind of justice to the fractal intricacies of language, skin, migration, state. Thereby, the gravity (I mean _gravitas_ , the meaning, but also the _center_ of gravity) of the term 'queer' itself deepens and shifts\" ( _Tendencies_ , 9). Such constant shifting is a central focus of all four of the chapters that follow. Only through this constant shifting can the Queer Renaissance guard against the exclusionary practices that already endanger it and that have plagued\u2014and ultimately grounded\u2014earlier movements.\n\nHence, when practitioners of the contemporary Queer Renaissance avail themselves of movements such as the New Negro or the Chicano Renaissance, they resist the tendency to contain fluidity. At the same time, they reinvoke those earlier moments' sharp critique of American culture. Cherr\u00ede Moraga, for instance, writes, \"For me, 'El Movimiento' has never been a thing of the past, it has retreated into subterranean uncontaminated soils awaiting resurrection in a 'queerer,' more feminist generation\" ( _Last Generation_ , 148). For Moraga, then, the contemporary moment signals a revitalization or rebirth of Chicano\/a political\/artistic practices _through the lens_ of queer and feminist thought. This revitalization sustains and extends the Chicano Renaissance's and _El Movimiento's_ indictment of America: _\"An art that subscribes to integration into mainstream Amerika is not Chicano art\"_ (Moraga, _Last Generation_ , 61).\n\nSince Moraga writes openly as Chicana, feminist, and queer, understanding those identities as inextricably imbricated, and since queer art similarly indicts \"mainstream Amerika,\" Moraga's assertion could as easily describe the Queer Renaissance of which she is a part as it does the Chicano art that is her immediate topic. In fact, the title of the collection of prose and poetry from which these quotations are taken seems to license such an ongoing linkage of \"queer\" and \"Chicano.\" Calling her collection _The Last Generation_ , Moraga insists that the reproduction stops here: both literally, since she is a lesbian who chooses not to have children, and figuratively, in that she is a Chicana resisting the reproduction of assimila-tionist \"Hispanic\" narratives. In short, Moraga reinvents Chicano\/a identity in and through queerness and vice versa. She describes her writing as \"a queer mixture of glyphs\" that \"shape[s] the world I know at the turn of this century.... Like the ancient codices, _The Last Generation_ begins at the end and moves forward\" (4). Judging by the writing that follows this description of her project, the \"end\" Moraga invokes here is the end of assimilation to Anglo-American, heterosexual, and patriarchal norms. Like all the \"queer glyphs\" that constitute the Queer Renaissance, Moraga's prose and poetry are dedicated to a dismantling of such norms, wherever they are encountered: \"I hold a vision requiring a radical transformation of consciousness in this country, that as the people-of-color population increases ... we will emerge as a mass movement of people to redefine what an 'American' is\" (61); \"We can work to teach one another that our freedom as a people is mutually dependent and cannot be parceled out\u2014class before race before sex before sexuality\" (174).\n\nOther American renaissances of this century (I think here especially of the Native American Renaissance and the Hawaiian Renaissance), despite important differences, share with the Queer Renaissance the commitment to a systemic critique of American culture. I would never argue, then, as David Bergman does in the introduction to his _Gaiety Transfigured: Gay Self-Representation in American Literature_ , that \"to be American is to be queer\" (10). Bergman jotted this observation down on a scrap of paper during a visit to Harper's Ferry with his lover, and he insists that \"in so doing I united the currents of the national spirit with my literary ancestry and my erotic desires\" (10). Of course, Bergman's national epiphany here has to be read in the context of Harper's Ferry: that is, in the context of John Brown's famous attempt to _change_ the America of his day. In this sense, \"America\" for Bergman, as for many people at various locations on the political spectrum, stands for \"liberation\" and \"freedom.\" The abstractness of Bergman's formulation and his celebratory lyricism, however, prevent him from moving beyond this oversimplified nationalism. Bergman's lyricism also prevents him from questioning the potential ramifications of his thesis about \"the national spirit.\" If \"to be American is to be queer,\" in what sense is it or is it not the case that \"to be queer is to be American\"? Could \"American\" simply replace \"Queer\" on the T-shirts that so delighted Sedgwick at the gay pride parade, since the two terms are interchangeable? Of course not: \"American\" carries too many other contradictory and reactionary meanings that are not easily dismissed (or, to use Bergman's language, \"transfigured\"). Rather than teasing out these contradictions and insisting\u2014as Moraga does\u2014that we continuously \"redefine what an 'American' is,\" however, Bergman uses this scrap of paper to pull together a sort of high-modernist unity from the disparate fragments of his life. Moraga, in contrast, advocates not unification but dissolution: ' _I the Soviet Union could dissolve, why can't the United States?\" (The Last Generation_ , 168).\n\nIf Bergman had more fully explored these contradictions, then perhaps he would have asserted that to be queer is to _contest_ America. In this formulation, \"Queer America\" is less a description than an ongoing imperative. It is this imperative, I think, that the Queer Renaissance performs and extends, and it at times shares this imperative with, and inherits it from, the New Negro and Chicano Renaissances.\n\n### **New Queer Writers**\n\n\"Queerness\" is not merely contestatory, however; over the past several years, activists, journalists, scholars, and many others have sharply contested the meanings of queerness itself. In many locations, and in gay and lesbian journalism in particular, _queer_ is still used as a simple synonym for embodied lesbians and gay men. And indeed, such an association should remain primary; as Sedgwick notes, \"Given the historical and contemporary force of the prohibitions against _every_ same-sex sexual expression, for anyone to disavow those meanings, or to displace them from the term's definitional center, would be to dematerialize any possibility of queerness itself\" _{Tendencies_ , 8). Still, _queer_ as I am using it in this project (and certainly, as Sedgwick deploys it in hers) should be seen as more expansive and not simplistically coextensive with fixed homosexual bodies.\n\nMy use of the term should be seen, for instance, as a deliberate contrast to Michelangelo Signorile's use of it in his national bestseller _Queer in America: Sex, the Media, and the Closets of Power_. Although Signorile explores a plethora of locations in his book, in each \"queers\" look exactly the same. Like many ethnographers before him, everywhere Signorile goes, he sees people just like himself. In sections such as \"Queer in New York,\" \"Queer in Washington,\" \"Queer in Hollywood,\" and \"Queer in America,\" Signorile positions those who engage in heterosexual activity as always automatically on the outside of \"queer\" (\"Because they've never experienced the closet, because they have no idea what it's like to be queer, sympathetic straights often see coming out or being outed as the most excruciating, most horrible thing imaginable\" [xx]), while those who engage in homosexual activity are automatically on the inside\u2014as if, were Jesse Helms and Robert Dornan to be caught in a tabloid love nest, it would indicate that all along they have been unacknowledged members of the queer club.\n\nTo me, Signorile's use of expressions such as \"all the president's queers\" (165), \"Pete Williams['s] . . . queerness\" (104), or \"closeted queer\" (71) verge on the oxymoronic. In fact, I would go so far as to say that, given the current structure of the American political system \"inside the Beltway\" and in contrast to what Signorile implies throughout _Queer in America_ , there are no \"queers in power,\" only queer moments: Maryland senator Barbara Mikulski's characterization of the Senate Judiciary Committee as an \"inquisition\" and Republican senators as \"grand inquisitors,\" an accusation she made before delivering the resounding \"No!\" vote that registered her opposition to the confirmation of Clarence Thomas to the Supreme Court; Surgeon General Joycelyn Elders's assertion that we should understand masturbation as part of human sexuality and even that we\u2014gasp!\u2014should communicate that to our children. The cross-identifications and commitments that led to these moments and many others like them indicate, certainly, that queer _processes_ are alive and well on the inside, disrupting the system; but embodied \"queers in power,\" whatever such creatures might look like, are not\u2014Representatives Barney Frank, Gerry Studds, and, especially, Steve Gunderson notwithstanding. Indeed, I would argue that Elders, complete with sparkling heterosexual and Christian credentials, is the closest we have come to a highly visible queer in power, and that her rapid dismissal (Elders: \"[The conversation] was more than one minute and less than two\") attests to the difficulty of articulating queerness to the current political system, given that queerness is so intent on _changing_ that system.\n\nMy use of _queer_ in this project has more in common with David Wojnarowicz's uses of the term than with Signorile's. For Wojnarowicz, queerness is fluid and disruptive. Wojnarowicz's use of _queer_ shifts constantly throughout his _Close to the Knives: A Memoir of Disintegration_ , so that at various points in the text it describes fags, dykes, junkies, hustlers, the homeless, people living with AIDS, people of color, women, and the occasional straight man\u2014all of those who stand in opposition to what Wojnarowicz calls throughout \"this pre-invented existence\" (174), \"the illusion of the ONE-TRIBE NATION ... this illusion called AMERICA\" (153), or more globally, this \"shit planet\" (168). To Wojnarowicz, \"Being Queer in America\" (the title of one of his essays) means constantly living \"close to the knives.\" Living close to the knives, in turn, means dealing with danger and rage as basic components of everyday life: Will you suddenly face the knives, literally or metaphorically pinned to the wall by queer bashers, or will you instead suddenly grab the knives yourself, to use against the bashers? In short, one lives close to the knives because this is a dangerous and infuriating world in which queer lives of all sorts are expendable:\n\nI thought of the neo-nazis posing as politicians and religious leaders and I thought of my genuine fantasies of murder and wondered why I never crossed the line. It's not that I'm a _good_ person or even that I am afraid of containment in jail; it may be more that I can't escape the ropes of my own body, my own flesh, and bottom line in the pyramids of power and confinement one demon gets replaced by another in a moment's notice and no one gesture can erase it _all_ that easily. (Wojnarowicz, 33)\n\nOn a journey through the American West, Wojnarowicz is not surprised to read in the newspaper of one city\u2014a city he describes as \"a government war town filled with a half million workers employed in the various research centers attempting to perfect a president's dream of laser warfare from the floating veil of outerspace\" (30)\u2014about a Native American teenage boy who is terrorizing civilians by driving his car onto the sidewalk. Instead, given \"this pre-invented existence\" and \"the illusion of the ONE-TRIBE NATION,\" Wojnarowicz is surprised by others' surprise: \"I thought of the face of our current president floating disembodied and ten stories tall over the midnight buildings. I wondered why any of these things, like the kid in his camaro, are a surprise. Why weren't more of us doing this?\" (32).\n\nWojnarowicz carries his outrage to his art and, in photographs and paintings such as _Fuck You Faggot Fucker_ (1984), translates that outrage into a queer challenge to those in \"the pyramids of power.\" _Fuck You Faggot Fucker_ displays a collage of images, including a crude drawing of anal sex flanked by the words \"Fuck You Faggot Fucker.\" The drawing appears to be graffiti, perhaps pulled from the walls of a public restroom. Wojnarowicz places a much larger drawing of two men kissing, waist-deep in water, in the center of the collage and overlays the men's bodies with a color map of North America. The graffiti beneath these men suggests that they are embattled, surrounded by\u2014indeed, almost drowning in\u2014hostility, yet their kiss remains the most prominent feature of the collage. Moreover, Wojnarowicz defiantly maps queer acts and eroticisms onto countless locations everywhere, since all of North America is incorporated into the kiss between these two men. A far cry from Signorile's desire to \"out\" Malcolm Forbes or Jodie Foster, Wojnarowicz's desire is to represent queer identity otherwise: not as an identity one might discover tucked away in the \"closets of power\" but rather as a fluid and disruptive identity openly constructed and reconstructed in innumerable _pockets_ of power by the many who share Wojnarowicz's opposition to a society \"pre-invented\" to sanction \"the legalized murder of those who are diverse in their natures\" (154)-\n\nIn the pages that follow, I consider the various ways other poets, playwrights, essayists, and novelists have represented such queer identities. I have specifically started here with Signorile and Wojnarowicz, two white and male writers, in order to tease out differences within that group over and around queerness. Another difference worth pointing out is that some white men (in opposition to both Signorile and Wojnarowicz) have often been among the concept's greatest detractors: Bruce Bawer and Rich Tafel, for instance, are among the many white gay men who think the movement would gain more respect if we _queers_ would only behave.\n\nI highlight this disjunction because there are some who would turn the most recent history of the term into the story of a split between white gay men (a group that is assumed to be monolithic) and everyone else. Consider, for instance, Ruth Novazcek's concerns: \"I worry about the word queer. I still have this image of the gay clone in a black leather jacket and shaved 'blonde' head and I worry that it will perpetuate that aesthetic\" (qtd. in Smyth, 34). Or take Robyn Wiegman's too-easy dismissal of queer theory: \"The turn toward the word _queer_ is intended to displace [the] displacement of difference, to mark a mutual eccentricity ... as central. While I certainly agree with the political imperative of such a project, it seems to me that the deployment of _queer_ so far . . . actually neutralizes difference\" (\"Introduction,\" _iyn_. 1). Yet rejection of _queer_ might also neutralize difference; in fact, Bawer's repeated rejections of the concept suggest as much. Bawer is the editor of _Beyond Queer: Challenging Gay Left Orthodoxy_ , a collection that he hopes will convey the message that it's time \"for the gay rights movement to grow up\" (xiv). For Bawer and his cohorts, however, growing up \"beyond queer\" apparently means moving _away_ from difference and diversity: the contributors to _Beyond Queer_ are overwhelmingly white and male; it is, indeed, difficult for me to think of a recent gay and\/or lesbian collection that is less inclusive than Bawer's.\n\n_Pace_ Wiegman, I certainly agree that _queer_ has the _potential_ to neutralize difference. My point here is simply that resistance to or dismissal of the concept also has the potential to neutralize difference, and one goal of our inquiries might be to negotiate these tensions. Rather than simply assuming that _queer_ perpetuates a white male aesthetic, we might consider more critically how and why the concept has _failed_ to speak to some white men and how it has already been deployed by many different gay and lesbian writers of color. Many white men do not want to be queer, and many women and men of color were queer before queer was hip.\n\nIt is this belief that makes _The Queer Renaissance_ a project that is more about writers such as Audre Lorde, Randall Kenan, and Gloria Anzald\u00faa than about the group that has recently been labeled explicitly the \"New Queer Writers,\" The New Queer Writers (Wojnarowicz, Dennis Cooper, Eric Latzky, Kevin Killian, Dodie Bellamy, and several others) are a loosely associated group of writers who nonetheless share a commitment to interrogating sexual and social identities that are made marginal by contemporary society, even (and in some cases, especially) by contemporary lesbian and gay culture. I think this group of writers is very much part of the Queer Renaissance (much more so than, say, a mainstream gay writer such as David Leavitt), which is why I have focused on Wojnarowicz here. In this project, however, I am interested in bringing together \"related but disparate views [that] inform and corrupt each other,\" if I may appropriate one recent description of the New Queer Writing (qtd. in Latzky, 14). I want, in short, to corrupt queer writing by examining an even wider variety of \"margins\": not simply the high-literary \"bad boys\" who look back to Jean Genet and William S. Burroughs as their forebears (important as that group may be) but also those writers who look back to the various liberation movements I have surveyed in this introduction and who draw their commitment to the representation of boundary-crossing and systemic critique from them. To my mind, Wojnarowicz, Cooper, Latzky, and the rest have produced some of the most provocative and exciting contemporary queer writing to date, and I do not mean to suggest that they do not recall or cite in their writing the movements I have considered. New Queer Writers, however, come in a wide variety of forms, and the bad boys and girls I have nodded to here are only a small part of a \"vast, diverse, and growing anti-assimilationist queer movement\" (Cooper, xi). Hence, in the chapters that follow, I consider novelists, poets, and playwrights whose work might not, at first, be thought to represent the New Queer Writing. The theories they advance and the unruly identities they depict, however, ensure that their work is very much in concert with the work of this school.\n\nChapter 1 thus reads Audre Lorde's _Zami: A New Spelling of My Name_ alongside Edmund White's _A Boy's Own Story_ and considers how, at the outset of the Queer Renaissance, _Zami_ redefines the parameters of the coming-out story. Although the Gay Liberation Front intended \"coming out\" to provide a starting point for radical political action, by the late 1970s the coming-out story had evolved into a predictable narrative, detailing a protagonist's teleological journey toward an essential wholeness. Examining coming out alongside feminist theories of positionality, I suggest that _Zami_ resists this individual and essentialist understanding of gay identity and constructs instead an\/other myth of queer positionality\u2014not because Lorde attempts, as White does in _A Boy's Own Story_ , to construct her protagonist as an alienated, marked \"other\" but because _Zami_ argues for collective identity, for a \"self\" defined in and through others.\n\nChapter 2 carries queer theory to another region, where \"community\" is not always the sustaining force it is in _Zami_. I analyze Randall Kenan's _A Visitation of Spirits_ \u2014which focuses on a black gay youth growing up in the rural African American town of Tims Creek, North Carolina\u2014in order to consider what happens when queer desire turns up in apparently unlikely and inhospitable places. This chapter engages African American theories of signification and suggests that _A Visitation of Spirits_ effects a queer trickster identity, able to reverse hierarchies of power and transform even the supposed \"margins\" of the queer world.\n\nChapter 3 focuses on Gloria Anzald\u00faa's _Borderlands I La Frontera_ and the current academic penchant for transgressing disciplined\/disciplinary boundaries. I argue that there's more than one way to cross a border: border crossings may challenge the mechanisms of power that produce borders, divided identities, and histories of exploitation, but they may also uncritically posit cosmopolitan consumers\/theorists, able to move anywhere and everywhere with ease. Anzald\u00faa's \"new mestiza\" crosses the U.S.-Mexican and other borders not to escape identity and history but rather to challenge the unchecked mobility of oppressive systems of power. The new mestiza turns the searchlight away from those border residents deemed \"illegal\" and onto those who delimit and police the border in order to secure unlimited mobility for themselves.\n\nChapter 4 examines lesbian and gay work generated by the AIDS crisis. I begin by considering how Tony Kushner, in his critically acclaimed play _Angels in America_ , queers the concept of perestroika in order to promote the performance of a disruptive, coalition-based identity. I then provide a genealogy of Kushner's aesthetic: using Sarah Schulman's fictionalization of ACT UP in her novel _People in Trouble_ , I put Kushner's play into conversation with the queer AIDS activist identities that were being performed in the streets during the writing and production of _Angels in America_. This final chapter considers more directly how the critical aesthetic that is at play in the works that constitute the Queer Renaissance shapes, and has been shaped by, liminal sexual and activist \"identities in a crisis.\"\n\nFinally, in the epilogue, I conclude that the Queer Renaissance is ongoing, despite\u2014and in the face of\u2014the recent rise of gay neoconservatism and of studies that seek to ground homosexuality in biology. Queer inquiry commences by challenging the boundaries established by others and thus thrives despite attempts by some to fix us with a well-behaved neoconser-vatism or an underdeveloped hypothalamus. Activists and writers in the Queer Renaissance continue to refuse such constricting identities. The queer trickster, the AIDS activist, the new mestiza, Zami\u2014these are the identities I analyze here. Born of a wide variety of twentieth-century liberation movements, such identities have been reborn in the contemporary Queer Renaissance.\n\n## **_Chapter One_  \nBoys' Own Stories and New Spellings of My Name: Coming Out and Other Myths of Queer Positionality**\n\n### **Myths of Queer Positionality**\n\nIn _The Beautiful Room Is Empty_ , Edmund White's nameless narrator envisions a day when gay people will claim the right to define themselves: \"Then I caught myself foolishly imagining that gays might someday constitute a community rather than a diagnosis\" (226). This exhilarating thought comes to White's protagonist as he finds himself in the middle of an uprising at the Stonewall Inn Bar in Greenwich Village on the night of June 27, 1969. Drawing on Civil Rights rhetoric, the protagonist and his friends reclaim and reposition their own experiences with chants such as \"Gay is good\" and \"We're the Pink Panthers\" (226).\n\nAlthough White's account is fictional, the riots outside the Stonewall Inn are generally considered the beginning of the contemporary gay liberation movement. They did indeed usher in a decade of redefinition by lesbian and gay communities. Within weeks, the Gay Liberation Front (GLF) had formed, employing the slogan \"Out of the Closets and into the Streets!\" Within a year, the Radicalesbians, influenced by both gay liberation and the women's movement, had presented feminists with the \"woman-identified woman,\" a position that they hoped would facilitate the formation of challenging, politicized coalitions among women. By 1974, activists had successfully removed homosexuality from the American Psychiatric Association's (APA's) list of mental disorders. In short, the new names and identities embraced by White's protagonist and his friends were high on the agenda for early gay liberationists.\n\nThese newly available gay and lesbian identities were claimed and proclaimed through the act of \"coming out.\" This act provided lesbians and gay men with positions that could serve as starting points for the radical political action the early gay liberationists believed was necessary to reconfigure the systems of capitalism and patriarchy responsible for gay and lesbian oppression. Indeed, the very slogan of the gay liberationists (and the title of a 1972 essay by Allen Young), \"Out of the Closets, into the Streets,\" suggests not simply that one claims a position (\"out of the closet\") but that one moves from that position to effect radical social change. Young writes, \"Of course, we want to 'come out\/. . . But the movement for a new definition of sexuality does not, and cannot, end there. . . . The revolutionary goals of gay liberation, including the elimination of capitalism, imperialism and racism, are premised on the termination of the system of male supremacy\" (10). Similarly, \"woman-identification,\" according to the Radicalesbians, could be \"develop [ed] with reference to ourselves, and not in relation to men. This consciousness is the revolutionary force from which all else will follow\" (Radicalesbians, 176). Like the identity positions (pro)claimed by all the so-called new social movements, the identities into which gay and lesbian activists \"came out\" were collective identities meant to generate radical social change based on new and different ways of understanding the world.\n\nComing out generally does not have the same radical edge for the new generation of queer activists, or for the writers of the Queer Renaissance, that it had for their gay liberationist forebears. On the contrary: coming out, along with its product\u2014one's \"coming-out story\"\u2014has been thoroughly critiqued by many contemporary lesbian and gay writers. In particular, theorists have critiqued coming-out stories for their emphasis on the \"discovery\" of an individual and essential gay identity, unmarked by other categories of difference, such as race or class. This chapter briefly surveys these and other criticisms but simultaneously attempts to lay the theoretical groundwork for a reclamation of coming out's radical potential. Specifically, through readings of Edmund White's _A Boy's Own Story_ and Audre Lorde's _Zami: A New Spelling of My Name_ , I consider whether a feminist-informed and antiracist analysis might redefine the parameters of the coming-out story, shaping it into a myth of what I call \"queer (op)positionality.\"\n\nThrough the term _(op)positionality_ , I intend to invoke the \"opposition\" to established and oppressive systems of power that was voiced by the GLF, the Radicalesbians, and members of all the new social movements, and that has been rearticulated in the contemporary Queer Renaissance. I also intend to invoke \"positionality\" theory (or \"standpoint epistemology\") as it has evolved in recent feminist writing. Standpoint theorists, often explicitly citing both the strengths and the weaknesses of the social movements of the 1960s and early 1970s, argue for a nonessentialized \"position\" from which to forge coalition-based political action. Linda Alcoff, for example, argues, \"If we combine the concept of identity politics with a conception of the subject as positionality, we can conceive of the subject as nonessentialized and emergent from a historical experience and yet retain our political ability to take gender as an important point of departure\" (433). As Alcoff sees it, this new position is both fluid and relational: \"being a 'woman' is to take up a position within a moving historical context and to be able to choose what we make of this position and how we alter this context,\" so that \"women can themselves articulate a set of interests and ground a feminist politics\" (435).\n\nSimilarly, Donna Haraway's feminist redefinition of \"objectivity\" argues for an openly acknowledged, although partial, position or perspective. Haraway writes, \"Feminist objectivity is about limited location and situated knowledge, not about transcendence and splitting of subject and object\" (190). This _partial_ perspective is necessitated since, not unlike Edmund White's narrator in _The Beautiful Room Is Empty_ , with his concern about psychiatric diagnoses, Haraway confesses that she has occasional paranoid fantasies about so-called _impartial_ , \"objective\" discourses that appropriate \"embodied others\" as \"objects\" of knowledge:\n\nAcademic and activist feminist enquiry has repeatedly tried to come to terms with the question of what _we_ might mean by the curious and inescapable term \"objectivity.\" We have used a lot of toxic ink and trees processed into paper decrying what _they_ have meant and how it hurts _us_. The imagined \"they\" constitute a kind of invisible conspiracy of masculinist scientists and philosophers replete with grants and laboratories; and the imagined \"we\" are the embodied others, who are not allowed _not_ to have a body. (183)\n\nBecause of these fears, Haraway's redefinition of \"objectivity,\" like the rhetoric of gay liberation, gives preference to other ways of seeing, particularly those ways of seeing that emerge from what she calls the \"standpoints of the subjugated.\" She writes, \"'Subjugated' standpoints are preferred because they seem to promise more adequate, sustained, objective, transforming accounts of the world\" (191). Such standpoints are actually _more_ \"objective\" because they do not claim to see, simultaneously, \"everything from nowhere\" (189) or \"to be,\" as Richard Dyer puts it in his analysis of the social construction of whiteness, \"everything and nothing\" (Dyer, 45). Such standpoints are more \"transforming\" because a coalition politics that emphasizes working together across difference is fundamental to what theorists of positionality envision. Indeed, this model would not have been shaped in the first place if feminists of color had not called for a more rigorous analysis of the differences within feminism and for an ongoing interrogation of the ways in which feminist concerns overlap with the concerns of other groups. Both Haraway and Alcoff, consequently, place race as well as gender at the center of their analyses: to Haraway, the category \"women of color\" \"marks out a self-consciously constructed space that cannot affirm the capacity to act on the basis of natural identification, but only on the basis of conscious coalition, of affinity, of political kinship\" (Haraway, 156); to Alcoff, positionality theory \"can be readily intuited by people of mixed races and cultures who have had to choose in some sense their identity\" (Alcoff, 432).\n\nOther theorists, recognizing the value of such coalition-based and self-reflexive positionality, have attempted to link gay male and feminist standpoint theory. Earl Jackson, Jr., for one, begins his study of Robert Gl\u00fcck by acknowledging, \"One of the most important things gay men can learn from feminist and lesbian-feminist discursive practices is how to read and write from responsibly identified positions\" (112). Nevertheless, there are drawbacks to any attempt to link, specifically, _coming out_ to feminist theories of positionality; these drawbacks become evident from the critiques of coming out I alluded to earlier. The position \"out of the closet,\" much more than the \"standpoint\" of recent feminist theory, has become in the past two decades a mandated and delimited position, for both men and women. In the process of forging the imperative to come out, unfortunately, some lesbian and gay communities lost the sense that coming out was, as feminist theorists now argue about the feminist standpoint, only a beginning point from which to launch political action. The collective rallying cry \"Out of the Closets, into the Streets\" quickly became the demand that individuals simply \"Come out of the closet.\" Assimilation, rather than transformation, became the goal; increased visibility, it was thought, would lead to gay civil rights and acceptance into mainstream society. Martin Duberman explains that the Gay Activists Alliance emerged\n\nas a breakaway alternative to the Gay Liberation Front, and would shortly supercede it.. . . Whereas GLF had argued that sexual liberation had to be fought for in conjunction with a variety of other social reforms and in alliance with other oppressed minorities, GAA believed in a single-minded concentration on gay civil rights and eschewed \"romantic\" excursions into revolutionary ideology ( _Cures_ , 213\u201314)\n\nMaintaining the GLF's insistence that gay men and lesbians speak up but abandoning their politics of alliance, the GAA redirected the movement and circumscribed the meaning of coming out.\n\nIn addition to the dangers of quietism, coming out as a focus for gay and lesbian theory could also underwrite an apolitical essentialism. The narrowing of vision Duberman recounts could, and often does, narrow even further, so that coming out comes to signify solely the assertion of one's (supposedly long-repressed) identity. This model of coming out, by itself, exhibits little concern for how lesbian or gay identities are socially constituted, for how they are intersected by other arenas of difference, or for what sort of collective political action might develop from an assertion of one's gay or lesbian identity. Coming out here becomes a suspiciously white and middle-class move toward \"self-respect,\" not revolutionary social change, and many contemporary coming-out narratives might be seen as products of this shift toward individualism and essentialism. To be fair, however, John D'Emilio writes, in reference to the early gay liberationists:\n\nFor a gay man or lesbian of that time, I don't think that it was possible to experience anything of comparable intensity. In a psychological sense it was an act of \"revolutionary\" import. No manner of political analysis could convince someone who had come out that he or she wasn't turning the world inside out and upside down. ( _Making Trouble_ , 249)\n\nStill, D'Emilio's comment is less a trumpeting of the benefits of coming out than it is a note of caution. He is attempting to contextualize, but also look critically at, the psychological empowerment that coming out could bring: \"Only later, as the movement matured, would it become clear that coming out was a first step only. An openly gay banker is still a banker\" ( _Making Trouble_ , 249). By the end of the 1970s and throughout the 1980s, many lesbians and gay men no longer had the sense that coming out was a first step only. Coming out came to have one meaning, across all social locations: announcing one's homosexuality. The act no longer necessarily carried the sense that lesbians and gays should collectively move to _new_ locations; one could come out, and stay out, anywhere. Although \"coming out conservative\" would have been a logical impossibility to members of the GLF, by 1992, because of the ways in which the act had been redefined as the assertion of one's essential, no longer repressed identity, it was the title of a popular book about Marvin Liebman, a formerly closeted anticommunist and conservative activist.\n\nIn a slightly different vein, Haraway, too, stresses that essentialism in its many guises is a pitfall for feminist theories of positionality, especially those, like hers, that foreground the \"standpoints of the subjugated\": \"But here lies a serious danger of romanticizing and\/or appropriating the vision of the less powerful while claiming to see from their positions. ... A commitment to mobile positioning and to passionate detachment is dependent on the impossibility of innocent 'identity' politics\" (191\u201392). Haraway's disclaimers suggest that these essentializing tendencies might not engulf feminist theories of positionality, as long as \"situating knowledge\" entails continually and collectively repositioning identity: \"The knowing self is partial in all its guises, never finished, whole, simply there and original; it is always constructed and stitched together imperfectly, and _therefore_ able to join with another, to see together without claiming to be another\" (193).\n\nAn \"innocent identity politics,\" however, has already engulfed the coming-out narrative, according to many lesbian and gay critics. Diana Fuss sees even in the Radicalesbians a \"tension between the notions of 'developing' an identity and 'finding' an identity\" that \"points to a more general confusion over the very definition of 'identity' and the precise signification of 'lesbian'\" ( _Essentially Speaking_ , 100). Biddy Martin discusses how, by the end of the 1970s, the imperative to come out is evolving into a predictable (and white) narrative, and how many coming-out stories \"are tautological insofar as they describe a process of coming to know something that has always been true, a truth to which the author has returned\" (89). Jeffrey Minson takes this accusation of tautology a step further, suggesting that \"far from constituting a break from a repressive, closetted past, coming out might be situated as the latest in a long line of organised rituals of confession. . . . Sexual avowal therefore is a mode of social regimentation\" (37). In Minson's ominous scenario, coming out simply reproduces and undergirds the homophobic notion that homosexuality wholly explains a lesbian or gay person's identity.\n\nFinally, although Fuss's, Martin's, and Minson's criticisms might also be leveled at some feminist theories of positionality, coming out may be problematic for a unique, more mundane reason: the imperative \"Come out!\" is by now, for many, a worn-out refrain. \"National Coming Out Day\" is at this point televised yearly on _The Oprah Winfrey Show_ , and encouragement, such as the exhortation in my own campus newspaper to \"come out, wherever you are, and friends won't turn away\" (Behrens, 19), often sounds like pandering for heterosexual \"compassion.\" Since coming out, according to this mainstream model, is virtually synonymous with a call to \"respect yourself,\" many gay and lesbian people are understandably bored or irritated with this focus on coming out and its product, the coming-out narrative. For example, David Van Leer insists that for White, coming out is \"the quintessential gay experience\"; but in a review of _The Faber Book of Gay Short Stories_ , which White edited, Van Leer suggests that, in its \"preoccupation with the opinion of others,\" coming out \"sometimes looks like a bid for heterosexual sympathy, even for absolution\" (50). Sarah Schulman states more forcefully, \"The coming out story should be permanently laid to rest. ... It was a defining stage we had to go through, but it doesn't help us develop a literature true to our experience\" (qtd. in Fries, 8).\n\nAlthough I have no doubt that \"we\" will never construct a literature \"true to our experience,\" since that \"experience\" is multiple and that \"truth\" is always socially constituted and continually shifting, I am nonetheless sympathetic with Schulman's frustration over the primacy given the coming-out narrative, especially when this focus comes at the expense of attention to other queer stories. Richard Hall, in a review of White's _The Beautiful Room Is Empty_ , is more tentative than Schulman, suggesting, \"Not that the coming-out novel in its pristine liberationist form is dead. Maybe it's just weary. . .. Here is another coming-up-and-out story, taking the narrator into adolescence and young manhood...\" (27). Although Hall feels White's novel is told with \"wit, humor and aphoristic elegance\" (27), his reservations about the coming-out novel are standard fare these days in reviews of contemporary lesbian and gay literature.\n\nThus, despite a possible affinity with recent feminist theory, as a myth of \"queer positionality,\" coming out can be read as a worn-out concept. In the remainder of this chapter I explore more thoroughly why this is the case, but I also use the insights offered by feminist positionality theory to present an analysis that considers ways to revise or reclaim the coming-out story. Myths of queer positionality\/identity need not be hopelessly lost on teleological journeys toward essential wholeness; \"noninnocent\" myths of queer positionality can be shaped in a queer world that is about \"lived social and bodily realities [and] in which people are not afraid ... of permanently partial identities and contradictory standpoints,\" as in the cyborg world Haraway envisions and argues for (154). _A Boy's Own Story_ and _Zami: A New Spelling of My Name_ were both published in 1982, when the cultural phenomenon I call \"the Queer Renaissance\" had only just begun. Through an examination of these two very different coming-out stories, I want to flaunt the ways coming out has been reinvented in the Queer Renaissance as a myth of queer (op)positionality.\n\n### **Queer Oppositionality\/Queer Apositionality**\n\nEdmund White's _A Boy's Own Story_ is the first novel in a planned three-novel series. In each text, White traces the development of his nameless, faceless narrator. _A Boy's Own Story_ focuses on this development during the 1950s; _The Beautiful Room Is Empty_ , on this development during the 1960s. The novel as yet unfinished ( _The Farewell Symphony)_ will carry the protagonist through the 1970s and 1980s.\n\nThe \"invisibility\" used as a mechanism in these texts works as a metaphor for the pain the protagonist goes through as a gay youth in heterosexist America. Ed Cohen writes that\n\nthe structuring stories we commonly use in order to make sense of our daily lives provide us with very few plots that do not emplot us in normative versions of gender and sexuality. . . . [These normative narratives] have special consequences for those of us whose movements appear to transgress the possibilities of such acceptable representations, effectively rendering us \"unrepresentable.\" (\"Constructing Gender,\" 545)\n\nThus, like many other gay men and lesbians, White's protagonist feels that none of the people around him knows who he \"really\" is, and that he must consequently wear a (heterosexual) mask. The young narrator of _A Boy's Own Story_ muses, \"What if I could write about my life exactly as it was? What if I could show it in all its density and tedium and its concealed passion . . . ?\" (41). The point is, however, that he cannot, given the normative versions of gender and sexuality available to him, and hence invisibility works to underscore that the narrator is forced to \"live a lie\" at the same time as it works to stress more generally the pain and isolation of being marginalized in and by American society. Like the narrator of Ralph Ellison's _Invisible Man_ , White's protagonist might say \"I am invisible . . . because people refuse to see me\" (Ellison, 3).\n\nI mention Ellison at this point because White's use of the trope of invisibility in _A Boy's Own Story_ and elsewhere is not unlike Ellison's use of the same trope in his 1952 novel: both writers use invisibility to comment on the exclusionary logic at the heart of American culture. Perhaps not surprisingly, then, White's narrators often see parallels between their experiences and the experiences of African Americans in the United States. In _A Boy's Own Story_ , for example, the narrator finds himself pressured by his friends to accompany them to a whorehouse staffed by two black women and one white woman. As the protagonist sits in the waiting room, one of the black prostitutes engages him in conversation. The narrator admits, \"I felt sorry for her. I thought she might really need my ten dollars. After all this was Saturday night, and yet she didn't have any customers. Somehow I equated her fatness, her blackness, her unpopularity with my own outcast status\" (183). White's narrator goes on to imagine a marriage between the two, \"she a Negro whore and I her little protector.... If this fantasy kept me a pariah by exchanging homosexuality for miscegenation, it also gave me a sacrifice to make and a companion to cherish. I would educate and protect her\" (183). Since both characters are invisible \"outcasts,\" White's narrator imagines that they might more effectively face the world together.\n\nThe identifications here and elsewhere in White with African American (and with \"fat\" and \"unpopular\") identity, however, are suspect for several reasons, and my main argument in this section is that the construction of sexuality in _A Boy's Own Story_ is in tension with the construction of race in ways that forestall the possibility that the story might serve as a myth of queer (op)positionality. Although the novel is in some ways \"oppositional,\" I argue that any \"oppositionality\" is ultimately undercut by the \"apositionality,\" or invisibility, of whiteness in the text. It is not White's own story, necessarily, that produces this tension: by 1982, the standard coming-out story, with its single-minded focus on the discovery of one's essential identity, required such apositionality. The remainder of this section shows how white apositionality fixes _A Boy's Own Story_ as a representative\u2014or _the_ representative\u2014coming-out story and considers what ramifications the effacement of whiteness has for contemporary gay male writing generally. In the next section, however, I turn to an analysis of the ways in which Audre Lorde reconceives the coming-out story as a fluid and relational myth of queer (op)positionality\n\nInitially, White's novel may be read as \"oppositional\" for its disruption of linear models of sexual development in which heterosexuality is both the ultimate goal and the mark of maturity. This disruption of heterosexual telos is underscored by the very form of the text, which represents the protagonist's story in a nonlinear fashion (he is fifteen in the first chapter, seven at the beginning of the third, fifteen again by the end of the sixth, and so on). Yet, despite this \"opposition\" to heterosexual telos, White's novel nonetheless teleologically represents his protagonist's coming to a racially unmarked gay consciousness. Although overt acts of coming out into this new gay consciousness are deemphasized in the text (the narrator confesses to a friend that he is gay only once, offhandedly [88]), it is still clear throughout _A Boy's Own Story_ that homosexuality is precisely and primarily what White's nameless narrator must confront. Indeed, he uses a homosexual encounter with and betrayal of a \"straight\" teacher at the very end of the novel to confront this identity, which, he says, is \"at once my essence and also an attribute I was totally unfamiliar with . . . this sexual allure so foreign to my understanding yet so central to my being\" (198). In the end, the betrayal of his teacher finally allows White's narrator to work through the contradiction inherent in his \"impossible desire to love a man but not to be a homosexual\" (218). Thus, although _A Boy's Own Story_ certainly disrupts the cultural mandate to develop heterosexu-ally it also institutes a coming-out narrative as necessary for understanding one's (essential) gay identity. In fact, White later asserted that the novel \"succeeded partly because it seemed to fill an empty niche in the contemporary publishing ecology, the slot of the coming-out novel\" ( _Burning Library_ , 372).\n\nWhite's discussion of niches and slots implies that _A Boy's Own Story_ was an easy fit; the publishing world was ready for a novel with a unitary focus on coming out. I would argue, however, that at the time the \"contemporary publishing ecology\" could provide a home for White's novel only on two, unspoken conditions. First, gay identity had to be understood, as it is in _A Boy's Own Story_ , as an \"essence . . . central to [one's] being.\" Such an understanding is what enabled the empty niche to be filled so exactly. Second, gay identity could not be explicitly intersected by other facets of identity, such as gender or race: in other words, this \"gay\" slot was white and male, although not _openly_ white and male. Lisa Duggan's comments suggest that these two points are, in fact, interrelated: _\"Any_ gay politics based on the primacy of sexual identity defined as unitary and 'essential' . . . ultimately represents the view from the subject position '20th-century Western white gay male'\" (\"Making It Perfectly Queer,\" 18). Duggan outs the subject position as white and male, however; generally, such representations are advanced without being explicitly marked as such. And indeed, in _A Boy's Own Story_ , a unitary, essential gay subject position is achieved through the mechanism of invisibility.\n\nThe mechanism of invisibility in White's novel is not intentionally deployed to cover the main character's gender and race; White's narrator is \"invisible\" throughout the novel precisely because homosexuality is an identity that he cannot openly embrace. Yet the construction of sexuality here inadvertently colludes with hegemonic constructions of whiteness, which maintains its power precisely to the extent that it is able to remain hidden from view. As Richard Dyer argues, \"White power secures its dominance by seeming not to be anything in particular\" (44). What this effects in _A Boy's Own Story_ is contradictory: on the one hand, the story is one of marginalization and oppression; on the other hand, the story is _representative_. White himself recently gave voice to this contradiction, insisting, \"When I was growing up, I felt I was a totally freaky person . . . and then later I came to realize that my life, which I had thought was the most _exceptional_ imaginable, was actually the most _representative_. All I needed to do was to say what I went through in order to say what gay people went through in their evolution toward freedom\" (in Avena, 224). In the end, even White's title conveys this dual sense of the exceptional and the representative, suggesting that this \"boy's own story\" is as much about any (gay) boy as about White's specific protagonist. Indeed, I was struck, when I attempted to teach one of White's texts, by a student's response to my very first question, \"Why do you think White chooses not to give his narrator a name or face?\" A Filipino American student of mine responded, \"So we can put ourselves into the story?\"\n\nMy student was not \"wrong\" to identify with White's protagonist; he is, in fact, like White's narrator, a gay man living in a homophobic society. Yet the interrogative inflection my student gave to his response suggests that he suspected his resolution to my question might be a bit problematic. For White's character is not simply a representative \"Everyboy\": his mother and he play games with the classical radio station, guessing whether the composer is Haydn, Mozart, or early Beethoven (80); a (black female) maid, a (white male) therapist, and a private school are all part of his childhood; and he admits, however ironically, \"Even as I made much of my present miseries I was cautiously planning my bourgeois future\" (178). In short, aspects of the protagonist's identity can be read as race- and class-coded, despite his \"invisible\" gay identity. Indeed, his gay identity is rendered representative precisely because the \"naturalness\" of his racial identity is maintained through White's \"god-trick\" of \"invisibility\"\n\nThe term _god-trick_ is Haraway's, and she uses it to refer to seemingly innocent perspectives that claim to see the world more comprehensively while actually \"being nowhere\" (191). This may seem to be an unfair charge to level at White's story of gay development, but it is harder to dismiss in the context of Dyer's analysis of whiteness. Dyer suggests that\n\nwhite people . . . are difficult, if not impossible, to analyse _qua_ white. The subject seems to fall apart in your hands as soon as you begin. Any instance of white representation is always immediately something more specific\u2014 _Brief Encounter_ is not about white people, it is about English middle-class people; _The Godfather_ is not about white people, it is about Italian-American people; but _The Color Purple_ is about black people, before it is about poor, southern US people. (46)\n\nSimilarly, _A Boy's Own Story_ is not about white people, it is about gay people; but _Zami_ is the autobiography of a black lesbian, not of the gay community more generally. _A Boy's Own Story_ posits a seemingly innocent perspective that implicitly claims to see gay identity more comprehensively, but it is able to do so because white identity is nowhere to be seen.\n\nThe very landscape of _A Boy's Own Story_ underwrites white apositionality. The story takes place, alternately, in Illinois, Michigan, and Ohio. This is not Toni Morrison's Ohio, however, in which racial divisions are graphically (and in _Sula_ , geographically) represented. Like the protagonist, the various settings for the novel are unmarked: a \"boy's own story\" presumably takes place in Anywhere, U.S.A. David Bergman argues that \"the importance of Cincinnati .. . cannot be underestimated in White's fiction,\" but it is Bergman, not White, who actually names the \"Queen City\" that plays such a \"prominent role in [White's] autobiographical novel _A Boy's Own Story\"_ (\"Edmund White,\" 387). Of course, the Midwest has long had a reputation for being the blank \"nonregion\" of the United States, the land\u2014as in Don DeLillo's _White Noise_ \u2014of supermarkets, station wagons, and \"an expressway beyond the backyard\" (DeLillo, 4). This unmarked regional identity, however, does not preclude White in his novel from marking _other_ parts of the country regionally: the protagonist and his sister make fun of \"hillbillies\" from Kentucky (73), and the protagonist himself dreams of escaping to the \"charm\" and sophistication of New York City (52\u201357).\n\nAs in DeLillo, certainly, the blankness of White's Midwest might be read as simply a metaphor for the region's supposed cultural sterility. This interpretation would elide, however, the ways in which regional blankness underwrites racial invisibility in _A Boy's Own Story_. Like Kentuckians and New Yorkers, African Americans in White's Cincinnati are embodied as such, and throughout this text, it is the location of regional and racial identity in embodied \"others\" that enables the protagonist _not_ to have a regional and racial identity of his own. This disembodiment, in turn, ensures that the protagonist's coming-out story can be read as representative. The \"other,\" marked identities allow White's narrator to negotiate a problematic sleight of hand: while he effaces differences of race and region, he simultaneously appropriates representative \"outcast\" status for himself.\n\nIn _Playing in the Dark: Whiteness and the American Literary Imagination_ , Toni Morrison argues that white American writers have often used their African American characters to perform such sleights of hand. Morrison's study emerges from her interest \"in the way black people ignite critical moments of discovery or change or emphasis in literature not written by them\" (viii). Insisting that an understanding of American literature is incomplete without an understanding of the central role an African American presence has played in the American literary imagination, she concludes, \"What became transparent were the self-evident ways that [white] Americans choose to talk about themselves through and within a sometimes allegorical, sometimes metaphorical, but always choked representation of an Africanist presence\" (17). In _A Boy's Own Story_ , this metaphorical process begins in the second chapter. At first, the chapter might be read as starkly exposing the mechanisms of power that enable and ensure white privilege: the unmarked and privileged white identity is depicted here as depending for its very existence on the labor of a marked and African American identity. The protagonist admits:\n\nAs a little boy, I'd thought of our house ... as the place God had meant us to own, but now I knew in a vague way that its seclusion and ease had been artificial and that it had strenuously excluded the city at the same time we depended on the city for food, money, comfort, help, even pleasure. The black maids were the representatives of the city I'd grown up among. I'd never wanted anything from them\u2014nothing except their love. To win it, or at least to ward off their silent, sighing resentment, I'd learned how to make my own bed and cook my own breakfast. But nothing I could do seemed to make up to them for the terrible loss they'd endured. (36)\n\nAlthough he realizes his knowledge is \"vague,\" White's protagonist is able to recognize here that race and geography shape subjectivity, including his own.\n\nEvents later in the chapter corroborate the possibility that, at least initially, the narrator is gaining insight into the unequal distribution of power and wealth. When their maid's daughter survives a bloody fight and needs help, the protagonist's father takes him along on a journey to her home in the \"dangerous\" section of town. This journey forces the protagonist to confront the poverty of the African American section of their town:\n\nThat had been another city\u2014Blanche's two rooms, scrupulously clean in contrast to the squalor of the halls, her parrot squawking under the tea towel draped over the cage, the chromo of a sad Jesus pointing to his exposed, juicy heart as though he were a free-clinic patient with a troubling symptom, the filched wedding photo of my father and stepmother in a nest of crepe-paper flowers, the bloody sheet torn into strips that had been wildly clawed off and hurled onto the flowered congoleum floor. (50)\n\nNonetheless, the wedding photo \"filched\" by Blanche from her white employer is the first sign, I think, of what Morrison might label the \"choked representation\" in this scene. White specifically chokes, or checks, the \"silent, sighing [and potentially threatening] resentment\" of \"the black maids\" through the representation of Blanche's shrine, which certainly suggests anything but resentment. Moreover, the identification of an\/other city is already appropriated by the end of the chapter to facilitate the protagonist's _own_ developing sense of self. The very use of the name Blanche foreshadows the possibility that this scene could be as much about \"White\" as it is about the maid herself. And indeed, when he is considering running away to New York City, the protagonist appropriates\u2014without reference to Blanche herself\u2014the rhetoric he had earlier used to identify her: \"I'd go hungry! The boardinghouse room with the toilet down the hall, blood on the linoleum, Christ in a chromo, crepe-paper flowers ...\" (56). In this entire section, difference is not so much \"exposed\" and challenged as it is safely contained\u2014in White's Cincinnati, in this section of his novel, and, ultimately, in his own \"outcast\" protagonist.\n\nThis is White's second chapter, and racial difference becomes a crucial factor only once more in the novel: when White's protagonist and his friends, in the final chapter, visit the whorehouse. Despite the reemergence of race, though, the relations of power that determine and maintain white dominance remain safely behind in chapter 2. What was earlier an identification _of_ racial difference and discrimination becomes, at the whorehouse, an identification _with_ racial difference (White's protagonist, as I mentioned earlier, equates the woman's situation with his own). It is precisely the \"invisibility\" of the protagonist's racial identity (as well as the \"blankness\" of the regional scene upon which all of this is played out) that allows for this slippage and appropriation to go unnoticed. Since the narrator's regional and racialized body is invisible, he can safely appropriate \"other\" identities for his own limited\u2014I would not necessarily call them \"queer\"\u2014uses. Thus the blankness of the Midwestern landscape and the invisibility of whiteness in _A Boy's Own Story_ are more than simply metaphors for cultural sterility. Unmarked and dislocated racial and regional identities are exactly what enable this to be a \"representative\" story about _gay_ people, rather than about white Midwesterners.\n\nHence queer (op)positionality in _A Boy's Own Story_ is choked by white apositionality. As I suggested, this is as much a function of how the coming-out story had evolved as it is of White's particular novel. However, in some other contexts, the ways in which the identity \"Edmund White\" is constructed replicates this pattern. For example, after White edited an anthology of gay fiction ( _The Faber Book of Gay Short Fiction)_ that included only one black writer (James Baldwin) and no women, the controversy was reported in a publication no less mainstream than _USA Today_. In the article, Kent Fordyce explicitly spells out the tension between sexuality and race: \"I think Faber flubbed the title.... I think it should be 'Edmund White's Anthology of White Short Story Writers'\" (qtd. in C. Wilson, 8D). _The book isn't about white people_ , White himself seems to be saying when he explains, \"[I] read dozens of stories by dozens of gay black writers and I didn't find anything too suitable. And I thought it was wrong to include them just because they were black\" (qtd. in C. Wilson, 8D).\n\nHis dismissal of race notwithstanding, in the foreword to _The Faber Book_ , White continues to appropriate racial identity for its comparative value: \"Do gays really constitute something like an ethnic minority? Does an author's sexuality represent a more crucial part of his identity than his social class, generation, race or regional origins?\" (xvii). White's persistent blindness to race (and to other arenas of difference) except when he is appropriating it to talk about his own oppression is surely what led Essex Hemphill to signify on White in his own introduction to _Brother to Brother: New Writings by Black Gay Men:_ \"When black gay men approached the gay community to participate in the struggle for acceptance and to forge bonds of brotherhood ... we discovered that the beautiful rhetoric was empty\" (xix).\n\nIn _Essentially Speaking: Feminism, Nature and Difference_ , Diana Fuss admits to implying throughout that \"the adherence to essentialism is a measure of the degree to which a particular political group has been culturally oppressed\" (98). And yet, in her chapter on lesbian and gay identity politics, Fuss spends only a page early on \"exposing\" the \"essentialism\" of the Combahee River Collective, Cherr\u00ede Moraga, and Barbara Smith (99). Although Audre Lorde is mentioned in passing in an earlier chapter (44), no other openly gay or lesbian writers of color are engaged in the chapter on identity politics, or in the entire book, for that matter. To me, this hardly provides enough material to justify a sweeping statement the logical conclusion of which would be that lesbians of color are the _most_ essentialist of all. In fact, in contrast to Fuss, I would propose that those who are oppressed in only one facet of their identity often stand the most to gain from essentialism. As White writes in introducing his anthology of (white) gay short story writers, \"Most gay men believe they did not choose to be homosexual, that this orientation was imposed on them, although whether by nature or nurture they have no way of knowing\" (ix). White does not and need not necessarily speak for all white gay men here, but as in _A Boy's Own Story_ , coming out into an essential and essentially oppressed gay identity is exactly what allows White to mask white and male power and to assume a voice that purports to speak for \"most gay men.\"\n\nThe subordination of racial identity to an unmarked (white) gay identity has ramifications far beyond White's texts. It is evident in some recent criticism of gay male fiction that likewise tends not to notice the apositionality of White and whiteness. David Bergman, in his study of gay self-representation in American literature, _Gaiety Transfigured_ , examines Baldwin in one chapter on \"The Agony of Gay Black Literature,\" while White and other whites provide the material for the next chapter on \"Alternative Service: Families in Recent American Gay Fiction.\" Yet under what discursive regime can Baldwin be understood as _not_ about \"family\"? Indeed, I have a hard time thinking of many twentieth-century American writers more interested in exploring the family than James Baldwin. Nonetheless, Bergman positions White under the sign _family_ , while Baldwin is positioned and contained under the sign _race_.\n\nBergman explains, in the introduction to _Gaiety Transfigured_ , that he added a chapter on \"race and homosexuality\" only after the \"excellent suggestions\" of Robert K. Martin (24), and he also concedes, \"I felt some . . . reluctance when I began to explore black gay literature, namely, that as a white man I would fail to grasp the subtle\u2014and not so subtle\u2014differences between the black and gay experiences\" (13). Although he admits that it was a challenge, he reports that he now feels \"a greater sympathy with the gayness of black men than many heterosexual blacks have expressed\" (13). In short, Bergman may not feel competent to write much about black gays, but he is certainly competent to make sweeping dismissals of \"many\" black heterosexuals. Bergman's naive oversensitivity to the positionality of blacks and his reinscription of White\/white apositionality result in a chapter on \"family\" in which he examines without critical comment such questionable scenes from contemporary gay fiction as Andrew Holleran's description of two white men in Union Square observing a \"cocoa-colored youth\" whom they feel they have a right to \"have\" ( _Gaiety Transfigured_ , 188) and White's narrator's own fantasy of an expatriate white gay friend living as a \"garden god\" among a tribe in Mexico ( _Gaiety Transfigured_ , 199).\n\nBaldwin's ghettoization under the sign _race_ here connects, moreover, to the ways in which \"race\" is contained in the larger narrative that some critics, including Bergman, are beginning to tell us about post-Stonewall gay male literature. Around 1978 (or so the story goes), gay literature began to come of age. Since 1978, Bergman explains, \"when Edmund White's _Nocturnes for the King of Naples_ , Larry Kramer's _Faggots_ , and Andrew Holleran's _Dancer from the Dance_ gained critical and commercial success\u2014gay books have become a regular and increasingly large portion of trade publishers' lists\" ( _Gaiety Transfigured_ , 9\u201310). This version of the story is underwritten by White: \"It wasn't until 1978 that three gay novels came out: Larry Kramer's _Faggots; Dancer from the Dance_ , by Andrew Holleran; and my _Nocturnes for the King of Naples_. Those three books gave the impression of a new wave, of a new movement coming along\" (in Bonetti, 95). White modestly appends a disclaimer\u2014\"Mine was probably the least important of those three, as a publishing event\" (in Bonetti, 95)\u2014but just as for Bergman, 1978 is the banner year for White, the turning point for post-Stonewall gay male literature. As White asserts elsewhere, \"Gay male fiction was suddenly on the map\" (\"On the Line,\" xii).\n\nYet where was Baldwin during all of this critical and commercial success ? Although Baldwin was never wholly comfortable with the term _gay_ , his most comprehensive study of _black_ gay desire is nonetheless a product of exactly this period, _Just Above My Head_ was published in 1979, but portions of the novel had begun to appear in that \"banner year,\" 1978. My objection here could certainly be qualified (Baldwin was nearing the end of his career, whereas the three white authors were at the beginning of theirs; and _Just Above My Head_ first began to appear in _Penthouse_ , hardly the premier gay venue), but this has not kept some critics from telling the White\/Bergman \"banner year\" story otherwise. Joseph Beam, for example, in his introduction to _In the Life: A Black Gay Anthology_ , positions Baldwin's novel as one of the few landmarks: \"More and more each day, as I looked around the well-stocked shelves of Giovanni's Room, Philadelphia's gay, lesbian, and feminist bookstore where I worked, I wondered where was the work of Black gay men. . . . How many times could I reread Baldwin's _Just Above My Head_ and Yulisa Amadu Maddy's _No Past, No Present, No Future?\"_ (14). The year 1978 may have been commercially successful for some gay writers, but the narrative that Bergman, White, and others tell depends on a unitary notion of what \"gay\" literature is. In contrast, for Beam gay literature was not necessarily \"on the map,\" since so much territory still remained uncharted. Baldwin and Maddy, however, not White, Holleran, and Kramer, provide Beam with a place to begin the journey.\n\nLike Beam, Emmanuel Nelson, in his preface to _Contemporary Gay American Novelists: A Bio-Bibliographical Critical Sourcebook_ , makes Baldwin (and John Rechy) central to his own coming-of-age as well as to the story we might tell ourselves about contemporary gay literature. In contrast to the deracinated narratives Bergman and White provide, Nelson's narrative begins with the overbearing presence of \"whiteness\":\n\nBarely twenty years old, I had just arrived in the United States from India to work toward a doctorate in twentieth-century American literature. Before long I grew uncomfortable and impatient with a good deal of American literature that was the staple of my graduate courses: the works of white, straight authors. . . . The exclusive focus on white writers .. . was tiresome and frustrating, (xi)\n\nGay literature provided Nelson with alternatives. \"Gay literature,\" however, signifies much differently for Nelson than it does for Bergman and White:\n\nIt was then that I started to discover, on my own, those literary territories whose existence was either unacknowledged or derisively dismissed in the classrooms. ... I began to seek reflections of my own realities within the ethnic and gay spaces of American literature. In particular, I was drawn to the works of James Baldwin and John Rechy. I was drawn to Baldwin because of his elegant prose, his expansive humanity, his sharp challenges to the logic of racism, and his uncompromising deconstructions of conventional sexual assumptions. I was drawn to Rechy because of his authentic style and his rebellious stance; moreover, I imagined an affinity with his dark, Latino protagonists and their familiar and frantic journeys through the anarchic sexual underworlds. That Baldwin and Rechy were, like me, ethnic as well as sexual outsiders in American culture made their perspectives recognizable; their voices and visions became reassuring, even liberating. Their widely different styles of managing their competing ethnocultural and homosexual subjectivities offered me potential models to reconcile the conflicting claims of my own multiple identities. Above all, Baldwin and Rechy enabled me to rediscover American literature, (xi\u2013xii)\n\nOf course, there is more than a little irony in this alternative narrative, in which a scholar from India \"rediscovers\" America and its literature. In contrast to the \"discoverer\" of America who came to this continent and saw \"Indians,\" Nelson comes to America, looks around at its literature, and sees nothing but white folks. By putting a face to the unmarked (white) \"American\" identity of his graduate studies (\"I grew uncomfortable and impatient with .. . the works of white, straight authors. . . . The exclusive focus on white writers . . . was tiresome and frustrating\"), Nelson is able to resist and move beyond the hegemonic narrative being told not only about contemporary gay literature but about American literature more generally.\n\n\"One of the signs of the times is that we really don't know what 'white' is,\" Kobena Mercer writes (\"Skin Head Sex Thing,\" 204). Whiteness, after all, maintains its hegemony by passing itself off as no-thing. As _A Boy's Own Story_ and some recent gay male criticism indicate, whiteness is apositionality; it denies \"the stakes in location, embodiment, and partial perspective [and makes] it impossible to see well\" (Haraway, 191). \"Every gay man has polished his story through repetition,\" White writes in the foreword to _The Faber Book_ , \"and much gay fiction is a version of this first tale\" (ix). Perhaps. Yet White's celebration of the coming-out story as the original gay tale obscures the ways in which the coming-out story positions some gay people as more \"polished\" or representative than others. In _A Boy's Own Story_ and _The Faber Book_ , White disavows the appropriations and erasures that enable him to transform _his_ story into \"a boy's own.\" Denying its own racial situatedness, White's coming-out novel fails as a \"noninnocent\" myth of queer (op)positionality. Like the texts Biddy Martin examines, White's understanding of the \"story\" \"reproduces the demand that women [and men] of color . . . abandon their histories, the histories of their communities, their complex locations and selves, in the name of a [gay] unity that barely masks its white, middle class cultural reference\/referent\" (Martin, 93). The Queer Renaissance requires another myth of queer positionality, one that renames \"gay unity\" by continually reimagining and relocating the complexity of queer histories, communities, and selves.\n\n### **An\/other Myth of Queer Positionality**\n\nAudre Lorde's _Zami: A New Spelling of My Name_ is set in roughly the same time as Edmund White's _A Boy's Own Story_. At about the same time that White's protagonist is learning the difference between Mozart and Haydn, Audre, the persona at the center of what Lorde calls her \"biomy-thography,\" goes to Washington, D.C., to celebrate her graduation from the eighth grade. Stopping at a Breyer's ice-cream and soda fountain, Audre and her family are told they can get their dessert to \"take out,\" but they cannot eat the ice cream on the premises. The bitter episode ends with Audre thinking, \"The waitress was white, and the counter was white, and the ice cream I never ate in Washington, D.C. that summer I left my childhood was white, and the white heat and the white pavement and the white stone monuments . . . made me sick to my stomach for the whole rest of that trip and it wasn't much of a graduation present after all\" (71). In stark contrast to _A Boy's Own Story_ , the mechanisms of white power are all too visible in _Zami_.\n\nClearly, despite nominal similarities (the 1950s setting, the 1982 publication date, the autobiographical elements, homosexuality), _A Boy's Own Story_ and _Zami_ are extremely different texts. Their publication history reflects their difference as well: whereas White's novel was published in hardcover by E. P. Dutton and in paperback by Plume, both divisions of New American Library, _Zami_ was rejected by a dozen or more mainstream publishing houses, including, as Barbara Smith reports, a house known for publishing gay titles (\"Truth That Never Hurts,\" 123). Smith explains,\n\n'The white male editor at that supposedly sympathetic house returned the manuscript saying, 'If only you were just one\/Black or lesbian\" (123). So much for the \"empty niche in the contemporary publishing ecology\" in 1982, the \"slot\" waiting to be filled by the coming-out novel! To my knowledge, Lorde was never interviewed by _USA Today_.\n\n_Zami_ was eventually published by Persephone Press and the Crossing Press Feminist Series, and Donna Haraway, Katie King, and others have already noted _Zami's_ importance for feminist theory. Haraway includes _Zami_ in her discussion of \"feminist cyborg stories,\" which \"have the task of recoding communication and intelligence to subvert command and control\" (175). King is more specific, suggesting:\n\nIt is in this currently contested time\/place [the lesbian bar of the 1950s] where \"the passing dreams of choice\" are mobilized that Lorde looks for the secrets of the making of her personal identity; the passing dreams of choice, where sexual identity is neither an existential decision nor biochemically\/psychoanalytically programmed, but instead produced in the fields of difference individually _and_ collectively (332).\n\nIn this section, taking Haraway's and King's observations as a starting point, I suggest further that, in these \"fields of difference\" where sexual identity is \"produced .. . individually _and_ collectively,\" _Zami_ also allows for a \"recoding\" of the coming-out story, a recoding along the lines of what I call \"queer (op)positionality\" In _Zami_ , coming out can be seen as an\/other myth of queer positionality\u2014not because Lorde makes, as White does in _A Boy's Own Story_ , an attempt to construct her protagonist into an alienated, marked \"other\" but rather because _Zami_ is concerned, as King notes, with collective identity. A humanist, unified self is not Lorde's objective, but rather a definition of \"self\" as defined in and through others, particularly those \"who work together as friends and lovers\" (Lorde, _Zami_ , 255).\n\nAlready this goes against White's objectives in _A Boy's Own Story_ , which ends, after all, with a betrayal that enables the protagonist to self-define _as against_ a lover. Moreover, Lorde's biomythography improves on Linda Alcoff's theory of positionality, which fears that \"poststructuralism's negation of the authority of the subject coincides nicely with classical liberal views that human particularities are irrelevant\" (Alcoff, 420). Indeed, the fiction of identity Lorde constructs in _Zami_ goes beyond the poststructuralism Alcoff fears, since a negation of the authority of the subject underscores here the _relevance_ of human particularities. Taking its protagonist through approximately two decades of development, _Zami_ is framed by a prologue and an epilogue that are meditations on the _particular_ women Audre has loved. In the prologue, Lorde poses the question \"To _whom do I owe the woman I have become?\"_ (4) and proceeds to answer it by naming and describing those who have shaped her identity. Lorde concludes the prologue by apostrophizing:\n\nTo the battalion of arms where I often retreated for shelter and sometimes found it. To the others who helped, pushing me into the merciless sun\u2014I, coming out blackened and whole.\n\n_To the journeywoman pieces of myself_.  \n _Becoming_.  \n _Afrekete_. (5)\n\nOf course, the penultimate stanza of the prologue might read like another teleological myth of essential wholeness, but Lorde immediately undercuts this with fragmentation and open-endedness (\"pieces of myself,\" \"becoming\"). Moreover, even the \"wholeness\" into which Lorde \"comes out\" here is unlike the myth of identity represented in White. Particular \"others\" have helped forge this identity, and the metonymic reference to baking underscores Lorde's emphasis on the construction, not the preexistence, of identity.\n\n_Zami_ constructed a collective \"new spelling of my name\" that was subsequently taken up and reshaped by other readers and writers in the Queer Renaissance. Lorde promoted a similar identity in her _Chosen Poems\u2014Old and New_ , which were also published in 1982. The rest of this section first overviews exactly how identity is reinvented in these two texts. Then, after examining the ways in which both Lorde and White have responded to being cast, in various contexts, as \"representative,\" I conclude by reconnecting _Zami_ to Stonewall and the myths of queer (op)positionality with which I began.\n\nIn the body of the text of _Zami_ , Audre has a number of \"friends and lovers,\" black and white, beginning with Genevieve, a friend who commits suicide while the two are still in high school. Both in her biomythography and elsewhere, Lorde stresses how important this event was for her. _\"Yes, I see Gennie often,\"_ she acknowledged in an interview twenty-five years after the suicide. \"I'll never forget what it is to see young waste and how painful it is. And I never got over wanting to help so that it would not happen again\" (in Corn well, 43). In _Zami_ , the placement of the suicide in the text makes it appear that the event inaugurates Audre into an identity separate from others; after the chapter detailing Gennie's suicide concludes, the next chapter opens, \"Two weeks after I graduated from high school, I moved out of my parents' home\" (103).\n\nThe separation from home and parents, however, does not mark the end of Audre's development; on the contrary, this emergence into a separate identity initiates a cycle of desire that takes Audre through a series of lovers over the course of the text. Although after Gennie's suicide, the protagonist _\"decided that I would never love anybody else again for the rest of my life\"_ (141), she admits the loss of the wholeness she felt with Gennie was actually the commencement, not the end, of desire: \"It is the last dream of children, to be forever untouched\" (141). That this loss was a commencement is underscored by the fact that, in the text, Lorde positions the admission that after Gennie she \"would never love anybody else again\" _after_ she details Audre's first sexual affair with _another_ woman, Ginger. Ginger helps Audre recognize that her earlier resolve to separate from others was both misguided and untrue to what she had learned from and with Gennie. After this affair, Genevieve is not often invoked in _Zami;_ in fact, Audre is caught off guard at one point when she mentions the girl to a lover: \"I surprised myself; usually I never talked about Gennie\" (185). In general, the formation of Audre's identity and her explorations of desire proceed without explicit reference to Genevieve.\n\nNear the end of the text, however, Audre meets Afrekete (\"Kitty\") at a party for black women. When the two women arrive at Afrekete's apartment, Audre notes they are \"in Gennie's old neighborhood\" (247). Indeed, the two are in Gennie's old neighborhood in more than one sense: Afrekete is Audre's final lover in _Zami_ , and she provides a fitting conclusion\/nonconclusion to the problematics of identity and desire that commenced with the loss of Gennie. On one level, the nickname Kitty recalls Audre herself, since\u2014as AnnLouise Keating points out\u2014Ginger had repeatedly labeled Audre the \"slick kitty from the city\" (Keating, 29). This is only the first link in a chain of associations, however; the identity of this final lover is highly unstable and merges freely with others. Not only does the epilogue explain what the prologue did not\u2014that Afrekete is the name of a goddess and of a _\"mischievous linguist, trickster, best-beloved, whom we must all become\"_ (255)\u2014but even as Audre and Kitty make love, the scene shifts seamlessly from the present to memories of Genevieve, so that the identity of the first lover reemerges in the identity of the last. Afrekete thus completes a circle for Audre\/Lorde, bringing her to the point where she can write that her \"life had become increasingly a bridge and field of women\" (255). In other words, whether she understands her life linearly or circularly, as a bridge over which she crosses or as a field that surrounds her on all sides, connections between women are what give that life its shape. Furthermore, in the epilogue, although \"human particularities\" (women's roles and names, both \"real\" and mythical) are present, \"Audre,\" in her specificity as a named, individual subject, is not. In fact, the \"new spelling of my name\" envisioned in the title is finally explained in the epilogue, and it turns out not to be about individuality at all: _\"Zami. A Carriacou name for women who work together as friends and lovers_ \" (255).\n\nAlcoff wants to reclaim an \"identity,\" fictional though it may be, from which women can construct a feminist politics (435). The queer position _Zami_ establishes, however, disclaims Lorde's individual identity. Earlier in the text, Audre realizes that, for her, the passage \"beyond childhood\" entails recognizing herself as \"a woman connecting with other women in an intricate, complex, and ever-widening network of exchanging strengths\" (175). Others have thus authored \"Zami,\" this new identity, with her. Sagri Dhairyam argues, _\"Zami_. . . calls itself 'biomythography\/a description which explicitly . . . recognizes the tactical uses of fictional identity, but refuses to grant the author primacy over the textuality of her life\" (231). In the end, this refusal to grant primacy to any concept of the supposedly individual author ensures that, in _Zami_ , fictional identity and _nonidentity_ alike construct \"the very house of difference rather than the security of any one particular difference\" ( _Zami_ , 226).\n\nAt the same time, this is not some White\/white \"god-trick\" that disavows its own situatedness. It may be impossible to read \"Audre\" as a self-identical, unified individual, but the identities \"black,\" \"lesbian,\" and \"woman\" are all present in the identity \"Zami.\" In a sense, Lorde's persona comes out into a fiction of nonidentity not unlike what Trinh T. Minh-ha envisions:\n\nA critical difference from myself means that I am not i, am within and without i. I\/i can be I or i, you and me both involved. . . . \"I\" is, therefore, not a unified subject, a fixed identity, or that solid mass covered with layers of superficialities one has gradually to peel off before one can see its true face. \"I\" is, itself, _infinite layers_. Its complexity can hardly be conveyed through such typographic conventions as I, i, or I\/i. Thus, I\/i am compelled by the will to say\/unsay, to resort to the entire gamut of personal pronouns to stay near this fleeing _and_ static essence of Not-I. (90, 94)\n\nThis, it seems to me, is queer (op)positionality at its best: an effacement of, and in-your-face-ment to, the liberal humanist God\/man\/subject, with its notions of separation, individualism, and fixity.\n\nAs much as _Zami_ works as a realization of Trinh's unstable i\/I\/Not-I, it also, _pace_ Alcoff, maintains in its very self-definition a commitment to feminist political action; these are, after all, women actively _working_ and _loving_ together. _Zami_ constructs a nonessentialized identity position from which to forge a coalition-based, and oppositional, politics. In fact, King and Haraway both position _Zami_ as an example of Chela Sandoval's \"oppositional consciousness\" (K. King, 338; Haraway, 174). Sandoval herself explains that the notion of oppositional consciousness provides feminists with a new, more fluid definition of \"unity\": \"These constantly speaking differences stand at the crux of another, mutant unity . . . mobilized in a location heretofore unrecognized. . . . This connection is a mobile unity, constantly weaving and reweaving an interaction of differences into coalition\" (18). The act of reading _Zami_ stands as a figure for the \"weaving and reweaving ... of differences into coalition\": since the \"new spelling of my name\" in Lorde's biomythography is not defined until the epilogue, one must read the entire text, with all of its \"constantly speaking differences\" and \"location[s] heretofore unrecognized,\" before one can begin to understand that new spelling.\n\n_Zami_ does not end with the epilogue, however. Indeed, there is a sense in which Lorde's biomythography cannot end, even with her death. By remembering the identities envisioned in _Zami_ , readers have attested to its ongoing vitality and success at achieving an\/other, mutant, mobile unity. The editors of _Afrekete: An Anthology of Black Lesbian Writing_ , for instance\u2014Catherine E. McKinley and L. Joyce Delaney\u2014frame their collection with selections from Audre Lorde. The first selection, \"Tar Beach,\" is the excerpt from _Zami_ wherein Audre is transformed through her encounter with Afrekete. The final selection, \"Today Is Not the Day,\" is a poem in which Lorde calls on Afrekete and resolves to continue working and loving in the face of mounting challenges (particularly cancer). Afrekete and _Zami_ , then, both set the stage for the explorations of identity furthered by this anthology and sustain the writer(s) through the process.\n\nIn her introduction to _Afrekete_ , McKinley explains, \"This is a story at once familiar and new. You may find yourself in it\" (xii). McKinley goes on to detail her own experience of first reading _Zami_ and explains how and why Lorde's text provides the writers included with a useful myth of positionality:\n\nAfrekete, in _Zami_ , is Audre's last embrace. Afrekete is a child of the South, a migrant to Harlem. She is someone you may know. She is both wonderfully common and of the substance from which myths are spun: 'round the way girl, early banjee, roots daughter, blues singer. . . . AFREKETE is many women. With contradictory selves. And while AFREKETE troubles identity politics\u2014her vision stretches much wider, (xiii-xv)\n\nAfrekete, then, is both ancient and new; she is mythological, historical, and visionary. With this contradictory and impetuous figure as their muse, the contributors to _Afrekete_ , according to McKinley, are committed to shaping new visions, selves, and communities:\n\nThe contributors and these editors identify as lesbian, gay, zamis, dykes, queers, Black, African, African-American, biracial\u2014and often may use these terms and others interchangeably. And while sexuality, or race for that matter, is and is not always at the center of their work, both deeply inform the writer's vision. The work featured is written in a range of styles, a breadth of aesthetics reflecting the birthing and meshing of seemingly disparate artistic sensibilities and traditions: Black and queer, as well as others. (xvi)\n\nMany other readers have similarly reinvented themselves and their communities because of the queer identities posited in and by Lorde and _Zami. 14_\n\nLorde's _Chosen Poems\u2014Old and New_ were published in the same year as her biomythography, and the narrative she constructs about her life in this collection of poetry is similar to the narrative she constructs in _Zami_ , in the sense that both texts posit a shifting positionality and a self ultimately defined in and through others. The volume opens with four poems depicting the poet's attempt to come to terms with Genevieve's death. In \"Memorial II,\" the poet approaches her mirror and sees not her own face but Gennie's. Despite this merging of the two girls' identities, however, \"Memorial II\" also represents the poet's attempt to recognize that she does indeed have an identity that is separate from Genevieve's. By the last stanza, the speaker acknowledges that any vision of Genevieve in the poet's mirror can only be a fantasy In this early poem, the poet must recognize herself, as she gazes into her mirror, as a subject separate from Gennie; for if Genevieve were to see her again, she would not recognize the young woman the poet has become: \"Are you seeking the shape of a girl \/ I have grown less and less \/ to resemble\" (5). Although the separation from Gennie is painful and difficult\u2014at the end the poet laments that \"your eyes \/ are blinding me \/ Genevieve\" (5)\u2014it is, at this point, nonetheless inevitable.\n\n\"Memorial II,\" then, with its emphasis on the formation of an individual identity, might allow for a reading of identity that is opposed to the collective identity represented in the epilogue to _Zami_. Yet, as in _Zami_ , this assumption of a separate identity in _Chosen Poems_ is only the beginning of the story. The poet's new, autonomous identity never quite seems to fit. In \"Change of Season,\" for example, she complains, \"Am I to be cursed forever with becoming \/ somebody else on the way to myself?\" (40). Beyond this, the figure of Gennie and that original loss continue to haunt \"Change of Season\" and many other poems: \"I was so terribly sure I would come to april \/ with my first love who died on a Sunday morning \/ poisoned and wondering \/ was summer ever coming\" (41). \"Memorial III: From a Phone Booth on Broadway\" shows how easily the poet's supposedly stable world is thrown into disarray by the memory of that loss:\n\nyou will blossom back into sound\n\nyou will answer\n\nmust answer\n\nanswer me answer me\n\nanswer goddammit\n\nanswer\n\nplease . . . (89)\n\nIn short, the death of Gennie in _Chosen Poems_ provokes the poet's assumption of a separate identity, but with all the insecurities and instabilities specific to a poststructuralist account of the subject incompletely sutured into an identity. In _Chosen Poems_ and _Zami_ alike, however, inauguration into identity for the poet is much more than this: inauguration into identity is simultaneously inauguration into a social system dependent on racism, sexism, and homophobia, and hence into a system intent on defining and controlling all that is black, female, and queer. \"Good Mirrors Are\n\nNot Cheap\" captures the lack of control the poet feels because the cultural context in which she finds herself allows only for deceitful, masked representations of her identity:\n\ndown the street\n\na glassmaker is grinning\n\nturning out new mirrors that lie\n\nselling us\n\nnew clowns\n\nat cut rate. (44)\n\nLorde is thus caught, in _Chosen Poems_ , in the paradox of recognizing the instability of any subject position and yet desperately needing to articulate an identity that has been systematically distorted. Gennie's suicide itself, in both _Chosen Poems_ and _Zami_ , illustrates this central paradox. In _Zami_ , before detailing in prose the events surrounding Gennie's death, Lorde includes a poem that foregrounds the ways in which the event is simultaneously an effacement and an assumption of identity:\n\nBut we wept at the sight of two men standing alone\n\nflat on the sky, alone,\n\nshoveling earth as a blanket\n\nto keep the young blood down.\n\nFor we saw ourselves in the dark warm mother-blanket\n\nsaw ourselves deep in the earth's breast-swelling\u2014\n\nno longer young\u2014\n\nand knew ourselves for the first time\n\ndead and alone. (97)\n\nWe \"knew ourselves for the first time \/ dead and alone\": obliteration of identity and assumption of identity come together in the same moment. All of this, at the same time, occurs in an oppressive system intent on \"keep[ing] the young blood down.\"\n\nIn \"Need: A Choral of Black Women's Voices,\" the final entry in _Chosen Poems_ and one newly written for Lorde's collection, the poet further probes this paradox, reproducing on a more urgent level the exploration of identity\/nonidentity found in the epilogue to _Zami_. Coming together with others is never easy (in fact, another selection in _Chosen Poems_ poses the insistent question \"Who Said It Was Simple\" [49]), but from the title of \"Need\" on, Lorde implies that a collective voice is necessary for survival. \"Need\" opens by giving voice to that which has been silenced and by making visible that which would otherwise be effaced:\n\nThis woman is Black\n\nso her blood is shed into silence\n\nthis woman is Black\n\nso her death falls to the earth\n\nlike the dripping of birds\n\nto be washed away with silence and rain. (111)\n\nThe poem has three speakers, \"I,\" \"P.C.,\" and \"B.J.G.,\" the latter two representing the voices of Patricia Cowan and Bobbie Jean Graham, two women murdered in Detroit and Boston in 1978 and 1979, respectively. As in _Zami_ , naming is a central preoccupation in \"Need.\" After P.C. and B.J.G. describe their violent deaths, the \"I\" of the poem rages, \"I do not even know all their names. \/ My sisters deaths are not noteworthy \/ nor threatening enough to decorate the evening news .. . blood blood of my sisters fallen in this bloody war \/ with no names no medals no exchange of prisoners\" (112). The three voices weave in and out in this prolonged meditation on violence and oppression, until the final stanza of the poem, when the \"I\" transforms into an \"All\": \" _'We cannot live without our lives' \/ 'We cannot live without our lives'\"_ (115). A note explains that the italicized quotation is from a poem by Barbara Deming. The words are therefore Lorde's and not Lorde's, and the individual identities of Lorde\/I\/Patricia Cowan\/Bobbie Jean Graham\/Barbara Deming, along with all the women named and not named in _Chosen Poems_ , coalesce, as they do in the epilogue to _Zami_ , into a collective and threatening identity that depends for its existence on the foregrounded yet constantly shifting positionality of the identity \"black\"\/\"lesbian\"\/\"woman.\" The poem, once again, belies any notion of an essentialized self existing apart from others, and indeed, in the face of such violence and destruction, such separation seems not only unproductive but absurd.\n\n\"Need\" itself is a particularly useful example of the ways in which Lorde comes out into a myth of collective identity. Lorde's _Chosen Poems\u2014Old and New_ was originally published by W. W. Norton and Company. \"Need,\" however, was reissued as a pamphlet in 1990 by Kitchen Table: Women of Color Press as part of their Freedom Organizing Series. In the preface to this new, revised version of _Need_ , Lorde traces the poem's genealogy. As 1978 was for white gay men, 1979 is a \"banner year\" for black and Latina lesbians. The latter banner year, however, looks significantly different from the former:\n\n\"Need\" was first written in 1979 after 12 Black women were killed in the Boston area within four months. In a grassroots movement spearheaded by Black and Latina Lesbians, Women of Color in the area rallied. . . . My lasting image of that spring, beyond the sick sadness and anger and worry, was of women whom I knew, loved, and trembled for: Barbara Smith, Demita Frazier, Margo Okazawa-Rey, and women whose names were unknown to me, leading a march through the streets of Boston behind a broad banner stitched with a line from Barbara Deming: \"WE CANNOT LIVE WITHOUT OUR LIVES. (3)\n\nLorde again attributes the words to Barbara Deming, but the identity\/nonidentity articulated in _Need_ and in Lorde's genealogy of \"Need\" belies any unified authorial consciousness. Instead, the identity is an example of Haraway's \"contradictory\" standpoint, which gains its strength precisely because authorial consciousness is \"permanently partial.\" Deming does not, therefore, become so much the \"source\" here as another element in the collective and shifting identity into which the women behind the banner come out. This collective identity is a powerful and threatening one, made more so by its lack of fixity. Indeed, the (re)issuance of _Need_ reaffirms and deploys that lack of fixity: each pamphlet includes a button with the line \"We Cannot Live without Our Lives\" printed on top of the pan-African colors and beside the symbol for female. Neither Lorde nor Kitchen Table makes any attempt here to suture, within the reissued text, the identity being articulated; on the contrary, the button and a \"Resources for Organizing\" section that follows the poem encourage women (and apparently, men) reading the text to join and hence continually reshape this collective identity. Lorde herself, in fact, traces the 1990 revisions of the poem to the ways in which others had used and reshaped it since its publication: \"Alterations in the text since the poem was originally published are a result of hearing the poem read aloud several times by groups of women\" ( _Need_ , 3).\n\nThe Kitchen Table version of _Need_ seems at least to allow for male inclusion: the \"Resources for Organizing\" section that follows the poem includes the addresses for organizations such as NCBLG: The National Coalition of Black Lesbians and Gays, the National Black Men's Health Network, Men Stopping Rape, and the Oakland Men's Project ( _Need_ , 16\u201317). Identity, as Lorde constructs it in _Zami, Need_ , and elsewhere, however, while not exclusionary, does not always simplistically include men or white people; connection is not necessarily easy or automatic. Hence \"Zami\" is _an_ \"other myth of queer positionality,\" not _the_ necessary corrective to versions of coming out such as White's. As Elizabeth Alexander notes, \"'A' new spelling (as opposed to 'the') means there is probably more to come\" (704). In the Queer Renaissance, there have indeed been many more \"new spellings,\" as subsequent chapters of this book will demonstrate.\n\nHowever, Lorde often does connect with men\u2014particularly men of color\/gay men\u2014throughout her work. In fact, one of Lorde's earliest uses of the phrase that would become the subtitle to _Zami_ occurs in a poem from the late 1970s to \"Brother Alvin,\" a boy from Lorde's second-grade class who suddenly died of tuberculosis:\n\nI search through the index\n\nof each new book\n\non magic\n\nhoping to find some new spelling\n\nof your name.( _The Black Unicorn_ , 54)\n\nA more recent and particularly poignant example of connection with a man comes in her poem to the late Joe Beam. In \"Dear Joe,\" Lorde uses as an epigraph words from \"Sister, Morning Is a Time for Miracles\" ( _Chosen Poems_ , 109\u201310). The words that had signified Lorde's attempt to connect with another woman, her sister, are here re-signified in her memorial to Beam: \"if you have ever tried to reach me \/ and I could not hear you \/ these words are in place of the dead air \/ still between us\" (\"Dear Joe,\" 47). \"Zami,\" then, may signify \"women working together as friends and lovers,\" but the new ways of spelling identity that Lorde envisions do not preclude working with and loving others.\n\nAs _Need_ ,\"Brother Alvin,\" and \"Dear Joe\" illustrate, and as with White in and out of _A Boy's Own Story_ , the construction of identity in _Zami_ connects to the construction of \"Audre Lorde\" in other contexts. I should note before continuing, however, that my earlier account of White as the one of these two authors more likely to be published by houses such as New American Library, while Lorde is consigned to lesser-known houses such as Persephone or Crossing Press, was somewhat unfair. White's publication history, like Lorde's, has at times been rocky, and Lorde has indeed been published by both small feminist presses and mainstream houses such as Norton. Both authors are therefore made available for representation in a variety of contexts. I do not want to institute an argument in this chapter that implies too much about either White's or Lorde's canonical or precanonical status: for example, that White is \"more canonical\" than Lorde because he is oppressed in only one facet of his identity and is hence closer to what Lorde called the \"mythical norm\" of American society ( _Sister Outsider_ , 116). The \"canonicity\" of either author is actually quite difficult to assess, and not only because both are contemporary authors. It would appear that both White and Lorde are fairly canonical, but in two different contexts. Although _Time_ magazine declared White \"America's most influential gay writer\" (L. Schulman, 58), it is Lorde who is more likely to be taught in college classrooms or to be the subject of scholarly articles. The MLA (Modern Language Association) Bibliography, for instance, lists forty entries for Lorde between 1981 and 1994 and twelve for White. Ten of White's entries, however, are articles he himself has written, compared to only four such articles for Lorde. Of Lorde's forty entries, moreover, seven are dissertations, underscoring her importance to the generation of scholars who will be shaping the academy over the next few decades.\n\nLorde's burgeoning academic reputation is highly contestable, of course; other contemporary writers, such as Toni Morrison and Thomas Pynchon, have garnered much more attention during the same time period (Morrison, 329 MLA Bibliography entries; Pynchon, 514 entries). Nonetheless, Lorde is better known than White in academe (in part because of women's studies departments and academic feminism), while White, in contrast, is becoming ensconced as the gay author mainstream readers need to know. The Quality Paperback Book Club (QPB), for example, \"proud to announce the launch of Triangle Classics, a series of landmark books illuminating the gay and lesbian experience,\" initially included one writer from the 1980s in their new series: Edmund White. The flyer for the series declares that QPB's edition \"brings together Edmund White's landmark novels of coming-of-age and coming out,\" that is, _A Boy's Own Story_ and _The Beautiful Room Is Empty_. The series included one black author, James Baldwin; but significantly, the Baldwin novel included, _Giovanni's Room_ , is about white Europeans.\n\nIf, as Richard Ohmann suggests, contemporary canon formation is the result of \"both large sales .. . and the right kind of critical attention\" (384), then it would seem that White is cornering the market on one necessary qualification and Lorde on the other. Michael B\u00e9rub\u00e9 complicates Oh-mann's thesis, suggesting that \"we are no longer confronting Ohmann's mid-1970s landscape\" and that academic critics \"now represent contemporary writers to different audiences from those of the nonspecialist press\" (31). To B\u00e9rub\u00e9, then, the Toni Morrison of academic criticism is not the same as the Toni Morrison of nonacademic journals and reviews; indeed, there is a competition between members of these two groups for what \"Toni Morrison\" will signify. In contemporary lesbian and gay writing, this competition, I would argue, is conducted not so much over individual authors as over the very sign _gay\/lesbian literature_ , and moreover, this competition tends to be split along gender lines. Hence, and very generally, the contemporary \"gay(\/lesbian) literature\" represented in nonacademic journals and reviews is not the same contemporary \"lesbian(\/gay) literature\" of academic criticism.\n\nConsider, for instance, the very different observations of Ed Cohen, a professor of English at Rutgers University, and Victoria Brownworth, a lesbian journalist and fiction writer. Cohen justifies his focus on gay men in his essay on \"Constructing Gender,\" in _The Columbia History of the American Novel_ , in this way:\n\nIf I focus now on the former [gay male writing] rather than the latter [writings by women of color\/lesbians, which he acknowledges have also been important to his intellectual development] it is because I know that in a volume like this one it is likely that the works of women of many races and ethnicities will have been addressed heretofore, while the works of men who are exploring the possibilities for sexual and emotional intimacies with other men will most probably remain eccentric. (557)\n\nIn contrast, Brownworth insists:\n\nUnlike gay men . . . lesbians have not been part of the big queer book boom of the last few years. We haven't received the same advances as our gay male counterparts. So while there may be more lesbian writers than ever before, few of us are making a living by writing alone\u2014most of us supplement our income with teaching or lecturing. For lesbians there's no money in being a writer. (49)\n\nLesbian writers and readers have witnessed a few major \"publishing events\" over the past few years (most notably Dorothy Allison's _Bastard out of Carolina_ , which was a finalist for the National Book Award), but some have claimed that these events were made possible because the lesbian content was not always explicit in such books.\n\nThis is not to argue, by any means, that academic criticism is somehow inherently more progressive because lesbians are on top; on the contrary, in both arenas, the uses to which the winner of the gendered competition might be put are more important than who, specifically, wins. Publishers take advantage of White's purchase _outside_ the academy, for example, to market his texts. The cover of White's 1978 novel _Nocturnes for the King of Naples_ includes _Newsweek's_ assessment of the author, which is identical to that of _Time_ magazine: \"White is unquestionably the foremost American gay novelist.\" _Inside_ the academy, in contrast, Lorde is able to cash in on a desire for \"difference\" that can be, nonetheless, safely contained. Anna Wilson, discussing Lorde's increasing canonization in the academy, argues:\n\nFor feminist academia Lorde is particularly effective as a token: since she is Black, lesbian and a mother, her work compactly represents that generally repressed matter towards which white feminists wish to make a gesture of inclusion\u2014but since Lorde conveniently represents so much at once, she can be included without her presence threatening the overall balance of the white majority vision. (77)\n\nCertainly, the majority vision in literary studies, if not in feminist academia, is also heterosexual, or at least heterosexualized. Thus, in literary studies, Lorde can \"conveniently represent,\" along with other \"differences,\" a homosexuality that nonetheless does not threaten to disrupt the straight narrative.\n\nOf course, according to Ohmann's thesis, openly lesbian and gay literature will never be canonized, since the two necessary ingredients of canon formation are not really coalescing for any individual author. I have already mentioned B\u00e9rub\u00e9's complication of Ohmann, but I include here a third, mediating context that emerged during, and even before, the Queer Renaissance and that particularly complicates the canonicity question for queers: the gay and lesbian marketplace. Lorde and White may not be \"canonical\" in quite the same way as other contemporary writers are, but because of the existence of a community-based marketplace, as White himself points out, \"even quite celebrated heterosexual authors\u2014watching their books go out of print or out of stock\u2014might well envy the longevity of books written by lesbians and gay men\" (\"Twenty Years On,\" 4). Lesbian and gay bookstores, literary reviews, award ceremonies, and the like ensure that canonization is an extremely complex affair for openly gay and lesbian writers. In this third context, the community context, White and Lorde are both among the most canonical of contemporary writers.\n\nThus both Edmund White and Audre Lorde are positioned by others, in various ways, as \"representative,\" and as representative of overlapping and competing constituencies with varying degrees of access to the \"center.\" Each author, however, responds differently to the ways in which he or she has been represented by others, and in general, White's and Lorde's responses parallel the ways in which each constructs identity within the texts I have been examining. In 1990, when Lorde was presented with the second annual Bill Whitehead Memorial Award, in recognition of outstanding contributions to lesbian and gay literature, she informed the audience, \"One award will not counterbalance a continuing invisibility of Lesbian and Gay writers of color\" (\"What Is at Stake,\" 66). Using her individual location to emphasize how her identity had been shaped in and through others, Lorde went on to explain that the best way to honor her was to honor those she had loved and worked with: \"If this group wishes to truly honor my work, built upon the creative use of differences for all our survivals, then I charge you, as a group, to include and further expose the work of new Lesbian and Gay writers of color within the coming year, and to report on what has been done at next year's award ceremony\" (\"What Is at Stake,\" 66). In contrast, Edmund White, the recipient of the first annual Bill Whitehead Memorial Award in 1989, concluded his speech with the reflection, \"Oddly enough, what literature has always taught us is that only in tracing our individuality can we become universal\" (\"Twenty Years On,\" 5). \"We,\" \"our,\" and \"us,\" of course, signify quite differently in White's speech from how \"our\" signifies in Lorde's \"creative use of differences for all our survivals.\" Despite his stress on \"the recording of our differences\" in gay and lesbian literature, when White himself acknowledges that \"it has struck me as no coincidence that many of the most original writers of this century have been gay\" (5), the list he produces in support of this claim includes no people of color.\n\n_Then I caught myself foolishly imagining that gays might someday constitute a community rather than a diagnosis_. White's protagonist's thoughts during the Stonewall Riots, with which I began this chapter, apparently contradict the points I have been making about White\/white apositionality and appear to participate in a more productive myth of identity, akin to those articulated in _Zami_ and \"Need.\" This moment of potentiality in _The Beautiful Room Is Empty_ , however, is trumped by the novel's notorious ending:\n\nI stayed over at Lou's [one of the nameless narrator's friends]. We hugged each other in bed like brothers, but we were too excited to sleep. We rushed down to buy the morning papers to see how the Stonewall Uprising had been described. \"It's really our Bastille Day,\" Lou said. But we couldn't find a single mention in the press of the turning point of our lives.\" (227\u201328)\n\nAt the end of _The Beautiful Room Is Empty_ , bittersweet isolation and invisibility triumph over the possibility of community.\n\nYet the story of Stonewall has been told otherwise. Martin Duberman's historical overview, _Stonewall_ , interweaves the stories of six people who were active in lesbian and gay communities during the time of the Stonewall Riots. Craig Rodwell was one of the men actually present when the riots broke out, and Duberman details his reaction: \"Craig dashed to a nearby phone booth. Ever conscious of the need for publicity\u2014for _visibility_ \u2014and realizing that a critical moment had arrived, he called all three daily papers, the _Times_ , the _Post_ , and the _News_ , and alerted them that 'a major story was breaking\/Then he ran to his apartment a few blocks away to get his camera\" ( _Stonewall_ , 198; emphasis mine). Rodwell's photographs never came out, but Duberman's \"day after\" is nonetheless not characterized by the existential alienation White's protagonist and his friend feel:\n\nWord of the confrontation spread through the gay grapevine all day Saturday. Moreover, all three of the dailies wrote about the riot (the _News_ put the story on page one), and local television and radio reported it as well. The extensive coverage brought out the crowds, just as Craig had predicted (and had worked to achieve). All day Saturday, curious knots of people gathered outside the bar to gape at the damage and warily celebrate the fact that, for once, cops, not gays, had been routed. ( _Stonewall_ , 202)\n\nAt Stonewall and in _Stonewall_ , gay men and lesbians \"come out\" into a myth of collective identity, and the ramifications of that collective act are still being felt today.\n\nDuring high school, the protagonist of _A Boy's Own Story_ and Tommy, his current obsession, go slumming: \"He and I had trekked more than once downtown ... to listen, frightened and transported, to a big black Lesbian with a crew cut moan her way through the blues\" (120\u201321). Of course, this exoticization of the \"big black Lesbian\" is only a minor incident for the narrator, unconnected to the larger project of coming out into his own, individual, gay consciousness. And yet the black lesbian singing the blues in _A Boy's Own Story_ is not as out of place as she might at first appear. Teleological and essential (white) \"boys' own stories\" at this point offer feminism and queer politics little in the way of queer (op)positionality. Like the protagonist of Lorde's _Zami_ , whose \"heart ached and ached for something [she] could not name\" (85), the blues singer in White's novel needs an\/other myth of queer positionality. Coming out into an essential wholeness may be the myth that lesbians and gay men are told they must embrace, but as Audre's teacher declares early on, when the young protagonist of _Zami_ refuses to take dictation in the same manner as the rest of the class, Audre is \"a young lady who does not want to do as she is told\" (26). Lorde responds instead with a nonessentialized, non-self-identical \"new spelling of my name\" in _Zami_ , and only through this new construction is she able to envision a queer and powerful community of women, whose new identities are permanently partial and whose coalitions are conscious.\n\n## **_Chapter Two_  \nQueer Locations\/Queer Transformations**\n\n\"Vito Russo pointed out in cinema . . . that historically, the gay character always had to end up with his head in the oven or in some similar state,\" Henry Louis Gates, Jr., explained in a 1991 interview. \"It was like a Hays rule that you had to come to a bad end. _Giovanni's Room_ isn't really an exception to this; and in Randall Kenan's book you get a brilliant tormented homosexual, Horace, who commits suicide\" (in Rowell, 454). Gates praised Kenan's novel _A Visitation of Spirits_ but was nonetheless wary of the suicidal ending: \"There's another way of reading this [suicide]: which is just as a way of registering some pretty tragic facts of history. . . . But I want Randall Kenan to, as it were, take Horace to the big city in his next novel\" (in Rowell, 454).\n\nGates's prescription for Kenan is in many ways predictable; the \"migration to the big city\" is a widely available trope in contemporary lesbian and gay literature, with a long and illustrious history And yet I find the need to transport characters like Horace off to \"the big city\" symptomatic of a regional elision in queer theory generally. What Gates elides in his suggestion to Kenan is the fact that taking Horace _to_ anywhere also entails taking him _from_ somewhere. In this case, the unmentioned \"somewhere\" is the fictional Fundamentalist Christian, rural, African American community of Tims Creek, North Carolina. Not the most conducive atmosphere for the expression of queer desire, certainly; but as liberal lesbian and gay thought likes to remind us, \"we are everywhere,\" and rather than concede that \"everywhere\" actually means New York and San Francisco, I am interested in the (perhaps more radical) implications of recognizing that \"everywhere\" includes such an apparently marginal and inhospitable place.\n\nGates's vision of Horace in the big city is understandable given the tragic suicide that ends the novel, yet the desire to fix Horace's situation is not exactly true to Gates's own theories of signification. In _The Signifying Monkey_ (and elsewhere), Gates examines the rhetorical process of \"Signifyin(g)\" as it has evolved in diverse African Caribbean and African American communities. \"The Afro-American concept of Signifying),\" Gates writes, involves \"formal revision that is at all points double-voiced\" (22). Gates sees this double-voiced rhetorical principle at play in the Signifying Monkey tales, which have been passed on and revised by African American speakers in \"barrooms, pool halls, and [on] street corners\" (54). The Signifying Monkey tales present listeners with a master of trickery: the Signifying Monkey bests his opponent, the Lion, by skillfully opening up a play of meaning. Through a series of insults directed at the Lion and attributed to the Elephant, the Signifying Monkey tricks the Lion into sparring with the Elephant, who\u2014in turn\u2014always physically defeats this supposed \"king of the jungle.\" The Signifying Monkey succeeds because the Lion, who always equates the figurative with the literal, is unable to see through the Monkey's linguistic games. \"Another way of reading\" _A Visitation of Spirits_ could position the text within the tradition of Signifyin(g), considering how the novel shapes a queer trickster identity that rewrites both the \"pretty tragic facts of history\" that Gates acknowledges and the \"corrective\" he offers: the migration to the big city.\n\nContemporary gay fiction that deals with \"family\" or \"community\" often exposes the ways those concepts cover over difference: the group achieves a cohesive identity through disavowal of \"aberrant\" individual identities. Thus, as Gates suggests, \"One thing that a good deal of contemporary fiction that deals realistically with gay themes achieves, which I think is very important, is to desentimentalize the notion of 'community' as an unadulterated good\" (in Rowell, 454). In my mind, this is precisely what Kenan does with the story of Horace in _A Visitation of Spirits;_ yet Gates is uncomfortable enough with Horace's suicide to envision _another_ community for Horace in \"the big city.\" Certainly, the narrative in which Gates places Horace allows for the possibility of an _alternative_ community, perhaps one more akin to the sustaining community envisioned in _Zami: A New Spelling of My Name_. Difference, however, is again suppressed by transporting Horace off to a community of others \"like him,\" and away from the community he threatens. Locating Horace in an urban area where, presumably, a \"black gay identity\" is more developed and secure effaces the possibility of transforming the community in which Horace is already located and\u2014more important\u2014undermines Kenan's critique of the \"regime of sameness\" embodied by the people of Tims Creek.\n\nThe term _regime of sameness_ is Marcos Becquer's, and before I proceed with my reading of Horace and \"transformation\" in Tims Creek, North Carolina, I want to use Becquer's analysis to center my own. Becquer analyzes \"snapping\" and \"vogueing,\" two of the black gay discursive practices celebrated in and by Marlon Riggs's critically acclaimed film _Tongues Untied. 5_ Since the discursive practices in Riggs's film \"emerge both from within and against the cultural and historical discourses operating around them\" (8), Becquer argues that _\"Tongues Untied_ can confront and condemn the regime of sameness which alienates black gays from the black community, the white gay community, and discourse\/representation in general\" (14). In other words, black gays, confronted by black heterosexual or white gay communities with the compulsion to be \"the same,\" can, in turn, use the discourses made available by those very communities to contest such a compulsion. Becquer's analysis thus foregrounds \"difference\" while nonetheless arguing for the connective, political importance of a subverted and subversive \"sameness\":\n\nIt is, then, however ironically or heroically, just that differentiated voice within sameness which _Tongues Untied_ attempts to distill, so as to ensure not only that black gays speak up, but that they remain audible in discourse. It proceeds, then, not by nullifying the value of sameness within difference . . . but by acknowledging its political importance and admitting, within the logic of \"constructed identity,\" that sameness is always already a part of difference, as well as vice versa. (15)\n\nBecquer's theory, with its focus on snapping and vogueing, is particularly relevant when discussing the tremendous outpouring of urban and secular black gay cultural production represented in films such as Riggs's _Tongues Untied_ and Jennie Livingston's _Paris Is Burning_. Indeed, Becquer explicitly elaborates on \"black gays' _secular_ use of snapping as a means by which to metaphorically awaken one out of the codes of (discursive) domination\" and on vogueing's \"reconstitut[ion of] the literal _urban_ battlefield of bloody violence . . . into a figurative arena upon which these confrontations between images are played out\" (9, 12; my emphasis). In this urban and secular context, snapping and vogueing address the exclusionary practices of black heterosexual and white gay communities: Becquer concludes, for instance, that the segment on snapping in _Tongues Untied_ \"depicts the snap precisely in its ability to overcome the discursive mechanisms which position black gays beneath both black heterosexuals and white gays\" (9). Through the snap, black gays reposition themselves as apart from and yet a part of, on the one hand, white gays, and on the other hand, black heterosexuals, and \"the very binarism of sameness\/difference\" (15) is thereby deconstructed. In other words, snapping (and elsewhere in Becquer's article, vogueing) works as an empowering and signifying difference for black gays, but this signifying difference nonetheless forges connections with the groups it critiques. With this revised idea of sameness in mind, Becquer concludes, _\"Tongues Untied_ is not a separatist film\" (15).\n\nBecquer's analysis is fueled by \"recent revisions of identity politics,\" which understand identity as a construction and thus allow for \"the hope of deconstructing the binarism of otherness which marks discursive alienation and domination by acknowledging that the other is always already a part of ourselves and vice-versa\" (7). Yet, although putting sameness back into difference forges connections and hence undermines, as Becquer argues, black homophobia and white gay racism, this queer theoretical move should not overshadow the ongoing need for a queer theory that challenges the \"regime of sameness,\" even when that regime is reproduced _inside_ the cultural category \"black gay.\" Despite Becquer's best efforts, indeed, his article concludes with the inscription of a fairly monolithic, snapping and vogueing \"black gay identity\" singular (16). Although he begins the article decrying \"the essentialism inherent in notions of _the_ black subject or _the_ gay sensibility\" (7), Becquer himself subtly moves from plurality to singularity in his discussion of black gay identities: _Tongues Untied_ is, at the beginning, \"a condensed version of black gay (collective) experiences\" (8) but has become, by the end, a celebration of \"the emergence of a black gay difference that is unique\" (15).\n\nThis slippage does not invalidate Becquer's argument; it simply demonstrates, as Ed Cohen suggests, that \"no matter how sensitively we go about it, 'identity politics' has great difficulty in affirming difference(s)\" (\"Who Are 'We'?\" 76). Becquer's article, with its critique of white gay and black heterosexual hegemony, recognizes the difficulty of affirming difference but simultaneously affirms the \"sameness\" that is always already present within \"difference.\" Cohen argues, however, with a nod to Diana Fuss, that \"identity politics is predicated on denying the difference that is already there in 'the same'\" (\"Who Are 'We'?\" 76). This predication ensures that difference can be denied or repressed even when identity politics is grounded in sophisticated poststructuralist attempts, such as Becquer's, to move beyond the sameness\/difference binarism.\n\nWhile black gays are undeniably marginalized by black heterosexuals and by both heterosexual and gay whites, and hence are strategically positioned to disrupt and decenter heterosexual and white hegemony, a focus on _urban_ black gays will always, in turn, produce other margins. Still, I want to extend rather than disarm Becquer's analysis. By pushing his ideas further, I hope to create a space in which to consider black gay cultural production (and perhaps queer cultural production generally) _outside_ an urban, secular arena. It was no accident that I used Becquer's analysis to \"center\" my own. Until quite recently, queer theory has predominantly \"centered\" on urban areas. Such a focus is in some ways inevitable: after all, as John D'Emilio and Estelle B. Freedman explain, it was initially in American cities that a gay subculture flourished in the middle of the twentieth century (288). The urban \"center\" that D'Emilio, Freedman, and other queer theorists and historians have analyzed, though, might be productively understood as part of a complex array of \"centers\" and \"margins,\" since both concepts emerge relationally. Moreover, in the Queer Renaissance, critical interrogation of the processes of marginaliza-tion might proceed hand in hand with the establishment of unexpected, even unlikely, new centers. Thus we would do well to consider\u2014before placing Kenan's Horace \"safely\" (or more \"appropriately\"?) in an urban \"center\"\u2014just what black queer desire is doing in, or does to, rural North Carolina. Essex Hemphill's words, with their subtle promise\/threat that black gays will transform _whatever_ community they are in, seem appropriate to me here: \"I ask you brother: Does your mama _really_ know about you? Does she _really_ know what I am?... I hope so, because I _am_ coming home\" ( _Ceremonies_ , 42).\n\n### **Transformations (I): Horace Thomas Cross**\n\n_A Visitation of Spirits_ concerns itself with several members of the Cross-Greene clan, an African American family living in Tims Creek, North Carolina, in the mid-1980s. The novel focuses particularly on Horace, the youngest member of the Cross family, and on Jimmy Greene, Horace's cousin, the young minister at the First Baptist Church of Tims Creek and the first black principal of Tims Creek Elementary School. _A Visitation of Spirits_ is organized around two days in the life of the Cross-Greene family: December 8, 1985, and April 29\u201330, 1984. The text moves back and forth between these two days, and each section heading further specifies the exact placement of events in time: for instance, \"December 8, 1985 \/ 8:45 A.M. . .. April 30, 1984 \/ 1:15 A.M. (3, 66). This temporal precision gives each section of the novel the appearance of measurable, scientific \"fact\"; the events of _A Visitation of Spirits_ , however, belie any easy distinction between \"fact\" and \"fiction.\" Unable to live up to the (heterosexual) expectations of the community and consequently dissatisfied with his life, Horace Cross attempts to use a magic spell to transform himself into a bird. When the transformation fails, \"spirits\" and \"demons\" reveal themselves to Horace in order to lead him on a whirlwind journey through his own life. Past, present, and future blur together as freely as fact and fantasy as Horace's journey progresses. Even Horace himself is confused as to whether what he is seeing is \"real\" or not. One of the demons attempts to explain: \"Ghosts? Yeah, you might call them ghosts. Ghosts of the past. The presence of the present. The very stuff of which the future is made. This is the effluvium of souls that surround men daily\" (73).\n\nThe echoes here are of the humanistic, transformative experience of Ebenezer Scrooge, and in fact, Kenan begins his novel with an epigraph from _A Christmas Carol_. In _A Visitation of Spirits_ , however, Horace is not in a position to experience the same sort of happy ending as Scrooge does in _A Christmas Carol_. In contrast to the miserly gentleman of Charles Dickens's tale, Horace, after observing the constraint and confusion he has endured throughout his young life, does not undergo some humanistic \"redemption\"; instead, he commits suicide. In the end, it is not Horace the individual but the position and the place in which he finds himself that are in need of transformation.\n\nAfter spending weeks preparing the spell, Horace decides that his transformation into a bird will occur at midnight on April 29, 1984. Horace performs the necessary incantation, and a few minutes after midnight, someone or something begins to call: \"The voice said: Come\" (27). The remainder of Horace's story, like Ebenezer Scrooge's, is told in three episodes, roughly corresponding to Horace's past, present, and future. In the first episode, the voice\/demon shows Horace the \"ghosts\" of his own past. Horace finds himself in front of the First Baptist Church of Tims Creek, where a service is in progress. His entire family (including a five-year-old version of Horace himself) is present in the congregation. This community, to Horace, is the embodiment of what he is attempting to escape:\n\nThey were fat and thin, light and dark, tall and short, farmers, salesmen, mechanics, barbers, nurses, mothers, fathers, aunts, uncles, cousins, lovers, friends. Here was community, not a word but a being. Horace felt it as though for the first time. Here, amid these singing, fanning, breathing beings were his folk, his kin. Did he know them? Had they known him? It was from them he was running. Why? (73)\n\nClearly, \"community\" in this episode signifies much differently than it did in Audre Lorde's _Zami: A New Spelling of My Name_. In my previous chapter, I used _Zami_ to argue for \"other\" identities; that is, for identities defined in and through others. The community re-presented to Horace by the demon, however, is a far cry from Lorde's sustaining vision of women working together as friends and lovers. The enumeration of various identities here suggests that this is another \"house of difference,\" but\u2014 _pace Zami_ \u2014Horace at this point wants to define himself only _against_ , not _through_ , this group.\n\nIn this chapter, consequently, I look more critically at the idea of community. I explore the ways in which this community incites the \"transformation\" of Horace the _individual_ , but I also consider, conversely, how the location of Horace and his queer story in rural North Carolina puts into play a critique that has the potential to transform this _community_ , and others like it, on the supposed margins of the queer world. After examining the importance of North Carolina to Horace's story, I look specifically at the institution in Tims Creek most in need of transformation: the church. I then expand my argument, looking to the community more generally, to consider some strategies (\u00e0 la Becquer and Gates) that Horace might use to begin the queer work of transforming this community and its institutions. Although this section concludes with some of the ways in which Horace apparently fails to effect change, the next section insists that failure is always _only_ apparent or temporary. Through a reading of Horace's cousin Jimmy Greene, I argue that _A Visitation of Spirits_ ultimately succeeds in establishing a queer trickster identity intent on exposing the queerness at the center of identification and desire, disrupting the exclusionary logic on which \"community\" is often founded, and sustaining a transformative critique of this community.\n\nHorace himself underscores the urgency of transformation occurring in rural North Carolina: \"Cats had a physical freedom he loved to watch, the svelte, smooth, sliding motion of the great cats of Africa, but he could not see transforming himself into anything that would not fit the swampy woodlands of Southeastern North Carolina. He had to stay here\" (11). Horace is imagining his own transformation, not the community's, but his thoughts here nonetheless provide a point of departure for an interrogation of this queer location. For some reason, Horace feels compelled to stay in Tims Creek; something about the location proscribes the possibility that his story take place elsewhere.\n\nHorace never solves the riddle of why he feels this compulsion to stay, but near the end of the story there is an allusion to a text that could provide some clues. Horace\n\nwrote his autobiography, without stopping, one long suspended effort, words upon words flowing out of him, expressing his grief. But he never read what he had written, hoping rather to exorcise his confusion. So strong was his belief in words\u2014perhaps they would lead him out of this strange world in which he had suddenly found himself. In the end, after reams and reams of paper and thousands of lines of scribble, he had found no answers. In frustration he burned it. (239)\n\n_In frustration he burned it;_ the text that could facilitate an understanding of Horace's position in \"this strange world\" is apparently lost. Nothing in this magical realist novel mandates that this \"autobiography\" _remain_ destroyed, however, and near the end, readers are presented with a short section narrated in Horace's own voice. This section, \"Horace Thomas Cross: Confessions\" (245\u201351), is the only part of the novel in which Horace speaks in the first person. Although it is not \"reams and reams of paper,\" it does read like the autobiography described above. Almost every sentence of this section in which Horace meditates on his life in Tims Creek begins with the phrase \"I remember\": \"I remember the first time I saw Granddaddy kill a chicken. I remember it, dirty-white and squawking, and Granddaddy putting it down on a stump\" (245). Similar vignettes fill out this representation of life in rural North Carolina as Horace's \"confessions\" continue.\n\nHorace himself \"never read what he had written,\" but it quickly becomes clear to those who do that, despite the hardships Horace endured, this rural setting is not simply the site of \"backwardness\" or \"repression.\" Like other communities in which black gay men find themselves, Tims Creek, North Carolina, is a site of struggle and possible transformation. And like the snapping and vogueing black gay men of Becquer's analysis, Horace's own queer sense of self emerges both from within and against the community around him:\n\nI remember my Aunt Ruthester's chocolate-chip cookies and how she would make an extra batch for me. ... I remember the way it made my mouth happy, dissolving almost as soon as I ate it, buttery and hot.\n\nI remember finally touching a man, finally kissing him. I remember the surprise and shock of someone else's tongue in my mouth. I remember the taste of someone else's saliva. I remember actually feeling someone else's flesh, warm, smooth. ... I remember being happy that I was taking a chance with my immortal soul, thinking that I would somehow win in the end and live still, feeling immortal in a mortal's arms. I remember then regretting that it was such a sin. I remember the feeling I got after we climaxed, feeling hollow and undone, wishing I were some kind of animal, a wolf or a bird or a dolphin, so I would not have to worry about wanting to do it again; I remember worrying how the other person felt.\n\nI remember church and praying. I remember revival meetings and the testifying of women who began to cry before the congregation and ended their plea of hardships and sorrow and faithfulness to the Lord with the request for those who knew the word of prayer to pray much for me. I remember taking Communion and wondering how the bread was the body and the grape juice was the blood and thinking how that made us all cannibals. . . . Then I remember the day I realized that I was probably not going to go home to heaven, cause the rules were too hard for me to keep. That I was too weak.\n\nI remember me. (250\u201351)\n\nI quote at length from this passage because, as I suggested, this autobiography might provide clues as to what queer desire is doing in, and does to, this community. Clearly, Horace's desire arises in opposition to the mores of this community; as far as they are concerned, queer desire is simply \"sinful.\" Because of their stringent moral codes, Horace feels like an outcast and wants to escape by transforming himself into an animal. At the same time, however, Horace's desire emerges from _within_ this community. The very language of the fire-and-brimstone sermons he has endured (\"I remember being happy that I was taking a chance with my immortal soul\") heightens the eroticism of his encounters and helps solidify his developing queer identity. Moreover, despite the fact that Horace ultimately feels weak in the face of such a powerful religious institution, his confessions highlight his attempts to appropriate and re-signify the language of that institution (e.g., \"feeling immortal in a mortal's arms\"). In short, this may be a sin, but it is also, nonetheless, a contestation\u2014however temporary\u2014of the community's ideas about \"sin\" and \"mortality.\" As Lisa Duggan writes, the \"project of constructing identities\" is \"a historical process in which contrasting 'stories' of the self and others\u2014stories of difference\u2014are told, appropriated, and retold as stories of location in the social world of structured inequalities\" (\"Trials,\" 793). In the fictional social world that is the First Baptist Church of Tims Creek, North Carolina (a \"social world of structured inequalities,\" certainly), Horace must learn, somehow, to make sense of his identity, and he does so by appropriating the language of this institution for his own queer uses.\n\nOther elements of this autobiography underscore the extent to which Horace uses this particular location to construct and make sense of his own identity. A description of the sumptuous tastes and smells of a country home, for example (the batch of cookies \"made my mouth happy .. . buttery and hot\"), immediately precedes and influences Horace's attempt to find a language for making sense of his erotic encounters with men (\"I remember the surprise and shock of someone else's tongue in my mouth. I remember the taste of someone else's saliva\"). And if this sharp juxtaposition were not enough to demonstrate how intertwined the various elements of Horace's life are, food and flesh come together yet once more in the Communion, as Horace muses on the \"cannibalism\" of that familiar ritual.\n\nImmediately before the final sentence of these confessions, Horace asserts, \"I was too weak.\" And yet his \"I remember me,\" with its placement in a paragraph of its own at the very end of his \"Confessions,\" overrides such an assertion of weakness and implies that he will not relinquish any part of his identity. The sentence \"I remember me\" solidifies the confessions that have preceded it as indelible parts of who Horace is. Despite his difference(s), Horace's identity has been shaped not simply against but also within the community of Tims Creek, North Carolina. Duggan writes that \"identity\" can be understood \"as a narrative of a subject's location within social structure. . .. Never created out of whole cloth, never uniquely individual, each narrative is a retelling, an act of social interaction, a positioned intervention in the shared, contested narratives of a given culture\" (\"Trials,\" 793\u201394). Horace concludes that he has lost this \"contest\"; he feels he must transform himself, since he is \"too weak\" to live up to the community's expectations. Yet Horace's \"positioned intervention\" might be read differently, in a way that is less \"uniquely individual.\" Since Horace's story is an act of social interaction, we might ask ourselves what significance that story and its violent conclusion have for this particular North Carolina community. I contend that Kenan makes the \"margins\" central to his vision of a transformed community: through and against the tragic trajectory of _A Visitation of Spirits_ , a queer trickster identity materializes. Although such an identity is seemingly deferred during Horace's brief life, it nonetheless survives after Horace's death to begin the work of transforming the social structure in which his story is located.\n\nThe foundation of the social structure of Tims Creek is the church, and not surprising, the church is the institution sending Horace the loudest message that he, as an individual, is in need of transformation. I have already mentioned Horace's discomfort at the vision of the congregation of the First Baptist Church of Tims Creek. His discomfort is compounded by the sermon he hears during that vision. The Reverend Hezekiah Barden, minister of the church in the days before Horace's cousin Jimmy, informs the congregation that he is \"gone step on some toes this morning\" (77) and proceeds to read from the first chapter of Romans:\n\nThey are without excuse: Because that, when they knew God, they glorified him not as God, neither were thankful ... for even their women did change the natural use into that which is against nature:\n\nAnd likewise also the men, leaving the natural use of the woman, burned in their lust one toward another; men with men working that which is unseemly, and receiving in themselves that recompense of their error which was meet. (77)\n\nAs the sermon continues, the opposition between Horace and this religious community is spelled out in stark terms: \"Now you can say, 'Well, Brother Barden, you ain't _liberated_. You ain't up with the _times'. . . Liberated? Behind the times?_ Brothers and sisters, there is no time but now, and now I am telling you: It's unclean. You heard what Paul wrote to the Romans: Unclean\" (78\u201379). The voice of the congregation serves as a chorus behind Barden's sermon, giving its approval to everything he says: \"Unclean. . . . Go head, Reverend, and preach, now. ... Go on ahead. Tell it. . . .\" (79\u201380). The community\/chorus positions the Reverend Barden as its voice or representative, and as far as he is concerned, homosexuality and Christianity are mutually exclusive; being \"clean\" means _not_ being gay. Hence, when the queer individual confronts this religious institution, something clearly has to give.\n\nThat religion, and not Horace, should be what gives is corroborated by Kenan's signification on other texts. I have already mentioned how _A Visitation of Spirits_ reverses the individualistic equation of Dickens's _A Christmas Carol_. Beyond this, however, Kenan's novel is a complex signification on James Baldwin's 1953 _Go Tell It on the Mountain. 11_ The story of a young boy torn between sin and the church, after all, has been told before. Through _A Visitation of Spirits_ , Kenan rewrites Baldwin's story of John Grimes, bringing it\u2014if you will\u2014out of the closet. Even the epigraphs to the various sections of Kenan's novel occasionally repeat, with a difference, the epigraphs in Baldwin's; for example, both authors employ, in some way, Revelation 22:17: \"And the Spirit and the bride say, Come. And let him that heareth say, Come. And let him that is athirst come. And whosoever will, let him take the water of life freely.\" As an epigraph to the first section of his novel, Baldwin simply repeats verbatim the King James Version of this text. As an epigraph to the second section of _his_ novel, Kenan reworks and shortens the passage: \"Whosoever will, let him come . . .\" (29). This epigraph not only echoes Baldwin; it also echoes the voice Horace hears after his attempt to turn himself into a bird fails. In this context, Kenan's epigraph repeats with a difference: \"let him come\" takes on a queer new meaning. In _A Visitation of Spirits_ , which openly details one young boy's confrontation with his homosexuality and in which a deep voice repeatedly tells him to \"come,\" the homoerotic pun on the word is difficult to miss. Kenan's signification, in turn, makes it well-nigh impossible to read Baldwin's epigraph \"straight.\" Beginning with the epigraphs, then, the two texts come together, and Baldwin's \"closeted\" text is, in effect, \"outed\" by Kenan's.\n\nKenan's signification on Baldwin is much more complex than a simple reworking of epigraphs, however. _A Visitation of Spirits_ in general reads like an openly gay version of _Go Tell It on the Mountain_ , and it is Kenan's more general signification on Baldwin that suggests the church and community in _A Visitation of Spirits_ , not Horace, are in need of transformation. In _Go Tell It on the Mountain_ , John\u2014like Horace\u2014struggles against sin: \"all the pressures of church and home [were] uniting to drive him to the altar\" (13). Moreover, just as it does for Horace and the community of Tims Creek, North Carolina, \"sin\" for John and the others in _Go Tell It on the Mountain_ particularly connotes \"sins of the flesh.\" John watches the pastor's nephew Elisha and his girlfriend, Ella Mae, for instance, when they are brought before the church for a \"public warning\" designed to keep them from having sex, which would be the \"ultimate\" transgression: \"Sin was not in their minds\u2014not yet; yet sin was in the flesh; and should they continue with their walking out alone together, their secrets and laughter, and touching of hands, they would surely sin a sin beyond all forgiveness\" (17). In his own life, likewise, John's fear of sin and damnation is always connected to the \"flesh\":\n\nHe had sinned. In spite of the saints, his mother and his father, the warnings he had heard from his earliest beginnings, he had sinned with his hands a sin that was hard to forgive. In the school lavatory, alone, thinking of the boys, older, bigger, braver, who made bets with each other as to whose urine could arch higher, he had watched in himself a transformation of which he would never dare to speak. (18\u201319)\n\nAs this passage makes clear, however, even though both young men struggle against the \"flesh,\" there is a significant difference between the \"temptations\" Elisha faces and those facing John. Although it is not as explicitly gay as Horace's story in _A Visitation of Spirits_ , John's struggle against \"sin\" has an implicit homoerotic dimension. In fact, the \"un-speakability\" of John's desires here only confirms this particular dimension of his struggle. As Eve Sedgwick makes clear, there is a \"centuries-long historical chain of substantive uses of space-clearing negatives to void and at the same time to underline the possibility of male same-sex genitality\" ( _Epistemology_ , 202). According to Sedgwick, \"the speakable nonmedical terms, in Christian tradition, for the homosexual possibility for men\" were (and are), paradoxically, such terms as _unspeakable, unmentionable, things fearful to name_ , and _the love that dare not speak its name_ (202\u20133). Hence John's attempt to shield himself here is ironically his undoing; he speaks (queer) volumes in his very efforts to remain silent.\n\nIn addition to the masturbatory fantasies growing out of his observation of the genital play of the \"older, bigger, braver\" boys, John is virtually obsessed with Elisha, and that obsession is almost always detailed in _physical_ terms: \"John stared at Elisha all during the [Sunday school] lesson, admiring the timbre of Elisha's voice, much deeper and manlier than his own, admiring the leanness, and grace, and strength, and darkness of Elisha in his Sunday suit, wondering if he would ever be holy as Elisha was holy\" (13). Baldwin goes no further with John's obsession than this, but in his signification on _Go Tell It on the Mountain_ , Kenan does. Horace in _A Visitation of Spirits_ is much more specific about the ways in which such homoerotic preoccupations bring discomfort to a young boy in the church: \"He would be lying to himself if he said he had not been attracted to Gideon. . . . Why couldn't the Lord take this bit of torture, this careful trap away? Give him larceny to fight instead. Let his piety be questioned. Try to force him to lie, to worship false gods, to dishonor his mother and father, to covet his neighbor's home . . . but why Gideon?\" (146).\n\nThe differences between the two texts are not reducible to their relative openness or secrecy about homoerotic desire, however. Horace's story diverges from John's at key points, and it is ultimately in the two texts' divergence, more than in their convergence, that Kenan's sharp indictment of the church comes out. Like Horace throughout _A Visitation of Spirits_ , John goes through a harrowing, \"supernatural\" experience in the last section of _Go Tell It on the Mountain_. John falls to the floor of the church in some sort of ecstatic, spiritual trance, and like Horace, begins to hear voices calling to him. Eventually, one voice in particular pulls John out of this trance: \"'Rise up, Johnny,' said Elisha. .. . 'Are you saved, boy?' 'Yes,' said John, 'oh, yes!'\" (206). Like Horace, John is \"transformed\" by his supernatural experience; unlike Horace, John is \"saved.\" Of course, there is a fair amount of irony in Baldwin's novel; Baldwin is by no means uncritical of the church, and consequently, exactly how or whether John's life will be different remains unclear in the last line of the novel: \"'I'm ready,' John said, 'I'm coming. I'm on my way'\" (221). Still, John himself claims to be positively transformed by his experience. He ultimately submits to the church's authority, even though that submission is channeled through his homoerotic obsession for Elisha: \"Elisha ... no matter what happens to me, where I go, what folks say about me, no matter what _anybody_ says, you remember\u2014please remember\u2014I was saved. I was _there\"_ (220).\n\nIn Kenan's re-vision of the confrontation between a young boy and the church, however, accommodation to this religious institution, as it is, is no longer possible. Through Horace's suicide, _A Visitation of Spirits_ suggests, in contrast to _Go Tell It on the Mountain_ , that the gay individual cannot and does not survive such attempts at accommodation with the church. In the end, the religious institution, and not John\/Horace, needs to be \"saved.\" Salvation, certainly, is a seemingly difficult concept to queer, but _A Visitation of Spirits_ nonetheless sets just such a queering into motion. If the straight definition of the term fixes the meaning of actions (as, for instance, \"sin\") or individuals (as \"redeemed\" or \"damned\"), then the queer concept opens up a play of meaning that disrupts these fixations and establishes another sort of church altogether.\n\nReligion is just one facet\u2014the most prominent and oppressive facet\u2014of community life in Tims Creek. The church is not the only place, however, where Horace feels he does not belong, and the church is hence not the only part of the community in need of transformation. Indeed, Horace confronts the \"regime of sameness\" almost everywhere he goes in Tims Creek. None of the subcommunities of which Horace is a part is comfortable with \"difference\" within its ranks; thus, in none of the locations in which Horace finds himself is he able to be comfortable with \"sameness.\" The compulsion to be \"the same,\" even as it is reproduced within the cultural category \"black gay,\" invalidates any of Horace's attempts to come to terms with his own identity. \"You black, ain't you?\" Horace's aunt asks him (186). One of Horace's lovers from the community theater where he works taunts him, \"Faggot.... What's the matter? Don't like to be called what you are?\" (225). Even Gideon, another black gay character who is Horace's first affair, says to Horace in the heat of an argument, \"But remember, black boy, you heard it here first: You're a faggot, Horace. . . . At least I know what I am\" (164).\n\nIn each confrontation, it is not that the labels are wholly inappropriate for Horace; it is just that every question of identity in _A Visitation of Spirits_ needs to be followed by a \"yes, but. . .\" Judith Butler's comments are particularly relevant to Horace's situation here: \"The prospect of _being_ anything . . . seems to be more than a simple injunction to become who or what I already am\" (\"Imitation,\" 13). Throughout _A Visitation of Spirits_ , the discordant and demanding chorus of voices surrounding Horace immobilizes him, apparently preventing any strategic resistance of the compulsion to be the same.\n\nAs a black gay teenager in Tims Creek, Horace always finds himself embodying what are\u2014at least, to the other members of the groups of which he is a part\u2014contradictory identities: in his own family, as in the church, he is \"black,\" but not \"gay\"; at the community theater where he works, he (along with many of the other actors) is openly \"gay,\" but his \"blackness\" is rendered invisible (particularly by the production itself, which is about the history of the Cross family\u2014the _white_ Cross family\u2014in North Carolina); with his \"alternative\" and white high school friends, he is \"smart and black\" (237), but he is not \"gay,\" and he feels his \"blackness\" is tokenized. Only with Gideon does Horace find a \"niche,\" where he should \"fit\" exactly. But, although their relationship is consummated, Horace and Gideon do not embrace a \"black gay identity\" together, an identity with which they subvert and expose the contradictions of the various communities of which they are constituents. Instead, the pressure from each of these communities precludes the possibility of Gideon and Horace coming together. Horace, caught up in the belief that he is in need of transformation, is ultimately unable to imagine that he and Gideon might work together to transform others. Gideon comes to signify, for Horace, everything about himself that must be obliterated. In his final scene with Gideon, Horace literalizes this belief by attacking Gideon: \"Horace hit Gideon. Full square in the mouth, so quickly he himself did not realize what he had done, so hard he could not doubt he meant to do it. But had he wanted to hit Gideon, or himself for not wanting to hit him?\" (163).\n\nThe reflection of himself that Horace sees in Gideon is, in an earlier scene, even more explicit: \"Gideon turned around and looked at Horace. He paid no attention to the snickering boys who stood about Horace; he just fixed Horace with a gaze whose intensity frightened him. Now, looking at the phantoms, Horace realized how it had seemed to be more than an angry glare. It was more of a curse. A prophecy\" (100). This defiant gaze links Gideon, and the potential for a black gay identity that Gideon embodies, to Horace. Horace (mis)recognizes himself in Gideon, and this event\u2014in a world where every community appears to legislate against the very possibility of such an identity\u2014propels Horace forward onto a path fraught with anxiety. Throughout _A Visitation of Spirits_ , Horace is caught in a cycle of recognition and disavowal, which his confrontation with Gideon instigates. Horace does not see his identification with Gideon as a locus of possibility; he cannot comprehend \"black gay identity\" as a multivalent sign that has the potential to open up or disrupt the meanings of \"community\" that debilitate him. Instead, believing he has seen the final \"truth\" about himself in Gideon, Horace feels compelled to repudiate such a truth, due to the prohibitions he feels from various communities against embracing such an identity.\n\nIn a pivotal scene near the end of the novel (in a section where, a la Dickens, the voice of the ghost has disappeared, even though a presence remains behind to show Horace visions of things yet to come), although Gideon is no longer present, there is a final repetition of what has become for Horace the terrifying cycle of recognition and disavowal. On entering a room at the theater, Horace sees a black man, dressed as a clown, applying white makeup. As Horace comes closer to the figure, he realizes that it is himself: \"The double stood up. He was exactly the same height as Horace, the same build. In his sparkling color, he turned to look at both their reflections in the mirror: Horace in his brown nakedness, covered with dirt and ash and grass in his hair, a gun in his hand, and the other Horace, white-faced, dressed as a clown\" (220). Horace (mis)recognizes himself and himself and himself, but like his encounters with Gideon, this recognition is not ultimately reassuring but rather the sign of some terrible inadequacy: \"Of all the things he had seen this night, all the memories he had confronted, all the ghouls and ghosts and specters, this shook him the most. Stunned, confused, bewildered, he could only stare at his reflection, seeing him and him and him\" (219).\n\nHorace's reaction here is, not surprisingly, a far cry from the strategic, subversive use to which some (urban) black gay men put mirrors. Becquer writes that\n\nthe voguer creates a scenario in which s\/he first carefully arranges him\/herself before a mirror. . . . Once satisfied with his\/her own constructed image, the voguer then turns the mirror onto his\/her opponent. This is done in the hopes of making evident how badly the opponent is in need of a dramatic make-up job, that is, of a drastic reconstruction and reconceptual-ization of his\/her own identity. . . . While the Lacanian subject razsrecognizes him\/herself as a whole before the mirror, the voguer recognizes him\/herself as a construction. In this sense the voguer can be seen to make use of the very simulacrum by which the Lacanian subject is duped. (13)\n\nIn _A Visitation of Spirits_ , exactly the opposite happens to Horace. Turning to the mirror, he can see only how badly he himself is in need of reconstruction\/reconceptualization. The terror for Horace in all of this doubling and redoubling lies in the recognition of what he believes is the \"truth\" about himself, which he can no longer escape: he is a clown, he wears a mask, he only plays a role, he betrays the communities of which he is a part.\n\nIn _The Signifying Monkey_ , Gates suggests, \"Thinking about the black concept of Signifyin(g) is a bit like stumbling unaware into a hall of mirrors: the sign itself appears to be doubled, at the very least, and (re)doubled upon ever closer examination\" (44). But although Becquer's voguers\u2014like, indeed, the Signifying Monkey himself, who might be understood as their analogue\u2014would be able to \"work\" the indeterminacy of the signifier that such a stumbling into a hall of mirrors would entail (Becquer writes, \"The voguer 'reads' the simulacrum\" [13]), Horace, even as he literally stumbles into just such a hall, is not. Despite the literal proliferation of meanings and identities in this scene, Horace believes he is stuck with one wholly inadequate identity, which is always and only \"disgusting,\" \"unclean,\" an \"abomination\" (100). Equating the figurative with the literal (or the reflection\/representation with the essence) and reading the literal as inescapable, Horace is unable to recognize and deploy the constructedness of identity against the multiple communities that surround him.\n\nThe Signifying Monkey of Gates's analysis succeeds precisely because of his ability to luxuriate in the indeterminacy of meaning. Horace, however, explicitly rejects such a potentially liberating strategy. In the end, like the lion who is always signified on in the Signifying Monkey tales, Horace is undone by the indeterminacy of meaning:\n\nThen ... he saw what he had led himself to see, the reason, the logic, the point. It was round and square. It was hard and soft, black and white, cold and hot, smooth and rough, young and old. It had depth and was shallow, was bright and dull, took light and gave light, was generous and greedy. Holy and profane. Ignorant and wise. Horace saw it and it saw Horace, like the moon, like the sea, like the mountain\u2014so large he could not miss it, so small he could barely see it. The most simple, the most complex, the most wrong, the most right. . . . Horace saw clearly through a glass darkly and understood where he fit. Understood what was asked of him.\n\nHorace shook his head. No. He turned away. No. He turned his heart away. No.\n\nThis had been Horace's redemption, and Horace said no. (232, 234)\n\nExactly what \"it\" is that might have been Horace's redemption is unclear in this list of oxymorons, although Horace's rejection of this redemption is clear. The \"meaning,\" or signified, of this vision, however, may be beside the point. What Horace refuses by turning away from this list of oxymorons is not the signified but\u2014as Gates might put it\u2014the \"sheer materiality, and the willful play, of the signifier itself\" ( _Signifying Monkey_ , 59). Horace is apparently unable to deploy the instability of the signifier to his advantage, in contrast to secular and urban black gays who \"read\" the inadequacies of the communities in which they are positioned by snapping and vogueing, and in contrast to that exemplary trickster figure, the Signifying Monkey. Instead, Horace turns away from such strategies and maintains until the end that he himself, and not his location, needs transformation.\n\n_A Visitation of Spirits_ is, after all, a tragedy, and from this point in the text Horace's story moves with apocalyptic rapidity toward its suicidal conclusion. In my placement of voguers and the Signifying Monkey alongside Horace's final vision, however, I mean to suggest that the text points to another possibility: a queer trickster identity might make evident to the communities of Tims Creek how badly they are in need of a \"dramatic make-up job.\" In the Signifying Monkey tales, the Monkey repeatedly trounces his opponents by luxuriating in the indeterminacy of meaning. Gates writes, \"Motivated Signifyin(g) is the sort in which the Monkey delights; it functions to redress an imbalance of power, to clear a space, rhetorically. To achieve occupancy in this desired space, the Monkey rewrites the received order by exploiting the Lion's hubris and his inability to read the figurative other than as the literal\" ( _Signifying Monkey_ , 124). Horace's vision likewise affords him the opportunity to rewrite the received order; the string of oxymorons directs attention away from the signified and toward the sheer materiality of the signifier, providing Horace with a glimpse of a world where what is profane is holy, and what is most wrong is most right. Gates explains, \"The Signifying Monkey tales .. . can be thought of as versions of daydreams, the Daydream of the Black Other, chiastic fantasies of reversal of power relationships\" ( _Signifying Monkey_ , 85). Horace's final vision is a textbook example of such a daydream in which power relationships are reversed, and although he refuses to embrace and deploy what had been his \"redemption,\" his vision at the very least suggests that these fantasies are always already in circulation, even within a community as apparently \"marginal\" and \"stable\" as Tims Creek, North Carolina.\n\nThe Signifying Monkey's strategy depends on repetition, with a difference, of the signifier; in other words, the Monkey understands and exploits the impossibility of attaching the signifier to one fixed signified. This \"repetition with a difference\" might be productively linked to Judith Butler's idea of \"subversive repetition\" (in fact, Butler's subversive repetition can be said to depend on repetition with a difference). Particularly in light of Horace's \"inevitable\" tragedy, Butler's comments seem appropriate:\n\nHeterosexuality offers normative sexual positions that are intrinsically impossible to embody, and the persistent failure to identify fully and without incoherence with these positions reveals heterosexuality itself not only as a compulsory law, but as an inevitable comedy. Indeed, I would offer this insight into heterosexuality as both a compulsory system and an intrinsic comedy, a constant parody of itself, as an alternative gay\/lesbian perspective. ( _Gender Trouble_ , 122)\n\nTo Butler, identity is constituted through a series of attempts to embody normative gender and sexual identities. She argues that the compulsive repetition these normative positions engender exposes the \"truth\" of identity as a fiction: \"The injunction _to be_ a. given gender produces necessary failures, a variety of incoherent configurations that in their multiplicity exceed and defy the injunction by which they are generated\" ( _Gender Trouble_ , 145). Butch\/femme and drag performances are among the \"subversive bodily acts\" that particularly foreground this defiance. Such practices parody the idea of an original gender identity and in the process _\"reveal . . . the imitative structure of gender itself\u2014as well as its contingency\" (Gender Trouble_ , 137). The attention to style over substance suggests an affinity between the cultural practice of drag or butch\/femme and the practice of Signifyin(g), which similarly turns on an awareness of the distinction between manner and matter (Gates, _Signifying Monkey_ , 70). Horace refuses to admit such a distinction and is thus unable to claim an \"alternative gay\/lesbian perspective,\" such as the perspective Butler envisions. Instead, the compulsion to be the same has engendered in Horace the belief that his identity must be fixed, once and for all. Neither a single individual nor the regime of sameness with which he contends, however, can entirely contain the gender trouble Horace's story spotlights. As the next section argues, gender trouble keeps in play not only the meaning of Horace's identity but even the meaning, or the supposed certainty, of his inevitable \"tragedy\" and \"death.\"\n\nAt this point in the novel, however, it would _seem_ that Horace's journey\u2014in contrast to Ebenezer Scrooge's\u2014does indeed end in death. The neo-Dickensian specter finally leads Horace to a cemetery: \"Convincing himself he knew the outcome of this story, he fully expected to see his own grave. Though he did not understand the point of this transparent charade, he was convinced that would be the best end\" (231). Prior to his suicide, and standing before the grave, Horace confronts his double one last time. Following Gates and Butler, I propose that this double, because of the \"redemptive\" vision he provokes, embodies the \"sheer materiality\" and \"willful play\" of the signifier. Face-to-face with the slipperiness of signification, however, Horace determines to end it once and for all:\n\nHis reflection stood there, his hand extended. I'm your way, he said. . . . You can follow the demon if you want. It's your choice.\n\nHorace looked at his hand. His hand. Never had he felt such self-loathing, and by and by, his depression became anger as he glared at the spirit. . . .\n\nIn such a rage he could barely see, Horace raised his gun and fired. The report was not as loud as he had expected. But there on the ground he lay, himself, a gory red gash through his chest. His face caught in a grimace, moaning and speaking incoherently. Please. No. No. He looked at his hand, covered in blood, and Horace looked up at Horace, his eyes full of horror, but in recognition too, as if to say: You meant it, didn't you? You actually hate me? (234\u201335)\n\nGates writes that \"the Signifying Monkey is often called the Signifier, he who wreaks havoc upon the Signified\" ( _Signifying Monkey_ , 52). In stark contrast, Horace in the end wreaks havoc on the Signifier, his \"redemption,\" and consequently, it is he himself who is undone.\n\nEven as it questions it (\"Why? Why. You didn't have to\"), the text itself thus graphically stages Horace's rejection of the Signifyin(g) alternative. In some ways, then, neither Gates's nor my vision of and for Horace is realized: Horace is not transported \"safely\" off to the big city, and neither is he able to begin a much-needed transformation of the place where he is already located. Indeed, as early as the scene in which he hits Gideon, Horace\u2014confirmed in the belief that he, and not his surroundings, must change\u2014dismisses the idea of \"another world, another place\": \"He imagined another world, another place, in which he could gladly have complied with Gideon's wish and fallen into lusty, steamy, lascivious abandon\u2014but no\" (164).\n\nHis inability to imagine such a place, along with his subsequent suicide, demonstrates what the compulsion to be the same has done to Horace. In the context of all the demands put on him by various communities, Horace's suicide can be seen as an apt re-presentation of the violence involved in the attempt to \"alienate conclusively, _definitionally_ , from anyone on any theoretical ground the authority to describe and name their own sexual desire\" (Sedgwick, _Epistemology_ , 26), or indeed, any component of their own identity. That _A Visitation of Spirits_ appeared in the same year (1989) that right-wing religious and political leaders attempted to suppress the findings of the Department of Health and Human Services' Report of the Secretary's Task Force on Youth Suicide, which claimed that gay teenagers account for 30 percent of all teenage suicides (Ruta, 12), only underscores the validity of representing the compulsion toward sameness as violence. Horace's suicide is detailed in stark, scientific (and hence \"real\") prose and is juxtaposed to the \"fantastic\" events in this postmodern, magical realist text. The realness of the suicide moves the theoretical opposition \"Is It Real?\"\/\"Is It Fantasy?\" to a more urgent level and, in the process, starkly confronts and condemns the community where Horace is located: What if it is real? How is the community implicated in such violence?\n\nIn spite of Horace's violent rejection of the Signifyin(g) possibilities embodied in his double, then, the text itself provokes a critique of the community. Immediately after the suicide, the question of what queer desire (and its violent extermination) is doing in and to rural North Carolina is foregrounded. As the novel concludes, the narration shifts to the second person in a nostalgic section called \"Requiem for Tobacco\":\n\nYou remember, though perhaps you don't, that once upon a time men harvested tobacco by hand. There was a time when folk were bound together in a community, as one, and helped one person this day and that day another, and another the next, to see that everyone got his tobacco crop in the barn each week, and that it was fired and cured and taken to a packhouse to be graded and eventually sent to market. But this was once upon a time. (254)\n\nThe section continues, lamenting the tragic loss of this idyllic way of life. But this section is already and inescapably in dialogue with the suicide that immediately precedes it. Because of this dialogue, the \"time when folk were bound together in a community, as one,\" is exposed even as it is being constructed. The mythical, pastoral wholeness of this \"community\" is ripped apart as surely as \"the bullet did break the skin of his forehead, pierce the cranium, slice through the cortex and cerebellum, irreparably bruising the cerebrum and medulla oblongata, and emerge from the back of the skull, all with a wet and lightning crack. This did happen\" (253). Mikhail Bakhtin argues that \"sexuality is almost always incorporated into the idyll only in sublimated form,\" since the idyllic form in literature demands a unity of time and place that smooths over differences and avoids the \"naked realistic aspect\" of life (226). Kenan's juxtaposition here of idyll and suicide foregrounds the murderous consequences of such a sublimation. Kenan himself insists, in a recent interview, \"It seemed, and it seems . .. that for that community to change they have to understand the devastation that they're wreaking on certain people\" (in Hunt, 416). In the end, Horace's story highlights the need for transformation of this community on the margins of the queer world. Kenan's shift to second-person narration in the conclusion further emphasizes this need: although \"you remember\" what this community was like, you should not, after _A Visitation of Spirits_ , be able to consider this or any community without a queer sense that something is amiss.\n\nThe \"Requiem for Tobacco\" section that concludes _A Visitation of Spirits_ is not the first time in the text that the narrator addresses the reader directly. The very first section of the novel\u2014\"December 8, 1985 \/ 8:45 A.M. (3)\u2014concludes with a similar account of a \"time when folk were bound together in a community as one.\" This section, \"ADVENT (or The Beginning of the End)\" (6), describes a community coming together for a hog killing and, like \"Requiem for Tobacco,\" includes direct appeals to the reader:\n\nYou'Ve been to a hog killing before, haven't you? They don't happen as often as they once did. People simply don't raise hogs like they used to.\n\nOnce, in this very North Carolina town, practically everyone with a piece of land kept a hog or two, at least. And come the cold months of December and January folk would begin to butcher and salt and smoke and pickle. In those days a hog was a mighty good thing to have, to see you through the winter. But you know all this, don't you? (6)\n\nNostalgia for \"those days\" permeates the section, yet the jeering tone of the second-person appeals undercuts any sense of security that the nostalgia might have brought. As in \"Requiem for Tobacco,\" the second-person narration taunts \"you,\" instilling a queer sense of uneasiness, suggesting that something is wrong with this idyllic communal picture and that \"you\" are somehow complicit in the problem.\n\nEveryone in the community has an expected role to play in the hog killing, as they do at the end of the text, in \"Requiem for Tobacco.\" Women and men alike know their place, and each group entrusts to the other the socialization of girls and boys, respectively. \"Advent\" is associated with the beginning of a new cycle, and this Advent is no exception. Through shared and recurrent rituals such as this hog killing, the socialization of the community's youth into fixed gender roles commences:\n\nBeneath the shed, the women would be busy, with knives, with grinders, with spoons and forks; the greasy tables littered with salts and peppers and spices, hunks of meat, bloody and in pans to be made into sausages, pans of cooked liver to be made into liver pudding. Remember the odor of cooking meats and spices, so thick, so heady?. . . Some older man will give a young boy a gun, perhaps, and instruct him not to be afraid, to take his time, to aim straight. The men will all look at one another and the boy with a sense of mutual pride, as the man goes over to the gate and with some effort moves the three slats that close off the hogpen. (8)\n\nThe boy _cum_ man then takes his gun and slaughters the hog, whose sacrifice will guarantee the continued vitality of this community and its rituals.\n\n\"But,\" as the narrator says, \"I'm sure youVe witnessed all this, of course\" (8). And in a sense, this community has indeed witnessed such a \"sacrifice\" before: even though Horace has not yet been introduced into the text, chronologically this hog killing (December 8, 1985) _follows_ Horace's suicide (April 30, 1984). Kenan's manipulation of time makes each of these scenes echo the other; \"you\" should hear echoes of the hog killing in the suicide\/idyll at the end of the novel, but since the suicide actually predates the hog killing, the latter already contains echoes of the former. The hog killing is, after all, not the first time this community has \"instructed\" a young boy to \"aim straight\" and has given him a gun.\n\nOnce the suicide\/idyll is put into conversation with the hog killing, it is impossible not to suspect that Horace's death, like the hog's, is a sacrifice carried out for the \"good of the community.\" That is, this community can exist only in the sort of (heterosexual) pastoral wholeness represented in \"ADVENT\" and \"Requiem for Tobacco,\" through disavowal or outright elimination of some of its members. The hog and Horace both play a sacrificial role for this community, and the passages detailing their individual moments of death underscore this similarity of purpose:\n\nThe hog rears up on its hind legs like a horse, bucking, tossing its head, but only once, twice. It seems to land miraculously on its front legs, but only for a split second. It topples, hitting the ground with a thud, and lets out a sound that you might call a death rattle\u2014all in a matter of seconds. Its eyes fix intently on nothing. (9)\n\nHis entire body convulsed several times; it excreted urine. Defecated. The tongue hung out of the mouth and during the convulsions was clamped down upon, releasing blood to be mixed with the ropes of saliva stringing down. His heartbeat slowly decreased in pressure and intensity, soon coming to a halt; the arteries, veins, and capillaries slowly collapsed. The pupils of his eyes, now tainted in a film of pink, stopped dilating, resting like huge drops of ink surrounded by brown liquid in a pool of milk. Finally the eyes themselves rolled back, staring up, as though examining the sun through the canopy of tree limbs. (253\u201354)\n\nAfter both deaths, the community is once again free to celebrate a mythical wholeness.\n\nKenan's placement of the events on December 8 and on April 29\u201330, respectively, corroborates this reading of each as a sacrifice intended for the good of the community In Christian mythology, Advent is the beginning, in early December, of the church year. Like the hog killing, it is supposedly a happy time, when Christians celebrate the arrival of the one who is born to die so that the community can live. The \"salvation\" of the Christian community is contingent on the death of Christ, and in _A Visitation of Spirits_ , Kenan uses this sacrificial idea to sharply indict the Christian community of Tims Creek, North Carolina. Nonetheless, as with Kenan's use of _Go Tell It on the Mountain_ and _A Christmas Carol_ , this is no simple appropriation of someone else's story. Horace's very name\u2014Horace \"Cross\"\u2014links him to the \"sacrificial lamb\" of Christianity, but Kenan repeats even the Christian story with a difference: although April 30, 1984, did indeed fall during the Easter season (\"This,\" as the narrator of _A Visitation of Spirits_ might say, \"is a fact\"), this Easter is characterized not by a resurrection but rather by a graphic and apparently irreversible suicide. Horace Cross might die for this community's sins, but\u2014it would seem\u2014he does not rise again.\n\n### **Transformations (II): James Malachai Greene**\n\nThrough both the hog killing and the tobacco harvest, it is presumably the reader who recognizes the disavowal of difference on which this community is founded; the community itself simply proceeds with its celebration of mythical wholeness, oblivious to the violence its rituals demand and effect. In fact, the reader's transformation is virtually contingent on the community's nontransformation: because the community is unable to adapt and is consequently responsible for Horace's death, readers learn to view such communities askance, to perceive something queer in their very disavowal of queerness. Yet analyzing Horace as simply dead fails to account for death's ironic lack of finality in the Kenan corpus, and understanding the community as simply untransformed disregards how Horace, even in death, ultimately does become the sort of trickster figure Gates finds in the Signifying Monkey tales in particular and in the African American literary tradition in general. At the end of the novel, Horace\u2014apparently \"possessed\" by a visitation of spirits\u2014emerges from the woods and shoots himself in front of his cousin Jimmy Greene. Jimmy, who at the time of Horace's suicide is both the minister at the First Baptist Church of Tims Creek and the first black principal of Tims Creek Elementary\n\nSchool, had earlier attempted to transform Horace, explaining to him that being gay is \"wrong\" (113). In the end, however, it is Jimmy who is transformed by Horace, and since he is the head of two community institutions, the ramifications of this transformation are far-reaching. The transformations Horace initiates are always incomplete (or, in the language of the previous chapter, \"permanently partial\"), but they are simultaneously inevitable and ongoing: the closet that the regime of sameness requires and constructs can never fully contain the \"visitation of (queer) spirits\" its exclusionary logic unleashes.\n\nThe very title of Kenan's second work suggests that we look carefully at \"death\" in his works. _Let the Dead Bury Their Dead_ is a collection of twelve stories, most of them once again set in Tims Creek, North Carolina. The collection takes its title from Christ's injunction, in Matthew 8:22, to a would-be disciple: \"But Jesus said unto him, Follow me; and let the dead bury their dead\" (KJV). Christ's words in this passage are simply a poetic way of saying \"drop everything and look to the future with me\" or \"put what's dead and gone behind you.\" In Kenan's rewriting of the Christian injunction, however, such an interpretation is no longer possible; the \"dead\" in Kenan's work tend not to _stay_ dead. In both _A Visitation of Spirits_ and _Let the Dead Bury Their Dead_ , the dead are constantly coming back, in various and perturbing ways, to haunt and disrupt the present. In fact, the very first story of Kenan's collection underscores the futility of attempts to \"let the dead bury their dead.\"\n\n\"Clarence _and_ the Dead ( _And_ What Do They Tell You, Clarence? _And_ the Dead Speak to Clarence),\" the first selection in _Let the Dead Bury Their Dead_ , is the story of a young boy who conveys cryptic, unsettling messages to the friends and neighbors who come to visit his grandparents' home. What is especially peculiar about these particular messages is that they come from people in the community who have long since died. For example, Clarence tells Emma Chaney, whose mother died long before Clarence was born, \"Your mama says Joe Hattan is stepping out on you with that strumpet Viola Stokes\" (5). Clarence and his \"queer knowledge\" (12) wreak havoc on the comfortable security of the present-day Tims Creek community, reversing people's assumptions about their pasts, their families, their marriages. Anyone who wanted to abide by the biblical dictum is effectively precluded from doing so: \"Clarence told people things a four-year-old boy ain't had no business knowing the language for, let alone the circumstances around them. All from people dead five, six, ten, twenty, and more years\" (8).\n\nHence, even though I have just argued that there is no resurrection in _A Visitation of Spirits_ , we would do well to view the apparent \"irreversibility\" of Horace's suicide with suspicion. Granted, Horace doesn't return to Tims Creek in exactly the same way as the dead in Clarence's story, but I contend that he nonetheless wreaks havoc on the (heterosexual) stability of the community. Ultimately, Horace does become the sort of trickster figure Gates locates throughout the African American literary tradition, \"surfacing when we least expect him, at a crossroads of destiny\" ( _Signifying Monkey_ , 64). Like the dead in Clarence's story, Horace (re)surfaces in order to disrupt the complacent security of Tims Creek, to render unstable the binary oppositions (center\/margin, clean\/unclean, etc.) that ground the life of this community. And because Kenan relates _A Visitation of Spirits_ in a nonlinear fashion, with the events of April 29\u201330, 1984 (the day of Horace's suicide), interwoven with the events of December 8, 1985, Horace's (comedic) postsuicide trickster role wreaks havoc even as the (tragic) events leading to his death are teleologically represented.\n\nThe first and apparently unlikely power relation that Horace reverses is the \"opposition\" between the \"ghosts\" and \"Ebenezer Scrooge.\" As I have suggested throughout this chapter, Kenan's novel is a complex signification on Dickens's famous story, with Horace cast in the role of Scrooge. As early as the epigraph from _A Christmas Carol_ , however, there is evidence that Horace has been miscast:\n\n\"Are spirits' lives so short?\" asked Scrooge.\n\n\"My life upon this globe is very brief,\" replied the Ghost. \"It ends tonight.\"\n\n\"To-night!\" cried Scrooge.\n\n\"To-night at midnight. Hark! The time is drawing near.\" ( _A Visitation of Spirits_ , epigraph page)\n\nThe epigraph introduces Kenan's text and thereby suggests that _A Visitation of Spirits_ will be some sort of retelling of _A Christmas Carol_ At the same time, this epigraph complicates any easy equivalency between Dickens's characters and Kenan's. Although Horace's whirlwind journey through past, present, and future apparently links him to Scrooge, the epigraph belies such a reading, linking him instead to the spirit. After all, it is Horace, in _A Visitation of Spirits_ , whose \"life upon this globe is very brief,\" and at the stroke of midnight on April 30, 1984, it is Horace's life that rapidly heads toward its conclusion.\n\nAs the text continues, Horace's metaphorical link to Scrooge is obvious; nonetheless, his metonymic link to the ghosts (which, since it is introduced in the epigraph, actually precedes the metaphorical link to Scrooge) continues to surface when we least expect it. For instance, the hog killing of the first section ends with the assertion \"But the ghosts of those times are stubborn; and though the hog stalls are empty, a herd can be heard, trampling the grasses and flowers and fancy bushes, trampling the foreign trees of the new families, living in their new homes. A ghostly herd waiting to be butchered\" (10). As I have said, Horace is linked to the hog in that both serve as sacrifices for the good of the community. This passage replays that connection, underscoring as well how both disrupt the idyllic stability of the community, \"trampling the grasses and flowers and fancy bushes . . . living in their new homes.\" Beyond that, however, the linkage here of hogs and ghosts extends the metonymic chain: the ghosts call to mind the hogs who call to mind Horace who calls to mind the ghosts. And if this were not enough, the next section graphically links both the hogs and the ghosts to Horace. This section, the first in which Horace appears, opens with an ellipsis: \". . . What to become?\" (11). The ellipsis serves a dual function here. On the surface, it stands in for the process of thought Horace is going through as he decides to turn himself into a bird. On another level, however, the ellipsis serves as a visual link to what has immediately preceded, and Horace is once again connected to both the hogs and the ghosts.\n\nThe identification of Horace with the ghosts, as opposed to identification with Scrooge, is crucial for understanding Horace's reemergence as a trickster in _A Visitation of Spirits_ , since this identification means that, in the end, he may not be so much the _object_ as the _agent_ of transformation. (Scrooge's ghosts, after all, are responsible for his own radical transformation.) Perhaps, then, we are wrong to read Horace as simply \"dead\" after his suicide, despite the fact that his death is detailed with such graphic and scientific exactness. Perhaps\u2014to repeat (with a difference) the Gates quotation with which I began this chapter\u2014 _A Visitation of Spirits_ is _not_ so much about \"a brilliant tormented homosexual, Horace, who commits suicide\" (in Rowell, 454). Perhaps such a description particularly overlooks the cryptic pronouncement by the narrator that concludes the suicide scene: \"Ifs and maybes and weres and perhapses are of no use in this case. The facts are enough, unless they too are subject to doubt\" (^4 _Visitation of Spirits_ , 254). Indeed, the \"fact\" of Horace's complete annihilation should very well be subject to doubt, given how Horace\u2014even after his suicide\u2014continues to \"live\" as the Signifying Monkey _cum_ queer agent of transformation, \"haunting\" the community in general and Jimmy in particular.\n\nJimmy's story unfolds in two ways over the course of _A Visitation of Spirits_. First, he is one of the main characters in the December 8, 1985, sections of the novel. These sections ostensibly focus on Jimmy's journey, with his Aunt Ruth and Uncle Zeke (two elderly members of the Cross clan), to visit a cousin who is in the hospital in Fayetteville, North Carolina. More precisely, however, these sections focus on the internal monologues of various characters as they sort out their conflicts with and connections to other members of the family (this is Kenan's answer to _Go Tell It on the Mountain's_ \"Prayers of the Saints\" section).\n\nSecond, and more important, Jimmy's character comes out in all his complexity in three personal \"Confessions\" sections. These confessions are not dated, but since Horace's suicide haunts each of them, they must be considered as written after Horace's death and are probably roughly contemporaneous with the events of late 1985. Overall, the novel is divided into five sections, each of which is then divided into even smaller sections (\"December 8, 1985\"; \"April 29, 1984\"; etc.). Three of the five main sections open with these \"Confessions\" from Jimmy; the final section opens with Horace's own \"Confessions\" (which I analyzed in the previous section). As is often the case with events in _A Visitation of Spirits_ , Horace's confessions actually _predate_ Jimmy's, despite the fact that they are not included until the novel's final section. Of course, there is no way Jimmy could have seen Horace's \"Confessions\"; as we know, Horace burned his autobiography in frustration. Nonetheless, Jimmy's own writings betray an ironic anxiety about Horace's influence. The very heading of Jimmy's text is, stylistically, a direct echo of Horace's own: \"Horace Thomas Cross: Confessions\" (245); \"James Malachai Greene: Confessions\" (31, 107, 171). Hence, even though Jimmy could not possibly have read Horace's text, Horace's story appears to have profoundly influenced the way Jimmy documents his own private thoughts.\n\nThere is at least one major difference between Horace's confessions and Jimmy's. Although Horace's \"Confessions\" section is an extended prose meditation on his own life, Jimmy's three \"Confessions\" sections are not\u2014so to speak\u2014straight prose. Each of Jimmy's \"Confessions\" sections is interrupted by a short script, complete with stage directions. Significantly, each of these dramatic episodes is a re-presentation of a critical moment in _Horace's_ life: the first sketches out the strange conversation between Jimmy and Horace a few minutes before the suicide; the second relates an earlier scene, in June 1983, when Horace first \"comes out\" to Jimmy, looking to his cousin for pastoral guidance; the third, later in 1983, replays a Thanksgiving dinner where Horace offends the entire family by showing up with a pierced ear. Although \"confessions\" tend to signify uncensored, \"unmediated\" prose meditations for some reason, Jimmy cannot face his most intense, gay-identified memories of Horace so directly. These memories demand another form; Jimmy desires or needs to distance himself from them. Perhaps Jimmy indulges a fantasy that these dramatic passages somehow position him on the outside, passively observing events onstage from a comfortable seat in the audience. Nonetheless, these passages are contained within his \"autobiographical\" sections and are thus framed by highly personal revelations of Jimmy's weaknesses and fears. No matter how much Jimmy would like to distance himself from these memories, they are implicated in his most private and personal attempts to define himself.\n\nThese dramatic interruptions are early indications of how Jimmy has been and continues to be transformed by Horace, who, in the role of trickster, delights in reversing the received order of things. In particular, these dramatic passages, and Jimmy's \"Confessions\" in general, illustrate how decidedly unstable the heterosexual\/homosexual binary has become for Jimmy. Kenan's revision of Baldwin helps illustrate this point: if Horace can be compared to John Grimes, then Jimmy can be compared to Elisha, the young religious figure whom John admires and counts on for spiritual guidance. Yet to the degree that identity in _A Visitation of Spirits_ , as opposed to _Go Tell It on the Mountain_ , is more explicitly homosexual for Horace, so too is identity less convincingly heterosexual for Jimmy. Of course, Elisha's performance of heterosexuality is a tough act to follow: Elisha is, after all, a spectacular heterosexual, in the sense that the church makes a spectacle of him and Ella Mae, so convinced are they of the couple's (hetero)sexual designs. But whereas in Go _Tell It on the Mountain_ the community is convinced in advance of Elisha's heterosexual prowess (Elisha is, in a way, _too_ heterosexual), in _A Visitation of Spirits_ Jimmy has trouble convincing anyone, including himself, that he is unequivocally heterosexual; instead, in each of Jimmy's \"Confessions,\" Horace's haunting of Jimmy foregrounds the irresolvable instability of the heterosexual\/homosexual binary which Jimmy thought he was on top of. Sedgwick's analysis of the ironies of _Billy Budd_ is apropos here: \"The death of the text's homosexual marks . . . not a terminus but an initiation . . . into the narrative circulation of male desire\" ( _Epistemology_ , 99\u2013100). For Jimmy, this \"initiation\" transforms his life on two levels, private and public, and since Jimmy is representative of the larger community, the transformation of his public role ensures that, in the end, Horace's story does indeed set into play a transformation of this community on the supposed \"margins\" of the queer world.\n\nThe ending of Jimmy's first \"Confessions\" section serves as a good introduction to the ways in which Horace, after his suicide, haunts Jimmy's private life. The dramatic interruption of this particular section replays the conversation between the two on the morning of Horace's death. After observing this drama, Jimmy returns again to his private thoughts. These thoughts are decidedly restless, however, and are made more so by their placement beneath Jimmy's recollection of his earlier conversation with Horace:\n\n( _Jimmy's eyes grow wide with horror. He swallows.)_\n\nJIMMY: I'm supposed to believe my cousin has been possessed? By a demon?  \nHORACE: Yeah. Something like that.\n\nI don't prepare elaborate meals for myself. Many is the night that I have eaten from a can. On the nights that I have no church business, no deacon board meetings, no prayer meetings, no auxiliary meetings or trustee board meetings, I try to read. I still enjoy Augustine and Erasmus. Maybe Freud, or Jung, or Foucault. Black history: Franklin, Quarles, Fanon. Occasionally fiction. But invariably I wind up asleep after about ninety minutes, only to awaken without fail around eleven o'clock to watch the late news. Then it's back to bed, which in many ways seems the object of leaving bed in the first place. (43\u201344)\n\nCall me queer, but the question \"Why is Jimmy reading Foucault?\" is too tempting to pass up here. After Horace's death, Jimmy clearly has trouble focusing his attention; consequently, he drifts from activity to activity. His scurrying from text to text, in particular, comes across as a restless search for \"meaning.\" Horace's undead intrusion into Jimmy's private life, however, has destabilized the secure meanings around which Jimmy's life was organized. The inclusion of Foucault on Jimmy's reading list especially highlights the process of transformation that is at work in Jimmy. And particularly in the context of a list beginning with Freud, it is difficult not to imagine that Jimmy is turning to _The History of Sexuality_ for answers. The imagination runs first, perhaps, to some of Foucault's more famous formulations:\n\nThe nineteenth-century homosexual became a personage, a past, a case history, and a childhood, in addition to being a type of life, a life form, and a morphology, with an indiscreet anatomy and possibly a mysterious physiology, Nothing that went into his total composition was unaffected by his sexuality. ... It was a secret that always gave itself away. . . . The sodomite had been a temporary aberration; the homosexual was now a species. (Foucault, 43)\n\nBut why limit Jimmy? Perhaps he has long since mastered the basic texts of Foucault and is now reading, in the French original, some of the more obscure interviews. The possibilities are titillating:\n\nI should like to say \"it is necessary to work increasingly at being gay,\" to place oneself in a dimension where the sexual choices that one makes are present and have their effects on the ensemble of our life. I should like to say also that sexual choices must be at the same time creators of ways of life. To be gay signifies that these choices diffuse themselves across the entire life; it is also a certain manner of refusing the modes of life offered; it is to make a sexual choice the impetus for a change of existence. (Foucault, qtd. in Cohen, \"Foucauldian Necrologies,\" 91)\n\nOf course, the Foucauldian fantasies I have conjured up for Jimmy here are not intended to name, definitively, the specific text or texts Jimmy may be reading. Quite the contrary: my intent is not to have the final say about what Jimmy's experiences mean but rather to foreground the ways in which Horace, as trickster, has set into motion an inescapable play of meaning, so that the very _impossibility_ of fixing the specific text effectively produces _possibilities_ in Jimmy that he has, until now, consistently (and homophobically) disavowed. Hence neither of the Foucault selections above should be taken as my vote for what Jimmy is reading, as if I could somehow fix Kenan's intention. Likewise, I do not intend here to label Jimmy\u2014in a minoritized sense\u2014as \"gay.\" Instead, I propose that Jimmy's own attempt to fix meaning has redounded on him, destabilizing particularly the heterosexual securities around which his life (and the life of the community) is organized. Before the suicide, gay identity\/desire was \"safely\" embodied by Horace, who was\u2014a la Foucault\u2014a personage, a past, a case history, and the like. After the suicide, Jimmy wants to maintain some sort of security by \"containing\" gay identity\/desire in the dramatic supplements to his \"Confessions.\" Yet, immediately after coming out of one of these dramatic interruptions, it is clear that Jimmy is unable to leave queerness behind; as Sedgwick suggests, \"In the wake of the homosexual, the wake incessantly produced since first there _were_ homosexuals, every human relation is pulled into its shining representational furrow\" ( _Epistemology_ , 128). In this case, Jimmy attempts to leave Horace behind in the dramatic supplement, only to stumble into Foucault.\n\nFurthermore, even _within_ the Foucault passages I have included above, a movement from a minoritized and contained gay identity to a more pervasive, inescapable queerness is reproduced: what is, in the first passage, a discrete and embodied identity, located in others, has become, in the second, more free-floating, \"diffusing] across the entire life.\" One of the _Oxford English Dictionary's_ definitions of \"spirit\" is \"a particular character, disposition, or temper existing in, _pervading_ , or animating a person or set of persons\" (emphasis mine). In this sense, then, perhaps \"a visitation of spirits\" refers less to the ghostly crew that leads Horace on his whirlwind journey than to the queer spirit pervading Jimmy's life after Horace's death. Ironically, it is Jimmy, not Horace, who seems to have \"a secret that always gives itself away.\" In the end, Jimmy's attempts to \"contain\" a homosexual are undone by the juxtaposition (or is it the coming together?) of Horace and Foucault. The queerness Jimmy is trying to \"fix\" may very well be within himself.\n\nAn episode from one of the other dramatic supplements to Jimmy's \"Confessions\" illustrates even more clearly how freely queer meanings circulate beyond Jimmy's control, despite his best efforts to contain them. It is Horace's last Thanksgiving, November 1983. Horace arrives onstage late, as the Reverend Barden, the family's guest, is praying over the food. Jonnie Mae (Jimmy's grandmother and Horace's great-aunt) scolds the boy lightly for being late, but when she realizes he has pierced his ear, her scolding becomes severe. She drops a spoon loudly into a dish of corn, and everyone at the table looks up in horror at Horace. The stage directions explain that \"JIMMY _seems bewildered by the reaction of the women\"_ (183). Nonetheless, Jimmy has trouble finding the appropriate language to counter his aunts' and grandmother's reactions:\n\nJONNIE MAE: (to ZEKE): Uh-huh. You see, Zeke? You see? What did I tell you? Now it starts this way, but how will it end? ( _Stands_.) No better sense than to go on and follow whatever them white fools do. You'd follow them to hell, wouldn't you? I\u2014\n\nRUTHESTER ( _moving to comfort her_ ): Mamma, it ain't that bad. He just\u2014JONNIE MAE: He _just_ pierced his ear. Like some little girl. Like one of them perverts.\n\nJIMMY: Mamma, it's really not that big a deal. Boys pierce their ears nowadays all the time. It's not thought of as\u2014(184)\n\nCan you say \"gay\"? For some reason, Jimmy can not; but then again, he can't entirely _not_ say it, either. Certainly, everyone in this scene is having trouble finding the right words, but it is to Jimmy, more than anyone else, that a trace of queer meaning attaches. Indeed, the very unspeakability of the word (and Jimmy's is the only fragmented sentence here that could possibly be completed by a single word) speaks queerness.\n\nMoreover, although at the time Jimmy argues that \"it's really not that big a deal,\" Horace and his earring come back here to haunt Jimmy's private confessions. In _The Signifying Monkey_ , Gates traces a \"signifying chain\" from the slave narrative of Ukawsaw Gronniosaw to Ralph Ellison's _Invisible Man_. In Gronniosaw's narrative, this signifying chain is a literal gold chain that serves as a marker for the African heritage Gronniosaw seeks to discard; in Ellison, the literal chain from Brother Tarp's days on the prison gang \"signifies a heap more\" and helps Ellison's narrator \"remember what we're really fighting against\" ( _Signifying Monkey_ , 135\u201336). In Kenan's novel, Gates's signifying chain becomes a signifying earring, a marker of both a particular \"cultural heritage\" and\u2014in an ironic inversion of Ellison\u2014the disruptive force that Jimmy and the others are \"really fighting against.\"\n\nThe dash that signifies the presence-absence of gayness in Jimmy's speech is a graphic representation of the queer meanings that, once again, Jimmy is unable to contain \"safely\" in the dramatic supplements to his \"Confessions.\" The play of meaning that Horace initiates again spills over into the prose sections of these \"Confessions,\" wresting authoritative control away from Jimmy. Ironically, Horace's conspirator in this disruptive project is Jimmy's wife, Anne, whose death comes a year or two before Horace's. Jimmy often invokes the two characters at the same point in his narrative (\"I sit on the porch and wait for the sun. All the while pondering dreams of Anne. And Horace\" [36]), and his memories of Anne are often as disruptive as his memories of Horace. Indeed, although one might expect that his wife would help stabilize Jimmy's heterosexual identity, it is with Anne that the inevitable failure of that identity is often most pronounced.\n\nThe possibility of failure is evident in Jimmy's recollections of the couple's first years together:\n\nI met Anne during the second semester of my sophomore year. I didn't sleep with her for almost two years; bedding her had not been my original goal. She fascinated me. . . .\n\nOne autumn day, one of the few days in which there were no meetings for her to rush to, she turned to me as we stood beneath a huge cottonwood tree on campus and asked, \"Why haven't you tried to screw me?\"\n\nI didn't know whether to laugh, cry, or just stare. She had light brown eyes and sometimes she focused them with such unbridled sincerity that they belied her canniness. I kissed her long and hard.\n\n\"So you aren't a faggot?\"\n\nI chuckled.\n\n\"Good.\"\n\nThe first time she took me into her bed I was impotent. \"Are you sure you're not a faggot?\" she asked. We laughed at the entire situation and went out for Chinese food. I never had that problem again. (174, 175\u201376)\n\nJimmy never had \"that problem\" with Anne again, but as this passage makes clear, he can not seem to shake that _other_ problem: even when he does not speak them, gay meanings attach themselves to Jimmy. Indeed, as with the dash in the Thanksgiving scene, the chuckle and the laugh speak most explicitly in their very refusal to speak. Once again Jimmy cannot say (he is) \"gay,\" but then again, he can't entirely _not_ say it, either. Or, more precisely, Jimmy relies on a (sexual) performance to solidify what he cannot convincingly put into words: his gendered heterosexual identity. As Butler might put it, \"The abiding gendered self [is] . .. structured by repeated acts that seek to approximate the ideal of a substantial ground of identity, but which, in their occasional discontinuity, reveal the temporal and contingent groundlessness of this 'ground'\" ( _Gender Trouble_ , 141). In this sense, the chuckle and the laugh might be seen as only part of a more general laugh _track_ for this inevitable heterosexual sitcom, and Jimmy and Anne's affair here may be just one more episode in which heterosexuality attempts to cover its all-too-obvious discontinuities.\n\nAnne has joined with Horace, then, in a monstrous, unnatural coupling that re-members the stable heterosexual meanings of Jimmy's life for him and flaunts the impossibility of embodying normative heterosexual identities. The opening of Jimmy's third and final \"Confessions\" confirms Anne in this role. Jimmy's narration (at this point, a rhapsody about Anne's physical beauty and about her hold on Jimmy's life) is interrupted by a series of direct challenges from Anne: \"Look at me.... What do you see?... Do you really see me?. .. Me?... Do you?... Do you even want to, really?.. . Are you even capable?\" (171\u201372). Like Anne's suggestion that he is a \"faggot,\" these challenges push Jimmy toward (hetero)sexual assertion, and he concludes: \"The metaphysics called it a little death. I think they were right\" (172). Yet heterosex for Jimmy is never quite as much fun as it is for John Donne, and Jimmy's reiteration of the metaphysical poets' metaphor for sex is not without irony; in what way, exactly, does Jimmy \"die a little death\" every time he is with Anne? The next paragraph opens with Jimmy's confession \"I have lied. To myself\" (172), and although Jimmy has ostensibly moved on to a new topic, it is difficult not to see this admission as a commentary on what has immediately preceded: Anne's \"Are you even capable?\" and Jimmy's defensive answer in the form of a sexual performance.\n\nAll of Jimmy's sexual memories, then, suggest that he is \"in the closet\"\u2014not the gay closet, exactly, but a closet of crisis, what Sedgwick might label \"the closet of, simply, the homosexual secret\u2014the closet of imagining _a_ homosexual secret\" ( _Epistemology_ , 205). This queer possibility\u2014never substantiated\u2014lurks behind all of Jimmy's recollections. For example, he writes of the wild adventures he had in college before meeting Anne:\n\nI slept with anything that was willing. I got drunk almost every weekend of my freshman year, many times during the week as well . . . my only regret being that my aunts and my grandmother never seemed to suspect that I was a hypocrite, a liar. For when I came home I still read the Scriptures in church and taught Bible school, only to return to school and recruit the first co-ed who gave me a willing glance. I realize that this was the true sin. (174)\n\nAs usual, Jimmy's \"double life\" here is not _necessarily_ gay; his \"co-eds,\" after all, effectively gender that which is so ambiguous in \"anything that was willing.\" Yet the \"secret\" that is produced by Jimmy's closet is never definitely _not_ a homosexual secret, \"so permeative has the suffusing stain of homo\/heterosexual crisis\" been to crises of masculine definition more generally (Sedgwick, _Epistemology_ , 72\u201373). Indeed, like the dash or the chuckle or the laugh, Jimmy's \"closet\" in this memory takes on queer meanings not because it specifically articulates such possibilities but rather because the metaphor of the closet that he implicitly invokes can never entirely disarticulate them.\n\nMoreover, Jimmy's prose memories are always read in the context of those \"other\" memories, the memories of Horace, which he can never completely contain. Horace, as trickster, ensures that queerness can never be entirely cleaned out of any of Jimmy's closets, and the one dramatic supplement to Jimmy's \"Confessions\" that I have not yet discussed demonstrates this most clearly. In this scene, Jimmy is confronted by Horace's explicitly \"gay\" closet and, in his ministerial role, confidently admonishes the boy, \"You know as well as I what the Bible says\" (113). Despite his pastoral assurance that homosexuality is \"wrong,\" however, Jimmy is again unable to keep traces of gay meaning from attaching to himself. Furthermore, this scene, in turn, transforms how straight we read Jimmy's other private memories; in particular, \"anything that was willing\" is thrown, decidedly, into gay relief. This time, the semantic marker for the presence-absence of gayness is not the dash, the chuckle, the laugh, or the \"double life,\" but, well, \".. . you know\":\n\nHORACE ( _quickly):_ I think I'm a homosexual.\n\nJIMMY ( _smiling, puts his hand on Horace's shoulder):_ Horace, we've all done a little . . . you know .. . experimenting. It's a part of growing up. It's .. . well, it's kind of important to\u2014\n\nHORACE: But it's not experimenting. I like men. I don't like women. There's something wrong with me.\n\nJIMMY: Horace, really. I have reason to believe it's just a phase. I went through a period where I. . . you know, experimented.\n\nHORACE: Did you enjoy it?\n\nJIMMY ( _slightly stunned):_ En . . . Enjoy it? Well. . . I. .. you know. Well, the  \nphysical pleasure was ... I guess pleasant. I really don't remember. HORACE: Did you ever fall in love with a man?\n\nJIMMY: Fall in love? No. ( _Laughs.)_ Oh, Horace. Don't be so somber. Really. I think this is something that will pass. I've known you all your life. You're perfectly normal. (112\u201313)\n\n\"Like me\" is the implicit subtext here\u2014\"You're perfectly normal, _like me.\"_ Horace, however, reverses this compulsion toward sameness, substituting instead what Kobena Mercer has called the \"challenge of sameness\" (\"1968,\" 427) Maybe, just maybe, Jimmy, _you_ are like _me:_ Did you enjoy it (as I have enjoyed it)? Did you ever fall in love with a man (as I have fallen in love with a man) ? In spite of Jimmy's best efforts, Horace transforms Jimmy's understanding of himself, and after Horace's suicide, that transformation grows even more pronounced. \"I have never lost my fear of the dead,\" Jimmy admits in one of his \"Confessions\" sections. \"No matter how many funerals I attend, perhaps no matter how old I get, I will fear the dead. In my dreams the dead rise\" (189).\n\nThe challenge of sameness embodied by Horace transforms Jimmy's private life; Horace destabilizes the heteronormative laws around which Jimmy's life is organized. This transformation, however, has far-reaching ramifications: Jimmy, as minister, is representative of and for this community. And although Jimmy feels that this community could never accept Horace's \"deviation,\" he nonetheless writes that Horace's \"reason for existing, it would seem, was for the salvation of his people\" (188). Consequently, the transformation Horace brings about is not merely the private transformation of Jimmy. If Horace exists \"for the salvation of his people,\" then Jimmy is the minister entrusted to bring Horace's message to the rest of the community.\n\nOne of Jimmy's professors at seminary inadvertently prepares him for this public role by facilitating the connection between Horace and Jimmy. The professor's name is Philip Schnider: \"We called him Rabbi. He got a kick out of it too, he being one of the rarest of birds, a Christian Jew\" (31). This unorthodox character\u2014the very first person introduced into Jimmy's \"Confessions\"\u2014tells Jimmy \"You, Greene, will make a great theologian\" (33). Rabbi's opinion does not come from his belief that Jimmy has the ability to adhere to a strict code of conduct or to expound how others ought to live their lives. Instead, Rabbi's opinion comes from his observation of Jimmy's \"Curiosity\": \"Curiosity with a capital C. Understand what I'm saying? This is a gift, now. Don't get me wrong. There are plenty of guys here who've got curiosity. But they don't have it with a capital C. Understand? They're bright, sure. Some maybe brighter than you.. .. [But] they don't have desire. A _real_ desire. Few people nowadays do\" (33). Jimmy can not understand what Rabbi means by this cryptic observation, and Rabbi's subsequent attempts to elaborate do little to clear up confusion. Yet, later in the same \"Confessions\" section, Jimmy returns to Rabbi's words. Thinking, as usual, about Anne and Horace, Jimmy concludes, \"Anne was not a romantic; I am. So was Horace. But he had something more, that damned curiosity . .. with a capital C\" (36). Suddenly, what was so enigmatic to Jimmy earlier exactly describes his connection to Horace. Apparently, after Horace's suicide, the transformation Jimmy undergoes enables him to understand what had only baffled him before.\n\n\"Curiosity\" and \"desire\" are not ingredients that normally go into the making of a Tims Creek minister; desire, in fact, because of the unlikely, unsanctioned, and queer connections it affirms, is usually equated with sin in this community. Still, although Rabbi never clearly explains what he means by these concepts, something that might be called \"Curiosity\" or _\"real_ desire\" indeed appears to fuel Jimmy's ministry, particularly after Horace's death. After the dramatic episode detailing Horace's conflict with the family over the earring, for instance, Jimmy returns to his prose \"Confessions\" with more questions than answers:\n\n[Horace] didn't quite know who he was. That, I don't fully understand, for they had told him, taught him from the cradle on. I guess they didn't reckon the world they were sending him into was different from the world they had conquered, a world peopled with new and hateful monsters that exacted a different price.\n\nWhat has happened to us? Can I cry out like the prophet Jonah and ask God to guide my hand and direct me toward the proper remedy? Once, oh once, this beautiful, strong, defiant, glorious group could wrestle the world down, unshackle themselves, part seas, walk on water, rise on the winds. What happened? Why are we now sick and dying?... How, Lord? How? How can we defend ourselves and grow strong again? How can we regain the power to lift up our heads and sing? When will the Host of Hosts visit us with favor and strength? (188\u201389)\n\nBehind this string of questions is a desperate curiosity about why the world is as it is, and an urgent desire for it to be better, freer, more loving. The group had assumed that the \"proper remedy\" was expulsion of the \"abomination.\" This expulsion was to guarantee a return to mythical wholeness. Jimmy realizes at this point that such attempts at \"purification\" actually guarantee not health and wholeness but sickness and death. \"Salvation\" will not entail securing the distinction between \"clean\" and \"unclean\" and expelling the latter. Instead, Jimmy realizes, these fixed and exclusionary understandings of salvation are precisely the problem. \"He was a son of this community,\" Jimmy writes of Horace, \"more than most\" (188). With this simple recognition, Jimmy questions and reverses the community's easy assumptions that queerness belongs elsewhere.\n\nQuestions such as these were far from the mind of the Reverend Barden, the minister who preceded Jimmy Rev. Barden had only answers, no questions. I have already mentioned, for example, his unequivocal use of Paul's letter to the Romans to condemn homosexuality. In the same sermon, Barden turns Jimmy's questions about weakness into a quick and easy answer:\n\n\"I can't look at evil and turn my back. It saddens me. . . . And you know what it is, don't you? You know what causes it?. .. Why a weak spirit. Ain't nothing more. Can't say the folks don't know, cause they do. But it's like the writer said another time: The spirit is willing\u2014O my Jesus, help me this morning\u2014but the flesh is weak. Ain't nothing else but weak-willed people. What else is gone cause this mess, children?\" ... Go head, Reverend, and preach now. (79)\n\nTo Barden\u2014and the congregation, which is more than willing to jump on the bandwagon of moral superiority\u2014the answer to the world's ills lies in overcoming \"weakness.\" Success in such an effort will be measured by the extent to which one can stand apart from and condemn those who fail to live up to a legalistic, heteronormative code of conduct. To a transformed Jimmy, in contrast, this moral code is the very _mark_ of weakness, and his searching questions signal a queer desire to move away from such an impoverished worldview.\n\nAfter Horace's death, this shift is most apparent in a scene that ostensibly has nothing to do with Horace. Despite Horace's absence, however, Jimmy's interpretation of this scene (and of the ways in which he is complicit in the community's repeated failures) has everything to do with how his life has been transformed by Horace. The summer after Horace's suicide, Jonnie Mae dies, and Jimmy's mother, Rose, returns to Tims Creek for the funeral. The family sees Rose as an ungrateful \"tramp,\" a woman who loved pleasure more than she loved her mother, Jonnie Mae. Rose's three sisters had been the \"good\" girls; Rose, in contrast, was the pariah. After returning home three times, pregnant and alone, she finally left Tims Creek for good, unable to bear the community's disdain. Jonnie Mae's funeral is the first time in his adult life that Jimmy has seen the woman. During the service, she stands apart from the family: \"Rose stood at the graveside ceremony as Reverend Raines eulogized my grandmother, reading Paul's letters to the Corinthians ('Though I speak with the tongues of men and of angels, and have not charity, I am become as sounding brass, or a tinkling cymbal')\" (120). When Rose is overcome with grief, all but one in the community refuse to comfort her. The haunting refrain from First Corinthians measures this community against a more inclusive vision of society, and the community is found wanting:\n\nWhen she began to cry, a calm and low sob, she looked suddenly forlorn and forsaken, a child lost and alone. And even now I cannot believe that no one, not Aunt Rebecca, not Aunt Ruthester, not Aunt Rachel [Rose's sisters], not Isador, not Franklin, not I [Rose's children], especially not I, turned to comfort her. (\"Charity suffereth long, and is kind; charity envieth not; charity vaunteth not itself, is not puffed up.\") Until finally, in a moment of such black tension, huddled over the lowering casket, a thin, invisible line separating the prodigal from the faithful, my Uncle Lester stepped over that line and put his hand on her shoulder. (\"For we know in part, and we prophesy in part. But when that which is perfect is come, then that which is part shall be done away\") It was a crude and clumsy gesture, but in all its lack of grace it was full of grace. (\"For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known.\") Seeing that, I knew my sin, but was unrepentant. (120\u201321)\n\nThe passage from First Corinthians that serves as a refrain here is an ironic commentary on the other Pauline passage that so sharply condemned Horace. At Jonnie Mae's funeral, the individual \"pariahs\" (Horace, Rose) are no longer the ones who need transforming, as Rev. Barden's interpretation of that passage from Romans implied. Without love (\"charity\" in the King James Version), all the community's pompous words are no better than \"sounding brass, or a tinkling cymbal,\" and suddenly, Jimmy sees this clearly, despite his inability to act. When Horace was first introduced into the text, the narrator explained, \"More than anything else, he wanted to have grace\" (11). Rose's brother Lester, in contrast to Jimmy and the rest of the community, acts with grace in this scene and, in so doing, allies himself with Horace. Ironically, Lester is only a peripheral character in _A Visitation of Spirits;_ he is repeatedly depicted as inept, hardly worthy of notice. Rose, Lester, and Horace reverse such characterizations, however, making the periphery central to a transformed\/transforming vision of a new community.\n\nThe \"Confessions\" section in which the story of Rose appears is also the section in which Horace \"comes out\" during the dramatic interruption to Jimmy's prose. The two episodes are related in that both represent the failure of Jimmy's ministry. They are \"confessions\" in the limited sense; Jimmy's inclusion of them is an admission that he has done something wrong. The episode that opens this \"Confessions\" section, however, suggests that Jimmy has learned his lesson and that Horace and Rose both have disrupted and transformed not only his private life but his public role as the community's minister, the community's \"representative.\" In the opening scene, Jimmy goes to visit a woman in his congregation: Margur-ette Honeyblue, a seventy-three-year-old woman who is presumably on her deathbed. This scene opens the section, but it actually follows the other two episodes chronologically. More important, however, it follows the other two episodes _developmentally:_ gone is the Jimmy who feels it his duty to explain the \"rules\" to Horace or who\u2014along with the rest of the family\u2014ostracizes Rose, the \"sinner.\"\n\nJimmy arrives at \"Miss Margurette's\" a few minutes before she is scheduled to take her medicine. Looking directly at Jimmy, Miss Margurette instructs her daughter to bring her a beer so she can wash down her pills. It is a minor incident, but Jimmy recognizes it as a test, and the thoughts this scene elicits confirm not only that Jimmy is a changed man, individually, but also that he has a different vision of how members of the community might relate to one another:\n\nHow could I communicate that I was not, did not want to be the holy and pious dictator of a pastor they had been used to for all their lives, that my very presence had nothing to do with my condemnation of their way of life, that I couldn't give a flying fuck about the still [Miss Margurette's son-inlaw] Lucius had out in the woods behind the house, or how [the family] sold all sorts of regulated beverages illegally from their kitchen, or that the last time they had been to church was to funeralize Margurette's husband twenty years ago. There was no way to say: I have not come here to judge you. To say: I want to introduce a new way of approaching Christian faith, a way of caring for people. I don't want to be a watchdog of sin, an inquisitor who binds his people with rules and regulations and thou shalts and thou shalt nots. (110)\n\nHumbled by the incident, Jimmy feels inadequate to bring this new way of living into being: \"But looking past those eyes so full of past hurt and past rejection and past accusation, I could only smile and let be what was\" (110). \"Letting be,\" however, in this instance is precisely the place to begin, and Miss Margurette's reaction to Jimmy's nonchalance indicates that the process of healing has already begun: \"She lay there quiet and as still as the bed. A warm breeze troubled the curtains and she gave me a wink, took a deep long breath, and sighed\" (110). Outside the window, Jimmy notes, \"a small bird flapped its wings against the wind\" (110). Horace, of course, had earlier hoped to escape the regime of sameness in Tims Creek by transforming himself into a bird, and this bird recalls that attempt at transformation. The literal success or failure of that particular transformation, however (\"Is it real?,\" \"Is it fantasy?\"), is at this point irrelevant next to other, more far-reaching transformations. After his suicide, Horace survives to haunt Jimmy and open up the meanings that have hitherto grounded the life of this community.\n\nThus, Horace begins the work of transforming this community by first transforming Jimmy's public role as the community's minister. Although the transformation\u2014of this community, of the meanings of \"community\"\u2014is (always) incomplete, Jimmy's \"new way of approaching Christian faith,\" which resists founding a community on exclusion but which no longer resists queer identifications and desires, assures that transformation will be ongoing. The re-vision of Christianity that Horace incites in Jimmy is the flip side of Kenan's signification on Baldwin. Jimmy Greene, after all, is not the first \"Jimmy\" in the African American tradition to envision a spirituality more dedicated to justice and love. Baldwin himself writes:\n\nIt is not too much to say that whoever wishes to become a truly moral human being (and let us not ask whether or not this is possible; I think we must _believe_ that it is possible) must first divorce himself from all the prohibitions, crimes, and hypocrisies of the Christian church. If the concept of God has any validity or any use, it can only be to make us larger, freer, more loving. If God cannot do this, then it is time we got rid of Him. ( _Fire Next Time_ , 67)\n\nJimmy Greene's Christianity, which rejects the God of \"thou shalt nots\" and seeks to transform the community into a freer, more caring place, is at least one possible resolution of Baldwin's injunction here. Through Jimmy, Kenan introduces a crucial distinction between \"religiosity\" and \"faith.\" Rejecting the pomposity and homophobia of religiosity Kenan nonetheless validates faith as a genuine force, capable of bringing about connections between human beings (McRuer, \"Randall Kenan,\" 234).\n\nBaldwin's desire is to forge connections between people who do not necessarily share identificatory locations, or who share some locations and not others. Hence, like Kenan's Jimmy, Baldwin does not need to count on \"Christians\" to be the group that of necessity institutes this freer, more loving society. In fact, Kenan's Rabbi is prefigured in Baldwin's essay by another Jewish character: the young James Baldwin's best friend, who inadvertently teaches him a lesson about the amount of hate in the \"Christian\" world. Baldwin writes:\n\nMy best friend in high school was a Jew. He came to our house once, and afterward my father asked, as he asked about everyone, \"Is he a Christian?\"\u2014by which he meant \"Is he saved?\" I really do not know whether my answer came out of innocence or venom, but I said coldly, \"No. He's Jewish.\" My father slammed me across the face with his great palm, and in that moment everything flooded back\u2014all the hatred and all the fear. ... I wondered if I was expected to be glad that a friend of mine, or anyone, was to be tormented forever in Hell, and I also thought, suddenly, of the Jews in another Christian nation, Germany. They were not so far from the fiery furnace after all, and my best friend might have been one of them. I told my father, \"He's a better Christian than you are,\" and walked out of the house. ( _Fire Next Time_ , 54\u201355)\n\nIn _A Visitation of Spirits_ , Baldwin's best friend grows up and goes to\u2014of all places\u2014a Christian seminary, where he continues teaching \"Christians\" about _\"real_ desire\": the desire for justice, the desire for love\u2014subjects about which they appear to know so very little.\n\nKenan's tribute to Baldwin complicates the theories of signification Gates develops in _The Signifying Monkey_. In a review of Mary Helen Washington's anthology _Invented Lives_ , Gates writes, \"[Black] female authors claim other black women as their ancestors (such as Zora Neale Hurston and Ann Petry) whereas most older black male writers denied any black influence at all\u2014or worse, eagerly claimed a white paternity\" (qtd. in V. Smith, 66). In other words, women writers' desire to bond lovingly with female ancestors\u2014a characteristic of what Gates calls \"unmotivated\" signification\u2014stands in stark contrast to what most male writers tend to feel: an anxiety of influence that leads them to critique other (male) writers who have preceded them, repeating and revising those writers' tropes and narrative strategies. Kenan, however, tends to fit neither of Gates's gendered categories; he rewrites\/reverses Baldwin's story of a young boy in conflict with the church, but he also repeats, _without_ negative critique, Baldwin's vision of a more just society Contrary to Gates's implication, then, some men \"lovingly bond\" with other men as readily as women bond with women. Thus, although I agree with Gates's reading in _The Signifying Monkey_ of the profundity of Alice Walker's bond with Hurston, I propose that the Kenan-Baldwin relation disrupts the gendered dichotomy at work in Gates's theories. Kenan's embrace of Baldwin, like Walker's of Hurston, is one of \"the most loving revision[s] ... we have seen in the tradition\" ( _Signifying Monkey_ , 255).\n\n### **\"It rit quar. Aint spose ta be lak dat\":  \nOuting the Signifying Monkey**\n\nIn the title story to Kenan's collection _Let the Dead Bury Their Dead_ , a young slave girl named Phoebe\u2014one of many slaves living on the antebellum North Carolina plantation (Canaan) of U.S. Senator Owen Alexander Cross\u2014complains to her mistress about a mysterious slave named Menes: _\"Misress . .. He Big an Black an Hateful and der aint no cause fuh such ta be ober usens_ in de Big House. _It rit quar. Aint spose ta be lak dat\"_ (308). Despite Phoebe's concerns, the crisis that commences with this big, dark slave's control over the \"Big House\" escalates as the story continues: Menes (also known as Pharaoh) burns down the house and leads a band of slaves into the North Carolina bogs, where they found a \"maroon society\" dedicated to the perpetual disruption of \"business as usual\" on the Cross plantation. Although Phoebe initially has misgivings about Menes, eventually she too joins the ranks of those who escape from Canaan.\n\nThis story is discovered among the Greene papers after Jimmy's death in 1998. In the introduction to \"Let the Dead Bury Their Dead,\" dated August 5, 2000, Reginald Kain\u2014or \"RK\"\u2014of Brooklyn, New York, informs readers: \"On March 12, 1998, the Reverend James Malachai Greene died in a car accident on the way home from a conference of ministers in Atlanta; he had just entered the town limits of Tims Creek, North Carolina\" (277). According to RK, the story of Menes and the maroon society is part of an \"oral annotated history\" of Tims Creek and is one of the more interesting documents to be discovered among the Greene papers after the accident.\n\nJimmy's death suggests that Kenan has a habit of killing off characters not long after they come to an awareness of the ways in which their community needs transforming. One might expect, after _A Visitation of Spirits_ , that Jimmy would begin to promote radical change in Tims Creek and elsewhere, but instead, Kenan kills him in a car accident. As in _A Visitation of Spirits_ , however, the \"facts\" may be \"subject to doubt,\" and I argue in conclusion that the existence of Menes\/Pharaoh marks yet another resurrection of the queer trickster embodied first by Horace and later\u2014because of his desire to reverse the received meanings of Christianity\u2014by Jimmy. Menes\/Pharaoh, in fact, \"literally\" rises from the dead: after his death, Menes\/Pharaoh's grave is discovered to be empty, and later in the story, he once again returns to transform the community which has fallen into complacency. In Menes\/Pharaoh, Kenan draws on characteristics from both Jimmy and Horace. For example, Menes\/Pharaoh actually carries out the work Jimmy envisions: the maroon society would\n\ncome to Pharaoh and he'd . . . tell em to keep themselves ready, to look out for one another, not to be like the white man, reaching and grabbing and trying to own everything, even people. Told em to remember that they come from a great land and a great people and such-like. Wont preaching he done, more like learning, learning em to love themselves and the world round em. (305)\n\nMoreover, Menes\/Pharaoh's very appearance carries on the work of \"queering\" the Signifying Monkey begun by Horace in _A Visitation of Spirits_. As the notice Senator Cross circulates for Menes\/Pharaoh's apprehension describes him: \"MENES, African-born, about 35 to 40 Years of Age, about Six-Feet-One Inches High, Square made, Left Ear pierced with Silver Metal Ring, Queer Markings about Face\" (302).\n\nOf course, you might argue, the fact that Menes wears an earring and is \"marked\" as \"queer\" means nothing, particularly in this nineteenth-century context. Lots of African slaves wore earrings. It's not such a big deal. It's not thought of as\u2014. Yet I submit that Kenan's work as a whole repeatedly inscribes the presence-absence of queerness into the Signifying Monkey as decidedly as the evasions in _A Visitation of Spirits_ attached gay meanings to Jimmy Certainly, the signifying earring that reappears here need not be, in Jimmy's words, \"thought of as\u2014\"; yet even when Kenan's tricksters are not _exactly_ \"gay,\" they are not exactly \"straight\" either. Menes\/Pharaoh may not embody a twentieth-century category such as \"gay,\" but the signifying earring ensures that we nonetheless read him (and his commitment to disrupting, from the \"margins,\" the hegemonic \"center\") _through_ Horace. The heterosexual\/homosexual binary is always included among the hierarchies of power that Kenan's characters have the ability to upset, and in this sense, Kenan's work \"outs\" the Signifying Monkey.\n\nSince he is already the African American \"figure of indeterminacy\" (Gates, _Signifying Monkey_ , n), it is more than a little redundant to \"out\" as \"queer\" the Signifying Monkey. Still, I think Kenan actualizes, in both _A Visitation of Spirits_ and _Let the Dead Bury Their Dead_ , what is only potentially queer in Gates's analysis of the Signifying Monkey tales. \"Outing\" the Signifying Monkey entails explicitly adding the heterosexual\/homosexual binary to the repertoire of reversals the Signifying Monkey effects; it entails envisioning the sort of \"necessarily and desirably queer world\" that Michael Warner suggests will overcome the totalizing tendencies of heteronormativity (8). In the Queer Renaissance, the \"queer world\" has not been \"contained\" in New York and San Francisco; indeed, the term _queer_ suggests that resistance to heteronormativity will continue to turn up and transform even the most apparently \"inappropriate\" places. Kenan does not hustle Horace off to the big city in his collection of stories, but despite his initial misgivings about _A Visitation of Spirits_ , Gates does not complain. Instead, on the back cover of _Let the Dead Bury Their Dead_ , Gates calls Kenan, with his \"generous moral imagination,\" a \"fabulist for our times.\" Queer transformations can begin in the queerest of locations; as \"fabulist for our times,\" Kenan transforms a place on the so-called margins into a center of the queer world.\n\n## **_Chapter Three_  \nUnlimited Access? Queer Theory in the Borderlands**\n\nI want to begin in a queer place. My central text in this chapter is Gloria Anzald\u00faa's eclectic mix of theory poetry and nonfiction prose in her 1987 collection _Borderlands\/La Frontera: The New Mestiza_ , and my central subjects are those Anzald\u00faa calls _los atravesados:_ \"the squint-eyed, the perverse, the queer, the troublesome, the mongrel, the mulato, the half-breed, the half dead: in short, those who cross over, pass over, or go through the confines of the 'normal'\" (3). _Los atravesados_ are those who live in the \"borderlands\" of Anzald\u00faa's title; caught between two colliding cultures, this queer group of \"outcasts\" emerges from attempts to separate, definitively, \"us\" from \"them.\" Before discussing _los atravesados de Anzald\u00faa_ , however, I examine briefly another group of fictional \"outcasts\" who \"crossed over\" earlier in the century in Jack Kerouac's _On the Road_.\n\nAlthough Kerouac might likely play a leading role in a study of homo- _sociality_ in the American canon, he is certainly an unlikely candidate for inclusion in a study of the contemporary Queer Renaissance. Yet I want to begin this chapter elsewhere, temporarily outside of the Queer Renaissance and the queer communities that are my subject. The two chapters that have preceded this one have examined variations on the idea of queer \"community,\" focusing on, respectively, what queers do in and to the communities of which they are a part. This chapter initially approaches the issue from a different direction, considering what _other_ \"communities\" might do with and to queerness.\n\nKerouac's appropriation of the trope of \"the border\" sets the stage for my analysis of contemporary deployments of borders and of queerness. Using Kerouac, I make the somewhat unremarkable assertion that there is more than one way to cross a border. Using Anzald\u00faa, who stands at the crossroads where \"border theory\" and queer theory intersect, I expand on this assertion in order to consider how there are different ways to cross the theoretical border. Queer theory is definitely part of what might be called the \"theoretical border-transgressing boom,\" but instead of deploying queer theory as some sort of theoretical NAFTA that provides cultural critics with \"unlimited access\" to new markets, Anzald\u00faa's particular brand of queer\/Chicana theory challenges the unchecked mobility of oppressive systems of power. Some overly celebratory understandings of queerness, not unlike Kerouac's \"outcasts\" in _On the Road_ , tend to efface the ways in which identities and histories are structured in domination, so that some identities are immobilized while white, male, and heterosexual power is able to travel anywhere and everywhere with ease. Anzald\u00faa's work undermines this structural domination by insistently foregrounding a \"mestiza queer\" identity.\n\nFor Anzald\u00faa, \"the border\" and \"queerness\" stand as figures for the failure of easy separation. Rather than establishing two discrete entities, each attempt at separation actually produces (mestiza\/queer) identities that do not wholly fit in either location. Such identities are consequently marked as \"other\" (undocumented, illegal, perverse) by those with the power to police the border. The marking and fixing of \"other\" identities directs attention away from the border guards themselves, who in turn pass unmarked and unrestrained. The opposition of \"unmarked\" and \"marked\" is, at times, far from metaphorical. In a recent article in the _Arizona Daily Star_ , physicist Bill Wattenburg suggests that a \"florescent glowing dust\" be used on the border to police \"illegals\"; those who cross over ( _los atravesados)_ would pass through the dust, and authorities could then use \"ultraviolet lights and lasers [to] track down the marked people\" (qtd. in Day an, 812). The mestiza queer identity theorized by Anzald\u00faa, however, counters the border guards' absolute mobility with another, critical mobility.\n\nThe Kerouacian hipsters with whom I begin my analysis are hardly a pure coextension of these border guards or of ominous, all-pervasive \"systems of power.\" I begin with _On the Road_ , however, in an attempt to situate the history of borders and border crossings as multifaceted, involving more than two sides. Relations between the multiple players are often contradictory. There is doubtless much in the Beat project as a whole that is oppositional to the restrictive white middle-class mores of the 1950s, even while that project is simultaneously implicated in the very forces it opposes. In _On the Road_ , the hipsters' and border guards' locations are contiguous but are not reducible to each other; hence the hipsters' border crossing marks an extremely complex concourse of oppositional and hegemonic forces. So, too, with border crossings in the academy: there is not, on the one side, a group that establishes borders, and opposite it, another that transgresses them. The particular logic of the \"fad,\" for instance, demonstrates how complicated border crossings may be. Fads (such as the theoretical border-transgressing boom) may, in the first place, speak to the desire for community, the desire for transformation, or the desire to cross over to something new that opposes the status quo. At the same time, fads are part of a much larger network: they are marketable, and market forces demand and ensure that a fad will inevitably disappoint and eventually disappear, clearing the way for the next fad.\n\nThe mestiza queer locates and manipulates these and other contradictions effected by the border and by border crossings. Not content to remain a casualty of binaristic thinking or to be merely celebrated for her hip transgressions, Anzald\u00faa's mestiza queer redraws the boundaries, mobilizing a critique that spotlights the location and history of all the players in a given border conflict, and in the process unmasks the border guards' and others' attempts to escape identity and history.\n\n### **Escape from Identity and History:  \nCrossing the Border in _On the Road_**\n\nIn Kerouac's 1957 novel, the protagonist Sal Paradise is a young writer who \"want[s] to take off\": \"Somewhere along the line I knew there'd be girls, visions, everything; somewhere along the line the pearl would be handed to me\" (11). \"Beaten down\" by a world that lacks \"vision,\" Kerouac's protagonist takes to the road to find it, along with \"girls\" and \"everything.\" And indeed, in one of the final sections of the text, it seems as if this desire to have it all might actually be realized. Sal and his friends Stan Shephard and Dean Moriarty head south toward \"the magic land at the end of the road\" (276). As the trio approaches the U.S.-Mexican border (the \"magic border\" [273]), a shift occurs:\n\nTo our amazement, it looked exactly like Mexico. It was three in the morning, and fellows in straw hats and white pants were lounging by the dozen against battered pocky storefronts.\n\n\"Look\u2014at\u2014those\u2014cats!\" whispered Dean.. .. We couldn't take our eyes from across the street. We were longing to rush right up there and get lost in those mysterious Spanish streets. (274)\n\nKerouac's protagonists assume that \"crossing the border\" means one thing: the signifier \"Mexico\" and its signified fit together \"exactly.\" Mexico here, like the fellows \"lounging\" against the storefronts, is passive and immediately accessible; one can \"rush right up there and get lost in\" this magical union of desire and object. What these characters elide, however, is that theirs is a particular _construction_ of Mexico: it is the hip, exotic land of excess Kerouac's boys have dreamed about discovering.\n\nThe protagonists are held up by the border officials, but ultimately, even the police in this new world fit the exotic picture they are constructing:\n\nThey weren't like officials at all. They were lazy and tender. Dean couldn't stop staring at them. He turned to me. \"See how the _cops_ are in this country. I can't believe it!\" He rubbed his eyes. \"I'm dreaming.\" Then it was time to change our money. We saw great stacks of pesos on a table and learned that eight of them made an American buck, or thereabouts. We changed most of our money and stuffed the big rolls in our pockets with delight. (274\u201375)\n\nCrossing the border is a transcendent (or, we might say, \"beatific\") experience for Kerouac's characters\u2014an experience that finally allows them to escape the banality and rigidity of (white) American middle-class life. To them, the land they have discovered on crossing the border is \"magical\" and \"delightful.\" The powerful and bulging pockets with which this chapter of _On the Road_ concludes, however, suggest\u2014and perhaps not so subtly\u2014that someone, at least, is about to get fucked. As the journey continues, this connection between American buying power and white heterosexual male sexual power is literalized. Victor, the boys' hip new Mexican friend, takes them to a whorehouse where, as Sal Paradise explains, they can \"buy _se\u00f1orita\"_ (281). The protagonists spend several hours at the establishment to which Victor takes them, patronizing women from several Central and South American countries. The narrator proclaims, \"Through our deliriums we began to discern [the girls'] varying personalities\" (287), but none of the women is given a name. In fact, the \"wildest,\" who was \"half Indian, half white, and came from Venezuela\" (287), becomes, in only a few pages, coextensive with her supposed homeland: \"Venezuela clung about my neck and begged for drinks. . . . With Venezuela writhing and suffering in my arms I had a longing to take her in the back and undress her\" (289). Apparently, white male power in Kerouac's text buys not only individuals but entire nations. The reduction of this woman to Venezuela is particularly ironic, given that her mestiza identity and Mexican location complicate any easy notion of \"origin.\" Despite how \"wild\" she is, the boys \"keep her in her place\" with the epithet \"Venezuela.\"\n\nThis border crossing undoubtedly reveals more about the white male protagonists than it does about the inhabitants of Mexico; to these \"hip\" bohemians, \"all of Mexico [is] one vast Bohemian camp\" (302) that allows them to transcend their own rigid and restricting identities and cultural contexts. This \"transcendence\" is not entirely negative: Beat writers despised what they saw as the materialism and stultifying conformity of middle America, and, without question, they effectively shaped alternative visions that had a major impact on the next generation. Andrew Ross, examining a passage from _On the Road_ where Sal Paradise walks the streets of Denver \"wishing [he] were a Negro,\" suggests that \"[white] fantasies, however romantic ... had real and powerful social effects ... for the white students in SNCC [the Student Nonviolent Coordinating Committee] who participated in the struggle for civil rights, and who supported the black liberation movements of the sixties\" (68\u201369). Despite the potential positive impact of the Beat writers' racial fantasies, however, the example Ross chooses\u2014a fleeting fantasy on the streets of Denver\u2014pales in comparison to the extreme exploitation represented later in the novel, when the protagonists cross the border into Mexico.\n\nIn fact, Ross's observation might be inflected somewhat differently: if we admit that white fantasies such as Sal's had \"real and powerful social effects,\" then we would have to admit simultaneously that Kerouacian _male_ fantasies also had such real and powerful effects\u2014an admission that is borne out historically by the New Left's callous disdain for, even outright hostility toward, feminism in the 1960s. This misogyny, coupled with the romanticism Ross details (which is nonetheless racist), would have particularly disastrous consequences for women of color, who \"experience racism not as 'blacks' [or 'Latinos'], but as black women\" and Latinas (de Lauretis, \"Eccentric Subjects,\" 134). Sal Paradise, of course, walks the streets of Denver wishing he were a Negro _man_. In Denver, Sal's desire to escape his own identity can be read as simply a romantic fantasy that (perhaps) lays the groundwork for powerful, and potentially positive, social effects; in Mexico, when this fantasy is symbolically realized, the negative material consequences for the women on the other side of the border are far more difficult to dismiss.\n\nIronically, the constricting white American identities that Kerouac's protagonists try to escape are reproduced in the Mexican section of _On the_ _Road_. As John Tytell confirms, in Mexico \"Dean and Sal act without grace, as 'self-important moneybag Americans'\" (168). _On the Road_ demonstrates that the logic of a border crossing that seeks to escape identity actually facilitates the emergence of the self-important American. Dean insists that they are \"leaving everything behind us and entering a new and unknown phase of things ... so that we can safely think of nothing else and just go on ahead with our faces stuck out like this, you see, and _understand_ the world as, really and genuinely speaking, other Americans haven't done before us\" (276). However, his admission that they can and will \"safely think of nothing else\" guarantees that the boys never fully consider how \"other Americans\" before them _have_ understood the world, how they inadvertently replicate American perspectives, or how the path they have taken parallels the path taken earlier by others. The boys' \"wonderful Mexican money that went so far\" buys them anything they want and can take them \"right on to South America\" if they so desire (276, 277). As they get closer to Mexico City, Dean's wristwatch even buys them the authenticity they left home to discover. Encountering a three-year-old girl standing in front of a group of Indians, Dean exclaims, \"She'll never, never leave here and know anything about the outside world. It's a nation. Think of the wild chief they must have! . . . How different they must be in their private concerns and evaluations and wishes!\" (297\u201398). Not paying for this \"difference\" makes Dean uncomfortable, and he presents the little girl with his wristwatch: \"She whimpered with glee. The others crowded around with amazement. Then Dean poked in the little girl's hand for 'the sweetest and purest and smallest crystal she has personally picked from the mountain for me'\" (298). In a sense, Dean gives up his wristwatch and buys the \"timelessness\" and \"authenticity\" that countless American travelers before and after him have sought to discover.\n\nNothing could be more alien to Kerouac's protagonists, however, than the thought that they are participating in a history of exploitation of the region they are exploring. On the contrary, in their minds, they have left banal American identities behind them at the border and now freely merge with what they see as a hip new Mexican identity. This escape from their own identities, however, necessarily obscures that history of exploitation. Sal's thoughts as the other boys sleep illustrate just how glaring their effacement of history is:\n\nThe boys were sleeping, and I was alone in my eternity at the wheel, and the road ran straight as an arrow. Not like driving across Carolina, or Texas, or Arizona, or Illinois; but like driving across the world and into the places where we would finally learn ourselves among the Fellahin Indians of the world, the essential strain of the basic primitive, wailing humanity that stretches in a belt around the equatorial belly of the world from Malaya (the long fingernail of China) to India the great subcontinent to Arabia to Morocco to the selfsame deserts and jungles of Mexico and over the waves to Polynesia to mystic Siam of the Yellow Robe and on around, on around, so that you hear the same mournful wail by the rotted walls of Cadiz, Spain, that you hear 12,000 miles around in the depths of Benares the Capital of the World. These people were unmistakably Indians and were not at all like the Pedros and Panchos of silly civilized American lore\u2014they had high cheekbones, and slanted eyes, and soft ways; they were not fools, they were not clowns; they were great, grave Indians and they were the source of mankind and the fathers of it. The waves are Chinese, but the earth is an Indian thing. As essential as rocks in the desert are they in the desert of \"history.\" (280\u201381)\n\nIn this passage, specific histories and struggles are effaced as people of color the world over are blurred into one exotic, hip lump: Malaysia is China is India is Arabia is Morocco is Mexico is Polynesia is Siam. Ultimately, the narrator settles on Benares, a sacred Hindu city on the Ganges River in India, to stand in for the \"Capital of the World,\" Of course, it is more than a little ironic that this invocation of India is followed in the text by the narrator's observation that \"these people were unmistakably Indians,\" since the history of white exploitation of indigenous peoples on this continent begins with just such a mistake, or (mis)naming. As the passage concludes, history indeed requires the distancing quotation marks Kerouac employs, for this is not so much history as it is history's efface-ment. Centuries of exploitation are erased as Kerouac's neo-Columbian hipster once again \"discovers\" a New World full of abundant resources that are his for the taking.\n\nEven when _On the Road_ was published, however, the \"border\" was a contested site. In 1958, one year after the publication of _On the Road_ , Americo Paredes published his important study of Chicano _corridos_ (ballads, folk songs), _\"With His Pistol in His Hand\": A Border Ballad and Its Hero_. Paredes was the first of many Chicano scholars to respond to Walter Prescott Webb's studies of the Texas Rangers, which championed the supposed \"heroics\" of the Rangers and denigrated the Mexicans and Mexican Americans who lived within the Rangers' jurisdiction and who were often victims of their attempts to maintain 'Taw and order\" (J. Sald\u00edvar, 49\u201353). Through an extended analysis of \"El Corrido de Gregorio Cortez,\" Paredes contested Webb's characterization of the Texas Rangers and the U.S. Mexican border region. In contrast to both Webb's representation of the border region as the site of Anglo-American \"heroics\" and romanticized Anglo understandings of the region as \"Old Spain in Our Southwest,\" Paredes \"redefined the border . . . [as] a historically determined geopolitical zone of military, linguistic, and cultural conflict\" (Cald\u00e9ron and Sald\u00edvar, 5). \"El Corrido de Gregorio Cortez\" was, for Paredes, paradigmatic of early twentieth-century border _corridos_ in general, which he saw as tending \"toward one theme, border conflict,\" and \"toward one concept of the hero, the man fighting for his right with his pistol in his hand\" (Paredes, 149; qtd. in R. Sald\u00edvar, 28).\n\nFar from providing an escape from identity, the border thus becomes a metaphor for the complex and oppressive social and historical forces that have shaped and continue to shape Chicano identity. Teresa McKenna writes that the border between Texas and Mexico \"stands as figure and metaphor for the transition between nations and the complex of connections which continue to exist for all Mexicans whether border residents or not\" (qtd. in J. Sald\u00edvar, 54). The meanings borne by border _crossings_ , in turn, are also very different in Chicana\/o scholarship from the meanings borne by the \"magical\" border crossing in Kerouac's novel. In 1981, for example, Cherr\u00ede Moraga and Gloria Anzald\u00faa compiled a collection of prose and poetry that complicates easy, celebratory notions about the transgression of boundaries. \"Literally, for years,\" Moraga writes in her preface to that volume, \"I have dreamed of a bridge\" (xviii). That dream is realized with the publication of _This Bridge Called My Back: Writings by Radical Women of Color_ , but the editors' title complicates the idea of \"crossing over\" by introducing history and embodiment: this crossing is not simply celebratory but is rather inscribed onto the bodies of women of color; \"this bridge,\" according to Moraga and Anzald\u00faa, is \"called my back.\"\n\n_This Bridge Called My Back_ stands at the crossroads where border theory, feminism, and a nascent queer theory intersect. Indeed, I would argue that, because of its construction of relational and critical identities, the collection is one of the first texts of the Queer Renaissance. Moraga writes, \"But the passage is _through_ , not over, not by, not around, but through\" (xiv); she goes on to explain her hope that _This Bridge Called_ _My Back_ \"will function for others, colored or white, in the same way\" (xiv). In spite of the hope for coalition, however, Moraga's emphasis (\"the passage is _through_ \") suggests that _This Bridge Called My Back_ (and other books for which it will eventually function as an antecedent) cannot and will not advocate \"crossings\" that allow for an avoidance of identity and history Anzald\u00faa and she have put together a collection that is about border crossings and connections between women, Moraga explains, but connection is possible only if readers are willing to ask difficult questions about the many silences surrounding race, class, and sexuality in feminism, and to admit they do not have all the answers (xiv\u2013xv).\n\nIn the end, _On the Road_ in fact demonstrates that an avoidance of the difficult questions of history and identity cannot be easily sustained; ultimately, the hipsters succumb to the history they were trying to escape. Sal Paradise and his friends come down from the emotional, transcendental high that crossing the border facilitated, only to return once again, \"beaten down,\" to the United States. \"Crossing the border\" inevitably disappoints these American boys, since \"the magic land at the end of the road\" turned out to be little more than a passing fad.\n\n### **Queer Theory in the Borderlands**\n\nAcademic border crossings are all the rage. Cultural studies, promising to break down the borders between disciplines or between the academy and outside communities, trumpets the advent of \"postdisciplinary knowledge\"; in literary studies, the Modern Language Association (MLA) has recently published an assessment of the profession explicitly titled _Redrawing the Boundaries_. The editors of that collection, Stephen Greenblatt and Giles Gunn, write that \"our profession has become increasingly concerned with the problem of boundaries.. . . These boundaries can be crossed, confused, consolidated, and collapsed; they can also be revised, reconceived, redesigned, or replaced\" (4). Likewise, in the field of education, theorist Henry Giroux has developed the idea of \"border pedagogy\" in his book _Border Crossings: Cultural Workers and the Politics of Education_.\n\nQueer theory, fashionably late yet central to this theoretical border-crossing party, similarly champions the crossing and recrossing of borders. As queer theorist Lisa Duggan writes:\n\nThe notion of a \"queer community\" ... is often used to construct a collectivity no longer defined solely by the gender of its members' sexual partners. This new community is unified only by a shared dissent from the dominant organization of sex and gender. .. . The \"queer\" nation is a newly defined political entity, better able to cross boundaries and construct more fluid identities. (\"Making It Perfectly Queer,\" 20\u201321)\n\nI do not intend to be disingenuous toward any of the critical schools mentioned here or to dismiss what theorists are doing as \"trendy\" or unreflective. On the contrary: one of my intentions in this chapter is to question the ways in which queer theory in particular gets interpellated as a \"fad\" and to disarticulate it from that which is, supposedly, merely trendy. In this section of my chapter, I begin that disarticulation by considering briefly two types of theoretical border crossings: first, the crossing that interrogates the construction of borders, identities, and histories; and second, the crossing that implicitly posits a cosmopolitan consumer\/theorist, able to move everywhere with ease, and that is consequently more celebratory of transgression for transgression's sake. I then offer an extended reading of Gloria Anzald\u00faa's queer Chicana border theory and argue that Anzald\u00faa's work, which ironically both interrogates and celebrates critical border crossings, redefines mobility through its deployment of the mestiza queer. _Borderlands\/La Frontera_ does represent a mestiza queer subject able to cross borders skillfully, but her movement, even when it is celebratory, is always in the service of locating other identities and histories that seek to maintain power by escaping detection.\n\nThe theorists and critical schools mentioned above generally represent the first sort of border crossing. They constantly question what \"border crossings\" might mean; each seeks to examine how and where power is located in both the demarcation and transgression of boundaries. In fact, Greenblatt and Gunn's insistence that \"boundaries can be crossed, confused, consolidated, and collapsed\" syntactically links the consolidation\u2014or reestablishment\u2014of power to its confusion or collapse: as in Kerouac's work, the crossing of one boundary may entail the reinforcement of others. After their delineation of what can be done with and to boundaries, Greenblatt and Gunn go on to insist that we not take them lightly: \"The one thing they cannot be in literary studies is entirely abolished\" (4). Giroux likewise connects his theory of \"border pedagogy\" to an interrogation of configurations of power: border pedagogy allows for \"forms of transgression in which existing borders forged in domination can be challenged and redefined... . Border pedagogy must take up the dual task of not only creating new objects of knowledge but also addressing how inequalities, power, and human suffering are rooted in basic institutional structures\" (28\u201329). Finally, Duggan also attends to \"basic institutional structures\" when she stresses that queer border crossings are unified only by their \"shared dissent from the dominant organization of sex and gender.\" In short, scholars in cultural studies, radical pedagogy, queer theory, and the like are aware that the crossing of borders is no simple matter. Borders and boundaries are not inviolate, but they are almost always the products of configurations of power that are not easily displaced.\n\nAt the other end of the spectrum, seemingly, are border crossings that tend to affirm the transgression of boundaries for transgression's sake. Such celebratory transgressions are perhaps more in evidence outside the academy: as Lauren Berlant and Michael Warner point out, \"There are even components of the national mass media, such as _Details_ and MTV, that have cultivated a language of queerness in their highly capitalized forums\" (346). In these consumer contexts, queerness and the transgression of boundaries are marketable. It was such marketability that supposedly made 1993 the year of \"lesbian chic\" and that, oddly enough, packaged both Tony Kushner and Jack Kerouac as Gap ads (\"Kerouac,\" the ad insists, \"wore khakis\"). As Phillip Brian Harper writes, \"The easy appropriability of the signifiers of certain forms of social marginality makes them prime commodities in the mass-cultural drive to market the effects of disfranchisement for the social cachet that can paradoxically attach to it\" (188). The marketing of queerness puts transgressive \"new\" (in the case of Kerouac, reissued) identities up for sale, but what get elided in the transaction are the systemic forces that construct all identities, marking some as exotic and marketable while others are allowed to pass unmarked. At first, the consumption of a fashionable queerness in videos, in magazines, and on talk shows might seem far removed from a critical academic queerness. Yet the academy, certainly, is also a consumer context, and in the 1990s, queer theory inside the academy (like lesbian chic or _Details_ on the outside) is\u2014for many\u2014decidedly \"hip\"; it is often casually characterized as the \"latest thing\" or the \"next wave.\" One of the problems with such \"hipness\" is that\u2014as with Sal Paradise's border crossing or any \"passing fad\"\u2014inevitable disappointment and failure become structural components of queer theory.\n\nWe can look carefully at the contradictions inherent in queer theory's academic location and marketability without forgoing its critical promise, however. Academic and consumer border crossings overlap but are not absolutely reducible to each other, and we need not be, like Kerouac's hipsters, blind to contiguities and contradictions. Because of the existence of more celebratory variants of queer theory, however, some critics have been quick to deny queer theory's transformative potential. Such critics would see the critical type of theoretical border crossing as simply and totally reducible to the celebratory. Donald Morton, for instance, argues that the queer theory of Eve Sedgwick and others in the academy is \"basically nothing more than the politics of shocking the sensibilities of the bourgeois reader by 'performing difference'\" (134). Against what he sees as a \"ludic (post)modern theory [that] demands the de-essentializa-tion, de-naturalization, and de-centering of _signs_ and _representations_ in the name of personal libidinal liberation,\" Morton posits \"resistance (postmodernism [that] demands collective emancipation by the overthrow of existing exploitative and oppressive structures and a rebuilding of the space of the public sphere along new lines of coherence\" (138). Despite the fact that many practitioners of queer theory not only commit to the \"de-essentializations, de-naturalizations, and de-centerings\" he denounces but understand these as the precondition for shaping precisely such \"new lines of coherence,\" Morton\u2014because of what he sees, and demonizes, as pure \"performance\"\u2014is determined not to see queer theory as at all participating in this collective rebuilding. Instead, he homogenizes queer theorists into one apolitical, flaming lump. To paraphrase Emma Goldman: apparently one can't dance in Donald Morton's revolution.\n\nIn a way, Morton's denunciations of queer theory can be read as homophobic calls not to \"flaunt it.\" Moreover, Morton reduces the players in this border conflict to two (the \"redeemed\" and the \"damned,\" to borrow language from my last chapter). In contrast to Morton, we might recognize that, even though I have sketched out two types of theoretical border crossings, the boundary between these two types is not absolute. It is, indeed, impossible for any border crossing to be purely one or the other. In fact, it is the desire for a pure critical and political space (the magic land at the end of the road?) that both produces queer theory as the \"latest thing\" and allows for it to be dismissed as a \"passing fad.\" Rather than acceding to either characterization, however, queer theory might expose the desire for purity (as opposed to, say, perversion?) as a fantasy and as an impediment to the critical work it seeks to advance. We might envision the best queer theory, then, as aware of the tensions caused by its contradictory location and on guard against attempts to cast its critical transgressions as (merely) hip, even as it simultaneously puts the \"celebration\" or \"performance\" back into \"politics\"\u2014or rather, uses performance to spotlight and challenge, as Morton would have it, \"existing exploitative and oppressive structures.\"\n\nWith this goal in mind, I turn to the particular convergence of queer theory and border theory in the work of Anzald\u00faa. Anzald\u00faa's work puts into play a new and transgressive identity but at the same time resists mere transgression for transgression's sake. Anzald\u00faa exhibits an awareness of the complexity of \"existing borders forged in domination\" and controverts queer theorists who do not. She even queries \"queerness\" if and when it seeks to escape the complexities of history and identity There's more than one way to cross the theoretical border, and _Borderlands\/La Frontera: The New Mestiza_ crosses the border in ways that challenge the unchecked mobility of oppressive systems of power: Anzald\u00faa exposes the supposed \"authorities\" who mark her as \"other\" while attempting to pass unmarked themselves. More than just a passing fad for appropriation by disembodied theoretical hipsters, queer theory as practiced by Anzald\u00faa becomes a tool for the difficult work of shaping and performing what she calls \"\/a _concien-cia de la mestiza_... a new _mestiza_ consciousness, _una conciencia de mujer_ \" (77). This critical mestiza, woman-centered consciousness is the \"consciousness of the Borderlands\" that the text celebrates (77).\n\nThe remainder of this chapter explores how Anzald\u00faa shapes mestiza queer identity in _Borderlands I La Frontera_. First, I analyze the poem that opens Anzald\u00faa's first essay, since that poem provides a blueprint, of sorts, for the text as a whole. In her opening poem, Anzald\u00faa transforms the abject identity into which she has been interpellated into a resistant identity, intent on exposing and dismantling the history of oppression to which both her identity and the border stand as citations. Second, I examine how that transformation is reproduced on a larger scale in _Borderlands I La Frontera_ more generally. Third, and finally, I consider the ways in which Anzald\u00faa theorizes mestiza identity in and through queerness and vice versa. Although Anzald\u00faa has at times been relegated to the theoretical margins, this chapter redraws the boundaries in an attempt to position her (and the \"positions\" she advocates) as central to the theoretical agenda of queer theory. Like Zami and the theories of queer positionality I examined in my first chapter, the queer trickster identity I delineated in my second, and the bisexual artistic\/activist identity I explore in my last, the mestiza queer identity Anzald\u00faa posits exposes and undermines the dominant, normative organizations of sexuality, gender, and race that police _los atravesados_ in an attempt to \"keep them in their place.\"\n\nAnzald\u00faa divides _Borderlands\/La Frontera_ into two sections: \" _Atravesando Fronteras\/_ Crossing Borders,\" which consists of seven autobiographical\/poetical\/theoretical essays, and _\"Un Agitado Viento_ [An agitated wind]\/ _Ehecatl_ , The Wind,\" which is made up of six sections of Anzald\u00faa's poetry Border crossings are central to the entire text, but Anzald\u00faa's crossings are not simply hip migrations to an exotic land of excess. Anzald\u00faa's first representation of a border crossing comes in a poem she includes in her first essay, \"The Homeland, Aztl\u00e1n\/El _otro M\u00e9xico_ [The other Mexico].\" In this poem, Anzald\u00faa reinvents the marked identity the border compels her to embody and thereby establishes a pattern of transformation that is replicated in the text as a whole. The poem commences in Border Field Park, on the U.S.-Mexican border south of San Diego:\n\nI walk through the hole in the fence  \nto the other side.  \nUnder my fingers I feel the gritty wire  \nrusted by 139 years  \nof the salty breath of the sea.\n\nBeneath the iron sky\n\nMexican children kick their soccer ball across,\n\nrun after it, entering the U.S. (2)\n\nThis is not the \"end of the road\" for Anzald\u00faa; crossing the border here marks the _beginning_ of the poet's journey in _Borderlands\/La Frontera_ , and there is nothing very magical or transcendent about it. On the contrary: the children's soccer ball, accidentally kicked across this border, underscores the everyday triviality of this event. At the same time, the children's trivial \"accident\" contrasts ominously with the other historical \"accident\" that has occurred \"beneath the iron sky\": the accident that produced the artificial border in the first place.\n\nLike the soccer ball, Anzald\u00faa is no respecter of borders. The very text of \"The Homeland, Aztlan _El otro M\u00e9xico_ \" crosses back and forth freely between prose and poetry, between sections consisting of Anzald\u00faa's \"original\" writing and excerpts (of prose, poetry, song) from other writers whom she has found useful and pleasurable. Her lack of respect for the border, however, is accompanied by a profound respect for the people whose lives and histories have been shaped by the border, and who have resisted, in various ways, the oppression that this accident of history has generated. This multifaceted understanding of the border, introduced in her opening poem, fuels Anzald\u00faa's entire theoretical project. Her lack of respect is evident in the ease with which the poet walks across the border. Unlike Kerouac's protagonists, her crossing is not sanctioned by hip authorities; she is not likely to gape in amazement at the \"cats\" who patrol the border. At the same time, even as she crosses, she reveals her awareness of how long the border has engendered both oppression and resistance: \"I feel the gritty wire \/ rusted by 139 years \/ of the salty breath of the sea.\"\n\nThe border and its history in this text are not picturesque; Anzald\u00faa uses harsh adjectives (\"gritty,\" \"iron\") to describe the setting, and the wavelike structure of the poem itself reproduces the insistent pounding of the sea that has gouged the hole in this fence (\"silver waves marbled with spume \/ gashing a hole under the border fence\" [1]). Yet Anzald\u00faa's lack of respect for the border, coupled with her simultaneous awareness of its oppressive history, leads her to yearn for something different, and she begins to identify with the harsh, insistent pounding of the ocean. She begins to look\u2014as it were\u2014for other holes in the fence, for ruptures in the seemingly monolithic force field that would separate her from \"the other side.\" In this poem, and in Anzald\u00faa's theoretical project in general, mestiza queer identity comes out as one of those ruptures, and she repeatedly deploys this identity as a challenge to those who have created the border and who police it in order to maintain the division of \"us\" from \"them.\"\n\n\"I walk through the hole in the fence\" suggests nothing remarkable or dreamlike about crossing the border. Earlier in the poem, however, prior to the poet's crossing-over, there is at least one magical moment of sorts:\n\n_Miro el mar atacar_ [I watch the sea attack]\n\n_la cerca en_ [the fence in] Border Field Park\n\n_con sus buchones de agua_ [with its cleansing watery maw],\n\nan Easter Sunday resurrection\n\nof the brown blood in my veins. (2)\n\nIn this passage, the pounding of the sea\u2014though violent\u2014is far from negative, since it leads to a \"resurrection of the brown blood\" in the poet's veins. This resurrection, however, is different from the transcendence of identity experienced by Kerouac's protagonists; it actuates a renewed understanding of and appreciation for the poet's specific identity and history that contrast sharply with the hipsters' attempts to escape both. Furthermore, the poet conceives her mestiza identity here as something that defies the history of oppression that the border signifies, just as the powerful ocean gnaws away at the gritty wire fence.\n\nOne of the institutions that literally secures that fence, and the corresponding opposition between \"us\" and \"them,\" is the Immigration and Naturalization Service (INS), the branch of the U.S. Justice Department responsible for policing the border. Significantly, the INS has been exposed as one of the more discriminatory of government agencies; a group of black agents in Los Angeles filed a suit charging that the INS repeatedly denied them promotions and that people of color are extremely underrep-resented in positions of power within the agency. Joyce Jones reports that of approximately four thousand border patrol employees, only thirty-eight are black (18). Hence the current organization of the INS reproduces the \"us\" (white power structure)\/\"them\" (\"illegal\" brown or black \"others\") binary that Anzald\u00faa contests. If the department were less discriminatory presumably this division would be harder to uphold.\n\nHowever, neither in this poem nor elsewhere does Anzald\u00faa depict mestiza identity as that which _automatically_ disrupts institutions such as the INS. Instead, Anzald\u00faa's work represents a process, whereby the mestiza queer comprehends her multiple identities (mestiza, queer, \"outcast\") as results of unsuccessful attempts to divide people and transposes the meaning of those identities, turning them against ongoing attempts to maintain hierarchical divisions. Without such transpositions, mestizo\/a identity might trouble such institutions, but it will certainly not disrupt them.\n\nAnzald\u00faa makes this clear in her bilingual book for children, _Friends from the Other Side\/Amigos del otro lado_. Prietita is a young Chicana, living in Texas, who befriends an undocumented Mexican boy and his mother. When the border authorities search Prietita's town, one of these authorities is Chicano:\n\nFrom behind the curtains, Prietita and the herb woman watched the Border Patrol van cruise slowly up the street. It stopped in front of every house. While the white patrolman stayed in the van, the Chicano _migra_ got out and asked, \"Does anyone know of any illegals living in this area?\" Prietita and the herb woman saw a couple of people shake their heads and a few others withdraw into their houses.\n\nThey heard a woman say, \"Yes, I saw some over there,\" pointing to the _gringo_ side of town\u2014the white side. Everybody laughed, even the Chicano _migra_.\n\nDetr\u00e1s de las cortinas, Prietita y la se\u00f1ora miraban la camioneta de la migra pasar despacito por la calle. Se paraba frente a cada casa y uno de los dos agentes de la migra se bajaba. Casi todas las veces se bajaba el agente chicano para preguntar, \"\u00bfSaben ustedes d\u00f3nde se esconden los mojados?\" Algunos indicaban que no con la cabeza y otros se quedaban callados. Algunos se met\u00edan en sus casas.\n\nSiempre hab\u00eda una persona que contestaba, \"S\u00ed, v\u00ed a unos por ali\u00e1,\" apun-tando al lado del pueblo donde viv\u00edan los gringos. Todos se reian, hasta el agente chicano. (not paginated)\n\nThe Chicano's (uneasy?) laughter indicates that his presence complicates, but does not disrupt or disable, the power relations in this scene. The Chicano border guard still participates in the system that would mark as ''illegal\" and then deport Prietita's friend and his mother. Indeed, the Chicano character literally translates that system for the predominantly white, English-speaking agents, thereby securing the border patrol's ability to move freely. Another character, however, makes visible that which the Chicano border guard would mask: the \"illegal\" _gringo_ side of town. Hence disruption comes not from an essential mestiza\/o or Chicana\/o identity but from an identity committed to re-vision. Anzald\u00faa's new mestiza refuses to be kept in her place and instead spotlights the inconsistencies and elisions the border conceals.\n\nIn contrast to the Chicano border guard, who eventually gets back into his van and drives away with the other (white) officers, the speaker in the initial poem in _Borderlands\/La Frontera_ moves from the recognition of a history of oppression toward the strategic deployment of mestiza identity as that which challenges the attempt to maintain divisions between people. To illustrate this trajectory, I quote at length from the final part of Anzald\u00faa's poem:\n\n1,950 mile-long open wound\n\ndividing _a pueblo_ , a culture,  \nrunning down the length of my body,  \nstaking fence rods in my flesh,  \nsplits me splits me  \n _me raja me raja_\n\nThis is my home  \nthis thin edge of  \nbarbwire.\n\nBut the skin of the earth is seamless.\n\nThe sea cannot be fenced,  \n _el mar_ does not stop at borders.\n\nTo show the white man what she thought of his  \narrogance,  \n _Yemaya_ blew that wire fence down.\n\nThis land was Mexican once,  \nwas Indian always  \nand is.  \nAnd will be again.\n\n_Yo soy un puente tendido_  \n _del mundo gabacho al del mojado_ ,  \n _lo pasado me estird pa' 'tras_  \n _y lo presente pa' 'delante_.  \n _Que la Virgen de Guadalupe me cuide_  \n _Ay ay ay, soy mexicana de este lado_.\n\n[I am a bridge extending  \nfrom the world of the Spaniard\/European to that of the \"wetback,\"  \nThe past stretches me backward  \nand the present forward.\n\nThe Virgin of Guadalupe watches over me\n\nAy ay ay, I am a Mexican woman of this side.] (2\u20133)\n\nIn the first stanza of this passage, Anzald\u00faa foregrounds an ongoing history of oppression by reading the border as an open wound that divides a people, a culture, even individuals (\"splits me splits me \/ _me raja me raja\")_. But this history of oppression, inscribed onto the poet's very body, simultaneously secures the possibility\u2014indeed, the inevitability\u2014of resistance. Here and throughout _Borderlands I La Frontera_ , Anzald\u00faa aptly demonstrates that the attempt to divide absolutely \"us\" from \"them\" can never fully succeed, since it always produces groups of people who do not fit either category These groups find themselves in an in between position that could disrupt or destroy\u2014just as _Yemaya_ destroys the border fence\u2014the dualistic foundation on which binary thinking rests. Binary thinking casts the poet into an abject position here, but Anzald\u00faa works that abject, mestiza position against the very ideology that produces it.\n\nIn this poem, Anzald\u00faa reproduces a disruptive, in-between position in several ways. First, Anzald\u00faa writes bilingually or multilingually, switching codes without always bothering to translate. Diane Freedman points out that Anzald\u00faa's code switching signals for some privileged readers that there are \"borders they cannot cross\" (53); hence, even as Anzald\u00faa challenges the border patrol in the name of those whom the border would mark and separate, she implicitly insists that no one should have unlimited access to others' lives and histories. Throughout _Borderlands\/La Frontera_ , Anzald\u00faa employs not only Spanish and English but Chicano\/a and working-class variants of both, as well as an occasional word or phrase of Nahuatl. As she explains later in the text, _\"Deslenguadas. Somos los del espanol deficiente_ [Foul-mouthed, shameless. We speak the deficient Spanish]. We are your linguistic nightmare, your linguistic aberration, your linguistic _mestisaje_ [mongrels\/half-breeds], the subject of your _burla_ [joke]\" (58). Anzald\u00faa's switching of linguistic codes stands as the figure for a disruptive and vital mestiza\/o existence more generally: \"Chicano Spanish is a border tongue . . . _un nuevo lenguaje_ [a new language]. _Un lenguaje que corresponde a un modo de vivir_ [a language that corresponds to a way of life]. Chicano Spanish is not incorrect, it is a living language\" (55).\n\nSecond, Anzald\u00faa reproduces the in-between position in this poem by describing the land as simultaneously Mexican and Indian, despite attempts by Anglos (during the 139 years that have elapsed between the Treaty of Guadalupe-Hidalgo and the writing of this text) to confine Mexican identity to _el otro lado_ (the other side). The poet herself exclaims that she is a _\"mexicana de_ este _lado,\"_ in order to conclude the poem with literal evidence (again, inscribed on her body) that the divisions enacted by Anglo tyranny have not been ultimately successful.\n\nFinally, the poet deploys a cultural mestiza identity through her use of various mythical and religious figures, such as _la Virgen de Guadalupe_ (\"the single most potent religious, political and cultural image of the Chicano\/raexz _can_ 0\" [30]) and Yemaya, who is the Yoruban and Afro-Cuban goddess of the sea (Freixas, 27; Lorde, _Black Unicorn_ , 121\u201322). Later in the text, Yemaya is also identified as \"the wind,\" and Anzald\u00faa positions herself as the daughter of both Yemay\u00e1 and Oy\u00e1, \"the whirlwind\" (95n. 7). Both Anzald\u00faa and her text, then, embody this cultural mestiza identity\n\nSince she begins the poem with a metonymic linkage of woman and land (\"1,950 mile-long open wound . . . running down the length of my body\"), Anzald\u00faa also redeploys a masculinist and colonialist trope that would figure both (as in _On the Road)_ as passively waiting to be conquered. Anzald\u00faa links woman and land, however, not to reenact but to indict the colonial history of which such a trope is a part. Her indictment is sustained throughout the poem and can be heard in the poet's final anguished cry. Although _Ay ay ay_ can, on one level, be read as a simple lament, it is simultaneously a defiant, resistant cry. The poet's body, which had earlier been marked as divided, with an open wound running down its length, becomes in the end a bridge connecting past and present. The poem's trajectory works to transform mestiza identity from a casualty of oppression and division into a sign of resistance and connection.\n\nStill, Anzald\u00faa's bridge imagery in this conclusion is not simply celebratory, just as it was not in _This Bridge Called My Back_. That the conclusion can be read as a lament reminds readers that this transformation has come at some cost. Moreover, the image of a body connecting the \"Old\" and the \"New\" World recalls La Malinche, the Aztec woman whom some, such as Octavio Paz, have denigrated as La Chingada, \"the fucked one,\" who supposedly \"betrayed\" her people through her union with Hernan Cortes. Alfredo Mirande and Evangelina Enriquez dispute Paz and the others, emphasizing instead La Malinche's passivity:\n\nOne persistent indictment of La Malinche in folklore, concerning the fact that she was Cort\u00e9s's mistress or whore, implies active volition on her part, but this need not have been the case. Paz . . . suggests that Malinche's sexual transgression derives precisely from passive volition on her part, an openness or willingness to be violated. Whether active or passive, history does not ascribe volition to Malinche in becoming Cortes's mistress; she was presented to him as a slave along with nineteen other women when he arrived on the coast of Mexico. (24)\n\nLa Malinche thus passes from critic to critic and interpretation to interpretation in a bizarre parody of the transaction between Cortes and those who met him on the coast of Mexico.\n\nThrough her invocation of the La Malinche story, however, Anzald\u00faa intervenes in the myth on several levels, replacing captivity with freedom (\"The sea cannot be fenced\"), betrayal with steadfastness (\"This land was Mexican once, \/ was Indian always \/ and is\"), and passivity with agency (\"To show the white man what she thought of his arrogance, \/ _Yemaya_ blew that wire fence down\"). Most important, the poem's final line replaces the historical and critical objectification of La Malinche (and the nineteen other women) with a new subjectivity: _\"Ay ay ay, soy mexicana de este lado.\" Soy mexicana_ is already a subjective construction, but since the poem is both multilingual and constantly shifting between various linguistic codes, readers can hear (multiple) subjectivity in _Ay ay ay_ as well, since the phrase aurally recalls the English \"I I I.\" The end of Anzald\u00faa's poem thereby literalizes Sidonie Smith's argument: \"For Anzald\u00faa the topography of the borderland is simultaneously the suturing space of multiple oppressions and the potentially liberatory space through which to migrate toward a new subject position\" (200).\n\nAnzald\u00faa's first poem in _Borderlands I La Frontera_ , with its border crossing from one understanding of the borderlands and of mestiza identity to another, serves as a blueprint for the text as a whole. The desperate and oppressive \"borderlands\" that dominate \"The Homeland, Aztlan\/E\/ _otro Mexico,\"_ the first essay, have been transformed into _el camino de la mestiza_ (the mestiza way) in the final essay, _\"La conciencia de la mestiza\/_ Towards a New Consciousness.\" The six sections of _\"Un Agitado Vientol Eh\u00e9catl_ , The Wind\" that follow the final essay and that conclude _Borderlands\/La Frontera_ can then be read as poetic explorations\/performances of this new mestiza consciousness. Despite Anzald\u00faa's turn to the lyrical and poetic in the second half of her text, however, _Borderlands\/La Frontera_ contains no transcendent escape from identity and history a la Sal Paradise; rather, Anzald\u00faa's liminal and disruptive identity \"puts history through a sieve, winnows out the lies, looks at the forces that we as a race, as women, have been a part of\" (82). Over the course of the text, Anzald\u00faa's mestiza queer refocuses her (and her reader's) vision on the border conflicts that more celebratory travelers disavow.\n\nAfter the opening poem, which concludes with the defiant assertion that she is what Anglo-American binary thinking would like to disavow (a \"Mexican woman of _this_ side\"), Anzald\u00faa moves immediately into the borderlands. In the first essay, the region that is the borderlands remains a desperate and dangerous place, inhabited by the \"prohibited\" and \"forbidden\"; that is, by those whose identities are proscribed by dualistic thinking:\n\nThe U.S.-Mexican border _es una herida abierta_ [is an open wound] where the Third World grates against the first and bleeds. And before a scab forms it hemorrhages again, the lifeblood of two worlds merging to form a third country\u2014a border culture. Borders are set up to define the places that are safe and unsafe, to distinguish _us_ from _them. A_ border is a dividing line, a narrow strip along a steep edge. A borderland is a vague and undetermined place created by the emotional residue of an unnatural boundary. It is in a constant state of transition. The prohibited and forbidden are its inhabitants. _Los atravesados_ live here. . . . The only \"legitimate\" inhabitants are those in power, the whites and those who align themselves with whites. Tension grips the inhabitants of the borderlands like a virus. Ambivalence and unrest reside there and death is no stranger. (3\u20134)\n\nAnzald\u00faa's borderlands stand in stark contrast to the tension-free land that Kerouac's protagonists enter, where everything is pleasantly \"mysterious\" and even the authorities are \"lazy and tender\" ( _On the Road_ , 274). In fact, the passage immediately after this description of the borderlands in Anzald\u00faa's text highlights instead _conflict_ with the \"authorities.\" The passage consists of an anecdote about _la migra_ , who deport a character named Pedro to Guadalajara, despite the fact that Pedro is a fifth-generation American. In his terror, Pedro is unable to find the English words to explain his situation, and since he has not brought his birth certificate with him to work in the fields, the white authorities deport him ( _Borderlands\/La Frontera_ , 4).\n\nAnzald\u00faa's anecdote thus ironizes the Kerouac passage I examined earlier, in which \"these people were unmistakably Indians and were not at all like the Pedros and Panchos of silly civilized American lore\" ( _On the Road_ , 280). Although this passage criticizes Americans, Sal and his friends remain blind both to the ways in which their exoticization and primitiviza-tion of \"these people\" are implicated in \"silly civilized American lore\" and to the ways in which they themselves carry \"America\" with them. The effacement of an ongoing Anglo-American history of exploitation of Mexico inevitably effaces the material conditions that actual Pedros (and Panchos) face. Not surprising, this is not the last complication of celebratory border-crossing tropes that Anzald\u00faa effects. In _On the Road_ , Kerouac's hipsters are flabbergasted by the \"great stacks of pesos\" they receive in exchange for American dollars (275). Anzald\u00faa's essay in _Borderlands\/La Frontera_ reveals how such North-to-South border crossings are part of a larger exploitative economy:\n\n_Los gringos_ had not stopped at the border. . .. Currently, Mexico and her eighty million citizens are almost completely dependent on the U.S. market. The Mexican government and wealthy growers are in partnership with such American conglomerates as American Motors, IT&T and Du Pont which own factories called _maquiladoras_. One-fourth of all Mexicans work at _maquiladoras;_ most are young women. . . . The devaluation of the _peso_ and Mexico's dependency on the U.S. have brought on what the Mexicans call _la crisis. No hay trabajo_ [There is no work]. Half of the Mexican people are unemployed. In the U.S. a man or woman can make eight times what they can in Mexico. By March, 1987, 1,088 pesos were worth one U.S. dollar. I remember when I was growing up in Texas how we'd cross the border at Reynosa or Progreso to buy sugar or medicine when the dollar was worth eight _pesos_ and fifty _centavos_. (10)\n\nIn contrast to those moving North to South, those making the migration from South to North do so out of material necessity, brought on by the position of economic dependency into which the United States has cast Mexico. When these travelers cross the border, moreover, they do not receive great stacks of either pesos or dollars.\n\nThe bulk of \"The Homeland, Aztl\u00e1n _Elotro M\u00e9xico\"_ combines history, mythology, philosophy, and poetry in order to document the existence of those who reside in the borderlands. The title refers to Aztlan, the region of the southwestern United States that is the mythological homeland of Chicanos\/as. At the same time, Anzald\u00faa's theory of the borderlands is about other, less literally geographic borderlands; those who live in Anzald\u00faa's borderlands, to return to the quote with which I began this chapter, are \"the squint-eyed, the perverse, the queer, the troublesome, the mongrel, the mulato, the half-breed, the half dead; in short, those who cross over, pass over, or go through the confines of the 'normal'\" (3). This large cast of characters notwithstanding, Anzald\u00faa concludes \"The Homeland Aztl\u00e1n\/ _Elotro M\u00e9xico\"_ by focusing on how the U.S.-Mexican border and the borderlands provoke danger for one very specific figure:\n\n_La mojada, la mujer indocumentada_ [the female \"wetback,\" the undocumented woman], is doubly threatened in this country. Not only does she have to contend with sexual violence, but like all women, she is prey to a sense of physical helplessness. As a refugee, she leaves the familiar and safe homeground to venture into unknown and possibly dangerous terrain.\n\nThis is her home  \nthis thin edge of  \nbarbwire. (12\u201313)\n\nThe poetic echo at the end of the essay is from the poem that opens _Borderlands\/La Frontera_ , in which Anzald\u00faa asserted, \"This is my home \/ this thin edge of \/ barbwire\" (3). Although the experience of the borderlands extends beyond specific figures and geographic locations, Anzald\u00faa's pronominal shift as she returns to the language of her earlier poem (\"This is _my_ home .. . This is _her_ home\") simultaneously grounds borderlands existence materially in the endangered figure of _la mujer indocumentada_.\n\nThe end of \"The Homeland, Aztl\u00e1n _El otro M\u00e9xico_ ,\" with these images of tension and danger, places the essay firmly within the first part of the trajectory I have been sketching out: Anzald\u00faa demonstrates the ways in which subjects are cast into abject positions as a result of binary thinking and how identities emerge as casualties of oppression, but she does not yet indicate fully whether such locations might be transformed into sites of resistance. In the second essay, _\"Movimientos de rebeld\u00eda y las culturas que traicionan_ \"(Movements of rebellion and cultures of betrayal), however, that possibility once again emerges: \"There in front of us is the crossroads and choice: to feel a victim where someone else is in control .. . or to feel strong, and, for the most part, in control. My Chicana identity is grounded in the Indian woman's history of resistance\" (21). By the final essay, \"La _conciencia de la mestiza\/To_ wards a New Consciousness,\" the transformation that is initiated in the early essays (and that is presented in microcosm in Anzald\u00faa's first poem) is complete.\n\nAnzald\u00faa returns to material conditions such as those experienced by _la mujer indocumentada_ throughout _Borderlands\/La Frontera_. Hence the transformation represented in the text is not one that smooths over or escapes from conflict but rather one that locates and engages conflict. Anzald\u00faa's border crossings do not perpetuate but instead bring to light the effacement of identity and history that patriarchal, Anglo-American power encourages. _Borderlands I La Frontera_ also discourages purely celebratory appropriations (by theoretical hipsters) of borderlands existence, since such appropriations would universalize away specific experiences. Anzald\u00faa's pronominal shift, in fact, reverses the course of appropriation. Instead of taking what is \"yours\" and making it \"mine,\" Anzald\u00faa moves in the opposite direction: \"This is my home ... This is her home.\" This is not to say by any means, that Anzald\u00faa discourages critical dialogue or coalition; it is simply to say that Anzald\u00faa's text in many ways resists what Annamarie Jagose calls \"utopic undifferentiation\" (152) and, indeed, insists that critical dialogue proceed by specifying all the players in any given border conflict.\n\nJagose reads Anzald\u00faa's _liminal_ position as a _transcendent_ one and mistakes the critical mobility of the mestiza for an absolute mobility that would escape conflict. Jagose insists, _\"Borderlands_ installs and valorizes the _mestiza_ as a figure whose traversals of the border between inside and outside ensure only a utopic undifferentiation, merging and indistinguish-ability\" (152). Although I would not deny that _Borderlands\/La Frontera_ contains utopic moments, Jagose's argument necessarily discounts Anzald\u00faa's persistent attention to the material conditions of borderlands existence. Jagose's \"only\" purifies _Borderlands\/La Frontera_ of its perverse insistence on specificity. Jagose argues that the text \"ultimately represses\" its understanding of the border and the mestiza \"as the paradoxical site of undifferentiation and distinction\" (157); _only_ \"utopic undifferentiation\" remains in the end. According to Jagose's \"ultimate\" logic, Anzald\u00faa's concluding essay should valorize _nothing but_ \"merging and indistinguish-ability.\"\n\nIn actuality, even in the final pages of the essay, the mestiza advances by differentiating and distinguishing the particular histories and conflicts the border cites:\n\nThis land has survived possession and ill-use by five countries: Spain, Mexico, the Republic of Texas, the U.S., the Confederacy, and the U.S. again. It has survived Anglo-Mexican blood feuds, lynchings, burnings, rapes, pillage.\n\nToday I see the Valley still struggling to survive... . The borderlands depression that was set off by the 1982 peso devaluation in Mexico resulted in the closure of hundreds of Valley businesses. Many people lost their homes, cars, land. Prior to 1982, U.S. store owners thrived on retail sales to Mexicans who came across the border for groceries and clothes and appliances. While goods on the U.S. side have become 10, 100, 1000 times more expensive for Mexican buyers, goods on the Mexican side have become 10, 100, 1000 times cheaper for Americans. Because the Valley is heavily dependent on agriculture and Mexican retail trade, it has the highest unemployment rates along the entire border region; it is the Valley that has been hardest hit. (90)\n\n\"Spain, Mexico, the Republic of Texas, the U.S., the Confederacy\"\u2014these signs mark specific restructurings of the borderlands. Furthermore, each restructuring repositions, and even multiplies, the players in this border conflict\u2014players whom Anzald\u00faa's account, in turn, renders _distinguishable:_ the homeless, the unemployed, U.S. store owners, Mexican buyers, American shoppers. As this passage indicates, then, liminality might be understood as the opposite of transcendence: the mestiza sees more clearly not because she is \"above it all\" but because she is caught between colliding cultures.\n\nThus, in _\"La conciencia de la mestiza_ \/Towards a New Consciousness,\" Anzald\u00faa participates in the Chicano\/a critical tradition in which the border marks not a site of transcendence but a geographical and metaphorical site of struggle. The \"new consciousness\" that the text enacts, however, encourages the mestiza to see herself as enabled, not immobilized, by the struggle. The final essay makes the transformation explicit:\n\nThe new _mestiza_ copes by developing a tolerance for contradictions, a tolerance for ambiguity. She learns to be an Indian in Mexican culture, to be Mexican from an Anglo point of view. She learns to juggle cultures. She has a plural personality, she operates in a pluralistic mode\u2014nothing is thrust out, the good, the bad and the ugly, nothing rejected, nothing abandoned. Not only does she sustain the contradictions, she turns the ambivalence into something else. (79)\n\nAnzald\u00faa's \"something else\" is akin to the \"something else\" enacted by various texts in the Queer Renaissance: like Audre Lorde's Zami, Anzald\u00faa's mestiza queer understands identity through collectivity; like Randall Kenan's queer trickster, she is determined to disrupt and transform both a specific locale and the dualistic, hierarchical systems of power that oppress her. In one of Anzald\u00faa's earlier essays, included in _This Bridge Called My Back_ , the collectivity-transformation equation is made explicit:\n\nWe are the queer groups, the people that don't belong anywhere, not in the dominant world nor completely within our own respective cultures. Combined we cover so many oppressions. But the overwhelming oppression is the collective fact that we do not fit, and because we do not fit _we are a threat_. In El Mundo Zurdo [the Left-Handed World] I with my affinities and my people with theirs can live together and transform the planet. (\"La Prieta,\" 209)\n\nSix years later, in _Borderlands\/La Frontera_ , this disruptive figure emerges once again, more fully developed: \"Let us hope that the left hand, that of darkness, of femaleness, of 'primitiveness,' can divert the indifferent, right-handed, 'rational' suicidal drive that, unchecked, could blow us into acid rain in a fraction of a millisecond\" (69). Over the course of the text, the mestiza queer moves from being a casualty of dualistic thinking to being an agent of transformation: _\"Su cuerpo es una bocacalle_ [Her body is an intersection]. _La mestiza_ has gone from being the sacrificial goat to becoming the officiating priestess at the crossroads\" (80). Earlier, \"life in the borderlands\" had been characterized by paralysis: \"Petrified, she can't respond, her face caught between _los intersticios_ , the spaces between the different worlds she inhabits\" (20). In contrast, in the final essay, _la mestiza_ uses the seemingly interminable condition of being \"caught between\" identities and cultures to actively mediate with and contest the various cultures and identities that come together at the crossroads.\n\nThe terms _mestiza_ and _queer_ function similarly in Anzald\u00faa's work. Although she never obscures the specificity of oppressions that people experience (say, \"homosexuals and others who deviate from the sexual common\" [18], on the one hand, and people of color, such as _la mujer indocumentada_ , on the other), Anzald\u00faa nonetheless sees \"queer\" and \"mestiza\" identity coming together in similar liminal and disruptive ways. Both mestizas and queers, after all, are among _los atravesados_ ,\"those who cross over.\" In \"La _conciencia de la mestiza\/To_ wards a New Consciousness,\" Anzald\u00faa explicitly probes how these identities inform each other, and in her work after _Borderlands\/La Frontera_ , she continues, insistently, to link them.\n\nOne of the first ways in which Anzald\u00faa brings these two identities together is in their formation: both result from attempts to identify and police deviant \"others.\" Judith Raiskin explains how the conflation of a marked sexual and racial \"deviance\" occurred historically and argues that Anzald\u00faa rewrites that legacy:\n\nCategories of sexual behavior and identity created by nineteenth-and twentieth-century sexologists were also influenced by the classification systems of race, whereby people of color, particularly \"mixed race\" people, and homosexuals were conflated through the ideas of evolution and degeneration in the late nineteenth century. . . . Gloria Anzald\u00faa allows homosexuality and mixed-race identity to reflect each other by using and reworking the language of nineteenth-century evolutionary theory. (157; 159)\n\nIn the queer context, the \"marking\" of \"degeneration\" is most evident in \"La Prieta\" (The dark-skinned one), the essay from _This Bridge Called My Back_ that ends with the defiant assertion \"We are the queer groups ... and because we do not fit _we are a threat\"_ (209). In that essay, Anzald\u00faa subjects queerness to a transformation similar to the one she will later perform on mestiza identity in _Borderlands\/La Frontera_. In contrast to the defiant assertion with which the essay ends, \"La Prieta\" begins with an exclusionary marking. After explaining that her mother \"fucked before the wedding ceremony\" (199), Anzald\u00faa writes that her mother felt \"guilt at having borne a child who was marked 'con la se\u00f1a [with the sin],' thinking she had made me a victim of her sin. In her eyes and in the eyes of others I saw myself reflected as 'strange,' 'abnormal,' 'QUEER.' I saw no other reflection. Helpless to change that image, I retreated into books and solitude and kept away from others\" (199). Like Horace in Kenan's _A Visitation of Spirits_ , Anzald\u00faa has looked in various \"mirrors\" and seen only inadequacy. At the end of \"La Prieta,\" however, Anzald\u00faa moves from a debilitating solitude to an enabling collectivity, as \"we . . . the queer groups\" turn the reflection\/(mis)recognition against the glassmakers, whose ideology of \"normalcy\" requires and produces the \"abnormal\" and the \"strange.\"\n\nHence, and more important, Anzald\u00faa links mestiza and queer identities in terms of their function: to unmask Anglo, patriarchal, and heterosexual systems of power, which rest on binaristic foundations and which declare that some people \"fit\" while others do not. Nowhere, perhaps, is this similarity of purpose more clear than in \"La _conciencia de la mestiza\/_ Towards a New Consciousness,\" where mestiza identity is bolstered by queerness and vice versa. There, immediately after situating _la mestiza_ as the \"officiating priestess at the crossroads,\" Anzald\u00faa links mestiza identity to queerness. Furthermore, in this passage Anzald\u00faa reproduces even in her syntax the movement away from an understanding of mestiza and queer identity as individual and abject and toward an understanding of both as collective and transformative:\n\nAs a _mestiza_ I have no country, my homeland cast me out, yet all countries are mine because I am every woman's sister or potential lover. (As a lesbian I have no race, my own people disclaim me; but I am all races because there is the queer of me in all races.) . . . _Soy un amasamiento_ , I am an act of kneading, of uniting and joining that not only has produced both a creature of darkness and a creature of light, but also a creature that questions the definitions of light and dark and gives them new meanings. (80\u201381)\n\nFrom being outcast and disclaimed to being at the center of \"all countries ... all races\"\u2014Anzald\u00faa's syntax \"gives new meaning\" to \"deviant\" and individual mestiza\/queer identities, reinventing them as collective and ensuring thereby that echoes of _multiple_ subjectivity be heard around her ( _Yo) soy_ construction.\n\nAs the essay continues, the mediating, transformative role Anzald\u00faa assigns to _la mestiza_ is clearly delegated to queers as well:\n\nBeing the supreme crossers of cultures, homosexuals have strong bonds with the queer white, Black, Asian, Native American, Latino, and with the queer in Italy, Australia and the rest of the planet. Our role is to link people with each other.. . . People, listen to what your _joteria_ [queer contingent] is saying.\n\nThe mestizo and the queer exist at this time and point on the evolutionary continuum for a purpose. We are a blending that proves that all blood is intricately woven together, and that we are spawned out of similar souls. (84\u201385)\n\nSignificantly, the section heading that immediately follows this passage declares, _\"Somos una gente_ [We are a people]\" (85), emphasizing a connected collectivity that is then underscored by the epigraph Anzald\u00faa employs for the ensuing section, which she takes from Gina Valdes's _Puentes y Fronteras: Coplas Chicanas_ (Bridges and borders: Chicana ballads):\n\n_Hay tantisimas fronteras_\n\n_que dividen a la gente_ ,\n\n_pero por cada frontera_\n\n_existe tambi\u00e9n un puente_.\n\n[There are many, many borders\n\nwhich divide people, but for every border\n\nthere exists also a bridge.]\n\n( _Borderlands\/La Frontera_ , 85)\n\n\"What your _joter\u00eda_ is saying,\" then, is that attempts to separate absolutely \"us\" from \"them\" will fail, since bridges are always already conjoined to borders. The border crossing celebrated at the end of Anzald\u00faa's essay section certainly engages conflict and is still only the beginning of a process, the results of which cannot be wholly predicted in advance. \"Your _joter\u00eda,\"_ however, boldly envisions a transformed, less hierarchical world: \"A massive uprooting of dualistic thinking in the individual and collective consciousness is the beginning of a long struggle, but one that could, in our best hopes, bring us to the end of rape, of violence, of war\" (80).\n\nAnzald\u00faa's reference to rape, violence, and war suggests that her mestiza queer comprehends the dangers of nationalism. Moreover, perhaps because of those dangers, the mestiza queer does not seek a hip Queer or Chicano \"Nation\" that would supplant, however metaphorically, the existing nations she criticizes. In this respect, Anzald\u00faa's linkage of queer and mestiza identity differs from Cherr\u00ede Moraga's linkage of queer and Chicana identity in \"Queer Aztlan: The Re-formation of Chicano Tribe\" ( _Last Generation_ , 145\u201374). In my mind, Anzald\u00faa's mestiza queer has more theoretical promise than Moraga's Queer Aztlan, which Moraga describes as a \"Chicano homeland that could embrace _all_ its people, including its joteria\" (147). Moraga's theory, like Anzald\u00faa's, affirms identification across difference (\"Chicana lesbians and gay men ... seek a nation strong enough to embrace a full range of racial diversities, human sexualities, and expressions of gender\" [164]), but it nonetheless ends up locating (or \"fixing\") another \"nation,\" whereas Anzald\u00faa's theory, in the end, more rigorously queries both nations and nationalisms. ( _Borderlands I La Frontera_ , remember, moves _from_ \"The Homeland, Aztlan,\" and _to_ a new and flexible mestiza consciousness.)\n\nAlthough Anzald\u00faa understands where more nationalistic theories are coming from, she nonetheless is fairly consistent, in _Borderlands\/La Frontera_ and elsewhere, in her advocacy for more connections outside of oneself or one's groups:\n\nA bridge excludes racial separatism. So the concept has taken a beating recently because of the reactionary times we're going through and the upsurge in racism and white supremacy. But I can see that in the '90s a rainbow _serpent_ bridge composed of the new mestizas\/os, bi- and multiracial queer people who are mixed and politicized will rise up and become important voices in our gay, ethnic and other communities. (\"To(o) Queer,\" 260\u201361)\n\nIn this passage, the new mestiza again brings to light contradiction and conflict (racial separatism, reactionary times, racism, white supremacy),\n\nbut she simultaneously envisions the impossible \"something else\": a rainbow bridge, a bridge without borders.\n\nAfter mestiza queer identity emerges as a disruptive, transformative force at the end of the essay section of _Borderlands\/La Frontera_ , Anzald\u00faa celebrates\/performs this critical \"new consciousness\" in _\"Un Agitado Viento\/Ehecatl_ , The Wind,\" the poetry section that comprises the second half of that text. The poems in this section reproduce the liminal linguistic position Anzald\u00faa introduces in the essays: some are written entirely in English, some in Spanish, some in a combination of the two. Some of the Spanish poems are followed by English translations; some are not. Once again, the liminality of these poems is also not simply linguistic; Anzald\u00faa forges a cultural mestiza identity by continuing to mix various myths and traditions (Spanish, Mexican, Native American). Finally, as with the Lorde poems I examined in my first chapter, the poetic voice itself often represents a coming together of various voices. At times, it is ambiguous whether the speaker is Anzald\u00faa, the women she loves, or a larger group of oppositional voices.\n\nIn _\"Arriba mi gente\"_ (Rise up, my people), for example, Anzald\u00faa performs the mestiza queer consciousness that she advocated in her essays:\n\nChorus: _Arriba mi gente_ [Rise up, my people],\n\n_toda gente arriba_ [all people, rise up].\n\nIn spirit as one,\n\nall people arising\n\n_Toda la gente junta_ [all the people united]\n\n_en busca del Mundo Zurdo_ [in search of the Left-\n\nHanded World]\n\n_en busca del Mundo Zurdo_. (192)\n\nThe chorus to this poem is mestiza in its mixture of Spanish and English and queer in its invocation of _el Mundo Zurdo_ , which Anzald\u00faa previously identified as the world of \"the queer groups, the people that don't belong anywhere\" (\"La Prieta,\" 209). _El Mundo Zurdo_ is also the place where transformation is possible, and here again, queer mestiza identity and _el Mundo Zurdo_ are linked to collectivity and transformation: the poem ends with the imperative _\"Arriba, despierta mi gente_ [Rise up, awake my people] \/ _a liberar los pueblos_ [to liberate the masses] \/ _Arriba mi gente, despierta_ [Rise up my people, awake]\" (193). The dedication to the poem\u2014 _\"para_ Tirsa Quinones who wrote the music and Cherr\u00ede Moraga who sang it\" (192)\u2014underscores that this is a performance; mestiza queer identity is literally \"staged\" here and is at once individual and collective: Anzald\u00faa, Qui\u00f1ones, and Moraga come together as one mestiza queer voice. In this context also, \"chorus\" might be read as \"a group of people singing together\" or as what Lorde envisions in _Need_ as \"a chorale of women's voices\" (7), rather than simply as \"a refrain,\" so that the collective identity extends even beyond those Anzald\u00faa explicitly names.\n\nHence, in _Borderlands\/La Frontera_ , Anzald\u00faa\u2014\"trapped\" by identities (\"queer,\" \"mestiza\") that are not supposed to fit anywhere\u2014works those identities against the systemic forces that produce them as \"other,\" \"deviant.\" In the process, mestiza queer identity moves from being a site of individual abjection to one of collective transformation. The transformation mestiza queer identity advances is a transformation of hierarchies that perpetuate domination of individuals and groups on the basis of race, class, gender, and sexuality: \"mestiza\" speaks to and for those subordinated because of their race, class, and gender; \"queer,\" to those subordinated because of their sexuality, gender, and\u2014because Anzald\u00faa understands it as a \"working-class word\"\u2014class (\"To(o) Queer,\" 250). Even this schema-tization, however, is ultimately too easy, too neat. The mestiza queer as Anzald\u00faa deploys her moves away from easy categorizations, seeking instead to understand, for instance, how race complicates gendered issues or how class confounds simplistic, unexamined assumptions about sexuality. Anzald\u00faa contests the demarcation of borders intended to divide people neatly, highlighting instead the overlapping margins that belie such attempts to divide absolutely.\n\nIn her work after _Borderlands\/La Frontera_ , Anzald\u00faa continues to reinvent mestiza\/queer identity. In \"Bridge, Drawbridge, Sandbar or Island: _Lesbians-of-Color Hacienda Alianzas_ [Forging Alliances],\" she writes, \"The mestiza queer is mobile, constantly on the move, a traveler, _callejera_ [gadabout], a _cortacalles_ [jaywalker]. Moving at the blink of an eye, from one space, one world to another, each world with its own peculiar and distinct inhabitants, not comfortable in anyone of them, none of them 'home\/yet none of them 'not home' either\" (218). The mestiza queer \"mobility\" detailed here is quite different from various white, male, and heterosexual mobilities. In contrast to the extreme comfort Kerouac's boys feel among the \"peculiar and distinct inhabitants\" of Mexico (inhabitants who, despite their peculiarity, are nonetheless recognizably hip, as far as Sal and the boys are concerned), the mestiza queer feels discomfort everywhere she goes. There is no magical end of the road for this gadabout as she drifts from place to place, and her travels and crossings are, like jaywalking, usually unsanctioned by, and in opposition to, \"the law.\" Furthermore, as Anzald\u00faa's essay continues, it becomes clear that this traveler's goal is to identify those who claim unlimited access to her life and experience: those who attempt to \"fix\" her, to mark her as \"other,\" to name and dismiss her experience. She calls her family and Chicana\/o communities generally to task for their \"heterosexist bullshit\" and anti-feminism (218); she critiques white lesbians for their \"exclusionary or racist remarks\" (218); she feels empathy with men of color, \"only to be saddened that they need ... to be educated about women-only space\" (219).\n\nIn short, Anzald\u00faa counters the unchecked mobility of others with a mobility intent on unmasking the hierarchies and exclusions they perpetuate. This critical mobility animates a politics of alliance:\n\nIn alliance we are confronted with the problem of how we share or don't share space, how we can position ourselves with individuals or groups who are different from and at odds with each other, how can we reconcile one's love for diverse groups when members of these groups do not love each other, cannot relate to each other, and don't know how to work together. (219)\n\nAs \"Bridge, Drawbridge, Sandbar or Island\" continues, Anzald\u00faa emphasizes (literally) the importance of identifying and countering oppressive relations of power: _\"coalition work attempts to balance power relations and undermine and subvert the system of domination-sub ordination_ that affects even our most unconscious thoughts\" (224\u201325).\n\nIn \"To(o) Queer the Writer\u2014Loca, escritora y chicana [Crazy woman, writer, and Chicana],\" Anzald\u00faa again brings together queer and mestiza identities, asserting, \"The new mestiza queers have the ability, the flexibility, the malleability, the amorphous quality of being able to stretch this way and that way. We can add new labels, names and identities as we mix with others\" (279). This malleability and amorphism, used in the service of unmasking the operations of domination-subordination, is not unlike the image with which Anzald\u00faa closes the essay section of _Borderlands\/La Frontera_. Although the earlier image in _Borderlands\/La Frontera_ is not so much a \"working\" of identity as a more literal working of the land, both images emerge from the desire for a radically different world:\n\nThe Chicano and Chicana have always taken care of growing things and the land. ... Growth, death, decay, birth. The soil prepared again and again, impregnated, worked on. A constant changing of forms, _renacimientos de la tierra madre_ [renascences of Mother Earth].\n\nThis land was Mexican once  \nwas Indian always  \nand is.\n\nAnd will be again. ( _Borderlands\/La Frontera_ , 91)\n\nSignificantly, Anzald\u00faa's _renacimientos_ here intersect with the understanding of \"renaissances\" I am advancing as a critical context for all the writers of the Queer Renaissance. Anzald\u00faa does not seek an escape from history and location; instead, her _renacimientos_ , with their \"constant changing of forms,\" seek a re-vision of history, identity, and location. The reemergence of the refrain from Anzald\u00faa's earliest poem in _Borderlands\/La Frontera_ (\"This land was Mexican once \/ was Indian always \/ and is\") underscores her connection of these _renacimientos_ to alternative histories of resistance. Moreover, because of the transformation Anzald\u00faa puts into play over the course of the text, the refrain at this point is even more of a defiant assertion than it was in the earlier essay. It is, literally and symbolically, the last word of Anzald\u00faa's essay section. Because of Chicano\/a, mestizo\/a, and queer flexibility and malleability, Anzald\u00faa is able to shape life in the borderlands, where formerly \"ambivalence and unrest\" resided and death was \"no stranger\" (4), into something vibrant and triumphant, like the plants that emerge out of the dead ground.\n\n### **Mestiza Queer Agents**\n\nThe note of triumph sounds most clearly in \"La _conciencia de la mestizal_ Towards a New Consciousness,\" but it is foreshadowed as early as the preface to _Borderlands I La Frontera_ , where Anzald\u00faa asserts, \"There is an exhilaration in being a participant in the further evolution of humankind, in being 'worked' on. I have the sense that certain 'faculties'\u2014not just in me but in every border resident, colored or non-colored\u2014and dormant areas of consciousness are being activated, awakened. Strange, huh?\" (preface, no page number). Over the course of _Borderlands I La Frontera_ , Anzald\u00faa establishes that the evolutionary process is ongoing. The \"exhilaration\" she feels is not transcendence; \"work\" here, as the image of working the land much later in the text suggests, entails a grounding in location and history. Moreover, the mestiza queer's own work is not done once she has shed light on the border guards. Instead, the \"faculties being activated\" within the mestiza queer provide her with the mobility necessary to shed light on all the players and overlapping locations in this border struggle. Anzald\u00faa encourages those in all the geographical and theoretical communities through which she moves to consider the tensions and contradictions they embody and the ways in which their locations overlap with other locations, whether privileged or marginalized. Thus, Anzald\u00faa even queers \"queer\" if she sees the concept interpellated as a fad and appropriated by theoretical hipsters seeking escape from identity and history.\n\nThe work of Judith Butler can help both in considering how such a queering moves forward and in complicating questions of agency in _Borderlands\/La Frontera_. As the invocation of evolution should confirm, the mestiza queer's movement is not simplistically voluntaristic; instead, the queer agency Anzald\u00faa theorizes (where certain unexpected \"faculties\" are \"awakened\") exemplifies Butler's insistence that the subject \"is always the nexus, the non-space of cultural collision, in which the demand to resignify or repeat the very terms which constitute the 'we' cannot be summarily refused, but neither can they be followed in strict obedience\" (Butler, _Bodies That Matter_ , 124). Butler's understanding of performativity as _\"working the weakness in the norm\"_ (237), in turn, glosses Anzald\u00faa's own performance of mestiza queer identity as that which is produced but never fully contained by the border or border guards.\n\nIn a parody of the ways in which some have understood her notion of \"gender performativity\" from _Gender Trouble_ , Butler writes in _Bodies That Matter: On the Discursive Limits of \"Sex,\"_ \"If I were to argue that genders are performative, that could mean that I thought that one woke in the morning, perused the closet or some more open space for the gender of choice, donned that gender for the day, and then restored the garment to its place at night\" (x). Distancing herself from such voluntarism, Butler explores in _Bodies That Matter_ the possibility of a \"repetition that fails to repeat loyally\" the identities into which one has already been cast (220). If binaristic systems of power demand that some embody, over and over, abject identity positions (say, \"mestiza,\" \"queer\"), those same abjecting systems also provide \"the occasion to work the mobilizing power of injury, of an interpellation one never chose\" (123). Using recent theoretical and political citations of \"queer\" as an example, Butler explains:\n\nParadoxically, but also with great promise, the subject who is \"queered\" into public discourse through homophobic interpellations of various kinds _takes_ _up_ or _cites_ that very term as the discursive basis for an opposition. This kind of citation will emerge as _theatrical_ to the extent that it _mimes and renders hyperbolic_ the discursive convention that it also _reverses_. The hyperbolic gesture is crucial to the exposure of the homophobic \"law\" that can no longer control the terms of its own abjecting strategies. (232)\n\nLikewise, Anzald\u00faa, in the process of \"citing\" the mestiza identity she did not choose and that she experiences as an open wound at the beginning of _Borderlands\/La Frontera_ , renders hyperbolic and reverses the conventions of an Anglo-American discursive regime that would subordinate mestizas\/os, \"keeping them in their place\" while reserving unchecked mobility for itself. Anzald\u00faa's \"disloyal repetition\" of mestiza queer identity both figures the border as always already transgressed by those who were supposed to be immobilized by it and turns the searchlight on those who fixed the border in the first place.\n\nRepetitions can be \"disloyal,\" however, not only to dualistic ideologies that would compel mestizas, queers, and others to embody abjection but also to \"alternative\" discursive regimes that would compel them to embody \"hipness.\" Anzald\u00faa writes:\n\nIf I have to pick an identity label in the English language, I pick \"dyke\" or \"queer,\" though these working-class words (formerly having \"sick\" connotations) have been taken over by white middle-class lesbians in the academy. Queer is used as a false unifying umbrella which all \"queers\" of all races, ethnicities and classes are shoved under. At times we need this umbrella to solidify our ranks against outsiders. But even when we seek shelter under it we must not forget that it homogenizes, erases our differences. (\"To(o) Queer,\" 250).\n\nIn this passage, Anzald\u00faa repeats with a difference even her own previous endorsement of queerness; \"queers\" here are no longer simply \"the supreme crossers of cultures\" who have bonds with \"the rest of the planet.\" Still, despite the shift, Anzald\u00faa's cautions here reiterate one of the main tenets of her theory: what is needed is not an effacement of identity and history but a grappling with specific identities and histories, particularly histories of oppression and exploitation. The mestiza queer theory Anzald\u00faa advocates refuses to let dominant identities travel unmarked under the banner of a unifying hipness\u2014in this case, an overly celebratory, transcendent, and unmarked white queerness that elides its contiguity to privilege and disavows any policing role it might continue to play in and around the borderlands. Queerness may provide a useful bridge, but as always for Anzald\u00faa, a bridge is never _in and of itself_ cause for celebration. She writes elsewhere that \"being a bridge . . . may mean a partial loss of self. Being 'there' for people _all the time, mediating all the time_ means risking being 'walked' on, being 'used.' I and my publishing credentials are often 'used' to 'colorize' white women's grant proposals, projects, lecture series, and conferences. If I don't cooperate I am letting the whole feminist movement down\" (\"Bridge, Drawbridge,\" 223). As with feminism, so too with queer theory: even (or perhaps particularly) beneath the sign of queerness, whiteness must be marked and white privilege challenged. When white middle-class lesbians and gay men \"frame the terms of debate,\" Anzald\u00faa suggests, \"they police the queer person of color with theory . . . Their theories limit the ways we think about being queer\" (\"To(o) Queer,\" 251).\n\nAnzald\u00faa's cautions here, in her comments and in the title of this later essay (\"To(o) Queer the Writer\"), signal a certain retreat from the identity \"queer.\" Yet this is the essay that includes the bold assertion I have already quoted: \"The new mestiza queers have the ability, the flexibility, the malleability, the amorphous quality of being able to stretch this way and that way\" (249). Anzald\u00faa's retreat, then, is not from queerness per se but from monolithic and overly celebratory understandings of it. Indeed, Anzald\u00faa's title obviously anticipates this double move: \"to queer\" calls readers and writers to action at the same time that \"too queer\" cautions them. Anzald\u00faa shapes\u2014as she has since _This Bridge Called My Back_ , long before queerness was hip\u2014what Butler might call a _\"critical_ queerness\" ( _Bodies That Matter_ , 223): a queerness that is both critically necessary and critical of itself; a queerness that insists on foregrounding, and hence undermining, the ways in which identities (even, potentially, other queer identities) are structured in domination so that certain histories of exploitation or appropriation are effaced. As Butler explains:\n\nAs expansive as the term \"queer\" is meant to be, it is used in ways that enforce a set of overlapping divisions: in some contexts, the term appeals to a younger generation who want to resist the more institutionalized and reformist politics sometimes signified by \"lesbian and gay\"; in some contexts, sometimes the same, it has marked a predominantly white movement that has not fully addressed the way in which \"queer\" plays\u2014or fails to play\u2014within non-white communities; and whereas in some instances it has mobilized a lesbian activism, in others the term represents a false unity of women and men. Indeed, it may be that the critique of the term will initiate a resurgence of both feminist and anti-racist mobilization within lesbian and gay politics or open up new possibilities for coalitional alliances that do not presume that these constituencies are radically distinct from one another. ( _Bodies That Matter_ , 228\u201329)\n\nAnzald\u00faa's titular use of the phrase \"to(o) queer\" is a critical redeployment of the term precisely in the service of the \"feminist and anti-racist mobilization\" Butler envisions here. Butler's call for attention to how _queer_ may or may not play in nonwhite communities, in turn, is a recognition, \u00e0 la Anzald\u00faa, that it should not be solely white middle-class lesbians and gay men (such as Butler or myself) who \"frame the terms of the debate.\"\n\nAnzald\u00faa and Butler come together at this point not only in their critical understanding of queerness but also in their emphasis on coalition. Just as mestiza queer agency is not simplistically voluntaristic, neither is it simply individual. Indeed, as with Zami in my first chapter, mestiza queer identity is not simplistically voluntaristic precisely _because_ it is not individual. I have already examined some ways in which Anzald\u00faa performs a collective mestiza queer identity in the poetry section of _Borderlands\/La Frontera_. The emphasis on \"coalition and alliance building\" (\"Bridge, Drawbridge,\" 229) is evident throughout her work and is a central component of the mestiza queer identity she theorizes: _\"Dime con quien andas,\"_ she writes, quoting a Mexican saying, \"y _te dire quien eres_ [Tell me who your friends are and I'll tell you who you are]\" ( _Borderlands I La Frontera_ , 62). Her commitment to identity through collectivity is undoubtedly one reason why two of Anzald\u00faa's most important contributions to feminist, multiethnic, and lesbian\/gay studies have been anthologies: _This Bridge Called My Back_ , which she coedited with Cherr\u00ede Moraga, and _Making Face, Making Soul\/Haciendo Car as_. The former text has been credited by many with revolutionizing feminism, and although neither text is marketed as \"lesbian\/gay,\" I would argue that both\u2014because of the collective and disruptive queer positions they advance\u2014should be seen as central texts in queer studies and in the Queer Renaissance as well.\n\nLong before taking the magical journey across the border, Kerouac's Sal Paradise heads to San Francisco. He explains, \"There were plenty of queers. Several times I went to San Fran with my gun and when a queer approached me in a bar John I took out the gun and said 'Eh? Eh? What's that you say?' He bolted. I've never understood why I did that; I knew queers all over the country\" (On _the Road_ , 73). Despite Sal's incredulity at his own actions, queer readers of On _the Road_ might provide an answer to his quandary: Sal acts as he does to ensure an unlimited mobility for himself (indeed, \"all over the country\" and beyond), while simultaneously policing \"others\" who, with _their_ mobility, might expose and disrupt the privileged positions he occupies. Such a mobility, which ultimately reinforces relations of domination-subordination, stands in stark contrast to the mestiza queer mobility Anzald\u00faa champions. \"Allies,\" Anzald\u00faa writes, \"remember that the foreign woman, 'the alien,' is _nonacayocapo_ which in Nahuatl means one who possesses body (flesh) and blood like me. _Aliadas, recuerda que la mujer ajena tambien es nonacayocapo, la que tiene cuerpos y sangre como yo\"_ (\"Bridge, Drawbridge,\" 229). Queer mestizas cross borders not in order to consolidate definitions of \"self\" and \"other\" or to appropriate the \"exotic\" and transcend the \"self\" but rather to unmask the separations and disavowals the border perpetuates and to locate, in a transformed borderlands, others\/allies who can similarly imagine a queer, more humane, new world.\n\n## **_Chapter Four_  \nQueer Identities in a Crisis**\n\nAnd then came _Angels in America_. There is something predictable, perhaps, about ending with the play that has received more attention than any gay or lesbian work of the past fifteen years, and indeed, more attention than any American play of the past half century Tony Kushner's _Angels in America: A Gay Fantasia on National Themes_ is a two-part, seven-hour spectacle that has been showered with praise by gay and nongay critics alike. In 1993 the first part\u2014 _Millennium Approaches\u2014_ won the Pulitzer Prize for drama and received four Tony Awards, including the awards for best play, best director, and best actor. _Millennium Approaches_ was also presented with the Lambda Literary Award for gay and lesbian drama. The second part, _Perestroika_ , received three Tony Awards in 1994, including for best play and best actor. Undoubtedly, this unprecedented success contributed to John Clum's decision to put a still from _Angels in America_ on the cover of his study _Acting Gay: Male Homosexuality in Modern Drama_ and to label his discussion of Kushner's play \"The Culmination\" (313).\n\nI certainly agree that Kushner's play merits praise; nonetheless, this chapter attempts to resist such teleological thinking. I am not simply celebrating _Angels in America_ here, as if mainstream success were itself sufficient for ensuring a work's inclusion in the Queer Renaissance, and as if _Angels in America_ were thus somehow the crowning glory of this contemporary literary and cultural movement. As my previous chapter indicated, in fact, a purely celebratory stance is in many ways inimical to the critical work the Queer Renaissance advances. Hence this chapter begins by considering how _Angels in America_ stages queer identity and theory in ways akin to the other queer works I have been examining throughout. I argue that _Angels in America_ provides audiences with a demonstration of how contradiction and disruption can be deployed critically and communally. Focusing mostly on _Perestroika_ , I explore how Kushner queers that particular concept to promote the performance of an openly disruptive, contradictory mingling, and\u2014consequently\u2014hopeful identity. The performance of this identity secures hope by articulating the messiness and mingling that the characters ultimately champion to a collective identity that can and will be continually reshaped in the fight against AIDS.\n\nThe chapter then attempts to provide a genealogy, of sorts, for this queer aesthetic. The second half of the chapter surveys Sarah Schulman's fictionalization of the early days of the AIDS Coalition to Unleash Power (ACT UP) in her novel _People in Trouble_ , in order to put Kushner's theories more explicitly into conversation with other \"demonstrations.\" AIDS activism premiered in the streets during the writing and production of _Angels in America_ , and the queer identities that activists performed during their \"successful run\" parallel those performed onstage in Kushner's \"gay fantasia.\" If, as both _Webster's New World Dictionary_ and the _American Heritage Dictionary_ suggest, a fantasia is \"a medley of familiar tunes,\" then Kushner's gay fantasia, like any other, draws its themes and variations from the work of many different (mostly unnamed) composers. This chapter demonstrates how such an artistic collaboration proceeds.\n\nThe liminal queer identities that have been the focus of previous chapters are again central here, but I am particularly concerned at this point with the relationship between these new sexual and political identities and the aesthetic: with how those sexual and political identities incite the reinvention of artistic identities and visions, and vice versa. That relationship is one of the main foci of Schulman's novel, which depicts the exposure of Kate, a visual artist, to both the queer activist identities shaped by Justice (Schulman's fictional answer to ACT UP) and a liminal queer sexual identity (Kate is a bisexual woman migrating between life with Peter, her husband, and with Molly, her lesbian lover). Kate's negotiation of new sexual and political identities profoundly alters her understanding of art and leads her to a critical postmodern aesthetic intent on disrupting the harmonious use of space that her corporate sponsors exact. This chapter, then, is concerned less with applauding Kushner's mainstream success story than it is with examining the ways in which liminal sexual and activist identities\u2014the \"identities in a crisis\" of my title\u2014have shaped the critical aesthetic not only of _Angels in America_ but also of the many other works that constitute the contemporary Queer Renaissance.\n\n### **Staging the Queer Renaissance**\n\n_Angels in America_ is set in New York City during 1985 and 1986. This is, as the play repeatedly reminds audiences, Ronald Reagan's America\u2014a bleak world that offers its inhabitants little in the way of hope. As Louis Ironson, one of the central characters, muses:\n\nMaybe we are free. To do whatever.\n\nChildren of the new morning, criminal minds. Selfish and greedy and  \nloveless and blind. Reagan's children.\n\nYou're scared. So am I. Everybody is in the land of the free. God help us all. ( _MA_ , 74)\n\nDespite this sardonic cry for help, however, and despite the apocalyptic rumblings that pepper _Millennium Approaches_ , neither God nor Reagan appears to be coming to these characters' aid. Quite the opposite, in fact: _Millennium Approaches_ presents a world in disarray, moving further and further into chaos. Louis's lover, Prior Walter, is diagnosed with AIDS, and their relationship is strained to the breaking point when Prior is rushed to the hospital in the middle of the night. \"Tell him goodbye, tell him I had to go,\" Louis instructs the nurse before walking away from the hospital and from Prior ( _MA_ ,52).\n\nMeanwhile, Roy Cohn\u2014Kushner's dramatic representation of the arch-conservative lawyer and former aide to Senator Joseph McCarthy\u2014is also diagnosed with AIDS. He responds to this diagnosis by threatening his doctor:\n\nSay: \"Roy Cohn, you are a homosexual.\"\n\n( _Pause)_\n\nAnd I will proceed, systematically, to destroy your reputation and your practice and your career in New York State, Henry. Which you know I can do. . . . No, Henry, no. AIDS is what homosexuals have. I have liver cancer. ( _MA, 44_ , 46)\n\nFinally, the other focal relationship of _Millennium Approaches_ also fractures over the course of the play, as Joe Pitt, Roy's prot\u00e9g\u00e9 in the federal court of appeals, acknowledges his homosexuality and abandons his wife, Harper. Joe and Harper's Mormon marriage no longer serves as a protective covering, and Harper is left alone with her Valium addiction as Joe departs in search of sex in Central Park.\n\n_Millennium Approaches_ concludes with the arrival of an angel, who bursts into Prior's bedroom only to literalize the devastation that has been implicit throughout the play: _\"As the room reaches darkness, we hear a terrifying_ CRASH _as something immense strikes earth; the whole building shudders and a part of the bedroom ceiling, lots of plaster and lathe and wiring, crashes to the floor\" (MA_ ,118). The angel declares Prior a \"Prophet\" and informs him that \"the Great Work begins\" ( _MA_ ,119). The second half of _Angels in America_ , then, emerges out of this literal and metaphoric wreckage. _Perestroika_ , which Kushner dubs \"essentially a comedy\" ( _P_ ,8), reconfigures the relationships and resolves (in part) the crises of _Millennium Approaches:_ Louis, after an affair with Joe characterized by passionate sex and intense disagreements about politics, rejects Joe's conservatism and attempts reconciliation with Prior. Harper, too, self-confidently puts Joe behind her, along with the ugliness she associates with their New York life, and catches a night flight to San Francisco, which she has learned is \"unspeakably beautiful\" ( _P_ ,122). Prior assures Louis that he still loves him but insists nonetheless that Louis not move back in. Roy Cohn dies, and Prior continues to live with AIDS, developing along the way a rather eccentric friendship with Hannah, Joe's Mormon mother, who has flown in from Utah to attend to the crisis between her son and Harper.\n\nMost important, over the course of _Perestroika_ , Prior works to understand and reject the vision the angel brings him. \"YOU MUST STOP MOVING!\" the angel commands:\n\nForsake the Open Road:\n\nNeither Mix Nor Intermarry: Let Deep Roots Grow:\n\nIf you do not MINGLE you will Cease to Progress:\n\nSeek Not to Fathom the World and its Delicate Particle Logic:\n\nYou cannot Understand, You can only Destroy,\n\nYou do not Advance, You only Trample.\n\nPoor blind Children, abandoned on the Earth,\n\nGroping terrified, misguided, over\n\nFields of Slaughter, over bodies of the Slain:\n\nHOBBLE YOURSELVES! ( _P_ , 52)\n\nThe angel and her celestial colleagues feel that chaotic human movement, mingling, and progress have driven God away; Prior's role as prophet, consequently, is to preach resignation and stasis. After a struggle with the angel and the vision she brings, Prior follows her to heaven, where he at last refuses the death-in-life existence she and the other angels call for, demanding instead to be blessed with \"More Life\" (P, 135\u201336).\n\nIn this section, I consider the ways in which Kushner stages queer identity and theory in _Angels in America_ , and for this reason I focus on the Prior plot. The queer identities I have traced throughout this study have a clear analogue in the identity performed by Prior; through the Prior plot, Kushner probes perestroika in order to present audiences with a queer political theory advocating interdependence, increased openness, and ongoing democratic participation and contestation. Sexuality provides Kushner with the ground upon which to forge this theory, but sexuality also serves as a figure for his broader political vision. _Angels in America_ deploys the messiness and interconnectedness of sex as a metaphor for the collective and contested identities-in-process shaped and reshaped in the ongoing struggle against AIDS. Rather than renouncing the contradictions of human existence, as the angel demands, Prior embraces them and insists that they sustain a more radical democracy\u2014a radical democracy I call \"queer perestroika.\"\n\nPrior's story, based on the biblical story of Jacob's nocturnal struggle with an angel in the book of Genesis\u2014or more properly, the _Book of J_ , as interpreted by Harold Bloom\u2014is ultimately about the survival of a people who thrive not in spite of but because of their migrating and mingling. J's Jacob, alone at night in a strange land, encounters a mysterious figure who wrestles with him until daylight. At the end of this struggle, Bloom writes, \"the angel acknowledges the difference between the victim he had expected . . . and the hero he has encountered throughout the night\" (218). Jacob demands that the angel bless him, which he does by granting Jacob the name of Israel, \"God-clutcher.\" Bloom explains that one possible interpretation of this exchange is that the stranger is named Israel himself, and that he loses his name to Jacob (218). _Perestroika_ likewise enacts such a renaming. \"Angels in America,\" which at the end of _Millennium Approaches_ would appear to refer to supernatural beings such as the one who crashes into Prior's bedroom, seems, at the end of _Perestroika_ , a more apt designation for those like Prior, who cling to desire, hope, and life even as they are told by apparently all-powerful forces to \"Creep away to Death\" (P, 135). Prior resists this celestial injunction, ensuring that in the end, as with J's Jacob\/Israel, \"what matters . . . is not so much the identity of the more-than-human that could not hold on, but the new identity of the human that refused to let go\" (Bloom, 218).\n\nIn what follows, I trace this exchange of identities more carefully, focusing initially on what it is about this forbidding angel that would make an exchange desirable. I argue that the logic of the angel's identity undermines itself, since her reactionary message is in tension with her fabulous and queer sexuality. Prior clings to the latter component of the angel's identity and\u2014in the deconstructive struggle\u2014articulates that fabulous sexuality to a collective political vision that affirms queer perestroika rather than the isolation and individualism the angel preaches.\n\nInitially, the identity of the angel is enigmatic. In the first part of _Angels in America_ , she does not even appear onstage until the final scene, when she crashes through Prior's bedroom ceiling. Earlier in the play, however, in keeping with the apocalyptic rumblings throughout _Millennium Approaches_ , the angel's voice periodically interrupts Prior's scenes:\n\nLook up, look up,\n\nprepare the way\n\nthe infinite descent\n\nA breath in air\n\nfloating down\n\nGlory to . . . ( _MA_ , 35)\n\nStage directions indicate that this voice is _\"incredibly beautiful\"_ ( _MA_ , 34) and that its approach causes _\"dramatic change[s] in lighting\"_ ( _MA_ , 35). The angel is thus simultaneously beautiful and powerful, but beyond this, Kushner gives us few clues into the nature of her identity. Her role throughout _Millennium Approaches_ is apparently to announce the approach of someone or something\u2014though not necessarily of God, since here and elsewhere the refrain \"Glory to God\" is truncated. Audiences learn something about the angel's identity only from the effect she has on Prior: \"I want the voice,\" Prior tells Belize, an African American former drag queen who is Prior's best friend and Roy Cohn's nurse. \"It's all that's keeping me alive. . . . You know what happens ? When I hear it, I get hard\" ( _MA_ ,60). Neither Prior nor Belize can particularly explain such a reaction at this point, but Prior extracts a promise from Belize that he will not let the doctors change Prior's drugs.\n\nThe queer connections between desire, sexuality, and the angel solidify in _Perestroika_. Kushner titles the first act of _Perestroika_ \"Spooj\" (P, 13), and although audiences do not learn immediately what the term _Spooj_ (printed in their program) means\u2014or, more important, what occurs in the meeting between Prior and the angel after she bursts into his bedroom\u2014this act nonetheless picks up where _Millennium Approaches_ left off. Again we are introduced to the angel's identity indirectly by seeing the effect she has on Prior. Immediately after she leaves his apartment, Prior calls Belize at the hospital to report. The meaning of \"Spooj\" comes\u2014so to speak\u2014as soon as Belize picks up the phone:\n\nBELIZE: Ten East.\n\nPRIOR: I am drenched in spooj.\n\nBELIZE: Spooj?\n\nPRIOR: Cum. Jiz. Ejaculate.\n\nBELIZE: Spooj?\n\nPRIOR: Onomatapoetic, isn't it?  \nI've had a wet dream.\n\nBELIZE: Well about time. Miss Thing has been abstemious.  \nShe has stored up beaucoup de spooj.\n\nPRIOR: It was a woman.\n\nBELIZE: YOU turning straight on me?\n\nPRIOR: Not a _conventional_ woman. . . . An angel. (P, 23)\n\nWhen Belize and Prior meet in the second act, Prior explains in more detail exactly what took place during his meeting with the angel. Kushner uses a flashback scene in order to stage, finally, the actual encounter. The angel appears to Prior to declare him a prophet and to supply him with \"The Book\" (P, 47), whose contents he is to communicate to the people. These contents are revealed, however, only after a very queer and erotic episode that bears out the connections between the angel and sex that the audience has been primed to expect. I quote from this episode at length to stress how completely it satisfies these sexual expectations:\n\n( _The Angel's lines are continuous through this section. Prior's lines overlap_.  \nThey both get very turned-on.)\n\nPRIOR ( _Hit by a wave of intense sexual feeling):_ Hmmmm . . .\n\nANGEL: The Pulse, the Pull, the Throb, the Ooze . . .\n\nPRIOR: Wait, please, I. . . . Excuse me for just a minute, just a minute OK\n\nI. . .\n\nANGEL: Priapsis, Dilation, Engorgement, Flow:  \nThe Universe Aflame with Angelic Ejaculate . . .\n\nPRIOR ( _Losing control, he starts to hump the book):_ Oh shit. . .\n\nANGEL: The Feathery Joinings of the Higher Orders,\n\nInfinite, Unceasing, The Blood-Pump of Creation! PRIOR: OH! OH! I. . . OH! Oh! Oh, oh . . .\n\nANGEL ( _Simultaneously):_ HOLY Estrus! HOLY Orifice! Ecstasis in Excelsis! AMEN!\n\n( _Pause. If they had cigarettes they'd smoke them now.)_ (P, 47\u201348)\n\nThe angel and Prior's interaction here is, if you will, queer as fuck; as Prior explains to Belize, \"The sexual politics of this are _very_ confusing\" (P, 49). The Chicago production of _Angels in America_ staged this scene with the angel (Carolyn Swift) suspended just above the head of the bed on which Prior (Robert Sella) was lying. Prior initially retreated to the foot of the bed when the angel crashed through his ceiling, but as he was more and more overcome by the \"intense sexual feeling,\" he moved back onto the bed, turned over, and raised his ass up to the angel. The angel and Prior thus appeared to be engaging in anal sex, with the angel on top. Throughout this scene, Swift's angel maintained a thrusting motion as she delivered her lines, until the final \"AMEN,\" which was spoken\u2014or rather, ecstatically shouted\u2014with the first syllable drawn out: \"AAAAAAAA-MEN!\" The impression given was that the angel, with this exclamation, had ejaculated in Prior's ass, and this was substantiated by Prior's immediate response:\n\nPRIOR: What _was_ that?\n\nANGEL: Plasma Orgasmata.\n\nPRIOR: Yeah well no doubt. (P, 48)\n\nAfter this erotic encounter, Belize and the audience finally hear, for the first time, what the angel and her cohorts want from Prior. The angel explains to Prior that God apparently created in humankind the potential for change, for movement forward (P, 49). However, \"Bored with His Angels, Bewitched by Humanity, \/ In Mortifying Imitation of You,\" God Himself began to move and migrate, until one day\u2014the day of the Great San Francisco Earthquake of 1906\u2014He left and never came back (P, 5051). Since the angels feel that the messiness of human existence and progress drove God away, they are convinced that the only way to bring back God is through a cessation of movement and mingling: \"FOR THIS AGE OF ANOMIE: A NEW LAW!\" (P, 56). The angel even suggests, obliquely, that AIDS is to be a figure for this new law:\n\nOn you in you in your blood we write have written: STASIS!\n\nThe END. ( _P_ , 54)\n\nIn other words: your constant motion (one might even say \"promiscuity\") has brought you to this point and has marked you; there is nothing left for you to do but resign yourself to this conclusion, and tell others to do the same.\n\nGiven the foreplay audiences (and Prior himself) have experienced with this angel, the demand that he commence a sort of death-in-life existence should be a bit of a surprise. Prior is stunned by the contradiction: \"What are you? Did you come here to save me or destroy me?\" (P, 53). Belize, in contrast, forgets completely the sexual ecstasy that has preceded and tries to focus Prior's attention solely on the problematic content of this \"prophecy\": \"This is . . . worse than nuts, it's . . . well, don't migrate, don't mingle, that's . . . malevolent, some of us didn't exactly _choose_ to migrate, know what I'm saying. . . . This is just you, Prior, afraid of the future, afraid of time. Longing to go backwards so bad you made this angel up, a cosmic reactionary\" (P, 55). With its focus on malevolence, Belize's interpretation intersects with Bloom's rereading of the Jacob story. Bloom reads Jacob's angel as relatively hostile, in spite of normative Christian and Judaic interpretations that suggest otherwise. \"'Wrestling Jacob' is a powerful image, particularly in Protestantism,\" Bloom writes, \"where the agon is essentially seen as a loving struggle between Jacob and God. But . . . there is absolutely nothing loving about this sublime night encounter\" (216). Likewise, the angel's message in _Perestroika_ can be read as hostile to human vitality.\n\nSince she appears onstage only at the very end of the first play, there is not much in _Millennium Approaches_ that foreshadows the angel's gloomy message in _Perestroika_. Kushner does provide a few indirect hints in the first play of what is to come, however. In both _Millennium Approaches_ and _Perestroika_ , all of the minor characters are played by actors who are also playing major roles. Sister Ella Chapter, for instance, is the real estate agent who agrees to sell Hannah's house in Salt Lake City when Hannah decides to head to New York to help Joe and Harper. Kushner's instructions call for the actor playing the angel to play Sister Ella Chapter ( _MA_ , 4)\u2014and with good reason, as Sister Ella's philosophy is similar to what the angel will preach in _Perestroika_. Sister Ella explains to Hannah, \"I always thought: People ought to stay put. That's why I got my license to sell real estate. It's a way of saying: Have a house! Stay put! It's a way of saying traveling's no good\" ( _MA_ ,82). Prior's nurse, Emily, is played by the same actor, and Kushner likewise has this character, quite literally, speak for the angel. At one point Emily unexpectedly begins yelling at Prior in Hebrew:\n\nEMILY: There's really nothing to worry about. I think that shochen  \nbam-romim hamtzeh menucho nechono al kanfey haschino. ( _MA_ , 98)\n\nPrior becomes extremely agitated, but Emily denies that she has said anything out of the ordinary: _\"Hebrew? (Laughs)_ I'm basically Italian-American. No. I didn't speak in Hebrew\" ( _MA_ ,98). Emily's denial confirms for audiences that they are getting not _her_ personality but rather a literal foreshadowing of the angel's personality. In the Chicago production, when Swift delivered her Hebrew lines, she added to the impression that the angel's personality would be fairly hostile by having her Emily face Prior and scream.\n\nThe Hebrew words Emily screams here, and in two more outbursts in this scene, both compound and complicate the angel's hostility. The Hebrew, from the _Daily Prayer Book\/Ha-Siddur Ha-Shalem_ , asks the spirit of God to bring quietness and peace to Prior. These are not necessarily hostile sentiments, but since this is a prayer for the dead, and since Prior\u2014in this scene and throughout _Angels in America_ \u2014continues to _live_ with AIDS, the angel's words could be understood as betraying a death wish for Prior. Most American audiences would likely see only hostility if and when the actor playing Emily screamed these lines, since\u2014like Prior (or myself)\u2014they would be unlikely to have a working knowledge of Hebrew.\n\nYet despite this hostility, and despite the fact that\u2014given her message of resignation or death wish\u2014the angel is indeed a \"cosmic reactionary,\" the contradiction remains: she is repulsive because of her demand that human movement and interaction cease, but at the same time, she is eminently attractive in that she is responsible for not only the hottest sex Prior has had in months but possibly the queerest sex he has ever had. Moreover, later in the play, she has this effect on others: Hannah, who is sitting with Prior in his hospital room, _\"has an enormous orgasm as the Angel flies away to the accompanying glissando of a baroque piccolo trumpet\" (P_ ,120). This scene takes place after Prior has left the room, and again, the performance of sexuality is both queer and intense. Swift's angel in the Chicago production turned to Hannah (Barbara Robertson) with a lascivious look on her face and proceeded to rub the inside of her thigh invitingly, as Hannah watched from across the room, exclaiming, \"What? What? You've got no business with me, I didn't call you, you're _his_ fever dream not mine, and he's gone now and you should go too, I'm waking up right .. . NOW!\" (P, 120). Instead of waking up, however, Hannah is seduced; she walks over to the angel, and the two fall into a passionate kiss. After this kiss, Robertson performed her orgasm by falling back on Prior's bed and shaking all over as the angel disappeared in a flash of light. The kiss between Swift and Robertson and Robertson's exuberant orgasm elicited vigorous applause from Chicago audiences. Audiences may not have agreed with or may have been baffled by the hostile angelic message, but their applause clearly bespoke approval of the women's sexual performance.\n\nEarlier in _Perestroika_ , as Hannah struggles to come to terms with her son Joe's homosexuality, she explains to Prior, \"It just seems . . . ungainly. Two men together. It isn't an appetizing notion but then, for me, men in _any_ configuration . . . well they're so lumpish and stupid\" (P, 104). Hannah's statement is thrown into ironic relief when she eventually gets a taste of sex with the angel; not finding men palatable apparently does not forestall, and may even facilitate, Hannah's extreme satisfaction with this female angel. In the end, Hannah's sex scene only underscores the angel's contradictions: she brings intense pleasure to everyone, including the audience, while demanding, in effect, that all such pleasure cease.\n\nJohn Clum, working with an earlier draft of _Perestroika_ , glosses over this contradiction by reading the sexuality of the angels not as paradoxically similar to the acts they denounce but rather as a sort of safeguard against those very acts. For Clum, apparently, angelic sex itself is not movement and mingling but is, instead, that which keeps the angels from moving. It is this sex-as-stasis, Clum implies, that God ultimately rejects: \"Having tired of the eternal sexual bliss of paradise and its ever adoring angels, he followed the example of his creation, man\" (314). To support this opposition, Clum provides an earlier version of the lines Kushner has the angel say, which I quoted above:\n\nHis lust gave way to wanderlust  \nIn mortifying imitation of you, his least creation. (Clum, 314)\n\nGiven these lines, with their suggestion that lust gives way to wanderlust, Clum is entirely justified in the opposition he sets up between angelic sex and human movement\/migration. But Kushner removed this line from later versions, including the published version of _Perestroika;_ through this removal, Kushner makes it more difficult to naturalize away the contradiction I have been drawing out. Contradiction, for Kushner, is one of the angel's central components: \"She's the contradictory spirit of America. . . . She's the angel of history. She's the intermediary between the divine and the material world. She's also the angel of theatricality. And yes . . . she's a human body in God-like drag\" (qtd. in Kroll, 58). Hence, Kushner dispenses with a line that might position the angels' sexuality itself, and not simply the angelic message, in opposition to vitality and, through this alteration, explicitly heightens the tension between the angels' sexuality and their message.\n\nUltimately, this tension facilitates the exchange of identities and the queer theory that _Angels in America_ stages. That exchange of identities commences, I would argue, with another tension and another exchange that is performed in _Perestroika:_ an exchange of glances between Prior and the angel immediately after she demands that humankind stop moving. A lot is communicated through this exchange of glances, including, I think, Prior's acknowledgment to the angel that he sees the contradiction between the fatalistic prophecy she has just given and the hot sex that preceded it. The angel, in turn, sees this acknowledgment in Prior's glance and is, not surprisingly, uncomfortable with his recognition of her contradictions:\n\nPRIOR: Whatever you are, I don't under  \nstand this visitation, I don't understand what you want from me, I'm not a prophet, I'm a sick, lonely man,  \nI. . . Stop. Moving. That's what you want. Answer me! You want me dead.\n\n( _Pause. The Angel and Prior look at each other.)_\n\nANGEL: YES. NO. NO.  \n( _Coughs)_  \nYES.  \nThis is not in the Text, We _deviate_ . . . (P, 53)\n\nWe deviate, indeed. Prior's \"deviation\" here, with its complementary destabilizing gaze, is his first explicit challenge to the new order that the angels seek to fashion. In the Chicago production, Carolyn Swift sharply differentiated her \"yes\" and \"no,\" highlighting how little the angel expected this sudden alteration in \"the Text\"; her \"yes\" was delivered softly, as though she and Prior were locked together in a spell (as indeed they were a few moments earlier, as Robert Sella's Prior listened, entranced, to her demand that he stop moving), but her \"no\" was shouted, as she abruptly came to realize that Prior would resist her demands. Through such nascent acts of deviation, Prior begins to usurp the authority of the angel's position. As the angel acknowledges, Prior's performance marks a moment of textual transgression, or what we might call a moment of queering.\n\nIn this scene, the players present audiences with a literal dramatization of the transgressions the other texts in my study have written. In fact, the angel's italicized response to Prior's gaze (\"We _deviate_. . .\") could be a byword for my entire project, since in each chapter I have examined the ways in which lesbian and gay writers in the Queer Renaissance shape queer identities that, in their deviation from heteronormativity, are disruptive and transformative. _Collective_ deviation is again crucial here. As the play continues, Prior moves to a location where multiple actors envision a more radical democracy emerging from the contradiction and chaos the angel resists. Such movement from contradiction to community is not automatic: Roy Cohn, for instance, takes his understanding of chaos in the opposite direction. Roy says to Joe in his very first scene, \"I see the universe . . . as a kind of sandstorm in outer space with winds of mega-hurricane velocity, but instead of grains of sand it's shards and splinters of glass\" ( _MA_ ,13). Roy's understanding of chaos fuels ruthlessness and individualism, not community. In contrast, Prior's affirmation of the mess-iness of human existence encourages him to appropriate and disseminate not only the angel's authority but also her ironic and unintended acknowledgment that deviation is threatening when it takes place in the first-person plural. In the Queer Renaissance, multiple actors in many different locations have declared, \"We deviate.\" Hence, despite the fact that Prior himself suggests this is one of his weakest, most isolated moments (\"I'm a sick, lonely man\"), it is in this scene that _Angels in America_ effectively begins to stage the Queer Renaissance: not because it is clear to audiences at this point that Kushner's play is an individual artistic achievement but rather because Kushner and the actors, along with all of those involved in the \"production\" of queerness in _Angels in America_ , implicitly connect Prior's performance here to the widespread \"rebirth\" of radical gay and lesbian political analysis occurring offstage. Actors onstage and offstage authorize the Queer Renaissance; Prior's identity in this scene mingles with the new queer and resistant identities performed in the streets.\n\nThe political analysis that is central to the Queer Renaissance seeks to expose and undermine the ways in which domination is structurally secured in and through the construction of normative identities. In the preceding chapter, for Gloria Anzald\u00faa this involved working sexual and racial categories against themselves, so that mestiza queer identity was no longer simply a mark of abjection but rather a force that might unsettle the system of binaristic thinking on which normative identities depend. Similarly, in this scene, Prior's recognition of the angel's contradictions begins to undermine the identificatory regime she and her cohorts seek to institute. Initially, the angel, with her demand that human movement and migration cease, appears virtually all-powerful. Prior begins to recognize, however, that the angel attempts to deny or exclude what she has already incorporated into her own identity. The struggle between them, then, is a comically deconstructive one, through which Prior is able to effect both an explicit reversal of power relations and an implicit exchange of names.\n\nI call the struggle \"comic\" not only because Kushner himself dubs _Perestroika_ a \"comedy\" but also because of the ways in which it inscribes a deconstructive relationship onto the bodies of these characters: the struggle between the angel and Prior is an ironic _performance_ of deconstruction. Generally, deconstructive analyses expose how the logic of a system of meaning undermines itself. More specifically, binary oppositions that present themselves as apparently, \"naturally\" equal and opposite (man\/woman, straight\/gay) can be shown to exist actually in relationships of domination and subordination that valorize the first term and disparage the second. Despite this positioning of the second term as ancillary, however, the first term secures its identity and dominance only through the ongoing exclusion of the second term; the first term must repeatedly define itself in opposition to what it is not. Ironically, this act of exclusion ensures that the first term actually depends for its meaning on the second term, and this dependence, in turn, puts the supposed \"dominance\" of the first term into perpetual jeopardy. Hence the first term not only excludes but _incorporates_ the second term, and this simultaneous exclusion and incorporation makes the structuring of systems of meaning in terms of binary oppositions irresolvably unstable.\n\nIn _Perestroika_ , Kushner's performance of deconstruction is comic in that his characters literally re-present a deconstructive scenario. The angel seeks to establish an opposition between order and stasis, which are to be valorized, and mingling and migration, which are to be subordinated. However, this opposition is ripe for deconstruction, since the angel cannot even describe the new order she desires without opposing it to the human action she claims to repudiate: the angels value stasis precisely because it is _not_ movement or mingling. Yet Kushner takes this strict opposition, and the subsequent performance of deconstruction, one step further, since the angel literally incorporates, for the audience, the movement she would exclude from her identity: the angel calls for inaction but at the same time _embodies_ the hottest action Prior has had in months. The reversal of power relations that takes place will thus be a true disruption from within, as the logic of the angel's identity undermines itself.\n\nThe orgasmic yell the angel emits during her initial encounter with Prior (\"HOLY Estrus! HOLY Orifice! Ecstasis in Excelsis! AMEN!\" [P, 48]), juxtaposed with the prophecy she later gives him (\"STASIS!\/The END\" [P, 54]), spotlights the foundational illogic of her identity. She preaches a morbid stasis and yet both preaches and _performs_ ec-stasis (\"in the highest! So be it!\"). This contradiction is not lost on Prior, who explains to Belize that the angels are \"basically incredibly powerful bureaucrats . . . they're sort of fabulous and dull all at once\" (P, 49). _Fabulous_ , used for decades in gay male vernacular communities, is an important term in the Kushner lexicon. In \"Notes Toward a Theater of the Fabulous,\" Kushner's foreword to a recent anthology of contemporary gay plays, he writes:\n\n_Fabulous_ became a popular word in the queer community\u2014well, it was never unpopular, but for a while it became a battle cry of a new queer politics, carnival and camp, aggressively fruity, celebratory and tough like a streetwise drag queen:\"FAAAAABULOUS!\" . . . _Fabulous_ is one of those words that provide a measure of the degree to which a person or event manifests a particular, usually oppressed, subculture most distinctive, invigorating features, (vii)\n\nIt is to the most fabulous\u2014and thus, as Kushner's notes make clear, to the \"queerest\"\u2014aspects of the angel's identity that Prior eventually clutches, explaining to all the angels in the last act that although he rejects their prophecy of resignation, he nonetheless wants to be blessed, by them, with \"more life.\" Sensing a tradition of fabulousness beneath the angel's dullness and desire for stasis, Prior seizes on the former. As Walter Benjamin insists, \"In every era the attempt must be made anew to wrest tradition away from a conformism that is about to overpower it\" (255). The deconstructive struggle Prior wages against the angel wrests a fabulous tradition away from straight conformism and thereby implicitly responds to Benjamin's call to \"brush history against the grain\" (257).\n\nClum writes, \"Prior . . . wants more life and will, like Jacob, wrestle with the angel to get it\" (319); Clum does not clarify, however, _why_ Prior would want \"more life\" from this angel and her cohorts, who come across as fairly stuffy celestial prudes in his analysis. Since Clum never moves beyond the angels' calls for \"stasis, death\" (319), or their desire for \"an end to human movement, migration, change\" (314), Prior's motives remain unclear in Clum's interpretation, unless Prior wrestles a blessing from the angel simply because she is a powerful figure or\u2014more insipidly\u2014because Jacob did so. My argument, in contrast, is that queer sex generally and the angels' sexuality specifically are crucial to understanding not only the exchange between Prior and the angel but _Angels in America_ as a whole. Prior deconstructs the angel's message of doom by clinging to the fabulous and erotic sexuality he has shared with her, and in the end, this vitality wins out. Prior returns the Book containing the prophecy and informs the angels, \"We can't just stop. We're not rocks\u2014progress, migration, motion is . . . modernity. It's _animate_ , it's what living things do. We desire\" (P, 132). Prior's appropriation of the angel's \"we\" disregards the anxiety inherent in her \"We deviate\" and instead celebrates connection among \"living things.\" Prior's \"we,\" moreover, cannot but encompass the angel at this point, given that his only _performance_ of desire in _Perestroika_ is with her. And although the angel initially appears incredulous (\"Oh who asks of the Orders Blessing\/With Apocalypse Descending?\/Who demands: More Life?\" [P, 135]), her own internal contradictions ultimately sanction the reversal of her message. Indeed, even the casting and staging of this scene, as in many others, represent interconnectedness and mingling: just as the actor playing the angel takes on additional roles throughout _Angels in America_ , the actors playing Harper, Joe, Hannah, Belize, Louis, and Roy all take on additional roles as angels here.\n\n_Angels in America_ revels in the eros of this interconnectedness. Clum suggests, \"AIDS drama may not be erotic, but it is unabashedly romantic\" (74); this argument, however, is difficult to sustain given _Perestroika_ , which includes several onstage sex scenes in addition to the human and angelic orgasms I have detailed. Indeed, Kathleen Chalfant, who played Hannah on Broadway, actually suggested, \"An unmentioned subtext . . . is we'd like the whole audience to have a simultaneous orgasm\" (qtd. in Friend, 160). In my mind, it is this queer insistence on sex, sexuality, and interconnectedness that both sets _Angels in America_ apart from more \"unabashedly romantic\" and individualistic AIDS plays and makes it central to the Queer Renaissance. The outcome of _Perestroika_ both celebrates queer sexuality and, more important, deploys it as a figure for Kushner's broader political vision of queer perestroika.\n\nSome critics, however, deemphasize the physical in _Angels in America_ and focus instead on the spiritual, rebinarizing the oppositions the play deconstructs. Rob Baker, for example, reads the play as alluding to the alchemical process whereby \"the inner soul\" is transformed from a \"basic instinctual self into a higher spiritual being\" (214). When the angel announces the \"Great Work\" at the end of _Millennium Approaches_ , Baker argues, the capitalization particularly suggests that Kushner has in mind the alchemical \"Great Work,\" the transformation of base lead into pure gold that served as a symbol for the transformation of the soul (214). Baker's account of the breadth of Kushner's allusiveness is useful, but he does not go far enough. Although he denies his reading is connected to \"New Age religiosity\" (220), he nonetheless grounds his argument about spirituality with a predictable New Age emphasis on self-transformation: the angel \"is the force (and perhaps the only force) which can meet AIDS head-on and combat that other brutally real\/unreal mysterious phantom force in a fight to the finish. Once Prior has known her, he can face that same struggle in himself\" (221). The emphasis on personal struggle and transformation allows Baker, at his weakest, to endow AIDS itself with mystical qualities. It is one thing to say characters _use_ AIDS to achieve transformation but quite another to mystify AIDS by suggesting it is some sort of _prerequisite_ to transformation: \"AIDS itself, in all its horror, ironically may be the very Philosopher's Stone that the characters . . . must 'possess' in order to achieve the necessary transformation, be it political or spiritual\" (215).\n\nDespite Baker's nod here to the political as well as the spiritual, he emphasizes only the spiritual in his analysis. He insists that Kushner's play is both political and sexual, but he never explains how; nor does he consider how the various elements (spiritual, political, sexual) might be interdependent. By contrast, in my reading of the play, the sexual is both the ground for Prior's spiritual struggle with the angel and the element that ensures he will prevail against her prophecy of resignation. The result of this spiritual struggle is, in turn, a political theory that I label _queer perestroika:_ that is, a politics of sexuality emphasizing interdependence, coalition, openness, and democratic participation and contestation\u2014or, if you prefer more physical terms, coming together, mingling, messiness, and disruption. Not surprising, as Kushner's \"Notes Toward a Theater of the Fabulous\" acknowledges, numerous variations on this theory were playing successfully in the streets during the writing and production of _Angels in America_ , breeding such slogans as \"We're here, we're queer, we're fabulous, get used to it\" and \"Safe sex\u2014Do it!\" In fact, Michael Mayer, the director of the Chicago production, explicitly acknowledged that one of the messages of the play was \"We _are_ fabulous\u2014get used to it!\" (qtd. in Hamilton-Little, 18).\n\nClum suggests that the \"key words\" in _Perestroika_ are \"blessing\" and \"forgiveness\" (322). Without disputing this, I would add the titular term _perestroika_ or the concept _queer perestroika_ to Clum's list, despite the fact that \"perestroika\" itself is not spoken onstage until the play's epilogue, and \"queer perestroika\" is not spoken at all. The term _queer perestroika_ is actually mine, but I think it is nonetheless a key concept and a useful and logical label for the theory Kushner advances in _Angels in America_. The Russian word means \"a restructuring of all social relations\" (Miracky 12) and was used during the 1980s to refer to President Mikhail Gorbachev's comprehensive plan for reform of the Soviet Union. Gorbachev's goal was to build _glasnost_ , or \"openness,\" into the political process, increasing citizen participation and decreasing government bureaucratization. Political institutions were to be decentralized, and Soviet foreign policies were to include, overtly, recognition of international interdependence.\n\nIn the epilogue to the play, \"Bethesda,\" the characters comment directly on this process. It is February 1990, and four years have passed since the end of the last act; as Prior explains, \"I've been living with AIDS for five years. That's six whole months longer than I lived with Louis\" (P, 146). Prior, Louis, Belize, and Hannah (who is now fully transformed into a hip New Yorker) are gathered at the Bethesda Fountain in Central Park. Louis is celebrating global change:\n\nLouis: Gorbachev is the greatest political thinker since Lenin. . . . Remember back four years ago? The whole time we were feeling everything everywhere was stuck, while in Russia! Look! Perestroika! The Thaw! It's the end of the Cold War! The whole world is changing! Overnight!\n\nHANNAH: I wonder what'll happen now in places like Czechoslovakia and Yugoslavia. (P, 145\u201346)\n\nAs Hannah's comment makes clear, there is more than a little dramatic irony in this celebration of perestroika. Audiences know what the characters do not: \"places like Czechoslovakia and Yugoslavia\" no longer exist, and indeed, the Soviet Union itself has fallen; nationalistic fervor and ethnic conflict in those same places have brought about untold carnage;\n\nGorbachev himself has been removed from power, along with his plans for reform within socialism.\n\nThe dramatic irony of Hannah's musings about Yugoslavia and Czechoslovakia, however, intersects irony of another sort: reports of perestroika's death may have been exaggerated, since the queer perestroika that has already succeeded overshadows the perestroika that has already \"failed.\" The play juxtaposes Gorbachev's \"failure\" here with Prior's success. Prior's success, furthermore, is both ongoing and larger than Prior the individual:\n\nPRIOR: [to the audience] . . . This disease will be the end of many of us, but not nearly all, and the dead will be commemorated and will struggle on with the living, and we are not going away. We won't die secret deaths anymore. The world only spins forward. We will be citizens. The time has come.  \nBye now.  \nYou are fabulous creatures, each and every one.  \nAnd I bless you: _More Life_.  \nThe Great Work Begins. (P, 148)\n\n_Perestroika_ thus ends with Prior's appropriation not only of the angel's \"we\" but also of her pronouncement from the end of _Millennium Approaches:_ \"The Great Work begins.\" Rejecting the angel's demand for death-in-life and clinging instead to queer perestroika, Prior reverses the relations of power and appropriates for himself the most fabulous aspects of the angel's identity. The \"Angels in America\" are, at the end of the play, not those seemingly supernatural visitors who, in this play, crash into human lives and demand resignation but rather those fabulous (and entirely human) creatures who struggle on in the fight against AIDS\u2014moving forward, acting up, demanding that we not die secret deaths any longer. Moreover, Prior's \"Great Work\" here is no longer Baker's alchemical, spiritual, and individual process; instead, it is linked to the very political and public work of forging a more radically democratic citizenship.\n\nThis link explicitly puts _Angels in America_ into conversation with the new and hopeful queer identities being performed offstage, in groups such as ACT UP and Queer Nation. It is thus appropriate that Prior is, at this point, literally in conversation with those seated offstage, in the audience. In this scene, Kushner both commemorates the transformative queer work that angels in America have performed in the previous four years, and\u2014through Prior's direct address to the audience\u2014recruits others to carry on that work, which has only just begun. Such commemoration of the past, coupled with the recognition of an ongoing struggle for more life, is represented by the Bethesda Fountain itself, which depicts the Jewish myth of a fountain in Jerusalem that could cleanse those who bathed in it of pain and disease. As Prior explains about the Bethesda angels whose statues grace the fountain in Central Park, \"They commemorate death but they suggest a world without dying\" (P, 147). Certainly, Prior's comments about these mythical figures could be seen as a return to the supernatural, opposed once more to the earthly and the political. Yet the resignification of \"angels\" that has already occurred in this play complicates such a reading, focusing attention instead on those figures who, during the writing and production of _Angels in America_ , were indeed both commemorating death (in collective works such as the AIDS memorial quilt) and actively demanding (in demonstrations around the country) an end to the dying.\n\nAt one point in _Millennium Approaches_ , Louis\u2014wracked with guilt over his abandonment of Prior\u2014tells Belize, \"There are no angels in America\" ( _MA_ ,92). Of course, since Louis says this while performing in a play called _Angels in America_ , his observation is already a bit ironic. Yet the irony goes much further than this, given Prior's \"deviation\" from the celestial text of resignation\u2014a deviation which guarantees that the trajectory of the play will advance toward his appropriation and dissemination of the queer and fabulous aspects of the angel's identity. In fact, the very staging of this scene renders Louis's thoughts ironic by causing audiences to focus on Prior. The scene is split: Louis and Belize are in a coffee shop on one side of the stage, while Prior and his nurse Emily are in an examination room on the other. The split scene ensures that even as Louis denies their existence, audiences are confronted with at least two potential angels in America: Emily, who is played by the actor who plays the angel (and this is the scene that later underscores that connection, when Emily unexpectedly shouts at Prior in Hebrew), and Prior, who will eventually claim the title after wresting \"more life\" from her. In the Chicago production, the irony was compounded even further by Belize (Reg Flowers), who\u2014as he sat listening skeptically to Louis\u2014wore a denim jacket with an angel's wings embossed on the back. Hence there are multiple angels in America, even as Louis speaks. He simply cannot see them, so intent is he, after leaving Prior, on a performance of isolated individualism.\n\nIsolation and individualism are frequent targets of Kushner's wrath, so _Queer Identities in a Crisis_ it is not surprising that _Angels in America_ ultimately rejects both in favor of the vision of collectivity represented in the epilogue. In the afterword to _Perestroika_ \u2014called, appropriately enough, \"With a Little Help from My Friends\"\u2014Kushner writes, \"We pay high prices for the maintenance of the myth of the Individual: we have no system of universal health care, we don't educate our children, we can't pass sane gun control laws, we elect presidents like Reagan, we hate and fear inevitable processes like aging and death, and on and on\" (P, 150). _Angels in America's_ theory of queer perestroika, in contrast, like Audre Lorde's understanding of the identity \"Zami,\" ensures the survival of the group. Just as the name Jacob wins from the angel in _The Book of J_ \u2014Israel\u2014symbolizes the survival of a people, so, too, the exchange of identities performed in _Angels in America_ underscores ongoing solidarity and interconnectedness.\n\nScott Tucker is one of the only critics to address directly the issue of solidarity in _Angels in America_ , and\u2014in contrast to my reading here\u2014he does so negatively: \"Solidarity, if it means anything, must be harder to come by than this very kind of unearned inclusion\" (34). What elicits such a response from Tucker is Kushner's allusion, in a _Village Voice_ interview, to an earlier version of the epilogue at the Bethesda Fountain:\n\nPrior used to have a section in his final speech where he said these very confrontational things: \"We won't die for you anymore, and fuck you if you can't accept it.\" I changed it because all the straight people in the cast came to me and said, \"We feel hurt by this. You ask us to go on this journey with you, and we go, and at the end you turn it into us-and-them.\"\n\nI felt very angry at first: \"Come on, grow up.\" . . . It felt like the whiny American thing of \"We're one big family. We are the world.\" . . . On the other hand, these were political, deeply decent people who were feeling something I did not want people to be feeling at the end of the play. That kind of political note would only work if it could be understood in the context of an embracing gesture the play is making that I want the play to make. (qtd. in Tucker, 34)\n\nTucker concludes from this episode that \"Kushner responded like a social worker keeping harmony in a support group\" (34) and suggests that \"any playwright dealing with AIDS in America must lead the audience through a _convincing_ stretch of hell before revealing a bit of heaven. Or even before revealing a bit more decency on earth\" (35).\n\nYet Kushner's \"compromise\" might be read otherwise, as part and parcel of the queer theory of perestroika Kushner advances throughout _Angels in America_ , particularly in the second half. Queer perestroika admits a wide variety of transformations, in contrast to Tucker's attack on Kushner with its queerer-than-thou implication that there are acceptable and unacceptable transformations. Generally, the Queer Renaissance resists such fixed notions of queerness. As my previous chapters have demonstrated, different contexts and locations (rural North Carolina, the U.S.-Mexican border region) require and produce different understandings of queerness. Moreover, in coalitions across difference and in conversations with \"political, deeply decent people\" (even if they are, in one sense, \"the straight people in the cast\"), multiple actors continually reinvent these local queer identities.\n\nTucker admits that he has not yet seen _Perestroika_ (35)\u2014a significant admission, since his conclusions might differ considerably if the event he objects to were read in the context of the play as a whole. Prior's comments are staged, after all, and their meaning is contingent on that staging. In the epilogue, Prior is flanked by three characters, and three characters only: Belize, Louis, and Hannah. All three have come through Prior's struggle with him, and two of these characters (Louis and Hannah) have been substantially transformed in the process. Hannah, particularly, should be seen as someone who, over the course of the play, enters into a conversation with queerness, and whose personal and political identity is shaped in new and critical ways as a result of that conversation. If this is not\u2014 _pace_ Tucker\u2014\"meaningful\" solidarity, not to mention a primary goal of queer theory and activism, I'm not sure what is. Moreover, this scene obviously locates several characters _offstage_ \u2014most notably, Roy Cohn and his protege Joe Pitt. Certainly, _Angels in America_ affirms the \"humanity\" of both characters (no one deserves to die of AIDS, not even Roy Cohn, and the play maintains this profoundly), but neither Roy's nor Joe's philosophy admits either character to the vision of political solidarity Kushner stages in the end. In fact, in one of Louis's most important developmental scenes, he literally and _physically_ rejects Joe's politics. After researching and reading the opinions Joe wrote during the early 1980s for Judge Theodore Wilson, Second Circuit Court of Appeals, Louis confronts Joe, and a fight breaks out. Louis receives a cut eye and mouth in the process and says to Joe, who is mortified at having hit Louis, \"I could have you arrested. . . . They'd think I put you in jail for beating me up. . . . But it'd really be for those decisions\" (P, 112).\n\nTucker feels Kushner is premature in his celebration of \"decency on earth,\" and that the characters need to go through a \"convincing stretch of hell.\" Ironically, Louis opens this confrontation with Joe by quoting Joseph Welch, who, at the Army-McCarthy Hearings, exclaimed, _\"Have you no decency, at long last, sir, have you no decency at all?\"_ (P, no). In other words, Kushner\/Louis, far from celebrating decency on earth, invokes one of the most glaring and famous examples of political _indecency_ and implicitly suggests that those who engage in such politics\u2014whether in the 1950s or the 1980s\u2014cannot be part of the progressive vision of inclusion performed in the epilogue. These are not easy lessons for Louis to learn\u2014one might even say that he goes through a convincing stretch of hell\u2014but they are crucial for his development away from the individualism and betrayal he performs for the bulk of the play.\n\nGiven what Louis and some of the other characters _do_ go through, then, it is hard not to think that Kushner's straight cast members had a point when they insisted they were with him for the journey. And Kushner's acceptance of that point, I think, might be read as part of the queer perestroika staged by _Angels in America_ , if ongoing democratic contestation is indeed part of that queer theory. Significantly, although Tucker casts the actors in a relationship of dependency (Kushner as social worker, actors as support group), Kushner himself _refuses_ such a relationship: dispensing with his initial impulse to interpret his actors as whiny children (\"Come on, grow up\"), Kushner moves instead toward understanding them as adults with political commitments, including the commitment to fight alongside him. At the same time, Kushner rejects compromise when it obscures that commitment. When a Broadway producer asked him to delete the play's subtitle\u2014 _A Gay Fantasia on National Themes_ \u2014rather than risk alienating potential theatergoers, Kushner refused (Sandla, 31). With the Broadway producer, Kushner rightly saw an effort to mask _Angels in America's_ queer political perspective and to pander to those who might be made uncomfortable by that perspective; with the actors, who compose this \"gay fantasia\" with him, he acknowledged that gay and nongay others _share_ that critical queer perspective, and that queer perestroika demands a recognition of such interdependence.\n\n\"Political identities remaking identities are . . . productive of collective political struggle,\" Douglas Crimp writes, \"but only if they result in a broadening of alliances rather than an exacerbation of antagonisms. And the latter often seems to result when, from within development toward a politics of alliance based on relational identities, old antagonisms based on fixed identities reemerge\" (\"Right On,\" 16). In _Angels in America_ , Kushner attempts to negotiate this dilemma, using a constantly shifting yet insistent queer sexuality as a figure for the progressive theory his play stages. Moving characters toward a renunciation of stasis, isolation, and fixed identities, _Angels in America_ proffers as an alternative the performance of queer perestroika: messy, mingling, and ultimately hopeful. \"Marx was right,\" Kushner insists in his afterword, \"the smallest indivisible human unit is two people, not one; one is a fiction. From such nets of souls societies, the social world, human life springs. And also plays\" (P, 158). Kushner's staging of Marx's queer theory makes _Angels in America_ one of the central texts of the contemporary Queer Renaissance.\n\n### **The Queer Renaissance: Taking It to the Streets**\n\nEarlier in his analysis of \"political identities,\" Crimp cites Teresa de Laure-tis's observation that \"it takes two women, not one, to make a lesbian\" and concludes that \"we can begin to rethink identity politics as a politics of relational identities formed through political identifications that constantly remake those identities\" (Crimp, \"Right On,\" 12; de Lauretis, \"Film and the Visible,\" 232). Crimp's invocation of de Lauretis in an article on ACT UP links desire and activism; constantly shifting queer _sexual_ identities are here articulated to shifting, coalition-based _activist_ identities. As the previous section demonstrated, _Angels in America_ performs a similar articulation: Prior's understanding and appropriation of the angel's fabulous sexuality provides a basis for the collective queer political theory envisioned in the end.\n\nThe theater can be an important site both for the transmission of such queer theories and for the constant remaking of identities that Crimp advocates. David Roman suggests as much when he writes:\n\nFor critics who argue that gay and lesbian performance only preaches to the \"choir,\" it is important to recognize that such a choir does not exist unless one presupposes . . . that there is only possible a singular monolithic community of activists which reflect the \"ideal spectator.\" In any event conversion entails more than simply rhetoric; it demands conversion of ideas and belief systems into direct action, a possibility only beginning to come into fruition. (217\u201318)\n\nFor Roman, the \"community\" is neither monolithic nor necessarily shaped in advance. Furthermore, the \"conversion of ideas and belief systems into direct action\" Roman calls for implies that the theater is contiguous to other locations. Kushner himself champions that contiguity; in fact, for Kushner, conversion moves in both directions. _Angels in America_ reshapes the ideas _of_ gay and lesbian direct action and insists, in turn, that the ideas performed in the play be reshaped yet again _in and through_ direct action. As Kushner's closing remarks in the keynote address he delivered to the Fifth Annual OutWrite Conference suggest, the conversion of belief systems and identities takes place in many different and ever-multiplying locations: \"See you in Washington. See you on the Internet. See you on the streets\" ( _Thinking_ , 79). Hence, the reifying force of a Pulitzer Prize and Tony Awards notwithstanding, Kushner's achievement is not isolated: _Angels in America_ speaks directly to and with contemporaneous developments in queer theory and activism.\n\nThis is not to deny that powerful mainstreaming forces occlude such a connection. Indeed, the mainstream culture industry's mystification of social relations could be seen quite clearly when Kushner accepted his Tony Awards (particularly his second). As Kushner attempted to verbalize his debt to his \"lesbian and gay brothers and sisters\" who continued to struggle against AIDS and homophobia, the orchestral music swelled to indicate that his time was up and that the television broadcast was about to cut to a commercial. In this scenario (and in awards ceremonies generally), the supposed singularity of Kushner's achievement was the important thing, not the ongoing dialogue he wanted to highlight.\n\nIn this section, then, I aim to amplify, rather than drown out, the critical conversation between art and activism by turning to Sarah Schulman's novel _People in Trouble. 14_ I continue to argue here, through an examination of Schulman's character Kate, that various liminal (sexual, activist) \"identities in crisis\" can breed queer artistic identities and visions. Schulman places Kate in a context that specifically evokes the tax abatements granted by Ed Koch to Donald Trump for the construction of Trump Tower, his multimillion-dollar luxury shopping mall and apartment complex, but that\u2014with its representation of privatization and of tax cuts for wealthy entrepreneurs\u2014more generally records the mayoral strategy since Koch came to power. In Schulman's novel, the mayor of New York City offers real-estate developers millions in tax rebates if they agree to fund public artwork on their properties. Kate, a visual artist, suddenly finds herself commissioned to produce a major public artwork outside a former public building that the city is converting into private office space. Significant changes in Kate's private life, however, are simultaneous to this professional development: first, her identity as a married and heterosexual woman is disrupted as she begins to juggle life with her husband, Peter, and her lesbian lover, Molly; and second, her complacent political identity is thrown into crisis by her exposure to an ACT UP-like organization called Justice. Analyzing each of Kate's identities (sexual, political, artistic) in turn, I argue here that the critical postmodern aesthetic Kate eventually brings to her commission emerges directly from these new sexual and activist identities. This portrait of the artist as an emergent queer then sets the stage for some concluding thoughts on _Angels in America_ and\u2014in the epilogue that follows this chapter\u2014for some final reflections on the Queer Renaissance generally.\n\nAs with _Angels in America, People in Trouble_ rumbles with apocalyptic anxiety; like the first half of Kushner's play, Schulman's novel could easily be subtitled \"Millennium Approaches.\" The novel opens: \"It was the beginning of the end of the world but not everyone noticed right away Some people were dying. Some people were busy. Some people were cleaning their houses while the war movie played on television\" (1). In the midst of this chaos Schulman places Kate, who is happily buying two bras for Molly: \"Which one would Molly wear first?. . . There was the demure lace that opened from the front, like walking in through a garden gate. Then there was the really dirty push-up that didn't need to open. Kate could lift Molly's breasts right out over the top\" (2). The immediate juxtaposition of chaos and desire serves somewhat contradictory functions. On the one hand, Kate's desire keeps her from noticing right away that the world is spinning out of control. She is not one of those who is \"dying\" but rather one of those who is \"busy\": she is buying bras for her lover and \"enjoying herself thoroughly\" (1). On the other hand, _People in Trouble_ constructs desire as something that cannot and will not be eradicated; in the midst of the apocalypse, people are still having sex. As Prior comes to see in _Angels in America_ , and as the central argument of the first section of this chapter insisted, \"We can't just stop. . . . It's _animate_ , it's what living things do. We desire\" (P, 132).\n\nHence, although Kate is apparently oblivious in this scene to the chaos outside, her budding desires nonetheless connect her (along with almost everyone) to the ongoing messiness and vitality of human existence. Such contradictions are central to Schulman's characterization of Kate, and as Kate begins to face these contradictions, they propel her supposedly stable identity into crisis. In general, especially early in the novel, Kate and Peter are almost cartoons, expressing cartoon liberal views: \"They talked details. All details. The contents of that day's _Times_ , including which airlines had proposed merger. The plight of the American farmer. Something having to do with percentage points. Both Kate and Peter clearly believed in quoting statistics\" (54). About gay men, lesbians, and AIDS, in particular, the two are virtual parodies of themselves: _\"These are men with AIDS_ , Peter realized. . . . _But that one doesn't look like he has it. He looks like he works out. . . . That black man_ , thought Peter. _I wonder if he's gay or if he got it from drugs\"_ (58). Although Peter is by far the worst, Kate shares many of his naive conceptions. She worries, \"With Molly so many things could go wrong. . . . She'd trap me, try to turn me into a lesbian\" (21), and she insists defensively, even late in the novel, \"I don't want to live in a ghetto like lesbians do. I like men. I want to be universal\" (186). Both Peter and Kate repeatedly express regret that Molly \"hates men,\" despite overwhelming evidence to the contrary. In fact, Molly has many more male friends than either Kate or Peter; they are simply gay men, men with AIDS, men of color, homeless men. These men cannot disrupt Kate and Peter's thesis, since for this couple, \"man\" equals \"Peter\" and those like Peter and is unmediated by any other arena of difference.\n\nBut Kate's introduction and persistent connection to the vicissitudes of desire complicate Schulman's construction of the couple as liberal-hetero cartoon. Because of the queer changes occurring in Kate's life, profound differences between Peter and her eventually materialize. As the novel progresses, Peter shores up the unexamined liberal belief in a secure distinction between self and other, even as that belief disintegrates for Kate. Initially, both characters are presented behind walls or windows, with the world on the other side: \"It had been a hallucinatorily hot summer with AIDS wastes and other signs of the Apocalypse washing up on the beaches. Kate had spent it working in her studio, only in the evening\" (12). One evening, in particular, Kate witnesses the Tompkins Square Park Riots, in which four hundred police officers, in August 1988, rioted against the homeless, housing activists, punks, anarchists, and other locals, who were protesting a curfew. As she watches this struggle, Kate consciously decides to maintain the barrier between herself and the chaos outside: \"She stayed at the window watching and then made the decision not to enter into it\" (12). Peter, likewise, first appears with barriers between himself and the world outside; one early chapter has him safely ensconced in a coffee shop as he watches a gay funeral across the street. But whereas Peter uses \"the other side\" to reinforce his (heterosexual) sense of self, Kate probes queer desire in ways that begin to reshape her identity.\n\nI certainly do not mean to suggest, however, that Peter's heterosexuality is not (always already) an \"identity in crisis,\" or that his identity is not reshaped by the lesbians and gay men he encounters. In the coffee-shop scene, for example, as he watched the mourners file into the church, Peter noted\u2014\"with some relief\" that their life was so \"clearly defined\"\u2014that \"his and Kate's inner circle were all heterosexual couples. It had just turned out that way\" (31). Peter's remembrance of a brief gay affair from his past, though, immediately undercuts this observation. Peter, who is a lighting and set designer (or, we might say, a master at compelling others to believe in illusory identities and locations), dismisses the affair as \"part of being in theater\" (32), but his incessant need to situate himself _as against_ the queer other (\"that's not me\") nonetheless manifests his anxiety about his performance as \"heterosexual male.\" This definition as against the other resurfaces as Peter's curiosity gets the best of him and he crosses the street to the church where the funeral is taking place. Although crossing the street apparently belies the separation of self and other Peter seeks to maintain, he thinks of what he is doing as \"touring,\" and notes to himself that he \"had learned from traveling in Mexico that when you are watching another culture in church it is best to stand in back\" (33). The gay mourners thus become the foreign others who provide an identity for the nervous tourist, who can supposedly leave at any time.\n\nPeter's need to establish such a \"secure\" identity, repeatedly, suggests that in an important sense, Peter's identity, like Kate's, is unstable and \"in crisis.\" The couple diverge, however, in their responses to that instability Peter is always on the defensive and desirous of moving _away from_ queerness; he anxiously demands of a woman he meets after he and Kate have begun to drift apart, \"You're not gay, are you? You're not bi or unsure or in transition? You're heterosexual, right?\" (170). Kate, in contrast, although not entirely free of anxious disavowals of lesbian identity (\"I want to be universal\"), eventually moves _toward_ queerness and allows her \"heterosexual\" identity to be radically reshaped.\n\nIronically, the language describing Peter's visit to the gay church resurfaces near the beginning of Kate's affair, when Molly takes Kate on \"a guided tour of all the lesbian bars below Fourteenth Street\" (81). However, the effect this trip has on Kate, the \"tourist,\" helps illustrate her difference from Peter, since by the end of the chapter, the word no longer seems quite appropriate. At the first bar they visit, Kate encounters a woman she knows whose husband is a sculptor. Kate turns to Molly: \"Isn't that great? . . . That a straight woman like Susan can feel comfortable coming to a place like this\" (81\u201382). Kate's cheerful exuberance is checked, however, when she learns from Molly\u2014who goes on to imply that \"straight\" is as inadequate a descriptor for Kate as it is for Susan\u2014that Susan came to this bar to meet her female lover. Kate muses, \"How weird about Susan. . . . It makes me feel like I don't really know her. Like I don't have any idea of who she is\" (82). Kate's sense of dislocation soon escalates when, at the next bar, Molly attempts to explain butch and femme identities:\n\n\"Everyone here is femme.\"\n\n\"How can you tell who's femme?\"\n\n\"That's the question of the year. After a while you just know. Usually it's the one who puts her arms around the other woman's shoulders when they're dancing.\"\n\n\"Which one of us is femme?\"\n\n\"Neither. That's a joke. It's too early to tell because you still act straight.\n\nYou have to be out a little longer before these subtle nuances take shape.\"\n\n\"Why are you always telling me what I'm going to become and what I'm going to think? How do you know?\" (82)\n\nAlthough Kate's reaction again bespeaks some anxiety about new and queer identities, her curiosity gets the best of her and she pursues the butch\/femme issue further, telling Molly at the next bar that she still cannot see the difference. To clarify, Molly asks Kate which women she finds attractive and then informs her, \"Those are all butches\" (84). Kate then begins to sort this out on her own as Molly moves onto the dance floor with someone else. The crisis intensifies as Kate watches the two women:\n\nKate watched them for a while, the way they picked up each other's rhythm  \nand figured out how to move together. They figured it out rather quickly.\n\n_What do you know?_ she thought. _Molly dances hot with everyone. It's not just me. That's the way she dances_.\n\nAnd for the first time ever, Kate felt jealous.\n\n\"Excuse me,\" she said, cutting in abruptly, \"but I have to dance with my girlfriend.\" And took her to the floor, noticing immediately that Molly's arms went around her neck. (84)\n\nThe conclusion of this passage in particular suggests that Kate is no longer a tourist in the queer world she and Molly have been exploring; moving literally from the margins to the center, Kate declares aloud her implication in that world: \"Excuse me, but I have to dance with my girlfriend.\" Moreover, Kate \"notices immediately\" that her incorporation into this system of desire does not secure for her a fixed identity position but rather one that is constantly shifting: although her initial identifications of the women she finds attractive position her temporarily as \"femme,\" Molly's arms around her neck reposition Kate, a page later, as \"butch.\"\n\nUltimately, Kate's re-formation around butch and femme identities marks only a beginning for the larger transformation incited by her night out with Molly. Molly begins to trouble Kate's unconscious need to return to a default, normative position: her need to \"set things straight,\" as it were. The reshaping of this normative position, and of Kate's former identities (\"heterosexual,\" \"tourist\"), continues even as the evening concludes:\n\nMolly announced one last stop.\n\n\"Where could we possibly be going now?\"\n\n. . . \"It's a vintage gay porn store. You know the necessities stay open later than frivolous indulgences.\"\n\nKate felt one second of resistance before walking through the front door. (84)\n\nOnce inside the porn store, pictures of \"cocks . . . on beautiful young men\" entrance Kate (85). At first, this experience again appears to fix Kate's identity: \"This is how I know I'm not a lesbian. . . . Because I'm turned on by cock. I like cock\" (86). Molly, however, once again disrupts this identification, bringing attention not only to the normative voices _speaking through_ Kate but also to the voices _silenced_ by the compulsion to set things straight:\n\n\"Good for you. Does Peter have to hear you say, T like pussy'? Bet not. . . . Have you ever said to Peter, 'I like pussy'?\"\n\n\"No I haven't,\" [Kate] said. \"It has never occurred to me to say that. It wouldn't be appropriate.\"\n\nThen she felt uncomfortable. (85)\n\nMolly's challenge and Kate's discomfort spotlight the new location into which Kate has moved: \"it wouldn't be appropriate,\" any longer, for her to ground her sexual identity through a definition that positions her as against queerness. Kate's discomfort here attests to her nascent understanding that she has inappropriately used disclaimers to shore up an impenetrable heterosexual identity from which she excludes her love for women. It can certainly be true, given her desire for men and her ongoing relationship with Peter, that she is \"not a lesbian,\" but Molly's comments implicitly remind Kate that it is true because she is bisexual or because she is queer, not because her love of \"cock\" insulates her heterosexual life. Just as her love for men continues to resonate in her \"lesbian\" world, so, too, her love for women now resonates in and reshapes her \"straight\" world.\n\nThe chapter ends with images that serve as analogues for the \"in-between\" position into which Kate has moved. Alone in her studio, Kate thinks about the evening and masturbates: \"Her face showed great pleasure. She could rock down on her hips and swing into a low moan. She could dance around her studio being led by her own hand\" (86). This erotic moment is stimulated by a magazine\u2014a present Molly gave her at the end of their evening on the town. Molly purchased the magazine at the porn store because it made her think of Kate immediately, but she waited until they were alone to present it and insisted that Kate not open it until Molly was gone. \"That had been a week ago,\" the narrator explains. \"She'd opened it a dozen times since then. It was a collection of transsexuals in various poses. . . . It was packed with photos of euphorically happy men in sexy, slimy, girly getups with hard pricks and big boobs. They looked so turned on. They turned her on with their dicks and tits, how excited they were\" (86). The pictures of transsexuals underscore, especially in their depiction of these men as \"euphorically happy,\" the promise of the shifting location Kate now occupies. Appropriately enough, given the identity crisis that has commenced in this chapter, Kate masturbates in front of the mirror. At this point, both magazine and mirror reflect identities that are no longer fixed in any one gendered or sexual location.\n\nMolly's gift again unsettles Kate and Peter's easy assumption that \"Molly hates men.\" Quite the opposite is true, it would seem, since Molly is so comfortable giving a magazine full of \"men\" to her girlfriend, knowing all the while that the magazine will fuel Kate's erotic fantasies. Kate and Peter (mis)read as \"hate\" Molly's adeptness at making \"gender trouble,\" in Judith Butler's sense: \"subverting and displacing those naturalized and reified notions of gender that support masculine hegemony and het-erosexist power . . . through the mobilization, subversive confusion, and proliferation of precisely those constitutive categories that seek to keep gender in its place by posturing as the foundational illusions of identity\" ( _Gender Trouble_ , 33\u201334). Ironically the \"cock\" itself is a constitutive force here, since Kate uses it, when she enters the porn store, to shore up a safe location for herself (\"not a lesbian\"). This safety is exposed as an illusion by Molly's gift, however, which actually proliferates cocks, thereby detaching them from a fixed location and draining them of their ability to confer on Kate a rigid heterosexual identity.\n\nSignificantly, though, and in contrast to Peter, Kate begins to take pleasure in that loss of rigidity and moves toward queerness. While masturbating in front of the mirror, for instance, Kate \"could feel her orgone rushing inside her like a waterfall, like crowds of teenage girls held back from the Beatles, who suddenly in a tearful frenzy, break free of the police and lose control. She was open in every way. There were no obstacles. She was streaming\" (86). Kate's erotic thoughts reveal that she has learned the lesson of the porn store: there is no sign (whether that sign be \"cocks\" or \"teenage girls thrusting toward the Beatles\") that is established, absolutely, as heterosexual. Molly has taught Kate that signs can be \"queered,\" and Kate here demonstrates what a good student she is. Kate queers the ostensibly heterosexual image of the girls and the Beatles: her desire is directed much more toward the crowd of surging females than toward the Beatles, the pleasure in the image comes more from the idea of women collectively thwarting (presumably male) police control than from connection with the band, and the whole image is metonymically connected to the queer images of transsexuals to which Kate is masturbating. At this point, moreover, Molly no longer choreographs Kate's development alone. Instead, larger queer forces (happy men in slimy, sexy, girly getups; thrusting teenage girls), now welcomed by Kate herself, begin to move Kate's transformation forward.\n\nWhen Molly and her friend Pearl unexpectedly notice Kate walking past the park, Molly's surprise at what she sees shows how far Kate has come. Pearl's comments, in fact, suggest that Molly herself has not yet recognized the pleasure Kate takes in her new identities, or\u2014for that matter\u2014Kate's agency in the transformation process: \"You say she's not coming out, but take a look at that\" (89). The two women watch as Kate, dressed as a man, walks down the street:\n\nKate was a man. Anyone in the street would have thought so. But she was a better man than most because she was so strikingly handsome in her black suit. She strode powerful and erect like a well-bred charming man. A male model perhaps. A movie star. She didn't wear a white button-down. She was much too stylish for that. Kate, the man, wore a soft blue shirt designed for a sexy strong man's leisure. It was cut to hang from his neck and muscles. Kate was thrilling. She was the most handsome man on the street. (89\u201390)\n\nWhen Pearl asks Molly if she knows what's going on with Kate, Molly \"stop[s] one more time to take a hard last look\" and responds, \"I have no idea\" (90). Kate's willingness to move toward queerness reshapes her identity to such an extent that it cannot be contained by those observing her; neither side of the heterosexual\/homosexual binary can fully account for her emerging queer subjectivity.\n\nThese developments in Kate's \"personal\" life are simultaneous to, and intertwined with, new developments in her \"political\" life. Although Kate smugly understands herself as \"progressive\" or even \"radical,\" she is fairly complacent politically at the beginning of _People in Trouble_ , subscribing to little more than a vague humanism. An encounter she has in a coffee shop exemplifies her complacency. A woman in the coffee shop informs Kate that she has just bought a new raincoat. Kate's self-satisfied political identity is thrown into crisis by this encounter as decidedly as her \"heterosexual\" identity is by her travels with Molly:\n\n\"When you get something new,\" [the woman] said conspiratorially, \"you have to watch out that it doesn't get stolen. You have to avoid people who need money and people who need raincoats and keep them away from yours. But I feel bad being dry on the street when my brothers and sisters have nowhere to sleep.\"\n\n\"And contradictions are what let us know that we are fully human,\" Kate said.\n\n\"But?\" the woman answered, waiting.\n\n\"But?\"\n\n\"But,\" she said running old-looking hands over a young-looking face. \"But then what?\" (13)\n\nKate is bothered by the challenge, to which she can provide no response. She had wanted to use \"contradictions\" to _settle_ her identity: contradictions \"let us know that we are fully human.\" Yet, just as the contradictions in Kate's sexual identity begin to reshape that identity so, too, do contradictions in scenes such as this one unsettle and reshape Kate's \"human(ist)\" political identity. In the text, the encounter in the coffee shop follows Kate's memory of watching the Tompkins Square Riots from her studio. Unsettling or disruptive events are thus multiplying before Kate's very eyes, and at a rate that mocks her attempts to smooth over or harmonize the contradictions through appeals to the \"human\" or \"universal.\"\n\nAs unrest in the New York City of _People in Trouble_ ferments, Schulman again positions Kate initially on the outside: Kate is not present in the chapter that introduces Justice, the fledgling AIDS activist organization. Nonetheless, Justice brings attention to issues of crisis and liminality similar to those I have been examining in conjunction with Kate, and as Molly infers at the end of the chapter, Kate and Justice could go well together. Initially, Molly alone attends a vigil for those lost to AIDS. The vigil underscores both the ways in which Molly's life has been altered by AIDS and\u2014through its direct echo of Kate's appeal to the \"human\"\u2014the insufficiencies of Kate's political philosophy:\n\nIt made [Molly] feel something very human; a kind of nostalgia with public sadness and the sharing of emotions. But then what?\n\nThis dying had been going on for a long time already. So long, in fact, that there were people alive who didn't remember life before AIDS. And for Molly it had made all her relations with men more deliberate and detailed. First, the men changed. They were more vulnerable and open and needed to talk. So she changed. Passing acquaintances became friends. And when her friends actually did get sick there was a lot of shopping to do, picking up laundry and looking into each other's eyes. She had never held so many crying men before in her life. (44\u201345)\n\nThe vigil commemorates these friends of Molly's, and the many others, male and female, who have died. But as Molly watches balloons with the names of the dead rise to the sky, and as groups of people break into small groups and depart, her mourning is checked: \"Molly felt enormous anger. These were her friends. These were her dead friends. She saw their faces. Were their lives worth less than the lives of heterosexuals? Where was Kate? She should be here at a time like this\" (46). Molly's sense that her mourning is not validated represents a common response to societal complacency about AIDS. Crimp explains that \"for anyone living daily with the AIDS crisis, ruthless interference with our bereavement is as ordinary an occurrence as reading the _New York Times_. The violence we encounter is relentless, the violence of silence and omission almost as impossible to endure as the violence of unleashed hatred and outright murder\" (\"Mourning and Militancy,\" 237). In this scene, Schulman, like Crimp, connects complacency and quietism to an entire system that values only \"straight\" lives. In the process, she once again inverts the stereotype that lesbians hate men. As Crimp's placement of \"silence and omission\" on a continuum with \"unleashed hatred\" demonstrates, and as Molly's angry meditations confirm, it is not lesbians (who have, after all, \"never held so many crying men before\") but rather the heterosexual system that \"hates (gay) men\" and people with AIDS in general. Awareness of this systemic hatred unsettles Molly and elicits a string of questions for which she has no answers.\n\nLike Prior, however, Molly prepares to move from an uncertain position to a contiguous, communal location. As in _Angels in America_ (and many other works in the Queer Renaissance), it is not contradiction per se but contradiction articulated as a collective and ongoing challenge to oppressive systems of power (we\u2014first-person _plural_ \u2014deviate) that provides a basis for action and change. Schulman depicts this movement from contradiction to collectivity by turning directly from Molly's questions, in one paragraph, to Justice, in the next:\n\nMolly saw two men handing out leaflets. . . . The first thing she saw was that they were wearing black T-shirts. On their chests were large pink triangles with the word _justice_ scrawled, graffiti style. . . .\n\nDO YOU THINK IT'S RIGHT?\n\n_That people are dying and the government does nothing? If you do not think that this is right then do something about it_.\n\nThe flyer went on to invite people to a weekly meeting. Molly folded it four times and pushed it into her pocket. She missed Kate very much. She wished Kate were there. Molly walked home feeling open and vulnerable and then very angry with an energy that had nowhere to go. (47)\n\nThe chapter thus concludes with a return to Kate\u2014another character who might be described as having an energy with nowhere to go. At the same time, though, the chapter belies the idea that such queer energy need go nowhere. Kate and Molly can come together with other \"people in trouble\" in a group like Justice, and it is there that the Butlerian implications of \"trouble\" come out. Schulman's title, certainly, can call to mind weak or helpless people, caught or undone by trouble: \"victims\" in need of rescue. As the novel progresses, however, that passive and objective sense of the title gives way to a more active and subjective sense: these people collectively _make and mobilize_ trouble to disrupt governmental and societal complacency and bring an end to the dying.\n\nMolly carries her newfound uneasiness and dislocation back to her relationship with Kate: \"Here we are trying to have a run-of-the-mill illicit lesbian love affair. . . . And all around us people are dying and asking for money\" (113). Kate shares Molly's dismay at the suffering all around them, but she nonetheless tries to smooth over the contradiction Molly sees by invoking her own identity as \"artist\":\n\n\"I'm not protected,\" Kate said. \"I'm a poor artist. I am not a powerful person in this society. Don't be so self-deprecating, it's unbearably righteous.\"\n\n\"You're not poor. Neither am I.\"\n\n\"Listen, Molly, when I was your age I was a lot more radical than you are, so don't lay that on me. That's your trip.\"\n\n\"Let's go to the meeting,\" Molly said. ''Let's go there now.\" (114)\n\nThis scene between the two women generates a paradox. On the one hand, Kate\u2014up to her old tricks again\u2014disavows elements of her identity that might disrupt the ways in which she perceives herself. Ironically, she disavows the _privileged_ part of her identity here, but the effect is similar to her earlier disavowals of queerness: Kate constitutes her identity as \"artist\" and \"radical\" as fixed and insulated (in fact, with her appeal to the past, she practically _embalms_ her radical identity). On the other hand, Kate reveals, however tentatively, the impossibility of absolute fixity and insulation. She contends that AIDS or homelessness \"could happen to me. . . . It could happen to you\" (113). Although she makes this pronouncement with a fair amount of smugness and with the intent of \"winning\" a debate with Molly, Kate has a valid point here, if we take seriously her earlier negotiations of self and other in the lesbian bars, the porn store, and even the coffee shop, where the woman in the raincoat spoke of her \"brothers and sisters\" on the street. Kate learns that the separation between self and other is not absolutely secure, and in many ways, the queer identity she brings to this conversation\u2014an identity Schulman reinvokes by describing Kate here as \"a peacock in a man's overcoat\" (114)\u2014attests to that point. Perhaps this explains why Molly ends the conversation (and the chapter) not with an insistence that Kate is simply wrong or misguided but with the suggestion, \"Let's go to the meeting. . . . Let's go there now.\" Kate indeed may be somewhat misguided in her self-satisfied understanding of artists and radicals, but Molly also apparently recognizes in Kate's willingness to see herself as implicated in the struggles of the homeless and of people with AIDS an \"energy\" not unlike Molly's own. The repetition of _go_ (\"go to the meeting,\" \"go there now\") reanimates Molly's earlier \"energy that had nowhere to go.\" Molly comprehends her uneasiness and dislocation, along with Kate's naive but promising identity-in-process, as preconditions for involvement in Justice.\n\nIn her description of the direct action by Justice that follows, Schulman draws on more than one actual ACT UP event. In June 1987, when the Third International Conference on AIDS was set to open in Washington, D.C., activists descended on the White House to protest the Reagan administration's apathy toward AIDS issues. (Ronald Reagan had spoken the word _AIDS_ publicly only in the previous month.) Police officers wearing bright yellow gloves arrested sixty-four protesters at the White House. Activists, desirous of drawing attention to the officers' paranoia, which would feed public hysteria about \"catching\" AIDS through casual contact, chanted for the television cameras, \"You're gloves don't match your shoes! You'll see it on the news!\" (Crimp and Rolston, 32\u201333). Schulman relocates the yellow-glove incident and the chants to New York City.\n\nMore centrally, however, Schulman re-presents a November 1988 action, when ACT UP rallied to protest the fact that five thousand to ten thousand homeless New Yorkers were living with AIDS. ACT UP demonstrated at the Trump Tower in New York City on the day after Thanksgiving, \"the biggest shopping day of the year\" (Crimp and Rolston, 122). The flyer activists distributed during the demonstration explained the reasons for their anger:\n\n* Donald Trump received a tax abatement of $6,208,773 to build Trump Tower. This money could have rehabilitated about 1,200 city-owned apartments. Instead Trump gets richer while homeless people get sicker.\n\n* The city favors private developers like Trump over small community-based organizations that are committed to providing low-income housing and housing for PWAs. The AIDS Center of Queens County, for example, never even received an acknowledgment of the proposal it submitted for 20 scattered-site apartments for homeless PWAs.\n\n* Mayor Koch and the city allow Trump to displace people from their apartmerits when he wants to build a building. Trump is symbolic of a system that lets the rich do whatever they want while letting the poor die. (Crimp and Rolston, 122)\n\nThe Trump Tower was the ideal location for the points activists wanted to make. Containing both a luxury shopping mall and million-dollar apartments, the Tower stands, for a few, as the symbol of a \"healthy economy.\" For activists, however, the Tower stood and stands for the ways in which the economic system _denies_ health, well-being, and housing to many.\n\nACT UP's flyer exemplifies Rosalyn Deutsche's reworking of Walter Benjamin's famous dictum \"There is no document of civilization which is not at the same time a document of barbarism\" (Benjamin, 256). With a nod to Benjamin, Deutsche insists that \"New York's spatial documents of ascendancy\" and redevelopment are always at the same time \"documents of homelessness\" and eviction (\"Uneven Development,\" 123). In Schulman's re-presentation in _People in Trouble_ of ACT UP's demonstration, this central paradox fuels the action Justice takes. At the meeting, activists discuss Ronald Home (Schulman's fictional answer to Donald Trump), with no illusions about how his wealth requires and engenders other people's poverty:\n\n\"This week many of you received eviction notices from Ronald Home's development company. This is the man who has warehoused thousands of empty apartments while ninety thousand people live in the subways and stairwells and public bathrooms of this city. Now we have learned that he has purposely bought buildings with more than fifty percent gay tenants in the hope that we will drop dead and leave him with empty apartments. He files these eviction notices anticipating that some of us will be too ill to contest. Now let me ask you, what are we going to do to get _justice_?\"\n\nThere was a great steamy silence when [the speaker] finished, almost like _subito piano_ in music; the quiet after a crescendo, like falling off a cliff. . . .\n\n\"We should go there now,\" [one man] said. \"We're angry now, so we should go now.\" (118\u201319)\n\nLike the bass in a jazz ensemble, Schulman's throbbing \"go . . . go . . . go\" keeps the beat here, reappearing once again to channel the collective energy generated by the band. The \"quiet after a crescendo\" is like Molly's \"energy that had nowhere to go,\" and again Schulman implicitly suggests that such energy can produce a disruptive queer identity. Additionally, this scene lends credence to Kate's earlier points about the tenuousness of the separation between self and other: gay people can indeed be threatened with eviction and homelessness. Justice, however, takes that point much further than Kate\u2014or rather, Justice provides Kate with a \"demonstration\" of what might be done with those points. Through this demonstration, Schulman represents some of the basic tenets of ACT UP. First, it is an AIDS _Coalition:_ AIDS activism entails recognizing that discrete, or apparently discrete, identities (\"gay,\" \"lesbian,\" \"homeless,\" \"PWA [Person with AIDS]\") can come together to \"unleash power\" and create change. Second, the identity of this queer creature\u2014the \"AIDS activist\"\u2014is not located in advance but is, instead, as constantly shifting as the location of her politics: the Trump Tower, St. Patrick's Cathedral, the White House, the Food and Drug Administration, the streets.\n\nSchulman shifts the location of Justice's initial action from the Trump Tower of ACT UP fame to the fictional Home Castle, which the narrator describes as \"the biggest, lushest, most ostentatious and expensive hotel from the Eastern Seaboard to Rodeo Drive . . . renowned, not only for its lavishness, but also for the transplanted tropical rain forest that had been re-created inside the lobby to serve as a symbolic moat with actual crocodiles\" (119). Schulman's shift to Ronald Home's Castle makes even more explicit the systemic connections between homelessness and economic \"growth\" that ACT UP sought to spotlight: Home, in _People in Trouble_ , is not simply \"housed\" while evicting others but housed in a literal and luxurious _castle_. Furthermore, Schulman's ironic inclusion of a \"transplanted tropical rain forest\" places the crisis in an even larger, global system, since actual rain forests in the Southern Hemisphere are being, in effect, \"transplanted\" North, as the South labors to support the North's consumption habits. Activists in _People in Trouble_ descend on and occupy this \"castle\" to draw attention to these systemic inequities and to protest the city's housing policies. Fifty pairs of Playtex Living Gloves, \"which turned out to be an unfortunate lemon yellow\" (126), are ordered by the hotel overseers for use in removing the demonstrators, who immediately begin to chant, \"Your gloves don't match your shoes,\" for the television cameras (126\u201327).\n\nThe cameras capture the activists on film, but Schulman shifts the focus, exposing the ways in which the camera's eye is itself far from innocent. In the process, Schulman reveals how her own participation in experimental queer film and video production may have influenced _People in Trouble_. Schulman has, in fact, been centrally involved in contemporary film and video production: in 1987, she and Jim Hubbard founded the New York Lesbian and Gay Experimental Film Festival, and she is herself a member of ACT UP, which has produced several experimental videos. From its inception, ACT UP displayed a heightened awareness of media strategies; activists, particularly activists with video cameras, both critiqued the ways in which the mainstream media has written \"the story of AIDS\" and documented that story otherwise, using a variety of experimental techniques. Paula Treichler writes that AIDS activist video work \"demonstrates the manipulation of conventional cultural narratives and representations to tell an alternate story\" (79).\n\nIn her representation of Justice's action at the Castle, Schulman manipulates the \"conventional cultural narrative\" through the use of a jump cut. After the chapter describing the meeting that Molly and Kate attend, Schulman shifts abruptly to a chapter focusing on Peter, who is watching a newscast on TV at a bar on Second Avenue. The television thus mediates most of the action. As the anchorman tells the story, \"Hundreds of AIDS victims have occupied the restaurant and lobby of Ronald Home's Castle in midtown Manhattan. They are demanding that the superstar developer rescind eviction notices sent to homosexual men in Horne-owned buildings\" (123). By splicing preparations for the demonstration together with newscast, however, Schulman complicates the televisual mediation: although the news anchor tells the same old story, that story speaks differently next to Justice's rally, which readers have just witnessed. Schulman's jump cut from the rally to the television newscast brings out _contesting_ narratives. The TV reporter, whose sympathies clearly lie with the \"superstar developer,\" calls these people \"victims,\" but the group itself refuses to be victimized. Schulman's wide angle exposes the media: journalists do not \"objectively\" report preexisting stories; instead, they actively produce both the stories and the interpretations that should accompany them. In this scene, Schulman's own camera work produces a story that demonstrates how the news aligns itself with corporate interests and contains the threat of activism.\n\nKate's political identity, however, now resists containment. _People in Trouble_ at this point positions Kate as a central player in issues of representation and contestation. Peter in fact recognizes that Kate is in the middle of it all: though he is \"barely able to get out the words,\" he suddenly exclaims aloud to those in the bar, \"That's my wife\" (124). When the bartender challenges Peter with the suggestion that there are only men in the crowd on the screen, Peter insists, \"No, that's her. The redhead with the crew cut in a suit and tie. That's my wife\" (124). Kate appears on TV, in \"earnest conversation\" with another activist, her \"orange hair lookfing] more mandarin on the eerie color TV\" (124). Readers never learn the topic of Kate's \"earnest conversation,\" but Kate does inform Molly later that this has been an \"important night\" for her and that it has given her \"a lot of ideas\" (127). Perhaps, given that Schulman places this visual artist at the center of a televised event seen by the people of New York City, we might imagine Kate discussing the conjunction of representation and activism. The cameras transmitting her image to viewers all over the city momentarily fix Kate in earnest conversation. Kate herself, however, has recently received funding for the production of a \"public artwork\" that\u2014as her agent explains\u2014will likewise be \"seen by the people of New York City\" (99). The \"ideas\" to which Kate refers, then, are very likely ideas about how to refocus the city's gaze through her artwork, which she undoubtedly sees in a different light after her experience with Justice.\n\nAs _People in Trouble_ progresses, Justice engages in several more direct actions that shape and reshape a queer AIDS activist identity. Fury, a group focusing on the specific needs of women with AIDS, joins Justice, and both groups demonstrate, for various issues, at locations around the city. Kate is not directly involved in all of these actions, but the changes in both her \"personal\" and her \"political\" identity persist, transforming her aesthetic. Antifoundational sexual and activist communities and movements trouble the heterosexual and humanist foundations of Kate's identity. As Deutsche writes, \"To localize meaning and identity\u2014to assert that they are confined in discrete sites and determinate origins\u2014is to deny the worldly relations that, in the absence of transcendental sources of truth, both constitute meanings and, crucially, put them at risk\" (\"Art and Public Space,\" 44). Kate cannot deny the queer relations that reconfigure her world as the text continues; these relations delocalize her identity and consequently put at risk the meanings she has heretofore invested in her artwork.\n\nAt the beginning of the novel, Kate's aesthetic might be described as \"high modernist\": she strives in her artwork for purity of form; she wants her artwork to be \"universal,\" to \"let us know that we are fully human,\" and to somehow rise above the contingencies of history and politics. But, as the narrator explains early in the novel, \"something was changing in the way she was seeing and it had started to affect her drawing\" (14). People in trouble, identities in crisis\u2014these are the referents for the elusive \"something\" that alters Kate's aesthetic.\n\nKate's meditation on the changes in her drawing follows the troubling scene in the coffee shop where the woman with the raincoat challenges her: \"But then what?\" The narrator's comments after this challenge confirm that it has provoked a crisis for Kate: \"There were three or four things that terrified Kate and they came to her in moments. . . . Kate feared the consequences of chaos, but was comfortable with fragments, when they were freely chosen\" (14). Kate founds her comfort, and her artistic practices, on \"free choice,\" but terror threatens to overcome her comfort and alter her artistic practices as the inadequacies of her high-modernist aesthetic ideal of \"fragments freely chosen\" become more and more apparent. The motto for Kate's aesthetic at the beginning of the novel\u2014a motto that encapsulates both her fear of the consequences of chaos and her comfort with freely-chosen fragments\u2014might be T.S. Eliot's famous line \"These fragments I have shored against my ruins\" (Eliot, 50). Yet this motto encapsulates all that is proving to be inadequate about Kate's high modernism as _People in Trouble_ continues: Eliot's declaration implicitly invokes a fully formed and autonomous self, both in danger of ruin from without and capable of shaping art that \"transcends\" chaos and history by unifying or harmonizing fragments and contradictions. In Eliot's poem, the possibility of transcendence brings peace in the final line (\"Shantih shantih shan-tih\" [50]), which onomatopoeically intimates that rain finally arrives to settle the contradictions and chaos of the Waste Land. In _People in Trouble_ , in contrast, Kate's \"terror\" suggests that this peace cannot be sustained; instead, Kate's identities in crisis unsettle the possibility of reliance on a fully formed, autonomous self.\n\nKate thinks \"free choice\" can secure order and lock out chaos. As Linda Hutcheon writes in _The Politics of Postmodernism_ , however, \"Postmodernism has called into question the messianic faith of modernism, the faith that technical innovation and purity of form can assure social order\" (12). Hutcheon suggests that \"messianic faith\" in art is no longer possible precisely because postmodernism is always \"complicitous\" in the very systems of power (such as \"economic capitalism and cultural humanism\") it \"chooses\" to critique (13). Like the hipsters I considered in the previous chapter, Kate initially disclaims any complicity in systems of power. She argues instead for a pure artistic space outside such systems. Yet, as Hutcheon writes, \"unacknowledged modernist assumptions about closure, distance, artistic autonomy, and the apolitical nature of representation are what postmodernism sets out to uncover and deconstruct\" (99). In terms of _People in Trouble_ , this might suggest that the postmodern \"identities in crisis\" Kate embraces (identities that, following Hutcheon, are _not_ closed, distant, autonomous, or apolitical) will \"uncover and deconstruct\" the \"unacknowledged modernist assumptions\" Kate brings to her artwork.\n\nKate does not subscribe unequivocally to all of these high-modernist tenets; she does insist, for instance, that her artwork is \"political\": \"It teaches people to see things in a new way. . . . Form is content. New forms are revolutionary\" (113). Still, Kate grounds her \"politics\" on a disavowal of art's _complicity_ in the systems of power her \"revolution\" would supposedly topple. At the same time, the queer identities Kate contends with make it more difficult for her to deny that complicity. During Justice's action at Home's Castle, for instance, she experiences firsthand the ways in which that space has multiple meanings: the Castle is simultaneously a \"document of ascendancy and homelessness.\" Since Kate comes away from that action with \"a lot of ideas,\" and since the public artwork she is commissioned to produce will grace another of Home's buildings, it is not surprising that Kate begins to question the assumption that her artwork will have a unified and critical meaning (whether \"universally human\" or \"politically revolutionary\"). Indeed, she even begins to recognize how her desire for unity and transcendence is complicitous with the economic and social philosophies of her patron Home and those like him. Ultimately, Kate realizes, both she and Home are committed to a harmonious use of space that contains, rather than critically deploys, the threats posed by contradiction and difference.\n\nDeutsche's work on public art and public space is apropos here. Deutsche criticizes the way in which discourses about public art, like discourses of urban \"redevelopment,\" repress contradiction, even though, paradoxically, city officials, arts administrators, and neoconservative critics alike invoke contradiction and contestation as one of public art's strengths. Commentators champion the \"new public art\" of the 1980s and 1990s because it is \"for 'the people' . . . encourage[s] 'participation' . . . promises a democratic spirit of equality\" and promotes \"accessibility\" (\"Art and Public Space,\" 34). Yet, as Deutsche makes clear, such invocations of \"democracy,\" the \"public,\" and even \"contested space\" depend on \"repressing\u2014on establishing as external to the 'public'\u2014the differences and conflicts as well as the outright injustices of urban life\" (\"Art and Public Space,\" 38\u201339). As an example, Deutsche cites a conflict over Jackson Park, in which the \"Friends of Jackson Park\" (people housed in the vicinity) decided to put a lock on the park's gate at night in order to discourage homeless people from sleeping there. City officials and commentators, pleased that the \"public\" was \"contesting\" the meaning of urban space, praised the Friends of Jackson Park for being \"determined to keep a park a park\" (qtd. in \"Art and Public Space,\" 38). Yet, as Deutsche points out, such \"contestation\" establishes the homeless as always external to the public and naturalizes the relations of power that _produce_ \"public space.\" The demand to \"keep a park a park\" disallows the participation of the homeless in determining exactly what the space of Jackson Park will mean and assumes instead that the meaning of that particular urban space is secured in advance. Homeless people's presence in Jackson Park foregrounds the contradictions of an urban \"redevelopment\" that displaces those whose labor is no longer needed; a lock on Jackson Park, in the name of \"the public,\" displaces (yet again) that contradiction.\n\n\"Public art\" functions in similar ways to \"public space,\" according to Deutsche: public art mutes contradictions even as proponents trumpet \"democracy\" and universal \"accessibility\" as its central components. Deutsche writes, \"The new public art . . . moves 'beyond decoration' into the field of spatial design in order to affirm, rather than question, its space, to conceal its constitutive social relations\" (\"Uneven Development,\" 117). As _People in Trouble_ continues, Kate begins to realize that her corporate sponsors expect just such an affirmation of social relations as they are\u2014and indeed, that with her appeals to \"universality,\" she is complicitous in that affirmation. Although Kate's agent implicitly assures her that she sculpts on behalf of \"the people\" by suggesting that \"the work [will] be seen by people on the streets going to work . . . [and will] not be shut up in some exclusive, out-of-the-way gallery\" (99), Kate becomes increasingly suspicious of the supposed public and democratic nature of her project. Home's buildings are newly privatized spaces and his redevelopment of New York City is displacing more and more people from their homes, and Kate begins to comprehend that Home's support of \"public art\" outside his buildings smooths over those contradictions.\n\nIronically, Kate's agent, perhaps \"determined to keep a street a street,\" peoples the city with men and women on their way to work. In her migrations between her apartment and Molly's, however, Kate begins to notice the massive numbers of people _living_ on the streets. Once again, her exposure to new queer identities has begun to disrupt the ideas she carries to her artwork. Kate muses to herself, \"You see so much more when you walk down the street alone. That's why people work so hard to avoid walking alone too often. What people see when they're alone can drive them mad\" (67). To Kate's agent, the city runs like a well-oiled machine, and Kate's work can be part of the technology: people see Kate's sculpture and continue on to work. Stepping out of the herdlike procession, Kate finally notices the contradictions endemic to urban space.\n\nLike her shifting sexual identity, Kate's shifting political identity and her connection to a community of activists take her out of the blind stampede. Near the end of _People in Trouble_ , Kate believes she has finally finished her sculpture; then she suddenly learns of the death of Scott, one of the founders of Justice and a friend of hers and Molly's. Scott's funeral is scheduled for the same day as the sculpture's dedication and unveiling. Earlier in the novel, Kate\u2014working to complete her commission yet at the same time troubled by her new sexual and political experiences\u2014had confessed to Scott, \"Sometimes a person has to stop talking about art for a moment and take a look around\" (166). When she learns of Scott's death, this is precisely what happens:\n\nShe walked outside and _noticed everything_. The buses had been painted a new color. There was a new song on the radio. All the kids were singing it. She passed two parks filled with street people drinking or sleeping or smoking Coke or cigarettes or crying or talking to themselves and to others or dying. She sat with them for a while, once in each park, and smelled their urine and sweat. Every garbage can on Second Avenue had been picked through. She saw the headline on a newsstand: AIDS VICTIMS RIOT.\n\nThree elderly women asked her for money. She gave them everything she had. Then she went to the bank machine and got out more. Four times young men tried to sell her drugs. In each case she bought what they offered without inspection and dropped three bags of marijuana and one crack vial on the sidewalk. There was trash everywhere. The streets were broken and filled with holes. There was a hooker on Twelfth Street who was clutching her vagina and crying. (216\u201317, emphasis mine)\n\nSuddenly, Kate can no longer harmonize away the contradictions urban existence places before her; she cannot comprehend \"people in trouble\" (the poor women, the drug dealers, the crying hooker, Scott) through appeals to a vague humanism. Schulman herself insists, \"Society is so stratified that people can occupy the same physical space and never see each other, and also have completely different experiences of that space\" (qtd. in Munt, \"Somewhere,\" 43). As _People in Trouble_ concludes, Kate finally realizes her complicity in the production of space that represses contradictions and stratifications.\n\nKate's emerging queer consciousness thus changes her aesthetic. When Kate returns to her apartment, her interaction with Peter confirms that her identity as \"artist\" has been transformed as decidedly as her sexual and political identities. She listens silently to Peter's lament:\n\n\"I understand you feel a need to be politically active but I think that is something we can do together. Homosexuals don't have a monopoly on morality, you know. We have always agreed that our artwork is our political work. We have always agreed that challenging form is more revolutionary than any political organization ever can be. But if you feel a need to be part of a group, we can do that together. . . . We can be exactly like we were before.\"\n\nShe placed her fingers flat against his chest. It was a wall. It moved. There was hair underneath his shirt. She wanted to dig her fingernails in and tear him apart.\n\nHe spoke again. What did he say this time? (217\u201318)\n\nKate no longer hears Peter and the ideas he espouses. She cannot base her aesthetic on Peter's ideals; Peter's ideals are, in fact, a wall or barrier blocking her development as an artist. This wall or barrier, however, can be moved, if not completely torn apart. Peter's nostalgic desire for things \"exactly like they were before\" comes across as incredibly naive, since it endows Kate's previous sexual, political, and artistic identities with an essential wholeness and dismisses Kate's new queer identities as nonessential supplements. \"But,\" as Jacques Derrida insists, \"the supplement supplements. It adds only to replace\" (145). Kate and Peter _never_ were \"exactly like they were before,\" and Kate at this point recognizes that Peter's request is actually a heteronormative demand that she \"fix\" her identity and disavow the queer relations that are constantly reshaping it.\n\nShe and Peter cannot be exactly like they were before, and Kate cannot see space exactly as she did before. Kate's conception of Peter's ideas as a wall that can be moved, however, demonstrates Kate's acceptance that her aesthetic, like her sexual and political identities, need not be understood as static. A final scene\u2014the dedication of Kate's sculpture\u2014starkly represents how far Kate has come. Ronald Home will preside over the dedication, which takes place on the day of Scott's funeral. Although the narrator records silence as the scene opens, no one can mute the discord of this coincidence: \"Kate looked closely at the crowd. There were some people she knew from Justice meetings. . . . There were no words. They did not touch. There were no embraces, only anger and a shared determination that passed between them. . . . [Scott's coffin] was placed in the hearse without eulogy or speeches\" (219\u201320).\n\n\"There were no words\" because Schulman has by now firmly established the process this group goes through to form a collective and disruptive queer identity; as anger and shared determination come together, \"go . . . go . . . go\" again pulsates beneath this scene without needing to be spoken. And indeed, once in the streets, the group does begin to move, even stopping traffic: \"[Kate] climbed over cars, disregarded them. When there are that many people the traffic can't move. When that many people walk together the traffic has to stop\" (220). The group walks directly to the space outside the building where Home is beginning the speech that will dedicate Kate's artwork.\n\nAlthough homeless people occupy almost all urban locations in a \"redeveloped\" city such as New York, \"authorities\" generally round the homeless up and displace them for \"public\" events such as political rallies, holidays, and dedications. When Home sees Kate and her queer group approaching, he similarly tries to displace them and the contradictions they embody: \"He was searching for just the right throwaway comment to invalidate all the people in front of him and at the same time make great copy for the front page of the next day's _New York Post_. But nothing very clever seemed to come to mind\" (221). After the group surges forward, upsetting vehicles and cameras, Kate climbs atop an overturned CNN van and looks at the artwork spread out in front of her. The language that describes the confusion that follows, as Kate puts the \"finishing touches\" on her piece, suggests that she has moved far from her commitment to a universalizing, high-modernist aesthetic:\n\nKate pushed harder than she had ever pushed and clawed her way to the front of the stage, catching and tearing her flesh on the splintered police sawhorses that lay mangled everywhere. Then she climbed under it, crawling on the dirt and garbage over wires, rags, cans of paint and turpentine. She watched her own hands turn black and her arms cake with dirt and blood surrounded by the moving spikes of pants legs bobbing around her. Dragging the cans and power lines to the base of the collage's wooden frames, she looked back at the chaos behind her. Each gesture was too large and so unusual that the action passed before her like a high-speed silent film. Only there was no silence. (221\u201322)\n\nKate's artwork goes up in flames, engulfing Ronald Home, who will indeed make the news but in a way he never anticipated: \"Real-estate mogul Ronald Horne met a fiery death today when a freak accident occurred during a riot by AIDS victims. An art installation designed for the inauguration of a new health club caught fire and enveloped the billionaire developer in a flaming collage\" (224). Kate's corporate sponsors may require harmonious uses of space and containment of chaos and contradiction, but Justice displays a different aesthetic altogether, one that critiques what it was \"designed for.\" A \"club\" where the few pay for \"health\" simultaneously denies health and well-being to the many relegated to perpetual nonmember status. Critically and collectively _deploying_ chaos and contradiction, Kate and her cohorts insist that such contradictions be made public. Urban space is contested space, and Justice's \"flaming\" aesthetic works in the end to flaunt that dictum.\n\n### **Perestroika Redux**\n\nJust as the beginning of Schulman's novel could be subtitled, a la Kushner, \"Millennium Approaches,\" so too could the conclusion be subtitled \"Perestroika\": the mingling, messiness, collective contestation, and hope that characterize the theory staged in _Angels in America_ transform Kate in _People in Trouble_ , who in turn works with Justice to incorporate those ideas into her sculpture. Ironically, it is not at all clear in the novel whether Schulman endorses Kate's newfound aesthetic. The final chapter focuses on Molly and only obliquely and dismissively mentions Kate: \"Kate developed a high profile as a result of Home's death and could be read about in an essay by Gary Indiana in the _Village Voice_ and one by Barbara Kruger in _ArtForum_. In fact, Kate began working extensively in burning installations and quickly got commissions from a number of Northern European countries to come start fires there\" (225). There is a fair amount of cynicism in this removal of Kate from the scene, especially since Molly remains behind, with Justice, in New York City. Moreover, the geographic distance might stand as both figure for and confirmation of what various biphobic figures maintain throughout _People in Trouble:_ Kate is always already distanced from the gay community, not really committed to \"our\" struggle.\n\nIt is possible, however, to look critically at Schulman's perhaps unintentional slippage from \"flaming\" to \"burning.\" If we understand a \"flaming\" aesthetic as alive, brilliant, or\u2014as Kushner might note\u2014 _faaaaabulous!_ then a \"burning\" aesthetic, given Schulman's cynicism, might suggest the opposite: faded brilliance, world-weariness, resignation. Yet, although Kate's cooptation by \"the establishment\" may be inevitable, this need not mean that she loses her critical edge and forgets the lessons she has learned throughout _People in Trouble_ about complicitous critique; indeed, one might reasonably expect at this point that Kate will continue to contest from the \"inside,\" in queer ways, the meanings exacted by \"the establishment.\" The citation of Kruger and Indiana is, in this sense, particularly relevant: Hutcheon often uses Kruger's photographs, with their simultaneous use and abuse of high modernism, as examples of a critical postmodernism that parodies discourses about high art, the media, gender, and sexuality, and Indiana's very _name_ suggests complicity and critique, since it is at once the quintessential Midwestern town, immortalized in song and technicolor by a cherubic Ronnie Howard, _and_ the signature of a \"New Queer Writer\" who writes about children, sexuality, and violence. More important, though, the slippage from \"flaming\" to \"burning\" belies the ways in which \"complicitous critique\" describes ACT UP's and Schulman's (and Kushner's) _own_ aesthetic strategies.\n\nDouglas Crimp and Adam Rolston, for instance, draw attention to the strategies used by cultural workers in ACT UP:\n\nIf their sophisticated postmodern style has gained art world attention and much-needed funding for Gran Fury [a graphic arts group within ACT UP], the collective has accepted it only hesitantly, often biting the hand that feeds. Their first poster commission from an art institution was discharged with a message about art world complacency: WITH 42,000 DEAD, ART IS NOT ENOUGH. (19)\n\nACT UP, in short, reconfigures its complicity, using \"art world attention\" to spotlight \"art world complacency\" Similarly, Schulman's own \"flaming\" aesthetic should be seen as both complicitous and critical, especially since she links her work to Kate's by titling both her novel and Kate's sculpture _People in Trouble_. In the last sentence of the novel, Molly and her friends angrily head out to another demonstration, because\u2014as the narrator explains\u2014\"there was nothing more to say\" (228). Like Kate with her sculpture and ACT UP with their graphics, Schulman bites the hand that feeds (or reads) here, implicitly suggesting, as she lays down her smoking pen, that these words\u2014or, more specifically, the act of writing or reading them\u2014will never be enough. As with the conclusion to _Angels in America_ , so too with (both) _People in Trouble(s):_ \"The Great Work _Begins_ \" (P, 148, emphasis mine).\n\nIn the opening scene of _Perestroika_ , Kushner introduces a minor character who is played by the actor playing Hannah, Joe's Mormon mother who is transformed by queerness over the course of the play Aleksii Antedilluvianovich Prelapsarianov, \"the World's Oldest Living Bolshevik,\" speaks in \"the Hall of Deputies, the Kremlin,\" in January 1986 (P, 13). Stage directions explain that this character is _\"unimaginably old and totally blind\"_ (P, 13). He stands at the front of the stage and demands:\n\nWhat System of Thought have these Reformers to present to this mad swirling planetary disorganization, to the Inevident Welter of fact, event, phenomenon, calamity? Do they have, as we did, a beautiful Theory, as bold, as Grand, as comprehensive a construct. . . . You who live in this Sour Little Age cannot imagine the grandeur of the prospect we gazed upon: like standing atop the highest peak in the mighty Caucasus, and viewing in one all-knowing glance the mountainous, granite order of creation. You cannot imagine it. I weep for you. (P, 13\u201314)\n\nAccording to the World's Oldest Living Bolshevik, the \"next Beautiful Theory\" must be shaped and absorbed _before_ the people head to \"the barricades\": \"If the snake sheds his skin before a new skin is ready, naked he will be in the world, prey to the forces of chaos\" (P, 14).\n\nDirectly behind this speech, however, Kushner restages the ending to _Millennium Approaches:_ the angel has just crashed through Prior's ceiling and is hovering above his bed. Such a juxtaposition throws the World's Oldest Living Bolshevik's words into crisis: the chaos has already burst in upon Prior; he has had no opportunity to shape, in advance, an autonomous theory and identity. In short, the straight theory that the World's Oldest Living Bolshevik espouses no longer suffices in this queer world. What is needed instead\u2014what was playing in the streets and what _Angels in America_ and _People in Trouble_ affirm\u2014is queer perestroika: identities-and theories-in-process, not \"viewing in one all-knowing glance the mountainous, granite order of creation\" but rather collectively and continuously multiplying and reshaping locations for the production of identity, theory, activism, and art; not standing, in solitude, on \"the highest peak in the Caucasus\" but acting up, in solidarity, across differences in coalition with others.\n\n## **_Epilogue_  \nPost-Queer?**\n\nBy concluding with Sarah Schulman, ACT UP, and queers in the street, I have come full circle, since my first chapter concluded with Audre Lorde and the resistant identities she observed being shaped and reshaped in the streets a decade earlier: \"My lasting image of that spring .. . was of women whom I knew . . . and women whose names were unknown to me, leading a march through the streets of Boston behind a broad banner stitched with a line from Barbara Deming: 'WE CANNOT LIVE WITHOUT OUR LIVE'\" (Lorde, _Need_ , 3). The connection is perhaps more than coincidental, since Schulman studied under Lorde at Hunter College in the early 1980s. Schulman writes, \"She told us, 'That you can't fight City Hall is a rumor being circulated by City Hall'\" (My _American History_ , xvii). As far as Lorde was concerned, power was not concentrated on high; women working together as friends and lovers could both rewrite the rumors being circulated about their powerlessness and effect a new world.\n\nThe Lorde-Schulman connection is appropriate, given my intention in this project of reading the Queer Renaissance in and through, rather than above, the larger queer social context of the period. The classroom here, notably, facilitates such a reading: it provides neither a pure origin nor an ultimate end for the identities I have explored in this study. Instead, the classroom, for both Lorde and Schulman, is a location contiguous to other locations; both women are simultaneously coming from and going to someplace else. Schulman does not simply learn how to be queer when she gets to Hunter College and reads about it in a book, but neither do her books (or Lorde's, or those read in Lorde's classroom) simply \"reflect\" queer identities shaped elsewhere. The renaissance of queer creative work and the renaissance of queer identities and political analyses have been mutually constitutive. Each represents and fuels the other.\n\nIn the past few years, however, some gay writers have appeared who want queers to stop this flurry of reproductive activity. In this epilogue, I consider briefly whether the emergence of writers such as Bruce Bawer and Andrew Sullivan marks the beginning of a \"post-queer\" moment. Bawer, Sullivan, and others have gained a great deal of attention by calling for such a moment: their emphasis is on individuality and an abstract individual freedom rather than on relational, fluid identities; they stress not a critique of but an accommodation and assimilation to contemporary society and its institutions. If the Queer Renaissance constructs collective identities intent on disrupting dominant hierarchical understandings of sex, gender, sexuality, race, and class, does the emergence of neoconserva-tive gay writers committed to the status quo or gay scientists who want to ground homosexuality in biology mark a waning of the Queer Renaissance?\n\nBawer, for one, has virtually sounded the death knell for \"queerness\": \"Today, reasonable gay voices are beginning to be heard in the media. [My writing] is both a cause of this phenomenon and a symptom of it. Also symptomatic are the decline of ACT UP, the dissolution of Queer Nation, and the growing influence of the Log Cabin Republicans\" (\"No Truce,\" 74). I conclude, however, that the Queer Renaissance is ongoing despite, and indeed because of, the prominence given to such writers. When such writers can be taken as \"representative,\" it is clear that, in many ways, the queer work I have been examining, and that has been developing for more than fifteen years, has only just begun and is now needed more than ever.\n\nBawer has forcefully denounced the flamboyance of queers of all kinds: drag queens, leather folk, sex radicals. These groups, according to Bawer, stand between gay men (Bawer has very little to say about lesbians) and acceptance by heterosexuals: \"If the heterosexual majority ever comes to accept homosexuality, it will do so because it has seen homosexuals in suits and ties, not nipple clamps and bike pants; it will do so because it has seen homosexuals showing respect for civilization, not attempting to subvert it\" ( _A Place_ , 51). Gay Pride Day, especially, worries Bawer:\n\nIt seemed to me . .. that the sort of pride on display in the Gay Pride Day march was, in many cases, not so much pride in the sense of \"self-respect\" or \"dignity\" than pride in the sense of \"arrogance,\" \"conceit,\" \"hubris.\" Real pride, after all, is a hard-won individual attribute. It doesn't come from being gay, or from belonging to _any_ group. It can come, however, from dealing with the fact of your homosexuality in a responsible and mature manner, from not using it as a club to beat other people with or as an excuse to behave irresponsibly or unseriously. ( _A Place_ , 159)\n\nAs an antidote to this \"irresponsibility,\" Bawer offers his book _A Place at the Table: The Gay Individual in American Society:_ \"I felt that if the image of homosexuals that the march projected was ever to be set right, those of us who were displeased with it in its present form couldn't keep away entirely. We had to do _something_ \" (157). _A Place at the Table_ , then, purports to fix the image, to set it right, by distancing \"mainstream gays\" (35) from the antics of those pesky queers.\n\nOf course, Bawer's irritation with those who supposedly use their sexuality \"as a club to beat other people with\" completely mystifies the fact that it is precisely those people who really _have_ been beaten with clubs, in ever-increasing numbers, over the past few decades. Moreover, Bawer's disdain for groups and his deification of the individual are ahistorical: the space in which someone like Bawer can confidently be a \"gay individual in American society\" was forged over time by the very flamboyant groups he would disavow. The fact that Bawer's arguments are easy to counter, however, has not kept the straight media from showering _A Place at the Table_ with hyperbolic praise. On the front cover of Bawer's volume, the _New York Times's_ Christopher Lehmann-Haupt declares _A Place at the Table_ \"an intelligent and eye-opening book\" that \"smashes the common stereotypes of gay people to smithereens.\" On the back cover, the _Wall Street Journal's_ Jonathan Rauch writes, \"Of all the sinkholes in American politics, the debate over homosexuality is the rankest. . . . With the brac-ingly rational passion of a writer who can think and feel at the same time, [Bawer] charts a path out of the swamp.\" And on the inside, John Fink of the _Chicago Tribune_ insists, \"If there is one book about homosexuality and gay rights that everyone should read, it is probably this one.\" Bawer worries in _A Place at the Table_ that queers (swamp rats?) represent gay life to straight America. Fink's suggestion that Bawer's book is the one gay book everyone should read, however, implies that our threatening representation might be contained. Bawer's book alone, apparently, can speak for us all.\n\nMeanwhile, even as neoconservative writers like Bawer win acclaim by lambasting the \"rowdier\" elements of queer culture, biologists are searching for the gene that would tidy up the mess that is human sexuality, securing once and for all a biological origin for sexual orientation. Simon LeVay is the neurobiologist most associated with this movement. In _The Sexual Brain_ , LeVay writes:\n\nIt is not unrealistic to expect a gene or genes influencing sexual orientation to be identified within the next few years, since there are at least three laboratories in the United States alone that are working on the topic. If such genes are found, it will be possible to ask where, when, and how these genes exert their effects, and hence to gain a more basic understanding of the biological mechanisms that make us straight or gay. (127)\n\nWhereas the queer identities in this study have sought to unsettle and denaturalize the mechanisms of power that produce our current understandings of identity, the gay gene LeVay and his colleagues long for would fix and naturalize sexual identity.\n\nThese various developments indeed suggest that we are potentially moving or have moved to a post-queer moment. At the very least, these developments highlight the ongoing tension between liberal reformers and radical liberationists that I focused on, following John D'Emilio, in my introduction. Just as the later Mattachine Society and the Daughters of Bilitis advocated a professional, suit-and-tie approach to homosexual rights that discounted the structural critique of American society advanced by the early Mattachine Society, and just as the Gay Activists Alliance's single-minded focus on gay rights trumped the Gay Liberation Front's commitment to coalition with other movements for liberation, so too could the rise of gay neoconservatism and a renewed faith in \"experts\" be seen as a \"retreat to respectability\" (D'Emilio, _Sexual Politics_ , 75) that signals the end of the Queer Renaissance.\n\nBut, as I have already suggested, it does not have to be that way. We queers need not be contained, and one \"representative\" need not (and indeed, cannot) speak for us all. This is, in the final analysis, a _queer_ renaissance, and there is no reason to expect that it will follow a predictable (straight?) trajectory that simply reproduces the mistakes of the past. Furthermore, these developments might be read as constituting another beginning as much as an end. Queer analysis, after all, _commences_ by questioning the boundaries established by others: as Gloria Anzald\u00faa writes, \"The street reader looks at an experience as something that's alive and moving or about to move\"; \"Queer readers want to interact, to repeat back or reflect or mirror, but also do more than just reflect back or mirror\u2014to add to the dialogue\" (\"To(o) Queer,\" 259, 256). Anzald\u00faa's reflecting back or mirroring is not a pure reflection but rather a refraction, bringing into view those issues that are hidden or effaced by the border guards. In the contemporary context, then, a queer reading might begin by asking: Whom does the fixing of boundaries around a gay gene or of representatives for lesbians and gay men serve? Why are these boundaries and representatives being established at this particular time? How are these boundaries intersected by others, and what is to be gained from interrogating or crossing them?\n\nAs far as biology is concerned, the answer to the first question, to many, would seem self-evident: clearly (they would say), _gays and lesbians_ are served by the search for the gay gene, and by the positioning of an \"expert\" like LeVay as our representative. One of Bawer's main theses, for instance, is that (well-behaved) homosexuality should be accepted because it is most definitely _not_ a choice; he therefore loudly applauds \"a scientific study purportedly demonstrating a correlation between homosexuality and a certain pattern of genetic markers add[ing] to the growing body of evidence that sexual orientation is hereditary\" ( _A Place_ , 56). LeVay has been loudly applauded in other arenas as well, even being hailed as a hero at the Los Angeles Gay Pride Parade. And in a recent informal poll, 61 percent of readers of the lesbian and gay newsmagazine _The Advocate_ said that it would help the movement if homosexuality were found to be biologically determined (February 6, 1996, 8).\n\nA queer analysis, however, always attentive to connections across differences, might complicate the question: Why are sexual identities being fixed in the brain at this particular time? Or, more specifically (and perhaps more provocatively), does it matter that _The Sexual Brain_ emerged one year before the fanfare surrounding _The Bell Curve_ , Richard J. Herrnstein and Charles Murray's \"study\" that purportedly grounds poverty and other social inequities in biology? Siobhan Somerville implicitly makes this link when, after probing the connections between the \"scientific racism\" of the late nineteenth century and the contemporaneous emergence of the medicalized homosexual body, she suggests that a queer inquiry into these historical connections might also consider how \"these analogies [have] been used to organize bodies in other historical moments\" and how \"the current effort to rebiologize sexual orientation and to invoke the vocabulary of immutable difference reflect [s] or influence [s] existing cultural anxieties and desires about racialized bodies\" (265\u201366). Such a query might qualify the self-evidence of the assumption that lesbians and gay men are served by the pursuit of a gay gene.\n\nThere are several ways in which the effort to rebiologize sexual orientation reflects existing cultural anxieties about race. Most important, the two reflect each other in the ways in which they focus our vision. In an attempt to identify the anxieties centralized in _The Bell Curve_ , Steven Fraser writes:\n\nAt a time when most indices record expanding inequalities in American life\u2014not only in income and wealth distribution, but in public and private schooling, in matters of health care, even in our varying capacities to rear the newborn\u2014 _The Bell Curve_ naturalizes those phenomena, turns them into inescapable symptoms of a biological class fate. At the same time, by associating the \"cognitive underclass\" [Herrnstein and Murray's term for those \"genetically predisposed\" to lower IQs] with every grisly or disturbing form of social behavior, from crime to unwed teenage motherhood, the authors direct our gaze away from those institutional centers of power that in an earlier era might have had to shoulder the blame for our most grievous inequalities and social pathologies. (4\u20135)\n\nA similar refocusing of our gaze is at work in _The Sexual Brain_ , I would argue, despite LeVay's best intentions: the book directs our gaze away from the institutions that secure and maintain heterosexual dominance\u2014indeed, away from the institution of heterosexuality itself\u2014to a naturalized homosexuality. As Jonathan Katz writes in _The Invention of Heterosexuality_ (a book that is, I think, the antithesis\u2014or the Anzald\u00faan refraction\u2014of _The Sexual Brain_ ), \"Rarely do we focus for long on the riddle of heterosexuality\u2014our gaze turns quickly to 'the problem of homosexuality'\" (15). By focusing our gaze once again on \"the problem of homosexuality\" (indeed, focusing our gaze _microscopically_ on that \"problem\"), the search for a gay gene only accelerates the turn away from a critique of heterosexuality. And just as _The Bell Curve_ lets economic institutions off the hook, allowing those in power to regret (perhaps), but not change, the \"expanding inequalities in American life,\" so, too, _The Sexual Brain_ lets the institution(s) of heterosexuality off the hook. If a gay gene is secured, there is no reason to assume that homophobia will suddenly end. Indeed, in a culture already primed to see difference as a regrettable biological fact, there is nothing to keep heterosexuals from seeing homosexuality similarly as a natural but regrettable biological destiny, all the while doing nothing to change the institutions that secure heterosexual dominance. And whether or not they understand themselves as having a gay gene, in a world where homosexuality is the \"aberration\" and heterosexuality the \"norm,\" young gay men and lesbians will still kill themselves.\n\nThis is especially true given the ways that the \"gay gene\" has already been appropriated by the radical right. In a document that was used in the campaign for Colorado's Amendment 2, the measure that prohibited local communities from establishing ordinances that would protect lesbian and gay civil rights and that was later struck down by the U.S. Supreme Court, Tony Marco writes, \"Many 'natural' or possibly 'innate' behaviors of both animals and human beings remain unprotected or strictly controlled by society. The striking behavior of poisonous snakes is perfectly natural. Do we argue that 'because snakebites are natural' people ought to welcome and take no steps to protect themselves from venomous vipers?\" The right wing likes to disavow the connection between its rhetoric and violence, but this analogy, I think, makes the connection unavoidable: most people cornered by a venomous viper would strike to kill.\n\nAs I stressed in my introduction, however, the \"institutional centers of power\" did \"shoulder the blame\" in an earlier era: the early Mattachine Society launched a Marxist critique of the organization of American society, particularly the organization of the American family; the Gay Liberation Front made connections between homosexual oppression and a host of other oppressions that are secured in a capitalist and patriarchal society. It is these earlier moments and impulses that are implicitly and explicitly cited and reshaped in the contemporary Queer Renaissance. And at a time when studies such as _The Bell Curve_ and _The Sexual Brain_ would fix difference in biology before or instead of attending to ongoing structural inequalities, such a queer renaissance is needed more than ever. As Lisa Duggan points out in her afterword to Katz's book, with a nod to anthropologist Carole S. Vance:\n\nWe see headlines announcing the existence of a \"gay brain,\" but no such reporting on the multiplying historical studies that show sexual identity to be cross-culturally and historically variable. We wait to open a copy of _USA Today_ to read \"Heterosexuality Not 'Natural,' Not 'Normal,' Study Finds.\" Chances are, we'll be waiting a long time. (194)\n\nTo Duggan, that is why studies like Katz's, which contest heteronormative assumptions, are so vital (Duggan, 196). As Katz himself writes, \" _Heterosexual_ and _homosexual_ refer to a historically specific system of domination\u2014of socially unequal sexes and eroticisms. . . . Biological determinism is misconceived intellectually, as well as politically loathsome. For it places the problem in our bodies, not in our society\" (189). In contrast, Katz's study, along with the studies and identities multiplying in the Queer Renaissance, locates the problem in society, not in our queer bodies and eroticisms.\n\nIf, despite these cautions, _The Bell Curve_ and _The Sexual Brain_ still seem to be at opposite ends of the spectrum, we should be aware of at least one commentator for whom they are not. David Duke has recently stated:\n\nI think there is a genetic predisposition to it [homosexuality]. . . . I've read the genetic studies, and I think there is definitely a predisposition to it. But you know what is so funny about all of that? _The New York Times_ says stuff like, We can't blame gays for their behavior because they are born that way. That's the argument. Well, don't they realize that the same law of genetics, the same differences in the physical aspects of the brain, could account for differences in personality and races and so forth? I mean, it follows naturally. (qtd. in Boulard, 32)\n\n_I mean, it follows naturally_. Duke's \"straight talk\" here could be seen as the antithesis of the statement I identified in the last chapter as a byword for my project: \"This is not in the Text, We _deviate_.\" At the same time, Duke's statement makes clear that deviation from what supposedly \"follows naturally\" is crucial: queer identities are needed that refuse to acquiesce to a gay biologism or gay neoconservatism, especially when (or precisely because) those movements are used to underwrite the marking and policing of \"others.\" In the process of realizing that \"deviation,\" moreover, such queer identities can and should continue to challenge an array of other conclusions, from all points on the political spectrum, that likewise \"follow naturally\" from heteronormative \"common sense\": David Duke's conclusion, for instance, that an \"indelible, unwashable AIDS tattoo\" would protect people from \"AIDS carriers\" (qtd. in Boulard, 32), or Bruce Bawer's conclusion that but for the \"irresponsible antics\" of queers ( _A Place_ , 222), upstanding lesbians and gay men would be accepted into American society. Queer analyses demonstrate that the \"irresponsible antics\" of queers are responsible not only for forging a few spaces more conducive to \"upstanding\" gay and lesbian desire but also for exposing how exclusion of lesbians and gay men (and many, many others) is built into the very structure of American society, and for imagining a world shaped otherwise. The latter queer work is ongoing and has only just begun.\n\nQueer activists and theorists will undoubtedly continue to look critically at neoconservative rhetoric seeking to move \"beyond queer\"; after all, as a Queer Nation sticker that was ubiquitous a few years ago announces, \"Queers Bash Back.\" Of course, it is possible to argue that the search for the gay gene and recent gay neoconservative writing are not really queer bashing; that they are, instead, merely strategic moves; and that despite their flaws, they nonetheless help to advance the gay and lesbian movement. They are bridges. In response, I would turn again to Moraga and Anzald\u00faa's _This Bridge Called My Back:_ \"A bridge gets walked over,\" Moraga writes in her preface to that collection, \"over and over and over again\" (xv). Moraga's insight suggests that we ask ourselves: If the books that have been so championed by the mainstream press recently are bridges, on whose backs are those bridges built? Bawer's \"gay individual in American society\" requires the queer other for his self-definition. Ironically, the same might be said about many queer theorists and activists. The latter, however, try not to disavow the relationship. As Anzald\u00faa asserts, \"We have come to realize that we are not alone in our struggles nor separate nor autonomous but that we\u2014white black straight queer female male\u2014are connected and interdependent\" (\"La Prieta,\" iv).\n\nIn my analysis in this project of various queer identities (Zami, queer trickster, mestiza queer, AIDS activist\/artist), I have stressed that the work of imagining and shaping a queer world occurs both inside and outside the texts I have examined, and that \"inside\" and \"outside\" are mutually constitutive. Ironically, Bawer recognizes this, since the chapter in _A Place at the Table_ that criticizes parades and protests is the same chapter that condemns the queer literature of the past fifteen years. \"Poetry is not a luxury,\" goes the famous phrase by Audre Lorde ( _Sister Outsider_ , 36). This study insists that neither is poetry (or fiction or drama) an addendum, added onto or rising above queer activism or theory. Instead, the texts of the Queer Renaissance are intersections, or crossroads, where diverse queer communities both read and write, observe and shape, their own experiences and identities.\n\n## **_Notes_**\n\n### **Notes to the Introduction**\n\n. For an excellent overview of \"the gay and lesbian publishing boom,\" see William J. Mann. Mann cites a _Publishers Weekly_ estimate that about 3 percent of the fifty-eight thousand titles published in 1993, or just under two thousand titles, were lesbian and gay, though he also notes, \"Exact figures are hard to come by, if only because of the fluid nature of what constitutes a 'gay book'\" (25).\n\n_Lambda Book Report's_ 1991 article \"Decade Dance\" considered how \"gay and lesbian literature took off in the 1980s\" and included a profile of those whom _LBR_ saw as the fifty \"most influential people in the industry\" during that decade (10). Such a profile, especially of those \"in the _industry_ ,\" would have been virtually unthinkable fifteen years earlier. Gay novelist Richard Hall's article \"Gay Fiction Comes Home,\" which appeared on the front page of the _New York Times Book Review_ in 1988, was seen by some as an indication that mainstream journalism was finally acknowledging the incredible productivity of openly gay writers (Nelson, \"Towards a Transgressive Aesthetic,\" 16). The first Lambda Literary Awards (\"Lammies\") were presented in 1989. See Labonte, 60\u201361, for a discussion of how, by 1994, these awards were taken very seriously by mainstream publishers.\n\nMy use of \"rare\" to describe openly lesbian and gay work in the 1970s is relative to the boom in the 1980s and 1990s; many openly gay and lesbian works were, of course, written or performed in the 1970s. Newly formed feminist presses such as Daughters, Inc., were particularly important for lesbian writers, many of whom got their start in such presses during the 1970s (Rita Mae Brown is the most famous example). Still, as Bonnie Zimmerman makes clear, the 1980s represent a significant advance: \"Between 1973 and 1981, an average of only five lesbian novels was published each year; between 1984 and 1987, that number had jumped to twenty-three, and recent figures indicate that the pace is quickening\" (207). For men, the shift is perhaps even more apparent: of the nearly five hundred primary works mentioned in the various entries in Emmanuel Nelson's _Contemporary Gay American Novelists: A Bio-Bibliographical Critical Sourcebook_ , well over half were published in the 1980s and 1990s. About one-fifth were published in the 1970s, with another one-fifth published in all the decades prior to the 1970s. Even this calculation, however, may be a bit deceiving, since many gay writers who published prior to 1980 published more openly gay work after 1980.\n\n. Sedgwick's comments on renaissances make clear that scholars are beginning to see same-sex desires, affections, and activities as central to many literary periods; hence the \"cultural phenomenon\" I examine here is the explosion of _openly_ lesbian and gay writing. I do not mean to champion fixed identities through this focus (quite the opposite, in fact), only to note that something unprecedented has indeed occurred. Although \"lesbian\" desires, for instance, are certainly discernible in a novel such as Nella Larsen's _Passing_ (1929) and in the Harlem Renaissance in general, there is still a significant difference between Larsen's novel and openly lesbian work such as Audre Lorde's _Zami: A New Spelling of My Name_ (1982).\n\n. See, for example, Crimp, \"Right On, Girlfriend!\"; Duggan, \"Making It Perfectly Queer\"; Edelman, 112\u201317; Halperin, 62\u201367. _Out\/Look_ , the now defunct national lesbian and gay quarterly, published a special issue on the \"Birth of a Queer Nation\" in 1991. I discuss more thoroughly how my own use of \"queer\" fits into this collective project later in this introduction.\n\n. In 1992, Dennis Cooper published _Discontents: New Queer Writers_ , an anthology of angry, graphic, and often violent short stories, essays, comic strips, and interviews. _Discontents_ brought together many different queer writers, some of whom I mention briefly at the end of this introduction. Sarah Schulman is the only writer considered in _The Queer Renaissance_ whose work is consistently linked to this movement. For another discussion of the New Queer Writers, see Latzky.\n\n. On the relationship of urban renewal to dislocation and polarization of rich and poor, see Williams and Smith. Williams and Smith also mention, briefly, Detroit's Renaissance Center (204). See also my discussion of Schulman's novel _People in Trouble_ in chapter 4.\n\n. It is perhaps not surprising that I first formulated these thoughts in a rundown Admiral Benbow Inn in downtown Memphis, a city similarly beset by \"urban renewal.\" I have no doubt that the majority of Memphis's poor and African American residents are still waiting for the construction of the monstrous Great American Pyramid on the banks of the Mississippi River to improve their lives.\n\n. Although Crimp's article is concerned primarily with the visual arts, Michael Denneny's comment makes it clear that similar rhetoric has been used in discussions of the literature of AIDS. Denneny's ideas on the subject haven't gotten any better of late:\n\nGreat writers have always been reactionary.... It was true of 19th-century Russian literature and 18th-century English literature and almost assuredly will be true today. Political people are appalled by reality as it is and want to change it. Great artists are people who find intense beauty in the way life is and want to memorialize it. (Qtd. in DeLynn, 55)\n\nFor another literary example of appeals to the \"transcendent,\" consider David Leavitt's comments about Susan Sontag's short story \"The Way We Live Now.\" Leavitt writes, \"Sontag . . . had written a story that transcended horror and greed, and which was therefore redemptive, if not of AIDS itself, then at least of the processes by which people cope with it\" (28). For an alternative perspective on Sontag, see D. A. Miller, who criticizes what he sees as patronizing \"urbanity\" in Sontag's _AIDS and Its Metaphors:_ \"Sontag is right when she says that _AIDS and Its Metaphors_ is 'not another book about AIDS'; rather it is a book (to be set next to Allan Bloom's or E.D. Hirsch's) that defends this culture, whose value . . . the epidemic provides a usefully extreme opportunity for once again recommending\"\n\n. I consider the interrelation of art and activism more fully in chapter 4. On these issues, see Doug Sadownick, who asserts, \"ACT UP's grassroots war against AIDS is escalating, and the ammunition is art\" (26). Sadownick demonstrates that Crimp's ideas about activists and artists are shared by other members of ACT UP:\n\nVideo artist and ACT UP member Phil Zwickler finds it \"revolting\" and \"telling\" that art presenters would \"still try and shove 'art for art's sake' shit down our throats.\" Performance artist Tim Miller finds the art world still cloven between those who vie for transcendence and those who want to fix the world. \"The challenge of this time,\" he says, \"is for artists to also be citizens, social activists, journalists; the time when you just get to sit in your studio and make art product is over. This is no different from the Latin American model of the citizen artist, or a playwright soon to be president of Czechoslovakia or the student artists in Tienanmen Square.\" (29)\n\nMiller's comments already begin to indicate that the concept of a global renaissance, like the concept of the artistic generally, might be resignified, as something other than an escape from secular history.\n\n. Quotations from _Sexual Politics, Sexual Communities_ are cited parenthetically as _SP_. Quotations from _Making Trouble_ are cited parenthetically as _MT_. Other analyses of the 1950s and 1970s that I have found useful in compiling and contextualizing this section include Adam; Duberman, _Cures_ and _Stonewall;_ Kennedy and Davis; Kissack; Nestle. George Chauncey's _Gay New York: Gender, Urban Culture, and the Making of the Gay Male World, 1890\u20131940_ takes on what Chauncey labels \"the myth of invisibility\" (3): the belief that prior to World War II, gay men and lesbians were so isolated from one another that community formation was virtually impossible. Chauncey debunks this myth by providing extensive evidence for a vibrant and highly visible gay male culture in New York City in the first third of the twentieth century. _Gay New York_ suggests that the historical moments I have chosen to survey are not necessarily the only moments when lesbians and gay men shaped identities across differences and critiqued heteronormativity and thus are not the only two historical moments to be recalled in the Queer Renaissance. Future work in gay and lesbian history will undoubtedly uncover even more locations when and where lesbians and gay men forged fluid and disruptive identities.\n\n. On this topic, see Luis Valdez's play _Zoot Suit_ , which details the Zoot Suit Riots of the 1940s, when a group of Chicanos were sent to prison without substantial evidence for the murder of a Sleepy Lagoon man. Valdez's play was part of his Teatro Campesino (Farmer's Theater) and part of the Chicano Renaissance.\n\n. Compare, as well, Joan Nestle's memories: \"We needed the Lesbian air of the Sea Colony to breathe the life we could not anywhere else, those of us who wanted to see women dance, make love, wear shirts and pants. Here, and in other bars like this one, we found each other and the space to be a sexually powerful butch-femme community\" (37).\n\n. I discuss this shift, along with the GLF\/GAA strategy of \"coming out,\" in chapter 1.\n\n. For an excellent collection of essays on the sex wars of the early 1980s, see Duggan and Hunter. Analyses by writers of color and others sensitive to the divisions within both Anglo-American feminism and lesbian and gay politics include Beam; Bulkin, Smith, and Pratt; Hemphill, _Brother to Brother;_ Hull, Scott, and Smith; Lorde, _Sister Outsider_ and \"What Is at Stake\"; Martin; Moraga and Anzald\u00faa.\n\n. On the Lesbian Avengers, see Schulman, _My American History_ , 277\u2013324; Dewan. Schulman implies that the activist spirit in this group reinvigorated ACT UP ( _My American History_ , 311).\n\n. For overviews of the Harlem Renaissance that fill out some of the tensions I discuss in these paragraphs, see Lewis; Singh, 1\u201339. Gates's \"The Trope of a New Negro and the Reconstruction of the Image of the Black\" traces the vicissitudes of the term _New Negro_ from the nineteenth century through the Harlem Renaissance.\n\n. On the Native American Renaissance, see Lincoln; on the Hawaiian Renaissance, see Sumida.\n\n. At times Moraga's critique of America and of nationalism leads her to posit another, Chicano nationalism. In chapter 3, I am more critical of Moraga and of this alternative.\n\n. Senator Jesse Helms (R-N.C.) and Representative Robert Dornan (R-Calif.) are two of the most vocally homophobic members of Congress; Pete Williams was a spokesperson for the Pentagon in the Bush administration and was outed by Signorile and others in protest over the military's policy of discharging lesbians and gay men. Vigorous opposition to the nomination of Clarence Thomas to the Supreme Court in 1991 arose mainly because of charges brought by a former employee, Anita Hill, although both his record on civil rights and his judicial qualifications in general also were and are highly questionable. Senate hearings on the Thomas-Hill controversy were held in the fall of 1991. Barbara Mikulski was, at the time, one of only two women in the Senate; she was the only one of the two to support Hill (and civil rights generally) by opposing Thomas's nomination. Barney Frank (D-Mass.), Gerry Studds (D-Mass.), and Steve Gunderson (R-Wis.) were the first openly gay members of Congress. Dr. Joycelyn Elders served as surgeon general in the Clinton administration until she was dismissed in 1994 because of her comments on masturbation.\n\nElders's comments about the length of her telephone conversation with Bill Clinton were made on the _Larry King Live_ television show. There are numerous other examples of Elders's willingness to take on various queer identities, and there is no doubt that such a willingness hastened her dismissal. When asked by the newspaper _USA Weekend_ why she agreed to grant a \"controversial\" interview to _The Advocate_ , Elders asserted, \"Why shouldn't I talk to _The Advocate?_ I would do it again.\" And when critics of her opinions on the necessity of condom use began to call her the \"condom queen,\" she told the _New York Times Magazine_ , \"If I could be the condom queen and get every young person who is engaged in sex to use a condom in the United States, I would wear a crown on my head with a condom on it! I would!\" These facts about Elders were drawn from the \"Agenda\" section in the following issues of _The Advocate:_ March 8, 1994; July 12, 1994; March 7 and 12, 1995. Elders was criticized in particular by some religious leaders for talking to the gay and lesbian press; she insisted that she would do so again, and after her dismissal, she even contributed book reviews to _The Advocate_. See the article by Elders, \"Three-Ring Circus.\"\n\nMy comments in this section have obviously focused on the dearth of \"queers in Washington.\" Although I still disagree with Signorile's use of the concept, I would agree with his assertion that there are, indeed, \"queers in Hollywood.\" Signorile and I, however, would not necessarily have the same people in mind, since I would include in that category, for instance, television comedian Roseanne.\n\n. Rich Tafel is head of the gay Republican group known as the Log Cabin Republicans. Bruce Bawer is the author of _A Place at the Table: The Gay Individual in American Society_. I engage more directly with Bawer in my epilogue.\n\n. I also develop this argument in chapter 3. See Phelan, 151\u201354, for another discussion of these conflicts. It is interesting that Phelan is wary of how _queer_ might \"deny the colonization of lesbians and people of color that occurs within 'gay' politics\" (154), even though all the queer theorists she cites in her discussion are women (Arlene Stein, Gayle Rubin, Robyn Ochs, and Lisa Duggan).\n\nAlthough Wiegman criticizes \"the deployment of _queer_ so far,\" she provides no example of such a deployment. Wiegman thinks that Teresa de Lauretis coined the term _queer theory_ , which is perhaps part of the problem, since de Lauretis is so anxious in her introduction to the special issue of _differences_ on queer theory to distance the concept from actual queer activists (de Lauretis, \"Queer Theory,\" xvii n. 2). Since I was participating in a graduate seminar called \"Queer Theory\" at the time de Lauretis's volume appeared, I am skeptical about positioning her as the individual author of the concept. In other locations, Wiegman's thoughts on queer theory are more complicated. In a recent review of books by Judith Butler and Eve Sedgwick, for instance, she concludes approvingly that \"identity is always out of place\" in the queer theory of these two writers (Wiegman, review, 895). _Who Can Speak? Authority and Critical Identity_ , a collection of essays edited by Judith Roof and Wiegman, additionally contains several engagements with queer theory.\n\n. I am reminded here of the assertions of an audience member at the final plenary session of \"InQueery\/InTheory\/InDeed,\" the Sixth Annual Lesbian and Gay Studies Conference held at the University of Iowa in November 1994. Audience members' comments generally revolved around predictable issues of exclusion and marginalization at the conference. One woman, however, chose to register her criticism in a unique\u2014and to my mind, highly productive\u2014way: after reciting a list of authors, publication dates, and titles (\"Cherr\u00ede Moraga and Gloria Anzald\u00faa, 1981, _This Bridge Called My Back_ . . . Audre Lorde, 1982, _Zami: A New Spellingof My Name_ . . . Gloria Anzald\u00faa, 1987, _Borderlands\/La Frontera: The New Mes-tiza_ \"), the conference participant asserted, \" _That_ is queer theory.\" In other words, far from being mere \"additions,\" these women writers of color are always already on the inside, _authorizing_ queer theory\u2014not in the (patronizing) sense of merely looking on and giving approval to someone else's project but rather in the sense of writing (and living) the history of queer theory themselves.\n\n### **Notes to Chapter 1**\n\n. John D'Emilio explains that homophile activists adopted the slogan \"Gay is Good\" in 1968 ( _Making Trouble_ , 239). Barry Adam notes that Franklin Kameny of the Mattachine Society of Washington, D.C., had used the phrase as early as 1964: \"In [the] face of the MSNY [Mattachine Society of New York] president's traditional homophile contention that 'we must lose the label of homosexual organizations,' Kameny asserted simply that 'gay is good!'\" (71). Kameny was consciously referencing African American rhetoric, such as the slogan \"Black is Beautiful.\" For a discussion of the influence on the early gay movement of the rhetoric of African American and Latino groups such as the Black Panthers and the Young Lords, see D'Emilio, _Making Trouble_ , 240\u201341. On Stonewall generally, see Duberman, _Stonewall_.\n\n. See the essays included in Jay and Young. On the GLF, see D'Emilio, _Sexual Politics_ , 233\u201335, and _Making Trouble_ , 239\u201346; Adam, 73\u201389; Kissack. Adam specifically recounts the GLF and GAA (Gay Activists Alliance) zaps that helped pave the way for the APA shift (81\u201382). On the Radicalesbians, see Echols, 215\u201317, 232; Kissack, 121\u201323. The construction of the woman-identified woman enabled heterosexual feminists to identify more fully with their lesbian sisters, but it was, in many ways, a conservative move. Alice Echols argues that it was \"designed to assuage heterosexual feminists' fears about lesbianism\" (215); consequently, it ended up desexualizing lesbianism. Still, despite its evasion of \"the knotty problem of sexuality,\" the construction at least temporarily \"redefined lesbianism as the quintessential act of political solidarity with other women\" (Echols, 217).\n\n. I have already mentioned the re-presentation of Civil Rights rhetoric in Edmund White's fictional account of Stonewall. The name Gay Liberation Front itself was meant to give tribute to the liberation struggles in Vietnam and Algeria (Duberman, _Stonewall_ , 217).\n\n. For a good overview of feminist standpoint epistemology, as well as an \"archeology of standpoint theory\" (76), see Sandra Harding. Harding explains that feminist standpoint theory emerges from an engagement with Marxism (53\u201354). On this point, see Haraway, 186\u201387. By translating standpoint theory to a queer context, I am trying to further coalition building among theorists; like Harding, I recognize that \"even though standpoint arguments are most fully articulated as such in feminist writings, they appear in the scientific projects of all the new social movements\" (54). Echols's overview of radical feminism demonstrates that the boundaries between the different groups engaging in this self-reflexive constructionism were often blurred.\n\n. On this shift, see also D'Emilio, \"Foreword,\" xxiv\u2013xxviii, and _Making Trouble_ , 239\u201351.\n\n. I am grateful to Steve Amarnick for helping me sort out some of the issues I am working with here. I am not dismissing the psychological importance of coming out, nor am I denying that Oprah Winfrey's sensitivity to coming out is empowering to thousands of gay and lesbian individuals. Rather, I am suggesting that the \"coming out equals self-respect\" model alone can be equally disempowering for the gay and lesbian _movement_ , and for the many other lesbian and gay people who too infrequently see themselves represented in any other way.\n\n. For a counterexample, see John Preston. The very fact that Preston can say, \"I always shake my head in disbelief when I read critics who think the coming out novel is just a stage we're going through\" (39), suggests that many critics have expressed their reservations about the genre. Indeed, Preston's article is in the same issue of _Lambda Book Report_ that includes Sarah Schulman's derisive comments about the coming-out story (see Fries). In his introduction to the most recent edition of _A Boy's Own Story_ , White himself notes the reservations that have been expressed about the genre: \"Now [1994] there's an excess of coming-out novels, and critics talk of creating a ban against any further ones\" (\"On the Line,\" xv).\n\n. Neither White nor Lorde has necessarily identified with the concept of \"queer\" as it is deployed in this chapter and throughout this study, although White has begun to use the term _queer_ more frequently, generally as a synonym for gay men and lesbians. See White, \"The Personal Is Political: Queer Fiction and Criticism,\" in _The Burning Library_ , 367\u201377. The work of Lorde (and other women of color) in the early 1980s, however, helped usher in the queer theory and activism of the later 1980s by facilitating the emergence of the disruptive identities that are at the center of the Queer Renaissance\u2014identities that are not fixed in advance but are rather constantly reshaped in interaction with other identities. See chapter 3 for a fuller consideration of these issues in relation to Gloria Anzald\u00faa and the 1981 publication of _This Bridge Called My Back;_ see my introduction, for a consideration of these issues in relation to my own use of the term _queer_. Some of Lorde's readers, of course, do identify with the term _queer_ and deploy it in various ways, as my discussion of _Afrekete: An Anthology of Black Lesbian Writing_ later in this chapter highlights.\n\nFor another reading of _Zami_ through and as feminist positionality theory, see Carlston, 226, 231\u201332, 236. On positionality theory in relation to Lorde's 1986 collection of poetry _Our Dead Behind Us_ , see Hull, 155, 159.\n\n. My thoughts on the ways in which _A Boy's Own Story_ disrupts heterosexist understandings of linear sexual development are indebted to Kenneth Kidd.\n\n. White himself has at other times disavowed the possibility that _A Boy'sOwn Story_ might be somehow representative; he writes, \"The novel is not a political tract, nor is it meant to be representative or typical\" (\"On the Line,\" xii). My larger point here, however, is that the invisibility of whiteness allows for _ABoy's Own Story_ to be cast as representative regardless of White's (clearly contradictory) intentions. The packaging of the novel underscores this: in spite of White's disavowal, a blurb on the back cover from the _New York Times's_ Christopher Lehmann-Haupt announces, \"It is any boy's story. . . . For all I know, it may be any girl's story as well.\" The Plume paperback has gone through at least three cover designs since 1982, each utilizing a photograph of a different young (white) boy.\n\n. I am grateful to Michael Thurston for pointing out this connection to me.\n\n. Barbara Smith has made a similar point (Bulkin, Smith, and Pratt, _75\u201376_ ).\n\n. This interpretation attempts to resist what Sagri Dhairyam points out about critical responses to Lorde's poetry: \"The relational, shifting points of the politics of identity enacted by and through Lorde's poems are stilled by their recuperation into canons of feminine or lesbian identity\" (243). Nonetheless, I have framed this interpretation in a chapter about \"coming out,\" and this, coupled with my own gay male location, might still the \"shifting points of the politics of identity\" yet again, recuperating Lorde, this time, into a rapidly expanding canon of \"queer theory.\" This tension within \"queerness,\" which I discuss at greater length in chapter 3, is unavoidable, but I hope that a negotiation of the tension will fuel, rather than forestall, the work of queer theory.\n\n. Lorde's work has influenced the shaping and reshaping of so many readers, writers, and communities that it would be impossible to compile a comprehensive list of the locations where such reinvention has occurred. _Celebrate the Life and Legacy of Audre Lorde_ , a booklet distributed for Lorde's memorial service in New York City on January 18, 1993, particularly attests to the scope of her vision and influence. _Celebrate the Life and Legacy_ includes tributes from, among many others, Palestinian feminists, groups fighting for Hawaiian independence, the Organization of Women Writers of Africa, and Men of All Colors Together. I am grateful to Steve Amarnick for sending me a copy of this document.\n\n. In 1992 a revised edition of _Chosen Poems\u2014Old and New_ was published as _Undersong_. In that collection, Lorde reiterated her commitment to revision: \"The process of revision is, I believe, crucial to the integrity and lasting power of a poem. The problem in reworking any poem is always when to let go of it, refusing to give in to the desire to have that particular poem _do it all_ , say it all, become the mythical, unattainable Universal Poem\" (xiii).\n\n. For white gay male creative work that is more self-reflexive about whiteness or about positionality, respectively, see Allan Gurganus's 1990 collection of stories _White People_ and Tom Joslin and Peter Friedman's 1993 film _Silverlake Life: The View from Here_. Gurganus's stories explore both gay and nongay white life in North Carolina, and Joslin and Friedman's film is a collective effort (often depending on who had the strength to work the camera) at documenting Joslin's and his lover Mark Massi's deaths from AIDS. The position from which the camera \"sees\" is repeatedly foregrounded and complicated. For a discussion of embodied and communal identity in the work of another white gay male author, Robert Gliick, see Jackson. See also Crimp, \"Right On, Girlfriend!\" and my discussion of Tony Kushner and \"queer perestroika\" in chapter 4, for considerations of the sort of collective identity I discuss throughout this chapter.\n\n. Jewelle Gomez notes, significantly, that \"commercial companies . . . would publish Audre's poems but could not bring themselves to publish the more explicit ideas of her essays\" (7). On White's sometimes rocky experiences with publishers, see Bonetti, 101\u20132.\n\n. I am grateful to Stacy Alaimo for calling my attention to this series. The Book-of-the-Month Club instituted a similar series with fewer authors but\u2014initially\u2014the same problems: included are Baldwin ( _Giovanni's Room_ ), White ( _A Boy's Own Story_ and _The Beautiful Room Is Empty_ [\"available here in one exclusive volume\"]), and Rita Mae Brown ( _Rubyfruit Jungle_ , published in 1973). After Lorde's death in 1992, however, an \"exclusive three-in-one volume\" of _Zami, Sister Outsider_ , and _Undersong_ was the first new addition to the Book-of-the-Month Club series. This addition highlights the difficulty of making overly hasty generalizations about either White's or Lorde's canonicity. Also, I should note that, despite mainstream attempts to cast White as \"America's most influential gay writer,\" such attempts have hardly garnered him nationwide fame. White has found such fame instead in European countries: in France, White is considered by many to be the most important American writer since Henry James, and in England, as he himself explains, _A Boy's Own Story_ \"made me so well known that English fans are always astonished to learn that most Americans don't know who I am\" (\"On the Line,\" xix).\n\n. Since, at the time of this award, Lorde was in Berlin undergoing treatment for the cancer she had been battling for more than a decade (and which eventually took her life), Jewelle Gomez, another black lesbian writer, delivered Lorde's speech. The \"I\" of Lorde's speech is thus destabilized in a way similar to that of the \"I\" in \"Need,\" discussed above.\n\n. White's \"greatest hit\" list is as follows: among non-American writers, \"Marcel Proust, Virginia Woolf, Colette, Jean Genet, Thomas Mann, Christopher Isherwood, Ronald Firbank,\" and among Americans, \"Elizabeth Bishop, James Merrill, and John Ashbery\" (\"Twenty Years On,\" 5). Overall, I might add, White's 1989 acceptance speech is about two and a half times longer than Lorde's 1990 speech.\n\n. Such bittersweet isolation is conveyed by the very title of White's novel: _\"The Beautiful Room Is Empty_ . . . takes its title from one of Kafka's letters alluding to the unfortunate inability of two people (or perhaps two psyches) to inhabit a single space\" (Radel, 184). White's beautiful and empty room contrasts significantly with Lorde's \"house of difference,\" where cohabitation, though never easy, is both possible and necessary for survival.\n\n### **Notes to Chapter 2**\n\n. Vito Russo lists lesbian and gay characters, the films in which they appear, and the causes of their deaths in his \"Necrology\" in _The Celluloid Closet: Homosexuality in the Movies_ (347\u201349). In the 1930s the Hays Office, headed by former postmaster general Will Hays, was responsible for the creation of the Motion Picture Production Code. The motion picture industry used this code to regulate itself (Russo, 31). Under the Hays code, virtually all portrayals of homosexuality were negative, and many ended in death, so that audiences would see that \"vice\" was appropriately \"punished.\" My thanks to Ramona Curry for discussing the Hays code with me.\n\n. \"Widely available\" is an understatement; migration-to-the-big-city novels could compete against coming-out stories for the title of \"Most Common Lesbian\/Gay Genre.\" Pre-Stonewall works that turn upon migrations to the city include Ann Bannon's Beebo Brinker series (for example, _Beebo Brinker_ [1962]; the back cover declares, \"She landed in New York, fresh off the farm\") and John Rechy's _City of Night_ (1963). Post-Stonewall explorations of the trope include Armistead Maupin's Tales of the City series (for example, _Tales of the City_ [1978]; the front cover of the first paperback edition depicts a female figure with luggage, dwarfed and awed by the immensity of the Golden Gate Bridge), Andrew Holleran's _Dancer from the Dance_ (1978), and Edmund White's _The Beautiful Room Is Empty_ (1988). Ethan Mordden's _I've a Feeling We're Not in Kansas Anymore_ (1985) explicitly thematizes the disjunction between the city and the \"provinces.\"\n\nOf course, as Gates himself notes, the migration from South to North (and the corresponding shift from rural to urban) is also a trope in the African American literary tradition ( _Signifying Monkey_ , xxv). In a way, suggesting that Randall Kenan replicate this trope precludes him from signifying on it (see note 11, below).\n\nIn the Queer Renaissance, the dominance of the migration-to-the-big-city theme has been implicitly challenged by various Southern writers, including Kenan. For a few of the many texts in which Southern writers consider how queer subjectivity is shaped in the backwoods, marshes, and trailer parks of the South, see Allan Gurganus's _White People_ (1990), Blanche McCrary Boyd's _The Revolution of Little Girls_ (1991), and Dorothy Allison's _Bastard out of Carolina_ (1992). See also Kenan and Allison's conversation about Southern queer literature, \"Spies Like Us: Talking between the Lines.\"\n\n. The second chapter of _The Signifying Monkey_ , \"The Signifying Monkey and the Language of Signifyin(g): Rhetorical Difference and the Orders of Meaning\" (44\u201388), provides a thorough explanation of the black concepts of \"Signification\" and \"Signify in (g).\" \"The bracketed _g_ ,\" Gates writes, \"enables me to connote the fact that this word is, more often than not, spoken by black people without the final _g_ as 'signifyin'\" (46).\n\n. This presumption, of course, may be a bit hasty. Gates himself, earlier in the interview I have been citing, notes that \"a lot of gay black men in Harlem . . . are tired of being used for batting practice\" (in Rowell, 454).\n\n. \"Vogueing,\" developed by black and Latino gay men in New York City, is a form of dance in which dancers imitate and implicitly critique \"high-fashion\" styles and poses, such as those depicted in _Vogue_ magazine. \"Snapping\" is a gesture of pride and defiance used to \"read,\" punctuate, or invalidate another's discourse (Becquer, 8\u201312).\n\n. Michel Foucault's famous formulation is relevant here:\n\nThere is not, on the one side, a discourse of power, and opposite it, another discourse that runs counter to it. Discourses are tactical elements or blocks operating in the field of force relations; there can exist different and even contradictory discourses within the same strategy; they can, on the contrary, circulate without changing their form from one strategy to another, opposing strategy. (101\u20132)\n\n. Despite this disclaimer, I think an earlier article of mine on _A Visitation of Spirits_ did disarm Becquer's analysis (see McRuer, \"A Visitation of Difference\"); I hope that my indebtedness to Becquer's argument is clearer here (see note 22, below).\n\n. The theme of \"coming home\" runs throughout Hemphill's work. This quotation comes from his essay \"Does Your Mama Know about Me?\" Later in _Ceremonies_ , in \"Loyalty,\" Hemphill makes a similar point: \"We will not go away with our issues of sexuality. We are coming home\" (64). See also Joseph Beam, \"Brother to Brother: Words from the Heart\":\n\nWhen I speak of home, I mean not only the familial constellation from which I grew, but the entire Black community: the Black press, the Black church, Black academicians, the Black literati, and the Black left. Where is my reflection? I am most often rendered invisible, perceived as a threat to the family, or am tolerated if I am silent and inconspicuous. I cannot go home as who I am and that hurts me deeply. ( _In the Life_ , 231)\n\nHemphill's work, with its insistence that \" _I am_ coming home,\" explicitly responds to Beam's lament here.\n\n. For another brief overview of _A Visitation of Spirits_ , see my bio-bibliographical article \"Randall Kenan,\" 234\u201335.\n\n. Reverend Barden reappears in _Let the Dead Bury Their Dead_ , Kenan's collection of short stories. In \"Ragnarok! The Day the Gods Die,\" Barden's internal thoughts are juxtaposed with his pontifications from the pulpit. \"Ragnarok!\" is, on the surface, the record of Barden's eulogy to the late Sister Louise Tate of the First Baptist Church. This eulogy is interrupted, however, by Barden's parenthetical meditations on his decades-long affair with the woman. Barden's hypocrisy in this story helps to de-authorize the bigotry he preaches in _A Visitation of Spirits_.\n\n. Interestingly, and _pace_ Gates, both of Kenan's significations here move an \"urban\" story to the rural location of Tims Creek. _A Christmas Carol_ , of course, is set in nineteenth-century London. _Go Tell It on the Mountain_ is set primarily in New York. Several of Baldwin's characters come to New York _from_ the South, suggesting that moving the story from New York _to_ the South (which exactly reverses the situation) is part of Kenan's signification on Baldwin.\n\n. Similar passages occur elsewhere in _Go Tell It on the Mountain;_ for example, \"[John] wanted to stop and turn to Elisha, and tell him . . . something for which he found no words\" (219). Emmanuel Nelson discusses such passages in his examination of homoeroticism in _Go Tell It on the Mountain_ (\"James Baldwin,\" 9\u201311).\n\n. I have provided only a skeletal outline of the ways in which _A Visitation of Spirits_ signifies on _Go Tell It on the Mountain_ , since to do more would be beyond the scope of this chapter. But it is not just in the story of Horace that Kenan signifies on Baldwin. Several sections of _A Visitation of Spirits_ focus on older members of the community (Zeke and Ruth) who explore their pasts through internal monologue or through \"dialogue\" with God. As I mention briefly later, I read these sections as signifying on the second part of Baldwin's novel, \"The Prayers of the Saints\" (63\u2013189).\n\nFor an analysis of other black gay significations on the church, see Nero, 23843. As I suggested, Baldwin as well is quite critical of the church. For a more explicit and autobiographical critique, see Baldwin's _The Fire Next Time:_\n\nBeing in the pulpit was like being in the theatre; I was behind the scenes and knew how the illusion was worked. ... I knew, though I did not wish to know it, that I had no respect for the people with whom I worked. I could not have said it then, but I also knew that if I continued I would soon have no respect for myself. ... I really mean that there was no love in the church. It was a mask for hatred and self-hatred and despair. (55\u201358)\n\nSee Edelman, 68\u201371, for an analysis of the ways in which Baldwin resignifies the concept of salvation in his last novel, _Just above My Head_. I return to Baldwin in the next section of this chapter, where\u2014through a consideration of Jimmy Greene\u2014I am more explicit about how the church might be \"saved.\" The \"other sort of church\" _A Visitation of Spirits_ envisions through Jimmy might be understood as a local manifestation of the \"other myths of queer positionality\" I advocate in chapter 1.\n\n. A few pages before this scene, Kenan playfully underscores that his story should be seen not in a direct line of descent from but rather as a signification on Dickens. Horace is ostensibly discussing, with one of his lovers, the historical inaccuracies of the theatrical production of the history of the Cross family, but his comment ironically speaks to (and invalidates) the parallels between _A Visitation of Spirits_ and _A Christmas Carol:_ \"[This play] is more than a little inaccurate, to tell the truth. I didn't have a great-great-great-grandfather named Ebenezer\" (224).\n\n. Eve Sedgwick discusses the Department of Health and Human Services report in her article \"Queer and Now\":\n\nI think everyone who does gay and lesbian studies is haunted by the suicides of adolescents. To us, the hard statistics come easy: that queer teenagers are two to three times likelier to attempt suicide, and to accomplish it, than others; that up to 30 percent of teen suicides are likely to be gay or lesbian; that a third of lesbian and gay teenagers say they have attempted suicide; that minority queer adolescents are at even more extreme risk. ( _Tendencies_ , 1)\n\n. If we take into account the annual number of deaths from lung cancer that can be linked to cigarette use, then the tobacco idyll itself, even without the suicide that immediately precedes it, is dependent on numerous sacrifices for the \"good of the community.\" My thanks to Elizabeth Davies for suggesting this to me.\n\n. The other epigraph to _A Visitation of Spirits_ is from William Gibson's _Neuromancer:_ \"To call up a demon you must learn its name. Men dreamed that, once, but now it is real in another way\" (epigraph page; Gibson, 243).\n\nThe Dickens epigraph is a commentary on how Kenan's signification on _A Christmas Carol_ should be read. One way (among many) to read the Gibson epigraph is as a commentary on _A Visitation of Spirits's_ other major signification, on Baldwin. Since Kenan's novel \"outs\" _Go Tell It on the Mountain_ , it rejects the tradition of coding homosexuality as the \"unnameable\" or \"unspeakable\" and instead \"learns its names.\"\n\n. Actually, although I focus on Foucault, I do not mean to suggest that the other writers Jimmy is reading are insignificant. Foucault most clearly introduces my argument about the ways in which sexual meanings have become unstable for Jimmy; later in this section, I discuss the transformations of Jimmy's public life (asminister), and in that context it might be as appropriate to consider why Jimmy is reading Erasmus or Augustine ( _Confessions?_ ). Still, I focus on Foucault at this point because Jimmy's reasons for reading Foucault are a bit more ambiguous than his reasons for reading some of the others. After all, although John Hope Franklin (or C. L. Franklin), Benjamin Quarles, and Frantz Fanon are obviously important to Jimmy, he precedes their names with a touchstone that explains why: \"black history.\"\n\n. The translation is Ed Cohen's.\n\n. Perhaps I should say that my Foucauldian fantasies are not intended to \"name\" in the _standard_ English sense of that word. On the trope of naming in the black literary tradition, see Gates, _Signifying Monkey_ , 55, 69, 82, 87.\n\n. On the tension between minoritizing and universalizing views of homo\/heterosexual definition, see Sedgwick, _Epistemology of the Closet_ , 82\u201386.\n\n. Mercer writes:\n\nThe challenge of sameness entails the recognition that we share the same planet, even if we live in different worlds. We inhabit a discursive universe with a finite number of symbolic resources which can nevertheless be appropriated and articulated into a potentially infinite number of representations. Identities and differences are constructed out of a common stock of signs, and it is through the combination and substitution of these shared elements that antagonism becomes representable as such. (\"1968,\" 427)\n\nIn general, this section on Jimmy and the \"challenge of sameness\" that Horace poses is the part of my argument that differs most sharply from my earlier article on _A Visitation of Spirits_. In \"A Visitation of Difference: Randall Kenan and Black Queer Theory,\" my critique of identity politics tended to elide the theoretical potential of subversive sameness.\n\n. With this view, Kenan positions himself as part of a strong tradition that rejects the oppressive elements of the African American church but embraces its more progressive, visionary elements. Baldwin is part of this tradition as well. Cornel West would perhaps identify Baldwin's and Kenan's views as related to what he calls \"prophetic moral reasoning.\" According to West, prophetic moral reasoning\n\nencourages a coalition strategy that solicits genuine solidarity with those deeply committed to antiracist struggle. ... A prophetic framework replaces black cultural conservatism with black cultural democracy. Instead of authoritarian sensibilities that subordinate women or degrade gays and lesbians, black cultural democracy promotes the equality of black women and men and the humanity of black gays and lesbians. In short, black cultural democracy rejects the pervasive patriarchy and homophobia in black [and white] American life. (397\u201398)\n\n. In _A Visitation of Spirits_ , although Rabbi helps Jimmy rewrite the \"traditional\" Christianity of Tims Creek, North Carolina, it is possible to read this ostensible rewriting as closer to the original spirit of Christianity. Many of Christ's disciples, of course, called him \"Rabbi.\" I'm grateful to Amy Farmer for reminding me of this.\n\n. Indeed, Gates's own tributes to Houston Baker often suggest as much; see, for example, his aside to Baker in \"Canon-Formation, Literary History, and the Afro-American Tradition,\" 15.\n\n. In 1994, Chelsea House Publishers launched a series of biographies intended for young lesbians and gay men, Lives of Notable Gay Men and Lesbians, under the general editorship of Martin Duberman. Kenan's biography of James Baldwin was the first installment in this series. See Kenan, _James Baldwin_. See also V. Hunt's recent interview with Kenan, in which Kenan cites Baldwin's influence in his decision to become a writer and identifies Baldwin as \"the one other person [he] can really be paired with\" (Hunt 412, 415).\n\n. The complete title of the story is \"Let the Dead Bury Their Dead; Being the Annotated Oral History of the Former Maroon Society called Snatchit and then Tearshirt and later the Town of Tims Creek, North Carolina [circa 1854\u20131985]\" (271). \"Let the Dead Bury Their Dead\" was originally conceived as part of _A Visitation of Spirits_ (Hunt, 417).\n\n. I am suggesting that the emergence of Menes\/Pharaoh marks a \"resurrection,\" of sorts, for Jimmy, but Jimmy's death might be read ironically in another sense: Jimmy dies on March 12, 1998. March 12 is Kenan's birthday.\n\n### **Notes to Chapter 3**\n\n. The best study of male \"homosociality\" in literature is Eve Kosofsky Sedgwick's _Between Men: English Literature and Male Homosocial Desire_. In _BetweenMen_ , Sedgwick examines \"erotic triangles\" in which the bond between two (male) rivals for a (female) beloved's affection is often as strong, or stronger, than the bond between either rival and the beloved (21). This erotic economy facilitates the exchange of women but simultaneously requires a structural homophobia that sharply proscribes the men's relationships with each other, thereby ensuring their participation in the patriarchal system.\n\nAlthough Sedgwick argues that, at the end of the nineteenth century, this homosocial system began to give way \"to a discussion of male homosexuality and homophobia as we know them\" (202), a \"male bonding\" or homosociality that nonetheless requires a structural homophobia obviously continued to exist in various forms into and throughout the twentieth century. Robin Wood, for example, argues that the 1970s \"buddy film\" requires \"an explicitly homosexual character\" who \"has the function of a disclaimer\u2014our boys are not like _that_ \" (228\u201329). Kerouac's novels, which also make use of such queer disclaimers, are antecedents for such films. Of course, the homosocial system of the buddy film and of Kerouac's texts becomes somewhat difficult to sustain when, as in Kerouac's _The Subterraneans_ , the protagonist ends up sleeping with the queer disclaimer (72\u201374); drugs and alcohol, however, are always conveniently on hand to explain (away) the lapse. Although this structural homophobia makes Kerouac a fascinating object of analysis for contemporary queer inquiry, I think it also prohibits consideration of him as the \"classical gay writer\" Sarah Schulman considers him to be (My _American History, 167_ ); hence my description of him as an \"unlikely candidate\" for inclusion in my study.\n\n. The United States, Canada, and Mexico signed the North American Free Trade Agreement (NAFTA) in 1993. Many regulations on goods and services crossing the border were lifted; the agreement was intended to provide all three countries with increased access to the others' markets. Project Censored includes \"NAFTA's Broken Promises\" among its top ten censored stories of 1995. Less than two years after passage of the agreement, many American corporations have cut jobs in the United States\u2014more than 150,000 in 1994 alone\u2014and moved their plants to Mexico, where they have exploited both workers and the environment in blatant violation of labor rights and environmental regulations in that country.\n\n. Teresa de Lauretis's insight here comes from her analysis of Barbara Smith's landmark \"Toward a Black Feminist Criticism.\" Turning to Gloria T. Hull, Patricia Bell Scott, and Barbara Smith's 1982 anthology _All the Women Are White, All the Blacks Are Men, but Some of Us Are Brave_ , de Lauretis adds, \"The term 'blacks' does not include (comprehend) black women any more than the term 'man' (white men) includes or comprehends white women. The black feminist concept of a simultaneity of oppressions means that the layers are not parallel but imbricated into one another; the systems of oppression are interlocking and mutually determining\" (de Lauretis, \"Eccentric Subjects,\" 134). In general, Andrew Ross's analysis in _No Respect: Intellectuals and Popular Culture_ cannot accommodate such observations: Ross has his \"race\" chapter (read: black men) and his \"gender\" chapter (read: white women), but never do the two meet.\n\n. In Kerouac's work, the particular coming together of _attraction to_ and _revulsion for_ women of color is perhaps presented most starkly in _The Subterraneans_ , in the relationship between Leo Percepied and Mardou Fox, a young African American woman. Although his affair with Mardou is passionate, Leo repeatedly abandons her, often for explicitly racist reasons: \"I wake from the scream of beermares and see beside me the Negro woman with parted lips sleeping and little bits of white pillow stuffing her black hair, feel almost revulsion, realize what a beast I am for feeling anything near it\" (24). Despite the protagonist's apparent awareness of it, the attraction-revulsion pattern repeats throughout _The Subterraneans_.\n\n. For an incisive overview of the _corrido_ tradition, see R. Sald\u00edvar, 26\u201342. See also J. Sald\u00edvar, 49\u201384 (esp. 51\u201356).\n\n. Harper goes on to connect the elision of specificity that results from the mass-cultural marketability of marginality to the similar elision of specificity that can occur in academic analyses of social marginality (195). For another consideration of the marketability of a \"fashionable\" queerness, see Clark.\n\n. Of course, inevitable disappointment has been a component of many, many theoretical movements. Slavoj Zi\u017eek argues that identity becomes a rallying point in political movements when subjects phantasmatically invest identity with an expectation of wholeness. Such wholeness will always be exposed as fictive; hence inevitable disappointment occurs when the identificatory sign fails to live up to its original promise (cited in Butler, _Bodies That Matter_ , 209). As an alternative to this politicization of identification, Judith Butler offers the politicization of disidentifica-tion: \"What are the possibilities of politicizing d\/sidentification, this experience of _misrecognition_ , this uneasy sense of standing under a sign to which one does and does not belong?... It may be that the affirmation of that slippage, that failure of identification is itself the point of departure for a more democratizing affirmation of internal difference\" ( _Bodies That Matter_ , 219). My reading of Anzald\u00faa and this chapter in general are very much indebted to Butler's ideas; and as my conclusion should make clear, my reading of Butler is, in turn, indebted to Anzald\u00faa. As Butler herself notes, she is not the first to suggest such a politicization of disidentification:\n\nTo understand \"women\" as a permanent site of contest, or as a feminist site of agonistic struggle, is to presume that there can be no closure on the category and that, for politically significant reasons, there ought never to be. . .. The numerous refusals on the part of \"women\" to accept the descriptions offered in the name of \"women\" not only attest to the specific violences that a partial concept enforces, but to the constitutive impossibility of an impartial or comprehensive concept or category. (221)\n\nSee Riley, whom Butler herself cites, for another feminist argument suggesting a contestatory relation to the sign _woman;_ see also Snitow, 9.\n\n. Norma Alarcon writes:\n\nThe most remarkable tendency in the work reviewed in this essay [Alarc\u00f3n's \"The Theoretical Subject(s) of _This Bridge Called My Back_ and Anglo-American Feminism\"] is the implicit or explicit acknowledgment that, on the one hand, women of color are excluded from feminist theorizing on the subject of consciousness and, on the other, that though excluded from theory, their books are read in the classroom and\/or duly (foot)noted. (39)\n\nAlthough not specifically \"feminist\" or \"queer,\" Henry Giroux's work provides some of the more glaring examples of this relegation to the margins: although he wrote a book called _Border Crossings_ , he virtually ignored Chicana\/o scholarship. Anzald\u00faa shows up on one page, in an epigraph for Giroux's own thoughts on \"The Politics of Voice and Difference\" (168).\n\n. Through her representation of this process, Anzald\u00faa cites one of the legacies of the Chicano Renaissance. Luis Valdez's Brechtian _actos_ , or short one-act plays, were designed to incite audience members to action (such as joining the union or striking) through careful consideration of their social locations. _Los Vendidos_ (The sell-outs), in particular, encouraged audiences to resist an assimilationist Mexican American identity and to assume instead a critical Chicano identity. See Valdez and El Teatro Campesino. Anzald\u00faa improves on her predecessors in that the identity she posits is more fluid and multiple than the male-centered identity of the Chicano Renaissance (Yarbro-Bejarano, 12). For more on the Chicano Renaissance, see my introduction.\n\n. My interpretation in this section, with its emphasis on specificity over and against appropriation, intersects with Shane Phelan's; see Phelan, Phelan argues that Anzald\u00faa's mestiza consciousness \"has been an avenue for white women to develop a new understanding of alliance\" that precludes simply adopting another's perspective as one's own (73).\n\n. In _Lesbian Utopics_ , Jagose criticizes those who \"figure 'lesbian' as utopic and outside dominant conceptual frameworks\" (5). Jagose situates Anzald\u00faa's borderlands as a site for such lesbian utopic thinking. At the other end of the spectrum, Cherr\u00ede Moraga argues, \"Ironically, the most profound message of _La Frontera_ I believe has very little to do with lesbians. . . . Lesbian desire is not a compelling force in the book\" (\"Algo secretamente amado,\" 154\u201355). Too lesbian, not lesbian enough\u2014Anzald\u00faa is caught between colliding critics as well as cultures (and I am not innocent of participation in the conflict; see note 15, below).\n\n. According to Raiskin, the very term _mulatto_ reflects this conflation, since it is a racial categorization that draws its name from the sterile product (the mule) of an \"unnatural\" sexual coupling (157). For more on the historical linkage of the mulatto and the homosexual, see Somerville, 256\u201360.\n\n. Similar points, of course, could be made about the work of other women of color, such as Audre Lorde (see chapter 1). I have already discussed briefly Moraga and Anzald\u00faa; see, in addition, Sandoval; B. Smith, \"Toward a Black Feminist Criticism\"; Hull, Scott, and Smith.\n\n. Chela Sandoval sees this sort of mobility as central to \"U.S. Third World Feminism\" generally (10\u201317, esp. 14\u201315). For another analysis explicitly juxtaposing this sort of mobility to what I label an \"unlimited access\" mobility, which maintains relations of domination-subordination by allowing white, male, and heterosexual identities to go unmarked, see Dhaliwal, 87, 93\u201396. The mobility Anzald\u00faa performs has affinities with the affirmation of \"mixing and migration\" that Tony Kushner stages in _Angels in America_. See my discussion of _Angels in America_ and Sarah Schulman's _People in Trouble_ in chapter 4.\n\n. I am implicated in Anzald\u00faa's comments here, since I am a white and male writer whose identity as such often can and does go unmarked and unremarked. In this chapter, and in this project generally, I am attempting to take comments such as Anzald\u00faa's seriously; to counter the tendency to include lesbians of color as footnotes that validate a writer's ability to be \"inclusive\" while nonetheless maintaining a white center and colored margin (see, for example, my comments on Giroux in note 8, above), I am attempting to position the work of lesbian and gay writers of color as at the very center of the Queer Renaissance. However, despite my arguments against \"hipness\" in this chapter, my declared intentions and my commitment to the politics of alliance do not fully displace me from the potentially patronizing position of (white) hostess at a hip and inclusive queer party. I leave this as a necessary tension in my work; attempting to smooth away this tension would, I think, be an example of the very \"escapes from identity\" against which I have argued in this chapter.\n\n. Nor is Anzald\u00faa's lament that white gay men and lesbians \"police the queer person of color with theory\" a simple denunciation of \"theory.\" Elsewhere, Anzald\u00faa writes, \"Because . . . what passes for theory these days is forbidden territory for us, it is _vital_ that we occupy theorizing space, that we not allow whitemen and women solely to occupy it. By bringing in our own approaches and methodologies, we transform that theorizing space\" [ _Making Face_ , xxv). This quote suggests not only that theory is used to police the queer person of color but also that the queer person of color with theory is policed. Anzald\u00faa resists, _with theory_ , that policing.\n\n. For examples crediting _This Bridge Called My Back_ with revolutionizing feminism, see de Lauretis, _Technologies_ , 10; Sandoval, 5; Quintana, 112\u201315, 139\u201340. Lisa Duggan also writes, \"The elaboration of ... a ['queer'] locale within feminist theory could work a radical magic similar to that of the category 'women of color.' As many feminists have argued, the category 'women of color' as proposed in such ground-breaking anthologies as _This Bridge Called My Back_ , is a significant conceptual and political innovation\" (\"Making It Perfectly Queer,\" 25). I agree with Duggan here, while giving an added twist to her argument: \"women of color\" is not simply _like_ \"queer\"; it was theorized _as_ \"queer\" well before queer theory became hip (particularly, as I have suggested, in Anzald\u00faa's \"La Prieta\"). But this extension of Duggan's argument is actually indebted to Duggan herself, who in 1991 taught the first \"Queer Theory\" course at the University of Illinois at Urbana-Champaign, and whose syllabus placed _This Bridge Called My Back_ in just such an \"originary\" position within queer theory.\n\n_Making Face, Making Soul_ extends the disruptive and queer tradition of _This Bridge Called My Back_ from the first page of Anzald\u00faa's introduction, where she explores _\"gestos subversivos_ , political subversive gestures, the piercing look that questions or challenges, the look that says, 'Don't walk all over me,' the one that says, 'Get out of my face'\" (xv). Connection and coalition are central issues in _Making Face, Making Soul_ , but as with _This Bridge Called My Back_ , and as this quote illustrates, these connections are never simply celebratory.\n\n### **Notes to Chapter 4**\n\n. Quotations from _Angels in America: Millennium Approaches_ are cited parenthetically as _MA_. Quotations from _Angels in America: Perestroika_ are cited parenthetically as P.\n\n. Biblical scholars generally agree that the earliest strand in Genesis, Exodus, and Numbers was written by a single author, J, who lived around the tenth century B.C.E. Harold Bloom argues that J was a woman and that later authors seriously revised her work, tempering her irony and taming her impish Yahweh. More normative versions of Judaism and Christianity, which Bloom thinks would be unrecognizable to J, stemmed from these revisions. Kushner acknowledges his debt to Bloom's reading of J's Jacob story in his \"Playwright's Notes\" for _Perestroika_ ( _P_ , 7).\n\n. The first national tour of _Angels in America_ , directed by Michael Mayer, originated in Chicago at the Royal George Theatre, with a cast made up of both Chicago and New York actors. In the spirit of my second chapter's commitment to redefining centers and margins, I focus mostly on this national production\u2014rather than the \"original\" Broadway production\u2014when I comment on the performance of _Angels in America_. Mayer describes the production as \"different conceptually from the New York version,\" which was \"too complicated technically to be toura-ble.\" My theses about the performance of identity in _Angels in America_ actually work best in conjunction with the national production, which Mayer describes, alternatively, as \"actor-driven\" (qtd. in Rossen, 32).\n\n. Undoubtedly, the contradictory nature of this scene was appreciated more fully in the Tel Aviv production. My thanks to Dan Sharon of the Asher Library at Chicago's Spertus Institute of Jewish Studies, for providing me with translations of these Hebrew passages.\n\n. Kushner has Walter Benjamin's contradictory \"angel of history\" in mind here. Benjamin analyzes a Paul Klee painting, \"Angelus Novus,\" that \"shows an angel looking as though he is about to move away from something he is fixedly contemplating\" (257). Benjamin reads this angel, whose wings are open, as being blown backward into the future, toward which he refuses to look. The wind is so strong that the angel can no longer close his wings (257\u201358). Both Kushner's and Klee\/Benjamin's angels, then, simultaneously represent stasis and extreme movement.\n\n. I am thinking in this section not only of Eve Sedgwick's formulation of her own deconstructive project ( _Epistemology_ , 9\u201310) but also\u2014through my insistence that Prior moves to a new location\u2014of her recognition that \"a deconstructive understanding of these binarisms,\" although necessary, is not \"sufficient to disable them\" (10). With this in mind, it is important to point out that while each of my chapters might be said to perform a deconstructive analysis, each additionally attempts to supplement such an analysis with an examination of new queer identities that work to refuse the terms of the binary relations from which they emerge and which they cite. As the preceding chapter suggests, however, following Gloria Anzald\u00faa and Judith Butler, this is an ongoing project, in that each new performance of \"queerness\" can and will generate other exclusions that must be critically negotiated.\n\n. A local production of _Angels in America_ at the Station Theater in Urbana, Illinois, augmented the queerness of the angel by casting Shelley Holt in the role. In a small community such as Urbana-Champaign, many audience members would undoubtedly remember (and bring to their watching of _Angels in America_ ) Holt's performance of Benita in Brad Fraser's _Unidentified Human Remains and the True Nature of Love_. Benita is the queer psychic who seems to know everything about everyone in Edmonton, where Fraser's play is set.\n\n. Rob Baker credits Robert Chesley's 1984 _Night Sweat_ as the first AIDS play to present onstage gay sex scenes and positions _Angels in America_ as the next major play to do so (184). In general, most of the major plays about AIDS have presented a more subdued sexuality or have\u2014as with Larry Kramer's _The Normal Heart_ \u2014actually preached against gay male \"promiscuity.\" Kramer's semi-autobiographical character Ned Weeks opines, \"Why is it we can only talk about our sexuality, and so relentlessly? You know, Mickey, all we've created is generations of guys who can't deal with each other as anything but erections\" (Kramer, 58). For pointed critiques of _The Normal Heart_ , see Crimp, \"How to Have Promiscuity,\" 246\u201353; Roman, 209\u201311.\n\n. The audience for Kushner's attack on \"the myth of the Individual\" reaches beyond the readership of _Perestroika:_ Kushner originally published this piece in the _New York Times_ , and he republished it in his recent collection of essays. See Kushner, _Thinking_ , 33\u201340.\n\n. Although I have not focused much on the Roy Cohn plot in _Angels inAmerica_ , it is nonetheless an important element of the play. As Kushner explains, \"Part of the impulse to write _Angels in America_ came from the way this man who I hated got an obituary in the _Nation_ by Robert Sherrill that was completely homophobic. The question of forgiveness may be the hardest political question people face. . . . But forgiveness, if it means anything, has to be incredibly hard to come by\" (qtd. in Tucker, 33). My point here is that Kushner thematizes \"forgiveness\" and affirmation of the character's humanity in the play but does not uncritically absorb these themes into the vision of political solidarity in the end.\n\nScott Tucker has some potentially valid reservations about Kushner's inclusion of Cohn in the play. Tucker insists that, \"as an emblematic figure of reaction in the 1980s,\" Cohn\n\ncannot really bear the burden Kushner places upon him. . . . William F. Buckley, for example, is an equally grotesque and dramatic figure who once proposed tattooing all HIV-positive people for easy identification\u2014and his political power far exceeded Roy Cohn's during that decade. Introducing a _living_ public menace in the play would have made \"the question of forgiveness\" Kushner raises a great deal more challenging and pointed. . . . [Cohn's] _presence_ in the play is not the problem, but it underscores the complete _absence_ of the main players in power during those years. (33\u201334)\n\n\"Complete absence,\" however, is a misnomer. Reagan, in particular, is a constant and looming presence in these characters' lives\u2014a menacing presence that would have been _diluted_ , I think, had Kushner actually embodied Reagan as a character. Kushner is very much aware of \"the main players in power\" during these years, and it is these players against which some (and only some) of his characters are united in the end.\n\n. Kushner explains in his \"Playwright's Notes\" that the cases cited in this scene are actual cases with some names and circumstances changed (P, 8). Louis confronts Joe with only a few specific cases, but it is clear that Joe has written many more. The first involved plaintiffs on Staten Island who were suing a New Jersey toothpaste company because the smoke from the company's factory had led to the hospitalization of at least three children. The judge ruled that the plaintiffs had no case because the Air and Water Protection Act, which they were citing, did not apply to people, only to air and water. The second involved a gay soldier who filed an appeal when the army kicked him out and cheated him of his pension. Although the soldier got his pension again, the judge's decision stated that it was only because the army in this case had foreknowledge of the soldier's homosexuality, and that homosexuals are still not guaranteed to equal protection under the law (P, 109\u201310).\n\n. Moreover, conservative critics _did_ feel excluded: \"The crowd-pleasing swipes at conservatives, Republicans, and Mormons, an occasional irritant in Part I, have multiplied this time to fill the dramatic vacuum\" (Olson, 72); \"The playwright is not given to moderation in expression. He says Ronald Reagan will go down in history as a 'tremendously evil' man, who ran a 'closet-fascist government'\" (Grenier, 54).\n\n. Kushner, in turn, reiterates his commitment to solidarity over and over again in his interviews and writings. See, particularly, his response to neoconservative writers Andrew Sullivan and Bruce Bawer:\n\nWhat of the things gay children have to fear, in common with all children?  \nWhat of the planetary despoilment that kills us? Or the financial necessity that drives some of us into unsafe, insecure, stupid, demeaning and ill-paying jobs? Or the unemployment that impoverishes some of us? Or the racism some of us face? Or the rape some of us fear? What about AIDS? Is it enough to say, Not our problem? Of course gay and lesbian politics is a progressive politics: It depends on progress for the accomplishment of any of its goals. Is there any progressive politics that recognizes no connectedness, no border-crossings, no solidarity or possibility for mutual aid? (\"A Socialism of the Skin,\" 13)\n\n. Kushner himself has implied that \"ongoing dialogue\" for him includes dialogue with lesbian and gay novelists such as Schulman:\n\n[Interviewer]: Larry Kramer wrote in _The Advocate_ last year that the gay novel is a dead form; the only interesting work happening in gay literature is in the theater. What is your reaction to that?\n\nTony Kushner: Oh, Larry! I don't know what he's talking about, because he's gone off to write a novel as far as I know.\n\nI just read Michael Cunningham's new novel _[Flesh and Blood]_ , which is just astonishing. And Chris Bram has a new novel _[Father of Frankenstein]_ , which is amazing. I think Sarah Schulman writes amazing novels. Certainly there's Dorothy Allison\u2014I think _Bastard out of Carolina_ is one of the best things I've ever read. So I don't agree at all. (Lowenthal, 11)\n\nKushner provides an extended blurb on the back cover of Schulman's most recent novel, _Rat Bohemia_ , and quotes approvingly from _Rat Bohemia_ in the preface to his collection of essays ( _Thinking_ , ix).\n\n. On Donald Trump and the Trump Tower, see Crimp and Rolston, 122.\n\n. During the summer of 1988, medical waste (catheter bags, sutures, hypodermic needles, and vials of blood\u2014some of which tested positive for HIV) washed ashore on beaches from Long Island to Staten Island.\n\n. On the Tompkins Square Park Riots, see N. Smith, 59.\n\n. For a more extended consideration of the issues in this paragraph, see my discussion in chapter 2 of Randall Kenan's character Jimmy Greene from _A Visitation of Spirits_.\n\n. The dance floor is an appropriate location for this announcement, since it has been, historically, an important site for the (re) fashioning of queer identities. See, for example, my overview in chapter 2 of Marcos Becquer's reading of _Tongues Untied_ and _Paris Is Burning_ , as well as Becquer and Jose Gatti's more extensive analysis of vogueing.\n\n. Edward King, in _Safety in Numbers: Safer Sex and Gay Men_ , criticizes ACT UP and coalition work, suggesting that they have contributed to the contemporary \"degaying\" of AIDS (191\u201392). King's analysis focuses primarily on Britain, where the degaying of AIDS has proven disastrous for gay men: the British government has shifted funding away from organizations that address the specific needs of gay men, even though the vast majority of people living with AIDS in Britain are gay men.\n\nKing's study in general is important and convincing, but in his zeal to critique the degaying of AIDS, I think he conflates universalizing (\"AIDS is everyone's problem\") and coalition. Perhaps this conflation is necessary, given the particular problems in Britain that King addresses. Throughout my analysis, however, I hope it has been clear that \"coalition\" entails coming together to face, not efface, the specific needs of various groups. In the United States, moreover, many people living with AIDS are already located in multiple communities: gay and bisexual men of color, particularly, have been disproportionately impacted by the epidemic. Audre Lorde's comments from chapter 1 are again apropos here: \"It was a while before we came to realize that our place was the very house of difference rather than the security of any one particular difference\" ( _Zami_ , 226).\n\n. See especially Bordowitz, \"Picture a Coalition,\" which provides an overview of the Testing the Limits Collective, a group of video activists who documented many of ACT UP's early actions. See also Bordowitz, \"The AIDS Crisis Is Ridiculous\"; Saalfield; Saalfield and Navarro, _367\u201365_.\n\n. Schulman's own position on this issue is unclear, but she maintains that _People in Trouble's_ \"principal idea . . . was how personal homophobia becomes societal neglect\" (in Loewenstein, 220; see also Schulman, _My American History_ , xviii). Since it is unclear whether she is referring here to Molly's or Kate's \"personal homophobia,\" Schulman's statement is fairly ambiguous. Either way, however, it is problematic and biphobic: Does Molly's personal homophobia lead her to neglect her activism and \"the community\" by \"wasting time\" with a \"straight\" woman? Or does Kate's personal homophobia prevent her from making a \"real\" commitment to the gay and lesbian world?\n\nWhenever I have taught this novel, Kate's character initially has elicited the standard biphobic litany from students: she \"can't make up her mind,\" is \"indecisive and mean,\" treats both Molly and Peter \"unfairly,\" and so forth. Invariably, however, discussion gravitates toward Kate, who\u2014given the profound shifts she experiences throughout the novel\u2014is a more seductive character than either Molly or Peter. Schulman's attempts to contain Kate (by providing readers with a theme through which to read her: \"personal homophobia becomes societal neglect\") are never, I think, entirely successful.\n\n. I am thinking here of Gary Indiana's disturbing contribution to the anthology _Discontents: New Queer Writers:_ \"He goes to a party where a five year old child gets passed around for everyone to fuck, and then it's chopped up with steak knives and consumed\" (167). Schulman is also a contributor to this anthology, so if her citation of Indiana in _People in Trouble_ is parodic, she herself is nonetheless allied with him in at least one queer venue.\n\n## **_Works Cited_**\n\nAbelove, Henry, Mich\u00e8le Aina Barale, and David M. Halperin, eds. _The Lesbian and Gay Studies Reader_. 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Boston: Beacon Press, 1990.\n\n## _Index_\n\nactivism: AIDS, , \u201316, , , \u201372, \u201374, \u201395, \u2013202\n\nantiwar,\n\nfeminist, \u201362,\n\nhousing, , 191\u201393, \u201398\n\nqueer, , , \u201353, , , \u201372, \u201374, , \u201380, , , \u201313,  n. 20,  n. 8. _See also_ ACT UP; Black Panthers; Chicano movement; civil rights movement; Daughters of Bilitis; Gay Activists Alliance; Gay Liberation Front; Lesbian Avengers; Mattachine Society; Queer Nation; Radicalesbians; Student Nonviolent Coordinating Committee; Young Lords\n\nACT UP (AIDS Coalition to Unleash Power), \u20134, , \u201316, , , , , , , \u201394, , , ,  n. 8,  n. 14, \u201336 n. 20,  n. 21\n\nAdam, Barry,  n. 9,  nn. 1,\n\n_Advocate, The_ , , \u201319 n. 18\n\nAfrekete, , \u201355, \u201357\n\n_Afrekete: An Anthology of Black Lesbian Writing_ (McKinley and Delaney), \u201357,  n. 8\n\nAfrican American studies: theories of signification, , \u201370, \u201382, \u201390, , \u201397, , \u201313, \u201315,  nn. 2, ,  n. 11, \u201326 n. 13,  nn. 14, ,  n. 20. _See also_ Gates, Henry Louis, Jr. Kenan; queer trickster\n\nAIDS: in _Angels in America_ , \u201359, \u201363, , \u201374, , , \u201334 n. 10\n\n\"degaying\" of, \u201336 n. 20\n\nand medical waste, ,  n. 16\n\nin _People in Trouble_ , , , \u201395, \u2013202\n\nand political rebirth, \u201316\n\nquilt,\n\nand the radical right,\n\nand renaissance discourse, , , \u20139,  n. 7,  n. 8\n\nin _Silverlake Life_ ,  n. 16\n\nAIDS activist, as identity, , , , , \u201356, , \u201372, \u201374, \u201380, \u201395, \u2013202,\n\nAlarcon, Norma,  n. 8\n\nAlcoff, Linda, \u201335, \u201353, ,\n\nAlexander, Elizabeth,\n\nAllison, Dorothy, ,  n. 2,  n. 14\n\nAltman, Dennis,\n\nAmerican Psychiatric Association (APA), , ,  n. 2\n\nAmerican Renaissance (New England Renaissance), , \u20136,\n\n_Angels in America:_ Broadway production of, , ,  n. 3\n\nRoyal George Theatre (Chicago) production of, , \u201365, \u201367, , ,  n. 3\n\nStation Theater (Urbana, Ill.) production of,  n. 7\n\nTel Aviv production of,  n. 4\n\nAnzald\u00faa, Gloria, ix, , , \u201318, , ,  nn. 7, , , , \u201331 n. 11,  nn. 14, ,  n. 6\n\n_Borderlands\/La Frontera_ , , , , \u201331, \u201347, \u201349, \u201350, , ,  n.\n\n\"Bridge, Drawbridge, Sandbar or Island: _Lesbians-of-Color Hacienda Alianzas,\"_ \u201348, , ,\n\n_\"La conciencia de la_ mestiza\/Towards a New Consciousness,\" , , \u201342, \u201346, \u201349,\n\n_Friends from the Other SidelAmigos del otro lado_ , \u201332\n\n\"The Homeland, Aztl\u00e1n\/ _El otro M\u00e9xico,\"_ \u201331, \u201339,\n\n_Making Face, Making Soul\/Haciendo Caras_ , ,  n. 16,  n. 17\n\n\"La Prieta,\" , , , ,  n. 17\n\nand queerness, \u201317, , \u201353,  n. 8\n\n\"To(o) Queer the Writer\u2014Loca, escritora y chicana,\" \u201346, , , \u201353, \u20139. _See also_ borders; new mestiza; _This Bridge Called My Back_\n\nAshbery, John,  n. 20\n\nAugustine, ,  n. 18\n\nBaker, Houston, \u201328 n. 25\n\nBaker, Rob, , ,  n. 8\n\nBakhtin, Mikhail,\n\nBaldwin, James, , \u201350, \u201313,  n. 23,  n. 26\n\n_The Fire Next Time_ , \u201312, \u201326 n. 13\n\n_Giovanni's Room_ , , ,  n. 18\n\n_Go Tell It on the Mountain_ , \u201382, , ,  nn. 11, , ,  n. 17\n\n_Just Above My Head_ , ,  n. 13\n\nBannon, Ann,  n. 2\n\nBawer, Bruce, ,  n. 13\n\n_Beyond Queer_ ,\n\n_A Place at the Table_ , \u20137, , , ,  n. 19\n\nBeam, Joseph,\n\n_In the Life_ , ,  n. 13,  n. 8\n\nBeatles, the,\n\nBecquer, Marcos, \u201373, , , ,  nn. 5, ,  n. 19\n\nBehrens, Bill,\n\nBellamy, Dodie,\n\n_Bell Curve, The_ (Herrnstein and Murray), \u201312\n\nBenjamin, Walter, , ,  n. 5\n\nBergman, David, ,\n\n_Gaiety Transfigured_ , , \u201348\n\nBerlant, Lauren,\n\nB\u00e9rub\u00e9, Michael, ,\n\nBill Whitehead Memorial Award,\n\nbiology, \u201331, \u201313\n\nbisexuality, , , \u201387, ,  n. 22\n\nBishop, Elizabeth,  n. 20\n\nBlack Panthers, ,  n. 1\n\nBloom, Allan,  n. 7\n\nBloom, Harold, , ,  n. 2\n\n_Book of J_ , , ,  n. 2\n\nBook-of-the-Month Club, , \u201323 n. 18\n\nborders: academic, , , , \u201328\n\nand border guards, , \u201318, , , , \u201332, , , , \u201350, \u20139\n\nand Chicana\/o theory, \u201317, \u201324, ,  n. 8\n\nand queer theory, \u201317, , \u201328\n\nUnited States\u2013Mexican, , , \u201322, \u201341, . _See also_ Anzald\u00faa; Kerouac\n\nBordowitz, Gregg,  n.\n\nBoyd, Blanche McCrary,  n. 2\n\nBram, Christopher,  n. 14\n\nBrown, John,\n\nBrown, Rita Mae,  n. 1,  n.\n\nBrownworth, Victoria,\n\nBryant, Anita,\n\nBuckley, William R,  n.\n\nBulkin, Elly,  n. 13\n\nBurroughs, William S.,\n\nBush, George, ,  n. 18\n\nbutch\/femme identities, , \u201384,  n. 11\n\nButler, Judith, ,  n. 20,  n. 6\n\n_Bodies That Matter_ , \u201351, \u201353, \u201330 n. 7\n\n_Gender Trouble_ , \u201388, , \u201386, \u201390\n\nCalder\u00f3n, H\u00e9ctor,\n\ncanons: American, ; lesbian and gay, \u201366, \u201323 n.18\n\nCarlston, Erin G.,  n. 8\n\nCervantes, Lorna Dee,\n\nChalfant, Kathleen,\n\nChauncey, George,  n.\n\nChelsea House Publishers,  n.\n\nChesley, Robert,  n. 8\n\nChicano movement (El Movimiento), \u201320, , ,  n. 17\n\nChicano Renaissance, , \u201320, , , , ,  n. 10,  n.\n\nChicano\/a studies, \u201324, , , ,  n. 5,  nn. 8, . _See also_ borders\n\nChristianity: Fundamentalist, , , \u201378, \u201382,\n\nrevisions of, \u201378, , , , \u201312, \u201326 n. 13,  nn. 23, . _See also_ Baldwin; Kenan\n\ncivil rights movement, , , ,  n. 3\n\nClark, Danae,  n. 6\n\nClinton, Bill, vii,  n. 18\n\nClum, John, , \u201366, ,\n\nCohen, Ed, , , ,  n. 19\n\nCohn, Roy, , , , , , , \u201334 n. 10\n\nColette,  n. 20\n\nColorado,\n\nCombahee River Collective,\n\ncoming out, \u201330,  n.\n\nand apositionality, \u201346, \u201367\n\nand collective identity, \u201333, \u201357, \u201362, \u201368\n\ncritiques of, , \u201339, \u201321 n. 6,  n. 7\n\nand positionality theory, , \u201339, \u201353, \u201357, \u201362, \u201368. _See also_ coming-out stories\n\ncoming-out stories, \u201330, , , \u201339, \u201346, \u201351, \u201353, , \u201368,  n. 7,  n. 2. _See also_ Lorde; White\n\nconfession, \u201378, \u2013111,  n. 18\n\nCongress, United States, vii, \u201326,  n. 18\n\nCooper, Dennis, , ,  n. 4\n\nCort\u00e9s, Hern\u00e1n,\n\nCowan, Patricia,\n\nCrimp, Douglas, \u20139, \u201378, \u201389, \u201392, ,  nn. 3, ,  n. 8,  n. 16,  n. 8,  n. 15\n\nCrossing Press Feminist Series, ,\n\ncultural studies, \u20136, ,\n\nCunningham, Michael,  n. 14\n\n_Daily Prayer Book\/Ha-Siddur Ha-Shalem_ ,\n\nDaughters, Inc.,  n. 1\n\nDaughters of Bilitis (DOB), ,\n\nDavis, Madeline D., ,  n. 9\n\ndeconstruction, \u201373, , \u201369, , , \u201397, ,  n. 6\n\nDefense of Marriage Act (DOMA), vii\n\nDelaney, L. Joyce,\n\nde Lauretis, Teresa, , ,  n. 20,  n. 3,  n. 17\n\nDeLillo, Don, ,\n\nD'Emilio, John, \u201317, ,  n. 5\n\n_Making Trouble_ , \u201316, ,  n. ,  nn. 1, ,\n\n_Sexual Politics, Sexual Communities,\u201312_, ,  n. ,  n. 2\n\nDeming, Barbara, , ,\n\nDenneny, Michael, ,  n. 7\n\nDerrida, Jacques,\n\n_Details_ ,\n\nDetroit, \u20137,  n. 5\n\nDeutsche, Rosalyn, , , \u201398\n\nDewan, Shaila,  n. 14\n\nDhairyam, Sagri, , \u201322 n. 13\n\nDhaliwal, Amarpal,  n. 14\n\nDickens, Charles: _A Christmas Carol_ , , , , , \u201396,  n. 11,  nn. 14,\n\nDonne, John,\n\nDornan, Robert, ,  n. 18\n\nDuberman, Martin,  n.\n\n_Cures_ , \u201336,  n.\n\n_Stonewall,, _ n. ,  nn. 1,\n\nDuBois, W. E. B.,\n\nDuggan, Lisa, \u201342, \u201378, \u201325, , ,  n. 3, \u201318 n. 13,  n. 20,  n. 17\n\nDuke, David,\n\nDyer, Richard, , ,\n\nEchols, Alice,  nn. 2,\n\nEdelman, Lee,  n. 3,  n. 13\n\nElders, Joycelyn, , \u201319 n. 18\n\nEliot, T.S.,\n\nEllison, Ralph, ,\n\nEnr\u00edquez, Evangelina,\n\nErasmus, ,  n. 18\n\nessentialism, \u201337, , \u201342, , , , , ,\n\nFanon, Frantz, ,  n. 18\n\nfeminism, ,  n. 1, \u201330 n. 7\n\nacademic, , , \u201353\n\nand gay and lesbian politics, \u201315, \u201353\n\nlesbian, ,\n\nand the New Left,\n\nand positionality theory, , \u201335, \u201339, , \u201356, ,  n. 4,  n. 8\n\nand race, , , \u201324, , \u201353,  n. 13,  n. 3,  nn. 8, ,  nn. 13, , \u201332 n. 17\n\nradical, , , \u201333,  nn. 2,\n\nFink, John,\n\nFirbank, Ronald,  n. 20\n\nFlight of the Mind,\n\nFlowers, Reg,\n\nForbes, Malcolm,\n\nFordyce, Kent,\n\nFoster, Jodie,\n\nFoucault, Michel, \u2013101,  n. 6, \u201327 n. ,  n. 20\n\nFrank, Barney, ,  n.\n\nFranklin, C. L., ,  n. 18\n\nFranklin, John Hope, ,  n. 18\n\nFraser, Brad,  n. 7\n\nFraser, Steven,\n\nFrazier, Demita,\n\nFreedman, Diane,\n\nFreedman, Estelle B.,\n\nFreixas, Claudio,\n\nFreud, Sigmund,\n\nFriedman, Peter,  n. 16\n\nFung, Richard,\n\nFuss, Diana,\n\n_Essentially Speaking_ , ,\n\nGates, David,\n\nGates, Henry Louis, Jr., \u201370, , , \u201313, ,  n. 4,  n. 11\n\nand Houston Baker, \u201328 n. 25\n\n_The Signifying Monkey_ , \u201370, \u201389, , , , \u201313, ,  nn. 2, ,  n. 20\n\n\"Trope of a New Negro,\" ,  n.\n\nGatti, Jos\u00e9,  n. 19\n\nGay Activists Alliance (GAA), \u201314, \u201336, ,  n. ,  n. 2\n\nGay Liberation Front (GLF), , , \u201314, , , \u201333, \u201337, , ,  n. ,  nn. 2,\n\nGay Pride Day, \u20134, , \u20137,\n\nGenet, Jean, ,  n. 20\n\nGibson, William,  n. 17\n\nGiroux, Henry, , \u201326,  n. 8,  n.15\n\nGl\u00fcck, Robert, ,  n. 16\n\nGoldman, Emma,\n\nGoldstein, Richard,\n\nGomez, Jewelle,  n. 17,  n. 19\n\nGonzales, Rodolfo \"Corky\": _I Am Joaquin\/Yo Soy Joaquin_ , \u201320,\n\nGorbachev, Mikhail, \u201373\n\nGraham, Bobbie Jean,\n\nGreenblatt, Stephen, ,\n\nGrenier, Richard,  n.\n\nGronniosaw, Ukawsaw,\n\nGunderson, Steve, ,  n. 18\n\nGunn, Giles, ,\n\nGurganus, Allan,  n. 16,  n. 2\n\nHall, Richard, ,  n. 1\n\nHalperin, David,  n. 3\n\nHaraway, Donna, \u201335, , , , , , , ,  n. 4\n\nHarding, Sandra,  n. 4\n\nHarlem Renaissance (New Negro Renaissance), , , , \u201322, , ,  n. 2,  n.\n\nHarper, Phillip Brian, ,  n. 6\n\nHawaii, vii\u2013viii\n\nHawaiian Renaissance, ,  n. 16\n\nHay, Harry, \u201311\n\nHays, Will, ,  n. 1\n\nHealth and Human Services, U.S. Department of, ,  n.\n\nHelms, Jesse, ,  n. 18\n\nHemphill, Essex: _Brother to Brother_ , ,  n. 13\n\n_Ceremonies_ , , \u201325 n. 8\n\nHerrnstein, Richard J., \u201312\n\nheterosexuality: black gay critiques of, \u201373\n\ninstability of, \u201388, \u2013106, , \u201387, , , \u201327 n\u2013228 n. 1\n\ninvention of, \u201312\n\nand mobility, , , \u201348, \u201355,  n. 14\n\nand queer perestroika, \u201377. _See also_ queer trickster\n\nHill, Anita,  n. 18\n\nhipsters, \u201318, \u201322, , \u201327, , , , , , , , _\u201352_, \u201354, ,  n. ,  n. 17. _See also_ Kerouac\n\nHirsch, E.D.,  n. 7\n\nHolleran, Andrew, , ,  n. 2\n\nHolt, Shelley,  n. 7\n\nhomelessness, , , , \u201393, \u201399,\n\nhomosociality, ,  n. 1\n\nHoward, Ronnie,\n\nHubbard, Jim,\n\nHull, Gloria T.,  n. 13,  n. 8,  n. 3,  n. 13\n\nHunter, Nan, \u201318 n. 13\n\nHurston, Zora Neale, \u201313\n\nHutcheon, Linda, \u201397,\n\nidentity politics, , ,  n. 22\n\nImmigration and Naturalization Service, U.S.,\n\nIndiana, Gary, , ,  n. 23\n\nindividualism, , , , , , \u201378, \u20137, ,  n. , \u201335 n. 13\n\nInQueery\/InTheory\/InDeed Conference,  n.\n\ninvisibility, \u201340, \u201346, , \u201367,  n. 10\n\nIsherwood, Christopher,  n. 20\n\nJackson, Jr., Earl, ,  n. 16\n\nJagose, Annamarie, , \u201331 n. 11\n\nJames, Henry,  n. 18\n\nJay, Karla,  n. 2\n\nJohnson, Charles,\n\nJohnson, James Weldon, ,\n\nJones, Joyce,\n\nJoslin, Tom,  n. 16\n\nJulien, Isaac,\n\nJung, Carl,\n\nJustice, U.S. Department of,\n\nKafka, Franz,  n.\n\nKameny, Franklin,  n. 1\n\nKaplan, Carla,\n\nKatz, Jonathan Ned: _The Invention of Heterosexuality_ , \u201312\n\nKeating, AnnLouise,\n\nKenan, Randall, ,  n. 2\n\nand James Baldwin, \u201382, , , \u201313,  nn. 11, ,  n. 17,  n. 23,  n. 26\n\nand Charles Dickens, , , , , \u201396,  n. 11,  nn. 14,\n\nand critique of community, \u201371, , \u201375, \u201393\n\nand disruption of heterosexuality, , \u2013106\n\n_Let the Dead Bury Their Dead_ , , \u201315,  n. 10,  nn. 27,\n\nand transformation of community, , \u201379, \u201397, , \u201312, \u201315,\n\n_A Visitation of Spirits_ , , \u201371, \u2013113, , , , nn. 9, , \u201326 n. 13,  nn. 14, , \u201327 n.18,  nn. 22, ,  n.27,  n. 18. _See also_ queer trickster\n\nKennedy, Elizabeth Lapovsky, ,  n.\n\nKerouac, Jack,\n\n_On the Road_ , \u201322, , , , , \u201327, , , , , , \u201354\n\n_The Subterraneans_ ,  n. 1,  n. 4\n\nKillian, Kevin,\n\nKing, Edward, \u201336 n. 20\n\nKing, Katie, ,\n\nKing, Larry,  n. 18\n\nKissack, Terence, ,  n. ,  n. 2\n\nKitchen Table: Women of Color Press, \u201362\n\nKlee, Paul,  n. 5\n\nKoch, Ed, , \u201392\n\nKramer, Larry, , ,  n. 8,  n. 14\n\nKruger, Barbara, ,\n\nKushner, Tony, , , \u20133,  n.5,  n. 14\n\n_Angels in America_ , , \u201378, , , , , , ,  n. 14,  nn. 1, , \u201334 n. 10,  n. 11\n\ncriti\u2013cal response to, , \u201372, \u201377,  nn. 10,\n\n\"Notes Toward a Theater of the Fabulous,\" ,\n\nand queer activism, , \u201368, \u201372, \u201374, , \u201379,\n\nand queer perestroika, , \u201356, , , \u201371, \u201373, \u201378, ,  n. 16\n\nand queer sexuality, , \u201363, \u201366, \u201372, \u201378,  n. 8\n\n\"A Socialism of the Skin,\" \u201335 n. 13\n\n_Thinking about the Longstand\u2013ing Problems of Virtue and Happiness_ , ,  n. . _See also_ AIDS activist, asidentity\n\nLabonte, Richard,  n. 1\n\nLacan, Jacques,\n\n_Lambda Book Report_ ,  n. 1,  n. 7\n\nLambda Literary Awards, , ,  n. 1\n\nLarsen, Nella,  n. 2\n\nLatzky, Eric, ,  n. 4\n\nLeavitt, David, ,  n. 7\n\nLehmann\u2013Haupt, Christopher, ,  n. 10\n\nlesbian and gay studies, \u201312,  n. . _See also_ literary studies; queer theory\n\nLesbian Avengers, ,  n. 14\n\nlesbian chic,\n\nLeVay, Simon,\n\n_The Sexual Brain_ , \u201312\n\nLewis, David Levering,  n.\n\nLiebman, Marvin,\n\nLincoln, Kenneth,  n. 16\n\nliterary studies, \u201325, \u201316 n. 2\n\nAmerican, , , \u201350,\n\nlesbian and gay, , \u20133, \u201350, \u201366,  n. 1,  n. 14. _See also_ renaissance discourse\n\nLivingston, Jennie: _Paris Is Burning_ , ,  n. 19\n\nLocke, Alain,\n\nLog Cabin Republicans, ,  n. 19\n\nLorde, Audre, , , , _\u201366_, , \u201322 n. 13,  n. 17,  nn. 19, , ,  n. 13\n\n_The Black Unicorn_ , ,\n\n\"Brother Alvin,\"\n\n_Chosen Poems\u2014Old and New_ , , \u201360, ,  n.\n\n\"Dear Joe,\"\n\nmemorial service,  n. 14\n\n\"Need: A Choral of Black Women's Voices,\" \u201362, , , ,  n. 19\n\n_Our Dead Behind Us_ ,  n. 8\n\n_Sister Outsider_ , , ,  n. 13,  n. 18\n\n_Undersong_ ,  nn. 14,\n\n\"What Is at Stake in Lesbian and Gay Publishing Today,\" ,  n. 13\n\n_Zami: A New Spelling of My Name_ , \u201330, , , , \u201357, , , , , , , , , , ,  n. 2,  n. ,  n. 8,  n. ,  n. 20. _See also_ Zami, as identity\n\nMaddy, Yulisa Amadu,\n\nMalinche, La, \u201336\n\nMann, Thomas,  n. 20\n\nMann, William J.,  n. 1\n\nMarco, Tony,\n\nMarcus, Leah S., \u20138\n\nmarginality, \u201340, , \u201373, \u201379, , , , , ,  n. ,  n. 10,  n. 6,  n. 8,  n. ,  n. 3\n\nand New Queer Writers,\n\nand region, , , , , , , ,\n\nand renaissance discourse, \u201310\n\nmarriage, gay and lesbian, vii\u2013viii\n\nMartin, Biddy, , ,  n. 13\n\nMartin, Robert K.,\n\nMarx, Karl,\n\nMassi, Mark,  n. 16\n\nMattachine Society, , \u201311, , , , , , ,  n. 1\n\nMaupin, Armistead, \u201324 n. 2\n\nMayer, Michael, ,  n. 3\n\nMcCarthy, Joseph, ,\n\nMcKenna, Teresa,\n\nMcKinley, Catherine E., \u201357\n\nMcRuer, Robert, ,  n. 7,  n. ,  n. 22\n\nMen Stopping Rape,\n\nMercer, Kobena, , \u20136,  n. 22\n\nMerrill, James,  n. 20\n\nmigration, urban: in gay and lesbian literature, \u201370, , , \u201324 n. 2\n\nMikulski, Barbara, ,  n. 18\n\nMiller, D. A.,  n. 7\n\nMiller, Tim,  n. 8\n\nMinson, Jeffrey,\n\nMirande, Alfredo,\n\nmodernism, , \u201397, ,\n\nModern Language Association (MLA), ,\n\nMoraga, Cherr\u00ede, , \u201324, \u201347, , \u201331 n. 11\n\n_The Last Generation_ , \u201324, ,  n. 17. _See also This Bridge Called My Back_\n\nMordden, Ethan,  n. 2\n\nMorrison, Toni, , ,\n\n_Playing in the Dark_ , , ; _Sula_ ,\n\nMorton, Donald, \u201328\n\nMTV,\n\nMu\u00f1oz, Carlos, Jr.,\n\nMurray, Charles, \u201312\n\nMurrieta, Joaquin, \u201320\n\nNational Black Men's Health Network,\n\nNational Coalition of Black Lesbians and Gays (NCBLG),\n\nNational Coming Out Day,\n\nNative American Renaissance, ,  n. 16\n\nNavarro, Ray,  n.\n\nNazi Germany,\n\nNelson, Emmanuel, \u201350,  n. 1,  n.\n\nneoconservatism, gay, \u201331, \u201313,  n\u201319, \u201335 n. 13\n\nNero, Charles I.,\n\nNestle, Joan,  nn. 9,\n\nNew American Library, ,\n\nnew historicism, \u20136\n\nnew mestiza, , , \u201336, \u201354, , ,  n. 10\n\nand agency, \u201353\n\nand critical mobility, , , , , \u201341, \u201348, , ,  n. 14\n\nNew Negro identity, \u201321,  n.\n\nNew Queer Writers, , \u201331, ,  n. 4,  n. 3\n\nNew Right, , \u201312\n\n_Newsweek_ , ,\n\nNew York Lesbian and Gay Experimental Film Festival,\n\nNorth American Free Trade Agreement (NAFTA), , \u201329 n.2\n\nNorth Carolina, , , \u201371, , \u201379, \u201393, , , ,  n. 16,  n. . _See also_ Kenan\n\nNorton and Company, W. W., ,\n\nNovazcek, Ruth,\n\nOakland Men's Project,\n\nOchs, Robyn,  n. 20\n\nOhmann, Richard, \u201364,\n\nOkazawa\u2013Rey, Margo,\n\nOlson, Walter,  n.\n\nOther Countries,\n\n_Out\/Look_ ,  n. 3\n\nOutWrite Conference,\n\nOy\u00e1,\n\nParedes, Americo, \u201323\n\nPaz, Octavio,\n\nPease, Donald E., \u20136\n\n_Penthouse_ ,\n\nperformativity, , \u201388, \u20134, \u201328, , \u201347, \u201353, \u201356, , , \u201369, \u201375, \u201379,\n\nPersephone Press, ,\n\nPhelan, Shane,  n. 20,  n. 10\n\nPittsburgh,\n\npostmodernism, , , , , \u201397,\n\npoststructuralism, , \u201353, , \u201373, . _See also_ deconstruction\n\nPratt, Minnie Bruce,  n. 13\n\nPreston, John,  n. 7\n\nProust, Marcel,  n. 20\n\npublic art, \u201380, , \u201399, \u2013202\n\nPulitzer Prize, ,\n\nPynchon, Thomas,\n\nQuality Paperback Book Club (QPB),\n\nQuarles, Benjamin, ,  n. 18\n\nqueer, as concept, , , , \u201329, , , , \u201345, , \u201353, , , ,  n. 3,  n. 20,  n. 8,  n. 17\n\nqueer identities, \u20134, \u201317, , , , _\u201356_, , \u201368, \u201374, , \u201378, \u201380, \u201387, , \u20136, , \u201313, \u201319 n. ,  n. 8,  n. .,  n. 6,  n. . _See also_ AIDS activist, as identity; new mestiza; queer trickster; Zami, as identity\n\nQueer Nation, , , , , ,  n.\n\nQueer Renaissance: and biology, \u201331, , \u201313\n\nand Chicano Renaissance, , \u201320, , \u201324\n\nas confluence of cultural production and political analysis, , \u201310, , , , ,\n\nand cultural production, viii, \u20133, , ,  n.\n\nand Harlem Renaissance, , , \u201322, ,\n\nand gay neoconservatism, \u201331, \u201313\n\nand lesbian and gay marketplace, _\u201366;_\n\nand radical queer political analysis, viii, , , \u201368\n\nand region, , , , \u201324 n.\n\nand reinvention of identity, \u20134, , \u201339, , , , , , , , \u201368, , , \u201378, , , , ,  n. ,  n. 8\n\nand _This Bridge Called My Back_ , \u201324, ,  n. 8, \u201332 n.\n\n\"Queers Bash Back,\"\n\nqueer theory, ,  n. ,  nn. 20, ,  n. 4,  n. 13,  n. 16,  n. 17\n\nand borders, \u201317, \u201328\n\ncritiques of, \u201328, \u201353\n\nand identity, , \u201317, ,  n. 8\n\nand region, , \u201373\n\nstaging of, , , , \u201372, \u201378, \u201379,\n\nqueer trickster, , , , , , \u201379, \u201388, \u201397, , , , \u201315, , ,\n\nQui\u00f1ones, Tirsa, \u201347\n\nQuintana, Alvina E., \u201332 n.\n\nrace: and coming\u2013out stories, , , , \u201346, \u201351\n\nand essentialism, , \u201342,\n\nand gay male literary criticism, \u201350\n\nand the Justice Department,\n\nand lesbian and gay politics, , \u201373, \u201353,  n.\n\nand lesbian and gay anthologies, , \u201347,\n\nand lesbian and gay awards, ,\n\nand the New Left,\n\nand positionality theory,\n\nand queerness, \u201329, \u201353, ,  n. ,  n. 8\n\nand scientific racism, \u201312. _See also_ African American studies, Chicano\/a studies, feminism, new mestiza, whiteness\n\nRadel, Nicholas,  n.\n\nRadicalesbians, \u201333, ,  n.\n\nRaiskin, Judith, ,  n.\n\nRauch, Jonathan,\n\nReagan, Ronald, , , , ,  nn. 10,\n\nRechy, John, \u201350,  n.\n\nregion: and queerness, , \u201370, \u201373, , , \u201324 n.\n\nand whiteness, \u201344, . _See also_ borders; Hawaii; North Carolina; urban space\n\nRenaissance, the (Italian and English Renaissance), , \u20138\n\nRenaissance Center (Detroit), \u20137,  n.\n\nrenaissance discourse, , \u20135, \u201310, , ,  n. 8\n\nRiggs, Marlon: _Tongues Untied_ , \u201372,  n.\n\nRiley, Denise,  n.\n\nRobertson, Hannah,\n\nRodwell, Craig,\n\nRolston, Adam, \u201392, ,  n.\n\nRoman, David, \u201379,  n. 8\n\nRoof, Judith,  n.\n\nRoseanne,  n. 18\n\nRoss, Andrew, ,  n.\n\nRubin, Gayle,  n.\n\nRusso, Vito, ,  n.\n\nRuta, Suzanne,\n\nSaalfield, Catherine,  n.\n\nSadownick, Doug,  n. 8\n\nSald\u00edvar, Jos\u00e9 David, ,  n.\n\nSald\u00edvar, Ram\u00f3n,  n.\n\nSandoval, Chela, ,  nn. 13, ,\n\nSchulman, Sarah, ix, ,  n. ,  n.\n\nand artistic identity, , \u201380, \u201391, \u2013202, \u20133\n\nand film and video production, \u201394\n\n_My American History_ , \u201315, ,  n. ,  n. ,  n.\n\nand New Queer Writers,  n. 4,  n. 23\n\n_People in Trouble_ , , , \u2013202, \u20133, ,  n. ,  n. ,  n.\n\nand political identity, , \u201380, \u201395, , ,\n\n_Rat Bohemia_ ,  n.\n\nand sexual identity, , , \u201387, , , . _See also_ AIDS activist, as identity\n\nScott, Patricia Bell,  n. ,  n. ,  n.\n\nSedgwick, Eve Kosofsky, , , ,  n.\n\n_Between Men_ ,  n.\n\n_Epistemology of the Closet_ , , , , \u201399, , ,  n. ,  n. ,  n. 6\n\n_Tendencies_ , \u20134, \u201323, ,  n.\n\nSella, Robert, , \u201367\n\nSex Wars, , \u201318 n.\n\nSherrill, Robert,  n. 10\n\nSignorile, Michelangelo, \u201328\n\n_Queer in America_ , \u201327, \u201319 n. 18\n\n\"Silence = Death,\"\n\nSingh, Amritjit,  n.\n\nSmith, Barbara, , \u201352, ,  n. ,  n. ,  n. ,  n.\n\nSmith, Bessie,\n\nSmith, Neil,  n. ,  n.\n\nSmith, Sidonie,\n\nSmyth, Cherry, ,\n\nsnapping, \u201372, , ,  n.\n\nSnitow, Ann,  n.\n\nSomerville, Siobhan, ,  n.\n\nSontag, Susan,  n.\n\nStein, Arlene,  n.\n\nStonewall Inn Riots, , , , , \u201367,  nn. 1,\n\nStudds, Gerry, ,  n. 18\n\nStudent Nonviolent Coordinating Committee (SNCC),\n\nsuicide: gay and lesbian, , , , , , \u201390, \u201393, \u201397, \u201311,  nn. 15,\n\nSullivan, Andrew, ,  n.\n\nSumida, Stephen,  n.\n\nSupreme Court, U.S., , ,  n.\n\nSwift, Carolyn, , \u201365, \u201367\n\nTafel, Rich, ,  n.\n\nTaylor, Elizabeth,\n\nTeatro Campesino, ,  n. 10,  n.\n\nTexas Rangers, \u201323\n\n_This Bridge Called My Back_ (Moraga and Anzald\u00faa), \u201324, , , , , , ,  n. ,  n. ,  n. 8,  n. 8,  n. , \u201332 n.\n\nThomas, Clarence, ,  n. 18\n\n_Time_ , ,\n\nTompkins Square Park Riots, , ,  n.\n\nTony Awards, ,\n\nToomer, Jean,\n\ntranssexuality, \u201386\n\nTreichler, Paula,\n\nTrinh T. Minh\u2013ha, \u201356\n\nTrump, Donald, , \u201392,  n.\n\nTucker, Scott, _\u201377_,  n.10\n\nTytell, John,\n\nurban space, \u20137, \u201399, \u20132,  nn. 5,\n\n_USA Today_ , ,  Valdes, Gina,\n\nVald\u00e9z, Luis, ,  n. 10,  n.\n\nVance, Carole S.,\n\nVan Leer, David,\n\nViolet Quill Club,\n\nVirgin of Guadalupe, ,\n\nvogueing, \u201372, , , ,  n. ,  n.\n\nWalker, Alice, \u201313\n\nWarner, Michael, \u201317, ,\n\nWashington, Mary Helen,\n\nWattenburg, Bill,\n\nWebb, Walter Prescott, \u201323\n\nWest, Cornel,  n. 23\n\nWhite, Edmund, , , \u201367,  n.\n\nand gay male literary criticism, \u201351\n\n_The Beautiful Room Is Empty_ , , , , , , \u201367,  n. ,  n. ,  n. ,  n.\n\n_A Boy's Own Story_ , , , \u201330, , \u201346, , \u201351, , , , \u201368,  nn. 7, , , \u201323 n.\n\n_The Burning Library_ , ,  n. 8\n\n_The Faber Book of Gay Short Stories_ , , \u201347, \u201351\n\n_The Farewell Symphony_ ,\n\n_Nocturnes for the King of Naples_ , ,\n\n\"Twenty Years On,\" _65\u201366_ ,  n.\n\nwhiteness, , \u201330, , , , \u201351, \u201322, \u201332, \u201353,  n. 10,  n. ,  n. 10,  nn. 14, ,\n\nWiegman, Robyn, ,  n.\n\nWilliams, Pete, ,  n. 18\n\nWilliams, Peter,  n.\n\nWilson, Anna,\n\nWinfrey, Oprah, ,  n. 6\n\nWojnarowicz, David, \u201328,\n\n_Close to the Knives_ , \u201327\n\n_Fuck You Faggot Fucker_ ,\n\nWood, Robin,  n.  Woolf, Virginia,  n.\n\nYarbro\u2013Bejarano, Yvonne,  n.\n\nYemay\u00e1, \u201334,\n\nYoung, Allen, ,  n.\n\nYoung Lords,  n.\n\nZami, as identity, , \u201330, , \u201357, , , , ,\n\nZimmerman, Bonnie,  n.\n\n\u017di\u017eek, Slavoj,  n.\n\nZwiclder, Phil,  n. 8\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Contents\n\n  1. HOW TO USE\n  2. INTRODUCTION TO BARCELONA\n  3. ITINERARIES\n  4. BEST OF BARCELONA\n  5. PLACES\n    1. 1. Along the Ramblas\n    2. 2. Barri G\u00f2tic\n    3. 3. Port Vell and Barceloneta\n    4. 4. El Raval\n    5. 5. Sant Pere\n    6. 6. La Ribera\n    7. 7. Parc de la Ciutadella\n    8. 8. Montju\u00efc\n    9. 9. Port Ol\u00edmpic and Poble Nou\n    10. 10. Dreta de l'Eixample\n    11. 11. Sagrada Fam\u00edlia and Gl\u00f2ries\n    12. 12. Esquerra de l'Eixample\n    13. 13. Gr\u00e0cia and Parc G\u00fcell\n    14. 14. Camp Nou, Pedralbes and Sarri\u00e0-Sant Gervasi\n    15. 15. Tibidabo and Parc de Collserola\n    16. 16. Montserrat\n    17. 17. Sitges\n  6. ACCOMMODATION\n  7. ESSENTIALS\n  8. MAPS AND SMALL PRINT\n\n  1. How to Use\n  2. How to Use\n  3. Table of contents\n\n# How to use this Rough Guide ebook\n\nThis **Pocket Rough Guide to Barcelona** is one of a new generation of informative and easy-to-use travel-guide ebooks that guarantees you make the most of your trip, whether you're spending an afternoon or a few days away.\n\nFrom the table of contents, you can click straight to the main sections of the ebook. Start with the Introduction, which gives you a flavour of Barcelona, with details of when to visit and what to see, followed by helpful day-by-day and themed Itineraries. The Best of Barcelona picks out the highlights you definitely won't want to miss, from Gaud\u00ed's _modernista_ masterpieces to creative Catalan cuisine. The Places chapters are your comprehensive neighbourhood-by-neighbourhood guide to the city, with full-colour maps featuring all the sights and listings. Finally, Accommodation recommends the best hotels, B&Bs and hostels and Essentials covers all the practical information you'll need, from public transport to opening hours and festivals. A handy chronology and useful language list round off the guide.\n\nDepending on your hardware, you can double-tap on the maps to see larger-scale versions, or select different scales. There are also thumbnails below more detailed maps \u2013 in these cases, you can opt to \"zoom left\/top\" or \"zoom right\/bottom\" or view the full map. The screen-lock function on your device is recommended when viewing enlarged maps. Make sure you have the latest software updates, too.\n\n# Introduction to **Barcelona**\n\nIt's tempting to say that there's nowhere quite like Barcelona \u2013 there's certainly not another city in Spain to touch it for sheer style, looks or energy. The glossy mags and travel press dwell enthusiastically on its outrageous architecture, designer shopping, cool bars and vibrant cultural scene, but Barcelona is more than just this year's fad. It's a confident, progressive city, one that is tirelessly self-renewing while preserving all that's best about its past. As old neighbourhoods bloom, and landmark museums and sights are restored with panache, there's still an enduring embrace of the good things in life, from the daily market to the late-night caf\u00e9.\n\nRestaurants in the Pla\u00e7a Reial\n\nThe province of Catalunya (Catalonia in English), of which Barcelona is the capital, has a historical identity going back as far as the ninth century, and through the long period of domination by outside powers, as well as during the Franco dictatorship, it proved impossible to stifle the Catalan spirit. The city reflects this independence, being at the forefront of Spanish political activism, radical design and architecture, and commercial dynamism.\n\nThis is seen most perfectly in the glorious _modernista_ (Art Nouveau) buildings that stud the city's streets and avenues. Antoni Gaud\u00ed is the most famous of those who have left their mark on Barcelona in this way: his Sagrada Fam\u00edlia church is rightly revered, but just as fascinating are the (literally) fantastic houses, public buildings and parks that he and his contemporaries designed.\n\nThe city also boasts an extensive medieval Old Town \u2013 full of pivotal buildings from an earlier age of expansion \u2013 and a stupendous artistic legacy, from national (ie, Catalan) collections of Romanesque, Gothic and contemporary art to major galleries containing the life's work of the Catalan artists Joan Mir\u00f3 and Antoni T\u00e0pies (not to mention a celebrated showcase of the work of Pablo Picasso).\n\nCarved Door at Casa Batll\u00f2\n\nBarcelona is equally proud of its cutting-edge restaurants \u2013 featuring some of the best chefs in Europe \u2013 its late-night bars, even its football team, the mercurial, incomparable FC Barcelona. Add a spruced-up waterfront, five kilometres of resort-standard sandy beach, and Olympic-rated sports and leisure facilities, and you have a city that entertains and cossets locals and visitors alike.\n\nDespite its size, Spain's second city is a surprisingly easy place to find your way around. In effect, it's a series of self-contained neighbourhoods stretching out from the harbour, flanked by parks, hills and woodland. Much of what there is to see in the centre \u2013 Gothic cathedral, Picasso museum, markets, Gaud\u00ed buildings and art galleries \u2013 can be reached on foot, while a fast, cheap, integrated public transport system takes you directly to the peripheral attractions and suburbs. Meanwhile, bike tours, sightseeing buses and cruise boats all offer a different way of seeing the city.\n\n### Best places for a Barcelona picnic\n\nParc de la Ciutadella is the city centre's favourite green space, while the gardens of Montju\u00efc offer some fantastic views. Any time the sun shines, the beach between Barceloneta and Port Ol\u00edmpic makes for a great alfresco lunch, though for a real in-the-know experience stock up at the market and head for the Collserola hills.\n\nParc de la Ciutadella\n\nTrue, for all its go-ahead feel, Barcelona has its problems, not least a petty crime rate that occasionally makes the international news. But there's no need to be unduly paranoid, and it would be a shame to stick solely to the main tourist sights since you'll miss so much. Tapas bars hidden down decrepit alleys, designer boutiques in gentrified Old Town quarters, street opera singers belting out an aria, bargain lunches in workers' taverns, neighbourhood funicular rides, unmarked gourmet restaurants, craft workshops, restored medieval palaces and specialist galleries \u2013 all are just as much Barcelona as the Ramblas or Gaud\u00ed's Sagrada Fam\u00edlia.\n\nMercat de Sant Antoni\n\n### When to visit\n\nBarcelona is an established city-break destination with a year-round tourist, business and convention trade. Different seasons have different attractions, from spring dance festivals to Christmas markets, but there's always something going on. As far as the weather is concerned, the best times to go are spring and autumn, when the temperatures are comfortably warm and walking the streets isn't a chore. In summer, the city can be very hot and humid while August sees many shops, bars and restaurants close as the locals head out of the city in droves. It's worth considering a winter break, as long as you don't mind the prospect of occasional rain. It's generally still warm enough to sit out at a caf\u00e9, for example, even in December or January.\n\n< Back to Introduction to Barcelona\n\n# Barcelona at a glance\n\n## Eating\n\nIn the popular Old Town areas food and service can be indifferent and expensive. There are some great bars and restaurants in tourist-heavy **La Ribera** and the **Barri G\u00f2tic** , but you really need to explore the up-and-coming neighbourhoods of **Sant Pere** , **El Raval** and **Poble Sec** for the best local finds, from traditional taverns to chic contemporary tapas bars. Michelin stars and big bills are mostly found in the **Eixample** , while for the best fish and seafood head for harbourside **Barceloneta** or the **Port Ol\u00edmpic**. The suburb of **Gr\u00e0cia** is also a nice, village-like place to spend the evening, with plenty of good mid-range restaurants.\n\n## Drinking\n\nIt should probably be called Bar-Celona \u2013 whatever you're looking for, you'll find it here, from bohemian boozer to cocktail bar. **Passeig del Born** (La Ribera) is one of the hottest destinations, with Sant Pere close on its heels, while there's an edgier scene in **El Raval** and around **Carrer de Blai** (Poble Sec). The main concentration of designer bars (as well as the city's gay scene) is in the **Esquerra de l'Eixample** , while the theme bars of **Port Ol\u00edmpic** are mainstream summer-night playgrounds for locals and visitors. Bars usually stay open till any time between 11pm and 2 or 3am.\n\n## Shopping\n\nDesigner and high-street fashion can be found in the Eixample along **Passeig de Gr\u00e0cia** and **Rambla de Catalunya** , though for new names and boutiques the best hunting ground is in the Old Town streets around **Passeig del Born** (La Ribera). Secondhand and vintage clothing stores line **c\/de la Riera Baixa** (El Raval), there's music and streetwear along nearby **c\/dels Tallers** , and for antiques and curios it's best in the streets near **c\/Banys Nous** (Barri G\u00f2tic). The markets, meanwhile, are king, from the heavyweight **Boqueria** to lesser-known gems like the **Mercat Santa Caterina** in trendy Sant Pere or Gr\u00e0cia's **Mercat de la Llibertat**.\n\n## Nightlife\n\n **Clubs** in Barcelona start late and go on until 5 or 6am, and while Thursday to Sunday sees the most action, there are **DJs** on the decks every night. The big name venues tend to be found out in the old industrial zones like **Poble Nou** ; downtown music clubs are often jazz-orientated, though local rock, pop, indie and even flamenco get regular airings in venues across the **Barri G\u00f2tic** and **El Raval**. For typically Catalan surroundings, a concert at Sant Pere's **Palau de la M\u00fasica Catalana** can't be beaten, while the principal venue up in the Eixample is the contemporary city concert hall, **L'Auditori**.\n\n ****\n\nOur recommendations for where to eat, drink and shop are listed at the end Places chapter.\n\n < Back to Introduction to Barcelona\n\n# Itineraries\n\nDay one in Barcelona\n\nDay two in Barcelona\n\nModernista Barcelona\n\nBudget Barcelona\n\n## Day one in Barcelona\n\n **1** **The Ramblas  \n** Everyone starts with a stroll down Barcelona's most emblematic street.\n\n **2** **Mercat de la Boqueria  \n** Wander through the stalls of one of Europe's best markets and soak up the vibrant atmosphere.\n\n**3** **La Seu  \n** The calm cloister of Barcelona's cathedral is a haven amid the bustle of the Gothic Quarter.\n\n**4** **Museu d'Hist\u00f2ria de Barcelona  \n** This place holds the archeological history of Roman Barcelona \u2013 right under your feet.\n\n **Lunch**\n\nStop near the church of Santa Mar\u00eda del Pi for alfresco drinks and a market-fresh meal at _Taller de Tapas_.\n\n **5** **Museu Picasso  \n** Walk through the tight-knit medieval streets of La Ribera to the must-see museum, housed in the city where Picasso developed his inimitable style.\n\n **6** **Parc de la Ciutadella  \n** Take time out in the museums, palm-houses and gardens of the city's favourite park.\n\n **7** **Port Ol\u00edmpic  \n** The beach, boardwalk and seafront promenade set the scene for a sundowner drink.\n\n**Dinner**\n\nSome of the city's coolest places to eat and drink are in El Raval; try the creative cuisine at stylish _Biblioteca_ 's open kitchen.\n\n< Back to Itineraries\n\n## Day two in Barcelona\n\n **1** **Sagrada Fam\u00edlia  \n** To avoid the worst of the bustling crowds, arrive at Gaud\u00ed's masterpiece at opening time.\n\n **2** **Passeig de Gr\u00e0cia  \n** Europe's most extraordinary urban architecture decorates the modern city's main avenue.\n\n**3** **Museu Egipci de Barcelona  \n** A captivating collection including mummies, amulets and sarcophagi transports you back to ancient Egypt.\n\n **4** **Vin\u00e7on  \n** Design, style and shopping opportunities, all under one roof.\n\n **Lunch**\n\nThere's fabulous food at _Tapas, 24_, a classy uptown bar-diner.\n\n **5** **Museu Nacional d'Art de Catalunya  \n** The triumphant landscaped approach to Montju\u00efc culminates in the extraordinary Catalan National Art Gallery.\n\n **6** **Fundaci\u00f3 Joan Mir\u00f3  \n** The modernist building on the hill houses the life's work of Catalan artist Joan Mir\u00f3.\n\n**7** **Ride the Trasbordador Aeri  \n** A thrilling cable car sweeps you across the inner harbour from Montju\u00efc to Port Vell.\n\n **8** **Barceloneta  \n** For marina or beach views, grab a table at an outdoor caf\u00e9 in the old fishermen's quarter.\n\n**Dinner**\n\nAfter cold beer and tapas in up-and-coming Sant Pere, dine in Catalan style at La Ribera's _Senyor Parellada_.\n\n< Back to Itineraries\n\n##  _Modernista_ Barcelona\n\n **Visionary _modernista_ architects, like Antoni Gaud\u00ed, changed the way people looked at buildings. Their style, a sort of Catalan Art Nouveau, left Barcelona with an extraordinary architectural legacy that goes far beyond the famous sights of the Sagrada Fam\u00edlia church and Parc G\u00fcell.**\n\n **1** **Hospital de la Santa Creu i de Sant Pau  \n** Don't miss Llu\u00eds Dom\u00e8nech i Montaner's innovative public hospital, near the Sagrada Fam\u00edlia.\n\n **2** **La Pedrera  \n** Inventive design permeates every aspect of Gaud\u00ed's fantastical \"stone quarry\" apartment building.\n\n**3** **Casa Amatller  \n** Catch a guided tour of this stunning house belonging to a nineteenth-century chocolate manufacturer.\n\n**Lunch**\n\nTake a lunch stop at _Casa Calvet_, an early Gaud\u00ed townhouse, now a classy Catalan restaurant.\n\n **4** **Palau de la M\u00fasica Catalana  \n** Book in advance for a tour of this dramatic concert hall, or buy a ticket for an evening performance.\n\n **5** **Arc de Triomf  \n** Gateway to the Ciutadella park is this giant red-brick arch.\n\n**6** **Castell dels Tres Dragons  \n** The park's eye-catching \"castle\" is a _modernista_ showcase for the city's natural science museum.\n\n **Dinner**\n\nIn summer, Parc G\u00fcell is open until well into the evening \u2013 and the bars and restaurants of fashionable Gr\u00e0cia are close at hand.\n\n< Back to Itineraries\n\n## Budget Barcelona\n\n **Barcelona may be one of Europe's most fashionable cities, but it remains remarkably good value as far as most visitors are concerned. You certainly don't need to spend a fortune to eat well, see the major sights and enjoy yourself.**\n\n **1** **The Ramblas  \n** Barcelona's greatest show \u2013 a stroll down the Ramblas \u2013 is a free spectacle around the clock.\n\n**2** **MNAC  \n** The ticket for the showpiece National Art Gallery is valid for two full days, and there's free entry on the first Sunday of every month.\n\n **Lunch**\n\nVirtually every restaurant offers a weekday _men\u00fa del dia_ , so lunch is a bargain at places like _Caf\u00e9 de l'Acad\u00e8mia_ where dinner might cost three times as much.\n\n **3** **Relax at the beach  \n** Enjoy five miles of sand, boardwalks and promenades.\n\n**4** **Font M\u00e0gica  \n** There's no charge to watch this magnificent display of water and light.\n\n **5** **Caixa Forum  \n** Entry to this dazzling arts and cultural centre costs less than a cocktail.\n\n **6** **Parc de la Ciutadella  \n** Unfurl a picnic blanket in the city's green lung.\n\n**Dinner**\n\nMany of Barcelona's markets also have stylish restaurants attached \u2013 like _Cuines Santa Caterina_ in Sant Pere's dramatic Mercat Santa Caterina.\n\n< Back to Itineraries\n\n# Best of Barcelona\n\nBig sights\n\nShopping\n\nRestaurants\n\nMuseums\n\nArt galleries\n\nTapas bars\n\nBars and clubs\n\nMusic, dance and theatre\n\nOutdoor Barcelona\n\n# Big sights\n\n **1** **Sagrada Fam\u00edlia  \n** The most famous unfinished church in the world \u2014 the \"Sacred Family\" temple is the essential visit for Gaud\u00ed fans.\n\n**2** **La Seu  \n** Pride of the Gothic era, the city's majestic medieval cathedral anchors the Old Town.\n\n**3** **Parc G\u00fcell  \n** A public park without compare, where contorted stone pavilions, gingerbread buildings and surreal ceramics combine unforgettably.\n\n**4** **Museu Nacional d'Art de Catalunya (MNAC)  \n** The National Museum of Art celebrates the glories of a thousand years of Catalan painting and sculpture.\n\n**5** **The Ramblas  \n** The city's iconic central thoroughfare, where hawkers, stall-holders, eccentrics and tourists collide to gleeful effect.\n\n< Back to Best of Barcelona\n\n# Shopping\n\n **1** **Mercat de la Boqueria  \n** The city's finest food market is a show in its own right, busy with locals and tourists from dawn to dusk.\n\n**2** **Casa Gispert  \n** A hideaway treasure-trove of nuts, dried goods, coffee and spices in the heart of La Ribera.\n\n**3** **Vin\u00e7on  \n** The most eminent showcase in town for Catalan household style, design and furnishings.\n\n**4** **Custo Barcelona  \n** Barcelona's brightest designer brand injects colour and fun into shop windows across the city.\n\n**5** **L'Arca  \n** Vintage heaven in the Gothic Quarter.\n\n< Back to Best of Barcelona\n\n# Restaurants\n\n **1** **Biblioteca  \n** Creative, market-led Catalan cuisine is on the menu at fashionable _Biblioteca_.\n\n**2** **Flash, Flash  \n** A fun retro Gr\u00e0cia landmark that specializes in tortillas of all tastes, savoury to sweet.\n\n**3** **La Tomaquera  \n** Food cooked on the chargrill ( _a la brasa_ ) is the speciality at this bustling tavern.\n\n**4** **Cinc Sentits  \n** Delight all five senses at Jordi Artal's Michelin-starred restaurant.\n\n**5** **Senyor Parellada  \n** To eat home-style food with the locals, make time for this ravishingly handsome Old Town restaurant.\n\n< Back to Best of Barcelona\n\n# Museums\n\n **1** **Museu del Disseny  \n** The city's applied art collections have a shiny, new home at the sleek _Disseny Hub_ building.\n\n**2** **Museu Egipci  \n** Break off from your uptown shopping to view this select showing of fascinating Egyptian antiquities.\n\n**3** **Museu Frederic Mar\u00e8s  \n** Don't miss the museum dedicated to the \"mad collector\" to beat them all.\n\n**4** **CosmoCaixa  \n** A science museum complete with its own rainforest.\n\n**5** **Camp Nou and FC Barcelona  \n** Tour one of Europe's most magnificent stadiums, home to the local football heroes.\n\n< Back to Best of Barcelona\n\n# Art galleries\n\n **1** **Fundaci\u00f3 Joan Mir\u00f3  \n** There's no more beautiful gallery in the city than the house on the hill presenting the life's work of Joan Mir\u00f3.\n\n**2** **Caixa Forum  \n** There's always an exhibition worth seeing in the city's best arts and cultural centre \u2014 not to mention concerts, films and other events.\n\n**3** **Fundaci\u00f3 Antoni T\u00e0pies  \n** Acquaint yourself with the work of the master Catalan abstract artist, contained within a striking Eixample mansion.\n\n**4** **Museu d'Art Contemporani de Barcelona (MACBA)  \n** Post-war contemporary art (Spanish, Catalan and international) has a home in a stunning building in El Raval.\n\n**5** **Museu Picasso  \n** The city's most visited art collection traces Picasso's career in its entirety.\n\n< Back to Best of Barcelona\n\n# Tapas bars\n\n **1** **Tickets  \n** A chance to taste some of Ferran Adr\u00eca's famous dishes at the _elBulli_ chef's (relatively) affordable tapas joint.\n\n**2** **Bodega La Plata  \n** Stand-up snacks and wine from the barrel are the staples at this famous Old Town bar.\n\n**3** **Dos Palillos  \n** East meets West \u2013 and hipsters meet each other \u2013 in El Raval's cool Asian fusion tapas bar _Dos Palillos_.\n\n**4** **El Xampanyet  \n** Step into this La Ribera institution for a glass of Catalan fizz and a bite or two before dinner.\n\n**5** **Bar Pinotxo  \n** The best tapas bar in the best market \u2014 the Boqueria's _Pinotxo_ , no contest.\n\n< Back to Best of Barcelona\n\n# Bars and clubs\n\n **1** **Kahala  \n** The heady drinks at this Polynesian-themed cocktail bar will have you saying \"\u00a1 _Aloha, Barcelona_!\"\n\n**2** **Moog  \n** The city's techno club of choice for aficionados, open from midnight for nonstop dancing.\n\n**3** **Caf\u00e9 del Sol  \n** Sip _copas_ and people watch from the _terrassa_ of this Pla\u00e7a del Sol institution.\n\n**4** **Resolis  \n** Resolis is the signature boho bar in the thriving El Raval neighbourhood.\n\n**5** **Milk  \n** Mixology at its finest in the Barri G\u00f2tic's most welcoming bar and bistro.\n\n< Back to Best of Barcelona\n\n# Music, dance and theatre\n\n **1** **Sidecar  \n** This rootsy Pla\u00e7a Reial rock club presents a varied roster of gigs and club nights, with indie and global sounds to the fore.\n\n**2** **Palau de la M\u00fasica Catalana  \n** The city's finest concert hall, a _modernista_ classic encased in a contemporary shell.\n\n**3** **Tarantos  \n** The best place for flamenco sessions \u2014 the after-show dancing carries on into the small hours.\n\n**4** **Caf\u00e9 Teatre Llantiol  \n** It's mime, magic and much, much more at Barcelona's long-established cabaret venue.\n\n**5** **Gran Teatre del Liceu  \n** Book ahead for opera tickets at this renowned city landmark, though the guided tours are open to all.\n\n< Back to Best of Barcelona\n\n# Outdoor Barcelona\n\n **1** **Parc Zool\u00f2gic  \n** Barcelona's zoo packs the world's fauna into the rolling grounds of the Ciutadella park.\n\n**2** **City beaches  \n** The great urban escape is to the city's 5km of sand-fringed ocean, dotted with parks and playgrounds.\n\n**3** **Jard\u00ed Bot\u00e0nic de Barcelona  \n** The impressive botanical gardens spread across a hillside above the Olympic Stadium.\n\n**4** **Parc de la Ciutadella  \n** Whatever the season, the city's nicest park always springs a surprise.\n\n**5** **Port Ol\u00edmpic  \n** Twin towers and the landmark Frank Gehry fish dominate Barcelona's liveliest resort area.\n\n< Back to Best of Barcelona\n\n**1** **Along the Ramblas** Barcelona's most famous avenue.\n\n **2** **Barri G\u00f2tic** The \"Gothic Quarter\" preserves the city's historic core.\n\n **3** **Port Vell and Barceloneta** Top beaches and seafood restaurants.\n\n **4** **El Raval** Up-and-coming Old Town neighbourhood.\n\n **5** **Sant Pere** A great market plus fashionable bars, boutiques and restaurants.\n\n **6** **La Ribera** The medieval artisan's quarter is home to the Picasso Museum.\n\n **7** **Parc de la Ciutadella** A favourite park on the edge of the Old Town.\n\n **8** **Montju\u00efc** The art museums, castle and gardens make for a popular day out.\n\n **9** **Port Ol\u00edmpic and Poble Nou** Beaches, bars and urban recreation along the city's revitalised waterfront.\n\n **10** **Dreta de l'Eixample** The \"right-hand\" side of the modern city centre has an unparalleled display of _modernista_ architecture.\n\n **11** **Sagrada Fam\u00edlia and Gl\u00f2ries** Barcelona's most amazing building is Gaud\u00ed's unfinished masterwork of a church.\n\n **12** **Esquerra de l'Eixample** The city centre's \"left-hand\" side offers cool bars, top-end restaurants and the gay quarter.\n\n **13** **Gr\u00e0cia and Parc G\u00fcell** Visit Gr\u00e0cia to see Gaud\u00ed's inspiring urban park.\n\n **14** **Camp Nou, Pedralbes and Sarri\u00e0-Sant Gervasi** Shops, galleries and a magnificent football stadium.\n\n **15** **Tibidabo and Parc de Collserola** A mountain-top amusement park plus rambling forest walks.\n\n **16** **Montserrat** The crag-bound monastery makes a popular day-trip.\n\n **17** **Sitges** Escape to the beach at Spain's biggest gay resort.\n\n# Along the Ramblas\n\nSights\n\nShops\n\nCaf\u00e9s\n\nRestaurants and tapas bars\n\nBar\n\nNo day in the city seems complete without a stroll along the Ramblas, Spain's most famous thoroughfare. Cutting through Barcelona's Old Town areas, and connecting Pla\u00e7a de Catalunya with the harbour, it's at the heart of the city's self-image \u2013 lined with caf\u00e9s, restaurants, souvenir shops, flower stalls and newspaper kiosks. The name (from the Arabic _ramla_ or \"sand\") refers to a seasonal stream bed that was paved over in medieval times. Since the nineteenth century it's been a fashionable promenade, and today the show goes on, as human statues, portrait painters, buskers and card sharps add to the vibrancy of Barcelona's most enthralling street. There are metro stops at Catalunya (top of the Ramblas), Liceu (middle) and Drassanes (bottom), or you can walk the entire length in about twenty minutes.\n\n## Pla\u00e7a de Catalunya\n\n Catalunya.MAP\n\nThe huge formal square at the top of the Ramblas stands right at the heart of the city. It's not only the focal point of events and demonstrations \u2013 notably a mass party on New Year's Eve \u2013 but also the site of prominent landmarks like the main city tourist office, the white-faced El Corte Ingl\u00e9s department store and El Triangle shopping centre (with the popular _Zurich_ outdoor caf\u00e9 beneath it).\n\nThe Ramblas itself actually comprises five separate named sections, starting with the northern stretch, Rambla Canaletes, nearest Pla\u00e7a de Catalunya, which is marked by an iron fountain \u2013 a drink from this supposedly means you'll never leave Barcelona. Further down is the bird market on Rambla Estudis and the sudden profusion of flower stalls on Rambla Sant Josep, near the Boqueria market.\n\nPla\u00e7a de Catalunya\n\n## Esgl\u00e9sia de Betlem\n\nRamblas 107 Liceu.Daily 8am\u20131.30pm & 6\u20139pm.MAP\n\nIt seems hard to believe, but the Ramblas was a war zone during the Spanish Civil War as the city erupted into factionalism in 1937. George Orwell was caught in the crossfire (an episode recorded in his _Homage to Catalonia_ ) and, with anarchists sacking the city's churches at will, the rich interior of the Baroque Esgl\u00e9sia de Betlem was completely destroyed. However, the main facade on c\/del Carme still sports a fine sculpted portal.\n\nSculpted portal of the Esgl\u00e9sia de Betlem\n\n## Palau Moja\n\nRamblas 188 Liceu 933 162 740.Tours available by appointment only.MAP\n\nThe arcaded Palau Moja dates from the late eighteenth century and still retains an exterior staircase and elegant great hall. The palace's gallery (entrance is around the corner in c\/Portaferrissa) is occasionally open for exhibitions relating to all things Catalan. Take a look, too, at the illustrated tiles above the fountain at the start of c\/de la Portaferrissa, which show the medieval gate (the _Porta Ferri\u00e7a_ ) and market that were once sited here.\n\n### The Ramblas statues\n\nTime stands still for no man \u2013 not even for the famed human statues of the Ramblas, who make a living out of doing just that. A motley crew of figures once flanked the length of the street, but change is inevitable, and in 2012 \u2013 in an effort to keep pedestrian traffic moving, and prevent pickpockets from preying on the gathering crowds \u2013 the city moved the human statues to the wide stretch of Rambla de Santa M\u00f2nica. The number of statues was then capped to thirty a year, with performers being required to audition for the permit-only slots. Despite the cutback in territory and number, the remaining human statues are still an attraction. Be it Galileo or a horned demon, these stalwarts of the Ramblas continue to climb upon their plinths and strike a pose. What else is a statue going to do?\n\n## Palau de la Virreina\n\nRamblas 99 Liceu 933 161 000, www.bcn.cat\/cultura.Galleries Tues\u2013Sun noon\u20138pm.for current exhibitions see  lavirreina.bcn.cat; information office daily 10am\u20138.30pm.MAP\n\nGraceful eighteenth-century Palau de la Virreina is the HQ of the cultural department of the Ajuntament (city council), and there's a ground-floor information centre where you can find out about upcoming events and buy tickets. Various galleries and studios house changing exhibitions of contemporary art and photography, while at the back of the palace courtyard you can usually see the city's enormous Carnival giants ( _gegants_ ), representing the thirteenth-century Catalan king Jaume I and his wife Violant. The origin of these ornate, five-metre-high figures is unclear, though they probably first enlivened medieval travelling fairs and are now an integral part of Barcelona's festival parades.\n\n### An insider's guide to the Ramblas\n\nThere are pavement caf\u00e9s and restaurants all the way down the Ramblas, but the food can be indifferent and the prices high, so be warned. (For better value, go into the Boqueria market, where the traders eat.) The strolling crowds, too, provide perfect cover for pickpockets \u2013 keep a wary eye on your possessions at all times, especially when watching the buskers or shopping at the kiosks. And \u2013 however easy it looks to win \u2013 if you're going to play cards or dice with a man on a street, you've only yourself to blame if you get ripped off.\n\n## Mercat de la Boqueria\n\nRamblas 91 Liceu 933 182 584, www.boqueria.info.Mon\u2013Sat 8am\u20138.30pm.MAP\n\nOther markets might protest, but the city's glorious main food market really can claim to be the best in Spain. It's officially called the Mercat Sant Josep, though everyone knows it as La Boqueria. A riot of noise and colour, it's as popular with locals who come here to shop daily as with snap-happy tourists. Everything radiates out from the central fish and seafood stalls \u2013 bunches of herbs, pots of spices, baskets of wild mushrooms, mounds of cheese and sausage, racks of bread, hanging hams and overloaded meat counters. It's easy to get waylaid at the entrance by the fruit cartons and squeezed juices, but the flagship fruit and veg stalls here are pricey. It's better value further in, in particular in the small outdoor square just beyond the north side of the market where the local allotment holders and market-gardeners gather. Everyone has a favourite market stall, but don't miss _Petras_ and its array of wild mushrooms (it's at the back, by the market restaurant, _La Gardu\u00f1a_ ) or _Frutas y Verduras Jes\u00fas y Carmen_ , which is framed with colourful bundles of exotic chillies. And of course, there are some excellent stand-up tapas bars in the market as well, open from dawn onward for the traders.\n\nMercat de la Boqueria\n\n## Pla\u00e7a de la Boqueria\n\n Liceu.MAP\n\nThe halfway point of the Ramblas is marked by Pla\u00e7a de la Boqueria, with its large round pavement **mosaic by Joan Mir\u00f3**. It's become something of a symbol for the city and is one of a number of public works in Barcelona by the artist, who was born just a couple of minutes' away in the Barri G\u00f2tic. Over at Ramblas 82, **Casa Bruno Quadros** \u2013 the lower floor is now the Caixa Sabadell \u2013 was built in the 1890s to house an umbrella store, which explains its delightful facade, decorated with Oriental designs, dragons and parasols.\n\n## Gran Teatre del Liceu\n\nRamblas 51\u201359 Liceu www.liceubarcelona.cat.Box office  934 859 913; tours  934 859 914, daily (except Aug) at 10am, 11.30am, noon, 12.30pm & 1pm.Guided tour \u20ac11.50, other tours \u20ac5.50.MAP\n\nBarcelona's celebrated opera house was first founded in 1847 and rebuilt after a fire in 1861 to become Spain's grandest theatre. Regarded as a bastion of the city's late nineteenth-century commercial and intellectual classes, the Liceu was devastated again in 1893 when an anarchist threw two bombs into the stalls during a production of _William Tell_ \u2013 twenty people died. It then burned down for the third time in 1994, when a workman's blowtorch set fire to the scenery of an opera set. The latest restoration of the lavishly decorated interior took five years, and the opera house opened again in 1999, complete with a modern extension, the Espai Liceu, which also houses a music and gift shop and a caf\u00e9. You'll see and learn most on the more expensive, hour-long 10am guided tour (the other, shorter, cheaper tours are self-guided). Highlights include the Salon of Mirrors and the impressive gilded auditorium containing almost 2300 seats, making it one of the world's largest opera houses. Some tours also include the option of visiting the glorious _modernista_ -styled rooms of the Cercle del Liceu, the opera house's private members' club.\n\nFor Liceu performances, check the website for details and make bookings well in advance. The traditional meeting place for audience and performers alike, meanwhile, is the famous _Caf\u00e9 de l'Opera_ , just across the Ramblas.\n\nGran Teatre del Liceu\n\n## Arts Santa M\u00f2nica\n\nRamblas 7 Drassanes 935 671 110, www.artsantamonica.cat.Tues\u2013Sat 11am\u20139pm, Sun & hols 11am\u20135pm.Free.MAP\n\nDown from the Liceu, the bottom part of the Ramblas (Rambla de Santa M\u00f2nica) was historically a theatre and red-light district, and it still has a rough edge or two. Flagship building is the Augustinian convent of Santa M\u00f2nica, which dates originally from 1626, making it the oldest building on the Ramblas. It was remodelled in the 1980s as a contemporary arts centre, and hosts regularly changing exhibitions \u2013 it's an unusual gallery space dedicated to \"artistic creation, science, thought and communication\" so there's usually something worth seeing, from an offbeat art installation to a show of archive photographs. Meanwhile, pavement artists and palm readers set up stalls outside on the Ramblas, augmented on weekend afternoons by a street market selling jewellery, beads, bags and ornaments.\n\nStreet Performer, Ramblas\n\n## Museu de Cera\n\nRamblas 4\u20136, entrance on Ptge. de Banca Drassanes 933 172 649, www.museocerabcn.com.July\u2013Sept daily 10am\u201310pm; Oct\u2013June Mon\u2013Fri 10am\u20131.30pm & 4\u20137.30pm, Sat & Sun 11am\u20132pm & 4.30\u20138.30pm.\u20ac15.MAP\n\nYou'd have to be hard-hearted indeed not to derive some pleasure from the city's wax museum. Located in a nineteenth-century bank building, it presents an ever more ludicrous series of tableaux in cavernous salons and gloomy corridors, depicting recitals, meetings and parlour gatherings attended by an anachronistic \u2013 not to say perverse \u2013 collection of characters, from Hitler to Princess Diana. Needless to say, it's enormously amusing, culminating in cheesy underwater tunnels and space capsules and an unpleasant \"Terror Room\". Even if this doesn't appeal it is definitely worth poking your head into the museum's extraordinary grotto bar, the _Bosc de les Fades_.\n\n< Back to Along the Ramblas\n\n## Shops\n\n### Casa Beethoven\n\nRamblas 97 Liceu.Mon\u2013Fri 9am\u20132pm & 4\u20138pm, Sat 9am\u20132pm & 5\u20138pm; closed Aug.MAP\n\nWonderful old shop selling sheet music from wooden library stacks, plus CDs and music reference books \u2013 not just classical, but also rock, jazz and flamenco.\n\n### El Corte Ingl\u00e9s\n\nPl. de Catalunya 14 Catalunya www.elcorteingles.es.Mon\u2013Sat 9.30am\u20139.30pm.MAP\n\nThe city's biggest department store has nine retail floors (fashion, cosmetics, household goods, toys), a good basement supermarket and \u2013 best of all \u2013 a top-floor caf\u00e9 with terrific views. For music, books, computers and sports gear, head for the nearby branch at Av. Portal de l'Angel 19.\n\n### El Triangle\n\nPl. de Catalunya 4 Catalunya eltriangle.es.Mon\u2013Sat 10am\u201310pm.MAP\n\nShopping centre dominated by the flagship FNAC store, which specializes in books (good English-language selection), music, film and computer stuff. Also a Camper (for shoes), Habitat (homeware) and Sephora (cosmetics), plus lots of boutiques, and a caf\u00e9 on the ground floor next to the extensive newspaper and magazine section.\n\n## Caf\u00e9s\n\n### Caf\u00e9 de l'Opera\n\nRamblas 74 Liceu 933 177 585, www.cafeoperabcn.com.Daily 8.30am\u20132.30am.MAP\n\nIf you're going to pay through the nose for a seat on the Ramblas, it may as well be at this famous old caf\u00e9-bar opposite the opera house, which retains a _fin-de-si\u00e8cle_ feel. Surprisingly, it's not a complete tourist-fest and locals pop in day and night for drinks, cakes and tapas.\n\n### Caf\u00e9 Zurich\n\nPl. Catalunya 1 Catalunya 933 179 153.Mon\u2013Fri 8am\u201311pm, Sat 11am\u2013midnight, Sun 9am\u201311pm.MAP\n\nThe most famous meet-and-greet caf\u00e9 in town, right at the top of the Ramblas and underneath El Triangle shopping centre. It's good for croissants and breakfast sandwiches and there's a huge pavement terrace, but sit inside if you don't want to be bothered by endless rounds of buskers and beggars.\n\n### Escrib\u00e0\n\nRamblas 83 Liceu 933 016 027, www.escriba.es.Daily 9am\u20139pm.MAP\n\nWonderful pastries and cakes from the renowned Escrib\u00e0 family business in a classy Art Nouveau shop. Many rate this as the best bakery in Barcelona.\n\n## Restaurants and tapas bars\n\n### Amaya\n\nRamblas 20\u201324 Drassanes 933 021 037, www.restauranteamaya.com.Daily 1.30\u20134pm & 7\u201311.30pm.MAP\n\nThere aren't too many reliable places on the Ramblas itself, but _Amaya_ \u2013 a fixture since 1941 \u2013 fits the bill for a meal. The restaurant (its _terrassa_ offers front-row seats to the action on the Ramblas) serves Basque seafood specialities, from octopus to anchovies, as well as rice dishes, _canelons_ (Catalan\u2013style cannelloni) and rich oxtail stew. The lunch and tapas menus offer the cheapest and most enjoyable introduction to the cuisine; otherwise, main dishes cost \u20ac14\u201320.\n\n### Bar Pinotxo\n\nMercat de la Boqueria, Ramblas 91 Liceu 933 171 731, pinotxobar.com.Mon\u2013Sat 6.30am\u20134pm; closed Aug.MAP\n\nThe market's most renowned refuelling stop \u2013 just inside the main entrance on the right \u2013 attracts traders, chefs, tourists and celebs, who stand three deep at busy times. A coffee, a grilled sandwich and a glass of _cava_ is the local breakfast of choice, or let the cheery staff steer you towards the tapas and daily specials (\u20ac5\u201315), anything from a slice of tortilla to fried baby squid.\n\n### La Gardu\u00f1a\n\nMercat de la Boqueria, c\/Jerusalem 18 Liceu 647 223 776.Mon\u2013Sat 9am\u201311pm.MAP\n\nTucked away at the back of the frenetic Boqueria market, this is an especially enticing place for lunch when there's a good-value _men\u00fa del dia_ \u2013 basically, you'll be offered the best of the day's produce at pretty reasonable prices, and if you're lucky you'll get an outdoor seat with market views. At dinner, most mains are \u20ac9\u201320.\n\n## Bar\n\n### Bosc de les Fades\n\nPtge. de Banca 5 Drassanes 933 172 649.Mon\u2013Thurs 10am\u20131am, Fri 10am\u20131.30am, Sat 11am\u20132am, Sun 11am\u20131am.MAP\n\nDown an alley by the wax museum, the \"Forest of the Fairies\" is festooned with gnarled plaster tree trunks, hanging branches, fountains and stalactites. It's a bit cheesy, which is perhaps why it's a huge hit with the twenty-something crowd who huddle in the grottoes with a cocktail or two.\n\n< Back to Along the Ramblas\n\n# Barri G\u00f2tic\n\nSights\n\nShops\n\nCaf\u00e9s\n\nRestaurants and tapas bars\n\nBars\n\nClubs\n\nThe Barri G\u00f2tic, or Gothic Quarter, on the east side of the Ramblas forms the heart of Barcelona's Old Town. Its buildings date principally from the fourteenth and fifteenth centuries, and culminate in the extraordinary Gothic cathedral known as La Seu. Around here are hidden squares, some fascinating museums, the city's old Jewish quarter and the remains of the Roman walls. It takes the best part of a day to see everything \u2013 longer if you factor in the abundant caf\u00e9s, antique shops, boutiques and galleries. Note that the southern area, en route to the harbour, is rather less gentrified than the cathedral district \u2013 take care at night in the poorly lit streets. Metros Liceu (west), Jaume I (east) and Drassanes (south) provide access to the neighbourhood.\n\nzoom left\n\nzoom right\n\n## La Seu\n\nPl. de la Seu Jaume I.Mon\u2013Fri 8am\u201312.45pm & 5.15\u20137.30pm, Sat 8am\u201312.45pm & 5.15\u20137.30pm, Sun & hols 8am\u20131.45pm & 5.15\u20137.30pm.Free during general admission times, otherwise tourist admission charge obligatory (Mon\u2013Sat 1\u20135pm, Sun 2\u20135pm), \u20ac6, includes entrance to all sections.MAP\n\nBarcelona's cathedral is one of the great Gothic buildings of Spain, dedicated to Santa Eul\u00e0lia, who was martyred by the Romans for daring to prefer Christianity \u2013 her ornate tomb rests in a crypt beneath the high altar. A magnificent fourteenth-century **cloister** looks over a lush tropical garden complete with soaring palm trees and honking white geese. There's also glittering church treasure on show in the cathedral museum.\n\nPerformances of the Catalan national dance, the _sardana_ , take place in front of the cathedral (usually Sun at noon, plus Easter\u2013Nov Sat at 6pm), while the pedestrianized Avinguda de la Catedral hosts an antiques market every Thursday, and a Christmas craft fair in December.\n\nLa Seu\n\n## Museu Dioces\u00e0\n\nAv. Catedral 4 Jaume I 933 152 213.Tues\u2013Fri 11am\u20136pm, Sat 11am\u20132pm & 2.30\u20136pm, Sun 11am\u20132pm.\u20ac6.MAP\n\nStand back to look at the cathedral buildings and it's easy to see the line of Roman towers that originally stood on this spot incorporated into the later medieval structures. One such tower formed part of the cathedral almshouse, now the Museu Dioces\u00e0, whose soaring spaces have been beautifully adapted to show an impressive collection of religious art and church treasures. The ticket also includes entrance into the temporary art and architecture exhibitions held here.\n\n## Reial Cercle Art\u00edstic\n\nC\/Arcs 5 Jaume I 933 181 774, www.dalibarcelona.com.Daily 10am\u201310pm.\u20ac10.MAP\n\nThe handsome Gothic palace housing the Royal Art Circle hosts various free exhibitions and concerts, though the big draw is the collection of 44 original sculptures by Salvador Dal\u00ed, completed in the 1970s. A lovely terrace restaurant above the Gothic streets also offers a pricey lunch, while on your way to or from the cathedral spare a glance for the graffiti-like frieze surmounting the nearby **Collegi d'Arquitectes** on Pla\u00e7a Nova \u2013 designed by that other inimitable master, Pablo Picasso.\n\n## Pla\u00e7a del Rei\n\n Jaume I.MAP\n\nThe harmonious enclosed square of Pla\u00e7a del Rei was once the palace courtyard of the Counts of Barcelona. Stairs climb from here to the palace's main hall, the fourteenth-century **Sal\u00f3 del Tinell**. It was here that Ferdinand and Isabella received Christopher Columbus on his triumphant return from his famous voyage of 1492. At one time the Spanish Inquisition met in the hall, taking full advantage of the popular belief that the walls would move if a lie was spoken. Nowadays it hosts temporary exhibitions, while concerts are occasionally held in the hall or outside in the square. The palace buildings include the beautiful fourteenth-century **Capella de Santa Agata** , and the romantic Renaissance **Torre del Rei Mart\u00ed**. There's no public access to the tower, but the interiors of the hall and chapel can usually be seen during a visit to the adjacent Museu d'Hist\u00f2ria de Barcelona.\n\nPla\u00e7a del Rei\n\n## Museu Frederic Mar\u00e8s\n\nPl. de Sant Iu 5\u20136 Jaume I 932 563 500, www.museumares.bcn.cat.Tues\u2013Sat 10am\u20137pm, Sun & hols 11am\u20138pm.\u20ac4.20, free Sun after 3pm & first Sun of the month.MAP\n\nDon't miss a visit to one of the Old Town's most fascinating museums, which occupies a wing of the old royal palace. It celebrates the diverse passions of sculptor, painter and restorer Frederic Mar\u00e8s (1893\u20131991), whose beautifully presented collection of ancient and medieval sculpture does little to prepare visitors for Mar\u00e8s's true obsession, namely a kaleidoscopic array of curios and collectibles. Entire rooms are devoted to keys and locks, cigarette cards and snuffboxes, fans, gloves and brooches, walking sticks, dolls' houses, old gramophones and archaic bicycles \u2013 to list just a sample of what's in the collection.\n\n## Museu d'Hist\u00f2ria de Barcelona (MUHBA)\n\nPl. del Rei, entrance on c\/del Veguer Jaume I 932 562 100, www.museuhistoria.bcn.cat.Tues\u2013Sat 10am\u20137pm, Sun 10am\u20138pm.\u20ac7.MAP\n\nThe Barcelona History Museum comprises half a dozen sites across the city, though its principal hub is what's known as the \"Conjunt Monumental\" (or monumental ensemble) of Pla\u00e7a del Rei, whose crucial draw is an amazing underground archeological section \u2013 nothing less than the remains of the Roman city of Barcino (first century BC to the sixth century AD), which stretch under the surrounding streets as far as the cathedral. Excavations and explanatory diagrams show the full extent of the streets and buildings \u2013 from lookout towers to laundries \u2013 while models, mosaics, murals and finds help flesh out the reality of daily life in Barcino.\n\nThere's a well-stocked book and gift shop on site (entrance on c\/Llibreteria), while the museum ticket also allows entry to the other MUHBA sites, notably the Poble Sec air-raid shelter and Pedralbes monastery. Roman fans should also take the opportunity to see the four remarkable Roman columns of the Temple d'August in the courtyard of the nearby Centre Excursionista de Catalunya (c\/Paradis 10; closed Mon).\n\nExhibit at the Museu d'Hist\u00f2ria de Barcelona\n\n## Esgl\u00e9sia de Santa Mar\u00eda del Pi\n\nPl. Sant Josep Oriol Liceu.April\u2013Oct Mon\u2013Sat 10am\u20137pm, Sun & hols 4\u20138pm; Nov\u2013March Mon\u2013Sat 10am\u20136pm, Sun & hols 4\u20137pm.\u20ac4.MAP\n\nThe fourteenth-century church of Santa Mar\u00eda is known for its marvellous stained glass, particularly a 10m-wide rose window (often claimed, rather boldly, to be the largest in the world). The church flanks Pla\u00e7a Sant Josep Oriol, the prettiest of three delightful adjacent squares and an ideal place for an outdoor coffee and a browse around the weekend **artists' market** (Sat 11am\u20138pm, Sun 11am\u20132pm).\n\nThe church is named \u2013 like the squares on either side, Pla\u00e7a del Pi and Pla\u00e7eta del Pi \u2013 after the pine tree that once stood here. A **farmers' market** spills across Pla\u00e7a del Pi on the first and third Friday, Saturday and Sunday of the month, while the characteristic caf\u00e9s of narrow **Carrer de Petritxol** (off Pla\u00e7a del Pi) are the places to head to for a cup of hot chocolate \u2013 _Dulcinea_ at no. 2 is the traditional choice \u2013 and a browse around the street's commercial art galleries. The most famous is **Sala Pares** at c\/Petritxol 5, well known as the site of Picasso's first solo exhibition.\n\nEsgl\u00e9sia de Santa Mar\u00eda del Pi\n\n## Museu del Cal\u00e7at\n\nPl. Sant Felip Neri 5 Liceu 933 014 533.Tues\u2013Sun & hols 11am\u20132pm.\u20ac2.50.MAP\n\nThe former headquarters of the shoemakers' guild (founded in 1202) houses a one-room footwear museum, containing originals dating back to the 1600s as well as oddities like the world's biggest shoe, made for the city's Columbus statue at the bottom of the Ramblas. The museum flanks one side of pretty Pla\u00e7a Sant Felip Neri, where in summer you can eat outside at the restaurant of the boutique _Neri Hotel_ , which sets out candlelit tables in the square.\n\n## Antiga Sinagoga\n\nC\/Marlet 5, corner with c\/Sant Dom\u00e8nec del Call Liceu 933 170 790, www.calldebarcelona.org.June\u2013Aug Mon\u2013Fri 10.30am\u20136.30pm, Sat & Sun 10.30am\u20132.30pm; Sept\u2013May Mon\u2013Fri 11am\u20135.30pm, Sat & Sun 11am\u20133pm.\u20ac2.50.MAP\n\nBarcelona's medieval Jewish quarter was centred on c\/Sant Dom\u00e8nec del Call, where a synagogue existed from as early as the third century AD until the pogrom of 1391, but even after that date the building survived in various guises and has since been sympathetically restored. The city authorities have signposted a few other points of interest in what's known as \"El Call Major\", including the **Centre d'Interpretaci\u00f3 del Call** in nearby Pla\u00e7eta Manuel Rib\u00e9 (Tues\u2013Fri 11am\u20132pm, Sat & Sun 11am\u20137pm; free), whose informative storyboards (in English) shed more light on Barcelona's fascinating Jewish heritage.\n\n## Palau de la Generalitat\n\nPl. de Sant Jaume Jaume I 934 024 600.Tours on 2nd and 4th Sun of the month (not Aug), every hour, 10am\u2013noon; also on April 23, and Sept 11 & 24.Passport or ID required.Free.MAP\n\nThe home of the Catalan government presents its oldest aspect around the side, where the fifteenth-century c\/del Bisbe facade contains a medallion portraying Sant Jordi (St George, patron saint of Catalunya). Inside, there's a beautiful first-floor cloister, the intricately worked chapel and salon of Sant Jordi as well as an upper courtyard planted with orange trees. You can visit the interior on a one-hour guided tour on alternate Sundays (only one or two tours each day are in English), while the Generalitat is also open on public holidays, particularly April 23 \u2013 the **Dia de Sant Jordi** (St George's Day). A nationalist holiday in Catalunya, this is also a local Valentine's Day, when it's traditional to exchange books and roses.\n\nPalau de la Generalitat\n\n## Pla\u00e7a de Sant Jaume\n\n Jaume I.MAP\n\nThe spacious square at the end of the main c\/de Ferran is at the heart of city and regional government business, and the traditional place for demonstrations and local festivals. Whistle-happy local police try to keep things moving, while taxis and bike-tour groups weave between the pedestrians.\n\n## Ajuntament de Barcelona\n\nPl. de Sant Jaume Jaume I 934 027 000.Public admitted Sun 10am\u20131.30pm, entrance on c\/Font de Sant Miquel.Free.MAP\n\nOn the south side of Pla\u00e7a de Sant Jaume stands Barcelona's city hall. On Sundays you're allowed into the building for a self-guided tour around the splendid marble galleries and staircases. The highlights are the magnificent fourteenth-century council chamber, known as the **Sal\u00f3 de Cent** , and the dramatic historical murals in the **Sal\u00f3 de les Cr\u00f2niques** (Hall of Chronicles).\n\n## Palau Mox\u00f3\n\nPl. de Sant Just 4 Jaume I 933 152 238, palaumoxo.com.Guided tours for groups by reservation only.\u20ac13.MAP\n\nThe Palau Mox\u00f3 has been in the hands of the same family since 1770, which makes it unique in Barcelona \u2013 especially since most other palatial Baroque residences were destroyed during the Civil War. You'll see grand salons and intimate chambers on the weekly guided tours, while regular concerts provide another taste of the noble life.\n\n## Pla\u00e7a Reial\n\n Liceu.MAP\n\nThe elegant Pla\u00e7a Reial \u2013 hidden behind an archway off the Ramblas \u2013 is studded with palm trees and decorated iron lamps (designed by the young Antoni Gaud\u00ed), and bordered by pastel-coloured arcaded buildings. Sitting in the square certainly puts you in mixed company \u2013 buskers, eccentrics, tramps and bemused visitors \u2013 though most of the really unsavoury characters have been driven off over the years and predatory waiters are usually the biggest nuisance these days. Don't expect to see too many locals until night falls, when the surrounding bars come into their own. Passing through on a Sunday morning, look in on the **coin and stamp market** (10am\u20132pm).\n\nPla\u00e7a Reial\n\n## Carrer d'Aviny\u00f3\n\n Liceu.MAP\n\nCarrer d'Aviny\u00f3, running south from c\/de Ferran towards the harbour, cuts through the most atmospheric part of the southern Barri G\u00f2tic. Formerly a red-light district, it still looks the part \u2013 lined with dark overhanging buildings \u2013 but the funky caf\u00e9s, streetwear shops and boutiques tell the story of its creeping gentrification. A few rough edges still show, particularly around **Pla\u00e7a George Orwell** , a favoured hangout for locals with its cheap caf\u00e9s, restaurants and bars, some of which offer seating on the lively square.\n\n## La Merc\u00e8\n\n Drassanes.MAP\n\nIn the eighteenth century, the harbourside neighbourhood known as La Merc\u00e8 was home to the nobles and merchants enriched by Barcelona's maritime trade. Most moved north to the more fashionable Eixample later in the nineteenth century, and since then Carrer de la Merc\u00e8 and surrounding streets (particularly Ample, d'en Gign\u00e0s and Regomir) have been home to a series of old-style taverns known as _tascas_ or _bodegas_ \u2013 a glass of wine from the barrel and a plate of tapas here is one of the Old Town's more authentic experiences.\n\nAt Pla\u00e7a de la Merc\u00e8, the **Esgl\u00e9sia de la Merc\u00e8** is the focus of the city's biggest annual bash, the Festes de la Merc\u00e8 every September, dedicated to the co-patroness of Barcelona, whose image is paraded from the church. It's an excuse for a week of partying, parades, special events and concerts, culminating in spectacular pyrotechnics along the seafront.\n\n< Back to Barri G\u00f2tic\n\n## Shops\n\n### Almacenes del Pilar\n\nC\/Boqueria 43 Liceu www.almacenesdelpilar.com.Mon\u2013Sat 10am\u20132pm & 4.30\u20138pm; closed Aug.MAP\n\nA world of frills, lace, cloth and material used in the making of Spain's traditional regional costumes. You can pick up a decorated fan for just a few euros, though quality items go for a lot more.\n\n### L'Arca\n\nC\/Banys Nous 20 Liceu www.larcadelavia.com.Mon\u2013Fri 10am\u20132pm & 5\u20138pm, Sat 11am\u20132pm; closed Aug.MAP\n\nCatalan brides used to fill up their nuptial trunk ( _l'arca_ ) with embroidered linen and lace, and this shop is a treasure-trove of vintage and antique textiles. Period costumes can be hired or purchased as well \u2013 one of Kate Winslet's _Titanic_ costumes came from here.\n\n### Artesania Catalunya\n\nC\/Banys Nous 11 Liceu www.artesania-catalunya.com.Mon\u2013Sat 10am\u20138pm, Sun & hols 10am\u20132pm.MAP\n\nIt's always worth a look in the showroom of the arts and crafts promotion board. Exhibitions change but most of the work is contemporary in style, from basketwork to glassware, though traditional methods are still very much encouraged.\n\n### Cerer\u00eda Subir\u00e0\n\nBxda. Llibreteria 7 Jaume I.Mon\u2013Thurs 9am\u20131.30pm & 4\u20138pm, Fri 9.30am\u20138pm, Sat 10am\u20138pm.MAP\n\nBarcelona's oldest shop (it's been here since 1760) boasts a beautiful interior, selling unique handcrafted candles.\n\n### Espai Drap Art\n\nC\/Groc 1 Jaume I www.drapart.net.Tues\u2013Fri 11am\u20132pm & 5\u20138pm, Sat 6\u20139pm.MAP\n\nThe Drap Art creative recycling organisation has a shop and exhibition space for artists to show their wildly inventive wares, from trash bangles to tin bags.\n\n### Formatgeria La Seu\n\nC\/Daguer\u00eda 16 Jaume I www.formatgerialaseu.com.Tues\u2013Thurs 10am\u20132pm & 5\u20138pm, Fri & Sat 10am\u20133.30pm & 5\u20138pm; closed Aug.MAP\n\nSells the best farmhouse cheeses from independent producers all over Spain. Chatty Scottish owner Katherine is usually on hand to advise, and you can try before you buy with a \u20ac3 tasting plate \u2013 ask about the \"formatgelat\", a cheese-ice cream fusion that's unique to the shop.\n\n### Formista\n\nC\/Sagristans 9 Jaume I www.formista.com.Mon\u2013Sat noon\u20139pm.MAP\n\nGallery-shop hybrid selling unique handmade objects by international designers and artists. There are sleek leather handbags alongside porcelain jewellery, hand-printed textiles and more.\n\n### Herborista del Rei\n\nC\/del Vidre 1 Liceu 933 180 512, herboristadelrei.com.Tues\u2013Fri 4\u20138pm, Sat 10am\u20138pm.MAP\n\nA renowned early nineteenth-century herbalist's shop, tucked off Pla\u00e7a Reial, which stocks more than 250 medicinal herbs designed to combat all complaints.\n\n### La Manual Alpargatera\n\nC\/d'Aviny\u00f3 7 Liceu www.lamanualalpargatera.com.Mon\u2013Fri 9.30am\u20131.30pm & 4.30\u20138pm, Sat 10am\u20131.30pm & 4.30\u20138pm.MAP\n\nIn this traditional workshop they make and sell _alpargatas_ (espadrilles) to order, as well as producing other straw, rope and basket work.\n\n### El Mercadillo\n\nC\/Portaferrissa 17 Liceu.Mon\u2013Sat 11am\u20138.30pm.MAP\n\nThe camel at the entrance marks this hippy-dippy indoor street market of shops and stalls selling T-shirts, skate-wear, vintage gear and jewellery.\n\n### Papabubble\n\nC\/Ample 28 Jaume I www.papabubble.com.Mon\u2013Fri 10am\u20132pm & 4\u20138.30pm, Sat 10am\u20138.30pm, Sun 11am\u20137.30pm; closed Aug.MAP\n\nGroovy young things roll out home-made candy to a chill-out soundtrack. Come and watch them at work, sample a sweet, and take home a gorgeously wrapped gift.\n\n## Caf\u00e9s\n\n### Bar del Pi\n\nPl. Sant Josep Oriol 1 Liceu 933 022 123.Tues\u2013Fri 9am\u201311pm, Sat 9.30am\u201311pm, Sun 10am\u201310pm.MAP\n\nBest known for its terrace tables on one of Barcelona's prettiest squares. Linger over drinks and sandwiches as the Old Town reveals its charms, especially during the weekend artists' market.\n\n### Caelum\n\nC\/Palla 8 Liceu 933 026 993, www.caelumbarcelona.com.Mon\u2013Thurs 10am\u20138.30pm, Fri & Sat 10.30am\u201311.30pm, Sun 11.30am\u20138.30pm; open afternoons in Aug.MAP\n\nThe lovingly packaged confections in this upscale caf\u00e9-deli (the name is Latin for \"heaven\") are made in convents and monasteries across Spain. Choose from marzipan sweets from Seville, Benedictine preserves or Cistercian cookies.\n\n### Caj Chai\n\nC\/Sant Dom\u00e8nec del Call 12 Liceu 933 019 592, www.cajchai.com.Mon 3\u201310pm, Tues\u2013Sun 10.30am\u201310pm.MAP\n\nThis refined back-street boudoir offers a menu of painstakingly prepared teas, from Moroccan mint to organic Nepalese _oolong_ , plus brownies, _baklava_ and sandwiches.\n\n## Restaurants and tapas bars\n\n### Bodega La Plata\n\nC\/de la Merc\u00e8 28 Drassanes 933 151 009.Mon\u2013Sat 9am\u20133pm & 6.30\u201311pm.MAP\n\nA classic taste of the Old Town, with a marble tapas counter open to the street (anchovies are the speciality) and dirt-cheap wine straight from the barrel.\n\n### Caf\u00e9 de l'Acad\u00e8mia\n\nC\/Lled\u00f3 1 Jaume I 933 198 253.Mon\u2013Fri 1.30\u20134pm & 8.30\u201311.30pm; closed 2 weeks in Aug.MAP\n\nGreat for a date or a lazy lunch, with creative Catalan dishes served in a romantic stone-flagged restaurant or outside in the medieval square. Expect classy grills, fresh fish, rice dishes, seasonal game and a taste of local favourites like salt cod, wild mushrooms or grilled veg. Prices are pretty reasonable (mains \u20ac11\u201318) and it's always busy, so dinner reservations are essential. A no-choice _men\u00fa del dia_ is a bargain for the quality (and it's even cheaper eaten at the bar).\n\n### Ginger\n\nC\/Palma Sant Just 1 Jaume I 933 105 309, www.ginger.cat.Tues\u2013Sat 7.30pm\u20133am; closed 2 weeks in Aug.MAP\n\nCocktails and fancy tapas in a slickly updated 1970s-style setting. It's a world away from _patatas bravas_ and battered squid \u2013 think roast duck vinaigrette, stuffed aubergine rolls, tuna tartare and vegetarian satay for around \u20ac6.50\u20139 a pop.\n\n### Koy Shunka\n\nC\/Copons 7 Jaume I 934 127 939, www.koyshunka.com.Tues\u2013Sat 1.30\u20133pm & 8.30\u201311.30pm, Sun 1.30\u20133pm.MAP\n\nThe city's hottest Japanese chef, Hideki Matsuhisa, has branched out from his original _Shunka_ restaurant with a rather more hip, nearby sister joint, where peerless sushi and dishes like grilled Wagyu beef and roast black cod await. With pricey rice rolls, mains around \u20ac20, and \u20ac82 and \u20ac118 tasting menus, it's a more rarefied experience all round, and you'll definitely need to book.\n\n### Matsuri\n\nPl. Regomir 1 Jaume I 932 681 535, www.matsuri-restaurante.com.Daily 8pm\u2013midnight.MAP\n\nA soothing place for creative Southeast-Asian cuisine, with a large menu concentrating on Thai-style noodles, salads and curries, as well as sushi, sashimi and tempura dishes. A daily specials list has some more unusual choices, and staff offer friendly advice. Around \u20ac30 a head.\n\n### El Sal\u00f3n\n\nC\/l'Hostal d'en Sol 6\u20138 Jaume I 933 152 159, www.elsalon.es.Mon\u2013Sat 8pm\u2013midnight.MAP\n\nIt's easy to fall for the cosy charms of _El Sal\u00f3n_ , with its candlelit tables in a Gothic dining room and summer terrace in the nearby square. The contemporary Mediterranean menu changes every few months, with inventive salads giving way to things like grilled aubergine, pepper and onion served with goat's cheese, or lamb with mustard and honey sauce. Salads and starters are from \u20ac8, with most mains in the range of \u20ac10 to \u20ac16.\n\n### Sensi Tapas\n\nC\/Ample 26 Jaume I 932 956 588, www.sensi.es.Daily 7pm\u2013midnight.MAP\n\nBest to make reservations as this intimate space in stone and dark wood and splashes of red quickly fills with diners looking for tapas with an exotic spin. There are impeccably executed classics like buttery _patatas bravas_ , but the stars of the show, such as the tender Iberian pork _tataki_ , take their cues from further afield (\u20ac5\u201312).\n\n### Shunka\n\nC\/Sagristans 5 Jaume I 934 124 991.Tues\u2013Sun 1.30\u20133.30pm & 8.30\u201311.30pm; closed 4 weeks in Aug & Sept.MAP\n\nLocals think this is the best Japanese restaurant in the Old Town \u2013 it's certainly always busy, so advance reservations are essential, though you might strike lucky if you're prepared to eat early or late. The open kitchen and bustling staff are half the show, while the food \u2013 sushi to udon noodles \u2013 is really good. Around \u20ac40.\n\n### Taller de Tapas\n\nPl. Sant Josep Oriol 9 Liceu 933 018 020, www.tallerdetapas.com.Daily noon\u2013midnight (Fri & Sat until 1am).MAP\n\nThe fashionable \"tapas workshop\" sucks in tourists with its pretty location by the church of Santa Mar\u00eda del Pi. There's a year-round outdoor terrace, while the open kitchen turns out reliable, market-fresh dishes, with fish a speciality at dinner, from grilled langoustine to seared tuna (most tapas \u20ac4\u201312). There are other branches around town, (including one at c\/Argenteria 51 in the Born), though this was the first.\n\n### Venus Delicatessen\n\nC\/d'Aviny\u00f3 25 Liceu 936 760 315.Mon\u2013Fri 8.30am\u20131am, Sat & Sun 10am\u20132am.MAP\n\nNot a deli, despite the name, but it's a handy place serving Mediterranean bistro cuisine throughout the day and night. It's also good for vegetarians, with things like lasagne, couscous, moussaka and salads, mostly meat-free and costing around \u20ac7\u201310.\n\n## Bars\n\n### L'Ascensor\n\nC\/Bellafila 3 Jaume I 933 185 347.Daily 6pm\u20132am.MAP\n\nSliding antique wooden elevator doors signal the entrance to \"The Lift\", but it's no theme bar \u2013 just an easy-going local hangout, great for a late-night drink.\n\n### La Cerveteca\n\nC\/d'en Gign\u00e0s 25 Jaume I 933 150 407, www.lacerveteca.com.Mon\u2013Thurs & Sun 6\u201311pm, Fri & Sat 6pm\u2013midnight.MAP\n\n _La Cerveteca_ offers the city's best world beer selection, available to drink in or take out. Brews are taken seriously here, but it's not a beard-and-sandals beer-fest, more a modern art, Muddy Waters, Latino-beat kind of place for ale enthusiasts.\n\n### Glaciar\n\nPl. Reial 3 Liceu 933 021 163.Mon\u2013Thurs noon\u20132.30am, Fri & Sat noon\u20133am, Sun 11am\u20132.30am.MAP\n\nAt this traditional Barcelona meeting point the terrace seating is packed most sunny evenings and at weekends, and the comings and goings in the Old Town's funkiest square are half the entertainment.\n\n### Milk\n\nC\/d'en Gign\u00e0s 21 Jaume I 932 680 922, www.milkbarcelona.com.Daily 9am\u20132am.MAP\n\nIrish-owned bar and bistro that's carved a real niche as a welcoming neighbourhood hangout. Decor, they say, is that of a \"millionaire's drawing room\", with its sofas, cushions and antique chandeliers. Get there early for the famously relaxed brunch (daily 9am\u20134.30pm), or there's dinner and cocktails every night to a funky soundtrack.\n\n### Pipa Club\n\nPl. Reial 3 Liceu 933 024 732, www.bpipaclub.com.Daily 10pm\u20133am.MAP\n\nHistorically a pipe-smoker's haunt, it's a wood-panelled, jazzy, late-night kind of place \u2013 ring the bell to be let in and make your way up the stairs.\n\n### Schilling\n\nC\/de Ferran 23 Liceu 933 176 787, www.cafeschilling.com.Daily 10am\u20133am.MAP\n\nSomething of a haven on this heavily touristed drag, _Schilling_ has a certain European \"grand-caf\u00e9\" style, with its high ceilings, big windows and upmarket feel. It has a loyal gay following, but it's a mixed, chilled place to meet friends, grab a _copa_ and move on.\n\n### Zim\n\nC\/Daguer\u00eda 20 Jaume I.Mon\u2013Sat 6\u201311pm; closed Aug.MAP\n\nThe owner of the adjacent Formatgeria La Seu (cheese shop) offers up this tiny, hole-in-the-wall tasting bar for selected wines from boutique producers. It can be a real squeeze, and hours are somewhat flexible, but for a reviving glass or two accompanied by farmhouse cheese, cured meat and artisan-made bread, you can't beat it.\n\n## Clubs\n\n### Harlem Jazz Club\n\nC\/Comtessa de Sobradiel 8 Jaume I 933 100 755, www.harlemjazzclub.es.Closed Aug.MAP\n\nFor many years _the_ hot place for jazz, where every style gets an airing, from African and Gypsy to flamenco and fusion. Live music Tues\u2013Sun at 10.30pm and midnight (weekends 11.30pm & 2am). Entry \u20ac5\u201310, depending on the night and the act.\n\n### Jamboree\n\nPl. Reial 17 Liceu 933 191 789, www.masimas.com.MAP\n\nThey don't get the big jazz names here that they used to, but the nightly gigs (at 8pm & 10pm; from \u20ac12) still pull in the crowds, while the wild Monday night WTF jazz, funk and hip-hop jam session (from 8pm; \u20ac5) is a city fixture. Stay on for the club, which kicks in after midnight (entry \u20ac10) and you get funky sounds and retro pop, rock and disco until 5am.\n\n### La Macarena\n\nC\/Nou de Sant Francesc 5 Drassanes www.macarenaclub.com.Mon\u2013Thurs & Sun 11.45pm\u20135am, Fri & Sat 11.45pm\u20136am.MAP\n\nOnce a place where flamenco tunes were offered up to La Macarena, the Virgin of Seville \u2013 now, a heaving temple to all things electro. Entry free until around 1am, then from \u20ac5.\n\n### Sidecar\n\nPl. Reial 7 Liceu 933 177 666, www.sidecarfactoryclub.com.Mon\u2013Thurs 7pm\u20135am, Fri & Sat 7pm\u20136am.MAP\n\nHip music club \u2013 pronounced \"See-day-car\" \u2013 with gigs (usually at 10.30pm) and DJs (from 12.30am) that champion rock, indie, roots and fusion acts, so a good place to check out the latest Catalan hip-hop, rumba and flamenco sounds. Entry \u20ac7\u201310, though some gigs up to \u20ac20.\n\n### Tarantos\n\nPl. Reial 17 Liceu 933 191 789, www.masimas.com.MAP\n\n _Jamboree_ 's sister club is the place for short, exuberant flamenco tasters, where young singers, dancers and guitarists perform nightly at 8.30pm, 9.30pm & 10.30pm (with extra sessions in July and August). Purists are a bit sniffy, but it's a great introduction to the scene. Entry \u20ac10.\n\n< Back to Barri G\u00f2tic\n\n# Port Vell and Barceloneta\n\nSights\n\nRestaurants and tapas bars\n\nBars\n\nBarcelona has an urban waterfront that merges seamlessly with the Old Town, providing an easy escape from the claustrophobic medieval streets. The harbour at the bottom of the Ramblas has been thoroughly overhauled in recent years and Port Vell (Old Port), as it's now known, presents a series of heavyweight tourist attractions, from sightseeing boats and maritime museum to aquarium and IMAX cinema. By way of contrast, Barceloneta \u2013 the wedge of land to the east, backing the marina \u2013 retains its eighteenth-century character, and the former fishing quarter is still the most popular place to come and eat paella, fish and seafood. Metro Drassanes, at the bottom of the Ramblas, is the best starting point for Port Vell; Barceloneta has its own metro station.\n\nzoom left\n\nzoom right\n\n## Mirador de Col\u00f3n\n\nPl. Portal de la Pau Drassanes 933 025 224.March\u2013Sept daily 8.30am\u20138.30pm; Oct\u2013Feb daily 8.30am\u20137.30pm.\u20ac4.MAP\n\nThe monument at the foot of the Ramblas commemorates the visit made by Christopher Columbus in June 1493, when the navigator received a royal welcome in Barcelona. Columbus tops a grandiose iron column, 52m high, guarded by lions, and you can ride the lift up to the panoramic viewing platform at Columbus's feet. Meanwhile, from the quayside in front of the Columbus monument, Las Golondrinas sightseeing boats depart on regular trips throughout the year around the inner harbour.\n\nMirador de Col\u00f3n\n\n## Museu Mar\u00edtim\n\nAv. de les Drassanes Drassanes 933 429 920, www.mmb.cat.Daily 10am\u20138pm.MAP\n\nBarcelona's medieval shipyards, or Drassanes, were in continuous use \u2013 fitting and arming Catalunya's war fleet or trading vessels \u2013 until well into the eighteenth century. The stone-vaulted buildings make a fitting home for the city's excellent Maritime Museum, though a large-scale renovation project means access is restricted, probably until sometime in 2015. In the meantime, there's cut-price admission (free on Sun after 3pm) to one or two changing exhibitions that draw on the museum collections, and continued access to museum activities, shop and caf\u00e9. The ticket also includes a short tour of the _Santa Eul\u00e0lia_ , a vintage three-masted schooner moored down on the Moll de la Fusta harbourside (closed Mon, check hours at the museum).\n\nInside the Museu Mar\u00edtim\n\n## L'Aqu\u00e0rium\n\nMoll d'Espanya Drassanes 932 217 474, www.aquariumbcn.com.Daily: July & Aug 9.30am\u201311pm; June & Sept 9.30am\u20139.30pm; Oct\u2013May 9.30am\u20139pm; until 9.30pm at weekends.\u20ac20.MAP\n\nPort Vell's high-profile aquarium drags in families and school parties throughout the year to see \"a magical world, full of mystery\". Or, to be more precise, to see 11,000 fish and sea creatures in 35 themed tanks representing underwater caves, tropical reefs and other maritime habitats. It's vastly overpriced, and despite the claims of excellence it offers few new experiences, save perhaps the 80-metre-long walk-through underwater tunnel, which brings you face to face with gliding rays and cruising sharks.\n\n## Marem\u00e0gnum\n\nMoll d'Espanya Drassanes.Daily 10am\u201310pm.MAP\n\nFrom near the Columbus statue, the wooden Rambla de Mar swing bridge strides across the harbour to the Marem\u00e0gnum mall and leisure centre on Moll d'Espanya. It's a typically bold piece of Catalan design, its soaring glass lines tempered by the undulating wooden walkways that provide scintillating views back across the harbour to the city. Inside are two floors of gift shops and boutiques, plus a range of caf\u00e9s and fast-food outlets.\n\n## IMAX Port Vell\n\nMoll d'Espanya.Drassanes 932 251 111, www.imaxportvell.com.Screenings 11am\u201310pm, later at weekends.Tickets \u20ac9.75 or \u20ac13.70 for 2 films.MAP\n\nBarcelona's IMAX theatre has three screens showing films virtually hourly in 3D or giant-screen format. The themes are familiar \u2013 forces of nature, heroic exploration, alien adventure, etc, and some Hollywood 3D blockbusters are shown, but films are in Spanish or Catalan only.\n\n### It takes two\n\nYou want a seafood paella or an _arr\u00f2s negre_ (black rice, with squid ink), or maybe a garlicky _fideu\u00e0_ (noodles with seafood) \u2013 of course you do. Problem is, you're on your own and virtually every restaurant that offers these classic Barcelona dishes does so for a minimum of two people (often you don't find out until you examine the menu's small print). Solution? Ask the waiter upfront, as sometimes the kitchen will oblige single diners, or look for the dishes on a _men\u00fa del dia_ (especially on Thurs, traditionally rice day), when there should be no minimum. Probably best not to grab a stranger off the street to share a paella, however desperate you are.\n\n## Museu d'Hist\u00f2ria de Catalunya\n\nPalau de Mar, Pl. de Pau Vila 3 Barceloneta 932 254 700, www.mhcat.net.Tues\u2013Sat 10am\u20137pm (Wed until 8pm), Sun & hols 10am\u20132.30pm.\u20ac4.50; last Tues of month free.MAP\n\nA dramatic harbourside warehouse conversion contains a museum tracing the history of Catalunya from the Stone Age to the present day. Poke around the interior of a Roman grain ship or compare the rival nineteenth-century architectural plans for the Eixample. The top-floor caf\u00e9-bar boasts a glorious view of the city from its terrace \u2013 no museum ticket needed \u2013 while the flash restaurants in the **Palau de Mar** arcade below overlook the busy marina.\n\n## Barceloneta\n\n Barceloneta.MAP\n\nThere's no finer place for lunch on a sunny day than Barceloneta, an eighteenth-century neighbourhood of tightly packed gridded streets with bustling harbour on one side and sandy beach on the other. There's a local market, the **Mercat de la Barceloneta** (Mon\u2013Thurs & Sat 7am\u20133pm, Fri 7am\u20138pm), with a couple of excellent bars and restaurants, while Barceloneta's famous seafood restaurants are found across the neighbourhood, but most characteristically lined along the harbourside Passeig Joan de Borb\u00f3.\n\n### The cross-harbour cable car\n\nThe most thrilling ride in the city is across the harbour on the Trasbordador Aeri, or cable car, which sweeps from the Sant Sebasti\u00e0 tower at the foot of Barceloneta to Montju\u00efc. Departures are every 15min (daily: March\u2013May & Sept\u2013Oct 11am\u20137pm; June\u2013Aug 11am\u20138pm; Nov\u2013Feb 11am\u20135.30pm; \u20ac11 one-way, \u20ac16.50 return), but expect queues in summer and at weekends as the cars only carry about twenty people at a time.\n\n## Platja de Sant Sebasti\u00e0\n\n Barceloneta.MAP\n\nBarceloneta's beach \u2013 the first in a series of sandy city beaches \u2013 curves from the flanks of the neighbourhood, past the swimming pools of the Club Nataci\u00f3 and out to the landmark sail-shaped _W Barcelona_ hotel. Closer to the Barceloneta end there are beach bars, outdoor caf\u00e9s and sculptures, while a double row of palms backs the esplanade that runs above the sands as far as the Port Ol\u00edmpic (a 15min walk). Bladers, skaters, joggers and cyclists have one of the Med's best views for company.\n\nThe cross-harbour cable car\n\n< Back to Port Vell and Barceloneta\n\n## Restaurants and tapas bars\n\n### Can Maj\u00f3\n\nC\/Almirall Aixada 23 Barceloneta 932 215 455, www.canmajo.es.Tues\u2013Sat 1\u20134pm & 8\u201311.30pm, Sun 1\u20134pm.MAP\n\nThis ticks all the boxes for a quality seafood meal, with a bonus of a lovely summer _terrassa_ by the beach promenade where you can tuck into wonderful rice dishes, _fideu\u00e0_ (noodles with seafood), _suquet_ (fish stew) or grilled fish. Expect to spend \u20ac40\u201350 a head.\n\n### Can Ma\u00f1o\n\nC\/Baluard 12 Barceloneta 933 193 082.Mon 8\u201311pm, Tues\u2013Fri 8.15\u201311am, 12.15\u20134pm & 8\u201311pm, Sat 8.15\u201311am & 12.45\u20134pm; closed Aug.MAP\n\nThere's rarely a tourist in sight in this classic old-fashioned diner. Basically, your choice is fried or grilled fish, supplemented by a few daily seafood specials and basic meat dishes. It's an authentic, no-frills experience that's likely to cost you less than \u20ac15 a head.\n\n### Cova Fumada\n\nC\/Baluard 56 Barceloneta 932 214 061.Mon\u2013Wed 9am\u20133.20pm, Thurs & Fri 9am\u20133.20pm & 6\u20138.20pm, Sat 9am\u20131.30pm; closed Aug.MAP\n\nBehind brown wooden doors on the market square (there's no sign) is this rough-and-ready tavern with battered marble tables and antique barrels. That it's always packed is a testament to the quality of the market-fresh tapas (\u20ac2\u201310), from the griddled prawns to the _bomba_ (spicy potato meatball).\n\n### Jai-Ca\n\nC\/Ginebra 9 & 13 Barceloneta 932 683 265.No.9 Mon & Wed\u2013Sun noon\u2013midnight; No.13 Tues\u2013Sun 9am\u2013midnight.MAP\n\nA great choice for tapas (dishes up to \u20ac10), with bundles of razor clams, stuffed mussels, crisp baby squid and other seafood platters.\n\n### Kaiku\n\nPg. de Joan de Borb\u00f3 74 Barceloneta 932 219 082.May\u2013Sept Tues\u2013Sat 1.30\u20133.30pm & 7\u201310pm, Sun 1.30\u20133.30pm; Oct\u2013April Tues\u2013Sun 1.30\u20133.30pm; closed Aug.MAP\n\nGreat lunch-only joint on the seafront terrace for fantastic, Basque-influenced seafood meals (dishes \u20ac8\u201317, cheaper set menu served Tues\u2013Fri). Tastes are out-of-the-ordinary \u2013 think smoked vegetable rice with mushrooms and rocket, or steamed mussels with thyme \u2013 and you'll need to book.\n\n### Somorrostro\n\nC\/Sant Carles 11 Barceloneta 932 250 010, www.restaurantesomorrostro.com.Mon & Thurs\u2013Sat 8pm\u201311.30pm, Sun 2\u20134pm.MAP\n\nCreative \"boat-to-table\" Mediterranean cuisine with a mission: preserving and promoting the livelihood and traditions of this fishing community. The three-course evening menu (\u20ac17; available to the first 15 customers) is great value, and it's good fun watching the open kitchen in action.\n\n### Vaso de Oro\n\nC\/Balboa 6 Barceloneta 933 193 098.Daily 9am\u2013midnight.MAP\n\nAn old favourite for stand-up tapas (\u20ac4\u201315) \u2013 there's no menu, but order the _patatas bravas_ , some thick slices of fried sausage, grilled shellfish and a dollop of tuna salad and you've touched all the bases. Unusually, they also brew their own beer, light and dark.\n\n## Bars\n\n### Caf\u00e9 de Los Angelitos\n\nC\/l'Almirall Cervera 26 Barceloneta 932 211 224, www.cafedelosangelitos.es.Mon\u2013Thurs 5pm\u20132am, Fri 5pm\u20133am, Sat 3pm\u20133am, Sun 3pm\u20132am.MAP\n\nIt's easy to lose track of time at this congenial and cosy neighbourhood bar, where attentive bartenders craft classic cocktails, from martinis to manhattans (from \u20ac6), and serve up simple yet delicious tapas.\n\n### Can Paixano\n\nC\/de la Reina Cristina 7 Barceloneta 933 100 839, www.canpaixano.com.Mon\u2013Sat 9am\u201310.30pm; closed 2 weeks in Aug.MAP\n\nA must on everyone's itinerary is this counter-only joint where the drink of choice \u2013 all right, the only drink \u2013 is _cava_ by the glass or bottle. It's popular, so you may have to fight your way in.\n\n### Makamaka Beach Burger Caf\u00e9\n\nPg. de Joan de Borb\u00f3 76 Barceloneta 932 213 520.Sun\u2013Thurs noon\u20132am, Fri & Sat noon\u20132.30am).MAP\n\nYou really can't ask for more: creative cocktails and some of the city's finest burgers served on a large, beachside _terrassa_. It's Hawaii-meets-Barcelona \u2013 laidback, late night and lots of fun.\n\n< Back to Port Vell and Barceloneta\n\n# El Raval\n\nSights\n\nShops\n\nCaf\u00e9s\n\nRestaurants and tapas bars\n\nBars\n\nClubs\n\nThe Old Town area west of the Ramblas is known as El Raval (from the Arabic word for suburb), and has always formed a world apart from the nobler Gothic quarter. Traditionally a red-light area, and once notorious for its sleazy Barri Xin\u00e8s (China Town), it still has some very seedy corners (particularly south of Carrer de Sant Pau), though it's changing rapidly, notably in the \"upper Raval\" around Barcelona's contemporary art museum, MACBA. Cutting-edge galleries, designer restaurants and fashionable bars are all part of the scene these days, while an arty, affluent crowd rubs shoulders with the area's Asian and North African immigrants and the older, traditional residents. Metros Catalunya, Liceu, Drassanes and Paral.lel serve the neighbourhood.\n\nzoom left\n\nzoom right\n\n## Museu d'Art Contemporani de Barcelona (MACBA)\n\nPl. dels \u00c0ngels 1 Catalunya 934 120 810, www.macba.cat.Mon & Wed\u2013Fri 11am\u20137.30pm, Sat 10am\u20139pm, Sun & hols 10am\u20133pm; closed Tues all year.\u20ac10.MAP\n\nThe iconic contemporary art museum \u2013 with a stark main facade constructed entirely of glass \u2013 anchors the regenerated upper Raval. The collection represents the main movements in art since 1945, mainly (but not exclusively) in Catalunya and Spain, and depending on the changing exhibitions you may catch works by major names such as Joan Mir\u00f3, Antoni T\u00e0pies or Eduardo Chillida. Joan Brossa, leading light of the Catalan Dau al Set group of the 1950s, also has work here. There are free guided tours of the permanent collection (tour times vary; check website for details), and a good museum shop.\n\nMuseu d'Art Contemporani de Barcelona\n\n## Filmoteca de Catalunya\n\nPl. Salvador Segu\u00ed 1\u20139 Liceu www.filmoteca.cat.Cinema Tues\u2013Fri 5\u201310pm, Sat & Sun 4.30\u201310pm. Exhibition Hall Tues\u2013Sun 4\u20139pm.\u20ac4.MAP\n\nThe Josep Llu\u00eds Mateo-designed Filmoteca de Catalunya marks yet another step in the government's push to revitalize El Raval. Opened in early 2012, the building has two below-ground cinemas, as well as a film library, a bookshop and spaces for permanent and temporary cinema-related exhibitions. All the films are shown in their original language with Spanish or Catalan subtitles.\n\n## Centre de Cultura Contempor\u00e0nia de Barcelona (CCCB)\n\nC\/Montalegre 5 Catalunya 933 064 100, www.cccb.org.Tues\u2013Sun 11am\u20138pm.\u20ac6, free on Sun after 3pm.MAP\n\nThere's a wide range of city-related exhibitions on show at the contemporary culture centre (ranging from photography to architecture), as well as a varied cinema, concert and festival programme. The imaginatively restored building was once an infamous workhouse and asylum, and the main courtyard still retains its old tile panels and presiding statue of patron saint, Sant Jordi. At the back, the _C3_ caf\u00e9-bar makes the most of its _terrassa_ overlooking the modern square joining the CCCB to MACBA.\n\nCentre de Cultura Contempor\u00e0nia de Barcelona\n\n## Pla\u00e7a de Vicen\u00e7 Martorell\n\n Catalunya.MAP\n\nThe Raval's nicest traffic-free square lies just a few minutes' walk from MACBA. The small playground here is well used by local families, and the arcaded square features a first-rate caf\u00e9, the _Kasparo_ \u2013 a real find if you're looking for a break from sightseeing. Meanwhile, around the corner are several other caf\u00e9s, while the narrow **Carrer del Bonsucc\u00e9s** , **Carrer Sitges** and **Carrer dels Tallers** house a concentrated selection of the city's best independent music stores and urban and streetwear shops.\n\n### The beat from the street\n\nThe Barcelona sound \u2013 _mestiza_ \u2013 is a cross-cultural musical fusion whose heartland is the immigrant melting-pot of the Raval. Parisian-born Barcelona resident Manu Chao kick-started the whole genre, but check out the Carrer dels Tallers music stores for the other flag-bearers \u2013 Cheb Balowski (Algerian\u2013Catalan fusion), Ojos de Brujo (Catalan flamenco and rumba), GoLem System (dub\/reggae) and Macaco (rumba, raga, hip-hop).\n\n## Hospital de la Santa Creu\n\nEntrances on c\/del Carme and c\/de l'Hospital Liceu.Daily 10am\u2013dusk.Free.La Capella exhibition information available on  www.lacapella.bcn.cat,  934 427 171. MAP\n\nThe neighbourhood's most historic relic is the Gothic hospital complex founded in 1402. After the hospital shifted location in 1930, the buildings were subsequently converted to cultural and educational use (including the Catalan national library), and visitors and students now wander freely through the charming medieval cloistered garden. Inside the c\/del Carme entrance (on the right) you can see some superb seventeenth-century decorative tiles, while opposite is a remarkable eighteenth-century anatomical theatre inside the **Reial Acad\u00e8mia de Medicina** (open Wed only 10am\u2013noon; free; ring the bell). There's also the rather nice open-air _El Jard\u00ed_ caf\u00e9 at the c\/de l'Hospital side, while the hospital's former chapel, **La Capella** (entered separately from c\/de l'Hospital), is an exhibition space for new contemporary artists.\n\n## Rambla del Raval\n\n Liceu.MAP\n\nThe most obvious manifestation of the changing character of El Raval is the palm-lined boulevard that was gouged through former tenements and alleys, providing a huge pedestrianized space between c\/de l'Hospital and c\/de Sant Pau. The _rambla_ has a distinct character that's all its own, mixing kebab joints, phone shops and grocery stores with an increasing number of fashionable caf\u00e9s and bars. Signature building, halfway down, is the glow-in-the-dark, designer _Barcel\u00f3 Raval_ hotel, while kids find it hard to resist a clamber on the bulbous cat sculpture. A weekend street market (selling everything from samosas to hammocks) adds a bit more character.\n\nJust off the top of the _rambla_ , **Carrer de la Riera Baixa** is at the centre of the city's secondhand\/vintage clothing scene. A dozen funky little independent clothes shops provide the scope for an hour's browsing.\n\nCat scultpture on the Rambla de Raval\n\n## Palau G\u00fcell\n\nC\/Nou de la Rambla 3\u20135 Liceu 934 725 775, www.palauguell.cat.Tues\u2013Sun & hols 10am\u20138pm (Nov\u2013March until 5.30pm), last admission 1hr before closing.\u20ac12, free first Sun of the month.MAP\n\nEl Raval's outstanding building is the extraordinary townhouse designed (1886\u201390) by the young Antoni Gaud\u00ed for a wealthy industrialist. At a time when other architects sought to conceal the iron supports within buildings, Gaud\u00ed displayed them instead as decorative features. Columns, arches and ceilings are all shaped and twisted in an elaborate style that was to become the hallmark of Gaud\u00ed's later works, while the roof terrace culminates in a fantastical series of tiled chimneys. Visitor numbers are limited \u2013 expect to queue or to receive a time-specific ticket.\n\nPalau G\u00fcell\n\n## Esgl\u00e9sia de Sant Pau del Camp\n\nC\/de Sant Pau 101 Paral.lel 934 410 001.Mon\u2013Sat 10am\u20131.30pm & 4\u20137.30pm.Admission to cloister \u20ac3.MAP\n\nThe unusual name of the church of Sant Pau del Camp (St Paul of the Field) is a graphic reminder that it once stood in open countryside beyond the city walls. Sant Pau was a Benedictine foundation of the tenth century, and above the main entrance are primitive thirteenth-century carvings of fish, birds and faces, while others adorn the charming cloister.\n\n### High society hotel\n\nSome of the most influential names in Catalan _modernista_ design came together to transform the dowdy nineteenth-century **Hotel Espa\u00f1a** (c\/de Sant Pau 9\u201311,  www.hotelespanya.com; MAP, ) into one of the city's most lavish addresses. With a gloriously tiled dining room, an amazing marble fireplace and a mural-clad ballroom, the hotel was the fashionable sensation of its day. A contemporary restoration has done a wonderful job of showing off the classy interior \u2013 there are guided tours (\u20ac5) twice a day, Mon\u2013Fri; ask at the reception.\n\n## Mercat de Sant Antoni\n\nC\/del Comte d'Urgell 1 Sant Antoni www.mercatsbcn.cat.Mon\u2013Thurs 7am\u20132.30pm & 5\u20138.30pm, Fri & Sat 7am\u20138.30pm.MAP\n\nThe neighbourhood's impressive nineteenth-century produce market is another that's being entirely remodelled but a provisional market building continues in operation in the meantime (on Ronda Sant Antoni). Come on Sunday and there's a **book and coin market** (9am\u20132pm) here instead, with collectors and enthusiasts arriving early to pick through the best bargains. The traditional bolt hole is _Els Tres Tombs_ , the restaurant-bar on the corner of Ronda de Sant Antoni, open from 6am until late for a good-natured mix of locals, market traders and tourists.\n\n< Back to El Raval\n\n## Shops\n\n### La Central del Raval\n\nC\/d'Elisabets 6 Catalunya www.lacentral.com.Mon\u2013Fri 9.30am\u20139pm, Sat 10am\u20139pm.MAP\n\nOccupying a unique space in the former Misericordia chapel, this is a fantastically stocked arts and humanities treasure-trove, with books piled high in every nook and cranny and a big English-language section. There are further outlets in MACBA (contemporary art museum) and MUHBA (Barcelona History Museum).\n\n### Discos Castell\u00f3\n\nC\/Tallers 7 Catalunya www.castellodiscos.es.Mon\u2013Sat 10am\u20138.30pm.MAP\n\nYou can track down pretty much anything in this well-stocked shop (it has been around since 1928), including classical recordings and pop, rock, _mestiza_ , electronica, hardcore and Spanish and Catalan sounds \u2013 and if not, other nearby specialists should do the job.\n\n### Fantastik\n\nC\/Joaquin Costa 62 Universitat www.fantastik.es.Mon\u2013Fri 11am\u20132pm & 4\u20138.30pm, Sat noon\u20139pm.MAP\n\nBeguiling gifts, crafts and covetable objects from four continents. You'll never know how you lived without them, whether it's Chinese robots, African baskets, Russian domino sets or Vietnamese kitchen scales.\n\n### Gotham\n\nC\/Lle\u00f3 28 Universitat 934 124 647, gotham-bcn.com.Mon\u2013Fri 5\u20138pm, other times by appointment only.MAP\n\nThis stylish shop is the place to come for retro (1950s to 1980s) furniture, lighting, homeware and accessories, plus original designs.\n\n### Holala! Plaza\n\nPl. Castella 2 Universitat www.holala-ibiza.com.Mon\u2013Sat 11am\u20139pm.MAP\n\nVintage heaven in a warehouse setting (up past CCCB), for denim, flying jackets, Hawaiian shirts, baseball gear and much, much more. Also check out the other Raval stores at c\/Riera Baixa 11 and c\/Tallers 73.\n\n### Lailo\n\nC\/de la Riera Baixa 20 Liceu www.lailovintage.es.Mon\u2013Fri 10.30am\u20132pm & 5\u20138pm, Sat 10.30am\u20132.30pm & 5\u20138.30pm.MAP\n\nSecondhand and vintage clothes shop with a massively wide-ranging stock. If you're serious about the vintage scene, this is your first stop \u2013 and if you don't find what you want just move on down the street to the neighbouring outlets.\n\n## Caf\u00e9s\n\n### Caf\u00e8 de les Del\u00edcies\n\nRambla de Raval 47 Liceu 934 415 714.Mon\u2013Thurs 8.30am\u201311pm, Fri 8.30am\u20132am, Sat 11am\u20132am, Sun 11am\u2013midnight.MAP\n\nOne of the first off the blocks in this revamped part of the neighbourhood, and still perhaps the best, plonking thrift-shop chairs and tables beneath exposed pipes and girders and coming up with something cute, cosy, mellow and arty. Locals love to linger here, meeting for breakfast, sandwiches and tapas to share.\n\n### Granja M. Viader\n\nC\/Xucl\u00e0 4\u20136 Catalunya 933 183 486, www.granjaviader.cat.Mon\u2013Sat 9am\u20131.15pm & 5\u20139.15pm.MAP\n\nThe oldest traditional _granja_ (milk bar) in town is a real historical survivor \u2013 it even has a pavement plaque outside for services to the city. The original owner was the inventor of \"Cacaolat\" (a popular chocolate drink), but for a taste of the old days you could also try _mel i mat\u00f3_ (curd cheese and honey) or _llet Mallorquina_ (fresh milk with cinnamon and lemon rind).\n\n### El Jard\u00ed\n\nC\/de l'Hospital 56 Liceu 933 291 550, www.eljardibarcelona.es.Mon\u2013Fri 9am\u2013midnight, Sat noon\u2013midnight, Sun 11am\u2013midnight.MAP\n\nThe \"garden bar\", hidden in the elegant courtyard of the Gothic Hospital de la Santa Creu, is a real away-from-the-bustle find \u2013 a year-round covered deck offering drinks, snacks, salads and sandwiches during the day, plus a decent lunch menu and a changing list of market-fresh tapas.\n\n### Kasparo\n\nPl. Vicen\u00e7 Martorell 4 Catalunya 933 022 072.Daily 9am\u2013midnight; closed four weeks over Dec & Jan.MAP\n\nA place to relax, in the arcaded corner of a quiet square, with outdoor seating year-round. There's muesli, Greek yoghurt and toast and jam for early birds. Later, sandwiches, tapas and assorted _platos del dia_ (dishes of the day) are on offer \u2013 things like hummus and bread, vegetable quiche, couscous or pasta.\n\n### Mendiz\u00e1bal\n\nC\/Junta de Comer\u00e7 2 Liceu.Daily 10am\u2013midnight, June\u2013Sept until 1am.MAP\n\nDon't look for a caf\u00e9 because there isn't one. Instead, this cheery stand-up counter across from the Hospital de la Santa Creu dispenses juices, shakes, beer and sandwiches to passing punters. The lucky ones grab a table over the road in the shady little square.\n\n## Restaurants and tapas bars\n\n### Biblioteca\n\nC\/Junta del Comer\u00e7 28 Liceu 934 126 261, labibliotecagourmande.com.Mon\u2013Sat 1\u20134pm & 8\u201311.30pm; closed 2 weeks in Aug.MAP\n\nOne of the finest places to sample what Barcelona tends to call \"creative cuisine\". It's a stylish operation, with an open kitchen turning out fish dishes that might be cooked Japanese- or Basque-style, lamb given the local treatment (with parsnip and turnip), or the signature dish of venison pie served with a zippy veg pur\u00e9e of the day. Meals cost from around \u20ac40, and clued-up English-speaking staff make it a hassle-free dining experience. Reservations advisable.\n\n### Dos Palillos\n\nC\/d'Elisabets 9 Catalunya 933 040 513, www.dospalillos.com.Tues & Wed 7.30\u201311.30pm, Thurs\u2013Sat 1.30\u20133.30pm & 7.30\u201311.30pm; closed 2 weeks in Jan & 3 weeks in Aug.MAP\n\nFlag-waver for Asian fusion tapas is this hipster hang-out, which offers \u00e0 la carte dim sum in the front galley bar (steamed dumplings to grilled oysters and stir-fried prawns; average spend \u20ac30) and a back-room, counter-style Asian bar where tasting menus (\u20ac60, \u20ac70 and \u20ac85, reservations required) wade their way through the highlights.\n\n### Elisabets\n\nC\/d'Elisabets 2 Catalunya 933 175 826.Mon\u2013Sat 8am\u201311pm; closed Aug.MAP\n\nReliable Catalan home cooking served at cramped tables in a jovial brick-walled dining room. Everyone piles in early for breakfast, the hearty lunch (1\u20134pm) is hard to beat for price, or you can just have tapas, sandwiches and drinks at the bar.\n\n### Mam i Teca\n\nC\/de la Lluna 4 Sant Antoni 934 413 335.Mon, Wed\u2013Fri & Sun 1\u20134pm & 8pm\u2013midnight, Sat 8.30pm\u2013midnight; closed 2 weeks in Aug.MAP\n\nAn intimate (code for very small) place for superior tapas and fine wines, run by a wine-loving gourmet. All the meat is organic, the regional cheeses are well chosen, and market-fresh ingredients go to make up things like daily pasta dishes, a platter of grilled vegetables or a serving of lamb cutlets (most dishes \u20ac6\u201312, though some up to \u20ac20). Finish with chocolate truffles or home-made ice cream. There are only three or four tables, or you can perch at the bar.\n\n### Mes\u00f3n David\n\nC\/de les Carretes 63 Paral.lel 934 415 934, www.mesondavid.com.Daily 1\u20134pm & 7.30\u201311.30pm.MAP\n\nThis down-to-earth Galician bar-restaurant is a firm favourite with neighbourhood families, who bring their kids before they can even walk. The weekday _men\u00fa del dia_ is a steal \u2013 maybe lentil broth followed by grilled trout \u2013 while traditional Galician dishes like octopus or the _combinado Gallego_ (\"ham, salami, ear\") go down well with the regulars. Lunch is around \u20ac12, otherwise most dishes \u20ac7\u201315, and there's a good-natured bang on the clog gong for anyone who leaves a tip.\n\n### Romesco\n\nC\/Sant Pau 28 Liceu 933 189 381.Mon\u2013Fri 1\u201311.30pm, Sat 1\u20134.30pm; closed Aug.MAP\n\nOld Barcelona hands talk lovingly of the _Romesco_ , and as long as you accept its limitations (dining under strip-lights, gruff waiters) you can hardly go wrong, as the most expensive thing on the menu is a \u20ac10 grilled steak and most dishes go for \u20ac6 or less. It's basic but good, with big salads, country broths and grilled veg to start, followed by tuna steak, lamb chops or grilled prawns from the market, scattered with parsley and chopped garlic. If you spend more than \u20ac15, you've probably eaten someone else's dinner as well.\n\n### Sesamo\n\nC\/Sant Antoni Abat 52 Sant Antoni 934 416 411.Tues\u2013Sun 7.30pm\u2013midnight.MAP\n\nA classy fusion tapas place offering up a vegetarian-orientated chalkboard menu of innovative dishes. Small and not-so-small plates roll out of the open kitchen \u2013 veggie-stuffed courgette rolls (a sort of Catalan sushi), slow-roast tomato tart, coconut curry or a daily risotto and pasta dish, all in the \u20ac7\u201315 range. The Catalan wines and cheeses are a high point too.\n\n### Teresa Carles\n\nC\/Jovellanos 2 Catalunya 933 171 829, www.teresacarles.com.Daily 9am\u201311.30pm.MAP\n\nStylish vegetarian and vegan cuisine served in a hip \u2013 but most certainly not \"hippie\" \u2013 space with soaring ceilings, exposed bricks walls and soft, white lighting. The lunch menu (\u20ac9.50) is a great bargain, while \u00e0 la carte offerings like artisanal pastas, a hearty seitan burger and vegan ceviche cost \u20ac9\u201312. Brunch served until 2pm.\n\n### La Ver\u00f2nica\n\nRambla de Raval 2\u20134 Liceu 933 293 303.Mon\u2013Thurs 1pm\u2013midnight, Fri & Sat noon\u20131am, Sun noon\u201312.30am; closed 2 weeks in Aug.MAP\n\nFunky, retro pizzeria _La Ver\u00f2nica_ fits right into the new-look Rambla de Raval. Loads of crispy pizzas (mostly vegetarian, between \u20ac10 and \u20ac15) and inventive salads, enjoyed by a young crowd.\n\n## Bars\n\n### Almirall\n\nC\/de Joaquin Costa 33 Universitat 933 189 917, www.casaalmirall.com.Daily 6pm\u20132am (Fri & Sat until 3am).MAP\n\nDating from 1860, Barcelona's oldest bar is a _modernista_ classic \u2013 check out the ornate doors, counter and stupendous, glittering bar. It's long been a venerated leftist hangout and because it's not too young and not too loud, it's always good for a late-night drink away from the party crowd.\n\n### La Confiter\u00eda\n\nC\/de Sant Pau 128 Paral.lel.Mon\u2013Fri 6pm\u20133am, Sat & Sun 2pm\u20133am.MAP\n\nThis old bakery and confectioner's \u2013 carved wood bar, faded tile floor, murals, antique chandeliers and mirrored cabinets \u2013 is now a popular bar and meeting point. It's out on a limb in the Raval, but the glorious interior is certainly worth a detour, and it's a handy stop-off in any case on the way to a night out in Poble Sec.\n\n### Marmalade\n\nC\/de Riera Alta 4\u20136 Sant Antoni 934 423 966, www.marmaladebarcelona.com.Mon\u2013Wed 6.30pm\u20132am, Thurs\u2013Sun 10am\u20132.30am.MAP\n\nA hugely glam facelift for the old Muebles Navarro furniture store goes for big, church-like spaces and a back-lit Art Deco bar that resembles a high altar. Cocktails, bistro meals and gourmet burgers pull in a relaxed dine-and-lounge crowd, and there's a popular weekend brunch too. If you like the style, give the more informal Barri G\u00f2tic sister bar, _Milk_ , a whirl.\n\n### Marsella\n\nC\/de Sant Pau 65 Liceu 934 427 263.Daily 10pm\u20132.30am.MAP\n\nAuthentic, atmospheric, sleaze-period bar \u2013 named after the French port of Marseilles \u2013 where absinthe is the drink of choice. It's frequented by a spirited mix of oddball locals and young trendies, all looking for a slice of the old Barri Xines.\n\n### Muy Buenas\n\nC\/del Carme 63 Liceu 645 309 671.Tues\u2013Sat 9am\u20133am.MAP\n\nArguably the Raval's nicest traditional watering hole, with a restored _modernista_ interior and eager-to-please staff. A long marble trough does duty as the bar, and the beer's pulled from antique beer taps.\n\n### Resolis\n\nC\/Riera Baixa 22 Sant Antoni 934 412 948.Mon\u2013Thurs 6pm\u20131am, Fri & Sat 6pm\u20133am.MAP\n\nA decayed, century-old bar turned into a cool hangout with decent tapas, from veggie bruschetta to steamed mussels. They didn't do much \u2013 a lick of paint, polish the panelling, patch up the brickwork \u2013 but now punters spill out of the door onto \"secondhand clothes street\" and a good time is had by all.\n\n### Zelig\n\nC\/del Carme 116 Sant Antoni www.zelig-barcelona.com.Tues\u2013Thurs & Sun 11am\u20132am, Fri & Sat 11am\u20133am.MAP\n\nThe photo-frieze on granite walls and a fully stocked cocktail bar make it very much of its _barri_ , but _Zelig_ stands out from the crowd. It offers a chatty welcome, a tendency towards 1980s sounds and a slight whiff of camp.\n\n## Clubs\n\n### Caf\u00e9 Teatre Llantiol\n\nC\/Riereta 7 Sant Antoni 933 299 009, www.llantiol.com.Shows daily 9pm & 11pm (Sun 6pm & 9pm); late show Sat.MAP\n\nLocal-language theatre isn't accessible to non-speakers, but you might want to give this idiosyncratic caf\u00e9-cabaret a try. As well as Catalan-language plays, there are shows featuring a mix of mime, song, clowning, magic and dance, and sometimes there's English-language stand-up comedy by local and visiting acts. Check the website to see what's on when. Tickets \u20ac10\u201315.\n\n### La Concha\n\nC\/Guardia 14 Drassanes 933 024 118.Daily 5pm\u20133am.MAP\n\nThe Arab-flamenco fusion creates a great atmosphere, worth braving the slightly dodgy area for. It's a kitsch, gay-friendly joint, dedicated to the \"incandescent presence\" of Sara Montiel, Queen of Song and Cinema (a much-loved Spanish actress and singer), with uninhibited dancing by tourists and locals alike. Admission free.\n\n### Jazz S\u00ed Club\n\nC\/Requesens 2 Sant Antoni 933 290 020, www.tallerdemusics.com.MAP\n\nThis is a great place for inexpensive (\u20ac6\u20138) gigs in a tiny sweat-box of a club associated with the Taller de Musics (music school). Every night from around 7.45 or 8pm there's something different, from exuberant rock, blues, jazz and jam sessions to the popular weekly Cuban (Thursday) and flamenco (Friday) nights. There are usually a couple of sessions a night, with an interval in between \u2013 your first drink is included in the price, and the bar is as cheap as chips.\n\n### Moog\n\nC\/Arc del Teatre 3 Drassanes 933 191 789, www.masimas.com.Daily midnight\u20135am.MAP\n\nInfluential club with a minimalist look, playing techno, electro, drum 'n' bass and trance to a cool but up-for-it crowd. There's a second, less manic dancefloor as well. Admission \u20ac10.\n\n< Back to El Raval\n\n# Sant Pere\n\nSights\n\nCaf\u00e9\n\nRestaurants and tapas bars\n\nBar\n\nPerhaps the least visited part of the Old Town is the medieval _barri_ of Sant Pere, the area that lies immediately north of the Barri G\u00f2tic and across Carrer de la Princesa from La Ribera. It has two remarkable buildings \u2013 the _modernista_ concert hall, known as the Palau de la M\u00fasica Catalana, and the stylishly designed neighbourhood market, Mercat Santa Caterina. There's been much regeneration in the _barri_ over recent years, and it's well worth an afternoon's stroll or a night out, with new boulevards and community projects alongside DJ bars and designer shops. To walk through the neighbourhood, you can start at Metro Urquinaona, close to the Palau de la M\u00fasica Catalana, with Metro Jaume I marking the southern end of Sant Pere.\n\n## Palau de la M\u00fasica Catalana\n\nC\/Sant Pere M\u00e9s Alt Urquinaona 902 442 882, www.palaumusica.org.Guided tours daily 10am\u20133.30pm, plus Easter week & July 10am\u20136pm, in English every 30 mins.\u20ac18.MAP\n\nBarcelona's most extraordinary concert hall was built in 1908, to a design by visionary _modernista_ architect Llu\u00eds Dom\u00e8nech i Montaner. The elaborate exterior is simply smothered in tiles and mosaics, while a mighty bulbous stained-glass skylight caps the second-storey auditorium (which contemporary critics claimed to be an engineering impossibility). The more recent Petit Palau offers a smaller auditorium space, while to the side a contemporary glass facade and courtyard provide the main public access. Concerts here (throughout the year) include performances by the Orfeo Catal\u00e0 choral group and the Barcelona city orchestra, though you can also catch anything from flamenco to world music. Numbers are limited on the very popular fifty-minute guided tours, so it's best to buy a ticket a day or two in advance (by phone, online or at the box office).\n\nPalau de la M\u00fasica Catalana\n\n## L'Antic Teatre\n\nC\/Verdaguer i Callis 12 Urquinaona 933 152 354, www.anticteatre.com.MAP\n\nAn independent theatre with a wildly original programme, from video shows and offbeat cabaret to modern dance and left-field music. The best bit may just be the magical, light-strung garden-bar (daily 4pm\u2013midnight), a real insiders' place but open to all.\n\n## Pla\u00e7a de Sant Pere\n\n Arc de Triomf.MAP\n\nThe neighbourhood extends around three parallel medieval streets, carrers de Sant Pere M\u00e9s Alt (upper), Mitja (middle) and Baix (lower), which contain the bulk of the finest buildings and shops \u2013 a mixture of boutiques, textile shops, groceries and old family businesses. The streets all converge on the original neighbourhood square, Pla\u00e7a de Sant Pere, whose foursquare church flanks one side, overlooking a flamboyant iron drinking fountain. Originally built in the shape of a Greek cross, the church, **Sant Pere de les Puel.les** ( www.parroquiasantpere.org), dates back to 945 AD and was the city's first convent of Benedictine nuns. It occasionally hosts concerts.\n\n## Mercat Santa Caterina\n\nAv. Francesc Camb\u00f2 16 Jaume I 933 195 740, www.mercatsantacaterina.com.Mon 7.30am\u20132pm, Tues, Wed & Sat 7.30am\u20133.30pm, Thurs & Fri 7.30am\u20138.30pm; July & Aug open mornings only.MAP\n\nAn eye-catching renovation of the neighbourhood market retained its original walls but added slatted wooden doors and windows and a dramatic multicoloured wave roof. It's one of the best places in the city to shop for food, and its market restaurant and bar are definitely worth a visit in any case. During renovation work, the foundations of a medieval convent were discovered here and the excavations are visible in the **Espai Santa Caterina** (Mon\u2013Sat 10am\u20132pm; free) at the rear of the market.\n\nMercat Santa Caterina\n\n## Pla\u00e7a de Sant Agusti Vell\n\n Jaume I.MAP\n\nThe pretty, tree-shaded Pla\u00e7a de Sant Agusti Vell sits in the middle of an ambitious urban regeneration project, which has transformed previously crowded alleys. To the north, locals tend organic allotments in the middle of the landscaped Pou de la Figuera _rambla_ , while south down **Carrer d'Allada Vermell** are overarching trees, a children's playground and a series of outdoor caf\u00e9s and bars. Meanwhile, running down from Pla\u00e7a de Sant Agusti Vell, **Carrer dels Carders** \u2013 once \"ropemakers' street\" \u2013 is now a funky retail quarter mixing grocery stores, caf\u00e9s, boutiques and craft shops.\n\n## Centre C\u00edvic Convent de Sant Agusti\n\nC\/del Comer\u00e7 36 Jaume I 932 565 000, www.bcn.cat\/centrecivicsantagusti.Mon\u2013Fri 9am\u201310pm, Sat 10am\u20132pm & 4\u20139pm.Admission charges vary, some events free.MAP\n\nDriving many of the neighbourhood improvements is the community centre installed inside the revamped Convent de Sant Agusti, whose thirteenth-century cloister provides a unique performance space. There's a full cultural programme here, from workshops to concerts, with a particular emphasis on electronic and experimental music and art, and don't miss the excellent convent caf\u00e9.\n\n## Museu de la Xocolata\n\nC\/del Comer\u00e7 36 Jaume I 932 687 878, www.pastisseria.com.Mon\u2013Sat 10am\u20137pm, Sun 10am\u20133pm.\u20ac5.MAP\n\nPart of the Convent de Sant Agusti contains Barcelona's chocolate museum, which is a rather uninspiring plod through the history of the stuff. Whether you go in or not depends on how keen you are to see models of Gaud\u00ed buildings or religious icons sculpted in chocolate. There are some very nice chocs to buy in the shop though (free to enter), while at the adjacent Escola de Pastisseria, glass windows look onto the students learning their craft in the kitchens.\n\n< Back to Sant Pere\n\n## Caf\u00e9\n\n### Bar del Convent\n\nPl. de l'Acad\u00e8mia, C\/del Comer\u00e7 36 Jaume I 932 565 017, www.bardelconvent.com.Tues\u2013Thurs 10am\u20139pm, Fri & Sat 10am\u201310pm, Sun 11am\u20135pm.MAP\n\nThe cloister caf\u00e9-bar is a bargain for lunch and light meals, with soups, stir-fries, lasagne and couscous for \u20ac4\u20137. At night it's more of a bar, with a range of live shows, DJs and concerts.\n\n## Restaurants and tapas bars\n\n### El Atril\n\nC\/dels Carders 23 Jaume I 933 101 220, www.atrilbarcelona.com.Mon 6pm\u2013midnight, Tues\u2013Sun noon\u2013midnight.MAP\n\nThe \"Music Stand\" is a cosy bar-restaurant complete with summer _terrassa_ and an international menu, from Thai red curry to _moules frites_ (mains \u20ac10\u201315). There's a good-value lunch, and dinner from 7pm, otherwise it's modern tapas, drinks and a decent Sunday brunch.\n\n### Comer\u00e7 24\n\nC\/Comer\u00e7 24 Jaume I 933 192 102, carlesabellan.com.Tues\u2013Sat 1.30\u20133.30pm & 8.30\u201311pm; closed 2 weeks in Aug.MAP\n\nCarles Abellan calls his cuisine \"glocal\" (ie global + local) and in the oh-so-cool interior you're presented with tapas-style dishes mixing flavours and textures to calculated effect (foie gras and truffle hamburger, tuna sashimi on pizza). This is pricey, cutting-edge Barcelona dining (\u20ac80 or so a head); reservations advised.\n\n### Cuines Santa Caterina\n\nMercat Santa Caterina, Av. Francesc Camb\u00f3 16 Jaume I 932 689 918, grupotragaluz.com.Bar daily 9am\u201311.30pm (Thurs\u2013Sat until 12.30am), restaurant daily 1\u20134pm & 8pm\u2013midnight (Thurs\u2013Sat until 12.30am).MAP\n\nA ravishing open-plan tapas bar and market restaurant with tables under soaring rafters. Food touches all bases \u2013 pasta to sushi, Catalan rice to Thai curry \u2013 with most things costing \u20ac9\u201312.\n\n### Mosquito\n\nC\/dels Carders 46 Jaume I 932 687 569, www.mosquitotapas.com.Mon 7.30pm\u20131am, Tues\u2013Sun 1pm\u20131am.MAP\n\nAsian tapas bar with paper lanterns, artisan beers and an authentic, made-to-order dim sum menu (dishes \u20ac3\u20135), from shrimp dumplings to tofu rolls.\n\n## Bar\n\n### Casa Paco\n\nC\/d'Allada Vermell 10 Jaume I 933 149 320.Daily 9am\u20132am (Fri & Sat until 3am), Oct\u2013March opens 6pm.MAP\n\nThis cool music joint is a hit on the weekend DJ scene. There's a great _terrassa_ under the trees, but if you can't get a table here try one of several other alfresco bars down the boulevard.\n\n< Back to Sant Pere\n\n# La Ribera\n\nSights\n\nShops\n\nCaf\u00e9s\n\nRestaurants and tapas bars\n\nBars\n\nThe traditional highlights of the old artisans' quarter of La Ribera are the Museu Picasso (Barcelona's single biggest tourist attraction) and the graceful church of Santa Mar\u00eda del Mar. The cramped streets between the two were at the heart of medieval industry and commerce, and it's still the neighbourhood of choice for local designers, craftspeople and artists, whose boutiques and workshops lend La Ribera an air of creativity. Galleries and applied art museums occupy the medieval mansions of Carrer de Montcada \u2013 the neighbourhood's most handsome street \u2013 while the _barri_ is at its most hip in the area around the Passeig del Born, whose caf\u00e9s, restaurants and bars make it one of the city's premier nightlife centres. The most direct access point for La Ribera is Metro Jaume I.\n\nPasseig del Born\n\n## Museu Picasso\n\nC\/de Montcada 15\u201323 Jaume I 932 563 000, www.museupicasso.bcn.cat.Tues\u2013Sun & hols 9am\u20137pm.\u20ac11, Sun after 3pm & first Sun of the month free.MAP\n\nThe celebrated Museu Picasso is one of the most important collections of Picasso's work in the world, but even so some visitors are disappointed, since the museum contains none of his best-known pictures and few in the Cubist style. But there are almost 4000 works in the permanent collection \u2013 housed in five adjoining medieval palaces \u2013 which provide a fascinating opportunity to trace Picasso's development from his drawings as a young boy to the mature works of later years. Paintings from his art-school days in Barcelona (1895\u201397) show tantalizing glimpses of the city that the young Picasso was beginning to explore, while works in the style of Toulouse-Lautrec reflect his interest in Parisian art. Other selected works are from the famous Blue Period (1901\u201304) and Pink Period (1904\u201306), and from his Cubist (1907\u201320) and Neoclassical (1920\u201325) stages. The large gaps in the main collection only underline Picasso's extraordinary changes of style and mood, best illustrated by the jump to 1957, a year represented by his interpretations of Vel\u00e1zquez's masterpiece _Las Meninas_.\n\nAs well as showing changing selections of sketches, prints and drawings, the museum addresses Picasso's work as a ceramicist, highlighting the vibrantly decorated dishes and jugs donated by his wife Jacqueline.\n\nA free guided tour is the best way to get to grips with the collection \u2013 in English currently on Sundays at 11am (book in advance by phone or via the website). There's a courtyard caf\u00e9, and, of course, a shop full of Picasso-related gifts.\n\nMuseu Picasso\n\n## Museu Europeu d'Art Modern\n\nC\/Barra de Ferro 5 Jaume I 933 195 693, meam.es.Tues\u2013Sun 10am\u20138pm.\u20ac7, guided tour \u20ac2 (Sat & Sun noon).MAP\n\nThere is not one photograph on display at the Museu Europeu d'Art Modern, a fact you may find hard to believe considering how photorealistic many of the paintings are. Located in a renovated eighteenth-century palace \u2013 and just metres from the Museu Picasso \u2013 the museum focuses primarily on modern and contemporary figurative art. Its three floors brim with haunting, humorous and sometimes disturbing works by the likes of Eduardo Naranjo, Paul Beel and Carlos Saura Riaza. It's also home to modern, Art Deco and Catalan sculptures.\n\n### Picasso in Barcelona\n\nAlthough born in M\u00e1laga, **Pablo Picasso** (1881\u20131973) spent much of his youth \u2013 from the age of 14 to 23 \u2013 in Barcelona. This time encompassed the whole of his Blue Period (1901\u201304) and provided many of the formative influences on his art. Not far from the Museu Picasso you can see many of the buildings in which Picasso lived and worked, notably the Escola de Belles Arts de Llotja (c\/Consolat del Mar, near Estaci\u00f3 de Fran\u00e7a), where his father taught drawing and where Picasso himself absorbed an academic training. The apartments where the family lived when they first arrived in Barcelona were at Pg. d'Isabel II 4 and c\/Reina Cristina 3, both near the Escola, while Picasso's first real studio (in 1896) was located over on c\/de la Plata at no. 4. A few years later, many of his Blue Period works were finished at a studio at c\/del Comer\u00e7 28. His first public exhibition was in 1901 at the extravagantly decorated _Els Quatre Gats_ tavern (c\/Montsi\u00f3 3, Barri G\u00f2tic;  www.4gats.com); you can still have a meal there today.\n\n## Museo del Mamut\n\nC\/Montcada 1 Jaume I 932 688 520, museomamut.com.Daily: April\u2013Oct 10am\u20139pm; Nov\u2013March 10am\u20138pm.\u20ac7.50.MAP\n\nThough by no means mammoth in size, the Museu del Mamut (Mammoth Museum) will appeal to Ice Age film fans and anyone with an interest in prehistoric animals. As well as a shaggy, life-sized mammoth reaching to within an inch of the ceiling, there's a sabre-toothed tiger and a musk buffalo along with plenty of authentic skeletons and several reproductions of cave paintings. The experience is refreshingly hands-on without the \"no touching\" signs seen at most of the city's museums. Displays are in Spanish and Catalan with English-language notes available at the ticket counter.\n\n## Esgl\u00e9sia de Santa Mar\u00eda del Mar\n\nPl. de Santa Maria Jaume I 933 102 390.Daily 9am\u20131pm & 5\u20138pm.Free during general admission times, otherwise \u20ac5\/10 guided tours 1\u20135pm.MAP\n\nThe Ribera's flagship church was begun on the order of King Jaume II in 1324, and finished in only five years. Built on what was the seashore in the fourteenth century, Santa Mar\u00eda was at the centre of the medieval city's trading district (nearby c\/Argenter\u00eda, for example, is named after the silversmiths who once worked there), and it came to embody the commercial supremacy of the Crown of Aragon, of which Barcelona was capital. It's an exquisite example of Catalan-Gothic architecture \u2013 as all the later Baroque trappings were destroyed during the Civil War, eyes instead are concentrated on the simple spaces of the interior, especially the beautiful stained glass.\n\n## Passeig del Born\n\n Jaume I.MAP\n\nFronting the church of Santa Mar\u00eda del Mar is the fashionable Passeig del Born, once the site of medieval fairs and entertainments (\"born\" means tournament) and now an avenue lined with a parade of plane trees shading a host of classy bars, delis and shops. At night the Born becomes one of Barcelona's biggest bar zones, as spirited locals frequent a panoply of drinking haunts \u2013 from old-style cocktail lounges to thumping music bars. Shoppers and browsers, meanwhile, scour the narrow medieval alleys on either side of the _passeig_ for boutiques and craft workshops \u2013 carrers Flassaders, Vidreria and Rec, in particular, are noted for clothes, shoes, jewellery and design galleries.\n\nShop display on the Passeig del Born\n\n## El Born Centre Cultural\n\nPl. Comercial 12 Jaume I\/Barceloneta 932 564 190, elborncentrecultural.cat.Tues\u2013Sun 10am\u20138pm.Free access to the centre, \u20ac5.50 guided tours.MAP\n\nThe Antic Mercat del Born (1873\u201376) was the biggest of Barcelona's nineteenth-century market halls. It was the city's main wholesale fruit and veg market in 1971, and was then due to be demolished, but was saved thanks to local protests. It lay empty for decades, but in 2013 reopened as El Born Centre Cultural, where the extensive archeological remains of eighteenth-century shops, factories, houses and taverns are showcased inside the building's restored glass-and-cast iron frame.\n\n< Back to La Ribera\n\n## Shops\n\n### Almacen Marabi\n\nC\/Cirera 6 Jaume I www.almacenmarabi.blogspot.com.Mon\u2013Sat noon\u20132.30pm & 5\u20138.30pm.MAP\n\nMariela Marabi, originally from Argentina, makes handmade felt finger dolls, mobiles, puppets and animals of extraordinary invention. She's often at work at the back, while her eye-popping showroom also has limited-edition pieces by other selected artists and designers.\n\n### La Botifarreria de Santa Maria\n\nC\/Santa Maria 4 Jaume I labotifarreria.com.Mon\u2013Fri 8.30am\u20132.30pm & 5\u20138.30pm, Sat 8.30am\u20132.30pm.MAP\n\nIf you ever doubted the power of the humble Catalan pork sausage, drop by this designer temple-deli where otherwise beautifully behaved locals jostle at the counter for the day's home-made _botifarra_ , plus rigorously sourced hams, cheese, p\u00e2t\u00e9s and salamis. True disciples can even buy the T-shirt.\n\n### Bub\u00f3\n\nC\/Caputxes 10 Jaume I www.bubo.es.Daily 10am\u201310pm (Fri & Sat until 1am).MAP\n\nThere are chocolates and then there are Bub\u00f3 chocolates \u2013 jewel-like creations and playful desserts by pastry maestro Carles Mampel. This very classy shop is complemented by their minimalist new-wave tapas place, _Bub\u00f3bar_ , a couple of doors down at no.6.\n\n### La Campana\n\nC\/Princesa 36 Jaume I turronelslacampana.com.Daily 9am\u201311pm; closed Feb.MAP\n\nThis lovely old shop from 1890 presents handmade pralines and truffles, but is best known for its beautifully packaged squares and slabs of _turr\u00f3n_ , traditional Catalan nougat.\n\n### Casa Gispert\n\nC\/Sombrerers 23 Jaume I www.casagispert.com.Tues\u2013Fri 9.30am\u20132pm & 4\u20138.30pm (also Mon in Oct, Nov & Dec), Sat 10am\u20132pm & 5\u20138.30pm.MAP\n\nRoasters of nuts, coffee and spices for over 150 years. It's a truly delectable store of wooden boxes, baskets, stacked shelves and tantalizing smells, and there are organic nuts and dried fruit, teas and gourmet deli items available too.\n\n### Custo Barcelona\n\nPl. de les Olles 7 Barceloneta www.custo-barcelona.com.Mon\u2013Sat 10am\u20139pm, Sun noon\u20138pm.MAP\n\nWhere the stars get their T-shirts \u2013 hugely colourful (highly priced) designer tops and sweaters for men and women. There are other branches around town (at Ramblas 109, c\/de Ferran 36, and at L'Illa shopping), while last season's gear gets another whirl at Pl. del Pi in the Barri G\u00f2tic.\n\n### Iriarte Iriarte\n\nC\/Cotoners 12 Jaume I www.iriarteiriarte.com.Tues\u2013Sat 4.30\u20138.30pm.MAP\n\nAtelier-showroom for sumptuous handmade leather bags and belts. The alley (off c\/Cotoners) has several other interesting craft workshops and galleries to browse.\n\n### U-Casas\n\nC\/Espaseria 4 Jaume I www.casasclub.com.Mon\u2013Sat 10.30am\u20138.30pm (Fri & Sat until 9pm).MAP\n\nCasas has four lines of shoe stores across Spain, with the U-Casas brand at the young and funky end of the market. Never mind the shoes, the stores are pretty spectacular, especially here in the Born where an enormous shoe-shaped bench-cum-sofa takes centre-stage. Other branches are at c\/Tallers 2 (Raval), c\/Portaferrissa 25 (Barri G\u00f2tic) and L'Illa and Marem\u00e0gnum shopping centres.\n\n### Vila Viniteca\n\nC\/Agullers 7 & 9 Jaume I www.vilaviniteca.es.Mon\u2013Sat 8.30am\u20138.30pm (closes Sat at 2.30pm in July & Aug).MAP\n\nA very knowledgeable specialist in Catalan and Spanish wines. Pick your vintage and then nip over the road for the gourmet deli part of the operation.\n\n## Caf\u00e9s\n\n### La Bascula\n\nC\/Flassaders 30 Jaume I 933 199 866.Wed\u2013Sat 1pm\u2013midnight, Sun 1\u20138pm.MAP\n\nA hippy-chic makeover for an old backstreet chocolate factory. It's a welcoming veggie place, serving pastas, sandwiches, turnovers, crepes, dipping platters and salads (dishes \u20ac9), and there are dozens of organic teas, coffees, wines, juices and shakes.\n\n### Caf\u00e9 del Born\n\nPl. Comercial 10 Barceloneta 932 683 272.Mon\u2013Thurs 8am\u20131am, Fri 8am\u20133am, Sat 9am\u20133am, Sun 9am\u20131am.MAP\n\nNo gimmicks, no fusion food, and dodgy local art kept to a bare minimum \u2013 the recipe for success at this ever-popular neighbourhood caf\u00e9-bar. There's a simple Mediterranean menu on offer, while Sunday brunch is the big weekend draw.\n\n### Pim Pam Burger\n\nC\/Sabateret 4 Jaume I 933 152 093, pimpamburger.com.Daily 1pm\u2013midnight (Fri & Sat until 12.30am).MAP\n\nThe go-to choice for a quick burger and fries, or franks or sandwiches (\u20ac2.50\u20136). There are a few stools and tables if you'd rather not eat on the hoof, while _Pim Pam Plats_ , around the corner on c\/del Rec, is their outlet for budget-beating take-home meals.\n\n## Restaurants and tapas bars\n\n### Cal Pep\n\nPl. de les Olles 8 Barceloneta 933 107 961, www.calpep.com.Mon 7.30\u201311.30pm, Tues\u2013Fri 1\u20133.45pm & 7.30\u201311.30pm, Sat 1.15\u20133.45pm; closed Easter week & Aug.MAP\n\nThere's no equal in town for off-the-boat and out-of-the-market tapas. You may have to queue, and prices are high for what's effectively a bar meal (up to \u20ac60), but it's definitely worth it for the likes of impeccably fried shrimp, grilled sea bass, Catalan sausage, or squid and chickpeas \u2013 all overseen by Pep himself, bustling up and down the counter.\n\n### Euskal Etxea\n\nPl. de Montcada 1\u20133 Jaume I 933 102 185, www.euskaletxea.cat.Daily 10am\u201312.30am (Fri & Sat until 1am).MAP\n\nThe bar at the front of the local Basque community centre is great for sampling _pintxos_ \u2013 elaborately fashioned pint-sized tapas, held together by a stick. Just point to what you want (and keep the sticks so the bill can be tallied).\n\n### Kiosko Burger\n\nAv. del Marqu\u00e8s de l'Argentera 1 Barceloneta 933 107 313, kioskoburger.com.Daily 1pm\u20131.30am.MAP\n\nIn Barcelona's best gourmet burger outlet, great-tasting artisan bread rolls and homemade sauces set the tone, while a dozen superb types of burger (\u20ac6\u20139) come any way you like, from Catalan (with a roast garlic alioli) to Japanese (with teriyaki sauce).\n\n### Mercat Princesa\n\nC\/Flassaders 21 Jaume I 932 681 518, mercatprincesa.com.Daily 9am\u2013midnight (Thurs\u2013Sat until 1am).MAP\n\nEnjoy food from more than a dozen gourmet stalls (think plump Chinese dumplings, grilled artisan sausages and artfully mounded _montaditos_ ) at communal tables in a restored fourteenth-century palace's interior courtyard. It's a great option, especially if your taste buds are pulling you in multiple directions. From \u20ac2.50 a dish.\n\n### Salero\n\nC\/Rec 60 Barceloneta 933 198 022, www.restaurantesalero.com.Mon\u2013Sat 1.30\u20134pm & 8.30pm\u2013midnight, Sun 8.30pm\u2013midnight; closed 2 weeks in Aug.MAP\n\nA crisp, modern space fashioned from a former salt-cod warehouse \u2013 if white is your colour, you'll enjoy the space. The food's Mediterranean-Asian, presenting delights like vegetable tempura or a _mee goreng_ (fried noodle) of the day (most dishes \u20ac12\u201318).\n\n### Senyor Parellada\n\nC\/Argenteria 37 Jaume I 933 105 094, www.senyorparellada.com.Daily 1\u20133.45pm & 8.30\u201311.30pm.MAP\n\nA gorgeous renovation of an eighteenth-century building is the mellow background for genuine Catalan food \u2013 home-style cabbage rolls, duck with figs, a papillote of beans with herbs \u2013 served from a long menu that doesn't bother dividing starters from mains. Most mains cost \u20ac8\u201319.\n\n### El Xampanyet\n\nC\/de Montcada 22 Jaume I 933 197 003.Tues\u2013Sun noon\u20133.30pm & 7\u201311.30pm; closed two weeks in Aug.MAP\n\nTraditional blue-tiled bar doing a roaring trade in sparkling _cava_ , cider and traditional tapas (anchovies are the speciality). The drinks are cheap and the tapas turn out to be rather pricey, but there's usually a good buzz about the place.\n\n## Bars\n\n### Espai Barroc\n\nPalau Dalmases, c\/de Montcada 20 Jaume I 933 100 673, www.palaudalmases.com.Daily 7pm\u20132am.MAP\n\nThe handsome Baroque mansion of Palau Dalmases is open in the evenings for wine, champagne or cognac in refined surroundings or, once a week, the billowing strains of live opera (Thurs at 11pm; \u20ac20, first drink included).\n\n### La Fianna\n\nC\/Manresa 4 Jaume I 933 151 810, www.lafianna.com.Daily 6pm\u20132am.MAP\n\nFlickering candelabras, rough plaster walls and deep colours set the Gothic mood in this stylish lounge-bar. Relax on the chill-out beds and velvet sofas, or book ahead to eat \u2013 the fusion-food restaurant is open from 8.30pm.\n\n### Mudanzas\n\nC\/Vidrier\u00eda 15 Barceloneta 933 191 137.Daily 10am\u20132.30am, Aug opens at 6pm.MAP\n\nLocals like the relaxed atmosphere, while those in the know come for the wide selection of rums, whiskies and vodkas from around the world.\n\n### El Nus\n\nC\/Mirallers 5 Jaume I 933 195 355.Mon\u2013Thurs 7.30pm\u20132.30am, Fri & Sat 7.30pm\u20133am.MAP\n\nStill has the feel of the shop it once was, down to the antique cash register, though now it's a kind of jazz-bar-cum-gallery \u2013 a quiet, faintly old-fashioned, late-night place.\n\n### La Vinya del Senyor\n\nPl. Santa Maria 5 Jaume I 933 103 379.Mon\u2013Thurs noon\u20131am, Fri & Sat noon\u20132am, Sun noon\u2013midnight.MAP\n\nA great wine bar with front-row seats onto the lovely church of Santa Mar\u00eda del Mar. The wine list is really good \u2013 with a score available by the glass \u2013 and there are oysters, smoked salmon and other classy tapas available.\n\n< Back to La Ribera\n\n# Parc de la Ciutadella\n\nSights\n\nRestaurants and tapas bars\n\nBar\n\nClub\n\nWhile you might escape to Montju\u00efc or the Collserola hills for the air, there's no beating the city's green lung, Parc de la Ciutadella (open daily 10am until dusk), for a break from the downtown bustle. Though the park holds a full set of attractions, on lazy summer days you may simply want to stroll along the garden paths and row lazily across the ornamental lake. The name of the park recalls a Bourbon citadel which used to occupy the site, the building of which caused the brutal destruction of a great part of La Ribera neighbourhood. This symbol of authority survived uneasily until 1869; after this the area was made into a park. It was subsequently chosen as the site of the 1888 Universal Exhibition, from which period dates a series of eye-catching buildings and monuments by the city's pioneering _modernista_ architects.\n\n## Arc de Triomf\n\nPg. Llu\u00eds Companys Arc de Triomf.MAP\n\nA giant brick arch announces the architectural splendours to come in the park itself. Conceived as a bold statement of Catalan intent, it's studded with ceramic figures and motifs and topped by two pairs of bulbous domes. The reliefs on the main facade show the city of Barcelona welcoming visitors to the 1888 Universal Exhibition, held in the park to the south. Connecting the arch to the park is a gorgeous promenade flanked by linden and palm trees and ornate lamp posts. Just before the park's entrance on Passeig de Pujades stands the monument to Francisco de Paula Rius i Taulet, the four-time mayor of Barcelona credited with helping bring the Universal Exhibition to the city.\n\nParc de la Ciutadella\n\n## Cascada\n\nParc de la Ciutadella Arc de Triomf.MAP\n\nThe first of the major projects undertaken inside the park was the Cascada, the monumental fountain in the northeast corner. It was designed by Josep Fontser\u00e9 i Mestr\u00e8s, the architect chosen to oversee the conversion of the former citadel grounds into a park, and his assistant was the young Antoni Gaud\u00ed, then a student. The Baroque extravagance of the Cascada is suggestive of the flamboyant decoration that was later to become Gaud\u00ed's trademark. The best place to contemplate the fountain is from the small open-air caf\u00e9-kiosk. Near here you'll also find a small lake, where you can rent a rowing boat and paddle about among the ducks.\n\n## Hivernacle and Umbracle\n\nPg. de Picasso Arc de Triomf.Under renovation at the time of writing.Free.MAP\n\nThe two unsung glories of Ciutadella are its plant houses, arranged either side of the Museu Martorell. The larger Hivernacle (conservatory) features enclosed greenhouses separated by a soaring glass-roofed terrace. If anything, the Umbracle (palm-house) is even more imposing, with a vaulted wood-slat roof supported by cast-iron pillars, which allows shafts of light to play across the assembled palms and ferns. Traditionally, there's always been a caf\u00e9-bar in the Hivernacle, set among the plants and trees, though it was closed at the time of writing while restoration work continued on the building.\n\n## Museu de Ci\u00e8ncies Naturals\n\nPg. de Picasso Arc de Triomf museuciencies.bcn.cat.MAP\n\nThe city's Natural Science Museum has its public showcase, the Museu Blau, over at the Diagonal Mar F\u00f2rum site, but its genesis lies in two interesting buildings in Ciutadella park that are currently undergoing major renovation (and will be for some time). The Neoclassical **Museu Martorell** , which opened in 1882, was actually the first public museum to be built in the city, designed by leading architect of the day Antoni Rovira i Trias. For decades this housed the city's geological collections; the new permanent exhibition here will concentrate on the development of the natural sciences in Barcelona. The other building has always been a city favourite, a whimsical red-brick confection that was long the zoology museum. It's universally known as the **Castell dels Tres Dragons** (Three Dragons Castle), designed by _modernista_ architect Llu\u00eds Dom\u00e8nech i Montaner and originally intended for use as the caf\u00e9-restaurant for the 1888 Universal Exhibition. It's going to become the research, study and conservation centre for the Natural Science Museum's geology and zoology collections, and will be known as the Laboratori de Natura (Laboratory of Nature).\n\nParc Zool\u00f2gic\n\n## Parc Zool\u00f2gic\n\nC\/de Wellington Ciutadella-Vila Ol\u00edmpica 902 457 545, www.zoobarcelona.com.Daily: Jan\u2013March & late Oct\u2013Dec 10am\u20135pm; April\u2013mid-May & mid-Sept\u2013late late Oct 10am\u20136pm; mid-May\u2013mid-Sept 10am\u20137pm.\u20ac17.90.MAP\n\nCiutadella's most popular attraction by far is the city zoo, which takes up most of the southeastern part of the park (main entrance on c\/Wellington, signposted from  Ciutadella-Vila Ol\u00edmpica). It boasts more than 2000 animals from over 400 different species \u2013 which is seen by some as too many for a zoo that is still essentially nineteenth century in character, confined to the formal grounds of a public park. Nonetheless it's hugely popular with families, as there are mini-train and pony rides, a petting zoo and daily dolphin shows alongside the main animal attractions. The many endangered species on show include the Iberian wolf, and big cats such as the Sri Lankan leopard, snow leopard and the Sumatran tiger. The zoo's days in its current form are numbered: over the next few years parts of it will be completely remodelled as it attempts to expand its facilities and modernize.\n\n< Back to Parc de la Ciutadella\n\n## Restaurants and tapas bars\n\n### Ikibana\n\nPg. Picasso 32 Barceloneta 932 956 732, www.ikibana.es.Daily: June\u2013Sept 1pm\u20131am; Oct\u2013May 1\u20134pm & 8.30pm\u201312.30am (Fri & Sat until 1am).MAP\n\nRiding the fusion wave is this glam Japanese-Brazilian with a lounge-bar setting, just a step or two out of the park, which offers a great weekday lunch deal. A series of dainty little dishes presents a mix of zingy rice- and seaweed-roll combos and exotic tempuras, with the full menu extending to ceviches and Kobe beef burgers (dishes \u20ac6\u201313, tasting menu \u20ac33).\n\n### Picnic\n\nC\/Comer\u00e7 1 Arc de Triomf 935 116 661, www.picnic-restaurant.com.Mon 1\u20134pm, Tues\u2013Fri 1\u20134pm & 8pm\u201312.30am (cocktails until 2am), Sat & Sun 11.30am\u20135pm.MAP\n\nJust a short stroll from the park, this lovely little spot serves classic brunch food such as eggs Benedict, pancakes and French toast, plus more creative dishes like duck hash and fried green tomatoes (brunch Fri\u2013Sun; \u20ac5.50\u201314). The dinner menu focuses on tapas, with offerings that include oysters in tempura, grilled kangaroo and Myanmar pickled tealeaf salad. There's also a weekday lunch menu (Tues\u2013Thurs; from \u20ac12), as well as thirst-quenching drinks such as the refreshingly tart pink lemonade.\n\n## Bar\n\n### Inercia Classic\n\nPg. Picasso 20 Barceloneta 933 107 207, www.inerciagroup.com.Daily 8am\u20133am.MAP\n\nDuring the summer heat, a cold drink on the shaded _terrassa_ \u2013 which sits under the arcaded walkway of Passeig de Picasso across from the park \u2013 is just the ticket. It's a great spot to start the evening before diving into Born's vibrant nightlife.\n\n## Club\n\n### Magic\n\nPg. Picasso 40 Barceloneta 933 107 267, www.magic-club.net.Thurs\u2013Sat 11pm\u20136am.MAP\n\nA Barcelona classic that's been rocking out since the mid-1970s. While first and foremost a rock 'n' roll club, _Magic_ doesn't take itself too seriously. The usual suspects (Ramones, AC\/DC and Iggy Pop) are played alongside hits from the likes of the Beastie Boys, the Violent Femmes and more.\n\n< Back to Parc de la Ciutadella\n\n# Montju\u00efc\n\nSights\n\nCaf\u00e9\n\nRestaurants and tapas bars\n\nBars\n\nClubs\n\nFor art and gardens you need to head across the city to the verdant park area of Montju\u00efc, site of the 1992 Olympics. The hill is topped by a sturdy castle and anchored around the heavyweight art collections in the Museu Nacional d'Art de Catalunya (MNAC). Two other superb galleries also draw visitors, namely Caixa Forum and the celebrated Fundaci\u00f3 Joan Mir\u00f3, not to mention a whole host of family-oriented attractions, from the open-air Poble Espanyol (Spanish Village) to the cable-car ride to the castle. Meanwhile, the various gardens that spill down the hillsides culminate in Barcelona's excellent botanical gardens. For Caixa Forum, Poble Espanyol and MNAC use Metro Espanya; the Trasbordador Aeri (cable car from Barceloneta) and Funicular de Montju\u00efc (from Metro Paral.lel) drop you near the Fundaci\u00f3 Joan Mir\u00f3.\n\n## Pla\u00e7a d'Espanya\n\n Espanya.MAP\n\nMontju\u00efc's characteristic gardens, terraces, fountains and monumental buildings were established for the International Exhibition of 1929. Gateway to the Exhibition was the vast Pla\u00e7a d'Espanya and its huge Neoclassical fountain, with striking twin towers, 47m high, standing at the foot of the imposing Avinguda de la Reina Maria Cristina. This avenue heads up towards Montju\u00efc, and is lined by exhibition halls used for trade fairs. At the end, monumental steps (and modern escalators) ascend the hill to the Palau Nacional (home of MNAC), past water cascades and under the flanking walls of two grand Viennese-style pavilions. The higher you climb, the better the views, while a few caf\u00e9-kiosks put out seats on the way up to MNAC.\n\nPla\u00e7a d'Espanya\n\n## Monument de les Quatre Columnes\n\n Espanya.MAP\n\nSimplest of architect Josep Puig i Cadafalch's ideas for the ceremonial gateway to Montju\u00efc at Pla\u00e7a d'Espanya were four 20m-high columns, erected in 1919 on a raised site below the future Palau Nacional. Who could possibly object? The authoritarian government of General Primo de Rivera, as it happened, knowing perfectly well that the architect, a Catalan nationalist, meant the four columns to represent the four stripes of the Catalan flag. Down they came in 1928, to be replaced by the Magic Fountain, and not until 2010 were the columns (reconstructed using the original plans) to be seen again in public \u2013 erected across from the fountain by the city government as \"an act of memory\" and symbol of freedom and democracy.\n\n## Caixa Forum\n\nAv. Francesc Ferrer i Gu\u00e0rdia 6\u20138 Espanya 934 768 600, www.fundacio.lacaixa.es.Sept\u2013June daily 10am\u20138pm; July & Aug Thurs\u2013Tues 10am\u20138pm, Wed 10am\u201311pm.\u20ac4.MAP\n\nThe former Casaramona textile factory (1911) at the foot of Montju\u00efc conceals a terrific arts and cultural centre. The exhibition halls were fashioned from the former factory buildings, whose external structure was left untouched \u2013 girders, pillars, brickwork and crenellated walls appear at every turn. The undulating roof (signposted \"terrats\") offers unique views, while the high Casaramona tower, etched in blue and yellow tiling, is as readily recognizable as the huge Mir\u00f3 starfish logos emblazoned across the building. The contemporary art collection focuses on the period from the 1980s to the present, and works are shown in partial rotation alongside an excellent programme of changing exhibitions across all aspects of the arts. There's also the Mediateca multimedia space, plus an arts bookshop, children's activities and a 400-seat auditorium for music, art and literary events.\n\n## Font M\u00e0gica\n\nPl. de Carles Buigas Espanya.April\u2013Oct Thurs\u2013Sun 9\u201311.30pm; Nov\u2013March Fri, Sat & hols 7\u20138.30pm.Free.MAP\n\nOn selected evenings, the \"Magic Fountain\" at the foot of the Montju\u00efc steps becomes the centrepiece of an impressive, if slightly kitsch, sound-and-light show, as the sprays and sheets of brightly coloured water dance to the music.\n\nThe Font M\u00e0gica in action\n\n## Pavell\u00f3 Mies van der Rohe\n\nAv. Francesc Ferrer i Gu\u00e0rdia 7 Espanya 934 234 016, www.miesbcn.com.Daily 10am\u20138pm; guided visits in English on Sat at 10am.\u20ac5.MAP\n\nThe German contribution to the 1929 International Exhibition was a pavilion designed by Mies van der Rohe (and reconstructed in 1986 by Catalan architects). It's considered a major example of modern rationalist architecture \u2013 a startling conjunction of dark-green polished onyx, shining glass and watery surfaces. Unless there's an exhibition in place (a fairly regular occurrence) there is little to see inside, though you can buy postcards and books from the small shop and debate quite how much you want a Mies mousepad or a \"Less is More\" T-shirt.\n\n## Poble Espanyol\n\nAv. Francesc Ferrer i Gu\u00e0rdia 13 Espanya 935 086 300, www.poble-espanyol.com.Mon 9am\u20138pm, Tues\u2013Thurs 9am\u2013midnight, Fri 9am\u20133am, Sat & Sun 9am\u20134am.\u20ac9.50, family ticket \u20ac30, night ticket \u20ac6.50, combined ticket with MNAC \u20ac18.MAP\n\n\"Get to know Spain in one hour\" is what's promised at the Spanish Village \u2013 an open-air park of reconstructed Spanish buildings, such as the medieval walls of \u00c1vila, through which you enter. The echoing main square is lined with caf\u00e9s, while the surrounding streets and alleys contain around forty workshops, where you can witness crafts like engraving, weaving and pottery. Inevitably, it's one huge shopping experience, and prices are inflated, but children will love it (they can run free as there's no traffic) and there are plenty of family activities. Get to the village early to enjoy it in relatively crowd-free circumstances \u2013 once the tour groups arrive, it becomes a bit of a scrum. You could always come instead at the other end of the day, to venues like _Tablao de Carmen_ or _La Terrrazza_ , when the village transforms into a vibrant centre of Barcelona nightlife.\n\n## Museu de Carrosses F\u00fanebres\n\nC\/Mare de D\u00e9u de Port 56\u201358.Bus #21 from  Paral.lel. 934 841 999, cbsa.cat.Wed\u2013Sun 10am\u20132pm.Free\n\nOne of the city's more esoteric attractions, the Funerary Carriage Museum is fittingly located at the entrance to Montju\u00efc Cemetery. The horse-drawn carriages on display were used for city funeral processions from the 1830s until the service was mechanized in the 1950s, when the silver Buick (also on display) came into use. Most of the carriages and hearses are extravagantly decorated, and some carried dignitaries, politicians and big-name bullfighters to their final resting places. There are also plenty of old photographs of them in use in the city's streets, alongside antique uniforms, mourning wear and formal riding gear.\n\n## Museu Nacional d'Art de Catalunya (MNAC)\n\nPalau Nacional Espanya 936 220 376, www.mnac.cat.Tues\u2013Sat 10am\u20138pm (Oct\u2013April until 6pm), Sun & hols 10am\u20133pm.\u20ac12, ticket valid 48hr, first Sun of the month free; special exhibitions, varied charges apply.MAP\n\nCatalunya's national art gallery is one of Barcelona's essential visits, showcasing a thousand years of Catalan art in stupendous surroundings. For first-time visitors it can be difficult to know where to start, but if time is limited it's recommended you concentrate on the medieval collection. It's split into two main sections, one dedicated to Romanesque art and the other to Gothic \u2013 periods in which Catalunya's artists were pre-eminent in Spain.\n\nThe collection of Romanesque frescoes in particular is the museum's pride and joy \u2013 removed from churches in the Catalan Pyrenees, and presented in a reconstruction of their original setting. MNAC also boasts an unsurpassed nineteenth- and twentieth-century Catalan art collection (until the 1940s \u2013 everything from the 1950s onwards is covered by MACBA in the Raval). It's particularly strong on _modernista_ and _noucentista_ painting and sculpture, the two dominant schools of the period, while there are some fascinating diversions into subjects like _modernista_ interior design, avant-garde sculpture and historical photography.\n\nBlockbuster exhibitions, and special shows based on the museum's archives are popular (separate charges may apply).\n\nMuseu Nacional d'Art de Catalunya\n\n## Museu Etnol\u00f2gic\n\nPg. Santa Madrona 16\u201322 Espanya www.museuetnologic.bcn.cat.The museum is currently closed for renovation, expected to be completed in late 2014.MAP\n\nThe Ethnological Museum boasts extensive global cultural collections and puts on excellent exhibitions, which usually last for a year or two and focus on a particular subject or geographical area. Refreshingly, pieces close to home aren't neglected, which means that there's also often a focus on local and national themes, like rural life and work or Spanish carnival celebrations.\n\n## Museu d'Arqueologia\n\nPg. Santa Madrona 39\u201341 Espanya 934 232 129, www.mac.cat.Tues\u2013Sat 9.30am\u20137pm, Sun & hols 10am\u20132.30pm.\u20ac4.50, free last Tues of each month Oct\u2013June.MAP\n\nThe city's main archeological collection spans the centuries from the Stone Age to the time of the Visigoths, with the Roman and Greek periods particularly well represented. Finds from Catalunya's best-preserved archeological site \u2013 the Greek remains at Emp\u00faries on the Costa Brava \u2013 are notable, while on an upper floor life in Barcino (Roman Barcelona) is interpreted through a vivid array of tombstones, statues, inscriptions and friezes.\n\n### Teatre Grec and the Barcelona festival\n\nCentrepiece of Barcelona's annual summer cultural festival ( grec.bcn.cat) is the **Teatre Grec** (Greek theatre), cut into a former quarry on the Montju\u00efc hillside. Starting in late June (and running throughout July and sometimes early Aug), the festival incorporates drama, music and dance, with some of the most atmospheric events staged in the Greek theatre, from Shakespearean productions to shows by avant-garde performance artists.\n\n## La Ciutat del Teatre\n\nMercat de les Flors Poble Sec 932 562 600, www.mercatflors.org.Teatre Lliure 932 289 747,  www.teatrelliure.cat; Institut del Teatre  932 273 900,  teatrelliure.cat. MAP\n\nAt the foot of Montju\u00efc the theatre area known as La Ciutat del Teatre (\"Theatre City\") occupies a corner of the old working-class neighbourhood of Poble Sec. Here, off c\/de Lleida, you'll find the **Mercat de les Flors** \u2013 once a flower market, now a centre for dance and the \"movement arts\" \u2013 and the progressive **Teatre Lliure** (\"Free Theatre\"), while the sleek **Institut del Teatre** brings together the city's major drama and dance schools.\n\nPerformance at the Barcelona festival\n\n## Poble Sec\n\n Poble Sec.MAP\n\nThe Poble Sec neighbourhood provides a complete contrast to the landscaped slopes of Montju\u00efc. The name (\"Dry Village\") is derived from the fact that this working-class neighbourhood originally had no water supply. Today, the hillside grid of streets is lined with down-to-earth grocery stores and good-value restaurants, while Poble Sec is also emerging as an \"off-Raval\" nightlife destination, with its fashionable bars and music clubs \u2013 pedestrianized Carrer de Blai is the epicentre of the scene. It has its own metro station, or it's an easy walk from El Raval, while the Montju\u00efc funicular has its lower station at nearby  Paral.lel.\n\n## Refugi 307\n\nC\/Nou de la Rambla 175 Paral.lel 932 562 100, www.museuhistoria.bcn.cat.Guided tours Sun at 10.30am, 11.30am & 12.30pm.\u20ac3.40.MAP\n\nFor a fascinating look at one of the city's hidden corners, visit Poble Sec's old Civil War air-raid shelter, dug into the hillside by local people in 1936. The tunnels could shelter up to 2000 people from Franco's bombs \u2013 you follow your guide into the labyrinth to the sound of screaming sirens, which at the time gave the locals just two minutes to get safely underground. Storyboards and photographs by the entrance explain the gripping history of the Civil War in Barcelona. Tours are in Spanish or Catalan, though someone usually speaks English, and you can just turn up on the day.\n\n## Estadi Ol\u00edmpic\n\nMuseu Ol\u00edmpic i de l'Esport, Av. de l'Estadi 60 Espanya 932 925 379, www.museuolimpicbcn.cat.Tues\u2013Sat 10am\u20136pm (April\u2013Sept until 8pm), Sun & hols 10am\u20132.30pm.\u20ac5.10.MAP\n\nThe 65,000-seater Olympic Stadium was the ceremonial venue for the 1992 Barcelona Olympics. From the front, a vast _terrassa_ provides one of the finest vantage points in the city, while the space-age curve of Santiago Calatrava's communications tower dominates the skyline.\n\nAround the other side, across the road from the stadium, the history of the Games \u2013 and Barcelona's successful hosting \u2013 are covered in the Olympic and Sports Museum. It's a fully interactive experience, with lots of sports gear and memorabilia displayed, but even so it's probably one for hardcore Olympics fans only.\n\n## Fundaci\u00f3 Joan Mir\u00f3\n\nParc de Montju\u00efc Espanya 934 439 470, www.fundaciomiro-bcn.org.Tues\u2013Sat 10am\u20137pm (July\u2013Sept until 8pm), Thurs until 9.30pm, Sun & hols 10am\u20132.30pm.\u20ac11, exhibitions from \u20ac7.MAP\n\nBarcelona's most adventurous art museum houses the life's work of the great Catalan artist Joan Mir\u00f3 (1893\u20131983). Inside the stark white building is a permanent collection of works largely donated by Mir\u00f3 himself and covering the period from 1914 to 1978. The paintings and drawings in particular are instantly recognizable, being among the chief links between Surrealism and abstract art, while there's also a selection of fascinating original sketches \u2013 Mir\u00f3's enormous tapestries and outdoor sculptures, for example, often started life as a doodle on a scrap of notepaper. The museum's Sala K provides a rapid appraisal of Mir\u00f3's entire _oeuvre_ in a representative selection of works. Elsewhere are pieces by other artists in homage to Mir\u00f3, and exhibitions by young experimental artists in the Espai 13 gallery.\n\nThe museum sponsors temporary exhibitions, film shows, lectures and children's theatre. There's also a caf\u00e9-restaurant with outdoor tables on a sunny patio \u2013 and you don't need a museum ticket to go in.\n\nHanging tapestry at the Fundaci\u00f3 Joan Mir\u00f3\n\n## Funicular de Montju\u00efc\n\nAv. del Paral.lel Paral.lel www.tmb.cat.Every 10min, Mon\u2013Fri 7.30am\u20138pm, Sat, Sun & hols 9am\u20138pm (April\u2013Oct daily until 10pm).\u20ac2.15, transport tickets and passes valid.MAP\n\nThe quickest way to reach the lower heights of Montju\u00efc is to take the funicular, from inside the station at  Paral.lel. At the upper station you can switch to the Montju\u00efc cable car, or you're only a few minutes' walk from the Fundaci\u00f3 Joan Mir\u00f3.\n\n## Telef\u00e8ric de Montju\u00efc\n\nAv. de Miramar Paral.lel, then Funicular www.tmb.cat.Daily: April, May & Oct 10am\u20137pm; June\u2013Sept 10am\u20139pm; Nov\u2013March 10am\u20136pm.\u20ac7.50 one-way, \u20ac10.80 return.MAP\n\nThe cable car up to the castle and back is an exciting ride, and the views, of course, are stupendous. There's an intermediate station, called Mirador, where you can get out and enjoy more sweeping vistas.\n\n## Castell de Montju\u00efc\n\nCarretera de Montju\u00efc 932 564 445, www.bcn.cat\/castelldemontjuic.Daily: April\u2013Sept 10am\u20138pm; Oct\u2013March 10am\u20136pm.\u20ac5, free Sun after 3pm.MAP\n\nBarcelona's fortress served as a military base and prison for decades, and was where the last president of the prewar Catalan government, Llu\u00eds Companys, was executed on Franco's orders on October 15, 1940. However, in 2008 the castle was symbolically handed over to the city and restoration work is transforming the site into a combined peace museum, memorial space and Montju\u00efc interpretation centre. Exhibitions may take place here in the meantime, though the cable-car ride and dramatic location merit a visit in any case. The rampart views are magnificent, while below the walls the panoramic **Cam\u00ed del Mar** pathway looks out over port and ocean. It runs for 1km to the Mirador del Migdia viewpoint, where there's a great open-air bar called _La Caseta_ (weekends from noon, plus summer weekend DJ nights).\n\nJard\u00ed Bot\u00e0nic de Barcelona\n\n## Jard\u00ed Bot\u00e0nic de Barcelona\n\nC\/Dr Font i Quer 2 Espanya 932 564 160, www.museuciencies.bcn.cat.Daily: April\u2013Sept 10am\u20137pm; Oct\u2013March 10am\u20135pm.\u20ac3.50, or combined ticket with Museu Blau \u20ac7; free Sun after 3pm & first Sun of month.MAP\n\nPrincipal among Montju\u00efc's many gardens is the city's Botanical Garden, laid out on terraced slopes offering fine views of the city. The Montju\u00efc buses run here directly, or it's a five-minute walk around the back of the Olympic Stadium. The beautifully kept contemporary garden has landscaped zones representing the flora of the Mediterranean, Canary Islands, California, Chile, South Africa and Australia. Just don't come in the full heat of the summer day, as there's very little shade.\n\n< Back to Montju\u00efc\n\n## Caf\u00e9\n\n### Federal\n\nC\/del Parlament 39 Sant Antoni 931 873 607, federalcafe.es.Mon\u2013Thurs 8am\u201311pm, Fri 8am\u20131am, Sat 9am\u20131am, Sun 9am\u20135.30pm.MAP\n\nSunday brunch is the hottest ticket in town at this effortlessly cool Aussie-owned caf\u00e9 with a great little roof garden, squished into a corner townhouse. Whether you're looking for a flat white and French toast, or a bacon butty and a glass of Kiwi Sauv Blanc, there's nowhere else quite like this in Barcelona.\n\n## Restaurants and tapas bars\n\n### Bella Napoli\n\nC\/Margarit 14 Poble Sec 934 425 056.Daily 1.30\u20134pm & 8.30pm\u2013midnight.MAP\n\nAuthentic Neapolitan pizzeria serving the city's finest pizzas straight from a beehive-shaped oven. Or there's a huge range of pastas, risottos and veal _scaloppine_ (most dishes between \u20ac9 and \u20ac15).\n\n### Quimet i Quimet\n\nC\/Poeta Cabanyes 25 Paral.lel 934 423 142.Tues\u2013Sat noon\u20134pm & 7\u201310.30pm, Sun noon\u20134pm; closed Aug.MAP\n\nPoble Sec's cosiest tapas bar is a place of pilgrimage where classy finger food (dishes \u20ac3\u201310) is served from a minuscule counter.\n\n### Taper\u00eda Lolita\n\nC\/Tamarit 104 Poble Sec 934 245 231, www.lolitataperia.com.Tues & Wed 7pm\u2013midnight, Thurs 7pm\u20132am, Fri 7pm\u20132.30am, Sat 1\u20134pm & 7pm\u20132.30am.MAP\n\nA hip bar which serves classic tapas to tuned-in city folk and in-the-know tourists. You'll eat for around \u20ac25 \u2013 don't miss the signature-dish _patatas bravas_ or the deep-fried _bombas_ (meatballs) and _croquetas_.\n\n### Tickets\/41\u00b0\n\nAv. Paral.lel 164 Poble Sec ticketsbar.es. 41grados.es.Online reservations only. _Tickets_ Tues\u2013Fri 7\u201311pm, Sat 1\u20133.30pm & 7\u201311.30pm. _41\u00b0_ Daily opens to non-reservation-holders at midnight; closed 2 weeks in Aug.MAP\n\nStar-studded tapas bar by _El Bulli_ -famed chef Ferran Adri\u00e0, where amazingly inventive dishes (\u20ac5\u201320 each, expect to spend \u20ac70) mix impeccably sourced ingredients with sheer flights of fancy. At the adjacent and very sleek cocktail and oyster bar _41\u00b0_ , even more outr\u00e9 snacks and canap\u00e9s are served. Reservations are hard to get, and are taken up to two months in advance.\n\n### La Tomaquera\n\nC\/Margarit 58 Poble Sec 675 902 389.Tues\u2013Sat 1\u20133.45pm & 8\u201311pm, Sun 1\u20133.45pm; closed Sun in June & July; closed Aug.MAP\n\nChatter-filled traditional tavern where the chefs set to hacking steaks and chops from great hunks of meat. It's not for the faint-hearted, but the grilled chicken is sensational and the _entrec\u00f4tes_ enormous (most mains \u20ac8\u201315).\n\n## Bars\n\n### Bar Seco\n\nPg. Montju\u00efc 74 Paral.lel 933 296 374, bar-seco.com.Mon\u2013Wed 9am\u20135.30pm, Thurs 9am\u20131am, Fri 9am\u20132am, Sat 10am\u20132am, Sun 10am\u20136pm.MAP\n\nThe \"Dry Bar\" is a local hit, with its mellow vibe, fresh juices and artisan beers.\n\n### Cervecer\u00eda Jazz\n\nC\/Margarit 43 Poble Sec 934 433 259, cerveceriajazz.com.Tues\u2013Sat 7pm\u20133am.MAP\n\nGrab a stool at the carved bar and shoot the breeze over a Catalan craft beer. It's an amiable joint with great music, jazz to reggae, and locals swear that the burgers are the best in town.\n\n### La Tieta\n\nC\/de Blai 1 Paral.lel 600 742 532.May\u2013Nov Mon 6pm\u2013midnight, Tues\u2013Thurs noon\u2013midnight, Fri & Sat noon\u20131am; Dec\u2013April Tues\u2013Thurs noon\u2013midnight, Fri & Sat noon\u20132am, Sun noon\u20134pm.MAP\n\nSmall but perfectly formed, \"The Aunt\" is a cool drinks and tapas place with an open window onto the street and just enough room for a dozen or so good friends.\n\n### Xix Bar\n\nC\/Rocafort 19 Poble Sec 934 234 314, www.xixbar.com.Mon 6.30pm\u20132.30am, Tues\u2013Sat 5pm\u20133am.MAP\n\nAn old _granja_ (milk bar) turned candlelit cocktail bar. It's big on gin, boasting over a hundred varieties.\n\n## Clubs\n\n### Maumau\n\nC\/Fontrodona 33 Paral.lel 934 418 015, www.maumaunderground.com.Thurs\u2013Sat 9pm\u20132.30am.MAP\n\nUnderground lounge-club, cultural centre and chill-out space, with nightly video projections, all sorts of exhibitions and guest DJs playing deep, soulful grooves.\n\n### Sala Apolo\n\nC\/Nou de la Rambla 113 Paral.lel 934 414 001, www.sala-apolo.com.MAP\n\nOld-time ballroom turned hip concert venue with gigs on two stages (local acts to big names) and an eclectic series of club nights, from punk or Catalan rumba sounds to the weekend's long-running _Nitsa Club_ ( www.nitsa.com). Gigs \u20ac10\u201335, club nights \u20ac10\u201315.\n\n### Tablao de Carmen\n\nPoble Espanyol Espanya 933 256 895, www.tablaodecarmen.com.Tues\u2013Sun, shows at 7pm & 9.30pm.MAP\n\nPoble Espanyol's famous flamenco club features a variety of shows from seasoned performers and new talent. From \u20ac41, rising to \u20ac70 for the show plus dinner. Reservations required.\n\n### La Terrrazza\n\nPoble Espanyol Espanya 932 724 980, laterrrazza.com.May\u2013Oct Thurs\u2013Sat 11.45pm\u20136am.MAP\n\nOpen-air summer club for nonstop dance, house and techno. Don't get there until 3am and be prepared for the style police. Admission \u20ac15\u201320.\n\n< Back to Montju\u00efc\n\n# Port Ol\u00edmpic and Poble Nou\n\nSights\n\nRestaurants\n\nClubs\n\nThe main waterfront legacy of the 1992 Olympics was the Port Ol\u00edmpic, the marina development which lies fifteen minutes' walk along the promenade from Barceloneta. Locals make full use of the beach and boardwalks, descending in force at the weekends for a leisurely lunch or late drink in one of the scores of restaurants and bars. There are also fine beaches further north near the old working-class neighbourhood of Poble Nou, while the impressive Museu Blau anchors the waterside zone known as the Parc del F\u00f2rum. Access to the area is by metro to Ciutadella-Vila Ol\u00edmpica or Poble Nou, or bus #59 runs from the Ramblas through Barceloneta and out to Port Ol\u00edmpic.\n\n## Port Ol\u00edmpic\n\n Ciutadella-Vila Ol\u00edmpica.MAP\n\nApproaching the Olympic port, the golden mirage above the promenade slowly reveals itself to be a huge **copper fish** (courtesy of Frank Gehry, architect of the Bilbao Guggenheim). It's the emblem of the seafront development constructed for the 1992 Olympics, incorporating the port itself \u2013 site of many of the Olympic watersports events \u2013 which is backed by the city's two tallest buildings, the **Torre Mapfre** and the steel-framed **Hotel Arts Barcelona** , both 154m tall. Two wharves contain the bulk of the action: the Moll de Mestral has a lower deck by the marina lined with bars, while the Moll de Gregal sports a double-decker tier of seafood restaurants. The beach, meanwhile, turns into a full-on summer resort, backed by class-conscious clubs along Passeig Mar\u00edtim appealing to local rich kids and A-list celebs.\n\nPort Ol\u00edmpic\n\n## City beaches\n\nFrom  Ciutadella-Vila Ol\u00edmpica it's a 15min walk along the promenade to Bogatell beach.MAP\n\nA series of sandy beaches stretches for 5km north of Port Ol\u00edmpic. Split into different named sections (Nova Ic\u00e0ria, Bogatell etc), which boast showers, playgrounds and open-air caf\u00e9-bars, it's a pretty extraordinary facility to find so close to a city centre. A sunny day, even in winter, brings the locals out in force, and the sands are regularly swept and replenished.\n\n## Centre de la Platja\n\nPg. Mar\u00edtim Ciutadella-Vila Ol\u00edmpica 932 210 348, www.bcn.cat\/platges.June\u2013Sept Tues\u2013Sun 10am\u20137pm, plus March\u2013May weekends only.MAP\n\nOn the boardwalk arcade, in front of the Hospital del Mar, the city council has opened a beach visitor centre. There's a full programme of walks, talks and sports, as well as buckets and spades for kids, and frisbees, volleyball and beach tennis gear available for pick-up games on the sands.\n\n### Diagonal Mar\n\nThe waterfront district north of Poble Nou was developed in the wake of the Universal Forum of Cultures Expo (held in 2004). It's promoted as **Diagonal Mar** , anchored by the Diagonal Mar shopping mall ( El Maresme F\u00f2rum or tram T4) and with several classy hotels, convention centres and exhibition halls grouped nearby. The dazzling **Edifici F\u00f2rum** building is the work of Jacques Herzog (architect of London's Tate Modern), while the main open space is claimed to be the second-largest square in the world after Beijing's Tiananmen Square. This immense expanse spreads towards the sea, culminating in a giant solar-panelled canopy that overlooks the marina, beach and park areas. In summer, temporary bars, dancefloors and chill-out zones are established at the **Parc del F\u00f2rum** , and the city authorities have shifted some of the bigger annual music festivals and events down here to inject a bit of life outside convention time. At other times it can be a bit soulless, but it's definitely worth the metro or tram ride if you're interested in heroic-scale public projects.\n\n## Rambla del Poble Nou\n\n Poble Nou.MAP\n\nA twenty-minute walk up the beach from Port Ol\u00edmpic, the Rambla de Poble Nou runs through the most attractive part of **Poble Nou** (\"New Village\"). The old industrial neighbourhood is at the heart of a huge city regeneration scheme, but the local avenue remains unchanged \u2013 a run of modest shops, caf\u00e9s and restaurants, including the classic juice and milk bar of _El Tio Che_ (Rambla de Poble Nou 44\u201346).\n\n## Cementiri de Poblenou\n\nAv. d'Icaria Bogatell.Daily 8am\u20136pm.MAP\n\nThis vast nineteenth-century mausoleum has its tombs set in walls 7m high. With birdsong accompanying a stroll around the flower-lined pavements, quiet courtyards and chapels, this village of the dead is a rare haven of peace in the city.\n\nThe Edifici F\u00f2rum\n\n## Museu Blau\n\nPl. Leonardo da Vinci 4\u20135, Parc del F\u00f2rum El Maresme F\u00f2rum, or tram T4 932 566 002, www.museuciencies.bcn.cat.Tues\u2013Sat 10am\u20137pm, Sun 10am\u20138pm.\u20ac7, includes entrance to Montju\u00efc botanical garden; free first Sun of the month & every Sun after 3pm\n\nThe Natural Science Museum's million-strong collection of rocks, fossils, plants and animals has a state-of-the-art home in the visually stunning Blue Museum, housed in the F\u00f2rum building, whose permanent exhibition \u2013 Planeta Vida (Planet Life) \u2013 plots a journey through the history of life on earth. It's heavily focused on evolutionary, whole-earth, Gaia principles, with plenty of entertaining, interactive bells and whistles to guide you through topics as diverse as sex and reproduction and conservation of the environment.\n\n< Back to Port Ol\u00edmpic and Poble Nou\n\n## Restaurants\n\n### Agua\n\nPg. Mar\u00edtim 30 Ciutadella-Vila Ol\u00edmpica 932 251 272, grupotragaluz.com.Daily 1\u20133.45pm & 8\u201311.30pm (Fri & Sat until 4.30pm & 12.30am).MAP\n\nNicest boardwalk restaurant on the strip, with a contemporary Mediterranean menu, from salads to grills. Prices are pretty fair (most dishes \u20ac10\u201325), so it's usually busy.\n\n### El Cangrejo Loco\n\nMoll de Gregal 29\u201330 Ciutadella-Vila Ol\u00edmpica 932 210 533, www.elcangrejoloco.com.Daily 1pm\u2013midnight.MAP\n\nThe terrace at the \"Crazy Crab\" offers ocean views, and the food is first rate. A mixed fried-fish plate is a typically Catalan starter, and the rice dishes are thoroughly recommended. From around \u20ac40.\n\n### Els Pescadors\n\nPl. Prim 1 Poble Nou 932 252 018, www.elspescadors.com.Daily 1\u20133.45pm & 8\u201311.30pm; closed two weeks in Dec.MAP\n\nThe best top-class fish restaurant in Barcelona? It's a tough call, but many would choose this hideaway place in a pretty Poble Nou square. The menu offers daily fresh fish dishes, and plenty more involving rice, noodles or salt-cod. Don't go mad and you'll escape for around \u20ac60.\n\n### Xiringuito Escrib\u00e0\n\nRonda del Litoral 42, Platja Bogatell Ciutadella-Vila Ol\u00edmpica 932 210 729, xiringuitoescriba.com.Daily: Mon\u2013Fri 1\u20134.30pm & 8\u201311pm, Sat & Sun 1\u20135pm & 8\u201311pm.MAP\n\nBeachfront restaurant (really a glorified beach shack) that's enough off the beaten track (a 15min walk from Port Ol\u00edmpic) to mark you out as in the know. High points are paellas and daily fish specials (around \u20ac20), followed by sensational cakes and pastries from the Escrib\u00e0 family patisserie.\n\n## Clubs\n\n### Club Catwalk\n\nC\/Ramon Trias Fargas 2\u20134 Ciutadella-Vila Ol\u00edmpica 932 216 161, www.clubcatwalk.net.Tues\u2013Sun midnight\u20135am.MAP\n\nPortside club of choice playing house, funk, soul and r'n'b for well-heeled locals and visitors. It's under the _Hotel Arts_ ; admission \u20ac12\u201315.\n\n### Razzmatazz\n\nC\/Pamplona 88 Bogatell 933 208 200, www.salarazzmatazz.com.Thurs (Razz club only), Fri & Sat 1\u20136am.MAP\n\nRazzmatazz hosts the biggest in-town rock gigs, and at weekends turns into \"five clubs in one\", spinning mixed sounds in variously named bars. Admission \u20ac17.\n\n< Back to Port Ol\u00edmpic and Poble Nou\n\n# Dreta de l'Eixample\n\nSights\n\nShops\n\nCaf\u00e9s\n\nRestaurants and tapas bars\n\nThe nineteenth-century street grid north of Pla\u00e7a de Catalunya is the city's main shopping and business district. It was designed as part of a revolutionary urban plan \u2013 the Eixample in Catalan (\"Extension\" or \"Widening\") \u2013 that divided districts into regular blocks, whose characteristic wide streets and shaved corners survive today. Two parallel avenues, Passeig de Gr\u00e0cia and Rambla de Catalunya, are the backbone of the Eixample, with everything to the east known as the Dreta de l'Eixample (the right-hand side). It's here that the bulk of the city's famous _modernista_ (Catalan Art Nouveau) buildings are found, along with an array of classy galleries and some of the city's most stylish shops. Start your exploration from either Metro Passeig de Gr\u00e0cia or Metro Diagonal.\n\n## Passeig de Gr\u00e0cia\n\n Passeig de Gr\u00e0cia.MAP\n\nThe prominent avenue, which runs northwest from Pla\u00e7a de Catalunya, was laid out in its present form in 1827. It later developed as a showcase for _modernista_ architects, eagerly commissioned by status-conscious merchants and businessmen. Walk the length of Passeig de Gr\u00e0cia from Pla\u00e7a de Catalunya to Avinguda Diagonal (a 25min stroll) and you'll pass some of the city's most extraordinary architecture, notably the famous group of buildings (including casas Amatller and Batll\u00f2) known as the Mansana de la Disc\u00f2rdia, or \"Block of Discord\", as they show off wildly varying manifestations of the _modernista_ style and spirit. Further up is Antoni Gaud\u00ed's iconic apartment building La Pedrera, while, in between, wrought-iron Art Nouveau street lamps, fashion stores and designer hotels set the tone for this resolutely upscale avenue.\n\nPasseig de Gr\u00e0cia\n\n## Museu del Perfum\n\nPg. de Gr\u00e0cia 39 Passeig de Gr\u00e0cia 932 160 121, www.museodelperfume.com.Mon\u2013Fri 10.30am\u20138pm, Sat 11am\u20132pm.\u20ac5.MAP\n\nAt the back of the Regia perfume store is a private collection of over five thousand perfume and essence bottles from Egyptian times onwards. There are some exquisite pieces, including Turkish filigree-and-crystal ware and bronze and silver Indian elephant flasks, while more modern times are represented by scents made for Brigitte Bardot, Grace Kelly and Elizabeth Taylor.\n\n## Casa Amatller\n\nFundaci\u00f3 Amatller, Pg. de Gr\u00e0cia 41 Passeig de Gr\u00e0cia www.amatller.org.MAP\n\nJosep Puig i Cadafalch's striking Casa Amatller (1900) was designed for Antoni Amatller, a Catalan chocolate manufacturer, art collector, photographer and traveller. It's awash with coloured ceramic decoration, while inside the hallway twisted stone columns are interspersed with dragon lamps. Guided tours are occasionally available (check website for details) while renovation works continue and usually include a visit to Amatller's photographic studio as well as chocolate-tasting in the original kitchen. The house also displays temporary exhibitions under the auspices of the Amatller Institute of Hispanic Art.\n\nCasa Amatller\n\n## Casa Batll\u00f2\n\nPg. de Gr\u00e0cia 43 Passeig de Gr\u00e0cia 932 160 306, www.casabatllo.es.Daily 9am\u20139pm, access occasionally restricted.\u20ac18.50.Advance-purhase tickets advised, in person, by phone or online.MAP\n\nThe most extraordinary creation on the \"Block of Discord\" is the Casa Batll\u00f2, designed for the industrialist Josep Batll\u00f2 and finished in 1907. Antoni Gaud\u00ed created an undulating facade that Salvador Dal\u00ed later compared to \"the tranquil waters of a lake\". The sinuous interior, meanwhile, resembles the inside of some great organism, complete with meandering, snakeskin-patterned walls. Self-guided audio tours show you the main floor, the patio and rear facade, the ribbed attic and celebrated mosaic roof-top chimneys. Advance tickets are recommended (by phone, in person or online); the scrum of visitors can be a frustrating business at peak times.\n\nCasa Batll\u00f2\n\n## Fundaci\u00f3 Antoni T\u00e0pies\n\nC\/Arag\u00f3 255 Passeig de Gr\u00e0cia 934 870 315, www.fundaciotapies.org.Tues\u2013Sun 10am\u20137pm.\u20ac7.MAP\n\nThe definitive collection of the work of Catalan abstract artist Antoni T\u00e0pies i Puig (who died in 2012) is housed in _modernista_ architect Llu\u00eds Dom\u00e8nech i Montaner's first important building, the Casa Montaner i Simon (1880). You can't miss it \u2013 the foundation building is capped by T\u00e0pies's own striking sculpture, _N\u00favol i Cadira_ (\"Cloud and Chair\", 1990), a tangle of glass, wire and aluminium. The artist was born in Barcelona in 1923 and was a founding member (1948) of the influential avant-garde Dau al Set (\"Die at Seven\") artists' group. T\u00e0pies's abstract style matured in the 1950s, with underlying messages and themes signalled by the inclusion of everyday objects and symbols on the canvas. Changing exhibitions focus on selections of T\u00e0pies's work, while other shows highlight works by other contemporary artists.\n\n## Museu Egipci de Barcelona\n\nC\/de Val\u00e8ncia 284 Passeig de Gr\u00e0cia 934 880 188, www.museuegipci.com.Mon\u2013Sat 10am\u20138pm, Sun 10am\u20132pm.\u20ac11.MAP\n\nBarcelona's Egyptian Museum is an exceptional private collection of over a thousand ancient artefacts, from amulets to sarcophagi \u2013 there's nothing else in Spain quite like it. Visitors are given a detailed English-language guidebook, but the real pleasure is a serendipitous wander, turning up items like cat mummies or the rare figurine of a spoonbill (ibis) representing an Egyptian god. There are temporary exhibitions, plus a good shop and terrace caf\u00e9, while the museum also hosts children's activities and themed events.\n\n### Modernisme\n\nThe Catalan offshoot of Art Nouveau, **modernisme** , was the expression of a renewed upsurge in Catalan nationalism in the 1870s. Its most famous exponent was **Antoni Gaud\u00ed i Cornet** (1852\u20131926), whose buildings are apparently lunatic flights of fantasy, which at the same time are perfectly functional. His architectural influences were Moorish and Gothic, while he embellished his work with elements from the natural world. The imaginative impetus he provided inspired others like **Llu\u00eds Dom\u00e8nech i Montaner** (1850\u20131923) \u2013 perhaps the greatest _modernista_ architect \u2013 and **Josep Puig i Cadafalch** (1867\u20131957), both of whom also experimented with the use of ceramic tiles, ironwork, stained glass and stone carving. This combination of traditional methods with modern technology became the hallmark of _modernisme_ \u2013 producing some of the most exciting architecture to be found anywhere in the world.\n\n## Jardins de les Torres de les Aig\u00fces\n\nC\/Roger de Ll\u00faria 56, between c\/Consell de Cent and c\/Diputaci\u00f3 Girona.Daily 10am\u2013dusk.Free.MAP\n\nThe original nineteenth-century Eixample urban plan \u2013 by utopian architect Ildefons Cerd\u00e0 \u2013 was drawn up with local inhabitants very much in mind. Space, light and social community projects were part of the grand design, and something of the original municipal spirit can be seen in the Jardins de les Torres de les Aig\u00fces, an enclosed square (reached down a herringbone-brick tunnel) centred on a Moorish-style water tower. It has been handsomely restored by the city council, who turn it into a backyard family beach every summer, complete with sand and paddling pool. Another example of the old Eixample lies directly opposite, across c\/Roger de Ll\u00faria, where the cobbled **Passatge del Permanyer** cuts across an Eixample block, lined by candy-coloured single-storey townhouses.\n\n## Mercat de la Concepci\u00f3\n\nBetween c\/de Valencia and c\/d'Arag\u00f3 Girona www.laconcepcio.com.Mon & Sat 8am\u20133pm, Tues\u2013Fri 8am\u20138pm.MAP\n\nFlowers, shrubs and plants are a Concepci\u00f3 speciality (the florists on c\/Valencia are open 24 hours a day), and there are some good snack bars inside the market and a few outdoor caf\u00e9s to the side. The market takes its name from the nearby church of **La Concepci\u00f3** (entrance on c\/Roger de Ll\u00faria), whose quiet cloister is a surprising haven of slender columns and orange trees.\n\nFlowers at the Mercat de la Concepci\u00f3\n\n## Palau Montaner\n\nC\/de Mallorca 278 Passeig de Gr\u00e0cia 933 177 652, www.rutadelmodernisme.com.Guided visits: for groups only; reservations required.\u20ac6.MAP\n\nThe Palau Montaner (1896) has a curious history \u2013 after the original architect quit, Llu\u00eds Dom\u00e8nech i Montaner took over halfway through construction, and the top half of the facade is clearly more elaborate than the lower part. Meanwhile, the period's most celebrated craftsmen were set to work on the interior, which sports rich mosaic floors, painted glass, carved woodwork and a monumental staircase.\n\nThe building is now the seat of the Madrid government's delegation to Catalunya, but it is possible to arrange **guided tours** that explain something of the house's history and show you the lavish public rooms, grand dining room and courtyard. It's unusual to be able to get inside a private _modernista_ house of the period, so it's definitely worth the effort.\n\n## La Pedrera\n\nPg. de Gr\u00e0cia 92, entrance on c\/Proven\u00e7a Diagonal 902 400 973, www.lapedrera.cat.March\u2013Nov daily 9am\u20138pm; Dec\u2013Feb daily 9am\u20136.30pm.\u20ac16.50.Nits d'estiulast week June to first week Sept Thurs\u2013Sat 9\u201311.30pm; \u20ac25, advance sales online. MAP\n\nAntoni Gaud\u00ed's weird apartment building at the top of Passeig de Gr\u00e0cia is simply not to be missed \u2013 though you can expect queues whenever you visit. Popularly known as La Pedrera, \"the stone quarry\", its rippled facade, curving around the street corner in one smooth sweep, is said to have been inspired by the mountain of Montserrat, while the apartments themselves resemble eroded cave dwellings. Indeed, there's not a straight line to be seen \u2013 hence the contemporary joke that the new tenants would only be able to keep snakes as pets. The self-guided visit includes a trip up to the extraordinary _terrat_ (roof terrace) to see at close quarters the enigmatic chimneys \u2013 you should note that the roof terrace is often closed if it's raining. In addition, there's an excellent exhibition about Gaud\u00ed's life and work installed under the 270 curved brick arches of the attic. **El Pis** (\"the apartment\"), on the building's fourth floor, re-creates the design and style of a _modernista_ -era bourgeois apartment in a series of extraordinarily light rooms that flow seamlessly from one to another. The apartment is filled with period furniture and effects, while the moulded door and window frames, and even the brass door handles, all follow Gaud\u00ed's sinuous building design. During the early summer's **Nits d'estiu** (advance booking essential) you can enjoy the amazing rooftop by night with a complimentary glass of _cava_ and music, while other concerts are also held at La Pedrera at various times.\n\nThrough the grand main entrance of the building there's access to the Pedrera **exhibition hall** , which hosts temporary art shows of works by major international artists.\n\nLa Pedrera\n\n## Vin\u00e7on\n\nPg. de Gr\u00e0cia 96 Diagonal 932 156 050, www.vincon.com.Mon\u2013Fri 10am\u20138.30pm, Sat 10.30am\u20139pm.MAP\n\nThe Vin\u00e7on store emerged in the 1960s as the country's pre-eminent purveyor of household furniture and design, pioneered by Fernando Amat, the \"Spanish Terence Conran\". There are various separate street entrances, including Carrer de Proven\u00e7a 273 and de Carrer Pau Claris 175, while the extraordinary furniture floor gives access to a terrace with views of La Pedrera. The _Sala Vin\u00e7on_ gallery (same hours as shop; admission free) puts on shows of graphic and industrial design and contemporary furniture.\n\nDisplay at Vin\u00e7on\n\n## Palau Robert\n\nPg. de Gr\u00e0cia 107 Diagonal 932 388 091, www.gencat.cat\/palaurobert.Mon\u2013Sat 10am\u20138pm, Sun 10am\u20132.30pm.Free.MAP\n\nVisit the information centre for the Catalunya region for regularly changing exhibitions on all matters Catalan, from art to business. There are several exhibition spaces, both inside the main palace \u2013 built as a typical aristocratic residence in 1903 \u2013 and in the old coach house. The centre is also an important concert venue for recitals and orchestras, while the gardens around the back are a popular meeting point for local nannies and their charges.\n\n## Palau Bar\u00f3 de Quadras\n\nAv. Diagonal 373 Diagonal 934 678 000, www.llul.cat.Mon\u2013Fri 8am\u20137.30pm.MAP\n\nThe beautifully detailed Palau Bar\u00f3 de Quadras (a Josep Puig i Cadafalch work from 1904) now serves as the headquarters of the Institut Ramon LLull, an organization that promotes Catalan language studies at universities worldwide. Though most of the building is closed to the public, visitors can look around its stunningly ornate ground floor during the institute's opening hours. If you first see the building from the Avinguda Diagonal side, be sure walk around to Carrer del Rossell\u00f3 \u2013 this side of the building is decorated in a more subdued, but very lovely, _modernista_ style.\n\nPalau Bar\u00f3 de Quadras\n\n## Casa de les Punxes\n\nAv. Diagonal 416\u2013420 Diagonal.No public access.MAP\n\nCadafalch's largest work, the soaring Casa Terrades, is more usually known as the Casa de les Punxes (\"House of Spikes\") because of its red-tiled turrets and steep gables. Built in 1903 for three sisters, and converted from three separate houses spreading around an entire corner of a block, the crenellated structure is almost northern European in style, reminiscent of a Gothic castle.\n\n< Back to Dreta de l'Eixample\n\n## Shops\n\n### Antonio Mir\u00f3\n\nRambla de Catalunya 125 Diagonal www.antoniomiro.es.Mon\u2013Sat 10.30am\u20138.30pm.MAP\n\nThe showcase for Barcelona's most innovative designer, now also branding accessories and household design items.\n\n### Bulevard dels Antiquarius\n\nPg. de Gr\u00e0cia 55\u201357 Passeig de Gr\u00e0cia bulevarddelsantiquaris.com.Mon\u2013Sat 10am\u20138.30pm; closed Sat in Aug.MAP\n\nAn arcade with over seventy shops full of antiques, from toys and dolls to Spanish ceramics and African art.\n\n### Casa del Llibre\n\nPg. de Gr\u00e0cia 62 Passeig de Gr\u00e0cia www.casadellibro.com.Mon\u2013Sat 9.30am\u20139.30pm.MAP\n\nBarcelona's biggest book emporium, with lots of English-language titles and Catalan literature in translation.\n\n### Colmado Quilez\n\nRambla de Catalunya 63 Passeig de Gr\u00e0cia www.lafuente.es.Mon\u2013Fri 9.30am\u20132pm & 4.30\u20138.30pm, Sat 9.30am\u20132pm, plus Sat afternoon Sept\u2013Dec.MAP\n\nClassic Catalan grocery piled high with tins, jars and packets, overseen by sober gents in collar and tie and blue smocks.\n\n### Cubi\u00f1\u00e0\n\nC\/Mallorca 291 Verdaguer www.cubinya.es.Mon\u2013Sat 10am\u20132pm & 4.30\u20138.30pm.MAP\n\nThe building is stupendous \u2013 Dom\u00e8nech i Montaner's _modernista_ Casa Thomas \u2013 while inside holds the very latest in household design.\n\n### Mango\n\nPPg. de Gr\u00e0cia 36, plus others Passeig de Gr\u00e0cia www.mango.com.Mon\u2013Sat 10am\u20139.30pm.MAP\n\nBarcelona is where high-street fashion chain Mango began. For last season's gear at unbeatable prices, head to Mango Outlet (c\/Girona 37).\n\n### Purificacion Garcia\n\nC\/de Proven\u00e7a 292 Diagonal www.purificaciongarcia.es.Mon\u2013Sat 10am\u20138.30pm.MAP\n\nA hot designer with an eye for fabrics \u2013 Garcia's first job was in a textile factory. She's also designed clothes for film and theatre, and her costumes were seen at the opening ceremony of the Barcelona Olympics.\n\n### Reserva Ib\u00e9rica\n\nRambla de Catalunya 61 Passeig de Gr\u00e0cia 902 112 641, reservaiberica.com.Mon\u2013Sat 9.30am\u20139pm.MAP\n\nA ham wonderland specializing in _jam\u00f3n ib\u00e9rico de bellota_ , the finest of all of the Spanish cured hams, which comes from acorn-fed pigs. Pick up pre-packaged samplers or tuck into a plate of paper-thin slices at one of the marble-topped tables.\n\n## Caf\u00e9s\n\n### Caf\u00e9 del Centre\n\nC\/Girona 69 Girona 934 881 101.Mon\u2013Fri 8am\u201311pm; closed Aug.MAP\n\nThis quiet coffee stop is only four blocks from the main drag of Passeig de Gr\u00e0cia. It's been here since 1873 (a plaque outside honours its service to the city) and, with its timeworn _modernista_ decor, seems largely unchanged.\n\n### Forn de Sant Jaume\n\nRambla de Catalunya 50 Passeig de Gr\u00e0cia 932 160 229.Mon\u2013Sat 9am\u20139pm.MAP\n\nUptown _pastisseria_ whose glittering windows are piled high with croissants, cakes, pastries and sweets. The small adjacent caf\u00e9 has _rambla_ seats, or you can take away your goodies for later.\n\n### Laie Llibreria Caf\u00e9\n\nC\/Pau Claris 85 Passeig de Gr\u00e0cia 933 027 310, www.laie.es.Mon\u2013Fri 9am\u20139pm, Sat 10am\u20139pm.MAP\n\nThe city's first and best bookshop-caf\u00e9 is known for its popular weekday buffet breakfast spread, set lunch deals and \u00e0 la carte dining.\n\n## Restaurants and tapas bars\n\n### La Bodegueta\n\nRambla Catalunya 100 Diagonal 932 151 725, www.labodegueta.com.Mon\u2013Fri 7am\u20131.45am, Sat 8am\u20131.45am, Sun & hols 6.30pm\u20131.45am; closed mornings in Aug.MAP\n\nLong-established _bodega_ with _cava_ and wine by the glass, as well as ham, cheese, anchovies and other tapas to soak it up.\n\n### Casa Calvet\n\nC\/de Casp 48 Catalunya 934 124 012, www.casacalvet.es.Mon\u2013Sat 1\u20133.30pm & 8.30\u201311.30pm; closed Mon June\u2013Aug.MAP\n\nAntoni Gaud\u00ed's earliest commissioned townhouse is a marvel of interior decoration, making for a truly glam night out. It offers a seasonally changing, modern Catalan menu, with desserts that are artworks in themselves, though with mains around the \u20ac30 mark (or tasting menus from \u20ac49 to \u20ac70) expect it to be a wallet-emptying experience.\n\n### Ciudad Condal\n\nRambla de Catalunya 18 Passeig de Gr\u00e0cia 933 181 997.Daily 7.30am\u20131.30am.MAP\n\nThis handy city-centre pit stop is the best of the uptown tapas-hall-style places. Breakfast sees the bar groan under the weight of a dozen types of crispy baguette sandwich, plus croissants and pastries, while the daily changing tapas selection ranges far and wide \u2013 _patatas bravas_ to octopus. It can be standing room only at lunch (and not much of that either), so get there early. Dinner doesn't really have the same buzz, but it's a useful stop for a drink at any time.\n\n### Elj apo n\u00e9s\n\nPtge. de la Concepci\u00f3 2 Diagonal 934 872 592, grupotragaluz.com.Daily 1\u20134pm & 8.30\u201311.30pm (Fri & Sat until 12.30am).MAP\n\nDesigner style \u2013 gunmetal grey interior, black-clad staff, sharp service \u2013 at moderate prices gives this minimalist Japanese restaurant the edge over its more traditional rivals. Tick your choices from the long menu of sushi, sashimi, noodles and tempura and hand it to the waiter; average cost is around \u20ac25 a head.\n\n### El Mussol\n\nC\/Arag\u00f3 261 Passeig de Gr\u00e0cia 934 876 151.branch at c\/de Casp 19 933 017 610.Daily 1pm\u20131am (c\/Arag\u00f3 branch from 8am).MAP\n\nChain of big rustic diners, known for their meat and vegetables _a la brasa_ (on the grill), most of which run between \u20ac6 and \u20ac12. They're good places to sample hearty Catalan country cooking, with snails and wild mushrooms on the menu all year round and _cal\u00e7ots_ (big spring onions) a spring speciality.\n\n### Tapas, 24\n\nC\/Diputaci\u00f3 269 Passeig de Gr\u00e0cia 934 880 977, www.carlesabellan.com.Mon\u2013Sat 9am\u2013midnight.MAP\n\nCarles Abellan, king of pared-down designer cuisine at his restaurant _Comer\u00e7, 24_ , offers a simpler tapas menu at this retro basement bar-diner. There's a reassuringly traditional feel that's echoed in the menu \u2013 _patatas bravas_ , Andalucian-style fried fish, _bombas_ (meatballs), chorizo sausage and fried eggs. But the kitchen updates the classics too, so there's also _calamares romana_ (fried squid) dyed black with squid ink or a burger with _foie gras_. Most tapas dishes cost around \u20ac4 to \u20ac16. And there's always a rush and bustle at meal times, so be aware that you might well have to queue.\n\n### Tragaluz\n\nPtge. de la Concepci\u00f3 5 Diagonal 934 870 621, www.grupotragaluz.com\/tragaluz.Daily 1.30\u20134pm & 8.30pm\u2013midnight.MAP\n\nA stylish uptown standby that attracts beautiful people by the score, and the classy Mediterranean-with-knobs-on cooking, served under a glass roof ( _tragaluz_ means \"skylight\"), doesn't disappoint. The menu ranges from seasonal salads to grilled seabass, with mains costing \u20ac16\u201330. It's a relaxing stop for those fresh off the _modernista_ trail (La Pedrera is just across the way).\n\n< Back to Dreta de l'Eixample\n\n# Sagrada Fam\u00edlia and Gl\u00f2ries\n\nSights\n\nShops\n\nCaf\u00e9s\n\nRestaurants and tapas bars\n\nIf there's one building that is an essential stop on any visit to Barcelona it's Antoni Gaud\u00ed's great church of the Sagrada Fam\u00edlia (Metro Sagrada Fam\u00edlia). In many ways this has become a kind of symbol for the city, representing the glory of Catalan design and endeavour. Most visitors make a special journey out to see the church and then head back into the centre, but it's worth taking in the few blocks south of the area known as Gl\u00f2ries for a further set of attractions, including the city's main concert hall and music museum, and Catalunya's flagship national theatre building.\n\nzoom left\n\nzoom right\n\n## Sagrada Fam\u00edlia\n\nC\/Mallorca 401 Sagrada Fam\u00edlia 935 132 060, www.sagradafamilia.org.Daily, April\u2013Sept 9am\u20138pm; Oct\u2013March 9am\u20136pm. Tours in English April\u2013Oct 11.15am, 12.30pm, 1.45pm & 3pm; Nov\u2013March 10.15am, 12.15pm & 3pm.\u20ac14.80 (under-10s free), or \u20ac19.30 including tour; combination ticket with Casa Museu Gaud\u00ed at Parc G\u00fcell \u20ac18.30.MAP\n\nThe metro drops you right outside the overpowering church of the Sagrada Fam\u00edlia (\"Sacred Family\"). Begun in 1882 on a modest scale, the project changed the minute that 31-year-old architect Antoni Gaud\u00ed took charge in 1884 \u2013 he saw in the Sagrada Fam\u00edlia an opportunity to reflect his own deepening spiritual feelings. Gaud\u00ed spent the rest of his life working on the church and was adapting the plans right up to his untimely death. Run over by a tram on June 7, 1926, his death was treated as a Catalan national disaster, and all of Barcelona turned out for his funeral.\n\nAlthough the building survived, Gaud\u00ed's plans were mostly destroyed during the Spanish Civil War. Work restarted in the 1950s amid great controversy, and has continued ever since \u2013 as have the arguments. Some maintained that the Sagrada Fam\u00edlia should be left incomplete as a memorial to Gaud\u00ed, others that the architect intended it to be the work of several generations. Either way, based on reconstructed models and notes, the project is now moving towards completion (within the next twenty years, it's said), and the building is beginning to take final shape.\n\nThe size alone is startling (Gaud\u00ed's original plan was to build a church to seat over 10,000 people), while the carved spires, monumental bronze doors and vibrant facades are an imaginative and symbolic _tour de force_. Gaud\u00ed made extensive use of human, plant and animal models to exactly produce the likenesses he sought \u2013 the spreading stone leaves of the roof in the church interior, for example, were inspired by the city's plane trees. A **lift** (\u20ac4.50) up one of the towers provides an unforgettable close-up view of the work, while in the **crypt** (where Gaud\u00ed is buried) a fascinating museum traces the construction of the church \u2013 you can also view sculptors and model-makers at work.\n\nSagrada Fam\u00edlia\n\n## Hospital de la Santa Creu i de Sant Pau\n\nC\/de Sant Antoni Maria Claret 167 Hospital de Sant Pau 935 537 801, www.santpaubarcelona.org.Mon\u2013Sat 10am\u20136.30pm (Nov\u2013March until 4.30pm), Sun & hols 10am\u20132.30pm. English-language tours Mon\u2013Sat noon, 1pm, 4pm & 5pm (Nov\u2013March noon, 1pm & 4pm), Sun & hols noon & 1pm.\u20ac8; tours \u20ac14.MAP\n\nLlu\u00eds Dom\u00e8nech i Montaner's _modernista_ public hospital is possibly the one building that can touch the Sagrada Fam\u00edlia for size and invention, its whimsical pavilions, turrets and towers adorned with sculpture, mosaics, stained glass and ironwork. The old buildings have been superseded by the modern hospital behind, but there are interesting guided tours of the historical complex \u2013 and it's an easy stroll up here from the Sagrada Fam\u00edlia.\n\n## Casa Macaya\n\nPg. de Sant Joan 108 Verdaguer.MAP\n\nJust four blocks from the Sagrada Fam\u00edlia, Josep Puig i Cadafalch's Casa Macaya (1898\u20131900) is well known for its imaginative exterior carvings by craftsman Eusebi Arnau \u2013 look for the angel holding a camera or the sculptor himself on his way to work by bike.\n\nCasa Macaya\n\n## Pla\u00e7a de les Gl\u00f2ries Catalanes\n\n Gl\u00f2ries.MAP\n\nBarcelona's major avenues all meet at Pla\u00e7a de les Gl\u00f2ries Catalanes, dedicated to the Catalan \"glories\", from architecture to literature. Gl\u00f2ries is at the centre of the city's latest bout of regeneration. Signature buildings are Jean Nouvel's cigar-shaped **Torre Agbar** , a remarkable aluminium-and-glass tower inspired by Montserrat, and the sleek new Disseny Hub, future home to the city's applied art collections. Nearby **Parc del Clot** shows what can be done in an urban setting within the remains of a razed factory site. Trams speed down Avinguda Diagonal to the Diagonal Mar district, passing the **Parc del Centre del Poble Nou** (10min walk, or tram stop Pere IV), another park on an old industrial site.\n\n## Museu del Disseny and Disseny Hub\n\nPl. de les Gl\u00f2ries Catalanes 37\u201338 Gl\u00f2ries museudeldisseny.cat.Check museum website for hours and entrance fee. Disseny Hub  932 566 713,  dhub.cat. Tue\u2013Sun 10am\u20138pm. Hub free. MAP\n\nThe new Museu del Dissney brings together Barcelona's applied art collections inside the Disseny Hub building. At the time of writing the museum was still transitioning into its new home, but in the meantime, the zinc-plated building alone is worth seeing. In addition to the museum, the Hub is also home to temporary exhibition spaces, a public library and the headquarters of local design institutions such as the Foment de les Arts i del Disseny (FAD).\n\n## Teatre Nacional de Catalunya\n\nPl. de les Arts 1 Gl\u00f2ries 933 065 700, www.tnc.cat.MAP\n\nCatalunya's National Theatre features an enterprising programme of classics, original works and productions by guest companies. The building itself makes a dramatic statement, presenting a soaring glass box encased within a Greek temple, and there are guided **tours** for anyone interested in learning more (currently Wed & Thurs; \u20ac8; reservations required).\n\nTeatre Nacional de Catalunya\n\n## L'Auditori\n\nC\/Lepant 150 Gl\u00f2ries 932 479 300, www.auditori.cat.Museu de la M\u00fasica, c\/Padilla 155  932 563 650,  www.museumusica.bcn.cat. Tues\u2013Sat & hols 10am\u20136pm, Sun 10am\u20138pm. \u20ac5, free first Sun of the month and every Sun after 3pm. MAP\n\nThe city's main contemporary concert hall is home to the Barcelona Symphony Orchestra (OBC), though the programme also includes chamber, choral, jazz and world concerts. There's also the city's entertaining Museu de la M\u00fasica (Music Museum), which displays a remarkable collection of historic instruments and musical devices.\n\n< Back to Sagrada Fam\u00edlia and Gl\u00f2ries\n\n## Shops\n\n### Centre Comercial Barcelona Gl\u00f2ries\n\nAv. Diagonal 208 Gl\u00f2ries www.lesglories.com.Mon\u2013Sat 10am\u201310pm.MAP\n\nAnchoring the neighbourhood is this huge 230-store mall with all the national high-street fashion names represented (H&M, Zara, Bershka, Mango) as well as a big Carrefour supermarket and an eight-screen cinema complex.\n\n### Els Encants Vells\n\nAv. Meridiana 69 Gl\u00f2ries encantsbcn.com.Mon, Wed, Fri & Sat 9am\u20138pm.MAP\n\nThe city's flea market has a shiny new home next to the Teatre Nacional de Catalunya. The site sports multiple levels of open-air treasure-hunting \u2013 all protected by a large canopy whose metallic underside reflects the bustling market below.\n\n## Caf\u00e9s\n\n### El Rac\u00f3 del Mercat\n\nMercat de la Sagrada Fam\u00edlia, c\/de Padilla 255 Sagrada Fam\u00edlia 934 363 452, www.mercatsagradafamilia.com.Tues\u2013Thurs 7am\u20132.30pm & 5.30\u20138.30pm, Fri 7am\u20138.30pm, Sat 7am\u20132pm.MAP\n\nOnly two blocks east of the Sagrada Fam\u00edlia \u2013 and not a tourist in sight. Browse the stalls and pick up your picnic lunch, and make for the stand-up market bar, which has pastries, sandwiches and tapas at local prices.\n\n## Restaurants and tapas bars\n\n### Alkimia\n\nC\/Ind\u00fastria 79 Sagrada Fam\u00edlia 932 076 115, www.alkimia.cat.Mon\u2013Fri 1.30\u20133.30pm & 8\u201310.15pm; closed Easter week and 3 weeks in Aug.MAP\n\nAsk Barcelona foodies which is the best Catalan new-wave restaurant in town, and once they've all stopped bickering this is the one they'll probably plump for. \"Alchemy\" is what's promised by the name, and that's what chef Jordi Vil\u00e0 delivers in bitingly minimalist style. It's a Michelin-starred operation, so reservations are vital \u2013 expect a bill north of \u20ac100 a head too (though the \u20ac40 lunch is a comparative bargain).\n\n### La Taquer\u00eda\n\nPtge. del Font 5 Sagrada Familia 931 26 13 59, lataqueria.eu.Tues\u2013Sun 1\u20134.30pm & 8.30\u201311.30pm (Fri & Sat until midnight).MAP\n\nWhen it comes to Mexican food in Europe, throwing the word \"authentic\" around can get you in trouble with the purists, but it's an apt description for the menu here, which ranges from _tacos al pastor_ to fresh-made guacamole (\u20ac5\u201310).\n\n< Back to Sagrada Fam\u00edlia and Gl\u00f2ries\n\n# Esquerra de l'Eixample\n\nSights\n\nShops\n\nRestaurants and tapas bars\n\nBars\n\nClubs\n\nThe long streets west of Rambla de Catalunya as far as Barcelona Sants station are perhaps the least visited on any city sightseeing trip. With all the major architectural highlights found on the Eixample's eastern (or right-hand) side, the Esquerra de l'Eixample (left-hand side) was intended by its nineteenth-century planners for public buildings and institutions, many of which still stand. However, the Esquerra does have its moments of interest \u2013 not least a couple of excellent museums and an eye-catching public park or two \u2013 while it's here that some of the city's best bars and clubs are found, particularly in the gay-friendly streets of the so-called Gaixample district, near the university.\n\nzoom left\n\nzoom right\n\n## Universitat de Barcelona\n\nGran Via de les Corts Catalanes 585, at Pl. de la Universitat Universitat.MAP\n\nBuilt in the 1860s, the grand Neoclassical university building is now largely used for ceremonies and administration purposes, but you can visit the main hall or the fine arcaded courtyards and extensive gardens. The traditional student meeting point is the _Bar Estudiantil_ , outside in Pla\u00e7a Universitat, where you can usually grab a pavement table.\n\nUniversity courtyards\n\n## Fundaci\u00f3 Francisco Godia\n\nC\/Diputaci\u00f3 250 Passeig de Gr\u00e0cia 932 723 180, www.fundacionfgodia.org.Mon & Wed\u2013Sun 10am\u20138pm.\u20ac6.MAP\n\nThe private art collection of aesthete and 1950s racing driver Francisco Godia springs a real surprise, concentrating on medieval art, ceramics and modern Catalan art. Not all of the collection can be shown at any one time, so pieces are rotated on occasion, while special exhibitions run in tandem, for which there's usually no extra charge.\n\n## Museu del Modernisme Catal\u00e0\n\nC\/Balmes 48 Passeig de Gr\u00e0cia 932 722 896, mmcat.cat.Mon\u2013Sat 10am\u20138pm, Sun & hols 10am\u20132pm.\u20ac10.MAP\n\nBarcelona's traditional \"gallery district\", around c\/Consell de Cent, is a fitting location for the stupendous _modernista_ collection housed in the Museu del Modernisme Catal\u00e0. It's the private enterprise of the celebrated Gothsland antiques gallery, and displays a collection forty years in the making \u2013 including the famous marble decorative vase by craftsman Eusebi Arnau that was the gallery's symbol for over thirty years. This is just one of 350 works on show across two exhibition floors in a restored former textile warehouse \u2013 the grand, vaulted basement contains paintings and sculpture while on the ground floor is _modernista_ furniture, from screens to sofas. There are paintings and works by many famous names, from oils by Ramon Casas i Carb\u00f3 to sinuous mirrors and tables by Antoni Gaud\u00ed (originally made for the casas Batll\u00f3 and Calvet). But above all, this a rare opportunity to examine extraordinary Art Nouveau fixtures and fittings by less familiar artists \u2013 wonderful creations by pioneering cabinet-maker Joan Busquets i Jan\u00e9, for example, the dramatic carved headboards of Gaspar Homar i Mezquida, or the expressive terracotta sculptures of Lambert Escaler i Mil\u00e0. As a crash course in the varied facets of Catalan _modernisme_ , beyond the iconic buildings themselves, it's invaluable.\n\nRamon Casas I Carb\u00f3, _Dama con Foulard Rojo_ , Museu del Modernisme Catal\u00e0\n\n## Mercat del Ninot\n\nC\/Mallorca 133 Hospital Clinic 934 536 512, www.mercatsbcn.com.Mon\u2013Fri 8am\u20138pm, Sat 8am\u20133pm.MAP\n\nOne of the oldest markets in the city is currently undergoing a major refurbishment, but there's a large temporary market building on c\/Casanova, in front of the hospital. You won't find many tourists here and the shops around the market are refreshingly down-to-earth places to buy clothes, jewellery, accessories and homeware.\n\nAround the back of the hospital, it's also worth having a look at the **Escola Industrial** , formerly a textile mill, which boasts a 1920s chapel by Joan Rubi\u00f3 i Bellv\u00e9r, who worked with Antoni Gaud\u00ed. Students usually fill the courtyards, and you're free to take a stroll through to view the highly decorative buildings.\n\n## Parc Joan Mir\u00f3\n\nC\/de Tarragona Tarragona.Daily 10am\u2013dusk.MAP\n\nParc Joan Mir\u00f3 was laid out on the site of the nineteenth-century municipal slaughter-house. It features a raised piazza whose only feature is Joan Mir\u00f3's gigantic mosaic sculpture _Dona i Ocell_ (\"Woman and Bird\"), towering above a shallow reflecting pool. The rear of the park is given over to games areas and landscaped sections of palms and firs, with a kiosk caf\u00e9 and some outdoor tables among the trees. The children's playground here is one of the best in the city.\n\nDona i Ocell at Parc Joan Mir\u00f3\n\n## Arenas de Barcelona\n\nGran Via de les Corts Catalanes 373\u2013385, at Pl. d'Espanya Espanya 932 890 244, www.arenasdebarcelona.com.Daily 10am\u201310pm.MAP\n\nThe landmark building on the north side of Pla\u00e7a d'Espanya is the fabulous Moorish-style bullring, the Arenas de Barcelona, originally built in 1900 but re-imagined as a swish shopping and leisure centre that opened in 2011. On top is a walk-around promenade circling the **dome** that offers 360-degree views, while inside are four floors of shopping and entertainment, including cinema, gym and various restaurants (some on the top-floor promenade).\n\n### On the Mir\u00f3 trail\n\nWhen you've seen one Mir\u00f3 \u2013 well, you start to see them everywhere in Barcelona, starting with the large ceramic mural visible on the facade at the airport. The towering _Dona i Ocell_ in the Parc Joan Mir\u00f3 is unmissable, but you should also look down at your feet on the Ramblas for the pavement mural at Pla\u00e7a de la Boqueria. Mir\u00f3 also designed the _Caixa de Pensions_ starfish logo splashed across the Caixa Forum arts centre. In many ways, it's a Mir\u00f3 city, whatever Picasso fans might think.\n\n## Parc de l'Espanya Industrial\n\nC\/de Sant Antoni Sants-Estaci\u00f3.Daily 10am\u2013dusk.MAP\n\nIf you have time to kill at Barcelona Sants station, nip around the south side to Basque architect Luis Pe\u00f1a Ganchegui's urban park. Built on the site of an old textile factory, there's a line of concrete lighthouses contrasting with an incongruously classical Neptune, as well as boating lake, caf\u00e9 kiosk, playground and sports facilities. It's a decent attempt at reconciling local interests with the otherwise mundane nature of the surroundings, typical of the city's approach to revitalizing unkempt urban corners.\n\n< Back to Esquerra de l'Eixample\n\n## Shops\n\n### Alta\u00efr\n\nGran Via de les Corts Catalanes 616 Passeig de Gr\u00e0cia www.altair.es.Mon\u2013Sat 10am\u20138.30pm.MAP\n\nEurope's biggest travel superstore has a massive selection of travel books, guides, maps and world music, plus a programme of travel- related talks and exhibitions.\n\n### Jean-Pierre Bua\n\nAv. Diagonal 469 Hospital Clinic www.jeanpierrebua.com.Mon\u2013Sat 10am\u20138.30pm.MAP\n\nThe city's high temple for fashion victims: a postmodern shrine for Yamamoto, Gaultier, Miyake, Galliano, McQueen, McCartney, Westwood and other international stars.\n\n## Restaurants and tapas bars\n\n### Cervesaria Catalana\n\nC\/Mallorca 236 Passeig de Gr\u00e0cia 932 160 368.Daily 9am\u20131am.MAP\n\nA place that is serious about its tapas and beer \u2013 the counters are piled high, supplemented by a blackboard list of daily specials, while the walls are lined with bottled brews from around the world. It gets busy after work and at meal times, and you might have to wait for a table.\n\n### Cinc Sentits\n\nC\/Aribau 58 Universitat 933 239 490, www.cincsentits.com.Tues\u2013Sat 1.30\u20133.30pm & 8.30\u201310pm; closed two weeks in Aug.MAP\n\nJordi Artal's \"Five Senses\" wows diners with its contemporary Catalan cuisine \u2013 and the restaurant now has a Michelin star to boot, so you'll need to book. Six- and eight-course \"tasting menus\" (\u20ac65 and \u20ac80, matching wines available) use rigorously sourced ingredients (wild fish, mountain lamb, seasonal vegetables, farmhouse cheeses) in elegant, pared-down dishes that are all about flavour.\n\n### Fast V\u00ednic\n\nC\/Diputaci\u00f3 251 Passeig de Gr\u00e0cia 934 873 241, www.fastvinic.com.Mon\u2013Sat noon\u2013midnight; closed Aug.MAP\n\nEmphasis in this designer sandwich bar is on top-of-the-range ingredients from sustainable sources (gourmet sandwiches, wraps and salads \u20ac5\u201310), plus a range of more than twenty Catalan wines starting at pocket-money prices. It's the more democratic outpost of the hallowed wine-and-foodie temple next door that is _Monv\u00ednic_ ( www.monvinic.com).\n\n### La Flauta\n\nC\/Aribau 23 Universitat 933 237 038.Mon\u2013Sat 8am\u20131am; closed 3 weeks in Aug.MAP\n\nOne of the city's best-value lunch menus sees diners queuing for tables early \u2013 get there before 2pm to avoid the rush. While the name is a nod to the house speciality gourmet sandwiches (a _flauta_ is a crispy baguette) there are also tapas-style meals served day and night (dishes \u20ac4\u201310).\n\n### Me\n\nC\/Paris 162 Proven\u00e7a 934 194 933, www.catarsiscuisine.com.Tues & Sat 8.45\u201311.30pm, Wed\u2013Fri 1.45\u20133.30pm & 8.45\u201311.30pm.MAP\n\nAll the rage for its clever fusion of cuisines from Vietnam and New Orleans by way of Barcelona \u2013 all places dear to owner Javier's heart. Expect gumbo and marinated shrimp alongside papaya salad or grilled Saigon rib-eye with lemongrass, all in a stylish neo-Colonial setting. Mains are \u20ac15\u201323 though lunch is a cheaper, simpler affair.\n\n### Out of China\n\nC\/Muntaner 100 Proven\u00e7a 934 515 555, www.outofchinabarcelona.com.Mon\u2013Sat 1\u20134pm & 8pm\u2013midnight, Sun 1\u20134pm.MAP\n\nMost Chinese restaurants in Barcelona are pretty bland, but not _Out of China_. Black-and-red decor, lanterns and jazz-lounge sounds set the tone for a contemporary Chinese menu that's particularly hot on veggie options, from fried aubergine to tofu curry. Lunch is a good deal but even at night prices won't break the bank, with most dishes around \u20ac8\u201310.\n\n### Soda Bus\n\nC\/d'Aribau 125 Diagonal 934 531 044, balta1900.com.Mon\u2013Wed 9am\u20135pm, Thurs 9am\u20135pm & 10pm\u20132.30am, Fri 9am\u20135pm & 10pm\u20133am, Sat 10pm\u20133am; closed 2 weeks in Aug.MAP\n\nFormerly known as _El Rac\u00f3 d'en Balta_ (it has now joined forces with the century-old _Balta Bar_ next door), this funky hangout serves a popular weekday lunch, while at night you can eat for around \u20ac30 from a Mediterranean market-led menu, chowing down things like vegetarian ceviche or fish and hand-cut chips.\n\n### La Taverna del Cl\u00ednic\n\nC\/Rossell\u00f3 155 Hospital Clinic 934 104 221, latavernadelclinic.com.Mon\u2013Sat 1\u20135pm & 8\u201311.30pm.MAP\n\nThis sleek _taverna_ , named for the hospital over the road, is a gourmet tapas spot that concentrates on rigorously sourced regional produce. Snacks at the solid marble bar come in at just a few euros, but the serious food, accompanied by artisan-made olive oil and a high-class wine list, is served at one of the ten tables (book in advance) and runs to more like \u20ac10\u201330 a dish, from crispy suckling pig to sea urchins artfully arranged atop mounds of sea salt.\n\n## Bars\n\n### Aire Sala Diana\n\nC\/de Valencia 236 Passeig de Gr\u00e0cia 934 878 342, www.arenadisco.com.Thurs\u2013Sat 11pm\u20132.30am.MAP\n\nThe hottest, most stylish lesbian bar in town is a relaxed place for a drink and a dance to pop, house and retro sounds. Gay men are welcome too.\n\n### Belchica\n\nC\/Villaroel 60 Urgell 625 814 001.Mon\u2013Thurs 6pm\u20132.30am, Fri & Sat 6pm\u20133am, Sun 6pm\u20132am.MAP\n\nBarcelona's first Belgian beer bar guarantees a range of decent brews (including Trappist beers) that you can't get anywhere else. It's a cosy haunt, playing electronica, new jazz, lounge, reggae and other left-field sounds, and there are music and poetry nights.\n\n### Dietrich\n\nC\/Consell de Cent 255 Universitat 934 517 707.Thurs 10pm\u20132.30am, Fri & Sat 10pm\u20133am.MAP\n\nCornerstone of the Gaixample scene is this well-known music bar and \"teatro-caf\u00e9\" \u2013 _tranquilo_ during the week, but ever more hedonistic as the weekend wears on, with drag shows, acrobats and dancers punctuating the DJ sets.\n\n### Dry Martini\n\nC\/Aribau 166 Proven\u00e7a 932 175 072, www.drymartinibcn.com.Mon\u2013Fri 1pm\u20132.30am (opens 6.30pm in Aug), Sat & Sun 6.30pm\u20133am.MAP\n\nWhite-jacketed bartenders, dark wood and brass fittings, a self-satisfied air \u2013 it could only be Barcelona's legendary uptown cocktail bar. To be fair, though, no one in town mixes drinks better.\n\n### Punto BCN\n\nC\/Muntaner 63\u201365 Universitat 934 878 342, www.arenadisco.com.Daily 6pm\u20132.30am.MAP\n\nGaixample classic that attracts a lively crowd for drinks, chat and music. Wednesday happy hour is a blast, while Friday night is party night.\n\n### Quilombo\n\nC\/Aribau 149 Proven\u00e7a 934 395 406.Tues\u2013Thurs 9.30pm\u20133am, Fri & Sat 8.30pm\u20133.30am.MAP\n\nUnpretentious music bar that's rolled with the years since 1971, featuring live guitarists, Latin American bands and a clientele that joins in enthusiastically, maracas in hand.\n\n### Vel\u00f3dromo\n\nC\/Muntaner 213 Hospital Clinic 934 306 022.Daily 6am\u20132.30am (Fri & Sat until 3am).MAP\n\nA gleaming facelift has put the glam back into this lofty, Parisien-style Art Deco gem. It's ideal for swish drinks and cocktails, though with a breakfast, tapas and bistro menu by renowned chef Jordi Vil\u00e0 it's also made for early starts and later dinners.\n\n## Clubs\n\n### Antilla BCN Latin Club\n\nC\/Arag\u00f3 141\u2013143 Urgell 934 514 564, www.antillasalsa.com.Wed 10pm\u20134am, Thurs\u2013Sat 11pm\u20136am, Sun 7pm\u20133am.MAP\n\nLatin and Caribbean tunes galore \u2013 rumba, son, salsa, merengue, mambo, you name it \u2013 for out-and-out good-time dancing. There are live bands, killer cocktails and dance classes most nights.\n\n### Arena Madre\n\nC\/Balmes 32 Passeig de Gr\u00e0cia 934 878 342, www.arenadisco.com.MAP\n\nThe \"mother\" club sits at the helm of _Arena_ 's gay empire, all within a city block (pay for one, get in to all) \u2013 frenetic house at _Arena Madre_ (daily 12.30\u20135.30am), high-disco antics at _Arena Classic_ (c\/de la Diputaci\u00f3 233; Fri & Sat 2.30\u20135.30am), more of the same plus dance, r'n'b, pop and rock at the more mixed _Arena VIP_ (Gran Via de les Corts Catalanes 593; Fri & Sat 1\u20136am), and vintage chart hits at _Arena Dandy_ (same address and hours). Admission is usually \u20ac6 on weekdays, including Fridays, and \u20ac12 on Saturdays.\n\n### Danzarama\n\nGran V\u00eda de les Corts Catalanes 604 Universitat 933 019 743, www.danzarama.com.Daily 7am\u20132am (Fri & Sat until 3am).MAP\n\nOne of the stalwarts of the uptown gastro-club scene, with a flashy fusion restaurant (open day and night), summer _terrassa_ and cool bar and lounge, great for starting the night before some serious dancing elsewhere.\n\n### Luz de Gas\n\nC\/Muntaner 246 Proven\u00e7a 932 097 711, www.luzdegas.com.Daily 11.30pm\u20135am, occasional gigs from 9.30pm.MAP\n\nSmart live music venue, housed in a former ballroom, which is popular with a slightly older crowd and hosts live bands (rock, blues, soul, jazz and covers) every night around midnight. Foreign acts appear regularly too, mainly jazz-blues types but also old soul acts and up-and-coming rockers. Admission up to \u20ac20.\n\n### Metro\n\nC\/Sep\u00falveda 158 Universitat 933 235 227, www.metrodiscobcn.com.Daily 12.15\u20135am (Fri & Sat until 6.30am).MAP\n\nA gay institution in Barcelona, with cabaret nights and other events midweek, and extremely crowded club nights at weekends in its two rooms playing either current dance and techno or retro disco.\n\n< Back to Esquerra de l'Eixample\n\n# Gr\u00e0cia and Parc G\u00fcell\n\nSights\n\nShop\n\nCaf\u00e9s\n\nRestaurants and tapas bars\n\nBars\n\nClubs\n\nGr\u00e0cia was a village for much of its early existence, before being annexed as a suburb in the late nineteenth century. It still feels set apart from the city in many ways, and though actual sights are few and far between it's well known for its cinemas, bars and restaurants. The one unmissable attraction, meanwhile, is just on the neighbourhood fringe, nearby Parc G\u00fcell, an extraordinary flight of fancy by architectural genius Antoni Gaud\u00ed. To get to Gr\u00e0cia take the FGC train from Pla\u00e7a de Catalunya to Gr\u00e0cia station, or the metro to either Diagonal (south) or Fontana (north). From any of the stations, it's around a 500m walk to Gr\u00e0cia's main square, Pla\u00e7a del Sol, hub of the neighbourhood's renowned nightlife.\n\nView from Parc G\u00fcell\n\n## Mercat de la Llibertat\n\nPl. Llibertat 27 Fontana 932 170 995, www.mercatsbcn.com.Mon\u2013Fri 8am\u20138.30pm, Sat 8am\u20133pm.MAP\n\nYou may as well start where the locals start, first thing in the morning, shopping for bread and provisions in the neighbourhood market. The red-brick and iron structure has been beautifully restored and at _El Tast de Joan Noi_ (next to the Joan Noi fish counter) you can sample the breakfast of champions \u2013 oysters, grilled razor clams and a glass of _cava_.\n\n## Casa Vicens\n\nC\/de les Carolines 24 Fontana.No public access.MAP\n\nAntoni Gaud\u00ed's first major private commission (1883\u201385) took its inspiration from the Moorish style, covering the facade of the house in green and white tiles with a flower motif. The decorative iron railings are a reminder of Gaud\u00ed's early training as a metalsmith (and he also designed much of the mansion's original furniture).\n\n## Pla\u00e7a de la Virreina\n\n Fontana.MAP\n\nThis pretty square, backed by the parish church of Sant Joan, is one of Gr\u00e0cia's favourites, with a couple of bars providing a place to rest and admire the handsome houses, most notably the Casa Rubinat (1909). Nearby streets, particularly **Carrer Verdi** , contain many of the neighbourhood's most fashionable boutiques, galleries and caf\u00e9s.\n\n## Verdi and Verdi Park\n\nC\/Verdi 32 and c\/Torrijos 49 Fontana 932 387 990, www.cines-verdi.com.MAP\n\nBarcelona's favourite art-house cinemas have sister locations in adjacent streets, with nine screens showing original-language movies from around the world. Tickets are \u20ac9 at the weekend, \u20ac6 on Monday and \u20ac8 from Tuesday to Friday (\u20ac7 for the first screening of the day).\n\n## Pla\u00e7a de la Vila de Gr\u00e0cia\n\n Diagonal.MAP\n\nThe 30m-high clock tower in the heart of Gr\u00e0cia was a rallying point for nineteenth-century radicals \u2013 whose twenty-first-century counterparts prefer to meet for brunch at the square's popular caf\u00e9 _terrassas_.\n\nPillared walkway in Parc G\u00fcell\n\n## Parc G\u00fcell\n\nC\/d'Olot Vallcarca\/Lesseps www.parkguell.cat.Daily: April 8am\u20138pm; May\u2013Oct 8am\u20139pm; Nov\u2013March 8.30am\u20136pm.\u20ac7 online or \u20ac8 ticket office\/ATMs at Lesseps and Vallcarca metro stations. Advance bookings recommended.MAP\n\nGaud\u00ed's Parc G\u00fcell (1900\u201314) was his most ambitious project after the Sagrada Fam\u00edlia, conceived as a \"garden city\" of the type popular at the time in England, but opened as a public park instead in 1922. Laid out on a hill, which provides fabulous views back across the city, the park is an almost hallucinatory expression of the imagination. Pavilions of contorted stone, giant decorative lizards, meandering rustic viaducts, a vast Hall of Columns, carved stone trees \u2013 all combine in one manic swirl of ideas and excesses, like the famous meandering ceramic bench that snakes along the edge of the terrace above the columned hall. Your ticket grants access to these sites, now part of the **monumental zone**. Only 400 visitors are allowed inside this zone each half-hour (once inside you can stay as long as you like), so it's advised to book ahead. The area outside the monumental zone is free.\n\nThe most direct route to Parc G\u00fcell is on bus #24 from Pla\u00e7a de Catalunya, Passeig de Gr\u00e0cia or c\/Gran de Gr\u00e0cia, which drops you at the eastern side gate. From  Vallcarca, walk a few hundred metres down Avinguda de Vallcarca until you see the mechanical escalators on your left, ascending Baixada de la Gl\u00f2ria \u2013 follow these to the western side park entrance (15min in total). From  Lesseps, turn right along Travessera de Dalt and then left up steep c\/Larrard, which leads (10min) straight to the main entrance on c\/Olot.\n\n## Casa-Museu Gaud\u00ed\n\nParc G\u00fcell Vallcarca\/Lesseps 932 193 811, www.casamuseugaudi.org.Daily: April & May 10am\u20138pm; June\u2013Sept 9am\u20138pm; Oct\u2013March 10am\u20136pm.\u20ac5.50, combination ticket with Sagrada Fam\u00edlia \u20ac18.30.MAP\n\nOne of Gaud\u00ed's collaborators, Francesc Berenguer, designed and built a turreted house within Parc G\u00fcell for the architect (though he only lived in it intermittently). This contains a diverting collection of some of the furniture Gaud\u00ed designed for other projects \u2013 a typical mixture of wild originality and brilliant engineering \u2013 as well as plans and objects related to the park and to Gaud\u00ed's life. His study and bedroom have been preserved and there's an inkling of his personality, too, in the displayed religious texts and pictures, along with a silver coffee cup and his death mask, made at the Santa Pau hospital where he died.\n\nGaud\u00ed's desk in the Casa-Museu Gaud\u00ed\n\n## Parc de la Creueta del Coll\n\nPg. de la Mare de Deu del Coll 89 Vallcarca.Daily 10am\u2013dusk.Free.MAP\n\nFor a different kind of experience altogether, combine a trip to Gaud\u00ed's extravagant park with this contemporary urban space laid out on the site of an old quarry, whose sheer walls were retained in the landscaping. You're greeted at the top of the park steps by an Ellsworth Kelly metal spike, while suspended by steel cables over water is a massive concrete claw by the Basque artist Eduardo Chillida. There are also palms, promenades and a kiosk-caf\u00e9.\n\nBus V17 from near Pla\u00e7a de Catalunya at Via Laietana and Metro Urquinaona, stops 100m from the park, or you can walk from  Vallcarca in about twenty minutes (there's a map of the neighbourhood at the metro station). It's worth knowing that if you visit Creueta del Coll first and then take the main Passeig de la Mare de Deu del Coll, there are signposts leading you into Parc G\u00fcell the back way.\n\n< Back to Gr\u00e0cia and Parc G\u00fcell\n\n## Shop\n\n### Hibernian Books\n\nC\/Montseny 17 Fontana www.hibernian-books.com.Mon 4.30\u20138.30pm, Tues\u2013Sat 10.30am\u20138.30pm.MAP\n\nBarcelona's only secondhand English bookstore has around 40,000 titles in stock \u2013 you can part-exchange, and there are always plenty of giveaway bargains available.\n\n## Caf\u00e9s\n\n### Gelateria Caffetteria Italiana\n\nPl. de la Revoluci\u00f3 2 Fontana 932 102 339.Sun\u2013Thurs 2pm\u20131am, Fri & Sat 2pm\u20132am; Nov\u2013April opens at 4pm.MAP\n\nFor real hand-made Italian ice cream (the shop has been churning it out since 1881), and a stroll around a pretty square in the sun. Expect queues at peak times and then more waiting as you struggle to choose from the twenty-odd flavours.\n\n### La Nena\n\nC\/Ramon i Cajal 36 Joanic 932 851 476.Mon 8.30am\u20132pm & 4\u201310.30pm, Tues\u2013Thurs 8.30am\u201310.30pm, Fri 8.30am\u201311pm, Sat 9am\u201311pm, Sun 9.30am\u201310.30pm; closed Aug.MAP\n\nGreat for home-made cakes, waffles, quiches, organic ice cream, squeezed juices and the like. But parents also like the \"little girl\" as it's very child-friendly, from the changing mats in the loos to the little seats, games and puzzles.\n\n## Restaurants and tapas bars\n\n### Flash, Flash\n\nC\/de la Granada del Pened\u00e8s 25 Diagonal 932 370 990, www.flashflashbarcelona.com.Daily 1pm\u20131.30am (bar open 11am\u20132am).MAP\n\nA classic 1970s survivor with a keen sense of style, _Flash, Flash_ does tortillas (\u20ac6\u20139) served any time you like, any way you like, from plain and simple to elaborately stuffed, with sweet ones for dessert. If that doesn't grab you, try the reasonably priced menu of salads, steaks, burgers and fish. Either way, you'll love the original white leatherette booths and monotone photo-model cutouts \u2013 very Austin Powers.\n\n### Kuai Momos\n\nC\/Mart\u00ednez de la Rosa 71 Diagonal 932 185 327, momosbcn.com.Mon\u2013Wed 8\u201311.30pm, Thurs\u2013Sat 8pm\u2013midnight.MAP\n\n\"Tapas with chopsticks\" reads the motto of this restaurant, which has an intimate vibe best described as sushi bar meets Catalan tavern. The kitchen rolls out quite the tour of Asia's greatest culinary hits, from pork-and-vegetable _momos_ (dumplings) and lacquered duck to fragrant Thai curries and plump fishcakes. Tapas run \u20ac5\u201314 (you'll eat well for around \u20ac30).\n\n### Nou Candanchu\n\nPl. de la Vila de Gr\u00e0cia 9 Diagonal 932 377 362.Mon, Wed, Thurs & Sun 7am\u20131am, Fri & Sat 7am\u20132am.MAP\n\nGood for lunch on a sunny day or a leisurely night out on a budget, when you can sit beneath the clock tower and soak up the atmosphere in the ever-entertaining local square. There's a wide menu \u2013 tapas and hot sandwiches, but also steak and eggs, steamed clams and mussels, or cod and hake cooked plenty of different ways. It's managed by an affable bunch of young guys, and there's a lot of choice for \u20ac8\u201312.\n\n### La Pepita\n\nC\/C\u00f2rsega 343 Diagonal\/Verdaguer 932 384 893, lapepitabcn.com.Mon 8pm\u20131.30am, Tues\u2013Sat 9am\u20131.30am, kitchen open 1\u20134.30pm & 7.30pm\u2013midnight; closed 2 weeks in Aug\/Sept.MAP\n\nThere's usually a queue out the door, and deservedly so. The tapas, like roasted chicken croquettes with romesco sauce (\u20ac4) or aubergine fritters with goats' cheese, honey and apples (\u20ac8), are fantastic, and the atmosphere is chatty and convivial. Hundreds of \"love notes\" scrawled by customers on the white-tiled walls hint at its popularity. It's a good place for drink if you show up outside of kitchen hours, too.\n\n### La Singular\n\nC\/Francesc Giner 50 Diagonal 932 375 098.Mon\u2013Thurs 1\u20134pm & 8.30\u201311.30pm, Fri 1\u20134pm & 8.30pm\u201312.30am, Sat 8.30pm\u201312.30am.MAP\n\nThe tiny kitchen turns out refined Mediterranean food at moderate prices (most dishes \u20ac5\u201315) \u2013 think aubergine and smoked fish salad or chicken stuffed with dates and ham. There's always something appealing on the menu for veggies, too. It's a cornerstone of the neighbourhood, with a friendly atmosphere, but there are only nine tables, so go early or reserve.\n\n### Taverna El Glop\n\nC\/Sant Llu\u00eds 24 Joanic 932 137 058, www.tavernaelglop.com.Daily 1pm\u20131am.MAP\n\nThe rusticity (stone-flagged floors, baskets of garlic) stops just the right side of parody and the lunch _men\u00fa del dia_ is one of the city's best deals; otherwise expect to spend around \u20ac25 a head for grills and other tavern specials prepared in front of you on the open kitchen ranges.\n\n## Bars\n\n### La Baignoire\n\nC\/Verdi 6 Fontana 677 408 993.Daily 7pm\u20131am (Fri & Sat until 3am).MAP\n\nCosy little wine bar offering a small corner of sophistication on an otherwise busy street \u2013 Ella Fitzgerald on CD, a dozen good wines by the glass and cheesy nibbles.\n\n### Caf\u00e9 Salambo\n\nC\/Torrijos 51 Fontana 932 186 966, www.cafesalambo.com.Mon\u2013Thurs noon\u20131am, Fri & Sat noon\u20133am, Sun noon\u2013midnight.MAP\n\nWhere the pre- and post-cinema crowd meets (both Verdi cinemas are on the doorstep). It's a long-standing neighbourhood hangout, with something of a colonial feel, and there are lots of wines and _cava_ by the glass, and good food too.\n\n### Caf\u00e9 del Sol\n\nPl. del Sol 16 Fontana 934 155 663, cafedelsol.cat.Daily 1pm\u20132.30am (Fri & Sat until 3am).MAP\n\nThe grandaddy of the Pla\u00e7a del Sol scene sees action day and night. On summer evenings, when the square is packed with revellers, there's not an outdoor table to be had, but even in winter this is a popular drinking den \u2013 the pubby interior has a back room and gallery, often rammed to the rafters.\n\n### Canig\u00f3\n\nC\/Verdi 2 Fontana 932 133 049, barcanigo.com.Mon\u2013Thurs 10am\u20132am, Fri 10am\u20133am, Sat 8pm\u20133am.MAP\n\nFamily-run neighbourhood bar now entering its third generation. It's not much to look at, but the drinks are cheap and it's a Gr\u00e0cia institution with a loyal following, packed out at weekends especially with a young, largely local crowd.\n\n### El\u00e8ctric Bar\n\nTrav. de Gr\u00e0cia 233 Joanic www.electricbarcelona.com.Daily 7pm\u20132am (Fri & Sat until 3am).MAP\n\nThe regular gigs and eclectic live programming may have stopped, but Gr\u00e0cia's original grunge-feel boho bar is still a great place for a drink, and Friday night is always a wild Brazilian music thrash. There's an occasional concert too, so keep checking the website or their Facebook page.\n\n### Heliogabal\n\nC\/Ramon i Cajal 80 Joanic www.heliogabal.com.Wed\u2013Sat 9.30pm\u20132am (Fri & Sat until 3.30am).MAP\n\nNot much more than a boiler room given a lick of paint, but filled with a cool, twenty-something crowd, here for the live poetry and music \u2013 expect something different every night (Catalan versifying, jazz jam sessions and earnest singer-songwriters), starting at 10pm. Admission is usually \u20ac5\u201310, depending on the act, and drinks aren't expensive.\n\n### Samsara\n\nC\/Terol 6 Fontana 932 853 688.Mon\u2013Thurs 8.30pm\u20131am, Fri 7.30pm\u20132am, Sat 12.30\u20133.30pm & 7.30pm\u20132am, Sun 12.30\u20133.30pm & 7.30pm\u20131am.MAP\n\nIt's totally Gr\u00e0cia \u2013 low tables, low lighting and painted concrete walls, plus a chillout soundtrack and a projection screen above the bar. There's also contemporary tapas and \"platillos\" (little plates) but it's a bar first and foremost, with DJs cracking out house and techno sets at the weekend.\n\n### Vinilo\n\nC\/Matilde 2 Diagonal 626 464 759.Mon\u2013Thurs 8pm\u20132am, Fri & Sat 8pm\u20133am.MAP\n\nWear a beret, surgically attached to your iPad? Favour _Blade Runner_ , Jeff Buckley and Band of Horses? This bar's for you \u2013 a dive bar with the lighting set at perpetual dusk, where time slips easily away.\n\n### Virreina\n\nPl. de la Virreina 1 Fontana 932 379 880.Daily 9am\u20132am.MAP\n\nAnother real Gr\u00e0cia favourite, on one of the neighbourhood's prettiest squares, with a very popular summer _terrassa_. Cold beer and sandwiches are served to a laidback crowd \u2013 it's one of those places where you drop by for a quick drink and find yourself staying for hours.\n\n## Clubs\n\n### Centre Artes\u00e0 Tradicion\u00e0rius\n\nTrav. de Sant Antoni 6\u20138 Fontana 932 184 485, www.tradicionarius.cat.MAP\n\nThe best place in town for folk, traditional and world music by Catalan, Spanish and visiting performers, including some occasional big names. Admission is usually \u20ac5\u201315, and you can expect anything from Basque bagpipes to Brazilian singers. There are also music and instrument workshops, while CAT sponsors all sorts of outreach concerts and festivals, including an annual international folk and traditional dance festival between January and April.\n\n### Otto Zutz\n\nC\/de Lincoln 15 Fontana 932 380 722, www.ottozutz.com.Tues\u2013Thurs 11pm\u20133am, Fri & Sat midnight\u20136am.MAP\n\nIt first opened in 1985, and has lost some of its erstwhile glam cachet, but this three-storey former textile factory still has a shedload of pretensions. The sounds are basically hip-hop, r'n'b and house, and with the right clothes and face you're in (you may or may not have to pay, depending on how impressive you are, the day of the week, the mood of the door staff, etc).\n\n< Back to Gr\u00e0cia and Parc G\u00fcell\n\n# Camp Nou, Pedralbes and Sarri\u00e0-Sant Gervasi\n\nSights\n\nShopping\n\nRestaurants and tapas bars\n\nBars\n\nClubs\n\nOn the northwestern edge of the centre, the city's famous football stadium, Camp Nou, draws locals and visitors alike, both to the big game and to the FC Barcelona museum. Nearby, across Avinguda Diagonal, the Palau Reial de Pedralbes is home to serene public gardens (the lush vegetation hides an early work by Gaud\u00ed), while a half-day's excursion can be made by walking from the palace, past the Gaud\u00ed dragon gate at Pavellons G\u00fcell, to the calm cloister at the Gothic monastery of Pedralbes. You can complete the day by returning via Sarri\u00e0, to the east, more like a small town than a suburb, with a pretty main street and market to explore. At night, the focus shifts to the bars and restaurants of neighbouring Sant Gervasi in the streets north of Pla\u00e7a de Francesc Maci\u00e0.\n\nzoom left\n\nzoom right\n\n## Avinguda Diagonal\n\n Maria Cristina.MAP\n\nThe uptown section of Avinguda Diagonal runs through the heart of Barcelona's flashiest business and shopping district. The giant **L'Illa** shopping centre flanks the avenue \u2013 the stepped design is a prone echo of New York's Rockefeller Center. Designer fashion stores are ubiquitous, particularly around **Pla\u00e7a de Francesc Maci\u00e0** and Avinguda Pau Casals \u2013 at the end of the latter, **Tur\u00f3 Parc** (daily 10am\u2013dusk) is a good place to rest weary feet, with a small children's playground and a caf\u00e9-kiosk. Meanwhile, behind L'Illa, it's worth seeking out **Pla\u00e7a de la Concordia** , a surprising survivor from the past amid the uptown tower blocks \u2013 the pretty little square is dominated by its church belltower and ringed by local businesses (florist, pharmacy, hairdresser), with an outdoor caf\u00e9 or two for a quiet drink.\n\nCamp Nou stadium\n\n## Camp Nou and FC Barcelona\n\nAv. Ar\u00edstides Maillol Collblanc\/Maria Cristina. 902 189 900 or  934 963 600 from outside Spain, www.fcbarcelona.com.Match tickets (\u20ac50\u2013100) also from Ticketmaster  ticketmaster.es. MAP\n\nIn Barcelona, football is a genuine obsession, with support for the local giants FC (Futbol Club) Barcelona raised to an art form. \"More than just a club\" is the proud boast, and during the dictatorship years the club stood as a Catalan symbol around which people could rally. Arch rivals, Real Madrid, on the other hand, were always seen as Franco's club. The swashbuckling team \u2013 past European champions and darling of football neutrals everywhere \u2013 plays at the magnificent Camp Nou football stadium, built in 1957, and enlarged for the 1982 World Cup semi-final to accommodate 98,000 spectators. A new remodelling (by architect Norman Foster) plans to update the stadium over the next few years, but even today Camp Nou provides one of the world's best football-watching experiences.\n\nThe **football season** runs from August until May, with league games usually played on Sundays. Tickets are relatively easy to come by, except for the biggest games, and go on general sale up to a month before each match \u2013 buy them online, at the ticket office, or by calling ServiCaixa.\n\nThe stadium complex hosts basketball, handball and hockey games with FC Barcelona's other professional teams, and there's also a public ice rink, souvenir shop and caf\u00e9.\n\n## Camp Nou Experience\n\nCamp Nou, Av. Ar\u00edstides Maillol, enter through Gates 7 & 9 Collblanc\/Maria Cristina. 902 189 900 or  934 963 600, www.fcbarcelona.com.April & mid-Oct to Dec Mon\u2013Sat 10am\u20136.30pm, Sun & hols 10am\u20132.30pm; May to mid-Oct daily 9.30am\u20137.30pm; tours until 1hr before closing.\u20ac23, children 6\u201313 years \u20ac17.MAP\n\nNo soccer fan should miss the Camp Nou stadium tour and museum, billed as the \"Camp Nou Experience\". The self-guided tour winds through the changing rooms, onto the pitch and up to the press gallery and directors' box for stunning views. The museum, meanwhile, is jammed full of silverware and memorabilia, while displays and archive footage trace the history of the club back to 1901. Finally, you're directed into the massive **FC Botiga** , where you can buy anything from a replica shirt to a branded bottle of wine.\n\nMuseu del Futbol\n\n## Palau Reial de Pedralbes\n\nAv. Diagonal 686 Palau Reial.Gardens daily 10am\u2013dusk.Free.MAP\n\nOpposite the university on Avinguda Diagonal, formal grounds stretch up to the Italianate Palau Reial de Pedralbes \u2013 basically a large villa with pretensions. It was built for the use of the royal family on their visits to Barcelona, with funds raised by public subscription, and received its first such visit in 1926. However, within five years the king had abdicated and the palace somewhat lost its role. Franco kept it on as a presidential residence and it later passed to the city. The rooms had been used to show off the city's applied art collections, but those collections are being moved to the new Museu del Disseny near Pla\u00e7a de les Gl\u00f2ries Catalanes. Although the palace is now closed to the public, the gardens \u2013 a breezy oasis of Himalaya cedars, strawberry trees and bougainvillea \u2013 are worth a visit. Hidden in a bamboo thicket, to the left-centre of the fa\u00e7ade, is the \"Hercules fountain\" (1884), an early work by Antoni Gaud\u00ed. He also designed the parabolic pergola, which is covered in climbing plants and is a nice place to sit and rest your feet. In late June, a music festival ( festivalpedralbes.com) takes place in the palace's gardens.\n\nThe Palau Reial de Pedralbes gardens\n\n## Pavellons G\u00fcell\n\nAv. de Pedralbes 7 Palau Reial 933 177 652, www.rutadelmodernisme.com.Tours Sat & Sun at 10.15am & 12.15pm in English, plus 11.15am & 1.15pm in Spanish\/Catalan.\u20ac6.MAP\n\nAs an early test of his capabilities, Antoni Gaud\u00ed was asked by his patron, Eusebi G\u00fcell, to rework the entrance, gatehouse and stables of the G\u00fcell summer residence. The resultant brick and tile buildings are frothy, whimsical affairs, though it's the gateway that's the most famous element. An extraordinary winged dragon of twisted iron snarls at the passers-by, its razor-toothed jaws spread wide in a fearsome roar. During the week you can't go any further than the gate, but guided visits show you the grounds and Gaud\u00ed's innovative stables, now used as a library by the university's historical architecture department.\n\n### Here be dragons\n\nThe slavering beast on Gaud\u00ed's dragon gate at the Pavellons G\u00fcell is not the vanquished dragon of Sant Jordi (St George), the Catalan patron saint, but the one that appears in the Labours of Hercules myth, a familiar Catalan theme in the nineteenth century. Gaud\u00ed's design was based on a work by the Catalan renaissance poet Jacint Verdaguer, a friend of the G\u00fcell family, who had reworked the myth in his epic poem, _L'Atl\u00e0ntida_ \u2013 thus, the dragon guarding golden apples in the Gardens of Hesperides is here protecting instead an orange tree (considered a more Catalan fruit). Gaud\u00ed's gate indeed can be read as an homage to Verdaguer, with its stencilled roses representing those traditionally given to the winner of the Catalan poetry competition, the Jocs Floral, which the poet won in 1877.\n\n## Sarri\u00e0\n\nFGC Sarri\u00e0, or bus #64 from Pl. Universitat or Pedralbes.MAP\n\nThe Sarri\u00e0 district was once an independent small town and still looks the part, with a narrow, traffic-free main street \u2013 c\/Major de Sarri\u00e0 \u2013 at the top of which stands the much-restored church of Sant Vicen\u00e7. The church flanks the main Passeig de la Reina Elisenda de Montcada, across which lies the neighbourhood market, Mercat Sarri\u00e0, housed in a 1911 _modernista_ red-brick building. You'll find a few other surviving old-town squares down the main street, prettiest of which is Pla\u00e7a Sant Vicen\u00e7 de Sarri\u00e0 (off c\/Ma\u00f1e i Flaquer), where there's a statue of the saint.\n\nMonestir de Pedralbes\n\n## Monestir de Pedralbes\n\nBiaxada del Monestir Palau Reial and 20min walk, or FGC Reina Elisenda and 10min walk, or bus #64 from Pl. Universitat 932 563 434, www.museuhistoria.bcn.cat.April\u2013Sept Tues\u2013Fri 10am\u20135pm, Sat 10am\u20137pm, Sun 10am\u20138pm; Oct\u2013March Tues\u2013Fri 10am\u20132pm, Sat & Sun 10am\u20135pm.\u20ac7.MAP\n\nFounded in 1326 for the nuns of the Order of St Clare, this is in effect an entire monastic village set within medieval walls on the outskirts of the city. The cloisters in particular are the finest in Barcelona, built on three levels and adorned by the slenderest of columns. Side rooms and chambers give a clear impression of medieval convent life, and also display a selection of the monastery's treasures, while the adjacent church contains the carved marble tomb of the convent's founder, Elisenda de Montcada, wife of King Jaume II. After 600 years of isolation, the monastery was sequestered by the Generalitat during the Civil War. It was turned into a museum in 1983, and a new adjacent convent was built, where the Clare nuns still reside.\n\n< Back to Camp Nou, Pedralbes and Sarri\u00e0-Sant Gervasi\n\n## Shopping\n\n### L'Illa\n\nAv. Diagonal 555\u2013559 Maria Cristina www.lilla.com.Mon\u2013Sat 10am\u20139.30pm.MAP\n\nThe landmark uptown shopping mall is stuffed full of designer fashion, plus Camper (shoes), FNAC (music, film and books), Sfera (cosmetics), Decathlon (sports), El Corte Ingl\u00e9s (department store), Caprabo (supermarket), food hall and much more.\n\nYou can get here by metro (Maria Cristina) or tram (from Pl. de Francesc Maci\u00e0), or on the **Tomb Bus shopping line service** , which also visits other uptown stores on a circular route from Pla\u00e7a de Catalunya (departures every 6\u20138min; tickets available on board).\n\n## Restaurants and tapas bars\n\n### Bar Tom\u00e1s\n\nC\/Major de Sarri\u00e0 49, FGC Sarri\u00e0 932 031 077.Mon\u2013Sat noon\u20134pm & 6\u201310pm; closed Aug.MAP\n\nThe best _patatas bravas_ in Barcelona? Everyone will point you here, to this unassuming, white-Formica-table bar in the suburbs (a 12min train ride from Pla\u00e7a de Catalunya FGC) for their unrivalled spicy fried potatoes with garlic mayo and _salsa picante_. It's not all they serve, but it might as well be.\n\nThey fry noon to 3pm and 6pm to closing, so if it's _patatas bravas_ you want, be sure to take a note of the hours.\n\n### Can Punyetes\n\nC\/Mari\u00e0 Cub\u00ec 189, FGC Muntaner 932 009 159, www.canpunyetes.com.Daily noon\u20133.45pm & 8pm\u2013midnight.MAP\n\nTraditional grillhouse-tavern that offers diners a taste of older times. Simple salads and tapas, open grills turning out _botifarra_ (sausage), lamb chops, chicken and pork \u2013 accompanied by grilled country bread, white beans and char-grilled potato halves. It's cheap (almost everything under \u20ac10) and locals love it.\n\n### Casa Fernandez\n\nC\/Santal\u00f3 46, FGC Muntaner 932 019 308, www.casafernandez.com.Mon\u2013Sat 1\u20135pm & 8pm\u2013midnight, Sun noon\u20134.30pm.MAP\n\nThe long kitchen hours are a boon for the bar-crawlers in this neck of the woods. It's a contemporary place featuring market cuisine, though they are specialists in \u2013 of all things \u2013 fried eggs, either served straight with chips or with Catalan sausage, garlic prawns or other variations. Most dishes are in the range of \u20ac6 to \u20ac15.\n\n### Fragments Caf\u00e8\n\nPl. de la Conc\u00f3rdia 12 Les Corts 934 199 613, fragmentscafe.com.Tues\u2013Wed 12.30pm\u20131am, Thurs & Fri 12.30\u20132.30am, Sat 11.30am\u20132.30am, Sun 11.30am\u20131am; closed 2 weeks in Aug.MAP\n\nA classy yet casual bistro popular with locals for its fresh, classic food that's served in the charming dining room or in the shaded garden. The lunch menu is a steal (\u20ac12), with the rest of the dishes, from tapas to fresh pasta, around \u20ac5\u201314.\n\n## Bars\n\n### Gimlet\n\nC\/Santal\u00f3 46, FGC Muntaner 932 015 306, www.gimletbcn.com.Mon\u2013Wed 6pm\u20131am, Thurs 6pm\u20132.30am, Fri & Sat 6pm\u20133am.MAP\n\nThis favoured cocktail joint is especially popular in summertime, when the streetside tables offer a great vantage point for watching the party unfold. There are also two or three other late-opening bars on the same stretch.\n\n### Universal\n\nC\/Mari\u00e0 Cub\u00ec 182, FGC Muntaner 934 136 362, www.universalbcn.com.Mon\u2013Sat 11pm\u20133.30am (Fri & Sat until 4.30am).MAP\n\nA classic designer music bar that's been part of the Barcelona style scene since 1985. Sounds range from house to back-to-the-80s, but be warned: drinks are fairly pricey and they operate a strict door policy; if your face doesn't fit you might not get in.\n\n### Kahala\n\nAvgda. Diagonal 537 Maria Cristina 934 309 026, www.kahalabarcelona.com.Mon 10am\u20136pm, Tues\u2013Fri 10am\u20131am, Sat 10am\u20132am.MAP\n\nOpen since 1971, this Hawaiian-themed bar is a treasure trove of Polynesian kitsch: gurgling waterfalls, bamboo furniture and grimacing tiki masks abound. The drinks \u2013 from the _Perla de Vicio_ (\"Pearl of Vice\") to the classic Mai Tai \u2013 pack quite the punch and are certain to ready you for an evening of island \u2013 pardon \u2013 _club_ hopping.\n\n## Clubs\n\n### Bikini\n\nC\/Deu i Mata 105 Les Corts 933 220 800, www.bikinibcn.com.Thurs\u2013Sat midnight\u20135am.MAP\n\nThis traditional landmark of Barcelona nightlife (behind the L'Illa shopping centre) offers a regular diet of great indie, rock, roots and world gigs followed by club sounds, from house to Brazilian, according to the night. Admission usually \u20ac15\u201325, though some big-name gigs up to \u20ac40.\n\n### Sala BeCool\n\nPl. Joan Llongueras 5 Hospital Clinic 933 620 413, www.salabecool.com.MAP\n\nThumping uptown club venue for local and national rock, indie and electro\/techno bands and DJs. Gigs usually Thursday to Saturday nights at around 10pm, followed by DJ sessions from midnight or 1am. Admission for either runs from \u20ac10 to \u20ac20, depending on who's appearing. They also sponsor Friday-night acoustic nights at the next-door Irish bar _Dublin_ (music Fri at 11pm, free admission, bar open daily until 3am).\n\n< Back to Camp Nou, Pedralbes and Sarri\u00e0-Sant Gervasi\n\n# Tibidabo and Parc de Collserola\n\nSights\n\nBar\n\nThe views from the heights of Tibidabo (550m), the peak that signals the northwestern boundary of the city, are legendary. On a clear day you can see across to the Pyrenees and out to sea even as far as Mallorca. However, while many make the tram and funicular ride up to Tibidabo's amusement park, few realize that beyond stretches the Parc de Collserola, an area of peaks, wooded river valleys and hiking paths \u2013 one of Barcelona's best-kept secrets. You can walk into the park from Tibidabo, but it's actually better to start from the park's information centre, across to the east above Vallvidrera, where hiking-trail leaflets are available. Meanwhile, families won't want to miss CosmoCaixa, the city's excellent science museum, which can easily be seen on the way to or from Tibidabo.\n\n## Parc d'Atraccions\n\nPl. del Tibidabo 932 117 942, www.tibidabo.cat.Days and hours vary, but basically June\u2013Sept Wed\u2013Sun, rest of the year Sat, Sun & hols only, closed Jan & Feb. Park open from noon until 7\u201311pm depending on season.Skywalk ticket \u20ac12.70, full admission \u20ac28.50, plus family\/discount tickets.MAP\n\nBarcelona's self-styled \"magic mountain\" amusement park takes full advantage of its hillside location to offer jaw-dropping perspectives over the city. Some of the most famous rides (like the aeroplane \u2013 spinning since 1928 \u2013 and the carousel) are grouped under the discounted \"Skywalk\" ticket. Summer weekends finish with parades, concerts and a noisy _correfoc_ , a theatrical fireworks display.\n\nView over the Parc d'Atraccions\n\n## Sagrat Cor\n\nElevator operates daily 10am\u20138pm & 3\u20137pm.\u20ac2.MAP\n\nNext to Tibidabo's amusement park climb the shining steps of the Templo Expiatorio de Espa\u00f1a \u2013 otherwise known as the Sagrat Cor (Sacred Heart). This is topped by a huge statue of Christ, and inside the church an elevator climbs to a viewing platform from where the city, surrounding hills and shimmering sea glisten in the distance.\n\n### Getting to Tibidabo\n\nReaching the heights of Tibidabo takes up to an hour, all told, from the city centre. First, take the FGC train (line 7) from Pla\u00e7a de Catalunya station to **Avinguda Tibidabo** (the last stop), where you cross the road to the tram\/bus shelter (the Bus Turist\u00edc stops here too). The **Tramvia Blau** , an antique tram service (Jan\u2013April & mid-Oct to Dec Sat, Sun & hols 10am\u20136pm; May\u2013June & mid-Sept to mid-Oct Sat, Sun & hols 10am\u20137.30pm; July to mid-Sept & Easter week daily 10am\u20137.30pm; \u20ac4.20 one-way) then runs you up the hill to Pla\u00e7a Doctor Andreu; out of season there's a bus service instead during the week. Here, you change to the **Funicular del Tibidabo** , with connections every 15min to Tibidabo (operates when the Parc d'Atraccions is open; \u20ac7.70 without park admission, \u20ac4.10 with park admission). Alternatively, the **Tibibus** runs direct to Tibidabo from Pla\u00e7a de Catalunya, outside El Corte Ingl\u00e9s (from 10.15am every day the park is open; \u20ac2.95, reimbursed with park admission).\n\n## Torre de Collserola\n\nCarretera de Vallvidrera al Tibidabo 932 117 942, www.torredecollserola.com.March, April, Nov & Dec Sat & Sun noon\u20131.45pm & 3.30\u20135.45pm (May & Oct until 6.45pm); July\u2013Sept Wed\u2013Sun noon\u20131.45pm & 3.30\u20137.45pm; closed Jan & Feb.\u20ac5.MAP\n\nFollow the road from the Tibidabo car park and it's only a few minutes' walk to Norman Foster's soaring communications tower, built for the 1992 Olympics. This features a glass elevator that whisks you up ten floors (115m) for extensive views \u2013 70km, they claim, on a good day. Note that there's a combo ticket for the tower available at the Tibidabo amusement park.\n\n## Parc de Collserola\n\nCentre d'Informaci\u00f3, FGC Baixada de Vallvidrera (on the Sabadell or Terrassa line from Pl. de Catalunya; 15min) 932 803 552, www.parcnaturalcollserola.cat.Daily 9.30am\u20133pm.MAP\n\nGiven a half-decent day, local bikers, hikers and outdoor enthusiasts all make a beeline for the city's ring of wooded hills beyond Tibidabo. The park information centre lies in oak and pinewoods, an easy ten-minute walk up through the trees from the FGC Baixada de Vallvidrera train station. There's a bar-restaurant here with an outdoor terrace, plus an exhibition on the park's history, flora and fauna, while the staff hand out English-language leaflets detailing the various park walks. Some of the well-marked paths \u2013 like the oak-forest walk \u2013 soon gain height for marvellous views over the tree canopy, while others descend through the valley bottoms to springs and shaded picnic areas. Perhaps the nicest short walk from the information centre is to the **Font de la Budellera** (1hr 15min return), a landscaped spring deep in the woods. If you follow the signs from the _font_ to the Torre de Collserola (another 20min), you can return to Barcelona on the funicular from the nearby suburban village of **Vallvidrera** (daily 6am\u2013midnight; every 6\u201310min), which connects to Peu del Funicular, an FGC train station on the line from Pla\u00e7a de Catalunya.\n\nTorre de Collserola\n\n## Museu-Casa Verdaguer\n\nVilla Joana, Carretera de l'Esgl\u00e9sia 4 932 047 805, www.verdaguer.cat.Mon & Wed\u2013Sun 10am\u20131.30pm, Tues 5\u20137pm (April\u2013Nov Sat 5\u20137pm).Free.MAP\n\nIf you're up at the park at the weekend, it's worth having a quick look inside the country house that sits just below the Collserola information centre. Jacint Verdaguer (1845\u20131902), the Catalan Renaissance poet, lived here briefly before his death, and the house has been preserved as an example of well-to-do nineteenth-century Catalan life.\n\n## CosmoCaixa\n\nC\/Isaac Newton 26 932 126 050, www.cosmocaixa.com.Tues\u2013Sun 10am\u20138pm.\u20ac4, under-16s free.MAP\n\nA dramatic refurbishment in 2005 turned the city's science museum into a must-see attraction, certainly if you've got children in tow \u2013 it's an easy place to spend a couple of hours and can break the journey on your way to or from Tibidabo. Partly housed in a converted _modernista_ hospice, the museum retains the original building but has added a light-filled public concourse and a huge underground extension with four subterranean levels, where hands-on experiments and displays investigate life, the universe and everything, \"from bacteria to Shakespeare\". The two big draws are the hundred tonnes of \"sliced rock\" in the Geological Wall and, best of all, the Bosc Inundat \u2013 nothing less than a thousand square metres of real Amazonian rainforest, complete with croc-filled mangroves, anacondas and giant catfish. Other levels of the museum are devoted to children's and family activities, which tend to be held at weekends and during school holidays \u2013 pick up a schedule when you arrive. There are also daily shows in the planetarium (in Spanish and Catalan only), a great gift shop and a caf\u00e9-restaurant with outdoor seating.\n\nThe easiest way to reach CosmoCaixa is by FGC train from Pla\u00e7a de Catalunya to Avinguda del Tibidabo station, and then walk up the avenue, turning left just before the ring road (10min) \u2013 or the Tramvia Blau or Bus Turist\u00edc can drop you close by.\n\nCosmoCaixa\n\n< Back to Tibidabo and Parc de Collserola\n\n## Bar\n\n### Mirablau\n\nPl. del Dr. Andrea, Av. Tibidabo 934 185 879.Daily 11am\u20135am.MAP\n\nUnbelievable city views from a chic bar near the Tibidabo funicular that fills to bursting at times. By day, a great place for coffee and views, by night a rich-kid disco-tunes stomping ground.\n\n< Back to Tibidabo and Parc de Collserola\n\n# Montserrat\n\nSights\n\nThe mountain of Montserrat, with its rock crags, vast monastery and hermitage caves, stands just 40km northwest of Barcelona. It's the most popular day-trip from the city, reached in around ninety minutes by train and then cable car or rack railway for a thrilling ride up to the monastery. Once there, you can visit the basilica and monastery buildings and complete your day with a walk around the woods and crags, using the two funicular railways that depart from the complex. There are caf\u00e9s and restaurants at the monastery, but they are relatively pricey and none too inspiring \u2013 you may wish to take a picnic instead.\n\n## Aeri de Montserrat\n\nMontserrat Aeri 938 350 005, www.aeridemontserrat.com.Departures every 15min, March\u2013Oct daily 9.40am\u20132pm & 2.35\u20137pm; Nov\u2013Feb daily 10.10am\u20132pm & 2.35\u20135.45pm.MAP\n\nFor the cable-car service, get off the train from Barcelona at Montserrat Aeri station (52min). You may have to wait in line fifteen minutes or so, but then it's only a five-minute swoop up the sheer mountainside to a terrace just below the monastery \u2013 probably the most exhilarating ride in Catalunya. Returning to Barcelona, the line R5 trains depart hourly from Montserrat Aeri (from 9.37am).\n\nMontserrat\n\n## Cremallera de Montserrat\n\nMonistrol de Montserrat 932 051 515, www.cremallerademontserrat.com.Departures every hour, daily 8.48am\u20135.38\/7.38pm (later services at weekends April\u2013Oct, plus daily July\u2013Sept).MAP\n\nThe alternative approach to the monastery is by the Montserrat rack railway, which departs from Monistrol de Montserrat station (the next stop after Montserrat Aeri, another 4min), and takes twenty minutes to complete the climb. The original rack railway on Montserrat ran between 1892 and 1957, and this modern replacement recreates the majestic engineering that allows the train to climb 550m in 4km. Returning to Barcelona, the line R5 trains depart at least hourly from Monistrol de Montserrat (from 9.21am).\n\n### Getting to Montserrat\n\nTo reach the Montserrat cable-car\/rack-railway stations, take the **FGC train** (line R5, direction Manresa), which leaves daily from **Pla\u00e7a d'Espanya** ( Espanya) at hourly intervals from 8.36am. All fare options are detailed at **Pla\u00e7a d'Espanya** , including return through-tickets from Barcelona (around \u20ac20) either for the train\/cable car or train\/rack railway. There are also two combination tickets available: the **Trans Montserrat** (\u20ac27.50), which includes all transport services, including unlimited use of the mountain funiculars, and the audiovisual show; and the **Tot Montserrat** (\u20ac43.70), which includes the same, plus monastery museum and a cafeteria lunch. Both tickets are also available at the Pla\u00e7a de Catalunya tourist office.\n\n## Monestir de Montserrat\n\nVisitor centre 938 777 701, www.montserratvisita.com.Daily 9am\u20135.30pm (Sat, Sun & July\u2013Sept until 6.45pm). Walking maps and accommodation advice available.MAP\n\nLegends hang easily upon the monastery of Montserrat. Fifty years after the birth of Christ, St Peter is said to have deposited an image of the Virgin (known as La Moreneta), carved by St Luke, in one of the mountain caves. The icon was lost in the early eighth century after being hidden during the Moorish invasion, but reappeared in 880, accompanied by the customary visions and celestial music. A chapel was built to house it, and in 976 this was superseded by a Benedictine monastery, set at an altitude of nearly 1000m. Miracles abounded and the Virgin of Montserrat soon became the chief cult image of Catalunya and a pilgrimage centre second in Spain only to Santiago de Compostela \u2013 the main pilgrimages to Montserrat take place on April 27 and September 8.\n\nThe monastery's various outbuildings \u2013 including hotel, post office, souvenir shop and bar \u2013 fan out around an open square, and there are extraordinary mountain views from the terrace. The best restaurant is inside the _Hotel Abat Cisneros_ (meals around \u20ac40), though the finest views are from the cliff-edge _Restaurant de Montserrat_ (around \u20ac25) \u2013 the self-service cafeteria, one floor up, is where you eat with the all-inclusive _Tot Montserrat_ ticket.\n\nMonastery building at Montserrat\n\n## Bas\u00edlica\n\nBasilica daily 7.30am\u20138pm. Access to La Moreneta 8\u201310.30am & noon\u20136.30pm (mid-July to Sept also 7.30\u20138pm).Free.MAP\n\nOf the religious buildings, only the Renaissance basilica, dating largely from 1560 to 1592, is open to the public. **La Moreneta** stands above the high altar \u2013 reached from behind, by way of an entrance to the right of the basilica's main entrance. The approach to this beautiful icon reveals the enormous wealth of the monastery, as you queue along a corridor leading through the back of the basilica's rich side chapels. Signs at head height command \"SILENCE\" in various languages, but nothing quietens the line which waits to kiss the image's hands and feet.\n\nThe best time to be here is when Montserrat's world-famous **boys' choir** sings (Mon\u2013Fri at 1pm & 6.45pm, Sun at noon & 6.45pm; not Sat; performance times may vary during school holidays at Christmas\/New Year and from late June to mid-Aug). The boys belong to the Escolania, a choral school established in the thirteenth century and unchanged in musical style since its foundation.\n\nArtwork in the Bas\u00edlica\n\n## Museu de Montserrat\n\n 938 777 745.Daily 10am\u20135.45pm.\u20ac7.MAP\n\nThe monastery museum presents a few archeological finds brought back by travelling monks, together with valuable painting and sculpture dating from the thirteenth century onwards, including works by Old Masters, French Impressionists and Catalan _modernistas_. There's also a collection of Byzantine icons, though other religious items are in surprisingly short supply, as most of the monastery's valuables were carried off by Napoleon's troops who sacked the complex in 1811. For more on the history, and to learn something of the life of a Benedictine community, visit the **Espai Audiovisual** (Mon\u2013Fri 9am\u20135.30pm, Sat & Sun 9am\u20136.45pm; \u20ac5), near the information office.\n\n## Mountain walks\n\nFunicular departures vary by season, but mostly every 20min, daily 10am\u20136pm, weekends only Oct\u2013March.Santa Cova \u20ac3.50 return, Sant Joan \u20ac9 return, combination ticket \u20ac10\n\nFollowing the mountain tracks to the caves and hermitages, you can contemplate Goethe's observation of 1816: \"Nowhere but in his own Montserrat will a man find happiness and peace.\" The going is pretty good on all the tracks and the signposting is clear, but you do need to remember that you are on a mountain. Take water if you're hiking far and keep away from the edges.\n\nTwo separate funiculars run from points close to the cable-car station. One drops to the path for **Santa Cova** , a seventeenth-century chapel built where the Moreneta icon is said to have been found. It's an easy walk of less than an hour there and back. The other funicular rises steeply to the hermitage of **Sant Joan** , from where it's a tougher 45 minutes' walk to the **Sant Jeroni** hermitage, and another 15 minutes to the Sant Jeroni summit at 1236m. Several other walks are also possible from the Sant Joan funicular, perhaps the nicest being the 45-minute circuit around the ridge that leads all the way back down to the monastery.\n\nThe hermitage of Sant Joan\n\n< Back to Montserrat\n\n# Sitges\n\nSights\n\nShopping\n\nRestaurants and tapas bars\n\nBars\n\nThe seaside town of Sitges, 36km south of Barcelona, is definitely the highlight of the local coast \u2013 a great weekend escape for young Barcelonans, who have created a resort very much in their own image. It's also a noted gay holiday destination, with an outrageous annual carnival (February\/March) and a summertime nightlife to match. During the heat of the day, though, the tempo drops as everyone hits the beach. Out of season Sitges is delightful: far less crowded, and with a temperate climate that encourages promenade strolls and Old Town exploration.\n\n## The Beaches\n\nMAP\n\nThere are clean sands on either side of the Old Town, though they become extremely crowded in high season. For more space keep walking west from Passeig de la Ribera along Passeig Mar\u00edtim promenade, past eight interlinked beaches that run a couple of kilometres down the coast as far as the _Hotel Terramar_. Many of the handsome, nineteenth-century seafront mansions were built by successful local merchants (known as \"Americanos\") who had returned from Cuba and Puerto Rico.\n\nOn the beach at Sitges\n\n### Sitges information\n\nTrains to Sitges leave Passeig de Gr\u00e0cia or Barcelona Sants stations every twenty minutes, more frequently at peak times (destination Vilanova\/St Vicen\u00e7), and it's a thirty- to forty-minute ride depending on the service. The main Oficina Turisme (Pl. Eduard Maristany 2  938 944 251,  www.sitgestur.cat; Mon\u2013Fri 10am\u20132pm & 4\u20136.30pm, Sat 10am\u20132pm & 4\u20137pm, Sun 10am\u20132pm) is adjacent to the train station. Note that Monday isn't the best day to come, as the museums and many restaurants are closed. As well as Carnival, Sitges is known for its celebrated annual sci-fi, horror and fantasy fest, the Festival Internacional de Cinema ( www.sitgesfilmfestival.com) every October.\n\n## The Old Town\n\nMAP\n\nThe knoll overlooking the town beaches is topped by the landmark Baroque parish church dedicated to Sant Bartolomeu, whose festival is celebrated in the last week of August. The views from the terrace sweep along the coast, while behind in the narrow streets of the Old Town you'll find whitewashed mansions, as well as the town hall and the **Mercat Vell** (Old Market), the latter now an exhibition hall. The pedestrianized shopping street, **Carrer Major** , is the best place for browsing boutiques.\n\n### The gay scene and Carnival\n\nThe Sitges gay scene is frenetic and ever-changing, but the bulk of the nightlife is centred on Pla\u00e7a de l'Industria. Summer, of course, sees one long nonstop party, but Carnival time (Feb\/March) is also notoriously riotous, with a full programme of parades, masked balls, concerts and beach parties. Highlights are Sunday night's Debauchery Parade and Tuesday's Extermination Parade, in which exquisitely dressed drag queens twirl lacy parasols, while bar doors stand wide open and the celebrations go on till dawn. The other big bash is Gay Pride Sitges ( www.gaypridesitges.com), a weekend of events plus street parade every July.\n\n## Cau Ferrat, Maricel and Rom\u00e0ntic museums\n\nTues\u2013Sat 9.30am\u20132pm & 3.30\u20136.30pm (July\u2013Sept afternoons 4\u20137pm), Sun 10am\u20133pm.\u20ac3.50 each.MAP\n\nThree museums showcase the town's artistic heritage, not least **Museu Cau Ferrat** (c\/Fonollar; closed for renovations), the former house of _modernista_ artist Santiago Rusi\u00f1ol (1861\u20131931). Next door, the **Museu Maricel** (closed for renovations) contains minor art works, ceramics and sculpture, while in July and August (usually two evenings a week) the main part of the mansion itself is open for guided tours and concerts. Occupying a stately bourgeois house of 1793, the **Museu Rom\u00e0ntic** (c\/Sant Gaudenci 1) shows the lifestyle of a rich, nineteenth-century Sitges family (guided tours every hour).\n\n< Back to Sitges\n\nOld Town mansion\n\n## Shopping\n\n### Mercat Municipal\n\nAv. Artur Carbonell 938 949 777.Mon, Wed & Sat 8.30am\u20132pm, Tues, Thurs & Fri 8.30am\u20132pm & 5.30\u20138.30pm.MAP\n\nThe town market is a great place to put together a picnic of cured meats, olives, cheese, fresh bread and fruit.\n\n## Restaurants and tapas bars\n\n### Alfresco\n\nC\/Pau Barrabeitg 4 938 940 600, alfrescorestaurante.es.Daily 8.30pm\u2013midnight (May\u2013June closed Mon).MAP\n\nExuding romance \u2013 from its tucked-away location off a stepped alley to its trellised patio \u2013 the restaurant serves Catalan cuisine with Asian influences (green Thai curry, duck breast with glass noodles), with meals for \u20ac25\u201335.\n\n### Beach House\n\nC\/Paseo de la Ribera 33 935 168 136, www.beachhousesitges.com.Daily 9.30am\u20132.30am.MAP\n\nAussie owners have created a highly relaxed seafront restaurant, offering both lunch and dinner menus (from \u20ac13 and \u20ac20) of the best Mediterranean-Asian fusion food, from green Thai curry to black rice with squid and bonito fish flakes. It's a cocktail joint too, while the open-air terrace adds a touch of seaside romance.\n\n### Bonita Bonita\n\nC\/Major 17 932 228 540, bonitabonita.es.Mon & Wed\u2013Sun 4pm\u20132am.MAP\n\nThis vibrantly styled and very friendly tapas bar gives small plates an English twist with dishes like chorizo Scotch eggs. Its most memorable offering (though you may not remember much after ordering it) is the giantini, which is big enough for five people to share.\n\n### Fragata\n\nPg. de la Ribera 1 938 941 086, www.restaurantefragata.com.Daily 12.30\u20134pm & 8.30pm\u201312.30am.MAP\n\nTypical of the new wave of classy seafood places in town, where catch-of-the-day choices like grilled scallops or wild sea bass cost \u20ac17 to \u20ac28.\n\n### La Salseta\n\nC\/Sant Pau 35 938 110 419, lasalseta.com.Tues\u2013Sat 1.15\u20133.30pm & 8.30\u201311pm, Sun 1.15\u20133.30pm.MAP\n\nClassic, unpretentious Catalan dishes (cod with garlic confit, seafood paella) made from locally sourced ingredients have kept this 23-year-old restaurant's cosy dining room full of tourists and locals alike.\n\n## Bars\n\n### Baron\n\nC\/Sant Gaudenci 17. Mon\u2013Fri 9am\u20133pm & 6\u201311pm, Sat 10am\u20133pm & 6\u201311.30pm, Sun (June\u2013Aug only) 11.30am\u20133pm.MAP\n\nDefinitely not a style bar, but an old tavern that's a real slice of Sitges nonetheless. It's nowhere near the sea, so there's more of a down-to-earth local crowd.\n\n### Parrots Pub\n\nPl. de l'Industria 938 941 350, parrotspub.com.Daily 9pm\u20132am.MAP\n\nThe stalwart of the gay bar scene in Sitges, with front-row seats on all the action.\n\n### Vikingos\n\nC\/Marqu\u00e9s de Montroig 7\u20139 938 949 687, www.losvikingos.com.Mon\u2013Wed 1pm\u201312.30am Fri 1pm\u20131am, Sat 11am\u20131.30am, Sun 11am\u201312.30am.MAP\n\nLong-standing party-zone bar with a huge air-conditioned interior and streetside terrace. This and the similar _Montroig_ next door serve drinks, snacks and meals from morning until night to a really mixed crowd.\n\n### Voramar\n\nC\/Port Alegre 55. Daily 5pm\u2013midnight (Sat & Sun until 12.30am); closed Wed.MAP\n\nCharismatic seafront bar, away from the main crowds, just right for an ice-cold beer or sundowner cocktail.\n\n< Back to Sitges\n\n# Accommodation\n\nAlong the Ramblas\n\nBarri G\u00f2tic\n\nPort Vell and Barceloneta\n\nEl Raval\n\nSant Pere\n\nLa Ribera\n\nPort Ol\u00edmpic\n\nDreta de l'Eixample\n\nSagrada Fam\u00edlia and Gl\u00f2ries\n\nEsquerra de l'Eixample\n\nGr\u00e0cia\n\nLes Corts\n\nMontju\u00efc\n\nTibidabo\n\nHorta\n\nFinding a hotel vacancy in Barcelona at any time of year can be very difficult, so it's best to book in advance. The absolute cheapest rooms in a simple family-run _hostal_ or _pension_ , sharing a bathroom, cost around \u20ac50 (singles from \u20ac30), though if you want private facilities \u20ac70\u201380 a night is more realistic. There's a fair amount of choice around the \u20ac100 mark, while up to \u20ac200 gets you the run of decent hotels in most city areas. For Barcelona's most fashionable hotels count on \u20ac250\u2013400 a night, while dorm beds in youth hotels go for \u20ac20 to \u20ac30 depending on the season. An eight-percent tax (IVA) is added to all accommodation bills (though it's sometimes included in the quoted price), and a new \"tourist tax\" adds \u20ac0.65 to \u20ac2.25 per person per night to your bill for stays of up to seven days. Under 17s are exempt, and most tourist establishments in Barcelona fall into the lowest tax band.\n\nBreakfast isn't usually included, unless specifically stated in the reviews. Credit cards are accepted almost everywhere (though American Express isn't always). There's a lot of street noise in Barcelona, so bring earplugs if you're at all concerned.\n\nYou can reserve accommodation online with the city tourist board ( www.hotelsbcn.com) or make same-day bookings in person only at their tourist offices ( www.barcelonaturisme.com) at Pla\u00e7a de Catalunya, the airport and elsewhere. Many agencies offer apartment rental, which can be good value for couples and friends.\n\n### Along the Ramblas\n\n### Hostal Benidorm\n\nRamblas 37 Drassanes 933 022 054, www.hostalbenidorm.com.MAP\n\nRefurbished _pension_ that offers real value for money, hence the tribes of young tourists. Rooms available for one to five people, and a balcony with a Ramblas view if you're lucky (and prepared to pay a bit more). \u20ac75\n\n### Hotel 1898\n\nRamblas 109 Catalunya 935 529 552, www.hotel1898.com.MAP\n\nThe former HQ of the Philippines Tobacco Company got an eye-popping boutique refit, adding four grades of rooms (the standard is \"Classic\") in deep red, green or black, plus sumptuous suites and dramatic public areas, including neocolonial lobby-lounge bar, heated rooftop pool and glam spa facilities. Special rates from \u20ac160, otherwise \u20ac250\n\n### Hotel Eurostars Ramblas Boqueria\n\nRamblas 91\u201393 Liceu 933 435 461, www.eurostarsramblasboqueria.com.MAP\n\nSnappy little boutique rooms in a small three-star hotel right outside the Boqueria market. There's not much space, but all you need is on the doorstep, and the soundproofing is good so you get a street view without the racket. \u20ac130\n\n### Hotel Oriente\n\nRamblas 45 Liceu 933 022 558, www.orienteatiramhotels.com.MAP\n\nFor somewhere on the Ramblas that's traditional but not too pricey, this historic three-star is your best bet \u2013 nineteenth-century style in the grand public rooms and tastefully updated bedrooms, some with Ramblas views (though the quieter ones face inwards). \u20ac100\n\n### Hotel Rivoli Ramblas\n\nRamblas 128 Catalunya 934 817 676, www.hotelserhsrivolirambla.com.MAP\n\nThe elegant rooms in this four-star hotel are imaginatively furnished (Art Deco to contemporary), and all come with spacious bathrooms, while the front ones have floor-to-ceiling windows and classic Ramblas views. There's a lovely rooftop terrace and bar. Special rates from \u20ac99, otherwise \u20ac150\n\n< Back to Accommodation\n\n### What's the neighbourhood like?\n\nIf you hanker after a **Ramblas** view, you'll pay for the privilege \u2013 generally speaking, there are better deals to be had either side of the famous boulevard, often just a minute's walk away. Alongside some classy boutique choices, most of Barcelona's cheapest accommodation is found in the Old Town, principally the **Barri G\u00f2tic** and **El Raval** neighbourhoods, which both still have their rough edges \u2013 be careful (without being paranoid) when coming and going after dark. North of Pla\u00e7a de Catalunya, the **Eixample** \u2013 split into Right ( **Dreta** ) and Left ( **Esquerra** ) \u2013 has some of the city's most fashionable hotels. Those near **Sants** station are convenient for Montju\u00efc and the metro system, and those further north in **Les Corts** for the Avinguda Diagonal shopping district. For waterfront views look at **Port Vell** at the end of the Ramblas, and at the **Port Ol\u00edmpic** southeast of the Old Town \u2013 while new four- and five-star hotels abound further out on the metro at the **Diagonal Mar** conference and events site. If you prefer neighbourhood living then **Gr\u00e0cia** is the best base, as you're only ever a short walk away from its excellent bars, restaurants and clubs.\n\n### Barri G\u00f2tic\n\n### Hotel El Jard\u00ed\n\nPl. Sant Josep Oriol 1 Liceu 933 015 900, www.eljardi-barcelona.com.MAP\n\nLocation is all \u2013 overlooking the charming Pla\u00e7a del Pi \u2013 and though rooms can seem a bit bare and plain, some look directly onto the square. You can have breakfast at the hotel, though the _Bar del Pi_ below is nicer. \u20ac90, terrace or balcony \u20ac105\n\n### Hotel Rac\u00f3 del Pi\n\nc\/del Pi 7 Liceu 933 426 190, www.hotelh10racodelpi.com.MAP\n\nA stylish three-star hotel in a great location. Rooms \u2013 some with balconies over the street \u2013 have wood floors and granite-and-mosaic bathrooms. There's a glass of _cava_ on check-in and free coffee and pastries during the day in the bar. \u20ac125\n\n### Itaca Hostel\n\nc\/Ripoll 21 Jaume I 933 019 751, www.itacahostel.com.MAP\n\nBright and breezy hostel close to the cathedral offering spacious rooms with balconies. Dorms are mixed, though you can also reserve a private room or apartment (sleeps up to six), and with a capacity of only 30 it feels more house party than hostel. Dorm beds \u20ac27, rooms \u20ac70, apartments \u20ac120\n\n### Neri Hotel\n\nc\/de Sant Sever 5 Liceu\/Jaume I 933 040 655, www.hotelneri.com.MAP\n\nA delightful eighteenth-century palace close to the cathedral houses this stunning boutique hotel of just 22 rooms and suites, featuring swags of flowing material, rescued timber and granite-toned bathrooms. Catalan designers have created eye-catching effects, like a tapestry that falls four floors through the central atrium, while a beamed library and stylish roof terrace provides a tranquil escape. Breakfast is served bento-box style, either out in the courtyard in summer or in the fine contemporary Mediterranean restaurant. \u20ac285\n\n### Pensi\u00f3 Alamar\n\nc\/Comtessa de Sobradiel 1 Liceu\/Jaume I 933 025 012, www.pensioalamar.com.MAP\n\nIf you don't mind sharing a bathroom, this simply furnished _pension_ makes a convenient base. There are twelve rooms (including singles, doubles and triples), most with little balconies, and there's a friendly welcome, laundry service and use of a kitchen. No credit cards. \u20ac45\n\n### Pensi\u00f3n Mari-Luz\n\nc\/del Palau 4, 2\u00b0 Liceu\/Jaume I 933 173 463, www.pensionmariluz.com.MAP\n\nThis old mansion, on a quiet Barri G\u00f2tic street, offers inexpensive rooms (most share bathrooms), plus a more personal touch than many other places of its kind. It can be a tight squeeze when full, but a dozen slick apartments ( www.apartmentsunio.com) a few minutes' walk away in the Raval offer more space. Rooms \u20ac70, apartments \u20ac95\n\n< Back to Accommodation\n\n### Port Vell and Barceloneta\n\n### Bonic Barcelona\n\nc\/Josep Anselm Clav\u00e9 9 Drassanes 626 053 434, www.bonic-barcelona.com.MAP\n\nThis chic and charming \"urban guesthouse\" is located just a few steps from port and Ramblas, and features Gothic-Moorish decor and gorgeous tiled floors. The eight rooms are simple, and the three renovated bathrooms are shared. Advance reservations essential; minimum two-night stay required. \u20ac90\n\n### Equity Point Sea\n\nPl. del Mar 1\u20134 Barceloneta. www.equity-point.com; online bookings only.MAP\n\nThe budget beachside choice is this neat little hostel with modern en-suite rooms sleeping four to eight people. The attached caf\u00e9 looks right out onto the boardwalk. Dorm beds \u20ac26, includes breakfast\n\n### Hotel Duquesa de Cardona\n\nPg. de Colom 12 Drassanes\/Barceloneta 932 689 090, www.hduquesadecardona.com.MAP\n\nStep off the busy harbourfront highway into this soothing four-star haven, set in a remodelled sixteenth-century mansion. The rooms are calm and quiet, decorated in earth tones and immaculately appointed. Although not all of the rooms have harbour views, all guests have access to the stylish roof-deck overlooking the harbour. It's great for sundowner drinks, and boasts (if that's the word) probably the city's smallest outdoor pool. \u20ac160\n\n### Marina View B&B\n\nPg. de Colom Drassanes 678 854 456, www.marinaviewbcn.com.MAP\n\nSix classy rooms featuring bold colours, excellent bathrooms and nice touches, from minibars with normal drinks prices to hospitality trays. Breakfast included (served in the room), and advance reservations essential. \u20ac120, harbour views \u20ac140\n\n### W Barcelona\n\nPl. de la Rosa dels Vents Barceloneta 932 952 800, www.w-barcelona.com.MAP\n\nSignature building on the Barceloneta seafront is the stupendously cool, wave-shaped _W Barcelona_. The open-plan designer rooms offer fantastic views through floor-to-ceiling windows, while facilities are first rate, from iPod docks to an infinity pool. There's a hip, resort feel, with direct beach access, a chill-out lobby bar and rooftop lounge, while dining in _Bravo24_ is courtesy of hot Catalan chef Carles Abell\u00e1n. \u20ac260\n\n< Back to Accommodation\n\n### El Raval\n\n### Barcel\u00f3 Raval\n\nRambla del Raval 17\u201321 Liceu\/Sant Antoni 933 201 490, www.barcelo.com.MAP\n\nNeighbourhood landmark is this glow-in-the-dark tower whose USP is the 360-degree top-floor terrace with plunge pool and city views. Sophisticated, open-plan rooms have a crisp, space-station-style sheen, plus iPod docks, coffee-makers and other cool comforts, while the slinky lobby \"B-lounge\" is the place for everything from breakfast to cocktails. \u20ac120\n\n### Casa Camper\n\nc\/Elisabets 11 Universitat\/Liceu 933 426 280, casacamper.com.MAP\n\nSynonymous with creative, comfy shoes, Barcelona-based Camper has taken a bold step into the hospitality business with this sleek, minimalist hotel. All the rooms are divided by a corridor: the \"sleeping\" side faces a six-story tall vertical garden; the other part is a \"mini-lounge\" with a flat-screen TV, hammock and street-facing balcony. Breakfast is included. \u20ac215\n\n### Hostal C\u00e8ntric\n\nc\/Casanova 13 Universitat 934 267 573, www.hostalcentric.com.MAP\n\nA good upper-budget choice a couple of minutes' walk from the Raval proper. The recently refurbished rooms (2013) offer plenty of light, plus a\/c and private bathrooms. Some include balconies. \u20ac94\n\n### Hostal Grau\n\nc\/Ramelleres 27 Catalunya 933 018 135, www.hostalgrau.com.MAP\n\nA really friendly _pension_ with attractive, colour-coordinated rooms \u2013 superior rooms also have balconies and a touch of modern Catalan style. Two small private apartments in the same building (sleeping two to four, available by the night) offer a bit more independence. \u20ac85, en-suites \u20ac105, apartments \u20ac150\n\n### Hotel Espa\u00f1a\n\nc\/de Sant Pau 9\u201311 Liceu 935 500 000, hotelespanya.com.MAP\n\nThis _modernista_ icon has been sumptuously restored, and the gem-like interior \u2013 colourful mosaics, sculpted marble, iron swirls and marine motifs \u2013 has no equal in Barcelona. Guest rooms are a boutique blend of earth tones and designer style, with rain-showers, iPod docks and the like, while the handsome house restaurant \u2013 known as _Fonda Espa\u00f1a_ \u2013 offers contemporary Catalan bistro dishes by renowned chef Mart\u00edn Berasategui. \u20ac165\n\n### Hotel Onix Liceo\n\nc\/Nou de la Rambla 36 Liceu\/Drassanes 934 816 441, onixliceohotel.com.MAP\n\nSteps from Palau G\u00fcell, this four-star hotel features minimalist decor that melds nicely with the building's older architectural elements, such as the grand marble staircase that curves up from the lobby to the second floor. There's a tropical patio and big-for-Barcelona pool on the ground floor and an airy Mozarab-influenced lounge area. \u20ac130\n\n### Hotel Peninsular\n\nc\/de Sant Pau 34 Liceu 933 023 138, www.hotelpeninsular.net.MAP\n\nThe interesting old building originally belonged to a priestly order, which explains the slightly cell-like rooms. However, there's nothing spartan about the galleried courtyard (around which the rooms are ranged), hung with tumbling houseplants, while breakfast (included) is served in the arcaded dining room. \u20ac70\n\n### Hotel Sant Agust\u00ed\n\nPl. Sant Agusti 3 Liceu 933 181 658, www.hotelsa.com.MAP\n\nBarcelona's oldest hotel occupies a former convent, with front balconies overlooking a restored square and church. It's of three-star standard, with the best rooms located in the attic, from where there are rooftop views. \u20ac120\n\n### Market Hotel\n\nc\/Comte Borrell 68, at Ptge. Sant Antoni Abad Sant Antoni 933 251 205, www.markethotel.com.es.MAP\n\nThe designer-budget _Market_ makes a splash with its part Japanese, part neocolonial look \u2013 think jet-black rooms with hardwood floors and boxy wardrobes topped with travel trunks. \u20ac100\n\n< Back to Accommodation\n\n### Sant Pere\n\n### Pensi\u00f3 2000\n\nc\/Sant Pere M\u00e9s Alt 6, 1\u00b0 Urquinaona 933 107 466, www.pensio2000.com.MAP\n\nAs close to a traditional family-style B&B as Barcelona gets \u2013 seven rooms in a welcoming mansion apartment strewn with books, plants and pictures. A third person could easily share most rooms (\u20ac20 extra). \u20ac90\n\n< Back to Accommodation\n\n### La Ribera\n\n### Chic & Basic\n\nc\/de la Princesa 50 Jaume I 932 954 652, www.chicandbasic.com.MAP\n\nFrom the babbling blurb (\"it's fresh, it's cool, it's fusion\") to the all-in-white rooms with adjustable mood lighting, everything is punchily boutique and in-your-face. Chic, certainly \u2013 basic, not at all, though the concept eschews room service, mini-bars and tons of staff at your beck and call. Meals are courtesy of the attached \u2013 also effortlessly cool \u2013 restaurant. There's a more budget _Chic & Basic_ on c\/Tallers (near Pl. Universitat, El Raval) and apartments peppered around the city centre too (details on the website). \u20ac120\n\n### Equity Point Gothic\n\nc\/Vigatans 5 Jaume I. www.equity-point.com; online bookings only.MAP\n\nBackpacker heaven, not far from the Picasso museum. There's a great roof terrace, and all sorts of tours available, while each bunk bed gets its own cabinet and reading light. Dorm beds \u20ac25, includes breakfast\n\n### Hostal Nuevo Col\u00f3n\n\nAv. Marqu\u00e8s de l'Argentera 19, 1\u00b0 Barceloneta 933 195 077, www.hostalnuevocolon.com.MAP\n\nRun by a friendly family, featuring 24 spacious rooms painted yellow and kitted out with directors' chairs and double glazing. Sunny front rooms all have side views to Ciutadella park, while Fran\u00e7a station is right opposite. \u20ac55, en-suites \u20ac70\n\n### Hotel Banys Orientals\n\nc\/de l'Argenteria 37 Jaume I 932 688 460, www.hotelbanysorientals.com.MAP\n\nFunky boutique hotel with 43 minimalist rooms, plus some duplex suites in a nearby building. Hardwood floors, sharp marble bathrooms and urban-chic decor \u2013 not to mention bargain prices for this sort of style \u2013 make it a hugely popular choice. The attached restaurant, _Senyor Parellada_ , is a great find too. \u20ac115, suites \u20ac143\n\n< Back to Accommodation\n\n### Port Ol\u00edmpic\n\n### Hotel Arts Barcelona\n\nc\/Marina 19\u201321 Ciutadella-Vila Ol\u00edmpica 932 211 000, www.hotelartsbarcelona.com.MAP\n\nStill the city benchmark for five-star designer luxury, service and standards. Effortlessly classy rooms feature floor-to-ceiling windows with fabulous views. Stunning duplex apartments have their own perks (24hr butler service, personal Mini Cooper), and dining options range from the terrace restaurant to Michelin-starred chef Sergi Arola's contemporary tapas place, _Arola_. Seafront gardens encompass an open-air pool and hot tub, while the jaw-dropping Six Senses spa occupies the two top floors. \u20ac295\n\n< Back to Accommodation\n\n### Dreta de l'Eixample\n\n### Equity Point Centric\n\nPg. de Gr\u00e0cia 33 Passeig de Gr\u00e0cia. www.equity-point.com; online bookings only.MAP\n\nThe biggest hostel in the city occupies a refurbished _modernista_ building in a swish midtown location. Private twins, doubles, triples and quads available, all with shower room, balcony and views, while dorms (all en suite) sleep up to fourteen. Excellent facilities (bar, laundry, free wifi, common room with TV, music and Wii) include a spectacular roof terrace with views of the famous boulevard and its buildings. Dorm beds \u20ac13, rooms \u20ac130, breakfast included\n\n### Hostal L'Antic Espai\n\nGran Via de les Corts Catalanes 660 Passeig de Gr\u00e0cia 933 041 945, www.lanticespai.com.MAP\n\nCamp and cosy, this beautifully ornate period piece springs a surprise in every room, from mosaic tile floors to antique pendants, with candles and flowers at every turn. Room 102 has an original glassed-in balcony, and 107 opens onto an internal terrace with a candelabra-topped table. Modern bathrooms and DVD players keep comforts up to date. \u20ac120\n\n### Hostal Girona\n\nc\/Girona 24, 1\u00b0 Urquinaona 932 650 259, www.hostalgirona.com.MAP\n\nDelightful family-run _pension_ with a wide range of cosy, traditional rooms (some sharing a bathroom, others with a shower or full bath) \u2013 the best and biggest have balconies, though you can expect some noise. \u20ac60, full en-suites \u20ac85\n\n### Hostal Goya\n\nc\/de Pau Claris 74, 1\u00b0 Urquinaona 933 022 565, www.hostalgoya.com.MAP\n\nBoutique-style _pension_ that offers stylishly decorated rooms on two floors of a mansion building. There's a fair range of options, with the best rooms opening onto a balcony or terrace. Comfortable sitting areas, and free tea and coffee, are available on both floors. \u20ac95, balcony\/terrace \u20ac110\n\n### Hotel Condes de Barcelona\n\nPg. de Gr\u00e0cia 73\u201375 Passeig de Gr\u00e0cia 934 450 000, www.condesdebarcelona.com.MAP\n\nStraddling two sides of c\/Mallorca, the _Condes_ is fashioned from two former palaces, its rooms all turned out in contemporary style, some with jacuzzi, balcony or private terrace, some with views of Gaud\u00ed's La Pedrera. Meanwhile, Michelin-starred Basque chef Mart\u00edn Berasategui offers fine dining in the hotel's acclaimed _Lasarte_ restaurant and a more informal bistro menu in _Loidi_. \u20ac148\n\n### Hotel Majestic\n\nPg. de Gr\u00e0cia 68 Passeig de Gr\u00e0cia 934 881 717, www.hotelmajestic.es.MAP\n\nTraditional grande-dame hotel, first opened in 1918, though refitted in contemporary style and muted colours to provide a tranquil city-centre base. The absolute clincher is the rooftop pool and deck, with amazing views over to the Sagrada Fam\u00edlia, while the excellent _Drolma_ restaurant is a boon. \u20ac200\n\n### Hotel Omm\n\nc\/Rossell\u00f3 265 Diagonal 934 454 000, www.hotelomm.es.MAP\n\nThe glam designer experience that is _Omm_ means minimalist, open-plan rooms, a studiously chic bar, the Michelin-starred _Moo_ restaurant, terrace, pool and Spaciomm \"relaxation centre\", not to mention fearsomely handsome staff. It's not to everyone's taste \u2013 it's probably fair to say that the less annoyed you are by the endless Omm\/Moo tagging of services and facilities, the more you'll like the hotel. \u20ac200\n\n### Mandarin Oriental\n\nPg. de Gr\u00e0cia 38\u201340 Passeig de Gr\u00e0cia 931 518 888, www.mandarinoriental.com\/barcelona.MAP\n\nThe super-sleek _Mandarin Oriental_ fills the premises of a former bank building with a soaring white atrium and a serene selection of gorgeously light rooms. There's the obligatory superstar restaurant, _Moments_ (under the direction of the world's most Michelin-starred female chef, Carme Ruscalleda), while bar, spa, mimosa garden and rooftop \"dipping pool\" combine oriental tranquility and Euro cool. \u20ac400\n\n### The5rooms\n\nc\/Pau Claris 72, 1\u00b0 Urquinaona 933 427 880, www.thefiverooms.com.MAP\n\nThe owner's impeccable taste and fashion background are evident in gorgeous contemporary-styled B&B rooms that are spacious and light-filled, with original artwork above each bed, exposed brick walls and terrific bathrooms. Breakfast is served whenever you like, drinks are always available, and Jessica is happy to sit down and talk you through her favourite bars, restaurants and galleries. Suites and apartments are also available. \u20ac145, apartments \u20ac175 for 2 people\n\n< Back to Accommodation\n\n### Sagrada Fam\u00edlia and Gl\u00f2ries\n\n### Barcelona Urbany\n\nAv. Meridiana 97 Clot 932 458 414, www.barcelonaurbany.com.MAP\n\nBumper steel-and-glass 400-bed hostel that's a bit off the beaten track, but on handy metro and airport train routes and with amazing views of Torre Agbar. The rooms are like space-shuttle pods \u2013 boxy en-suites with pull-down beds (sleeping two to eight), power-showers and key-card lockers \u2013 that are just as viable for couples on a budget as backpackers. There's a bar and terrace, plus free gym, jacuzzi and pool entry in the same building. Dorm beds \u20ac10\u201330, private rooms \u20ac50\u201385, breakfast included\n\n### Hotel Eurostars Monumental\n\nc\/Consell de Cent 498\u2013500 Monumental 932 320 288, www.eurostarshotels.com.MAP\n\nAn excellent-value four-star choice within walking distance of the Sagrada Fam\u00edlia. The 45 rooms are crisply appointed in dark wood and earth tones, staff are really helpful and the top-floor suites boast terrace views of the Gaud\u00ed church. \u20ac120, suites \u20ac200\n\n< Back to Accommodation\n\n### Esquerra de l'Eixample\n\n### Alternative Creative Youth Home\n\nRonda Universitat 17 Universitat\/Catalunya 635 669 021, www.alternative-barcelona.com.MAP\n\nThe hostel hangout for an art crowd who love the laid-back vibe, projection lounge, cool music and city-savvy staff. The regular hostel stuff is well designed too, with a walk-in kitchen and a maximum of 24 people spread across three small dorms. Dorm beds \u20ac20\u201336\n\n### Casa de Billy\n\nGran Via de les Corts Catalanes Rocafort 934 263 048, casabillybarcelona.com.MAP\n\nA welcoming guesthouse in a restored nineteenth-century building near Pla\u00e7a d'Espanya, with spotless rooms (some en suite) elegantly decorated by Billy himself. It's a rich, refined and utterly charming experience \u2013 for an extremely reasonable price. Small breakfast included, two-night minimum, advance reservations essential (contact for directions), over-18s only. \u20ac70\n\n### Expo Hotel Barcelona\n\nc\/Mallorca 1\u201323 Sants-Estaci\u00f3 936 003 020, www.expohotelbarcelona.com.MAP\n\nBright, spacious rooms at a good-value four-star hotel. Each has a sliding window onto a capacious terrace and the best have views across to Montju\u00efc. \u20ac90\n\n### Gran Hotel Torre Catalunya\n\nAv. Roma 2\u20134 Sants-Estaci\u00f3 936 006 966, www.torrecatalunya.com.MAP\n\nThe landmark four-star deluxe hotel outside Sants station features sweeping views from all sides. Breakfast on the 23rd floor is a buzz; there's also a spa with indoor pool, and guests can use the nearby sister _Expo_ hotel's outdoor pool. \u20ac110, superior \u20ac135\n\n### Hotel Inglaterra\n\nc\/Pelai 14 Universitat 935 051 100, www.hotel-inglaterra.com.MAP\n\nThe boutique little three-star sister to the Dreta's _Hotel Majestic_ has harmoniously toned rooms and snazzy bathrooms. Space is at a premium, but some rooms have cute private terraces, others street-side balconies, while best of all are the romantic roof terrace and pool. \u20ac125\n\n### Somnio Barcelona\n\nc\/Diputaci\u00f3 251 2o Passeig de Gr\u00e0cia 932 725 308, www.somniohostels.com.MAP\n\nSisters Lauren and Lee from Chicago bring their passion for Barcelona right into their upscale _pension_ , dropping \"tips for the day\" into your room each morning. Simple but smart rooms with wood-block floors cater for singles, couples and friends. There are four spacious twin rooms, four double rooms (two en suite) and a single. Some have balconies. Singles \u20ac44, doubles \u20ac78, en-suites \u20ac87\n\n< Back to Accommodation\n\n### Gr\u00e0cia\n\n### Casa Gracia\n\nPg. de Gr\u00e0cia 116 Diagonal 931 874 497, casagraciabcn.com.MAP\n\nA vibrant and stylish space spread over six floors in a _modernista_ building, with bonuses like a concierge, themed dinners and evening concerts. The rooms (from dorms to doubles to six-bed private rooms) have a\/c and are en suite, while the deluxe suite pampers with a spa bath, slippers and bathrobes. Though _Casa Gracia_ is technically a hostel, you'll feel like you're staying in a (pretty good) hotel. Breakfast included. Dorms \u20ac25, rooms \u20ac90\n\n### Hotel Casa Fuster\n\nPg. de Gr\u00e0cia 132 Diagonal 932 553 000, www.hotelcasafuster.com.MAP\n\n _Modernista_ architect Llu\u00eds Dom\u00e8nech i Montaner's magnificent Casa Fuster (1908) is the backdrop for five-star deluxe luxury with service to match. Rooms are in earth tones, with huge beds, smart bathrooms, and remote-controlled light and heat, while public areas make full use of the architectural heritage \u2013 from the magnificent pillared lobby bar, the _Caf\u00e9 Vien\u00e9s_ , to the panoramic roof terrace and pool. There's also a contemporary restaurant, _Galax\u00f3_ , plus fitness centre, sauna and 24hr room service. \u20ac176\n\n< Back to Accommodation\n\n### Les Corts\n\n### Gran Hotel Princesa Sofia\n\nPl. Pius XII 4 Maria Cristina 935 081 050, www.princesasofia.com.MAP\n\nA classic \u2013 one of the first five-star hotels in town thirty years ago \u2013 and well placed for shoppers, with wide-ranging city views from the upper floors. It still exudes old-school charm (the concierges know everything) though the warm-toned rooms, massages and treatments in the Aqua Diagonal Wellness Centre, pool (with retractable roof) and superior club rooms and lounges offer a more contemporary experience \u2013 the Barcelona football team stays and eats here before every home match. An immense buffet breakfast is served in the _Contraste_ restaurant \u2013 which also features an attractive patio for dining in the summer. \u20ac140, club rooms \u20ac210\n\n< Back to Accommodation\n\n### Montju\u00efc\n\n### Hotel Miramar\n\nPl. Carlos Iba\u00f1ez 3 Paral.lel and Funicular de Montju\u00efc 932 811 600, www.hotelmiramarbarcelona.com.MAP\n\nThe remodelled _Miramar_ has 75 stylish rooms with sweeping views over the city. From the architecture books in the lounge to the terrace jacuzzis, you're in designer heaven, augmented by a stunning pool and tranquil gardens. \u20ac199\n\n< Back to Accommodation\n\n### Tibidabo\n\n### Gran Hotel La Florida\n\nCtra. Vallvidrera a Tibidabo 83\u201393, 7km from the centre 932 593 000, www.hotellaflorida.com.MAP\n\nThis five-star place on Tibidabo mountain recreates the glory days of the 1950s, when _La Florida_ was at the centre of Barcelona high society. Its terraces and pools have amazing views, while some of the seventy rooms and suites have private gardens or terraces and jacuzzis. There's also a spa, restaurant, poolside bar and shuttle-bus service to town. Online advance bookings from \u20ac198, otherwise \u20ac225\n\n< Back to Accommodation\n\n### Horta\n\n### Alberg Mare de D\u00e9u de Montserrat\n\nPg. de la Mare de D\u00e9u del Coll 41\u201351 Vallcarca. 932 105 151, reservations on  934 838 363, , www.xanascat.cat.MAP\n\nA popular hostel, set in a converted mansion with gardens, terrace and great city views, with dorms sleeping six, eight or twelve. It's a long way from the centre (though it's close to Parc G\u00fcell) \u2013 from the metro, follow Av. Rep\u00fablica d'Argentina, c\/Viaducte de Vallcarca and then the signs, while buses (V17 from Pla\u00e7a d'Urquinaona, plus night buses) stop just across the street. Reception open 8am\u201311pm; main door closes at midnight, but opens every 30min thereafter. Dorms \u20ac22, includes breakfast\n\n[< Back to Accommodation](PRG_BARCEL156-165ACCOMM.xhtml\n            )\n\n# Essentials\n\nArrival\n\nGetting around\n\nDirectory A\u2013Z\n\nFestivals and events\n\nChronology\n\nCatalan\n\n# Arrival\n\nBy air\n\nBy train\n\nBy bus\n\nIn most cases, you can be off the plane, train or bus and in your central Barcelona hotel room within the hour. Note that Ryanair flights (and some others) to Barcelona are actually to Girona (90km north) or Reus (110km south), and though there are reliable connecting bus and train services this means up to a 90-minute journey from either airport to Barcelona city centre.\n\n### By air\n\nBarcelona's airport ( 902 404 704,  www.aena.es) is 18km southwest of the city. A **taxi** to the centre costs up to \u20ac30, including the airport surcharge (plus other surcharges for travel after 9pm, at weekends or for luggage in the boot). Far cheaper is the **airport train** (5.42am\u201311.38pm; journey time 18min; \u20ac4.10; info on  902 240 202), which runs every thirty minutes to Barcelona Sants station (see \"By train\") and then continues on to Passeig de Gr\u00e0cia (best stop for Eixample, Pla\u00e7a de Catalunya and the Ramblas). It departs from Terminal T2, and there's a free shuttle bus to the station from T1 which takes around ten minutes. City travel passes and the Barcelona Card (available at the airport) are valid on the airport train.\n\nAlternatively, the **Aerob\u00fas** service (Mon\u2013Sat 6am\u20131am; \u20ac5.90, \u20ac10.20 return, departures every 5\u201310min;  www.aerobusbcn.es) from T1 and T2 stops in the city at Pla\u00e7a d'Espanya, Gran Via \u2013 Urgell, Pla\u00e7a Universitat and Pla\u00e7a de Catalunya (travel time 30min). Aerobus departures back to the airport leave from in front of El Corte Ingl\u00e9s in Pla\u00e7a de Catalunya \u2013 note that there are separate services to Terminals T1 and T2.\n\n### By train\n\nThe national rail service is operated by RENFE  902 320 320  www.renfe.com). The city's main station is **Barcelona Sants** , 3km west of the centre, with a metro station ( Sants Estaci\u00f3) that links directly to the Ramblas ( Liceu), Pla\u00e7a de Catalunya and Passeig de Gr\u00e0cia. The high-speed AVE line between Barcelona and Madrid has cut the fastest journey between the cities to under three hours. These services also arrive at and depart from Barcelona Sants, though a second high-speed station for the city is under construction in the north of the city.\n\nSome Spanish intercity services and international trains also stop at **Estaci\u00f3 de Fran\u00e7a** , 1km east of the Ramblas and close to  Barceloneta.\n\nRegional and local commuter train services are operated by Ferrocarrils de la Generalitat de Catalunya, or **FGC** ( 932 051 515,  www.fgc.cat), with stations at **Pla\u00e7a de Catalunya** , at the top of the Ramblas (for trains from coastal towns north of the city); **Pla\u00e7a d'Espanya** (for Montserrat); and **Passeig de Gr\u00e0cia** (Catalunya provincial destinations).\n\n### By bus\n\nThe main bus terminal is the **Estaci\u00f3 del Nord** ( 902 260 606,  www.barcelonanord.com;  Arc de Triomf) on c\/Ali-Bei, three blocks north of Parc de la Ciutadella. Various companies operate services across Catalunya, Spain and Europe from here \u2013 it's a good idea to reserve a ticket in advance on long-distance routes (a day before at the station is usually fine, or buy online). Some intercity and international services also make a stop at the bus terminal behind Barcelona Sants station. Either way, you're only a short metro ride from the city centre.\n\n< Back to Essentials\n\n# Getting around\n\nTickets and travel passes\n\nThe metro\n\nTrams\n\nBuses\n\nTrains\n\nFuniculars and cable cars\n\nTaxis\n\nCycling and bike rental\n\nCity tours\n\nBarcelona has an excellent integrated transport system which comprises the metro, buses, trams and local trains, plus a network of funiculars and cable cars. The local transport authority has a useful website ( www.tmb.cat, English-language version available) with full timetable and ticket information, while a city transport map and information is posted at major bus stops and all metro and tram stations.\n\n### Tickets and travel passes\n\nOn all the city's public transport (including night buses and funiculars), you can buy a **single ticket** every time you ride (\u20ac2.15), but it's much cheaper to buy a **targeta** \u2013 a discount ticket card. They are available at metro, train and tram stations, but not on the buses.\n\nBest general ticket deal is the T-10 (\"tay day-oo\" in Catalan) _targeta_ (\u20ac10.30), valid for ten separate journeys, with changes between methods of transport allowed within 75 minutes. This card (also available at newsagents' kiosks) can be used by more than one person at a time \u2013 simply validate it the same number of times as there are people travelling.\n\nOther useful (single-person) _targetes_ include the T-Dia (1 day's unlimited travel; \u20ac7.60), and there are also multi-day combos (HolaBCN!) for up to five days (\u20ac30.50). Prices given are for passes valid as far as the Zone 1 city limits, which in practice is everywhere you're likely to want to go except Montserrat and Sitges. For trips to these and other out-of-town destinations, buy a specific ticket.\n\n### The metro\n\nThe quickest way of getting around Barcelona is by **metro** , which currently runs on six main lines. A few stations on a new line, L9, are also now open \u2013 on its completion, this will be the longest underground line in Europe (almost 50km) and will run between the airport, city centre and high-speed Sagrera train station.\n\nThere's a limited network of stations in the Old Town, but you can take the metro directly to the Ramblas (Catalunya, Liceu or Drassanes), and to the edge of the Barri G\u00f2tic, El Raval and La Ribera.\n\nMetro entrances are marked with a red diamond sign with an \"M\". Its **hours of operation** are Monday to Thursday, plus Sunday and public holidays 5am to midnight; Friday 5am to 2am; Saturday and the day before a public holiday, 24hr service. The system is safe, but some of the train carriages are heavily graffitied, and buskers and beggars are common.\n\n### Trams\n\nThe **tram** system ( www.tram.cat) runs on six lines, with departures every eight to twenty minutes throughout the day from 5am to midnight. Lines **T1, T2 and T3** depart from Pla\u00e7a Francesc Maci\u00e0 and run along the uptown part of Avinguda Diagonal to suburban destinations in the northwest \u2013 useful tourist stops are at L'Illa shopping and the Maria Cristina and Palau Reial metro stations. **Line T4** operates from Ciutadella-Vila Ol\u00edmpica (where there's also a metro station) and runs up past the zoo and TNC (the National Theatre) to Gl\u00f2ries, before running down the lower part of Avinguda Diagonal to Diagonal Mar and the F\u00f2rum site. You're unlikely to use the more suburban lines T5 and T6.\n\n### Buses\n\nMost **buses** operate daily, roughly from 4am or 5am until 10.30pm, though some lines stop earlier and some run on until after midnight. Night bus services fill in the gaps on all the main routes, with services every twenty to sixty minutes from around 10pm to 4am. Many bus routes (including all night buses) stop in or near Pla\u00e7a de Catalunya, but the full route is marked at each bus stop, along with a timetable.\n\n### Trains\n\nThe FGC **commuter train line** has its main stations at Pla\u00e7a de Catalunya and Pla\u00e7a d'Espanya, used when going to Sarri\u00e0, Vallvidrera, Tibidabo and Montserrat. The national rail service, RENFE, runs all the other services out of Barcelona, with local lines designated as **Rodiales\/Cercan\u00edas**. The hub is Barcelona Sants station, with services also passing through Pla\u00e7a de Catalunya (heading north) and Passeig de Gr\u00e0cia (south). Arrive in plenty of time to buy a ticket, as queues are often horrendous, though for most regional destinations you can use the automatic vending machines instead.\n\n### Funiculars and cable cars\n\nAs well as the regular city options, Barcelona also has some fun transport trips and historic survivors. A few **funicular railways** are still widely used, particularly up to Montju\u00efc and Tibidabo, while summer and year-round weekend visits to Tibidabo also combine a funicular trip with a ride on the clanking antique tram, the **Tramvia Blau**. Best of all, though, are the two **cable car** ( _telef\u00e8ric_ ) rides: from Barceloneta across the harbour to Montju\u00efc, and then from the top station of the Montju\u00efc funicular right the way up to the castle.\n\n### Taxis\n\nThere are taxi ranks outside major train and metro stations, in main squares, near large hotels and at places along the main avenues. To call a taxi in advance (few of the operators speak English, and you'll be charged an extra \u20ac3 or \u20ac4), try: Barna Taxis  933 222 222; Radio Taxi  933 033 033; Servi-Taxi  933 300 300; or Taxi Amic  934 208 088.\n\nA fun way to get around the Old Town, port area and beaches is by **Trixi** ( www.trixi.com), a kind of love-bug-style bicycle-rickshaw. They tout for business between 11am and 8pm near the Columbus statue at the bottom of the Ramblas, and outside La Seu cathedral in the Barri G\u00f2tic, though you can also flag them down if one cruises by. Fares are fixed (from \u20ac15 for 30min, longer tours also available) and the _trixistas_ are an amiable, multilingual bunch for the most part.\n\n### Cycling and bike rental\n\nThe city council is investing heavily in cycle lanes and bike schemes, notably the **Bicing** pick-up and drop-off scheme ( www.bicing.cat), which is touted as Barcelona's new public transport system. You'll see the red bikes and bike stations all over the city, but Bicing is not aimed at tourists, rather at locals who are encouraged to use the bikes for short trips (users can register online or at the **Oficina de la Bicicleta** , Pl. Carles Pi i Sunyer 8\u201310, Barri G\u00f2tic, between c\/Canuda and c\/Duran i Bas).\n\nThere are plenty of **bike rental** outfits more geared to tourist requirements. Rental costs around \u20ac20 a day with companies all over town, including Un Coxte Menys ( www.bicicletabarcelona.com) and Biciclot ( www.bikinginbarcelona.net).\n\nCurrently there are around 200km of **cycle paths** throughout the city, with plans to double the network in the future. Not all locals have embraced the bike yet, and some cycle paths are still ignored by cars or clogged with pedestrians, indignantly reluctant to give way to two-wheelers. But, on the whole, cycling around Barcelona is not the completely hairy experience it was just a few years ago, while you can always get **off-road** in the Parc de Collserola, where there are waymarked trails through the woods and hills.\n\n### City tours\n\nThe number of available tours is bewildering, and you can see the sights on anything from a Segway to a hot-air balloon. A good place to start is the official Barcelona Turisme website ( www.barcelonaturisme.com), which has a dedicated tours section offering online sales and discounts.\n\nHighest profile are the two tour-bus operators with daily board-at-will, open-top services (1 day \u20ac27, 2 days \u20ac35), which drop you outside every attraction in the city. The choice is between **Barcelona City Tour** ( www.barcelonatours.es) or the **Bus Tur\u00edstic** ( www.barcelonaturisme.com), with frequent departures from Pla\u00e7a de Catalunya and many other stops \u2013 tickets are available on board.\n\nAdvance booking is advised (at Pl. de Catalunya tourist office) for **Barcelona Walking Tours** ' two-hour historical Barri G\u00f2tic tour (daily all year, in English at 9.30am; \u20ac15.50). There are also \"Picasso\", \" _Modernisme_ \" and \"Gourmet\" walking tours on selected days.\n\nThe guides at **My Favourite Things** ( www.myft.net) reveal Barcelona in a new light, particularly on the signature tour \"My Favourite Fusion\", which gives an insider's view of the city. Tours (in English, flexible departures) cost from \u20ac26 per person and last around four hours, and there's always time for diversions, workshop visits and caf\u00e9 stops. Altogether more idiosyncratic is **Follow the Baldie** (contact through website,  www.followthebaldie.com), with whom you can variously tour anarchist Barcelona, track tarantulas near Sitges or stagger from bar to tavern in rural Catalunya.\n\n**Bike tours** now infest the city, with follow-the-leader cycle packs careering through the Old Town alleys. There are flyers and bike outfits everywhere and you'll pay around \u20ac25 for a guided 3hr tour.\n\nAt any time of year, the sparkling harbour waters invite a cruise and **Las Golondrinas** ( 934 423 106,  www.lasgolondrinas.com) daily sightseeing boats depart (at least hourly June\u2013Sept, less frequently Oct\u2013May) from the quayside opposite the Columbus statue, at the bottom of the Ramblas ( Drassanes). Two separate services visit either the port (40min; \u20ac7.20), or port and local coast (1hr 30min; \u20ac15).\n\nThere are also afternoon catamaran trips around the port with **Catamaran Orsom** ( www.barcelona-orsom.com; Easter week & June\u2013Sept daily, May & Oct daily except Tues & Wed; \u20ac15.50) plus summer evening jazz cruises (daily June\u2013Aug; \u20ac17.50).\n\n< Back to Essentials\n\n# Directory A\u2013Z\n\n### Addresses\n\nThe main address abbreviations are Av. (for Avinguda, avenue), c\/ (Carrer, street), Pg. (Passeig, boulevard\/street), Bxda. (Baixada, alley), Ptge. (Passatge, passage) and Pl. (Pla\u00e7a, square). The address \"c\/Picasso 2, 4\u00b0\" means: Picasso street, number two, fourth floor.\n\n### Crime\n\nTake all the usual precautions and be on guard when on public transport or on the crowded Ramblas and the medieval streets to either side. Easiest place to report a crime is the G\u00f9ardia Urbana (municipal police) station at Ramblas 43, opposite Pl. Reial ( Liceu;  932 562 430,  www.bcn.cat\/guardiaurbana; 24hr, English spoken). For a police report for your insurance go to c\/Nou de la Rambla 80, El Raval ( Paral.lel  933 062 300).\n\n### Electricity\n\nThe electricity supply is 220v and plugs come with two round pins \u2013 bring an adapter (and transformer) to use UK and US cell phone chargers etc.\n\n### Embassies and consulates\n\nAustralia, Av. Diagonal 433, Eixample,  Diagonal  933 623 792,  www.spain.embassy.gov.au; Britain, Av. Diagonal 477, Eixample,  Hospital Clinic  902 109 356, or from outside Spain  913 342 194,  www.ukinspain.fco.gov.uk; Canada, Pl. Catalunya 9,  Catalunya  932 703 614,  www.canadainternational.gc.ca; Republic of Ireland, Gran Via Carles III 94, Les Corts,  Maria Cristina\/Les Corts  934 915 021,  irelanda.es; New Zealand, Trav. de Gr\u00e0cia 64, Gr\u00e0cia, FGC Gr\u00e0cia,  932 090 399,  www.nzembassy.com; USA, Pg. de la Reina Elisenda 23, Sarri\u00e0, FGC Reina Elisenda,  932 802 227,  barcelona.usconsulate.gov.\n\n### Gay and lesbian travellers\n\nEpicentre of the gay scene is the so-called Gaixample, an area of a few blocks near the university in the Esquerra de l'Eixample. The annual Pride festival runs for ten days in June ( www.pridebarcelona.org). General listings magazine _Guia del Ocio_ can put you on the right track for bars and clubs. The single best English-language website is  60by80.com\/Barcelona. For other information, contact the lesbian and gay city telephone hotline on  900 601 601 (daily 6\u201310pm only).\n\n### Health\n\nThe following central hospitals have 24hr accident and emergency services: Centre Perecamps, Av. Drassanes 13\u201315, El Raval,  Drassanes  934 410 600; Hospital Clinic i Provincial, c\/Villaroel 170, Eixample,  Hospital Clinic  932 275 400; Hospital del Mar, Pg. Marit\u00edm 25\u201329, Vila Ol\u00edmpica,  Ciutadella-Vila Ol\u00edmpica  932 483 000.\n\nUsual pharmacy hours are 9am to 1pm and 4 to 8pm. At least one in each neighbourhood is open 24hr (and marked as such).\n\n### Emergency numbers\n\nCall  112 for emergency ambulance, police and fire services; for the national police service call  091.\n\n### Lost property\n\nAnything recovered by police, or left on public transport, is sent to the Oficina de Troballes (municipal lost property office) at Pl. Carles Pi Sunyer 8\u201310, Barri G\u00f2tic  Jaume I\/Catalunya (Mon\u2013Fri 9am\u20132pm;  010). You could also try the transport office at  Universitat.\n\n### Money\n\nSpain's currency is the euro (\u20ac), with notes issued in denominations of 5, 10, 20, 50, 100, 200 and 500 euros, and coins in denominations of 1, 2, 5, 10, 20 and 50 cents, and 1 and 2 euros. Normal banking hours are Monday to Friday from 8.30am to 2pm, and there are out-of-hours exchange offices down the Ramblas, as well as at the airport, Barcelona Sants station and the Pl. de Catalunya tourist office. ATMs are available all over the city, and you can usually withdraw up to \u20ac300 a day.\n\n### Museums and passes\n\nMany museums and galleries offer free admission on the first or last Sunday of the month, and most museums are free on the saints' days of February 12, April 23 and September 24, plus May 18 (international museum day). The useful Barcelona Card (2 days \u20ac34, 3 days \u20ac44, 4 days \u20ac52 or 5 days \u20ac58;  www.barcelonaturisme.com) offers free public transport, plus museum and attraction discounts. The Articket (\u20ac30; valid three months;  www.articketbcn.org) covers free admission into six major art galleries, while the _Ruta del Modernisme_ (\u20ac12; valid one year;  www.rutadelmodernisme.com) is an excellent English-language guidebook and discount-voucher package that covers 116 _modernista_ buildings, plus other benefits.\n\n### Opening hours\n\nBasic working hours are Monday to Saturday 9.30 or 10am to 1.30pm and 4.30 to 8 or 9pm, though many offices and shops don't open on Saturday afternoons. Local caf\u00e9s, bars and markets open from around 7am, while shopping centres, major stores and large supermarkets tend to open all day from 10am to 9pm, with some even open on Sunday. Museums and galleries often have restricted Sunday and public holiday hours, while on Mondays most are closed all day.\n\n### Phones\n\nPublic telephones accept coins, credit cards and phone cards (the latter available in various denominations in tobacconists, newsagents and post offices). The cheapest way to make an international call is to go to a _locutorio_ (phone centre); these are scattered throughout the old city, particularly in the Raval and Ribera. You'll be assigned a cabin to make your calls, and afterwards you pay in cash.\n\n### Post\n\nThe main post office (Correus) is on Pl. d'Antoni L\u00f3pez, at the eastern end of Pg. de Colom, in the Barri G\u00f2tic (Mon\u2013Fri 8.30am\u20139.30pm, Sat 8.30am\u20132pm;  Barceloneta\/Jaume I). For stamps it's easier to visit a tobacconist (look for the brown-and-yellow _tabac_ sign), found on virtually every street.\n\n### Public holidays\n\nOfficial holidays are: Jan 1 (Cap d'Any, New Year's Day); Jan 6 (Epifan\u00eda, Epiphany); Good Friday & Easter Monday; May 1 (Dia del Treball, May Day\/Labour Day); June 24 (Dia de Sant Joan, St John's Day); Aug 15 (L'Assumpci\u00f3, Assumption of the Virgin); Sept 11 (Diada Nacional, Catalan National Day); Sept 24 (Festa de la Merc\u00e8, Our Lady of Mercy, Barcelona's patron saint); Oct 12 (D\u00eda de la Hispanidad, Spanish National Day); Nov 1 (Tots Sants, All Saints' Day); Dec 6 (Dia de la Constituci\u00f3, Constitution Day); Dec 8 (La Imaculada, Immaculate Conception); Dec 25 (Nadal, Christmas Day); Dec 26 (Sant Esteve, St Stephen's Day).\n\n### Tickets\n\nYou can buy concert, sporting and exhibition tickets with a credit card using the ServiCaixa ( www.servicaixa.com) automatic dispensing machines in branches of La Caixa savings bank. You can also order tickets online through ServiCaixa or TelEntrada ( www.telentrada.com). For advance tickets for all city council (Ajuntament) sponsored concerts visit the Palau de la Virreina, Ramblas 99.\n\n### Time\n\nBarcelona is one hour ahead of the UK, six hours ahead of New York and Toronto, nine hours ahead of Los Angeles, nine hours behind Sydney and eleven hours behind Auckland. This applies except for brief periods during the change-overs to and from daylight saving (in Spain the clocks go forward in the last week in March, back again in the last week of Oct).\n\n### Tipping\n\nLocals leave only a few cents or round up the change for a coffee or drink, and a euro or two for most meals, though fancier restaurants will expect ten to fifteen percent. Taxi drivers normally get around five percent.\n\n### Tourist information\n\nThe city's tourist board, Turisme de Barcelona ( 807 117 222 from within Spain,  932 853 834 from abroad,  www.barcelonaturisme.com), has its main office in Pla\u00e7a de Catalunya (daily 9.30am\u20139.30pm;  Catalunya), down the steps in the southeast corner of the square, where there's a tours service and accommodation desk. There's also an office in the Barri G\u00f2tic at Pla\u00e7a de Sant Jaume, entrance at c\/Ciutat 2 (Mon\u2013Fri 8.30am\u20138.30pm, Sat 9am\u20137pm, Sun & hols 9am\u20132pm;  Jaume I), and staffed information booths dotted across the city. The city's  010 telephone enquiries service (available 24hr; some English-speaking staff available) can help with questions about transport, public services and other matters. The city hall (Ajuntament;  www.bcn.cat) and regional government (Generalitat;  www.gencat.cat) websites are also mines of information about every aspect of cultural, social and working life in Barcelona. Concerts, exhibitions and festivals are covered in full at the walk-in office of the Institut de Cultura at the Palau de la Virreina, Ramblas 99,  Liceu ( 933 161 000,  www.bcn.cat\/cultura; daily 10am\u20138.30pm). The most useful listings magazine is the Spanish-language _Guia del Ocio_ (out every Thursday;  www.guiadelociobcn.es), available at any newspaper kiosk.\n\n### Travellers with disabilities\n\nBarcelona's airport and Aerob\u00fas are fully accessible to travellers in wheelchairs. On the metro only lines 1, 2 and 11 are fully accessible, with elevators at major stations (including Pl. de Catalunya, Universitat, Pg. de Gr\u00e0cia and Sagrada Fam\u00edlia) from the street to the platforms. However, all city buses have been adapted for wheelchair use, while the city information line \u2013  010 \u2013 has accessibility information for museums, galleries, hotels, restaurants, museums, bars and stores. Many Old Town attractions have steps, cobbles or other impediments to access.\n\n### Water\n\nWater from the tap is safe to drink, but generally doesn't taste very nice. You'll be given bottled mineral water in a bar or restaurant.\n\n< Back to Essentials\n\n# Festivals and events\n\nAlmost any month you visit Barcelona you'll coincide with a festival, event or holiday. The best are picked out here, but for a full list check out the Ajuntament (city hall) website  www.bcn.cat\/cultura.\n\n### Festes de Santa Eul\u00e0lia\n\nMid-February www.bcn.cat\/santaeulalia\n\nWinter festival around February 12 in honour of the young Barcelona girl who suffered a beastly martyrdom at the hands of the Romans. There are parades, concerts, fireworks and _sardana_ dancing.\n\n### Carnaval\/Carnestoltes\n\nWeek before Lent (Feb or March)\n\nCostumed parades and other carnival events across every city neighbourhood. Sitges, down the coast, has the most outrageous celebrations.\n\n### Dia de Sant Jordi\n\nApril 23\n\nSt George's Day celebrates Catalunya's patron saint, with hundreds of book and flower stalls down the Ramblas and elsewhere.\n\n### Primavera Sound\n\nUsually late May www.primaverasound.com\n\nThe city's hottest music festival attracts top names in the rock, indie and electronica world.\n\n### S\u00f3nar\n\nJune www.sonar.es\n\nEurope's biggest, most cutting-edge electronic music, multimedia and urban art festival presents three days of brilliant noise and spectacle.\n\n### Verbena\/Dia de Sant Joan\n\nJune 23\/24\n\nThe \"eve\" and \"day\" of St John herald a \"night of fire\", involving bonfires and fireworks (particularly on Montju\u00efc) and watching the sun come up on the beach.\n\n### Festival de Barcelona Grec\n\nFrom end June to August grec.bcn.cat\n\nThis is the city's main performing arts festival, with many events staged at Montju\u00efc's Teatre Grec.\n\n### Festes de la Merc\u00e8\n\nEnd September www.bcn.cat\/merce\n\nThe city's main festival is celebrated for a week around September 24, with costumed giants, firework displays and human tower competitions.\n\n### Festival Internacional de Jazz\n\nOctober\/November www.barcelonajazzfestival.com\n\nThe annual jazz festival attracts big-name artists to the clubs, as well as smaller-scale street concerts.\n\n### Celebrating Catalan-style\n\nCentral to any traditional Barcelona festival is the parade of _gegants_ , five-metre-high giants with papier-m\u00e2ch\u00e9 or fibreglass heads. Also typically Catalan is the _correfoc_ (\"fire-running\"), where drummers, dragons and demons cavort in the streets. Meanwhile, teams of _castellers_ \u2013 \"castle-makers\" \u2013 pile person upon person to see who can construct the highest tower.\n\n< Back to Essentials\n\n# Chronology\n\n###  ** **c230 BC****\n\nCarthaginians found the settlement of \"Barcino\", probably on the heights of Montju\u00efc.\n\n###  ** **218\u2013201 BC****\n\nRomans expel Carthaginians from Iberian peninsula in Second Punic War. Roman Barcino is established around today's Barri G\u00f2tic.\n\n###  ** **304 AD****\n\nSanta Eul\u00e0lia \u2013 the city's patron saint \u2013 is martyred by Romans for refusing to renounce Christianity.\n\n###  ** **c350 AD****\n\nRoman city walls are built, as threat of invasion grows.\n\n###  ** **415****\n\nVisigoths sweep across Spain and establish temporary capital in Barcino (later \"Barcelona\").\n\n###  ** **711****\n\nMoorish conquest of Spain. Barcelona eventually forced to surrender (719).\n\n###  ** **801****\n\nBarcelona retaken by Louis the Pious, son of Charlemagne. Frankish counties of Catalunya become a buffer zone, known as the Spanish Marches.\n\n###  ** **878****\n\nGuifr\u00e9 el Pel\u00f3s (Wilfred the Hairy) declared first Count of Barcelona, founding a dynastic line that was to rule until 1410.\n\n###  ** **985****\n\nMoorish sacking of city. Sant Pau del Camp \u2013 the city's oldest surviving church \u2013 built after this date.\n\n###  ** **1137****\n\nDynastic union of Catalunya and Arag\u00f3n established.\n\n###  ** **1213\u201376****\n\nReign of Jaume I, \"the Conqueror\", expansion of empire and beginning of Catalan golden age.\n\n###  ** **1282\u20131387****\n\nBarcelona at the centre of an aggressively mercantile Mediterranean empire. Successive rulers construct most of Barcelona's best-known Gothic buildings.\n\n###  ** **1348****\n\nThe Black Death strikes, killing half of Barcelona's population.\n\n###  ** **1391****\n\nPogrom against the city's Jewish population.\n\n###  ** **1410****\n\nDeath of Mart\u00ed el Hum\u00e0 (Martin the Humane), last of Catalan count-kings. Beginning of the end of Catalan influence in the Mediterranean.\n\n###  ** **1469****\n\nMarriage of Ferdinand of Arag\u00f3n and Isabel of Castile.\n\n###  ** **1479****\n\nFerdinand succeeds to Catalan-Arag\u00f3n crown, and Catalunya's fortunes decline. Inquisition introduced to Barcelona, leading to forced flight of the Jews.\n\n###  ** **1493****\n\nChristopher Columbus received in Barcelona after his triumphant return from New World. The shifting of trade routes away from Mediterranean and across Atlantic further impoverishes the city.\n\n###  ** **1516****\n\nSpanish crown passes to Habsburgs and Madrid is established as capital of Spanish empire.\n\n###  ** **1640\u201352****\n\nThe uprising known as the \"Wars of the Reapers\" declares Catalunya an independent republic. Barcelona is besieged and eventually surrenders to the Spanish army.\n\n###  ** **1714****\n\nAfter War of Spanish Succession, throne passes to Bourbons. Barcelona subdued on September 11 (now Catalan National Day); Ciutadella fortress built, Catalan language banned and parliament abolished.\n\n###  ** **1755****\n\nBarceloneta district laid out \u2013 gridded layout is an early example of urban planning.\n\n###  ** **1778****\n\nSteady increase in trade; Barcelona's economy improves.\n\n###  ** **1814****\n\nAfter Peninsular War (1808\u201314), French finally driven out, with Barcelona the last city to fall.\n\n###  ** **1859****\n\nOld city walls demolished and Eixample district built to accommodate growing population.\n\n###  ** **1882****\n\nWork begins on Sagrada Fam\u00edlia; Antoni Gaud\u00ed takes charge two years later.\n\n###  ** **1888****\n\nUniversal Exhibition held at Parc de la Ciutadella. _Modernista_ architects start to make their mark.\n\n###  ** **1893****\n\nFirst stirrings of anarchist unrest. Liceu opera house bombed.\n\n###  ** **1901****\n\nPablo Picasso's first public exhibition held at _Els Quatre Gats_ tavern.\n\n###  ** **1909****\n\nSetmana Tr\u00e1gica (Tragic Week) of rioting. Many churches destroyed.\n\n###  ** **1922****\n\nParc G\u00fcell opens to the public.\n\n###  ** **1926****\n\nAntoni Gaud\u00ed is run over by a tram; Barcelona stops en masse for his funeral.\n\n###  ** **1929****\n\nInternational Exhibition held at Montju\u00efc.\n\n###  ** **1936\u201339****\n\nSpanish Civil War. Barcelona at the heart of Republican cause, with George Orwell and other volunteers arriving to fight. City eventually falls to Nationalists on January 26, 1939.\n\n###  ** **1939\u201375****\n\nSpain under Franco. Generalitat president Llu\u00eds Companys executed and Catalan language banned. Emigration encouraged from south to dilute Catalan identity. Franco dies in 1975.\n\n###  ** **1977****\n\nFirst democratic Spanish elections for 40 years.\n\n###  ** **1978\u201380****\n\nGeneralitat re-established and Statute of Autonomy approved. Conservative nationalist government elected.\n\n###  ** **1992****\n\nOlympics held in Barcelona. Rebuilding projects transform Montju\u00efc and the waterfront.\n\n###  ** **1995****\n\nMACBA (contemporary art museum) opens, and signals the regeneration of El Raval district.\n\n###  ** **2004****\n\nDiagonal Mar hosts Universal Forum of Cultures, heralding the transformation of the Poble Nou district.\n\n###  ** **2006****\n\nNew statute of autonomy agreed with Spain.\n\n###  ** **2008****\n\nMontju\u00efc castle symbolically handed over from state to city, to become a peace museum.\n\n###  ** **2011****\n\nFC Barcelona are European football champions for fourth time.\n\n###  ** **2014****\n\nConstitutional Court of Spain deems the Catalan government's 2013 declaration of sovereignty to be unconstitutional.\n\n< Back to Essentials\n\n# Catalan\n\nIn Barcelona, Catalan (Catal\u00e0) has more or less taken over from Castilian (Castellano) Spanish as the language on street signs and maps. On paper it looks like a cross between French and Spanish and is generally easy to read if you know those two. Few visitors realize how important Catalan is to those who speak it: never commit the error of calling it a dialect. Despite the preponderance of the Catalan language you'll get by perfectly well in Spanish as long as you're aware of the use of Catalan in timetables, on menus, and so on. However you'll generally get a good reception if you at least try communicating in the local language.\n\n### Pronunciation\n\nDon't be tempted to use the few rules of Spanish pronunciation you may know \u2013 in particular the soft Spanish Z and C don't apply, so unlike in the rest of Spain, the city is not Barthelona but Barcelona, as in English.\n\n **a** as in hat if stressed, as in alone when unstressed.\n\n **e** varies, but usually as in get.\n\n **i** as in police.\n\n **ig** sounds like the \"tch\" in the English scratch; lleig (ugly) is pronounced \"yeah-tch\".\n\n **o** a round, full sound, when stressed, otherwise like a soft U sound.\n\n **u** somewhere between the U of put and rule.\n\n **\u00e7** sounds like an English S; pla\u00e7a is pronounced \"plassa\".\n\n **c** followed by an E or I is soft; otherwise hard.\n\n **g** followed by E or I is like the \"zh\" in Zhivago; otherwise hard.\n\n **h** is always silent.\n\n **j** as in the French \"Jean\".\n\n **l.l** is best pronounced (for foreigners) as a single L sound; but for Catalan speakers it has two distinct L sounds.\n\n **ll** sounds like an English Y or LY, like the \"yuh\" sound in million.\n\n **n** as in English, though before F or V it sometimes sounds like an M.\n\n **ny** corresponds to the Castilian \u00d1.\n\n **qu** before E or I sounds like K; before A or O, or if the U has an umlaut (\u00dc), sounds like KWE, as in quit.\n\n **r** is rolled, but only at the start of a word; at the end it's often silent.\n\n **t** is pronounced as in English, though sometimes it sounds like a D; as in viatge or dotze.\n\n **v** at the start of a word sounds like B; in all other positions it's a soft F sound.\n\n **w** is pronounced like a B\/V.\n\n **x** is like SH or CH in most words, though in some, like exit, it sounds like an X.\n\n **z** is like the English Z in zoo.\n\n### Words and phrases\n\n### Basics\n\n **Yes, No, OK**\n\nSi, No, Val\n\n **Please, Thank you**\n\nSi us plau, Gr\u00e0cies\n\n **Hello, Goodbye**\n\nHola, Ad\u00e9u\n\n **Good morning**\n\nBon dia\n\n **Good afternoon\/night**\n\nBona tarde\/nit\n\n **See you later**\n\nFins despr\u00e9s\n\n **Sorry**\n\nHo sento\n\n **Excuse me**\n\nPerdoni\n\n **I (don't) understand**\n\n(No) Ho entenc\n\n **Do you speak English?**\n\nParleu angl\u00e8s?\n\n **Where? When?**\n\nD\u00f3nde? Cuando?\n\n **What? How much?**\n\nQu\u00e8? Quant?\n\n **Here, There**\n\nAqu\u00ed, All\u00ed\/All\u00e1\n\n **This, That**\n\nAix\u00f2, All\u00f2\n\n **Open, Closed**\n\nObert, Tancat\n\n **With, Without**\n\nAmb, Sense\n\n **Good, Bad**\n\nBo(na), Dolent(a)\n\n **Big, Small**\n\nGran, Petit(a)\n\n **Cheap, Expensive**\n\nBarat(a), Car(a)\n\n **I want**\n\nVull (pronounced \"vwee\")\n\n **I'd like**\n\nVoldria\n\n **Do you know?**\n\nVost\u00e8s saben?\n\n **I don't know**\n\nNo s\u00e9\n\n **There is (Is there?)**\n\nHi ha(?)\n\n **What's that?**\n\nQu\u00e8 \u00e9s aix\u00f2?\n\n **Do you have...?**\n\nT\u00e9...?\n\n **Today, Tomorrow**\n\nAvui, Dem\u00e0\n\n### Accommodation\n\n **Do you have a room?**\n\nT\u00e9 alguna habitaci\u00f3?\n\n **...with two beds\/double bed**\n\n...amb dos llits\/llit per dues persones\n\n **...with shower\/bath**\n\n...amb dutxa\/bany\n\n **It's for one person (two people)**\n\nPer a una persona (dues persones)\n\n **For one night (one week)**\n\nPer una nit (una setmana)\n\n **It's fine, how much is it?**\n\nEsta b\u00e9, quant \u00e9s?\n\n **Don't you have anything cheaper?**\n\nEn t\u00e9 de m\u00e9s bon preu?\n\n### Directions and transport\n\n **How do I get to...?**\n\nPer anar a...?\n\n **Left, Right**\n\nA la dreta, A l'esquerra\n\n **Straight on**\n\nTot recte\n\n **Where is...?**\n\nOn \u00e9s...?\n\n **...the bus station**\n\n...l'estaci\u00f3 de autobuses\n\n **...the train station**\n\n...l'estaci\u00f3\n\n **...the nearest bank**\n\n...el banc m\u00e9s a prop\n\n **...the post office**\n\n...l'oficina de correus\n\n **...the toilet**\n\n...la toaleta\n\n **Where does the bus**\n\nDe on surt el\n\n **to...leave from?**\n\nautob\u00fas a...?\n\n **Is this the train for Barcelona?**\n\nAquest tren va a Barcelona?\n\n **I'd like a (return) ticket to...**\n\nVoldria un bitlet (d'anar i tornar) a...\n\n **What time does it leave (arrive in)?**\n\nA quina hora surt (arriba a)?\n\n### Numbers\n\n **1**\n\nun(a)\n\n **2**\n\ndos (dues)\n\n **3**\n\ntres\n\n **4**\n\nquatre\n\n **5**\n\ncinc\n\n **6**\n\nsis\n\n **7**\n\nset\n\n **8**\n\nvuit\n\n **9**\n\nnou\n\n **10**\n\ndeu\n\n **11**\n\nonze\n\n **12**\n\ndotze\n\n **13**\n\ntretze\n\n **14**\n\ncatorze\n\n **15**\n\nquinze\n\n **16**\n\nsetze\n\n **17**\n\ndisset\n\n **18**\n\ndivuit\n\n **19**\n\ndinou\n\n **20**\n\nvint\n\n **21**\n\nvint-i-un\n\n **30**\n\ntrenta\n\n **40**\n\nquaranta\n\n **50**\n\ncinquanta\n\n **60**\n\nseixanta\n\n **70**\n\nsetanta\n\n **80**\n\nvuitanta\n\n **90**\n\nnovanta\n\n **100**\n\ncent\n\n **101**\n\ncent un\n\n **200**\n\ndos-cents (dues-centes)\n\n **500**\n\ncinc-cents\n\n **1000**\n\nmil\n\n### Days of the week\n\n **Monday**\n\ndilluns\n\n **Tuesday**\n\ndimarts\n\n **Wednesday**\n\ndimecres\n\n **Thursday**\n\ndijous\n\n **Friday**\n\ndivendres\n\n **Saturday**\n\ndissabte\n\n **Sunday**\n\ndiumenge\n\n### Months of the year\n\n **January**\n\nGener\n\n **February**\n\nFebrer\n\n **March**\n\nMar\u00e7\n\n **April**\n\nAbril\n\n **May**\n\nMaig\n\n **June**\n\nJuny\n\n **July**\n\nJuliol\n\n **August**\n\nAgost\n\n **September**\n\nSetembre\n\n **October**\n\nOctobre\n\n **November**\n\nNovembre\n\n **December**\n\nDesembre\n\n### Menu reader\n\n### Basic words\n\n **Esmorzar**\n\nTo have breakfast\n\n **Dinar**\n\nTo have lunch\n\n **Sopar**\n\nTo have dinner\n\n **Ganivet**\n\nKnife\n\n **Forquilla**\n\nFork\n\n **Cullera**\n\nSpoon\n\n **Taula**\n\nTable\n\n **Ampolla**\n\nBottle\n\n **Got**\n\nGlass\n\n **Carta**\n\nMenu\n\n **Sopa**\n\nSoup\n\n **Amanida**\n\nSalad\n\n **Entremesos**\n\nHors d'oeuvres\n\n **Truita**\n\nOmelette\n\n **Entrep\u00e0**\n\nSandwich\n\n **Torrades**\n\nToast\n\n **Tapes**\n\nTapas\n\n **Mantega**\n\nButter\n\n **Ous**\n\nEggs\n\n **Pa**\n\nBread\n\n **Olives**\n\nOlives\n\n **Oli**\n\nOil\n\n **Vinagre**\n\nVinegar\n\n **Sal**\n\nSalt\n\n **Pebre**\n\nPepper\n\n **Sucre**\n\nSugar\n\n **El compte**\n\nThe bill\n\n **S\u00f3c vegetari\u00e0\/vegetariana**\n\nI'm a vegetarian\n\n### Cooking terms\n\n **Assortit**\n\nAssorted\n\n **Al forn**\n\nBaked\n\n **A la brasa**\n\nChar-grilled\n\n **Fresc**\n\nFresh\n\n **Fregit**\n\nFried\n\n **A la romana**\n\nFried in batter\n\n **All i oli**\n\nGarlic mayonnaise\n\n **A la plantxa**\n\nGrilled\n\n **En escabetx**\n\nPickled\n\n **Rostit**\n\nRoast\n\n **Salsa**\n\nSauce\n\n **Saltat**\n\nSaut\u00e9ed\n\n **Remenat**\n\nScrambled\n\n **Del temps**\n\nSeasonal\n\n **Fumat**\n\nSmoked\n\n **A l'ast**\n\nSpit-roasted\n\n **Al vapor**\n\nSteamed\n\n **Guisat**\n\nStewed\n\n **Farcit**\n\nStuffed\n\n### Fish and seafood\/Peix i marisc\n\n **Anxoves\/Seitons**\n\nAnchovies\n\n **Calamarsets**\n\nBaby squid\n\n **Orada**\n\nBream\n\n **Clo\u00efses**\n\nClams\n\n **Cranc**\n\nCrab\n\n **Sipia**\n\nCuttlefish\n\n **Llu\u00e7**\n\nHake\n\n **Llagosta**\n\nLobster\n\n **Rap**\n\nMonkfish\n\n **Musclos**\n\nMussels\n\n **Pop**\n\nOctopus\n\n **Gambes**\n\nPrawns\n\n **Navalles**\n\nRazor clams\n\n **Salm\u00f3**\n\nSalmon\n\n **Bacall\u00e0**\n\nSalt cod\n\n **Sardines**\n\nSardines\n\n **Llobarro**\n\nSea bass\n\n **Llenguado**\n\nSole\n\n **Calamars**\n\nSquid\n\n **Peix espasa**\n\nSwordfish\n\n **Tonyina**\n\nTuna\n\n### Meat and poultry\/Carn i aviram\n\n **Bou**\n\nBeef\n\n **Embotits**\n\nCharcuterie\n\n **Pollastre**\n\nChicken\n\n **Xori\u00e7o**\n\nChorizo sausage\n\n **Pernil serr\u00e0**\n\nCured ham\n\n **Llonganissa**\n\nCured pork sausage\n\n **Costelles**\n\nCutlets\/chops\n\n **\u00c0nec**\n\nDuck\n\n **Pernil dol\u00e7**\n\nHam\n\n **Xai\/Be**\n\nLamb\n\n **Fetge**\n\nLiver\n\n **Llom**\n\nLoin of pork\n\n **Mandonguilles**\n\nMeatballs\n\n **Porc**\n\nPork\n\n **Conill**\n\nRabbit\n\n **Salsitxes**\n\nSausages\n\n **Cargols**\n\nSnails\n\n **Bistec**\n\nSteak\n\n **Llengua**\n\nTongue\n\n **Vedella**\n\nVeal\n\n### Vegetables\/Verdures i llegums\n\n **Carxofes**\n\nArtichokes\n\n **Alberg\u00ednia**\n\nAubergine\/eggplant\n\n **Faves**\n\nBroad\/lima beans\n\n **Carbass\u00f3**\n\nCourgette\/zucchini\n\n **All**\n\nGarlic\n\n **Mongetes**\n\nHaricot beans\n\n **Llenties**\n\nLentils\n\n **Xampinyons**\n\nMushrooms\n\n **Cebes**\n\nOnions\n\n **Patates**\n\nPotatoes\n\n **Espinacs**\n\nSpinach\n\n **Tom\u00e0quets**\n\nTomatoes\n\n **Bolets**\n\nWild mushrooms\n\n### Fruit\/Fruita\n\n **Poma**\n\nApple\n\n **Pl\u00e0tan**\n\nBanana\n\n **Ra\u00efm**\n\nGrapes\n\n **Mel\u00f3**\n\nMelon\n\n **Taronja**\n\nOrange\n\n **Pera**\n\nPear\n\n **Maduixes**\n\nStrawberries\n\n### Desserts\/Postres\n\n **Past\u00eds**\n\nCake\n\n **Formatge**\n\nCheese\n\n **Flam**\n\nCr\u00e8me caramel\n\n **Gelat**\n\nIce cream\n\n **Arr\u00f2s amb llet**\n\nRice pudding\n\n **Tarta**\n\nTart\n\n **Yogur**\n\nYoghurt\n\n### Drinks\n\n **Cervesa**\n\nBeer\n\n **Vi**\n\nWine\n\n **Xampan\/cava**\n\nChampagne\n\n **Caf\u00e8 amb llet**\n\nLarge white coffee\n\n **Caf\u00e8 tallat**\n\nSmall white coffee\n\n **Descafeinat**\n\nDecaf\n\n **Te**\n\nTea\n\n **Xocolata**\n\nDrinking chocolate\n\n **Granissat**\n\nCrushed ice drink\n\n **Llet**\n\nMilk\n\n **Orxata**\n\nTiger nut drink\n\n **Aigua**\n\nWater\n\n **Aigua mineral**\n\nMineral water\n\n **Zumo**\n\nJuice\n\n### Catalan specialities\n\n **Amanida Catalana** Salad served with sliced meats (sometimes cheese)\n\n **Arr\u00f2s a banda** Rice with seafood, the rice served separately\n\n **Arr\u00f2s a la marinera** Paella: rice with seafood and saffron\n\n **Arr\u00f2s negre** \"Black rice\", cooked in squid ink\n\n **Bacall\u00e0 a la llauna** Salt cod baked with garlic, tomato and paprika\n\n **Botifarra (amb mongetes)** Grilled Catalan pork sausage (with stewed haricot beans)\n\n **Cal\u00e7ots** Large char-grilled spring onions\n\n **Canelons** Cannelloni, baked pasta with ground meat and b\u00e9chamel sauce\n\n **Conill all i oli** Rabbit with garlic mayonnaise\n\n **Crema Catalana** Cr\u00e8me caramel, with caramelized sugar topping\n\n **Escalivada** Grilled aubergine\/eggplant, pepper\/capsicum and onion\n\n **Espinacs a la Catalana** Spinach cooked with raisins and pine nuts\n\n **Esqueixada** Salad of salt cod with peppers\/capsicums, tomatoes, onions and olives\n\n **Estofat de vedella** Veal stew\n\n **Faves a la Catalana** Stewed broad beans, with bacon and botifarra\n\n **Fideu\u00e0** Short, thin noodles (the width of vermicelli) served with seafood\n\n **Fuet** Catalan salami\n\n **Llenties guisades** Stewed lentils\n\n **Mel i mat\u00f3** Curd cheese and honey\n\n **Pa amb tom\u00e0quet** Bread (often grilled), rubbed with tomato, garlic and olive oil\n\n **Pollastre al cava** Chicken with _cava_ sauce\n\n **Pollastre amb gambes** Chicken with prawns\n\n **Postres de m\u00fasic** Cake of dried fruit and nuts\n\n **Salsa Romesco** Spicy sauce (with chillis, nuts, tomato and wine), often served with grilled fish\n\n **Samfaina** Ratatouille-like stew (onions, peppers\/capsicum, aubergine\/eggplant, tomato), served with salt cod or chicken\n\n **Sarsuela** Fish and shellfish stew\n\n **S\u00edpia amb mandonguilles** Cuttlefish with meatballs\n\n **Suquet de peix** Fish and potato casserole\n\n **Xat\u00f3** Mixed salad of olives, salt cod, preserved tuna, anchovies and onions\n\n< Back to Essentials\n\n# MAPS AND SMALL PRINT\n\nMaps\n\nSmall print\n\n###\n\n## Maps\n\nBarcelona\n\nBarcelona transport\n\nThe Ramblas\n\nBarri G\u00f2tic\n\nPort Vell and Barceloneta\n\nEl Raval\n\nSant Pere\n\nLa Ribera\n\nParc de la Ciutadella\n\nMontju\u00efc\n\nPoble Sec\n\nPort Ol\u00edmpic and Poble Nou\n\nDreta de l'Eixample\n\nSagrada Fam\u00edlia and Gl\u00f2ries\n\nEsquerra de l'Eixample\n\nGr\u00e0cia\n\nParc G\u00fcell\n\nCamp Nou, Pedralbes and Sarri\u00e0-Sant Gervasi\n\nTibidabo and Parc de Collserola\n\nMonestir de Montserrat\n\nSitges\n\n### Map symbols\n\n< Back to Maps and Small print\n\n## Publishing information\n\nThis third edition published April 2015 by\n\n**Rough Guides Ltd**\n\n80 Strand, London, WC2R 0RL\n\n11, Community Centre, Panchsheel Park, New Delhi 110017, India\n\n**A Penguin Random House Company**\n\nPenguin Books Ltd. 80 Strand, London, WC2R 0RL\n\nPenguin Group (USA) 345 Hudson Street, NY 10014, USA\n\nPenguin Group (Australia) 250 Camberwell Road, Camberwell, Victoria 3124, Australia\n\nPenguin Group (NZ) 67 Apollo Drive, Mairangi Bay, Auckland 1310, New Zealand\n\nPenguin Group (South Africa) Block D, Rosebank Office Park, 181 Jan Smuts Avenue, Parktown North, Gauteng, South Africa 2193\n\nRough Guides is represented in Canada by Tourmaline Editions Inc., 662 King Street West, Suite 304, Toronto, Ontario, M5V 1M7\n\n\u00a9 Rough Guides, 2015\n\nMaps \u00a9 Rough Guides\n\nNo part of this ebook may be reproduced in any form without permission from the publisher except for the quotation of brief passages in reviews.\n\nThe publishers and authors have done their best to ensure the accuracy and currency of all information in **Pocket Rough Guide Barcelona** ; however, they can accept no responsibility for any loss or inconvenience sustained by any reader as a result of its information or advice.\n\nISBN: 9780241009345\n\nThis digital edition published 2015.  \nISBN: 9780241214497\n\n**ROUGH GUIDES CREDITS** **Updater:** Laurie Isola **Text Editors:** Rebecca Hallett, Lara Kavanagh, Edward Aves **Layout:** Nikhil Agarwal, Sachin Tanwar **Photography:** Roger d'Olivere Mapp, Chris Christoforou, Tim Kavenagh **Cartography:** James MacDonald, Ed Wright, Katie Bennett **Picture Editors:** Michelle Bhatia, Nicole Newman, Rhiannon Furbear **Proofreader:** Susannah Wight **Production:** Linda Dare **Cover Design:** Nikhil Agarwal, Dan May, Nicole Newman, Chlo\u00eb Roberts\n\n**DIGITAL PRODUCTION TEAM** **Digital Producer:** Vikki Nousiainen **Head of Digital Media, Delhi:** Manjari Hooda **Producer:** Rahul Kumar **Assistant Editor:** Etika Kapil **Senior Software Engineer:** Ravi Yadav **Senior Digital Designer:** Susant Pati **Operations Assistant:** Tauhid Nasir\n\n## The Authors\n\n**Jules Brown** first visited Barcelona in 1985. Apart from this book he has also written half a dozen other Rough Guides, and contributed as researcher and editor to many others. But he's beginning to think he's left it too late to play for Huddersfield Town.\n\n**Laurie Isola** is a California-based writer and researcher whose work has appeared in various magazines and newspapers, as well as online. During the nearly three years she lived in Barcelona, she could be found treasure hunting at Els Encants Vells or sipping a _tallat_ on a sunny _terrassa_.\n\n## Photo credits\n\nAll images \u00a9 Rough Guides except the following:\n\n_Alkimia_ \u00a9 courtesy of _Alkimia_\n\nBar, Passeig del Born \u00a9 Travel Pictures\/Alamy\n\n_Caf\u00e9 del Sol_ \u00a9 Travel Pictures\/Alamy\n\nCamp Nou \u00a9 Gregorio\/Alterphotos\/EXPA\/NewSport\/Corbis\n\nCasa Calvet \u00a9 Courtesy of Casa Calvet\n\nCastellers by La Seu \u00a9 Stefano Politi Markovina\/AWL Images\n\n_Cinc Sentits_ \u00a9 Jordi Artal\n\nCloister, La Seu \u00a9 Juergen Richter\/LOOK\/Getty Images\n\nDisseny Hub\/Museu del Disseny \u00a9 Juan Moyano\/Dreamstime.com\n\n_Dos Palillos_ \u00a9 courtesy of _Dos Palillos_\n\n_El Xampanyet_ \u00a9 travelstock44\/Alamy\n\nGran Teatre del Liceu \u00a9 LOOK Die Bildagentur der Fotografen GmbH\/Alamy\n\nInterior, Sagrada Fam\u00edlia \u00a9 Eurasia\/Robert Harding Picture Library\n\n_Kahala_ \u00a9 courtesy of _Kahala_\n\nLa Seu \u00a9 Image Broker\/Robert HardingPicture Library\n\nMercat de Sant Antoni \u00a9 Karsol\/Dreamstime.com\n\nMonastery building, Montserrat \u00a9 Chris Farrugia\/iStock\n\nMontju\u00efc cable car \u00a9 Kalman89\/Dreamstime.com\n\nMontserrat \u00a9 Shurkva\/iStock\n\nMuseu d'Hist\u00f2ria de Barcelona (MUHBA) \u00a9 Ramon Manent\/Corbis\n\nParc de la Ciutadella \u00a9 Mel Longhurst\/Robert Harding Picture Library\n\nParc G\u00fcell \u00a9 Ivern Photo\/Robert Harding Picture Library\n\nParc G\u00fcell \u00a9 Luciano Mortula\/123RF.com\n\nPicasso Museum \u00a9 Quim Roser\/Robert Harding Picture Library\n\n_Picnic_ \u00a9 courtesy of _Picnic_ \/Christian Schriefer\n\nPla\u00e7a del Rei \u00a9 Xavier Subias\/SuperStock\n\nPla\u00e7a Reial \u00a9 Steven Vidler\/Corbis\n\nRamon Casas i Carb\u00f3, _Dama con Foulard Rojo_ \u00a9 Museu del Modernisme Catal\u00e0\n\nSculpture in Parc Joan Mir\u00f3 \u00a9 Sandro Vannini\/Corbis\n\nSitges \u00a9 lillisphotography\/iStock\n\n_Taverna del Clinic_ \u00a9 courtesy of _Taverna del Clinic_\n\nThe Ramblas \u00a9 Jose Fuste Raga\/Corbis\n\nTickets \u00a9 courtesy of Tickets\n\n_Tickets_ \u00a9 Randolph Images\/Alamy\n\nTorre de Collserola, Tibidabo \u00a9 Pere Sanz\/iStock\n\n**Front cover** Colourful mosaic chimney, Palau G\u00fcell \u00a9 Stefano Politi Markovina\/Alamy\n\n## Acknowledgements\n\nLaurie would like to thank Montse Planas of Turisme de Barcelona, and Jules Brown for all his hard work on the previous edition of this guide.\n\n## Help us update\n\nWe've gone to a lot of effort to ensure that the third edition of the **Pocket Rough Guide Barcelona** is accurate and up-to-date. However, things change \u2013 places get \"discovered\", opening hours are notoriously fickle, restaurants and rooms raise prices or lower standards. If you feel we've got it wrong or left something out, we'd like to know, and if you can remember the address, the price, the hours, the phone number, so much the better.\n\nPlease send your comments with the subject line \" **Pocket Rough Guide Barcelona Update** \" to  mail@roughguides.com. We'll credit all contributions and send a copy of the next edition (or any other Rough Guide if you prefer) for the very best emails.\n\nFind more travel information, connect with fellow travellers and book your trip on  www.roughguides.com\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nForeword\n\nI joined Microsoft at a time when the development of SharePoint 2013 was well underway. My first week there, I was told that I owned the Design Manager and Device Channel user experiences, which you'll learn about in just a matter of pages. As quickly and thoroughly as possible, I had to ramp up on a set of features that would later be part of the keynote demos at SharePoint Conference 2012 in Las Vegas. My starting point was the previous version of this book, written for SharePoint 2010. But the world changed significantly between these releases, and the authors went along for that ride, becoming my friends along the way as we worked together to make designing websites on SharePoint a great experience all around.\n\nSharePoint has a rich history as a document management tool, but branding was often an afterthought. Although it was possible to create beautiful SharePoint sites, it wasn't a familiar process for designers. It was with this in mind that we created a new way to approach design in SharePoint Server 2013. We didn't want you to have to make the trade-off between an easy, gorgeous design and a SharePoint site. Whether you're a small business owner who can't afford to hire a designer, or a large enterprise with the ability to hire an entire design team, SharePoint Server 2013 provides a number of site design capabilities to match every customer.\n\nPeople with no previous SharePoint experience should be pleased with the revamped and reenergized theming experience that gives you access to a swath of full-blown composed looks, which you should think of as a restaurant meal with whatever substitutions you'd like! From one place, you can mix and match fonts, color palettes, layouts, and background images to your heart's content. If that's not enough, you can also build your own set of composed looks for your company, using the existing ones as a template. More advanced users are able to break down the wall between HTML mockups and SharePoint implementation; with a couple of button clicks, you can convert your HTML design into something SharePoint understands while continuing to work in whatever code editor you like best. You'll find a set of common HTML controls you'll use to hook up SharePoint functionality, as well as the ability to easily create mobile- or tablet-specific designs.\n\nAll these improvements amount to an experience where you are no longer working from scratch or having to worry too much about how SharePoint works. Instead, you get to focus on adjusting and improving your design to make full use of the features and power SharePoint offers, and I can't wait to see what you do with it. The authors of this book will help get you there. They've been exploring these tools since we could first let them. They know the ins and outs of branding in SharePoint as well as, if not better than, anyone. In this book, they'll share with you their tips and tricks so you can go forth and build great sites!\n\n\u2014Alyssa Levitz, SharePoint Program Manager, Microsoft Corporation\n\n# Contents\n\nForeword\n\nIntroduction\n\nPart 1: The Basics\n\nChapter 1: What is SharePoint Branding and UI Design?\n\nChapter 2: SharePoint Overview\n\nChapter 3: Working with the SharePoint 2013 User Interface\n\nPart 2: Planning a Design and Getting Started\n\nChapter 4: Planning for Branding\n\nChapter 5: Using the Design Manager to Start a Design in SharePoint\n\nChapter 6: Cascading Style Sheets and SharePoint\n\nPart 3: Advanced SharePoint Branding\n\nChapter 7: Creating Custom Master Pages and Page Layouts\n\nChapter 8: Advanced SharePoint Branding Tasks\n\nChapter 9: Creating Content Rollups with SharePoint WCM\n\nChapter 10: Composed Looks and Custom Branding\n\nPart 4: Other Branding Concepts\n\nChapter 11: Modern Web Design and SharePoint\n\nChapter 12: Designing Apps\n\nIntroduction\n\nThank you for picking up this book! We are truly experiencing an interesting step in the evolution of SharePoint. For the first time since SharePoint was invented, Microsoft is embracing the idea that creating beautiful and engaging SharePoint sites should be within reach of anyone from business users to developers, and of course, traditional web designers. With the creation of new features such as the Design Manager and composed looks, Microsoft has lowered the barriers to creating branding in SharePoint.\n\nThis book is intended to explain the beginner features but also to provide knowledge of the underlying SharePoint branding technology so that you can build a complete solution for branding your SharePoint site, whether it is an internal intranet site or a public-facing Internet site. Making the SharePoint user interface look good requires a designer that is comfortable with design theory and traditional web technologies, as well as having the ability to deal with topics traditionally handled by developers.\n\nNot everyone who picks up this book is looking to become the next Picasso of SharePoint. For these readers, the first two sections of the book are dedicated to understanding just enough about SharePoint branding to plan, create, and apply custom designs to their sites. After you become comfortable with the basics of SharePoint branding, the second two sections of the book take you on a tour of creating custom SharePoint branding from scratch, providing the background knowledge needed to understand how the new, easy branding features work behind the scenes.\n\nHow This Book Is Organized\n\nThe primary goal is to provide the best source of knowledge for SharePoint 2013 branding no matter what your specific skill level is. Although you can certainly use the book as a reference for specific topics, some of the examples throughout the book do build on each other. By the end of the book, you will have learned how to work with all the technology needed to create a fully branded SharePoint site.\n\nThe book is divided into four sections:\n\nThe Basics\n\nIntroductory topics are explored, such as understanding what SharePoint branding means, understanding the basics of how SharePoint works behind the scenes, and an overview of how the SharePoint user interface can be used to edit the page-level branding.\n\nPlanning a Design and Getting Started\n\nAfter explaining the basics, the book dives into planning for branding and starting a design in SharePoint. This section includes topics such as requirements gathering, wireframes, using the new SharePoint 2013 Design Manager and composed looks features, and a deep discussion on how CSS works in SharePoint.\n\nAdvanced SharePoint Branding\n\nThe second half of the book goes into more intricate topics starting with advanced SharePoint branding. This includes a deep dive into creating master pages and page layouts from scratch, and a chapter that focuses on many of the common tasks associated with creating a fully branded SharePoint site. Rollup Web Parts such as the Content Query Web Part and the Content Search Web Part are covered as well as information on creating composed looks.\n\nOther Branding Concepts\n\nThe final section of the book discusses using modern design techniques with SharePoint such as HTML5, CSS3, jQuery, web fonts, and responsive web design. Finally, you learn about using simple online tools to create and style SharePoint apps.\n\nWho Should Read This Book\n\nOne of the most common SharePoint requests you might hear is to make a site \"not look like SharePoint.\" In most organizations this request might be made to someone who wears many hats but isn't specifically a web designer. In other cases, the request might be made to a web designer or even a developer. To many of those users, SharePoint branding might simply mean that you want to change some colors and put your company header at the top of the page, whereas other users want to create a highly branded corporate intranet portal with a cutting-edge design.\n\nThis book is intended for a wide range of readers and skill levels. There is a basic assumption that you have some understanding of how modern websites are created. This includes knowledge of HTML, CSS, and some understanding of the creative design process.\n\nTools You Need\n\nHaving a SharePoint 2013 installation available to follow along with the examples can definitely make a big difference when you read this book. If you don't have access to a dedicated SharePoint 2013 server, you can install SharePoint 2013 on a virtual machine to try it out locally, or you can look into cloud-based options such as Office 365's SharePoint Online.\n\nYou will also want to have a code-editing tool such as either Visual Studio, Adobe Dreamweaver, Expression Web (recently discontinued, but a free download), or even the free Notepad++. Along with these, you may also need to have some traditional web design programs, such as Adobe Photoshop or Microsoft Expression Design (also discontinued and free). The following list can get you started with software for following along with the book:\n\n\u2022 SharePoint 2013 Download: `<http:\/\/technet.microsoft.com\/en-us\/evalcenter\/hh973397.aspx>`\n\n\u2022 Install and Configure SharePoint 2013: `<http:\/\/technet.microsoft.com\/en-us\/library\/cc262957.aspx>`\n\n\u2022 Office 365 Trial: `<http:\/\/www.microsoft.com\/en-us\/office365>`\n\n\u2022 Visual Studio 2012: `<http:\/\/www.microsoft.com\/visualstudio\/eng\/downloads>`\n\n\u2022 Adobe Photoshop: `<http:\/\/www.adobe.com\/products\/photoshop.html>`\n\n\u2022 Adobe Dreamweaver: `<http:\/\/www.adobe.com\/products\/dreamweaver.html>`\n\n\u2022 Notepad++: `<http:\/\/notepad-plus-plus.org\/>`\n\n\u2022 Microsoft Expression Web and Design: `<http:\/\/www.microsoft.com\/expression\/>`\n\nWhat's on the Website\n\nAs you work through the examples in this book, you may choose either to type in all the code manually or to use the source code files that accompany the book. All the source code used in this book is available for download at http:\/\/www.wrox.com. When at the site, simply locate the book's title (either by using the Search box or by using one of the title lists) and click the Download Code link on the book's detail page to obtain all the source code for the book.\n\nThe Road Ahead\n\nAs you start to learn about branding a SharePoint site, it's not uncommon to get frustrated. Designing for SharePoint is different than designing for your own website. It involves overriding and adjusting a design to fit within someone else's code base, in this case Microsoft's out-of-the-box SharePoint code. But don't get discouraged; with a little patience and the appropriate help, you can apply branding to SharePoint. This book aims to provide you with all the knowledge and techniques required to bridge this gap.\n\n1\n\nThe Basics\n\n1. What is SharePoint Branding and UI Design?\n\n2. SharePoint Overview\n\n3. Working with the SharePoint 2013 User Interface\n1\n\nWhat Is SharePoint Branding and UI Design?\n\nWhat's in this Chapter\n\n1. Reasons for SharePoint Branding\n\n2. Exploring Intranet, Internet, and Extranet sites\n\n3. Branding SharePoint at Low, Medium, and Full Effort\n\nWhen most organizations think about SharePoint, style and design for the user interface (UI) traditionally hasn't been one of the first things to come to mind. But as SharePoint has evolved from its humble beginnings as a pure collaboration tool, the UI design has slowly moved up the list with every release. What was once a luxury for most SharePoint sites, custom branding and design are now an important part of every implementation. The old adage is, \"You can't judge a book by its cover,\" and although that may be true, you can tell a lot by comparing the various versions of SharePoint to the earlier editions of this book. Beginning with the first edition of this book on SharePoint 2007 design, each edition has evolved from just another typical looking technical book to the full-color book you are reading today.\n\nChances are that if you are reading this book, you've already decided to take control of the way SharePoint looks. Perhaps you have a dated intranet that you want to replace with SharePoint 2013. Maybe you want to make your site look more modern and improve the function with better design. Or maybe you just want your site to not look like a SharePoint site.\n\nThere are plenty of reasons why people end up on the branding and UI design path, but no matter your reason, it can be difficult to know where to start. The goal of this book is to provide you with all the tools you need to brand your SharePoint sites. This book is structured into four sections, starting with easier concepts and moving to more advanced branding concepts. The first section is an overview of concepts that you must understand before diving into branding topics; as you progress through the book, the topics continue to become more granular. Whether you are new to design work, new to SharePoint design work, or someone who does SharePoint design full-time, there is something in this book for you.\n\nThis chapter starts at the highest level and discusses SharePoint branding: what it is and why it is important. You'll get an idea of exactly what branding means in the world of SharePoint. Later, the chapter discusses the branding features in SharePoint 2013 and explains what options are available.\n\nIntroducing SharePoint Branding\n\nThe textbook definition of branding is the act of building a specific image or identity that people recognize in relation to your company or product. That's quite a mouthful! In more simple terms, branding refers to the thoughts and feelings conveyed by a company or product. For example, one of the most iconic brands is Coca-Cola. When you simply see the logo, you can associate feelings or recall good times you've had in affiliation with the product\u2014at least that's what the company hopes. There are plenty of other examples of companies with well-known brands such as Nike, McDonald's, UPS, Walt Disney, and others. These companies have chosen a marketing identity that enables the public to quickly and easily recognize them. This is branding and is one of the most important things to a company.\n\n\"A word is a word, and a picture is worth a thousand...but a brand is worth a million.\"\n\n\u2014Tony Hsieh, CEO Zappos in Delivering Happiness\n\nA company's branding is applied in many different ways, one of which typically includes a website. On the web, conveying a corporate brand usually involves the colors, fonts, logos, and supporting graphics all pulled together with HTML and CSS to provide the branded look and feel for a site.\n\nYou have likely visited a company's website before, so the idea of what it means to apply branding to a website shouldn't be a new concept. But when you start to think about SharePoint branding, additional elements are used to create the branding on the screen. The traditional aspects of branding on the web still apply; however, there are more moving parts in SharePoint. Master pages, page layouts, Web Parts, and other pieces are all involved. If you are a traditional web designer, you might be used to approaching a new site as if it were a blank canvas. SharePoint, however, was created to be used out of the box with no changes\u2014applying branding and design essentially require an understanding of how to work with someone else's code, which in this case is the code generated by SharePoint. This can be a challenge for some who might be new to the world of SharePoint. For example, in an HTML site, if there's something on the page that you want to remove\u2014say you want to hide the search box\u2014the typical behavior might be to just remove it from the code. One of the most common stories told by designers who are new to SharePoint is just this: they want to remove something from the page and they try to just remove the code. In many cases this simple and familiar act creates an undesirable effect, such as the entire page failing to load and displaying an error message. The point here isn't to scare anyone, but rather to explain that branding in SharePoint is a little different. Most people don't consider driving to be too difficult, but you wouldn't advocate someone who has never driven just to hop behind the wheel without a little guidance.\n\nWhy Brand SharePoint?\n\nIf you are reading this book, it is likely that you don't need to be convinced to brand SharePoint. Organizations decide to brand SharePoint for a number of reasons, such as to match their corporate brand or even simply to make the new site \"not look like SharePoint.\"\n\nConsider that SharePoint wasn't created with a look and feel that necessarily works best for your organization. Branding isn't just about aesthetics; it also plays an important role in usability and user adoption. Part of the equation is helping users to feel comfortable using a site, which always seems to be an easier proposition when it feels familiar. In a corporate setting, users are inundated with corporate branding, so having a site that matches certainly helps. Branding also helps organizations that have employees geographically dispersed. If all employees log in to the same site, it helps if that corporate brand is reinforced, no matter where the user is located. All that might sound a bit touchy-feely but consider the alternative\u2014a site with little or no branding that doesn't reflect your company's corporate image. In that case, the sky isn't going to fall; however, it isn't too difficult to see how a well-branded site can have a positive impact on its users. It is the same type of feeling you get walking into a well-designed and decorated room. You just get an extra welcoming and harmonious feeling.\n\nAs mentioned earlier, branding is a marketing term that has been also applied to SharePoint to refer to changing the look and feel of something. In a more complete sense, SharePoint branding is also often referred to as User Experience (UX) development. That term implies a deeper meaning than aesthetics and starts to show the complete value of why an organization might want to brand its SharePoint site. Although the value aesthetics bring is difficult to quantify, when you start to think about improving the UX for a site, there most definitely is measurable value. As a simple example, think about your company's current intranet or public website. It isn't uncommon for people to say that they think that it is difficult to find things on the current site or that it needs to be redesigned.\n\nSo why brand with SharePoint? The answer is going to be different for every organization, but the basic goal is to create a great user experience for anyone that visits the site. There's going to be many factors that ultimately determine what a \"great user experience\" means for your users. As you read through this book, you'll get a good idea of the options available and what is involved to deliver on those options to create the best possible experience for your users!\n\nIs SharePoint Beautiful?\n\nWhen you look at SharePoint, what do you think? When you create your first site, is what you see on the screen beautiful? Depending on what template you look at, you might see something different, but in general SharePoint has a distinct out-of-the-box look. Over the years, most would agree that the distinct SharePoint look has improved; although the most common branding request has remained unchanged: Can you make the site not look like SharePoint?\n\nCompanies spend lots of money developing their brand. Companies want to stand out in the crowd, which makes sense for many reasons. It isn't much different from how people feel about their personal \"brand.\" Imagine if everyone in the world who worked with Microsoft technologies had to wear the same uniform. Some folks might be perfectly content, but many would prefer to wear whatever makes sense to them. For some that might mean suits; for others it might mean a shirt, shorts, and flip-flops. Therefore, out-of-the-box SharePoint may be beautiful to you but not to others. However, one thing most can agree on is that it is nearly impossible to come up with a single look for a product that'll work for every organization on the planet using SharePoint. Customization is inevitable.\n\nWith the idea that custom branding is a common desire, it begs the question, \"How much can I customize the design?\" In truth, you'll most likely be more limited by time, budget, and technical resources than you would by SharePoint capabilities to customize a design.\n\nCommunication Versus Collaboration\n\nAlthough SharePoint sites are unique, they all fall into one of three categories: intranet, Internet, or extranet. Each of these SharePoint sites has a different audience and design considerations as part of the planning phase. It is important to note that the intended purpose for your site will have a major impact on your ultimate design.\n\nThe next few sections discuss the typical considerations for each of the three environments. At a higher level, each environment consists of many SharePoint sites, and each of these sites is usually designed to primarily facilitate either communication or collaboration. It is certainly possible to do a little bit of both, but for the core decisions made about branding, most sites favor one more than the other. For example, most of the sites on an organization's intranet fall under the category of a collaboration site because this is where most users store content and collaborate with others on their day-to-day tasks. The intranet home page for most companies is usually also designed as a place to convey information to employees, such as the latest company news, announcements, or events. When users first visit the site, they are taken to this homepage where they are presented with all this information, and from there they navigate to another area of the intranet to do work and collaborate. Effectively harnessing all SharePoint's capabilities into a seamless experience for users is part science and part art.\n\nBranding projects often overlook the importance of determining whether the intended purpose of a site is either collaboration or communication. From a technical standpoint, sites designed primarily for communication or collaboration require different SharePoint templates, which require different approaches to branding. Consider that the approach for branding a SharePoint site based on the Publishing Portal template (which is designed for public-facing sites) is different from a branding site, which is designed purely for internal collaboration.\n\nConsidering who uses the site and how they are supposed to use it should be key in creating the design for your site. There's a distinct difference in the intended purpose of a site designed purely for communicating information in a one-way fashion verses a collaboration site designed for a two-way flow of information.\n\nIntranet Sites\n\nIntranet sites are typically available only to employees and partners who connect locally to the network or use a virtual private network (VPN). The focus of intranet sites is to facilitate information delivery and collaboration for specific sets of users. These sites often have multiple content authors, as well as many users who consume the content and collaborate on new content.\n\nUnlike public Internet sites, the browsers and system capabilities of intranet sites are usually controlled by the IT department. This makes designing a SharePoint intranet easier because fewer variables need to be considered. For example, if your organization supports only one browser, your need to design and test is reduced to only that one browser.\n\nAs mentioned in the previous section, most intranets are designed to facilitate communication, but the vast majority of sites that are created are of the collaboration variety. Usually, this necessitates a highly customized homepage for the site with subsites that are focused on pure collaboration. Often, intranet sites must be customized to match the look and feel of a company's corporate branding. Following is an example of a custom-branded SharePoint intranet site that uses a custom master page, CSS, and images to create a new look and feel. Note that the version of SharePoint you use impacts the templates you can choose from (discussed more in Chapter 2, \"SharePoint Overview\").\n\nInternet Sites\n\nInternet sites are public-facing and typically have anonymous users visiting them using a variety of Internet browsers and devices. These sites are usually driven by marketing, with only a few content authors and tightly controlled content.\n\nTypically, public-facing Internet sites offer the opportunity to create highly stylized designs. They pose a greater design challenge than internal-facing sites because it is more difficult to control the technology being used by visitors who access the site. In other words, additional effort must be taken to ensure that the site displays properly across all types of browsers and conforms to whatever compliance standards need to be met for the given site. Not only is the creative effort usually greater for an Internet site than for an intranet site, but the actual implementation effort is also typically higher because of the added complexity.\n\nOne example of a corporate-branded Internet site that was built with SharePoint is Ferrari.com. It has a highly customized user interface, and without some poking around in the HTML source, it is hard to see any evidence that it uses SharePoint.\n\nExtranet Sites\n\nExtranet sites combine the security and collaboration of an intranet site with the more heavily emphasized branding found in Internet sites. The goal for most extranet sites is to enable external partners to collaborate with an organization. This is usually accomplished by having a public-facing Internet site that users can access initially. When on the site, users enter a username and password to access a secure site, where they can collaborate with users from inside the company. For example, a manufacturing organization might have an extranet site to enable distributors to log in and place orders or to obtain other information to help them sell the organization's products.\n\nThe biggest challenge with extranets is usually security. Most organizations want external users to log in to see what they need to see, but no more. Maintaining this balance of security can be tricky. From a branding perspective, it usually means that you must ensure that your branding is consistent across all areas of the site, especially those that extranet users will be accessing.\n\nUnderstanding the Levels of Branding\n\nWhen you are thinking about your SharePoint branding project, understanding the environment you'll be customizing is the first step. As discussed in the previous section, each environment presents unique challenges. Another consideration is deciding what it is that you plan to actually implement. For instance, have you ever tried to do a home improvement project? Most people have no trouble changing a light bulb; a smaller number of people would be comfortable laying tile in a room; and even a smaller group would be comfortable adding on a new room to their house. The simplest tasks around the house can take a few minutes and are done by just about anyone, but other tasks take more time and require a more complex skillset. Customizing and branding SharePoint is similar. Depending on what you want to accomplish, some tasks require more skill or effort.\n\nThe various levels of branding for SharePoint have changed somewhat for SharePoint 2013 because the overall approach to branding has changed a bit more with this release than in previous versions. This section helps set expectations for some of the varied branding tasks in SharePoint and what's required for each level.\n\nFollowing are three different approaches to create branding in SharePoint:\n\n\u2022 Low effort\u2014Typically, this approach includes all the branding tasks that an end user with limited training can perform. By using out-of-the-box branding, even someone with little knowledge of traditional website development and design can create a customized site. With out-of-the-box functionality, users can select master pages or composed looks. It is even possible to add a logo image to the top of every site by simply uploading an image and changing the reference in one of the menus (see the example \"Replacing the Default Logo on a Composed Look\" in Chapter 3, \"Working with the SharePoint User Interface.\") All these changes can be made through SharePoint menus and be completed in a matter of minutes.\n\n\u2022 Medium effort\u2014This approach is good to add some level of unique branding to a site. A medium level of effort would typically require users who have an understanding of CSS and HTML. These users can make changes to the CSS or HTML of a site or even copy and modify out-of-the-box master pages or other SharePoint pages to create custom branding for their site. Medium-effort users would typically make their changes with a design tool such as Dreamweaver or Expression Web. This approach usually takes days or weeks depending on the specific tasks.\n\n\u2022 Full effort\u2014If your design requirements are highly customized, your only option is the full-effort approach. This involves creating custom master pages, custom CSS, and potentially some custom page layouts. This approach is good for those who are experienced with traditional website design and have some knowledge of SharePoint or ASP.NET master pages. It is also well suited for public Internet sites and highly styled internal employee portals. These types of highly customized efforts can take weeks or months.\n\nWhich option you choose is likely going to depend on the skill set of the people doing the work and the budget for the project. Throughout this book you see different examples about what is possible for each of the approaches.\n\nSummary\n\n\u2022 Branding is the act of building a specific image or identity that people recognize in relation to your company or product. It applies to both traditional websites as well as SharePoint.\n\n\u2022 The basic goal for branding SharePoint is to create a great user experience for anyone that visits the site.\n\n\u2022 Every site created in SharePoint is focused on either communication or collaboration. The types of sites you have impact the approach to branding you take.\n\n\u2022 SharePoint 2013 has new options that have changed the ways that many approach branding. Whether you are a new SharePoint user or a seasoned SharePoint branding veteran, anyone can brand SharePoint!\n2\n\nSharePoint Overview\n\nWhat's In This Chapter\n\n1. SharePoint Basics\n\n2. Master Pages and the Master Page Gallery\n\n3. Tools for Applying Branding in SharePoint\n\n4. New Branding Features in SharePoint 2013\n\nFor those of you who are familiar with SharePoint, much of this chapter might be a review. If you are new to SharePoint, however, this chapter is designed to give you a broad overview of the product\u2014especially the elements relevant to the branding process. This chapter sets the stage for many of the concepts and terminology used throughout the book.\n\nWhat You Need to Know About SharePoint\n\nSharePoint is a vast product, and there are likely several dozen books written on a wide variety of topics pertaining to it. The focus of this book is obviously SharePoint branding and design\u2014the cover probably gave that away. However, as a designer, you must have a fundamental understanding of SharePoint and the various pieces you'll be working with that can ultimately impact the type of work you create.\n\nVersions\n\nThe most basic piece of information you need to understand is which version of SharePoint you'll be using. SharePoint 2013 is available at the time of this writing in a few different varieties:\n\n\u2022 SharePoint Foundation 2013\u2014Sometimes referred to as the free version of SharePoint. SharePoint Foundation provides the core collaboration and document management capabilities.\n\n\u2022 SharePoint Server 2013\u2014SharePoint Server is the licensed version of SharePoint and comes in two varieties: Standard and Enterprise.\n\n\u2022 Office 365 (SharePoint Online)\u2014SharePoint Online is the SharePoint component of the Office 365 suite. A number of plans are available, each with its own set of features and functionality.\n\nFor a more detailed comparison of the different versions of SharePoint, see http:\/\/technet.microsoft.com\/en-us\/library\/jj819267.aspx.\n\nBoth SharePoint Server 2013 and SharePoint Online include the set of functionality referred to as publishing, which has several useful additions for branding projects:\n\n\u2022 The Web Content Management (WCM) functionality provides users with a robust publishing platform. This means that users can author pages with rich and structured content and publish the pages in a controlled way using out-of-the-box workflows.\n\n\u2022 Publishing provides more robust options for navigation. This gives users more control and provides greater flexibility than what is provided out-of-the-box by SharePoint Foundation.\n\n\u2022 Publishing also enables site administrators to easily change a master page for any site, and all its subsites, from a SharePoint site's settings page.\n\nNOTE\n\nFrom a branding perspective, the publishing functionality in SharePoint Server 2013 and SharePoint Online provides many more options for creating highly styled sites than SharePoint Foundation 2013. Therefore, many of the examples throughout this book are geared toward sites with publishing enabled. However, in most cases the examples still apply to a nonpublishing site with minimal changes to the steps. The only exception to this would be when the examples cover functionality that is not available with SharePoint Foundation 2013. Where possible throughout the book, different approaches for achieving the same results in SharePoint Server, SharePoint Online, and SharePoint Foundation are highlighted.\n\nAlthough the public site in SharePoint Online does use publishing, its features are slightly different from typical publishing sites. For more information on this, see Chapter 8, \"Advanced SharePoint Branding Tasks.\"\n\nUnderstanding the Site Structure in SharePoint\n\nThe version of SharePoint you use helps to determine what specific functionality is available to you, but some concepts in SharePoint are universal across every version. The overall structure of SharePoint objects is one of these concepts. The following list gives an extremely high-level overview of these major objects. These objects are listed in order of the biggest container to the smallest:\n\n\u2022 Web applications\u2014Essentially the same thing as a website in Internet Information Services (IIS). By default, each SharePoint web application has a unique URL and also a separate content database in SQL.\n\n\u2022 Site Collections\u2014Each web application contains one or more site collections. These are containers that define permissions and other functionality for the sites within the site collections.\n\n\u2022 Sites\u2014A site (developers sometimes refer to them as webs\u2014short for website) is a collection of lists, libraries, and other content usually around a specific topic.\n\n\u2022 Lists and Document Libraries (Apps)\u2014Specific containers where content resides. Terminology for SharePoint 2013 had changed, so adding a new list or document library is under the Add an App menu of the Settings menu  . Apps can also be added from the Site Contents page.\n\n\u2022 Items and Documents\u2014Individual pieces of content created in lists and libraries.\n\nFrom a practical perspective, there are a few things to remember about the site structure. The first thing is that there are permissions at each level and how tightly (or loosely) controlled they are can cause unique challenges. This is usually something outside the realm of a typical SharePoint designer, but it is important to consider.\n\nThe second thing is that to create your branding in SharePoint you need to create things that interact with each of these objects to some degree. Which files need to be put where? What needs to be done at the site collection versus the site? The answers to these types of questions will become more obvious as you proceed through the book.\n\nMaster Page Gallery\n\nThe Master Page Gallery is a specialized document library. In SharePoint 2013, the Master Page Gallery has become the default location for all SharePoint branding assets. If you are used to creating websites with a technology that isn't SharePoint, this concept of all branding assets living under the same hierarchy is probably familiar. In previous versions of SharePoint, master pages and page layouts were stored in the Master Page Gallery, and this is still the case for SharePoint 2013. Other assets such as CSS, Images, and JavaScript were usually stored in a separate document library. The SharePoint 2013 approach is just simpler, which is often a good thing.\n\nEvery site has its own master page gallery that you can access by going to Settings menu \u21d2 Site Settings and then clicking on the Master Pages and Page Layouts link under the Web Designer Galleries section. At this point the details of the contents in the Master Page Gallery aren't too important, but you'll become familiar with this library as you progress.\n\nHow Branding Works in SharePoint\n\nBefore diving into the specifics of creating branding in SharePoint 2013, you must understand some of the key ways in which branding can be applied in SharePoint. The following sections examine these key concepts in detail.\n\nComposed Looks\n\nImagine that you want to make some changes to spruce up your home that can be done relatively quickly and easily. Your budget is limited, so you decide to paint the walls with a brighter and more appealing color. In SharePoint terms, this would be equivalent to applying a theme. Technically speaking, themes can be thought of as changes that are applied to the existing look and feel through the use of CSS.\n\nOver the past few versions of SharePoint, themes have evolved more than any other branding function. In SharePoint 2007, themes were stored on the file system of the SharePoint server and consisted of mostly XML, CSS, and images that were applied to a master page. The bigger issue was that although themes were considered to be one of the simplest approaches to branding, creating custom ones was actually quite complicated and required changes to files on the server, which required direct access to the file system. SharePoint Server Administrators aren't usually too fond of making changes to files on the file system, so to address this problem SharePoint 2010 took a new approach to themes and used the same mechanism used by Microsoft Office to create .THMX files that enabled users to select from twelve colors and two fonts. This could all be done through Office clients or through the SharePoint UI, which made custom theme creation much easier than with SharePoint 2007. When created, these files could be uploaded and applied to any site. This meant that themes could be applied without making any changes to the file system, but the downside here was that themes in SharePoint 2010 essentially just recolored sites. They didn't provide a way to modify CSS or add images. Many users felt this iteration of themes was almost too simplistic.\n\nSharePoint 2013 introduces another approach to themes that is noninvasive like SharePoint 2010 but enables more flexibility to add background images and other customizations similar to the types of customization typically made with SharePoint 2007 themes. This new approach to themes is called composed looks. A composed look consists of a color palette, fonts, background image, and associated master page. When a user selects a composed look, all these associated design elements are applied to the site. There are several composed looks available to choose from out-of-the-box, but you can also customize these or even create your own.\n\nComposed looks are a simple way to brand your SharePoint sites, especially collaboration sites. Larger implementations will likely have both publishing and collaboration sites. In these cases a combination of approaches that include composed Looks and custom master pages could be used to present a unified look and feel.\n\nNOTE\n\nYou can learn more about using composed looks in Chapter 3, \"Working with the SharePoint User Interface,\" and about creating composed looks in Chapter 7, \"Creating Custom Master Pages and Page Layout.\"\n\nMaster Pages\n\nRevisiting the home improvements metaphor, if applying a composed look is comparable to painting the walls, then using master pages is comparable to altering the physical structure of the house. By changing the physical structure of your house, nearly anything is possible. Want to add a new room or knock down some walls? No problem. The sky is the limit as long as you have the right skills or can find a good general contractor to help you get the job done. The same is true with master pages in SharePoint.\n\nRemember the good old days of classic web design, all the way back to the late 1990s and early 2000s, when web pages were created with the look and feel hard-coded in every page? Changing the footer, for example, typically required you to access every page in the site, repeating the same change on each one, and then uploading them back to the server. This tedious manual process created many opportunities to introduce errors.\n\nWith the advent of ASP.NET 2.0, master pages were introduced to alleviate this problem. Just like in a typical ASP.NET website, master pages enable designers and developers to create a consistent look and feel for all the pages in a SharePoint website. Every page on a SharePoint site references a master page. When a page is loaded in a browser, SharePoint merges the master page with the page, and the resulting styled page is returned to the user. Master pages enable organizations to create a consistent look and feel across all sites, which is a far better approach than was available in the past.\n\nIn the typical SharePoint site, master pages define the outer shell of the website. Sometimes called the chrome, this shell defines much of the overall look for every page loaded in the site. Master pages include HTML; SharePoint-specific controls (navigation, search, and so on); and content placeholders, containers used to load specific pieces of content from the referring content. A content placeholder is essentially a named container used to render various pieces of content. The most common example of this is `PlaceHolderMain`, which exists on every page layout. This is the content placeholder where field controls, Web Part zones, and anything else that is to be rendered in the central area of the page typically would exist.\n\nSharePoint comes with a few out-of-the-box master pages that can be used for website branding right away. Unlike previous versions of SharePoint, the same out-of-the-box master pages are available for all versions:\n\nContent Pages\n\nAs discussed in the previous section, master pages define the outer shell of a SharePoint page, but the body of the page itself is also important. Depending on the purpose of your site (collaboration versus communication) and the site template you choose, you can have different types of content pages available to you. There are three main types of content pages in SharePoint 2013:\n\n\u2022 Publishing pages\n\n\u2022 Web Part pages\n\n\u2022 Wiki pages\n\nEach type of page has different options and a different intended use, so your functional requirements dictate which page is right for each scenario.\n\nPublishing Pages\n\nPublishing pages are available only in SharePoint on sites where the publishing feature is enabled. Publishing enables authors to create pages that have an approval workflow so that content can be reviewed and approved before being published. For example, you might create a page announcing a new policy, but the new page would need to be approved by your manager before it could be viewed by others on the website.\n\nPublishing pages are created by using page templates called page layouts. If master pages create the outer shell of a SharePoint page, then page layouts define the body of a page. They enable content authors to create pages that contain text, HTML, graphics, rich media, and more.\n\nTo continue with the house metaphor used throughout the chapter, imagine a room with furniture in it\u2014perhaps a couch and a few chairs and tables. The room would be similar in concept to a page in SharePoint; the pieces of furniture would be the various fields on the page. Applying a new page layout would be similar to rearranging the furniture in the room. The room is still the same and the pieces of furniture are still the same, but they are laid out differently and ultimately have a different look and feel.\n\nSeveral out-of-the-box page layouts can be used right away in a SharePoint Server site, but also remember that designers and developers can always create their own custom page layouts. For example, when a user creates a new page in SharePoint Server, the same content can be arranged as a news article or as a welcome page, based on the page layout that is selected.\n\nNOTE\n\nChapter 7 discusses page layouts in greater detail.\n\nAlong with defining how content is arranged on a page, page layouts also define the location of editable fields and Web Parts. Web Parts, which can be thought of as self-contained widgets of functionality, can be arranged in pages through the use of Web Part zones that are defined in page layouts. Web Part zones enable content authors to add and arrange multiple Web Parts vertically or horizontally.\n\nPublishing pages are the most highly structured of the SharePoint content pages. Not only do they use page layouts that provide a fairly strict template for creating content, but the approval workflow helps to prevent unwanted content from ever being seen. Because of the structure provided by publishing pages, they are best-suited for sites where communication is the primary goal. For situations that require a highly stylized layout, a publishing page with custom page layout is ideal.\n\nWeb Part Pages\n\nThe function of Web Part pages is fairly obvious by their name. They are pages that contain Web Part zones, where Web Parts can be placed that display things like list data, images, rich media, or other functionality. Any user with the proper permissions, usually a user with at least Contribute permissions, can create and edit Web Part pages.\n\nWeb Part pages are most appropriate when you simply need a page to display some information via Web Parts. Because the goal of Web Parts is to deliver functionality to the page, often by displaying content from lists and libraries, in most cases the Web Part page itself wouldn't be the place where actual collaboration occurs. In addition, it lacks the structure and formalized publishing process that you get with a publishing page. All versions of SharePoint 2013 support Web Part pages.\n\nWiki Pages\n\nThe goal of a wiki is to enable users to quickly create content and collaborate with other users; thus, wikis tend to be informal and unstructured.\n\nOne of the most commonly created types of site across most implementations is the Team site, which is intended to be a place where teams can collaborate. Beginning with SharePoint 2010, the Team site template came with wiki functionality. From a branding perspective this is important because Team sites will likely be the template used for the greatest number of sites across your organization\u2014assuming you use SharePoint for your intranet.\n\nWhen you create a Team site, the first page you see when you go to the site is an example of a wiki page. In addition, every time you choose to Add a New Page from the Settings menu, a new wiki page is created. Wiki pages are unique because they have been designed to support this idea of quickly adding content to the page. For example, if you click the Edit button at the top right of a Team site, you see a few different areas that initially look like Web Part zones because they contain Web Parts. What you are actually looking at is a Text Layout, which arranges content areas on the page. In these content areas you can put text, HTML, rich media, or even Web Parts. You can also easily change the layout for the page by clicking the Text Layout button on the ribbon to choose from eight different options.\n\nWiki pages are available in all versions of SharePoint. Specifically for SharePoint versions with publishing available, the Enterprise Wiki template supports the capability to create and use page layouts with wiki pages.\n\nDisabling Wiki Pages\n\nText Layouts are useful because of how they can be used to quickly create content. Designers, however, seem to have diverse opinions about wiki pages. Some love the flexibility, whereas others feel that because of the flexibility and unstructured nature of the wiki pages, it causes more problems than it solves. The purpose here isn't to pick one side or the other\u2014but to point out the various opinions.\n\nIf you do find yourself in the camp that doesn't like wiki pages, the good news is that the functionality is easy to disable. All you need to do is click the Settings menu \u21d2 Site Settings and click Manage Site Features, which is found under the Site Actions heading (not to be confused with the old Site Actions menu). From here, click the Deactivate button next to the Wiki Page Home Page feature. Now when you go back to the homepage of your site, you'll notice that when you put the page into edit mode, you have a Web Part page instead of a text layout page. If you prefer this approach to the default Team site with the wiki, you could save your new site as a template, which would enable you to quickly create more non-wiki Team sites.\n\nCascading Style Sheets\n\nCSS is pervasive in SharePoint branding. Almost every aspect of SharePoint is styled by CSS. All the SharePoint controls loaded by a master page are styled by CSS, as well as many Web Parts\u2014even composed looks use CSS. Because of the importance of CSS in SharePoint, a sound understanding of it is crucial to becoming skilled at branding in SharePoint. Chapter 6, \"Cascading Style Sheets and SharePoint,\" explains the typical SharePoint elements and what is involved with styling them using CSS.\n\nSharePoint 2013 splits its default CSS across several smaller CSS files that are loaded according to what controls are available on a given page; however, much of the main CSS for SharePoint still resides in one CSS file named corev15.css. The intent of this division of labor is to load only the CSS that's necessary to render a given page. This approach reduces the total amount of data that needs to be downloaded with every page load compared to loading everything, regardless of whether it was needed.\n\nWhat's New for Branding in SharePoint 2013\n\nWith more than 100 million SharePoint users worldwide, it is probably fair to assume that a good portion of those who are reading this book have at least some familiarity with SharePoint and its features. Whether you are a seasoned SharePoint design pro or someone who is just learning the ropes, there's been a large number of changes to branding and the processes related to branding in SharePoint 2013. Starting with SharePoint 2007, branding was something that started growing in importance, and by SharePoint 2010 custom branding seemed like it was a key part of every implementation project. Microsoft made a lot of investments in improving not only the features related to branding and design, but also around the process of creating custom branding.\n\nNew Approach to Branding\n\nOne of the most common perceptions people have about SharePoint branding is that it is a specialized skill that is difficult to master. One problem is that many traditional web designers struggle with some of the technical aspects of SharePoint. On the flip side, there are also plenty of developers who understand the technical aspects of branding but don't have design skills. This gap between designers and developers is an issue that Microsoft has attempted to address in SharePoint 2013 by looking at the way traditional web designers work and trying to adjust the way branding is created in SharePoint.\n\nWhen you look at how websites are usually created, the design process across most platforms is the same. Most people start creating a design in whatever tool you are familiar with, such as Adobe Photoshop. From there the next step is to turn that design into HTML and CSS. However, turning your HTML and CSS into a SharePoint master page can be a tricky process, and this caused some confusion for traditional web designers. There are several tales of talented web designers who were brought to their knees the first time they tried to brand SharePoint. You'd usually hear them yelling things like \"I don't understand why I can't make this work\" or \"[Other web technology] doesn't make me have to do it that way!\"\n\nTo make the process of branding SharePoint 2013 sites less daunting, one of Microsoft's goals with this release was to create an experience that removed some of the technical hurdles many designers faced. Traditional web designers have no trouble creating working HTML and CSS-based mockups of a site design, but the process of converting that working mockup into a master page needed to be addressed.\n\nDesign Manager\n\nThe answer to this dilemma was to create a tool in SharePoint that did all the dirty work for you. The Design Manager is collectively a set of pages and functionality in SharePoint 2013 that can help guide you through the design process. Basically, the design manager gives you the ability to map a local drive to a site's Master Page Gallery and import HTML and CSS files into SharePoint. From there, SharePoint actually converts the HTML files into a master page. This helps to address one of the biggest pain points that designers had with SharePoint, but that is just scratching the surface of the possibilities. What if you don't have a designer? No worries. A quick search on the Internet can turn up a number of sites that offer various HTML design templates, many of which are free. You could even use one of these templates as the starting point for your SharePoint design.\n\nNOTE\n\nChapter 5, \"Using the Design Manager to Start a Design in SharePoint,\" offers details about the Design Manager.\n\nHTML-based Master Pages and Page Layouts\n\nThe Design Manager has a number of other pages that give you the ability to further customize your design. One of the other important things to remember is that after you import your HTML and convert it into a SharePoint master page, there is still a corresponding HTML version of that master page. The same thing is true if you have any page layouts. Initially, this might seem strange\u2014why would you want to have two versions of the same file?\n\nThe answer to that question is another big change to the branding approach for SharePoint 2013. Web designers usually have their favorite tools when it comes to building HTML and CSS, such as Adobe Dreamweaver, Microsoft Expression Web, or something else. In the past, the SharePoint design process would force them out of the tools that they were the most comfortable with and into others such as SharePoint Designer. So in SharePoint 2013, after HTML and CSS are imported, designers can still edit HTML and CSS files with the tools they are most familiar with, even after they've been converted into a master page. As designers make changes to the HTML files, SharePoint automatically makes the appropriate changes to the corresponding master page or page layout.\n\nCross-site Publishing\n\nOne of the more common complaints with previous versions of SharePoint was about the limited options for sharing content. For example, although it was possible to share content in SharePoint 2010, there wasn't an out-of-the-box option to create content in one site collection and have it roll up to another site collection. The result was that often trade-offs were needed for certain scenarios.\n\nA new feature of SharePoint 2013 is called cross-site publishing, which uses search to surface content\u2014this is referred to as search-driven content. This concept might seem a little strange at first, but there are a number of examples across the web where search-driven content is already used. Perhaps you've been to a site such as Amazon.com and noticed that after buying a birthday present for your 4-year-old daughter the site is kind enough to show you other items it thought you might enjoy, such as various dolls and princesses. The site kept track of the items you had looked at in the past and, using search-driven content, displayed those items to you on the site's homepage the next time you visited. In a scenario like this, a lot of content that appears personalized to you is likely displayed via search-driven content.\n\nA number of pieces work together to make the functionality known as cross-site publishing, which are discussed in this section. At the most basic level, this includes the site where the content is authored, the search piece that crawls and indexes the content, and a Web Part to query and display the content. It might all seem a bit complicated, but the goal for this section is to provide some basic information on the functionality; if your project needs cross-site publishing, you'll most certainly be interested to know how to brand and style it.\n\nNOTE\n\nFor more information about cross-site publishing, see the following articles:\n\n\u2022 \"SharePoint 2013 Cross-site Publishing Overview: Part 1\" (<http:\/\/sharepoint.microsoft.com\/blog\/Pages\/BlogPost.aspx?pID=1038>)\n\n\u2022 \"Plan for Cross-site Publishing in SharePoint Server 2013\" (<http:\/\/technet.microsoft.com\/en-us\/library\/jj635876(v=office.15).aspx>)\n\n\u2022 \"How to set up a Product-centric Website in SharePoint Server 2013\" (http:\/\/blogs.technet.com\/b\/tothesharepoint\/archive\/2013\/02\/14\/how-to-set-up-a-product-centric-web-site-in-sharepoint-2013.aspx).\n\nCatalogs\n\nA catalog is basically just an attribute that can be used to specify that a list or library contains content that you'd want to make accessible to be shared by other site collections. For example, imagine you have a public-facing product site. You might have all the information about your products in an internal site collection that only specific authors have access to. From there they can create, edit, and manage all the details of the products, which would be presented on your public-facing Internet site. This all might sound like an obvious scenario, but this wasn't easily possible in SharePoint before this version.\n\nThis new approach essentially enables the separation of content from the presentation. Content can be authored in a list as fields of data that are indexed by search and then presented to the user where they are needed and presented in a way that makes the most sense depending on the location. That means that the same content can be presented in different ways if needed. So if you consider the example of a product site, you might have a camera that has a corresponding page that provides all the usual details you might expect on a website. But if that camera goes on sale or you want to highlight it in a special way, you can present that same camera in a different way on a different page somewhere on the site\u2014all you need to do is style the presentation accordingly. This is made possible through display templates, which are discussed later in this section.\n\nYou might want to use catalogs in several different ways. The following is a list of some common scenarios where this tool can be useful:\n\n\u2022 Intranet\u2014Authoring content in one site collection that is presented to another site collection. In this case, marketing or another department might be creating all the content in a central site that would then be displayed on the main intranet pages.\n\n\u2022 Internet\u2014Authoring content and products could be done in separate internal sites and syndicated to specific external sites as needed. This would be similar to the intranet scenario but would add a separate catalog for products in addition to the content. Content from multiple catalogs could be shared to one or more public-facing Internet sites.\n\n\u2022 Extranet\u2014Authoring for content could be done in internal sites and then shared with specific external sites. In this scenario, content could be shared from a single catalog to many external sites.\n\nThere is more involved with catalogs than just purely sharing content. The whole subject of catalogs could easily fill an entire chapter in this book. This section provided a high-level overview on the subject, but a deeper dive is needed to cover all the specific details. A good starting point is the Microsoft TechNet article, \"Configure cross-site publishing in SharePoint 2013\" (http:\/\/technet.microsoft.com\/en-us\/library\/jj656774(v=office.15).aspx.aspx)).\n\nContent Search Web Part\n\nIn the past, the primary means of rolling up content was using the Content Query Web Part (CQWP), which basically sent out a query across all the objects in a site collection to find items that matched a specified set of criteria. Rolling up content using search is a more efficient way to do things because it pulls results from an index that is optimized for serving results. A more practical way to describe the difference between the two approaches is to do a search for a file on your computer. If the computer has been indexed, the results display almost immediately, but if the computer must search folder by folder, it takes significantly longer. In the case of SharePoint, that isn't to say using the CQWP was a bad thing; it just had limits to its scalability. The CQWP is still available in SharePoint 2013 and will remain an important way to roll up content for certain scenarios.\n\nThe Content Search Web Part is the primary mechanism for surfacing search-driven content to the page in SharePoint 2013. It can show any content as long as SharePoint search can crawl it and add it to the index. This is a huge change from the past because it means that not only is this approach far more scalable, but it also enables a tremendous amount of flexibility. From a planning perspective, this means that there are a number of new options available. For example, in the past you were limited to rolling up content within a site collection with the CQWP, but now with the Content Search Web Part, you can create content wherever you want and surface where it is needed.\n\nTypically, when you type in a search query to try to find something, you enter a set of terms or criteria to make sure that your results are relevant to what you are looking for. The Content Search Web Part is no different, but instead of typing in a new set of search terms every time you go to the page, it uses a predefined query. To specify the query, the Content Search Web Part has a query builder that enables you to create a search query that can surface the results you're looking for. There are several queries that are predefined, or more advanced users can configure custom query parameters. There's even a results preview pane where you can see the items that are coming back to further refine the Web Part.\n\nAfter the results come back from the Web Part, they are presented to the user, but by themselves those results would look quite rough. The Content Search Web Part uses a new mechanism called Display Templates to present the results to the user. Display templates are discussed in the next section.\n\nThere are a few things to keep in mind about the Content Search Web Part. The first thing is that content needs to be in the search index before it can be displayed by this Web Part. That means that your content won't display until the next search crawl happens. In most cases this won't be too big of an issue, but it is worth mentioning because you might be working on something and scratching your head when you aren't seeing a piece of content that you just know should be showing up.\n\nAnother consideration is that to take full advantage of this Web Part, you need to make sure that any content you want to surface has the right metadata associated with it. This is another case in which it doesn't sound like it is a big deal, but if your environment involves a large amount of content potentially spread across many site collections (or more), this can be a little more involved than it might initially seem. For any project that leverages the Content Search Web Part, you should take the time to carefully consider and plan accordingly for content and the metadata that's required.\n\nFinally, the Content Search Web Part isn't available in all versions of SharePoint. As of this writing it is only available in SharePoint Server 2013 Enterprise. It is not currently available for SharePoint Online; however, Microsoft has stated that it is planned to be added in the future.\n\nNOTE\n\nFor more information about the Content Search Web Part, see \"Introducing the Content Search Web Part\" (<http:\/\/sharepoint.microsoft.com\/blog\/Pages\/BlogPost.aspx?pID=1053>).\n\nDisplay Templates\n\nDisplay templates control the presentation of the results that come back from the Content Search Web Part as well as other search Web Parts, such as the Core Search Results and Refinement Panel. In the past, styling the results of Web Parts required XSL, which it seemed that no one was actually an expert with. The experience of trying to customize XSL to meet specific design requirements was often tedious and frustrating. For many it is probably good news that display templates use HTML and JavaScript as the way to customize results.\n\nDisplay templates are stored in the Master Page Gallery of your site collection, and there are several available out-of -the-box. Even without making any changes to SharePoint, it is easy to understand the potential flexibility of display templates if you consider the different Web Parts, such as the Search Results and Refinement Web Parts that also use display templates in SharePoint 2013. At this point, the important thing to remember is that display templates should provide a more familiar way for designers to style rollups. They are extremely versatile and can be used for everything from styling simple search results all the way to dynamic animated rotators. Display templates are discussed in more detail in Chapter 9, \"Creating Content Rollups with SharePoint WCM.\"\n\nThis shows a Content search Web Part using a custom display template.\n\nDevice Channels\n\nSupport for mobile devices, tablets, and other screen sizes have become an important design for modern websites. In previous versions of SharePoint, supporting different devices was possible, but building and customizing the experience for these devices wasn't as straightforward as many designers would have liked. SharePoint 2013 has addressed this need with a new feature of publishing called device channels. Device channels enable a site to be rendered in different ways depending on the type of device that is requesting the page. For example, if you browse to a website whatever browser you use can pass along some information called a user agent string that identifies the software making the web request. If you've never seen a user agent string, here are a few common ones:\n\n`Internet Explorer 9: Mozilla\/5.0 (compatible; MSIE 9.0; Windows NT 6.1; WOW64; Trident\/5.0)`\n\n`iPhone 5: Mozilla\/5.0 (iPhone; CPU iPhone OS 5_0 like Mac OS X) AppleWebKit\/534.46 (KHTML, like Gecko) Version\/5.1 Mobile\/9A334 Safari\/7534.48.3`\n\n`Lumia 920: Mozilla\/5.0 (compatible; MSIE 10.0; Windows Phone 8.0; Trident\/6.0; IEMobile\/10.0; ARM; Touch; NOKIA; Lumia 920)`\n\nDevice channels enable you to filter based on any piece of information that is available in the user agent string. From here you can assign a different master page to the respective channel. This enables you to create a master page with associated CSS that is more optimal for the targeted device.\n\nSummary\n\n\u2022 The version of SharePoint you use determines the features and functions that you'll be able to use, which could impact your branding approach.\n\n\u2022 Composed looks, master pages, page layouts, and pages are the major components used in SharePoint branding. Cascading Style Sheets (CSS) are used to style all the elements of the SharePoint pages that are rendered in the browser.\n\n\u2022 SharePoint 2013 enables new approaches to branding. This includes the capability for users to import HTML and CSS, which get converted into a working SharePoint master page using the Design Manager.\n\n\u2022 Designers are able to map the Master Page Gallery to a local drive and use familiar tools such as Dreamweaver to edit SharePoint branding.\n\n\u2022 The addition of cross-site publishing and other improvements such as device channels make it easier to create custom-designed dynamic sites in SharePoint 2013.\n3\n\nWorking with the SharePoint 2013 User Interface\n\nWhat's In This Chapter\n\n1. Using the Ribbon to Edit Pages\n\n2. Adding Images and Embedding Videos\n\n3. Introduction to Managed Navigation\n\n4. Applying Themes with Composed Looks\n\n5. Setting Master Pages\n\n6. Working with Page Layouts\n\nIt's hard to create a design for SharePoint without at least a basic understanding of how it works from an end-user perspective. In this chapter, you will take a quick tour of the SharePoint 2013 user interface and learn how to work with content on a page, add apps and Web Parts, and modify the navigation. Then you'll take a look at the out-of-the-box ways to change the look and feel of your site with composed looks, master pages, and page layouts.\n\nPage Editing and the Ribbon\n\nIf you haven't heard of the ribbon by now, put this book down immediately and upgrade your old copy of Office 2003 to something from this decade\u2014say, Office 2013. In short, the ribbon has been present in the Office suite of products since the 2007 release and found its way prominently into the SharePoint 2010 interface to make working with sites easier and more akin to its desktop counterparts such as Word and Excel.\n\nThe ribbon has made a triumphant return in SharePoint 2013 to continue on the legacy of its ancestors. It acts much in the same way that the ribbon acts in the Office desktop products and in SharePoint 2010, with a few refinements.\n\nIf you put a SharePoint page into edit mode you'll see the ribbon in all its glory\u2014providing you with an interface similar to Word for formatting your text, adding pictures and other content, and even spell checking your web page.\n\nThe ribbon's contextual nature hides (or disables) options that don't apply to what you are working on at any given moment. For instance, if you click a picture you've added to a page, the Image Formatting and Design tabs appear. Otherwise, they are hidden because they don't apply to the immediate application. Likewise, you see a different set of ribbon options when you edit a page than when you work with a list.\n\nThe concept behind the ribbon is to put more options in front of the user rather than hide them in menus and dialogs. This enables you to accomplish more without changing screens or hunting for options; the most common tasks you need are right in front of you.\n\nThe behavior of the ribbon is similar to SharePoint 2010, but you may find that some items that you've been accustomed to have a new home. In addition, you'll notice that the ribbon is divided into two sections. The colored section at the top is called the Suite Bar. A closer look at the SharePoint 2013 ribbon's bits and pieces should clear up any inconsistencies between these two versions.\n\nOn the left you see the ribbon conveniently reminding you that you are using SharePoint with the text in the upper-left corner. In farm installations of SharePoint, the text says \"SharePoint.\" If you're using Office 365, you'll see the Office 365 logo. It can't be changed or hidden from any site setting in the browser, but it could be removed on a custom master page or hidden with some custom CSS. Below that you see the available tabs that the ribbon offers at the moment. When you want to open different sections of the ribbon, this is where you click (just like in the Office clients). If you are used to SharePoint 2010 you may be wondering where the Site Actions button is; read on to solve the mystery.\n\nMoving on, in the middle of the ribbon is where you can see additional links to commonly accessed areas in SharePoint. These include SkyDrive, Newsfeed, and Sites. If you use Office 365, you see a few extra links here including a link to Outlook. It's worth noting that you can create a custom delegate control to add additional links to the Suite Bar.\n\nThe right side of the ribbon is where things start to get interesting. On the top right of the ribbon, you see the name of the currently logged-in user, a gear icon, and a question mark icon. Below those icons is another row of icons and links, starting with Share, Follow, Sync, Edit, and a Full Screen toggle icon. These are described in the following table.\n\nAlthough the ribbon sports a slightly updated look from its 2010 counterpart, you'll find that the functionality is largely the same. Sure, things have been moved around some, but you should find that the ribbon is intuitive and easy to use, especially after you start to use its functionality to edit pages.\n\nCreating Sites and Subsites\n\nThroughout this chapter and the rest of the book you work with sites that are based on different templates such as team sites and publishing sites. If you have access to Central Administration on your SharePoint server you can create top-level site collections using the various site templates that are discussed in the book (to learn more about creating site collections see: http:\/\/technet.microsoft.com\/en-us\/library\/cc263094.aspx). However, if you don't have access to Central Administration or you want to create subsites that are based on a template other than the root site collection, you can easily use the SharePoint web interface to create as many different types of subsites as you like.\n\nIf you have worked with SharePoint at all before you are probably comfortable creating many types of sites, but if you are new to SharePoint you should take a moment to review the following example.\n\nExample: Creating a Subsite\n\n1. From the homepage of your site, click the Settings menu \u21d2 Site contents.\n\n2. Scroll to the bottom of the Site contents page to the Subsites section. Any existing subsites are listed here, with the option to create new subsites. Click the new subsite link.\n\n3. Type a Title for the new subsite and optionally a description.\n\n4. Type the end of the URL for the site you'll be creating. For example, if you are creating a Syrup subsite for Randy's Waffles, your URL might be <http:\/\/randyswaffles\/syrup>.\n\n5. Select the template to be used for the site. The type of site you choose will generally be dictated by the work done on it. For this example, click the Collaboration tab and choose the Team Site template.\n\n6. Select whether you want to use the same permissions as the parent site (in which case the site will inherit the parent's permissions) or whether you want to use unique permissions for the site. For this example, select Use same permissions as parent site.\n\n7. Choose whether the top link bar (top navigation) will be the same as the parent site. This setting also depends on what you are using the subsite for. If you are creating a site for a team to work together on a project, it might make sense for them to create a unique navigation structure that they can customize. If you're creating an Internet-facing site you would probably want to keep the navigation consistent throughout the site. For this example, select Yes.\n\n8. Click the Create button to create your subsite.\n\nKeep in mind that subsites can contain other subsites. The process is exactly the same, no matter if you are creating subsites from the root of the site collection or from within other subsites.\n\nwarning\n\nIf you notice that you are unable to create common types of sites there is a menu you can check: Site Settings \u21d2 Look and Feel \u21d2 Page layouts and site templates. Sometimes the Subsite Templates section is set to only allow you to create sites based on certain site templates. You can simply add back Site Templates here or you can change the setting to Subsites can use any site template.\n\nThe Page Editing Experience in SharePoint 2013\n\nSome minor editing differences exist between the various types of SharePoint sites, but there are a few basic concepts that apply to page editing no matter what type of site you use. These basics are important to know before getting into specifics.\n\nTo access the basic tools to begin editing your page, click the Edit link in the upper right of the ribbon. (If you're working on a publishing site and don't see the Edit icon, click the Settings menu, then click Show ribbon.) The page switches to Edit mode. This is where you can add and format text, add media (such as images and video), add apps and Web Parts, and generally lay out your page the way you want it to look.\n\nNOTE\n\nIf you're familiar with SharePoint 2010's editing experience, rest assured that not a whole lot has changed this time around. A few buttons have been added or moved here and there, but overall if you knew the ribbon in SharePoint 2010, you aren't going to be in for much of a shock here.\n\nEditing and Formatting Text\n\nWhen SharePoint 2010 arrived, it drastically improved the page-editing experience for users. It was much easier to quickly add content to a page and format it because of the ribbon. It wasn't without its quirks, however, but overall the editing experience was a good one. SharePoint 2013 is no exception; it makes editing and formatting text on a SharePoint page easy and even improves the experience introduced in SharePoint 2010.\n\nEditing a page in SharePoint 2013 is similar to editing a Word document. When you put your page into Edit mode, you'll be primarily working with the Format Text tab on the ribbon. In this tab, you have the ability to change fonts, change the font style (such as bold, italic, and underline), color your text, create bulleted and numbered lists, align your text, choose from predefined styles for page elements (such as Heading 1 and Heading 2), or choose between built-in text styles. This Styles section replaces the two drop-down options called Styles and Markup Styles from SharePoint 2010. You can also add custom styles to this section, which is detailed in Chapter 6, \"Cascading Style Sheets and SharePoint.\"\n\nNo matter if you work on a collaboration site (such as a site built from the Team site template) or a publishing site (such as a site created using the publishing site template or a collaboration site with the publishing features enabled), the left side of the Format Text tab has Save and Check In buttons. If your site is a publishing site, you will find an additional Publish button under the Publish tab in the ribbon. More about these buttons and what they do is covered in the \"Saving Your Page\" section later in this chapter. For now, just remember where they are.\n\nThere are several other similarities between editing SharePoint pages using the ribbon and editing documents using Word. You can use the ribbon to check the spelling on your page (though this option is limited to sites with publishing features enabled). If you're so inclined, you can also edit the HTML source for your page's content if you need to fine-tune something. Also, just like in Word, you can copy, cut, and paste content, as well as undo any mistakes you make.\n\nYou may recall in past versions of SharePoint many users had a hard time copying content created in a Word document and pasting it into SharePoint. Often, content pasted into SharePoint from Word just didn't look right, especially if the Word document had numbered or bulleted lists, different fonts, or a stylized layout, and a lot of time and effort had to be spent cleaning up the look of the page. Sometimes users found it easier to manually re-create the contents of the Word document in SharePoint rather than spend the time cleaning up the look of the page. Fortunately, the copy-paste functionality from Word to SharePoint has been drastically improved and content now appears much cleaner in SharePoint 2013 than in previous versions.\n\nTo get a better feel for the page-editing experience in a couple of different site templates, try creating and editing a page in a site based on the Team Site template. (If you don't have a Team site created, refer to the \"Creating Sites and Subsites\" section of this chapter.) The following example applies to all collaboration sites (without publishing features enabled). Collaboration sites in SharePoint 2013 utilize text layouts for holding the text and content on a page. These are essentially templates that provide areas to add text on a page. You can choose from any of the eight available text layouts, but there isn't a way to add additional layouts. If your branding requires a layout that differs from what is available in the text layouts, you will need to look at enabling publishing features and creating custom page layouts for that.\n\nExample: Getting Started Editing a Page\n\n1. Click the Settings menu \u21d2 Add a page.\n\n2. In the New page name field, type Why I Like Waffles.\n\n3. Click Create. The page opens in Edit mode. The default text layout is the One column layout that spans the entire body of the page.\n\n4. In the Format Text tab of the ribbon, click the Text Layout drop-down and select One column with sidebar. The text layout changes so there are two areas for adding content now.\n\n5. Click the cursor in the larger text area and type Waffles are Delicious.\n\n6. Select the text with your mouse cursor, and click Heading 1 in the Styles section of the ribbon. You see a live preview of what the text will look like as you hover over each option.\n\n7. Click the mouse cursor after Waffles are Delicious, press Enter to go to a new line, and type the following text: Here are three reasons I like waffles:\n\n8. Select the text you just typed and change the font by clicking the font drop-down (next to the font size drop-down) and selecting Trebuchet MS. Press Enter to go to a new line.\n\n9. In the Paragraph section of the ribbon, click the Bulleted List button and type the following text (press Enter after each to go to a new line):\n\nYou can smother them in syrup and butter\n\nThey go well with bacon\n\nThey come in a variety of flavors\n\n10. Click the Save button on the left side of the ribbon to see how the page looks.\n\n11. On the right side of the ribbon, click the Edit icon to open the page in Edit mode again and on a new line below the bulleted items, type the following text: I think I should go eat some waffles now!\n\n13. Select the text you just typed and make the text bold by clicking the B button under the font selection drop-down. Then click the Save button on the left side of the ribbon.\n\nThese steps are pretty basic to show a simple example of how page editing works in SharePoint 2013. When working with a site that has the publishing features enabled (such as a publishing site template) the editing experience is similar to editing pages on a collaboration site, except that publishing sites use page layouts instead of text layouts. Page layouts have different SharePoint field controls associated with them to store and display information. They are more flexible than text layouts in that designers can create custom page layouts to accommodate just about any design need. Page layouts can also utilize Web Part zones for storing Web Parts and apps.\n\nExample: Editing a Page on a Publishing Site Template\n\n1. From the homepage of the publishing site, click the Settings menu \u21d2 Add a page.\n\n2. In the Give it a name field, type Why I Like Waffles, then click Create. Your new page layout opens in Edit mode. By default, SharePoint publishing sites use a page layout that has a page content field control that spans the width of the page (you'll learn how to change this setting later in the section \"Working with Page Layouts.\")\n\nYou might notice that the Format Text tab in a publishing site is identical to the Format Text tab in a collaboration site with one exception: the Text Layouts button has been replaced with a Check Spelling button.\n\n3. Click within the Page Content field control and type Why I Like Waffles.\n\n4. In the ribbon, click the Page tab, and then click the Page Layout drop-down. In the Article Page section, choose the Image on left layout.\n\n5. Your page's content remains the same, but now you can see extra fields built in to the page that you can fill out, such as an image field, an article date field, and a byline field. You won't be doing anything with these fields at the moment; for now, just note that this is one of the differences between page layouts and text layouts. You'll learn more about fields on page layouts later in the chapter.\n\n6. Select the Why I Like Waffles text you just typed, and then select the Heading 1 style from the ribbon.\n\n7. Click the mouse cursor after Why I Like Waffles, press Enter to go to a new line, and type the following text: Here are three reasons I like waffles:\n\n8. Select the text you just typed and change the font by clicking the font drop-down (next to the font size drop-down) and selecting Trebuchet MS. Press Enter to go to a new line.\n\n9. In the Paragraph section of the ribbon, click the Bulleted List button. Type the following text (press Enter after each to go to a new line):\n\nWaffles are delicious\n\nThey go well with bacon\n\nThey come in a variety of flavors\n\n10. Click the Save button on the left side of the ribbon to see how the page looks.\n\n11. On the right side of the ribbon, click the Edit icon to open the page in Edit mode again and on a new line below the bulleted items, type the following text: I think I should go eat some waffles now!\n\n12. Select the text you just typed and make the text bold by clicking the B button under the font selection drop-down. Then click the Save button on the left side of the ribbon.\n\nAs you can see, the processes for editing pages in a collaboration site and a publishing site are virtually identical. The major difference is the fact that publishing sites utilize page layouts, which can feature additional fields and Web Part zones, on the page that you can take advantage of.\n\nWorking with Images\n\nIf you spend any amount of time browsing the web, you have probably noticed that almost all websites use some sort of imagery on their pages. So it's likely that you'll want to add images to your page as well. There are several ways to add images to a page.\n\n\u2022 In a Page Content control or text field on a Wiki page\u2014The Page Content control found on page layouts in publishing sites and the wiki page text areas in collaboration sites not only enables you to insert text, but also enables you to insert images. You can choose to insert an image in SharePoint from your computer or from another URL.\n\n\u2022 Using an Image Viewer Web Part\u2014You can use a Content Editor Web Part or Image Viewer Web Part to add images to your page in a zone or in a page content area. This option is more limiting (and cumbersome to work with) than using other methods. You need to have your image uploaded to SharePoint and know the URL (or at least have the URL copied to the clipboard) before you use the Image Viewer Web Part; it curiously doesn't have a Browse feature to let you find the image. In addition, you can't take advantage of the new image renditions feature or resize the image in any way, so you are stuck using the original image size. In other words, if you need to add an image to your page, there aren't many compelling reasons to use the Image Viewer Web Part from a branding perspective.\n\n\u2022 Using a Content Editor Web Part\u2014The Content Editor Web Part is sometimes referred to as the Swiss army knife of Web Parts, and with good reason. It enables you to add text, images, scripts, and styles to your page. Content Editor Web Parts are kind of like mini-page content field controls. You can add images, select where the image comes from, and format the image to fit your needs.\n\n\u2022 In an image control on a publishing page\u2014Some page layouts have a built-in image field control that enables you to add an image. Although the link in the control says to Click Here to Insert an Image from SharePoint, you can also upload an image file from your computer as you browse the site's contents.\n\nEach of these ways follows the same basic steps to get an image on your page. Generally, you can choose to add an image that's already been uploaded to your site somewhere, or you can browse your computer for an image you want to upload and add. These steps can apply to either a collaboration site or a site with publishing enabled.\n\nExample: Adding an Image to a Page\n\n1. Click the Settings menu \u21d2 Add a page.\n\n2. In the New page name field (or Give it a name field if you're on a publishing site), type Waffle Picture.\n\n3. Click Create. The page opens in Edit mode.\n\n4. On the ribbon, click the Insert tab; then click the Picture drop-down.\n\n5. In the Picture drop-down, select From Computer.\n\n6. Click the Browse button to browse your computer for an image. If you have downloaded the files for this chapter, look for WaffleWithSyrup.jpg, and click Open.\n\n7. Select the Destination Library you'd like your image to go in (Images for publishing sites or Site Assets for collaboration sites are the default locations, but you can change this if you wish). If your site has publishing enabled, optionally type some comments in the Version Comments field.\n\n8. Click OK. If your site has publishing enabled, you'll be prompted to fill in additional metadata about the image. Optionally do so, and click the Check In button in the ribbon in the dialog, or the Save button at the bottom of the dialog. Your image is added to the page. .\n\n9. Click the image to select it. In the ribbon, click the Image tab that has appeared.\n\n10. In the Horizontal Size field in the ribbon, type 500px and press Enter. The image is resized.\n\n11. Click the Image Styles drop-down and select Light border.\n\n12. Toward the left side of the ribbon, change the Alt Text field value to A Picture of a Waffle and Syrup. This text will appear when the mouse cursor is hovered over the image.\n\n15. Click the Save button on the right side of the ribbon.\n\nYou'll also notice in the Image tab on the ribbon that there are a few other options you can take advantage of when working with images, such as changing out the image with the Change Picture button, changing the position of the image in relation to the text with the Position drop-down, and increasing the horizontal and vertical spacing between the image and text with the Vertical Space and Horizontal Space options. Finally, if your site has publishing features enabled, you can use image renditions to resize\/crop your images.\n\nImage Renditions\n\nA new concept to SharePoint 2013 is image renditions. The basic purpose of image renditions is to offer differently sized versions of images uploaded into SharePoint. To take advantage of this useful feature, you need to be working on a site with publishing features enabled, and also have BLOB caching enabled for your web application.\n\nExample: Enabling the BLOB Cache\n\n1. Log in to your SharePoint server as an administrator.\n\n2. Click Start \u21d2 Computer and navigate to the following directory: `C:\\inetpub\\wwwroot\\wss\\VirtualDirectories`.\n\n3. Open the folder of the web application in which BLOB caching will be enabled. For instance, if your site's URL is <http:\/\/wafflenet\/> the folder you want is most likely called `wafflenet80`.\n\n4. Create a backup copy of the `web.config` file. This is always recommended if you ever need to edit or modify this file, so you can always revert back in the event that something happens.\n\n5. Open the `web.config` file in Notepad or another editor.\n\n6. Press Ctrl+F to open the Find dialog and type <BlobCache.\n\n7. Press Enter to perform the search, which brings up the following line:\n\n`<BlobCache location=\"C:\\BlobCache\\14\" path=\"\\.(gif|jpg|jpeg|jpe|jfif|bmp|dib|tif|tiff|themedbmp|themedcss|themedgif|themedjpg|themedpng|ico|png|wdp|hdp|css|js|asf|avi|flv|m4v|mov|mp3|mp4|mpeg|mpg|rm|rmvb|wma|wmv|ogg|ogv|oga|webm|xap)$\" maxSize=\"10\" enabled=\"false\" \/>`\n\n8. At the end of the line, change `enabled=\"false\"` to `enabled=\"true\"`.\n\n9. Save the `web.config` file. You can log out of the server now.\n\nWhen using image renditions, your image will be cropped and resized according to a handful of preset sizes, including any custom sizes you create. You can then further customize how each image is cropped and resized for each rendition, and add those versions of the image to the page. In past versions of SharePoint, it was possible to resize an image on a page; however, the image's file size remained the same no matter how large or small the image actually displayed on the page. Image renditions aren't just the original image displayed smaller or larger. They are entirely unique image files stored on the SharePoint server's hard drive. When image renditions are configured, SharePoint automatically crops and sizes the original image to the specified dimensions and saves those files as individual images on the hard drive. Those newly created images are then referenced back to the original image.\n\nWhen you add an image onto a page, you can click the Pick Rendition button in the Image tab on the ribbon to choose the size (or image rendition) of the image. SharePoint swaps out your original image with its image rendition version, which can reduce the overall size of the page weight. Image renditions are also useful when leveraged in Device Channels, so the same image can be used on mobile sites without having to resize everything manually. The other major benefit of image renditions is that you can use it to enforce consistency among the images on a site.\n\nIf one of the renditions of a particular image doesn't look right, you can adjust how the image displays in each rendition. Keep in mind that editing the renditions for an image affects all instances of that image rendition throughout the site.\n\nExample: Creating and Using an Image Rendition\n\n1. Click the Settings menu \u21d2 Site Contents.\n\n2. Click the Images library. (If you don't have an Images library, click the Add an App link, select the Picture Library app, and name it Images.)\n\n3. Click the \\+ New item link to browse your computer for an image to upload, or you can simply drag an image from your desktop to the image library to quickly upload it. (This example uses EggsAndBacon.jpg from the chapter downloads.)\n\n4. Click the Settings menu \u21d2 Add a page.\n\n5. In the Give it a name field, type Image Rendition Page and click Create.\n\n6. Click the Insert tab in the ribbon, click the Picture drop-down, and select From SharePoint.\n\n7. In the left pane, click the Images library, click the EggsAndBacon image to select it, and click Insert at the bottom of the Select an Asset dialog.\n\n8. Click the Save button on the right side of the ribbon. The image you just added is pretty big and a little busy. Your image rendition would be better if it was smaller and focused on a specific part of the image; namely, the delicious-looking plate of food.\n\nPhoto Credit: Tom Drisgill\n\n9. Click the Settings menu \u21d2 Site Settings.\n\n10. Under the Look and Feel header, click Image renditions. (If you don't have the BLOB cache enabled, you'll see an error on this page, so make sure it is enabled.)\n\n11. Above the existing image renditions, click the Add new item link.\n\n12. In the Name field, type WaffleNet Image Rendition. This is what will show up in the Pick Rendition drop-down, so be descriptive. Here you can also specify the width and\/or the height of the image, in pixels. In this example, type 500 in the Width field. This results in a 500-pixel-wide image. You must specify either the width or height of an image rendition, or both.\n\n13. Click Save.\n\nWhen specifying both a height and width for an image rendition, SharePoint crops images down to the specified size. You might lose part of your pictures, which is why it's a good idea to preview your images and adjust the rendition for an image as needed. This is explained in greater detail in the next example. If you specify only the width or height, the image rendition resizes the image and shows the entire image at the specified width or height, keeping the image proportions the same. No cropping occurs here, unless you manually crop the image when you edit the rendition for a particular image. After you set up your image rendition, you can try it out.\n\nExample: Editing an Image Rendition\n\n1. Click the Settings menu \u21d2 Site contents.\n\n2. Click the Images app and hover the mouse cursor over the EggsAndBacon.jpg image preview.\n\n3. Click the... (ellipsis) in the lower-right corner of the thumbnail.\n\n4. In the pop-out menu, click Edit Renditions.\n\nPhoto Credit: Tom Drisgill\n\n5. On the Edit Renditions page, look at the various renditions that SharePoint has put together. The various renditions are ordered by size. Chances are some of the cropping doesn't look great on some of them, which is why it's not a bad idea to preview the image renditions when you upload an image to the site. Click the Click to change link under the image preview for the WaffleNet image rendition you created.\n\nPhoto Credit: Tom Drisgill\n\n6. If you are prompted to check out the image, click OK; otherwise continue to the next step.\n\n7. In the Crop Rendition window, click and drag the corner handles of the image to change the cropping area. Crop the image so only the plate of food is visible. You can also click and drag the highlighted section around on the image to change what area of the image will show in the image rendition. A preview of your rendition appears at the bottom of the dialog window.\n\nPhoto Credit: Tom Drisgill\n\n8. Above the image rendition preview, click Save.\n\n9. Scroll to the bottom of the Edit Rendition page and click OK to save your changes to the image renditions.\n\n10. If you checked out the image, be sure to check it in so other users can use the image renditions you set up.\n\nNow that you've created and edited your rendition, it's time to put it to use.\n\nExample: Adding an Image Rendition\n\n1. Return to the image rendition page you created earlier. (SharePoint added a link to the page on the top and left navigation areas of the site, or you could view the site contents and find the page in the Pages library.)\n\n2. On the right side of the ribbon, click the Edit button.\n\n3. With the page in Edit mode, click the image you added earlier.\n\n4. Click the Image tab that appears on the ribbon.\n\n5. Click the Pick Rendition drop-down. (Note that this drop-down will not show if BLOB Cache is not enabled.)\n\n6. Select the WaffleNet image rendition you created and your image will be resized and cropped to the dimensions specified in the image rendition.\n\n7. Click the Save button in the ribbon to save your page. As you can see, the image is much smaller on the page and has been cropped to show only a specific area of the overall image.\n\nPhoto Credit: Tom Drisgill\n\nWorking with Videos\n\nVideo has now become an almost essential part of many websites and you will likely want to add one to your site one day. There are two ways you can add video to your site: You can use the built-in Media Web Part to play videos stored on the site, or you can embed a video from an external video provider (such as YouTube or Vimeo). Both ways work well, although there is some consideration to be made when storing videos in the SharePoint site because videos tend to take up a lot of disk space. Storing on an external video hosting site can save you a lot of disk space, but the downside is that the video needs to load from an external site, which can potentially cause a slight increase in page load times.\n\nThe Media Web Part was introduced in SharePoint 2010, and it's largely the same in SharePoint 2013. Embedding a video is a different story, however. Although you could technically embed externally hosted videos in previous versions of SharePoint using the Content Editor Web Part to edit the HTML source, it was always more trouble than it felt like it should have been and it wasn't a very intuitive process for many users. Luckily, Microsoft has made the process much easier by adding functionality to embeded code. When you want to embed a video (or other code, for that matter), you simply paste in the embed code provided from the video provider site, and SharePoint does the rest. Just like that, you have added a video to your page.\n\nExample: Uploading a Video Using Media Web Part\n\n1. Click the Settings menu \u21d2 Site contents.\n\n2. Click the Documents library to open it. You'll upload the video here.\n\n3. Drag a video file from your computer to the document library to upload it. If you're using the file downloads for this chapter, you'll be using WaffleTower.mp4.\n\n4. Click the Settings menu \u21d2 Add a page.\n\n5. Give the page a name, such as Waffle Video, and click Create.\n\n6. When the new page opens, it is in Edit mode. Click the Insert tab on the ribbon.\n\n7. Click the Video and Audio drop-down and select From SharePoint. (Note that there is also a From Computer option in this drop-down. You can use this to upload and add videos on-the-fly without leaving your page.)\n\n8. In the Select an Asset dialog window, click the Documents library where you uploaded your video, and click WaffleTower.mp4 so that its URL appears in the Location (URL) field at the bottom of the dialog window.\n\n9. Click Insert. The Web Part is added to your page. Click the Play button or anywhere in the Web Part to view the video.\n\nPhoto Credit: Tom Drisgill\n\n10. Click the Media Web Part title to open the Web Part tab on the ribbon. (Note that two titles appear: the Web Part title only appears in Edit mode by default, and the video title appears on the video itself when you hover the mouse over the video.).\n\n11. On the Media tab, change the Title field from Media Web Part to Waffle Tower Video.\n\n12. Take note of the options you have in the Web Part tab of the ribbon for the Media Web Part. These include the ability to change the video, set basic playback options such as automatically starting the video on page load, looping the video continuously until it is stopped by the user, changing the style of the video player (you can choose between a dark or light player), and how large (in pixels) the video should display on the screen. For example, if you wanted to increase the video size to take more horizontal space than the default size, you could increase the horizontal width to your desired size. Below the sizing fields is a check box that allows you to preserve the aspect ratio of the video, meaning that SharePoint automatically adjusts the height as you adjust the video's width (and vice versa) in order to maintain the same proportions of the video in the Web Part.\n\n13. Click the Page tab on the ribbon, and click Save.\n\nThe Media Web Part's default look is a little bland, but you can choose a different image to display as the preview image of the video. The bad news is that if you want it to be a still frame from the video, you need to capture that still frame, save it as an image, and upload it to SharePoint. The process is a little complicated.\n\nExample: Adding a Preview Image to a Media Web Part\n\n1. Start playing the video and pause it on the scene you want to capture.\n\n2. Open a screen capture tool such as the built-in Snipping tool in Windows 7 and 8.\n\n3. Grab the frame from the video and save it to your computer. If you're following along using the files for this chapter, you can use the WaffleTowerPreview.jpg image instead of getting a screenshot yourself.\n\n4. On the Waffle Video page, click the Edit button on the ribbon.\n\n5. Click the Waffle Tower Video Web Part title to open the Media tab on the ribbon.\n\n6. Click the Change Image drop-down and select From Computer.\n\n7. Browse your computer for the image you captured from your video (or WaffleTowerPreview.jpg if you are using the files for this chapter).\n\n8. Select the destination library to which you want to upload the image and click OK. If you're on a publishing site, you can use the Images library. If you are using a collaboration site, you can use the Site Assets library.\n\n9. If necessary, fill in any of the metadata information about the image, and click the Save or Check In button. (Depending on your site type or library settings, you may not be prompted for this step.)\n\n10. Click the Save button in the ribbon to preview your page.\n\nThe Web Part now displays your preview image and looks much nicer than the default. Users will have a better idea of what they can expect from the video.\n\nPhoto Credit: Tom Drisgill\n\nNow that you know how to use the built-in Media Web Part, take a look at embedding external videos onto your page.\n\nExample: Embedding External Videos\n\n1. On your favorite video site, find the video you want to add.\n\n2. Many video sites provide you with a code snippet to embed the video on another site. In YouTube and Vimeo, you can click the Share button. (At the time of this writing, you can find these buttons below the video or to the right of the video, respectively.)\n\n3. Copy the embed code provided by the video site.\n\n4. Return to your Waffle Video page in SharePoint, and make sure it's in Edit mode.\n\n5. Click the Insert tab on the ribbon.\n\n6. Click the Embed Code button.\n\nNOTE\n\nYou may have noticed earlier that the Video and Audio drop-down also contains an Embed option for adding video to the site. These two options open the same dialog.\n\n7. On the Embed dialog, paste your copied code into the input field. SharePoint gives you a preview of your embedded object.\n\nPhoto Credit: Tom Drisgill\n\n8. Click the Insert button to add the video to the page and click Save.\n\nYou can now play the video pulled in from an external site directly on your page with minimal effort required to set it up.\n\nAdding Web Parts and Apps\n\nWeb Parts and apps can be thought of as blocks of functionality that can be added to the page to accomplish a certain task. You just learned how to add images and video to your page using various Web Parts. The only difference between adding regular Web Parts and adding those for pictures or videos is that the Web Parts for pictures and videos get their own special shortcut buttons in the Insert tab on the ribbon. Web Parts and apps can also be views of the contents in a list or library, a rollup Web Part for aggregating content from across your site, and other collaborative components.\n\nNOTE\n\nMany places in SharePoint 2013 the term \"app\" is sometimes used generically to describe a Web Part for a list or library. There are also actual apps (similar to the type of app that can be purchased on a mobile device) that can be downloaded and installed to SharePoint 2013 to be used on their own or surfaced on a page with an App Part.\n\nWeb Parts and apps can be added just about anywhere on a SharePoint page:\n\n\u2022 In a text layout column (on collaboration sites)\n\n\u2022 In a page content control (on publishing sites)\n\n\u2022 In a Web Part zone (on publishing sites)\n\nText layouts, used by the wiki features, enable users to quickly insert text, images, and Web Parts\/apps. Publishing sites feature page layouts that can have control blocks for specific types of content. One of these content controls is called the Page Content control and is similar to a text layout because it enables users to insert text, images, and Web Parts\/apps. Finally, page layouts can also contain Web Part zones, which are sections of the page specifically for adding Web Parts and Apps.\n\nNOTE\n\nYou can also add some Web Parts to a custom page layout outside of the areas previously mentioned; however, the Web Part needs to be configured directly on the page layout in these instances because its properties won't be editable in the browser. Working with page layouts is covered in more detail in Chapter 5, \"Using the Design Manager to Start a Design In SharePoint\" and Chapter 7, \"Creating SharePoint Branding\"; this chapter is all about what you can do in the user interface of SharePoint 2013.\n\nIf you're working in a content field or text layout, position the cursor where you want the Web Part to be added. On the ribbon, click the Insert tab \u21d2 Web Part. (You can also click App Part on the ribbon to insert existing lists or libraries, now called Apps in SharePoint 2013, to your page.)\n\nIf you want to add a Web Part to a Web Part Zone on the page, simply click the Add a Web Part link in the zone.\n\nWhen you start adding a Web Part, the first thing you see is a big Web Part menu appear at the top of your page. On the left of this menu is a Categories section, which divides up the Web Parts into groupings. Select a category and you can see all the different Web Parts that it contains. Then select the Web Part you want, click the Add button, and you're on your way. (This process is identical to SharePoint 2010, so if you've upgraded to SharePoint 2013 you'll feel at home here.)\n\nYou can click through the various categories to see the different types of Web Parts available. Depending on your needs, chances are there's a Web Part that can help you. For branding purposes, here are some of the most common Web Parts you're likely to encounter:\n\n\u2022 Content Editor Web Part\u2014Similar to a text box in Word. You can add a CEWP to a zone for an extra area of content that you can format and edit, or even add styles and scripts to. The CEWP can be found in the Media and Content category in the Web Part picker menu.\n\n\u2022 Content Query Web Part\u2014Aggregates content (such as news articles or items in a list) and displays them in predefined styles. You can also create custom styles to use for the CQWP, which is covered in Chapter 9, \"Creating Content Rollups with SharePoint WCM.\" The CQWP is found in the Content Rollup category, and is only available on sites with publishing enabled.\n\n\u2022 Content Search Web Part\u2014New to SharePoint 2013, this is similar to the CQWP but more powerful because it can show content from search results based on a query you specify. Like the CQWP, the Content Search Web Part is found in the Content Rollup category and is also only available on publishing sites. As of this writing, the CSWP is not available on SharePoint Online sites; however, Microsoft has indicated it may become available at some point. You can find more information on the CSWP in Chapter 9.\n\n\u2022 Script Editor Web Part\u2014You can insert snippets of HTML or scripts in this Web Part. This Web Part is found in the Media and Content category.\n\nOf course, this list barely scratches the surface of the available Web Parts in SharePoint 2013, but these are the most common ones you'll likely encounter.\n\nSaving Your Page\n\nEarlier you were introduced to the Save, Check In, and Publish buttons found on the ribbon. These buttons are your way of saving and sharing your formatted page with your team or the world. The Save and Check In buttons appear in the Format Text tab in the ribbon, as well as in the Page tab. The Publish button, unique to publishing sites, appears in its own publishing tab.\n\nWhen you put a publishing page into Edit mode, it is automatically checked out to you, but to share the page with others, it needs to be checked in and\/or published, depending on the type of site you work with. Collaboration sites don't automatically check out the page to you unless you specifically click the Check Out button. Checking out a page prevents someone else from editing the page at the same time as you. When you have a page checked out, you can see the latest changes you've made, but other users still see the page as it appeared before you checked it out.\n\nTo exit Edit mode and save your changes, you can click Save. In collaboration sites, simply saving your changes without having the page checked out enables others to see the changes you've made. If you have the page checked out, clicking Save enables you to save your latest changes while retaining your check out. Only you can see the latest changes on the page as long as you have it checked out. If you finish making changes you can click the Check In button. This saves your edits and allows others to see your new content and make their own edits.\n\nOn a site with publishing features enabled, checking in pages allows users with certain edit rights to see your latest changes, but everyone else will still see the previously published version of a page. If the page has never been published, users without edit rights won't be able to see the page at all. Not all users can see changes unless a page has been published, which gives you a buffer between making changes and sharing the final version with your team or the world. The Publish button is found within the Publish tab on the ribbon. As you might guess, it's only available on sites with publishing features enabled.\n\nNOTE\n\nIf you are using the Publishing Site with Workflow site template (or have enabled content approval requirements in the site's Pages library), your Publish tab won't have a Publish option. Instead, you'll see various buttons such as Submit, Schedule, Approve, Reject, and Start a Workflow. These enable you to control approval for a page to be published and to schedule when that publication will happen. For more information on publishing workflows, visit <http:\/\/office.microsoft.com\/en-us\/sharepoint-server-help\/work-with-a-publishing-approval-workflow-HA102891575.aspx>\n\nWorking with Navigation in SharePoint 2013\n\nOne of the most important parts of any website is how users can find their way around. A common way they do this is using the site's navigation, which either organizes the various areas of the site into categories or displays the site structure in an easy-to-follow format. Some navigation may be simple, whereas other sites may sport more complex navigation. Although users will spend plenty of time admiring the beautiful branding on your site, eventually they are going to want to dig in and find some content, which is what they'll be using the navigation for. You'll likely want to set up a navigation that makes sense for your site; therefore, you must understand how navigation in SharePoint 2013 works. If you already know this, and are just looking for information on how to style the navigation, you can find that topic detailed in the \"Styling Key Areas of SharePoint\" and \"Putting CSS to Work in SharePoint\" sections of Chapter 6.\n\nTypes of Navigation\n\nAmong the other improvements and changes in SharePoint 2013 is navigation. Now, there are two distinct types of navigation in SharePoint:\n\n\u2022 Structured Navigation\n\nStructural navigation is the same navigation structure that SharePoint 2007 and SharePoint 2010 used and is sometimes referred to as classic navigation. This navigation structure can be automatically or manually built out by accessing the Navigation page from the Site Settings screen. Basically, you can have SharePoint automatically build out the site's navigation to match the site hierarchy, or you can manually add headers and links to replace or supplement the automatically built navigation.\n\n\u2022 Managed Navigation\n\nManaged navigation is a new concept in SharePoint 2013 that enables you to build navigation based on managed metadata. Your site's navigation can be derived from a managed term set in the metadata term store. There are a few major benefits to using managed navigation. You can structure the navigation to give your SharePoint pages and sites \"friendly URLs,\" which are easier to remember than the more traditional URLs SharePoint generally creates. For example, where you would normally see the URL http:\/\/www.randyswaffles.com\/marketing\/pages\/About-Us.aspx, you could use managed navigation to create a friendly URL of http:\/\/www.randyswaffles.com\/AboutUs. Another benefit of managed navigation is that friendly URLs are easily read and ranked by search engines, so if you run a public-facing site in SharePoint 2013, using managed navigation may make more sense than traditional navigation. When setting up managed navigation, you also have several options for further refining the search engine optimization. Managed navigation is also used by the new product catalog feature in SharePoint 2013.\n\nIt's worth noting that you can use both types of navigation in your site at the same time. You can set up the global navigation (also known as top navigation or main navigation) to use managed metadata navigation and have the current navigation (also known as left navigation or contextual navigation) be built using the structural navigation. You might consider this type of approach if you wanted to maintain a consistent navigation across the top of your site with the benefits of managed metadata navigation but with the ease of updating and adding additional links to the current navigation using the structural navigation.\n\nModifying the Navigation\n\nSharePoint 2013 gives you quite a bit of flexibility to modify the navigation. You can sit back and let SharePoint handle building out the navigation for you, or you can take a more hands-on approach and build the whole thing out yourself. You can even take a hybrid approach in which you let SharePoint handle most of the heavy lifting while you add a few links here and there to suit your needs. An example of this approach in an intranet scenario would be allowing SharePoint to automatically list all of the subsites in a site collection in the navigation as they are built, while you manually add a navigation header with links to other non-SharePoint internal systems within your organization.\n\nBecause there are two styles of navigation this time, you need to decide which method you want to use and how you want the navigation to be structured. Using the classic structural navigation takes a more straightforward approach to the site's navigation, whereas the managed navigation offers some additional flexibility but requires more work up front to configure, mainly due to the many options that are available, such as the ability to specify friendly URLs.\n\nOne of the first things you'll notice on your site is that in the navigation areas, you get a distinct Edit Links option. This enables you to edit the navigation inline by adding, removing, or editing links on-the-fly. Editing this navigation ties directly into the method of navigation you use.\n\nExample: Editing Navigation\n\n1. On the homepage of your site, click Edit Links in the top navigation area. The Edit Links link is replaced with a \\+ link option, as well as Save and Cancel buttons and an area for dragging and dropping links.\n\n2. Click \\+ link, which opens an Add a link dialog.\n\n3. In the Text to display field, type WaffleNet Home. In the Address field, type \/ and click OK. This creates a link back to the home of the current site. The link is added to your navigation.\n\nNOTE\n\nThe address field accepts absolute hyperlinks, such as `http:\/\/wafflenet\/`, as well as relative hyperlinks, such as `\/pages\/default.aspx`.\n\n4. Click the WaffleNet Home link and drag it to the first position in the row of navigation items, just to the right of the SharePoint site logo.\n\n5. When you finish working with the navigation, click the Save button to the right of your navigation (depending on your browser window's size you may need to scroll to the right to see the button).\n\nIf you are working on a collaboration site without publishing features enabled, there is another way you can add and manage the links in your site's navigation; however, your options for creating a robust navigation are somewhat limited compared to sites with the publishing features enabled.\n\nWorking with Navigation Items\n\nDepending on the type of site you're working with and that site's navigation settings, you'll see one of two icons next to each navigation item. In a collaboration site, you'll see an X to the right of each navigation item. Clicking this X removes the item from the navigation. If you're working on a publishing site, you'll see an icon that looks like a stylized eye to the right of each navigation item. Clicking this icon hides the item from appearing in the navigation (but doesn't delete it). In a publishing site, you might also see navigation items with an X next to them, which lets you delete the item. The difference is that on sites with publishing enabled, SharePoint can automatically build out a navigation structure as you add sites and\/or pages. On these automatically added links, you have the ability to show or hide them without completely deleting the item from the navigation. If you have set up your navigation to have drop-down items, you can click the drop-down and hide or show links in the drop-down by clicking the icon next to each navigation item. (How to build out drop-down items is covered in the next section.)\n\nExercise: Editing Navigation in a Collaboration Site without Publishing Features\n\n1. From the site's homepage, click the Settings menu \u21d2 Site settings.\n\n2. Under the Look and Feel header, you'll see two links related to the navigation: Quick launch, which lets you manage the navigation elements along the left of the page, and Top link bar, which lets you manage the links at the top of the page. Click Top link bar.\n\n3. On this page, you can edit the properties of the existing links by clicking the Edit icon next to each item. You can also add a new link by clicking on New Navigation link and reorder the links by clicking Change Order. These options are pretty self-explanatory.\n\n4. Return to the Site Settings page and click Quick Launch.\n\n5. Managing the navigation for the quick launch is nearly identical to the Top link bar, except you have the ability to add heading links to group your navigation items. Click the New Heading link to do so. Headings need a URL and a description. When you add a new item to your navigation, you have the option to select the heading it will appear under.\n\nAs you can see, it's pretty easy to set up a basic navigation structure on a collaboration site. If your navigation needs are more robust than what is available in a collaboration site, you'll need to consider using a publishing site template or enable the SharePoint Publishing Infrastructure feature in your site.\n\nUsing the inline editing feature on a publishing site lets you set up a basic navigation quickly, but you need to put in a little more effort if you want your navigation to have drop-down menus or non-linked headers. This example is designed to give a quick overview of the navigation options available on the Navigation Settings page.\n\nExample: Adding Drop-Down Menus to Navigation\n\n1. Click the Settings menu \u21d2 Site settings.\n\n2. Under the Look and Feel header, click Navigation. (If you don't see a link for Navigation, but instead see links for Quick Launch and Top Link Bar, you aren't on a site with publishing enabled.)\n\n3. Begin by selecting which type of navigation you want to use for the global navigation: structural or managed. If you are on a subsite, you can also select to use the same navigation as the parent site.\n\n4. Next, select the type of navigation you want to use for the current navigation: structural or managed.\n\n\u2022 There are two types of structural navigation to choose from here: You can have the navigation show the current site, the contents of the current site, and the site's siblings, or you can show only the contents of the current site. If you choose structural navigation for either the global or current navigation areas, you have the additional options of showing only subsites, only pages, or both. In addition, you can specify how many dynamic items the navigation should be limited to showing.\n\nNOTE\n\nDepending on your master page selection, the current navigation may be positioned toward the top of your page instead of its usual position on the left. Oddly, Oslo.master, one of the master pages included out-of-the-box, moves the current navigation into the position of the top navigation area. This is something to keep an eye out for as you build out your navigation.\n\n\u2022 If you use managed navigation for either the global navigation or current navigation, a section called Managed Navigation: Term Set appears. This is where you choose an existing term set that has been created on the site or in Central Administration, or you can create a new term set in the term store for your current site collection. Take a look at the next section, \"Creating a Term Set,\" for details on how to create a new navigation term set.\n\nWARNING\n\nBy default, SharePoint 2013 should automatically create a term set named Site Navigation for use with managed navigation. There are instances, however, where SharePoint fails to automatically create this term set. In these cases, SharePoint may also prevent you from creating new term sets, responding with an error that says: \"Failed to create term set. A default managed metadata service connection hasn't been specified.\"\n\nWhile it may seem like the service application wasn't set up at all, there is actually a simple change that can be applied from Central Administration to fix this issue.\n\nTo learn more about this fix, see the following blog post on the subject by Todd Klindt: `http:\/\/www.toddklindt.com\/SP2013ManagedNav`.\n\n5. In the Managed Navigation: Default Page Settings section, select whether you want to add newly created pages to the navigation automatically (which is why the pages you have created thus far in the chapter have been appearing automatically in the navigation). Also choose whether the navigation should create friendly URLs automatically. If you select this option, SharePoint creates pages in the default Pages library, however, instead of a URL like <http:\/\/wafflenet\/pages\/WaffleOfTheMonth.aspx>, it will be <http:\/\/wafflenet\/WaffleOfTheMonth>. This is known as a friendly URL (or FURL).\n\n6. If you use structural navigation, you see the Structural Navigation section near the bottom of the Navigation settings page. Set options for whether you want to manually sort the navigation items or have SharePoint automatically sort them for you, choose how the items should be automatically sorted, and edit the existing navigation structure.\n\n7. To edit the structural navigation, use the Editing and Sorting section. You can move links around in the global navigation or current navigation by clicking the Move Up and Move Down buttons on the tool bar. You can hide and show links that SharePoint has automatically created in your navigation for you by clicking the Show or Hide buttons. You can manually add, delete, and edit headings and links by clicking the Add Heading and Add Link buttons. Headings are top-level items in your navigation that can be used to group links under.\n\n8. To add a heading, click the Add Heading button on the tool bar. In the dialog window, give the heading a title, optionally give it a URL, select whether the link should open in a new window, and optionally provide a description, which is used as alternative text when a user hovers the mouse over the navigation item. You can also use audience targeting on a link, which shows or hides the heading and anything under it based on the audience that a user is a part of. Click OK to create your header.\n\n9. Click the Add Link button to create links to content in your SharePoint site or to external websites. Adding a link is exactly the same process as adding a header. The main difference is that a link can be moved under a header for grouping.\n\n10. Finally, at the bottom of the Navigation settings page, choose whether the Show ribbon and Hide ribbon commands are available on the Settings menu.\n\n11. When you have made your changes to the navigation, click OK to commit your changes. This step is important. None of the changes will be applied until you click OK, so make sure you don't navigate away from this page while building your navigation, otherwise you will lose your changes.\n\nAlthough the Navigation setting screen is straightforward, you need to spend some time thinking about how your navigation will be set up on your site. Think about how your audience will use your site and whether the navigation should actually mimic the physical hierarchy of your site, be completely separate from the physical hierarchy of the site, or follow a kind of hybrid approach where you add or remove some items to the automatically created navigation. In reality, any of these scenarios are perfectly valid, though one will likely make the most sense based on knowledge of your requirements, your site, and your site's audience. Chapter 4, \"Planning for Branding,\" discusses many other topics to consider when planning out your SharePoint site.\n\nIf you decide that you want to use managed navigation for either the global or current navigation, there are some additional configurations that need to be made for each navigation item. Read on to find out how to set up and configure a navigation term set.\n\nCreating a Navigation Term Set\n\nThere are a couple ways you can open up your term store to edit the navigation. On the Navigation settings page, you can click the Term Store Management Tool link under the Managed Navigation Term Set section. Or from the Site Settings page, you can click the Term Store Management link under the Site Administration header. Alternatively, if you are in Central Administration, you can open your Managed Metadata service application and create your term set there.\n\nWhen you're in the Term Store Management Tool, use the tree on the left to navigate the various term sets. On the right side of the Term Store page is a tabbed interface for modifying the settings of the item you have selected.\n\nNOTE\n\nTo create term sets in the term store, you may need to set up as a term store administrator in the Managed Metadata service application. For more information on this, you can visit <http:\/\/office.microsoft.com\/en-us\/office365-sharepoint-online-enterprise-help\/assign-roles-and-permissions-to-manage-term-sets-HA102771983.aspx>.\n\nAlso, be sure to check the warning in the previous section for information on a problem that could affect your ability to create new term sets.\n\nKeep in mind that this is a chapter covering the basics of the SharePoint 2013 UI in a branding book and not a deep-dive into setting up an advanced, complex metadata navigation structure, so every item and option isn't covered in detail.\n\nAs you learned earlier, when configured properly, SharePoint automatically creates a term store named Site Navigation, which is also the default setting for managed navigation. SharePoint automatically adds new terms to this Site Navigation term store for every publishing page that is created. If you want to create your own pre-configured managed navigation hierarchy, you can follow this example, which walks you through the basics of creating a new navigation term store.\n\nExample: Creating a Term Set\n\n1. Click the Settings menu \u21d2 Site settings, and then click the Term store management link under the Site Administration header.\n\n2. Click the drop-down on the right side of the Site Collection - <your site collection> node to open the menu, and select New Term Set. A new term set is created and is waiting for you to enter a name. Type WaffleNet Nav for the name of your term set, and then press Enter.\n\n3. With the WaffleNet Nav Term Set selected, click the Intended Use tab at the top of the right pane.\n\n4. Check the box next to Use This Term Set for Site Navigation, and click Save. This (as you might suspect) enables you to use the term set you are about to create as your site's navigation. Optionally, you can uncheck the Available for Tagging check box if you don't want your navigation items to also be available as terms for tagging content. For this example, you can just leave the item checked.\n\n5. Click the drop-down on the right side of the WaffleNet Nav term set, and select Create Term. If you don't see your term set, just click the arrow to the left of the Site Collection - <your site collection> node to expand it and show the tem sets on the site collection, including the WaffleNet Nav term set.\n\n6. Now you'll begin building out the top level of your navigation. Type WaffleNet Home and press Enter. After each navigation item, press Enter to automatically create a new term, which is used as an item in the navigation. After each term is created, you can start typing the next term. Create the following terms in the term set:\n\nWaffle Departments\n\nWaffle News\n\nWaffle Projects\n\nWhen you have entered the last term, press Enter twice to stop entering terms.\n\n7. Terms can have child terms nested below them, which appear as drop-down items in your navigation. Click the drop-down arrow to the right of the Waffle Departments term and select Create Term. Then build out the following subterms exactly as you did the top-level terms:\n\nWaffle Baking\n\nFood Tasting\n\nSyrup Engineering\n\n8. Repeat this same process to build out the following terms under the Waffle Projects term:\n\nBacon Flavored Syrup\n\nGiant Waffle Oven\n\nWaffle Food Truck\n\n9. When you finish building out your navigation, click the WaffleNet Nav Term Set name.\n\n10. In the right pane, click the Custom Sort tab. This is where you can either have the navigation items sort alphabetically or choose a custom order for your navigation items. Click the Use custom sort order option. Change the sort order drop-down of the Waffle Projects term to number 2, then click Save.\n\n11. Click the Term-Driven Pages tab in the right pane. If your site is going to feature dynamically driven content on templates or if you are designing a navigation for a catalog of items, you can set the target page for term set items or catalog item page items. For this example, you won't be utilizing dynamic page content, so leave these items unchecked.\n\n12. Click the Waffle Departments term and then click the Custom Sort tab. Notice that you can change the sort order of all the child tags underneath a tag.\n\n13. With the Waffle Departments term still selected, click the Navigation tab at the top of the right pane and in the Navigation Node section, click the check box next to Customize, then in the field change the navigation node title to Waffle Depts. This enables individual navigation items to display differently than when you set it up in the term set. In this case, the term \"Waffle Departments\" will actually display as \"Waffle Depts\" in the navigation.\n\n14. Next click the Term-Driven Page tab and in the Configure Friendly URL settings for child terms section, click the check box next to Customize and change the field from waffle-departments to depts.\n\n15. Click Save to save the changes to your navigation.\n\nBecause the Waffle Departments term was set to display as a link header, step 14's setting affects the child items under the navigation. Items under the Waffle Depts header in the navigation appear in the navigation as \/depts\/<nav item> due to changing the friendly URL setting here. For instance, your navigation URL might display as http:\/\/wafflenet\/depts\/bacon-flavored-syrup. If your navigation item was set on the previous tab to be a term-driven page with a friendly URL instead of a header, you could configure the page's friendly URL here. As another example, your page could be physically located at www.randyswaffles.com\/pages\/waffle-recipes.aspx, but you can configure the URL to appear as www.randyswaffles.com\/recipes, which is much easier to remember and type.\n\nThe Term-Driven Pages tab also lets you specify a target page for a navigation item if necessary. In other words, you can specify which page each navigation item is supposed to go to on this tab.\n\nExercise: Adding Custom Navigation\n\n1. Click the Settings menu \u21d2 Site settings.\n\n2. Under the Look and Feel header, click Navigation.\n\n3. In the Global Navigation section, make sure the navigation type selected is Managed Navigation. The navigation items will be represented using a Managed Metadata Term Set.\n\n4. In the Managed Navigation: Term Set section, click the WaffleNet Nav term set. Scroll to the bottom of the Navigation Settings page and click OK. Your custom navigation now shows up at the top of the site.\n\nThe example you just worked through was meant to serve as an introduction to setting up a managed navigation term set. As you probably noticed during the exercise, there are many other settings that can be changed and utilized to create a far more complex navigation scheme than what you just set up. None of these navigation items actually takes you anywhere other than a page not found error though. This is because although you created the navigation itself, it's not directly tied to any existing pages or sites yet.\n\nOne important thing to note is that although it seems that creating a central, out-of-the-box cross-site collection navigation is possible with managed navigation (something that has long been on many SharePoint site collection admins' wish lists), sadly this isn't the case. A managed navigation term set is \"owned\" by the site collection it's being used by, and trying to use it on another site collection results in the ownership of the term set being transferred to the other site collection. This breaks the navigation for the original site collection until the term set is associated with the original site collection again or a new navigation scheme is used.\n\nEach type of navigation has its benefits and drawbacks, and although managed navigation took more effort to set up, you can see there is a lot of potential benefit and customization that can be achieved if you want to take the time to work through setting up all the various properties for each navigation item. To really get familiar with managed navigation and all it has to offer, spend some time building corresponding pages and subsites within your site, and use the skills you have learned so far, associating them with your navigation terms. Experiment with the various options available in the managed navigation settings.\n\nUsing Composed Looks\n\nAs you learned in Chapter 2, the idea of applying themes to a SharePoint site isn't exactly a new concept. It actually dates all the way back to SharePoint 2003. SharePoint 2007 offered theming capabilities to change the look of your site through the use of images and CSS. This approach offered more flexibility but it required access to the file system of the SharePoint server and changes to system files, which many admins had reservations about. SharePoint 2010 changed course for how themes behaved and how they were applied to a site, but creating and applying themes in the browser was considered too simplistic by many users. Now, in SharePoint 2013, the idea of changing the look of the site through a theme-type engine is quite different from the themes of yesteryear. This version introduces a concept called composed looks, which takes theming to a new, modern level. The approach here is a happy medium between the 2007 and 2010 approaches to themes.\n\nChapter 2 compared the use of themes to decorating a house. Another way to think of composed looks is in their similarity to how WordPress themes work. If you run a blog in WordPress, you may have used custom WordPress themes to change the look of your site to set it apart from others on the web. Even if you aren't a blogger using WordPress, you likely have come across WordPress blogs online. Composed looks are similar; they can take a default SharePoint 2013 site and give it a customized look and feel with minimal effort.\n\nAs noted in Chapter 2, composed looks consist of a background image, a master page selection, a color palette, and a font set. When you choose a composed look, SharePoint applies the styles and master page associated with the composed look, and with minimal effort you have created a customized site.\n\nOut-of-the-box, you get 14 prebuilt composed looks to choose from in SharePoint 2013. At the time of this writing, public sites running in Office 365 have several more options than an on-premise SharePoint installation has. You can use the included composed looks as-is, customize them a little to put your own twist on your site, or create your own custom composed look. You learn more about creating a completely custom composed look in Chapter 10, \"Composed Looks and Custom Branding.\" We could talk all day (okay, maybe not all day, but at least for a few hours) about composed looks, but they'll make more sense if you actually try it out for yourself.\n\nChoosing a Composed Look\n\nLike many things in SharePoint, there is more than one way to get to the Composed Look settings page. If you use a SharePoint Team site template, the Getting Started app includes a helpful tile called What's Your Style? that can take you to the Composed Look screen. If you're on a publishing site, in the set of links for Visual Designers, the link Design Your Site takes you to the Design Manager where you can click the Pick a Pre-Installed Look link. Alternatively, you can click the Settings menu \u21d2 Site settings \u21d2 Change the look. No matter how you get there, the Change the Look screen is where the real fun begins.\n\nExample: Setting up a Composed Look\n\n1. Browse through the available composed looks.\n\n2. Select the Immerse composed look to see a sample preview of the look.\n\n3. Click the Try It Out link to see a preview of your site with your site's actual content. This look might be too dark for what you're after, so you can change it up.\n\n4. Click the No, Not Quite There link to go back to the sample preview screen.\n\n5. Click the Start Over link on the left to return to the selection screen.\n\n6. Select the Sketch composed look.\n\n7. Click the Try It Out link.\n\n8. Click the Yes, Keep It link to apply the composed look to the site.\n\nChanging a Composed Look\n\nThe composed look you've chosen isn't bad, but it's not close to your company's colors or overall corporate branding. You'll likely want to modify this composed look to get the site to look more personalized to the company.\n\nExample: Modifying a Composed Look\n\n1. Return to the composed look screen. The currently applied Composed Look appears first in the grid of available composed looks. Click the Current composed look (or you could select Sketch again if you want) to open the quick preview of the composed look.\n\n2. On the left, click the Change link underneath the thumbnail of the background image; then browse your computer for an image to use as a background image. If you're following along with the files for this chapter, use WafflesIntranetBG.jpg. The preview sample updates with your new background.\n\nNOTE\n\nYou can also drag-and-drop an image file from Windows Explorer to the thumbnail area of the composed look to swap out the background image faster. It's also important to note that if you want to use a .png or .gif file for the background, the file size needs to be 150 kilobytes or less.\n\nPhoto Credit: Robert S. Donovan\n\n3. The color palette needs to be changed, too. In the Colors drop-down, hover over the colors for a brief description of the palette. Select the color palette that is Primarily White with Grey-80% and Black. It's the last option in the Colors list.\n\n4. Leave the site layout as Oslo (the master page setting) and leave the Font drop-down set to Rockwell Light\/Segoe UI.\n\nNOTE\n\nThe fonts that are set here apply to the headers and default body font for the site. The first listed font in the pair is applied to headers in the site, whereas the second font is used as the body font.\n\n5. Click the Try It Out link for a preview of the actual site.\n\n6. Click Yes, Keep It to apply the revised composed look to the site.\n\nPhoto Credit: Robert S. Donovan\n\nIf you want to give the site an even more complete, customized look, upload the company logo to replace the out-of-the-box site icon.\n\nExample: Replacing the Default Logo on a Composed Look\n\n1. From the site homepage, click the Settings menu \u21d2 Site Settings.\n\n2. Under the Look and Feel header, click Title, description, and logo.\n\n3. In the Logo and Description section, you can choose to upload a logo file from your computer or browse for one already in your SharePoint site. For now, click From Computer.\n\n4. Browse your computer for a logo file and click OK. If you're using the file downloads for this chapter, use the WafflesIntranetLogo.png image file.\n\n5. The logo is placed in the Site Assets folder by default. If you don't want to use this location upload the logo to a different location first. If you decide to change the logo later, click in the Insert Logo field, delete the path, and start again.\n\n6. Type a description of the logo that will be used as alternative text for the logo and click OK.\n\n7. Pat yourself on the back. Your site is now branded with a custom background, color palette, and now a custom logo!\n\nPhoto Credit: Robert S. Donovan\n\nSharePoint Master Pages\n\nAs discussed in the previous chapter, SharePoint 2013 uses master pages to control the overall look and placement of the major elements on the page. You can think of master pages as the overall template of a SharePoint site's look and feel. Elements that are common throughout the site will be found on a master page, such as the site heading, search, navigation, and maybe a footer.\n\nSharePoint master pages utilize content placeholders for most of these elements. For example, instead of coding a search box onto the master page, you would insert a content placeholder control, and SharePoint injects the necessary code into the placeholder to render the search box when each page is loaded. Master pages also include a placeholder for rendering the content on a page layout.\n\nAll versions of SharePoint 2013 utilize master pages, but the ability to change a master page directly (without using a composed look) within the browser is only available on SharePoint sites with the publishing features enabled. It is possible to change the master page on a non-publishing site using SharePoint Designer 2013 or PowerShell.\n\nSharePoint 2013 comes with a couple master pages to get you started:\n\n\u2022 Seattle.master\n\n\u2022 Oslo.master\n\nAs of this writing, SharePoint Online includes a couple extra master pages not available in stand-alone installations of SharePoint 2013, although this could change at any point:\n\n\u2022 Berlin.master\n\n\u2022 Lyon.master\n\n\u2022 Tokyo.master\n\nThere are a few ways to set the master page on your site. If you work on a publishing site (or a site with the publishing features enabled), you can set the master page from the Site Settings page.\n\nExample: Setting the Master Page from Site Settings\n\n1. Click the Settings menu \u21d2 Site Settings.\n\n2. Under the Look and Feel header, click Master Page. (This link is not available on sites without publishing enabled.)\n\n3. On the Master Page settings screen, choose the Site master page (the master page that will be used on all content pages). If you have set up device channels for your site to utilize alternative master pages, you can choose which master page is applied to each device channel here as well.\n\n4. In the System master page section, choose the master page you want to use on non-content pages (generally pages served from the `_layouts` directory such as Site Settings pages).\n\n5. Two other options are available toward the bottom of the Site Settings page: Theme and Alternate CSS URL. These options are collapsed by default, but to expand their respective sections click the arrow to the left of the section titles. Theme enables you to control whether your current site inherits the parent site's composed look and enables you to apply the composed look of the current site to all child sites. Alternate CSS URL enables you to apply a custom CSS file to override the out-of-the-box SharePoint styles.\n\n6. Click OK to set your master page.\n\nIf you work on a site without publishing enabled, you can still set a different master page, but it can be done only through SharePoint Designer or a custom SharePoint feature. You'll learn more about creating branding features in Chapter 7.\n\nExample: Setting the Master Page from SharePoint Designer 2013\n\n1. Open SharePoint Designer 2013 (SPD).\n\n2. If you haven't opened your site in SPD before, click the Open Site button. If you have, look for your site in the Recent Sites list, and click it.\n\n3. If you clicked the Open Site button, type the full URL of your site, and click Open.\n\n4. If prompted, log in with your credentials.\n\n5. When the site opens, click Master Pages in the navigation pane in SPD.\n\n6. In the main window, right-click the master page you want to set, and select Set as Default Master Page.\n\n7. Repeat step 6, but select Set as Custom Master Page.\n\n8. Return to your browser and refresh the page to see your new master page applied to the site.\n\nWorking with Page Layouts\n\nIn a publishing site, page layouts are essentially the templates that contain the page content. Although the master page controls the overall positioning of the major site elements, a page layout controls the overall look of each page that is built using its template. Custom page layouts can be created and applied to your site, which you'll learn more about in Chapter 7.\n\nA page layout can consist of various Web Part zones and field controls, which define the areas of the page that can be edited. The availability of the various field controls are defined by the content type the page layout is based on. Content types are a collection of site columns used to store information about an item in SharePoint that describes what that item is.\n\nIn SharePoint 2013, there are two main out-of-the-box page layout types to choose from: article layouts and welcome layouts. The functions of these layouts are pretty similar, but the fields they each contain vary because page layouts based on the Article Page content type contain one set of fields, while page layouts based on the Welcome Page content type contain another set. There are some common field controls between the two content types, such as the Page Content field control you learned about earlier in the chapter. You'll learn more about content types in Chapter 7.\n\nIn Chapter 2, page layouts were compared to a room in your house, where the content fields in the page layout were represented by the furniture in the room. Rearranging the furniture makes the room look different, but it ultimately contains the same furniture. Similarly, changing a page layout rearranges the content on the page, which makes it look different. In some instances (such as changing from an article page layout to a welcome page layout) you might see additional fields that you didn't see before. This is like rearranging the furniture in your room and replacing an ottoman with a coffee table.\n\nPage layouts also provide the strictest control over page editing. As discussed earlier in the \"Saving Your Page\" section, one of the benefits of using page layouts is that you have a buffer between your edits and the audience for your page. When editing a page created using a page layout, you can make changes and check the page in, providing only a small set of users with edit rights to see the page. When you are ready to share your page, you can publish it. Although page layouts give site owners strict control over the look of pages, they also provide designers with the most flexibility because they can create completely custom page layouts for a site using additional fields, custom content types, additional HTML, CSS, and images. For instance, a site could have a completely customized landing page layout that sets it apart from the rest of the site.\n\nOn publishing sites, you can change the page layout of your page with just a couple clicks. The availability of page layouts depends on what settings you have selected for your site. You can choose to make all page layouts available or restrict the available page layouts to a select few for consistency throughout your site.\n\nExample: Setting Available Page Layouts\n\n1. Click the Settings menu \u21d2 Site Settings.\n\n2. On the Site Settings page, under the Look and Feel header, click Page layouts and site templates.\n\n3. Scroll down to the Page Layouts section of the page. Notice that you have three options for controlling which page layouts are available: You can inherit the available page layout settings from the parent site; you can make all page layouts available; or you can choose which specific page layouts you want to make available to use on the site.\n\n4. Select the option Pages in this site can only use the following layouts, then select the following layouts to be available on your site (you can select multiple items at once by holding down Ctrl and clicking the items in the list). When you've selected all the items, click the Add > button.\n\n\u2022 (Article Page) Body only\n\n\u2022 (Article Page) Image on left\n\n\u2022 (Article Page) Image on Right\n\n\u2022 (Article Page) Summary Links\n\n\u2022 (Welcome Page) Blank Web Part page\n\n\u2022 (Welcome Page) Splash\n\n\u2022 (Welcome Page) Summary Links\n\n5. Scroll down to the New Page Default Settings section. This section allows you to change the default page layout for the site, which is the page layout that is used when you create a new page from the Settings menu. Select (Article Page) Image on Left.\n\n6. When you finish, click OK.\n\nExample: Using the Default Page Layout\n\n1. Navigate to the homepage of your site.\n\n2. Click the Settings menu \u21d2 Add a page.\n\n3. In the Add a page dialog, type Types of Waffles, then click Create. Your new page is created using the (Article Page) Image on Left layout you specified.\n\n4. On the new page, type Here are my favorite kinds of waffles. Select this text and click the Heading 2 style in the ribbon. Then place the cursor after the Types of Waffles text, and press Enter.\n\n5. Type the following on your page:\n\n\u2022 Original\n\n\u2022 Belgian\n\n\u2022 Blueberry\n\n\u2022 Buttermilk\n\n6. In the Article Date field, type today's date.\n\n7. In the Byline field, type your name.\n\n8. Add an image to the page in the Page Image field control, as you learned earlier in the chapter. Don't forget about image renditions! If you are working with the chapter downloads, you can use the WaffleBreakfast.jpg image. (Need a refresher on images? Revisit the \"Working with Images\" section earlier in the chapter.)\n\n9. Click the Save button in the ribbon.\n\nPhoto Credit: Robert S. Donovan (background); Tom Drisgill (foreground)\n\nOf course, it is also possible to change the layout of a page after it has been created, as you know from working with page layouts earlier in the chapter. Here's a refresher on how that's done. You'll also take a look at the various properties of the page you just created so you can see how page layouts reference columns to store information.\n\nExample: Changing an Existing Page Layout\n\n1. Put your page into Edit mode. (Remember, you can click the Edit icon on the ribbon or click the Settings menu \u21d2 Edit Page.)\n\n2. Click the Page tab on the ribbon.\n\n3. Click the Page Layout drop-down.\n\n4. Notice that only the page layouts that you selected earlier (the article page layouts and welcome page layouts) are available to choose from now. Select the Summary Links layout in the Welcome Page section.\n\n5. Notice that your text and image are still preserved in the field controls on the page, but the Article Date and Byline fields are gone. Click the Page tab in the ribbon and change the Page Layout to the Image on Right page layout. Your Article Date and Byline fields return, and they still contain the same information you populated them with.\n\n6. Save the page, then click the Settings menu \u21d2 Site contents \u21d2 Pages library.\n\n7. Click the... (ellipses) to the right of the Types-of-Waffles page, then in the pop-up menu that appears, click the... (ellipses) on that menu. Finally, click the View Properties option.\n\n8. Scroll through the various properties associated with this page. Because the page is currently set to use an article page layout, all the available columns are coming from the article page content type. Note that all the fields that appeared on your page, such as Article Date and Byline, display here. These fields are all stored in the Pages library as columns from the site content types added to the library. Page layouts reference these columns and can display them as field controls that users can fill out. Click the Close button to return to the Pages library.\n\n9. Click Types-of-Waffles to return to this page. Switch the page layout back to Image on Left.\n\n10. Click the Publish tab in the ribbon, then click the Publish button to publish your page. In the Publish dialog, click Continue to finish the publish process. (If you don't see a Publish button, you are probably working on a publishing site with workflow site template. In this case, just click the Save or Check In button on the Page tab in the ribbon.)\n\nThis exercise shows you how the content that you have on your page remains associated with your page, even if the corresponding field or Web Part zones do not appear on the new layout. This is because the information is stored in the columns of the Pages library, even if those fields aren't actively being displayed. Another thing to note, if your page contains Web Parts in Web Part zones, and the layout you change the page to doesn't have the same zones, your Web Parts will all be preserved on the page, but will likely all appear in the same zone, so you probably need to reorder them on your page.\n\nBefore continuing on to the next chapter, you should change the page layout options to use any available page layout again. That way you won't run into any issues completing the exercises in the rest of the book.\n\nExercise: Reverting Page Layout Options to Default\n\n1. Click the Settings menu \u21d2 Site settings.\n\n2. Under the Look and Feel header, click Page layouts and site templates.\n\n3. In the Page Layouts section, select the option Pages in this site can use any layout.\n\n4. Scroll down and click OK.\n\nSummary\n\n\u2022 The page-editing experience in SharePoint 2013 is similar to SharePoint 2010. The ribbon has a new look and some items such as the Settings menu (formerly known as the Site Actions menu) and some buttons have moved around, but the experience is largely the same as SharePoint 2010.\n\n\u2022 Adding images, videos, and other Web Parts can add to the functionality of your page. Image renditions should save SharePoint administrators (and end-users) some headaches by displaying physically smaller, resized, and\/or cropped versions of images on pages in the site.\n\n\u2022 The new embed functionality also makes embedding code and external videos on a page much easier than in past versions of SharePoint.\n\n\u2022 SharePoint 2013 introduces managed navigation to go along with the traditional structural navigation. This new navigation type introduces much more robust features, such as the ability to utilize friendly URLs, but also takes some extra effort to set up.\n\n\u2022 Composed looks are one of the most noticeable changes on the branding front in SharePoint 2013. This is the new way of applying a theme to your site and allows you to apply a custom look and feel with minimal effort.\n\n\u2022 SharePoint 2013 still makes use of master pages and page layouts, with many of the same concepts found in past versions. The two out-of-the-box master pages, Seattle.master and Oslo.master, are similar but each has some unique characteristics. Master pages can also be applied using composed looks.\n\n2\n\nPlanning a Design and Getting Started\n\n4. Planning for Branding\n\n5. Using the Design Manager to Start a Design in SharePoint\n\n6. Cascading Style Sheets and SharePoint\n4\n\nPlanning for Branding\n\nWhat's In This Chapter\n\n1. Analyzing Functional and Non-functional Requirements\n\n2. Organizing Data\n\n3. Estimating Project Constraints and Considerations\n\n4. Planning with Wireframes and Prototypes\n\nThis chapter discusses the process of properly planning for SharePoint branding. In many ways, this topic is similar to the planning that goes into any creative project, but some of the steps are unique for SharePoint. Even if you are experienced with planning typical design projects, this chapter provides an overview for what it takes to get your SharePoint branding project started.\n\nWhy Plan for Branding?\n\nPlanning is important for just about any type of project, regardless if it is technical. But all too often people and organizations fail to do proper planning. Most seem to be eager to start and decide to figure out specific details along the way.\n\nThe value of planning seems to be easier to understand when the context is personal. Imagine you are renovating your house and you want to convert a den into a bedroom, but you have a fixed budget and want to make sure you are adding the most value to your home for the money you are spending. Given those parameters it should seem more logical to talk with a designer or architect to get a sense of what you want to accomplish instead of just sending in a construction crew to start knocking down walls and building things in hopes that the final product is good.\n\nBy skipping the planning, you run the risk of putting a wall in the wrong place, so a door won't open properly, or maybe you end up with a project that instead of adding value becomes an eyesore in the rest of your house. Mistakes like that tend to be expensive to fix. In the digital world, many times people take this type of situation for granted because they usually deal in pixels or other digital assets that seem less permanent than the physical walls of a home. But just because you don't live in the websites you create doesn't mean that you shouldn't plan.\n\nWhether your project is a construction project or a new SharePoint site, the first part of the process should be to put together a plan. Making mistakes on paper (or whatever your medium of choice is) will always be less expensive to address compared to making those same mistakes during construction.\n\nGiven that upfront planning clearly makes a lot of sense for most SharePoint websites, how much planning is needed? Do you need to spend months and months of daylong meetings? Not necessarily. Depending on the size of the project, the intricacy of the SharePoint site, and the amount of decision makers involved (often known as stakeholders), planning for branding could take months, just a few days, or even hours. The next few sections discuss some of the key steps you should take when planning for branding. Although it may seem like a lot, keep in mind that smaller projects can get away with doing only a portion of them. For example, if your branding project requires only small changes to the out-of-the-box user interface, there is probably no need to go through the full process of making design comps. In many ways, the decision for how much planning is needed for a project will be unique for every project. The key to this process is to carefully consider your branding before actually executing it in SharePoint.\n\nPerforming Requirements Analysis\n\nWhether your SharePoint branding project is a site for 10 users or 10,000 users, before the project can be considered complete, certain requirements must be met. This is why the first step of almost any project, regardless if it is a SharePoint branding project, should include some amount of requirements analysis. Requirements analysis involves gathering and understanding the specific needs of a project.\n\nIf you look back on the construction example in the previous section, you can imagine that as homeowners who want some new changes to your house, there are some basic preferences about what you'd be looking for. For example, what is the intended use for the room? How much storage is required? What are the power and lighting needs? These are just a few of the questions that need to be asked before designing what the project will ultimately look like. In a sense, these questions help to define the functional requirements for the project. Functional requirements define what the project is supposed to accomplish but not necessarily how those things will be achieved.\n\nNOTE\n\nFunctional requirements are usually defined as something specific to the way the site will look or behave. Nonfunctional requirements refer to the more overarching requirements, such as cost and reliability.\n\nRequirements analysis involves gathering and understanding the specific needs of a project. Typically, this process includes asking a lot of questions and breaking larger problems into more manageable pieces to gain a better understanding of them. Requirements should not be vague or lofty ideas; instead, they should be both measurable and actionable. To put it another way, for a requirement to be useful, you should be able to tell when it has been accomplished successfully. If the requirement has no success criteria, it should be broken down into smaller, more discrete requirements that do.\n\nConsider the following two example requirements. The first is ambiguous and not measurable, which makes it difficult to know when the task is accomplished; whereas the second can easily be measured for successful completion:\n\n\u2022 Requirement #1\u2014 The design shouldn't have horizontal scroll bars for most users.\n\n\u2022 Requirement #2\u2014 The site will support a resolution of 1280 x 720 without horizontal scroll bars.\n\nIn general, gathering requirements isn't too difficult. The challenge is determining which of the requirements are the most important and which ones should take the highest priority. Because every project and organization is different, there isn't one sure-fire answer to this dilemma, but here are a few tips that can help you choose:\n\n\u2022 For each requirement, consider the overall value to the business and the cost to implement. You could even create a spreadsheet and assign scores to each requirement based on each attribute. For example, a requirement could have a high business value and be rated a 9 on that dimension but be costly to implement and therefore be rated a 3. The total \"score\" for this requirement would be 12. This approach is a more unbiased way of determining priority.\n\n\u2022 Be realistic about what is possible given timelines and the skillset of your team. It is always easier to add items when a project is ahead of schedule rather than trying to remove items that are too complex.\n\n\u2022 Using a phased implementation is a great way to break up a more complex project. Stakeholders tend to prefer seeing smaller pieces sooner as compared to waiting a long time to see the \"perfect\" solution. A phased implementation also enables minor course corrections along the way, which ultimately leads to a better solution.\n\nWhen gathering project requirements, you must involve all the project stakeholders. Stakeholders are people or groups that have a vested interest in the project's success. They may be affected by the project directly, such as an executive or manager, or indirectly, such as users who will depend on the final product.\n\nAdding stakeholders to the requirements gathering process can mean more time to complete the process and increased complexity. In fact, depending on the project, some stakeholders can actually be detrimental to successful requirements analysis. For example, the owner of a company may be too abstracted from day-to-day operations to give meaningful feedback, or certain employees, such as those recently hired, may not offer a valuable opinion regarding the project. Determining the right stakeholders can be a difficult task, but at the end of the day, if someone has enough influence to change the project in later stages, it's a good idea to include that person early. The reason is obvious: By involving the appropriate stakeholders early in the requirements analysis process, you can avoid making those dreaded, and costly, last-minute changes to a project to meet a critical need.\n\nThe following sections discuss the most important aspects of the requirements gathering that are necessary before starting any SharePoint branding project.\n\nSharePoint Version\n\nOne of the first decisions to make is which version of SharePoint to use for the project. The major decision for most is choosing between SharePoint Server 2013 and SharePoint Foundation 2013. Many organizations also choose to use SharePoint Online as part of Office 365 from Microsoft. From a branding standpoint, the biggest difference between these products is the availability to use a set of features called publishing. Both SharePoint Server 2013 and Office 365 provide the ability to use publishing.\n\nFor highly customized sites, publishing is preferred because it has several useful additions for branding projects:\n\n\u2022 Enabling publishing gives the ability to use Design Manager, which is a central hub to manage most aspects of a custom design. It provides the ability to import HTML and CSS, which can then be converted into a SharePoint master page. Additional tasks such as creating and editing page layouts, managing device channels, and importing\/exporting design packages can also be managed through the Design Manager. For more information on the Design Manager see Chapter 5, \"Using the Design Manager to Start a Design in SharePoint.\"\n\n\u2022 Publishing sites enable you to select a new master page by using the web user interface in Site Settings. Although you could use custom code to perform this task in SharePoint Foundation, it is not included out-of-the-box.\n\n\u2022 Publishing sites enable developers and designers to create page-level templates by using page layouts.\n\n\u2022 Advanced Web Content Management functionality is available with publishing. This includes Web Parts such as the Content Query Web Part (CQWP) and Content Search Web Part (CSWP), which can be used for delivering a large amount of content to highly customized sites.\n\nAlthough it is still possible to create a custom look and feel to your website if you don't have access to a version of SharePoint with publishing available, it does limit your options, and from a planning perspective, it makes the overall design process more challenging. Some tasks that are simple otherwise become more difficult without publishing. A good example of this is switching master pages\u2014in a site with publishing, there is a menu where you can just choose a master page. In a non-publishing site there is no menu so you must set the master page with either SharePoint Designer or PowerShell. Not impossible, but not preferable. There are a number of similar examples to this, along with the fact that some functionality just isn't available without Publishing. The key is just to understand what version you are working with and align your plan accordingly. You can make a beautiful and engaging site with any version of SharePoint!\n\nType of SharePoint Website\n\nTypically, SharePoint sites are set up as one of three types of websites: public-facing Internet sites, internal-facing intranet sites, and extranet sites, which have aspects of both Internet and intranet sites. The users of each of these types of websites differ vastly. Here is a quick breakdown of some of the differences between them:\n\n\u2022 Internet sites\u2014These sites are usually marketing-driven and typically have tightly controlled content with few content authors. They tend to be more stylistic than intranet sites and are targeting a much wider spectrum of users. Because Internet sites are public-facing, developers have no control over the type of browser, device, or the screen resolution that will be used to visit the site.\n\n\u2022 Intranet sites\u2014These sites typically sit behind a corporate firewall and are specifically tailored to help internal users work in a more collaborative and efficient manner. Intranet sites usually have many content authors and numerous employees who are consuming documents and collaborating. Because these sites are internal-facing, companies can control browser and screen-resolution requirements if wanted.\n\n\u2022 Extranet sites\u2014These sites are a hybrid of the previous two. They are typically intranet sites that have a separate area for external users to authenticate into. From a branding perspective, the extranet area can be similar to an Internet site or an intranet site, depending on the particular objective for the extranet. Sometimes, an extranet can be collaborative, and sometimes it can be simply information published to external users. For example, a company may work with external partners and want a place to securely collaborate without giving these partners access to the internal network. An extranet is the ideal place to do this.\n\nTargeted Browsers\n\nAnother important decision to make is which browsers will be targeted by your branding. Although many people may say that a website should support all browsers equally, it is often impractical to test each browser for pixel-perfect display. This is why it's a good idea to decide early what browsers and operating systems will be supported by your SharePoint site. As mentioned earlier, this decision is typically more important for public-facing Internet sites, for which you have no control over which browser is used to access the site. Conversely, corporations can dictate strict browser requirements for intranet sites and prohibit unsupported browsers from accessing the system.\n\nOne typical way of choosing a level of supported browser is to consult industry websites that track this information across extremely large usage statistics. One such site is W3Counter.com; here are its published browser statistics as of January 2013 (http:\/\/w3counter.com\/globalstats.php ):\n\nIn a perfect world scenario, determining which browsers to target would be as simple as checking the current statistics for what browsers visit your site. Or if you have an internal site, perhaps you only need to account for whatever browsers are supported by your IT department. But if you're looking for more general information, the preceding statistics for usage among Internet sites can be a helpful guideline when determining which browsers to target for your project, even if it is for an internal site. Especially considering that an increasing number of companies use SharePoint to collaborate with others outside of their organization.\n\nScreen Resolution\n\nYou also need to decide what screen resolution will be targeted for your SharePoint branding. Until a few years ago, computer monitors supported only a limited number of screen resolutions, mostly 800 x 600 or even 640 x 480. Nowadays, it's not uncommon to see website visitors browsing at resolutions of 1920 x 1200 or higher. Most web designers currently consider 1366 x 768 to be the most common screen resolution. According to the most recent survey on W3Counter.com (January 2013), more than 55 percent of users are running a resolution of 1280 x 800 or higher.\n\nAs you create your branded SharePoint website, you must balance the desire to display large amounts of information with common user screen resolutions. This can be even more challenging depending on how you choose to support mobile devices. For more information on mobile support in SharePoint see Chapter 8, \"Advanced SharePoint Branding Tasks\" and Chapter 11, \"Modern Web Design and SharePoint.\"\n\nFixed-Width vs. Full-Width\n\nOne of the most common questions about screen resolutions is, \"Should I use a fixed-width design or full-width design?\" The out-of-the-box SharePoint master pages are full width, which means that the design expands to take up 100 percent of the space available in the browser window. A fixed-width design is set to a specific page width. There's no right or wrong answer for which one is better. The general rule of thumb is that highly customized designs usually work better with fixed-width designs, and sites where collaboration is performed seem to work better with full-width designs. Perhaps the best option is to do multiple versions of your branding to support both fixed and full width, which allows for a consistent look and feel but provides flexibility to support all scenarios.\n\nInformation Architecture and Organization\n\nWhen you think of any of the iconic buildings of the world such as the Sydney Opera House, Taj Mahal, or the Pyramids, it is usually the physical appearance that is so memorable. However, those buildings also required proper engineering and architecture to achieve their respective greatness.\n\nA SharePoint site is similar in that it requires a good foundational architecture and organization system to make that memorable physical appearance possible. There are few examples of web design on any platform in which you talk only about pure design. Most SharePoint projects include functional requirements beyond simply looking good. The way that you achieve these requirements is to create a well-organized information architecture that gives you the ability to create a site that functions as good as it looks.\n\nInformation Architecture\n\nInformation architecture is a general term that describes the organization of data in a website. From a SharePoint perspective, information architecture refers to the structure of the various SharePoint objects in a farm, which could include web applications, content databases, service applications, site collections, lists and document libraries, metadata, permissions, and so on. Based on the requirements for a given effort, an information architecture is created that supports the overall business goals. For example, the information architecture needed for a public-facing Internet site to sell products is fundamentally different than one for an internal corporate site focused on collaboration.\n\nAs a rule of thumb, it is a good idea as you create your mockups and creative designs to look at the elements on a given page and ask, \"What is that and where does it come from?\" For example, if you have a creative design and show news items, you need to determine where that content was created? How was it created? Is it intended to be shown on the page via a web part? If so, which one and what customizations are needed to display the web part on the page? As you start thinking about your SharePoint design in this way, it becomes easier to see the relationship that information architecture plays in relation to SharePoint branding and design.\n\nNOTE\n\nInformation architecture is important, but don't forget that the amount of effort you put into a project should be proportional to the actual project. If you want to create a basic site, there's no need to overcomplicate things.\n\nTaxonomy\n\nTaxonomy is the science of classifying and organizing things and is a core component of a site's architecture. In science it is often used to organize, or classify, plants and animals in a hierarchy based on their relationships. You may remember a famous taxonomy from science class: kingdom, phylum, class, order, family, genus, and species.\n\nFrom a web perspective you can think of taxonomy in two different ways. Developers think about the structure of the data, and designers typically think of the visual aspect of the taxonomy and how users would navigate through a site. Both are correct, and for planning your site, you must consider both approaches.\n\nSharePoint can understand only one structure for how things are organized, but that structure might not be the most intuitive for visitors to your site. The good news is that you can present whatever view of the taxonomy you want to your users to help make the site as usable as possible.\n\nThere are a number of books and theories for creating taxonomies, and this text just scratches the surface. When creating a taxonomy, think about the website's audience. How will they use the site? Are they familiar with your internal organizational structure, or will they be expecting information to be organized differently? People with different roles inside of an organization may use information differently. Executives may look for strategic information, whereas workers on the floor may need specific information to do their job.\n\nThis structure can eventually manifest itself in the SharePoint site's navigation. Chapter 3's section, \"Working with Navigation in SharePoint 2013,\" looks at SharePoint navigation in detail. Providing multiple forms of navigation could be beneficial. Some users may feel comfortable using the primary navigation, whereas others may prefer to search. Your site may even contain information that is so important it should \"roll-up\" to a top-level page from where the content is located. In these cases, a SharePoint Web Part would be perfect.\n\nUsability\n\nArchitecture and taxonomy are both key parts in creating SharePoint sites, but by themselves the greatest information architecture, taxonomy, or creative design cannot guarantee a successful project. The ultimate measure of success for any project is user adoption and whether those users feel like they are benefiting from the site. The most technically complex effort or most innovative creative design won't be appreciated if no one wants to use it.\n\nThe best sites strike a balance between form and function. Again, every site is going to be different, but as long as you've been listening to your users' needs, you'll have a good understanding of the problems that need to be solved. After you've determined these problems, the next step in the design process is to design a site that enables those same users to get to what they need in an efficient way:\n\n\u2022 Keep your design clean and simple\u2014There is plenty of research that shows that people become less efficient when presented with too many options. A clean and simple design with clear options can make it easier for users to get where they are going. This is often one of the most challenging usability rules for an organization to follow.\n\n\u2022 The mythical three-click rule\u2014In web design you often hear people say that you need to access more than 80 percent of content in a website within three clicks. This is a good rule of thumb, but more than three clicks is okay as long as users still feel they know where they are and where to go next. This relates to the clean and simple design rule, too. For example, every piece of content doesn't need to be linked directly from the homepage as long as users know where to look.\n\n\u2022 Navigating your site shouldn't take special training\u2014This might seem like an obvious rule, but if you take users and sit them down in front of your design, would they know where to go to find common pieces of content? Whether you do something basic such as ask the opinion of co-workers, or perform formal usability testing, this is a good way to check if your new site is as easy to use as you think it is.\n\n\u2022 Users seek content in different ways\u2014There are four primary ways that users find content in a website. Every user has a different preference, but an effective usability strategy should consider these different approaches:\n\n\u2022 Using the site's navigation\n\n\u2022 Clicking on the content on the page\n\n\u2022 Searching for content\n\n\u2022 Using a site map or other direct link\n\nThe topic of usability is another deep topic. Large complex implementations will probably go through a more formal exercise of usability planning. But for most projects just being aware of some basic usability principles can provide massive benefits. One of the simple things you can do is when you design your site, imagine people that you know who aren't as computer savvy as the users. Consider how difficult it would be for them to find things in your design. All usability theory aside, that is usually a good place to start with a design of any size.\n\nAsking the Right SharePoint Branding Questions\n\nIt is often helpful to start the planning process with a set of simple standard questions. These questions are best answered by those who are familiar with business need driving a new implementation. This section provides some common questions that are a good starting point for most projects:\n\nNOTE\n\nIn practice, you may not proceed through your requirements gathering and analysis process in the order listed. The order is less important than making sure that you capture enough information to help you make the correct decisions during the design process.\n\n\u2022 Is there an existing website or SharePoint site that the new SharePoint site will replace? If so, will the content need to be migrated?\n\n\u2022 Will this be a new design or will it be based on some existing branding?\n\n\u2022 Is there an existing corporate style guide that needs to be followed?\n\n\u2022 What is the time frame to deliver the new branding?\n\n\u2022 Is there any existing branding or marketing material that can or should be used, such as corporate logos?\n\n\u2022 Are there any stylistic requirements, such as preferred or disliked colors, fonts, or imagery? (Example: \"Our competitor's website is red, so we'd prefer to avoid red for ours.\")\n\n\u2022 Who will the audience be? What will they be looking for in the website? Are they corporate users? Are they teenagers? Would a cutting-edge, modern design speak to them or would a tried-and-true corporate layout be better?\n\n\u2022 Will the site be primarily used for communication? Collaboration? Both?\n\n\u2022 Who will approve the new branding? Is it a single person or committee?\n\n\u2022 How will the branding files be deployed to the server? What is the plan for deploying updates and maintenance of the branding files?\n\n\u2022 Will master pages or composed looks be used to create the branding? Will both be required?\n\n\u2022 How will the pages of the site be created? Is there a person or team responsible for creating the content and entering data?\n\n\u2022 Are there different requirements for displaying the site on different devices such as tablets or mobile phones?\n\n\u2022 According to what criteria will the branding be judged a success? What goals does it need to address?\n\n\u2022 What sort of navigation is required? Horizontal navigation? Vertical navigation? Both? Are dynamic drop-downs or fly-outs wanted for submenu items?\n\n\u2022 Are there any specific types of content that require company-specific information that needs to be accounted for? (Example: Internal news item with metadata to identify which products it is related to).\n\n\u2022 Are there any specific requirements for Web Parts to roll up content? Will they require custom development or just styling?\n\n\u2022 Will any third-party Web Parts or controls be needed? Will any of the third-party Web Parts need to be migrated to the new site?\n\nRequirements analysis can be a formal process, involving scheduled meetings and full documentation, or a brief process, involving just a few key stakeholders and some quick decisions. Either way, the result should be a better understanding of what will actually be built\u2014in this case, how SharePoint will look and behave.\n\nProject Estimation\n\nIn many ways, project estimation for SharePoint is the same as any other IT project. Constraints such as due date, feature scope, and budget impact the estimation process greatly. These three common constraints are often interconnected; when one becomes more aggressive, it could affect the other two.\n\nFor instance, the budget could be too low to accommodate all the wanted requirements. This would indicate that the scope needs to decrease. Alternatively, the timeline for delivery could be aggressive, which could translate into more hours or workers assigned to the project.\n\nThe topic of project estimation, in theory and in practice, could fill an entire book. A comprehensive discussion is therefore impossible within one chapter. Your organization may follow a particular development approach such as Waterfall or Agile. But regardless of your approach the process should generally involve the following steps:\n\n1. Decide on a high-level project timeline. When will the project start and when do you estimate it will end?\n\n2. Break down the requirements into specific tasks.\n\n3. Estimate the amount of time each of those tasks will take.\n\n4. Determine which resources will work on each task. Will certain tasks require specialized skills? If so, be sure to account for the availability of someone with those skills.\n\n5. To the extent possible, determine any roadblocks or risks you might face, and attempt to troubleshoot them proactively.\n\n6. Allow enough time to include all phases of traditional development, including any necessary extra planning, building the branding and the SharePoint site, testing and review, making any necessary adjustments, and deploying the project to a production environment.\n\nSharePoint projects have specific factors that could impact project estimation. The following is a list of some things that should be considered:\n\n\u2022 Will the project involve creating custom master pages and page layouts, or will it just use a composed look?\n\n\u2022 What is the skill level of the designers and developers involved in the project? Do they have experience with SharePoint?\n\n\u2022 How will the project be deployed to a production environment, and how will updates be applied?\n\n\u2022 How complex is the design that is being attempted?\n\n\u2022 How many different areas of SharePoint will be branded?\n\n\u2022 How many Web Parts will need custom styling?\n\n\u2022 How flexible is the due date and budget?\n\n\u2022 How many different browsers will you be targeting?\n\n\u2022 Will mobile devices be considered?\n\n\u2022 Will any third-party Web Parts or apps need to be branded?\n\n\u2022 Will the effort require custom development work?\n\nEven after thinking through all of this, estimates can still be off. SharePoint projects tend to have a lot of moving parts, and it is almost impossible to anticipate every minor detail. Team members are often dependent on each other's specific skill sets. Rarely in a SharePoint project will every task be accomplished by the same person; the server administrator who installs SharePoint will most likely not be the same person who designs the branding. Whether this is your first SharePoint project or your fiftieth, be sure to assume extra time in your estimate to account for the unknown. It is usually advisable to plan for potential variance to both the effort level as well as the duration of the project. In the event that everything goes exactly as planned, then you can use any extra planned variance to do even more great things!\n\nCreating Wireframes\n\nThere are many different aspects of planning a branding project. So far you've seen the technical, architectural, and organizational sides of planning, but there is still a side remaining: creative planning. The creative planning process for larger websites often begins with the creation of several black-and-white wireframes. Wireframes are skeletal page designs; they capture the layout and flow of a website without focusing on colors and graphics.\n\nThe major reason for creating wireframes before full-color comps is to ensure that all the stakeholders and decision makers focus on the content, layout, and page flow, rather than get hung up on whether emerald green is better than mint green. This allows for quicker iteration between ideas because creating black-and-white wireframes is faster than creating full-color designs. In some cases, wireframes should be built for every page on a website, whereas in other cases you can get by with just making wireframes for the major sections of the site. The decision for whether wireframes should be made for every page will be different for every project, but it generally is based on how different all the pages will be and how granular you want to get with the planning process.\n\nBefore delving into the process of creating a wireframe, however, take a look at a sample wireframe for Randy's Waffles (the fictitious company used in several examples throughout this book).\n\nNotice that the focus of a wireframe is the placement of page content, user behavior, and functionality, not stylistic colors or fonts. By creating wireframes such as this, you enable the project's stakeholders to quickly visualize, and hopefully approve, the site's general layout. This gives designers a lot of latitude to create a branded SharePoint design using whatever styles they like because as long as it matches the approved wireframes, the design will meet the project's needs.\n\nWireframes can be created in a number of ways, from simple pen and paper diagrams to dedicated software programs such as Microsoft Visio, Adobe Illustrator, or Balsamiq Mockups. Several of these tools have prebuilt shapes or stencils that can be used to piece together typical web user interfaces.\n\nWhen creating a brand for SharePoint, a key consideration is what pieces of SharePoint functionality will be supported by the design. A typical SharePoint page is made up of several controls and other pieces of functionality. Some of these functional controls are required for SharePoint to be used, but others are purely optional, based on your own project requirements. This is particularly true for public-facing Internet sites; several pieces of SharePoint functionality are not appropriate to show to anonymous public website viewers. For instance, by default, the help button in the top-right corner of the typical SharePoint user interface contains help for SharePoint itself, not the website that users are currently viewing.\n\nHere you can see the default SharePoint user interface (based on seattle.master), with each of the major areas of functionality labeled.\n\n1. The Ribbon The ribbon is the contextual menu at the top of the page used to interact with the current page or activity. Technically the blue bar at the top is actually the suite navigation, but the two are grouped together in this diagram because visually they form the ribbon area that \"sticks\" to the top of the browser window and does not scroll with the page content.\n\n2. Suite Navigation Also known in code as the suite bar, this provides consistent navigation across the suite of applications available in Office 365 as well as surfacing branding and links for traditional farm installs of SharePoint. On the left it includes Microsoft branding for either SharePoint or Office 365. On the right it includes several of the other links labeled here.\n\n3. Suite Links This includes links to My Sites functionality and, depending on how the server is configured and whether it is living in Office 365, there may be several links to other applications like Outlook and Calendar.\n\n4. Welcome Menu This control shows the current username and has a drop-down menu to go to the About Me page (which links to your My Site) and a Sign Out button. When there is no authenticated user, this control shows the Sign In link.\n\n5. Settings Menu or Gear Formerly known as the Site Actions button, this drop-down menu displays options for managing many aspects of the SharePoint site.\n\n6. Help This links to the SharePoint help pop-up window.\n\n7. Ribbon Contextual Tabs These tabs enable the user to switch between major sections of the ribbon. They are contextual based on what activity the user is currently performing on the site.\n\n8. Quick Access Toolbar This toolbar is a selection of quickly accessible menu items for performing various tasks on your SharePoint site. The items vary by the type of SharePoint site and what activity the user is currently performing on the site. For instance, you can quickly edit and save the pages from the quick access toolbar. This section also includes the Focus on the Content button, which shows a minimal view of just the main areas of the SharePoint site.\n\n9. Site Logo The site logo shows by default in SharePoint. The logo can be changed by clicking Site Settings \u2192 Look and Feel\u2192 Title, description, and logo.\n\n10. Top Navigation Also known as the top link bar or global navigation, this is the primary horizontal navigation for the SharePoint site.\n\n11. Page Title This shows the page title, as well as other related page information, sometimes including an icon that shows the description in a pop-up.\n\n12. Search Box Here you see a search box and button that will launch SharePoint site searches. It can be configured to show more options such as a drop-down of search scopes.\n\n13. Left Navigation Also known as the Quick Launch or current navigation, it is typically used for vertical secondary navigation to show pages related to the current location.\n\n14. Tree View This displays a Windows Explorer style tree view representation of the current site. It also shows metadata navigation and filtering if the appropriate feature is activated.\n\n15. Page Content This is the main content placeholder for SharePoint. It is required for rendering the actual content of pages.\n\nWhen creating wireframes for your SharePoint site, you must remember all the different types of content that are supported in SharePoint. It is easy to forget that content authors could be creating community sites, wiki sites, meeting workspaces, and many other types of content other than just the traditional content pages.\n\nNOTE\n\nLooking back at the wireframes for Randy's Waffles, NOTE that not every single piece of SharePoint functionality is going to be included on this site.\n\nYou can see that some of the functionality that won't be included in this design is the left navigation and the Tree View. Because this wireframe is diagramming the public Internet site for Randy's Waffles, this functionality would probably confuse anonymous Internet visitors who are not familiar with SharePoint. Conversely, if you were to create wireframes for Randy's Waffles intranet pages, they would probably include most of these functional elements because intranet users may need them to collaborate more efficiently with each other.\n\nAnother thing to notice in these wireframes is that the body content areas are filled with fake text. This technique is known as greeking because it uses text in Greek or Latin (as shown in the figure) in a pseudorandom pattern to fill in the blocks of text. Much like other aspects of the wireframes, the goal here is to get stakeholders to focus on the general layout, rather than the actual text in the document. Greeking works better than just filling in the text blocks with \"Content goes here\" because it more accurately mimics the size and spacing of actual body content. Rather than make up greeked text by hand, resources such as Lorem2 (http:\/\/lorem2.com) provide stock greeked passages of varying size for anyone to copy and paste into wireframes and design comps.\n\nBy creating wireframes early in a SharePoint branding project, decisions about what should be on the page and how it should be generally arranged on the screen can be discussed and agreed upon. When it's time to actually create the user interface, the designer can focus purely on the creative aspects of the design rather than get mired in functional requirements.\n\nCreating Realistic Design Comps\n\nCreating wireframes is certainly a valuable process for planning a new SharePoint website; however, for any serious branding effort, you need to make decisions based on creative concepts such as colors, font, and images as well. Before actually creating a brand in SharePoint, much like with wireframes, it would be good to have a way of creating realistic mockups of what the final website will look like. This is where design comps (sometimes known as prototypes) come into play. Unlike wireframes, design comps are intended to mimic the final branding of the actual SharePoint site as closely as possible without actually creating any code. Design comps should include all the things that make up a final design, such as colors, fonts, form elements, photos, and anything else that will appear on the final rendered website page.\n\nMany programs are available to create realistic design comps. For instance, you could use something such as Microsoft Paint, which has been included with every copy of Windows since version 1.0. You could even use a pencil and paper if you want a sense of nostalgia. However, there are some compelling reasons to seek out a more advanced solution. Design programs such as Microsoft Expression Design and Adobe Photoshop are shining examples of modern software that is geared toward creating design comps (among many other things). Both programs provide capabilities that aid in creating and maintaining reusable, realistic design comps.\n\nWith all the amazing things you can do in Expression Design and Photoshop, it's tempting to go crazy creating the world's most uniquely branded website. However, remember that the design comp is meant to imitate what can actually be created in a SharePoint website. Ultimately, learning whether something is easily creatable in SharePoint will take some time to get the hang of. One good method for learning this is to look at the functionality provided in the out-of-the-box master pages, both seattle.master and oslo.master. For example, if you plan to use SharePoint's top navigation functionality, it will be difficult to support long navigation item titles. This is because the horizontal navigation rendering in SharePoint does not wrap multiple lines of text per item by default.\n\nAs you create your design comp, it's a good idea to consider how the various elements in your design will be created in SharePoint. Will the elements use out-of-the-box SharePoint functionality with some styles applied to them, or will they require some amount of custom code? If they can be accomplished with Web Parts, will they require custom styling? All the answers to these questions aren't needed immediately while creating the design comp, but the project's schedule and budget should be considered, and the design's complexity may need to be scaled back to accommodate these factors.\n\nAs the design comps are completed, key stakeholders may need to be engaged again to sign off on the final look and feel. Are the colors and the design in line with what they were hoping for? If not, you might need a few iterations of the design comps before they are fully agreed upon and finalized. Although this can certainly be time consuming, ultimately it's much better to work these issues out earlier in a design tool than try to adjust master pages and CSS later.\n\nNOTE\n\nThe layered Photoshop design comp for Randy's Waffles is available for download with the rest of the examples in this book at Wrox.com.\n\nThe design comp shown here is highly stylized relative to the out-of-the-box SharePoint branding. If your project needs to make only minor changes to the seattle.master master page styling, you could create a design comp by starting with a screen shot of an existing SharePoint page. From there, you could use a design program to replace only the areas that are going to change. This technique could be used to quickly and easily create realistic design comps for the kind of simple branding discussed in Chapter 5.\n\nSummary\n\n\u2022 Planning for any project is all about clarifying your goals and mitigating changes and problems early before they become more difficult to deal with during implementation.\n\n\u2022 Ask questions up front to gather requirements early in the process.\n\n\u2022 Based on the purpose of your site, create an information architecture that is functional and highly usable.\n\n\u2022 Wireframes enable you to plan by focusing on the page structure and user behavior before creative elements are applied.\n\n\u2022 Create realistic design comps based on the wireframe, which can then be used as you implement your design.\n\n\u2022 In the following chapters you start with more basic SharePoint branding and work toward fully branded SharePoint sites with custom master pages and page layouts.\n5\n\nUsing the Design Manager to Start a Design in SharePoint\n\nWhat's In This Chapter\n\n1. Understanding the Design Manager\n\n2. Converting an HTML file to a SharePoint Master Page\n\n3. Using the Snippet Gallery\n\n4. Creating a Page Layout using the Design Manager\n\n5. Publishing and Applying Your Master Page\n\n6. Creating and Importing a Design Package\n\nThe reason you picked up this book and have made it this far is to learn how to make SharePoint look, well, not like SharePoint. You've learned what it means to brand SharePoint, some of the basics of SharePoint branding, and how to actually use SharePoint. You've gone through the planning process and created your design. This is where things get interesting! In this chapter, you will learn how to turn all your hard work, planning, and HTML design into a SharePoint site using the Design Manager.\n\nIntroducing Design Manager\n\nFirst things first: What exactly is the Design Manager? If you skipped over Chapter 2, \"SharePoint Overview,\" here's a brief refresher. The Design Manager is a new addition to SharePoint 2013 publishing sites. At its most basic level, the Design Manager is essentially a central hub for helping designers create and apply SharePoint branding. This can be accomplished by creating a blank master page that can be edited later or by converting an existing HTML file into a master page. There's a little more to it than that, which you learn as you dig deeper into each step in the process, but that is the main purpose.\n\nNOTE\n\nA SharePoint Online Public Site in Office 365 has different Design Manager steps, but they are similar to those in publishing sites.\n\nLimitations of Design Manager\n\nThe Design Manager is a great tool for starting a SharePoint branding project, but chances are you still need to do some tweaking and adjusting to the design after it's been converted to ensure what you see in SharePoint is what your original design looked like. The next chapter covers these tweaks and explains how CSS affects the final look and feel of your site and how you can override the default look and feel of many of the SharePoint components you'll add to your master page. Highly complex designs may be better suited for starting from a starter master page and skipping the Design Manager altogether.\n\nThe Design Manager consists of eight easy steps, and step 1 barely counts because it is just a quick Welcome screen with a couple useful links to import a design package (covered in the \"Creating a Design Package\" section of this chapter) or to change the composed look of the site. You walk through these steps as the chapter progresses.\n\nOne of the most significant changes to SharePoint 2013's approach to branding is that the Design Manager facilitates the use of any HTML editor for maintaining your page layouts and master pages, such as Dreamweaver, Notepad++, Expression Web, and even SharePoint Designer 2013 if you're so inclined. After you run your HTML page through the Design Manager to convert it to a master page, SharePoint creates a corresponding `.master` file and syncs it with the `.htm` or `.html` file that you started from. This is a big change from previous versions in that you edit the HTML file directly and let SharePoint automatically make the changes to the master page or page layout files.\n\nAnother thing to consider is that the Design Manager is only available on sites with the publishing features enabled. Branding created with the Design Manager will work on team\/collaboration sites, but there could be modifications that are needed for the branding to fully support all of the site features. Check out Chapter 7, \"Creating SharePoint Branding,\" for the lowdown on creating SharePoint branding from scratch. Once you have the Design Manager there are several ways to start using it on publishing sites: From the homepage of a freshly created publishing site, you can click the Design Your Site link under the I'm the Visual Designer header. Alternatively, you can click Settings menu \u21d2 Design Manager. Finally, if you are on the Site Settings page, under the Look and Feel header, you can click the Design Manager link. However you get there, the first thing you see is the helpful welcome page. There's not that much going on here, so just click on Manage Device Channels on the left to move onto the next step.\n\nStep 2 of the Design Manager is very similar to the Device Channels page that you can access from the Site Settings page.\n\nIt shows the device channels already set up on your site and gives you the option to set up additional device channels. By using device channels, you can opt to apply different master pages for different devices or browsers through the use of keywords that SharePoint looks for in a requesting browser's user agent string. For instance, a mobile device may receive a master page optimized for mobile viewing, whereas a desktop browser would get the site's default master page. If you'd like more information on building and using device channels, see Chapter 8, \"Advanced SharePoint Branding Tasks.\"\n\nUploading Design Files\n\nIn many cases, step 3 is actually the first step used in Design Manager. This is where you'll map a network drive to your Master Page Gallery and move your branding assets into the SharePoint site. Once you map a network drive to this URL, you can lay the foundation for working with your SharePoint branding going forward.\n\nExample: Mapping a Network Drive to the Master Page Gallery\n\n1. If you're not already in the Design Manager, click the Settings menu \u21d2 Design Manager \u21d2 Upload Design Files.\n\n2. Copy the URL provided in step 3 of the Design Manager. (Note: If you use Internet Explorer, don't right-click the link and select Copy Shortcut. This copies the URL in a format that drive mapping can't read. Instead, simply select the text and press Ctrl+C to copy it.)\n\n3. Depending on your version of Windows, the steps to start mapping a drive are slightly different:\n\n\u2022 In Windows 7 or Windows Server 2008\/2008 R2, click Start \u21d2 right-click Computer \u21d2 map network drive.\n\n\u2022 In Windows 8 or Windows Server 2012, open the Start Screen, type Computer. Right-click Computer and select Map Network Drive at the bottom of the screen.\n\n4. Choose your drive letter.\n\n5. Paste the URL you copied from the Design Manager into the Folder field.\n\n6. Select whether you want the drive to reconnect at logon, and also whether you would like to connect to the drive using different credentials than what you are currently logged into the machine with. You need to log on to the server with an account that has at least the Design permission level to work with the Design Manager and files in the Master Page Gallery. Click Finish when done and Windows Explorer opens and displays the contents of the Master Page Gallery in all its glory.\n\nNOTE\n\nYou can find out more about the drive mapping process at <http:\/\/support.microsoft.com\/kb\/2616712>.\n\nWhen building custom branding for previous versions of SharePoint, you would have put your master pages and page layouts in the Master Page Gallery and store all your custom CSS, images, and scripts in the Style Library or other document library. Things have changed slightly in the latest version. In SharePoint 2013, you can put your CSS, images, scripts, master pages, and page layouts all in the Master Page Gallery. The Style Library is still available, but now that you can map a drive directly to the Master Page Gallery to manage your branding assets from one location, it makes more sense to keep everything together.\n\nBecause you have just mapped a drive to the Master Page Gallery, uploading all your design assets is as simple as dragging and dropping from one window to another. The following steps walk you through uploading your branding assets to the Master Page Gallery:\n\nExample: Uploading Your Branding Assets to the Master Page Gallery\n\n1. If your drive isn't already open, navigate to your newly mapped drive. In Windows 7\/Server 2008\/2008 R2, click Start \u21d2 Computer. In Windows 8\/Server 2012, open the Start screen \u21d2 type Computer and click Computer to open it. In the list of drives, open your mapped drive.\n\n2. Create a folder within the mapped Master Page Gallery to hold all your images and styles. Because you're working with the Randy's Waffles site, call the folder Waffles.\n\n3. Open the Chapter 5 HTML folder on your computer, select all the files, and drag them into the Waffles folder in the Master Page Gallery. This should include the Assets folder and all its contents, the Font folder and all its contents, the Products folder and all its contents, seven image files, `favicon.ico`, `style.css`, and `Waffles2013.html`.\n\nAt this point, you're ready to convert your master page and start editing.\n\nEditing Master Pages\n\nNow that you've uploaded your design files to the Master Page Gallery, the next step is to convert your HTML to a SharePoint master page. SharePoint does all the heavy lifting here, so all you need to do is point it in the right direction and sit back and relax. The next sections walk you through this process and the basics of working with the Snippet Gallery to create your master page masterpiece.\n\nNOTE\n\nOne thing that's strongly recommended here is creating a backup of your original HTML file. After you go through the conversion process, you may notice that SharePoint added a number of commented blocks of code in it. It may initially cause some concern, but this is all expected. By making a backup of your original file, you can rest easy if something goes horribly wrong later in the process, knowing that you can do another conversion process from your original HTML file and start over.\n\nConverting an HTML Design into a Master Page\n\nThe process of converting HTML to a SharePoint master page is the cornerstone of the Design Manager. Everything you've done up to this point has led to this moment\u2014it's time to turn your plain HTML into something more: a SharePoint master page.\n\nClick step 4 of the Design Manager, Edit Master Pages.\n\nWhat you see here are options for converting an HTML file to a master page, or creating a blank starter master page. Below that, there are two master page files (`oslo.master` and `seattle.master`, SharePoint 2013's out-of-the-box master pages) that have been successfully converted.\n\nNOTE\n\nBecause you're interested in converting your `Waffles2013.html` file into a SharePoint master page, you can ignore the Create a Minimal Master Page link. Just know that clicking it opens a dialog window where you can give your minimal master page a name, and it'll create a corresponding HTML version for you to edit using your web design tool of choice.\n\nExample: Converting an HTML File to a Master Page\n\n1. Click Convert an HTML file to a SharePoint master page.\n\n2. In the Select an Asset dialog, browse to your Waffles folder within the Master Page Gallery, and select the `Waffles2013.html` file. The Location (URL) field at the bottom of the Select an Asset dialog shows the filename of the selected file.\n\n3. Click the Insert button.\n\n4. After a few seconds the Design Manager refreshes, your Waffles2013 file is listed, and its status reads Conversion Successful.\n\nThat's it! Not overly exciting, but that's actually a good thing\u2014an uneventful conversion generally means a successful conversion. What actually happened in the background was impressive. SharePoint parsed through your HTML file, adding commented code blocks and snippets throughout, turning it from a simple HTML file into a master page file that it can read and understand.\n\nIf for some reason your conversion is unsuccessful, you'll receive a status of Warnings and Errors. If you click the Warnings and Errors link, SharePoint will try to point out where you went wrong. Generally, it's due to malformed HTML. Because SharePoint parses through the document, your HTML needs to be XML compliant\u2014make sure you have no stray, unclosed tags; make sure your tags are closed in the proper order and tag properties are properly enclosed in quotes; and just generally make sure your HTML has the correct syntax.\n\nA few other things you'll need to check as you go through the conversion process:\n\n\u2022 Your DOCTYPE needs to be listed in all uppercase letters.\n\n\u2022 Don't include any `<form>` tags in your HTML document; SharePoint adds a form automatically during the conversion.\n\n\u2022 If you decide to use a modern template such as the HTML5 Boilerplate, there are some \"gotchas\" to be aware of. These are covered in detail in Chapter 11, \"Modern Web Design and SharePoint.\"\n\n\u2022 The Design Manager has a bug where apps cannot be added to sites using custom branding. As of this writing the issue has not been corrected, although Microsoft has indicated it will address the issue in a future patch. For more information on the issue and a fix, see <http:\/\/blog.drisgill.com\/2012\/12\/design-manager-bug-sharepoint-2013-rtm.html>.\n\nAfter the conversion process has been completed, click the Waffles2013 filename and you can see a preview of your master page.\n\nWARNING\n\nIf your design has a background image applied to the `<body>` tag with CSS, there are some potential \"gotchas\" that you may need to look out for. SharePoint 2013 can be picky about how it displays background images on the `<body>` tag. If your background image isn't showing, you might try applying the background to another element on the page. Microsoft has added a CSS class called `.ms-backgroundImage` to the `<body>` tag that you can add properties to, including the background-image property. Or, you can try applying the background image to the `.s4-workspace` div. In the case of this design, the background image displays on the `<body>` tag without using any of these other methods.\n\nAs you can see from the screen shot there are a couple things that just don't quite look right. For one, the search box isn't lined up correctly. (This is OK for now; you'll replace the placeholder HTML search box with a SharePoint search box code snippet in the next section and style it in the next chapter.) The other important thing to point out is that at the bottom of the page, below the footer, a yellow box has been added, containing the text \"This Div, Which You Should Delete, Represents the Content Area That Your Page Layouts and Pages Will Fill. Design Your Master Page Around This Content Placeholder.\" This yellow box is important; it represents the area in which SharePoint displays page content within the master page. You eventually want to move this yellow div to its proper place within the master page, which you learn to do in the next section.\n\nWorking with Design Tools\n\nNow that the conversion process is finished, it's time to look at the converted HTML. Earlier the chapter mentioned that with the new approach to branding SharePoint 2013, virtually any web design tool can be used, and there are several design tools of the trade that are available to you. In SharePoint 2010, just about the only program you could use for working with master pages was SharePoint Designer 2010. You can still work with master pages in SharePoint Designer 2013, but you must use the Code view; the WYSIWYG Design view has been removed. Otherwise, the interface is nearly identical, so if you're familiar and comfortable working with SharePoint Designer, you certainly can.\n\nNOTE\n\nSharePoint Designer can't access mapped drives, but it has the ability to browse to the Master Page Gallery within the program.\n\nMany designers may prefer something more traditional though, such as Dreamweaver or Expression Web. Even a text editor such as Notepad++ (<http:\/\/notepad-plus-plus.org>) can be used if you don't need a design preview. This was the idea of changing how SharePoint branding works in 2013: By expanding the available tools for creating and maintaining branding for SharePoint, it is easier to create branding because designers can use familiar tools.\n\nBecause you've mapped a drive to the Master Page Gallery, you can browse to that location, open the `Waffles2013.html` file, and begin the editing process, just as you would with any other HTML file. The HTML is a bit different than the last time you looked at it. Because of the conversion process, it looks at first glance like you sent your HTML through a digital meat grinder. SharePoint added all sorts of commented code blocks and snippets, increasing the 160-line HTML file to a 344-line file. Initially, you might be a little lost and disoriented among all the clutter. After all, it was easy to find your way through your nice clean HTML file. But if you look through everything SharePoint has added, you'll actually find your original HTML, largely intact, and you'll see that the converted file isn't so scary.\n\nMany web design tools such as Dreamweaver and Expression Web have preview modes that give you an idea of what your site will look like in the browser. If you're using one of these kinds of web design tools you've probably noticed that things look a little off. Part of it may be the way the program renders the preview, but some of it has to do with the fact that SharePoint branding relies on a lot of moving parts, which make rendering a challenge. You can see the preview of the `Waffles2013.html` file in Dreamweaver in the following figure. You will also notice that in the preview of your site, SharePoint has added some snippet blocks for some elements of the UI\u2014notably the block where the ribbon will be positioned, and the aforementioned yellow block below your main content.\n\nA good starting point for editing your converted master page is moving the yellow block of content to the correct location. Right now it's sitting below the content of your page, which means that all page layout content will appear here as well if you apply this master page to your site. It needs to be moved into the main body area of the page.\n\nConsider that your initial HTML design includes all elements you will see on the page including what will eventually become a master page and a page layout. Your job at this point is to start figuring out which part of the overall page layout is related to page content.\n\nNOTE\n\nWhen creating your own HTML design, you might find it helpful to add a comment to the beginning of the content that will be a page layout and another comment at the end, so it's easy to discern where you will move the placeholder at this point.\n\nIn the `Waffles2013.html` file, the area you want to replace is the content within the `<div class=\"customBodyHolder\"> `tag. Complete the following steps to remove the HTML mockup and move the page content placeholder to its correct location. These steps were created using Adobe Dreamweaver CS5.5. They are as generic as possible, but you may find some slight difference in your editing program:\n\nExample: Moving a Content Placeholder\n\n1. If you haven't opened `Waffles2013.html` from the mapped drive in your web design program, do so now. You should also see a corresponding `Waffles2013.master` file, but you'll only be working with the HTML file. SharePoint automatically updates the master page file with any changes you make in the HTML file. If you don't see the master page file, you may just need to refresh the Windows Explorer window by pressing F5.\n\n2. In the HTML, find the `<div class=\"customBodyHolder\">` tag. It should start around line 158.\n\n3. Select all the content inside the `customBodyHolder` div tag; there are two comment tags to help you find the areas you should select. Select all of the content from `<!-- START Page Content -->` to `<!-- END Page Content -->`, which should be near lines 159\u2013306.\n\n4. Cut (Ctrl+X) these lines from the HTML file. (Don't delete them!) Create a new page in your web editing program (usually, pressing Ctrl+N will do this, though your program may differ) and paste (Ctrl+V) the lines in the HTML for later. If your web design program adds starter HTML such as a DOCTYPE and `<head>` and `<body>` tags, you can delete everything and simply paste your block of cut code. If you want, you can save this HTML page somewhere easily accessible, such as your desktop. It's just a temporary page that you're using to keep this HTML set aside for the time being.\n\n5. Save the `Waffles2013.html` file, and switch back to the preview of your master page in the browser. Refresh the page to see your progress. You should see the master page with the now-empty main body. The yellow div block is still positioned below the footer.\n\n6. Now you're ready to move the placeholder into the main body area of the page. Switch back to your web design program. Scroll to the bottom of the HTML and find the following block of code (approximately lines 182\u2013194, though your line numbers may differ if you left any empty lines when you cut the body content in step 4).\n\n`<div data-name=\"ContentPlaceHolderMain\">`\n\n` <!--CS: Start PlaceHolderMain Snippet-->`\n\n` <!--SPM:<%@Register Tagprefix=\"SharePoint\" Namespace=\"Microsoft.SharePoint.WebControls\"`\n\n` Assembly=\"Microsoft.SharePoint, Version=15.0.0.0, Culture=neutral,`\n\n` PublicKeyToken=71e9bce111e9429c\"%>-->`\n\n` <!--MS:<SharePoint:AjaxDelta ID=\"DeltaPlaceHolderMain\" IsMainContent=\"true\" runat=\"server\">-->`\n\n` <!--MS:<asp:ContentPlaceHolder ID=\"PlaceHolderMain\"`\n\n` runat=\"server\">-->`\n\n` <div class=\"DefaultContentBlock\" style=\"border:medium black solid;`\n\n` background:yellow; color:black; margin:20px; padding:10px;\">`\n\n` This div, which you should delete, represents the content area`\n\n` that your Page Layouts and pages will fill. Design your Master`\n\n` Page around this content placeholder.`\n\n` <\/div>`\n\n` <!--ME:<\/asp:ContentPlaceHolder>-->`\n\n` <!--ME:<\/SharePoint:AjaxDelta>-->`\n\n` <!--CE: End PlaceHolderMain Snippet-->`\n\n`<\/div>`\n\n7. Cut this code from the HTML (Ctrl+X) and place the cursor within the `<div class=\"customBodyHolder\"><\/div> `tags. Press Enter to create an empty line between the opening `<div> `and closing `<\/div>`.\n\n8. Paste the code snippet (Ctrl+V) in the empty line between the `customBodyHolder` div tags. If you use a design tool with a preview window, the preview should update with the yellow block in the body area of the HTML file. Save `Waffles2013.html`.\n\n9. Switch back to the browser, and refresh the preview window. The yellow block is correctly positioned in the body area. (There is now a gap between the navigation and the white body of the page. Don't worry about this. It goes away when you clean up the yellow div tag in the next couple steps.)\n\n10. Within the placeholder code you just pasted is a `<div>` tag that needs to be deleted. The text and yellow box simply help you locate the div to position it properly; it is not needed for any other purpose, so you can safely delete it. Switch back to your design program and in the code you just pasted, select and delete the following content (approximately lines 164-167):\n\n`<div class=\"DefaultContentBlock\" style=\"border:medium black solid; background:yellow; color:black; margin:20px; padding:10px;\">`\n\n` This div, which you should delete, represents the content area that your Page Layouts and pages will fill. Design your Master Page around this content placeholder.`\n\n` `\n\n` <\/div>`\n\n11. Save `Waffles2013.html`.\n\n12. Return to the browser and refresh the master page preview. You should see your master page with an empty body area, which is exactly what you would like to see at this point. You now see what the master page should look like after the yellow div block is removed. (The gap between the body area and the navigation is fixed, too.)\n\nAt this point you've mapped a drive to the Master Page Gallery, uploaded your design files, converted your HTML file into a SharePoint master page, and have moved the page content placeholder to its correct location. Not bad, but there's still more to come.\n\nIn the next few examples, you'll use the Snippet Gallery to put actual SharePoint controls onto the page, such as the search box and top navigation. When doing so you will be copying and pasting code around your HTML file. If you get lost along the way you can download the completed HTML master page and page layout examples from the downloadable code for this chapter.\n\nUsing the Snippet Gallery\n\nUp to now you've made your changes to the `Waffles2013.html` file using the design tool of your choice. When SharePoint converted the HTML file into a master page, it added a placeholder for the main content of the site to the file, albeit not where you would have liked it to end up. In the previous section you walked through the process of working with the HTML file in your web design tool to make some changes to the HTML file and moving the placeholder to its correct location.\n\nBut what about other areas of the site? As you may know, SharePoint uses a number of content placeholders for various elements that you see on the site, such as the navigation, search box, header, and a host of other content. When a page is requested by a browser, SharePoint injects the proper code into each placeholder, which shows up as the HTML in the final rendered version that you see. In the `Waffles2013.html` file, several areas of the page need the placeholder HTML you used for your design purposes to be replaced with SharePoint placeholders. Two of the areas are the search box and top navigation. You can find the SharePoint placeholders used for these areas in the Snippet Gallery.\n\nThe Snippet Gallery is a collection of code snippets for various pieces of SharePoint functionality that you can use in your master page. Each snippet is found on the gallery's ribbon. Clicking a snippet button on the ribbon presents you with a preview window of what that snippet will look like when it's rendered, the actual snippet of code that you can copy and paste into your master page, and an expanding accordion set of options that you can use to make changes to the code snippet if you want.\n\nThere are actually two versions of the Snippet Gallery: one for master pages and one for page layouts. Because you're working with a master page here, you're presented only with those code snippets that can be added to a master page.\n\nWARNING\n\nSeveral of the examples in this chapter include approximate line numbers to look for in your HTML file. Your line numbers may differ some based on how your editor works, and if you have published the file in SharePoint. When HTML master pages and page layouts are published, some of the snippet code is expanded automatically from single long lines of code to several tabbed out lines.\n\nExample: Adding Snippets to a Master Page\n\n1. You should have the master page preview opened in your browser from the previous section. If you closed your browser or navigated away from the preview, simply open Design Manager and click Edit Master Pages and click Waffles2013.html.\n\n2. Toward the upper-right corner of the page, above the ribbon preview, click the Snippets link to open the Snippet Gallery. The Snippet Gallery opens in a new browser tab, so you can easily switch back and forth between the gallery and the master page preview.\n\n3. In the Snippet Gallery, click the Search Box button in the Navigation section on the ribbon. The Preview window shows what the default search box looks like, and you see a code snippet in the area directly below the preview.\n\n4. On the right in the Customization - Search Box `SearchBoxScriptWebPart `section, click some of the various headers to explore the options you can set on the search box. You won't make any changes here, so just leave the fields blank, but if you were making changes, you must click the Update button for them to be reflected in the code snippet. If you accidentally make a change, simply click the Reset button above the Customization section. This resets the code snippet back to its default state.\n\n5. In the HTML Snippet section, select and copy all the text in the scrolling window. (If you use Internet Explorer, you have a Copy to Clipboard button below the HTML snippet window that can be used to easily copy the snippet to the clipboard. If IE asks if you want to allow the webpage to copy the code after you've clicked this button, just click Allow).\n\n6. Return to your design program and find the `<div class=\"customSearchHolder\">` tag (around line 121).\n\n7. Select and delete the contents within the `customSearchHolder` div, which should be the `<input>`, `<a>`, and `<img>` tags. The highlighted text in the screenshot shows the content that should be deleted.\n\n8. Place the cursor inside the `<div class=\"customSearchHolder\"><\/div>` tags, press Enter to create a blank line, and paste in the snippet you copied from the Snippet Gallery onto the blank line between the `customSearchHolder` divs.\n\n9. Save the `Waffles2013.html` file, return to the master page Preview tab in your browser, and refresh the page to see your change.\n\nWhen you started with your HTML design, some items were essentially HTML placeholders that were going to be replaced with SharePoint controls. The search box is one of them. The original HTML file used a simple HTML `<input>` field to represent the search box. SharePoint's search control is more complex than that and as such requires a bit of extra work to style to match the original design. You'll learn more about that in Chapter 6, \"Cascading Style Sheets and SharePoint.\"\n\nAs for the code snippets, there is actually some logic to how they are laid out, despite looking like a bunch of gibberish at first glance. You'll notice that every snippet is made up of commented blocks of code. Each comment begins with a set of letters, which helps you figure out what each line means. The general structure of a code snippet is as follows:\n\n\u2022 Snippet Start\u2014Starts the code snippet with an opening `<div>` tag and comments.\n\n\u2022 SharePoint markup\u2014The code that contains the namespace registration tag (if one is required) and the corresponding control that renders the item in SharePoint.\n\n\u2022 HTML Preview\u2014A read-only preview that web design programs can read and generate an approximate look for the control.\n\n\u2022 Snippet End\u2014Ends the snippet with comments and usually a closing `<\/div>`.\n\nSome snippets may have more than one section of SharePoint markup and HTML Preview sections, but the preceding list provides the basic structure. Each of the comments that appear in each section begin with various abbreviations and have specific meaning to help you understand the markup a little better. These abbreviations represent the following:\n\n\u2022CS\/CE (Comment Start\/Comment End)\u2014Starts and ends the code snippet.\n\n\u2022 SPM (SharePoint markup)\u2014Denotes a single line of code and in some cases is used as registration for a namespace.\n\n\u2022 MS\/ME (Markup start\/Markup end)\u2014Starts and ends a SharePoint control snippet.\n\n\u2022 PS\/PE (Preview Start\/Preview End)\u2014The code inside these comment blocks is the HTML preview file that help design tools render the look of the SharePoint control.\n\nGenerally speaking, you shouldn't need to modify anything in the code snippets outside of the Snippet Gallery. Any styling you need to do can generally be done by overriding the out-of-the-box styles. Again, this will be covered in the next chapter.\n\nNow you can return to the Snippet Gallery to replace your static HTML navigation with a dynamic SharePoint navigation control.\n\nExample: Replace Navigation Placeholder with SharePoint Navigation Snippet\n\n1. Return to the Snippet Gallery in the browser. If it's no longer open, you can get there by opening Design Manager \u21d2 Edit Master Pages \u21d2 Waffles2013.html \u21d2 Snippets.\n\n2. On the ribbon, click the Top Navigation button on the left. In some snippet previews, the preview window renders a live preview of the control. For example, if you have already built your site's navigation, your navigation structure displays in the preview window instead of a generic preview.\n\n3. By default, the navigation snippet doesn't show drop-downs, so you need to adjust its properties. Scroll down the page to the Customization - Top Navigation (AspMenu) section. Change the StaticDisplayLevels value to 1. In this same section, click the Behavior section to expand it, and change the MaximumDynamicDisplayLevels value to 1.\n\n4. Scroll up the page and click the Update button to update the HTML snippet.\n\n5. In the HTML Snippet section, select and copy all the text from the scrolling text field.\n\n6. Switch to your web design program with `Waffles2013.html` open and find `<div class=\"customTopNavHolder\">` (around line 130). Within that tag, select and delete the entire `<ul class=\"customTopNav\">` tag and all its contents (around lines 131-152).\n\n7. Place the cursor within the `<div class=\"customTopNavHolder\"><\/div>` tags, and paste in the Top Navigation snippet you just copied from the Snippet Gallery.\n\n8. Save `waffles2013.html`.\n\nOne last thing you'll want to do to make navigating the site a little easier is to replace the site logo link and image with the Site Logo snippet, which makes the logo a link back to the homepage of the site.\n\nExample: Replacing Placeholder Site Logo with SharePoint Site Logo Snippet\n\n1. Switch to the Snippet Gallery and click the Site Logo snippet in the ribbon.\n\n2. Scroll down the page to the Customization - Site Logo (SPSimpleSiteLink) section and click the Appearance header to expand it. In the CssClass properties, replace the default text, ms-siteicon-a, with customLogo.\n\n3. Still in the Customization - Site Logo (SPSimpleSiteLink) section, expand the Navigation section and in the NavigateUrl field, `type `~sitecollection\/.\n\n4. In the Customization - Site Logo (SiteLogoImage) section, expand the Appearance section and delete the default text from the CssClass field. Then expand the Misc section, and in the LogoImageUrl field, replace the default URL in the field with \/_catalogs\/masterpage\/waffles\/logo.png. Be sure to change the path if your site collection is not at the root of your web application (Example: \/sites\/SubSiteCollection\/_catalogs\/masterpage\/waffles\/logo.png).\n\n5. Scroll up and click the Update button, then select and copy the HTML snippet.\n\n6. Switch to your web editing program and replace the following HTML with the Site Logo snippet you just copied, which should be around lines 118-120:\n\n`<a href=\"#\" class=\"customLogo\">`\n\n` <img src=\"logo.png\" alt=\"Back to Home\" \/>`\n\n`<\/a>`\n\n7. Save `Waffles2013.html`, return to the master page preview in the browser, and refresh the page to see your changes. The site logo should look exactly the same as it did before. You can see the SharePoint navigation in place of the HTML navigation. If you don't have any navigation links created, you only see an Edit Links link in the navigation. The link is highlighted in the screenshot.\n\nNOTE\n\nYou also could have left the original HTML for the logo and simply changed the `<a href=\"#\"`... to `<a href=\"\/\"`... to link back to the root of the site. However, using the Site Logo control gives you the ability to easily switch out the site's logo in the SharePoint UI by clicking the Settings menu \u21d2 Site settings \u21d2 Title, description, and logo. In the Logo and Description section, you can add a new logo image from your computer or from within SharePoint, which replaces the logo currently displaying in the header area. This can be helpful if your design utilizes variations of the same logo, such as for different department sites or sites for different branches of the same company.\n\nThe search box, top navigation, and site logo are just three of the many snippets available in the Snippet Gallery to add to your custom master page. In addition to these, you can add a variety of other controls and Web Parts to your page. The following table details the available controls you can add via the Snippet Gallery.\n\nThe process for adding these snippets is exactly the same as the process for the ones you just added to the Waffles2013 file. When you work with your own design, select the snippet from the ribbon, copy the code, and paste it into the appropriate location in the HTML within your design tool.\n\nIn the `Waffles2013.html` file, the `<div>` tags you selected were conveniently named to make it as easy as possible to find the blocks of code you needed to work with. This is a fine option as you build out your own HTML file, or you could add comment blocks around the various elements on your HTML page so it's easier to find what you need to replace after your file is converted.\n\nOne last important thing to note is that if you were to apply this master page to the site as the system master page and open a SharePoint dialog box, many of the branding elements will show in the dialog, which isn't always desirable. This issue discussed in more depth and remedied in Chapter 7 in the section \"Using a Starter Master Page.\"\n\nSkipping Step 5\n\nThe next step in the Design Manager is step 5: Edit Display Templates. In a nutshell, display templates are used in search results and Web Parts that derive their contents from search data (such as the Content Search Web Part) to display search content. In other words, they are templates for displaying search results and data. The Design Manager provides you with a view into the display templates folder in the Master Page Gallery, so you can see the available templates that can be used as a starting point for creating any custom template you want.\n\nChapter 9, \"Creating Content Rollups with SharePoint WCM,\" covers creating and using Display Templates, so you can actually skip over this step for now. Instead, move on to step 6 in the Design Manager, \"Edit Page Layouts.\"\n\nEditing Page Layouts\n\nYou'll recall from Chapter 2, \"SharePoint Overview,\" that page layouts are basically layout templates for displaying content on each page in SharePoint. All publishing pages are based on a page layout, and each page layout is based on a content type. Microsoft has a handful of page layouts you can use when building your sites, but you also have the option to create custom page layouts. Just like the out-of-the-box page layouts, your custom page layouts need to be based on a content type. A content type is simply a collection of site columns that SharePoint uses to hold information about a particular item. There are a couple kinds of content types that are applied to page layouts, including Article Page and Welcome Page. These two content types share some similarities, but differ in some of the columns that they contain. Page layouts based on the Article Page content type are generally used for creating content pages, whereas page layouts based on the Welcome Page content type are often used for site or subsite homepages. There's no hard rule that says you have to use one or the other, but the different fields that appear on each page layout may help influence your preference depending on your needs.\n\nOn step 6 of the Design Manager you see a Create a Page Layout link and a list of all the page layouts available in the site. If you are just starting out and haven't created any page layouts yet, this list will be blank, letting you know that there are no HTML Page Layouts to show. You can also see which content type each page layout is associated with. In the last section, you worked with your web design tool to make some changes to the `Waffles2013.html` file and turn it into a useable master page. One of those changes was to remove the content from the body of the design because it would normally appear as a page layout. This section walks through adding that content back as a page layout, stripping out some of the extra placeholder content and replacing it with Web Part zones for adding SharePoint Web Parts.\n\nExample: Creating a Page Layout and Adding Placeholder Content\n\n1. In step 6 of the Design Manager, click the Create a Page Layout link. You may need to open Design Manager again because the preview page doesn't provide any link to head back to the Design Manager.\n\n2. In the Create a Page Layout dialog, type WafflesHome in the Name field. This becomes the filename that appears in the Master Page Gallery when editing the page layout.\n\n3. Select Waffles\/Waffles2013 in the Master Page drop-down to associate this page layout with your custom master page.\n\n4. Select Welcome Page in the Content Type drop-down. Because you're going to be creating the page layout that you'll use for the homepage of your site, it makes sense to use the Welcome Page content type. Click OK to create your new page layout. The Edit Page Layouts page refreshes and lists the newly created WafflesHome page layout.\n\n5. Switch to your mapped drive. When SharePoint creates a new page layout, it puts it in the root of the Master Page Gallery. You can leave it there, or if you'd like to keep all your related files together, you can move the HTML file into the Waffles folder with your custom master page. Open the drive mapped to the Master Page Gallery, and find `WafflesHome.html`. Drag and drop this file into the Waffles folder. (If you receive a warning that the file might be harmful, you can safely click OK.) You don't need to move the associated `WafflesHome.aspx` page because after the HTML file is moved, SharePoint moves the .aspx file automatically, although you may need to refresh your mapped drive by pressing F5 to see the change right away.\n\n6. Return to the Design Manager in the browser and refresh the step 6 page. Click WafflesHome to open a preview of the page layout within your custom master page.\n\nNOTE\n\nOn the preview page for the custom page layout, SharePoint has already added some sample content. This set of controls appears because you are basing your layout on the Welcome Page content type. If you had selected the Article Page content type, you would have seen a different set of controls. Some fields are common among the various content types, including the Page Image and Page Content fields.\n\n7. Near the upper right of the page, click the Location link. A dialog opens telling you the location of the file you're previewing. Verify that it appears as `[YourMappedDrive]:\\Waffles\\WafflesHome.html`.\n\n8. Switch to your web editor, and open the `WafflesHome.html` file from within the Waffles directory in the mapped drive.\n\n9. Look at the HTML code and find the line `<!--MS:<asp:ContentPlaceHolder ID=\"PlaceHolderMain\" runat=\"server\">--> `(line 70). There are several lines that begin with `<asp:ContentPlaceHolder`, so make sure you find the correct one with the ID of `PlaceHolderMain`. This is the placeholder for the main body of the page layout. All the controls and content that appear on the page should be within this tag. Look through the various `<div>` tags within the `PlaceHolderMain` control to familiarize yourself with how the page layout is built. Each `<div>` contains a snippet for the various components on the page that SharePoint added for you.\n\n10. Select all the content within the `PlaceHolderMain` control (all the divs and code snippets) and delete it (approximately lines 71\u2013112). You will add some of these controls back in the coming steps. For now, it's just easier to start from a blank slate.\n\n11. Position the cursor at the end of the line `<!--MS:<asp:ContentPlaceHolder ID=\"PlaceHolderMain\" runat=\"server\">-->` and press Enter to create a new line below.\n\n12. Switch to the document where you pasted the HTML of the body of the mockup from the last section. Select all of the HTML and copy it. (If you didn't do this step or closed the new page without saving the content, you can just open your original unconverted Waffles2013 file and copy the HTML between `<!-- START Page Content -->` and `<!-- END Page Content -->`.)\n\n13. Return to your editor, and paste in the HTML you just copied within the empty `PlaceHolderMain` control. Save the file. Return to the page layout preview in the browser and refresh the page.\n\nRight now things look a lot like the original file you converted, but you've only completed moving the SharePoint placeholder for the main content area of the page.\n\nThe next stage of this process requires you to start removing some of your placeholder HTML to make room for Web Part zones and other SharePoint control snippets.\n\nExample: Replacing Page Layout Placeholder Content with Snippets\n\n1. Switch back to your design program. In the WafflesHome.html file, find the following block of code:\n\n`<div class=\"customHeroLeft\">`\n\n` <a href=\"#\" class=\"customHeroPhoto\"><img src=\"hero.jpg\" alt=\"\" \/>`\n\n` <\/a>`\n\n`<\/div>`\n\n2. Delete the lines `<a href=\"#\" class=\"customHeroPhoto\"><img src=\"hero.jpg\" alt=\"\" \/><\/a>. `The highlighted code shown in the screenshot is what should be removed from the page.\n\n3. A couple lines down, find the `<div class=\"customHeroRight\">` tag and delete all the contents within the tag (which should include the `<h1>`, `<p>`, and `<a>` tag). The `customHeroRight` div should be empty. Go ahead and save your page now.\n\n4. Switch back to the preview page in the browser, and click the Snippets link near the upper right of the page. This opens the Snippet Gallery for page layouts in a new Browser tab. The page layout Snippet Gallery looks and behaves identically to the master page Snippet Gallery, with the exception of the available controls. These are specific only to page layouts.\n\n5. By default, the Snippet Gallery opens to the Page Image control, which is actually the snippet you're going to add first. This snippet is located under the Page Fields drop-down on the ribbon. Just like you did in the master page Snippet Gallery, select all the text in the scrolling text field in the HTML Snippet section and copy it.\n\n6. Return to the design program, and paste the snippet within the `<div class=\"customHeroLeft\">` tag, where the picture of the waffles used to be. Save your page.\n\n7. Switch back to the Snippet Gallery tab in the browser, and click the Page Fields drop-down; then select and copy the Page Content snippet from the HTML Snippet section. (This field and the Page Image field are ones that were already included on the page layout by default when you created it. You could have simply moved these snippets around instead of deleting them, but for this example it was easier to delete everything and add the fields back in.)\n\n8. In your design program, paste the snippet code inside the `<div class=\"customHeroRight\">` tag. Save the file and return to the page layout preview in the browser. Refresh the page and see your changes.\n\n9. Switch back to the design program, and delete the content within the` <div class=\"customColLeft\">` and `<div class=\"customColRight\">` tags because this was simply placeholder HTML for the design. You are going to replace the content with Web Part zones, which can allow you to add in Web Parts later to fill in the content.\n\n10. Return to the Snippet Gallery, and click the Web Part Zone button in the Containers section on the ribbon.\n\n11. In the Customization area on the snippet page, the Important section is expanded, showing the ID field. By default SharePoint generates a unique ID for each Web Part zone. You can change the long string of random numbers and letters to something more digestible. Delete the text from the ID field, and type LeftCol as the ID.\n\n12. Click the Update button above the Customization area to update the code snippet. Look at the snippet and notice that your new ID of LeftCol appears in the tag that begins `<!--MS:<WebPartPages:WebPartZone...`.\n\n13. Just like before, select and copy the HTML snippet code, and in your design program, paste in the copied snippet within the `<div class=\"customLeftCol\">` tag.\n\n14. Switch back to the Snippet Gallery. Change the ID of the Web Part zone to RightCol, click the Update button to update the code snippet with the new ID, and select and copy the HTML snippet.\n\n15. In your design program, paste the snippet in the `<div class=\"CustomRightCol\">` tag. Save your changes.\n\n16. Return to the page layout preview and refresh the page. You can't see the zones because you're not in edit mode, but if you've followed these steps, rest assured they are there.\n\nIn the next section you will look at publishing and applying your custom master page, and you also see how to apply your custom page layout so that you can see the results of your hard work.\n\nPublishing and Applying a Master Page\n\nNow it's time to actually publish your master page, page layout, and your design elements so that you can apply them and see how it looks and behaves on the site, outside of a preview page. In a real-world scenario, you'd probably want to spend some more time tweaking the look and feel of some of the SharePoint controls before getting to this point, although you can still make changes to your files even after they've been published. When you publish your files, you are checking them in and making them available to anyone who has read access to the site. In the next chapter, you'll learn about styling the SharePoint controls to match the original design.\n\nWhen you click on step 7 in the Design Manger, you're presented with some rudimentary instructions for publishing your custom designs and a couple links: one for opening the Master Page Gallery and another for opening the master page settings. You can also get to either of these places from the Site Settings page under the Galleries heading and the Look and Feel heading, respectively.\n\nNow you'll walk through the process of publishing your design assets, page layout, and master page. Then you'll apply the master page to the site.\n\nExample: Publishing your Design Files\n\n1. Click on Publish and Apply Design on the left side of the Design Manager.\n\n2. Click the link Go to the Master Page Gallery. The Master Page Gallery opens in a new browser tab. (Alternatively, if you were outside the Design Manager, you could simply click the Settings menu \u21d2 Site settings \u21d2 Master pages and page layouts.)\n\n3. Click the Waffles folder to display its contents and put a check in the box that appears next to the `Waffles2013.html` file when you hover over the filename.\n\n4. On the ribbon, click the Files tab; then click the Publish button toward the right side.\n\n5. In the dialog that opens, optionally type a comment in the Comments field; then click OK to publish your master page. SharePoint automatically publishes the corresponding `Waffles2013.master `file.\n\n6. In the column headings, check the box in the leftmost column to select all the items in the Waffles folder. Notice that the Publish button isn't available to click. That's because there's no way to mass publish all the items in this folder from within the Master Page Gallery. (Okay, technically there is; you can actually mass check out every file in the Waffles folder, with the exception of folders, and then check in all the files at once, electing to publish a major version of each. But for now you're going to take a different approach.)\n\n7. Click the Settings menu \u21d2 Site settings.\n\n8. Under the Site Administration header, click Content and Structure to open the Content and Structure manager. This page, which has been present in SharePoint since the 2007 release, still retains the same basic look and feel it had in SharePoint 2007.\n\n9. In the left pane of the Content and Structure page, click the + next to Master Page Gallery. Then click the Waffles folder to display its contents in the main window of the page.\n\n10. At the top of the main body, click the icon of stacked check boxes to select all items in the folder.\n\n11. Click the Actions drop-down in the toolbar, and select Publish. In the dialog that appears, optionally type a comment, and then click OK. This publishes all items in the Waffles folder.\n\n12. Click the Assets folder within the Waffles folder to open it up. Then repeat steps 10 and 11 to publish the items inside this folder.\n\n13. In the toolbar, click the Navigate Up button to return to the Waffles directory.\n\n14. Repeat the publish process for the items in the remaining folders (Font and Products).\n\n15. When everything is published, click the Back to '[Site name]' link in the upper left of the Content and Structure window to return to your site.\n\nNow that your custom master page, custom page layout, and all associated styles and images have been published, you can apply the master page to the site.\n\nExample: Apply Your Custom Master Page to the Site\n\n1. Return to the Design Manager tab in the browser.\n\n2. Still on step 7, click the link Assign Master Pages to Your Site Based on Device Channels. This opens the Master Page Settings in a dialog. (When you open this page from the Site Settings page, it doesn't display in a dialog.)\n\n3. If you have any device channels set up, the device channel name appears with a drop-down to select an alternative master page in the Site Master Page settings section, so set whatever master page you'd like for your device channels, or set it to Follow Default Channel. For the Default drop-down, select Waffles\/Waffles2013. (If you don't have any device channels set up, just apply the Waffles\/Waffles2013 master page to the only drop-down that appears.)\n\n4. Change the System Master Page setting drop-down to Waffles\/Waffles2013.\n\nNOTE\n\nIn case you are wondering what the difference is between the two master page settings, the Site master page setting applies the selected master page to all publishing pages, while the System master page setting applies the selected master page to all other page types, including settings pages.\n\n5. Click OK to apply your custom master page to your site. If you are changing the master page from within the Design Manager, you need to refresh the page to see the newly applied master page. This is because the master page settings opened in a modal dialog and didn't refresh the original page when it closed. When you access the master page settings from the Site Settings page, the branding is applied as soon as you click OK because you'll be redirected back to the Site Settings page immediately after.\n\nAnd voil\u00e0! Just like that, you're looking at your master page on a live SharePoint site! You can see the Design Manager with your custom design applied.\n\nNow it's time to try out your custom page layout:\n\nExample: Applying Your Custom Page Layout\n\n1. Click the Settings menu \u21d2 Add a Page. In the Add a page dialog, type Home in the Give It a Name field and click Create.\n\n2. Your page opens up in Edit mode. Click the Page tab on the ribbon; then click the Page Layout drop-down. Scroll to the bottom of the Page Layout drop-down, and select your custom page layout, WafflesHome.\n\n3. Your custom page layout displays, with the fields and Web Part zones you added in. The page may look a little \"off\" at first, but once you start adding some content it will display correctly. In the Page Content field control, click the text Click here to add new content. Use the Format Text tab on the ribbon to make some text changes if you want. Alternatively, you can edit the HTML source for this field control by clicking the Edit Source button in the Format Text tab of the ribbon. If you are so inclined, you could add in the HTML from the original design:\n\n`<h1>Your Source for Quality Breakfast Foods<\/h1>`\n\n`<p> `\n\n`Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Phasellus hendrerit. Pellentesque aliquet nibh nec urna. In nisi neque, aliquet vel, dapibus id, mattis vel, nisi. Sed pretium, ligula sollicitudin aoreet viverra, tortor libero sodales leo, eget blandit nunc tortor eu nibh. Nullam ollis. Ut justo. `\n\n`<\/p>`\n\n`<a class=\"customLearnMore\" href=\"#\">Learn More<\/a>`\n\n4. In the Page Image field control, click the link Click here to insert a picture from SharePoint.\n\n5. In the Edit Image properties dialog that opens, click the Browse button next to the Selected Image field.\n\n6. At the bottom of the Select an Asset dialog, type \/_catalogs\/masterpage in the Location (URL) field, and press Enter to open the Master Page Gallery.\n\nNOTE\n\nYou can't browse directly to the Master Page Gallery using the folder structure UI in the Select an Asset dialog, which is why you are opening it up by typing the folder path. In a real-world scenario you would probably want to consider moving some of the images used in the body of the page from the Master Page Gallery into a more easily accessible location such as an image library on the site.\n\n7. In the Master Page Gallery, click the Waffles folder to open it up; then click the hero.jpg file to select it. Its URL should appear in the Location (URL) field. Click Insert to close the Select an Asset dialog.\n\n8. Optionally fill in the rest of the fields on the Edit Image Properties dialog, such as selecting an Image Rendition, adding Alternate text (which shows up when you hover the mouse over the image and is also used by screen readers as a description of the image), and a hyperlink to go to a different page if you'd like. In addition, you have the option to set some layout properties and resize the image if you want. For now, you can just click OK to add the image to the page and then click the Save button on the right side of the ribbon to see how your page is coming along.\n\n9. Open the page in Edit mode again by clicking the Edit button on the right side of the ribbon, where the Save button was previously.\n\n10. In Zone 1 on the page, click the Add a Web Part link. The Web Part menu opens.\n\n11. In the Categories section, scroll through the categories, and click Media and Content.\n\n12. In the Parts section, Content Editor should be selected. (If not, click it.) Click Add to add this Web Part to Zone 1. This Web Part will be used as a placeholder for now; you'll replace it later when you go through the examples in Chapter 9, \"Creating Content Rollups with SharePoint WCM.\"\n\n13. Hover over the Web Part's title, and click the drop-down arrow that appears at the far-right side of the Web Part. Select Edit Web Part from the menu.\n\n14. In the tool pane that opens on the right side of the page, expand the Appearance section, and change the Title field to Shop Our Sale Products. Click OK at the bottom of the tool pane.\n\n15. In the newly added Content Editor, click the link Click here to add new content and type some text in the Web Part.\n\n16. Repeat steps 10 to 15 to add a Content Editor Web part to Zone 2. Change the title of this Content Editor to Breakfast Talk. (This Web Part is also just a placeholder for now.)\n\n17. Click the Save button on the ribbon.\n\nIn a real-world scenario, you'd probably want to change the page layout of your site's actual home page (`default.aspx`) and just delete the Web Parts Microsoft has added to help you start. For this example, you may still want to keep the original home page intact for now if you're still new to SharePoint 2013 because it provides some useful links.\n\nNOTE\n\nWhen you are finished with the default links Microsoft has provided on the publishing site's default homepage, you can delete them all at once instead of one by one. To do this, simply append `?contents=1` to the end of the URL in the browser's address bar to open the Web Part maintenance for the current page. Then you can check the Select All check box to select all the Web Parts on the page, and click Delete to remove them.\n\nThe page layout now works as expected, and the master page looks decent. It's not perfect, but that will be fixed in the next chapter. It's now time to finish using the Design Manager by creating a design package.\n\nCreating a Design Package\n\nNow that you've (mostly) completed the design for your site, you can package up your changes into a design package using the Design Manager. A design package is a Windows Solution Package (WSP), sometimes called a SharePoint Solution, which contains your custom branding assets from the site and instructions on where those assets should be deployed. When a design is packaged up in a WSP, it can be imported to other SharePoint sites or farms, and the custom branding you've built can be applied to other sites. Generally, you wouldn't want to package up your design until it's complete and free of some of the style issues you're seeing. Although you won't be fixing some of the style issues (such as the search box and top navigation) until the next chapter, you can still walk through the process of packaging your design now so that you'll be familiar with it later.\n\nExample: Creating a Design Package\n\n1. In the Design Manager, click Create Design Package.\n\n2. In the Design Name field, SharePoint has provided a name for you; it uses the site's name by default. You can change it if you want. This becomes part of the filename for the design package. The other part is the version number of the design that appears directly under the name field. As SharePoint points out, you can always change this filename later, and it will still be recognized as the same design package. This is due to the unique identifiers included in the package.\n\n3. You have the option to include the site's search configuration in the design package. This can be useful if you've spent some time customizing the search experience in the site with custom query rules, result types, result sources, and other search-related assets. For this example, leave the box next to Include Search Configuration in This Package unchecked.\n\n4. Click Create to create your design package. You see a message that SharePoint is working on creating the package, and after a few moments, the Design Manager reappears with a link underneath the Create button.\n\n5. Click the Your package is ready. Click here to download link to save your WSP to your computer for importing into another site. Save your WSP file to a location on your computer, such as your desktop.\n\nKey Facts about the WSP\n\nThere are several things worth noting about the WSP. First, the WSP includes your custom assets only if they have been published. If a file is checked out to you, it won't be included in the WSP. In addition, any file that has been published but checked out again to be modified won't have the latest version. Only the last major published version of a file is included. For example, if your master page is checked in as version 2, and you check it out to make a few additional changes and check it in as minor version 2.1, the Design Manager includes only version 2.0 in the WSP.\n\nThe other thing you should know about creating a design package is that it includes all files that have been customized. This means that if you've created a page layout that uses a custom content type it will get the content type and all custom site columns. If you had previously applied a composed look to your site, the Design Manager includes several files related to that as well. This is just something to keep in mind as you create design packages so that you don't mistakenly package files that shouldn't be deployed.\n\nYou can actually open the WSP file in a few steps because it's really just a Windows cabinet file (.cab) with a different extension.\n\nExample: Exploring the Contents of a WSP File\n\n1. Find the WSP file you just saved on your computer.\n\n2. Right-click the file and select Rename.\n\n3. Replace `.wsp` with `.cab` as the file's extension. (Alternatively, you could juts append `.cab` after `.wsp`.) If you receive a warning about changing the file's extension, you can click Yes. The file's icon should change to the `.cab` file icon.\n\n4. Double-click the file to open it, and browse its contents. Scrolling through you can notice files and images related to your branding. You may also notice quite a few files with file extensions such as `.themedcss` and `.themedpng`. The Design Manager actually collects files related to composed looks and includes them as well if you applied a composed look to the site before starting your custom branding. If you've uploaded any files to a site assets library, the style library, and anything else you've included in the Master Page Gallery, they are in here as well.\n\nYou have little control over what the Design Manager includes in the WSP. However, you also had to do little to create the WSP. If you'd prefer to have more control over which files get deployed or to add custom behaviors when the branding is applied to the site (such as automatically switching the master page), it is possible to manually create a WSP using Visual Studio. You will learn more about creating custom branding packages in Chapter 7.\n\nLimitations of Design Packages\n\nThere are some things you need to consider when using the Design Manager to create a design package. For starters, the Design Manager only creates sandbox solutions. This is perfect for deploying code to Office 365 (where sandbox solutions are the only option), or between site collections, but for larger implementations that have more complex requirements, it might be limiting. Chapter 7 covers the differences between sandbox solutions and the other type, farm solutions.\n\nAnother thing to consider is that Design Manager is only available in sites with publishing enabled. It works well to create a design package and import it into another publishing site, but it can be a bit trickier if the intent is to use that branding in a site that doesn't have publishing. Although it is possible, some extra work needs to be done to make sure the design is compatible, and different steps need to be taken to apply the branding to the new site.\n\nThe design export functionality of Design Manager is a great way to quickly move a custom design to another site, but it isn't a silver bullet for every scenario. Chapter 7 provides alternatives, including creating a SharePoint design package from scratch. It is good to understand both approaches to determine which way is right for you.\n\nFinal Steps\n\nYou might be wondering what happens now that you've packaged up your design into a WSP. Well, for starters, you can import it to another site collection to apply your branding. If you want to try it out, you can try creating a new site collection based on the Publishing Site template to import the branding package.\n\nExample: Importing Your Design Package\n\n1. If you've just created a fresh publishing site collection, you can click the Import a design package link on the home page. If your site is already established and you've removed the default content from the home page, click the Settings menu \u21d2 Site settings \u21d2 Import Design Package (found under the Look and Feel header). Or if you remember all the way back to step 1 of the Design Manager, there's a link there to import a package as well.\n\n2. On the Import Design Package page, click the Browse button. Then browse your computer for your WSP file that you saved, and click OK.\n\n3. Click the Import button to begin importing your design package. SharePoint lets you know that it's working on it, and after it's finished, the page reappears with the design applied, just like magic.\n\nWhen you imported the design package, SharePoint added the WSP to the Solutions gallery, which you can access by clicking the Settings menu \u21d2 Site settings \u21d2 Solutions (found under the Web Designer Galleries header). In here, you can see your solution name and notice that its status is listed as Activated. Part of the import process is the automatic activation of the feature.\n\nSummary\n\n\u2022 The new SharePoint 2013 Design Manager was introduced to lower the bar of entry for SharePoint designers. The Design Manager can be used to convert static HTML files into SharePoint master pages.\n\n\u2022 You now have the ability to map a drive to access the HTML file and make changes to the look and feel of the master page. When you save the HTML file, SharePoint automatically updates the corresponding .master file. Just about any web design tool can be used to edit SharePoint 2013 master pages because you are editing an HTML file.\n\n\u2022 The Snippet Gallery enables you to add in snippets of SharePoint code for elements on SharePoint master pages and page layouts, such as the top navigation, left navigation, search box, and site image, among others.\n\n\u2022 The Design Manager can be used to create custom page layouts, which can also be edited in the web design tool of your choice.\n\n\u2022 SharePoint 2013 can export design files as a sandbox solution that can be imported and applied to other SharePoint locations.\n6\n\nCascading Style Sheets and SharePoint\n\nWhat's in this Chapter\n\n1. Understanding CSS\n\n2. How CSS Works in SharePoint\n\n3. Styling the SharePoint User Interface\n\n4. Styling Custom SharePoint Branding\n\nIn the last chapter you learned about features that Microsoft created to help with arranging and styling SharePoint sites, such as the Design Manager, master pages, page layouts, and composed looks. Unlike those features, Cascading Style Sheets (CSS) is one key area in which Microsoft has leveraged existing industry technology to provide you with a lot of power to change the look of a page. Behind the scenes SharePoint relies on a lot of CSS to make the user interface look like it does; there are more than 13,000 lines of uncommented CSS in SharePoint 2013 (up from 7000+ in SharePoint 2010). Being good at SharePoint branding often means having a thorough understanding of CSS; there is no other technology that makes a more profound impact on the rendered interface.\n\nThis chapter is not intended as a reference manual for coding CSS\u2014to do so would take too many pages. Instead, you get a brief overview of some of the key CSS concepts. If you are experienced with CSS, you may want to still review this chapter because SharePoint tends to push the limits of normal CSS expertise. Unlike when you work on your own websites, in many cases with SharePoint branding you need to override HTML and CSS written by engineers at Microsoft whose first priority is performance, not your ability to easily restyle their markup. If you are new to CSS or feel your skills aren't as strong as they need to be, be sure to study this section carefully. There is a list of more in-depth resources for learning CSS at the end of the following section; read through them to learn more about CSS before diving into more intricate SharePoint styling.\n\nA Brief CSS Primer\n\nCSS is a style sheet language that tells web browsers how to present a document, whether that document is HTML or any technology based on XML. CSS was created as a way to separate the presentation from the content or data in a document. This split allows for much greater reuse of styles and permits an easier and more maintainable editing process than the old way of adding presentation directly to each page's HTML markup. In many ways CSS is used in SharePoint exactly the same as any other website; this is because SharePoint uses ASP.net to ultimately render HTML, which displays when a page is visited.\n\nNOTE\n\nCSS has gone through a few iterations since it was introduced more than 15 years ago. Modern browsers (as of this writing) primarily support CSS3 and all versions prior, although many times browser manufacturers disagree on the appropriate response for a given CSS3 code snippet. Because of this, it's a good idea to test all your CSS in a few different browsers. SharePoint 2013 makes use of several newer CSS3 concepts, which is likely the reason why versions of Internet Explorer prior to IE8 are not fully supported for SharePoint 2013.\n\nTo understand CSS properly, you must start with the basic CSS rule syntax. Rules control how specific HTML elements will be styled by CSS. At first glance CSS seems simple; the rules are based on only two things: a selector and a declaration. Don't be fooled by this simplicity; creating pixel-perfect web design with CSS can be challenging.\n\nSelector { Declarations }\n\nDeclarations are surrounded by braces (or curly brackets) and are divided into two parts, Properties and Values, which are separated by a colon. Multiple declarations are separated by a semicolon. Properties represent the visual attribute of the element that you want to style based on the setting specified by the Value.\n\nSelector { Property: Value; Property: Value; }\n\nNOTE\n\nYou can omit the final semicolon after the last declaration in a rule, but many designers include it for ease of maintenance. Adding the last semicolon is helpful when you make quick additions to CSS; this way you don't need to remember to add one back before adding new declarations.\n\nKeep in mind that the examples in this section of the chapter are specifically talking about traditional web pages, not CSS as it would apply in an actual SharePoint site where there are other factors that would affect the results.\n\nCSS\n\n`p, h1 { font-family: \"Segoe UI\"; color: Red; `\n\n` } `\n\nHTML\n\n`<div>`\n\n` <h1>Waffles<\/h1>`\n\n` <p>Sweet delicious waffles<\/p>`\n\n` I would eat them every day if I could.`\n\n`<\/div>`\n\nThe CSS rule can cause all <p> and <h1> tags to render in the Segoe UI font with a red color. Note that font family values that include more than one word, like Segoe UI here, should be surrounded by quote marks.\n\nResult\n\nWARNING\n\nA quick word about case-sensitivity and CSS: The W3C says: \"The case-sensitivity of document language element names in selectors depends on the document language. For example, in HTML, element names are case-insensitive, but in XML they are case-sensitive.\" Because of this inconsistency, it's often a good idea to code your CSS with case-sensitivity in mind.\n\nFor more information on this topic see SitePoint's article on case-sensitivity: `http:\/\/reference.sitepoint.com\/css\/casesensitivity`.\n\nCSS\n\n`span { color: red; }`\n\n`.myClass { color: green; }`\n\n`#myID { color: purple; }`\n\n`.myClass2 a { color: orange; }`\n\n`.myClass2 > a { color: lightblue; }`\n\n`.myClass3 + div { color: blue; } `\n\n`.myClass4 ~ div { color: lime; }`\n\n`* { color: silver; }`\n\nHTML\n\n`<span>This Type selector will be Red<\/span> `\n\n`<div class=\"myClass\">`\n\n` This Class selector will be Green`\n\n`<\/div>`\n\n`<div id=\"myID\">`\n\n` This ID selector will be Purple`\n\n`<\/div>`\n\n`<div class=\"myClass2\">`\n\n` <a href=\"#\">This is a Child of myClass2, it will be Light Blue<\/a>`\n\n` <div>`\n\n` <a href=\"#\">This is a Descendent of myClass2 but not a direct child, it will be Orange<\/a>`\n\n` <\/div>`\n\n`<\/div>`\n\n`<div class=\"myClass3\"><\/div>`\n\n`<div>This is Adjacent to myClass3 so it will be Blue<\/div>`\n\n`<div>This is not Adjacent to myClass3. It is nothing but it will be styled with the Universal selector, it will be Silver<\/div>`\n\n`<div class=\"myClass4\"><\/div>`\n\n`<div>This is a General Sibling of myClass4 so it will be Lime<\/div>`\n\n`<div>This is also a General Sibling of myClass4 so it will be Lime<\/div>`\n\nResult\n\nPseudo Classes\n\nYou can expand upon the basic set of selectors by using pseudo classes to apply styles based on the state of specific elements. For example, with pseudo classes you can apply style based on whether a link has been visited previously or if the mouse hovers over an element. This table shows some of the most common pseudo classes you can find in SharePoint.\n\nAttribute Selectors\n\nAttribute Selectors were added with CSS2 and expanded in CSS3. They enable rules to match HTML elements that have specific attributes defined. For example, an attribute selector could match a specific type of `<input>` element such as `<input type=\"submit\">`.\n\nCSS\n\n`div[title] { color: red; }`\n\n`div[title=\"myTitle\"] { color: green; }`\n\n`div[title~=\"myTitle2\"] { color: purple; }`\n\n`div[lang|=\"en\"] { color: orange; }`\n\n`div[title*=\"waffle\"] { color: lightblue; }`\n\n`div[title^=\"waffles\"] { color: blue; } `\n\n`div[title$=\"waffles\"] { color: lime; }`\n\nThis shows how the concepts from the preceding table would look in a modern web browser.\n\nHTML\n\n`<div title=\"\">This will be Red<\/div>`\n\n`<div title=\"myTitle\">This will be Green<\/div>`\n\n`<div title=\"myTitle2 myTitle3\">This will be Purple<\/div>`\n\n`<div lang=\"en-us\">This will be Orange<\/div>`\n\n`<div title=\"I like waffles too much\">This will be Light Blue<\/div>`\n\n`<div title=\"waffles are great\">This will be Blue<\/div>`\n\n`<div title=\"My favorite meal is waffles\">This will be Lime<\/div>`\n\nResult\n\nUnderstanding CSS Specificity\n\nIf more than one rule applies to an element, a set of guidelines decides which style to apply. This concept is important in SharePoint because you often need to override existing out-of-the-box CSS rules to apply your own style to SharePoint.\n\nTypically, if a selector has more than one rule applied to it, the most specific rule takes precedence and has its styles applied to the selector. This concept is known as specificity and is probably one of the most challenging things to understand in CSS. The following details all the different rules and lists them in order of specificity.\n\n\u2022 The universal selector `*` is the least specific.\n\n\u2022 Elements such as `H1` are more specific.\n\n\u2022 If a rule has more than one selector, it is more specific.\n\n\u2022 Class selectors are more specific than elements.\n\n\u2022 ID selectors are more specific than class selectors.\n\n\u2022 Inline styles are styles applied directly to an HTML element (rather than assigned from an internal or external style sheet) and are the highest normal specificity.\n\n\u2022 Applying `!important` to the end of an attribute trumps everything else, always taking precedence unless a later `!important` attribute overrides it.\n\nIf you want to dig into the specifics of specificity (try saying that ten times fast) you can use the following equation to calculate specificity:\n\n(If there is an inline style applied add 1000)\n\n+\n\n(The number of ID selectors) x 100\n\n+\n\n(The number of Class selectors) x 10\n\n+\n\n(The number of HTML element selectors) x 1\n\nThis gives you the specificity of a given selector; the selector that is most specific applies its style. If two selectors are otherwise equivalently specific, the selector that comes last has its style applied to the element. Adding `!important` to an attribute immediately changes this equation, making that attribute the most important style to apply. For example, in the following CSS, the first style takes precedence because it has a specificity of 11, whereas the second has a specificity of 1.\n\nCSS:\n\n`<style type=\"text\/css\">`\n\n` .myClass div { color: blue; } `\n\n` div { color: red }`\n\n`<\/style>`\n\nHTML:\n\n`<div class=\"myClass\">`\n\n` <div>What color will this be?<\/div>`\n\n`<\/div>`\n\nResult\n\nNOTE\n\nOften when working with CSS and SharePoint, you run into instances in which a simple CSS rule won't apply the style you want to an element. Typically in these cases, the solution is to try to make your CSS rule more specific by adding parent classes or an `!important` to the end of the declaration.\n\nInheritance\n\nCertain CSS properties (like fonts and colors) pass on their value to child elements. This process is known as inheritance, and it allows for smaller CSS files because you don't need to manually call out simple settings for every element in your HTML file. For example, you can set the font-family for the `body` tag, and it is passed on to all the child elements automatically.\n\nNot all CSS properties pass on inherited values to their children. For a full list of which CSS properties have inheritance, see the W3C article, \"Cascading Style Sheets Level 2 Revision 1 (CSS 2.1) Specification \u2013 Appendix F\" at http:\/\/www.w3.org\/TR\/CSS2\/propidx.html. If a property doesn't have inheritance though, you can often force specific child elements to inherit the value of the parent by adding the `inherit` value in CSS.\n\nCSS\n\n`div {border: 1px solid red;}`\n\n`p {border: inherit;}`\n\nHTML\n\n`<div>`\n\n` <p> Help, I'm trapped inside a box inside another box!<\/p>`\n\n`<\/div>`\n\nResult\n\nApplying CSS to a Web Page\n\nIn standard web development you can apply CSS to an HTML page in three ways:\n\n\u2022 Inline Styles\u2014CSS is applied to elements with the style attribute defined inside individual HTML elements. This option should be avoided in most scenarios because of the specificity rules described earlier; inline styles remove the ability to easily override styles without using the `!important` tag.\n\n`<div style=\"color: red;\">Hey I'm going to be red<\/div>`\n\n\u2022 Internal Style Sheet\u2014CSS is added directly to the HTML file via the `<style>` tag, typically in the `<head>` of a document. This is a good option for adding styles to individual pages, but ultimately this method lacks the benefit of reuse and ease of maintenance. To reuse these styles you would have multiple versions of the internal style sheets applied to multiple pages.\n\n`<head>`\n\n` <style type=\"text\/css\">`\n\n` div {color: red;}`\n\n` <\/style>`\n\n`<\/head>`\n\n\u2022 External Style Sheet\u2014CSS lives in its own file and is loaded into an HTML page with the `<link>` tag in the `<head>` section of the HTML. This is the best option for applying a lot of styles across an entire site.\n\n`<head>`\n\n` <link rel=\"stylesheet\" type=\"text\/css\" href=\"style.css\" \/>`\n\n`<\/head>`\n\nAdditional Resources for Traditional CSS\n\nThe first part of this chapter provides a crash course in some of the basics of CSS, but it can't do justice to all that there is to know about the topic. The following is a selection of good websites for learning more about CSS:\n\n\u2022 W3C\u2014www.w3.org\/Style\/CSS\/\n\n\u2022 W3Schools\u2014http:\/\/w3schools.com\/css\/)\n\n\u2022 Mozilla Developer Network\u2014https:\/\/developer.mozilla.org\/en-US\/docs\/CSS\/Getting_Started\n\nIf you would rather have an actual book by your side, here are some good options:\n\n\u2022 Beginning HTML and CSS (Wrox, 2013)\n\n\u2022 Beginning CSS: Cascading Style Sheets for Web Design (Wrox, 2011)\n\nCSS Inside of SharePoint\n\nIf you are comfortable with CSS as it applies to traditional web design, you may wonder how CSS is different in SharePoint. It was mentioned earlier that SharePoint has a lot of CSS applied to it by default. You can learn a lot about how to style SharePoint just by looking at the default SharePoint CSS. Take a moment and view the source of the main SharePoint CSS file `COREV15.CSS` by opening it in a text editor (from the server in the SharePoint 15 folder under `\\TEMPLATE\\LAYOUTS\\1033\\STYLES\\Themable\\COREV15.CSS`).\n\nWhen you open `COREV15.CSS` in a text editor such as Notepad++, you can quickly see that there are a lot of lines of CSS with few helpful comments. You can also notice that a large portion of the CSS rules apply to classes that start with `.ms-`. Although there are some other class prefixes such as `.s4-`, they are in the minority. These prefixes can be helpful when you inspect the CSS of a page because you can scan for `.ms-` classes quickly and see styles that you know were created by Microsoft. You will also notice that there are many child and descendent selectors used. When you try to override a style rule, you want to make sure your rule is at least as specific as the original, or it won't apply to these elements.\n\nAnother thing that jumps out immediately when looking at `COREV15.CSS` is that there are many comments that look like this:\n\n`\/* [ReplaceFont( themeFont:\"body\")] *\/`\n\nAs you may have guessed, these aren't your typical CSS comments; instead, they are related to composed looks, which is the new version of SharePoint 2010 themes. When a composed look is applied, these comments tell SharePoint to change the attribute of the CSS that immediately follows the comment. You can see more about creating composed looks and using these comments in your own custom CSS in Chapter 10, \"Composed Looks and Custom Branding,\" but for now just know that when a composed look is applied to SharePoint, many of the default colors, fonts, and even background images will change from how they are defined in the SharePoint CSS file.\n\nModern CSS in SharePoint\n\nSharePoint 2013 takes advantage of some modern CSS techniques, including CSS3, vendor prefixes, and media types. Some common CSS3 techniques that you see in SharePoint 2013 include: box-shadow to apply a shadow behind elements; border-radius to round the corners of elements; word-wrap to break long words onto a new line; text-overflow to show ellipses when text would overflow out of its container; and RGBa colors, which define the red, green, and blue values (0-255) and also a decimal number (0-1) to apply alpha transparency to the color, like this:\n\n`background-color:rgba( 255,255,255,0.85 ) `\n\nMicrosoft also uses vendor prefixes in its CSS to push the envelope of web design and take advantage of features that some modern browsers support beyond the agreed upon CSS specification. You can see vendor prefixes that start with specific prefixes, for example: Microsoft (`-ms-`), Mozilla browsers (`-moz-`), WebKit browsers (`-webkit-`), and Opera (`-o-`). Some common uses for vendor prefixes in SharePoint 2013 are for gradient fills, alpha fills, and rotation.\n\nNOTE\n\nFor more information on these advanced techniques, check out these resources:\n\nCSS3.Info\u2014CSS3 Previews at `http:\/\/www.css3.info\/preview\/`\n\nWC3\u2014Media types at `http:\/\/www.w3.org\/TR\/CSS2\/media.html`\n\nAbout.com\u2014CSS Vendor Prefixes at `http:\/\/webdesign.about.com\/od\/css\/a\/css-vendor-prefixes.htm`\n\nTools for Working with CSS\n\nBesides using dedicated web design tools such as Dreamweaver and Expression Web, browser-based CSS tools are some of the most powerful ways to work with large amounts of CSS. The following tools can help you analyze a SharePoint site and identify the CSS classes so that you can override them. In general, these tools can often be launched via the F12 key when browsing a website:\n\n\u2022 Internet Explorer F12 Developer Tools\u2014Named after the key combination that opens it, this plug-in was previously a separate download for IE7 called IE Developer Toolkit. Now it is included with every Internet Explorer.\n\n\u2022 Firebug for Firefox\u2014A third-party add-on for Firefox that can be downloaded from `www.getfirebug.com`.\n\n\u2022 Chrome Developer Tools\u2014Included with modern versions of the Chrome browser.\n\nOne useful feature of these tools is the capability to point to an area of the rendered page inside the browser to see a breakdown of what CSS is being applied to that area and how CSS rules are being overridden. Because of minor differences between the ways that different browsers render pages, you may need to use each tool to truly understand all the styles applied to a SharePoint site. After your selected style overrides are identified, you can add them into a custom CSS file and apply them to SharePoint to make the overrides permanent.\n\nExample: Using Firebug to Work with CSS\n\nFor this example you use Firebug for Firefox to change the header of the team site with the out-of-the-box `seattle.master` applied to it. The concept is similar for the other browser-based CSS tools.\n\n1. Ensure that the Firebug add-on has been installed in Firefox.\n\n2. Open Firefox and browse to your SharePoint site.\n\n3. Press F12 or click Tools \u21d2 Web Developer \u21d2 Firebug \u21d2 Open Firebug to open Firebug.\n\n4. Click the inspect icon at the top of the Firebug panel or right-click on an element inside your page and select Inspect Element with Firebug.\n\n5. Try to highlight the entire header portion of the page (the horizontal area below the ribbon that contains the site icon, navigation, and site name). One easy way to select the entire header is to click the mouse to the left of the SharePoint site icon. After clicking the area, you can see (on the right side of the Firebug window) that the CSS class #`s4-titlerow` has no background-color applied yet.\n\n6. Firebug enables you to manipulate the CSS values to immediately see the results. To try this, make sure #`s4-titlerow` is still selected, and on the right panel, click `64px` next to `height`. Press the up and down arrows on your keyboard to see the height changes in real time in the browser.\n\n7. To add a new CSS declaration, double-click the selector's closing bracket and a new row will be added. Then type a new rule, such as `background-color: silver;`. Notice that as you type values Firebug tries to guess what you are going to type based on the available options.\n\nAny changes made here are only temporary; they disappear when the browser is refreshed. You can, however, select classes in the right panel and copy and paste them into your own custom CSS to make them permanent.\n\nApplying Custom CSS to SharePoint\n\nIn SharePoint 2013 applying custom CSS to a page works similarly to how you would do it for traditional HTML pages (discussed earlier in the \"Applying CSS to a Web Page\" section). If you want to include any of these types of CSS, you can add them manually to master pages and page layouts, and in some cases you can even add references to them within the page content. However, SharePoint also provides its own mechanisms for adding CSS that you can see in the next few sections.\n\nWARNING\n\nOne common mistake is to just log in to the server and find and edit the core SharePoint CSS file in the SharePoint root. This should be avoided because it is problematic for supportability. If Microsoft releases an update for SharePoint (which is inevitable), the `COREV15.CSS` file in the SharePoint root is likely to be overwritten. Also, changes to this file will apply branding to all web applications on the server, which is probably not desirable.\n\nAdding CSS to Page Content\n\nIf you need to add an internal style sheet to a single page in SharePoint, it would be tempting to put the page in edit mode and, from the ribbon, click Format Text \u21d2 Edit Source, or even to use a Content Editor Web Part. However, SharePoint's handling of both of these is quirky, depending on which edition of SharePoint you have (Server or Foundation) and whether you have a publishing or team site.\n\nSharePoint 2013 includes some more elegant solutions for adding CSS to a rich text field or a Web Part zone. Adding CSS to a rich text field is as simple as putting the page in edit mode and, from the ribbon, clicking Insert \u21d2 Embed Code. This opens a dialog box where you can add CSS that will be applied to the page when you click Insert. For Web Part zones you can use the new Script Editor Web Part (SEWP), which enables content authors to add HTML snippets or scripts to the page, and it's also a good option to add internal style sheets easily. One common page level CSS change that people find useful with the SEWP is hiding the Quick Launch navigation on the left side of the default SharePoint UI.\n\nExample: Hiding the Quick Launch with the Script Editor Web Part\n\n1. Open a SharePoint page in Edit mode.\n\n2. Find a Web Part Zone and click Add a Web Part.\n\n3. When the Web Part menu opens at the top of the page, click Categories \u21d2 Media and Content \u21d2 Script Editor \u21d2 Add.\n\n4. When the Script Editor Web Part is added to the page, click the link on the right side called Edit Snippet. Sometimes the link won't show until you first click Edit Web Part from the drop-down at the top right of the Web Part.\n\n5. When the dialog box opens, add the following code:\n\n`<style type=\"text\/css\">`\n\n`#sideNavBox {`\n\n` display: none;`\n\n`}`\n\n`#contentBox {`\n\n` margin-left: 20px;`\n\n`}`\n\n`<\/style>`\n\n6. Click the Insert button on the right of the dialog box.\n\n7. You should immediately see that Quick Launch is hidden. You can save the page and publish so that other users can see the change.\n\nUsing Alternate CSS\n\nAlternate CSS is a feature that enables you to quickly and easily apply custom CSS to a particular site and all its subsites. Alternate CSS is available only from the SharePoint web user interface in a SharePoint Server site with publishing activated. That said, the same techniques could be used in a nonpublishing site with custom code or a Windows PowerShell command.\n\nThe following steps describe how to add an external style sheet to a SharePoint Server 2013 publishing site using Alternate CSS. Before trying these steps, be sure you have a custom CSS file added to your SharePoint site in a subfolder under the master page gallery. If you need a refresher on adding a CSS file, see the \"Uploading Design Files\" section in Chapter 5, \"Using the Design Manager to Start a Design in SharePoint.\"\n\nExample: Using Alternate CSS\n\n1. Load the SharePoint 2013 site in your web browser and navigate to Site Settings \u21d2 Look and Feel \u21d2 Master Page.\n\n2. To test with the default SharePoint UI, make sure Site and System master pages are both set to `seattle.master`; then expand the section titled Alternate CSS URL.\n\n3. Next, choose Specify a CSS file to be used by this site and all sites that inherit from it, and then click the Browse button.\n\n4. From the Select an Asset dialog box, navigate to your custom CSS file, and select it and click Insert. If you placed your CSS in the master page gallery you may have to type \/_catalogs\/masterpage in the Location (URL) box and press Enter to start browsing in the correct location.\n\n5. If you want to apply the style sheet to all subsites, you can check Reset all subsites to inherit this alternate CSS URL.\n\n6. From the master page menu, click OK, and the site refreshes with the CSS styles applied to it.\n\nAdding External Style Sheets with CssRegistration\n\nIf you view the source of the default SharePoint 2013 master page `seattle.master`, you can see a reference to `<SharePoint:CssRegistration>` that points to `COREV15.CSS`. This is how SharePoint loads the primary CSS that is applied to most of the default user interface. The same control is used to add your own custom CSS to override `COREV15.CSS`.\n\nWhen you use the Design Manager to make an HTML master page like you did in the previous chapter, SharePoint converts your external CSS links behind the scenes into valid `<SharePoint:CssRegistration>` controls that point to the custom CSS files in the Master Page Gallery.\n\nFor example, if your HTML file has the following reference:\n\n`<link rel=\"stylesheet\" type=\"text\/css\" href=\"style.css\" \/>`\n\nThe corresponding master page would look like this:\n\n`<SharePoint:CssRegistration name=\"&lt;% $SPUrl:~sitecollection\/`\n\n` _catalogs\/masterpage\/DemoBranding\/style.css %&gt;\" runat=\"server\" `\n\n` after=\"SharepointCssFile\" \/>`\n\nSharePoint knows where the CSS file is located in the master page gallery (`_catalogs\/masterpage\/DemoBranding`) and makes a site collection relative reference. The `After` property is used to add the CSS link after the `COREV15.CSS` file to ensure your custom CSS always loads after the out-of-the-box CSS. By applying the CSS with `CssRegistration`, the master page and CSS are always applied together.\n\nYou will learn more about using the `CssRegistration` control in master pages in Chapter 7, \"Creating SharePoint Branding.\"\n\nEditor Styles\n\nMuch like the previous version, SharePoint 2013 enables you to add specific styles to your custom CSS that can be accessed through the ribbon and used by content authors on your site. In this section, you learn about adding styles to the following areas of the Format Text tab on the ribbon:\n\n\u2022 Font Face\u2014The font face used on the selected text.\n\n\u2022 Font Size\u2014The size of the selected text.\n\n\u2022 Highlight Color\u2014Background color behind the selected text.\n\n\u2022 Font Color\u2014The selection text color.\n\n\u2022 Styles \u21d2 Page Element\u2014An HTML element will be injected around the selection and a style will be applied to that element.\n\n\u2022 Styles \u21d2 Text Style\u2014A specific style will be applied to the selection.\n\nFor these styles, the selector always begins with `.ms-rte` (for Rich Text Editor), then the ribbon area that is being updated (like `FontFace`), followed by a dash and a custom name that you provide. For example, `.ms-rteFontFace-WafflesFontFace` adds a new `FontFace` to the ribbon. The declaration often includes a special Microsoft code such as `-ms-name` to tell the ribbon how the item should be named when the user hovers over visual options, such as `-ms-name:\"Custom Comic Sans\";`.\n\nThe following CSS shows some example editor styles. If this CSS were added to your SharePoint site, the rich text editor in the ribbon would update to include these new style options. The table that follows the code shows how the CSS would work and look, from top to bottom.\n\n`\/*--Font Face--*\/`\n\n`.ms-rteFontFace-WafflesFontFace {`\n\n` -ms-name:\"Custom Comic Sans\";`\n\n` font-family: 'Comic Sans MS';`\n\n`}`\n\n`\/*--Font Size--*\/`\n\n`.ms-rteFontSize-WafflesFontSize {`\n\n` font-size: 50px;`\n\n`}`\n\n`\/*--Highlight Color--*\/`\n\n`.ms-rteBackColor-WafflesBackColor {`\n\n` background-color: purple;`\n\n` -ms-name: \"\";`\n\n` -ms-color:\"CustomPurple\";`\n\n`}`\n\n`\/*--Font Color--*\/`\n\n`.ms-rteForeColor-WafflesForeColor {`\n\n` color: lime;`\n\n` -ms-name: \"\";`\n\n` -ms-color:\"CustomLime\";`\n\n`}`\n\n`\/*--Styles - Page Element--*\/`\n\n`DIV.ms-rteElement-WafflesElement {`\n\n` -ms-name:\"My Custom Element\";`\n\n`}`\n\n`.ms-rteElement-WafflesElement {`\n\n` border: 1px solid red;`\n\n`}`\n\n`\/*--Styles - Text Styles--*\/`\n\n`.ms-rteStyle-WafflesStyle {`\n\n` -ms-name:\"CustomStyle\";`\n\n` color: lime;`\n\n` font-size: 50px;`\n\n`}`\n\nFont\n\nSelector\n\n`.ms-rteFontFace-WafflesFontFace`\n\nDeclaration Description\n\n`-ms-name:\"Custom Comic Sans\";`\n\nThe name that shows in the font options in the ribbon.\n\n`font-family: 'Comic Sans MS';`\n\nThe font applied to the selection.\n\nFont Size\n\nSelector\n\n`.ms-rteFontSize-WafflesFontSize`\n\nDeclaration Description\n\n`font-size: 50px;`\n\nThe font size applied to the selection.\n\nNOTE\n\nNo `-ms-name` is used for this because the ribbon just shows the option.\n\nHighlight Color\n\nSelector\n\n.ms-rteBackColor-WafflesBackColor\n\nDeclaration Description\n\n`background-color: purple;`\n\nThe background color applied to the selected text.\n\n`-ms-color:\"CustomPurple\";`\n\nThe name that shows when the user hovers over the option on the ribbon.\n\n`-ms-name: \"\";`\n\nWhatever text is listed here shows after the -ms-color name. Behind the scenes Microsoft uses this to tell you how the color relates to the theme option it might represent.\n\nFont Color\n\nSelector\n\n`.ms-rteForeColor-WafflesForeColor`\n\nDeclaration Description\n\n`color: lime;`\n\nThe color applied to the selected text.\n\n`-ms-color:\"CustomLime\";`\n\nThe name that shows when the user hovers over the option on the ribbon.\n\n`-ms-name: \"\";`\n\nWhatever text is listed here shows after the -ms-color name. Behind the scenes Microsoft uses this to tell you how the color relates to the theme option it might represent.\n\nPage Element Style\n\nSelector\n\n`DIV.ms-rteElement-WafflesElement`\n\nDeclaration Description\n\n`-ms-name:\"My Custom Element\";`\n\nThe name that shows in the option on the ribbon.\n\nNOTE\n\nThis one works a little differently than the others: The first selector has an HTML element before it. (In the example this is DIV.) When this option is selected from the ribbon, SharePoint injects this HTML element around the selected text and then applies the style declared next to that element.\n\nSelector\n\n`.ms-rteElement-WafflesElement`\n\n`border: 1px solid red;`\n\nDeclaration Description\n\nThis is the style applied to the element defined in the previous item.\n\nText Style\n\nSelector\n\n`.ms-rteStyle-WafflesStyle`\n\nDeclaration Description\n\n`-ms-name:\"CustomStyle\";`\n\nThe name that shows in the option on the ribbon.\n\n`color: lime;`\n\n`font-size: 50px;`\n\nAny styles listed are applied to the selected element.\n\nStyling Key Areas of SharePoint\n\nIt's been said before, there is a lot of CSS in SharePoint 2013, and because of that, there are a lot of elements that need to be styled in any given SharePoint branding project. The following sections describe the most common areas you might want to style with CSS. Think of this section as an X-ray that shows you the underlying CSS behind what appears on the page.\n\nNOTE\n\nMany styles have several selectors and are specific. It is impossible to cover all the styles in SharePoint 2013, but you can use tools such as Firebug for Firefox, which was discussed earlier in this chapter, to figure out any complex styling challenge with some effort and a good bit of patience.\n\nBody and Ribbon\n\nNOTE\n\nThe body of a SharePoint page can be confusing. You can certainly apply styling to the body element tag and have it show for the entire page. However, because of how the ribbon applies to SharePoint, you can also think of the area below the ribbon as the \"body\" of the page because it is the main scrollable container for all non-ribbon content. This area has an ID of `s4-workspace` applied to it.\n\nSearch Box\n\nTop Navigation\n\nTop Navigation in Edit Mode\n\nQuick Launch or Left Navigation\n\nNOTE\n\nQuick Launch navigation works essentially the same as Top Navigation but instead of being prefixed with `.ms-core-listMenu-horizontalBox`, it is prefixed with `.ms-core-listMenu-verticalBox`.\n\nSite Logo, Title, and Description\n\nWeb Part Zone and Web Part\n\nPutting CSS to Work in SharePoint\n\nNow that you have a better understanding of how the out-of-the-box styles work in SharePoint 2013, you can apply this knowledge to an actual SharePoint site. If you have worked through the examples in Chapter 5, you remember converting the Randy's Waffles HTML design into a functioning master page using the Design Manager and snippets.\n\nWARNING\n\nIf you haven't finished the Randy's Waffles example from the previous chapter, the following examples may not make a lot of sense. If you want to jump directly into the examples anyway, you can download the finished files for Chapter 5 from the companion website for this book on Wrox.com. But be sure to see the section \"Uploading Design Files\" in Chapter 5 to learn how to upload the files into the Master Page Gallery. If you'd rather not type each of these examples, you can copy them from the finished code included with this chapter's download files.\n\nThis master page was functional, but it wasn't very pretty. You can update the Randy's Waffles CSS to appear more aesthetically pleasing in SharePoint. To make the necessary updates, use a mapped drive to your master page gallery and open the Randy's Waffles style sheet at `\/_catalogs\/masterpage\/waffles\/style.css` in your editor. The following sections guide you through which changes to make and how. After each CSS listing, a screenshot is shown of how each change would affect the result in a browser.\n\nBackground Image\n\nThe first thing you might notice about the converted HTML master page is that the background image isn't showing properly. This has to do with a unique style that SharePoint adds to the markup for use with composed looks. By default, without a composed look applied, CSS is set to override the body background with a blank image. To add the wood grain background image back, add the following line below the `body` styles in the style sheet:\n\n`.ms-backgroundImage {`\n\n` background-image: url(bg.jpg) !important;`\n\n`}`\n\nSearch Box\n\nTo make the search box look like the original design, you need to add the rounded box background image behind it, hide the borders and background color of the input box, and hide and replace the original search button. You can replace the entire search section of the style sheet with this code.\n\n`\/* add rounded background behind search box *\/`\n\n`.customSearchHolder {`\n\n` background: url(\"searchbox.png\") no-repeat scroll center top transparent;`\n\n` float: right;`\n\n` height: 38px;`\n\n` position: relative;`\n\n` right: 0;`\n\n` top: 50px;`\n\n` width: 246px;`\n\n`}`\n\n`\/* style input box *\/`\n\n`.customSearchHolder input {`\n\n` background: none repeat scroll 0 0 transparent;`\n\n` border: medium none;`\n\n` color: #585858;`\n\n` margin: 0 0 0 16px;`\n\n` padding-top: 10px;`\n\n` width: 173px;`\n\n` font-weight:normal;`\n\n`}`\n\n`\/* box around search *\/`\n\n`.customSearchHolder .ms-srch-sb {`\n\n` background: none no-repeat scroll 0 0 transparent;`\n\n` margin-top: 0px;`\n\n` width: 226px;`\n\n`}`\n\n`\/* search box border *\/`\n\n`.customSearchHolder .ms-srch-sb-border, .customSearchHolder .ms-srch-sb-borderFocused {`\n\n` border-style: none;`\n\n`}`\n\n`\/* add new search button image *\/`\n\n`.customSearchHolder .ms-srch-sb > .ms-srch-sb-searchLink {`\n\n` background: url(\"search-button.png\") no-repeat scroll left top transparent;`\n\n` height: 18px;`\n\n` left: 5px;`\n\n` margin: 0;`\n\n` padding: 0;`\n\n` position: relative;`\n\n` top: 10px;`\n\n` width: 18px;`\n\n` float: right;`\n\n`}`\n\n`\/* hide the original search button *\/`\n\n`.customSearchHolder .ms-srch-sb-searchImg {`\n\n` display: none;`\n\n`}`\n\n`\/* hide search dropdown *\/`\n\n`.customSearchHolder .ms-srch-sb > .ms-srch-sb-navLink { `\n\n` top: 10px;`\n\n` display: none;`\n\n}\n\nTop Navigation\n\nThe top navigation consists of a few different states; you can start with the main navigation holder and items. You must style the holder, UL, LIs, links, the link selected state, and the link hover state, and you also need to hide the drop-down arrow that shows for items that have subnavigation flyouts.\n\nLike the previous section, just replace the entire top navigation section of the style sheet with this new CSS:\n\n`\/* add style behind top navigation *\/`\n\n`.customTopNavHolder {`\n\n` height: 35px;`\n\n` background-image: url('nav-bg.png');`\n\n` background-repeat: repeat;`\n\n` padding-left: 23px;`\n\n`}`\n\n`\/* top nav ul *\/`\n\n`.customTopNavHolder ul {`\n\n` margin: 0px;`\n\n` list-style-type: none;`\n\n` padding: 0px;`\n\n`}`\n\n`\/* blue divider *\/`\n\n`.customTopNavHolder .root > li {`\n\n` border-right: 2px solid #66a2c4;`\n\n`}`\n\n`\/* no border on edit link *\/`\n\n`.ms-listMenu-editLink {`\n\n` border-right: 0px none !important;`\n\n`}`\n\n`\/* top nav item *\/`\n\n`.customTopNavHolder .static > li > .ms-core-listMenu-item {`\n\n` font-family: 'ArvoRegular',arial,sans-serif;`\n\n` font-size: 15px;`\n\n` height: 33px;`\n\n` line-height: 31px;`\n\n` padding: 0 15px;`\n\n` margin-right: 0px !important;`\n\n` color: white;`\n\n`}`\n\n`\/* selected *\/`\n\n`.customTopNavHolder .root > li > .ms-core-listMenu-selected {`\n\n` color: #403a35;`\n\n` background-color: white;`\n\n`}`\n\n`\/* top nav hover *\/`\n\n`.customTopNavHolder a.ms-core-listMenu-item:hover {`\n\n` text-decoration: none; `\n\n` background-color: white;`\n\n` color: #403a35;`\n\n`}`\n\n`\/* top nav selected hover *\/`\n\n`.customTopNavHolder .ms-core-listMenu-selected:hover {`\n\n` color: #403a35 !important;`\n\n` background-color: white !important;`\n\n`}`\n\nNext, style the flyout holder, flyout items, and their hover state. You see these only if you have subnavigation flyout items for your site. Add the following lines to the previously added navigation styles:\n\n`\/* flyout arrow *\/`\n\n`.customTopNavHolder .additional-background {`\n\n` padding-right: 0px !important;`\n\n` background-image: none !important;`\n\n`}`\n\n`\/* flyout holder *\/`\n\n`.customTopNavHolder ul.dynamic {`\n\n` min-width: 200px;`\n\n` border:0px none;`\n\n` padding-top:1px;`\n\n` margin-left: 0px;`\n\n` margin-top: 1px;`\n\n`}`\n\n`\/* flyout item *\/`\n\n`.customTopNavHolder ul.dynamic .ms-core-listMenu-item {`\n\n` display: block;`\n\n` border: 1px solid #bbbbbb;`\n\n` border-top: 0px;`\n\n` color: #827C70;`\n\n` font-family: 'ArvoRegular',arial,sans-serif;`\n\n` font-size: 14px;`\n\n` padding: 10px;`\n\n`}`\n\n`\/* flyout item hover *\/`\n\n`.customTopNavHolder li.dynamic > a:hover {`\n\n` text-decoration:none;`\n\n` background-color: #EEEEEE;`\n\n` color: #827C70;`\n\n`}`\n\nLastly, you may find that the navigation looks bad when a user clicks on the Edit Links button. The following style forces the navigation items and background colors to match the nonedited styles:\n\n`\/* edit and edit mode links *\/`\n\n`.ms-navedit-linkCell a, `\n\n`.ms-navedit-linkCell .ms-core-listMenu-selected:hover,`\n\n`.customTopNavHolder .ms-navedit-editLinksText > span > .ms-metadata, `\n\n`.ms-core-listMenuEdit:hover > tr > `\n\n`.ms-navedit-linkCell > .ms-core-listMenu-item,`\n\n`.ms-core-listMenu-horizontalBox div.ms-navedit-hiddenAppendArea, `\n\n`.ms-navedit-addLinkText {`\n\n` color: white !important;`\n\n` background-color: transparent !important;`\n\n` height: 33px;`\n\n` border: 0px none;`\n\n`}`\n\nPage Content\n\nFor the page content the big change that needs to be made is how SharePoint styles the title text \"Your Source for Quality Breakfast Foods.\" You need to add some default styles that override the H1 and H2 elements. For this and the remaining sections, you can place the new code at the end of the style sheet.\n\n`\/* main page title colors *\/`\n\n`.customHeroHolder h1, .customHeroHolder h2{`\n\n` color: #535353;`\n\n` font-weight: normal;`\n\n`}`\n\nWeb Parts\n\nBecause SharePoint uses Web Parts for a lot of the user interface, you must be careful here. You can decide to apply custom styles to all these instances, but if you make a big change, it might be better to limit the styling to only Web Parts displayed on your custom publishing pages. This can be achieved by adding a parent `DIV` around content in a custom page layout and then prefixing all your CSS styles with the class assigned to this `DIV`. In the Randy's Waffles example, the Web Part Zone already had a parent `DIV` with a class of `customColumns` that can be used for these styles.\n\nTo finish the styles for the Web Parts, style the Web Part title text, URLs in these titles, padding around the Web Part zones, and padding on the actual Web Part body.\n\n`\/* web part title color on custom page layout *\/`\n\n`.customColumns .ms-webpart-titleText.ms-webpart-titleText, .customColumns .ms-webpart-titleText > a {`\n\n` background-color: #80AFCB;`\n\n` font-size: 20px;`\n\n` padding: 5px 15px;`\n\n` font-weight: normal;`\n\n` color: white; `\n\n`}`\n\n`\/* web part zone on custom page layout *\/`\n\n`.customColumns .ms-webpart-zone {`\n\n` padding: 0px 0px 10px;`\n\n`}`\n\n`\/* web part zone on custom page layout *\/`\n\n`.customColumns .ms-WPBody {`\n\n` padding: 15px;`\n\n`}`\n\nThe Ribbon\n\nThe ribbon at the top of the page is best left in its default arrangement, but there is no reason why you can't change some of the colors that clash with the new look and feel. You can change the links on the top right to white, the top bar to black, and the bottom bar to white. You must also make changes for IE8 because it uses special CSS prefixed with `.ms-core-needIEFilter`.\n\n`\/* suitebar link color *\/`\n\n`.ms-core-suiteLink a {`\n\n` color: white;`\n\n`}`\n\n`\/* suitebar left *\/`\n\n`#suiteBarLeft {`\n\n` background-color: black;`\n\n`}`\n\n`\/* handle ie8 *\/`\n\n`.ms-core-needIEFilter #suiteBarLeft{`\n\n` filter: none;`\n\n` background-color: black;`\n\n`}`\n\n`\/* ribbon bar and suite bar right *\/`\n\n`#globalNavBox, #suiteBarRight {`\n\n` background-color: white;`\n\n`}`\n\n`\/* handle ie8 *\/`\n\n`.ms-core-needIEFilter #suiteBarRight {`\n\n` background-color: white;`\n\n`}`\n\nDialogs\n\nThe dialog boxes that SharePoint presents for activities such as creating a new page can pose unique styling challenges. For instance, if you went through the example in the previous chapter, you know that your master page needs to have `.ms-dialogHidden` added to any container elements that you don't want to show in a dialog box.\n\nThat works easy enough, but there may be times when you have parent HTML elements that pass down inherited styles to child elements (specifically the main page content area) that don't look nice in a dialog box. This typically happens with container elements that apply decorative styles like backgrounds, padding, margin, width, height, and so on. In these instances, you can't just hide the parent element because it would also hide the page content of the dialog box.\n\nYou can knock out these rogue styles by resetting the parent element's CSS only when it shows inside of a dialog box. By including `.ms-dialog` before the CSS declaration, the CSS rules apply their style only when the elements are in a dialog.\n\n`\/* fix margins when dialog is up *\/`\n\n`.ms-dialog .customWidth, .ms-dialog .customBodyHolder, .ms-dialog body {`\n\n` margin:0 !important;`\n\n` min-height:0 !important;`\n\n` min-width:0 !important;`\n\n` width:auto !important;`\n\n` height:auto !important;`\n\n` background-color: white !important;`\n\n` background-image: none !important;`\n\n` padding: 0px !important;`\n\n` overflow:inherit;`\n\n` box-shadow: none;`\n\n`}`\n\nAt this point the CSS is complete for the Randy's Waffles example. Save the file and refresh your browser to see the results. Don't forget that you need to publish the CSS file if you want anonymous users to see the changes. In Chapter 9, \"Creating Content Rollups with SharePoint WCM,\" you learn more about creating rollup Web Parts to complete the bottom half of the original design.\n\nSummary\n\n\u2022 The role of CSS is significant in the understanding of how branding and styles are applied to SharePoint. A strong understanding of CSS helps you create custom SharePoint branding.\n\n\u2022 CSS is made up of rules that have selectors for matching HTML elements and declarations that define how the elements will be styled.\n\n\u2022 When multiple CSS rules are applied to an HTML element, the most specific rule overrides other less specific styles.\n\n\u2022 Inheritance causes some CSS properties to pass their style along from parent elements to their children.\n\n\u2022 CSS can be applied to a web page in three different ways. Inline styles are applied directly to HTML elements, internal style sheets are embedded inside of an HTML page, and external style sheets are separate CSS files that are linked from the HTML page.\n\n\u2022 Tools like Internet Explorer's F12 Developer Tools, Firebug for Firefox, and Chrome Developer Tools are helpful for determining how CSS is applied to complex HTML like you find in SharePoint 2013.\n\n\u2022 CSS can be applied to SharePoint on pages themselves using rich text fields or Web Parts; applied via the Alternate CSS feature in publishing sites or with code; or directly linked from a custom master page using the CssRegistration tag.\n\n\u2022 Applying branding to SharePoint can involve overriding many default SharePoint styles. Take your time and work through each section of the page including the page body, navigation, search, Web Parts, and more until you have SharePoint branding that achieves your design goals.\n\n3\n\nAdvanced SharePoint Branding\n\n7. Creating Custom Master Pages and Page Layouts\n\n8. Advanced SharePoint Branding Tasks\n\n9. Creating Content Rollups with SharePoint WCM\n\n10. Composed Looks and Custom Branding\n7\n\nCreating Custom Master Pages and Page Layouts\n\nWhat's In This Chapter\n\n1. ASP.NET Master Pages\n\n2. SharePoint Master Pages\n\n3. Starter Master Pages\n\n4. Custom Page Layouts\n\n5. SharePoint Solution Packages\n\nIn Chapter 5, \"Using the Design Manager to Start a Design in SharePoint,\" you learned about how the SharePoint 2013 Design Manager can help you easily create master pages and page layouts from standard HTML and CSS designs. Although this option is fine for many scenarios, there may be times when creating SharePoint branding from scratch is a better option. One major reason for this alternative might be if you have SharePoint Foundation or don't have access to a SharePoint Server Publishing site. (The Design Manager feature is only available in sites with the Publishing feature activated.)\n\nEven if you do have access to the Design Manager feature, after you become accustomed to how master pages and page layouts work, you may find it easier to skip the Design Manager and go right to creating custom branding by hand. Think of it like this: if you want to create an iPhone or Windows Phone app, you could use a code generator to make a simple app, but this approach is limited. When you feel comfortable coding for the phone, you would probably find that you have more power and control when coding the app by hand in the native coding language for the device. SharePoint branding is similar; if you are building a highly stylized website, you could build it with Design Manager, but you will likely find it easier to just learn the underlying technology and code it with master pages and page layouts. This is especially true if you already have experience branding in a previous version of SharePoint.\n\nNOTE\n\nSharePoint 2013 can use branding created from HTML by using the Design Manager features or custom branding that has been created by hand, like in previous versions of SharePoint. There isn't really a good word to describe these handmade master pages and page layouts; some people call them custom branding and others call them classic branding. In this chapter, any master page or page layout that is not created from HTML with the Design Manager is referred to as a custom master page.\n\nAnother reason why you might decide to code master pages by hand is that the Design Manager master pages were not built with SharePoint collaboration sites in mind (such as Team and Blog sites), and as such, there may be issues that crop up. For example, there is a SharePoint Feature activated on many collaboration sites called the Minimal Download Strategy (MDS). The MDS feature enables collaboration sites to load pages a lot faster than SharePoint 2010. The feature accomplishes this by running all page requests through a single page, `\/_layouts\/15\/start.aspx`, which looks for visual differences between new page requests and the previously loaded page, and it refreshes only the content that is different between the two pages.\n\nThe problem here is that Design Manager master pages were designed to work with publishing sites, and the master pages have code that turns off the MDS caching behavior. When the MDS functions are turned off, you end up with every page on your site loading twice; first loading `start.aspx` and then a second time just loading the normal non-MDS URL. This double page load is not a great user experience, so you have two options to mitigate this issue when creating collaboration sites. You can either deactivate the MDS feature from Site Settings \u21d2 Site Actions \u21d2 Manage site features and lose that performance gain, or you can code your master page by hand with the MDS feature in mind.\n\nThe following chart gives a breakdown of when the Design Manager is appropriate and when custom master pages should be made. Of course, not everything is cut and dry, and there are situations in which you could use one or the other. These instances are indicated with a solid dot in the recommended column, and a lightly shaded dot in the acceptable column.\n\nIn this chapter, you learn how master pages work in traditional ASP.NET applications and, more importantly how they work in SharePoint 2013. You will learn how to create custom master pages and page layouts by hand and to work with the MDS feature. Lastly, you learn about the process for deploying branding files to a production environment with a SharePoint Solution.\n\nUnderstanding Traditional ASP.NET Master Pages\n\nIf you remember back to Chapter 2, you learned about how master pages were introduced in ASP.NET 2.0 to solve the problem of web pages having styling, content, and code commingled on every page. Master pages made the process of maintaining websites much easier and less error-prone by separating the styling and layout from content and code.\n\nMaster pages are used to create a template that controls many aspects of the overall layout of multiple pages in an ASP.NET website. They are typically used as a shell to hold all the common HTML content, such as the DOCTYPE, meta information, CSS, navigation, footers, and the general layout of the chrome of the site. A website may have one master page that controls all the pages, or it could have several master pages that apply different layouts to different areas of the site. In addition to enabling easier website maintenance, master pages are especially helpful for creating a consistent look for large websites with many content authors of varying skill levels.\n\nThe next few sections discuss traditional ASP.NET master pages before you dive into the deep end to see how master pages work in SharePoint.\n\nMaster Page Structure\n\nFrom a coding perspective, a master page is a type of ASP.NET page written in either Visual Basic (VB) or C# and has a file extension of `.master`. Although you can use either language to create traditional ASP.NET applications, SharePoint uses only C# master pages. The following example illustrates a simple master page:\n\n`<%@ Master language=\"C#\"%>`\n\n`<!DOCTYPE HTML>`\n\n`<html>`\n\n`<head>`\n\n` <meta charset=\"utf-8\">`\n\n` <title>Demo Title<\/title>`\n\n`<\/head>`\n\n`<body>`\n\n`<form runat=\"server\">`\n\n` <div class=\"mainbody\">`\n\n` <asp:ContentPlaceHolder ID=\"MainBody\" runat=\"server\" \/>`\n\n` <\/div>`\n\n` <div class=\"footer\">`\n\n` <asp:ContentPlaceHolder id=\"Footer\" runat=\"server\">`\n\n` Copyright 2013 - Randy's Waffles`\n\n` <\/asp:ContentPlaceHolder>`\n\n` <\/div>`\n\n`<\/form> `\n\n`<\/body>`\n\n`<\/html>`\n\nYou can see that master pages begin with the `@ Master` directive that declares the language in which the master page will be written. In this case the master page is written in C#. There is a `<form>` tag with the `runat=\"server\"` property, which is a common attribute that tells ASP.NET that it needs to do some server processing on this control. There are also two content placeholders in the body of the master page: One has no content and is closed immediately but the other has some default footer content. Otherwise, the rest of the master page code is composed of standard HTML used to lay out the page.\n\nContent Pages\n\nMaster pages on their own don't actually produce anything useful; browsing to a master page's URL directly can cause the web server, in this case Internet Information Services (IIS), to display an error. In traditional ASP.NET websites, content pages (files with an `.ASPX` extension) can refer to a master page for their layout. When users browse these pages, IIS merges the page content with the master page, and the resulting web page displays to the user.\n\nContent pages declare a master page via an `@ Page` directive at the top of their code. Following is a simple content page:\n\n`<%@ Page Language=\"C#\" MasterPageFile=\"demo.master\" %>`\n\n`<asp:Content ContentPlaceHolderID=\"MainBody\" Runat=\"Server\" >`\n\n` Hello World`\n\n`<\/asp:content>`\n\n`<asp:Content ContentPlaceHolderID=\"Footer\" Runat=\"Server\" ><\/asp:content>`\n\nNotice that the `@ Page` directive `MasterPageFile` property defines this page's master page as `demo.master`. Assuming the master page shown earlier was named `demo.master`, this content page would load that master page and use it to display its content.\n\nContent Placeholders\n\nContent placeholders define areas of a master page that can be replaced by information located on a content page. The master page's content placeholders receive their content from controls placed on content pages. The content pages can override content in as many or as few of the master page's content placeholders as desired; this means that not every content placeholder in a master page needs to be overridden by content pages. Because of this, content placeholders can define some default content to show whenever a content page does not override them. In the \"Master Page Structure\" section example the `Footer` content placeholder provided the default content `Copyright` `2013` `-` `Randy's` `Waffles`.\n\nContent pages that use master pages, like the one in the preceding \"Content Pages\" section, require all content after the `@ Page` directive be located inside of content controls that start with `<asp:Content>`. If anything is placed outside a content control, an error occurs.\n\nThe content controls are tied to specific content placeholders by a matching `ContentPlaceHolderID`. In the \"Content Pages\" section's example, the content page would add the text Hello World where the `MainBody` content placeholder is located in the master page, and it would override the default text in the `Footer` content placeholder with nothing. When loaded in the browser, the resulting page would have the following HTML source:\n\n`<!DOCTYPE HTML>`\n\n`<html>`\n\n`<head>`\n\n` <meta charset=\"utf-8\">`\n\n` <title>Demo Title<\/title>`\n\n`<\/head>`\n\n`<body>`\n\n`<form>`\n\n` <div class=\"mainbody\">`\n\n` Hello World`\n\n` <\/div>`\n\n` <div class=\"footer\">`\n\n` <\/div>`\n\n`<\/form> `\n\n`<\/body>`\n\n`<\/html>`\n\nWARNING\n\nAlthough not every content placeholder in a master page needs to have a corresponding content control applied in a content page, the opposite is not true. Every content control defined in the content page must have a matching content placeholder in the applied master page; otherwise, an error occurs. In traditional ASP.NET applications this is easy to manage because you are in complete control over how master pages and content pages are set up. Because SharePoint is a dynamic environment with many different site templates, from blogs to wikis and everything in between, it is not uncommon to have all kinds of pages that use content placeholders in unexpected ways or not at all.\n\nUser Controls and Server Controls\n\nAnother important concept related to master pages is how controls are used to take complex code functionality and reduce it down to a few lines of code. Developers use controls to bundle HTML, existing ASP.NET server controls, and custom functionality into reusable components that can be added to master pages and content pages. User controls have a file extension of .`ascx` and perform more simple tasks than server controls. Server controls are coded and compiled into DLLs to be loaded on the web server. These controls enable master pages to provide large amounts of custom functionality while containing a relatively small amount of code for maintainability.\n\nTo use a control, you must first register it at the top of a content page or master page. This assigns a `TagPrefix` and a `TagName`, both of which are used to refer to the control in the page, as well as an `src` that points to the .`ASCX` control on the web server, or both a `Namespace` and an `Assembly` to point to a compiled DLL. Imagine that you had some search functionality prebuilt as a custom user control named `searchbox.ascx`. To include this functionality on a master page or content page, you would first register the control near the top of the page, below the language declaration, as shown here:\n\n`<%@ Master language=\"C#\"%>`\n\n<%@ Register TagPrefix=\"Custom\" TagName=\"Search\" src=\"searchbox.ascx\" %>\n\nTo actually use the registered control in the master page, you use a combination of the `TagPrefix` and the `TagName`. The following example shows the control added to the page inside of a `<div>` tag:\n\n`<div id=\"header\">`\n\n` <Custom:Search ID=\"mySearchControl\" runat=\"server\" \/>`\n\n<\/div>\n\nWhen the page is loaded in a browser, whatever content the control is programmed to display will be added to the page inside of the `<div>` tags. Also, if the control was set up to receive custom properties, they could be applied when the control is added to the page. For example, if the search control has a property named `ButtonImage` to set the search button image, it could be added like this:\n\n`<Custom:Search ID=\"mySearchControl\" ButtonImage=\"go.png\" runat=\"server\" \/>`\n\nSharePoint leverages the concept of controls heavily to encapsulate much of the functionality in a SharePoint site. In the next section, you look at SharePoint master pages; you can probably notice that many controls are used.\n\nMaster Pages in SharePoint\n\nMaster pages in SharePoint 2013 work much the same way that they do in traditional ASP.NET applications because SharePoint is essentially one big ASP.NET application. Every page browsed in a SharePoint site uses a master page to lay out its content. One striking difference with master pages in SharePoint is that master page references aren't hard-coded in pages. Instead, SharePoint pages point dynamically to either the Site or System master page applied to the SharePoint site that contains the page. The site master page is used by publishing pages while the system master page is generally applied to all nonpublishing pages such as forms, views, and settings pages. By pointing dynamically to these two master pages, SharePoint sites can have their master page easily switched for all their pages via the web interface.\n\nApplying Master Pages in SharePoint\n\nBesides using custom code or PowerShell, there are two primary ways to apply master pages in SharePoint 2013: by using the SharePoint Server master page setting screen or using SharePoint Designer 2013.\n\nUsing SharePoint Server Publishing to Apply Master Pages\n\nIf you have a SharePoint Server publishing site, the web interface provides an easy method for applying master pages. With publishing enabled on the root site collection, you can apply master pages to all subsites in the site collection, including subsites that do not have publishing enabled. To change the master pages in the web interface navigate to Site Settings \u21d2 Look and Feel \u21d2 Master page. From this page you can change both the Site Master Page and System Master Page, and you can also apply these selections to any subsites below the current site. Remember that the Site master page is used primarily by publishing pages, whereas the System master page is used throughout SharePoint, including nonpublishing pages and the application pages, such as Site Settings.\n\nUsing SharePoint Designer 2013 to Apply Master Pages\n\nIf you don't have SharePoint Server or don't have access to a publishing site, the next easiest option for applying master pages is to use SharePoint Designer 2013.\n\nExercise: Applying Master Pages with SharePoint Designer 2013\n\n1. Open SharePoint Designer 2013, and load a SharePoint Server 2013 site.\n\n2. From the Site Objects menu on the left, click Master Pages.\n\n3. In the main area of SharePoint Designer, click next to the name of a master page.\n\n4. On the ribbon in the Actions group, you can click either Set as Default or Set as Custom. These options correspond to the System and Site master pages, respectively. If the master page is currently set as either of these options, the icon will be grayed out for setting the master page on the ribbon.\n\nIn SharePoint Designer, you can also right-click on the master page name in the Site Objects menu and select either Set as Default Master Page or Set as Custom Master Page. After setting the master page as either Default or Custom, you can browse back to your SharePoint site and see that the master page has changed.\n\nSharePoint Master Page Structure\n\nEarlier you saw a sample ASP.NET master page; you can compare this to a SharePoint master page to better understand the differences. To do so, map a drive to your Master Page Gallery and open `seattle.master` in your editor of choice. In the interest of space, the following is a portion of `seattle.master` with some of the lines of code truncated to fit better on the pages:\n\n`<%@Master language=\"C#\"%>`\n\n`<%@ Register Tagprefix=\"SharePoint\" Namespace=\"Microsoft...`\n\n`<%@ Register Tagprefix=\"Utilities\" Namespace=\"Microsoft...`\n\n`<%@ Import Namespace=\"Microsoft.SharePoint\" %> `\n\n`<%@ Assembly Name=\"Microsoft.Web.CommandUI...`\n\n`<%@ Import Namespace=\"Microsoft.SharePoint.ApplicationPages\" %>`\n\n`<%@ Register Tagprefix=\"WebPartPages\" Namespace=\"Microsoft...`\n\n`<%@ Register TagPrefix=\"wssuc\" TagName=\"Welcome\"`\n\n` src=\"~\/_controltemplates\/15\/Welcome.ascx\" %>`\n\n`<!DOCTYPE html PUBLIC \"-\/\/W3C\/\/DTD XHTML 1.0 Strict\/\/EN\"`\n\n` \"http:\/\/www.w3.org\/TR\/xhtml1\/DTD\/xhtml1-strict.dtd\">`\n\n`<SharePoint:SPHtmlTag`\n\n` dir=\"<%$Resources:wss,multipages_direction_dir_value%>\"`\n\n` ID=\"SPHtmlTag\" runat=\"server\" >`\n\n`<head runat=\"server\">`\n\n` <meta name=\"GENERATOR\" content=\"Microsoft SharePoint\" \/>`\n\n` <meta http-equiv=\"Content-type\" content=\"text\/html;`\n\n` charset=utf-8\" \/>`\n\n` <meta http-equiv=\"X-UA-Compatible\" content=\"IE=10\"\/>`\n\n` <meta http-equiv=\"Expires\" content=\"0\" \/>`\n\n` `\n\n` <SharePoint:SPPinnedSiteTile runat=\"server\"`\n\n` TileUrl=\"\/_layouts\/15\/images\/SharePointMetroAppTile.png\"`\n\n` TileColor=\"#0072C6\" \/>`\n\n` <SharePoint:RobotsMetaTag runat=\"server\"\/>`\n\n` `\n\n` <SharePoint:PageTitle runat=\"server\">`\n\n` <asp:ContentPlaceHolder id=\"PlaceHolderPageTitle\" runat=\"server\">`\n\n` <SharePoint:ProjectProperty Property=\"Title\" runat=\"server\" \/>`\n\n` <\/asp:ContentPlaceHolder>`\n\n` <\/SharePoint:PageTitle>`\n\n` `\n\n`...`\n\nThe first thing you may notice looking at the SharePoint master page is that it is generally much larger than the simple ASP.NET master page listed earlier. Like ASP.NET, it begins with an `@ Master` directive that declares the language as C#, and then several controls and assemblies are registered or imported. These controls and assemblies are what make SharePoint work, and they are sprinkled throughout the master page and arranged with HTML and ASP.NET to create all the SharePoint functionality that appears on the page. After the controls and assemblies you see the Document Type Declaration or DOCTYPE, which tells the browser how to render the HTML markup. SharePoint 2013 can use a variety of DOCTYPEs, but it has been tuned to work best with either XHTML 1.0 Strict or HTML5 DOCTYPES. The previous example uses the XHTML 1.0 Strict DOCTYPE.\n\nOne prevalent control is registered as `SharePoint`, and you can see tags throughout the master page listed as `<SharePoint:`XYZ`>`, where XYZ describes the control's functionality. The `<SharePoint:SPHtmlTag>` is one of those controls; it injects the page's HTML tag dynamically based on various page criteria. Along with controls like this, you see that a lot of standard HTML, JavaScript, and CSS are also declared, such as the four meta tags at the top. The other major feature of the master page is a series of content placeholders, such as `PlaceHolderPageTitle`, which injects the title of the SharePoint pages into the HTML `<title>` tag. Many of these content placeholders are required by SharePoint and you can see an error if you simply remove them.\n\nThe following content placeholders are required in SharePoint 2013. Their names provide a fairly accurate description of what they are typically used for, but SharePoint pages can use them for anything, and they may not always contain the content you would expect.\n\n`PlaceHolderAdditionalPageHead` | `PlaceHolderPageDescription`\n\n---|---\n\n`PlaceHolderBodyAreaClass` | `PlaceHolderPageImage`\n\n`PlaceHolderBodyLeftBorder` | `PlaceHolderPageTitle`\n\n`PlaceHolderBodyRightMargin` | `PlaceHolderPageTitleInTitleArea`\n\n`PlaceHolderCalendarNavigator` | `PlaceHolderQuickLaunchBottom`\n\n`PlaceHolderFormDigest` | `PlaceHolderQuickLaunchTop`\n\n`PlaceHolderGlobalNavigation` | `PlaceHolderSearchArea`\n\n`PlaceHolderGlobalNavigationSiteMap` | `PlaceHolderSiteName`\n\n`PlaceHolderHorizontalNav` | `PlaceHolderTitleAreaClass`\n\n`PlaceHolderLeftActions` | `PlaceHolderTitleAreaSeparator`\n\n`PlaceHolderLeftNavBar` | `PlaceHolderTitleBreadcrumb`\n\n`PlaceHolderLeftNavBarBorder` | `PlaceHolderTitleLeftBorder`\n\n`PlaceHolderLeftNavBarDataSource` | `PlaceHolderTitleRightMargin`\n\n`PlaceHolderLeftNavBarTop` | `PlaceHolderTopNavBar`\n\n`PlaceHolderMain` | `PlaceHolderUtilityContent`\n\n`PlaceHolderMiniConsole` | `SPNavigation`\n\n`PlaceHolderNavSpacer` | `WSSDesignConsole`\n\nNOTE\n\nWhen you use the Design Manager to create HTML-based master pages, the required content placeholders are handled for you. This is not the case when creating a custom master page by hand.\n\nThe Ribbon\n\nAnother major aspect of SharePoint master pages is the inclusion of the ribbon. The ribbon is the area at the top of the page that includes all the contextual buttons that help content authors and site owners use SharePoint.\n\nIf you look at the source of a Design Manager-created master page, you see that Microsoft uses a one-line control to include the ribbon:\n\n``<PublishingRibbon:PublishingRibbon runat=\"server\" \/>``\n\nAlthough this control is extremely helpful, it is specific to SharePoint Server publishing and is not intended to be used with SharePoint Foundation sites. If you need to create a master page that will be used in one of these scenarios you can actually use the full ribbon, which can be seen by opening `seattle.master`. The ribbon actually includes all the code from line 72 through 343...a good portion of the 616-line master page. Luckily, for the most part you don't need to worry about the actual ribbon code in a custom master page; you can just copy the code as it's provided in one of the out-of-the-box master pages. This is also true for creating master pages for special SharePoint features like My Sites, which require a special ribbon like the one in `mysite.master` that is created automatically with My Sites.\n\nwarning\n\nThe way the ribbon is implemented in SharePoint can often give designers grief. This is because the SharePoint master page turns off the normal browser scrollbars. SharePoint then uses JavaScript to calculate the remaining browser height and adds the scrollbars back manually in a way that enables the ribbon to stay \"pinned\" to the top of the browser.\n\nAjaxDelta Controls and Minimal Download Strategy\n\nYou can also see a lot of instances of a specific control, `<SharePoint:AjaxDelta>`, in SharePoint master pages. The AjaxDelta control is directly related to how the Minimal Download Strategy (MDS) knows which parts of the layout to refresh before the page is rendered. More specifically, anything not wrapped in an AjaxDelta control will not refresh between pages of a site with the MDS feature activated.\n\nFor example, if you have some dynamic page element, such as the page title, and it is not wrapped in an AjaxDelta control, every page on a site with MDS enabled will show the same title from page to page. This is because the first page routed through `start.aspx` causes MDS to cache everything on the page that isn't wrapped in an AjaxDelta; every other page will show the same title. On the other hand, if it were wrapped in an AjaxDelta control, MDS would know it needs to render any differences from the previously loaded page.\n\nHere is an example of a control wrapped in an AjaxDelta control:\n\n`<SharePoint:AjaxDelta ID=\"DeltaSearch\" BlockElement=\"true\" runat=\"server\">`\n\n` <asp:ContentPlaceHolder id=\"PlaceHolderSearchArea\" runat=\"server\">`\n\n` <SearchWC:SearchBoxScriptWebPart `\n\n` UseSiteCollectionSettings=\"true\" `\n\n` EmitStyleReference=\"false\" `\n\n` ShowQuerySuggestions=\"true\" `\n\n` ChromeType=\"None\" `\n\n` UseSharedSettings=\"true\" `\n\n` TryInplaceQuery=\"false\" `\n\n` ServerInitialRender=\"true\" `\n\n` runat=\"server\" \/>`\n\n` <\/asp:ContentPlaceHolder>`\n\n`<\/SharePoint:AjaxDelta>`\n\nEach AjaxDelta control has a unique ID and an optional attribute `BlockElement`. When `BlockElement` is `true` the AjaxDelta wraps the control in an HTML `<div>`; otherwise it uses a `<span>`.\n\nNOTE\n\nAt some point in your SharePoint branding life, chances are good that you will forget to put an AjaxDelta around something important on your master page. If you see an issue in which something isn't updating from page to page or notice that some users are experiencing different rendering than you would expect for the same control, it's a good idea to double-check that the control is wrapped in an AjaxDelta.\n\nEditing Custom Master Pages\n\nTo edit a custom master page, you can still use a mapped drive to the site collection master page gallery much like you do with the Design Manager. The major difference is that because you are working directly with a `.master` file, WYSIWYG editors such as Dreamweaver and Expression Web cannot show a design preview. This isn't a big problem if you are comfortable editing code; you can easily have your favorite web browser open in another window and just refresh every time you make a change. Because your changes are saved directly to the server, you can see the updates immediately if you are authenticated as the same user to browse the site as you are to edit with. If for some reason you need to test with a different user, you need to publish and approve a major version of your `.master` file before the user can see the changes. Some common tools for working with custom master pages are Notepad++, Adobe Dreamweaver, Microsoft Expression Web, Microsoft Visual Studio, or even SharePoint Designer 2013.\n\nWARNING\n\nAlthough you can work on custom master pages with SharePoint Designer 2013, you cannot edit files from the mapped drive. SharePoint Designer is limited to connecting directly to the site collection using its own methods. The Site Objects pane on the left side of the screen is a view of your site collection much like you would see in a mapped drive.\n\nThis is probably a great time to remind you not to make edits like this directly on your production server. Best practice dictates that you should have a development environment, whether it is a cloud-based server, local server, or even a virtual machine, as long as it closely replicates the intended production environment. Later in the \"Understanding SharePoint Solution Packages\" section you learn about packaging your changes into a SharePoint Solution (WSP) with Visual Studio for deploying to production environments.\n\nCreating a Custom Master Page\n\nBecause of all the required content placeholders and other functionality that is loaded by default in SharePoint master pages, it is often impractical to just start a custom master page from a blank file. Instead, you can choose between a few different options when starting a new custom master page that includes much of this required code.\n\nDesign Manager\n\nYou could create a master page with the Design Manager and then decouple the `.html` and `.master` file and just start editing the `.master` file. To do this, follow these steps:\n\n1. From the Design Manager click Edit Master Pages.\n\n2. Find the master page, and click the Ellipsis menu.\n\n3. In the pop-up menu, click the next Ellipsis menu, and in that pop-up menu, click Edit Properties.\n\n4. From there you can uncheck Associate File, and click Save.\n\nAlthough this is a tempting and easy way to start, there are some downsides to this method. First, the master page created from the Design Manager is often tabbed and spaced poorly, and not easy to read. Perhaps more important though, as mentioned earlier, master pages created with this method are not intended to be used with collaboration sites with MDS activated.\n\nEditing an Existing Master Page\n\nIf you want to make a custom master page that is only slightly different from the out-of-the-box master pages, you can make a copy of `seattle.master` or `oslo.master` and make any changes you would like. This method is also easy, but it is only appropriate for making lightly branded master pages. Anything more stylized would require a lot of work to undo the layout that is already created. Also, these master pages are fairly lengthy and are not commented to help you with the process.\n\nThe easiest way to try this method is to open a mapped drive to your Master Page Gallery and simply copy, paste, and rename `seattle.master`. From there you can edit away to your heart's content.\n\nStarter Master Page\n\nProbably the best way to start a new custom master page is to use a starter master page. The concept of starter master pages has been around since SharePoint 2007, although they were known as minimal master pages back then. A starter master page is a community-based solution that is freely downloadable and includes only the bare minimum of code to make SharePoint work properly, with nice comments and tabbing. Otherwise, it is just a blank canvas for you to build branding around. You can download my starter master pages from http:\/\/startermasterpages.codeplex.com.\n\nUsing a Starter Master Page\n\nThe process of implementing a starter master page is quite similar to using the Design Manager to create one. Here is a quick overview of the steps:\n\n1. Create a traditional HTML and CSS design.\n\n2. Map a drive to the site collection's Master Page Gallery.\n\n3. Copy the HTML design's images and CSS to a folder in the Master Page Gallery.\n\n4. Download and copy a starter master page to the Master Page Gallery.\n\n5. Edit the starter master page with your editor of choice, and add the HTML from the original design to the starter master page. Arrange it around the key areas of the starter master page.\n\n6. Instead of adding snippets, add and arrange the SharePoint controls and content placeholders that the Design Manger would have created.\n\n7. Reference the custom CSS, and modify it to adjust for how SharePoint is showing the page.\n\n8. Create page layouts with SharePoint Designer 2013.\n\n9. Package as SharePoint Solution WSP with Visual Studio and deploy to production.\n\nAs you can see, many of these steps are similar to using the Design Manager except you aren't relying on it to generate much of the code for you. The upside to this is that you have maximum control over the readability and functionality of the master page.\n\nNow take a look at an example of using a starter master page to create a branded intranet design for your friends at Randy's Waffles. The following example uses a predesigned HTML site for Randy's Waffles intranet known as WaffleNET. Because this design will be used in a collaborative intranet, the body content wasn't mocked up in the HTML; instead, you can just use the page content that SharePoint creates and style it as needed.\n\nThe design for this example is similar to the design for Randy's Waffles used in Chapter 5, but there are some notable differences. Obviously, there is the addition of an intranet logo, in this case WaffleNET. The public-facing Randy's Waffles logo has been made smaller and moved to the right. Also the width of the page has been changed to a variable size that will expand to 90 percent of the browser screen width. A subtle background was added to the header area to make it more interesting at this larger size, and the footer has been removed to allow for more content. Lastly, the font for the top navigation has been decreased in size because intranets often have a lot of navigation. There are also some important concepts that are not shown in the HTML design: the intranet site will include Quick Launch navigation on the left and the out-of-the-box SharePoint site title, page, and description information will be added above the body content.\n\nYou can download the HTML for this example from the downloadable code for this chapter. For this example you use a SharePoint Server site with publishing turned on. If you don't have SharePoint Server or the ability to activate the publishing feature, you can still follow along, but there will be some differences.\n\nExample: Creating a Custom Master Page with a Starter Master Page\n\n1. Start by downloading the starter master pages from <http:\/\/startermasterpages.codeplex.com>. This includes several starter master pages, but for this example you want `Starter PubCollab.master`. If you don't have publishing, you can follow along instead with `Starter Foundation.master`.\n\n2. Like in previous chapters, map a drive to your SharePoint site collection, and navigate to your Master Page Gallery. Make a subfolder here called WaffleNET, and drag in the starter master page and all the files from the WaffleNET HTML (from the chapter downloads) except `WaffleNet.html`. You don't need the HTML file because you'll be merging the contents with the starter master page.\n\n3. Rename the starter master page to `WaffleNet.master`, and open it and the `WaffleNet.html` file in your editor of choice.\n\n4. Next, you need to begin merging the layout of the HTML design with the functions of the starter master page. Start by editing a few key areas in the `<head>` section of the starter master page. In the Favicon section, update the `SPShortcutIcon`'s `IconUrl` and `SPPinnedSiteTile`'s `TileUrl` to match the corresponding folder and filename. Be sure to change the path if your site collection is not at the root of your web application (Example: `\/sites\/SubSiteCollection\/_catalogs\/masterpage\/WaffleNet\/favicon.ico`):\n\n`<SharePoint:SPShortcutIcon runat=\"server\" `\n\n` IconUrl=\"\/_catalogs\/masterpage\/WaffleNet\/favicon.ico\" \/>`\n\n`<SharePoint:SPPinnedSiteTile runat=\"server\" `\n\n` TileUrl=\"\/_catalogs\/masterpage\/WaffleNet\/MetroAppTile.png\" `\n\n` TileColor=\"#2184bc\" \/>`\n\n5. Update the preconfigured `CssRegistration` at the bottom of the `<head>` to match the branding CSS. The `CssRegistration` tag is smart enough to allow `$SPUrl`, which uses the `~sitecollection` token to point to the root of the current site collection. This was not usable in the previous step.\n\n` <SharePoint:CssRegistration ID=\"CssRegistration1\" name=\"&lt;% `\n\n` $SPUrl:~sitecollection\/_catalogs\/masterpage\/WaffleNet\/style.css`\n\n` %&gt;\" runat=\"server\" after=\"SharepointCssFile\" \/>`\n\n6. You've now completed the head section; of course, more complicated designs might require more work in the preceding steps. Next, you look at using the body of the HTML design to lay out SharePoint functionality: copy all the content of `WaffleNet.html` between the opening and closing `<body>` tags. Paste this into the `WaffleNet.master` after `<div id=\"s4-bodyContainer\">`. This is the main container for the scrolling area of the page below the ribbon.\n\n7. Replace the static HTML logo with the actual SharePoint `SiteLogoImage`. Scroll down to `<SharePoint:AjaxDelta ID=\"Custom_Logo\"...>` and cut it out along with all the included controls, and replace the static HTML image. You will be replacing this:\n\n`<a href=\"#\" class=\"customLogo\"><img src=\"wafflenet.png\"`\n\n` alt=\"Back to Home\" \/><\/a>`\n\nwith:\n\n`<SharePoint:AjaxDelta ID=\"Custom_Logo\" BlockElement=\"true\"`\n\n` runat=\"server\">`\n\n` <SharePoint:SPLinkButton runat=\"server\"`\n\n` NavigateUrl=\"~sitecollection\/\">`\n\n` <SharePoint:SiteLogoImage ID=\"x63829de2201a4365a3904788f682d0a3\"`\n\n` LogoImageUrl=\"&lt;% $SPUrl:~sitecollection\/_catalogs\/masterpage\/`\n\n` StarterBranding\/logo.png %&gt;\" AlternateText=\"Back to Home\"`\n\n` runat=\"server\" \/>`\n\n` <\/SharePoint:SPLinkButton>`\n\n`<\/SharePoint:AjaxDelta>`\n\n8. In the section you just replaced, update the placeholder logo with the actual branding logo:\n\n`<SharePoint:SiteLogoImage ID=\"x63829de2201a4365a3904788f682d0a3\"`\n\n` LogoImageUrl=\"&lt;%`\n\n` $SPUrl:~sitecollection\/_catalogs\/masterpage\/WaffleNet\/wafflenet.png`\n\n` %&gt;\" AlternateText=\"Back to Home\" runat=\"server\" \/>`\n\n9. Replace the static HTML search with the SharePoint search control. This requires you to replace this line\n\n`<input type=\"text\" \/>`\n\n`<a href=\"#\" class=\"customSearchButton\">`\n\n` <img src=\"search-button.png\" alt=\"Search\" \/>`\n\n`<\/a>`\n\nwith:\n\n`<SharePoint:AjaxDelta ID=\"DeltaSearch\" BlockElement=\"true\"`\n\n` runat=\"server\">`\n\n` <asp:ContentPlaceHolder id=\"PlaceHolderSearchArea\" runat=\"server\">`\n\n` <SearchWC:SearchBoxScriptWebPart UseSiteCollectionSettings=\"true\" `\n\n` EmitStyleReference=\"true\" ShowQuerySuggestions=\"true\"`\n\n` ChromeType=\"None\" UseSharedSettings=\"true\"`\n\n` TryInplaceQuery=\"false\" ServerInitialRender=\"true\" `\n\n` runat=\"server\" \/>`\n\n` <\/asp:ContentPlaceHolder>`\n\n`<\/SharePoint:AjaxDelta>`\n\n10. Replace the entire static HTML top navigation with the SharePoint top navigation. You will be replacing this\n\n`<ul id=\"customTopNav\">`\n\n` <li><a href=\"#\" class=\"customTopNavItem\">Home<\/a><\/li>`\n\n` <li><a href=\"#\" class=\"customTopNavItem\">Sales<\/a><\/li>`\n\n` <li><a href=\"#\" class=\"customTopNavItem\">Marketing<\/a><\/li> `\n\n` <li><a href=\"#\" class=\"customTopNavItem\">HR<\/a><\/li>`\n\n` <li><a href=\"#\" class=\"customTopNavItem\">IT<\/a><\/li>`\n\n` <li><a href=\"#\" class=\"customTopNavItem\">Help<\/a><\/li>`\n\n`<\/ul>`\n\nwith this:\n\n`<SharePoint:AjaxDelta ID=\"DeltaTopNavigation\" BlockElement=\"true\" `\n\n` CssClass=\"ms-displayInline ms-core-navigation ms-dialogHidden\"`\n\n` runat=\"server\">`\n\n` <SharePoint:DelegateControl runat=\"server\"`\n\n` ControlId=\"TopNavigationDataSource\" Id=\"topNavigationDelegate\">`\n\n` <Template_Controls>`\n\n` <asp:SiteMapDataSource ShowStartingNode=\"False\"`\n\n` SiteMapProvider=\"SPNavigationProvider\" id=\"topSiteMap\"`\n\n` runat=\"server\" StartingNodeUrl=\"sid:1002\"\/>`\n\n` <\/Template_Controls>`\n\n` <\/SharePoint:DelegateControl>`\n\n` <SharePoint:AspMenu ID=\"TopNavigationMenu\" Runat=\"server\"`\n\n` EnableViewState=\"false\" DataSourceID=\"topSiteMap\"`\n\n` AccessKey=\"<%$Resources:wss,navigation_accesskey%>\" `\n\n` UseSimpleRendering=\"true\" UseSeparateCss=\"false\"`\n\n` Orientation=\"Horizontal\" StaticDisplayLevels=\"2\"`\n\n` AdjustForShowStartingNode=\"true\" MaximumDynamicDisplayLevels=\"2\"`\n\n` SkipLinkText=\"\" \/>`\n\n`<\/SharePoint:AjaxDelta>`\n\n11. You now move the major body functions from the starter master page up into the `customBodyHolder` of the HTML design. This starts with the `sideNavBox`, which holds the Quick Launch menu and the `contentBox`, which holds the main body content placeholder `PlaceHolderMain`: Select `<div id=\"sideNavBox\"... >` and everything inside of it, as well as `<div id=\"contentBox\">` and everything inside of it. Cut and paste this code between the starting <`div class=\"customBodyHolder\">` tag and closing `<\/div>` tag.\n\n12. Now you need to move the page title, breadcrumbs, and description to the area at the top of the `<div id=\"contentBox\">`. First, however, you must remove some extra markup that is left over from all your previous cutting and pasting. Inside of `<div id=\"s4-titlerow\">`, you can see three extra sets of comments and `<div class=\"ms-dialogHidden\"><\/div>` tags. These won't be needed, so you can remove them.\n\n13. Now all that remains inside of `<div id=\"s4-titlerow\">` is the out-of-the-box page title, breadcrumbs, and description. You can select the entire section including the start and end tags for `<div id=\"s4-titlerow\">` and paste that right below `<div id=\"contentBox\">`.\n\nAt this point the basic master page structure is complete, but if you applied this master page now, you would notice that all the SharePoint dialogs (such as the Settings menu \u21d2 Add a page) include parts of the branding that are unwanted.\n\nThis is because SharePoint uses the System master page for the chrome of all dialogs. To remove the extra branding inside of dialogs, you can add the CSS class `ms-dialogHidden` to a few of the existing tags. Any element with this CSS class is not rendered inside of a dialog but shows fine in a normal page. To do so simply update both the `customTopHolder` and `customTopNavHolder` `<div>` tags to include `ms-dialogHidden` like so:\n\n`<div class=\"customTopHolder ms-dialogHidden\">`\n\n`<div class=\"customTopNavHolder ms-dialogHidden\">`\n\nThat's all that needs to happen for the WaffleNET master page. At this point none of the CSS changes have been applied to handle SharePoint styling, but the process for updating the CSS is similar to what you saw in the previous chapter. The final CSS is actually quite similar to the Randy's Waffles CSS with just a few subtle tweaks to make the design more intranet-friendly. You can download the final master page and CSS from the downloadable code for this chapter.\n\nNOTE\n\nAs you were moving SharePoint functionality around in the master page, did you notice that most of the controls and content placeholders in the starter master page are surrounded by AjaxDelta controls? This is meant to handle the Minimal Download Strategy feature that was discussed earlier. Any control or content placeholder that can have dynamic content needs to be wrapped in a unique AjaxDelta control.\n\nTaking the Master Page Further\n\nBefore you can apply the master page to your site, you first need to publish a major version. In Chapter 5, you learned how to publish an HTML-based Design Manager master page; the process for publishing a custom master page is only slightly different.\n\nExample: Publishing a Custom Master Page\n\n1. Navigate to Site Settings, and under Web Designer Galleries, click Master pages and page layouts.\n\n2. Click the `WaffleNet` subfolder, and then click `WaffleNet.master`.\n\n3. You see an arrow appear to the right of the filename; expand that and then click Publish a Major Version.\n\n4. When the Publish a Major Version dialog opens, click OK.\n\nTo apply this master page to your site, you can follow the steps shown earlier in the \"Applying Master Pages in SharePoint\" section.\n\nIf you intend to use this branding on a public-facing site, all the assets including master pages, page layouts, CSS, images, and JavaScript need to be checked in as major versions and published before anonymous users can load the site. This can take a long time if you use the same method that you published the master page with earlier. Instead, an easier way to publish a lot of files is to use the old Content and Structure page. This is one area of SharePoint that has had little enhancement since SharePoint 2007, which is a shame because there are some useful features hidden in here.\n\nExample: Publishing Files Using the Content and Structure Page\n\n1. From Site Settings under Site Administration, click Content and structure.\n\n2. On the left side of the screen, you see a view of your site collection. Expand Master Page Gallery, and then click the `WaffleNet` folder.\n\n3. Now on the right side of the screen, you can see the contents of the WaffleNet folder. You can click the check boxes next to each item, or you can click the Select All box at the top of the grid.\n\n4. With all the files selected, click the Actions menu, and select Publish.\n\n5. From the Publish dialog, simply click OK.\n\nIf you want to add more out-of-the-box SharePoint functionality to the master page, one way to find out what code to add is to make a quick HTML-based master page with the Design Manager, and use the Snippet gallery to see what the resulting SharePoint code looks like in the related master page. You learn some more advanced tips and tricks for working with specific master page tasks in the next chapter. If you want to remove functionality from the master page, see the next section to learn how to remove required content placeholders.\n\nWorking with Required Content Placeholders\n\nWith all these required content placeholders, you are likely wondering how you can remove areas of functionality that aren't required for a specific branding project. Rather than delete required content placeholders (which causes an error), you can hide them on the master page. If you look at the code for either a Design Manager master page or one of the starter master pages, you see that unused content placeholders are relegated to the bottom of the master page and they are hidden by using the attribute `Visible=\"False\"`. Content placeholders with the visibility turned off can still function properly in SharePoint, but any content they generate will be omitted from the HTML source seen by browsers.\n\nThe following code shows an example of some hidden content placeholders:\n\n`<asp:ContentPlaceHolder id=\"PlaceHolderBodyAreaClass\" Visible=\"False\" runat=\"server\" \/>`\n\n`<asp:ContentPlaceHolder id=\"PlaceHolderTitleAreaClass\" Visible=\"False\" runat=\"server\" \/> `\n\nDealing with Errors\n\nOne of the first things you will realize when working on your custom master page is that when you make a mistake, SharePoint's default error messages are rather unhelpful. If you omit one of the required content placeholders and browse a site that uses your master page, SharePoint replies with simply Sorry, Something Went Wrong. An Unexpected Error Has Occurred.\n\nAlthough newer versions of SharePoint include a correlation ID that can be looked up on the server, that process can be daunting. For more information on looking up correlation IDs, see http:\/\/zimmergren.net\/technical\/sp-2010-find-error-messages-with-a-correlation-id-token-in-sharepoint-2010.\n\nFor a more useful solution, SharePoint can actually return descriptive error messages, but for security reasons they are turned off by default. If you work in a development environment, though, it can be quite helpful to enable these full error messages.\n\nExample: Turning on the Full Error Messages\n\n1. Log in to the physical SharePoint Server machine, and navigate to the IIS directory that holds your SharePoint website. It will most likely be located at `C:\\inetpub\\wwwroot\\wss\\VirtualDirectories\\` and will be in a subdirectory with the port number of your SharePoint site. If you have trouble finding it, you can open IIS, and in the Sites folder, right-click your SharePoint site, and then select Explore. This takes you directly to the directory that holds your SharePoint website.\n\n2. Locate the file named `web.config`, and open it in a file editor such as Notepad++.\n\n3. Press Ctrl+F, and find the line of code that contains the word callstack. You should find a line that looks like this:\n\n`<SafeMode MaxControls=\"200\" CallStack=\"false\" DirectFileDependencies=\"10\" `\n\n` TotalFileDependencies=\"250\" AllowPageLevelTrace=\"false\">`\n\nChange both the `CallStack` and `AllowPageLevelTrace` attributes from `false` to `true`.\n\n4. Press Ctrl+F again, and this time search for the word errors. You should find a line that looks like this:\n\n`<customErrors mode=\"On\" \/>`\n\n5. Change the mode from `On` to `Off`. This tells IIS not to show its customary basic error messages and to instead display the raw detailed error messages.\n\n6. Save and close `web.config`. This change causes IIS to restart the web application in which your SharePoint site resides. Be careful when making edits to this file in production or shared development environments.\n\nWith these changes in place, browsing a page with an error will now reveal the complete error message.\n\nCreating Custom Page Layouts\n\nPage layouts are critical to web content management because they enable content authors to create pages with specific editable fields and predefined HTML for content arrangement. Compared to traditional ASP.NET, you can think of page layouts as templates for creating content pages with specific layouts. This section focuses on providing a deep understanding of how page layouts work and how you can create your own page layouts for use in SharePoint 2013. To follow along with the examples in the chapter, you need to have SharePoint Server 2013 with a publishing site; SharePoint Foundation and Team sites without publishing activated do not include the capability to use page layouts.\n\nUnderstanding Content Types and Site Columns\n\nAlthough content types are certainly used for many other things in SharePoint, such as lists, they are critical to the creation of page layouts as well. One particularly tricky concept to understand with page layouts is that they are always based on exactly one content type, which in turn is made up of site columns.\n\nIf you look at a publishing page in Edit mode, you can see several editable field controls.\n\nThese editable fields are defined by which site columns are available in the underlying content type. To put it another way, editable field controls on page layouts have their data stored in the site columns that define them. You see an example of creating a custom page layout in the next section, but for now it's useful to understand that when you create a custom page layout, you first select the content type on which it is based. From there you can select from the available site columns to be used as field controls.\n\nTo gain a better understanding of how all these concepts are related, the out-of-the-box page layout Image on Left can be analyzed. This page layout is based on the Article Page content type. The content type has a Publishing HTML site column named Page Content. The Image on Left page layout has an editable field control in it called `PublishingPageContent`, which is directly related to this site column. In turn, any page that uses the Image on Left page layout, when edited, can have Page Content information entered and saved from the field control.\n\nAll page layouts in SharePoint are based on the Page content type or another content type that is inherited from the Page content type. When creating your own page layouts, the simplest way to start is by basing them on any of the out-of-the-box Page Layout content types. If you find that you need to have different editable fields in your page layout than those available in the out-of-the-box content types, you can create your own content type (inheriting from either the Page Layout or Page content type), add extra site columns, and base your page layout off of it.\n\nPage Layout Structure\n\nTo better understand the structure of page layouts, here is a sample of a simple page layout. This page layout includes just a rich HTML field and a Web Part zone:\n\n`<%@ Page language=\"C#\" `\n\n` Inherits=\"Microsoft.SharePoint.Publishing.PublishingLayoutPage,`\n\n` Microsoft.SharePoint.Publishing,Version=15.0.0.0,Culture=neutral,`\n\n` PublicKeyToken=71e9bce111e9429c\" %>`\n\n`<%@ Register Tagprefix=\"SharePointWebControls\"`\n\n` Namespace=\"Microsoft.SharePoint.WebControls\"`\n\n` Assembly=\"Microsoft.SharePoint, Version=15.0.0.0, Culture=neutral,`\n\n` PublicKeyToken=71e9bce111e9429c\" %>`\n\n`<%@ Register Tagprefix=\"WebPartPages\"`\n\n` Namespace=\"Microsoft.SharePoint.WebPartPages\" `\n\n` Assembly=\"Microsoft.SharePoint, Version=15.0.0.0, Culture=neutral, `\n\n` PublicKeyToken=71e9bce111e9429c\" %> `\n\n`<%@ Register Tagprefix=\"PublishingWebControls\" `\n\n` Namespace=\"Microsoft.SharePoint.Publishing.WebControls\" `\n\n` Assembly=\"Microsoft.SharePoint.Publishing, Version=15.0.0.0,`\n\n` Culture=neutral, PublicKeyToken=71e9bce111e9429c\" %> `\n\n`<%@ Register Tagprefix=\"PublishingNavigation\" `\n\n` Namespace=\"Microsoft.SharePoint.Publishing.Navigation\" `\n\n` Assembly=\"Microsoft.SharePoint.Publishing, Version=15.0.0.0,`\n\n` Culture=neutral, PublicKeyToken=71e9bce111e9429c\" %>`\n\n`<asp:Content ContentPlaceholderID=\"PlaceHolderPageTitle\"`\n\n` runat=\"server\">`\n\n` <SharePointWebControls:FieldValue id=\"PageTitle\" FieldName=\"Title\"`\n\n` runat=\"server\"\/>`\n\n`<\/asp:Content>`\n\n`<asp:Content ContentPlaceholderID=\"PlaceHolderMain\" runat=\"server\">`\n\n` <div class=\"MainContent\">`\n\n` <PublishingWebControls:RichHtmlField`\n\n` FieldName=\"PublishingPageContent\" runat=\"server\"\/>`\n\n` <\/div>`\n\n` <div class=\"WebParts\">`\n\n` <WebPartPages:WebPartZone runat=\"server\" Title=\"Web Parts\"`\n\n` ID=\"WPZ1\"><\/WebPartPages:webpartzone>`\n\n` <\/div>`\n\n`<\/asp:Content>`\n\nLike ASP.NET pages, the page layout begins with `@ Page`, but instead of referencing a master page, it simply states that the page layout is written in C# and inherits code from the SharePoint Server `PublishingPageLayout` code. Then there are several controls registered that are used throughout the page layout. After those Register tags, everything else in the page layout must be contained inside of a content placeholder. In this code, there are two content placeholders: `PlaceHolderPageTitle`, which overrides content in the master page's HTML `<title>` section, and `PlaceHolderMain`, which is the main content section in the master page.\n\nInside the first placeholder you see `<SharePointWebControls:FieldValue>`, which is a value control and is used to simply display a field (rather than allow it to be editable as you see next). In this case, `PageTitle` displays inside the HTML `<title>` section. In the next placeholder you see a `<PublishingWebControls:RichHtmlField>`, which is a field control that will be editable when the page is in Edit mode. Lastly, there is a Web Part zone, which is defined as `<WebPartPages:WebPartZone>`; this enables the content author to add any number of Web Parts to this page.\n\nNOTE\n\nWeb Part zones are not the only place in which content authors can place and configure Web Parts. Content authors can place Web Parts anywhere within Wiki page content areas and in publishing HTML field controls. This is a powerful feature, but Web Part zones provide a nice way to have a designated area that arranges Web Parts horizontally or vertically in a controlled fashion.\n\nExample: Creating a Custom Page Layout with SharePoint Designer 2013\n\n1. Open your SharePoint Server site in SharePoint Designer 2013.\n\n2. From the Site Objects menu on the left side, click Page Layouts.\n\n3. From the ribbon, click New Page Layout.\n\n4. From the New dialog box, use the following settings, and then click OK.\n\n\u2022 Content Type Group\u2014Page Layout Content Types\n\n\u2022 Content Type Name\u2014Article Page\n\n\u2022 URL\u2014`DemoLayout.aspx`\n\n\u2022 Title\u2014Demo Page Layout\n\n5. Select the Toolbox pane on the right, scroll down to the bottom, and expand the section named SharePoint Controls.\n\n6. If the Toolbox pane is not shown, click View from the ribbon and then Task Panes \u21d2 Toolbox.\n\n7. Expand Page Fields and Content Fields. Page Fields shows all the site columns that were inherited from the parent content type. Content Fields shows all the site columns that were added to the actual content type from which the page layout was created.\n\nReplacing the GUID\n\nWhen you drag fields from the Toolbox, SharePoint Designer unfortu-nately adds them with the behind-the-scenes SharePoint unique identifier (GUID) for the `FieldName`. So you end up with something like `f55c4d88-1f2e-4ad9-aaa8-819af4ee7ee8` instead of `PublishingPageContent`. If you want to improve your code readability, you can manually replace the GUID with the actual field name. To find the behind-the-scenes field name, navigate to Site Settings, and under Web Designer Galleries, click Site content types. Then click the content type that you are using for the page layout (in this case Article Page). From there, click the column that you are using for your field, and in your browser's URL, look for a section that starts with `Field=` . In this case you see `Field=PublishingPageContent`, so replace the GUID with `PublishingPageContent`.\n\n8. Add a field control to the page layout to allow authors to add content. Drag Page Content from the Content Fields into the `PlaceHolderMain`. Add an HTML `<div>` around the field control like this:\n\n`<div>`\n\n` <PublishingWebControls:RichHtmlField FieldName=\"f55c4d88-1f2e-4ad9-aaa8-`\n\n`819af4ee7ee8\" runat=\"server\"><\/PublishingWebControls:RichHtmlField>`\n\n`<\/div>`\n\n9. Add a Web Part zone to the page layout. For SharePoint Designer 2013 this can be done from the ribbon by clicking Insert \u21d2 Web Part Zone. If the button is grayed out, refresh the SharePoint Designer interface by pressing F5. Add the Web Part Zone in a `<div>` below the Page Content field like this:\n\n`<div>`\n\n` <WebPartPages:WebPartZone id=\"g_9A81CAACC0F1428EAEE87D345D2093C4\" `\n\n` runat=\"server\" title=\"Zone 1\">`\n\n` <\/WebPartPages:WebPartZone>`\n\n`<\/div>`\n\n10. Type Ctrl+S to save the page layout.\n\nUsing Edit Mode Panels to Hide and Show Page Layout Sections\n\nSometimes you may want to have a field control that shows only when a content author is editing the page; this is useful if you want to keep track of certain information on a page but not show it to regular users browsing the site. For example, a field control like that would come in handy if you want the author to be able to add and edit the article's date so that you can roll it up with a Web Part on another page and sort by date. Fortunately, doing this is a simple process.\n\nExample: Adding an EditModePanel around the Field Control\n\n1. Below the Web Part zone in the page layout from the previous example, add the following code:\n\n`<PublishingWebControls:EditModePanel runat=\"server\">`\n\n`<\/PublishingWebControls:EditModePanel>`\n\n2. Add the article date field control to the page layout. Drag Article Date from the Content Fields between the opening and closing `<PublishingWebControls:EditModePanel>` like this:\n\n`<PublishingWebControls:EditModePanel runat=\"server\">`\n\n` <SharePointWebControls:DateTimeField`\n\n` FieldName=\"71316cea-40a0-49f3-8659-f0cefdbdbd4f\" runat=\"server\">`\n\n` <\/SharePointWebControls:DateTimeField>`\n\n`<\/PublishingWebControls:EditModePanel>`\n\n3. Press Ctrl+S to save the page layout.\n\nCompleting your Custom Page Layout\n\nEven though your page layout is created, you still need to check in a major version and test it before content authors can use it. This section describes how to do this.\n\nExample: Publishing a Page Layout\n\n1. Click back on the Page Layouts item in the Site Objects menu in SharePoint Designer 2013, and then click the icon for `DemoLayout.aspx`.\n\n2. On the ribbon, click Check In, select Publish a Major Version, and click OK.\n\nFinally, you can test the new page layout by creating a page and switching its layout to the newly created page layout.\n\nExample: Testing a Page Layout\n\n1. Click the Settings menu \u21d2 Add a page.\n\n2. Give the page a name, and then click Create.\n\n3. Select the new page layout from the ribbon by clicking Page \u21d2 Page Layout \u21d2 Demo Page Layout.\n\n4. Edit the content and click Save. If you want other users to see the content, you also need to click Publish.\n\nNOTE\n\nThe great thing about page layouts is if you make future changes to the page layout, all pages based on it will be updated automatically.\n\nUpgrading a SharePoint 2010 Master Page to 2013\n\nAlthough creating custom SharePoint 2013 master pages from a starter master page is cool, you may be wondering what to do if you already have existing SharePoint 2010 branding that you want to use in SharePoint 2013. Unfortunately, you can't just apply a SharePoint 2010 master page to a 2013 site and have it function as expected. The master pages are set up similarly between the two versions though; for instance, they are both a similar file size and contain a ribbon that takes up a good portion of the code. That being said, there is a little work that needs to be done to properly update a 2010 master page to 2013. The easiest way to start is to compare your SharePoint 2010 master page to one of the starter master pages and make changes as needed.\n\nHere are some of the high-level things that you need to think about changing for SharePoint 2013 master pages:\n\n\u2022 The DOCTYPE should be XHTML 1.0 Strict or HTML5.\n\n\u2022 The metatag that tells IE which version to target should be updated to IE10: `<meta http-equiv=\"X-UA-Compatible\" content=\"IE=10\" \/>`.\n\n\u2022 Branding assets can be moved to the Master Page Gallery if you want, but it is also acceptable to keep them in the Style Library or the `_layouts` folder. Obviously, if you move the assets, their paths need to change in the master page.\n\n\u2022 In SharePoint 2013, the main CSS file `corev15.css` is not automatically loaded, so you want to load it with `<SharePoint:CssRegistration Name=\"corev15.css\" runat=\"server\" \/>`.\n\n\u2022 Many of the SharePoint controls in the `<head>` section and the beginning of the `<body>` section have been deprecated and\/or replaced with new controls.\n\n\u2022 CSS needs to be updated to reflect the new SharePoint 2013 CSS class naming used throughout the markup.\n\n\u2022 The ribbon needs to be updated with either the one-line publishing ribbon or the much larger one from either an out-of-the-box or starter master page.\n\n\u2022 The out-of-the-box search box control has been updated to: `<SharePoint:DelegateControl runat=\"server\" ControlId=\"SmallSearchInputBox\" Version=\"4\"\/>`.\n\n\u2022 The available navigation `SiteMapProviders` have been updated including two that work with both structured and managed navigation: `GlobalNavigationSwitchableProvider` and `CurrentNavigationSwitchableProvider`.\n\n\u2022 Be sure to check the list of required content placeholders from the \"SharePoint Master Page Structure\" section and update your master page accordingly.\n\nThese are just a few of the most obvious changes you need to make to a SharePoint 2010 master page to get it to work with SharePoint 2013. Because every master page is unique, there may be more issues that need to be addressed in your 2010 master page.\n\nUnderstanding SharePoint Solution Packages\n\nThe SharePoint 2013 Design Manager includes a feature to export design packages, which is a great option for moving branding assets between site collections or environments. However, the feature itself doesn't provide much flexibility, and understanding its limitations can help you determine if the export design packages functionality will work well for your project. For example, the feature does not allow any flexibility for what gets packaged and instead includes all files that have been edited from the Master Page Gallery, Style Library, Theme Gallery, and Device Channels. The design package does not include content pages, navigation settings, or anything from the term store. Another consideration is that this design package needs to be applied one site collection at a time, which can make applying updates more difficult if you have an implementation with more than one site collection. In general, many larger or more custom projects require additional functionality as part of the deployment, such as the ability to include custom code along with the branding assets. To truly create a customized branding solution, the best option is to make a SharePoint Solution or WSP and deploy it to production servers. The upcoming sections discuss some of the most common concepts related to branding deployments.\n\nCustomized and Uncustomized Files\n\nAlthough the terms are generic sounding, customized and uncustomized files mean something specific in the SharePoint vernacular. If you've been around SharePoint for a while, you might have heard the terms customized and uncustomized as well as ghosted and unghosted.\n\nAll the files used to create the out-of-the-box sites are uncustomized and ghosted. Uncustomized is an attribute associated with a file that indicates that a file hasn't been modified. Ghosted means that the source of the file lives on the file system of the server and not in the content database. There is still a pointer in the content database (the ghost of the file) that tells SharePoint to look for the source of a file on the server's file system. In this case, when the page is requested, SharePoint still looks to the content database, but because the instance of the page does not include a source, the file is rendered from the file system.\n\nUncustomized files make it possible to have many copies of a file point to the same source\u2014similar to how a template for a document works. For example, if you have many sites using the same page layout, you can make modifications to a single file, and every place that uses the uncustomized page layout is updated. This is helpful from a design perspective because it allows you to make changes easily.\n\nA customized file is one that has been modified in any way through a mapped drive or by using SharePoint Designer. After a file has been customized, it becomes more difficult to apply updates on a broad scale. Because customized files are created by editing and because they limit the flexibility of applying updates, it is advisable to use care when making edits, especially in the production environment.\n\nAn unghosted file refers to a file whose source lives in the content database. In earlier versions of SharePoint, the terms customized and unghosted referred to the same concept, but with the introduction of sandbox solutions in SharePoint 2010, the meaning of the terms began to refer to uniquely different aspects.\n\nFor example, the export design functionality of Design Manager creates a sandbox solution to package branding files. The sandbox solution is never added to the file system of the server, therefore by definition its files are considered ghosted. But, the files it deploys are still in an uncustomized state. This is a bit tricky to follow at first since the terminology is vague. Technical aspects aside, the following list summarizes the most important considerations around this concept:\n\n\u2022 Out-of-the-box files deploy in an uncustomized and ghosted state, which provides the greatest flexibility for updating on a broad scale. This can only be achieved with a full trust farm solution.\n\n\u2022 Sandbox solutions deploy files in an uncustomized and unghosted state. These files must be deployed separately to each site collection, but updates can be broadly applied to files within the site collection.\n\n\u2022 Once a file has been customized, it won't be updated with updates applied via a farm or sandbox solution.\n\n\u2022 The export design functionality of Design Manager deploys files as a sandbox solution.\n\nFor more information on customized and uncustomized files, see Andrew Connell's MSDN article here: http:\/\/msdn.microsoft.com\/en-us\/library\/cc406685.aspx. It was written for SharePoint 2007 but the concepts still apply for SharePoint 2013.\n\nFeatures and Solutions\n\nThe primary mechanism used to provision uncustomized instances of files in SharePoint is called a feature. A feature enables a developer to deploy files and other site customizations or functionality to a site collection or individual site. When activated, the feature deploys the files to SharePoint in an uncustomized state.\n\nAlthough they have a generic-sounding name, features refer to something specific in SharePoint. Every feature includes a `Feature.xml` file that provides basic information (title, description, scope of deployment, and more), a unique ID, the name and location for an assembly, or one or more element manifest files. The element manifest file is what describes to SharePoint where to deploy the various files contained in the feature. When an assembly is referenced, usually it deploys files the same way that the element manifest would, except it requires compiled code instead of the Collaborative Application Markup Language (CAML) used by the elements file.\n\nBy themselves, features enable you to deploy branding assets in an uncustomized state, but the files related to the feature (`feature.xml`, element manifests, and the branding files to be deployed) ideally need a structured way to move everything around. SharePoint Solutions or, more simply, solutions are a sealed cabinet (`.CAB`) file with a `.WSP` extension. Solutions help to get files where they need to be in a safe way to help ensure the reliability of the SharePoint servers. They provide a mechanism for moving files between server farms like between development and production environments. In addition, solutions also provide a way to simultaneously deploy files across multiple servers in the farm.\n\nSharePoint Branding Solutions\n\nIn SharePoint 2013 there are two types of solutions: farm solutions and sandboxed solutions. Farm solutions, which were available starting in SharePoint 2007, are typically deployed by server administrators and are definitely a preferred approach to the alternative of manually deploying files to multiple servers. However, the challenge is that farm solutions can deploy almost any type of file to the server with full trust\u2014from an administrator's perspective it is difficult to ensure that a farm solution isn't going to negatively impact a SharePoint farm. For this reason it is important to thoroughly test all farm solutions that are deployed.\n\nFrom a branding perspective, farm solutions present a small challenge. Most custom branding files such as master pages, page layouts, CSS, or images are relatively harmless and easily could be deployed through the user interface or SharePoint Designer. But if you want to avoid customized files, sandboxed solutions are an option that can be uploaded and activated by Site Collection Administrators without a Farm Administrator's intervention.\n\nIn concept, a sandboxed solution is a safe way to deploy files to a specific site collection on your SharePoint server. Farm administrators can actually validate and monitor a sandboxed solution, which allows them to define automated policies about what is allowed and not allowed in the sandbox. Sandboxed solutions deploy in a partial trust state, so there are some limitations to the type of code that can be run in the sandbox. For the most part though, this shouldn't limit the ability to deploy most branding files. For more information on sandboxed solutions, see Developing, Deploying, and Monitoring Sandboxed Solutions in SharePoint 2010 by Paul Stubbs (http:\/\/msdn.microsoft.com\/en-us\/magazine\/ee335711.aspx).\n\nBoth farm solutions and sandboxed solutions are the preferred ways to deploy your branding files, but which one should you choose? Sandboxed solutions provide an additional layer of protection, which allows files to deploy to the server while minimizing the impact to the server. Because of this, they are the only option for deploying a custom branding solution to Office 365 SharePoint Online sites. However, sandbox solutions have limitations; they can't do some powerful things that farm solutions can do. For instance, you cannot create a feature receiver to automatically apply your branding when new sites are created. For this reason, typically larger or more complex branding solutions that aren't installed into the cloud are set up as farm solutions.\n\nThe following table shows the three options for getting files onto a server and their pros and cons:\n\nYou can download the completed Visual Studio solution for the WaffleNET example you saw earlier from the downloadable code for this chapter. This can serve as both a great educational tool for learning how to create a SharePoint branding solution, or it could be modified and used directly in your own SharePoint projects.\n\nNOTE\n\nFor a detailed explanation of creating a branding solution for SharePoint 2013, you can read Deploying Branding Solutions for SharePoint 2010 Sites using Sandboxed Solutions by Ted Pattison (<http:\/\/msdn.microsoft.com\/en-us\/library\/gg447066.aspx>).\n\nSummary\n\n\u2022 HTML-based master pages created with the Design Manager feature in SharePoint 2013 were intended to be used with publishing sites. For truly custom branding that applies across SharePoint including collaboration sites, it may be necessary to create custom master pages by hand the classic way.\n\n\u2022 In traditional ASP.NET websites, ASPX content pages refer to master pages that provide a unified look and feel. Content pages have content controls that can override content placeholders in the master page.\n\n\u2022 SharePoint master pages work similarly to ASP.NET except that pages don't refer directly to a specific master page; instead, they reference Site or System master pages that can be applied from either the Site Settings menu in publishing sites or with PowerShell or SharePoint Designer 2013. SharePoint master pages also have a set of required content placeholders that are used throughout the various pages of a SharePoint site.\n\n\u2022 Collaboration sites have the Minimal Download Strategy (MDS) feature turned on by default. This feature speeds up page loading but requires the use of AjaxDelta controls placed throughout your master page, surrounding areas that dynamically change from page to page.\n\n\u2022 Starter master pages like those available at <http:\/\/startermasterpages.codeplex.com> are useful for creating new custom SharePoint branding.\n\n\u2022 Page layouts are another big part of SharePoint branding. They are based on content types, which define site columns that can eventually be used as editable field controls in pages.\n\n\u2022 SharePoint branding should be packaged into a solution package known as a WSP for deployment to production servers. Solutions can be created as either farm (which runs at the whole farm) or sandboxed (which runs at the site collection). If you are deploying to Office 365 and SharePoint Online you should use a solution marked as sandboxed.\n8\n\nAdvanced SharePoint Branding Tasks\n\nWhat's In This Chapter\n\n1. Code-Based Navigation Settings\n\n2. Traditional Breadcrumbs\n\n3. Device Channels\n\n4. My Sites\n\n5. SharePoint Online\n\nWith any SharePoint branding project, most people think only about the major tasks, such as creating the master page and working with the HTML and CSS. Although these tasks are certainly the biggest ones, a number of other important tasks are usually required to fully achieve the intended creative design. This chapter is a collection of topics to help you create a refined and polished final product.\n\nWorking with Navigation\n\nSharePoint 2013 provides many valuable options for controlling the way navigation looks using just the web user interface. However, several things can be changed only by editing controls directly in master pages. Whether you look at the global navigation at the top of the page or the current navigation, (Quick Launch) that shows on the left, SharePoint primarily uses two controls to show navigation: `SiteMapDataSource` and `AspMenu`. If you want to have better control of how navigation shows in your branding, it's a good idea to understand these two controls.\n\nSiteMapDataSource\n\nThe `SiteMapDataSource` control is essentially a data source that can be used by controls in ASP.NET to display information about the hierarchical arrangement of sites and pages throughout your SharePoint site. The most typical use for `SiteMapDataSource`s is to show navigation information, but they can also be used for tree views and breadcrumbs. A typical `SiteMapDataSource` might look like this:\n\nLooking at the out-of-the-box Seattle master page, you can see that Microsoft surrounds the `SiteMapDataSource` with a delegate control like this:\n\n`<SharePoint:DelegateControl runat=\"server\" `\n\n` ControlId=\"TopNavigationDataSource\" Id=\"topNavigationDelegate\">`\n\n` <Template_Controls>`\n\n` <asp:SiteMapDataSource ShowStartingNode=\"False\" `\n\n` SiteMapProvider=\"SPNavigationProvider\" id=\"topSiteMap\" `\n\n` runat=\"server\" StartingNodeUrl=\"sid:1002\"\/>`\n\n` <\/Template_Controls>`\n\n`<\/SharePoint:DelegateControl>`\n\nIt does this so that `Seattle.master` can be applied to any site in SharePoint, whether you have SharePoint Foundation or SharePoint Server, and even if the site is a publishing site or a team site. The delegate control swaps out the sitemap with the appropriate choice for that site automatically. If you want to have more control over this process, you can remove the delegate control from around the `SiteMapDataSource` and set the `SiteMapProvider` to match your wanted navigation.\n\nYou can see all the available Sitemaps for your SharePoint server by looking at the `web.config` file in the root of the web server. (Typically this would be located on the server in a subfolder of `C:\\inetpub\\wwwroot\\wss\\VirtualDirectories\\`.) If you search the `web.config` file for `<providers>`, you see a section that includes many different sitemaps and a brief description of each. The sitemaps are different depending upon what version of SharePoint is installed and what site template was used to create the site you are looking at. Many of these sitemaps aren't appropriate for either the latest version of SharePoint or even for navigation controls.\n\nSharePoint Foundation 2013 uses the sitemap named `SPNavigationProvider` to show basic navigation in the top link bar and Quick Launch navigation. SharePoint Server, on the other hand, has more options available, which include the old 2010-style Structured Navigation and the new Managed Navigation based on Managed Metadata term sets. Here are the various publishing navigation sitemaps:\n\n\u2022 `GlobalNavigation`\u2014Shows Structured Navigation items in the global navigation for this website. This navigation is shown at the top of the page in most websites.\n\n\u2022 `CurrentNavigation`\u2014Shows Structured Navigation items to display in current navigation for this website. This navigation is shown on the side of the page in most websites.\n\n\u2022 `G``lobalNavigationTaxonomyProvider`\u2014Similar to `GlobalNavigation` but used for Managed Navigation instead of Structured Navigation.\n\n\u2022 `CurrentNavigationTaxonomyProvider`\u2014Similar to the `CurrentNavigation` but used for Managed Navigation instead of Structured Navigation.\n\n\u2022 `GlobalNavigationSwitchableProvider`\u2014Shows `GlobalNavigation` or `GlobalNavigationTaxonomyProvider` depending on which is selected in the Navigation Settings.\n\n\u2022 `CurrentNavigationSwitchableProvider`\u2014Shows `CurentNavigation` or `CurrentNavigationTaxonomyProvider` depending on which is selected in the Navigation Settings.\n\nAspMenu\n\nThe `AspMenu` control is what actually renders navigation items based on a selected `SiteMapDataSource`. Here is a sample `AspMenu` control from a typical SharePoint master page:\n\nThis control has a lot of different options available to change its behavior, but the following two properties are probably the most important to understand. They indicate how many levels of navigation to show normally and how many will show when you hover over an item and show its children as fly-outs. These are two settings that many designers would like Microsoft to include in the web interface's navigation settings because it would be useful for site owners to be able to change them easily as opposed to them being editable only through code.\n\n\u2022 `StaticDisplayLevels`\u2014This property represents the number of levels of hierarchical navigation nodes that SharePoint shows by default in the navigation. For example, if the `AspMenu` orientation is set to Horizontal and this property is set to 2, you can see the first and second levels of navigation shown in a line horizontally by default on the page.\n\n\u2022 `MaximumDynamicDisplayLevels`\u2014This property represents the number of levels of hierarchical navigations that show dynamically as fly-outs. The levels that show here are counted after the number of levels that are set to show for `StaticDisplayLevels`. For example, if `StaticDisplayLevels` is set to 3 and `MaximumDynamicDisplayLevels` is set to 2, you see the first three levels of navigation shown on the page automatically; then you see the next two levels of navigation as dynamic fly-outs only when you mouse over their parent nodes. The actual navigation that shows is determined by a number of other factors, such as if you use structured or managed navigation and whether the navigation settings in the UI are set to show subsites.\n\nWARNING\n\nBoth instances of `AspMenu` that are used in the default `Seattle.master` are surrounded by content placeholders. This means that various pages throughout SharePoint can replace what shows in the top and left navigation. If you want to have more control over this process, you may want to move the `AspMenu` tags out of the content placeholders. After moving the `AspMenu` tags, you need to hide the content placeholders, as you learned in Chapter 7, \"Custom Master Pages and Page Layouts.\"\n\nTree View\n\nThe tree view is an interesting feature that provides an expandable view of the site structure shown as a hierarchical tree. You have probably seen it used in collaboration sites, in the bottom of the left navigation.\n\nIf you don't see the tree view on your site, there is an easy menu for showing or hiding it available from Site Settings \u21d2 Look and Feel \u21d2 Tree view. From this menu you can hide or show either the tree view or the entire Quick Launch menu for the SharePoint site.\n\nIn a lot of ways, the tree view is difficult to work with from a branding perspective. This is mostly because it adds an expandable area inside of a scrolling iFrame to an already cramped place on the page, but also because there is a lot going on dynamically in the HTML for the tree view. It's tempting to just say you aren't going to support the tree view in a custom branded SharePoint site, but if you work on a collaborative intranet site, this option might not be wise. Many business users find value in the tree view because it adds in metadata navigation for navigating or filtering the navigation options using metadata terms.\n\nBecause the tree view is used primarily as a self-contained piece of functionality, you can add the tree view to your master page simply by copying the default delegate control from either the default `Seattle.master` or a starter master page. Here is the standard code that creates the tree view:\n\n`<SharePoint:DelegateControl runat=\"server\" `\n\n` ControlId=\"TreeViewAndDataSource\">`\n\n` <Template_Controls>`\n\n` <SharePoint:SPHierarchyDataSourceControl `\n\n` runat=\"server\" `\n\n` id=\"TreeViewDataSourceV4\" `\n\n` RootContextObject=\"Web\" `\n\n` IncludeDiscussionFolders=\"true\" \/>`\n\n` <SharePoint:SPRememberScroll runat=\"server\" `\n\n` id=\"TreeViewRememberScrollV4\" `\n\n` onscroll=\"javascript:`\n\n` _spRecordScrollPositions(this);\" `\n\n` style=\"overflow: auto;\">`\n\n` <SharePoint:SPTreeView id=\"WebTreeViewV4\" `\n\n` runat=\"server\" ShowLines=\"false\" `\n\n` DataSourceId=\"TreeViewDataSourceV4\"`\n\n` ExpandDepth=\"0\" `\n\n` SelectedNodeStyle-CssClass=\"ms-tv-selected\" `\n\n` NodeStyle-CssClass=\"ms-tv-item\" `\n\n` SkipLinkText=\"\" NodeIndent=\"12\" `\n\n` ExpandImageUrl=\"\/_layouts\/15\/images\/`\n\n` tvclosed.png?rev=23\" `\n\n` ExpandImageUrlRtl=\"\/_layouts\/15\/images\/`\n\n` tvclosedrtl.png?rev=23\" `\n\n` CollapseImageUrl=\"\/_layouts\/15\/images\/`\n\n` tvopen.png?rev=23\" `\n\n` CollapseImageUrlRtl=\"\/_layouts\/15\/images\/`\n\n` tvopenrtl.png?rev=23\" `\n\n` NoExpandImageUrl=\"\/_layouts\/15\/images\/`\n\n` tvblank.gif?rev=23\">`\n\n` <\/SharePoint:SPTreeView>`\n\n` <\/SharePoint:SPRememberScroll>`\n\n` <\/Template_Controls>`\n\n`<\/SharePoint:DelegateControl>`\n\nUsing Traditional Breadcrumbs\n\nBreadcrumbs\u2014those fun, little navigational links frequently seen at the top of websites\u2014are one thing that SharePoint 2013 doesn't handle well out-of-the-box. For SharePoint 2013, Microsoft decided to remove the useful (but fairly ugly) hierarchical breadcrumbs that were available in a pop-out menu at the top left of the ribbon for SharePoint 2010. However, by default there is something similar to a breadcrumb shown in a large font near the top of the page body.\n\nThis section is rendered using the `SPTitleBreadcrumb` control as well as a content placeholder. Here is the actual code as it is seen in `Seattle.master`:\n\n`<SharePoint:AjaxDelta id=\"DeltaPlaceHolderPageTitleInTitleArea\" `\n\n` runat=\"server\">`\n\n` <asp:ContentPlaceHolder id=\"PlaceHolderPageTitleInTitleArea\" `\n\n` runat=\"server\">`\n\n` <SharePoint:SPTitleBreadcrumb runat=\"server\" `\n\n` RenderCurrentNodeAsLink=\"true\" `\n\n` SiteMapProvider=\"SPContentMapProvider\" `\n\n` CentralAdminSiteMapProvider=\"SPXmlAdminContentMapProvider\">`\n\n` <PATHSEPARATORTEMPLATE>`\n\n` <SharePoint:ClusteredDirectionalSeparatorArrow `\n\n` runat=\"server\" \/>`\n\n` <\/PATHSEPARATORTEMPLATE>`\n\n` <\/SharePoint:SPTitleBreadcrumb>`\n\n` <\/asp:ContentPlaceHolder>`\n\n`<\/SharePoint:AjaxDelta>`\n\nAlthough this is useful for showing list and library information and looks a lot like breadcrumbs, the user actually isn't provided with traditional breadcrumb functionality that shows the parent sites above the current page.\n\nFortunately, there is a traditional ASP.NET control that you can use in your master pages, which is a close approximation of typical website breadcrumbs, known as `SiteMapPath`. Here is the simplest way to use it in your master page and an example of the result:\n\n`<SharePoint:AjaxDelta ID=\"SiteMap\" BlockElement=\"true\" runat=\"server\">`\n\n` <asp:SiteMapPath runat=\"server\"\/>`\n\n`<\/SharePoint:AjaxDelta>`\n\nThe control also takes a property `SiteMapProvider` that enables you to pass in a specific provider, much like the `AspMenu` you learned about earlier. However, it is typically best to leave this property blank because you are essentially telling SharePoint to use the default `SiteMapProvider` for the specific type of site that is viewed.\n\nCreating a Dynamic Back to Home Link\n\nIf you followed the custom master page example in Chapter 7, you have seen the use of `$SPUrl` tokens to make a dynamic link back to the top of the site collection. This token can be useful when creating branding that applies to larger SharePoint sites, especially those with multiple sites or site collections.\n\nWARNING\n\nThe `$SPUrl` token is only available to SharePoint Server because it is related to the publishing features. If you use a Foundation site, you must manually make relative links such as `\/_catalogs\/masterpage`, which would go to the top of the web application root and then navigate down to `_catalogs` and `masterpage`.\n\n`$SPUrl` has two available tokens, `~site` and `~sitecollection`, which can navigate to either the top of the current site or the top of the current site collection, respectively. Here is an example of using `$SPUrl` to find a CSS file by looking at the root of the site collection:\n\n`<SharePoint:CssRegistration ID=\"CssRegistration1\"`\n\n` name=\"&lt;% $SPUrl:~sitecollection\/_catalogs\/masterpage\/`\n\n` WaffleNet\/style.css %&gt;\" `\n\n` runat=\"server\" after=\"SharepointCssFile\" \/>`\n\nNotice that the entire `$SPUrl` is surrounded by `&lt;%` and `&gt%`, which are essentially HTML escaped versions of the ASP.NET inline open and close tags `<%... %>`. They tell ASP.NET that there is some code that needs to be interpreted in this section.\n\nAlthough this usage of `$SPUrl` is fairly straightforward, you can also see some instances in which the token shortcut is used without the `$SPUrl` section. This is because some controls are wired to understand the tokens by themselves. `SPLinkButton` is one such control. Here is an example of an `SPLinkButton` wrapped around a `SiteLogoImage` to create a logo that links back to the top level of a site collection:\n\n`<SharePoint:SPLinkButton runat=\"server\" `\n\n` NavigateUrl=\"~sitecollection\/_catalogs\/masterpage\/waffles\/logo.png\">`\n\n` <SharePoint:SiteLogoImage ID=\"myLogo\" LogoImageUrl=\"&lt;% `\n\n` $SPUrl:~sitecollection\/_catalogs\/masterpage\/waffles\/logo.png %&gt;\" `\n\n` AlternateText=\"Back to Home\" runat=\"server\" \/>`\n\n`<\/SharePoint:SPLinkButton>`\n\nNotice that the `NavigateUrl` property of `SPLinkButton` only has the `~sitecollection` token, whereas the `LogoImageUrl` property of `SiteLogoImage` needs the full `$SPUrl` and the beginning and end tags to function properly.\n\nHandling Anonymous Users\n\nAnonymous users, those visitors to your site that are just browsing the pages or that have no credentials to authenticate, have unique needs that may need to be addressed in your SharePoint branding. You must first decide if your site will support anonymous browsing. By default SharePoint has anonymous access turned off for web applications, so if you decide to enable it there are a couple things you need to do to provide access.\n\nExample: Turning on Anonymous Access\n\n1. From Central Administration on the server, click Manage web applications.\n\n2. Select the desired web application, and from the ribbon click Authentication Providers.\n\n3. From the list of authentication providers, click Default. If you have more than one authentication provider, you must click the zone that anonymous users would be accessing the site from.\n\n4. Check the box for Enable Anonymous Access, and click Save. The page does not refresh when you click Save in this dialog box, so just close the box after clicking Save.\n\nAt this point the web application permits anonymous users to reach the site, but you still need to allow anonymous access for the particular SharePoint site.\n\n5. On your SharePoint site, click Site Settings \u21d2 Users and Permissions \u21d2 Site permissions.\n\n6. From the ribbon, click Anonymous Access.\n\n7. In the Anonymous Access dialog, select Anonymous users can access: Enter Web site and click OK.\n\nYour site is now enabled for anonymous access, but if you have custom branding assets, they all need to be checked in as major versions and published before anonymous users can see the site. You can learn more about this process from Chapter 7.\n\nNow that you have a site that is viewable anonymously, there are a few more things to consider, namely whether the ribbon should show and how anonymous users can authenticate into the site. Typically, anonymous users are consumers of content on your site and won't need to edit pages or lists. Because of this, they usually don't need to see the ribbon at the top of SharePoint. This is especially true for public-facing SharePoint sites, where the ribbon is not only unnecessary but also changes the typical scrolling behavior of the website.\n\nHiding the ribbon is already handled if you use the Design Manager to create an HTML master page, or if you use the shorter `PublishingRibbon` control in a custom master page. This is not the case if you use SharePoint Foundation or the longer ribbon code included with the out-of-the-box master pages. If you want to manually hide the ribbon, you can use `SPSecurityTrimmedControl` to show and hide content based on whether the user is anonymous or authenticated via the `AuthenticationRestrictions` property. This property can be set to either `AnonymousUsersOnly` or `AuthenticatedUsersOnly` to show content to users based on their authentication status. Here is an example that would hide the ribbon for anonymous users. It could be placed anywhere in the master page, but for readability it might be best to add this code right before the ribbon code:\n\n`<SharePoint:AjaxDelta id=\"DeltaLogin\" runat=\"server\">`\n\n` <SharePoint:SPSecurityTrimmedControl runat=\"server\" `\n\n` AuthenticationRestrictions=\"AnonymousUsersOnly\">`\n\n` <style type=\"text\/css\">`\n\n` #s4-ribbonrow, #suiteBar { visibility:hidden;display:none; }`\n\n` <\/style>`\n\n` <\/SharePoint:SPSecurityTrimmedControl>`\n\n`<\/SharePoint:AjaxDelta>`\n\nIn the middle of the control is a style that can hide the ribbon; the style will be added only to the page for anonymous users.\n\nAnother thing to consider when hiding the ribbon for anonymous users is that you will also be hiding the default login control, which is included at the top right of the ribbon. This can be a problem for users that do need to authenticate because they will see the site in anonymous mode first and not have a way to log in. One way to handle this is to educate your users about how they need to navigate manually to `\/_layouts\/15\/Authenticate.aspx` (or any URL in Site Settings) to launch the authentication process. Although this would save you some effort, it's not the most elegant solution. Luckily, you can use the same `SPSecurityTrimmedControl` to add in the welcome menu (which is also the login control when you aren't yet authenticated) for anonymous users only. Here is the update code that includes a style to hide the ribbon and the welcome control:\n\n`<SharePoint:AjaxDelta id=\"DeltaLogin\" runat=\"server\">`\n\n` <SharePoint:SPSecurityTrimmedControl runat=\"server\" `\n\n` AuthenticationRestrictions=\"AnonymousUsersOnly\">`\n\n` <style type=\"text\/css\">`\n\n` #s4-ribbonrow, #suiteBar { visibility:hidden;display:none; }`\n\n` <\/style>`\n\n` <wssuc:Welcome runat=\"server\" EnableViewState=\"false\" \/>`\n\n` <\/SharePoint:SPSecurityTrimmedControl>`\n\n`<\/SharePoint:AjaxDelta>`\n\nThis simply adds a Sign In link to the page rather haphazardly; you may want to spend some more time styling it to match with the rest of your branding.\n\nWARNING\n\nDepending on the master page that is used, particularly master pages that are created with the Design Manager, the welcome control is registered at the top of the master page with the prefix `wssucw` and other times it is registered to the prefix `wssuc`. This frustrating inconsistency can cause the preceding code to have an error. If you run into this error, you can change the line to `<wssucw:Welcome runat=\"server\" EnableViewState=\"false\">`.\n\nWorking with Device Channels\n\nIf you've read Chapter 5, \"Using the Design Manager to Start a Design in SharePoint,\" or played with the Design Manager, you have probably seen the option (typically listed as step 2) for Managing Device Channels. This topic was moved from Chapter 5 because in a lot of ways it doesn't make sense to think about device channels until after you understand the basics of making master pages.\n\nDevice channels are a feature of SharePoint Server publishing, so you cannot use them with Foundation or basic team sites. They can be thought of simply as a way of declaring rules for specific devices that can then be used to decide which master page to apply to a site or even to change certain parts of a page only for those devices. For example, you could set up a device channel for Apple iOS devices and another for Microsoft Surface tablets and show different branding, either subtly changed or completely different, depending upon which device is used to browse your SharePoint site.\n\nDevice channels work by looking for keywords in the browser's user agent, a string of text that is sent to the server by every browser to identify its characteristics. For example, here is the user agent string for the Safari browser on Apple's iPad:\n\n`Mozilla\/5.0 (iPad; U; CPU OS 3_2_1 like Mac OS X; en-us) AppleWebKit\/531.21.10 (KHTML, like Gecko) Mobile\/7B405`\n\nBecause user agent strings can change as new OS versions are released, you can use a portion of the string (like \"iPad\") as a wildcard keyword when targeting browsers with device channels.\n\nNOTE\n\nDevice channels are not the same thing as media queries in CSS3. Device channels enable you to change the display based on the browser that is visiting your site, whereas media queries enable you to change styles based on browser capabilities such as resolution, orientation, width, height, and so on. You can actually use both of these technologies together to create truly responsive mobile experiences in SharePoint.\n\nCreating a Device Channel\n\nCreating Device Channels in SharePoint 2013 is actually simple; you can access the menu from either the Design Manager or through Site Settings \u21d2 Look and Feel \u21d2 Device Channels. If you haven't created any device channels yet, you see the list contains only the default device channel applied to all of SharePoint. The following fields are used to define Device Channels:\n\n\u2022 Active\u2014Use this check box to mark device channels as activated or deactivated on your site.\n\n\u2022 Name\u2014The name that shows in the user interface for this specific device channel.\n\n\u2022 Alias\u2014The identifier that can be used with Device Channel panels, which you learn more about in a bit.\n\n\u2022 Description\u2014Optionally, you can give the device channel a description.\n\n\u2022 Device Inclusion Rules\u2014Enables you to specify user agent substrings that SharePoint can use to decide what device channel to apply. You can enter several substrings on separate lines, and SharePoint will look through all of them.\n\nYou could try to create a device channel for the iPad but that device would be a little difficult for everyone to test. Instead, you can create a device channel for a freely downloadable browser such as Google's Chrome browser. Device channels can be created for any browser that sends a user agent string, not just mobile devices. If you want to find out what the user agent string looks like for a particular browser, you can use a free web service such as http:\/\/whatsmyuseragent.com\/.\n\nThe user agent string for the latest version of Google Chrome is as follows:\n\n`Mozilla\/5.0 (Windows NT 6.1; WOW64) AppleWebKit\/537.11 (KHTML, like Gecko) Chrome\/23.0.1271.97 Safari\/537.11`\n\nTo make a device channel that matches this, you need to pick a substring inside of this text that you think would be unique to the browser you are targeting. In this case the word \"Chrome\" appears in the user agent so that should work well.\n\nTo create a new one click New Item and fill in the following fields; then click Save:\n\n\u2022 Name\u2014Chrome\n\n\u2022 Alias\u2014Chrome\n\n\u2022 Device Inclusion Rules\u2014Chrome\n\n\u2022 Active\u2014Checked\n\nAt this point you have a device channel created, but you need to apply a master page to the device channel before you can see it in action.\n\nNOTE\n\nIf more than one device channel could match a specific browser, SharePoint applies whichever channel appears first in this list. If you need to change the order of the device channels, there is an easy method via the Reorder Channels button on the ribbon for the device channel page.\n\nApplying a Master Page to a Device Channel\n\nApplying a master page to a specific device channel is also easy. Start by clicking Site Settings \u21d2 Look and Feel \u21d2 Master page. Now that you have a device channel created, you can see more than one option under the Site Master Page settings. In this case you have one for Chrome, and you always see the Default device channel. The default behavior is to have all new device channels set to follow the default channel; in other words, they will show whatever master page is already set for the site.\n\nFrom here you can set the master page from the Chrome device channel to a unique master page and have everything else use the default setting. For example, you could select the WaffleNet master page from the previous chapter to display only for Chrome browsers, and have everyone else continue to see the default Seattle master page.\n\nYou have a few options for testing your newly applied device channel. Obviously, the easiest way to test would be to just download Chrome and visit the site to see a different master page than you would from Internet Explorer. But what if you don't have Chrome, or more commonly, what if you don't have the expensive mobile device you are trying to make a device channel for? Microsoft gives you an easy way to test any device. Simply append `DeviceChannel=YourDeviceChannel` to the end of the URL like this:\n\n`http:\/\/RandysWaffles\/Pages\/default.aspx?DeviceChannel=Chrome`\n\nThis allows you to choose a device channel and see the site through that browser's \"eyes.\" You can also access this feature from any page that you are editing: from the ribbon click the Page tab and then in the Page Actions section click Preview.\n\nDevice Channel Panels\n\nApplying master pages to device channels is a great way to completely change the look of a site based on the browsing device, but in some cases it can introduce more maintenance overhead to manage multiple master pages. What if you want to have one master page but change out specific parts of the UI based on the browsing device? Microsoft has provided a control for this purpose, the Device Channel Panel. (Say that three times fast!)\n\nDevice Channel Panels are available from the Snippet Gallery for adding to HTML-based Design Manager master pages, but for this chapter you will focus on using the actual control in custom master pages that you create without the Design Manager. Device Channel Panels can be used in either master pages to affect all pages in a site, or they can be added to individual page layouts to affect certain pages. Here is what the code looks like for a typical Device Channel Panel. First, you need to add a register tag at the top of the master page or page layout to bring in the publishing web controls:\n\n`<%@Register Tagprefix=\"Publishing\" `\n\n` Namespace=\"Microsoft.SharePoint.Publishing.WebControls\" `\n\n` Assembly=\"Microsoft.SharePoint.Publishing, Version=15.0.0.0, `\n\n` Culture=neutral, PublicKeyToken=71e9bce111e9429c\"%>`\n\nThen wherever you want the special content to appear, add the Device Channel Panel like this:\n\n`<Publishing:DeviceChannelPanel runat=\"server\" `\n\n` IncludedChannels=\"Chrome\">`\n\n` This will only show for Chrome!`\n\n`<\/Publishing:DeviceChannelPanel>`\n\nThe key property for the panel is the `IncludedChannels`, which tells SharePoint which device channel aliases to look for to include this content. You can list more than one channel alias, separating them by commas. In this case `This will only show for Chrome!` would show only when the Chrome device channel is applied. After you do any testing with this, be sure to change the master page settings back to Follow Default Channel at least until you get the hang of things; otherwise, you might have a different master page applied for the channel when working on future examples.\n\nThe previous code example was rather arbitrary; a more useful exercise for Device Channel Panels would be to change the arrangement of the page based on the device channel. For example, you could hide the Quick Launch left menu for devices such as the iPhone only. In the following code, the Device Channel Panel is set to override the applied styles if an iPhone device channel is being applied. Keep in mind you would need to set up a corresponding device channel with an alias of \"iPhone\" that targets an iPhone Safari browser user agent before you could use this panel. This code could be placed anywhere in a master page, but for readability, it might make sense to place it directly above the Quick Launch controls.\n\n`<Publishing:DeviceChannelPanel runat=\"server\" `\n\n` IncludedChannels=\"iPhone\">`\n\n` <style type=\"text\/css\">`\n\n` #sideNavBox { display: none; }`\n\n` #contentBox { margin-left: 20px; }`\n\n` <\/style>`\n\n`<\/Publishing:DeviceChannelPanel>`\n\nWhen this Device Channel Panel code displays, the styles hide the Quick Launch menu and decrease the left margin on the main content area to expand to fill the extra room. If you have an iPhone, you can test the Device Channel Panel; if not, you can use one of the methods previously shown to preview the site as an iPhone.\n\nWARNING\n\nOne thing that you can't use Device Channel Panels for is Web Part zones. You can embed preconfigured Web Parts in Device Channel Panels, but zones are not supported. If you need Web Parts to leverage device channels, you can actually use JavaScript to check which panel is being applied and react accordingly. For more information about this, see \"Plan device channels in SharePoint Server 2013\" at `http:\/\/msdn.microsoft.com\/en-us\/library\/jj862343.aspx`.\n\nWorking with My Sites\n\nIf you have done any work with previous versions of SharePoint, you have probably heard of My Sites, which enable users to easily share information about themselves and their work in a central location. After My Sites are configured, you can access your My Site from several of the links on the top right of the ribbon including Newsfeed, SkyDrive, Sites, and from USER NAME \u21d2 About Me.\n\nMy Sites Architecture\n\nArchitecturally speaking, My Sites are a bit different than dealing with a normal SharePoint site. At the highest level you must know that My Sites have both a My Site host, which is a shared Site Collection between all users, and a personal site, which is a separate site collection created for each My Site user. Pages such as the Newsfeed and About Me are served from the shared My Site host, whereas pages such as SkyDrive and Sites are served from your personal site.\n\nYou can tell the difference between them by looking at the URL; pages that use the personal site typically have `\/personal\/<name of user>\/` in the URL. You can learn more about how My Sites are configured by reading \"Plan for My Sites in SharePoint Server 2013\" at http:\/\/technet.microsoft.com\/en-us\/library\/cc262500.aspx.\n\nBecause of the way My Sites are set up with a shared Site Collection for the host and individual site collections for the personal sites, actually applying branding to these site collections becomes a bit more challenging than a typical SharePoint site. If you had rights to the My Site host site collection, you could certainly still map a drive to the master page gallery and create and edit master pages, and they would affect all of the user's Newsfeeds. But for the personal sites, you would be editing only one user's master page. Any new personal site would not show the branding because new personal sites are generated on-the-fly when a user first clicks on one of the My Sites links. In general, applying branding to My Sites is better left to a SharePoint Solution (WSP) created with Visual Studio. For more information on creating a Visual Studio solution for My Sites, see the Starter SharePoint My Site Customization CodePlex project at http:\/\/drisgill.com\/go\/mysites-solution. This project is currently set up for 2010 but the concepts should be similar for 2013.\n\nNOTE\n\nMy Sites is an area of SharePoint in which the Design Manager cannot be used to create HTML-based master pages. If you want to brand My Sites, you must create a custom master page.\n\nMy Sites 2010 and 2013 Differences\n\nIn SharePoint 2010, branding My Sites involved two master pages; the My Site host used a special master page called `mysite.master`, whereas the personal site used the typical `default.master`. In SharePoint 2013, both sections use `mysite15.master` for almost everything specific to My Sites. However, there are still some instances inside of My Site in which you can create typical SharePoint sites. For example, if you create a personal team site or blog site inside of your My Site, these sites would use the typical `seattle.master` rather than the specific `mysite15.master`. Another key difference between SharePoint 2010 and 2013 master pages is that in 2010, Microsoft added a global navigation bar across the top of the My Sites pages. Alternatively, SharePoint 2013 My Sites shares the same top Suite Bar as a typical SharePoint site would. This section focuses on the `mysite15` master page because anything that uses the default seattle master page can be branded using methods learned throughout the other chapters in this book.\n\nIn many ways, the default My Site branding is similar to typical SharePoint branding with a few key changes. The following diagram highlights the major differences between a My Sites master page and a typical SharePoint master page.\n\nTo achieve these changes, mysite15.master has different code from seattle.master. To brand SharePoint 2013 My Sites, you must understand these differences to create a master page that reflects the rest of your site's branding.\n\nBranding My Sites in SharePoint 2013\n\nMuch like creating branding for traditional SharePoint sites, you have some options for actually creating the My Sites branding. With My Sites you shouldn't work directly with a production server. Instead, you should have a local development environment, either a physical or virtual machine, for quickly making changes and seeing the results in real time. When you are happy with your My Sites branding, refer to the link in the previous section for instructions on packaging them as a SharePoint solution.\n\nYou can easily map a drive to the Master Page Gallery of the My Site host and work directly with master pages. Just like with typical SharePoint branding, if you want to make something that looks almost exactly like the out-of-the-box My Site branding, you may want to start by making a copy of `mysite15.master` and making changes there. However, most designers find it easier to use a starter master page to create more complex My Site branding. You can download a starter master page for My Sites from http:\/\/startermasterpages.codeplex.com. Work with it the same way you did for the custom master pages in Chapter 7.\n\nChances are good you will already have existing SharePoint 2013 branding by the time you need to make My Site branding. You can reuse most of your HTML markup, as well as any CSS and images that you used for your standard SharePoint branding. Just remember that the page content is driven by the My Sites experience. Specifically for My Sites, the branding should avoid sweeping changes to the main content of the page, and instead focus mostly on color and font changes. Unless you are getting into creating custom-developed Web Parts or controls, most My Site branding initiatives leave the arrangement of the My Site pages as they are by default. The following completed My Site branding could match with the WaffleNet branding from Chapter 7.\n\nApplying a My Site master page is a bit more challenging, though. By default, publishing isn't turned on for My Sites, so you won't see the typical Site Settings \u21d2 Master page option. To apply the master page, you can either activate publishing features for the My Site host, use SharePoint Designer 2013 (by right-clicking the master page \u21d2 Set as Default Master Page \u21d2 Set as Custom Master Page), or you can use Visual Studio or PowerShell code to apply the master page. For information on using PowerShell to apply a master page, see Todd Klindt' s blog post \"How to use PowerShell to set the Master Page in SharePoint 2010\" at http:\/\/drisgill.com\/go\/powershell-masterpage.\n\nUnderstanding SharePoint Online\n\nThroughout this book, branding SharePoint 2013 is mostly discussed when it is installed in your own farm. As of the writing of this book, that is by far the most common way SharePoint is used, but Microsoft is keen to move SharePoint installations into its cloud-based offering known as Office 365. Microsoft refers to SharePoint in Office 365 as SharePoint Online.\n\nAs a designer, you may wonder how SharePoint Online affects the branding process. Interestingly enough, the SharePoint Online differences are drawn along a division between whether the site is public-facing or an intranet site.\n\nWARNING\n\nWriting about SharePoint Online is a lot like trying to hit a moving target. Microsoft treats the 2013 version of SharePoint Online as a versionless product. What you see today in SharePoint Online for the user interface, menus, and even functionality could be vastly different than how it looked at the time of this writing.\n\nSharePoint Online Intranet Sites\n\nThe great thing about branding intranet sites in SharePoint Online is that there isn't much difference from branding traditional intranet sites for SharePoint installed in the farm. For instance, see if you can tell which of the following two screen shots is a team site in SharePoint Online and which is a farm-based team site.\n\nIf you said the second one is the SharePoint Online site, you are correct! Pretty much the only thing that visually differentiates a SharePoint online intranet site from a farm-based one is the Suite Bar at the top of the ribbon. In SharePoint Online, this bar shows a logo for Office 365 on the left, and you can see more options related to navigating to the other sections of Office 365 on the right side of the bar. That being said, under the covers there are some significant differences in SharePoint Online worth mentioning.\n\nPublishing Sites in SharePoint Online\n\nAs you have learned in previous chapters, SharePoint branding is often enhanced by the abilities that come with the publishing features that are included with SharePoint Server in a typical farm-based install. For SharePoint Online, there are some subtle nuances that dictate which sites support the publishing features based on what license edition you have. This can be confusing because all editions of SharePoint online include the ability to have one site designated as a public website, which has some aspects of publishing, as you will learn about in the next section.\n\nSmall Business editions of Office 365 include just one team site, which does not include publishing, plus the public website. Enterprise editions of Office 365 come with a team site collection at the root web application for the domain that does not include publishing; however, they do include the ability to create new publishing site collections underneath that root web application. For more information on Office 365 editions and SharePoint Online, see: http:\/\/www.microsoft.com\/en-us\/office365\/compare-plans.aspx\n\nMapping a Drive to SharePoint Online\n\nYou can create SharePoint Online branding much like you have in the other chapters of this book for traditional SharePoint branding. In previous chapters you used the Design Manager and worked with custom master pages; in both cases mapping a drive to the Master Page Gallery was an important step. You can still use Design Manager (if you have a publishing site) and you can still map drives to SharePoint Online, but the process of getting authenticated before mapping the drive can be a little challenging in SharePoint Online. One trick is to make sure to check the boxes to both Remember Me and Keep Me Signed In, before you click the Sign In button. This ensures that you can seamlessly map a drive from Windows to the SharePoint site.\n\nIf you still run into trouble logging in and mapping a drive to SharePoint Online, see \"How to configure and to troubleshoot mapped network drives that connect to SharePoint Online sites in Office 365 for enterprises\" at http:\/\/support.microsoft.com\/kb\/2616712.\n\nSandboxed Solutions\n\nProbably the biggest difference for working with SharePoint Online is the lack of ability to add farm-based Visual Studio solutions. SharePoint 2013 adds the concept of apps to help traditional developers build powerful solutions and still deploy them to SharePoint Online, but unfortunately the SharePoint App model doesn't make it easy to deploy and apply branding to SharePoint Online. Instead, to deploy branding to a SharePoint Online site, you can use a sandboxed solution. For more information on sandboxed solutions see \"Sandboxed solutions overview\" at http:\/\/technet.microsoft.com\/en-us\/library\/ee721992(v=office.14).aspx. This article talks about creating solutions for SharePoint 2010 but the concepts are similar.\n\nSharePoint Online Public Sites\n\nAs previously mentioned, all SharePoint Online editions have the ability to create one public-facing website ideal for small businesses that is based on SharePoint 2013. These public sites are worth noting because this is one area of SharePoint Online in which SharePoint deviates a lot from what is available in a farm install. If you have played with public sites in SharePoint Online prior to 2013, you probably were less than thrilled with the implementation. Public sites in SharePoint Online 2010 were based on an older product named Office Live Small Business, which was wedged into the SharePoint 2010 interface and lacked many of the useful features used for SharePoint 2010 branding. For SharePoint 2013, Microsoft completely retooled the public site experience to more closely match the typical SharePoint 2013 branding experience. For instance, it includes the ability to work with master pages, page layouts, composed looks, and even some aspects of the Design Manager.\n\nNOTE\n\nFor more information on creating a public site in SharePoint online, see Microsoft's article, \"Set Up a Public Website\" at `http:\/\/drisgill.com\/go\/spo-public-site`.\n\nThe first time you see a public site in SharePoint Online 2013, you can immediately notice that something is different from a traditional default SharePoint 2013 homepage. Although you can certainly see the hallmarks of a normal SharePoint 2013 site such as the ribbon and site logo, the bulk of the design looks nothing like the default Seattle master page or any of the usual composed looks. The UI is sparse, and there is a large background image floating behind the text.\n\nNot only does the site look different, but also the menus and options for SharePoint online have been changed some to focus the user on activities geared toward working with a public-facing website for small businesses. These changes include different ribbon menus, extra out-of-the-box master pages named after cities (such as Berlin, Lyon, and Tokyo), and new composed looks for small businesses, such as hotels and restaurants.\n\nSharePoint Online public sites even include some custom controls such as persistent headers and footers that can be edited in one place and shown on many pages.\n\nProbably the most interesting difference in SharePoint Online public sites is the Site tab on the ribbon. This tab includes several helpful items for working with a small business website. If you browse to the homepage of your public site, and click the Site tab, you can see the following options.\n\n\u2022 Change the Look\u2014A shortcut to the Change the Look menu from Site Settings.\n\n\u2022 Edit Title\u2014A dialog menu to quickly edit the site title and description. This menu is similar to the Title, Description, and Logo menu from Site Settings but without the logo options.\n\n\u2022 Change Logo\u2014A dialog menu to quickly change the site logo. Also similar to the Title, Description, and Logo menu from Site Settings.\n\n\u2022 Edit Menu\u2014Opens the quick editing of the top navigation. This is the same as clicking the Edit Links button next to the navigation.\n\n\u2022 Edit Site Elements\u2014One of the more interesting options in this menu. Clicking this puts the page into a mode where you can edit some common elements that all pages share; this includes the Header and Footer sections. Any edits made here will persist across all pages. This item has no similar functionality in a farm install of SharePoint.\n\n\u2022 Search Box\u2014A dialog menu for hiding the search box.\n\n\u2022 Edit Style Sheet\u2014A dialog box for editing the already applied Alternate CSS style sheet. You learned about Alternate CSS in Chapter 6, \"Cascading Style Sheets and SharePoint,\" but this option is different than any available in a farm install of SharePoint. This enables you to quickly edit and save changes to the Alternate CSS in the web UI without having to use a text editor.\n\nAfter you get past some of these changes for SharePoint Online public sites, you can use many of the same techniques you learned about throughout this book to create branding for them. For example, you can still use the Design Manager to create HTML master pages and page layouts, and you can still map a folder to the master page gallery and create custom master pages. Just remember that public-facing sites have different needs than internal intranet sites. Be sure to also take into account that the default pages in a SharePoint Online public site are set up with different page layouts than farm installs of SharePoint, and they can include things such as editable persistent headers and footers. Be sure to thoroughly test your pages in your new branding before letting the public view the site.\n\nNOTE\n\nPublic Sites in SharePoint Online include a button at the top right labeled either Make Website Online or Make Website Offline. These options can be useful when you are first building the site or if you make large sweeping changes. You can edit the site when it's offline and then bring it back online when your edits are finished.\n\nSummary\n\n\u2022 SharePoint navigation has several settings that are only available by editing code in master pages. This includes the AspMenu, which actually shows the navigation and the SiteMapDataSource, which provides data to the AspMenu.\n\n\u2022 Two of the most commonly changed properties of AspMenu are StaticDisplayLevels and MaximumDynamicDisplayLevels, which control how many levels of navigation show normally and as fly-outs when you hover over parent items.\n\n\u2022 The tree view is useful specifically on collaboration sites for showing a scrollable hierarchy of the site contents as well as providing metadata navigation and filtering.\n\n\u2022 Traditional breadcrumbs were not included with SharePoint 2013 out-of-the-box. The SiteMapPath control can be used to show breadcrumbs in your SharePoint site.\n\n\u2022 The $SPUrl tokens can be used in SharePoint Server websites to create links that dynamically point back to the top of a site or site collection.\n\n\u2022 Anonymous users have unique needs in SharePoint sites. You can hide the ribbon from displaying to anonymous users, but be sure to also include a link for them to log in if they have credentials on the site.\n\n\u2022 Device Channels can be used to show certain master pages or even parts of a master page to devices based on their browsers' user agent strings.\n\n\u2022 My Sites use a unique master page named mysite15.master to apply its branding. This master page is similar to the default SharePoint branding but is simplified considerably. Also, My Sites branding is typically packaged in a solution package (WSP) so that it can be applied programmatically as new personal sites are created.\n\n\u2022 Office 365 includes SharePoint Online intranet sites, which are very similar to traditional farm-based SharePoint sites. You can also create one public site in SharePoint Online that can be used to easily style a public-facing website for small businesses. This concept is unique to Office 365 and not included with a farm install of SharePoint.\n9\n\nCreating Content Rollups with SharePoint WCM\n\nWhat's In This Chapter\n\n1. Understanding SharePoint Rollups\n\n2. Working with the Content Query Web Part (CQWP)\n\n3. Configuring and using the Content Search Web Part (CSWP)\n\n4. Creating and Applying Display Templates\n\nMuch of this book has focused on the specific ways to customize the look and feel of a SharePoint site. For branding and designing in SharePoint, master pages, CSS, and page layouts are usually the first things that someone thinks about. However, to implement a custom design means that in many instances you have to also consider how to style some of the functional areas of the site. This chapter discusses the basics of creating one of the most common functional areas that SharePoint branders and designers will encounter\u2014rolling up and styling content with SharePoint WCM Web Parts.\n\nAn Introduction to Rollups in SharePoint\n\nChapter 4, \"Planning for Branding,\" talks about creating an information architecture, taxonomy, wire frames, and realistic design comps. That process should be fairly technology-agnostic and similar to designing almost any type of site. Some SharePoint specifics are discussed, but for the most part that process should be fairly common to any design project. The chapters following Chapter 4 walk you through the various techniques required to create the major elements of your design within SharePoint, which to this point have been mostly static. This chapter discusses how to customize the rollup Web Parts that deliver dynamic content to SharePoint pages. The ability to combine a custom look and feel with dynamic elements is one of the more powerful aspects of SharePoint. For example, think back to the design for Randy's Waffles where a section on the homepage displayed products. You could have just hard-coded the HTML for those products in a page layout or a Content Editor Web Part (CEWP). But imagine if one of the products changed pricing and you needed to update the information. In a static website, that means you'd have to go to every page where that product price is displayed and make the change. In large sites this could mean updating several pages. The good news is that there's a better way!\n\nUnderstanding WCM Capabilities\n\nSharePoint is a robust Web Content Management (WCM) platform that enables you to efficiently manage content. One of the most basic concepts of WCM is that content can be created in a single place and surfaced throughout the site wherever it is needed. In other words, rather than needing to copy the same piece of content to make it show in several places, WCM enables you to create it once and control the rollup of that content to make it display where you need it. In the example for Randy's Waffles, this means creating products and surfacing the content for them wherever the design dictates, such as on the homepage and potentially other pages throughout the site. If you need to update a product's pricing, you can simply edit the single product details page, and then wherever that product is displayed the updated pricing will appear, whether it is on one page or several pages.\n\nThis has far-reaching practical implications beyond editing products on fictional websites. An example of a product-centric site is one of the easiest to demonstrate, but here are a few other common examples of rollups that might apply in your environment:\n\n\u2022 News\u2014Different departments often create content to share with the rest of an organization. Content can be actual press releases or just general information. You can use rollups so that these pieces of content can be displayed in a common location on the homepage. Similarly, if content is tagged with metadata, you can roll them up to other more specific pages as needed.\n\n\u2022 Events\u2014Similar to the news scenario, events occur at specific dates and times. As sites across your organization create their own events, it is common to have these events roll up to a central place so that users can get a consolidated view of what is happening in an organization.\n\n\u2022 Related documents\u2014One of the more complex rollup scenarios, related documents requires you to tag content with metadata so you can roll up a specific tag as needed. For example, if you have a Project Management Office site and want to add a Web Part to roll up all content tagged as a Project Plan and related to a SharePoint Project, you can create content organically but make it surface to the website without someone from within the department specifically seeking it out.\n\nThese are just a few of the more common scenarios you might consider as you plan your site. SharePoint 2013's WCM capabilities have significantly improved over previous versions, and this opens up a huge number of possible scenarios. The topics covered in this book provide a foundation for styling simple scenarios as well as give you a starting point for more complex scenarios.\n\nBefore Getting Started with Rollups\n\nThere are a few concepts that might be worth reviewing before diving into rollups. Chapter 2, \"SharePoint Overview,\" discusses the various versions of SharePoint along with a brief overview of Web Content Management (WCM), also referred to as publishing in SharePoint. The techniques referenced in this chapter require publishing functionality, so a basic understanding of these concepts covered in Chapter 2 will be helpful.\n\nTo further complicate things, although the rollup options discussed in this chapter are all considered publishing functionality, not all options are available across all versions of SharePoint with publishing. Therefore, each section in this chapter briefly covers in which SharePoint versions the respective functionality is available. This is just a reminder to keep your eyes open because the techniques and functionality covered in this chapter might not be available to everyone based on the version of SharePoint you are using.\n\nNOTE\n\nFor a more detailed comparison of different versions of SharePoint, see the following: <http:\/\/technet.microsoft.com\/en-us\/library\/jj819267.aspx>. Although the title references SharePoint Online, all versions, including on-premises versions, are covered.\n\nUsing the Content Query Web Part (CQWP)\n\nSince SharePoint Server 2007, the Content Query Web Part (CQWP) has been the primary way to roll up content. The CQWP is included in all versions of SharePoint 2013 where it is possible to enable publishing. This Web Part is fairly straightforward and does exactly what its name implies: It rolls up content based on a specific query. This is a common task on just about any dynamic website. The CQWP Web Part was designed to be reusable so that whenever a need to display content rolled up from within the site arose, a user could add the Web Part to a page, define a specific query, and have the content presented on the page. This is obviously much more efficient than writing a custom Web Part every time you want to display some rolled up content. The user can also change the query, as well as the results it produces, without the need of a developer.\n\nHow the CQWP Works\n\nAfter you define your query, the CQWP looks for content matching your query throughout the site collection. Then it displays the results, which are presented using predefined XSL templates. You can style these XSL templates or even create your own for further flexibility. The Web Part results are security trimmed, meaning that CQWP respects SharePoint user security permissions. This means that if a user doesn't have access to view the content, the Web Part won't display it.\n\nAlthough displayed CQWP results are customizable via XSL, few folks consider themselves XSL experts. Working with XSL can be tricky and difficult to debug when you make a mistake. For example, if you make a typo in your XSL, you'll quickly realize that something isn't correct. But the challenge is that big mistakes fail the same way that minor typos do, making it tedious to find your exact error. Those mistakes will break (until it's fixed) every web part that uses the XSL. Patience is important when creating custom XSL for the CQWP; even experienced designers deal with many of the same challenges.\n\nUnlike the Content Search Web Part (CSWP), discussed later in the section \"Using the Content Search Web Part (CSWP),\" the CQWP doesn't rely on search, and instead queries down through the site structure of a site collection. This means that returned results display as soon as content is modified. Note that the CQWP can display results only from within the site collection where it is being used.\n\nFor many sites, the CQWP is perfectly suited to meet your content rollup needs. However, you might find the Web Part limiting for larger implementations with more complex requirements. No matter the size of your implementation, the CQWP is likely to be to used. Understanding how to customize the CQWP is a useful skill for any SharePoint designer.\n\nCustomizing the Content Query Web Part\n\nYou can customize your CQWP in many ways, but the general customization process begins with creating custom content for your rollup, then moves into creating custom styles, and ends with applying these custom styles to your CQWP.\n\nExample: Creating the Content for your CQWP\n\n1. From the homepage of your newly branded Randy's Waffles site that you set up in Chapters 5 and 6, click the Settings menu \u21d2 Site contents.\n\n2. At the bottom of the site contents page, click the New subsite link.\n\n3. On the New SharePoint Site menu, enter the following settings:\n\n\u2022 Title\u2014Breakfast Talk\n\n\u2022 Description\u2014You can optionally type a description for the new site.\n\n\u2022 URL name\u2014`BreakfastTalk`\n\n\u2022 Select a template\u2014Click the Publishing tab and choose the Publishing Site template.\n\n\u2022 User Permissions\u2014Leave the default setting Use Same Permissions As Parent Site.\n\n\u2022 Navigation Inheritance\u2014Select Yes to use the same items in the top navigation as the parent.\n\n4. Click Create.\n\n5. When the site is created, click the Settings menu \u21d2 Add a Page. In the Add a Page dialog box, enter Article1 for the name and click Create.\n\n6. When the page is created, click the Page tab on the ribbon; then click the Page Layout drop-down and select the Image on Left layout option.\n\n7. In the Page Image field control, click the Click here to insert a picture from SharePoint link, and on the Edit Image Properties dialog, click the Browse button for the Selected Image input field.\n\n8. On the Select an Asset dialog, click in the Location (URL) field and type http:\/\/<`yoursite`>\/_catalogs\/masterpage\/Waffles then press Enter. (Replace `<yoursite>` with the actual URL of your site, such as <http:\/\/wafflenet>.) The Location (URL) field will change to a relative URL to the Waffles folder in the Master Page Gallery.\n\n9. Within the Select an Asset dialog box, click the Assets folder to open it; then select one of the images in the folder (BlogPhoto1.jpg, for example) and click Insert.\n\n10. On the Edit Image Properties dialog, click OK.\n\n11. Fill in the remaining fields on the Article1 page.\n\n\u2022 Type today's date in the Article Date field.\n\n\u2022 Type a name in the Byline field.\n\n\u2022 Add some content to the Page Content field as well.\n\n\u2022 (Optional) You can fill in the Image Caption if you want, but this content won't be included in the rollup later.\n\nNOTE\n\nIf you need to insert some placeholder text quickly, you can take advantage of a couple hidden features in Microsoft Word that generate placeholder text. In a Word document, type =lorem() and press Enter to generate several paragraphs of Lorem Ipsum text that you can copy and paste into SharePoint as filler text. If you want more control over the content of the placeholder text, you can specify how many paragraphs and sentences you want to include. For example, if you typed =lorem(5,4) and press Enter, you generate five paragraphs of lorem ipsum text, each containing four sentences. (You can also use =rand() to the same effect, but this generates actual words and sentences instead of the fake lorem ipsum text.)\n\n12. After you add content to your page, click the Publish tab on the ribbon and click Publish.\n\n13. Create three more pages in the Breakfast Talk site, using different images, article dates, bylines, and content.\n\nNow that you have created some content that you can roll up, it's time to get to work on styling the CQWP. For this particular rollup, you add the CQWP to the homepage of your site and have it roll up the articles from the newly created Breakfast Talk site. Because space on the homepage is limited, you need to truncate the text of the articles so that just a snippet of the article text shows. The remainder of the article is truncated and is followed by an ellipsis (...). You also need to include a picture of the post author, the author's name, and the date of the post.\n\nYou first want to edit the XSL style sheet containing the different templates that you can apply to the CQWP. Then add a CQWP to your site's homepage, configure it to roll up the content you just created, and apply your item style to the rollup, as shown in the following steps:\n\nExample: Styling your CQWP\n\n1. Navigate back to the homepage of your site collection, and click the Settings menu \u21d2 Site Contents.\n\n2. Click Style Library; then click the XSL Style Sheets folder to open it.\n\n3. Click the ellipsis (...) to the right of ItemStyle, and then click the ellipsis in the menu after Follow. Click Check Out.\n\nNOTE\n\nYou can also select ItemStyle by clicking the check box to the left of the filename and using the Check Out button in the Files tab on the ribbon.\n\nNOTE\n\nFor a refresher on how to map a drive, refer to Chapter 5.\n\n4. Map a network drive to the root of your site (for example, <http:\/\/wafflenet\/>).When your mapped drive opens, browse to the Style Library \u21d2 XSL Style Sheets folder.\n\n5. Before going further, make a backup copy of ItemStyle.xsl and save it somewhere on your computer. This way, you can quickly revert to the original contents of the file if you make a mistake. Keep in mind that if the custom XSL you add to ItemStyle.xsl doesn't have the correct syntax, none of the Content Query Web Parts that use ItemStyle.xsl can display any content in the site, so it's a good idea to keep a backup copy of this file somewhere. You can always revert to a previous version of this file using SharePoint's version history, or you can copy the contents from another copy of ItemStyle.xsl from a different SharePoint site collection. Generally, making a backup is considered the best option.\n\n6. Open ItemStyle.xsl in the editor of your choice.\n\n7. At the top of ItemStyle.xsl in the `<xsl:stylesheet`... declaration section, place the cursor after the line `xmlns:cmswrt=\"http:\/\/schemas.microsoft.com\/WebParts\/v3\/Publishing\/runtime\"` and press Enter to create a new line. Then add the following to the new line:\n\n`xmlns:ddwrt=\"http:\/\/schemas.microsoft.com\/WebParts\/v2\/DataView\/runtime\"`\n\n8. Scroll to the end of the ItemStyle.xsl file. In the second to last line, place the cursor in front of the closing `<\/xsl:stylesheet>` tag and press Enter to create a new line above it and add the following:\n\nNOTE\n\nThis step uses templates from Waldek Mastykarz's blog post at http:\/\/blog.mastykarz.nl\/generating-short-description-content-query-web-part\/. Although the blog post references SharePoint 2007, the concepts and code still work perfectly fine in SharePoint 2013. You can copy the templates with names that start with `Imtech` from this blog post if you would rather not type them out.\n\n`<xsl:template name=\"CustomPostRollup\" `\n\n` match=\"Row[@Style='CustomPostRollup']\" mode=\"itemstyle\">`\n\n` <xsl:variable name=\"SafeLinkUrl\">`\n\n` <xsl:call-template name=\"OuterTemplate.GetSafeLink\">`\n\n` <xsl:with-param name=\"UrlColumnName\" select=\"'LinkUrl'\"\/>`\n\n` <\/xsl:call-template>`\n\n` <\/xsl:variable> `\n\n` `\n\n` <div class=\"customBlogItem\">`\n\n` <div class=\"customBlogImage\">`\n\n` <xsl:value-of select=\"@PublishingRollupImage\" `\n\n` disable-output-escaping=\"yes\"\/>`\n\n` <\/div>`\n\n` <xsl:value-of select=\"@Title\"\/> - <xsl:value-of select=\"`\n\n` ddwrt:FormatDateTime(string(@ArticleStartDate),1033,`\n\n` 'MMM dd yyyy')\" \/><br\/>`\n\n` <a href=\"{$SafeLinkUrl}\">`\n\n` <xsl:call-template name=\"Imtech.GenerateSummary\">`\n\n` <xsl:with-param name=\"Content\" `\n\n` select=\"@PublishingPageContent\" \/>`\n\n` <xsl:with-param name=\"Length\" select=\"80\" \/> `\n\n` <xsl:with-param name=\"Suffix\" select=\"'...'\"\/>`\n\n` <\/xsl:call-template>`\n\n` <\/a>`\n\n` <div class=\"customClear\"><\/div>`\n\n` <\/div>`\n\n`<\/xsl:template> `\n\n` `\n\n`<xsl:template name=\"Imtech.GenerateSummary\">`\n\n` <xsl:param name=\"Content\"\/>`\n\n` <xsl:param name=\"Length\"\/>`\n\n` <xsl:param name=\"Suffix\"\/>`\n\n` <xsl:variable name=\"cleanContent\">`\n\n` <xsl:call-template name=\"Imtech.RemoveHtml\">`\n\n` <xsl:with-param name=\"String\" select=\"$Content\"\/>`\n\n` <\/xsl:call-template>`\n\n` <\/xsl:variable>`\n\n` <xsl:call-template name=\"Imtech.SubstringBeforeLast\">`\n\n` <xsl:with-param name=\"String\"`\n\n` select=\"substring($cleanContent, 1, $Length)\"\/>`\n\n` <xsl:with-param name=\"Char\" select=\"' '\"\/>`\n\n` <\/xsl:call-template>`\n\n` <xsl:if test=\"string-length($cleanContent) &gt; $Length\">`\n\n` <xsl:value-of select=\"$Suffix\" disable-output-escaping=\"yes\"\/>`\n\n` <\/xsl:if>`\n\n`<\/xsl:template>`\n\n` `\n\n`<xsl:template name=\"Imtech.RemoveHtml\">`\n\n` <xsl:param name=\"String\"\/>`\n\n` <xsl:choose>`\n\n` <xsl:when test=\"contains($String, '&lt;')\">`\n\n` <xsl:value-of select=\"substring-before($String, '&lt;')\"\/>`\n\n` <xsl:call-template name=\"Imtech.RemoveHtml\">`\n\n` <xsl:with-param name=\"String\" select=\"substring-after($String,`\n\n` '&gt;')\"\/>`\n\n` <\/xsl:call-template>`\n\n` <\/xsl:when>`\n\n` <xsl:otherwise>`\n\n` <xsl:value-of select=\"$String\"\/>`\n\n` <\/xsl:otherwise>`\n\n` <\/xsl:choose>`\n\n`<\/xsl:template>`\n\n` `\n\n` <xsl:template name=\"Imtech.SubstringBeforeLast\">`\n\n` <xsl:param name=\"String\" \/>`\n\n` <xsl:param name=\"Char\" \/>`\n\n` <xsl:param name=\"subsequent\"\/>`\n\n` <xsl:choose>`\n\n` <xsl:when test=\"contains($String, $Char)\">`\n\n` <xsl:if test=\"$subsequent = 1\">`\n\n` <xsl:value-of select=\"$Char\"\/>`\n\n` <\/xsl:if>`\n\n` <xsl:value-of select=\"substring-before($String, $Char)\"\/>`\n\n` <xsl:call-template name=\"Imtech.SubstringBeforeLast\">`\n\n` <xsl:with-param name=\"String\" select=\"substring-after($String,`\n\n` $Char)\" \/>`\n\n` <xsl:with-param name=\"Char\" select=\"$Char\" \/>`\n\n` <xsl:with-param name=\"subsequent\" select=\"1\"\/>`\n\n` <\/xsl:call-template>`\n\n` <\/xsl:when>`\n\n` <xsl:otherwise>`\n\n` <xsl:if test=\"$subsequent != 1\">`\n\n` <xsl:value-of select=\"$String\"\/>`\n\n` <\/xsl:if>`\n\n` <\/xsl:otherwise>`\n\n` <\/xsl:choose>`\n\n` <\/xsl:template>`\n\n9. Save the ItemStyle.xsl.\n\n10. Return to the browser and refresh the Style Library\/XSL Style Sheets page. Select the check box next to ItemStyle.xsl; click the Files tab on the ribbon and click Check In. Choose to publish a major version and click OK.\n\nYou have just added some custom styles to the ItemStyle.xsl. First, you added a line in the declaration section that helps the XSL to properly format dates. Then you added several templates to the ItemStyle file. The first one (the one you use to apply to a CQWP) is called `CustomPostRollup` and tells the CQWP how to format the data it rolls up. The additional styles that begin with `<xsl:template name=\"Imtech...>` are referenced by the `CustomPostRollup` to truncate the text the rollup displays and also to help clean up the display by stripping out the HTML markup.\n\nNow it's time to put all these pieces together and add a CQWP to your homepage and apply your custom item style template. If you added any placeholder Web Parts to your homepage while following the examples in Chapter 5, you will delete them in the following example.\n\nExample: Adding and Configuring the CQWP\n\n1. Navigate to the homepage of your site and click the Edit button on the ribbon. (If you don't see the Edit button, click the Settings menu \u21d2 Show Ribbon.)\n\n2. If you added a Web Part in Zone 2 in Chapter 5 you can delete it by hovering over the Web Part title, clicking the drop-down arrow that appears toward the right side, and selecting Delete from the Web Part menu. Click OK when you are prompted to delete the Web Part.\n\n3. In Zone 2, click the Add a Web Part link.\n\n4. From the Web Part menu at the top under Categories, click Content Rollup, and then under Parts click Content Query. Then click Add.\n\n5. In the Content Query Web Part that you just added, click the open the tool pane link. Alternatively, you can click the drop-down arrow that appears toward the right side of the Web Part header when you hover over it, and select Edit Web Part.\n\n6. In the tool pane that appears to the right of your page content, expand the Query section by clicking the plus symbol or the Query header, and select the option to Show items from the following site and all subsites; then click the Browse button.\n\n7. In the Select Site dialog that appears, click the Breakfast Talk site, and then click the OK button.\n\n8. In the Content Type section, change the first drop-down to Page Layout Content Types, and then make sure the second drop-down is set to Article Page.\n\n9. Expand the Presentation section of the tool pane, and in the Grouping and Sorting section, change the Sort items by drop-down to Article Date.\n\n10. Below the Sort items by section, change Limit the number of items to display from 15 to 4, and in the Styles section, change the Item style drop-down to CustomPostRollup.\n\n11. In the Fields to display section, fill in the fields as follows:\n\n\u2022 Link\u2014URL Path;\n\n\u2022 Title\u2014Byline;\n\n\u2022 PublishingRollupImage\u2014Page Image;\n\n\u2022 ArticleStartDate\u2014Article Date;\n\n\u2022 PublishingPageContent\u2014Page Content;\n\n12. Expand the Appearance section of the tool pane, and change the Title field to Breakfast Talk and then click OK.\n\n13. Click the Publish tab on the ribbon, then click the Publish button to publish the page.\n\nThere you have it! Your custom CQWP now displays the page image, the author, the article date, and a limited text preview of the content.\n\nNOTE\n\nThis particular rollup is also set up to utilize the rollup image field that is built in to the out-of-the-box article pages in SharePoint 2013, which enable you to display a different picture on your rollup than you do on the actual article page, just as you could in previous versions of SharePoint. If you want to use the rollup image field to display in your rollup instead of the page field, you would change the value of the `PublishingRollupImage` field in the Fields to display the section to Rollup Image instead of Page Image. If you aren't sure if content authors on your site are going to use the Page Image or the Rollup Image, you can set the rollup to look for both images by typing Page Image; Rollup Image; in the field. The page image takes precedence over the rollup image in this case (if you reversed the order, the rollup image would take precedence), but if there isn't a page image set on the page, the rollup image will display.\n\nUsing the Content Search Web Part (CSWP)\n\nAs mentioned in the previous section, the CQWP became the primary way to do rollups in SharePoint since its introduction in SharePoint Server 2007. Chapter 2 covered several of the new enhancements to SharePoint. One of the biggest changes to SharePoint 2013 is the addition of the Content Search Web Part (CSWP), which leverages SharePoint search to roll up the content.\n\nNOTE\n\nAs of this writing, the CSWP is only available in SharePoint Server 2013 Enterprise with sites that have enabled publishing. It is currently not available in any of the SharePoint Online SKUs from Office 365, but Microsoft has stated that it intends to add the functionality in the future.\n\nThe CSWP can show any content as long as SharePoint search can crawl the content and add it to the index. There are a number of benefits to this approach:\n\n\u2022 You can roll up content across site collection boundaries. In SharePoint, a single search service can crawl, index, and provide results for whatever web applications are associated with it. This means that if you can find a piece of content with a search, the CSWP can surface the content. When you perform a search, you type in a search query and click the Search button before receiving results. For the CSWP, you specify the search query in the Web Part configuration settings to return the content you want to roll up.\n\n\u2022 The CSWP has better performance than the CQWP. Because the CSWP is based on the SharePoint search feature, it has excellent performance even when querying with complex rules and filters.\n\n\u2022 You can style the CSWP using HTML, CSS, and JavaScript. While the CQWP uses XSLT for styling the results, that is often difficult to debug, so the CSWP uses display templates for styling. Display templates are based on HTML and they can be used with CSS and JavaScript to easily style results.\n\nKeep in mind that, similar to CQWP, the content that the CSWP returns is security trimmed. If a user doesn't have access to a specific piece of content then it isn't included within the results.\n\nNOTE\n\nOne important thing to consider about this Web Part is that just as with SharePoint search, content needs to be checked in, published, and crawled by the search index before it is available to be returned by the Web Part.\n\nPrepping for the CSWP\n\nBecause the CSWP shows content that has been indexed by the SharePoint search crawl, it is a good idea to ensure you have some content indexed before looking at the options available in the CSWP. In the following example you create content that can be used in later examples, in which you will finish the original Randy's Waffles example (from Chapter 5 and Chapter 6) by adding a CSWP to the bottom-left corner that rolls up publishing pages in a subsite.\n\nThe CSWP examples in this chapter assume you are using managed navigation on your site. In general the examples work with structured navigation just fine, but the steps may be different for you. If you haven't worked with managed navigation before you may want to review Chapter 3, \"Working with the SharePoint 2013 User Interface.\" Pay particular attention to the warning in the section titled \"Working with Navigation in SharePoint 2013\"; this describes an issue you might run into when first configuring managed navigation on your site.\n\nChecking your FURL Setting\n\nThe following example contains one tricky step in relation to how pages are created in SharePoint 2013. Essentially, if you have friendly URLs (FURLs) turned on in the navigation settings, the URL for a subsite behaves differently than if you have friendly URLs turned off. This becomes important in later examples because when FURLs are on, newly created pages under a subsite have URLs that look like they belong to the root site, for example: http:\/\/rootsite\/subpage. Alternatively, if FURLs are turned off, the pages look like they normally do in SharePoint: http:\/\/rootsite\/subsite\/pages\/subpage.aspx.\n\nYou can check your FURL setting by navigating to Site Settings and under Look and Feel, click Navigation. Scroll down to Managed Navigation: Default Page Settings. From there you can check the setting for Create Friendly URLs for New Pages Automatically to see whether FURLs are turned On or Off. The default behavior is for SharePoint to automatically create friendly URLs for all new pages.\n\nYou can use whichever setting you prefer here, but make note of it as there will be extra steps to follow if you have FURL's turned On.\n\nExample: Creating Content for the CSWP\n\n1. Create a publishing-based subsite named SaleItems.\n\n2. Inside of the new subsite create a new publishing page.\n\n3. Switch the page layout to Article Page - Image on Left.\n\n4. Add a Page Image, a product price for the Byline, and Page Content. You can use fake content if you'd like, but make the image fairly small in size\u2014about 150px wide by 100px tall.\n\nNOTE\n\nFor a refresher on how to map a drive, refer to Chapter 5.\n\n5. After entering the page content, save, check-in, and publish the page.\n\n6. The example later in the chapter looks best with six results, so repeat steps 2 through 5 for five more pages; try to enter different content for each page so that you see different results when the CSWP rolls them up.\n\nNow that the pages are created, it's a good time to consider the previous warning about FURLs. If you have FURLs turned Off, you don't need to do anything extra at this time. If FURLs are turned On (the default setting), there are some extra steps that need to be performed. Follow these steps after you have created all the publishing pages.\n\nExample: Organizing Pages with FURLs\n\n1. Click the Settings menu \u21d2 Site Settings \u21d2 Site Administration \u21d2 Term Store Management.\n\n2. In the menu on the left, expand the folder for your site collection, and then expand the node for your navigation. If SharePoint created the node for you the node is called Site Navigation.\n\n3. Click the drop-down next to your navigation node and select Create Term, and type Sale-Items. You'll use this term as a parent for the subpage terms that are auto-generated. It's important not to name this term exactly the same as the name of your subsite, otherwise SharePoint will have problems navigating to the node.\n\n4. Click on the new term Sale-Items and then on the right side of the page click the Navigation tab. Change the Navigation Node Type radio button to Simple Link or Header and click the Save button.\n\n5. For each publishing page that you created earlier, click the drop-down that is next to the name of the corresponding node, and then select Move Term. In the dialog that appears, select Sale-Items and click OK.\n\nNow that you have content created and arranged properly under the subsite you need to get the content indexed by SharePoint search. If you have continuous crawling turned on for your SharePoint search, the new pages will eventually be crawled automatically. Otherwise, follow the next example to include the new pages in the search index.\n\nExample: Crawl the New Publishing Pages\n\n1. On your SharePoint server, open SharePoint 2013 Central Administration.\n\n2. Under Application Management, click Manage Service Applications.\n\n3. Scroll down and click the first entry for Search Service Application.\n\n4. From the left menu under Crawling, click Content Sources. The page will refresh quickly, and then click Start All Crawls.\n\n5. The status updates to Starting.... Wait a minute or two and then check the status by clicking Refresh. Eventually, the crawl finishes and the status changes to Idle.\n\nWARNING\n\nIf you have never run a crawl on your site before you must run a full crawl, which may take a while to complete.\n\nWhen the status changes to Idle, your search index should be updated to include the new pages. You can check to see if they have been indexed by searching for the new pages' content on your SharePoint site. If the search results include your new pages, they were successfully added to the index. If your new pages aren't in the results then you need to go back and check to make sure they were published and ensure that SharePoint search is set up to crawl the site properly.\n\nNow that you have publishing pages created and indexed by SharePoint search you can add a CSWP to a publishing page to look at the available options the CSWP provides. Like the Content Query Web Part example earlier, this example uses the homepage design of Randy's Waffles from Chapter 5. If you haven't been following along with the previous chapters you can create a new publishing page, but the example steps will be different. Because the CSWP is an Enterprise feature, you must have a publishing site on SharePoint Server with an Enterprise License to follow along with the example.\n\nExample: Adding a CSWP to a Publishing Page\n\n1. Navigate to the homepage of your site and put the page in Edit mode.\n\n2. If you added a Web Part in Zone 1 in Chapter 5 you can delete it by hovering over the Web Part title, clicking the drop-down arrow that appears toward the right side, and selecting Delete from the Web Part menu. Click OK when you are prompted to delete the Web Part.\n\n3. In Zone 1, click the Add a Web Part link.\n\n4. From the Web Part menu at the top under Categories, click Content Rollup, and then under Parts click Content Search. Click Add. The CSWP is loaded on the page and by default shows recently updated pages.\n\nWhew, that was a lot of work just to look at a Web Part, but it is necessary to fully understand how queries work for the CSWP. In the next section you take a look at some of the options that are available for queries in the CSWP.\n\nUnderstanding CSWP Queries\n\nTake a quick look at the Web Part properties for the CSWP by clicking the little arrow drop-down on the right side of the CSWP title and selecting Edit Web Part. The very first option in the Web Part properties menu under the Search Criteria section is the Change query button.\n\nClicking this button opens a dialog with a powerful query builder that includes several tabs with options for querying content. The next few sections look at the most commonly used options that are available in these tabs.\n\nBasics Tab\n\nThe Build Your Query dialog opens to the first tab called Basics. Under the Basics section you are provided with a few initial options for querying content:\n\n\u2022 Select a query \u2014 Provides a drop-down of content sources that can be selected for querying content.\n\n\u2022 Restrict by app \u2014 Filters the query to only return results from a specific location.\n\n\u2022 Restrict by tag \u2014 Filters the query to only return results tagged with specific metadata.\n\nAs you select and make changes to your options, the Search Results Preview pane on the right updates automatically to show the relevant results. You can check this preview after trying new options and see if the query returns results that you expected.\n\nYou might also notice the Switch to Advanced Mode link in the upper-right corner of the left section of the query builder. By default, the query builder opens in Quick Mode, which is more simplistic and designed for users who aren't intimately familiar with the Keyword Query Language (KQL) used by SharePoint search. You can learn more about KQL by viewing the links at the end of this section.\n\nAdvanced Mode has two major features beyond the default Quick Mode:\n\n\u2022 You can view and modify the KQL that is being built for you. If you'd like to add additional keyword or property filters, the Advanced Mode provides an interface for adding these to your query to bring back more specific results.\n\n\u2022 Advanced Mode actually adds another tab at the top of the query builder for sorting. The sorting tab enables you to control the order of the results that are shown in the CSWP.\n\nRefiners Tab\n\nThe Refiners tab works similar to the SharePoint refinement panel when you do a search. Here you can filter results based on whether content matches one or more managed properties that are enabled as refiners in the SharePoint search settings. You can choose multiple values to refine if necessary; as you add refiners, the result preview to the right changes.\n\nSettings Tab\n\nThe Settings tab has four options that go a bit beyond the typical rollup scenarios you might encounter, but essentially they help define the performance behavior of the CSWP. This is useful in more complex or high-traffic implementations as well as the URL Rewriting behavior that's used with the new SharePoint 2013 Catalog feature.\n\nTest Tab\n\nThe test tab displays the full KQL for the query you've built. This can be helpful with learning how KQL works. You can see the KQL that is generated from all of the settings you've selected in the other tabs and potentially use similar KQL in Advanced Mode for future queries.\n\nNOTE\n\nFor more information about the Content Search Web Part, see \"Introducing the Content Search Web Part\" at <http:\/\/sharepoint.microsoft.com\/blog\/Pages\/BlogPost.aspx?pID=1053>.\n\n\"Configure Search Web Parts in SharePoint Server 2013\" gives additional information on working with search Web Parts including the Content Search Web Part at http:\/\/technet.microsoft.com\/en-us\/library\/jj679900.aspx.\n\n\"Building Search Queries in SharePoint 2013\" from MSDN provides additional details about building queries with Keyword Query Language (KQL) and FAST Query Language (FQL) at http:\/\/msdn.microsoft.com\/en-us\/library\/jj163973.aspx.\n\nTo learn more about building queries for the CSWP, check out Chris O'Brien's post on the topic at <http:\/\/www.sharepointnutsandbolts.com\/2012\/10\/using-content-search-web-part-and.html>.\n\nCustomizing the CSWP\n\nNow that you have a better understanding of how queries are made for the CSWP, you can put the knowledge to use by rolling up the publishing pages you created earlier. This example uses the master page and styles from the Randy's Waffles branding that was created in Chapters 5 and 6. Follow along with these steps without using the Randy's Waffles branding, but the styles will be different from the screen shots. You can download the completed Randy's Waffles master page and CSS from the code downloads for this chapter.\n\nAlthough this is a simple example, it illustrates some key points; you could use these same methods to roll up pages across site collections or even do things such as filter the pages based on meta tags.\n\nNOTE\n\nThis example continues from the previous one so if you don't already have the Build Your Query dialog open, be sure to edit your homepage, put the CSWP in Edit More, and click the Change query button at the top of the Web Part properties on the right side of the page.\n\nExample: Rolling up Publishing Pages with the CSWP\n\n1. Make sure the Basics tab is set to Quick Mode. If it's set to Advanced Mode, switch it back.\n\n2. From the Basics tab, choose these options:\n\n\u2022 Select a query\u2014Items matching a content type (System).\n\n\u2022 Restrict by app\u2014Specify a URL .\n\n\u2022 For the URL box, think back to the \"Prepping for the CSWP\" section when you looked at whether FURLs were turned on or off. If FURLs were left on, enter the parent term you created http:\/\/`YOURSERVER`\/Sale-Items. If FURLs were turned off, enter the name of the subsite you created http:\/\/`YOURSERVER`\/SaleItems.\n\n\u2022 Restrict by tag\u2014Don't restrict by any tag.\n\n\u2022 Restrict by content type\u2014Article Page.\n\n\u2022 Add additional filters\u2014Leave this set to the default or clear out the field.\n\nNOTE\n\nAs you make changes, the Search Results Preview window updates to reflect these. By choosing to query only items matching a content type and by restricting the content type to Article Page, your CSWP shows only pages that use page layouts that are based on the Article Page content type like Article Page - Image on Left. You also restricted the query to only show items under the subsite by entering the URL for that site or the parent term.\n\nWARNING\n\nRestricting queries to specific URLs can be difficult to work with depending on how your server is configured. If you are running into problems getting the query to return results in the preview window you may want to change the setting for Restrict by app back to Current site collection. Your results won't be as specific, but at least you can complete the example properly.\n\n3. Change the sorting to arrange the pages by the date and time that the page was created. To do this, click the Switch to Advanced Mode link.\n\n4. Click the Sorting tab and change the Sort by drop-down to Created. Then change the direction to Ascending. When you are in Advanced Mode, the Search Results Preview doesn't automatically update. To see your changes, click on the Test tab.\n\n5. At this point the query is complete; click OK at the bottom of the dialog window.\n\n6. Next, you can change some of the display properties to see what the CSWP would look like with paging turned on. To do so, from the Web Part properties menu on the right side of the page, under Display Templates, switch Control to List with Paging. For Item select Picture on left, 3 lines on right\n\n.\n\nNOTE\n\nBy default, the Web Part properties menu has the Number of Items to Show option set to show three items at a time. Earlier you created six pages that can display in this query result. Therefore the CSWP shows the pages three items at a time with a next and previous arrow.\n\n7. Click OK on the Web Part properties menu.\n\n8. Save the page to see the end result.\n\nThe CSWP has a handy paging menu at the top right that enables you to scroll through the result pages. At this point you have a working CSWP querying the correct results, but the display doesn't really match the original example. In the next section, you will learn how to create a display template that makes some simple changes to the display of the query results.\n\nWorking with Display Templates\n\nDisplay templates are made up of HTML and JavaScript files that control how results are presented when the content is returned from the CSWP query or other search Web Parts, such as the Search Results and the Refinement Panel. As you learned earlier in the chapter, styling the CQWP involves working with XSL, a language many people find challenging to work with. The switch to an HTML- and JavaScript-based approach is probably a welcomed change for many SharePoint veterans.\n\nDisplay templates are stored in the Master Page Gallery of your site collection. Microsoft was kind enough to include several out-of-the-box display templates that can be used to style Web Parts like the CSWP (they are also great for learning how to create your own display templates). At the end of the previous example you actually used out-of-the-box display templates to show the items arranged in pages. If you edit the Web Part again and look at the Web Part properties under Display Templates, you can see two drop-downs: Control and Item. These correspond to two different types of display templates that are used together to style results:\n\n\u2022 Control Template\u2014Defines the overarching behavior of the group of items that are returned. By default there are three options: List, List with Paging, and Slideshow. List is the most straightforward of the options, simply showing the results one after each other. You saw List with Paging in action in the previous example, and Slideshow shows off the results using a fancy style with the items dynamically rotating from one to the next on the page.\n\n\u2022 Item Template\u2014Defines the style for each of the individual items that are returned from the query. There are several out-of-the-box options available to show items in various styles.\n\nAlthough the out-of-the-box templates are a good starting point, many custom branding projects require the creation of your own custom display templates. Because the CSWP uses HTML and JavaScript to style results, customizing the display of CSWPs should be more approachable by designers than styling CQWPs using XSL.\n\nNOTE\n\nBy default the CSWP uses client-side rendering to present results; however, this is less than ideal for situations where automated systems process your page content; for example, when a search engine like Bing or Google is indexing a public-facing website. In these cases, it may be beneficial to use server-side rendering for the CSWP. For more information on this, see Waldek Mastykarz's post \"Using server-side rendering with Content Search Web Part in SharePoint 2013\" at <http:\/\/blog.mastykarz.nl\/server-side-rendering-content-search-web-part-sharepoint-2013\/>.\n\nManually Changing Property Mappings\n\nIn the previous example, you used the List with Paging Control display template and the Picture on left, 3 lines on right Item display template; together these created a display with a paging control. In addition, for each item, the photo was on the left and the title was on the right. Notice that the byline you added to all the pages (which had the sale price) doesn't show up in the CSWP at this point. Although you set the display template to show three lines on the right, only the title shows. This is because of property mappings, which are how display templates receive content from the results of the query.\n\nIf you expand the Property Mapping section of the CSWP Web Part properties menu you can actually see what shows for each of the three lines. By default, Line 1 is set to show the Title of the content and Line 2 is set to show the Description. However, in the previous example nothing was shown on the second line because the publishing pages you created earlier didn't have descriptions set. One way of correcting this is to change Line 2 from showing the description to showing the Article Byline. If you wanted to do this, you would check the box for Change the Mapping of Managed Properties for the Fields in the Item Display Template and then change the drop-down for Line 2 to ArticleByLineOWSTEXT (which is the behind-the-scenes name for the managed property that corresponds to the Article Byline).\n\nInstead of changing the property mappings manually, in the next few sections, you learn how to create a new display template that displays items in rows of three as well as changing the Line 2 property mapping to automatically pull the byline information by default.\n\nCreating a Display Template\n\nWhen you work with display templates, it rarely makes sense to start one from scratch; instead, you would typically modify a copy of whichever out-of-the-box display template is most similar to your intended display. It's also worth noting that every display template includes both an HTML file and a JavaScript file that share the same root filename. You always want to edit the HTML file, because SharePoint automatically generates the JavaScript file by interpreting the HTML file. The following example walks you through starting a new display template by copying an existing one.\n\nExample: Creating a New Display Template\n\n1. From a drive you've mapped to your Master Page Gallery, navigate down to the Display Templates subfolder, where all the display templates live. Each subfolder holds different types of display templates, such as templates for searches and filters. The CSWP uses display templates in the Content Web Parts folder.\n\n2. Navigate down to the Content Web Parts folder. There are many default display templates listed in here that you can use to learn from. In the next step, you will make a copy of the Picture on left, 3 lines on right display that you selected previously when you configured your CSWP.\n\n3. Make a copy of `Item_Picture3Lines.html` and rename it WaffleDisplay.html. When you do this SharePoint automatically generates a corresponding JavaScript file.\n\n4. Open `WaffleDisplay.html` in your editing program of choice and you should see the following display template code.\n\nNow that you have made a new display template by copying an existing one, you are almost ready to update it with your own custom display template code. But before you start making changes, the next section walks you through the copied display template code and explains how it works.\n\nUnderstanding the Display Template\n\nA lot goes on inside a typical display template, and the one just created is no exception; some areas of the code are noteworthy for customization and others can typically be ignored. The following list looks at the display template code from top to bottom, describing the areas that you should keep in mind when creating a custom display template and skipping over the unimportant sections:\n\n\u2022 The title of the HTML page (between `<title>` and `<\/title>`) is used to surface the name of the display template in the Web Part properties menu. In this case the title is Picture on left, 3 lines on right. This is what you selected in the Display Templates \u21d2 Item drop-down on the Web Part properties menu in the CSWP example.\n\n\u2022 Next, you see a group of XML tags that start with `<mso:`\u2014Most of these should not be changed. The exception is `ManagedPropertyMapping`, which you use to surface extra property mappings on the Web Part properties menu. The syntax for adding new mappings looks like this (note that each property mapping is separated by a comma):\n\n`'AuthorVar'{Author Name}:'Author',`\n\nIn this case `AuthorVar` is used in the actual display template HTML file, `Author Name` shows as a select box in the Web Part properties menu for property mappings, and `Author` is the default option that was selected in that drop-down.\n\n\u2022 Next, there is a section that has a comment warning you not to add HTML to this part of the display template.\n\n\u2022 After that, you will see a `<script>` tag that includes a call to `$includeLangageScript`. This section is used to call in other JavaScript files. You will not be making any changes to this section.\n\n\u2022 Next, you will see a div tag. In this case `<div id=\"Item_Picture3Lines\">`. This is the main container for the editable section of the display template. You MUST leave this div tag here, but you can change the ID that is associated with it.\n\n\u2022 Inside of this div you see several lines of HTML and JavaScript code. All JavaScript is escaped by surrounding it with `<!--#_` and `_#-->` and all display variables are surrounded with `_#=` and `=#_`. Later, when you edit and save this file, SharePoint will convert these escaped pieces of JavaScript into functioning code.\n\nIn addition to the aforementioned escaped characters, this section of the file consists of typical JavaScript that you would use within traditional web pages including some extra SharePoint fanciness added in. You will see a few SharePoint functions used throughout the sample display templates; here is a description of a few of the more common functions used:\n\n\u2022 `$getItemValue`\u2014Retrieves the value of managed properties that were defined in the MSO section at the top of the display template. For example, `$getItemValue(ctx, \"Line 2\");` retrieves whatever managed property is applied for the display template from Line 2 in the Web Part properties menu.\n\n\u2022 `$htmlEncode`\u2014Encodes a given string so that it is safe for displaying as HTML\n\n\u2022 `$urlHtmlEncode`\u2014Encodes a given string so that it is safe for using as a URL.\n\n\u2022 `Srch.ContentBySearch.getPictureMarkup`\u2014Returns HTML markup for a given image including references to the proper image rendition.\n\nAlong with these functions there is an object named `ctx` (short for context) used throughout the display templates. This object refers back to the result set returned from the CSWP query. Here is a list of some of the more important uses for the `ctx` object in the default display templates:\n\n\u2022 ctx.ClientControl\u2014Returns information from the actual Web Part on the page that shows the display template. Frequently used to retrieve a unique ID for this Web Part instance. This ID is then appended to HTML elements in the display template to give the elements their own unique IDs. Without this, each item might use the same exact HTML IDs in its HTML elements, making styling and scripting difficult.\n\n\u2022 ctx.ListData\u2014Returns information about the set of results provided by the query. Used in the control display templates because they show the group of items returned from the query.\n\n\u2022 ctx.CurrentItem\u2014Returns the data for the current item being processed by an item display template.\n\n\u2022 ctx.CurrentItemIdx\u2014Returns the index number of the current item being processed by an item display template. This number starts from zero for the first item instead of one, so if you do any sort of conditional formatting, you may need to increment the index value by 1 to get the actual position starting from 1.\n\nCustomizing the Display Template\n\nNow that you have some background on how the display template is set up, you can look at the changes required for the Waffle Display template to work properly. If you get lost following this example, you can download the completed display template from the downloads for this chapter:\n\nExample: Creating a Custom Display Template\n\n1. Change the title of the display template to the following:\n\n`<title>Waffle Display<\/title>`\n\n2. Update the `ManagedPropertyMapping` for Line 2 to include a default setting that points to the byline. To do this, scroll down to the MSO section and find the `ManagedPropertyMapping` line. Replace this part:\n\n`'Line 2'{Line 2}: 'Description'`\n\nWith this code:\n\n`'Line 2'{Line 2}:'ArticleByLineOWSTEXT'`\n\n3. Add some JavaScript to insert a CSS break after every third item. To do this, find the first instance of `<!--#_` and add the following code inside of it (add the code before the line that starts with `var encodedID`).\n\n`var customRowEnd = \"\";`\n\n`if((ctx.CurrentItemIdx + 1) % 3 == 0){`\n\n` customRowEnd = \"<div style='clear:both;'><\/div>\";`\n\n`}`\n\nThis code sets an initial string variable called `customRowEnd` and then looks at the index counter for the current item and decides if it is divisible evenly by 3. If the item is divisible by 3, `customRowEnd` will be set to show a div with a CSS style that adds a break.\n\n4. Remove the default display; you will replace it with a new display in step 6. To do this, find the section of code that begins with:\n\n`<div class=\"cbs-picture3LinesContainer\"`\n\n5. Remove this entire section of code, including the start and end tags, leaving only the following code at the bottom of the display template:\n\n` <\/div>`\n\n` <\/body>`\n\n`<\/html>`\n\n6. Add in the custom display code before the closing `<\/div>:`\n\n`<div class=\"customProductsItem\">`\n\n` <!--#_if(!linkURL.isEmpty){_#-->`\n\n` <!--#_if(!pictureMarkup.isEmpty){_#-->`\n\n` <a class=\"customProductsImage\" href=\"_#= linkURL =#_\">`\n\n` _#= pictureMarkup =#_`\n\n` <\/a>`\n\n` <!--#_}_#-->`\n\n` `\n\n` <!--#_if(!line1.isEmpty){_#-->`\n\n` <a href=\"_#= linkURL =#_\">_#= line1 =#_<\/a><br\/>`\n\n` <!--#_}_#-->`\n\n` `\n\n` <!--#_if(!line2.isEmpty){_#-->`\n\n` _#= line2 =#_`\n\n` <!--#_}_#-->`\n\n` <!--#_}_#-->`\n\n`<\/div>`\n\n` `\n\n`<!--#_if(!customRowEnd.isEmpty) {_#-->`\n\n` _#= customRowEnd =#_`\n\n`<!--#_}_#-->`\n\nNOTE\n\nAlthough this code may look confusing, it's actually not hard to understand once you get used to the markup. Everything surrounded by `<!--# #-->` is JavaScript, and everything surrounded by `#= =#` is a managed property or variable being output by the display template. You will see several places in which conditional JavaScript is used to determine whether a variable is empty before it is displayed.\n\nWhen this display template is rendered in the browser, the resulting code shows the product image, line 1, line 2, and then `customRowEnd`, which is a variable that adds a break after every third item. The following pseudocode shows an example of what could be output from this display template for each item:\n\n`<div class=\"customProductsItem\">`\n\n` <a class=\"customProductsImage\" href=\"LINK TO ITEM\">`\n\n` <img src=\"IMAGE SOURCE\" \/>`\n\n` <\/a>`\n\n` <a href=\"LINK TO ITEM\">TITLE<\/a><\/br>`\n\n` BYLINE`\n\n`<\/div>`\n\n`<div style=\"clear: both;\"><\/div>`\n\n7. These are all the changes needed for the display template. Save the file and SharePoint can regenerate the corresponding JavaScript file.\n\n8. To finish the example, edit the SharePoint page that has your CSWP, and bring up the Web Part properties menu by clicking Edit Web Part. Make the following changes:\n\n\u2022 Search Criteria \u21d2 Number of items to show`\u20146`\n\n\u2022 Display Templates \u21d2 Control\u2014`List`\n\n\u2022 Display Templates \u21d2 Item`\u2014Waffle Display`\n\n9. You should also check to make sure the Property Mappings now shows `ArticleByLineOWSText `for Line 2.` `When all those settings are in place, click OK.\n\n10. Assuming no errors display, Save and Publish the page.\n\nIf you have any errors, SharePoint tries to guide you in the right direction to show which part of the display template caused the error. Like all debugging in SharePoint, it pays to go slow and make small changes when you work on a new display template. The nice thing about display templates compared to XSL and the CQWP is you can see changes to the display templates by just saving the HTML file and refreshing your browser.\n\nIf you have the completed Randy's Waffles master page from Chapter 5 and have applied the CSS from Chapter 6 to your site, you should already see custom styling for the products. The CSS from the Randy's Waffles branding that applies to the display template is as follows:\n\n`.customProductsItem {`\n\n` padding-right: 14px;`\n\n` float: left;`\n\n` padding-bottom: 15px;`\n\n`}`\n\n`.customProductsImage {`\n\n` display: block;`\n\n` padding-bottom: 2px;`\n\n`}`\n\n`.customProductsImage img {`\n\n` border: 1px solid #dddddd;`\n\n`}`\n\nThe finished product uses the query from the CSWP and the custom display template you created in the previous example to show two rows of sale products with images, titles, and a price (which you entered into the byline field).\n\nIf you are comfortable working with JavaScript, you can easily take this example to the next level. Perhaps you create a highlight effect for the photos or maybe add code that parses the value of the product's price and conditionally highlights heavily discounted prices in the display.\n\nCQWP Versus CSWP\n\nThroughout this chapter we've covered the two primary Web Parts that roll up content in WCM. There's quite a bit of information to take in, both for using the Web Parts and for customizing them. But the big question is how to best apply all of this new information to the various scenarios you might find in your implementation. For many readers, the decision regarding which method you use for rolling up content is made for you, simply because the CQWP is available in more versions of SharePoint whereas the CSWP is (as of this writing) only available with SharePoint Server 2013 Enterprise. If do find yourself in a position where you need to decide which approach is better, the following chart breaks down some of the major factors you might want to consider:\n\nNOTE\n\nAlthough currently not available with O365 as of this writing, Microsoft has stated it plans to make the CSWP available in the future.\n\nThere is also a possibility that you end up using both Web Parts in your SharePoint implementation. For simpler rollup scenarios, the CQWP is useful. Although the CSWP is flexible and powerful, it also requires an additional level of complexity because it leverages search. In practice, there tends to be a bit more effort involved to configure and customize the CSWP, which could be overkill if you're just rolling up a few pages.\n\nSo overall, which one is best? As with many things in SharePoint, the ultimate answer is, \"It depends.\" The good news is that SharePoint 2013 provides more options than ever for rolling up your content. It is now possible to do things that previously would have required a significant custom effort to re-create. Whether it is search-driven content, supporting mobile sites, or something else, SharePoint 2013 provides a powerful and flexible set of Web Content Management capabilities. And if you are familiar with the way SharePoint rolled up content in the past, the classic approach to content rollup is still available! There's something for everyone to create beautiful and engaging SharePoint sites with dynamic content.\n\nSummary\n\n\u2022 SharePoint is a robust Web Content Management (WCM) platform that enables you to more efficiently manage content. The most basic principle of WCM is the idea of creating a single piece of content and being able to surface that content wherever it might be needed.\n\n\u2022 The CQWP is included in all versions of SharePoint 2013 where it is possible to enable publishing. The Web Part is fairly straightforward and does exactly what its name implies: it rolls up content based on a specific query. This is a common task on just about any dynamic website.\n\n\u2022 The results returned from the CQWP are styled with XSL, which can be customized.\n\n\u2022 One of the biggest changes to SharePoint 2013 is the addition of the Content Search Web Part (CSWP), which leverages SharePoint search to roll up the content.\n\n\u2022 Display templates are made up of HTML and JavaScript files that control how results are presented when the content is returned from the CSWP query or other search Web Parts, such as the Core Search Results and the Refinement Panel.\n\n\u2022 When it comes to deciding between CQWP and CSWP you may end up using both. Or the version of SharePoint you are using might make the choice for you.\n10\n\nComposed Looks and Custom Branding\n\nWhat's In This Chapter\n\n1. Configuring Composed Looks\n\n2. Themable Custom Master Pages\n\n3. Master Page Previews\n\nIn Chapter 3, \"Working with the SharePoint 2013 User Interface,\" you learned about how to use composed looks to apply style to a SharePoint 2013 site without a lot of effort. Although it's great that Microsoft provides several composed looks with SharePoint, you may want to create your own composed looks and make them available to your site owners for use on their sites. The cool thing about composed looks is you can provide a base master page design, and users can change the colors, fonts, and background images without knowing anything about CSS or master pages.\n\nUnderstanding Composed Looks\n\nTo begin understanding composed looks, you should check out the composed looks gallery by clicking Site Settings \u21d2 Web Designer Galleries \u21d2 Composed looks. By default, this page has a list of all the composed looks included with SharePoint.\n\nIn this list, all the column names are behind-the-scenes names that correspond to friendly labels that you would see when you go to the Change the Look menu.\n\nThe following shows the new item menu from the composed looks gallery (Site Settings \u21d2 Web Designer Galleries \u21d2 Composed looks) on the left and the menu you see when you select a composed look (Site Setting \u21d2 Look and Feel \u21d2 Change the Look) on the right:\n\nThe following list details what each setting in the composed looks gallery represents and how it relates to the Change the Look menu.\n\n\u2022 Title\u2014The title for the list item that is created for each composed look.\n\n\u2022 Name\u2014The name or label that appears next to the look in the Change the Look menu.\n\n\u2022 Master Page URL\u2014A reference to the preferred master page that goes with the composed look. In the Change the Look menu, this shows in the Site Layout drop-down. This can be confusing terminology because Site Layout could be confused with page layouts, which are a different concept.\n\n\u2022 Theme URL\u2014Another confusing option because themes were the overarching name for functionality that was similar to composed looks in SharePoint 2010. In the composed looks gallery, Theme URL refers to an XML file that lists preferred colors for the composed look. This is shown in the Change the Look menu as Colors.\n\n\u2022 Image URL\u2014References the URL for a background image to be used in the composed look. On the Change the Look menu, this is represented by the image thumbnail in the top left of the screen.\n\n\u2022 Font Scheme URL\u2014Refers to an XML file that lists preferred fonts for this composed look. On the Change the Look menu, this is simply shown as Fonts.\n\n\u2022 Display Order\u2014This number represents the sort order from low to high that the composed look shows on the first page of the Change the Look menu with all the thumbnail previews.\n\nOne interesting thing about the composed looks gallery is that each item simply represents a suggestion for the Change the Look menu. In reality the user can select a composed look and then change the background image, colors, master page, and fonts from all the available options. This can be a good thing because it gives the user a much larger array of options from which to mix and match.\n\nCreating SPFont and SPColor Files\n\nBefore you learn about using composed looks with your own master page, you should create a simple composed look to use with one of the default master pages. To get started with this you'll want to make some simple font and color XML files for the composed look. These files live in `\/_catalogs\/theme\/15`, which is actually one level above the master page gallery that you may have mapped earlier in the book. If you haven't already done so, go ahead and map a drive to the root of your SharePoint server instead of the master page gallery. From there, simply navigate to the `_catalogs\/theme\/15` folder. You can see a long list of files that end with the extensions `.spfont` and `.spcolor`; these represent all the default color and font options for composed looks.\n\nNOTE\n\nDepending on your Explorer window settings you may have to turn off hidden files in order to see the `_catalogs` folder.\n\nYou need to make copies of `fontscheme001.spfont` and `Pallete001.spcolor`. Name the copies `BreakfastFonts.spfont` and `BreakfastColors.spcolor`, respectively.\n\nSPColor\n\nStart by opening `BreakfastColors.spcolor` in a text editing program.\n\nThe bulk of the file is color definitions that start with `<s:color...` and include the name of the color and a hexadecimal value like you would use to represent colors in CSS. One thing to note in these colors is that they will sometimes be listed with eight digits instead of the normal six. These are 8-bit aRGB colors, where the first two digits represent the alpha (or transparency) value in hex, followed by the normal CSS hex color digits.\n\nThe other thing to note is the second line in the file, which defines some important parameters for this color file:\n\n\u2022 `isInverted`\u2014Used primarily by the color drop-downs in the SharePoint ribbon. If `isInverted` is set to `True`, SharePoint knows to invert the color options in the color dropdowns so that the default text colors start lighter and go darker to let them show better on a dark background.\n\n\u2022 `PreviewSlot1`, `PreviewSlot2`, and `PreviewSlot3`\u2014These refer to the colors drop-down that shows in the Change the Look menu. Each slot points to a color value in the preceding list of colors. For example, `Pallete001.spcolor` uses `BackgroundOverlay`, `BodyText`, and `AccentText` for its color slots. In the Change the Look menu you can see the colors white, gray, and blue, respectively, because they are defined by the hex settings for each of those colors in the color list.\n\nWhen you understand the way the file is set up, changing colors is actually not that complicated. You can make a simple edit to the color file by changing the value for `BodyText` to a noticeable red color.\n\nReplace this line:\n\n`<s:color name=\"BodyText\" value=\"444444\" \/>`\n\nWith this color and save the file:\n\n`<s:color name=\"BodyText\" value=\"B22222\" \/>`\n\nBecause you are changing a color that is referenced in a `PreviewSlot`, in this case `BodyText`, you can see this color in the drop-down when selecting the composed look.\n\nSPFont\n\nTo familiarize yourself with an SPFont file, look at the `BreakfastFonts.spfont` file:\n\nThis file has a little extra going on. On line two the `<s:fontScheme> `tag is set up similarly to the `<s:color> `tag in the SPColor file in that it defines some `previewSlots` referenced in the following list of `fontSlots`. Inside of the `fontSlots` node, you see several `fontSlot` child nodes, each with a \"name\" that defines the slot. Inside of each `fontSlot` node you see an `<s:latin>` node that configures the actual fonts that will be used for this slot plus several nodes that tell SharePoint what fonts to use if the default language is set to a foreign language that doesn't use Latin type faces. Focusing on the `latin` node, you can see that it defines the following properties:\n\n\u2022 `Typeface`\u2014The name of the font that shows in the Change the Look menu.\n\n\u2022 `EOTSRC`\u2014Points to the URL for an .EOT font file.\n\n\u2022 `WOFFSRC`\u2014Points to the URL for a .WOFF font file.\n\n\u2022 `TTFSRC`\u2014Points to the URL for a .TTF font file.\n\n\u2022 `SVGSRC`\u2014Points to the URL for an .SVG font file.\n\n\u2022 `Largeimgsrc`\u2014Points to the URL for an image that represents the font in the Change the Look menu. This image shows the user a sample of what the font will look like for the first `previewSlot`.\n\n\u2022 `Smallimgsrc`\u2014Points to the URL for an image that represents the secondary font in the Change the Look menu. This image shows the user a sample of what the font will look like for the second `previewSlot`.\n\nIf you have never created fonts for use on the web, you may be confused by the EOT, WOFF, TTF, and SVG settings here. They represent the four common font formats that the different browsers use to show custom fonts. If you want more information on how web fonts work, check out http:\/\/www.fontsquirrel.com, which has many web fonts available for download and a generator that creates the EOT, WOFF, TTF, and SVG files for you.\n\nTo make life easy, a sample web font is provided that you can download from the code for this chapter to create your SPFont file. Download the Arvo font and then copy the entire Arvo folder into `\/_catalogs\/theme\/15\/Arvo`. This includes EOT, SVG, TTF, and WOFF fonts as well as two images you can use for the `largeimgsrc` and `smallimgsrc`. With these files in place, you can change the first `latin` slot on line 5 to match this and then save the file:\n\n`<s:latin typeface=\"ArvoRegular\" `\n\n`eotsrc=\"\/_catalogs\/theme\/15\/Arvo\/Arvo-Regular-webfont.eot\" `\n\n`woffsrc=\"\/_catalogs\/theme\/15\/Arvo\/Arvo-Regular-webfont.woff\" `\n\n`ttfsrc=\"\/_catalogs\/theme\/15\/Arvo\/Arvo-Regular-webfont.ttf\" `\n\n`svgsrc=\"\/_catalogs\/theme\/15\/Arvo\/Arvo-Regular-webfont.svg\" `\n\n`largeimgsrc=\"\/_catalogs\/theme\/15\/Arvo\/ArvoRegularLarge.png\" `\n\n`smallimgsrc=\"\/_catalogs\/theme\/15\/Arvo\/ArvoRegularSmall.png\" \/>`\n\nCreating a Simple Composed Look\n\nNow that you have created SPFont and SPColor files, you can use them to build a simple custom composed look. For this example, you can use an existing background image and one of the existing out-of-the-box master pages.\n\nExample: Creating a Simple Composed Look\n\n1. Navigate to Site Settings \u21d2 Web Designer Galleries \u21d2 Composed Looks.\n\n2. Click new item at the top of this page to create a new composed look.\n\n3. Enter the following options for the composed look:\n\n\u2022 Title\u2014Breakfast\n\n\u2022 Name\u2014Breakfast Theme\n\n\u2022 Master Page URL\u2014`\/_catalogs\/masterpage\/oslo.master`\n\n\u2022 Theme URL\u2014`\/_catalogs\/theme\/15\/BreakfastColors.spcolor`\n\n\u2022 Image URL\u2014`\/_layouts\/15\/images\/image_bg117.jpg`\n\n\u2022 Font Scheme URL\u2014`\/_catalogs\/theme\/15\/BreakfastFonts.spfont`\n\n\u2022 Display Order\u20145\n\nThe master page URL points to the out-of-the-box Oslo master page, which shows off a little more of the background image than the Seattle master page does. Theme URL and Font Scheme URL point to the new color and font files you created in the previous section. Rather than make your own image for this example, you can use one that comes with SharePoint. In this case, Image URL points to a photo of some food on a table that is included in the SharePoint install folder on the server. Finally, Display Order tells SharePoint to show this theme near the top of the Change the Look menu.\n\n4. When you have the options set to match these, click Save.\n\nNOTE\n\nYou could browse a selection of background photos that are included with SharePoint by looking on your server in the SharePoint 15 folder under `\\TEMPLATE\\IMAGES`. The background images are all of the format `image_bg*.jpg`.\n\nAt this point the composed look is created, and you can select it from the Change the Look menu by clicking Site Settings \u21d2 Look and Feel \u21d2 Change the Look. Because Display Order was set to 5, the new composed look should be either the first or second option in the list. If you have already applied a com-posed look to your SharePoint site, you see that the composed look shown first is labeled Current. You can already see that the Breakfast Theme is showing the breakfast table background image that was set earlier.\n\nClick the thumbnail for the newly created Breakfast Theme and you will see that all the settings for the composed look show as the default settings on this screen. The breakfast table image shows on the top left; the `BodyText` red color you added shows in the drop-down. If you click the color drop-down and hover over this color, you can see some text that tries to describe the applied colors. In this case SharePoint says, \"This palette is primarily White with Red and Dark Blue.\" Interesting enough, you entered a hex value for the color but SharePoint knows to show the user-friendly text that says the color is red. The Site Layout also shows that the Oslo master page is selected, and then the Fonts drop-down shows the font you added earlier, Arvo Regular, as the default primary font. This drop-down shows an image thumbnail of the font, not the actual font rendering in the browser. On the right side of the menu, you see a preview of what everything should look like when it's put together. You'll learn more about that preview later in the \"Creating a Master Page Preview\" section.\n\nIf you want to change any of the options, you can do that now. All the available values are made up of options created for the other available composed looks.\n\nExample: Choose and Apply a Composed Look\n\n1. When you have the options set how you like them, click Try It Out.\n\n2. From here click Yes, Keep It to apply the composed look to your SharePoint site.\n\nWithout doing a lot of work, you now have a SharePoint site that shows a custom background image, some custom colors, and even custom fonts. Although this is handy, you can actually push the envelope even further by applying composed looks to your own customer master pages.\n\nApplying a Composed Look to Custom Branding\n\nWhen a composed look is applied to a SharePoint 2013 site, SharePoint makes a copy of the applied CSS and updates the colors, fonts, and background image to match the composed look settings. When this happens, behind the scenes SharePoint looks for specific CSS comments that describe these changes. If you ever looked at how SharePoint 2010 applied theme comments to custom CSS, this concept is virtually identical for SharePoint 2013 composed looks.\n\nSharePoint looks for and applies the composed look settings to any CSS, whether it is out-of-the-box SharePoint CSS or your own custom CSS, as long as the CSS meets two key criteria:\n\n\u2022 Your custom CSS must be located in a \"themeable\" location. The following locations and any subfolders beneath them are valid themable locations:\n\n\u2022 In the SharePoint server's root folder:\n\n`15\\TEMPLATE\\LAYOUTS\\1033\\STYLES\\Themable`\n\n\u2022 At the top level of a site collection in one of these folders:\n\n`\/Style Library\/Themable\/`\n\n`\/Style Library\/en-us\/Themable\/`\n\nWARNING\n\nDepending upon your SharePoint edition and the site template used to create your SharePoint site, these folders may not be created for you automatically. In those cases you can simply create the folder yourself and themes will work.\n\n\u2022 Your custom CSS must be loaded from a master page using the CssRegistration tag like this:\n\n`<SharePoint:CssRegistration ID=\"CssRegistration1\" `\n\n` name=\"\/Style Library\/en-us\/Themable\/style.css\" runat=\"server\" `\n\n` after=\"SharepointCssFile\" \/>`\n\n\u2022  SharePoint will not apply a composed look to your CSS if you try to load it with a standard HTML link tag, like this:\n\n`<link rel=\"stylesheet\" href=\"\/Style Library\/en-us\/Themable\/style.css\" `\n\n` type=\"text\/css\" media=\"screen\" \/>`\n\nNOTE\n\nIf you use the Design Manager to create an HTML master page, SharePoint typically converts an HTML `<link>` tag to the appropriate CssRegistration tag automatically for you.\n\nThese two criteria are what SharePoint looks for to apply composed looks to custom CSS called from a master page. However, there is one more thing required to actually select a master page from the Site Layout drop-down in composed looks: a preview for the master page. This is what shows in the list of composed look thumbnails as well as the large preview section on the right of the Change the Look menu. Although it may seem like these previews are showing static thumbnail images or even functioning master pages, in actuality they are generated from a special type of preview file. You learn more about creating this preview file later in the \"Creating a Master Page Preview\" section.\n\nComposed Look Theme Tokens\n\nWhen a composed look is applied to a site, SharePoint finds and replaces special CSS comment tokens. Here is a list of the tokens that SharePoint looks for. In each case, SharePoint will find the token and modify the next line of CSS, injecting a value derived from the composed look that is applied to the SharePoint site:\n\n\u2022 `ReplaceBGImage`\u2014Replaces a background image with the image assigned in the applied composed look image URL.\n\n`\/* [ReplaceBGImage] *\/ `\n\n`background-image:url(\"background.png\");`\n\n\u2022 `ReplaceFont`\u2014Replaces a font with one of the fonts from the Font Scheme URL defined in the applied composed look.\n\n`\/* [ReplaceFont(themeFont:\"body\")] *\/ `\n\n`font-family: sans-serif;`\n\n\u2022 `ReplaceColor`\u2014Replaces a color with one of the colors from the Theme URL defined in the applied composed look.\n\n`\/* [ReplaceColor(themeColor:\"BodyText\")] *\/`\n\n`color:#444;`\n\n\u2022 `RecolorImage`\u2014Recolors images using either Tinting or Filling. Tinting is useful for shifting the colors of a multicolor image, whereas Filling is more useful for changing a single, solid color icon with transparency like a PNG.\n\n`\/* [RecolorImage(themeColor:\"BodyText\",method:\"Filling\")] *\/ `\n\n`background-image:url(\"icon.gif\");`\n\nIf you want to get a good idea of how these tokens work, you can see how Microsoft uses them by viewing the source for `COREV15.CSS` on your server. You can see several hundred of these comments sprinkled throughout the CSS file; each one represents a change that would occur in the master page when a composed look is applied to it.\n\nNOTE\n\n`COREV15.CSS` is located on the server in the SharePoint 15 folder under: `\\TEMPLATE\\LAYOUTS\\1033\\STYLES\\Themable\\COREV15.CSS`\n\nNow that you've learned about how composed looks work and what token can be applied to custom CSS, it's time to give it a try. You can use the WaffleNet branding created in Chapter 7, \"Creating SharePoint Branding,\" to do so. If you haven't finished that example yet, you can get the completed code from the Chapter 7 downloads file. You also need a drive mapped to the root of your SharePoint site, much like the earlier example for creating SPFont and SPColor files.\n\nWARNING\n\nThe examples in this chapter assume you have a SharePoint publishing site in a site collection created at the root of your SharePoint server. If your site is in a site collection that was not created at the root, some of the paths to files may need a reference like `\/sites\/sitename`.\n\nExample: Adding Composed Look Tokens to Custom CSS\n\n1. Using a mapped drive, navigate to `_catalogs\/masterpage`. Make sure you already have a subfolder named WaffleNet with the WaffleNet master page example loaded into it.\n\n2. Edit WaffleNet.master in your editor of choice. You will be adding a CSS reference to a new themeable CSS file. After the CssRegistration tag add the following line, and save the file:\n\n`<SharePoint:CssRegistration ID=\"CssRegistration2\" name=\"&lt;% `\n\n` $SPUrl:~sitecollection\/style library\/themable\/WaffleThemed.css %&gt;\" `\n\n` runat=\"server\" after=\"style.css\" \/>`\n\nAt this point the referenced CSS file doesn't exist, but you will create it in the next few steps. Since you are already working in the WaffleNet folder, now is a good time to create a preview file for the master page. Remember that a preview file is required to select the master page from the Change the Look menu. In this example, you can cheat a little and use one of the existing preview files. You learn more about creating a real preview file later in the \"Creating a Master Page Preview\" section.\n\n3. From your mapped drive navigate up one folder from the WaffleNet directory to the masterpage root. Find Oslo.preview and copy it; then navigate back to the WaffleNet folder and paste it in. Rename the file WaffleNet.preview.\n\n4. To publish both the master page and the preview file, navigate to Site Settings \u21d2 Web Designer Galleries \u21d2 Master Pages and Page Layouts. Navigate to the WaffleNet folder; then click the drop-down next to each file, and select Publish a Major Version. Click OK on the dialog window.\n\nNOTE\n\nPreview files always match the corresponding master page filename exactly, only with a .preview extension instead of a .master extension. If you ever change the master page filename, you should also change the corresponding preview filename and vice versa.\n\nExample: Creating the WaffleThemed.css File\n\n1. Using the drive mapped to the root of your SharePoint site, navigate to Style Library. If there is not a folder named Themable there, create it now, and then navigate to that folder.\n\n2. Create a new file named WaffleThemed.css and open in it your favorite editor.\n\n3. Add CSS and composed look tokens only for areas of the design that you want to be specifically controlled by composed looks. There may be other out-of-the-box styles that pick up on composed look changes, but the ones defined here will be the only custom areas in your design that can purposely use the composed look. Add the following lines of code and then save the file:\n\n`.ms-backgroundImage {`\n\n` \/* [ReplaceBGImage] *\/`\n\n` background-image: url(`\n\n` \/_catalogs\/masterpage\/wafflenet\/bg.jpg) !important;`\n\n`}`\n\n`.customBread .ms-core-pageTitle {`\n\n` \/* [ReplaceColor(themeColor:\"BodyText\")] *\/`\n\n` color: #00b3ff !important;`\n\n` \/* [ReplaceFont(themeFont:\"title\")] *\/ `\n\n` font-family: \"Segoe UI Semilight\",\"Segoe UI\",\"Segoe\",`\n\n` Tahoma,Helvetica,Arial,sans-serif !important;`\n\n` font-size: 30px !important; `\n\n`}`\n\nThe first rule uses a `ReplaceBGImage` token to replace the default wood grain background image if the composed look applies one. If not, the line after the `ReplaceBGImage` will continue to apply the standard background image.\n\nThe second rule changes the page title that appears in the breadcrumb area of the page. `ReplaceColor` first changes the color to match the composed look `BodyText` color setting. Then `ReplaceFont` changes the font to the title font from the composed look font setting. Finally, the font size is changed to 30px; this has nothing to do with composed looks, but it can help you see the changes over the original smaller font size.\n\nWARNING\n\nBecause CSS references can sometimes load out of order, it's a good idea to add `!important` to the end of the styles to ensure the composed look takes precedent over other CSS code. Each line in the previous example already has `!important` added to it.\n\nExample: Checking in the WaffleThemed.css file\n\n1. Because the WaffleThemed.css file lives in the Style Library, it needs to be checked in as a major version before you can use it. From the SharePoint Site click the Settings gear, and from the drop-down click Site contents.\n\n2. From the Site contents menu, click the blue square for Style Library and then click Themable.\n\n3. Click the ellipses \"...\" next to WaffleThemed and then the second set of ellipses \"...\" in the pop-up menu; then click Check In. Next to Version select 1.0 Major Version (Publish) and click OK.\n\nWARNING\n\nBecause of how the Style Library is set up, you need to check out the file before you make edits to WaffleThemed.css, as well as check it in again when you are done. You must do this every time you want to work with the file.\n\nExample: Applying your Composed Look to your Custom Theme\n\n1. Navigate to Site Settings \u21d2 Look and Feel \u21d2 Change the Look and then from the list of thumbnails select Breakfast Theme.\n\n2. The default Site Layout is set to the Oslo master page. Now that you have a preview file created for WaffleNet, you can select the drop-down and choose WaffleNet. Remember you used a copy of Oslo's preview file, so the preview on the right will look like that master page.\n\n3. Click Try It Out. Unlike the preview on the previous page, this preview window actually shows your custom master page with the composed look applied to it. Click Yes, Keep It to apply the composed look to your custom branding.\n\nAfter SharePoint is done \"thinking,\" you see that the composed look is applying the custom background image of the breakfast table and the page title is showing with the red `BodyText` color and uses the custom Arvo font.\n\nBecause SharePoint enables you to change the default settings that are applied with a composed look, having this preview file allows your custom master page to be selectable with the other composed looks. This makes it possible to mix and match settings to make a variety of new looks.\n\nCreating a Master Page Preview\n\nWhen you applied the composed look in the previous section, you saw a preview for the Oslo master page in both the composed look thumbnail and the larger preview when selecting the look. This was because you copied the preview file from the Oslo master page. These thumbnail and larger previews are actually generated by iFramed HTML created from that preview file. Because those two pages show the thumbnail and preview at different sizes, SharePoint is actually scaling the iFramed HTML as needed to fill the necessary sizes.\n\nThis preview file for Oslo looked nothing like your custom master page in the previews from the previous example. To get your custom master page to show, you need to create a custom preview file to match your custom master page. The process of creating a custom preview file is actually kind of tricky because it is mostly an HTML file, not a master page file, yet it has special markup that is unlike any traditional HTML file or even a typical master page. To make matters worse, recall that the same preview file shows as a small thumbnail and a larger preview, which means the markup needs to stretch and compress according to the size of the preview that is shown.\n\nPreview File Structure\n\nBefore getting too deep into creating a preview file, you should look at the basic structure of one of the default previews. Use your mapped drive to navigate to `_catalogs\/masterpage` and open Oslo.preview in your text editor. You can see that the file is broken into four areas by three `[SECTION]` identifiers.\n\nThroughout the CSS and HTML sections of the preview file, you can notice bracketed tokens that are used to inject strings into the markup. The following sections cover some of the key tokens that you see in previews.\n\nReaders may need to zoom in on this image and view it in landscape\n\nCSS Tokens\n\n\u2022 `[ID]`\u2014As you just learned, this token injects a unique identifier before each CSS rule to make them stand out when shown in a grid of composed look thumbnail previews.\n\n\u2022 `[T_HEIGHT]` and `[T_WIDTH]`\u2014The height and width of the preview window.\n\n\u2022 `[T_IMAGE]`\u2014The URL of the optional background image.\n\n\u2022 `[T_IMGHEIGHT] `and `[T_IMGWIDTH]`\u2014The height and width of the background image.\n\n\u2022 `[T_THEME_COLOR_<SLOTNAME>]`\u2014The color value for a given slot name in the SPColor file.\n\n\u2022 `[T_THEME_COLOR_<SLOTNAME>_AA]`\u2014Like the previous item but shows the color including the two extra hex digits needed to show alpha transparency for the color.\n\n\u2022 `[T_<SLOTNAME>_FONT]`\u2014The font value for a given slot name in the SPFont file.\n\nHTML Tokens\n\nThe HTML tokens are a little different from the CSS tokens. Where the CSS tokens refer to settings from the composed look like fonts, color, and a background image, the HTML tokens are strings that are predefined by Microsoft to be used as placeholders in the HTML design of the preview. For instance, one token injects fake top navigation, whereas another injects a fake search box.\n\nNOTE\n\nThe previous figure shows `Seattle.preview`, which includes most of the available HTML tokens. The Oslo.preview actually includes one extra token named `[SITETITLE]` which appears above the Quick Launch on the left.\n\nCreating a Custom Preview File\n\nThe process of creating the actual preview file for the WaffleNet master page would take up quite a few pages and take a lot of time to follow. Instead this section discusses the general steps for making your own preview, and includes some tips for making the task a little easier. You can also download the completed preview file from the chapter downloads for studying later\n\nExample: Making Your Own Preview\n\n1. Begin by copying one of the out-of-the-box master page previews, either `Seattle.``preview` or `Oslo.preview` from `_catalogs\/masterpage`. The two are subtly different, so check out a preview of both from the Change the Look menu before deciding which is the most similar to your custom design. Copy the preview file to the same folder as your custom master page, and rename it to the same exact filename with a `.preview` extension.\n\n2. Open the file in a text editor, and in the first two sections enter default SPColor and SPFont filenames. For example, the WaffleNet preview starts with this code:\n\n`BreakfastColors.spcolor`\n\n`[SECTION]`\n\n`BreakfastFonts.spfont`\n\n`[SECTION]`\n\nNOTE\n\nThe references do not include a path; SharePoint knows to look for them in `_catalogs\/theme\/15`.\n\n3. Skip over the next section with all the CSS rules, and go to the final section with the HTML; you'll come back to the CSS after the HTML. The default HTML in the Oslo and Seattle previews is fairly easy to read and modify; it represents a much more simplified version of the corresponding master pages. The preview HTML has a parent element of `<div id=\"dgp-pageContainer\">` and contains many child divs with style classes that describe their contents, including divs for the suite bar \/ ribbon, top navigation, page title, logo, left navigation, search box, and some fake page content.\n\n4. Modify the out-of-the-box preview file HTML to look similar to your custom branding. You want to try to get the broad strokes right; the preview does NOT need to be an exact replica of the master page. Ultimately, you must decide how close you want the preview to look to your custom branding. Use the HTML tokens that are already in the out-of-the-box preview; arrange them with your custom HTML to get a nice preview.\n\n5. Jump back up in the code to the previous CSS section. For this section there are a couple things that you need to handle.\n\n\u2022 First, update any of the existing out-of-the-box preview CSS that represents colors, fonts, and images from composed looks that you are NOT using in your custom branding. For instance, most of the WaffleNet page text is dark gray no matter what composed look is applied. Because of that the WaffleNet preview has changes like the following:\n\n`[ID] .dgp-contentBoxTableHeader`\n\n`{`\n\n` color: [T_THEME_COLOR_BODYTEXT];`\n\n` font-family: [T_BODY_FONT];`\n\n` font-size:1.125em;`\n\n` margin:1% 0 2% 0;`\n\n`}`\n\nIs changed to:\n\n`[ID] .dgp-contentBoxTableHeader`\n\n`{`\n\n` font-size:1.125em;`\n\n` margin:1% 0 2% 0;`\n\n`}`\n\nFor WaffleNet the color and font-family were removed; they instead inherit a color and font-family from the body.\n\n\u2022 Next, after you make any necessary changes to the default preview CSS, add any custom CSS you need to make your preview look like the master page. In some ways you can leverage copy and pasting from your existing custom CSS file styles, but remember a couple things:\n\n\u2022 You don't want to just paste in all your custom CSS; you need only a small percentage of the CSS to style this much simpler preview file.\n\n\u2022 Every CSS rule you add should not only begin with the unique identifier token `[ID]` but you also want to change the selectors to be unique from your typical custom branding. This may be a little confusing, but the need for uniqueness is because the previews are shown inside of iFrames, inside of whatever master page is currently applied to your site. If you use the same exact selector names as you did in the applied custom branding like the WaffleNet master page, it can make inspecting and debugging CSS tricky. In the WaffleNet preview, all the custom CSS classes begin with `prvCustom`.\n\n\u2022 You should strive to have all the height and width references use percentage-based measurements so that the preview scales properly. Having said that, you can certainly get by with some less important measurements set as pixels, especially if they are small pixel sizes.\n\n\u2022 In general, background images besides the main background image should be avoided or used sparingly. The out-of-the-box preview files for Oslo and Seattle use little, colored rectangles in place of most images. This is because if you want to use custom background images you need to create smaller versions of the originals in an image-editing program so that the smaller-scaled versions look good\u2014plus they still need to have CSS that allows them to scale appropriately. This means having a special `-ms-filter` in CSS to handle IE8's lack of CSS3 support. The WaffleNet preview includes a few custom background images; you can download the final preview code to see how they are handled.\n\n6. When you think you have the preview file set up the way you like, save the file. Before you test the previews though, create a new composed look that uses the WaffleNet master page. Follow the steps from earlier in this chapter, but use the following settings for the composed look:\n\n\u2022 Title\u2014WaffleNet\n\n\u2022 Name\u2014WaffleNet Theme\n\n\u2022 Master Page URL\u2014`\/_catalogs\/masterpage\/WaffleNet\/WaffleNet.master`\n\n\u2022 Theme URL\u2014`\/_catalogs\/theme\/15\/BreakfastColors.spcolor`\n\n\u2022 Image URL\u2014`\/_catalogs\/masterpage\/WaffleNet\/bg.jpg`\n\n\u2022 Font Scheme URL\u2014`\/_catalogs\/theme\/15\/BreakfastFonts.spfont`\n\n\u2022 Display Order\u20142\n\n7. With this composed look created, you can then go to the Change the Look menu and see the thumbnail preview in the list of composed looks. Compare it to your custom SharePoint branding. Be sure to also click the composed look and see the bigger preview in the next menu. Make sure the preview looks good at both scales.\n\nRemember the goal here is to get it close to the master page look and feel, but it doesn't need to be an exact mirror image. If you need to debug the HTML or CSS, you can use tools such as IE's F12 toolbar or Firefox's Firebug, but be aware that the HTML is buried deep in the source, and the inspection tool often does not bring up the exact location in the HTML source. If you need to make changes to the preview file, open it in your text editor and make changes at any time. Whenever you click Save, the composed look preview should update accordingly.\n\nSummary\n\n\u2022 Composed looks in SharePoint 2013 enable users to change the look and feel of a site without a lot of work.\n\n\u2022 Composed looks set default options for a master page, background image, colors, and fonts, which can then be changed by a user before the composed look is applied to a site.\n\n\u2022 Composed look fonts and colors are defined by XML files that end in .spfont and .spcolor, accordingly.\n\n\u2022 Master pages can use CSS comments that have specific tokens, which will receive values from an applied composed look in order to change background images, colors, and fonts. For these tokens to work, the CSS must live in a themable location.\n\n\u2022 Custom master pages will not show up as options in the Change the Look menu unless they have a corresponding .preview file with a matching filename.\n\n\u2022 Composed look preview files are based on a unique file structure that defines default colors, fonts, preview-specific CSS and HTML markup, as well as special tokens that will inject helpful values. The HTML and CSS in previews represent a simpler version of the master page branding that they represent. The CSS in previews should be based on percentage sizes to allow for scaling between two different preview sizes that SharePoint uses in the Change the Look menu.\n\n4\n\nOther Branding Concepts\n\n11. Modern Web Design and SharePoint\n\n12. Designing Apps\n11\n\nModern Web Design and SharePoint\n\nWhat's in this Chapter?\n\n1. HTML5 and CSS3\n\n2. jQuery in SharePoint\n\n3. Open Source Templates\n\n4. Designing Typography\n\n5. Structuring Content with Grids\n\n6. Designing for Mobile Devices\n\nOne of the most frequently asked user interface questions that came up for SharePoint 2010 was, \"Does it support HTML5?\" Chances are most people asking this question are interested in both HTML5 and CSS3, which are two technologies almost always lumped together, probably because they both entered the web development scene around the same time. When designers talk about modern web design techniques, they are usually referring to HTML5, CSS3, and other technology such as web fonts and jQuery.\n\nThe answer to the question of whether SharePoint 2010 supported HTML5 was always yes and no because SharePoint is browser-based software. Ultimately, whether a website shows HTML5 and CSS3 depends on browser support. SharePoint 2010 was basically coded to meet IE8 HTML standards, and IE8 was not known for its excellent support for modern HTML techniques. The out-of-the-box SharePoint 2010 master pages even had code in them to instruct Internet Explorer 9 to render the page in IE8 mode. With a little work, you could certainly make modern HTML techniques work in SharePoint 2010, but ultimately you weren't guaranteed a good experience throughout a SharePoint site.\n\nFast forward to SharePoint 2013 and the story has changed for the better. SharePoint 2013 is coded with IE9 and IE10 in mind, using standards-based HTML. IE8 is the lowest IE browser version supported in SharePoint 2013, and many special tweaks were added to the HTML and CSS to support IE8's lesser rendering capabilities. This is good news for those that want to use modern design techniques such as HTML5 and CSS3.\n\nYou may not realize it, but all HTML master pages created with Design Manager are HTML5, and almost all the custom master page examples in this book are based on HTML5 as well. This is all great for fans of modern web design, but unfortunately the two default out-of-the-box master pages (Seattle and Oslo) use the old XHTML 1.0 Strict DOCTYPE instead of the new HTML5 one. Interesting enough, even without a valid HTML5 DOCTYPE, Oslo and Seattle still show HTML5 and CSS3 content properly in most modern browsers. When it comes to IE rendering, SharePoint 2013 includes a key meta tag in the out-of-the-box master pages:\n\n`<meta http-equiv=\"X-UA-Compatible\" content=\"IE=10\" \/>`\n\nThis tag essentially tells IE that the markup was written for IE10 and to use that rendering mode if it's available.\n\nThis chapter looks at some of the ways that modern design techniques can be used with SharePoint 2013. This includes discussing HTML5, CSS3, web fonts, jQuery, and more.\n\nHTML5\n\nWhen you look at modern web design, between HTML5 and CSS3, much of the exciting new features are found in CSS3. That said, some new features in HTML5 are worth discussing:\n\n\u2022 New tags for `<video>` and `<audio>` to embed various types of media on the page without a plug-in such as Silverlight or Flash\n\n\u2022 A `<canvas>` tag that enables you to draw on the page using JavaScript\n\n\u2022 Several new semantic HTML markup tags including `<section>`, `<article>`, `<header>`, `<nav>`, and `<footer>`, which can be used to organize HTML code better\n\n\u2022 New form elements and input types such as color and date fields\n\n\u2022 Support for Scalable Vector Graphics (SVG)\n\n\u2022 Geolocation to identify the physical location of the browser\n\n\u2022 New concepts such as drag and drop and persistent local storage\n\nFor a full list of new features included with HTML5, see http:\/\/www.w3schools.com\/html\/html5_form_input_types.asp.\n\nOut of these noteworthy new features, two of the more visual additions that come with HTML5 are particularly interesting because they add features that formerly required plug-ins like Flash or Silverlight. These features are videos and canvases, and they are discussed in the following sections.\n\nNOTE\n\nSome of the examples in this chapter that show off HTML5 and CSS3 techniques require a modern browser such as IE9. If you need to use these features with older browsers such as IE8, you should look into technology such as Modernizr and polyfills. Modernizr is a JavaScript library that can detect browser capabilities as well as add support for new semantic HTML5 elements, whereas polyfills are JavaScript libraries that add in functionality that mimics some of the more advanced HTML5 and CSS3 features such as the video and canvas tags.\n\nFor more information on Modernizr and polyfills, check out <http:\/\/modernizr.com\/docs\/>.\n\nHTML5 Video and SharePoint\n\nThe basic markup for adding video in HTML5 looks like this:\n\n`<video poster=\"thumbnail.png\" width=\"320\" height=\"240\">`\n\n` <source src=\"movie.mp4\" type=\"video\/mp4\">`\n\n` Your browser does not support the video tag.`\n\n`<\/video>`\n\nTo try out the new HTML5 video features in SharePoint, it might be tempting to start a new HTML5 master page and add a Script Editor Web Part for adding the video markup, but none of this is actually needed. This is because SharePoint 2013 uses HTML5 by default when you use the ribbon to add video to a page. If you remember, Chapter 3, \"Working with the SharePoint 2013 User Interface,\" has a section titled \"Working with Videos,\" which includes an example for adding video to the page.\n\n\u00a92008, Blender Foundation www.bigbuckbunny.org\n\nIf you view source on a SharePoint 2013 page that has video added, you will find code similar to this buried in the HTML:\n\nThis includes the standard HTML5 `<video>` tag with some options such as a poster or preview image and the height and width of the video. Inside of the HTML5 video tag is where you would normally put some messaging to notify users that their browser can't display HTML5 video properly. Instead of doing that, SharePoint includes an `<object>` tag, which loads a Silverlight player. Although not everyone in the world has the Silverlight player, this is a nice feature to include as a fallback in case HTML5 doesn't work in the user's browser.\n\nFor more information on the specifics of how video works in HTML5, check out http:\/\/www.w3schools.com\/html\/html5_video.asp.\n\nAdding an HTML Canvas to SharePoint\n\nAnother interesting feature of HTML5 is the `<canvas>` tag. The canvas enables you to draw in two dimensions using standard JavaScript. For example, the following code would draw a blue square inside of an HTML5 canvas:\n\n`<script>`\n\n` var ctx=document.getElementById(\"myCanvas\").getContext(\"2d\");`\n\n` ctx.fillStyle=\"#0fff00\";`\n\n` ctx.fillRect(0,0,150, 150);`\n\n`<\/script>`\n\nFor more information on drawing with the HTML5 canvas, see https:\/\/developer.mozilla.org\/en-US\/docs\/HTML\/Canvas\/Tutorial\/Drawing_shapes.\n\nThe following example uses code from this Mozilla Developer Network article to draw a smiley face in a canvas tag added to SharePoint 2013. The example uses the default Seattle master page, which displays HTML5 fine in most modern browsers. If you are a purest, you might consider creating a new master page with the Design Manager because it creates master pages that use the HTML5 DOCTYPE.\n\nExample: Using HTML Canvas to Draw a Smiley Face\n\n1. Create a new page by clicking Settings Menu \u21d2 Add a page. In the dialog that opens, give the new page a name such as Canvas Example and click Create.\n\n2. When the new page opens, leave the cursor in the Page Content field. From the ribbon click Insert \u21d2 Web Part.\n\n3. Navigate to Categories \u21d2 Media and Content \u21d2 Parts \u21d2 Script Editor, and then click Add.\n\n4. From the Script Editor Web Part, click EDIT SNIPPET.\n\n5. Add the canvas tag, give it an ID of canvas for referencing, and give it a width and height of 200 pixels:\n\n`<canvas width=\"200\" height=\"200\" id=\"canvas\"><\/canvas>`\n\n6. Next, add code from the smiley face example from the URL earlier. This code makes a reference to the canvas tag's ID and then creates a drawing path using arcs and `moveTo` to move the \"pen\" around the canvas. The code also applies a stroke to the path; without this stroke the path would be transparent.\n\n`<script type=\"text\/javascript\">`\n\n` var ctx = document.getElementById('canvas').getContext('2d');`\n\n` ctx.beginPath();`\n\n` ctx.arc(75,75,50,0,Math.PI*2,true); \/\/ Outer circle`\n\n` ctx.moveTo(110,75);`\n\n` ctx.arc(75,75,35,0,Math.PI,false); \/\/ Mouth (clockwise)`\n\n` ctx.moveTo(65,65);`\n\n` ctx.arc(60,65,5,0,Math.PI*2,true); \/\/ Left eye`\n\n` ctx.moveTo(95,65);`\n\n` ctx.arc(90,65,5,0,Math.PI*2,true); \/\/ Right eye`\n\n` ctx.stroke();`\n\n`<\/script>`\n\n7. With the code complete, click Insert.\n\n8. Save and publish the page.\n\nWARNING\n\nSometimes SharePoint won't show the canvas drawing immediately after the page is saved. If this happens, just refresh your browser, and the smiley face drawing should show.\n\nCSS3\n\nYou can learn a lot about working with CSS in Chapter 6, \"Cascading Style Sheets and SharePoint,\" including a brief overview of how SharePoint 2013 uses a lot of CSS3 in the out-of-the-box CSS. This chapter just highlights the fact that SharePoint 2013 can show almost any CSS3 that your browser can actually render. You can see this in action in the following example.\n\nExample: Adding CSS to Script Editor Web Part\n\n1. Edit the page from the previous example, and then from the Script Editor Web Part, click EDIT SNIPPET.\n\n2. Add `<div class=\"shadowbox\">` around the `<canvas>` tag like this:\n\n`<div class=\"shadowbox\">`\n\n`<canvas width=\"200\" height=\"200\" id=\"canvas\"><\/canvas>`\n\n`<\/div>`\n\n3. Add the following CSS to the previous section. This styles the shadowbox div by adding a height, width, and border as well as a border-radius to round the corners and a box-shadow to add a drop shadow behind the div:\n\n`<style type=\"text\/css\">`\n\n` .shadowbox {`\n\n` width: 200px;`\n\n` height: 200px;`\n\n` border: 1px solid black;`\n\n` border-radius: 10px;`\n\n` box-shadow: 5px 5px 10px #999999;`\n\n` }`\n\n`<\/style>`\n\n4. Click Insert and then save and publish the page.\n\nThis was a simple example of using CSS to style a div; for more CSS3 ideas, you can explore the CSS3 examples at` `http:\/\/www.css3.info\/preview\/.\n\njQuery\n\nIf there is one technology that has forever changed the face of web development over the past decade it would have to be the release of the jQuery JavaScript library. You have probably heard a lot about jQuery over the past couple years, but if you don't know what it is, here is a description from the official jQuery website (http:\/\/jquery.com\/):\n\n\"jQuery is a fast, small, and feature-rich JavaScript library. It makes things like HTML document traversal and manipulation, event handling, animation, and Ajax much simpler with an easy-to-use API that works across a multitude of browsers. With a combination of versatility and extensibility, jQuery has changed the way that millions of people write JavaScript.\"\n\nBefore jQuery existed most JavaScript was written by hand, and any visual quirks between different browsers had to be handled manually with fixes and workarounds. Dynamic websites were more difficult to code and had to be updated constantly to support new browser versions as they were released. jQuery changed all this by providing one open source code base for JavaScript that enables anyone to quickly and easily control many aspects of an HTML page. Today, if you can think of a visual technique, someone has probably already coded an open source jQuery plug-in that can implement it quickly and easily. Sites such as Facebook and Twitter use technology similar to jQuery to provide dynamic user experiences that many users expect in modern web design.\n\nAlthough there were some quirks when using jQuery with SharePoint 2010, SharePoint 2013 works great with jQuery. Microsoft has actually based a lot of the SharePoint 2013 App model on the idea that client-side scripting such as jQuery can be used to create rich applications that interact with SharePoint data without needing to understand the intricacies of C# development. You will learn more about how apps work with SharePoint 2013 in Chapter 12, \"Designing Apps.\"\n\nLoading jQuery in SharePoint\n\nWhen using jQuery with SharePoint 2013, the first step is to figure out how to load the jQuery library. Certainly you could load jQuery for just one page if you want to with a Content Editor or Script Editor Web Part, but chances are you want to use jQuery in multiple pages throughout your site. The following sections highlight a few ways you can load jQuery into SharePoint along with some pros and cons for each option.\n\nWARNING\n\nIf jQuery is loaded more than once on the same page, you could run into errors. Keep this in mind if you are including custom controls or Web Parts that might include jQuery themselves. Some of the following methods listed can be used to make sure jQuery loads only once. You can also use JavaScript to check if jQuery is not already loaded via this conditional: `if ``(typeof jQuery == 'undefined')`\n\nLink to Content Delivery Network (CDN) in the Master Page\n\nContent Delivery Networks, or CDNs, are a collection of servers located around the world providing centralized hosting for commonly used Internet files. Two of the most popular CDNs for jQuery are hosted by Microsoft and Google. Here is an example of what the reference would be to either of them; the most common location for this reference would be near the bottom of the `<head>` section of your master page:\n\n`<script type=\"text\/javascript\" `\n\n` src=\"http:\/\/ajax.aspnetcdn.com\/ajax\/jQuery\/jquery-1.9.0.min.js\">`\n\n` <\/script>`\n\n`<script type=\"text\/javascript\"`\n\n` src=\"http:\/\/ajax.googleapis.com\/ajax\/libs\/jquery\/1.9.0\/jquery.min.js\">`\n\n` <\/script>`\n\nBoth these links point to the minified or compressed version of the library for use in production environments; if you want to use the more human readable uncompressed version for debugging, you can remove `.min` for either reference. Also, keep in mind that as jQuery evolves, new version numbers will be released, and you may someday want to update your references to reflect a newer version.\n\nLink to a Local Copy from the Master Page\n\nThis method is similar to the CDN method except you download the latest jQuery library from `jQuery.com` and add the file to your site collection. There are many places you could put the file, but with the new mapped drive features of SharePoint 2013, it would probably be easiest to add the library to the same place your master page is located in the Master Page Gallery. After the library is added to the same location as your master page, you can reference it much like these code examples do:\n\n\u2022 From a custom master page located in a site collection at the web root, use this code:\n\n`<script type=\"text\/javascript\"`\n\n` src=\"\/_catalogs\/masterpage\/MyMasterPageFolder\/jquery-1.9.0.min.js\">`\n\n`<\/script>`\n\n\u2022 If you use the Design Manager to create an HTML-based master page and your master page and jQuery are located in the same folder, you can reference jQuery without a path, and SharePoint can handle making the correct reference on conversion to a master page.\n\n`<script type=\"text\/javascript\" src=\"jquery-1.9.0.min.js\">`\n\n`<\/script>`\n\nUse a ScriptLink Tag in the Master Page\n\nScriptLink is a tag that SharePoint uses to load JavaScript while ensuring that a script is loaded only once as well as ensuring that any dependencies have been loaded first.\n\nThis is what a reference would look like to the jQuery library if it were copied to the Master Page Gallery of your site collection:\n\n`<SharePoint:ScriptLink ID=\"ScriptLink1\" runat=\"server\"`\n\n` Name=\"~sitecollection\/_catalogs\/masterpage\/jquery-1.9.0.min.js\" \/>`\n\nNOTE\n\nIf you use the Design Manager to create an HTML master page, you would need to surround the line with `<!--SPM:` and `-->`.\n\nUse a Custom Action to Load jQuery\n\nCustom actions are a feature of SharePoint that enables you to extend the default user interface with custom code, such as adding a button to the ribbon programmatically. Similarly, a custom action can be created with Visual Studio, which automatically loads jQuery for an entire site collection. For more information on using Visual Studio to create this feature, see the blog post by Bijay Kumar, \"Call jQuery to SharePoint page using Custom Action\" at http:\/\/drisgill.com\/go\/jquery-action.\n\nUsing jQuery with SharePoint\n\nNow that you know how to load jQuery into SharePoint, you can try a quick example. This example uses a SharePoint Server publishing site and the Design Manager feature to quickly create an HTML-based master page. If you don't have SharePoint Server or access to Design Manager, the steps may be different, but the concepts remain the same. For this example, you will add a button that hides and shows the SharePoint search box with a simple fade in and out effect.\n\nExample: Using jQuery to Show and Hide a Search Box\n\n1. Start by navigating to Site Settings \u21d2 Look and Feel \u21d2 Design Manager.\n\n2. On the left menu, click Edit Master Pages \u21d2 Create Minimal Master Page. Give it a name such as jQueryDemo and click OK.\n\n3. From a mapped drive to your Master Page Gallery, open the new HTML master page for editing.\n\n4. In your web browser, from the master page list in Design Manager, click Conversion Successful next to the new master page. Then at the top of the next page, click Snippets.\n\n5. From the ribbon click Search Box; then copy and paste the snippet into the master page above the `PlaceHolderMain` snippet.\n\n6. Navigate to the Edit Master Pages menu in Design Manager, and next to the master page, click the ellipsis (...) and then the second ellipsis in the dialog window. Click Publish a Major Version and then click OK.\n\n7. Apply the master page from Site Settings \u21d2 Look and Feel \u21d2 Master Page. There is no navigation in the master page, so you may have to move around the site by typing in URLs.\n\n8. To see the homepage, navigate back to the top-level URL for your site. At this point the master page isn't special, just a simple HTML5 master page with a search box.\n\n9. In your editor, before the closing `<\/head>` tag, add the following line:\n\n`<script type=\"text\/javascript\" `\n\n` src=\"http:\/\/ajax.aspnetcdn.com\/ajax\/jQuery\/jquery-1.9.0.min.js\">`\n\n`<\/script>`\n\n10. Now add an ID to the div around your search box so that jQuery can reference it properly. Update `<div data-name=\"SearchBox\">` by adding `id=\"SearchBox\"`, like this:\n\n`<div data-name=\"SearchBox\" id=\"SearchBox\">`\n\n11. Add a button above the search box div:\n\n`<button id=\"SearchToggle\" type=\"button\" style=\"margin:10px 0px;\">`\n\n` Toggle Search`\n\n`<\/button>`\n\n12. Now all you need is some jQuery to handle the button click. Add the following code to the bottom of the page, before the closing `<\/body>` tag:\n\n`<script>`\n\n`$(document).ready(function(){`\n\n` $(\"#SearchToggle\").click(function(){`\n\n` $(\"#SearchBox\").toggle(500);`\n\n` });`\n\n`});`\n\n`<\/script>`\n\nThis jQuery code waits until the HTML document is fully loaded and then it adds a click function to the SearchToggle button. When the button is clicked the `SearchBox` div toggles between a hidden and showing state. The 500 inside of the toggle function tells jQuery to fade the search box in and out over 500 milliseconds.\n\nThis is a rather simple example of the power of jQuery; for more information on how jQuery works, check out http:\/\/docs.jquery.com\/.\n\nHTML5 Boilerplate\n\nOne of the great things about modern web design is that folks are embracing the idea of creating free community resources. In the past few years, a new open source template concept has emerged that provides designers with a common standard starting point for projects that use HTML5, CSS3, and jQuery.\n\nOne popular open source template for modern web design is the HTML5 Boilerplate (sometimes abbreviated as H5BP). According to its website:\n\n\"HTML5 Boilerplate helps you build fast, robust, and adaptable web apps or sites. Kick-start your project with the combined knowledge and effort of 100s of developers, all in one little package.\"\n\nhttp:\/\/html5boilerplate.com\n\nHere is a list of some of the features included with HTML5 Boilerplate:\n\n\u2022 A barebones starter HTML template that is mobile friendly\n\n\u2022 Placeholder icons for both a favicon and Apple touch-enabled devices\n\n\u2022 Google Analytics code\n\n\u2022 The latest version of the minified jQuery library\n\n\u2022 Modernizr to help support older browsers such as IE8\n\n\u2022 Special open source CSS called Normalize.css to help render HTML elements consistently across different browsers\n\n\u2022 CSS styles for media queries (for more information see the upcoming section on responsive design) as well as print styles\n\nThese features are great but you may wonder how well HTML5 Boilerplate plays with SharePoint 2013. Because HTML5 Boilerplate uses a lot of new and advanced techniques, there are actually a few things in it that don't work well with the SharePoint 2013 Design Manager HTML import process. Because of this you need to make some changes to the HTML5 Boilerplate to be able to use it with SharePoint Server 2013's Design Manager feature.\n\nNOTE\n\nThe HTML5 Boilerplate codebase is constantly changing. This chapter was written with version 4.1.0. If you download a more recent version, the HTML-to-master page conversion process may require some adjustment before it works properly.\n\nTo follow along with this example, browse to http:\/\/html5boilerplate.com\/ and download the latest version of HTML5 Boilerplate.\n\nNOTE\n\nAlthough it would be tempting to just ignore HTML5 Boilerplate completely for SharePoint, the implications actually go further than you might think. Many modern web design templates, whether open source or for pay, are actually based on the HTML5 Boilerplate codebase. Keep this in mind if you ever work with one of these design templates; they may require the same modifications before converting them to working master pages.\n\nExample: Changing HTML5 Boilerplate to Work with the SharePoint 2013 Design Manager\n\n1. From a mapped drive to your Master Page Gallery, make a new folder called HTML5Boilerplate and copy in all the files from the HTMLBoilerplate zip file.\n\nNOTE\n\nSome of the files that are unique to non-Windows servers will fail to copy over, so just skip these files.\n\n2. Rename index.html to HTML5Boilerplate.html so that you can find it easily in the list of master pages and open it in an editor.\n\n3. Take a look at the HTML source; there aren't a lot of lines of code, but there are some unique techniques used. If you try to convert the HTML to a master page now, you would see some strange behavior. The Design Manager would fail to give you a status for the conversion, and the details would say, \"We couldn't find the converted file. After editing and saving your HTML design file, refresh this page.\"\n\n4. This Design Manager confusion is caused by the code at the top of the HTML5 Boilerplate. Namely, the lines before the `<head>` tag that start with `<!--[if lt IE7>...` are lines used to add special CSS classes for old versions of IE. Although this technique is technically valid HTML, it causes issues for the Design Manager. To handle the issue, edit the top section so that it looks like this:\n\n`<!DOCTYPE html>`\n\n` <html class=\"no-js\">`\n\n` <head>`\n\n5. With that change, the HTML would convert to a working master page in Design Manager, but you need to make a couple other minor changes to how the jQuery is loaded. Scroll down to the first reference to `jquery.min.js`, which is the code for loading jQuery from the Google CDN. The line sometimes gets changed automatically when the file is copied into SharePoint. Make sure the line doesn't say this:\n\n`<script src=\"\/ajax\/libs\/jquery\/1.9.0\/jquery.min.js\"><\/script>`\n\nInstead the line should be\n\n`<script `\n\n` src=\"\/\/ajax.googleapis.com\/ajax\/libs\/jquery\/1.9.0\/jquery.min.js\">`\n\n`<\/script>`\n\n6. If the line is incorrect, change it back at this time. The syntax for this URL may seem odd, but it is used to load either https or http depending on the protocol used on the site.\n\n7. The last changes that need to be made are to the next line after the one you just looked at. By default it looks like this:\n\n`<script>`\n\n` window.jQuery || document.write('<script src=\"js\/vendor\/jquery-`\n\n` 1.9.0.min.js\"><\\\/script>')`\n\n`<\/script>`\n\nThis line is used as a fallback in case the CDN can't be loaded for some reason; in those cases jQuery will be called from a local directory. This line is technically correct, but SharePoint gets confused by it. Two changes need to be made:\n\n\u2022 Split into three lines so that SharePoint can parse it properly\n\n\u2022 Change the jQuery path to include the path from the top of your site collection to the HTML5Boilerplate folder in your master page gallery. The following code shows how the code needs to change. If your site collection is not at the root, your reference may be different:\n\n`<script>`\n\n` window.jQuery || document.write('<script `\n\n` src=\"\/_catalogs\/masterpage\/HTML5Boilerplate\/js\/vendor\/jquery-`\n\n` 1.9.0.min.js\"><\\\/script>')`\n\n`<\/script>`\n\nSave these changes; at this point the HTML file is ready to be converted to a master page. If you want to make more changes to the file, you can do them now, or you could wait until later after the conversion process.\n\nConverting an HTML File to a Master Page\n\n1. From your SharePoint site navigate to Site Settings \u21d2 Look and Feel \u21d2 Design Manager\n\n2. On the left menu, click Edit Master Pages \u21d2 Convert an HTML File to a SharePoint Master Page.\n\n3. In the Select an Asset window, navigate to the HTML5BoilerPlate folder; select HTML5Boilerplate.html and click Insert.\n\n4. When the page refreshes the master page should have converted successfully, so you can publish it and apply the master page to your site if you want.\n\nAt this point, HTML5 Boilerplate code is loaded into SharePoint, and you can take advantage of many of its features. The look and feel isn't pretty; essentially it just says, \"Hello world! This is HTML5 Boilerplate.\" But that's the beauty of templates like this; you can use them as a jumping-off point to create anything you want using modern web design standards in SharePoint 2013.\n\nWARNING\n\nOne thing to watch out for in templates such as HTML5 Boilerplate is code that is known as a CSS reset. These CSS resets help standardize the rendering of HTML elements across all the different browser platforms. Although this helps maintain a standard look across browsers, it can affect the out-of-the-box SharePoint styles as well. Be sure to test sites with CSS resets thoroughly before using in production. Usually you can see changes in the way SharePoint styles out-of-the-box menus, toolbars, and the ribbon.\n\nIn HTML5 Boilerplate this code is known as `normalize.css`. For the most part`,` `normalize.css` is less aggressive than CSS resets that are used in some other open source projects. Be sure to test them, especially in collaborative sites where users need to interact with menus and the ribbon.\n\nWeb Fonts\n\nWhen creating a great design, typography is one of the main concepts that comes into play. Outside of the web world, traditional and print designers have spent the last 30 years using thousands of different styles and types of fonts to create interesting and engaging content. Web developers have been trying to include aesthetically pleasing typography through the use of fonts since the beginning of HTML; however, until recently this has been limited to mostly relying on the default fonts installed on users' computers. With the advent of CSS3 and modern browser technology, however, web designers finally have a reliable mechanism for showing many fonts that go beyond those that the users already have installed on their computers.\n\nThrough the use of the CSS3 `font-face` rule, web designers can include licensed fonts with their CSS code. Using this new method, browsers can download the font source files and display text using the styles defined by the particular font. The following is a simple example of how `font-face` is used to show a unique font:\n\n`@font-face {`\n\n` font-family: 'ArvoRegular';`\n\n` src: url('fonts\/Arvo-Regular-webfont.eot');`\n\n` src: url('fonts\/Arvo-Regular-webfont.eot?#iefix') `\n\n` format('embedded-opentype'),`\n\n` url('fonts\/Arvo-Regular-webfont.woff') format('woff'),`\n\n` url('fonts\/Arvo-Regular-webfont.ttf') format('truetype'),`\n\n` url('fonts\/Arvo-Regular-webfont.svg#ArvoRegular') format('svg');`\n\n` font-weight: normal;`\n\n` font-style: normal;`\n\n`}`\n\nThis code was generated by a useful, free service available from http:\/\/www.fontsquirrel.com. .\n\nFont Face Basics\n\nThe font face begins with `font-family: 'ArvoRegular'`. This is how you would refer to this font throughout your CSS file. After that, there are URLs listed for four different types of font sources. Although it would be a lot easier to include only one font source file, unfortunately, each web browser vendor uses different font types, and these four cover the most common browser needs:\n\n\u2022 EOT\u2014Embedded Open Type (EOT)\n\n\u2022 WOFF\u2014Web Open Font Format (WOFF)\n\n\u2022 TTF\u2014True Type (TTF)\n\n\u2022 SVG\u2014Scalable Vector Graphics (SVG)\n\nGathering all the font formats for each font you want to use can be a pain. Luckily, tools like the aforementioned Font Squirrel @font-face Generator can create the four formats for you based on whatever type you upload.\n\nWARNING\n\nNot every font creator enables their fonts to be embedded as web fonts. Be sure to check the licensing of all fonts before including them in your SharePoint site.\n\nAlong with the ability to add fonts directly to your server for users to see with the font-face rule, there are also services that can handle the entire process for you. Google Web Fonts (http:\/\/www.google.com\/webfonts) is a free service that includes many open source fonts, even providing the hosting of these files and sample code to include them with your design. Another service called Typekit (https:\/\/typekit.com) provides a similar service but for licensed fonts. It has a fee associated with it but offers a free trial option.\n\nUsing Web Fonts with SharePoint 2013\n\nUsing web fonts with SharePoint 2013 is similar to how you would use them in traditional HTML. If you have followed along with the examples in this book, you have already used web fonts with the WaffleNet example from Chapter 7, \"Creating SharePoint Branding.\" The following example includes a web font that is used for the body text.\n\nExample: Using a Web Font with SharePoint\n\n1. Acquire the source files, EOT, WOFF, TTF, and SVG for the font you want to display on the web. The aforementioned Font Squirrel can help you get the files that you need.\n\n2. From a mapped drive to your Master Page Gallery, copy the images into the same folder as your master page. You may want to create a subfolder for the fonts to keep them organized and away from the rest of the site files.\n\n3. If you don't already have a CSS file for the master page, create one and reference it in the master page or the HTML file. After you have a CSS file associated to the master page, open it for editing.\n\n4. Add the font-face rule for your new font. Again, Font Squirrel can help you with the syntax if you aren't sure. Here is a sample font-face rule; in this example the font was copied to a subfolder named fonts. The code is exactly the same as the earlier example because web fonts work the same in SharePoint as they do in standard HTML.\n\n`@font-face {`\n\n` font-family: 'ArvoRegular';`\n\n` src: url('fonts\/Arvo-Regular-webfont.eot');`\n\n` src: url('fonts\/Arvo-Regular-webfont.eot?#iefix') `\n\n` format('embedded-opentype'),`\n\n` url('fonts\/Arvo-Regular-webfont.woff') format('woff'),`\n\n` url('fonts\/Arvo-Regular-webfont.ttf') format('truetype'),`\n\n` url('fonts\/Arvo-Regular-webfont.svg#ArvoRegular') format('svg');`\n\n` font-weight: normal;`\n\n` font-style: normal;`\n\n}\n\n5. Use the font-family reference in your CSS styles; for example, you could set the `<body>` to use the custom font. Here the font size is also increased so that the change is easier to see:\n\n`body {`\n\n`font-family: 'ArvoRegular',arial,sans-serif;`\n\n` font-size: 120%;`\n\n`}`\n\n6. Save the CSS file and you should see the custom font load in your SharePoint site. Remember ,if you want anonymous users to see the styles, you need to publish all the related assets including the master page, CSS, and all the font files.\n\nNotice that much of the body content is now using the larger custom font; in this case the font is Arvo Regular.\n\nGrid-based Design\n\nThe use of grids in traditional print design is another concept, like typography, that has been around for a long time. The idea is that it's easier to create an appealing page layout if you use tools such as a grid overlay to arrange content in a structured pattern. By overlaying the page with a grid, designers could arrange content into columns and rows (either symmetrically or asymmetrically) and as long as the grid was used, the page ended up looking nice to the reader.\n\nOne of the founders of grid-based design, Josef M\u00fcller-Brockmann, provides a good explanation about grids:\n\n\"The grid system is an aid, not a guarantee. It permits a number of possible uses and each designer can look for a solution appropriate to his personal style. But one must learn how to use the grid; it is an art that requires practice.\"\n\nThis image, provided by the Wikimedia Commons (<http:\/\/en.wikipedia.org\/wiki\/File:Grid2aib.svg>), shows a typical grid that might be used to align text in a page layout.\n\nAs web design has evolved over the years, so has the level of sophistication that has been applied to web page layouts. In recent years, the idea of using grids in web design has become popular. This popularity has increased since the introduction of open source web grid systems such as the 960 Grid System (http:\/\/960.gs). The 960 Grid System was created to target 1,024-pixel-wide displays, the most common low-end resolution for users at that time. After accounting for scroll bars and other things, the team settled on the standard web page size of 960-pixel-wide, which was easily divided by many numbers making it a good width from which to apply a grid system. Essentially, after 960 Grid is added to a website, the designer can easily create rows and columns that conform to a large variety of appealing combinations. You can learn more about working with 960 Grid from the articles linked from http:\/\/960.gs\/.\n\nMuch like HTML5 Boilerplate that you learned about earlier, there are some gotchas to using grid systems with SharePoint. Most notably is that many grid systems such as 960 Grid use CSS that aggressively resets the standard rendering for HTML elements across browser platforms. Although this helps the grid stay consistent, be sure to test the result on your SharePoint site.\n\nExample: Using 960 Grid with SharePoint 2013\n\n1. Begin by downloading the code from <http:\/\/960.gs> by clicking the Big ol' DOWNLOAD button :).\n\n2. After the file is downloaded, you can see a folder inside the zip file called code. Map a drive to your master page gallery, make a subfolder for 960Grid, and copy in all the files from the code folder. If you want you can rename demo.html to 960Grid.html so that it's easier to find as a master page. Then just use the SharePoint Server Design Manager feature to convert the html file into a functioning master page without changing any of the code. These steps are similar to any of the previous Design Manager examples.\n\n3. After the master page has been converted, you can make some changes by editing the HTML master page. The default 960Grid.html file (originally demo.html) includes some example styles and HTML markup that isn't needed in actual usage. Remove this reference: `<link rel=\"stylesheet\" href=\"css\/demo.css\" \/>`.\n\n4. Remove most of the sample 960 Grid content that is inside of <div id=\"s4-bodyContainer\"> leaving just the `<div class=\"container_12\"><\/div>`. Then move the `ContentPlaceHolderMain` `DIV` and all its contents into the `container_12 DIV`. The section should look like this when you are done:\n\n`<div id=\"s4-bodyContainer\">`\n\n` <div class=\"container_12\">`\n\n` <div data-name=\"ContentPlaceHolderMain\">`\n\n` <!--CS: Start PlaceHolderMain Snippet-->`\n\n` <!--SPM:<%@Register Tagprefix=\"SharePoint\" `\n\n` Namespace=\"Microsoft.SharePoint.WebControls\" `\n\n` Assembly=\"Microsoft.SharePoint, Version=15.0.0.0, `\n\n` Culture=neutral, PublicKeyToken=71e9bce111e9429c\"%>-->`\n\n` <!--MS:<SharePoint:AjaxDelta ID=\"DeltaPlaceHolderMain\" `\n\n` IsMainContent=\"true\" runat=\"server\">-->`\n\n` <!--MS:<asp:ContentPlaceHolder ID=\"PlaceHolderMain\" `\n\n` runat=\"server\">-->`\n\n` <!--ME:<\/asp:ContentPlaceHolder>-->`\n\n` <!--ME:<\/SharePoint:AjaxDelta>-->`\n\n` <!--CE: End PlaceHolderMain Snippet-->`\n\n` <\/div>`\n\n` <\/div>`\n\n`<\/div>`\n\nAt this point the 960Grid is clean and ready to use with SharePoint; however, nothing is actually using the grid yet. You can save the master page, publish it, and apply it to your SharePoint site. Keep in mind that you didn't include any navigation snippets, so the interface will be minimal.\n\nTo take advantage of the grid though, you could create a new page layout using the Design Manager as you learned in Chapter 5, \"Using the Design Manager to Start a Design in SharePoint.\" The downloads for this chapter include a completed Design Manager page layout that is arranged like this pseudo code:\n\n`<div class=\"grid_4\">`\n\n` Page Image Snippet`\n\n` Image Caption Snippet`\n\n`<\/div>`\n\n`<div class=\"grid_8\">`\n\n` Title Snippet`\n\n` Page Content Snippet`\n\n`<\/div>`\n\n`<div class=\"clear\"><\/div>`\n\n`<div class=\"grid_4\">`\n\n` Web Part Zone 1 Snippet`\n\n`<\/div>`\n\n`<div class=\"grid_4\">`\n\n` Web Part Zone 2 Snippet`\n\n`<\/div> `\n\n`<div class=\"grid_4\">`\n\n` Web Part Zone 3 Snippet`\n\n`<\/div>`\n\nAlthough you can't see the gridlines, notice that the fields are arranged in a grid pattern. The first row contains a smaller column on the left with the Page Image, Image Caption, and a larger column on the right with the Title and Page Content. Then there is a second row split into three equal-sized columns with Web Part zones in them. In the page layout code, 960 Grid uses the classes `grid_4` and `grid_8` to set the grid column widths, and it uses `<div class=\"clear\"><\/div>` to start new rows.\n\nResponsive Design\n\nFor modern web design, the hottest new topic is responsive design. It's probably no surprise to you that mobile devices have become incredibly popular in recent years. Almost everyone has a smartphone or tablet device, especially in the business world. In turn, web designers have had to struggle to not only keep up with designing websites to target all these small screens, but to also keep mobile style and content consistent with the traditional full-sized desktop browsing experience.\n\nThe idea of responsive design is one potential answer to this problem, providing a means of designing a website so that it looks good on a widescreen desktop browser but also scales intelligently to look right on the smallest of modern mobile devices. The topic of responsive design even came up during the 2012 U.S. presidential election, as one candidate's website used a responsive design to handle all browsers, and the other used two targeted designs split between desktop and mobile experiences. At the time of this writing, there isn't a clear-cut winner in this debate\u2014even though the presidential race was decided. The decision on whether one method or the other is right for a particular project is left to the designer.\n\nThe key to allowing responsive design to become as popular as it is today comes from the addition of media queries in CSS3. With media queries it's possible to target portions of a CSS style sheet to different devices based on the device's capabilities. Media queries consist of a media type (screen, print, and so on) and at least one expression that can check to see if a browsing device supports a specific media feature (device-width, resolution, device-aspect-ratio, and so on). For example, the following CSS would target the features of an iPad because its resolution falls between 768 and 1,024 pixels wide depending on the orientation of the device:\n\n`@media only screen and (min-device-width : 768px) and (max-device-width : 1024px) {`\n\n` body {`\n\n` font-size: 16px;`\n\n` }`\n\n`}`\n\nAny device that meets those screen-width specifications has the style rule inside the media query applied; in this case, the body has its font size changed to 16px.\n\nYou can learn more about media queries from https:\/\/developer.mozilla.org\/en-US\/docs\/CSS\/Media_queries.\n\nFor a list of media queries for common mobile devices, see http:\/\/css-tricks.com\/snippets\/css\/media-queries-for-standard-devices\/.\n\nNOTE\n\nMedia queries are different than device channels. You can learn more about how these two concepts differ by reviewing device channels in Chapter 8, \"Advanced SharePoint Branding Tasks.\"\n\nIn the next section, you will look at an existing responsive HTML design and use the Design Manager to create a simple master page that uses media queries to respond to mobile devices.\n\nCreating a Responsive Master Page\n\nYou can use the Design Manager feature to convert a responsive HTML design into a SharePoint master page. Keep in mind that you need SharePoint Server with publishing enabled to use the Design Manager.\n\nWARNING\n\nThe responsive master page that you create in this chapter is meant as an exercise in showing how responsive design can work in SharePoint. As such, it has been tested only minimally for the homepage of a publishing site. If you want to use this code in production\u2014particularly if you want to use it for collaboration sites\u2014more testing and changes may need to be made to accommodate all usage scenarios.\n\nThe bulk of the responsive awesomeness that you will see later in the example starts with an HTML5 design that has responsive CSS applied to it. You can get the initial HTML design from the chapter downloads. Before you try to convert this HTML into a working master page you should understand how it's set up. To do so, open the HTML page `Responsive.html` in a text editor.\n\nFor the most part, this is a straightforward HTML5 page, with header and body sections, and placeholders for a logo, top navigation, title, search box, left navigation, and main body content.\n\nIn the `<head>` section you can see a few interesting lines. First, there is this tag:\n\n`<meta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\" \/>`\n\nThe viewport tag is used for mobile devices such as smartphones, which have the capability to load a page initially larger than the actual browser size. These devices enable the user to zoom in and out with finger gestures to see more or less of the page. The viewport tag tells these devices how to load the page, including the initial width and zoom scale. On the `Responsive.html` page, this is set to the most basic option of zooming into the page to show the entire contents of the page on first load. When you work with a responsive design, this tag is necessary because you want to control the mobile browsing experience at different sizes.\n\nAfter the viewport tag, the style sheet is loaded for the page, and after that you will see JavaScript that includes `respond.min.js`. Respond.js is the name of a lightweight polyfill that adds support for CSS3 media queries to older browsers such as IE8. This is needed because the responsive HTML leverages media queries extensively, and you need something to handle IE8's lack of media query support. For more information on Respond.js see https:\/\/github.com\/scottjehl\/Respond.\n\nEverything else in the HTML file is normal HTML5 markup; most of the magic happens in the included CSS style sheet.\n\nResponsive CSS\n\nThe style sheet `Responsive.css` (available for download) is broken into three major sections to handle the three supported browsers sizes. When you are looking at making responsive CSS there are two basic approaches you can decide between. You can make the CSS either desktop first or mobile first; in other words, the design will either start out as a desktop design that is reduced as the browser gets thinner, or it is a mobile design that expands as the browser gets wider. Although you could use either method, `Responsive.css` uses a desktop first design because SharePoint is primarily a desktop web application. (At least in 2013\u2014for future versions this could change.) Because of this design decision the CSS file starts out with desktop styles, then has styles for a medium browser width, and it ends with mobile styles.\n\nLooking at `Responsive.css` you see that the first section of CSS is the default desktop CSS. There are styles that set a minimum and maximum width for the page content to allow the page to be centered and look good, but not stretch too far if the resolution is high. The elements are arranged as you would normally see them in SharePoint; they are in two main rows, the header section with the site logo, top navigation, site title, search, and a body section with the left navigation and main body content. Many of the elements float on the same line as other elements and spread across the width of the page.\n\nThe next section begins with this line:\n\n`@media only screen and (max-width: 767px) {`\n\nThis is a media query that looks only at devices smaller than the width of a tablet, such as the iPad when it is in portrait mode. Anything above that width shows the standard desktop CSS, but devices smaller than 768 pixels wide show this CSS (unless they are even smaller and fall into the next category). In this media query the example CSS does a few things. Most notably, the body style font size is reduced a little, the left navigation is hidden while the main content area is increased to 100 percent, and the logo and navigation are split into separate rows to allow for more room for navigation to show across the screen.\n\nThe last section begins the mobile media query like this:\n\n`@media handheld, only screen and (max-width: 480px), only screen and (-webkit-min-device-pixel-ratio: 2) and (max-device-width: 1500px) {`\n\nThis media query is a bit more complex than the previous one. It starts by checking for devices that describe themselves as handheld, which is typically an older designation for phones that aren't considered smartphones. Next, the media query checks for devices that are no bigger than the typical mobile phone screen width of 480 pixels wide. Lastly, the media query checks for devices that have high-pixel density such as the retina iPhone and iPad as well as some Android devices, but will target them only if they are less than 1,500 pixels wide. If a device meets any of those criteria, the mobile media query styles will be rendered. The goal here is to target smaller mobile devices without catching bigger tablet devices.\n\nThe CSS in this last mobile section makes a few changes to the display. First, the body size is increased because mobile devices are a little harder on the eyes. Next, the page width is changed to `auto` to allow the page to expand and contract as needed, and most of the padding, margin, and floating are turned off for the major sections. The biggest change is the navigation, which is changed from displaying inline to block, placing them vertically down the page\u2014an orientation that lends itself well to mobile device browsing. Also, the navigation items are increased in size and gain a background color to make them more finger-friendly.\n\nExample: Using a Responsive Design in SharePoint\n\n1. Map a drive to your Master Page Gallery and create a subfolder named Responsive. Then copy in all the HTML, CSS, and JavaScript from the example.\n\n2. Use the Design Manager to convert the HTML into a functioning master page. If you need a refresher on these steps, see the previous examples in this chapter or Chapter 5.\n\n3. After you have the HTML file converted to a master page, you can click its name in the list of master pages in the Design Manager under the Edit Master Pages step. This brings up the preview screen; if you resize your browser, you see that the media queries are already working and applying different styles to the different browser sizes.\n\n4. Although the media queries work at this point, you can do a few things to make the experience nicer. Start by adding the following snippets to the HTML master page. Each snippet can replace the comment that corresponds to it in the HTML master page:\n\n\u2022 Site Logo\n\n\u2022 Site Title\n\n\u2022 Search Box\n\n\u2022 Vertical Navigation\n\n\u2022 Replace Main should be replaced with the `PlaceHolderMain` snippet that is already included on the page.\n\n5. The last snippet to add is the Top Navigation, but you need to change two options before you cut and paste it. Under Customization - Top Navigation (AspMenu) change Important \u21d2 StaticDisplayLevels to 1; expand Behavior and change MaximumDynamicDisplayLevels to 1 as well. Click Update before copying the code. Then in the HTML master page, replace the entire `<UL>` structure with the snippet. There is also a rogue `<A HREF>` in this snippet that has no navigation item; you should remove this because otherwise it will be styled with an empty blue box like the other navigation items in the mobile view. Do a Find on your HTML master page to find and delete this line:\n\n` <a name=\"startNaviation\"><\/a>`\n\n6. At this point the HTML master page is ready to go, but there are a couple changes to make to `Responsive.css` for SharePoint-specific styling. Rather than list the code here, replace your existing CSS with the finished `Responsive.css` from the chapter downloads.\n\nBy replacing this CSS you have added several things. In the desktop section, the styles are updated for the body container, top navigation, left navigation, and the welcome page to allow them to look nice in SharePoint. This section is fairly similar to what you would do after converting any HTML design to a master page. For the next section, the medium-sized media query, there are no updates needed for SharePoint.\n\nThe last section, the mobile media query, gets the bulk of the updates to handle SharePoint-specific elements that need to look different on the smaller screen. Here is a list of some of the main styling changes that were made:\n\n\u2022 The ribbon is hidden because it would be difficult to use at this size.\n\n\u2022 The search box size is increased.\n\n\u2022 All the top navigation styles gain the `!important` property to ensure they get applied in SharePoint. This is a quick-and-easy way to override the more specific SharePoint styling.\n\n\u2022 For the top navigation, the fly-outs and their corresponding arrow indicators are hidden. Also, the Edit button that shows if you have navigation editing rights is hidden. These concepts would have behaved incorrectly in the mobile view, so it's best to hide them.\n\n\u2022 Some page-level styles are changed, specifically for the default homepage of a publishing site to allow the page layout columns to adjust from being side by side to displaying vertically down the page.\n\nWith all these changes in place, the responsive master page is completed. The final result shown here is the default homepage of a publishing site at different browser sizes.\n\nDesktop\n\nMedium\n\nMobile\n\nSummary\n\n\u2022 SharePoint 2013 was created with modern web design techniques in mind. It even uses HTML5 and CSS3 concepts in the default code base. For example, HTML5 is used for loading videos and CSS3 is used for adding shadows behind elements.\n\n\u2022 The jQuery JavaScript library can be used with SharePoint 2013 to perform a variety of dynamic user interface effects including animations.\n\n\u2022 Modern web design templates can be used with SharePoint 2013, but there are some gotchas when it comes to using certain templates with the Design Manager, specifically in regard to some of the techniques used in the common open source HTML5 Boilerplate template.\n\n\u2022 Web fonts can easily be referenced in custom SharePoint 2013 branding to add unique typography.\n\n\u2022 Grid-based designs like the one provided with 960 Grid can be used with SharePoint 2013 to enable easier creation of layouts with uniformly aligned rows and columns.\n\n\u2022 SharePoint 2013 lends itself well to the use of responsive design in custom branding. Responsive design is a technique that enables website layouts and styles to adapt based on different browser features, such as the height and width of mobile devices like tablets and phones. \n12\n\nDesigning Apps\n\nWhat's in this Chapter:\n\n1. SharePoint-hosted Apps\n\n2. Cloud-hosted Apps\n\n3. CSS for Apps\n\n4. Autohosted Apps\n\n5. Chrome Control\n\nSharePoint 2013 and Office 2013 introduce a new app model fit for the cloud. In this new app model you can write code in any language, framework, or platform that you choose. This is possible because your app does not actually run on the SharePoint server. The core design principal in SharePoint 2013 is that no user code runs on the server. This chapter teaches you what you need to design and brand apps that look and feel like the SharePoint site on which they are hosted.\n\nSharePoint Cloud App Model\n\nIn the past, you could write full trust applications, sandboxed solutions, and applications that ran on the client, either WPF or mobile devices. Although full trust applications and sandboxed solutions (including Silverlight) are still supported, they are not the future direction of SharePoint development and are deprecated in Office 365.\n\nThe key to the SharePoint app model and the flexibility it delivers comes from the fact that the code is not running on the SharePoint server. That means the SharePoint server is free to change as frequently or infrequently as the SharePoint product team wants to with less impact on your app. This is especially important as SharePoint Online in Office 365 becomes a more popular way that users deploy SharePoint. If your app is not running on the SharePoint server, where is it running? The answer is any place you want. You learn about the various hosting options for your SharePoint app in this chapter.\n\nThere are a number of ways that you can surface your app in SharePoint, even though it runs outside of SharePoint. This chapter covers the three user experience options for your app and provides more detailed examples of each of these in action.\n\nThe last piece of the app puzzle is branding. If your app runs outside of SharePoint, how do you make your app look and feel like the SharePoint site it is installed in? There are a number of ways to pull context and style information from SharePoint and use this to style your app. You see examples of these in this chapter.\n\nThis chapter offers only a brief look into the new app model focusing on what you need to get started designing and branding apps. You learn how to use the new browser developer tool Napa and dive a little into Visual Studio to build apps. Although this chapter is on the SharePoint app model, the Office 2013 app model is nearly identical, and the same tools and techniques you learn for SharePoint also apply to building Office apps.\n\nNOTE\n\nThe SharePoint app model is similar conceptually to developing Facebook applications. If you have built Facebook applications, you can leverage those skills to build SharePoint apps. Similarly, after you learn how to build SharePoint apps, you can then leverage those skills to build Facebook apps.\n\nHosting Options\n\nA SharePoint app is a web application hosted outside of the SharePoint server. There are two basic hosting options: SharePoint-hosted and cloud-hosted (or a combination of the two).\n\nSharePoint-Hosted\n\nSharePoint-hosted apps are a bit of a misnomer because the app is not actually hosted in SharePoint. These are browser-only apps. That is, the web pages are stored in SharePoint and delivered to the browser as HTML pages. All the code runs on the client side. There is no server-side code allowed. These are simple apps to create from a designer's perspective because there is only HTML, CSS, and JavaScript code involved.\n\nLater in the \"Creating and Branding SharePoint-Hosted Apps\" section of this chapter you use the new browser-based Napa tools to create and brand a SharePoint-hosted app. The SharePoint-hosted app can also include other declarative SharePoint items such as list templates and modules. But the primary UI is an HTML web page that uses JavaScript to interact with the SharePoint server's REST API or the client-side object model (CSOM).\n\nCloud-Hosted\n\nCloud-hosted apps are SharePoint apps hosted outside of SharePoint. This may or may not be in a cloud-hosting environment. The only requirement is that it is not running on SharePoint. Therefore, this could be a cloud-hosting environment such as Windows Azure, Amazon AWS, or Rackspace. It could also be on-premise in your enterprise environment. For example, you can run SharePoint in your own data centers. You can have another server in the same data center running Internet Information Server (IIS) and hosting the apps for your SharePoint farm.\n\nThe general term used by SharePoint to describe the hosting models is cloud-hosted. Microsoft is working to make Windows Azure be the first and best option for hosting your SharePoint and Office apps. You see how this is critical in the autohosted option in the next section.\n\nProvider-Hosted\n\nThe first cloud-hosting model is provider-hosted. Provider-hosted means that you as the provider of the application are responsible to host the application some place. Windows Azure is probably the best choice to do this, but any platform capable of serving up web pages can work. This can be Windows Azure, Amazon AWS, Google Compute, or another web-hosting provider like Rackspace.\n\nThe benefit of using the provider-hosted option is that you control the development platform. You can use any framework and any version, such as .Net, PHP, Java, Ruby, and so on. You also control the software installed on the server and the versioning. The downside to provider-hosted apps is that you own the server and must maintain and service the server and applications. Also, you are responsible for the costs associated with hosting the app.\n\nIn this book, you do not build provider-hosted apps. You can learn all the branding techniques using SharePoint-hosted and autohosted apps. Then apply that knowledge when you build a provider-hosted app.\n\nAutohosted\n\nAutohosted apps are cloud-hosted apps that are automatically hosted for you by Office 365. Autohosted apps have the ease of deployment of a SharePoint-hosted app but have the power of a provider-hosted app. In an autohosted app Office 365 works under the covers to create a Windows Azure Web Sites instance and a Windows Azure SQL Database on-the-fly for you and deploys your app and data to this newly provisioned site.\n\nThe benefit to this model is that you do not need to worry about finding and maintaining a hosting environment for your apps. One interesting aspect for autohosted apps is that for every app install there is a single and unique Windows Azure Web Sites site and Windows Azure SQL Database created. When the app is uninstalled, the instance of the website and database are also destroyed. This differs from the provider-hosted model where every installation of your app points back to a single instance of your app, which would potentially isolate users and customers in a multitenant app.\n\nNOTE\n\nTo learn more about the hosting option for SharePoint apps visit MSDN at `http:\/\/msdn.microsoft.com\/en-us\/library\/fp179887.aspx`.\n\nUser Experience Options\n\nAs you've seen, there are several options for hosting your app. There are also various user experience options available. The user can interact with your app in three ways: through an immersive full-page app, as an App Part, or as a custom action. You could also use any combination of these three user experiences together in a single app. The following sections explore what these mean for users and designers. Later in the \"Branding Autohosted Apps\" section of this chapter you learn to build all three of these user experiences.\n\nImmersive Full-Page App\n\nThe immersive full-page app is the default entry point for SharePoint apps. In this type of app, the user is redirected to the URL that is defined as the start page for the app. The start page is located outside of SharePoint, and the user navigates to this web page. As a SharePoint user and developer, this is probably the hardest idea to understand. The experience is as if you typed a new address in the browser and navigated to a new site. You may wonder, how is this a SharePoint app?\n\nThe glue that holds the SharePoint site and the full-page app together is the branding. When the app is started by SharePoint, it not only navigates to the start page URL, it also sends along some other information about the user and the site that launched the application. Using this additional information you can brand the site to look like the SharePoint farm that it belongs to. Later in the \"Branding Autohosted Apps\" section of this chapter you learn how to use branding to make the full-page app tie back to the SharePoint site it was launched from.\n\nApp Part\n\nThe traditional way to surface applications in SharePoint has been through Web Parts. In the new app model there is a different way called App Parts. App Parts look and behave similar to Web Parts except they are actually just an iFrame window to a web page hosted by your app. Your app is still hosted some place depending on the hosting model you chose. The App Part enables you to surface a page to the user in a frame on the SharePoint site.\n\nThe branding of App Parts is important because the App Part sits on a branded page of the SharePoint site. It will be painfully obvious when the App Part is not branded or branded incorrectly. The App Part can reference the styles of the host SharePoint site to correctly style all the elements in the App Part.\n\nCustom Action\n\nThere are two ways for your app to interact with custom actions: using the ribbon or using a menu item. Both methods provide entry points for your app to interact with the user. The branding associated with custom actions enables you to add ribbon items and launch dialog windows. These are not visual branding elements though so they will not be covered in this book. You can read more about creating custom actions on the MSDN site at http:\/\/msdn.microsoft.com\/en-us\/library\/jj163954.aspx.\n\nIn the next section, you learn how to bring all these concepts together to create apps for SharePoint 2013 using the Napa browser tool and Visual Studio 2012.\n\nCreating and Branding SharePoint-Hosted Apps\n\nBuilding and branding SharePoint-hosted apps using the Napa online code editor is the best way to start branding. Napa is itself a SharePoint app that enables you as a designer or power user to quickly build SharePoint-hosted apps using only your web browser. After you add the Napa app to your SharePoint developer site, you are ready to begin creating SharePoint apps. Apps created with Napa are SharePoint-hosted apps that are based on a SharePoint Master Page that contains a reference to the SharePoint-themed CSS file. You can use this CSS to brand your app. The following sections walk you through the process of installing Napa and creating your first SharePoint app. The Napa tool is rapidly adding new features. You explore only a few of these features in this book, but you can to learn more about the tool from http:\/\/dev.office.com.\n\nNapa Office 365 Development Tools\n\nYou can create SharePoint-hosted apps using Visual Studio, but as a designer you may feel more comfortable building apps using only your browser. Napa makes it incredibly easy to start. First, you need to create a developer site in Office 365.\n\nYou can sign up for a free site at http:\/\/msdn.microsoft.com\/en-us\/library\/fp179924.aspx. The sign-up process is straightforward.\n\nWARNING\n\nBoth Napa and Office 365 are frequently updated by Microsoft. The code shown in this chapter may appear differently for you depending on any changes that may have occurred since this book has been published.\n\nExample: Creating a Napa Account\n\n1. Enter your basic contact information.\n\n2. Specify a new Office 365 user ID. This is not only the tenant administrator for your new site, but it is also the name of your new domain. For example, you could create a user called `Administrator` and a new domain called `WaffleApps`. This results in a login id of `administrator@WaffleApps.OnMicrosoft.com` for the site `https:\/\/WaffleApps.SharePoint.com`.\n\n3. Enter a code that will be sent to your mobile phone.\n\n4. Click Create My Account to start the provisioning process.\n\nThis is the site that will be used throughout this chapter. If you work along with the book, replace the administrator and WaffleApps names with the user and domain name that you used when you signed up for your Office 365 account.\n\nInstalling Napa\n\nWhen your new Office 365 developer site has been provisioned, you can download the Napa app. If you start from an existing SharePoint tenancy, remember that you can install the Napa app only in a site collection based on the developer site template.\n\nThere are two common ways to add an app to your site. The first is to choose Add an App from the Settings menu in the top right of the site page located between your username and the help icon.\n\nThe second way to add an app is to click on the Site Contents link on the left side of the site page. This opens the Site Contents page. The Site Contents page shows all installed apps and has an Add an App icon as the first item on the page. Clicking this takes you to the same place as clicking the Add an App from the Settings menu.\n\nThe Add an App page, `\/_layouts\/15\/addanapp.aspx`, lists the apps you can add. These are basic apps that come with SharePoint, such as a Document Library, Custom List, and Task. If you are new to SharePoint 2013, remember that to simplify the concept count, all these items like Document Libraries and Lists are considered apps. There is also a filtered list of apps from your organization. These are apps that have been installed in the Corporate Catalog. The Corporate Catalog is a central store of apps approved by your administrators. The Corporate Catalog also enables enterprises to side load apps without installing them in the SharePoint Store, which is run by Microsoft.\n\nOn the Add an App page, you want to install the Napa app, which is located in the SharePoint Store. The SharePoint Store is Microsoft's online store for developers to publish apps and for consumers to discover, purchase, and install apps. Think of this just like other apps stores you are familiar with such as Apple's iTunes Store, Google's Play Store, or the Microsoft Store for Windows 8 or Windows Phone 8. You can click the SharePoint Store link on the left to browse to the SharePoint Store website.\n\nExample: Installing the Napa App\n\n1. On the SharePoint Store website, type Napa in the Find an App search box on the upper-right side of the page. There will most likely be only one result.\n\n2. Click the Napa icon to open the app details page. The app details page shows you a number of screen shots of the app. It also has some details such as the version, rating, and release date. There is also a link to the app support site, `http:\/\/go.microsoft.com\/fwlink\/?LinkId=258346`, which is simply a tutorial article on the MSDN website.\n\n3. Install the Napa Office 365 Development Tools by clicking the blue Add It button.\n\n4. Sign in with a Microsoft account (previously known as a Live ID). Note that this cannot be your `.onmicrosoft.com` account.\n\n5. Click the green Continue button to confirm the installation of the Napa app. After the confirmation screen you will be prompted to choose the sites to install the app on. In this case you may have only the one developer site, which is selected by default.\n\n6. Click Return to Site to complete the installation. The Trust It prompt appears and is a standard OAuth style prompt that gives the user the ability to approve what the app can do. This is an all-or-nothing prompt. If you cancel and do not trust the app, the app will not be installed. The Napa app requires full control on the site collection and access to basic user information.\n\n7. Click Trust it and the Napa app appears in your Site Contents page as an available app. The Napa app icon appears with the new tag, indicating it was recently installed. If you hover over the app icon, you see an ellipses appear. Clicking the ellipses shows you more details about the app.\n\nCreating Your First SharePoint App\n\nThe Napa app enables you to create SharePoint-hosted apps. As you learned earlier, SharePoint-hosted apps are client-side only and contain no server-side code. When you click the Napa app to launch it you see that the app is a full-page immersive app.\n\nThis starting page is the page where you can select the type of app to build. Currently there are four app types supported by Napa: App for SharePoint, Task Pane App for Office, Content App for Excel, and Mail App for Office. This book focuses on building apps for SharePoint, but many of the techniques you learn here also apply to building apps for Office.\n\nNOTE\n\nThe Napa app is a provider-hosted SharePoint app hosted in Windows Azure. The URL for the app is `http:\/\/www.napacloudapp.com` and there is a chrome control at the top of the page.\n\nFor your App for SharePoint project you can either change the project name or keep it as the default, in this case SharePointApp1. You then simply click Create and in a few seconds your new SharePoint app project is created and ready for editing directly in the browser. The Napa editor is amazing for a browser-only code editor. Many of the conveniences of a full Visual Studio editor are available, such as color coding, instance highlighting, indenting, and bracket matching. There is even JavaScript code completion and snippets. And because this is a cloud-based tool, expect these features to improve and more features to be added frequently.\n\nThe project structure of the SharePoint-hosted app consists of two web pages. The Default.aspx is the starting page for the app and provides the immersive full-page experience. The second page is the ClientWebPart.aspx page. This provides the client App Part experience. The other pages are common pages to most web projects. There is a CSS page, an application icon image, and a JavaScript file. You can explore these in more detail.\n\nThe Default.aspx page should be familiar to you if you have done SharePoint development before. You can see in the following code that the page has a number of SharePoint assembly references. The most interesting one is a Page reference to the SharePoint master page, `~masterurl\/default.master`.\n\n`<%@ Page Inherits=\"`\n\n` Microsoft.SharePoint.WebPartPages.WebPartPage,`\n\n` Microsoft.SharePoint,`\n\n` Version=15.0.0.0,`\n\n` Culture=neutral,`\n\n` PublicKeyToken=71e9bce111e9429c\"`\n\n` MasterPageFile=\"~masterurl\/default.master\"`\n\n` language=\"C#\" %>`\n\nThe next two items in the page are content controls. The first content control is for the header information and has the ID of `PlaceHolderAdditionalPageHead`. This header control contains the references to the JQuery JavaScript library, your CSS file in your project called App.css, and a reference to your JavaScript file called App.js.\n\n`<%-- The markup and script in the following Content element will be`\n\n` placed in the <head> of the page --%>`\n\n`<asp:Content ContentPlaceHolderId=\"PlaceHolderAdditionalPageHead\"`\n\n` runat=\"server\">`\n\n` <script src=\"https:\/\/ajax.aspnetcdn.com\/ajax\/`\n\n` jQuery\/jquery-1.6.2.min.js\"`\n\n` type=\"text\/javascript\"><\/script>`\n\n` `\n\n` <!-- Add your CSS styles to the following file -->`\n\n` <link rel=\"Stylesheet\" type=\"text\/css\"`\n\n` href=\"..\/Content\/App.css\" \/>`\n\n` `\n\n` <!-- Add your JavaScript to the following file -->`\n\n` <script type=\"text\/javascript\" src=\"..\/Scripts\/App.js\"><\/script>`\n\n`<\/asp:Content>`\n\nThe second content control has the ID of `PlaceHolderMain` and represents the body or main content of the app. The default project template outputs the username in a div tag. This is where you put the UI for your application.\n\n`<%-- The markup and script in the following Content element will be`\n\n` placed in the <body> of the page --%>`\n\n`<asp:Content ContentPlaceHolderId=\"PlaceHolderMain\" runat=\"server\">`\n\n` <div>`\n\n` <p id=\"message\">`\n\n` <!-- The following content will be replaced with the`\n\n` user name when you run the app - see App.js -->`\n\n` initializing...`\n\n` <\/p>`\n\n` <\/div>`\n\n` <\/asp:Content>`\n\nThe ClientWebPart.aspx page appears as an App Part when the app is installed. In the following code, you can see that the App Part is a simple html page. It contains a head section with references to the SharePoint client-side object model (CSOM). It also contains a reference to your app.css style sheet. Inside the body tag there is a div that contains a span. You put the contents of your app in the span.\n\n`<!-- The following tells SharePoint to allow this page to be hosted`\n\n` in an IFrame -->`\n\n`<WebPartPages:AllowFraming runat=\"server\" \/>`\n\n`<html>`\n\n` <head>`\n\n` <!-- Add your CSS styles to the following file -->`\n\n` <link rel=\"Stylesheet\" type=\"text\/css\"`\n\n` href=\"..\/Content\/App.css\" \/>`\n\n` <!-- The following scripts are needed when using the`\n\n` SharePoint object model -->`\n\n` <script type=\"text\/javascript\"`\n\n` src=\"https:\/\/ajax.aspnetcdn.com\/ajax\/4.0\/1\/MicrosoftAjax.`\n\n` js\"><\/script>`\n\n` <script type=\"text\/javascript\"`\n\n` src=\"\/_layouts\/15\/sp.runtime.debug.js\"><\/script>`\n\n` <script type=\"text\/javascript\"`\n\n` src=\"\/_layouts\/15\/sp.debug.js\"><\/script>`\n\n` <\/head>`\n\n` <body class=\"partBody\">`\n\n` <div class=\"partDiv\">`\n\n` <span class=\"partContent\">`\n\n` Your content goes here...`\n\n` <\/span>`\n\n` <\/div>`\n\n` <\/body>`\n\n`<\/html>`\n\nThe App.js file contains some sample code that you can use to start creating your app. The most interesting code is the initialization code. You want to make sure that the SharePoint client object model is loaded and that the client context is ready. When this is ready you can execute your initialization code; in this case it is the function called `sharePointReady()`. The `sharePointReady()` function gets a reference to the SharePoint client context and sets the global variable called context. Next, the sample code calls the `getUserName() `function. The `getUserName()` function gets the user object from the server asynchronously and sets the paragraph object that has the ID equal to \"message\" with the user title.\n\n`var context;`\n\n`var web;`\n\n`var user;`\n\n`\/\/ This code runs when the DOM is ready. It ensures the SharePoint`\n\n`\/\/ script file sp.js is loaded and then executes sharePointReady()`\n\n`$(document).ready(function () {`\n\n` SP.SOD.executeFunc('sp.js', 'SP.ClientContext', sharePointReady);`\n\n`});`\n\n`\/\/ This function creates a context object which is needed to use the`\n\n`\/\/ SharePoint object model`\n\n`function sharePointReady() {`\n\n` context = new SP.ClientContext.get_current();`\n\n` web = context.get_web();`\n\n` `\n\n` getUserName();`\n\n`}`\n\n`\/\/ This function prepares, loads, and then executes a`\n\n`\/\/ SharePoint query`\n\n`\/\/ to get the current users information`\n\n`function getUserName() {`\n\n` user = web.get_currentUser();`\n\n` context.load(user);`\n\n` context.executeQueryAsync(onGetUserNameSuccess,`\n\n` onGetUserNameFail);`\n\n`}`\n\n`\/\/ This function is executed if the above call is successful`\n\n`\/\/ It replaces the contents of the 'helloString' element with the`\n\n`\/\/ user name`\n\n`function onGetUserNameSuccess() {`\n\n` $('#message').text('Hello ' + user.get_title());`\n\n`}`\n\n`\/\/ This function is executed if the above call fails`\n\n`function onGetUserNameFail(sender, args) {`\n\n` alert('Failed to get user name. Error:' + args.get_message());`\n\n`}`\n\nThe last file in the default project is the App.css style sheet. You can see in the following code that there are three classes defined that set some basic attributes. These are actually applied in the ClientWebPart.aspx page.\n\n`.partBody`\n\n`{`\n\n` margin: 0;`\n\n` padding: 0;`\n\n` border: 0;`\n\n`}`\n\n`.partDiv`\n\n`{`\n\n` margin: 0;`\n\n` padding: 0;`\n\n` border: 0;`\n\n` position: relative;`\n\n` width: 300px;`\n\n` height: 200px;`\n\n` background-color: #0000FF;`\n\n` font-family: 'Segoe UI', Arial, Helvetica, sans-serif;`\n\n` color: #FFFFFF;`\n\n` font-size: 14pt;`\n\n`}`\n\n`.partContent`\n\n`{`\n\n` position: absolute;`\n\n` top: 5px;`\n\n` left: 5px;`\n\n`}`\n\nThe default project template from Napa is just a starting point that shows some examples of what is possible. You are free to expand the project in any way that makes sense for your app. This code runs in the browser, so you could write to any version of HTML you prefer that is supported by SharePoint. The best way may be to push forward with modern browsers that support HTML 5 or above, but that is a decision for you to make.\n\nThe manifest properties are the last important piece of the project. You can access these properties by clicking the Properties icon on the bottom left. The manifest Properties dialog contains information about the project, such as the Name, Title, Icon, and Start Page URL. You can see these same properties again when you open the project in Visual Studio.\n\nAt this point you have not made any changes to the default project, but it is ready to run. Click the Play icon on the bottom left to compile and package, install, and run the project. Notice that the app has a blue menu bar at the top. This menu bar is provided by the master page and, among other things, provides a hook or anchor back to the SharePoint site that the app belongs to. Without this menu bar, which is a suggested best practice that all apps have, there would be no way to know this app was part of the SharePoint site.\n\nNOTE\n\nThe URL for this app is `https:\/\/waffleapps-d67e55a03c4bf8.sharepoint.com\/SharePointApp1\/`. SharePoint creates a new site-to-host app, which has the site name plus a random number. This provides a separate security domain enforced by the browser.\n\nSo far you have created an App Part as part of the default project. Just like you do with Web Parts, you can insert an App Part onto a SharePoint page. The App Part shows your app in a frame on an existing SharePoint page that may contain a combination of Web Parts and App Parts.\n\nExample: Insert an App Part on a SharePoint page\n\n1. Using the link on the top left of the app, navigate back to the SharePoint site. On the homepage, you see a document library called Apps in Testing, which now contains at least one project, called SharePointApp1. This is the project you just created.\n\n2. You can add the App Part to this page as well. Click Edit on the top right of the page to put the page in edit mode. When in edit mode, click the Insert Ribbon tab.\n\n3. Click the App Part icon to open the Parts dialog window. The Parts dialog lists a number of parts installed with SharePoint by default. You also see your SharePointApp1 App Part.\n\n4. Select the SharePointApp1 App Part on the left pane to see the details in the right pane. These details come from the manifest properties.\n\n5. Click Add to add the SharePointApp1 App Part to the page. The App Part is inserted into the page, and the page is still in edit mode.\n\n6. Click the Save button on the ribbon to save the page and exit edit mode.\n\nThe Napa tool is a great tool for web developers to easily create SharePoint-hosted apps. The tool doesn't require installing Visual Studio and reduces the concepts to get started building apps. Now that you have created your app you are ready to start branding.\n\nBranding your App with CSS and Styles\n\nOne of the first things you want to do after creating your app is brand it to match the branding of the host website, which is the site where the app is installed. To do this you need to use the style sheet from the host web, which is part of the app template. The SharePoint-hosted app derives from a simple master page; this is the app template. The app template provides some basic features such as the navigation bar at the top of the page. It also provides the basic placeholders for the two content controls: `PlaceHolderAdditionalPageHead` and `PlaceHolderMain`.\n\nTo achieve the look and feel of the host site for your app using the style sheet, use the host's style definitions in your app. For example, add the style called `ms-accentText` to your paragraph tag with the id of message. It will look like the following code.\n\n`<asp:Content ContentPlaceHolderId=\"PlaceHolderMain\" runat=\"server\">`\n\n` <div>`\n\n` <p id=\"message\" class=\"ms-accentText\">`\n\n` <!-- The following content will be replaced with the`\n\n` user name when you run the app - see App.js -->`\n\n` initializing...`\n\n` <\/p>`\n\n` <\/div>`\n\n`<\/asp:Content>`\n\nYou can see how the `ms-accentText` changes the Hello `administrator@waffleapps.onmicrosoft.com` message when you run the app.\n\nMany of the host web style sheet classes like the one you just used are also defined on MSDN at http:\/\/msdn.microsoft.com\/en-us\/library\/jj220046. Although it is outside the scope of this book to enumerate all the available styles, you will see a few common examples here. You should become familiar with the standard styles and guidelines for creating SharePoint apps. The goal of any app design is to be consistent with the host web styles. See Chapter 6, \"Cascading Style Sheets and SharePoint,\" for a detailed CSS explanation.\n\nTo see examples of how some of the styles render when you change the look and feel of your site, run the following code to your app:\n\n`<div>`\n\n` <p class=\"ms-core-pageTitle\">Page Title (ms-core-pageTitle)<\/p>`\n\n` <h1>Heading 1<\/h1>`\n\n` <h2>Heading 2<\/h2>`\n\n` <h3>Heading 3<\/h3>`\n\n` <p class=\"ms-webpart-titleText\">`\n\n` Web Part Title (ms-webpart-titleText)<\/p>`\n\n` <p class=\"ms-dlg-heading\">Dialog Heading (ms-dlg-heading)<\/p>`\n\n` <p class=\"ms-commandlink\">`\n\n` Command Link (ms-commandLink)<\/p>`\n\n` <p class=\"ms-accentText\">Accent Text (ms-accentText)<\/p>`\n\n` <p class=\"ms-error\">Error Text (ms-error)<\/p>`\n\n` <p class=\"ms-disabled\">Disabled Text (ms-disabled)<\/p>`\n\n`<\/div>`\n\nBranding Autohosted Apps\n\nIn the previous section you learned how to use Napa to build SharePoint-hosted apps. Although this was a good starting point for learning, you actually want to build your real-world apps as cloud-hosted apps. Cloud-hosted apps give you room to grow and fully leverage not only the SharePoint platform, but also the hosting platform where the app resides.\n\nIn this section, you learn how to use Visual Studio to develop autohosted SharePoint apps. You also learn how to apply branding to these apps that complement the SharePoint site they are surfaced in.\n\nVisual Studio 2012\n\nVisual Studio is an amazing development tool for building SharePoint applications. However, this book is not about Visual Studio, so it doesn't go into great detail about the tool. The goal is to show you enough so that you can understand the branding parts.\n\nInstalling Visual Studio\n\nThe first thing you need to do is install Visual Studio 2012. You can find detailed instructions on doing so at http:\/\/msdn.microsoft.com\/en-us\/library\/e2h7fzkw.aspx. After you have Visual Studio installed, you need to install the Microsoft Office Developer Tools for Visual Studio 2012. The Office tools are a separate add-on for Visual Studio that enable you to create Office 2013 and SharePoint 2013 apps. The tools download page has a link under the tools section at http:\/\/msdn.microsoft.com\/en-us\/office\/apps\/fp123627 to download the tools. The tools are actually installed using the Microsoft Web Platform Installer (WebPI) tool.\n\nCreating your Second SharePoint App\n\nYou can now try your hand at creating the same app you just built in the previous section, but this time using Visual Studio. The great thing about developing SharePoint 2013 apps is that you do not need SharePoint installed on the same machine. Unlike SharePoint 2010, you can now develop applications without ever installing SharePoint. In this case, you can use the Office 365 site that you created in the beginning of this chapter.\n\nExample: Creating an App for SharePoint using Visual Studio\n\n1. Launch Visual Studio 2012 with elevated permissions. Building apps for SharePoint 2013 requires administrator rights.\n\n2. Create a new App for SharePoint 2013 called SharePointApp2.\n\n3. Tell Visual Studio where the SharePoint developer site is that you want to use to build the app.\n\n4. Choose the hosting option for the app. By default it is set to Autohosted, which is the type of hosting you want to use for this chapter.\n\nProject Structure\n\nThe Autohosted solution created by Visual Studio has two projects. The first project is the actual SharePoint app project. This has the same name you provided when you created the solution. In this case it is called SharePointApp2. It contains an AppIcon.png image and an AppManifest.xml file.\n\nThe second project is your actual web project. This is the project that contains your app logic and pages. In this case, Visual Studio creates a standard web project called SharePointApp2Web. The project has the Default.aspx page as the start page for your application. You learn more about how to brand this page in the next section. The web project also contains a Scripts folder with jQuery JavaScript files. You want to replace this with the latest version of jQuery from http:\/\/www.jquery.org. There is one last file called TokenHelper.cs. The TokenHelper file contains helper code to help you do advanced coding using the OAuth security tokens passed by SharePoint. As a designer, the discussion of OAuth is outside the scope of the book.\n\nNOTE\n\nBy default, Visual Studio uses an ASP.NET Web Forms Application project when creating the autohosted app. You can simply delete this web project and add another project type, such as an ASP.NET MVC 4 Web Application.\n\nRunning and Debugging\n\nYou can start debugging your SharePoint app just like any other Visual Studio project by pressing F5. When you press F5, Visual Studio does a number of steps to make the experience easy. First, the project is compiled and packaged. Next, you need a temporary place to host the web app while debugging, so Visual Studio starts a local web server and deploys your web project. Then Visual Studio automates the deployment and starts the app on the SharePoint server. All these steps take a few minutes the first time to complete, and you will be prompted to trust the app as well. Note that there is no branding whatsoever on the page. Remember that the SharePoint-hosted app you created in Napa had a blue navigation bar at the top and the host CSS styles. In the cloud-hosted apps, you are responsible for providing the branding.\n\nIn the cloud-hosted apps you can run server-side code. Note that the title of the host web displays here. In the following code, which is the Default.aspx code, there is no code or HTML to display the host web title.\n\n`<%@ Page Language=\"C#\" AutoEventWireup=\"true\"`\n\n` CodeBehind=\"Default.aspx.cs\"`\n\n` Inherits=\"SharePointApp2Web.Pages.Default\" %>`\n\n`<!DOCTYPE html`\n\n` PUBLIC \"-\/\/W3C\/\/DTD XHTML 1.0 Transitional\/\/EN\"`\n\n` \"http:\/\/www.w3.org\/TR\/xhtml1\/DTD\/xhtml1-transitional.`\n\n` dtd\">`\n\n` <html xmlns=\"http:\/\/www.w3.org\/1999\/xhtml\">`\n\n` <head runat=\"server\">`\n\n` <title><\/title>`\n\n` <\/head>`\n\n` <body>`\n\n` <form id=\"form1\" runat=\"server\">`\n\n` <div>`\n\n` <\/div>`\n\n` <\/form>`\n\n` <\/body>`\n\n`<\/html>`\n\nThe code behind for the default.aspx page contains code on the load event that retrieves the SharePoint client context from the page request and parses it with the token helper. The client context enables you to use the client-side object model (CSOM) to retrieve the Title property of the app web. You can see the complete code in the following listing.\n\n`using System;`\n\n`using System.Collections.Generic;`\n\n`using System.Linq;`\n\n`using System.Web;`\n\n`using System.Web.UI;`\n\n`using System.Web.UI.WebControls;`\n\n`namespace SharePointApp2Web.Pages`\n\n`{`\n\n` public partial class Default : System.Web.UI.Page`\n\n` {`\n\n` protected void Page_Load(object sender, EventArgs e)`\n\n` {`\n\n` \/\/ The following code gets the client context and`\n\n` \/\/ Title property by using TokenHelper.`\n\n` \/\/ To access other properties, you may need to`\n\n` \/\/ request permissions on the host web.`\n\n` var contextToken = TokenHelper.`\n\n` GetContextTokenFromRequest(Page.Request);`\n\n` var hostWeb = Page.Request[\"SPHostUrl\"];`\n\n` using (var clientContext =`\n\n` TokenHelper.GetClientContextWithContextToken(`\n\n` hostWeb, contextToken, Request.Url.Authority))`\n\n` {`\n\n` clientContext.Load(clientContext.Web, web => web.`\n\n` Title);`\n\n` clientContext.ExecuteQuery();`\n\n` Response.Write(clientContext.Web.Title);`\n\n` }`\n\n` }`\n\n` }`\n\n`}`\n\nChrome Control\n\nThe chrome control is a control provided by SharePoint to display the navigation bar at the top of your application. The chrome control visually ties the app with the site from which the app was launched. Remember that your app is just a redirect to another web page outside of SharePoint, and the chrome control provides a hyperlink back to the SharePoint site. Without the chrome control, the user would not know that the app is related to the host SharePoint site in any way. You are not required to use the chrome control, but you must provide a link back to the SharePoint if you want to conform to the SharePoint UX guidelines at `http:\/\/msdn.microsoft.com\/en-us\/library\/jj220046.aspx`. The chrome control also provides a custom drop-down menu and a link to a help page, as well as a link to the host SharePoint site's CSS file, just like the App Template did for SharePoint-hosted apps. It is a good idea to add the chrome control to your app.\n\nExample: Adding the Chrome Control to your Web Page\n\n1. Load the JavaScript libraries that are dependencies for the chrome control. You can do this declaratively in HTML by pulling the libraries from a content delivery network (CDN) endpoint. Add the following code to the head section of your Default.aspx page:\n\n`<head runat=\"server\">`\n\n` <title><\/title>`\n\n` <script`\n\n` src=\"\/\/ajax.aspnetcdn.com\/ajax\/4.0\/1\/MicrosoftAjax.js\">`\n\n` <\/script>`\n\n` <script`\n\n` src=\"\/\/ajax.aspnetcdn.com\/ajax\/jQuery\/`\n\n` jquery-1.7.2.min.js\">`\n\n` <\/script>`\n\n` <script src=\"\/Scripts\/ChromeControl.js\">`\n\n` <\/script>`\n\n`<\/head>`\n\nNotice in the previous code that there is also a script reference to the ChromeControl.js. This is just a helper script that you write to dynamically load the script from the host SharePoint site. You can name this script anything you want. You can find more examples of this script on MSDN at `http:\/\/msdn.microsoft.com\/en-us\/library\/jj163201`.\n\n2. Create a new file under the Scripts folder called ChromeControl.js and add the following code. The code simply loads the `SPHostUrl\/_layouts\/15\/SP.UI.Controls.js` file.\n\n`var hostweburl;`\n\n`\/\/ Load the SharePoint resources.`\n\n`$(document).ready(function () {`\n\n` \/\/ Get the URI decoded app web URL.`\n\n` hostweburl =`\n\n` decodeURIComponent(`\n\n` getQueryStringParameter(\"SPHostUrl\")`\n\n` );`\n\n` \/\/ The SharePoint js files URL are in the form:`\n\n` \/\/ web_url\/_layouts\/15\/resource.js`\n\n` var scriptbase = hostweburl + \"\/_layouts\/15\/\";`\n\n` \/\/ Load the js file and continue to the`\n\n` \/\/ success handler.`\n\n` $.getScript(scriptbase + \"SP.UI.Controls.js\")`\n\n`});`\n\n`\/\/ Function to retrieve a query string value.`\n\n`\/\/ For production purposes you may want to use`\n\n`\/\/ a library to handle the query string.`\n\n`function getQueryStringParameter(paramToRetrieve) {`\n\n` var params =`\n\n` document.URL.split(\"?\")[1].split(\"&\");`\n\n` var strParams = \"\";`\n\n` for (var i = 0; i < params.length; i = i + 1) {`\n\n` var singleParam = params[i].split(\"=\");`\n\n` if (singleParam[0] == paramToRetrieve)`\n\n` return singleParam[1];`\n\n` }`\n\n`}`\n\n3. After all the required JavaScript libraries are loaded, declaratively add the chrome control to your page. There are more options available to the chrome control that you see next, but for now here is the shortest amount of code to display the chrome control.\n\n`<div id=\"PlaceHolderChromeControl\"`\n\n` data-ms-control=\"SP.UI.Controls.Navigation\"`\n\n` data-ms-options=`\n\n` '{`\n\n` \"appTitle\" : \"SharePointApp2\"`\n\n` }'>`\n\n`<\/div>`\n\n4. Run the project to see the chrome control.\n\nThere are other options on the chrome control that are worth examining. The `data-ms-options` property is an array of option properties. In the previous example you used only the `appTitle` property. In the following code you can see that there are many more available. First, there are two icons that you can provide. The first is the `appIconUrl`, which is the icon that appears under the colored menu bar at the top of the page next to the `appTitle`. The second is the `appTitleIconUrl`, which is a small 24-by-24 pixel icon that appears in the menu bar at the top. In this example, you see a small waffle icon. Both of these icons are available with the downloads for this chapter.\n\n`<div id=\"PlaceHolderChromeControl\"`\n\n` data-ms-control=\"SP.UI.Controls.Navigation\"`\n\n` data-ms-options=`\n\n` '{`\n\n` \"appIconUrl\" : \"\/Images\/AppIcon.png\",`\n\n` \"appTitleIconUrl\" : \"\/Images\/WaffleIcon.png\",`\n\n` \"appTitle\" : \"SharePointApp2\",`\n\n` \"appHelpPageUrl\" : \"\/Pages\/Help.aspx\",`\n\n` \"settingsLinks\" : [`\n\n` {`\n\n` \"linkUrl\" : \"\/Pages\/About.aspx\",`\n\n` \"displayName\" : \"About\"`\n\n` },`\n\n` {`\n\n` \"linkUrl\" : \"\/Pages\/Config.aspx\",`\n\n` \"displayName\" : \"App Config\"`\n\n` }`\n\n` ]`\n\n` }'>`\n\n`<\/div>`\n\nThe `appHelpPageUrl` enables you to provide a link to a help page. The last property is the `settingsLinks`. The `settingsLinks` is an array of menu items that appear on the right side of the menu bar as a drop-down menu of links. In the previous example there are two menu items, one for the About page and one for a Configuration page. The chrome control renders with these additional properties.\n\nBuilding App Parts\n\nAs mentioned previously, you can think of App Parts as Web Parts for Apps. They are created and used just like Web Parts, but they are actually just an iFrame that points to your App Part page in your app. Because the App Part is hosted in a frame on an existing page that does not require a visual control linking it back to the host site, you do not use the chrome control. In other words, the App Part lives on a page that already has the branding and navigation at the top of the page. You do not want to duplicate the navigation again inside of your App Part.\n\nTo the user, App Parts and Web Parts are the same. To a developer, they achieve the same goal but are implemented differently. You can create an App Part by adding a Client Web Part project item to your app solution in Visual Studio. The following steps will walk you through the process of creating an App Part and adding code to the part.\n\nExample: Creating an App Part\n\n1. In Visual Studio, add a new Client Web Part project item to the SharePoint app project. Remember there are two projects in your solution: the SharePoint app project and the web project.\n\n2. Use the default name of ClientWebPart1.\n\n3. The project system of Visual Studio creates the ClientWebpart.aspx page in your web project for you. Visual Studio creates a new ClientWebPart1 element in the manifest of the SharePoint app project as well.\n\n4. The aspx file that Visual Studio creates for you has a script block to reference the style sheet of the SharePoint host that the App Part is installed. Use the following code to add the `link` element to reference the `\/layouts\/15\/defaultcss.ashx` CSS file. This is the same file that is added when you use the chrome control.\n\n`<script type=\"text\/javascript\">`\n\n` \/\/ Set the style of the client web part page to be`\n\n` \/\/ consistent with the host web.`\n\n` function setStyleSheet() {`\n\n` var hostUrl = \"\"`\n\n` if (document.URL.indexOf(\"?\") != -1) {`\n\n` var params =`\n\n` document.URL.split(\"?\")[1].split(\"&\");`\n\n` for (var i = 0; i < params.length; i++) {`\n\n` p = decodeURIComponent(params[i]);`\n\n` if (\/^SPHostUrl=\/i.test(p)) {`\n\n` hostUrl = p.split(\"=\")[1];`\n\n` document.write(`\n\n` \"<link rel=\\\"stylesheet\\\" href=\\\"\" +`\n\n` hostUrl +`\n\n` \"\/_layouts\/15\/defaultcss.ashx\\\" \/>\");`\n\n` break;`\n\n` }`\n\n` }`\n\n` }`\n\n` if (hostUrl == \"\") {`\n\n` document.write(\"<link rel=\\\"stylesheet\\\"`\n\n` href=\\\"\/_layouts\/15\/1033\/styles\/`\n\n` themable\/corev15.css\\\" \/>\");`\n\n` }`\n\n` }`\n\n` setStyleSheet();`\n\n`<\/script>`\n\n5. Now that the host CSS file is referenced in your App Part, you can use any of the style elements on the page. Add the following div snippet to your page. This is the same code that you used in the SharePoint-hosted page that was created with Napa.\n\n`<body>`\n\n` <form id=\"form1\" runat=\"server\">`\n\n` <div>`\n\n` <p class=\"ms-core-pageTitle\">`\n\n` Page Title (ms-corepageTitle)<\/p>`\n\n` <h1>Heading 1<\/h1>`\n\n` <h2>Heading 2<\/h2>`\n\n` <h3>Heading 3<\/h3>`\n\n` <p class=\"ms-webpart-titleText\">`\n\n` Web Part Title (ms-webpart-titleText)<\/p>`\n\n` <p class=\"ms-dlg-heading\">`\n\n` Dialog Heading (ms-dlg-heading)<\/p>`\n\n` <p class=\"ms-commandlink\">`\n\n` Command Link (ms-commandLink)<\/p>`\n\n` <p class=\"ms-accentText\">Accent Text (ms-accentText)<\/p>`\n\n` <p class=\"ms-error\">Error Text (ms-error)<\/p>`\n\n` <p class=\"ms-disabled\">Disabled Text (ms-disabled)<\/p>`\n\n` <\/div>`\n\n` <\/form>`\n\n`<\/body>`\n\n6. Press F5 to run the project. When the project starts you will be on the Full Page of your app.\n\n7. Navigate back to the host site, put the page in edit mode, and insert the App Part.\n\nSummary\n\n\u2022 The new SharePoint app model opens new opportunities.\n\n\u2022 Branding apps is more important when the apps are hosted outside of SharePoint.\n\n\u2022 The chrome control visually ties the app back to the host site.\n\n\u2022 SharePoint-hosted apps are client-side only apps hosted in SharePoint.\n\n\u2022 Provider hosted apps are cloud-hosted web apps that run outside of SharePoint.\n\n\u2022 Provider-hosted apps can be written in any language using any framework, such as ASP.Net, MVC, Ruby on Rails, or PHP.\n\n\u2022 SharePoint apps can leverage the host's CSS files to look and feel like the host site.\n\n\u2022 SharePoint apps can appear as full-page immersive apps, as App Parts on a SharePoint page, or as a Dialog or Ribbon item.\nSharePoint\u00ae 2013 Branding and User Interface Design\n\nPublished by  \nJohn Wiley & Sons, Inc.  \n10475 Crosspoint Boulevard  \nIndianapolis, IN 46256  \nwww.wiley.com\n\nCopyright \u00a9 2013 by John Wiley & Sons, Inc., Indianapolis, Indiana\n\nPublished simultaneously in Canada\n\nISBN: 978-1-118-49567-4ISBN: 978-1-118-49561-2 (ebk) ISBN: 978-1-118-71073-9 (ebk)\n\nManufactured in the United States of America\n\n10 9 8 7 6 5 4 3 2 1\n\nNo part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http:\/\/www.wiley.com\/go\/permissions.\n\nLimit of Liability\/Disclaimer of Warranty: The publisher and the author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation warranties of fitness for a particular purpose. No warranty may be created or extended by sales or promotional materials. The advice and strategies contained herein may not be suitable for every situation. This work is sold with the understanding that the publisher is not engaged in rendering legal, accounting, or other professional services. If professional assistance is required, the services of a competent professional person should be sought. Neither the publisher nor the author shall be liable for damages arising herefrom. The fact that an organization or Web site is referred to in this work as a citation and\/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or website may provide or recommendations it may make. Further, readers should be aware that Internet websites listed in this work may have changed or disappeared between when this work was written and when it is read.\n\nFor general information on our other products and services please contact our Customer Care Department within the United States at (877) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.\n\nWiley also publishes its books in a variety of electronic formats and by print-on-demand. Not all content that is available in standard print versions of this book may appear or be packaged in all book formats. If you have purchased a version of this book that did not include media that is referenced by or accompanies a standard print version, you may request this media by visiting http:\/\/booksupport.wiley.com. For more information about Wiley products, visit us at www.wiley.com.\n\nLibrary of Congress Control Number: 2012956398\n\nTrademarks: Wiley, the Wiley logo, are trademarks or registered trademarks of John Wiley & Sons, Inc. and\/or its affiliates, in the United States and other countries, and may not be used without written permission. SharePoint is a registered trademark of Microsoft Corporation. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc. is not associated with any product or vendor mentioned in this book.\n\nDedicated to waffles (thank you for staying delicious) and to Jackie (the love of my life) for always being there to eat them with me.\n\n\u2014Randy Drisgill\n\nTo my mom, thanks for always correcting my grammar.\n\n\u2014John Ross\n\nTo my daughter, Julia, the artistic one in the family.\n\n\u2014Paul Stubbs\n\nAbout the Authors\n\nRandy Drisgill has more than a decade of experience developing, designing, and implementing web technologies for clients ranging from small businesses to Fortune 500 companies. For the past 5 years, he has been working exclusively with SharePoint branding projects at SharePoint911, which was subsequently acquired by Rackspace in 2012. During this time, Randy has worked on more than 50 internal and public-facing SharePoint branding projects.\n\nRandy is an active member of the SharePoint community, having contributed to several articles and books on the topic, as well as being the cofounder\/comanager of the Orlando SharePoint User Group (OSPUG). In 2009, Randy was recognized by Microsoft as an authority on SharePoint branding by being awarded MVP status for SharePoint Server.\n\nRandy lives in Orlando, Florida, with his wife. You can find Randy online on Twitter as @Drisgill or at his blog at <http:\/\/blog.drisgill.com>.\n\nJohn Ross is a Senior Consultant with Rackspace and has more than eight years of experience implementing solutions for clients ranging from small businesses to Fortune 500 companies as well as government organizations. He has worked with all project phases from analysis to implementation and has been involved with a wide range of SharePoint solutions that include public-facing Internet sites, corporate intranets, and extranets.\n\nJohn is an active member of the community and has presented at conferences all around the world. He is the cofounder\/comanager of the Orlando SharePoint User Group (OSPUG). In 2009, John was awarded Microsoft's MVP award for SharePoint Server for his community contributions.\n\nJohn lives in Orlando, Florida with his wife and two children. Visit his blog at http:\/\/johnrossjr.wordpress.com or follow his SharePoint adventures on Twitter @JohnRoss.\n\nPaul Stubbs is a Microsoft Architect for Windows Azure focusing on cloud solution development and SharePoint. Previously, Paul worked with the SharePoint community developing SharePoint, Office, and Silverlight solutions and training. Paul has also worked as a Senior Program Manager on the Visual Studio team in Redmond, Washington. Paul is a Microsoft Certified Trainer (MCT) and has received Microsoft Certified Applications Developer (MCAD) and Microsoft Certified Solution Developer (MCSD) certifications. He has authored several books on solution development using Microsoft Office, SharePoint, and Silverlight, wrote several articles for MSDN Magazine, and has also spoken at many conferences such as Tech-Ed, Build, MIX, and the Professional Developers Conference (PDC) and Tech-Ready. Paul has a passion for new technologies and sharing those experiences with the community on his blog at `http:\/\/blogs.msdn.com\/pstubbs``.`\n\nAbout the Contributor\n\nRyan Keller has been working with SharePoint technologies since 2007, and he has worked as a SharePoint consultant since 2009 when he joined SharePoint911, a consulting company from Cincinnati, Ohio. In 2012, Rackspace Hosting acquired SharePoint911, and he is now part of the SharePoint team at Rackspace. He has helped write and edit material related to SharePoint 2010 for Microsoft, wrote as a contributing author to the popular Professional SharePoint 2010 Administration, and served as a technical editor for Beginning SharePoint Designer 2010, Professional SharePoint 2010 Branding and User Interface Design, and for this book, Professional SharePoint 2013 Branding and User Interface Design. He is also a contributing author on Professional SharePoint 2013 Administration. Since 2011, Ryan has focused his consulting efforts exclusively on SharePoint branding and has worked on projects ranging from simple designs to complex custom branding solutions. Ryan lives in Firestone, Colorado, with his wife Brittany and two kids, Kylie and Nathan.\n\nAbout the Technical Editors\n\nHeather Waterman is the Director of Synteractive Studio, a subsidiary of Smartronix based in Hollywood, Maryland. She is responsible for leading the designers and developers with an emphasis on web design for SharePoint. She has over 12 years of web design and development experience, the past six with a primary focus on SharePoint branding. With these skills she has quickly become a leader in the SharePoint branding community. Her past SharePoint Branding projects include the award-winning Recovery.gov and Treasury.gov among others. Prior to joining Synteractive, she was the president and CEO of the Waterman Design Group, during which time she developed website templates for resell and provided SharePoint branding for clients including Shell and Merck. When she is not working on client projects, she actively contributes design and branding time to the community by developing blogs and sites for other community leaders. You can find her on twitter `@hwaterman` or via her blog `www.heatherwaterman.com`.\n\nLarry Riemann has over 18 years of experience architecting and creating business applications for some of the world's largest companies. Larry is an independent consultant who owns Indigo Integrations and SharePoint Fanatics. He is an author, contributing author, and technical editor on three other SharePoint books and is an occasional speaker at conferences. For the last several years he has focused on SharePoint, creating and extending functionality where SharePoint leaves off. In addition to his expertise with SharePoint, Larry is an accomplished .Net Architect and has extensive expertise in systems integration, enterprise architecture, and high-availability solutions. You can contact Larry at `larry@spfanatics.com`.\n\nCredits\n\nAcquisitions Editor\n\nMary James\n\nProject Editor\n\nVictoria Swider\n\nContributing Author\n\nRyan Keller\n\nTechnical Editors\n\nHeather Waterman\n\nRyan Keller\n\nLarry Riemann\n\nProduction Editor\n\nDaniel Scribner\n\nCopy Editor\n\nSan Dee Phillips\n\nEditorial Manager\n\nMary Beth Wakefield\n\nFreelancer Editorial Manager\n\nRosemarie Graham\n\nAssociate Director of Marketing\n\nDavid Mayhew\n\nMarketing Manager\n\nAshley Zurcher\n\nBusiness Manager\n\nAmy Knies\n\nProduction Manager\n\nTim Tate\n\nVice President and Executive Group Publisher\n\nRichard Swadley\n\nVice President and Executive Publisher\n\nNeil Edde\n\nAssociate Publisher\n\nJim Minatel\n\nProject Coordinator, Cover\n\nKatie Crocker\n\nInterior Designer\n\nEd Cross\n\nCompositor\n\nLissa Auciello-Brogan, Abshier House\n\nProofreader\n\nLowell Kim, Abshier House\n\nIndexer\n\nKelly Dobbs-Henthorne, Abshier House\n\nCover Designer\n\nElizabeth Brooks\n\nPhotographer\n\nErik Wieder\n\nAcknowledgments\n\nWell, here we are again, a new version of SharePoint and a new branding book to go along with it. This is the third SharePoint branding book I've worked on, and every time I forget exactly how much work goes into putting these things together. With each new book there are, of course, new concepts to learn in a short amount of time, but new challenges also arise. For this book, the new challenge began when Jim Minatel from Wrox came to me with the crazy idea that we could produce a highly designed, full-color SharePoint branding book. The idea of making something that looks entirely different from any other SharePoint book on the market got me excited to jump back in to the writing process. There is absolutely no way this full-color book could be a reality without the help of many people that I need to thank.\n\nFirst, I want to thank the author team for helping me put together the best collection of SharePoint 2013 branding knowledge that exists today. This includes my coworker, copresenter, and good friend, John Ross, as well as my buddy, Paul Stubbs, Sr. Architect at Microsoft. We also had assistance from my co-worker, Ryan Keller, who worked both as a contributing author and technical editor. Another big thanks goes out to Heather Waterman, who tech edited some of the more advanced branding chapters and went above the call of duty to make sure the book turned out awesome. Thanks to Larry Riemann and Mark Watts for tech editing chapters as well.\n\nAlong with the writing team, another obvious big thanks goes out to everyone at Wrox Press who helped us get this book to you. This includes Jim Minatel, Mary James, Victoria Swider, San Dee Phillips with Apostrophe Editing Services, Abshier House team of Debbie Abshier, Lissa Auciello-Brogan, Kelly Henthorne, Lowell Kim, the eBook wizards and probably several several other people behind the scenes. They not only helped us sound intelligent, but also put together a really great-looking book. We also owe a great deal of thanks to our designer Ed Cross for creating the amazing look of the chapters and Erik Wieder for lending his photography skills to take that look even further. Special thanks to Jon Duckett for paving the way for a full-color technical book at Wiley.\n\nI want to personally thank several folks from Microsoft without whom this book may have never seen the light of day. These people provided some of the best help and information that I have ever seen for a product that was still being developed. Primarily, this help came from the wonderfully talented Alyssa Levitz (SharePoint Program Manager at Microsoft), who never failed to find us the answers no matter how stupid the question may have seemed at the time, even if it meant answering a question over Facebook while she was on vacation! Topics like the Design Manager, composed looks, and the minimal download strategy would not have been described nearly as well without her help. Some of the other Microsoft folks that helped us understand SharePoint 2013 better include, in no particular order: Mark Kashman, Ethan Gur-esh, Kevin Gjerstad, Josh Stickler, Lionel Robinson, Jeremy Kelley, Jonathan Kern, Manfred Berry, Cindy Liao, Arye Gittelman, Petru Moldovanu, Tom Werner, Matt Evans, Reagan Templin, Nina Ruchirat, Stephen Howard, Kevin Davis (AWESOME), and anyone else I'm forgetting. These people all helped make SharePoint 2013 the best version of SharePoint ever.\n\nSpecial thanks go out to everyone I work with at Rackspace, both new coworkers and the old SharePoint911 family. I don't think anyone could ask for a better set of coworkers or a more talented team of SharePoint experts. Also, thanks to my bosses Jeff DeVerter, Shane Young, and Walt Leddy for all their support during the transition to SharePoint 2013 and their support of this writing effort.\n\nLastly, I need to thank all my friends and family who put up with me working long hours throughout several months to put together this book. Most of them have no idea what I do for a living and will probably never read past this paragraph, but I couldn't have done it without all your friendship and support throughout the years. This includes my beautiful wife Jackie Drisgill, my parents Pat and Tom Drisgill, my in-laws Debbie and Dave Auerbach, my grandparents George and Mary Shea and Thomas and Elsie Drisgill, my friends Adam McCard and Marcela Errazquin, Jenn and Mark Clemons, John and Vanessa Ross, Josh and Rachel Witter, and all my friends from Orlando, New York City, San Antonio, Boston, and throughout the SharePoint community; you know who you are!\n\nThe soundtrack for the writing of this book can be found at http:\/\/drisgill.com\/go\/spotify-playlist.\n\n\u2014Randy Drisgill\n\nWhen we first learned that the next version of SharePoint was going to place a big focus on enhancing the design experience, and when Jim Minatel from Wiley presented the idea of doing a highly designed, full-color SharePoint book, it seemed like the stars were aligning. Bringing all of the new technical concepts together in a new format wouldn't have been possible without the help and support of many people.\n\nI'd like to thank the rest of the author team; this book has your blood, sweat, and tears all over it. Big thanks to my good friend Randy Drisgill for being the driving force behind this book and dragging it across the finish line. It has been a wild and strange journey since our days back on the 8th floor to today. To Paul Stubbs, I appreciate all of your help and insight throughout the years. Big thanks to our coworker Ryan Keller for all his hard work as both a contributing author and technical editor.\n\nTo the technical editors, Heather Waterman and Larry Rieman, thanks once again for keeping us honest. This book is better because of your efforts. Thank you both!\n\nWith this release, I was completely blown away by the help and support from so many people at Microsoft; I hope we've done you proud. To Alyssa Levitz, I don't think anything written here in these acknowledgments could convey everything you've done to help make this book what it is. You're the best! There are so many others at Microsoft who have been generous with their time, including: Mark Kashman, Ethan Gur-esh, Josh Stickler, Lionel Robinson, Manfred Berry, Tom Werner, Reagan Templin, Nina Ruchirat, Stephen Howard, Kevin Davis, and I'm sure many others I've forgotten. It has been fun working with you all during this process. Let's do it again sometime!\n\nThanks to the team at Wrox Press for giving us the opportunity to write this book. Jim Minatel, Mary James, Victoria Swider, San Dee Phillips (Apostrophe Editing Services), Abshier House team of Debbie Abshier, Lissa Auciello-Brogan, Kelly Henthorne, Lowell Kim, eBooks wizards and many others I'm sure. Thanks for taking a chance on doing something different with a SharePoint book and helping us make it the best it can be.\n\nTo everyone at Rackspace, I'm so proud to work with the best SharePoint team in the universe! Thanks to Shane Young, Jeff DeVerter, and Walt Leddy for the support and encouragement throughout the entire process.\n\nLast but certainly not least, I'd like to say a special thank you to all of my friends and family. To my lovely wife Vanessa, thanks for putting up with all the late nights, long weekends, and my general crankiness that goes along with the writing process. To my kids, Ben and Julia, thanks for all the hugs and smiles! And to all the rest of my friends and family, unfortunately you'll be seeing and hearing from me more often now!\n\n\u2014John Ross\n\nThanks to Randy Drisgill and John Ross, my coauthors, for driving this book and for asking me to join them in writing it. Working with Randy and John has been a great experience for me, and I have learned a lot about the design space from them. I also really appreciate their vision for this book to be more than just another reference book, but a book that can not only teach you, but inspire you. I also want to thank our project editor, Victoria Swider, for her infinite patience and understanding as we tried to balance the writing of the book with our day jobs, family, and SharePoint schedule. And finally, I would like to thank Mary James, our Acquisitions Editor, for making this book possible and enabling us to realize our vision.\n\n\u2014Paul Stubbs\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable des Mati\u00e8res\n\nPage de Titre\n\nTable des Mati\u00e8res\n\nPage de Copyright\n\nDU M\u00caME AUTEUR\n\nD\u00e9dicace\n\nPR\u00c9FACE\n\n1 - G\u00c9OGRAPHIE SENTIMENTALE\n\n2 - POUR CYTH\u00c8RE EMBARQU\u00c9\n\n3 - LES VENGEANCES DE LA PERDRIX\n\n4 - VOIR L'IC\u00d4NE PA\u00cfENNE\n\n5 - SEPT P\u00c9CH\u00c9S CAPITEUX\n\n6 - AFIN DE MOURIR VIVANT\n\n7 - LA BR\u00dbLURE DES CORPS PLI\u00c9S\n\n8 - LE MONDE DES VOLEURS DE FEU\n\n9 - LES VERTUS DE LA FOUDRE\n\n10 - POLITIQUE DE LA GASTRONOMIE\n\n11 - IDENTIT\u00c9 DE LA PHOTOGRAPHIE\n\n12 - TIRER LA BARBE \u00c0 TOUTE MAJEST\u00c9\n\n13 - LA GAIET\u00c9 ENNUYEUSE\n\n14 - FRAN\u00c7AISES, ENCORE UN EFFORT...\n\n15 - C\u00c9L\u00c9BRATION DU GAZ LACRYMOG\u00c8NE\n\n16 - ENTRE DEUX N\u00c9ANTS\n\n17 - L'INNOCENCE DU DEVENIR\n\n18 - QU'IL FAUT \u00c9TRANGLER LA MORT\n\n19 - LE SALON VERDURIN, ENCORE\n\n20 - BRUITS DU PLIAGE DES NUAGES\n\n21 - GYN\u00c9COLOGIE DES PR\u00c9CIEUSES\n\n22 - DES MACHINES \u00c0 RENVERSER LE PLATONISME\n\n23 - LE D\u00c9SIR DES CHOSES INEXISTANTES\n\n24 - LES HOMMES AUX PAUPI\u00c8RES COUSUES\n\n25 - ESTH\u00c9TIQUE DE L'UBIQUIT\u00c9\n\n26 - LE CONTRAIRE D'UNE CHOSE MENTALE\n\n27 - LA PHILOSOPHIE DANS LE PARLOIR\n\n28 - ANATOMIE D'UNE MACHINE D\u00c9SIRANTE\n\n29 - POUR UN MAT\u00c9RIALISME PO\u00c9TIQUE\n\n30 - LE SEXE, LE SANG, LA MORT\n\n31 - LES MIROIRS DE NARCISSE AU TH\u00c9\u00c2TRE\n\n32 - CONSID\u00c9RATIONS SUR LES ROGNONS DU PHILOSOPHE\n\nCOLLECTION \u00ab FIGURES \u00bb DIRIG\u00c9E PAR BERNARD-HENRI L\u00c9VY\n\u00a9 \u00c9ditions Grasset & Fasquelle, 1998.\n\n978-2-246-78038-0\nDU M\u00caME AUTEUR\n\nGEORGES PALANTE, Essai sur un nietzsch\u00e9en de gauche, Folle-Avoine, 1989.\n\nLE VENTRE DES PHILOSOPHES, Critique de la raison di\u00e9t\u00e9tique, Grasset, 1989.\n\nCYNISMES, Portrait du philosophe en chien, Grasset, 1990.\n\nL'ART DE JOUIR, Pour un mat\u00e9rialisme h\u00e9doniste, Grasset, 1991\n\nL'\u0152IL NOMADE, La peinture de Jacques Pasquier, Folle-Avoine, 1993.\n\nLA SCULPTURE DE soi, La morale esth\u00e9tique, Grasset, 1993 (Prix M\u00e9dicis de l'essai).\n\nARS MORIENDI, Cent petits tableaux sur les avantages et les inconv\u00e9nients de la mort, Folle Avoine, 1994.\n\nLA RAISON GOURMANDE, Philosophie du go\u00fbt, (Prix Libert\u00e9 litt\u00e9raire) Grasset, 1995.\n\nM\u00c9TAPHYSIQUE DES RUINES, La peinture de Monsu Desiderio, Mollat, 1995.\n\nLES FORMES DU TEMPS, Th\u00e9orie du sauternes, Mollat, 1996.\n\nLE D\u00c9SIR D'\u00caTRE UN VOLCAN, Journal h\u00e9doniste, Grasset, 1996.\n\nPOLITIQUE DU REBELLE, Trait\u00e9 de r\u00e9sistance et d'insoumission, Grasset, 1997.\n\nA C\u00d4T\u00c9 DU D\u00c9SIR D'\u00c9TERNIT\u00c9. Fragments d'\u00c9gypte, Mollat, 1998.\n\nTous droits de traduction, de reproduction et d'adaptation   \nr\u00e9serv\u00e9s pour tous pays.\n\u00ab Vivre \u2013 cela signifie pour nous : changer constamment en lumi\u00e8re et en flamme tout ce que nous sommes. \u00bb NIETZSCHE, Le Gai Savoir, pr\u00e9face, 6.3.\nPR\u00c9FACE\n\nAutobiographie de ma m\u00e8re\n\nJe me souviens de chiens qui hurlaient \u00e0 la mort, comme aux portes d'un enfer ou d'un \u00e9quarrissage, rendus furieux par l'odeur du sang, de la charogne ou de l'abattage imminent. J'imagine la peau sur les os, le cou entrav\u00e9 par une cha\u00eene, pel\u00e9, et les longues plaintes lanc\u00e9es \u00e0 la face du soleil qui br\u00fble les peaux en cette apr\u00e8s-midi de printemps insolemment voluptueux. Les hurlements n'en finissent pas, r\u00e9it\u00e9r\u00e9s jusqu'\u00e0 l'\u00e9c\u0153urement. Dans ces cris, j'entends parfois des g\u00e9missements d'enfants ou des plaintes d'adultes riv\u00e9s sur un lit de douleur. J'aspire \u00e0 ne plus entendre ces appels au secours, ces souffrances modul\u00e9es en voix animales. Je me rappelle \u00e9galement la lumi\u00e8re crue, violente, blanche, le soleil aveuglant et les nuages satur\u00e9s de violences lumineuses.\n\nJ'accompagne ma m\u00e8re aux services des archives de l'assistance publique o\u00f9 l'administration a fini par lui donner l'autorisation de prendre connaissance du contenu de son dossier silencieux et myst\u00e9rieux depuis plus de soixante ann\u00e9es. De son enfance, je n'ai jamais su que les mauvais traitements, les coups, les s\u00e9vices, les vexations, les placements, les privations, la faim et le froid, puis cet abandon qui, racont\u00e9 par elle dans mes premi\u00e8res ann\u00e9es, avait pris pour moi les formes d'un cageot laiss\u00e9 au pied d'une \u00e9glise de village en Normandie, ma m\u00e8re \u00e0 l'int\u00e9rieur, confi\u00e9e au bon vouloir d'un destin qui se r\u00e9v\u00e9la maussade. Elle proc\u00e9dait d'une b\u00e9ance qu'elle voulait combler avec des noms, des histoires, un roman familial semblable \u00e0 celui des autres ; elle me demanda d'\u00eatre \u00e0 son c\u00f4t\u00e9 sur ce chemin g\u00e9n\u00e9alogique. Mon p\u00e8re est l\u00e0, mon fr\u00e8re non, les chiens aboient toujours, la gueule en direction du n\u00e9ant.\n\nEn entrant dans le b\u00e2timent administratif, je sens ma m\u00e8re tendue, remplie de larmes, la voix f\u00e9brile, un peu tremblante. Elle sait qu'on va la mettre en face d'informations susceptibles, peut-\u00eatre, de devenir une chair, une voix, des corps, une pr\u00e9sence, des visages, des gestes. Derri\u00e8re ces murs, elle trouvera vraisemblablement mati\u00e8re \u00e0 renouer des fils cass\u00e9s, \u00e0 d\u00e9couvrir et tracer son chemin dans un labyrinthe sombre o\u00f9 se m\u00e9langent des monstres et des chim\u00e8res, des peurs et des angoisses, des promesses et des souhaits, des d\u00e9sirs et des inqui\u00e9tudes. Enfant abandonn\u00e9e \u00e0 l'\u00e2ge o\u00f9 l'on ne parle pas, elle va peut-\u00eatre, adulte, enfin se trouver et se retrouver, ou bien se perdre d\u00e9finitivement. Je comprends, alors, que son corps chavire, bruisse sous l'\u00e9motion et chaloupe devant l'immensit\u00e9 du large.\n\nMon p\u00e8re est silencieux, comme \u00e0 chaque fois que l'essentiel advient, car, me semble-t-il, il croit moins aux mots qu'\u00e0 la pr\u00e9sence min\u00e9rale des \u00eatres. Mais je sais ce que disent son dos un peu vo\u00fbt\u00e9, sa t\u00eate courb\u00e9e, son regard braqu\u00e9 sur le sol et sa d\u00e9marche empes\u00e9e dans une gl\u00e8be imaginaire. Devant le spectacle de ces deux corps, de ces deux chairs, de mes parents troubl\u00e9s, je deviens l'adulte auquel ils demandent la prise en charge de ce qu'il faudra assumer apr\u00e8s avoir entendu et vu. Soumis au destin, pareils \u00e0 des enfants, ils me demandent d'inverser les r\u00f4les, de les conduire, de les guider comme jadis ils ont vraisemblablement dirig\u00e9 mes premiers pas. Je n'ignore pas qu'en sortant de ce b\u00e2timent, j'aurais moi aussi d\u00e9couvert un fragment de mon identit\u00e9, des grands-parents, des r\u00e9f\u00e9rences, une g\u00e9ographie familiale, un monde, et qu'il faudra m\u00eamement, pour mon compte, m'arranger d'ombres et de menaces, de promesses ou de surprises.\n\nLes lieux, semblables \u00e0 toute entreprise administrative, transpirent le carc\u00e9ral et le p\u00e9dagogique, entre commissariat de police et rectorat acad\u00e9mique, un genre de direction de l'\u00e9quipement qui traiterait des identit\u00e9s ainsi que des chemins vicinaux et des contre-all\u00e9es foresti\u00e8res, en transfigurant le r\u00e9el, les vies, les existences singuli\u00e8res, les romans biographiques en notes, fiches, archives, et autres documents class\u00e9s, r\u00e9pertori\u00e9s, num\u00e9rot\u00e9s. L\u00e0 o\u00f9 des chairs ont fr\u00e9mi, des corps exult\u00e9, des vies exsud\u00e9, des violences perc\u00e9, des douceurs ravi, l'alchimie bureaucratique g\u00e9n\u00e8re des dossiers et r\u00e9duit l'existentiel incarn\u00e9 \u00e0 du papier pelure entrelard\u00e9 de carbones \u00e9puis\u00e9s, p\u00e2lis et attach\u00e9s par des trombones rouill\u00e9s. Mon arbre g\u00e9n\u00e9alogique plantait ses racines dans ces odeurs fades.\n\nLe bureau du fonctionnaire \u00e9tale le bonheur adipeux et suspect des familles qu'on exhibe photographiquement, dans un cadre kitsch et des postures niaises. Je n'aime pas l'\u00e9talement des f\u00e9tiches bromur\u00e9s rappelant subrepticement \u00e0 l'\u0153il ce qui fondamentalement ne devrait jamais quitter l'\u00e2me. Le p\u00e8re de famille satisfait, la sueur perlant au front, le ventre d\u00e9bordant une chemise \u00e0 carreaux platement color\u00e9s, annonce que lui aussi fait partie de cette grande famille des orphelins. De la sorte, il semble dire qu'\u00e0 d\u00e9faut d'un lignage noble, on pourra toujours se contenter et se consoler de l'avoir pour parent\u00e8le. Je crains le pire \u00e0 la lecture du dossier...\n\nLa fen\u00eatre ouverte, on entend, bien s\u00fbr, le hurlement des chiens entrav\u00e9s. Leur pr\u00e9sage me rappelle la travers\u00e9e du Styx, la proximit\u00e9 de Charon, les compagnonnages ind\u00e9sirables. Ma m\u00e8re fait face au compulseur de dossier, mon p\u00e8re la flanque sur sa droite, moi sur sa gauche. Sa main se serre sur elle-m\u00eame, comme le poing d'un nouveau-n\u00e9 agripp\u00e9 sur ses ongles que l'angoisse enfonce dans la paume. Son sac \u00e0 main sur les genoux, assise tout en raideur sur sa chaise, elle attend les mots de cet homme qui, ce jour ordinaire, fait son travail. Elle allait au-devant d'une lumi\u00e8re \u00e0 diriger vers les ab\u00eemes o\u00f9 grouillent ses douleurs, et le fonctionnaire, fort des cours de psychologie qu'il a d\u00fb subir et supporter pour en \u00eatre l\u00e0, au contact de ce public particulier et fragile, montre toute l'incomp\u00e9tence affective dont on peut \u00eatre capable en pareille occasion.\n\nJe hais la lenteur, les circonlocutions, les d\u00e9tours, l'embarras, les atermoiements ; t\u00e9tanis\u00e9, emprunt\u00e9, il s'emp\u00eatre dans les p\u00e9riphrases, pratique les chemins de traverse, incapable de mouvement direct, net et pr\u00e9cis. Les chiens hurlent, il transpire. Ma m\u00e8re se tend comme un arc de verre, pr\u00eat \u00e0 se briser, en mille \u00e9clats. Mon p\u00e8re recouvre le mutisme que je lui connaissais, enfant, quand les journ\u00e9es de travail harassantes le condamnaient au silence impressionnant des grandes fatigues. Le dossier, maigre, repose sur le bureau du fonctionnaire ; ses doigts, boudin\u00e9s, p\u00e8sent sur le papier dont je me dis qu'il a l'\u00e2ge de ma m\u00e8re.\n\nApr\u00e8s les propos d'usage, le cat\u00e9chisme psychologique, les consid\u00e9rations humanistes, le verbiage administratif, l'homme au front perl\u00e9 se rend compte qu'il lui faut bien arriver au fait. Vaillamment, il commence par annoncer la date d'une premi\u00e8re dent, des premiers pas, une scarlatine, une otite, des v\u00e9g\u00e9tations, une appendicite, des tailles et des poids d\u00fbment consign\u00e9s en encres de diff\u00e9rentes couleurs, le tout syst\u00e9matiquement estampill\u00e9, tamponn\u00e9, contresign\u00e9. Je vois et lis \u00e0 l'envers les documents qu'il manipule tels des manuscrits pr\u00e9cieux. Ragaillardi par son audace, en feuilletant le carnet de sant\u00e9, il lit : \u00ab para\u00eet intelligente \u00bb, puis : \u00ab Petite nature. \u00bb Il sourit, content de lui, avant de tr\u00e9bucher sur une bl\u00e9pharite qui r\u00e9duit \u00e0 n\u00e9ant tous ses efforts pour para\u00eetre \u00e0 la hauteur. Ne sachant rien de cette affection, ignorant sa nature banale ou mortelle, l'\u00e2ge de ma m\u00e8re aidant, il conclut qu'on ne doit plus craindre grand-chose et s'autorise \u00e0 reprendre son sourire niais.\n\nJ'ai envie de le jeter aux chiens, dehors, lui et sa sueur, lui et son art de parler pour ne rien dire, d'\u00e9luder et de pratiquer savamment les \u00e9vitements essentiels. Je vois sur la main de ma m\u00e8re la blancheur trahissant le sang qui ne circule plus \u00e0 force de contraction, de crispation. Il annonce piteusement la minceur du dossier, la seule information inutile puisque le temps pass\u00e9 au spectacle de ses tergiversations laisse tout le loisir d'en constater l'\u00e9vidence. Les doigts translucides de ma m\u00e8re, le silence de mon p\u00e8re, l'embarras du fonctionnaire, et je m'entends exiger d'en venir \u00e0 l'essentiel, d'arr\u00eater les multiples occasions de diff\u00e9rer. Il avoue que le dossier contient des informations d\u00e9licates ; je dis que nous sommes venus pour entendre tout ce qu'il peut nous apprendre. D\u00e9li\u00e9, investi, enfin d\u00e9barrass\u00e9 de toute responsabilit\u00e9 propre, il devient parfait en lecteur neutre.\n\nEntre ricanements de hy\u00e8nes et longues plaintes de loup, sous le soleil d'un mauvais d\u00e9miurge au rire sarcastique, il m\u00eale sa voix \u00e0 l'aboiement des chiens. Je vois en lui, fils de l'Assistance publique, fr\u00e8re en cela de ma m\u00e8re, une ombre soumise \u00e0 la n\u00e9cessit\u00e9 condamn\u00e9e \u00e0 rejouer devant nous ce qu'il ne cesse vraisemblablement de ressasser pour lui : l'\u00e9nonc\u00e9 des raisons pour lesquelles un jour des parents abandonnent leur enfant, le d\u00e9laissent, l'oublient, le refusent, le rejettent. La prosodie du fonctionnaire, sur le mode psalmodi\u00e9, tente de faire s'effondrer les murs d'incompr\u00e9hension dans lesquels, autiste, demeure celui qu'on a une fois r\u00e9pudi\u00e9. Derri\u00e8re son bureau, le dossier en main, lisant, il semble raconter \u00e0 son inconscient une histoire toujours aussi incompr\u00e9hensible, de ces romans et de ces trames confuses dont on emporte l'obscurit\u00e9 avec soi, dans la tombe.\n\nD\u00e9laissant les papiers jaun\u00e2tres, les notes de frais pour une paire de galoches et ses clous, une petite robe ou un tricot, il avoue sottement le caract\u00e8re anecdotique de ce qu'il \u00e9carte \u2013 apr\u00e8s avoir pris soin d'en donner le d\u00e9tail. Vient alors le moment o\u00f9 il entreprend de lire un proc\u00e8s-verbal, un document de tribunal. Le texte dit que le p\u00e8re de ma m\u00e8re, longuement hospitalis\u00e9, mais conscient malgr\u00e9 quatre ann\u00e9es d'alitement, renonce \u00e0 exercer la paternit\u00e9 sur ses enfants et qu'il d\u00e9sire le placement de son fils a\u00een\u00e9 dans un orphelinat de pr\u00eatres sal\u00e9siens, \u00e0 Giel, dans l'Orne.\n\nLe document pr\u00e9cise trois informations coupantes comme un rasoir et qui tranchent l'\u00e2me sur-le-champ : le motif invoqu\u00e9 pour que sa propre m\u00e8re cesse d'en avoir la garde est la prostitution; il signale les d\u00e9m\u00eal\u00e9s de cette femme avec la justice, notamment pour une affaire de recel; il pr\u00e9cise que l'enfant ne proc\u00e8de pas du p\u00e8re officiel, mais d'un homme dont il donne le nom \u2013 Aurelio Sanchez. Du trou noir dans lequel ma m\u00e8re veut porter la lumi\u00e8re comme une occasion de d\u00e9couvrir des princes et des rois, des ch\u00e2teaux et des f\u00e9es, des tr\u00e9sors et des promesses, il ne sort soudainement que des rats, de la vermine et des cancrelats. L\u00e0 o\u00f9 elle esp\u00e9rait un peu de magie et de r\u00eave, de myst\u00e8re et de joie, la possibilit\u00e9 d'un peu de douceur et de paix, elle re\u00e7oit en plein visage la boue et la terre dont on fait les tombes. Elle pleure doucement; je prends sa main, froide. Mon p\u00e8re regarde les siennes pos\u00e9es sagement sur ses deux genoux. J'entends encore la vitesse bruyante du silence qui passa, bruissant dans les oreilles au rythme du sang qui fouette l'int\u00e9rieur du corps.\n\nLes chiens hurlent toujours quand nous sortons. Ma m\u00e8re flageole et tient mon bras comme lorsqu'en sortant du cimeti\u00e8re on s'\u00e9prouve d\u00e9s\u00e9quilibr\u00e9 par la douleur, chancelant d'avoir port\u00e9 en terre une personne aim\u00e9e qu'on ne verra plus. Elle avait toujours ignor\u00e9 ce dont elle proc\u00e9dait et m'avait confi\u00e9 vouloir savoir et pr\u00e9f\u00e9rer tout plut\u00f4t que l'ignorance, la plus insupportable parce que la plus fantasmatique des douleurs. Elle me dit : \u00ab Maintenant je sais. \u00bb Je comprends, alors, quels ab\u00eemes l'ont travaill\u00e9e pendant des ann\u00e9es, jusqu'\u00e0 ce jour d'animaux hyst\u00e9riques et de printemps br\u00fblant ; je saisis distinctement dans sa silhouette fatigu\u00e9e la somme cristallis\u00e9e des souffrances, des angoisses, des culpabilit\u00e9s, des d\u00e9nis, des hyst\u00e9ries, des peines, les nuits d'insomnie et les heures de haine de soi transfigur\u00e9es en violences \u00e0 l'endroit du monde; je la d\u00e9couvre brutalement, comme d\u00e9voil\u00e9e, apparue dans la cruaut\u00e9 d'une lumi\u00e8re noire; j'entr'aper\u00e7ois combien elle paya, en victime \u00e9missaire, donc innocente, l'aveugle veulerie inflig\u00e9e par la pulsion de mort de tiers inconscients depuis le d\u00e9but de son existence.\n\nNous avons parl\u00e9. Et je fus, \u00e0 nouveau, p\u00e8re de mes parents, investi du verbe et de ses pouvoirs pour panser les blessures et r\u00e9\u00e9crire une narration brutale, la transfigurer, la r\u00e9tablir, l'authentifier dans la logique d'un roman personnel, d'une autobiographie o\u00f9 il fallait r\u00e9injecter des sentiments, de l'affection, de l'amour\u2013 tant ma m\u00e8re en avait manqu\u00e9. Sur le trajet du retour, elle d\u00e9sira s'arr\u00eater sur la tombe d'une vieille dame chez qui elle avait \u00e9t\u00e9 longuement plac\u00e9e par l'Assistance publique. Dans le cimeti\u00e8re, nous avons rencontr\u00e9 une de ses amies d'enfance. Aupr\u00e8s du parterre des morts, sur le gravier qui crissait, elle retrouvait sa jeunesse, son enfance. Il me fallut recourir pour son usage aux mots avec lesquels on reconstitue les gestes qui ont manqu\u00e9.\n\nLa moralit\u00e9 douteuse, le vol, la prostitution, les mauvais traitements ? Je t\u00e2chais de r\u00e9duire tout cela \u00e0 de nouvelles perspectives, moins moralisatrices que m\u00e9caniques. La r\u00e9putation d\u00e9plorable ? du papier \u00e9crit \u00e0 l'encre morale destin\u00e9 \u00e0 stigmatiser et fustiger une femme dont la faute consiste \u00e0 continuer de vivre quand son mari attend la mort depuis des ann\u00e9es sur un lit d'h\u00f4pital, atteint de ces \u00e9tranges et fatales maladies \u00e9volutives, qui, bien avant le tr\u00e9pas, s\u00e9parent le malade du monde des vivants. Le recel ? l'\u00e9poque n'est gu\u00e8re avare en jugements de valeurs et la brique vol\u00e9e sur un chantier pour faire une cage \u00e0 lapin au fond du jardin devient chez les gens ordinaires prompts \u00e0 juger, un vol monstrueux. La prostitution ? la pr\u00e9sence d'un ouvrier espagnol dans la maison, ou d'un autre, sinon d'autres, transforme la m\u00e8re seule et sans ressources pour \u00e9lever ses trois enfants en femme de mauvaise vie qui monnaie ses charmes alors qu'elle accepte de quoi remplir tout bonnement les assiettes. J'essaie de formuler dix fois ces arguments en derviche tourneur d\u00e9sireux de sortir ma m\u00e8re de sa tristesse et de sa prostration. Je la sens soulag\u00e9e, tout de m\u00eame, d'avoir une histoire, m\u00eame celle-l\u00e0.\n\nDu dossier et des informations qu'il donne, je vois appara\u00eetre un arbre g\u00e9n\u00e9alogique tout entier r\u00e9v\u00e9l\u00e9 : une cour des miracles qui me pla\u00eet bien et fait le pendant aux origines normandes et aux \u00e9tymologies patronymiques scandinaves de mon p\u00e8re. De son c\u00f4t\u00e9, un lignage d'hommes de la terre et du cheval, ouvriers agricoles et charrons, mar\u00e9chaux-ferrants, l'aspect chthonien et hyperbor\u00e9en, le regard bleu de mon p\u00e8re, sa solidit\u00e9 de roc, ma grand-m\u00e8re aveugle et pieuse, diseuse de chapelets, douce et bonne, mon grand-p\u00e8re, cavalier d'infanterie, gaillard moustachu, gaz\u00e9 sur le front des Ardennes ; du c\u00f4t\u00e9 de ma m\u00e8re, une trace espagnole, sauvage, hors-la-loi, puis bretonne, un terre-neuvas en mer lors de la naissance de son fils, une juive alsacienne, un grabataire cachexique, puis ce qui raconte la fin des terres, la haute mer, les br\u00fblures andalouses, les passions arabes et la m\u00e9moire juive. J'ai aim\u00e9 d'un seul coup ma nouvelle famille.\n\nEt puis, \u00e0 ma m\u00e8re d\u00e9sireuse de changer la noirceur d'un pass\u00e9 sans trace pour un peu de la lumi\u00e8re des histoires qu'on peut raconter, je dis que, quoi qu'il en soit, son abandon proc\u00e8de d'un acte d'amour, car son p\u00e8re a voulu la prot\u00e9ger, elle et sa s\u0153ur, les \u00e9carter d'une femme que de son lit d'h\u00f4pital, entre d\u00e9sappointement et ressentiment l\u00e9gitime, il jugeait indigne, \u00e0 tort ou \u00e0 raison. Quand ma m\u00e8re imaginait devoir l'Assistance publique \u00e0 la d\u00e9saffection de ses parents, je la persuadais qu'il fallait croire au d\u00e9sir ultime d'un p\u00e8re de prot\u00e9ger ses enfants de ce qu'il savait ou croyait savoir du dehors \u2013 inform\u00e9 par ces seuls avis toujours autoris\u00e9s des donneurs de le\u00e7on et des vendeurs de morale. Ma m\u00e8re sembla consentir \u00e0 ma proposition.\n\nIl me fallut rentrer chez moi et laisser mes parents \u00e0 leur face-\u00e0-face. Sur le seuil de la maison, ravag\u00e9e mais apais\u00e9e, douloureuse mais cheminant vers un peu plus de paix, ma m\u00e8re m'avoua que les choses \u00e9taient bien ainsi. Sa m\u00e8re \u00e9tait d\u00e9c\u00e9d\u00e9e r\u00e9cemment. Remari\u00e9e peu apr\u00e8s la guerre, fid\u00e8le jusqu'\u00e0 la mort et pendant un demi-si\u00e8cle, semble-t-il, \u00e0 l'homme de ce second mariage dont je n'ai pas retrouv\u00e9 la trace, elle donnait l'impression d'avoir laiss\u00e9 derri\u00e8re elle les orages de son avant-guerre. Peut-\u00eatre son second \u00e9poux vit-il encore, non loin... Ma m\u00e8re voyait dans ces personnages anim\u00e9s un mouvement pr\u00e9f\u00e9rable au n\u00e9ant qui triomphait pr\u00e9c\u00e9demment dans son existence. Plut\u00f4t une danse macabre o\u00f9 se tr\u00e9moussent ces acteurs bariol\u00e9s que l'\u00e9paisseur noire d'une ignorance angoissante. Elle pleura, m'embrassa, m'\u00e9treignit, se vo\u00fbta, se retourna, ferma la porte derri\u00e8re elle. Je pris la route.\n\nEn voiture, roulant vite, tr\u00e8s vite, sur mon chemin de retour je d\u00e9couvris que je venais de rencontrer ma m\u00e8re, pour la premi\u00e8re fois apr\u00e8s l'avoir ignor\u00e9e et m\u00e9connue pendant presque quatre d\u00e9cennies. Que son pass\u00e9 ignor\u00e9 me l'avait rendue invisible, autant qu'elle l'avait \u00e9t\u00e9 \u00e0 elle-m\u00eame; d\u00e9voil\u00e9, il m'apparaissait dans l'\u00e9vidence de ce que Nietzsche appelle avec justesse l'innocence du devenir. Que ce que ma m\u00e8re m'avait fait vivre, enfant, parfois douloureusement \u2013 de ce que l'on dit et de ce que l'on ne dit pas \u2013 elle l'avait fait en aveugle, ob\u00e9issant \u00e0 d'obscures impulsions dont son \u00e2me depuis toujours fait les frais, \u00e0 son corps d\u00e9fendant. Que ce qui constitua son quotidien pendant mes vingt premi\u00e8res ann\u00e9es r\u00e9sultait d'un pur mais vain d\u00e9sir de rendre visible ce qui en elle agissait en pr\u00e9curseur sombre. Que son absence d'enfance lui interdisait de regarder la mienne en face et qu'elle avait inconsciemment tent\u00e9 de m'en priver pour n'avoir pas \u00e0 \u00eatre seule au monde dans cette posture d'animal d\u00e9chiquet\u00e9. Que m'envoyer en pension, dans un orphelinat \u2013 sans jamais savoir qu'il avait \u00e9t\u00e9 celui dans lequel son fr\u00e8re avait \u00e9chou\u00e9 un demi-si\u00e8cle auparavant \u2013 \u00e0 l'\u00e2ge de dix ans, et pour sept ann\u00e9es d'internat, lui permettait de me demander d'assumer, moi, ce qu'elle n'avait pu assumer, elle.\n\nDans le silence et la vitesse, je songeais \u00e0 cette phrase de Nietzsche qui enseigne que tout ce qui ne tue pas fortifie et \u00e0 l'usage que j'en fais depuis ma premi\u00e8re lecture, adolescent. Je pensais au d\u00e9terminisme, \u00e0 la n\u00e9cessit\u00e9, aux puissances auxquelles chacun ob\u00e9it avec plus ou moins de bonheur. Je constatais la mis\u00e8re et la d\u00e9tresse de ma m\u00e8re, je songeais au g\u00e2chis qu'il y a toujours pour des parents \u00e0 demander \u00e0 leurs enfants d'\u00eatre \u00e0 la hauteur de vieilles \u00e9minences \u00e0 partir desquelles le mauvais \u0153il les regarde et les souffrances pointent sur eux depuis leurs g\u00e9n\u00e9alogies nocturnes. J'\u00e9valuais les d\u00e9g\u00e2ts des effets de la pulsion de mort quand elle habite une chair, de son in\u00e9vitable transmission, d'une m\u00e8re \u00e0 un fils, d'un \u00eatre \u00e0 un autre, d'un adulte \u00e0 un enfant, quand un jour la folie de grandes personnes l'a d\u00e9clench\u00e9e dans l'\u00e2me puis le corps d'un enfant. Je me sentais plein d'une vraie compassion sereine pour ma m\u00e8re.\n\nEn qu\u00eate d'une identit\u00e9 \u00e0 asseoir, afin de disposer des preuves qu'on existe bel et bien, pour se persuader d'\u00eatre v\u00e9ritablement visible, r\u00e9el, pr\u00e9sent aux yeux d'autrui dont l'adoubement devient essentiel et fondateur, malgr\u00e9 le fantasme de l'abandon qui ne d\u00e9serte jamais la chair un jour sinistrement \u00e9lue, je sais pouvoir disposer, pour ma part, de l'\u00e9criture dans laquelle j'essaie de saisir et structurer une forme viable, la mienne, de donner consistance \u00e0 un caract\u00e8re jamais acquis, toujours \u00e0 fortifier. Ne disposant pas des mots, je sais ma m\u00e8re d\u00e9finitivement vou\u00e9e \u00e0 une errance dont pour ma part le verbe me dispense en partie. Mais d\u00e9sormais je crois savoir o\u00f9, quand et pourquoi, la n\u00e9cessit\u00e9 m'a fait nomade l\u00e0 o\u00f9 je le suis. L'\u00e9criture de mes livres m'emp\u00eache une giration d\u00e9sesp\u00e9r\u00e9e et je t\u00e2che de me mouvoir entre d\u00e9sir de volcan et vertus de foudre, frayant mon chemin dans une perp\u00e9tuelle apocalypse d'\u00e9l\u00e9ments, aspirant dans l'incandescence \u00e0 des archipels de com\u00e8tes o\u00f9 solitude et abandon s'\u00e9loigneraient \u00e0 des vitesses sid\u00e9rales.\n1\n\nG\u00c9OGRAPHIE SENTIMENTALE\n\nAux temps b\u00e9nis o\u00f9 les Lumi\u00e8res \u00e9taient cens\u00e9es \u00e9clairer l'Europe tout enti\u00e8re, quelques-uns se demandaient comment on peut \u00eatre persan? Dufresny avait inaugur\u00e9 la m\u00e9thode, quelques ann\u00e9es avant Montesquieu, en convoquant des Siamois pour avancer des perspectives \u00e9trang\u00e8res sur un monde dont il voulait faire le portrait. Deux si\u00e8cles plus tard, \u00e0 l'\u00e8re du village plan\u00e9taire, du cosmopolitisme int\u00e9gral et des fronti\u00e8res pr\u00e9tendument pulv\u00e9ris\u00e9es, nombreux sont ceux qui, \u00e0 Paris, centre du monde, me demandent : comment peut-on \u00eatre normand ? Pourtant, \u00e0 eux qui vivent dans la capitale comme des caricatures de provinciaux, accroch\u00e9s \u00e0 leurs quartiers, confin\u00e9s dans leurs zones, terroris\u00e9s d\u00e8s qu'ils mettent un pied en dehors de leur arrondissement, p\u00e9trifi\u00e9s d\u00e8s qu'il s'agit de franchir la Seine, il devrait \u00eatre facile de trouver la r\u00e9ponse.\n\nMalgr\u00e9 tout, r\u00e9tifs et born\u00e9s, ils persistent \u00e0 faire les ahuris, \u00e0 \u00e9carquiller les yeux, et refuser mon parall\u00e8le lorsque j'associe ma province \u00e0 la leur. Protestations, r\u00e9criminations, d\u00e9n\u00e9gations, voire haussement de ton : il n'en est rien, la comparaison est impossible, on ne saurait affirmer pareille ineptie, etc. Et de poursuivre en faisant de la province un tableau aussi ridicule que parfois les provinciaux le font de la capitale. Rats des villes contre rats des champs. Il faut imaginer les arguments et les caricatures abonder : la ville du c\u00f4t\u00e9 de la civilisation et de la culture, la campagne du c\u00f4t\u00e9 de la nature et de la barbarie. Le b\u00e9ton, le bitume et les beaux-arts contre la terre, l'herbe et l'arri\u00e9ration mentale.\n\nQuand j'en \u00e9tais \u00e0 mes premiers livres, ces arguments ne manquaient pas de m'inqui\u00e9ter et je sentais mon provincialisme comme une contradiction, une erreur dont ici et l\u00e0 on me faisait sentir la port\u00e9e. N'\u00eatre pas parisien ! J'ai m\u00eame le souvenir de s\u0153urs journalistes politiquement correctes (je crains le pl\u00e9onasme) qui, du haut de je ne sais quelle intuition ou perversion talmudique, avaient forg\u00e9 le concept de goy-terroir pour stigmatiser ceux qui, comme elles, ne jouissaient pas de l'avantage d'\u00eatre bien n\u00e9s. Tout ce qui n'\u00e9tait pas Paris valait jungle, savane, barbarie, monstruosit\u00e9, pr\u00e9histoire. Longtemps je me suis demand\u00e9 si la province \u00e9tait seulement ce cloaque que disent les urbains radicaux avant de trouver ridicules ceux qui fustigent les racines des autres pour mieux vanter les leurs sous pr\u00e9texte de n'en point avoir : on ne vit pas sans elles, car elles sont mentales. Ni sol, ni terre, ni terroir, qui rappellent trop les heures o\u00f9 l'on cachait sous ces vocables les plus pitoyables id\u00e9ologies dont les finalit\u00e9s \u00e9taient toujours d'ajouter du sang \u00e0 ces gl\u00e8bes-l\u00e0. Aucune terre ne me para\u00eet promise, \u00e9lue ou sacr\u00e9e, aucun coin de plan\u00e8te plus qu'un autre sous pr\u00e9texte qu'il abriterait des souvenirs ou nourrirait des m\u00e9moires. Les collines ne sont pas inspir\u00e9es, pas plus que la terre d'hypoth\u00e9tiques r\u00e9gions transfigur\u00e9es par le doigt de Dieu. Racines et races sont trop parentes pour que j'y trouve plaisir.\n\nEn revanche, je crois \u00e0 l'Histoire, aux lieux de m\u00e9moire, aux r\u00e9gions habit\u00e9es par des fulgurances antiques qui fr\u00e9missent encore longtemps apr\u00e8s que le bruit et la fureur se sont tus. Strates et couches, depuis longtemps confondues, de passages, de langages et de coutumes, superpositions d'\u00e2mes aguerries et de temp\u00e9raments volcaniques aux esprits furtifs d'anc\u00eatres modestes, discrets et laborieux : les pays constituent des g\u00e9ographies mentales et sentimentales dans lesquelles la terre compte pour rien, sinon comme \u00e9crin.\n\nPour la Normandie qu'on dit aujourd'hui Basse, un duch\u00e9 et des conqu\u00e9rants, des paysans et des navigateurs, des laboureurs et des marins, des femmes ardentes et courageuses, des hommes peu causants et endurants. Mais aussi des peintres de ciels magiques et des litt\u00e9rateurs rebelles, des philosophes ind\u00e9pendants et des soldats d\u00e9cid\u00e9s : Eug\u00e8ne Boudin pour les nuages, Octave Mirbeau pour les invectives, Alain ou Tocqueville pour la libert\u00e9 de l'esprit, Guillaume le Conqu\u00e9rant pour les campagnes de conqu\u00eates. Et tous ceux qui de Barbey le Dandy \u00e0 Poulet-Malassis l'insoumis ont install\u00e9 la libert\u00e9, l'ind\u00e9pendance, l'autonomie au-dessus de tout.\n\nLa Normandie est moins une terre de g\u00e9ographes qu'une g\u00e9ographie de terriens, laboureurs de plaines autant que de mers, courb\u00e9s sous la pluie fine qui mouille jusqu'\u00e0 l'os, ou tremp\u00e9s par les embruns, le pied ferme sur les pontons autant que dans le rayon du labour, les Normands paraissent des \u00e9l\u00e9ments d'un paysage o\u00f9 l'eau se module sous toutes ses formes : les vagues de la Manche, les flux et reflux doux sur les c\u00f4tes du Calvados, les escarpements et les blocs effondr\u00e9s ordonnant les paysages de la Hague dans le Cotentin, mais aussi les pluies trombales, les gr\u00eales violentes ou les flocons \u00e9pais, les eaux tenaces quand elles font gouttes en pluies de printemps, les brouillards qui n'en finissent pas lors des automnes aux odeurs de sous-bois, les for\u00eats tremp\u00e9es et les humus satur\u00e9s, gorg\u00e9s, les ruissellements dans les pommiers, les herbages, le bocage, les secrets du Pays d'Auge et les chaumes au fa\u00eetage des anciens manoirs ou des fermes d\u00e9labr\u00e9es. L'eau triomphe partout, dans la boue o\u00f9 sont les vaches indolentes, dans les chemins o\u00f9 passent les gibiers silencieux, dans les champs trou\u00e9s par une mare, dans les paysages qui cachent des rivi\u00e8res aux abords bois\u00e9s. Eau d'acier des \u00e9tangs aussi, eau brune des flaques, eau argent\u00e9e des fontaines.\n\nPaysage doux, climat rude, disons sans concession, sans demi-mesure : ici plus qu'ailleurs, il me semble, les lumi\u00e8res sont uniques parce que liquides, lav\u00e9es, lustr\u00e9es par les eaux. Les ciels impressionnistes fr\u00e9missent l\u00e0, en g\u00e9sine depuis que la terre s'\u00e9panouit, contemporaine des temps g\u00e9ologiques. Le bleu n'y \u00e9clate que par la magie d'une luminosit\u00e9 cristalline, comme concentr\u00e9e par une immense masse d'eau qui se serait faite loupe dans l'azur. Les vents sculptent ces masses lumineuses, eux aussi charg\u00e9s d'eau. Moutons en profusion ou longues tra\u00een\u00e9es neigeuses, volumes parfois pr\u00e9textes \u00e0 toutes les variations chromatiques possibles sur le th\u00e8me des gris, bleus et noirs. Ici, l'espace fournit un palimpseste aux jeux que font les \u00e9l\u00e9ments.\n\nDans ces lieux-l\u00e0, satur\u00e9s par la souvenance des saisons, les villes subissent les m\u00e9tamorphoses inflig\u00e9es par le temps, d'autant que la Seconde Guerre mondiale a fait basculer la r\u00e9gion dans l'urbanisme d'une reconstruction de type g\u00e9n\u00e9ralement sovi\u00e9tique. B\u00e9ton, volumes in\u00e9l\u00e9gants, grossiers, les architectures de cette p\u00e9riode, priv\u00e9es autant que publiques, semblent proc\u00e9der d'une d\u00e9bauche dans l'usage dispendieux de l'espace. Contre les r\u00e9duits, les petites maisons ancestrales, les constructions naines des si\u00e8cles pass\u00e9s, les urbanistes et autres b\u00e9tonneurs de l'imm\u00e9diat apr\u00e8s-guerre ont fait prolif\u00e9rer une \u00e9pid\u00e9mie de cubes et parall\u00e9l\u00e9pip\u00e8des en b\u00e9ton massif qui accroche mal la lumi\u00e8re et se sature rapidement des humidit\u00e9s d'ici, rendant les cit\u00e9s grises comme les hivers scandinaves.\n\nLes quelques colombages ou pierres blanches, maintenant l\u00e9preuses, des carri\u00e8res proches d'Argentan, sont noy\u00e9s dans les b\u00e2tisses qui ont suivi, des ann\u00e9es cinquante \u00e0 nos jours, avec pour seul fil conducteur l'accumulation, le bouchage de trous, le remplissage d'espaces vides. A d\u00e9faut d'urbanisme coh\u00e9rent, les villes normandes sont devenues des m\u00e9tastases o\u00f9 l'on reconna\u00eet ici ou l\u00e0 les d\u00e9jections d'architectes de sous-pr\u00e9fectures qui s'essaient, depuis cinquante ans, \u00e0 reproduire grandeur nature les modes telles qu'elles apparaissent dans leurs pitoyables revues sp\u00e9cialis\u00e9es : n\u00e9o-constructivistes \u00e0 la Le Corbusier pendant les Trente Glorieuses, n\u00e9o-minimales \u00e0 la Jean Nouvel pendant la d\u00e9cennie mitterrandienne, les trouvailles architecturales normandes depuis le passage des tanks am\u00e9ricains font penser, c'est selon, au monumental sovi\u00e9tique, aux sinistres baraquements des espaces concentrationnaires nazis, aux cit\u00e9s radieuses n\u00e9o-Brasilia, aux villages pavillonnaires des phalanst\u00e8res utopistes. Rien de r\u00e9jouissant.\n\nRestent des t\u00e9moignages de ce que fut la Normandie en ses abbayes, ses \u00e9glises, ses donjons, ses ch\u00e2teaux, ses manoirs, quand ils ont \u00e9t\u00e9 \u00e9pargn\u00e9s par les \u00e9tats-majors du g\u00e9n\u00e9ral Eisenhower. Dira-t-on, un jour, combien les destructions inflig\u00e9es aux villes bas-normandes lors des nettoyages massifs qui suivirent les semaines du D\u00e9barquement, lorsqu'elles ont \u00e9t\u00e9 parachev\u00e9es par les r\u00e9alisations architecturales de la reconstruction, demeurent des punitions quotidiennes pour l'oeil ? La guerre moderne \u00e9radique les traces du pass\u00e9, elle plonge dans une in\u00e9vitable amn\u00e9sie, emp\u00eache le lignage architectural des formes et des esth\u00e9tiques particuli\u00e8res, puis, finalement, condamne \u00e0 un ersatz de sens sous forme de pis-aller constructeur.\n\nCe qui donc s'\u00e9l\u00e8ve dans le ciel normand, ce sont ces villes qui ne parviennent pas \u00e0 retrouver une \u00e2me, comme si, \u00e0 jamais, elle leur avait \u00e9t\u00e9 vol\u00e9e, confisqu\u00e9e lors des bombardements, des combats, des incendies qui ont ravag\u00e9, br\u00fbl\u00e9 ici comme l\u00e0-bas Carthage, Varsovie ou Berlin. Rien, il ne reste rien, sinon la fuite en avant pour faire des cit\u00e9s un genre de laboratoire o\u00f9 se v\u00e9rifient des hypoth\u00e8ses d'\u00e9cole. D\u00e9monstration est faite, grandeur nature, de l'incapacit\u00e9 de ce si\u00e8cle en g\u00e9n\u00e9ral, et de notre \u00e9poque en particulier, \u00e0 produire un style, pas m\u00eame un grand style, mais assez pour qu'on y lise et d\u00e9couvre un temp\u00e9rament, un caract\u00e8re, un cachet.\n\nLe temps des cath\u00e9drales et celui des abbayes avait permis un art normand, celui des conqu\u00eates en Italie du Sud et en Sicile avec son roman sp\u00e9cifique, ses volumes et son bestiaire, ses \u00e9quilibres et ses significations. De m\u00eame, les villes et villages de Flaubert, Maupassant ou Barbey d'Aurevilly, autant que la campagne de Charlotte Corday, s'\u00e9taient reproduits pendant des si\u00e8cles, inscrivant l'architecture et l'urbanisme normands dans ce qui est devenu maintenant r\u00e9sidu pour cartes postales, syndicats d'initiative et promotions des alliances touristiques. La Seconde Guerre mondiale a violemment pulv\u00e9ris\u00e9 le pass\u00e9, rendu pr\u00e9caire le pr\u00e9sent, hypoth\u00e9qu\u00e9 l'avenir et pour le temps que dureront les \u00e9poques sans colonnes vert\u00e9brales id\u00e9ologiques, donc esth\u00e9tiques. De l'\u00e2me des Gabriel, architectes argentanais auxquels on doit la sublime place de la Concorde, \u00e0 Paris, autant que le superbe clocher de l'une des deux \u00e9glises de la sous-pr\u00e9fecture, il ne reste plus rien. Des formes sollicit\u00e9es par Fernand L\u00e9ger, constructeur, b\u00e2tisseur et coloriste lui aussi compatriote des contemporains de Mansart, il n'est plus question. Pas plus d'\u00e2me ni de formes dans la pierre et dans la ville que dans le si\u00e8cle qui se termine ou celui qui s'annonce.\n\nLes cit\u00e9s de l'ancien duch\u00e9 ravag\u00e9es par la guerre ne s'en sont pas remises, elles claudiquent depuis. La Basse-Normandie s'est install\u00e9e dans une \u00e9trange culture de mort qui fait cro\u00eetre les vivants d'aujourd'hui \u00e0 l'ombre des victimes du D\u00e9barquement. Ces sinistres perspectives actuelles qui c\u00e9l\u00e8brent Thanatos, sont devenues la seule manne des acteurs sociaux, politiques, locaux et r\u00e9gionaux qui rentabilisent le tragique \u00e9v\u00e9nement par un tourisme de masse insolent \u00e0 l'endroit des milliers de soldats tomb\u00e9s sur les plages. Minable recyclage qui renforce d'autant l'effacement de la m\u00e9moire que se d\u00e9ploient les strat\u00e9gies mercantiles gravitant autour du d\u00e9barquement des Alli\u00e9s.\n\nEntre l'image d'Epinal lait-beurre-cr\u00e8me-vaches-pommiers et celle du soldat-parachutiste-h\u00e9ros-victime-lib\u00e9rateur, il est bien pitoyable d'avoir \u00e0 choisir. La Normandie r\u00e9side en dehors de cette alternative satisfaisante pour le seul folklore avec son cort\u00e8ge et ses cohortes de sous-entendus sinon r\u00e9actionnaires, du moins conservateurs. Le Normand aux doigts crochus, avare, oscillant n\u00e9vrotiquement entre le oui et le non, le tout distill\u00e9 dans une prudence exacerb\u00e9e, doubl\u00e9e d'un mutisme matois, demeure un clich\u00e9 \u00e0 disposition du thurif\u00e9raire des arts et traditions populaires.\n\nLes r\u00e9gions qui ont perdur\u00e9, de l'Antiquit\u00e9 m\u00e9morable aux lendemains de la R\u00e9volution fran\u00e7aise, sont d\u00e9finitivement mortes avec l'av\u00e8nement du chemin de fer, \u00e9quivalent technologique et industriel de l'\u00e9dit royal sur la langue de Villers-Cotter\u00eats. La Nation se constitue moins avec les Jacobins de 1789 qu'avec les locomotives \u00e0 vapeur qui modifient les distances et rendent caduques, totalement impossibles, les identit\u00e9s s\u00e9culaires. S'attacher \u00e0 ces \u00e9chouages folkloriques rend toute Normandie contemporaine improbable.\n\nQuelle Normandie, alors, pour sortir des impasses dans lesquelles elle croupit ? La mienne, au quotidien, \u00e0 savoir un luxe. Car j'aime en elle le temps qui s'y d\u00e9roule, d\u00e9veloppe et \u00e9tale. Un temps g\u00e9n\u00e9alogique, panth\u00e9iste, g\u00e9ologique, climatologique mais aussi et surtout mystique. Pour le circonscrire de fa\u00e7on analogique, je songe au d\u00e9sert de Charles de Foucauld, aux vastes \u00e9tendues de sable qui montrent d'abord un \u00e9tat d'\u00e2me, un esprit, une m\u00e9ditation \u00e0 l' \u0153uvre. D\u00e9sert de stylite parent des p\u00e2turages du paissant, des espaces parcourus par le gyrovague et autres m\u00e9ditatifs extravagants du d\u00e9but de notre \u00e8re. D\u00e9sert de nomade familier des p\u00e9r\u00e9grinations conceptuelles de Deleuze. D\u00e9sert de Palestine, de Syrie, de Samarie, d'Egypte, et d'Anatolie o\u00f9 se sont foment\u00e9s les pens\u00e9es et les actes du gnosticisme. D\u00e9sert libyen de Cyr\u00e9na\u00efque, patrie des inventeurs de l'h\u00e9donisme. D\u00e9sert mental des m\u00e9ditatifs et des po\u00e8tes, des for\u00eats de Thoreau ou des hautes mers au large, tr\u00e8s loin du monde, l\u00e0 o\u00f9 les cargos croisent sur des routes hant\u00e9es par le souvenir des conqu\u00e9rants du monde moderne. D\u00e9serts africains, jaunes et br\u00fbl\u00e9s par le soleil. D\u00e9sert amazonien, satur\u00e9 de verts et d'humidit\u00e9, de pourritures et de grouillements. D\u00e9serts intellectuels qui se nourrissent du silence et de l'absence, de la m\u00e9ditation et du retrait.\n\nCar on ne pense ni ne r\u00e9fl\u00e9chit dans la ville, sinon ce qui la nourrit, la c\u00e9l\u00e8bre et entretient son temps apocalyptique, son mode et son efficace. Temps c\u00e9r\u00e9bral, intellectuel jusqu'au paroxysme, au point qu'on y vide les dur\u00e9es pour les remplir de n\u00e9ant, de nihilisme et de bile noire. Temps de l'acc\u00e9l\u00e9ration et de la consumation en pure perte, temps de la dispersion des \u00e9nergies, de leur d\u00e9pense gratuite, et de la c\u00e9l\u00e9bration des futilit\u00e9s constitutives de l'instant, du moment, de la mode : le temps des villes se nourrit de la substance des \u00eatres qui l'habitent. Lautr\u00e9amont raconte des m\u00e9gapoles comme des monstres, des animaux fantastiques qui, pareils au catobl\u00e9pas de Pline et Flaubert, se r\u00e9v\u00e8lent autophages. Plus grandes elles sont, plus imp\u00e9rieuses sont leurs volont\u00e9s, plus exigeants leurs commerces. Verhaeren raconte leurs tentacules, Nietzsche y stigmatise les \u00e9piciers en abondance, les \u00e2mes d\u00e9prim\u00e9es et les poitrines \u00e9troites, les yeux envieux et les doigts gluants, les importuns et les impertinents, les \u00e9crivassiers et les braillards, les ambitieux exasp\u00e9r\u00e9s.\n\nLa Normandie, d'aucuns le d\u00e9plorent, ne peut \u00eatre un lieu de pouvoir : y vivre, y habiter suppose qu'on ait fait son deuil de la volont\u00e9 de puissance sociale. Rien de plus ridicule qu'un d\u00e9sir de position dominante quand on \u00e9volue dans un d\u00e9sert. A quoi bon du strass et des paillettes quand on dispose de tout le mica des sables ? A qui ou \u00e0 quoi commander ? A des ombres, des fant\u00f4mes, des moulins \u00e0 vent ? On ne trouvera sur ces terres ni argent ni centre de d\u00e9cision, ni secteurs sociaux n\u00e9vralgiques ni espaces culturels cons\u00e9quents.\n\nLe d\u00e9sert n'est doux qu'\u00e0 ceux qui optent radicalement pour l'\u00eatre contre l'avoir. Pour les autres, les aspirants \u00e0 quelque domination, les postes paraissent ridicules, les fonctions supposent toujours des grenouilles voulant se faire aussi grosses que le b\u0153uf. Les types mis en sc\u00e8ne par Flaubert sont toujours l\u00e0 : Bouvard et P\u00e9cuchet, Charles et Emma Bovary, Monsieur Homais, autant dire la sottise, la fatuit\u00e9, l'arrogance, la pr\u00e9tention, la b\u00eatise, l'envie, la libido, les m\u00eames acteurs se partagent le m\u00eame monde sur une m\u00eame sc\u00e8ne.\n\nEt toujours il y a le dandy, type cher au c\u0153ur de Baudelaire, fou de Honfleur, sachant ce que ses Fleurs du mal doivent \u00e0 Alen\u00e7on, la ville bas-normande, et \u00e0 son fameux \u00e9diteur. Et Brummell, praticien de cette esth\u00e9tique du d\u00e9sespoir et de la forme contre l'avachissement consubstantiel \u00e0 toute \u00e9poque, \u00e9lisant Caen pour y jeter ses derniers feux, puis mourir. Et Barbey d'Aurevilly th\u00e9orisant Brummell, r\u00e9digeant un manifeste pour la singularit\u00e9, la r\u00e9volte, la r\u00e9sistance, la r\u00e9bellion, l'autonomie, le geste seigneurial et \u00e9l\u00e9gant, dont le dandy anglais fournit le pr\u00e9texte et l'occasion. Et Proust \u00e9lisant Cabourg. Or le dandy triomphe en moine solitaire dans son d\u00e9sert, en m\u00e9ditatif impassible et distant, en sage averti de la vanit\u00e9 de presque tout, sauf du style.\n\nAux heures des villes tentaculaires et des gr\u00e9garismes c\u00e9l\u00e9br\u00e9s, quand chacun d\u00e9sire le d\u00e9sir de l'autre, ali\u00e8ne son temps, sa vie, son \u00eatre pour quelques mis\u00e9rables fr\u00e9tillements sociaux qui lui donnent l'illusion d'\u00eatre un peu moins seul, le d\u00e9sert \u00e9lu \u2013 et non subi \u2013 d\u00e9signe le lieu des \u00e2mes rebelles. On n'y souffre pas de proximit\u00e9s qui sont des promiscuit\u00e9s, on n'y rencontre pas de philosophe vertueux vendant des arri\u00e8re-mondes tout en comptant ses louis pour son bas de laine, on n'y fr\u00e9quente pas de ministres ou d'importants, de gens remplis d'eux-m\u00eames et de leur suffisance, on n'y c\u00f4toie pas Rastignac, d\u00e9j\u00e0 en route pour la capitale \u2013 mais seulement leurs caricatures. Dans cette solitude, la seule compagnie passable, c'est soi-m\u00eame. A l'heure des bilans, on conclut que le d\u00e9sert est mieux habit\u00e9 et mieux fr\u00e9quent\u00e9, parce que peupl\u00e9 d'ombres, que tous les lieux o\u00f9 la chaleur animale tient lieu de viatique.\n\nDe la Normandie, je retiens finalement qu'elle est d'abord, et surtout, une occasion de ciels dans lesquels on peut surprendre les nuages, les merveilleux nuages de l'homme \u00e9nigmatique baudelairien. Qu'on peut y lire, en familier de mantiques transfigur\u00e9es en occasions de po\u00e9sie v\u00e9cue, des histoires de libert\u00e9 o\u00f9 n'ont part ni p\u00e8res ni m\u00e8res, ni femmes ni enfants, ni amis ni patries, ni or ni beaut\u00e9. Le jeu des nuages, seulement, tout simplement. A qui voudrait se persuader de l'excellence, ici, des \u00ab merveilleuses constructions de l'impalpable \u00bb \u2013 comme l'\u00e9crit toujours l'auteur du Spleen de Paris \u2013, il reste la possibilit\u00e9 de succomber \u00e0 la tentation de cet exil volontaire au d\u00e9sert, car il rend possible la m\u00e9ditation du b\u00e9n\u00e9dictin, figure sans laquelle il n'est pas de libertin possible, car tous deux, n\u00e9cessaires, indispensables l'un \u00e0 l'autre, s'obligent et s'exigent.\n2\n\nPOUR CYTH\u00c8RE EMBARQU\u00c9\n\nBordeaux a longtemps \u00e9t\u00e9, \u00e0 mes yeux, l'impossible ville hant\u00e9e par les m\u00e2nes de H\u00f6lderlin fou ou le souvenir d'un Goya habit\u00e9 par les sorci\u00e8res, les boucs tragiques, les sabbats, les cannibales, criminels et autres anthropophages. L\u00e0, le po\u00e8te d'Hyp\u00e9rion a entam\u00e9 un basculement de l'autre c\u00f4t\u00e9 du miroir, lieu de songes goyesques plus vrais que nature. Du peintre espagnol en exil, il me semblait que la ville avait conserv\u00e9 le talent pour les rictus, les visages effrayants, les faci\u00e8s \u00e9pouvantables, les portraits de vivants plus m\u00e9caniques que poup\u00e9es et pantins de Kleist. Mes errances dans Bordeaux tenaient des seuls cauchemars dus \u00e0 la passion amoureuse, celle qui interdit tout pluriel f\u00e9minin dans une existence et fait basculer les vies du c\u00f4t\u00e9 o\u00f9 la foudre aura renvoy\u00e9 les deux morceaux, soudainement devenus distincts, d'une m\u00eame existence. Pour mieux cacher les d\u00e9tails de ces temps-l\u00e0, je m'\u00e9vertue \u00e0 montrer les temps autres, avec force pr\u00e9cisions qui, toutefois, ne parviennent pas \u00e0 me convaincre du caract\u00e8re efficace de mon projet.\n\nFaut-il alors que je me demande plus avant pour quelles raisons le d\u00e9sir d'un voyage d'\u00e9criture au c\u0153ur du monde apocalyptique des peintures de Monsu Desiderio s'est initi\u00e9 l\u00e0, dans les rues de la ville de Goya exil\u00e9, dans les g\u00e9ographies alentour que Denis Mollat m'a fait sillonner? Car Desiderio excelle en peintre des connaisseurs d'ab\u00eemes, des familiers de raz-de-mar\u00e9e, le fixateur des lendemains de fin du monde, le contemporain de tous les brasiers, de tous les autodaf\u00e9s, de toutes les d\u00e9collations, de tous les martyres, de toutes les passions.\n\nDans le monde de feu de ces toiles peintes \u00e0 deux et sign\u00e9es d'un m\u00eame nom, il y a, en raccourci, deux ou trois des obsessions de mon existence : les catastrophes, qu'elles soient tremblements de terre qui engloutissent ou volcans qui purifient par l'incandescence, et les consolations sublimes qu'elles induisent, l'amiti\u00e9, l'\u00e9criture, l'art, la conversation, et les occasions d'un corps voluptueux. A Bordeaux, j'ai respir\u00e9 les odeurs m\u00e9phitiques de soufre qu'on d\u00e9couvre au bord des ab\u00eemes de l'enfer et, m\u00eamement, j'ai abord\u00e9 les contr\u00e9es magiques famili\u00e8res aux peintures du XVIIIe si\u00e8cle qui pr\u00e9c\u00e8dent la R\u00e9volution fran\u00e7aise.\n\nEn plus de la peinture joyeuse et libertine de cette \u00e9poque, j'aime aussi les toiles de Joachim Patinir, les paysages magnifiques dans lesquels \u00e9voluent des hommes ignorants du destin qu'ils subissent en toute innocence. Dans Le Passage du Styx, un enfant est conduit par Charon, entre deux rives, sans qu'on sache, par la direction de la barque qui vient vers le regardeur, l'origine du voyage ou son aboutissement : va-t-on vers le paradis, avec ses anges, ses fontaines, sa douceur, son calme, ou vers les enfers avec ses monstres, ses guerres, ses brasiers, ses incendies, ses animaux monstrueux qui guettent, embusqu\u00e9s ? Le destin des hommes se r\u00e9sume l\u00e0 : quelles rives aborder? Du pire au meilleur, ou l'inverse ? A Bordeaux, j'ai connu les territoires d\u00e9sol\u00e9s, les champs de bataille et les guerres meurtri\u00e8res. C'\u00e9tait \u00e0 l'H\u00f4tel Gambetta. Je devais aborder un continent r\u00e9dempteur de l'autre c\u00f4t\u00e9 de la rue dont l'h\u00f4tel tient son nom. C'\u00e9tait la librairie Mollat.\n\nD'un c\u00f4t\u00e9, le Bordeaux de H\u00f6lderlin, de Goya, sombre et nocturne, de l'autre, celui de Montaigne et de Sollers, solaire et lumineux, v\u00e9nitien \u00e0 sa mani\u00e8re, italien dans l'all\u00e9gresse. Un Bordeaux qui, pour moi, venant de Normandie, raconte sans ambages la parent\u00e9 de la ville avec le Sud, ce Sud magnifique qui enthousiasmait Nietzsche pour ses vertus alcyoniennes, la l\u00e9g\u00e8ret\u00e9, l'\u00e9l\u00e9gance, le raffinement, le style. Ce Bordeaux-l\u00e0, Denis Mollat s'en fit pour moi le cic\u00e9rone.\n\nBordeaux des eaux terreuses de la Garonne, des pierres grises de la ville que l'on voit sur le pont Gustave-Eiffel, lorsque le train arrive du Nord. Bordeaux des \u00e9glises aux clochers qui trouent le ciel comme des ifs flamboyants. Bordeaux des vins, bien s\u00fbr, mais aussi de Michel Eyquem, libertin, libertaire et singulier en tout. Bordeaux de La Bo\u00e9tie et de cette amiti\u00e9 sublime parmi toutes, derni\u00e8re corne pouss\u00e9e par l'Antiquit\u00e9 romaine dans le ventre de la France des pr\u00e9mices modernes. Bordeaux des all\u00e9es Tourny et des entrep\u00f4ts Lain\u00e9 o\u00f9 j'ai laiss\u00e9 aussi un peu de mon ombre, de l'esplanade des Quinconces et de l'H\u00f4tel Normandie o\u00f9 j'ai mes quartiers depuis qu'en mon \u00e2me a br\u00fbl\u00e9 le Gambetta. Place de la Com\u00e9die, les quais, la tour saint Michel, porte Dijeaux, un op\u00e9ra de Rossini entendu au th\u00e9\u00e2tre Victor-Louis et le tombeau de Montaigne. Un caf\u00e9 o\u00f9 je me suis r\u00e9chauff\u00e9 un hiver sali de neige fondue, un autre o\u00f9 je me suis rafra\u00eechi l'un de ces apr\u00e8s-midi d'\u00e9t\u00e9 br\u00fblants qui rendent aveuglantes les pierres blanches de l\u00e0-bas. Et puis la librairie Mollat.\n\nMon premier passage dans les salons Albert-Mollat s'est fait sur le mode du presque malentendu : j'y \u00e9tais attendu, je ne le savais pas et j'errais alentour, soucieux de chasser les d\u00e9mons qui m'accablaient. Au d\u00e9tour de la rue, je suis retourn\u00e9 vers un champ de bataille d\u00e9sert\u00e9 par son ennemi disparu. Le go\u00fbt d'un caf\u00e9 kenyan dans une br\u00fblerie, non loin, remplissait ma bouche en m\u00eame temps que l'amertume des jours perdus pour la jubilation. J'ai lu mon nom sur une affiche. D\u00e9couvrant ainsi les obligations auxquelles j'avais souscrit sans m'en souvenir, me sachant bordelais pour d'autres raisons ces deux jours-l\u00e0. Je suis entr\u00e9 dans la librairie comme on p\u00e9n\u00e8tre dans un havre. Je me souvenais de Montesquieu, voisin par La Br\u00e8de de Denis Mollat \u00e0 Saucats, qui avouait n'avoir jamais connu de peine qu'une heure de lecture n'ait pulv\u00e9ris\u00e9e. Et l\u00e0, dans un sanctuaire sublime pour les livres, j'ai d\u00e9couvert un magasin, une usine, un paradis pour la chose imprim\u00e9e.\n\nTout fut comme en un r\u00eave. Je ne me souviens plus de rien, aucun visage, aucun mot, aucun public, aucun sujet. Des journalistes comme les fid\u00e8les d'aujourd'hui qui sont l\u00e0 en signe d'une belle connivence silencieuse \u00e0 chacun de mes passages ? Je ne sais plus. Quel complice sur la tribune sobre et sombre? Pas de souvenir. Quel regard ce visage antique de bronze toujours braqu\u00e9 dans le dos du conf\u00e9rencier ? Quels mouvements dans cette petite pi\u00e8ce \u00e0 la gauche des intervenants, l\u00e0 o\u00f9 veillent, discrets et efficaces, celles et ceux qui rendent possibles ces rencontres semblables \u00e0 celles des salons de Mme de Genlis ou de l'\u00e9pouse du fermier g\u00e9n\u00e9ral Helv\u00e9tius? M\u00e9moire vide, absence. D\u00e9j\u00e0 l\u00e0, je suppose, les acteurs de ce qui permet \u00e0 chacun de mes passages ce ballet dans un genre de th\u00e9\u00e2tre du XVIIIe si\u00e8cle.\n\nCar Bordeaux reste, contre vents et mar\u00e9es, une ville dont l'histoire et la g\u00e9ographie rel\u00e8vent indiscutablement du si\u00e8cle des Lumi\u00e8res avant 1789. Les ors et brocarts, les lustres et verroteries, les chaises et les boudoirs, les sofas et les jardins, la conversation, le port, le maintien, l'allure, les jeux avec les silences et les secrets, tout proc\u00e8de de Beaumarchais et de Marivaux, sinon de Sade et de Montesquieu. Alors que mes premiers contacts avec la librairie Mollat sont effondr\u00e9s dans les sables de l'amn\u00e9sie, tous ceux qui suivirent sont grav\u00e9s, comme une eau-forte dans les moindres d\u00e9tails, \u00e0 la place m\u00eame laiss\u00e9e vide par l'engloutissement originaire. Bordeaux est devenue pour moi une enclave dix-huiti\u00e9miste avec Denis Mollat pour intercesseur.\n\nLa date de naissance de cette r\u00e9demption dans la cit\u00e9 est dans ma m\u00e9moire. Le lieu aussi : sur le trottoir, en sortant des salons Albert-Mollat, dans la rue Vital-Carles. La lumi\u00e8re dansait, blanche et fragile, douce et moins violente qu'en Normandie. Des images pr\u00e9cises se m\u00e9langent \u00e0 ce souvenir : la sensation physique de l'acc\u00e9l\u00e9ration du train en quittant Paris pour Bordeaux, le passage de la Loire, ses eaux qui fl\u00e2nent entre les bancs de sable, les modifications infinit\u00e9simales des qualit\u00e9s de la lumi\u00e8re en m\u00eame temps que l'adoucissement de la pente des toits, les tuiles fa\u00eeti\u00e8res, les pierres joint\u00e9es \u00e0 sec, blanches, qui accrochent le soleil. Des noms de ville : Poitiers, Angoul\u00eame, puis Libourne et les portes mentales qui s'ouvrent. Un rideau dans lequel s'engouffrait le vent, \u00e0 Libourne, justement, \u00e0 la devanture d'un magasin, la respiration du tissu, les drap\u00e9s beaux comme dans une peinture du Greco, mobiles comme dans le ralenti du cin\u00e9ma.\n\nBordeaux Saint-Jean. Arr\u00eat. Descente. Emotions, maintenant, \u00e0 chaque fois, car je sais Denis Mollat m'attendant dans la gare moins douloureuse des fant\u00f4mes anciens. La carte du Sud de la France dessin\u00e9e sur un mur, la verri\u00e8re, les lacis et entrelacs de fer, les premiers mots de mon complice retrouv\u00e9, les premi\u00e8res minutes en direction du restaurant. La Garonne long\u00e9e, ses micro-tourbillons dans lesquels une \u00e2me peut se perdre, le ciel lav\u00e9 et doux, les vignes et les acteurs de cette pi\u00e8ce de Marivaux jou\u00e9e pour moi par Denis Mollat et les siens : Alexandre de Lur-Saluces et Suzanne Abadie, Jean-Marie Plane et Jean-Marie Amat, Raymonde et Jean Ramet, le sommelier du Saint-Jammes, l'h\u00f4telier du Normandie, trois ou quatre d\u00e9miurges de la librairie.\n\nSur cette sc\u00e8ne qui semble une prolongation dans toute la ville et alentour de ce qui pourrait se jouer au th\u00e9\u00e2tre Victor-Louis, on trouve des mill\u00e9simes historiques d'Yquem, des d\u00e9jeuners en t\u00eate \u00e0 t\u00eate, Denis Mollat et moi, des projets qui se formulent et d'autres qui aboutissent, une amiti\u00e9 moins exprim\u00e9e dans son essence que dans mille occasions incarn\u00e9es, des vignobles travers\u00e9s, des disques \u00e9cout\u00e9s en voiture, une fois Biber, une autre la voix de Marcel Duchamp, presque toujours des musiciens aim\u00e9s par l'un ou l'autre, baroques ou contemporains, des visites dans le Sauternais ou \u00e0 Saint-Emilion, des pibales et un rioja, de la lamproie et un saint-julien, sinon un porto sublime \u00e0 Madrid.\n\nA chaque fois, comme en t\u00e9moins d'un monde qui aurait conserv\u00e9 l'excellence des usages contemporains de Montesquieu, d'antiques vertus sont c\u00e9l\u00e9br\u00e9es : pr\u00e9venance et d\u00e9licatesse, politesse et magnificence, douceur et silence, conversation et gentillesse. Amiti\u00e9, donc, comme fleurissent les plantes rares dans les circonstances exceptionnelles. L\u00e0-bas, \u00e0 Bordeaux, j'ai l'impression de rentrer de plain-pied dans une soci\u00e9t\u00e9 qui aurait conserv\u00e9 les principes chers au c\u0153ur de Grimod de La Reyni\u00e8re : les microsoci\u00e9t\u00e9s h\u00e9donistes, les affinit\u00e9s \u00e9lectives, les plaisirs g\u00e9n\u00e9ralis\u00e9s, les soucis d'alt\u00e9rit\u00e9s voluptueuses, la c\u00e9l\u00e9bration sensualiste du monde, le mat\u00e9rialisme dionysien, l'amiti\u00e9.\n\nLes salons Albert-Mollat, en appendice jubilatoire \u00e0 la librairie, me sont devenus, depuis et \u00e0 chaque fois, des occasions de dire en quoi, comment et pourquoi je me sens, en m\u00eame temps que nietzsch\u00e9en, et parce que tel, contemporain de Fragonard, Watteau, Lancret et autres sp\u00e9cialistes des embarquements pour Cyth\u00e8re. Dans mon panth\u00e9on de lieux magiques, Bordeaux est ma Venise o\u00f9 l'on parle fran\u00e7ais, Denis Mollat, un certain condottiere dont j'ai \u00e9crit, ailleurs, combien la figure m'a \u00e9t\u00e9 essentielle pour formuler, venant de la piazza San Zanipolo, une \u00e9thique esth\u00e9tique \u00e0 l'usage d'aujourd'hui. Pour le reste, le silence suffira.\n3\n\nLES VENGEANCES DE LA PERDRIX\n\nPourquoi faut-il, chez les oiseaux comme chez les hommes, que certains soient dou\u00e9s d'\u00e9l\u00e9gance, d'autres pas ? Quelles raisons font l'un pataud, l'autre vif et dou\u00e9 pour les arabesques des plus fantaisistes ? Qui pr\u00e9f\u00e9rerait pour lui-m\u00eame le dandinement ridicule des oies aux z\u00e9brures d'acier inflig\u00e9es par les hirondelles aux ciels orange des fins de journ\u00e9es d'\u00e9t\u00e9 ? Ici, la basse-cour et ses volatiles emprunt\u00e9s, l\u00e0 les azurs d\u00e9gag\u00e9s \u00e0 perte de vue. Le choix est-il compliqu\u00e9 ? Car, enfin, qui voudrait \u00eatre pintade chez Aristote plut\u00f4t qu'albatros chez Baudelaire? Bien s\u00fbr, on dit la premi\u00e8re ardente \u00e0 l'amour, agressive, m\u00e9chante et rus\u00e9e, c'est du moins ce que rapporte le philosophe stagirite dans son Histoire des animaux. De m\u00eame s\u00e9vit-elle depuis toujours en parangon d\u00e9moniaque, en incarnation de la tentation et de la perdition. En revanche, albatros chez Baudelaire, aigle chez Lautr\u00e9amont ou alcyon chez Nietzsche, voil\u00e0 qui ne manque pas d'allure.\n\nPourquoi donc la pintade semble-t-elle si ridicule, tr\u00e9buchant dans ses tentatives de vol en forme de courses cahotantes ? Quelles raisons la condamnent \u00e0 installer son nid par terre, \u00e0 m\u00eame le sol, plut\u00f4t que dans la cime des arbres ? Ovide donne la r\u00e9ponse quand il envoie la pintade dans les parages de D\u00e9dale et d'Icare lors m\u00eame que le pire arrive puisque le p\u00e8re enterre son fils. On conna\u00eet l'histoire qui fait de D\u00e9dale, artiste en m\u00e9taux et fabricant de labyrinthe, le prisonnier d'un roi de Cr\u00e8te qui l'a \u00e9loign\u00e9 d'Ath\u00e8nes, son pays natal. Minos a pr\u00e9vu que, par la route ou par les mers, D\u00e9dale envisagera la fuite. Pas par le ciel.\n\nPour cette simple raison l'Ath\u00e9nien en mal de libert\u00e9 et de retrouvailles avec le sol natal d\u00e9cide de s'enfuir par les airs et avec lui son fils Icare. Voil\u00e0 pourquoi l'on peut lire dans Les M\u00e9tamorphoses : \u00ab Il tourne son esprit vers l'\u00e9tude d'un art inconnu, ouvrant de nouvelles voies \u00e0 la nature. \u00bb Autant dire que D\u00e9dale annonce ce qui devient sous la plume de Descartes l'art de se rendre ma\u00eetre et possesseur de la nature. L\u00e0 o\u00f9 s\u00e9vit la contrainte, l'homme du labyrinthe veut la libert\u00e9; l\u00e0 o\u00f9 triomphent la prison, l'exil et la soumission, il projette la fuite, l'\u00e9vasion et l'autonomie retrouv\u00e9e. Pour son dessein, l'oiseau fournit le mod\u00e8le animal des volutes libres et des trajets sans entraves. Gageons que dans sa prison au bord de la mer, l'alcyon, compagnon ins\u00e9parable du p\u00e8re de Zarathoustra, a servi de mod\u00e8le \u00e0 D\u00e9dale qui aspirait justement \u00e0 la l\u00e9g\u00e8ret\u00e9, l'\u00e9l\u00e9gance et la libert\u00e9 de l'animal ail\u00e9. Toujours est-il qu'il fomente le projet de se faire lui-m\u00eame volatile. Lui-m\u00eame et son jeune fils Icare avec lequel il entend bien quitter son lieu d'exil, par les airs, avant de prendre la direction de la cit\u00e9 aim\u00e9e.\n\nD\u00e9dale ramasse les plumes abandonn\u00e9es par les oiseaux de passage, en transit sur le territoire maudit, lors de leurs trajets libertaires. R\u00e9miges et pennes, longues et courtes, tout est collect\u00e9 puis agenc\u00e9 pour imiter l'aile d'oiseau. Les plus solides, puis les plus fragiles, enfin des fils ou lacets de lin pour arrimer l'ensemble. Le tout est courb\u00e9 pour mieux ressembler aux architectures portantes des b\u00eates ail\u00e9es. Cet artifice r\u00e9alis\u00e9, comme pr\u00e9lev\u00e9 sur un grand et gros oiseau, reste \u00e0 l'associer au corps, telle une proth\u00e8se n\u00e9cessitant l'ajustement. Le point de jonction entre l'outil et les membres suppose une suture essentielle : d'elle d\u00e9pend le bon fonctionnement de l'appareillage. La fixation se fait avec de la cire.\n\nVoici le projet : transformer le corps pesant et mat\u00e9riel, lourd et prisonnier du sol, en m\u00e9canique plus l\u00e9g\u00e8re que l'air, gr\u00e2ce \u00e0 l'\u00e9nergie apport\u00e9e par le battement des ailes. D\u00e9dale essaie le m\u00e9canisme au sol. Dans ses bras, il sent la force d\u00e9cupl\u00e9e, dans son torse, l'\u00e9nergie ramass\u00e9e, dans ses jambes, la portance sublim\u00e9e. A ce moment, il r\u00e9ussit un double exploit : d\u00e9passer le platonisme en l'incarnant paradoxalement et inventer les principes de toute a\u00e9ronautique future.\n\nD\u00e9passement du platonisme ? Certes, en descendant le ciel sur terre, en effa\u00e7ant le registre d'un \u00e9ther noir seulement travaill\u00e9 par les vents. Car rien n'est plus farcesque qu'un ciel atopique, rempli d'id\u00e9es invisibles, ou de formes intelligibles, qui n'apparaissent \u00e0 aucun endroit. Le ciel de D\u00e9dale est un azur vivant dans lequel les hommes peuvent trouver place, au c\u00f4t\u00e9 des oiseaux. Et s'affranchir de la terre, soit pour entrer dans le secret des profondeurs abyssales, avec le poisson, soit pour conna\u00eetre les plaisirs sid\u00e9raux, avec les alcyons, les albatros ou les aigles. Voil\u00e0 un authentique projet prom\u00e9th\u00e9en.\n\nL'exercice de cette libert\u00e9 radicale exige des pr\u00e9cautions, comme toujours en pareil cas. D\u00e9dale les formule pour Icare : ne pas voler trop pr\u00e8s de l'onde, ni trop familier du soleil, car, dans les deux cas, on risque de chuter, ici, \u00e0 cause d'une voilure alourdie par les embruns et l'eau, l\u00e0, parce que l'extr\u00eame chaleur ferait fondre la cire, rendant l'appareillage absolument caduc. Dans ces mots du p\u00e8re \u00e0 son fils, il faut entendre un genre de conseil aristot\u00e9licien de prudence, car on ne s'empare pas du ciel pour la premi\u00e8re fois sans un minimum de pr\u00e9cautions \u00e9thiques : ni morgue, ni exc\u00e8s d\u00e9mesur\u00e9 pouvant laisser croire qu'on va se mesurer aux dieux. Il s'agit de voler pour s'affranchir et non pour narguer les divinit\u00e9s. La d\u00e9cision est prise. Icare et D\u00e9dale se lancent dans le vide et t\u00e2chent d'exp\u00e9rimenter cet \u00ab art dangereux \u00bb, pour le dire comme Ovide.\n\nOn conna\u00eet l'histoire : les deux hommes r\u00e9ussissent leur tentative. Au sol, laboureurs, p\u00e2tres et p\u00eacheurs voient ces deux \u00e9tranges cr\u00e9atures \u00e9voluer dans l'espace. Leur stup\u00e9faction est telle qu'ils prennent ces formes ail\u00e9es pour quelque divinit\u00e9 en transit. Puis, enivr\u00e9 par le succ\u00e8s de l'op\u00e9ration, le fils n\u00e9glige les invites paternelles. Sans souci des points cardinaux balis\u00e9s par l'ancien, sans respect des interdits salutaires, Icare m\u00e9sestime les mises en garde et s'approche du soleil. La pr\u00e9diction se r\u00e9alise : la cire fond, l'aile se d\u00e9fait, l'appareillage se d\u00e9mantibule, et le jeune homme est pr\u00e9cipit\u00e9 dans l'onde par son orgueil et son arrogance m\u00e9lang\u00e9s. On ne brave pas impun\u00e9ment les dieux et l'on ne chasse pas sans risque sur leurs territoires.\n\nOn sait le tr\u00e9pas dans \u00ab les flots c\u00e9rul\u00e9ens \u00bb, la peine et la souffrance du p\u00e8re. De m\u00eame, on constate une nouvelle fois que le d\u00e9sir de fonder de nouvelles possibilit\u00e9s, d'inventer, de bouleverser le cours des choses et d'opposer la volont\u00e9 des hommes \u00e0 l'ordre naturel induit dans le cours du monde des perturbations dont on ne mesure pas toujours a priori les effets. En inaugurant l'art de voler, D\u00e9dale montre la puissance du vouloir et ses effets possibles sur ce qui, de toute \u00e9ternit\u00e9, rel\u00e8ve de la n\u00e9cessit\u00e9. Le geste de D\u00e9dale, en son genre, est ath\u00e9e, iconoclaste, mat\u00e9rialiste et fondateur de modernit\u00e9. La libert\u00e9 par l'artifice contre la soumission et l'ob\u00e9issance aux exigences naturelles, voil\u00e0 la le\u00e7on \u00e0 retenir de cette histoire d'ailes.\n\nEt la perdrix, direz-vous ? Elle est l\u00e0, embusqu\u00e9e, au moment o\u00f9 D\u00e9dale enterre son fils. A l'instant m\u00eame o\u00f9 Icare est inhum\u00e9, dans cette pointe du temps dans laquelle, normalement, tout s'arr\u00eate, y compris la haine et la guerre, la perdrix \u00ab applaudit en battant des ailes et chanta pour t\u00e9moigner sa joie\u00bb. Qu'est-ce qui peut ainsi justifier la cruaut\u00e9 du volatile que le chagrin d'un p\u00e8re compte pour rien dans la manifestation de son enthousiasme? Ovide raconte l'histoire et, l'ayant lue, l'on se prend \u00e0 craindre que les hommes aient \u00e0 payer toujours le prix du ressentiment et de la vengeance de la perdrix. Qu'on en juge : D\u00e9dale n'est pas un inconnu pour cet oiseau dont le nom proc\u00e8de du patronyme d'un jeune homme que le p\u00e8re d'Icare eut \u00e0 \u00e9duquer sur l'invitation de sa propre s\u0153ur. Perdrix \u00e9tait le nom du neveu de D\u00e9dale. Brillant, inventeur de la scie et du compas, le gar\u00e7on prodige fait na\u00eetre une jalousie exacerb\u00e9e chez son oncle. Un jour qu'il a conduit Perdrix dans la citadelle de Minerve, D\u00e9dale le pr\u00e9cipite du haut des murailles pour en finir avec lui. Dans sa bienveillance, Pallas m\u00e9tamorphose l'enfant pendant le temps que dure sa chute : elle en fait l'oiseau \u00e9ponyme.\n\nDepuis, le g\u00e9nie vigoureux de l'enfant se transmet aux pattes et aux ailes de l'animal, les instruments de son salut. Il a conserv\u00e9 son nom. Toutefois, malgr\u00e9 le talent qu'il met dans ses extr\u00e9mit\u00e9s, il demeure en fait pi\u00e8tre voleur ou planeur. Les cimes lui sont interdites, il d\u00e9pose ses \u0153ufs dans les haies, car le souvenir de sa chute lui fait redouter les hauteurs. Voil\u00e0 pour quelles raisons il faut craindre le ressentiment de l'animal, t\u00e9moignage toujours actif de la haine qu'on peut entretenir pour un homme. D\u00e9dale l'inventeur du vol \u00e9tait donc \u00e9galement l'homicide que l'on sait. Que faut-il en conclure? Qu'une libert\u00e9 acquise ici est pay\u00e9e l\u00e0 de n\u00e9gatif; que chaque progr\u00e8s dans un lieu se double d'une r\u00e9gression dans un autre ; que la dialectique des contraires impose cette perp\u00e9tuelle oscillation entre un affranchissement et un asservissement ; que circuler dans les airs un jour suppose souffrir sur terre un autre jour ; que le vol appelle la chute. En un mot, qu'Icare et D\u00e9dale forment un Janus bifrons.\n\nA cette lumi\u00e8re, on peut chercher des prog\u00e9nitures \u00e0 ces cr\u00e9atures ail\u00e9es des origines mythologiques. Des fr\u00e8res et s\u0153urs \u00e0 Icare et D\u00e9dale. Peut-\u00eatre m\u00eame une descendance \u00e0 la perdrix joyeuse les jours d'enterrement. Car apr\u00e8s Minos et Pallas, la Cr\u00e8te et Ath\u00e8nes, le souffle de l'histoire se fait sentir plut\u00f4t dans les contr\u00e9es du Proche et du Moyen-Orient, non loin d'un autre bestiaire qui convoque un b\u0153uf, un \u00e2ne, une vierge et un charpentier. Une com\u00e8te balaie le ciel et indique la voie \u00e0 des marchands flanqu\u00e9s d'anges. Ces cr\u00e9atures ail\u00e9es prennent historiquement le relais des h\u00e9ros mythologiques. Anges gardiens ou porteurs de bonnes nouvelles, anges messagers ou proph\u00e8tes p\u00e9r\u00e9grins, ils hantent les cosmogonies jud\u00e9o-chr\u00e9tiennes et remplacent celles des Grecs. D\u00e9dale ici, Rapha\u00ebl l\u00e0, Icare en un lieu, Gabriel en un autre. Tous stigmatisent l'arrachement \u00e0 la condition humaine. Voler comme un ange, c'est appara\u00eetre dans la revendication d'alt\u00e9rit\u00e9 radicale \u00e0 l'endroit de l'humaine condition : \u00eatre dans un autre monde avec un autre corps, r\u00e9aliser une autre chair pour d'autres desseins. L'oiseau, dans tous les cas de figure, vaut parangon libertaire en animal libre comme l'air. Sans limite et sans contrainte, le champ qui s'ouvre \u00e0 lui est illimit\u00e9. Son domaine? L'infini toujours recommenc\u00e9.\n\nL'ange se taille dans une mati\u00e8re diff\u00e9rente de celle des hommes. D\u00e9dale se veut ailleurs que l\u00e0 o\u00f9 il croupit en exil. Chaque fois, les ailes soulignent l'envie d'un autre lieu, d'un autre temps, d'une autre condition. L\u00e0 o\u00f9 triomphe avec indolence ce qui s'apprivoise sous la rubrique de la chute des corps et de la gravitation universelle, les anges magnifient l'id\u00e9al d'une existence \u00e9mancip\u00e9e des n\u00e9cessit\u00e9s physiques, mais aussi biologiques, spatiales ou temporelles. Immortel, \u00e9ternel, l'ange ignore les d\u00e9sirs triviaux de soif, de faim ou de sexualit\u00e9, pas plus il ne ressent l'obligation de parler pour se faire comprendre : il se nourrit de manne, communique par impulsion volontaire avec ses semblables, co\u00efncide avec les processus intellectuels. Ses ailes racontent tout cela.\n\nLa pesanteur ? Il ignore. La pourriture, la d\u00e9composition, la g\u00e9n\u00e9ration et la corruption ? Il n'en sait rien. Le manque? L'incompl\u00e9tude ? des mots, de simples mots vides de sens. Quand les exigences de la terre requi\u00e8rent l'homme \u2013 y vivre, la travailler, la fouiller, la retourner, la parcourir, puis, finalement, l'ouvrir pour s'y reposer et y dispara\u00eetre, confondu \u00e0 la gl\u00e8be, d\u00e9vor\u00e9 par la vermine \u2013, les anges se suffisent d'air.\n\nDans l'imaginaire des hommes, ces cr\u00e9atures aux ailes chatoyantes \u2013 Fra Angelico a tout montr\u00e9 sur ce sujet \u2013 proc\u00e8dent de ce qu'apr\u00e8s Feuerbach on pourrait appeler des hypostases : ce que l'homme n'est pas, il le projette dans une cr\u00e9ature investie de puissances d\u00e9sir\u00e9es, il les structure dans une identit\u00e9 factice qu'il se met \u00e0 v\u00e9n\u00e9rer et \u00e0 laquelle il finit par donner une autonomie dont pourtant elle ne dispose pas. Ce rien avec quoi il fait un tout signale et d\u00e9signe ses manques, ses impuissances, ses limites. Les anges existent pour dire l'intraitable m\u00e9lancolie des hommes, leur soumission \u00e0 l'ordre naturel, leur arraisonnement \u00e9ternel et douloureux \u00e0 la terre, au sol, au globe terrestre. Voler, pour l'homme, permet d'imaginer qu'il peut n'\u00eatre pas homme, n'\u00eatre plus homme. Ne plus d\u00e9sirer, travaill\u00e9 par le manque ; ne plus marcher, courb\u00e9 par la fatigue ; ne plus peiner, encha\u00een\u00e9 par la faute. Voler, aller et venir entre les astres, passer d'une plan\u00e8te \u00e0 l'autre, \u00eatre un oiseau, voil\u00e0 qui d\u00e9multiplie l'humain en l'homme, voil\u00e0 qui \u00e9largit les possibles aux limites de l'inimaginable.\n\nQui, ici, entend d\u00e9j\u00e0 les sarcasmes de la perdrix? Quelle ou\u00efe suffisamment fine d\u00e9c\u00e8le la pr\u00e9sence de l'animal dont Aristote dit que, mauvais voilier, il ne quitte pas le sol et se roule dans la poussi\u00e8re ? Car les anges disposent de leur revers : des cr\u00e9atures ail\u00e9es n\u00e9gatives tout enti\u00e8res vou\u00e9es au mal, des alli\u00e9es de la perdrix dans leur vol contrari\u00e9 par la chute. Il faut nommer Lucifer, le porteur de lumi\u00e8re, et autres archanges d\u00e9chus qui pr\u00e9f\u00e8rent d\u00e9sob\u00e9ir pour \u00eatre libres, plut\u00f4t que l'inverse. Eux aussi paient leur impi\u00e9t\u00e9, leur volont\u00e9 libertaire, d'une chute qui leur interdit les \u00e9volutions ang\u00e9liques dans l'espace.\n\nFinies les cabrioles c\u00e9lestes, les arabesques stellaires et les ailes dans le dos, car les anges qui optent pour la libert\u00e9 perdent leurs r\u00e9miges au profit d'un phallus. A moins que ce ne soit l'inverse : en laissant pousser le phallus en eux, en acceptant de s'unir \u00e0 des mortelles, comme le rapportent les \u00e9crits intertestamentaires, les anges voient tomber leurs ailes. Voler est donc le propre de qui est sans libido? Sinon le contraire : \u00eatre sans libido apparente \u00e0 l'oiseau ? Vraisemblablement. Et il faut imaginer, au c\u0153ur m\u00eame de la cr\u00e9ature qui glisse dans les airs, une chair minimale, des poches d'air, des creux, des espaces nourris de vide, des interstices satur\u00e9s de rien, un squelette fait d'os aussi vides que les tuyaux d'un syrinx.\n\nLorsqu'ils choisissent de vivre parmi les hommes, les anges d\u00e9chus retrouvent un corps, aussi ne volent-ils plus. L'inverse vaut tout autant : ils ne volent plus, donc ils retrouvent un corps et par l\u00e0 m\u00eame, ils d\u00e9choient. Le corps arrogant, orgueilleux, d\u00e9sob\u00e9issant d'Icare retrouve celui de Lucifer, dot\u00e9 des m\u00eames qualit\u00e9s. Ces chairs-l\u00e0 chutent pour avoir pr\u00e9f\u00e9r\u00e9 un usage maximal de leur libert\u00e9 \u00e0 la soumission aux ordres, ici de prudence grecque, l\u00e0 d'ob\u00e9issance chr\u00e9tienne. Qui pr\u00e9f\u00e9rerait \u00eatre une machine volante sans d\u00e9sir \u00e0 ce qu'il est d\u00e9j\u00e0, \u00e0 savoir une machine d\u00e9sirante sans ailes ? Vaut-il mieux poss\u00e9der un phallus tout en \u00e9tant sans ailes ou des plumes en voilures et un p\u00e9nis envol\u00e9 ? Car l'\u00e9poque b\u00e9nie est r\u00e9volue qui, dans les cultes orgiaques et ithyphalliques, permettait des phallus ail\u00e9s pour magnifier l'\u00e9nergie, le dynamisme et la puissance de toute libido digne de ce nom. Nos temps sacrifient au d\u00e9passement dialectique : plus d' alternative entre machine volante et machine d\u00e9sirante, car l'une permet maintenant l'autre. Des m\u00e9ditations de L\u00e9onard de Vinci \u00e0 la conqu\u00eate effective de l'espace, les hommes se lib\u00e8rent progressivement de la pesanteur en m\u00eame temps qu'ils acc\u00e8dent \u00e0 des vitesses consid\u00e9rables. Le temps et l'espace ainsi ma\u00eetris\u00e9s, le vol devient une occasion m\u00e9taphysique hors pair.\n\nPar-del\u00e0 les anges, d\u00e9chus ou non, le d\u00e9sir de voler cher \u00e0 D\u00e9dale ne cesse de traverser les \u00e9poques. Au deuxi\u00e8me si\u00e8cle de notre \u00e8re, Lucien de Samosate raconte dans son Icarom\u00e9nippe, sous-titr\u00e9 le voyageur a\u00e9rien, combien le vol icarien est pr\u00e9cieux pour obtenir un point de vue sur le monde. En l'occurrence sur les occupations \u00e0 contre-emploi des philosophes : l'\u00e9picurien se b\u00e2frant, le sto\u00efcien se f\u00e2chant, le cynique courant apr\u00e8s les honneurs. Ou sur les mesquines activit\u00e9s humaines : se disputer, plaider, intriguer, aspirer au pouvoir, copuler, prier, travailler. M\u00e9nippe volant dans les airs s'en va sur la lune. Et le d\u00e9sir de fouler le sol aux crat\u00e8res engendre \u00e0 lui seul une immense litt\u00e9rature. Parmi l'abondance des r\u00e9f\u00e9rences, peu rel\u00e8vent directement de l'histoire de la philosophie. Sinon les voyages de Cyrano de Bergerac.\n\nJe ne m'\u00e9tonne pas, d'ailleurs, qu'il faille attendre les libertins du Grand Si\u00e8cle pour faire du vol et du voyage dans la lune un exercice philosophique \u00e9quivalent, \u00e0 tout prendre, \u00e0 celui des penseurs qui mettent en sc\u00e8ne des Persans ou des \u00eeliens imaginaires pour obtenir un meilleur angle de vue que celui des autochtones sur le monde r\u00e9el. Le vol n'est plus symbole de libert\u00e9, d'affranchissement ou d'autonomie, mais occasion d'obtenir une th\u00e9orie du monde \u2013 et l'on se souviendra que l'\u00e9tymologie installe la th\u00e9orie du c\u00f4t\u00e9 de la contemplation, du regard avis\u00e9 et scrutateur. M\u00e9nippe dans l'Antiquit\u00e9 tardive, Cyrano en plein si\u00e8cle cart\u00e9sien, Niels Holgersson \u00e0 l'aube de ce si\u00e8cle, Mafarka la cr\u00e9ation futuriste de Marinetti, voyagent dans l'espace pour fournir des points de vue critiques sur le r\u00e9el. Leurs motivations sont philosophiques : il s'agit d'embrasser d'un seul coup d'\u0153il l'ensemble de ce monde qu'on regarde en satiriste, \u2013 qualit\u00e9 dont ne devrait jamais se d\u00e9partir le philosophe digne de ce nom.\n\nArt\u00e9midore l'\u00e9non\u00e7ait dans sa Cl\u00e9 des songes : \u00abTous les volatiles sont ind\u00e9pendants et n'ont pas de ma\u00eetres \u00bb, et voler manifeste une r\u00e9elle ind\u00e9pendance d'esprit l\u00e0 o\u00f9 le commun des penseurs se satisfait de la terre, du grouillement sur le globe. Cyrano \u00e9crit dans L'Autre Monde qu'il a \u00abjou\u00e9 le personnage d'oiseau\u00bb. Et nombre d'extravagances montrent un individu t\u00e2chant d'\u00eatre plus l\u00e9ger que l'air pour grimper dans l'azur. Mais tous les stratag\u00e8mes mis en \u0153uvre dans le livre proc\u00e8dent du m\u00eame souci d'exploiter cette hypoth\u00e8se scientifique partag\u00e9e alors par tous : la nature a horreur du vide. D\u00e9j\u00e0 l'artifice sollicit\u00e9 par le philosophe du Grand Si\u00e8cle montre une machine qui prend en compte les trouvailles de la physique du moment.\n\nDans ses premi\u00e8res tentatives, Cyrano s'entoure le corps de fioles remplies de ros\u00e9e en attendant que le soleil, rar\u00e9fiant le liquide et r\u00e9alisant le vide d'air, agisse en moteur attractif. Une autre fois, et parce qu'il soigne avec de la moelle de b\u0153uf les blessures occasionn\u00e9es par une chute exp\u00e9rimentale, le philosophe libertin d\u00e9colle du sol pour la raison que la lune, passant pour se nourrir de la moelle des animaux, aspire et attire goul\u00fbment le cosmonaute imp\u00e9trant. Pour finir, c'est une machine v\u00e9ritable, caisson de bois et d\u00f4me en icosa\u00e8dre, vaisseau de cristal et miroirs ardents, qui permet, via le jeu de la condensation et des d\u00e9penses d'\u00e9nergies r\u00e9sultant de la rencontre de l'air chaud et de l'air froid, un authentique envol conclu par un alunissage une fois et un asoleillage, cr\u00e9ons le n\u00e9ologisme, une autre fois.\n\nDe l'aile toute simple, copi\u00e9e sur l'oiseau, aux machines folles c\u00e9l\u00e9br\u00e9es par la plume d'un penseur ou par le trait d'un dessinateur de g\u00e9nie comme L\u00e9onard de Vinci, le d\u00e9sir d'Icare est bien vite oublieux des vols ang\u00e9liques au profit des vols r\u00e9els. On conna\u00eet l'histoire qui, des essais ridicules de prototypes en forme d'oiseaux aux r\u00e9ussites confirm\u00e9es au-dessus de la Manche ou de l'Atlantique, conduisent les hommes du r\u00eave icarien \u00e0 la r\u00e9alit\u00e9 volante. Des moines t\u00e9m\u00e9raires qui se jettent du haut de la tour de leur couvent, avec un empennage rocambolesque, aux essais de Montgolfier confirm\u00e9s par les ascensions de Pil\u00e2tre de Rozier, jusqu'aux a\u00e9roplanes, la ma\u00eetrise de l'air se fait crescendo avec un point culminant le 21 juillet 1969 \u00e0 3 heures 56 minutes et 20 secondes, heure de Paris.\n\nA ce moment pr\u00e9cis, pulv\u00e9risant les d\u00e9sirs de D\u00e9dale, surclassant l'arrogance d'Icare, renvoyant \u00e0 n\u00e9ant l'hypoth\u00e8se d'une Voie lact\u00e9e peupl\u00e9e d'anges, rendant caduques les hypoth\u00e8ses \u00e9pist\u00e9mologiques et m\u00e9caniques de Cyrano, Armstrong pose le pied sur le sol lunaire, non loin de la mer de la Tranquillit\u00e9. A cette heure, entre les bips crachouillants, les images noir et blanc aux d\u00e9finitions incertaines, la po\u00e9tique des mouvements en apesanteur et cette \u00e9trange \u00e9pop\u00e9e racont\u00e9e en temps r\u00e9el sur le petit \u00e9cran, qui a entendu le ricanement de la perdrix, toujours in\u00e9vitablement embusqu\u00e9e ?\n\nCar il faut l'imaginer, l\u00e0 encore, battant des ailes, caquetant comme une vieille poule \u00e0 l'annonce de cette information. Une ascension r\u00e9ussie ? Un sommet atteint ? Un cap franchi dans les trajets vers plus de verticalit\u00e9 ? Oui, certes. Mais aussi, et comme toujours en pareil cas, une chute annonc\u00e9e. Laquelle ? Moins spectaculaire que celle d'Icare, moins manifeste que celle des archanges d\u00e9chus, r\u00e9volt\u00e9s, elle est pourtant radicale, fondamentale, pareille \u00e0 celle qui, si l'on en croit Alexandre Koyr\u00e9, suivit le passage du monde clos \u00e0 l'univers infini lors des p\u00e9riodes contemporaines de la d\u00e9monstration de l'h\u00e9liocentrisme des physiciens d'alors, Copernic, Kepler ou Galil\u00e9e, contre le g\u00e9ocentrisme du bon sens populaire, de l'Eglise et des coutumes.\n\nQuitter la terre, voler au-dessus d'elle, puis malgr\u00e9 elle, contre elle, en la laissant derri\u00e8re soi, comme M\u00e9nippe, Cyrano, les h\u00e9ros de Jules Verne, Nodier, Restif de la Bretonne ou Mafarka, c'est une chose. Une autre est de s'installer sur une plan\u00e8te \u00e9trang\u00e8re et de regarder la terre comme une alt\u00e9rit\u00e9 radicale, une inqui\u00e9tante \u00e9tranget\u00e9, une particule en suspension, une poussi\u00e8re parmi des milliers de galaxies. Vue des monts Riph\u00e9es, du golfe Torride ou de la mer des Vapeurs, notre plan\u00e8te dut para\u00eetre bien petite, sinon ridicule \u2013 et le destin des hommes qui grouillent sur elle, singuli\u00e8rement microscopique, voire d\u00e9risoire. Si Koyr\u00e9 a raison, et que des r\u00e9volutions mentales se trouvent induites par le passage de Ptol\u00e9m\u00e9e \u00e0 Copernic, alors il faut affirmer que cette journ\u00e9e de juillet 1969 fonctionne dans l'esprit des nations industrialis\u00e9es comme une fracture m\u00e9taphysique radicale. L'oiseau tient ses promesses plus loin et mieux encore qu'on pourrait l'imaginer.\n\nQuelles cons\u00e9quences pour cette r\u00e9volution silencieuse ? L'acc\u00e8s pour tout un chacun \u00e0 un monde sans points cardinaux, sans bornes, apparemment infini, o\u00f9 la relativit\u00e9 triomphe en tout apr\u00e8s avoir seulement r\u00e9gn\u00e9 sur le terrain de la physique et des sciences dures. D'un seul coup, on acc\u00e8de \u00e0 l'av\u00e8nement d'un r\u00e9el plus ind\u00e9fini qu'infini, plus incompr\u00e9hensible que d\u00e9termin\u00e9. En posant le pied sur la lune, Armstrong annonce un petit pas pour l'homme, un grand pour l'humanit\u00e9 : mesurait-il qu'il renvoyait la m\u00e9taphysique, la pens\u00e9e, l'id\u00e9ologie dans des ab\u00eemes que depuis, rien ni personne n'a r\u00e9ussi \u00e0 \u00e9clairer de quelque lumi\u00e8re que ce soit ? Pour parvenir \u00e0 cet autre monde, qui relativise d\u00e9finitivement le n\u00f4tre, il faut jeter dans des brasiers tout ce qui constitue l'id\u00e9ologie depuis des si\u00e8cles : la religion, la cosmogonie, la philosophie, en un mot, le sens.\n\nCe que d\u00e9couvrent les cosmonautes, comme on ouvre une bo\u00eete de Pandore ou comme on d\u00e9voile une oeuvre cach\u00e9e derri\u00e8re un tissu, c'est le nihilisme dans lequel, depuis, notre \u00e9poque croupit. Le rire de la perdrix se fait encore entendre. Pour tous ceux qui croient aux d\u00e9sirs prom\u00e9th\u00e9ens de D\u00e9dale, il est temps de tordre le cou du volatile et de r\u00e9investir intellectuellement les immenses zones de libert\u00e9 laiss\u00e9es vierges par la d\u00e9couverte du plus r\u00e9cent des nouveaux mondes. Alors la perdrix ne sera plus qu'un antique souvenir et l'heure sera aux alcyons n\u00e9cessaires.\n4\n\nVOIR L'IC\u00d4NE PA\u00cfENNE\n\nL'inventeur de la feuille de vigne \u00e0 des fins pudiques fut un formidable acc\u00e9l\u00e9rateur de pornographie auquel on doit \u2013 l'aurait-il imagin\u00e9 ? \u2013 une incroyable avanc\u00e9e sur le terrain qu'il entendait combattre. Car il faut bien en convenir, rien ne s\u00e9duit plus que montrer ce qui est cach\u00e9, rien n'est plus sinistre \u00e0 regarder que ce qui s'exhibe commun\u00e9ment. D'o\u00f9 l'int\u00e9r\u00eat majeur des d\u00e9voilements rares de ce qui est devenu obsc\u00e8ne aux yeux des moralistes de tous bords, lecteurs de Bible et de Coran, de Marx et de Freud, et autres d\u00e9c\u00e9r\u00e9br\u00e9s qui font le nihilisme de notre \u00e9poque.\n\nBraguetteurs de la Renaissance, inventeurs de machines en fer-blanc pour couvrir les parties dites honteuses des sculptures et statues aux phallus flaccides et p\u00e9trifi\u00e9s, allumeurs de b\u00fbchers et promoteurs de fatwas, psychiatres convertissant aux valeurs de la famille, du couple, de la monogamie, de la fid\u00e9lit\u00e9 et de la procr\u00e9ation, tous travaillent dans le m\u00eame sens : \u00e9teindre tout d\u00e9sir socialement inacceptable, r\u00e9duire tout corps indocile aux injonctions castratrices du temps dans lequel il \u00e9volue. L'\u00e9poque sacrifie ardemment \u00e0 l'hygi\u00e8ne, au calibre, \u00e0 la r\u00e9duction du corps \u00e0 une peau de chagrin.\n\nD'aucuns veulent au contraire \u00e9tendre le corps aux dimensions sup\u00e9rieures, en faire d'immenses voilures pour des vaisseaux nouveaux. La chair, la viande, les formes, l'\u00e9nergie libidinale, les forces, voil\u00e0 qui les sollicite avec excitation, jusques et y compris dans l'exc\u00e8s n\u00e9cessaire pour r\u00e9pondre aux tyrannies du moment. Et, sur le terrain de l'art contemporain, je ne crois aux esth\u00e9tiques engag\u00e9es que lorsqu'elles impliquent le corps dans le dessein de nouvelles combinatoires \u00e9thiques, donc, au sens primitif, politiques.\n\nDans ce cercle des acteurs de l'autre corps, je voudrais m'attarder sur le travail photographique radical d'Ariane Lopez-Huici. Aux c\u00f4t\u00e9s de celui de Serrano qui saisit la mort et la morgue, de Doningan Cumming qui fige la vieillesse et ses ravages, de Cindy Sherman qui radicalise la mise en sc\u00e8ne et l'existence de l'\u00eatre comme repr\u00e9sentation pour autrui, Ariane Lopez-Huici montre la chair qu'en des cat\u00e9gories actuelles il faut appeler impudique. Chairs onanistes, ob\u00e8ses, m\u00e9tiss\u00e9es, ludiques, toutes ont en commun d'\u00eatre habituellement cach\u00e9es, renvoy\u00e9es dans les mondes nocturnes du priv\u00e9.\n\nQui ne s'est masturb\u00e9 ? Qui le dirait ? Qui le montrerait ? Et comment le dire, le montrer, le raconter dans l'\u00e9vidence d'une simplicit\u00e9 qui, sollicitant la crudit\u00e9 impudique de ce qui appara\u00eet d'abord, d\u00e9bouche sur une autre pudeur, plus grande, plus exigeante, plus radicale? Qui n'a d\u00e9sir\u00e9 le grain d'une peau color\u00e9e, africaine, ou celle, incestueuse, d'un parent? Quel homme n'a souhait\u00e9 la chair abondante, g\u00e9n\u00e9reuse, d'une poitrine, d'un ventre, la largeur d'un bassin ou le dos vaste et les fesses lourdes d'une femme qui le renvoie ainsi \u00e0 l'enfance, aux limbes de son existence et de l'humanit\u00e9 ? Dans l'intimit\u00e9 des r\u00eaves, des d\u00e9sirs libres ou des fantasmes, ces images ont \u00e9t\u00e9 fugaces ou puissantes, r\u00e9currentes ou rares, mais toujours vivantes.\n\nEt si l'art ne trouvait sa justification que dans la sublimation, au sens freudien, de ces pulsions naturelles et primitives ? Si des premiers temps \u00e0 aujourd'hui le dessein des artistes n'\u00e9tait, \u00e0 leur corps plus ou moins d\u00e9fendant, de d\u00e9voiler un peu de ces images qui assaillent les \u00e2mes les plus dissemblables ? Si la production esth\u00e9tique n'avait cess\u00e9 d'\u00eatre une catharsis, une purification, au sens philosophique, de ces images qui assaillent les nuit\u00e9es de l'\u00e2me?\n\nPhallus et vagins pari\u00e9taux, d\u00e9jeuners sur l'herbe, crucifixions, mythologies de nudit\u00e9s aux taureaux, aux pluies d'or, aux cygnes libidineux, corridas, Vierges \u00e0 l'enfant, martyrs, bains turcs, couple primitif, origine du monde des corps, racontent le d\u00e9sir et la souffrance, le plaisir et l'\u00e9nergie, le luxe et la volupt\u00e9, la jubilation et la force. D'une fois l'autre, des viandes surgissent, form\u00e9es, sculpt\u00e9es, contenues pour exprimer la n\u00e9cessit\u00e9 apollinienne dans les crit\u00e8res esth\u00e9tiques et \u00e9thiques du moment.\n\nPour autant, l'exhib\u00e9 est codifi\u00e9, justifi\u00e9 par une occasion de culture et de civilisation : la mythologie, l'histoire, les textes sacr\u00e9s. Restent un impens\u00e9 de l'histoire de l'art, une zone d'ombre, un silence. Car on cherchera un dieu Onan dans l'action, ou m\u00eame une chair tellement g\u00e9n\u00e9reuse que Boucher et Rubens para\u00eetront \u00e9tiques, on ne trouvera rien en dehors des cartons qui circulent sous le manteau. Rien qui c\u00e9l\u00e8bre la masturbation comme une annonce faite au d\u00e9sir, une crucifixion du plaisir, une c\u00e8ne de chair ; rien qui \u00e9l\u00e8ve le corps monstrueux, anormal, atypique, mais de mani\u00e8re outr\u00e9e, au rang d'objet pour une passion. Pas de turgescence, pas d'\u00e9jaculation, pas de graisse ni de cellulite accumul\u00e9es comme pour une richesse vertueuse appelant un jour d\u00e9pense ou consumation h\u00e9donistes.\n\nDans ce d\u00e9sert des corps intimes, les photographies d'Ariane Lopez-Huici agissent en catharsis et g\u00e9n\u00e8rent dans l'instant l'\u00e9trange sentiment d'assister \u00e0 la d\u00e9charge d'un \u00e9clair qui illumine des v\u00e9rit\u00e9s violentes. L'\u0153il qui pr\u00e9side \u00e0 ces traces de foudre et les th\u00e9\u00e2tralise en clich\u00e9s ne s'encombre pas de Praxit\u00e8le ou des formes canoniques de la statuaire classique. La proportion id\u00e9ale, le nombre et le chiffre qui commandent l'agencement des volumes, des forces, des plans, voil\u00e0 qui n'est pas d'actualit\u00e9. Les corps montr\u00e9s s'installent aux antipodes des corps apolliniens parce qu'ils c\u00e9l\u00e8brent sans vergogne la v\u00e9rit\u00e9 nue et crue, naturelle.\n\nD\u00e9bordantes, expansives, conqu\u00e9rantes, sinon arrogantes, les chairs expos\u00e9es, exhib\u00e9es, sont antiplatoniciennes et f\u00eatent un mat\u00e9rialisme radical. Elles ne tiennent pas leur existence, ni leur l\u00e9gitimit\u00e9, d'une id\u00e9e intelligible qui leur pr\u00e9existerait et les d\u00e9passerait dans la perfection, le qualitatif. Leur existence se confond \u00e0 leur essence. Rien ne trahit l'hypoth\u00e8se d'une d\u00e9perdition d'excellence dans le processus d'incarnation descendant du ciel id\u00e9al vers la mati\u00e8re peccamineuse. La chair montr\u00e9e n'est pas p\u00e9cheresse car elle jubile par-del\u00e0 le bien et le mal, au-del\u00e0 des cat\u00e9gories perverties par ce que Nietzsche appelait la moraline.\n\nLes corps f\u00eat\u00e9s par Ariane Lopez-Huici ne sont pas polic\u00e9s. Tout autant, ils ignorent la police : celle des m\u0153urs, des religions, des philosophies qui communient dans l'id\u00e9al asc\u00e9tique, la haine de soi, la n\u00e9gligence et le refus de la chair. De m\u00eame, ils ignorent cette autre autorit\u00e9 id\u00e9ologique contemporaine du politiquement correct. Les femmes, les gros, les Noirs, les incestueuses, les homosexuelles sont exacerb\u00e9s sans les pr\u00e9cautions d'usage famili\u00e8res aux verbeux et aux donneurs de le\u00e7ons. Pas de religion, de l'ath\u00e9isme ; pas de spiritualisme, du mat\u00e9rialisme; pas de transcendance, de l'immanence : la photographie d'Ariane Lopez-Huici est rebelle, combattante et r\u00e9sistante. Radicale en tout.\n\nUn homme se masturbe, deux s\u0153urs se touchent, s'effleurent, se m\u00ealent, une femme ob\u00e8se pose, s'\u00e9tire, se tord, se plie, s'expose, une femme blanche s'offre au corps de l'homme noir, et vice versa, l'artiste elle-m\u00eame en sc\u00e8ne, dansant, appelant, sollicitant et obtenant la transe sous l'objectif de son appareil photo : chaque fois se stylisent des stations pour une Passion H\u00e9doniste \u2013 comme il y eut une Passion Christique. Des moments pour un mouvement, des instants fig\u00e9s, fix\u00e9s, pour la narration d'une histoire nouvelle des corps \u2013 ou pour une histoire des corps nouveaux.\n\nIl semble \u00e9vident, en m\u00e9ditant ce travail d'artiste, qu'un corps c'est d'abord la cristallisation d'une mati\u00e8re \u2013 la viande \u2013 dans des formes socialement acceptables ou d\u00e9fendables. Apr\u00e8s le regard, il s'agit d'inventer de nouvelles chairs enfin dissoci\u00e9es des conditions sociales et culturelles qui les permettent. L'esth\u00e9tique propos\u00e9e par Ariane Lopez-Huici n'est pas c\u00e9r\u00e9brale, intellectuelle ou conceptuelle mais sensuelle, h\u00e9doniste et factuelle. Ses propositions fournissent des invitations pour chacun \u00e0 se r\u00e9concilier avec soi-m\u00eame au d\u00e9triment du social, malgr\u00e9 lui, contre lui. Elles proposent une chair sans le regard moral, donc sans la mythologie des m\u0153urs de l'\u00e9poque.\n\nLe corps c\u00e9l\u00e9br\u00e9 par l'artiste ne co\u00efncide pas avec le corps d\u00e9sir\u00e9, esp\u00e9r\u00e9, voulu dans une tension \u00e9puisante et g\u00e9n\u00e9ratrice de douleurs, de malheurs, de souffrances. Au contraire. Il est le corps r\u00e9el, travaill\u00e9 par la cellulite, le gras, la mati\u00e8re, le sperme, la libido, l'\u00e9nergie, le d\u00e9sir, les pigments, les muscles. Corps ob\u00e8se ou masturb\u00e9, pr\u00eat\u00e9 \u00e0 sa s\u0153ur ou donn\u00e9 \u00e0 sa ma\u00eetresse, irradi\u00e9 par la danse ou c\u00e9l\u00e9br\u00e9 par la transe, chaque fois se manifeste une \u00e9piphanie des chairs lib\u00e9r\u00e9es des carcans, des arch\u00e9types du moment esth\u00e9tique que sont pour nous, le march\u00e9 publicitaire des corps lisses, sans asp\u00e9rit\u00e9, sans surcharge pond\u00e9rale, sans flux, sans mati\u00e8re, presque sans organes.\n\nFi des constructions virtuelles qui supposent des formes pures, puisque n'existant que sous la trame fictive des pixels ; fi des peaux sous lesquelles rien n'existe, puisqu'elles sont des palimpsestes pour les offres et les demandes toujours transfigur\u00e9es du march\u00e9 plan\u00e9taire des d\u00e9sirs hygi\u00e9niques et des plaisirs prot\u00e9g\u00e9s; fi des relations \u00e0 autrui m\u00e9diatis\u00e9es par un regard charg\u00e9 du poids d'une civilisation travaill\u00e9e par les ic\u00f4nes publicitaires : les corps affich\u00e9s revendiquent une opposition radicale \u00e0 ce que produit le social. Ils sont ceux de l'\u00e9ternelle intimit\u00e9, ceux de la pudeur red\u00e9finie.\n\nAu-dessus des \u00e9poques, comme un arc tendu entre les peintures pari\u00e9tales et la modernit\u00e9 contemporaine, ignorant les scories grecques, romaines et chr\u00e9tiennes d\u00e9pos\u00e9es par les vents glac\u00e9s de l'id\u00e9al asc\u00e9tique, Ariane Lopez-Huici aspire \u00e0 une \u00e9thique solaire, l'autre face des puissances nocturnes qui depuis trop longtemps ravagent les corps. D'une certaine mani\u00e8re, entre f\u00eates des fous et processions ithyphalliques, non loin des cort\u00e8ges bachiques et des saturnales empourpr\u00e9es, ses images expriment la quintessence de Dionysos : saper autant que possible les injonctions mortif\u00e8res d'Apollon.\n\nEn \u0153uvrant du c\u00f4t\u00e9 de ces zones d'ombre avec le dessein de les dissiper, de les trouer avec la lumi\u00e8re esth\u00e9tique, Ariane Lopez-Huici reprend \u00e0 son compte l'invitation nietzsch\u00e9enne enjoignant \u00e0 l'invention de nouvelles possibilit\u00e9s d'existence. Son cheminement suppose une dialectique : tant que demeure, lourd, le voile de l'ancienne pudeur, tant que persistent les brouillards antiques de la moraline, ses photographies choquent et d\u00e9stabilisent d'autant le regardeur qu'il est plus ou moins victime de l'id\u00e9al asc\u00e9tique.\n\nLa capacit\u00e9 \u00e0 supporter, \u00e0 voir et regarder ces \u0153uvres s\u00e9lectionne et distingue, au degr\u00e9 z\u00e9ro \u2013 ceux qui n'en supportent ni la vue ni l'id\u00e9e \u2013 celui qu'habite avec trouble et douleur le malaise et l'obsession d'un corps id\u00e9al. Au sommet \u2013 ceux pour qui cette exhibition offre une occasion de m\u00e9ditation \u2013 se rencontreront les m\u00e9taphysiciens dont d\u00e9pend la formulation d'une \u00e9thique de philosophe-artiste plus que jamais n\u00e9cessaire en nos temps lest\u00e9s par les chapes de plomb moralisatrices. Le regard errant, apr\u00e8s avoir tout vu de ce que montre Ariane Lopez-Huici, on n'\u00e9vite pas de s'installer, corps et \u00e2me, quelque part entre ces deux infinis dont t\u00e9moignent ses ic\u00f4nes pa\u00efennes.\n5\n\nSEPT P\u00c9CH\u00c9S CAPITEUX\n\nJ'avais vu un lapin jouer du cornet \u00e0 bouquin, des chiens capara\u00e7onn\u00e9s comme des tatous d\u00e9chirer le corps d'une charogne, une lame d'acier \u00e9merger d'un couple d'oreilles, une truie coiff\u00e9e telle une nonne, un oiseau royal ingurgiter un homme dont les vents donnaient naissance \u00e0 des hirondelles. Je n'avais pas manqu\u00e9 le corps d'un chevalier d\u00e9pec\u00e9 par des chiens, un d\u00e9mon jouer de son long nez comme d'une fl\u00fbte, un poisson bip\u00e8de tracter une charrette de foin en compagnie d'une horde bestiale travestie avec des costumes d'hommes. Je m'\u00e9tais attard\u00e9 sur des potences o\u00f9 guettaient des loups, des gibets o\u00f9 pendait, loqueteuse, la peau des hommes, des brasiers qui r\u00e9duisaient des villes \u00e0 la poussi\u00e8re. Je me souvenais d'un ornithorynque mangeant une fraise, d'un rat dans un tube de verre, d'une moule o\u00f9 reposait le corps d'un homme, puis de porcs et d'ours chevauch\u00e9s.\n\nIci, au Prado, comme l\u00e0, au mus\u00e9e national de Lisbonne, ou ailleurs \u00e0 Vienne, Copenhague ou S\u00e2o Paulo, J\u00e9r\u00f4me Bosch m'avait coup\u00e9 le souffle. Son monde sublime se refuse encore, et vraisemblablement pour longtemps, \u00e0 toute r\u00e9duction intellectuelle. D'o\u00f9 le b\u00e9n\u00e9fice et l'int\u00e9r\u00eat consid\u00e9rables de ces images expressives en de\u00e7\u00e0 et au-del\u00e0 de ce que vraisemblablement elles signifient : la redoutable efficacit\u00e9 des songes enfant\u00e9s par l'\u00e2me pour figurer les tentations, les damnations, les plaisirs, la cr\u00e9ation, la folie, ou les implacables cons\u00e9quences du p\u00e9ch\u00e9 originel.\n\nAussi, pour illustrer les sept p\u00e9ch\u00e9s capitaux, Bosch choisit un dessus de table circulaire, comme pour exprimer l'\u00e9ternel retour des choses, l'incessante ronde des passions humaines, des vices r\u00e9p\u00e9t\u00e9s et des pratiques qui en participent : l'\u00e9p\u00e9e d'un buveur qui va bient\u00f4t trancher le corps d'une \u00e9pouse peut-\u00eatre infid\u00e8le; la complaisance pour elle-m\u00eame d'une femme devant le miroir qu'un singe diabolique lui tend; l'avachissement d'un riche en pr\u00e9sence du fou qui fait sa cour; l'endormissement d'un paresseux comme un chien face \u00e0 l'\u00e2tre; la gloutonnerie triviale d'un mangeur de cochonnailles; le rendeur de justice corrompu par la bourse d'un plus v\u00e9nal que lui ; et toutes les autres postures sociales qui trahissent l'\u00e9ternel empire des passions, des d\u00e9sirs, des pulsions, des instincts sur l'homme du commun.\n\nBien s\u00fbr, pour les besoins apolog\u00e9tiques, Bosch installe quatre figurations des \u00e9tapes humaines comme des points cardinaux : la mort, le Jugement dernier, l'enfer et le paradis. En enfer, l\u00e0 o\u00f9 il excelle, le peintre ramasse, en une presque vignette, toutes les mani\u00e8res communes \u00e0 l'\u00e9poque de stigmatiser les p\u00e9ch\u00e9s : le gourmand mange des serpents et des l\u00e9zards, le luxurieux d\u00e9couvre le sexe de sa partenaire couvert par un crapaud, les autres p\u00e9cheurs sont \u00e9mascul\u00e9s, br\u00fbl\u00e9s, tourment\u00e9s, d\u00e9chiquet\u00e9s, tortur\u00e9s. On conna\u00eet la logique, elle anime Dante en litt\u00e9rature, Thomas d'Aquin en philosophie, Paschal de l'Estochart en musique et tous les autres qui s'installent sur le terrain de l'\u00e9dification religieuse : les p\u00e9ch\u00e9s conduisent en enfer, les vertus au paradis.\n\nEt si c'\u00e9tait l'inverse ? Si le Jardin des D\u00e9lices triomphait en lieu o\u00f9 s'expriment les prosp\u00e9rit\u00e9s du vice en m\u00eame temps que les malheurs de la vertu ? Si Sade avait raison de pr\u00e9f\u00e9rer Juliette la perverse \u00e0 Justine la trop sage? Qu'on imagine, en effet, un monde duquel soudain dispara\u00eetrait tout ce qui est stigmatis\u00e9 comme vicieux, notamment par l'entremise des sept p\u00e9ch\u00e9s capitaux. Plus de gourmandise ni de luxure, plus d'envie ni de col\u00e8re, plus d'orgueil, d'avarice ni de paresse. Alors, tout serait d'une exceptionnelle fadeur, d'un intraitable ennui, d'un \u00e9pouvantable gris. Morne la vie quotidienne, sinistre l'existence, aussi d\u00e9sesp\u00e9rante qu'une plan\u00e8te o\u00f9 aurait triomph\u00e9 la peste ou le chol\u00e9ra.\n\nTous les thurif\u00e9raires de la vertu devraient y songer. Tous les rab\u00e2cheurs de morale, les vendeurs d'arri\u00e8re-mondes, les vertueux contamin\u00e9s par la moraline seraient bien inspir\u00e9s d'y r\u00e9fl\u00e9chir \u00e0 deux fois. Apr\u00e8s quoi les fabricants de petits trait\u00e9s des grandes vertus feraient mieux de concocter des grands trait\u00e9s des petites vertus, les seules qui donnent du piment et du sens \u00e0 l'existence. Car quoi, que veut-on? Une vie quotidienne dans laquelle on n'ait plus de plaisir \u00e0 boire et \u00e0 manger, \u00e0 d\u00e9sirer l'ivresse, \u00e0 pratiquer l'\u00e9bri\u00e9t\u00e9, sinon l'ivret\u00e9? Une table sur laquelle on ne puisse plus voir que le sinistre verre d'eau, le cro\u00fbton de pain et de petit fromage rassis du vieil Epicure, commensal d'un Nazar\u00e9en amateur d'eau plate et de poissons parfois multipli\u00e9s par ses soins ? Plus de bons vins ni de tables joyeuses, plus de cuisines raffin\u00e9es, de festins et banquets de Trimalcion ? Laissons cela aux moines et aux philosophes devenus leurs aides de camp.\n\nLes tristes sires vaticanesques et les contempteurs de la tyrannie du plaisir conspuent ce qui montre trop la chair en l'homme, ce qui rappelle sa nature exclusivement mat\u00e9rielle, naturelle, instinctive, pulsionnelle. Car le corps a des besoins que les religions s'\u00e9vertuent toutes \u00e0 n\u00e9gliger, \u00e0 ne pas satisfaire, comme s'il suffisait de ne pas vouloir entendre une demande pour qu'elle cesse sur-le-champ de se formuler. Ne pas manger ni boire, sauf pour permettre la survie du corps, puisque cela demeure le seul moyen de rendre encore possible l'\u00e2me et que seule cette derni\u00e8re importe : voici la doctrine officielle des promoteurs de l'id\u00e9al asc\u00e9tique. De m\u00eame, pour les usages de la chair, du corps sensuel et qu\u00eateur de plaisirs. Car, dans notre civilisation jud\u00e9o-chr\u00e9tienne, il faut soumettre le besoin sexuel \u00e0 la n\u00e9cessit\u00e9 g\u00e9n\u00e9sique de perp\u00e9trer l'esp\u00e8ce et de faire durer les g\u00e9n\u00e9rations afin que des millions d'hommes puissent encore, comme en un immense holocauste, continuer d'expier le p\u00e9ch\u00e9 originel. Souffrir, travailler, peiner, autant de destins auxquels il faudrait se soumettre comme si devoir mourir un jour ne suffisait pas.\n\nL'Eglise condamne la gourmandise parce qu'elle se r\u00e9v\u00e8le prop\u00e9deutique aux plaisirs sexuels. Chacun savait, bien avant l'\u00e9l\u00e9gante et souriante d\u00e9monstration de Brillat-Savarin, les plaisirs de la table avant-coureurs de ceux de l'alc\u00f4ve. Et si Grimod de La Reyni\u00e8re vante les avantages de la bonne ch\u00e8re sur les femmes, c'est, au-del\u00e0 de l'exercice de style, parce qu'il n'ignorait pas que la poularde farcie aux truffes qu'on accompagne d'un superbe champagne en compagnie galante finit toujours par faire le lit de la luxure. Qu'on se le dise...\n\nTrop manger, trop boire, aimer les d\u00e9lices de la chair permet l'empire du corps sur la raison, la d\u00e9mission des organes de la soumission sociale, le syst\u00e8me nerveux, au profit de ceux de la subjectivit\u00e9 radicale \u2013 le reste. Comment s'y prendre pour ne pas p\u00e9cher, ne pas succomber au vice de la gourmandise ? En multipliant les mortifications, les mac\u00e9rations, les m\u00e9ditations sur les suites funestes associ\u00e9es \u00e0 la dilection pour les plaisirs de bouche. Avec le renoncement aux \u00e9nergies constitutives de son corps, le sujet contribue \u00e0 \u00e9difier le social, \u00e0 structurer et faire fonctionner les machines de la civilisation. D'o\u00f9 les invectives des moralisateurs \u00e0 l'endroit de la sexualit\u00e9.\n\nCharg\u00e9s d'\u00e9nergie captive, les corps annoncent des dangers pour le social qu'ils mettent sans cesse en p\u00e9ril. La famille, le mariage, la fid\u00e9lit\u00e9, la monogamie sont les trouvailles de la civilisation pour capter, d\u00e9tourner et an\u00e9antir les puissances fabuleuses de la libido. La temp\u00e9rance pour les plus ardents, la continence pour les plus dou\u00e9s, la virginit\u00e9 pour les plus saints, voil\u00e0 les solutions possibles au probl\u00e8me de la sexualit\u00e9. C'est du moins ce que disent les promoteurs de grandes vertus, les ennemis du vice. Et pour bien montrer que l'Eglise se nourrit du sacrifice des individus et vit des formes sociales qu'elle g\u00e9n\u00e8re \u00e0 partir de ces holocaustes quotidiens et permanents, les th\u00e9ologiens sp\u00e9cialistes en cat\u00e9chisme expriment sans vergogne les limites dans lesquelles la sexualit\u00e9 semble possible : copuler est pensable et chaste quand l'accouplement vise exclusivement la procr\u00e9ation et que les corps ne consentent ni au d\u00e9sir ni au plaisir susceptibles d'appara\u00eetre en cours de route. Est chaste celui qui, dans l'\u0153uvre de chair, soumet son geste \u00e0 la seule n\u00e9cessit\u00e9 de continuer \u00e0 pourvoir le troupeau des expiateurs de la faute originaire.\n\nD\u00e9j\u00e0 les moralistes d\u00e9barrassent les tables des assiettes et des bouteilles, des verres et des plats. Voil\u00e0 qu'ils installent des lits s\u00e9par\u00e9s dans les chambres, fourbissent avec ardeur les ceintures de chastet\u00e9, enfilent les chemises perc\u00e9es et calculent qu'en s'y prenant bien, ils comptabiliseront une douzaine de relations sexuelles dans leur existence \u2013 soit autant que d'enfants con\u00e7us. Car il faut signaler que, pour ces deux p\u00e9ch\u00e9s capitaux, l'insupportable c'est le corps qui jouit d'\u00eatre tout simplement ce qu'il est. L'anath\u00e8me lui-m\u00eame concerne l'essence de la chair, sa nature, sa pure et simple existence. Ne pas manger, ne pas boire, ne pas toucher le corps de l'autre, voil\u00e0 un id\u00e9al pour l'usage des chairs. Autant dire qu'on ne signifie pas mieux la n\u00e9cessit\u00e9 d'\u00eatre pareil \u00e0 un cadavre.\n\nPas de corps, pas de chair, pas d'usage h\u00e9doniste de soi. Tout doit \u00eatre r\u00e9alis\u00e9 dans l'optique d'une subsomption de l'\u00e9nergie individuelle sous le pavillon des grandes formes sociales : ici le travail qui permet la prosp\u00e9rit\u00e9 de la patrie, l\u00e0 une paternit\u00e9 qui rend possible la famille, tout le temps, une sensualit\u00e9 confin\u00e9e et sublim\u00e9e dans les \u00e9difices sociaux. La sexualit\u00e9 libre, la chair entrav\u00e9e par les vapeurs alcooleuses, voil\u00e0 qui interdit toute fixit\u00e9 et immobilit\u00e9 sociale. Rien n'est plus antisocial que l'amour de soi, rien n'est plus social que la haine de soi. L'h\u00e9donisme affiche radicalement un individualisme, une option philosophique qui avance l'existence et l'int\u00e9r\u00eat des seuls individus, au d\u00e9triment de toute relation, secondaire et soumise \u00e0 une perp\u00e9tuelle r\u00e9\u00e9valuation. Le r\u00e9seau vaut moins que les monades qui le constituent. Le chr\u00e9tien, amateur d'id\u00e9al asc\u00e9tique, veut, quant \u00e0 lui, le corps disponible pour les travaux et les jours, les \u0153uvres sociales.\n\nLa haine de soi, en m\u00e9pris d'une partie de soi pour une autre partie de soi, est aussi sur\u00e9valuation d'autrui, quel qu'il soit. Son existence est toujours entendue comme sup\u00e9rieure \u00e0 la mienne, plus importante. L'autre comme donneur de sens \u00e0 ma vie, voil\u00e0 un nouveau leitmotiv des brandisseurs de p\u00e9ch\u00e9s capitaux. Autrui vaut plus que moi qui ne suis que dans, par, et pour lui. Aimer son prochain, lui pardonner ses fautes, pratiquer la charit\u00e9 dispense de consid\u00e9rer toute intersubjectivit\u00e9 l\u00e0 pourtant o\u00f9 elle se trouve : sur un perp\u00e9tuel terrain de violence. Inutile d'inviter \u00e0 aimer son prochain s'il est aimable. Rien que de tr\u00e8s normal. En revanche, pour ceux qui ne le sont pas ? les marchands du temple, les tortionnaires, les dictateurs, les tyrans? Puisque la col\u00e8re est un p\u00e9ch\u00e9 et qu'il faut pratiquer la temp\u00e9rance, l'amour, la douceur, comment ne pas imaginer qu'ainsi vertueux on va directement \u00e0 l'abattoir?\n\nA quoi ressemblerait un monde sans col\u00e8re ? A un chantier plus sanglant encore qu'au quotidien, \u00e0 une jungle o\u00f9 les plus forts ne feraient qu'une bouch\u00e9e des plus faibles, car la b\u00eate qui sort les dents en l'homme, m\u00eame chez le plus d\u00e9muni, induit parfois l'occasion de reculades, y compris chez les grands fauves. Le temps de diff\u00e9rer d'une seconde le coup de patte fatal peut suffire pour \u00e9quilibrer, dans la guerre de tous contre tous, les pertes dans le camp des forts autant que dans celui des faibles. La col\u00e8re est pensable comme d\u00e9monstration d'une force \u00e9vitant la violence. Le tribun et le pamphl\u00e9taire, le harangueur et le revendicateur, le tyrannicide et le partisan illustrent ces figures de la saine col\u00e8re. La tol\u00e9rance et la politesse sont utiles quand elles sont n\u00e9cessaires, autant dire tant que la col\u00e8re et la force ne sont pas vitales pour assurer la survie.\n\nSi le pr\u00e9dicateur interdit la col\u00e8re, il veut aussi bannir l'envie qu'il int\u00e8gre dans les modalit\u00e9s destructrices du rapport \u00e0 autrui. Envier, c'est viser la place de l'autre, son statut, son avoir, son pouvoir, son \u00eatre. D'une certaine mani\u00e8re, c'est aussi se d\u00e9tester que de vouloir \u00eatre un autre \u2013 voil\u00e0 qui devrait plaire au christianisme : une occasion suppl\u00e9mentaire de ne pas s'aimer, une aubaine ! Pourtant, ce qui motive la condamnation de l'envie, c'est que le d\u00e9sir qui la caract\u00e9rise a pour objet une maximalisation de soi, une augmentation de soi. Envier suppose vouloir plus, davantage, revendiquer plus et mieux pour une subjectivit\u00e9 croupissant en de\u00e7\u00e0 de ce qu'elle imagine d'elle.\n\nDe cela m\u00eame qui justifie la condamnation de l'envie par les pr\u00eatres, je conclus \u00e0 la n\u00e9cessit\u00e9 de la ch\u00e9rir : elle fournit le moteur actif et architectonique de toute sculpture de soi, de toute expansion de soi, de toute formulation et formation d'identit\u00e9. Sans elle, pas de d\u00e9sir mim\u00e9tique, cher au c\u0153ur de Ren\u00e9 Girard, pas de mod\u00e8le, d'id\u00e9al du moi, d'objectif \u00e0 viser pour bander les forces qui permettent d'atteindre un but autorisant une vitalit\u00e9 surabondante du moi. Sans envie, pas de Schubert se structurant selon Beethoven, pas de Sartre se r\u00e9alisant par rapport \u00e0 Flaubert, pas de Michel-Ange se d\u00e9ployant en regard de Vinci. Tout projet originel, en mati\u00e8re de psychanalyse existentielle, suppose des fins qu'on approche seulement gr\u00e2ce \u00e0 l'envie. Un monde dans lequel on n'envierait rien ni personne, aucun autre \u00e9tat, aucun autre statut, aucune autre existence, serait aussi d\u00e9sert qu'une plan\u00e8te ravag\u00e9e par la collision avec une m\u00e9t\u00e9orite. L'envie suppose le mod\u00e8le sans lequel rien de vertical n'est possible.\n\nPour la m\u00eame raison, l'orgueil que stigmatisent les rh\u00e9teurs en chaire agit en formidable agent de projection de soi vers le haut. L'humilit\u00e9, que vantent ceux-l\u00e0 m\u00eames en antidote, invite \u00e0 ramper, \u00e0 l\u00e9cher le sol, \u00e0 se contenter des reptations auxquelles nous condamne habituellement la n\u00e9cessit\u00e9 m\u00e9taphysique doubl\u00e9e de l'ordre social. L'estime de soi suppos\u00e9e par l'orgueil, qui n'est pas vanit\u00e9, fatuit\u00e9, ni pr\u00e9somption, permet de consid\u00e9rer que toute alt\u00e9rit\u00e9, toute intersubjectivit\u00e9, sont redevables d'une double n\u00e9cessit\u00e9 : \u00eatre, soi, \u00e0 la bonne place pour, ensuite, installer autrui dans celle qui lui revient dans un \u00e9change voulu selon l'ordre h\u00e9doniste.\n\nL'orgueil est condition de possibilit\u00e9 d'une juste installation de soi dans le r\u00e9el : s\u00fbrement pas aussi bas que le souhaiteraient les chr\u00e9tiens, pas forc\u00e9ment aussi haut que le voudraient les vaniteux. Sans orgueil, les simples, les modestes, les petits, les pauvres ne pourraient jamais escompter rien d'autre que les raclures, les d\u00e9chets, les reliefs qui jonchent le sol. D\u00e9pourvu de cette vertu cardinale, pas d'\u00e9l\u00e9gance possible ni de condensation ou de vaporisation du moi sur le mode baudelairien, pas de bip\u00e8des dans un monde de quadrup\u00e8des, pas de tenue dans une \u00e9poque avachie. L'humilit\u00e9 agit en auxiliaire du contentement de soi dans l'ordre du monde quand on occupe la place des moucherons. Ni Dieu, ni la n\u00e9cessit\u00e9 ne justifient qu'on se satisfasse du pire quand il est notre lot, de l'invivable quand il est de fait.\n\nDans la gourmandise et la luxure, un certain usage de soi et de son corps est combattu ; dans la col\u00e8re, l'envie et l'orgueil, un certain usage de soi dans la relation \u00e0 autrui, au monde. Dans les deux cas, ce qui est fustig\u00e9, c'est qu'on puisse ne pas \u00eatre content de son sort et de sa place, qu'on d\u00e9sire un autre lieu, un autre destin, qu'on s'en fasse le promoteur, le d\u00e9cideur, le ma\u00eetre. La col\u00e8re contre l'\u00e9tat des choses, l'envie d'un autre monde, l'orgueil pour en dessiner les contours plus am\u00e8nes, voil\u00e0 aux yeux des moralisateurs qui montre trop l'homme soucieux de son expansion et d'un empire sur soi puis sur le monde. Aussi faut-il condamner, associer ces puissances \u00e0 la faute, au p\u00e9ch\u00e9. Soyez temp\u00e9rants, pratiquez l'humilit\u00e9, excellez dans l'amour du prochain. En fait, ne souhaitez rien d'autre que ce qui advient. L'ordre du monde est bon puisqu'il est ainsi, voulu par Dieu autant que par les hommes qui s'en r\u00e9clament.\n\nNe pas \u00eatre pour soi, ne pas \u00eatre autrement que dans la perspective d'une relation ir\u00e9nique et \u00e9th\u00e9r\u00e9e avec autrui, malgr\u00e9 ce qu'enseigne tous les jours le r\u00e9el, voil\u00e0 qui conduit, en une ultime salve, \u00e0 distinguer deux derniers p\u00e9ch\u00e9s qui trahissent, une fois encore, la n\u00e9cessit\u00e9 imp\u00e9rieuse d'\u00eatre pour le social et de sacrifier toute vell\u00e9it\u00e9 subjective et singuli\u00e8re au profit du L\u00e9viathan collectif. Tout ce qui autorise une inertie et pourrait entra\u00eener une paralysie du social devient peccamineux. Ainsi de l'avarice et de la paresse.\n\nLes th\u00e9ologiens d\u00e9finissent l'avarice comme un attachement excessif \u00e0 l'argent, certes, mais aussi aux biens de la terre. O\u00f9 l'on retrouve la haine chr\u00e9tienne pour l'ici-bas, l'immanence, le monde sensible, toujours sacrifi\u00e9s au profit de l'au-del\u00e0, de la transcendance et du monde intelligible. Pourquoi l'avarice est-elle un p\u00e9ch\u00e9? Parce qu'elle traduit l'amour de l'argent pour soi et non pour le monde. Toute confiscation des richesses pour son propre compte entra\u00eene l'immobilisation des capitaux et le ralentissement \u00e9conomique, jusqu'\u00e0 une \u00e9ventuelle immobilit\u00e9 compl\u00e8te.\n\nL'argent doit viser la production des richesses collectives, l'individu n'est qu'un rouage dans une machine \u00e9conomique et politique. Or, malgr\u00e9 l'invitation faite aux chr\u00e9tiens par le plus c\u00e9l\u00e8bre des charpentiers \u00e0 pr\u00e9f\u00e9rer le royaume de Dieu \u00e0 celui de C\u00e9sar, le pouvoir temporel des propri\u00e9taires a toujours fait bon m\u00e9nage avec celui, pr\u00e9tendument spirituel, des pr\u00eatres. D\u00e9tachement des biens de ce monde, disent-ils, et g\u00e9n\u00e9rosit\u00e9... Faut-il pr\u00e9ciser que, dans une perspective h\u00e9doniste, l'usage des richesses ne contribue en rien aux n\u00e9cessit\u00e9s du social et du collectif, que la prosp\u00e9rit\u00e9 individuelle n'est pas un instrument au service des r\u00e9sultats g\u00e9n\u00e9raux, que l'avarice devient plut\u00f4t vertu lorsqu'elle est entendue comme l'art de g\u00e9rer pour soi, et de mani\u00e8re capricieuse, son rapport au monde mat\u00e9riel, sans le moindre souci des effets collectifs, nationaux, \u00e9conomiques et patriotiques ?\n\nAsocial dans son avarice, parce que soumettant ses richesses \u00e0 son pur et simple plaisir, l'h\u00e9doniste soucieux de cumuler les vices chr\u00e9tiens pour devenir un vertueux jubilatoire t\u00e2chera de pratiquer le dernier p\u00e9ch\u00e9 avec une all\u00e9gresse consomm\u00e9e : \u00e0 savoir la paresse. Car dans l'arsenal des fautes capitales, elle revendique radicalement l'usage libertaire de soi, de son temps et de sa personne. Lever et coucher \u00e0 son heure, loisir \u00e0 convenance, et moins de travail salari\u00e9 possible, car les vertus propos\u00e9es par les chr\u00e9tiens sont les plus susceptibles de parachever la soumission de l'\u00e9nergie individuelle et libertine aux forces collectives et sociales : le courage auquel ils invitent vaut auxiliaire de la n\u00e9gation de soi et de la mise de son corps, de son temps, de son \u00e9nergie, de sa vie, de ses forces au service des causes qui ali\u00e8nent la singularit\u00e9, l'individualit\u00e9, la subjectivit\u00e9. Ne pas \u00eatre paresseux suppose se sacrifier totalement aux imp\u00e9ratifs sociaux.\n\nA quoi donc pourrait bien ressembler cette plan\u00e8te sur laquelle, enfin devenus vertueux, les hommes se seraient mis \u00e0 ne plus boire autre chose que de l'eau, \u00e0 ne plus manger que du pain sec, \u00e0 ne plus toucher au corps d'autrui autrement que pour faire des enfants? Un univers o\u00f9 l'on ne trouverait plus de col\u00e8re, d'envie et d'orgueil permettant le mouvement, la vie, le changement, la transformation du r\u00e9el aux dimensions de ses d\u00e9sirs ? Une terre sur laquelle l'intersubjectivit\u00e9 serait de tol\u00e9rance et de politesse, de temp\u00e9rance et de douceur, de prudence et d'amour, d'humilit\u00e9 et de simplicit\u00e9 ? De quoi serait fait un monde o\u00f9 tous rivaliseraient de g\u00e9n\u00e9rosit\u00e9 sociale et de courage, de go\u00fbt pour le travail et de passion pour l'entreprise? Compassion ici, mis\u00e9ricorde l\u00e0, puret\u00e9 une fois, bonne foi une autre. Amour partout. L'ensemble serait sinistre et le go\u00fbt du paradis serait d'une extr\u00eame fadeur.\n\nAu moins, avec l'avarice, les all\u00e9gories de la peinture et de la sculpture nous l'enseignent depuis toujours, on rencontre les harpies aux t\u00eates et seins de femmes, ailes et serres d'oiseau ; avec la col\u00e8re, toutes les vari\u00e9t\u00e9s de poignards, cimeterres, couteaux, \u00e9p\u00e9es et autres kriss, toutes les variations sur le th\u00e8me du coup de poing, de la gifle et du soufflet, toutes les invectives, les insultes et les quolibets ; avec l'envie, des femmes qui se nourrissent de serpents, et dont le visage est malsain, les yeux louches, les dents g\u00e2t\u00e9es, le corps d\u00e9charn\u00e9, la langue infest\u00e9e de venin; d'autres qui rongent leur c\u0153ur ou leurs visc\u00e8res pendant qu'un gros chien aboyant les accompagne; avec la gourmandise, des b\u00e2freurs couronn\u00e9s de feuilles de vigne et de pampres, portent des plateaux de fruits pendant que courent dans leurs jambes loups voraces, porcs infects, ours morfals et h\u00e9rissons gav\u00e9s; avec la luxure, on croise des femmes au sexe et aux seins rong\u00e9s par des crapauds et des serpents, elles chevauchent de temps en temps des boucs lubriques suivis de sangliers, coqs, li\u00e8vres et lapins, porcs \u00e0 nouveau, parfois elles se mirent dans des psych\u00e9s qui cachent mal le singe derri\u00e8re elles ; avec l'orgueil, on voit une femme accompagn\u00e9e d'un lion et d'un aigle, animaux souverains sur terre et dans les airs, un paon, un miroir ; enfin, la paresse aidant, le cort\u00e8ge se termine avec un homme ob\u00e8se brinquebalant sur un \u00e2ne flanqu\u00e9 d'un b\u0153uf et d'une truie.\n\nFinalement, tout cela fournit un bestiaire bien plus jubilatoire que celui des \u00e9dens vertueux o\u00f9 l'on mourrait d'ennui, o\u00f9 l'on succomberait \u00e0 la perspective de se r\u00e9veiller chaque matin en sachant que rien n'arrive en dehors de l'\u00e9ternelle r\u00e9p\u00e9tition d'un ti\u00e8de bonheur sans asp\u00e9rit\u00e9, lisse comme la surface d'une flaque d'eau croupie. La vie vertueuse, telle que l'entendent les pr\u00eatres et les philosophes qui vont avec, serait mortelle : tout ce qui donne du go\u00fbt \u00e0 la vie en serait retranch\u00e9. Car rien d'autre que ces p\u00e9ch\u00e9s capiteux ne pr\u00e9sente d'int\u00e9r\u00eat, \u00e0 moins de pr\u00e9f\u00e9rer l'immobilit\u00e9 au mouvement, la collectivit\u00e9 \u00e0 soi-m\u00eame, l'ordre fig\u00e9 au r\u00e9jouissant d\u00e9sordre, la froideur des cadavres raidis dans l'\u00e9ternit\u00e9 \u00e0 la ti\u00e9deur des peaux qui s'\u00e9changent et se pr\u00eatent \u2013 autant dire la mort \u00e0 la vie.\n6\n\nAFIN DE MOURIR VIVANT\n\nO\u00f9 ai-je lu, d\u00e9tail d'apocalypse et de bord de tombe, que dans telle ou telle chambre \u00e0 gaz, \u00e0 deux doigts de p\u00e9rir sous les vapeurs mortelles, certains s'\u00e9taient ru\u00e9s sur les murs, d\u00e9chirant le b\u00e9ton de leurs doigts, y laissant des traces encore aujourd'hui visibles, alors que d'autres se vidaient de toutes leurs mati\u00e8res pendant que d'aucuns s'adonnaient une derni\u00e8re fois \u00e0 une copulation tragique? Avant de p\u00e9rir, dans l'imminence du n\u00e9ant, les comportements s'exacerbent, les folies se d\u00e9clenchent, les hyst\u00e9ries du corps triomphent. Ces le\u00e7ons-l\u00e0 sont celles des vanit\u00e9s en peinture ou des all\u00e9gories des relations qu'entretiennent Eros et Thanatos. Pourquoi faut-il que les hommes soient si peu conscients du fait que, dans quelques secondes ou dans quelques dizaines d'ann\u00e9es, ce qui est identique en regard de l'\u00e9ternit\u00e9, ils sont tous condamn\u00e9s \u00e0 mort et qu'ils devraient vivre en cons\u00e9quence?\n\nPascal donne de la condition humaine une image saisissante et juste : des hommes encha\u00een\u00e9s les uns aux autres, dans une cave sombre et humide, attendant que la porte s'ouvre et qu'un bourreau s'empare de l'un d'entre eux, avant de l'\u00e9gorger sous le regard des autres, impuissants. De la m\u00eame mani\u00e8re, il avait bien vu, et racont\u00e9, combien dans cette situation il n'\u00e9tait qu'une solution possible : le divertissement. Tout divertissement suppose consolation et toute consolation proc\u00e8de d'une peine \u00e0 d\u00e9passer, d'un chagrin, mineur ou majeur, dont il faut faire son deuil. S'occuper \u00e0 autre chose, d\u00e9tourner son regard, concentrer son attention sur un autre objet : jouer aux cartes ou au billard, chasser ou babiller, s'\u00e9tourdir, tromper la n\u00e9cessit\u00e9, \u00e9carter les ombres d\u00e9l\u00e9t\u00e8res et t\u00e2cher de ne pas se souvenir de la perte d'un enfant ou du doigt lourd d'un mauvais d\u00e9miurge annon\u00e7ant le tr\u00e9pas pour bient\u00f4t.\n\nAu sortir de mes vingt ans, l'\u00e2ge auquel on ne veut pas s'entendre dire qu'on devrait se r\u00e9jouir d'avoir toute la vie devant soi, je ne prenais plaisir \u00e0 lire que Schopenhauer ou Cioran. Toute autre pens\u00e9e m'incommodait ou me semblait d'une incroyable inefficacit\u00e9, sinon d'une redoutable inad\u00e9quation \u00e0 l'\u00e9vidence : le pire \u00e9tait toujours certain, \u00e0 quoi bon autre chose qu'un regard braqu\u00e9 sur cette certitude effrayante comme le regard du basilic ? Spinoza et sa joie, bien s\u00fbr, mais je n'y adh\u00e9rais gu\u00e8re. Kant, Platon, Descartes ? Inutiles et incertains. Nietzsche me plaisait plus que tous, mais pour de mauvaises raisons, celles qu'on avance \u00e0 cet \u00e2ge-l\u00e0 et auxquelles s'accrochent leurs contempteurs aujourd'hui.\n\nSoucieux de donner forme \u00e0 mes angoisses, mes craintes ou mes peurs, travaill\u00e9 par la mort comme par une maladie qui progresse chaque jour, j'avais r\u00e9solu d'\u00e9crire quelque chose qui, sur deux cents pages, aurait pu ressembler \u00e0 un livre. Son titre devint Permanence de l'apocalypse, car on ne plaisante pas avec les choses s\u00e9rieuses. J'y repris, peu ou prou, ce qui fait le credo pessimiste depuis Qohelet jusqu'\u00e0 Cioran : l'inconv\u00e9nient d'\u00eatre n\u00e9, le triomphe in\u00e9luctable du pire, la haine des consolations et des divertissements, la mort comme g\u00e9nie musag\u00e8te et unique mesure de toute chose, la souffrance consubstantielle \u00e0 l'\u00eatre, l'\u00e9ternel travail de l'entropie, la sup\u00e9riorit\u00e9 in\u00e9vitable du n\u00e9gatif, les m\u00e9tamorphoses toutes douloureuses du temps, les variations sur le th\u00e8me des ruines et des champs de bataille, des d\u00e9combres et des effondrements, la destination de tout au n\u00e9ant, et autres douceurs du m\u00eame genre.\n\nAutant dire qu'on riait peu en traversant mes contr\u00e9es et que ma g\u00e9ographie \u00e9tait plut\u00f4t sinistre. Mais, en ayant dit tout cela, que restait-il ? La bouche du revolver ou les pieds du crucifix ? La corde pour se pendre ou l'enjambement des anciens parapets? Car les familiers du pire enseignent tous qu'en vertu du fait qu'il faudra mourir un jour, autant mourir un peu, sinon beaucoup, tout de suite. Le corps va p\u00e9rir demain? Tuons-le \u00e0 petit feu aujourd'hui. Les d\u00e9sirs dispara\u00eetront avec les passions et le reste ? \u00c9tranglons tout cela d\u00e8s \u00e0 pr\u00e9sent. Chacun est-il d\u00e8s sa naissance, d\u00e8s sa conception m\u00eame, en marche vers le n\u00e9ant? Acc\u00e9l\u00e9rons le mouvement, donnons-lui plus de pouvoir encore en le pr\u00e9sentifiant, en le c\u00e9l\u00e9brant, en lui vouant un culte. Vivons dans l'intimit\u00e9 du pire, couchons avec des cadavres, embrassons sur la bouche la t\u00eate glac\u00e9e des d\u00e9capit\u00e9s. Demandez la recette \u00e0 l'abb\u00e9 de Ranc\u00e9.\n\nJe cherchais des consolations, je ne trouvais que d\u00e9pr\u00e9ciations. Le christianisme et le sto\u00efcisme, l'\u00e9picurisme et le bouddhisme consolent en c\u00e9l\u00e9brant l'extinction, l'amoindrissement de soi, la course \u00e0 l'ab\u00eeme : aller au-devant du pire \u00e0 vive allure, pour se donner l'impression de choisir l\u00e0 o\u00f9 l'on se contente d'ob\u00e9ir \u00e0 la n\u00e9cessit\u00e9. J\u00e9sus, Marc Aur\u00e8le, Epicure et Bouddha, finalement, formulent ce que Schopenhauer et Cioran reprennent. Haine du corps, haine de soi, vindicte contre le fait d'\u00eatre, d\u00e9consid\u00e9ration de ce qui est, hic et nunc, au profit d'un au-del\u00e0 salvateur : le paradis, l'ataraxie, le nirvana, \u00e0 chaque fois des consolations pires que le mal qu'elles sont cens\u00e9es soigner.\n\nLisant ma Permanence de l'apocalypse, une belle \u00e2me bien inspir\u00e9e ce jour-l\u00e0 me demanda pourquoi, \u00e9crivant ce que j'\u00e9crivais, pensant ce que je pensais, affirmant ce que j'affirmais, je ne m'\u00e9tais pas encore mis une balle dans la t\u00eate. La remarque, juste, m\u00e9ritait une r\u00e9ponse appropri\u00e9e... Sinon sur le terrain balistique, du moins sur celui des id\u00e9es. Pourquoi, en effet, ne pas se faire sauter la cervelle quand on pense que tout est \u00e0 ce point apocalyptique ? Quelles raisons font qu'on continue de vivre dans un monde si on le trouve \u00e0 ce point r\u00e9pugnant, immonde, insupportable ? Alors que le n\u00e9ant dans l'instant s'offre \u00e0 la port\u00e9e du premier venu, qu'est-ce qui justifie un d\u00e9sir de vivre plus vivace que celui de mourir ? Thanatos aurait-il un ennemi, une puissance lui r\u00e9sistant, plus forte et plus efficace que lui ?\n\nJe cherchai la r\u00e9ponse en moi plus qu'en ma biblioth\u00e8que. Et la trouvai. Si le r\u00e9el est v\u00e9ritablement tel, noir comme la plus profonde des nuits de haute mer, on peut tout de m\u00eame faire de son existence quelque chose de lumineux comme une trace de com\u00e8te dans le ciel z\u00e9br\u00e9, le temps bref d'un passage d'\u00e9toile filante. Ainsi le pessimisme est-il sublim\u00e9, d\u00e9pass\u00e9 par le tragique. Le pessimiste sacrifie au pire et s'aveugle sur le seul fond de toile; le tragique, quant \u00e0 lui, se persuade que la noirceur vaut seulement d\u00e9cors pour une histoire susceptible d'\u00eatre solaire. Un amateur de dialectique dirait que le pessimiste fait d'un moment dans le mouvement la seule instance d'un processus qui, aux yeux du penseur tragique, se compose d'au moins deux temps.\n\nEn fait, les pessimistes pr\u00e9tendent ignorer th\u00e9oriquement la consolation quand leur vie tout enti\u00e8re t\u00e9moigne de leur pratique farouche et ardente, assidue et r\u00e9guli\u00e8re, de ce divertissement g\u00e9n\u00e9rateur d'\u00e9thique. En revanche, les tragiques revendiquent la consolation comme ce qui justifie leur existence malgr\u00e9 la certitude entropique. Pour autant, la vie quotidienne des grands contempteurs du monde ne cesse de fournir un r\u00e9servoir de consolations : le vin, le haschisch, la musique, la table, les femmes, la lecture, les voyages, l'\u00e9criture, la conversation, l'amiti\u00e9, le soleil, ont permis \u00e0 nombre de temp\u00e9raments familiers du pire, de ne pas finir leur existence au bout d'une corde ou \u00e9cras\u00e9s au pied d'une falaise, mais gentiment, vieillards, dans leur lit, sous la couette o\u00f9 ils s'\u00e9teignent diminu\u00e9s jusqu'\u00e0 la derni\u00e8re extr\u00e9mit\u00e9, consum\u00e9s \u00e0 la mani\u00e8re des vieilles chandelles.\n\nLe suicide est rare chez les philosophes, encore plus chez ceux qui ont fait profession de c\u00f4toyer les ab\u00eemes en dandies imp\u00e9nitents. Seuls quelques nietzsch\u00e9ens, de Michelstaedter \u00e0 Deleuze en passant par Sarah Kaufmann, d\u00e9montrent l'impossibilit\u00e9 d'une existence solaire, sans fa\u00e7on tout aussi solaire d'en sortir quand la pi\u00e8ce devient insupportable, injouable. Aux antipodes existentiels, j'imagine Leopardi ex\u00e9crant le monde dans chaque page du Zibaldone et se r\u00e9conciliant avec force sorbets comme il avait coutume de le faire ; je songe \u00e0 Schopenhauer conspuant le monde dans son ouvrage majeur, mais se reposant de ses invectives th\u00e9oriques et m\u00e9taphysiques en jouant quotidiennement de la fl\u00fbte sous un portrait de Goethe; j'ai une pens\u00e9e pour Rabbe, d\u00e9laissant quelque temps son Album d'un pessimiste pour se pr\u00e9parer une bouffarde fum\u00e9e dans le recueillement ; je sais Cioran pointant les millions de raisons de l'inconv\u00e9nient d'\u00eatre n\u00e9 avant de rejoindre ses amis, eux aussi pessimistes, soit pour boire un bon bordeaux, sacrifier \u00e0 Comus apr\u00e8s Bacchus, rire ou converser, soit, redevenu seul, \u00e9couter un quatuor de Brahms, lire une page de Nietzsche \u2013 ou \u00e9crire, cordial souverain.\n\nEt je me demande pour quelles raisons, vendant du pire \u00e0 longueur de pages, ne choisissant pas la mort volontaire, pratiquant la consolation comme un viatique, aucun n'ait eu l'honn\u00eatet\u00e9 de passer la vitesse sup\u00e9rieure pour s'extraire des couleurs sombres du fond avant d'inviter chacun \u00e0 se faire trace de feu pour illuminer la Voie lact\u00e9e le temps d'un souffle avant extinction. Car tous les pessimistes ont ainsi pratiqu\u00e9 pour leur compte. Certes, en \u00e9crivant : \u00abJe ne pense pas toujours, donc je ne suis pas toujours triste\u00bb, Cioran montrait que la complaisance dans le pire est presque \u00e0 chaque fois un effet de la pens\u00e9e qui se prend n\u00e9vrotiquement pour objet en oubliant qu'elle ferait mieux d'\u00e9lire la vie. De m\u00eame, on sait qu'en augmentant son savoir, on augmente sa tristesse. Pour autant, la solution n'induit pas le refus de penser ou l'abandon de toute m\u00e9ditation. Je crois plut\u00f4t qu'elle r\u00e9side dans la r\u00e9duction du n\u00e9gatif \u00e0 n'\u00eatre qu'un moment dans un mouvement qui doit conduire \u00e0 la positivit\u00e9. Dans cette dialectique achev\u00e9e, accomplie, la consolation, synth\u00e8se gouvernementale, vaut puissance cardinale.\n\nAinsi, apr\u00e8s la question qui m'avait \u00e9t\u00e9 pos\u00e9e \u2013 pourquoi continuer de vivre malgr\u00e9 une vision aussi sombre du monde ? \u2013je me suis mis en qu\u00eate d'ajouter cent ou deux cents pages \u00e0 Permanence de l'apocalypse. Elles firent l'objet de ce que j'appelais alors un Trait\u00e9 des consolations. Depuis, je n'effectue que des variations sur ce th\u00e8me. Bien s\u00fbr, tout ce qui constituait l'objet du premier temps ne s'\u00e9tait pas \u00e9vapor\u00e9 soudain et miraculeusement. Tout restait pertinent : la mort, l'entropie, la souffrance, la douleur, \u00e9videmment, continuaient de triompher et continueront. Toutefois, devenues couleurs sombres pour le fond de la peinture h\u00e9doniste, ces \u00e9vidences \u00e9taient renvoy\u00e9es l\u00e0 o\u00f9 elles n'auraient jamais d\u00fb cesser d'\u00eatre : trouant le bric-\u00e0-brac des d\u00e9cors dans lesquels se joue la pi\u00e8ce. Ensuite, ce qui devenait important \u00e9tait cette pi\u00e8ce : comment la jouer, en attendant le baisser de rideau et l'extinction de tous les feux ? De quelle mani\u00e8re tenir la bonne distance m\u00e9taphysique entre le d\u00e9ni, l'oubli, la n\u00e9gligence du pire et l'interdiction, l'inhibition qu'il suscite parfois ? Ni trop amn\u00e9sique, ni traumatis\u00e9.\n\nL'\u00e9vidence de la mort, son caract\u00e8re in\u00e9luctable, l'obligation pour tout et pour tous de sombrer dans le n\u00e9ant ne doivent pas paralyser mais dynamiser. La consolation est l'art de cette dynamique. Les th\u00e9oriciens du pire d\u00e9laissant la corde racontent donc leurs variations sur ce th\u00e8me : l'humour, l'ironie et le rire, la conversation, l'amiti\u00e9 et le divertissement, l'exacerbation des sensations, des \u00e9motions, des perceptions, l'esth\u00e9tique d'un corps r\u00e9concili\u00e9 tout entier avec lui-m\u00eame, l'art de jouir et de faire jouir, les oeuvres jubilatoires de la chair, la musique et les beaux-arts, le souci des paysages et de la contemplation, de la m\u00e9ditation, les lectures, l'\u00e9criture. Et tout ce qui donne \u00e0 la vie du go\u00fbt, de la forme et de la consistance.\n\nEn mati\u00e8re de r\u00e9el, d'existence, l'avers de la m\u00e9daille est la mort, son revers, l'h\u00e9donisme : la consolation fonctionne, structure l'unit\u00e9 et g\u00e9n\u00e8re la compl\u00e9mentarit\u00e9 d'une face et d'une autre, d'un recto et d'un verso, d'un pile et d'une face. Il faudra mourir, le temps est l'acteur de cette \u00e9vidence, alors jubilons et travaillons le temps comme une mati\u00e8re. Ceux qui s'arr\u00eatent au premier moment de cette affirmation croupissent dans le pessimisme ; en revanche, ceux qui poursuivent, installent leur h\u00e9donisme dans le tragique avec lequel on sublime les n\u00e9cessit\u00e9s pour produire le plus de feu possible avant les \u00e8res glaciaires.\n\nLorsque l'on a choisi l'h\u00e9donisme tragique contre le pessimisme mortif\u00e8re, le corps demeure l'ultime raison. Lui seul peut \u00eatre l'acteur de cette transfiguration. Il n'y a que lui, et c'est lui qui dispara\u00eetra, englouti dans l'oubli. Donc, il reste la seule instance concern\u00e9e par la consolation. Le probl\u00e8me est finalement moins d'avoir \u00e0 mourir que de vivre, et de bien vivre, en attendant. Lorsque la mort viendra et que le coup de faux partira, il n'aura servi \u00e0 rien de s'\u00eatre taillad\u00e9 le corps toute l'existence dans l'attente de ce jour-l\u00e0 et de ce moment-l\u00e0. Mieux vaut \u00eatre debout, en ayant consum\u00e9 son existence en de perp\u00e9tuelles occasions de volupt\u00e9s.\n\nAlors, quand la mort gagnera, du moins nous n'aurons pas \u00e9t\u00e9 sans lui r\u00e9sister, sans la narguer et lui refuser plus de pouvoir qu'elle n'en dispose d\u00e9j\u00e0. Ceux qui auront voulu pactiser avec elle, s'en faire une complice, une alli\u00e9e se retrouveront \u00e0 la m\u00eame enseigne, car \u00e0 cette heure, la diff\u00e9rence compte pour peu. Du moins, pour le reste, les premiers succomberont vivants alors que les seconds tr\u00e9passeront d\u00e9j\u00e0 morts par leur faute, depuis longtemps, pareils \u00e0 des cadavres. Dans leur chair blanche semblable \u00e0 celle des tr\u00e9pass\u00e9s, le sang ne coulera pas quand l'outil funeste les tranchera comme on taille dans une pierre.\n7\n\nLA BR\u00dbLURE DES CORPS PLI\u00c9S\n\nLa danse est l'art de musiquer le r\u00e9el. En son temps, Littr\u00e9 signalait la d\u00e9su\u00e9tude du mot. Pourtant, dans une th\u00e9orie g\u00e9n\u00e9ralis\u00e9e des correspondances, il n'est pas sans int\u00e9r\u00eat de voir comment communiquent les arts, de quelle fa\u00e7on ils correspondent, sur le mode contrapuntique ou contradictoire, dans des logiques de compl\u00e9mentarit\u00e9 ou selon des modalit\u00e9s exclusives. Chacun gagnerait \u00e0 \u00e9couter ceux qui travaillent \u00e0 la jonction des flux, l\u00e0 o\u00f9 s'\u00e9laborent les sens nouveaux et leurs promesses de jubilation. Nietzsche appelait de ses v\u0153ux le philosophe-artiste, il importe, pour sa g\u00e9n\u00e9alogie et son av\u00e8nement, de prendre des le\u00e7ons chez les artistes-philosophes. Dont les danseurs. Eux seuls apportent aux penseurs qui s'interrogent sur ce que peut le corps, des r\u00e9ponses autorisant enfin le d\u00e9passement des seules s\u00e9cheresses th\u00e9oriques d'\u00e9criture.\n\nParmi les danseurs de la sc\u00e8ne contemporaine, Fran\u00e7ois Raffinot pense et agit en promoteur d'une grammaire singuli\u00e8re, d'un langage avec sa syntaxe, d'un vocabulaire avec ses codes, le tout contribuant \u00e0 l'expression d'une vision du monde, ce qui, finalement, suffit \u00e0 d\u00e9finir et caract\u00e9riser l'\u0153uvre philosophique propre. La philosophie est moins un art de cr\u00e9er des concepts que la proposition d'une perspective sur le r\u00e9el. Pour sa part, d\u00e9passant sa formation de philosophe logicien, Fran\u00e7ois Raffinot \u00e9labore une m\u00e9taphysique et propose une \u00e9thique exprim\u00e9e par une esth\u00e9tique. Donc une politique.\n\nDans son monde, on trouve tout ce qui permet de formuler et de circonscrire ce qu'apr\u00e8s Gilles Deleuze et selon ses termes on pourrait appeler \u00ab un n\u00e9o-baroque moderne \u00bb. On sait le passage de l'artiste dans le monde baroque sp\u00e9cifique, ses mises en sc\u00e8ne d'op\u00e9ra, la cr\u00e9ation et la direction d'une compagnie sp\u00e9cialis\u00e9e dans les registres du Grand Si\u00e8cle, ses chor\u00e9graphies de ballets dans des \u0153uvres de Purcell, sa dilection pour Rameau. Et son Alceste de Lully, presque une invitation \u00e0 pratiquer la politique du personnage \u00e9ponyme devenu le parangon que l'on sait. Apr\u00e8s quoi, en bonne logique, Fran\u00e7ois Raffinot donne l'impression de tourner le dos \u00e0 ce monde-l\u00e0 pour un autre.\n\nOr, il n'en est rien. Le passage du Rameau de Plat\u00e9e au Dusapin de Medea-Material n'induit en rien une r\u00e9volution copernicienne. Au-del\u00e0 du fait que cette pi\u00e8ce du compositeur contemporain a \u00e9t\u00e9 pens\u00e9e et compos\u00e9e pour instruments anciens, puis dirig\u00e9e et enregistr\u00e9e par Philippe Herreweghe, dont c'est l'univers de pr\u00e9dilection, il faut aussi pr\u00e9ciser que le baroque, tel que le d\u00e9finit Gilles Deleuze, est moins un monde dat\u00e9, fix\u00e9 et fig\u00e9 dans l'histoire de l'Occident chr\u00e9tien, qu'une fa\u00e7on d\u00e9ploy\u00e9e dans le temps. Est baroque ce qui magnifie des variations sur le th\u00e8me du pli. A cet effet, Fran\u00e7ois Raffinot explore un langage tout entier relevant de cette fa\u00e7on particuli\u00e8re : pli des corps dans la lumi\u00e8re, pli des peintures et des couleurs du d\u00e9cor dans la toile, pli de la lumi\u00e8re dans l'espace sc\u00e9nique, pli de la chor\u00e9graphie dans le temps de la repr\u00e9sentation, pli de la musique dans l'espace et dans le corps des danseurs. En ce sens, n\u00e9e du concept exprim\u00e9 et formul\u00e9 dans le baroque du Grand Si\u00e8cle, la danse semble ne jamais avoir quitt\u00e9 cet \u00e9trange et singulier registre de l'art du pliage.\n\nAussi, pour marquer sa discipline, un artiste doit-il en utiliser les codes, certes, mais aussi s'en \u00e9manciper. Pas de cr\u00e9ation dans le psittacisme, ou d'inscription dans l'histoire avec la seule r\u00e9p\u00e9tition. Fran\u00e7ois Raffinot propose un monde et une vision du monde qui ne n\u00e9gligent en rien l'histoire : celle, \u00e9ternelle, de la mort, mais aussi celle, particuli\u00e8re, du sida ; celle, universelle, de la pers\u00e9cution des artistes par les pouvoirs, mais \u00e9galement celle, particuli\u00e8re, de l'auteur des Versets sataniques dont le corps est revendiqu\u00e9 par les ayatollahs ; celle, plan\u00e9taire, de l'opposition entre la forme et l'\u00e9nergie, mais m\u00eamement, celle des musiques apolliniennes conceptuelles et des oeuvres dionysiennes r\u00e9p\u00e9titives, Pascal Dusapin d'un c\u00f4t\u00e9, Louis Andriessen de l'autre. Adieu dit aussi l'amiti\u00e9, Scandal Point l'engagement, Sin arrimo y con arrimo le duel. De sorte qu'\u00e0 chaque fois, c'est l'occasion d'un tombeau pour Dominique Bagouet, Salman Rushdie, Apollon ou Dionysos.\n\nFran\u00e7ois Raffinot cite saint Jean de la Croix : \u00ab Sans appui et avec appui, sans lumi\u00e8re et vivant \u00e0 l'obscur, tout entier m'en vais en me consumant. \u00bb Ce pourrait \u00eatre aussi bien sign\u00e9 Georges Bataille, Guy Debord ou Gilles Deleuze. Triomphant et claudiquant, nocturne et solaire, forme rigoureuse et \u00e9nergie pure, \u00e0 chaque fois le sujet \u00e9volue dans le registre du feu, des brasiers, de l'incandescence. Entre ces deux instances, comme dans le baroque la lumi\u00e8re et l'obscurit\u00e9, le haut et le bas, Fran\u00e7ois Raffinot ne traque pas les zones interm\u00e9diaires et leurs variations possibles. Pas de souci des micro-intervalles entre les composantes exclusives baroques : l'\u00e9picentre de la chor\u00e9graphie, c'est la consumation. Dans la mort, la douleur, la souffrance, la pers\u00e9cution, la maladie, la sexualit\u00e9, l'\u00e9rotisme, dans tous les cas de figure prime l'art paroxystique de la br\u00fblure.\n\nLe lieu de ces br\u00fblures, c'est le corps. D'o\u00f9 son art consomm\u00e9 de les plier, de les d\u00e9plier. La danse voulue par l'artiste suppose la d\u00e9monstration magistrale que le pli d\u00e9finit la forme d\u00e9velopp\u00e9e sur sc\u00e8ne ; que l'int\u00e9rieur et l'ext\u00e9rieur \u2013 de la sc\u00e8ne, du th\u00e9\u00e2tre aussi bien que des corps qui y \u00e9voluent \u2013 sont les deux instances dont le jeu oppositionnel produit une nouvelle harmonie, celle qui fait du corps un v\u00e9hicule de ce parti pris de musiquer le monde ; que le haut et le bas, dans une m\u00eame juxtaposition dialectique, trouvent leur conciliation dans l'\u00e9piphanie de la forme en force ; que le d\u00e9pli suppose, dans une logique revendiqu\u00e9e, la possibilit\u00e9 d'un recommencement du pli ; que la texture vaut mode d'apparition du pliage de la mati\u00e8re et qu'en l'occurrence, cette texture se d\u00e9ploie avec celle des chairs montr\u00e9es sous la protection et le voile de leur peau ; enfin que le paradigme absolu est le corps. Dans ces six modes reposent comme en un \u00e9crin l'identit\u00e9 du n\u00e9o-baroque moderne dont Fran\u00e7ois Raffinot se fait le d\u00e9miurge.\n\nLe diptyque appuie le signe du go\u00fbt particulier qu'a le chor\u00e9graphe pour les formes. Sa lecture musicologique des partitions le rend sensible \u00e0 l'architecture qui est aussi architectonique. La science de ces formes qui sont \u00e9galement des forces est prol\u00e9gom\u00e8ne \u00e0 l'incarnation. Car musiquer le monde en \u00e2me de la compagnie, en paraclet, c'est r\u00e9ussir le passage de la mat\u00e9rialit\u00e9 du son \u00e0 celle des corps. Entre le corps du son, sa mati\u00e8re, et le son du corps, ses effets via les mouvements, s'installe tout le travail d\u00e9miurgique superbement r\u00e9capitul\u00e9.\n\nDans cet ordre d'id\u00e9es, Scarcdal Point propose un travail embl\u00e9matique, puisqu'il s'agit de donner un corps \u00e0 Salman Rushdie qui est quasi d\u00e9poss\u00e9d\u00e9 du sien sur le mode public. La voix priv\u00e9e de corps, telle une parole mystique venue du ciel, sans qu'une chair la supporte, se transfigure en corps esth\u00e9tique. Face \u00e0 la d\u00e9sincarnation voulue par les mollahs, contre cette volont\u00e9 des religieux int\u00e9gristes de priver de corps, d'arracher sa chair \u00e0 un \u00eatre qui pense et \u00e9crit, Fran\u00e7ois Raffinot r\u00e9agit en r\u00e9alisant l'incarnation. L\u00e0 o\u00f9 le musulman radical veut faire taire une voix en pourchassant et condamnant un corps, le chor\u00e9graphe veut faire advenir un corps \u00e0 partir d'une voix qu'il musique.\n\nDans cette op\u00e9ration de transmutation, Salman Rushdie pr\u00eate sa voix quand les danseurs, simultan\u00e9ment, et par cette op\u00e9ration particuli\u00e8re, se trouvent dot\u00e9s, quant \u00e0 eux, de la parole qui leur fait d\u00e9faut lorsqu'ils \u00e9voluent habituellement sur sc\u00e8ne. D\u00e9sincarn\u00e9e, la voix sans corps devient sur la sc\u00e8ne d'Avignon un corps sans organe, pour le dire comme Artaud, puis Deleuze, une esp\u00e8ce mutante nourrie de la dialectique des combats, baroques eux aussi, entre la voix lumineuse de l'un et les corps dansant dans la nuit du silence des autres.\n\nD'Adieu, je retiendrai l'amiti\u00e9, le travail en forme d'hommage pour un homme, sa douleur et ses souffrances ; de Sin arrimo y con arrimo, la n\u00e9cessit\u00e9 de combinatoires multiples pour exprimer les relations dialectiques imbriqu\u00e9es entre les formes destin\u00e9es \u00e0 dire le r\u00e9el ; de Scandal Point, un genre de synth\u00e8se de ces deux temps t\u00e9moigne que le souci pour Apollon n'exclut pas la passion pour Dionysos, ni l'inverse. Ainsi appara\u00eet la possibilit\u00e9, en musiquant le monde, de transfigurer des nuits qui, comme le d\u00e9sert dans les pages de Nietzsche, donnent sans cesse l'impression de cro\u00eetre.\n8\n\nLE MONDE DES VOLEURS DE FEU\n\nO\u00f9 sont les voleurs de feu, chers au c\u0153ur d'Arthur Rimbaud? Dans quelque Harrar, du c\u00f4t\u00e9 de l'Ethiopie, dans tel club anglais au c\u0153ur d'Oxford, en un jardin zen non loin de Kobe, ou dans tout autre lieu o\u00f9 l'on a fait du raffinement une \u00e9thique en m\u00eame temps qu'une esth\u00e9tique? Cheminent-ils dans le d\u00e9sert, consum\u00e9s par le vent ? Conversent-ils dans des fauteuils de cuir profonds ? Boivent-ils le th\u00e9 br\u00fblant apr\u00e8s les c\u00e9r\u00e9monies qui le rendent possible ? O\u00f9 sont-ils en tout endroit o\u00f9 la fum\u00e9e triomphe en art avec ses r\u00e8gles, ses lois, ses principes et ses codes jamais \u00e9crits ? Je penche pour les g\u00e9ographies et les \u00e9thiques o\u00f9 se sont \u00e9panouies les civilisations d'herbes br\u00fbl\u00e9es, celle de l'opium oriental, du tabac des conquistadors ou du haschisch des po\u00e8tes de l'Europe fatigu\u00e9e.\n\nAussi, si l'on doit avancer une petite th\u00e9orie du tabac, Baudelaire est-il plus pertinent que Freud et Sartre, les deux seuls pratiquants qui formulent peu ou prou quelques consid\u00e9rations sur le sujet. Pour le p\u00e8re de la psychanalyse, grand fumeur de cigares au quotidien, la tabagie offre purement et simplement un substitut \u00e0 l'onanisme, la grande histoire de tout un chacun \u2013 selon lui ; chez Sartre, du moins dans L'Etre et le N\u00e9ant, le fait de fumer est lui aussi substitut, mais au d\u00e9sir de cr\u00e9mation du monde entier. Dans les deux cas, \u00e0 d\u00e9faut d'une masturbation en bonne et due forme ou d'une consumation g\u00e9n\u00e9ralis\u00e9e et plan\u00e9taire, on se contente d'un analogon, bien utile si l'on en juge par les extr\u00eames auxquels, \u00e0 d\u00e9faut, on se trouverait r\u00e9duit.\n\nMais si l'on doit demander secours \u00e0 Baudelaire, c'est moins au po\u00e8te ou au familier des paradis artificiels qu'au penseur et th\u00e9oricien du dandysme. Car le tabac, quand il est pratique du cigare, et non de la cigarette, illustre l'art dandy par excellence. Et toute th\u00e9orie de la tabagie qui \u00e9conomise le complexe d'Onan le solipsiste ou de N\u00e9ron l'incendiaire doit passer par la m\u00e9taphysique de Brummell. Pour quelles raisons ? S\u00fbrement pas pour le plaisir du souverain art de d\u00e9plaire que suppose et affiche tout fr\u00e8re de Baudelaire. Pas plus pour le narcissisme flamboyant et les occasions de para\u00eetre qu'il pourrait fournir. Ni m\u00eame en vertu de l'injonction \u00e0 vaporiser et concentrer le moi ins\u00e9parable du codex dandy. Non. L'art du cigare est baudelairien quand il exprime une m\u00e9taphysique de l'objet, du temps et de l'infinit\u00e9simal qui d\u00e9bouche comme dans une clairi\u00e8re sur une politique.\n\nLe dandysme a c\u00e9l\u00e9br\u00e9 l'objet pour y concentrer nombre d'informations philosophiques. De la cravate \u00e0 la canne et \u00e0 ses variations dans l'usage, en passant par la tabati\u00e8re l\u00e9gendaire du Beau, le dandy \u00e9lit une chose banale, sinon triviale, et, lui donnant son onction dans l'usage, en fait une occasion d'exprimer une vision du monde. L'art de nouer sa cravate est moins une anecdote qu'une d\u00e9monstration par les faits de la n\u00e9cessit\u00e9 d'exigence, de perfection, d'excellence et de singularit\u00e9 dans chaque moment de la vie quotidienne, f\u00fbt-il le plus apparemment d\u00e9nu\u00e9 d'int\u00e9r\u00eat. La tension, ma\u00eetresse du dandy, s'exprime tout enti\u00e8re par l'intercession de ce morceau de tissu qui acc\u00e8de de la sorte \u00e0 la dignit\u00e9 m\u00e9taphysique. Les plus audacieux diraient ontologique...\n\nLe cigare est un objet aux variations multiples car aucun, dans le registre du module identique, ne ressemble \u00e0 son double apparent. Couleur de cape, conditions multiples de provenance, de travail, de conditionnement, de conservation, le cigare se lit en artifice embl\u00e9matique de Baudelaire : l'antinature c\u00e9l\u00e9br\u00e9e, l'artefact dans sa quintessence. Produit manufactur\u00e9, voulu par le g\u00e9nie des hommes, \u00e9l\u00e9ment de culture et de civilisation, il est en lui-m\u00eame un raccourci de la n\u00e9cessit\u00e9 formul\u00e9e par tous les dandies de s'arracher \u00e0 la nature pour r\u00e9aliser une multiplicit\u00e9 de signes et de gestes ouvrag\u00e9s. Devenu f\u00e9tiche, le cigare, comme tous les objets soumis \u00e0 ce traitement, s'installe du c\u00f4t\u00e9 du sacr\u00e9 et exige le rite.\n\nEn ce sens, transfigur\u00e9 en pr\u00e9texte \u00e0 rituel, le cigare persiste dans l'illustration de la m\u00e9taphysique envisag\u00e9e. F\u00e9tiche des bois pr\u00e9cieux pour la bo\u00eete dans laquelle seront entrepos\u00e9s les modules, ronce de noyer ou acajou, loupe d'orme ou toute autre essence exotique, rare ou extravagante ; f\u00e9tiche des guillotines et autres instruments s\u00e9cateurs guilloch\u00e9s, sign\u00e9s, dessin\u00e9s, avec lesquels se tranchent les t\u00eates; f\u00e9tiche des allumettes et de leurs bois d'essences appropri\u00e9es ; f\u00e9tiche des cendriers dans lesquels s'abandonnent les reliefs d'une combustion qui fut consommation et consumation, verreries anciennes ou porcelaines d\u00e9licates. Chacun de ces registres peut \u00eatre \u00e0 lui seul l'occasion d'un mus\u00e9e extravagant \u00e9chapp\u00e9 d'un roman symboliste de Huysmans.\n\nLe fumeur de cigares promeut un syst\u00e8me d'objets \u00e9lus le transfigurant en m\u00e9taphysicien dont le monde se r\u00e9v\u00e8le par l'ensemble de ces petits embl\u00e8mes charg\u00e9s de sens et d'histoire. Le r\u00e9el est somm\u00e9 de passer par l'individuation dans ces fragments symboliques. Autant que les miroirs, les v\u00eatements, les tabati\u00e8res, les tenues de roulier color\u00e9es, les cannes, les fleurs v\u00e9n\u00e9neuses et extravagantes, les gilets de soie et autres brocarts \u00e0 ramages, les cravates et les gants beurre frais, les cigares sont promus dans l'ordre d'une mystique qui trahit une vision du monde. L'objet interc\u00e8de en interm\u00e9diaire pour une po\u00e9tique du r\u00e9el qui est \u00e9galement po\u00e9tique de la r\u00eaverie. Car le cigare induit aussi une m\u00e9taphysique du temps.\n\nFumer apparente aux d\u00e9miurges qui sculptent le temps selon leur vouloir. Autant que la musique et la d\u00e9gustation des vins extraordinaires, le cigare permet une authentique c\u00e9l\u00e9bration du temps. Le feu qui rend possible la combustion du module vaut m\u00e9taphore de l'\u00e9nergie, de la vitalit\u00e9 qui consume m\u00eamement un corps et une existence. De l'allumage \u00e0 l'extinction en passant par les variations de dur\u00e9e \u2013 accelerando, presto, andante, etc. \u2013 sublim\u00e9es entre ces deux extr\u00e9mit\u00e9s, fumer un cigare est un raccourci m\u00e9taphysique tout aussi symbolique que le sont les Vanit\u00e9s en peinture. De tout feu port\u00e9 au pied du cigare, il faudra conclure t\u00f4t ou tard \u00e0 la r\u00e9duction pure et simple de l'ensemble du module \u00e0 de la cendre. Avant le n\u00e9ant et les poudres qui font s'en retourner la mati\u00e8re \u00e0 la terre s'expriment mille occasions de volupt\u00e9s, de plaisirs sculpt\u00e9s, sollicit\u00e9s, attendus, cr\u00e9\u00e9s, d\u00e9clin\u00e9s.\n\nLa cr\u00e9mation implique une ma\u00eetrise du feu dans l'optique exigeante de la conservation du principe \u00e9nerg\u00e9tique. Tous les fumeurs le savent qui ont entendu un jour d\u00e9composer le cigare en trois tiers : le foin, le c\u0153ur et le fumier. Qui ne songerait ici \u00e0 la question pos\u00e9e par le Sphinx sur les \u00e2ges de l'existence? Toute vie se d\u00e9compose en ces trois temps qui s\u00e9parent un mouvement ascendant et un autre, descendant, avec en son milieu une \u00e9poque profitant de l'excellence des deux ab\u00eemes qui le bordent. Enfant, adulte et vieillard, l'homme va vers les cendres apr\u00e8s avoir br\u00fbl\u00e9, plus ou moins incandescent, dans une dur\u00e9e qui aura permis plus ou moins de surprises, de volutes, d'extravagances en arabesques, de folies en fum\u00e9es. D'aucuns m\u00eame, par d\u00e9faut d'int\u00e9r\u00eat, par n\u00e9gligence, verront s'\u00e9teindre ce qui n'aurait donn\u00e9 que plaisirs et satisfactions si d'aventure la passion et la sollicitude h\u00e9donistes avaient prim\u00e9. Quiconque aspire \u00e0 l'existence d\u00e9multipli\u00e9e doit viser l'incandescence c\u00e9l\u00e9br\u00e9e. Aucun voleur de feu ne subsiste sans cette volont\u00e9 d\u00e9lib\u00e9r\u00e9e, \u00e0 chaque instant, de faire de la br\u00fblure une invite \u00e0 la densit\u00e9 de la vie quotidienne.\n\nDu temps saisi par le registre des vanit\u00e9s on peut passer au temps h\u00e9doniste dont la quintessence s'exprime dans le pr\u00e9sent. Le cigare fournit un pr\u00e9texte \u00e0 co\u00efncider avec le moment idoine dans l'expression pure et simple d'une c\u00e9l\u00e9bration de l'instant. Le feu, la fum\u00e9e, le go\u00fbt, sont des instances qui, conjugu\u00e9es, produisent la volupt\u00e9 consubstantielle \u00e0 la tabagie. Ici et maintenant, dans l'\u00e9vidence d'un temps magnifi\u00e9, fumer ralentit les dur\u00e9es pour faire \u00e9clore une p\u00e9riode sp\u00e9cifiquement voluptueuse, vraisemblablement celle de la pri\u00e8re pour les mystiques, s\u00fbrement celle de la m\u00e9ditation pour les pa\u00efens. Quand la coul\u00e9e du temps se fait plus lente, plus sereine, plus douce, apparaissent des sentiments, des sensations et des \u00e9motions qui installent le fumeur dans un monde aux perceptions d\u00e9cupl\u00e9es. D'o\u00f9 la troisi\u00e8me proposition de cette m\u00e9taphysique dandy : cette ontologie de l'objet doubl\u00e9e d'une th\u00e9orie du temps h\u00e9doniste se compl\u00e8te par un art du d\u00e9codage des signes infinit\u00e9simaux.\n\nL\u00e0 encore, sur ce terrain de la pratique de l'infinit\u00e9simal, je veux filer la m\u00e9taphore esth\u00e9tique au sens large du terme. Pr\u00e9cisons. Aucune \u00e9thique n'existe sans un souci perp\u00e9tuel et exacerb\u00e9 de l'autre. Pas de morale sans pr\u00e9venance, sans pr\u00e9voyance, sans calcul des effets possibles, sans arithm\u00e9tique des plaisirs pensables. Pour ce faire, il faut un savoir sans d\u00e9faillance de la variation microscopique : en mati\u00e8re d'\u00e9thique, c'est une inflexion dans la voix, un d\u00e9bit particulier pour une parole, une musique plus lente que d'habitude dans l'exposition ou le d\u00e9roulement d'une phrase, mais aussi un port de t\u00eate singulier, un corps plus raide que d'habitude, un visage plus ferm\u00e9, une bouche moins souple pour le verbe. Le rapport \u00e0 l'autre passe essentiellement par le savoir de ces \u00e9tats changeants et riches d'enseignement. Le cigare condense des le\u00e7ons possibles pour cette sapience.\n\nComment? En enseignant \u00e0 qui sait l'entendre l'art du d\u00e9codage des mille variations constitutives de ce plaisir. Fumer un cigare occasionne des exercices de style qui tous autorisent la perception et la sensation des variables pensables. D'abord, les cinq sens renseignent sur la nature du cigare. Le regarder d\u00e9couvre l'aspect, l'allure, formes du module et couleurs de la cape, diam\u00e8tres et longueurs. On apprend sous les doigts le bruissement de la tripe sous la cape, la souplesse, la ductilit\u00e9, le grain, sa finesse, la tension, la compacit\u00e9. Le fumage \u00e0 cru dira a priori les notes de corne, de musc, de miel, de r\u00e9glisse, de terre, de truffe, d'herbe, ou encore de cannelle ou de cacao. Les subtilit\u00e9s en bouche se modulent suivant l'avancement de la consumation. Parfums grill\u00e9s, torr\u00e9factions et odeurs de br\u00fbleries, capiteuses ou ent\u00eatantes variations avant l'empyreumatique.\n\nL'esprit et l'intelligence, sinon la m\u00e9moire et le jeu avec les images mentales, permettent l'exercice sur les degr\u00e9s de richesse ou de g\u00e9n\u00e9rosit\u00e9 de la fum\u00e9e, sur les agr\u00e9ments, les \u00e9quilibres et la persistance des saveurs. Comme en pr\u00e9sence de grands flacons de vins, on peut interroger les fum\u00e9es en bouche afin de remarquer et pr\u00e9ciser richesse et intensit\u00e9, \u00e9toffe et corpulence, \u00e9l\u00e9gance et distinction, gr\u00e2ce et amabilit\u00e9, style et race, mais aussi harmonie et rondeur, complexit\u00e9 et \u00e9quilibre, pr\u00e9sence et discr\u00e9tion. On agit de m\u00eame pour isoler les d\u00e9fauts : \u00e2cret\u00e9, amertume, rudesse, grossi\u00e8ret\u00e9, acidit\u00e9, piquant. Autant de cat\u00e9gories qui, de plein droit, ressortissent \u00e0 l'\u00e9thique, car elles peuvent signifier aussi justement le caract\u00e8re ou le temp\u00e9rament d'un sujet.\n\nCet art de lire les micro-intervalles dans le r\u00e9el conduit \u00e0 l'\u00e9laboration d'une morale sensualiste, aristocratique, \u00e9lective, h\u00e9doniste et jubilatoire. La th\u00e9orie du temps transfigur\u00e9 suppose une \u00e9thique et une m\u00e9taphysique qui d\u00e9bouchent sur une politique au sens pr\u00e9cis du terme. En fumant ses cigares, le voluptueux oppose un refus de l'unidimensionnel que c\u00e9l\u00e8bre le fumeur de cigarettes. En effet, la cigarette est embl\u00e9matique du calibrage auquel aspire notre civilisation marchande : toujours semblable \u00e0 sa voisine, identique de l'une \u00e0 l'autre en forme, couleur et saveur, insipide dans l'asepsie suppos\u00e9e, on l'imagine volontiers comme un raccourci du standard auquel tend toute civilisation. L'identit\u00e9 se d\u00e9finit dans la ressemblance. La possibilit\u00e9 de duplications infinies trahit l'assemblage de m\u00eames comme seule hypoth\u00e8se pour construire un monde.\n\nLa cigarette est pathologique, d\u00e9mocratique au sens donn\u00e9 par Baudelaire \u00e0 ce terme. G\u00e9n\u00e9ratrice d'assu\u00e9tude, elle entretient plus de rapports avec Thanatos que le cigare, auxiliaire d'Eros. Le fumeur de cigare est acteur, il choisit et ne subit pas sa relation \u00e0 la fum\u00e9e. Le premier est requis par une d\u00e9pendance physiologique, le second consent \u00e0 une histoire m\u00e9taphysique. La cigarette est la production ind\u00e9finie d'un objet type de l'\u00e8re m\u00e9canique pour laquelle les sensations et les \u00e9motions sont calibr\u00e9es, reproductibles et voulues dans les limites de la pure et simple r\u00e9it\u00e9ration ; le cigare offre \u00e0 chaque fois un produit diff\u00e9rent, un compos\u00e9 complexe d'\u00e9l\u00e9ments qui r\u00e9agissent \u00e0 toutes les variations possibles, il est la revendication de la diff\u00e9rence cultiv\u00e9e dans une \u00e8re aristocratique qui suppose les micro-soci\u00e9t\u00e9s h\u00e9donistes \u00e9lectives. La cigarette se fume partout, n'importe comment, avec n'importe qui, elle fournit la dose n\u00e9cessaire de nicotine comme d'autres ont besoin de leur dose d'alcool.\n\nLe cigare suppose une \u00e9thique et une conception h\u00e9doniste de l'intersubjectivit\u00e9 : th\u00e9oriquement, et en vertu des d\u00e9ontologies qui agissent comme des lois non \u00e9crites sur ce sujet, on ne le fume qu'avec l'assentiment des convives, des proches \u2013 de l'autre. Pas question de pratiquer le m\u00e9pris en infligeant son plaisir comme un d\u00e9sagr\u00e9ment. Choisissant son module, son temps, son heure, sujet de son d\u00e9sir, et non objet, le fumeur peut diff\u00e9rer sa volupt\u00e9 s'il sait devoir la payer du d\u00e9plaisir d'autrui. Au contraire, bien souvent, du fumeur de cigarette, objet de sa dilection, presque toujours requis en urgence par les n\u00e9cessit\u00e9s physiologiques de retrouver la nicotine indispensable.\n\nPour autant, le cigare est \u00e9galement devenu l'objet de reconnaissance sociale des castes d\u00e9tentrices de pouvoirs, des politiciens aux journalistes en passant par les repr\u00e9sentants du monde des affaires. Autant pr\u00e9ciser d\u00e8s \u00e0 pr\u00e9sent que, dans pareil cas, l'h\u00e9donisme dispara\u00eet devant le d\u00e9sir d'appara\u00eetre ce que l'on est : un acteur social qui reproduit le capitalisme dans ses travers et son arrogance infatu\u00e9e. La m\u00e9taphysique en jeu vaut purement et simplement signe de reconnaissance des pr\u00e9dateurs. Je n'ai gu\u00e8re de sympathie pour cette pratique-l\u00e0. La m\u00e9ditation, la pratique m\u00e9taphysique des voleurs de feu disparaissent alors. Reste un accessoire ridicule, comme parfois peuvent l'\u00eatre les grands crus de vins, quand ils sont utilis\u00e9s et sollicit\u00e9s pour c\u00e9l\u00e9brer les fins triviales de la reconnaissance sociale et de la distinction tribale.\n\nFumer un cigare en public, ostensiblement, \u00e9quivaut \u00e0 boire un superbe mill\u00e9sime dans des conditions d\u00e9plorables. Le cigare suppose la proximit\u00e9 amicale, complice, le lieu ad hoc, le retrait, l'\u00e9tat d'esprit libre et disponible pour des errances m\u00e9ditatives, le corps en \u00e9tat, l'\u00e2me libre pour le nomadisme spirituel, voire une solitude dense et joyeuse. Sinon, il trahit le snob dont l'\u00e9tymologie caract\u00e9rise \u00e0 dessein et fort \u00e0 propos l'individu d\u00e9finitivement sans noblesse. Car le voleur de feu, aux antipodes de l'animal pr\u00e9dateur en qu\u00eate de marquage de territoire, est plut\u00f4t familier de ces plan\u00e8tes o\u00f9 \u00e9voluent les artistes, l\u00e0 o\u00f9 les d\u00e9sirs et les plaisirs se r\u00e9concilient pour le grand devoir qu'est la jubilation.\n9\n\nLES VERTUS DE LA FOUDRE\n\nLa lenteur m'\u00e9nerve. Autant que la prudence, sa forme \u00e9thique. Elle m'ennuie, \u00e9corche ma patience et trouble mes rythmes. J'aime la vitesse \u2013 celle de la foudre et des \u00e9clairs, des \u00e9ruptions volcaniques et des raz-de-mar\u00e9e, celle des fulgurances et des ravissements. J'aime le temps acc\u00e9l\u00e9r\u00e9 jusqu'aux extr\u00e9mit\u00e9s productrices de dur\u00e9es magnifiques. Le temps qui tend, resserre et comprime, densifie, raccourcit et sculpte l'espace pour g\u00e9n\u00e9rer des densit\u00e9s extraordinaires. La vitesse, c'est du temps qui concentre l'espace.\n\nO\u00f9 sont les lents ? Chez les m\u00e9ditatifs, les patients, les mystiques ? Je ne crois pas. Ceux-l\u00e0 ouvragent leur r\u00e9el dans l'optique des \u00e9piphanies qu'ils appellent en br\u00fblant. Les r\u00e9v\u00e9lations, les conversions se font toutes sur le mode de la foudre qui tombe, de la lumi\u00e8re qui irradie dans l'insolence les nuits o\u00f9 croupissent les d\u00e9c\u00e9r\u00e9br\u00e9s. L'illumination d\u00e9gage toujours une \u00e9nergie fulgurante qui coupe en deux, sectionne aussi bien les pierres, les arbres que les existences. Tous les fous de vitesse sont fanatiques de la br\u00fblure et de la consumation de leur existence. Leur \u00e9l\u00e9ment de pr\u00e9dilection est le feu, mena\u00e7ant quand le jeu outrepasse et ouvre des perspectives sur l'apocalypse.\n\nO\u00f9 donc, alors ? Chez les \u00e9conomes, les amateurs de th\u00e9saurisation, les anxieux \u00e9tourdis par l'envergure. Ceux que l'avenir effraie, qui sont inquiets des formes prises par le temps dans ses \u00e9coulements. Le pass\u00e9 les comble et leur fournit une dimension de pr\u00e9dilection, l\u00e0 tout est rang\u00e9, calme, immobile. Ce qui advient est fort de toutes les surprises possibles : la vitesse sollicite ces probabilit\u00e9s sur le mode ludique. Ludique et m\u00e9taphysique, car les d\u00e9sirs d'acc\u00e9l\u00e9ration du temps, jusqu'aux bords les plus fragiles, d'\u00e9vidence, approchent de la mort et du temps pulv\u00e9ris\u00e9.\n\nLa vitesse triomphe en conqu\u00eate de l'esprit et de la civilisation. Je remarque l'absence singuli\u00e8re de dieu pour la dire, l'exprimer et la caract\u00e9riser dans l'Antiquit\u00e9 grecque ou romaine. Herm\u00e8s signifie la circulation, la communication, certes, mais sur un mode temporel non pr\u00e9cis\u00e9. Les ailes aux chevilles disent l'attache d\u00e9li\u00e9e permettant le vol, le transport c\u00e9leste. Plus vite que le passant, bien s\u00fbr, qui \u00e9volue en p\u00e9r\u00e9grin sur les routes poussi\u00e9reuses, mais moins que les dieux apparemment seuls \u00e0 disposer de la vitesse. Car ils peuvent la puissance, la force et l'\u00e9nergie dans la clart\u00e9 violente d'un instant lumineux : ils d\u00e9chirent, d\u00e9chiquettent, punissent, vengent, tuent, veulent, mais \u00e0 chaque fois leur d\u00e9sir co\u00efncide dans le temps avec leur d\u00e9cision. La vitesse r\u00e9duit au maximum la distance entre le d\u00e9sir et sa r\u00e9alisation. Sur ce terrain les seules cr\u00e9atures divines r\u00e9ussissent la pure et simple confusion d'une volont\u00e9 et d'un acte.\n\nDu fait que la vitesse caract\u00e9rise exclusivement les dieux, il faut vraisemblablement conclure \u00e0 la divinit\u00e9 qu'elle permet. Dans le projet cart\u00e9sien fondateur de modernit\u00e9 \u2013 se rendre comme ma\u00eetre et possesseur de la nature \u2013, la vitesse agit en puissance cardinale. Par elle se dit l'\u00e9mancipation humaine des \u00e9l\u00e9ments. L\u00e0 o\u00f9 les oiseaux volent, dans un azur satur\u00e9 d'\u00e9ther, les hommes ont essay\u00e9 tout ce qui permet la portance avant de vaincre d\u00e9finitivement la r\u00e9sistance de l'air; dans les eaux froides o\u00f9 \u00e9voluent les poissons, les imaginaires fabuleux o\u00f9 la pieuvre g\u00e9ante croise la raie magnifique, des inventeurs ont sollicit\u00e9 leurs folies les plus po\u00e9tiques avant de parvenir \u00e0 troubler le monde du silence ; sur les terres poudreuses, dans les steppes aussi bien que les savanes, dans les champs ou sur les chemins trac\u00e9s par la patience, les ing\u00e9nieurs mettent au point toutes les machines qui ont un jour permis d'en finir avec le seul pas des hommes.\n\nChaque fois, il a fallu une \u00e9nergie transfigur\u00e9e, une m\u00e9canique suffisamment serve pour reconvertir des forces musculaires en mouvements a\u00e9riens, aquatiques ou terrestres. Le moteur a \u00e9t\u00e9 l'auxiliaire de ces captations d'\u00e9nergie et il appara\u00eet peu ou prou d\u00e8s que surgissent dans les ateliers les cardans, les rouages et les m\u00e9caniques de d\u00e9multiplication. Avec le moteur commence, au-del\u00e0 du pur et simple affranchissement \u00e0 l'endroit des contraintes naturelles, la possibilit\u00e9 de modifier le rapport de l'homme au temps, donc \u00e0 l'espace. La vitesse fournit le suppl\u00e9ment d'\u00e2me de cette conqu\u00eate et l'occasion de m\u00e9taphysiques nouvelles.\n\nAvant l'apparition effective du moteur m\u00e9canique \u2013 1859 \u2013 le mot rel\u00e8ve d'une occurrence ancienne puisqu'il caract\u00e9rise ce qui donne le mouvement, mais exclusivement dans le registre th\u00e9ologique. Aristote recourt au premier moteur immobile pour caract\u00e9riser Dieu. Jusqu'\u00e0 la r\u00e9volution industrielle, dans l'usage habituel, le terme rel\u00e8ve du seul monde ontologique. L'usage vernaculaire du substantif induit une fracture dans la lecture du temps comme un ph\u00e9nom\u00e8ne dont la physique rendait compte avec les cat\u00e9gories newtoniennes. La th\u00e9orie de la relativit\u00e9 d\u00e9montre enfin la liaison du temps, de l'espace et de la vitesse.\n\nL'invention de la vitesse est contemporaine des recherches en physique sur la th\u00e9orie de la relativit\u00e9 g\u00e9n\u00e9ralis\u00e9e. Paul Langevin avance cette hypoth\u00e8se, en 1911, que deux jumeaux voyageant l'un \u00e0 une vitesse proche de la lumi\u00e8re, l'autre \u00e0 celle du pas, le premier vieillirait plus lentement que le second. Autant dire sur le terrain m\u00e9taphysique que la vitesse se pense en auxiliaire de vitalit\u00e9 l\u00e0 o\u00f9 la lenteur semble un facteur plus activement entropique. Etre vite, c'est \u00eatre plus longtemps, plus vivement. La vitesse para\u00eet bien l'une des modalit\u00e9s de la variation sur le th\u00e8me du temps, donc de la mort.\n\nDevenus semblables aux dieux, affranchis plus nettement des contraintes de temps et d'espace, cr\u00e9ateurs de nouvelles dur\u00e9es, les hommes apprivoisent et r\u00e9alisent la vitesse domestique. Apr\u00e8s l'aristocratie de ses usages vint la d\u00e9mocratisation de ce qu'il convient d'appeler la dromotique. Et ce si\u00e8cle a vu l'homme passer du monde des marcheurs du n\u00e9olithique \u00e0 celui des humains prolong\u00e9s par les proth\u00e8ses supersoniques qui mettent chaque point du globe \u00e0 sa port\u00e9e quasi imm\u00e9diate. Son monde r\u00e9el se limitait aux mesures des pas accumulables dans une journ\u00e9e, il devient un univers dont la rotondit\u00e9 efface les points de rep\u00e8re qui permettaient le sens. Entre ces deux extr\u00eames, la lenteur du marcheur et la vitesse du voyageur a\u00e9rien, on enregistre une formidable acc\u00e9l\u00e9ration qui colle n'importe qui le dos au sol. Ne sommes-nous pas d'ailleurs encore allong\u00e9s, terrass\u00e9s par cette formidable force, gisant la face tourn\u00e9e vers les \u00e9toiles ? Autant que le premier pas de l'homme sur la lune, il faut voir dans la vitesse un sympt\u00f4me, un signe, sinon une occasion de nouveau monde. La m\u00eame fracture avait s\u00e9par\u00e9 le monde clos de l'univers infini, elle \u0153uvre aujourd'hui entre le monde lent et l'univers acc\u00e9l\u00e9r\u00e9.\n\nCette physique d\u00e9bordant son registre et produisant une m\u00e9taphysique g\u00e9n\u00e8re ses cultes, ses rites et ses sacrifices. En l'occurrence les f\u00eates o\u00f9 l'on c\u00e9l\u00e8bre les noces de feu entre l'homme et le moteur, la conscience et la machine, le cerveau et la m\u00e9canique. L'Antiquit\u00e9 disposait de dionysies, de bacchanales, de saturnales, elle f\u00eatait la f\u00e9condit\u00e9, la floraison, la germination et toutes les manifestations possibles de vitalit\u00e9. La modernit\u00e9 ouverte par les temps m\u00e9caniques rend possibles des comp\u00e9titions sous forme de f\u00eates dans lesquelles on c\u00e9l\u00e8bre la vitesse, l'endurance ou les acc\u00e9l\u00e9rations maximales. Chaque fois, on y installe en ma\u00eetre et en h\u00e9raut \u2013 sinon en h\u00e9ros \u2013 le centaure des Temps modernes dessin\u00e9 par le pilote accoupl\u00e9 \u00e0 sa machine.\n\nSur la terre br\u00fbl\u00e9e du Nevada, \u00e0 Black Rock, Richard Noble d\u00e9passe la vitesse du son et franchit la barre des 1000 km\/h. Le 4 ao\u00fbt 1983, les hommes avancent d'autant leur course vers la lumi\u00e8re. Centaure des Temps modernes, il r\u00e9alise le r\u00eave cher aux Futuristes italiens : c\u00e9l\u00e9brer les noces, les \u00e9pousailles du fer et de la chair, de la m\u00e9canique et du vivant, de l'intelligence humaine et de la technologie m\u00e9canique, du corps et du moteur.\n\nMarinetti appelait de ses v\u0153ux une nouvelle religion de la vitesse, il proph\u00e9tisait des grands rendez-vous d'automobiles et d'a\u00e9roplanes, annon\u00e7ant par l\u00e0 les concentrations de sports m\u00e9caniques et le gigantisme voluptueux des a\u00e9roports. Pareillement, il vantait les m\u00e9rites du train et du transatlantique, tous deux ap\u00f4tres de cette \u00abnouvelle religion morale de la vitesse\u00bb. En 1916, il \u00e9crivait : \u00ab Il faut pers\u00e9cuter, fouetter, aiguillonner, torturer tout ce qui p\u00eache contre la vitesse. \u00bb\n\nDans le v\u00e9hicule qui trace comme la queue d'une com\u00e8te balaie la Voie lact\u00e9e, le corps se transfigure. Les f\u00eates m\u00e9caniques th\u00e9\u00e2tralisent ces \u00e9thiques ludiques. J'ai souvenir, dans l'habitacle d'une Spice qui avait couru les 24 heures du Mans, des m\u00e9tamorphoses qui travaillaient ma chair dans les modalit\u00e9s de cette c\u00e9l\u00e9bration : passage du silence dans le volume habit\u00e9 au bruit du moteur br\u00fblant derri\u00e8re le dos, harnachement et invitation \u00e0 faire corps avec le si\u00e8ge profil\u00e9 du v\u00e9hicule, allumage des voyants sur le tableau de bord, r\u00e9servoirs, contact, vibrations qui co\u00efncident avec le hurlement du moteur, odeurs d'essence, de gaz br\u00fbl\u00e9s, de gomme, de caoutchouc. Et d\u00e9part de la voiture.\n\nLes acc\u00e9l\u00e9rations, les freinages, les virages, les trajectoires dessin\u00e9es au plus pr\u00e8s, les gestes du pilote, le passage des vitesses qui craquent, puis la plainte des engrenages, tout contribue \u00e0 d\u00e9multiplier l'incarnation. A l'entr\u00e9e d'un virage, la d\u00e9c\u00e9l\u00e9ration concentre le corps dans le ventre puis les jambes, enfin les pieds. A la sortie, l'acc\u00e9l\u00e9ration r\u00e9installe le centre de gravit\u00e9 dans le ventre, dans les reins, autour de la colonne vert\u00e9brale o\u00f9 s'enroulent les \u00e9motions, les sensations et les perceptions. L'habitacle se sature d'odeurs m\u00e9lang\u00e9es, de vacarme, de vibrations et de chaleur. La temp\u00e9rature monte jusqu'\u00e0 cinquante degr\u00e9s. Le paysage se fond dans des optiques semblables \u00e0 celles des a\u00e9ropeintres futuristes, voire dans les lumi\u00e8res qui caract\u00e9risent Turner. Les \u00e9nergies musiqu\u00e9es installent dans un temp\u00e9rament sonore et \u00e9nerg\u00e9tique proche des compositions d'Edgard Var\u00e8se.\n\nLe v\u00e9hicule arr\u00eat\u00e9, le moteur coup\u00e9, je sens encore les vibrations dans l'\u00e9picentre des muscles. La chair para\u00eet trembler encore, oublieuse du sol retrouv\u00e9. Le temps de l'exp\u00e9rience g\u00e9n\u00e8re des dur\u00e9es singuli\u00e8res dans lesquelles la mati\u00e8re du corps enregistre variations aigu\u00ebs, tensions et rel\u00e2chements, contractions et d\u00e9tentes. L'ensemble de la chair a v\u00e9cu un autre temps dans lequel l'int\u00e9grit\u00e9 du corps se joue. La vitesse excessive, ludique et tragique, installe la vie plus en proximit\u00e9 avec l'accident, sinon la catastrophe. Le geste est essentiel, on lui doit le p\u00e9ril ou la s\u00e9curit\u00e9. En acc\u00e9l\u00e9rant le corps, en le portant \u00e0 une vitesse qui le modifie, l'affecte et le travaille, le pilote tutoie le pire, plus ou moins. Parfois plus, voire trop. Seule est riche l'existence jou\u00e9e. Risquer de perdre seul rend possibles les gains jubilatoires. Cette \u00e9thique de la course vaut pour l'existence dans sa nudit\u00e9 radicale.\n\nJ'aime les vitesses excessives pour les frissons donn\u00e9s, pour l'acuit\u00e9 n\u00e9cessit\u00e9e, celle des b\u00eates de proie dans la savane, pour l'attention d\u00e9cupl\u00e9e qu'elles exigent \u2013 toujours la lutte pour la survie. Tout alors est plus important. Chaque variation musculaire infinit\u00e9simale se multiplie. L'anodin dans une logique lente devient crucial et d\u00e9terminant dans la vitesse. Le c\u0153ur acc\u00e9l\u00e8re, la respiration se modifie, plus courte et plus profonde. Parfois le souffle se retient comme avant le bondissement du fauve. La tension nerveuse et l'\u00e9motion musculaire ressemblent vraisemblablement \u00e0 celles qui hantaient les hommes primitifs dans leurs cavernes face au p\u00e9ril. Les f\u00eates c\u00e9l\u00e9br\u00e9es sont celles des temps furieux et violents.\n\nDans l'habitacle de la Spice, dans ma voiture, la nuit, dans le fuselage des planeurs o\u00f9 parfois je prends place, pendant l'atterrissage et le d\u00e9collage des avions qui me conduisent en Afrique ou en Am\u00e9rique et qui sont gorg\u00e9s de k\u00e9ros\u00e8ne, derri\u00e8re la vitre des trains \u00e0 grande vitesse lorsqu'ils d\u00e9passent les 300 km\/h, sur une moto au compteur presque bloqu\u00e9, je songe toujours \u00e0 ces f\u00eates modernes qui c\u00e9l\u00e8brent \u00e0 chaque fois l'empire et le triomphe des hommes sur les \u00e9l\u00e9ments.\n\nLa vitesse occasionne des bacchanales soucieuses d'exprimer la densit\u00e9 des volupt\u00e9s et des jubilations quand elles se prennent non loin de Thanatos tenu \u00e0 distance comme une meute en furie. Parfois, imprudence ou chance qui tourne, force du destin ou caprices de l'al\u00e9a, comme pour se venger d'\u00eatre ainsi tutoy\u00e9e et moqu\u00e9e, d\u00e9fi\u00e9e et jou\u00e9e, la mort reprend ses droits et cueille l'insolent qui a voulu la toiser en s'accouplant \u00e0 la vitesse. On conna\u00eet la punition, digne de l'Enfer de Dante : ceux qui auront voulu aller vite, le plus vite possible, pareils \u00e0 la foudre qui tombe, ceux qui auront d\u00e9sir\u00e9 les fulgurances et les vertus de l'\u00e9clair, les z\u00e9brures de l'acier et les plaisirs de la vitesse seront arr\u00eat\u00e9s dans leur course, foudroy\u00e9s par cette lumi\u00e8re qu'ils d\u00e9siraient. Dans le n\u00e9ant dont les portes s'ouvrent alors pour toujours, le nautonier employ\u00e9 aux basses besognes abandonnera la d\u00e9pouille au dieu qu'on c\u00e9l\u00e8bre en pareil endroit : la lenteur, cette singuli\u00e8re lenteur qui, l\u00e0-bas, a \u00e9pous\u00e9 l'\u00e9ternit\u00e9.\n10\n\nPOLITIQUE DE LA GASTRONOMIE\n\nLa m\u00e9moire n'est pas la chose du monde la mieux partag\u00e9e et le long enfouissement du pass\u00e9 vichyste fran\u00e7ais semble pour beaucoup dans l'actuelle r\u00e9\u00e9mergence des id\u00e9es de cette \u00e9poque, sinon dans le recyclage pur et simple de cette id\u00e9ologie honteuse \u2013 y compris dans nombre d'endroits o\u00f9 l'on croirait la neutralit\u00e9 de mise. Ainsi la gastronomie et l' \u0153nologie, domaines dans lesquels on imagine mal la politique avoir son mot \u00e0 dire. Et pourtant. Ici comme ailleurs, dans la philosophie ou les beaux-arts, la musique et les arts plastiques, nombreux sont ceux qui distillent des pens\u00e9es qui, peu ou prou, doucement, lentement, mais s\u00fbrement, font le lit des id\u00e9ologies d'une extr\u00eame droite soucieuse d'asseoir intellectuellement sa l\u00e9gitimit\u00e9 et sa respectabilit\u00e9 avant une conqu\u00eate plus agressive du pouvoir. Aucune parole n'est neutre, aucune prise de position l\u00e9g\u00e8re lorsqu'on donne des armes \u00e0 ceux qu'on dit honnir. Il ne faut pas m\u00e9riter des amis dont on refuserait la complicit\u00e9, et la prudence devrait \u00eatre de mise quand on signe ici ou l\u00e0 des lignes qui r\u00e9jouissent les r\u00e9visionnistes soucieux de c\u00e9l\u00e9brer sans cesse les vertus mar\u00e9chalistes travesties ou en pleine lumi\u00e8re.\n\nQuelles sont-elles ? Car on finit par ignorer l'existence dans l'Hexagone, au pays du coq agressif, d'un fascisme sp\u00e9cifiquement fran\u00e7ais dont les lignes de force furent nettes, alors, avant de devenir floues pour la plupart des citoyens d'aujourd'hui. On se souvient de l'antis\u00e9mitisme et des initiatives collaborationnistes actives ; on sait peut-\u00eatre, aussi, la politique familialiste, nataliste, patriotique, nationaliste, le tout doubl\u00e9 par l'\u00e9loge du travail, la F\u00eate des m\u00e8res, les chantiers de jeunesse, la c\u00e9l\u00e9bration des travaux des champs ; on ignore plus nettement le contenu radicalement traditionaliste de la r\u00e9volution nationale et sa comm\u00e9moration perp\u00e9tuelle des r\u00e9gions, de la terre, du folklore, des paysans, des terroirs, des campagnes ; sait-on, enfin, ce que tout cela supposait comme m\u00e9taphysique ? Haine de la raison et des rationalistes, m\u00e9pris des intellectuels et des cit\u00e9s qui les abritent, \u00e9loge de l'instinct et des \u00e9motions, v\u00e9n\u00e9ration des terriens et des campagnes o\u00f9 ils travaillent la terre. Condamnation radicale et syst\u00e9matique de la modernit\u00e9 et c\u00e9l\u00e9bration automatique de la tradition : voil\u00e0 les ma\u00eetres mots de ce qui fit globalement la pens\u00e9e de Vichy.\n\nComment, d\u00e8s lors, peut-on reprendre ces id\u00e9es-l\u00e0 sans un quelconque malaise ? Sans une once de frissonnement ? Trop peu de cadavres, pas assez de boucherie dans l'Europe, \u00e0 cette \u00e9poque, pour justifier si peu de sensibilit\u00e9, actuellement, aux d\u00e9sirs gramsciens de l'extr\u00eame droite obs\u00e9d\u00e9e par la reconqu\u00eate du pouvoir intellectuel en forme de prop\u00e9deutique \u00e0 la reconqu\u00eate pure et simple? La le\u00e7on \u00e9tait-elle si peu \u00e9vidente, hier, qu'un acad\u00e9micien d'aujourd'hui puisse, l\u00e9ger et incons\u00e9quent, maintenant, vanter les m\u00e9rites des manieurs de pelles et d'avirons cens\u00e9ment plus vertueux que les intellectuels et les philosophes de cette seconde moiti\u00e9 du si\u00e8cle? Les cons\u00e9quences furent-elles si b\u00e9nignes, si indiff\u00e9rentes, qu'on distribue aujourd'hui des satisfecit aux musiciens et peintres d'avant la modernit\u00e9 pour mieux fustiger ceux qui travaillent depuis Sch\u00f6nberg et Boulez, quand on est auteur de livres philosophiques \u00e0 succ\u00e8s ? L'\u00e9cho est-il \u00e9teint d\u00e9finitivement qu'on pr\u00e9tende faire de la philosophie en recyclant \u00e0 longueur de pages les vertus qui font depuis toujours la famille, le travail et la patrie ? O\u00f9 donc en est la pens\u00e9e, \u00e0 cette heure empuantie par les miasmes de ceux qui, consciemment ou non, redonnent sa dignit\u00e9, pourtant d\u00e9finitivement perdue, \u00e0 un Mar\u00e9chal de France d\u00e9liquescent?\n\nCar la gastronomie et l'\u0153nologie ont aussi des auxiliaires aux c\u00f4t\u00e9s des id\u00e9ologues de la r\u00e9volution nationale. Voulez-vous du terroir, des r\u00e9gions, des paysans, de la terre, de l'authenticit\u00e9? Souhaitez-vous des produits sains, tatou\u00e9s avec leur provenance d'origine, fran\u00e7ais, enracin\u00e9s? Aspirez-vous \u00e0 vous d\u00e9barrasser du cosmopolitisme, de l'internationalisation, de l'argent, des marchands ? En avez-vous assez des m\u00e9t\u00e8ques que sont les \u00e9pices basan\u00e9es aux odeurs fortes, conqu\u00e9rantes et imp\u00e9rieuses? Aimeriez-vous jeter par-dessus bord les intellectuels qui cuisinent, les conceptuels, les c\u00e9r\u00e9braux, pour leur pr\u00e9f\u00e9rer la plat\u00e9e des grand-m\u00e8res, nourrissante, bourrative, adapt\u00e9e \u00e0 la m\u00e9taphysique du trou \u00e0 boucher selon l'expression de Sartre? Que vous dirait la fin des ploutocrates et des \u00e9litistes, des francs-ma\u00e7ons et des mondialistes dans les cuisines ?\n\nSi d'aventure vous avez souscrit, alors vous aimerez le manifeste sign\u00e9 par Marc de Champ\u00e9rard et intitul\u00e9 Terroirs d'en France. Passons sur la syntaxe approximative du titre. Arr\u00eatons-nous \u00e0 la forme : sans conteste, la page se veut un contrepoint aux deux que sign\u00e8rent Christian Gault et Henri Millau pour le bapt\u00eame de la nouvelle cuisine en 1971. L'ennemi est d'ailleurs nettement point\u00e9 comme tel. Or les deux textes agissent chacun \u00e0 leur mani\u00e8re en discours de la m\u00e9thode de ce que peuvent \u00eatre deux types de cuisine, autant dire deux sympt\u00f4mes d'une m\u00eame \u00e9poque. D'une part, dans l'\u00e8re des revendications conceptuelles et minimales du moment, la nouvelle cuisine promeut le r\u00f4le cardinal, majeur, de la pens\u00e9e dans son domaine : associations, proximit\u00e9s, quintessences, pr\u00e9sentations, exp\u00e9rimentations, raret\u00e9s, extravagances, jeux, elle signe une volont\u00e9 esth\u00e9tique de rompre avec la cuisine bourgeoise h\u00e9rit\u00e9e de Car\u00eame. D'autre part, en pleine \u00e9poque fin de si\u00e8cle caract\u00e9ris\u00e9e par le retour tous azimuts aux s\u00e9curit\u00e9s traditionnelles, la cuisine de terroir invite \u00e0 la r\u00e9p\u00e9tition, \u00e0 l'\u00e9ternelle duplication, \u00e0 la fixit\u00e9 : bouillir, mijoter, noyer dans les sauces, c\u00e9l\u00e9brer l'\u00e9paisseur et la lourdeur pourvu qu'elles soient r\u00e9gionales et de terroir. Elle aspire \u00e0 restaurer un ordre ancien \u2013 ce qui illustre l'\u00e9tymologie de r\u00e9actionnaire.\n\nQue stigmatise Marc de Champ\u00e9rard ? Le cosmopolitisme am\u00e9ricain : toujours utile pour servir de repoussoir : une vertu sur les plages du d\u00e9barquement et un vice dans notre mode d'existence ; l'internationalisation de la gastronomie : les produits standards, sans identit\u00e9, voyageurs dont on ignore la provenance ; le fantasme de la st\u00e9rilit\u00e9, de l'insipide; les grandes surfaces de distribution. L'Europe, bien \u00e9videmment. O\u00f9 donc, pourtant mieux qu'en dehors de France, constate-t-on l'unicit\u00e9 et la singularit\u00e9 de la gastronomie fran\u00e7aise ? Car elle seule a connu une r\u00e9volution copernicienne avec la nouvelle cuisine et l'on reconna\u00eet n'importe o\u00f9 sur la plan\u00e8te les cuissons et les appr\u00eats fran\u00e7ais. La France n'est pas contamin\u00e9e par le monde entier car elle exporte sur ce terrain un savoir-faire \u00e9vident. Loin d'\u00eatre menac\u00e9e, la gastronomie fran\u00e7aise travaille toujours en pointe, avec ses artistes et gr\u00e2ce \u00e0 eux.\n\nQuels sont les principes du manifeste traditionaliste ? La revendication de l'authenticit\u00e9, de la v\u00e9rit\u00e9, des saveurs, du patrimoine alimentaire des r\u00e9gions ; l'investiture populiste donn\u00e9e au grand public pour juger de l'excellence d'une cuisine ; le m\u00e9pris des laboratoires d'artistes au profit des fermes auberges et des chambres d'h\u00f4tes associ\u00e9es ; la sant\u00e9, l'\u00e9quilibre et la m\u00e9moire restaur\u00e9s. Qui croit que les artistes de la gastronomie fran\u00e7aise pratiquent une cuisine inauthentique, fausse, insipide, avec des produits de synth\u00e8se ? Qui voudrait mettre aux voix de la d\u00e9mocratie et du plus grand nombre la d\u00e9cision d'appartenance \u00e0 l'histoire de la musique et la qualit\u00e9 esth\u00e9tique d'une sonate de Dutilleux? Qui pourrait bien imaginer que la nouvelle cuisine et ses suivantes induisent maladie, d\u00e9s\u00e9quilibre ou proc\u00e8dent d'une quelconque amn\u00e9sie?\n\nQu'on prenne garde, car tout cela permet un r\u00e9pertoire des id\u00e9aux de Vichy : la terre qui, elle, ne ment pas ; le jugement de go\u00fbt confin\u00e9 aux borborygmes du bon sens populaire ; le tropisme pour les m\u00e9taphores hygi\u00e9nistes, de la st\u00e9rilit\u00e9 \u00e0 la maladie, de l'insipide au corrompu; l'opposition des intellectuels, les artistes, et des manuels, les artisans, pour \u00e9lire les seconds au d\u00e9triment des premiers ; la haine de ce qui n'est pas indig\u00e8ne, fix\u00e9, enracin\u00e9; le discr\u00e9dit jet\u00e9 sur les avant-gardes modernistes et la mise en avant des conservations du patrimoine ; la trompette de la d\u00e9cadence embouch\u00e9e et le jeu avec les peurs identitaires ; les instincts agoniques modul\u00e9s dans les appels au sursaut et \u00e0 la restauration ; la nostalgie excit\u00e9e, le futurisme fustig\u00e9. Cessons l\u00e0.\n\nLa rh\u00e9torique binaire et manich\u00e9enne de Marc de Champ\u00e9rard trahit une simplicit\u00e9 qui confine au simplisme. Car de quelle cuisine s' agit-il ? Peut-on en parler au singulier, comme si toutes se r\u00e9sumaient \u00e0 une seule rubrique ? De la cuisine artisanale et r\u00e9p\u00e9titive des grand-m\u00e8res, dont le lieu est familial, aux laboratoires des artistes gastrosophes dont la topique est m\u00e9taphysique, il existe un nombre incalculable de variations. Et pour quelles raisons, diable, faudrait-il choisir l'une pour mieux fl\u00e9trir l'autre ? La restauration affective et quotidienne ne co\u00efncide pas avec la gastronomie esth\u00e9tique et ponctuelle. A chaque moment, pour chaque occasion correspondent un lieu, une \u00e9thique, une cuisine.\n\nPourquoi d\u00e9truire la gastronomie dont l'essence se r\u00e9v\u00e8le diff\u00e9rente ? Car la cuisine haut de gamme est \u00e0 la restauration quotidienne ce que la haute couture est au pr\u00eat-\u00e0-porter : un lieu d'\u00e9laboration de formes et de pens\u00e9es nouvelles, une occasion de productions et de concepts novateurs, un laboratoire d'exp\u00e9rimentations existentielles. La grande cuisine fran\u00e7aise exige cette recherche commune \u00e0 tous les espaces esth\u00e9tiques du moment : la nouvelle cuisine co\u00efncidait, sur le terrain artistique et \u00e9thique, avec le go\u00fbt pour le minimal perceptible alors dans les arts plastiques, la litt\u00e9rature ou la musique. Celle d'aujourd'hui semble se formuler dans ce qui me para\u00eet caract\u00e9riser notre \u00e9poque : la grande signature, l'individualit\u00e9 forte et singuli\u00e8re, la revendication d'un temp\u00e9rament et d'un style subjectifs.\n\nQu'on songe \u00e0 Marc Veyrat, Claude Gagnaire ou Michel Bruneau qui d\u00e9veloppent le concept d'une cuisine h\u00e9doniste dans laquelle les recettes d'antan, les sp\u00e9cialit\u00e9s locales, les produits r\u00e9gionaux ou du cru sont sublim\u00e9s par leur g\u00e9nie propre qui permet, par exemple chez Michel Bruneau, un pigeonneau de Cr\u00e8vec\u0153ur en cro\u00fbte de sel et gousses de vanille, un civet de morilles et foie gras, truffes himalayennes ou une soupe de fraises aux piments de Jama\u00efque, \u00e0 chaque fois, une citation de la r\u00e9gion normande, voire une c\u00e9l\u00e9bration de ses produits, sans pour autant n\u00e9gliger l'apport cosmopolite des vanilles, truffes et piments.\n\nFaut-il montrer la Normandie dans sa superbe ? Michel Bruneau \u00e9vitera le poulet Vall\u00e9e d'Auge, la tarte aux pommes ou la teurgoule arros\u00e9s de cidre. Sa g\u00e9ographie sentimentale suppose plus de finesses. La Normandie est-elle une r\u00e9gion rude, sauvage, o\u00f9 les pluies le disputent aux vents et aux ciels c\u00e9l\u00e9br\u00e9s par Eug\u00e8ne Boudin ? Est-elle rigoureuse, mais lav\u00e9e de belles lumi\u00e8res, aust\u00e8re, mais rehauss\u00e9e par une infinie gamme de verts ? Brille-t-elle par ses registres m\u00e9lang\u00e9s, la terre et la mer, les c\u00f4tes pour la p\u00eache et le pays de l'int\u00e9rieur pour les produits plus typiques ? Le Cotentin travaill\u00e9 comme un paysage romantique et le Pays d'Auge indolent dans ses clart\u00e9s ? Alors il faut magnifier tout cela dans des plats qui exprimeront ces vertus et ces qualit\u00e9s : un grillage de viande sur des galets r\u00e9cup\u00e9r\u00e9s au bord de mer et port\u00e9s \u00e0 l'incandescence, un poisson cuit sur des os de b\u0153uf, un homard au lait et foin, des hu\u00eetres au parfum de camembert, un d\u00e9lice caf\u00e9, tabac, whisky. Rien qui suppose la le\u00e7on restrictive prise chez les anc\u00eatres, tout qui d\u00e9montre un talent extraordinaire pour la sublimation, dans ses acceptions freudiennes et chimiques.\n\nDans la po\u00e9tique gastronomique, au sens grec des termes, on retrouve tout ce qui fait l'artiste dans son domaine : la subjectivit\u00e9 par laquelle s'expriment et se cristallisent des intuitions, le temp\u00e9rament et le caract\u00e8re au service d'une vision du monde \u2013 la Weltanschauung des philosophes allemands ; la cr\u00e9ativit\u00e9 se nourrissant d'imaginaire, de r\u00eave et d'audaces ; la convergence des instincts dans des formes \u00e0 chaque fois nouvelles qui expriment la quintessence d'une \u00e9poque, l'esth\u00e9tique au service d'une \u00e9thique, voire l'\u00e9thique dite par des moyens esth\u00e9tiques, l'h\u00e9donisme jubilatoire et ludique. En fait, tout ce qui autorise la lecture du travail des cuisiniers comme une variation possible sur un th\u00e8me m\u00e9taphysique.\n\nLa cuisine bourgeoise de Car\u00eame exprime la passion pour les travestissements, les enrobages, les disparitions de l'essentiel sous le monumental clinquant, et jusqu'\u00e0 Escoffier, cet art particulier fournit la preuve rutilante de l'existence du monde triomphant aux affaires; la nouvelle cuisine signifie la fin d'un monde \u2013 celui qu'inaugure et cl\u00f4t Mai 68 \u2013 et l'\u00e9mergence d'une sensibilit\u00e9 plus conceptuelle, plus c\u00e9r\u00e9brale, plus directement intellectuelle ; la cuisine h\u00e9doniste d'aujourd'hui propose un univers habit\u00e9 par des singularit\u00e9s joyeuses, puissantes et riches, tourn\u00e9es vers une imagination qui d\u00e9passe l'aspect apollinien radical des pr\u00e9d\u00e9cesseurs des ann\u00e9es 1970-80 pour c\u00e9l\u00e9brer plus nettement la folie et l'invention dionysiennes. La cuisine traditionnelle a toujours \u00e9t\u00e9 la n\u00e9gation de ces cuisines qui sont aussi miroirs d'une histoire mobile, dialectique, au profit d'une conception fixiste, immobile et anhistorique du monde. L\u00e0 s'opposent la conception g\u00e9ographioue de la tradition et la conception historique de la modernit\u00e9.\n\nLa perspective artistique et historique n'induit pas pour autant l'oubli pur et simple de la g\u00e9ographie. Mais loin de r\u00e9duire les dimensions de celle-ci aux r\u00e9gions vers lesquelles le monde viendrait, les actions gastronomiques modernistes \u00e9tendent l'influence de la cuisine fran\u00e7aise \u00e0 la plan\u00e8te enti\u00e8re : la premi\u00e8re est centrip\u00e8te et suppose la fermeture sur soi, le monde clos, le local ; la seconde, centrifuge, autorise l'ouverture sur l'univers, le global. L'une est enracin\u00e9e, immobile, l'autre nomade, mondiale. La garbure, le hochepot, la bourride, le cassoulet, proc\u00e8dent des r\u00e9gions o\u00f9 elles satisfont superbement le tourisme. Les cr\u00e9ations des cuisiniers-artistes fournissent des occasions de dire l'esprit fran\u00e7ais \u00e0 Hong Kong, New York, Osaka, Berlin, Londres & Cie. Et toutes deux ont leur fonction, leur utilit\u00e9. Mais s\u00fbrement pas l'une contre l'autre.\n\nToute vision binaire, manich\u00e9enne et exclusive sert des int\u00e9r\u00eats id\u00e9ologiques n\u00e9fastes et partisans. La cuisine de tradition sert \u00e0 exprimer les parties d'une entit\u00e9 nationale, la modernit\u00e9, le tout de cette globalit\u00e9. Or le tout ne se r\u00e9duit pas \u00e0 la somme de ses parties en vertu d'une simple arithm\u00e9tique, car la transcendance proc\u00e8de d'artistes dans leur domaine de comp\u00e9tence privil\u00e9gi\u00e9. En cuisine, l'artisan g\u00e9ographe de la nostalgie n'est pas l'artiste inventeur du futur, mais tous deux demandent au m\u00eame mat\u00e9riau d'exprimer une m\u00e9taphysique. Le d\u00e9sir de guerroyer, d'imposer un monde contre l'autre, la volont\u00e9 terroriste, imp\u00e9rieuse et int\u00e9griste des cuisines de r\u00e9p\u00e9tition ou des cuisines d'invention, cachent mal une aspiration \u00e0 mettre la gastronomie au service de causes qui oublient la vis\u00e9e h\u00e9doniste de toutes les cuisines. Il n'existe pas de petite ou de grande cuisine, de n\u00e9gligeable ou d'essentielle, si elle est bonne, c'est-\u00e0-dire si elle procure \u00e0 celui qui l'\u00e9lit, au moment o\u00f9 il la choisit, les plaisirs qu'il attend d'elle. L\u00e0 encore se v\u00e9rifie la validit\u00e9 des th\u00e8ses nietzsch\u00e9ennes sur le perspectivisme en esth\u00e9tique. Mais si l'on doit prendre parti dans un combat dont les initiateurs sont vivement agressifs, alors on saura que la tradition port\u00e9e en \u00e9tendard a toujours \u00e9t\u00e9, contre la modernit\u00e9 vilipend\u00e9e, une occasion de faire le lit des conservateurs, puis des r\u00e9actionnaires, enfin de ceux qui communient et esp\u00e8rent dans les droites les plus infr\u00e9quentables.\n\nDans le m\u00eame ordre d'id\u00e9es, je m'inqui\u00e8te de ceux qui font subir pareil sort aux liquides, les parties les plus intellectuelles et spirituelles du repas selon Brillat-Savarin. Car les pass\u00e9istes et r\u00e9actionnaires en mati\u00e8re de gastronomie solide disposent aussi de thurif\u00e9raires et de pendants en mati\u00e8re de gastronomie liquide. L'\u0153nologie dispose \u00e9galement d'une aile droite qui communie dans les valeurs de la r\u00e9volution nationale des terroirs, des r\u00e9gions et de tout l'attirail conceptuel associ\u00e9. Les ma\u00eetres d'\u0153uvre de ces vins dits naturels et authentiques proc\u00e8dent tous de l'id\u00e9ologie de Jules Chauvet, Constant Bourquin et Max L\u00e9glise, trois fieff\u00e9s droitiers dans leurs domaines de pr\u00e9dilection \u2013 et ailleurs.\n\nLe pr\u00e9facier \u00e0 l'un des livres de L\u00e9glise, un manuel de d\u00e9gustation \u00e0 l'usage des d\u00e9butants, ne cache pas l'objectif de l'auteur : travailler avec des pr\u00e9suppos\u00e9s aristot\u00e9liciens \u00e0 la r\u00e9habilitation de la m\u00e9decine de Paracelse et de la chimie spagirique, c'est-\u00e0-dire de la tradition herm\u00e9tique. Nul doute que l'Aristote convoqu\u00e9 pour les besoins est celui qui travaille tout le Moyen Age, celui qu'a confisqu\u00e9 l'Eglise pour lui faire rendre gorge en le flanquant du tuteur thomiste. Sans oublier pour autant, chez ces sectateurs, le filtre des \u0153uvres d'Herm\u00e8s Trism\u00e9giste, contempteur de choix des rationalit\u00e9s qui feront l'Occident.\n\nL'option aristot\u00e9lico-thomiste de l'\u0153nologue bourguignon n'embarrasse pas plus que de raison ses recherches, car lorsqu'il pr\u00e9tend fonder philosophiquement une esth\u00e9tique sensorielle, il se r\u00e9clame sans ambages des pr\u00e9suppos\u00e9s de Kant et de Hegel, rien de moins. Il faudrait entrer dans les d\u00e9tails pour expliquer d'abord en quoi une esth\u00e9tique sensorielle induit un genre de pl\u00e9onasme puisque l'\u00e9tymologie du premier terme suppose celle du second : une esth\u00e9tique, dans ce qu'en dit l'origine du mot, c'est une th\u00e9orie de la sensation. On n'\u00e9vite pas qu'apr\u00e8s la perception, la sensation et l'\u00e9motion, l'esth\u00e9tique n\u00e9cessite une conceptualisation qui suppose la raison, le concept, l'analyse. Aussi devient-elle rationnelle, quoi qu'on en dise ou pense. On ne saurait se contenter d'une sensation de la sensation. Ensuite, il semble souhaitable de dire tout ce qui s\u00e9pare Kant et Hegel avant de conclure, apr\u00e8s lecture pointilleuse de la Critique de la facult\u00e9 de juger, \u00e0 l'impossibilit\u00e9 radicale de faire du vin, et de la gastronomie, un objet d'esth\u00e9tique ou de critique de la raison transcendantale. Une critique de la raison di\u00e9t\u00e9tique ou oenologique suppose un recours au plus antikantien, a fortiori au plus anti-thomiste des philosophes, \u00e0 savoir Nietzsche.\n\nMais L\u00e9glise besogne loin de toute modernit\u00e9 puisqu'en passant, apr\u00e8s des consid\u00e9rations sur regarder, sentir, go\u00fbter en mati\u00e8re de d\u00e9gustation de vins, il \u00e9gratigne Van Gogh dans une phrase qui m\u00e9rite une station : \u00abIl faut tout le d\u00e9sarroi mental de l'establishment culturel contemporain pour porter au pinacle les violences color\u00e9es d'un Van Gogh ou les d\u00e9lires color\u00e9s de quelques schizophr\u00e8nes. \u00bb Pr\u00e9cisons que le texte date de 1991. Dans la foul\u00e9e, le rock and roll en prend pour son grade et ailleurs la nouvelle cuisine, on s'en serait dout\u00e9, r\u00e9duite \u00e0 n'\u00eatre que l'amusement de snobs confits dans la lassitude, banalement promoteurs de platitudes et de fadeurs alimentaires dirig\u00e9es contre la gastronomie classique. Qu'on se le dise...\n\nQue propose donc cette esth\u00e9tique sensorielle, apr\u00e8s qu'on a d\u00e9cid\u00e9 d'oublier la scolastique traditionnelle, l'alchimie, les approximations philosophiques et les jugements de valeurs r\u00e9actionnaires en mati\u00e8re de modernit\u00e9 ? Rien qui rel\u00e8ve du m\u00e9morable ou r\u00e9volutionne l'esth\u00e9tique en la prenant l\u00e0 o\u00f9 elle a \u00e9t\u00e9 port\u00e9e \u00e0 son point d'incandescence : chez Marcel Duchamp. Car si le vin peut \u00eatre dit une \u0153uvre d'art, ce n'est certainement pas parce qu'il rel\u00e8ve sp\u00e9cifiquement des proc\u00e9d\u00e9s de fabrication ancestraux plus en relation avec l'alchimie et la tradition qu'avec les m\u00e9thodes de l'\u0153nologie moderne. Le vin est une \u0153uvre d'art si et seulement si l'on consent au pr\u00e9suppos\u00e9 esth\u00e9tique de Duchamp en vertu de quoi l'acteur, le regardeur ou le go\u00fbteur fondent la validit\u00e9 artistique d'un geste, d'un mot, d'une \u0153uvre ou d'un objet. Et pour ce faire, aucune esth\u00e9tique n'est pensable en dehors du nietzsch\u00e9isme libertaire de l'inventeur des ready-made. La seule esth\u00e9tique capable de rendre compte d'un grand vin comme d'une sonate de Beethoven est celle du perspectivisme dont proc\u00e8de toute l'avant-garde artistique du si\u00e8cle. En dehors de cette option philosophique, on condamne le vin \u00e0 rester la production d'un viticulteur r\u00e9ductible \u00e0 ses compos\u00e9s chimiques et organoleptiques.\n\nLa p\u00e9roraison de Max L\u00e9glise est loin de parvenir \u00e0 ses fins sur le terrain g\u00e9n\u00e9alogique. D'autant, qu'outre l'apparente ignorance du terrain conceptuel sur lequel il s'installe sans vergogne, l'\u0153nologue ne cache pas l'aune \u00e0 laquelle il mesure les productions esth\u00e9tiques du moment et \u00e9crit : \u00ab Face aux exub\u00e9rances passag\u00e8res des minorit\u00e9s agissantes (ou agit\u00e9es), le calme et la s\u00e9r\u00e9nit\u00e9 conservatrice des majorit\u00e9s silencieuses assurent la continuit\u00e9. \u00bb Dont acte.\n\nA ce point d'avancement, on pourra alors mettre en perspective : la passion immod\u00e9r\u00e9e pour les terroirs ; la v\u00e9rit\u00e9 et l'authenticit\u00e9 de la terre; la pr\u00e9f\u00e9rence de l'alchimie, de la biodynamique et de la tradition \u00e0 toute chimie mol\u00e9culaire et g\u00e9n\u00e9tique de la modernit\u00e9 en acte ; la confiance dans le jugement du peuple ; la m\u00e9fiance \u00e0 l'endroit des sp\u00e9culations intellectuelles de personnes toujours cr\u00e9dit\u00e9es de snobisme et qu'on fl\u00e9trit \u00e0 l'aide du sempiternel vocabulaire hygi\u00e9niste de la maladie, de la fadeur, de la platitude, de l'\u00e9puisement; le culte pour la seule \u00e9motion non doubl\u00e9e d'une r\u00e9flexion, d'une conceptualisation \u2013 chaque fois on retrouve le registre qui fait en toute \u00e9poque les belles heures de la pens\u00e9e r\u00e9actionnaire et conservatrice.\n\nMax L\u00e9glise engendre des \u00e9mules dans un nombre incalculable de clubs qui tous relaient cette pens\u00e9e complice des id\u00e9ologies que l'on sait. Ecologistes int\u00e9gristes, sectateurs forcen\u00e9s de la nature et de ses bienfaits, proph\u00e8tes des naturoth\u00e9rapies et autres hom\u00e9opathies ou m\u00e9decines dites douces, gnostiques et adeptes des microsoci\u00e9t\u00e9s traditionalistes, tous se retrouvent dans ces c\u00e9nacles o\u00f9 les pratiquants officient sous la bienveillante autorit\u00e9 d'un initi\u00e9 qui, en coll\u00e8ge, bien s\u00fbr, d\u00e9cide du bien et du mal, du bon et du mauvais, de ce qu'il faut \u00e9lire ou rejeter. Reste pour les fid\u00e8les, \u00e0 boire la parole du ma\u00eetre et les vins qui vont avec.\n\nExemple en Basse-Normandie autour de Pierre Paillard \u2013 et de son Club du vin authentique \u2013 qui distille sa bonne parole en feuillets \u00e0 usage interne, dans des conf\u00e9rences donn\u00e9es avec l'aval des autorit\u00e9s culturelles du cru. L'ensemble de ces prestations id\u00e9ologiques aboutit \u00e0 un livre, La Qu\u00eate du vin, qui ne dissimule en rien l'id\u00e9ologie du mouvement. Le vin est un myst\u00e8re, la tradition alchimique seule peut en rendre compte; la biodynamie et l'\u00e9cologie sont les seuls moyens d'exprimer son essence; le g\u00e9nie des terroirs seul g\u00e9n\u00e8re la v\u00e9rit\u00e9 ; la d\u00e9cadence a assez dur\u00e9, le temps vient d'organiser \u00ab la renaissance du vin fran\u00e7ais \u00bb.\n\nQuelles sont les b\u00eates noires de Pierre Paillard ? Les m\u00eames que celles de son ma\u00eetre Max L\u00e9glise : la modernit\u00e9, la technologie, l'\u0153nologie interventionniste (sic), la mondialisation, les marchands, la chimie, le progr\u00e8s, les m\u00e9dias. Hier \u00e9tait superbe et la nostalgie ravive les plaies ; aujourd'hui est laid, corrompu, d\u00e9cadent. Hier ? les gens d'Eglise et les premiers vignobles acclimat\u00e9s afin d'accueillir et de pratiquer la charit\u00e9 \u00e0 l'endroit des p\u00e8lerins ou des gens de passage. Aujourd'hui ? les marchands destructeurs de v\u00e9rit\u00e9 des vins. Les premiers avaient pour eux l'honneur, les seconds la malice.\n\nEn vertu de ces principes, on enseigne que l'imp\u00e9rialisme marchand, relay\u00e9 en cela par l'Europe, a rendu impossible un vin issu traditionnellement des vignes. Seuls les adeptes du club savent et peuvent r\u00e9sister \u00e0 ce cosmopolitisme mondialiste en c\u00e9l\u00e9brant des vins non filtr\u00e9s, non soufr\u00e9s, travaill\u00e9s selon les m\u00e9thodes alchimiques et naturelles. D'o\u00f9 la promotion de bouteilles choisies, apr\u00e8s avoir \u00e9t\u00e9 coll\u00e9gialement go\u00fbt\u00e9es, via une Chronique d'avant boire dans laquelle les flacons s\u00e9lectionn\u00e9s peuvent, au choix \u2013 j'ai relev\u00e9 ces parfums dans l'une des parutions \u2013 lib\u00e9rer des senteurs de saindoux, de produits de beaut\u00e9, d'encaustique, de pot-au-feu, de camphre, de loukoum. Sinon de lutherie et de vieille rose. Bigre ! \u00e0 quoi peuvent donc bien ressembler les vins qui n'ont pas \u00e9t\u00e9 retenus ?\n\nLa logique du club est parano\u00efaque : une conspiration mondiale, relay\u00e9e par les m\u00e9dias incapables de lever le tabou, impose des vins frelat\u00e9s \u2013 les bordeaux faisant superbement l'affaire en mati\u00e8re de victime sacrificielle \u2013 , inauthentiques, imbuvables et ind\u00e9fendables de quelque mani\u00e8re que ce soit. On risque le discours r\u00e9sistant pour faire \u00e9chec \u00e0 la ghetto\u00efsation dans laquelle les amateurs de vins biologiques sont pr\u00e9tendument confin\u00e9s. Contre la dictature du mental exerc\u00e9e par les conspirateurs, les affid\u00e9s du vin r\u00e9g\u00e9n\u00e9r\u00e9 c\u00e9l\u00e8brent Dionysos, en oubliant que, l\u00e2ch\u00e9 tout seul dans la nature, sans l'ombre ni le g\u00e9nie tut\u00e9laire d'Apollon, le dieu de la forme et des structures, on peut craindre tout, et le pire, de la divinit\u00e9 orgiaque. La c\u00e9l\u00e9bration des cultes dionysiaques, associ\u00e9e \u00e0 une critique de la raison occidentale, du rationalisme, \u00e0 un \u00e9loge sans retenue de la pens\u00e9e traditionnelle, alchimique et irrationaliste, produit les plus graves d\u00e9rapages id\u00e9ologiques et historiques dans les droites fascisantes de ce si\u00e8cle.\n\nBien s\u00fbr, pers\u00e9cut\u00e9s et r\u00e9sistants, les amateurs de vins authentiques sont accul\u00e9s \u00e0 des positions agoniques. Dans les follicules de l'association, on peut lire ceci : \u00ab Si notre club ressemble un peu \u00e0 une machine de guerre, c'est qu'il est temps d'organiser l' insurrection de l'esprit.\u00bb Les choses, ainsi, ont le m\u00e9rite d'\u00eatre clairement dites. Aussi, lorsque l'on r\u00e9torque \u00e0 Pierre Paillard qu'en proc\u00e9dant de la sorte il pourrait avoir des amis ind\u00e9sirables ou faire le jeu d'une extr\u00eame droite qui sait pouvoir compter sur l'effet gramscien de ce genre de discours, il feint l'\u00e9tonnement sans pour autant oublier que ses propos, son travail et son club ont d\u00e9j\u00e0 obtenu les faveurs et les \u00e9loges appuy\u00e9s de Pr\u00e9sent, le quotidien de l'extr\u00eame droite fran\u00e7aise... Faut-il aller plus avant?\n\nLes travaux de Bourdieu montrent ce que le jugement de go\u00fbt gastronomique exprime d'appartenance sociale, de provenance culturelle ou d'occupation de secteurs \u00e9conomiques. Les arts de la table, la gastronomie et l'\u0153nologie plus particuli\u00e8rement, sont susceptibles de lectures politiques aussi bien que philosophiques ou m\u00e9taphysiques. Les styles, les caract\u00e9ristiques d'une \u00e9poque, d'un temps, s'expriment aussi bien dans une architecture que dans une \u0153uvre symphonique, dans un roman ou une peinture, une sculpture ou un type particulier d'urbanisme. M\u00eamement dans une cuisine o\u00f9 se disent des visions du monde pass\u00e9istes ou modernistes, nostalgiques ou futuristes. Le sens r\u00e9side aussi bien dans le contenu d'une assiette que dans une architrave de cath\u00e9drale gothique.\n\nLes enjeux sont lourds en mati\u00e8re de culture : toutes les civilisations totalitaires ont voulu r\u00e9glementer, r\u00e9gimenter, d\u00e9cider et promouvoir des oukases pr\u00e9sent\u00e9s comme jugements esth\u00e9tiques convenables, pr\u00e9sentables. L'autodaf\u00e9 a toujours \u00e9t\u00e9 le mode d' action des manich\u00e9ens qui jouent une vision du monde contre une autre. Les r\u00e9f\u00e9rences \u00e0 l'authentique, \u00e0 la tradition, \u00e0 la v\u00e9rit\u00e9 du sol et du terroir, sinon du sang et de la race, ne cessent de faire le jeu des amateurs de r\u00e9volution nationale, de redressement id\u00e9ologique, de r\u00e9novation et de restauration identitaire.\n\nLes plus na\u00effs ignorent qu'ils travaillent pour de plus avertis, les moins inform\u00e9s ne savent pas qu'ils d\u00e9blaient le terrain pour de plus cyniques. L'innocence n'est plus d\u00e9fendable ni supportable quand la volont\u00e9 de conqu\u00eate nationaliste de l'extr\u00eame droite s'affiche publiquement et que la culture fait l'objet des premiers assauts. La gastronomie dans ses usages n'est ni neutre ni apolitique. Sans toujours le savoir, elle sert des int\u00e9r\u00eats en \u00e9tant telle plut\u00f4t qu'une autre.\n\nAussi faut-il conna\u00eetre la pertinence de tout combat quand il s'agit d'augmenter la libert\u00e9 et de multiplier les possibles. A d\u00e9faut, quand on a install\u00e9 une nation tout enti\u00e8re dans la c\u00e9l\u00e9bration des terroirs, des r\u00e9gions, de la terre, de l'authentique, de la tradition, de la nature, \u00e7a n'est plus la cuisine des provinces que l'on retrouve dans sa vie quotidienne ou dans son assiette, mais celle des topinambours et des rutabagas, des ersatz de caf\u00e9 et de la saccharine, le tout distribu\u00e9 et r\u00e9glement\u00e9 avec les tickets de rationnement. Adieu veaux, vaches, cochons, couv\u00e9es, adieu artistes aux fourneaux, bien s\u00fbr, mais aussi grand-m\u00e8res au coin de leurs chemin\u00e9es. Reste un peu de pain sec et d'eau. Trop tard car, alors, les seules d\u00e9bauches de viandes consenties par les agueusiques au pouvoir sont celles de la chair humaine alors devenue la moins co\u00fbteuse des denr\u00e9es.\n11\n\nIDENTIT\u00c9 DE LA PHOTOGRAPHIE\n\nOn sait la dilection toute particuli\u00e8re du philosophe Z\u00e9non d'El\u00e9e pour les tortues aussi rapides que les coureurs, les athl\u00e8tes qui ne parviennent pas \u00e0 doubler lesdites bestioles, le bruit que fait un tas de mil en tombant, les groupes lanc\u00e9s \u00e0 vive allure dans la direction l'un de l'autre et qui ne font que se croiser, ou encore les fl\u00e8ches qui, bien que tir\u00e9es par un archer talentueux, n'arrivent jamais sur leur cible. Et alors, diront les impatients ? Alors Z\u00e9non a raison lorsque, dans son c\u00e9l\u00e8bre argument logique, il affirme que le mouvement n'\u00e9tant constitu\u00e9 que d'instants immobiles about\u00e9s, il n'y a pas de dynamisme possible. Voil\u00e0 pourquoi les fl\u00e8ches n'arrivent jamais \u00e0 bon port. Et pour quelles raisons, en dehors des instants immobiles compos\u00e9s, rien n'existe.\n\nEt qui donc a v\u00e9rifi\u00e9 exp\u00e9rimentalement la pertinence des arguments de l'El\u00e9ate? Nic\u00e9phore Niepce et F\u00e9lix Tournachon, dit Nadar, les deux protagonistes de la g\u00e9n\u00e9alogie de la photographie. Pas besoin de convoquer un professeur au Coll\u00e8ge de France, Bergson en l'occurrence, pour r\u00e9fl\u00e9chir sur les relations entre la pens\u00e9e et le mouvant, le temps et le mouvement, l'instant mobile et l'\u00e9ternit\u00e9 immobile avant de conclure \u00e0 l'invalidit\u00e9 des arguments de Z\u00e9non. Les photographes font tous les jours la d\u00e9monstration du contraire : Z\u00e9non pense juste quand il scrute le mouvement, comme d'autres la mati\u00e8re, et qu'il isole l'instant en guise de composante essentielle. D'autres d\u00e9couvraient simultan\u00e9ment l'atome, l'ins\u00e9cable. L'instant est au mouvement ce que l'atome est \u00e0 la mati\u00e8re : une quintessence irr\u00e9ductible, le noyau dur du r\u00e9el envisag\u00e9.\n\nLe photographe agit en grand qu\u00eateur de cet \u00e9picentre. Son d\u00e9sir consiste \u00e0 fixer l'un de ces milliers d'instants constitutifs du temps pour en extraire mati\u00e8re \u00e0 sens et \u00e0 figure, \u00e0 perception, sinon \u00e0 \u00e9motion. Dans l'acte de photographier g\u00eet tout ce qui permet l'incarnation du propos de Z\u00e9non : au creux de ce qui bouge, il y a l'immobile, tapi, fig\u00e9, et il s'agit de le traquer, puis de s'en rendre ma\u00eetre afin de le montrer, de l'exhiber comme un troph\u00e9e. Cette op\u00e9ration fait de son entrepreneur un acteur op\u00e9rant aux carrefours de la d\u00e9miurgie et de la m\u00e9taphysique, de la ph\u00e9nom\u00e9nologie et de la dialectique, de l'\u00e9thique et de l'esth\u00e9tique, autant dire qu'en sculptant ainsi le temps, en congelant la m\u00e9moire, en ouvrageant sa conscience, le photographe incarne un souci proprement philosophique.\n\nD\u00e9miurge est le cr\u00e9ateur d'images, le fabricateur d'ic\u00f4nes qui veut arr\u00eater le temps, s'en rendre ma\u00eetre et possesseur dans sa proposition d'un instant saisi en son essence. Sa proie? le Ka\u00efros des Grecs, l'instant propice, celui en de\u00e7\u00e0 et au-del\u00e0 duquel rien n'est possible ou pensable : soit parce que l'int\u00e9r\u00eat r\u00e9side dans un moment particulier du mouvement encadr\u00e9 par des p\u00e9rip\u00e9ties sans importance, soit parce qu'il est dans la persistance d'une figure qu'on veut faire durer dans une variation sur le th\u00e8me de la nature morte. Photographier l'instant d'un mouvement ou l'\u00e9ternit\u00e9 d'une nature qu'on pourrait dire morte, \u00e0 chaque fois, ce qui est vis\u00e9, c'est l'immobile qui r\u00e9side dans le mouvement ou dans le temps.\n\nMuybridge et Marey exp\u00e9rimentent les formes du temps, les successions de la dur\u00e9e, les d\u00e9compositions et d\u00e9constructions du divers, du multiple, en des images entre lesquelles il y a place, encore, pour d'autres images. Tous deux t\u00e2chent, dans leurs obsessions chronoscopiques, de montrer ce que Z\u00e9non enseigne : le mouvement r\u00e9duit \u00e0 la somme des instants fixes qui le composent, la vie dynamique d\u00e9montr\u00e9e dans ses constructions statiques. Eux seuls d\u00e9plient, d\u00e9ploient, ce qui fait le d\u00e9tail de la qu\u00eate de tout photographe : partir \u00e0 la recherche de la fixit\u00e9 avec laquelle on fait la mobilit\u00e9, tenter d'isoler et de montrer l'immarcescible.\n\nEn jouant avec le temps, en allant chercher ce qui le constitue, de quoi il est fait, du moins sur quels modes il appara\u00eet, le photographe op\u00e8re comme le philosophe qui t\u00e2che de r\u00e9duire et de r\u00e9unir sous le registre de l'un ce qui se montre exclusivement sous le mode du divers. L'un et l'autre veulent le clich\u00e9 ou le concept uniques pour rendre compte du r\u00e9el multiple. Cette unicit\u00e9 \u00e0 laquelle tend l'homme de la photographie en fait un adepte de Parm\u00e9nide contre le cin\u00e9aste affid\u00e9 d'H\u00e9raclite. Le premier pr\u00e9socratique, lui aussi \u00e9l\u00e9ate, comme Z\u00e9non, philosophe sur l'un qui se suffit, parfait, sans commencement ni fin, parce que proc\u00e9dant de lui-m\u00eame dans la plus absolue des incorruptibilit\u00e9s : cet un pourrait \u00eatre la photographie. Le second, \u00e9ph\u00e9sien, pense le fleuve qui coule \u00e9ternellement, et dans lequel on ne saurait se baigner deux fois : cette fluidit\u00e9, cet \u00e9coulement semblent caract\u00e9riser le cin\u00e9ma.\n\nDans l'un et l'autre cas, le temps id\u00e9al se con\u00e7oit diff\u00e9remment : arr\u00eat\u00e9 pour le photographe, fix\u00e9, fig\u00e9, entrav\u00e9. Reconstruit, reconsid\u00e9r\u00e9 pour le cin\u00e9aste, mais tout entier r\u00e9v\u00e9l\u00e9 dans le d\u00e9roulement, le d\u00e9veloppement. D'une certaine mani\u00e8re, Platon r\u00e9concilie les deux pens\u00e9es en faisant du temps l'image mobile de l'\u00e9ternit\u00e9 immobile, de sorte que le photographe appara\u00eet seulement pr\u00e9occup\u00e9 par la qu\u00eate et l'obtention d'une preuve de sa capture de l'\u00e9ternit\u00e9 immobile en une image. Le d\u00e9miurge r\u00e9alise cette magie : concentrer dans un clich\u00e9 la quintessence de ce qui structure le mouvement. Au milieu du fleuve, et pour l'exprimer : la sph\u00e8re...\n\nDans son cheminement, le photographe-d\u00e9miurge se fait aussi m\u00e9taphysicien cr\u00e9ateur de signes g\u00e9n\u00e9rateurs de sens dans le temps, malgr\u00e9 lui, contre lui et avec lui. Les images confectionn\u00e9es sont parentes de celles qui, il y a des millions d'ann\u00e9es, ornaient les murs des cavernes. Une photo de graffiti par Brassa\u00ef l'exprime \u00e0 merveille et constitue un manifeste \u00e0 elle seule. Griffures, mains n\u00e9gatives, figures g\u00e9om\u00e9triques, traces, entailles, tout affirme la puissance de ce que l'on sait depuis Malraux \u00eatre un antidestin. Les g\u00e9ographies lisibles sur les surfaces pari\u00e9tales se superposent \u00e0 celles qu'on d\u00e9chiffre sur les papiers aux sels d'argent : le contraire de palimpsestes. Car dans l'\u00e9poque pr\u00e9historique, malgr\u00e9 le temps qui passe, et contre celui-l\u00e0 m\u00eame, s'appuyant sur lui pour mieux le d\u00e9passer, les artistes install\u00e9s aux deux extr\u00e9mit\u00e9s de cette cha\u00eene humaine actualisent sans cesse la qu\u00eate de signes opposant leur force au n\u00e9ant du monde.\n\nL\u00e0 o\u00f9 l'entropie fait son \u0153uvre, face \u00e0 la destruction, aux ruines qui menacent, devant l'\u00e9vidence d'une mort annonc\u00e9e - de l'individu et des civilisations \u2013 l'homme des cavernes et celui des pellicules photosensibles affirment, sublime r\u00e9volte, une volont\u00e9 d'\u00e9ternit\u00e9. Leurs \u0153uvres sont des r\u00e9sistances, des vols faits au d\u00e9triment du temps et de ses dommages. Ce jeu de l'instant destin\u00e9 \u00e0 l'\u00e9ternit\u00e9 suppose paradoxalement l'emprunt des trajets de la mort : la pose exig\u00e9e du sujet est p\u00e9trification, mise en \u00e9tat de rigidit\u00e9 ou d'immobilit\u00e9 suppos\u00e9e par le tr\u00e9pas. D'o\u00f9 l'invitation de sourire pour conjurer le s\u00e9rieux habituellement prop\u00e9deutique au n\u00e9ant. Sur les suaires, ou les photographies, s'impriment la m\u00e9moire et le souvenir, ces deux modalit\u00e9s du temps cruel.\n\nPhotographier, c'est congeler du temps, immortaliser les hapax qui structurent une existence, un paysage, une \u00e9poque, une situation, un personnage. C'est, par le regard jet\u00e9 sur la photographie tir\u00e9e, viser la r\u00e9it\u00e9ration ind\u00e9finie d'un instant capt\u00e9 un jour. Dans le clich\u00e9 o\u00f9 g\u00eet le moment p\u00e9trifi\u00e9 se trouve ce qui permet \u00e0 n'importe quel instant la chaleur d'un regard et la d\u00e9miurgie d'un nouveau temps, d'un acc\u00e8s neuf \u00e0 des moments anciens. Jadis et nagu\u00e8re deviennent de la sorte ici et maintenant. D'o\u00f9, apr\u00e8s une op\u00e9ration effectu\u00e9e par le photographe sur le temps, la constitution par ses tirages d'une m\u00e9moire, d'une aide apport\u00e9e, par des points de rep\u00e8re fixes, \u00e0 toute volont\u00e9 mn\u00e9monique.\n\nLe temps propose une organisation de la m\u00e9moire. Dans la Voie lact\u00e9e o\u00f9 se perdrait n'importe quel astronome, les instants vol\u00e9s constituent des points lumineux brillants comme des rep\u00e8res cardinaux. Les photographies scintillent en morceaux choisis de r\u00e9el destin\u00e9s, par leur concision, leur charge, leur puissance, leur force, leur originalit\u00e9, leur singularit\u00e9, \u00e0 dire en un \u00e9clair ce qui \u00e9conomise de longs discours. De Lascaux \u00e0 l'\u00e9preuve d\u00e9velopp\u00e9e \u00e0 l'instant, la volont\u00e9 de quintessencier le monde, de le r\u00e9duire \u00e0 deux ou trois images qui en exprimeraient autant, voire plus, que les infinies modulations d'un temps d\u00e9pli\u00e9, hante l'op\u00e9rateur et le laisse sans repos. Dans le fouillis du monde, les photographies sont des lumi\u00e8res.\n\nDurant le trajet qui conduit du temps \u00e0 la m\u00e9moire, toutes les distorsions possibles et imaginables ont l'occasion de s'\u00e9noncer. La photographie entretient une \u00e9trange relation avec la v\u00e9rit\u00e9. Le photographe qui sait, lui, ce qu'il veut fixer, donc montrer, dire, est \u00e9galement \u00e9thicien, au sens qu'on trouve sous la plume de Kierkegaard \u2013 combattant pour un syst\u00e8me de valeurs. O\u00f9 r\u00e9side le vrai dans un clich\u00e9 qui se contente de montrer? Un photographe de guerre (Cecil Beaton) propose un cadavre ou un soldat qui s'\u00e9croule sous le feu : ennemi, ami ? Bourreau, victime ? Vrai mort qui repose ou acteur qui pose ? Belle ou vraie ? Juste ou fausse ? Information ou propagande? Brutalit\u00e9 de la situation crue ou mise en sc\u00e8ne machiav\u00e9lique? Que dire de l'intervention sur le cadre, le tirage? Sur la dissimulation par retouche ou montage, voire sur les magies rendues possibles par la num\u00e9risation qui, \u00e9trange paradoxe, replace aujourd'hui la photographie dans la situation de concurrence o\u00f9 elle se trouvait \u00e0 sa naissance avec la peinture. Le pixel aurait r\u00e9joui les pointillistes...\n\nLa photographie est un fragment de r\u00e9el \u00e0 lire, comme les arch\u00e9ologues pratiquent avec les pi\u00e8ces d\u00e9couvertes \u00e0 partir de quoi ils reconstituent l'ensemble de la forme et de la figure dont elles proc\u00e8dent. Ni vraie ni fausse, elle structure un sympt\u00f4me de ce qui, pour faire sens, m\u00e9rite lecture, mise en perspective, compr\u00e9hension. Elle n'est pas imm\u00e9diatement donn\u00e9e et suppose une culture pour le d\u00e9chiffrage. La m\u00e9taphysique et l'\u00e9thique du photographe rel\u00e8vent du perspectivisme nietzsch\u00e9en : il propose une lecture, une vision du monde, mais n'\u00e9nonce rien qui proc\u00e8de de la v\u00e9rit\u00e9. Un moment ontologique, en l'occurrence photographique, ne peut r\u00e9capituler un mouvement m\u00e9taphysique, celui du monde, qu'apr\u00e8s avoir pris connaissance de ce qui lie les deux instances. De quelle guerre, par quel photographe, dans quel camp, \u00e0 quel moment, dans quelles circonstances telle ou telle photographie a-t-elle \u00e9t\u00e9 prise ? Alors seulement on peut envisager le sens, apr\u00e8s la premi\u00e8re \u00e9motion due \u00e0 la seule mise en pr\u00e9sence avec l'image.\n\nTout clich\u00e9 se contentant de n'\u00eatre que lui-m\u00eame pour pr\u00e9tendre faire sens risque de dissimuler l'essentiel de son projet qu'une lecture avis\u00e9e met en p\u00e9ril par l'information apport\u00e9e. Photographier, ce peut \u00eatre aussi falsifier, mentir, servir une propagande politique ou id\u00e9ologique : l'un qui traque les Communards (Appert), l'autre les faci\u00e8s anthropom\u00e9triques (Bertillon), tel qui c\u00e9l\u00e8bre les jeux de Berlin ou l'Allemagne nazie (Riefenstahl), un autre la Chine populaire (Cartier-Bresson), un dernier les produits les plus embl\u00e9matiques de la soci\u00e9t\u00e9 de consommation, sinon l'usage \u00e9rotique ou sensuel des femmes \u00e0 destination du march\u00e9 (Newton), tous diront au moins deux choses, la premi\u00e8re qui montre une apparence, la seconde une r\u00e9alit\u00e9. L'une se donne imm\u00e9diatement, l'autre n'appara\u00eet qu'apr\u00e8s initiation \u00e0 la m\u00e9taphysique qui pr\u00e9side \u00e0 l'obturation du rideau. En la mati\u00e8re, notre \u00e9poque est d'un illettrisme int\u00e9gral doubl\u00e9 d'un nihilisme sans fond.\n\nLe temps sculpt\u00e9 du d\u00e9miurge, la m\u00e9moire constell\u00e9e du m\u00e9taphysicien, la v\u00e9rit\u00e9 ou la falsification de l'\u00e9thicien supposent chaque fois chez le photographe la pratique d'un ph\u00e9nom\u00e9nologue. Se mettre derri\u00e8re un bo\u00eetier et un objectif install\u00e9 entre soi et le monde, projeter le regard, viser, \u00e9lire un sujet, cadrer, faire entrer dans un espace telle substance du monde, \u00e9lue, exclure ce qui n'est pas elle, op\u00e9rer une dialectique de l'\u00eatre et du n\u00e9ant sur le r\u00e9el, \u00e9quivaut sans conteste \u00e0 pratiquer, r\u00e9aliser, incarner, une ph\u00e9nom\u00e9nologie de la perception. Tout ce qui rel\u00e8ve du vocabulaire sartrien de L'Etre et le N\u00e9ant, sinon de la Critique de la raison dialectique, fonctionne \u00e0 merveille sur le terrain photographique : r\u00e9ification, ontologie et dimensions de la temporalit\u00e9, preuve ontologique, th\u00e9orie du regard structurant l'identit\u00e9, modalit\u00e9s de l'Etre-pour-Autrui, de l'Etre-dehors-pour-l'Autre, dialectique de l'En-Soi et du Pour-Soi, envers du pratico-inerte et situation, regard temporalisant, bien d'autres cat\u00e9gories fonctionneraient dans le cadre d'une lecture de la photographie comme acte ph\u00e9nom\u00e9nologique.\n\nRetenons les op\u00e9rations de cadrage et de vis\u00e9e comme ce qui permet une th\u00e9orie du regard par lequel advient l'\u00eatre d'une situation. Ce que j'\u00e9lis dans le viseur d\u00e9coupe sp\u00e9cifiquement dans le r\u00e9el une figure qui acc\u00e8de \u00e0 l'\u00eatre, alors que l'ensemble dans lequel a \u00e9t\u00e9 effectu\u00e9e cette taille est imm\u00e9diatement renvoy\u00e9 du c\u00f4t\u00e9 du n\u00e9ant. Ce qui acc\u00e8de \u00e0 l'\u00eatre l'est par un projet, une volont\u00e9. L'\u0153il, le doigt et le d\u00e9clencheur permettent \u00e0 la conscience de se projeter et de contribuer \u00e0 une logique de l'av\u00e8nement de l'\u00e9v\u00e9nement : \u00e9lection d'un instant, isolement d'un moment, fixation d'un fait photographi\u00e9 comme atteinte et saisie d'une quintessence.\n\nEn c\u00e9l\u00e9brant les noces de la conscience et de l'objet qui la l\u00e9gitime, la ph\u00e9nom\u00e9nologie photographique \u00e9nonce la radicalit\u00e9 du mat\u00e9rialisme qu'elle suppose. Ce qui, d'ailleurs, provoque les furies de Baudelaire f\u00e2ch\u00e9 qu'une industrie dont le seul talent consiste en l'exactitude de pure et simple reproduction puisse s'installer en pr\u00e9tendante de l'antique peinture soucieuse de magnifier l'imagination, le r\u00eave, l'impalpable et la po\u00e9sie. La photographie vise la r\u00e9alit\u00e9 sensible, elle et seulement elle. Rien, dans son dessein, pour r\u00e9actualiser une option id\u00e9aliste ou spiritualiste dans le genre n\u00e9oplatonicien : on ne photographie pas un monde pour vanter les m\u00e9rites d'un autre, sup\u00e9rieur, mais invisible, quand la peinture se pense, se voit et se pratique comme l'occasion d'une perp\u00e9tuelle intercession en faveur du monde c\u00e9leste. La photo montre exclusivement le visible. L'Eglise ne s'y trompe pas : d\u00e8s les limbes de cet art nouveau, jamais en retard d'une sottise r\u00e9actionnaire, elle condamne l'invention impie, coupable de se pr\u00e9occuper du seul ici-bas.\n\nL'ic\u00f4ne pa\u00efenne qu'est toujours un tirage sur papier sensible apporte la preuve que la seule matrice possible de l'\u00eatre, c'est le monde, qu'il est causa sui et que la photographie seule est reflet, image sensible, ombre participative. Elle formule une anti-all\u00e9gorie de la Caverne o\u00f9 se jouent les jeux d'ombres et de lumi\u00e8res id\u00e9alistes avec lesquels l'esth\u00e9tique occidentale se dit depuis l'Antiquit\u00e9 grecque jusqu'aux \u00e9croulements rendus possibles par la modernit\u00e9.\n\nCette modernit\u00e9, d'ailleurs, n'est pas sans devoir \u00e0 l'invention de la photographie qui, comme tous les arts, joue un r\u00f4le dans la dialectique des esth\u00e9tiques vivantes. D'o\u00f9 l'ultime qualit\u00e9 d'esth\u00e9ticien du photographe. Certes, \u00e0 l'origine, elle fournit d'abord une technique avant d'\u00eatre un art. Industrie de la reproductibilit\u00e9 m\u00e9canique et artisanale d'une partie de la r\u00e9alit\u00e9 color\u00e9e et en trois dimensions, transform\u00e9e en image plane et bicolore, la photographie devient vite l'auxiliaire des voyageurs et des ethnologues, des scientifiques et des juges, des journalistes et des historiens, du publiciste et du g\u00e9ographe, du militaire et du p\u00e8re de famille. Seul l'usage d\u00e9termine l'appartenance de telle ou telle au monde de l'art. Et ce pourra \u00eatre le clich\u00e9 d'un criminologue, d'un touriste ou d'un soldat.\n\nLes mus\u00e9es, galeries et figures d'embrayeurs jouent leur jeu et font, ou non, acc\u00e9der certaines photographies au panth\u00e9on artistique quand elles abandonnent les autres \u00e0 la solitude des bo\u00eetes cartonn\u00e9es o\u00f9 elles attendent un meilleur destin que le banal entrep\u00f4t domestique. Nadar ici, Monsieur Prudhomme l\u00e0. Mais dans l'un et l'autre cas, on conviendra que la photographie porte une puissance artistique tout autant soumise aux caprices de l'histoire et de l'arbitraire pour sa l\u00e9gitimation que n'importe quelle autre \u0153uvre d'art peinte ou sculpt\u00e9e.\n\nLe photographe-d\u00e9miurge, m\u00e9taphysicien, ph\u00e9nom\u00e9nologue, \u00e9thicien, esth\u00e9ticien, ne manque pas d'\u00eatre un artiste car il philosophe en acte \u2013 ce qui, \u00e0 mes yeux, d\u00e9finit la fonction. Pour cela, il rel\u00e8ve des m\u00eames cat\u00e9gories que le peintre ou le sculpteur, le musicien ou le po\u00e8te. Dans une esth\u00e9tique exprimant enfin la caducit\u00e9 de Kant et de Hegel, il y aurait place \u00e0 part enti\u00e8re pour la photographie au c\u00f4t\u00e9 d'autres disciplines encore suspectes pour les esprits chagrins. Le premier chapitre de l'histoire de cet art nouveau commencerait avec la narration de ce que les autres lui doivent, \u00e0 savoir l'essentiel de leurs trajets depuis un si\u00e8cle et demi.\n\nEn prenant sa place, la photographie \u00e9puise la peinture de l'\u00e9poque et exige d'elle, dans une impitoyable logique darwinienne, une adaptation aux nouvelles conditions \u00e9dict\u00e9es : ce que peut mieux qu'elle l'art nouveau, il faut l'abandonner. Repr\u00e9senter fid\u00e8lement la r\u00e9alit\u00e9, viser l'objectivit\u00e9 maximale, la co\u00efncidence la plus absolue entre le r\u00e9el et sa repr\u00e9sentation, tout cela devient lettre morte. Ingres et Puvis de Chavannes ne s'y trompent pas qui luttent f\u00e9rocement contre la photographie afin de pouvoir continuer \u00e0 peindre des sc\u00e8nes que Nadar aurait pu composer et reproduire m\u00e9caniquement avec la m\u00eame fid\u00e9lit\u00e9.\n\nLe r\u00e9alisme mourant, reste \u2013 et Baudelaire aurait d\u00fb mieux qu'un autre s'en apercevoir \u2013 la possibilit\u00e9 d'une autre voie, royale pour les puissances de l'imagination et du r\u00eave qu'il ch\u00e9rissait tant. L'esth\u00e9tique de la modernit\u00e9 r\u00e9side l\u00e0, en germe. Apr\u00e8s l'abandon des canons classiques, il faut inventer l'avenir en consid\u00e9rant la nouvelle donne : plus d'obligation \u00e0 respecter le sacro-saint sujet, le motif et l'objectivit\u00e9. L'id\u00e9ologie id\u00e9aliste, spiritualiste et platonicienne, fauch\u00e9e, g\u00eet \u00e0 terre. A sa place, on constate l'av\u00e8nement de l'imaginaire lib\u00e9r\u00e9, de la subjectivit\u00e9 radicale, du perspectivisme int\u00e9gral. Autant dire de la modernit\u00e9 absolue.\n\nDans ces temps o\u00f9 la photographie montre des contours nets, des formes pr\u00e9cises, des compositions \u00e9labor\u00e9es, la peinture propose le triomphe de l'impression, de la division, du point, puis des subjectivit\u00e9s avec lesquelles se d\u00e9fait le classicisme et se structure la modernit\u00e9 : Turner et Monet, puis C\u00e9zanne. La suite est connue. Et Duchamp vient, appelant de ses v\u0153ux un art encore nouveau qui serait \u00e0 l'ensemble des productions du moment ce que les beaux-arts ont \u00e9t\u00e9 apr\u00e8s l'invention de la photographie : un vieux monde. Cette r\u00e9volution est encore \u00e0 venir.\n12\n\nTIRER LA BARBE \u00c0 TOUTE MAJEST\u00c9\n\nDans le fouillis des papiers qui envahissent et assaillent mon bureau, j'ai retrouv\u00e9 nagu\u00e8re une \u00e9ph\u00e9m\u00e8re revue d\u00e9partementale dans laquelle j'ai publi\u00e9 mon premier article \u00e0 pr\u00e9tention quelque peu philosophique. Son titre \u00e9tait \u00ab Tirer la barbe \u00e0 toute majest\u00e9 \u00bb et j'y envisageais la question du politique chez Alain. Ne l'ayant jamais relu, j'avais envie de persister dans cette perspective et laisser \u00e0 hier ce qui rel\u00e8ve d'hier. Mais la curiosit\u00e9 a emport\u00e9 mes r\u00e9solutions et j'ai repris les quatre pages pour les r\u00e9\u00e9valuer. J'avais envie de me souvenir des raisons pour lesquelles j'aimais alors la politique d'Alain et celles qui auraient pu faire que je l'appr\u00e9cie encore.\n\nToutes les maladresses susceptibles d'\u00eatre attendues dans pareil exercice de style s'y trouvaient : j'avais vingt-trois ans, je subissais aupr\u00e8s d'une directrice de th\u00e8se la technique laborieuse qui r\u00e9duit le s\u00e9rieux \u00e0 la citation, aux r\u00e9f\u00e9rences et \u00e0 la preuve de l'existence qu'on a plus lu que compris. Pas d'audaces, peu de subjectivit\u00e9, un mod\u00e8le du genre dormitif universitaire. Quelques photos, dont certaines prises par mes soins \u00e0 Mortagne, dans le mus\u00e9e ou la maison natale, une bibliographie, m\u00eame, adornaient l'ensemble. Finalement, l'esprit universitaire n'y trouvait pas plus son compte que l'esprit journalistique, le premier visant toujours plus d'obscurit\u00e9, le second toujours plus d'inconsistance. Restait l'hommage rendu \u00e0 une pens\u00e9e libre.\n\nJ'ai aim\u00e9, j'aime et aimerai Alain pour son ind\u00e9fectible ironie \u00e0 l'endroit des puissants, des grands et de ceux qu'il appelait les importants, tout autant que son art de philosopher sans oublier l'humour et la dr\u00f4lerie, le cynisme ou la litote sans, non plus, sacrifier et n\u00e9gliger la vie quotidienne \u2013 seul objet philosophique qui vaille. Dans le bestiaire politique des animaux d\u00e9sireux de se faire aussi gros que le b\u0153uf, j'avais relev\u00e9 chez lui : un troupeau de militaires flanqu\u00e9 d'une colonie de diplomates, un paquet d'administrateurs aux c\u00f4t\u00e9s de bureaucrates et de fonctionnaires, une cohorte de ministres et de d\u00e9put\u00e9s, des amateurs de glands sur les k\u00e9pis, du sous-pr\u00e9fet au polytechnicien, des tra\u00eeneurs de sabre au Quai Conti, des infatu\u00e9s de l'Education nationale, de l'inspecteur au recteur en passant par les professeurs, des banquiers et des directeurs, des critiques \u2013 ah, les critiques, engeance sublime ! Sans oublier les mitr\u00e9s d'\u00e9v\u00each\u00e9s, abbayes et autres pr\u00e9bendes sacerdotales. Il est sain de ne pas aimer ces gens-l\u00e0 quand on a vingt ans. Il semble d'autant plus r\u00e9jouissant de s'apercevoir qu'on ne les prise pas plus vingt ans apr\u00e8s.\n\nDe plus, Alain me r\u00e9jouissait par la contrepartie ajout\u00e9e \u00e0 ces d\u00e9testations, \u00e0 savoir une infinie tendresse pour les autres, ceux qui subissent le pouvoir de ceux-l\u00e0, les petits, les humbles, les modestes, les sans-parole, les d\u00e9munis, les priv\u00e9s de tout \u00e0 qui il ne reste que le d\u00e9sespoir d'une existence consacr\u00e9e \u00e0 r\u00e9p\u00e9ter les occasions de douleur et de tristesse. D'autant que dans la corporation de ceux qu'il est convenu d'appeler les intellectuels, ou les philosophes, les sans-grade sont purement et simplement sans existence, puisque sans importance et sans utilit\u00e9 aucune pour leurs carri\u00e8res. A l'inverse, et pour trop leur ressembler, les gens de lettres c\u00e9l\u00e8brent et d\u00e9fendent les parasites contre lesquels peste Alain.\n\nEnfin, Alain me plaisait pour son esprit libertaire, \u00e0 l'oppos\u00e9 de tout dogmatisme, dans la plus pure des traditions qui, dans cette galaxie de fous furieux de libert\u00e9, me font pr\u00e9f\u00e9rer F\u00e9lix F\u00e9n\u00e9on, Zo d'Axa ou Georges Palante \u00e0 S\u00e9bastien Faure, Jean Grave ou Han Ryner, encore trop impr\u00e9gn\u00e9s par l'esprit pr\u00eatre. Chez ceux-l\u00e0, animaux solitaires bien que solidaires, la passion pour la libert\u00e9 induit une redoutable m\u00e9fiance \u00e0 l'endroit des d\u00e9tenteurs du pouvoir, puis du pouvoir lui-m\u00eame, corrupteur dans son essence. M\u00e9pris pour l'Etat \u2013 L\u00e9viathan pour Alain, monstre froid pour Fr\u00e9d\u00e9ric Nietzsche \u2013, ind\u00e9fectible confiance en l'individu doubl\u00e9e par un pessimisme \u00e0 l'endroit du social, du groupe et du pluriel, condamnation de toutes les cristallisations gr\u00e9gaires qui se nourrissent de la force et de l'\u00e9nergie des singularit\u00e9s rebelles, des partis aux soci\u00e9t\u00e9s diverses en passant par les syndicats et autres manifestations de l'instinct soci\u00e9taire, la pens\u00e9e libertaire d'Alain m\u00e9ritait d'autant cette \u00e9pith\u00e8te que le penseur a plusieurs fois d\u00e9clin\u00e9 l'invitation d'\u00eatre un affid\u00e9 des id\u00e9es anarchistes.\n\nUn mot encore sur mes raisons d'aimer Alain : cette volont\u00e9 de promouvoir une forme en philosophie qui soit, comme le dialogue ou le discours, le trait\u00e9 ou les lettres, le journal ou le po\u00e8me, une occasion singuli\u00e8re d'exprimer une pens\u00e9e singuli\u00e8re. Les cinq mille Propos, dans leur diversit\u00e9 aussi bien que leur r\u00e9p\u00e9tition, leur opacit\u00e9 ou leur luminosit\u00e9, leur fra\u00eecheur ou leur usure, permettent \u00e0 chaque fois des possibilit\u00e9s de r\u00e9it\u00e9rer une pens\u00e9e sur le mode musical de la variation \u00e0 partir du m\u00eame th\u00e8me. J'ai parl\u00e9 ailleurs de mon go\u00fbt pour une pens\u00e9e en archipel. Elle est l\u00e0, tout enti\u00e8re en efflorescence. Et j'aime les formes qui t\u00e2chent de co\u00efncider avec leur fond, les expressions qui essaient de fonctionner en harmonie avec les id\u00e9es qu'elles servent.\n\nD'o\u00f9 le recours permanent d'Alain \u00e0 la vie quotidienne, au r\u00e9el de tout un chacun, \u00e0 l'exp\u00e9rience, aux anecdotes constitutives des travaux et des jours et surtout \u00e0 l'usage de la premi\u00e8re personne. Qu'un philosophe revendique la libert\u00e9 jusque dans l'expression d'une subjectivit\u00e9 annonc\u00e9e comme telle m'a toujours s\u00e9duit. Et la tradition philosophique fran\u00e7aise, de Montaigne \u00e0 Bachelard, de Descartes \u00e0 Pascal, de Rousseau \u00e0 Maine de Biran, n'a cess\u00e9 d'\u00eatre travers\u00e9e par ce tropisme qui, \u00e0 tout prendre, vaut probit\u00e9 l\u00e0 o\u00f9 d'autres travestissent leur moi pour mieux le v\u00e9n\u00e9rer tout en le pr\u00e9tendant ha\u00efssable.\n\nDans cette profusion de propos, j'ai \u00e9lu l'un d'entre eux comme une cl\u00e9 pour la pens\u00e9e politique d'Alain. Il me para\u00eet d'autant plus puissant qu'il raconte une situation dont je pense pouvoir dire, comme tant d'autres qui furent pensionnaires, que j'en proc\u00e8de radicalement. Ces deux pages ont pour titre \u00abL'odeur du r\u00e9fectoire\u00bb. Elles avancent cette id\u00e9e qu'\u00e0 l'origine de toute pens\u00e9e raisonnable et rationnelle on trouve toujours une \u00e9motion, une passion, une blessure d'enfance. Alain y \u00e9voque cette odeur bl\u00eame, m\u00e9lange de fadeur et de traces grasses, ces effluves \u00e9c\u0153urants qui participent de la moisissure des choses et des hommes, et que l'on ne sent que dans les r\u00e9fectoires. Quelle qu'elle soit, de pensionnaires ou de militaires, de pr\u00eatres ou de collectivit\u00e9s m\u00e9dicales, l'odeur du r\u00e9fectoire, c'est d'abord l'impression d'une lame sale qui tranche la gorge et emporte l'\u00e2me aussi.\n\nCette lame qui a coup\u00e9 l'existence en deux, on lui doit tous les partages essentiels qui suivront entre les ob\u00e9issants et les r\u00e9volt\u00e9s, les dociles et les rebelles. Les familiers de l'odeur du r\u00e9fectoire sont des chiens fous, des amateurs d'errance et de libert\u00e9, ces fameux \u00e2nes rouges impossibles \u00e0 atteler dans le bestiaire d'Alain. L'ordre, la r\u00e8gle, le respect, les lois, les r\u00e8glements, la politesse, la morale, les classiques, la p\u00e9dagogie et les palmes acad\u00e9miques, parce que tout cela sent trop mauvais, ne soumettra jamais les \u00e2mes fid\u00e8les \u00e0 l'enfant bless\u00e9 qu'ils furent. Et cette maladie de l'odorat se rappelle aux \u00e2mes fragiles et dures \u00e0 la fois, tous les ans, \u00e0 la m\u00eame \u00e9poque, contemporaine des ciels tristes de septembre. Depuis plus de trente ans, cette odeur m'insupporte, me renverse le c\u0153ur et me rappelle \u00e0 chaque fois aux souvenances n\u00e9cessaires. Elle est celle qui fait chavirer les b\u00eates qu'on m\u00e8ne \u00e0 l'abattoir. Voil\u00e0 autant de raisons qu'on voudra pour garder au philosophe toute sa tendresse : l'ironie, la distance \u00e0 l'endroit des puissants, la proximit\u00e9 avec les sans-grade, l'esprit libertaire, l'invention formelle d'une exposition rh\u00e9torique, le souci d'une vie philosophique et d'une pens\u00e9e enracin\u00e9e dans le pragmatisme, la r\u00e9bellion, l'individu d\u00e9fendu bec et ongles. Et c'est d\u00e9j\u00e0 tant dans le trajet d'un penseur.\n\nPour autant, apr\u00e8s mes lectures pour ce travail, je n'ai gu\u00e8re retrouv\u00e9 Alain. Ni dans mes travaux universitaires, ni dans les articles, conf\u00e9rences, ou livres \u00e0 venir. D\u00e8s que je sortais du registre politique o\u00f9 le philosophe m'avait plu, je ne rencontrais que des occasions de passer mon chemin. A chaque fois que j'ai avanc\u00e9 dans ma connaissance d'Alain, j'ai constat\u00e9 avec un peu de tristesse qu'il avait perdu mille occasions de rencontrer son temps. Pas forc\u00e9ment d'y adh\u00e9rer, mais au moins de s'attaquer aux probl\u00e8mes de son \u00e9poque. Ses dates le font contemporain d'\u00e9v\u00e9nements essentiels pour l'\u00e9mergence de la modernit\u00e9. Il a toujours d\u00e9fendu le pass\u00e9, et c\u00e9l\u00e9br\u00e9 les vertus d'antan. La photographie voit le jour en 1826, le cin\u00e9ma en 1895 : Alain passe \u00e0 c\u00f4t\u00e9 et, soit en dit du mal, soit des sottises. En mati\u00e8re scientifique, on sait sa d\u00e9fense v\u00e9h\u00e9mente des erreurs de Kant et son refus d'adh\u00e9rer \u00e0 la th\u00e9orie de la relativit\u00e9 d'Einstein. Passons sur la psychanalyse dont il fustige la vision du monde simiesque quand il aurait pu lire avec un immense profit Malaise dans la civilisation ou L'Avenir d'une illusion qui lui auraient \u00e9vit\u00e9 tant de d\u00e9convenues th\u00e9oriques et personnelles puis permis les id\u00e9es les plus fortes sur la guerre, son origine, l'instinct de mort, les mouvements de foule et autres th\u00e8mes familiers du philosophe.\n\nLaissons de c\u00f4t\u00e9 Les Demoiselles d'Avignon de Picasso (1907), Am\u00e9riques d'Edgard Var\u00e8se (1921), Ulysse de Joyce (1922) : Alain \u00e9crit sur le parangon de la peinture classique qu'est Jean Auguste Dominique Ingres, ne joue au piano et n'aime que Bach, Mozart et Chopin, et ne c\u00e9l\u00e8bre le roman que chez Balzac, Dickens ou Stendhal. N'attendons rien sur le terrain de la modernit\u00e9, Alain va jusqu'\u00e0 pester contre le v\u00e9lo, le t\u00e9l\u00e9phone et l'automobile. Il est d\u00e9sesp\u00e9r\u00e9ment du si\u00e8cle dernier, nostalgique et irr\u00e9ductible contemporain d'Auguste Comte qu'il aimait tant. La philosophie d'Alain me semblait autant d'hier que sa r\u00e9flexion politique me paraissait une occasion de penser cette question pour demain.\n\nQuand parut la biographie d'Alain par Maurice Savin, me souvenant de mon ancien travail sur la politique du philosophe, j'ai souhait\u00e9 mettre en perspective, ce que j'aime pratiquer pour tous les penseurs, la r\u00e9flexion th\u00e9orique et la conduite dans l'existence. Fid\u00e8le au principe nietzsch\u00e9en qu'une vie \u00e9claire une \u0153uvre autant que l'inverse, j'ai cherch\u00e9 une occasion de mieux comprendre le travail, l'homme et leur relation. Ce que je d\u00e9couvris des engagements politiques d'Alain fut consternant. Non pas dans sa jeunesse, o\u00f9 il ne commit pas l'erreur funeste de l'antidreyfusisme, mais dans sa pleine maturit\u00e9, puis sur les temps de la vieillesse annonc\u00e9e. En effet, les deux guerres mondiales de ce si\u00e8cle faisant partie de son paysage politique, il m'\u00e9tait apparu que des informations sur ce sujet ne manqueraient pas d'\u00e9clairer ma lanterne sur la question du politique.\n\nQuelle ne fut pas ma stup\u00e9faction de d\u00e9couvrir dans le journal du philosophe, non pas l'engag\u00e9 volontaire de 14-18, ce que tout un chacun conna\u00eet, mais la position munichoise et attentiste, le pacifisme int\u00e9gral du penseur, mais aussi des consid\u00e9rations bienveillantes sur Adolf Hitler et d'autres, plut\u00f4t n\u00e9gatives sur le g\u00e9n\u00e9ral de Gaulle ? Comment le penseur avis\u00e9 des pouvoirs et des cinq cents propos politiques pouvait-il se tromper avec autant de nettet\u00e9 ? De quelle mani\u00e8re une r\u00e9flexion libertaire a-t-elle pu sombrer dans une pratique de vie quotidienne semblable \u00e0 celle des Fran\u00e7ais les plus moyens, voire les plus douteux ? O\u00f9 \u00e9tait l'auteur des pages magnifiques sur l'esprit de r\u00e9sistance ? Qu'\u00e9tait devenu l'homme lucide sur les agissements de ceux qui se prennent pour les importants ?\n\nJ'ai repris alors tout ce qu'il \u00e9tait loisible de lire pour comprendre la politique d'Alain : les \u00c9l\u00e9ments d'une doctrine radicale (1925), Le Citoyen contre les pouvoirs (1926), Propos de politique (1934), Politique (posthume, 1962), et enfin Propos sur les pouvoirs (1985). Laissons de c\u00f4t\u00e9 ce qui a vieilli trop nettement sur le scrutin d'arrondissement, les vertus compar\u00e9es de Combes, Briand, ou Millerand, la loi des trois ans. Et retenons l'essentiel du discours d'Alain sur la politique afin de t\u00e2cher de comprendre ce qui explique ce grand \u00e9cart entre une th\u00e9orie lucide et une pratique fourvoy\u00e9e.\n\nAlain revendique l'\u00e9pith\u00e8te de radical. Qu'est-ce qu'un radical ? Dans son esprit, c'est moins un thurif\u00e9raire des politiques concr\u00e8tes de Clemenceau ou de Caillaux, de Herriot ou de Daladier, qu'un homme qui pense seul contre les pouvoirs et les partis, pour la R\u00e9publique et le respect des r\u00e8gles de droit. Ce qui, bien s\u00fbr, ne va pas sans un pied de nez \u00e0 l'endroit de l'Eglise, de sa puissance et de ses repr\u00e9sentants. C'est un homme qui n'aime pas l'Etat, ce monstre qui agit avec la lenteur d'un saurien et l'efficacit\u00e9 n\u00e9gative d'un animal dangereux, le L\u00e9viathan sorti de son zoo par Hobbes. C'est aussi une personne qui se d\u00e9fie du pouvoir, de tous les pouvoirs, et de tous ceux qui le veulent, s'en emparent ou s'en servent, fussent-ils radicaux au sens \u00e9troit du terme. C'est enfin un individu solitaire, taiseux \u00e0 l'endroit des groupes, distant devant toute manifestation de l'instinct gr\u00e9gaire, et promoteur d'un \u00e9ternel esprit de r\u00e9sistance, sinon de r\u00e9bellion.\n\nLe radical selon le souhait du philosophe n'aime pas les partis et leur pr\u00e9f\u00e8re les id\u00e9es, il ne sacrifie \u00e0 aucun esprit de secte, de caste, pour la bonne et simple raison qu'il c\u00e9l\u00e8bre et pr\u00e9serve l'individu contre vents et mar\u00e9es. D'o\u00f9 une m\u00e9taphysique c\u00e9l\u00e8bre du citoyen contre les pouvoirs et du sujet contre les structures qui proc\u00e8de philosophiquement d'une anthropologie. Or, sa conception de l'homme recycle clairement un platonisme orthodoxe : l'homme est une passion que doit travailler et r\u00e9duire sans cesse la raison. Tiraill\u00e9 par le sentiment, min\u00e9 par les d\u00e9sirs, les instincts et les pulsions, requis par leur formidable puissance, l'homme ne m\u00e9rite sa valeur et sa noblesse qu'en \u00e9chappant \u00e0 cette mal\u00e9diction des tropismes n\u00e9gatifs au profit d'une volont\u00e9 d'empire sur soi qui r\u00e9ussit. La v\u00e9rit\u00e9 de la politique est donc anthropologique.\n\nTr\u00e8s vite, ce dualisme platonicien devient un manich\u00e9isme ordonnateur de sa vision du monde. Du c\u00f4t\u00e9 des passions on trouvera la guerre, l'injustice, la force, l'agressivit\u00e9, le n\u00e9gatif, du c\u00f4t\u00e9 de la raison, la paix, la justice, l'esprit, le droit, la bienveillance, le positif. Ce qui divise et ce qui r\u00e9unit, ce qui veut le sang et ce qui veut la fraternit\u00e9, tout ob\u00e9it aux oscillations voulues par l'empire des instincts ou par le r\u00e8gne de la raison. Dans l'esprit d'Alain, le radical d\u00e9finit l'homme pour lequel la politique est l'art de cette raison gouvernante. La passion la plus dangereuse conduit les hommes \u00e0 vouloir le pouvoir, aspiration d\u00e9l\u00e9t\u00e8re par excellence.\n\nLe pouvoir, quant \u00e0 lui, donne \u00e0 un homme la possibilit\u00e9 d'en asservir et d'en soumettre un autre, sous le pr\u00e9texte qu'en abandonnant sa libert\u00e9, son autonomie, voire son jugement \u00e0 l'arbitraire, l'individu soumis obtiendra une r\u00e9compense, un b\u00e9n\u00e9fice de nature politique, sociale ou id\u00e9ologique : un travail, un salaire, de la consid\u00e9ration, le salut, des avantages divers et multiples. Selon cette logique, le pouvoir suppose et l\u00e9gitime la mise au service de l'un par l'autre : le soldat et l'officier, l'enfant et ses parents, le fid\u00e8le et le pr\u00eatre, l'\u00e9tudiant et le professeur, la femme et l'homme, le sujet et le roi, l'\u00e9lecteur et son d\u00e9put\u00e9, le citoyen et le chef de l'Etat. Chaque fois, le pouvoir devient l'agent de l'assomption de l'un et de la d\u00e9fection de l'autre. La raison et l'autonomie c\u00e8dent le pas \u00e0 l'autorit\u00e9.\n\nDans toute hi\u00e9rarchie, l'\u00e9tymologie nous en donne la preuve et la d\u00e9monstration, se dissimule un sacr\u00e9 qui suppose la soumission. Alain veut la r\u00e9duction de ce sacr\u00e9 \u00e0 la portion congrue. Le pouvoir est la\u00efcis\u00e9 seulement s'il cesse d'\u00eatre v\u00e9cu sur le principe du droit divin : on doit pouvoir le mettre en cause, l'inqui\u00e9ter, l'examiner et demander des comptes \u00e0 quiconque en use plus \u00e0 ses fins propres que dans l'ordre pour lequel il devrait \u00eatre utilis\u00e9 : la paix sociale et le r\u00e8gne du droit et de la justice.\n\nL'instrument qui autonomise le pouvoir, en fait dispara\u00eetre les rouages aux yeux du simple citoyen, c'est l'Etat et son arm\u00e9e de fonctionnaires, d'administrateurs et de bureaucrates. En tant que tel, il se constitue et continue de vivre sur le mode de la cristallisation : nourri de l'abdication des forces individuelles dans l'op\u00e9ration du contrat social, l'Etat tient sa puissance de l'abandon auquel consentent les citoyens. D\u00e9poss\u00e9d\u00e9s au profit de m\u00e9canismes et de machines qui vident les revendications l\u00e9gitimes de leurs contenus, les sujets sont dup\u00e9s dans le fondement de leur l\u00e9gitimit\u00e9. La repr\u00e9sentation politique double l'effet n\u00e9gatif de l'Etat et renvoie tout un chacun \u00e0 la situation du vassal.\n\nQuiconque s'oppose \u00e0 cette m\u00e9canique d\u00e9vorante se voit rapidement annihil\u00e9, soit sur le mode violent et policier, soit sur le mode non moins funeste mais plus fourbe de la r\u00e9cup\u00e9ration, de la transmutation des \u00e9nergies revendicatrices en passivit\u00e9s obtenues contre des avantages symboliques. Tout opposant devient un auxiliaire z\u00e9l\u00e9 de ce contre quoi il lutte quand on lui offre les moyens de changer de camp. Les sources du ressentiment taries, restent de nouveaux domestiques g\u00e9n\u00e9r\u00e9s par le syst\u00e8me. Alain d\u00e9crit finement le processus de r\u00e9cup\u00e9ration des \u00e9lites et leur domestication. Les penseurs, les intellectuels, les \u00e2mes rebelles sont emp\u00each\u00e9s de nuire quand ils sont recycl\u00e9s dans le syst\u00e8me moyennant paiement en monnaies sociales. Honneurs, respectabilit\u00e9, s\u00e9curit\u00e9, reconnaissance, fonctions, gratifications, argent, tout est bon qui permet pouvoirs r\u00e9els et pouvoirs symboliques en \u00e9change de la docilit\u00e9 et de la soumission utiles \u00e0 la permanence du syst\u00e8me. Et l'on retiendra les L\u00e9gions d'honneur, les tribunes journalistiques, les r\u00e9compenses acad\u00e9miques, les bicornes dor\u00e9s, les missions aupr\u00e8s des ministres comme autant de signes qui trahissent la collaboration des \u00e9lites avec les pouvoirs en place. Restent les incomp\u00e9tents, les eunuques, les valets et les plumes serviles, tous agents de la plus \u00e9vidente reproduction sociale. Le pouvoir transfigure leur \u00e9nergie; l'Etat fournit le m\u00e9canisme dans lequel elle s'exprime et se perd.\n\nSoit, donc, le radical d\u00e9fini comme l'homme averti et affranchi sur les passions, le pouvoir, l'Etat, et leurs agencements n\u00e9gateurs, quels moyens peut-il mettre en \u0153uvre pour aller au-del\u00e0 de cette pure et simple connaissance du n\u00e9gatif et pratiquer un radicalisme actif? Alain avance l'id\u00e9e int\u00e9ressante du pouvoir contr\u00f4leur. Apr\u00e8s la critique du pouvoir, voici venu le temps du pouvoir de la critique. L'id\u00e9e, ancienne \u00e0 gauche, traverse l'histoire politique depuis les Sans-Culottes de 1793 jusqu'aux Enrag\u00e9s de Mai 68 via les Communards de 1871. Et dans son hypoth\u00e8se, elle constitue le noyau dur de la pens\u00e9e politique d'Alain. Le contr\u00f4le, c'est le gouvernement contre l'Etat, le droit pour les gouvern\u00e9s de dire \u00e0 leurs gouvernants ce qu'ils pensent de la fa\u00e7on dont les seconds s'acquittent de leur t\u00e2che.\n\nQue veut Alain? La possibilit\u00e9 permanente de demander des comptes aux \u00e9lus, quelle que soit leur place dans la machine politique. Afin d'\u00e9viter le r\u00e8gne de la d\u00e9magogie et la surench\u00e8re d\u00e9brid\u00e9e qui fleurit dans les p\u00e9riodes \u00e9lectorales, l'\u00e9lu aurait \u00e0 justifier en permanence, et non pas seulement aux moments charni\u00e8res des \u00e9ch\u00e9ances, ce qu'il fait, comment, pourquoi, et pour qui. De sorte qu'un Propos dat\u00e9 du 12 juillet 1910 envisage sans ambages le pouvoir de d\u00e9poser sur-le-champ quiconque faillit dans son mandat repr\u00e9sentatif.\n\nCar un radical sait la nature entropique du mouvement politique : des d\u00e9clarations retentissantes et ronflantes, quand il s'agit d'obtenir les suffrages, aux reniements, en passant par les mille et une compromissions qui justifient \u00e0 chaque fois le recentrage puis la droitisation du propos de l'\u00e9lu, on doit toujours craindre la disparition des bonnes intentions et des professions de foi au profit de la pure et simple composition cynique avec le pire. Il y eut m\u00eame de belles \u00e2mes, r\u00e9cemment, pour th\u00e9oriser et l\u00e9gitimer les options ren\u00e9gates en distinguant la culture d'opposition et la culture de gouvernement. Alain souhaite qu'aux commandes un \u00e9lu choisi pour ses id\u00e9es d'opposition n'oublie pas qu'il est l\u00e0 pour les r\u00e9aliser et non pour jouir purement et simplement des avantages de la fonction. Le pouvoir contr\u00f4leur rappelle \u00e0 l'ordre l'\u00e9lu afin qu'il n'oublie pas le citoyen.\n\nCertes, l'intention est louable et l'id\u00e9e redoutable pour donner ou redonner sa dignit\u00e9 \u00e0 la politique. De la sorte, \u00e9thique et art de gouverner ne seraient pas s\u00e9par\u00e9s et entretiendraient une relation \u00e9quilibr\u00e9e. Avec sa poign\u00e9e de propos sur ce sujet, Alain brille \u00e0 ce moment de l'analyse plus dans la logique du Discours de la servitude volontaire de La Bo\u00e9tie que du Contrat social de Rousseau. D'o\u00f9 la force de sa pens\u00e9e politique. Comment donne-t-il \u00e0 son id\u00e9e les moyens th\u00e9oriques de devenir une r\u00e9alit\u00e9 ? Que faut-il mettre en \u0153uvre pour exercer de fait ce pouvoir contr\u00f4leur ?\n\nFid\u00e8le \u00e0 ses options et pr\u00e9suppos\u00e9s anthropologiques, Alain se m\u00e9fie toujours des passions. Aussi ne faut-il pas attendre chez lui une justification de ce pouvoir contr\u00f4leur quand il s'exerce dans la rue, sur les barricades, sur le mode des gr\u00e8ves ou des revendications agressives. Encore moins, bien s\u00fbr, s'il doit s'appuyer sur les \u00e9nergies consubstantielles aux r\u00e9volutions. Le droit, la loi et le seul pouvoir de la raison justifient la r\u00e9alisation de cette force \u00e9thique, de cette puissance morale. Et cette grande id\u00e9e aboutit \u00e0 une petite proposition : le pouvoir contr\u00f4leur devrait se suffire de l'hypoth\u00e8se de l'interpellation \u00e0 la Chambre dans le strict cadre l\u00e9gal et juridique de l'assembl\u00e9e parlementaire. Quoi, cette formidable occasion de rendre son pouvoir au citoyen avorte dans la maigre possibilit\u00e9 de poser des questions \u00e0 un d\u00e9put\u00e9, voire \u00e0 un ministre ? Lui demander des comptes, voil\u00e0 la r\u00e9volution \u00e9thique appel\u00e9e de ses v\u0153ux par Alain ? Il faut malheureusement y consentir : les promesses sont plus jubilatoires que ce qui peut \u00eatre tenu. La force de la pens\u00e9e politique d'Alain est l\u00e0, l\u00e0 aussi sa faiblesse. L'id\u00e9e, d'une incontestable modernit\u00e9, et d'une perp\u00e9tuelle actualit\u00e9 depuis les d\u00e9bats entre les Girondins et les Montagnards, n'a d'int\u00e9r\u00eat que pass\u00e9e dans les faits. Et le r\u00f4le du philosophe, son travail, consiste \u00e9galement \u00e0 dire selon quelles modalit\u00e9s ce passage du papier aux faits devient possible et pensable. La mise en sc\u00e8ne du pouvoir contr\u00f4leur m\u00e9rite mieux que cet effondrement au berceau.\n\nPour quelles raisons cette proposition politique qui fournit aussi une occasion \u00e9thique devient-elle un pur et simple brimborion ? La r\u00e9ponse se trouve dans un propos dat\u00e9 du 7 septembre 1912. La date a son importance car elle permet de constater que la pens\u00e9e d'Alain est formul\u00e9e avant la Premi\u00e8re Guerre mondiale et que ses choix bellicistes en 14-18 et munichois, puis attentistes en 1940, proc\u00e8dent de cette \u00e9trange prise de position formul\u00e9e en 1912.\n\nQue dit ce texte? Qu'exprime cette position? Que les deux vertus du citoyen sont la r\u00e9sistance et l'ob\u00e9issance. Il faut bien lire r\u00e9sistance et ob\u00e9issance et non pas r\u00e9sistance ou ob\u00e9issance. N'importe qui verrait dans cette double invite une contradiction : r\u00e9sister, c'est ne pas ob\u00e9ir, tout autant qu'ob\u00e9ir suppose ne pas r\u00e9sister. Alain, non. Au contraire, il veut les deux. Les mauvaises langues diraient : par atavisme normand. L'ob\u00e9issance, parce qu'il ne veut pas faire l'\u00e9conomie de l'ordre; la r\u00e9sistance, pour ne pas passer la libert\u00e9 par-dessus bord. La rh\u00e9torique ne suffit pas et l'on ne peut gu\u00e8re imaginer comment, dans les faits, s'incarne cette id\u00e9e en forme d'oxymore id\u00e9ologique : \u00ab Ob\u00e9ir en r\u00e9sistant. \u00bb Il ne veut ni la tyrannie, ni l'anarchie qu'il voit pourtant dans la destruction de l'ob\u00e9issance et dans celle de la r\u00e9sistance. Dans les deux cas, les pouvoirs en profitent pour exacerber leur nature dangereuse. Si l'on ach\u00e8ve de suivre Alain, d\u00e9sob\u00e9ir fortifie la tyrannie.\n\nQuid de la r\u00e9sistance ? Qu'en est-il de ce fameux esprit de r\u00e9sistance dont parle le philosophe? Alain le confine \u00e0 l'avis et \u00e0 l'opinion qu'on peut garder par-devers soi. Tant qu'il pense, le citoyen demeure libre, l'individu souverain et l'ob\u00e9issance n'est pas oblig\u00e9e. En revanche, dans les faits et gestes, dans les actes, il est tenu d'ob\u00e9ir. Inflexible d'esprit et soumis de corps, r\u00e9volt\u00e9 dans l'\u00e2me, mais docile en chair et en os, le sujet platonicien d'Alain peut r\u00e9server ses \u00e9tats d'\u00e2me \u00e0 ses propres divagations mentales pourvu qu'il ait le doigt sur la couture du pantalon dans la caserne o\u00f9 il ne manque pas de se rendre quand on l'y appelle. Gageons que chez les pr\u00eatres qui humiliaient les enfants dans les r\u00e9fectoires, parmi les patrons qui proc\u00e8dent de m\u00eame avec leurs ouvriers dans les usines, du c\u00f4t\u00e9 des garde-chiourmes dans les prisons, ou dans la forteresse les officiers avec leurs soldats, aucun ne souhaite \u00eatre aim\u00e9 dans son for int\u00e9rieur par sa victime pourvu qu'elle ob\u00e9isse, consente et ne se rebelle pas.\n\nDans son existence, et venons-y enfin, le philosophe pratique cette discipline \u00e0 deux reprises. R\u00e9sultats d\u00e9sastreux ! On sait son pacifisme th\u00e9orique avant la Premi\u00e8re Guerre mondiale, ses appels, ses propos, ses textes, ses \u00e9crits de toute nature qui tous vont dans le m\u00eame sens : la guerre induit une d\u00e9bauche de passions, elle est mauvaise, toujours. Et n'importe quelle paix est pr\u00e9f\u00e9rable \u00e0 n'importe quelle guerre. Il n'en est aucune de bonne. On sait la mont\u00e9e des p\u00e9rils, les pr\u00e9mices, la catastrophe annonc\u00e9e, l'occupation de la Ruhr, les Sud\u00e8tes, \u00e0 chaque fois, Alain \u00e9crit, dit non, refuse l'id\u00e9e de la guerre. Et l'on conna\u00eet la suite, elle \u00e9clate tout de m\u00eame.\n\nR\u00e9sistance, ob\u00e9issance : Alain s'engage alors que son \u00e2ge le dispense d'aller au front. Il fait trois ann\u00e9es de guerre et sa correspondance, son Journal \u00e9galement, t\u00e9moignent tristement du fait qu'il aime ce monde-l\u00e0, viril, fraternel, dangereux, solidaire, authentique. Il confie des bonheurs d'artilleur dont les tirs bien ajust\u00e9s font mouche sur les lignes ennemies ; il maudit les permissions d\u00e9pourvues de sens, parce que d\u00e9mobilisatrices de la tension; il avoue un plaisir d'esth\u00e8te, une curiosit\u00e9 d'entomologiste dans les raisons de son engagement ; il avoue \u00eatre \u00ab ivre de joie\u00bb, c'est son expression, en voyant l'artillerie fran\u00e7aise dominer celle des Allemands ; il confie m\u00eame dans une lettre dat\u00e9e du 22 juillet 1915 \u00e0 Marie Salomon qu'il tuerait cinq cents fantassins de sa main si on lui en donnait l'ordre. Et en m\u00eame temps que son corps ob\u00e9it aux ordres de l'arm\u00e9e fran\u00e7aise, son esprit se rebelle et dit non ?\n\nPendant qu'il ajuste le tir sur le front ennemi et se r\u00e9jouit de voir passer les obus de mortier guid\u00e9s par ses conseils, son esprit r\u00e9siste? Qui peut adh\u00e9rer \u00e0 cette dichotomie? Qui souscrit \u00e0 ce platonisme \u00e0 la petite semaine ? A qui fera-t-il croire qu'un obus envoy\u00e9 par quelqu'un s'y refusant mentalement mais y consentant dans les faits fera moins de victimes qu'envoy\u00e9 avec l'assentiment intellectuel et spirituel de l'artilleur? On ne peut pousser le platonisme et le dualisme aussi loin. Le corps et l'\u00e2me entretiennent une relation intime, puisque l'un et l'autre apparaissent en modalit\u00e9s diff\u00e9rentes d'une m\u00eame entit\u00e9 : la mati\u00e8re.\n\nLes officiers fran\u00e7ais se moquent \u00e9perdument de l'assentiment moral d'Alain s'il fait un soldat efficace, ob\u00e9issant aux ordres, comme il pratique pendant ses trois ann\u00e9es de guerre. Et l'\u00e9tat-major se satisfait de la seule r\u00e9sistance mentale d'Alain pourvu qu'il puisse compter sur son engagement actif, efficace et effectif, puisque le 16 avril 1916, le philosophe pacifiste obtient, et ne refuse pas, la Croix de guerre avec citation \u00e0 l'ordre de l'arm\u00e9e. D'autres, plus r\u00e9sistants, paient leur engagement antimilitariste, pacifiste et rebelle d'un peloton d'ex\u00e9cution et d'une balle dans la peau. Alain dira son refus des promotions dans l'ordre militaire et son grade de simple brigadier \u00e0 la fin des hostilit\u00e9s, il n'en a pas moins contribu\u00e9, par son savoir-faire d'artilleur, \u00e0 la plus grande boucherie de ce si\u00e8cle. Ceux qui le d\u00e9corent ne s'y trompent pas.\n\nJe retiens, en forme de premi\u00e8re le\u00e7on donn\u00e9e par cette schizophr\u00e9nie du philosophe, cette id\u00e9e qu'une r\u00e9sistance seulement spirituelle est nulle et non avenue, \u00e9gale \u00e0 z\u00e9ro et qu'une ob\u00e9issance, quelle qu'elle soit, quelles qu'en soient les raisons, satisfait toujours celui qui l'exige du moment qu'il l'obtient. D'o\u00f9 le corr\u00e9lat induit : il faut r\u00e9sister ou ob\u00e9ir, soit l'un, soit l'autre, mais pas les deux. A d\u00e9faut, install\u00e9 dans le sophisme et la rh\u00e9torique, on se contente de jouer avec les mots : le diable n'a pas de ces d\u00e9licatesses et prend ce qu'on lui donne, m\u00eame si le don s'effectue avec une arri\u00e8re-pens\u00e9e r\u00e9sistante.\n\nUne autre fois, en vertu du m\u00eame principe sophistique qu'on peut ob\u00e9ir et r\u00e9sister, mais cette fois-ci galvanis\u00e9 par les le\u00e7ons de la Premi\u00e8re Guerre mondiale, Alain d\u00e9montre par les faits l'inanit\u00e9 de son platonisme th\u00e9orique. L\u00e0 encore, on sait la mont\u00e9e des p\u00e9rils, les propos et discours de Hitler, les invasions. Alain, toujours pacifiste, est bien s\u00fbr munichois. Il croit que tout doit \u00eatre mis en \u0153uvre pour \u00e9viter la guerre, une nouvelle guerre. Tout, c'est-\u00e0-dire tout. Y compris le pire. Ainsi, dans son Journal, \u00e0 la date du 16 janvier 1940, Alain \u00e9crit ceci de Hitler : \u00ab il \u00e9prouvait l'humiliation et l'amour pour ses compatriotes; il t\u00e9moignait pour le citoyen \u00bb. Et plus loin, pendant la d\u00e9b\u00e2cle, au hasard d'une phrase, il constate que \u00ables chefs Allemands furent de bons administrateurs \u00bb.\n\nDans l'ensemble, \u00e0 la lecture des pages de son texte intime, Alain para\u00eet r\u00e9joui que l'\u00e9ternel diff\u00e9rend entre les Fran\u00e7ais et les Allemands soit enfin r\u00e9solu, f\u00fbt-ce au prix de la soumission de la France. A l'\u00e9poque, il imagine une revanche plus co\u00fbteuse pour la paix que la situation de fait, \u00e0 savoir l'occupation militaire allemande et la politique collaborationniste de la France. Ob\u00e9issance aux faits, cette fois-ci, et pas m\u00eame, semble-t-il, r\u00e9sistance spirituelle.\n\nOn voudrait pouvoir s'arr\u00eater l\u00e0. Mais il faut encore ajouter ceci : dans le m\u00eame Journal, Alain \u00e9crit : \u00abJ'esp\u00e8re que l'Allemand vaincra; car il ne faut pas que le g\u00e9n\u00e9ral de Gaulle l'emporte chez nous. Il est remarquable que la guerre revient \u00e0 une guerre juive, c'est-\u00e0-dire \u00e0 une guerre qui aura des milliards et aussi des Judas Maccab\u00e9es. \u00bb Ajoutons \u00e0 cela qu'\u00e0 cette \u00e9poque, et dans les ann\u00e9es suivantes, la revue de la Nouvelle Revue Fran\u00e7aise \u00e9tant dirig\u00e9e par Drieu La Rochelle, collaborateur notoire, Alain ne refuse pas d'y publier quelques textes.\n\nO\u00f9 donc se dissimulait le philosophe si lucide quant aux pouvoirs, \u00e0 l'Etat et aux importants ? Que faisait le penseur jubilant de fustiger les militaires, les policiers, les commis de l'administration qui tous \u0153uvraient aux c\u00f4t\u00e9s d'un Mar\u00e9chal sp\u00e9cialiste des boucheries de 1914 et pourvoyeur des ex\u00e9cutions de rebelles en nombre en 1917 ? O\u00f9 avait-il la t\u00eate, Alain, pour ne pas voir dans le g\u00e9n\u00e9ral de Gaulle l'incarnation de ce qu'il appelait de ses v\u0153ux : l'esprit de r\u00e9sistance, la passion r\u00e9publicaine, les vertus de l'opposition, le m\u00e9pris des partis, et la mise au service de la politique \u00e0 des fins \u00e9thiques? Que restait-il d'Alain pourtant tellement sain de corps et d'esprit qu'il fait para\u00eetre huit livres entre 1939 et 1945 ?\n\nPar la suite, dans les six ann\u00e9es d'existence qui lui restent apr\u00e8s guerre, il n'y eut pas d'ajouts aux propos politiques. Rien qui, \u00e0 la lumi\u00e8re de ces deux tragiques errances, ait pouss\u00e9 le philosophe \u00e0 reprendre sa pens\u00e9e pour biffer et raturer cette id\u00e9e qu'on pouvait, qu'on devait ob\u00e9ir et r\u00e9sister en m\u00eame temps. Rien pour retrouver le chemin th\u00e9orique du pouvoir contr\u00f4leur afin d'en repr\u00e9ciser les contours et d'en formuler de nouvelles modalit\u00e9s d'application. Rien pour c\u00e9l\u00e9brer la r\u00e9sistance contre l'ob\u00e9issance, malgr\u00e9 elle. Rien pour saluer les hommes qui avaient incarn\u00e9 cette r\u00e9sistance, qui avaient fait du concept philosophique le substantif que l'on sait.\n\nAlain \u00e9non\u00e7ait son programme d\u00e8s 1931 : \u00ab Tirer la barbe \u00e0 toute majest\u00e9. \u00bb D\u00e8s 1933, il aurait eu tout le loisir de tirer la moustache de Hitler, sinon celle de Staline, avant que de plus s\u00e9rieuses occupations ne le requi\u00e8rent si d'aventure il avait voulu donner chair et consistance \u00e0 ses propos politiques. Ils sont rest\u00e9s textes, mots, livres, d\u00e9clarations d'intention, v\u0153ux pieux. D'aucuns trouveront ici, et malheureusement, qu'une fois de plus les philosophes se contentent de parler, d\u00e9connect\u00e9s du monde r\u00e9el, quand d'autres, moins berc\u00e9s de dialectique, de rh\u00e9torique et de pens\u00e9es grandiloquentes, incarnent dans l'anonymat le v\u00e9ritable esprit radical, celui de la r\u00e9sistance, de toutes les r\u00e9sistances. Ceux-l\u00e0 donneront raison aux textes d'Alain et inviteront \u00e0 ne c\u00e9l\u00e9brer les id\u00e9es que lorsqu'elles sont effectives, en actes, dans les faits. Platonicien jusque dans les impasses, Alain se limite \u00e0 l'homme de mots, d'id\u00e9es, au philosophe comme disent les gens modestes aux yeux desquels s'exprime de la sorte une partie de l'inanit\u00e9 de la fonction. Sinon au pur et simple professeur de philosophie, lecteur de textes, entregloseur patent\u00e9, fonctionnaire de la pens\u00e9e, besogneux du commentaire d'\u0153uvres estampill\u00e9es.\n\nEncore un mot, pour conclure : les pages sur ob\u00e9issance et r\u00e9sistance, ce fameux propos du 7 septembre 1912, s'il a \u00e9t\u00e9 vain, aussi bien pour Alain que pour tous les hommes d'action int\u00e9ress\u00e9s par la politique dans leur chair et leur existence, du moins n'aura pas \u00e9t\u00e9 inutile pour tout le monde puisque, dans leur immense sagesse et leur sagacit\u00e9 sans limite, dans leur clairvoyance av\u00e9r\u00e9e et leur lucidit\u00e9 sans \u00e9gale, les gens de l'administration de l'Education nationale, les fameux importants, inspecteurs, professeurs p\u00e9dagogues et recteurs, ont choisi pour la session du baccalaur\u00e9at technologique 1996, Acad\u00e9mie de Caen, rien moins que ces lignes redoutables. Si les importants, eux, n'ont rien compris \u00e0 ce qui se disait de grave, d'essentiel et de majeur derri\u00e8re ce texte, les \u00e9l\u00e8ves, eux, dans toutes leurs copies \u2013j'en ai corrig\u00e9 150 \u2013 ont bien compris ce qu'il fallait voir : soit on ob\u00e9it, soit on r\u00e9siste. Mais ils n'ont pas os\u00e9 \u00e9crire que peut-\u00eatre un philosophe, un important comme aurait dit Alain, pouvait se tromper ou tenir des propos insanes. Preuve s'il en est qu'on a toujours et plus que jamais raison d'enseigner la r\u00e9volte et de c\u00e9l\u00e9brer les rebelles, les boutefeux et tous ceux qui refusent de composer en invitant \u00e0 joindre le geste \u00e0 la parole.\n13\n\nLA GAIET\u00c9 ENNUYEUSE\n\nPourquoi faut-il que les philosophes de ce si\u00e8cle ajoutent \u00e0 leur surdit\u00e9 habituelle et cong\u00e9nitale des consid\u00e9rations \u00e9logieuses sur des musiciens soucieux de consacrer leur existence \u00e0 c\u00e9l\u00e9brer laborieusement la gaiet\u00e9, la joie, le badin et le primesautier? Faut-il convoquer pour cette pathologie un syndrome de Nietzsche et consid\u00e9rer l'\u00e9loge du l\u00e9ger en musique comme une incapacit\u00e9 pour un temp\u00e9rament \u00e0 rencontrer en face le s\u00e9rieux, la gravit\u00e9 puissante et tragique, sinon la monumentalit\u00e9 de l'\u00e9nergie sculpt\u00e9e par certains compositeurs ? Car Bizet ne peut tenir t\u00eate \u00e0 Wagner, pas plus que Federico Mompou, le client de Jank\u00e9l\u00e9vitch, ne rivalise s\u00e9rieusement avec Beethoven ou l'Offenbach de Cl\u00e9ment Rosset avec Berlioz. Alors, coquetterie, go\u00fbt de la provocation, int\u00e9r\u00eats id\u00e9ologiques ou affections nerveuses sur le mode nietzsch\u00e9en ?\n\nOn sait la part majeure prise par l'int\u00e9r\u00eat id\u00e9ologique et la musique mise au service d'une cause qui n'est pas elle. Nietzsche voulait faire payer \u00e0 Wagner un trop d'amour m\u00e9tamorphos\u00e9 un jour en haine. Jank\u00e9l\u00e9vitch entendait rayer de la carte intellectuelle tout ce qui, de pr\u00e8s ou de loin, avait pu \u00eatre allemand. Ce qui l'a amen\u00e9, lui, le philosophe de la morale sp\u00e9cialis\u00e9e en amour du prochain, \u00e0 c\u00e9l\u00e9brer l'Espagne, l'Italie, la Russie et la France, pour n'avoir pas \u00e0 \u00e9crire ou dire du bien d'un seul Germain, f\u00fbt-il n\u00e9 deux si\u00e8cles avant la barbarie nazie.\n\nEt Cl\u00e9ment Rosset ? Etrange cette passion pour le Bol\u00e9ro de Ravel et l'\u0153uvre compl\u00e8te de Jacques Offenbach. Mais cons\u00e9quente si l'on se souvient que, pour lui, l'essence de la musique co\u00efncide avec celle du r\u00e9el. Schopenhau\u00e9rien, il veut l'\u0153uvre musicale comme une occasion d'assister au spectacle du monde pour mieux \u00e9chapper \u00e0 sa cruaut\u00e9. Soit. Mais Offenbach? Offenbach fournit l'arch\u00e9type du compositeur qui cr\u00e9e une \u0153uvre enti\u00e8rement d\u00e9volue \u00e0 la joie, \u00e0 la gaiet\u00e9, \u00e0 la jubilation. Musique gaie et rien que gaie, elle r\u00e9duit l'effet \u00e0 l'expression souveraine et irr\u00e9sistible de l'all\u00e9gresse insouciante. Les partitions laiss\u00e9es par le p\u00e8re de La Fille du tambour-major, doubl\u00e9es de livrets d\u00e9c\u00e9r\u00e9br\u00e9s, dispensent de penser, r\u00e9fl\u00e9chir, et invitent \u00e0 oublier la nature tragique du r\u00e9el pour mieux se complaire, divertir, dans l'\u00e9cume des choses.\n\nPeut-on mieux soumettre la musique et lui demander de procurer du divertissement au sens pascalien du terme ? Pendant que j'\u00e9coute une musique joyeuse, je suis la joie, le tragique semble se d\u00e9sengager du monde qui appara\u00eet m\u00e9diatis\u00e9, m\u00e9tamorphos\u00e9 par la plus pure des logiques consolantes. Pour les tenants du divertissement, la musique cesse d'\u00eatre int\u00e9ressante d\u00e8s qu'elle revendique une part c\u00e9r\u00e9brale essentielle. Et l'on ne s'\u00e9tonnera pas de voir ceux qui attendent d'elle la consolation, le divertissement, et rien d'autre, consid\u00e9rer avec m\u00e9pris toutes les cr\u00e9ations musicales qui proc\u00e8dent des r\u00e9volutions de la seconde Ecole de Vienne ou de Var\u00e8se. Combien de philosophes avertis, auteurs de trait\u00e9s d'esth\u00e9tique ou de consid\u00e9rations publi\u00e9es sur tel ou tel musicien qui fait le bonheur des salons et festivals bourgeois, ont stigmatis\u00e9 ces derni\u00e8res ann\u00e9es toute musique contemporaine au nom du sentiment, du plaisir facile, du Beau id\u00e9al, d'un pr\u00e9tendu h\u00e9donisme, toutes raisons qui cachent mal les int\u00e9r\u00eats id\u00e9ologiques r\u00e9actionnaires ou conservateurs du moment ?\n\nJ'aime la musique contemporaine, car plus qu'une autre elle touche \u00e0 l'essence de la musique, elle approche au plus pr\u00e8s le noyau dur interdisant qu'on parle d'elle avec des cat\u00e9gories humanistes, anthropologiques ou sentimentalistes. Pas de subjectivisme ou d'\u00e9panchements dans l'ordre autobiographique \u2013 du genre : j'aime Schubert, parce qu'il m'a fait pleurer quand j'avais dix ans \u2013, la musique contemporaine exprime une violence \u00e0 l'endroit du r\u00e9el pour en dire la consistance tragique et chiffr\u00e9e, \u00e9nergique et structur\u00e9e, mat\u00e9rielle et \u00e9vanescente. Le musicien est m\u00e9taphysicien, philosophe mat\u00e9rialiste qui construit et propose un monde sonore susceptible d'\u00eatre abord\u00e9 sur le registre exclusif de la variation sur le th\u00e8me de la force musiqu\u00e9e.\n\nFaut-il donc en rester \u00e0 la vision utilitariste et divertissante de la musique? Ou la fustiger d\u00e8s qu'elle cesse d'\u00eatre un support \u00e0 pratiques sociales de reconnaissance de classes ? Ici on se condamne \u00e0 c\u00e9l\u00e9brer des \u0153uvres inconsistantes et le z\u00e8le voudrait m\u00eame qu'\u00e0 l'exc\u00e8s, on finisse par vanter les m\u00e9rites de la musique de vari\u00e9t\u00e9s, plus divertissante encore que les moins s\u00e9rieux des musiciens classiques; l\u00e0 on s'interdit de p\u00e9n\u00e9trer dans ce si\u00e8cle presque achev\u00e9 en trouvant que le maximum de tol\u00e9rance peut s'effectuer jusqu'\u00e0 Debussy, mais pas au-del\u00e0. A moins qu'on trouve dans ces cent derni\u00e8res ann\u00e9es des occasions de d\u00e9roger \u00e0 bon compte en croyant aimer une musique d'aujourd'hui quand les seuls compositeurs sont de ce si\u00e8cle alors que leurs compositions rel\u00e8vent du si\u00e8cle dernier.\n\nJe songe ainsi \u00e0 Jean Fran\u00e7aix que je connaissais par le disque. Je n'aimais gu\u00e8re cette musique qui aurait pu r\u00e9jouir Jank\u00e9l\u00e9vitch \u2013 elle est fran\u00e7aise \u2013 et Cl\u00e9ment Rosset \u2013 elle est d\u00e9lib\u00e9r\u00e9ment joyeuse. Et je dirai jusqu'\u00e0 la naus\u00e9e. Un festival donn\u00e9 tous les mois de septembre dans mon Orne natale me permit d'assister \u00e0 un hommage au musicien en sa pr\u00e9sence, l'ann\u00e9e qui pr\u00e9c\u00e9da sa disparition. Concert touchant, le vieil homme est en forme, ses doigts et son intelligence d\u00e9li\u00e9s comme en temps de prime jeunesse. N\u00e9 en 1912, \u00e9l\u00e8ve de Nadia Boulanger, il avait quatre-vingt-quatre ans au jour du concert et paraissait hors du temps, contemporain de D'Indy et Roussel, Chabrier et Ibert.\n\nOn donnait des \u0153uvres de musique de chambre, dont le quintette pour clarinette et cordes qui fut pr\u00e9sent\u00e9, selon des termes tenus un jour par Pierre-Jean R\u00e9my, comme un monument dans la musique de ce si\u00e8cle. Le pire \u00e9tait donc \u00e0 craindre. La musique qui m\u00e9rite les \u00e9loges de l'Acad\u00e9mie fran\u00e7aise et de l'Institut ne saurait \u00eatre rien moins qu'accorte, polie, pr\u00e9sentable, propre, peu susceptible de d\u00e9ranger qui ou quoi que ce soit. Elle arbore la M\u00e9daille des arts et lettres, sinon la L\u00e9gion d'honneur, et d'elle il ne faut pas attendre grand-chose d'autre. En effet, elle semble faite pour appeler les glands dor\u00e9s et les bicornes, les dipl\u00f4mes et le bravo des bourgeois.\n\nLes histoires de la musique, qui comme toutes les encyclop\u00e9dies et ouvrages de ce genre fournissent souvent des occasions pour appointer des compilateurs, disent toutes la m\u00eame chose de la musique de Jean Fran\u00e7aix : fra\u00eeche et tonique, naturelle et spontan\u00e9e, gracieuse et ing\u00e9nieuse, l\u00e9g\u00e8re et raffin\u00e9e, \u00e9l\u00e9gante et vivace, burlesque et fantaisiste, charmante et ironique, espi\u00e8gle et divertissante, badine et narquoise, facile et simple, enjou\u00e9e et agr\u00e9able, claire et insouciante. Tant de qualit\u00e9s enfil\u00e9es finissent par faire craindre le d\u00e9faut. D'ailleurs, je ne suis pas bien s\u00fbr de pointer l\u00e0 tant de qualit\u00e9s : pour des boissons, peut-\u00eatre, des produits d'entretien, certes, des soir\u00e9es mondaines, \u00e9videmment, des conversations bourgeoises, s\u00fbrement, mais pour la musique, toutes ces vertus sont-elles tellement vertueuses ?\n\nPlus dr\u00f4le, les compilateurs de sous-pr\u00e9fecture qui avaient r\u00e9dig\u00e9 les notices luxueuses du festival parlaient sans craindre le ridicule d'une \u00ab musique fruit\u00e9e et savoureuse \u00bb avant d'ajouter, plus s\u00e9rieusement, qu'en la personne de Jean Fran\u00e7aix \u00abl'esprit de Mozart continue \u00e0 souffler\u00bb. Je ne commenterai pas au-del\u00e0 d'une simple remarque : si Mozart, ce parangon de musicien pour bourgeois sourds, pouvait \u00eatre convoqu\u00e9 dans pareille entreprise pour solliciter et motiver le client payant, il n'y avait rien \u00e0 craindre, on pouvait se rendre au concert sans risque, malgr\u00e9 le vingti\u00e8me si\u00e8cle affich\u00e9 par la carte d'identit\u00e9 du compositeur.\n\nJamais je ne me suis tant ennuy\u00e9. La gaiet\u00e9 devient p\u00e9nible quand elle est assen\u00e9e comme un devoir, un pensum, et qu'elle flotte en \u00e9tendard pour masquer le r\u00e9el qui la d\u00e9ment en permanence. La joie en musique produit chez moi le m\u00eame effet que celle des pr\u00eatres qui b\u00ealent en chaire et invitent \u00e0 un onanisme \u00e9thique avec pr\u00e9texte esth\u00e9tique \u2013 ou th\u00e9ologique. La gaiet\u00e9 coule comme un robinet d'eau ti\u00e8de, \u00e0 sati\u00e9t\u00e9. Et l'on retrouve dans la facture habile du musicien de quoi c\u00e9l\u00e9brer le vide, l'air, le courant d'air m\u00eame, sinon l'inconsistance dans sa quintessence. Rien fait musique, un authentique pari pour les tenants et adeptes de la th\u00e9ologie n\u00e9gative.\n\nJ'ai cette sensation, souvent, en \u00e9coutant de la musique baroque ou de la musique contemporaine exp\u00e9rimentale \u2013 des \u0153uvres au kilom\u00e8tre \u2013, qui pourraient durer des heures, et dont on ne comprend pas ce qui les fait commencer ni m\u00eame ce qui justifie leur interruption. Musique \u00e0 programme, en fait, charg\u00e9e de d\u00e9livrer un message ou d'illustrer un propos, de montrer un savoir-faire ou de proposer une comp\u00e9tence dans l'art du remplissage et de l'\u00e9criture acad\u00e9mique. Pas de temp\u00e9rament propre ni d'individualit\u00e9 souveraine, mais le p\u00e2le reflet de l'\u00e9poque qui permet de parler de l'esprit d'un temps.\n\nLes artifices apparaissent aussi nettement que le nez au milieu de la figure, les clins d'\u0153il et proc\u00e9d\u00e9s humoristiques agissent avec la m\u00eame finesse, la citation rythmique issue des reliquats du jazz semble tout juste dans le retrait n\u00e9cessaire permettant de montrer qu'on a quitt\u00e9 le si\u00e8cle de M\u00e9hul ou Gossec, enfin le registre est nettement revendiqu\u00e9 comme na\u00eff. C'est-\u00e0-dire? On pourrait entendre les m\u00e9lodies, les phrases musicales de Jean Fran\u00e7aix, dans la cour d'une \u00e9cole, dans la balade scolaire d'un mercredi apr\u00e8s-midi de pensionnaire ou dans le voyage scout. Rondes enfantines, comptines pr\u00e9-pub\u00e8res ou m\u00e9lop\u00e9es destin\u00e9es \u00e0 l'\u00e9cran des films en noir et blanc des ann\u00e9es cinquante o\u00f9 l'on voit passer, indolentes et solitaires, des 4 CV au pied de la tour Eiffel.\n\nNa\u00efve, cette musique excelle dans le registre de la simplicit\u00e9 enfantine, mais \u00e9galement dans celui du folklore fran\u00e7ais, villageois et paysan, rural et familial. On entend, ici ou l\u00e0 dans les pi\u00e8ces de chambre, des fantaisies \u00e0 leur place un jour de f\u00eate, puis des ritournelles, \u00e0 l'occasion d'un mariage ou d'un \u00e9v\u00e9nement gr\u00e9gaire. Les racines populaires de la musique dite classique s\u00e9duisent toujours les gens qui refusent peu ou prou les difficult\u00e9s du conceptuel ou du c\u00e9r\u00e9bral affich\u00e9 comme tel. Ceux qui, en ce si\u00e8cle, revendiquent Bart\u00f3k ou Stravinski, le font souvent pour les complaisances de ces compositeurs \u00e0 l'endroit des musiques populaires de leurs pays.\n\nMusique ? Oui, musique fran\u00e7aise quintessenci\u00e9e. Celle qui permet au m\u00eame plumitif inspir\u00e9 \u2013 celui des musiques fruit\u00e9es \u2013 d'\u00e9crire dans sa pr\u00e9sentation : \u00abJean Fran\u00e7aix compose comme il respire, c'est-\u00e0-dire avec naturel, spontan\u00e9it\u00e9, sans apparemment se poser les questions d'esth\u00e9tique et de m\u00e9taphysique qui ont conduit quelques-uns de ses confr\u00e8res \u00e0 une st\u00e9rilit\u00e9 navrante. Jean Fran\u00e7aix a r\u00e9alis\u00e9 son but : faire de la musique s\u00e9rieuse, sans gravit\u00e9.\u00bb Peut-\u00eatre, mais j'aurais pr\u00e9f\u00e9r\u00e9 qu'il fasse de la musique grave sans se prendre au s\u00e9rieux. Pire, qu'il ait justement consenti \u00e0 mettre un peu de souci esth\u00e9tique et un soup\u00e7on de m\u00e9taphysique dans sa musique. Gageons qu'elle ne s'en serait pas moins bien port\u00e9e.\n\nD'ailleurs, composer comme on respire, suppose le risque soit de mal composer, soit, plus grave, de mal respirer. Le naturel ne convient jamais \u00e0 l'art qui vit essentiellement d'artifice et de faux-semblants, de ruses et de sublimations, de transcendances et de culture. Et puis la gravit\u00e9 n'est pas un d\u00e9faut. La l\u00e9g\u00e8ret\u00e9 me semble un plus grand p\u00e9ch\u00e9 quand on ne la pratique pas tel un art supr\u00eame, comme chez Chamfort ou Nietzsche. Enfin, refuser d'\u00eatre grave, \u00e0 mon avis, rend difficile la possibilit\u00e9 d'\u00eatre s\u00e9rieux. Qu'on se contente d'\u00eatre grave, le s\u00e9rieux viendra de surcro\u00eet, s'il doit venir. Car il n'y a pas p\u00e9ch\u00e9, sauf pour les sots, d'\u00eatre l'un et l'autre.\n\nPas de concept, mais la v\u00e9rit\u00e9 de la tradition, pas de complexit\u00e9, mais la simplicit\u00e9 des m\u00e9lodies de toujours, pas de m\u00e9taphysique ou d'esth\u00e9tique, mais les bonnes vieilles recettes de l'assentiment populaire et du grand public : voil\u00e0 qui fait un ensemble seyant et vertueux pour les amateurs bourgeois de musique classique. Pas de profondeur ou de tragique, pas de gravit\u00e9, pas de cette grossi\u00e8ret\u00e9 qui suppose un auditeur cultiv\u00e9 ou amateur \u00e9clair\u00e9 ou \u00e9duqu\u00e9 : l'all\u00e9gresse et la joie, l'humour et la l\u00e9g\u00e8ret\u00e9 se comprennent par tous, et sans d\u00e9tour.\n\nAlors le bourgeois peut s'encanailler, il peut croire \u00e0 moindres frais qu'il adh\u00e8re \u00e0 la musique contemporaine, la vraie, la seule, celle qu'on comprend tout de suite, mais dont l'acc\u00e8s semble d'autant facile et simplifi\u00e9 qu'elle est sommaire, simple, vide et creuse. Le prix \u00e0 payer. Et une bonne partie de la musique fran\u00e7aise n\u00e9oclassique de ce si\u00e8cle, fi\u00e8re de sa tradition l\u00e9g\u00e8re et diaphane, claire et distincte, simple et gracieuse, ne doit son existence qu'\u00e0 cette complaisance \u00e0 l'endroit des bons sentiments, la joie, l'all\u00e9gresse, la beaut\u00e9, la joliesse. Certes, ils permettent la facilit\u00e9 d'acc\u00e8s, mais la brillance de surface se fait payer par une absence de profondeur dommageable. On ne fait pas de musique essentielle avec des bons sentiments, tout juste une composition superficielle disparue au moindre souffle.\n\nCe jour-l\u00e0, les bourgeois trouvant leur compte, debout, ont ovationn\u00e9 cet homme et ces \u0153uvres qui leur permettaient de continuer \u00e0 croire que la musique doit \u00eatre plaisante et non profonde, agr\u00e9able et non s\u00e9rieuse, aimable et divertissante et non inqui\u00e9tante et m\u00e9taphysique. Tous ces m\u00e9lomanes applaudissent depuis bient\u00f4t quinze ans de festival, Sauguet et Landowski, G\u00e9rard Cond\u00e9 et Jean Fran\u00e7aix en imaginant qu'il n'est de musique que celle-l\u00e0, flatteuse et complaisante \u00e0 l'endroit de celles et ceux qui, encore et toujours, pratiquent le concert classique comme une occasion de reconnaissance sociale, un comportement de classe.\n\nOr, leur religion sociale ignore 1789 et l'ind\u00e9pendance acquise des compositeurs d'aujourd'hui \u00e0 l'endroit des bourgeois qui persistent \u00e0 penser que leurs anc\u00eatres nourrissaient Mozart et qu'\u00e0 ce titre l'artiste leur est toujours redevable de la pitance qu'ils lui permettent avec la circulation de leurs richesses. En donnant ces musiques-l\u00e0 dans un concert, le jeu social jubile d'\u00eatre tel : des compositeurs offrent \u00e0 des bourgeois l'impression qu'ils sont intelligents \u2013 occasion rare et sublime. Ils ne sont jamais aussi g\u00e9n\u00e9reux de leurs bravos et vivats qu'en de pareils moments.\n\nCe soir-l\u00e0, l'auteur du programme pouvait rentrer chez lui satisfait du travail accompli : Jean Fran\u00e7aix, il avait raison, n'a ennuy\u00e9 personne avec de l'esth\u00e9tique ou de la m\u00e9taphysique, du sens ou de la profondeur. Il a eu la politesse bien fran\u00e7aise, fruit\u00e9e si j'en crois les nouvelles fa\u00e7ons de parler musique, d'\u00e9pargner tout effort \u00e0 son auditoire en restant simple et na\u00eff, frais et \u00e9l\u00e9gant, disons-le en un mot que n'aura pas os\u00e9 l'apprenti critique musical : divertissant.\n\nPour leur part, les familiers du festival ont ressenti la m\u00eame satisfaction narcissique et, aux c\u00f4t\u00e9s de leurs amis bien-n\u00e9s et bien d\u00e9cor\u00e9s acoquin\u00e9s, fr\u00e9missant d'avoir applaudi de la musique contemporaine, ils sont rentr\u00e9s chez eux, encore \u00e9moustill\u00e9s par tant d'audace. Une pareille charge d'\u00e9motion a suffi pour un soir. Ils ont dormi du sommeil du juste en ayant bais\u00e9 leur \u00e9pouse sur le front, avant de se retourner pour sombrer dans la lourde inconscience des nuits qui se ressemblent. Peut-\u00eatre m\u00eame que dans leurs r\u00eaves, fruit\u00e9s eux aussi, ils ont fredonn\u00e9 les m\u00e9lodies de Jean Fran\u00e7aix, encore \u00e9tonn\u00e9s d'avoir vibr\u00e9 \u00e0 des accents de musique contemporaine.\n\nUne anecdote, pour conclure : en sortant du concert, dans les lumi\u00e8res d'un automne qui s'annon\u00e7ait sans ambages, un jeune gar\u00e7on handicap\u00e9 mental, habill\u00e9 comme un clone par ses parents \u00e0 qui il avait \u00e9chapp\u00e9, le temps pour eux de glousser et piailler avec leurs semblables, se pr\u00e9cipita sur un quidam. La main tendue, avec un sourire d'ange triste, tout entier m\u00e9tamorphos\u00e9 et habit\u00e9 par l'essence de cette musique qui ravissait ses parents, il tonitrua, d\u00e9sarticul\u00e9, un \u00abBonne ann\u00e9e\u00bb qui, encore aujourd'hui, n'en finit pas de me r\u00e9jouir.\n14\n\nFRAN\u00c7AISES, ENCORE UN EFFORT...\n\n(Pour un libertinage solaire, II.) D\u00e9j\u00e0 je sens chez quelques femmes les faux sorties de la cave et les piques astiqu\u00e9es pour une nouvelle confrontation avec les hommes, ces ennemis de toujours. J'avais bien song\u00e9 intituler mon texte Nos amies les femmes, puis je me suis dit qu'il fallait attendre un peu pour l'ironie, ne pas commencer bille en t\u00eate, laisser un peu le temps de chauffer l'atmosph\u00e8re avant les premi\u00e8res reparties glaciales. Mais je n'arrive pas \u00e0 me retenir, c'est plus fort que moi. J'avais aussi pens\u00e9 \u00e0 Sur les femmes, en forme de clin d'\u0153il \u00e0 Diderot et son superbe texte sur ce sujet. Et puis, l\u00e0 encore, j'ai jug\u00e9 qu'on pourrait voir ce titre comme une autre occasion de me brocarder et que, pareil \u00e0 Valmont dans les Liaisons dangereuses, c'est sur l'arri\u00e8re d'une V\u00e9nus callipyge qu'on m'aurait imagin\u00e9 \u00e9crivant les lignes \u00e0 suivre. Afin d'\u00e9viter le reproche d'avoir utilis\u00e9 m\u00e9taphoriquement le bas du dos des dames en \u00e9critoire pour mes consid\u00e9rations sur elles, j'ai pr\u00e9f\u00e9r\u00e9 placer mon titre sous les m\u00e2nes et le patronage du Sade le moins nocturne, celui qui dans La Philosophie dans le boudoir invite \u00e0 des efforts pour \u00eatre r\u00e9publicain...\n\nJe voudrais ici r\u00e9pondre aux accusations diverses, multiples et fr\u00e9quentes qui m'ont \u00e9t\u00e9 faites par tels ou telles, pensant qu'\u00e0 ne pas b\u00ealer derri\u00e8re la prosodie classique d'Aragon pour lequel \u00abla femme est l'avenir de l'homme \u00bb, on devient de facto un fieff\u00e9 misogyne. Hol\u00e0 ! un gros mot. Car il est gros, tr\u00e8s gros. De sens d\u00e9couvert et cach\u00e9, de lourds reproches et de sympt\u00f4mes, de malentendus aussi. Car misogyne est insultant. Comme raciste, sexiste, colonialiste, sinon, parfois, f\u00e9ministe, tant sont grandes les parent\u00e9s entre ceux qui veulent ha\u00efr d'abord en imaginant pouvoir comprendre ensuite.\n\nA ne pas communier dans la guimauve aragonesque, \u00e0 lui pr\u00e9f\u00e9rer la jubilation de Charles Fourier d\u00e9vou\u00e9 \u00e0 la cause papillonnante ou l'acier tremp\u00e9 d'Andr\u00e9 Breton c\u00e9l\u00e9brant l'amour fou, on est bien souvent renvoy\u00e9 du c\u00f4t\u00e9 des ennemis des femmes. Ils sont pourtant d\u00e9j\u00e0 trop nombreux pour y ajouter le renfort de ceux qui demandent seulement que le f\u00e9minisme ne soit pas aussi ridicule que le sexisme auquel il s'oppose et dont, si souvent, il emprunte les armes, les techniques et les proc\u00e9d\u00e9s. Ha\u00efr ou d\u00e9tester les femmes, voil\u00e0 ce que dit Littr\u00e9 non loin de misogame \u2013 qui n'aime pas le mariage \u2013 et miss, que l'on a reconnu. Qui donc, peut \u00eatre dit misogyne ? Et quand ?\n\nL'accusation s'appuie sur un dossier que les commissaires du peuple ont instruit avec la m\u00eame ardeur et la m\u00eame probit\u00e9 que Fouquier-Tinville. D'abord dans La Sculpture de soi qu'on a vu, et non pas lu, quatre lignes suffisaient \u00e0 me disculper, comme un manuel du parfait macho \u00e0 l'usage de ceux qui n'en avaient pas. Jugez-en ! Un condottiere c\u00e9l\u00e9br\u00e9, transform\u00e9 en h\u00e9ros pour une \u00e9thique moderne ! Un soudard, un soldat, un mercenaire ! Et un livre tout entier \u00e9crit au masculin, triomphant de toute sa superbe, \u00e9non\u00e7ant sa masculinit\u00e9 comme on porte un braquemart triomphant dans une procession ithyphallique ! Oui, je confesse, il est \u00e9crit \u00e0 la premi\u00e8re personne et, jusqu'\u00e0 nouvel ordre, non pas en vertu du fait que, g\u00e9n\u00e9ralement, le masculin l'emporte sur le f\u00e9minin, mais seulement chez moi, particuli\u00e8rement, sur le terrain physiologique et hormonal.\n\nToutefois, dans la seule note en bas de page de ce livre, les quatre lignes pr\u00e9cit\u00e9es, j'ai invit\u00e9 les lectrices chatouilleuses \u00e0 remplacer si besoin \u00e9tait tout ce qui appara\u00eet au masculin dans l'ouvrage par un f\u00e9minin pur et dur, orthodoxe \u00e0 souhait. J'ai \u00e9galement signal\u00e9, en pr\u00e9ambule \u00e0 toute analyse, que le condottiere selon mon souhait n'\u00e9tait ni le personnage r\u00e9el, historique, ni m\u00eame tel ou tel, qui fut nomm\u00e9 \u2013 Bartolomeo Colleoni \u2013, mais d'abord la transfiguration qu'en avait faite Verrocchio dans sa statue \u00e9questre, ensuite ce que je souhaitais de mes v\u0153ux : le passage \u00e9pur\u00e9 une fois encore \u00e0 une figure \u00e9thique proc\u00e9dant d'une figure esth\u00e9tique de la Renaissance.\n\nPar ailleurs, lorsque j'ai c\u00e9l\u00e9br\u00e9 la virilit\u00e9 que j'estimais plus que tout en l'homme, j'ai imm\u00e9diatement signal\u00e9 que cette vertu \u00e9tait \u00e9galement possible, indistinctement, chez les femmes dans la mesure o\u00f9, \u00e0 mes yeux, elle signifie l'humain en l'homme, ce qui le tient debout, droit, quand tout dans le social l'invite \u00e0 l'abaissement, \u00e0 la reptation. Comment aurais-je exclu les femmes de ce projet \u00e9thique ? La tension, la force, la volont\u00e9, la d\u00e9cision \u2013 vertus par moi choy\u00e9es \u2013 sont possibles et pensables au m\u00eame titre et au m\u00eame degr\u00e9 chez les hommes et chez les femmes.\n\nMais voil\u00e0, pour entendre ces pages-l\u00e0, il faudrait les avoir lues, avoir un talent plus grand pour la compr\u00e9hension patiente que pour le d\u00e9ni et le proc\u00e8s sans appel. Il est m\u00eame des hommes, des journalistes d\u00e9c\u00e9r\u00e9br\u00e9s (trois au moins), s'il ne fallait craindre le pl\u00e9onasme, qui ont vu dans ces pages de quoi m'installer du c\u00f4t\u00e9 des nazis. Rien moins. Parler de force ou de virilit\u00e9 suppose immanquablement se faire le fourrier des fascistes au regard des reproducteurs de clich\u00e9s. Quand ceux-ci n'ont pour ennemis que ceux-l\u00e0, ils sont pr\u00e8s de triompher. Qu'on cesse de r\u00e9p\u00e9ter des lieux communs, qu'on arr\u00eate de b\u00ealer avec le troupeau des conformistes de tous bords, des penseurs unidimensionnels, et alors seulement on pourra envisager une pens\u00e9e solaire, radieuse, libertine et libertaire, radicalement antifasciste, qui rende caduque toute entreprise misogyne, sexiste et discriminatoire. Le fascisme sous toutes ses formes, y compris le sexisme, m\u00e9rite de meilleurs ennemis que ces grenouilles voulant se faire aussi grosses que le b\u0153uf.\n\nCeux-l\u00e0 m\u00eames, qui, oublieux des progr\u00e8s de leurs ennemis, pr\u00e9f\u00e8rent donner l'\u00e9trivi\u00e8re \u00e0 leurs amis, ont cherch\u00e9 de quoi filer la m\u00e9taphore misogyne dans Le D\u00e9sir d'\u00eatre un volcan. Bien leur en a pris ! Ils trouv\u00e8rent P\u00e9n\u00e9lope et les autres, un mod\u00e8le du genre sexiste et phallocrate, bien s\u00fbr. Un monument \u00e9lev\u00e9 \u00e0 la gloire du machisme occidental, \u00e9videmment ! Enfin une r\u00e9cidive ! Du grain \u00e0 moudre pour l'instruction du dossier. Qu'ai-je \u00e9crit dans ces pages que j'ai voulues avant tout un \u00e9loge de la diversit\u00e9, de la multiplicit\u00e9 des r\u00f4les et registres f\u00e9minins? Qu'ai-je avanc\u00e9 de coupable en disant qu'il ne fallait pas r\u00e9duire les femmes \u00e0 la monogamie conjugale d\u00e9bouchant sur la seule et in\u00e9luctable maternit\u00e9? Qu' ai-je dit de coupable en avan\u00e7ant l'hypoth\u00e8se de relations complices amicales, intellectuelles, c\u00e9r\u00e9brales avec des femmes auxquelles on ne demanderait rien de sexuel \u2013 ou l'inverse ? En quoi me suis-je damn\u00e9 pour avoir \u00e9nonc\u00e9 le souhait d'un nouveau contrat sensuel h\u00e9doniste, d'une nouvelle intersubjectivit\u00e9 sexuelle et amoureuse?\n\nJ'y racontais, trop succinctement, bien s\u00fbr, on l'est toujours trop pour ses ennemis, jamais assez pour ses amis, quelles figures parmi tant d'autres avaient travers\u00e9 l'imaginaire occidental f\u00e9minin : Eve et Marie, Omphale et Carmen, M\u00e9d\u00e9e et Elvire. Puis P\u00e9n\u00e9lope. J'y d\u00e9veloppais des consid\u00e9rations sur le savoir et la maternit\u00e9, le d\u00e9sir et la passion, la conjugalit\u00e9 et la s\u00e9duction, la fid\u00e9lit\u00e9 aussi. Et je me retrouvais dans la peau d'un homme qui voulait des cases, bien rang\u00e9es, bien nettes, dans lesquelles installer toutes les femmes comme dans des r\u00e9duits carc\u00e9raux, humides et sinistres, avec impossibilit\u00e9, en \u00e9tant ici, d'\u00eatre ailleurs.\n\nAu contraire, j'ai d\u00e9sign\u00e9 ces figures comme des embl\u00e8mes de la raison sexu\u00e9e occidentale, des arch\u00e9types, sinon, pour le dire comme Deleuze, des personnages conceptuels. Elles supposaient, en contrepoint, Adam et Joseph, Hercule et Don Jos\u00e9, Jason et Don Juan, sinon Ulysse, toutes entit\u00e9s m\u00eamement charg\u00e9es de sens sans que pour autant j'aie aspir\u00e9 une seconde \u00e0 enfermer l'humanit\u00e9 masculine dans ces sept r\u00f4les et eux seuls. Mais elles sont figures qui transpercent l'Occident et signent l'issue, parfois tragique, des rapports entretenus par les hommes avec les femmes. Quel homme n'a exp\u00e9riment\u00e9 l'ob\u00e9issance, la paternit\u00e9, la soumission, le d\u00e9semparement, la rupture, le libertinage, sinon l'errance? Faut-il pour autant conclure chez moi \u00e0 une misanthropie g\u00e9n\u00e9ralis\u00e9e, \u00e0 une haine des hommes, un m\u00e9pris visc\u00e9ral ? Je ne crois pas. Ces figures n'acc\u00e8dent pas \u00e0 l'embl\u00e9matique sans signifier profond\u00e9ment dans l'inconscient d'une civilisation. Les pointer comme telles, comme sch\u00e8mes, cat\u00e9gories, concepts op\u00e9ratoires \u2013 masculins ou f\u00e9minins \u2013, \u00e7a n'est rien d'autre que poursuivre, avec des points de rep\u00e8re partag\u00e9s par tout un chacun, la qu\u00eate d'un sens qui, pour ma part, me r\u00e9siste encore. Et c'est heureux.\n\nOn m'a beaucoup reproch\u00e9 P\u00e9n\u00e9lope, figure du sacrifice, embl\u00e8me de l'automutilation ou de l'asservissement. Je ne veux rien dire pour elle, \u00e0 sa place, ni me justifier, ni m'excuser, ni donner arguments pour qu'on comprenne. On m'a parl\u00e9 de servitude. Je sais. Mais j'ai lu La Bo\u00e9tie et n'ignore pas sa nature myst\u00e9rieuse, toujours, parce que proc\u00e9dant d'une \u00e9trange et puissante volont\u00e9. De celle-ci, je n'ai rien \u00e0 dire, pour ne pas avoir \u00e0 la trahir. Ni m\u00eame en donner les quelques mis\u00e9rables cl\u00e9s \u00e0 ma disposition. Je me souviens de ces mots, dans le Discours de la servitude volontaire : \u00ab Soyez r\u00e9solus de ne plus servir, et vous voil\u00e0 libre. \u00bb Si la r\u00e9solution fait d\u00e9faut, c'est qu'existent des raisons pour cela. S\u00fbrement. Pour ce que j'en sais, elles resteront priv\u00e9es. Dans mon panth\u00e9on, P\u00e9n\u00e9lope si\u00e8ge au plus haut : elle aime, comme peu, Ulysse comme il est. Je souhaite \u00e0 tous et toutes une figure \u00e0 ce point salvatrice. Sans elle, il y a bien longtemps que Thanatos, le compagnon le plus proche des hommes, aurait triomph\u00e9 sans partage. D'aucuns qui cherchent \u00e0 quoi ressemble l'amour imaginent mal qu'il puisse avoir cette forme. Elle est l'expression quintessenci\u00e9e de la conjugalit\u00e9 qui ne donne que ce qu'elle a, ce qui n'\u00e9puise pas tout, mais rend possible tout le restant.\n\nDe cette figure, les amazones tireront argument pour mon accusation. Elles auront raison, je ne les suivrai pas sur ce terrain. On ne d\u00e9teste pas les femmes quand on ne sait pas les aimer comme elles voudraient l'\u00eatre et qu'on les c\u00e9l\u00e8bre comme on peut, maladroitement, confus\u00e9ment, en donnant \u00e0 l'une ce que l'on ne peut ni ne sait donner \u00e0 l'autre, quand on ne veut pas pour m\u00e8re de ses enfants ou compagne pr\u00e8s de soi celle qu'on aime d\u00e9sesp\u00e9r\u00e9ment, d'un amour fou, telle Nadja, ou celle avec laquelle on c\u00e9l\u00e8bre les fastes des d\u00e9sirs et des plaisirs qui aspirent \u00e0 leur satisfaction jubilatoire dans l'innocence du tragique conjur\u00e9 pour un soir. Dans le fouillis des \u00e2mes qui se perdent et tentent de se retrouver en m\u00e9langeant leurs peaux, P\u00e9n\u00e9lope brille en l'\u00e9toile polaire qui permet de lire dans le ciel \u00e0 peu pr\u00e8s comme dans une carte. Est-on coupable de s'\u00eatre perdu et d'errer sans points de rep\u00e8re?\n\nPour n'\u00eatre pas en reste, les guetteuses n\u00e9gatives ajout\u00e8rent au dossier apr\u00e8s m'avoir entendu \u00e0 France Culture o\u00f9 je revenais sur cette question. J'avais \u00e9crit un livret d'op\u00e9ra radiophonique pour Eric Tanguy. Son titre \u00e9tait Le Libertin foudroy\u00e9. J'y mettais en sc\u00e8ne un homme, une femme et un tiers dans un \u00e9change \u00e9pistolaire qui t\u00e2che de montrer l'entropie d'une relation amoureuse, de la passion des premiers jours \u00e0 la d\u00e9liquescence finale, en passant par le d\u00e9sir d'enfant chez la femme. Ces pages \u00e9taient pour moi l'occasion de dire la disparit\u00e9 des d\u00e9sirs et des plaisirs, la diff\u00e9rence des aspirations et des destins envisag\u00e9s \u00e0 leurs histoires d'amour par les hommes et les femmes.\n\nA l'antenne, on lut un passage d'une lettre \u00e9crite par l'homme \u00e0 la femme et qui racontait son pass\u00e9 libertin. J'en fus rendu responsable et auteur \u00e0 titre priv\u00e9. Cet homme, c'\u00e9tait moi. Ce qu'il disait, c'\u00e9tait ce que je pensais. J'avais beau r\u00e9torquer qu'il s'agissait d'une \u0153uvre de fiction, qu'une cr\u00e9ature imaginaire est un mixte d'invention, de r\u00e9alit\u00e9, de distorsions, que j'\u00e9tais tout autant dans le personnage f\u00e9minin, ou dans le tiers r\u00e9citant, rien n'y fit. Il fallait que je fusse libertin, donc misogyne, d\u00e9testant les femmes, les utilisant, les r\u00e9duisant \u00e0 l'\u00e9tat d'objets. Les pr\u00e9mices br\u00fblantes, le d\u00e9sir avant le plaisir, la demande de maternit\u00e9, l'incandescence de la passion, la trag\u00e9die entropique, la disparit\u00e9 des attentes, tout cela est \u00e9ternel. Dans le livret, d'ailleurs, le personnage f\u00e9minin \u00e9tait Elle, le masculin, Lui, arch\u00e9types par excellence. Mais je devais \u00eatre ce misogyne qu'on a vu ailleurs.\n\nJe r\u00e9pondis aux lettres, sinon aux demandes d'explications qui me furent faites. De bons ap\u00f4tres, certainement lecteurs d'Aragon, celui d'avant les petits gar\u00e7ons, me firent l'\u00e9loge de la maternit\u00e9, un p\u00e8re me c\u00e9l\u00e9bra par lettre les quasi-couvades reprises \u00e0 son compte pendant la grossesse de sa femme, d'autres me plaignaient et appelaient la psychanalyse pour faire r\u00e9gner l'ordre et la discipline sociale dans laquelle tous ou presque communient. On n'aime pas les pratiquants de l'ath\u00e9isme familialiste. Enfin, il fallait bien que ma misogynie f\u00fbt prouv\u00e9e : on voyait dans mon refus du mariage, de la paternit\u00e9, de la fid\u00e9lit\u00e9 sexuelle, de la monogamie, et de tout l'attirail bourgeois convoqu\u00e9 pour faire tenir droit et debout l'entreprise familiale, une s\u00e9rie de raisons majeures et d\u00e9finitives \u00e0 ma haine des femmes. En effet, ne pas les vouloir m\u00e8res, cuisini\u00e8res, au foyer, dociles et soumises, propri\u00e9t\u00e9s d'un seul, engag\u00e9es pour la vie, voil\u00e0 qui trahit le phallocrate, le f\u00e9odal utilisant les femmes comme un seigneur son bien.\n\nEn revanche, l'homme digne, celui que les femmes z\u00e9l\u00e9es appellent de leurs v\u0153ux, c'est vraisemblablement l'\u00e9poux, le p\u00e8re, le fr\u00e8re et le copain m\u00e9lang\u00e9s en une seule personne vid\u00e9e de sa substance, de son \u00e9nergie, pour en faire une m\u00e9canique sociale susceptible d'\u00eatre d\u00e9cor\u00e9e de la M\u00e9daille de la Famille Fran\u00e7aise. Ceux-l\u00e0 fredonnent Aragon, mis en musique par Jean Ferr\u00e2t, ils fr\u00e9tillent et gloussent, se font h\u00e9ro\u00efques, et portent en banderoles \u00e9clatantes leur go\u00fbt pour l'androgynat en confessant leur ineffable tristesse de ne pouvoir eux-m\u00eames syndicalement porter l'enfant, quatre mois et demi, pour partager les t\u00e2ches m\u00e9nag\u00e8res.\n\nDe tels piliers de l'ordre moral m\u00e9ritent les applaudissements des f\u00e9ministes. Qu'on lise ou relise Deleuze et Guattari qui disent dans L'Anti-\u0152dipe tout ce que le familialisme doit aux secours d'une psychanalyse tout autant m\u00e9sus\u00e9e qu'un certain f\u00e9minisme fanatique et agressif. Qu'on lise ou relise Annie Le Brun, et plus particuli\u00e8rement L\u00e2chez tout et Vagit Prop. On y d\u00e9couvrira une pens\u00e9e libre, radicalement libre, d\u00e9barrass\u00e9e de ces hochets revendicatifs qui emp\u00eachent toute action individuelle v\u00e9ritablement subversive. Annie Le Brun, au-del\u00e0 du f\u00e9minisme apostolique et romain, propose une f\u00e9minit\u00e9 solaire et libertaire, grande et g\u00e9n\u00e9reuse, forte de Sade et de Breton, de Roussel et de Cravan. Loin du ressentiment agressif et pourvoyeur de la permanence du malentendu entre les sexes, elle agit en femme, ne devant rien aux hommes. Son ardeur et sa vivacit\u00e9 br\u00fblent dans la c\u00e9l\u00e9bration de la puissance libertaire et l'art de magnifier les subjectivit\u00e9s radieuses \u2013 qu'elles soient m\u00e2les ou femelles.\n\nQue peut-on dire ou \u00e9crire sur les femmes en cette fin de si\u00e8cle quand on est homme et qu'on aspire \u00e0 une pens\u00e9e r\u00e9g\u00e9n\u00e9r\u00e9e sur ce sujet? Pas grand-chose en dehors de la vulgate et de ce qu'il est convenu de croire en terme d'id\u00e9ologie consensuelle. Poser la question : qu'est-ce qu'une femme ? est d\u00e9j\u00e0 suspect. On flaire le pi\u00e8ge, le traquenard misogyne, si l'interrogation ne vient pas d'un lieu d\u00fbment r\u00e9pertori\u00e9 comme pr\u00e9tendument ami, alli\u00e9. Or, parfois, les ennemis les plus \u00e0 craindre sont dans le camp m\u00eame des combattants, dans les rangs, parmi les plus sinc\u00e8res. Au milieu m\u00eame des femmes. Annie Le Brun l'a superbement d\u00e9montr\u00e9 : des transfuges, des ren\u00e9gates, des agents doubles servent plus la cause phallocrate que l'inverse.\n\nComment ? En installant le sexisme partout, surtout l\u00e0 o\u00f9 il n'est pas. En faisant de tout homme qui ne r\u00e9cite pas le cat\u00e9chisme f\u00e9ministe un ennemi d\u00e9clar\u00e9, pas m\u00eame potentiel. En refusant qu'on examine autrement la question, qu'on aille au-del\u00e0 de l'orthodoxie pr\u00e9sent\u00e9e comme telle. En \u00e9vitant le concours et la collaboration des hommes dans le combat libertaire sous pr\u00e9texte qu'on ne travaille pas avec l'ennemi de classe. Or, les vieux sch\u00e9mas chr\u00e9tiens et marxistes ne cessent d'impr\u00e9gner le discours f\u00e9ministe \u00e9troit : religion, orthodoxie, h\u00e9r\u00e9sie, vulgate, \u00e9vangiles, lutte des classes, r\u00e9volution permanente, avant-garde \u00e9clair\u00e9e, restent chez elles des concepts op\u00e9ratoires alors que tout cet attirail a suppos\u00e9, et suppose encore, une reconsid\u00e9ration th\u00e9orique partout ailleurs.\n\nComment, alors, d\u00e9poussi\u00e9rer la cause, \u0153uvrer \u00e0 une mise \u00e0 jour du combat dans les perspectives contemporaines ? En d\u00e9pla\u00e7ant les doutes sur un autre terrain, notamment l\u00e0 o\u00f9 s'installent les hommes qui revendiquent haut et clair un f\u00e9minisme semblable comme deux gouttes d'eau au bon vieux machisme. Exemple ? Le lendemain de mon passage \u00e0 France Culture, un sp\u00e9cialiste m\u00e2le du politiquement correct, des roucoulements voluptueux dans la voix, affirmait sans une ride dans le ton que tout ce qui s'\u00e9tait fait de grand dans l'humanit\u00e9 l'avait \u00e9t\u00e9 par les femmes et seulement gr\u00e2ce \u00e0 elles. Pas de rires, l'acquiescement b\u00e9at et convenu des amateurs d'id\u00e9ologies correctes. Un ami des femmes ce flagorneur qui soumet sans difficult\u00e9 les amazones et r\u00e9duit leur esprit critique \u00e0 rien ! Pas suspect, cet homme de gauche, parangon de vertus socialistes, \u00e9tait cr\u00e9dible en assenant cette contrev\u00e9rit\u00e9 que tout un chacun disposant d'encore un peu de sant\u00e9 mentale aurait rep\u00e9r\u00e9e. Les racistes disent qu'on ne doit rien aux peuples en dehors de la race blanche. Sont-ils antiracistes ceux qui, aujourd'hui, aux Etats-Unis, affirment sans broncher, que toute la civilisation occidentale proc\u00e8de de la seule Afrique noire ? Qu'on ne desserve pas sa cause, quand elle est juste, par une r\u00e9volution si radicale qu'elle ram\u00e8ne, apr\u00e8s l'illusion d'un mouvement violent, \u00e0 l'endroit m\u00eame d'o\u00f9 tout \u00e9tait parti.\n\nLa femme n'est pas l'avenir de l'homme, article num\u00e9ro un. Le second? l'homme, pas plus, n'est l'avenir de la femme. Chacun demeure le propre inventeur de son existence, sans qu'il soit besoin de se reposer sur l'un ou l'autre en lui demandant de faire le travail de sculpture de soi \u00e0 notre place. L'avenir de chacun r\u00e9side dans son vouloir et sa r\u00e9sistance \u00e0 l'endroit de ce qu'on veut qu'il soit socialement, \u00e0 l'exclusion de tout autre dessein.\n\nDevant pareille t\u00e2che, les hommes et les femmes doivent accomplir le m\u00eame trajet, avec vraisemblablement pour les dames plus de difficult\u00e9s \u00e0 s'\u00e9manciper encore, plus et mieux, dans cette civilisation faite par les hommes et pour eux depuis si longtemps. A \u00e9viter : un projet de culture amazone o\u00f9 ce qui \u00e9tait regrettable hier, la soumission d'un genre \u00e0 l'autre, deviendrait d\u00e9fendable le lendemain sous pr\u00e9texte que les victimes auraient le droit de devenir bourreaux.\n\nQuelle chance pour un f\u00e9minisme h\u00e9doniste ? La chance \u00e9galitaire par excellence. Celle qui permettrait un m\u00eame combat pour les m\u00eames valeurs : la libert\u00e9, le souci et l'usage de soi enti\u00e8rement soumis au pur et simple caprice subjectif. Hommes ou femmes, sans distinction aucune. Le progr\u00e8s ne se trouve pas dans la possibilit\u00e9, pour une femme de devenir b\u00fbcheron, ou pour un homme d'ouvrir un salon de toilettage pour chiens, pour une femme directrice d'un office quelconque d'\u00eatre appel\u00e9e Madame le directeur ou pour quelque virago, de r\u00e9\u00e9crire le Dictionnaire de la langue fran\u00e7aise en le faisant passer par les fourches Caudines d'un f\u00e9minisme linguistique pu\u00e9ril. Il triomphe dans la revendication solaire et solitaire de sa subjectivit\u00e9, pas en tant que femme ou homme, mais dans la perspective d'un individualisme v\u00e9ritablement r\u00e9volutionnaire.\n\nPour c\u00e9l\u00e9brer cet h\u00e9donisme g\u00e9n\u00e9ralis\u00e9, il faut mettre en \u0153uvre une m\u00e9thode bien oubli\u00e9e, \u00e0 tort, car elle permet un travail radicalement philosophique : celle que Remy de Gourmont, qu'on ne lit pas assez, propose et illustre dans La Culture des id\u00e9es, un livre qui ouvre ce si\u00e8cle. Gourmont, franc-tireur, penseur libre, form\u00e9 \u00e0 l'\u00e9cole de Schopenhauer et de Nietzsche, propose la dissociation d'id\u00e9es. C'est-\u00e0-dire ? Partant du constat que les mots fonctionnent toujours associ\u00e9s, en couple, l'un appelant toujours l'autre en compl\u00e9ment oblig\u00e9, il se demande quelles raisons pr\u00e9sident \u00e0 ces \u00e9tranges assemblages. Pourquoi, par exemple, l'arriv\u00e9e du mot Honneur suppose \u00e0 chaque fois celui de Patrie ? Selon quels m\u00e9canismes la convocation de Art induit-elle l'apparition de Beaut\u00e9? Justice supposant Ch\u00e2timent, Instruction appelant Intelligence ou Devoir sollicitant Bonheur.\n\nEntre ces couples s'installe le b\u00e9n\u00e9fice social ou culturel de la civilisation. La soci\u00e9t\u00e9 a int\u00e9r\u00eat \u00e0 ce que tel ou tel mot fonctionne en bin\u00f4me pour produire une id\u00e9e g\u00e9n\u00e9ratrice d'ordre. L'\u00e9ducation scolaire, la promotion des pens\u00e9es serviles dans une \u00e9poque, le relais familial, l'univers m\u00e9diatique aujourd'hui, tout va dans le m\u00eame sens et enseigne ces couples n\u00e9fastes comme des v\u00e9rit\u00e9s auxquelles il faut sacrifier, sous peine d' avoir des comptes \u00e0 rendre. Le lieu commun ressass\u00e9 devient \u00e0 force une v\u00e9rit\u00e9 sociale enseign\u00e9e sans vergogne pour le plus grand bien du monde comme il va.\n\nLa dissociation d'id\u00e9es vise une destruction de ces mots d'ordre sociaux au profit d'une pens\u00e9e libre qui g\u00e9n\u00e8re ses associations dans le souci de c\u00e9l\u00e9brer d'autres int\u00e9r\u00eats que ceux de la civilisation dominante. Les illusions sociologiques qui agissent comme les imp\u00e9ratifs cat\u00e9goriques sont \u00e0 l'origine des pens\u00e9es gr\u00e9gaires et communautaires, donc communes. Parmi celles-ci, l'association p\u00e9renne entre l'Amour et le Couple qui suppose, en \u00e9toile, des renvois \u00e0 Fid\u00e9lit\u00e9, Paternit\u00e9, Maternit\u00e9, Sexualit\u00e9, Conjugalit\u00e9. Le jeu associatif ne permet pas d'envisager l'un sans que l'autre apparaisse comme une n\u00e9cessit\u00e9 ontologique, m\u00e9taphysique. Dire l'un suppose nommer l'autre. Supposer ici, c'est exiger l\u00e0.\n\nQui \u00e9chappe \u00e0 la tyrannie de ces associations ? Ceux que leur marginalit\u00e9 revendiqu\u00e9e ou v\u00e9cue comme telle installe en fer de lance, \u00e0 l'avanc\u00e9e du savoir des effets de cette rh\u00e9torique associative \u00e0 vis\u00e9es politiques. Par exemple ? Les homosexuels, les transsexuels, les libertins, les couples dont l'un des deux est nettement plus jeune. Tous rencontrent les difficult\u00e9s et r\u00e9ticences impos\u00e9es par le social en vertu des religions profess\u00e9es via les associations d'id\u00e9es : un Homme ? Donc une Femme. Un Couple ? Donc Paternit\u00e9 et Maternit\u00e9. Un Enfant? Donc une Famille, un Foyer. En dehors de ces logiques sociales pourvoyeuses, lorsqu'on y ob\u00e9it, de compensations en reconnaissances r\u00e9elles et symboliques, il n'est que nomadisme, libert\u00e9 et caprice, ce que le social ne saurait accepter. Le mod\u00e8le offert par l'homosexuel illustre en acte la dissociation d'id\u00e9es : l'homme n'appelle pas la femme, ni le couple la procr\u00e9ation, pas plus l'absence d'enfants l'inexistence du couple. Chaque registre est v\u00e9cu s\u00e9par\u00e9ment, de fa\u00e7on autonome, r\u00e9gi par le seul d\u00e9sir, la seule volont\u00e9 de jouissance, sans consid\u00e9ration aucune des effets sociaux.\n\nEt les femmes ? Quand donc dira-t-on, seulement dire, que leur \u00e9panouissement, leur r\u00e9alisation, ne passe pas obligatoirement par les chemins trac\u00e9s pour elles par les soci\u00e9t\u00e9s exclusivement nourries de Famille, cellule de base \u00e9l\u00e9mentaire \u00e0 tout l'\u00e9difice soci\u00e9tal ? Quand donc ach\u00e8vera-t-on cette r\u00e9volution commenc\u00e9e par la dissociation d'id\u00e9es qui, dans le registre mental, a permis la distinction de la Sexualit\u00e9, de l'Amour et de la Procr\u00e9ation, avant que, dans les faits, les aides m\u00e9dicales, de la contraception \u00e0 la conception assist\u00e9e, permettent l'incarnation de cette dissociation id\u00e9ologique? De sorte qu'il ne vient plus \u00e0 l'id\u00e9e de personne, aujourd'hui, de contester l'autonomie et l'ind\u00e9pendance de chacun de ces registres, si ce n'est le militant int\u00e9griste de la droite la plus r\u00e9actionnaire et conservatrice.\n\nLa sexualit\u00e9 libre peut d\u00e9sormais se pratiquer sans qu'on envisage de subir les enfants ou de les craindre comme une punition, un ch\u00e2timent au plaisir; pas plus elle n'exige, dans son exercice, la corr\u00e9lation d'un sentiment exclusivement amoureux; l'amour libre : on peut aimer sans \u00e9pouser, sans passer \u00e0 l'acte sexuel, sans cohabiter, sans envisager la conjugalit\u00e9 ; la procr\u00e9ation libre : on peut, revers oblig\u00e9 \u00e0 la possibilit\u00e9 d'une sexualit\u00e9 sans maternit\u00e9, envisager une maternit\u00e9 sans sexualit\u00e9, et m\u00eame une procr\u00e9ation sans amour du g\u00e9niteur. Et toutes ces libert\u00e9s sont des conqu\u00eates qui proc\u00e8dent d'une entreprise longue et lente de dissociation d'id\u00e9es. Aucune n'oblige, mais toutes ouvrent plus large encore l'\u00e9ventail des possibles.\n\nQuels seraient les combats d'aujourd'hui pour parachever cette r\u00e9volution singuli\u00e8re ? La dissociation radicale de toutes ces possibilit\u00e9s sexuelles et sensuelles d'avec tout social possible, donc de toute emprise familialiste. Que disparaisse le noyau dur de la famille nucl\u00e9aire qui inspire encore actuellement, pour leur plus grande part, les modalit\u00e9s de la relation sexu\u00e9e. Certes le mariage a recul\u00e9, mais la cohabitation sur le m\u00eame mod\u00e8le est de plus en plus pris\u00e9e et de plus en plus t\u00f4t. La s\u00e9dentarit\u00e9 sensuelle, la fixit\u00e9 amoureuse, la monogamie conjugale, sont propos\u00e9es comme des vertus, des mod\u00e8les, le nomadisme, l'errance, comme des vices, des perversions. Globalement, l'intersubjectivit\u00e9 sexuelle se vit encore et toujours sur le mode monogame, fid\u00e8le, h\u00e9t\u00e9rosexuel et durable. La soci\u00e9t\u00e9 qui trouve son compte \u00e0 sauver ce qui peut encore l'\u00eatre de la famille a int\u00e9r\u00eat \u00e0 faire perdurer ces sch\u00e9mas et les associations d'id\u00e9es qui le permettent. De sorte que la libido, v\u00e9ritable \u00e9nergie libertaire, subit une fois de plus l'encagement dans des formes qui, peu ou prou, rappellent les vieilles modalit\u00e9s de la famille n\u00e9olithique pour lequel les pratiques s\u00e9dentaires remplacent les chasses, les cueillettes et les partenaires nomades.\n\nToutes les formes qui visent \u00e0 r\u00e9duire le nomadisme libidinal \u00e0 la s\u00e9dentarit\u00e9 proviennent pour la plupart du principe tribal dont Engels a \u00e9tabli la g\u00e9n\u00e9alogie consubstantielle \u00e0 celle de l'Etat et de la propri\u00e9t\u00e9 priv\u00e9e. L'errance supporte la d\u00e9pr\u00e9ciation, elle g\u00eane ceux qui aspirent au fixisme social et aux \u00e9dens sociologiques install\u00e9s sur la caste familiale et enracin\u00e9s dans la r\u00e9partition formelle des r\u00f4les sexuels. Elle est dangereuse pour la p\u00e9rennit\u00e9 sociale. Hegel ne s'y est pas tromp\u00e9 et pense la famille comme premier maillon d'une cha\u00eene qui, via toutes les stations de la soci\u00e9t\u00e9 civile \u2013 contrat, mariage, propri\u00e9t\u00e9, \u00e9ducation des enfants, travail \u2013 culmine dans l'Etat.\n\nBien s\u00fbr, tous ceux qui voudront la dissociation d'id\u00e9es, l'autonomie des registres tenus habituellement group\u00e9s sous la houlette gr\u00e9gaire, se verront \u00e9vit\u00e9s, malmen\u00e9s, fustig\u00e9s, exclus apr\u00e8s avoir fait l'objet de tentatives de r\u00e9cup\u00e9ration, d'int\u00e9gration. Vouloir l'autonomie radicale et libertaire de sa libido, conduit \u00e0 rencontrer les puissances castratrices sous toutes ses formes : les instances morales sociales par la voie la plus directe, l'opinion publique, le jugement de proximit\u00e9, voire celui de celles ou ceux qui, form\u00e9s au psittacisme sociologique, iront r\u00e9p\u00e9tant ce qu'on n'aura cess\u00e9 de leur enseigner ici, l\u00e0, ailleurs.\n\nL'h\u00e9donisme ne se pratique que dans une logique duelle et contractuelle. A d\u00e9faut, il quitte le terrain jubilatoire pour glisser dans les culs-de-basse-fosse des modalit\u00e9s f\u00e9odales. Faire jouir n'est pas pensable en dehors d'un consentement du tiers concern\u00e9 par l'initiative. D'o\u00f9 le nombre important d'\u00e9chouages quand tel ou telle use de sa libido non pas souverainement, librement, individuellement, mais socialement, culturellement et id\u00e9ologiquement. Ici, une volont\u00e9 solaire et libertaire, l\u00e0 un d\u00e9sir gr\u00e9gaire et mortif\u00e8re : dans la rencontre se fomentent les occasions de malentendus. L'un aspire \u00e0 un amour libre et \u00e0 la conservation par-devers soi d'une possibilit\u00e9 d'errance, l'autre veut la s\u00e9dentarit\u00e9 et la cristallisation morphologiquement conjugale.\n\nLa dissociation d'id\u00e9es doit \u00e9galement viser la d\u00e9construction de l'\u00e9toile form\u00e9e autour de la notion de f\u00e9minit\u00e9 et qui dispose en branches les notions de s\u0153ur, amie, camarade, amante, m\u00e8re, fille. Sur le terrain des m\u0153urs, les \u00e9volutions sociologiques r\u00e9centes entretiennent la confusion de ces modes et autorisent \u00e0 croire la possibilit\u00e9, pour une femme, d'\u00eatre toutes ces figures, \u00e0 la fois, en m\u00eame temps, avec le m\u00eame homme qui, \u00e0 son tour, pourrait jouer les r\u00f4les de fr\u00e8re, ami, p\u00e8re, amant, ou fils avec elle, le tout sans dommages pour la relation sexu\u00e9e. Les nouveaux p\u00e8res c\u00e9l\u00e8brent cette synth\u00e8se o\u00f9 tout dispara\u00eet et rien n'appara\u00eet qui pourrait signifier une nouvelle masculinit\u00e9. Car cette entreprise de fusion du divers dans l'un se fait au seul profit de l'\u00e9mergence d'un nouvel animal : l'androgyne.\n\nCet \u00eatre total, \u00e9mergeant d'une boue antique et pointant les oreilles jusque chez Freud, se pr\u00e9vaut de la bisexualit\u00e9 qui, yin et yang psychanalytique, \u00e9nonce la f\u00e9minit\u00e9 dans l'homme et la masculinit\u00e9 dans la femme, non pas comme un ver dans le fruit, mais comme une chance de conduire l'un \u00e0 l'autre en rendant moins hommes les hommes, moins femmes les femmes, plus autre l'un, et vice versa, de sorte qu'on aboutirait \u00e0 la naissance d'un tiers qui r\u00e9jouit les amateurs contemporains de formes asexu\u00e9es, hygi\u00e9niques, simples et sommaires. Enfin, on aurait tu\u00e9 le m\u00e2le, \u00e9touff\u00e9 la femelle et mis bas un monstre heureux de n'\u00eatre plus ni l'un ni l'autre, mais jubilant de sa nature monstrueuse.\n\nL\u00e0 encore triomphe le social qui dispose d'une forme th\u00e9orique de plus en plus susceptible de s'incarner. Elle est le mixte enfin r\u00e9ussi de tout ce qui permet une d\u00e9virilisation du monde, un asexualisme g\u00e9n\u00e9ralis\u00e9 : l'homme androgyne, ami et p\u00e8re, fils et copain de celle qui est pour lui sa m\u00e8re et sa fille, sa copine et sa s\u0153ur. Embrassons-nous Folleville ! Plus besoin d'aller chercher ailleurs ce dont on dispose chez soi : tout est dans le m\u00eame ou la m\u00eame personne. Et tout s'y trouve si bien qu'on ne peut rep\u00e9rer rien qui ressemble, de pr\u00e8s ou de loin, \u00e0 un homme ou \u00e0 une femme.\n\nVoulez-vous encore de ces vieilles lunes ? Parlez-vous toujours de ces pr\u00e9historiques anc\u00eatres, disparus dans la nuit des temps sociologiques ? On vous taxera de misogynie, l\u00e0 encore. D'homme refusant de se d\u00e9mettre de ses pr\u00e9rogatives de m\u00e2le, r\u00e9sistant et ne se faisant pas \u00e0 l'abandon de ses antiques privil\u00e8ges machistes. Soit, allons-y, donnons dans la moralisation et la psychologisation du d\u00e9bat. Je ne veux pas, en effet, d'une civilisation dans laquelle on int\u00e9grerait et ferait dig\u00e9rer par la machine sociale toutes les diversit\u00e9s sexuelles, de couleurs, d'id\u00e9es ou de religion, d'esprit ou de pens\u00e9e.\n\nSacrifi\u00e9s au creuset conformiste et niveleur, les hommes et les femmes jet\u00e9s dans le brasier des confusions, on n'aurait plus qu'une figure, seule et unique, caract\u00e9ris\u00e9e par le refus de l'identit\u00e9, l'incapacit\u00e9 \u00e0 assumer son \u00eatre et la soumission de son d\u00e9sir propre au d\u00e9sir de l'autre, miroir et reflet fid\u00e8le du d\u00e9sir social relay\u00e9 par les instances de reproduction id\u00e9ologique. L'\u00e9cole a distribu\u00e9 et r\u00e9parti les r\u00f4les sexuels et sociaux, la famille a repris la litanie en ch\u0153ur, le monde m\u00e9diatique a grossi l'ensemble en fournissant les ic\u00f4nes et autres images pieuses c\u00e9l\u00e9brant la nouvelle divinit\u00e9 androgyne. Quiconque r\u00e9siste m\u00e9rite l'\u00e9pith\u00e8te et le jugement infamants.\n\nNous sommes loin des jubilations propos\u00e9es par Fourier dans son Nouvel Ordre amoureux et qui souhaitait moins aligner l'individu sur le social que l'inverse. Les passions, chez lui, fournissent moins un carburant \u00e0 br\u00fbler pour offrir des parfums agr\u00e9ables \u00e0 la divinit\u00e9 sociale que des \u00e9nergies \u00e0 utiliser en synergie avec le collectif. Son \u0153uvre, introuvable et abandonn\u00e9e par tous, \u00e9diteurs et enseignants, \u00e9crivains et philosophes, propose une dissociation d'id\u00e9es radicale qui lib\u00e8re l'amour des pr\u00e9jug\u00e9s habituellement associ\u00e9s : fid\u00e9lit\u00e9, exclusivit\u00e9, monogamie, procr\u00e9ation, couple, dur\u00e9e, fixit\u00e9, hygi\u00e9nisme, et autres calibrages sensuels, au profit du seul imp\u00e9ratif h\u00e9doniste pensable : le ludisme. La nouvelle association d'id\u00e9es propos\u00e9e par le p\u00e8re de l'attraction passionn\u00e9e est la suivante : amour et ludisme. Autant dire qu'il propose une authentique r\u00e9volution en attaquant la famille classique s\u00e9dentaire au profit des affinit\u00e9s \u00e9lectives nomades.\n\nL'androgynat suppose moins chez lui une solution collective et g\u00e9n\u00e9ralisable qu'une hypoth\u00e8se parmi une infinit\u00e9. Fourier propose la c\u00e9l\u00e9bration de toutes les manies lubriques dans une \u00e9conomie g\u00e9n\u00e9ralis\u00e9e des jouissances. Et toutes, elles proc\u00e9deraient d'une \u00e9lection libre, d'un caprice individuel et singulier. Au lieu de cette errance libidinale, de ce nomadisme sensuel, on c\u00e9l\u00e8bre aujourd'hui le triomphe de la disparition des diff\u00e9rences au profit de l'unification. Le f\u00e9minisme a plus \u00e0 perdre de la fin de l'identit\u00e9 f\u00e9minine que de la confusion des genres o\u00f9 ne s'associent que des faiblesses soci\u00e9taires, celles d'hommes qui voudraient \u00eatre femmes, celles de femmes qui d\u00e9sirent \u00eatre des hommes, chacun aspirant \u00e0 n'\u00eatre rien d'autre que les propositions d'une \u00e9poque.\n\nLa femme virilis\u00e9e et l'homme f\u00e9minis\u00e9 ravissent et s\u00e9duisent avec bonheur Elisabeth Badinter qui voit dans la fin des sexes la disparition de la cause de la guerre entre eux. L'id\u00e9e est courte, et je ne suis pas bien s\u00fbr que la meilleure fa\u00e7on de soigner une maladie consiste \u00e0 souhaiter la mort du malade. C'est radical, s\u00fbr, mais n'y a-t-il pas d'autre solution en dehors de cette apocalypse du divers ? Faut-il un athanor aussi destructeur pour une pierre philosophale aussi indigente ? Vaut-il mieux une guerre des sexes avec la diversit\u00e9 et les richesses que suppose chaque partie prenante de la relation agonique plut\u00f4t que la paix avec l'uniformit\u00e9 induite, le combat cessant faute de combattants ?\n\nSur le champ de bataille o\u00f9 se prom\u00e8ne Elisabeth Badinter avec le sourire b\u00e9at des pacifistes amoureux de la paix \u00e0 tout prix, il n'y a plus que des cadavres. Les survivants se ressemblent, androgynes, au sexe indistinct, m\u00e9lange de fantasme p\u00e9dophile pour une humanit\u00e9 avant la d\u00e9termination adulte et de d\u00e9sirs r\u00e9gressifs subsistant miraculeusement apr\u00e8s la castration. Faut-il, pour lutter contre l'in\u00e9galit\u00e9 des sexes avec efficacit\u00e9 \u00e9radiquer purement et simplement la diff\u00e9rence des sexes? Elisabeth Badinter r\u00e9pond oui et montre un go\u00fbt indigne d'elle pour la confusion entre ces deux registres qui ne se recoupent absolument pas : une diff\u00e9rence ne devient une in\u00e9galit\u00e9 que dans un usage sociologique, puis id\u00e9ologique, enfin politique. Intrins\u00e8quement, l'alt\u00e9rit\u00e9 ne peut \u00eatre la cause de la r\u00e9partition hi\u00e9rarchique c\u00e9l\u00e9br\u00e9e par telle ou telle soci\u00e9t\u00e9. S'attaquer \u00e0 l'in\u00e9galit\u00e9 demeure une t\u00e2che f\u00e9ministe, se tromper d'ennemi en concentrant ses foudres sur la diff\u00e9rence est une erreur tactique et strat\u00e9gique, de celle avec lesquelles on offre la victoire \u00e0 ses ennemis.\n\nLa diff\u00e9rence m'importe, ici, l\u00e0 et partout. Tout ce qui vise la disparition du divers vise le monolithique dont r\u00eavent, in fine, les r\u00e9gimes autoritaires et fascistes. Les \u00e9trangers et les \u00e9tranget\u00e9s, quelles qu'elles soient, d'o\u00f9 qu'elles viennent, me plaisent comme elles sont, et je ne les veux pas int\u00e9gr\u00e9es, dig\u00e9r\u00e9es par le Moloch social et le L\u00e9viathan \u00e9tatique. La n\u00e9gritude me pla\u00eet noire dehors et dedans, et non pas blanchie int\u00e9rieurement \u00e0 force de conformismes culturels et d'invitations \u00e0 l'int\u00e9gration qui d\u00e9truit plus s\u00fbrement que n'importe quelle autre violence. Il en va pareillement de la diff\u00e9rence des sexes qu'on a int\u00e9r\u00eat \u00e0 entretenir, d'abord pour maintenir en \u00e9tat un r\u00e9el divers, multiple, foisonnant, riche, chatoyant, bariol\u00e9 quand tout vise \u00e0 l'uniformit\u00e9, ensuite pour mieux lutter contre l'in\u00e9galit\u00e9 l\u00e0 o\u00f9 elle persiste, sans que soit entretenue une confusion d\u00e9sastreuse entre les deux id\u00e9es \u00e0 dissocier.\n\nLe culte de la diff\u00e9rence est consubstantiel \u00e0 la r\u00e9sistance plus que jamais n\u00e9cessaire contre les tentatives g\u00e9n\u00e9ralis\u00e9es pour imposer l'homme unidimensionnel dont le portrait fut camp\u00e9 avec justesse par Marcuse. Un f\u00e9minisme h\u00e9doniste proposerait les modalit\u00e9s de cette r\u00e9sistance. La misogynie s'appuie sur la consid\u00e9ration de la diff\u00e9rence comme d'une aubaine pour accuser, fomenter et g\u00e9n\u00e9raliser des in\u00e9galit\u00e9s. Les deux sexes sont n\u00e9cessaires sans qu'on doive parler d'un deuxi\u00e8me sexe ou d'un second, voire d'un troisi\u00e8me comme occasion d'en finir avec les deux premiers.\n\nL'androgyne g\u00e9n\u00e9ralis\u00e9 est une chance pour les niveleurs, les amateurs de consensus plan\u00e9taire, une plaie pour les libertaires. Elisabeth Badinter dit en filigrane tout ce qui l'a t\u00e9tanis\u00e9e jusqu'alors puisqu'elle se r\u00e9jouit, avec la fin des sexes, d'assister \u00e0 la fin du d\u00e9sir, de la guerre et de la passion. En m\u00eame temps, elle applaudit aux m\u00e9diocrit\u00e9s de substitution qui apparaissent simultan\u00e9ment : la tendresse et la complicit\u00e9. Elle c\u00e9l\u00e8bre les liens qui unissent les membres d'une m\u00eame famille et souhaite l'extrapolation jusqu'au noyau dur du couple. Son mod\u00e8le de relation sexu\u00e9e ? celle de la m\u00e8re et de son enfant, du fr\u00e8re et de la s\u0153ur.\n\nQuel programme ! Exit l'amour fou d'Andr\u00e9 Breton, bienvenue \u00e0 la paternit\u00e9, h\u00e9ro\u00efsme des Temps Modernes selon Charles P\u00e9guy. Fallait-il tant de bruit pour en arriver l\u00e0 ? Tant de fureur et de feu pour pareil r\u00e9sultat minable? Se r\u00e9jouir de ce qui fait la joie des vieux couples, au bord de la tombe ! le programme est mortif\u00e8re \u00e0 souhait... La conjugalit\u00e9 a ses vertus, Charles Fourier y verrait l'occasion d'une manie lubrique parmi des milliers d'autres, pourquoi pas. Mais la penser en seul mod\u00e8le r\u00e9ussi d'intersubjectivit\u00e9 sexu\u00e9e et sexuelle !\n\nLes pr\u00eatres et ceux qui les soutiennent aujourd'hui, les philosophes compagnons de route du christianisme, peuvent se r\u00e9jouir du r\u00e9sultat. Des ann\u00e9es apr\u00e8s l'incarnation des dissociations d'id\u00e9es en actes pratiqu\u00e9es par Mai 68, on c\u00e9l\u00e8bre, via l'androgynat, les vertus du couple bourgeois tel qu'il appara\u00eet dans La Com\u00e9die humaine. Fin du d\u00e9sir incandescent et de l'amour fou, fin des brasiers amoureux et des plaisirs libertaires, fin des passions qui travaillent le corps et l'\u00e2me comme des br\u00fblures, des vertiges sensuels et des d\u00e9lires conjugu\u00e9s. Au revoir Nadja.\n\nNombreux sont ceux qu'effraient leurs d\u00e9sirs et leurs plaisirs et qui aiment les solutions radicales : supprimer la diff\u00e9rence sexuelle, f\u00e9miniser les hommes, masculiniser les femmes, culpabiliser la jubilation, c\u00e9l\u00e9brer les vertus qui ti\u00e9dissent et refroidissent puis rapetissent. C\u00e9der sur son d\u00e9sir, reculer devant l'ivresse, ne pas consentir aux jubilations libertaires, s'interdire de vivre, se priver, mourir avant l'heure. Dans cette logique hivernale, Charles Fourier, Andr\u00e9 Breton, Gilles Deleuze, Annie Le Brun fonctionnent comme des garde-fous, des invitations \u00e0 allumer de grands feux pour y jeter toutes les id\u00e9es associ\u00e9es qui servent la s\u00e9dentarit\u00e9 amoureuse et agissent de conserve avec le social tout-puissant. Tous, ils aident \u00e0 poser les lin\u00e9aments pour un f\u00e9minisme libertaire qui laisse aux amateurs de hochets leurs petites excitations sur l'art de diagnostiquer la misogynie l\u00e0 o\u00f9 elle ne s\u00e9vit pas et qui font son lit l\u00e0 o\u00f9 elle triomphe. Que viennent les incandescences avec lesquelles pourrait s'\u00e9tablir un nouvel ordre amoureux.\n\nEt en attendant? Les lendemains qui chantent sont si souvent longs \u00e0 advenir qu'une morale provisoire peut sembler n\u00e9cessaire pour prix de toute patience. Pour ma part, je n'ai jamais consenti \u00e0 la morale par provision sur le mode cart\u00e9sien. Trop hypocrite, elle permet d'\u00eatre d'hier ou de demain selon la prudence et les n\u00e9cessit\u00e9s sociales du moment. Plus scolastique, mais pas encore moderne, tout en \u00e9tant l'un et l'autre, selon l'int\u00e9r\u00eat, voil\u00e0 qui m'a toujours sembl\u00e9 un manque de probit\u00e9. Alors ?\n\nAlors il faut se rendre \u00e0 l'\u00e9vidence qu'on peut \u00eatre seul, et longtemps, \u00e0 pratiquer une morale, un langage, une rh\u00e9torique, un monde, qu'on est seul \u00e0 hanter, voire habiter. Le d\u00e9ni aidant, le malentendu ajoutant, l'incompr\u00e9hension peut ravager toute intersubjectivit\u00e9 un temps beaucoup trop long pour qu'il soit loisible d'envisager une couture plus fine, plus pr\u00e9cise et pr\u00e9cieuse dans la relation. Finalement, il me semble n'avoir entendu l'\u00e9pith\u00e8te de misogyne que pour mettre fin \u00e0 un dialogue impossible, pour terminer une discussion qui n'en finit pas de chercher ses marques sans les trouver, pour \u00e9viter une conversation sur l'usage h\u00e9doniste de soi. Impossibilit\u00e9 d'un langage commun, les mots ne signifient plus la m\u00eame chose pour l'un et pour l'autre. Dans l'\u00e9tranget\u00e9 creus\u00e9e comme un ab\u00eeme, tout donne l'impression d'ajouter du n\u00e9ant au n\u00e9ant.\n\nDans l'inflexion de la voix et le silence, derri\u00e8re les mots, juste l\u00e0 o\u00f9 ils sont fragiles et branlants, dans une attention, une intention, dans le d\u00e9faut de pr\u00e9venance, aux c\u00f4t\u00e9s du signe infinit\u00e9simal qui fait sens, je cherche une information permettant de trouver une volont\u00e9 manifeste, s\u00fbre d'elle et pr\u00e9cise, inform\u00e9e et d\u00e9cid\u00e9e. Je qu\u00eate un d\u00e9sir clair et distinct, sans travestissement, sans duplicit\u00e9, puis t\u00e2che de formuler le mien avec la m\u00eame clart\u00e9 et la m\u00eame distinction, avant d'attendre les complicit\u00e9s ou les d\u00e9fiances, les confiances ou les distances. J'attends souvent en vain.\n\nL'h\u00e9donisme que je propose, en mati\u00e8re d'intersubjectivit\u00e9 sexuelle et sensuelle, suppose chez l'autre le m\u00eame acquis que chez moi : une dissociation d'id\u00e9es radicale des notions d'amour et de cohabitation, de paternit\u00e9 et de fid\u00e9lit\u00e9, de monogamie et de dur\u00e9e. Chaque information par moi donn\u00e9e vaut proposition faite \u00e0 autrui pour une cartographie \u00e0 son usage. Un balisage du terrain, la d\u00e9signation des fleuves et des pr\u00e9cipices, des ravins et des voies navigables, des zones interdites et des secteurs possibles, ce que les g\u00e9ographes appellent des taches blanches, puis un genre de Carte du Tendre. Enfin, de quoi d\u00e9terminer des points cardinaux et une lecture dans la Voie lact\u00e9e si d'aventure les rep\u00e8res s'effacent ou se brouillent sur terre.\n\nEst-ce suffisant? Pour m'endormir en paix avec moi-m\u00eame, certainement. Pour \u00e9viter les \u00e9cueils, les emb\u00fbches et les malentendus, s\u00fbrement pas. Souvent, si souvent, j'ai l'impression d'une errance en solitaire, parlant seulement pour moi-m\u00eame un genre de volap\u00fck dont je d\u00e9sesp\u00e8re qu'il devienne un jour, et un tant soit peu, une langue commune. Trop de difficult\u00e9s \u00e0 partager, \u00e0 faire comprendre ou \u00e0 vivre sans qu'autrui n'ait envie de ses propres divagations et de ses points de rep\u00e8re particuliers qui, parfois, vont contre la complicit\u00e9, l\u00e0 m\u00eame o\u00f9 l'on envisageait et croyait \u0153uvrer pour le bien commun. Bien s\u00fbr, il est facile, dans ce cas de figure, quand le malentendu culmine, quand le d\u00e9ni a puissamment construit des digues qui rompent un jour, de couper court et de renvoyer l'autre \u00e0 sa misogynie. C'est facile, d'acc\u00e8s imm\u00e9diat et blessant autant que peu efficace pour comprendre, se comprendre un peu. L'invective vaut comme un \u00e9chec, une fin de non-recevoir, un succ\u00e8s pour les t\u00e9n\u00e8bres.\n\nLe monde n'est pas fait pour ceux qui veulent vivre le plus possible selon leurs propres r\u00e8gles, leurs lois, si tout cela ne s'embourbe pas dans le sillon d'une morale sociale conventionnelle. On peut lutter longtemps, seul, contre vents et mar\u00e9es, arborant une r\u00e8gle du jeu qu'on propose et dont autrui dispose selon ses propres habitudes. L'\u00e9ternelle antinomie entre l'individu et la soci\u00e9t\u00e9 s'illustre et triomphe l\u00e0 plus que nulle part ailleurs.\n\nLe registre amoureux est celui du malentendu le plus cruel lorsque les aspirations de l'un sont la libert\u00e9 solaire et radieuse quand l'autre vise les voies royales qui m\u00e8nent aux r\u00e9compenses sociales et symboliques. L'h\u00e9donisme agit en force antisociale, nourrie de la r\u00e9sistance aux \u00e9difices communautaires cristallis\u00e9s. Souvent, nous appelons misogyne celui dont le discours sur les femmes ne co\u00efncide pas avec celui auquel on sacrifie quand il est s\u00e9curisant, correct pour l'id\u00e9ologie dominante, reposant pour l'esprit.\n\nPar paresse, facilit\u00e9 ou gr\u00e9garisme, on adh\u00e8re bien souvent \u00e0 un corpus d'id\u00e9es prises pour des v\u00e9rit\u00e9s et qui se r\u00e9duisent aux lieux communs d'une \u00e9poque transform\u00e9s en mensonges sociaux auxquels on voue un culte pour l'\u00e9dification d'une religion soci\u00e9taire. Les individus pieux s'y croient libres alors qu'ils se contentent d'ob\u00e9ir aux mots d'ordre id\u00e9ologiques. En attendant, les existences singuli\u00e8res se sacrifient au profit des b\u00e9n\u00e9fices sociaux collectifs. Et L\u00e9viathan continue de prosp\u00e9rer.\n\nCrucifi\u00e9s, \u00e9cartel\u00e9s, ceux qui voudraient une logique moins mortif\u00e8re pour les subjectivit\u00e9s n'ont plus qu'\u00e0 faire leur deuil de toute intersubjectivit\u00e9 r\u00e9ellement radieuse sur le terrain des sexes et des \u00e2mes qui voudraient \u00eatre s\u0153urs. Reste l'intraitable m\u00e9lancolie pour les soirs o\u00f9 l'on se demande comment on pourrait mettre encore la main un peu \u00e0 son langage nouveau, pour le parfaire, le rendre plus compr\u00e9hensible, praticable. Et pourvoyeur d'effets.\n15\n\nC\u00c9L\u00c9BRATION DU GAZ LACRYMOG\u00c8NE\n\nL'\u00e9poque est b\u00e9nie pour les philosophes compagnons de route du christianisme qui fol\u00e2trent entre L\u00e9gion d'honneur et grandes vertus, conseils de proximit\u00e9 aux ministres de droite et c\u00e9l\u00e9bration d'un catholicisme repeint aux couleurs fadasses de la la\u00efcit\u00e9 kantienne. Elle ch\u00e9rit tout autant les thurif\u00e9raires du nihilisme int\u00e9gral, les donneurs de le\u00e7ons, pleureuses patent\u00e9es du si\u00e8cle, jeteurs de poudre aux yeux et autres vendeurs d'arri\u00e8re-mondes qui font commerce de la na\u00efvet\u00e9 et de la sottise aujourd'hui g\u00e9n\u00e9ralis\u00e9es. Entre un Acad\u00e9micien dont la prose fait p\u00e2lir et para\u00eetre sobre celle des d\u00e9cadents du si\u00e8cle dernier et de pr\u00e9tendus intellectuels libres arrimant leurs avis dans le seul air du temps, il reste peu de place pour une pens\u00e9e singuli\u00e8re et affranchie des mirages contemporains.\n\nPour ceux qui r\u00e9sistent encore au recyclage des vertus du Vatican et ne se contentent pas d'un caf\u00e9 avec Socrate, peu de possibilit\u00e9s subsistent. Peu, mais assez pour qu'on puisse s'arr\u00eater quelque temps sur Raoul Vaneigem que n'ont pas encore r\u00e9cup\u00e9r\u00e9 ceux qui d\u00e9vitalisent le dernier Debord dont la complaisance nihiliste tombe fort \u00e0 propos pour leur entreprise de d\u00e9pe\u00e7age. Depuis son Trait\u00e9 de savoir-vivre \u00e0 l'usage des jeunes g\u00e9n\u00e9rations (1967), il approfondit et pr\u00e9cise une pens\u00e9e qui reste droite, franche et nette quand tout autour de lui c\u00e9l\u00e8bre le torse, le torve, sinon le virage \u00e0 angle droit.\n\nVaneigem ne se contente pas de raconter le r\u00e9el comme il va pour en souligner la complexit\u00e9, voire pour faire des livres d\u00e9veloppant \u00e0 sati\u00e9t\u00e9 cette id\u00e9e que, trop complexe, il n'est rien de dicible \u00e0 son propos. Il fustige un monde, celui du capitalisme plan\u00e9taire, pour en proposer un autre, au grand dam de ceux qui communient dans l'id\u00e9al d'un renoncement \u00e0 toute utopie au profit d'un r\u00e9alisme pr\u00e9sent\u00e9 comme de bon aloi et seul tenable. Et cet univers appel\u00e9 de ses v\u0153ux dans Nous qui d\u00e9sirons sans fin (1996) invite tout bonnement \u00e0 achever Mai 68 comme on parach\u00e8ve une entreprise qui n'a pas \u00e9t\u00e9 men\u00e9e \u00e0 bien compl\u00e8tement.\n\nA l'heure du recyclage massif des anciens soixante-huitards dans la banque, l'inspection g\u00e9n\u00e9rale, la presse douteuse, le conseil aux chefs d'Etat de droite et de gauche, le narcissisme m\u00e9morialiste ou l'\u00e9vocation tremblante sur le mode de l'ancien combattant, Raoul Vaneigem persiste et signe. Toute son \u0153uvre \u2013 aujourd'hui plus d'une dizaine de livres \u2013 est construite en forme de variation sur le th\u00e8me de la subversion. Elle propose un \u00e9difice consacr\u00e9 \u00e0 la c\u00e9l\u00e9bration du gaz lacrymog\u00e8ne. L'ensemble utilise des registres divers, du libelle \u00e0 l'avertissement en passant par la lettre, le manifeste ou le trait\u00e9. Pour autant, elle ne n\u00e9glige pas les moyens de l'\u00e9rudition qui l'apparentent, sur ce terrain \u00e0 l'arch\u00e9ologie du meilleur Michel Foucault. Et l'on peut lire le Mouvement du libre esprit (1986) comme inaugurant une nouvelle fa\u00e7on de pratiquer l'histoire telle qu'on la retrouve dans La R\u00e9sistance au christianisme (1993) ou Les H\u00e9r\u00e9sies (1994). Pr\u00e9cisons.\n\nL'historien peut mettre son savoir et sa recherche au service d'une pure et simple saisie du pass\u00e9, pour le seul plaisir de comprendre ce qui a eu lieu, les encha\u00eenements, le jeu des causes et des cons\u00e9quences, la rh\u00e9torique des faits, la logique seule. A cette enseigne, on trouve tous les historiens que le pass\u00e9 int\u00e9resse de fa\u00e7on intrins\u00e8que, comme on se d\u00e9couvre f\u00e9tichiste ou collectionneur, sp\u00e9cialiste de l'embaumement, des momies et de la mort. D\u00e9vou\u00e9s \u00e0 Thanatos, fossoyeurs et croque-morts du pass\u00e9 soumettent l'histoire au remplissage des cimeti\u00e8res. Combien doit-on, \u00e0 ceux-l\u00e0, d'\u00e9tudes sinistres, utiles seulement \u00e0 ceux qui se reproduisent sur ce m\u00eame sujet, dans la seule atmosph\u00e8re confin\u00e9e des biblioth\u00e8ques ?\n\nFoucault inaugure une pratique id\u00e9ologique et philosophique de l'histoire. Le savoir combatif vise la mise \u00e0 mal des imp\u00e9ratifs autoritaires au profit d'un id\u00e9al libertaire. Si la recherche sur la folie, l'enfermement, ses modalit\u00e9s, ses d\u00e9finitions, ses acceptions, si les analyses sur la prison, la maladie, la sexualit\u00e9, la psychiatrie, l'\u00e9cole, la discipline, si ces pratiques intellectuelles, donc, doivent tendre au seul \u00e9paississement des rayonnages de librairie, alors elles ne m\u00e9ritent pas une seconde de peine. En revanche, si l'histoire d'hier vise \u00e0 la transformation du pr\u00e9sent pour un avenir plus radieux, alors on peut demander \u00e0 l'historien de quitter la Sorbonne pour rejoindre, dans la rue, dehors, les lieux o\u00f9 leur culture peut servir de levier aux subversions n\u00e9cessaires.\n\nRaoul Vaneigem pratique ainsi l'histoire, en philosophe engag\u00e9, et non en chien de garde, en acteur politique d\u00e9couvert, et non en pr\u00e9tendu chercheur masquant son hypoth\u00e9tique objectivit\u00e9 derri\u00e8re une perp\u00e9tuelle ardeur \u00e0 c\u00e9l\u00e9brer le monde comme il va. Son Mouvement du libre esprit montre qu'\u00e0 c\u00f4t\u00e9 de l'histoire et de l'historiographie officielles, celles des universit\u00e9s et de l'\u00e9dition, des productions intellectuelles et livresques, il existe une histoire parall\u00e8le, celle des marges et des ombres, \u00e0 laquelle de belles \u00e2mes int\u00e9ress\u00e9es n'accordent que blancs et silences. Aux c\u00f4t\u00e9s de l'id\u00e9al asc\u00e9tique chemine, parall\u00e8le, mais ne se rencontrant jamais \u2013 par l'effet des cloisonnements dus aux officiels, et entretenus par eux \u2013, une tradition de l'id\u00e9al h\u00e9donique.\n\nL\u00e0 o\u00f9 les lieux communs proposent une chronologie allant de Platon \u00e0 Heidegger via saint Augustin, saint Thomas d'Aquin, Descartes et autres maillons n\u00e9cessaires dans la cha\u00eene de la vision du monde spiritualiste et monoth\u00e9iste, id\u00e9aliste et asc\u00e9tique, il faut proposer Aristippe de Cyr\u00e8ne et l'\u00e9cole cyr\u00e9na\u00efque, Diog\u00e8ne et les cyniques, les gnostiques licencieux, les libertins \u00e9rudits, Luillier et des Barreaux, La Mothe le Vayer et autres esprits libres. De m\u00eame pour la constellation, riche et inconnue, des Fr\u00e8res et Soeurs du Libre Esprit dont Raoul Vaneigem a \u00e9tabli la g\u00e9n\u00e9alogie et le portrait dans une recherche rest\u00e9e sans descendance.\n\nLa th\u00e8se de l'ouvrage est radicale et tellement forte de cons\u00e9quences : \u00abLe Moyen Age a \u00e9t\u00e9 chr\u00e9tien comme les Pays de l'Est sont communistes\u00bb (publi\u00e9 en 1986), de sorte qu'il faut en finir avec \u00ab la l\u00e9gende saint-sulpicienne d'un Moyen Age baignant dans la foi chr\u00e9tienne comme la sardine dans l'huile \u00bb. Qui donc, aujourd'hui, commerce autre chose que des images d'Epinal habituelles sur le sujet m\u00e9di\u00e9val? Qui travaille sur les textes de Willem Corneliz d'Anvers ou de Bentivenga Da Gubbio? De Marguerite Por\u00e8te ou de Heilwige Bloemardinne ? De s\u0153ur Katr\u00e9e ou de Walter de Hollande ? Qui lit ou fait lire, commente et diffuse, les \u0153uvres de ces moines et moniales h\u00e9donistes, de ces b\u00e9gards et b\u00e9guines, amauriciens et autres picards, adamites de Boh\u00eame, alumbrados et lo\u00efstes ? Six pages de bibliographie, plus de quatre cents noms brillants dans cette constellation inconnue, et toujours autant d'universitaires pour se croire singuliers en faisant travailler leurs \u00e9tudiants sur leurs propres th\u00e8ses et travaux en cours consacr\u00e9s aux sujets les plus \u00e9cul\u00e9s ?\n\nQuel int\u00e9r\u00eat ? O\u00f9 donc, diront les impatients, peut-on flairer dans ce travail les effluves du gaz lacrymog\u00e8ne ? Vaneigem montre que l'id\u00e9ologie dominante se nourrit toujours des orthodoxies promues par elles dans le dessein de c\u00e9l\u00e9brer les pulsions mortif\u00e8res et de circonscrire ou d\u00e9truire les vitalit\u00e9s foisonnantes. Toute civilisation se fonde sur le sacrifice de la pulsion de vie et la g\u00e9n\u00e9ralisation de la pulsion de mort. Depuis la conversion de Constantin, l'Occident se constitue sur ce principe : haine de la vie, \u00e9loge de la mort, m\u00e9pris des d\u00e9sirs, des plaisirs, des jouissances, des jubilations, et promotion de l'id\u00e9al asc\u00e9tique sous toutes ses coutures. En se faisant l'arch\u00e9ologue des formes historiques prises par l'h\u00e9donisme dans une \u00e9poque \u2013 entre le XIIIe et le XVIe si\u00e8cle \u2013, Raoul Vaneigem d\u00e9montre que le principe de r\u00e9alit\u00e9 et le principe de plaisir ne s'opposent pas fatalement, mais que l'un doit pouvoir informer l'autre. L'oubli de ces preuves d\u00e9signe symptomatiquement les int\u00e9r\u00eats id\u00e9ologiques du syst\u00e8me qui pratique l'amn\u00e9sie. A l'historien, philosophe subversif, d'en fournir \u00e0 nouveau la publicit\u00e9.\n\nCe travail s'\u00e9largit dans la somme qu'est La R\u00e9sistance au christianisme (1993), un ouvrage magistral qui analyse la nature, la forme, la dynamique des h\u00e9r\u00e9sies dans l'histoire depuis les origines jusqu'au XVIIIe si\u00e8cle. En s'attachant \u00e0 d\u00e9montrer la g\u00e9n\u00e9alogie et le fonctionnement des h\u00e9t\u00e9rodoxies, Raoul Vaneigem indique les modalit\u00e9s de formation d'une orthodoxie. A qui sait extrapoler, cette grille vaut pour toutes les entreprises de quadrillage et d'empire id\u00e9ologique : tous les totalitarismes, religieux avec le christianisme, politiques avec les fascismes bruns et rouges, ou verts d'aujourd'hui, \u00e9conomiques, avec le capitalisme plan\u00e9taire, ob\u00e9issent aux m\u00eames lois et m\u00e9canismes.\n\nD'o\u00f9 l'int\u00e9r\u00eat, chez ceux qui c\u00e9l\u00e8brent le conformisme dans l'\u00e9poque, de faire silence sur ces marges o\u00f9 s'expriment les nouvelles possibilit\u00e9s d'existence. Qui conna\u00eet, parmi les chiens de garde contemporains, universitaires et penseurs mondains confondus, les doctrines dosith\u00e9ennes et nazor\u00e9ennes, s\u00e9thiennes, naas\u00e8nes, \u00e9bionites, malchis\u00e9d\u00e9quiennes, elchasa\u00eftes, carpocratiennes, basilidiennes, marcionites, montanistes, valentiniennes, marcosiennes, bard\u00e9saniennes, navatiennes ? Lequel avoue une familiarit\u00e9 avec les serpents qui ondulent si bien, comme parfois le ventre des femmes, les sectateurs du phallus magnifique, des adorateurs des myst\u00e8res du sexe f\u00e9minin, b\u00e9ance sur l'absolu et la magie int\u00e9grale ? O\u00f9 se cachent les commensaux des mangeurs de sperme et des orgiaques, des fornicateurs mystiques et autres p\u00e9dagogues du salut par l'extase sexuelle ? Et dans cette galerie de pens\u00e9es alternatives, on d\u00e9couvre \u00e9galement des promoteurs de cosmologie po\u00e9tique, des dispensateurs de visions du monde enflamm\u00e9es, des mystagogues d'univers r\u00eav\u00e9s, des philosophes proposant d'autres fa\u00e7ons de d\u00e9finir la mati\u00e8re ou de penser le rapport \u00e0 la divinit\u00e9. Il y a l\u00e0 assez de dynamite entrepos\u00e9e pour miner l'\u00e9difice chr\u00e9tien dans son ensemble. Autant dire qu'on peut comprendre le soin que mettent les tenants de l'ordre \u00e0 garder leurs ouailles \u00e0 l'abri de ces pens\u00e9es funestes...\n\nOn remarque, ici comme \u00e0 chaque fois que la philosophie peut \u00eatre radicale et subversive, l'organisation du silence et l'entreprise de d\u00e9tournement \u00e9difi\u00e9es autour de ces travaux priv\u00e9s de tous les soutiens m\u00e9rit\u00e9s : ni dans l'\u00e9dition, ni dans la presse, ni dans le d\u00e9bat, ni ailleurs. Ne parlons ni de l'Universit\u00e9, ni des lieux o\u00f9 sont cens\u00e9es se faire les r\u00e9flexions alternatives. Raoul Vaneigem est laiss\u00e9 \u00e0 son \u00e9rudition, seul, parce qu'il propose un usage politique du savoir et ne se contente pas de faire des livres pour le plaisir d'allonger une bibliographie personnelle. Nizan a tout dit des usages du chien de garde, de ses coutumes et de ses habitudes : il ne mord rien ni personne, en dehors des errants inutiles \u00e0 sa pitance. Il lui suffit d'une gamelle r\u00e9guli\u00e8re et de pitoyables servitudes en rapport.\n\nLa culture au service de la subversion, voil\u00e0 une th\u00e9orie pendable dans une \u00e9poque o\u00f9 le savoir chemine presque toujours \u00e0 la solde du march\u00e9. Voil\u00e0 aussi qui est impardonnable, p\u00e9ch\u00e9 mortel dans le monde des lettres. Quand le savoir agit si souvent en pouvoir \u00e0 l'usage du monde dominant et au service des gens en place, il est roboratif de lire des penseurs dont le projet revendique absolument l'inverse : Vaneigem veut donner \u00e0 la culture le pouvoir prop\u00e9deutique \u00e0 la subversion que lui donnait Gramsci. Foucault et Deleuze agissaient et pensaient de m\u00eame. Et avant eux, Marx et Engels. Sortie des usages de reproduction sociale, elle transfigure la vie quotidienne. Si le savoir est seulement occasion de pouvoir dans la conservation du monde comme il va, il m\u00e9rite le b\u00fbcher. Rien d'autre.\n\nQuand il d\u00e9laisse l'\u00e9critoire historique, Raoul Vaneigem peut \u00eatre philosophe impr\u00e9cateur, et faire de l'\u00e9poque son objet de pr\u00e9dilection. Ainsi, Nous qui d\u00e9sirons sans fin (1996) offre une occasion pour ses lecteurs de voir r\u00e9appara\u00eetre les th\u00e8ses qui ont fait le bonheur d'autres g\u00e9n\u00e9rations et qu'on peut encore et toujours lire dans le Trait\u00e9 de savoir-vivre \u00e0 l'usage des jeunes g\u00e9n\u00e9rations (1967), le Livre des plaisirs (1979) ou l'Adresse aux vivants sur la mort qui les gouverne et l'opportunit\u00e9 de s'en d\u00e9faire (1990). Ces cris identiques diff\u00e9remment modul\u00e9s expriment \u00e0 chaque fois la m\u00eame haine du Vieux Monde et la m\u00eame urgence de s'en d\u00e9faire au profit d'un univers moins soucieux de Thanatos que d'Eros.\n\nLe temps passe et l'art des ren\u00e9gats s'affine avec toujours autant de talent, les fum\u00e9es de Mai 68 dispers\u00e9es, les lendemains \u00e9thyliques d'apr\u00e8s-mitterrandisme ayant succ\u00e9d\u00e9 aux triomphes de la soci\u00e9t\u00e9 de consommation, pr\u00e9c\u00e9d\u00e9 le retour d'une r\u00e9action qui a du mal \u00e0 se contenir, puis annonc\u00e9 un socialisme d\u00e9lav\u00e9 et insipide, Raoul Vaneigem maintient son cap. Dans le paysage id\u00e9ologique du moment, hygi\u00e9niste \u00e0 souhait, aseptis\u00e9 comme une morgue, il ose des concepts qui font sourire les cyniques et, l\u00e0 encore, fleurent bon le gaz lacrymog\u00e8ne, car il ose parler de marchandise et de march\u00e9, de valeur d'usage et de valeur d'\u00e9change, de capital et de baisse tendancielle des taux de profit, de prol\u00e9tariat et de paup\u00e9risation, d'argent et d'accumulation mon\u00e9taire. Des notions d'autant \u00e9vit\u00e9es aujourd'hui qu'on \u00e9vacue efficacement les situations et les \u00e9tats de fait qu'elles caract\u00e9risent et auxquels elles se rapportent. Combien de philosophes triomphent en autruche dans un monde abandonn\u00e9 aux loups ?\n\nAvec ce mat\u00e9riel conceptuel, Raoul Vaneigem analyse les perversions du moment : des produits toxiques industriels \u00e0 ceux de l'agro-alimentaire en passant par les violences urbaines ou scolaires, les ravages \u00e9cologiques et les d\u00e9lires de la manipulation g\u00e9n\u00e9tique, sinon les folies g\u00e9nocidaires et autres hyst\u00e9ries dues au complexe militaro-industriel, tout ce qui montre l'emballement d'un monde derri\u00e8re la mort et la religion du capitalisme plan\u00e9taire est point\u00e9, lu et d\u00e9cod\u00e9 dans une logique critique \u2013 tel qu'on entendait ce terme dans l'Ecole de Francfort. Tout ce qui permet de sacrifier aux pulsions de mort m\u00e9rite les fl\u00e8ches du philosophe.\n\nDans sa critique de l'h\u00e9donisme et de la volont\u00e9 de puissance, je regrette pour ma part qu'il utilise ces deux notions comme les conservateurs en usent \u2013 dont Jean-Paul II... L'h\u00e9donisme peut \u00eatre un auxiliaire du consum\u00e9risme capitaliste, certes, tout autant que la volont\u00e9 de puissance. Dans l'un et l'autre cas, il s'agit de c\u00e9l\u00e9brer la marchandise et les pulsions destructrices, v\u00e9ritables f\u00e9tiches du lib\u00e9ralisme. Mais dans une logique non capitaliste, ces deux concepts peuvent signifier autre chose et m\u00e9riter une d\u00e9finition r\u00e9nov\u00e9e, sinon r\u00e9volutionn\u00e9e. Leur usage sans pr\u00e9caution et sans distinction risque de valoir \u00e0 Raoul Vaneigem des amis qu'il ne m\u00e9rite pas.\n\nD'autant que son projet de soci\u00e9t\u00e9 rel\u00e8ve depuis le d\u00e9but de l'h\u00e9donisme, sans conteste, et proc\u00e8de radicalement d'une volont\u00e9 de puissance co\u00efncidant avec une volont\u00e9 de jouissance. L'ensemble se trouve plac\u00e9 sous le signe de la po\u00e9sie v\u00e9cue, de la production de soi-m\u00eame comme seule \u0153uvre qui vaille, de l'invitation \u00e0 accomplir ses d\u00e9sirs, \u00e0 leur donner toute l'expansion possible et imaginable, afin de r\u00e9concilier l'homme avec lui-m\u00eame, ses semblables et la plan\u00e8te. Le tout servant les desseins de la jubilation d\u00e9cupl\u00e9e.\n\nEn contrepoint \u00e0 cette lecture du r\u00e9el inform\u00e9 par un n\u00e9gatif envahissant, Raoul Vaneigem propose une soci\u00e9t\u00e9 construite sur la c\u00e9l\u00e9bration des pulsions de vie. Le principe f\u00e9d\u00e9rateur de son utopie ? il faut mettre l'\u00e9conomie au service de l'homme et, de la sorte, op\u00e9rer une r\u00e9volution copernicienne qui permettrait une c\u00e9l\u00e9bration des d\u00e9sirs l\u00e0 o\u00f9 toute civilisation moderne vit du renoncement, de la castration et de l'automutilation. D'o\u00f9 des pages radieuses sur le travail et le ch\u00f4mage, le loisir et la cr\u00e9ation de soi. Voire l'\u00e9ducation.\n\nCe nouveau contrat social suppose la fin du r\u00e8gne de la quantit\u00e9 et l'av\u00e8nement de l'\u00e8re de la qualit\u00e9, la disparition de la p\u00e9nurie et l'\u00e9mergence de l'abondance, l'abolition de la valeur d'\u00e9change et la c\u00e9l\u00e9bration de la valeur d'usage, la condamnation du tertiaire bureaucratique et la promotion du primaire producteur de richesses, la coop\u00e9rative rempla\u00e7ant le mode de production gigantesque, les \u00e9nergies douces se substituant aux \u00e9nergies polluantes, la pr\u00e9servation de la plan\u00e8te primant son exploitation, la technique servant la lib\u00e9ration des hommes plut\u00f4t qu'asservissant leurs corps et leurs vies. On reconna\u00eet l\u00e0, bien \u00e9videmment, le socle th\u00e9orique et prom\u00e9th\u00e9en d'un Mai 68 \u00e0 parachever.\n\nSur le terrain du volontarisme politique, et sur la question des moyens, Raoul Vaneigem a \u00e9crit il y a peu un superbe petit livre qui surclasse nombre de trait\u00e9s de p\u00e9dagogie et s'intitule Avertissement aux \u00e9coliers et lyc\u00e9ens (1995). Dans la tradition des pens\u00e9es libertaires de S\u00e9bastien Faure \u00e0 A.S. Neill, Raoul Vaneigem formule une protestation qui me fait penser au film de Jean Vigo que je place plus haut que tout : Z\u00e9ro de conduite. Que disent ces pages? Qu'il faut conjurer la mal\u00e9diction des \u00e9coles o\u00f9 l'on apprend l'ob\u00e9issance, le dressage, la discipline, l'ordre, la soumission, la docilit\u00e9 et autres perversions relevant de la pulsion de mort au profit de tout ce qui permet de c\u00e9l\u00e9brer la pulsion de vie : le d\u00e9sir de savoir, le plaisir d'apprendre, la jubilation de comprendre, la puissance de cr\u00e9er, construire, composer. Je n'ai cess\u00e9 de songer, le lisant, au Faux Principe de notre \u00e9ducation de Max Stirner, un ouvrage de la m\u00eame puissance, de la m\u00eame \u00e9nergie rebelle et vivante.\n\nQue proposent ces soixante pages ? Qu'il vaut mieux, ici comme ailleurs, c\u00e9l\u00e9brer la vie plut\u00f4t que la mort. L'id\u00e9e para\u00eet simple, bien que subversive dans un monde soucieux de produire quotidiennement l'inverse. On risque de mal mesurer combien cette proposition renverse les valeurs. Sur le terrain de l'\u00e9cole, donc, puisque c'est l'objet de ce manifeste, l'id\u00e9e induit des r\u00e9flexions radicales et capitales. Par exemple, on comprend pour quelles raisons les architectures scolaires font si souvent songer \u00e0 celle des \u00e9difices carc\u00e9raux : on y pratique selon les m\u00eames m\u00e9thodes et en vertu des m\u00eames principes, il para\u00eet normal que ces deux institutions r\u00e9pressives aient un m\u00eame \u00e9crin.\n\nQuelles m\u00e9thodes, en l'occurrence? La castration, l'enseignement de l'ennui, la culture de la violence, la mise \u00e0 mort du vivant en chacun au profit du cadavre instaur\u00e9 en mod\u00e8le. Selon le principe fouri\u00e9riste de l'attraction passionnelle, Vaneigem constate que l'enfant pose des questions auxquelles on ne donne pas de r\u00e9ponse alors qu'adolescent, \u00e0 l'inverse, il doit ing\u00e9rer des r\u00e9ponses aux questions qu'il ne se pose pas, ou plus. Entre les deux \u00e9tats, la soif de savoir non \u00e9tanch\u00e9e, et l'abreuvage de qui n'a plus soif, il y a le travail et l'\u0153uvre de l'\u00e9ducation d'aujourd'hui, semblable \u00e0 celle d'hier. Dans ces lieux de mort, on c\u00e9l\u00e8bre la structure contre l'\u00e9l\u00e8ve, le programme contre la cr\u00e9ativit\u00e9, le dressage contre l'\u00e9ducation, la servitude contre la libert\u00e9, l'ennui contre la jubilation, l'ob\u00e9issance contre l'autonomie, l'\u00e2me contre le corps.\n\nD'o\u00f9 le r\u00f4le r\u00e9pressif et mortif\u00e8re des enseignants, leur inscription dans une logique o\u00f9 l'on pratique moins le contr\u00f4le des connaissances que \u00abla mise en examen \u00bb, l'instrumentalisation de l'\u00e9l\u00e8ve \u00e0 des fins gr\u00e9gaires contre son d\u00e9veloppement singulier, la c\u00e9l\u00e9bration de la quantit\u00e9 de savoir inutile contre la qualit\u00e9 des connaissances utiles, l'id\u00e9ologie dominante contre une pens\u00e9e alternative, la m\u00e9moire docile contre l'intelligence inventive. L'\u00e9cole s'aligne ouvertement sur l'entreprise o\u00f9 l'on apprend la rentabilit\u00e9, le march\u00e9, la comp\u00e9tition et toutes les logiques aff\u00e9rentes aux lois de l'offre et de la demande. D'o\u00f9 la g\u00e9n\u00e9alogie des exclusions, de la violence, de la force et de la ruse utilis\u00e9es comme m\u00e9thodes, d'o\u00f9 les \u00e9checs scolaires inflig\u00e9s aux rebelles du syst\u00e8me, alors qu'on gratifie et r\u00e9compense la r\u00e9ussite des domestiques dociles consentant aux r\u00e8gles du march\u00e9. Le syst\u00e8me scolaire vaut antichambre du syst\u00e8me social, avec son cort\u00e8ge de corruption, de mafia, de d\u00e9composition.\n\nRaoul Vaneigem veut une \u00e9cole o\u00f9 l'on n'apprenne pas \u00e0 v\u00e9n\u00e9rer la mort et ses accessoires, l'ennui et le gr\u00e9garisme. Selon le principe de l'attraction passionnelle, il appelle de ses v\u0153ux une p\u00e9dagogie du corps tout entier r\u00e9concili\u00e9 avec lui-m\u00eame, un souci des sens et de la sensualit\u00e9 en contrepoids \u00e0 la volont\u00e9 exclusive de s'adresser \u00e0 la seule intelligence conceptuelle, une particularisation de l'enseignement qui prenne en consid\u00e9ration l'individu pour lui-m\u00eame, et non le membre d'une communaut\u00e9 dans laquelle il devrait se fondre, puis se perdre. L'ensemble tend \u00e0 un gai savoir ludique moins soucieux d'un mod\u00e8le \u00e9conomique, lib\u00e9ral et marchand, que d'une r\u00e9f\u00e9rence h\u00e9doniste, libertaire et jubilatoire.\n\nContre la haine de soi, enseign\u00e9e par la frustration de d\u00e9sirs inassouvis, transform\u00e9e en haine d'autrui, donc en violence, la p\u00e9dagogie de Raoul Vaneigem vise l'\u00e9panouissement, antidote radical aux pulsions n\u00e9gatives dirig\u00e9es contre soi et contre autrui. Le principe de cette \u00e9ducation est simple : \u00ab Il n'y a pas d'enfants stupides, il n'y a qu'\u00e9ducations imb\u00e9ciles.\u00bb D'o\u00f9 l'invite faite aux \u00e9ducateurs et aux \u00e9l\u00e8ves \u00e0 prendre en main leur destin, sans attendre une solution venue de ministres, charg\u00e9s de missions et autres d\u00e9l\u00e9gu\u00e9s \u00e0 la question scolaire dans les bureaux dor\u00e9s de la R\u00e9publique. Que vienne le temps des mon\u00f4mes et des chahuts r\u00e9parateurs...\n\nVoil\u00e0 donc quelques traits pour le portrait d'un philosophe intempestif et d'actualit\u00e9, un historien et un pamphl\u00e9taire, un po\u00e8te et un r\u00eaveur, un chercheur et un r\u00e9volt\u00e9, un r\u00e9volutionnaire et un \u00e9crivain. Bien s\u00fbr, les cyniques souriront. Ceux que le monde satisfait pour les avantages qu'ils en retirent aussi. Les endeuill\u00e9s des id\u00e9es de gauche, fossoyeurs de toutes les r\u00e9actions d'aujourd'hui, crieront \u00e0 l'utopie. Evidemment, et l'on n'attendra rien d'autre de ceux-l\u00e0, depuis longtemps couch\u00e9s sous le ventre de plus perfides qu'eux. Raoul Vaneigem n'est pas lu, cit\u00e9 et comment\u00e9 comme on pourrait l'attendre en une \u00e9poque d\u00e9c\u00e9r\u00e9br\u00e9e et pauvrissime en mati\u00e8re de penseurs et de pens\u00e9es critiques.\n\nOn encense et c\u00e9l\u00e8bre Guy Debord qui a fait se rejoindre le situationnisme de ses origines et l'\u00e9loge de l'alcoolisme dans un raccourci qui pla\u00eet aux amateurs de pens\u00e9es affadies ou \u00e9mascul\u00e9es. D'aucuns y voient le signe d'un \u00e9chec des id\u00e9es critiques et se rallient \u00e0 la plume, la prose, le talent litt\u00e9raire de Debord, ils c\u00e9l\u00e8brent son go\u00fbt pour les auteurs peu lus ou peu cit\u00e9s, son savoir des strat\u00e8ges militaires et des penseurs orientaux, son talent pour le secret et la dissimulation. Pire, la soci\u00e9t\u00e9 spectaculaire et le spectaculaire int\u00e9gr\u00e9 deviennent objets de palabres dans les \u00e9coles de journalisme. Alors ?\n\nAlors reste Vaneigem, le m\u00eame homme qui formule les barricades et se fait familier des parfums de gaz lacrymog\u00e8ne en Mai 68, le m\u00eame que n'auront atteint ni le pessimisme, ni le nihilisme, ni le pr\u00e9tendu r\u00e9alisme des pr\u00e9tendus pragmatiques. Lui qui n'a pas accept\u00e9 les ponts d'or qu'une exploitation de son statut d'ancien th\u00e9oricien de 68 et d'ex-situationniste lui auraient fournis depuis, s'il l'avait voulu. Lui qu'auront refus\u00e9 les universitaires n'acceptant pas qu'il vienne, ici, enseigner l'histoire des h\u00e9r\u00e9sies dans une chaire pr\u00e9vue \u00e0 cet effet, ou, l\u00e0, les distributeurs de pr\u00e9bendes qui grenouillent au CNRS o\u00f9 l'on fonctionnarise les impuissants dont la sp\u00e9cialit\u00e9 consiste \u00e0 chercher toujours et ne trouver jamais.\n\nAux derni\u00e8res nouvelles, jeune p\u00e8re de famille ayant pass\u00e9 la soixantaine, il a toujours la main \u00e0 la plume. Le nom de Chiara, son enfant, et de Brigitte, la maman, ouvraient en exergue et d\u00e9dicace Nous qui d\u00e9sirons sans fin. Plac\u00e9e sous le signe de la vie, de la naissance, des forces et des \u00e9nergies qui la manifestent, la pens\u00e9e de Vaneigem, d\u00e9lib\u00e9r\u00e9ment du c\u00f4t\u00e9 de la r\u00e9sistance, se d\u00e9voue tout enti\u00e8re \u00e0 la cause d'Eros, de Bacchus, de Dionysos et d'un Prom\u00e9th\u00e9e qui mettrait sa puissance au service des causes libertaires.\n\nLa derni\u00e8re fois que nous nous sommes rencontr\u00e9s, nous avons bu un pauillac, la prochaine fois, nous nous sommes propos\u00e9 un margaux. Nous referons le monde, une fois de plus, je l'\u00e9couterai comme j'aurais vraisemblablement \u00e9cout\u00e9 Charles Fourier si j'en avais \u00e9t\u00e9 le contemporain. Au milieu de l'apr\u00e8s-midi, quand nous nous quitterons pour rejoindre chacun nos destins, l'\u00e2me un peu chavir\u00e9e par les vapeurs de l'ivret\u00e9, je sais que le monde m'appara\u00eetra plus que jamais comme un vaste chantier \u00e0 ensemencer de d\u00e9sirs et de plaisirs. Et je me sentirai moins seul, d\u00e9bord\u00e9 sur ma gauche par une vitalit\u00e9 sup\u00e9rieure.\n16\n\nENTRE DEUX N\u00c9ANTS\n\n(Pour Marie-Claude) \u2013 B\u00e9nissons les rencontres avec le public de lecteurs, car elles fournissent bien souvent l'occasion de mesurer, via les questions pos\u00e9es, l'\u00e9tendue des d\u00e9g\u00e2ts en mati\u00e8re de vision du monde ordinaire. Oublions les galimatias et autres amphigouris qui permettent \u00e0 leurs auteurs des th\u00e9rapies au moindre co\u00fbt ; laissons de c\u00f4t\u00e9 les interventions faussement savantes dont l'int\u00e9r\u00eat ne d\u00e9borde pas le questionneur fr\u00e9tillant de trouver ainsi une occasion de briller \u00e0 peu de frais, du moins \u00e0 ses propres yeux ; passons notre chemin quand il s'agit d'entendre des sophistes et autres faux logiciens qui traquent la contradiction et promulguent des arr\u00eat\u00e9s repeints aux couleurs d'une dialectique imparable, alors que tout \u00e0 leur impatience d'intervenir, ils n'ont rien \u00e9cout\u00e9 du contenu de l'intervention \u00e0 la tribune; n\u00e9gligeons les savants, doctes et universitaires qui \u00e9crivent, publient, pr\u00e9parent ou pr\u00e9voient plus s\u00fbrement une brochure sur le sujet qui, elle, r\u00e9glera d\u00e9finitivement le probl\u00e8me. Et arr\u00eatons-nous sur l'homo sapiens plein de ses certitudes admirables, infatu\u00e9 mais ne le sachant pas, transformant les lieux communs en pens\u00e9es profondes, habillant les propos normalement confin\u00e9s au zinc des comptoirs avec de mis\u00e9rables et loqueteuses pr\u00e9tentions th\u00e9oriques. Redoutons celui-l\u00e0.\n\nLa question me fut pos\u00e9e \u00e0 l'universit\u00e9 de Brasilia par un individu persuad\u00e9 que je n'aime pas les enfants parce que je me refuse \u00e0 la paternit\u00e9 : \u00ab Pensez-vous, dit-il, que les enfants sont autre chose qu'un tube digestif?\u00bb Silence dans la salle, et dans l'instant qui suit, profitant du micro, il ajoute : \u00ab Moi pas. \u00bb J'eus t\u00f4t fait de lui dire que, s'il consid\u00e9rait un enfant selon ce principe, il ne manquerait pas d'en faire un repr\u00e9sentant id\u00e9al du r\u00e8gne v\u00e9g\u00e9tal, ou un mammif\u00e8re inf\u00e9rieur de son style. On devient souvent ce que l'inconscient de nos parents veut pour nous, malheureusement. Et dans cet ordre d'id\u00e9es, il para\u00eet plus facile de fabriquer des monstres que des g\u00e9nies, des imb\u00e9ciles que des artistes. Regardant cet homme, au visage lisse comme son intelligence, je me demandais combien ils \u00e9taient de milliers, sinon de millions, \u00e0 prof\u00e9rer pareille sottise.\n\nL'\u00e9tymologie d'enfant suppose l'absence de parole, l'incapacit\u00e9 \u00e0 parler, \u00e0 communiquer sur le terrain exclusivement verbal. Pour autant, combien de fois faudra-t-il dire que les communications non verbales vont plus loin, au-del\u00e0 de la m\u00e9canique langagi\u00e8re, bien souvent efficace et pertinente pour un certain nombre d'op\u00e9rations intellectuelles, mais tout aussi inappropri\u00e9e pour la plupart des projets communicationnels. Qui dira, une fois encore, qu'on \u00e9change avec le corps tout entier, bien en de\u00e7\u00e0 et bien au-del\u00e0 des compromis linguistiques qui offrent toujours une occasion de laisser une moindre place \u00e0 la chair pour lui pr\u00e9f\u00e9rer la seule \u00e2me ? Du silence de l'enfant, de son incapacit\u00e9 \u00e0 parler, faut-il d\u00e9duire son inhumanit\u00e9 ? La r\u00e9duction \u00e0 un pur et simple syst\u00e8me physiologique peut-elle se pr\u00e9valoir du silence commun aux b\u00eates, aux fous, aux morts, aux sages, aux m\u00e9ditatifs, aux renon\u00e7ants ? Cet homme r\u00e9pondait oui. Et combien avec lui ?\n\nJ'ai racont\u00e9 la sexualit\u00e9 des enfants, les pouces suc\u00e9s, les \u00e9rections dans le ventre de la m\u00e8re, le traumatisme de la naissance, les dynamiques d'une chair dans le liquide amniotique, les informations transmises via les peaux qui se touchent, le dos de l'enfant et la paroi de l'ut\u00e9rus; j'ai dit les parfums et les odeurs comme des marqueurs qui structurent les identit\u00e9s, les saveurs, les go\u00fbts et les d\u00e9go\u00fbts d\u00e9j\u00e0 donn\u00e9s, d\u00e9j\u00e0 transmis, les modifications \u00e9thiques, donc physiologiques, spirituelles et m\u00e9taphysiques, donc chimiques, de la m\u00e8re et le savoir qu'en a l'enfant via le sang, l'influx, la lymphe, le rythme cardiaque, le souffle de la respiration ; j'ai continu\u00e9 avec les vibrations de la voix et les fr\u00e9quences basses ou aigu\u00ebs qui travaillent le corps en suspension dans l'eau magique; j'ai avanc\u00e9 l'hypoth\u00e8se d'inconscients qui communiquent, leurs jouissances et leurs d\u00e9sirs, leurs craintes, leurs peurs et leurs angoisses, leurs \u00e9motions et leurs sensations, voire leurs perceptions.\n\nEt ajout\u00e9 que toutes ces apocalypses \u00e9taient le fait de tubes digestifs, certes, mais en pr\u00e9cisant qu'il en va de cette fonction comme des autres : chacune permet le trajet d'un \u00eatre dans un monde, d\u00e9j\u00e0. Trajet dans lequel s'ouvrent des routes, se dessinent des sentiers, murmurent des hypoth\u00e8ses, s'annoncent des destins. Sinon se proposent des drames et des trag\u00e9dies. Et ce avant toute parole prof\u00e9r\u00e9e, avant tout papa ou tout maman flattant le narcissisme des parents et permettant des satisfactions primaires du c\u00f4t\u00e9 des seuls adultes. D\u00e8s qu'apparaissent ces syllabes affectives, il faut craindre que l'essentiel ait \u00e9t\u00e9 dit, car l'existence ne sert qu'\u00e0 r\u00e9v\u00e9ler, au sens photographique, un clich\u00e9 d\u00e9j\u00e0 pris depuis longtemps.\n\nMon interlocuteur d'un soir semblait \u00e9tonn\u00e9, interdit. \u00abEn vous lisant, pourtant, j'ai bien vu que vous n'aimiez pas les enfants... \u00bb O\u00f9 ai-je \u00e9crit une seule ligne qui ressemble \u00e0 cela, de pr\u00e8s ou de loin, ou permette ce genre d'extrapolation? M\u00eame de tr\u00e8s loin? Mon d\u00e9sir d'\u00eatre st\u00e9rile \u2013je veux cette formulation qu'entendent mal les m\u00e2les \u2013 rel\u00e8ve d'une volont\u00e9 ontologique. Avoir un enfant devrait proc\u00e9der d'un projet m\u00e9taphysique radical et non d'une pure et simple ob\u00e9issance aveugle aux imp\u00e9ratifs de libido r\u00e9cup\u00e9r\u00e9s et structur\u00e9s par les besoins sociaux. Faire surgir \u00e0 l'Etre ce qui, en dehors d'un vouloir, n'en proviendrait sinon jamais doit supposer des raisons, et plus sp\u00e9cifiquement de bonnes raisons. Lesquelles ? Je n'en vois pas. A bien r\u00e9fl\u00e9chir, je vois m\u00eame plut\u00f4t des raisons de ne pas procr\u00e9er que l'inverse.\n\nD'abord, il faudrait proc\u00e9der \u00e0 une dissociation d'id\u00e9es : la conception d'un enfant n'a rien \u00e0 voir avec l'amour ou les sentiments que se portent les deux qui font un couple. Pourquoi diable faudrait-il consentir \u00e0 cette association comme \u00e0 une \u00e9vidence ? On ferait des enfants parce qu'on s'aime? Ce serait la finalit\u00e9, le sens de tout couple ? Il faudrait laisser un nom? Pour ma part, je ne vois pas l'amour n\u00e9cessitant ou exigeant un tiers comme preuve de son existence; pas plus je n'imagine qu'un enfant agisse en occasion de c\u00e9l\u00e9brer une vie \u00e0 deux : un plus un \u00e9galant trois, voil\u00e0 une arithm\u00e9tique dont j'attends qu'on me d\u00e9montre la validit\u00e9; enfin, la trace me para\u00eet moins imp\u00e9rieuse dans un nom si fragile qu'il dispara\u00eet lorsqu'il est port\u00e9 par une fille \u00e9pousant un gar\u00e7on, que dans toute autre occasion d'incarnation ou de m\u00e9moire.\n\nAssoci\u00e9es, ces id\u00e9es le sont depuis que la soci\u00e9t\u00e9 a int\u00e9r\u00eat \u00e0 contenir, retenir, voire arr\u00eater la libido dans son \u00e9nergie rebelle et libertaire. Les enfants mis en perspective avec l'amour et le couple, servent seulement \u00e0 produire des clich\u00e9s destin\u00e9s au culte des v\u00e9rit\u00e9s sociales utiles \u00e0 l'instinct gr\u00e9gaire : en proc\u00e9dant de la sorte, on rattache la sexualit\u00e9 aux n\u00e9cessit\u00e9s collectives, on pose la premi\u00e8re pierre de l'\u00e9difice social en fondant et l\u00e9gitimant la famille qui absorbe les singularit\u00e9s, les individualit\u00e9s, dans des machines transform\u00e9es en rouages pour l'usine soci\u00e9taire. A chaque fois qu'on sacrifie \u00e0 ce lieu commun \u2013 associer la procr\u00e9ation et l'amour \u2013, on contribue \u00e0 l'enrichissement d'une mythologie, on fait cro\u00eetre la somme des croyances utiles \u00e0 l'existence et \u00e0 la permanence des collectivit\u00e9s.\n\nAinsi, l'amour se met au service des int\u00e9r\u00eats gr\u00e9gaires quand il ne devrait jamais cesser d'\u00eatre une occasion asociale ou antisociale d'exprimer une \u00e9nergie fabuleuse et de rendre deux existences sublimes. A d\u00e9faut, l'enfant install\u00e9 en tiers exige et n\u00e9cessite \u2013 c'est d'ailleurs son droit le plus strict et le plus absolu quand il est l\u00e0, n'ayant rien demand\u00e9 \u2013, l'essentiel, sinon la totalit\u00e9 de l'\u00e9nergie de ceux qui l'ont appel\u00e9 et con\u00e7u. Soit on y consent, et les deux existences singuli\u00e8res des parents culminent dans la perspective de l'\u00e9dification d'un \u00eatre ; soit on n'y consent pas, et les libert\u00e9s sont sauves. L'amour conserve alors ses chances. Sinon...\n\nJ'ai toujours \u00e9t\u00e9 \u00e9tonn\u00e9 de ces remarques faites sur mon refus de paternit\u00e9 comme d'une marque d'\u00e9go\u00efsme caract\u00e9ris\u00e9, quand nombre de ceux qui font des enfants les \u00e9duquent, disons plut\u00f4t les n\u00e9gligent ou les oublient dans leurs projets, en parangons d'\u00e9gocentrisme. Faut-il rappeler presque en forme de boutade que le mariage demeure la principale cause des divorces ? Que nombre de p\u00e8res ou de m\u00e8res se comportent dans l'existence comme s'ils ou elles \u00e9taient absolument d\u00e9pourvus de devoir, \u00e0 l'endroit de leur prog\u00e9niture? Et que je trouve plus \u00e9go\u00efste de faire des enfants qu'on n'int\u00e8gre pas dans ses projets de vie au quotidien, vingt-quatre heures sur vingt-quatre, plut\u00f4t que de ne pas en faire ?\n\nL'entreprise de paternit\u00e9 \u2013 de maternit\u00e9 aussi, bien entendu \u2013 est radicalement m\u00e9taphysique. Elle suppose qu'on ait r\u00e9fl\u00e9chi, au-del\u00e0 des clich\u00e9s \u00e0 jeter par-dessus bord, sur la nature d'un projet d'existence, l'engagement dans le temps, l'\u00e9dification existentielle, l'action irr\u00e9m\u00e9diable, la nature de la libert\u00e9, les formes du contrat, le statut d'un tiers, le destin d'un sentiment, le trajet d'une affection, les m\u00e9tamorphoses du r\u00e9el, les risques de l'entropie, une identit\u00e9 assur\u00e9e, la fragilit\u00e9 du d\u00e9sir. Dans tous les cas de figure, on devrait savoir qu'il s'agit d'op\u00e9rer sur la mati\u00e8re la plus noble \u2013 une existence.\n\nLas ! les hommes vivent comme s'ils ne devaient jamais vieillir, comme si leurs d\u00e9sirs et leurs plaisirs allaient se cristalliser dans l'\u00e9ternit\u00e9 d'une jeunesse et d'une sant\u00e9 qui dispara\u00eetront, quoi qu'on fasse. De l'instant ils induisent l'immortalit\u00e9 \u2013 de leurs sentiments, de leur \u00eatre, de leurs aspirations, de leurs identit\u00e9s, ils supposent l'\u00e9ternit\u00e9. Chacun croyant rester le m\u00eame alors que le temps \u00e9coul\u00e9 les modifie sans cesse jusqu'\u00e0 rendre radicalement \u00e9trangers ceux qui se croyaient d'une m\u00eame galaxie. Le tiers qui, lui, proc\u00e8de d'une plan\u00e8te une et unie, avant mitose ou m\u00e9iose m\u00e9taphysique, a le droit de demander que tout soit fait pour p\u00e9renniser ce dont il est issu. Les enfants devraient pouvoir attendre ce que personne n'est en droit de leur proposer, encore moins de leur offrir : l'\u00e9ternit\u00e9 de l'\u00e9tat dans lequel ils ont \u00e9t\u00e9 th\u00e9oriquement con\u00e7us \u2013 la ferveur.\n\nOr qui croit possible l'entretien d'une permanence de la ferveur ? Les sots. Car d\u00e8s la naissance, la vie est assez entam\u00e9e pour avoir \u00e0 mourir, et d\u00e9j\u00e0, ils sont susceptibles d'\u00eatre socialis\u00e9s. L'entreprise commence alors : du marquage et du ravage, du num\u00e9rotage et de l'\u00e9tiquetage, de l'emballage et du pesage, du gavage et de l'apprentissage. Il faut tout inculquer : l'art d'user de soi, d'employer son temps, de d\u00e9ployer et d\u00e9placer son corps, celui de tout faire comme il faut, de penser comme il se doit, de s'exprimer comme on l'entend. Puis, le corps dress\u00e9, on s'occupe m\u00eamement de l'\u00e2me, et les deux instances, d'ailleurs, seront sollicit\u00e9es conjointement : la contrainte de l'un induit la contrainte de l'autre. Les institutions, la scolarisation, la famille, les cr\u00e8ches, tout dispense un savant art de renoncer \u00e0 soi-m\u00eame au profit d'une int\u00e9gration dans le social dont je me demande d'ailleurs ce qui la distingue v\u00e9ritablement de la fusion.\n\nJe n'aurais pas la force, ni le d\u00e9sir, ni la cruaut\u00e9 de me faire l'instrument de cette domestication. Je ne vois aucune occasion de refouler la na\u00efvet\u00e9, la simplicit\u00e9 et la franchise naturelles des enfants au profit d'un enseignement de la ruse, des ficelles et du mensonge, des strat\u00e9gies de la dissimulation et des masques de l'hypocrisie, puis de tout ce qui permet, via l'artifice et la tromperie, de favoriser pr\u00e9tendument l'art de vivre en soci\u00e9t\u00e9. Je ne vois aucune bonne raison pour emp\u00eacher l'empire des d\u00e9sirs et des plaisirs chez un \u00eatre \u00e0 qui je devrais enseigner le renoncement, la contention, la r\u00e9tention, l'abandon des chim\u00e8res au nom d'un principe de r\u00e9alit\u00e9 destin\u00e9 \u00e0 le rendre docile, soumis, ob\u00e9issant, \u00e9teint. Je n'ai aucune envie de pulv\u00e9riser une propension \u00e0 la libre disposition de soi, \u00e0 la jouissance du commerce pris \u00e0 soi-m\u00eame, sous pr\u00e9texte qu'il faudrait apprendre \u00e0 c\u00e9der sur son d\u00e9sir pour ne pas avoir \u00e0 trop souffrir dans l'existence. Je ne veux pas couper, tailler, arracher, \u00e9masculer et contraindre, comme d'aucuns agissent sur les bonsa\u00efs en visant le nanisme tel un id\u00e9al. Je n'aspire pas \u00e0 ce talent o\u00f9 excellent les pr\u00eatres et les bouchers, les fossoyeurs et les moralistes.\n\nJ'entends d\u00e9j\u00e0 les d\u00e9fenseurs des mensonges sociaux, des clich\u00e9s et images d'Epinal qu'ils serrent dans leurs mains, affal\u00e9s sur le prie-Dieu sociologique. Je les vois, excellents dans le d\u00e9ni et l'aveuglement comme on se mutile, en h\u00e9raut de la scolastique contemporaine. Je les connais, pour les avoir entendus, si souvent d\u00e9j\u00e0 sur ce sujet : on a stigmatis\u00e9 mon manich\u00e9isme, trop caricatural; on a fustig\u00e9 mon abord c\u00e9r\u00e9bral du monde, vous pensez trop \u2013 comme s'il \u00e9tait possible de trop penser quand l'alternative est penser ou ne pas penser; on a regrett\u00e9 l'univocit\u00e9 de mon propos, les enfants \u00e7a n'est pas seulement \u00e7a. Quoi, alors ? Qu'est-ce donc ? Un r\u00e9confort pour sa vieillesse, m' a-t-on dit... Quelle \u00e9trange id\u00e9e ! Je n'ai jamais imagin\u00e9 cet \u00e2ge-l\u00e0 pour moi, encore moins, l' imaginant, souhaiterais-je mes enfants en ma compagnie dans ce naufrage. Ce me semble d\u00e9j\u00e0 bien assez que nous ayons \u00e0 y tendre. Des enfants pour son vieil \u00e2ge, comme une poire pour la soif... De quoi pousser la chaise roulante qui nous m\u00e8ne \u00e0 l'ab\u00eeme, nos enfants aux commandes ! Plut\u00f4t mourir seul, dans une chambre d'h\u00f4tel, en n'infligeant rien \u00e0 personne.\n\nL'entreprise g\u00e9n\u00e9sique proc\u00e8de si souvent de raisons aveugles, dont celles qui permettent d'associer paternit\u00e9 et maternit\u00e9 \u00e0 des exp\u00e9riences initiatiques, que je n'ai gu\u00e8re envie de consentir \u00e0 ces passages oblig\u00e9s, en mati\u00e8re de religion sociale, pour imaginer qu'hier imp\u00e9trant parce que c\u00e9libataire et solitaire, je deviendrais aujourd'hui initi\u00e9 dans le monde adulte du simple fait d'avoir donn\u00e9 naissance \u00e0 un enfant. Dans toutes les soci\u00e9t\u00e9s, traditionnelles, primitives ou se croyant plus \u00e9volu\u00e9es, demeure cette croyance au fait que des signes autorisent et l\u00e9gitiment socialement une onction communautaire. Avoir des enfants passe pour l'un de ces signes.\n\nCombien, pourtant, d\u00e9couvrent que la magie n'op\u00e8re pas et que, devenus p\u00e8res ou m\u00e8res, ils demeurent emp\u00eatr\u00e9s dans leurs histoires existentielles, se refusant \u00e0 devenir vraiment adultes, c'est-\u00e0-dire \u00e0 regarder, voir en face, puis accepter la nature tragique du r\u00e9el ? Tels qui imaginaient les parures de la chefferie ou les plumes de l'autorit\u00e9 m\u00e9taphysique en donnant la vie \u00e0 un \u00eatre se retrouvent Gros-Jean comme devant avec leurs illusions perdues et leurs espoirs \u00e9vanouis. On ne devient pas adulte en faisant des enfants, en acc\u00e9dant soi-m\u00eame au statut de parent, ce serait trop simple. Mais c'est assez simpliste pour \u00eatre enseign\u00e9 comme une vertu soci\u00e9taire.\n\nD'o\u00f9 les malentendus de ceux qui \u00e9l\u00e8vent, plut\u00f4t qu'ils n'\u00e9duquent, des enfants avec lesquels ils entretiennent de path\u00e9tiques et pathologiques relations ludiques, sans souci d'une t\u00e2che \u00e9ducative authentiquement \u00e9difiante. Enfants eux-m\u00eames au milieu d'enfants, les leurs, ils n'ont de prog\u00e9niture que pour retrouver \u00e0 bon compte et sans complexes un monde duquel ils ne sont pas sortis. Plut\u00f4t que tendre vers l'\u00e9thique avec leurs fils ou filles, ils sacrifient \u00e0 l'\u00e9tique, au r\u00e9gressif et \u00e0 l'involution. Etre p\u00e8re ou m\u00e8re suppose sa propre enfance apprivois\u00e9e, ma\u00eetris\u00e9e. A d\u00e9faut, l'abstention me semble pr\u00e9f\u00e9rable.\n\nD'aucuns, bien s\u00fbr, voient dans mon refus d'enfant un sympt\u00f4me de mon \u00e9chec \u00e0 domestiquer mes premi\u00e8res ann\u00e9es, mon enfance et mon adolescence v\u00e9cues dans la douleur. Forts des arguments d'autorit\u00e9 que permet le recours facile et grossier aux cat\u00e9gories freudiennes, les amateurs de religion familialiste et soci\u00e9taire pointent l'\u0152dipe mal r\u00e9solu, les comptes \u00e0 r\u00e9gler avec soi-m\u00eame, un p\u00e8re pas assez tu\u00e9, une m\u00e8re pas assez conquise, une fixation sur le stade oral, voire une homosexualit\u00e9 refoul\u00e9e, car il ne faut pas s'interdire ce qui permet l'usage policier des lectures psychanalytiques. Et pourquoi pas ? Ma m\u00e8re fut abandonn\u00e9e, puis plac\u00e9e, battue, mal aim\u00e9e, et voulut pour moi, quand j'avais dix ans, un internat dans un orphelinat o\u00f9 j'ai appris \u00e0 survivre quand tout \u00e9tait fait pour que nous mourions. Proc\u00e9dant de cet abandon par procuration, d\u00e9coulant de cette douleur qu'une m\u00e8re me contraint \u00e0 partager et assumer avec elle, dans la violence d'une demande qui s'est faite pour elle dans le plus total des aveuglements, je ne peux qu'avoir \u00e9t\u00e9 sensible, et le rester, \u00e0 cette question : pourquoi fait-on des enfants ?\n\nQuand la plupart nagent dans la na\u00efvet\u00e9 et l'incons\u00e9quence, agissent sans r\u00e9fl\u00e9chir, consentent \u00e0 l'usage gr\u00e9gaire et social de leur libido, je ne parviens pas \u00e0 me d\u00e9partir d'un souci interrogateur : pourquoi la paternit\u00e9 ? Quelles raisons \u00e0 la maternit\u00e9 ? Quelles obligations \u00e0 s'installer sur ce terrain? Quelles libert\u00e9s, aussi, en la mati\u00e8re? Quelles latitudes? Bien s\u00fbr, les enfants que l'on fait, ou que l'on s'abstient de faire, donnent tous, dans leur pr\u00e9sence ou leur absence, des occasions de r\u00e9ajuster et de compenser le d\u00e9calage et le ratage des embo\u00eetages de nos enfances. Tous demandent \u00e0 leurs prog\u00e9nitures de prendre en charge une dette qui n'est pas la leur : les grands-parents \u00e9duquent leurs enfants. Ou se fabriquent un avenir sans petits-enfants.\n\nQuand aux heures silencieuses on eut la peau arrach\u00e9e et la chair \u00e0 vif, l'\u00e2me d\u00e9chir\u00e9e et le corps naus\u00e9eux \u00e0 cause de la dose de haine \u00e0 ingurgiter, on a vu, dans son malheur, les nerfs et les m\u00e9canismes qui structurent les enfants \u00e0 la mani\u00e8re des pantins articul\u00e9s ou d\u00e9sarticul\u00e9s. Et quand on est homme, on sait, pour son bonheur ou son malheur, que le d\u00e9sir d'enfant proc\u00e8de souvent du d\u00e9sir de l'autre auquel on consent, plus que celui qui nous travaille en propre le ventre et la chair. Et l'on peut vouloir une femme tellement femme qu'on n'aurait pas besoin, pour l'aimer, qu'elle devienne m\u00e8re et nous fasse des enfants.\n\nEt je songeai, les remarques s'encha\u00eenant, au corps d'une vieille dame que j'avais aim\u00e9e et visit\u00e9e \u00e0 la morgue, la semaine pr\u00e9c\u00e9dant cette soir\u00e9e. Dans la pi\u00e8ce aux odeurs trop fades pour \u00eatre honn\u00eates, j'ai regard\u00e9 ce petit corps reposant, comme un enfant dort, l'\u00e2me \u00e9teinte. J'avais aim\u00e9 un visage, une allure, une fa\u00e7on de marcher, le timbre d'une voix, un sourire permanent, un rire, souvent, la douceur de la joue que j'embrassai \u2013 et que je n'embrasserai plus. Son visage \u00e9tait serein, mais je devinais d\u00e9j\u00e0 sous lui le cr\u00e2ne qui s'annon\u00e7ait et poussait comme exigeant son tribut. Le silence, enfin, ce silence des enfants qui ne parlent pas encore, et celui de ma vieille amie qui ne parlerait plus.\n\nDans son habit rose, sous la lumi\u00e8re fade elle aussi, aupr\u00e8s des gerbes de fleurs, son thorax ne se soulevant plus pour trahir une respiration qui l'avait d\u00e9sert\u00e9e, j'ai soudain song\u00e9 \u00e0 son enfance : elle avait \u00e9t\u00e9 enfant. Son adolescence, ses parents, ce que je savais de sa jeunesse, sa fa\u00e7on de boire du champagne, des photos anciennes regard\u00e9es en sa compagnie, ses doigts qui portaient de si beaux bijoux, tout convergea vers cette \u00e9vidence : elle avait \u00e9t\u00e9 enfant. Sous le m\u00eame visage, l\u00e0 o\u00f9 avait tent\u00e9 de percer le cr\u00e2ne, je vis aussi, lumineuse, la figure de son enfance. Tout cela pour \u00e7a. Ce trajet de quatre-vingt-quatre ans pour aboutir l\u00e0, dans cette morgue, \u00e0 deux pas du cimeti\u00e8re o\u00f9 elle allait se d\u00e9composer, se min\u00e9raliser, et laisser les seules traces d'un nom et de dates qui, eux aussi, dispara\u00eetraient un jour.\n\nBien s\u00fbr, il y avait eu sa fille, son petit-fils, et son arri\u00e8re-petite-fille, une libraire que j'aime et dont j'aimais le mari, lui aussi disparu, un fils traducteur de po\u00e8tes japonais qui gagne sa vie en commer\u00e7ant \u00e0 des niveaux vertigineux, et une arri\u00e8re-petite-fille eurasienne qu'elle aura aim\u00e9e, embrass\u00e9e, touch\u00e9e, caress\u00e9e. Entre ces \u00e9pisodes, deux n\u00e9ants, celui dont elle provenait, celui dans lequel elle g\u00eet maintenant : la vie se r\u00e9duit \u00e0 la course qui nous conduit du n\u00e9ant dont nous proc\u00e9dons vers celui qui nous engloutit. L'\u00e9ternit\u00e9 est pour lui seul. La vie passe en souffle, rien, presque rien susceptible d'\u00eatre un tout par la transfiguration h\u00e9doniste, certes, mais qui restera malgr\u00e9 tout une mis\u00e8re tant reste grande la solitude des morts que nous ferons.\n\nEt il faudrait nourrir cette farce ? Enrichir ce th\u00e9\u00e2tre cruel comme si rien de tout cela n'\u00e9tait ? Il faudrait vouloir l'aveuglement, s'entretenir dans l'innocence, ne pas r\u00e9fl\u00e9chir, ob\u00e9ir aux caprices d'une libido imp\u00e9rieuse pour se donner l'illusion de lutter efficacement contre les triomphes perp\u00e9tuels de la mort? Je ne me sens pas de talent pour la d\u00e9n\u00e9gation ou la complaisance dans l'illusion, pour l'ivresse qui me ferait mettre mon \u00e9nergie sexuelle au service de cette vaste entreprise de conjuration, impuissante, perdue d'avance, qui partout exige de la chair \u00e0 cercueil comme d'aucuns veulent de la chair \u00e0 canon.\n\nJe revins \u00e0 mes interlocuteurs, aux questions de la salle, et t\u00e2chai de pr\u00e9ciser que n'avoir pas d'enfants proc\u00e8de non pas d'une haine, d'un m\u00e9pris de ce qu'ils sont, mais d'un amour trop grand \u00e0 donner pour ce que permet le monde, trop \u00e9troit, trop vain, trop cruel. D'une autre, dans l'assembl\u00e9e, alors que se dissipait le visage de ma vieille amie qui poursuivait son trajet en direction des ab\u00eemes o\u00f9 triomphe le silence, cette remarque : et le plaisir que j'ai, moi, \u00e0 vivre, \u00e0 \u00eatre, bien que n'ayant rien demand\u00e9 \u00e0 mes parents, qu'en faisais-je? Il me fallut poser l'\u00e9quation du pr\u00e9f\u00e9rable : une existence qu'on essaie de rendre vivable, au jour le jour, en \u00e9laborant une vision h\u00e9doniste du monde, ou les plaisirs du non-\u00eatre?\n\nCertes, il fallut pr\u00e9ciser la nature de ce non-\u00eatre torturant les lecteurs du Parm\u00e9nide de Platon tout autant qu'il apporte r\u00e9jouissance et consolation aux amateurs nihilistes de pens\u00e9es orientales. Le non-\u00eatre n'est pas l'\u00eatre qui n'est plus, apr\u00e8s la mort, choisie ou subie, mais l'\u00e9tat de ce qui n'a jamais \u00e9t\u00e9 et n'a pas \u00e0 proc\u00e9der d'une modalit\u00e9 (l'\u00eatre) supposant la transition vers une autre (le n\u00e9ant). Pas plus il ne s'obtient n\u00e9gativement, en retirant tout ce qui faisait la substance de l'\u00eatre pour ne plus laisser qu'une enveloppe vide, mais paradoxalement pleine encore du non-\u00eatre. Il n'est pas le concept, ou l'id\u00e9e pure, quintessenci\u00e9 par rar\u00e9faction de l'\u00eatre jusqu'\u00e0 obtention d'un reliquat conceptuel rendu possible par la chimie proc\u00e9dant d'un genre de th\u00e9ologie n\u00e9gative. On songe au fameux couteau sans lame auquel il manque le manche de Lichtenberger et qui, lui non plus, ne suffit \u00e0 donner une id\u00e9e approchante ou approximative du non-\u00eatre. Car le non-\u00eatre demeure inconcevable, au sens \u00e9tymologique, par qui que ce soit, puisqu'il proc\u00e8de, pour exister intellectuellement, de l'\u00eatre d'un sujet, et qu'une mise en sc\u00e8ne de l'absence n'est pas possible pour ce qui se vivifie de pr\u00e9sence. Vivant, je ne peux rien savoir de ce qu'est le non-\u00eatre dont nous proc\u00e9dons et vers lequel nous allons \u2013 ni s'il est pr\u00e9f\u00e9rable \u00e0 l'\u00eatre, ou l'inverse. Je sais la douleur qu'il y a, non pas \u00e0 mourir, mais \u00e0 vivre en se sachant mortel. Et cette douleur est consubstantielle \u00e0 l'\u00eatre, pourvu qu'il soit conscient, lucide et d\u00e9barrass\u00e9 de tout type d'\u0153ill\u00e8res m\u00e9taphysiques, ontologiques, mystiques ou religieuses.\n\nJubiler, donc, du simple fait d'\u00eatre ? Je crois malheureusement qu'il n'y a pas de jouissance sans conscience et qu'elle s'envisage exclusivement par un travail de l'esprit et de la volont\u00e9. Remercier mes parents ? S\u00fbrement pas. Les accabler non plus. Faire avec. Malgr\u00e9 les perspectives et le destin in\u00e9vitable. Du moins, ne pas contribuer, par une g\u00e9nitalit\u00e9 complice du vaste mouvement entropique de l'univers, \u00e0 perp\u00e9tuer, pour ma part, l'\u00e9ternit\u00e9 de la cruaut\u00e9 du monde et du destin d'un \u00eatre que j'aurais arrach\u00e9 aux brumes du non-\u00eatre o\u00f9 il n'a m\u00eame pas \u00e0 se trouver, \u00e0 l'instant, pour \u00eatre heureux \u2013 puisqu'il est au-del\u00e0, en de\u00e7\u00e0.\n\nEt pour ceux qui sont l\u00e0? Qui n'ont rien demand\u00e9, mais sont l\u00e0? Parce que la vie n'est pas belle comme un donn\u00e9, mais comme une construction, pour la raison qu'elle n'est pas sublime a priori, mais dans l'a posteriori d'une volont\u00e9 \u2013 c'est tout le sens de l'h\u00e9donisme \u2013, il faut leur rendre l'existence la plus jubilatoire possible. Avant le cadavre, malgr\u00e9 lui, sans obsession pour le n\u00e9gatif qui partout triomphe, il faut \u00e9laborer une r\u00e9sistance, une \u0153uvre, proposer les moyens d'une sculpture de soi dont la philosophie permet l'occasion. Si je refuse la paternit\u00e9 pour savoir \u00eatre incapable de donner tout l'amour que j'estime n\u00e9cessaire \u00e0 un \u00eatre pour qu'il soit, je pense qu'on peut offrir aux \u00eatres qui naissent les pitoyables instruments d'une conjuration du n\u00e9gatif\u2013 bricoler dans l'incurable, aurait dit Cioran.\n\nC'est-\u00e0-dire? Fournir d\u00e8s que possible les moyens intellectuels de savoir la nature cruelle du r\u00e9el, et, conjointement, ce que peuvent \u00eatre les possibilit\u00e9s d'en conjurer au maximum les effets. L\u00e0 o\u00f9 les enfants, naturellement enthousiastes et interrogateurs, questionnent et disent encore leur proximit\u00e9 avec l'intelligence consubstantielle aux \u00e9nergies qui structurent l'\u00e9volution de la plan\u00e8te, il faut entretenir la ferveur. Comment ? En c\u00e9l\u00e9brant le savoir et la r\u00e9ponse, en fournissant d\u00e8s qu'ils les demandent les solutions ou les fa\u00e7ons d'y parvenir.\n\nLes visions du monde qui architecturent et soutiennent notre civilisation sont d\u00e9liquescentes et ne subviennent plus \u00e0 nos besoins m\u00e9taphysiques. M\u00eame dans ses formulations n\u00e9o-philosophiques du moment, l'id\u00e9al asc\u00e9tique chr\u00e9tien est exsangue. Il a fait son temps. L'eschatologie et l'ontologie associ\u00e9es ont perdu toute cr\u00e9dibilit\u00e9. Fin de la m\u00e9taphysique, fin, aussi, de l'\u00e9thique et de sa forme popularis\u00e9e, la morale. L'univers reste toujours aussi cruel, m\u00e9taphysiquement, et les armes anciennes sont \u00e9mouss\u00e9es. Restent de nouvelles possibilit\u00e9s d'existence qui supposent la promotion d'une pens\u00e9e agressive au service de ces combats \u00e0 mener.\n\nQue les enfants jet\u00e9s sur cette plan\u00e8te aient les moyens de la comprendre et d'en combattre les noirceurs. Que la ferveur, que ne sauraient entretenir ou solliciter des parents d\u00e9faillants, soit c\u00e9l\u00e9br\u00e9e, osons le paradoxe, dans les lieux habituellement consacr\u00e9s \u00e0 faire des natures \u00e9teintes : les \u00e9coles. Qu'on enseigne la philosophie, pas l'histoire de la philosophie, mais la philosophie, \u00e0 ceux qui disposent d\u00e9j\u00e0 de toutes les questions possibles et \u00e0 qui, bien souvent, on ne sait, ne peut, ou ne veut, faire l'aum\u00f4ne des r\u00e9ponses dignes de ce nom et qu'ils sont pourtant en droit d'attendre. Qu'on ne s'imagine pas r\u00e9volutionnaire en souhaitant cet enseignement l'ann\u00e9e qui pr\u00e9c\u00e8de la classe terminale, mais qu'on l'installe d\u00e8s qu'un enfant se trouve propuls\u00e9 dans le syst\u00e8me scolaire : dans le primaire. Qu'on y ait souci de ce qui fuse chez eux, et immanquablement, chez tous, sur l'origine de notre syst\u00e8me solaire, des plan\u00e8tes, les raisons pour lesquelles on vieillit, on meurt, ce qui se passe apr\u00e8s le tr\u00e9pas, pourquoi il faut travailler, ce qui se dit dans des mots comme hier, aujourd'hui et demain, pour quelles raisons la pluie tombe du ciel, ce qu'il en est du corps et des d\u00e9sirs qui le travaillent, des plaisirs qui le consument, et ces mille questions qui dans leur simplicit\u00e9 na\u00efve et fruste, po\u00e9tique, trahissent la pr\u00e9sence d'une \u00e2me de philosophe que toujours on tue.\n\nEux qui naturellement se portent sur les questionnements ontologiques, th\u00e9ologiques, m\u00e9taphysiques, philosophiques, ils subissent bien souvent l'aveuglement et le d\u00e9sint\u00e9r\u00eat, voire la d\u00e9sinvolture de ceux qui infligent l'existence comme on impose une peine, un pensum. Dans nos civilisations, \u00e9duquer \u2013 on \u00e9l\u00e8ve d'ailleurs des enfants, disent certains, comme d'autres du b\u00e9tail \u2013, \u00e9quivaut \u00e0 \u00e9teindre les incandescences et an\u00e9antir les ferveurs. En pratiquant l'\u00e9teignoir, les parents et les \u00e9ducateurs avec eux, contribuent \u00e0 rendre l'existence sinistre et tragique quand seulement devrait \u00eatre sinistre et tragique le savoir du n\u00e9ant dont nous proc\u00e9dons et vers lequel nous allons. Le n\u00e9ant est la loi dans la mort, il ne devrait pas l'\u00eatre tant que dure la vie. Mais r\u00e9sister \u00e0 la furie meurtri\u00e8re de ceux qui nourrissent le social, le monde, le r\u00e9el, et la cruaut\u00e9 de tout cela, avec les corps et les \u00e2mes de leurs enfants, est au-dessus de mes forces.\n\nLe lendemain, dans l'avion qui me ramenait, je lisais Michelet, loin des pr\u00e9occupations de la veille au soir. Pr\u00e8s de moi, une enfant parlait, jouait, passait le temps. Elle se mit dans la t\u00eate, devant son fauteuil, d'infliger la discipline de la ceinture \u00e0 son biberon qui roulait \u00e7\u00e0 et l\u00e0. Elle y parvint, amarrant l'objet qui fuyait avec plus ou moins de bonheur. Triomphante, elle leva la t\u00eate, son regard rencontra le mien, elle eut un superbe sourire. Dans son visage magique et magnifique, je vis les traits de la morte qu'elle serait, un jour, reposant dans une morgue, apr\u00e8s un qui l'aurait pr\u00e9c\u00e9d\u00e9e, avant l'autre qui la suivrait. Moi prenant mon tour, je ne sais quand, ni o\u00f9. Mais le sachant, d\u00e8s aujourd'hui. Je repris ma lecture, entre deux n\u00e9ants, comme elle.\n17\n\nL'INNOCENCE DU DEVENIR\n\nDebussy ha\u00efssait les mots. Faut-il lui en vouloir tant il para\u00eet en France, sur le terrain de la composition, le parangon du musicien clair et distinct, \u00e0 l'\u00e9criture simple, naturelle, \u00e9l\u00e9gante et h\u00e9doniste ? Cette haine du verbe n'aurait-elle pour raison que de tenir fermement compagnie \u00e0 un amour immod\u00e9r\u00e9 du son, du seul son ? On pourrait le croire tant triomphe le culte vou\u00e9 au silence chez Claude Achille. Ecrire sur Debussy devient par la m\u00eame occasion une gageure dont il faut craindre les pires cons\u00e9quences.\n\nSe taire, alors? Ou persister dans ce qui, finalement, pourrait bien \u00eatre une th\u00e9orie des correspondances \u00e0 sa mani\u00e8re en imaginant que le mot peut dire le son, que le verbe est capable de rendre compte d'une partie de la musique? Le pari me tente trop et les philosophes s'y sont trop peu risqu\u00e9s pour n'y pas trouver un peu de l'ivresse qui fait le fond de toute \u0153uvre musicale. Le propos est net, sans appel : \u00ab il est m\u00eame inutile que la musique fasse penser \u00bb, \u00e9crit-il un soir de f\u00e9vrier 1901 \u00e0 Paul Dukas. Son r\u00f4le, sa nature et sa fonction? Exiger l'\u00e9coute, demander et obtenir l'audition, l'attention, requ\u00e9rir l'ou\u00efe seule. Pas de commentaire ou de critique, pas de glose ou de th\u00e9orie, seulement une oreille bien dispos\u00e9e et d\u00e9sireuse d'\u00e9prouver du plaisir avec de belles compositions. En ce sens, loin des sp\u00e9culations de Sch\u00f4nberg, Debussy revendique une option intenable, car aucune oreille ne tient seule, dans le vide, ni ne fonctionne sans un minimum de cerveau \u00e0 disposition. Un son sans conscience serait ruine de l'\u00e2me, bruit.\n\nL'ou\u00efe est un sens qui, comme tous les autres, dispose d'une m\u00e9moire, d'une culture, d'un savoir inducteurs de potentialit\u00e9s et de possibilit\u00e9s. Une \u00e9coute suppose un travail intellectuel, un mode de penser impossible \u00e0 dissocier radicalement de l'op\u00e9ration mentale qu'est toujours une sensation si l'on souhaite sa transformation en perception. Et la sensation ne suffit pas \u00e0 la musique qui exige des perceptions. Debussy veut une \u00e9coute divertissante qui conduise \u00e0 un pays chim\u00e9rique, introuvable, et dispense de se souvenir qu'un \u00eatre, c'est avant tout une pr\u00e9sence au monde dans sa totalit\u00e9, pas seulement sur le mode sonore.\n\nLe mot qu'il n'aime pas pour dire la musique, il ne l'aime pas non plus pour dire le monde. Finalement, Debussy semble un nominaliste qui se d\u00e9fie du signifiant avec pour seul souci de substituer du son puissant et efficace au verbe impuissant et limit\u00e9. L\u00e0 o\u00f9 se termine la pertinence du mot commence le pouvoir de la musique. D'o\u00f9 l'impossibilit\u00e9 radicale, pour le compositeur, de permettre \u00e0 l'un de dire l'autre \u2013 ce qui n\u00e9cessite des am\u00e9nagements avec la m\u00e9taphysique quand il faut songer \u00e0 un livret pour une musique et \u00e0 une musique pour un livret.\n\nL\u00e0 o\u00f9 le verbe d\u00e9faille, impuissant, ridicule, contraint, Debussy jubile en composant et en formulant l'hypoth\u00e8se d'un relais sonore musical pour conjurer les d\u00e9faillances langagi\u00e8res. L'impuissance du mot lib\u00e8re un plaisir \u00e0 musiquer. D'o\u00f9 une proximit\u00e9 avec Mallarm\u00e9 obs\u00e9d\u00e9 par l'ellipse, la m\u00e9taphore, l'abord, le contour, tout ce qui permet d'utiliser encore les mots, mais pour \u00e9viter d'autres mots, ceux de la tribu. Debussy cong\u00e9die le verbe et veut une m\u00e9taphysique du son pur, du seul son pour dire l'indicible, pour tenter d'approcher ce qui se refuse aux logorrh\u00e9es et \u00e0 la dialectique, \u00e0 la rh\u00e9torique et aux phrases.\n\nDe sorte que le musicien formule en acte, sans le secours d'une quelconque th\u00e9orie verbalis\u00e9e, bien s\u00fbr, une m\u00e9taphysique de la musique parente de celles que proposent Schopenhauer et Nietzsche, alors d\u00e9couverts et \u00e0 la mode en France : la musique n'exprime pas des sentiments, elle n'est pas r\u00e9v\u00e9lation sensible, sur le mode platonicien, d'une r\u00e9alit\u00e9 intelligible, pas plus qu'elle ne fournit d'instrument pour dire le r\u00e9el, elle est ce r\u00e9el, elle s'y confond, car l'un et l'autre formulent diff\u00e9remment une m\u00eame apparition du monde, ici selon des modalit\u00e9s sp\u00e9cifiquement sonores. Elle n'exprime pas la vie, elle est la vie.\n\nLa musique propose donc la modification sonore de la seule substance qui soit : le r\u00e9el. Pas besoin, donc, pour l'\u00e9crire, de songer \u00e0 des chiffres, des nombres, des combinaisons math\u00e9matiques pour appr\u00e9hender au plus juste, au plus proche, le r\u00e9el qu'on souhaite \u00e9clairer. Il suffit d'\u00e9lire ce que les mots ne diront pas \u00e0 l'intelligence et que seules les oreilles entendront pour informer la sensibilit\u00e9, puisque le compositeur croit \u00e0 la dissociation des deux registres. D'o\u00f9 une phrase embl\u00e9matique sur la musique de son temps et celle qui suit : \u00ab On cherche trop \u00e0 \u00e9crire, on fait de la musique pour le papier alors qu'elle est faite pour les oreilles. \u00bb Il faut ici se souvenir d'un Mallarm\u00e9 \u00e9crivant : \u00abla po\u00e9sie n'est pas faite avec des id\u00e9es mais avec des mots \u00bb pour induire ce qu'aurait pu \u00e9crire Debussy dans son registre : la musique n'est pas faite avec des id\u00e9es mais avec des sons. Les mots m\u00e9pris\u00e9s peuvent alors rejoindre les id\u00e9es, elles aussi vilipend\u00e9es, au profit d'un hypoth\u00e9tique son qui n'entretiendrait aucune relation avec le langage ou avec le sens. Cette religion du sonore pur suppose chez son proph\u00e8te \u2013 qui en ce sens va plus loin que Sch\u00f6nberg qui associe sans cesse le son, le sens, le verbe, l'id\u00e9e et promeut une musique impure, au sens \u00e9tymologique \u2013 une mystique associ\u00e9e dont la pratique est l'\u00e9coute.\n\nBien s\u00fbr, cette religion sonore debussyste a fait l'objet d'approches et de qualifications en son temps, et nombreux se sont battus, alors, pour savoir si elle relevait du dogme impressionniste ou des encycliques symbolistes. Monet pour les couleurs et les cama\u00efeux ? Moreau pour les symboles et les signes ? Debussy avait r\u00e9pondu, d\u00e9j\u00e0, \u00e0 l'\u00e9poque, que ces querelles n'\u00e9taient que de mots, tout justes bonnes \u00e0 satisfaire les journalistes, les barbouilleurs de revue, soucieux d'user de termes commodes pour discr\u00e9diter la nature profonde d'un travail et m\u00e9priser l'auteur. Or Debussy impressionniste ou symboliste vaut tout autant que Debussy familier du style pictural troubadour, sinon nazar\u00e9en ou pr\u00e9rapha\u00e9lite. Et \u00e0 ce jeu, on pourrait tout autant songer \u00e0 Jean-Louis Ducis, Friedrich Overbeck ou Dante Gabriel Rossetti. De religion, il fallut faire son deuil car Debussy n'\u00e9tait d'aucune mani\u00e8re un sectaire des pratiques esth\u00e9tiques communautaires de son temps, trop individu pour cela. Il rel\u00e8ve moins d'une religion r\u00e9v\u00e9l\u00e9e que d'une th\u00e9ologie.\n\nCette m\u00e9taphysique au service d'une mystique, sinon l'inverse, a son thurif\u00e9raire en Vladimir Jank\u00e9l\u00e9vitch qui en assure les formules. Il faut bien ce sp\u00e9cialiste en th\u00e9ologie n\u00e9gative, ce digne \u00e9mule de la scolastique du Pseudo-Denys l'Ar\u00e9opagite, pour promouvoir un langage destin\u00e9 \u00e0 dire l'impuissance du langage, pour proposer une pens\u00e9e vou\u00e9e \u00e0 l'expression de l'inefficacit\u00e9 de la pens\u00e9e. Car Jank\u00e9l\u00e9vitch vaut moins en philosophe qu'en th\u00e9ologien si le premier d\u00e9finit celui qui aime et c\u00e9l\u00e8bre les lumi\u00e8res crues et nettes du savoir, puis fait confiance au pouvoir des mots et de la raison, l\u00e0 o\u00f9 le second v\u00e9n\u00e8re et cultive les obscurit\u00e9s, les ombres, les flous, les impr\u00e9cisions relevant des registres irrationnels. Le th\u00e9ologien jouit des fum\u00e9es et des brouillards, jubile des trembl\u00e9s et des myst\u00e8res, il aime les brumes, et l'\u00e9clairage philosophique lui semble trop brutal, trop douloureux pour les v\u00e9rit\u00e9s et certitudes qu'il met en \u00e9vidence, car elles paraissent souvent, voire toujours, en contradiction avec le clinquant mystique assurant la permanence de son commerce th\u00e9ologique.\n\nA quoi ressemble l'appareil conceptuel de cette th\u00e9ologie n\u00e9gative au service de la musique de Debussy ? On trouve en elle des expressions \u00e9l\u00e9gantes qui ne manquent pas de po\u00e9sie, certes, et avalisent comme essentielles et fondatrices les impuissances du verbe, du mot et du langage, mais toujours \u00e0 l'aide du verbe, des mots et du langage. Jank\u00e9l\u00e9vitch excelle dans l'art de l'oxymore en lieu et place de pens\u00e9e, d'o\u00f9 l'abondance de cr\u00e9atures hybrides tels la pr\u00e9sence lointaine ou l'absence pr\u00e9sente, l'espressivo inexpressif ou le pianissimo sonore, le mouvement immobile ou la v\u00e9locit\u00e9 stationnaire, ici la suggestion apr\u00e8s coup, l\u00e0 l'expression de l'inexprimable, ailleurs le s\u00e9rieux frivole, ou encore la superficie profonde, l'apparition disparaissante, le sommet \u00e9tant bien \u00e9videmment, saint du saint et s\u00e9pulcre \u00e0 lui tout seul, la musique silencieuse. De m\u00eame, en sus des expressions, on trouvera des notions, des pseudo-concepts, de fausses cat\u00e9gories tels les c\u00e9l\u00e8bres je-ne-sais-quoi et presque-rien, les r\u00e9jouissants myst\u00e8res et les troublants ineffables, les redoutables indicibles et les intempestifs quelque part. On aura compris : les scolastiques s\u00e9vissent en toutes \u00e9poques et il n'est pas que les Teutons pour en promouvoir de superbes.\n\nPourquoi donc th\u00e9ologie n\u00e9gative? Pour cet art savant et consomm\u00e9 de ne jamais rencontrer en face le noyau dur d'une probl\u00e9matique. Cette pr\u00e9f\u00e9rence av\u00e9r\u00e9e pour les bords, les marges, les contours, flous de pr\u00e9f\u00e9rence, suppose une pr\u00e9dilection pour les nuages, les brouillards, les parfums, les brises, les nuits, la neige, l'eau, la pluie, le sable, le r\u00eave, la lune, sinon les enfants. Autant dire l'ensemble de la grammaire et de la syntaxe th\u00e9matique du compositeur. Or, ce monde ne semble rep\u00e9rable qu'en p\u00e9riph\u00e9rie, en surface, comme des preuves sensibles de l'existence intelligible d'une instance plus centrale ou cardinale dans l'\u0153uvre du musicien. Et cet \u00e9picentre, ce sur quoi tient tout l'\u00e9difice musical debussyste, c'est le silence.\n\nDebussy l'avait lui-m\u00eame pr\u00e9cis\u00e9 dans une lettre \u00e0 Ernest Chausson : le silence donne l'occasion de l'\u00e9motion, en agent d'expression, il invite son correspondant \u00e0 ne pas rire de cet apparent paradoxe pour un musicien : construire et \u00e9difier sur ce qui para\u00eet le plus antinomique de la musique. Car on oppose la musique au silence, comme le jour \u00e0 la nuit, la vie \u00e0 la mort. Mais chacun voit combien l'un devient le moyen de l'autre et quel int\u00e9r\u00eat il y a pour Debussy \u00e0 s'installer juste aux franges, aux cr\u00e9puscules, aux aurores, aux naissances et aux tr\u00e9pas, aux premiers et aux derniers souffles. Le monde du musicien s'\u00e9panouit \u00e0 la crois\u00e9e des chemins, l\u00e0 o\u00f9 le verbe se tait et o\u00f9 la musique commence \u00e0 parler. En ce sens, son \u0153uvre se fait contemporaine des naissances du son, consubstantielle aux balbutiements musicaux. D'o\u00f9 une perp\u00e9tuelle volont\u00e9 d'en d\u00e9coudre avec le silence, de se mesurer \u00e0 lui, et \u00e0 lui seul. Dans cet ordre d'id\u00e9es, la lecture faite par Pierre Boulez de Pell\u00e9as et M\u00e9lisande comme d'un op\u00e9ra o\u00f9 triomphe le caract\u00e8re f\u00e9lin fournit une redoutable efficacit\u00e9 conceptuelle. L'oeuvre g\u00eet aux aguets, en arr\u00eat, embusqu\u00e9e avant le geste fatal.\n\nDans sa correspondance, Debussy rapporte les moments o\u00f9 il a plaisir \u00e0 travailler sur la partition : il signale les bruits inqui\u00e9tants faits par un brin d'herbe d\u00e9rang\u00e9 dans son sommeil, le mutisme de Pell\u00e9as et M\u00e9lisande lorsqu'ils d\u00e9couvrent par une grotte le monde tellurique et les entrailles de la terre. Ailleurs il soigne et cis\u00e8le le son de l'eau dormante dans la fontaine des aveugles, celui d'un baiser donn\u00e9 le long d'une chevelure, celui d'une respiration, d'un battement de c\u0153ur, ou d'une marche lointaine. Dans le ch\u00e2teau, on chuchote pour ne pas troubler la mort proche et les voix les plus assur\u00e9es viennent du bout du monde apr\u00e8s avoir vol\u00e9 et plan\u00e9 au-dessus des eaux. La raison de ce culte du silence install\u00e9 dans le ch\u0153ur de la cath\u00e9drale d\u00e9di\u00e9e \u00e0 la religion du son pur ? Arkel la donne : \u00ab L'\u00e2me humaine est tr\u00e8s silencieuse. \u00bb\n\nSur la question du silence, Jank\u00e9l\u00e9vitch disserte avec tout le brillant qu'on lui conna\u00eet : les degr\u00e9s de l'inanition et le non-\u00eatre virtuel, l'intramusical et la possibilit\u00e9 pour le silence d'\u00eatre primordial ou terminal, la transparence sans profondeur, puis le myst\u00e8re ineffable et diaphane. Pour t\u00e2cher d'approcher ce monde sans le dire, pour \u00e9viter le vacarme fait par le silence quand on cherche \u00e0 en p\u00e9n\u00e9trer le secret, il convoque l'haleine d'une jeune fille, la brise de Dieu ou les t\u00e9l\u00e9grammes d'amour port\u00e9s par ladite brise. Puis il encha\u00eene sur \u00ab la pr\u00e9sence de l'absence et l'absence pr\u00e9sente, l'existence inexistante et l'inexistence de l'existence, la pr\u00e9sence invisible de celui qui n'est pas l\u00e0\u00bb. Enfin, il propose un d\u00e9tour du c\u00f4t\u00e9 de la myst\u00e9riologie, de la th\u00e9osophie des ann\u00e9es 1880, du s\u00e9raphisme, des activit\u00e9s m\u00e9diumniques et de tout l'attirail d\u00e9cadentiste symboliste de l'\u00e9poque qui auraient passionn\u00e9 Debussy. Pour conclure, il reconvoque le myst\u00e8re et ajoute que sur lui, il n'y a presque rien \u00e0 dire. Sur ce presque-rien, il \u00e9crit deux livres.\n\nHeureusement, du silence on peut parler autrement que sur le terrain mystique de la th\u00e9ologie n\u00e9gative et sa pr\u00e9sence ne requiert pas qu'on y adh\u00e8re \u00e0 ce point qu'il submerge, an\u00e9antisse et condamne soit au mutisme, soit aux jubilations troubles de constater l'impuissance des logorrh\u00e9es. Que les po\u00e8tes, les th\u00e9ologiens, les mystiques, voire les musiciens, qui consacrent leur existence \u00e0 c\u00e9l\u00e9brer les modes d'acc\u00e8s irrationnels au monde reculent devant le c\u0153ur du silence, rien de plus normal.\n\nEn revanche, que les philosophes refusent d'en faire un objet de pens\u00e9e, voil\u00e0 qui peut troubler. On a pu dire que le silence des espaces infinis produisait tel ou tel type d'effet, l'associer \u00e0 la m\u00e9ditation, \u00e0 la r\u00e9flexion, comme une condition de possibilit\u00e9, le montrer comme proc\u00e9dant d'un renoncement au monde, voire d'un signe d'acc\u00e8s \u00e0 un autre univers, l'installer aux c\u00f4t\u00e9s du sage ou de l'initi\u00e9, chaque fois, on \u00e9vite de dire sa nature et sa d\u00e9finition. Si d'aventure il occupe bien le parfait milieu de la musique de Debussy, quel est-il ?\n\nOn pourrait dire qu'il s'\u00e9panouit dans les limbes de la dur\u00e9e indissociable de tout son, dans l'\u00e9picentre du temps d\u00e9velopp\u00e9, tapi dans l'indivisible, \u00e0 la mani\u00e8re dont l'instant immobile compose le d\u00e9roulement mobile, ou comme le vide structure en son sein la mati\u00e8re et les plus petites particules ins\u00e9cables. Paradoxalement, \u00e0 l'endroit du mouvement, on trouve l'immobile qui le rend possible ; au lieu de la mati\u00e8re, on rencontre le vide en garantie d'av\u00e8nement ; l\u00e0 o\u00f9 se d\u00e9ploie le son, on d\u00e9couvre le silence transfigur\u00e9 en occasion d'existence. Le silence rar\u00e9fie le son jusqu'\u00e0 l'absence \u00e9pingl\u00e9e au revers d'une m\u00e9daille dans le c\u0153ur m\u00eame du son : involution paradoxale, toute \u00e9volution post\u00e9rieure s'y nourrit. La m\u00e9canique quantique, avec ses \u00e9nergies minimales, puissantes et furieuses, fonctionne en m\u00e9taphore du silence.\n\nQuand Debussy veut une musique pure, il vise l'approche maximale de ce point, jusqu'au bord ultime au-del\u00e0 duquel le silence fait suite au son, comme pr\u00e9c\u00e9demment la musique suppl\u00e9ait les manques du verbe. De sorte que la m\u00e9lodie de Debussy se d\u00e9plie dans le temps \u00e0 la fa\u00e7on d'une dur\u00e9e qui cache en son sein le silence et s'en nourrit. Et si les abords religieux, mystiques et th\u00e9ologiques d\u00e9\u00e7oivent, concernant cette musique, on peut avancer l'id\u00e9e d'une lecture philosophique possible pour rendre compte de la sp\u00e9cificit\u00e9 de ces compositions hant\u00e9es par le silence.\n\nS'il faut un cic\u00e9rone pour conduire cette recherche, je veux \u00e9lire sans conteste Henri Bergson penseur d'un monde hant\u00e9 par le temps, la dur\u00e9e, la m\u00e9moire, l'\u00e9nergie spirituelle, les \u00e9tats de conscience, l'\u00e9volution cr\u00e9atrice, le mouvant, le souvenir, la dur\u00e9e, la simultan\u00e9it\u00e9, mais aussi l'id\u00e9e, l'image, le pr\u00e9sent, l'intuition, la continuit\u00e9, les formes, la r\u00e9alit\u00e9, les forces et tout un riche appareil conceptuel susceptible d'affectation \u00e0 une lecture coh\u00e9rente du monde. Gilles Deleuze a pu parler du bergsonisme, en faire la th\u00e9orie, puis exp\u00e9rimenter sa validit\u00e9 sur le terrain du cin\u00e9ma, et l'on imagine ce qu'aurait pu \u00eatre un ouvrage similaire sur la question de la musique.\n\nL'\u0153uvre de Bergson ne contient pas \u00e0 proprement parler une th\u00e9orie de la musique, encore moins du silence. M\u00eame si l'ensemble des travaux est constell\u00e9 de r\u00e9f\u00e9rences \u00e0 ce domaine, le philosophe se soucie de cet univers sp\u00e9cifique plut\u00f4t pour recourir \u00e0 des images ou des m\u00e9taphores. Pour autant, il ne cesse de dire que la philosophie a plus \u00e0 voir avec l'art qu'avec les sciences et, plus sp\u00e9cifiquement avec la musique. Pour quelles raisons ? Vraisemblablement parce que la musique r\u00e9v\u00e8le un monde qui, de conserve avec la philosophie, n\u00e9cessite la puissance et le recours de l'intuition plus que des op\u00e9rations de la d\u00e9duction. L\u00e0 o\u00f9 l'homme de raison d\u00e9duit, analyse, structure, le penseur, et le musicien se mettent \u00e0 l'\u00e9coute de vibrations fragiles, d'\u00e9motions et de sensations imperceptibles. Tous deux se nourrissent de variations et de modulations infinit\u00e9simales du r\u00e9el.\n\nDans un entretien avec la presse souvent pass\u00e9 inaper\u00e7u, Bergson r\u00e9pond \u00e0 quelques questions concernant la musique. L'article para\u00eet dans Paris-Journal le 11 d\u00e9cembre 1910 sur le principe d'un entretien avec Georges Amiel. Debussy entreprend alors l'\u00e9criture de la musique de sc\u00e8ne du Martyre de saint S\u00e9bastien, sur un texte de Gabriele D'Annunzio, et il vient d'achever La plus que lente, une \u0153uvre pour piano. Dans ces pages du journal, Bergson dit son go\u00fbt pour le symbolisme en g\u00e9n\u00e9ral et l'\u00e9criture de Maeterlinck en particulier. Il ajoute : \u00abOn m'a aussi signal\u00e9 combien la musique de M. Debussy et de son \u00e9cole est une musique de la dur\u00e9e, par l'emploi de la m\u00e9lodie continue qui accompagne et exprime le courant unique et ininterrompu de l'\u00e9motion dramatique. J'ai d'ailleurs une pr\u00e9dilection instinctive pour l'\u0153uvre de M. Debussy. Voil\u00e0. Nulle part ailleurs dans les traces laiss\u00e9es par le philosophe on ne trouve un quelconque approfondissement de cette id\u00e9e. Pas plus dans la longue correspondance du musicien o\u00f9 l'on cherchera en vain une mention du nom seul du prix Nobel fran\u00e7ais.\n\nBergson en qu\u00eate d'\u00e9lucidation de la nature du temps rencontre la dur\u00e9e et t\u00e2che d'en p\u00e9n\u00e9trer le m\u00e9canisme. De la m\u00eame mani\u00e8re, en s'inqui\u00e9tant de la sp\u00e9cificit\u00e9 de la musique du compositeur, on achoppe immanquablement sur la question du silence. Aussi peut-on oser l'\u00e9quivalence et affirmer que la dur\u00e9e est au temps ce que le silence est \u00e0 la musique : une condition de possibilit\u00e9, presque une forme a priori de la sensibilit\u00e9 musicale. Que dit Bergson du temps? Qu'il est \u00able fant\u00f4me de l'espace obs\u00e9dant la conscience r\u00e9fl\u00e9chie \u00bb. Je dirais volontiers de cette musique qu'elle d\u00e9plie m\u00eamement le fant\u00f4me du silence obs\u00e9dant la conscience r\u00e9fl\u00e9chie.\n\nEt de la dur\u00e9e ? Le philosophe pr\u00e9cise qu'elle est \u00ab la forme que prend la succession de nos \u00e9tats de conscience quand notre moi se laisse vivre, quand il s'abstient d'\u00e9tablir une s\u00e9paration entre l'\u00e9tat pr\u00e9sent et les \u00e9tats ant\u00e9rieurs. En projetant du temps dans l'espace, nous obtenons une figuration de la dur\u00e9e sur le seul mode de l'entendre \u00bb. Peut-on mieux d\u00e9finir la musique, toute musique? L'art de projeter du temps dans l'espace ? Aussi, pour en revenir plus sp\u00e9cifiquement \u00e0 Debussy, on pourrait avancer, selon le principe \u00e9nonc\u00e9 dans l'Essai sur les donn\u00e9es imm\u00e9diates de la conscience, que sa musique excelle dans l'art d'injecter dans ce temps puls\u00e9, d\u00e9velopp\u00e9, d\u00e9pli\u00e9, une dose maximale de silence, la dose la plus tenable \u00e0 cette \u00e9poque avant d\u00e9s\u00e9quilibre du c\u00f4t\u00e9 du mutisme.\n\nQuand le philosophe du Coll\u00e8ge de France analyse les op\u00e9rations qui conduisent \u00e0 la construction d'un chiffre, je ne peux \u00e9galement m'emp\u00eacher de penser \u00e0 celles qui pr\u00e9sident \u00e0 l'\u00e9laboration d'un son : l'intuition d'un milieu homog\u00e8ne o\u00f9 peuvent se d\u00e9ployer, distincts, des termes pr\u00e9cis, puis l'organisation dynamique de ces termes dans l'optique d'une multiplicit\u00e9 qualitative. Dans le temps, elles s'ajoutent, dans l'espace, elles se distinguent. Avan\u00e7ons l'hypoth\u00e8se que le silence s'installe dans l'interstice, entre les dur\u00e9es temporelles et dans l'\u00e9tendue des intervalles. Soit cette progression dans la r\u00e9flexion sur la nature du silence : il g\u00eet dans l'\u00e9picentre de la dur\u00e9e, certes, mais aussi dans ce qui permet la jonction avec ce qui n'est pas lui, le son. Debussy op\u00e8re en d\u00e9miurge de ce perp\u00e9tuel arrangement avec le silence : assez pour conjurer la musique, jamais trop pour \u00e9viter qu'elle sombre, inutile, comme les mots avant elle. Au point de jonction, \u00e0 l'endroit m\u00eame de la suture, il joue \u00e0 vaporiser le son, comme Baudelaire vaporisait le moi pour r\u00e9aliser la chimie du dandysme.\n\nAfin de saisir la complexit\u00e9 et le mouvant qui habitent et hantent la philosophie de Bergson, il faut s'arr\u00eater sur un concept majeur dans son \u0153uvre : celui de d\u00e9roulement. A premi\u00e8re vue, le temps et la dur\u00e9e ne semblent pouvoir s'offrir \u00e0 la conscience que sur le mode spatial, selon le principe de la succession. De sorte que l'instant n'appara\u00eet et ne trouve son identit\u00e9 que pr\u00e9c\u00e9d\u00e9 par un avant, suivi par un apr\u00e8s, le tout install\u00e9 dans un d\u00e9veloppement lin\u00e9aire. Depuis saint Augustin, chacun, en Occident, sacrifie \u00e0 cette lecture. Appr\u00e9hender le temps prioritairement sur le mode du d\u00e9ploiement dans l'espace suppose le souci d'un seul moi ext\u00e9rieur qui se donne \u00e0 voir dans les trois dimensions.\n\nPour autant, on peut \u00e9galement \u00eatre attentif aux lois d'un moi int\u00e9rieur pour lequel l'aperception se fait non plus sur le mode de l'espace, mais sur celui de la dur\u00e9e pure. Le musicien s'adresse \u00e0 ce moi, et plus particuli\u00e8rement Debussy, dans un langage qui donne au silence les moyens et les occasions de retrouvailles avec l'essence de soi. La musique appara\u00eet telle une voie d'acc\u00e8s \u00e0 la vie intime, magnifi\u00e9e dans la co\u00efncidence de l'action et de l'\u00e9coute. La simultan\u00e9it\u00e9, la dur\u00e9e, la qualit\u00e9 prennent le pas sur la succession, l'\u00e9tendue et la quantit\u00e9. Ainsi, le temps projet\u00e9 dans l'espace, pour le moi ext\u00e9rieur, peut \u00eatre saisi par le moi int\u00e9rieur et devenir une dur\u00e9e \u00e9prouv\u00e9e. Les ponctuations de celle-ci d\u00e9finissent le silence.\n\nL'ensemble de l'\u0153uvre de Bergson fournit des moyens de rentrer plus encore dans la musique de Debussy, moins musique du myst\u00e8re que m\u00e9canique de la dur\u00e9e, moins cr\u00e9ation destin\u00e9e \u00e0 c\u00e9l\u00e9brer l'ineffable et l'indicible que proposition pour saisir l'essence du temps et de la musique, sinon du temps musical \u2013 \u00e0 savoir le silence comme contrepoint n\u00e9cessaire au son. Cette volont\u00e9 de composer une musique qui soit le r\u00e9el, d'\u00e9crire musicalement l'essence du monde telle qu'il l'entend, installe Debussy dans l'obligation de tenir compte de la nature du temps, des frasques de la dur\u00e9e, de ses modalit\u00e9s d'exposition et de d\u00e9veloppement, afin de proposer une \u0153uvre dont la m\u00e9lodie se soumet \u00e0 l'imp\u00e9ratif lin\u00e9aire auquel on n'\u00e9chappe pas. A ceux qui reprochaient une absence de m\u00e9lodie dans Pell\u00e9as, Debussy r\u00e9torquait que l'op\u00e9ra n'\u00e9tait que m\u00e9lodie. \u00abLa m\u00e9lodie, ajoutait-il, est ininterrompue, sans nulle tr\u00eave, car elle vise \u00e0 reproduire la vie m\u00eame (...). Il n'y a pas de chant dans la vie : il y a des rythmes, des atmosph\u00e8res, des couleurs, mais ceux-ci, bien que variant sans cesse, se succ\u00e8dent sans solution de continuit\u00e9, pour l'\u00e9ternit\u00e9. \u00bb La variation, dans le temps, la succession, dans l'espace, et l'\u00e9ternit\u00e9, dans le silence, voil\u00e0 mati\u00e8re \u00e0 aborder les rives de l'\u0153uvre debussyste.\n\nLa pr\u00e9f\u00e9rence pour une m\u00e9lodie lin\u00e9aire exclut donc la volont\u00e9 de produire une trame. Le musicien s'en explique dans une lettre \u00e0 Edwin Ewans. Opter pour une trame donne \u00e0 entendre un monde complexe, \u00e9labor\u00e9, reconstruit par une op\u00e9ration de l'esprit. Ce que ne veut pas Debussy. La co\u00efncidence du monde avec sa musique, et vice versa, suppose une simplicit\u00e9, du moins un registre unique -un fil plus qu'une trame. Le temps tel qu'il se laisse comprendre, ne peut supposer, comme chez Wagner, la simultan\u00e9it\u00e9 ou l'ubiquit\u00e9 : les personnages wagn\u00e9riens expriment en m\u00eame temps leurs sentiments et les r\u00e9flexions sur les raisons qui les font agir. Debussy ne croit pas \u00e0 la simultan\u00e9it\u00e9, mais \u00e0 la succession \u2013 comme dans le registre du temps r\u00e9el. D'o\u00f9 une revendication clairement affich\u00e9e musicalement, du d\u00e9roulement et de sa simplicit\u00e9 radicale.\n\nOr la forme prise par le d\u00e9roulement dans le monde, c'est le devenir, ce que d'aucuns appellent le destin. Et le silence vaut marque de ce devenir, il en est l'identit\u00e9. Inconnu, pas m\u00eame voil\u00e9, car n'\u00e9tant pas encore, mais advenant seulement, le devenir suppose une r\u00e9v\u00e9lation proc\u00e9dant d'un \u00e9tat qu'au mieux le silence caract\u00e9rise. L'\u00e9ternel pr\u00e9sent est le lieu dans lequel il s'incarne, c'est aussi la modalit\u00e9 debussyste par excellence. Au point d'attente, avant r\u00e9v\u00e9lation, incarnation, av\u00e8nement, juste au seuil du devenir, il n'existe qu'un grand, immense et assourdissant silence.\n\nQuelles issues pour un Debussy hant\u00e9 par le silence pour l'exprimer quintessenci\u00e9 dans un op\u00e9ra o\u00f9 les mots, quoi qu'on fasse, quoi qu'on dise, jouent un r\u00f4le majeur, sinon architectonique ? Lui qui fustigeait le verbe, comment a-t-il pu vouloir composer avec lui, sinon pour lui? Ses m\u00e9lodies et cantates furent l'occasion de musiquer plus d'une vingtaine de po\u00e8tes, des meilleurs aux pires, de Mallarm\u00e9 \u00e0 Paul Bourget. Il y eut des tentatives et des tentations pour l'op\u00e9ra, certes, mais un seul, on le sait, aboutit : Pell\u00e9as et M\u00e9lisande. Pour quelles raisons ? Vraisemblablement parce que Maeterlinck, le librettiste, est aux mots ce que Debussy est \u00e0 la musique : un d\u00e9vot du silence, un amateur de longueurs blanches entre deux demi-teintes.\n\nFid\u00e8le \u00e0 sa conception lin\u00e9aire du temps et de la dur\u00e9e, le compositeur appelait de ses v\u0153ux un talentueux qui saurait \u00abtrouver l'interview en musique \u00bb, autant dire quelqu'un qui r\u00e9ussirait \u00e0 rester au plus proche de la conversation normale, loin des amplitudes habituelles du bel canto. Maurice Maeterlinck lui donna l'impression d'\u00eatre cet homme qui, disant les choses \u00e0 demi, lui permettrait d'exploiter ce demi-silence pour y greffer sa musique et l'en nourrir. Du jeu entre les mots et les sons, le musicien produira une forme dont je me plais \u00e0 penser que, loin de toutes les trames, au plus proche de la simple variation sur le fil, elle emprunte \u00e0 l'art complexe et subtil de la dentelle.\n\nQu'est-ce, en effet, que la dentelle ? L'art de conduire un fil, un simple et modeste fil, dans un entrelacs suffisamment \u00e9crit pour qu'en naisse un motif tout entier construit sur des trous, des blancs, du vide, de l'espace, et autres modalit\u00e9s m\u00e9taphoriques et spatiales du silence. Au c\u0153ur m\u00eame de la dentelle, on peut lire une multitude d'alv\u00e9oles dont l'agencement permet le surgissement d'architectures, de g\u00e9om\u00e9tries et de formes. Le fil de la dentelle debussyste est la m\u00e9lodie, fragile et t\u00e9nue. L'\u0153uvre appara\u00eet comme une broderie qui tient son identit\u00e9 de cette m\u00eame fragilit\u00e9 et de cette m\u00eame t\u00e9nuit\u00e9. La pi\u00e8ce de Maeterlinck semble elle aussi une dentelle, autant sur le terrain de l'\u00e9criture que de l'ambiance dans laquelle elle installe \u2013 annon\u00e7ant Beckett pour qui lirait sans pr\u00e9jug\u00e9s. Rappelons que le dramaturge belge plaisait \u00e0 Bergson vraisemblablement pour ces raisons.\n\nLa compl\u00e9mentarit\u00e9 du librettiste et du musicien se fait m\u00eamement en mati\u00e8re de vision du monde. Maeterlinck, symboliste pour ceux qui aiment les \u00e9tiquettes faciles, a aussi et surtout \u00e9crit en 1898 un ouvrage intitul\u00e9 La Sagesse et la Destin\u00e9e dans lequel il synth\u00e9tise des lectures mystiques, Ruysbroek, po\u00e9tiques, Novalis, ou philosophiques, Emerson. L'ensemble est plac\u00e9 sous le patronage tr\u00e8s convenu \u00e0 l'\u00e9poque d'un Schopenhauer dont on d\u00e9couvre l'existence et qui fait des ravages dans les milieux symbolistes. Debussy raconte dans une lettre \u00e0 Robert Godet dat\u00e9e du 25 d\u00e9cembre 1898 qu'il a assist\u00e9 au caf\u00e9 Vachette \u00e0 une empoignade entre Jean Mor\u00e9as et Henri Mercier, un ami de Rimbaud, qui avait Schopenhauer pour enjeu et objet.\n\nLe livret de Pell\u00e9as et M\u00e9lisande est constell\u00e9 de r\u00e9f\u00e9rences, sinon au penseur pessimiste, du moins au schopenhau\u00e9risme de l'\u00e9poque. Le philosophe allemand affirmait, comme Debussy, que la musique \u00e9tait le monde, qu'elle formulait dans son mode sonore l'essence du r\u00e9el, \u00e0 savoir le vouloir. Dans l'op\u00e9ra se r\u00e9p\u00e8tent une douzaine de leitmotive \u2013 malgr\u00e9 le mal que Debussy a prof\u00e9r\u00e9 sur cette fa\u00e7on de proc\u00e9der \u2013 dans lesquels on retrouve une th\u00e9matique proche de celle qui appara\u00eet dans Le Monde comme volont\u00e9 et comme repr\u00e9sentation : l'Anneau pour dire la tyrannie de la destin\u00e9e et l'\u00e9ternel retour du m\u00eame, le Destin et l'Enfant comme occasions d'exprimer l'absence de libre arbitre et l' innocence du devenir, le Pardon telle une fa\u00e7on de signifier le r\u00f4le m\u00e9taphysique et \u00e9thique de la piti\u00e9, l'Eveil et l'Accomplissement de l'Amour, enfin, pour caract\u00e9riser l'\u00e9nergie et la puissance du vouloir-vivre.\n\nUn seul personnage \u00e9chappe \u00e0 la tyrannie du devenir : Arkel, pour la raison qu'il est un vieillard. Non que la sagesse accompagne de fait les hommes arriv\u00e9s \u00e0 l'extr\u00e9mit\u00e9 de leur existence, mais parce qu'ils sont lib\u00e9r\u00e9s de l'emprise et de l'empire du vouloir dans ses modalit\u00e9s g\u00e9n\u00e9siques. Enfin devenus vieux, leur destin est presque accompli, leur devenir se r\u00e9duit \u00e0 rien. Aussi savent-ils la v\u00e9rit\u00e9 de la nature cruelle du r\u00e9el et du monde. Loin d'\u00eatre un drame de l'adult\u00e8re bourgeois, comme le pensent les bourgeois eux-m\u00eames, fid\u00e8les en cela aux projections qui les rassurent et les dispensent de r\u00e9fl\u00e9chir, Pell\u00e9as incarne par excellence l'op\u00e9ra tragique qui met en sc\u00e8ne ce que dans une formule de Nietzsche on pourrait appeler l'innocence du devenir.\n\nNon loin d'un Pierre Boulez qui voyait dans l' \u0153uvre une illustration du Th\u00e9\u00e2tre de la cruaut\u00e9, il faut lire l'op\u00e9ra comme une catharsis, au sens que lui donne Aristote, qui permet d'assister \u00e0 la repr\u00e9sentation spectaculaire d'un destin, le n\u00f4tre, mais qu'on voit transfigur\u00e9, travaillant les autres : un homme, une femme, un fr\u00e8re, un enfant, un vieillard, son \u00e9pouse. Et tous les ingr\u00e9dients qui font l'\u00e9ternel retour des choses : l'amour, l'inceste, l'adult\u00e8re, la jalousie, le mensonge, la passion, la mort, la violence, l'angoisse, la peur, la solitude, le solipsisme. Ces \u00eatres sont des masques sous lesquels nous sommes, ces passions, les n\u00f4tres v\u00e9cues, endoss\u00e9es par les autres avant qu'elles ne redeviennent nos d\u00e9froques. Arkel confesse avoir acquis une foi dans le cours des \u00e9v\u00e9nements, il suppose que tout ce qui advient proc\u00e8de d'une n\u00e9cessit\u00e9 et d\u00e9fend \u00e0 chacun, bourreau ou victime, Golaud ou M\u00e9lisande, de croire qu'il a d\u00e9sir\u00e9, choisi et voulu ce qui est arriv\u00e9. Pas plus l'enfant n\u00e9e du myst\u00e8re, d'un p\u00e8re l\u00e9gal, mais sans amour, ou d'un amoureux, mais sans l\u00e9gitimit\u00e9. Et quand tout se sera effondr\u00e9, que la mort aura ravag\u00e9 et laiss\u00e9 orphelins deux enfants, Yniold et cette petite dont on ignore jusqu'au nom, Arkel dira qu'ils doivent vivre. A leur tour, maintenant, de subir et d'assister en spectateur au d\u00e9ploiement de leur destin.\n\nDe sorte que l'op\u00e9ra peut se voir et s'entendre comme la mise en sc\u00e8ne, ludique, d'un autre spectacle, tragique celui-ci : celui de la condition humaine. Dans l'\u00e9cart susceptible d'\u00eatre enregistr\u00e9 entre ces deux instances, on trouve, en compagnie du silence, les lin\u00e9aments d'une catharsis en bonne et due forme, telle une illustration des th\u00e8ses aristot\u00e9liciennes : voir ce qui ne se regarde pas en face s'il n'est m\u00e9diatis\u00e9 par une \u0153uvre. L'art permet de supporter une dose de tragique insupportable sans elle, on se souvient d'Arkel appelant de ses v\u0153ux la beaut\u00e9 d'autant plus ardemment qu'il se trouve proche de la mort.\n\nO\u00f9 l'on retrouve le silence en musique comme occasion de faire entendre le d\u00e9roulement du devenir et l'innocence qui accompagne ce d\u00e9ploiement. \u00abSi j'\u00e9tais Dieu, se lamente Arkel, j'aurais piti\u00e9 du c\u0153ur des hommes.\u00bb R\u00e9miniscences schopenhau\u00e9riennes et Th\u00e9\u00e2tre de la cruaut\u00e9, l\u00e0 aussi, dans le spectacle de cette douleur qui frappe la plan\u00e8te \u00e0 la fa\u00e7on de la foudre ou des com\u00e8tes : une immense tra\u00een\u00e9e de feu et de lumi\u00e8re, l'incendie, l'apocalypse, des brasiers, des incendies partout, puis le silence qui troue le vide et dure, remplissant le temps jusqu'aux limites o\u00f9 s'\u00e9nonce l'\u00e9ternit\u00e9 comme un genre d'\u00eatre proc\u00e9dant du n\u00e9ant. Apr\u00e8s les dur\u00e9es fragiles de Debussy demeur\u00e9es sans \u00e9cho, tombe, ainsi qu'une nuit, l'\u00e9ternit\u00e9 d'un silence qui, lui, ne nourrira aucune musique.\n18\n\nQU'IL FAUT \u00c9TRANGLER LA MORT\n\n(Lettre \u00e0 Jean-Claude Camus pour tenter de le consoler) \u2013. Bien Cher Ami, J'ai sous les yeux le paquet de lettres re\u00e7ues de vous depuis six ann\u00e9es. Toutes elles font sur mon bureau un petit \u00e9difice au format du pliage en quatre de chacune de vos missives. Dans cette centaine d'envois, cent treize tr\u00e8s exactement, se trouvent en grande partie votre existence et la mienne crois\u00e9es, des peines et des douleurs, des joies et des plaisirs, des silences aussi, des confidences et des chuchotements adress\u00e9s \u00e0 une oreille aim\u00e9e en attendant un mot de consolation, un geste de compassion ou n'importe quoi nous permettant un condouloir \u00e0 deux. J'ai retrouv\u00e9 aussi des traits, des fus\u00e9es, des plaisanteries, des clins d'\u0153il de vous \u00e0 moi, invisibles au tiers qui chercherait sens entre les mots, entre les phrases.\n\nJe ne sais plus ce que je vous ai \u00e9crit, ce que vous avez re\u00e7u, ce que j'ai envoy\u00e9. Un long trait modul\u00e9 comme une m\u00e9lodie baroque, avec ses volutes, ses grotesques, ses d\u00e9pliages, ses d\u00e9ploiements et ses r\u00e9it\u00e9rations. Une couleur et ses tonalit\u00e9s, un m\u00e9lange de le\u00e7on de t\u00e9n\u00e8bres et de monodies sur l'inconstance et la vanit\u00e9 du monde, une fresque en forme de pi\u00e8ce musicale o\u00f9 se c\u00f4toieraient tombeaux, regrets et danses joyeuses. De la m\u00e9lancolie, souvent, s\u00fbrement, des confidences, tout le temps. Mais je n'en sais plus la teneur, d'autant qu'en vous relisant, je n'imaginais pas que nous nous soyons autant racont\u00e9s. Et je suis heureux de cette nudit\u00e9 \u00e9chang\u00e9e \u00e0 la mani\u00e8re de deux amis romains trait\u00e9s dans le marbre apr\u00e8s avoir \u00e9t\u00e9 familiers des bruissements et des vibrations du forum.\n\nPlus d'une centaine de lettres, donc, o\u00f9 vous m'avez racont\u00e9 l'Alg\u00e9rie et le m\u00e9tier d'instituteur, votre fils et vos lectures, des d\u00e9m\u00e9nagements et la retraite, votre m\u00e8re et vos proches, le vin de Reuilly et les livres, le tabac fum\u00e9 \u00e0 la pipe et quelques conversations avec des amis, le plaisir des journaux et la place accord\u00e9e \u00e0 mes livres dans l'\u00e9conomie de vos affinit\u00e9s \u00e9lectives. Et puis Anne, votre fille, morte le 20 mai 1986 dans un accident o\u00f9 vous avez p\u00e9ri vous aussi, me semble-t-il. Dans cette b\u00e9ance ouverte sous vos pieds, pareil aux grands seigneurs de jadis, vous avez tout jet\u00e9 de votre vie. Dans l'ab\u00eeme, vous avez \u00e9galement ajout\u00e9 ce qui pourrait encore faire votre existence, apr\u00e8s.\n\nDepuis, vous attendez la mort qui, vous le savez, ne vous permettra pas de la rejoindre, car l\u00e0 plus qu'ailleurs, on boit la solitude jusqu'\u00e0 la lie et l'on ne rejoint personne. Tout juste peut-on permettre \u00e0 un squelette de se m\u00e9langer \u00e0 un autre squelette, ou \u00e0 des cendres de ne faire plus qu'une seule poudre. Mais les retrouvailles de charniers n'en sont pas. Je sais que vous et moi partageons un irr\u00e9ductible ath\u00e9isme conjurateur des hypoth\u00e8ses id\u00e9alistes, spiritualistes ou religieuses. Notre mat\u00e9rialisme, \u00e0 tous les deux, nous oblige \u00e0 \u00e9viter les compromissions l\u00e0 o\u00f9 pourrait appara\u00eetre la tentation d'un mensonge apaisant. Reste la consolation philosophique, \u00e0 l'exclusion de toute autre. Et je veux, aujourd'hui, vous adresser quelques pages qui offriraient le pendant des rh\u00e9toriques sto\u00efciennes antiques avec lesquelles on proposait aux gens aim\u00e9s des arguments pour qu' au-del\u00e0 de la douleur qui les frappe, ils aient encore envie de vivre, malgr\u00e9 tout.\n\nJe suis arriv\u00e9 \u00e0 la philosophie par l'Antiquit\u00e9, comme vous le savez. Voie royale s'il en est une, car en mati\u00e8re de pens\u00e9e, il vaut mieux commencer dans la lumi\u00e8re du forum romain et des arcs de Tib\u00e8re ou de Titus que dans les quartiers des villes chauff\u00e9es de ressentiment o\u00f9 se font les r\u00e9putations d'aujourd'hui. Mon vieux ma\u00eetre, Lucien Jerphagnon, me rendait les prostitu\u00e9es des quartiers chauds, les embarras de Rome ou les d\u00e9bats du S\u00e9nat, plus aimables que toutes les vacations farcesques contemporaines. Et Lucr\u00e8ce fut pour moi une illumination, lui qui tend la totalit\u00e9 de sa philosophie comme un arc en direction de la consolation. Car toute pens\u00e9e digne de ce nom est proposition de compassion. En dehors de quoi, elle ne m\u00e9rite pas une seconde de peine.\n\nEn lisant ou relisant, pour cette lettre que je vous destine, les consolations de Cic\u00e9ron, Plutarque ou S\u00e9n\u00e8que, j'ai mesur\u00e9 tout ce que le christianisme leur devait : le renoncement, la haine de la vie c\u00e9l\u00e9br\u00e9e en occasion de faire para\u00eetre la mort plus p\u00e2le, moins douloureuse, la n\u00e9cessit\u00e9 de se soumettre au destin et aux coups du sort, le subterfuge d'un arri\u00e8re-monde peupl\u00e9 d'\u00e2mes lib\u00e9r\u00e9es, l'obligation de faire avec les cycles et les alternances qui r\u00e8glent autant le cours des existences singuli\u00e8res que celui des plan\u00e8tes. Mais rien de tout cela ne m'a convaincu car tout semble chez eux effet de rh\u00e9torique et pr\u00e9texte \u00e0 exposition de dext\u00e9rit\u00e9 philosophique.\n\nCombien de talent d\u00e9pens\u00e9 par ces philosophes de l'id\u00e9al asc\u00e9tique, pour honnir l'existence, accus\u00e9e d'\u00eatre un r\u00e9ceptacle \u00e0 tous les maux, afin de faire para\u00eetre douce la mort qui lib\u00e9rerait d'avoir \u00e0 souffrir plus longtemps? Quel temps perdu pour conduire les imp\u00e9trants \u00e0 la consolation de l'existence d'un inconv\u00e9nient d'\u00eatre n\u00e9, que la mort est pr\u00e9f\u00e9rable \u00e0 la vie, que le r\u00e9el n'offre que des occasions de souffrance ! Que d'ardeur \u00e0 c\u00e9l\u00e9brer le nihilisme, \u00e0 faire d\u00e9sirer la mort ici et maintenant comme seule fa\u00e7on d'apprendre \u00e0 mourir demain ! Que de rh\u00e9torique pour nous persuader que l'individu ne vaut rien, que l'esp\u00e8ce est tout, et qu'il faut se r\u00e9jouir d'\u00eatre maillon dans l'\u00e9conomie d'une cha\u00eene !\n\nJe ne veux plus souscrire au versant nocturne de la philosophie antique, ni \u00e0 une autre qui soit complaisante \u00e0 l'endroit du pire. La p\u00e9dagogie du d\u00e9sespoir, les vertus du n\u00e9gatif, la m\u00e9lancolie amoureuse \u00e0 l'endroit de la mort, la m\u00e9taphysique des m\u00e9tempsycoses, le recyclage tardif et romain de la plupart des th\u00e8ses platoniciennes du Ph\u00e9don, la consid\u00e9ration du corps comme un tombeau de l'\u00e2me que lib\u00e9rerait Thanatos, tout cela ne me convainc pas. Et je crois savoir que vous n'y trouvez pas plus de plaisir ou de satisfaction que moi pour trop y sentir et subodorer ce qui fera les beaux jours du christianisme.\n\nJe voudrais pouvoir lire avec vous une consolation \u00e9crite par Diog\u00e8ne de Sinope ou Aristippe de Cyr\u00e8ne, mes pr\u00e9f\u00e9r\u00e9s, comme vous ne l'ignorez pas. J'aimerais que le temps ait \u00e9pargn\u00e9 un trait\u00e9, un dialogue, des fragments de consid\u00e9rations \u00e9crits par un cyr\u00e9na\u00efque ou un cynique pour les citer dans les lettres que je vous envoie, vous les offrir en viatique, les conseiller tel un cordial. On y lirait une invitation \u00e0 savoir l'entropie et la mort comme loi du monde et la n\u00e9cessit\u00e9 de r\u00e9sister \u00e0 ces forces qui conduisent au n\u00e9ant. On y trouverait s\u00fbrement un \u00e9loge de la tension, de la raideur devant Thanatos, une c\u00e9l\u00e9bration de l'\u00e9l\u00e9gance et un balbutiement du dandysme entendu comme une \u00e9thique de la verticalit\u00e9 en face du tr\u00e9pas. Peut-\u00eatre y d\u00e9busquerait-on des phrases avec lesquelles nous ferions, vous et moi, des citations qui nous permettraient de conduire notre existence.\n\nDe toute fa\u00e7on, et presque certainement, on lirait chez un cyr\u00e9na\u00efque une invitation \u00e0 m\u00e9priser la mort sous toutes ses formes et un \u00e9loge d\u00e9sesp\u00e9r\u00e9 de la vie dans toutes ses modalit\u00e9s. Il fustigerait tout ce que nous appelons masochisme depuis que la psychanalyse a donn\u00e9 un nom \u00e0 cet art consomm\u00e9 qu'ont les hommes de retourner la pulsion de mort contre eux, en doses infinit\u00e9simales, non mortelles, mais suffisantes pour empoisonner leur existence pendant toute sa dur\u00e9e. Il donnerait dans l'incantation pour inviter \u00e0 l'incandescence de la vie quotidienne. Et pour ce faire, nul doute qu'il appellerait \u00e0 la mobilisation g\u00e9n\u00e9rale et \u00e0 la guerre sans merci contre toutes les formes prises par l'id\u00e9al asc\u00e9tique. Il faut \u00e9trangler la mort.\n\nCar l'h\u00e9donisme suppose un volontarisme, et toutes les \u00e9coles de vie antiques, sto\u00efciennes et \u00e9picuriennes, cyniques ou cyr\u00e9na\u00efques me plaisent quand elles proposent l'apologie des pouvoirs de la volont\u00e9 sur le monde. M\u00eame si l'on peut constater souvent, trop souvent, les limites de ce vouloir en face de la toute-puissance de la n\u00e9cessit\u00e9, du moins peut-on en savoir le rayon d'action et les possibilit\u00e9s. Aristippe n'aurait certainement pas d\u00e9savou\u00e9 Plutarque quand il \u00e9crit, dans sa Consolation \u00e0 Apollonius sur la mort de son fils : \u00abLe deuil s'attache \u00e0 ceux qui le servent. Si vous le m\u00e9prisez, il s'\u00e9loignera de vous. Si, au contraire, vous lui rendez avec soin les honneurs auxquels il pr\u00e9side, c'est-\u00e0-dire les regrets et les larmes, il vous aimera et vous enverra sans cesse de quoi fournir son culte. \u00bb De m\u00eame, il aurait vraisemblablement souscrit \u00e0 S\u00e9n\u00e8que pr\u00e9cisant dans sa Consolation \u00e0 Marcia : \u00ab Quelle est cette d\u00e9mence que de se ch\u00e2tier soi-m\u00eame de ses infortunes, et d'augmenter la source de ses maux ? \u00bb Dites-moi, Cher Ami, que vous ne consentez pas \u00e0 ces deux citations ?\n\nToutes deux arc-boutent la philosophie h\u00e9doniste qui peut \u00eatre positive ou n\u00e9gative suivant qu'on cherche le plaisir ou l'\u00e9vitement du d\u00e9plaisir. Sur la question de la mort, l'h\u00e9doniste \u00e9vite tous les d\u00e9plaisirs qu'elle occasionne, quelle qu'en soit la nature. Le deuil est de ceux-l\u00e0, avec son cort\u00e8ge de douleurs, de souffrances, de peines et de larmes. Le volontarisme exige la conjuration de l'affliction et n\u00e9cessite qu'on cherche et trouve le moyen de pulv\u00e9riser le n\u00e9gatif. Lequel? En d\u00e9couvrant d'abord que nous ne sommes affect\u00e9s, troubl\u00e9s ou perturb\u00e9s que par des id\u00e9es, des repr\u00e9sentations, car le r\u00e9el se donne sur le seul mode de la m\u00e9diatisation intellectuelle. S\u00e9n\u00e8que \u00e9crit : \u00ab Si le destin se laisse vaincre par les larmes, pleurons ensemble. L'homme aide sa douleur, et son affliction n'est pas telle qu'il l'\u00e9prouve, mais telle qu'il se la propose. \u00bb Peut-on mieux dire ? La douleur triomphe avant tout dans l'id\u00e9e qu'on s'en fait, la repr\u00e9sentation qu'on en a. Pas de v\u00e9rit\u00e9 d'une sensation sans subjectivit\u00e9 de celle-ci. Elle n'est que mise en forme d'une perception en conformit\u00e9 avec une sensation par un sujet qui a le pouvoir de majorer ou de minorer l'\u00e9cho de ses affects.\n\nQuelles repr\u00e9sentations fautives g\u00e9n\u00e8re la douleur dans un deuil ? L'id\u00e9e que, par exemple, nous croyons pleurer le mort quand nous nous contentons de pleurer sur nous-m\u00eames. Car l\u00e0 o\u00f9 il repose, ni nos larmes ni nos joies ne lui sont de quelque secours ou de quelque usage. Ce que nous d\u00e9plorons? la privation dans laquelle nous sommes de ne plus pouvoir jouir de lui, de sa pr\u00e9sence, de sa conversation, de sa proximit\u00e9. Ce qui nous manque ? qu'il ne puisse plus jouer le r\u00f4le du miroir dans nos transports. Car lui, dans l'\u00e9tat qui est le sien, il n'a que faire de notre \u00e9motion ou de notre d\u00e9sint\u00e9r\u00eat : toute sympathie, toute empathie, toute communication demeurent d\u00e9sormais nulles et non avenues. Notre peine se nourrit de l'id\u00e9e fausse et magique qu'un \u00eatre dure bien qu'il ne soit plus. Mais laissons les Anciens et leurs rh\u00e9toriques de consolation pour en venir plus directement \u00e0 vous.\n\nJ'ai d\u00e9cid\u00e9 de vous \u00e9crire une lettre plus sp\u00e9cifiquement destin\u00e9e \u00e0 tenter de vous consoler le jour o\u00f9, en public, une vieille femme me questionna sur la mort, me demandant ce que pouvait bien la philosophie devant celle d'un enfant. J'ai t\u00e2ch\u00e9 de r\u00e9pondre. On me dit, \u00e0 l'issue de la rencontre, cette dame \u00e9vanouie dans les rues de la ville, qu'elle ne parvenait pas \u00e0 mettre au-dessus d'elle la mort de sa fille, r\u00e9cemment suicid\u00e9e, et qu'elle allait \u00e0 toutes les rencontres o\u00f9 se succ\u00e9daient les philosophes aux tribunes de conf\u00e9rences pour tenter d'obtenir une r\u00e9ponse qui vaille et lui rende un peu de go\u00fbt \u00e0 l'existence. J'ai song\u00e9 \u00e0 vous, sachant que vous ne m'aviez jamais demand\u00e9 ce que je veux aujourd'hui vous offrir : une consolation comme on pouvait les lire dans les temps b\u00e9nis o\u00f9 la philosophie aidait \u00e0 vivre, \u00e0 mieux vivre. Quelque chose qui ressemble \u00e0 un essai de morale cyr\u00e9na\u00efque \u00e0 votre endroit et pour nos temps m\u00e9lancoliques.\n\nVous souvenez-vous de cette premi\u00e8re lettre par vous envoy\u00e9e ? Elle donnait l'impression que vous n'appeliez pas de r\u00e9ponse, du moins que vous n'en attendiez pas plus que ce que vous esp\u00e9riez encore de l'existence. Vos initiales, votre nom et le cachet de la poste suffisaient pour que je d\u00e9sire vous faire une r\u00e9ponse quand, si souvent, les exigences comminatoires et indigentes du courrier des lecteurs me font la plupart du temps opter pour une politesse courtoise et minimale. D\u00e8s votre deuxi\u00e8me lettre, dat\u00e9e du 18 ao\u00fbt 1990, vous m'appreniez la mort de votre fille : \u00abVoici quatre ans, ma fille de seize ans a \u00e9t\u00e9 tu\u00e9e accidentellement. \u00bb D'autres courriers ont suivi et tous portaient mention d'une date, sauf celle dans laquelle vous m'avez donn\u00e9 les d\u00e9tails de l'accident. Comme si elle transcendait le calendrier, susceptible d'\u00eatre \u00e9crite tous les jours, et de toute \u00e9ternit\u00e9, sans cesse \u00e0 r\u00e9\u00e9crire, donc n'ayant pas besoin de proc\u00e9der d'un jour en particulier. Vous en \u00eates-vous aper\u00e7u ?\n\nVous m'y racontez une jeune et jolie jeune fille, brillante, dou\u00e9e, aim\u00e9e par ceux qui l'approchaient, amis, parents, voisins et professeurs dont certains, me dites-vous, demeurent hant\u00e9s par son sourire. Une \u00e2me fra\u00eeche et r\u00e9volt\u00e9e, indign\u00e9e par les injustices et les brutalit\u00e9s du monde comme il va. Vous me rapportez des papiers et sa voix sur une cassette, ses propos consign\u00e9s l\u00e0, par-del\u00e0 le silence, sur la vieillesse qu'elle pensait comme un naufrage et ne voulait pas conna\u00eetre, sur la mort et la m\u00e9tempsycose. Vous ajoutez qu'elle y parlait aussi de cr\u00e9mation et de dons d'organes, puis qu'elle disait son d\u00e9sir de pouvoir pratiquer ces deux m\u00e9taphysiques appliqu\u00e9es. Votre lettre continue : \u00abPr\u00e9monitions? Nous avons respect\u00e9 ses choix. Le 21 mai, elle l\u00e9guait ses yeux, son c\u0153ur, etc. Le 23, cr\u00e9mation. \u00bb\n\nPlus loin, vous dites le visage \u00e9pargn\u00e9 dans l'accident, la blessure infime mais fatale, le choc sur l'asphalte, le traumatisme cr\u00e2nien, la tempe, l'\u0153d\u00e8me au cerveau, pas d'\u00e9gratignure, l'int\u00e9grit\u00e9 d'un corps fauch\u00e9 l\u00e0, doucement. Puis les d\u00e9g\u00e2ts mat\u00e9riels quasi nuls, sa bicyclette, la voiture. La somme des co\u00efncidences funestes : le chauffeur qui tous les jours empruntait un autre trajet d\u00e9rogeant ce jour-l\u00e0, le d\u00e9part anticip\u00e9 d'Anne pour \u00eatre plus t\u00f4t au lyc\u00e9e afin d'y rendre un service. Votre lettre, encore : \u00ab Au moment pr\u00e9cis de sa mort, j'\u00e9tais dans ma classe, j'ai v\u00e9cu des minutes blanches, vides, impossibles \u00e0 reconstituer. Etranges. \u00bb Et vous me rapportez, ensuite, des signes troublants, une sortie en ville et des mots entendus dans un cin\u00e9ma qui vous semblaient venus d'elle; ou encore, \u00e0 sept cents kilom\u00e8tres de l'accident, ce mouvement de t\u00eate requis vers la devanture d'un magasin dont l'enseigne affichait l'initiale du pr\u00e9nom et le nom du chauffeur \u00e0 l'origine de l'accident.\n\nVoil\u00e0, tout \u00e9tait dit de la trag\u00e9die, vous m'aviez racont\u00e9 cette journ\u00e9e dans laquelle s'engloutissait une partie de votre existence. Quel avait \u00e9t\u00e9 ce jour pour moi ? J'ai repris quelques carnets o\u00f9, en dehors de tout autre d\u00e9tail, je notais alors le titre des livres que je lisais : mardi 20 mai 1986, je finissais le Journal du s\u00e9ducteur de Kierkegaard, je sortais des lectures de Derrida et de deux Lyotard. Rien d'autre, pas de souvenirs pr\u00e9cis. J'avais vingt-sept ans et ce jour-l\u00e0, je devais donner mes cours dans le lyc\u00e9e o\u00f9 j'enseigne encore aujourd'hui. Journ\u00e9e banale pour moi, journ\u00e9e tragique pour vous. Nos deux existences, alors, entamaient peut-\u00eatre ce jour un rapprochement, imperceptiblement, comme deux continents qui, s\u00e9par\u00e9s ici, s'acheminent l\u00e0 vers une rencontre.\n\nD'autres lettres suivent qui rapportent un geste, un trait, un mot, un souvenir d'Anne. Sur son calepin, cette quintessence de sapience h\u00e9doniste : \u00abNe perdez pas votre temps, car c'est la substance m\u00eame de la vie. \u00bb Etrange phrase, na\u00efve et profonde, \u00e9vidente et radicale, vraie dans sa substance tant elle r\u00e9sume l'imp\u00e9ratif cat\u00e9gorique de l'h\u00e9donisme : l'usage de son temps, seul capital dont on soit s\u00fbr de n'\u00eatre pas l\u00e9s\u00e9 avant le point final. Ce temps, il vous est devenu un calvaire et, ne croyez-vous pas, du moins il me semble, vous en faites un usage qu'aurait r\u00e9prouv\u00e9 Anne?\n\nQuand vous m'\u00e9crivez, le 20 septembre 1995 : \u00abCette obsession de compter. Depuis le 20 mai 1986. Pour ne jamais pouvoir effacer. Soit neuf ans et un trimestre ou cent douze mois ou quatre cent quatre-vingt-sept semaines ou trois mille quatre cent neuf jours. \u00bb Si Anne avait raison, et elle avait raison, elle a raison, et de tout temps elle aura toujours raison, alors l'usage que vous faites de votre temps appara\u00eet un suicide d'autant plus vain qu'il ne rach\u00e8tera en rien, ni jamais, la mort de votre fille. Le triomphe de Thanatos sur sa chair suffit bien pour que vous n'ayez pas \u00e0 vous associer corps et \u00e2me \u00e0 cette puissance mal\u00e9fique. Vous lui devez d\u00e9j\u00e0 l'absence d'Anne, n'ajoutez pas de mort \u00e0 la mort.\n\nVous-m\u00eame confiez que votre ardeur \u00e0 comptabiliser se superpose \u00e0 une volont\u00e9 de ne pas oublier. Pourtant, vous d\u00e9tournez de la sorte la seule pharmacop\u00e9e dont vous disposez : le vouloir. Tant que vous soumettrez celui-ci aux puissances n\u00e9gatives de Thanatos, Eros et la vie ne pourront pas prendre le dessus, et vous donnerez deux corps \u00e0 la mort pour laquelle un est toujours mille fois trop. Et rien ne se trouvera chang\u00e9 du n\u00e9ant dans lequel Anne repose, car on ne comble pas un vide avec l'offrande de sa propre existence : elle compte pour rien dans le pr\u00e9cipice o\u00f9 vous la jetez. Le n\u00e9ant absorbe tout et ne restitue rien, jamais. Le rachat, quelles qu'en soient les formes, est une invention des religions pour ajouter \u00e0 l'id\u00e9al asc\u00e9tique et aux renoncements, un subterfuge \u00e9labor\u00e9 pour faire triompher les puissances mortif\u00e8res et salir toutes celles qui montrent la vie en acte, en \u0153uvre et en puissance.\n\nEn 1991, au deux milli\u00e8me jour de son absence, nous ne sommes pas loin des f\u00eates que j'ai toujours abhorr\u00e9es, vous me dites votre d\u00e9testation des fins d'ann\u00e9e. Enfant, elles \u00e9taient pour moi l'occasion de mesurer la pauvret\u00e9 et l'indigence financi\u00e8re dans laquelle se trouvaient mes parents. Adulte, elles me semblent des circonstances permettant de saisir toute l'inanit\u00e9 du jeu social, l'\u00e9tendue des hypocrisies familiales et la puissance des conformismes gr\u00e9gaires. Je sais aussi que, pour vous, et ceux dont l'\u00e2me n'est pas rabougrie, No\u00ebl est un moment in\u00e9vitable o\u00f9 l'on pense \u00e0 ceux qui nous manquent. Mais l\u00e0 encore, il faut \u00e9viter les comptages : ceux des f\u00eates sociales et officielles, tout autant que celles de son anniversaire. Chaque ann\u00e9e, quand vous me dites l'\u00e2ge qu'Anne aurait, il faut vous dire qu'elle a d\u00e9sormais seize ans pour toujours. L'usage du temps, n'oubliez pas \u2013 elle avait tellement raison.\n\nUne autre fois, vous me racontez cette photo de classe aper\u00e7ue puis d\u00e9couverte chez une amie, et Anne surprise par votre regard dans le clich\u00e9. Ailleurs, apr\u00e8s l'image silencieuse, la voix sans pr\u00e9sence ou le d\u00e9sir d'une voix qui dise autre chose, et continue. Je vous comprends pour avoir perdu aussi une femme qui comptait dans mon existence et, en quelque sorte, avait \u00e9t\u00e9 ma m\u00e8re : c'est sa voix qui me fait le plus cruellement d\u00e9faut maintenant. Mais ce visage et cette voix ne sont plus pensables, vous le savez : ce qui les rendait possibles a disparu, et il ne sert \u00e0 rien que vous d\u00e9siriez l'impossible, sinon \u00e0 entretenir la douleur comme un feu perp\u00e9tuellement attis\u00e9 pour qu'il ne vienne pas \u00e0 s'\u00e9teindre.\n\nLa le\u00e7on des Anciens appara\u00eet une fois encore des plus pr\u00e9cieuses : sur quoi nous n'avons aucun pouvoir, il n'est aucune sagesse \u00e0 vouloir lutter. Ce qui d\u00e9pend de nous seul m\u00e9rite les efforts et le travail de notre volont\u00e9. L'objet du vouloir, pour faire son deuil, aux antipodes de ce sur quoi nous sommes impuissants, doit \u00eatre ce sur quoi nous avons puissance : l'absence ne se conjure pas, la douleur occasionn\u00e9e par cette absence, si. L'\u00e9nergie qui ne vise pas ce comble travaille en pure perte, abandonn\u00e9e sans frais \u00e0 l'holocauste de l'entropie g\u00e9n\u00e9ralis\u00e9e.\n\nEt puis, apr\u00e8s la parole, vous me dites le f\u00e9tiche, au sens o\u00f9 les Africains le pratiquent et dans l'acception donn\u00e9e \u00e0 l'analogon chez les ph\u00e9nom\u00e9nologues : un objet charg\u00e9 d'un sens, une chose \u00e0 laquelle on pr\u00eate une \u00e2me et \u00e0 laquelle on demande de jouer un r\u00f4le dans l'\u00e9conomie de nos douleurs. Une ann\u00e9e, je vous avais envoy\u00e9 un bouquet de roses pour ce que je croyais en quelque lieu \u00eatre la tombe d'Anne et vous m'aviez r\u00e9pondu : \u00abJ'en d\u00e9tacherai une et, rite annuel, je la d\u00e9poserai \u00e0 l'heure dite sur la route, \u00e0 l'endroit de l'accident, et regarderai les autos la mutiler sans qu'aucune cherche \u00e0 l'\u00e9viter. Mais ce bouquet s'\u00e9panouira dans ma chambre sur le gu\u00e9ridon o\u00f9 repose l'urne et les cendres d'Anne. Elle a \u00e9t\u00e9 incin\u00e9r\u00e9e (au cr\u00e9matorium de Bordeaux). A mon chevet donc, le gu\u00e9ridon, l'urne et les fleurs, la photo prise le 9 mai pour ses seize ans, la boucle de cheveux et sa cha\u00eene en or. \u00bb\n\nIl m'arrive, \u00e0 moi aussi, de pratiquer les bonheurs de la pens\u00e9e magique et les plaisirs de rendre vivants les objets auxquels on pr\u00eate un pouvoir : des fragments d\u00e9tach\u00e9s sur la plaie vive d'une \u00e2me et qui ont forme de vieilles lettres, de photos, de notes de restaurant, de billets d'avion, de papiers et d'enveloppes \u00e0 en-t\u00eates r\u00e9cup\u00e9r\u00e9s dans un h\u00f4tel \u00e0 l'autre bout du monde. Aussi je comprends qu'on veuille pr\u00eater aux choses une \u00e9nergie, une \u00e2me, une puissance qu'on ne conc\u00e9derait pour rien \u00e0 ce qu'on imagine d\u00e9sert\u00e9 par la force po\u00e9tique. Mais vos gestes ressemblent \u00e0 ceux qui lac\u00e8rent, mutilent, d\u00e9truisent. Pourquoi solliciter l'\u00e2me de votre fille pour la faire \u00e9craser \u00e0 nouveau ? Que pensez-vous conjurer ou apaiser, \u00e9vacuer ou calmer en organisant ces r\u00e9it\u00e9rations sacrificielles ? Quelles raisons avez-vous de rejouer sans cesse cette sc\u00e8ne de carnage? A quelle fin? Aucune, vous le savez pourtant pour l'avoir exp\u00e9riment\u00e9 mille fois en n'ayant finalement obtenu qu'une douleur surmultipli\u00e9e.\n\nJe vous crois plus en peine, alors, plus \u00e9corch\u00e9 encore, plus dilac\u00e9r\u00e9 que jamais, plus \u00e0 vif, l'\u00e2me plus sanguinolente que n'importe quand. Plus proche de la mort que jamais \u2013 mais vivant encore. Votre douleur, m\u00eame mille fois multipli\u00e9e, ne rach\u00e8tera pas la mort de votre jeune fille. S'il suffisait de quelques h\u00e9catombes pour que se rel\u00e8vent des tombeaux ceux qui s'y trouvent couch\u00e9s, alors oui, on ne devrait pas s'en priver. Si d'aventure trancher la t\u00eate d'un coupable pouvait redonner vie \u00e0 sa victime, lui permettre de sortir de son cercueil, alors oui, il ne faudrait pas h\u00e9siter une seule seconde, et nous devrions jubiler du couteau de la guillotine, en jouir, s'en repa\u00eetre, go\u00fbter, boire le sang, et revivre avec le ressuscit\u00e9. Si, par bonheur, mourir un peu, soi, tous les jours, \u00e0 petit feu, par sa propre main, en avivant les plaies, en fouillant les blessures, en fouaillant les b\u00e9ances, devait amoindrir, effacer, supprimer une douleur, alors, \u00e9videmment, nous n'aurions qu'\u00e0 commencer le bal avec Thanatos. Puis danser comme des fous jusqu'\u00e0 l'aube de l'humanit\u00e9 nouvelle r\u00e9g\u00e9n\u00e9r\u00e9e par ces festivit\u00e9s macabres.\n\nMais vous le savez bien, la mort n'accepte rien, aucun cadeau, aucun sacrifice ; elle ne joue pas ce jeu-l\u00e0, autiste, silencieuse, elle ignore tout ce qui n'est pas elle : elle prend, avale, se nourrit de tout ce qu'on lui offre, mais ne r\u00e9gurgite rien ; incapable de reconnaissance, elle prendrait l'univers entier avant son heure si d'aventure on ne lui r\u00e9sistait pas, toutes forces tendues, toute vitalit\u00e9 band\u00e9e comme un c\u00e2ble ; elle ne veut rien savoir, rien entendre de votre douleur; indiff\u00e9rente, elle ne s'en r\u00e9jouit pas, ne s'en afflige pas. Et, en tout, vous n'aurez r\u00e9ussi qu'\u00e0 vous faire complice d'elle, aucunement d'Anne et de son souvenir.\n\nVos nuits pass\u00e9es en compagnie de ses cendres, \u00e0 proximit\u00e9 de l'urne, vos r\u00eaves et vos cauchemars non loin de ses m\u00e8ches, votre respiration et votre souffle tout pr\u00e8s de la photo, son bracelet tourn\u00e9 et retourn\u00e9 mille fois dans vos doigts, tenu dans le creux de votre main, tout cela ne fera qu'ajouter de la peine \u00e0 la peine. Rien de ce calcul ne se retranchera du compte que votre fille avait d\u00e8s sa naissance avec le n\u00e9ant : on ne peut d\u00e9livrer l'autre de ses peines, de ses fardeaux, en prendre une part, m\u00eame minime, et le travail de deuil n'est pensable qu'entre soi et soi. On ne doit esp\u00e9rer aucun arrangement avec la mort des autres. La consolation n'est pensable que si elle invite l'autre \u00e0 se ressaisir de ses propres forces. L'ouvrage ne rel\u00e8ve de personne d'autre que soi.\n\nNe comptez plus les jours, les mois, les ann\u00e9es, n'attendez plus le retour d'un visage ou la renaissance d'une voix, n'\u00e9coutez plus cette cassette, ne regardez plus les photos, ne relisez pas ses carnets, ne c\u00e9l\u00e9brez pas les dates anniversaires de sa naissance, ni celles de sa mort, ne caressez plus ces cheveux, ne touchez plus ce bracelet, ne jetez plus de fleurs sur la route en attendant leur destruction comme fut d\u00e9truit le corps de votre fille, ne portez plus le regard sur l'urne qu'il faut mettre en terre, ne dormez plus pr\u00e8s des cendres de votre enfant, reposez-vous, rien ne fera que cette trag\u00e9die n'ait pas eu lieu.\n\nVous avez retrouv\u00e9 des po\u00e8mes \u00e9crits par l'une de ses amies pour accompagner le cercueil dans sa cr\u00e9mation et qui avaient \u00e9t\u00e9 \u00e9gar\u00e9s avant de devoir \u00eatre br\u00fbl\u00e9s dans le cercueil ? Alors portez-y le feu vous-m\u00eame pour ne plus c\u00e9l\u00e9brer de tombeau, au sens antique du terme, qu'en donnant raison \u00e0 Anne qui \u00e9crivait \u2013 vous \u00e9crivait \u2013 que l'emploi du temps seul m\u00e9rite souci. Vous ne pouvez lui donner tort, ce serait manquer \u00e0 son souvenir, en g\u00e2chant votre temps, le brisant, le d\u00e9truisant, le d\u00e9testant comme ce qui vous serait p\u00e9nible \u00e0 vivre pour la seule raison qu'il a \u00e9t\u00e9 refus\u00e9 \u00e0 Anne. Ce que vous infligez \u00e0 votre \u00e2me ne r\u00e9parera jamais son corps perdu. A vous faire souffrir, car je ne crois pas que vous ayez du plaisir \u00e0 c\u00e9l\u00e9brer la mort, vous n'entamerez en rien l'in\u00e9luctable dont la nature est d'\u00eatre imperm\u00e9able \u00e0 toute attaque, toute agression. Vous, en revanche, vous allez au-devant du sort que la mort vous r\u00e9serve et nous r\u00e9serve \u00e0 tous. Vous faites son travail avant l'heure, comme si vous vouliez lui donner raison de tailler dans le vif. Vous vous offrez en complice \u00e0 celle qu'il faut honnir de vous avoir d\u00e9j\u00e0 trop vol\u00e9.\n\nEntre ce que vous vous infligez et la mort de votre enfant, n'existent aucune causalit\u00e9, aucune relation. Rien ne relie cette trag\u00e9die et l'enfer dans lequel vous vous \u00eates emmur\u00e9. Vous ne faites que rendre un culte \u00e0 ce qui vous a enlev\u00e9 Anne. A aucun moment, en aucune mani\u00e8re, la mort ne m\u00e9rite qu'on lui donne, de plein accord, avec consentement, plus qu'elle ne prend d\u00e9j\u00e0. L'id\u00e9al asc\u00e9tique se quintessencie dans le deuil : il trouve en lui mati\u00e8re \u00e0 triompher sans quartiers.\n\nLa fid\u00e9lit\u00e9 \u00e0 ceux qui ne sont plus ne doit pas se faire dans la mutilation et le sacrifice, on n'y gagne rien, sinon faire triompher deux fois la mort. En revanche, elle peut se faire dans la mise en conformit\u00e9 de son existence avec une maxime qui \u00e9tait celle du disparu ou une id\u00e9e qu'on lui doit comme un h\u00e9ritage d'amour transmis. Je songe une fois encore \u00e0 cette phrase qui ouvrait son calepin. Savez-vous que vous me l'avez rapport\u00e9e deux fois ? Le 9 septembre 1991 et le 5 f\u00e9vrier 1996 o\u00f9 vous me dites m\u00eame qu'elle l'avait soulign\u00e9e d'un triple trait. Vous n'\u00e9crivez jamais deux fois la m\u00eame chose, dans ces cent treize lettres, vous ne vous r\u00e9p\u00e9tez jamais. Je voudrais voir dans le signe de cette deuxi\u00e8me apparition ce qui pr\u00e9c\u00e9dera la troisi\u00e8me et vaudrait comme le troisi\u00e8me trait sous la phrase, une invitation \u00e0 vivre selon son souhait comme seule fa\u00e7on de lui rendre hommage, aujourd'hui, et de lui permettre de c\u00e9l\u00e9brer, avec elle, pour elle, avec vous, pour vous, la vie, rien que la vie.\n\nVous me r\u00e9pondrez. J'attends d\u00e8s \u00e0 pr\u00e9sent votre r\u00e9ponse, avec un peu d'inqui\u00e9tude, car je sais avoir soulev\u00e9 des monceaux de peines et de douleurs, avoir raviv\u00e9 des blessures, rouvert des plaies, peut-\u00eatre. J'enverrai cette lettre pour que vous la receviez \u00e0 No\u00ebl prochain, comme on embrasse ceux qui nous sont chers, malgr\u00e9 toute l'affectation de ces soirs-l\u00e0, en occasion de rappeler qu'avant l'hypoth\u00e9tique et peu probable naissance d'un messie devenu c\u00e9l\u00e8bre le 25 d\u00e9cembre, ce que l'on f\u00eatait cette nuit-l\u00e0, dans les cultes pa\u00efens, c'\u00e9tait la grande f\u00eate solaire du solstice d'hiver dans le calendrier julien, la naissance de Mithra, la lumi\u00e8re invincible.\n\nCes heures-l\u00e0, pour mon vieux ma\u00eetre et moi, sont l'occasion de nous dire combien nous y sommes sensibles, loin des colifichets qui, sur cette date, excitent les nostalgiques de l'Occident pa\u00efen : le maximum de clart\u00e9, en juin, me r\u00e9jouit pour la dur\u00e9e la plus longue du jour, mais m'attriste parce que, d\u00e8s le lendemain, nous commen\u00e7ons \u00e0 perdre du temps en lumi\u00e8re. En revanche, le 21 d\u00e9cembre, l\u00e0 o\u00f9 nous touchons le fond avec un maximum de nuit et d'obscurit\u00e9, nous savons que le jour suivant magnifie la reconqu\u00eate de la lumi\u00e8re. Et je me r\u00e9jouis alors de remonter vers de plus en plus de clart\u00e9. Je voudrais que vous m'accompagniez dans cette remont\u00e9e vers la surface. Je vous embrasse.\n19\n\nLE SALON VERDURIN, ENCORE\n\nTous deux ont lu La Recherche du temps perdu, elle d'abord, lui ensuite, parce que femme au foyer, elle a le loisir de lui pr\u00e9parer ses lectures avec la componction de ceux qui coupent la viande \u00e0 leur chien. De la m\u00eame mani\u00e8re, elle a dig\u00e9r\u00e9 la p\u00e2t\u00e9e en lui distillant les grands moments, en pointant les morceaux de bravoure, ceux qu'on ne peut pas ne pas conna\u00eetre si l'on veut dire qu'on l'a lue, qu'on en est, qu'on conna\u00eet, \u00e9videmment. Vinteuil, elle sait, donc il sait maintenant ; Elstir et Swann, Odette de Cr\u00e9cy et Charlus, elle a racont\u00e9, donc il conna\u00eet dor\u00e9navant.\n\nElle pratique de la m\u00eame mani\u00e8re avec deux ou trois auteurs qu'ils resservent en permanence avec un art particulier du recyclage, comme ces viandes qui r\u00e9apparaissent en entr\u00e9es, en plats de r\u00e9sistance, voire en sorbets, avant de finir dans la p\u00e2t\u00e9e du chien pr\u00e9cit\u00e9. Proust offre un morceau de choix, car il permet le th\u00e9, les petits g\u00e2teaux, l'art de la conversation m\u00e9disante et fielleuse qui comble les dimanches apr\u00e8s-midi, sinon d'une mortelle longueur. \u00d4, la tragique affliction vesp\u00e9rale !\n\nSavent-ils que depuis leur d\u00e9couverte ils jouent en plus ridicules encore le salon Verdurin et tous ses travers mesquins ? Ignorent-ils leur talent consomm\u00e9 dans l'art de d\u00e9marquer au plus pr\u00e8s le d\u00e9calogue du petit noyau, de la pitoyable Eglise qui, ayant \u00e9dict\u00e9 ses lois, produit ses h\u00e9t\u00e9rodoxies, ses inquisitions, ses b\u00fbchers et ses piloris, ne se nourrit et ne vit que du sang qu'elle jouit de faire couler? Min\u00e9s par la jalousie, taraud\u00e9s par le ressentiment, d\u00e9vor\u00e9s par l'envie comme d'aucuns sont rong\u00e9s par la l\u00e8pre, ils subissent leurs affects telles de vieilles poules de basse-cour furieuses de n'\u00eatre pas des oiseaux de proie et conscientes de finir leur existence dans la peau de caricatures rempla\u00e7ables. Leur \u00e9tat affectif, intellectuel et mental n'est pas enviable.\n\nLui est \u00e0 elle ce qu'ils lui sont aussi : des caniches en mal de ma\u00eetre, des \u00e9nergies fragiles en qu\u00eate d'une forme. Et elle, alors ? Elle fr\u00e9tille en copie conforme des avis donn\u00e9s dans le journal du soir, arriv\u00e9 dans la sous-pr\u00e9fecture le lendemain. Mais son statut confortable de femme \u00e0 la maison lui donne l'avantage, sur les autres membres du salon, d'un l\u00e9ger temps d'avance obtenu au d\u00e9savantage de ceux qui peinent au labeur pour gagner leur pain quotidien dans l'administration, pour ceux dont la langue est la plus pendante, dans leurs officines lib\u00e9rales, pour les bourgeois les mieux aguerris.\n\nSi l'on veut savoir ce qu'il faut penser, ce qu'elle pensera donc, il faut lire son journal avant elle, ce qui n'a de cesse d'\u00eatre dr\u00f4le quand on a pratiqu\u00e9 une fois ou deux de cette mani\u00e8re. Si d' aventure des gr\u00e8ves ou des probl\u00e8mes d'acheminement la privent de ses journaux, quotidiens ou hebdomadaires, alors elle ne pense plus, le temps qu'arrivent les suivants. Aussi serait-elle dans l'obligation de diff\u00e9rer, reporter ou annuler tous ses d\u00eeners et tous ses th\u00e9s le temps que durerait une gr\u00e8ve, si par malheur elle devait \u00eatre priv\u00e9e longtemps de ses sources d'information, donc de ses br\u00e9viaires et autres cat\u00e9chismes \u00e0 l'usage de ceux qui pensent par procuration.\n\nDes expositions \u00e0 voir, des livres \u00e0 lire, des musiques \u00e0 \u00e9couter, des villes \u00e0 visiter, elle ne dit que ce qu'elle peut r\u00e9p\u00e9ter. Il ne r\u00e9p\u00e8te que ce qu'elle aura dit. Et bient\u00f4t ils ne rapporteront que ce qu'ils auront entendu de sa bouche, \u00e0 lui, s\u00fbrs de ne pas commettre d'impairs en jugeant avant de l'avoir vue une r\u00e9trospective admirable, sublime ou g\u00e9niale, car ils c\u00e9l\u00e8brent l'hyperbole, ou minable, nulle, d\u00e9plorable, car ils aiment aussi les lieux communs affil\u00e9s en lame de guillotine. Sous leur r\u00e9gime, comme on le disait de Staline ou de P\u00e9tain plus que de Fran\u00e7ois Ier, on peut se dispenser de r\u00e9fl\u00e9chir, il suffit d'ob\u00e9ir \u2013 et de r\u00e9p\u00e9ter. La religion de ce monde-l\u00e0 est le psittacisme port\u00e9 \u00e0 son incandescence.\n\nDonc, la presse parisienne a parl\u00e9. Puis, la pythie souffle aux oreilles de son mari. Lui, peu embarrass\u00e9 par les d\u00e9tails, les fioritures ou la d\u00e9licatesse, ne conna\u00eet que les trompettes violemment embouch\u00e9es et les grosses caisses particuli\u00e8rement maltrait\u00e9es. Apr\u00e8s lui, il ne faut plus juger de rien, c'est cacophonie et mod\u00e8le wagn\u00e9rien pour une fanfare municipale, c'est la convocation des moyens pour une symphonie des mille qui avorterait dans un orph\u00e9on poussif. Et tous les \u00e9lus vont, grenouillant comme d'autres grouillent, portant la bonne parole, pr\u00eachant la bonne nouvelle, enseignant les populations ignares, sottes et demeur\u00e9es. Ce qu'on avait lu la semaine pr\u00e9c\u00e9dente \u00e0 Paris devient huit jours plus tard une p\u00e2le tisane, une d\u00e9coction insane, resservie de moins en moins pr\u00e9sentable \u00e0 l'usage des soucieux de culture qui croupissent en sous-pr\u00e9fecture.\n\nDans les soir\u00e9es o\u00f9 l'invitation vaut adoubement, aupr\u00e8s duquel la tradition chevaleresque f\u00e9odale para\u00eet embl\u00e9matique du plus grossier des arts de vivre et du plus rustre des usages, on pratique avec bonheur l'art de la citation et des r\u00e9f\u00e9rences, un genre de dominos, en plus trivial, \u00e0 l'usage de ceux dont la culture proc\u00e9derait \u00e9troitement d'un jeu de soci\u00e9t\u00e9. A-t-on lu tel dernier roman? Conna\u00eet-on telle r\u00e9cente pi\u00e8ce de th\u00e9\u00e2tre ? A-t-on visit\u00e9 telle r\u00e9trospective majeure? S'est-on rendu ici ou l\u00e0, dans une capitale d'Europe ou dans un pays jamais visit\u00e9 par personne avant qu'en propre ils n'y aient mis les pieds? S'est-on souci\u00e9 de rapporter des diapositives ? Quand les verra-t-on ? Bient\u00f4t ? Qu'on ne fasse pas languir ceux qui se nourrissent de culture comme d'autres d'anabolisants et d'amph\u00e9tamines.\n\nMais qui n'aurait rien lu, rien vu ou rien visit\u00e9, qui n'aurait pas mis les pieds dans une salle obscure ou dans une ville d\u00fbment chroniqu\u00e9 par un guide de voyage passerait pour le dernier des provinciaux, un individu invert\u00e9br\u00e9 ou d\u00e9c\u00e9r\u00e9br\u00e9, quelqu'un \u00e0 qui l'on pourrait dire en public combien il est ridicule, d\u00e9plorable, pitoyable. Reste une possibilit\u00e9 de se racheter : en compensant, en comblant les manques, en invitant ceux qui savent, c'est-\u00e0-dire lui, donc elle, \u00e0 nous faire l'aum\u00f4ne de leur savoir, de leur science, de leur immense culture et de leur g\u00e9n\u00e9rosit\u00e9 intellectuelle sans bornes. Alors elle affectera une allure pensive et il parlera, racontant ce qu'il faut dire, penser, croire et d\u00e9fendre en croisade.\n\nDe sorte que les nouveaux convertis se pr\u00e9cipiteront dans la nouvelle religion r\u00e9v\u00e9l\u00e9e et communieront le temps qu'il, c'est-\u00e0-dire elle, aura imparti \u00e0 l'assembl\u00e9e. Car il faudra aimer ce qu'ils aiment le temps qu'ils l'aimeront, et n'avoir pas le mauvais go\u00fbt d'appr\u00e9cier un peu avant ou un peu apr\u00e8s eux, ce qui produirait un casus belli sans possibilit\u00e9 de n\u00e9gociation, une occasion d'hostilit\u00e9 pas pr\u00e8s de s'\u00e9teindre. On n'y gagnerait qu'une invective, un mot de m\u00e9pris ou une consid\u00e9ration arrogante.\n\nTel musicien qui appelait qu'on s'agenouille pour une t\u00e9tralogie en direct, devant son poste de radio, devenu persona non grata l'ann\u00e9e suivante, d\u00e9routera alors celui qui aura fait l'effort de s'y mettre, en croyant sans cesse y parvenir, quand il n'aura fait que se contenter de simuler \u2013 ce qui suffit pour l'usage domestique et social. Devenu wagn\u00e9rien quand son ma\u00eetre est pass\u00e9 aux joies du shamisen, l'apprenti n'aura de recours qu'\u00e0 mettre les bouch\u00e9es doubles, \u00e0 jeter les livrets d'op\u00e9ra par-dessus bord, et \u00e0 se fournir dans les meilleurs magasins o\u00f9 l'on diffuse des disques de musique traditionnelle orientale.\n\nAu d\u00eener, o\u00f9 les le\u00e7ons sont donn\u00e9es aux imp\u00e9trants, les repas sont des occasions de pratiquer le laboratoire et de mesurer le degr\u00e9 de vassalit\u00e9 des invit\u00e9s, sinon les limites de leur bonne volont\u00e9. Les ann\u00e9es fastes, ou n\u00e9fastes, comme on voudra, donnent \u00e0 la terrine le r\u00f4le que jouait pr\u00e9c\u00e9demment Wagner. Aussi trouvera-t-on dans l'assiette l'\u00e9quivalent de l'instrument \u00e0 cordes japonais : de la coquille saint-Jacques crue, avec un filet d'huile, un tour de moulin \u00e0 poivre et une feuille de lierre dans l'\u00e9picentre de l'assiette. Terreur : faut-il consommer ou est-ce pour d\u00e9corer ?\n\nL'existence bascule soudain : mange-t-on, au pays du soleil levant, de la feuille de lierre crue, telle quelle, avec la queue qui nous nargue, mortelle peut-\u00eatre? N'est-on pas en pr\u00e9sence d'une sublimation d'aquarelle? N'est-ce pas une citation d'Hiroshige? Un clin d'\u0153il en amande au dernier roman de Tanizaki? On mobilise ses forces, on t\u00e2che de se souvenir de ce qu'on a lu la semaine pr\u00e9c\u00e9dente, mais c'\u00e9tait sur le cin\u00e9ma japonais, on a manqu\u00e9 le livre de cuisine. On t\u00e2che de r\u00e9viser, on passe \u00e0 toute allure dans sa t\u00eate les r\u00e9f\u00e9rences accumul\u00e9es : et que dirait un pr\u00eatre shinto\u00efste ? Un philosophe de la tradition zen ? Ou les fr\u00e8res Troisgros ? Et toujours cette feuille de lierre... On se prend \u00e0 r\u00eaver d'un navarin d'agneau avec un saint-\u00e9milion pour lesquels on donnerait toute sa fortune \u2013 avant d'\u00eatre sauv\u00e9 par la ma\u00eetresse de maison qui plie sa serviette avant de rejoindre l'office : elle a laiss\u00e9 la feuille, gisante comme une st\u00e8le fun\u00e9raire bris\u00e9e.\n\nC'est bon : \u00e7a ne se mange pas. Passe pour cette fois. Reste \u00e0 montrer un talent consomm\u00e9 pour avaler en deux bouch\u00e9es la totalit\u00e9 de l'entr\u00e9e \u00e0 peine effleur\u00e9e, attendant les autres pour, l\u00e0 encore, prendre avis et le\u00e7on, aligner, s'aligner sur leurs usages. Le deuxi\u00e8me plat sera du fugu s\u00e9ch\u00e9 \u2013 suit le commentaire avis\u00e9 du domestique uxoral expliquant comment, quand il est mal cuisin\u00e9, ce poisson envoie les gastronomes directement au cimeti\u00e8re sans avoir pris la peine de leur laisser esp\u00e9rer quelque chose \u00e0 l'h\u00f4pital, au service des empoisonnements. Suite du repas.\n\nQuand \u00e0 leur table apparaissent les vins, c'est, pour qui \u00e9tait au-del\u00e0 de l'\u00e2nonnement sur ce sujet, une occasion de mesurer combien leur cave, comme les fleurs qu'ils offrent, leur ressemble : petits vins frais sans vertus, mill\u00e9simes insipides dans des registres qui, de toute fa\u00e7on, \u00e9taient plus appel\u00e9s \u00e0 satisfaire le gastrologue un jour que l' \u0153nologue toujours. La Loire y tr\u00f4ne sans partage, l\u00e0 aussi parce qu'on affiche un snobisme construit contre celui qui affecte pr\u00e9tendument ceux qui c\u00e9l\u00e8brent les grands crus de Bordeaux. Contre les seuls snobs qui, bien s\u00fbr, ne voient que par Yquem ou Ausone, on vante les m\u00e9rites de Bourgueil ou de Sancerre. Pour se d\u00e9marquer, para\u00eetre autre, \u00e0 d\u00e9faut d'\u00eatre quelque chose, puisque quelqu'un \u00e9tait impensable, on se rafra\u00eechit avec des vins, comme d'autres en des lieux r\u00e9serv\u00e9s \u00e0 cet effet. Finalement, pingres en tout, ils sont bien incapables d'investir dans les vins avec la magnificence exig\u00e9e par les grands crus. Les bourgeois pr\u00e9f\u00e8rent les valeurs capitalisables et recyclables dans autre chose que le pur et simple plaisir.\n\nD'ailleurs, pour se dispenser de penser ou de r\u00e9fl\u00e9chir, l\u00e0 aussi, ils faisaient confiance \u00e0 des clubs qui leur envoyaient des caisses de bouteilles comme d'autres re\u00e7oivent \u00e0 lire la s\u00e9lection des livres du mois. Les picrates de Corse ou des C\u00f4tes-du-Rh\u00f4ne, les poisons de Cahors ou du Sud-Ouest d\u00e9cr\u00e9t\u00e9s magnifiques, restait aux imp\u00e9trants le seul devoir d'opiner. A d\u00e9faut, ils croulaient sous une avalanche de critiques et se trouvaient affubl\u00e9s des \u00e9pith\u00e8tes infamantes que, ne voulant pas m\u00e9riter, ils faisaient tout pour \u00e9viter. Y compris prendre le risque d'une perforation stomacale. Le nez dans leur verre comme dans un calice, l'\u0153il perdu sur la ligne bleue de la tapisserie, grumelant comme des apprentis fiers de leur savoir neuf, ils avalaient la potion pharmaceutique avec la m\u00eame componction gr\u00e9gaire que s'il se f\u00fbt agi d'un nectar descendu de l'Olympe.\n\nQuand on ne mange pas, disons plut\u00f4t entre les bouch\u00e9es, on m\u00e9dit \u2013 le strict n\u00e9cessaire. Juste pour activer la salivation et faciliter la digestion. Pour un peu, un jet de fiel lanc\u00e9 contre celui-ci ou celui-l\u00e0 passerait pour une prescription de la facult\u00e9 de m\u00e9decine : \u00e7a ne peut pas faire de mal, globalement, ce serait m\u00eame plut\u00f4t conseill\u00e9 pour des raisons di\u00e9t\u00e9tiques et hygi\u00e9niques. Se vider de sa haine, c'est laisser place \u00e0 l'immense vacuit\u00e9 avec laquelle on peut ensuite se gonfler comme des baudruches, aux bords de l'\u00e9clatement, tant la suffisance est grande dans ce monde embl\u00e9matique du nanisme. La dose de violence dont on se d\u00e9fait, en la modulant, car on festoie entre gens bien \u00e9lev\u00e9s, est proportionnelle \u00e0 l'\u00e9tendue du d\u00e9sert affectif qu'on lib\u00e8re en soi, quand on a laiss\u00e9 la place \u00e0 ce qui nous habite g\u00e9n\u00e9ralement si l'on est nourri de ressentiment.\n\nSi elle devait \u00eatre portraitur\u00e9e, c'est simple, elle ressemblerait \u00e0 la fleur qu'elle offre toujours en cas d'invitation : un unique strelitzia, dit aussi bec-de-perroquet, qui fait impression par son allure tarabiscot\u00e9e et qui a l'immense avantage de ne pas co\u00fbter cher. Une longue tige, droite comme la justice, s\u00e9v\u00e8re comme un barreau de prison, allong\u00e9e \u00e0 la mani\u00e8re de ces piques au bout desquelles on fichait les t\u00eates \u00e0 la bouche bourr\u00e9e de foin, non loin de la Bastille. Une tige identique \u00e0 une lance et une fleur semblable \u00e0 une gueule d'oiseau avec un bec effil\u00e9 pareil \u00e0 une aiguille, coupant comme une lame, tranchant tel un rasoir.\n\nOn chercherait en vain un oeil dans le brouillage d'\u00e9corces dures, de plaques de k\u00e9ratine, comme celles qui recouvrent les insectes, cuirass\u00e9s de carapaces et d'armures rigides. La fleur est aveugle. Est-ce d'ailleurs encore une fleur? Ne s' agit-il pas plut\u00f4t, quand on l'arbore en l'offrant, d'une m\u00e9taphore du sexe qu'on r\u00eave? L\u00e0 o\u00f9 certains jubilent des orchid\u00e9es annonciatrices de promesses de bonheur, d'autres disent leur sexe froid comme celui du diable, leur ventre scell\u00e9, coi, telle leur \u00e2me incapable de s'offrir \u2013 car on ne donne que si l'on a. Sexe m\u00e9tallique et m\u00e9canique, dur et glac\u00e9, dangereux, coupant, castrateur d\u00e8s qu'on s'y fait annoncer, fleur v\u00e9n\u00e9neuse cachant un vagin dent\u00e9, le cadeau est fait en toute innocence. Et il ne co\u00fbte rien en argent, donc en affection.\n\nLa fleur unique dans son papier d'emballage s'offre avec le m\u00eame c\u00e9r\u00e9monial qu'une momie dans un sarcophage. Oiseau d\u00e9charn\u00e9, insecte pr\u00e9dateur, elle semblait pr\u00eate \u00e0 ouvrir sa bouche, d\u00e9couvre des rasoirs \u00e0 la place des dents, l'ensemble attendant une occasion propice pour se jeter au visage de qui n'aurait pas ob\u00e9i \u00e0 la ma\u00eetresse ne se d\u00e9pla\u00e7ant qu'accompagn\u00e9e de ce genre d'animal pr\u00e9historique et violent. A c\u00f4t\u00e9 de cette fleur tenue en laisse, la feuille de lierre n'\u00e9tait qu'un amuse-bouche, une bluette dont on regrettait l'aspect sympathique.\n\nLes ans, sinon les si\u00e8cles passent, mais chez eux, la table reste la m\u00eame, seuls changent les invit\u00e9s, ceux qui revendiquent l'insigne honneur d'y \u00eatre appel\u00e9s, convi\u00e9s. Pour avoir leur rond de serviette, d'aucuns se sont damn\u00e9s et ont attendu plus que de raison, le dos cass\u00e9 d'avoir pratiqu\u00e9 la courbette, la r\u00e9v\u00e9rence et la posture humiliante. J'ai connu des petits bourgeois rong\u00e9s par l'envie, qui auraient donn\u00e9 la virginit\u00e9 de leur fille, ce qui \u00e9tait nul et non avenu, l'intelligence de leur fils, ce qui \u00e9tait n\u00e9gligeable, une partie de leur fortune, ce qui \u00e9tait minable, tout l'honneur dont ils \u00e9taient capables, c'est-\u00e0-dire rien, ou n'importe quoi d'autre dont ils se seraient crus propri\u00e9taires, pour pouvoir se retrouver un jour devant la feuille de lierre, le poisson douteux ou avoir \u00e0 plonger dans un vase inappropri\u00e9 le strelitzia offert dans son linceul transparent. A force de courbettes, avec une constance qui les honore, en for\u00e7ant la porte, ils ont r\u00e9ussi \u00e0 obtenir leur droit de cit\u00e9 dans le cercle et se retrouvent avec de pr\u00e9tendus plasticiens qui se poussent du col, offrant leurs d\u00e9jections comme des cadeaux ou un potlatch qui, sinon, n'auraient jamais trouv\u00e9 preneurs, ni acheteurs, bien s\u00fbr.\n\nEvidemment, ceux-l\u00e0, tant qu'ils \u00e9taient \u00e0 la porte, hurlaient avec les loups et crachaient sur ceux-ci qui le leur rendaient bien. Les exclus d'hier font d'excellents convives quand on a \u00e9puis\u00e9 toutes les bonnes volont\u00e9s et qu'il faut bien finir par accepter \u00e0 sa table, outre ceux dont on s'\u00e9tait priv\u00e9 nagu\u00e8re, ceux aussi qu'on a conchi\u00e9s et compiss\u00e9s jadis, et pendant des ann\u00e9es. Mais les uns ont la m\u00e9moire aussi courte que l'intelligence, les autres le pardon aussi facile que le reniement. Pour qui aurait compt\u00e9 les passages, les glaires ont \u00e9t\u00e9 un sport pendant des ann\u00e9es. Quand on attend que l'huis s'ouvre pour acc\u00e9der \u00e0 la pitance, il faut bien s'occuper. Combien d'insultes, de haine, de m\u00e9pris, de m\u00e9disances, de part et d'autre, avant que tout ce monde se retrouve, craignant plus que tout la solitude, pour des d\u00eeners o\u00f9 ils prennent la place encore chaude du dernier d\u00e9go\u00fbt\u00e9 ayant quitt\u00e9 les lieux ! Quelle liste il faudrait dresser pour comptabiliser ceux qui, hier, faisaient les frais des conversations o\u00f9 ils \u00e9taient salis et qui, le lendemain, se sont retrouv\u00e9s convi\u00e9s \u00e0 tenir leur r\u00f4le dans l'assassinat des convives de la veille ! Avec les cadavres encore frais on faisait bombance les soirs de r\u00e9veillon...\n\nEntre les mauvais vins servis \u00e0 contretemps, aux mauvaises temp\u00e9ratures, et les nourritures sinon dangereuses, du moins exotiques, entre deux conversations viles et m\u00e9disantes, mesquines ou malveillantes, lui, donc elle, formulait l'hypoth\u00e8se qu'il faudrait faire passer des examens aux gens, comme il disait, pour leur autoriser ici la visite d'une ville, l\u00e0 celle d'un mus\u00e9e, ailleurs le visionnage d'un film. Il ignorait certainement que, pris \u00e0 son propre jeu, recal\u00e9 la plupart du temps, il aurait d\u00fb, elle donc, se contenter des murs de la sous-pr\u00e9fecture, des expositions de peinture foment\u00e9es par le syndicat intercommunal ou je ne sais quelle soci\u00e9t\u00e9 acad\u00e9mique locale tant ses lieux communs ne pouvaient \u00eatre pris au s\u00e9rieux par quelque commission d'examen pensable. Lui qui c\u00e9l\u00e8bre aujourd'hui, en se croyant moderne, les cin\u00e9astes d'avant-garde d' avant-hier, n'aurait eu qu'\u00e0 enregistrer son recalage au premier tour.\n\nMais les minables du petit noyau, qui avaient attendu leur tour pendant des ann\u00e9es, frigorifi\u00e9s dans le chambranle de la porte, acquies\u00e7aient et opinaient du chef en ignorant, eux aussi, qu'en obtenant m\u00eame le certificat, l'autorisation, l'Ausweis, ils n'auraient pas les moyens que leur laissaient leurs pouvoirs d'achat respectifs d'envisager autre chose qu'une partie de p\u00e9tanque sur la C\u00f4te d'Azur. A ce prix, tous aspirent aux soci\u00e9t\u00e9s o\u00f9 le ressentiment permet aux m\u00e9diocres de s'installer dans les miradors o\u00f9 ils r\u00e9gentent la vie des autres. D'aucuns, d'ailleurs, avaient fait sans vergogne le trajet qui les conduisait d'un stalinisme de bon aloi \u00e0 cet \u00e9litisme de d\u00e9g\u00e9n\u00e9r\u00e9s o\u00f9 l'on n'est jamais aussi bien servi que par soi-m\u00eame.\n\nL'intelligence ne brillait pas dans le c\u00e9nacle. En revanche, il faut c\u00e9l\u00e9brer la m\u00e9moire \u00e0 toute \u00e9preuve, la redoutable et formidable m\u00e9moire qui permet aux Guides bleus et verts, aux encyclop\u00e9dies de voyage et d'oenologie, aux articles de presse, m\u00eame des mois apr\u00e8s leur parution, surtout des mois apr\u00e8s leur parution, de faire la loi, et en m\u00eame temps de faire impression, aupr\u00e8s de ceux dont la biblioth\u00e8que est vide de ces monuments \u00e0 l'aide desquels on brille \u00e0 moindres frais. L\u00e0 o\u00f9 la r\u00e9flexion aurait permis un temp\u00e9rament, une personnalit\u00e9, un caract\u00e8re, la r\u00e9p\u00e9tition fabriquait des perroquets, de ridicules machines \u00e0 r\u00e9citer leurs le\u00e7ons. Une heure de lecture d'avance sur son voisin, et c'\u00e9tait le triomphe assur\u00e9 pendant des lustres. Ce qui, revers de la m\u00e9daille, pouvait valoir au trop brillant deux ou trois paquets d'injures pour avoir remontr\u00e9 \u00e0 l'un, donc \u00e0 l'autre, un talent certain pour lire, donc luire en amont.\n\nBien s\u00fbr, trop bien \u00e9lev\u00e9s pour \u00eatre un tant soit peu courageux, ces esth\u00e8tes affichaient en mati\u00e8re politique un anarchisme de droite de bon aloi leur permettant, avec les bourgeois, d'appuyer sur l'\u00e9pith\u00e8te dextre, pour les gens de gauche, de bien faire remarquer leur ind\u00e9fectible option libertaire. Dans tous les cas, ils se retrouvaient au centre, autant dire pr\u00eats \u00e0 se vendre aux plus offrants. Quoi qu'il en soit, ils n'auraient pas cill\u00e9 devant le choix entre le salut de l'humanit\u00e9 tout enti\u00e8re et la sauvegarde d'une estampe d'un ma\u00eetre japonais. Et l'holocauste leur aurait paru plus d\u00e9fendable que le papier br\u00fbl\u00e9, tel N\u00e9ron incendiant Rome et jouissant de voir se consumer sous ses yeux un peuple qu'il a toujours consid\u00e9r\u00e9 comme de la vermine.\n\nToute r\u00e9f\u00e9rence \u00e0 l'histoire ou \u00e0 la politique leur semblait grossi\u00e8re, vulgaire, tellement malvenue qu'il ne restait qu'\u00e0 plonger le nez dans l'assiette ou \u00e0 oser je ne sais quelle contorsion pour se faire pardonner pareilles fautes de go\u00fbt si par malheur la question avait \u00e9t\u00e9 effleur\u00e9e. Que p\u00e9risse le monde, que cr\u00e8vent les pauvres, si d'aventure on leur laissait, au milieu des ruines et des d\u00e9combres, leur salon minable, leur c\u00e9nacle haineux. Dans les apocalypses encore fumantes, ils trouveraient encore moyen de lancer leur fiel sur les cadavres et les charognes de ceux qu'ils envient et qui font l'objet de leur ressentiment.\n\nJe songe \u00e0 la fin de La Recherche, aux perruques poudr\u00e9es, aux visages effondr\u00e9s, aux rides, aux embonpoints de ceux qui, jadis, contenaient leur haine dans une maigreur d\u00e9laiss\u00e9e au profit d'une ob\u00e9sit\u00e9 qui dit leur d\u00e9sir de poids social et ne trahit qu'une comp\u00e9tence dans l'art de g\u00e9n\u00e9rer le superf\u00e9tatoire. Infatu\u00e9s, s\u00fbrs d'eux, arrogants, ils sont satur\u00e9s de fiel, pr\u00eats \u00e0 d\u00e9gorger des flots de bile. Lorsque la faux les d\u00e9capitera pour les emporter dans le trou du cimeti\u00e8re, ils cracheront encore leur venin et ils expireront couverts de leur propre bave. Bien \u00e9videmment, loin d'avoir v\u00e9cu d'art et d'amour, ils auront croupi dans la haine, cuits par elle, confits dans le ressentiment. Et leurs traces se comptabiliseront sur le seul march\u00e9 des mesquineries, l\u00e0 o\u00f9 se battent les m\u00e9diocres, les rat\u00e9s, les demi-soldes et les existences avort\u00e9es gisant sur des terrains vagues o\u00f9 l'on se contente d'\u00eatre encore vivant, bien que d\u00e9membr\u00e9, n'ayant plus m\u00eame figure humaine.\n\nBien s\u00fbr, et l'on m'aura compris, cette permanence du salon Verdurin n'est qu'une vue de l'esprit et, comme on dit ailleurs, toute ressemblance avec des personnages ayant exist\u00e9 ne serait que fortuite et pure co\u00efncidence. Pour que je prenne la plume en de pareilles pages, il aurait fallu, par exemple, que je sois abus\u00e9 en sortant de l'adolescence, que je r\u00e9alise mon erreur deux ou trois ann\u00e9es plus tard. Mais \u00e7a n'aurait pas suffi. Si tel avait \u00e9t\u00e9 le cas, je me serais content\u00e9 de silence et j'aurais mis toute mon ardeur \u00e0 oublier mes sottises pub\u00e8res. On n'est jamais \u00e0 l'abri d'une erreur quand il faut changer de monde, quitter celui de l'innocence, aborder celui du cynisme et qu'on ignore tout de La Rochefoucauld ou de Baltasar Graci\u00e2n.\n\nPour justifier de tels mots, il aurait fallu que ceux-l\u00e0 sortent, eux, du silence dans lequel il y avait un peu de l'\u00e9l\u00e9gance requise \u00e0 demeurer. S'ils avaient d\u00e9rog\u00e9, consacr\u00e9 leur temps \u00e0 chercher l'affrontement, accumul\u00e9 les piques, les occasions, les salet\u00e9s, alors ils auraient provoqu\u00e9 mon courroux, et seulement \u00e0 ce moment j'aurais consenti \u00e0 aller sur le terrain, juste une fois, pour leur montrer qu'en dehors du code de l'honneur qui voudrait d'abord le silence, ensuite l'acceptation du combat seulement avec des \u00e9gaux, il existe des fa\u00e7ons de dire, une fois, juste une fois, qu'on aimerait un minimum de la courtoisie qui r\u00e9glait jadis les rapports entre gens debout. Mais comme de pareils avortons sont impensables sur le terrain r\u00e9el, je me suis content\u00e9 d'un jeu th\u00e9orique. Ceux qui n'en croiraient rien diraient qu'ils savent possible une pareille engeance et que les fictions sont toujours tr\u00e8s en de\u00e7\u00e0 de la r\u00e9alit\u00e9. Proust sociologue ? Pourquoi pas...\n20\n\nBRUITS DU PLIAGE DES NUAGES\n\nDira-t-on assez qu'il faut se m\u00e9fier de tous ceux qui avouent sans complexe aimer les nuages au point qu'ils les fixent, les suivent du regard, les interrogent, les sollicitent, les accompagnent dans leurs trajets nomades et les chevauchent parfois au point de devenir eux-m\u00eames des figures tiss\u00e9es de l'\u00e9toffe fractale de ces moutonnements ou de ces striages po\u00e9tiques ? Baudelaire a \u00e9crit la passion de l'homme \u00e9nigmatique qui n'aime personne, ne tient \u00e0 rien, ou \u00e0 si peu, mais r\u00e9torque \u00e0 son questionneur \u00e9tonn\u00e9 qu'il affectionne par-dessus tout les nuages, les merveilleux nuages. Un archipel des amateurs de nuages devrait d'ailleurs voir le jour, autant qu'un autre compos\u00e9 des fous de pliages ou des points, des fanatiques de lignes ou de cris, d'une couleur ou d'un son. Pascal Dusapin photographie les nuages et ne cache pas sa folie pour leurs fa\u00e7ons d'incarner le mouvement, le d\u00e9placement, le dynamisme des trajets de lumi\u00e8re, sinon les variations sur le th\u00e8me du temps, celui des chronom\u00e8tres tout autant que des m\u00e9t\u00e9orologues.\n\nForme de nuages, construction et vitesse, volumes et dynamismes, tout exprime la passion pour l'objet fractal, celui dont le bord, quel qu'il soit, dit l'ensemble, raconte la totalit\u00e9. L\u00e0 o\u00f9 niche l'infime g\u00eet l'infini, principe de base, premier donc, de l'amateur de nuages. La dentelure, les d\u00e9coupages f\u00e9briles, les fissures ouvrageant les limites expriment et rendent possibles des images qui contiennent d'autres images, exactement les m\u00eames. Le nuage fournit aussi l'occasion d'exprimer une m\u00e9taphysique du M\u00eame et de l'Autre, second principe de l'homme \u00e9nigmatique, en vertu de quoi le semblable diff\u00e8re du dissemblable dans la co\u00efncidence des identit\u00e9s. Et l'on pourrait aussi raconter le pliage acteur de nuage, r\u00e9troversion de cette mati\u00e8re qui n'en est pas une, involution, circonvolution, retournement, pli, d\u00e9pli, surpli. Troisi\u00e8me principe du m\u00e9taphysicien des cumulo-nimbus : une ontologie du pli. Cessons l\u00e0. La richesse de ces blocs d'\u00e9ther mobile montre qui les aime en d\u00e9miurge d'un genre particulier parce qu'il travaille des mati\u00e8res qui n'en sont pas, ou si peu de temps.\n\nDu nuage on pourrait dire la forme et la force conceptuelle. Pour ce faire, il faudrait recourir \u00e0 Gilles Deleuze dont les incursions dans les territoires o\u00f9 la pens\u00e9e se d\u00e9place en puissance nomade rendent pensable une th\u00e9orie o\u00f9 correspondent, au sens baudelairien, les nuages, le concept, le temps et la musique. De sorte que Pascal Dusapin propose en musique ce que l'on retrouve en philosophie chez Gilles Deleuze. Pour autant, le glissement serait-il possible qui autorise \u00e0 parler de musique deleuzienne pour celle de Pascal Dusapin comme celle de Debussy me semble bergsonienne et celle de Var\u00e8se nietzsch\u00e9enne ? Il faudrait imaginer Itou (1985), une pi\u00e8ce pour clarinette basse, Time Zones (1989), le deuxi\u00e8me quatuor \u00e0 cordes ou Rom\u00e9o & Juliette (1985-1988), le premier op\u00e9ra, comme des points de rep\u00e8re dans l'\u0153uvre et la soixantaine de num\u00e9ros d'opus \u00e0 ce jour (mars 1997), tels des embl\u00e8mes o\u00f9 se disent les fa\u00e7ons deleuziennes de Dusapin en musique.\n\nOublions les consid\u00e9rations sur la nature du temps telles qu'elles apparaissent chez Deleuze et Boulez, l'un m\u00e9ditant l'autre, et retour. Temps puls\u00e9, double sur son terrain de l'espace stri\u00e9 dans celui de la peinture, ou temps non puls\u00e9, double de l'espace lisse, temps du tempo et des hypoth\u00e8ses sonores en efflorescences qui permettent d'accrocher l'intelligence et de fournir des points de rep\u00e8re \u00e0 l'oreille, ou temps amorphe, voil\u00e0 qui n'entre plus ici en ligne de compte. La musique moderne et contemporaine vit immerg\u00e9e dans ce plan liss\u00e9 qui d\u00e9classe et surclasse le quadrillage m\u00e9trique et formel. Dans le temps o\u00f9 elle se d\u00e9ploie, la musique d'aujourd'hui s'\u00e9tend dans la pl\u00e9nitude sur cet espace o\u00f9 rien n'adh\u00e8re et o\u00f9 tout vit de mouvement, de dynamisme, de glissements, de d\u00e9placements, de condensations. Dans l'\u00e8re des intensit\u00e9s, des densit\u00e9s et des flux sonores, la composition de ce si\u00e8cle qui s'ach\u00e8ve se sera d\u00e9ploy\u00e9e dans un monde dont le cartographe le plus avis\u00e9 est sans conteste Deleuze.\n\nHier, la musique \u00e9tait un art du seul temps. Elle renvoie aujourd'hui \u00e0 un autre temps, celui d'apr\u00e8s la relativit\u00e9 g\u00e9n\u00e9rale, doubl\u00e9 d'un souci majeur pour l'espace. L\u00e0 o\u00f9 les forces errent sans formes, les formes ne peuvent \u00eatre sans forces : cette ultime d\u00e9cennie musicale exprime le d\u00e9sir de r\u00e9concilier ces deux instances dans l'av\u00e8nement d'un langage dont la coh\u00e9rence se montre enfin. Dans Mille Plateaux, Deleuze et Guattari avancent que cette r\u00e9volution sonore ach\u00e8ve l'\u00e2ge des oiseaux et rend possible celui des insectes. Le son capt\u00e9 par Pascal Dusapin suppose une patience pour les chasses subtiles qui d\u00e9couvrent un insecte d'un genre nouveau, d'un embranchement particulier, proc\u00e9dant d'un ensemble nettement domin\u00e9 par le type deleuzien. Et l'on est deleuzien en faisant primer les agencements, en architecte qui n'oublie pas, pour en inverser le sens, l'id\u00e9e de Schopenhauer selon laquelle l'architecture, c'est de la musique congel\u00e9e. Puisque la musique devient de l'architecture congel\u00e9e, il faut la penser en terme de style, et moins de grammaire, pour dire ce qui, au-del\u00e0 des objets avec lesquels on compose, r\u00e9git les relations entre ces formes. La lettre importe moins que le mot, moins le mot que la phrase, moins la phrase que ce qu'elle permet de composer, paragraphes, parties, ensembles et sous-ensembles, le tout culminant dans une \u0153uvre valant par les agencements qui l'auront rendue possible et qu'elle aura rendus possibles elle aussi.\n\nBien s\u00fbr, le mat\u00e9riau de Pascal Dusapin est la force, et l'ensemble des variations envisageables sur ce th\u00e8me : \u00e9nergie et puissance, organisation de flux, mat\u00e9rialisation de pulsions, vouloir d'intensit\u00e9s, dialectique de la concentration et de la dilatation, logique de v\u00e9locit\u00e9 et principe lyrique, tout ce qui permet une force, \u00e0 savoir une violence sachant o\u00f9 elle va \u2013 le rem\u00e8de m\u00eame \u00e0 la violence \u2013 devient un temps de culture, d'artifice musical et d'antinature. L'essence de la musique, depuis les premi\u00e8res percussions hasardeuses et al\u00e9atoires de l'\u00e9poque pr\u00e9historique aux si\u00e8cles \u00e9lectriques et \u00e9lectroniques, de Lascaux \u00e0 l'IRCAM, g\u00eet dans ce corps \u00e0 corps d'un sujet et d'un mat\u00e9riau immat\u00e9riel. Mais la force commune \u00e0 tous, pourvu qu'ils composent de la musique, ne permet en rien d'exprimer la sp\u00e9cificit\u00e9 des formes : la grammaire en tant que telle ne formule rien du style comme signature.\n\nOr la signature de Pascal Dusapin suppose un d\u00e9tour par le genre de graphologie pratiqu\u00e9 par Deleuze et qui renvoie \u00e0 des concepts op\u00e9ratoires pour rendre compte d'un \u00e9difice, de sa coh\u00e9rence, d'une architecture singuli\u00e8re et de ce qui la rend reconnaissable. Producteur de formes et architecte d'un agencement de celles-ci, le compositeur \u00e9quilibre son travail dans les trois temps d'une dialectique qui suppose l'embo\u00eetement des moments successifs pour produire l'\u00e9v\u00e9nement dans lequel culmine la complexit\u00e9 de son talent \u2013 l'op\u00e9ra. Premier temps : le monde du point et de son expansion dans l'espace, la ligne. O\u00f9 l'on retrouve l'ensemble des pi\u00e8ces \u00e9crites pour des instruments seuls. Deuxi\u00e8me temps : le monde des plans et de leur d\u00e9ploiement dans un volume, les blocs. O\u00f9 l'on rencontre les pi\u00e8ces qui n'existent que dans la logique d'un agencement dont la complexit\u00e9 va croissant de deux \u00e0 une multitude de lignes \u00e0 enchev\u00eatrer, du duo aux compositions orchestrales dans toutes les modalit\u00e9s concertantes, au sens \u00e9tymologique. En mati\u00e8re de temps brut, chronom\u00e9trable, ce deuxi\u00e8me moment constitue l'essentiel du temps musical ouvrag\u00e9 par l'artiste, un peu plus de la moiti\u00e9 de sa masse globale. Troisi\u00e8me temps : le monde des r\u00e9seaux et de leurs maillages agenc\u00e9s en textures qui produisent les figures les plus complexes o\u00f9 se disent le pli et le devenir, concepts majeurs de la philosophie deleuzienne. O\u00f9 l'on sait pouvoir ranger les op\u00e9ras dont l'\u00e9tymologie latine exprime la co\u00efncidence avec ce qui fait \u0153uvre. L'ensemble, un peu plus de mille minutes, pas encore vingt heures, se profile sur le mode rhizomatique illustrant \u00e0 ravir les multiplicit\u00e9s et la libert\u00e9 de mouvement g\u00e9n\u00e9ralis\u00e9e qui traversent m\u00eamement la musique de Pascal Dusapin et la philosophie de Deleuze et Guattari.\n\nDe la ligne, donc. El\u00e9mentaire, primitive, au sens imm\u00e9diat, premi\u00e8re, elle renvoie au minimum requis en mati\u00e8re musicale pour \u00eatre. Sur cette expansion du point dans l'espace, il faut rappeler la formation d'organiste de Pascal Dusapin. Et son enfance. Une double confidence faite \u00e0 Danielle Cohen-Levinas permet de savoir une ballade dans la prime jeunesse effectu\u00e9e le long de la Moselle, le vent qui souffle dans la fut\u00e9e de roseaux, puis une punition au lyc\u00e9e, un week-end, qui oblige l'amateur de rives \u00e0 se rendre \u00e0 l'\u00e9glise pour assister \u00e0 une messe alors qu'il est \u00e9lev\u00e9 dans la plus pure tradition ath\u00e9e. Premi\u00e8re rencontre avec l'orgue qui devient son instrument de pr\u00e9dilection. Sc\u00e8nes originaires telles qu'on les d\u00e9couvre dans la mythologie grecque ou chez Virgile pour expliquer la naissance de la musique avec les roseaux ; exp\u00e9riences fondatrices rep\u00e9r\u00e9es dans la psychanalyse existentielle sartrienne pour raconter la r\u00e9currence d'un registre et la permanence d'une facture : l'autobiographie d'un corps, ici comme ailleurs, pr\u00e9lude aux lev\u00e9es de brumes sur le sens.\n\nL'absence d'\u00e9criture, soliste ou autre, pour ce qui ressemble de pr\u00e8s ou de loin \u00e0 un orgue, se double et se compl\u00e8te par une pratique de certains instruments en organiste. Car Pascal Dusapin revendique l'installation de son travail dans cet espace de musique pure ouvert gr\u00e2ce \u00e0 la r\u00e9sonance laiss\u00e9e par l'\u00e9cho de l'accord final qui troue le silence pendant quinze \u00e0 vingt secondes durant lesquelles la musique tourne dans l'espace, l'envahit, le hante, l'habite et se d\u00e9ploie dans son essence. Ce pur objet sonore g\u00e9n\u00e9r\u00e9 par l'orgue qui vient de se taire, mais n'a pas encore \u00e9t\u00e9 recouvert par le silence, produit \u00e0 lui seul une fixation de tension \u00e0 partir de laquelle advient le d\u00e9sir du compositeur d'entrer en r\u00e9sonance avec le monde tout entier. De cet objet sonore pur, on peut dire ce que Boulez raconte de la musique : elle se fait avec des sons, certes, mais aussi avec des \u00e9critures, des gestes et des concepts. Pascal Dusapin dirait certainement avec des affects et des percepts, deux notions deleuziennes. Affects et percepts transfigur\u00e9s en concepts, voil\u00e0 de quoi suivre \u00e0 la trace les premi\u00e8res lignes de force du travail du compositeur.\n\nCette citation se rep\u00e8re \u00e0 plusieurs reprises dans l'ensemble du travail \u00e9crit pour des instruments solistes. Mais aussi, de la fa\u00e7on la plus extravagante et la plus manifeste, dans les quarante-cinq minutes d'une note tenue \u00e0 la p\u00e9dale d'un positif, annonc\u00e9es avant toute ouverture de rideau, en prol\u00e9gom\u00e8ne \u00e0 la repr\u00e9sentation au concert de Medeamaterial, \u2013 comme l'anachrousis des anciens Grecs rendait possible l'ordre du discours musical propos\u00e9 en suite du son primitif, primordial. Son d'avant le son, parole informe d'avant le verbe, r\u00e9miniscence d'un propyl\u00e9e de T\u00e9tralogie o\u00f9 se fomente le monde avant le monde, le r\u00e9el avant le r\u00e9el. Ce paysage sonore d'un chaos pas m\u00eame chaotique dessine les contours de ce qui est sans bord mais d'o\u00f9 advient la forme primitive, la premi\u00e8re manifestation d'artifice dans un univers de pure nature.\n\nCe souci d'un son d'avant le son, puis d'un affect r\u00e9current transfigur\u00e9 en concept, le souffle d'un vent tendu pour une tension mentale, voil\u00e0 qui installe le compositeur au c\u0153ur du point \u00e9tir\u00e9 dans l'espace jusqu'\u00e0 la ligne. Le son est moins la mati\u00e8re trait\u00e9e que l'id\u00e9e. D'ailleurs l'onde sonore a la m\u00eame consistance que le concept, sinon le nuage : visible dans son \u00eatre, ou dans ses effets, mais imperceptible autrement que sur le mode de l'\u00e9vanescence. Le temps musical lisse de la musique contemporaine ne laisse aucune chance \u00e0 l'accrochage de rep\u00e8res autres que des mouvements. Pas de points fixes, mais des dynamiques, pas de rochers pour un arraisonnement, mais des nuages identiques, tels qu'en eux-m\u00eames l'\u00e9ternit\u00e9 les change.\n\nApr\u00e8s le point, cette note tenue sans modulation, dans la pure essence de son \u00eatre, dans son existence nue, la ligne advient et s'anime de vibrations. D'o\u00f9 le d\u00e9but des agencements, la naissance de ce qui se trouve modul\u00e9, par la suite, \u00e0 des degr\u00e9s de complexit\u00e9 infinis. L'onde sonore, la tension, le concept, l'id\u00e9e supposent une mise en perspective. Adviennent aussi les agencements constituant le centre de l'\u0153uvre de Deleuze et de Dusapin. Moins l'\u00eatre que ce qui se passe entre ce qui est : voil\u00e0 ce qui doit soucier et pr\u00e9occuper un amateur de m\u00e9taphysique immanente et d'ontologie mat\u00e9rialiste radicale.\n\nPascal Dusapin confie : \u00abLe sens en litt\u00e9rature, comme le son en musique ou la couleur en peinture ne rel\u00e8vent pas pour moi d'un ordre hi\u00e9rarchique premier. C'est pourquoi j'ai coutume de dire que je ne m'int\u00e9resse pas au son bien que j'\u00e9crive des partitions qui sonnent. J'aime lorsque l'objet dispara\u00eet derri\u00e8re le concept, tout en assumant sa fonction d'oeuvre \u00e0 voir, \u00e0 lire ou \u00e0 \u00e9couter.\u00bb Faire dispara\u00eetre l'objet derri\u00e8re le concept, voil\u00e0 qui installe l'id\u00e9e en musique au premier plan et suppose l'agencement comme ce qui rend possible un ordre d'exposition et de proposition. Point, puis ligne, voil\u00e0 les premiers pas du concept, ses balbutiements.\n\nOutre l'annonce sonore primitive faite \u00e0 M\u00e9d\u00e9e, dans l'op\u00e9ra, la ligne distingue la forme prise par le concept d\u00e8s son tra\u00e7age musical. R\u00e9sonance du point d'orgue, tournoiements du son dans l'architecture, volutes dans le roman, le gothique ou le caisson neutre de l'euphonie des ing\u00e9nieurs, le concept jubile d'un pur d\u00e9ploiement lin\u00e9aire dans l'espace. L'expansion du point structure une grammaire d'objets stylistiques et initie un temp\u00e9rament, un ton reconnaissable, un d\u00e9bit de vent singulier dans le tuyau des roseaux mosellans. Chaque r\u00e9it\u00e9ration s'appuie sur une diff\u00e9renciation : le M\u00eame musical se dit dans les instruments Autres, Diff\u00e9rents \u2013 une petite quinzaine, mais toujours pas d'orgue. Clarinette basse, alto, violoncelle, contrebasse : peut-on mieux signifier le registre rond, grave, masculin, profond, voluptueux tout en \u00e9vitant de dire nettement le clavier et le buffet ?\n\nLes titres eux-m\u00eames co\u00efncident avec la forme et le contenu, le concept et sa bri\u00e8vet\u00e9 : monosyllabes ou doubles, mais jamais au-del\u00e0 de trois : Inside (1980), If (1984), Item (1985), Itou (1985), Mimi (1986), Ici (1987), Indeed (1997). Le minimalisme en tout, l'\u00e9conomie, le titre bref autant que la dur\u00e9e de la pi\u00e8ce, entre cinq et quatorze minutes avec une moyenne de neuf. L'aphorisme plus que la nouvelle, avec ce que cela suppose d'obligation dans la quintessence, la concentration, la force et la densit\u00e9. Cordes et vents, essentiellement, pour ce monde simple o\u00f9 l'on retrouve, la mythologie aidant, sinon la po\u00e9tique des \u00e9v\u00e9nements, des visc\u00e8res animaux tendus sur des caisses, des carapaces, des bois creus\u00e9s, des troncs \u00e9vid\u00e9s, qui laissent, comme avec la lyre et les roseaux, une fois encore le vent produire les vibrations n\u00e9cessaires aux m\u00e9tamorphoses des affects de l'enfance en concepts d'aujourd'hui destin\u00e9s \u00e0 devenir des percepts pour autrui.\n\nLa ligne conna\u00eet des variations dans le processus m\u00eame de son av\u00e8nement : \u00e0 chaque avanc\u00e9e du point, \u00e0 chacun de ses d\u00e9placements dans l'espace se constituent des variations de hauteur et des m\u00e9tamorphoses d'intensit\u00e9, des infl\u00e9chissements dynamiques et des ruptures rythmiques. Toujours, c'est l'occasion d'exp\u00e9rimenter une gamme de possibles, une \u00e9tendue d'hypoth\u00e8ses auditives. Parfois m\u00eame, le ludisme c\u00f4toie l'essai d'une int\u00e9gration sonore transversale, par exemple le bruit des cl\u00e9s de la clarinette basse propos\u00e9 en compl\u00e9ment musical contrapuntique, sinon polyphonique \u00e0 l'ensemble de la pi\u00e8ce.\n\nLes vents jouent avec les percussions dans le dessein de constituer un animal sonore et un instrument musical mutant illustrant la th\u00e9orie deleuzienne du devenir-enfant ou du devenir-animal, du devenir-insecte de tel ou tel objet d'orchestre habituel. Ailleurs, l'agencement des lignes produit des \u00e9bauches de plans o\u00f9 se c\u00f4toient des registres parents, jazz ou free-jazz dans Itou, silences et blancs contrast\u00e9s par le son et la note qui d\u00e9limitent des zones dans Item, parl\u00e9 et chant\u00e9 dans Il-Li-Ko, couleurs instrumentales et couleurs vocales m\u00eamement trait\u00e9es avec le seul trombone dans Indeed.\n\nLe monde sonore du point dans le temps devenant ligne dans l'espace fournit les formes a priori de la sensibilit\u00e9 du compositeur. A partir de celles-ci se structure un autre monde, pas obligatoirement plus complexe, ou moins, mais s\u00fbrement plus satur\u00e9, moins habit\u00e9 ou hant\u00e9 par le vide, le silence. De l'aphorisme \u00e0 la nouvelle, c'est moins, chez le m\u00eame auteur, le monde exprim\u00e9 qui se transforme, se m\u00e9tamorphose, que la fa\u00e7on diff\u00e9rente d'exprimer un m\u00eame univers. Des pages du Gai Savoir de Nietzsche o\u00f9 le philosophe use des \u00e9clairs, des traits, de la foudre, \u00e0 celles de Par-del\u00e0 le bien et le mal o\u00f9 il juxtapose des formes plus longues, il y a le m\u00eame rapport qu'entre le monde de la ligne, chez Pascal Dusapin, et celui des plans et des blocs qui rend les agencements possibles et manifestes \u2013 avant le monde des textures, des maillages et des plis qui offre chez le compositeur l'\u00e9quivalent, chez le philosophe, d'Ainsi parlait Zarathoustra. Chaque fois, les moyens diff\u00e8rent pour produire une m\u00eame vision du monde, coh\u00e9rente et styl\u00e9e dans la perfection d'une \u0153uvre classique.\n\nDes blocs, pour suivre. Le point d\u00e9pli\u00e9 dans l'espace donne une ligne, c'est entendu. Cette m\u00eame ligne d\u00e9ploy\u00e9e dans le m\u00eame \u00e9ther fournit l'occasion de plans, de surfaces d\u00e9limit\u00e9es qui, agenc\u00e9es, pourvu qu'elles proc\u00e8dent d'au moins deux couleurs, deux mondes, deux instruments ou deux registres, sans limitation de nombre, structurent et g\u00e9n\u00e8rent des blocs. Du dessin on passe au plan de coupe et d'un monde en deux dimensions \u00e0 un autre, en trois. L\u00e0 o\u00f9 l'enfance des roseaux donnait sa teinte au monde primitif du solo lin\u00e9aire, la r\u00e9f\u00e9rence \u00e0 l'architecture fournit son mod\u00e8le, son \u00e9pist\u00e9m\u00e9, aurait dit Foucault, \u00e0 un Pascal Dusapin dont l'ontogen\u00e8se r\u00e9capitule la phylogen\u00e8se : trajet de toute la musique dans celui d'un seul individu.\n\nLe compositeur ne cache pas une formation d'architecte qui le rend familier des agencements o\u00f9 s'expriment lignes de force, lignes de fuite, pouss\u00e9es et r\u00e9partition des forces, cl\u00e9s de vo\u00fbte, arcatures et voussures, crois\u00e9es et montants, ordres, styles, corps d'\u00e9difice ou m\u00eame ornements. De quoi fournir un second registre pour une autre grammaire d'objets susceptibles de mieux saisir le langage de Pascal Dusapin. Dans ce deuxi\u00e8me temps de sa pens\u00e9e musicale, on peut suivre \u00e0 la trace la m\u00e9tamorphose des forces en formes, le devenir g\u00e9ographe de l'historien, le passage du temps d\u00e9pli\u00e9 \u00e0 l'espace d\u00e9ploy\u00e9.\n\nLes blocs, qui furent d'ab\u00eeme chez Sade, sont ici d'\u00e9nergie pure et de variations sur le th\u00e8me apollinien de la forme. La quintessence de ce moment peut se lire dans le deuxi\u00e8me quatuor \u00e0 cordes dont le titre, Time Zones (1989), dit assez la compl\u00e9mentarit\u00e9 des d\u00e9sirs de temps et d'espace conjugu\u00e9s. Ces deux formes, r\u00e9unies sans ambages depuis la Critique de la Raison pure, rendent possibles, ici comme dans le criticisme kantien, l'intuition sensible et le principe de la connaissance a priori. En tant que formes du sens interne, elles autorisent l'intuition de soi et de notre \u00e9tat int\u00e9rieur. Sans elles, pas de sujet, donc pas de musique pensable. Le deuxi\u00e8me quatuor offre un moment embl\u00e9matique de philosophie \u00e0 l'\u00e9tat pur, l'\u00e9quivalent, dans la profusion et la richesse des Mille Plateaux de Deleuze et Guattari, dans la production de ce si\u00e8cle : une somme ath\u00e9ologique, une encyclop\u00e9die de l'immanence, un registre non exhaustif, mais d'une vastitude sans nom, du plus grand nombre d'agencements imaginables.\n\nAu degr\u00e9 z\u00e9ro de l'agencement, on trouve le duo. Deux points en mouvement dans un espace qui, sous forme de lignes semblables \u00e0 celles qui \u00e9voluent dans les toiles de Mir\u00f3, parfois se coupent, se rapprochent, s'\u00e9loignent, s'ignorent, se d\u00e9sirent et finissent par dessiner tout un monde sonore. Le registre solo oblige \u00e0 la ligne, exige le dessin au trait. Le registre multiple r\u00e9v\u00e8le des entrecroisements qui architecturent le r\u00e9el musiqu\u00e9 en plans. La couleur d'une ligne autonome, solitaire, d\u00e9limite des espaces qui entrent en relation sous la rubrique desdits plans. Ceux-l\u00e0 m\u00eames, agenc\u00e9s eux aussi, donnent naissance \u00e0 des blocs, morceaux \u00e9paissis dans le volume sonore jusqu'\u00e0 donner l'occasion de mat\u00e9riaux plus susceptibles d'agencements que des lignes.\n\nDe deux au multiple par dizaines r\u00e9unies en groupes, un ensemble, un orchestre et autres combinatoires chiffr\u00e9es, tout semble possible en mati\u00e8re de nombre. Le son s'en trouve d'autant induit. L\u00e0 o\u00f9 Deleuze d\u00e9passe philosophiquement le platonisme et ce qui le caract\u00e9rise radicalement, \u2013 l'opposition binaire, le dualisme, l'intelligible oppos\u00e9 au sensible \u2013, il d\u00e9cr\u00e8te l'av\u00e8nement de la multiplicit\u00e9 et du rhizomatique. Pascal Dusapin s'engage sur le m\u00eame terrain, versant musical. Constructions libres en arborescences sauvages, croissances en devenirs divers, multiplications fractales et d\u00e9multiplications en \u00e9cho, les voies du compositeur conduisent dans des clairi\u00e8res o\u00f9 priment la connexion, la relation, la mise en perspective. Le rhizome suppose l'art de connecter les points et d'agencer les lignes. Voil\u00e0 tout le programme de Pascal Dusapin dans ce deuxi\u00e8me temps de sa dialectique intellectuelle.\n\nQue dit Time Zones ? Quelles pouss\u00e9es rhizomatiques peut-on rep\u00e9rer, ici ou l\u00e0, subjectivement, \u00e0 l'\u00e9coute? D'abord que se compl\u00e8tent, s'ajoutent et se pr\u00e9cisent des lieux et des dates : Tokyo, Paris, Rotterdam, New York, Bruxelles, S\u00e2o Paulo, Gen\u00e8ve, La Napoule, puis un \u00e9tagement de dates entre 1988 et 1990, l'\u00e9cart dans lequel s'\u00e9crit l'\u0153uvre. Agencement temporel et agencement spatial, zone de Time et Time de la zone. Vingt-quatre pi\u00e8ces pour quatuor \u00e0 cordes constituent cet objet musical, soit autant que de fuseaux horaires et d'heures dans une journ\u00e9e, la plan\u00e8te, l\u00e0 encore, comme m\u00e9taphore sph\u00e9rique de l'espace et la journ\u00e9e en \u00e9quivalent circulaire du temps. Ces fuseaux se pointent au nord et au sud en un lieu nodal, les deux p\u00f4les, qui diraient le souci binaire du platonicien quand Pascal Dusapin opte pour la multiplicit\u00e9 rhizomatique des chiffres du monde.\n\nAu commencement semble une respiration surprise dans un sommeil que hanteraient des songes et des r\u00eaves. La musique habit\u00e9e par le silence ponctue d'autres zones et rend possibles d'autres temps. De la m\u00eame mani\u00e8re, les tensions, les \u00e9tirements de lignes m\u00e9lodiques, les \u00e9longations et le p\u00e9trissage de la mati\u00e8re sonore montrent une fa\u00e7on d'habiter l'espace lisse (Time zone 1, 2). Expansions et silences, striages et pointages, \u00e9tagements et s\u00e9quences de plans multiples ou de hauteurs diversifi\u00e9es (3) ici. L\u00e0, bourdonnements, bruissement d'ailes d'insectes, k\u00e9ratine musiqu\u00e9e, vibrations, stridences, \u00e9chelonnements et production de plans, soit en avant, soit en retrait, circulation du son en dynamiques de proximit\u00e9 ou de distance (4). Ailleurs, travail de la puissance qui chemine vers l'acte et progresse du c\u00f4t\u00e9 de l'av\u00e8nement, disposition de points sur le mode pictural (5). Une fois, langueurs, expansion en directions multiples, procession rhizomatique, dialectique ludique d'\u00e9preuve avec la mati\u00e8re, sa souplesse, sa ductilit\u00e9, sa r\u00e9sistance aussi (7). Une autre, exp\u00e9rimentation d'intensit\u00e9s et de couleur dans les aigus, virtuosit\u00e9 et d\u00e9monstrations d'aptitudes du son (8). Ou rondeurs, graves et m\u00e9tamorphoses d'une ligne, progression et efforts accomplis pour un autre statut (9, 10, 11, 12). Respirations entrav\u00e9es, m\u00e9lancolie, emp\u00eachement d'expansion, reptation (13). Aurores musicales, levers du jour, clart\u00e9 irradiante, saturation lumineuse (14). Consonances, complicit\u00e9s (15, 16, 17), stries, entailles, coupures (18), \u00e9tiage, jeux de lignes parall\u00e8les et de plans superpos\u00e9s, m\u00e9moire de point d'orgue (19). Puissance et violence sourde, rondeur, expansion d'une masse (20). Enfin, et pour le dire dans un mot de F\u00e9lix Guattari, chaosmose dont \u00e9mergent une ligne tendue et une autre, musiqu\u00e9e en hauteurs multiples. Ostinato, basse continue, hauteurs variables, intensit\u00e9s conjugu\u00e9es et final en point d'orgue (21, 22, 23, 24). Autant dire que ces trajets dans le temps et dans l'espace, sur des continents divers et en des heures changeantes autorisent une mise en ordre sonore du monde parente de celle qu'on trouve expos\u00e9e dans Mille Plateaux, une somme o\u00f9 se raconte la multiplicit\u00e9 des r\u00e9seaux et des agencements, multiplicit\u00e9, mais \u00e0 aucun moment complexit\u00e9 dont il faudrait constater l'\u00e9vidence avant de renoncer \u00e0 savoir ou \u00e0 comprendre. Au contraire, Pascal Dusapin compose en gnostique et propose en m\u00e9taphysicien qui reprend \u00e0 son compte la m\u00e9taphore rhizomatique afin de suivre quelques-uns des trajets sonores qui expriment au mieux le r\u00e9el modifi\u00e9 sous l'angle du son.\n\nTime Zones \u00e9quivaut aux projets encyclop\u00e9diques de Diderot, Novalis ou Hegel dans lesquels sont propos\u00e9es des odyss\u00e9es de la conscience et des p\u00e9r\u00e9grinations organis\u00e9es en fonction d'une cl\u00f4ture qui comprend en son principe le paradoxe d'une possibilit\u00e9 d'ouverture. Achev\u00e9, mais mobile, complet, mais dynamique, l'\u00e9tat du savoir, ici sonore, cristallise ce qui peut \u00eatre dit d'un monde dans une forme propre, un style, un temp\u00e9rament. Il n'est pas anecdotique de savoir ce monde des plans et des agencements o\u00f9 d\u00e9j\u00e0 s'\u00e9panche le point en ligne confin\u00e9 dans une musique dite de chambre. La chambre vaut comme la tour montanienne, le po\u00eale cart\u00e9sien, le boudoir libertin, le caf\u00e9 des Lumi\u00e8res, elle annonce un lieu o\u00f9 l'\u00e2me du monde co\u00efncide avec celui qui le pense. Camera oscura dans laquelle se pr\u00e9pare une image sonore qui peut \u00eatre \u00e9pique et monumentale quand les agencements se font moins dans la musique que de la musique \u00e0 d'autres secteurs.\n\nPascal Dusapin agit au contraire de l'insulaire confin\u00e9 dans son univers et promeut des archipels : il cr\u00e9e des passages et, disons-le dans les concepts chers \u00e0 Deleuze et Guattari, il d\u00e9territorialise. Comment ? En ouvrant la musique sur le monde et non en la fermant sur elle-m\u00eame, en la faisant cheminer, nomader, pourrait-on dire, aux c\u00f4t\u00e9s de la peinture, des arts plastiques, de la chor\u00e9graphie, de l'\u00e9criture, de la philosophie, de la danse. De sorte que son d\u00e9sir se fait politique, dans le sens le plus noble du terme. Des artistes conceptuels et minimalistes am\u00e9ricains aux pages de Deleuze, bien s\u00fbr, en passant par Beckett ou Cadiot, son librettiste, Dante ou Lucr\u00e8ce, sans oublier Sarajevo ou le g\u00e9nocide arm\u00e9nien, sinon les conditions de production du monde capitaliste \u2013 ce que dit aussi Rom\u00e9o & Juliette \u2013 Pascal Dusapin produit une \u0153uvre d'art totale dans le sillage moins de Wagner que de Duchamp pour lequel la fronti\u00e8re entre la vie et l'art dispara\u00eet au profit soit d'une esth\u00e9tisation de l'existence, soit d'une vitalisation de l'art. Les agencements dans la musique se doublent, chez lui, d'un agencement avec la musique de toutes les autres disciplines qui permettent le nomadisme, la d\u00e9territorialisation et la libert\u00e9 des mouvements rhizomiques.\n\nAussi, \u00e0 l'extr\u00e9mit\u00e9 inverse du point et de la ligne, dont proc\u00e8de l'ancrage originel de la racine, apr\u00e8s passage au plan et blocs, on trouve une multiplicit\u00e9 d'aboutissements de radicelles et autres pouss\u00e9es qui proc\u00e8dent du rhizome. Le genre qui concentre le mieux cette figure du rhizome est sans conteste l'op\u00e9ra et Pascal Dusapin propose pour ce monde sublime et ce genre majeur, des hypoth\u00e8ses qui disent, une fois encore, sa parent\u00e9 avec les concepts de la philosophie deleuzienne, en l'occurrence ceux de devenir et de pli. Musique de maillages et de r\u00e9seaux, de filets tendus et d'agencements conceptuels, l'\u0153uvre du compositeur suppose une lecture et un abord en forme de d\u00e9pliage.\n\nDes plis, pour finir. La parution du Pli sous-titr\u00e9 Leibniz et le baroque, de Gilles Deleuze, date de 1988. C'est tr\u00e8s exactement la date d'ach\u00e8vement de Rom\u00e9o & Juliette, le premier op\u00e9ra de Pascal Dusapin commenc\u00e9 en 1985, date \u00e0 laquelle le philosophe proposait au lecteur le second volume de sa r\u00e9flexion sur le cin\u00e9ma, l'image, le temps et le mouvement. L'oeuvre du compositeur vaut aussi comme une proposition sur les relations entretenues par le son avec ces temps et ces mouvements. De m\u00eame, dans Mille Plateaux, Deleuze et Guattari ont mis au point une th\u00e9orie du devenir parente de ce que Medeamaterial propose en 1991 : une m\u00e9ditation sonore et conceptuelle sur le devenir-animal de M\u00e9d\u00e9e, sinon le devenir-hyst\u00e9rique ou le devenir-fou.\n\nDans Le Pli Deleuze \u00e9crit : \u00ab La forme qui r\u00e9v\u00e8le ses plis devient force. \u00bb Jamais mieux que dans Rom\u00e9o & Juliette Pascal Dusapin ne porte \u00e0 son paroxysme l'art du pliage, voire celui de m\u00e9tamorphoser les formes conceptuelles de son travail en pures forces musicales. Le temps de l'op\u00e9ra est celui de l'aboutissement des points transfigur\u00e9s en lignes, des plans m\u00e9tamorphos\u00e9s en blocs et des r\u00e9seaux agenc\u00e9s en plis. O\u00f9 l'on assiste aux trois op\u00e9rations de la dialectique nietzsch\u00e9enne qui conduisent du chameau au lion et du lion \u00e0 l'enfant marqu\u00e9 par l'innocence du devenir. L'av\u00e8nement des forces est consubstantiel \u00e0 celui de l'op\u00e9ra. En amont, tr\u00e8s en amont, se rep\u00e8rent les lignes solistes, les m\u00e9lodies, les voix, les mots aussi, les traitements et les exp\u00e9rimentations, tout ce qui d\u00e9bouche, comme le livre de Mallarm\u00e9, dans cette forme absolue qu'est l'op\u00e9ra, le Livre du Po\u00e8te. De l'aphorisme \u00e0 la nouvelle, avais-je \u00e9crit : ici le trajet se pr\u00e9cise et conduit de la nouvelle au roman.\n\nRom\u00e9o & Juliette rel\u00e8ve de l'origami \u00e0 l'\u00e9tat pur. Ecouter cette \u0153uvre, c'est viser le d\u00e9pliage, mettre le nuage \u00e0 plat, lisser ses formes fractales et d\u00e9couvrir le souffle qui les habite. L'ensemble vit des pliages : pliage de la forme, d'abord, o\u00f9 se trouvent modifi\u00e9s la facture, le temps et les figures ; pliage des sens, ensuite, o\u00f9 l'on rep\u00e8re le pliage des r\u00e9f\u00e9rences et des registres ; pliage, enfin, des sons o\u00f9 s'agencent en plis les dictions, les phon\u00e8mes et les langues. Pr\u00e9cisons.\n\nPliage de la forme : l'op\u00e9ra lui-m\u00eame appara\u00eet telle une composition organis\u00e9e autour d'un noyau, la R\u00e9volution, une pi\u00e8ce exclusivement musicale, \u00e0 partir de quoi s'organise la totalit\u00e9 en un rhizome qui n'exclut pas la sym\u00e9trie, sur le mode du chandelier. Cette R\u00e9volution est une ligne de force nette et pr\u00e9cise en vertu de quoi s'organise le pliage avec d'un c\u00f4t\u00e9 le prologue, le d\u00e9but, le matin, l'avant et, de l'autre, en correspondance \u00e9quilibr\u00e9e, l'apr\u00e8s, le soir, la fin, l'\u00e9pilogue. Ce qui constitue une s\u00e9rie selon le principe aussi bien philosophique, je pense \u00e0 Fourier, que musical, sur le mode du miroir \u2013 je songe \u00e0 la composition en arche du Vin des amants de Berg.\n\nLe pliage de la facture renvoie au pliage du temps : l'ensemble de l'op\u00e9ra s'organise autour de la R\u00e9volution. Que va-t-il se passer ? Que se passe-t-il? Que s'est-il pass\u00e9? Du futur au pass\u00e9 en passant par le pr\u00e9sent, la r\u00e9ponse se montre, tranchante : rien, \u00ab il ne s'est rien pass\u00e9, le futur est au pr\u00e9sent, la fin \u00e9tait d\u00e9j\u00e0 au d\u00e9but \u00bb. Pli\u00e9 sur lui-m\u00eame, le temps lin\u00e9aire occidental dispara\u00eet au profit d'un temps oriental, circulaire, bouddhiste, sinon nietzsch\u00e9en et cosmologique. O\u00f9 l'on retrouve l'hypoth\u00e8se deleuzienne d'une lecture de l'\u00e9ternel retour de Nietzsche comme principe s\u00e9lectif permettant l'av\u00e8nement du r\u00e9el et la conservation des forces.\n\nDe la m\u00eame mani\u00e8re, facture et temps renvoient au pliage des personnages conceptuels que sont Rom\u00e9o et Juliette. L'essence shakespearienne et le devenir mythologique dans l'histoire des repr\u00e9sentations occidentales rendent possible une multiplicit\u00e9 de variations sur ce th\u00e8me. Les deux personnes \u2013 les amoureux embl\u00e9matiques que l'on sait \u2013 deviennent \u00e9galement autres, transfigur\u00e9es sinon par l'amour, du moins par leur devenir-amoureux commun et leur volont\u00e9 de \u00ab r\u00e9ussir une aventure extraordinaire une extraordinaire aventure \u00bb. D'o\u00f9 une nouvelle identit\u00e9 qui permet de conserver leur ressemblance tout en pr\u00e9cisant les dis-semblances : m\u00eames et autres, ils deviennent respectivement Rom\u00e9o 2 et Juliette 2, voix diff\u00e9rentes, registres et tessitures autres, bien qu'identit\u00e9s presque semblables. Pareillement, l'Ancien Monde peut se plier sur le Nouveau, l'Europe aux anciens parapets recouvrir l'Am\u00e9rique nouvelle des Indiens.\n\nPliage du sens : les r\u00e9f\u00e9rences elles aussi subissent ce traitement et l'on peut rep\u00e9rer des phrases relevant soit du bon sens et de la sagesse populaire, soit du patrimoine des grandes citations occidentales : un \u00abtout sourit \u00e0 la jeunesse\u00bb venu de nulle part et de partout c\u00f4toie \u00able bonheur est une id\u00e9e neuve en Europe\u00bb ou un \u00ab On a raison de se r\u00e9volter \u00bb que pr\u00e9cise un \u00ab 2 + 2 = 4 \u00bb, sinon un \u00absang impur, etc.\u00bb O\u00f9 l'on aura reconnu Monsieur Homais, ou quelqu'un qui lui ressemble, Saint-Just, Mao, Sartre, Moli\u00e8re, Rouget de l'Isle, et deux ou trois autres complices feuillet\u00e9s dans le texte m\u00eame d'Olivier Cadiot.\n\nM\u00eamement, les registres pli\u00e9s trahissent les centres d'int\u00e9r\u00eat de Pascal Dusapin qui d\u00e9sire, veut et voit tous azimuts. Politiques : lorsque surgissent dans le livret des consid\u00e9rations, mi-Marx, mi-Debord, sur le capital, la f\u00e9tichisation de la marchandise, l'argent; \u00e9pist\u00e9mologiques : d\u00e8s que suivent des propos sur l'ajout et le retranchement en math\u00e9matiques, les in\u00e9galit\u00e9s et les nombres, les produits compos\u00e9s et les facteurs ; po\u00e9tiques : ici ou l\u00e0 quand se disent un dragon ou des fleurs, des gazouillis d'oiseaux, le printemps, des jardins, sinon, sur le mode toujours parodique des \u00abbois couronn\u00e9s d'un reste de verdure\u00bb; philosophiques : \u00e0 chaque digression sur l'identit\u00e9, l'abstraction absolue, l'ali\u00e9nation, la suppression de soi, les moments du processus dialectique et autres r\u00e9f\u00e9rences h\u00e9g\u00e9liennes ; astronomiques : \u00e0 la fa\u00e7on blanquiste soucieuse d'\u00e9ternit\u00e9 par les astres, de vitesse des com\u00e8tes, de r\u00e9volution des plan\u00e8tes, de mar\u00e9es, d'antipodes et de jour sid\u00e9ral. O\u00f9 l'on retrouve l'encyclop\u00e9disme deleuzien et la volont\u00e9 d'un embrassement rhizomatique du monde.\n\nPliage du son, enfin : d'abord dans celui des langues o\u00f9 se c\u00f4toient, m\u00ealent et m\u00e9langent l'anglais de Shakespeare, le cr\u00e9ateur des personnages conceptuels, le fran\u00e7ais de Berlioz, l'un de leurs intercesseurs sur le terrain musical, pr\u00e9d\u00e9cesseur de Pascal Dusapin dans le registre musical, et une langue qui, m\u00e9taphorique ou r\u00e9elle, imaginaire ou vraie, est celle des Indiens, des habitants du Nouveau Monde. Le Quatuor o\u00f9 chacun s'essaie, apr\u00e8s la R\u00e9volution, confronte ses ressemblances et ses diff\u00e9rences, vaut patchwork de toutes ces langues qui sont autant de continents, de peaux, de cultures et d'hypoth\u00e8ses pour un traitement musical de la question du cosmopolitisme.\n\nUne autre fois, ce sont les phon\u00e8mes qui font eux-m\u00eames l'objet des pliages. D'o\u00f9 une floraison joyeuse d'onomatop\u00e9es : tagada et tralala, takatakatakataka ou hihihihi, ouououou et autres hm hm. Toutes voisinent avec la prose lue sur un mode lisse, stri\u00e9, hachur\u00e9, une fois chant\u00e9e, une autre dite. Pliage des dictions qui installent une musique dans les mots, non pas seulement \u00e0 c\u00f4t\u00e9 d'eux, mais avec eux, trait\u00e9s en objets sonores purs, ces \u00e9l\u00e9ments possibles pour le point d'orgue et ses r\u00e9sonances d\u00e9j\u00e0 rencontr\u00e9es.\n\nFormes, sens et sons, \u00e0 savoir figures, temps, facture, r\u00e9f\u00e9rences, registres, dictions, phon\u00e8mes, langues : peut-on imaginer complexit\u00e9 plus achev\u00e9e, multiplicit\u00e9 mieux sollicit\u00e9e ? D'o\u00f9 une proximit\u00e9, l\u00e0 plus que jamais, entre cet op\u00e9ra et l'\u0153uvre majeure de Deleuze et Guattari. Car Rom\u00e9o & Juliette offre \u00e9galement autant de plateaux que possible pour envisager la musique du compositeur en contrepoint du travail deleuzien. Tous les concepts philosophiques majeurs fonctionnent \u00e0 ravir dans l' \u0153uvre musicale, au point qu'on imagine le texte de l'un pli\u00e9 dans le son de l'autre, sinon l'inverse.\n\nLa correspondance peut aussi se faire avec Medeamaterial, une \u0153uvre plus tardive qui, elle, illustre \u00e0 souhait ce que peut donner un traitement musical du concept de devenir. Cet op\u00e9ra peut se lire comme un essai, r\u00e9ussi, de montrer de mani\u00e8re sonore le devenir-hyst\u00e9rique de la ma\u00eetresse, le devenir-infanticide de la m\u00e8re, le devenir-fou de la femme. On sait l'histoire de M\u00e9d\u00e9e, fille du roi de Colchide, petite-fille du soleil, magicienne enti\u00e8rement d\u00e9vou\u00e9e \u00e0 Jason, lui permettant de conqu\u00e9rir la Toison d'Or gr\u00e2ce \u00e0 des mal\u00e9fices. M\u00e9d\u00e9e sacrifie son fr\u00e8re pour que l'entreprise de Jason r\u00e9ussisse avant que ledit Jason, inconstant et infid\u00e8le, ne lui pr\u00e9f\u00e8re Cr\u00e9use. M\u00e9d\u00e9e d\u00e9cide alors de tuer sa rivale et \u00e9gorge les enfants qu'elle a eus de Jason. L'op\u00e9ra, \u00e0 partir du texte de Heiner M\u00fcller, se concentre sur cet \u00e9pisode singulier du basculement de la raison de M\u00e9d\u00e9e, son devenir-fou, ou encore, en termes deleuziens, son devenir-animal.\n\nVocalement et musicalement, l'\u0153uvre fournit l'occasion de pointer le moment o\u00f9 le chant laisse place au cri, o\u00f9 la scansion tragique remplace la diction, syllabe par syllabe, en un registre blanc et d\u00e9j\u00e0 voil\u00e9 par la mort annonc\u00e9e. Dans le silence qui gagne, la folie s'exprime dans les hachurages, les \u00e9clatements, les respirations musicales haletantes des cordes, les pulsions sanguines, voire sanguinaires ou sanguinolentes, le rythme cardiaque devenu sonore, ou l'inverse. L\u00e0 o\u00f9 la voix semble pacifi\u00e9e, seule et parl\u00e9e, se profilent l'aveuglement, la d\u00e9termination et l'ob\u00e9issance aux pures lois de la n\u00e9cessit\u00e9. Paroles translucides, aigus des violons, coups d'archet en stridence, \u00e9puisement, fragmentation du verbe \u2013 et av\u00e8nement de la folie. Le personnage conceptuel se replie : de voix et cri qu'elle fut, M\u00e9d\u00e9e devient silence, comme se ferme le temps en boucle. Dans l'autisme, elle sombre, un voile de sang devant les yeux, le monde disparaissant soudain de sa conscience quand triomphe le devenir-fou.\n\nReste la musique du compositeur, l'\u00e9cho qu'elle laisse, comme la trace d'un nuage et son bruit d\u00e8s lors qu'on envisage de le plier ou de le d\u00e9plier, de scruter le myst\u00e8re de la mati\u00e8re, f\u00fbt-elle vapeur d'eau, onde sonore ou n'importe quoi signifiant l'\u00e9vanescence. Tels ces philosophes orientaux qui ouvrent les oignons, en qu\u00eate de la v\u00e9rit\u00e9 de leur essence, et se condamnent \u00e0 seulement d\u00e9poser sur la natte les pelures qu'ils \u00f4tent \u00e0 chacun de leur geste, \u00e0 chaque progression vers le c\u0153ur et l'\u00e9picentre de ce qui les requiert et les inqui\u00e8te : ils ne sauront rien du myst\u00e8re qui les fascine. En cherchant le style d'un musicien, on se met en passe de devenir, soi aussi, un sage d\u00e9sabus\u00e9 ne trouvant au milieu de ces plis et replis, apr\u00e8s d\u00e9pli des surplis, qu'un vide, un immense silence sur quoi se construit l'ensemble de l'\u00e9difice musical.\n\nGisant comme des fragments de raku apr\u00e8s la col\u00e8re, on trouve, la musique de Pascal Dusapin \u00e9pluch\u00e9e, le nuage d\u00e9compos\u00e9, des forces et des formes, des points et des lignes, des plans et des blocs, des agencements et des r\u00e9seaux, des textures et des plis, des devenirs et des maillages, des \u00e9nergies et des dynamiques, tout ce qui fait un monde, une pr\u00e9sence. Pour autant, Deleuze pli\u00e9 dans Dusapin, ou l'inverse, l'origami n'a lieu que dans un grand et long silence qu'il faut vouloir troubler, d\u00e8s que possible, par l'\u00e9coute de quelques phrases musicales du compositeur. Dans ce monde de temps g\u00e2ch\u00e9 et perdu, dans cet univers de dur\u00e9es prostitu\u00e9es et n\u00e9glig\u00e9es de notre quotidien, Pascal Dusapin a ouvrag\u00e9 bient\u00f4t une vingtaine d'heures qui font un paysage. Il faut le parcourir en nomade et le vouloir en s\u00e9dentaire.\n21\n\nGYN\u00c9COLOGIE DES PR\u00c9CIEUSES\n\nFichtre ! j'\u00e9tais dans l'appartement d'une femme sublime, belle \u00e0 tordre les nerfs et le ventre, \u00e0 vriller l'\u00e2me et cheviller le corps, accessoirement de passage dans la chambre de sa jeune fille pr\u00e9nomm\u00e9e Cl\u00e9lia, et je voyais affich\u00e9e sur le mur une planche anatomique figurant un ut\u00e9rus, ouvert comme un mollusque, \u00e9tendu pareil \u00e0 un Christ, arrogant tel un chancre dans une consultation gyn\u00e9cologique... La dame que je savais digne, habituellement effray\u00e9e par tous les mots en provenance du Larousse m\u00e9dical, dont je n'ignorais pas non plus la bonne \u00e9ducation dans les \u00e9glises la\u00efques de la L\u00e9gion d'honneur \u2013 ce qui vous rend incapable, me semble-t-il, de trop plaisanter avec les choses de l'alc\u00f4ve \u2013, cette dame, donc, consentait ainsi, d\u00e9sinvolte, aux affichages subversifs de sa jolie prog\u00e9niture? Toute la chambre d'une adolescente \u00e9tait l\u00e0, bien en vue, et j'avais l'impression d'\u00eatre dans la carr\u00e9e embl\u00e9matique d'une locataire de son \u00e2ge. Sauf cet ut\u00e9rus...\n\nCl\u00e9lia, donc, parce que Stendhal. Et la maman de me raconter, le regard p\u00e9tillant, Parme et le palais Contarini, la beaut\u00e9 singuli\u00e8re de la marquise Crescenci, n\u00e9e Conti, sa blondeur, sa timidit\u00e9, sa nature romanesque et, bien s\u00fbr, sa passion pour Fabrice del Dongo. Sublime Cl\u00e9lia qui risque de perdre la raison sous l'effet d'une \u00e9motion trop violente, se donne \u00e0 l'homme qu'elle aime dans la cellule d'une citadelle, puis renonce brutalement \u00e0 la suite d'un v\u0153u. Femme des sensations violentes, jalouse et pieuse, hautaine et lucide, passionn\u00e9e et tendre, elle fait un parangon pour une f\u00e9minit\u00e9 qui traverse le temps. Cl\u00e9lia aussi, ajouta la dame radieuse et solaire, parce que Mlle de Scud\u00e9ry. Et son regard de se tourner vers la gravure expos\u00e9e sur le mur d'une mani\u00e8re convenue, entendue, guettant l'approbation.\n\nJe m'approche vaguement de l'ut\u00e9rus et mon regard entre dans une zone d'ombre. Un moment intellectuellement flou, puis la planche anatomique cesse de ressembler \u00e0 une gravure de V\u00e9sale pour appara\u00eetre plus distinctement et enfin nettement : la Carte du Tendre de Madeleine de Scud\u00e9ry comme elle illustre Cl\u00e9lie, histoire romaine ! Quasiment personne n'a lu cette suite romanesque en dix volumes, mais tout le monde la conna\u00eet en biais pour l'image d'Epinal de ladite Carte que tout lyc\u00e9en a parcourue d'un \u0153il distrait et d'une \u00e2me envieuse dans les manuels du temps de sa formation litt\u00e9raire, entre les guerres picrocholines et la photographie d'un manuscrit du M\u00e9morial retrouv\u00e9 dans les plis du v\u00eatement de Pascal.\n\nIl me revenait qu'adolescent j'avais aim\u00e9 cette g\u00e9ographie sentimentale qui a le m\u00e9rite d'\u00e9vacuer le corps dans une p\u00e9riode o\u00f9 il fait naturellement probl\u00e8me, la pubert\u00e9. Pas de chair, pas de sexe, pas de sang, mais des mots, du verbe, de la rh\u00e9torique, du discours amoureux en lieu et place de l'amour, sinon des preuves d'amour ou des corps qui se cherchent, se trouvent ou se manquent. De la casuistique comme on n'en fait plus pour produire une \u00e9bri\u00e9t\u00e9 verbale, une l\u00e9g\u00e8re ivresse limit\u00e9e au seul pouvoir des mots. Je fais un pas dans la direction de l'objet. Evidemment, l'ut\u00e9rus a disparu, du moins, disons-le ainsi, je l'ai enfoui dans une zone plus en retrait de mon esprit pour m'abandonner aux d\u00e9lices d'une p\u00e9r\u00e9grination rh\u00e9torique en compagnie de la dame toujours \u00e0 mes c\u00f4t\u00e9s.\n\nTendresse et Assiduit\u00e9, G\u00e9n\u00e9rosit\u00e9 et Probit\u00e9. Certes, voil\u00e0 des vertus attendues de tous et de toutes, ind\u00e9pendamment d'une relation sinon sexuelle, du moins sexu\u00e9e. Pas obligatoirement amoureuse. Vertus de politesse et de proximit\u00e9, pratiques h\u00e9donistes d'une intersubjectivit\u00e9 digne de ce nom, je ne vois rien l\u00e0 de sp\u00e9cifique au terrain amoureux. Lac d'indiff\u00e9rence, Mer d'inimiti\u00e9, sinon flots dangereux ou Terres inconnues, Madeleine de Scud\u00e9ry et son fr\u00e8re, associ\u00e9 dans l'entreprise, donnent dans les balbutiements de la cartographie quand la po\u00e9sie tient lieu de science \u2013 sinon l'inverse \u2013, lorsque le romanesque conserve l'avantage et qu'une \u00e9pist\u00e9mologie du local dans le global n'est pas encore pensable. Mais pourquoi pas ?\n\nJe ne me souvenais pas que la carte f\u00fbt aussi niaise. De l'ut\u00e9rus \u00e0 cette philosophie amoureuse pour adolescente anxieuse des choses du corps ou femme r\u00e9gressive honteuse des probl\u00e8mes du sexe, il y a un ab\u00eeme. Les villages : Tendre-sur-Estime ou Tendre-sur-Reconnaissance, Nouvelle Amiti\u00e9, inaugurent, d\u00e9j\u00e0, les utopies et les uchronies des sp\u00e9cialistes en cit\u00e9s radieuses et en hypoth\u00e8ses de paradis sur terre \u00e0 venir. Un monde o\u00f9 les hommes et les femmes, purs ectoplasmes, ne disposeraient pas de corps : le fantasme ang\u00e9lique chr\u00e9tien ! Madeleine de Scud\u00e9ry semble appeler cette litt\u00e9rature du r\u00eave \u00e9veill\u00e9 qui, politiquement, conduit de Thomas More \u00e0 Etienne Cabet.\n\nPour autant, le flou se fait \u00e0 nouveau. Le mouvement r\u00e9tinien agissant \u00e0 la mani\u00e8re d'une mise au point photographique, je revois, le nez sur la gravure, l'ut\u00e9rus prendre le dessus et s'imposer \u00e0 nouveau, telle la figure du refoul\u00e9 de son auteur. D'aucuns virent bien un milan d\u00e9ployer son vol \u00e0 l'envers dans une peinture de L\u00e9onard, pourquoi donc ne trouverait-on pas le muscle triangul\u00e9 dans cette cartographie de la Pr\u00e9cieuse pas encore ridicule ? Et, nettement, les villages de Tendre sur ceci ou cela dessinent d'excellents ovaires, r\u00e9partis de fa\u00e7on sym\u00e9trique, alors que le Fleuve Inclination a tout, le malheureux, d'un banal conduit urinaire, pendant que la suite de ma p\u00e9r\u00e9grination anatomique installe non loin de la vessie, qu'on me pardonne la d\u00e9gringolade des registres, le village de Nouvelle Amiti\u00e9, point de d\u00e9part oblig\u00e9 de tout voyage souhait\u00e9 par Madeleine. Au Nord, l\u00e0 o\u00f9 la mer est dite dangereuse, \u00e0 l'endroit o\u00f9 s'annoncent les Fleuves Reconnaissance et Estime, s'imposaient, imp\u00e9rieuses, de triviales trompes de Faloppe d\u00e9ploy\u00e9es comme les cuivres d'une fanfare antique. Et l\u00e0 o\u00f9 se jettent les trois fleuves, il m'appara\u00eet, je rougis d'avoir \u00e0 filer la m\u00e9taphore, quelque chose comme, disons, le vestibule de ce que le po\u00e8te appelle un bijou rose et noir. Faut-il continuer ?\n\nNon. Cessons l\u00e0 une lecture de m\u00e9decin l\u00e9giste qui pr\u00e9cipite \u00e0 terre toute l'entreprise de Madeleine de Scud\u00e9ry et interdit toute rh\u00e9torique pr\u00e9cieuse sur la question de l'amour pour lui substituer une anatomie des passions, une pathologie des \u00e9motions, sinon une gyn\u00e9cologie du discours amoureux. Encore que celles-ci peuvent faire sympt\u00f4me de ce qui sous-tend le propos pr\u00e9cieux sur l'amour et ce qui le rend possible et imaginable : la pr\u00e9sence, l'existence et la hantise d'un ut\u00e9rus dont on sait depuis Hippocrate, jusqu'\u00e0 Charcot et Freud, qu'il est le lieu o\u00f9 se fomentent l'hyst\u00e9rie \u2013 et son \u00e9tymologie. Que faire d'un muscle qui sature toute g\u00e9ographie symbolique sur le principe d'un retour du refoul\u00e9 ? R\u00e9tractile, certes, mais pas au point de dispara\u00eetre, disons corps et biens.\n\nLes sp\u00e9cialistes en neurosciences verraient l\u00e0 l'excellente occasion d'une biologie des passions dans lesquelles rates en rut et copulations derri\u00e8re les barreaux de cages valent moins m\u00e9taphores ironiques ou ludiques de l'amour que r\u00e9ductions, quintessences, expressions minimales, locales, d'une v\u00e9rit\u00e9 maximale, globale. Les discours philosophiques sur l'amour, de l'androgyne de Platon \u00e0 la cristallisation de Stendhal, de la Fin' Amor de Bernard de Ventadour au propyl\u00e9e du surhumain chez Nietzsche, des consid\u00e9rations sur l'Eros de Lou Salom\u00e9 au ludisme de la triade organis\u00e9e autour de Helen Hessel, tout cela sombre, chez les blouses blanches, dans des consid\u00e9rations sur les neurotransmetteurs, les marqueurs chimiques et autres puissances de ph\u00e9romones.\n\nRevenons \u00e0 nos moutons. La Carte de Mlle de Scud\u00e9ry peut se lire comme le sympt\u00f4me d'une r\u00e9gression qui hante son propos \u00e0 chaque instant : haine du corps et de la chair, m\u00e9pris des d\u00e9sirs et des passions, volont\u00e9 nette, bien que peut-\u00eatre inconsciente, de prendre parti, dans le Grand Si\u00e8cle, pour la r\u00e9action, la tradition et les valeurs platoniciennes ou chr\u00e9tiennes du pass\u00e9 qui subissent alors une attaque en r\u00e8gle. Peu de temps avant sa naissance, on br\u00fble Giordano Bruno \u00e0 Rome. Son enfance est contemporaine des proc\u00e8s de Galil\u00e9e et du supplice inflig\u00e9 \u00e0 Vanini sur la place publique \u00e0 Toulouse. L'\u00e9poque correspond au progr\u00e8s des sciences et de la raison que pers\u00e9cute l'Eglise avec de gros moyens et une d\u00e9termination sanglante. Campanella, contemporain de Mlle de Scud\u00e9ry, paie assez de sa personne, entre tortures et incarc\u00e9rations, pour le savoir et en faire la d\u00e9monstration.\n\nLe savoir se d\u00e9plie en m\u00eame temps que le libertinage se d\u00e9ploie, l'un permettant l'autre. Sur le terrain litt\u00e9raire para\u00eet en Espagne un Abuseur de S\u00e9ville sous la plume de Tirso de Molina doubl\u00e9 bient\u00f4t en France d'un Don Juan sign\u00e9 Moli\u00e8re. Madeleine de Scud\u00e9ry a vingt-trois ans lorsqu'appara\u00eet le premier texte, cinquante-huit le second. La litt\u00e9rature libertine de Th\u00e9ophile de Viau, Vauquelin des Yvetaux, Cyrano de Bergerac ou Saint-Evremond, fournit du fil \u00e0 retordre aux sectateurs chr\u00e9tiens de l'id\u00e9al asc\u00e9tique. De la m\u00eame mani\u00e8re, l'\u00e9poque compte avec des trouvailles scientifiques de taille, \u00e0 savoir la m\u00e9thode exp\u00e9rimentale de Bacon, le mat\u00e9rialisme moniste de Galil\u00e9e, l'h\u00e9liocentrisme de Kepler, la gravitation universelle de Newton.\n\nLe libertinage \u00e9rudit travaille le XVIIe si\u00e8cle en profondeur. Et l'on pourrait m\u00eame \u00e9crire l'histoire de ces cent ann\u00e9es comme celles d'un unique combat r\u00e9actif men\u00e9 \u00e0 l'endroit des avanc\u00e9es libertines : le travail cart\u00e9sien qui la\u00efcise la pens\u00e9e dans le respect de la religion de son Roi et de sa nourrice, les \u00e9crits apolog\u00e9tiques de Pascal et Malebranche ou encore ceux, mystiques, de Fran\u00e7ois d'Assise, Mme Guyon ou F\u00e9nelon, tout cela semble fr\u00e9mir devant les progr\u00e8s de l'ath\u00e9isme et du mat\u00e9rialisme qui s'imposent, sans arrogance, mais s\u00fbrement, et joueront leur r\u00f4le dans les fractures politiques du si\u00e8cle suivant.\n\nMadeleine de Scud\u00e9ry n'est pas cart\u00e9sienne, sa pathologie rel\u00e8ve moins des analyses de Descartes que de la tradition de la Renaissance et plus particuli\u00e8rement de Marsile Ficin. On sait l'\u00e9tendue des lectures et de la culture de la Pr\u00e9cieuse. Nul doute qu'elle conna\u00eet le Commentaire du \u00ab Banquet \u00bb de Platon du philosophe n\u00e9o-platonicien, ou Les Passions de l'\u00e2me, qui viennent de para\u00eetre chez Elzevier. Sa Carte du Tendre semble illustrer Ficin en m\u00eame temps que condamner Descartes. Marsile Ficin reprend les distinctions faites par Platon dans le Banquet entre l'aphrodite c\u00e9leste et l'aphrodite vulgaire, l'ouranienne et la pand\u00e9mienne, celle des \u00e2mes pures et celles des corps sales. Bien s\u00fbr, pareille distinction permet de fustiger les chairs sensibles et c\u00e9l\u00e9brer les esprits intelligibles.\n\nQue dit Descartes, en revanche, dans le livre o\u00f9 il analyse les passions ? Que celles-ci n'existent qu'incarn\u00e9es. Les Passions de l'\u00e2me paraissent en 1649 et Cl\u00e9lie, le roman dans lequel se trouve la Carte du Tendre, en 1654, du moins pour le premier volume. Le philosophe disserte sur ce qui distingue un corps mort et un corps vivant, ce qui n'est pas peu dans un si\u00e8cle o\u00f9 la maxime j\u00e9suite de saint Ignace de Loyola, perinde ac cadaver, se pratique \u00e0 l'envi et o\u00f9 il semble n'y avoir de v\u00e9rit\u00e9 de la chair vivante que dans son impassibilit\u00e9, sa neutralit\u00e9, sa froideur, ce qui l'apparente \u00e0 celle des morts. Haine des passions, conjuration des \u00e9motions, m\u00e9pris des sensations, discr\u00e9dit jet\u00e9 sur les perceptions, tout ce qui met en jeu un corps suppose pour la plupart de ceux qui \u00e9crivent et pensent alors, de l'erreur, de l'approximation et de la fausset\u00e9. En t\u00e9moignent des planches install\u00e9es entre les tours de Notre-Dame autant que des b\u00e2tons cass\u00e9s d\u00e8s leur plong\u00e9e dans un liquide.\n\nDescartes parle chaleur et mouvements, esprits animaux et organes des sens, il disserte sur le c\u0153ur et les muscles, fouille le cerveau et d\u00e9couvre la glande pin\u00e9ale sous un scalpel. Loin de Constante Amiti\u00e9 et Complaisance, aux antipodes de Grand Esprit et autres douceurs qui jalonnent les parcours de la Carte, l'auteur du Trait\u00e9 de l'homme entretient sans ambages de tremblements, de p\u00e2moisons, de langueurs et de larmes, de g\u00e9missements, de soupirs, de rire, voire d'all\u00e9gresse. Le Bernin inscrit tout cela en 1644 dans les plis de l'extase que l'on sait \u2013 le contraire de Madeleine.\n\nL'option r\u00e9gressive de l'auteur de Cl\u00e9lie se manifeste \u00e9galement dans la pr\u00e9f\u00e9rence accord\u00e9e au verbe sur l'action, \u00e0 la casuistique et \u00e0 la rh\u00e9torique sur la pragmatique exp\u00e9rimentale. Elle excelle dans le formalisme traditionnel, ce qui la rend aujourd'hui illisible, quand Descartes veut une pens\u00e9e enracin\u00e9e dans la vie quotidienne. L\u00e0 r\u00e9side d'ailleurs sa modernit\u00e9. Le philosophe assiste \u00e0 des le\u00e7ons d'anatomie, pratique lui-m\u00eame et exp\u00e9rimente, regarde, observe et pense \u00e0 partir de ce qu'il constate. Madeleine de Scud\u00e9ry veut un monde conforme \u00e0 l'id\u00e9e qu'elle s'en fait, elle souhaite faire rentrer le r\u00e9el dans la g\u00e9ographie qu'elle dessine, en petite fille oublieuse de son corps. Elle pr\u00e9f\u00e8re changer l'ordre du monde plut\u00f4t que d'envisager une r\u00e9volution copernicienne pour son propre compte. Famili\u00e8re des m\u00e9thodes d'exposition scolastiques, elle remplit son roman du quotidien des salons o\u00f9 elle th\u00e9\u00e2tralise son \u00e9criture et r\u00e9duit son existence \u00e0 l'art baroque des pointes, des madrigaux, bouts rim\u00e9s et autres jeux de soci\u00e9t\u00e9.\n\nEt l'on s'interroge sur la question des pr\u00e9f\u00e9rences amoureuses : vaut-il mieux s'\u00e9prendre d'une enjou\u00e9e, d'une m\u00e9lancolique ou d'une capricieuse? Comme si la r\u00e9ponse faisait un doute. Est-il d\u00e9fendable d'avoir un second amant si le premier est mort? Il me semble que dans les boudoirs o\u00f9 l'on cause, et m\u00eame ailleurs, on n'attend pas souvent le tr\u00e9pas du premier pour envisager le second, sinon le deuxi\u00e8me. La gloire est-elle l'apanage de l'amour? En ce lieu o\u00f9 se recrutent si peu de h\u00e9ros... Un amant doit-il d\u00e9sob\u00e9ir \u00e0 sa ma\u00eetresse si l'honneur l'y invite? Ou encore : doit-on mettre de l'esprit dans ses lettres d'amour? Pour ceux qui le peuvent, la question semble \u00e9tonnante. Vaut-il mieux \u00eatre prisonnier de guerre ou prisonnier d'amour? les familiers des premi\u00e8res ge\u00f4les jugeront...\n\nToutes ces interrogations n'obtiennent de r\u00e9ponse que dans la mesure o\u00f9 elles permettent un \u00e9loge en bonne et due forme du pur amour, de la vertu transparente et des intentions magnifiques. La chronologie rend ignorante Mlle de Scud\u00e9ry, et pour cause, mais elle est rien moins que kantienne... L'amour dont elle fait l'\u00e9loge, c'est le contraire de l'amour. Bluette, fleur bleue, vision rose des relations sexu\u00e9es, lecture r\u00e9gressive et enfantine du r\u00e9el, sa g\u00e9ographie du Tendre invite \u00e0 \u00e9viter le corps de l'autre, elle \u00e9vacue tout d\u00e9sir, elle \u00e9radique ce qui, de pr\u00e8s ou de loin, suppose un contact, un corps qui touche un autre corps. Dans son monde, la chair se vit sur le mode autiste et la seule figure de rapport possible est rh\u00e9torique.\n\nBien s\u00fbr, on se r\u00e9jouit de Jolis Vers et Billet doux ou galant tout autant que de Petits Soins, Empressement, Grands Services et autres guimauves qui, d\u00e8s 1654, constituent d\u00e9j\u00e0 un r\u00e9jouissant et exhaustif petit trait\u00e9 des grandes vertus. De la m\u00eame mani\u00e8re, \u00e9videmment, on b\u00e9nit Grand C\u0153ur, on c\u00e9l\u00e8bre Exactitude et l'on d\u00e9teste avec une v\u00e9h\u00e9mence appuy\u00e9e autant M\u00e9disance et Perfidie, Orgueil et M\u00e9chancet\u00e9 \u2013 ovaire droit sur la gravure \u2013 que Ti\u00e9deur et L\u00e9g\u00e8ret\u00e9, Oubli et In\u00e9galit\u00e9 \u2013 ovaire gauche. Mais par ailleurs, que faire de Soumission et Ob\u00e9issance, stations oblig\u00e9es pour aller de Nouvelle Amiti\u00e9 \u00e0 Tendre Reconnaissance ? Etranges exigences qui supposent les hommes au service, en domestiques des femmes... L\u00e0 o\u00f9 l'on veut d\u00e9licatesse et courtoisie h\u00e9rit\u00e9es du monde f\u00e9odal, on recourt \u00e0 suzerainet\u00e9 et vassalit\u00e9. Madeleine de Scud\u00e9ry veut une r\u00e9actualisation des pratiques chevaleresques contemporaines des \u00e9poques les plus platoniciennes, quand triomphaient la haine de la chair et la suspicion g\u00e9n\u00e9ralis\u00e9e \u00e0 l'endroit des corps. La pens\u00e9e pr\u00e9cieuse n\u00e9glige les sublimes formes de l'\u00e9rotique courtoise du d\u00e9but (XIe et XIIe si\u00e8cles) pour c\u00e9l\u00e9brer une variation sur le culte marial de la Renaissance. La Carte du Tendre propose une g\u00e9ographie \u00e0 l'usage des vierges qui entendent le rester.\n\nDans cette cartographie virginale, les eaux pr\u00e9sentent un r\u00e9el danger. Des lacs aux mers, dormantes ou furieuses, profondes et mortes, associ\u00e9es \u00e0 l'indiff\u00e9rence, ou inhumaines et violentes, complices de l'inimiti\u00e9, toutes d\u00e9plaisent \u00e0 Madeleine qui triomphe en femme de terre. Bachelard met au net les relations de l'eau et des songes puis associe cet \u00e9l\u00e9ment au f\u00e9minin et \u00e0 la maternit\u00e9. On mesure combien Madeleine de Scud\u00e9ry pouvait \u00eatre f\u00e2ch\u00e9e avec sa f\u00e9minit\u00e9, ou du moins avec ce qui en elle aurait pu entrer en contact avec les hommes, d\u00e8s le pointage du pseudonyme qu'elle se choisit pour intervenir dans son roman ou dans les samedis de son salon : Sapho...\n\nLa b\u00e9ance constitu\u00e9e par le Lac d'indiff\u00e9rence dans le relev\u00e9 topographique \u00e9tabli par ses soins, en appelle aussi \u00e0 la symbolique tant ce trou dessin\u00e9, grav\u00e9, inflig\u00e9 \u00e0 la terre, induit des \u00e9quivalences et des correspondances du c\u00f4t\u00e9 des orifices pratiqu\u00e9s dans une peau, celle du palimpseste g\u00e9ographique, certes, mais celle, \u00e9galement, d'un hymen malheureusement d\u00e9chir\u00e9 ou tristement pr\u00e9serv\u00e9, mais de toute fa\u00e7on associ\u00e9 \u00e0 une indiff\u00e9rence d\u00e9plorable dans un cas comme dans l'autre. Dans le graphisme, l'eau morte croupit dans les bourrelets d'un sphincter tragique.\n\nDe la m\u00eame mani\u00e8re que Madeleine de Scud\u00e9ry et son fr\u00e8re affichent une r\u00e9gression dans la chose dite par la Carte, ils trahissent le m\u00eame esprit r\u00e9actionnaire, au sens \u00e9tymologique, dans la fa\u00e7on de dire la chose, dans le traitement cartographique de l'information. Plut\u00f4t Marsile Ficin que Ren\u00e9 Descartes, platonicienne de pr\u00e9f\u00e9rence et s\u00fbrement pas libertine, scolastique dans l'expos\u00e9 et \u00e0 aucun moment moderne dans la facture, Madeleine de Scud\u00e9ry pratique \u00e9galement l'art des cartes avec plusieurs si\u00e8cles de retard.\n\nDepuis Mercator et la projection qui porte son nom, on ne peut plus repr\u00e9senter le monde comme Ptol\u00e9m\u00e9e ou Strabon sans revendiquer une pr\u00e9f\u00e9rence nette pour les formes anciennes, les choses anciennes, les id\u00e9es anciennes. L'option de la Pr\u00e9cieuse appara\u00eet l\u00e0 pour ce qu'elle ne cesse d'\u00eatre : une lecture r\u00e9actionnaire du monde, anim\u00e9e par une association du pass\u00e9 mythique au positif et doubl\u00e9e par une connotation n\u00e9gative du pr\u00e9sent.\n\nBonnes les pens\u00e9es, repr\u00e9sentations, id\u00e9es d'hier; mauvaises celles d'aujourd'hui, et qui plus est de demain. L'\u00e9pist\u00e9m\u00e9, dirait Foucault, de la Carte du Tendre co\u00efncide avec le devenir dominant des bourgeois dans la soci\u00e9t\u00e9 fran\u00e7aise. Recul de l'aristocratie bient\u00f4t achev\u00e9 dans un retrait d\u00e9finitif. Les valeurs de la Scud\u00e9ry supposent une qu\u00eate dans le pass\u00e9 de ce qui peut donner naissance \u00e0 un pr\u00e9sent et un futur. D'aucuns voient chez les Pr\u00e9cieuses un f\u00e9minisme en puissance, sinon en acte, parce que dans leurs productions litt\u00e9raires les femmes entendent soumettre leurs relations amoureuses au sentiment et non plus aux usages. Certes. Mais ce sentiment brille moins par l'innovation h\u00e9doniste, \u00e0 la Ninon de Lenclos, que dans la r\u00e9cup\u00e9ration et le recyclage des vieilles vertus chevaleresques mariales.\n\nCe regard en arri\u00e8re para\u00eet manifeste dans le recours \u00e0 une pratique rendue franchement caduque par Ort\u00e9lius un si\u00e8cle avant les exercices d'\u00e9criture de Madelon \u2013 dirait Moli\u00e8re. Les historiens pourraient classer la Carte du Tendre dans le registre p\u00e9rim\u00e9 de toutes celles qui ont \u00e9t\u00e9 dessin\u00e9es entre la chute de l'Empire romain et l'apparition de la technique de projection et d'\u00e9criture arabes. Dans cette longue dur\u00e9e, ce que l'on cartographie exprime une vision du monde, une conception subjective du r\u00e9el, une m\u00e9taphysique. Le dessin, la perspective, le trait, le mode de figuration racontent une po\u00e9tique moins soucieuse d'objectivit\u00e9 absolue ou de scientificit\u00e9 radicale que d'images suggestives co\u00efncidant avec une repr\u00e9sentation mentale.\n\nAvant Mercator, la carte propose la possibilit\u00e9 et l'occasion d'un itin\u00e9raire dans une g\u00e9ographie habitable. Terre plane ovale, en forme d' \u0153uf, le premier principe dont proc\u00e8de le r\u00e9el, carr\u00e9e ou rectangulaire, selon la rigueur math\u00e9matique d'une figure \u00e9quilibr\u00e9e en vertu du chiffre et du nombre, \u00eele circulaire entour\u00e9e d'oc\u00e9ans, comme une m\u00e9taphore du solipsisme, les formes propos\u00e9es expriment chaque fois des peurs, des craintes et des angoisses, des aspirations, des obsessions, ou des croyances. De toute fa\u00e7on, plus po\u00e9tique qu'\u00e9pist\u00e9mologique, la cartographie propose moins un monde de latitudes et de longitudes, de m\u00e9ridiens et de triangulations qu'un univers hant\u00e9 par des cr\u00e9atures myst\u00e9rieuses, voire inqui\u00e9tantes. Il n'y a pas de carte innocente.\n\nLa carte ancienne, d'hier et d'avant-hier, m\u00eame pour l'\u00e9poque de Mlle de Scud\u00e9ry, propose le dessin d'un espace dans lequel on pr\u00e9cise les emb\u00fbches et les points de rep\u00e8re afin de rendre possible le d\u00e9placement d'un point \u00e0 un autre : villages travers\u00e9s, risques g\u00e9omorphologiques, lacs ou for\u00eats, chemins ou montagnes, trac\u00e9s de vall\u00e9es ou \u00e9minences, escarpements et d\u00e9viations, c\u00f4tes ou bosquets. Les distances se calculent en jours de marche ou en pas. En fait, tout ce qui caract\u00e9rise la carte ancienne fournit le mod\u00e8le de la g\u00e9ographie sentimentale propos\u00e9e dans le premier livre de Cl\u00e9lie.\n\nMadeleine de Scud\u00e9ry pense, \u00e9crit et dessine dans l'obsession r\u00e9gressive. La modernit\u00e9 ne lui agr\u00e9e jamais : ni philosophique, ni litt\u00e9raire, ni cartographique, ni id\u00e9ologique, \u2013 son id\u00e9al \u00e9rotique co\u00efncide avec celui de l'id\u00e9al asth\u00e9nique en faveur depuis Platon et revitalis\u00e9 par le christianisme. Ninon de Lenclos avait raison d'appeler les Pr\u00e9cieuses des \u00abjans\u00e9nistes de l'amour\u00bb. Quand les bourgeoises singent les valeurs de l'aristocratie, elles r\u00e9actualisent de vieux habits, enfilent les antiques d\u00e9froques sur lesquelles elles ont fantasm\u00e9 leur vie durant. L'\u00e9chec de la Fronde avalise celui de l'aristocratie qui perd ainsi tout espoir de recouvrer une grandeur d\u00e9finitivement perdue. Ces heures-l\u00e0 pr\u00e9parent la R\u00e9volution fran\u00e7aise, annoncent des grondements et des mouvements souterrains.\n\nLa Carte du Tendre propose une th\u00e9\u00e2tralisation de l'intersubjectivit\u00e9 bourgeoise sur laquelle l'Occident a v\u00e9cu id\u00e9alement et sans discontinuer jusqu'\u00e0 la fracture de Mai 68. L'absence de corps, qu'il fallait \u00e9viter dans les parcours du Tendre, l'\u00e9radication de tout ce qui rappelle le d\u00e9sir, le plaisir, la suspicion g\u00e9n\u00e9ralis\u00e9e \u00e0 l'endroit des passions et le perp\u00e9tuel hallali lanc\u00e9 contre la chair, le triomphe sans partage de l'id\u00e9al asc\u00e9tique sous toutes ses formes, tout cela a re\u00e7u de formidables coups de boutoir lors du printemps de Mai qui formula une prop\u00e9deutique pour une nouvelle Carte du Tendre. Cette ouverture de lumi\u00e8re dans une histoire d\u00e9volue au masque, \u00e0 l'hypocrisie, au mensonge et \u00e0 la haine de soi vaut seulement comme une aurore. La le\u00e7on donn\u00e9e par Madeleine de Scud\u00e9ry, \u00e0 son corps d\u00e9fendant, ne manque pas d'int\u00e9r\u00eat. A son corps d\u00e9fendant? Oui, d'une certaine mani\u00e8re, son corps s'en d\u00e9fendait. Car je n'ai rien lu sur le compte de Madeleine qui n\u00e9glige de mentionner, \u00e0 chaque fois, son exceptionnelle laideur.\n\nDepuis Nietzsche, on conna\u00eet la part majeure prise par le ressentiment et les sentiments r\u00e9actifs dans l'\u00e9laboration des morales du m\u00e9pris de soi, de la haine du corps et du renoncement. Nombre de pages de la G\u00e9n\u00e9alogie de la morale disent avec la lucidit\u00e9 d'une cruaut\u00e9 jamais \u00e9gal\u00e9e combien le refus de la vie, des d\u00e9sirs et des plaisirs qui la magnifient, l'expriment et la subliment, proc\u00e8dent d'une f\u00e2cherie avec soi. La morale classique agrandit les failles qui hantent et s\u00e9parent le moi. Loin de combler les b\u00e9ances creus\u00e9es au c\u0153ur de l'\u00eatre entre la r\u00e9alit\u00e9 de soi et l'id\u00e9e de soi, le principe de plaisir et le principe de r\u00e9alit\u00e9, elle approfondit le n\u00e9gatif. Toute r\u00e9conciliation avec soi est rendue improbable, voire impossible par les \u00e9thiques du ressentiment.\n\nJolis Vers et Billet doux, Constante Amiti\u00e9 et Tendresse valent \u00e0 d\u00e9faut, dans l'impossibilit\u00e9 d'une carte du tendre moins d\u00e9volue \u00e0 l'aphrodite c\u00e9leste qu'\u00e0 l'autre, la seule, celle qui veille aux br\u00fblures des corps. Les passions de l'\u00e2me, quand celle-ci reste d\u00e9sesp\u00e9r\u00e9ment immat\u00e9rielle, croupiront, confin\u00e9es dans les g\u00e9ographies roses des petites filles amoureuses des satisfactions de la r\u00e9gression, joyeusement \u00e9conomes du corps de l'autre. Madeleine de Scud\u00e9ry est moins contemporaine des banquets \u00e9picuriens et sceptiques de La Mothe Le Vayer que de Fran\u00e7ois d'Assise et de ses invitations \u00e0 la vie d\u00e9vote. Sa laideur l'installe devant le refus de la chair dans l'amour comme le renard au pied des raisins renon\u00e7ant moins pour leur verdeur, c'est son discours, que par leur impossible acc\u00e8s.\n\nDans Cl\u00e9lie, elle fustige ceux qui, dans l'amour, demandent au corps qu'il tienne sa part et \u00e0 la beaut\u00e9 qu'elle ne soit pas compt\u00e9e pour rien. Son analyse est simple, voire transparente. Tant de limpidit\u00e9 ferait sourire s'il n'y avait l\u00e0 quintessence de ce qui anime les vendeurs d' arri\u00e8re-mondes et ceux qui obligent autrui \u00e0 se plier aux cons\u00e9quences de leurs propres n\u00e9vroses. Elle \u00e9nonce que la beaut\u00e9 dans l'amour n'attire que les habiles et les stupides, les sots et les imb\u00e9ciles. Beaut\u00e9 et jeunesse seraient de fausses valeurs car elles dispara\u00eetraient avec le temps, ainsi, face \u00e0 face, devenus vieux, ceux qui auraient cru en ces trompeuses vertus, n'auraient plus qu'\u00e0 se d\u00e9tester, se ha\u00efr, sinon se m\u00e9priser. Voil\u00e0 pourquoi il vaut mieux \u00eatre vieille et laide, le plus t\u00f4t possible, dans l'all\u00e9gresse, en \u00e9conomisant un d\u00e9tour par la fra\u00eecheur d\u00e9sesp\u00e9rante des jeunes ann\u00e9es...\n\nPour \u00e9viter le malheur de n'\u00eatre pas courtis\u00e9e, il lui para\u00eet plus simple de changer les r\u00e8gles du jeu et d'appeler amour ce qui tout simplement d\u00e9signe l'amiti\u00e9, cet amour qui fait l'impasse sur le corps. Nietzsche savait qu'une relation entre un homme et une femme ne peut oublier le corps, le d\u00e9sir et le plaisir, hormis dans l'hypoth\u00e8se d'une quelconque antipathie physique. Qui lui donnerait tort? Sapho se d\u00e9guise \u00e0 peine en apparaissant sous le travestissement de la plus c\u00e9l\u00e8bre des lesbiennes, celle d'ailleurs dont le nom servira \u00e0 la cr\u00e9ation d'un substantif, le saphisme, que, pour son malheur, Madeleine n'exp\u00e9rimenta m\u00eame pas.\n\nD'aucuns racontent la g\u00e9n\u00e9alogie de la Carte du Tendre en rappelant que Pellisson, son cadet de dix-sept ans, \u2013 on dit qu'il l'\u00e9galait en disgr\u00e2ce \u2013, qui malgr\u00e9 sinon sa laideur, du moins son physique ingrat, s'inqui\u00e9tait aupr\u00e8s de Madeleine des moyens de conqu\u00e9rir son c\u0153ur, et de parcourir le chemin qui conduisait de Particulier \u00e0 Tendre, se fit r\u00e9pondre par l'envoi de la cartographie que l'on sait. Que n'a-t-elle \u00e9tabli, cette Pr\u00e9cieuse heureusement courtis\u00e9e, un autre relev\u00e9 dans lequel le corps aurait eu sa place ! De sorte que saphique ou non, c\u00e9r\u00e9bral ou charnel, sensuel ou spirituel, l'amour aurait pu \u00eatre montr\u00e9 dans sa modernit\u00e9 comme une occasion de perp\u00e9tuelle volupt\u00e9 pour un corps qui se donne, on non, se pr\u00eate, ou pas, s'\u00e9conomise ou se d\u00e9pense, c'est selon, pourvu qu'\u00e0 chaque fois, il soit la grande raison, la seule et unique.\n\nJe sortis de ma r\u00eaverie, oublieux de l'ut\u00e9rus qui m'avait conduit jusque-l\u00e0, par la gr\u00e2ce de la voix de celle qui pr\u00e9nomma son enfant Cl\u00e9lia. Le temps d'un \u00e9clair qui br\u00fble la pens\u00e9e qu'on a, et la consume pour la d\u00e9truire, la seconde qui suivit, mon regard rencontra le sien. Face \u00e0 cette planche qui fut un temps pour moi d'anatomie et me semblait dire les fant\u00f4mes de Mlle de Scud\u00e9ry en m\u00eame temps que ceux d'un continent intellectuel, le n\u00f4tre, devant ce muscle qui jouait en contrepoint ironique avec la g\u00e9ographie dessin\u00e9e par la femme de lettres et son fr\u00e8re, il me semblait qu'il conviendrait de conjurer la laideur autrement qu'en \u00e9crivant des trait\u00e9s de morale, comme a fini par le faire la romanci\u00e8re de Cl\u00e9lie dans ses vieux jours. Par exemple, sur le mode fouri\u00e9riste, en c\u00e9l\u00e9brant l'absolu de la chair loin des canons occidentaux constitutifs de beaut\u00e9 et de laideur, de gr\u00e2ce ou de disgr\u00e2ce, de jeunesse d\u00e9sirable ou de vieillesse repoussante, et d\u00e9cidant par l\u00e0 m\u00eame, comme on fait tomber la lame d'une guillotine, de qui grossira le rang minoritaire des \u00e9lus et celui, majoritaire, des damn\u00e9s.\n\nLa m\u00e8re de Cl\u00e9lia souriait, interrogative et superbe. Moi qui entends la beaut\u00e9 comme une mal\u00e9diction pour les malentendus qu'elle g\u00e9n\u00e8re, j'eus soudain l'impression que j'aurais bien pu m'\u00eatre tromp\u00e9, le temps que se suspendent quelques secondes de gr\u00e2ce. Alors, je me souvins qu'en la g\u00e9ographie de Madeleine l'In\u00e9galit\u00e9 apparaissait en direction du trajet qui conduit \u00e0 l'Indiff\u00e9rence. Et qu'en de pareilles contr\u00e9es ce sont seulement les diff\u00e9rences exploit\u00e9es id\u00e9ologiquement qui g\u00e9n\u00e8rent et constituent les in\u00e9galit\u00e9s. Pour le bonheur des uns et le malheur des autres. Qu'advienne enfin le temps d'une cartographie libertaire o\u00f9 seules les peaux diraient singuli\u00e8rement une \u00e2me et non sociologiquement une figure. Alors Madeleine de Scud\u00e9ry et Ninon de Lenclos seraient \u00e0 \u00e9galit\u00e9 devant les plaisirs.\n22\n\nDES MACHINES \u00c0 RENVERSER LE PLATONISME\n\nSi d'aventure la modernit\u00e9 consiste \u00e0 renverser le platonisme, ce que je crois, alors il faut se soucier vivement du travail esth\u00e9tique de Fran\u00e7ois Loriot et Chantal M\u00e9lia, car il se pourrait bien qu'ils fournissent la m\u00e9taphore de ce qui, depuis Nietzsche et jusqu'\u00e0 Deleuze, anime les rebelles d\u00e9sireux de ce renversement de valeurs. Leur \u0153uvre r\u00e9ussit un singulier paradoxe : d\u00e9passer le platonisme en recourant aux cat\u00e9gories de Platon. Tour de force ironique, persuasion m\u00e9taphysique d'une rh\u00e9torique ludique, le travail ne manque pas d'int\u00e9r\u00eat. Que vise le renversement du platonisme? l'abolition de la transcendance, le d\u00e9passement du dualisme, la fin du discr\u00e9dit jet\u00e9 sur le sensible au profit d'une exacerbation du seul intelligible, l'ensemble rendant possible l'av\u00e8nement d'un monisme mat\u00e9rialiste radical. Rude t\u00e2che pour qui accepte de s'en charger...\n\nLes deux artistes qui n'en font qu'un, comme l'androgyne retrouv\u00e9 du discours d'Aristophane dans Le Banquet du d\u00e9j\u00e0 nomm\u00e9 Platon, r\u00e9ussissent leur pari d\u00e8s le rire caract\u00e9ristique des nietzsch\u00e9ens qui, depuis le vieux boutefeu jusqu'aux arch\u00e9ologies de Foucault ou les plateaux de Deleuze, savent la r\u00e9ussite de l'entreprise aux premi\u00e8res lueurs de ces \u00e9clats. Car leur travail esth\u00e9tique ne cesse de jouer, de sourire ou de rire, pas seulement avec les mots, mais aussi avec ce degr\u00e9 suppl\u00e9mentaire de complexit\u00e9 que fournissent les agencements avec lesquels se dit le monde. Ce qui, chez Platon, enseigne une participation du r\u00e9el sensible \u00e0 l'existence d'un monde intelligible devient l'inverse chez Loriot-M\u00e9lia : la forme intelligible doit son existence \u00e0 un sensible trivial et volontairement \u00e9lu comme tel.\n\nO\u00f9 donc, et quand, les artistes antiplatoniciens se font-ils platoniciens pour mieux r\u00e9ussir leur pari? Dans leur recours \u00e0 l'usage m\u00e9taphorique de la lumi\u00e8re, en r\u00e9actualisant la rh\u00e9torique all\u00e9gorique de la caverne, en proposant une nouvelle th\u00e9orie de la participation, en affirmant une nouvelle causalit\u00e9 entre l'objet r\u00e9el et le r\u00e9el virtuel, enfin en illustrant une th\u00e9orie du temps qui, via la lumi\u00e8re, exprime la permanence de la v\u00e9rit\u00e9 d\u00e9finie comme \u00ab image mobile de l'\u00e9ternit\u00e9 immobile \u00bb. L'ensemble s'affiche aux antipodes du s\u00e9rieux philosophique classique pour \u00e9lire et constituer une m\u00e9taphysique ludique et joyeuse.\n\nAvant d'entreprendre un voyage dans l'anticaverne esth\u00e9tique, je voudrais donner le mode d'emploi du dispositif des artistes. Eux aussi racontent, d'une certaine mani\u00e8re, la caverne et le feu, des hommes encha\u00een\u00e9s et une route ascendante, un petit mur qui coupe ce chemin et des figurines assimil\u00e9es \u00e0 des marionnettes, des ombres projet\u00e9es sur les parois de la caverne et des simulacres esquiss\u00e9s \u00e0 la surface des eaux. De sorte que leur mise en sc\u00e8ne propose la th\u00e9\u00e2tralisation all\u00e9gorique d'une m\u00e9taphysique de l'immanence. L'antre souterrain devient chambre noire, camera oscura sous les auspices de quoi se place l'ensemble du processus. Non pas un appareil photographique, ou m\u00e9taphorique, mais un monde tout entier limit\u00e9 au registre de l'aperception ph\u00e9nom\u00e9nale. L'\u00eatre, c'est le visible. Dans l'\u0153uvre on assiste au processus d'\u00e9criture de la lumi\u00e8re, \u00e0 sa gen\u00e8se, son av\u00e8nement et son expansion. Sa nature et son essence co\u00efncident avec son existence. Le tout propose l'occasion de machines \u00e0 r\u00e9fl\u00e9chir, aussi bien la lumi\u00e8re, antique m\u00e9taphore de sapience, que le monde, lui-m\u00eame all\u00e9gorie sur le mode baroque.\n\nLe dispositif se suffit des limites de cette chambre noire. Au sol, on d\u00e9couvre un ensemble d'objets h\u00e9t\u00e9roclites. La plupart du temps, leur registre est quotidien, banal, sinon trivial : bouteille d'eau en mati\u00e8re plastique, brosse \u00e0 dents, balle de ping-pong, poils de moquette, jouets d'enfant, plateau de petit-d\u00e9jeuner et autres babioles issues du petit monde des objets utiles. Je me souviens d'un passage consacr\u00e9 \u00e0 l'all\u00e9gorie de la caverne o\u00f9 Platon pr\u00e9cise qu'on trouve en elle \u00abtoutes sortes d'objets fabriqu\u00e9s, des statues, ou encore des animaux en pierre, en bois, fa\u00e7onn\u00e9s en toute sorte de mati\u00e8re \u00bb. De la m\u00eame mani\u00e8re, chez Loriot-M\u00e9lia on remarque la diversit\u00e9 des mati\u00e8res, des formes, des couleurs, des structures ou des agencements.\n\nLe point commun de tous ces objets semble leur capacit\u00e9 \u00e0 mettre en cause le registre intelligible platonicien. A savoir ? Le philosophe se demande quel peut bien \u00eatre le statut des ongles, des poils, de la crasse, et autres joyeuset\u00e9s, dans leur rapport \u00e0 l'univers intelligible. De quel absolu id\u00e9al proc\u00e8dent ces d\u00e9gradations quintessenci\u00e9es ? R\u00e9ponse : d'aucun. Ils posent probl\u00e8me et fournissent un genre d'objet r\u00e9tif \u00e0 la th\u00e9orie platonicienne de la participation du sensible \u00e0 l'intelligible. Confin\u00e9s \u00e0 l'entropie, au relatif, \u00e0 la d\u00e9gradation, au n\u00e9gatif, ces rebuts ne disposent pas d'un statut m\u00e9taphysique. D'o\u00f9 l'int\u00e9r\u00eat, pour les antiplatoniciens, de saisir cette opportunit\u00e9 afin d'en rire et d'en faire la source d'un monde id\u00e9al...\n\nL'ensemble de ces objets triviaux s'agence avec de la colle, fig\u00e9, fix\u00e9 sur une surface plane, horizontale, au sol. Au milieu de ce joyeux capharna\u00fcm, des miroirs orient\u00e9s captent une source lumineuse artificielle, une lampe \u00e0 basse tension, ou naturelle, le bon vieux soleil cher au c\u0153ur des penseurs id\u00e9alistes de toujours. Au millim\u00e8tre pr\u00e8s, tout dans sa place contribue \u00e0 la suite des \u00e9v\u00e9nements dont le th\u00e9\u00e2tre des op\u00e9rations suppose le passage au registre vertical. La lumi\u00e8re, aid\u00e9e de miroirs, autorise la transfiguration des plans et rend possible la transcendance \u00e0 partir de l'immanence. Ici r\u00e9side la transmutation esth\u00e9tique de la position philosophique de Platon. Pour le philosophe, le r\u00e9el sensible, immanent, tient son essence et son existence d'un monde intelligible, transcendant; \u00e0 l'inverse, pour les artistes, le plan vertical, id\u00e9al, proc\u00e8de de l'\u00e9talage horizontal, trivial.\n\nLe paradoxe appara\u00eet dans l'inscription sur cette surface verticale. L'alchimie est la suivante : avec des objets divers, h\u00e9t\u00e9roclites, des miroirs, de la colle et de la lumi\u00e8re on obtient une image, une ic\u00f4ne, un dessin virtuel. Du banal au merveilleux, le trajet s'effectue \u00e0 l'aide d'une volont\u00e9 m\u00e9taphysique qui emprunte le flux d'une lumi\u00e8re. D'o\u00f9 la transfiguration : avec des d\u00e9tritus la m\u00e9tamorphose esth\u00e9tique produit un oiseau \u00e0 roulette, une colonne sur un socle, un verre de vin, une machine \u00e0 \u00e9crire, un avion qui largue des bombes, un arc de triomphe, un \u0153il, et autres formes sculpt\u00e9es dans la lumi\u00e8re, avec elle et par elle. Noms de bapt\u00eame ? Le Chercheur d'\u00e9nigme, La Solitude du vainqueur, L'Aur\u00e9olus dit le loriot ou l'invention de la bicyclette, ou encore Sol\u0153il.\n\nRetenons donc ce jeu entre deux registres, deux r\u00e9els : l'un, apparemment chaotique, d\u00e9sordonn\u00e9, immanent, horizontal, au sol, trivial \u2013 le monde sensible des cat\u00e9gories platoniciennes \u2013; l'autre r\u00e9ellement figural, ordonn\u00e9, transcendant, vertical, au mur, sublime \u2013 le monde intelligible selon le philosophe. O\u00f9 donc est Platon ? Dans la mise en perspective de ces deux r\u00e9els \u00e0 la relation \u00e9crite sur le mode lumineux, dans la logique du d\u00e9coulement, de la participation, de la causalit\u00e9, dans l'interaction qui d\u00e9signe l'essence dont proc\u00e8de toute existence, dans le dualisme spiritualiste renvoyant m\u00e9taphoriquement \u00e0 l'\u00e9criture par la lumi\u00e8re. O\u00f9 donc est-il d\u00e9pass\u00e9, renvers\u00e9 ? Dans l'affirmation, soutenue par une d\u00e9monstration de visu, que l'intelligible proc\u00e8de du sensible, que l'image, l'ic\u00f4ne, la lumi\u00e8re sculpt\u00e9e, la forme, participent du sensible et des objets triviaux, que l'objet r\u00e9el rend possible l'av\u00e8nement du r\u00e9el virtuel.\n\nApr\u00e8s avoir racont\u00e9 son all\u00e9gorie de la caverne, Platon demande : qu'est-ce qui est le plus vrai ? avant de r\u00e9pondre, on s'en souvient, en c\u00e9l\u00e9brant le seul intelligible et en discr\u00e9ditant le r\u00e9el sensible. Eloge de l'id\u00e9e pure, du concept, de la forme id\u00e9ale, haine de la mati\u00e8re, du corps, de la chair, du monde concret. Nous vivons encore sur cette logique duelle et selon ce discr\u00e9dit jet\u00e9 sur le ph\u00e9nom\u00e9nal, le palpable, le perceptible qu'on supporte transfigur\u00e9, amoindri, cach\u00e9, recycl\u00e9. Loriot-M\u00e9lia affirment sans ambages la co\u00efncidence du monde vrai avec le visible. Pour eux, ce que l'on peut voir, c'est aussi bien le sensible que l'intelligible, puisque l'un est l'autre. Fid\u00e8les \u00e0 leur parti pris baroque dans l'art des pointes et l'excellence dans le jeu oxymorique, les artistes mettent en sc\u00e8ne un intelligible sensible, ou bien un sensible intelligible. D'o\u00f9 la promotion philosophique, dans cette esth\u00e9tique, d'un monisme mat\u00e9rialiste r\u00e9v\u00e9l\u00e9 gr\u00e2ce \u00e0 un jeu dualiste spiritualiste.\n\nEt l'on suivra sans difficult\u00e9 les odyss\u00e9es de l'oxymore, et de la rh\u00e9torique associ\u00e9e, en pointant ici un mou-dur en regard d'une gomme sensible contribuant au devenir d'un arc de triomphe intelligible, d'un trivial-noble permettant la transfiguration d'une bouteille en mati\u00e8re plastique en \u0153il, d'un divers-un d\u00e8s la multiplicit\u00e9 rep\u00e9r\u00e9e d'un bric-\u00e0-brac et l'unit\u00e9 de la forme qui s'en d\u00e9gage, d'un \u00e9clat\u00e9-harmonieux \u00e9vident entre l'horizontal chaotique et le vertical \u00e9quilibr\u00e9 dans une forme \u2013 l'ensemble produisant un r\u00e9el-id\u00e9al ou un id\u00e9al-r\u00e9el. Au-del\u00e0 de l'oxymore, du paradoxe, du d\u00e9passement d'\u00e9ventuelles contradictions, d'un genre d'Aujhebung plastique, le travail esth\u00e9tique produit une m\u00e9taphysique panth\u00e9iste en vertu de quoi, dans le monde, il n'existe qu'une seule substance diversement modifi\u00e9e.\n\nD'o\u00f9 la possibilit\u00e9 de d\u00e9passer les interrogations sur la nature ou le genre d'oeuvres propos\u00e9 par Loriot-M\u00e9lia : on l'a dite photographique. Certes, la chambre noire et la production d'une image sculpt\u00e9e par la lumi\u00e8re rend possible, voire facile, ce type de rapprochement. Pour autant, on peut tout aussi bien parler d'hologramme, d'anamorphose, de r\u00e9bus, de collage, de sculpture, d'installation, de vid\u00e9o d'un genre particulier o\u00f9 se croiseraient artefacts et simulacres, images virtuelles ou de synth\u00e8se. L'ensemble rel\u00e8ve indiscutablement de la d\u00e9marche conceptuelle, mais de beaucoup d'autres cat\u00e9gories habituellement utilis\u00e9es dans le registre de l'histoire de l'art contemporain. Pour ma part, j'aime croire qu'appara\u00eet dans ce travail d'artiste la g\u00e9n\u00e9alogie d'une m\u00e9taphysique iconique parente de la philosophie du soup\u00e7on qui se propose d'actualiser le renversement du platonisme, toujours urgent, et plus que jamais en ces temps d'intraitable m\u00e9lancolie, de spiritualisme d\u00e9l\u00e9t\u00e8re et de nihilisme g\u00e9n\u00e9ralis\u00e9. Une fois de plus, les philosophes voient leur chemin balis\u00e9 par les artistes \u2013 et c'est heureux.\n23\n\nLE D\u00c9SIR DES CHOSES INEXISTANTES\n\n(Lettre \u00e0 Eric Tanguy) \u2013. Connais-tu, cher Eric, ces pages de Marc Aur\u00e8le, au d\u00e9but des Pens\u00e9es pour moi-m\u00eame, dans lesquelles le philosophe fait le bilan, en de brefs textes, de ses dettes \u00e0 l'endroit de ceux qui lui sont \u00e9minemment chers ? J'aime toujours lire et relire ces lignes \u00e9crites par le vieil homme sous la toile de tente, le soir, apr\u00e8s les combats contre les Barbares arr\u00eat\u00e9s aux portes de l'Empire qu'il dirige alors. A mon p\u00e8re, qui m'a montr\u00e9, sans le savoir, ce que signifie vivre debout, \u00e0 mon vieux coiffeur, le confident de mes premi\u00e8res pens\u00e9es quand j'\u00e9tais enfant puis adolescent, \u00e0 mon vieux ma\u00eetre, professeur \u00e0 l'universit\u00e9, qui m'a le premier tendu la main en m'invitant \u00e0 passer de l'autre c\u00f4t\u00e9, l\u00e0 o\u00f9 s'\u00e9crivent les livres, \u00e0 ma compagne depuis vingt ans, je sais devoir l'essentiel de ce qui me constitue aujourd'hui. Sans eux, je ne serais pas ce que je suis. Moins ou plus, diff\u00e9rent ou radicalement autre, de toute fa\u00e7on, sur une autre route, d\u00e9j\u00e0 mort ou en cours de destruction, emp\u00eatr\u00e9 dans d'insolubles contradictions, interdit dans des impasses, \u00e0 terre, le genou, pour la meilleure hypoth\u00e8se, le visage et le corps enti\u00e8rement recouvert de poussi\u00e8re, pour la pire...\n\nSurprises de rencontres, n\u00e9cessit\u00e9s de d\u00e9terminations puissantes, force du destin et autres croisements singuliers qui tracent les hasards objectifs d'un Andr\u00e9 Breton dont j'aime le nomadisme po\u00e9tique : les proximit\u00e9s, autant que les \u00e9loignements, contribuent au portrait de soi, \u00e0 la figure qu'on porte et conduit doucement chaque jour vers la tombe. A toi, donc, je veux dire aujourd'hui ce que je dois en te l'\u00e9crivant ce soir, apr\u00e8s les \u00e9motions de la cr\u00e9ation d'une de tes pi\u00e8ces, les Huit Tableaux pour Orph\u00e9e, la troisi\u00e8me dont tu m'aies demand\u00e9 le texte. Dans la nuit qui a suivi cette soir\u00e9e, longue et blanche, comme tu t'en souviens, entre le champagne de l'imm\u00e9diat et le caf\u00e9 du petit matin, avant le retour et les pas fatigu\u00e9s dans les rues de Paris, dans ma voiture, alors que je quittais ta ville pour retrouver ma campagne, en roulant \u00e0 vive allure, heureux des lumi\u00e8res de cette journ\u00e9e qui s'annon\u00e7ait, fourbu par l'absence de sommeil, la conversation, les rires, les vapeurs d'alcool, j'ai \u00e9crit cette lettre, morceau par morceau, mentalement, entre la jubilation de co\u00efncider vivement avec l'existence et le plaisir fou que j'avais alors \u00e0 vivre, en bouff\u00e9es violentes et sublimes.\n\nQuelles probabilit\u00e9s avais-je, sans toi, de rentrer dans ce monde de la musique contemporaine autrement qu'en auditeur, en spectateur, en amateur passif? Que je puisse, de pr\u00e8s ou de loin, me trouver de plain-pied avec cet univers constitutif d'une enclave dans la g\u00e9ographie du monde intellectuel de ce si\u00e8cle, voil\u00e0 un effet de ta seule entremise, de ton unique entregent. Ma passion de m\u00e9lomane n'aurait pas suffi, surtout doubl\u00e9e par mon incomp\u00e9tence musicale et mon inscience musicologique. De l'Ecole de Vienne aux tendances actuelles de la musique contemporaine, du disque \u00e0 la modulation de fr\u00e9quence radiophonique, loin des concerts et de la relation directe, toujours difficile quand on a choisi la province, qu'on y vit, qu'on y travaille, ma d\u00e9couverte de ce continent s'est faite dans le d\u00e9sordre, en sauvage, comme j'aime le faire quand j' ai d\u00e9cid\u00e9 de d\u00e9fricher une for\u00eat dans laquelle je veux cesser de me perdre.\n\nEn te rencontrant, j'ai dispos\u00e9 d'un fil d'Ariane, lumineux et efficace. Bien en amont de Henri Demarquette au violoncelle et de Fran\u00e7ois-Fr\u00e9d\u00e9rique Guy au piano, dans les caves d'Yquem o\u00f9 furent donn\u00e9es quelques-unes de tes pi\u00e8ces, bien avant les applaudissements que tu me demandes de partager avec toi, sur sc\u00e8ne, apr\u00e8s la cr\u00e9ation d'une de tes \u0153uvres, quand j'en ai \u00e9crit le texte, bien loin des ajustements que nous r\u00e9alisons pour Chronos, l'op\u00e9ra auquel tu travailles et Le Libertin foudroy\u00e9, l'op\u00e9ra radiophonique dont tu m'as aussi demand\u00e9 les mots, je m'\u00e9tais r\u00e9joui des trajets complices que tu m'invitais \u00e0 parcourir avec toi, \u00e0 tes c\u00f4t\u00e9s, pour cette exp\u00e9rience d'amiti\u00e9 en acte. J'\u00e9tais loin d'imaginer que je te devrais ces plaisirs-l\u00e0...\n\nJe me souviens encore de ces approches mutuelles de part et d'autre, toi aimant mes livres, moi ton \u0153uvre musicale, et notre ardeur \u00e0 nous manquer au d\u00e9part, quand nous savions si peu l'un de l'autre. Adresses fautives, d\u00e9m\u00e9nagements, Villa M\u00e9dicis pour toi, courriers qui n'arrivaient pas, envois vol\u00e9s dans ta bo\u00eete aux lettres, et autres lapsus du monde comme il va, presque pour nous signifier qu'il n'\u00e9tait pas l'heure de notre rencontre, trop t\u00f4t. Je me souviens \u00e9galement, \u00e0 la mairie de Caen, d'une petite heure, pour moi entre deux cours, pour toi entre deux avions, consacr\u00e9e \u00e0 applaudir ton succ\u00e8s au prix de Rome. Une \u0153uvre de toi jou\u00e9e par une fl\u00fbtiste solo, les officiels, que je fuis comme la peste, et ton regard me cherchant pour accrocher un peu nos deux conversations. Ce ne fut pas, non plus, ce jour-l\u00e0.\n\nJe savais, par France Musique, que tu projetais un op\u00e9ra. J'ai souvenir, \u00e9galement, d'y avoir entendu pour la premi\u00e8re fois, la voix velout\u00e9e, chaude, puissante de Delphine Collot \u00e0 qui l'on doit la cr\u00e9ation de nos \u2013 comme tu dis \u2013 pi\u00e8ces pour voix soliste. J'imaginais que tu avais pressenti un librettiste ou que de toute fa\u00e7on, tes succ\u00e8s aidant, tu avais plut\u00f4t l'embarras du choix. Ce qui, d'une certaine mani\u00e8re, n'\u00e9tait pas \u00e9loign\u00e9 de la v\u00e9rit\u00e9. Comment aurais-je imagin\u00e9 que tu n'osais pas me demander? Surtout ce soir o\u00f9 nous nous sommes rencontr\u00e9s chez l'un de tes amis dont je savais le d\u00e9sir d'\u00e9crire ce texte dont tu avais besoin. La soir\u00e9e fut consacr\u00e9e pour toi \u00e0 \u00e9viter de me demander si le projet me tentait, pour moi \u00e0 faire silence sur le sujet pour qu'on n'imagine pas ma pr\u00e9sence justifi\u00e9e par opportunisme. Le registre du livret me semblait tellement aux antipodes de ce que je croyais pouvoir faire...\n\nTu n'as gu\u00e8re d\u00een\u00e9, d'ailleurs, me confiant, plus tard, que la r\u00e9p\u00e9tition de l'une de tes pi\u00e8ces \u00e0 laquelle tu venais d'assister t'avait sembl\u00e9 si d\u00e9plorable que, dehors, retrouvant la nuit, tu avais vomi dans la rue. Tes lunettes en \u00e9cailles sur le dessus de la t\u00eate, ta signature, v\u00eatu tout de jean, le sourire masquant \u00e0 ravir ta d\u00e9convenue, tu fus affable et n'importunas personne avec tes soucis, comme \u00e0 l'accoutum\u00e9e. Le repas avait pour basse continue ce projet d'op\u00e9ra qui me voyait silencieux et admiratif des \u00e9tats dans lesquels il semblait alors se trouver.\n\nPour compenser et d\u00e9passer ce silence sur le sujet du livret, tu m'as demand\u00e9 des retrouvailles \u00e0 Paris, sur ton terrain. Non loin de la rue des Abbesses, o\u00f9 j'avais des obligations ce jour-l\u00e0. Nous nous sommes retrouv\u00e9s, en t\u00eate \u00e0 t\u00eate, dans un restaurant o\u00f9, d'embl\u00e9e, nous nous sommes dit l'essentiel, ce qui fait le fond existentiel d'une vie : l'essence d'un trajet au quotidien, la tonalit\u00e9 m\u00e9taphysique d'un \u00eatre. Te souviens-tu de ce que je t'ai alors confi\u00e9? A l'instant, je crois, nous sommes devenus complices, ins\u00e9parables. L'ensemble fut scell\u00e9 de vin, de rire et de lumi\u00e8re. Puis d'autres secrets ont \u00e9t\u00e9 \u00e9chang\u00e9s, partag\u00e9s, offerts.\n\nCe jour, tu me demandas le livret de ton premier op\u00e9ra. Tu souhaitais une adaptation d'une nouvelle de Sade extraite des Crimes de l'amour car tu appr\u00e9ciais plus que tout la structure et la composition litt\u00e9raire de l'\u0153uvre. Heureusement, tu as ajout\u00e9 que tu attendais aussi que, sur sc\u00e8ne, l'effet soit des plus orgiaques. Je n'ai pas eu envie d'\u00e9crire pour adapter, encore moins pour permettre \u00e0 la sc\u00e8ne les th\u00e9ories sadiennes que je trouve en de\u00e7\u00e0 de toute jubilation v\u00e9ritable. Je t'ai propos\u00e9 un livret sur mesure dans lequel nous pourrions plus librement exprimer ce que nous avions envie, toi et moi, de dire, un genre d'autobiographie intellectuelle \u00e0 deux. Et ce fut Chronos aujourd'hui sur ton bureau en attente de l'av\u00e8nement, par toi, sur les planches.\n\nAvant de passer \u00e0 l'acte, d'oser \u00e9crire un texte destin\u00e9 \u00e0 l'op\u00e9ra, quelque chose entre le long po\u00e8me lyrique et la petite pi\u00e8ce de th\u00e9\u00e2tre, je me suis demand\u00e9 ce qui m'avait laiss\u00e9 croire que je pouvais en \u00eatre capable. Par-devers moi, je ne m'imaginais pas travaillant sur ce registre. Et si je veux dire aujourd'hui le premier savoir que je te dois, c'est celui-ci : qu'il faut parfois la main d'un tiers aimant et aim\u00e9 pour passer le pas, franchir une limite, accepter d'aller sur des terres o\u00f9, sinon, on n'oserait jamais mettre les pieds. Que la ma\u00efeutique est vertu d'ami, qu'elle fait l'ami, le rend possible, lui donne ses lettres de noblesse, co\u00efncide avec elle. Que la r\u00e9v\u00e9lation d'une puissance propre suppose parfois le d\u00e9tour par un tiers \u2013 et que ce tiers devient partie prenante de l'\u00e2me alors venue et qu'on lui doit.\n\nNos conversations sur les femmes, nos nuits de mots et de rire, nos \u00e9changes t\u00e9l\u00e9phoniques, nos t\u00e9l\u00e9copies nocturnes et intempestives, nos verres de bordeaux bus dans les soir\u00e9es de la rue Saint-Blaise, chez toi, nos complicit\u00e9s lilloises, r\u00e9moises, caennaises, parisiennes, argentanaises, nous ont conduit \u00e0 Marie-Madeleine. Quand tu m'as demand\u00e9, pour l'exercice d'\u00e9criture vocale que suppose l'op\u00e9ra, en amont, un texte pour une cantate, je d\u00e9sirai une figure f\u00e9minine de d\u00e9sir et de p\u00e9ch\u00e9 \u2013 d'une certaine mani\u00e8re une femme qui suppose ton go\u00fbt pour les personnages de Marivaux et le mien pour le Don Juan de Moli\u00e8re. La cr\u00e9ation devait avoir lieu dans une \u00e9glise. Le sujet, disais-tu, aurait gagn\u00e9 \u00e0 \u00eatre religieux, mais pa\u00efen aussi, un peu dionysien, un m\u00e9lange de dignit\u00e9 conf\u00e9r\u00e9e par l'histoire sainte et de soufre respir\u00e9 dans les vapeurs d'une femme dangereuse.\n\nC\u00e9l\u00e9bration de Marie-Madeleine t'a donn\u00e9 l'occasion de musiquer cette femme singuli\u00e8re avec un ch\u0153ur et un orchestre, cette voix une faite de diversit\u00e9s exprimant moins l'ing\u00e9nuit\u00e9 que le d\u00e9sir d'une femme imp\u00e9rieuse pour un homme dont l'histoire s'\u00e9vertue \u00e0 gommer le corps \u2013 car, en Occident le corps du Christ n'existe que sous forme p\u00e2tissi\u00e8re ou assimil\u00e9e. Marie-Madeleine me pla\u00eet pour l'abandon sensuel aux parfums et la violence de la passion, la fid\u00e9lit\u00e9 des peaux et la proximit\u00e9 aux heures tragiques, la magnanimit\u00e9 et la capacit\u00e9 \u00e0 la d\u00e9pense magnifique, l'ind\u00e9fectible pr\u00e9sence et l'amour immortel. Elle te pla\u00eet pour ce qu'elle dit de f\u00e9minin, d'\u00e9ternel et d'inqui\u00e9tant. Il fallait bien qu'elle nous s\u00e9duis\u00eet tous les deux. Finalement la cr\u00e9ation n'eut pas lieu dans une \u00e9glise, mais dans le cirque d'hiver de Reims dont la rotondit\u00e9 exprimait \u00e0 ravir l'\u00e9ternel retour du m\u00eame, et les folies circulaires de nos \u00e2mes virevoltantes.\n\nSur les femmes, il y eut encore et encore des conversations nocturnes, des soir\u00e9es longues qui entamaient la nuit tr\u00e8s avant. Je te racontais le libertinage, tu me parlais marivaudage, je te disais le catalogue et les corps d\u00e9sesp\u00e9r\u00e9s chez Don Juan, tu me rappelais les mots, la parole, la strat\u00e9gie, la tactique, le d\u00e9sir confin\u00e9 au seul verbe chez Marivaux \u2013 ce qui m'\u00e9nerve et te ravit. Il y avait l\u00e0 un bordeaux superbe et la musique s'\u00e9tait tue. Ni Sibelius ni Vivaldi, retourn\u00e9s \u00e0 l'ineffable, mais le silence que soutenaient en contrepoint des minutes lourdes dans lesquelles notre amiti\u00e9, finalement, se nourrissait de mani\u00e8re substantielle. En fait, Marie-Madeleine exprime de fa\u00e7on radicale ce qui fait nos diff\u00e9rences sur ce sujet. Toi, inquiet du d\u00e9sir, moi d\u00e9sesp\u00e9r\u00e9 du plaisir. Deuxi\u00e8me le\u00e7on que je te dois : apprendre que pour comprendre, projeter ses cat\u00e9gories sur la logique d'autrui pr\u00e9pare les catastrophes. Je l'ai appris ce soir-l\u00e0, saisissant qu'\u00e0 force de se croire image invers\u00e9e d'autrui plut\u00f4t qu'autre figure on forcit les vents avec lesquels se font les tornades.\n\nPlus tard, apr\u00e8s Marie-Madeleine, comme en \u00e9cho \u00e0 cette soir\u00e9e de silence dans les rues nocturnes de Paris, d\u00e8s que je sus qu'apr\u00e8s ces douleurs il n'y aurait plus de malentendu, je ressentis en moi la paix qui nous envahit aux petits matins blancs, d\u00e8s le lever du jour, et qu'on sait exaltante la t\u00e2che attendue pour les heures \u00e0 venir. Plus qu'un tour de cadran, je savais que nous attendaient de plus longues aventures rendues possibles par des minutes de serment silencieux, celles aux cours desquelles, sans se l'avouer, on se promet de ne plus jamais donner prise aux m\u00e9chantes blessures.\n\nApr\u00e8s le silence qui suivit la voix de Marie-Madeleine en \u00e9cho, il y eut ton d\u00e9sir d'un autre texte. Autant les mots de la cantate relevaient d'une fa\u00e7on de dire en une figure unique deux fa\u00e7ons d'appr\u00e9hender le monde des femmes, autant je ne voyais pas, a priori, quelles lignes je pourrais t'\u00e9crire pour un cycle de m\u00e9lodies avec violoncelle, fl\u00fbte et piano. Je me rendais en Espagne, tu me sollicitais \u00e0 quelques minutes de mon d\u00e9part. Je te promis d'y songer dans le train et de te voir, \u00e0 l'arriv\u00e9e, \u00e0 la gare, avant la route de l'a\u00e9roport \u00e0 destination de Madrid o\u00f9 j'avais rendez-vous avec Fernando Savater. Je venais de finir Les Formes du temps et songeais \u00e0 un portrait de Bacchus puisque je te sais, comme moi, plus amateur de Dionysos que d'Apollon, aussi bien dans ton existence que dans ta musique.\n\nA Madrid, je laissais vagabonder mon esprit en direction du texte que je te destinais. Je faisais confiance une fois de plus au hasard objectif de Breton. Les choses furent bien faites : au Prado, et dans deux ou trois mus\u00e9es o\u00f9 je traquais passionn\u00e9ment J\u00e9r\u00f4me Bosch, je d\u00e9cidais de t'\u00e9crire quelques lignes \u00e0 partir de cette \u00e9motion qui, au moment o\u00f9 elle eut lieu, me ramena imp\u00e9rieusement vers toi et les lignes \u00e0 \u00e9crire. Devant Le Jardin des d\u00e9lices terrestres, je fus saisi d'\u00e9motion comme nagu\u00e8re \u00e0 Lisbonne devant La Tentation de saint Antoine et \u00e0 Vienne en pr\u00e9sence du Jugement dernier. Le foisonnement, la folie, le d\u00e9lire, le r\u00eave, le tragique, le cocasse, l'abondance, le codage, l'ironie, tout me ravit, et j'y trouvais l'\u00e9quivalent de ta musique avec en t\u00eate les violences et l'\u00e9nergie de ton Jubilate.\n\nEt ce fut Hommage \u00e0 J\u00e9r\u00f4me Bosch que tu titras Le Jardin des d\u00e9lices. Comme pour la cantate, le travail en commun fut facile : le texte \u00e9crit, il me semble qu'il t'appartient et que je ne peux que consentir \u00e0 ce qui te sera n\u00e9cessaire. Que d'ailleurs tu me demanderas seulement ce qui te sera utile pour faire aboutir ton d\u00e9sir : \u00e9teindre une consonne, solliciter une voyelle, inverser une phrase, d\u00e9placer un mot, modifier une assonance ou une autre op\u00e9ration utile \u00e0 la mise en son. Je venais au mus\u00e9e pour les derniers Goya, et je fus saisis par Le Greco et ces toiles sans pareil de Bosch. En l'amiti\u00e9 dont je parle, l'un sommeille toujours dans l'\u00e2me de l'autre, au point qu'une \u00e9motion forte, une sensation violente, un plaisir, ne peuvent s'entendre qu'en commun, partag\u00e9s, du moins racont\u00e9s.\n\nEnfin, apr\u00e8s la cantate consacr\u00e9e \u00e0 Marie-Madeleine, puis la pi\u00e8ce c\u00e9l\u00e9brant l'onirisme de Bosch, tu me fis part d'un nouveau d\u00e9sir. N'importe quoi, pour une voix qu'accompagneraient huit violoncelles. Pas plus en pr\u00e9sence d'un travail tout fait que les fois pr\u00e9c\u00e9dentes, toujours autant attentif aux hasards objectifs, je re\u00e7us, dans les jours suivant ta demande, une proposition d'\u00e9criture dans une revue qui consacrait sa livraison \u00e0 Orph\u00e9e. J'aime Berlioz, \u00e9crivant sur ce sujet, un genre de poncif \u00e0 sa mani\u00e8re qui suppose la mise en ab\u00eeme du traitement musical des pouvoirs de la musique. Or, le mythe me pla\u00eet moins pour cette histoire que pour la constance amoureuse d'Orph\u00e9e dont la t\u00eate coup\u00e9e, tra\u00eenant dans l'eau apr\u00e8s le massacre, prononce encore le nom d'Eurydice. Comme dans Le Libertin foudroy\u00e9, ou m\u00eame Chronos, je dis ma passion pour les figures d\u00e9sesp\u00e9r\u00e9es qu'un amour a marqu\u00e9es pour l'\u00e9ternit\u00e9 et qui ne se remettent pas de l'avoir connu. Et ce furent Huit Tableaux pour Orph\u00e9e.\n\nPour Bosch et Orph\u00e9e, Delphine Collot fit la cr\u00e9ation. A tes c\u00f4t\u00e9s, lors des r\u00e9p\u00e9titions, l'apr\u00e8s-midi, dans le silence du th\u00e9\u00e2tre, alors que le monde va son rythme et que les auditeurs du soir sont encore \u00e0 leur travail, l'\u0153uvre sort des limbes et passe du noir et blanc de la partition \u2013 aux couleurs, volumes et dimensions rendues possibles par la cantatrice. Etrange sensation : \u00eatre \u00e0 tes c\u00f4t\u00e9s, voir, regarder, \u00e9couter, entendre, guetter le premier son, basculer dans le silence qui suit la premi\u00e8re exposition int\u00e9grale, sombrer dans le d\u00e9licieux vertige d'apr\u00e8s le d\u00e9ploiement de ton \u0153uvre, puis sentir la voix se d\u00e9placer dans sa propre gorge, le souffle s'acc\u00e9l\u00e9rer, la respiration trahir la rar\u00e9faction de l'air, les muscles se t\u00e9taniser \u2013 et se demander comment fait la chanteuse, elle, pour moduler un son chaud, puissant, voluptueux, tranchant et net. Debout, le corps \u00e9trangement pos\u00e9 en une posture baroque qu'elle affectionne, les pieds et les jambes en une sorte de pas de danse, le buste l\u00e9g\u00e8rement en arri\u00e8re, la t\u00eate port\u00e9e avec \u00e9l\u00e9gance, Delphine Collot donne \u00e0 ce moment une tension singuli\u00e8re. Je me demande toujours, dans cet instant pr\u00e9cis, \u00e0 quoi tu penses, ce qui traverse ton esprit, ce qui se passe dans ta t\u00eate...\n\nTroisi\u00e8me le\u00e7on : sans toi, je n'aurais jamais os\u00e9 aborder ce registre que je dirai po\u00e9tique. La po\u00e9sie d'aujourd'hui me laisse souvent sur ma faim dans son parti pris \u00e9tique, minimal, abscons, maigre, obscur, \u00e9sot\u00e9rique, ou pire, verbeux. J'aime la chair, les images baroques et le lyrisme, la pr\u00e9sence saturante des \u00e9motions et des sensations dans l'\u00e9criture, les sens exacerb\u00e9s, les perceptions magnifi\u00e9es, les parfums et les go\u00fbts, les images et les volumes, les sons et les correspondances baudelairiennes qui font vibrer, danser et r\u00e9agir ces vitalit\u00e9s d\u00e9bordantes. Dans le registre philosophique o\u00f9 j'officie habituellement, l'\u00e9criture soucieuse d'une prose litt\u00e9raire induit la condamnation \u00e0 mort inflig\u00e9e par les donneurs de le\u00e7ons.\n\nDe m\u00eame, pour les amateurs de seule \u00e9criture que contrarie dans leur lecture la pr\u00e9sence d'une seule id\u00e9e. Souvent ceux qui entretiennent l'antinomie souffrent \u00e0 produire quoi que ce soit en un seul domaine \u2013 la prose pure ou les id\u00e9es seules. J'aime, en po\u00e9sie, les baroques et les mani\u00e9ristes, les symbolistes et les surr\u00e9alistes, Jean de Sponde et Th\u00e9ophile de Viau, Saint-Pol Roux et Annie Le Brun, le contraire des \u00e2neries contemporaines qui oscillent entre l'herm\u00e9tisme radical et le simplisme d\u00e9liquescent, l'\u00e9sot\u00e9risme sans concessions et la banalit\u00e9 sid\u00e9rale. Je n'imagine pas la pertinence d'une po\u00e9sie philosophique. En revanche, il me plairait d'\u00e9voluer non loin des philosophies po\u00e9tiques, celles qu'inaugurent les pr\u00e9socratiques par exemple.\n\nDe sorte qu'en essayant de dire Marie-Madeleine, les cr\u00e9atures de J\u00e9r\u00f4me Bosch, les aventures d'Orph\u00e9e ou les tribulations d'un Libertin, je m'engage sur une voie, pour toi, par toi. Par ailleurs, si je consens \u00e0 ce registre pour le plaisir d'un texte \u00e0 te donner, il me faut te dire tout de m\u00eame l'\u00e9trange impression de malaise que j'ai \u00e0 m'aventurer sur pareilles terres. Ecrites pour toi, ces lignes d'abord destin\u00e9es \u00e0 un dialogue avec toi, un genre de conversation en t\u00eate \u00e0 t\u00eate, deviennent soudain, via la voix de la cantatrice et ta musique, une proposition faite \u00e0 un public. Loin du silence o\u00f9 j'imagine habituellement la fr\u00e9quentation de mes pages, dans le dialogue intime, il me faut d\u00e9couvrir la sensation d\u00e9sagr\u00e9able de l'exposition publique.\n\nPourquoi d\u00e9sagr\u00e9able? D'abord parce que j'ai l'impression que mon travail vaut pour rien, ou si peu, dans l'\u00e9laboration et l'ach\u00e8vement du tien. Mais aussi car je crains, malgr\u00e9 tout, qu'il soit remarquable, au sens \u00e9tymologique, par son indigence, qu'il attire l'attention par ses d\u00e9fauts. Sur le fil du rasoir, non loin du pr\u00e9cipice, il me semble que ce que j'\u00e9cris alors \u2013 des po\u00e8mes? \u2013 c\u00f4toie de tr\u00e8s pr\u00e8s le ridicule, qu'un souffle, un grain de poussi\u00e8re s\u00e9pare le s\u00e9rieux et le d\u00e9risoire, le tragique et le comique, la mine afflig\u00e9e et l'\u00e9clat de rire r\u00e9probateur. Emport\u00e9 au-del\u00e0 de ce que je fais habituellement, d\u00e9port\u00e9 sur des terres o\u00f9 je ne me suis jamais aventur\u00e9 seul, je ressens, lors des concerts, l'impression douloureuse que j'aurais mieux fait de m'abstenir, de ne pas consentir.\n\nA une vieille dame qui me demandait \u00e0 l'issue d'un concert l'effet de la mise en musique de mes textes, ce passage de l'intime, du registre secret et silencieux, \u00e0 celui, tapageur, bruyant, expos\u00e9 de la repr\u00e9sentation, je n'ai pas trouv\u00e9 d'autre mot qu' obsc\u00e8ne. Je ne sais pas pourquoi. Moi qui pratique l'autobiographie philosophique comme une hygi\u00e8ne nietzsch\u00e9enne, un d\u00e9sir et une volont\u00e9 de probit\u00e9 \u00e0 l'endroit de mes lecteurs, moi qui pense que toute id\u00e9e avanc\u00e9e se doit d'\u00eatre montr\u00e9e dans la logique existentielle l'ayant rendue possible, moi qui revendique l'inscription de tout concept dans le terreau d'une vie propre, d'une subjectivit\u00e9 qui doit s'annoncer et s'\u00e9noncer comme telle, je me retrouve en situation d'exhibitionnisme d\u00e8s lors que chantent ces pages \u00e9crites pour toi, loin de la simplicit\u00e9 silencieuse d'une amiti\u00e9 qu'on entretient.\n\nPar toi, je laisse mes habitudes d'\u00e9criture de c\u00f4t\u00e9 et je renverse les valeurs : dans mes livres je fais primer le fond sans n\u00e9gliger la forme, et je tiens aux id\u00e9es sans pour autant m\u00e9priser le style ; dans les po\u00e8mes, disons-le ainsi, je veux d'abord la forme, la f\u00eate des mots, la c\u00e9l\u00e9bration h\u00e9doniste de la langue, le jeu avec elle, sans pour cela faire fi de deux ou trois choses \u00e0 dire. Le d\u00e9sir et le plaisir, le r\u00eave et la r\u00e9alit\u00e9, la f\u00e9minit\u00e9 et la fid\u00e9lit\u00e9, la souffrance et la jubilation, la guerre et la mort dans l'amour, par exemple, traversent de part en part, comme des fl\u00e8ches le corps de saint S\u00e9bastien, les portraits de Marie-Madeleine, de Bosch et d'Orph\u00e9e. Voire plus encore Chronos et Le Libertin foudroy\u00e9. Mais dans ce jeu avec les mots, je m'aventure dissimul\u00e9 par ta musique, cach\u00e9 derri\u00e8re elle, prot\u00e9g\u00e9.\n\nSais-tu donc, ou l'imagines-tu, qu'\u00e0 chaque fois que tu me demandes de t'accompagner sur sc\u00e8ne pour partager les applaudissements avec toi, je m'\u00e9prouve toujours intrus et inopportun, mal venu, ind\u00e9sirable, indigne d'\u00eatre f\u00eat\u00e9 \u00e0 tes c\u00f4t\u00e9s? Car pour toi, je voudrais n'\u00e9crire que des textes sans signature, aussit\u00f4t oubli\u00e9s, dissoci\u00e9s de mon nom. Je sais d\u00e9j\u00e0 combien tu refuses ce que je t'\u00e9cris ici et quelle g\u00e9n\u00e9rosit\u00e9 tu mets \u00e0 me dire \u00e0 l'origine du contenu musical de ton travail dans la musicalit\u00e9 de mon \u00e9criture. Je ne peux pas le croire. Enfin je veux cesser l\u00e0-dessus pour ne retenir qu'une chose, et il s'agit d'une nouvelle le\u00e7on : tu m'as appris \u00e0 travailler \u00e0 deux. Ce qu'avant toi je croyais absolument impossible, en individualiste forcen\u00e9 et farouche, en sauvage int\u00e9gral. Je sais, depuis toi, qu'\u00e0 deux se pensent intellectuellement des choses impossibles seul. Quatri\u00e8me le\u00e7on, et de taille...\n\nJ'ai compris cela d\u00e8s qu'une question m'a \u00e9t\u00e9 pos\u00e9e, avant la cr\u00e9ation de l'octuor de violoncelles et voix. Apr\u00e8s qu'on m'a demand\u00e9, tu t'en souviendras, quelles \u00e9taient mes relations de travail avec toi, comment nous nous y prenions au quotidien, j'ai dit, en le d\u00e9couvrant au fur et \u00e0 mesure du d\u00e9pliage de ma pens\u00e9e, qu'on ne devenait pas ami en collaborant, mais qu'on \u00e9tait ami, puis qu'on collaborait. Je date la naissance de notre amiti\u00e9 ce jour de confidence, non loin de la rue des Abbesses. Le reste illustre les formes prises par elle.\n\nJ'ai aim\u00e9 d\u00e9couvrir en m\u00eame temps que je le pensais, ou penser en m\u00eame temps que je le d\u00e9couvrais, ce qui d\u00e9finit vraiment l'amiti\u00e9 : moins une religion qu'une foi, moins un pari susceptible d'\u00eatre perdu qu'une sapience certaine : conna\u00eetre le plaisir de cr\u00e9er des choses ensemble, de participer \u00e0 des projets communs. Car j'ai assez pay\u00e9 pour le savoir, je me suis assez fourvoy\u00e9 pour en \u00eatre averti aujourd'hui, l'amiti\u00e9 existe moins que les preuves d'amiti\u00e9 : ces preuves se donnent et se trouvent dans l'action commune. La scansion, le chant, le d\u00e9coupage, les \u00e9nergies, les violences, les silences, les images lanc\u00e9es dans l'obscurit\u00e9, les mots transfigur\u00e9s, la parole musiqu\u00e9e, puis offerte au public silencieux et anonyme, la confusion des registres et la production d'une m\u00e9lodie qui sature la salle obscure avant les applaudissements, je sais trouver l\u00e0 toutes les preuves de l'amiti\u00e9 incarn\u00e9e.\n\nApr\u00e8s, d\u00e8s que fusent les applaudissements, pire encore quand il faut parler, se montrer, aux antipodes de ce que je ressens habituellement apr\u00e8s une conf\u00e9rence en solitaire, je voudrais prendre la fuite et retrouver le silence, un immense et majestueux silence, \u00e9norme et voluptueux. Gabriel Faur\u00e9 affirmait de la musique qu'elle donnait \u00able d\u00e9sir des choses inexistantes \u00bb \u2013je songe, en mati\u00e8re de chose inexistante, \u00e0 la satisfaction donn\u00e9e par un point d'orgue aussi long qu'une existence, et dispensant, apr\u00e8s l'ach\u00e8vement d'un plaisir, de la douleur d'en chercher un nouveau.\n\nUltime le\u00e7on, que je te dois : avoir compris le Nietzsche qui affirme que l'amiti\u00e9 d\u00e9finit ce qui permet \u00e0 l'autre d'aller au-del\u00e0 de lui, de conna\u00eetre ses limites, mais aussi d'exp\u00e9rimenter, de savoir et de d\u00e9couvrir \u00e0 quoi ressemblent les g\u00e9ographies o\u00f9 l'on n'avait pas pr\u00e9vu de s'aventurer, celles qui installent en face de soi avec des peurs, des craintes, des angoisses, des sensations, des exp\u00e9riences limites inconnues en regard de quoi il s'agit de mener un combat pour s'apprivoiser, s'apprendre, aller au-del\u00e0 de ses certitudes afin de plonger un peu dans les nuits qui bordent les \u00e2mes, l\u00e0 o\u00f9 parfois se nourrissent les angoisses, les insomnies et les monstres qui peuplent les cauchemars. Pour avoir avanc\u00e9 sur ces terres-l\u00e0, accompagn\u00e9 par l'\u00e9cho de ta musique, et sachant alors ta pr\u00e9sence radieuse, il me faudra longtemps te remercier.\n24\n\nLES HOMMES AUX PAUPI\u00c8RES COUSUES\n\nIl m'a fallu un temps fou pour savoir l'importance de l'envie dans les relations avec autrui. Et encore, je ne suis pas toujours s\u00fbr d'\u00eatre v\u00e9ritablement affranchi sur ce sujet tant pullulent les malentendus qui font croire sinc\u00e8res des histoires \u00e9crites sous la rubrique des perversions, du ressentiment contenu, voire de la haine dissimul\u00e9e, travestie. Je comptabilise et totalise un nombre incroyable de d\u00e9ceptions en mati\u00e8re d'amiti\u00e9 pour avoir m\u00e9sestim\u00e9 la part majeure de cette passion mortif\u00e8re dans la motivation de ceux qui m'ont approch\u00e9, fr\u00e9quent\u00e9, et dont certains avaient obtenu ma confiance. J'ai plus souffert de peines de c\u0153ur en amiti\u00e9 qu'en amour et je crois savoir aujourd'hui leurs g\u00e9n\u00e9alogies dans l'envie install\u00e9e en tiers entre ceux \u00e0 qui je donnais mon amiti\u00e9 et moi.\n\nIl faut avoir v\u00e9cu un peu pour mesurer les d\u00e9g\u00e2ts qu'elle occasionne, avoir v\u00e9cu et, vraisemblablement, s'\u00eatre quelque peu \u00e9cart\u00e9 des sentiers balis\u00e9s o\u00f9 le plus grand nombre chemine. De sorte que la plupart du temps on devient un objet digne d'envie lorsque l'on a trac\u00e9 son parcours en individu libre, de mani\u00e8re oblique, en traverse. Tous ha\u00efssent la diff\u00e9rence car elle distingue, s\u00e9pare, isole et r\u00e9v\u00e8le des hi\u00e9rarchies dont l'\u00e9vidence brutalise l'inconscient des m\u00e9diocres. Car on envie ce que l'on croit \u00eatre la jouissance de l'autre pour \u00e9viter de regarder en face ses fiascos, ses limites ou ses propres incapacit\u00e9s \u00e0 jouir ou \u00e0 conna\u00eetre la volupt\u00e9. L'envie est passion d'impuissant.\n\nBien s\u00fbr, il existe des envies constructives, positives, avec lesquelles on architecture un vouloir, on \u00e9labore un trajet singulier. Victor Hugo voulant \u00eatre Chateaubriand ou rien, par exemple, voil\u00e0 un d\u00e9sir qui d\u00e9passe de loin celui du rond-de-cuir aspirant au statut de son sup\u00e9rieur hi\u00e9rarchique, sous-responsable aux commissions d'\u00e9tudes dans un bureau de seconde zone. J'\u00e9carte celles qui permettent aux grands de se mesurer \u00e0 \u00e9galit\u00e9, elles sont moins alors envie d'inf\u00e9rieurs qu'\u00e9mulations d'\u00e9gaux.\n\nL'envie suppose la jouissance prise \u00e0 la douleur d'autrui, voire, inversement, la peine d\u00e8s que se montre le bonheur ou la satisfaction de l'autre. Max Scheler donne la formule de cette vilaine passion : \u00ab Je peux tout te pardonner, sauf le fait que tu existes et que tu sois justement l'\u00eatre que tu es ; sauf le fait que je ne sois pas ce que tu es ; oui : que je ne sois pas toi. \u00bb Dans cette joie mauvaise \u2013 l'expression est de Nietzsche \u2013 scintille, noire, une dose massive de haine de soi impossible \u00e0 supporter et, pour cette raison, transform\u00e9e en m\u00e9pris de l'autre. L'envie proc\u00e8de de la transmutation d'une valeur n\u00e9gative, au creux de soi, en puissance destin\u00e9e \u00e0 d\u00e9truire autrui. Elle suppose la transfiguration d'une \u00e9nergie mortif\u00e8re interne en force haineuse lanc\u00e9e \u00e0 vive allure contre un autre qu'on jubile de salir pour mieux se d\u00e9faire des sanies port\u00e9es en soi. Un envieux, c'est avant tout un impuissant \u00e9panoui par son geste homicide contre ce qui constitue en l'autre son id\u00e9al du moi.\n\nL'homme du commun fait un envieux sur mesure : petit, \u00e9troit, mesquin, r\u00e9duit \u00e0 reproduire l'existence de la plupart, il grossit le rang des afflig\u00e9s de cette pathologie avec une facilit\u00e9 d\u00e9concertante. Semblable \u00e0 tous, unidimensionnel, souffrant d'un ego gonfl\u00e9 aux limites du possible et de l'acceptable, il d\u00e9teste ce qui, tous les jours, montre son inutilit\u00e9 et creuse l'\u00e9cart entre ce qu'il est, rien, et l'id\u00e9e qu'il se fait de lui, tout. Les m\u00e9diocres, dont la vie s'aligne en tout sur les mod\u00e8les offerts par la bourgeoisie occidentale, ne souffrent pas ce qui leur inflige par le gouffre et l'ab\u00eeme les preuves de la banalit\u00e9 de leur existence. Indigents, neutres, blancs, transparents, interchangeables, reproductibles, semblables, ils ne disposent que d'un recours pour vivre encore et malgr\u00e9 tout : l'envie, l'oubli de leur \u00e9chec transform\u00e9 en m\u00e9pris de la r\u00e9ussite des autres.\n\nL'envieux brise le miroir qui le renseigne sur sa laideur et se condamne, d\u00e8s sa rencontre avec une prochaine psych\u00e9, \u00e0 lancer une nouvelle fois ses projectiles contre la glace et son tain. En travaillant son corps, la passion vicieuse lui permet de vivre habit\u00e9 par une n\u00e9gativit\u00e9 \u00e0 l'\u0153uvre qui, sinon, le ferait imploser. C'est parce qu'il envie que l'impuissant peut encore continuer de vivre dans la peau de cet \u00eatre aux certitudes infimes, aux grandeurs improbables, aux petitesses \u00e9videntes. Le fiel lanc\u00e9 \u00e9conomise l'usage contre soi du produit corrosif. Il en va de la survie de son existence d\u00e9bile.\n\nTous les envieux que j'ai rencontr\u00e9s affichent une nature m\u00e9diocre. Ils montrent autant de hargne et d'activit\u00e9, de d\u00e9termination et d'arrogance, que le noyau dur de leur existence est endommag\u00e9, fissur\u00e9. A leur sujet, je veux m'arr\u00eater sur un singulier constat : leur commerce est d'autant plus insidieux qu'ils cherchent la compagnie de ceux qu'ils pourront le mieux d\u00e9tester et envier, salir et conchier, plus tard, d\u00e8s l'assurance de provisions faites de fausses raisons pour leur v\u00e9ritable haine. Leur tropisme pathologique les conduit vers ceux qui, loin d'eux, au-dessus, ou au-del\u00e0, leur fournissent l'occasion d'une belle haine, d'un excellent m\u00e9pris, d'un ressentiment de qualit\u00e9, le tout gonfl\u00e9 par une envie de premi\u00e8re consistance. Les envieux portent en eux une sapience obscure, enracin\u00e9e dans leur inconscient, en vertu de quoi ils sont inform\u00e9s souterrainement, profond\u00e9ment, mais aveugl\u00e9ment, de l'existence de cette rivi\u00e8re d'eau sale qui les traverse. Leur d\u00e9sir vise une vie sublime, et ils savent leur existence nulle.\n\nLes jets de bile, les crachats, les tra\u00een\u00e9es d'ordures rep\u00e9rables derri\u00e8re eux trahissent l'indigence de leur temp\u00e9rament, l'inconsistance de leur caract\u00e8re. Je mesure aujourd'hui combien j'\u00e9tais sot de croire tel ou tel dans mes parages pour une autre raison que l'attente d'une occasion de d\u00e9clencher un jour leur m\u00e9pris. Le jour venu se met \u00e0 couler, sur le mode de la fange, ce que j'avais pris pr\u00e9c\u00e9demment pour de l'affection, de la sympathie, voire des signes d'amiti\u00e9. Or leur approche, leur proximit\u00e9, leur man\u0153uvre de s\u00e9duction, leur passivit\u00e9 calcul\u00e9e, tout cela annonce une strat\u00e9gie, suppose la circonscription d'une cible sur laquelle ajuster son tir.\n\nLeur inconscient parle pour eux. A leur corps d\u00e9fendant, ils consentent \u00e0 \u00eatre le jouet de leurs pulsions mortif\u00e8res : du moins, sorties d'eux, \u00e9ject\u00e9es, extirp\u00e9es de leur ventre, ces violences cessent d'\u00eatre douloureuses et n\u00e9gatives \u00e0 leur endroit. Branch\u00e9s sur leur libido destructrice, les envieux \u00e9vitent leur consumation en travaillant \u00e0 la destruction d'autrui. Je vois d'ailleurs dans la raideur apparente de ceux-l\u00e0, dans leur arrogance de coq et de paon qui fr\u00e9tillent, ondulent et caquettent, jacassent, le sympt\u00f4me d'efficacit\u00e9 de ce dispositif qui les dote d'une hypoth\u00e9tique allure masquant mal le fond de leur mis\u00e8re.\n\nIl m'a fallu bien des interrogations, des stup\u00e9factions, des errances, des surprises et des douleurs, pour comprendre que l'envie peut expliquer la transformation soudaine d'une amiti\u00e9 en haine, ou justifier le violent et brutal devenir m\u00e9connaissable d'une sympathie sous le rictus d'une antipathie. J'ai d\u00e9couvert ce qui s'apparente \u00e0 un recyclage, \u00e0 une diversion, \u00e0 une m\u00e9tamorphose salutaire pour l'envieux ainsi d\u00e9barrass\u00e9 de sa n\u00e9gativit\u00e9 d\u00e9tourn\u00e9e en devenant violence concentr\u00e9e sur autrui. En griffant l'autre, l'envieux ignore qu'\u00e0 moindres frais, il se dispense d'une automutilation.\n\nLongtemps j'ai \u00e9t\u00e9 d\u00e9rout\u00e9, d\u00e9concert\u00e9, par les revirements spectaculaires qui transformaient en ennemi celui qui, la veille encore, du moins le croyais-je, avait \u00e9t\u00e9 mon ami. Dans le cas o\u00f9 le reniement est monstrueux, les anecdotes prises pour d\u00e9clencheurs ou t\u00e9moins brillent toujours dans le grotesque et le saugrenu. D'infimes d\u00e9tails deviennent les raisons majeures qui provoquent le bouleversement. Je m'\u00e9tonnais d'autant que ces causes relevaient presque toujours d'inventions, d'hypoth\u00e8ses, de travestissements, en un mot, de pr\u00e9textes. Il faut opposer une raison vraisemblable \u00e0 ce qui cache la v\u00e9rit\u00e9 du revirement : l'incapacit\u00e9 \u00e0 tenir plus longtemps une relation dans le faux registre o\u00f9 elle dure, puisque nourrie depuis l'origine pour servir un jour de chair \u00e0 d\u00e9chiqueter.\n\nAmour la veille, d\u00e9clarations dithyrambiques doubl\u00e9es de professions de foi merveilleuses le jour qui pr\u00e9c\u00e8de, puis, le lendemain, dans un basculement d'une violence insoup\u00e7onn\u00e9e, haine, m\u00e9pris, malveillance, m\u00e9disances, travestissements, torsions invraisemblables et r\u00e9\u00e9criture de l'histoire pour m\u00e9priser et salir ce qui avait \u00e9t\u00e9 port\u00e9 au pinacle vingt-quatre heures plus t\u00f4t. A-t-on chang\u00e9 si vite ? Est-on devenu si brutalement un autre ? Quelle m\u00e9tamorphose justifierait ainsi qu'on soit m\u00e9connaissable? Rien, v\u00e9ritablement, sinon le franchissement par autrui d'un seuil dans sa logique. Sa patience \u00e9puis\u00e9e, il ne peut plus attendre pour vider une humeur qui le ruine depuis longtemps. Guettant le moment propice pour d\u00e9clencher la fermeture des m\u00e2choires de son pi\u00e8ge, il a tout mis en \u0153uvre depuis des mois, voire des ann\u00e9es, pour jouir de ce moment-l\u00e0 : vider sur le corps et l'\u00e2me de l'autre ce trop-plein de haine de soi devenue insupportable en lui.\n\nL'\u00e2me de l'envieux est d'un chasseur embusqu\u00e9 en attente d'une seule occasion : an\u00e9antir la proie circonscrite, s\u00e9duite. Ses man\u0153uvres d'approche visent exclusivement la proximit\u00e9 avec la future cible pour mieux mettre en \u0153uvre un processus de destruction qu'il tient pr\u00eat \u00e0 d\u00e9clencher en permanence. L\u00e0 o\u00f9 l'on imagine sympathie, amabilit\u00e9, amiti\u00e9, proximit\u00e9, confidence, il n'y a qu'investissements dans le dessein, le jour du grand retournement, d'un usage belliqueux. Toute intimit\u00e9 sollicit\u00e9e sur le mode amical devient une arme dans les mains de l'envieux qui t\u00e2che de s'en rendre le confident par tous les moyens. Ces strat\u00e9gies guerri\u00e8res se mettent en place au corps d\u00e9fendant de leur acteur : une part aveugle en lui vise une circonscription pour mieux tuer demain alors qu'une autre entretient aujourd'hui l'allure des meilleurs sentiments. En saisissant cette v\u00e9rit\u00e9 devenue \u00e9vidence par exp\u00e9rience, j'ai compris pourquoi et comment La Bruy\u00e8re peut \u00e9crire, malheureusement, qu'il faut toujours voir dans son ami d'aujourd'hui un ennemi possible demain.\n\nLa coupure \u00e0 l'\u0153uvre et en acte dans l'incendiaire br\u00fblant ce qu'il a ador\u00e9, travaill\u00e9 par les feux de l'envie, sectionne son \u00e2me et inflige une blessure dont la douleur inconsciente justifie l'excitation mauvaise. D'aucuns qui m'ont vivement d\u00e9nigr\u00e9 un jour m'avaient souvent pr\u00e9c\u00e9demment v\u00e9n\u00e9r\u00e9 plus que de raison : des lecteurs accoutum\u00e9s \u00e0 des courriers obsc\u00e8nes o\u00f9 l'admiration se r\u00e9pand comme un chancre deviennent un jour, subitement, fanatiques d'un d\u00e9samour sans limites.\n\nPour cela, ils prennent pr\u00e9texte du retard dans une r\u00e9ponse ou d'une ti\u00e9deur dans le consentement \u00e0 leurs brass\u00e9es d'\u00e9loges, sinon de n'importe quelle autre faute morale (une apparition m\u00e9diatique faisant excellemment l'affaire), alors ils inversent le sens de leurs sentiments et passent de l'amour \u00e0 la haine. L\u00e0 ils avouent ce qui motivait leur flamme : une envie contre laquelle soudain ils se retournent. Vex\u00e9s de l'avoir ressentie substantiellement, ils d\u00e9cident de pratiquer la politique de la terre br\u00fbl\u00e9e laissant derri\u00e8re eux cendres et terres brunes dans lesquelles ils veulent la confusion et le n\u00e9ant.\n\nF\u00e2ch\u00e9s de s'\u00eatre d\u00e9nud\u00e9s, malheureux de leur transparence d\u00e8s l'aveu de raisons identiques pour leur haine et leur passion, ils avouent malgr\u00e9 eux une b\u00e9ance qui les travaille, un vide qui les habite, un immense manque \u00e0 combler. Leurs vies insanes, insipides, les conduisent au bord de pr\u00e9cipices o\u00f9 ils jouissent par procuration du spectacle de la vie d'autrui. Une correspondance, ou toute autre relation \u00e9pistolaire, verbale, r\u00e9elle ou fantasm\u00e9e, leur donne l'impression d'une complicit\u00e9 pulv\u00e9ris\u00e9e d\u00e8s la mise \u00e0 l'\u00e9preuve excessive de leur patience \u2013 alors d\u00e9borde la bile noire qu'ils remuent en eux.\n\nEn se faisant de l'autre une id\u00e9e hypostasi\u00e9e, l'envieux imagine se grandir en vertu du principe qu'en fr\u00e9quentant une personne transform\u00e9e en individu exceptionnel, par capillarit\u00e9, on b\u00e9n\u00e9ficie de son aura. D\u00e8s que possible, il reproche \u00e0 cette fiction sans consistance de ne pas \u00eatre telle qu'il l'avait voulue, imagin\u00e9e, d\u00e9sir\u00e9e, souhait\u00e9e, r\u00eav\u00e9e. Le r\u00e9el et l'imaginaire, dans cette affaire, sont intimement confondus, or dans ce registre plus rien ne correspond \u00e0 la r\u00e9alit\u00e9. Tout proc\u00e8de de la distorsion inflig\u00e9e par l'envie. Au lieu de consentir \u00e0 l'\u00e9vidence qu'il disserte sur des phantasmes, l'envieux pr\u00e9f\u00e8re rendre l'autre responsable, ou coupable, de son propre strabisme intellectuel.\n\nQuand il faudrait de la modestie dans le jugement, du moins de la prudence, triomphe le mensonge du grossissement des affects : autant son admiration rel\u00e8ve d'une erreur d'appr\u00e9ciation, autant son m\u00e9pris proc\u00e8de d'une m\u00eame incapacit\u00e9 \u00e0 voir. Ses compliments, dithyrambiques, valent autant que ses m\u00e9disances, hyst\u00e9riques, l'un et l'autre s'appliquent \u00e0 des fictions, des fant\u00f4mes. L'envieux use d'autrui tel l'initi\u00e9 vaudou avec les statuettes qu'il perfore en croyant que le f\u00e9tiche sur lequel il concentre sa violence co\u00efncide v\u00e9ritablement avec l'essence de l'\u00eatre vis\u00e9.\n\nL'absence de prudence conjugu\u00e9e au narcissisme de la victime rend possible cette \u00e9trange pathologie. Car on consent plus volontiers aux compliments qu'aux critiques. Et l'on imagine d'autant fond\u00e9s les premiers qu'on trouve malvenues les secondes, alors que tous deux d\u00e9coulent des strat\u00e9gies affectives consubstantielles \u00e0 n'importe quelle logique intersubjective. A s'imaginer plus digne de louanges que de critiques, on laisse l'envieux avancer, conqu\u00e9rir du terrain. En consentant aux flatteries, on baisse la garde, on se fait plus complice, plus abandonn\u00e9, donc plus vuln\u00e9rable sous la main de l'autre qui cache le poignard appel\u00e9 \u00e0 servir bient\u00f4t.\n\nEn levant la main sur son ami d'hier devenu l'ennemi d'aujourd'hui, l'envieux invoque des pr\u00e9textes utiles pour \u00e9tablir son propre portrait. Il reproche \u00e0 l'autre, \u00e9trangement, tout ce qu'il pourrait se reprocher \u00e0 lui-m\u00eame \u2013 ce que d'ailleurs son inconscient sait. Combien me reprochent tr\u00e8s exactement ce qu'ils ne veulent ni ne peuvent assumer de voltes en eux ? La lucidit\u00e9 leur est interdite, elle se paierait, pour eux, du prix fort : le craqu\u00e8lement de leur travestissement. D'aucuns, devenus coqs et paons locaux, englu\u00e9s dans la repr\u00e9sentation municipale ou districale, choisissant m\u00eame leurs amis en fonction du retentissement social induit dans la sous-pr\u00e9fecture, oublieux qu'il y a peu ils disaient pis que pendre de deux ou trois qui, alors, le leur rendaient bien, ceux-l\u00e0 m\u00eames qui d\u00e9fendent aujourd'hui publiquement et officiellement les couleurs d'un parti politique sur lequel je les ai entendus tenir nagu\u00e8re des propos m\u00e9prisants, ceux-l\u00e0, donc, ont sacrifi\u00e9 sur l'autel de la repr\u00e9sentation sociale et du ressentiment pathologique une histoire ancienne qui aurait pu s'arr\u00eater l\u00e0, sans autre dommage qu'un point pos\u00e9 \u00e0 la fin d'un mauvais roman. Las ! les choses auraient \u00e9t\u00e9 beaucoup trop simples...\n\nJe n'aurai gu\u00e8re eu la cruaut\u00e9 de rappeler leurs revirements et leurs trahisons, leurs reniements et leurs m\u00e9tamorphoses. A quoi bon, et pour quelles fins ? Quel int\u00e9r\u00eat ? La vie me requiert ailleurs, j'ai autre chose \u00e0 faire que calculer le degr\u00e9 de leurs changements de cap. Mais eux, pour \u00e9viter d'avoir \u00e0 supporter leur image, pour contourner leur m\u00e9moire, pour asseoir un d\u00e9ni magistral, pour s'\u00e9pargner des contorsions \u00e9normes s'il leur fallait aujourd'hui justifier l'excellence de la soupe dans laquelle ils crachaient abondamment la veille, par confort, donc, pr\u00e9f\u00e8rent maintenant fustiger ceux dont ils savent, inconsciemment, la m\u00e9moire intacte. Notamment la m\u00e9moire de leurs errances, de leurs jeux pitoyables et opportunistes, puis de tous leurs arrangements minables pour t\u00e2cher, en vain, de prendre cong\u00e9 avec leur m\u00e9diocrit\u00e9.\n\nEnvieux d'un destin auquel ils ne peuvent plus aspirer que sur le mode d\u00e9grad\u00e9, conscients qu'ils devront, dor\u00e9navant, se contenter d'un festin de miettes, ils reconstruisent l'existence autour de leurs petits mensonges. L'adult\u00e8re inv\u00e9t\u00e9r\u00e9 de la veille vante maintenant les m\u00e9rites de la famille et de la patrie dans les colonnes du journal local avant, vraisemblablement, de divorcer comme tout un chacun; au moment des \u00e9lections, la suffragette d'avant-hier, la f\u00e9ministe d'hier s'est m\u00e9tamorphos\u00e9e en sp\u00e9cialiste du point compt\u00e9 et brode en mots mi\u00e8vres l'amour qu'elle aimerait avoir pour son mari, elle excelle d\u00e9sormais en th\u00e9s mondains o\u00f9 l'on aff\u00fbte ses couteaux avant de les lancer au visage de tout ce qui n'est pas aussi m\u00e9diocre que soi. Le mim\u00e9tisme triomphe \u00e0 ce point chez ces errants en qu\u00eate d'identit\u00e9 \u00e0 parasiter que la couleur et le tomb\u00e9 des v\u00eatements, les coupes de cheveux, les int\u00e9r\u00eats culturels, les pratiques sociales, les go\u00fbts les plus sommaires, sont devenus interchangeables \u00e0 s'y m\u00e9prendre, tant le d\u00e9sir de se fondre, de se confondre, de s'indiff\u00e9rencier en croyant se distinguer, est manifeste. Ils n'existent qu'en pr\u00e9dateurs d'animaux eux-m\u00eames nourris du sang vici\u00e9 pr\u00e9lev\u00e9 sur le corps des autres \u2013 qu'une in\u00e9vitable rupture renverra un jour dans les basses fosses d'o\u00f9 ils viennent.\n\nCeux-l\u00e0, pr\u00e9cis\u00e9ment, et quelques autres sp\u00e9cimens rencontr\u00e9s dans les colloques, conf\u00e9rences et autres lieux de joutes publiques o\u00f9 se pressent dans la salle les gloires culturelles locales, les philosophes de canton, les professeurs \u00e9triqu\u00e9s dans leur statut, les \u00e9crivains \u00e0 compte d'auteur, les gribouilleurs de manuscrits en souffrance, les dipl\u00f4m\u00e9s du sup\u00e9rieur confin\u00e9s dans l'inf\u00e9rieur, les auteurs \u00e9puis\u00e9s par leur seul livre publi\u00e9, les frustr\u00e9s de temps m\u00e9diatique et autres hyst\u00e9riques \u00e9trangl\u00e9s par leurs ratages, m'ont appris combien l'envie tient un r\u00f4le majeur dans l'existence, elle et ses vices associ\u00e9s, le ressentiment, la mauvaise foi, l'hypocrisie, la jalousie, la fatuit\u00e9, l'arrogance, la vanit\u00e9, et autres sympt\u00f4mes qui montrent \u00e0 l'\u0153uvre le travail et le triomphe des id\u00e9ologies de la mortification, de la haine et du m\u00e9pris de soi impossibles \u00e0 vivre autrement que sur le mode du transfert et de la projection.\n\nL'amour et le d\u00e9samour entretiennent des relations d'une \u00e9trange proximit\u00e9, l'un poussant sa corne dans l'art\u00e8re de l'autre au moment le moins attendu. Coup d'\u00e9p\u00e9e qui sectionne un \u00eatre en deux, lame de sabre embroch\u00e9e dans le ventre pour y perforer les entrailles, dague plant\u00e9e dans le centre du coeur, \u00e0 chaque fois, l'\u00eatre, le ventre et le c\u0153ur qu'on avait donn\u00e9s se trouvent d\u00e9truits, salis, pi\u00e9tin\u00e9s, mortifi\u00e9s. Autant de gestes qu'on se dispense d'avoir \u00e0 retourner contre soi. Leur violence \u00e0 l'endroit d'autrui renseigne sur la brutalit\u00e9 des passions qui les rendent possibles au tr\u00e9fonds du corps de l'envieux.\n\nJe n'ai commenc\u00e9 \u00e0 saisir le fonctionnement de ces logiques qu'apr\u00e8s avoir entendu les raisons pour lesquelles tels ou tels justifiaient les ruptures qui scindent les vies en deux. Souvent d'ailleurs ces reproches ne s'adressent pas directement aux int\u00e9ress\u00e9s \u2013 qu'ils savent avertis de leurs jeux maladifs \u2013 mais \u00e0 tous les autres qui veulent bien les entendre, voire les rapporter. Celui qui subit ne comprend pas, ne saisit rien de ce qui advient. Les causes, les raisons, les explications, tout cela manque; celui qui agit, qui initie la rupture, s'appuie sur des pr\u00e9textes utiles pour renseigner rapidement sur ses v\u00e9ritables motivations. Il fait endosser \u00e0 l'autre ce qu'il ne peut, par impuissance, endosser lui-m\u00eame. La blessure narcissique serait trop importante. Il d\u00e9cr\u00e8te la rage de son chien parce qu'il a d\u00e9cid\u00e9 de l'abattre, sans autre motivation que le besoin de conduire jusqu'\u00e0 son extr\u00e9mit\u00e9 une pulsion de mort qui le hante et l'ab\u00eeme.\n\nLes illustres traducteurs d'un jour devenus maires de leur commune, par d\u00e9pit, les philosophes appel\u00e9s \u00e0 r\u00e9volutionner le monde des id\u00e9es transform\u00e9s en conseillers municipaux, ou en surveillants g\u00e9n\u00e9raux des lyc\u00e9es, les universitaires destin\u00e9s \u00e0 refondre, de fond en comble, la r\u00e9flexion dans leur si\u00e8cle devenus r\u00e9dacteurs des pages culturelles du bulletin municipal, d\u00fbment sign\u00e9es, les grands destins d\u00e9sign\u00e9s par les dieux pour laisser leur nom dans la post\u00e9rit\u00e9 litt\u00e9raire sur le monument en marbre du roman occidental devenus bouquinistes sur les march\u00e9s, les agr\u00e9g\u00e9s d\u00e9sagr\u00e9g\u00e9s, les docteurs sachant si peu, les certifi\u00e9s si conformes, les enseignants qui se voyaient d\u00e9j\u00e0 au Coll\u00e8ge de France et croupissent, en attendant, dans l'Education nationale, rabougris dans leur mesquinerie, confits dans leur ressentiment recuit, et j'allais oublier tous ceux qui m\u00e9tamorphosent leurs \u00e9chouages, leurs ratages, leurs fiascos, leurs impuissances dans le devenir de la critique, institutionnalis\u00e9e ou non, professionnelle ou d'amateur, litt\u00e9raire ou musicale, philosophique ou esth\u00e9tique, gastronomique ou m\u00e9diatique, \u00e0 tous les sens du terme, voil\u00e0 qui remplit et sature en envieux une arche de No\u00e9 abandonn\u00e9e aux flots d'un monde soumis au roulis.\n\nPitoyables, pour \u00e9viter d'avoir \u00e0 dig\u00e9rer ce mauvais brouet qu'est leur existence, ils se composent des visages de fortune, des allures de th\u00e9\u00e2tre et de com\u00e9diens identifi\u00e9s \u00e0 leur r\u00f4le pour se dispenser \u00e0 moindres frais de n'\u00eatre rien en dehors de la sc\u00e8ne. La fonction sociale leur fournit des b\u00e9quilles magnifiques. Dissimul\u00e9s derri\u00e8re elle, ils jouissent de pouvoir se dispenser d'\u00eatre en se contentant du para\u00eetre \u2013 car le social vit du recyclage de ces passions mauvaises. Le pouvoir, quel qu'il soit, propose un exutoire parfait \u00e0 ces \u00e9nergies qui, sinon, travaillent le corps et l'\u00e2me. L\u00e0 o\u00f9 les volont\u00e9s d\u00e9biles, les identit\u00e9s fragiles, les temp\u00e9raments branlants cherchent appui, on trouve toujours la fonction sociale et la repr\u00e9sentation qui l'accompagne.\n\nD'o\u00f9 la fascination exerc\u00e9e sur ces \u00e2mes en peine par ce qu'ils appellent la c\u00e9l\u00e9brit\u00e9 \u2013 cette situation dans laquelle se trouvent ceux que reconnaissent certains qu'ils ne connaissent pas. Quand on ne peut voir les envieux comme singularit\u00e9s manifestes, en tant qu'individualit\u00e9s magnifiques, et pour cause, on les remarque sous l'habit pr\u00eat\u00e9 par les institutions. Ils imaginent une volupt\u00e9 \u00e0 \u00eatre public quand, la plupart du temps, il faut en supporter les seuls d\u00e9sagr\u00e9ments, car toujours ce sont la libert\u00e9, l'autonomie, l'ind\u00e9pendance qui se trouvent sacrifi\u00e9es au regard de l'autre sous lequel il faut vivre. Devenir public, ou c\u00e9l\u00e8bre, pire encore, c'est ali\u00e9ner une libert\u00e9 dont se d\u00e9feraient volontiers ceux qui n'en disposent pas et, par cons\u00e9quent, ne peuvent en jouir \u2013 ni s'en d\u00e9faire...\n\nAli\u00e9n\u00e9 \u00e0 la repr\u00e9sentation sociale, tout de fa\u00e7ade et d'apparence, l'envieux qui recycle ses pulsions dans le jeu social, ce th\u00e9\u00e2tre de m\u00e9diocres, n'aime rien moins que le paravent qui le dispense de mesurer l'\u00e9tendue des d\u00e9g\u00e2ts au creux de son \u00eatre intime. Les professions, fonctions et activit\u00e9s d'autorit\u00e9, de jugement, de pouvoir, de critique, de sanction, d'arbitraire, tout ce qui permet l'empire et l'emprise sur quelqu'un qui n'en peut mais, trahit un fonds sp\u00e9cieux dont l'\u00e9nergie est l'envie. Professeur, politicien, \u00e9lus quelconques, critique litt\u00e9raire ou autre, policier, militaire, pr\u00eatre, juge, procureur, contr\u00f4leurs de tous acabits, inspecteurs divers, membres de jurys, b\u00e2tonniers, pr\u00e9sidents de tribunaux et autres fonctions aff\u00e9rentes \u00e0 la com\u00e9die sociale, rendent possibles des jouissances sombres dans l'exercice facile de la critique quand l'art leur est impossible. Ceux-l\u00e0 vivent et jubilent de ch\u00e2trer les d\u00e9sirs, les plaisirs et les jouissances de ceux qu'ils imaginent plus \u00e9panouis qu'eux.\n\nL'envieux au teint plomb\u00e9 m\u00e9lang\u00e9 de jaune et de noir, au sang meurtri \u2013 dit Descartes \u2013 vit dans un monde inexistant, fabriqu\u00e9 \u00e0 sa mesure, dans de petits formats qui permettent l'expansion facile de projets \u00e9troits. Il a cong\u00e9di\u00e9 toute esp\u00e8ce de r\u00e9el enracin\u00e9 dans l'\u00e9vidence au profit d'un pur imaginaire utile \u00e0 son sc\u00e9nario. Tout baigne chez lui dans le vert et le jaune maussades de la bile. Le bovarysme le travaille au ventre et l'on m\u00e9sestime le r\u00f4le tenu par cette modalit\u00e9 de la schizophr\u00e9nie. On doit son \u00e9tiologie \u00e0 Jules de Gaultier qui la d\u00e9finit comme la facult\u00e9 de l'homme \u00e0 se croire autre qu'il n'est. Entre son id\u00e9e de lui, et celle des autres, il creuse un foss\u00e9 infranchissable, un ab\u00eeme maintenu en l'\u00e9tat jour apr\u00e8s jour, car c'est son travail au quotidien : entretenir les raisons fictives de sa haine r\u00e9elle.\n\nAmateur de d\u00e9nis, de torsions, de distorsions, de travestissements, de m\u00e9tamorphoses, d'imaginaire, de fictions, l'envieux souffre de troubles de la perception qu'aucune correction ne peut conjurer. Dans une formule simple, on pourrait dire qu'il prend ses d\u00e9sirs pour la r\u00e9alit\u00e9, qu'il construit et fabrique de toutes pi\u00e8ces un r\u00e9el aux dimensions de ses fantasmes. La haine inconsciente de soi devient chez lui un amour de soi expansif manifeste dans l'op\u00e9ration de d\u00e9tournement des fleuves mauvais sur le corps de ceux qui passent \u00e0 sa port\u00e9e. Elues pour \u00eatre salies, leurs relations sont choisies pour le degr\u00e9 d'investissement n\u00e9gateur qu'elles permettent. Plus on d\u00e9forme la v\u00e9rit\u00e9 dans le dessein de la rendre extraordinaire, merveilleuse, fantastique, plus on aura \u00e9lev\u00e9 haut la cath\u00e9drale d'amour \u00e9rig\u00e9e pour sa future victime, plus il y a de jubilation \u00e0 inverser le fantasme, plus ils jouissent de miner, d\u00e9truire, casser, briser le monument.\n\nBien s\u00fbr, l'\u00e9dification furieuse est tout aussi ill\u00e9gitime que la destruction forcen\u00e9e, les deux op\u00e9rations proc\u00e9dant de la m\u00eame hyst\u00e9rie, mais corrompu par ce mauvais sang, l'envieux n'a pas le choix ou la possibilit\u00e9 de r\u00e9sister \u00e0 la n\u00e9cessit\u00e9 qui le ronge. Son destin se d\u00e9plie exclusivement dans le consentement : il ne sera jamais autre chose qu'un jouet, de lui et du monde. Sa seule fa\u00e7on de vivre avec cette damnation suppose cette transformation des autres en victimes. En lui, la pulsion bovaryque creuse un trou dans lequel il p\u00e9rira s'il n'y jette les victimes \u00e9lues pour son sacrifice propitiatoire. Sa vie se consume dans la qu\u00eate de perp\u00e9tuelles occasions d'holocaustes. Je n'envie pas pareils infirmes g\u00e2ch\u00e9s par une telle mis\u00e8re.\n\nJe me demande ce qui m'anime devant ces vies-l\u00e0, piteuses et vaines. De l'amertume quand la haine me vient au visage, lanc\u00e9e par eux ou d'autres qui les relaient, ces bonnes consciences toujours r\u00e9jouies de pouvoir rapporter un propos fielleux, ce qui les dispense du risque de le tenir eux-m\u00eames, et les assure du vilain b\u00e9n\u00e9fice de ceux qui le tiennent ; de la haine, au moment o\u00f9 la br\u00fblure est intense, dans le feu de l'action, \u00e0 l'instant o\u00f9 le crachat atteint son but ; du m\u00e9pris d\u00e8s que je retrouve sur mon passage ceux que j'\u00e9vite pour les dispenser de me supporter tout autant que pour m'\u00e9viter d'avoir \u00e0 les croiser.\n\nOr, je n'aime ni l'amertume, ni la haine, ni le m\u00e9pris, pas pour des raisons morales ou moralisantes, mais tout simplement parce qu'elles sont passions mauvaises qui g\u00e2chent mon existence et contrarient ma volont\u00e9 h\u00e9doniste de chercher les plaisirs, certes, mais plus encore d'\u00e9viter les d\u00e9plaisirs. Dante voulait que dans son Enfer les envieux aient les paupi\u00e8res cousues avec un fil d'acier. Je n'ai gu\u00e8re d'aspiration pour toute logique o\u00f9 il faut expier ses fautes. Je souhaite seulement tout mettre en \u0153uvre pour \u00e9viter cette engeance, par la pure vertu d'une arithm\u00e9tique des plaisirs bien entendus. Mais on ne peut gu\u00e8re demander \u00e0 ces jouets d'eux-m\u00eames, \u00e0 ces malades en quelque sorte, d'aspirer \u00e0 toute la politesse ou toute l'intelligence qui rendraient la vie moins insupportable. Il en va de leur nature, de leur essence, car leur destin consiste \u00e0 chercher des proies, les conqu\u00e9rir, et s'en servir pour assouvir un besoin de b\u00eate haineuse et f\u00e9roce.\n\nLe temps passant, leur man\u00e8ge m'\u00e9tant devenu moins obscur, plus familier, il me reste de la piti\u00e9, une immense piti\u00e9. Le spectacle de leurs machinations infantiles, r\u00e9gressives, minables, pitoyables, l'impression d'y voir de plus en plus clair dans ce processus de haine de soi m\u00e9tamorphos\u00e9 en ressentiment contre autrui, l'\u00e9tat des lieux mis\u00e9rables qui les fait se mouvoir dans la boue et les sanies parce qu'ils sont incapables d'un autre terrain de jeu pour leurs existences m\u00e9diocres, tout cela m'invite \u00e0 tout mettre en \u0153uvre pour parvenir, certains matins heureux, au grand rire nietzsch\u00e9en annonciateur des sorties de convalescence et du recouvrement de la sant\u00e9 \u2013 le go\u00fbt de terre ayant quitt\u00e9 la bouche.\n25\n\nESTH\u00c9TIQUE DE L'UBIQUIT\u00c9\n\nEn compulsant l'un de mes vieux dictionnaires, comme j'aime souvent le faire pour me dispenser de l'ennui qui m'assaille d\u00e8s que je commence la lecture d'un roman d'aujourd'hui, j'ai d\u00e9couvert que la photocopie naissait \u00e0 la fin du xixe si\u00e8cle. Du moins le terme qui signifie la reproduction d'un phototype sur une surface sensibilis\u00e9e. Etrangement, j'imaginais pour Nietzsche et ses contemporains la possibilit\u00e9 d'utiliser le mot. Plus loin, je d\u00e9couvrais que photocopier date tr\u00e8s pr\u00e9cis\u00e9ment de 1907. Entre cette fa\u00e7on de signifier le photogramme, un genre d'\u00e9criture de la lumi\u00e8re, ou avec elle, une sorte de petite s\u0153ur de la photographie, et sa banalisation quotidienne, il s'est \u00e9coul\u00e9 un si\u00e8cle, fort de toutes les r\u00e9volutions esth\u00e9tiques possibles et imaginables. De la g\u00e9n\u00e9alogie du mot \u00e0 l'usage esth\u00e9tique du proc\u00e9d\u00e9 en passant par le devenir vernaculaire de la technique on peut rep\u00e9rer un grand nombre des \u00e9tapes \u00e9crites de notre modernit\u00e9 artistique.\n\nLa travers\u00e9e du si\u00e8cle montre \u00e0 l'envi que Walter Benjamin, apr\u00e8s Val\u00e9ry, a raison d'affirmer que la transformation des techniques induit des changements dans la pratique des beaux-arts. Mais on peut \u00e9galement avancer l'hypoth\u00e8se inverse, car les m\u00e9tamorphoses de l'art produisent aussi des effets dans l'usage des techniques et le mouvement s'effectue moins dans l'unilat\u00e9ral que la sym\u00e9trie. Que la photographie r\u00e9volutionne la peinture, voil\u00e0 ce que plus personne ne conteste. Pour autant, on conclut plus rarement au fait que les modifications induites dans la peinture agissent \u00e0 nouveau et de mani\u00e8re essentielle sur la pratique de la photographie. Il en va, ici comme ailleurs, d'une dialectique \u00e9l\u00e9mentaire \u00e0 l'\u0153uvre dans toute r\u00e9alit\u00e9 vivante et dynamique.\n\nDepuis le cin\u00e9ma et la photographie, d\u00e9classant d'une certaine mani\u00e8re le th\u00e9\u00e2tre et la peinture, on assiste \u00e0 l'\u00e9mergence de techniques nouvelles encore \u00e0 m\u00eame d'exprimer tout leur suc. Je songe \u00e0 la vid\u00e9o et \u00e0 la photocopie qui naissent d\u00e8s que se popularisent et se banalisent dans le quotidien les pratiques cin\u00e9matographiques et photographiques. En outrant ces techniques du d\u00e9but du si\u00e8cle, les vid\u00e9astes et les plasticiens photocopieurs abordent de nouveaux continents qui m\u00e9ritent r\u00e9flexion. Nous \u00e9voluons dans l'\u00e8re de la reproduction technique et de l'extr\u00eame souplesse technologique que le cam\u00e9scope et le photocopieur, ces clones mutants de la chambre obscure photographique, mettent \u00e0 disposition de tout un chacun \u2013 pour le meilleur et pour le pire.\n\nSelon Walter Benjamin, le d\u00e9clin et la fin de l'aura caract\u00e9risent ce si\u00e8cle. Qu'est-ce que cette aura dont il est question chez le penseur de l'ange n\u00e9cessaire ? Un genre de nimbe qui enveloppe toute \u0153uvre unique, originale et authentique, un halo de lumi\u00e8re mentale et intellectuelle fabriqu\u00e9 \u00e0 l'aide du temps, du social, de la r\u00e9putation, une esp\u00e8ce de brillance spirituelle et conceptuelle qui \u00e9mane d'un objet ayant re\u00e7u son onction de l'histoire. L'aura caract\u00e9rise la singularit\u00e9, l'existence sans duplication, sans double. Elle synth\u00e9tise, dans le moment pr\u00e9sent, la charge \u00e9motionnelle et culturelle accumul\u00e9e en traversant les \u00e2ges. De sorte que par elle le lointain devient pr\u00e9sent. Sous la plume de son inventeur, elle est \u00abunique apparition d'un lointain, si proche qu'elle puisse \u00eatre \u00bb. En ce sens, elle rappelle l'ind\u00e9fectible lien de l'\u0153uvre d'art et du rituel, du sacr\u00e9, de la magie. Plus grande est la valeur cultuelle d'une \u0153uvre, plus grande est son aura, et vice versa.\n\nNos temps sacrifient la valeur cultuelle \u00e0 la valeur d'exposition en vertu de quoi ce qui pla\u00eet dans l'\u0153uvre c'est sa capacit\u00e9 \u00e0 d\u00e9placer un temps concentr\u00e9, sinon un concentr\u00e9 de temps, dans un espace perp\u00e9tuellement changeant. L\u00e0 o\u00f9 l'\u0153uvre a \u00e9t\u00e9 faite, le lieu pour lequel elle a \u00e9t\u00e9 pens\u00e9e, cr\u00e9\u00e9e, ses conditions pr\u00e9cises et uniques d'exposition valent maintenant moins que son trajet plan\u00e9taire dans des espaces neutres, blancs. Le d\u00e9clin et la fin de l'aura, selon Walter Benjamin, co\u00efncident avec une massification grandissante des jugements de go\u00fbt coextensifs au plaisir de consommer l'art en dilettante distrait, avec un investissement intellectuel minimal.\n\nLes productions de vid\u00e9astes et les cr\u00e9ations photocopiques d'artistes contemporains jettent la confusion dans les registres habituellement pratiqu\u00e9s. Elles revendiquent souvent le trouble dans les d\u00e9finitions de la valeur d'exposition et dans celle de la valeur cultuelle. La photocopie exprime de mani\u00e8re r\u00e9currente la reproductibilit\u00e9 et son extr\u00eame facilit\u00e9. Elle suppose, dans une logique benjaminienne extrapol\u00e9e, la disparition totale de l'aura, son an\u00e9antissement. De sorte que son apparition et son usage esth\u00e9tique induisent un nihilisme radical, quel que soit le projet de l'artiste d\u00e9sireux d'un pareil av\u00e8nement.\n\nOr il me semble que l'aura d'une \u0153uvre n'a rien \u00e0 craindre de la reproductibilit\u00e9, mais plut\u00f4t de l'inexistence d'une signature pour la soutenir avec assez de force, de d\u00e9termination et d'autorit\u00e9. L'absence d'aura est manif\u00e8ste dans l'\u0153uvre que ne porte pas une signature. Toute photocopie se trouve investie de puissance, comme une photographie, sinon un autre multiple, d\u00e8s qu'elle est sign\u00e9e. De la m\u00eame mani\u00e8re que l'aura d'un roman n'a pas grand-chose \u00e0 voir avec son support papier, ce qui nimbe une \u0153uvre, quelle qu'elle soit \u2013 photocopie ou autre \u2013, c'est son message, ce qu'elle dit et transporte, le sens qu'elle d\u00e9clenche en avalanche, la signification qu'elle induit comme une vague ouverte en mur et fendue par la nage d'un poisson en surface. L'aura jaillit de la puissance qu'elle rend possible.\n\nLa le\u00e7on d\u00e9coule de Duchamp. En son temps, elle \u00e9tait susceptible d'\u00eatre entendue par Walter Benjamin car L'\u0152uvre d'art \u00e0 l'\u00e9poque de sa reproduction m\u00e9canis\u00e9e para\u00eet en 1936, soit plus de vingt ans apr\u00e8s le premier ready-made. La signature vaut \u00e9quivalent du vouloir et de la singularit\u00e9 de qui la trace. Elle propose une figure rep\u00e9rable \u00e9crite avec l'encre de l'aura. Si l'individualit\u00e9 qui l'ose est vide, creuse, sans \u00e9paisseur intellectuelle, conceptuelle ou philosophique, alors aucune aura ne nimbe l'\u0153uvre propos\u00e9e au regardeur. Car si ce dernier est bien celui qui fait le tableau aussi bien que le contenu du travail, il vaut aussi en lieu travers\u00e9 par une histoire, des r\u00e9f\u00e9rences, une culture et une tradition. Le manque de volont\u00e9 critique d\u00e9virilise l'acte esth\u00e9tique, le d\u00e9faut de vouloir conceptuel occasionne les \u00e9clipses dans l'histoire de l'art \u2013 et probablement dans toute autre histoire partielle.\n\nLa photocopie multiplie les reproductions, noie et dilue volontairement l'original dans l'ind\u00e9fini et l'infini du divers. Elle confond \u00e0 dessein les registres du m\u00eame et de l'autre, de l'un et du multiple, de la diff\u00e9rence et de la r\u00e9p\u00e9tition, sinon de l'Un Bien et de ses hypostases au profit d'un monde \u00e9clat\u00e9, fragment\u00e9, complexe dans ses apparitions impossibles \u00e0 d\u00e9nombrer. L\u00e0 o\u00f9 le r\u00e9el vrai, l'authentique, l'original et le singulier triomphent \u2013 par exemple dans la peinture \u2013 en auxiliaires du monoth\u00e9isme et du dualisme, du spiritualisme et de l'id\u00e9alisme, l'artiste oppose la photocopie et revendique la copie, le double, le faux, le mensonge comme autant de composantes essentielles pour constituer le monde tel qu'il l'entend : impur, m\u00e9lang\u00e9, multiple, \u00e9clat\u00e9, rhizomique, diffus, voire confus.\n\nDe la m\u00eame mani\u00e8re, les photocopieurs r\u00e9investissent, en apaches, les domaines de l'esth\u00e9tique contemporaine o\u00f9 brillaient nagu\u00e8re les conceptuels, les tenants de l'art pauvre, les amateurs d'esth\u00e9tique de la communication, ceux qui, aujourd'hui, pratiquent la photographie esth\u00e9tisante ou manipul\u00e9e, sinon les artistes s\u00e9mioticiens, ces abstracteurs de quintessence ou encore les primitifs et na\u00effs que sont les dessinateurs de graffitis. A la jonction de toutes les fulgurances autoris\u00e9es par les nouvelles technologies de communication, ils investissent un espace libre, l'une de ces zones autonomes temporaires balis\u00e9es et rep\u00e9r\u00e9es par les cyber-acteurs libertaires contemporains. Leur projet autorise une esth\u00e9tique de l'ubiquit\u00e9 dans laquelle \u00e9voluent les signatures multiples desquelles \u00e9mergeront celles qui, de mani\u00e8re parfois inconsciente et aveugle, d\u00e9gageront l'aura des temps \u00e0 venir.\n\nLeurs travaux entament le cr\u00e9dit accord\u00e9 aux f\u00e9tiches c\u00e9l\u00e9br\u00e9s dans la tradition de l'histoire de l'art occidental. Ils d\u00e9mystifient, sapent de vieilles idoles, incendient les vieilles \u00e9glises o\u00f9 psalmodient des fid\u00e8les encore pieux. En \u00e9lisant pour cible tout ce qui permet l'usage marchand de l'\u0153uvre d'art, en fustigeant les m\u00e9tamorphoses qui transfigurent la valeur d'usage des \u0153uvres en valeur d'\u00e9change, les artistes agissent sur le terrain politique en libertaires anim\u00e9s par un nouveau souffle esth\u00e9tique. Si l'aura doit r\u00e9appara\u00eetre, elle surprendra en \u00e9mergeant du lot, l\u00e0 o\u00f9 une tentative co\u00efncidera avec les besoins de l'\u00e9poque en sens, en signification susceptible de d\u00e9passer le nihilisme. Alors une signature s'ajoutera \u00e0 l'histoire, exigeant plus tard qu'une autre la remplace, non pas sur un palimpseste, mais sur un genre de rouleau sacr\u00e9.\n\nMonistes, mat\u00e9rialistes, philosophes de la seule immanence, tenants d'une inversion des valeurs qui les \u00e9tablit en pr\u00eatres d'un panth\u00e9isme dans lequel on dirait le r\u00e9el dans la multiplicit\u00e9 de ses modifications \u2013 photocopiques en l'occurrence \u2013, les artistes arrim\u00e9s \u00e0 leurs machines avides de reproduire ind\u00e9finiment un m\u00eame qui, dans ses manifestations, est sans cesse un autre, agissent en techniciens, certes, mais leur vouloir les propulse sur le terrain d'un art que leur signature authentifie, certifie, atteste. Jusque dans leur \u00e9ventuel refus de signer, leur d\u00e9ni de paraphe, leur revendication d'anonymat, ils illustrent cette v\u00e9rit\u00e9 qu'un vouloir n'est jamais neutre, car toujours il fait sens : il passe outre l'hypoth\u00e8se d'un art pour l'art au profit d'un r\u00e9investissement social, sociologique et politique de la d\u00e9marche esth\u00e9tique.\n\nL\u00e0 o\u00f9 Walter Benjamin oppose la politisation marxiste de l'art \u00e0 l'esth\u00e9tisation fasciste de la politique, il n'y a qu'aporie, jeu intellectuel et rh\u00e9torique confin\u00e9 au ludisme verbal. Car tout art est politique, y compris \u2013 et peut-\u00eatre surtout \u2013 celui qui se pr\u00e9sente comme d\u00e9sengag\u00e9, apolitique ou ne visant que des fins esth\u00e9tiques. Un geste artistique ici induit l\u00e0 des tremblements, des frissons dans le social, sinon pr\u00e9pare des fissures et inaugure des brisures, des effondrements. En mati\u00e8re id\u00e9ologique, toute th\u00e9orie proc\u00e9dant de la tectonique des plaques est valide et op\u00e9ratoire.\n\nLes dada\u00efstes et les situationnistes auraient aim\u00e9 la charge offerte par les possibilit\u00e9s photocopiques d'un art de masse, subversif, transversal et paradoxalement aristocrate dans le refus de l'\u00e9litisme. Collages, affiches lac\u00e9r\u00e9es, photomontages, s\u00e9rigraphies sauvages, calligraphies triviales ou graffitis intempestifs et inactuels, photocopies, aujourd'hui, expriment un art vivant qui n'\u00e9lit pas le mus\u00e9e, la salle d'exposition mais les lieux de vie vivante \u2013 comme Rimbaud parlait de libert\u00e9 libre. Par-del\u00e0 les productions esth\u00e9tiques soutenues et foment\u00e9es par le social qui les rend possibles, les gestes photocopiques revendiquent l'illustration de nouvelles possibilit\u00e9s d'existences imaginaires. Rues et bureaux, trottoirs et march\u00e9s, palissades et barricades, couloirs de m\u00e9tro et murs l\u00e9preux, entr\u00e9es d'immeubles et b\u00e9tons de banlieue, beaux quartiers et architectures ludiques aussi bien que carc\u00e9rales, tous les lieux o\u00f9 passent des hommes et des femmes, o\u00f9 ils vivent, vont et viennent, loin des affectations mus\u00e9ales et des pratiques religieuses de masse des acteurs et des consommateurs d'art contemporain, tous ces lieux, donc, sont bienvenus pour que circule la photocopie.\n\nConnue ou inconnue, c\u00e9l\u00e8bre ou modeste, la signature de l'artiste photocopieur s'ajoute \u00e0 celle d'un Lautr\u00e9amont qui, au si\u00e8cle dernier, aspirait \u00e0 ce que la po\u00e9sie soit faite par tous. Le projet coexiste avec celui qui permet \u00e0 l'art d'\u00eatre fait, aussi, par quelques-uns. D'ailleurs, les fleuves qui, en surface, semblent radicalement s\u00e9par\u00e9s, tant leurs eaux paraissent impossibles \u00e0 m\u00e9langer, pourraient bien, en des couches plus souterraines, dans le secret de la terre et des g\u00e9ologies myst\u00e9rieuses, finir par se retrouver et se m\u00e9langer. La vie de la rue s'ach\u00e8ve toujours aux cimaises officielles \u2013 alors il faut changer de vie, encore et toujours. Rattrap\u00e9 par le mus\u00e9e, l'art se dess\u00e8che, se fossilise, s'effrite et meurt. Vivant, inventif et qu\u00eateur de nouvelles possibilit\u00e9s, de nouvelles combinaisons, il exprime ce qui ne devrait jamais cesser d'\u00eatre sa raison : un \u00e9loge du mouvement.\n26\n\nLE CONTRAIRE D'UNE CHOSE MENTALE\n\nSi d'aventure il me fallait expliquer la musique \u00e0 un sourd, nul doute que j'aborderais le sujet de mani\u00e8re oblique, transversale, en accostant sur des berges habituellement vierges. Car la plupart du temps les philosophes qui se sont essay\u00e9s \u00e0 parler de musique l'ont fait \u00e9trangement, en partant du principe qu'ils avaient affaire \u00e0 des gens sachant d\u00e9j\u00e0 par exp\u00e9rience ce qu'elle est essentiellement. Restait pour eux \u00e0 trouver les mots qui permettent le passage entre la th\u00e9orie et la pratique, entre le discours et les connaissances musicales de leurs lecteurs. De la m\u00eame mani\u00e8re qu'on aborde mieux une d\u00e9finition de la couleur en la destinant \u00e0 un aveugle, il me semble qu'il faut penser \u00e0 un sourd de naissance quand, r\u00e9fl\u00e9chissant sur la musique, on t\u00e2che d'en appr\u00e9cier l'essence.\n\nLe silence des philosophes sur cette question ne cesse de m'\u00e9tonner. Le silence ou les exercices inad\u00e9quats, voire les choses non dites, tels les oublis de ce sujet dans l'ensemble d'une \u0153uvre ou d'une pens\u00e9e. Quid, par exemple, du silence de Bergson quand on attendrait du philosophe du temps, de l'intuition, de la dur\u00e9e v\u00e9cue, de la mati\u00e8re et de la m\u00e9moire un ouvrage tout entier consacr\u00e9 au sujet? Pourquoi cette singuli\u00e8re aversion d'un Freud, cette m\u00e9fiance, sinon ce refus de la musique chez le th\u00e9oricien des pulsions magiques et myst\u00e9rieuses, chez le d\u00e9couvreur de l'inconscient et des logiques de sublimation? Qu'est-ce qui explique l'absence de consid\u00e9rations sur ce sujet chez Heidegger, penseur de l'oubli de l'\u00eatre qui aurait trouv\u00e9 avec ce souci l'occasion de suivre \u00e0 la trace la permanence de l'objet de sa passion ?\n\nIl y a pire que le silence ou l'\u00e9vitement. A savoir l'usage de la musique par un philosophe \u00e0 des fins d'illustration de la validit\u00e9 et de la pertinence de ses pr\u00e9suppos\u00e9s th\u00e9oriques. L'assujettissement int\u00e9ress\u00e9 installe la musique en p\u00e9riph\u00e9rie, et fournit des occasions de v\u00e9rifier la solidit\u00e9 du restant de l'\u00e9difice. Exemple ? Alain et ses consid\u00e9rations qui lui permettent, avec force p\u00e9dagogie, d'envisager le sujet comme on le ferait dans un manuel destin\u00e9 au grand public : d\u00e9finitions des sons, de la m\u00e9lodie, des instruments, de l'harmonie, des ornements, des timbres, des modes et de ce qui constitue le d\u00e9tail d'une le\u00e7on de musique. Le dessein de l'auteur consiste \u00e0 formuler les lin\u00e9aments d'une esth\u00e9tique classique o\u00f9 sym\u00e9trie, \u00e9quilibre, harmonie tiennent une place centrale, comme si l'Ecole de Vienne n'avait rien produit \u00e0 l'heure o\u00f9 il \u00e9crit.\n\nDe la m\u00eame mani\u00e8re Adorno th\u00e9orise aux antipodes d'Alain, mais selon un mode identique : le premier c\u00e9l\u00e8bre les valeurs et les vertus classiques, le second n'a de cesse de les d\u00e9truire pour vanter les m\u00e9rites de ce qu'il est alors convenu d'appeler la nouvelle musique, \u00e0 savoir le dod\u00e9caphonisme et le s\u00e9rialisme. Litt\u00e9rature militante, la Philosophie de la nouvelle musique, \u00e9crite par un philosophe qui est aussi un compositeur, \u00e9vite l'essence de la musique pour s'attarder sur les conditions sociologiques et politiques de sa r\u00e9ception. Sur le m\u00eame principe, quittant le camp des indiens Nambikwara, Claude L\u00e9vi-Strauss fait l'\u00e9loge de Rameau parce qu'il propose une musique toute de structures, de formes agenc\u00e9es sur un quadrillage qu'on pourrait dire structural.\n\nOublieux et militants ne sont gu\u00e8re sauv\u00e9s par le dilettantisme des contemporains qui effleurent le sujet et l'abordent sur le mode circonstanci\u00e9, presque hasardeux. Sartre, par exemple, donne un jour un entretien sur la musique contemporaine, mais on sent l'improvisation de ses interlocuteurs pour obtenir du grand homme un parach\u00e8vement de la statue : un tel monument se doit d'avoir abord\u00e9 le sujet et donn\u00e9 son avis sur la musique de son temps. Las, en bourgeois de province qu'il n'a cess\u00e9 d'\u00eatre sur cette question, le vieil homme dit sa passion pour Chopin mais s'emm\u00eale un peu d\u00e8s qu'il disserte sur Boulez ou Pousseur. Restons charitable et ne nous attardons pas aux avis du ma\u00eetre sur John Cage, un plaisantin, Xenakis, un scientifique, Kagel, un accompagnateur de cin\u00e9ma, ou Sch\u00f6nberg, d\u00e9cevant et maladroit. Sartre confirme qu'il reste un classique et d\u00e9sesp\u00e8re, avec l'argument des r\u00e9actionnaires de toujours, qu'il n'y ait plus de statut pour la beaut\u00e9 classique dans la musique du XXe si\u00e8cle.\n\nLa g\u00e9n\u00e9ration suivante n'ira gu\u00e8re au-del\u00e0 de l'allusion : Foucault r\u00e9dige quelques pages sur Jean Barraqu\u00e9 du temps qu'il est son compagnon ; Fran\u00e7ois Ch\u00e2telet exprime sa passion pour Verdi en m\u00eame temps que son go\u00fbt pour les viandes en sauce ; Deleuze et Guattari cr\u00e9ent quelques concepts, le temps lisse ou stri\u00e9, l'\u00e9poque des musiques d'oiseaux et celle des musiques d'insectes, les ritournelles, et puis plus rien. Il faut se tourner vers deux philosophes atypiques, Vladimir Jank\u00e9l\u00e9vitch et Cl\u00e9ment Rosset, pour trouver des consid\u00e9rations plus cons\u00e9quentes sur la musique. Mais ils ajoutent au cort\u00e8ge des oublieux, des militants, des dilettantes et des opportunistes la cat\u00e9gorie tout aussi peu r\u00e9jouissante des th\u00e9ologiens. A savoir ?\n\nApr\u00e8s l'effondrement du Reich nazi, Jank\u00e9l\u00e9vitch ne cesse de refuser, pour des raisons moralisatrices plus que m\u00e9taphysiques ou \u00e9thiques, la philosophie et la musique allemandes. Ce qui, convenons-en, laisse peu de place pour les choses essentielles. Restent les musiques italiennes, fran\u00e7aises, espagnoles, disons celles qui, latines, dispensent d'aborder le continent germanique. D'o\u00f9 son affection particuli\u00e8re pour Mompou et Albeniz, D\u00e9odat de S\u00e9verac et Debussy, Ravel et Faur\u00e9, et tous les autres qui, de Moussorgsky \u00e0 Rachmaninov via Scriabine, autorisent une m\u00e9ditation sur le mode et le principe de la th\u00e9ologie n\u00e9gative. Car Jank\u00e9l\u00e9vitch n'aborde la musique que par la p\u00e9riph\u00e9rie, les alentours, les lisi\u00e8res et les franges, les bords. Ses concepts de pr\u00e9dilection r\u00e9actualisent ceux de la th\u00e9ologie n\u00e9gative du Pseudo-Denys : l'ineffable et l'indicible, l'intemporel et l'intraduisible, l'in\u00e9vidence et le myst\u00e8re. De sorte que tout noyau dur est interdit au profit d'une perp\u00e9tuelle et inconsistante fuite r\u00e9flexive \u00e0 l'origine d'un genre de d\u00e9tumescence mentale.\n\nEn complice des sophistiques de l'auteur de Quelque part dans l'inachev\u00e9, Cl\u00e9ment Rosset aborde le probl\u00e8me avec la m\u00eame ardeur conceptuelle \u00e0 d\u00e9samorcer le vif du sujet, en familier des usages tautologiques. C'est-\u00e0-dire ? Rosset \u00e9tablit, en disciple av\u00e9r\u00e9 de Schopenhauer plus que de Nietzsche, une \u00e9quivalence entre la musique et le r\u00e9el, avant de s'en aller voir ailleurs, pour la raison que du r\u00e9el, on ne peut rien dire, sinon, \u00e0 la limite, qu'il est. Et encore, cette audace para\u00eet possible dans des moments de luxe intellectuel et seulement. Langage universel qui n'exprime rien, signifiant sans signifi\u00e9, objet singulier destin\u00e9 \u00e0 demeurer intact, inentam\u00e9, la musique est une \u00e9nigme qui co\u00efncide avec celle du r\u00e9el. Voil\u00e0 pourquoi, parfois, il vaut mieux \u00eatre sourd que d'entendre cela...\n\nQue pourrais-je donc dire, en prol\u00e9gom\u00e8nes, \u00e0 mon sourd de service ? Que les philosophes s'y connaissent pour \u00e9viter la musique et ne pas entendre le bruit de ses sir\u00e8nes, qu'ils agissent comme Ulysse en se bouchant les oreilles avec des tampons de cire, afin d'\u00e9viter de succomber \u00e0 la tentation et d'\u00eatre d\u00e9livr\u00e9s du mal. Ainsi font-ils. Comment donc expliquer, justifier cette ardeur \u00e0 l'\u00e9vitement, cette perp\u00e9tuelle fuite en avant, ces trajets symptomatiques ? Car la musique justifie le monde, fournit une dimension consolatrice essentielle et produit une forme singuli\u00e8re de l'h\u00e9donisme qui permet, malgr\u00e9 le tragique, de vivre encore. Et il me semble qu'avec Nietzsche la philosophie s'ouvre nettement en deux et se s\u00e9pare en installant d'un c\u00f4t\u00e9 les vieilles barbes sacrifiant \u00e0 l'id\u00e9al asc\u00e9tique et de l'autre les penseurs qui proposent le mod\u00e8le du philosophe-artiste. Pour ceux-l\u00e0, la musique joue un r\u00f4le capital, au statut central dans l'\u00e9conomie de la vision du monde.\n\nLoin des p\u00e9riph\u00e9ries famili\u00e8res aux pens\u00e9es de l'\u00e9vitement ou de l'oubli, le philosophe-artiste installe la musique \u00e0 l'\u00e9picentre de son projet et d\u00e9clare la n\u00e9cessit\u00e9 d'en finir avec les contempteurs du corps. Car ceux qui font l'\u00e9conomie de la musique, philosophes ou non, sont bien souvent ceux qui luttent contre leur corps et fustigent, p\u00eale-m\u00eale les d\u00e9sirs et les plaisirs, les passions et les pulsions, les instincts et les \u00e9motions. Le corps devient la grande raison de ceux qui, dans tous les sens du terme, entendent la musique. Et le sourd sait plus que tout autre que le corps est bien cette grande raison. Certes, il lui manque l'oreille, l'ou\u00efe et, au sens classique, l'image auditive du monde. Mais sa sagesse s'appuie sur ce manque et ce qui permet de le pallier : le toucher. Car la musique est l'art de toucher les corps. Et voil\u00e0 vraisemblablement l'une des raisons pour lesquelles elle fait peur. Voil\u00e0 pourquoi on l'\u00e9vite, poliment, avec d\u00e9f\u00e9rence, voil\u00e0 les raisons qui justifient la n\u00e9gligence.\n\nLe sourd n'a que faire de ce qui s'\u00e9crit sur la musique avant Nietzsche : bruit des sph\u00e8res en errance dans l'\u00e9ther, v\u00e9hicule complice des formes politiques, m\u00e9diation sp\u00e9cieuse du diabolique, occasion d'une querelle sur les pr\u00e9s\u00e9ances du texte ou du son, expression tautologique du vouloir, elle fournit le support \u00e0 mille digressions qui, pour la plupart, rendent possibles la mise \u00e0 l'\u00e9cart du corps, sa rel\u00e9gation dans les soupentes th\u00e9oriques, l\u00e0 o\u00f9 s'entassent les objets qui troublent l'id\u00e9e que la philosophie classique se fait d'elle-m\u00eame. Car la musique est le contraire d'une casa mentale, cette expression que Vinci r\u00e9servait et appliquait \u00e0 la peinture. Elle propose une activit\u00e9 radicalement corporelle dans une civilisation qui craint les possibilit\u00e9s de la chair.\n\nL'Eglise ne s'y trompe pas, ni les gouvernements totalitaires de ce XXe si\u00e8cle, qui ont violemment assur\u00e9 leur mainmise sur cette activit\u00e9 dangereuse qui affirme les droits de la chair, la l\u00e9gitimit\u00e9 des revendications du corps et la possibilit\u00e9 de solliciter directement une peau, un syst\u00e8me nerveux, une viande, une ossature, un squelette, un syst\u00e8me pileux et tout ce qui trahit l'immanence radicale de l'objet auquel s'adresse la musique. Les contempteurs du corps fustigent presque syst\u00e9matiquement la musique. Descartes, auteur en 1650 d'un Compendium musicae n'\u00e9chappe pas \u00e0 la r\u00e8gle et persiste, malgr\u00e9 ses avanc\u00e9es par ailleurs, son audace conceptuelle ici ou l\u00e0, \u00e0 agir et penser en scolastique. Dans ses Confessions, saint Augustin dit tout ce qu'il doit de jouissances et de plaisirs coupables \u00e0 la musique. Entre larmes de contrition et sanglots de regrets, il confesse comment il a \u00e9t\u00e9 envelopp\u00e9 et subjugu\u00e9 par \u00ables volupt\u00e9s de l'oreille\u00bb, de quelle mani\u00e8re, dans l'\u00e9coute de la musique, il accepte de mettre en avant le plaisir des sens qui cong\u00e9die la puissance de la raison, ce qui le rend p\u00e9cheur. Danger de la jubilation, \u00e9motion dans le chant, primaut\u00e9 du corps, retrait de l'\u00e2me, le P\u00e8re de l'Eglise \u00e9nonce la r\u00e8gle sur ce sujet : il faut se m\u00e9fier de la musique, cet auxiliaire des sens.\n\nDescartes proc\u00e8de m\u00eamement dans son \u00e9tude en privil\u00e9giant les aspects physico-acoustiques et math\u00e9matiques des sons et des intervalles. Puis il oppose l'harmonie, r\u00e9gie par la raison, \u00e0 la m\u00e9lodie qui, elle, rel\u00e8ve de la sensibilit\u00e9. La premi\u00e8re pouvant faire seule l'objet d'un discours rationnel, donc raisonnable, le philosophe met de c\u00f4t\u00e9 la m\u00e9lodie et r\u00e9ussit ce fameux tour de force de traiter dans la musique de ce qui trouble le moins le corps, \u00e0 savoir ce qui r\u00e9jouit le plus l'\u00e2me. Trop de corps et de passion. Pour la raison qu'il sollicite directement la chair, le chant est souvent mis en perspective avec la voix du diable. L'\u00e9vitement du corps fournit un exercice d'\u00e9cole pour la corporation des philosophes classiques. O\u00f9 donc trouve-t-on penseurs qui faillissent et osent, sur ce sujet, s'aventurer sur les terres r\u00e9jouissantes de l'immanence corporelle ?\n\nOn ne s'\u00e9tonnera pas de devoir recourir \u00e0 un homme singulier, un individu solitaire et libre, une conscience avertie de la jubilation pour aborder le sujet comme il m\u00e9rite de l'\u00eatre. Et ce penseur, c'est Roland Barthes auquel on doit de belles pages, profondes, p\u00e9n\u00e9trantes et convaincantes sur la question. Musicien, m\u00e9lomane, amateur de d\u00e9sirs et de plaisirs, \u00e9crivain de la passion amoureuse, plume fine et d\u00e9licate, Barthes a pens\u00e9 la musique \u00e0 plusieurs occasions dans son \u0153uvre. Pour c\u00e9l\u00e9brer Schumann ou analyser la diction de Panzera, quand il se demande ce qui d\u00e9finit le grain d'une voix, ou lorsqu'il r\u00e9fl\u00e9chit sur l'usage bourgeois de la musique de chambre, le chant de Fischer-Dieskau ou le toucher de piano de Dinu Lipatti, d\u00e8s qu'il t\u00e2che de comprendre pourquoi il aime la premi\u00e8re phrase d'un trio de Schubert, ce que signifie \u00e9couter, sinon ce qu'il faut comprendre et retenir de la surdit\u00e9 de Beethoven, Roland Barthes n'\u00e9vite pas le retour, le passage, le d\u00e9tour et le souci du corps.\n\nLoin des r\u00e9ticences de la corporation philosophique, au-del\u00e0 de ses a priori, Barthes en appelle \u00e0 un corps r\u00e9el, non pas une id\u00e9e, une figure conceptuelle, une construction spirituelle ou mentale sublim\u00e9e, mais un corps mat\u00e9riel, immanent, r\u00e9el, un physique authentique. Pour parler de Duparc ou de Faur\u00e9, du lied ou de la m\u00e9lodie, de Schumann ou de Beethoven, du piano ou de la sonate, il convoque muscles et muqueuses, cartilages et nez, poumon et gosier, glotte et langue, dents aussi, bras, doigts, et plus particuli\u00e8rement coussinets des doigts.\n\nPuis il extrapole sur le corps qui \u00e9coute et le corps interne, le corps qui joue et le corps pulsionnel, le corps qui bat, sans cesser de se r\u00e9f\u00e9rer \u00e0 \u00ab la v\u00e9rit\u00e9 sensuelle de la musique \u00bb. Voix qui bande ou voix nue, fa\u00e7on dont elle se tient dans le corps, voire fa\u00e7on dont le corps se tient en elle, il n'\u00e9vite rien de ce qui exprime la chair. Du grain, Barthes dit qu'il \u00abest le corps dans la voix qui chante, dans la main qui \u00e9crit, dans le membre qui ex\u00e9cute \u00bb. De sorte que dans sa d\u00e9monstration, il permet de conclure, \u00e0 destination du sourd que je n'oublie pas, que la musique est moins affaire d'oreille, ce lieu sp\u00e9cifique et partiel du corps, que de corps tout entier.\n\nAu sourd de naissance ne peut faire d\u00e9faut cet autre sens, la plupart du temps n\u00e9glig\u00e9, qu'est le toucher. D'abord on le mutile souvent en limitant \u00e0 la pulpe des doigts et en imaginant que nous ne toucherions qu'avec les mains, et encore, avec leur extr\u00e9mit\u00e9. Or nous touchons avec l'ensemble de la peau, cette immense voilure de plusieurs m\u00e8tres carr\u00e9s. Le sourd n'est pas exclu du monde de la musique parce que l'oreille lui fait d\u00e9faut, en revanche il le serait si d'aventure le toucher lui \u00e9tait interdit. On touche avec la bouche, bien s\u00fbr, mais aussi avec le ventre et le sexe, avec les volumes et les protub\u00e9rances, les \u00e9minences et les cavit\u00e9s, on touche avec les bras et les jambes, les reins et les \u00e9paules, car rien de ce qui d\u00e9finit un corps et ses orifices ne s'exclut de cette entreprise sensuelle.\n\nArt de toucher les corps, la musique qui aborde le sourd utilise sa peau comme celle d'un instrument \u00e0 percussion, certes, mais aussi \u00e0 vent ou \u00e0 cordes, car tous vibrent, anim\u00e9s d'un souffle, d'une caresse, d'un frottement, d'un contact, toutes op\u00e9rations justifiables \u00e9galement entre deux corps amoureux. Celui qui n'a pas d'oreille aborde presque la musique en privil\u00e9gi\u00e9, en sauvage, au contact direct de la mat\u00e9rialit\u00e9 brutale du son. Car l'oreille met plus \u00e0 distance que la peau, elle affine, donne les d\u00e9tails, fournit les variations infimes, les modulations presque infinit\u00e9simales, d'o\u00f9 la spiritualisation du son et son devenir immat\u00e9riel via le colima\u00e7on, le marteau et l'enclume avant le cerveau qui convertit l'ensemble de ces \u00e9nergies m\u00e9caniques en \u00e9nergies \u00e9lectriques destin\u00e9es \u00e0 fournir des informations, des images mentales, des concepts sonores. Le sourd entend la musique en \u00e9conomisant le filtre de l'oreille, il en exp\u00e9rimente directement la mat\u00e9rialit\u00e9.\n\nDe sorte qu'aid\u00e9 par le d\u00e9faillant auditif, le malentendant comme on dit sottement aujourd'hui, on peut affirmer, pour t\u00e2cher de circonscrire un peu le noyau dur de la musique, son essence, qu'en tous lieux, tout temps, elle se d\u00e9finit d'abord par des ondes constitutives d'une physiologie. Cette derni\u00e8re met en contact des \u00e9l\u00e9ments \u00e9pars dans une zone proche : des instruments avec des corps, des corps \u2013 de musiciens ou d'interpr\u00e8tes \u2013 avec d'autres corps \u2013 ceux des auditeurs. L'onde sonore relie le s\u00e9par\u00e9, elle associe pour des r\u00e9seaux et maillages alternatifs, al\u00e9atoires, mobiles et changeants, des corps qui exp\u00e9rimentent de la sorte une communication, une intersubjectivit\u00e9 physiologique.\n\nCette lecture mat\u00e9rialiste qui cong\u00e9die toute po\u00e9sie et tout lyrisme m\u00e9taphysique, voire tout \u00e9panchement spiritualiste, \u00e9vite l'\u00e9cueil des th\u00e9ologies n\u00e9gatives et permet la proposition d'une pathologie positive. La m\u00e9canique ondulatoire confine l'interpr\u00e9tation \u00e0 un mat\u00e9rialisme sensualiste, moins appuy\u00e9 sur une d\u00e9finition du son comme mati\u00e8re immat\u00e9rielle que comme mati\u00e8re d'une densit\u00e9 mesurable, d'une \u00e9paisseur susceptible d'\u00eatre \u00e9prouv\u00e9e par le d\u00e9codage d'un corps tout entier, et pas seulement par une oreille. Le son ne se voit pas, certes, mais loin d'\u00eatre invisible, on peut mat\u00e9rialiser son trajet, ses effets.\n\nPas plus que le vent dont on conna\u00eet l'existence par les seuls effets, les branches qui ploient, les feuilles qui tourbillonnent, la surface rid\u00e9e des eaux, le son ne se prouve qu'en s'\u00e9prouvant. Un obscur curieux du si\u00e8cle des Lumi\u00e8res, Louis Roger, remarque que dans une \u00e9glise, le souffle des bougies se modifie en fonction du jeu des tuyaux d'orgue. Il conclut, fort judicieusement, \u00e0 la mat\u00e9rialit\u00e9 invisible du son, mais \u00e0 son existence physique. Pareil \u00e0 la flamme d'une chandelle, le corps sur lequel viennent s'\u00e9craser les sons oscille, tremble, fr\u00e9mit, s'allonge, se r\u00e9tr\u00e9cit, se d\u00e9plie, se d\u00e9ploie, vibre, danse, il tend vers les sommets ou se recroqueville vers sa base.\n\nLes ondes qui \u00e9viteront l'oreille du sourd attaqueront sa peau, se fraieront un passage de l'ext\u00e9rieur du corps \u00e0 son int\u00e9rieur, en p\u00e9n\u00e9trant la surface et en inqui\u00e9tant les os, la moelle, les nerfs, les muscles, les syst\u00e8mes respiratoires, digestifs et sanguins. La preuve de la mat\u00e9rialit\u00e9 du son et de son efficacit\u00e9, c'est le c\u0153ur qui bat plus vite, le souffle qui s'acc\u00e9l\u00e8re, les reins qui semblent donner de l'expansion \u00e0 une pointe de jouissance sollicitant le bas du dos, les frissons qui travaillent l'\u00e9chine et la sculptent. A chaque fois, des organes, des visc\u00e8res, de la chair, de la lymphe, du corps.\n\nLes physiologues rep\u00e8rent les lieux de passage des effets de la musique dans le corps, les endroits de pr\u00e9dilection o\u00f9 agissent ces ondes. Ils isolent et distinguent le sternum, en quelque sorte la pointe avanc\u00e9e de la poitrine et du corps dans l'espace, l'ensemble thoracique et abdominal, la r\u00e9gion pelvienne, autant dire ce qui dispense d'une m\u00e9diatisation par la t\u00eate et le cerveau puis revendique l'acc\u00e8s direct et imm\u00e9diat aux zones les plus sensibles du tronc. Les sonorit\u00e9s basses d\u00e9clenchent des vibrations nettement localisables dans les zones \u00e9rog\u00e8nes internes alors que les m\u00e9lodies r\u00e9p\u00e9titives et les bourdonnements produisent de l\u00e9gers \u00e9tats hypnotiques. L\u00e0 o\u00f9 l'on croit l'oreille indispensable pour \u00e9couter la musique, il faut donc affirmer l'ensemble du corps mobilis\u00e9 pour l'entendre.\n\nPeut-on mieux expliquer pour quelles raisons, peut-\u00eatre \u00e0 leur corps d\u00e9fendant \u2013 allons jusque-l\u00e0 dans la magnanimit\u00e9 ! \u2013 les philosophes classiques avouent tous un m\u00e9pris, une haine de la musique, voire sont coupables de n\u00e9gligence ou perp\u00e9tuent son oubli dans leur champ d'activit\u00e9? Le ventre, le bas-ventre, les visc\u00e8res, l'int\u00e9rieur chaud et humide du corps, son animalit\u00e9 cach\u00e9e, sa brutalit\u00e9 dissimul\u00e9e, les flux d'humeurs, les all\u00e9es et venues de vents, de sang, de lymphe modifi\u00e9s et induits par l'autorit\u00e9 d'ondes ma\u00eetris\u00e9es par quelques-uns, voil\u00e0 de quoi effrayer tout contempteur du corps toujours \u00e0 l'\u0153uvre dans la confr\u00e9rie philosophique traditionnelle. Jeux de sensualit\u00e9, \u00e9motions \u00e9chang\u00e9es, passions partag\u00e9es, corps \u00e9branl\u00e9s, sensibilit\u00e9s sollicit\u00e9es, voil\u00e0 qui est trop au regard, sinon \u00e0 l'oreille, de qui porte la raison en sautoir et c\u00e9l\u00e8bre la m\u00e9thode, la r\u00e9flexion, la d\u00e9duction, la logique pour amasser aujourd'hui quelque v\u00e9rit\u00e9 d\u00e9mon\u00e9tis\u00e9e demain.\n\nM\u00e9lomanes et musiciens, donc, constituent une micro-soci\u00e9t\u00e9 produisant des affinit\u00e9s \u00e9lectives vivant selon cette logique. Si l'oubli de la musique est coextensif \u00e0 celui du corps, inversement on peut affirmer que le souci de la musique vaut comme preuve de l'int\u00e9r\u00eat qu'on lui porte. Il faut compl\u00e9ter le mat\u00e9rialisme de la m\u00e9canique ondulatoire par quelques consid\u00e9rations sur la mystique immanente permise par la musique. La logique sensualiste se prolonge dans une pratique du corps h\u00e9doniste, voire une pratique h\u00e9doniste du corps. O\u00f9 il est donc question de la transe ou de la possession et de ce que, il me semble, on peut appeler le principe d'Orph\u00e9e.\n\nCar le mythe grec d'Orph\u00e9e exprime ce qui traverse les temps et les histoires pour dire les puissances d'un certain sens magique de la musique sur les corps. Que dit le mythe? Que la musique dispose de pouvoirs tellement absolus qu'elle peut contrevenir \u00e0 la nature des choses, qu'elle peut faire de telle sorte que ce qui est soit autrement, aux antipodes. La cruaut\u00e9 d'un animal f\u00e9roce? La musique la transfigure en douceur, en gentillesse. L'insensibilit\u00e9 du r\u00e8gne min\u00e9ral ou la n\u00e9cessit\u00e9 du mouvement des cours d'eau ? Tout cela dispara\u00eet au profit de modifications radicales dans l'essence, alors les pierres deviennent sensibles et les fleuves d\u00e9bordent leur cours.\n\nEn Inde, des ragas font dispara\u00eetre le soleil et appara\u00eetre les t\u00e9n\u00e8bres, d'autres consument les musiciens qui s'aventurent \u00e0 les interpr\u00e9ter, m\u00eame s'ils ont pris la pr\u00e9caution de jouer quasi couverts d'eau dans l'onde d'une rivi\u00e8re : ils se trouvent transform\u00e9s en cendres ; certains, aussi, permettent \u00e0 la pluie de tomber, comme nagu\u00e8re dans une Bretagne o\u00f9 l'on pratiquait encore les processions... Chez les Grecs, mille histoires bien connues corroborent la puissance magique de la musique : Terpandre ou Tyrt\u00e9e apaisent les s\u00e9ditions ou enflamment les courages alors que Timoth\u00e9e fait se lever et marcher Alexandre en direction de ses armes, puis se rasseoir, passant de la col\u00e8re \u00e0 l'apaisement, le tout uniquement en variant les modes musicaux jou\u00e9s \u00e0 la lyre. Amphion ne verrait rien \u00e0 redire \u00e0 ces \u00e9vidences, lui qui fit se d\u00e9placer les pierres pour construire le mur de Th\u00e8bes rien qu'en jouant de son instrument. Et Sa\u00fcl, et David. Toutes les histoires vont en ce sens : la musique peut l'extravagant et agit sur les corps en les modifiant substantiellement. Ce qui est dispara\u00eet au profit d'une nouvelle modalit\u00e9. Tous les corps connaissent donc cette m\u00e9tamorphose, du min\u00e9ral \u00e0 l'humain en passant par le v\u00e9g\u00e9tal et l'animal, la musique transforme en profondeur la nature des choses.\n\nLe corps touch\u00e9 par l'onde, la physiologie modifi\u00e9e, produisent une mystique, induisent une logique de transe. Voici la deuxi\u00e8me le\u00e7on susceptible d'\u00eatre enseign\u00e9e au sourd : la musique parle au corps, certes, mais elle l'informe aussi, agit sur lui dans le sens d'une r\u00e9duction de la libert\u00e9 et d'une augmentation de la n\u00e9cessit\u00e9. Une fois de plus, les philosophes de profession et de caste ne peuvent que s'effrayer, s'inqui\u00e9ter. D\u00e8s que l'identit\u00e9 se trouve entam\u00e9e, \u00e0 partir du moment o\u00f9 l'int\u00e9grit\u00e9 d'un \u00eatre se modifie, la raison n'a plus aucune prise et il n'est plus que des corps soumis \u00e0 des magies, d\u00e9pendants de myst\u00e9rieuses op\u00e9rations. Le sourd enregistre ces m\u00e9tamorphoses, son corps comprend cette interaction entre l'onde et la modification des \u00e9tats de conscience. Malgr\u00e9 l'oreille qui fait d\u00e9faut, le corps du sourd entend l'appel pour d'autres mondes \u2013 invitation au voyage.\n\nDans son Dictionnaire de musique, Rousseau rapporte une \u00e9trange histoire de vaches, de Suisses, et de soldats devenus impropres au combat. Il me pla\u00eet de la rapporter : le philosophe pr\u00e9cise que lors des guerres qui supposaient des troupes fra\u00eeches, aguerries et courageuses, il \u00e9tait interdit aux fanfares militaires de jouer aux soldats suisses un air appel\u00e9 le ranz des vaches pour la raison qu'il induisait imm\u00e9diatement des pleurs, des larmes, des d\u00e9missions, des d\u00e9sertions, voire des suicides dans le rang des combattants helv\u00e8tes. Rousseau y voit le signe sonore auquel les Suisses associaient des souvenirs agr\u00e9ables d'enfance, des images joyeuses du pass\u00e9, des m\u00e9moires affectives voluptueuses, et autres mauvaises \u00e9motions pour le moral des troupes. Bien \u00e9videmment, n'importe quel \u00e9tranger au territoire helv\u00e9tique peut entendre le ranz sans trouble ni \u00e9motion, dans la plus absolue des int\u00e9grit\u00e9s mentales. Mais voil\u00e0...\n\nIl faut conclure \u00e0 cette \u00e9vidence que le corps comprend les messages que la musique envoie, les d\u00e9code, les int\u00e8gre et modifie ses \u00e9tats en fonction des informations donn\u00e9es. Des tarentelles aux saturnales en passant par les f\u00eates orgiaques et autres dionysies, sans oublier les c\u00e9r\u00e9monies traditionnelles du rite vaudou, les concerts rock ou pop, les repr\u00e9sentations th\u00e9\u00e2trales des op\u00e9ras de ce XXe si\u00e8cle finissant, la musique agit sur le corps de mani\u00e8re \u00e9vidente et produit des affections, des \u00e9motions, des sensations et autres transformations qui affectent les rythmes du corps, son fonctionnement habituel. D\u00e9bordements, transes, exc\u00e8s, possessions, d\u00e9pense, voil\u00e0 qui exc\u00e8de les limites supportables pour le philosophe de la tradition.\n\nEn plus des visc\u00e8res, du ventre, de l'action directe sur le syst\u00e8me nerveux et sur les \u00e9tats de conscience r\u00e9duits \u00e0 des modalit\u00e9s de la fonction corporelle, les pouvoirs et la magie de la musique outrepassent ce que ne peuvent tol\u00e9rer les ennemis du corps. Une chair qui ne s'appartient plus, voire s'abandonne et s'offre \u00e0 la volont\u00e9 d'un tiers, voil\u00e0 plus que ne peuvent en supporter les thurif\u00e9raires de la ma\u00eetrise, de l'asc\u00e8se, de la domination de soi sur soi. Indocile et libre, le corps abandonn\u00e9 aux rythmes et aux incantations musicales \u00e9prouve son essence et jouit de sa substance en manifestant une existence h\u00e9doniste. Logique de transe et mystique immanente, la musique offre donc l'occasion d'une expression sensualiste et mat\u00e9rialiste de l'existence sur un mode jubilatoire qui convoque la totalit\u00e9 du corps.\n\nPhysiologie et path\u00e9tique, onde et langage, la musique transporte le corps sur un terrain dangereux. Voil\u00e0 pourquoi elle se fait confisquer par les acteurs politiques et sociaux pour devenir une activit\u00e9 socialement acceptable \u00e0 int\u00e9grer dans le jeu communautaire habituel. La dimension sociologique de la musique est incontestable. En elle d'ailleurs s'enracinent sa puissance politique, son usage et son utilit\u00e9 dans la cit\u00e9. Car tout \u00e9difice social vit de cristallisations singuli\u00e8res, en l'occurrence celles qui supposent pour mati\u00e8re le renoncement des individus et des singularit\u00e9s, des subjectivit\u00e9s radieuses, \u00e0 leur pouvoir et \u00e0 leur capacit\u00e9 d'opposer le principe de plaisir au principe de r\u00e9alit\u00e9.\n\nAu sourd qui nous entendrait encore \u2013 via le langage des signes, \u00e9videmment \u2013, il faudrait pr\u00e9ciser o\u00f9 nous en sommes : la musique se d\u00e9finit comme la conjonction d'une m\u00e9canique ondulatoire, d'un mat\u00e9rialisme sensualiste et d'une mystique immanente, d'une logique de la transe \u00e0 quoi s'ajoute maintenant une religion identitaire fournissant l'occasion d'une pratique rituelle productrice d'un classement social. Physiologie, plus path\u00e9tique, plus sociologie, pour suivre \u00e0 la trace une onde devenant transe m\u00e9tamorphos\u00e9e en messe, voil\u00e0 de quoi approcher au mieux le noyau dur de la musique \u2013 tout en conservant sa surdit\u00e9.\n\nR\u00e9cup\u00e9r\u00e9e par le social, la musique devient un auxiliaire de coh\u00e9sion communautaire, un facteur gr\u00e9gaire de religion cultuelle et culturelle. L\u00e0 o\u00f9 les corps se vivent sur le mode expansif et jubilatoire, l\u00e0 o\u00f9 l'on sollicite leur pure dimension mat\u00e9rielle par l'onde et le mouvement, la civilisation exige une mise en forme, une cristallisation de ces \u00e9nergies trop puissantes pour \u00eatre tol\u00e9r\u00e9es. Leur charge pourrait mettre en p\u00e9ril l'\u00e9quilibre de l'\u00e9difice social obtenu \u00e0 force d'arrangements et de sacrifices consentis par ceux qui ont renonc\u00e9 \u00e0 leur puissance, \u00e0 leur propri\u00e9t\u00e9 et \u00e0 la dimension solaire de leur subjectivit\u00e9. De sorte que le recyclage de ces forces sauvages en disciplines sociales an\u00e9antit tout destin libertaire et ouvre la voie aux pratiques bourgeoises du concert, du disque et de la consommation culturelle, sacralis\u00e9e, f\u00e9tichis\u00e9e.\n\nLa musique perd son pouvoir incantatoire sur les corps d\u00e8s qu'elle devient un argument f\u00e9d\u00e9rateur du corps social. Elle s'adoucit, perd de sa violence et de sa brutalit\u00e9, de son efficace aussi, en contribuant au devenir social. Le corps sollicit\u00e9 sensuellement acc\u00e8de au devenir neutre d\u00e8s qu'il engrosse les masses, les foules, les classes qui consomment de la musique. Les hi\u00e9rarchies, les distinctions, entre musiques populaires et \u00e9litistes, de table ou s\u00e9rieuses, de vari\u00e9t\u00e9 ou classiques, anciennes ou contemporaines, europ\u00e9ennes ou ethnologiques, pop, folk, jazz ou rap, funk, techno, tous ces classements autorisent une asepsie des logiques h\u00e9donistes. Ils assurent l'an\u00e9antissement de ces \u00e9nergies sauvages quand elles travaillent les corps individualis\u00e9s au lieu d'assurer la coh\u00e9rence du corps social.\n\nDans son registre individuel, la musique est radicalement mat\u00e9rielle ; dans son usage collectivis\u00e9, socialis\u00e9, elle se spiritualise et subit les distorsions avec lesquelles elle entre dans le d\u00e9ploiement d'un rituel extr\u00eamement pr\u00e9cis \u2013 de Woodstock \u00e0 Bayreuth, de Memphis \u00e0 La Nouvelle-Orl\u00e9ans, de la Scala au Petit Conservatoire \u2013 de sorte que se trouve assur\u00e9e collectivement la d\u00e9virilisation de sa puissance et de ses effets sur l'individu. Au concert, sinon dans le confinement de la pratique marchande, la musique assure des fonctions de classement, de distinction sociale, elle rend possibles des appartenances identitaires et des communions cultuelles. Avant, il y avait des corps \u00e9branl\u00e9s et des \u00e2mes en \u00e9moi, des sens exacerb\u00e9s et des \u00e9motions vivement sollicit\u00e9es ; apr\u00e8s, d\u00e8s le devenir social de ces passions, il ne reste que coh\u00e9sions communautaires, identit\u00e9s tribales, \u00e9quilibres sociaux et occasions de gr\u00e9garismes constitu\u00e9s.\n\nLe sourd \u2013 s'il a consenti \u00e0 suivre ces consid\u00e9rations \u2013 dispara\u00eet en tant que tel pour acc\u00e9der \u00e0 une communaut\u00e9 d'\u00e9lection qu'il choisit en m\u00eame temps qu'elle le classe, d\u00e9signe et distingue. Son oreille d\u00e9faillante n'a pas emp\u00each\u00e9 un corps r\u00e9ceptif et, suivant les lieux o\u00f9 il pratique cette sollicitation de son corps tout entier, il b\u00e9n\u00e9ficie d'un paiement symbolique en satisfaction sociale. Pour avoir sacrifi\u00e9 la parole libre de son corps, il dispose d'occasions de c\u00e9l\u00e9brer son culte \u00e0 plusieurs. C'est le d\u00e9but de la fin pour ses \u00e9motions singuli\u00e8res. Vite elles deviendront des passions sociales.\n\nLe syst\u00e8me nerveux, les \u00e9tats de transe, le corps qui jubile, les ondes qui ravissent le ventre, les vibrations qui travaillent la libido, les substances sonores d\u00e9riv\u00e9es vers les possessions rituelles, tout dispara\u00eet corps et biens dans la g\u00e9n\u00e9alogie des \u00e9difices sociaux et collectifs. La soci\u00e9t\u00e9 de consommation, fid\u00e8le en cela \u00e0 son r\u00f4le perp\u00e9tuellement contre-r\u00e9volutionnaire et r\u00e9actionnaire, fournit les moyens de canaliser les \u00e9nergies rebelles et solitaires pour en faire des cristallisations gr\u00e9gaires, inoffensives d\u00e8s qu'elles contribuent \u00e0 un \u00e9difice collectif.\n\nDe l'\u00e9tat du corps sous l'emprise de la musique au corps de l'Etat soutenu par son action en mati\u00e8re de construction sociale, se dessine le trajet qui conduit de l'id\u00e9al h\u00e9doniste \u00e0 l'id\u00e9al asc\u00e9tique, du principe de plaisir au principe de r\u00e9alit\u00e9. On sait depuis Freud combien cette m\u00e9tamorphose, surtout dans ce sens, g\u00e9n\u00e8re des malaises dans notre civilisation. Dans la cit\u00e9, la musique fournit au corps sa dose d'\u00e9nergie socialement acceptable. Ni trop, ni trop peu. L'oubli de la musique, s'il est chez les philosophes consubstantiel \u00e0 celui du corps suppose, chez les politiques, un souci d'autant plus prononc\u00e9 qu'il trahit une volont\u00e9 d'emprise et d'empire sur les corps. Voil\u00e0 pourquoi tout acad\u00e9misme d'Etat suppose, derri\u00e8re les travestissements dont il se pare, une id\u00e9ologie ennemie des corps, des d\u00e9sirs et des pulsions. Toute musique qui permet au corps de se r\u00e9approprier et de recouvrer soit une \u00e9nergie perdue, soit une force en p\u00e9ril, guett\u00e9e pour \u00eatre mieux dig\u00e9r\u00e9e, m\u00e9rite d'\u00eatre plac\u00e9e sous les auspices de Dionysos, le seul dieu justifiant qu'on lui sacrifie.\n27\n\nLA PHILOSOPHIE DANS LE PARLOIR\n\nR\u00e9volution fran\u00e7aise ou non, droits de l'homme ou pas, lib\u00e9ration des m\u0153urs annonc\u00e9e ou presque, peu importe : les amateurs de sexualit\u00e9 libre paient toujours cher leur revendication de singularit\u00e9. Le marquis de Sade au si\u00e8cle dit des Lumi\u00e8res, Oscar Wilde dans une Angleterre qui a toujours clam\u00e9 haut et clair, avec une fiert\u00e9 et une arrogance jamais dissimul\u00e9es, son respect profond et radical des individus \u2013 tous les deux ont croupi et moisi en prison pour avoir pratiqu\u00e9 l'un une sexualit\u00e9 de groupe avec mise en sc\u00e8ne f\u00e9odale, l'autre une simple homosexualit\u00e9 fustig\u00e9e en la disant sodomite.\n\nPour ajouter \u00e0 ce triste tableau de sc\u00e8ne de chasse sexuelle, il faut citer \u00e9galement le nom d'Otto Muehl, artiste sans concession de l'Actionnisme viennois, enfant terrible de l'esth\u00e9tique radicale autrichienne, engag\u00e9, corps et \u00e2me, dans la confusion de son existence et de ses revendications artistiques fondamentales. Car il s'agit de ne pas oublier que de 1991 \u00e0 1997 Otto Muehl a v\u00e9cu plus de deux mille cinq cents jours dans les prisons autrichiennes pour avoir eu des relations sexuelles avec deux jeunes filles, alors consentantes, de quatorze et quinze ans.\n\nPr\u00e9cisons que celles-ci eurent lieu dans le cadre d'une communaut\u00e9 artistique, \u00e9thique et politique o\u00f9 tout \u00e9tait commun y compris la sexualit\u00e9. Ce que savaient pertinemment les parents de ces jeunes filles, et ces jeunes filles elles-m\u00eames, puisque tous en \u00e9taient membres depuis plusieurs ann\u00e9es. Ajoutons, pour faire bonne mesure, que lesdits parents, soudains prudes et moraux, cherchaient depuis longtemps un moyen de contrer l'autorit\u00e9 r\u00e9elle de Muehl au sein de la communaut\u00e9... L'Etat, lass\u00e9 des frasques de l'actionniste depuis des d\u00e9cennies, trouva l'occasion belle et d\u00e9cida, par ses tribunaux requis, de sept ann\u00e9es d'incarc\u00e9ration.\n\nAujourd'hui, il est temps de faire le point sur les \u0153uvres de prison d'Otto Muehl. Et elles sont nombreuses : lettres \u00e0 des amis, des intimes, \u00e0 sa famille, \u00e0 ses enfants, \u00e0 des artistes, dessins sur ces courriers ou sur de petits formats cartonn\u00e9s, textes d'entretiens donn\u00e9s \u00e0 Daniele Roussel, sa complice depuis toujours et la plus aboutie des historiennes de l'Actionnisme viennois. Des toiles aussi, sorties de prison gr\u00e2ce au stratag\u00e8me qui permettait d'entoiler deux fois le ch\u00e2ssis : une fois avec la toile \u00e0 faire passer de l'autre c\u00f4t\u00e9 des murs, une autre avec ce qui pla\u00eet \u00e0 l'administration p\u00e9nitentiaire \u2013 c'est d'ailleurs ce qui s\u00e9duit encore et toujours les bourgeois, et pour longtemps : natures mortes, bouquets de fleurs, paysages, et autres excipients.\n\nEn revanche, au contact direct du ch\u00e2ssis, on retrouve tous les sujets de Muehl : des femmes nues, des corps dans des postures libidineuses et triomphales, des agencements sensuels et sexuels, des situations scatologiques ou scabreuses, des collages intellectuels et plastiques, des images classiques et des objets aux sens d\u00e9tourn\u00e9s, des rats, des hy\u00e8nes, des chimpanz\u00e9s, et autres animaux du bestiaire carc\u00e9ral, des urinoirs, \u00e9chiquiers, jocondes et autres meules de foin, tous objets cod\u00e9s dans l'histoire de l'art.\n\nPendant sept ann\u00e9es, l'artiste a partag\u00e9 son existence avec des criminels, des tueurs, des psychopathes dont l'un a tu\u00e9 sa femme \u00e0 coups de hache. Et durant tout ce temps, il n'a pas baiss\u00e9 la garde, ni perdu un gramme de sa vitalit\u00e9 intellectuelle. Le but \u00e9tait de briser son corps, sa chair et son \u00e2me. Pour le corps, l'administration a presque r\u00e9ussi, avec l'aide bienveillante et complaisante d'un m\u00e9decin qui a sans cesse refus\u00e9 les demandes de soins oculaires sollicit\u00e9s par le peintre. Le r\u00e9sultat ? Otto Muehl a presque perdu l'usage d'un \u0153il. Puis il y a eu des probl\u00e8mes \u00e0 la prostate et le d\u00e9clenchement d'une maladie de Parkinson. Rien n'a fait envisager la r\u00e9vision de sa peine.\n\nSa m\u00e8re, dans ses derniers jours, aurait encore pu voir son fils. Son agonie le lui permettait. Otto Muehl avait obtenu une autorisation de se faire soigner non loin du domicile o\u00f9 elle se mourait. Une visite aurait pu \u00eatre organis\u00e9e, il aurait suffi d'une autorisation administrative et d'un l\u00e9ger \u00e9cart sur le trajet. Il n'a pas revu sa m\u00e8re. Elle repose aujourd'hui sous terre. Rappelons, pour m\u00e9moire, qu'\u00e0 l'\u00e9poque, le pr\u00e9sident de la R\u00e9publique au pouvoir, Kurt Waldheim, \u00e9tait mondialement connu pour sa participation \u00e0 des op\u00e9rations militaires nazies... L'un voyageait alors en fourgon cellulaire, menottes aux mains, l'autre circulait librement recevant les honneurs des chefs d'Etat de la plan\u00e8te.\n\nThomas Bernhard a longuement stigmatis\u00e9 la vilenie de l'Au-triche officielle, celle qui fit son miel du catholicisme et du nazisme sans imaginer une seule seconde une \u00e9ventuelle incompatibilit\u00e9 d'humeur entre ces deux visions du monde. Tout le travail d'Otto Muehl, avant l'emprisonnement, s'apparente \u00e0 un essai de th\u00e9rapie collective de cette faute majeure, \u00e0 une tentative de catharsis, de purification \u2013 au sens grec \u2013, de cette p\u00e9riode tout enti\u00e8re soumise au principe de mort. L\u00e0 o\u00f9 la Nation s'est donn\u00e9e sans retenue \u00e0 Thanatos, l'artiste revendique et c\u00e9l\u00e8bre Eros, le principe de l'\u00e9nergie vitale et de la libido libertaire.\n\nLors de son incarc\u00e9ration, Muehl restera soucieux d'une pareille \u00e9thique. Ses \u0153uvres de prison suivent la m\u00eame pente, traversent les m\u00eames contr\u00e9es, s'attaquent aux m\u00eames bastilles. Toujours il hait les d\u00e9sirs mortif\u00e8res, les mac\u00e9rations, les peines et les souffrances, les brutalit\u00e9s et les violences, les exploitations et l'ali\u00e9nation ; toujours il c\u00e9l\u00e8bre les plaisirs sensuels, sexuels, les \u00e9panouissements corporels, les \u00e9nergies radieuses, les jouissances et les jubilations de la chair. Les lettres, les billets, les dessins, les toiles, les entretiens qui ont franchi les murs de la prison exposent radicalement et absolument une r\u00e9elle th\u00e9orie anarchiste solaire.\n\nLa pens\u00e9e de l'artiste est franchement libertine, selon le sens originaire du terme au XVIIe si\u00e8cle : affranchie, sans souci des convenances sociales, \u00e9thiques, m\u00e9taphysiques, religieuses, dominantes et triomphantes. Ni Dieu, ni ma\u00eetre, sinon ni dieux, ni ma\u00eetres. Muehl reconna\u00eet pour seule autorit\u00e9 celle de l'\u00e9nergie libidinale, de la vitalit\u00e9 d\u00e9bordante des puissances en exc\u00e8s. L'\u0153uvre de prison radicalise cette option libertaire.\n\nL'incarc\u00e9ration prouve que l'Etat cr\u00e9e les d\u00e9viances qui justifient son existence pour, pr\u00e9tendument, les \u00e9radiquer. Il invente de toutes pi\u00e8ces cette justification, via son travail n\u00e9gateur. Son inexistence suffirait, car, sans lui, rien des d\u00e9chets qu'il produit n'encombrerait la vie commune. L'exploitation, la violence, l'ali\u00e9nation, pour les brutalit\u00e9s les plus voyantes, proc\u00e8dent de l'Etat, qui trouve ainsi mati\u00e8re \u00e0 l\u00e9gitimiter sa pr\u00e9sence : il est l\u00e0 pour les combattre. La d\u00e9shumanisation carc\u00e9rale pr\u00e9tend viser l'humanisation d'humains r\u00e9duits \u00e0 l'\u00e9tat animal. L'\u0153uvre de prison illustre ce paradoxe politique dans une crudit\u00e9 esth\u00e9tique qui n'\u00e9pargne rien du spectacle de la cruaut\u00e9 sociale.\n\nLes productions carc\u00e9rales s'affichent clairement ath\u00e9es, anticl\u00e9ricales, mat\u00e9rialistes et antichr\u00e9tiennes. Autant dire immanentes, libres, jubilatoires et dionysiennes. Le corps y prend toute la place. Le sacr\u00e9 y compte pour rien. Muehl s'\u00e9tonne, dans l'optique du rire nietzsch\u00e9en qu'il pratique en virtuose, que la farce chr\u00e9tienne ait pu durer si longtemps et que, vingt si\u00e8cles plus tard, on subisse encore et toujours les imp\u00e9ratifs mortif\u00e8res de la religion du crucifi\u00e9. La passion christique agit chez lui en embl\u00e8me sur lequel se fixent l'ironie, le d\u00e9tournement esth\u00e9tique et l'abord ludique.\n\nPour lutter contre les n\u00e9vroses sociales produites par le christianisme, le peintre attaque de plein fouet la Famille qu'il aborde \u2013 \u00e0 juste titre \u2013 comme le premier maillon de la cha\u00eene asc\u00e9tique. Contre le couple, le mariage, la fid\u00e9lit\u00e9, la monogamie, il avance la sexualit\u00e9 libre ob\u00e9issant aux seuls caprices de ceux qui y consentent. Si la paternit\u00e9 et la maternit\u00e9, en r\u00e9gime capitaliste, autorisent l'assise et le triomphe du patriarcat, en r\u00e9gime h\u00e9doniste, elles permettent une communaut\u00e9 \u00e9lective en devenir et en expansion.\n\nDe m\u00eame, lorsque la sexualit\u00e9 se soumet au r\u00e9gime de production marchand, on voit appara\u00eetre le sexisme, la pornographie, voire l'adult\u00e8re, pratiqu\u00e9 dans des conditions d'hypocrisie et de dissimulation qui entra\u00eenent la culpabilit\u00e9, ou de plus d\u00e9plorables cons\u00e9quences \u2013 dont la plupart du temps les enfants font les frais. Une libido libertaire renvoie chacun dans l'\u00e9panouissement individuel et singulier, dans l'autocr\u00e9ation de soi.\n\nD'o\u00f9 cette invite \u00e0 une sexualit\u00e9 ludique et d\u00e9barrass\u00e9e de tout complexe dont les \u0153uvres de prison t\u00e9moignent : jeu avec le sang, le sperme, l'urine, la salive, les f\u00e8ces, la bouche, le sexe, le ventre, les hanches, les seins, le phallus et leurs m\u00e9taphores fil\u00e9es, la lave des volcans, le feu des incendies, l'\u0153uf cher \u00e0 Georges Bataille, les mati\u00e8res associ\u00e9es aux performances de Hermann Nitsch \u2013 son comp\u00e8re actionniste.\n\nEn h\u00e9ritier du freudo-marxisme viennois, celui de Wilhelm Reich en l'occurrence, Otto Muehl d\u00e9veloppe l'id\u00e9e, ch\u00e8re au dernier Freud, que le malaise dans la civilisation provient de la gestion mortif\u00e8re des instincts. L\u00e0 o\u00f9 la soci\u00e9t\u00e9 se construit par captation et recyclage de l'\u00e9nergie provenant du renoncement des individus \u00e0 leur identit\u00e9, l'artiste propose la communaut\u00e9 fond\u00e9e sur le contrat h\u00e9doniste, r\u00e9vocable \u00e0 tout moment, et les affinit\u00e9s \u00e9lectives. De sorte que disparaissent les auxiliaires du malaise que sont le patriarcat, la propri\u00e9t\u00e9 priv\u00e9e, la famille, le colonialisme, et toutes les formes de suj\u00e9tion imagin\u00e9es et promues par le social.\n\nL'histoire s'\u00e9crit toujours avec le sang des victimes au nom desquelles on pr\u00e9tend combattre. La pens\u00e9e de Muehl est anhistorique, voire antihistorique. Dans sa prison tout autant que dans sa communaut\u00e9, il sait que la vie ne peut \u00eatre radicalement modifi\u00e9e par la r\u00e9volution. En revanche, il croit au devenir r\u00e9volutionnaire des individus. Tout sursaut subversif dans l'histoire devient tr\u00e8s vite une puissance sociale, r\u00e9cup\u00e9r\u00e9e par l'Etat, et d\u00e9virilis\u00e9e dans les meilleurs d\u00e9lais. D'o\u00f9 sa revendication nettement affirm\u00e9e d'une antinomie entre l'artiste et la soci\u00e9t\u00e9.\n\nLes lettres et les dessins, les entretiens et les toiles racontent tous \u00e0 leur mani\u00e8re les limites d'une communaut\u00e9 \u00e9largie sans bornes et sans souci des individus qui la composent. La prison lui a donn\u00e9 les moyens de r\u00e9fl\u00e9chir \u00e0 ce que fut l'exp\u00e9rience de Friedrichauf, ses qualit\u00e9s, mais aussi ses d\u00e9fauts, ses succ\u00e8s et ses \u00e9checs. La communaut\u00e9 a \u00e9chou\u00e9, pense-t-il, parce qu'elle finissait par ressembler \u00e0 un micro-Etat dans lequel agissaient des forces n\u00e9gatives \u2013 Guattari aurait parl\u00e9 de micro-fascismes consubstantiels \u00e0 l'exercice du pouvoir. L'incarc\u00e9ration a \u00e9t\u00e9 quelque temps l'occasion, pour son fondateur, de se faire l'historien de la communaut\u00e9. Ses conclusions ? L'impossibilit\u00e9 de la solitude et le danger de la prolif\u00e9ration des immensit\u00e9s collectives. Entre ces deux extr\u00e9mit\u00e9s \u00e0 lire comme des exc\u00e8s, il aspire \u00e0 une association n'exc\u00e9dant pas douze personnes.\n\nAvec cette cellule de base, contre l'Etat, la soci\u00e9t\u00e9 et la famille, l'artiste avance l'id\u00e9e d'une communaut\u00e9 int\u00e9grale de biens, de d\u00e9sirs et de plaisirs, de sexualit\u00e9 et de projets intellectuels. La n\u00e9gativit\u00e9 d'Otto Muehl vise une positivit\u00e9 revendiqu\u00e9e : inventer de nouvelles possibilit\u00e9s d'existence, cr\u00e9er de nouvelles fa\u00e7ons de vivre \u00e0 deux et \u00e0 plusieurs, agencer de nouvelles esth\u00e9tiques vitales dans le creuset d'une micro-soci\u00e9t\u00e9 h\u00e9doniste o\u00f9 tout serait mis en \u0153uvre pour emp\u00eacher l'\u00e9mergence et l'exercice d'une forme apparent\u00e9e \u00e0 un micro-Etat.\n\nSa pens\u00e9e de prison, radicale et sans dissimulation, s'illustre dans une iconographie violente. Couleurs, formes, dessin, sujets, traitement de l'information, tout trahit la seule divinit\u00e9 \u00e0 laquelle Otto Muehl avoue vouer un culte : la vitalit\u00e9. Ses lettres renvoient souvent \u00e0 cette \u00e9nergie primordiale et primitive. Il c\u00e9l\u00e8bre la grande sant\u00e9 nietzsch\u00e9enne, le rire, l'\u00e9normit\u00e9 et le disproportionn\u00e9, la libido et son expansion, la puissance des corps m\u00eal\u00e9s, enchev\u00eatr\u00e9s, p\u00e9n\u00e9tr\u00e9s, offerts, donn\u00e9s, abandonn\u00e9s, les vertus du cri.\n\nEn relation avec cet imp\u00e9ratif cat\u00e9gorique vitaliste, il propose une relecture audacieuse de certaines valeurs culturelles occidentales. La musique, par exemple, ne vaut pour lui que si elle offre au corps des occasions de jubiler. Autant dire qu'il associe la musique dite classique \u00e0 des logiques criminelles dans la mesure o\u00f9 elle propose un message cod\u00e9 et intellectuel en direction de l'esprit, de l'intelligence, alors qu'elle n\u00e9glige de s'adresser au corps, \u00e0 la chair, aux visc\u00e8res. Bach, Mozart, Schubert et Beethoven sont vou\u00e9s aux g\u00e9monies, coupables d'entretenir la s\u00e9paration entre le corps et l'\u00e2me, voire de servir les int\u00e9r\u00eats de la bourgeoisie, une fois par le biais de l'Etat, une autre par celui de l'Eglise.\n\nLa seconde Ecole de Vienne, avec Sch\u00f6nberg, Berg et Webern, re\u00e7oit pareillement sa vol\u00e9e de bois vert. D'autant que cette r\u00e9volution musicale n'a pas cess\u00e9, selon lui, dans le XXe si\u00e8cle, de se prolonger avec l'aide d'\u00e9mules acharn\u00e9s \u00e0 d\u00e9sincarner la musique, \u00e0 la r\u00e9duire \u00e0 une pure combinaison math\u00e9matique de sons aux antipodes de toute sensualit\u00e9. D'o\u00f9 son refus d'adh\u00e9rer \u00e0 cette branche de l'histoire de la musique pour lui pr\u00e9f\u00e9rer celle qui aboutit au tango ou au jazz, musiques populaires qui donnent au corps toute la place qui lui revient.\n\nEn mati\u00e8re de litt\u00e9rature, il aborde les longs romans, les cycles, les \u0153uvres qui exigent du temps de lecture \u2013 ironisant parfois et esp\u00e9rant qu'il restera assez longtemps emprisonn\u00e9 pour pouvoir achever la lecture int\u00e9grale de La Recherche du temps perdu. Il c\u00e9l\u00e8bre plus que tout l'\u0153uvre de James Joyce dont il appr\u00e9cie particuli\u00e8rement l'art de composer, de dessiner ou de peindre des situations. La narration propre \u00e0 l'\u00e9crivain irlandais lui fait dire qu'en lisant Ulysse, il a saisi qu'il avait affaire directement et nettement a un peintre excellant dans l'art de d\u00e9crire et composer.\n\nDans ses choix, dans ses revendications litt\u00e9raires, Otto Muehl balaie d'un revers de la main toute l'antilitt\u00e9rature d'aujourd'hui, maigre, \u00e9tique, satisfaite de son incapacit\u00e9 au style et fi\u00e8re de fabriquer des livres avec un nombre de signes tout juste suffisant pour faire une nouvelle. La litt\u00e9rature qu'il appr\u00e9cie est dense, forte, puissante, travers\u00e9e d'\u00e9clairs et de longueurs utiles, fouill\u00e9e par la vie et habit\u00e9e par le r\u00e9el, toujours pr\u00e9sent, jamais cong\u00e9di\u00e9, ni c\u00e9l\u00e9br\u00e9 sous ses formes banales et neutres, blanches et pauvres. Ce que veut Muehl, c'est une prose et une narration pleines de sang, de s\u00e8ve et d'\u00e9nergie \u2013 le contraire des tisanes ti\u00e8des contemporaines.\n\nSur la question de la peinture, l'artiste fustige le lignage esth\u00e9tique qui va de Paul C\u00e9zanne \u00e0 Marcel Duchamp \u2013 et ses suiveurs \u2013, du conceptuel au minimal via le pop art. Cette voie veut la forme pour elle-m\u00eame et c\u00e9l\u00e8bre l'agencement gratuit, le jeu de combinaison formel, loin de tout effet sensuel, voire libidinal que Muehl, en revanche, rep\u00e8re chez ceux qu'il c\u00e9l\u00e8bre : Michel-Ange, Van Gogh et Picasso, tous ma\u00eetres du corps, du muscle, du visage, du portrait, de l'\u00e9nergie, de la force, de la puissance, en un mot de l'incarnation.\n\nVan Gogh lui fournit un mod\u00e8le appropri\u00e9 pendant ses sept ann\u00e9es de prison. Les sujets de la folie, de la solitude et de l'enfermement, de l'errance illumin\u00e9e et de la couleur jaculatoire, du trait travaill\u00e9 par une force proc\u00e9dant de l'inconscient, du ventre, de la douleur int\u00e9rieure, voil\u00e0 qui le ravit, au sens premier. Par ailleurs, il pense Arles et Saint-R\u00e9my-de-Provence comme des lieux embl\u00e9matiques de son destin carc\u00e9ral. Enfin, il installe Van Gogh en pr\u00e9curseur radical de l'expressionnisme qu'il revendique pour la chair jamais cong\u00e9di\u00e9e, le sujet et le th\u00e8me toujours pr\u00e9sents.\n\nRien ne f\u00e2che plus Otto Muehl que l'abstraction, cette occasion esth\u00e9tique de d\u00e9r\u00e9aliser le monde, de le vider de toute consistance, de toute chair et de toute vie. Chez Duchamp, dont il ne conteste pas l'importance th\u00e9orique et philosophique pour le si\u00e8cle, voire au-del\u00e0, il voit pourtant le fossoyeur d'un certain type d'esth\u00e9tique et le g\u00e9n\u00e9alogiste de l'art confondu et r\u00e9duit \u00e0 la publicit\u00e9, au slogan, au march\u00e9. Le pop art fournirait, \u00e0 qui en douterait, la preuve manifeste : la soci\u00e9t\u00e9 de consommation, le capitalisme, la civilisation lib\u00e9rale font leurs d\u00e9lices de Warhol et de ses \u00e9pigones.\n\nL'expressionnisme de Muehl, son go\u00fbt pour la chair exacerb\u00e9e, s'exprime \u00e0 travers des moyens limit\u00e9s par son statut de prisonnier. R\u00e9duit aux 8 m2 de sa cellule, il peint de petits formats (50x35) dans lesquels il concentre une \u00e9nergie maximale. Il recourt \u00e0 la craie grasse, mais aussi \u00e0 tous les mat\u00e9riaux et d\u00e9chets qu'il peut recycler dans son \u00e9tat : la cendre de ses cigares, l'huile de ses bo\u00eetes de sardines, la poussi\u00e8re qui tra\u00eene sous sa paillasse. Il fait ses m\u00e9langes pauvres, comme dans la plus grande tradition de la peinture de chevalet \u00e0 travers les si\u00e8cles. Puis, sur la toile travaill\u00e9e cinq heures d'affil\u00e9e, les couleurs explosent comme des soleils au milieu de la nuit.\n\nLoin du march\u00e9 et de la d\u00e9coration, de la publicit\u00e9 et des convenances, Otto Muehl peint sa r\u00e9volte et son insoumission. Il s'attaque \u00e0 des id\u00e9es et veut un art de combat. On salit son corps, on m\u00e9prise sa chair ? Des n\u00e9gligences m\u00e9dicales avou\u00e9es \u00e0 l'unique douche hebdomadaire, des repas d\u00e9risoires \u00e0 la seule heure de d\u00e9tente en ext\u00e9rieur, le corps de l'artiste est sap\u00e9, min\u00e9, comme on truffe une falaise de dynamite en attendant qu'elle explose, s'effondre et s'affaisse, r\u00e9duite \u00e0 n\u00e9ant, pulv\u00e9ris\u00e9e.\n\nMuehl r\u00e9siste, tient et concentre toute sa force dans ses \u0153uvres de prison. Jamais sublimation n'a mieux m\u00e9rit\u00e9 son nom, jamais processus n'a \u00e9t\u00e9 aussi pertinent pour exprimer la puissance, la force et la g\u00e9n\u00e9alogie d'une \u0153uvre, sa forme et son contenu, sa charge et son \u00e9motion. Toujours ennemi des lois, encore, et plus que jamais rebelle et r\u00e9volt\u00e9, apr\u00e8s sept ann\u00e9es de prison, \u00e0 l'heure o\u00f9 la reptation de ceux qui sont dehors semble une seconde nature, il sort affaibli, malade, entam\u00e9 physiquement, certes, mais debout. Dans cette verticalit\u00e9, il y a une ind\u00e9niable le\u00e7on. Qui l'entendra encore ? Qui a les oreilles pour cela ?\n28\n\nANATOMIE D'UNE MACHINE D\u00c9SIRANTE\n\nFaut-il aller chercher ses secrets dans le ventre et l'\u00e2me du libertin en risquant de proc\u00e9der comme d'aucuns qui trucid\u00e8rent un jour la poule aux \u0153ufs d'or? Peut-on, sans risque ni danger de voir la montagne accoucher d'une souris, interroger le corps et l'enfance d'un homme \u00e0 femmes pour t\u00e2cher de trouver et comprendre ce qui fait cet \u00e9trange tropisme, cette singuli\u00e8re machine qu'est un s\u00e9ducteur ? Je ne sais. D'autant que les psychanalyses freudiennes ou existentielles ne permettent jamais d'obtenir autre chose qu'une approche, une fr\u00e9quentation des franges et des bords l\u00e0 o\u00f9 il faudrait viser le noyau dur et la forteresse.\n\nJe me suis toujours demand\u00e9 pourquoi Federico Fellini montre peu d'am\u00e9nit\u00e9 \u2013je ne parle ni de tendresse, ni d'affection, a fortiori d'amour \u2013 pour Giacomo Casanova auquel il a pourtant consacr\u00e9 un long film baroque et triste. Nombre de sc\u00e8nes montrent le libertin en m\u00e9canique d\u00e9sincarn\u00e9e, en machine de guerre sexuelle, en pantin d\u00e9sarticul\u00e9 excit\u00e9 sur le corps des femmes comme un m\u00e9canisme froid ob\u00e9issant \u00e0 un moteur d\u00e9bile mais in\u00e9puisable. Casanova, premier moteur immobile, cause incaus\u00e9e de la galaxie des libertins, s\u00e9ducteur embl\u00e9matique ? A voir...\n\nJe m'\u00e9tonne de l'animosit\u00e9 du cin\u00e9aste pour le h\u00e9ros v\u00e9nitien. Car, apr\u00e8s tout, il ne d\u00e9pare pas dans son capharna\u00fcm mythique de nains grima\u00e7ants, de g\u00e9ants emprunt\u00e9s, de femmes aux mamelles hom\u00e9riques, de figures grotesques, de pr\u00e9lats libidineux, d'antih\u00e9ros monstrueux, de putains magnifiques, de V\u00e9nitiens myst\u00e9rieux ou de Romains faussement h\u00e9ro\u00efques. Casanova stigmatise la n\u00e9cessit\u00e9 sexuelle faite loi, la machine d\u00e9sirante port\u00e9e \u00e0 son point d'incandescence, v\u00e9cue jusqu'au paroxysme. C'est aussi une figure cardinale de la destin\u00e9e tragique et malheureuse des hommes qu'aucune femme n'arr\u00eate ou ne retient.\n\nQui donc serait assez sot pour ha\u00efr le poisson qui remonte le courant du fleuve pour aller pondre ? Qui d\u00e9testerait le chant des rossignols ou leur poitrine tach\u00e9e de rouge? Qui fustigerait la cruaut\u00e9 des oiseaux de proie et la candeur des ramiers ? Quiconque ob\u00e9it \u00e0 la n\u00e9cessit\u00e9 ne saurait \u00eatre tenu pour responsable de son \u00eatre, ni de son devenir. Le libertin d\u00e9ploie ses strat\u00e9gies de s\u00e9duction comme le ver \u00e0 soie file, ni l'un ni l'autre ne choisissent cette destin\u00e9e et ces logiques auxquelles ils sont assujettis. D\u00e9tester Casanova pour ses frasques, c'est se rebeller contre l'ordre des choses satur\u00e9 de victimes, du c\u00f4t\u00e9 des femmes s\u00e9duites aussi bien que du s\u00e9ducteur imp\u00e9nitent.\n\nJe trouve d'autant \u00e9trange ce refus de Casanova que, pour la plupart, il s'accompagne d'un coefficient de sympathie non dissimul\u00e9 pour Don Juan. Alors que le premier rel\u00e8ve d'une r\u00e9alit\u00e9 historique, qu'il figure authentiquement une \u00e9paisseur biographique, avec ses limites et ses puissances humaines, sa nature et ses trajets essentiellement immanents, Don Juan triomphe en cr\u00e9ature th\u00e9orique, sous forme d'id\u00e9al de la raison ou d'arch\u00e9type constitu\u00e9 par les besoins de l'histoire. Il se pavane en mythe stratifi\u00e9 par les histoires et les g\u00e9ographies de l'Occident. Faudrait-il conclure que la complicit\u00e9 et la fascination des hommes vont plus facilement \u00e0 l'id\u00e9e pure qu'aux traces concr\u00e8tes, troubles, au fantasme int\u00e9griste qu' \u00e0 la r\u00e9alit\u00e9 ondoyante ?\n\nPour ma part, Casanova me pla\u00eet parce qu'il offre les traits et les d\u00e9tails d'une machine d\u00e9sirante telle qu'elle appara\u00eet dans Mille Plateaux, l'\u0153uvre de Deleuze et Guattari, entre machines de guerre et devenir-animal. Casanova ou le devenir-libertin, Casanova ou la machine d\u00e9sirante caract\u00e9ristique, voil\u00e0 qui fournit au philosophe, au s\u00e9ducteur, ou au quidam, d'in\u00e9puisables occasions d'\u00e9mulation pour comprendre, au lieu de pr\u00e9textes \u00e9ternels et infinis pour juger.\n\nLe V\u00e9nitien, touch\u00e9 par le contraire d'une gr\u00e2ce \u2013 disons par une gr\u00e2ce n\u00e9gative \u2013 vit sous le signe et l'empire de la mal\u00e9diction, car tout libertin traverse son existence en locataire d'un \u00eatre qui pourrait bien n'\u00eatre pas le sien, en usurpateur d'une figure qui joue sur sc\u00e8ne une pi\u00e8ce dont il conna\u00eet par c\u0153ur le d\u00e9tail et le d\u00e9roulement, et contre laquelle, pourtant, il ne peut rien. Pas plus qu'un quartz n'opte pour les angles et les ar\u00eates qui le d\u00e9finissent, le libertin ne choisit le libertinage. Trop de douleurs, d'ob\u00e9issances et de souffrances caract\u00e9risent cet \u00e9tat. Si d'aventure il pouvait choisir quelque chose, nul doute qu'il opterait pour de moins \u00e9prouvantes odyss\u00e9es et de moins tragiques destin\u00e9es. La nuit venue, sans r\u00e9pit, le s\u00e9ducteur d\u00e9plie sa cape et glisse son ombre sous les balcons \u00e0 d\u00e9faut d'\u00eatre amoureux.\n\nLes M\u00e9moires de Casanova m'\u00e9tonnent toujours dans leur attaque, notamment par cette pr\u00e9face sertissant avec insistance une revendication d'autoportrait intellectuel sinc\u00e8re qui expose une mise en garde trop nette et pr\u00e9cise pour \u00eatre honn\u00eate. De sorte que si l'on accorde cr\u00e9dit aux premi\u00e8res pages de cet immense livre, il faut croire son auteur quand il se pr\u00e9tend libre, monoth\u00e9iste, chr\u00e9tien et croyant en la Providence d'un Dieu immat\u00e9riel. J'aspire \u00e0 penser tr\u00e8s exactement le contraire, et les trois mille pages du texte qui suivent vont constamment en ce sens.\n\nCasanova me semble plut\u00f4t ath\u00e9e, ou sinon d'un d\u00e9isme qui d\u00e9fie toute concurrence et laisse tranquilles les cr\u00e9atures abandonn\u00e9es \u00e0 leur destin. Il me para\u00eet d\u00e9vou\u00e9 corps et \u00e2me \u00e0 un h\u00e9donisme aux antipodes des enseignements du nazar\u00e9en. Et, plus important, il m'appara\u00eet qu'au fur et \u00e0 mesure de son existence, il ob\u00e9it plus qu'il n'est libre. En permanence, dans l'ensemble et dans le d\u00e9tail, il se plie \u00e0 une n\u00e9cessit\u00e9 qui le d\u00e9borde, le d\u00e9passe et le conduit. Son essence et son devenir libertin rel\u00e8vent d'une m\u00e9canique contre laquelle il aurait mauvaise gr\u00e2ce \u00e0 se rebeller, car il n'y peut pas grand-chose...\n\nEn ouverture aux M\u00e9moires, donc, on aborde une question m\u00e9taphysique essentielle. A l'heure des bilans, des fins d'existence qui rendent les comptes possibles, le m\u00e9morialiste pose le probl\u00e8me des rapports entre n\u00e9cessit\u00e9 et libert\u00e9. Une existence est-elle fabriqu\u00e9e par un individu, ou l'inverse? A d\u00e9faut d'une r\u00e9ponse normande, le libertin tranche : il n'y a pas de n\u00e9cessit\u00e9, pas de force du destin. Rien qui ressemble au fatum des Anciens ou \u00e0 une puissance entra\u00eenant les individus dans la suj\u00e9tion d'une existence d\u00e9j\u00e0 \u00e9crite. Contre les th\u00e8ses sto\u00efciennes, qui r\u00e9apparaissent dans les panth\u00e9ismes de Spinoza ou de Schopenhauer, dans les m\u00e9taphysiques mat\u00e9rialistes ou nietzsch\u00e9ennes, Casanova revendique une adh\u00e9sion franche et massive \u00e0 l'hypoth\u00e8se catholique de la Providence. Il sait, il sent que l'affirmation philosophique de la n\u00e9cessit\u00e9 conduit immanquablement \u00e0 l'ath\u00e9isme, auquel il se d\u00e9fend d'adh\u00e9rer, par prudence, me semble-t-il. Son souvenir de la prison des Plombs devait encore le cuire...\n\nCasanova de confesser n'avoir jamais demand\u00e9 \u00e0 Dieu une requ\u00eate qui n'ait \u00e9t\u00e9 entendue et satisfaite. D'o\u00f9 sa d\u00e9fense de la pri\u00e8re, de la relation avec la divine Providence et la croyance en un destin de chacun voulu par la seule puissance divine. Ce qui advient, Dieu le veut. Si l'on suit la th\u00e8se du V\u00e9nitien un jour tonsur\u00e9, il faut imaginer le libertinage d\u00e9coulant d'une volont\u00e9 de Dieu \u2013 voil\u00e0, au demeurant, qui ferait une divinit\u00e9 sympathique \u00e0 laquelle je sacrifierais sans complexe et avec un v\u00e9ritable plaisir.\n\nOr je crains les professions de foi claironn\u00e9es avec force d\u00e9monstrations, si elles ouvrent un livre qu'on peut alors, quand on travaille au bureau de l'Inquisition, ne pas feuilleter plus avant, convaincu de la sorte par sa moralit\u00e9 \u2013 m\u00eame si le contenu, \u00e0 chaque page qui suit, d\u00e9ment l'avertissement. D'autant que les autorit\u00e9s du XVIIIe si\u00e8cle ne plaisantent pas avec les penseurs aux vell\u00e9it\u00e9s d'ath\u00e9isme, mais plus encore de mat\u00e9rialisme. Galil\u00e9e en a fait l'exp\u00e9rience et d'autres, plus tard, dont certains contemporains du V\u00e9nitien. Je suppose la prudence du libertin plus grande encore que sa foi. Son talent pour les discours opportunistes, sa capacit\u00e9 \u00e0 adapter le propos au moment pour pr\u00e9server sa libert\u00e9 d'aller et venir \u2013 son bien le plus cher \u2013 me paraissent trop souvent d'actualit\u00e9 chez lui pour \u00eatre ici absents.\n\nCasanova raconte mieux qu'il ne pense ou d\u00e9montre. Il brille dans la narration, l'histoire rapport\u00e9e, enjoliv\u00e9e, il \u00e9crit une langue fran\u00e7aise exquise, mais sa pens\u00e9e s'emm\u00eale et son propos, sa d\u00e9monstration sacrifient toute pr\u00e9cision. Quand il recourt aux notions de libert\u00e9, de n\u00e9cessit\u00e9, de caract\u00e8re ou de temp\u00e9rament, c'est bien souvent pour en faire un usage tout aussi libre et d\u00e9li\u00e9 que sa libido. Parfois, m\u00eame, l'un semble utilis\u00e9 dans l'acception de l'autre.\n\nLa dissertation philosophique n'est pas le fort du s\u00e9ducteur. Sa pr\u00e9face, ses lignes ap\u00e9ritives, semblent plut\u00f4t un masque, un loup comme on en voit dans les th\u00e9\u00e2tres v\u00e9nitiens et les peintures de Pietro Longhi. Il s'agit moins de th\u00e9orie que de dissimulation, moins d'analyse et de propos m\u00e9taphysique que d'art baroque de travestir, de m\u00e9tamorphoser, de dissimuler, de cacher l'essentiel \u2013 en l'occurrence l'existence peu catholique d'un libertin soumis \u00e0 la n\u00e9cessit\u00e9 d'un corps imp\u00e9rieux.\n\nEn sophiste convaincu, en amateur de paradoxes \u2013 ce qui \u00e9pate toujours le bourgeois de tous les temps \u2013 , il fait de la libert\u00e9 une r\u00e9alit\u00e9 qui dure tant qu'on y croit, une certitude fond\u00e9e autant qu'on y sacrifie. Il suffirait de n'y plus croire pour que cette v\u00e9rit\u00e9 essentielle passe de mode et disparaisse. Lisons Giacomo Casanova : \u00abL'homme est libre, mais il cesse de l'\u00eatre s'il ne croit pas \u00e0 sa libert\u00e9 ; et plus il suppose de force au destin, plus il se prive de celle que Dieu lui a donn\u00e9e en le douant de raison. \u00bb Quelle \u00e9trange objet \u00e0 l'existence soumise au cr\u00e9dit de sa pr\u00e9sence...\n\nSit\u00f4t la foi refus\u00e9e, disparu l'objet... Que serait une libert\u00e9 dont l'\u00eatre se limiterait au seul acte de foi d'un hypoth\u00e9tique usager? Soit elle est, dans l'absolu, soit elle n'est pas. Mais comment pourrait-elle \u00eatre, r\u00e9duite au pur acte de foi, sur le principe du postulat de la raison pure pratique cher aux pens\u00e9es d\u00e9sireuses d'une adh\u00e9sion discr\u00e8te et hypocrite aux principes chr\u00e9tiens. Casanova mobilise son talent pour la rh\u00e9torique, la dialectique, la sophistique, mais ne convainc pas sur l'existence pure d'une libert\u00e9 pure. Croyez donc \u00e0 la puissance de votre libert\u00e9, et vous voil\u00e0 libre ? Ce serait donc la fa\u00e7on libertine de r\u00e9soudre la troisi\u00e8me antinomie de la raison pure kantienne ? Malgr\u00e9 les M\u00e9moires du chevalier, la d\u00e9monstration reste \u00e0 faire...\n\nJe souscris d'autant moins \u00e0 la bonne foi du m\u00e9morialiste que tr\u00e8s vite, dans le corps de la pr\u00e9face, il recule sur l'\u00e9vidence de cette libert\u00e9. D\u00e9j\u00e0 fragile parce que seulement soumise au bon vouloir de la foi du raisonneur, elle trouve ses limites et ses bornes d\u00e8s l'apparition d'une passion. Casanova l'exp\u00e9rimente dans sa chair, il le consigne sur le papier : un homme soumis \u00e0 une passion perd toute libert\u00e9. Il devient esclave et se laisse entra\u00eener \u00e0 des actions qu'il ne veut ni ne d\u00e9cide. Pourtant, rien ne para\u00eet plus \u00e9vident aux yeux du libertin de carri\u00e8re que cette sagesse : les hommes ne vivent que de passions, sans cesse soumis \u00e0 elles, pieds et poings li\u00e9s.\n\nDescartes pointe six passions fondamentales et propose une path\u00e9tique organis\u00e9e selon l'ordre de leur composition et de leur agencement. A partir de l'admiration, de l'amour et de la haine, du d\u00e9sir, de la joie et de la tristesse, il retrouve des chatouillements dans le cerveau, de la fixit\u00e9 musculaire, de l'impulsion ou de l'inhibition de glande pin\u00e9ale, du gonflement ou r\u00e9tr\u00e9cissement des poumons, de la jouissance et de la langueur, des agitations et des \u00e9motions, des surprises de l'\u00e2me et de l'estime du possible, du sang \u00e9paissi ou rar\u00e9fi\u00e9. Ce qui est beaucoup, convenons-en, et t\u00e9moigne en faveur d'un monde path\u00e9tique g\u00e9n\u00e9ralis\u00e9 auquel croient tous ceux qui n'ont pas l'heur d'\u00eatre des chr\u00e9tiens forcen\u00e9s.\n\nEnfin, Descartes aidant, voil\u00e0 assez de troubles dans un corps pour savoir que tout est passion, m\u00eame la raison, et qu'on n'\u00e9chappe pas \u00e0 cet ordre des choses. Casanova n'ignore rien, en son temps, des avanc\u00e9es du cart\u00e9sianisme europ\u00e9en. Il sait la passion fournissant la grande raison, et toute sa vie t\u00e9moigne en ce sens. D'autant qu'il affirme sans se dissimuler : \u00able seul syst\u00e8me que j'aie eu, si toutefois c'en est un, fut celui de me laisser aller au gr\u00e9 du vent qui me poussait. \u00bb Avant de pr\u00e9ciser, un peu plus loin : \u00abj'ai \u00e9t\u00e9 toute ma vie la victime de mes sens. \u00bb Le masque tombe, on ne croit plus gu\u00e8re que le libertin ait sacrifi\u00e9 th\u00e9oriquement \u00e0 l'existence de la libert\u00e9 tant sa vie fut pratiquement d\u00e9monstration de l'empire de la n\u00e9cessit\u00e9.\n\nPour aborder les contr\u00e9es en brouilleur de cartes, en joueur pipart les d\u00e9s, le m\u00e9morialiste entreprend de distinguer le temp\u00e9rament du caract\u00e8re. Distinction oiseuse qui lui permet de faire du premier un invariant inn\u00e9 contre lequel on ne peut rien, puis du second un trait modulable selon l'\u00e9ducation et s'inscrivant tr\u00e8s rapidement sur le visage de son heureux propri\u00e9taire. D'o\u00f9 un savant m\u00e9lange de n\u00e9cessit\u00e9 physiologique et de libert\u00e9 m\u00e9taphysique, l'ensemble autorisant la lecture d'une biographie comme une perp\u00e9tuelle tentative de compromis entre les deux puissances.\n\nLe temp\u00e9rament bilieux, sanguin, flegmatique ou col\u00e9reux \u2013 on ne quitte pas l'antique \u00e9tiologie hippocratique \u2013 produit des indi vidus enti\u00e8rement soumis \u00e0 son registre. Casanova dit avoir par couru tout le spectre des temp\u00e9raments pendant ses ann\u00e9es de foi mation, d'enfance et de jeunesse, pour finalement s'arr\u00eater au sanguin auquel il dit devoir la tyrannie de sa puissance g\u00e9n\u00e9sique. Il confesse avoir plus ob\u00e9i \u00e0 \u00abl'impulsion du sentiment\u00bb qu'\u00e0 \u00abl'effet de [s] es r\u00e9flexions \u00bb. Je veux bien le croire, mais il faut alors convenir que sa th\u00e9orie de la libert\u00e9 tient de moins en moins au fur et \u00e0 mesure de l'avanc\u00e9e dans la lecture de la seule pr\u00e9face...\n\nD'o\u00f9, dans la machine du libertin, et selon ses propres cat\u00e9gories, une domination du temp\u00e9rament au d\u00e9triment du caract\u00e8re. Mais c'est pour le s\u00e9ducteur une occasion d'affirmer aussit\u00f4t sa vie tout enti\u00e8re sous la coupe du caract\u00e8re. C'est \u00e0 n'y rien comprendre. A moins qu'on fasse son deuil du s\u00e9rieux de l'auteur en mati\u00e8re de recours au concept et d'usage des notions. Ce \u00e0 quoi je tends... Ainsi quand le libertin \u00e9crit : \u00abj'aime \u00e0 me reconna\u00eetre toujours pour la cause principale du bien ou du mal qui m'arrive \u00bb, il faut presque syst\u00e9matiquement conclure qu'il aimerait bien, certes, mais qu'il ne le peut, car il doit subir sans broncher la loi de la n\u00e9cessit\u00e9, la d\u00e9b\u00e2cle du caract\u00e8re sous l'empire du temp\u00e9rament.\n\nEn lecteur dilettante, \u00e9clair\u00e9 sur ce qui circulait sous le manteau pendant ce XVIIIe si\u00e8cle, Casanova sait que Spinoza fonctionne en ennemi d\u00e9clar\u00e9 des chr\u00e9tiens et des spiritualistes de tout poil. Aussi, pour donner des gages aux pouvoirs qu'il n'oublie jamais de m\u00e9nager, il fustige l'auteur de l'Ethique en lui d\u00e9cernant un brevet de mauvaise conduite de ses affaires intellectuelles. Pourtant, \u00e0 lire attentivement une belle lettre que le philosophe hollandais destine \u00e0 G.H. Schuller \u00e0 la fin de l'ann\u00e9e 1674, le libertin aurait pris une excellente le\u00e7on de philosophie tragique, car il aurait saisi les raisons pour lesquelles les hommes se croient libres alors qu'ils se contentent d'ob\u00e9ir \u00e0 la n\u00e9cessit\u00e9.\n\nJ'aime cette lettre par-dessus tout, car elle exprime en une \u00e9conomie de moyens redoutable la profession de foi de tout penseur convaincu de la r\u00e9duction d'une existence \u00e0 la somme des traces laiss\u00e9es par les manifestations de la n\u00e9cessit\u00e9. Spinoza recourt \u00e0 la m\u00e9taphore d'une pierre jet\u00e9e en l'air. Ob\u00e9issant \u00e0 la loi de la chute des corps et \u00e0 la gravitation universelle, qui r\u00e9gissent l'ensemble des solides sur la plan\u00e8te, elle grimpe, ralentit, culmine, amorce une descente, puis chute, le tout, bien \u00e9videmment, en vertu de la pure et simple soumission \u00e0 la n\u00e9cessit\u00e9 physique.\n\nDotez-la, pr\u00e9cise le philosophe hollandais, de conscience et de langage, elle vous dira avoir choisi l'ensemble des op\u00e9rations auxquelles, pourtant, elle est tragiquement soumise. Elle ne veut rien, car elle ne peut rien vouloir. Il en va de m\u00eame pour Casanova et l'esp\u00e8ce humaine dans sa totalit\u00e9 : \u00ab Telle est cette libert\u00e9 humaine, \u00e9crit Spinoza \u00e0 Schuller, que tous les hommes se vantent d'avoir et qui consiste en cela seul que les hommes sont conscients de leurs d\u00e9sirs et ignorants des causes qui les d\u00e9terminent. \u00bb Radicale, forte, \u00e9mouvante, impressionnante et sans appel, la pens\u00e9e propos\u00e9e dans l'Ethique fonde toute philosophie tragique digne de ce nom.\n\nO\u00f9 l'on d\u00e9couvre le moteur de cette machine d\u00e9sirante : l'inscience en mati\u00e8re de motifs, la m\u00e9connaissance des raisons profondes qui nous font \u00eatre tels que nous sommes sans que nous n'en sachions jamais rien. Ou si peu. La libert\u00e9 m\u00e9taphysique est une illusion, seule triomphe la n\u00e9cessit\u00e9. Pour l'essentiel, une existence rel\u00e8ve de forces aveugles, de hasards, de rencontres opportunes, ou non, de croisements inopin\u00e9s, et de toute autre combinaison en rapport avec des instants propices.\n\nDans ce th\u00e9\u00e2tre d'ombres o\u00f9 se d\u00e9sarticule chaque \u00eatre, on ignore la g\u00e9n\u00e9alogie, car elle co\u00efncide avec les ann\u00e9es de la prime enfance, de la jeunesse et de l'adolescence qui supposent alors l'innocence et l'absence de moyens intellectuels efficaces. L'\u00e2me des enfants est blanche avant l'inscription sur elle du texte \u00e9crit par des mains aveugles et peu d\u00e9licates. Casanova devint libertin pour ne pas p\u00e9rir d'avoir \u00e9t\u00e9 abandonn\u00e9 \u00e0 lui-m\u00eame, comme tous ceux qu'anime cette passion douloureuse de la s\u00e9duction. Pour trouver une compagnie qui toujours lui fera d\u00e9faut. Pour n'avoir pas \u00e0 \u00eatre cong\u00e9di\u00e9 une autre fois, il ne s'attarde pas ; afin de ne plus jamais \u00eatre quitt\u00e9, il quitte le premier.\n\n\u00ab Cultiver le plaisir des sens fut toujours ma principale affaire : je n'en eus jamais de plus importante \u00bb, \u00e9crit le s\u00e9ducteur devenu vieux. Et l'on retrouve la pr\u00e9dominance du temp\u00e9rament sanguin et le peu de puissance du caract\u00e8re contre la pr\u00e9d\u00e9termination physiologique. Mais peut-on imaginer une g\u00e9n\u00e9alogie de ce temp\u00e9rament ? De quoi proc\u00e8de-t-il ? Rel\u00e8ve-t-il purement et simplement d'un corps, d'une seule physique de la chair? Dans le langage ancien et dualiste, on pourrait se demander s'il est inn\u00e9 ou acquis, s'il se transmet ou s'acquiert.\n\nMon hypoth\u00e8se est que l'enfance, les premi\u00e8res ann\u00e9es et les exp\u00e9riences originaires le constituent, le fabriquent, le structurent. Loin des d\u00e9finitions anciennes qui d\u00e9coulent de la m\u00e9decine hippocratique, tout aussi loin des antiquit\u00e9s de la caract\u00e9rologie, le temp\u00e9rament peut se d\u00e9finir comme le noyau dur de l'identit\u00e9, le noeud serr\u00e9 autour duquel un \u00eatre se cristallise, se durcit, s'agence. Il constitue un bloc impossible \u00e0 fracasser et \u00e0 p\u00e9n\u00e9trer. Ses lois restent inconnues et on ne le conna\u00eet que par les effets induits. Le temp\u00e9rament demeure quand tout dans une vie a disparu, il fournit la couleur dominante d'une individualit\u00e9 \u00e0 partir de quoi s'irisent de l\u00e9gers cama\u00efeux.\n\nLe temp\u00e9rament libertin peut m\u00eame proc\u00e9der du ventre maternel, l\u00e0 o\u00f9 les \u00e9thologues racontent l'impr\u00e9gnation, les traumatismes, les signes qui d\u00e9j\u00e0 produisent du sens et des messages d\u00e9cod\u00e9s par le syst\u00e8me nerveux primal. Vraisemblablement, \u00e0 la naissance, le corps de l'enfant supporte des charges, positives ou n\u00e9gatives, inductrices des premiers tropismes. Mais tout semble alors confus, impr\u00e9cis, vague. Car l'\u00e2me arrive neutre et vide de toute information dans un \u00e9tat qui dure peu, puis tout ce qui fait sens se m\u00e9morise dans la chair, le ventre, le corps, et ce dans les premiers temps, sauvages et primitifs.\n\nLe fantasme d'abandon qui, \u00e0 mon avis, induit toutes les virtualit\u00e9s libertines depuis toujours, peut tr\u00e8s bien se trouver activ\u00e9 dans les premiers mois d'une existence, a fortiori dans les premi\u00e8res semaines d'une vie intra-ut\u00e9rine. La qu\u00eate du s\u00e9ducteur vise le comble d'un manque violent, l'acquisition d'une certitude qui fait d\u00e9faut : il s'agit moins de savoir la possibilit\u00e9 de plaire et de s\u00e9duire que d'apprendre de la bouche ou du corps d'une femme, de ses yeux m\u00eames, qu'on peut exister pour elle.\n\nLe regard f\u00e9minin structure l'existence et l'identit\u00e9 d'un homme autrement et mieux que n'importe quel autre regard masculin. Si celui-l\u00e0 a fait d\u00e9faut, hier, dans les heures o\u00f9 se constitue un temp\u00e9rament, nul doute qu'une vie ne suffit pas pour obtenir des preuves. Le libertinage co\u00efncide avec une volont\u00e9 d'exister dans le regard f\u00e9minin qui manque un jour. La m\u00e9canique du s\u00e9ducteur est mue par ce moteur imp\u00e9rieux contre lequel il ne peut rien. Casanova en t\u00e9moigne avec une ardeur jamais d\u00e9mentie.\n\nLa psychanalyse officielle, qui r\u00e9duit tout au trio \u0153dipien, veut voir dans le s\u00e9ducteur un homme qui collectionne et accumule les femmes pour la raison qu'il rechercherait inconsciemment, dans ces num\u00e9ros successifs, la m\u00e8re, seule et unique cr\u00e9ature inaccessible. De la m\u00eame mani\u00e8re, en trompant les maris, en bafouant les adversaires, les concurrents, les hommes de ces femmes l\u00e9galement poss\u00e9d\u00e9es, le libertin s'opposerait au seul ennemi immortel et invincible, le p\u00e8re, bien s\u00fbr. Voil\u00e0 pourquoi, d\u00e9sirant les faveurs de la m\u00e8re en m\u00eame temps que la mort du p\u00e8re, le petit \u0152dipe devenu grand fait un excellent Don Juan \u2013 dont, au passage, il convient de dire qu'on ignore toujours \u00e0 quoi peuvent bien ressembler ses parents, sa m\u00e8re surtout.\n\nLes marxistes, qui eurent trop \u00e0 faire, et plus s\u00e9rieux, n'ont jamais, me semble-t-il, propos\u00e9 d'hypoth\u00e8se pour lire et d\u00e9chiffrer le libertinage. Roger Vailland lui-m\u00eame s'attaquant au probl\u00e8me dans Le Regard froid cesse soudain, et avec bonheur, de recourir aux cat\u00e9gories du mat\u00e9rialisme dialectique... Mais nul doute que sur cette question de sp\u00e9cieux doctrinaires verraient la n\u00e9gativit\u00e9 en acte d'une aristocratie d\u00e9class\u00e9e prenant sa revanche sur la bourgeoisie arrogante, sinon le triomphe des vertus f\u00e9odales exerc\u00e9es contre le prol\u00e9tariat et les classes laborieuses. Ou quelque chose d'approchant qui expliquerait le s\u00e9ducteur par ses conqu\u00eates titr\u00e9es et ses amours ancillaires.\n\nNul doute que des raisons freudiennes tout autant que des imp\u00e9ratifs de classes sociales entrent en jeu dans la qu\u00eate du libertin. Dans les deux cas, il ob\u00e9it \u00e0 des n\u00e9cessit\u00e9s qui le d\u00e9passent, il est le jouet de son inconscient tout autant que des conditions d'exercice social de son jeu sensuel. Mais les deux lectures passent sous silence ce qui sort de leurs grilles et n\u00e9gligent aussi bien la psychologie classique, l'\u00e9thologie contemporaine, que les m\u00e9caniques philosophiques propos\u00e9es \u2013 Nietzsche \u00e9tant bien souvent rel\u00e9gu\u00e9 derri\u00e8re Freud en la mati\u00e8re, bien qu'il agisse et pense souvent en pr\u00e9curseur de nombre de th\u00e8ses psychanalytiques.\n\nJe voudrais relire un aphorisme du Voyageur et son ombre qui me semble utile pour saisir l'\u00e9conomie de tout libertin. Nietzsche \u00e9crit : \u00abChacun porte en soi une image de la femme tir\u00e9e d'apr\u00e8s sa m\u00e8re : c'est par l\u00e0 qu'il est d\u00e9termin\u00e9 \u00e0 respecter les femmes en g\u00e9n\u00e9ral ou \u00e0 les m\u00e9priser ou \u00e0 \u00eatre au total indiff\u00e9rent \u00e0 leur \u00e9gard. \u00bb J'ajouterai, pour compl\u00e9ter un Nietzsche ayant eu beaucoup \u00e0 faire avec sa m\u00e8re, qu'on pourrait \u00e9tendre la port\u00e9e de cette id\u00e9e en pr\u00e9cisant que le rapport futur aux femmes tient bien \u00e9videmment de la relation \u00e0 la m\u00e8re, mais aussi \u00e0 toutes les femmes qui traversent une existence dans les premi\u00e8res ann\u00e9es, jusqu'\u00e0 la premi\u00e8re rencontre sexuelle.\n\nLa s\u0153ur, s'il y a lieu \u2013 et pour Nietzsche il y avait lieu, malheureusement \u2013, mais aussi les cousines, les institutrices, les voisines, les tantes, les \u00e9coli\u00e8res, les amies de la m\u00e8re ou des parents, les grand-m\u00e8res, les belles-s\u0153urs, et toutes les femmes qui assurent des fonctions dans l'\u00e9panouissement et le d\u00e9veloppement d'une vie d'enfant. Sans oublier, bien \u00e9videmment, la premi\u00e8re femme bibliquement connue, celle des premiers baisers, des premiers \u00e9mois, des premi\u00e8res caresses, celle avec qui l'on perd sa virginit\u00e9 et qui donne l'impression, soudain, magiquement, qu'on est devenu adulte.\n\nToutes ces femmes donnent du bonheur ou du malheur, du plaisir ou du d\u00e9plaisir, des baisers ou des gifles, de l'affection ou de l'indiff\u00e9rence, de la tendresse ou de la violence, de la douceur ou des coups. Certaines, des m\u00e8res parfois, cumulent un talent consomm\u00e9 pour le malheur, le d\u00e9plaisir, les gifles, l'indiff\u00e9rence, la violence et les coups. D'autres, exactement l'inverse. La plupart, un m\u00e9lange aux proportions variables. Chaque fois le temp\u00e9rament, la machine d\u00e9sirante, se trouvent model\u00e9s, structur\u00e9s, construits. Le m\u00e9canisme se met au point par \u00e0-coups, par t\u00e2tonnements, dans une totale improvisation, et surtout dans la plus absolue des ignorances, de part et d'autre de la relation.\n\nEt pour Casanova? Ce parangon du libertinage m\u00e9rite un arr\u00eat. Car ses M\u00e9moires racontent le nombre de femmes connues entre 1735, l'\u00e2ge de son premier \u00e9moi, et 1774, celui de la r\u00e9daction des premi\u00e8res pages de son livre majeur. Or les chiffres sont loin d'\u00eatre impressionnants : cent vingt-deux conqu\u00eates, soit une moyenne de trois par an dans la p\u00e9riode consid\u00e9r\u00e9e. Somme toute, c'est peu. Il n'y a pas de liste chez Casanova, rien qui ressemble \u00e0 un tableau de chasse arrogant et pr\u00e9tentieux, d\u00e9monstratif et paonnant. Rien d'espagnol chez cet Italien absolu.\n\nVoluptueux, certes, h\u00e9doniste, \u00e9videmment, libertin, s\u00fbrement, mais certainement pas collectionneur. Tout juste jubilant d'\u00eatre un homme de sensualit\u00e9 et de sensation, de sensualisme et de joies sans complications. Ce qui le d\u00e9finit ? son amour immod\u00e9r\u00e9 pour la libert\u00e9, la possibilit\u00e9 d'\u00eatre ici, l\u00e0 ou ailleurs, sans entrave, sans raison qui le retienne. Tout lui est bon pour pr\u00e9server ce seul bien absolu, y compris le mensonge, le travestissement, l'hypocrisie, la ruse \u2013 voire la profession d'indicateur, pourvu qu'elle assure d'une efficace impunit\u00e9. On peut lire dans les M\u00e9moires : \u00abJ'ai aim\u00e9 les femmes \u00e0 la folie, mais je leur ai toujours pr\u00e9f\u00e9r\u00e9 ma libert\u00e9. \u00bb Voil\u00e0, il me semble, la d\u00e9finition absolue du libertin.\n\nDe sorte qu'il est moins un pur homme \u00e0 femmes qu'un homme qui, dans son rapport aux femmes, tient avant tout \u00e0 pr\u00e9server et conserver sa libert\u00e9 intacte. Et si le d\u00e9sir du libertin consiste \u00e0 ne pas se laisser arr\u00eater dans sa course, \u00e0 ne pas consentir \u00e0 quelque d\u00e9sir f\u00e9minin fixiste que ce soit, c'est, me semble-t-il, parce qu'il n'entend pas conna\u00eetre \u00e0 nouveau des exp\u00e9riences d\u00e9j\u00e0 v\u00e9cues avec d'autres figures f\u00e9minines, celles de son enfance et de ses primes ann\u00e9es. Il ne veut ni couple, ni mariage, ni enfants, ni foyer, ni rien de ce qui attache \u00e0 un lieu pr\u00e9cis. Il hait les racines, en nomade inv\u00e9t\u00e9r\u00e9.\n\nQuelles femmes le libertin a-t-il rencontr\u00e9es dans ses ann\u00e9es v\u00e9nitiennes ? Sa m\u00e8re, bien \u00e9videmment, premi\u00e8re parmi les autres, aussi bien dans le temps que dans les effets produits. Actrice, com\u00e9dienne, madame Casanova m\u00e8re, c'est avant tout une grande absence et un abandon signifi\u00e9 \u00e0 son enfant dans les meilleurs d\u00e9lais \u2013 la premi\u00e8re ann\u00e9e. La prog\u00e9niture est confi\u00e9e par les parents, tous deux com\u00e9diens, \u00e0 la grand-m\u00e8re, illettr\u00e9e, mais apparemment affectueuse. Elle s'en occupe d\u00e9sormais.\n\nLa vieille femme fait face aux saignements de nez monstrueux et r\u00e9p\u00e9titifs de son petit-fils. Bien s\u00fbr, en lecteurs serviles de Fliess, les psychanalystes ou affid\u00e9s soucieux du cas Casanova n'ont pas manqu\u00e9 de mettre en perspective ce sympt\u00f4me h\u00e9morragique et une intense activit\u00e9 masturbatoire ! Je crains que pour ces tenants de l'hypoth\u00e8se int\u00e9griste, les saignements de nez ne soient une pathologie famili\u00e8re. Toujours est-il que Casanova confie que ces probl\u00e8mes constituent son premier souvenir. Il pr\u00e9cise qu'il a alors huit ans \u2013 ce qui correspond tr\u00e8s exactement \u00e0 l'\u00e2ge qu'il a quand son p\u00e8re meurt.\n\nPour le soigner, sa grand-m\u00e8re le conduit, avec force d\u00e9corum, dans une \u00eele de la lagune o\u00f9 il rencontre la m\u00e9decine et la sant\u00e9 recouvr\u00e9e sous la forme d'une sorci\u00e8re. Enduit d'un onguent singulier, on l'enferme dans une bo\u00eete qui me fait songer \u00e0 un cercueil. Du fond de son coffre sinistre, il entend les g\u00e9missements, gesticulations et autres invocations de la vieille femme. Sorti de ses planches, on lui assure la gu\u00e9rison s'il garde le silence sur ce qui adviendra au cours de sa nuit : en l'occurrence, l'apparition d'une femme superbe. On lui intime le silence absolu, sous peine de mort.\n\nEvidemment, la belle femme vient dans son sommeil. Casanova confie qu'il ne sait plus s'il dut cette venue \u00e0 un r\u00eave sollicit\u00e9 par une puissante attente ou si, tout bonnement, la grand-m\u00e8re n'a pas elle-m\u00eame organis\u00e9 la th\u00e9\u00e2tralisation de l'\u00e9v\u00e9nement. Casanova confirme que cette histoire constitue ses premiers souvenirs. En de\u00e7\u00e0, il ne se rappelle plus rien. Abandon des parents, rencontre d'une sorci\u00e8re doubl\u00e9e d'une f\u00e9e, magie noire et magie blanche, saignements de nez et mort du p\u00e8re, promesse de gu\u00e9rison et menaces de mort, l'enfance du libertin s'installe et se place sous le signe de la culpabilit\u00e9.\n\n\u00ab On croyait mon existence passag\u00e8re, et quant aux auteurs de mes jours, ils ne me parlaient jamais \u00bb, lit-on dans les premi\u00e8res pages des M\u00e9moires. Casanova avoue une immense solitude, et une souffrance croissante : sa grand-m\u00e8re le place dans une pension, \u00e0 Padoue \u2013 pour son bien, certainement, comme toujours en pareil cas. Mais il y souffre le martyre : \u00abC'est ainsi, pr\u00e9cise-t-il, qu'on se d\u00e9barrassa de moi. \u00bb Moquerie des pensionnaires, froid vif et incisif, rats sous les paillasses et vermine dans les draps, faim sans fond, l'enfant pleure de chagrin et de col\u00e8re, apprend la souffrance et le monde, la cruaut\u00e9 du r\u00e9el, l'injustice, la cupidit\u00e9, la fourberie et la vilenie des gens.\n\nDes ann\u00e9es apr\u00e8s, r\u00e9digeant ses souvenirs, il \u00e9crit : \u00ab Je me voyais maltrait\u00e9, grond\u00e9, quoiqu'il me par\u00fbt impossible d'\u00eatre coupable. \u00bb Malgr\u00e9 l'absence de raisons \u00e9videntes au traitement qu'on lui inflige, il n'emp\u00eache, l'abandon fait toujours surgir cette question au milieu des nuits de souffrance extr\u00eame : que faut-il avoir fait pour m\u00e9riter pareils traitements ? De quels crimes est-on coupable pour devoir expier pareillement? Quel p\u00e9ch\u00e9 mortel et infamant a-t-on commis un jour? Rien, sinon celui d'\u00eatre n\u00e9.\n\nBien s\u00fbr, il n'existe pas d'autres raisons \u00e0 cette logique que la n\u00e9gligence et l'imp\u00e9ritie des parents, leur l\u00e9g\u00e8ret\u00e9 et leur froid \u00e9go\u00efsme \u2013 leur absence d'amour. Un libertin qu\u00eate sous toutes ses formes, sous toutes les latitudes, dans toutes les circonstances, quelles que soient les occasions, y compris de triviales situations, cet amour un jour d\u00e9faillant. La preuve donn\u00e9e une fois ne suffit jamais \u00e0 qui connut pareille mal\u00e9diction. Dans l'abandon qui lui fut signifi\u00e9, dans ce cong\u00e9 qu'on lui donna, dans ce sevrage affectif originaire, j'aper\u00e7ois la source du tropisme g\u00e9n\u00e9rateur en lui du s\u00e9ducteur qu'on sait.\n\nEnfant, alors qu'il ne peut compenser ce manque par une qu\u00eate d\u00e9sesp\u00e9r\u00e9e de femmes sous forme de fuite en avant, il d\u00e9tourne ses douleurs dans des sublimations singuli\u00e8res : une passion pour la nourriture, une boulimie de lectures, un surcro\u00eet d'investissement dans les activit\u00e9s intellectuelles, la d\u00e9monstration, qui assure en m\u00eame temps la dissimulation de ses blessures, d'une intelligence pr\u00e9coce et d'un go\u00fbt pour le verbe ou les choses de l'esprit. Pour l'heure, alert\u00e9e par son petit-fils de ses conditions de d\u00e9tention, la grand-m\u00e8re d\u00e9place l'enfant. Il quitte la pension et se trouve plac\u00e9 dans une famille d'accueil o\u00f9 les conditions paraissent nettement meilleures.\n\nChez le Docteur Gozzi, il apprend la musique, le violon, les sciences, la logique d'Aristote, la cosmographie ptol\u00e9ma\u00efque, mais aussi le d\u00e9sir pour une jeune fille l\u00e9g\u00e8rement plus \u00e2g\u00e9e que lui, Bettine, la s\u0153ur du docteur. Ce sera son premier amour. Il a dix ans, elle en a treize. Un matin, elle entend lui passer des bas de laine tricot\u00e9s \u00e0 son intention. Experte en mailles et autres tricotages, elle ajuste si bien ses gestes pour l'enfilage du cadeau qu'elle obtient de lui qu'il se r\u00e9pande en bonne et due forme. Confusion de part et d'autre. Silence quelque temps. L'enfant culpabilise : \u00abIl me semblait que je l'avais d\u00e9shonor\u00e9e, que j'avais trahi la confiance de sa famille, viol\u00e9 les lois sacr\u00e9es de l'hospitalit\u00e9, que j'avais enfin commis un crime horrible que je ne pouvais effacer qu'en l'\u00e9pousant... \u00bb Mais le temps fait bien les choses. Bettine et lui se retrouvent sans crainte apr\u00e8s l'aveu de ses inqui\u00e9tudes \u00e0 la jeune fille.\n\nAmoureux, Casanova demande et obtient un rendez-vous dans la chambre de la talentueuse tricoteuse. Accord\u00e9. A l'heure dite, puis un peu plus tard, voire beaucoup plus tard, il attend, mort de froid, sous la neige, dehors, recroquevill\u00e9 dans les escaliers qui conduisent \u00e0 la chambre. L\u00e0, il se prend un coup de pied dans le ventre d\u00e9coch\u00e9 par un autre pr\u00e9tendant qui quitte Bettine apr\u00e8s une nuit pass\u00e9e dans ses draps. Premi\u00e8re trahison, premi\u00e8res sagesses douloureuses, premi\u00e8res blessures, premi\u00e8res d\u00e9convenues. Il \u00e9crira : \u00abj'\u00e9tais pire que mort. \u00bb\n\nPour \u00e9viter la chose d\u00e9voil\u00e9e dans toute la maison, Bettine entreprend une th\u00e9\u00e2tralisation hyst\u00e9rique : mensonges, affabulations, crises nerveuses, simulation de possession, convulsions r\u00e9ussies \u00e0 la perfection, larmes en quantit\u00e9, cris \u00e0 profusion, s\u00e9ances d'exorcisme, sc\u00e8nes d'insultes et de grossi\u00e8ret\u00e9s, elle fait le maximum \u2013 tout en n'oubliant pas de montrer, au passage, une pr\u00e9f\u00e9rence lucide et avertie pour un jeune pr\u00eatre exorciste qui a remplac\u00e9 l'ancien, peu rago\u00fbtant.\n\nCasanova, philosophe, toujours affubl\u00e9 de ses cat\u00e9gories mentales, conclut : \u00abJe ne voyais en elle qu'une cr\u00e9ature s\u00e9duite par son temp\u00e9rament. Elle aimait l'homme et elle n'\u00e9tait \u00e0 plaindre qu'\u00e0 cause des cons\u00e9quences. \u00bb Lorsqu'elle conna\u00eetra la fi\u00e8vre, puis la petite v\u00e9role, quand devenue bouffie par la maladie et puante \u00e0 force de transpiration malsaine, elle marinera dans les miasmes de son lit, Casanova la veillera, amoureux transi et convaincu de l'impossibilit\u00e9 de lutter contre son temp\u00e9rament. Lors de la r\u00e9daction de ses souvenirs, il consigne : \u00ab malgr\u00e9 cette belle \u00e9cole qui a pr\u00e9c\u00e9d\u00e9 mon adolescence, et qui aurait d\u00fb me servir d'\u00e9gide pour l'avenir, j'ai continu\u00e9 \u00e0 \u00eatre toute ma vie la dupe des femmes.\u00bb Finalement, il perd sa virginit\u00e9 \u00e0 dix-sept ans, manquant une femme qu'il avait \u00e9lue, pour se consoler entre les quatre bras de deux s\u0153urs imp\u00e9tueuses.\n\nJusqu'au jour de Nanette et Marton, Casanova n'a connu des femmes que l'abandon glacial d'une m\u00e8re, la vieillesse d\u00e9solante d'une grand-m\u00e8re, la laideur inqui\u00e9tante d'une sorci\u00e8re, le sadisme d'une logeuse de jeunes pensionnaires et l'hyst\u00e9rie caract\u00e9ris\u00e9e d'une jeune fille. Ce qui, pour le moins, constitue un bestiaire singulier et un bagage encombrant. De quelle image f\u00e9minine positive pouvait-il disposer pour vivre dans la qui\u00e9tude son rapport aux femmes ? Des mar\u00e2tres, des hyst\u00e9riques, des folles, des fourbes, des maritornes, des laiderons, des viragos, rien qui ressemble \u00e0 une nymphe ou une dryade. Pas d'affection, de tendresse, de douceur, de pr\u00e9sence, de sinc\u00e9rit\u00e9, d'authenticit\u00e9. Comment le libertin aurait-il pu trouver un jour ce dont il a toujours ignor\u00e9 l'existence ?\n\nDans le restant de sa vie, il exp\u00e9rimente ce que Nietzsche appelait \u00abl'innocence du devenir\u00bb, la soumission de sa vie tout enti\u00e8re \u00e0 ce principe de compensation qui lui fait chercher partout une preuve de son existence dans le lit des femmes, de toutes les femmes. Sa m\u00e9canique r\u00e9active ob\u00e9it \u00e0 l'absolue n\u00e9cessit\u00e9. Impuissant, il assiste aux mouvements de son corps qui le conduisent en direction des draps d'une femme o\u00f9 il sait ne pas pouvoir trouver ce que pourtant il feint toujours d'aller chercher. Casanova est condamn\u00e9 \u00e0 l'errance, au nomadisme, au mouvement perp\u00e9tuel, jusqu'\u00e0 la tombe.\n\nLe moteur de son \u00eatre, le noyau dur de son identit\u00e9, sont \u00e0 chercher du c\u00f4t\u00e9 de l'abandon : comment \u00eatre s\u00fbr et certain de son existence si d'aventure, un jour, une m\u00e8re nous a abandonn\u00e9? De quelles preuves peut-on disposer qu'on est bien autre chose qu'un ectoplasme si l'on a connu une fois, enfant, la rel\u00e9gation de son corps, de sa chair et de son \u00e2me dans des lieux o\u00f9 rien d'autre que la souffrance ne tient compagnie? Dans les bras des femmes, le libertin croit, un temps, aveugl\u00e9, puis cesse de croire, ensuite, d\u00e9niais\u00e9, qu'il trouvera de la paix, du repos ou quelque chose d'apparent\u00e9 \u00e0 une preuve de son existence r\u00e9elle.\n\nFiction, mensonge et damnation, dans cette qu\u00eate d\u00e9sesp\u00e9rante et triste, on ne trouve rien tout en d\u00e9couvrant qu'on cherchera longtemps, sinon toujours, un objet impossible. Si d'aventure quelque chose lui ressemblant venait \u00e0 tomber entre les mains du libertin, nul doute qu'il l'\u00e9craserait pour n'avoir pas \u00e0 s'encombrer d'un pareil cadeau empoisonn\u00e9. Que ferait-il d'un r\u00f4le dans lequel il lui faudrait jouer Tristan, son double n\u00e9gatif, sa contradiction, son antith\u00e8se ? Rien, ou un pitoyable usage qui le conduirait \u00e0 nouveau dans les pas de celui qu'il est condamn\u00e9 \u00e0 \u00eatre : un qu\u00eateur d'absolu qui sait l'inexistence de l'absolu.\n29\n\nPOUR UN MAT\u00c9RIALISME PO\u00c9TIQUE\n\nLa premi\u00e8re fois qu'on entend prononcer le nom d'un philosophe, et le titre de l'un de ses livres, ne comptent pas pour peu. Il me semble que l'enfant mis au contact d'un fragment de l'univers conceptuel, m\u00eame entr'aper\u00e7u, ressent les m\u00eames frayeurs et les m\u00eames fascinations que devant un gouffre ou aux pieds des douves d'un ch\u00e2teau fort, d'une imprenable forteresse. Longtemps, d'ailleurs, le continent philosophique m'a attir\u00e9 pour ce m\u00e9lange de crainte et d'int\u00e9r\u00eat puissant, d'inqui\u00e9tude sombre et pour ces promesses de bonheur dont j'ignorais tout, mais que je pressentais. Bachelard fut le premier dont j'ai d\u00e9couvert et m\u00e9moris\u00e9 le nom, je devais avoir huit ou neuf ans et il m'appara\u00eet aujourd'hui, plus le temps passe, que sa proximit\u00e9 me g\u00eanerait moins que toute autre dans la pens\u00e9e fran\u00e7aise de ce si\u00e8cle.\n\nDonc, je n'avais pas dix ans. Ma m\u00e8re \u00e9tait \u00e0 La Baule o\u00f9 elle suivait en vacances, mais pour y travailler, la famille o\u00f9 elle \u00e9tait femme de m\u00e9nage. J'avais \u00e9t\u00e9 envoy\u00e9 chez ma marraine qui avait un tout petit haras de quelques chevaux. Ma m\u00e8re y avait \u00e9t\u00e9 employ\u00e9e dans ses jeunes ann\u00e9es et j'avais l\u00e0 tout ce qui me servait de famille. La fille a\u00een\u00e9e de la maison avait quelques ann\u00e9es de plus que moi, peu, mais assez pour qu'\u00e0 cet \u00e2ge la diff\u00e9rence f\u00fbt grande entre l'enfant que j'\u00e9tais et l'adolescente qu'elle allait bient\u00f4t cesser d'\u00eatre. Bien s\u00fbr, je lui trouvais toutes les qualit\u00e9s et tous les charmes. Elle me semblait une femme \u00e0 conqu\u00e9rir d'autant plus s\u00e9duisante qu'elle me paraissait inaccessible.\n\nA quelques dizaines de m\u00e8tres du domicile familial, on avait abandonn\u00e9 aux deux s\u0153urs, dont mon idylle \u00e9tait l'a\u00een\u00e9e, une vieille habitation minuscule ayant longtemps servi de domicile \u00e0 l'ouvrier agricole. Elle \u00e9tait connue sous le nom de la petite maison. L\u00e0, j'y ai trouv\u00e9 des livres, de la musique, un peu d'alcool, les premi\u00e8res cigarettes, la tendresse de cette jeune femme au seuil de son existence. Et la proximit\u00e9 d'un ami du m\u00eame \u00e2ge que la fille de ma marraine. J'ignorais tout de son homosexualit\u00e9 et il me semblait qu'il lui revenait de droit, par la vertu de son \u00e2ge, la possibilit\u00e9 de conqu\u00e9rir et d'aimer la demoiselle en question.\n\nLeurs conversations ne m'excluaient jamais. Parfois, je sortais dans la campagne, sur le pas de la porte, et, la t\u00eate dans les \u00e9toiles, le visage lanc\u00e9 \u00e0 vive allure dans la direction de la Voie lact\u00e9e, je sentais mes jambes flageoler, mon corps fr\u00e9mir, et mes id\u00e9es virevolter dans la nuit. Les grillons chantaient encore, les odeurs de foin coup\u00e9 et d'herbes m'enivraient, les parfums du soir laissaient place \u00e0 ceux de la nuit, le vent dans les arbres agitait les cimes et faisait bruire les feuilles en longs g\u00e9missements, les chevaux frappaient leurs sabots contre la porte des boxes, leurs naseaux fr\u00e9missaient, en dehors des bruits de la nature, le silence \u00e9tait sid\u00e9ral. J'exp\u00e9rimentais ma chair et mon \u00e2me fondues dans les secrets de la vie nocturne.\n\nDedans la petite maison, la lumi\u00e8re \u00e9lectrique \u00e9tait bannie, on s'\u00e9clairait aux chandelles fix\u00e9es avec leur cire sur le goulot d'antiques bouteilles aux formes et couleurs baroques. Et puis le feu br\u00fblait toujours dans la chemin\u00e9e. La conversation dont j'\u00e9tais le t\u00e9moin silencieux se d\u00e9ployait comme se d\u00e9plie un chat, avec volupt\u00e9 et d\u00e9licatesse, bonheur et simplicit\u00e9. De longs silences laissaient au feu tout le loisir d'occuper l'espace sonore : les b\u00fbches de pommier cr\u00e9pitaient, explosaient parfois, suaient de temps en temps, sifflaient comme un animal qui aurait voulu dire sa souffrance et s'en serait trouv\u00e9 r\u00e9duit \u00e0 ces extr\u00e9mit\u00e9s. Elles d\u00e9gageaient de longs jets de vapeur, pareilles \u00e0 des machines sous pression. Des gouttes de s\u00e8ve tombaient sur les braises apr\u00e8s avoir bouillonn\u00e9, bave d'escargot sortie des myst\u00e8res du bois.\n\nPlus tard dans la soir\u00e9e, j'attendais le bruit que font les b\u00fbches calcin\u00e9es qui s'effondrent dans un m\u00e9lange de scintillements mats et de craquements nets. Les flammes diff\u00e9rentes dans leurs grandeurs induisaient sur les murs des dessins plus ou moins signifiants. Des figures trac\u00e9es dans les nuages, des reflets dessin\u00e9s \u00e0 la surface des eaux dormantes ou vives, des modulations sonores dans l'espace immense de la campagne normande, je connaissais toutes ces variations, ces modulations que j'aimais traquer, poursuivre, d\u00e9couvrir et qui n'ont pas, depuis, quitt\u00e9 ma m\u00e9moire.\n\nDans la pi\u00e8ce sombre, tout juste \u00e9clair\u00e9e par les bougies et la chemin\u00e9e, engonc\u00e9s dans de vieux fauteuils clubs en cuir anglais, tous les deux continuaient leur conversation. Apr\u00e8s un silence dont je saurais encore aujourd'hui indiquer la longueur, il lui demanda si elle avait lu La Psychanalyse du feu de Gaston Bachelard. Ce fut pour moi plus impressionnant encore que le spectacle de la vo\u00fbte \u00e9toil\u00e9e ou celui des nuits myst\u00e9rieuses : je ne comprenais pas m\u00eame ce que pouvait bien signifier ce titre ni quel livre se cachait derri\u00e8re. Elle dit que non, il pr\u00e9cisa que c'\u00e9tait superbe. Je n'en sus pas plus.\n\nCe soir-l\u00e0 \u2013 ou un autre, car il y en eut d'autres, mais il me plairait que ce soit le m\u00eame \u2013, je sortis un peu dans la campagne. Devant la petite maison, une route passait qui conduisait \u00e0 la ferme o\u00f9 mon p\u00e8re a travaill\u00e9 toute son existence. Sur le goudron, j'y lisais r\u00e9guli\u00e8rement les saisons : la terre des betteraves arrach\u00e9es en automne, la paille s\u00e9ch\u00e9e des foins en \u00e9t\u00e9, les traces de ma\u00efs ensil\u00e9 l'hiver, le lisier du printemps. De l'autre c\u00f4t\u00e9, \u00e0 trois ou quatre m\u00e8tres, j'avais avis\u00e9 une luciole, un ver luisant comme on disait alors. Avan\u00e7ant la main pour la prendre et tenir dans ma paume une clart\u00e9 animale, je re\u00e7us une sorte de d\u00e9charge \u00e9lectrique.\n\nJ'en conclus que ces bestioles avaient la facult\u00e9 de punir ainsi les ennuyeux pour pr\u00e9server de la sorte leur intimit\u00e9 au creux des foss\u00e9s. Le lendemain seulement, quand le jour fut revenu, je vis que j'avais plong\u00e9 la main et le bras dans un bouquet d'orties au beau milieu duquel luisait l'insoup\u00e7onnable animal. Je fis alors, en l'ignorant, un genre d'exp\u00e9rience de l'obstacle \u00e9pist\u00e9mologique cher au philosophe des sciences qu'est Gaston Bachelard. Entre animisme, substantialisme, g\u00e9n\u00e9ralit\u00e9 de premier aspect, fausse rigueur, observation premi\u00e8re et autres mises en garde faites par le penseur de la connaissance objective, j'avais navigu\u00e9 en eaux troubles.\n\nLa luciole m'avait moins tromp\u00e9 que l'usage de mon entendement. J'allais comprendre la chose bien plus tard, en classe terminale, quand je retrouvai le nom du philosophe associ\u00e9 \u00e0 La Formation de l'esprit scientifique. Le professeur pr\u00e9pos\u00e9 aux litt\u00e9raires avait le don pour endormir toute vell\u00e9it\u00e9 philosophique et mettait en garde dans la classe quiconque aurait pris trop de plaisir \u00e0 la discipline et l'aurait envisag\u00e9e en dehors du strict espace scolaire. Il invitait \u00e0 investir dans tout et n'importe quoi, mais surtout pas dans la mati\u00e8re qu'il professait. Bachelard lui fut l'occasion de ressasser son cours d'\u00e9pist\u00e9mologie \u2013 sa fiert\u00e9 \u00e9tant de reprendre chaque ann\u00e9e le cahier du meilleur \u00e9l\u00e8ve pour le dicter l'ann\u00e9e suivante \u00e0 de nouvelles victimes.\n\nJe retrouvais donc trace de mon souvenir ancien et me r\u00e9jouissais de voir r\u00e9appara\u00eetre dans mon existence la figure du philosophe de La Psychanalyse du feu. Consciencieuse et appliqu\u00e9e, ma voisine qui mettait au propre ses notes sur un immense cahier, le soir, chez elle, avait us\u00e9 et abus\u00e9 du stylo-quatre-couleurs pour peindre, ou presque, sur le papier blanc de son livre d'heures, un immense titre que je lus, effar\u00e9 : elle avait calligraphi\u00e9 un monumental Gaston Vachelard qui mettait toutes mes certitudes \u00e0 mal. Soudain, je doutai de moi. Avais-je mal entendu le nom en question? L'avais-je mal compris? Pire : mal retenu, d\u00e9form\u00e9, oubli\u00e9, maltrait\u00e9 ? Je me vexais d'avoir pu contrefaire des souvenirs et de me trouver tromp\u00e9 par ma m\u00e9moire. V\u00e9rification faite, il me fut agr\u00e9able de constater que la premi\u00e8re lettre du patronyme incrimin\u00e9 inaugurait l'alphabet plus qu'elle ne le concluait.\n\nJe lus donc, attentivement, cette Formation de l'esprit scientifique en lui trouvant des qualit\u00e9s d'\u00e9criture, la clart\u00e9, l'\u00e9l\u00e9gance, la pr\u00e9cision, une plume libre, rigoureuse et efficace. Le style de Bachelard, comme celui de son ennemi th\u00e9orique Bergson, propose une esth\u00e9tique \u00e0 lui seul. A l' \u00e9poque, j'avais renonc\u00e9 \u00e0 une carri\u00e8re de biologiste, tr\u00e8s t\u00f4t envisag\u00e9e, dans les premi\u00e8res ann\u00e9es du coll\u00e8ge, et je trouvais heureux ce point de jonction entre science et philosophie qui me permettait d'ajuster mes plaisirs et de les affiner. L'\u00e9pist\u00e9mologie enseign\u00e9e par Bachelard s\u00e9duit l'esprit et conquiert l'intelligence.\n\nA l'universit\u00e9, j'eus affaire \u00e0 un pitoyable enseignant pr\u00e9tendument sp\u00e9cialis\u00e9 en philosophie des sciences. Ses valeurs d'\u00e9pist\u00e9mologie, quels qu'en aient \u00e9t\u00e9 les niveaux, se noyaient dans le psittacisme des th\u00e8ses marxistes de Dominique Lecourt sur la question. Obstacles et ruptures \u00e9pist\u00e9mologiques, psychanalyse de la connaissance objective et rationalisme dynamique, Bachelard fut mis en morceaux, d\u00e9coup\u00e9, servi \u00e0 toutes les sauces, ici pass\u00e9 \u00e0 la moulinette pseudo-freudienne, l\u00e0 au hachoir marxiste, mais toujours superbement ignor\u00e9, n\u00e9glig\u00e9, incompris. Une fois de plus, l'universit\u00e9 brillait dans l'insipide et l'incolore, elle affadissait une pens\u00e9e r\u00e9ellement savoureuse.\n\nBachelard me ravissait, le soir venu \u2013 serait-il d'une plus grande efficace dans les heures vesp\u00e9rales, exclusivement? \u2013 quand je d\u00e9cidais d'oublier le travail de la journ\u00e9e avec des lectures plus jubilatoires et moins utilitaires. Je lus alors, enfin, cette fameuse Psychanalyse du feu \u2013 avec crainte et inqui\u00e9tude, puis bonheur et contentement. Loin des tristesses scolaires et universitaires, Bachelard propose toute la dimension subversive d'une pens\u00e9e \u00e0 la premi\u00e8re personne, radicalement subjective, revendiquant la r\u00eaverie comme une m\u00e9thode presque plus s\u00fbre pour parvenir \u00e0 la certitude que n'importe quel artifice rationnel. Du feu, que j'ai toujours aim\u00e9 avec passion, il donne une vision apparent\u00e9e \u00e0 celle de pr\u00e9socratiques qui n'auraient rien ignor\u00e9 des d\u00e9couvertes effectu\u00e9es depuis leurs temps lointains.\n\nLa mode dans l'enseignement sup\u00e9rieur, alors, s\u00e9vit c\u00f4t\u00e9 mat\u00e9rialisme dialectique revu et corrig\u00e9 par Althusser. Il est de bon ton d'\u00e9crire sur la haute, grande et claire pens\u00e9e de L\u00e9nine ou de s'interroger sur l'excellence des usages de la dialectique chez Mao, \u00e0 moins qu'on ne sacrifie \u00e0 de longues analyses sur Jdanov et la science bourgeoise. Toujours est-il que Bachelard passe pour le promoteur d'une philosophie id\u00e9aliste et que les marxistes ne savent comment s'y prendre avec un philosophe aussi insaisissable. D'autant qu'Althusser avait donn\u00e9 ses lettres de noblesse au vieil homme en lui empruntant les notions de rupture ou de coupure \u00e9pist\u00e9mologique. Embarras des nouveaux scolastiques...\n\nAinsi, pour marxiser Bachelard, rien ne fut plus simple que de le pr\u00e9senter coup\u00e9 en deux, comme une pomme tranch\u00e9e : une moiti\u00e9 \u00e9pist\u00e9mologique et une autre po\u00e9tique, sans qu'on sache comment faire pour associer, r\u00e9unir ou rapprocher les deux morceaux. Les dates, tout simplement, un souci b\u00eatement chronologique, auraient suffi pour \u00e9viter d'enseigner pareille sottise. Le vieil homme n'a pas subi ou exp\u00e9riment\u00e9 une rupture \u00e9pist\u00e9mologique entre le philosophe des sciences qu'il aurait commenc\u00e9 \u00e0 \u00eatre et le penseur de la r\u00eaverie qu'il aurait fini par incarner. Car les deux registres s'interp\u00e9n\u00e8trent \u00e0 partir de 1938 et jusqu'\u00e0 la fin. Bachelard \u00e9crit dans les deux mondes sans que jamais l'un supplante l'autre.\n\nLa bibliographie sur l'\u0153uvre du personnage, assez peu abondante, excelle singuli\u00e8rement en ouvrages et articles imb\u00e9ciles. Des souvenirs niais et \u00e9gotiques d'un mauvais po\u00e8te \u00e0 la biographie assommante et politiquement correcte o\u00f9 l'on n'apprend rien, ni sur le terrain de la biographie, ni sur celui de l'analyse des \u0153uvres, en passant par les resuc\u00e9es de travaux universitaires constitu\u00e9s de fiches mises bout \u00e0 bout, comme toujours, le constat est sinistre, l'\u00e9tat des lieux affligeant. Rien ne brille ou se d\u00e9tache dans la production sur le sujet; rien, de toute fa\u00e7on, qui serait un abord bachelardien du philosophe, de sa vie et de sa pens\u00e9e, puis de sa propre existence philosophique.\n\nBachelard m\u00e9rite mieux que ces \u00e9coliers infatu\u00e9s, ces laborieux du vers de mirliton ou ces moralisateurs donneurs de le\u00e7ons. Il suffirait qu'on veuille \u00e9couter le vieux sage, le lire, le suivre \u00e0 la trace, ou comme en une balade faite en sa compagnie dans les vallons champenois, pour entendre sa constante revendication : le philosophe des sciences illustre l'avers d'une m\u00e9daille au revers frapp\u00e9 par le penseur de l'imaginaire. Recto \u00e9pist\u00e9mologique, verso po\u00e9tique \u2013 mais pour une seule et m\u00eame feuille de papier. L'\u00e9pist\u00e9mologue contraint \u00e0 la rigueur analytique, au carcan conceptuel, n'a cess\u00e9 de se reposer en \u00e9crivant des r\u00eaveries, des textes o\u00f9 il est libre de r\u00e9injecter ce que le premier type d'exercice lui interdit : une philosophie du corps soutenue par la revendication nostalgique d'une subjectivit\u00e9 affirm\u00e9e gr\u00e2ce \u00e0 un d\u00e9tour philosophique du c\u00f4t\u00e9 d'exp\u00e9riences provinciales anciennes et fondatrices.\n\nLoin de toute rupture \u00e9pist\u00e9mologique, de toute contradiction, de toute antinomie si souvent point\u00e9es par les sorbonagres dans l'\u0153uvre, Bachelard \u00e9volue en philosophe qui n'oublie pas d'\u00eatre un homme, voire en homme qui ne renonce pas \u00e0 philosopher. La dizaine d'ouvrages qui fonde un mat\u00e9rialisme singulier, en forme de po\u00e9tique des \u00e9l\u00e9ments, lui permet d'aborder ce que l'enfant retient farouchement de ses promenades dans son village de Bar-sur-Aube : la maison natale et la symbolique de la cave et du grenier, l'eau dormante d'une mare aper\u00e7ue et d\u00e9couverte dans un sous-bois, lors d'une promenade en solitaire, le premier voyage fait sur les c\u00f4tes et la d\u00e9couverte de la mer \u00e0 l'\u00e2ge de trente ans, puis l'analyse du coquillage qui s'ensuit, alors que le nid fait l'objet d'un autre d\u00e9veloppement, en relation avec les balades sur les sentiers champenois.\n\nIci, ce sont les nuages, les vents, les nu\u00e9es, la pluie, l\u00e0, le ruisseau, la racine, le serpent, ailleurs, comme apr\u00e8s passage dans l'athanor, des pages sur la vigne qu'on taille et le vin qu'on boit. Puis une question \u00e0 mes yeux fondamentale, plac\u00e9e en toute extr\u00e9mit\u00e9 de La Terre et les R\u00eaveries du repos : Comment trouver un philosophe qui sache le boire ? Chaque fois, les analyses de Bachelard, pour pointues, fines et pr\u00e9cises qu'elles sont, n'oublient jamais le souci de la vie quotidienne, de l'existence, du monde r\u00e9el et de ceux qui le font. La po\u00e9tique des \u00e9l\u00e9ments, les m\u00e9ditations sur la mati\u00e8re, l'imagination, la r\u00eaverie, les forces et les relations entretenues par toutes ces instances, tout cela constitue un mat\u00e9rialisme po\u00e9tique dont il donne moins la formule pr\u00e9cise qu'il n'en indique les contours. A charge pour le lecteur d'effectuer ses propres r\u00eaveries sur le sujet.\n\nQuand il \u00e9crit ses ouvrages d'\u00e9pist\u00e9mologie pure et dure, il sait devoir se soumettre \u00e0 d'autres obligations m\u00e9thodologiques. Les imp\u00e9ratifs de pens\u00e9e, d'analyse et d'\u00e9criture contraignent sa nature ind\u00e9pendante. Ses livres sur la connaissance approch\u00e9e, la structure ondulatoire et corpusculaire de la lumi\u00e8re, ses pages sur l'induction, la mati\u00e8re et le rayonnement, la propagation thermique dans les solides, ses ouvrages sur la relativit\u00e9, la physique contemporaine lui co\u00fbtent en construction, en composition, en \u00e9laboration mentale. Son travail rel\u00e8ve alors de l'universit\u00e9 qu'il incarne, des trajets honorifiques habituels, aux fonctions d'autorit\u00e9 dans les jurys, en passant par les participations aux instances officielles. Bachelard ne refuse ni la L\u00e9gion d'honneur ni l'Acad\u00e9mie des sciences morales et politiques, ni le Grand Prix national des Lettres, mais c'est prioritairement l'auteur du Pluralisme coh\u00e9rent de la chimie moderne qu'on f\u00eate de la sorte. Car l'auteur du Lautr\u00e9amont, ou celui de l'imagination de la mati\u00e8re ou du mouvement, fait sourire un peu partout dans les milieux qu'on pr\u00e9tend autoris\u00e9s.\n\nBachelard se repose de la com\u00e9die humaine, parisienne et mondaine, dont il n'est pas dupe, dans sa r\u00e9gion de Champagne, \u00e0 Dijon. D'ailleurs, personne ne souligne jamais que, la plupart du temps, seuls ses livres sur les \u00e9l\u00e9ments sont dat\u00e9s et sign\u00e9s de la capitale de la Bourgogne. Dans La Po\u00e9tique de l'espace, il \u00e9crit : \u00abPour moi, je ne sais m\u00e9diter que les choses de mon pays. \u00bb Et ses pages ressortissant au registre de la po\u00e9tique s'enracinent toutes dans des exp\u00e9riences autobiographiques plus ou moins dissimul\u00e9es, ou oubli\u00e9es. De sorte que la po\u00e9tique soul\u00e8ve un coin du voile qui recouvre les choses obligatoirement cach\u00e9es lors du travail \u00e9pist\u00e9mologique.\n\nPourtant, Bachelard laisse parfois poindre un genre de retour du refoul\u00e9 dans chacun des registres. Ainsi, on peut lire dans La Formation de l'esprit scientifique des pages sur l'\u00e9ponge ou la digestion qui ne d\u00e9pareraient pas dans les ouvrages de po\u00e9tique ; de la m\u00eame mani\u00e8re, quand il \u00e9crit L'Air et les Songes, il ne recule pas devant des d\u00e9veloppements sur \u00ab la philosophie cin\u00e9matique et la philosophie dynamique \u00bb. Tao\u00efste \u00e0 sa mani\u00e8re, il pointe finement de blanc ses zones noires et de noir ses espaces blancs pour rappeler que l'animus habite aussi d\u00e9licatement l'anima \u2013 et vice versa.\n\nSa po\u00e9tique, dans son registre sp\u00e9cifique, ose une authentique habilitation de ce qui fait bondir les philosophes de profession classiques, id\u00e9alistes, spiritualistes, form\u00e9s \u00e0 la bonne \u00e9cole n\u00e9o-kantienne universitaire \u2013 c'est-\u00e0-dire presque tous. Bachelard, en effet, propose une philosophie sensuelle qui proc\u00e8de nettement du corps. De m\u00eame, il avance l'autorit\u00e9 sur laquelle il s'appuie : rien moins que lui dans sa plus triomphante et absolue libert\u00e9. Son Je et son Moi, librement mis en sc\u00e8ne, structurent et soutiennent ses r\u00eaveries. Le corps de Gaston Bachelard, ses souvenirs, sa m\u00e9moire, ses nostalgies, ses douleurs pass\u00e9es, ses moments heureux, ses images mentales, les arch\u00e9types sollicit\u00e9s dans le tr\u00e9fonds de son \u00e2me \u00e0 partir d'exp\u00e9riences singuli\u00e8res, les \u00e9motions subjectives, les perceptions individuelles, les affects propres, tout est bien, \u00e0 ses yeux, qui autorise une philosophie \u00e0 la premi\u00e8re personne.\n\nDans l'ordre universitaire, et selon ses principes policiers, le Moi parasite la connaissance et fournit un obstacle majeur au savoir, au jugement. Il trouble les certitudes rationnelles, c'est du moins ce qu'enseignent les traditions philosophiques depuis toujours : les sens sont trompeurs, la perception g\u00e9n\u00e8re des illusions, la subjectivit\u00e9 est de mauvais conseil, l'objectivit\u00e9 doit \u00eatre r\u00e9solument recherch\u00e9e. Or, chez le penseur bourguignon, ce qui ne cesse de me ravir, c'est que le professeur de la Sorbonne n'a pas tu\u00e9 l'enfant de Champagne. Le petit provincial \u00e9lev\u00e9 au contact d'une nature qui s'\u00e9panouit, radieuse et douce, demeure, solaire, loin de l'imp\u00e9rialisme technologique qui s'annonce et gronde dans ce milieu de xxe si\u00e8cle.\n\nLe mat\u00e9rialisme po\u00e9tique bachelardien proc\u00e8de d'une intuition heureuse : il s'agit pour le philosophe de t\u00e2cher de mettre un peu d'ordre th\u00e9orique dans le d\u00e9sordre du monde. Le divers, le multiple, la pluralit\u00e9, les fragments, tout cela rel\u00e8ve d'un certain nombre de principes ou de complexes, nombreux chez Bachelard : Jonas, Lautr\u00e9amont, Prom\u00e9th\u00e9e, Emp\u00e9docle, Novalis, Hoffmann, Caron, Oph\u00e9lie fournissent chaque fois des rubriques sous lesquelles la pluralit\u00e9 devient unique en se rangeant sagement sous une th\u00e9orie \u00e9labor\u00e9e pour l'occasion.\n\nMais l'abondance des tiroirs dans le texte du philosophe ne trompe pas sur son impuissance \u00e0 r\u00e9duire totalement les sujets abord\u00e9s \u00e0 une ou deux certitudes in\u00e9branlables. Et il le sait, tout en s'en moquant. La r\u00eaverie n'entend pas fournir des v\u00e9rit\u00e9s absolues, tout juste des impressions, des confidences, des propos \u00e0 b\u00e2tons rompus qui pourraient s'\u00e9changer dans les conditions qu'il examine : un verre de vin ou de punch \u00e0 la main, \u00e0 proximit\u00e9 de la flamme d'une chandelle, en devisant dans la campagne, le long d'une rivi\u00e8re, ou la t\u00eate en l'air, le regard perdu dans les nuages. L'efflorescence du r\u00e9el lui convient trop pour qu'il d\u00e9sire l'\u00e9teindre dans des formules, des id\u00e9es, des concepts.\n\nSa po\u00e9tique s'expose donc dans un grand d\u00e9sordre o\u00f9 les r\u00e9f\u00e9rences, les citations, les notes de lecture confondent les registres et les qualit\u00e9s. Combien de mauvais faiseurs de rimes pour illustrer ses intuitions ? Et quels grands po\u00e8tes oubli\u00e9s, n\u00e9glig\u00e9s ! Combien de mauvais peintres, plut\u00f4t artisans de l'outil \u2013 pinceau, burin \u2013 qu'artistes \u00e0 proprement parler, pour entretenir d'esth\u00e9tique? Au prix de quels \u00e9vitements, quand on aurait aim\u00e9 les consid\u00e9rations du philosophe sur les Futuristes, les Surr\u00e9alistes, ou des figures puissantes tel Picasso. Bachelard s'amuse, met en forme des fiches, collectionne, butine pour son plaisir et r\u00e9gurgite un miel sous forme de livres prioritairement publi\u00e9s chez Jos\u00e9 Corti, l'\u00e9diteur des r\u00eavenes.\n\nAlbert Flocon, Simon Segal, Henri de Waroquier, Louis Marcoussis pour les beaux-arts, Victor Poucel, Pierre Gu\u00e9guen, Maryse Choisy, No\u00ebl Bureau pour la po\u00e9sie, Yvonne Caroutch, Louis Chadourne, Louis Ulbach, Henri Bachelin pour le roman, et combien d'autres doivent d'avoir laiss\u00e9 un nom dans l'histoire parce que Gaston Bachelard leur a consacr\u00e9 une notice, une pr\u00e9face ou qu'il leur a emprunt\u00e9 une citation, un mauvais vers, quelques pitoyables alexandrins... Ren\u00e9 Char disait, acerbe, que, parmi les po\u00e8tes, Bachelard n'avait retenu que les m\u00e9diocres. Certes, bien souvent, mais le philosophe n'avait pas en t\u00eate de recourir aux seules valeurs s\u00fbres, estampill\u00e9es socialement par le petit monde intellectuel. En homme libre, il a \u00e9lu des amis, choisi des proches, sinon consenti de bonne gr\u00e2ce \u00e0 quelques demandes venues de ces chronophages qui pullulent dans le sillage des gens c\u00e9l\u00e8bres ou en vue.\n\nPas plus qu'il ne cite de r\u00e9f\u00e9rences garanties, Bachelard ne compose dans les usages avec le nombre d'or en t\u00eate. Son propos est libre, les longueurs de ses consid\u00e9rations varient selon le caprice. Les fiches de lecture juxtapos\u00e9es ne g\u00eanent pas le philosophe qui compose sur le principe surr\u00e9aliste du collage, de l'agencement capricieux de formes, d'images, de propos et de mots. La po\u00e9tique s'\u00e9nonce dans un fouillis, un d\u00e9sordre qui, finalement, reproduit celui du monde. Et l'on se prend parfois \u00e0 r\u00eaver du livre majeur d'un Gaston Bachelard qui aurait \u00e9lagu\u00e9 les citations, les r\u00e9f\u00e9rences, les manies d'universitaire s'appuyant sur des autorit\u00e9s tierces pour avancer ses propres conclusions, pour leur donner un semblant de s\u00e9rieux par le vernis de l'\u00e9rudition.\n\nCe livre aurait pu \u00eatre l'occasion d'une philosophie des \u00e9l\u00e9ments et d'un authentique mat\u00e9rialisme po\u00e9tique. Dans la langue claire qu'il pratiquait avec bonheur, Bachelard aurait pu formuler une esth\u00e9tique alternative \u00e0 celle des marxistes ou des freudiens, des id\u00e9alistes ou des structuralistes. R\u00eavons un peu, l'auteur du Droit de r\u00eaver ne saurait le d\u00e9conseiller. Cette esth\u00e9tique aurait mis le corps en son centre, elle aurait appel\u00e9 la nostalgie et l'enfance \u00e0 la rescousse, elle se serait appuy\u00e9e sur la subjectivit\u00e9 et sa capacit\u00e9 \u00e0 la m\u00e9ditation libre, elle aurait habilit\u00e9 les sens, les perceptions, l'imagination, les affections pour leur donner une autorit\u00e9 architectonique, elle aurait suppos\u00e9 un panth\u00e9isme singulier \u2013 qu'il appelait pancalisme \u2013 qui donne \u00e0 la nature un r\u00f4le primordial dans l'\u00e9conomie de toute culture, elle aurait investi le Moi, le Je, la subjectivit\u00e9 de l'individu pour les transfigurer en force cardinale, majeure, elle aurait racont\u00e9, enfin, la philosophie et la vie r\u00e9concili\u00e9es.\n\nEn reprenant les livres po\u00e9tiques, en feuilletant, en cheminant de mani\u00e8re capricieuse, en allant et venant dans les pages, je retrouve ses consid\u00e9rations sur la joie de go\u00fbter les fruits de son verger, l'art de cuire les confitures ou de confectionner une tarte au citron, les angoisses du solitaire travaill\u00e9 par les nuits d'insomnie, le plaisir sensuel de la lecture et de l'\u00e9criture, la promenade dans la campagne, le long d'un ruisseau, la flamme d'une chandelle ou le feu dans la chemin\u00e9e, la m\u00e9lancolie devant les eaux dormantes d'une mare dans la for\u00eat, l'importance du pays natal, l'odeur du pain et celle d'une volaille cuisant au tournebroche, et tout ce qui fait l'existence au quotidien. Avec cette esth\u00e9tique bachelardienne constitu\u00e9e, les scintillements du ver luisant conduiraient s\u00fbrement les enfants au paradis des philosophes.\n30\n\nLE SEXE, LE SANG, LA MORT\n\nQuand je mets en perspective tout ce que j'ai vu dans les mus\u00e9es depuis des ann\u00e9es que je parcours le monde en qu\u00eate d'images magiques, mythiques et fabuleuses, il me vient \u00e0 l'esprit qu'en tous les temps et tous les lieux, toutes histoires et toutes g\u00e9ographies confondues, ce qui fait l'essence de l'art pictural voyage bien souvent entre les trois pointes de ce triangle noir que sont le sexe, le sang et la mort. Des premi\u00e8res traces laiss\u00e9es par l'homme dans les cavernes d'Europe aux derni\u00e8res peintures, encore fra\u00eeches, entrepos\u00e9es dans les ateliers du m\u00eame ancien continent, ce sont la plupart du temps les variations sur ces trois instances qui dessinent et produisent l'histoire de l'art.\n\nAussi, je ne souscris pas, de quelque mani\u00e8re que ce soit, aux discours n\u00e9o-h\u00e9g\u00e9liens \u2013 ou construits sur le mode spengl\u00e9rien du D\u00e9clin de l'Occident \u2013, qui cherchent, et finissent par trouver, des cycles, des p\u00e9riodes, des mouvements rationnels dans une histoire qui ne cesse d'\u00eatre le lieu de l'irrationnel. D'autant que les qu\u00eateurs de pareilles logiques finissent toujours par r\u00e9injecter dans leurs propos les notions aff\u00e9rentes de naissance, de croissance, d'apog\u00e9e, de d\u00e9cadence, tous concepts souteneurs des fantasmes purificateurs actifs dans les p\u00e9riodes interm\u00e9diaires \u2013 comme en cette fin de cycle annonc\u00e9e.\n\nLes mill\u00e9naristes contemporains crient au retour du laid, du bizarre, du grotesque, de l'effroyable, de l'horrible, ou encore du mauvais go\u00fbt ou de l'anormal. Or tout l'art s'est nourri, depuis toujours, de ces figures de la n\u00e9gativit\u00e9. Du cabinet des F\u00e9lins dans la grotte de Lascaux aux morgues dans lesquelles Andrea Serrano photographie le d\u00e9tail de cadavres administrativement \u00e9tiquet\u00e9s, des salles de la n\u00e9cropole de Cheik Abd El-Gourna en Egypte \u00e0 l'atelier parisien de Vladimir V\u00e9licovick, des gisants gothiques dans les cath\u00e9drales europ\u00e9ennes aux installations d'Asta Gr\u00f6ting, des cires anatomiques de Fragonard aux derni\u00e8res gravures sur cuivre de Picasso, c'est toujours le sexe, le sang, la mort que l'on raconte, que l'on traque, trouve, fixe et montre.\n\nA ses sources, l'art se nourrit toujours, et de toute \u00e9ternit\u00e9, de bizarre, de laideur et d'horreur, avant que le temps fasse son effet en permettant d'apprivoiser et d'acclimater le n\u00e9gatif \u00e0 l'aide de cristallisations culturelles sociales. De sorte que, sublim\u00e9es, les ombres qui effraient deviennent des lumi\u00e8res avec lesquelles on \u00e9claire l'\u00e2me des errants m\u00e9taphysiques que sont toujours les hommes. Les fins de si\u00e8cle, qui plus est celles des mill\u00e9naires, suscitent et produisent en nombre les fantasmagories th\u00e9oriques, les constructions intellectuelles ponctuelles et les d\u00e9lires habituels. Chaque fois le discours sur la d\u00e9cadence suit, comme preuve de la difficult\u00e9 ou de l'incapacit\u00e9 des contemporains qui vieillissent \u00e0 accepter avec s\u00e9r\u00e9nit\u00e9 l'\u00e9coulement du temps, des secondes aussi bien que des si\u00e8cles \u2013 puisqu'apr\u00e8s tout La Palice aurait confirm\u00e9, c'est \u00e0 l'aide des secondes minuscules d'une br\u00e8ve existence que se font ces si longs si\u00e8cles constitutifs des plus durables civilisations.\n\nJe tiens pour l'inexistence des cycles et la permanence du chaos, du d\u00e9sordre dans l'histoire. Le temps \u00e9coul\u00e9 ne s'organise pas sur le mode de la ligne droite, de la fl\u00e8che dirig\u00e9e vers le futur, selon l'ordre du progr\u00e8s et de l'\u00e9volution, il se d\u00e9ploie plut\u00f4t sur le principe de l'arborescence et de la variation infinie sur les m\u00eames th\u00e8mes. Le r\u00e9el se d\u00e9voile sur le mode fractal et non lin\u00e9aire. Classique, Gothique, Baroque, Mani\u00e9risme, Rococo, Modernit\u00e9, Avant-Garde, et autres artifices d'historiens, signent moins des \u00e9poques dat\u00e9es que des caract\u00e8res, des temp\u00e9raments, des fa\u00e7ons pr\u00e9sentes dans toutes les \u00e9poques. Le traitement de l'information se modifie, mais l'information demeure. Le style ondoie, la chose exprim\u00e9e jamais.\n\nD'abord, du sexe. Qu'on ne se m\u00e9prenne pas en imaginant, avec ce mot, une seule perspective r\u00e9ductionniste limit\u00e9e et confin\u00e9e \u00e0 la sexualit\u00e9 g\u00e9nitale, aux petites affaires humaines qui g\u00e9n\u00e8rent de petites passions pour de petits soucis quotidiens. Je songe ici \u00e0 la vitalit\u00e9 partout \u00e0 l'\u0153uvre l\u00e0 o\u00f9 des forces r\u00e9sistent \u00e0 la mort. Il est \u00e9nergie et puissance, force qui permet l'\u00eatre d'une forme et sa dur\u00e9e. Le sexe vaut en ce sens comme l'un des principes chers aux pr\u00e9socratiques \u2013 l'Amour et la Haine d'Emp\u00e9docle par exemple \u2013 agissant comme des concepts op\u00e9ratoires \u00e0 l'aide desquels le divers se con\u00e7oit mieux, plus ramass\u00e9, mieux r\u00e9uni.\n\nLe sexe renvoie \u00e0 la vitalit\u00e9 aveugle, \u00e0 ce vouloir qui habite le vivant et lui fait d\u00e9sirer l'\u00eatre, la dur\u00e9e, la persistance dans une structure, un agencement, une composition. Le latin disposait, avant l'usage qu'en fit Freud, d'une libido mieux \u00e0 m\u00eame de signifier ce que j'entends par Sexe. Energie, envie, d\u00e9sir de volupt\u00e9, fantaisie, caprice, tyrannie de l'arbitraire, mais \u00e9galement opposition de soi au monde comme apposition, je vois l\u00e0 l'indispensable impetus sans lequel le r\u00e9el n'aurait aucune consistance, aucune existence. Le sexe est l'\u00e2me de ce qui vit.\n\nLes mains n\u00e9gatives trouv\u00e9es partout dans les grottes pr\u00e9historiques comme premi\u00e8res signatures de toute humanit\u00e9, m\u00eame si l'on est confondu en interrogations devant leur signification v\u00e9ritable, expriment ce vouloir poss\u00e9der, cette volont\u00e9 de marquer un territoire, de d\u00e9limiter, de s'approprier et de mettre en perspective le temps d'un homme, son histoire, et l'espace dans lequel il \u00e9volue, sa g\u00e9ographie. Sexe ce mobile originaire et cette signature sur le mode de l'empire; sexe aussi le devenir chose du monde sous l'emprise d'un vouloir d\u00e9cid\u00e9; sexe enfin l'immobilisation du temps \u00e9ternel dans l'entreprise et la d\u00e9cision d'un acte individuel. Bizarre, d\u00e9j\u00e0, ce premier vouloir.\n\nAu commencement \u00e9tait la main n\u00e9gative. Ensuite advinrent les signes qui tout de suite furent binaires, f\u00e9minins et masculins, comme si, d\u00e9j\u00e0, le sexe devait se dire dans le dualisme, l'opposition, la s\u00e9paration et peut-\u00eatre, \u00e0 terme \u2013 mais lequel ? \u2013 le d\u00e9sir d'une r\u00e9conciliation. Triangles, ovales et autres traces claviformes ou quadrangulaires, les symboles f\u00e9minins appellent ceux qui disent le masculin : crochets et propulseurs, barbel\u00e9s, b\u00e2tonnets, points et lignes. L'alphabet du sexe s'y \u00e9nonce tout entier. Seule manque la grammaire.\n\nElle prendra forme avec les vulves, les fesses, les seins, les hanches, les bassins exacerb\u00e9s. Et les phallus d\u00e9mesur\u00e9s, triomphants, puis les accouplements, parfois au partenaire zoomorphe. V\u00e9nus de Lespugue ou de Willendorf ici, humano\u00efde rupestre et libyen au visage de lapin et au sexe disproportionn\u00e9 l\u00e0, la grande ronde se joue sur le mode des principes oppos\u00e9s appel\u00e9s \u00e0 se retrouver, \u00e0 se rejoindre pour que le monde persiste dans sa forme et dans la dynamique qui le structure. Le d\u00e9fi au bon sens anatomique installe ces traces primitives sur le terrain de l'art o\u00f9 ce qui se dit est moins la reproduction fid\u00e8le du d\u00e9tail d'un corps que la m\u00e9tamorphose incarn\u00e9e d'une m\u00e9taphysique : maternit\u00e9, f\u00e9condit\u00e9, abondance, nourriture, vie et survie \u2013 les soucis du premier homme animeront et pr\u00e9occuperont le dernier.\n\nCe sexe-l\u00e0 ne cesse de travailler l'\u00e2me de tout un chacun. Comment vivre et survivre ? De quelle mani\u00e8re durer encore, malgr\u00e9 la toute-puissance de l'entropie? De quels subterfuges dispose qui doit mourir et le sait? L'art g\u00eet l\u00e0, dans ce d\u00e9sir de repousser les bornes et les limites de la mort, de conjurer aujourd'hui, dans l'instant, notre disparition tout \u00e0 l'heure, demain, ou apr\u00e8s-demain. L'artiste veut garrotter la mort, la ligoter et lui faire prendre patience.\n\nSurgissent chairs et corps, d\u00e9sirs et plaisirs sur les vases grecs o\u00f9 se pratique l'amour du m\u00eame nom, dans les maisons patriciennes de Pomp\u00e9i, sur les temples indiens o\u00f9 sont figur\u00e9es les postures amoureuses, dans les toiles de Lancret, Fragonard ou Watteau, chez Picasso qui peint des sexes f\u00e9minins comme on les tra\u00e7ait dans les cavernes pr\u00e9historiques. Ou, embl\u00e9matique, comme il se doit, dans L'Origine du monde de Gustave Courbet. D'aucuns y virent, en leur temps, la laideur, l'anormal, le d\u00e9go\u00fbtant, ou bien le bizarre. Certains, encore, il y a peu...\n\nUne toison pubienne, une fente comme une blessure m\u00e9taphysique, des cuisses ouvertes, \u00e9cart\u00e9es, un ventre, un nombril, une chair, une carnation, un grain, des seins, un cadrage qui exclut aussi bien le visage que les bras et les jambes, un tissu comme une antith\u00e8se de saint suaire, voil\u00e0 pour le contenu. Un genre de gros plan sur ce qui traverse l'histoire et la g\u00e9ographie, le temps et l'espace, une quintessence qui contient potentiellement toutes les Annonciations, Nativit\u00e9s, Vierges \u00e0 l'enfant, et autres s\u00e9quences christiques de la vie de J\u00e9sus \u2013 de la fuite en Egypte aux le\u00e7ons donn\u00e9es aux docteurs du Temple \u2013, voil\u00e0 pour la mise en page et la mystique.\n\nDe la m\u00eame mani\u00e8re, cette toile appelle les s\u00e9ances, nombreuses dans l'histoire des images, et plus particuli\u00e8rement de la peinture, de viols et d'enl\u00e8vements de Proserpine ou d'Europe, des Sabines ou de Ganym\u00e8de, et autres brutalit\u00e9s ch\u00e8res \u00e0 la gent masculine qui, depuis l'homme des cavernes, ne s'encombre gu\u00e8re de principes en mati\u00e8re de suj\u00e9tion. Lucr\u00e8ce viol\u00e9e, c'est l'objet sur lequel Sextus, fils de Tarquin le Superbe, appose sa main n\u00e9gative en signe de trace, d'appropriation, de possession, de soumission. Il inscrit, d\u00e9limite et marque son territoire sur le corps d'une femme comme le conqu\u00e9rant sur la terre d'un pays. Plus civilis\u00e9s, les hommes invitent aux embarquements pour Cyth\u00e8re et font aborder des rives o\u00f9 Eros triomphe moins violemment. Apparaissent les logiques de s\u00e9duction, alors la sexualit\u00e9 se dissimule sous les pr\u00e9textes de l'\u00e9rotisme.\n\nD\u00e9j\u00e0 l\u00e0 se disent la puissance des corps soumis \u00e0 la libido, l'empire des sens et des passions sur une chair confin\u00e9e \u00e0 l'enregistrement des vibrations et des soubresauts de la n\u00e9cessit\u00e9. Mains n\u00e9gatives, vulves, phallus, puis viols, enl\u00e8vements et perp\u00e9tuation de la chair, tout ce qui court de Lascaux \u00e0 Picasso exhibe, montre \u00e0 l'envi l'empire de cette \u00e9nergie premi\u00e8re, de cette puissance g\u00e9n\u00e9alogique de tout le r\u00e9el. Or cette force est rien moins que bizarre, \u00e9trange, inqui\u00e9tante, sinon, dans ses effets violents, laide, horrible, anormale et sanglante. Sans elle, pas de monde, pas de r\u00e9el, pas d'existence, rien qu'un vide sid\u00e9ral, et le silence des plan\u00e8tes anim\u00e9es de mouvements inutiles.\n\nDu sang, pour suivre. Dans la symbolique, depuis toujours, le sang vaut principe de vie. Il est le vecteur du mouvement, du souffle, de la fertilit\u00e9, de l'abondance, mais aussi des passions, des chaleurs qui traversent le corps. Dans notre xxe si\u00e8cle, il est devenu aussi le v\u00e9hicule de la mort et de l'infection par le sida. Eros et Thanatos jou\u00e9s dans le m\u00eame flux, dans la m\u00eame rougeur vive appel\u00e9e \u00e0 s\u00e9cher, noircir, s'\u00e9cailler et devenir poussi\u00e8re. Les m\u00e9tamorphoses du sang ne manquent pas dans l'histoire de l'art, de la chasse \u00e0 la guerre en passant par le crime, l'assassinat et autres occasions hyst\u00e9riques ou politiques de le faire couler.\n\nDu sexe au sang, il y a peu, gu\u00e8re plus que du sang \u00e0 la mort. D\u00e9j\u00e0 la grotte pr\u00e9historique accueille cette v\u00e9rit\u00e9 : des phallus sont transform\u00e9s en armes, sinon des armes prennent une forme phallique. Les pointes de sagaies, les armes de poing, les harpons de p\u00eacheurs, les ramures de cervid\u00e9s sont d\u00e9cor\u00e9s avec des sc\u00e8nes de chasse o\u00f9 appara\u00eet la figure r\u00e9currente de l'homme bless\u00e9 \u2013 dont le sang coule. Poursuivi par un bison, dressant les bras devant sa charge, frapp\u00e9 par lui ou attaqu\u00e9 par un ours qui l\u00e8ve la patte, voire perc\u00e9 par de longues sagaies, le corps de l'homme pr\u00e9historique conna\u00eet le p\u00e9ril au quotidien. Il risque sa peau et le percement de celle-ci au moindre moment de la journ\u00e9e, m\u00eame en dehors d'une p\u00e9riode sp\u00e9cifique de chasse.\n\nLe combat semble la forme la plus ancienne de l'intersubjectivit\u00e9. Son mode le plus ancien, avant les joutes oratoires ou les oppositions m\u00e9taphoriques qui s'y substituent, c'est vraisemblablement le corps \u00e0 corps, le d\u00e9sir de tuer l'autre, d'en finir avec sa chair. Cette volont\u00e9 traverse les si\u00e8cles avec les guerres, mais aussi avec le sacrifice propitiatoire exigeant l'\u00e9lection d'une victime \u00e9missaire. Le combat des Horaces et des Curiaces illustre magnifiquement cette r\u00e9duction du divers \u00e0 l'un, cette m\u00e9taphysique brutale de la lutte \u00e0 mort pour la reconnaissance et la puissance triomphante.\n\nMais plus encore, et mieux, du moins avec un succ\u00e8s plus absolu, je veux retenir la passion du Christ qui fut l'occasion d'une multiplication consid\u00e9rable des traitements de ce sujet sanglant par excellence. A l'origine, la mont\u00e9e au calvaire s'accompagne de coups de fouet \u00e0 lani\u00e8res de cuir, de crachats, de souffrances associ\u00e9es au port de la croix, de flagellations, d'\u00e9ponges imbib\u00e9es de vinaigre. Et l'on sait la crucifixion, entre deux larrons, sur le Golgotha : couronne d'\u00e9pines et front perl\u00e9 de gouttes de sang, flanc perc\u00e9 \u00e0 coups de lance et lymphe \u00e9coul\u00e9e, clous enfonc\u00e9s dans les mains et dans les pieds, le sujet fournit les prototypes dans l'histoire de l'art des Ecce homo et autres Crucifixions.\n\nLes si\u00e8cles qui suivent l'installation du christianisme au pouvoir offrent, d'un point de vue pictural, une litt\u00e9rature consid\u00e9rable dans la martyrologie. Nicolas fouett\u00e9, Thomas perc\u00e9 par des lances, S\u00e9bastien travers\u00e9 par des fl\u00e8ches, Jean-Baptiste, Julienne, Longin, Timoth\u00e9e et tant d'autres d\u00e9capit\u00e9s, Christine la langue coup\u00e9e, des fl\u00e8ches dans le c\u0153ur et le c\u00f4t\u00e9, Laurent sur son grill, tous sont mis \u00e0 mal et sanguinolents pour le bonheur et l'\u00e9dification du croyant de base. Les \u00e9glises en sont pleines, les plus grandes, magnifiques et symboliques, Saint-Pierre de Rome par exemple, mais aussi les plus modestes, dans les villages de campagne partout en Europe. Du sang, encore et toujours, de la souffrance c\u00e9l\u00e9br\u00e9e, v\u00e9n\u00e9r\u00e9e, mise en sc\u00e8ne et offerte en exemple.\n\nJe songe, en forme de peinture embl\u00e9matique, \u00e0 la toile de Poussin intitul\u00e9e Le Martyre de saint Erasme dans laquelle, attach\u00e9, allong\u00e9 de dos sur un banc, les mains li\u00e9es, tendu en arc par des tortionnaires sans haine apparente sur le visage, le supplici\u00e9 se fait vider le ventre de ses visc\u00e8res, attach\u00e9s \u00e0 une poulie. Un homme entra\u00eene son m\u00e9canisme et enroule l'intestin sur un cylindre, comme ceux qu'on trouve au-dessus des margelles de puits. Le ventre vid\u00e9, l'\u00e2me pure, le corps tortur\u00e9, le sang r\u00e9pandu, tout vise l'\u00e9dification des croyants.\n\nHorreur, l\u00e0 encore, sauvagerie et laideur, esth\u00e9tique de la douleur. Le progr\u00e8s depuis les cavernes se fait attendre : on v\u00e9n\u00e8re toujours autant la n\u00e9gativit\u00e9 pour en faire une positivit\u00e9 pr\u00e9sentable sous forme d'art, de sacr\u00e9, de religieux, de mystique, de m\u00e9taphysique ou de sociologie. Le mat\u00e9riau de base demeure la chair ha\u00efe, maltrait\u00e9e, destin\u00e9e au p\u00e9ril, \u00e0 la mort et \u00e0 la d\u00e9composition. Ce que signifient avec redondance les peintures d'histoire, un genre aux sc\u00e8nes de guerre nombreuses.\n\nSang des batailles annonc\u00e9es, en cours ou termin\u00e9es \u2013 Uccello, Breughel de Velours, Delacroix, David, Goya, Picasso, Otto Dix et tant d'autres \u2013, les champs guerriers \u00e0 San Romano, le massacre des Innocents, Issus ou Scio, les fusillades d'opposants \u00e0 Madrid, l'assassinat d'un tribun r\u00e9volutionnaire dans sa baignoire, le passage du col du Grand-Saint-Bernard, la retraite de Russie, les conflits d'int\u00e9r\u00eats entre royaut\u00e9s et empires d'Europe, les guerres r\u00e9volutionnaires, celles des conqu\u00eates napol\u00e9oniennes, victoires ou d\u00e9faites, fascismes et totalitarismes du XXe si\u00e8cle : le sang ne cesse d'\u00eatre mis en sc\u00e8ne, comme une variation sur le th\u00e8me pr\u00e9historique de la chasse. Pr\u00e9dateurs et proies s'opposent depuis toujours sur des fronts militaires o\u00f9 coule \u00e0 flots le liquide pr\u00e9cieux.\n\nDe la m\u00eame mani\u00e8re que jubilent les artistes dans le sang que font couler les hommes, ils aiment celui que verse la nature. Nombre de peintres sp\u00e9cialis\u00e9s dans les catastrophes excellent dans la mise en sc\u00e8ne de l'horreur consubstantielle au d\u00e9cha\u00eenement des \u00e9l\u00e9ments : tremblements de terre et leurs cons\u00e9quences, raz-de-mar\u00e9es, \u00e9ruptions volcaniques, villes d\u00e9sert\u00e9es, paysages foudroy\u00e9s, ruines, incendies dans les temples et les \u00e9glises, cit\u00e9s embras\u00e9es, pillages et massacres commis par les survivants hyst\u00e9riques. Qu'on se souvienne de l'\u0153uvre enti\u00e8re de Monsu Desiderio, de John Martin ou les toiles de P.H. de Valenciennes. Sang, encore et toujours.\n\nDe la mort, pour finir. Apr\u00e8s le sexe et ses dynamiques, le sang et ses coulures, la mort et ses cadavres, en abondance. Le trajet de toute existence s'inscrit dans ces stations : na\u00eetre, vivre, se reproduire, souffrir, mourir. D\u00e9j\u00e0 Lascaux montre des animaux d\u00e9capit\u00e9s, des bouquetins ou cervid\u00e9s sans t\u00eate. Magie de chasseur ? Animaux pi\u00e9g\u00e9s et envo\u00fbtements r\u00e9ussis? Rite de conjuration mystique ? S\u00e9paration du corps et de l'\u00e2me, d\u00e9j\u00e0? Cadavres de b\u00eates avec lesquels on pr\u00e9pare la nourriture des vivants ? De toute fa\u00e7on, l'int\u00e9grit\u00e9 du corps \u00e9tant atteinte, ce qui est propos\u00e9 vaut figure amput\u00e9e, existence entam\u00e9e.\n\nL'iconographie chr\u00e9tienne aime les cadavres, les mourants et les morts. Le Christ descendu de croix, faisant l'objet de lamentations collectives, allong\u00e9 sur les jambes de sa m\u00e8re, en Piet\u00e0, chaque fois on montre un corps mutil\u00e9, bless\u00e9 mortellement. Plaies au c\u00f4t\u00e9, larmes de sang sur le visage, pieds et mains d\u00e9fonc\u00e9s, le Dieu fait homme triomphe d'abord en homme, en chair martyris\u00e9e. L\u00e0 encore, je songe \u00e0 une \u0153uvre embl\u00e9matique, en l'occurrence \u00e0 la c\u00e9l\u00e8bre Crucifixion de Matthias Gr\u00fcnewald peinte sur un panneau du retable d'Issenheim.\n\nLes mains tordues de souffrance, comme des araign\u00e9es, les pieds contraints dans une torsion douloureuse, le thorax t\u00e9tanis\u00e9 par l'\u00e9touffement mortel toujours induit par la mise en croix, le sang, bien s\u00fbr, en abondance l\u00e0 o\u00f9 les clous et la lance ont \u00e9t\u00e9 port\u00e9s. Et puis cette chair d\u00e9j\u00e0 pourrie, corrompue, tach\u00e9e, verte, aux reflets de la d\u00e9composition annonc\u00e9e. Enfin ces piq\u00fbres qui constellent le corps en nombre infini et transforment la peau du Christ en pauvre v\u00eatement mit\u00e9. Quant au visage, il est construit sur un cri, comme presque toutes les \u0153uvres de Francis Bacon. A cette diff\u00e9rence pr\u00e8s : chez le Ma\u00eetre d'Issenheim, le cri n'a pas m\u00eame la force d'\u00eatre lanc\u00e9, il semble s'\u00e9baucher et s'achever aussit\u00f4t en une plainte \u00e9chou\u00e9e, un r\u00e2le long et lent. Le corps du Christ mort est bien celui d'une divinit\u00e9 qui a choisi d'endosser le destin des hommes.\n\nNombre de martyrs ont souhait\u00e9 ce destin, cette confusion de leur trajet avec celui de leur inspirateur mort sur la croix. Et \u00e0 leurs c\u00f4t\u00e9s ceux qui constituent le panth\u00e9on des martyrs la\u00efcs, Nastagio degli Onesti chez Botticelli, le fils de Saturne ou les cannibales de Goya, Bara ou Marat et autres h\u00e9ros de la R\u00e9volution fran\u00e7aise chez David. Mais d\u00e9j\u00e0 V\u00e9sale avait inaugur\u00e9 un genre qui surclassait le gisant ou l'\u00e9corch\u00e9 en pratiquant le relev\u00e9 anatomique et \u00e0 sa suite, tous les sp\u00e9cialistes italiens en c\u00e9roplastie dont l'anatomie est le pr\u00e9texte \u2013 ainsi Mascagni ou Fontana.\n\nRembrandt peint plus tard deux superbes toiles d'un cours de dissection, dont La Le\u00e7on d'anatomie du Docteur Joan Deyman qui propose un cadavre dont les pieds au premier plan et le cerveau fendu, le cr\u00e2ne d\u00e9fonc\u00e9, les m\u00e9ninges coup\u00e9es, puis ouvertes comme un linge, le ventre \u00e9visc\u00e9r\u00e9, ne parviennent pas \u00e0 effacer mon impression qu'on pourrait, ici, autopsier un Christ apr\u00e8s la descente de croix. Le traitement de l'information installe cette toile aux c\u00f4t\u00e9s de La Mise au tombeau qu'on peut voir \u00e0 Munich aussi bien que du Quartier de b\u0153uf expos\u00e9 \u00e0 Paris. Il y a peu entre la nature morte qui prend pour objet le cadavre, ou n'importe quelle autre chair d\u00e9sert\u00e9e par la vie, et les peintures classiques qui traitent de la crucifixion et de l'apr\u00e8s-mort du Christ.\n\nLe b\u0153uf de Rembrandt, donc, mais aussi le saumon de Melendez, le hareng de Claesz, la raie de Chardin, le gibier d'Oudry, le poulet de Soutine, la t\u00eate de mouton de Picasso, voire toutes les toiles dans lesquelles Francis Bacon \u00e9tudie une crucifixion, un portrait, un autoportrait ou peint tout simplement une Figure avec viande, tout exprime la parent\u00e9 de toutes les viandes, celle du plus humble des poissons et celle du plus c\u00e9l\u00e8bre des crucifi\u00e9s. Il n'y existe que des natures mortes, fussent-elles portraits, figures solitaires royales, imp\u00e9riales ou anonymes, ou en groupe, agr\u00e9gats de familles bourgeoises dans la p\u00e9riode classique, paysages romains ou autres, sc\u00e8nes de fumerie hollandaises. Le Lorrain tout autant que Le Brun.\n\nNature morte, le portrait de Louis XIV par Hyacinthe Rigaud ; nature morte, celui des \u00e9poux Arnolfini de Jan Van Eyck; nature morte aussi celui de Monsieur Leblanc puis de son \u00e9pouse immortalis\u00e9s s\u00e9par\u00e9ment, en bons bourgeois, par Ingres ; natures mortes, enfin, sinon Vanit\u00e9s, Le Serment du Jeu de paume, Le Sacre de Napol\u00e9on par David, Guernica de Picasso, ou toute autre peinture dite d'Histoire. Car chaque fois ce qui est fix\u00e9, fig\u00e9, immobilis\u00e9 dans l'illusion de l'\u00e9ternit\u00e9 ce sont des moments fugaces, pass\u00e9s et qui ne reviendront pas.\n\nTout est vanit\u00e9, aussi bien le genre pratiqu\u00e9 comme tel, sous ce nom et cette rubrique, que tout ce qui cache et travestit la mouche habitu\u00e9e des charognes, le ver dans le fruit, le sablier qui coule, le papillon, les fleurs appel\u00e9es \u00e0 se faner, les cr\u00e2nes et autres symboles de la pr\u00e9carit\u00e9 du temps. Parce qu'il faudra mourir, l'artiste court \u00e0 l'ab\u00eeme, hant\u00e9, habit\u00e9 par le sexe, le sang, la mort. Mais il mourra, bient\u00f4t, avant que d'autres prennent sa suite, en obs\u00e9d\u00e9s du bizarre, en fous de la laideur, en fantasques travaill\u00e9s par l'horrible.\n\nCertes le sexe, le sang, la mort n'\u00e9puisent pas l'ensemble des productions artistiques. D'autant moins depuis que la peinture a abandonn\u00e9 le sujet et s'est achemin\u00e9e \u2013 \u00e0 partir de l'impressionnisme jusqu'\u00e0 l'abstraction pure \u2013 vers le seul culte de la forme ou de la lumi\u00e8re, c'est-\u00e0-dire du trait et de la couleur gratuits. Depuis les d\u00e9constructions des Nymph\u00e9as de Monet, la peinture a la plupart du temps cong\u00e9di\u00e9 le sujet. Mais ceux qui persistent dans son traitement, en l'empoignant \u00e0 bras-le-corps, Pablo Picasso et Henri Matisse avant-hier, Francis Bacon et Lucian Freud hier, Rustin et Vladimir V\u00e9likovick aujourd'hui, tous retrouvent le chemin de Lascaux et des cavernes pr\u00e9historiques pour exprimer la permanence de l'apocalypse, l'\u00e9ternel empire sur les hommes du bizarre et de la laideur, de la mort et du sang, du sexe et de l'horrible.\n\nGuerres et corridas, maladies et vieillesse, d\u00e9luges et naufrages, catastrophes et cataclysmes, enl\u00e8vements de vierges et \u00e9tals de bouchers, assassinats et maternit\u00e9s, \u00e9rotisme et sexualit\u00e9, cannibales et monstres, moutons d\u00e9capit\u00e9s et t\u00eates align\u00e9es en salles d'anatomie, sorci\u00e8res hyst\u00e9riques et femmes lascives, phallus triomphants et tortures inflig\u00e9es, m\u00eame m\u00e9taphoriques, surtout m\u00e9taphoriques, voil\u00e0 qui n'a cess\u00e9 de nourrir la peinture. Voil\u00e0 qui ne cessera de la nourrir tant la figuration pure, tout autant que l'abstraction pure \u2013 pour autant qu'elles aient jamais exist\u00e9 ainsi, sans m\u00e9langes \u2013 affichent aujourd'hui les impasses dans lesquelles elles se trouvent. Basquiat l'a montr\u00e9, Keith Harring aussi, et d'autres qui \u0153uvrent dans le silence o\u00f9 les ateliers fonctionnent en athanor : dans le creuset dont ils ont le secret, les artistes fouleront toujours le sexe, le sang, la mort pour transfigurer ces mati\u00e8res \u00e9ternelles en peintures magnifiques qui nous permettent d'attendre la mort en la mettant quelque peu \u00e0 distance \u2013 avant qu'elle ne s'en vienne enfin nous chercher.\n31\n\nLES MIROIRS DE NARCISSE AU TH\u00c9\u00c2TRE\n\n(Commentaires sur le spectacle de la soci\u00e9t\u00e9) \u2013. En tous lieux et temps, l'art est conceptuel. Nul besoin d'attendre, au xxe si\u00e8cle, celui qui se pr\u00e9sente sp\u00e9cifiquement comme tel pour arr\u00eater une date de naissance. Des premi\u00e8res traces aux derni\u00e8res, ce que dit l'artiste propose toujours un miroir dans lequel il avance l'id\u00e9e qu'il se fait de lui-m\u00eame. Une \u00e9poque agit pareillement en offrant au regard ce qu'elle pense d'elle sous forme d'artifice ouvrag\u00e9. Que ce soit \u00e0 son corps d\u00e9fendant ou \u00e0 dessein. De sorte qu'il n'est pas de lecture possible, d'une toile de Poussin aussi bien que d'une \u0153uvre de Carl Andr\u00e9, sans une grille, un d\u00e9codage d'une symbolique sp\u00e9cifiquement adapt\u00e9e \u00e0 l'\u0153uvre et \u00e0 ses conditions de production.\n\nL'illusion de ceux qui prisent l'art ancien et fustigent l'art contemporain vient de ce qu'ils croient comprendre une \u0153uvre d'art classique quand ils se contentent d'un plaisir pris \u00e0 la ressemblance, au m\u00e9tier, \u00e0 l'habilet\u00e9 de l'artisan. Pourtant, qui peut se limiter \u00e0 la seule surface des peintures du Grand Si\u00e8cle ? Qui se satisfait de l'\u00e9cume, de l'apparence, de l'illusion, de l'impression d\u00e9corative premi\u00e8re? Ceux qui demandent \u00e0 l'art de se cantonner aux fonctions du mobilier bourgeois. Et ils sont de plus en plus nombreux \u00e0 penser pareillement avec les productions les plus r\u00e9centes. Apr\u00e8s le devenir petit-bourgeois de l'Impressionnisme, on assiste actuellement \u00e0 une m\u00e9tamorphose du m\u00eame genre pour les \u0153uvres de ce si\u00e8cle qui s'ach\u00e8ve.\n\nPour autant, le temps passant, les \u0153uvres paraissent de moins en moins lisibles et ce d'autant qu'elles s'\u00e9loignent de leurs conditions d'av\u00e8nement. Un contemporain du Grand Si\u00e8cle, inform\u00e9 de la mythologie, de la symbolique, des usages, des coutumes, des r\u00e9f\u00e9rences de l'\u00e9poque, un commanditaire ou un spectateur affranchis des codes, comprenaient vraisemblablement mieux l'art de leur temps que nous, en cette fin de mill\u00e9naire inculte et barbare, le leur \u2013 et aussi le n\u00f4tre... Ce que dit la peinture vaut toujours comme un concentr\u00e9 de mythologie sociale, un succ\u00e9dan\u00e9 des mensonges sociaux auxquels une civilisation sacrifie pour s'asseoir, exister et durer.\n\nL'id\u00e9e me venait, en rentrant de l'exposition que le Mus\u00e9e des Beaux-Arts de Nantes a consacr\u00e9e au portrait dans la peinture contemporaine de Louis XIV, que peut-\u00eatre jamais on avait autant charg\u00e9 l'art pictural d'une \u00e9poque de symboles, de signes et de connivences. Le Grand Si\u00e8cle excelle dans la volont\u00e9 de se mettre en sc\u00e8ne, de se repr\u00e9senter, de se donner en spectacle, et d'aimer l'image qu'il donne de lui-m\u00eame. Narcisse au th\u00e9\u00e2tre. Je songeais plus particuli\u00e8rement \u00e0 cette toile de Hyacinthe Rigaud repr\u00e9sentant Louis XIV dans la posture officielle qui a travers\u00e9 les si\u00e8cles, pour finir sur les bons points de mon enfance, entre autres images d'Epinal, ou dans les chapitres du livre d'histoire consacr\u00e9 au XVIIe si\u00e8cle. Chacun conna\u00eet cette peinture reproduite \u00e0 des millions d'exemplaires, des gravures de l'\u00e9poque aux copies de toujours en passant par les \u00e9puisements dans les chromos utiles aux encyclop\u00e9dies d'histoire de l'art. J'ai voulu, apr\u00e8s avoir vu l'exposition, comprendre plus pr\u00e9cis\u00e9ment ce que dit cette toile, ce qu'elle a de conceptuel, et aller au-del\u00e0 de l'image banale.\n\nTout, dans cette \u0153uvre inaugurale du si\u00e8cle (1701) exprime l'essence et la nature divines du pouvoir royal. La mise en sc\u00e8ne, la mise en page, la grammaire de tous les objets requis et plac\u00e9s, la composition, la structure, les citations, les r\u00e9f\u00e9rences, les couleurs, les v\u00eatements, les drap\u00e9s, les plis, la posture du corps, tout fait signe pour souligner la participation divine du Roi au Ciel, la relation intime et g\u00e9n\u00e9tique du pouvoir royal, terrestre, et du pouvoir divin, c\u00e9leste. La m\u00e9taphysique du si\u00e8cle de Louis XIV est radicalement religieuse et plus particuli\u00e8rement chr\u00e9tienne dans sa version f\u00e9odale.\n\nLa grammaire d'objets symboliques suppose le sceptre, la main de justice, la couronne, l'\u00e9p\u00e9e, la m\u00e9daille, l'estrade, le tr\u00f4ne et la fleur de lys. L'ensemble se r\u00e9sume \u00e0 la mise en sc\u00e8ne de ce mobilier conceptuel. J'ai recherch\u00e9 dans ma biblioth\u00e8que le texte de rituel du sacre royal tel qu'il est rapport\u00e9 par Du Tillet et Godefroy dans leur version fran\u00e7aise. J'y ai retrouv\u00e9 ce qui appara\u00eet dans le portrait de Hyacinthe Rigaud et fait le symbolisme du pouvoir royal.\n\nLe sceptre, par exemple, que le Roi re\u00e7oit des mains de l'archev\u00eaque le jour du couronnement, \u00e9quivaut, on le sait, \u00e0 la puissance royale. Il est \u00abdroite verge du royaume, verge de vertu \u00bb \u00e0 l'aide de quoi le Roi s'engage \u00e0 d\u00e9fendre le royaume de toutes les agressions possibles et imaginables, surtout celles qui viseraient \u00e0 s\u00e9parer la France et l'Eglise, la premi\u00e8re \u00e9tant dite fille a\u00een\u00e9e de la seconde. Avec le sceptre, on peut redresser, indiquer la bonne direction ou confirmer la bonne et droite voie. Sa forme synth\u00e9tise l'axe du monde, une image \u00e0 laquelle toute civilisation a toujours sacrifi\u00e9.\n\nLes cosmogonies traditionnelles supposent un point de jonction entre le ciel et la terre. Le sceptre fournit l'occasion de cette r\u00e9union. Voil\u00e0 pourquoi, chez Rigaud, le Roi tient cet objet verticalement, de sorte que dans sa main, par sa puissance et son intercession, le terrestre et le c\u00e9leste se relient dans l'essentiel. La liaison se fait au lieu m\u00eame o\u00f9 est le sceptre. Cet endroit fixe et d\u00e9signe la capitale th\u00e9ocratique, la ville c\u00e9leste incarn\u00e9e, indissociable de la personne et de la figure royales.\n\nDans la toile, la pointe de l'objet, une fleur de lys, touche le bleu d'un coussin fleurdelis\u00e9 o\u00f9 reposent d'autres attributs de la royaut\u00e9. Entre autres une main de justice qui vaut symboliquement pour la main gauche de Louis associ\u00e9e aux arcanes de la vie spirituelle. \u00abVerge de vertu et d'\u00e9quit\u00e9 \u00bb, elle rend possible l'exercice des vertus cardinales sur le principe d'une cl\u00e9 de David \u2013 David signifiant d'ailleurs \u00ab vaillante main \u00bb. Son pouvoir ne cesse d'\u00eatre grand puisqu'elle gu\u00e9rit les maladies et les infirmit\u00e9s spirituelles des hommes comme individualit\u00e9s profanes, certes, mais aussi comme fragments du corps social.\n\nTraditionnellement, cette main est un b\u00e2ton d'or d'une coud\u00e9e de longueur. A son extr\u00e9mit\u00e9, la main est d'ivoire. Le pouce, l'index et le majeur sont dress\u00e9s ; l'annulaire et l'auriculaire pli\u00e9s. L'annulaire est p\u00e9nitence, le m\u00e9dius charit\u00e9, l'index raison d\u00e9monstrative, le pouce, divinit\u00e9. L'agencement de ces doigts dans la main propose \u00e0 lui seul un discours th\u00e9ologique et politique. Le pliage signifie la cl\u00e9mence et le retour sur soi, l'auriculaire annonce la disponibilit\u00e9 du Roi \u00e0 entendre. L'ensemble exprime \u00e0 qui peut, veut et sait lire, la charit\u00e9 et l'intelligence qui fait comprendre, la foi et la p\u00e9nitence en retrait. L'ivoire vaut pour la puret\u00e9 et l'inflexibilit\u00e9 de la justice.\n\nSur le m\u00eame coussin, en retrait du Roi, on peut aussi remarquer la couronne. Le jour du sacre, pos\u00e9e sur la t\u00eate royale, elle signifie \u00ab mis\u00e9ration et mis\u00e9ricorde \u00bb. Ses rayons expriment la puissance de diffraction universelle du principe monoth\u00e9iste th\u00e9ologique et de sa forme politique, la monarchie. L'objet, structur\u00e9 en trois registres, autorise une lecture horizontale et verticale. Chaque bande signifie un ordre, l'agencement signalant un principe g\u00e9n\u00e9ral, global. Les trinit\u00e9s correspondent aussi bien aux trois \u00e2ges de la vie, de l'homme ou de l'humanit\u00e9, qu'aux trois \u00e9tages de l'univers, aux trois ordres sociaux, ou autres variations sur le th\u00e8me, cher \u00e0 Dum\u00e9zil, de la trifonctionnalit\u00e9.\n\nL'ensemble symbolise l'\u00eatre absolu, la substance la plus subtile de la cr\u00e9ation. L'objet rappelle la J\u00e9rusalem c\u00e9leste dans laquelle l'homme b\u00e9n\u00e9ficie des avantages non n\u00e9gligeables de la vie \u00e9ternelle. Le couronnement, comme c\u00e9r\u00e9monie, permet de r\u00e9aliser en l'homme singulier (Louis le quatorzi\u00e8me) l'homme total (le Roi) en qui, gr\u00e2ce \u00e0 qui et avec qui, chacun trouve sa place. Par la couronne, l'universel s'incarne dans le particulier qui, d\u00e8s lors, et ainsi transform\u00e9, m\u00e9tamorphos\u00e9, devient divin. La t\u00eate humaine du monarque est alors en contact avec la vo\u00fbte c\u00e9leste et la totalit\u00e9 de l'univers. Cette ouverture sur l'immensit\u00e9 de la vie se retrouve dans la tonsure des moines, ou la kippa des Juifs.\n\nA son c\u00f4t\u00e9 gauche le Roi arbore une \u00e9p\u00e9e visible par son pommeau et la naissance du fourreau. La garde est ouvrag\u00e9e et la gaine sertie de pierres pr\u00e9cieuses rouges et bleues. Ce qui suppose vraisemblablement rubis et saphirs, \u00e0 savoir la pierre la plus ardente, m\u00e9taphore de la lumi\u00e8re dont aucune t\u00e9n\u00e8bre ne parvient \u00e0 venir \u00e0 bout, et gemme c\u00e9leste dont la contemplation m\u00e8ne aux cieux. Le saphir, symbole d'esp\u00e9rance et de justice divine, suppose la pr\u00e9vention de la pauvret\u00e9, la protection contre la col\u00e8re, la trahison, les mauvais jugements. Il augmente courage, joie, vitalit\u00e9 et dissipe les humeurs.\n\nL'arme en tant que telle pr\u00e9sente une utilit\u00e9 manifeste quand il s'agit de repousser les adversaires, aussi bien de l'Eglise que du Royaume \u2013 mais ce sont les m\u00eames. En toute civilisation, l'\u00e9p\u00e9e signifie la puissance supr\u00eame. Par elle, la puissance divine devient puissance humaine, la justice de Dieu, justice des hommes. Avec elle se s\u00e9parent les imperfections et s'obtient la qualit\u00e9 spirituelle. Verbe \u00e9clair de la parole du Christ, elle remplit l'homme qui en dispose gr\u00e2ce \u00e0 la puissance primordiale. De m\u00eame la tradition pense l'\u00e9p\u00e9e en m\u00e9taphore de l'androgynat primitif, car en elle la lame une, bien que constitu\u00e9e de deux fils tranchants, permet la r\u00e9conciliation du masculin et du f\u00e9minin.\n\nDans la c\u00e9r\u00e9monie du sacre, le Roi re\u00e7oit l'\u00e9p\u00e9e de la main de l'archev\u00eaque. On ne peut mieux dire que le pouvoir royal tient sa puissance du pouvoir divin, que le Roi est sur terre le repr\u00e9sentant de Dieu, son envoy\u00e9, celui qu'il a d\u00e9sign\u00e9 pour conduire tr\u00e8s chr\u00e9tiennement les affaires des hommes. L'\u00e9p\u00e9e obtenue du religieux est pos\u00e9e sur l'autel, puis rendue au Conn\u00e9table de France, s'il existe, sinon \u00e0 l'un des Barons choisis qui la portera et la tiendra rigoureusement et scrupuleusement droite pendant toute la c\u00e9r\u00e9monie : sa verticalit\u00e9 annonce la droiture du r\u00e8gne qu'on inaugure. Il s'agit de montrer \u00e0 tous l'axe \u00e9ternellement stable du monde.\n\nAvec le sceptre, la main de justice, la couronne et l'\u00e9p\u00e9e, arborant le collier du Saint-Esprit \u2013 une longue et lourde cha\u00eene d'or au bout de laquelle pend une d\u00e9coration \u00e9toil\u00e9e \u2013, le Roi r\u00e8gne sur un ensemble d'objets fleurdelis\u00e9s : le coussin, son v\u00eatement, le sceptre, le tr\u00f4ne, tout est marqu\u00e9 aux fleurs de lys. Les symboliques classiques font proc\u00e9der la fleur de lys d'hypoth\u00e9tiques d\u00e9formations d'abeilles qui, nombreuses, \u00e9taient cens\u00e9es se grouper en essaim autour du Roi pour souligner sa nature divine en m\u00eame temps que l'unit\u00e9 du peuple autour de lui.\n\nA l'origine de la figuration, au d\u00e9but, et avant toute stylisation, la fleur de lys, constitu\u00e9e de six p\u00e9tales, exprime le chiffre de la perfection, de l'union du ciel et de la terre. Clovis, dit-on, l'aurait choisie comme embl\u00e8me de puret\u00e9 lors de son bapt\u00eame. En se simplifiant, elle se montre sous la forme de deux virgules, l'une orient\u00e9e vers la droite, l'autre vers la gauche, agenc\u00e9es autour d'un axe. L\u00e0 se manifeste la double nature du Roi, sa participation au divin et sa dimension humaine. Lorsqu'elle surmonte le sceptre, elle rappelle l'archange Gabriel, notamment lorsqu'il annonce la bonne nouvelle \u00e0 Marie.\n\nLe Roi et ses attributs apparaissent sur un genre d'estrade, elle-m\u00eame essentiellement visible par une marche de trois degr\u00e9s : celui o\u00f9 officie le peintre, celui, aussi, des autres hommes et du peuple ; celui o\u00f9 le Roi se montre, lui et les signes de son pouvoir ; celui o\u00f9 se trouve son tr\u00f4ne, vide. On songe, bien s\u00fbr, dans cet ordonnancement, \u00e0 un genre d'\u00e9chelle symbolique dont les degr\u00e9s conduisent \u00e0 des certitudes d'autant plus essentielles qu'elles se situent au sommet. La dynamique induite par les marches suppose l'initiation et rend possible l'acc\u00e8s \u00e0 des connaissances sup\u00e9rieures.\n\nDe Hyacinthe Rigaud, r\u00e9duit et contraint \u00e0 une station sur le premier degr\u00e9, au tr\u00f4ne, vide, mais triomphant au sommet de cette gradation, il y a tout ce qui permet de mettre une fois de plus en \u00e9vidence l'intercession du Roi entre les deux ordres : terrestre, humain, trop humain, et c\u00e9leste, divin, tr\u00e8s divin. Un monde sensible o\u00f9 croupissent le r\u00e9el, la mati\u00e8re, le monde, les gens, le quotidien, s'oppose \u00e0 un monde intelligible o\u00f9 para\u00eet le symbole essentiel, le vide \u00e9quivalent \u00e0 la pr\u00e9sence absolue, supr\u00eame.\n\nL'organisation du r\u00e9el en trois temps occupe un cinqui\u00e8me de la toile, celui de sa base : le tiers qui concerne l'ensemble des hommes, utilise une partie ridicule de la toile, dans le coin inf\u00e9rieur gauche ; la partie m\u00e9diane, centrale, celle du Roi dont on fait le portrait, la plus importante en espace occup\u00e9, utilise les trois quarts de la surface totale au sol ; en revanche, le troisi\u00e8me temps, celui du c\u00e9leste, n'est qu'entr'aper\u00e7u picturalement et ne se r\u00e9v\u00e8le que par la position sur\u00e9lev\u00e9e du si\u00e8ge royal qui s'y trouve.\n\nLe tr\u00f4ne, donc, est en retrait sur la figure et la personne royale. Il se dissimule en partie dans l'ombre donn\u00e9e par une tenture. Pourtant, il est l'objet essentiel en tant que matrice dont proc\u00e8de tout pouvoir royal. Si l'on peut dire \u00ab Vive le Roi \u00bb quand le Roi meurt, c'est parce que malgr\u00e9 la mort physique d'un homme, un Louis quelconque, la royaut\u00e9, elle, perdure. Et si elle dure, c'est gr\u00e2ce au tr\u00f4ne, centre de l'\u00e9difice sacr\u00e9 qui engendre le souverain. Les Rois passent, le tr\u00f4ne demeure. Son nom latin, cathedra, donnera l'\u00e9tymologie de l'\u00e9difice chr\u00e9tien que l'on sait. Symboliquement, le tr\u00f4ne exprime la cl\u00e9 des potentialit\u00e9s divines et r\u00e9gule l'\u00e9nergie universelle. G\u00e9n\u00e9ralement, les artisans menuisiers qui le r\u00e9alisent ne sont pas avares de symbolique chiffr\u00e9e pour exprimer l'excellence des proportions de l'univers. On laisse aux secrets g\u00e9om\u00e9triques le soin de raconter la divine harmonie, l'\u00e9quilibre cosmique.\n\nLe portrait du Roi met en sc\u00e8ne ses attributs, ses pr\u00e9rogatives, sa force et sa puissance. L'exercice propose une apolog\u00e9tique. La mise en page de la peinture de Rigaud se fait sous un dais tendu de riches brocarts. Sceptre et main de justice, couronne et \u00e9p\u00e9e, estrade et tr\u00f4ne, m\u00e9daille et fleur de lys, tout se montre sur sc\u00e8ne, le dais valant rideau d'un th\u00e9\u00e2tre ouvert sur le spectacle de la com\u00e9die royale. Mais le tissu pr\u00e9cieux de ce rideau de sc\u00e8ne en forme de citation la\u00efque et pa\u00efenne, renvoie, par sa nature, sa texture et sa composition, puis ses couleurs, \u00e0 une symbolique tout aussi charg\u00e9e.\n\nCar aucune couleur, dans cette toile, ne rel\u00e8ve bien \u00e9videmment du hasard. Toutes expriment l'excellence, la virginit\u00e9, la puissance, le pouvoir et le c\u00e9leste. Le pourpre, bien s\u00fbr, du lourd dais de velours, et les ors qui le frangent et festonnent, symbolisent le pouvoir et le luxe. Dans l'Antiquit\u00e9 romaine, le coquillage qui permettait d'extraire cette couleur \u00e9tait rare, donc pr\u00e9cieux. Elle fut utilis\u00e9e par les Empereurs romains pour signifier leur puissance avant que l'Eglise, elle aussi, n'en fasse un signe de pouvoir pour ses cardinaux.\n\nPourpre du sang christique, sang vers\u00e9 par J\u00e9sus sur la croix pour le rachat des p\u00e9ch\u00e9s du monde, couleur du liquide de r\u00e9demption, et de purification pour les croyants catholiques, le rouge signifie aussi la Pentec\u00f4te et l'Esprit Saint. Dans l'\u0153uvre de Hyacinthe Rigaud, l'\u00e9piphanie du sacr\u00e9 se fait sur le corps du Roi, en un enveloppement annonc\u00e9 par le haut, du ciel vers la terre, du sacr\u00e9 vers l'humain, du divin vers le sensible. Lumi\u00e8re, souffle, il brille et r\u00e9chauffe, \u00e9claire comme l'astre solaire \u2013 et l'on sait la dilection de Louis pour la plan\u00e8te aux rayons dor\u00e9s.\n\nL'or se lit sur les m\u00eames principes symboliques que le pourpre rare et pr\u00e9cieux. Le plus introuvable des m\u00e9taux fait songer, par ses reflets, \u00e0 la lumi\u00e8re qui, elle, ne cesse d'\u00eatre associ\u00e9e \u00e0 la parole divine, au verbe christique, au propos biblique \u2013 celle de saint Jean Bouche d'Or par exemple. En alchimie, bien s\u00fbr, l'or est finalit\u00e9 de la transmutation, aboutissement des m\u00e9tamorphoses, signe d'une op\u00e9ration r\u00e9ussie de conversion des forces modestes, des natures viles, des structures courantes en r\u00e9duction d'excellence. De m\u00eame, dans les r\u00e9cits mythologiques fondateurs, l'Age d'Or correspond au moment de perfection, avant toute entropie, il caract\u00e9rise les temps paradisiaques, les \u00e9poques glorieuses et b\u00e9nies \u2013 ceux qu'imagine Louis XIV pour son si\u00e8cle.\n\nChez les Egyptiens aussi, grands pourvoyeurs de mythes pour l'Occident du Grand Si\u00e8cle, l'or est la chair du soleil, la mati\u00e8re du Roi-Pharaon. Dans le portrait de Hyacinthe Rigaud, les cordeli\u00e8res \u00e0 glands, les revers de tenture, le fonds du dais, les broderies du coussin, celles des fleurs de lys, les boiseries du tr\u00f4ne, tout est d'or. Et l'on songe, bien \u00e9videmment, aux prosp\u00e9rit\u00e9s associ\u00e9es, notamment \u00e0 l'or des \u00e9pis de bl\u00e9, dans un monde enti\u00e8rement rural o\u00f9 les moissons constituent les richesses des nations, puis \u00e0 celui des pi\u00e8ces de monnaie des tr\u00e9sors du royaume terrestre.\n\nDans la logique chromatique signifiante, le rouge et or dispose d'un pendant contrapuntique dans le bleu et blanc. Le premier groupe vaut pour la puissance et la richesse, le second pour la virginit\u00e9, la puret\u00e9 et les vertus associ\u00e9es. Le bleu, couleur de l'azur et de l'infini, ne s'obtient dans la nature que par la transparence qui, elle, se r\u00e9alise par l'intercession de l'air ou de l'eau, \u00e9l\u00e9ments de la puret\u00e9. Entre le n\u00e9ant de l'espace sid\u00e9ral ou celui des fonds sous-marins, lorsque s'interpose la transparence, on obtient la couleur bleue.\n\nD'o\u00f9 son association \u00e0 l'immat\u00e9rialit\u00e9, donc \u00e0 la perfection sainte qu'on pr\u00eate \u00e0 la Vierge Marie. Couleur mariale, elle devient sans difficult\u00e9 celle de l'innocence et de la puret\u00e9, de la spiritualit\u00e9 quintessenci\u00e9e apr\u00e8s toute op\u00e9ration de d\u00e9mat\u00e9rialisation. Kandinsky \u00e9crit de cette couleur qu'elle \u00abattire l'homme vers l'infini et \u00e9veille en lui le d\u00e9sir de puret\u00e9 et une soif de surnaturel \u00bb. Le manteau du Roi, celui du sacre, mais aussi celui du portrait, le tr\u00f4ne, le matelassage de la couronne, le coussin sur lequel reposent les attributs sacr\u00e9s du monarque, tout est bleu.\n\nQuant au blanc, il caract\u00e9rise les dentelles, fines broderies pr\u00e9cieuses en point de Venise, d'Alen\u00e7on ou d'Argentan, milliers d'heures de labeur d\u00e9licat pour de fragiles pi\u00e8ces qui fournissent un genre de vanit\u00e9s, de m\u00e9taphores du temps travaill\u00e9, ourl\u00e9, cousu, agenc\u00e9 en figures singuli\u00e8res et symboliques, rares et pr\u00e9cieuses. Blanches la cravate de soie, la veste et la chemise, le tout festonn\u00e9 ou rehauss\u00e9 de dentelles. Blanche la culotte avec \u00e9galement des raffinements de broderies vaporeuses. Blancs aussi les bas de soie institu\u00e9s par les Valois pour le costume d'apparat. Blanches enfin les jarreti\u00e8res fix\u00e9es sous le genou royal avec une boucle sur le c\u00f4t\u00e9.\n\nLe v\u00eatement de Louis XIV justifie, explique ou explicite les ateliers mis en place par Colbert qui a impuls\u00e9 une production importante de soie, de dentelles et autres fabriques de tissus pr\u00e9cieux Mais la blancheur absolue et symbolique triomphe dans le manteau \u00e0 doublure d'hermine. L'animal passait, depuis l'Antiquit\u00e9 \u2013 chez Elien en particulier \u2013, pour parangon de puret\u00e9. On racontait que si elle tombait dans une orni\u00e8re o\u00f9 elle risquait de tacher son pelage immacul\u00e9, l'hermine pr\u00e9f\u00e9rait se laisser mourir : plut\u00f4t morte et blanche que vivante et sale. On comprend que les hommes politiques en fassent si t\u00f4t un symbole, mais il vaut plus en v\u0153u pieux qu'en fid\u00e8le autoportrait.\n\nChastet\u00e9, puret\u00e9, innocence, couleur de l'agneau et de l'ange, de l'aurore et des commencements, de la gr\u00e2ce et de la transfiguration, le blanc qui \u00e9blouit symbolise toute th\u00e9ophanie. Le Roi, \u00e0 lui seul, vaut \u00e9piphanie du sacr\u00e9 et du divin. Pourpre du pouvoir et or de la richesse, bleu marial et blanc virginal, les couleurs dans lesquelles se dit et se montre Louis XIV expriment sans conteste l'excellence de sa fonction et de sa personne, la nature sacr\u00e9e de son \u00eatre et le caract\u00e8re divin de sa singularit\u00e9 sur terre.\n\nAussi, quand il arbore des chaussures, c'est pour porter un mod\u00e8le \u00e0 haut talon rouge surmont\u00e9 d'une boucle et d'un ruban de la couleur de l'artifice rehaussant. Dans la c\u00e9r\u00e9monie du sacre, les souliers \u00e9taient remis par le Grand Chambellan de France. L'Ep\u00eetre de Paul aux Eph\u00e9siens enseignait d\u00e9j\u00e0 que les chaussures repr\u00e9sentaient l'empressement \u00e0 propager l'Evangile de la Paix. De m\u00eame, elles s'associent \u00e0 la personnalit\u00e9 agissante, \u00e0 la cr\u00e9ation permanente de soi dans l'objectif du perfectionnement de sa spiritualit\u00e9. Pour \u00eatre Roi, il faut \u00eatre chauss\u00e9. Une fois consacr\u00e9, le monarque ne touche plus jamais la terre de ses pieds nus. Marcher avec des chaussures, et ce depuis les traditions \u00e9gyptiennes les plus anciennes, \u00e9quivaut \u00e0 prendre possession d'une terre, d'un terrain, d'un territoire.\n\nLe talon, quand il n'est pas l'artifice du bottier, d\u00e9signe la partie du corps par o\u00f9 l'on imagine ancestralement que la vie et l'\u00e2me s'\u00e9chappent. C'est par l\u00e0 aussi que la mort peut donc entrer dans le corps d'un homme. Si le talon, lieu de tous les dangers, au moins depuis Achille, doit \u00eatre et peut \u00eatre prot\u00e9g\u00e9, c'est par la chaussure. Et le talon est rouge \u2013 ce qui fit le d\u00e9lice des dandys au XIXe si\u00e8cle \u2013 parce qu'\u00e0 l'\u00e9poque, les talons \u00e9tant de couleurs diff\u00e9rentes, la noblesse porte exclusivement celle-ci. Le contact avec le sol ne se fait pas chez le Roi ind\u00e9pendamment de cette m\u00e9diation qui interpose la naissance, l'origine, la qualit\u00e9 du sang et les forces telluriques primitives.\n\nLes objets du Roi, les couleurs du Roi, les v\u00eatements du Roi, il faut bien que nous en arrivions au corps du Roi, \u00e0 sa chair, \u00e0 son allure. Dans le portrait de Hyacinthe Rigaud, Louis XIV triomphe en une \u00e9trange posture assez peu naturelle, parce que tout enti\u00e8re cod\u00e9e. Comme les pas dans la danse baroque, le compas des jambes \u00e9quivaut chaque fois \u00e0 une signification : ouverture, fermeture, croisement, \u00e9cartement, tout raconte une histoire. Le ciseau des membres inf\u00e9rieurs du roi se constitue avec une jambe droite avanc\u00e9e en direction du regardeur de la toile et une jambe gauche dont le pied se dirige presque en arri\u00e8re. L'angle est difficile \u00e0 appr\u00e9cier v\u00e9ritablement, mais il para\u00eet \u00e9vident que les droites sur lesquelles chaque pas s'aligne tracent deux lignes dont le croisement, comme celui du cardo et du decumanus des villes romaines, d\u00e9signe un point central, un axe du monde sur lequel est ent\u00e9 le corps du Roi.\n\nDans cette posture, aussi peu anatomique que celle dans laquelle les Egyptiens anciens repr\u00e9sentent leurs figures humaines, le Roi tourne son bassin, au-dessus de ce ciseau de jambes, et son torse offre au portraitiste une importante surface, un genre de plastron sur lequel se remarquent le bleu du manteau, le blanc de l'hermine et l'or de la croix du Saint-Esprit. Le bras droit tendu et la main pos\u00e9e sur le sceptre, puis le bras gauche, repli\u00e9 pour permettre \u00e0 la main de s'enfoncer et de se perdre dans les plis du tissu royal, dessinent une ligne horizontale qui croise la ligne verticale sur laquelle le portrait de Louis en pied s'organise.\n\nDe sorte que le corps royal semble s'inscrire dans un genre de triangle ainsi \u00e9nonc\u00e9 : la main droite pour l'extr\u00e9mit\u00e9 sup\u00e9rieure gauche (pour qui regarde), le coude des membres sup\u00e9rieurs pour l'extr\u00e9mit\u00e9 sup\u00e9rieure droite, l'avanc\u00e9e du pied droit pour pointe. Dans cette figure, la t\u00eate arrive comme un objet pos\u00e9 pour parachever l'ensemble. Ainsi, dans cet agencement singulier appara\u00eet la posture christique. Le corps du Roi qui pose se confond avec celui du Christ en croix. Le triangle de feu, pour les alchimistes, se superpose \u00e0 celui des blessures du Fils de Dieu sur son instrument de torture.\n\nLe visage, enfin. Car l'\u00e9poque sacrifie \u00e0 cette id\u00e9e \u2013 d\u00e9velopp\u00e9e plus tard par la physiognomonie \u2013 que l'int\u00e9rieur s'inscrit sur l'ext\u00e9rieur, que l'\u00e2me s'\u00e9crit sur la face, le dedans sur le dehors. Dans le m\u00eame temps, Le Brun propose des s\u00e9ries de dessins dans lesquels les hommes finissent par ressembler aux figures animales qui leur correspondent m\u00e9taphoriquement. Ce travail singulier, \u00e0 mi-chemin entre l'ironie et le s\u00e9rieux, dispose \u00e0 la m\u00eame \u00e9poque de son pendant versifi\u00e9 dans l'\u0153uvre de La Fontaine.\n\nD'un cardinal \u00e0 un oiseau de proie et d'un oiseau de proie \u00e0 la m\u00e9chancet\u00e9, d'un marquis \u00e0 un \u00e2ne et d'un \u00e2ne \u00e0 la b\u00eatise, d'une pr\u00e9cieuse \u00e0 une pie et d'une pie \u00e0 la sottise pr\u00e9tentieuse, voil\u00e0 les trajets qui conduisent de l'homme \u00e0 la b\u00eate. D'un Louis XIV, donc, \u00e0 quel animal ? On songe \u00e0 un rongeur, un blaireau, une fouine, une musaraigne, ou quelque chose d'approchant : de petits yeux pliss\u00e9s, allong\u00e9s comme ceux d'un animal malin, un nez fort, busqu\u00e9, qui ressemble \u00e0 un museau, une petite bouche avec une moue qui h\u00e9site entre m\u00e9pris et morgue, la l\u00e8vre sup\u00e9rieure mang\u00e9e, disparaissant dans la l\u00e8vre inf\u00e9rieure, l'ensemble s'inscrit dans un ovale adipeux, le tout surmont\u00e9 par l'in\u00e9vitable perruque exhibant le syst\u00e8me pileux comme un syst\u00e8me social.\n\nLe grand peintre, c'est celui qui, apr\u00e8s avoir install\u00e9 partout les signes, les symboles, le sens, trouve moyen de r\u00e9aliser un excellent compromis entre la v\u00e9rit\u00e9 et le d\u00e9sir social, la ressemblance et l'attente du commanditaire. Comment ne pas rendre la laideur trop criarde, trop insoutenable ? De quelle fa\u00e7on \u00e9viter les d\u00e9fauts d'un physique sans pour autant tomber dans l'hagiographie picturale imm\u00e9diatement rep\u00e9rable en regard du mod\u00e8le ? Entre la r\u00e9alit\u00e9 d'un visage ingrat et le canon social du moment, l'artiste doit pouvoir obtenir le meilleur \u00e9quilibre. Ni trop, ni trop peu. De la mesure avant toute chose, de la gr\u00e2ce et de l'\u00e9l\u00e9gance, de la politesse entendue comme un art de la bonne distance, voil\u00e0 les principes de l'\u00e9thique picturale de cour.\n\nLe portrait de Hyacinthe Rigaud a connu une fortune consid\u00e9rable. On l'a reproduit grav\u00e9, peint, copi\u00e9 de mani\u00e8re impensable. Mais surtout, il a fourni un canon esth\u00e9tique duquel d\u00e9coule un nombre incroyable de portraits. Les historiens de la p\u00e9riode en comptabilisent trente mille effectu\u00e9s en vingt-cinq ans dans le seul Paris. L'exercice du peintre a stimul\u00e9, excit\u00e9, encourag\u00e9 et inspir\u00e9 un grand nombre d'autres artistes qui, singuli\u00e8rement, font de ce portrait un genre d'id\u00e9e de la raison dont participe presque toute la peinture contemporaine du Roi-Soleil.\n\nLes portraits de nobles, d'eccl\u00e9siastiques, de militaires, de courtisanes, de rois \u00e9trangers, d'artistes, d'hommes politiques au service de la cour, de chasseurs, d'hommes ou de femmes jouant avec leur chien, de magistrats, d'enfants, tous citent cet exercice embl\u00e9matique de Rigaud : r\u00e9f\u00e9rences pr\u00e9cises, renvois discrets, soulignages et surlignages de traits plus nets, artifices parlants, d\u00e9cors affich\u00e9s, mise en sc\u00e8ne revendiqu\u00e9e, th\u00e9\u00e2tralisation du sujet, esth\u00e9tique soutenue par une m\u00e9taphysique, beaux-arts proc\u00e9dant d'une th\u00e9ologie, d'une mystique, tout converge vers le sens \u2013 ce narcissisme d'une \u00e9poque se mirant dans les figures et les individus qui assurent les fonctions de pouvoir.\n\nLa peinture est conceptuelle, chose mentale, plus encore que ne le disait L\u00e9onard, parce qu'elle suppose le discours de l'\u00e9poque qui l'accompagne, ce que les Allemands appellent justement la Weltanschauung, la vision du monde. Il lui faut, pour \u00eatre vue et comprise, per\u00e7ue, saisie, les tenants et les aboutissants philosophiques. Hyacinthe Rigaud, mais aussi Jean Jouvenet, Nicolas de Largillierre, Pierre Mignard, Charles Le Brun, Pierre Mosnier, Fran\u00e7ois de Troy, Claude Lefebvre, Joseph Vivien, et tant d'autres, dont certains n'ont laiss\u00e9 qu'un nom, malgr\u00e9 leurs centaines de toiles peintes, tous racontent une \u00e9poque dans le d\u00e9sir et l'id\u00e9e qu'elle a d'elle-m\u00eame. Sa passion pour les h\u00e9ros mythologiques, les figures de l'Antiquit\u00e9 romaine, les saints et les saintes du christianisme, son go\u00fbt pour le th\u00e9\u00e2tre, expriment le temps du Grand Si\u00e8cle : il est d'un narcissisme exacerb\u00e9.\n\nLes fleurs, les perles, les animaux, les d\u00e9corations, les bijoux, le tableau dans le tableau, les tissus, les v\u00eatements, les drap\u00e9s, les postures, les portraits histori\u00e9s, tout exprime un monde dans le plaisir qu'il a de se contempler, dans la sienne assurance d'illustrer un moment glorieux dans l'histoire de l'humanit\u00e9, \u00e0 la hauteur des Egyptiens, des Grecs ou des Romains. Le si\u00e8cle de Louis XIV se dit et se veut de grandeur et de satisfaction non contenue. Il se pense, se voit et s'imagine grand, immense, magnifique, majestueux, triomphant, d\u00e9mesur\u00e9 par la trace laiss\u00e9e dans l'histoire.\n\nLa royaut\u00e9, fi\u00e8re de ce qui la fait, la constitue, montrant son Homme de cour comme le parangon de l'ach\u00e8vement de l'humanit\u00e9, passe autant de temps \u00e0 s'aimer, \u00e0 se c\u00e9l\u00e9brer, \u00e0 se montrer, \u00e0 se mettre en sc\u00e8ne qu'elle met d'ardeur \u00e0 oublier le monde, les autres, la r\u00e9alit\u00e9 qui n'est pas de cour, le peuple, sa mis\u00e8re et sa pauvret\u00e9. Quelques ann\u00e9es apr\u00e8s ces visages frais, ronds, roses et blancs, rubiconds, quelques d\u00e9cennies apr\u00e8s ces ors et brocarts, ces narcisses au th\u00e9\u00e2tre et ces individus confits en d\u00e9votion pour les fonctions dans lesquelles ils jouissent, plastronnant avec les titres qu'ils alignent et les quartiers de noblesse qu'ils revendiquent, cinquante ans plus tard, ou gu\u00e8re plus, l'ombre de la guillotine appara\u00eetra pour couper court \u00e0 ce fantasme d'\u00e9ternit\u00e9. Car tout est vanit\u00e9, et poursuite du vent : Thanatos a toujours raison de Narcisse.\n32\n\nCONSID\u00c9RATIONS SUR LES ROGNONS DU PHILOSOPHE\n\nApr\u00e8s l'apparition d'une com\u00e8te radieuse et jubilatoire dans le ciel des id\u00e9es, plus rien ne subsiste comme avant. La r\u00e9volution sous la vo\u00fbte induit un bouleversement \u00e0 la suite duquel une grande partie des valeurs du pass\u00e9 sombrent dans la caducit\u00e9. Des pans entiers de choses hier importantes paraissent alors billeves\u00e9es ridicules. Ce qui faisait v\u00e9rit\u00e9 devient erreur, l\u00e0 o\u00f9 l'on voyait certitudes il n'y a plus que gravats. Les monuments croulent et font ruines, les \u00e9difices majestueux survivent en repaires pour chiens galeux, pauvres h\u00e8res et rats affam\u00e9s. Les soleils br\u00fblants de la veille refroidissent, les empires s'effondrent, les anciens princes errent en mendiants, vagabonds et pitoyables rebuts. Ils perdent leur superbe, leur arrogance et grelottent d'avoir quitt\u00e9 les feux de la sc\u00e8ne. Montaigne fut et demeure l'une de ces com\u00e8tes qui flambent et tracent une fois tous les deux ou trois si\u00e8cles. J'aime la br\u00fblure singuli\u00e8re et profonde qu'il inflige \u00e0 l'histoire de la philosophie.\n\nG\u00e9nie et philosophe, visionnaire et penseur, d\u00e9couvreur de nouveau monde et \u00e9crivain, anatomiste lucide de l'\u00e2me et po\u00e8te, au sens grec du terme, il joue tous les r\u00f4les \u00e0 la fois. Je ne cesse de me demander, en le lisant, comment il a pu ainsi sur tous les sujets, ou presque, inventer notre modernit\u00e9 sans coup f\u00e9rir. Pr\u00e9cisons d'ailleurs que toutes ses th\u00e8ses, quand elles ne sont pas devenues v\u00e9rit\u00e9s d'\u00e9vidence, supposent une v\u00e9ritable r\u00e9volution mentale pour \u00eatre seulement accept\u00e9es au tournant de notre mill\u00e9naire \u2013 sur l'\u00e9ducation, le mariage, le colonialisme ou le tragique, par exemple. Je tiens que rien de l'Occident, dans sa version th\u00e9orique et pratique, n'aurait \u00e9t\u00e9 tel sans l'exception que fut cet homme. Et je me r\u00e9jouis de pouvoir y recourir chaque fois qu'une pharmacop\u00e9e intellectuelle m'est n\u00e9cessaire.\n\nDans le registre de la pens\u00e9e, car il \u0153uvre autant en \u00e9crivain qu'en philosophe, \u2013 seuls certains universitaires et quelques cuistres le nient encore, et ce sont souvent les m\u00eames \u2013 on lui doit les intuitions parmi les plus fulgurantes. A savoir? Que le corps, dans ses heurs et malheurs, vaut g\u00e9n\u00e9alogie de toute pens\u00e9e ; que le moi suppose la narration de ses aventures sur le mode du roman picaresque afin qu'on en sache plus sur la constitution d'une subjectivit\u00e9 dans l'absolu; que l'individu doit s'inventer de toutes pi\u00e8ces pour contrevenir \u00e0 l'immobilit\u00e9 qui le contraint quand Dieu prend toute la place; que l'eud\u00e9monisme implique une vision du monde et des pr\u00e9suppos\u00e9s tragiques, l'un et l'autre fonctionnant en contrepoint; que des formes nouvelles sont n\u00e9cessaires pour exprimer des fonds nouveaux ; mais aussi que la volupt\u00e9 peut tenir lieu de cap dans cette odyss\u00e9e du Moi et de la raison moderne ; que la chair ne m\u00e9rite s\u00fbrement pas autant de mal\u00e9diction et de m\u00e9pris que le christianisme le pense, le dit et l'enseigne; enfin que la vie et la philosophie connaissent une intime couture au point que l'une sans l'autre ne m\u00e9rite gu\u00e8re qu'on s'y arr\u00eate, encore moins qu'on s'y attarde.\n\nPour toutes ces raisons, Montaigne me ravit, me touche et m'\u00e9meut avec une ind\u00e9fectible constance. Chacun de mes p\u00e8lerinages \u00e0 sa tour, \u00e0 son tombeau, sur ses terres, chaque fois que je monte les escaliers de sa librairie, que j'\u00e9coute le silence du paysage d\u00e9ploy\u00e9 au pied du ch\u00e2teau, d\u00e8s que dans ces pi\u00e8ces je regarde la campagne qu'il voyait, quand je m'attarde sur les poutres grav\u00e9es de citations au point que j'en ai mal au cou, dans le dos et les reins, lorsque je m'installe dans le r\u00e9duit o\u00f9 il \u00e9coutait la messe, dans la salle au-dessus de l'oratoire, d\u00e8s que j'imagine la biblioth\u00e8que et la disposition des livres, leur nombre, quand je songe au philosophe marchant, dictant debout, lorsque j'effleure la selle ou la malle de ses voyages expos\u00e9s dans la chambre, chaque fois, je me sens envahi de pi\u00e9t\u00e9, comme lors d'une visite \u00e0 un ma\u00eetre dont l'existence rend le monde plus aimable et d\u00e9sirable.\n\nDehors, dans le parc, assis sur la rambarde, guettant les bruits qui viennent de loin, de si loin, je m'abandonne \u00e0 la r\u00eaverie, avant de mettre dans ma poche un marron qui ne quittera d\u00e9sormais plus mon bureau. D\u00e9ambulant dans le bourg, m'arr\u00eatant \u00e0 l'\u00e9glise, devant les fonts baptismaux o\u00f9 il a \u00e9t\u00e9 re\u00e7u par le pr\u00eatre, retrouvant l'activit\u00e9 du village, je pense au trajet plan\u00e9taire des Essais, \u00e0 leur pr\u00e9sence sur l'ensemble du globe, \u00e0 leur influence sur la destin\u00e9e de l'Europe, puis de l'Occident. Jamais peut-\u00eatre autant qu'avec ce livre des id\u00e9es n'ont \u00e9t\u00e9 prolong\u00e9es par des faits, des gestes, des actions, des r\u00e9volutions dans le quotidien, des s\u00e9ismes dans des \u00e2mes, des ruptures dans des modes de pens\u00e9e, des d\u00e9veloppements dans nombre d'oeuvres, mineures et majeures. En contemporain des nouveaux continents d\u00e9couverts, Montaigne offre \u00e0 notre vieux monde de quoi poursuivre dans la voie qu'il s'est trac\u00e9e pour permettre d'ajouter des com\u00e8tes aux com\u00e8tes, puis de constituer un archipel de celles-ci.\n\nLe fondement de la m\u00e9taphysique des Essais suppose le corps en raison majeure et arch\u00e9ologie des principes. Je crois que cette pens\u00e9e fournit les moyens d'une herm\u00e9neutique mat\u00e9rialiste pour tout d\u00e9chiffrage philosophique. Dans l'ensemble de l'\u0153uvre, du livre majeur aux notes de voyage en passant par l'\u00e9ph\u00e9m\u00e9ride ou la correspondance, tout fait trace en ce sens : il n'existe que le corps, la chair, ses histoires, ses aventures, son odyss\u00e9e et son roman. Le reste advient en variation sur ce th\u00e8me : \u00e9criture, pens\u00e9e, vision du monde, syst\u00e8me, philosophie, tout exprime la souverainet\u00e9 de la viande. J'aspire d'ailleurs \u00e0 lire l'\u0153uvre de Montaigne comme une s\u00e9rie de consid\u00e9rations sur ses propres rognons puisque ceux-ci \u2013 qu'il nomme ainsi \u2013 occasionnent chez lui la plupart des douleurs dont proc\u00e8dent ses tentatives de conjuration pendant plus des trois lustres de la patiente mise au point d'une version d\u00e9finitive du livre.\n\nCombien, pendant tout le temps que dure la r\u00e9daction, de d\u00e9tails sur les vents, pets et autres flatulences, sur les grains de coriandre pour vesser; sur les urines, troubles, claires, rouges, brunes, sombres, limpides, \u00e9paisses ; sur les selles molles, parfum\u00e9es, souples, abondantes ; sur la forme des pierres \u00e9ject\u00e9es, du grain de millet aux pommes de pin en passant par une, plus extravagante, qui rappelle un sexe miniature ; sur les maux de t\u00eate et les n\u00e9vralgies dentaires ; sur les vomissements, naus\u00e9es et autres bouff\u00e9es de chaleur ou rougeurs au visage ? La chair du philosophe souffre et ne cesse de souffrir sans autre th\u00e9rapie que l'exercice d'une volont\u00e9 eud\u00e9moniste, plus efficace que toute m\u00e9decine ou cure thermale. La sagesse du philosophe suppose une algodic\u00e9e, une sapience connue par le corps douloureux et qu'il veut lib\u00e9rer de cet \u00e9tat. Je ne vois pas d'autre fonction \u00e0 la pratique de la philosophie depuis qu'elle existe.\n\nLes Essais proposent un bulletin de sant\u00e9 transcend\u00e9, d\u00e9pass\u00e9, sublim\u00e9. Ils constituent le journal d'une conscience acc\u00e9dant \u00e0 son autonomie par le travail op\u00e9r\u00e9 sur une chair qui supporte les violences de la colique n\u00e9phr\u00e9tique et des autres affections. Pas de pens\u00e9e sans un corps qui les produit, pas de chair neutre, ind\u00e9pendante des troubles qui la d\u00e9stabilisent ou l'\u00e9quilibrent. Tenir le compte de ses \u00e9tats de sant\u00e9, consentir au narcissisme de l'hypocondriaque, se faire scrutateur des signes avant-coureurs de la souffrance, pr\u00e9voir la douleur pour la conjurer, l'emp\u00eacher, et finalement m\u00e9priser la mort travestie en maladie, voil\u00e0 le projet de Montaigne, en quoi il co\u00efncide avec les constructions sot\u00e9riologiques de l'Antiquit\u00e9 : bien vivre, mieux vivre, vivre autrement, diff\u00e9remment, en paix avec soi-m\u00eame. La sagesse se d\u00e9voie, me semble-t-il, quand elle n\u00e9glige de proposer une alternative pa\u00efenne au salut des croyants.\n\nLes bulletins de sant\u00e9 ne cessent d'affluer dans les pages du grand livre. Ici, bien s\u00fbr, des consid\u00e9rations sur la gravelle lui font pester contre la m\u00e9decine et les m\u00e9decins, les charlatans dissimul\u00e9s derri\u00e8re le pr\u00e9texte de la science pour se contenter d'une promotion de l'illusion, moyennant honoraires qui, eux, ne trompent pas, ce qui induit une critique s\u00e9v\u00e8re, ironique et sans appel, de ceux qui usent du corps d'autrui comme d'une chose sur laquelle on peut asseoir une puissance, exercer un abus ; l\u00e0, une m\u00e9moire d\u00e9faillante, une r\u00e9sistance entam\u00e9e, une endurance \u00e9court\u00e9e et autres signes prop\u00e9deutiques \u00e0 la vieillesse le font m\u00e9diter sur la corruption de la sant\u00e9, la fragilit\u00e9 d'une chair, la soumission de tous les organes \u00e0 la tyrannie du temps et de l'entropie, \u2013 ce qui nous vaut des consid\u00e9rations nominalistes sur le temps, moins diss\u00e9qu\u00e9 comme tel, dans sa nature et sa d\u00e9finition, que vu et compris en biais par un discours sur ses effets, ses pouvoirs.\n\nOutre la sant\u00e9 et la maladie, ces figures naturelles de l'entropie, Montaigne envisage le corps dans les situations artificielles de l'accident. Ainsi une agression commise par des malfrats lors d'une travers\u00e9e de for\u00eat ou la visite muscl\u00e9e \u00e0 son domicile d'une bande de soudards tr\u00e8s nettement mal intentionn\u00e9s. Chaque fois, ce sont pour lui occasions d'exp\u00e9rimenter l'usage des passions sur son propre corps . la peur, la crainte, l'angoisse ne sont pas de son lot, inutiles et dangereuses, trop g\u00e9n\u00e9ratrices de complications et de d\u00e9plaisirs. Il disserte alors sur la confiance, la s\u00e9r\u00e9nit\u00e9, la placidit\u00e9 et fait l'\u00e9loge de vertus apaisantes \u00e0 l'aide desquelles on apprivoise le d\u00e9rangement, on le conjure et le cong\u00e9die. Comment mieux dire l'utilit\u00e9 de la philosophie? Je ne vois gu\u00e8re meilleure fa\u00e7on.\n\nDe la m\u00eame mani\u00e8re, on sait combien une chute de cheval fut pour lui ce que dans L'Art de jouir j' ai appel\u00e9 un hapax existentiel, \u00e0 savoir un \u00e9v\u00e9nement majeur et fondateur dans une vie puisqu'il induit ind\u00e9finiment, pendant le restant de l'existence, une tentative de r\u00e9ponse aux questions pos\u00e9es dans ce moment dense et exceptionnel. Se voyant mourir, analysant ce qui advient dans une circonstance cruciale comme celle-ci, Montaigne exp\u00e9rimente sa pens\u00e9e, t\u00e2che de voir si ce qui fonctionne th\u00e9oriquement, lorsqu'il est sain et sauf, agit m\u00eamement quand il se trouve sur le point de d\u00e9faillir ou de tr\u00e9passer.\n\nUne autre fois, pendant une fameuse \u00e9pid\u00e9mie de peste, les processions macabres, les morts abandonn\u00e9s partout dans la campagne, pourrissant \u00e0 l'air libre, les miasmes et la panique des familles en d\u00e9route, la sienne propre, lui fournissent un mod\u00e8le qui l'incite \u00e0 r\u00e9fl\u00e9chir sur l'humaine condition et sur la mort comme v\u00e9rit\u00e9 ultime de tout ce qui vit. Malade, affect\u00e9, accident\u00e9, vieillissant, expos\u00e9, toujours il pense, via sa propre exp\u00e9rience, la relation du corps et des passions, de la chair et des \u00e9motions qui la traversent. D'o\u00f9 les pages des Essais consacr\u00e9es \u00e0 la peur, au caract\u00e8re, au temp\u00e9rament, \u00e0 la volont\u00e9, \u00e0 l'imagination. Puis les extensions existentielles de cette m\u00e9ditation permettent d'aborder les questions m\u00e9taphysiques de la libert\u00e9, du libre arbitre, du d\u00e9terminisme, de la conscience, de la n\u00e9cessit\u00e9, de la mort, bien s\u00fbr. Le corps, toujours, initie la pens\u00e9e. Ce qui lui advient fournit les occasions du roman philosophique qu'il \u00e9crit \u00e0 son sujet. Pareil projet ne cesse de me r\u00e9jouir.\n\nDe la m\u00eame mani\u00e8re, son corps bien portant lui sert \u00e9galement d'astrolabe fournissant les points de rep\u00e8re n\u00e9cessaires pour naviguer en pleine mer. Hors maladie, d\u00e9p\u00e9rissement, \u00e9pid\u00e9mie ou accident, Montaigne \u00e9coute son corps comme on guette une brise pour cingler vers le large. D'o\u00f9, parfois, une autobiographie crue, cruelle et sans concession avec soi-m\u00eame. Ainsi lorsqu'il confie disposer d'un sexe tr\u00e8s en dessous de la taille r\u00e9glementaire et d'une libido tout aussi capricieuse qu'est petit son instrument. Qui, de la sorte, prouvera mieux qu'une m\u00e9taphysique, voire une mystique, c'est avant tout le travestissement d'une physiologie ? Et qu'on a la th\u00e9orie de ce que le corps exige aveugl\u00e9ment? Ou qu'un philosophe doit se contenter de formuler verbalement ce que la chair a d\u00e9j\u00e0 obtenu par ses voies propres ?\n\nLibidinal contrari\u00e9 par une chair r\u00e9calcitrante, Montaigne prend soin de noter tout ce que son corps r\u00e9tif dit, raconte, montre ou trahit : sa petite taille, son physique ingrat, son allure forte, son visage plein, son oreille inepte, sa voix haute, son manque d'adresse, sa pr\u00e9sence emprunt\u00e9e. Puis il confie le d\u00e9tail de son dormir et conc\u00e8de aux r\u00eaves une place qui d\u00e9classe, et de loin, les approximations alors cr\u00e9dit\u00e9es d'Art\u00e9midore d'Eph\u00e8se. Enfin, il d\u00e9taille ses go\u00fbts alimentaires. enseigne sa fa\u00e7on de couper son vin puis de le boire, regrette que son plaisir d'hu\u00eetres ne puisse \u00eatre support\u00e9 par un corps rebelle, s'attarde sur sa passion des viandes tr\u00e8s faisand\u00e9es. De sorte que sa di\u00e9t\u00e9tique \u00e9nonce l'une des modalit\u00e9s de son \u00e9thique \u2013 et j'ai toujours souscrit \u00e0 cette collision intellectuelle malgr\u00e9 les malentendus induits tous azimuts.\n\nCorps sexuel et sexu\u00e9, mobile et vivant, dormeur et r\u00eaveur, mangeur et dig\u00e9rant, Montaigne n'\u00e9vite pas la chair, il ne la sublime ni ne la pr\u00e9sente sous les formes \u00e9th\u00e9r\u00e9es du concept, de l'id\u00e9e. Au contraire des classiques qui le pr\u00e9c\u00e8dent en spiritualisant la chair, il s'\u00e9vertue \u00e0 mat\u00e9rialiser l'esprit. Le philosophe c\u00e9l\u00e8bre le corps d\u00e9sirant, le corps souffrant, le corps malade, le corps soucieux, le corps voluptueux, le corps jubilant et discr\u00e9dite toute acception d'un corps glorieux au sens chr\u00e9tien du terme. C'est en marcheur, en chasseur, en cavalier, en homme de la terre que Montaigne met le corps \u2013 son corps \u2013 au centre du monde. Contre la scolastique conceptuelle, l'id\u00e9alisme philosophique, le spiritualisme catholique, les Essais inventent le corps physiologique et l'installent \u00e0 l'\u00e9picentre de la philosophie. Il en est pour lesquels cette audace vaut toujours h\u00e9r\u00e9sie \u2013 je les plains...\n\nL'\u00e9criture du grand livre de Montaigne proc\u00e8de d'une s\u00e9cr\u00e9tion et fournit le sympt\u00f4me d'une pens\u00e9e moderne \u00e9non\u00e7ant les nouvelles conditions de possibilit\u00e9 de la pens\u00e9e. Dieu n'est pas encore mort \u2013 les libertins \u00e9rudits entament le travail de sape au si\u00e8cle suivant, notamment sous l'influence des Essais \u2013 mais les impulsions sont donn\u00e9es. D\u00e8s que le corps s'\u00e9nonce et s'avance comme la grande raison, la g\u00e9n\u00e9alogie de toute morale et le point d'arraisonnement d'une vision du monde, je pense que les jours de toute transcendance sont compt\u00e9s. Alors est proche l'av\u00e8nement de l'immanence, seul rem\u00e8de philosophique pour \u00e9viter les illusions et les parh\u00e9lies mentales ou intellectuelles.\n\nLa pens\u00e9e de Montaigne permet une cartographie tr\u00e8s pr\u00e9cise de la modernit\u00e9. Du corps en acte au livre en puissance, puis des pages r\u00e9elles \u00e0 la chair transfigur\u00e9e, s'effectuent de perp\u00e9tuels aller et retour pendant les vingt ann\u00e9es que durent la r\u00e9daction, l'ajustement, la fabrication de l'\u0153uvre. Son \u00e9thique est une di\u00e9t\u00e9tique \u2013 comme est tout aussi juste l'inverse \u2013, sa physiologie ouvre les possibilit\u00e9s d'une philosophie, sa discipline d'\u00e9criture transfigure la mati\u00e8re et lui donne un statut cardinal. Avec l'av\u00e8nement de la chair humaine se pr\u00e9parent in\u00e9luctablement la mort des anges, puis celle des dieux.\n\nPas plus qu'un corps ne se r\u00e9duit \u00e0 la somme des organes qui le composent, les Essais ne se suffisent d'\u00eatre un ensemble de fonctions, d'organes eux aussi, de syst\u00e8mes \u2013 respiratoires, digestifs, excr\u00e9tionnaires, circulatoires. Chercher des strates, des \u00e9poques, des p\u00e9riodes, des couches, des \u00e9paisseurs, des dates, des versions dans le livre, \u00e9quivaut \u00e0 vouloir d\u00e9couvrir le myst\u00e8re de la vie d'un \u00eatre en pratiquant l'anatomie et l'autopsie d'un mort. On ne qu\u00eate pas le souffle et l'\u00e9nergie d'une chair en d\u00e9sirant la d\u00e9couper, puis la fouiller. Le livre de Michel de Montaigne, c'est son corps transfigur\u00e9, et vice versa\u2013je ne vois d'ailleurs pas d'autre fonction au livre digne de ce nom. Avec lui peut se dire l'alchimie en vertu de quoi un verbe se fait chair ou l'inverse. En dehors des corps, on n'enregistre que vide sid\u00e9ral et musique des plan\u00e8tes.\n\nLe premier continent d\u00e9couvert par Montaigne \u2013 le fondement physiologique de toute pens\u00e9e \u2013 appelle de nouvelles terres et annonce de nouvelles rives \u00e0 aborder, car il n'est pas homme \u00e0 se contenter d'un abordage, il lui faut d\u00e9fricher un continent tout entier. Sur les plages vierges de toute pr\u00e9sence, avant la d\u00e9couverte du monde clos et de l'univers infini, puis des inqui\u00e9tudes m\u00e9taphysiques aff\u00e9rentes, Montaigne \u00e9nonce de quoi r\u00e9dimer avant tout nouveau p\u00e9ch\u00e9 m\u00e9taphysique : il invente l'individu occidental. En Orient, en Afrique o\u00f9 je voyage souvent, cette figure ins\u00e9cable et indivisible, cet atome irr\u00e9ductible ne dispose toujours pas d'une existence aussi nette, et le g\u00e9n\u00e9ral, l'universel, le tout priment absolument. Bien avant Descartes, qu'on cr\u00e9dite habituellement et faussement de cette innovation, Montaigne jette par-dessus bord les vieux outils de la scolastique, les formes et les substances, les essences et les attributs, les puissances et les actes, puis il pose, souverain, l'individu au centre de la philosophie. L\u00e0 o\u00f9 triomphaient des concepts, il habilite le corps, la chair en sa formule individuelle.\n\nCette trouvaille sublime, Montaigne l'effectue apr\u00e8s un long voyage, une p\u00e9r\u00e9grination d'\u00e9criture qui transfigure les Essais en genre d'odyss\u00e9e de la conscience, sinon en roman picaresque du Je, du Moi. Pareil aux h\u00e9ros grouillants dans la litt\u00e9rature picaresque espagnole, le philosophe narre ses aventures, raconte ses heurs et malheurs, conte les p\u00e9rip\u00e9ties de son trajet, prend \u00e0 t\u00e9moin, sectionne son histoire en chapitres o\u00f9 l'on rencontre un androgyne, un voleur, un paysan, un monstre, une veuve, un ami, un roi, une courtisane, un philosophe, un vagabond, un cavalier, un militaire, un artisan, un p\u00e8re, une \u00e9pouse, une confidente, une fille, un \u00e9cuyer, un instructeur, un th\u00e9ologien, un latiniste, une disciple, et tant d'autres figures constitutives du th\u00e9\u00e2tre baroque de ses m\u00e9ditations, \u2013 \u00ab guette-chemin de vie humaine \u00bb, \u00e9crivait Mateo Alem\u00e0n de son h\u00e9ros Guzm\u00e0n de Alfarache.\n\nPour la premi\u00e8re fois, sur cette sc\u00e8ne du monde philosophique, la pens\u00e9e s'exprime \u00e0 la premi\u00e8re personne et montre \u00e0 l'\u0153uvre une conscience constitu\u00e9e pour elle-m\u00eame, un Je dessin\u00e9 par petites touches jusqu'\u00e0 faire na\u00eetre la peinture d'un Moi. Loin de l'entreprise augustinienne d'une autobiographie au service de l'apolog\u00e9tique chr\u00e9tienne et de la conversion, aux antipodes d'une confession destin\u00e9e \u00e0 fournir un mod\u00e8le d'adh\u00e9sion au projet catholique de dilution de la subjectivit\u00e9 en lui, le roman picaresque de Montaigne veut l'\u00e9mergence d'une figure et son autonomie, sa vie pour elle-m\u00eame. Le Moi n'est pas exacerb\u00e9 pour \u00eatre mieux dissous, ou montr\u00e9 pour \u00eatre mieux subtilis\u00e9, il acc\u00e8de pour lui-m\u00eame \u00e0 l'excellence, \u00e0 la souverainet\u00e9 et \u00e0 la majorit\u00e9. Les sommit\u00e9s catholiques et bien-pensantes saisiront bien la diff\u00e9rence lorsqu'elles condamneront le grand livre coupable \u2013 la mise \u00e0 l'index date de 1669 \u2013 de montrer le chemin d'une conscience libre et lib\u00e9r\u00e9e, d'une autonomie gracieuse et voluptueuse, d'une subjectivit\u00e9 ind\u00e9pendante et radieuse. C'\u00e9tait trop, bien \u00e9videmment, pour les tristes figures catholiques.\n\nQuand il d\u00e9couvre sa propre individualit\u00e9, Montaigne met au jour l'individualit\u00e9 universelle. De sorte qu'il invente l'Autre en m\u00eame temps que le Moi. Comment s'y prend-il ? D'abord, il dissocie dans le corps une chair qui vit et souffre, jouit et p\u00e2tit, puis une conscience qui sait et peut mettre \u00e0 distance cette partie essentiellement charnelle de l'\u00eatre. Le Je et le Moi acc\u00e8dent au statut d'objets s\u00e9par\u00e9s, susceptibles d'un regard tiers, autre, d\u00e9doubl\u00e9, bien que provenant intellectuellement du Moi en question. Ainsi, une partie de soi pourra dire \u00e0 l'autre partie qu'elle sait, peut et veut. La construction de soi par soi devient \u00e9pist\u00e9mologiquement et philosophiquement possible. L'\u00eatre du sujet peut faire l'objet d'un investissement, d'une volont\u00e9, d'une d\u00e9cision, d'une sculpture, d'un projet. La saisie de soi tel un autre rend pensable une vie entendue comme une \u0153uvre d'art. Ainsi une esth\u00e9tique de l'existence o\u00f9 l'\u00e9thique agisse en cl\u00e9 de vo\u00fbte devient possible sous forme de chef-d'\u0153uvre.\n\nL'autoportrait du philosophe au miroir, avec le d\u00e9doublement impliqu\u00e9, fait des Essais une peinture mani\u00e9riste o\u00f9 le sujet se confond \u00e0 l'objet, o\u00f9 l'\u0153il du peintre veut la retranscription du corps et de l'\u00e2me dudit peintre. De sorte que le Moi qui pense pense le Moi qui est, en quoi l'on d\u00e9couvre le cogito de Montaigne et la formulation, en dehors de tout vocabulaire n\u00e9o-scolastique, d'un dualisme subtil dissociant ce que deviendront une substance \u00e9tendue et une substance pensante. Le roman du Je suppose un perp\u00e9tuel aller et retour entre ces deux instances dont le point de jonction ne saurait \u00eatre l'artifice physiologique d'une quelconque glande pin\u00e9ale. L\u00e0 o\u00f9 le corps et l'\u00e2me se rejoignent, il n'est qu'un point m\u00e9taphysique : l'\u00eatre, l'identit\u00e9 du sujet, sa nature, sa d\u00e9finition. Pour Montaigne, il n'existe qu'une seule v\u00e9rit\u00e9, le corps, dont les modalit\u00e9s apparaissent sous de multiples chatoiements. Une fois la conscience, une autre l'\u00e2me, ici la chair, ailleurs les organes. Ici la sant\u00e9, l\u00e0 la jouissance, ailleurs la maladie, une fois la gourmandise, une autre la peur \u2013 ce Montaigne-l\u00e0, vivant, fr\u00e9missant, me parle en intime.\n\nL'\u0153uvre effectue la liaison, aussi bien sur le mode livresque que selon le principe existentiel. La chair de l'\u00e2me du philosophe, sinon l'\u00e2me de sa chair, s'exposent dans son livre autant que dans son quotidien, l\u00e0 o\u00f9 les deux entreprises se m\u00ealent intimement, au point qu'on ne fait plus la part de l'un et de l'autre. Le dualisme n'est pas d'ordre m\u00e9taphysique mais \u00e9pist\u00e9mologique : la conscience agit en organe et se distingue du c\u0153ur autant que ce dernier du cerveau. Leur s\u00e9paration ne vaut pas opposition irr\u00e9ductible, mais compl\u00e9mentarit\u00e9 positive. La pens\u00e9e moniste et mat\u00e9rialiste du philosophe proc\u00e8de de sa philosophie du corps et de sa th\u00e9orie de l'individu.\n\nLe livre achev\u00e9 t\u00e9moigne de ces mouvements entre les organes, il \u00e9pouse sur le papier les circulations, les flux, les mobilit\u00e9s, les soubresauts du corps. L\u00e0 o\u00f9 se manifeste une odyss\u00e9e, celui qui la consigne jubile des trajets, des d\u00e9placements, des forces en jeu. Dans ce roman picaresque, les h\u00e9ros sont aussi la conscience, la sant\u00e9, le corps, la maladie, l'imagination, les passions, les affections, la volont\u00e9, et toute la grande distribution du th\u00e9\u00e2tre classique des moralistes du Grand Si\u00e8cle \u00e0 venir. Car la vie, sous toutes ses formes, pr\u00e9occupe le philosophe. Voil\u00e0 pourquoi il traque des efflorescences, des effervescences, des fourmillements, des foisons, des profusions, des d\u00e9bordements, des pullulements, des fermentations, des bouillonnements. Les Essais captent la vitalit\u00e9 dans ses perp\u00e9tuelles expansions. Je crois qu'avant lui personne n'a \u00e9cout\u00e9 la vie avec autant d'attention et de curiosit\u00e9 \u2013 en m\u00e9decin qui philosophe, sinon en philosophe qui m\u00e9dicalise. A l'usage de l'astrolabe pour l'infiniment grand, il annonce et ajoute celui du st\u00e9thoscope pour l'infiniment petit.\n\nPar ailleurs, le roman du Moi suppose que du Je on puisse parler de mani\u00e8re romanesque, entre p\u00e9rip\u00e9ties et m\u00e9ditations, aventures et repos. Ainsi, en se faisant le romancier de sa subjectivit\u00e9, Montaigne passe de son individualit\u00e9 propre \u00e0 celle de tout un chacun. L\u00e0 o\u00f9 il m\u00e9dite, dans sa tour, \u00e9loign\u00e9 du monde, il formule des v\u00e9rit\u00e9s universelles susceptibles d'\u00e9largissements aux hermaphrodites et aux monstres, aux cannibales et aux \u00e9trangers, aux femmes et aux enfants, aux Noirs et aux Jaunes, aux paysans et aux rois, aux sots et aux sages \u2013 aux autres, qu'il invente \u00e9galement comme semblables. Ce qui vaut pour lui vaut pour tous et toutes, ici comme ailleurs, et de toute \u00e9ternit\u00e9.\n\nLoin de r\u00e9duire Montaigne au scepticisme m\u00e9thodologiquement abondant dans l'Apologie de Raimond Sebond, je veux me souvenir combien sa pens\u00e9e \u00e9nonce de certitudes absolues et non relatives, comment son jugement est habituellement ferme, et non vague ou suspendu. Du corps et de l'individu, il se demande ce qu'il peut savoir, certes, mais ce qu'il sait, il n'en doute aucunement. Le perspectivisme qu'il invente n'est point relativisme mais conflit d'interpr\u00e9tations opposant des subjectivit\u00e9s diff\u00e9remment avanc\u00e9es sur le chemin du savoir et de la connaissance. L'immanence sur laquelle joue le philosophe suppose le combat de forces, le conflit, la diversit\u00e9, les diff\u00e9rences, les divergences \u2013 le mouvement. Mais loin de penser que tout vaut tout, indiff\u00e9remment, indistinctement, Montaigne invite \u00e0 douter des certitudes \u00e9tablies, s\u00fbrement pas de celles qui peuvent encore l'\u00eatre, voire qui le seront.\n\nDans le roman picaresque, certains bataillent contre des monstres quand d'autres, dessill\u00e9s, savent qu'il en va seulement de paisibles moulins. Montaigne transfigure les chim\u00e8res, les d\u00e9truit, et, lucide, invite \u00e0 voir le r\u00e9el comme il est. De sorte qu'il avance s\u00fbrement dans son odyss\u00e9e : du corps au Je, du Je au Moi, du Moi \u00e0 l'individu, il parvient au tragique qui suppose le monde vu au-del\u00e0 de l'optimisme et du pessimisme, dans sa r\u00e9alit\u00e9 crue, nette, coupante, aux ar\u00eates vives. Ni d\u00e9sesp\u00e9r\u00e9, ni enthousiaste, ni effondr\u00e9 ni excit\u00e9, il veut et voit le r\u00e9el \u00e0 partir des v\u00e9rit\u00e9s d'\u00e9vidence : le triomphe du temps, l'in\u00e9luctabilit\u00e9 de la mort, le travail perp\u00e9tuel de l'entropie, le mouvement partout, le changement sans cesse, et l'homme solitaire au milieu du fleuve d'H\u00e9raclite, insoucieux du rire et des larmes, optant pour la fermet\u00e9 d'un gai savoir tragique. L\u00e0 encore, je me sens chez moi dans les Essais.\n\nApr\u00e8s avoir abord\u00e9 des terres festives et riches de promesses, Montaigne mouille son vaisseau sur un nouveau continent dont il est, une fois de plus, le d\u00e9couvreur, l'inventeur : il formule un eud\u00e9monisme dont il th\u00e9orise voluptueusement le contenu pendant deux d\u00e9cennies. De l'eud\u00e9monisme, je dirai qu'il est la version douce de l'h\u00e9donisme qui, lui, revendique un radicalisme \u00e9thique. L'eud\u00e9monisme suppose que le souverain bien \u00e0 rechercher en priorit\u00e9, c'est le bonheur; l'h\u00e9doniste, lui, opte pour le plaisir. L'\u00e9tymologie t\u00e9moigne et avalise la diff\u00e9rence. Dans les deux cas prime la soumission de la philosophie \u00e0 l'art de bien vivre, aux techniques et moyens pour construire et produire une existence susceptible de conjurer le tragique.\n\nAristote a fait de ce souci le principe cardinal de ses deux \u00e9thiques : comment d\u00e9finir, pr\u00e9ciser et circonscrire un bien absolu ? De quelle mani\u00e8re r\u00e9aliser cette esth\u00e9tique du quotidien? Qu'est-ce qui distingue la sagesse, le bien, la vertu, le bonheur et le plaisir? Que retenir de s\u00fbr et certain pour distinguer la connaissance, l'acquisition et la pratique de cette sagesse ? Montaigne reprend la philosophie sur ce terrain et propose la formulation compl\u00e8te, homog\u00e8ne, d'un eud\u00e9monisme tragique moderne. La vie de tous les jours, l'emploi de son corps et de son temps, l'usage des plaisirs et les techniques de soi, pour le dire dans la terminologie contemporaine, voil\u00e0 le fond de toute qu\u00eate philosophique. Pour qui douterait de la coh\u00e9rence des Essais et chercherait un ordre dans l'apparent et hypoth\u00e9tique fouillis, l'eud\u00e9monisme fournit un fil conducteur, une trame, un axe pour tout ce qui s'est \u00e9crit entre la premi\u00e8re et la derni\u00e8re \u00e9dition des trois livres, entre le propos d'ouverture du premier et les mots de conclusion de l'ouvrage. Je ne sache pas qu'on ait d\u00e9j\u00e0 bross\u00e9 le portrait de ce Montaigne-l\u00e0.\n\nL'eud\u00e9moniste et l'h\u00e9doniste construisent leur propos sur l'\u00e9vidence \u2013 trop souvent n\u00e9glig\u00e9e \u2013 qu'il faudra mourir, que la vie demeure br\u00e8ve quelle qu'en soit la longueur, et que le p\u00e9ch\u00e9 mortel consiste \u00e0 faire un mauvais usage de son existence. Le corps individuel va vers le n\u00e9ant, la chair est appel\u00e9e \u00e0 se fl\u00e9trir, \u00e0 vieillir, \u00e0 p\u00e9rir, \u00e0 pourrir. Le sage \u00e9tablit les r\u00e8gles qui permettent une vie douce et heureuse, un quotidien apais\u00e9, harmonieux, autorisant la volupt\u00e9, la joie, le bonheur, la jubilation, le plaisir. L\u00e0 o\u00f9 le n\u00e9gatif fait son travail, quand la mort avance, progresse, le philosophe oppose son style, son art, son \u00e9l\u00e9gance, sa vertu, sa grandeur, son empire, son pouvoir, sa volont\u00e9. Quand tout tend \u00e0 la disparition, il c\u00e9l\u00e8bre l'apparition, l'av\u00e8nement de ce qui rend possible un r\u00e9el solaire. Vie philosophique, pens\u00e9e existentielle et vision tragique du monde se r\u00e9pondent, s'appellent, se compl\u00e8tent. C'est pour oublier ces v\u00e9rit\u00e9s d'\u00e9vidence que la philosophie s'est perdue, fourvoy\u00e9e, a vendu son \u00e2me et s'est constitu\u00e9e bien souvent sur le mod\u00e8le scolastique des anciennes th\u00e9ologies. Je crains que ce si\u00e8cle philosophique en passe de s'achever, symptomatiquement silencieux sur la figure du philosophe de la tour, n'ait \u00e9t\u00e9 lui aussi trop travaill\u00e9 par les p\u00e9dants.\n\nL'eud\u00e9monisme, pour se constituer, met en avant une pens\u00e9e de la nature et c\u00e9l\u00e8bre tout ce qui en d\u00e9coule : rien de naturel ne peut \u00eatre mauvais. M\u00eame si le corps agit en grande raison, si la chair oblige, les pulsions contraignent, le sage sait que, malgr\u00e9 la n\u00e9cessit\u00e9, le d\u00e9terminisme, il faut mettre sa volont\u00e9 au service du consentement \u00e0 ce qui advient d\u00e8s lors qu'on ne peut rien contre. En revanche, lorsqu'il y a m\u00eame une infime possibilit\u00e9 d'agir, il faut vouloir infl\u00e9chir le cours de ce qui peut l'\u00eatre. Contre ce \u00e0 quoi on ne peut \u00e9chapper, le philosophe ne se r\u00e9volte pas, il ne se rebelle pas inutilement et en pure perte contre l'ordre des choses. Mieux, il adh\u00e8re, il \u00e9pouse le mouvement, il aime son destin. Le volontarisme eud\u00e9moniste suppose la jubilation au simple fait d'\u00eatre et au libre exercice de son corps dans la vie quotidienne.\n\nLa sapience tragique induit un gai savoir actif. Et le premier travail qui attend le philosophe sur le chemin de cette sagesse \u00e0 fonder, c'est la critique des valeurs de l'asc\u00e9tisme sous toutes ses formes. Je laisse aux onanistes du concept, de l'entreglose et de la citation le d\u00e9bat d'un Montaigne sto\u00efcien, sceptique, \u00e9picurien, voire cynique, et pr\u00e9f\u00e8re lire les Essais comme le livre d'un eud\u00e9moniste produisant une th\u00e9orie coh\u00e9rente. Ni cyr\u00e9na\u00efque, ni pyrrhonien, ni socratique, ni rien de ce qui rel\u00e8ve de la philosophie antique, Montaigne formule une pens\u00e9e moderne abandonnant derri\u00e8re elle aussi bien les grandes \u00e9coles gr\u00e9co-romaines que leurs avatars chr\u00e9tiens.\n\nCertes, le christianisme de Montaigne est av\u00e9r\u00e9. D'abord dans ce que l'on peut lire au fil des pages de son livre majeur, mais aussi dans ce que rapporte le Journal de voyage qui montre le philosophe \u00e0 Notre-Dame de Lorette, o\u00f9 il effectue un p\u00e8lerinage et offre un ex-voto le repr\u00e9sentant avec sa femme et sa fille, \u00e0 Rome, o\u00f9 il sollicite une audience papale et une b\u00e9n\u00e9diction, pour sa personne et pour son \u0153uvre, enfin dans nombre d'\u00e9glises visit\u00e9es. Son oratoire personnel, sa messe du matin dans la tour, ses attaques mesur\u00e9es, mais pourtant r\u00e9elles, contre les protestants, tout montre une foi priv\u00e9e toujours \u00e9pargn\u00e9e par l'anatomie critique. Le chapitre consacr\u00e9 \u00e0 Raimond Sebond expose un christianisme temp\u00e9r\u00e9, entre d\u00e9isme et panth\u00e9isme, familier des preuves de l'existence de Dieu par la finalit\u00e9, \u00e9pargnant la question des dogmes ou la nature de la foi et dans l'ensemble des \u00e9crits, je ne vois rien plaidant pour l'agnosticisme, encore moins pour l'ath\u00e9isme.\n\nEn inventeur de la pens\u00e9e la\u00efque, Montaigne dissocie sa foi priv\u00e9e et sa pens\u00e9e publique. Il retient du catholicisme qu'il est la pratique ancestrale de ce qu'un autre appellera la religion de son roi et de sa nourrice. A ce titre, il laisse aux croyants la possibilit\u00e9 de pratiquer leur foi en paix tout en les invitant \u00e0 penser en hommes libres. Par ailleurs, dans la religion du Nouveau Testament, il n'entend pas consentir \u00e0 tout ce qui brime le corps, m\u00e9prise la chair ou fustige d\u00e9sirs, plaisirs et passions. D'o\u00f9 la possibilit\u00e9 de rendre \u00e0 Dieu ce qui est \u00e0 Dieu et au philosophe ce qui lui revient de droit, \u00e0 savoir le reste : l'\u00e9thique, la morale, la vertu, le bien et le mal, les valeurs, l'art de bien conduire son existence. La voie ainsi trac\u00e9e conduit \u00e0 une pure et simple \u00e9conomie de Dieu. Les suivants de Montaigne, dans son esprit, et via Marie de Gournay, ne failliront pas dans ce travail. Je songe \u00e0 La Mothe le Vayer et Saint-Evremond, par exemple, et \u00e0 tous ceux qui \u00e9voluent dans leur constellation.\n\nLe corps n'est pas p\u00e9cheur, la volupt\u00e9 n'est pas coupable, le plaisir n'est pas r\u00e9pr\u00e9hensible. D'ailleurs Montaigne pratique volontiers un certain nombre de p\u00e9ch\u00e9s capitaux avec une r\u00e9elle satisfaction : de la luxure \u00e0 la gourmandise, de la paresse \u00e0 l'orgueil \u2013 au fil de mes lectures et relectures, je ne lui constate aucun penchant pour l'envie, la col\u00e8re ou l'avarice \u2013, il t\u00e2te de tout. Les femmes lui plaisent, leur compagnie, leur pr\u00e9sence, qu'elles soient v\u00e9nales ou non, faciles ou pas, il est sensible \u00e0 leur charme, leur beaut\u00e9, leur proximit\u00e9 ; le vin, les viandes, le gibier, la table le ravissent ; le sommeil, le dormir, le repos, la pleine et enti\u00e8re disposition de soi, le loisir, l'otium, la chasse, la marche, l'ind\u00e9pendance, la paix, rien ne lui va mieux, ni ne le charme autant ; enfin, il s'aime assez, m\u00eame dans la d\u00e9testation, et je crois savoir qu'il enjolive parfois les moments o\u00f9 il se montre en h\u00e9ros devant l'adversit\u00e9 l\u00e0 o\u00f9, peut-\u00eatre, l'occasion ne lui a pas \u00e9t\u00e9 donn\u00e9e de l'\u00eatre \u00e0 ce point. Nul n'est parfait... En un mot, il traite son corps en ami et ne le mortifie pas en ennemi. Il aime la vie et ne fait aucune concession \u00e0 la mort. Ce en quoi il ne communie pas dans le christianisme paulinien, m\u00eame transfigur\u00e9 sur le mode la\u00efc \u2013 que j'aime la souverainet\u00e9 de cette libert\u00e9 rare...\n\nS'il go\u00fbte les plaisirs de la chair, il ch\u00e9rit m\u00eamement ceux que, par commodit\u00e9, je dirais de l'\u00e2me. De l'amiti\u00e9 \u00e0 la lecture en passant par l'\u00e9criture, la conversation, le voyage, la m\u00e9ditation, il jubile \u00e0 l'exercice de la pens\u00e9e. Mettre en sc\u00e8ne le plaisir lui donne \u00e9galement du plaisir. Il sait qu'aller au-devant d'une jouissance ou s'en souvenir c'est l'actualiser ou la r\u00e9actualiser. De la m\u00eame mani\u00e8re qu'il cherche les occasions d'aller vers la joie, il s'\u00e9vertue \u00e0 conjurer tout ce qui peut produire du d\u00e9sagr\u00e9ment, du d\u00e9plaisir, de la peine, de la douleur, aussi bien v\u00e9cu dans le moment que sollicit\u00e9 par la pens\u00e9e, soit \u00e0 venir, soit \u00e9coul\u00e9e. Le volontarisme eud\u00e9moniste veut ma\u00eetriser le pass\u00e9 aussi bien que l'avenir en r\u00e9duisant ces formes du temps au statut de fictions destin\u00e9es \u00e0 perturber ou r\u00e9jouir l'instant. Et il opte pour la r\u00e9jouissance. Montaigne excelle en ma\u00eetre du moment \u00e0 vivre, dense et joyeux. Ni ce qui a eu lieu ni ce qui va venir ne doit troubler la pointe de temps qu'on habite. Le\u00e7on cardinale pour lutter contre les douleurs. Je ne cesse d'y consentir, puis d'y tendre.\n\nHabiter le pr\u00e9sent; vouloir chaque instant comme s'il devait se multiplier et engrosser des potentialit\u00e9s d'avenir; aspirer \u00e0 l'\u00e9ternit\u00e9 en instillant dans le moment une dose diamantine de sagesse, de pens\u00e9e; faire co\u00efncider la vie et la pens\u00e9e, et retour : voil\u00e0 diverses formules d'un m\u00eame imp\u00e9ratif cat\u00e9gorique. A priori, le pr\u00e9sent offre une forme vide \u00e0 remplir philosophiquement, en jouissant de la vie, en jubilant d'\u00eatre au monde. Le pr\u00e9sent doit donner l'occasion de fomenter une pr\u00e9sence, de structurer une subjectivit\u00e9. Soumettre le pass\u00e9 et le futur \u00e0 l'ordre dans lequel le corps et la chair sont install\u00e9s, voil\u00e0 qui permet de conjurer la mort. Finalement, rien ne produit de d\u00e9plaisir que ce \u00e0 quoi on a consenti. En souvenance de La Bo\u00e9tie, j'\u00e9crirais volontiers : soyez r\u00e9solu de ne plus souffrir et vous voil\u00e0 libre.\n\nPour l'instant, je ne distingue pas l'h\u00e9doniste et l'eud\u00e9moniste : vision tragique du r\u00e9el, r\u00e9habilitation du corps charnel, confiance en la nature quand l'exercice du libre arbitre fait d\u00e9faut, le pr\u00e9sent apprivois\u00e9 quand l'usage de la libert\u00e9 est possible, promotion d'un gai savoir actif, lutte contre la mort et ses complices dans l'asc\u00e9tisme ou la mac\u00e9ration, go\u00fbt des plaisirs du corps et de l'\u00e2me sans les hi\u00e9rarchiser, voil\u00e0 mati\u00e8re \u00e0 \u00e9thique commune. En revanche, je dissocie les deux complices en pr\u00e9cisant que seul l'h\u00e9doniste, plus radical, revendique la brutalit\u00e9 de la foudre et des \u00e9clairs, la force et la puissance des volcans, l'\u00e9nergie des tremblements de terre et le lyrisme d'une vitalit\u00e9 d\u00e9bordante ; il veut sans concession le plaisir fort, violent, tel un alcool pur, sans souci de l'\u00e9conomie, sans calcul du risque, ignorant volontairement le danger, insolent \u00e0 l'endroit de la mort \u2013 d\u00fbt-il payer le prix fort.\n\nPour sa part, l'eud\u00e9moniste revendique et pratique la douceur, l'\u00e9quilibre et la paix, la s\u00e9r\u00e9nit\u00e9 et le bien-\u00eatre, un bonheur soyeux, poli, rabot\u00e9, une volupt\u00e9 am\u00e8ne, accorte, sinon humble ; il est poli face \u00e0 la mort, civil, voire affable. L'h\u00e9doniste, s'il pratique volontiers l'eud\u00e9monisme, \u00e0 ses heures, \u2013 l'inverse ne me semblant pas possible \u2013 sait aussi et peut parfois d\u00e9border les limites imparties par ce parti pris calme et bourgeois. Montaigne n'ira pas au-del\u00e0 de cette \u00e9thique douce. Pas d'\u00e9l\u00e9ments d\u00e9cha\u00een\u00e9s, de catastrophes, d'hyst\u00e9ries, d'\u00e9normit\u00e9s, de violences radieuses chez lui, pas de d\u00e9sir de r\u00e9volution ou de volont\u00e9 destructrice, mais la tranquillit\u00e9 d'un homme que le romantisme n'aurait s\u00fbrement pas tent\u00e9. Il aime, vit et pratique en vieil homme. M\u00eame quand il se met \u00e0 la r\u00e9daction de ses Essais, \u00e0 trente-neuf ans, il pense comme fatigu\u00e9, las et \u00e9conomise son \u00eatre en \u00e9vitant de l'exposer.\n\nLes vertus sp\u00e9cifiquement eud\u00e9monistes de Montaigne sont la douceur, la prudence et la discr\u00e9tion, \u2013 quoi qu'on pense pour cette derni\u00e8re. De la douceur, on peut dire qu'elle ne cesse de transpara\u00eetre dans son livre, par exemple sous la forme d'un \u00e9loge appuy\u00e9 de la tol\u00e9rance : jamais il ne pose la question des limites de celle-ci, aucune fois il ne s'interroge sur l'intol\u00e9rable, question bien plus puissante que l'habituelle logorrh\u00e9e sur le devoir de tol\u00e9rer. Cet \u00e9loge de la douceur lui fait critiquer nettement la torture pratiqu\u00e9e dans les prisons du royaume, sinon l'acharnement sur le corps des supplici\u00e9s, spectacle des rues jug\u00e9 par lui barbare et hors de propos. Pareillement, il met en cause les politiques imp\u00e9riales, imp\u00e9rieuses, dominatrices et coloniales : il n'a pas de mots assez durs pour fustiger l'entreprise des conquistadors, leur volont\u00e9 d\u00e9lib\u00e9r\u00e9e d'en finir avec une culture, une civilisation pour imposer le pouvoir de l'or et des \u00e9pices, puis du christianisme. Qui d'autre que lui parmi les philosophes \u00e0 cette heure, et plus tard ? Peu de monde, je le crains...\n\nToutefois, en pleine p\u00e9riode de guerre de religion, silencieux sur le massacre de la Saint-Barth\u00e9lemy, \u00e0 \u00e9gale distance des catholiques et des protestants, mais plut\u00f4t du c\u00f4t\u00e9 des premiers, faisant ses hommages au pape avec le cadeau de son livre, remis en main propre \u2013 c'est une formule \u2013, n\u00e9gligeant fort opportun\u00e9ment d'\u00e9diter les po\u00e8mes de La Bo\u00e9tie dans ses Essais pour ne pas donner de gages \u00e0 ceux des huguenots qui tiennent l'auteur du Discours de la servitude volontaire pour l'un des leurs, le philosophe manifeste une douceur un peu suave quand un peu de tenue h\u00e9doniste aurait justifi\u00e9 la position nette, tranch\u00e9e, l'esclandre ou le refus de composer avec le souverain pontife de la Saint-Barth\u00e9lemy, des royaumes tr\u00e8s chr\u00e9tiens et des exactions coloniales. Mais je crains que le chr\u00e9tien ne soit d\u00e9finitivement et structurellement interdit d'h\u00e9donisme alors que l'eud\u00e9monisme, dans les meilleures hypoth\u00e8ses, lui reste encore possible...\n\nLa douceur le conduit \u00e0 la prudence et celle-ci \u00e0 la discr\u00e9tion. Aussi, pour un homme qui a tant dit, \u00e9crit, racont\u00e9, qui s'est tellement mis en sc\u00e8ne, les silences propagent un bruit \u00e9trange que j'ai envie de troubler. Quid de sa m\u00e8re et de sa femme ? On croit savoir, par l'\u00e9pisode du collier, la complicit\u00e9 de ces deux-l\u00e0 contre Michel de Montaigne, sinon l'aide nette et franche de l'ancienne pour disculper sa belle-fille dans ce qui semble \u00eatre un adult\u00e8re entre sa bru et son autre fils. Quid de l'hypoth\u00e9tique frigidit\u00e9 de sa femme ? De ses amours tarif\u00e9s ici ou l\u00e0, \u00e0 Bordeaux, Paris ou dans les villes qu'il traverse pendant les dix-huit mois de son voyage ? Quid de la nature de sa relation avec l'excellente Marie de Gournay \u2013 amie, amante, ma\u00eetresse, disciple, fille sublim\u00e9e? Quid de l'homosexualit\u00e9 quand des falsifications volontaires et av\u00e9r\u00e9es de citations latines dans les Essais semblent fournir \u00e0 d'\u00e9ventuels enqu\u00eateurs contemporains une piste pour r\u00e9pondre \u00e0 cette question autrement qu'en recyclant l'\u00e9ternelle pr\u00e9sentation des manuels litt\u00e9raires ? Quid de son Dieu, de sa forme, de sa nature, de son existence? Quid r\u00e9ellement de ses positions sur les dogmes du catholicisme quand le grand livre montre un homme libre et critique l\u00e0 o\u00f9 la pratique, sinon le Journal de voyage, laissent entrevoir une foi de charbonnier ?\n\nDoux, prudent et discret, Montaigne ne s'expose pas, ne veut rien craindre. Son eud\u00e9monisme lui fait renoncer \u00e0 prendre le risque de choquer publiquement son Roi, son Pape, et, privativement, sa M\u00e8re, sa Femme, toutes figures de l'autorit\u00e9 auxquelles il consent, ce qui pourrait se comprendre, mais aussi auxquelles il se soumet \u2013 pour avoir la paix. L'h\u00e9doniste pr\u00e9f\u00e8re parfois la guerre. Du monarque et du successeur de saint Pierre, il risquait la disgr\u00e2ce, l'excommunication, l'embastillement; des matrones, une vindicte qui, de toute fa\u00e7on, semblait n'avoir besoin d'aucun pr\u00e9texte pour se manifester jour apr\u00e8s jour. D'o\u00f9, cons\u00e9quence de son eud\u00e9monisme, la c\u00e9l\u00e9bration d'un irr\u00e9fragable conservatisme revendiqu\u00e9 et pratiqu\u00e9 sans vergogne. Partout amoureux du mouvement, Montaigne vante sur le terrain politique, au sens large, les m\u00e9rites de l'immobilit\u00e9, de la fixit\u00e9, du statisme et de la r\u00e9p\u00e9tition. Il veut que rien ne change. Seul l'h\u00e9doniste, parce qu'il outrepasse, peut et veut en appeler \u00e0 la nouvellet\u00e9.\n\nDu corps \u00e0 l'individu, de l'individu \u00e0 l'eud\u00e9monisme, Montaigne conduit son lecteur sur des continents sublimes o\u00f9 l'on peut encore d\u00e9couvrir de nouvelles perspectives, de nouveaux paysages, en l'occurrence une \u00e9pist\u00e9mologie intempestive, une m\u00e9thode inactuelle susceptibles d'une perp\u00e9tuelle actualit\u00e9, quels que soient les histoires et les g\u00e9ographies, les temps et les lieux. Il invente une forme, une fa\u00e7on, un style, il promeut un mode de penser, une th\u00e9orie, au sens \u00e9tymologique, un regard. L'ensemble se place sous le signe du changement, de la plasticit\u00e9 aussi bien du monde que du livre charg\u00e9 d'en rendre compte. Rien ne dure, sauf le passage. Avec son livre, Montaigne \u00e9labore une machine \u00e0 capturer ces flux, une technique pour immobiliser, le temps d'un propos, l'allure et la nature ondoyante du r\u00e9el. Pour ce faire, il entreprend d'essayer \u2013 d'o\u00f9 le titre \u00e9ponyme.\n\nLa m\u00e9taphore esth\u00e9tique correspondant \u00e0 Montaigne n'est pas celle du sculpteur qui enl\u00e8ve des fragments, supprime des morceaux, taille et coupe pour retrancher dans la masse brute. Les Essais sont et font peinture, du moins pour la technique : ajouts, pr\u00e9cisions, superpositions, compl\u00e9ments, raccords, repentirs, corrections. La composition ressortit au caprice, \u00e0 la volont\u00e9 libre. Chez lui, pas de livres multiples dont les uns amenderaient les autres, confirmeraient ou infirmeraient le propos des pr\u00e9c\u00e9dents, il veut un seul et m\u00eame livre. Comment d'ailleurs pourrait-on vouloir plusieurs vies, diff\u00e9rentes, s\u00e9par\u00e9es? Qu'on puisse conna\u00eetre mille existences en une, voil\u00e0 qui est possible, souhaitable m\u00eame, qu'on puisse lire mille livres en un, voil\u00e0 ce que propose ce grand \u0153uvre. Montaigne invente le livre unique pour une existence sans double, puis il installe l'auteur \u00e0 l'\u00e9picentre de cette jonction.\n\nLe trajet solitaire du philosophe co\u00efncide avec la forme sans duplication de son ouvrage. De la tour am\u00e9nag\u00e9e du ch\u00e2teau aux Essais en passant par l'existence propre, le monolithe triomphe, le monument et l'\u00e9difice priment. J'aime cette fa\u00e7on d'\u00eatre sans pr\u00e9curseur, sans mod\u00e8le, sans successeur. Apr\u00e8s le gothique des sommes th\u00e9ologiques et des argumentations sp\u00e9cieuses de la patrologie universitaire, pour en finir avec le latin et la langue des doctes, les questions, articles, objections, r\u00e9ponses, solutions et autres artifices destin\u00e9s \u00e0 donner une allure scientifique \u00e0 des propos hypoth\u00e9tiques, Montaigne fait table rase, r\u00e9dige en fran\u00e7ais, dans une langue savoureuse et composite o\u00f9 le gascon chatoie tout aussi bien que la langue vernaculaire. Les philosophes de m\u00e9tier et de profession s'\u00e9tranglent..\n\nIl d\u00e9truit l'\u00e9difice du savoir traditionnel, la plupart du temps universitaire ou d'Eglise, puis b\u00e2tit une architecture dans laquelle il suffit du bon vouloir pour se mouvoir. Contre les ghettos, les p\u00e9dants, les discours herm\u00e9tiques, le vocabulaire des sectes et des castes visant la reproduction, il met au jour une nouvelle fa\u00e7on de philosopher, ni populaire ni populiste, ni complaisante ni racoleuse, soucieuse de permettre \u00e0 ceux qui la d\u00e9sirent d'y acc\u00e9der sous b\u00e9n\u00e9fice d'un investissement intellectuel digne de ce nom. A l'heure des offres indignes faites aux demandes l\u00e9gitimes de pens\u00e9e et de r\u00e9flexion, je pense cette option plus d'actualit\u00e9 que jamais.\n\nL'id\u00e9e lui serait inacceptable d'une pens\u00e9e au rabais faite pour le plus grand nombre afin de laisser croire qu'il suffirait, pour philosopher, de recycler les lieux communs, les propos du temps ou les id\u00e9es fixes de la majorit\u00e9. Il donne \u00e0 chacun les moyens d'acc\u00e9der au monde des id\u00e9es en faisant, pour sa part, ce qui rel\u00e8ve de son seul pouvoir : d\u00e9graisser le conceptuel, cong\u00e9dier la sottise pr\u00e9tentieuse, abolir le charabia, interdire l'incantation verbeuse, d\u00e9lier la langue philosophique, faire proc\u00e9der la pens\u00e9e de la vie, \u00e9viter la casuistique pour elle-m\u00eame, \u00e9crire en romancier, philosopher en artiste, dissoudre le mod\u00e8le scientifique, adouber la litt\u00e9rature, sinon la po\u00e9sie. A charge pour le lecteur d'accomplir l'autre moiti\u00e9 du chemin et d'exiger de soi l\u00e0 o\u00f9 le philosophe a offert sa complicit\u00e9 et sa proximit\u00e9. La conversation avec les livres d'un auteur exige du talent chez le lecteur, Montaigne le lib\u00e8re de toutes les t\u00e2ches rebutantes ou ennuyeuses pour lui donner vraiment la possibilit\u00e9 d'exercer le talent en question.\n\nL'\u00e9pist\u00e9mologie des Essais me pla\u00eet \u00e9galement parce qu'elle conf\u00e8re \u00e0 l'anecdote un statut philosophique. La dissertation, la glose, le commentaire s'effectuent moins sur un morceau de livre, un extrait de biblioth\u00e8que, que sur un fragment de r\u00e9el pr\u00e9lev\u00e9 dans la vie quotidienne. La citation illustre et ne tient pas lieu de r\u00e9flexion par procuration. La rencontre d'un monstre, un accident d'\u00e9quitation, un voyage en bateau, une conversation avec l'un de ses employ\u00e9s, le spectacle d'un jardinier dans son potager, un d\u00eener d'amis, la ressemblance d'un p\u00e8re avec son fils, une soir\u00e9e bien arros\u00e9e, l'arriv\u00e9e de Br\u00e9siliens dans un port normand, le croisement de sa fille dans la cour du ch\u00e2teau, l'usage du mouchoir ou des couverts, les parfums venus d'une cuisine, un r\u00eave dont on se souvient au matin, une promenade \u00e0 pied, une propension nocturne au priapisme, tout cela fournit des pr\u00e9textes, des occasions, des accroches pour une dissertation, une analyse, une m\u00e9ditation, un d\u00e9veloppement plus ou moins long et plus ou moins d\u00e9bordant.\n\nLes snobs, les p\u00e9dants, les universitaires et les gens s\u00e9rieux r\u00e9pugnent \u00e0 cela. Ils pr\u00e9f\u00e8rent l'\u00e9th\u00e9r\u00e9, le pr\u00e9tentieux, l'incompr\u00e9hensible, l'obscur, l'alambiqu\u00e9, et tout ce qui permet, sous une forme ou une autre, de r\u00e9actualiser le vieux phantasme d'une th\u00e9ologie terrorisant ceux qui n'ont pas l'heur d'\u00eatre d\u00e9j\u00e0 croyants, affid\u00e9s ou soumis. Ils vilipendent un manque de rigueur proc\u00e9dant chez eux du regret qu'il n'y ait pas l\u00e0 quelque belle confusion ou sublime fum\u00e9e qu'ils se proposeraient d'\u00e9claircir, le dipl\u00f4me \u00e0 la main, alors qu'ils se contenteraient d'ajouter de l'obscurit\u00e9 et de la crasse, voire de cr\u00e9er la confusion l\u00e0 o\u00f9 elle n'est pas. Il me faudrait nommer l\u00e0 une grande partie de la corporation...\n\nDonneurs de le\u00e7ons, maniaques du crayon rouge, professeurs de ressentiment, cr\u00e9ateurs impuissants, distributeurs de bons points se cachent derri\u00e8re l'institution pour interdire cette philosophie, la d\u00e9valuer ou lui d\u00e9nier le droit de se parer des plumes de la fonction, car elle les d\u00e9poss\u00e8de de leurs ridicules pr\u00e9rogatives de traducteurs et d'adaptateurs immodestes de pens\u00e9es pr\u00e9tendument n\u00e9buleuses en brouets verbeux et insipides. Montaigne discr\u00e9dite cette caste et appelle \u00e0 l'\u00e9conomie de tout tiers entre lui, son \u0153uvre et son lecteur. Tout ce qui s'interpose est suspect\u00e9, et avec raison. En n\u00e9gligeant la caste des derviches penseurs, des voyageurs de commerce du concept d'autrui, il se pr\u00e9pare nombre d'ennemis. Et c'est tant mieux. Que durent les anecdotes, elles cr\u00e9ditent le s\u00e9rieux d'une entreprise de gai savoir.\n\nLa langue, la forme et l'anecdote visent la fabrication d'un registre sp\u00e9cifique \u00e0 la philosophie. Sous de fausses allures patelines, les Essais proposent une psychagogie, une mise \u00e0 mal de tous les \u00e9difices illusoires, phantasmes, dogmes, mensonges sociaux et autres cristallisations gr\u00e9gaires au profit d'une pens\u00e9e critique la\u00efque, dissoci\u00e9e du monde religieux renvoy\u00e9, quant \u00e0 lui, \u00e0 l'univers propre au th\u00e9ologien et au pr\u00eatre d'o\u00f9 est exclu le philosophe. Apr\u00e8s avoir distingu\u00e9 et sp\u00e9cifi\u00e9 que les deux mondes fonctionnent suivant des principes diff\u00e9rents, la foi pour l'un, l'examen critique pour l'autre, Montaigne assoit d\u00e9finitivement la pens\u00e9e sur l'unique plan de l'immanence.\n\nEn d\u00e9couvreur imp\u00e9nitent, il invente l'intempestivit\u00e9 qui lui permet d'\u00eatre lisible depuis quatre si\u00e8cles, et pour d'autres encore vraisemblablement. Ses intuitions ne cessent de m'\u00e9tonner sur le terrain de l'\u00e9thologie, de la g\u00e9n\u00e9tique, de la p\u00e9dagogie, de la psychosomatique, de la sociologie, de l'ethnologie. Je persiste dans l'expectative admiratrice... De son \u0153uvre d\u00e9coulent en ligne plus ou moins directe le si\u00e8cle qui suit, le cart\u00e9sianisme, le libertinage \u00e9rudit, puis le sensualisme, le mat\u00e9rialisme, l'esprit encyclop\u00e9dique \u2013 que le Journal de voyage invente \u2013, voire le criticisme et le perspectivisme. Rien de philosophique n'a vu le jour sans lui ou malgr\u00e9 lui. Peut-on imaginer post\u00e9rit\u00e9 plus r\u00e9ussie ? Je ne crois pas.\n\nDe Montaigne, je veux retenir la le\u00e7on majeure que l'\u00e9criture, la vie et la philosophie ont partie li\u00e9e; qu'une r\u00e9flexion n'ayant pas pour point de d\u00e9part et d'arriv\u00e9e l'existence singuli\u00e8re et la possibilit\u00e9 de se comporter en singularit\u00e9 absolue ne m\u00e9rite aucune heure de peine; que le corps subsiste apr\u00e8s l'effondrement de tout; que les livres \u2013 \u00f4 combien ! \u2013 sont affaires essentielles dans une vie ; que la mort triomphe quoi qu'on fasse, du moins qu'il est possible, en attendant, de construire un \u00e9difice r\u00e9sistant et superbe ; que le moi n'est pas ha\u00efssable ; qu'une pens\u00e9e doit \u00eatre praticable ; qu'une vie philosophique vaut autant, sinon plus, qu'une pens\u00e9e vivante; que les plaisirs consolent d'avoir \u00e0 dispara\u00eetre; que la mort n'est rien, finalement, ou pas grand-chose \u2013 et que le seul probl\u00e8me qui m\u00e9rite d'\u00eatre pos\u00e9 et r\u00e9solu renvoie \u00e0 l'usage de soi entre deux n\u00e9ants.\n\nPendant les trois jours qui pr\u00e9c\u00e8dent sa mort, Montaigne souffre d'une esquinancie de la bouche qui lui interdit toute parole, laissant d\u00e8s lors au livre le soin de dire l'essentiel. A la plume, pour en finir avec des milliers d'heures d'une pratique autrement g\u00e9niale, il \u00e9crit ses derni\u00e8res volont\u00e9s. Parmi elles, le d\u00e9sir de prendre cong\u00e9 avec un certain nombre des gentilshommes de ses voisins, puis de payer en main propre quelques-uns des legs annonc\u00e9s dans son testament, car, trop avis\u00e9 de la nature humaine, il sait ce que p\u00e8sent aux yeux des h\u00e9ritiers les souhaits ultimes d'un mort enseveli sous la dalle. La tradition veut que son dernier soupir se soit fait fort opportun\u00e9ment contemporain de l'\u00e9l\u00e9vation, au moment de la messe dite pour lui dans sa chambre. S'il faut en croire Pierre de Brach, sa mort fut \u00e0 la hauteur de ce que proposent les Essais, de ce qu'enseigne son livre, de ce qu'il en a dit. Son existence aussi. L'\u0153uvre \u00e9tait achev\u00e9e, accomplie, aussi bien le livre que la vie.\n\nMadame l'\u00e9pouse, Fran\u00e7oise de la Chassaigne, fait alors une veuve exemplaire et contribue, avec Marie de Gournay, \u00e0 la mise au point d\u00e9finitive du grand \u0153uvre de son \u00e9poux disparu. Puis elle commande qu'on fouille le thorax du philosophe pour en arracher le c\u0153ur \u2013 \u00e0 d\u00e9faut des rognons \u2013 et d\u00e9cide du transport \u00e0 la chapelle Saint-Michel de Montaigne. Le reste du corps est inhum\u00e9 dans l'\u00e9glise des Feuillants \u00e0 Bordeaux avant que le c\u00e9notaphe, dont on ne sait s'il contient vraiment le squelette du philosophe, ou s'il est vide, ne tr\u00f4ne fort singuli\u00e8rement au milieu d'une pi\u00e8ce du Mus\u00e9e des beaux-arts, entre deux gardiens dolents et abrutis. Je trouve peu philosophiques le heaume, l'armure, les gantelets, l'\u00e9p\u00e9e, les mains jointes et la raideur du gisant; en revanche j'aime assez le petit lion \u00e0 ses pieds, gros comme un chien de compagnie, affectueux, pr\u00e9sent comme un chat, silencieux et fid\u00e8le, loin du roi des animaux habituellement repr\u00e9sent\u00e9.\n\nLe corps du philosophe, loin des tombeaux, des \u00e2mes frisson-nantes, des m\u00e2nes de la tour, des \u00e9motions qu\u00eat\u00e9es, des souvenirs sollicit\u00e9s, je l'ai moins retrouv\u00e9 dans le muscle cardiaque dess\u00e9ch\u00e9, dans l'hypoth\u00e8se d'un ensemble d'ossements plus ou moins certifi\u00e9s ou sous le couvercle qu'un quelconque monument fun\u00e9raire aux cr\u00e2nes ail\u00e9s, aux grotesques et arabesques mani\u00e9ristes sculpt\u00e9es dans le marbre, ni dans le silence recueilli qui accompagne les lieux du p\u00e8lerinage philosophique. Le corps de Michel de Montaigne, il m'a presque \u00e9t\u00e9 donn\u00e9 de le toucher, de le prendre dans mes mains, d'en sentir le poids, d'en \u00e9prouver la pr\u00e9sence \u2013 comme je fus corps jadis dans les bras d'un infirmier un jour que je manquai un premier rendez-vous avec la mort \u2013, gr\u00e2ce \u00e0 l'amiti\u00e9, quand, par elle, il me fut offert de feuilleter l'exemplaire de Bordeaux \u00e0 la biblioth\u00e8que municipale. Moment \u00e9mouvant. On l'avait sorti d'un coffre-fort, extrait d'un bo\u00eetier de cuir, pos\u00e9 sur support de velours noir, puis ouvert. Je d\u00e9couvris sur les folios secs et souples, l'\u00e9criture, fine, serr\u00e9e, r\u00e9guli\u00e8re, l'impression des caract\u00e8res, grav\u00e9s dans le papier, creus\u00e9s dans le v\u00e9lin craquant, la fra\u00eecheur de la reliure et des ors, la qualit\u00e9 des nerfs et du cuir, la tranche color\u00e9e et les goutti\u00e8res comme une ride unique sur un visage. Les parfums s\u00e9culaires et les odeurs d'\u00e9ternit\u00e9 \u2013 l'odeur de Michel de Montaigne \u2013, je les \u00e9prouvais dans une \u00e9motion qui ne me quitte pas. Le livre \u00e9quivaut c\u00e9notaphe et tombeau, la vie ne vaut que transfigur\u00e9e en livre.\n\nFIN DU DEUXI\u00c8ME TOME\n\n***\n\nTOME TROISI\u00c8ME :\n\nL'ARCHIPEL DES COM\u00c8TES\nCOLLECTION \u00ab FIGURES \u00bb DIRIG\u00c9E PAR BERNARD-HENRI L\u00c9VY\n\nLes Aristocrates libertaires, Manifeste.\n\nJean-Paul Aron et Roger Kempf, le P\u00e9nis et la D\u00e9moralisation de l'Occident.\n\nDominique Auffret, Alexandre Koj\u00e8ve.\n\nJean Baudrillard, la Gauche divine.\n\nJean Baudrillard, les Strat\u00e9gies fatales.\n\nJean-Marie Benoist, la R\u00e9volution structurale.\n\nClaudie et Jacques Broyelle, Apocalypse Mao.\n\nClaudie et Jacques Broyelle, les Illusions retrouv\u00e9es.\n\nMadeleine Chapsal, Envoyez la petite musique.\n\nFran\u00e7ois Ch\u00e2telet, Jacques Derrida, Michel Foucault, Jean-Fran\u00e7ois Lyotard, Michel Serres, Politiques de la philosophie (textes r\u00e9unis par Dominique Grisoni).\n\nCatherine Cl\u00e9ment, le Go\u00fbt du miel.\n\nCatherine Cl\u00e9ment, Les fils de Freud sont fatigu\u00e9s.\n\nCatherine Cl\u00e9ment, l'Op\u00e9ra ou la D\u00e9faite des femmes.\n\nCatherine Cl\u00e9ment, la Syncope.\n\nCatherine Cl\u00e9ment, Vies et l\u00e9gendes de Jacques Lacan.\n\nBernard Cohen, Portes de J\u00e9rusalem.\n\nAnnie Cohen-Solal, Paul Nizan, communiste impossible.\n\nChristian Delacampagne, Antipsychiatrie. Les voies du sacr\u00e9.\n\nGalvano Della Volpe, Rousseau et Marx.\n\nJean-Toussaint Desanti, R\u00e9flexions sur le Temps. Variations philosophiques 1. Conversations avec Dominique-Antoine Grisoni.\n\nLaurent Dispot, la Machine \u00e0 terreur.\n\nLaurent Dispot, Manifeste archa\u00efque.\n\nJean-Paul Doll\u00e9, Danser maintenant.\n\nJean-Paul Doll\u00e9, Fureurs de ville.\n\nJean-Paul Doll\u00e9, l'Odeur de la France.\n\nJean-Paul Doll\u00e9, Voie d'acc\u00e8s au paisir.\n\nUmberto Eco, Lector in fabula.\n\nLuc Ferry et Alain Renaut, Heidegger et les Modernes.\n\nMichel Gu\u00e9rin, Lettres \u00e0 Wolf ou la R\u00e9p\u00e9tition.\n\nMichel Gu\u00e9rin, Nietzsche, Socrate h\u00e9ro\u00efque.\n\nG\u00e9rard Haddad, les Biblioclastes.\n\nG\u00e9rard Haddad, Manger le livre.\n\nHeidegger et la question de Dieu (sous la direction de R. Kearney et J.S. O'Leary).\n\nJacques Henric, la Peinture et le mal.\n\nJacques Henric, le Roman et le sacr\u00e9.\n\nL'Identit\u00e9, s\u00e9minaire dirig\u00e9 par Claude L\u00e9vi-Strauss, 1974-1975.\n\nChristian Jambet, Apologie de Platon.\n\nChristian Jambet et Guy Lardreau, l'Ange.\n\nChristian Jambet et Guy Lardreau, le Monde.\n\nFran\u00e7ois Jullien, Figures de l'immanence. Pour une lecture philosophique du Yi king.\n\nGuy Konopnicki, l'\u00c2ge d\u00e9mocratique.\n\nGuy Konopnicki, l'Amour de la politique.\n\nGuy Lardreau, la Mort de Joseph Staline.\n\nMichel Le Bris, l'Homme aux semelles de vent.\n\nMichel Le Bris, le Paradis perdu.\n\nDominique Lecourt, Bachelard. Le jour et la nuit.\n\nEmmanuel Levinas, Entre nous. Essais sur le penser-\u00e0-l'autre.\n\nEmmanuel Levinas, Dieu, la Mort, le Temps.\n\nBernard-Henri L\u00e9vy, la Barbarie \u00e0 visage humain.\n\nBernard-Henri L\u00e9vy, \u00c9loge des intellectuels.\n\nBernard-Henri L\u00e9vy, l'Id\u00e9ologie fran\u00e7aise.\n\nBernard-Henri L\u00e9vy, le Testament de Dieu.\n\nThierry L\u00e9vy, le Crime en toute humanit\u00e9.\n\nClaude Lorin, l'Inachev\u00e9 (Peinture-Sculpture-Litt\u00e9rature).\n\nClaude Lorin, Pour saint Augustin.\n\nJean-Luc Marion, l'Idole et la distance.\n\nAnne Martin-Fugier, la Bourgeoise.\n\nAnne Martin-Fugier, la Place des bonnes.\n\nJacques Martinez, Moderne for ever.\n\nJacques Martinez, Par hasard et par exemple.\n\nG\u00e9rard Miller, Du p\u00e8re au pire.\n\nPhilippe Muray, la Gloire de Rubens.\n\nMarc Nacht, A l'aise dans la barbarie.\n\nSami Na\u00efr, Le Regard des vainqueurs. Les enjeux fran\u00e7ais de l'immigration.\n\nPhilippe Nemo, l'Homme structural.\n\nMichel Onfray, l'Art de jouir.\n\nMichel Onfray, Cynismes. Portrait du philosophe en chien.\n\nMichel Onfray, le D\u00e9sir d'\u00eatre un volcan. Journal h\u00e9doniste I.\n\nMichel Onfray, Politique du rebelle. Trait\u00e9 de r\u00e9sistance et d'insoumission.\n\nMichel Onfray, la Sculpture de soi. La morale esth\u00e9tique.\n\nMichel Onfray, le Ventre des philosophes.\n\nPasolini, s\u00e9minaire dirig\u00e9 par Antonietta Macciochi.\n\nFran\u00e7oise Paul-L\u00e9vy, Karl Marx, histoire d'un bourgeois allemand.\n\nPhilippe Roger, Roland Barthes, roman.\n\nPhilippe Roger, Sade. La philosophie dans le pressoir.\n\nGuy Scarpetta, l'Age d'or du roman.\n\nGuy Scarpetta, l'Artifice.\n\nGuy Scarpetta, Brecht ou le Soldat mort.\n\nGuy Scarpetta, \u00c9loge du cosmopolitisme.\n\nGuy Scarpetta, l'Impuret\u00e9.\n\nMichel Serres, Zola. Feux et signaux de brume.\n\nDaniel Sibony, l'Amour inconscient.\n\nDaniel Sibony, Avec Shakespeare.\n\nDaniel Sibony, Entre dire et faire.\n\nDaniel Sibony, Jouissances du dire.\n\nDaniel Sibony, la Juive : une transmission d'inconscient.\n\nDaniel Sibony, Perversions.\n\nBernard Sich\u00e8re, \u00c9loge du sujet.\n\nBernard Sich\u00e8re, Merleau-Ponty ou le Corps de la philosophie.\n\nBernard Sich\u00e8re, le Moment lacanien.\n\nAlexandre Soljenitsyne, l'Erreur de l'Occident.\n\nPhilippe Sollers, Vision \u00e0 New York.\n\nGilles Susong, la Politique d'Orph\u00e9e.\n\nArmando Verdiglione, la Dissidence freudienne.\n\nArmando Verdiglione, Fondation de la psychanalyse. I. Dieu.\n\nArmando Verdiglione, la Conjuration des idiots.\n\nGiambattista Vico, Vie de Giambattista Vico \u00e9crite par lui-m\u00eame.\n\nClaude Vig\u00e9e, l'Extase et l'Errance.\n\nClaude Vig\u00e9e, le Parfum et la Cendre.\n\nElie Wiesel, Signes d'exode.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by Al Haines\n\n\n\n\n\n\n\n\n\n\nPOEMS OF LONDON\n\nAND OTHER VERSES\n\n\n\nBY\n\nJOHN PRESLAND\n\n\n\n\nMACMILLAN AND CO., LIMITED\n\nST. MARTIN'S STREET, LONDON\n\n1918\n\n\n\n\nCOPYRIGHT\n\n\n\n\nCONTENTS\n\n\nPOEMS OF LONDON\n\n  London Dawn\n  Spring in Oxford Street\n  Judd Street, St. Pancras\n  Sparrows\n  The Moon in January\n  An August Night, 1914\n  Counted out--Olympia\n  The German Band\n  Street Music--I\n  Street Music--II\n  Piccadilly\n  In the Tube\n  London Idyll--I\n  A London Idyll--II\n  Finis\n\n\n  OTHER VERSES\n\n  In Early Spring\n  A Ballad of the Fall of Knossos\n  A Sun-Dial in a Garden\n  \"Two Only\"\n  The Saint's Birthday\n  Rupert Brooke\n  \"Comfort be with Apples, for I am sick of Love\"\n  Of England\n  Question\n  Leonardo to Monna Lisa.\n  The Eternal Flux\n  Love is the Ultimate Measure\n  November 8\n  The Lovers\n  The Gentle Heart\n  A Ballad for Herman\n  France\n  Ilgar's Song\n  The Inn--I\n  The Inn--II\n  \"To-Day I miss You\"\n  \"How Small the Thread\"\n  \"In all Things gracious there is a Thought of You\"\n  \"There's Duty, Friend\"\n  \"Evening\"\n  Finis\n\n\n\n\n  POEMS OF LONDON\n\n\n\n\n  LONDON DAWN\n\n  Dawn over London; all the pearly light\n  Trembles and quivers over street and park,\n  The houses are a strange, unearthly white;\n  Pavement and roof grow slowly, palely bright;\n  There is no shadow, neither light nor dark\n  But everything is steeped in glimmering dawn.\n\n  Oh, purity of dawn; oh, milk-and-pearl\n  Translucent splendour, spreading far and wide,\n  As on a yellow beach the small waves curl\n  --Almost as noiselessly as buds unfurl--\n  On windless mornings with the rising tide,\n  So flows the dawn o'er London, all asleep.\n\n  Indeed, I think that heaven is a sea,\n  And London is a city of old rhymes\n  Sunk fathoms deep in its transparency,\n  That folk of living lands may dream they see\n  And muse on, and have thoughts about our times,\n  How we were great and splendid, and now gone.\n\n  For never light the common earth has born,\n  This crystalline pale wonder that so falls\n  On streets and squares the daily toil has worn,\n  On blind-eyed houses, holding lives forlorn,\n  For the grey roads and wide, blank, grey-brick walls\n  Shine with a glory that is new and strange.\n\n  And not more wonderful, nor otherwise\n  Shall dawn come up upon the dewy hills,\n  Nor in the mountains, where the rivers rise\n  That water Eden; and no lovelier lies\n  The dawn on Paradise, than this that fills\n  The space 'twixt house and house with tremulous light.\n\n  Yet, on the pavement, huddled fast asleep,\n  A thing of dusty, ragged misery,\n  Grotesque in wretchedness, from London's deep\n  Spumed off, a strange, distorted thing to creep\n  From God knows where, and lie, and let all be\n  Unheeding, whether of the day or night.\n\n  Such tired, hopeless angles of the knees\n  And neck and elbows--and the dawning grey\n  Trembling to sunrise; in the park the trees\n  Begin to shiver lightly in a breeze,\n  And turning watchful kindly eyes away\n  The policeman passes slowly on his beat.\n\n\n\n\n  SPRING IN OXFORD STREET\n\n  A dash of rain on the pavement,\n  In the air a gleam of sun,\n  And the clouds are white, and rolling high\n  From Marble Arch all down the sky\n  --And that's the spring begun!\n\n  The sky is all a-shining\n  With sunniest blue and white,\n  The flags are streaming out full cry\n  As the crisp North wind comes bustling by,\n  And all the roofs are bright.\n\n  And all the shops and houses\n  Of sunlit Oxford Street,\n  --Pearl behind amber, gold by rose--\n  To grey the long perspective goes;\n  Till all the houses meet.\n\n  And there, in every gutter,\n  The glory of spring flowers\n  The whole long street with colour fills,\n  And across the yellow daffodils\n  Sharp sunshine and soft showers.\n\n  And among the drabs and greys and browns\n  Of folk going to and fro\n  Are trays of violets, darkly bright,\n  And yellow, like the spring moon's light,\n  Pale primrose-bunches show.\n\n  There's blue in every puddle,\n  And every pane of glass\n  Has a thousand little dancing suns,\n  --And up and down the glad news runs,\n  That spring has come to pass.\n\n\n\n\n  JUDD STREET, ST. PANCRAS\n\n  My dwelling has a courtyard wide\n  Where lord with lady well might pace,\n  --Such silks and velvets side by side,\n  And she a fan to shield her face!--\n  It's fine as any king's;\n  For there I see on either hand\n  The whole great stretch of London lie;\n  --Just so as any king might stand\n  Upon his roof, to watch go by\n  The flashing pigeon wings.\n\n  Just so a king might look abroad:\n  \"And this is all my own,\" says he,\n  And then he'd turn to some great lord,\n  Who'd acquiesce with gravity\n  --But that I do without,\n  For all of lord there is up here\n  Is this impassive chimney-stack,\n  And cloudy be my view or clear\n  My courtier will not answer back;\n  All silent I look out,\n\n  And see the flight of roofs that fade\n  Towards the West in golden haze,\n  And all this work men's hands have made\n  Like jewels in the sun's last rays--\n  I have a dwelling wide;\n  Three rooms are mine, but I can go\n  Up to this roof in shade or shine,\n  And watch all London change and glow\n  Rose, purple, gold; three rooms are mine--\n  And all of heaven beside.\n\n\n\n\n  SPARROWS\n\n  Brown little, fat little, cheerful sparrows!\n  I like to think, when I hear them chatter,\n  How, when the brazen noise was gone\n  Of the chariot-wheels, with the sparks a-scatter,\n  Their chirp was heard in old Babylon.\n\n  In Babylon, and more ancient Memphis,\n  They chattered and quarrelled, pecked and fumed,\n  And loved their loves, and flew their ways,\n  Where the royal Pharaohs lay entombed\n  Deep from the daylight's vulgar gaze.\n\n  Then, just such little homely fellows\n  (When the angry monarch, terrible,\n  Watched his curled Assyrians writhe)\n  They sat, on a carven granite bull\n  Unheeding of anguish, feathered and blithe.\n\n  So did they sit, on the roofs of Rome,\n  And preen themselves in the morning sun;\n  And Caesar saw them, brown and grey,\n  Whisk in the dust, when his course was run\n  And he took to the Forum his fated way.\n\n  Oh, changing time; oh, sun and birds\n  How little changing.  In the Square\n  This winter morning I have met\n  Old Egypt's grandson, stopped him there,\n  And \"Sir, you will outlive me yet,\"\n  Said I politely, \"mark my words.\"\n\n\n\n\n  THE MOON IN JANUARY\n\n  Sharp and straight are the scaffold poles,\n  Black on a delicate sky;\n  Upright they stand, across they lie,\n  In changeless angles fixed and bound,\n  The sunset light in mist is drowned,\n  And the moon has risen high;\n\n  High above houses, high and clear\n  Above the scaffolding,\n  So exquisite, so faint a thing,\n  The young moon's silver curve that shines\n  Above the fretting, tangled lines,\n  With the old moon in her ring.\n\n  The young moon holds the old black moon\n  In a sky all grey with frost,\n  By cable wires barred and crossed,\n  And below, the haze of purplish-brown\n  Smokes upward from the lamp-lit town\n  Where outlines all are lost.\n\n  The pure pale arch of windless sky,\n  The pure bright young moon's thread,\n  These wide and still are overhead;\n  And in the dusky glare below\n  The lamps go dotting, row on row,\n  And there is movement, to and fro,\n  Where far the pavements spread.\n\n\n\n\n  AN AUGUST NIGHT, 1914\n\n  The light has gone from the West; the wind has gone\n  From the quiet trees in the Park;\n  From the houses the open windows yellowly shine,\n  The streets are softly dark;\n\n  Row upon row the twisted chimneys stand,\n  Each angle sharply lined,\n  And the mass of the Institute rises, tower and dome,\n  Black on the sky behind;\n\n  Green is the sky, like some strange precious stone,\n  Dark, it yet holds the light\n  In its depths, like a bright thing shrouded over or veiled\n  By the creeping shadow of night;\n\n  And whiter than any whiteness there is upon earth\n  A faint star throbs and beats--\n  And the hurrying voices cry the news of the war,\n  Below, in the quiet street.\n\n\n\n\n  COUNTED OUT--OLYMPIA\n\n  The small white space roped off; the hard blue light\n  Burning intensely on the narrow ring,\n  And every muscle's movement sculpturing\n  Harshly, of those two naked men who fight;\n  Beyond, the yellow lights that seem to swing\n  Across abysmal darkness; and below,\n  Tier upon tier, all silent, row on row\n  The dense black-coated throng, and all a-strain\n  White faces, turned towards the narrow stage,\n  Watching intently; watching, nerves and brain,\n  As those two men, cut off in that blue glare\n  From all reality of place and age\n  Wherein our common being has a share,\n  Together isolated, watch and creep\n  --Sunk head, hunched shoulders, light of foot and swift,\n  Deadly of purpose--in that ancient game,\n  Which was not otherwise in forests deep\n  Of earth primeval: that light tread the same,\n  The same those watchful eyes, and those quick springs\n  Of a snake uncoiling; underneath the skin,\n  Glistening with sweat in that unearthly blaze,\n  The muscles run and check, like living things.\n  And then, the hot air tremulous with the din,\n  And all the great crowd surging to its feet,\n  Yet like a wave arrested, while the hands\n  Of the referee allot the moments' beat;\n  The seconds, strung like greyhounds on a leash\n  Await the signal; and there's one who stands\n  Still guarding, watchful, tense, while all around\n  Lamp-light and darkness seem to rock and spin\n  In one wild clamour; and upon the ground,\n  Beneath the stark blue light, the beaten man!\n\n\n\n\n  THE GERMAN BAND\n\n  When I was a little child\n  And lived very near the sky,\n  A German band was wonderful music\n  That could almost make me cry.\n\n  It was to me of a beauty\n  That I could not understand,\n  Though I dimly guessed at sorrow and joy\n  In a grown-up distant land.\n\n  All that I know with the years,\n  Much that I never shall know,\n  Was in my heart when the music came\n  In such guise, years ago.\n\n  And now when on Friday mornings\n  I hear my own child run,\n  When the German band in the street starts playing,\n  The wonder is never done;\n\n  The wonder at ways that our spirit\n  May take for itself to rise,\n  How a puddle may be a silver lake,\n  And a chimney touch the skies.\n\n  All the forms through which spirit\n  Yearns and strives to be known\n  Are only a little greater or less,\n  For great is the Spirit alone.\n\n\n\n\n  STREET MUSIC\n\n  I\n\n  There comes an old man to our street,\n  Dragging his knobby, lame old feet,\n  Once a week he comes and stands,\n  A concertina in his hands,\n  There in the gutter stops and plays,\n  No matter fine or rainy days\n  --Very humble and very old--\n  Pavement's for them who make so bold!\n  Prim, starched nurses, and ladies fair\n  With taffeta dresses and shining hair,\n  And gay little children, who break and run\n  To give him a penny--he seems to feel\n  (Out-at-elbows and out-at-heel)\n  That they've a right to the morning sun;\n  And so with gnarled old hands he'll play\n  For an hour, perhaps, then take his way,\n  Dragging his knobby, lame old feet\n  In the gutter of this quiet street.\n\n  There is no grudging in his eyes,\n  Nor anger, nor the least surprise\n  At the uneven scales of fate:\n  Glad of the sun, against the rain\n  Hunching his shoulders, age and pain\n  He takes as his appointed state,\n  And stands, like Lazarus, at the door\n  With the dread humility of the poor.\n\n\n\n\n  STREET MUSIC\n\n  II\n\n  I've heard a mad old fiddler play\n  Harsh, discordant, broken strains,\n  Down the wet street on a winter's day\n  When the rain was speckling the window-panes,\n\n  And though it was middle afternoon\n  And none of the lamps were lighted yet,\n  The night had settled down too soon\n  And the sky was low and dark and wet.\n\n  In a cracked old voice I've heard him sing,\n  Strangely capering to and fro,\n  Sawing his fiddle on one worn string,\n  A grotesque and desolate thing of woe,\n\n  Wagging his head and stamping his feet\n  (Unwitting of the passers-by\n  Hurrying through the gloomy street)\n  His shoulders hunched and his head awry.\n\n  The children would laugh when they saw him pass,\n  And \"Look,\" they'd say, \"at Crazy Joe!\"\n  And press their faces against the glass\n  To watch him--leering and lurching--go.\n\n  Where he comes from, nobody knows,\n  But he, being mad, is in God's hand,\n  And sacred upon his way he goes;\n  And his music--God will understand.\n\n\n\n\n  PICCADILLY\n\n  Above, the quiet stars and the night wind;\n  Below, the lamp-lit streets, and up and down\n  The tired, stealthy steps of those who walk\n  When the just sleep, at night, in London town.\n\n  Poor garish ghosts that haunt the yellow glare,\n  Wan spectres, lurking in the alleys dark\n  Among the tainted night-smells, while the wind\n  Is whispering to the trees across the Park;\n\n  For it is summer, may be, and the scent\n  Of new-mown hay is sweet across the fields,\n  But neither summer, nor the gleaming spring\n  One breath of healing to this dark life yields;\n\n  No morning sunshine greets these sidelong eyes\n  With blessings, daughters as they are of gloom,\n  Ghosts only, such as seem to have a shape\n  At night in some old evil, haunted room.\n\n  Would that they were indeed to be dissolved\n  At every sunrise!--they are living souls\n  Dragging mortality about foul streets\n  While overhead the star-lit heaven rolls.\n\n  Living souls are they, and they have their share\n  In seed and harvest, and the round world's boon\n  Of changing seasons, and the miracle\n  Of each month's waxing and waning of the moon.\n\n  Living souls are they, prisoned in a net\n  Of stealthy streets--age after age they've gone\n  Bearing the burden of a city's sin,\n  In London, and old Rome, and Babylon.\n\n\n\n\n  IN THE TUBE\n\n  A tired, working woman, draggle-tailed,\n  Came in, harsh-featured in the yellow glare\n  Of electricity; an urchin trailed\n  Clumsily after her, with towsled hair,\n  And sharp, pale features, and a vacant stare,\n  And in her arms she bore another child.\n\n  A sick child, doubtless, where all three looked sick;\n  The poor legs hanging limply, lean and blue,\n  Dangled grotesquely, for the boots, too thick\n  For such frail bones a touch could snap in two,\n  Like clock-weights seemed to swing, as staggered through\n  The burdened mother, till she found a seat.\n\n  Through dark unnatural to unnatural blaze\n  Of stations rocked the train; it tore the air\n  To shreds and tatters in the tunnelled ways\n  With such a noise as when hell's trumpets blare;\n  We, swaying, faced our fellow-creatures there\n  Each mercilessly pilloried in light.\n\n  The sick child lay against the woman's breast\n  Asleep, and she looked down on it and smiled,\n  And with her gaunt arms made her bird a nest\n  Against her poor worn bosom--sad and mild\n  In such wise looked Madonna at her Child\n  Where old saints worshipped, round the altar set.\n\n  Such glory of the spirit shone and streamed\n  In that brief moment, that her form and face\n  Were rags of vesture only, through which gleamed\n  The splendour; something of wonder and of grace\n  Making the poor flesh lovely--all the place\n  Grew holy with the Mother and the Child.\n\n\n\n\n  A LONDON IDYLL\n\n  I\n\n  A heavy sky, and a drizzling rain\n  And the lamps in rigid rows;\n  Long smears of light all down the street\n  Where a lean cat stalking goes;\n\n  Blank, save a glimmer here and there\n  The gaunt dark houses stand--\n  And a man and a girl against the gate\n  Whispering, hand in hand.\n\n  There is a little dripping sound\n  Of rain from off the roof;\n  And gleaming like black armour goes\n  The policeman's waterproof.\n\n  He crosses the road to give them room\n  As he takes his evening beat;\n  He also knows that heaven may look\n  Like a rainy London street.\n\n\n\n\n  A LONDON IDYLL\n\n  II\n\n  Just to all of us once there comes\n    This splendour and wonder of love,\n  When the earth is transmuted to silver and gold,\n    And heaven opens above;\n\n  When all we have ever seen with our eyes,\n    Daily, under the sun,\n  Seems like a miracle, happening again\n    To us two, instead of to one.\n\n  When there is nothing so ugly or mean,\n    But somehow shimmers and glows\n  In that light, whose spring is within our hearts\n    And whose stream o'er the wide earth flows.\n\n  When the spirit of us that is prisoned within\n    Seems at last to have wings,\n  And, soaring, looks with no common eyes\n    On no other than common things;\n\n  When we may freely enter and share\n    Heaven's splendour and mirth--\n  Just for a moment to all of us comes\n    This glory of love upon earth.\n\n\n\n\n  FINIS\n\n  S.C.K.S.\n\n  A book's end is the end of many hopes;\n  Much good endeavour; certain hours of stress\n  When brain and spirit fail, and laziness\n  Thralls the poor body--yet the purpose gropes\n  Athwart it all, and as the horseman cheers\n  His tired beast with chirrup, spur, and goad\n  Towards his home along the heavy road,\n  So drives us purpose till the end appears.\n  Read it who may!  Find more or less of good\n  Within its covers, but at least find this:\n  Glad service to a great and noble aim\n  That may be striven for, and understood,\n  And fallen short of--so not quite we miss\n  In our small lamp of clay Truth's very flame.\n\n\n\n\n  OTHER VERSES\n\n\n\n  IN EARLY SPRING\n\n  There's a secret, have you guessed   it, you with human eyes\n      and hearing--\n  Which the birds know, which the trees know, and by which\n      the earth is stirred,\n  Stirred through all her deep foundations, where the water-springs\n      are fastened,\n  Where the seed is, and the growth is, and the still blind life is heard?\n\n  There's a miracle, a miracle--oh mortal, have you seen it?\n  When the springs rise, and the saps rise, and the gallant cut-and-thrust\n  Of the spear-head bright battalions of the little green things growing\n  (Crocus-blade or grass-blade) pierce the brown earth's sullen crust?\n\n  Oh, wonder beyond speaking in the daily common happening;\n  But the little birds have known it, and the evening-singing thrush,\n  In the cold and pearly twilights that are February's token\n  Speaks of revelation through the falling day-time's hush.\n\n\n\n\n  A BALLAD OF THE FALL OF KNOSSOS\n\n  (_Circa_ 1400 B.C.)\n\n  Is it a whisper that runs through the galleries?\n  Is it a rustle that stirs in the halls?\n  Is it of mortals, or things that are otherwise\n  This sound that so haltingly, dreadfully falls,\n  Pauses, and hurries, and falls?\n\n  No moon, and no torches; not even a glimmer\n  To pin-prick the darkness that weighs like a sin,\n  And nothing is breathing, and nothing is stirring,\n  And hushed are the small owls without, and within\n  The mice to their holes have run in.\n\n  It is not the step of a foot on the pavement;\n  It is not the brush of a wing through the air;\n  It is not a passing, it is not a presence,\n  But the ghost of the fate that this palace must bear,\n  Of the ruin of Knossos goes there.\n\n       *     *     *     *     *\n\n  For on such a night, when the moon is dark,\n  And all of the stars are dumb,\n  With a sudden flare by the sea-ward gate\n  Shall the doom of Knossos come;\n  For a cry will shatter the brooding hush,\n  And the crickets and mice shall wake\n  To clatter and clash and shout and cry,\n  And the stumble of frenzied feet going by\n  Death's stride will overtake.\n\n  For into the glare of a new-lit torch\n  That shakes in a shaking grasp,\n  Sweat-streaked, wild-eyed, and dark with blood\n  Shall a runner break, and gasp\n  Of a burning harbour, of silent ships,\n  Of men sprung out of the night--\n  Is it men or devils?--He moans, and reels\n  Shoulder to wall, and a red stain steals\n  Down the frescoes gay and bright.\n\n  And hard on the word they hear approach\n  The surge of the battle near,\n  And to whistle of arrows, and clang of bronze\n  The palace awakes in fear.\n  Light!  Light! and torches, like waking eyes\n  Leap from each darkened door;\n  And the guard at the sea-ward gate go down\n  In the vast black sea of men, and drown,\n  While sweeps the torrent o'er.\n\n  What door shall hold, or what walls withstand\n  The roll of a full spring-tide,\n  With an on-shore wind?  And the gates of bronze\n  Ring, rock, and are flung aside;\n  And a myriad unknown raiders burst\n  Into the hall of the King,\n  Where Minos on his carved, stone seat\n  Beheld the nations at his feet,\n  Watched each its tribute bring.\n\n  Minos is slain; his guards are slain;\n  Which of his sons shall live\n  In this pillared Hall of the Double Axe\n  The word of the Kings to give?\n  Which of his sons?  Shall they know his sons\n  In this sudden terror sprung\n  On sleeping men?  Half-armed they stand,\n  Foot pressed to foot, hand tense to hand,\n  And muscles iron-strung.\n\n  The flame of the torches in the wind\n  Of their struggle blackens the wall,\n  And the floor is sticky with blood, and heaped\n  With the bodies of those that fall.\n  What if a son of Minos live?\n  In that horror of blood and gloom,\n  What of the noble, what of the brave?\n  Better to die, than endure as a slave\n  The days after Knossos' doom.\n\n  But above the scuffle of sandalled feet,\n  And the breath of men hard-pressed,\n  And the clash of bronze, and the gasp and thud\n  As the point goes through the breast,\n  And above the startled hoot of owls,\n  And the rattle of shield and spear,\n  The wailing voices of women rise\n  As their men are stricken before their eyes\n  And they huddle together in fear.\n\n  Slow comes the dawning in the East;\n  Pale light on the earth is shed,\n  And cool and dewy blows the wind\n  Over the writhen dead;\n  Pale light, which fades in the growing glare\n  Of the flames that swirl and leap\n  Through corridor, and bower, and hall,\n  On carven pillar and painted wall;\n  The flames that like sickles reap\n\n  A barren harvest of kingly things,\n  To be bound in ashy sheaves,\n  While driven forth by the work of his hands,\n  Stumbles the last of the thieves.\n  Behind him is fire, ruin, and death,\n  Before him the kine-sweet morn,\n  But vases of silver and cups of gold\n  And hoarded treasures fashioned of old\n  On his blood-stained back are borne.\n\n       *     *     *     *     *\n\n  Is it the night-wind alone that blows shuddering\n  Down the dim corridors, tangled with weeds;\n  Is it a bat's wing, or is it an owl's wing\n  That silently passes, as thistledown seeds,\n  In the Hall, where the small owlet breeds?\n\n  Here do the moonbeams come, slithering, wandering\n  Over the faded, pale frescoes that stand\n  Faint and remote on the walls that are mouldering,\n  Crowned with a King's crown, or flowers in hand,\n  --Pale ghosts of a gaily-dressed band.\n\n  Faintly they gaze on the wide desolation;\n  Faintly they smile when the white moonbeams play\n  Over the dust of the throne-room of Minos,\n  Over the pavements where small creatures stray,\n  The humble small things of a day.\n\n  But there are other nights, moonless and starless,\n  When no moth flutters, no bat flits, owl calls,\n  Something is stirring, something is rustling,\n  Something that is not of mortals befalls\n  In galleries, cellars, and halls.\n\n  Soundless and viewless, a strange ghostly happening,\n  Life, long since ashes, and flames, long since dead;\n  For the Angel of Time goes relentlessly, steadily\n  Over dark places that mankind has fled;\n  And the dust is not stirred by that tread.\n\n\n\n\n  A SUN-DIAL IN A GARDEN\n\n  Across the quiet garden sunlight flows\n  In wave on wave like water, heavy bees\n  Hang drowsily upon the drowsy flowers,\n  For it is very still, and all the trees\n  Are pyramided high in green and gold.\n  There is a sun-dial there to mark the hours\n  Where time is not, where time has grown so old\n  It does not move now; yet the shadow goes\n  Across the dial that's so warm to feel\n  Like a cold, stealthy, creeping, living thing.\n  You cannot see it steal\n  Minute from minute of the golden day\n  Till all are eaten away,\n  You cannot press it back with both your hands,\n  And, on the shadowed stone\n  Laying your cheek, you never warmth can bring\n  To what beneath the sad triangle stands,\n  Solitary in sunlight: for we know,\n  It takes the whole great swinging earth to throw\n  The little shadow on the little stone.\n\n\n\n\n  \"TWO ONLY\"\n\n  Only two hearts shall understand the sea\n  That speaks at nightfall, in the wash and lap\n  Of windless evenings under flaming skies;\n  Only two hearts shall hear the rising sap\n  In wet spring woods; and two alone, grown wise\n  In union, shall make discovery\n  Of what lies hidden, though before our eyes.\n\n  Oh, core of wonder in familiar things:\n  Magic of evening, and of early morn\n  But just created, with the dew of birth\n  All fresh upon it, heaven itself new-born\n  O'er the green splendour of the quiet earth\n  And like a just-awakened bird that sings\n  Because of sunlight, is the spirit's mirth.\n\n  All forms of beauty but express the soul\n  As in a looking-glass; the wind that goes\n  Low-talking to the trees beneath the stars,\n  Or the small sound of water, as it flows\n  Under old bridges, where the ivy mars\n  The sharp stone outline--these are in the whole\n  Of the World-Symphony small, tuneful bars.\n\n  And human beings in the span of years\n  Some part of all the world-wealth may receive,\n  More, less, but never all; and with dismay\n  We see slow Time his net of hours weave\n  To catch from us dear mortal night and day,\n  Ere we have taken in our eyes and ears\n  Beauty that lies around, beyond, away.\n\n  We, singly, feel a sudden sharp regret\n  Behind all beauty, but we--two in one,\n  As white and blue are separate in a flame\n  Yet mingled--we shall watch the hours run\n  Seeing with surer knowledge how the same\n  Eternal splendour for the soul is set,\n  And the day comes again from whence day came.\n\n\n\n\n  THE SAINT'S BIRTHDAY\n\n  One of God's blessed pitying saints one day,\n  Reaching out hands to touch the azure throne:\n  \"Because it is my birthday, Lord,\" he said,\n  \"That I was born in heaven, when I was known\n  By an earthly name, and stoned and left for dead,\n\n  \"Because it is the custom, Lord, of men\n  To keep their birthdays gladly, and with gifts\n  Grant me a blessing from your blessed stores.\"\n  And from the cloudy rose and amber drifts\n  About the Throne, God answered: \"It is yours.\"\n\n  Then sprang the glad Saint earthwards; at his feet\n  Were little golden flames, and all his hair\n  Was blown about his head like tongues of fire,\n  And like a star he burned through the dark air,\n  And came, and stood by farm and shed and byre\n\n  Before the earliest grey was in the East,\n  Or the first smoke above the chimney-stack\n  From earliest-rising housewife, yet the cheep\n  And twitter of birds did gladly welcome back\n  Him who such love for earth in heaven could keep,\n\n  And who on earth such love had had for men\n  And bird and beast, and all that lived and grew:\n  The sparrows in the eaves remembered him\n  And chirrupped in the gables, while the dew\n  Was dark still, and the day below the rim.\n\n  He stood there, in the village of his life\n  Ere he won heaven, and the breath of cows\n  Came as a benediction, and the smell\n  Of rain-sweet copses, and, where cattle browse,\n  Long grass, and running water in the dell.\n\n  And his heart opened with the love he had\n  For the dear toil-worn dwelling-place of men;\n  To hear the sheep crop, see the glimmering grey\n  Lighten the waiting windows once again,\n  And garden roses opening to the day.\n\n  Not otherwise was Eden once--he thought--\n  And by God's blessing it may be anew:\n  And so put forth the power God had lent\n  And took away all labour, and he drew\n  Heaven to earth, till earth and heaven were blent.\n\n  Time ceased to be; and yet the sun and shade\n  Shifted to make new beauty with the hours,\n  And the ripe earth, unlaboured, gave her yields,\n  No pain there was, no age, and all the flowers\n  Unwitheringly lovely filled the fields.\n\n  And all day long the birds in ecstasy\n  Sang without shadow of hawk or thought of death,\n  And the saint happily went about the ways\n  Filling each home with plenty--his very breath\n  Was like a little thrilling note of praise.\n\n  When all was done he stepped back, childish-wise,\n  To see and love his handiwork, and then\n  Came a sharp pain, and pierced him through and through;\n  He had wrought lovingly for the days of men,\n  But the heart of men his love could not renew:\n\n  The weary heart, the ever-questioning,\n  The loving, lacking, lonely, incomplete\n  For ever longing to be merged in one\n  With something other than itself; to beat\n  To another's pulse; to be for ever done\n\n  With its sad weight of personality.\n  Then God leaned down to his poor saint, and said:\n  \"Dear soul, would you make heaven upon the earth:\n  Nor know indeed My purpose in all birth,\n  Nor that My blessing is upon the dead?\"\n\n\n\n\n  RUPERT BROOKE\n\n  _April_ 1915\n\n  You that are gone into the dark\n  Of unknowing and unbeing;\n  You that have heard the song of the lark,\n  You that have seen the joy of the spring;\n  You have I seen, you have I known\n  --The word you have written, your pictured head--\n  And they say you are laid at Lemnos among the English dead.\n\n  Soul that is gone--is gone--\n  Whether into the dark,\n  Or into knowledge complete and the blinding light;\n  Soul that was swift and free,\n  Passionate, eager, bright,\n  Armed with a weapon for shams,\n  And set with wings for flight;\n  Soul that was questioning, restless, and all at odds with life,\n  Greedy for it, yet satiate, and sick with the shows of things\n  --And all laid down at Lemnos, the hunger, the love, the strife,\n  And the youthful grace of body, and the body's ministerings.\n\n  Darkness, darkness, or light!\n  You have leapt from the circle of sense,\n  And only your dust remains and the word you said:\n  \"If I should die,\" ... and we name you among the dead.\n  Yet have I a hope at heart\n  That somewhere away, apart,\n  Knowledge is yours and joy of the act fulfilled\n  To still your fever of soul as your fever of blood is stilled;\n  So shall you soar and run\n  In water and wind and air,\n  With your old clean joy of the sun,\n  And your gladness in all things fair,\n  Untouched by mortality's sadness, simple, perfect, at one.\n\n\n\n\n  \"COMFORT ME WITH APPLES, FOR I AM SICK OF LOVE\"\n\n  Red lilies under the sun,\n  Red apples hanging above,\n  And red is the wine that is spilled\n  On your bare white feet, O Love.\n\n  The poppies sullenly glow\n  In the smouldering red from the West,\n  And black are the dregs of the wine,\n  O Love, on your bare, white breast.\n\n  Aie! aie! when the wild swan flies\n  Lonely and dark is the place\n  That the white wings lightened, and death\n  Will cover your glowing face.\n\n  O thief that is night, O thieves!\n  Cold years that devour us all;\n  The lilies blossom and wilt,\n  The apples ripen and fall,\n\n  The apples, the apples of Love!\n  --Lo, where we have spilled the wine,\n  This quenchless earth is agape,\n  O Love, for your body and mine.\n\n\n\n\n  OF ENGLAND\n\n  White is for purity, blue for heaven's grace,\n  Purple is for Emperors, sitting in their place,\n  Yellow is for happiness, rose for Love's embrace,\n  But green--oh green, the green of England--that's for Paradise!\n\n  From seashore to seashore races the green tide;\n  With the pricking green of hedges by the wet roadside\n  --Or ever March triumphant comes with great, glad stride--\n  There is green, there's green in England, and a tale of Paradise.\n\n  Then the hawthorns flush and tremble in their early wondrous green,\n  And the willows are resplendent in a green-and-golden sheen,\n  Like the golden tents of princes, Babylonish, Damascene,\n  Or enchanted silent fountains of a Persian Paradise.\n\n  There are beech and birch and elm-tree--evening-still or\n        morning-tossed--\n  And the splendid generous chestnuts with their flame-like\n        blooms embossed,\n  There are oak and ash and elder, till the very sun is lost\n  In the green, delicious gloaming that's the light of Paradise.\n\n  Deeper, wider, steadier this beauty ever grows,\n  And from field-side up to tree-top the endless colour flows,\n  Till road and house and wayside, in the first days of the rose,\n  Are fathoms deep in waves of green, submerged in Paradise.\n\n  Oh dim and lovely hollows of all the woods that be;\n  Oh sunlight on the uplands, like a calm, great sea;\n  I think indeed the souls of those from circumstance set free\n  Look down, look down on England, saying: \"Ah, dear Paradise!\"\n\n\n\n\n  QUESTION\n\n  What of this gift of Life?\n  Passionate, swift, and rife\n  With pleasure or pain in the hand of the hurrying hours?\n  Oh little moment of space,\n  Oh Death's averted face,\n  How shall we grasp, shall we grasp what still is ours?\n\n  Chill, chill on either hand\n  Eternities must stand,\n  And pants between them, passionate and brief,\n  The moment's self, to make\n  Or unmake, but to take\n  Just here, just now, before death turns the leaf.\n\n  Ah, if the leaf but turn,\n  And if the soul discern\n  Another message on another page!\n  But if death shuts the book?\n  We may not know nor look;\n  We are fenced in upon a narrow stage;\n\n  While, splendid and intense,\n  Quick-strung in every sense\n  Life burns in us, and earth lies all around--\n  Far blue of summer seas,\n  Young green of age-old trees--\n  Bound by the season, by the horizon bound.\n\n  Oh colour, sound, and light,\n  Oh wondrous day and night,\n  Pale dawns, and evenings' splendid stretch of gold;\n  Keen beauty like a spear,\n  Half pleasure and half fear,\n  Goes through us for the things we may not hold.\n\n  Hot blood, hot noons, hot youth--\n  When Life seems all the truth,\n  And Death a mumbled far old fairy-tale;\n  When just the splendid days\n  Suffice our eager gaze,\n  The wondrous present that will never fail.\n\n  Then one day, with a fierce\n  Clamour of heart, we pierce\n  The light and see the shadows all behind,\n  And then, and not till then,\n  By the brief graves of men\n  The utter loveliness of flowers we find.\n\n  So little stretch of days,\n  And earth, with all her ways\n  Lovely enough, I think, for Paradise;\n  And body, mind, and heart,\n  Each separate complex part,\n  Wondrously made, and never quite made twice.\n\n  What of this gift of Life?\n  Shall it be worn in strife?\n  Shall it be idly spent, or idly stored?\n  Each for himself must dare\n  If the answer is here--or there,\n  Here for regret--or there for hope, O Lord?\n\n\n\n\n  LEONARDO TO MONNA LISA\n\n  I wish you were a beaker of Venetian glass\n  That I might fill you with most precious wine\n  And drink it, breathless--lo! the moments pass\n  Of that subliminal communion.\n  I take you from my lips, and crush you--so!--\n  Into a thousand shining particles;\n  So, at the last, my passionate greed shall know\n  That you were wholly mine.\n\n  I wish you were a rare, stringed instrument\n  Beneath my hand, and from you I would wring\n  Such unimagined music, as was sent\n  Never before, along the quivering nerves;\n  Such strange, sharp discords, out of which I'd mould\n  Music more sweet than the spring nightingale's;\n  Then, ere the magic of the sound was old,\n  Would I not rend each string?\n\n  Possess you?  Ah, not with the world's possession,\n  You still, strange creature; neither force nor will\n  Could make you serve a man's mere earthly passion.\n  I would dissolve you, in one blinding flash,\n  Into a drop of elemental dew,\n  And let you trickle down the barren rock\n  Into the black abyss, if so I knew\n  That you henceforth were powerless to mock\n  My spirit with your smile.\n\n\n\n\n  THE ETERNAL FLUX\n\n  Let us hold April back\n  One splendid hour\n  To bless the passionate earth\n  With golden shower\n  Of sunlight from the blue;\n  Oh April skies,\n  That earth yearns up to; blue has burned to gold,\n  Gold pales and dies\n  In delicate faint rose,\n  Oh flowing time, oh flux eternal.  Hold\n  The hour back.  The April hour goes.\n\n  Then, let it be of May,\n  When sound and sight\n  And all that's beauty manifest\n  Through all the day,\n  Of deep on deep with green,\n  Of light on light\n  Across the waves of blossom, when the white\n  Is lovelier than the rose, except the rose\n  Is loveliest of all;\n  When through the day the cuckoo calls unseen,\n  And at nightfall\n  The nightingale, whose music no man knows\n  The magic heart of, sitting in the dark\n  Sings still the world-old way;\n  When all of these,\n  Flowers and birds, and sunset and pale skies\n  Seem gathered up in scent,\n  And all of sound and sight\n  Dissolved, ethereal, not of ears and eyes\n  But only the soul-beauty of the brain\n  Flows, in such waves of perfume, over all\n  --Or like a song in colour, of such strain\n  As spirits finer than our own must hear\n  (The beautiful made clear);\n  Then, then, when it is May,\n  Surely our hand must touch eternity.\n  Day pales to night, stars pale upon the day,\n  And May's last blossoming hour flows away.\n\n  Not of June either, though the hanging skies\n  Make but a little span\n  'Twixt light and growing light;\n  And when through that short darkness palely flies\n  The silent great white moth\n  --A spirit lost in the night,\n  A soul, without will or way--;\n  When the arch of trees\n  Is duskily green, and close as a builded house\n  Where love with love might stay,\n  Guarded and still, from sight;\n  When the hay is sweet in the fields\n  And love is as sweet as hay;\n  When the life-impulse of the wonderful untamed earth\n  Has reached its fulness and height,\n  Is broad and steady and wide\n  As sweeps into splendid bays the flowing tide;\n  When God might look on the land,\n  When God might look on the sea,\n  And say: \"For ever be\n  Perfect, completed, achieved,\n  As now at this moment you stand.\"\n  Neither in June shall we stay the eternal flow\n  Nor grasp the present with pitiful, mortal hand,\n  For sliding past like water the June hours go.\n\n\n\n\n  \"LOVE IS THE ULTIMATE MEASURE OF THE SOUL\"\n\n  Love is the ultimate measure of the soul;\n  Love is the biting acid, the sure test\n  To strip the naked gold, discard the rest\n  Of earthly stuffs; Love is the one thing whole\n  In a world of broken parts, for Love is all.\n\n  Love is creation; Love is the low call\n  Of deep to deep; Love is the force that shapes\n  The thing that it believes, and while there gapes\n  The black earth-pit, where the poor flesh must fall,\n  Love builds on hope, and buds eternal life.\n\n  Love is a victory unsoiled by strife;\n  Who is there that shall adequately name\n  All that Love is, this thing as swift as flame\n  And vast as heaven, yet in every life\n  Tamed to the narrow needs of little men?\n\n  From humble love, that makes the partridge hen\n  Brave for her chickens, to the Love that shakes\n  The world from Calvary, all love partakes\n  Of immortality; one cannot pen\n  Divinity in words; Love is divine.\n\n  The very essence of God does Love enshrine;\n  For let the heart, however sorely tried,\n  Open itself to loving, and the wide\n  Earth is a home; love-lacking must decline\n  Where black fears crowd across the starless dark.\n\n  For Love is light; the faith that will embark,\n  Unpiloted, upon uncharted seas\n  Is Love alone; the fiery leap to seize\n  The splendid distant aim, the invisible mark,\n  What else but Love's?  Love is the thing that stands\n  Unchanged, on changing tides and shifting sands.\n\n\n\n\n  NOVEMBER 8\n\n  THE LITTLE SUMMER OF ALL SAINTS\n\n  The year stands still, the tearing winter winds\n  Hold off their claws a moment, that the trees\n  May keep the glory of their blended gold\n  A little minute; there's not so much breeze\n  As summer mornings hold.\n\n  Golden and still the hours; russet gold\n  The birch-leaves o'er the silver of the bark;\n  Pale gold the poplars, like a lady's hair,\n  And thunderous gold along the hollows dark\n  The sunlit brackens flare.\n\n\n\n\n  THE LOVERS\n\n  There are ghosts we walk with, lady of mine,\n  Arm in arm, and side by side,\n  Pallid ghosts, though the sun may shine,\n  Ghosts that are cold in the warmth of day,\n  And neither of us may fend them away,\n  But step by step they go with us, stride by stride.\n\n  There are doors in your heart that are shut to me,\n  And behind them dwellers I cannot know;\n  And my soul has windows that open wide\n  On a ghostly, memoried country-side,\n  That--lady of mine--you never will see,\n  Where your voice will never be heard, nor your footsteps go.\n\n  So we walk together, hand in hand,\n  While dark eyes peer at us, pale forms come,\n  And speak in my ear--or call your name\n  With a voice I hear not, for praise or blame,\n  And you walk alone with that ghostly band,\n  While I go by the side of you, pitying, powerless, dumb.\n\n\n\n\n  THE GENTLE HEART\n\n  What shall harm the gentle heart\n  In its purpose undefiled?\n  Even grief shall lose its smart\n  In some way becoming part\n  Of that nature, soothed and gentled,\n  As a sorrow to a child.\n\n  Through the blackness and the sin\n  Of the old world's wrongs and woes,\n  And through the greater dark within,\n  The gentle heart shall surely win,\n  As some bright angel, armed with mercy,\n  Swiftly on his errand goes.\n\n  All the body may have wrought,\n  All the energies of mind\n  That for its own purpose sought,\n  Make at length a little nought\n  Among the stars--the gentle heart\n  Death itself will leave behind.\n\n\n\n\n  A BALLAD FOR HERMAN\n\n  This is the ballad for Herman, the ballad of humble things,\n  The hedge-side thistles that flower, the small brown lark that sings,\n  And the stumbling flight of a beetle, and the dust\n        on a butterfly's wings.\n  The snails are out in the sunshine after the morning rain,\n  And the wasps are whirring and buzzing round the mulberry tree again,\n  And the ants are busy of course, working with might and main.\n\n  While the crickets leap, and rustle, and play at being blades of grass,\n  And humble-bumble the bees go, lurching as they pass,\n  And the flies are stupidly walking up the window-glass.\n\n  The sun is bright on the hedges, on thistle and bramble and briar,\n  The columbine leaves are heart-shaped, and shine as bright as fire\n  --And oh! the smell of the bracken, that's straight as Salisbury spire!\n\n  Life of the woods, life of the rivers, life of the trees,\n  Life of the rich plain-grasses that seed to the morning breeze,\n  And the thymy mountain-grasses June makes loud with bees.\n\n  This does not age nor alter; the low sharp song of the reeds\n  As the evening wind goes over, and the fishing heron feeds\n  On the still and shallow waters, salt with the floating weeds.\n\n  This does not change nor vanish; the mating calls of the springs,\n  When April's green on the copses, and bright on the shining wings\n  Of birds going backwards and forwards, while the whole green\n        forest sings.\n\n  All is our sister and brother, as once St. Francis said;\n  The little stones in the river, the bright sun overhead,\n  And newts, and the spawn of fishes, and the unnamed mighty dead.\n\n  This is the ballad for Herman.  O friend, may good befall!\n  There is never a star so distant, there is never a creature small,\n  But living and knowing and loving in our brain we hold them all.\n\n\n\n\n  FRANCE\n\n  _April_ 1915\n\n  Great ever, with the hope that seeks the stars;\n  The brain clear-cold, like ice; the soul like flame;\n  The spirit beating at the physical bars;\n  The reason guiding all--oh, there we name\n  France!\n\n  A country that can think, and thinking, acts;\n  A country that can act, and acting, dreams;\n  That neither bears the tyranny of facts,\n  Nor of its own dear hopes, nor of what seems,\n\n  But still, clear-visioned, treats with things that are;\n  Yet--seer, prophet, priest of life-to-be--\n  Leaps to the visionary days afar,\n  And all the splendour she will never see.\n\n  School of the spirit, chastening, yet a spur\n  For all that men aspire to: as of old\n  Athens held up the torch, and did incur\n  Persia, with her fierce armies manifold,\n\n  So France against the evil strikes and strives\n  For liberty, and we of island race,\n  --Humbled a little by our careless lives--\n  Glory to stand beside her in our place,\n\n  Glory that we are one in hope and aim\n  With her from whom in blood and agony\n  The second gift of human freedom came\n  Through Terror and the red Gethsemane.\n\n  On her fair, ravaged borders stand her guns,\n  She has thrown away the scabbards, bared the swords,\n  And, snatching laughter out of death, her sons\n  Challenge high Fate to show what life affords--\n  France!\n\n\n\n\n  ILGAR'S SONG\n\n  (From _King Monmouth_)\n\n  O love that dwells in the innermost heart of man\n  Secret and dark and still,\n  Like a bird in the core of a green mid-summer tree--\n  Height upon height and depth upon depth where never the eye can see\n  The brown bird, hidden and still.\n\n  O Love that is wild and eager, sun-lit and free\n  Like a seagull that turns in the sunlight above the sea;\n  Between the sea and the sky it flashes and turns,\n  And the sun on its wings is white,\n  While sharply and shrill by the headland the keen wind sings\n  Where the grass is salt and grey\n  With the beating winter spray,\n  And the seagull sweeps and soars on magnificent wings.\n\n  Love that is like a flame,\n  Held in the hollow hand,\n  So dear and precious a thing\n  As a light in a stranger land,\n  As a flickering candle to him who wanders by night.\n\n  Love that is wide as the dawn\n  To the eyes of night-bound men;\n  And the evil ghosts and the goblins it puts to flight,\n  And stealthy creatures of dark that rustle and creep,\n  And elfins and witches and all such devil's game\n  That cannot live in the light,\n  They squeak and gibber and cheep,\n  And vanish like shadows before the splendour of day.\n\n  Love that has wide, white wings like a flying swan\n  --Oh what a noble span,\n  From tip to tip they are more than the height of a man\n  And curved like the sails of a boat--\n  When over the evening river the wild swan flies\n  The curve of those wings is like the arch of the skies\n  Over the shielded earth.\n  Love is most like a bird,\n  For birds have least of the dust that gave them birth,\n  They soar and poise and float,\n  They wheel and swerve and skim,\n  And their wings are strong to the wind, and swift to the light,\n  And their voice is a promise of dawn while yet it is night,\n  And their song is a paean of hope before it is spring,\n  And the song of the bird to his mate is lyrical love.\n\n  Love is secret and holy, a spiritual thing,\n  Dark and silent and still\n  In the heart of man, as a treasure is hid in a shrine.\n  Love is splendid and fierce, as the summer sun\n  Drenches the sea and the sky with its blaze and shine,\n  Till every pebble is hot to the touch of the hand,\n  And the air is a-shimmer with heat o'er the hazy land--\n  Yet Love is not any of these things, Love is of one\n  With the strange, half-guessed at, vast, creative plan\n  We cannot see with our eyes nor understand--\n  Yet is Love pitiful too, for Love is of man.\n\n\n\n\n  THE INN\n\n  I\n\n  Friendship's an inn the roads of life afford\n  --I'll speak to you in metaphor, my friend--\n  And there a tired man his way may wend,\n  And, coming in, sit down beside the board,\n  Out of the dust and glare, and boldly send\n  For drink and victuals; haply cross his knees,\n  And in the cool dark parlour take his ease,\n  And gossip of his journey and its end.\n\n  That's friendship; there is neither right of place\n  Nor landlord duties, just the short hour's stay\n  From the sun and weariness between those kind\n  And quiet walls; and when the road's to face\n  Stony and long again, we take our way\n  Keeping that respite gratefully in mind.\n\n\n\n\n  THE INN\n\n  II\n\n  We take our pack, and jog our way again\n  Towards the windy sunset and the night;\n  The inn is now behind us, out of sight,\n  Showing no welcome shine of windowpane,\n  But dark and silent standing by the way\n  As we go forward, seeing mile on mile\n  Sink out of sight--just for a little while\n  We rested, in the middle of the day.\n\n  Is there an end at last, and shall we reach,\n  By the faint glimmer of new-risen stars,\n  Our house at last, and find the heart-repose\n  Which is the ultimate desire of each\n  Poor traveller--ah! shall they drop the bars,\n  And the doors open?  Dear my friend, who knows?\n\n\n\n\n  \"TO-DAY I MISS YOU\"\n\n  To-day I miss you ... \"Only for to-day,\n  Some little matter of hours and nothing more.\"\n  That at least the worldly-wise folk say,\n  Who've never waited for the opening door,\n  The greeting look, the known step on the floor;\n  Who've never missed a loved one like a lover.\n\n  To-day I miss you.  What to-morrow brings\n  Is the other side of all the stars, God knows!\n  Only to have you here, now evening swings\n  Its quiet shadow round the globe again,\n  And in our talk of old familiar things,\n  And in familiar gestures, turn of brain,\n  Looks, tone of voice, I may discern again\n  That union from which alone love grows.\n\n  We'd close the curtains;--while the world outside,\n  Noisily autumn, makes a sense of peace\n  Deeper within,--open the bookcase wide\n  And take a book out; then another book,\n  And then another....  \"Here's a favourite, look!\n  We cannot pass him.\" ... Then from reading cease,\n  Gossip and laugh, with finger in the page,\n  And challenge thought with thought, and mind with mind\n  Each speaking freely, that we might increase\n  Some knowledge to which, singly, we were blind.\n\n  So goes the evening.  Side by side we stand,\n  Dear friends and brothers, till, a sudden pause,\n  Or kindly, almost careless touch of hands,\n  Swings us to face each other, and we feel\n  Those deepest stirrings of the human heart\n  Man has no name for yet, those changeless laws\n  Of more than mating--that eternal part\n  Our body is aware of, and our brain,\n  Unchallenging with reason, must receive,\n  That sense of intimate wonder!--Now again,\n  The blinds are drawn; lamp, books, chairs, all retain\n  Familiar aspects, but, you absent, leave\n  The room all empty, empty all the day.\n\n\n\n\n  \"HOW SMALL THE THREAD THAT HOLDS UP HAPPINESS\"\n\n  How small the thread that holds up happiness;\n  But one frail life between the dark and me,\n  Your life, dear love--and here I seem to see\n  You whimsically smile, that I confess\n  The whole round world, with its vast energy,\n  Its summers, and its sunshine, and its aims,\n  Its splendid hopes, the faith that unquenched, flames\n  --All sunk into the compass of you and me.\n  Yes, you are right, the single leaves that fall\n  Mar not the summer; do I think one leaf\n  Denudes a forest?--We are nought at all.\n  Yet the bereaved small bird within the tree\n  May break its heart above its nest for grief\n  --And perhaps this must happen, love, to me.\n\n\n\n\n  \"IN ALL THINGS GRACIOUS THERE IS A THOUGHT OF YOU\"\n\n  In all things gracious there is a thought of you:\n  In the soft fall of April rain, the blue\n  Of April skies in the morning, the full moon\n  Of windless August nights, perfect and still,\n  When the white moonlight lies across the hill\n  Of new-cut stubble, where a little mist,\n  Flickering, rises.  In the song of birds\n  My heart turns to you, emptied all of words\n  By loveliness, and in the poise and swing\n  Of flowering grasses, and in the lingering\n  Grave, spacious fall of evening on the earth,\n  When the wide, liquid spaces of the sky,\n  Above the dewy fields and darkening lanes,\n  And windless water lying quietly,\n  Yield up the daylight, until none remains.\n\n  I could endure--or so it seems to me--\n  Without your presence, a life of winter days,\n  Stark, grey Novembers stretching endlessly,\n  Where I, forgetting laughter and bright things,\n  Might set my face to duty; but the stir,\n  The loveliness, the poignancy of springs,\n  The growth, the rise, the universal press\n  Up to sensation--ah, I could not bear\n  To live an April through, but must take wings\n  Out of a world too fair for loneliness.\n\n\n\n\n  \"THERE'S DUTY, FRIEND, TO JOG WITH ARM IN ARM\"\n\n  There's duty, friend, to jog with arm in arm\n  Through these dark streets; there's kindliness indeed,\n  And there's the hope a little more to weed\n  Our own small patch of life which the tares harm;\n  There's patience for the folly of the earth;\n  There's pity for the poor who suffer wrong;\n  There's honour for the striving and the strong\n  --But ah, dear friend of mine, where is the mirth?\n  Where's the old jollity of everyday\n  That makes a holiday of common things\n  Because they all are shared by us aright,\n  The trivial daily work and happenings\n  Having a sort of fervour and delight,\n  And the sun rising, even, a different way?\n\n\n\n\n  \"EVENING\"\n\n  Beloved of my soul, the day is done;\n  The busy noises cease, the lights are low;\n  Gently the doors shut to behind each one\n  Seeking his sleep; the fading embers glow\n  On silent hearths; the silent ashes fall--\n  Ah, absent spirit, do you hear me call,\n  Me, sitting waiting by the fireside?\n\n  This is the hour of all the night and day,\n  --This is the hour when, work put aside,\n  And all the talking, whether grave or gay,\n  For pleasure or for profit, hushed and dumb,\n  We used to, in the days before you died,\n  Seek out each other's mind for rest, and say:\n  \"Now am I home, and all is well with me;\n  To-day is gone, to-morrow is to come;\n  Here let us be.\"\n\n  Surely, for all the barriers of sense,\n  And the stark grossness of this flesh I wear,\n  For all the vacant distance of the skies\n  Between me here alone, and you, gone hence,\n  There must be some quick knowledge; I must hear\n  That dear familiar voice again, must see\n  Some semblance of you with my bodily eyes,\n  Now, now, when in the solitude I yearn\n  Towards your heart, my home; now when I turn\n  Humbly and searchingly towards that goal\n  That lies beyond the purchase of the world--\n  You again, you, dear comrade of my soul.\n\n\n\n\n  FINIS\n\n  Life, in its unimaginable heights,\n  When we may seize and apprehend the true\n  Soul essence, of one nature with the stars:\n  Rare moments when our senses are a mist\n  That the truth shines through:--oh, most strange and rare,\n  Such ecstasies as unimprisoned souls\n  Experience in that thin empyrean\n  Beyond the gross world; this we two have known\n  We two together.  There are memories\n  Of such high happiness in a fence of pain\n  As martyrs in their fiery heart of death\n  Have blessed their God for; passion and holiness,\n  When all the body (sinew, bone, and brain)\n  Are like a harp, from which the spirit makes\n  Marvels of harmony; some sense too rare\n  To be called happiness, not to be named indeed\n  In human speech--this we have touched and known\n  Together, at some thrilling edge of time.\n\n  I fall away from it; the barriers close\n  About me; I descend from the clear heights\n  Into the plains and valleys of the world.\n  The traffic of the market-place is mine,\n  The heat and dust, the jostling and the noise,\n  The kindly challenge and the neighbour-talk,\n  All these may claim me, so that I forget\n  To lift my eyes and see the far-off peaks,\n  And the eternal splendour of the stars.\n\n  So be it; let the tide of men's affairs\n  Carry me back and forward; let the rub\n  Of greasy ha'pence passed from hand to hand,\n  In humble traffic of a bunch of herbs\n  Not pass me by; let me jog arm in arm,\n  Or cheek by jowl, the shady side o' the street,\n  With friends and neighbours, glad to know them there,\n  Imperfect, human, kind, and tolerant.\n\n  So may the years go.  Yet, when the call comes,\n  And the world's colours fade before the eye\n  That turns for spiritual vision on itself;\n  When, from the four walls of the silent room,\n  The noises of the world fall back and fail\n  In that great silence which enrings the last\n  Ecstatic moment of experience,\n  Here on this earth--ah, then indeed I know\n  That I shall find you.  All that lies behind\n  (The years of trivial experience)\n  Shall open and fall back from off my soul,\n  As falls the brown sheaf from the opening bud;\n  And in that poignant moment, that mere breath\n  Of temporal time, that aeon of the soul,\n  I shall reach out and know you, mix with you\n  As flame with flame, as ray with ray of light,\n  Be perfectly yourself, as you are me,\n  With all else fallen, gone, dispersed away\n  Save the pure drop of spiritual essence--Then\n  Let come what may, light or oblivion.\n\n\n\n\n  _Printed by_ R. & R. CLARK, LIMITED, _Edinburgh_.\n\n\n\n\nRECENT POETRY\n\nPoems.  By RALPH HODGSON.  Fourth Thousand.  Fcap. 8vo.  3s. 6d. net.\n\nLivelihood: Dramatic Reveries.  By WILFRID WILSON GIBSON.  Crown 8vo.\n3s. 6d. net.\n\nWhin.  Poems by WILFRID WILSON GIBSON.  Crown 8vo.  3s. 6d. net.\n\nPoems of London and Occasional Verse.  By JOHN PRESLAND, author of\n\"Mary, Queen of Scots,\" \"Joan of Arc,\" \"Manin and the Defence of\nVenice,\" \"The Deluge and other Poems.\"  Crown 8vo.\n\nTwenty.  Poems by STELLA BENSON.  Crown 8vo.  3s. 6d. net.\n\nThe English Poets.  Selections with Critical Introductions by various\nwriters.  Edited by THOMAS HUMPHRY WARD, M.A.  Vol. V.  Crown 8vo. 10s.\n6d. net.\n\nThis volume contains Selections from Browning, Tennyson, Swinburne, W.\nMorris, Arnold, W. Barnes, R. H. Horne, Lord Houghton, E. FitzGerald,\nAubrey de Vere, Coventry Patmore, W. Johnson (W. Cory), Jean Ingelow,\nSir F. Doyle, Alex. Smith, George Meredith, T. E. Brown, C. G.\nRossetti, R. W. Dixon, A. L. Gordon, Lord de Tabley, Cardinal Newman,\nF. W. H. Myers, Philip Bourke Marston, R. L. Stevenson, Stephen\nPhillips, The Earl of Lytton, John Davidson, Ernest Dowson, Richard\nMiddleton, Rupert Brooke, etc.\n\nCANADIAN POETRY: Isabella Valancy Crawford, W. H. Drummond, Archibald\nLampman, Harold V. Wrong.\n\n\nMACMILLAN AND CO., LTD., LONDON.\n\n\n\nWORKS OF WILLIAM BUTLER YEATS\n\n  Plays for an Irish Theatre.  8vo.  8s. 6d. net.\n  Poems.  Second Series.  Crown 8vo.  6s. net.\n  The Celtic Twilight.  Crown 8vo.  6s. net.\n  Ideas of Good and Evil.  Crown 8vo.  6s. net.\n  Stories of Red Hanrahan, etc.  Crown 8vo.  6s. net.\n  Reveries over Childhood and Youth.  Illustrated.  Crown 8vo.  6s. net.\n  Responsibilities and other Poems.  Crown 8vo.  6s. net.\n  Per Arnica Silentia Lunae.  Crown 8vo.  4s. 6d. net.\n  The Tables of the Law.  Crown 8vo.  2s. net.\n  Deirdre.  Crown 8vo.  1s. net.\n  The King's Threshold.  Crown 8vo.  1s. net.\n  The Hour Glass.  Crown 8vo.  6d. net.\n  The Pot of Broth.  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+{"text":" \nPOMEGRANATE\n\nEdible\n\nSeries Editor: Andrew F. Smith\n\nEDIBLE is a revolutionary series of books dedicated to food and drink that explores the rich history of cuisine. Each book reveals the global history and culture of one type of food or beverage.\n\nAlready published\n\n_Apple_ Erika Janik _Banana_ Lorna Piatti-Farnell\n\n_Barbecue_ Jonathan Deutsch and Megan J. Elias\n\n_Beef_ Lorna Piatti-Farnell _Beer_ Gavin D. Smith\n\n_Brandy_ Becky Sue Epstein _Bread_ William Rubel\n\n_Cake_ Nicola Humble _Caviar_ Nichola Fletcher\n\n_Champagne_ Becky Sue Epstein _Cheese_ Andrew Dalby\n\n_Chillies_ Heather Arndt Anderson _Chocolate_ Sarah Moss  \nand Alexander Badenoch _Cocktails_ Joseph M. Carlin\n\n_Curry_ Colleen Taylor Sen _Dates_ Nawal Nasrallah\n\n_Doughnut_ Heather Delancey Hunwick _Dumplings_ Barbara Gallani\n\n_Edible Flowers_ Constance L. Kirker and Mary Newman\n\n_Eggs_ Diane Toops _Fats_ Michelle Phillipov _Figs_ David C. Sutton\n\n_Game_ Paula Young Lee _Gin_ Lesley Jacobs Solmonson\n\n_Hamburger_ Andrew F. Smith _Herbs_ Gary Allen _Honey_ Lucy M. Long\n\n_Hot Dog_ Bruce Kraig _Ice Cream_ Laura B. Weiss _Lamb_ Brian Yarvin\n\n_Lemon_ Toby Sonneman _Lobster_ Elisabeth Townsend\n\n_Melon_ Sylvia Lovegren _Milk_ Hannah Velten _Moonshine_ Kevin R. Kosar\n\n_Mushroom_ Cynthia D. Bertelsen _Nuts_ Ken Albala _Offal_ Nina Edwards\n\n_Olive_ Fabrizia Lanza _Onions and Garlic_ Martha Jay\n\n_Oranges_ Clarissa Hyman _Pancake_ Ken Albala\n\n_Pasta and Noodles_ Kantha Shelke _Pie_ Janet Clarkson\n\n_Pineapple_ Kaori O' Connor _Pizza_ Carol Helstosky\n\n_Pomegranate_ Damien Stone _Pork_ Katharine M. Rogers\n\n_Potato_ Andrew F. Smith _Pudding_ Jeri Quinzio _Rice_ Renee Marton\n\n_Rum_ Richard Foss _Salad_ Judith Weinraub _Salmon_ Nicolaas Mink\n\n_Sandwich_ Bee Wilson _Sauces_ Maryann Tebben _Sausage_ Gary Allen\n\n_Seaweed_ Kaori O'Connor _Soup_ Janet Clarkson\n\n_Spices_ Fred Czarra _Sugar_ Andrew F. Smith _Tea_ Helen Saberi\n\n_Tequila_ Ian Williams _Truffle_ Zachary Nowak\n\n_Vodka_ Patricia Herlihy _Water_ Ian Miller\n\n_Whiskey_ Kevin R. Kosar _Wine_ Marc Millon\n\n# Pomegranate\n\nA Global History\n\n_Damien Stone_\n\nREAKTION BOOKS\nPublished by Reaktion Books Ltd  \nUnit 32, Waterside  \n44\u201348 Wharf Road  \nLondon N1 7UX, UK  \nwww.reaktionbooks.co.uk\n\nFirst published 2017\n\nCopyright \u00a9 Damien Stone 2017\n\nAll rights reserved  \nNo part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publishers\n\nPage references in the Photo Acknowledgements and  \nIndex match the printed edition of this book.\n\nPrinted and bound in China in\/by\n\nA catalogue record for this book is available from the British Library\n\neISBN: 9781780237954\n\n## Contents\n\nIntroduction\n\n1 The Primordial Pomegranate: The Fruit in Myth\n\n2 Pomegranates in the Ancient World\n\n3 Jewish and Islamic Pomegranates\n\n4 Medieval Pomegranates\n\n5 Pomegranate Production and Culture Today\n\n6 The Pomegranate in Modern Literature, Art and Film\n\nAppendix: Pomegranate Cultivars\n\nRecipes\n\nReferences\n\nSelect Bibliography\n\nWebsites and Associations\n\nAcknowledgements\n\nPhoto Acknowledgements\n\nIndex\n\n## Introduction\n\nThe pomegranate bush raises its voice (tiny, insistent, and shrill): 'My seeds shine like the teeth of my mistress, the shape of my fruit is round like her breasts. I'm her favourite, I know, sweetest tree in the orchard, looking my best through every season.'\n\nTurin Papyrus, Egyptian, 12th century BC\n\nThe pomegranate is an unusual, alluring and ornamental fruit. Even though it can be rather awkward to eat, throughout human history it has remained a most desirable talismanic food item. Those who first wrote of the pomegranate, in ancient Mesopotamia, called it _nurma_ , but for us its name derives from the Latin _pomum_ (apple) and _granatum_ (seeded). It is from the fruit's distinctive shape, colour and seeds that its universal aesthetic appeal originates. The pomegranate embodies beauty, mystery and the female. Its evocative red juice has often been likened to blood.\n\nOne of the oldest foods in the world, the pomegranate seems to have come to us originally from the region that is now modern Iran, wild varieties being born of the bio-diverse Kopet Dag mountain range. It was first domesticated during the Neolithic Revolution, which began around 10,000 BC; human selection meant that the pomegranate took on appealing qualities it did not have in the wild. The pomegranate makes its first archaeological appearance at the start of the third millennium BC, in the form of carbonized pomegranate seeds excavated at the Middle Eastern sites of Jericho, Arad and Nimrud. In addition to the pomegranate ( _Punica granatum_ ), the genus _Punica_ contains one other species. The rare _Punica protopunica_ is a precursor from which the common pomegranate may derive. It is found only on the island of Socotra off the coast of Yemen. _P. protopunica_ is generally not eaten, however, as it is rather bitter. _Punica_ is from the same botanical family as myrtle, the Lythraceae.\n\nPomegranates ( _Punica granatum_ ) for sale, Mysore, India.\n\nThe part of the fruit that is of most interest to us is that which is edible: the seeds. Each seed is coated in an aril, that ever-so-desirable, blood-red and juicy, sour-sweet flesh. These are set close together in kaleidoscopic clusters embedded in a yellow or white leathery pulp. The versatility of the fruit, however, extends beyond its culinary function. Its skin, for example, is known to have been used to tan leather. The flowers of the pomegranate plant can produce a vibrant red dye for use in textiles, while a black dye can be extracted from the plant's roots. From the rind a yellow dye can be made. The pale wood of the pomegranate tree is very hard and durable. However, the small overall dimensions of the tree, with its thin trunk, makes it useless in construction (although an Old Babylonian text does refer to a 3-metre-long pomegranate beam used as a building edifice). Today pomegranate wood is really only used for crafting small-scale agricultural tools and some decorative items. Cups made of pomegranate wood have been made as far back as the Bronze Age.\n\nOtto Wilhelm Thom\u00e9, _Punica granatum_ , 19th-century botanical illustration.\n\nThe pomegranate tree, which is known to live for up to 300 years (although after fifteen years the flavour and production of the fruit declines), is shrub-like in appearance, closely branched and twiggy. Some varieties are evergreen while others are deciduous. The plant usually reaches a maximum height of about 8 metres. It can also be much smaller; dwarf pomegranates are a favourite variety of bonsai in Japan, owing to the plant's ability to achieve a desirable twist in the wood of its trunk, as well as the appeal of its miniature red fruits and flowers. Some pomegranate varieties are fruitless and grown simply for the appeal of their bell-shaped flowers. The pomegranate tree also works well as a hedging plant and has been used to reforest mountain slopes, improving the adverse effects of erosion.\n\nTo begin our story, I will make the intimidating pomegranate a little more approachable by explaining the best way to get rid of its off-putting skin and pith, so that the reader can enjoy eating some arils as they discover this fruit's colourful history. When purchasing a fresh pomegranate, note that the optimum fruit has smooth, glossy skin free from cracks and bruises. A pomegranate past its maturity will have dry-looking, wrinkled and tightened skin. There are over a thousand varieties of pomegranate (see Appendix for a selection of these). Although the outer skin is most commonly red, there is a full colour range from dark purple to yellow and green varieties. The inner arils likewise vary in their shade. Sometimes in a single pomegranate, arils of two different colours may be found. This is known as metaxenia, and is the result of a plant being fertilized by two different cultivars.\n\nPomegranate in growth.\n\nPomegranate tree exhibiting its flowers.\n\nTake your chosen pomegranate and cut off just enough of its crown (known as the calyx) to expose the yellow or white pith. Score the skin downwards in quarters and then allow the fruit to soak in a bowl of cold water for a few minutes. Holding the fruit under water, break it into sections with the fingers, separating the seeds from the pith. The seeds will sink to the bottom of the bowl, while the membrane will float to the top. After discarding the skin, drain the seeds and dry on paper towels. Alternatively, cutting the fruit in half and using a utensil to smack vigorously on the back of the skin will quickly dislodge the arils, although this tends to be a messier method. Your arils are ready for immediate consumption, and for use in cooking. Alternatively, they will last for up to one month in the refrigerator, or about three months in the freezer. Many people enjoy the juicy pulp of the aril but do not like consuming the central seed itself. To juice your fruit, you can use a blender to process the arils, afterwards using cheesecloth to separate the desired liquid from the leftover solids. Otherwise the pomegranate can be cut in half and have its juices squeezed out by means of a citrus press. When the juice is boiled it becomes concentrated, eventually developing into a thick, dark molasses.\n\n## 1\n\n## The Primordial Pomegranate: The Fruit in Myth\n\nBut secretly he slipped into my mouth a seed from a pomegranate, that honey-sweet food, and forced me, made me taste it against my will.\n\nHomeric hymn to Demeter, 7th\u20136th century BC\n\nAncient Greece\n\nOur story begins in the timeless realm of myth, through which people inspired by the pomegranate have composed fantastical accounts featuring the fruit. The pomegranate came to be an important figure in the literary lore passed down in both Eastern and Western cultures. Acting as a symbol of fecundity in both traditions, the pomegranate shows the dual nature of fertility, sometimes offering life, but also bringing about death.\n\nThe quotation at the beginning of this chapter is from one of the best-known stories about the pomegranate that comes to us from the classical world. This myth revolves around Persephone, the daughter of the Greek goddess of agriculture, Demeter. Persephone is kidnapped by Hades, the god of the dead, and taken to the underworld to be his wife and queen. Demeter, overwrought by her daughter's untimely descent into the land of the dead, renders the earth barren, destroying all of its vegetation. This in turn causes mankind to starve, and thus Zeus intervenes and demands that Hades return Persephone. Hades, unwilling to give up his new bride completely, tempts her with some pomegranate seeds. Although Persephone is released, because she tasted the food of the underworld, she is required to spend a third of each year there. She is permitted to spend the rest of the year with the other gods above. This myth was used by its classical audience as an explanation for the changing of the seasons: the portion of the year when Persephone was absent was winter, when Demeter once again stripped the earth of life and went into mourning. Persephone's resurrection came in the springtime, marked by rebirth in the natural world.\n\nA mystery religion devoted to Demeter and her daughter, centred on the site of Eleusis in Attica, seems to have promoted the idea of a pleasant existence after death for initiates. Ancient sources are quiet about what secret rituals occurred there, but it is speculated that a re-enactment of the myth took place, which probably required initiates to eat or drink from a pomegranate. The pomegranate in the Persephone myth symbolizes female fertility, as well as the loss of virginity that comes with the consummation of marriage, evoked by both its red stains and its rich seeds. Imagery of the fruit functions as a reminder of what it is to be a woman, the red seeds that spill from the pomegranate replicating the blood of the menstrual cycle. Greek medicine in fact prescribed pomegranate juice as a way of stopping menstrual bleeding.\n\nThe woman's monthly bleeding was seen as her equivalent of the man's fighting on the battlefield as a warrior, and thus the pomegranate was an attribute of the martial virgin goddess Athena, the protectress of young Athenian girls of a marriageable age. The Temple of Athena Nike atop the Acropolis housed a wooden cult statue that has long since been lost to decay, which depicted Athena with a pomegranate in her right hand (signifying the female battle), and a helmet in her left (the male battle).\n\nAthena is likewise represented with the pomegranate on several different ancient coins from Side in Pamphylia (on the southern Mediterranean coast of what is now Turkey). As well as giving its name to a place, _side_ is one of the ancient Greek words for pomegranate, and was also personified as the name of the wife of Orion. It is thought that the wedding of Orion to Side is a mythological explanation for the ripening of autumnal fruit, which occurs during the time of year when the Orion constellation can be seen in the night sky. Not all goes well for the happy couple, however, as Side commits a grave act of arrogance, comparing her ornamental beauty with that of the queen of the gods, Hera. Side, for punishment, is sent to the underworld for eternity, hence the association between the pomegranate and that place. Another tradition refers to Side as a young virgin who commits suicide on her mother's grave in order to avoid being raped by her father. Her blood gives birth to a pomegranate tree.\n\nAs a result of Hera's involvement in the myth of Side, she is often represented holding a pomegranate. One such representation was created by the sculptural master Polykleitos at Argos in the fifth century BC. Being made of ivory and gold, the sculpture did not survive antiquity, but sources tell us that in one of her hands Hera held a pomegranate as a globus cruciger. Pausanias, who saw the chryselephantine statue, writes: 'About the pomegranate I must say nothing, for its story is somewhat of a holy mystery.' This suggests that there existed a secret cult to Hera, requiring initiation, which revolved around the pomegranate, probably in connection with the Hera and Side myth. An important sanctuary to Hera is located in southern Italy at the Greek settlement of Paestum, where a chapel now stands to Madonna del Granato (Our Lady of the Pomegranate). The archaeologist Helmut Kyrieleis has noted that Mary, 'by virtue of her epithet and the attribute of a pomegranate must be the Christian successor of the ancient Greek goddess Hera'. The pomegranate that emphasized the fertility and domestic role of the ancient Greek mother of the gods thus came to be adopted by the virgin mother of the Christian God.\n\nBut the pomegranate in Greek myth wasn't just a symbol of fertility in the feminine sense. In ancient Greek society, a woman's body was traditionally seen as an empty vessel that needed to be filled by a man. A son was said to be a closer blood relation to his father, who planted his seed, than to his mother, who is simply the soil in which it was sown. Thus it was appropriate that the creation of the pomegranate plant, with its ever-fertile seeds, should be explained in masculine terms. The first pomegranate was consequently said to have been created by Aphrodite from the blood of her dying mortal lover Adonis. The fruit was one of the symbols of the sexually powerful Aphrodite, the goddess said to have first planted the pomegranate on the island of Cyprus. The fruit is often linked with depictions of Aphrodite, as on an Etruscan bronze mirror of 300 BC engraved with an image of the goddess holding a pomegranate-topped sceptre. The depiction of immortal figures on Etruscan mirrors is common, and an individual who used such a mirror would see the divine image of perfection imposed over his\/her own reflection.\n\nAdonis was not the only male figure from whose blood the pomegranate was perceived to have resulted. The same motif appeared in the myth surrounding the Phrygian great mother goddess, Cybele. Cybele is thought to derive from an earlier Hittite goddess named Kubaba, who in representations from the ninth to eight century BC is shown holding a pomegranate in one of her hands and a mirror in the other. Her tale begins with the birth of a wild man named Agdestis, who wreaks havoc among gods and mortals alike. The god of wine, Dionysus, decides to intervene and spikes the water fountain where Agdestis comes to drink. A drunken Agdestis falls into slumber, after which Dionysus uses a rope to bind his feet and makes a vine grow over his genitals. When Agdestis arises, stumbling to his feet in his hungover state, his genitals are ripped from his body by the entwined plant. From the blood that is spilled from his groin (replicating semen) onto the earth, a pomegranate tree grows. The story continues that a river nymph named Nana eats one of the pomegranates from the tree. Ripe with bloody seeds, the fruit in turn impregnates her. She gives birth to Attis, who then becomes Cybele's lover. Throughout this bizarre myth, the pomegranate is the generative and transformative force, literally taking the role of the man in procreation.\n\nHittite goddess Kubaba holding a pomegranate, 9th\u20138th century BC.\n\nAlthough grapes are the main attribute of the god of wine, Dionysus, the pomegranate is also associated with him in a number of myths. In a tale of his infancy, he is torn up and eaten by the Titans (though later reassembled). From the blood that spills onto the earth as he is cannibalized, a pomegranate tree grows. Thus certain ritual festivals banned the consumption of the pomegranate as being too sacred. Clement of Alexandria tells us: 'The women who celebrate the Thesmophoria are careful not to eat any pomegranate seeds which fall to the ground, being of opinion that pomegranates spring from the drops of Dionysus' blood.' Later, as an adult god, Dionysus seduces a maiden under the pretence that he will give her a crown. When she badgers the god for her reward, he turns her into a pomegranate, with its crown-shaped calyx.\n\nTwo final incidents in classical myth that allude to the pomegranate have the fruit as the catalyst for great suffering. The first concerns Tantalus, a mortal figure punished by the gods to suffer an eternal torment in the underworld. His name is the source of the English word 'tantalize'. Tantalus was forced to stand in a pool of water beneath a fruit tree with low branches, with the fruit ever eluding his grasp, and the water always receding before he could take a drink. Different sources depict him reaching out for different fruit; some (Homer's _Odyssey_ included) appropriately choose the pomegranate, the plentiful seeds of the desired fruit contrasting with Tantalus' torment of emptiness that makes him hunger and thirst eternally in the underworld. The other myth that contains a fruit which can be interpreted as a pomegranate is that of the Judgement of Paris. The story goes that in a beauty competition between the Greek goddesses, the Trojan prince, Paris, awards Aphrodite a prized 'golden apple', that is, a pomegranate. The ancient Greek word for apple, _m\u00ealon_ , was also sometimes used to describe pomegranates, which were much more common in the Mediterranean. The pomegranate, after all, was one of Aphrodite's attributes, as well as being associated with both Athena and Hera, who also vied for the fruit in the competition. The pomegranate was considered an aphrodisiac with magical properties, and thus, for Paris, the fruit allows for the wish-fulfilment of gaining Helen, the most beautiful woman in the world. Paris awards the fruit to Aphrodite after she offers him Helen, wife of the king of Sparta. Helen's abduction inevitably causes conflict. The fateful fruit, inscribed with the words _Ti Kallisti_ (For the Fairest), thus becomes responsible for the start of the Trojan War. The pomegranate's red, spilling seeds appropriately forecast the bloodshed that will result.\n\nArabia, Persia and Turkey\n\nCenturies later, another literary tradition that was similarly obsessed with the pomegranate developed in the Middle East. Under an Islamic backdrop, the pomegranate remains sexually charged.\n\nLimestone hand clutching a pomegranate, all that survives from a large Cypriot statue, 6th century BC.\n\n_The One Thousand and One Nights_ is an anthology of many tales from Arabian folklore, structured within the framework of the story of Scheherazade, who uses her ability as a storyteller to avoid execution. The pomegranate appears in many of her tales. One story concerns the son of the king of Yemen, Sayf, who on his adventures meets a man without arms and legs who has been fed pomegranate seeds by mice for 700 years while he waited for the prince's arrival. This link between the pomegranate and the infinite is also found in another tale relating to Hayid, a prophet to whom God grants a pomegranate with an unending amount of arils that will feed him perpetually. The 'Tale of the Young Nur and the Warrior Girl' contains several short songs in praise of various fruits. The hymn relating to the pomegranate compares the fruit's shape to the alluring female form:\n\nPolished delicates are we,  \nRuby mines in silver earth,  \nMaidens' blood of high degree  \nCurdled into drops of worth,  \nBreasts of women when they see  \nMan is near, and stand them forth.\n\nA very fertile pomegranate occurs in an Arabian Nights tale revolving around the sultan of Diyarbakr. The story begins by noting the sultan's inability to produce any children. He has a dream in which he is told to go to his garden and request a pomegranate from his gardener. He does as the dream commands. The Sultan eats fifty seeds from the pomegranate he is given and promptly all fifty of his wives become pregnant. One of the concubines, a woman named Pirouze, does not appear at first to be with child, and is sent away, but is later discovered to be pregnant anyway.\n\nAnother transformative pomegranate is to be found in the 'Twelfth Captain's Tale'. Our hero is a prince by the name of Muhammad who comes up against an evil sorcerer Moor. Muhammad is able to escape and overcome the Moor by transfiguring into a pomegranate:\n\n> To escape his persecutor, the prince used the virtue of the cord to change into a large pomegranate... as soon as the Moor touched it, it burst asunder, and all the grains were scattered on the floor. The sorcerer picked them up, one by one, until he came to the last grain... which contained the vital essence of Muhammad. As the vile magician stretched out his neck towards this final grain, a dagger came up out of it and stabbed him to the heart, so that he spat out his unbelieving soul in a stream of blood.\n\nThe blow to the Moor is fatal, the blood-evoking pomegranate bringing about his bloody end.\n\nThe Shahnameh, Iran's national epic by the poet Ferdowsi, narrates the lives of the Persian kings from the beginning of time until the seventh century AD. It is a conglomeration of a vast amount of mythology, legend and history that includes among its tales several references to the pomegranate. The fruit is used in a description of the physical beauty that first attracts the warrior Zal to his future wife Rudabeh: 'Her mouth resembles a pomegranate blossom, her lips are cherries and her silver bosom curves out into breasts like pomegranates.' Zal and Rudabeh have a son, the great Persian hero Rostam. Rostam would eventually take up arms against another Persian hero, Esfandiyar, who had previously been rendered invincible from eating a pomegranate. Esfandiyar, however, like Achilles, has one vulnerable spot: his eyes. The pomegranate's power cannot defend him from an arrow from Rostam's bow, shot straight through Esfandiyar's pupil. The Shahnameh again associates the fruit with death following the beheading of Siavash. The text notes how the pomegranate trees of the forest withered in sorrow after the wrongful execution of the Iranian hero.\n\nOne final story of the pomegranate in the Persian epic concerns a man named Farhad who becomes the lover of King Khosrow's wife, Shirin. As punishment for his transgression, he is exiled to Mount Behistun (known to us for its famed inscription by Darius the Great, which was later the key to the decipherment of cuneiform in the nineteenth century). He is given the task of clearing away the mountain with an axe, and told that if he successfully finds water he will be allowed to marry Shirin. Many years go by. It is not until Farhad has removed half the mountain that he does in fact find water. Khosrow approaches him with the (false) news that Shirin has died. Farhad goes mad with grief, throws his axe down and dies. Where the thrown axe falls, a pomegranate tree grows that bears fruit with the power to cure sickness.\n\nIn a fairy tale from Turkey, the son of a king desires a certain princess. After completing several challenges set by her father in order to gain an audience with the princess, she sets the youth a task herself: to retrieve a singing pomegranate branch from the Garden of Reh-Dew. Facing adversity along the way, the prince finally arrives at the garden and sees the pomegranates hanging from a tree like lamps. He picks off a branch with some fifty pomegranates on it, 'each of which sang a different song, as though all the music of the world were brought together there'. On retrieving the branch he is told by the guardian of the garden: 'never let it out of your sight. If you can listen to it throughout your wedding day the pomegranates will love you; you need fear nothing, for they will protect you in any distress.' The princess becomes aware of her champion's return when she hears the many tunes of the fifty fruits echoing through her city. As she meets the prince, 'the pomegranate branch chanted the union of their two hearts in such exquisite strains that they seemed to be lifted up from this earth to the Paradise of Allah.' Throughout their lengthy forty-day wedding, the couple continuously listen to the singing of the pomegranates. An imposter prince tries to claim the princess for himself, but the sound of the pomegranates that the prince and princess heard during their wedding keeps evil at bay. Although the true prince is beheaded, he is restored back to life and the pretender is overcome. In this tale, the princess's requirement for her suitor to retrieve the pomegranate branch is most interesting. As the branch has the power to guarantee a positive marriage experience, it functions like a dowry and alleviates the uncertainty that comes with being a bride. In traditional Turkish culture (and in other parts of this region, such as Armenia and Iran) the pomegranate is associated with weddings. Following the service, a bride may choose to toss a pomegranate on the floor. The number of seeds that spill out of the fruit are playfully considered to indicate the number of children she will have. Surely a choir of fifty pomegranates would grant the prince and princess an abundance of children.\n\nThe pomegranate occurs alongside this literary motif of the suitor's challenge in another fairy tale known as 'The Parrot Shah'. The story relates that an overprotective father makes use of a pomegranate to protect his daughter:\n\n> The king was very possessive and wanted to keep Gala all to himself, and with the help of a wizard, he had thought up a plan to discourage her suitors. A magic tree was planted in the garden, a huge pomegranate that had three fruits. At sunset, the branches bent over to touch the ground and the fruit split open. Inside each lay a soft feather bed. Gala, the princess, slept in the middle one, with her servants on each side. The fruit closed over the maidens and the branches swung back to the sky, carrying the princess high above all danger. Seven walls were built round the garden, each studded with thousands of spikes which nobody could ever cross.\n\nThe challenge of wooing the princess is eventually achieved by a shah of distant Persia who is able to transmigrate into the body of a parrot and fly off with the pomegranate. The sexual implications of the imagery in this tale are implicit. The desirable virgin, lying on a bed inside a red and seeded fruit, represents one thing for a man: sexual potential. The father's attempt to use a feminine fruit, which splits open and closes up again, to obstruct masculine advances on his daughter makes her all the more enticing. A similar story is found in the beginning of the Persian legend surrounding the Simurgh bird:\n\n> In the garden of the palace there grew a pomegranate tree with only three pomegranates; their seeds were fabulous gems that shone like lamps by night. When ripe, the pomegranates would turn into three beautiful girls who were to become the wives of the three princes.\n\nThe pomegranates are likewise stolen, only this time by a malevolent hand that descends from a dark cloud.\n\n## 2\n\n## Pomegranates in the Ancient World\n\nIf he is in his seventies and gasping respiration continually afflicts him and he asks for pomegranates, he will die. If he is in his seventies and he asks for dates and eats them, he will get well.\n\nMesopotamian omen-based medical text, 9th\u20137th century BC\n\nThe Ancient Near East: Mesopotamia, Egypt and Persia\n\nHistory begins with the invention of writing in Iraq in the late fourth millennium BC, and with it the history of the pomegranate. The creation of writing developed alongside the practice of marking possessions with seals. One Mesopotamian individual living in the first cities around 3500\u20133000 BC chose the motif of a pomegranate to mark ownership of property. Complete with engravings schematically representing the internal sections of the fruit, the seal is one of the earliest surviving artistic representations of a pomegranate. This seal would have functioned as a form of identification, acting as a personal signature on accounting\/trade documents written in clay, and ensuring the security of valued commodities sealed in vessels. The seal has perforations so that it could be worn on the body, either around the neck or on the wrist, as an item of jewellery. This steatite pomegranate was therefore inseparable from its owner's identity, embodying who they were. The fruit was probably chosen as the icon for a seal for its symbolic value. This pomegranate, stamped on daily business transactions, evoked the fruit's fertility, guaranteeing its owner fruitfulness in his affairs.\n\nMesopotamian stamp seal and its impression in the form of a pomegranate, 3500\u20133000 BC.\n\nA roughly contemporary artefact, an alabaster vessel known as the Uruk Vase, represents the levels of world order during this time. The temple, the plain on which divine and human interaction occurs, is represented in the topmost register. Underneath this a register bears a representation of the world of man. Further down is a register of domesticated animals. Out of the watery bottom layer, trees grow, some of which bear a trio of pomegranates. The pomegranate is among the vegetal base on which society rests, enjoyed by the upper human and divine realms.\n\nAlthough it is fortunate that recipes survive from ancient Mesopotamia, they are quiet as regards to the pomegranate's use in cooking. The fruit seems to have been a popular banquet item, one text recording the serving at a wedding of 'large pomegranate seeds plucked from their rinds'. Another manuscript notes that a hundred pomegranates were served as condiments among a long list of foods served at a feast held by the Assyrian king Ashurnasirpal II. Cuneiform tablets indicate that pomegranate was one of the various flavours added to the national and staple beverage invented by the Mesopotamian people: beer.\n\nThe pomegranate also makes an appearance in Mesopotamian narratives. Sumerian literature is dripping with plant-and agricultural-based allegories representing sexual intercourse. The pomegranate fits well among these. In one composition the historic king Shulgi (2094\u20132047 BC) approaches Inanna (also known as Ishtar), the Sumerian goddess of sex and war, with the following proposition: 'My sister, I would go with you to my pomegranate-tree. I would plant there my sweet, honey covered... '. A break in the text leaves it up to our imagination as to just how Shulgi spills the seed and juice of his pomegranate on the goddess. The Old Testament draws directly from this Near Eastern tradition, for the author of the Song of Songs (7:2) likewise exclaims: 'Let us go forth into the fields... where the pomegranates are in bloom. There I will give you my love.' The suggestiveness continues throughout this book: 'I would give you spiced wine to drink, the juice of my pomegranate (Songs 8:2).' Even later in Mesopotamia, during the Assyrian period of the first millennium BC, the fruit remains explicitly sexualized in texts. An incantation text of the period thus reads:\n\n> Incantation: A maid, beautiful, loving, has come forth. The goddess Ishtar, who loveth the apple and pomegranate, sexual strength, has come forth.\n> \n> Prayer for: If a woman has raised her eye to a man's penis.\n> \n> Ritual for it: Recite the incantation three times over either an apple or a pomegranate. Give it to the woman. Make her suck its juice. That woman will come and he will love her.\n\nThis is imitative magic, drawing on the fertile juice of the apple\/pomegranate in order to bring forth bodily fluids. Throughout the Mesopotamian textual tradition the fruit has strong ties with divinity, one Assyrian tablet describing a deity whose knees are pomegranates. It is represented as literally part of a god. The strong link between the pomegranate and Inanna\/Ishtar would influence the classical construction of female divinity.\n\nIconographically, the pomegranate is relatively rare in ancient Mesopotamian art. It does not occur in any sexualized images; rather, the fruit's powers are drawn upon in the more tranquil setting of the scenes featuring the Assyrian Tree of Life. This motif features on large-scale palatial reliefs as well as smaller decorative seals and ivories. The plant species that this tree is supposed to represent has long been debated over in scholarly circles (it is usually thought to be the date palm). Occasionally, however, the standard palmettes that are characteristic of representations of the sacred tree are replaced, undeniably so by pomegranates. Sometimes, the winged genii (known as _Apkallu_ ) who heraldically approach the tree in procession are also represented carrying a pomegranate branch. The pomegranate probably served an apotropaic function, absorbing the evil that might otherwise possess the human inhabitants of the palace, as well as representing the Assyrian empire as a fruitful garden through its connotations with fertility.\n\nMeanwhile, in Egypt, the pomegranate is remarkably absent from the Nile Valley region until the beginning of the New Kingdom period (sixteenth to eleventh century BC). By this time we have evidence for ancient intercontinental trade, or perhaps royal gift-giving, of the fruit, as attested by the presence of whole pomegranates found aboard the Ulu Burun shipwreck, a merchant vessel that sank with its vast international assemblage of cargo in the Mediterranean during the fourteenth century BC. A bronze cultic tripod of this period from Ugarit, Syria, is decorated with hanging pomegranate pendants, emphasizing the association the fruit had come to have across the ancient world with ritual, sacrifice and offering. The fruit was probably first brought back to Egypt as booty from the military conquests in Palestine and Syria that characterized the Egyptian experience of the early Eighteenth Dynasty. The pomegranate became a refreshing feature of Egyptian gardens, and, as highlighted in the quote from the Turin Papyrus featured in the Introduction, influenced the Egyptian concept of beauty. The pomegranate is included among the remedies of one of the oldest medical compilations, the Egyptian Papyrus Ebers of the sixteenth century BC, which suggests that it can be used as a treatment for the management of tapeworms. The pomegranate would have been quite effective against tapeworms, as its high alkaline value would paralyse the parasite's nervous system.\n\nEgyptian ointment spoon, 1336\u20131327 BC. |\n\n---|---\n\nAssyrian cylinder seal and its impression illustrating the Assyrian sacred tree with pomegranates (replacing the standard palmettes), 850\u2013825 BC. The king is represented symmetrically on both sides of the sacred tree, with two winged guardians behind him. The chief Assyrian god, Asshur, floats above the scene.\n\nAmarna relief depicting Nefertiti sitting on Akhenaten's lap and holding one of his daughters, 14th century BC. In front of the royal family is a container filled with fruit, including pomegranates.\n\nThe pomegranate appears in Egyptian art in both monumental temple reliefs and smaller, personal items. The Brooklyn Museum houses a colourful pomegranate-inspired ivory ointment spoon. The bowl of the ornamental spoon is crafted in the form of a large yellow-brown pomegranate. A second identical pomegranate covers the spoon bowl, acting as a lid that is slid open and closed by means of a pivot. Flowers, leaves and smaller pomegranates jut out from the stem that functions as the handle. The spoon demonstrates the Egyptian tendency to stylize the natural. The artisan who made it probably never saw a pomegranate tree in growth, for what is represented is an ideal rather than a reality: flowers and fruit never appear on the pomegranate plant at the same time. Even in Tutankhamun's famed tomb we find a representation of the fruit: a silver pomegranate-shaped repouss\u00e9 vessel. Such a vessel may have contained pomegranate juice. As in Mesopotamia, the juice was added to alcoholic beverages, although in Egypt a myth gave new significance to the pomegranate-beer cocktail: it was responsible for saving mankind. The tale tells us that the lion goddess Sekhmet, who has a taste for human blood, plans the mass destruction of all humanity. She is stopped by the sun god Re, who floods her path with a mixture of beer and pomegranate juice poured from 7,000 jugs. Sekhmet mistakes the red liquid for blood and quickly drinks it all. She is thus too drunk to pursue her slaughter. The pomegranate-shaped vases, charms and dried fruit specimens that have been found in many Egyptian burials were probably placed there as a symbol of the anticipated rebirth of the deceased into the afterlife. The fruit had ensured the survival of mankind once already in life, so why not in death?\n\n  | Hellenistic terracotta stamp seal from Egypt, illustrating a pomegranate in the centre of a basket composition, 2nd\u20131st century BC.\n\n---|---\n\nFrom 550\u2013330 BC, Egypt, Mesopotamia and the rest of the ancient Near East had come under the control of the Persian Empire. A cuneiform document from the Metropolitan Museum of Art records a Persian account of a rent payment made in pomegranates, attesting to their continued value and desirability. The Persian king Darius the Great is represented holding a pomegranate flower in a religious ritual at his capital of Persepolis. Alexander the Great would of course defeat the Persian Empire and absorb all this territory as his own; following his untimely death in his early thirties it was divided among his generals. Marking property was still practised at this time, with a stamp seal from Ptolemaic Egypt demonstrating the continued use of a pomegranate motif, although one that is much more elaborate, some 3,000 years later.\n\nFollowing this period of Greek rule, a second Persian empire arose in Iran, that of the Parthians (247 BC\u2013AD 224). The pomegranate features prominently on the decorative programme of their capital, Ctesiphon. The later Sassanian (AD 224\u2013651) site of Nizamabad depicts rows of pomegranates with wings in relief, a style that would influence Islamic designs. Numerous silver plates that survive from this period depict the exploits of the Sassanian kings. One of these features the pomegranate between a pair of rams' horns worn by a queen in what appears to be a marriage scene. She receives the ring of rulership from the king, who wears a pine-cone-topped crown equally evocative of fertility. Zoroastrianism was the religion of all these Persian kings, and its temples were traditionally surrounded by pomegranate trees. This religion is still practised today, and involves use of the pomegranate in certain rituals. The initiate of a Navjote, the Zoroastrian coming of age ceremony, is asked to sip a consecrated liquid known as _nirang_ in order to cleanse both body and soul. Traditionally this drink was the urine of a virile bull, but pomegranate juice is used as a substitute today. Celebration of _Nowruz_ , the Persian New Year, involves the presentation of a bowl of water containing a pomegranate into which coins have been inserted, or sometimes just a single pomegranate twig is used. The coin-encrusted fruit acts as a symbol for prosperity, longevity and good health. All these modern Iranian practices had their roots in the ancient Persian empires.\n\nSilver pomegranate vase from Tutankhamun's tomb, 14th century BC.\n\nFrieze from Nizamabad featuring winged pomegranates, 7th century AD.\n\nSassanian silver plate, 6\u20137th century AD.\n\nThe Classical World\n\nThere are only a few examples of the representation of pomegranates in the Bronze Age Aegean cultures of the Minoans and Mycenaeans. There is a similarity of shape between the pomegranate and the opium poppy, both plants having ancient connotations of fertility. There is no reason why the 'opium' crown of the Minoan 'poppy goddess' cannot be interpreted as a pomegranate. It is not until the advent of the Geometric period in Greece that we start to see a real explosion of clear material evidence relating to the pomegranate.\n\nFrom the tenth to eighth century BC a type of pomegranate-shaped vase was particularly prevalent, covered in geometric motifs: zigzags, triangles, meanders, swastikas, chequering. Undecorated ceramic pomegranates have likewise been found in the Levant from this period. The redness of the clay is effective in representing the fruit. The Geometric Greek vessels are wheel-made and have a small hole in the bottom, allowing liquid libations to be poured through them, or pebbles to be inserted to enable use as a rattle. This period was the time in Greece when myths started to be written down. The nature of the pomegranate came to define the powers of the gods in these stories. The fruit's ornamental form associated it with the beauty of Aphrodite. The likeness of pomegranate juice to blood represented life and death in the cult of Persephone. The abundance of seeds made it an appropriate symbol for marriage, presided over by the mother goddess Hera. Votive clay pomegranates as well as real fruits were left as offerings at tombs (possibly hung from the calyx), as well as the various sanctuaries of these divine figures. The pomegranate vessel, which died out after the Geometric Period, may have influenced the form of the standard _aryballos_ , a small spherical flask with a narrow neck used for holding perfume or oil.\n\nA rather unusual ceramic that features the pomegranate is a _kernos_ found at the temple of Hera at Samos. This unusual vessel of the sixth century BC is made up of a hollow circular tube affixed to smaller pots of various shapes that feed into it. These small vessels include the form of the pomegranate, standing out as the sole plant item, among both animal (such as a bull, shellfish, a toad and a ram) and human (a male warrior and a domestic female) shapes. The _kernos_ seems to have developed from an even older Near Eastern form: other examples of this type of vessel, although much less sophisticated (having only three feeder vessels: a pomegranate, a bull and a bird), have been excavated from sites such as the ninth-century BC Levant settlement of Tel es-Safi. The general theme seems to be one of plenty and the circular shape suggests that the whole ceramic is a representation of world order. The vessel may have been used to mix ritual liquids and grains, each being poured through the smaller vessel that functioned as a symbol for that ingredient, the strange cocktail being shared as a drink between individuals from the cup-shaped attachments. In the pomegranate vessel's case, the sacred juice of the fruit is a likely candidate.\n\nTerracotta vase from Attica in the form of a pomegranate, 8th century BC.\n\nClassical vase painting frequently places the pomegranate in scenes of a funerary nature (it should be remembered that most Greek vases of which the whole vessel survives are from funerary contexts, since they were buried as offerings with the dead). A red-figure _pelike_ illustrates the mythological figure Electra mourning at the tomb of her father Agamemnon, in front of which several offering items have been placed, including a pomegranate. In another vase scene, a figure in armour is being handed a pomegranate, the fruit reflecting the bloodiness he faces on the battlefield. The warrior, however, holds up his right hand in a gesture that seems to be declining the offered fruit, as if refusing to accept death.\n\n_Kernos_ from the Heraion of Samos, 7th century BC.\n\n_Kernos_ from Tel es-Safi, 9th century BC.\n\nIt is not only death, however, that is represented on vase illustrations featuring pomegranates. Other scenes portray women performing domestic and ritual tasks, such as wedding preparation. The pomegranate will often be depicted placed between individuals of the opposite sex as they gaze at each other, as if for the first time. Several vase paintings depict the winged god Eros with a pomegranate, linking the fruit with the personification of desire itself. The fruit also features commonly as a stylistic border decoration on vases.\n\nIn emotive grave stele from Greek cemeteries, the dead are either represented solo in a contemplative stance or saying goodbye to their living loved ones, who lose composure as they reach out, unwilling to let go. Both male and female dead are often depicted holding pomegranates. Examples of this include the stele of a certain Polyxena and of Megakles. On the fifth-century BC Harpy Tomb of Xanthus in Turkey, one frieze represents a procession of women towards a seated female who holds a pomegranate in one hand and a pomegranate flower in the other, which she raises to her nose. Each of the women holds various attributes. One of them carries the same fruit and flower as the seated woman. While the seated figure may be a deity, it is more likely that this is an heroic representation of an individual who was buried in the tomb being brought offerings by her living family members. The strong association between the pomegranate and such scenes of death suggests that the fruit was considered a chthonic symbol, which would ensure safe transition to the underworld for the deceased. After all, Persephone was restored to life for at least some of the year.\n\nRed-figure _pelike_ , 4th century BC. To the right of the mourning Electra, a pomegranate sits on her father's grave.\n\nBlack-figure amphora, 6th century BC. A warrior declines the offering of a pomegranate.\n\nIn a rare example of surviving Greek fresco paintings from the Lucanian tombs at Paestum, depictions of large red pomegranates are found in abundance. They hang rather conspicuously above a variety of scenes: the laying out of the dead body, people mourning the deceased, women doing textile work, agricultural labour, chariot racing, hunting, warriors in combat, banqueting and images of mythical creatures. The pomegranate fits well among the funerary rite scenes, but what of the domestic activities and sporting contests? The pomegranate may be physical, literally hung during the period of mourning, funerary games and feasts in memory of the deceased. Or it may be a symbol, repeated throughout this variety of different scenes, that hints at a belief in the promise of a continued pleasant existence after death. The fruit demarcates not real-world events, but heroic activities performed in the underworld, the pomegranate being the one constant that guides the deceased through these pursuits. It is likewise found elsewhere in Italian tomb painting, in the even older Etruscan cemetery at Cerveteri, where branches of the fruit held by women feature on decorative plaques.\n\nSeveral ancient Greek writers describe tombs on which the pomegranate grows of its own accord, as a living memorial of the dead buried there. Pausanias writes of how a pomegranate plant germinated on the tomb of the hero Menoikeus, who, according to legend, killed himself to fulfil a prophecy that guaranteed the survival of the city of Thebes at the price of his life. The bloody interior of the fruit reflects his self-sacrifice that saved the city. The same phenomenon occurred at the graves of two other individuals buried at Thebes, one of whom died defending the city, the other while attacking it. These were the brothers Eteokles and Polyneikes, who killed each other in combat. The Furies planted pomegranate plants on their tombs, which Philostratus' _Images_ tells us had split-open fruits that were forever dripping blood. The pomegranate is used to imbue real physical spaces with a mythic tradition.\n\nWest frieze from the Harpy Tomb of Xanthus, Turkey, 5th century BC. Note that two other individuals on the south (a male) and east (a female) sides of the tomb likewise hold pomegranates.\n\nRed-figure column krater, 4th century BC. Eros, wearing a pomegranate on his hip, approaches a woman. |\n\n---|---\n\nPomegranates feature in ancient Greek sculpture, largely in that civilization's Archaic period (eighth century BC to 480 BC), when they often appear held by kore statues. These statues portray young women with a braided hairstyle wearing heavy drapery. They are most commonly found in a sanctuary context. Kore statues typically hold an offering of either a flower, a fruit or a bird. Several examples hold a pomegranate: the notable Berlin Kore, for instance, is represented holding the fruit turned on its side with its calyx facing the viewer. Kore statues acted as an advertisement, to display and celebrate a family's daughter under the guise of a dedication to the gods. These sculptures were not portraits but the ideal. Indicating a girl's eligibility for marriage, they were used as vehicles for the creation of alliances with other elite families. The attribute of a pomegranate would accentuate her beauty and desirability. Like most sculpture from the classical world, these would originally have been painted vibrant colours. Red pigment poignantly survives on a pomegranate held to the breast of a now headless kore statue, Acr. 593, from the Athenian Acropolis. The kore statues were also sometimes used as grave markers. It is plausible that those of these statues that hold pomegranates indicate a girl who died young, portraying her in the likeness of Persephone. A kore portraying a girl named Phrasikleia holds out a single pomegranate aril and wears a necklace with beads shaped like the fruit. A heart-wrenching inscription on the statue reflects that she died before she had the opportunity to bear fruit: 'Grave marker of Phrasikleia. I shall be called a maiden forever, having been allotted this name by the gods in place of marriage.'\n\nDetail of Lucanian tomb painting showing a pomegranate and dog.\n\nLucanian tomb painting depicting a victorious warrior surrounded by floating pomegranates, 5th century BC.\n\nHowever, it was not only female statues that held pomegranates. The famed ancient Greek wrestler Milo of Croton had a statue of himself holding a pomegranate set up at Olympia. This was a reflection of Milo's position as a priest of Hera, and of a legend circulating about the wrestler that he was able to hold a pomegranate without it splitting open while competitors tried to pry it from his fingers.\n\nHomer recounts in the _Odyssey_ that the legendary eastern king Alcinous grew pomegranates in his garden. This is the beginning of a preoccupation in the accounts of ancient Greek writers with representing the East, several making use of a pomegranate in their construction of the non-Greek Other. These ancient historians saw Eastern luxury as decadent and effeminate, assuming that soft countries breed soft men. The father of history, Herodotus, tells us that the famous 10,000 'Immortals' of the Persian army (the unit's name stemming from the custom that every killed, wounded or sick member was immediately replaced with a new one, thus maintaining a constant number) carried spears which terminated in a golden or silver pomegranate at their lower end instead of a spike. This conspicuous display of wealth seems rather impractical for the battlefield. The multi-seeded and Eastern-originating pomegranate is being exploited to symbolize the immense and multi-ethnic Persian Empire. Herodotus has Darius the Great use a pomegranate to praise his most loyal subjects. As the Persian king sits eating a pomegranate, he is questioned as to what he wishes to have as many of as the number of seeds in the fruit. Darius replies 'Men like Megabazus,' complimenting his trusted military commander. Similarly Plutarch tells us that the Persian king Artaxerxes praised a man who gave him a single pomegranate, saying: 'By Mithra, this man would speedily make a city great instead of small were he entrusted with it.' The idea here is that the small pomegranate is used to represent the large one, its inner quantity of seeds making it far exceed its size.\n\nThe pomegranate was valued in Greek medicinal thought. The father of medicine, Hippocrates, recommended the fruit for a number of ailments, including the treatment of eye infections and morning sickness and to aid digestive health. In later Roman times, Pliny the Elder also noted the pomegranate's potential, recording many remedies that made use of the plant, from the treatment of scorpion stings to epilepsy. Pliny even noted that pomegranate rind could be burnt to repel gnats or ground down for use in the making of Roman perfume. The pomegranate is said to have arrived in Rome from Carthage, and was thus known to the Romans as _mala punica_ (the Punic apple). In Roman art the pomegranate is most commonly found among the fruits of the wreaths signifying plenty that can be found on the sides of tombs and altars, including the famed Ara Pacis of Augustus. The pomegranate may also be found as a design on ceramic oil lamps of the Roman Empire, as a motif type from Cyprus demonstrates, and another even further east in Israel.\n\nEven into late antiquity, the strong association between the pomegranate and female divinity was not forgotten. A sixth-century AD Byzantine tapestry from Egypt made of wool and linen depicts a very pagan theme, although by this time Christianity had become more predominant. It is a late representation of Hestia, the Greek goddess of the hearth. The goddess of the Hestia Tapestry wears a pomegranate headdress and earrings, emphasizing her role as the Polyolbos ('rich in blessings'). The fruit, with its many seeds, is appropriately evoked in this image of abundance. The blessings that Hestia gives out are in the form of pomegranates, each standing for a virtue such as wealth, prosperity or excellence. A Byzantine arch of the Monastery of Apa Apollo in Egypt, from roughly the same date, likewise features the pomegranate as the epitome of plenty, growing upon the running vines that form the arch. The theme is classical, although unlike the tapestry, the context is purely Christian. The pomegranate would come to play an important role at the Byzantine court during this late antique period. Part of an empress's marriage and coronation involved a ceremonial procession in which she was accompanied by dignitaries who carried three jewel-encrusted marble pomegranates.\n\nCypriot pomegranate lamp, 1st century AD.\n\nThe Byzantine Hestia Tapestry, made in Egypt in the 6th century.\n\nThe pomegranate still has very strong symbolic value in modern Greece, demonstrating just how enduring ancient ideas and traditions can be. The fruit thus continues to be a favourite decorative motif on houses around the Mediterranean, as well as on textiles and ceramics. The historian Efthymios G. Lazongas notes that in a local tradition from the Greek town of Epidaurus, 'peasants break open a pomegranate on the ploughshare of the plough and mingle the seeds of the pomegranate with the grains to be sown in order to achieve the desired prosperity.' The pomegranate still has funerary connotations in Greece today, as part of a _koliva_ offering prepared in commemoration of the dead. This is a sweet boiled wheat dish that contains pomegranate arils. Following a Greek wedding, a pomegranate is traditionally blessed and then broken on the threshold of the couple's home in the hope of a long and fruitful marriage. This also may occur at other times of transition, such as the purchase of a new house or at the beginning of a New Year.\n\n## 3\n\n## Jewish and Islamic Pomegranates\n\nOn its lower hem you shall make pomegranates of blue, purple and crimson yarns, all around the lower hem, with bells of gold between them all around \u2013 a golden bell and a pomegranate alternating all around the lower hem of the robe.\n\nExodus 28:33\u20134\n\nPomegranates in Ancient and Modern Judaism\n\nThe above quotation is part of a descriptive passage from Exodus that records how the God of the Israelites commanded the way in which high priests' garments were to be made. The pomegranate figures prominently in this design. But why would a fruit with such deep roots in pagan fertility cults decorate the garments of priests who worshipped a male monotheistic deity? Their appearance is probably a reference to sacrifice: the pomegranate is an elegant way of representing the blood of the offering, while the bells (also the shape of the pomegranate flower) recall the sounds that would accompany the religious experience. After all, the pomegranate plant already had an important role to play in sacrifice: a branch of pomegranate wood was traditionally used to make the skewer on which a paschal offering was roasted. The pomegranate garment is also likely to be a patriotic display of Jewish heritage, for in Deuteronomy 8:8 we are told that the pomegranate is one of the plant species identified with the Land of Israel. The fruit was brought to Moses by scouts in a demonstration of the fertility of the Promised Land, and it was under a pomegranate tree that the first king of Israel, Saul, sought shade from the harsh sun (1 Samuel 14:2). The fruit became inextricably linked with Jewish identity and experience, the Midrash reflecting: 'Thus were Israel in Egypt as a heap of stones... once they went out they became like a grove of pomegranates. All through the ages when mankind looked upon Israel they were praised.' The Hebrew Bible cites many localities in the Holy Land that came to be named after the fruit, such as Ain-Rimmon (Spring of the Pomegranate) and Sela Rimmon (Rock of the Pomegranate). The death of a pomegranate tree in ancient Jewish culture was considered to be a negative omen, indicating that God was unhappy. Today the pomegranate features as a ubiquitous motif on many items of Judaic art, from jewellery and utensils to protective, prayer-filled mezuzahs affixed to household doorways.\n\nJewish tradition teaches that the pomegranate is a symbol of righteousness, knowledge and wisdom because it is said to have 613 seeds, each representing one of the 613 _mitzvot_ (commandments) of the Torah. Torah scrolls are traditionally decorated with _rimmonim_ , a pair of pomegranate-shaped finials, usually made of silver, that are placed over the wooden scroll handles when the text is not in use. It is common practice among Jews to eat pomegranates on Rosh Hashanah, Jewish New Year, in reference to the fruit's erudite nature, but also as a guarantor of the productivity of the New Year. On this occasion a prayer is recited over the fruit: 'May it be Your will O Lord our God and the God of Our Fathers, that our good deeds will increase like the seeds of the pomegranate.'\n\nAlthough _rimmonim_ Torah covers are found in a variety of shapes today, this example by Itzhak Luvation retains a traditional pomegranate form.\n\nThe pomegranate is also included among the fruits presented for Tu Bishvat, a festival that celebrates trees and environmentalism. There is a saying in the Talmud that 'Even thy empty-headed ones are full of good deeds as a pomegranate is of seeds.' The Talmud also gives advice for the dream interpretation of pomegranates: 'Both Abiya and Raba dreamt about a ripening pomegranate. Abiya solved the dream \u2013 Thy transactions shall flourish like a pomegranate (meaning they shall be as many as the seeds of one).'\n\nThe Jewish physician Rabbi Shabtai Donolo noted the importance of the pomegranate in medicine, suggesting a gargle of pomegranate juice mixed with wine as a remedy for laryngitis. He advised that earache could be treated by grinding up dried pomegranate rinds, mixing with water and pouring this solution into the affected ear.\n\nThe pomegranate appears on ancient coins of Judea. It is one of only a few images that appear as a holy symbol on currency, the 'crowned' fruit taking the place of a crowned ruler's head. King Solomon is in fact alleged to have designed his crown based on the crown-like calyx of the pomegranate. The modern state of Israel still uses this pomegranate motif on coins and stamps.\n\nDetail of mosaic in the 6th-century Maon Synagogue.\n\nOne pomegranate in particular, made of hippopotamus ivory that originates from the age of biblical Israel, has caused a stir in modern times. The 44-millimetre-tall pomegranate bears an inscription in Hebrew circling around the neck of the fruit: 'holy to the priests, belonging to the House of Yahweh'. For this reason, it has been speculated that it originated from the sceptre of a high priest who resided at Solomon's Temple, which was said to use the fruit motif heavily in its decorative programme: biblical accounts tell us that two hundred representations of the pomegranates were depicted on each of the two bronze pillars, known as Boaz and Jachin, that marked the entrance to the holy space. The unprovenanced ivory pomegranate itself is generally accepted as an authentic thirteenth-century BC artefact. Uninscribed ivory pomegranates of the same style have been found at the Israeli site of Lachish and in Cypriot tombs. It is the inscription that fuels ongoing debate, since it is considered to be a modern forgery applied to a genuine artefact. The historian Mary Abram sums it up best when she comments: 'While a validated inscription might confirm its use in a temple setting, the fact that alleged forgers used the already-ancient ivory pomegranate to simulate a temple artefact supports its recognition (even by criminals) as a sacred symbol... no fruit but the pomegranate best combines the diversities of sensory pleasure, earth's seasonal cycles, worldly kingship, and holiness.' It is for this reason that the pomegranate is archaeologically represented in the Jewish art programmes of many ancient synagogues, notably in reliefs and mosaics at Capernaum, Hamat Tiberias, Beth Alpha and Maon.\n\nSome scholarly traditions have identified the pomegranate as the forbidden fruit of the Garden of Eden. It is the pomegranate that best represents decadent and sensuous beauty in passages from the Song of Songs, the narrator at one point flattering his beloved with the simile 'Your cheeks are like halves of a pomegranate behind your veil.' Another story in the Talmud also equates the pomegranate with temptation. This tale concerns Rabbi Hiyya ben-Abba who, being a pious man, prays daily for respite from temptation. His wife overhears him and wonders why he prays so, given that he has abstained from being with her sexually for many years. She thus decides to test him. The wife decks the disguise of a prostitute, walking to and fro in front of the rabbi in order to arouse his attention. She remarks that if the rabbi desires to be with her, her price is a pomegranate retrieved from the highest branch of a certain tree. The rabbi immediately climbs the tree, picks the pomegranate and brings it back to her. Only now, the wife reveals her true identity. Filled with shame at being led into temptation, the rabbi fasts for the rest of his life, until death. The pomegranate functions in the tale as the symbol of his transgression, marking his movement from the spiritual towards more bodily concerns.\n\nPomegranates for Muslims\n\nIslamic tradition shares with Judaism a great reverence for the pomegranate. Those who eat a pomegranate are considered to have their hearts cleansed and filled with _n\u00afur_ (light), rendering them free from sin and able to repel Satan's temptations for forty days. As representations of God, Muhammad and other religious personages are forbidden, stylized images from nature are common in religious art and architecture, and the pomegranate is a popular emblem. As well as being a favourite in manuscript border ornamentation, the pomegranate is a common iconographic trait of Islamic pottery. The styles of pomegranates on different types of Islamic pottery are demonstrated by three very different examples of vessels, all of which share the pomegranate motif. A common Islamic vessel type is a bronze ewer, characterized by its globular belly, tubular neck and slender beaded handle topped with a thumb-piece cast in the form of a pomegranate. The pomegranate also features on many painted Islamic blue-on-white glazed ceramics, both vessels and tiles. A third type is a unique mould-made vessel adorned with relief decoration featuring images of growing pomegranate trees beneath an upper register bearing a line of Qur'anic poetry. The pomegranate likewise enjoys popularity in Middle Eastern textiles, commonly appearing on both garments and carpets. The reason for the respect given to the fruit may be the result of a Hadith which claims that the Prophet Muhammad told his companions that each pomegranate contains an aril from heaven itself: 'There is not a pomegranate which does not have a pip from one of the pomegranates of the Garden (Jannah) in it.' The consumption of arils for the pre-dawn meal of _suhoor_ during Ramadan is thus considered to be an appropriate prelude to fasting.\n\nMould-made ceramic bowl from Susa, Iran, 7th\u20138th century.\n\nBronze ewer from Iran, 11th century.\n\nBlue and white ceramic bowl with a central motif of a pomegranate, Iran, _c._ 1500.\n\nLike Judaism, Islamic tradition attributes to the pomegranate a dangerous, seductive side. The transgressive effect of the fruit is found in a fable about the eighth-century AD caliph Yazid II. Being particularly fond of his singing slave girl Hababa, Yazid shuts himself away with her and orders that he not be disturbed. Together they spend the day eating until Hababa chokes to death on a pomegranate seed that had been playfully thrown into her mouth by the caliph. Abd al-Majid ibn Abdun's retelling of the tale speaks of the seed as being 'preordained'. Yazid is heartbroken and dies a few days later. The seductive nature of the fruit makes it a suitable means for the downfall of the caliph, who has rejected duty in favour of the company of a low woman. Although the Prophet Mohammad advised pregnant women to eat pomegranates if they wanted beautiful children, an anecdote concerning All\u0101mah Majlisi, the compiler of a Hadith collection, notes that the fruit needs to be from an appropriate source. When he was a child, All\u0101mah Majlisi was taken to a mosque with his father. While his father was inside praying, the young boy remained in the courtyard. On finishing his prayer, the father discovered that his son had poked a needle into a water skin that belonged to the mosque, and all the water had spilled out. On hearing of her son's behaviour, All\u0101mah Majlisi's mother related to her upset husband how this incident had came about. Years before, while she was pregnant with him, she had gone to a neighbour's house and seen a pomegranate tree. She made a prick in one of the pomegranates and sucked out the juice. It was this act that led to the later occurrence at the mosque. The pomegranate has the potential to be a powerful omen both in Islamic and Jewish tradition.\n\nIslamic carpet from the Silk Road settlement of Kashgar, 18th\u201319th century.\n\n## 4\n\n## Medieval Pomegranates\n\nGo to, sir; you were beaten in Italy for picking a kernel out of a pomegranate: you are a vagabond and no true traveller Lafeu in William Shakespeare's\n\n_All's Well that Ends Well_ , Act II Scene 3\n\nThe spread of the pomegranate benefited greatly from the Crusades, as fruit was brought back to various European nations by many returning knights. New dishes had to be devised in order to include the fruit in their respective cuisines. The _Modus viaticorum preparandorum et salsarum_ , an Occitan cookbook of the fourteenth century AD, includes among its recipes a dish known as _raymonia_. This recipe must have been considerably popular, as it even appears in the Middle English aristocratic cookbook _The Forme of Cury_. The instructions are as follows:\n\n> If you want to make a raymonia, take hens and cook them with salted meat. And take unblanched almonds, and wash them in lukewarm water, and grind them very strongly, and dilute with hen's broth, and strain. Afterwards, take pomegranate verjuice or pomegranate wine and add it. Then boil it and add enough sugar.\n\nThe central roundel from the Hinton St Mary mosaic showing Jesus Christ flanked by pomegranates, early 4th century AD.\n\nThis dish is a revision of the Arabic _rummaniya_ , a chicken bruet with pomegranate juice, ground almonds and sugar. However, it was much more than recipes that were adopted and adapted in the medieval period, as ancient notions were used to create new and powerful symbols surrounding the pomegranate.\n\nAs the medieval Christian world drew on many pagan traits adopted from their classical predecessors, the pomegranate retained the same ideology, borrowed from the Persephone myth. The fruit became associated with the Resurrection of Christ and the eventual resurrection of his believers, instead of the annual resurrection of crops. The central bust of Jesus in the Hinton St Mary mosaic, discovered in the county of Dorset, England, uses two pomegranates in conjunction with the Chi-Rho symbol to form a somewhat unconventional halo. It is from a Roman villa of the fourth century AD and is the earliest-known image of Christ in Britain and the only known portrait of Christ used in a mosaic pavement. To an audience of recently converted Roman pagans, the image of pomegranates would immediately conjure images of rebirth. The many seeds of the pomegranate protected inside a single rind came to stand for the Church itself, which unified people in faith. The pomegranate and its symbolism would survive in later icons and religious paintings. In the common scene of Mary holding the infant Jesus, one of the pair is sometimes represented holding a split pomegranate, or both have pomegranate seeds fall upon them. The most famous example of this type is by the Italian Renaissance master Sandro Botticelli. The open pomegranate, with its seeds bursting forth, is likened to Christ bursting forth from the tomb. The same themes are evoked in a fourteenth-century ivory. Pomegranates sprout from the central panel of a triptych in which the Crucifixion is depicted, forecasting the Resurrection.\n\nSandro Botticelli, _Madonna of the Pomegranate_ , 1487, tempera on wood.\n\nThree pomegranates top this 14th-century ivory triptych of the Passion of Christ. From the central split pomegranate pour arils that form a decorative border of the entire ivory.\n\nSouth Netherlandish Unicorn Tapestry, 15th century. The beast is chained to a pomegranate tree with wounds that drip pomegranate arils in place of blood. The meaning of the letters AE hanging from the tree branches are unknown, although it is speculated that these may be monogram initials of a pair of newlyweds.\n\nThe _Geoponica_ is a tenth-century Byzantine encyclopaedia of land cultivation, which includes mention of a practice involving pomegranates during this period. The text states that ancient Africans believed that a pomegranate branch was so powerful that any wild beast would become terrified in its presence. The author of the _Geoponica_ thus advises that a pomegranate branch should be laid at the entrance of the home for the protection of the household. This notion is based on an earlier Roman belief, for Pliny in his _Natural History_ claims that a pomegranate branch will repel snakes.\n\nOne mythical beast that the pomegranate had an effect on was the unicorn, popularized in medieval art as a mythical symbol of Christ. Known for its purity, the unicorn could only be captured by a virgin, after which it could be tamed by being tied to a pomegranate tree. A Netherlandish tapestry from the fifteenth century shows a wounded unicorn bleeding pomegranate seeds while chained to a pomegranate tree, further emphasizing the fertility potential already signified by the beast's prominent horn. Being symbolic of marriage, the image of the unicorn tied to the pomegranate tree, with the red seeds of the fruit spilling on him, functions as an allegory for the bridal sheets.\n\nThe twelfth-century historian William of Malmesbury took great interest in recording the ways in which people of the Middle Ages died. No anecdote he tells is stranger than one which includes a delightful image of a pomegranate. William relates that it is 'well known' that if any man were bitten by a leopard in Asia, a group of mice would subsequently plan an attack on his boat, with the objective of urinating on his wounds. This, according to William, would cause the man to die. The thousands of mice would row to meet their victim 'in the rinds of pomegranates, the insides of which they had eaten'. This is not the only instance of the pomegranate rind being associated with seafaring. Much more recently, in the nineteenth century, the archaeologist Sir Austen Henry Layard, while exploring ancient sites in Western Asia, observed that raft men on the Tigris preserved the suppleness of the skins used to build their boats by rubbing them with pomegranate rinds. William's tale probably refers to this technique of using the rind for buoyancy, a practice which most likely had much older origins.\n\nPomegranates featured in medieval architecture, the fruitfulness of the seeds making it a fitting symbol for evoking the Virgin Mary in church designs. The architectural example of the Pilastri Acritani demonstrates this. Its two pillars are carved with symmetrical and stylized pomegranate plants, each topped with a single fruit growing from vases. Now standing in Venice, they were traditionally regarded as trophies of the Venetian victory over the Genoese at Acre, Israel, in the thirteenth century. This however was proved false in 1960, with the discovery of the sixth-century church of St Polyeuctus in Istanbul, which bore identical stylistic traits. So how did they get to Venice? The pillars were probably part of the booty taken back after the sack of Constantinople by the Venetian-led Fourth Crusade in 1204. The church of St Polyeuctus, from which the pillars originate, dates from the reign of Justin I and was commissioned by the noblewoman Anicia Juliana. The choice of using pomegranates heavily in its architectural design is probably in tribute to her femininity. The imagery on the Pilastri Acritani is probably also an evocation of the Solomonic Temple, which, as discussed in Chapter Three, featured pomegranates prominently in its design.\n\nThe Pilastri Acritani in front of the Basilica di San Marco, originally part of the Church of St Polyeuctus in Constantinople.\n\nThe pomegranate was used to convey a message by many well-known figures of the Middle Ages. After conquering Granada in 1492 (and subsequently adding a pomegranate to her coat of arms), Isabella I is reported to have stood with a pomegranate in her hand and declared, 'Just like the pomegranate, I will take over Andalusia seed by seed.' Granada is of course the Moorish city in Spain, named after the Spanish word for our favourite fruit. From medieval times until today the pomegranate has featured prominently in the architecture of the city. Tourists exploring the city today will soon become aware of modern representations of the pomegranate on street signs, drains, church pews, fountains, bollards and fence spikes and as a logo of many shops. One of the gateways to the city's famed Alhambra is topped with the allegorical figures of Peace and Abundance alongside three split pomegranates. The coat of arms of the city likewise features the fruit.\n\nImperial crest giving the pomegranate a prominent position in the centre of the Cross of Burgundy and the Hapsburg firesteels, on the 16th-century Palace of Charles V at the Alhambra of Granada, Spain.\n\nThe pomegranate became an emblem on the coat of arms of Isabella's daughter Catherine of Aragon (1485\u20131536), the first wife of Henry VIII. Through the queen, the fruit was introduced into England at this time, and is often conjoined with the Tudor rose in iconography. The pomegranate featured prominently during the festivities held in honour of Catherine and Henry's marriage, such as in displays of gilded fruits. A field was planted with roses and pomegranates next to one another to denote the union of England and Spain. However, the pomegranate proved to be of no assistance to her fertility, and when Henry and Catherine were unable to produce a surviving male heir, the king eventually divorced her in favour of a new marriage to Anne Boleyn. Still today people honour Catherine, placing flowers and pomegranates on her grave in Peterborough Cathedral.\n\nWhen Anne Boleyn became queen, she adopted a new coat of arms showing a white falcon surrounded by Tudor roses. Anne's falcon is sometimes depicted hacking up Catherine's pomegranate (as in an illumination from page four of _Motets et Chansons_ , a music book owned by Anne). This image inspired the creation of a piece of silk embroidery by the artist Suky Best that she attributes to Anne Boleyn's mother, Elizabeth, which depicts Anne's falcon pecking at the pomegranate. Capturing the spirit of her sentiment towards Catherine, the French text around it contains Anne's personal motto: _Ainsi sera, groinge qui groinge_ ('That's the way it's going to be, however much people grumble').\n\n  | Manhole cover of a sewer in Granada, Spain.\n\n---|---\n\n  | Street sign in Granada, Spain.\n\n---|---\n\nPomegranate bollard in Granada, Spain. |\n\n---|---\n\nPomegranate fountain in Granada, Spain. |\n\n---|---\n\n  | Woodcut depicting a celebration of the joint coronation of Henry VIII and Catherine of Aragon, from a 16th-century manuscript. Henry sits beneath the Tudor Rose of England, while Catherine is under the pomegranate.\n\n---|---\n\nIn terms of textiles of the period, the pomegranate was a favoured motif embroidered on clothes during the medieval era and until the seventeenth century. It was adapted from the Islamic East, elaborate pomegranate patterns now status symbols adorning garments worn by elites of European society. Paintings show the fruit worn by both women (as in portraits of Mary Clopton and Eleonora di Toledo) and men (as worn by one of the Eastern kings in Gentile da Fabriano's _Adoration of the Magi_ ). The sixteenth-century Henry IV of France (who was known for passing the Edict of Nantes, which allowed for religious freedom, thus ending civil war between Catholics and Protestants) wore the pomegranate as a heraldic badge. He chose as his motto 'Sour, yet sweet,' comparing the nature of the pomegranate with his belief that a king should be firm but fair with his subjects. The Holy Roman Emperor Maximilian I of Austria likewise used the pomegranate as his emblem, and it is held by the monarch in a notable portrait of 1519 by Albrecht D\u00fcrer. In Maximilian's portrait, the pomegranate arils symbolize the ruler's subjects: diverse but held together under the rule of his empire.\n\n_Elizabeth Boleyns Embroidery, c. 1530 (A Fictional Embroidery That May or May Not Have Existed)_. Sewn, damaged and restored by the artist Suky Best in 2003, and presented as though it were a real artefact.\n\nRobert Peake the Elder, portrait of a woman traditionally identified as Mary Clopton wearing a pomegranate-textile dress, 16th century.\n\nHoly Roman Emperor Maximilian I holds a pomegranate as his globus cruciger in this portrait by Albrecht D\u00fcrer, 1519.\n\nMedieval medical thought held the pomegranate in high esteem. The fruit is the attribute of the fifteenth-century saint and medical doctor San Juan de Dios (St John of God), who devoted his life to the service of the poor and sick from his hospital in Granada. Statues of the saint depict him holding a pomegranate with a cross coming out of its calyx. The sixteenth century saw the Royal College of Physicians of London adopt the pomegranate into their coat of arms. The restorative fruit rests beneath an outstretched arm, gripped by a hand descending from the heavens. It is still their logo today. In medieval humour theory, the pomegranate was considered warm and moist in nature, the same traits attributed to the human male. Pomegranates were thus prescribed by medieval physicians to counterbalance phlegm and coughs, which were cold and dry in nature. When eaten before a meal, pomegranates supposedly stimulated the appetite, and were considered to have aphrodisiac properties. In medieval thought, the womb of a woman was a beast with a mind of its own, with the ability to wander around a woman's body causing ailments; for example, when it moved up into the head, it would cause headaches. The _Trotula_ , a medieval medical text, specifies that a wandering womb that had descended from the body could be restored by bathing in water that had been blended with various berries, nuts and seeds, including pomegranate arils and rinds. Throughout the _Trotula_ , the pomegranate is linked with conditions unique to the female experience, another passage recommending rubbing pomegranate on the feet to reduce the foot swelling that occurs during pregnancy. The text even notes the fruit's use in cosmetic treatments, claiming that a mixture of ground pomegranate, water, vinegar, oak apple and alum could be used to dye one's hair black.\n\nThe symbol of San Juan de Dios held by a statue in Granada, Spain.\n\n## 5\n\n## Pomegranate Production and Culture Today\n\nThe pomegranate is one of the sexiest foods on earth. Its crimson shade is the colour of desire. They're messy and sticky, and because there's no other way to eat a pomegranate than with your fingers, the act of consuming the juicy little seeds becomes a sensuous act of play.\n\nAmy Reiley, named Master of Gastronomy by Le Cordon Bleu\n\nToday's centres of the pomegranate industry are Iran, Afghanistan, Turkey, India, Israel, Spain and the United States, with its Californian plantations becoming increasingly significant on the world stage. Spain remains the only major producer of pomegranates for commercial export in the European Union. India exports heavily to the United Arab Emirates, Oman and the UK. The fruit holds a symbolic significance in the local economies of many Middle Eastern and Central Asian countries, as well as the Mediterranean. As of 2014 Iran and India remain the world's largest producers of the fruit. For many countries today it is the symbolic rather than the economic value of the pomegranate that has resulted in it becoming such a source of national pride. This is epitomized in Mexico, where the traditional dish _chiles en nagoda_ recreates the colours of the national flag. The green is rendered with chilli, the white with nut sauce and the red with pomegranate arils.\n\nAmerica and the Western Pomegranate Industry\n\nThe pomegranate arrived in the New World via Spanish missionaries. When Hern\u00e1n Cort\u00e9s conquered the Aztecs in the sixteenth century, pomegranate trees, the symbol of Spain, were planted in Mexico as a marker of the conquest. Spanish settlers later introduced the fruit to Texas, California and Arizona in the eighteenth century. Peter Collinson wrote of his high esteem for the fruit to his Philadelphian botanist friend John Bartram in 1762:\n\n> Don't use the Pomegranate inhospitably, a stranger that has come so far to pay his respects to thee. Don't turn him adrift in the wide world; but plant it against the side of thy house, nail it close to the wall. In this manner it thrives wonderfully with us, and flowers beautifully, and bears fruit... of all trees this is most salutiferous to mankind.\n\nBy 1771 even Thomas Jefferson had planted pomegranates at Monticello, his plantation in Virginia. Pomegranate-shaped beads are found on the distinctive Native American 'squash blossom' necklace design associated with the Navajo tribe. These were first made in the nineteenth century by Navajo silversmiths who learnt this art from the Spanish. The turquoise and silver beads in the necklace (round with a calyx), which have been misidentified as squash blossoms, were in fact adapted from the trouser and shirt buttons worn by the Spanish, which were in the form of their national motif, the Granadian pomegranate.\n\nToday the best known cultivar of pomegranate, the 'Wonderful' variety, originates from California, where it was first propagated in 1896 from a cutting brought west from Florida. The 'Wonderful' is known for its juiciness and sweetness. It is fairly resistant to rind cracking, the blight of the pomegranate industry. Especially after it rains, ripe pomegranates have a habit of cracking open and turning almost inside out. In nature, this splitting of the fruit is a biological device allowing for seeds to distribute themselves.\n\nDespite the pomegranate's appearance early in America's history, the fruit remained an oddity in the U.S. until recently, an exotic food item enjoyed mainly by immigrants and Americans who discovered it abroad. The debut of the Pom Wonderful juice company in 2002 (with their successful, if not sometimes misleading, advertising campaign promoting the fruit), and the publication of research that showed the high antioxidant, dietary fibre and vitamin value of pomegranate, as well as its potential anti-carcinogenic properties, have resulted in the fruit becoming available to a much wider Western audience. Studies done recently even showed that microbicides obtained from pomegranate juice are effective against HIV, preventing the entrance of the virus into the body. The fruit has the capability to absorb radiation and has been used in cancer research. The pomegranate received the hallowed 'super-food' status in an article published in _Men's Health_ in 2008, and suddenly demand for it escalated. It is now commonly found in sauces, soft drinks (sodas), ice creams, cakes, teas, chewing gum, jellies, chocolates, salad dressings, liqueurs and the famous syrup grenadine, produced from the fruit for use in drinks. Many brands have experimented with the release of limited-edition pomegranate-flavoured varieties of their product: pomegranate-flavoured liquorice produced by Darrell Lea confectioners, H\u00e4agen-Dazs's pomegranate ice cream, pomegranate-flavoured 7UP. Pomegranate-flavoured snack products have occasionally come under criticism, as the companies that create them tend to stress the high antioxidant and nutritional value of the pomegranate in their marketing campaigns despite fizzy drinks and sweets not being health foods.\n\nThe cosmetic industry has also begun to produce an increasing number of pomegranate-based soaps and body creams, the fruit being known for its sun-protection, anti-ageing, hydration and anti-inflammatory properties. A company devoted entirely to pomegranate skincare products, Pomega5, was established in California in 2005. The fruit retains its ancient connotations of sensuality in the modern market, pomegranate-flavoured personal lubricant being available from several companies (with charming brand names like 'Wet' and 'Sliquid'). Notable designers Jo Malone and Marc Jacobs have even produced pomegranate-scented lines of their fragrances. The desire for pomegranate flavouring has thus resulted in a large and growing market, which often cannot meet demand, for juice concentrate. Sixteen thousand acres of pomegranate plantation currently exist in California, but with the American pomegranate industry steadily on the rise, acreage is anticipated to increase considerably in coming years. The annual fresh harvest for the area currently yields about U.S.$60 million worth of fruit. Today the U.S. leads the world in increasing consumption of the fruit per capita: in 2002 Americans consumed ten times as many pomegranates annually as they did two years earlier in 2000. The USDA Agricultural Research Service at Davis, California, has a programme of distributing free cuttings of an assortment of cultivars to members of the general public interested in trying to grow their own pomegranate plant. They hold tasting events in which different pomegranate varieties are ranked on their flavour.\n\nThe Afghani Pomegranate Industry\n\nAnother country with a developing pomegranate industry, this time in the East, is Afghanistan. Recent war and political turmoil in the country have greatly disrupted this industry and had a serious impact on exportation of the fruit to an international audience. However, 2008 saw the beginnings of a re-emergence of the pomegranate industry in Afghanistan. In 2008 Afghanistan hosted its first international pomegranate fair in the hope that its farmers would become more renowned for producing the fruit than for opium poppy production. The fair has since become an annual event. Loren Stoddard of the American agency for international development U.S. Aid, which sponsored the fair, said that 'We want one product that could be the symbol of the new Afghanistan that will lead onto other products.' As part of the 2008 fair a tribal meeting covering the entire Nangarhar Province was called, with two hundred elders who agreed to finish poppy cultivation and switch to growing pomegranates. The tribal elders conducted a symbolic ceremony in a farmer's field, which they cleared of opium poppies, planting the first pomegranate tree saplings in their place. The year 2008 also saw increasing international demand, which resulted in the local market prices rising from 55 U.S. cents a kilogram to $1.60 a kilogram within a twelve-month period. The 2010 Afghan-Pak Trade and Transit Agreement has assisted the exportation of the pomegranate to its major buyers, permitting Afghan traders to export local agriculture produce to Indian and Pakistani markets via the Wagah border post. Of his country's fruit, the Afghani agricultural minister Mohammed Asif Rahini believes, 'there are other countries producing pomegranates but no other country can compete with ours and it's God's blessing that we have this quality.' The country now exports up to 80,000 tons of the fruit a year. The extent to which the pomegranate has come to symbolize this region of the world today has resulted in the fruit lending its name to the Middle Eastern affairs blog of the magazine _The Economist_.\n\nProduction Methods\n\nPomegranate plants generally will not grow fruit for the first three to five years, and for this to occur pruning is necessary, as fruit are produced at the tips of new growth. Pomegranate trees are capable of self-pollination, since they have a hermaphroditic flower. However, only 45 per cent of flowers on pomegranate trees that pollinate themselves will produce fruit. There is a great advantage in cross-pollinating with a second plant, as this increases fruit yield to 68 per cent. Wind is not an effective carrier of pollen, as it can be with other plants, and thus one must rely on either insects (mainly bees) to spread pollen or hand-pollination. This is usually achieved by applying a brush or swab to each yellowish, pollen-covered anther of the flower. The brush is then used to transfer the pollen grains to the sticky, receptive stigma on a flower of the desired pomegranate plant. Pollinated flowers lose their petals and mature into clusters of fruit that ripen within seven months. Productive plants can yield as many as 10 tons of fruit per plantation acre.\n\nThe pomegranate generally loves heat and is at optimum production in an arid climate. Pomegranate trees are reasonably hardy, and are fairly frost-, flood-, saline- and drought-tolerant. However the fruit can still suffer as a result of these factors, which may affect the taste, increase fall-off or stunt the growth of plants. The pomegranate plant has a xylopode, an underground stem that stocks nutrients, which is able to restore the bush if it dies back after harsh environmental conditions. Some cultivars are more adaptable to environmental conditions than others, such as the 'Agat' variety, which is capable of thriving in the snow of Russia. The people of Uzbekistan, to prevent their pomegranates from dying during their harsh winters, pile dirt over the bushes until they are completely buried and then wait until the following spring to unearth them. Other methods used in Europe to prevent pomegranates freezing include growing them in a polytunnel to keep the plants at a constant temperature. The pomegranate will grow in a variety of soil types, be they black, sandy, lime-rich or dry and rocky. It is best cultivated in alkaline soils, although those with heavy clay content may lighten fruit colour. Several insects pester plant growth, including _Virachola isocrates_ , which has come to be known as the pomegranate butterfly. It lays egg larvae on the plant, which when hatched burrow into the fruit.\n\nOnce the pomegranate is cut (not picked, as this will damage the plant) from the tree, it will cease ripening. Each fruit, which is in fact classed as a berry (defined as a fleshy fruit that encases seeds, produced from a single ovary), can contain 200 to 1,400 seeds. Often the fruit is grown for the express purpose of drinking rather than eating the arils. The traditional method of juicing in Iran resembles making wine: the cut-open fruits are stamped on in a clay tub by a person wearing special shoes. Popular Western methods for juicing the fruit include the use of a basket press or hydraulic extraction. The juice is usually filtered before consumption, due to the high tannin content of the fruit. The fruit is sometimes fermented to make pomegranate wine.\n\nDr Gregory Levin\n\nIf one man truly deserves a place in the history of the pomegranate, it is the Soviet scientist Dr Gregory Levin. Levin, who is somewhat of a hero in pomegranate circles, devoted his life to the study of the fruit from the agricultural research station of Garrigala in Turkmenistan. For the forty years he was at Garrigala, from 1961 to 2001, he achieved the world's largest pomegranate germplasm collection of 1,117 varieties. Levin went on hikes around Central Asia and the Caucasus in search of wild pomegranate varieties. Pomegranate varieties were also sent to Levin from contacts abroad, including Egypt, Algeria and the Himalayas. Levin bred the wild pomegranates in order to create new cultivars, using selection and mutation to develop desirable qualities. Frost-resistance, sweet flavour, high juice content, softness of seed and rind-cracking were particular concerns. Studies included the preservation and growth of pomegranate seeds stored in liquid nitrogen. He sent seedlings of the varieties he developed to a number of countries, including the U.S., where they are still grown today.\n\nHowever, his story became one of a paradise lost. The Eden that was Garrigala started its demise with the breakdown of the Soviet Union:\n\n> When Turkmenistan became an independent country in 1991, new governmental agencies began supervising us, replacing the USSR. They changed the name of the station. The new government of Turkmenistan apparently did not need science, and in particular, punicology \u2013 the study of pomegranates. They gradually stopped financing the station.\n\nThe pomegranates at Garrigala began dying from drought. For lack of a pump, workers carried water in cans from a nearby river, and even used sewage water to irrigate the fruit. In 2001 Levin was forced to immigrate to Israel. He discovered a few years later that his pomegranate plantations had been destroyed to make way for vegetables, on order of the Turkmenistan Ministry of Agriculture. Some of Levin's pomegranate cultivars were grown in his new home of Israel, where the water-saving technique of covering the plants in nets to reduce evaporation was utilized. But visiting the Israeli plantation in 2003 was not a positive experience for him. The pomegranate varieties that he developed, which were producing fruit with positive qualities, had been re-identified with numbers rather than the names he had given them. This limited researchers abroad, particularly in the United States, from having access to propagate the desirable pomegranate cultivars that Levin had written about. Today Levin voices his concern for the depleting numbers of wild pomegranates that are becoming endangered due to human pressures, stating that we must\n\n> preserve the biodiversity of the pomegranate. Cultivated varieties of pomegranate are artefacts, creations of civilization and evidence of a very long, incredible history, the way a Rafael [sic] or a Brueghel painting or a Bach fugue is an artefact to protect and preserve for the future.\n\nNurse Pomegranate\n\nIn his memoirs, Levin wrote of the pomegranate's use as a health tonic in his homeland. Pomegranate juice was administered to astronauts, miners, submariners and pilots in the USSR to sustain workers. Even the intergalactic research monkeys on Soviet satellites were fed a mixture of pomegranate and rosehips to maintain their health. We have seen that the pomegranate has been valued for its high vitamin content and medicinal traits from ancient times, and the restorative quality of the fruit in myth and legend has considerably influenced the perceptions that surround the pomegranate in medicine (and probably vice versa). Folk medicines hold the pomegranate in reverence as a form of treatment and prevention for a wide range of ailments; it is thought to accelerate wound healing as well as lessen pain and inflammation, and to help with diabetes, headaches, jaundice, cardiovascular disease, nausea, weight management, nosebleeds, asthma, gingivitis, ulcers, haemorrhoids, depression and respiratory infections, to name just a few. Certain properties are specific to particular pomegranate varieties. For example the Kara-Nar, a black fruit, is used to treat dysentery and vitiligo. In traditional Indian Ayurvedic medicine the pomegranate is a recommended food, its sour-sweetness establishing balance in the bodies of people who are ruled by the hot and fiery Pitta dosha type. It is commonly used in Indian cuisine in the form of medicinal Anardana powder, made from dried and ground pomegranate seeds. Ayurveda utilizes the fruit as a treatment for numerous conditions, such as using pomegranate juice eye drops to prevent cataracts. Not all research in the medicinal effects of the pomegranate has been positive, however, as consumption of the fruit is also known to interfere with certain pharmaceutical drugs.\n\nOne controversial use in folk medicine that has sparked scientific experimentation and much spilling of ink is that of the pomegranate's application as a contraceptive and abortifacient. Both taken orally and applied directly on the vagina, the pomegranate is said to trigger contractions in the uterus. One scientific study from the 1970s attempted to demonstrate this in animals. Female rats, fed pomegranate, had a 72 per cent decline in fertility rates, while guinea-pig subjects had a 100 per cent reduction. A forty-day withdrawal from pomegranate consumption saw the fertility of both species restored to normal. A 2003 report from Chhattisgarh, India, notes the use of pomegranate for birth control. The seeds of the fruit are ground into a powder and mixed with sesame oil, then placed in the vagina at the end of the menstrual cycle. Such a vaginal suppository has been known as far back as ancient Mesopotamian times, as demonstrated by an Assyrian cuneiform text that refers to the soaking of wool in pomegranate for placement inside a woman. The pharmacological historian John M. Riddle argues that the pomegranate's ancient use in birth control supports the idea that the pomegranate is the fruit of the biblical tree of knowledge. God's displeasure over the eating of the fruit was due to its contraceptive ability. Riddle believes that the pomegranate likewise became a symbol of the Mesopotamian sex goddess Inanna\/Ishtar for its ability to control fertility, being utilized by her sacred temple prostitutes (offering sex as a conduit to the divine) so they could avoid pregnancy. The pomegranate's ability to control fertility is attributed to its high concentration of the female sex hormone oestrogen, which today is used in the manufacture of pharmaceutical contraceptive pills. This whole idea suggests that the enduring symbol of the pomegranate, with its abundance of seeds, as the benevolent provider of fertility is a front for the fruit's darker nature as one that reverses fertility. Research in this area is far from conclusive, however: some prenatal specialists advise that pregnant women avoid pomegranate consumption, while others suggest it is beneficial to the developing foetus and may prevent pregnancy complications like eclampsia.\n\nChinese vase with modelled pomegranate around the neck, 18th century.\n\nPomegranate in Asia: Japan and China\n\nThe pomegranate arrived in China in the second century BC, via the Silk Road. In ancient Chinese poetry, it is not the fruit but the pomegranate flower that is the image of beauty. Many poets compare the flower to a skirt, as Wan Ch'u of the eighth century AD does: 'Her eyebrow paint eclipses daylily hues; The red skirt causes pomegranate flowers to die from envy.' The type of pomegranate grown in China is mainly a yellow-skinned variety. Yellow is an important colour in Chinese culture, and is considered the most beautiful, being symbolic of the earth, luck and royalty. Thus the Chinese consider the pomegranate one of the three blessed fruits, the others being the citron and the peach. The numerous seeds within the pomegranate are said to symbolize the good fortune of having many male offspring, as the Chinese word for the pomegranate seed, _zi_ , is also the word for sons. It has become associated with weddings, arils sometimes being scattered on the bedcovers of the bride and groom to guarantee conception. In traditional Chinese medicine, pomegranate is regarded as a yin tonic that cools and hydrates the body. A dessert of Thai origin that enjoys popularity all over Asia is known as mock pomegranate. It is made by boiling pieces of Chinese water chestnut covered in tapioca flour and red food colouring, and serving these in a dish of shaved ice, rose syrup and evaporated milk. The red chestnut pieces look just like arils, giving the dessert its name.\n\nIn Japan, the pomegranate, which is known as _zakuro_ , does not have much of a role in national cuisine. Although it is enjoyed in the form of _zakuro-shu_ , pomegranate seeds soaked in liquor, the main significance of the fruit in Japan falls in the religious sphere. The pomegranate is associated with Kishibojin, a goddess of children and motherhood. In Japanese iconography, Kishibojin is usually shown suckling an infant while holding a pomegranate in her right hand. Pomegranate imagery adorns the temples of this goddess, who is worshipped by women who are trying to have children. People leave wooden house-shaped plaques with pomegranates painted on them, known as _ema_ , at shrines for the goddess. Mythology, however, tells us that Kishibojin was not always benevolent. She was initially a violent Hindu she-demon known as Hariti who lived on the flesh of children. She is said to have been converted and cured of her child-eating habits by the Buddha, who taught her to redirect her bloodlust by eating the crunchy and bloody seeds of the pomegranate. Another divine figure, Kujaku Myoo, likewise holds a pomegranate as his attribute. He protects individuals from both physical and spiritual poisoning, using his pomegranate to repel evil spirits. Pomegranates often feature in Japanese woodblock prints and also as netsuke, miniature ivory or wooden sculptures that act as fasteners for pouches, hung from the waist by a cord. Notable are the Japanese woodblock artist Ohara Shoson's (1877\u20131945) two prints, both of which feature the motif of a cheeky cockatoo sitting on a branch of a pomegranate tree, some of the fruit of which have split open, exposing their seeds.\n\nKujaku Myoo holds a pomegranate across his chest in this 12th-century hanging scroll.\n\nOhara Shoson, _Cockatoo and Pomegranate_ , 1927.\n\nAnother Buddhist tale uses the pomegranate to teach an important lesson. This story begins by noting that the Buddha constantly carries a small drum with him that he claims will be played on the day a person comes to him with the greatest sacrifice of all. A rich maharaja, keen to earn such a boasting right, comes to the Buddha, offering him a vast treasure. The Buddha does not sound his drum. An old beggar woman then approaches and offers him a single pomegranate. The Buddha takes the fruit and without delay begins playing the little drum. The maharaja is angered by this, to which the Buddha replies:\n\n> It is natural for a Maharaja to offer gold. But what great sacrifice is made when a hungry old women offers the pomegranate fruit to the Guru despite her hunger. She did not care even for her life and gave the fruit. What greater sacrifice can there be? It is not sacrifice to offer what is superfluous for you. True sacrifice means giving up that which is most dear to you, that which you value most.\n\nThe inner plentifulness of the fruit exceeds the outer grandeur of material wealth.\n\nCelebrating the Pomegranate\n\nVarious peoples in modern times have honoured the pomegranate in festivals. One such group are the people of Azerbaijan, who hold an annual pomegranate fair in the town of Goychay. The pomegranate festival takes place every October, combining local cuisine featuring the pomegranate with traditional Azerbaijani music and dance. A favourite dish is _nasharab_ , a savoury sauce made of pomegranate that is usually served with fish. The festival typically features competitions, such as a prize for the largest pomegranate, a pomegranate costume contest and a timed pomegranate eating\/squeezing contest. The extent to which the pomegranate is linked with Azerbaijani identity was demonstrated by its use in the official logo for the 2015 European Games, held in Azerbaijan's capital, Baku. Designed by Adam Yunisov, the logo brings together five elements of the host country's culture: water, fire, a carpet and the legendary Simurg bird are all enclosed within the shape of a pomegranate. Trees covered in pomegranates feature prominently in the decorative programme of the eighteenth-century Azerbaijani Palace of Shaki Khans. In honour of the special place of the pomegranate in the hearts of the Azerbaijani people, a large novelty pomegranate has been built in Goychay. There are many other instances of pomegranate-shaped sculptural monuments around the world. A huge gold bowl stacked with a pyramid of red pomegranates features in Shahrdari Square in Saveh, Iran. Another example of the fruit used in architecture is a huge mosaic pomegranate by the artist Ruslan Sergeev, which is on public display in Jerusalem. The mosaic medium, using hundreds of tile shards affixed to the outer skin of the fruit, effectively evokes the idea of plenty that is normally associated with the multitudinous internal seeds.\n\nPomegranate tree illustration inside Azerbaijan's Shaki Khans Palace.\n\nTehran likewise celebrates its 'ruby from paradise' in festivities. Pomegranate producers from around Iran congregate at a festival allowing for the tasting and exchange of a variety of cultivars. Works of art featuring the pomegranate are put on display, and dances incorporating the fruit are performed. The 2011 ceremony featured the world's largest pomegranate cake, decorated with representations of the fruit and weighing 800 kg (1,764 lb). An Iranian postal stamp decorated with the fruit was also released for the event. Wish trees feature at the festival, to which are attached pomegranate ornaments. From these hang scrolls containing lines of Hafiz's poetry, which pose a solution to a problem faced by an individual. This classical Persian poet of the fourteenth century once wrote that 'when love mocks, ruby tears fall heavy as pomegranates.' The beloved Persian poet Rumi likewise praised the pomegranate in the thirteenth century, advising that 'If you buy a pomegranate, buy one whose ripeness has caused it to be cleft open with a seed-revealing smile... through its wide-open mouth it shows its heart, like a pearl in the jewel box of spirit.'\n\nCelebrations at the 2011 pomegranate festival in Goychay, Azerbaijan.\n\nMosaic pomegranate by Ruslan Sergeev, Jerusalem.\n\nPomegranate sculpture in Shahrdari Square, Saveh, Iran.\n\nAnother notable festival in celebration of the fruit occurs in the 'Heart of Pomegranate Country', Madera, California. As well as cooking demonstrations, a highlight of the fair is a competition known as the 'Pomegranate Grenade Launch' which involves firing pomegranates at targets from a large slingshot. Skydivers perform an airshow in which they hold pomegranates while parachuting into the event, which is held at the Madera airport in November. In 2013 the event was advertised in a humorous YouTube video featuring the mayor of Madera rapping to a parody cover of Macklemore's 2013 song 'Thrift Shop'.\n\nPop Culture\n\nThere are several references to the pomegranate in popular culture, an appropriate prelude leading us into our next chapter on the fruit in art. This list is continually growing as more people are inspired by the unusual nature of the pomegranate.\n\nPop singer Katy Perry uses a pomegranate as lipstick in the music video for her song 'Roar'. Adam Lambert's music video for the song 'Better than I Know Myself' features the musician crushing a pomegranate, causing the blood-like juice to spew from his hands. The pomegranate syrup grenadine is paid homage to in Lana del Rey's song 'Bel Air', and the character Michael Scott in the American TV show _The Office_ claims straight grenadine as his drink. Anne Baxter as Nefertiri in the 1956 film version of _The Ten Commandments_ attempts to seduce Moses, describing her own lips as 'moist and red like a pomegranate' in contrast to his wife's lips, which are 'chafed and dry as the desert sand'. The comedy film _Her Alibi_ (1989) features a struggling mystery novelist who writes the lines 'Her breasts squashed against him like ripe pomegranates,' creating a sensual scene using the same kind of phraseology that we have seen in Arabic folklore.\n\nThe pomegranate has even impacted the everyday lexicon. The French word for pomegranate, _grenade_ , was given to the hand-tossed explosive weapon, evoking the seed-scattering properties of our favourite fruit. The red seeds reflect the destructive blood-spilling caused by weaponry and war. The slang term 'pom' or 'pommy', used by Australians and New Zealanders to describe the British, is a contraction of pomegranate, and refers to the speedy reddening of the skin caused by sun exposure that occurs in people of English origin. The term may alternatively have developed from the use of the word 'pomegranate' as Australian rhyming slang for 'immigrant'.\n\nThe pomegranate features in the artwork of two cards from the Rider-Waite tarot deck, first published in 1910 and still the most popular deck available one hundred years later. In the High Priestess card a woman stands in front of a curtain that is covered in pomegranates in reference to the fact that she is seated inside Solomon's Temple, marked by the columns featuring the letters B and J. These are the pillars Boaz and Jachin (here representing light and dark), which, as discussed in Chapter Three, were also noted in the Bible to have been encrusted with pomegranates. The Empress card features a woman reclining amid a field of grain, wearing a robe embroidered in a repeated pomegranate motif. Both these cards use the pomegranate to evoke a scene of fertility. Like the fruit, the presence of a High Priestess in a reading concerns mystery and the divine, while the Empress is associated with sex, motherhood and growth.\n\nThe image of plentifulness that the pomegranate evokes is satirized in an amusing hoax advertisement for the all-in-one device the Pomegranate NS08 mobile phone. Launched in 2008, the website www.pomegranatephone.com claims that the ruby-red phone features a variety of inbuilt properties including a video projector, live voice translator, harmonica, coffee maker and razor. The website states that the phone has a battery the size and shape of a pomegranate aril that has the capacity to keep the phone running for five days. While exploring the features advertised, one is soon redirected to an information page about the region of Nova Scotia. The whole website is an elaborate advertisement, not for a new phone, but for tourism in the Canadian province (where 'you can get _everything_ you want in one place'). Although the advert features the fruit rolling around and the seeds splattering, Nova Scotia does not in fact have a pomegranate industry of its own. Elsewhere, in real electronics, the pomegranate has inspired researchers at Stanford University's SLAC National Accelerator Laboratory in the development of a new lithium ion battery that holds a cluster of tiny silicon particles in a hard carbon rind, in replication of arils in the fruit.\n\nMorris & Co., bird and pomegranate wallpaper, 1926.\n\n## 6\n\n## The Pomegranate in Modern Literature, Art and Film\n\nGuarded treasure, honeycomb partitions,  \nRichness of flavour,  \nPentagonal architecture.  \nThe rind splits; seeds fall \u2013  \nCrimson seeds in azure bowls,  \nOr drops of gold in dishes of enamelled bronze.\n\nAndr\u00e9 Gide, _Les Nourritures terrestres_ , trans. Dorothy Bussy\n\nThe use of the pomegranate in modern art can be exemplified in the contrast between two very different paintings depicting the same subject, the goddess Persephone. The first is the work of the nineteenth-century poet-painter and founder of the Pre-Raphaelites Dante Gabriel Rossetti. In this work, Persephone has bitten from the pomegranate and realizes its consequences. In the diagonal centre of the piece is the fateful fruit, whose flesh is the colour of the goddess's lips. One of her hands supports the other, which holds its weight: that heavy balance of life and death. In contrast, a recent painting by Cha Davis represents Persephone in a most unusual way. It is part of a series in which Davis represents her pet chicken Ester in a number of different guises. The traditional form of a woman is abandoned, Persephone now taking on a new animal form. Rather than being held by her, pomegranates now sprout from the goddess-bird's head. The fruit, with their tentacle calyces, are identified as literally part of her body. These two artworks show the traditional and diverse ways individuals have approached representing the nature of the pomegranate in the Persephone myth. The fruit, with all its mystery and sensuality, is what makes these unique scenes. It is the pomegranate's simple yet aesthetically pleasing appearance that appeals and has remained such a seductive icon.\n\nThe Armenian Arts\n\nThe films of Sergei Parajanov are charged with pomegranate imagery, reflecting his Armenian heritage. In _Sayat-Nova_ (The Colour of Pomegranates), the opening credits include a shot of juice spilling from pomegranates onto fabric, creating a stain in the shape of Armenia itself. Men in black robes are shown biting into pomegranates at a monastery. In the death scene of the poet protagonist we first see an ornamental dagger in the midst of several splattered pomegranates. A dreamlike sequence follows which features the poet having pomegranate juice (or is it blood?) poured on him by a white-faced woman wearing a green robe and a vegetal headdress (Death herself?). This draws on the long history of the pomegranate in Armenian art and architecture. The fruit appears on the thirteenth-century monastery of Geghard. A tympanum at one entrance is decorated with representations of trees from which hang entwined pomegranates and grape bunches. Two doves sit above the trees, looking inward towards the gateway. This pomegranate-grape-dove combination, all three of which represent life, is common in Armenian art. It appears among the mixture of biblical, mythological and rural-life-themed reliefs on the tenth-century Church of the Holy Cross at Akdamar Island. The pomegranate is also found as a column capital decoration at the seventh-century Zvartnots Cathedral. The fruit grows in the borders of pages from the Echmiadzin Gospel, a medieval Armenian illuminated bible from the tenth century. The longstanding fertility connotations of the pomegranate have resulted in the fruit's association with the Armenian wedding in modern times, pomegranate wine being the drink of choice for newlyweds. Armenians may wear a small dried pomegranate on a cord, known as a _taratosik_ ; this talismanic pendant is particularly associated with marriage.\n\n  | Cha Davis, _Ester Persephone_ , 2012.\n\n---|---\n\nIn modern times the pomegranate has come to represent the resilience of the Armenian people during the genocide of 1915. There is an Armenian saying that each pomegranate has 365 seeds, and that the people survived by eating one seed per day of their exile. The diaspora that resulted from the genocide saw the Armenian people spread across the world like the seeds of the pomegranate. The national importance of the fruit is honoured at the Armenian Genocide Monument of Larnaca, Cyprus, composed of a bronze sculpture surrounded by pomegranate and cypress trees. Armenian artists often use the pomegranate in their works to symbolize the suffering, hope, rebirth and survival that came with the genocide. Such an artist is the painter Rubik Kocharian, who was born in Armenia in 1940 and now lives in California. The pomegranate features in many of his works: being peeled by putti, split open in front of an Eastern lion relief, sitting next to a tablet inscribed with the Armenian alphabet, growing next to a classical sculpture. Kocharian's _Pomegranate Dance_ demonstrates very traditional Armenian imagery, illustrating a dance between two women before a pomegranate-bearing statue.\n\nDante Gabriel Rossetti, _Prosperina_ (the Roman name for the goddess). Rossetti has represented Persephone in a reflective pose, contemplating the effects of the fruit, 1874. |\n\n---|---\n\nChurch of the Holy Cross, Akdamar Island, 10th century.\n\nA fallen pomegranate at the 7th-century Zvartnots Cathedral.\n\nIllumination from the 10th-century Echmiadzin Gospel, Armenia.\n\nLiterature\n\nThe pomegranate has been used by many authors to emphasize an Eastern setting. This is the case for the pomegranate tree that plays a vital role in Khaled Hosseini's novel of sin and redemption, _The Kite Runner_ (2003). The changes in the relationship between childhood friends Amir and Hassan are reflected in changes in the appearance of the pomegranate tree. The tree is introduced as their shelter, isolated from the rest of Kabul and the rest of the world. It is there that they climb, play, carve their names on its trunk and eat of its fruit. Most importantly, the plant becomes a kind of tree of knowledge, under which Amir reads and teaches the illiterate Hassan. Although Amir and Hassan are from different social classes, they become equal at the pomegranate tree. This is not to last. An atrocity happens. Hassan is raped, and although Amir witnesses the act, he lacks the courage to intervene. When they return later to the pomegranate tree it is not the same. Their paradise has been lost. Amir begins pelting pomegranates at Hassan, attempting to provoke him to fight back. The bloody imagery recalls the earlier rape scene:\n\nRubik Kocharian, _Pomegranate Dance_ , 2010.\n\n> I hurled a pomegranate at him. It struck him in the chest, exploding in a spray of red pulp... I don't know how many times I hit him. All I know is that when I finally stopped, exhausted and panting, Hassan was smeared in red like he'd been shot by a firing squad.\n\nAmir wants to have his guilt lessened by being openly accused and punished for being the silent bystander. Instead, Hassan simply breaks a pomegranate over his own head. Throughout the rest of the novel Amir struggles for redemption. The pomegranate tree appears one more time, when Amir returns to Kabul much later as an adult. The tree no longer bears fruit. It has died, like his childhood friendship. All that remains is the memory of an idyllic time, embodied in those carvings made in the trunk long ago: 'Amir and Hassan, the sultans of Kabul.'\n\nThe Armenian-American writer William Saroyan wrote a short story called 'The Pomegranate Trees' in which a certain Uncle Melik buys a plot of land in the desert and attempts to grow an oasis of pomegranates on it. The sceptical nephew narrator remarks: 'He was too imaginative and poetic for his own good. What he wanted was beauty. He wanted to plant it and see it grow... It was pure aesthetics, not agriculture.' The trees are in constant battle with their environment, and get such little water they yield few fruit: 'My uncle harvested three small pomegranates. I ate one, he ate one, and we kept the other one up in his office.' A year later two hundred pomegranates are harvested, which are packed into eleven boxes and sent to a produce man for selling. The fruit won't even sell at $1 a box, and are thus sent back to Melik and his nephew, who eat them. Melik, from lack of profit, is forced to sell the desert land back, but pleas that the trees still need to be looked after. He and his nephew come back to the spot three years later to find all the trees dead. The tale concludes with the poignant statement: 'we didn't say anything because there was such an awful lot to say, and no language to say it in.' The pomegranate is instructional, functioning here as a symbol of the unattainable and unrealistic dream that will ultimately fail. It is the image of beauty, but not productivity. A similar perception is also found in Khalil Gibran's poem about the fruit, in which the author cannot bear the competing voices of the many seeds inside the pomegranate that he lives in, each expressing their hopes for the future. The parliament of seeds begin shouting their views simultaneously, becoming an indistinguishable noise. The author thus decides to move residence, to the quieter heart of a quince.\n\nPainting and Sculpture\n\nAs we have seen, mankind has enthusiastically included the pomegranate in its portrayal of the world, both physical and transcendent, since the beginning of history. The modern era sees a continuation of this pattern, often in new ways. It is common to find Graeco-Roman and Renaissance sculptures holding the pomegranate. True innovation comes with Alexander Calder, pioneer of the hanging mobile, who in the 1940s created an abstract kinetic sculpture from aluminium sheeting, inspired by the pomegranate tree. A single suspended sheet-cut pomegranate gives meaning to the rest of the piece, allowing us to identify the freeform shapes that hang around it as leaves that form the tree. Through Calder, the classical pomegranate-bearing sculpture was resurrected and reimagined. Likewise, in modern still-life paintings of the pomegranate, popularly used in such compositions, we again see tribute paid to the fruit's symbolic position in ancient art. With its antique notions of death, the fruit often features as a vanitas symbol in still-life arrangements, reminding the viewer of his or her own mortality and the brevity of life (sometimes appearing alongside a skull or being cut by a knife).\n\nMany notable modern artists have utilized the pomegranate in their own particular style: in Pablo Picasso's Cubism; in Henri Matisse's cut-outs; in the wallpapers of William Morris; in the glass lampshades of Louis Comfort Tiffany. The fruit plays a crucial role in the composition of Salvador Dal\u00ed's 1944 oil painting _Dream Caused by the Flight of a Bee around a Pomegranate a Second Before Awakening_. It illustrates a dream that was described to the artist by his wife Gala. This Surreal scene depicts two pomegranates. Gala's dream begins with a large broken-open pomegranate, two of its arils spilling out. From the fruit emerges a fish, from the fish emerges a tiger, and a second tiger emerges from the first. A rifle follows the tigers, the bayonet of which is about to pierce the naked body of Dal\u00ed's wife, which glows white in the centre of the scene. The pomegranate emphasizes the central theme of sexuality, the fruit being the original procreative force. Everything else emerges from its seeds in this Surrealist illustration of the Theory of Evolution. A second smaller pomegranate with a bee hovering around it gives the work its title. It casts a shadow in the shape of a heart. The fruit floats near Gala in recognition of the power of her female fertility. The pomegranate would also appear in later jewellery designed by Dal\u00ed. Rubies crafted to look like arils spill from his gold pomegranate heart brooch, with diamonds being used to represent the inner pith. In the ancient world, the pomegranate was a popular jewellery bead shape and, as Dal\u00ed's work demonstrates, it is still a fashionable ornament adorning brooches, necklaces, bracelets, rings and earrings today. The frequent use of garnet to represent the arils is appropriate in such jewellery, as this stone's name is derived from the Latin _granatum_ , because of its similarity of colour to the pomegranate aril.\n\n  | Silver pomegranate necklace with garnet crafted arils by Natalia Moroz and Sergey Zhiboedov of WingedLion. The naked figure evokes qualities of sensuality and femininity attributed to the pomegranate since ancient times.\n\n---|---\n\nSalvador Dal\u00ed, _Dream Caused by the Flight of a Bee around a Pomegranate a Second Before Awakening_ , 1944.\n\nIlya Zomb, _Fragment of Nightfall: Glimmer_ , 2001.\n\nIlya Zomb, _Mutual Admiration to Pomegranates_ , 2005.\n\nSurrealist painters of today, following Dal\u00ed's example, continue to utilize the pomegranate in creating phantasmagorical worlds. Such an artist is Ilya Zomb, who has used the fruit in many of his works, particularly in association with ballerina figures who dance upon the giant fruits. Other paintings feature pomegranate seeds being pecked at by birds, or being strung together and formed into a necklace. Particularly potent in their representation of fertility are his works _Fragment of Nightfall: Glimmer_ and _Mutual Admiration to Pomegranates_. In the former, two naked women recline against a giant pomegranate. Arils spill out, illuminating what is otherwise a dark and barren landscape (comparable to the underworld). Each of the women is holding and gazing into the large and luminous, jewel-like pomegranate seeds. _Mutual Admiration_ , in contrast, features several regular-sized pomegranates. One pomegranate is being balanced on a ballerina's head, while another is impaled before her on the horn of a rhinoceros. Seeds drip down from the penetrated fruit. Four more pomegranates rest on a bench in the foreground.\n\nOne final artwork, a true masterpiece, brings together the recurring key themes of beauty, mystery and femaleness that we have seen accompany the fruit throughout this book. This is William-Adolphe Bouguereau's 1875 painting _L'Orientale \u00e0 la grenade_ , which features a young girl decked in oriental garb peeling a pomegranate with her hands. The girl, like the fruit she holds, represents Eastern beauty. The blood-red jewels of her earrings replicate the arils that she picks from her pomegranate. There is no contextual detail, the painting focusing solely on the desirable girl whose eyes gaze out of the right-hand side of the frame. The painting has a twin, _Marchande de grenades_. It depicts the same girl in the same outfit, only this time Bouguereau reveals her to be a pomegranate seller on the streets of Cairo (the medieval gate of Bab Zuweila visible in the background), sitting in the dust beside her basket of fruit. Her straight and piercing stare beseeches the viewer to purchase her pomegranates. As the pomegranate is so strongly linked with fertility, the vending of the fruit may represent a loss of virtue, making us question what she is really selling on the street.\n\nWillem Kalf, _Still-life featuring Vanitas Motifs including a Pomegranate_ , 1640s.\n\nWilliam-Adolphe Bouguereau, _Girl with a Pomegranate_ , 1875.\n\nThroughout this work I have illustrated, using the example of one food item, how people of the past made sense of what they were eating. Food is a language; its consumption makes a statement about who we are. In both the East and the West, gesture and story evolved around the pomegranate, adding to the experience of its consumption. The fruit was even imagined to accompany an individual's journey beyond the grave. It's not just the pomegranate's wonderful taste and health benefits that appeal to us today, but this ancient aesthetic that has stimulated the imagination since people partook of the pomegranate's delights for the first time.\n\nNow it's time to take all that theory and put it into practice in the kitchen.\n\n## Recipes\n\nHistorical Recipes\n\nSausage on a Bed of Pomegranate and Grilled Plums\n\nIn the hilarious Roman mock epic the _Satyricon_ (Chapter 31), we are told of a wealthy freedman named Trimalchio who arranges an extravagant banquet for guests at his home. One of the courses described is 'hot sausages, on a silver grill with sliced damson plums and pomegranate seeds underneath'. The effect of the fruit bears the likeness to flames, beneath the sausages. This is our only source for this dish, from which I have created a simple recipe based on the description. A beautiful mix of colours and flavours, it would work well as an entr\u00e9e.\n\n1 cup (1 fruit) pomegranate arils  \n6 plums, cut in halves  \n4 pork sausages (if you want to be really authentic, you can make your own ancient Roman Lucanian sausages, via the recipe provided in Dalby and Grainger's _The Classical Cookbook_ )  \nmelted butter or vegetable oil\n\nCoat the plum halves in butter and grill on a barbecue until soft. Remove from heat and place in an oven to keep warm. Cook the sausages on a grill for about five to ten minutes until brown. When done, arrange the plums in rows on a platter. Scatter the pomegranate arils over the top. Cut the sausages into slices on the diagonal, and fan these over the top of the plums and pomegranate arils. Sprinkle with your choice of herbs and spices if desired.\n\nQueen's Pottage\n\nFrom John Nott's _The Cooks and Confectioners Dictionary_ (London, 1723), recipe courtesy of Ivan Day at www.historicfood.com\n\nThis almond-flavoured soup, garnished with pomegranate, originates from medieval France, and can be found in La Varenne's 1651 cookbook _Le Cuisinier fran\u00e7ois._ The dish came to enjoy popularity in England, and thus appears in an eighteenth-century English cookbook by John Nott. Alternate names for the dish include 'white soup' (the one frequently consumed by Jane Austen characters) and 'hedgehog soup' (it was sometimes served with small rolls spiked with almonds, giving the appearance of the animal). The earliest recipe, La Varenne's, instructs the chef to toast the surface of the soup with a red-hot fire shovel.\n\nBeat Almonds, and boil them in good Broth, a few Crums of Bread, the Inside of a Lemon, and a Bunch of sweet Herbs, stir them often, strain them, then soak Bread in the best Broth, which is to be thus made; Bone a Capon or Partridge, pownd the Bones in a Mortar, then boil them in strong Broth, with Mushrooms, then strain them through a Linnencloth; with this Broth soak your Bread; as it soaks, sprinkle it with the Almond-broth. Then put a little minced Meat to it, either of Partridge or Capon, and still as it is soaking, put in more Almond-broth, until it be full, then hold a red-hot iron over it; garnish the Dish with Pomegranates, Pistaches, and Cocks-combs.\n\nRoyal Salad\n\nFrom Antonio Latini, _Lo scalco moderna_ (Naples, 1692 and 1694), recipe courtesy of Ivan Day at www.historicfood.com\n\nCrunchy and fruity, the pomegranate does well in any salad. The popularity the pomegranate enjoys in the salad today is not a new phenomenon, however. A salad recipe from the seventh-century cookbook _Le Cuisinier m\u00e9thodique_ consisted simply of pomegranate arils, lemon slices and pistachios, sprinkled with sugar.\n\nTake endive, or scarola (another species of endive or chicory), mince it finely and put it to one side, until you have prepared a large basin, at the bottom of which are eight or ten biscottini, friselle or taralli (all three are types of hard breads), soaked in water and vinegar, with a little white salt. Put the said chopped endive on top, intermix with other salad stuff, albeit minced finely, make the body of the said salad on top at your discretion, intermix with radishes cut into pieces lengthways, filling in the gaps in the said basin with the ingredients listed below, all arranged in order. Pine nuts four ounces, stoned olives six ounces, capers four ounces, one pomegranate, white and black grapes ten ounces, twelve anchovies, tarantello (salted belly of tuna) four ounces, botargo three ounces, comfits, six ounces, preserved citron (and) preserved pumpkin twelve ounces, four hard-boiled eggs, whole pistachios four ounces, four ounces of raisins, other black olives six ounces. Caviar, four ounces, minced flesh of white fish, six ounces, little radishes, salt, oil and vinegar to taste, garnish the plate with slices of citrons, and citron flowers round about in order, take heed not to add salt or seasonings, until it goes to the table, and is about to be eaten.\n\nGranada Salad\n\nFrom _Los Angeles Times Cook Book_ , No. 21 (1905), submitted by S. Miller of 454 North Fair Oaks Avenue, Pasadena, California. A digital version of this cookbook is accessible online at <http:\/\/digital.lib.msu.edu>\n\nPlace the fruit (pomegranate) in the ice box until cold. Cut open with a sharp knife across the fruit division. Scrape out the red particles with a silver spoon. Place on shredded lettuce hearts and serve with the following salad dressing: Yolks of six eggs, one teaspoon mustard, one teaspoon salt, six teaspoons sugar, half cup of vinegar, half cup rich milk, one tablespoon butter and dash of cayenne pepper. Beat yolks, mustard, salt and sugar together; add vinegar and milk; then remaining ingredients. Cook in double boiler until thick. A cup of whipped cream added just before serving improves this dressing.\n\nModern Recipes\n\n_G\u00fclla\u00e7_\n\nRecipe courtesy of Zerrin Gunaydin at www.giverecipe.com\n\nThe pomegranate plays an important role in Middle Eastern sweets. The Turkish confection _g\u00fclla\u00e7_ is considered the forerunner from which baklava originates. It consists of thin layers of white pastry soaked in sweetened milk and rosewater. Between the layers are placed chopped walnuts, and a garnish of pistachio and pomegranate arils on top adds colour to the dessert. The dish originates from the Ottoman period, dating as far back as the fourteenth century, and is described in the Chinese food and health manual the _Yinshan Zhenyao_. It is especially popular during Ramadan, as it is light on the stomach after a full day's fasting. This dish relies on access to _g\u00fclla\u00e7_ wafers, which may be difficult to access in Western countries but should be available in Middle Eastern grocers.\n\n10 _g\u00fclla\u00e7_ sheets  \n6 cups (1\u00bd litres) milk  \n1\u00be cup (350 g) sugar  \n1 cup (150 g) finely chopped walnuts rosewater (optional)  \npomegranate arils and finely chopped pistachio for garnishing\n\nPut milk and sugar (and 1 tablespoon rosewater if you like) in a saucepan and apply heat until the sugar dissolves. Stir it occasionally and make sure it does not boil. Allow to cool; if you use the milk hot, your dessert will become mushy.\n\nLay your first _g\u00fclla\u00e7_ sheet in a square dish. Wet it with the warm milk. Repeat this with five sheets. Spread the crumbled walnut on the fifth. Then, lay another five sheets on top one by one, wetting each of them with milk. When you finish with the tenth one, pour the rest of the milk over the top. As the layers will combine when they are wet with the milk, they don't need to be perfectly arranged. After pouring the milk, the sheets will rise, but do not touch them. Cover the dish with plastic wrap and put it in the refrigerator for at least two hours. Leave the garnishing to just before you serve the dish, otherwise pomegranates and pistachio may change its colour. When you are ready to eat your _g\u00fclla\u00e7_ , remove it from the refrigerator, cut it into squares, garnish and serve.\n\nPomegranate Cobbler\n\nThe cobbler was created early on by settlers in American colonies. Depending on the region, the crust can be the consistency either of crisp biscuit or soft cake, and may be placed either above or below the fruit. Almost any fruit can be used as the base for this popular dessert, and the pomegranate is no exception.\n\n3 cups (2\u20133 fruit) pomegranate arils  \n\u00bd cup (100 g) caster sugar for fruit, and \u00be cup (150 g)  \ncaster sugar for batter  \n1 teaspoon baking powder  \n1 cup (115 g) plain flour  \n1 egg  \n\u00bd cup (125 ml) milk  \n1 tsp vanilla extract  \n\u00bc cup (60 ml) melted butter\n\nSift flour, baking powder and \u00be cup (150 g) of the sugar into a bowl. Add the vanilla, milk, egg and melted butter, and beat the mixture on high for about 5 minutes until a smooth consistency is achieved. In a baking dish, stir pomegranate arils and the \u00bd cup (100 g) sugar together. Pour the batter in an even layer over the fruit. Bake for 45 minutes at 180\u00b0C (360\u00b0F) or until the top is golden brown and springs back. Serve hot with vanilla ice cream.\n\nGrenadine\n\nGrenadine, a syrup made from the pomegranate, is very simple to make and is very useful. Today, most store-bought grenadine is artificially produced and contains no pomegranate. Grenadine is a common finishing ingredient in many classic cocktails including the Singapore Sling, Shirley Temple, Roy Rogers, Tequila Sunrise, Mimosa and Purple Rain. In mixology, the syrup is fairly interchangeable with fresh pomegranate juice or pomegranate liqueur, all three of which make an effective addition to a glass of champagne. Pomegranate is becoming a common addition in creating new takes on cocktail classics like Martinis, Cosmopolitans and Daiquiris. A much thicker version of grenadine, known in Middle Eastern cuisine as _nasharab_ , is served as a sauce on fish and meat.\n\njuice of 2 pomegranates (500 ml)  \n1 cup (200 g) sugar\n\nPour pomegranate juice and sugar into a small saucepan. Place on a medium heat on your stovetop. Allow sugar to dissolve and liquid to boil for about 10 minutes (or longer if you want a thicker sauce). A dash of lemon may be added for additional tartness if desired. Refrigerate in container before use.\n\nChocolate Pomegranate Drops\n\n1 200-g (8-oz) bar milk or dark chocolate  \n1 cup (1 fruit) pomegranate arils\n\nMelt the chocolate and stir in the pomegranate arils. Spoon portions onto baking paper or into patty pans, and allow to set in a refrigerator. A very simple yet delicious treat. Nuts, coconut and marshmallow can be added to create a pomegranate Rocky Road.\n\n_Kolliva_\n\nRecipe courtesy of Maria Benardis, from her book _My Greek Family Table_ (Sydney, 2009).\n\nThis traditional Greek dish, prepared following the death of a loved one, preserves the antique notion of the pomegranate as a memorial food item.\n\n4 cups (500 g) wheat kernels  \n2 tsp ground cinnamon, plus extra, for decorating  \n1 \u2154 cup (250 g) walnuts, chopped  \n\u00bd cup (75 g) raisins  \n2 cups (250 g) sesame seeds, toasted and crushed  \n\u00bd cup (60 g) icing sugar, sifted  \n\u00bd cup (75 g) blanched almonds, halved  \narils of 1 pomegranate\n\nCover the wheat kernels with water and soak overnight. Drain and rinse well.\n\nFill saucepan with water, add the soaked wheat and bring to a rapid boil. Reduce the heat and simmer for two hours until the wheat is tender and begins to split open. Stir frequently to prevent sticking, and add more water as required. When the wheat is cooked, remove the pan from the heat and leave to stand for about thirty minutes. Drain the wheat and rinse thoroughly in a colander. Spread it out on a clean tea towel and leave for about two hours, shaking occasionally. Place in a bowl, add the cinnamon, walnuts, raisins and half the sesame seeds and mix gently. Press the mixture down hard on a serving platter and spread the remaining sesame seeds evenly over the top. Dust with a thin layer of icing sugar. Sprinkle the cinnamon to form a cross on the _kolliva_ , then place the almonds on either side of the cinnamon to cross the cross. Place the pomegranate seeds over the exposed icing sugar to form a colourful design. To serve, simply spoon some of the mixture into small bowls. Serve at room temperature.\n\nPomegranate Macarons\n\nThe star of the pomegranate macaron lies in the fruity ganache filling, sandwiched between two almond macarons.\n\nThis recipe is contributed by blogger Barbara Chung of www.themacarondiaries.tumblr.com\n\n_For the pomegranate ganache filling_ :  \n\u00bc cup (60 ml) thickened cream  \njuice of 1 pomegranate (yields about \u00bc cup (50 ml))  \n1 cup (130 g) white chocolate, chopped\n\nChop up the white chocolate and put aside. Cut the pomegranate in half and, with great force, squeeze the juice out into a bowl. Allow the seeds to fall into the bowl as well. Pick out the extra seeds that don't easily fall in with a spoon and add them to the bowl. Pour the squeezed juice through a strainer into another bowl. Using a spoon, push the seeds into the strainer to extract as much juice as you can. Set aside.\n\nIn a small saucepan, on low heat, bring the cream to boil and remove from heat. Pour the hot cream over the chopped chocolate and mix until the chocolate has completely blended with the cream. Once the white chocolate and cream mixture has cooled, stir through the freshly squeezed pomegranate juice. Keep the mixture covered and refrigerate overnight.\n\n_For the macarons:_  \n1\u00bc cups (135 g) almond meal (ground almonds)  \n1\u00bd cups (180 g) confectioners' (icing) sugar  \n3 large egg whites aged at room temperature for a minimum of 24 hours  \n5 tablespoons (65 g) caster sugar  \n\u00bd tablespoon red powdered food colouring\n\nCombine the ground almonds with the icing sugar and red powdered macaron colouring and sift three times (or process the almonds and sugar in a food processor if you have one). Set aside. Beat the egg whites until foamy and then add the caster sugar one tablespoon at a time until mixture is thick, shiny and stiff. Fold in half of the dry mixture with the egg whites until combined. Then add about 80 per cent of the rest of the mixture (this is to make sure that the batter doesn't end up being too dry) and mix lightly in a circular motion. Commence folding the macaron mixture until it moves slowly when the bowl is rocked from side to side. A good way to check is by scooping a spoonful of batter and let it drop back into the bowl. If it slowly drops back into the bowl, lands on the rest of the batter in a small mound and slowly sinks back into the batter, then the process is complete. Do not over-fold your batter as this may cause your macaron batter to become overly runny and, consequently, have no feet.\n\nLine trays with double layers of baking paper. The double layer helps to prevent heat from passing through too quickly. Scoop the macaron mixture into a piping bag and pipe small mounds of macaron batter (about 2\u20132.5 cm or 1 inch diameter) onto the lined trays. Once you are done, tap the tray firmly against the counter or other flat surface to release any trapped air from the piped shells. This will help the macarons hold their round shape and enables the _pieds_ or little 'feet' to form. Set aside the trays to dry at room temperature. In order for macarons to form a smooth, shiny dome, the batter must not stick to your finger when touched.\n\nOnce the batter is dried, you may place them into the oven. Bake at 120\u2013130\u00b0C (250\u2013265\u00b0F) for 15\u201320 minutes; it is best to keep your eye on the macarons as each oven differs. Once you take them out of the oven, place the tray on a cooling rack and allow the shells to cool. Once the macarons have cooled, you may sandwich the shells with the pomegranate ganache filling to complete your macaron. Rest the macarons in the fridge overnight in order for the shells to absorb the moisture from the filling. Serve the completed macarons at room temperature.\n\n## References\n\nIntroduction\n\n David Zohary, with Maria Hopf and Ehud Weiss, _Domestication of Plants in the Old World: The Origin and Spread of Domesticated Plants in Southwest Asia, Europe, and the Mediterranean Basin_ (Oxford, 2013), pp. 134\u20135.\n\n John M. Riddle, _Goddesses, Elixirs, and Witches: Plants and Sexuality throughout Human History_ (New York, 2010), p. 17.\n\n Such as at Jericho. Other Bronze Age sites have yielded dried pomegranate fruit being used as cups or boxes themselves. Charles Singer, ed., _A History of Technology_ , vol. I: _From Early Times to Fall of Ancient Empires_ (Oxford, 1990), p. 372.\n\n1 The Primordial Pomegranate: The Fruit in Myth\n\n Efthymios G. Lazongas, 'Personification in Myth and Cult: Side, the Personification of the Pomegranate', in _Personification in the Greek World_ , ed. Emma J. Stafford and Judith Herrin (Farnham, 2005), p. 104.\n\n As described in one of the few surviving fragments of Heliodorus' extensive fifteen-book work on the Athenian Acropolis.\n\n Pausanias, _Description of Greece_ : 2.17.4, trans. W.H.S. Jones (Cambridge, MA, 1918), p. 335.\n\n Helmut Kyrieleis, 'The Heraion at Samos', in _Greek Sanctuaries: New Approaches_ (London, 1995), p. 106.\n\n Aeschylus, _Eumenides_ , 657\u201363.\n\n Or in some versions compared to the pomegranate as in Ovid, _Metamorphoses_ , 10.734\u20138.\n\n Clement of Alexandria, _Exhortation to the Greeks_ , 2.15, trans. G. W. Butterworth (Cambridge, MA, 1919), p. 177.\n\n The Golden Apple of Discord is one of the fruit from the Garden of the Hesperides, which were fabled to grant immortality. In the myth of Atalanta another golden 'apple' leads to sexual conquest.\n\n Powys Mathers, _The Book of the Thousand Nights and One Night_ , vol. III (London and New York, 2005), p. 301.\n\n Ibid., p. 401.\n\n Shahnameh 1.21\u20133.\n\n Ignacz Kunos, _Forty-four Turkish Fairy Tales_ (London, 1913), p. 171.\n\n Ibid.\n\n Ibid., p. 172.\n\n Reader's Digest, _Timeless Tales from Many Lands_ (New York, 2001), p. 330.\n\n Homa A. Ghahremani, 'Simorgh: An Old Persian Fairy Tale', _Sunrise Magazine_ (June\/July 1984).\n\n2 Pomegranates in the Ancient World\n\n Jean Bott\u00e9ro, _The Oldest Cuisine in the World: Cooking in Mesopotamia_ , trans. Teresa Lavender Fagan (Chicago, IL, 2004), pp. 101\u20132.\n\n Samuel Kramer, _The Sacred Marriage Rite: Aspects of Faith, Myth, and Ritual in Ancient Sumer_ (Bloomington, IN, 1969), p. 100.\n\n Sidney Smith, 'Pomegranate as a Charm', _Man_ , XXV (1925), p. 142.\n\n John M. Riddle, _Goddesses, Elixirs, and Witches: Plants and Sexuality throughout Human History_ (New York, 2010), p. 20.\n\n Sara Immerwahr, 'The Pomegranate Vase: Its Origins and Continuity', _Hesperia_ , LVIII\/4 (1989), p. 408.\n\n Pausanias, 9.25.1\n\n Philostratus, _Images_ , 2.29.\n\n Nikolaos Kaltsas, _Sculpture in the National Archaeological Museum, Athens_ (Los Angeles, CA, 2002), p. 48.\n\n Philostatus, _Vita Apollonii_ , 4.28.\n\n _Odyssey_ , 7.113.\n\n Herodotus 7.41.\n\n Herodotus 4.143.\n\n Plutarch, _The Parallel Lives: Artaxerxes_ , 4.3, trans Bernadotte Perrin (Cambridge, MA, 1923), p. 135.\n\n Pliny, _Natural History_ , 13.2, 20.82, 21.84, 22.70, 23.16, 23.42, 23.43, 23.57, 23.58, 23.60, 24.54, 29.11, 30.16.\n\n Found in the Byzantine book of court protocol the _De Ceremoniis_. Cited in Kathryn M. Ringrose, 'Women and Power at the Byzantine Court', in _Servants of the Dynasty: Palace Women in World History_ , ed. Anne Walthall (Berkeley, CA, 2008), p. 77.\n\n Efthymios G. Lazongas, 'Personification in Myth and Cult: Side, the Personification of the Pomegranate', in _Personification in the Greek World_ , ed. Emma J. Stafford and Judith Herrin (Farnham, 2005), p. 102.\n\n3 Jewish and Islamic Pomegranates\n\n Asaph Goor, 'The History of the Pomegranate in the Holy Land', _Economic Botany_ , XXI\/3 (1967), pp. 221\u20132.\n\n Ibid., p. 223.\n\n As exemplified in Joel 1:10, 12.\n\n Hebrew for 'pomegranate'.\n\n Goor, 'The History of the Pomegranate in the Holy Land', p. 222.\n\n Ibid., p. 223.\n\n Mary Abram, 'The Pomegranate: Sacred, Secular, and Sensuous Symbol of Ancient Israel', _Studia Antiqua_ , VII\/1 (Spring 2009), p. 27.\n\n Kiddushin 81b.\n\n Abu Nu'aim, _As-Suyuti's Medicine of the Prophet_ (Bloomington, IN, 1994), p. 63.\n\n4 Medieval Pomegranates\n\n Melitta Weiss Adamson, _Food in Medieval Times_ (Westport, CT, 2004), pp. 113\u201314.\n\n Pliny, _Natural History_ , 23.58.\n\n Christopher Daniell, _Death and Burial in Medieval England, 1066\u20131550_ (London, 1997), p. 76.\n\n Henry Layard, _A Popular Account of Discoveries at Nineveh_ (New York, 1854), p. 302.\n\n A local tradition of Andalusia.\n\n Bury Palliser, _Historic Devices, Badges, and War-cries_ (London, 1870), p. 380.\n\n Hildegard Schneider, 'On the Pomegranate', _Metropolitan Museum of Art Bulletin_ , IV\/4 (1945), p. 118.\n\n Monica H. Green, _The Trotula: A Medieval Compendium of Women's Medicine_ (Philadelphia, PA, 2001), pp. 87, 97 and 171.\n\n5 Pomegranate Production and Culture Today\n\n Peter Collinson and William Darlington, _Memorials of John Bartram and Humphry Marshall_ (Philadelphia, PA, 1849), p. 244.\n\n Note that many modern takes on grenadine substitute the pomegranate juice for artificial flavouring.\n\n Richard Ashton, with Barbara Baer and David Silverstein, eds, _The Incredible Pomegranate_ (Tempe, AZ, 2006), p. 76.\n\n Ben Farmer, 'Afghanistan Promotes Pomegranates over Opium Poppies in Farming Overhaul', _The Telegraph_ (20 November 2008).\n\n Julia Morton, _Fruit of Warm Climates_ (Miami, FL, 1987), pp. 352\u20135.\n\n Ashton, Baer and Silverstein, eds, _The Incredible Pomegranate_ , p. 21.\n\n Gregory Levin, _Pomegranate Roads: A Soviet Botanist's Exile from Eden_ (Forestville, CA, 2006), p. 25.\n\n Ibid., pp. 147\u20138.\n\n Ibid., p. 180.\n\n Ibid., p. 101.\n\n John M. Riddle, _Goddesses, Elixirs, and Witches: Plants and Sexuality throughout Human History_ (New York, 2010), p. 18.\n\n Ibid.\n\n Ibid.\n\n Ibid., p. 50.\n\n Donald Harper, 'Flowers in T'ang Poetry: Pomegranate, Sea Pomegranate, and Mountain Pomegranate', _Journal of American Oriental Studies_ , CVI\/1 (1986), p. 152.\n\n Sri Sathya Sai Baba, 'True Sacrifice', at <http:\/\/sssbpt.org\/index.html>, July 1988.\n\n Kabir Edmund Helminski, _Love's Ripening: Rumi on the Heart's Journey_ (Boston, MA, 2008), pp. 56\u20137.\n\n _Macquarie Dictionary_ (revised 3rd edn), p. 1478.\n\n6 The Pomegranate in Modern Literature, Art and Film\n\n Khaled Hosseini, _The Kite Runner_ (London, 2003), pp. 80\u201381.\n\n Ibid., p. 24.\n\n William Sayoran, _My Name is Aram_ (San Diego, CA, 1940), p. 27.\n\n Ibid., p. 39.\n\n Ibid., p. 42.\n\n## Select Bibliography\n\nAbram, Mary, 'The Pomegranate: Sacred, Secular, and Sensuous Symbol of Ancient Israel', _Studia antiqua_ , VII\/1 (2009)\n\nAshton, Richard, with Barbara Baer and David Silverstein, eds, _The Incredible Pomegranate_ (Tempe, AZ, 2006)\n\nGoor, Asaph, 'The History of the Pomegranate in The Holy Land', _Economic Botany_ , XXI\/3 (1967)\n\nImmerwahr, Sara, 'The Pomegranate Vase: Its Origins and Continuity', _Hesperia_ , LVIII\/4 (1989)\n\nLazongas, Efthymios G., 'Personification in Myth and Cult: Side, the Personification of the Pomegranate', in _Personification in the Greek World_ , ed. Emma Stafford and Judith Herrin (Farnham, 2005)\n\nLevin, Gregory, _Pomegranate Roads: A Soviet Botanist's Exile from Eden_ (Forestville, CA, 2006)\n\nMuthmann, Friedrich, _Der Granatapfel_ (Bern, 1982)\n\nRiddle, John M., _Goddesses, Elixirs, and Witches: Plants and Sexuality Throughout Human History_ (New York, 2010)\n\nSeeram, Navindra P., with Risa N. Schulman and David Heber, eds, _Pomegranates: Ancient Roots to Modern Medicine_ (Boca Raton, FL, 2006)\n\n## Websites and Associations\n\nA very informative website on pomegranates put out by the University of Florida\n\nwww.crec.ifas.ufl.edu\/extension\/pomegranates\n\nAril Systems\n\nPromotional website of a pomegranate seed-extractor machine\n\nwww.arilsystem.com\n\nPom Wonderful\n\nLeading American distributor\n\nwww.pomwonderful.com\n\nPomlife\n\nAustralian distributor\n\nwww.pomlife.com.au\n\nWebsite of the Pomegranate Council. Notable are pages on crafts (make your own pomegranate wreath!) and recipes\n\nwww.pomegranates.org\n\n## Acknowledgements\n\nThis book is dedicated to my mother, who always gave me books and took me to museums. Writing it wouldn't have been possible without the help and support of many people. A big thanks to editor Andy Smith, and everyone at Reaktion Books for being most helpful throughout this project. Many thanks to all of my family and friends who listened politely and made suggestions, while I rambled on about pomegranates. Of note are the contributions of Clementine Sugita, Wendy Stone, Geoffrey Page, Louise Pryke and Beatrice McLoughlin. A special thanks goes to Rebecca Psarras for pointing me in the direction of (and helping me consume) a variety of pomegranate products.\n\nThank you to the artists and museums who have allowed use of their images throughout this work. Gratitude is also due to recipe contributors Barbara Chung, Maria Benardis, Ivan Day and Zerrin Gunaydin.\n\n## Photo Acknowledgements\n\nThe author and the publishers wish to express their thanks to the below sources of illustrative material and\/or permission to reproduce it.\n\nAlamy: p. 35 (The Print Collector); Suky Best: p. 77; \u00a9 The Trustees of The British Museum, London: pp. 28, 31 bottom, 34, 43, 44, 45, 50, 65; Brooklyn Museum of Art, New York: p. 31 top right; Corbis: pp. 8 (ImageBROKER), 11 (Guenter Rossenbach), 12 (Steve Goossen), 46 (Mimmo Jodice), 66 (Summerfield Press), 108 (Cha Davis), 109 (Christie's Images), 117 (Museo Thyssen-Bornemisza, Madrid \u00a9 Salvador Dal\u00ed, Fundaci\u00f3 Gala-Salvador Dal\u00ed, DACS); Flickr: p. 110 bottom; Dumbarton Oaks Collection, Washington, DC: p. 51; Getty Images: p. 47 (Leemage); J. P. Getty Museum\/The Getty, Los Angeles: p. 120; Homonihilis: p. 18; iStockphoto: p. 6 (Emrah Turudu); Jebulon: pp. 72, 74, 75, 80; Kasir: p. 101 bottom; Rubik Kocharian: p. 113; Los Angeles County Museum of Art: p. 96; The Louvre Museum, Paris: pp. 42, 59, 61; Itzhak Luvation: p. 55; Aren Maerir: p. 40; Urek Meniashvili: p. 99; The Metropolitan Museum of Art, New York: pp. 21, 36 top, 39, 60, 62, 68; Moonsun1981: p. 100; Natalia Moroz and Sergey Zhiboedov: p. 116; Ruslan Sergeev: p. 101 top; Spoliast: p. 71; Tokyo National Museum: p. 95; Victoria & Albert Museum, London: p. 104; The Walters Art Gallery, Baltimore: pp. 36 bottom, 67, 92; Werner Forman Archive: p. 32 (The Louvre Museum, Paris); \u00a9 Paul E. Williams 2011 \u2013 All Rights Reserved: p. 110 top; Ilya Zomb: p. 118.\n\n## Index\n\n_italic_ numbers refer to illustrations; **bold** to recipes\n\nAfghanistan 86\u20137\n\nAmerica 83\u20136, 102, **131**\n\nAphrodite 17, 38\n\napple, interchangeability of pomegranates with 20, 30, 57\n\nArmenia 107\u20138, 110\u201312\n\nAshurnasirpal II, King of Assyria 29\n\nAthena 16\u201317, 38\n\nAyurveda 91\n\nAzerbaijan 98\u20139\n\nBest, Suky 76, _77_\n\nBoleyn, Anne _see_ Tudor England\n\nBotticelli, Sandro _66_ , 69\n\nBouguereau, William-Adolphe, _L'Orientale \u00e0 la grenade_ 119\u201320, _121_\n\nbreasts, pomegranates likened to 7, 22\u20133, 46, 102\n\nBuddha 97\u20138\n\nByzantine Empire 49\u201351, 69\n\nCalder, Alexander 115\n\nCatherine of Aragon _see_ Tudor England\n\nChina 92, 93\u20134\n\nChocolate Pomegranate Drops **133**\n\nClement of Alexandria 19\n\ncoins 16, 37, 56, 57\n\nCollinson, Peter 83\n\ncontraception 91\u20133\n\ncrown, pomegranate likened to 19, 56\n\nCybele\/Kubaba 17\u201319, _18_\n\nCyprus 17, 21, 49, _50_ , 111\n\nDal\u00ed, Salvador 115\u201317, _117_\n\nDarius the Great 34, 48\n\nDavis, Cha 106\u20137, _108_\n\ndeath 14\u201315, 23, 39\u201348, 54, 61, 107, 115, **133**\n\nDionysus 18\u201319\n\ndomestication 7\u20138\n\n_Economist_ magazine 87\n\nEgyptian civilization 7, 32\u20134\n\nEros 41\n\nEteokles and Polyneikes 45\n\nfestivals 98\u2013101\n\nGeometric Greek pomegranates 37\u20138, _39_\n\nGibran, Khalil 114\n\nGranada (Spanish city) 71\u20135\n\nGranada Salad **130**\n\ngrenadine 84, 102, **132\u20133**\n\nGreece, ancient 14\u201320, 37\u201350\n\nGreece, modern 51\u20132, **113\u201314**\n\n_G\u00fclla\u00e7_ **130\u201331**\n\nHafiz (Persian poet) 100\n\nHarpy Tomb of Xanthus 41, _44_\n\nHebrew bible 53\u20134\n\nHenry IV, King of France 77\n\nHera 16\u201317, 38\n\nHerodotus 48\u20139\n\nHestia 49\u201351, _51_\n\nHinton St Mary mosaic 65\u20136, _65_\n\nIndia 82, 91\u20133\n\nIran 7, 23\u20134, 34\u20137, 59, 60, 61, 82, 88, 99\u2013101\n\nIsabella I of Castile 73\n\nIshtar 29\u201330, 93\n\nIslam 20\u201322, 58\u201363\n\nivory pomegranate 57\n\nJapan 10, 94\u20137\n\nJefferson, Thomas 83\n\nJesus Christ 59, 62, 65\u20139\n\njewellery 28, 54, _116_ , 117, 119\n\nJudaism 53\u20138\n\njuicing 12\u201313, 88\n\n_kernos_ (ceramic form) 38\u20139, _40_\n\nKishibojin 94, 97\n\n_Kite Runner, The_ 112\u201314\n\nKocharian, Rubik 112, _113_\n\n_Kolliva_ **133**\n\nkore (Archaic maiden statue type) 45\u20138\n\nKujaku Myoo _95_ , 97\n\nLayard, Sir Austen Henry 70\n\nLevin, Gregory 89\u201390\n\nLucanian tomb paintings 43\u20134, _46_ , _47_\n\nmarriage 15\u201316, 24\u20135, 29, 35\u20136, 38, 46\u20138, 51\u20132, 69, 73, 94, 108\n\nMary, mother of Jesus 17, 66, 70\n\nMaximilian I of Austria, Holy Roman Emperor 78\u20139, _79_\n\nmedicine 33, 49, 56, 79\u201381, 84, 90\u201393\n\nMenoikeus 44\u20135\n\nmenstruation 15\u201316, 93\n\n_Men's Health_ magazine 84\n\nMesopotamian civilization 7\u20138, 10, 27\u201330\n\nMexico 82\u20133\n\nMilo of Croton 48\n\nNew Year 37, 52, 54\u20135\n\nnon-food uses of pomegranates 8, 10, 81\n\n_One Thousand and One Nights_ 21\u20133\n\n_Parrot Shah, The_ 25\n\nPersephone 14\u201315, 38, 43, 47, 65, 106\u20137, 108, 109\n\nPersian civilization _see_ Iran\n\nPhrasikleia 47\u20138\n\nPilsastri Acritani 70, _71_\n\nplant characteristics 8\u201310, 87\u20138\n\nPliny the Elder 49, 69\n\npomegranate butterfly 88\n\nPomegranate Cobbler **131\u20132**\n\nPomegranate Macarons **134\u20136**\n\nPom Wonderful 84\n\npop culture 102\u20135\n\npottery 34\u201341, 58\u201361, _92_\n\nproto-pomegranate 8\n\nQueen's Pottage **128**\n\nReiley, Amy 82\n\n_Rimmonim_ (Torah scroll decoration) 54, _55_\n\nRoman civilization 49, 66, 69, **127**\n\nRossetti, Dante Gabriel 106, _109_\n\nRoyal Salad **129**\n\nRumi (Persian poet) 100\n\nSan Juan de Dios 79\u201380, _80_\n\nSaroyan, William 114\n\nSassanids 34, _36_\n\nSausage on a Bed of Pomegranate and Grilled Plums **127\u20138**\n\n_Sayat-Nova_ ( _The Colour Pomegranates_ , film) 107\n\nseafaring 70\n\nseals 27, _28_ , _31_ , _34_\n\nSekhmet 33\u20137\n\nSergeev, Ruslan 99, 101\n\nsexual metaphors 17, 19, 22, 25\u20136, 29\u201330, 58, 82, 93, 103, 115\u201317\n\nShahnameh (Persian Book of Kings) 23\u20134\n\nShakespeare, William 64\n\nShoson, Ohara _96_ , 97\n\nShulgi, King of Ur 29\n\nSide (Greek personification of the pomegranate) 16\n\nSolomon (king of Israel) 56\u20137, 103\n\nsquash blossom necklace 83\u20134\n\nstatues 16, 21, 45\u20138, 79\u201380, 101, 112, 115\n\ntarot 103\n\nTantalus 19\u201320\n\ntextiles 10, 53, _62_ , 76\u20137, _78_\n\nTree of Life 24, 30\u201331, 93, 99\n\n_Trotula, The_ 81\n\nTudor England 73, 76, 77\n\nTurkey 24\u20135, 130\u201331\n\nTurkmenistan 89\u201390\n\nTutankhamun 33, _35_\n\nUlu Burun shipwreck 32\n\nunicorn 68, 69\n\nVanitas symbol 115, _120_\n\nvernacular 102\n\nwar, association with 15\u201316, 20, 41, 44\n\nWilliam of Malmesbury 69\u201370\n\nZomb, Ilya _118_ , 119\n\nZoroastrianism 37\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Die Autonomen\n\nvon Klaus Farin\nOriginalausgabe\n\n\u00a9 2015 Archiv der Jugendkulturen Verlag KG, Berlin;  \nprverlag@jugendkulturen.de  \nAlle Rechte vorbehalten  \n1. Auflage Januar 2015\n\nVertrieb f\u00fcr den Buchhandel\n\nBugrim (www.bugrim.de)\n\nAuslieferung Schweiz\n\nKaktus (www.kaktus.net)\n\nE-Books, Privatkunden und Mailorder\n\nshop.jugendkulturen.de\n\nLayout & Cover\n\nMaximilian Bornmann und Doroth\u00e9e Schwartzmann, sehen und ernten \u2013 Werkstatt f\u00fcr Konzeption und Gestaltung\n\nDruck\n\nwerbeproduktion bucher\n\nISBN\n\nPrint: 978-3-943774-39-9  \nPDF: 978-3-943774-40-5  \nEPUB: 978-3-943774-41-2\n\nDieses Buch gibt es auch als E-Book.\n\nUnsere B\u00fccher kann man auch abonnieren: shop.jugendkulturen.de\n\n# Die Autonomen\n\nvon Klaus Farin\nKlaus Farin, geboren 1958 in Gelsenkirchen, lebt seit 1980 \u2013 Punk sei Dank \u2013 in Berlin. Nach T\u00e4tigkeiten als Sch\u00fclerzeitungsredakteur und Fanzine-Macher, Konzertveranstalter und -Security, Buchh\u00e4ndler und Journalist nun freier Autor sowie Lehrbeauftragter und Vortragsreisender in Schulen und Hochschulen, Jugendklubs und Justizvollzugsanstalten, Akademien und Unternehmen. Diverse Ver\u00f6ffentlichungen \u00fcber Skinheads, Fu\u00dfballfans, Gothics, Karl May und andere (zuletzt: Frei.Wild. Deutschrock aus S\u00fcdtirol. Archiv der Jugendkulturen 2015).\n\nVon 1998 bis 2011 war Klaus Farin Leiter des auch von ihm gegr\u00fcndeten Archiv der Jugendkulturen (www.jugendkulturen.de), das Materialien jeglicher Art (Fanzines, Flyer, Tontr\u00e4ger, B\u00fccher, wissenschaftliche Studien usw.) \u00fcber & aus Jugendkulturen sammelt, analysiert, archiviert und der interessierten \u00d6ffentlichkeit zur Verf\u00fcgung stellt. Seit 2011 ist er Vorsitzender der Stiftung Respekt \u2013 Die Stiftung zur F\u00f6rderung von jugendkultureller Vielfalt und Toleranz, Forschung und Bildung.\n\n### Kontakt\n\nArchiv der Jugendkulturen Verlag, Fidicinstra\u00dfe 3, 10965 Berlin;  \nE-Mail: klaus.farin@jugendkulturen.de; Homepage: www.klaus-farin.de.\n\n## Inhalt\n\n### Die Emp\u00f6rten\n\n#### Geschichte\n\n#### Organisationsstrukturen\n\n#### Szene-Medien\n\n#### Theorie und Aktionsfelder\n\n#### Sexismus\n\n#### Militanz\n\n#### Antiimps und Antideutsche\n\n#### Autonome \u2013 eine Jugendszene?\n\n#### Literatur\n\n#### Fazit\n\n#### Literaturverzeichnis\n\n### \"Ruf niemals um Hilfe, es k\u00f6nnte die Polizei kommen!\"\n\n### \"Dieser Staat richtet sich gegen mich, und aus diesem Grund ist das Verh\u00e4ltnis zwischen mir und dem Staat ein feindliches.\"\n\n### \"Israel ist halt das Mallorca der Antideutschen.\"\n\n### \"Nat\u00fcrlich macht es mich w\u00fctend zu sehen, wie wir hier leben und wie andere woanders leben m\u00fcssen.\"\n\n### \"Ich m\u00f6chte, dass hier Menschen gleich behandelt werden.\"\n\n### \"Meine Zukunft? Definitiv keine schei\u00df-b\u00fcrgerliche Kleinfamilie!\"\n\n### Tierbefreierin\n\n### \"Privat kann man die Musik h\u00f6ren, die man will, es sie denn, sie ist frauenfeindlich, homophob, antisemitisch oder rassistisch.\"\n\n### \"Extreme Aktionen wie gegen die sogenannte Grauzone finde ich daneben. Bands verbieten, obwohl man sie nicht wirklich kennt und sich nicht mit ihnen unterhalten hat, ist schei\u00dfe.\"\n\n### \"Die Arroganz und Naivit\u00e4t der Menschen heute macht mir schon Angst.\"\n\n### \"Antifaschismus ist als Grundlage einer linksradikalen Organisierung nicht geeignet.\"\n\n### \"Wenn am 1. Mai sogar die Schaufenster linker Kollektive in Kreuzberg eingeschlagen werden, l\u00e4uft irgendwas falsch.\"\n\n### Live dabei am 1. Mai \u2013 Die \"Revolution\u00e4re 1. Mai Demo\" in Kreuzk\u00f6lln\n\n### \"Die Szene definiert sich ganz klar dar\u00fcber, dass sie sich als was Besseres f\u00fchlen als andere Leute.\"\n\n### \"Am 11. September hab ich mich total gefreut. Ich hab echt gejubelt.\"\n\n### \"Kopftuch oder Sturmmaske?\"\n\n### \"St\u00e4ndig ein Plenum nach dem anderen und immer Stellung nehmen. Es h\u00e4tte nur noch gefehlt, dass wir Antr\u00e4ge f\u00fcrs Molliwerfen stellen sollten.\"\n\n### \"Wenn Parteien mit Autonomen klark\u00e4men, w\u00e4re ich auch in einer Partei.\"\n\n### \"Gewalt sollte immer das Salz in der Suppe sein und nicht die Suppe selber!\"\n\n### Von der Sponti-Schule zum Antifa-Shopping\n\n### Die Zeitschrift radikal als Sprachrohr der linken autonomen Szene \u2013 Eine Medienanalyse\n\n#### Z\u00e4hne zeigen\n\n#### Das praktische Instrument\n\n#### Die Bewegung der Bewegung\n\n#### Der Kampf geht weiter, doch wohin? \u2013 Das Dilemma der Unintegrierbarkeit\n\n#### Die Flaute des Sturms und die L\u00f6cher im Segel\n\n#### Die Repression startet durch\n\n#### Going underground\n\n#### Es brodelt im Untergrund\n\n#### Radikal in der Krise\n\n#### Reset, Neustart \u2013 und Flop\n\n#### Die schwarze Serie\n\n#### Quo vadis, radikal?\n\n#### Lebt und lest radikal?\n\n### ExpertInneninterviews\n\n### \"Ein Maximum an Demokratie ist aus unserer Perspektive nicht erstrebenswert.\"\n\n#### Prof. Uwe Backes, 54 Jahre, Mitbegr\u00fcnder der vergleichenden Extremismusforschung.\n\n### \"Dass ihr deklariertes Ziel, den Kapitalismus und den Staat zu bek\u00e4mpfen, aussichtslos ist, ist offenkundig.\"\n\n#### Prof. Dr. Albert Scherr, 56 Jahre, Soziologe und Sozialp\u00e4dagoge.\n\n### Feindbeobachtung\n\n#### Felix Krautkr\u00e4mer, 35 Jahre, Redakteur der Junge Freiheit.\n\n### \"'Jugendkultur' ist ein Ausdruck wie ein Sack, in den man alles reinstecken kann, was man selbst, \u00e4lter geworden, nicht versteht.\"\n\n#### Prof. Dr. Wolf-Dieter Narr, 77 Jahre, Menschenrechtsaktivist und Demonstrationsbeobachter.\n\n### \"Das Potential f\u00fcr linksradikales Denken sehe ich in Deutschland nicht.\"\n\n#### Prof. Dr. Melanie Gro\u00df, 39 Jahre, Feministin.\n\n### Die Wut ist vorhersehbar\n\n### Bibliographie zum Buchprojekt Autonome\n\n#### Brosch\u00fcren des Verfassungsschutzes\n\n#### Weitere Literaturhinweise\n\n## Die Emp\u00f6rten\n\n\"Ich glaube, dass der Widerstand von heute an immer intensiver wird und in den 2020er Jahren in Europa und den Vereinigten Staaten einen H\u00f6hepunkt erreicht, um dann zwangsl\u00e4ufig in irgendeine Art Revolution zu m\u00fcnden. Dies ist unvermeidlich, da das alte System nicht von selbst verschwindet. [...] Diese Umw\u00e4lzung k\u00f6nnte nat\u00fcrlich auch durch friedvolle, parlamentarische Debatten auf den Weg gebracht werden, aber so wird es nicht kommen.\" (Karl Wagner 2012 in: 2052. Der neue Bericht an den Club of Rome. Eine globale Prognose f\u00fcr die n\u00e4chsten 40 Jahre)\n\n\"Mischt euch ein, emp\u00f6rt euch!\" \u2013 Diese Aufforderung eines Dreiundneunzigj\u00e4hrigen erregte im Jahr 2010 gro\u00dfes Aufsehen. Immerhin war der Aufrufer St\u00e9phane Hessel, der mit seinem Pamphlet gegen den Finanzkapitalismus kurzzeitig zum \"Gewissen der westlichen Welt\" geworden war, nicht irgendein linksradikaler No-Name, sondern \u00dcberlebender des Nazi-Konzentrationslagers Buchenwald, 1948 Mitautor der Allgemeinen Erkl\u00e4rung der Menschenrechte, die noch heute jeder Sch\u00fcler und jede Sch\u00fclerin irgendwann von ihren Politik- oder Geschichtslehrer_innen zu lesen bekommt, und langj\u00e4hriger Diplomat bei den Vereinten Nationen. \"Ich w\u00fcnsche allen, jedem Einzelnen von euch einen Grund zur Emp\u00f6rung\", schreibt er weiter. \"Das ist kostbar. Wenn man sich \u00fcber etwas emp\u00f6rt, wie mich der Nazi-Wahn emp\u00f6rt hat, wird man aktiv, stark und engagiert.\" (Hessel 2011a: 10).\n\nLeider ist es heute sehr schwer geworden, wahre Emp\u00f6rung hervorzurufen. Sicher: Wir regen uns gerne und viel auf, vermutlich mehr als in fr\u00fcheren Jahrzehnten \u2013 \u00fcber zu laut spielende Kinder, eine schlechte Internet-Verbindung, \u00fcber versp\u00e4tete Busse und Bahnen, die uns Minuten unseres Lebens sinn- und besch\u00e4ftigungslos warten lassen, \u00fcber M\u00fctter, die mit ihren Kindertransportkutschen die ganze Breite des B\u00fcrgersteigs belegen, \u00fcberhaupt \u00fcber schlendernde Menschen inmitten unserer Unrast. \u00dcber Kriegsfl\u00fcchtlinge und andere Fremde, die von \"unseren Steuergeldern\" leben, \u00fcber arbeitslose Hartz-4-Abh\u00e4ngige, die sich frech der oft menschenunw\u00fcrdigen Zwangsarbeit verweigern \u2013 und dies auch noch offen der Bildzeitung oder deren TV-\u00c4quivalent G\u00fcnther Jauch gestehen.\n\nDiese Art der Emp\u00f6rung, die der Logik des neoliberalen Gases folgt, das unsere soziale Umwelt durchtr\u00e4nkt und vergiftet, Konkurrenz, Neid, Rassismus und Raubtierkapitalismus als \"soziale Marktwirtschaft\" und \"moderne Leistungsgesellschaft\" verkl\u00e4rt, meint St\u00e9phane Hessel nicht, wenn er zu \"Emp\u00f6rung\" und \"Widerstand\" aufruft: \"Neues schaffen hei\u00dft Widerstand leisten. Widerstand leisten hei\u00dft Neues schaffen.\" (Ebd.: 21)\n\nDie Anl\u00e4sse, sich zu emp\u00f6ren und aus der Emp\u00f6rung Engagement wachsen zu lassen, liegen heute \"nicht mehr so offen zutage\" (ebd.: 19), schreibt er weiter. Das System der Machterhaltung ist geschmeidig geworden. \"Die systemerhaltende Macht der Disziplinar- und Industriegesellschaft war repressiv. Fabrikarbeiter wurden durch Fabrikeigent\u00fcmer brutal ausgebeutet. So f\u00fchrte die gewaltsame Fremd-Ausbeutung der Fabrikarbeiter zu Protesten und Widerst\u00e4nden. M\u00f6glich war hier eine Revolution, die das herrschende Produktionsverh\u00e4ltnis umst\u00fcrzen w\u00fcrde. In diesem repressiven System sind sowohl die Unterdr\u00fcckung als auch die Unterdr\u00fccker sichtbar. Es gibt ein konkretes Gegen\u00fcber, einen sichtbaren Feind, dem der Widerstand gilt. Das neoliberale Herrschaftssystem ist ganz anders strukturiert. Hier ist die systemerhaltende Macht nicht mehr repressiv, sondern seduktiv, das hei\u00dft, verf\u00fchrend. Sie ist nicht mehr so sichtbar wie in dem disziplinarischen Regime. Es gibt kein konkretes Gegen\u00fcber mehr, keinen Feind, der die Freiheit unterdr\u00fcckt und gegen den ein Widerstand m\u00f6glich w\u00e4re\", analysiert auch der Philosoph Byung-Chul Han in Ankn\u00fcpfung an ein Streitgespr\u00e4ch, das er mit Antonio Negri in der Berliner Schaub\u00fchne f\u00fchrte. \"Warum ist das neoliberale Herrschaftssystem so stabil? Warum gibt es so wenig Widerst\u00e4nde dagegen? Warum werden sie alle so schnell ins Leere gef\u00fchrt? Warum ist heute keine Revolution mehr m\u00f6glich trotz immer gr\u00f6\u00dfer werdender Schere zwischen Reichen und Armen? F\u00fcr eine Erkl\u00e4rung ist ein genaues Verst\u00e4ndnis notwendig, wie die Macht und Herrschaft heute funktioniert. [...] Der Neoliberalismus formt aus dem unterdr\u00fcckten Arbeiter einen freien Unternehmer, einen Unternehmer seiner selbst. Jeder ist heute ein selbstausbeutender Arbeiter seines eigenen Unternehmens. Jeder ist Herr und Knecht in einer Person. Auch der Klassenkampf verwandelt sich in einen inneren Kampf mit sich selbst. Wer heute scheitert, beschuldigt sich selbst und sch\u00e4mt sich. Man problematisiert sich selbst statt der Gesellschaft.\n\nIneffizient ist jene disziplinarische Macht, die mit einem gro\u00dfen Kraftaufwand Menschen gewaltsam in ein Korsett von Geboten und Verboten einzw\u00e4ngt. Wesentlich effizienter ist die Machttechnik, die daf\u00fcr sorgt, dass sich Menschen von sich aus dem Herrschaftszusammenhang unterordnen. Ihre besondere Effizienz r\u00fchrt daher, dass sie nicht durch Verbot und Entzug, sondern durch Gefallen und Erf\u00fcllen wirkt. Statt Menschen gef\u00fcgig zu machen, versucht sie, sie abh\u00e4ngig zu machen. [...] Die systemerhaltende Macht nimmt heute eine smarte, freundliche Form an und macht sich dadurch unsichtbar und unangreifbar. Das unterworfene Subjekt ist sich hier nicht einmal seiner Unterworfenheit bewusst. Es w\u00e4hnt sich in Freiheit. Diese Herrschaftstechnik neutralisiert den Widerstand auf eine sehr effektive Art und Weise. Die Herrschaft, die Freiheit unterdr\u00fcckt und angreift, ist nicht stabil. Das neoliberale Regime ist deshalb so stabil, immunisiert sich gegen jeden Widerstand, weil es von der Freiheit Gebrauch macht, statt sie zu unterdr\u00fccken. Die Unterdr\u00fcckung der Freiheit provoziert schnell Widerstand. Die Ausbeutung der Freiheit dagegen nicht.\" (Han 2014)\n\nWer sich trotzdem emp\u00f6rt, st\u00f6rt den Frieden. Macht sich unbeliebt. Gilt als \"Extremist\". \"Dreckaufw\u00fchler\" nannte man solche fr\u00fcher. Und es war nicht als Kompliment gedacht.\n\nDer Intellektuelle Hessel begr\u00fcndete sein Engagement nicht religi\u00f6s. Er brauchte nicht die Kr\u00fccke des Glaubens an eine \u00fcbernat\u00fcrliche Macht, um Verantwortung zu \u00fcbernehmen. \"Sartre lehrte uns, dass wir selbst, allein und absolut, f\u00fcr die Welt verantwortlich sind \u2013 eine fast schon anarchistische Botschaft. Verantwortung des Einzelnen ohne R\u00fcckhalt, ohne Gott. Im Gegenteil: Engagement allein aus der Verantwortung des Einzelnen.\" (Hessel 2011a: 11)\n\nAus autonomer Perspektive \u00fcbersetzt bedeutet dies: \"Es gibt kein richtiges Leben im falschen.\" Es ist nicht richtig zu versuchen, nur f\u00fcr sich selbst das Optimale aus den gesellschaftlichen Verh\u00e4ltnissen herauszupressen, koste es (den Rest der Welt), was es wolle. Es geht darum, die Existenzgrundlagen aller Lebewesen auf diesem Planeten zu optimieren, allen ein Leben ohne existenzielle Sorgen, in Freiheit und Selbstbestimmung, ohne Diskriminierungen aufgrund von Hautfarbe, Geschlecht, Religion, sexueller Orientierung usw. zu erm\u00f6glichen. Was jede_r damit anf\u00e4ngt, bleibt jedem\/r Einzelnen selbst \u00fcberlassen, aber jeder Mensch muss zumindest die materielle Chance haben, \"seines Gl\u00fcckes Schmied\" zu werden.\n\nDoch weil das gesamtgesellschaftliche Utopia zu verwirklichen keine realistische Option f\u00fcr die n\u00e4chsten Generationen ist, fangen wir eben schon einmal in unserem eigenen Lebensumfeld an. Immerhin mit der Hoffnung, um noch einmal St\u00e9phane Hessel zu Wort kommen zu lassen, dass, \"wenn eine aktive Minderheit sich erhebt, der Sauerteig seine Wirkung tun\" wird. \"Man verbindet sich mit dem Strom der Geschichte, und der gro\u00dfe Strom der Geschichte nimmt seinen Lauf dank dem Engagement der Vielen \u2013 zu mehr Gerechtigkeit und Freiheit.\" (Ebd.: 10)\n\nDas \"Lebensumfeld\", in dem gesellschaftliche Ver\u00e4nderungen angedacht, erkundet, exemplarisch verwirklicht werden k\u00f6nnen, definiert jede Autonomen-Generation und jede_r Autonome individuell immer wieder neu. Zur Geburtsstunde der autonomen Szene waren dies vor allem im konkreten Lebensbereich die Wiederaneignung von bezahlbarem Wohnraum (\"Instandbesetzungen\"), der zugleich auch als Basis zur Erprobung neuer Formen des Zusammenlebens jenseits der b\u00fcrgerlichen Kleinfamilie dienen sollte (Kommunen u. a. Wohngemeinschaften, Arbeits- und Lebenskollektive), die Schaffung eigener tempor\u00e4r befreiter Kulturzonen (autonome Jugendzentren etc.) sowie \u00fcber\u00f6rtlich die entstehende Anti-Kriegs- und Umwelt- bzw. Anti-Atomkraft-Bewegung.\n\n### Geschichte\n\nDas eigentliche Geburtsland der autonomen Bewegung (oder Szene \u2013 ich spare mir hier die artifizielle Debatte um die partiell unterschiedliche Bedeutung dieser beiden Begriffe und verwende sie synonym) ist Italien. In den dortigen Arbeitsk\u00e4mpfen der sp\u00e4ten Sechziger errang eine politisch-soziale Bewegung eine gro\u00dfe Bedeutung, die sich Autonomia Operaia (\"Arbeiterautonomie\") nannte. Die Autonomia Operaia f\u00fchrte Sabotageakte in Fabriken durch und auch ihre Demonstrationen waren von gewaltsamen Auseinandersetzungen mit der Polizei gepr\u00e4gt. Sie grenzte sich nicht nur von Staat und Kapital ab, sondern ebenso von der Kommunistischen Partei Italiens und den Gewerkschaften und entwickelte eine eigene Theorie, den Operaismus. Eines seiner zentralen Elemente war die Autonomie. Ende der 1970er Jahre verlor der Operaismus seine Bedeutung, aber dieses Verst\u00e4ndnis von Autonomie \u2013 selbstbestimmte politische K\u00e4mpfe unabh\u00e4ngig von Parteien und Gewerkschaften, spontane Bewegung unter Verzicht auf hierarchische Organisation und F\u00fchrung \u2013 pflanzte sich \u00fcber die Grenzen Italiens fort und geh\u00f6rt seitdem, zum Selbstverst\u00e4ndnis der Autonomen in ganz Europa.\n\nDas revolution\u00e4re Objekt der Begierde des Operaismus, aber eben auch des autorit\u00e4ren Kommunismus, die imaginierte \"Arbeiterklasse\", spielte im deutschsprachigen Raum allerdings keine bedeutende Rolle mehr. Stattdessen wurden die von den Patriarchen der K-Gruppen zu \"Nebenwiderspr\u00fcchen\" Degradierten nun unter der Vorgabe, auch das eigene Verhalten in den revolution\u00e4ren Kampf einzubeziehen, zu Querschnittsthemen, die alle angingen: die Unterdr\u00fcckung von Frauen (sp\u00e4ter auch Homosexuellen und anderen \"Geschlechtern\") und im 21. Jahrhundert schlie\u00dflich die Ausbeutung aller Lebewesen (Tierrechtsbewegung und Veganismus; vgl. hierzu auch Rinas 2012).\n\nAls eigenst\u00e4ndige Bewegung sichtbar wurden die Autonomen erstmals in den sp\u00e4ten 1970er Jahren als militanter Fl\u00fcgel der Anti-AKW-Bewegung und dann in den fr\u00fchen 1980er Jahren im Rahmen der Hausbesetzer_innen-Bewegung. Die ersten Zusammenh\u00e4nge und Aktivit\u00e4ten, bei denen sich selbst als \"autonom\" bezeichnende Gruppen auftraten, fielen in den Gro\u00dfst\u00e4dten Westdeutschlands und in West-Berlin ab 1980 auf. In Berlin bildete sich das erste Autonomen-Plenum im Sommer 1981. Das erste spektakul\u00e4re Datum autonomer Pr\u00e4senz ist jedoch schon der 6. Mai 1980: In Bremen kommt es zum ersten Mal zu massiven, auch militanten Protesten gegen eine Rekrutenvereidigung, ausgerechnet im Weserstadion. Der Staat wollte Flagge zeigen und wurde zum \u00f6ffentlichkeitswirksamen Geburtshelfer einer neuen Protestbewegung.\n\nSeitdem nahmen und nehmen autonome Gruppen regelm\u00e4\u00dfig an Demonstrationen und anderen Aktionen der Neuen Sozialen Bewegungen teil, vor allem der Friedensbewegung und der Anti-Atomkraft-Bewegung. Massiver beteiligt waren sie unter anderem am Widerstand gegen das Kernkraftwerk Brokdorf um 1981, die geplante Wiederaufarbeitungsanlage in Wackersdorf bis 1986, gegen den Bau der Startbahn West in Frankfurt am Main in den 1980er Jahren. Bis in die Gegenwart beteiligen sich auch viele Autonome an den Aktionen zur Behinderung der Castor-Transporte aus der franz\u00f6sischen Wiederaufarbeitungsanlage La Hague ins deutsche Atomm\u00fclllager Gorleben.\n\nIn der Anti-AKW-Bewegung stehen die Autonomen f\u00fcr eine undogmatische, linksradikale Position, die entstanden ist aus einer doppelten Abgrenzung gegen\u00fcber den gem\u00e4\u00dfigteren Forderungen der B\u00fcrgerinitiativen und daraus erwachsenen Gr\u00fcnen einerseits und eben gegen\u00fcber den K-Gruppen, deren hierarchische und autorit\u00e4re Strukturen sie ablehnten und denen sie ein taktisches Verh\u00e4ltnis zur Anti-AKW-Bewegung vorwarfen (vgl. A.G. Grauwacke 2003: 24). \"Die Anti-AKW-Bewegung w\u00e4re mit diesen gr\u00fcnen Langweilern nie dahin gekommen, wo sie heute ist! Wenn sie nicht massiv die Kosten in die H\u00f6he getrieben h\u00e4tte durch Konfrontation.\" (Jan)\n\nIn der Hausbesetzer_innen-Bewegung waren die Autonomen Teil eines vielf\u00e4ltigen Spektrums unterschiedlicher Aktivist_innen, die den Leerstand innerst\u00e4dtischen Wohnraums bei gleichzeitiger Wohnungsnot skandalisierten und die Hausbesetzungen nicht nur als symbolische Intervention, sondern auch als konkrete M\u00f6glichkeit der Schaffung von R\u00e4umen f\u00fcr die Realisierung alternativer Lebensentw\u00fcrfe verstanden. Dabei waren die Grenzen zwischen Punks, Alternativbewegung, Autonomen etc. flie\u00dfend, aber mitunter auch konfliktreich (siehe das ausf\u00fchrliche Interview in diesem Band dazu mit Michael Wildenhain).\n\n### Organisationsstrukturen\n\nDie Organisationsstrukturen der Autonomen sind im Kern basisdemokratisch und antiinstitutionell. Als explizites Gegenmodell zu den hierarchischen und parteif\u00f6rmigen Organisationsmodellen der K-Gruppen und -Parteien begr\u00fcndet, bilden lokale Kleingruppen die organisatorischen Kerne, in denen die politische Gruppe und der private Freundeskreis oft gro\u00dfe Schnittmengen haben. Die Koordination der politischen Arbeit findet in allen Aktivist_innen offenstehenden Plena statt. Vor allem in den 1980er Jahren gab es eine Reihe regelm\u00e4\u00dfiger regionaler und bundesweiter Treffen, die au\u00dferhalb konkreter Kampagnen der Koordination und dem Informationsaustausch dienten. In j\u00fcngster Zeit gibt es wieder verst\u00e4rkt, aber nicht immer erfolgreich Initiativen, nicht themengebundene Koordinationsstrukturen auf (\u00fcber-)regionaler Ebene in Form von autonomen \"Vollversammlungen\" zu etablieren.\n\nDa die Autonomen nicht durch eine formale Organisationsstruktur stabilisiert sind, stellen st\u00e4ndige, nicht enden wollende Diskussionen zur Selbstvergewisserung oder Aktualisierung der bestehenden Organisationsformen und der allgemeinen politischen Ziele ein identit\u00e4tsstiftendes Merkmal der Szene dar. In Phasen auslaufender Mobilisierungszyklen wird die Pr\u00e4misse der \"Politik der ersten Person\" regelm\u00e4\u00dfig infrage gestellt. \"Tenor der periodisch wiederkehrenden Organisationsdebatten war die Kritik an der Kampagnenf\u00f6rmigkeit autonomer Politik \u2013 dem schnellen Aufgreifen und dann wieder Fallenlassen aktueller Themen \u2013 und die Forderung nach einer gr\u00f6\u00dferen Verbindlichkeit autonomer Strukturen (z. B. Interim 162, 27.9.91; 'Wir sind doch kein Kampagnenheinz!'). Anstatt durch subjektive Betroffenheit solle sich autonome Politik aus der theoriegeleiteten Analyse gesellschaftlicher Widerspr\u00fcche begr\u00fcnden. Transparente Organisationsstrukturen sollten an Stelle der Plena und Kleingruppen mit wechselnder Beteiligung und unklarer Kompetenz treten. Gegen die Forderungen nach st\u00e4rkerer Formalisierung autonomer Strukturen wurde regelm\u00e4\u00dfig von Gegnern dieser Organisationsmodelle das autorit\u00e4re Scheitern der K-Guppen und die schon von Robert Michels (1911) konstatierte Eigendynamik der B\u00fcrokratisierung formaler Organisationen ins Feld gef\u00fchrt. Im Ergebnis blieben die Organisationsdebatten entweder folgenlos oder sie f\u00fchrten zu Ausgr\u00fcndungen, die allerdings nur im Antifa-Bereich eine relevante Mobilisierungsf\u00e4higkeit entfalten konnten. Dort spielen st\u00e4rker formalisierte Organisationsmodelle, wie sie 1991 die G\u00f6ttinger 'Antifa (M)' in der Interim gefordert hatte (Interim 161, 19.9.91), bis heute eine gewisse Rolle, wobei allerdings im Angesicht nachlassender Mobilisierungsf\u00e4higkeit auch auf klassische Basis- und Selbstmobilisierungsprozesse gesetzt wird, wie beispielsweise in der 'organize!'-Kampagne von Antifa-Aktivist_innen aus dem Ruhrgebiet.\" (Haunss 2013: 33f)\n\nBisher f\u00fchrten die Organisationsdebatten jedoch noch nie zu einer wirklich grundlegenden \u00c4nderung des autonomen Politikstils und der damit verbundenen, auf Basisdemokratie und Kleingruppen setzenden Organisationsstrategie. \"Das Festhalten am basisdemokratischen Ideal hat eine gro\u00dfe Bedeutung f\u00fcr die kollektive Identit\u00e4t der Autonomen.\" (Ebd.: 34) \"Das bedeutet eben, dass es keine Hierarchien gibt, dass es sehr wohl Regeln gibt, die Regeln aber im Konsens immer wieder neu diskutiert und entschieden werden m\u00fcssen, und nicht per Mehrheitswahlrecht oder Abstimmung. Nat\u00fcrlich ist Mehrheit immer wichtig, aber wichtiger ist die Argumentation. Also wenn zwanzig Leute gegen was abstimmen und es hat aber nicht wirklich Sinn, dann haben drei Leute immer noch die M\u00f6glichkeit, in einem demokratischen Basisgespr\u00e4ch zu erkl\u00e4ren, warum sich vielleicht die anderen zwanzig t\u00e4uschen, und sie zu \u00fcberzeugen, so dass dann trotzdem noch mal anders entschieden wird. [...] Manchmal ist es schon so, dass man sich gew\u00fcnscht h\u00e4tte, man h\u00e4tte so eine gewisse autorit\u00e4re Stimme darin gehabt, die sagt: So Leute, wir machen das jetzt so und Schluss. Und ihr werdet sehen, es wird gut f\u00fcr euch sein. [Lacht.] Aber das ging ja nicht, es mussten immer alle einverstanden sein.\" (Suzana) Wer damit nicht klarkommt, wandert ab in Parteien oder andere linke Organisationsstrukturen.\n\n### Szene-Medien\n\nNeben konkreten Treffen spielen \u2013 wie f\u00fcr alle Jugend- und Subkulturen und andere informelle Bewegungen \u2013 eigene Medien und Soziale Netzwerke eine wichtige Rolle f\u00fcr die Koordinierung der Bewegung \u00fcber den lokalen Handlungszusammenhang hinaus. Aus der kaum zu \u00fcberblickenden Vielzahl der regionalen und \u00fcberregionalen Zeitschriften, die in der autonomen Bewegung seit den 1980er Jahren entstanden (und wieder eingingen), sind von besonderer historischer und \u00fcberregionaler Bedeutung: Die Zeitschrift Autonomie (1975-79, sp\u00e4ter Autonomie \u2013 Neue Folge), sie stellte \"in der personellen Kontinuit\u00e4t einzelner Mitarbeiter_innen so etwas wie eine historische Br\u00fccke von der Studentenrevolte bis zur autonomen Szene in den 80er Jahren dar\" (Geronimo 1990: 61). Die Zeitschrift radikal war bis Mitte der 1990er Jahre eine der wichtigsten Plattformen f\u00fcr Diskussionspapiere, Berichte und Anschlagserkl\u00e4rungen, die einen gr\u00f6\u00dferen Leser_innenkreis als die diversen lokalen Bewegungszeitungen erreichen sollten. Die erste Ausgabe erschien bereits im Juni 1976 \u2013 damals noch als Sozialistische Zeitung f\u00fcr Westberlin. Doch schon bald wurde sie trotz \u2013 oder wegen \u2013 Verbotsverf\u00fcgungen, Beschlagnahmungen und ungez\u00e4hlter juristischer Verfahren f\u00fcr sicher zwei Jahrzehnte zum bedeutendsten Sprachrohr der autonomen Szene im deutschsprachigen Raum. (Deshalb widmet sich ihr beispielhaft f\u00fcr die Analyse der Autonomen-Medien ein eigener Beitrag in diesem Buch.) In den 1990er Jahren entwickelte sich die Berliner Autonomenzeitschrift Interim zum wichtigsten, auch \u00fcberregional bedeutsamen Kommunikationsmedium der Autonomen. Seit den 1990er Jahren spielen Internetportale eine immer wichtigere Rolle; vor allem indymedia, aber auch Termin- und Ank\u00fcndigungsportale wie Bewegungsmelder (Hamburg) oder Stressfaktor (Berlin).\n\n### Theorie und Aktionsfelder\n\nIm Verst\u00e4ndnis der Autonomen ist es entgegen der utopischen Selbstbezeichnung letztlich gar nicht m\u00f6glich, autonom (altgriechisch: \u03b1\u03cd\u03c4\u03cc\u03c2 autos = selbst und \u03bd\u03cc\u03bc\u03bf\u03c2 Nomos = Gesetz) zu sein. Jeder Mensch lebt nun einmal als soziales Wesen in einem Netz von Abh\u00e4ngigkeiten. Die entscheidende Frage ist, inwieweit diese Abh\u00e4ngigkeiten fremd- oder selbstbestimmt sind und wie es gelingen kann, die fremdbestimmten Abh\u00e4ngigkeiten zu minimieren. Der Weg ist das Ziel: \"Die Autonomen sind nichts Statisches, keine Partei, keine Denkschule und trotzdem eine politische Fraktion. Die Autonomen sind etwas sehr Nebul\u00f6ses, das sich zudem noch laufend ver\u00e4ndert. Sie sind ein politisches Milieu, ein Sammelbecken, eine Str\u00f6mung oder ein Spektrum. Es gibt darin verschiedene Verdichtungen, die sich manchmal sogar widersprechen. Autonom bedeutet nicht nur unabh\u00e4ngig, sondern selbstbestimmt. Gemeint ist damit, dass Autonome jeder paternalistischen Stellvertreterpolitik durch Parteien und Gewerkschaften eine Absage erteilen und sich lieber selbst und eigenverantwortlich organisieren. Die Autonomen sind also eine linksradikale, sozialrevolution\u00e4re und au\u00dferparlamentarische Bewegung.\" (Bj\u00f6rn)\n\nZu den theoretischen Grundlagen der Autonomen z\u00e4hlt die \"triple oppression\" (Rassismus, Sexismus und Klassismus, vgl. Strobl\/Viehmann u. a. 1993). Diese relativiert die monokausale Fokussierung sozialistischer und kommunistischer Klassiker, nach denen der Hauptwiderspruch im Kapitalismus liege, n\u00e4mlich der Konflikt zwischen gesellschaftlicher Produktion und privater Aneignung des Produkts, und andere Formen von Ausbeutung und Unterdr\u00fcckung wie Rassismus und Sexismus nur \"Nebenwiderspr\u00fcche\" darstellten. Vielmehr sei jeder Mensch Teil eines Netzes aus allen drei Gewaltformen, die sich nur je nach Lebenssituation unterschiedlich stark auspr\u00e4gten.\n\nJ\u00fcngere verbinden mit dem Begriff \"Autonome\" \u2013 neben \"Gewalt\" \u2013 vor allem das Thema Antifaschismus. In der Tat bildete die \"Antifa\"-Arbeit seit den 1990er Jahren, dem Mauerfall und in Folge der Wiedervereinigung dem Erstarken einer aggressiven jungen Neonazi-Szene, den zentralen Aktionsschwerpunkt der Autonomen. Doch die Themenvielfalt der autonomen Szene ist \u00fcberraschend gro\u00df, wie Sebastian Haunss in einer aktuellen Analyse der Interim festgestellt hat: \"Neben dem Thema der Anti-Atom-Politik ging es in den Debattentexten um Internationalismus und Globalisierung, Rassismus und Antirassismus, Antifaschismus, Antinationalismus, Stadtteilpolitik und Gentrifizierung, Militanz sowie an erster Stelle um Sexualit\u00e4t und das Verh\u00e4ltnis zwischen den Geschlechtern. In einer detaillierteren Analyse l\u00e4sst sich zeigen, dass diese Themen zum gr\u00f6\u00dferen Teil nicht episodisch einander abgel\u00f6st haben, sondern immer wieder aufgegriffen wurden. Bei aller Diskontinuit\u00e4t und Kampagnenhaftigkeit autonomer Politik gibt es daher ein klar sichtbares Set zentraler Themen, die die Bewegung seit ihren Anf\u00e4ngen begleiten.\" (Haunss 2013:31)\n\nSebastian Haunss' Analyse der Bewegungsdebatten kristallisierten einen \"Kern zentraler Elemente der Bewegungsidentit\u00e4t der Autonomen\" heraus, um die besonders intensive Auseinandersetzungen gef\u00fchrt worden sind:\n\n1. Politik der ersten Person: Der Anspruch einer subjektivistischen Politik, die individuelle Selbstver\u00e4nderung als mindestens ebenso wichtig ansieht wie die Ver\u00e4nderung der Gesellschaft. Die Einfl\u00fcsse der Frauen- und Sponti-Bewegung auf die Autonomen treten in diesem Punkt besonders deutlich zutage.\n\n2. Basisdemokratie: Eine Ablehnung traditioneller institutioneller Formen. Dieser antiinstitutionelle Impuls richtet sich sowohl gegen die Parteien und Institutionen des politischen Systems und der staatlichen Verwaltung als auch gegen Versuche, eher dem K-Gruppen-Modell folgende Organisationsstrukturen bei den Autonomen zu etablieren. Hier kommen Ideale zum Tragen, die aus einem meist diffusen Anarchismus und den Vorstellungen der Alternativbewegung gespeist sind.\n\n3. Systemopposition: Eine grunds\u00e4tzliche Gegnerschaft zur herrschenden (kapitalistischen) Ordnung der Gesellschaft (siehe Haunss 2013: 31f).\n\n### Sexismus\n\nIn der Zeit zwischen 1988 und 2001 wurden nach Haunss in der Interim \"341 Diskussionstexte und Positionspapiere ver\u00f6ffentlicht, in denen es um Sexismus, Sexualit\u00e4t, sexuelle Gewalt, Feminismus und die Geschlechterverh\u00e4ltnisse in der autonomen Bewegung ging. Es ist nach den Organisationsdebatten der zahlenm\u00e4\u00dfig bedeutendste Diskursstrang. Die Debatten liefen dabei entlang vierer Hauptlinien (f\u00fcr eine ausf\u00fchrliche Diskussion siehe Haunss 2004, S. 149ff):\n\na) Diskussionen \u00fcber die Notwendigkeit und Bedeutung separater Frauenorganisierung,\n\nb) Fragen der Parteilichkeit, der Definitions- und Entscheidungsmacht bei F\u00e4llen sexueller Gewalt und Vergewaltigung in der autonomen Szene,\n\nc) Sexualit\u00e4t und Begehren sowie\n\nd) allgemeine Diskussionen \u00fcber den Charakter und die Bedeutung des Patriarchats, besonders im Hinblick auf seine Verortung relativ und in Relation zu anderen Herrschaftsverh\u00e4ltnissen sowie \u2013 gegen Ende der 1990er \u2013 Diskussionen \u00fcber Feminismus, Postfeminismus und Dekonstruktion.\n\nDie H\u00e4ufigkeit und Intensit\u00e4t dieser Debatten verweist auf die zentrale Rolle dieses Themenfeldes in der autonomen Bewegung. Dabei handelt es sich vor allem um ein Thema mit 'interner' Bedeutung, d. h. Debatten zum Sexismus in der autonomen Bewegung \u00fcberwiegen bei Weitem gegen\u00fcber Beitr\u00e4gen und Mobilisierungen, die Sexismus in der Gesellschaft allgemein zum Thema haben.\" (Haunss 2013: 34f; vgl. auch seinen Beitrag zu \"M\u00e4nnlichkeitsbildern auf Plakaten der autonomen Bewegung\" in Farin\/M\u00f6ller 2014: 183-197)\n\nEnde der 80er Jahre stiegen viele Frauen aus der \"gemischten\" autonomen Szene aus. Das Thema \"Sexismus\" gewann an Brisanz, nachdem viele Frauen die an ihnen in der Szene begangenen \"sexualisierten Grenz\u00fcberschreitungen\" und Vergewaltigungen thematisierten und Konsequenzen einforderten bzw. zogen. So warfen die Frauen in Bochum die M\u00e4nner nach einer Sexismusdiskussion aus dem gemeinsamen autonomen Zentrum hinaus und erkl\u00e4rten dies zum \"autonomen Frauenzentrum\". Oftmals spaltete sich in den gef\u00fchrten \"Sexismusdiskussionen\" die Szene. \"In Berlin gibt's eigentlich so alle paar Jahre eine extreme Debatte \u00fcber irgendwas, die die Szene krass spaltet\", relativiert Leonie. Dennoch ist Sexismus und der Umgang zwischen den Geschlechtern vermutlich das brisanteste Thema f\u00fcr eine Szene, die nicht nur gesamtgesellschaftliche Ziele und Utopien formuliert, sondern an sich selbst den Anspruch stellt, schon heute im eigenen Alltag so weit wie m\u00f6glich hierarchiefrei und nicht-diskriminierend zu leben.\n\n\"Grunds\u00e4tzlich gibt es ja Sexismus \u00fcberall in allen Szenen so wie in der Restgesellschaft auch, das ist klar. Also auch in der autonomen Szene. Und nat\u00fcrlich waren es auch meistens die Frauen, die diese Themen eingebracht hatten, die gesagt haben, hier gibt es aber sexistisches Redeverhalten, weil die Typen reden schon wieder zwei Stunden, das ist auch Sexismus. Viele Intellektuelle sind verbale Macker, die reden drei Stunden und h\u00f6ren sich selber gerne reden. Aber es gibt auch eine autonome Queer-Szene, und die sehen das zum Beispiel als Hauptagitationsfeld. Aber nat\u00fcrlich gibt es Sexismus in der Szene. F\u00fcr mich gibt es \u00fcberall Sexismus, wie soll es dann in der autonomen Szene keinen Sexismus geben?! Sexismus, das sind doch Strukturen, die in allem drinstecken, also Machtstrukturen... Nat\u00fcrlich gibt es dieselben Dinge, die es woanders gibt, auch in der autonomen Szene, nur dass die Leute einen anderen Anspruch an sich selber haben und versuchen, genau diese Strukturen aufzubrechen und zu hinterfragen und zu reflektieren, auch, dass wir selber Teil dessen sind. Das habe ich oft in der autonomen Szene kennengelernt, da gab es eine hohe Selbstreflektion. Diese ganze Besch\u00e4ftigung mit sich selbst, das ist fast schon Psycho-Seminar. Kennzeichnend f\u00fcr die autonome Szene ist, dass alles und jedes immer stundenlang ausdiskutiert wird, weil selten etwas eindeutig ist. Wenn die Betreiber des Kneipenkollektivs X sagen, der Typ hat einen sexistischen Spruch gebracht, die Frau hat sich bel\u00e4stigt gef\u00fchlt, wird das normalerweise erst mal mit dem Typen ausdiskutiert und er muss Stellung dazu beziehen. Wenn er das aber wiederholt tut, dann fliegt er raus. Auf Konzerten, auf Partys, auf Veranstaltungen werden Leute bestimmt, die daf\u00fcr zust\u00e4ndig sind, so das Antisexismus-Team, die achten darauf und beobachten so ein bisschen. Und wenn die sehen, da gibt es einen Konflikt zwischen einem Typen und einer Frau oder der Typ macht die Frau dumm an und sie f\u00fchlt sich bel\u00e4stigt, dann gehen die hin und k\u00fcmmern sich darum. Also insgesamt hat Sexismus und Anti-Sexismus einen hohen Stellenwert in der autonomen Szene. Es wird sich viel damit auseinandergesetzt, das fand ich immer positiv.\" (Lena)\n\n\"Ich denke schon, dass die Rolle von Frauen in der autonomen Szene ganz klar ist, diesen Antisexismus zu vertreten. Das sind ja meistens Frauen, die dann sagen, so geht's aber nicht, und das wird auch ein bisschen erwartet, dass Frauen diese Meinung haben und diese Sachen durchk\u00e4mpfen und diese Konfrontation von Frauen initiiert stattfindet. Das ist selten so, dass M\u00e4nner das einbringen. Die M\u00e4nner in der Szene haben viele Themen und Rollen, die Frauen haben vor allem eine Hauptrolle: Antisexismus.\" (Leonie)\n\n\"Die Szene muss sich den Vorwurf der M\u00e4nnerdominanz schon vom Ursprung her machen lassen. Hat vielleicht auch schon mit dieser Rekrutierung \u00fcber Gewalt zu tun. Wenn ich so an uns denke, uns haben immer diese wilden Stra\u00dfenschlacht-Geschichten fasziniert. Autonome Politik ist nat\u00fcrlich auch viel R\u00e4uber und Gendarm spielen auf st\u00e4dtischem Niveau. Und das ist, auch wenn das jetzt sexistisch klingen mag, halt ein Jungs-Ding. Auch das \u00e4u\u00dfere Erscheinungsbild von Autonomen, der ber\u00fchmte Schwarze Block und so, ist nat\u00fcrlich m\u00e4nnerdominiert, das sind auch M\u00e4nneraktionsformen. Aber bei den J\u00fcngeren sind die M\u00e4dchen eher die, die, notgedrungen, das Ideologische mitbringen. Die bringen die anderen Themen rein, wo die Jungs nichts mit anfangen k\u00f6nnen. Zum Beispiel das Thema Frauen selbstverst\u00e4ndlich, das ist was, was Jungs gar nicht in den Kopf kommt, und dann auch diese Tierrechtsgeschichten und Veganismus. Das sind, wenn man ehrlich ist, schon eher die Frauenvarianten der autonomen Bewegung. Auch wenn die M\u00e4nner das im Regelfall nicht zugeben werden und dann sagen, nat\u00fcrlich, uns sind auch Tierrechte wichtig. Insofern haben die Frauenfraktionen da auch eine diskursive Macht bewiesen, indem sie solche Themen auf die Agenda gesetzt und durchgebracht haben, wo die Jungs nicht einmal auf die Idee gekommen w\u00e4ren, dass das Probleme sind.\" (Jan)\n\n\"Eine Bewegung, die die Verh\u00e4ltnisse, wie sie sind, nicht hinnimmt, ist nicht qua dieser Positionierung anders, sondern sie beschreibt nur einen Weg, auf dem sie entlanggehen will und auf dem sie sich ver\u00e4ndern wird. Privilegien abzugeben ist immer schwerer als Unterdr\u00fcckung abzuwerfen. Von daher war der 'Leidensdruck' unter M\u00e4nnern nicht so gro\u00df. Die H\u00e4userkampf-Bewegung oder die Bewegungen, die es \u00fcberhaupt gab, waren weder nur ein Abbild dessen, was sie angeblich bek\u00e4mpft haben, noch verk\u00f6rperten sie bereits den Himmel auf Erden. Sie haben darum gerungen.\" (Wolf Wetzel)\n\n### Militanz\n\nAutonome verstehen sich mehrheitlich als besonders entschlossene, radikale Teile anderer sozialer Bewegungen (Anti-AKW-Bewegung, Stadtteilinitiativen, Antifaschismus, Antirassismus etc.), die sich in ihren Aktionsformen nicht durch den Rahmen der Legalit\u00e4t einschr\u00e4nken lassen wollen. Militantes Handeln wird aus dieser Perspektive immer als eine gleichberechtigte Aktionsform unter anderen diskutiert. \"Es ist eines der Mittel, die genutzt werden. Es existiert eine gewisse Grundhaltung, was die Gewalt angeht, au\u00dferdem ist es schlicht medienwirksamer, um so auf Aktionen hinzuweisen. Wie w\u00e4re die Anti-Globalisierungsbewegung wohl aufgestellt, wenn es Seattle 1999 oder Genua 2001 nicht gegeben h\u00e4tte? Wie viele Menschen h\u00e4tten vom Naziaufmarsch j\u00e4hrlich im Februar in Dresden erfahren, wenn dieser nicht auch durch militante Mittel 2010 angegangen worden w\u00e4re? Gewalt gegen Sachen finde ich absolut in Ordnung, so lange der Kopf benutzt wird, das hei\u00dft, kein Familienwagen umgedreht oder der Tante-Emma-Laden entglast wird. Gewalt gegen Menschen hei\u00dfe ich nur im Falle von Gegenwehr f\u00fcr sinnvoll.\" (Henry) Diese Haltung, au\u00dferhalb der oft ritualisierten Auseinandersetzungen mit der Polizei bei Demonstrationen und bei den Versuchen, \u00f6ffentliche Auftritte von Neonazis zu verhindern, keine Gewalt gegen Personen anzuwenden, ist in der autonomen Szene weitgehend Konsens. \"Aktionen, die dieser Maxime widersprechen, ernten in der Regel scharfe Kritik in den Bewegungsmedien. Nur einmal wurde \u2013 1987 bei den t\u00f6dlichen Sch\u00fcssen auf Polizisten an der Startbahn West \u2013 das Tabu t\u00f6dlicher Gewalt gegen Personen gebrochen.\" (Haunss 2013: 37; vgl. auch ID-Archiv im IISG 1988)\n\nBei militanten Aktionen von Autonomen handelt es sich in der Regel um Sachbesch\u00e4digungen, entweder am Rande von Demonstrationen oder als gezielte Sabotage. Militanz wird in den Autonomen-Gruppen im Wortsinn als \"k\u00e4mpferisch\", nicht als \"milit\u00e4risch\" verstanden. Gewalt als Selbstzweck oder als inhaltsleeres Ritual wird von den meisten Autonomen prinzipiell abgelehnt, wenngleich die Praxis mancher autonomer Gruppen und Individuen immer wieder das Gegenteil zu bedeuten scheint. Dementsprechend hat Militanz f\u00fcr die autonome Bewegung auch vor allem eine symbolische Bedeutung. Sie dient in der Selbstinszenierung gegen\u00fcber der \u00d6ffentlichkeit und gegen\u00fcber anderen Bewegungen als Ausweis der Radikalit\u00e4t. Die Signalisierung der Bereitschaft, sich bei den Formen des Protests nicht an den legalen Rahmen zu halten, ist ein wichtiges Element der Bewegungsidentit\u00e4t.\n\n\u00c4hnlich wie die Organisationsdebatten folgten die Militanzdebatten bei den Autonomen oft abflauenden Mobilisierungswellen. Noch st\u00e4rker als erstere zeichnen sie sich durch eine hohe Zyklizit\u00e4t aus, d. h., mehr oder minder dieselben Argumente werden im Abstand weniger Jahre immer wieder aufs Neue diskutiert, ohne dass eine deutliche Weiterentwicklung der Bewegungspraxis und der Debatte zu beobachten w\u00e4re (siehe Haunss 2004: 169ff.).\n\n### Antiimps und Antideutsche\n\nDie Autonomen waren seit ihren Anf\u00e4ngen stets ein Schmelztiegel verschiedener Fraktionen der radikalen, au\u00dferparlamentarischen Linken. Je nach Region und Zeit dominierte dabei die eine oder andere Richtung. Zumeist waren es jene Gruppen und Individuen, die am \"Anarchosyndikalismus\" ausgerichtet waren, mitunter in bestimmten Regionen, St\u00e4dten oder auch nur Stadtbezirken aber auch die zeitweise RAF-nahen \"Antiimps\" (Antiimperialisten). Die Autonomen waren schon immer eine sehr heterogene Bewegung \u2013 jedoch war es den verschiedenen politischen Spektren m\u00f6glich, \"intern\" miteinander zu streiten und sich zumeist dennoch auf ein gemeinsames Handeln zu einigen.\n\nDie zwei gegens\u00e4tzlichen Pole werden grob als \"Antideutsche\" auf der einen Seite und eben \"Antiimps\" auf der anderen Seite bezeichnet. Die Spaltung entz\u00fcndete sich an der Debatte um Antisemitismus innerhalb der Linken und die \"Pal\u00e4stinenser-Solidarit\u00e4t\" bzw. die Haltung zu Israel. Die Positionen sind dabei so verh\u00e4rtet, dass es bei Zusammentreffen beider Fraktionen etwa in Hamburg und Berlin auch schon zu k\u00f6rperlichen Auseinandersetzungen und \u00dcberf\u00e4llen von Veranstaltungen und Kneipentreffen der \"Gegenseite\" kam.\n\nDie \"Antiimps\" betrachten die Politik Israels und der USA prinzipiell als imperialistisch. Sie solidarisieren sich so auch mit dem \"Befreiungskampf des pal\u00e4stinensischen Volkes gegen die israelische Unterdr\u00fcckung\". Zentral f\u00fcr ihre Weltsicht ist die Annahme, dass der Reichtum der Industrienationen auf der Ausbeutung der drei Kontinente S\u00fcdamerika, Afrika und Asien basiert und der Kapitalismus damit eine geostrategische Dimension bekommen hat: Da selbst die \u00e4rmsten Bewohner der Industriestaaten noch von der Ausbeutung der drei Kontinente profitieren, gibt es in den Industriel\u00e4ndern quasi au\u00dfer ihnen selbst und den politisch bewussten Einwander_innen aus ebenjenen L\u00e4ndern niemanden mehr, der ein materielles Interesse an einer sozialen Revolution hat. Also bleibt nur, die Befreiungsbewegungen und noch verbliebenen sozialistischen Regime in den \"Entwicklungsl\u00e4ndern\" zu unterst\u00fctzen und den \"milit\u00e4risch-industriellen Komplex\" in den Industriel\u00e4ndern durch Sabotage oder auch Intervention in politischen Bewegungen wie der Friedensbewegung zu bek\u00e4mpfen. Allerdings: Die meisten Antiimperialisten verstehen sich l\u00e4ngst nicht (mehr) als Autonome.\n\nDas gilt zunehmend allerdings auch f\u00fcr die \"Antideutschen\". Diese treten f\u00fcr die kritik- und bedingungslose Solidarit\u00e4t mit Israel ein und bewerten h\u00e4ufig auch die entsprechende Nahostpolitik der USA positiv. So ist es ihnen zu verdanken, dass nun auch auf linksradikalen Demonstrationen bisweilen USA-F\u00e4hnchen geschwungen werden.\n\nDie Mehrzahl der Autonomen sieht das freilich weniger amused: Sie lehnen die Verwendung von Nationalflaggen sowie Solidarit\u00e4tsbekundungen f\u00fcr Nationen oder nationale Befreiungsbewegungen prinzipiell ab, da Herrschaft und Unterdr\u00fcckung immanente Bestandteile des Konstruktes \"Nation\" seien. \"Mir gingen sie alle auf die Eierst\u00f6cke, die Selbstmordattent\u00e4ter genauso wie die fanatisierten Siedler. Weit mehr noch aber nervten mich diejenigen unter meinen Zeitgenossen, die diesen oder \u00e4hnliche Konflikte zum Anlass nahmen, Winkelemente der einen oder anderen Seite zu schwenken, ohne dass es daf\u00fcr auch nur einen nachvollziehbaren Grund gegeben h\u00e4tte. Egal, wie d\u00fcnn die Informationslage, wie gering die Kenntnisse der historischen Gegebenheiten oder das Wissen um das Geschehen vor Ort auch sein mochten, pawlowschen Hunden gleich f\u00fchlten sich diese neunmalklugen Tr\u00f6pfe im Krisen- oder Kriegsfall sogleich bem\u00fc\u00dfigt, Partei zu ergreifen und uns anderen ungebeten mitzuteilen, wer diesmal die Cowboys und wer die Indianer waren.\" (Off 2012: 106) \"Die Begriffe 'antideutsch' und 'Antiimperialist' sind heute innerlinke Fremdzuschreibungen, die nichts erkl\u00e4ren. Beispielsweise tauchen in Berlin periodisch diese ultranervigen zwanzigj\u00e4hrigen Politaktivisten auf, die mich wahlweise mit der Forderung nach uneingeschr\u00e4nkter Solidarit\u00e4t mit Israel oder der ebenfalls uneingeschr\u00e4nkten antiimperialistischen Solidarit\u00e4t mit Pal\u00e4stina nerven. Beide gehen mir auf den Keks. Beiden geht es nicht um die Sache, sondern um linke Identit\u00e4tsbildung, und beide nutzen daf\u00fcr den Nahostkonflikt. Meine Position als Autonomer war immer ganz klar antinational. Mit dem linksnationalistischen Antiimperialismusquatsch konnte ich nie etwas anfangen. Das ganze antiimperialistische Denkmuster, dieses manich\u00e4ische Weltbild, das den Globus in zwei Sph\u00e4ren aufteilt, ist mir fremd. F\u00fcr die klassischen Antiimps gibt es nur die Wei\u00dfen, die ausbeutenden Imperialisten der globalen Metropole, und die ausgebeuteten schwarzen, armen V\u00f6lker auf der anderen Seite. Dieses im kolonialistischen Befreiungskampf verhaftete Denken ebnet viele Widerspr\u00fcche v\u00f6llig ein. Die antideutsche Linke ist komplizierter. Ein Teil hat sich in der Auseinandersetzung um die zweite Intifada und den 11. September im Namen eines sehr fragw\u00fcrdigen Verst\u00e4ndnisses von Fortschritt und Moderne von der Linken verabschiedet. Sie glauben, dass die Alternative zum Elend des Kapitalismus nur noch die v\u00f6llige Barbarei sei. Mit dieser Position haben sie vor den Verh\u00e4ltnissen feige kapituliert und sind mit fliehenden Fahnen ins Lager der Liberalen und Neokonservativen gewechselt. Ein anderer Teil geh\u00f6rt nach wie vor zur radikalen Linken und widmet sich vor allem der Ideologiekritik. Das finde ich schon viel spannender! Dass es in der radikalen Linken heute \u00fcberhaupt eine so eine intensive Auseinandersetzung mit Antisemitismus, Antizionismus und Antiamerikanismus gibt, ist eben dieser Str\u00f6mung zu verdanken.\" (Bj\u00f6rn)\n\n### Autonome \u2013 eine Jugendszene?\n\nIn Verfassungsschutz-Publikationen wird immer wieder behauptet, die \"Anh\u00e4nger der Autonomen\" seien zwischen 15 und 30 Jahren alt (siehe Baron 2011: 239). Allerdings finden sich dort wie beim Verfassungsschutz \u00fcblich keinerlei Hinweise auf die empirische Basis und Quellen dieser Angaben. \"Die Teilnahme an einer beliebigen von Autonomen organisierten Veranstaltung oder Demonstration zeigt in jedem Fall, dass mit 30 bei den Autonomen noch lange nicht Schluss ist, es sich bei den Autonomen also in keinem Fall nur um eine Jugendbewegung handelt. Die Autonomen sind weder im demografischen noch im substanziellen Sinn eine Jugendbewegung. Das Altersspektrum der Aktivist_innen reicht weit \u00fcber die Jugendlichkeits-Grenze von \u00fcblicherweise 25 Jahren hinaus. Jugendkulturelle Stile und ein jugendkultureller Habitus spielen sicherlich f\u00fcr die kulturelle Selbstpositionierung vieler Aktivist_innen auch jenseits des Jugendalters eine wichtige Rolle. Allerdings gilt dies inzwischen f\u00fcr weite Teile der Gesellschaft, es handelt sich also nicht um ein Spezifikum der Autonomen, so dass in jedem Fall eine rein jugendkulturelle Perspektive zu kurz greift. Aus den Selbstzeugnissen von Aktivist_innen (Kongre\u00dflesebuchgruppe 1995; A.G. Grauwacke 2003) und in den Diskussionsbeitr\u00e4gen in autonomen Bewegungszeitschriften wird deutlich, dass die Attraktivit\u00e4t der Bewegung nicht in erster Linie in ihrem jugendkulturellen Appeal liegt, sondern daran, dass die Autonomen f\u00fcr eine radikale Kritik der bestehenden politischen und \u00f6konomischen Ordnung stehen. Wichtige Merkmale, die die anhaltende Attraktivit\u00e4t der autonomen Bewegung und damit auch ihre immer noch vorhandene Mobilisierungsf\u00e4higkeit ausmachen, sind sicherlich auch das Versprechen der Selbstbestimmung und Selbsterm\u00e4chtigung, die (relative) Abwesenheit von Hierarchien, die Bereitstellung (und Durchsetzung) von 'Freir\u00e4umen', die (nicht-kommerzielle) Orte f\u00fcr den Versuch der Verwirklichung alternativer Lebensweisen bieten, und sicherlich auch die Tatsache, dass die Aktionsorientierung der Autonomen einen Abenteuer- und Erlebnisraum jenseits der vorgegebenen Ordnung bietet.\" (Haunss 2013: 39f.)\n\nDennoch ist die autonome Szene stark jugendkulturell gepr\u00e4gt: War es in den 1980er Jahren noch der Punk, der Autonome in Bewegung setzte, eroberten in den Neunzigern parallel zum jugendkulturellen Mainstream elektronische Beats die Herzen der Szene. Heute spiegelt die autonome Szene l\u00e4ngst die Vielfalt und Multikulturalit\u00e4t der aktuellen Jugend wider: Von Hardcore\u00fcber Reggae- bis zu Schlagerpartys ist alles m\u00f6glich (nur Deutschrock geht irgendwie gar nicht...), Dreadlocks treffen auf Glatzen, Punks auf Hipster, in G-Star-Camouflage gewandete Antifas auf \"Bacon had a mom\"-T-Shirt-Tr\u00e4gerinnen. Schwarz ist immer noch mehr als eine Farbe, hat aber ihr Monopol verloren. Wie in allen anderen Jugendkulturen gibt es auch hier immer mehr Menschen \u00fcber 40, 50, 60 Jahre, die oft das organisatorische R\u00fcckgrat der Szenen bilden. Doch kaum jemand st\u00f6\u00dft erst im fortgeschrittenen Alter dazu. Insofern ist der Begriff Jugendszene oder -kultur vielleicht unvollkommen, aber nicht komplett falsch. Es sind fast ausschlie\u00dflich junge Menschen, die sich emp\u00f6ren, auf der Suche nach Gleichgesinnten in autonomen Zusammenh\u00e4ngen landen und dann gemeinsam mit anderen ihrer Emp\u00f6rung Taten folgen lassen.\n\n### Literatur\n\nDie Autonomen sind mittlerweile eine der am l\u00e4ngsten kontinuierlich aktiven Sozialen Bewegungen Nachkriegsdeutschlands. Sie k\u00f6nnen seit Ende der 1970er Jahre auf eine \u00fcber 30-j\u00e4hrige Geschichte zur\u00fcckblicken. Dennoch ist die Literatur, die sich mit dieser Bewegung besch\u00e4ftigt, recht \u00fcberschaubar. Neben Darstellungen von (ehemaligen) Aktivist_innen gibt es nur wenige Arbeiten, die sich wirklich ernsthaft, ergebnisoffen und multiperspektivisch mit der Szene besch\u00e4ftigen. Fast alle wissenschaftlichen und anderen Arbeiten fokussieren auf den Aspekt der Gewalt. \"Die dominante Perspektive, unter der die Autonomen in der \u00d6ffentlichkeit wahrgenommen werden, ist die der Gewalt. Kaum ein Zeitungsbericht oder Fernsehbeitrag \u00fcber die Autonomen kommt ohne Bilder brennender Barrikaden oder schwarz vermummter Demonstrant_innen aus. Dieser Ausgangspunkt strukturiert auch die Extremismusforschung (z. B. Pfahl-Traughber 1998; Baron 2011) und die Berichterstattung der Verfassungsschutz-\u00c4mter auf Bundes- und Landesebene \u2013 wobei beide Perspektiven sowohl inhaltlich als auch personell nicht immer klar zu unterscheiden sind.\" (Haunss 2013: 27) \u2013 Sprich: Hauptamtliche Mitarbeiter des Verfassungsschutzes firmieren in ihren Fachaufs\u00e4tzen, Rezensentent\u00e4tigkeiten etc. gerne als unabh\u00e4ngige Wissenschaftler. \"Auch in der sozialwissenschaftlichen Forschung bildet die offensiv propagierte Militanz der Autonomen oft den Anlass, sich mit der Bewegung zu besch\u00e4ftigen (Bock 1989; Busch 1989; Paris 1991; Rucht\/Teune 2008; Leach\/Haunss 2010). Neben vereinzelten Selbstzeugnissen von Aktivist_innen gibt es keine sozialwissenschaftliche Forschung \u00fcber die Gr\u00fcnde und Motive, sich den Autonomen anzuschlie\u00dfen und\/oder diese wieder zu verlassen. Es gibt kaum Untersuchungen, die sich mit der Interaktion von Autonomen und anderen Bewegungen besch\u00e4ftigt haben.\" (Haunss 2013: 27ff.) (Eine umfangreiche Literaturliste enth\u00e4lt dieses Buch im Anhang.) \"Der Versuch, die Autonomen zentral als gewaltbereit oder gewaltt\u00e4tig darzustellen, ist ersichtlich Bestandteil einer Kriminalisierungsstrategie und als solcher zu analysieren und zu kritisieren. Zudem verkennt dies die genuin politischen Motive, die f\u00fcr die autonome Szene grundlegend sind. An die Stelle einer politischen Auseinandersetzung mit den Autonomen tritt damit der Versuch, sie straf- und verfassungsrechtlich zu kontrollieren.\", meint der Soziologe Albert Scherr (das komplette Interview ist im Anhang dokumentiert).\n\nDieses im Kontext des Archiv der Jugendkulturen entstandene Buch ist ein Versuch, dies zu \u00e4ndern. \"Publikationen des Archiv der Jugendkulturen zeichnen sich \u00fcblicherweise dadurch aus, dass sie fr\u00fchzeitig \u2013 bevor die Wissenschaft und die seri\u00f6se Publizistik das Thema entdeckt \u2013 auf neue, gesellschaftlich relevante Entwicklungen im Umfeld jugendlicher Kultur- und Freizeitwelten aufmerksam machen\", schrieb einmal das Forschungsjournal Neue Soziale Bewegungen. \"Dass das kleine private Institut so schnell wie fundiert auf Trends reagieren kann, liegt an der immer wieder bewusst gesuchten N\u00e4he zu den Szenen selbst. Selbst Angeh\u00f6rige \u00fcblicherweise verdeckt agierender Szenen wie Satanisten oder Neonazis werden \u2013 trotz offen bekundeter inhaltlicher Distanz \u2013 mit dem Versprechen gek\u00f6dert, ausf\u00fchrlich und unzensiert zu Wort zu kommen. So sind die Ver\u00f6ffentlichungen des Archiv der Jugendkulturen stets sehr genaue, authentische Abbilder der subkulturellen Realit\u00e4t vor allem von Szenen, an die sich 'normale' ForscherInnen oft gar nicht herantrauen oder schon aufgrund ihres Habitus keine Chancen haben \u2013 und damit eine unersch\u00f6pfliche und unverzichtbare Quelle f\u00fcr die Arbeit eben dieser anderen WissenschaftlerInnen, aber auch f\u00fcr politische BildnerInnen, Medien- und Bewegungsmenschen.\"\n\n### Fazit\n\nKaum jemand \u2013 nicht einmal viele autonome Aktivist_innen selbst \u2013 glaubt, dass die Autonomen heute noch eine gesellschaftsver\u00e4ndernde Kraft darstellen. Zum Teil liegt das an der Szene selbst, meint Leonie: \"Die Szene definiert sich ganz klar dar\u00fcber, dass sie was Besseres sind als andere Leute, dass sie sich als was Besseres f\u00fchlen dadurch, dass sie sich so klar abgrenzen und dadurch einen bestimmten Status f\u00fcr sich innehaben, der \u00fcber ein innen\/au\u00dfen, gut\/schlecht funktioniert. Das klingt jetzt ganz b\u00f6se, aber ich find das schon ungew\u00f6hnlich in Deutschland, wie das hier in der autonomen Szene abl\u00e4uft zum Teil und wie die Szene auch so auftritt in der Art, dass Leute in der Nachbarschaft nicht unbedingt das Gef\u00fchl haben, mit denen will ich mich jetzt befreunden oder das sind ganz nette Jungs und M\u00e4dels. In anderen L\u00e4ndern ist der Schwerpunkt viel mehr darauf gerichtet, wie kommen wir r\u00fcber, wie k\u00f6nnen wir das vermitteln an Leute, dass die das verstehen, also viel mehr von der anderen Perspektive gedacht wird und nicht von der eigenen Perspektive, sondern gekuckt wird, wenn du jetzt die Oma von nebenan bist, wie soll die denn diese Parole verstehen und wie bereitet man die Sachen auf, dass da eine Kommunikation stattfindet. Deshalb frage ich mich auch manchmal bei der autonomen Szene, ist denn die Idee \u00fcberhaupt, die Gesellschaft zu ver\u00e4ndern, oder ist die Idee einfach nur, seinen eigenen Freiraum zu haben, aber auch nur f\u00fcr bestimmte Leute, die in erster Linie wei\u00df sind oder aus der Mittelklasse kommen, also das ist eine ganz klare demografische Tatsache. Diese ganze autonome Szene ist schon sehr privilegiert und ich finde, da fehlt manchmal so ein bisschen die Selbstkritik.\"\n\nF\u00fcr den Berliner Autonomen Bj\u00f6rn ist \"die Bildung von kritischem Bewusstsein\" von zentraler Bedeutung. \"Die gesellschaftlichen Verh\u00e4ltnisse werden heute von den meisten Menschen so selbstverst\u00e4ndlich wie ein Naturgesetz, wie die Schwerkraft hingenommen. Damit stirbt auch der Gedanke daran, dass es anders werden kann und dass Menschen um ihre Befreiung k\u00e4mpfen k\u00f6nnen. Ich glaube, dass es heute vor allem darum geht, die Idee der menschlichen Emanzipation am Leben zu halten. Etwas anderes als die Anziehungs- und \u00dcberzeugungskraft ihrer Ideen kann die radikale Linke eh nicht bieten. Die Autonomen k\u00f6nnen nur so etwas wie Impulsgeber sein. Sie sind ein Laboratorium subversiver Ideen und Praktiken. In bestimmten gesellschaftlichen Situationen, wie zum Beispiel in Griechenland vor drei Jahren, k\u00f6nnen sie zeitweise als Avantgarde von sozialen Bewegungen agieren. Aber gerade das Beispiel Griechenland zeigt die Begrenztheit der Autonomen. Jenseits vom Aufstand bekommen sie nichts hin. Es mangelt ihnen schlicht an einer langfristigen Strategie und einer Perspektive jenseits der Revolte. Johannes Agnoli hat mal \u00fcber die Autonomen gesagt, dass sie die Richtigen sind, die alles falsch machen. Ich glaube, das stimmt.\"\n\n\"Das Schlimmste ist die Gleichg\u00fcltigkeit. 'Ohne mich' ist das Schlimmste, was man sich und der Welt antun kann. Den 'Ohne mich'-Typen ist eines der absolut konstitutiven Merkmale des Menschen abhandengekommen: die F\u00e4higkeit zur Emp\u00f6rung und damit zum Engagement\" (Hessel 2011a: 13), mahnt St\u00e9phane Hessel, der als junger Widerstandsk\u00e4mpfer der Resistance miterleben musste, wie die Mehrzahl der Franzosen zur Zeit der Kollaborationsregierung von Vichy den Nazis gegen\u00fcber gleichg\u00fcltig sich ins \"unpolitische\" Private zur\u00fcckzogen \u2013 und damit alle Verbrechen zulie\u00dfen. \"Auch heute begegnen wir diesem Gegensatz zwischen militanten Pionieren des Widerstands und der passiven Masse. Ich bin versucht zu sagen, dass h\u00f6chstens 10 bis 20 Prozent der Menschen sich wirklich bewegen, um etwas Neues zustande zu bringen, und die anderen laufen dann eben mit. Und das ist bereits eine eher optimistische Aussage.\" (Hessel 2011b: 26)\n\nUnd doch steckt in den Worten von St\u00e9phane Hessel ganz nebenbei noch eine Optimismus rechtfertigende Aussage: \"... und die anderen laufen dann eben mit.\" Niemand kann heute bestreiten, dass die Bewegung der Achtundsechziger einigen L\u00e4ndern einen sp\u00fcrbaren evolution\u00e4ren Kick in Richtung Liberalisierung gegeben hat (auch wenn manches davon l\u00e4ngst wieder zur\u00fcckgedreht wurde). Und doch waren die linkspolitisierten Achtundsechziger seinerzeit realistisch betrachtet nur eine kleine Minderheit innerhalb der deutschen, franz\u00f6sischen etc. Gesellschaften. Die reichhaltige Landschaft zivilgesellschaftlichen Engagements w\u00e4re ohne den Aufbruch der Achtundsechziger nicht m\u00f6glich \u2013 war doch noch in den sp\u00e4ten Sechzigern die Mehrheit der westdeutschen Bev\u00f6lkerung der Meinung, Demonstrationen geh\u00f6ren nicht in eine Demokratie. Die deutsche Anti-AKW-Bewegung der 1980er Jahre war ebenso eine kleine, radikale, kriminalisierte Minderheit \u2013 doch ohne ihren militanten Widerstand gegen den atomaren Wahnsinn w\u00e4re der heute von einer konservativ gef\u00fchrten Regierung beschlossene (wenn auch nur halbherzig umgesetzte) Atomausstieg nicht denkbar. Dass sich heute liberal-konservative und populistische Politiker wie Guido Westerwelle oder Berlins Ex-B\u00fcrgermeister Klaus Wowereit als homosexuell outen und die gleichgeschlechtliche Partnerschaften diskriminierenden Gesetze ver\u00e4ndert wurden, ist ein Verdienst der radikalen Schwulen- und Lesbenbewegung fr\u00fcherer Jahrzehnte. So greift die autonome Bewegung auch heute zentrale Fragen unserer Zukunft auf und kritisiert unerm\u00fcdlich gegenw\u00e4rtige Fehlentwicklungen: Wie wollen wir zuk\u00fcnftig leben? Wie k\u00f6nnen wir heute schon besser miteinander leben und umgehen, ohne viele einzelne Menschen und die Erde als Ganzes zu zerst\u00f6ren? Warum seht Ihr weg, wenn vor Euren Augen Rassismus, Gewalt und anderes Unrecht geschieht?\n\nSie stellen diese Fragen nicht immer mit den richtigen Worten (so dass sie Menschen guten Willens, die nicht schon ein Dutzend autonome Plena besucht haben, auch verstehen k\u00f6nnen), und auch ihre Antworten sind nicht immer tauglich. \"Widerstand kann Berufung sein, doch Gestaltung geht dar\u00fcber hinaus. Aufbau ist die Fortsetzung des Widerstandes mit anderen Mitteln. Lebensweisheit ist das Gebot der Stunde. Man kann nicht nur Yin oder Yang sein; man muss sich im Rhythmus von beiden bewegen.\" (Hessel 2011b: 31) Sie zerfleischen sich bisweilen in absurden selbstreferenziellen Diskursen und begegnen sich und anderen in Formen, die weit vom \"Neuen Menschen\" entfernt sind. Sie sind oft dogmatisch, rechthaberisch, unsolidarisch mit Nicht-Szeneangeh\u00f6rigen, uniformierte Mitl\u00e4ufer, prollige Machos oder arrogante Intellektuelle. Von den eigentlichen Zielen einer gerechten Ver\u00e4nderung der Welt abgekapselt, haben sich in der autonomen Szene \"Wichtigtuer, Sektierer, Privilegienwahrer\" (Hessel 2011b: 31) breitgemacht. Sie gefallen sich oft so sehr in der Au\u00dfenseiter- und \"Randgruppe der Gesellschaft\"-Pose, dass sie gar nicht mehr wollen, dass andere ihrem Beispiel folgen, f\u00fcrchten sie doch den Verlust der liebgewonnenen Exklusivit\u00e4t.\n\nVieles davon sind schlicht jugendgem\u00e4\u00dfe \u2013 um nicht etwas despektierlicher \"pubert\u00e4re\" zu sagen \u2013 Verhaltensweisen, die schon lange \u00fcber das biologische Jugendalter hinausgewachsen sind. Im Gegenzug sind Junge noch zu einer Emp\u00f6rung und Radikalit\u00e4t f\u00e4hig, die sich auf Empathie \u2013 die F\u00e4higkeit, am Leid anderer Lebewesen mitzuleiden \u2013 und nicht blo\u00dfe Ratio st\u00fctzt, eine wichtige Ressource, die bei der Mehrheit der Erwachsenen schon verbraucht zu sein scheint, wie k\u00f6nnten sie \u2013 wir \u2013 sonst so viel Unrecht tagt\u00e4glich ertragen. Oder vielleicht sind auch nur durch zu viel Brot & Spiele und die notgedrungen konservative Hineinsozialisierung in die erwachsene Konkurrenz- und Spa\u00dfgesellschaft die Synapsen nicht mehr mit den entsprechenden Hirnregionen verbunden und es bedarf eines kr\u00e4ftigen Stromschlages, um die schlummernde Kraft der Emp\u00f6rung zu wecken?!\n\nOhnehin schon Reiche werden immer reicher, Arme immer \u00e4rmer, junge Menschen werden in sozialdarwinistischen Ausleseanstalten, genannt Schule, jetzt schon fast \u00fcberall ganzt\u00e4gig kaserniert, vitale Alte aus dem beruflichen und gesellschaftlichen Aktivleben in Wellnessrefugien abgeschoben, die Kindersterblichkeit liegt in den proletarischen Suburbs deutscher Gro\u00dfst\u00e4dte dreimal so hoch wie in den gutb\u00fcrgerlichen Vierteln \u2013 angesichts der horrenden strukturellen Gewalt in unserer Gesellschaft kann man selbst als Pazifist die Militanz der autonomen Szene ganz entspannt frei nach Bert Brecht bewerten: Was ist das Scheiben einwerfen bei einer Bank gegen\u00fcber der Gesch\u00e4ftsf\u00fchrung einer Bank? Oder, wie es der autonome Viktor formuliert: \"Wenn man in Amerika die Leute sieht, die mit 500 kg auf den H\u00fcften rumrennen, und dagegen die afrikanischen Kinder verhungern sieht, will man schon mehr ver\u00e4ndern.\"\n\n\"Die Welt ist keine Gerade\", las ich gestern auf einem T-Shirt. Sie hat Kurven und Buckel und gelegentlich un\u00fcberwindbare Gr\u00e4ben, die umgangen werden m\u00fcssen. Gelegentlich muss man erst zwei Schritte zur\u00fcckgehen, um vorw\u00e4rtszukommen. Autonome sind in vielerlei Hinsicht Experten darin, zwei Schritte zur\u00fcckzugehen, erreichte Ziele wieder einzurei\u00dfen, nach dem erfolgreichen Sprung \u00fcber die Barrikade deren Steine zu nutzen, um gleich vor sich eine neue aufzubauen. Doch vielleicht ist das einfach unabdingbar, um vorw\u00e4rtszukommen. Wer den vorgegebenen Pfad nicht verl\u00e4sst, entdeckt auch keine neuen Wege und verf\u00fcgt damit auch nicht \u00fcber einen Plan B, wenn sich der breite Pfad als Sackgasse entpuppt. Wie hei\u00dft es so sch\u00f6n: Nur wer nichts tut, macht keine Fehler. Doch im Nichtstun sind wir alle \u2013 die meisten von uns \u2013 schon selbst ganz gro\u00dfartig. Wenn uns eines Tages bewusst wird, dass dieses Nichtstun das Starren auf die Mauer am Ende der Sackgasse ist, wird es Stimmen wie die der Autonomen und anderer radikaler Abweichler_innen bed\u00fcrfen, uns alternative Wege aus der Falle zu offenbaren.\n\n### Literaturverzeichnis\n\nA.G. Grauwacke: Autonome in Bewegung. Aus den ersten 23 Jahren. Assoziation A, Berlin 2003.\n\nBaron, Udo: \"Die linksautonome Szene\", in: Dovermann, Ulrich (Hrsg.): Linksextremismus in der Bundesrepublik Deutschland. Bundeszentrale f\u00fcr Politische Bildung, Bonn (2011), S. 231\u2013245.\n\nBock, Marlene\/Reimitz, Monika\/Richter, Horst-Eberhard: Zwischen Resignation und Gewalt \u2013 Jugendprotest in den achtziger Jahren. Leske + Budrich, Opladen 1989.\n\nBusch, Heiner: \"Die B\u00fcrgerinitiative in der Lederjacke. Anmerkungen zu den Autonomen\", in: Vorg\u00e4nge 101, Jg. 28\/1989, Heft 5, S. 62\u201367.\n\nFarin, Klaus\/M\u00f6ller, Kurt (Hrsg.): Kerl sein. Kulturelle Szenen und Praktiken von Jungen. Archiv der Jugendkulturen, Berlin 2014.\n\nGeronimo: Feuer und Flamme. 1: Zur Geschichte und Gegenwart der Autonomen. ID-Verlag im IISG, Amsterdam 1990.\n\nHan, Byung-Chul: \"Warum heute keine Revolution m\u00f6glich ist\", in: S\u00fcddeutsche Zeitung vom 2. September 2014. Online unter: www.sueddeutsche.de\/politik\/2.220\/neoliberales-herrschaftssystem-warum-heute-keine-revolution-moeglich-ist-1.2110256 [21.11.2014].\n\nHaunss, Sebastian: Identit\u00e4t in Bewegung. Prozesse kollektiver Identit\u00e4t bei den Autonomen und in der Schwulenbewegung (B\u00fcrgergesellschaft und Demokratie Band 19). VS, Wiesbaden 2004.\n\nHaunss, Sebastian: \"Die Autonomen \u2013 eine soziale Bewegung zwischen radikaler Gesellschaftskritik und Subjektivismus\", in: Schultens, Ren\u00e9\/Glaser, Michaela (Hrsg.): \"Linke\" Militanz im Jugendalter. Befunde zu einem umstrittenen Ph\u00e4nomen. Deutsches Jugendinstitut e. V., Halle 2013, S. 26-46.\n\nHaunss, Sebastian: \"Wie male ich einen Autonomen? M\u00e4nnlichkeitsbilder auf Plakaten der autonomen Bewegung\", in: Farin, Klaus\/M\u00f6ller, Kurt (Hrsg.): Kerl sein. Kulturelle Szenen und Praktiken von Jungen. Archiv der Jugendkulturen, Berlin 2014, S. 183-197.\n\nHessel, St\u00e9phane: Emp\u00f6rt Euch! Ullstein, Berlin 2011a.\n\nHessel, St\u00e9phane: Engagiert Euch! Ullstein, Berlin 2011b.\n\nInterim, Berlin, Nr. 161 vom 19.9.91 und Nr. 162 vom 27.9.91.\n\nID-Archiv im IISG (Hrsg.): 2.11.87... Dokumentation: Berichte, Stellungnahmen, Diskussionen zu den Sch\u00fcssen an der Startbahn. Stichting Beheer IISG, Amsterdam 1988.\n\nKongresslesebuchgruppe: Der Stand der Bewegung \u2013 18 Gespr\u00e4che \u00fcber linksradikale Politik. Lesebuch zum Autonomie-Kongre\u00df 1995. Eigenverlag, Berlin 1995.\n\nLeach, Darcy K.\/Haunss, Sebastian: \"'Wichtig ist der Widerstand': Rituals of Taming and Tolerance in Movement Responses to the Violence Question\", in: He\u00dfd\u00f6rfer, Florian\/Pabst, Andrea\/Ullrich, Peter (Hrsg.): Prevent and Tame. Protest under (Self-)Control. Dietz, Berlin 2010, S. 73\u201398.\n\nMichels, Robert: Zur Soziologie des Parteiwesens in der modernen Demokratie. Untersuchungen \u00fcber die oligarchischen Tendenzen des Gruppenlebens. W. Klinkhardt, Leipzig 1911.\n\nOff, Jan: Happy Endstadium. Ventil, Mainz 2012.\n\nParis, Rainer: \"Vermummung\", in: Leviathan, Jg. 1\/1991, S. 117\u2013129.\n\nPfahl-Traughber, Armin: \"Die Autonomen. Portrait einer linksextremistischen Subkultur\", in: Aus Politik und Zeitgeschichte, B 9\u201310\/1998, S. 36-46.\n\nRinas, Bernd: Veganismus: Ein postmoderner Anarchismus bei Jugendlichen? Archiv der Jugendkulturen, Berlin 2012.\n\nRucht, Dieter\/Teune, Simon (Hrsg.): Nur Clowns und Chaoten? Die G8-Proteste in Heiligendamm im Spiegel der Massenmedien. Campus, Frankfurt am Main\/New York 2008.\n\nStrobl, Ingrid\/Viehmann, Klaus und GenossInnen\/autonome l.u.p.u.s.-Gruppe: Drei zu Eins. Metropolen(gedanken) und Revolution? ID-Archiv, Berlin\/Amsterdam 1993.\n\nWagner, Karl, in: 2052. Der neue Bericht an den Club of Rome. Eine globale Prognose f\u00fcr die n\u00e4chsten 40 Jahre. 2012. Online unter: www.clubofrome.org\/?p=703 [21.11.2014].\n\n## \"Ruf niemals um Hilfe, es k\u00f6nnte die Polizei kommen!\"\n\nWilly, 49 Jahre, lebt in Hamburg.\n\nWilly wurde mir von einem Freund vorgestellt. Wir treffen uns auf einem Bauwagenplatz; er wollte mich einen Tag vorher n\u00e4her kennenlernen, herausfinden, ob er mir vertrauen kann. Als mir Willy gegen\u00fcberstand, war ich erstaunt. So sieht also ein Autonomer aus?! Nach seinem \"Tach\" war es still zwischen uns, keiner wusste so recht, was er sagen sollte. Ich habe dann einfach vom Buchprojekt geplappert und wie toll ich das finde, dass er sich dazu bereit erkl\u00e4rt hat. Sein Kommentar: \"Ja ja!\"\n\nWilly ist ungef\u00e4hr 1,85 m gro\u00df, sein Haar ist raspelkurz und schwarz gef\u00e4rbt, er sieht aber eindeutig j\u00fcnger als 49 Jahre aus. Sein schlanker und sehniger K\u00f6rper steckt in einer schwarzen Dachdeckerhose, einem dunkelblauen Kapuzenpullover und einer schwarzen Jacke. Er hat ein sehr ernstes Gesicht und eine steile Falte zwischen den Augenbrauen. Seine Schuhe machen den Eindruck, als k\u00f6nnten sie einiges aushalten.\n\nWilly mustert mich ziemlich lange und meint dann nur: \"Meinetwegen k\u00f6nnen wir auch heute das Interview machen, morgen hab ich ein Konzert.\" Also machen wir uns auf den Weg in seine Wohnung. Bisschen mulmig ist mir schon, aber die Neugier siegt. Das Haus, in dem er wohnt, ist mit zusammengeschwei\u00dften Fahrradrahmen vergittert worden. Der Hausflur ist mit Plakaten beklebt und der Schriftzug \"No pasaran\" ist auf jeder Etage zu finden, ein Schlachtruf aus dem spanischen B\u00fcrgerkrieg.\n\nWilly wohnt im dritten Stock, auf dem Weg dahin bellt uns hinter jeder verschlossenen T\u00fcr ein Hund an. Als wir seine Wohnung betreten, bin ich schon wieder erstaunt: Er ist ordentlicher als ich. Willy kocht gleich zwei Kaffee, t\u00fcrkisch nat\u00fcrlich, alles andere ist Schnickschnack, wie er so sch\u00f6n sagt. Seine Wohnung besteht aus zwei R\u00e4umen, einer K\u00fcche mit einer Dusche in der Ecke und seinem Zimmer. Wir setzen uns in sein Zimmer und ich sitze seinem riesigen Plattenregal gegen\u00fcber. Es misst ungef\u00e4hr drei mal vier Meter. Die St\u00fchle knarren und anfangs sitzen wir sehr angespannt auf der Stuhlkante. Aber wir entspannen uns zunehmend, lehnen uns bald auch mal zur\u00fcck und sind uns nach dem Interview sogar ein bisschen vertraut; wir gehen von Kaffee und Zigaretten auf Bier und Joint \u00fcber.\n\n#### Meine erste Frage lautet: Wie bewertest du deine aktuelle Situation bzw. deine Zukunft?\n\nBin arbeitslos seit ein paar Monaten. Meine Zukunftspl\u00e4ne sind, einen neuen Job finden.\n\n#### Hast du da einen Traumberuf?\n\nT\u00e4towierer! Ich spiel noch in einer Band, nein, in zwei Bands, und dass da vielleicht mal was rumkommt. Also nix Ber\u00fchmtes, aber ausbauf\u00e4hig!\n\n#### Ihr habt Erfolg?\n\nJa.\n\n#### Gibt es etwas, wovor du Angst hast?\n\nKrankheiten. Mein Freund ist gerade sehr krank, er hat zum zweiten Mal Krebs und ihm geht's richtig schlecht. Das ist schwer zu ertragen, ihn so leiden zu sehen.\n\n#### Was \u00e4rgert dich am meisten, was macht dich so richtig w\u00fctend?\n\nIntolerante Menschen, schreiende Kinder am fr\u00fchen Morgen... Sonst f\u00e4llt mir gerade nichts ein.\n\n#### Wieso schreiende Kinder? Wohnen hier im Haus Kinder?\n\nN\u00f6, aber rings um unser Haus sind jetzt Kinderg\u00e4rten, da schreien schon morgens um acht die Kinder. Zuerst weil sie da nicht rein wollen, dann um zehn schreien die, weil sie nicht raus wollen, und nachmittags schreien sie, weil sie nicht nach Hause wollen. Das nervt!\n\n#### M\u00f6chtest du selbst Kinder haben?\n\nN\u00f6, ist mir zu viel Verantwortung, hab mit mir genug zu tun.\n\n#### Was w\u00fcrdest du anders machen als deine Eltern, wenn du Kinder h\u00e4ttest?\n\nToleranter sein. Ich durfte keine langen Haare haben und nicht anziehen, was ich wollte, bekam vorgeschrieben, mit wem ich nicht befreundet sein sollte. Selbst meine Musik, die ich so gerne geh\u00f6rt habe, hat die gest\u00f6rt. Und ich habe h\u00e4usliche Gewalt als Kind erfahren. Wir sind im Osten aufgewachsen, mein Vater war Bulle und wir waren f\u00fcnf Geschwister, ich war der J\u00fcngste und habe am meisten abbekommen.\n\n#### Wurdest du geschlagen?\n\nMein Vater hat gesoffen wie ein Loch und uns derbe verpr\u00fcgelt. Selbst meine Mutter. St\u00e4ndig mussten wir daf\u00fcr sorgen, dass genug Schnaps im Haus war.\n\n#### Musstet ihr seinen Schnaps kaufen? Als Kinder?\n\nJa, die kannten uns schon \u00fcberall. Ich habe mit acht Jahren schon Schnaps kaufen d\u00fcrfen, heute ein Unding!\n\n#### War das ein Ausl\u00f6ser f\u00fcr deine politische Entwicklung?\n\nIch denke schon, ja. Gekoppelt mit der Anti-Haltung, dass mein Vater Bulle war. Das hat mich schon fr\u00fch getrieben. Als ich alt und stark genug war, habe ich gelernt, mich gegen ihn zu wehren. Ich habe dann trotzdem meine Musik geh\u00f6rt, habe angezogen, was ich wollte, und war befreundet, mit wem ich wollte. Und ich habe auch zur\u00fcckgeschlagen. Als er einmal sturzbetrunken auf einem B\u00fcrgersteig lag, mitten in der Stadt, bin ich \u00fcber ihn r\u00fcbergestiegen und hab ihn angerotzt. Da war null Achtung und kein Mitleid bei mir, er war einfach Luft f\u00fcr mich! Zu der Zeit habe ich auch angefangen, mich gegen den Staat und dessen Diktatur zu wehren. Wir waren ziemlich viele Freunde und haben da schon Aktionen gestartet, wie Plakate gegen das atomare Wettr\u00fcsten kleben (wir haben die so gut angeklebt, dass die Drecksbullen die nicht abbekommen haben, da mussten sich echt jeweils zwei Bullen vor die Plakate stellen, dass die keiner lesen konnte), geheime Konzerte veranstaltet und immer \u00fcberlegt, wie wir dem Staat eins auswischen k\u00f6nnen. 1983 haben sie uns dann verhaftet. Alle. Wir sind zu Hause verhaftet worden mit dem Vorwurf der versuchten Republikflucht, das war nat\u00fcrlich ein Schock. Die Verh\u00f6re waren dementsprechend, und wir haben alle seelische und k\u00f6rperliche Sch\u00e4den davongetragen. Nat\u00fcrlich haben wir eine Entsch\u00e4digung bekommen, als wir in den Westen verkauft wurden, aber was n\u00fctzt das? Wir waren echt im Arsch!\n\n#### Bist du jetzt, nach so vielen Jahren, einigerma\u00dfen dar\u00fcber hinweg?\n\nDas Leben ging ja weiter. Wir sind fast alle nach Hamburg gezogen, und da war und ist die Hafenstra\u00dfe. [Grinst.]\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nEin bisschen angepunkt, ein alter Freak halt! Ich bin ein Schwuler, dem man das nicht gleich ansieht. Aber nicht missverstehen, ich versteck mich nicht, ich geh damit sehr offen um. Wer es unbedingt wissen will, bekommt es auch gesagt; aber ansonsten bin ich der Meinung, dass man Menschen nicht an ihrer Sexualit\u00e4t festmachen sollte.\n\n#### Hast du Hobbys?\n\n[Er lacht und grinst dann frech.] Das hab ich dir vorhin schon gesagt, nicht zugeh\u00f6rt, was? Ich spiele in zwei Bands und t\u00e4towiere.\n\n#### Bist du in einem Verein aktiv?\n\nJa. In \"Pappnasen e. V.\"\n\n#### Und was ist das?\n\nEin gemeinn\u00fctziger Verein, wir haben eine Kneipe mit humanen Preisen, humanen Eintrittspreisen. Wenn wir Konzerte veranstalten, arbeiten wir ehrenamtlich hinterm Tresen und kochen f\u00fcr die Bands.\n\n#### Ich hab geh\u00f6rt, dass ihr auch einmal w\u00f6chentlich ein \"perfektes Punker Dinner\" veranstaltet, wie kann ich mir das vorstellen?\n\nWir kochen ein Drei-G\u00e4nge-Men\u00fc mit den Streeties [Streetworkern] und laden ein; wer kommen mag, kann kommen.\n\n#### Coole Idee!\n\nJa, find ich auch.\n\n#### Bist du anderweitig gesellschaftlich bzw. politisch aktiv?\n\nNicht direkt in Gruppen. Ich geh auf Demos, sobald ich h\u00f6re, dass wieder eine Aktion am Start ist...\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nKeine so starke mehr wie fr\u00fcher, da war es sozusagen meine Lebensgrundlage. Im Osten gab es nicht so viele gute Bands, da mussten wir heimlich Musikladen schauen und West-Radio h\u00f6ren und die Musik dann irgendwie aufnehmen; es gab Bands, da hatten wir immer das Gef\u00fchl, die spielen ihre Musik extra f\u00fcr uns.\n\n#### Zum Beispiel?\n\nTon Steine Scherben zum Beispiel, die haben uns sehr gepr\u00e4gt. \"Keine Macht f\u00fcr Niemand\" war der Song f\u00fcr uns; wir haben mal drei Tage durchgefeiert und s\u00e4mtliche Parolen an die W\u00e4nde eines Hauses geschmiert, das gab Theater!\n\n#### Hast du eine bestimmte Lieblingsmusik?\n\nDa ich selbst gerne Punk spiele, h\u00f6re ich es auch gerne.\n\n#### Gehst du selbst auch zu Konzerten?\n\nJa, auf jeden Fall. Aber da bin ich nicht so festgelegt. Wenn mir etwas empfohlen wird, gehe ich auch mit Freunden hin, muss nicht immer Punk sein.\n\n#### Gehst du gezielt zu Konzerten, wo Bands spielen, die in der autonomen Szene aktiv sind?\n\nJa.\n\n#### Welche?\n\nMuss ich das sagen?\n\n#### Nein.\n\nM\u00f6chte ich auch nicht.\n\n#### Interessierst du dich f\u00fcr Mode oder speziell f\u00fcr K\u00f6rperschmuck?\n\nN\u00f6.\n\n#### Aber wenn ich deine Tattoos sehe... ?\n\nAch so, ja, ich wei\u00df jetzt nicht, ob das als Mode aufgefasst wird, aber f\u00fcr mich ist es keine Mode. Da ich selbst t\u00e4towiere und auch selbst gehackt bin, hat f\u00fcr mich alles eine Bedeutung. Ich mach das nicht, weil ich es schick finde. Schick ist f\u00fcr mich ein Schei\u00df!\n\n#### Hast du Vorbilder? Oder Leitbilder?\n\nNee.\n\n#### Da gibt's niemanden, an dem du dich orientierst oder so?\n\nN\u00f6. Es gibt Leute, vor denen ich ordentlich Respekt habe, die ich bewundere, die was in ihrem Leben geschafft haben, aber direkt als Vorbild w\u00fcrde ich die nicht bezeichnen.\n\n#### Schlie\u00dft du schnell neue Freundschaften?\n\nNein, gar nicht, das dauert ganz lange bei mir.\n\n#### Und welche Kriterien sind da f\u00fcr dich wichtig?\n\nToleranz, Ehrlichkeit, Sympathie und Aufgeschlossenheit. Ich m\u00f6chte mich auf meine Freunde verlassen k\u00f6nnen, ihnen vertrauen. Das geht bei Leuten nicht, die ich gerade erst kennengelernt habe; man wei\u00df ja nicht gleich, wie die ticken. Au\u00dferdem wei\u00df man auch nicht, ob das Spitzel sind, bin da aus der Zone gepr\u00e4gt, die hatten eine Menge Spitzel auf uns angesetzt und die sahen genauso aus wie wir, die waren unsere Freunde. Und warum sollte der VS anders ticken?\n\n#### Geh\u00f6rst du einer bestimmten Jugendszene an bzw. gibt es bestimmte Szenen, die dir sympathisch sind?\n\n[Z\u00f6gern und Nachdenken.] Nee. Ich h\u00e4ng halt viel mit Punks rum. Aber nicht nur! Also nein, eigentlich passe ich in keine Schublade. Dar\u00fcber habe ich mir noch nie Gedanken gemacht, du kannst ja echt Fragen stellen!\n\n#### Vielleicht kannst du eher sagen, was dir gegen den Strich geht?\n\nFaschos.\n\n#### Hast du da schlechte Erfahrungen gemacht?\n\nJa, reichlich! [Lacht ein bisschen, es impliziert, dass er nicht nur Pr\u00fcgel eingesteckt hat.] Auf Demos, auf Konzerten, auf der Stra\u00dfe.\n\n#### Und denn gibt's auch mal Dresche?\n\nKann passieren; wenn es sich nicht vermeiden l\u00e4sst, dann knallt es auch mal.\n\n#### Du hast vorhin angedeutet, dass du immer noch mit Leuten aus deiner Jugend befreundet bist. Hattet ihr immer Kontakt oder ist der zwischendurch abgebrochen?\n\nSowohl als auch. Mit meinen engsten Freunden habe ich sogar teilweise zusammengewohnt, mit anderen ist der Kontakt eingeschlafen, das ist doch normal, ist bei dir sicher auch so.\n\n#### Und deine Freunde gehen auch mit auf Demos?\n\nNein, das sind Stinos [Stinknormale]. Zu ihnen kann ich gehen, wenn ich mal \u00fcber ernste Themen oder Probleme reden m\u00f6chte.\n\n#### Sie wissen, dass du dich als Autonomen siehst?\n\n[Er schaut mich irritiert an.] Ja, das ist auch manchmal ein Streitpunkt. Sie machen sich Sorgen, wenn ich zu Demos gehe und verhaftet werde, und viele meiner Freunde lehnen Gewalt grunds\u00e4tzlich ab, da gibt's schon mal Streit.\n\n#### Woher beziehst du eigentlich deine Infos zum aktuellen Zeitgeschehen?\n\nVon Bekannten, Fernsehen, Internet und speziellen Zeitschriften.\n\n#### Welche denn?\n\nAntifa Nachrichten, Indymedia, Stressfaktor.\n\n#### Welche Rolle spielt die autonome Szene f\u00fcr dich?\n\nF\u00fcr mich pers\u00f6nlich eine sehr gro\u00dfe. Es ist grunds\u00e4tzlich sehr wichtig, in Deutschland eine autonome Szene zu haben. Nur die autonome Szene kann eigene R\u00e4ume erk\u00e4mpfen, und die Bewegung sollte auch wachsen, vor allem wieder zusammenwachsen, so viele Splittergruppen, das tut nicht not.\n\n#### Warum?\n\nDie fangen schon an, sich gegenseitig zu bek\u00e4mpfen, und ich finde es wichtig, dass man im Widerstand nicht alleine dasteht, sondern zusammenh\u00e4lt. Wenn man zum Beispiel gegen die bescheuerten Nazis oder irgendwelche intoleranten Schwachk\u00f6pfe, den homophoben, islamophoben Vollpfosten k\u00e4mpft. Und besonders, wenn ein paar Autos die stabile Seitenlage annehmen und man die Bullen an der Backe hat, gerade dann ist es wichtig, zusammenzuhalten und nicht sich gegenseitig zu bek\u00e4mpfen!\n\n#### Wie w\u00fcrdest du \"autonom\" bzw. die \"autonome Szene\" definieren? Wie w\u00fcrdest du es jemandem beschreiben, der das gar nicht kennt?\n\nAutonom bedeutet im weitesten Sinne Selbstbestimmung, f\u00fcr mich aber auch Selbstbestimmung gekoppelt mit massivem Widerstand oder besser noch Anarchie. Ja, Gesetzlosigkeit als eine Form der Ordnung, so haut das hin. Vor zwanzig Jahren haben wir H\u00e4user besetzt und Stra\u00dfenschlachten angezettelt, und wenn wir Steine und Schei\u00dfe auf die Bullen warfen, waren wir uns sicher, wir treffen keinen Falschen, wir haben sozusagen mit den Steinen auf die Misswirtschaft und die Machtstrukturen der Parteien und das verschissene sozialdemokratische Modell gezeigt.\n\n#### Bist du Mitglied in einer konkreten Gruppe?\n\nNein, mich kotzt diese Unterteilung total an. Schau dir die Vegan-Autonomen an. Wenn die eine Aktion starten, w\u00fcrde ich gerne mitmachen, weil ich das toll finde, aber die lassen mich nicht rein, weil ich ab und an einen D\u00f6ner verspeise.\n\n#### Wie war dein erster Kontakt zu den Autonomen?\n\nDas ist schon so lange her. Da f\u00fchlten sich Punks und Autonome irgendwie verbunden, jedenfalls haben sie zusammen Aktionen gestartet. Damals war der Anarchismus der Punks so sch\u00f6n destruktiv.\n\n#### Wie?\n\nDie wollten halt nur kaputt machen, ohne Botschaften, die lieferten dann wir.\n\n#### Gibt es deiner Meinung nach Ankn\u00fcpfungspunkte an andere Kulturen?\n\nJa, da gibt es in der HipHop-Szene eine gro\u00dfe Antifa, leider auch eine gro\u00dfe Fascho-Gruppe. Es gibt sicher auch andere, aber ich steig nicht mehr so durch, will ich auch gar nicht, ehrlich gesagt.\n\n#### Findest du denn, dass die autonome Szene auch eine Jugendkultur ist?\n\nJa, mittlerweile schon. Viele junge Leute schmei\u00dfen auf Demos Steine, weil sie gerade irgendeinen Hass auf jemanden haben, die schmei\u00dfen dann r\u00fccksichtslos in die Menge, oder sie ziehen sich so an, um vor wem auch immer zu gl\u00e4nzen oder auf die Kacke zu hauen.\n\n#### W\u00fcrdest du das als jugendkulturelle Elemente bezeichnen?\n\nJa. F\u00fcr mich ist die Kleidung ein Element, sich auszutauschen und zu organisieren. Wenn dann aber die Kiddies einfach nur so mit schwarzen Klamotten herumrennen, um vor ihren Freunden als coole Sau dazustehen... das finde ich doof. Auf Konzerten findest du die sehr h\u00e4ufig, sie geben sich da k\u00e4mpferisch und gehen danach wieder nach Hause zu Muddi.\n\n#### Also gibt's einen eigenst\u00e4ndigen autonomen Stil?\n\nAuf jeden Fall.\n\n#### Gibt es da bestimmte Codes, die zu beachten sind?\n\nJa, die Aufn\u00e4her zum Beispiel, die die Veganerfront benutzt, d\u00fcrfen nur schwarzwei\u00df auf ganz schwarzer Klamotte sein. Die \"Hasskappe\" ist ein Muss, auch oder gerade weil es verboten ist.\n\n#### Was spielt die Musik in der autonomen Szene f\u00fcr eine Rolle?\n\nSie spielt eine sehr gro\u00dfe Rolle, sie macht es uns leichter, Worte zu finden und Proteste zu formulieren, und l\u00e4sst uns \u00fcber die Texte der Musik auch mehr und unterschiedliche Menschen erreichen.\n\n#### Welche Themen sind dir jetzt speziell wichtig im Engagement?\n\nF\u00fcr mich ist die Toleranz eines der wichtigsten Themen, jeder soll sich und den andern so sein lassen, wie er ist und sein will.\n\n#### Und welche politischen Positionen vertrittst du?\n\nIch bin eher Anarcho, ich habe \u00fcberhaupt keinen Bock auf irgendwelche Parteien und ihr Gerede, ob das APPD oder FDP oder Linke ist, alles Gesocks und Schwachsinn.\n\n#### Gehst du w\u00e4hlen?\n\nJa, aber ich mache meine Stimme ung\u00fcltig.\n\n#### Wie siehst du den Antisexismus, den Antifaschismus, Antirassismus... die ganzen Antis eben?\n\nAn sich ist das eine sehr gute Sache, nur wie es praktiziert wird, ist mir manchmal zu \u00fcbertrieben. Wenn ich mal Fotze sage, dann muss ich nicht gleich ein Sexist sein, das ist mir echt zu \u00fcbertrieben. Auch die Vegan- oder Vegetarierfront \u00fcbertreibt und will alle anderen missionieren, das kotzt mich einfach an. Das empfinde ich als intolerant.\n\n#### Wie stehst du denn zum Konflikt der Antiimps und Antideutschen?\n\nDie Antideutschen finde ich ziemlich beh\u00e4mmert. Ich finde dieses Entweder-oder-Gefasel zum Kotzen. Entweder Israel oder Pal\u00e4stina, die sollen mir nicht so kommen! Predigen Solidarit\u00e4t und gehen untereinander gar nicht solidarisch miteinander um. Die Antiimps sind da nicht besser. Das ist f\u00fcr mich Schwarz-Wei\u00df-Denken.\n\n#### Ist autonom nicht schwarz-wei\u00df?\n\nFreund oder Feind? In gewisser Weise schon, aber nicht so extrem. Bei mir jedenfalls nicht, ich revidiere auch mal meine Meinung und schmei\u00df auch keine Mollis mehr, nur noch vereinzelt und gezielt Steine auf Bullen, wenn die wieder abnerven.\n\n#### Dein Verh\u00e4ltnis zum Staat und zur Polizei ist eher gespannt, oder?\n\nJa, f\u00fcr mich ist die Bullerei nur eine stumpfe Kn\u00fcppelgarde, die nehmen auf nichts und niemanden mehr R\u00fccksicht. Ruf niemals um Hilfe, es k\u00f6nnte die Polizei kommen!\n\n#### Spielen internationale Vernetzungen eine Rolle f\u00fcr dich? Hast du Kontakte?\n\nJa, hab ich, mehr m\u00f6chte ich nicht sagen.\n\n#### Hat sich die autonome Szene im Laufe der Jahrzehnte ver\u00e4ndert?\n\nSie hat sich in etliche Splittergruppen gespalten, mit verschiedenen Zielen. Im Grunde sind wir eine Front gegen den Staat. Aber irgendwie wollen alle ihr eigenes S\u00fcppchen kochen. Vor zwanzig Jahren war ein viel gr\u00f6\u00dferer Zusammenhalt, und es war auch lustiger. Heute ist alles so bierernst und p.c.-fuck, das nervt voll. St\u00e4ndig muss sich irgendwer auf einem Plenum rechtfertigen. Das ist mir alles zu verbissen.\n\n#### Fast alle Jugendkulturen sind m\u00e4nnlich dominiert. Gilt das auch f\u00fcr die autonome Szene?\n\nEs sind schon mehr M\u00e4nner als Frauen beteiligt, aber die Frauen haben genauso ein Mitspracherecht wie die M\u00e4nner. Wobei mir aufgefallen ist, dass bei Meinungsverschiedenheiten eher der Frau Recht gegeben wird. Sie nehmen die Meinung der Frau eher wahr. Also stell dir ein Plenum vor, f\u00fcnf Personen, drei M\u00e4nner, zwei Frauen. Die Frauen sind anderer Meinung als die M\u00e4nner, dann wird sich meist der Meinung der Frauen angeschlossen. Meine pers\u00f6nliche Erfahrung, das kann man sicher nicht auf die gesamte Bewegung herunterbrechen.\n\n#### Wie bewertest du die \u00f6ffentliche Wahrnehmung der autonomen Szene?\n\nEin gewisser Grundrespekt gegen\u00fcber unserer Risikobereitschaft und den daraus folgenden Konsequenzen ist schon sp\u00fcrbar. Durch die Pr\u00e4sentation in den Medien ist es allerdings nicht so erhellend. Obwohl das ja auch eine logische Schlussfolgerung ist, und im Grunde schei\u00dfen wir drauf, aber so manche Zeitung gibt dem Affen echt Zucker.\n\n#### Wie meinst du das?\n\nWenn wieder eine R\u00e4umung angesagt ist und wir mobilisieren wollen und beim Volk auch um Solidarit\u00e4t bitten, f\u00e4hrt die Presse, vornehmlich die Printmedien, echt scharfe Gesch\u00fctze auf. Die machen das ganz subtil, da werden krude Bezeichnungen f\u00fcr uns gefunden, die gleich klarmachen sollen, wer wir eigentlich sind: schlagende Raufbolde ohne Bildung, mit Migrationshintergrund und vor allem OHNE politische Motivation.\n\n#### Geh\u00f6rt denn Gewalt f\u00fcr dich nicht zur Szene?\n\nDoch, schon, aber eben gezielt.\n\n#### Welche Einstellung hast du denn zu militanten Aktionen?\n\nAlso in den 80ern waren militante Aktionen Gewalt gegen die politische Gewalt des Systems. Es war ein Erlebnis, Autos in Flammen aufgehen zu sehen und Superm\u00e4rkte zu pl\u00fcndern, das erzeugte so eine Art Zusammenhalt unter uns. Wir verhielten uns so, wie die Medien uns konstruierten, unser Hass war unermesslich. Ich wei\u00df nicht, ob das am Alter liegt, aber inzwischen steh ich dem Ganzen etwas zur\u00fcckhaltender gegen\u00fcber. Eine Radikalisierung des Protests ist sicher notwendig, aber mittlerweile hat das Ganze den Eindruck eines Wettr\u00fcstens f\u00fcr mich angenommen. Je mehr die Bullen sich panzern, desto ausget\u00fcftelter werden die Aktionen.\n\n#### Du w\u00fcrdest also militante Aktionen nicht grunds\u00e4tzlich ablehnen?\n\nNein, nicht grunds\u00e4tzlich, kommt darauf an, wof\u00fcr und mit wem und warum.\n\n#### Wie sch\u00e4tzt du die gesamtgesellschaftliche Bedeutung der Autonomen ein? Also welche M\u00f6glichkeiten hat die Szene, etwas zu ver\u00e4ndern?\n\nIch denke, ein bisschen was ver\u00e4ndert haben wir schon, sonst w\u00fcrde es die Hafenstra\u00dfe nicht geben oder ein paar noch ehemalig besetzte H\u00e4user in Berlin.\n\n#### Denkst du, dass es wirklich darum geht, etwas zu ver\u00e4ndern, oder eher angestrebt wird, nur die eigenen Freir\u00e4ume zu bewahren?\n\nGute Frage, im Grunde sind Karnickelz\u00fcchter und Kleing\u00e4rtner auch Autonome; jeder kocht sein eigenes S\u00fcppchen und will sich und seine Gruppe behaupten. Keine einheitliche Gruppe mehr zu sein, k\u00f6nnte das Aus f\u00fcr die autonome Szene bedeuten, aber ich kann es noch nicht so gut einsch\u00e4tzen, denke noch nicht lange dar\u00fcber nach.\n\n#### Wie gro\u00df ist die autonome Szene in deiner Stadt?\n\nSehr gro\u00df, eher sehr, sehr gro\u00df! Und es werden immer mehr. Mehr m\u00f6chte ich dazu nicht sagen.\n\n#### W\u00fcrdest du noch selbst etwas sagen?\n\nSagen m\u00f6chte ich nix mehr, schlie\u00dflich bist du auf mich zugekommen; ich will hier kein Statement abgeben, das ist nicht die richtige Plattform.\n\nDas Interview f\u00fchrte Bettina Thiele.\n\n## \"Dieser Staat richtet sich gegen mich, und aus diesem Grund ist das Verh\u00e4ltnis zwischen mir und dem Staat ein feindliches.\"\n\nPeter, 18 Jahre, lebt in Hamburg.\n\nWir sind in der Kleinen Pause, einem Imbiss auf St. Pauli, sitzen drau\u00dfen in einem Strandkorb. Essen Cheeseburger und Fritten. Die Kundschaft besteht aus Zecken, Yuppies und Jugendlichen aus dem Viertel, ortsbedingt viele St.-Pauli-Fans.\n\nPeter ist m\u00e4nnlich, gro\u00df und hat kurze blonde Haare. Er ist sportlich angezogen, Jogginghose, Adidas Sambas. Er wirkt aufgeregt, vielleicht ein bisschen breit, er riecht leicht nach Marihuana.\n\n#### Bist du in Hamburg geboren?\n\nJa, ich bin auch in Hamburg geboren.\n\n#### Was machen denn deine Eltern so beruflich?\n\nMein Vater hat ein Studium zum Lehrer gemacht, arbeitet aber derzeit nicht, meine Mutter arbeitet als ungelernte Kraft im p\u00e4dagogischen Bereich, also hat keine abgeschlossene Berufsausbildung. Also sie macht so Projekte mit Schulklassen und so was.\n\n#### Hat das einen politischen Hintergrund?\n\nDas hat keinen politischen Hintergrund, das ist mehr so allgemein... schon zu den Themen Globalisierung oder so, aber das ist nichts, was explizit einen politischen Charakter hat.\n\n#### Wie sieht denn deine gegenw\u00e4rtige Situation aus? Ich hab gerade mein Abitur gemacht und bewerbe mich jetzt um einen Studienplatz in Jura.\n\n#### Wie bewertest du deine aktuelle Situation?\n\n[Atmet aus.] Ist ja sehr allgemein gehalten, die Frage! Ich wei\u00df nicht. Die aktuelle Situation ist halt schei\u00dfe, also jetzt, wo man aus der Schule raus ist und dieser Rahmen nicht mehr gegeben ist, beginnt jetzt halt so die Zeit von dem Arbeitszwang. Also ich werd halt arbeiten m\u00fcssen, damit ich \u00fcber die Runden komme, oder erst mal studieren gehen, um dann halt sp\u00e4ter 'nen qualifizierten Beruf annehmen zu k\u00f6nnen, um da halt f\u00fcr mich pers\u00f6nlich einfach mehr rausschlagen zu k\u00f6nnen \u2013 'ne andere Wahl hab ich ja gar nicht, als meine Arbeitskraft zu verkaufen, nachdem ich sie hoffentlich bald weiter qualifiziert habe.\n\n#### Du hast gesagt, du willst Jura studieren. Kann man das so verstehen, dass Jurist dein Traumberuf ist?\n\nJa, zumindest in diesem System w\u00e4re das schon mein Traumberuf oder der Beruf, in dem ich mich noch am ehesten sehe.\n\n#### Und hast du sonst vielleicht noch irgendwelche privaten Tr\u00e4ume, so das typische Modell Haus, Kinder und Hof?\n\n[Lacht.] So weit denk ich nicht, also ich hab jetzt keine Pl\u00e4ne, so seh ich mich in zehn Jahren oder so was, solche Pl\u00e4ne hab ich noch gar nicht. Kinder kann ich mir vorstellen, auf jeden Fall, aber das h\u00e4ngt auch ein bisschen davon ab, wie sich das so weiterentwickelt und ob ich dann irgendwann sag, okay, das ist 'ne Welt, in die will ich Kinder setzen.\n\n#### Und wovor hast du am meisten Angst?\n\nAlso jetzt mit dem Studium, da kommt halt schon Druck, wenn man dann einen Kredit aufnehmen muss f\u00fcrs Studium und so was und da irgendwas schief geht, man schafft das Studium nicht und hat aber daf\u00fcr einen Kredit aufgenommen und solche Geschichten, dieser Druck, der halt automatisch auf einem lastet, wenn man in diese Arbeitswelt einsteigt, da kann Verschiedenes passieren, dass man dem nicht standh\u00e4lt, davor h\u00e4tte ich halt Angst. Oder auch, dass einem die Bullen dazwischen pfuschen und morgens vor der Haust\u00fcr stehen, solche Geschichten k\u00f6nnen einen auch ganz sch\u00f6n runterrei\u00dfen.\n\n#### Aber warum sollten die Bullen denn vor deiner Haust\u00fcr stehen?\n\n[Lacht.] Ja, weil ich halt zu den Leuten geh\u00f6re, mit denen man solche Interviews f\u00fchrt. Darauf antworte ich nicht, trotz Anonymisierung. Auf die Frage antworte ich nicht.\n\n#### Was \u00e4rgert dich denn am meisten so an der heutigen Gesellschaft?\n\nDass ich arbeiten gehen muss, nicht, um was zu tun, was meinen Bed\u00fcrfnissen und meinen Interessen entspricht, sondern f\u00fcr die Profite anderer.\n\n#### Du hast schon eine relativ fundierte politische Meinung. Wie weit w\u00fcrdest du denn den Einfluss deiner Eltern darauf einsch\u00e4tzen? Wie weit haben die deine politische Entwicklung beeinflusst?\n\nIch komme aus einem Haushalt, der schon so gewisse Werte immer vermittelt hat. Also ich hatte schon von Haus aus ein antirassistisches und antimilitaristisches Bewusstsein auf jeden Fall. Das kann man festmachen an so Dingen wie, dass ich keine Plastiksoldaten und \u00c4hnliches haben sollte. [Lacht.] Ich hab mich schon immer f\u00fcr Politik interessiert, das war etwas, was ich aus meinem Haushalt mitbekommen habe. Aber mein politischer Werdegang ist dann keiner, der von meinen Eltern gepr\u00e4gt wurde, mehr so der Blick, den man vielleicht auf gewisse Dinge hat, oder dass man \u00fcberhaupt den Blick auf politische Dinge richtet, das hab ich von zu Hause aus bekommen. Mein politischer Werdegang, darauf hatten meine Eltern dann allerdings keinen Einfluss mehr.\n\n#### Gibt es andere Personen, die vielleicht wichtig waren, Lehrer oder andere Verwandte?\n\nVerwandtschaft auf keinen Fall. Wenn ich mit Leuten aus meiner Familie \u00fcber Politik gesprochen habe, dann waren das wirklich meine Eltern. Also meine Verwandtschaft hat da keine Rolle gespielt, auch keine Geschwister oder so was. Bezugspersonen wie Lehrer oder so was in Sachen Politik hatte ich auch keine. Das kam dann mehr von anderen Leuten, die man halt so kennengelernt hat im Zuge von politischen Veranstaltungen.\n\n#### Was meinst du, was halten andere Leute von dir oder wie denken andere Leute \u00fcber dich?\n\nDie Leute kennen mich schon als einen sehr politischen Menschen, also das merken die Leute auf jeden Fall, dass ich jemand bin, der sich sehr f\u00fcr Politik interessiert. Wie ich sonst auf die Menschen wirke, kann ich schwer sagen; ich werd halt als ein politischer Mensch wahrgenommen, der gern diskutiert. Ich glaube, manche Leute finden das auch ziemlich cool, die begr\u00fc\u00dfen das, dass ich jemand bin, der auch die Diskussion sucht, der manchmal auch die Konfrontation sucht, w\u00e4hrend es auch Personen gibt, die noch nicht so den Blick auf politische Themen haben wie ich und die das dementsprechend irritiert, dass sich jemand so sehr um ein Thema bem\u00fcht, sich damit so viel auseinandersetzt.\n\n#### Und was machst du so in deiner Freizeit? Was hast du f\u00fcr Hobbys?\n\nIch spiel gern Fu\u00dfball, ich treib generell Sport, also Kampfsport auch, mache ich jetzt schon seit mehreren Jahren.\n\n#### Bist du dann auch in irgendeinem Sportverein aktiv oder in irgendeinem anderen Verein?\n\nNein. Ich hab fr\u00fcher Fu\u00dfball in einem Verein gespielt, aber ich hab jetzt keine Mitgliedschaft in irgendeinem Verein.\n\n#### Welche Rolle hat denn so Musik in deinem Leben?\n\nMusik ist wichtig, auf jeden Fall, also ich bin jetzt nicht so 'n Subkulturmusik-Typ, dass ich sag, ich bin jetzt der HipHopper oder so was und die Musik ist mein Leben... Ich h\u00f6re viel Musik auf jeden Fall, h\u00f6r eigentlich so ziemlich alle Genres auch durch. Ja, Musik ist schon Bestandteil von meinem Leben, aber nicht so, dass mich eine Musikrichtung oder die dazugeh\u00f6rige Subkultur besonders gepr\u00e4gt h\u00e4tte. Das ist halt so die Begleitmusik eher.\n\n#### Was h\u00f6rst du denn so f\u00fcr Musik?\n\nIch h\u00f6r viel Elektro, so Drum 'n' Bass, das ist die Musik, die ich \u00fcberwiegend h\u00f6re. Auf meinem MP3-Player sind Troopers, das ist mehr so Punk-Musik, aber auch so HipHop-Sachen wie K.I.Z. oder Haftbefehl.\n\n#### Was f\u00fcr sonstige kulturelle Sachen konsumierst du denn, also Literatur, Filme oder Serien?\n\nIch lese viel, also nicht nur politische Lekt\u00fcre, sondern auch einfach so Romane. Damit hab ich auch schon relativ fr\u00fch angefangen. Ansonsten guck ich halt Fernsehen; es gibt so zwei, drei Serien, die ich guck. Ich bin jetzt nicht so der Kinog\u00e4nger. H\u00e4ngt aber auch damit ein bisschen zusammen, dass das ganz sch\u00f6n teuer geworden ist.\n\n#### Wie informierst du dich \u00fcber das aktuelle Tagesgeschehen?\n\nDa spielt das Internet eine ganz entscheidende Rolle. Einfach, weil es der einfachste Weg ist von b\u00fcrgerlicher Presse bis hin zur linken Presse, von verschiedenen linken Str\u00f6mungen oder rechten Str\u00f6mungen, da hab ich einfach mit ein paar Klicks eine so gro\u00dfe Bandbreite an Meldungen, Informationen, Einsch\u00e4tzungen.\n\n#### Okay, und welche Medien au\u00dfer Internet konsumierst du noch?\n\nFernsehnachrichten auf jeden Fall, Tagesschau.\n\n#### Und liest du irgendwelche Zeitungen?\n\nJa, ich lese regelm\u00e4\u00dfig die Tagespresse, auch junge welt kaufe ich mir \u00f6fter mal, und wenn man mal in der S-Bahn die Mopo findet oder auch mal die Bild-Zeitung, dann les ich halt auch mal so was. Aber das sind keine Medien, die ich mir kaufen w\u00fcrde.\n\n#### Interessierst du dich f\u00fcr Mode?\n\nMode geht nicht komplett an mir vorbei, aber ich bin jetzt nicht so der modebewusste Mensch. Passiert auch schon, dass Leute mich darauf ansprechen, weil ich irgendwelche Begriffe nicht kenne, die in der Mode gerade ziemlich popul\u00e4r sind oder so was. Sicherlich pr\u00e4gt mich das auch bis zu einem gewissen Grad mit, aber ich bin jetzt nicht der Modeexperte.\n\n#### Und was h\u00e4ltst du von so Sachen wie K\u00f6rperschmuck oder T\u00e4towierungen?\n\nMenschen mit T\u00e4towierungen gefallen mir sehr gut, finde ich sehr h\u00fcbsch, aber es gibt auch Piercings, wo ich mich frag, wie man sich das blo\u00df antun kann. Aber das bleibt dann ja denen \u00fcberlassen, was sie mit ihrem K\u00f6rper machen. Nur weil ich mir selber Tattoos nicht stechen oder Piercings machen lassen w\u00fcrde, hei\u00dft das ja nicht, dass das f\u00fcr andere Menschen auch ein Tabu sein muss, das ist halt ein individuelles \u00c4sthetikempfinden, was da zum Ausdruck kommt.\n\n#### Was hast du so f\u00fcr Vorbilder? Hast du \u00fcberhaupt Vorbilder?\n\nAlso ein generelles Vorbild hab ich nicht so. Es gibt halt schon mal Menschen, an denen man sich orientiert, aber da hab ich jetzt nicht eine Person, an der ich das festmache. Ich sehe mal jemand, der besonders gut trainiert, zum Beispiel im Kampfsport, dass ich dann halt sag, der hat in dem Bereich so eine Vorbildfunktion oder an dem kann man sich orientieren. Und dann gibt es andere Menschen, die ich f\u00fcr sehr belesen halte, und orientiere mich in dem Bereich an denen. Aber ich hab nicht so ein zentrales Vorbild, wo ich sage, da finde ich alle diese Eigenschaften vereint.\n\n#### Vielleicht historische Figuren, die du bewunderst?\n\nZu Beginn meines politischen Werdegangs hab ich auch eine Che-Guevara-Biografie gelesen, das sind schon Dinge, die einen beeindrucken. Aber als ich dann selber mein eigenes politisches Bewusstsein mehr und mehr ausgepr\u00e4gt hatte, da f\u00e4ngt man dann an, auch solche Personen zu hinterfragen, und erkennt dann auch Dinge, die diesen Mythos auf jeden Fall brechen. Also ich w\u00fcrde ihn nicht mehr als ein Vorbild bezeichnen. Wenn ich mir die Dinge durchlese, die Che Guevara beispielsweise zum Thema Moral gesagt hat, da w\u00fcrde ich ihm fundamental widersprechen, so dass eigentlich diese Vorbildfunktion eher demontiert wurde, als dass sie sich weiter verst\u00e4rkte im Laufe meines politischen Werdegangs.\n\n#### Wie verbringst du denn so deine Freizeit?\n\nMeine Freizeitgestaltung ist relativ typisch f\u00fcr mein Alter, also ziemlich durchschnittlich gehalten. Ich treffe mich viel mit Freunden, geh auch mal feiern, mal ein bisschen was trinken, mal verbringt man auch einen Abend mit Freunden vor der Spielkonsole, mal geht man in verschiedene Clubs feiern oder in Bars oder in Kneipen. Dann verbringe ich nat\u00fcrlich auch Zeit im Internet, ich bin jetzt nicht so auf Facebook aktiv oder so was, aber trotzdem nutzt man nat\u00fcrlich auch das Internet zu verschiedensten Zwecken. Und ansonsten halt mit Sport, dass ich gern Fu\u00dfball spiele und Kampfsport mache.\n\n#### Und nach welchen Kriterien suchst du dir so deine Freunde aus?\n\nDas ist sehr durchmischt, also ich hab schon immer einen sehr durchmischten Freundeskreis gehabt, nicht nur von der Altersstruktur und von der sozialen Herkunft her. Das geht eigentlich so nach gemeinsamen Interessen. Es gibt nat\u00fcrlich viele Freunde, mit denen teile ich ein politisches Interesse; es gibt aber auch Freunde, da finde ich einfach, dass das nette sympathische Leute sind, ich teile nicht deren politische Ansichten und sie nicht zwangsl\u00e4ufig meine, aber trotzdem k\u00f6nnen das ja nette Leute sein, mit denen man sich so im pers\u00f6nlichen Umgang gut versteht.\n\n#### Du hast vorhin schon gesagt, dass du nicht so der Subkultur-Typ bist. Aber w\u00fcrdest du trotzdem sagen, dass du mit einer sympathisierst?\n\nN\u00f6, also Subkulturen sind noch nie so ganz meins gewesen. Weil ich immer das Gef\u00fchl hatte, da findet so 'ne Form von Abkapselung statt, was ich nicht f\u00fcr gut halte. Es gibt halt Subkulturen, f\u00fcr die habe ich eine gewisse Sympathie, und das ist halt dieses HipHop-Graffiti-Ding, weil ich einfach f\u00fcr die Graffiti-Kunst immer eine Begeisterung empfunden habe. Aber das ist jetzt nicht meine Subkultur.\n\n#### Hast du selber mal Graffiti gemacht?\n\nNein. Das war nie meine Welt.\n\n#### Gibt es Personen\/Gruppen, mit denen du \u00fcberhaupt nicht klarkommst?\n\nFaschisten, Rassisten, Sexisten, das sind Personen\/Gruppen, mit denen ich nicht klarkomme.\n\n#### Da sehe ich jetzt einen Widerspruch: Du sagst einerseits, man sollte sich nicht von der Gesellschaft abkapseln, aber andererseits sind Sexismus und Rassismus ja schon recht verbreitet oder nicht?\n\n[Lacht.] Nat\u00fcrlich bin ich von Menschen umgeben, wo ich merke, da ist ein sexistisches Denken vorhanden. Die Frage ist halt immer, wie doll sich das ausdr\u00fcckt. Also wenn jemand sein sexistisches Weltbild jeden Moment in alle Welt hinausposaunt und das sozusagen als Leitfaden in seinem Leben betrachtet, dann ist er einer Personengruppe zuzurechnen, die mir nicht imponiert. Wenn ich wei\u00df, bei jemandem ist halt ein gewisses Denkmuster vorhanden, aber der das nicht so aggressiv \u00e4u\u00dfert, kann ich auch mit so jemandem an einem Tisch sitzen und mich mit ihm normal unterhalten. Man versucht nat\u00fcrlich immer, den Leuten aufzuzeigen, wo halt die Fehler in ihrem Denken liegen und was sie f\u00fcr falsche Ansichten vertreten und woher diese Ansichten r\u00fchren. Also wenn ich beim Sparring einen Trainingspartner hab und der br\u00fcllt mir bei jedem Schlag \"Fotze\" ins Gesicht, dann w\u00fcrde ich ihn halt schon in der Umkleidekabine darauf ansprechen, was er sich eigentlich dabei denkt, warum er \"Fotze\" als die ultimative Beleidigung sieht.\n\n#### Welche Rolle spielen denn f\u00fcr dich Freundschaften?\n\nFreundschaften sind das Wichtigste mit im Leben, Menschen zu haben, auf die man sich verlassen kann, die sich auch auf einen selber verlassen k\u00f6nnen, mit denen man Spa\u00df hat, mit denen man gerne zusammen ist, das ist schon so das Wichtigste im Leben.\n\n#### Welche Rolle spielt denn Religion f\u00fcr dich?\n\nReligion ist f\u00fcr mich schlichtweg unwissenschaftlich; Religion hat keinen wissenschaftlichen Charakter, sie arbeitet mit Mythen und f\u00fchrt damit einfach zu einer Verblendung der Menschen. Die Motive, warum Menschen religi\u00f6s werden, sind verschiedene, aber der Ausdruck der Religion ist immer ein reaktion\u00e4rer, weil durch dieses G\u00f6tterbild findet ein Abgeben der Verantwortung eines Menschen statt, er gibt seine Verantwortung ab an ein h\u00f6heres Wesen. Das ist meine Grundkritik an der Religion. Und in der Praxis stellen Religionen ein sexistisches, rassistisches und einfach reaktion\u00e4res Gesellschaftsbild her, und deswegen bin ich auch nicht religi\u00f6s, und ich kritisiere auch Menschen, die sich in meinem Umfeld als religi\u00f6s betrachten.\n\n#### Was ist denn so deine Definition von \"autonom\", was bedeutet f\u00fcr dich \"autonome Szene\"?\n\nIch w\u00fcrde die autonome Szene heutzutage schon als Subkultur bezeichnen. Also die autonome Szene hat schon einen politischen Charakter, aber daraus hat sich mit der Weile auch eine Subkultur herausgebildet. Dieses anarchistische Denken, durch das ja die autonome Szene sehr gepr\u00e4gt ist, verschwindet eher und weicht einem unpolitischen Bewusstsein eigentlich, wo der Gedanke der Subkultur und sich abkapseln, Zugeh\u00f6rigkeitsgef\u00fchl mehr an Bedeutung gewinnt als die konkrete politische Arbeit. Ich hab das Gef\u00fchl, dass die autonome Szene nur noch ganz schwach auf gesellschaftliche Entwicklungen reagiert, weil man eigentlich zu so 'ner Subkultur geworden ist, so dass das Veranstalten von Konzerten und das Herausbringen neuer Klamotten-Labels die gr\u00f6\u00dfere Bedeutung gewinnen, nicht mehr das Aufgreifen aktueller Themen und das Suchen nach Debatten. Es ist mal wesentlich besser gewesen vor ein paar Jahren noch, dass auch bei Debatten, die in der Gesellschaft gef\u00fchrt wurden, autonome Positionen, also anarchistische Positionen deutlich mehr vertreten waren. Aber das schwindet nach und nach, wie man in der Presse sehen kann, wo autonome Positionen \u00fcberhaupt keine Rolle mehr spielen. Und das ist meiner Meinung nach darauf zur\u00fcckzuf\u00fchren, dass die autonome Szene mehr zu so 'ner Subkultur geworden ist.\n\n#### Was macht ihr denn so in eurer Gruppe?\n\nDer absolute Schwerpunkt ist erst mal die Bildungsarbeit, also dass man gemeinsam Texte liest, diskutiert, dass man \u00fcberhaupt ins Gespr\u00e4ch kommt. Das ist einfach das Wichtigste und dann auch das, was wir \u00fcberwiegend machen. Nat\u00fcrlich macht man auch so Randdinge, man organisiert mal 'ne Party, damit man Geld hereinbekommt f\u00fcr die Opfer von Repression, aber einfach das Wichtigste ist die Bildungsarbeit. Also wir lesen gerade einen Text zum Thema Rassismus, und dann liest man ihn gemeinsam, spricht dar\u00fcber, reflektiert das, das ist der Schwerpunkt.\n\n#### Wenn man sich die Medien anguckt, dann sind die Autonomen ja nun nicht f\u00fcr ihre Theorie-Arbeit bekannt, sondern mehr f\u00fcr ihre Praxis. Hast du da vielleicht Beispiele, wo du schon mal mitgewirkt hast? Oder Demos, an denen du dich beteiligt hast?\n\nIch hab mich schon an ganz vielen verschiedenen Demos beteiligt, da sind so Geschichten dabei wie die Revolution\u00e4re 1. Mai Demonstration in Hamburg, verschiedene Aktionen gegen das \u00f6ffentliche Auftreten von Faschisten in Form von NPD-Aufm\u00e4rschen... Ich glaube aber, dass die Bedeutung dieser Demonstrationen schwindet.\n\n#### Was f\u00fcr eine Zielsetzung verfolgst du denn, wenn du zum Beispiel bei so einer Demo mitmachst wie Revolution\u00e4rer 1. Mai?\n\nDer 1. Mai ist ein Sonderfall, zum einen hat er auch einen symbolischen Charakter, weil es einfach weltweit der Tag ist, an dem Revolution\u00e4re ihren Inhalt auf die Stra\u00dfe tragen. Generell bei Demonstrationen geht es darum, dass man erst mal eine \u00d6ffentlichkeit schafft. Wenn ich mit mehreren tausend Menschen auf der Stra\u00dfe bin, hab ich automatisch eine Plattform, weil ich halt Beachtung finde, die Menschen sehen mich auf der Stra\u00dfe und die Medien berichten anschlie\u00dfend dar\u00fcber. Da h\u00e4ngt nat\u00fcrlich viel davon ab, wie sie dar\u00fcber berichten.\n\n#### Was bedeutet denn \"autonom\" f\u00fcr dich genau? F\u00fcr dich und deine Gruppe?\n\nF\u00fcr mich ist der Begriff \"autonom\" eigentlich selbsterkl\u00e4rend. Er bedeutet ja, sich selbst Grenzen zu setzen, und das hei\u00dft einfach, dass wir uns als Gruppe die Grenzen, die es halt gibt, selber setzen aufgrund unseres politischen Bewusstseins: Das sind die Dinge, die wir tun, das sind die Dinge, die wir nicht tun, das sind die Positionen, die wir vertreten. Und auch in der Praxis beispielsweise, die Gesetze des Staates sind f\u00fcr uns keine Grenze; wir sagen nur, wir k\u00f6nnen sie nicht \u00fcbertreten, aber das ist nicht eine Grenze, die wir uns selber gesetzt haben, sondern die jemand anders uns gesetzt hat.\n\n#### Wird das immer im Konsens entschieden oder demokratisch oder wie kommt ihr zu solchen Positionen?\n\nDas sind Konsens-Entscheidungen \u00fcberwiegend, also bei Kleinigkeiten gibt es auch mal Unstimmigkeiten, aber der gro\u00dfe Rahmen an Positionen wird sowohl in der Theorie als auch in der Praxis mittels Konsens-Prinzip entschieden.\n\n#### Wenn du findest, dass die autonome Szene zu einer Art Subkultur geworden ist, w\u00fcrdest du sagen, dass es da auch bestimmte Dresscodes gibt oder so eine Art autonomen Stil?\n\nJa. Also das, was so als autonomer Stil interpretiert wird, ist dieses zu jeder Zeit in schwarzen Klamotten aufzutreten. Auch Dreadlocks erfreuen sich gro\u00dfer Beliebtheit. Mir begegnen auch \u00f6fter Menschen jetzt, die sagen, die und die w\u00fcrde ich automatisch der autonomen Szene zuordnen aufgrund ihres Erscheinungsbildes, bestimmte Kleidung, bestimmte Aufn\u00e4her, die man dann tr\u00e4gt.\n\n#### Was sind denn noch so Themen, die f\u00fcr dein Engagement besonders wichtig sind?\n\n\u00dcberwiegend ist es halt schon Klassenkampf, und da gibt es einfach verschiedene Ausdr\u00fccke, die das hat. Also wir haben in der Schule zum Beispiel viel diskutiert \u00fcber Bildung, wie funktioniert ein kapitalistischer Bildungsapparat, welchen Zweck erf\u00fcllt er. Dazu kommen nat\u00fcrlich auch noch so Ph\u00e4nomene wie Sexismus und Rassismus, wobei hinter vielen sexistischen und rassistischen Positionen auch ein Klasseninteresse steht, und dadurch ist das indirekt dann auch wieder eine Form von Klassenkampf.\n\n#### Und wie stehst du zu solchen Dingen wie Umweltschutz und Tierrechte?\n\nTierrechte, damit hab ich mich noch nicht so richtig besch\u00e4ftigt. Nat\u00fcrlich w\u00fcrde ich mich auch f\u00fcr eine artgerechte Tierhaltung einsetzen oder so was, aber ich bin kein Anh\u00e4nger dieser linken Str\u00f6mung, die sagt, ein vegetarischer oder sogar veganer Lebensstil ist das Einzige, was wirklich fortschrittlich ist. Umweltschutz ist halt auch Klassenkampf, das ist nat\u00fcrlich klar, wenn Profit im Vordergrund steht, wird halt die Umwelt zugunsten des Profits verschandelt. Also nat\u00fcrlich setzt man sich f\u00fcr den Umweltschutz ein, gleichzeitig darf man sich aber nicht einbilden, dass es einen gr\u00fcnen Kapitalismus gibt, weil jetzt ein Unternehmen einen Wald weniger rodet, wird es zu einer besseren Welt.\n\n#### Gibt es denn Tierrechtler, Vegetarier oder Veganer in deiner Gruppe?\n\nEs gibt Vegetarier in meiner Gruppe, aber die sind nicht wirklich politisch motiviert, die sind von Haus aus vegetarisch aufgewachsen oder welche meinen, das passt gerade in ihr aktuelles Ern\u00e4hrungskonzept, aber so als politische Motivation spielt Vegetarismus oder Veganismus bei uns \u00fcberhaupt keine Rolle.\n\n#### Wo stehst du denn im Konflikt zwischen Antiimps und Antideutschen?\n\n[Lacht.] Diese Antideutschen sind f\u00fcr mich Reaktion\u00e4re, die w\u00fcrde ich nicht als fortschrittlich im Sinne einer linken Szene sehen, weil sie meiner Meinung nach einfach Themen wie den Holocaust f\u00fcr eigene politische Interessen missbrauchen, die ich auch sehr wirr finde, dass da auch Positionen vertreten werden, die nicht mehr logisch sind, die meiner Meinung nach die Rationalit\u00e4t verlassen. Aber auf der Seite der Antiimps, auf die ich mich auch stellen w\u00fcrde, gibt es nat\u00fcrlich auch Positionen, die sind falsch. Bei manchen Positionen hat man das Gef\u00fchl, da werden seit \u00fcber 30 Jahren die immer gleichen Parolen hochgehalten. Aber generell glaube ich, dass die Wahrnehmung dieses Ph\u00e4nomens als innerlinker Konflikt falsch ist, weil ich diese Antideutschen nicht als eine linke Str\u00f6mung ansehe.\n\n#### Du sagst, Klassenkampf ist ein gro\u00dfes Ding f\u00fcr dich, Nationalstaaten willst du abschaffen... Das klingt alles so ein bisschen typisch Arbeiterbewegung. Welcher theoretischen Richtung w\u00fcrdest du dich denn zuordnen?\n\nIch bin Kommunist, also ich halte das, was Marx geschrieben hat, f\u00fcr wissenschaftlich richtig, das l\u00e4sst sich auch an Hand vieler Beispiele belegen. Und hinzu kommt dann noch die Imperialismus-Theorie von Lenin, also der Imperialismus als h\u00f6chstes Stadium des Kapitalismus, das ist auch etwas, das ich f\u00fcr richtig halte, wo es auch viele Beispiele und Belege gibt. Also wenn man mich in eine Schublade einordnen w\u00fcrde, w\u00e4re das wohl der Marxismus-Leninismus, weil der Begriff \"Kommunist\" alleine in Deutschland nicht mehr so viel aussagt, wenn man sich mal anguckt, was sich hier alles selber Kommunist nennt oder kommunistisch genannt wird.\n\n#### Hast du dich denn auch schon mal mit anarchistischen Theoretikern auseinandergesetzt?\n\nJa, es gibt da ja auch wissenschaftliche B\u00fccher dazu. Auf jeden Fall hab ich mich auch damit auseinandergesetzt. Das hat mich halt nie so \u00fcberzeugen k\u00f6nnen, wobei ich dazu sagen muss, dass es bei mir auch an der Praxis der Anarchisten lag, dass ich diese f\u00fcr grundlegend falsch halte. Also es gibt Anarchisten, mit denen unterhalte ich mich auch und merke, dass das Ziel \u00e4hnlich ist, aber wenn ich mir deren Weg dahin angucke, sehe ich einfach, dass der grundlegend falsch ist.\n\n#### Was meinst du da genau?\n\nDazu geh\u00f6rt zum einen der Gewaltfetisch bei manchen Leuten und dann dieser irrationale Glaube, man kann einfach einen einzelnen Staat abschaffen und dann sofort in friedlichen und anarchistischen Verh\u00e4ltnissen zusammen leben. Das halte ich f\u00fcr naiv, das halte ich f\u00fcr dumm, einfach fernab von der Realit\u00e4t, sodass ich nicht viel mit der politischen Str\u00f6mung anfangen kann.\n\n#### Gehst du w\u00e4hlen?\n\nNein, ich gehe nicht w\u00e4hlen.\n\n#### Warum nicht?\n\nIch halte von Parteien \u00fcberhaupt nichts. Wie gesagt, wir bezeichnen uns als autonom, als selbst Grenzen setzend, und in Deutschland werden Parteien Grenzen gesetzt: Die Grenze ist die freie Marktwirtschaft, das, was man Kapitalismus nennt, und weil wir genau diese Grenze \u00fcberschreiten wollen, aber die Parteien alle schon von der Verfassung her verpflichtet sind, innerhalb dieser Grenzen zu bleiben, deswegen kann ich mit keiner Partei etwas anfangen und gehe dementsprechend auch nicht zur Wahl.\n\n#### Hei\u00dft das, du kannst dich mit der Verfassung auch nicht so identifizieren?\n\nNein, ich kann mich mit der Verfassung \u00fcberhaupt nicht identifizieren.\n\n#### Wie w\u00fcrdest du dein Verh\u00e4ltnis zum heutigen Staat und zu seinen Institutionen wie der Polizei oder der Justiz bezeichnen?\n\nMit einem Wort: feindlich. Dieser Staat ist ein Klassenstaat, er vertritt die Interessen der profitbeherrschenden Klasse. Ich geh\u00f6re dieser Klasse nicht an, ich geh\u00f6re der Klasse an, die ihre Arbeitskraft verkauft. Das hei\u00dft, dieser Staat richtet sich gegen mich, und aus diesem Grund ist das Verh\u00e4ltnis zwischen mir und dem Staat ein feindliches.\n\n#### W\u00fcrdest du sagen, dass es Ver\u00e4nderungen in den letzten Jahren gab beim Umgang des Staates bzw. der Polizei mit Autonomen?\n\nMeine praktischen Erfahrungen dauern ja noch nicht so lange an, ich habe auf jeden Fall das Gef\u00fchl, dass die Gangart h\u00e4rter wird, was nicht verwunderlich ist, dass in Zeiten der Wirtschaftskrise und steigender Arbeitslosigkeit der Staat bem\u00fcht ist, sozialen Protest im Keim zu ersticken, und das passiert dann mittels Verfassung, Schlagstock und Schusswaffen. Also die Gangart hat sich auf jeden Fall versch\u00e4rft seitens des Staates.\n\n#### Du hast gesagt, dass du dich dem Leninismus zuordnen w\u00fcrdest; da ist ja auch Internationalismus wichtig. Welche Rolle spielen internationale Vernetzungen f\u00fcr dich?\n\nDie sind wichtig, weil man viel voneinander lernen kann: Welche Erfahrungen haben andere Menschen gemacht, in welchem Stadium des Klassenkampfes befinden sich andere L\u00e4nder, welche Fehler wurden da gemacht? Nat\u00fcrlich ist Internationalismus wichtig, ich orientiere mich nicht an Nationalstaaten, sondern an meiner Klasse, und meine Klasse gibt es \u00fcberall auf der Welt, und somit entsteht nat\u00fcrlich ein Zusammengeh\u00f6rigkeitsgef\u00fchl, weil es ja auch immer wieder zu Solidarit\u00e4tsaktionen kommt mit Menschen, die nicht in Deutschland aktiv sind, sondern im Ausland. Aber wichtig ist es eigentlich, den Kampf da zu f\u00fchren, wo man gerade ist, in der Schule, im Betrieb. Internationalismus kann diesen Kampf nicht ersetzen.\n\n#### Hast du dich mit der Geschichte und den Urspr\u00fcngen der autonomen Szene auseinandergesetzt?\n\nDamit hab ich mich schon besch\u00e4ftigt, aber das war nie mein Schwerpunkt; ich hab immer mehr versucht, mich mit den Positionen zu besch\u00e4ftigen.\n\n#### Du sagst selber, deine praktischen Erfahrungen reichen noch nicht so lange zur\u00fcck, aber konntest du vielleicht eine Ver\u00e4nderung in der autonomen Szene feststellen?\n\nSchwierig zu sagen, aber ich hab das Gef\u00fchl, dass die Bewegung seit dem G8-Gipfel im Aufwind ist, dass es mehr Aktionen gibt, mehr Demonstrationen, mehr Veranstaltungen. Aber so den gro\u00dfen Umschwung, gro\u00dfen Trend kann ich nicht erkennen.\n\n#### Wie bist du denn in die autonome Szene reingerutscht?\n\nBei mir lief das \u00fcber den Fu\u00dfball, muss ich ehrlich sagen. Ich bin schon sehr fr\u00fch zu den Spielen von St. Pauli gegangen und da sind einem dann so Symbole wie Hammer und Sichel begegnet, und dann hab ich halt angefangen, mich daf\u00fcr zu interessieren, und hab viel gelesen.\n\n#### Wie siehst du die \u00f6ffentliche Wahrnehmung der Autonomen?\n\nDie werden bewusst schlecht gemacht in der b\u00fcrgerlichen Presse, die Inhalte werden wenig aufgegriffen, meistens eher so Sachen wie Ausschreitungen, es wurden Autos angez\u00fcndet... Also wenn in der Presse \u00fcber Autonome berichtet wird, dann nur \u00fcber die Praxis.\n\n#### Gibt es noch etwas, was du gerne sagen w\u00fcrdest?\n\nMan soll nicht alles glauben, was man in den b\u00fcrgerlichen Medien \u00fcber Autonome und \u00fcber Linke und Kommunisten generell h\u00f6rt und liest, man soll lieber versuchen, mit den richtigen Leuten selber ins Gespr\u00e4ch zu kommen und mit denen zu diskutieren. Wenn man immer nur die Bild-Schlagzeilen sieht von Hass und Chaoten, das ist keine inhaltliche Auseinandersetzung.\n\n#### Aber es gibt solche Chaoten und Krawalltouristen?\n\nDie gibt es bestimmt und die wird es auch immer geben. Ich w\u00fcrde die nur nicht immer als unpolitisch darstellen, es wird schon einen Grund geben, warum manche Menschen gewaltt\u00e4tig gegen\u00fcber Polizisten sind. Man sollte sie nur nicht als Stellvertreter der autonomen Szene sehen.\n\n## \"Israel ist halt das Mallorca der Antideutschen.\"\n\nMichael Wildenhain, 56 Jahre, lebt in Berlin.\n\nMichael Wildenhain, Jahrgang 1958, ist der Sohn eines Schlossers und einer Erzieherin. Er wuchs in Westberlin auf. Nach dem Abitur absolvierte er ein Praktikum als Maschinenbauer und studierte anschlie\u00dfend u. a. Informatik, Philosophie, Mathematik und Wirtschaftsingenieurwesen. Er \u00fcbte diverse Hilfsjobs aus und geh\u00f6rte in den fr\u00fchen 80er Jahren zwei Jahre lang zu einer Gruppe jugendlicher Hausbesetzer in Berlin. Nach einer Zeit als Regie-Hospitant am Hamburger Thalia-Theater lebt er ab 1987 als freier Schriftsteller wieder in Berlin. Er ist Mitglied des Verbandes Deutscher Schriftsteller und des P.E.N.-Zentrums Deutschland und seit 2012 der Linkspartei.\n\nMichael Wildenhain ist Verfasser von Romanen, Erz\u00e4hlungen, Gedichten, Theaterst\u00fccken und Jugendb\u00fcchern. Im Mittelpunkt mehrerer seiner Werke stehen die Hausbesetzer- und Autonomenszene im Berlin-Kreuzberg der 80er Jahre sowie der \"Deutsche Herbst\" und Themen wie Gewalt und Rechtsradikalismus unter Jugendlichen.\n\nWerke (Auswahl): Zum Beispiel K., Berlin 1983; Prinzenbad, Berlin 1987; Das Ticken der Steine, Berlin 1989; Denn es ist die Maschine in ihnen, die von Z\u00e4rtlichkeiten tr\u00e4umt, Berlin 1991; Die Freiheit, die sie meinen, M\u00fcnchen 1991; Umstellt, Berlin 1991; Die kalte Haut der Stadt, Berlin 1991; Exit Berlin, Berlin 1994 (ein Kriminalroman, unter dem Pseudonym Carl Wille); Heimlich, still und leise, Frankfurt am Main 1994; Wer sich nicht wehrt, Ravensburg 1994; Die Zeit als Wolf, Weilerswist, 1995; Erste Liebe \u2013 deutscher Herbst, Frankfurt am Main 1997; Der deutsche Zwilling, Rom 1999; Wieland, der Meister oder Alle Schw\u00e4ne sind wei\u00df, Frankfurt am Main 2000; Der Augenblick des Absprungs, M\u00fcnchen 2001; Minna Musical, Heilbronn 2000 (zusammen mit Konstantin Wecker u. a.); Russisch Brot, Stuttgart 2005; Mit hei\u00dfem Herz, Berlin 2007; Die sch\u00f6nen scharfen Z\u00e4hne der Koralle, Bamberg 2007; Tr\u00e4umer des Absoluten, Stuttgart 2008; Blutsbr\u00fcder, Ravensburg 2011.\n\n#### Vielleicht kannst du erst mal kurz was zu dir sagen, zu deinem Hintergrund und zu den Themen, \u00fcber die du schreibst?\n\nIch schreibe im Prinzip seit Anfang der 80er Jahre. 1983 ist im Rotbuch-Verlag das erste Buch erschienen, Zum Beispiel K, und damals h\u00e4tte ich sicherlich gesagt, dass ich mich ganz explizit als politischer Schriftsteller verstehe. Dieses Buch ist entstanden aus einer existierenden Gruppe heraus, die eine Brosch\u00fcre erstellen wollte, um sich in einer politischen Stellungnahme abzugrenzen gegen\u00fcber solchen Organen wie dem Berlinteil der taz und dem Kursbuch. Es kam damals so ein Kursbuch heraus, Der gro\u00dfe Bruch 1981, Kursbuch '65, und die Leute, die da im Wesentlichen geschrieben haben, waren Benny H\u00e4rlin und Michael Sontheimer, die wir auch pers\u00f6nlich sehr gut kannten, weil die auch in Sch\u00f6neberg t\u00e4tig waren. Es gab einen Sch\u00f6neberger Besetzer-Rat, der sich recht regelm\u00e4\u00dfig traf, und da gab es halt zwei Fraktionen, die sogenannte \"Verhandler\"-Fraktion, der geh\u00f6rten auch Michael Sontheimer und Benny H\u00e4rlin an, die auch federf\u00fchrend den Aufbau der Berlin-taz vorangetrieben haben, und wir waren die sich etwas radikaler verstehende \"Nichtverhandler\"-Fraktion. Und wir fanden uns halt in unserem politischen Standpunkt nicht gen\u00fcgend gew\u00fcrdigt. Es gab zwar die radikal, aber die wurde zu dem Zeitpunkt ja schon stark verfolgt und ist auch nicht in so dichter zeitlicher Folge erschienen wie nat\u00fcrlich die Berlin-taz. Daraus ist also dieses Buch entstanden. Die Brosch\u00fcren-Gruppe zerfiel. Dieses Buch ist dann doch erschienen, aber eigentlich erst, als der H\u00f6hepunkt schon l\u00e4ngst vor\u00fcber war. Aber das war sozusagen der Beginn meiner ernsthaften T\u00e4tigkeit als Autor. Ich habe dann im Herbst '85 ein Informatikstudium begonnen, weil ich dachte, diese ganze Literatur, das bringt es nicht, habe dann aber weitergemacht und zwei weitere B\u00fccher bei Rotbuch ver\u00f6ffentlicht, eigentlich sogar noch einige mehr, ein Lyrikband und noch ein Gemeinschaftswerk mit zwei K\u00fcnstlern aus der DDR. 1987 ist Prinzenbad erschienen und 1991 Die kalte Haut der Stadt und in gewisser Weise stehen diese drei B\u00fccher in einer Reihe. Prinzenbad war f\u00fcr mich vor allem ein Buch der Niederlage und Die kalte Haut der Stadt ist entstanden sehr stark unter dem Eindruck des 1. Mai 1987. Und das sind halt auch die zwei Eckdaten, der Tod von Klaus J\u00fcrgen Rattay am 22.9.1981 nach der R\u00e4umung der acht H\u00e4user, vier waren in Sch\u00f6neberg, und zweites Eckdatum, 1. Mai 1987. Man k\u00f6nnte schon sagen, Die kalte Haut der Stadt ist so was wie ein literarischer Niederschlag der autonomen Bewegung mit Zentrum Westberlin. Im weiteren Verlauf habe ich eine ganze Reihe von B\u00fcchern geschrieben, vor allem dann auch Jugendromane. Ich bin dann 1997 noch mal weiter zur\u00fcckgegangen in Erste Liebe Deutscher Herbst und habe ein Buch geschrieben \u00fcber eine sich politisierende Gruppe von Sch\u00fclern unter dem Einfluss einer Referendarin beziehungsweise Lehrerin vor dem Hintergrund der Folie 1977, also \"Deutscher Herbst\", wobei das Buch im Fr\u00fchjahr '77 beginnt und ungef\u00e4hr ein Jahr sp\u00e4ter endet. Dann habe ich das Thema noch mal 2008 aufgenommen, ein Buch, das 35 Jahre umfasst und ganz stark konzentriert ist auf eine Person, Tariq hei\u00dft sie in meinem Buch, Tarek Mousli hei\u00dft sie real. Das ist der Mann, der bei den Revolution\u00e4ren Zellen war und die Leute reingeh\u00e4ngt hat... Das ist so eine Freundschaftsgeschichte, die in der sp\u00e4ten Kindheit an einer Grundschule beginnt und sich dann durchzieht bis fast in die Gegenwart 2004\/2005. Dann gibt es noch Theaterst\u00fccke, die sich haupts\u00e4chlich mit Rechtsradikalismus-Themen befassen, soweit sie explizit politisch sind, und es gibt eins, das aber nie aufgef\u00fchrt worden ist, mit dem Titel Rote Armee Fraktion.\n\nMeine B\u00fccher sind keine Recherche-B\u00fccher, sondern zun\u00e4chst mal habe ich mich politisch bet\u00e4tigt und dann erst sind diese B\u00fccher entstanden.\n\n#### W\u00fcrdest du sagen, dass die Hausbesetzer-Szene deckungsgleich war mit der autonomen Szene?\n\nBei den Leuten, die H\u00e4user besetzt haben Anfang der 80er Jahre, kamen h\u00f6chst unterschiedliche Gruppen zusammen. Das war eine ganz heterogene Sache. Das reichte bis hin zu Mieter- und Obdachloseninitiativen, es gab eine Gruppe von Rockern, die ein Haus kurze Zeit besetzt hatten in der Waldemarstra\u00dfe, und es gab nat\u00fcrlich auch Leute, die sich sehr politisch begriffen, sowohl in einer eher reformerischen Hinsicht, aus denen sind dann viele der Bezirksverordneten der Alternativen Liste hervorgegangen, und dann auch die Mietervereine und so was wie der Verein SO 36, und auf der anderen Seite gab es verschiedene Spektren der eher radikalen Seite. Und das war markiert durch die Trennungslinie verhandeln\/nicht verhandeln, also Bruch mit dem Staat \u2013 nicht verhandeln \u2013 oder doch das Haus eher erhalten \u2013 also verhandeln. Und die radikalen Teile umfassten nat\u00fcrlich auch ein gro\u00dfes Spektrum. Es gab Leute, die haben gesagt, was der Guerilla in El Salvador die Berge, das sollen die besetzten H\u00e4user der RAF sein [Lachen]. Und das hat sich im Verlauf dieses Jahres '81 so entwickelt und zugespitzt, und da sind die Autonomen in gewisser Weise entstanden, soweit man das so sagen kann. Es gab zwei Fraktionen: Die Autonomen und die Antiimps, also die Antiimperialisten, die sich zwar nicht als RAF-Sympathisanten erkl\u00e4rt haben, aber es letztlich doch waren. Die Antiimps hatten es in Berlin relativ schwer, weil n\u00e4mlich ihre K\u00f6pfe am 1. Mai 1980 fast geschlossen das Dach des Amerikahauses besetzt hatten, mit ein paar Mollis unter anderem, und deswegen fast alle eingefahren sind und zwei Jahre einsa\u00dfen und von daher die hei\u00dfe Phase der Hausbesetzung nicht richtig mitbekommen haben; und die wenigen von denen, die noch drau\u00dfen waren, die waren nicht die hellsten Kerzen im Leuchter. Die haben dauernd versucht, irgendwas zu entlarven, also die h\u00e4tten wahrscheinlich auch das Archiv der Jugendkulturen hier als Zentrale des CIA entlarvt, und ihre neuesten Entlarvungen haben sie dir dann immer [Lachen] aus klandestin anmutenden Plastikt\u00fcten heimlich zugesteckt bei irgendwelchen Versammlungen. Also die waren f\u00fcr uns nicht so richtig attraktiv. Die RAF war sowieso nicht besonders attraktiv, die war ja auch in Westberlin wenig pr\u00e4sent; au\u00dfer am Anfang, Baader-Befreiung und so, war die hier nicht so relevant, aufgrund der speziellen Lage von Berlin und auch wegen der St\u00e4rke des 2. Juni, der sich eher anarchistisch verstand und der eine gro\u00dfe Pr\u00e4senz zumindest in der Szene hatte und eine gro\u00dfe Sympathie. Wesentlich waren dann im weiteren Verlauf von diesen radikalen Gruppen allenfalls die Revolution\u00e4ren Zellen. Der 2. Juni war ja schon ein bisschen zerfallen, aber die Revolution\u00e4ren Zellen haben sich ja dann 1985 in Berlin noch mal neu gegr\u00fcndet. Ganz wichtig war aus meiner Sicht auch Die Autonomie \u2013 Neue Folge. Das ist eine Zeitschrift, die hatte einen Vorl\u00e4ufer, also es gab sozusagen \"Alte Folgen\", die sind in Frankfurt herausgegeben worden, einer der Chefkolumnisten war Joschka Fischer. In dieser \"alten\" Autonomie ist auch sein ber\u00fchmter Artikel erschienen, der sich an die RAF und Verwandte richtete, wo er, dem Sinn nach, sagte: \"Legt die Bomben beiseite, nehmt die Steine wieder in die Hand.\" Die Neue Folge ist dann in Hamburg gemacht worden, also die haben sich sp\u00e4ter auch auseinanderdividiert, aber so die ersten 14 Hefte circa, die f\u00fcr uns wichtig waren, sind in Hamburg erschienen und von Teilen der Hausbesetzer-Szene, und zwar insbesondere den Teilen, die sich als autonom oder Autonomen-nah verstanden, sehr stark rezipiert worden. Das waren sehr theoretische Hefte, mit viel Fu\u00dfnoten-Apparat und so weiter, die ausgesprochen unterschiedliche Themen behandelten, Anti-AKW usw.; ein ganz wichtiges Heft war f\u00fcr uns eins zum Thema Architektur als Aufstandsbek\u00e4mpfung. Da ging es darum, wie kann durch eine bestimmte Art von Stadtplanung von vornherein Rebellion im Keim erstickt werden. Es gab ein Heft \u00fcber die iranische Revolution, also sozusagen die Hoffnungen, die damit auch f\u00fcr die Linke verbunden waren [Lachen], und noch eine Reihe anderer Hefte. Die Autonomie Neue Folge wurde auch in der radikal diskutiert, bis fast Ende der 80er Jahre wurden noch in so Zirkeln die Artikel diskutiert und der Ansatz. Der Ansatz war im Kern ein antimodernistischer Ansatz, w\u00fcrde ich aus heutiger Sicht sagen. Also die Subsidiarit\u00e4t wurde gefeiert, und das ist ja auch f\u00fcr die Autonomen kennzeichnend und wichtig f\u00fcr ihr so langes \u00dcberleben: Man muss der gro\u00dfen Maschine Widerstand entgegensetzen, obwohl man eigentlich sp\u00fcrt, dass man sie nicht wirklich aufhalten kann.\n\nWir waren die J\u00fcngeren, wir waren die Radikaleren. Wir waren zwar von diesen Ideen affiziert, aber wir waren die, die nicht reformistisch abbiegen wollten, sondern so richtig revolution\u00e4r draufhauen. Und es passierte nichts, so war unsere Empfindung, und dann kam pl\u00f6tzlich mit den H\u00e4usern, die gro\u00dfe M\u00f6glichkeit.\n\nDie Autonomen hatten damals eigentlich keine langfristigen Perspektiven; sie waren schon in ihren Anf\u00e4ngen ganz stark gepr\u00e4gt von einem gewissen Antirealsozialismus, ich will jetzt nicht sagen Antikommunismus. Es gab so eine Parole im Herbst '81: \"Aufstand in Ost und West, nieder mit der Bonzen-Pest!\" Also man f\u00fchlte sich solidarisch mit Solidarnosc und es gab Anarchisten aus westeurop\u00e4ischen L\u00e4ndern, ich erinnere mich an einen Belgier, der war nach Polen gereist in dem Irrglauben, da irgendeine anarchistische Gruppe innerhalb von Solidarnosc unterst\u00fctzen zu k\u00f6nnen. Der Fall wurde publik, weil der hatte versucht, Teile von Radios dahin zu schmuggeln, um da einen illegalen Sender aufzubauen, und ist gekriegt worden. Mit illegalen Radios haben sich ja viele besch\u00e4ftigt. Zugleich gab es zu dem Zeitpunkt diese alternative Verweigerungshaltung gegen die Technologie. Es hatte etwas Maschinenst\u00fcrmerisches an sich. Oder es gab zumindest diese Aspekte, neben allem m\u00f6glichen anderen Fr\u00fchsozialistischen, Anarchistischen und so weiter.\n\n#### Wann ist dir denn zum ersten Mal bewusst die Selbstbezeichnung \"Autonome\" aufgefallen?\n\nAlso der Begriff bzw. die Bezeichnung als politische Gruppe oder Formation ist mir zum ersten Mal untergekommen und dann auch von mir weitergetragen worden so Mitte August 1981, denke ich. Und zwar wurde da ein Autonomen-Plenum gegr\u00fcndet in Berlin. Dieses Autonomen-Plenum setzte sich zusammen aus Leuten, die sich von den Besetzer-R\u00e4ten kannten. Viele von denen waren der Meinung, dass die Hochzeit der Besetzung vorbei ist, und sp\u00e4testens mit dem Herbst '81 hat sich die \u00dcberzeugung angefangen durchzusetzen. Es wurde teilweise sehr selbstironisch damit umgegangen, also dann haben die Leute immer von Automaten-Plenum oder so geredet. Man muss ja sowieso sagen, trotz aller Militanz waren die Anf\u00e4nge dieser Hausbesetzer-Bewegung eine Art Sp\u00e4thippie-Bewegung, auch im Outfit. Die meisten Leute hatten lange Haare, rannten teilweise sehr bunt rum, sehr zerfetzt und sehr disparat. Nur hat man nat\u00fcrlich schnell mitbekommen, dass man dadurch aufgefallen ist und f\u00fcr die Greiftrupps der Polizei leicht zu identifizieren war, und so hat sich nach und nach diese Uniformiertheit rausgebildet, also schwarze Lederjacke, schwarze Jeans, damals waren es eigentlich eher noch so Normaljeans und eine schwarze Kappe, und schon war man schwerer zu erkennen. Und au\u00dferdem hat man nat\u00fcrlich viel martialischer gewirkt. Aber die Anf\u00e4nge waren ganz klar Hippie, bis hin zu irgendwelchen Pluderhosen und Schals und so. Jedenfalls so tauchte dieser Begriff \"Autonome\" im Sommer '81 auf und er war von Anfang an diffus.\n\nWas die Zusammensetzung dieses Plenums betraf, waren es haupts\u00e4chlich Leute, die sich als militant verstanden und aus den Hausbesetzungen kamen. Viele von denen haben dann aber, wie gesagt, die These aufgebracht, dass man sich jetzt anderen politischen Inhalten zuwenden muss, weil die Hausbesetzungen ihren H\u00f6hepunkt \u00fcberschritten haben und klar ist, dass wir die H\u00e4user nicht gegen den staatlichen Zugriff verteidigen k\u00f6nnen, was ja am Anfang anders gesehen oder gehofft wurde. Der Kern dieser ganzen Sache war ja: Wir nehmen uns einen Teil des Wohlstands, zumal dort, wo er nicht pfleglich behandelt wird, in Form dieser leerstehenden H\u00e4user; wir begreifen uns als exterritorial, wenn man so will, als au\u00dferhalb der Gesellschaft, als Enklave, oder besser: Exklave \u2013 als exterritorial \u2013, und die verteidigen wir. Und das wurde sehr weit getrieben mit den ganzen Aufr\u00fcstungen der H\u00e4user. Bei H\u00e4usern, die durchsucht werden sollten, versammelten sich dann Gruppen mit Kn\u00fcppeln und Steinen und so weiter, um diesen Zugriff abzuwehren, also das war erstaunlich [Lachen]. Und diese ganze Sache fand im Herbst '81 noch mal einen H\u00f6hepunkt. Nach dem Tod von J\u00fcrgen Rattay gab es die Vermutung, dass der CDU\/FDP-Senat mit Heinrich Lummer als Innensenator weitere H\u00e4user r\u00e4umen w\u00fcrde, und zwar vor allem die, die sie als vermeintliche Schaltzentralen ausgemacht hatten in 36 [Kreuzberg], und dagegen gab es den Plan, diesen Stadtteil zu verteidigen, also an den Br\u00fccken und an den Zufahrtsstra\u00dfen, es waren so acht neuralgische Punkte. Dieser Plan wurde \u00fcber zwei Monate hinweg verfolgt. Verschiedene H\u00e4user, die au\u00dferhalb waren, wurden bestimmten Br\u00fccken zugeordnet, wir waren zum Beispiel der Kottbusser Br\u00fccke zugeordnet. Es gab ein Meldersystem vor den Polizeikasernen, sodass man wusste, wenn die anfangen auszur\u00fccken in gr\u00f6\u00dferem Umfang und mit schwerem Ger\u00e4t, dann geht es los. Und in dieser Zeit, die die brauchten, um anzur\u00fccken, h\u00e4tten diese Punkte verbarrikadiert werden sollen, um dann verteidigt zu werden. Das war nat\u00fcrlich etwas aberwitzig. Ich vermute, dass die andere Seite davon wusste, weil den ganzen Winter \u00fcber ist nicht ger\u00e4umt worden. Ich glaube, die wollten es einfach austrocknen und im Sande verlaufen lassen \u2013 und so war es dann auch. Der Winter hat im Prinzip die H\u00e4user zersetzt. Aber zu dem Zeitpunkt waren die Autonomen schon nicht mehr so stark mit den Hausbesetzungen konnotiert und setzten sich langsam davon ab. Das ging dann richtig los '82 mit der Vorbereitung der ersten gro\u00dfen Anti-Reagan-Demo am 11.06.82, wo auf dem Nollendorfplatz 3.000 Leute eingekesselt worden sind von der Polizei.\n\n#### Also ich h\u00e4tte gedacht, dass auch die Antiimps eigentlich sehr, sehr wichtig gewesen sind in der autonomen Szene in den 80er Jahren...\n\nDas werden verschiedene Leute unterschiedlich beschreiben. Zum Beispiel in Frankfurt waren die Antiimps extrem wichtig, aber in Westberlin, zumindest in Sch\u00f6neberg\/Kreuzberg, wurde die Sache '81 schon als Gegensatz diskutiert. Und zwar k\u00f6nnte man die Gegenposition festmachen als innenpolitisch\/au\u00dfenpolitisch. Die Autonomen setzten im Prinzip eher auf so eine Massenlinie, also Radikalisierung anhand von Problemen und Widerspr\u00fcchen und Antagonismen, wie es immer hie\u00df, die hier vorliegen, und f\u00fcr die Antiimps gab es vor allem die USA als Hauptfeind mit ihrem Werkzeug, der NATO, und dagegen musste man vorgehen. Und dem waren nat\u00fcrlich auch bestimmte politische Aktivit\u00e4ten untergeordnet, also zum Beispiel in Frankfurt die Startbahn. Und dadurch kann es sehr gut sein, dass die Autonomen dort eine st\u00e4rkere N\u00e4he hatten zu den Antiimps, weil die Startbahn wurde ganz klar als US-dominiertes NATO-Projekt betrachtet und entsprechend war der Kampf antiimperialistisch, aber das war bei den H\u00e4usern nicht so. Hingegen bei der Mobilisierung gegen Reagan schon. Und durch diese thematische Kehrtwendung haben nat\u00fcrlich auch viele Autonome, auch wenn sie sich nicht so verstanden haben wie die Antiimps in ihrer sehr dienenden Funktion, fast Abh\u00e4ngigkeit gegen\u00fcber bzw. von den gro\u00dfen Heroen der Roten Armee Fraktion, einen politischen Schulterschluss vollzogen insofern, als sie diesen Antiimperialismus sehr viel st\u00e4rker auf ihre Fahne geschrieben haben.\n\n#### Tauchte denn in der Zeit auch schon dieser Gegensatz, der heute ja so stark ist, zwischen den Antiimps und den Antideutschen auf?\n\nDas kam, meines Erachtens, viel sp\u00e4ter. Das ist eine Entwicklung, die eigentlich nach '89\/'90 erst losgegangen ist. Ab 1990, vor allem zum 3. Oktober, tauchte bei Demonstrationen diese Parole auf: \"Deutschland halt's Maul!\" Aber diese Form der Antideutschen gab es erst sp\u00e4ter. Es gab immer Auseinandersetzungen \u00fcber die Frage der Pal\u00e4stina-Solidarit\u00e4t, aber es war damals noch so, dass der gr\u00f6\u00dfere Teil der radikalen Linken, soweit er sich daf\u00fcr \u00fcberhaupt interessiert hat, zwar schon einerseits Solidarit\u00e4t \u00fcben wollte mit den Pal\u00e4stinensern, andererseits waren viele von denen im Kibbuz gewesen, das war ja ein fast prototypischer Punkt der Laufbahn. Nach der Schule ist man erst mal nach Israel in den Kibbuz gegangen. Aber ich glaube, was an diesen Kibbuzgeschichten damals stark wahrgenommen wurde, war dieses Alltagssozialistische. Der Staat Israel wurde zwar als imperialistisches Projekt und quasi als amerikanischer Satellit wahrgenommen, aber diese ganze Antideutschen-Diskussion hat sehr viel zu tun mit einer Verschiebung der Gro\u00dfkoordinaten, mit einem extremen Zweifel an so der sozialistisch-kommunistischen Zentralperspektive sp\u00e4testens nach dem Zusammenbruch des Ostblocks. Obwohl wir uns gar nicht als sympathisierend verstanden haben, w\u00fcrde ich schon sagen, dass es ein extremer Schlag war, sozusagen DER SCHLAG \u00fcberhaupt. Man hat da erst gemerkt, wie stark man von diesem Weltgef\u00fcge abh\u00e4ngig war als kleine Sumpfdotterblume \"Autonome\", als die ganze Sache in dieser Form zusammenbrach. Es kommt mir vor, als ob man von einem utopischen Fluchtpunkt Sozialismus\/Kommunismus umgeschwenkt ist auf einen negativ utopischen Fluchtpunkt, Verteidigung der Welt gegen ein neues Auschwitz. Und das w\u00e4ren dann die Antideutschen. Und dann hat es auch viel zu tun hat mit der Entwicklung im Nahen Osten. Da haben sich die Koordinaten ja extrem verschoben; die Hamas zum Beispiel ist erst '87 gegr\u00fcndet worden, und das ist eine ganz andere Organisation als die PLO, und vor diesem Hintergrund passt dann auch diese pl\u00f6tzliche Israel-Affinit\u00e4t und dieses militante Verteidigen des Staates Israel. Eine Praktikantin, sie war selbst eine Antideutsche, meinte mal ein bisschen ironisch: Na ja, Israel ist halt das Mallorca der Antideutschen [Lachen], und das ist eine wirklich h\u00fcbsche Formulierung. Selbst mein Sohn kannte schon mit 14 den Begriff \"antideutsch\" aus der Jugend-Antifa.\n\n#### Ist die Antifa also dann erst in diesem Zusammenhang wichtig geworden?\n\nDie Antifa ist sowieso lange nicht besonders wichtig gewesen, weil es, zumindest in Westberlin, lange keinen relevanten Neofaschismus gab. Es gab immer mal Antifa-Demonstrationen und Antifa-Aktivit\u00e4ten w\u00e4hrend der Zeit der besetzten H\u00e4user. Da gab es mal zum Anfang von \"Tuwat\" [Ende August 1981] irgend so eine Antifa-Demonstration gegen eine rechte Demonstration. Ich wei\u00df noch, wir hingen auf der Gotzkowsky-Br\u00fccke herum, die Polizei zwischen uns, es wurden auch ein paar Steine geschmissen, und das war's dann. Es gab immer mal \u00dcbergriffe auf besetzte H\u00e4user, zum Beispiel in der Maa\u00dfenstra\u00dfe am Nollendorfplatz, von so Teddyboys. Die sa\u00dfen immer gegen\u00fcber in einer Billardkneipe und kamen dann r\u00fcber und haben so ein bisschen Randale gemacht vor den besetzten H\u00e4usern. Aber die waren leicht abzuwehren. Die hatten ja alle so Teddytollen mit sehr viel Haarspray, und man brauchte nur von oben Wasser \u00fcber die auszukippen, dann sind ihre Haare kaputtgegangen und sie waren nicht mehr gesehen. [Lachen.]\n\nDie erste richtige Antifa-Mobilisierung, die ich mitgemacht habe, war zum 20. April 1983. Der dumme Fu\u00dfballverband hatte ausgerechnet an diesem Tag im Olympiastadion ein Freundschaftsspiel angesetzt zwischen Deutschland und der T\u00fcrkei. Und da gab es eine gro\u00dfe Nazi-Skinhead-Mobilisierung und so weiter und dagegen hatten wir ebenfalls mobilisiert. Da kamen zig Leute aus Westdeutschland, die hockten \u00fcberall rum, also Gruppen mit Kn\u00fcppeln, Punks \u2013 irgendwann alle betrunken. Und die sind immer rumgefahren in der Stadt und haben nach den Skinhead-Ansammlungen gesucht, die sich aber nicht einfanden. Stattdessen war die Stadt besetzt von der Polizei. Und solche Mobilisierungen in dieser Gr\u00f6\u00dfe und Form habe ich dann erst wieder Anfang der 90er erlebt. Die Antifa hat sich im Prinzip gegr\u00fcndet und ist wichtig geworden f\u00fcr die linke Szene, weil alles andere weggebrochen war, das war quasi ein neuer organisatorischer Kristallisationspunkt und eine gro\u00dfe Klammer, wo sich viele drunter finden konnten, die sich ansonsten nicht sehr einig waren, bis ins b\u00fcrgerliche Lager hinein. Und au\u00dferdem gab es ja auch pl\u00f6tzlich viele extreme Rechte, dieses uns\u00e4gliche nationale Erwachen, zun\u00e4chst \u00fcberwiegend in den sogenannten Neuen Bundesl\u00e4ndern, aber dann auch so ein Aufwallen der alten Bundesrepublik.\n\n#### Du hast schon die Skinheads und die Teddys erw\u00e4hnt, das waren wahrscheinlich so die Feindbilder, wenn es um jugendkulturelle Szenen geht, aber wie ist denn das Verh\u00e4ltnis zu anderen Szenen gewesen?\n\nAlso diese Sache mit den Teddys oder Rockabillies wurde eigentlich eher bel\u00e4chelt, das war nichts, was man besonders ernst nahm. Deren Hochzeit in Berlin war auch so '78\/'79\/'80, so als Jugendkultur in bestimmten Clubs, das hatte '81 schon wieder abgenommen. Bei den Skinheads war eigentlich von Anfang an klar, dass es auch Redskins gab, und die Leute, die sich mehr damit besch\u00e4ftigt hatten, wussten auch, wo das alles herkommt und dass es gar nicht urspr\u00fcnglich rechts war, aber dann quasi von rechts vereinnahmt wurde. Aber die Redskins waren mehr ein Spurenelement, die waren nie besonders stark. Normalerweise waren die Skinheads das Feindbild. Es gab sp\u00e4ter mal so einen Zwischenfall hier gleich um die Ecke vom Archiv in der Fidicinstra\u00dfe\/ Ecke Mehringdamm, da war damals eine Disco und da fanden auch Ska-Konzerte mit vielen Skinheads statt, und da ist dann eines Tages so eine Truppe aus dem Mehringhof nach irgend einem linken Konzert hochgezogen, hat den Notausgang mit M\u00fcllcontainern verbarrikadiert und Tr\u00e4nengas reingeworfen. Danach hat der Typ aufgeh\u00f6rt mit seinen Ska-Konzerten beziehungsweise den Club sogar abgegeben. Also es gab immer so einen Grundverdacht gegen Skinheads. Nat\u00fcrlich gab es auch Leute, die das Ganze besser musikkulturell interpretieren konnten, aber die Szene war ja \u00fcberwiegend intellektuellen- und kulturfeindlich. Punk-Musik war eine der wenigen kulturellen \u00c4u\u00dferungen, die nicht nur gestattet waren, sondern auch gefeiert wurden. Punk galt als links, irgendwie. Dieses ganze Elendspunks-Ambiente, was dann in den weiteren 80er Jahren immer mehr zunahm, war damals noch gar nicht so der Fall, sondern es gab eine sehr starke Rezeption von Stadtindianern, vor allem aus Italien herkommend, die sich nicht nur politisch, sondern teilweise auch sehr alltagsmilitant verstanden \u2013 und das war nat\u00fcrlich f\u00fcr die Autonomen das gro\u00dfe Vorbild. Also der ganze Begriff kommt ja sowieso aus Italien. Es gibt da ein sehr gutes Buch von Nanni Balestrini, Die Unsichtbaren, was bei Schwarze Risse [linker Buchladen und Verlag] immer wieder aufgelegt wird, da wird die autonome Bewegung Italiens gut beschrieben. Also diese Stadtindianer in Italien waren ein Ph\u00e4nomen, von dem wir immer mal gelesen hatten, die begriffen sich als Aufst\u00e4ndische, in so einer gewissen Totalit\u00e4t, also in ihrem gesamten Alltag, und rannten teilweise bewaffnet rum, auch auf den Demos. Es gab auch Schusswechsel bei diesen italienischen Demos und in Berlin gab es solche Leute auch, die sich stark so begriffen. Die waren allerdings meist mehr so hippiem\u00e4\u00dfig vom Outfit her, aber teilweise waren die auch bewaffnet; also es gab immer Leute, die rannten auch mit Knarren rum, aber die wurden gl\u00fccklicherweise nie eingesetzt. Jedenfalls war diese Punksache in Westberlin am Anfang sehr stark mit diesem Indianer-Gestus konnotiert. Aber unabh\u00e4ngig davon spielte Punk-Musik eine gro\u00dfe Rolle, nur f\u00fcr mich pers\u00f6nlich war Musik sehr zweitrangig.\n\n#### Warum?\n\nHat mich nicht so interessiert. Ich bin zwar gerne auf die Kiezdiskos gegangen, um da Pogo zu tanzen, aber es hatte keine Bedeutung. F\u00fcr mich war es nicht Jugendkultur, sondern eine politische Bewegung, die allerdings \u2013 wie alle rebellischen Bewegungen \u2013 davon lebte, dass sich ein gro\u00dfer Teil von Jugendlichen daran beteiligte, die ihre eigene Kraft extrem \u00fcbersch\u00e4tzen [lacht] und dadurch eine enorme Wucht entwickelten, zumindest f\u00fcr kurze Zeit.\n\n#### Also es gab auch einen gro\u00dfen Teil an Leuten, die ihre jugendliche Rebellionszeit in der Szene verbracht haben, aber danach wieder ausgestiegen sind und sich auch gar nicht so sehr mit den Inhalten besch\u00e4ftigt haben?\n\nJa. Das geh\u00f6rte und geh\u00f6rt bestimmt dazu. Aber ich empfinde das immer als ein bisschen diskreditierend, weil es so tut, als w\u00e4re es damit nicht ernst zu nehmen, und das empfinde ich als schwachsinnig, weil jede rebellische Bewegung in der gesamten Geschichte ist nicht von Opas gemacht worden [lacht], sondern von Leuten zwischen 18 und 23. Fr\u00fcher sind die in den Krieg gezogen, in irgendwelche K\u00e4mpfe Stamm gegen Stamm, Clan gegen Clan, und heutzutage, wenn sich die Chance bietet, dann kann man sich eben solchen Sachen zuwenden. Und f\u00fcr unsere Generation hat sich diese Chance eben pl\u00f6tzlich geboten. Es gab gewisse Vorl\u00e4ufer, also es gab ja schon spontane Organisierung an anderer Stelle, zum Beispiel das erste H\u00fcttendorf in Gorleben, es gab die Jugendzentrumsbewegung, auch in Kleinst\u00e4dten, da waren viele schon politisiert worden.\n\n#### Kannst du vielleicht noch was zur Rolle der Frauen in der Szene sagen?\n\nBei uns hat das Thema am Anfang keine gro\u00dfe Rolle gespielt, aber es gab H\u00e4user oder zumindest Wohnungen oder Teile von H\u00e4usern, die nur von Frauen besetzt waren. Es gab damals auch viel mehr Buchl\u00e4den und Discos nur f\u00fcr Frauen, und es gab halt auch Frauen innerhalb der Szene, die sich in einem feministischen Zusammenhang sahen und dies auch stark vertreten haben, und es gab in einigen anderen H\u00e4usern sehr harte und sehr oft auch zersetzende Diskussionen \u00fcber diese Problematiken. Und es gab immer wieder Diskussionen um Vergewaltiger, bis hin zu so Selbstjustiz-Sachen. Die wurden dann auch oft von den M\u00e4nnern unterst\u00fctzt, teilweise sehr vorauseilend. In der Zeit hat sich ja auch die Rote Zora gegr\u00fcndet, die Kampagnen gegen Sex-Gesch\u00e4fte bis hin zu Anschl\u00e4gen machten. Das war nat\u00fcrlich auch ein Niederschlag dieser Diskussion. Also am Anfang wurde die ganze Beziehungs- und Sexualit\u00e4tsgeschichte sehr frei verhandelt \u2013 was ich super fand. Es war immer mal wieder so, dass man mit einer Frau im Bett landete und die sagte dann: \"Aber keine Penetration.\" Dann hat man sich gesagt, okay, das ist jetzt hier die Regel, die aufgestellt wird. [Lacht.] Manchmal musste man auch noch ewig diskutieren, die ganze Nacht, was daran jetzt schlecht oder gut ist. Ich fand es nervig, aber es war nichts, was ich als ausschlie\u00dfend empfunden h\u00e4tte, es hatte noch nicht diesen bitteren Beigeschmack des absolut Prinzipiellen. Genauso in Ern\u00e4hrungsfragen: Wenn jemand statt Currywurst lieber Gr\u00fcnkernboulette gegessen hat, dann hat man gesagt: \"Gut, so what.\" [Lacht.]\n\n#### Heute wird wohl eher andersrum ein Problem daraus...\n\nDamals war auch das noch eine minorit\u00e4re Geschichte und die wurde nicht so prinzipienstarr verhandelt. Im Zuge der Verh\u00e4rtung der Auseinandersetzung konnte man dann merken, dass die eigentliche Hochzeit des Aufbruchs vorbei war. Wenn Diskussionen immer fundamentalistischer werden, immer gleich Prinzipien-Diskussionen sind, sind sie die Begleitmusik zum Niedergang.\n\n#### Hat sich die \u00f6ffentliche Wahrnehmung der autonomen Szene heute im Vergleich zu eurer Zeit eigentlich ge\u00e4ndert?\n\nJa, es hat einen Cut gegeben zwischen der b\u00fcrgerlichen \u00d6ffentlichkeit im weitesten Sinne und der Linken... Also die Hegemonie hat sich ge\u00e4ndert, es gibt keine linksliberale Hegemonie mehr \u2013 oder vielleicht sogar linke \u2013 in den meinungsmachenden Medien, sondern es gibt mittlerweile eine rechtsliberale, rechtskonservative. Es ist nicht ganz einheitlich, aber es hat sich doch sehr stark in diese Richtung verschoben. Die '68er damals haben von Anfang an auch in den gro\u00dfen Medien geschrieben, die waren pr\u00e4sent. Wir waren schon \u00fcberhaupt nicht mehr pr\u00e4sent. Als ich bei Rotbuch angefangen habe, da kam ich mir h\u00e4ufig vor wie der Vorzeige-Autonome... Die haben zwar sympathisiert, hatten aber im Alltag keinen Kontakt zu Leuten, die sich auf der Stra\u00dfe rumtummelten. Heute erreicht die Szene immer noch zu bestimmten Anl\u00e4ssen eine enorme Mobilisierungskraft, \u00fcber die man staunt. Aber dass diese dann auch politisch wahrgenommen wird in der Gesellschaft, nicht nur an den R\u00e4ndern der Gesellschaft, das ist einfach nicht der Fall. Da liest man im Tagesspiegel, dass der Verfassungsschutz dar\u00fcber erstaunt ist, dass die Autonomen da und dazu nichts sagen... Aber die Autonomen selbst werden gar nicht erst gefragt. Der Verfassungsschutz spricht sozusagen im Tagesspiegel und nicht nur dort f\u00fcr die Autonomen. Dass die radikale Linke in die Gesellschaft hineinwirken w\u00fcrde, diesen Eindruck habe ich nicht. Abgesehen von ihrer Mobilisierung zu punktuellen Anl\u00e4ssen und manchmal auch zu Kampagnen.\n\n#### War das wirklich anders in den 80er Jahren?\n\nEs war auf jeden Fall am Anfang anders. Da war es so, dass man mit den Hausbesetzungen durchaus Sympathien hatte in der Bev\u00f6lkerung. Die ging dann allerdings verloren, als sich die Sache immer militanter geb\u00e4rdete.\n\n#### Hat sich der Umgang mit Gewalt ver\u00e4ndert? Welche Rolle hat Gewalt fr\u00fcher gespielt?\n\nF\u00fcr uns war Gewalt im Wesentlichen instrumentell. Also es gab sehr moralische Diskussionen darum. Bei dieser Kreuzberg-Verteidigungsplan-Sache wurde ewig dar\u00fcber diskutiert, ob man einen Molli werfen darf gegen ein gepanzertes Polizeifahrzeug, wo jemand drin sitzt, oder nicht. Denn es gab meist so einen Konsens, dass die letztendliche Entscheidung von allen mitgetragen werden muss. Und diese ganzen Diskussionen waren extrem moralisch gepr\u00e4gt. Da gab es Leute, die waren gewaltlos, dann gab es Leute, die waren nicht unbedingt gewaltlos, haben aber gesagt, es ist nicht unsere Sache, wir k\u00f6nnen das nicht und wollen es auch irgendwie nicht, aber wir w\u00fcrden uns jetzt nicht als apodiktisch Gewaltlose bezeichnen. Die hatten sowieso einen schweren Stand, die absolut und programmatisch gewaltlos Auftretenden, die wurden immer bel\u00e4chelt. Es hatte nat\u00fcrlich viel damit zu tun, dass in der gesamten Anti-AKW-Bewegung und dann auch in der H\u00e4userbewegung durch die militanten Auftritte eine ganze Menge erreicht worden war, zumindest f\u00fcr eine gewisse Zeit, und das konnte man unmittelbar sehen \u2013 und das meine ich auch mit \"instrumentell\". Es gab halt so Parolen wie: \"Jede R\u00e4umung kostet Euch eine Million!\" Also da ging es dann immer darum, dass nach einer R\u00e4umung Scheiben eingeworfen wurden oder so. Ansonsten war ganz allgemein die Tendenz, keine Gewalt gegen Menschen anzuwenden, wobei das in direkten Auseinandersetzungen auf Demonstrationen mit der Polizei anders gesehen wurde. Aber ob man H\u00e4user gewaltsam verteidigt, da wurde von Anfang bis Ende ewig dr\u00fcber diskutiert. Es gab auch Vorbilder, weil in Amsterdam zum Beispiel waren die H\u00e4user teilweise mit Erfolg militant verteidigt worden. Es gab Leute, die waren sehr hart drauf, die sind aus dem Konsens ausgebrochen und w\u00e4ren auch bereit gewesen, Polizisten einzeln anzugreifen oder so. Aber solche Sachen wurden sehr kritisch gesehen, damals, sehr viele Leute haben sich auch absolut gegen Brandanschl\u00e4ge ausgesprochen, weil die so wenig kalkulierbar sind. Aber auch von den Leuten, die h\u00e4rter drauf waren, wurde immer gesagt, man muss damit verantwortlich umgehen und gucken, dass auch ja niemand da drin ist, und es darf nichts passieren und so weiter.\n\nEs gab ja einen extremen Bruch in Berlin durch den Tod des Demonstranten Klaus-J\u00fcrgen Rattay, und noch an demselben Abend gab es das Ger\u00fccht, dass nicht nur er gestorben ist, sondern mindestens zwei weitere Demonstranten. Und dann gab es von der Polizeiseite das Ger\u00fccht, dass ein Polizist erstochen worden ist. Das hat sich dann alles nicht verifiziert, aber das Empfinden war klar: Damit ist eine Grenze \u00fcberschritten, mit dem Tod von Klaus-J\u00fcrgen Rattay ist eine Grenze \u00fcberschritten. Bei so Auseinandersetzungen, so ist meine Erfahrung, werden ja immer Regeln etabliert, die nirgends fixiert sind, aber die trotzdem jeder kennt. Also zum Beispiel ist eine Regel: Es wird auf keinen Fall geschossen und es kommt auch keiner zu Tode. Und wenn so was verletzt wird, dann gibt es einen Bruch, und in Berlin f\u00fchrte der dazu, dass \u00fcber zwei Monate hinweg dieser Plan entwickelt wurde, den Stadtteil zu verteidigen. In fast allen politischen Bewegungen, die in irgendeiner Form relevant werden, spielt Gewalt eine Rolle. Da ist nur die Frage, in welcher Form. In \u00c4gypten, am Beginn des Aufstands, gab es viele Tote, und trotzdem redete man danach, in gewisser Weise zu Recht, siehe Syrien, davon, dass es eine gewaltlose Revolution gewesen sei. Also die Hausbesetzungen waren jedenfalls keine erkl\u00e4rterma\u00dfen gewaltfreie Bewegung. Das kann man nicht sagen.\n\n#### Wie beurteilst du den staatlichen Umgang mit der linken Szene? Auch den Kampf gegen den Linksextremismus und die Darstellung der Autonomen als Gefahr f\u00fcr die Demokratie?\n\n[Lacht.] Die Demokratie hat ja ein logisches Problem: Einerseits m\u00fcsste sie jede \u00c4u\u00dferung zulassen, wollte sie wirklich demokratisch sein, andererseits w\u00fcrde sie damit auch \u00c4u\u00dferungen zulassen, die sie selber abschaffen wollen. In diesem Spannungsfeld bewegt sich die Demokratie immer. Das andere Spannungsfeld ist das der politischen Hegemonie. Also die Linke und die Linksliberalen und die Linksb\u00fcrgerlichen haben weite Teile der gesellschaftlichen Hegemonie verloren, und insofern ist die obrigkeitsstaatliche Komponente, die mit Repressionen einhergeht, st\u00e4rker geworden. Dass die Autonomen irgendeine Gefahr f\u00fcr die Demokratie darstellen, halte ich f\u00fcr Schwachsinn, dazu haben sie schon mal viel zu wenig Ziele. Im Vergleich zu den Autonomen oder den gesamten linksradikalen Splittern aller Art sind ja die Rechten theoretisch extrem konsistent, also zu sagen, wir wollen einen nationalistischen Staat, wir wollen einen autochthonen Staat \u2013 Sarrazin redet ja immer von den Autochthonen \u2013, das ist ein politisches Programm. Und dieses politische Programm versuchen sie auch durchzusetzen, zum Gl\u00fcck mit wenig Erfolg. Wenn sie damit mehr Erfolg h\u00e4tten, dann w\u00e4ren sie tats\u00e4chlich eine Gefahr f\u00fcr die Demokratie. Und zwar selbst, wenn sie es mit gewaltfreien Mitteln machen w\u00fcrden, weil sie bestimmte Prinzipien aushebeln w\u00fcrden, die man als konstituierend begreift f\u00fcr unsere Demokratie. Das ist auf der linken Seite \u00fcberhaupt nicht zu beobachten, die Linke ist eigentlich nur in der Lage, punktuell zu wirken \u00fcber ihre Mobilisierungskraft anl\u00e4sslich bestimmter Punkte. Und damit ist sie schon mal per Definition nicht mehr revolution\u00e4r, sondern, allenfalls noch radikalreformerisch. Also das ist ein bisschen wie mit Kanonen auf Spatzen...\n\n#### Besteht die M\u00f6glichkeit, dass die radikale Linke tats\u00e4chlich noch mal zu einer Bedrohung f\u00fcr den Staat werden k\u00f6nnte?\n\nIch glaube schon, dass die radikale Linke eine Bedeutung bekommen kann, aber das kann sie nur, wenn sie politische B\u00fcndnisse eingeht mit anderen linken Kr\u00e4ften, dabei aber auch ihre politischen Positionen kenntlich macht. Also die Position: \"Wir sind h\u00e4rter drauf als ihr und wir bringen mehr Leute auf die Stra\u00dfe und wir hauen uns aufopferungsvoller mit der Polizei\", das ist keine politische Position. Man muss erkennen, wof\u00fcr sie stehen. Man muss erkennen, ob sie diese Dinge in der Lage sind durchzusetzen. Das wiederum k\u00f6nnen sie nur im B\u00fcndnis, sonst sind sie viel zu schwach. Wenn es der radikalen Linken nicht gelingt, Zugang zur migrantischen Jugend zu bekommen, hat sie keine Chance. Und der dritte Punkt: Es ist zwar ganz lustig, wenn irgendwelche griechischen oder italienischen Genossen zum Anti-G8 oder sonst wohin fahren und da mal zeigen, wie kampferprobt sie sind, aber das ist alles nutzlos, wenn es nicht gelingt, eine in Europa aufeinander abgestimmte Strategie zu entwickeln und auch taktisch sich auf bestimmte Dinge einzustellen. Wenn man nicht mal in den gro\u00dfen Gewerkschaftszeitungen von verdi oder so mitbekommt, wenn quasi Frankreich lahmgelegt ist oder in Griechenland der Aufstand tobt, dann funktioniert irgendwas nicht. Und dann wird die Linke auf keinen gr\u00fcnen Zweig kommen. Da wird dann immer mal so eine sozialdemokratische Formation irgendwo das Regierungsruder \u00fcbernehmen, aber das ist auch alles.\n\n#### Will die radikale Linke denn eigentlich \u00fcberhaupt noch \"die Welt\" ver\u00e4ndern und nicht nur die eigenen Refugien erhalten?\n\nEine interessante Frage. Es gibt ein Theorieproblem bei den Autonomen, eine logische Inkonsistenz, und zwar, dass man aus der Gesellschaft eben nicht aussteigen kann, wie behauptet, es sei denn, man zieht sich in ein verlassenes Dorf zur\u00fcck und macht da den Schafhirten... Die Autonomen haben ja im Prinzip in vieler Hinsicht nicht anders agiert denn als militanter Arm der Alternativen, und die Alternativen sind am Anfang eine R\u00fcckzugsbewegung gewesen. Jetzt sind sie halt, zumindest in ihrer parlamentarisch-parteilichen Form als Gr\u00fcne, so eine neokapitalistische Bewegung, die einen bisschen anderen Kapitalismus wollen, also ihr gro\u00dfer Erfolg hat damit zu tun, dass sie dem Kapitalismus einen Impuls geben \u00fcber diese \u00f6kologische Geschichte und dass sie sich darin mit gro\u00dfen Teilen des B\u00fcrgertums einig sind. Das muss nicht verwerflich sein, hat aber mit den Urspr\u00fcngen nicht mehr viel zu tun. Am Anfang war es eine starke R\u00fcckzugsbewegung, im Kern h\u00e4ufig antimodernistisch, siehe die Kommune-Bewegung; die Autonomen waren ein bisschen \u00e4hnlich: Wir ziehen in die besetzten H\u00e4user, dann machen wir die T\u00fcren zu. [Lacht.] Ich hab neulich mit einem Leipziger Studenten diskutiert, der findet diese ganzen Aussteiger-Sachen gut wie Containern und so, aber der Witz ist, wenn du so was machst, und das war auch unser theoretischer Fehler von Anfang an, bedienst du dich ja nur des Wohlstands, der da ist \u2013 aber der muss ja von jemandem geschaffen werden. Wir hatten diese ganzen Ideologie-Versatzst\u00fccke auch schon teilweise gelebt, also: Wir klauen nur noch in Superm\u00e4rkten, wir brauchen kein Geld. Aber irgendjemand muss die Superm\u00e4rkte ja beliefern.\n\nDas Interview f\u00fchrte Daniel Schneider.\n\n## \"Nat\u00fcrlich macht es mich w\u00fctend zu sehen, wie wir hier leben und wie andere woanders leben m\u00fcssen.\"\n\nW., 34 Jahre, lebt in Hamburg.\n\n#### Was ist dein Familienstand?\n\nLedig.\n\n#### Lebst du in einer festen Partnerschaft?\n\nNein.\n\n#### Welche Ausbildung haben deine Eltern?\n\nKaufm\u00e4nnischer Angestellter und Personalsachbearbeiterin. Mein Vater ist in Rente und meine Mutter arbeitet noch.\n\n#### Was machst du beruflich?\n\nIch arbeite halbtags im Gesundheitswesen.\n\n#### Wie bewertest du deine aktuelle Situation?\n\nIch arbeite in dem Beruf, den ich gelernt habe und den ich studiert habe, bin aber auch noch in der Weiterbildung und das ist auch der Beruf, den ich zuk\u00fcnftig machen m\u00f6chte.\n\n#### Gibt es etwas, wovor du Angst hast?\n\nAllein und einsam zu sterben.\n\n#### Und gibt es etwas, was dich w\u00fctend macht?\n\nUnmengen. Klar gibt's vieles, was mich w\u00fctend macht und an Punkten auch eher verzweifeln l\u00e4sst. In Bezug auf Arbeit: Zu sehen, wie im Gesundheitssektor Leute zu Nummern werden, dass man versucht, dagegen anzuarbeiten und denjenigen auch individuell zu sehen. Aber dass oftmals die Umst\u00e4nde haupts\u00e4chlich durch Kostendruck es einem immer schwerer machen, letztendlich das Beste f\u00fcr den einzelnen Menschen rauszuholen. Und ansonsten allgemein: Nat\u00fcrlich macht es mich w\u00fctend zu sehen, wie wir hier leben, und zu sehen, wie andere woanders leben m\u00fcssen, um es mal so ganz global zu sagen. Es macht mich w\u00fctend zu sehen, wie mit Menschen umgegangen wird, die nicht so privilegiert sind wie ich oder meine Familie, und das zeigt sich ja jeden Tag im Kleinen und auch in einer gr\u00f6\u00dferen politischen Entwicklung. Also was ist mit Leuten, die sich nicht eine Wohnung leisten k\u00f6nnen, die nicht dort leben k\u00f6nnen, wo sie wollen oder wie sie wollen oder wie es auch angemessen w\u00e4re. Die nicht f\u00fcr die Arbeit, die sie machen, gen\u00fcgend Geld bekommen. Wo es einfach nicht reicht zum Leben und zu viel ist zum Sterben. So was macht mich auf jeden Fall w\u00fctend.\n\n#### Was w\u00fcrdest du anders machen als deine Eltern?\n\nIch lebe insgesamt ein anderes Leben als meine Eltern, was nat\u00fcrlich auch generationsbedingt ist. Ich bin Mitte drei\u00dfig, ich bin nicht verheiratet, ich habe kein Eigenheim und das ist auch nicht in Planung, ich habe keine Kinder \u2013 ich bin generell in einer anderen Situation, als sie jemals waren. Ich muss mich nur um mich selber sorgen und bin dementsprechend nicht so vielen Zw\u00e4ngen ausgeliefert wie sie, als sie in meinem Alter waren. Ich finde es schwierig, das so generell zu sagen, wei\u00df nicht, ob ich heiraten w\u00fcrde. Kinder bekommen, ja. Aber so ein sehr auf Familie zentriertes Leben, sehr individualistisches Leben birgt aber auch viele Gefahren in sich \u2013 meine Eltern sind ja auch geschieden \u2013, deshalb wei\u00df ich nicht, ob ich das will. Das ist vielleicht etwas, was ich anders machen will: Wert drauf zu legen, dass nicht immer die Kinder oder die Familie einspringt, sondern dass man sich ein soziales Umfeld f\u00fcr sich auch die ganze Zeit erh\u00e4lt. Was nat\u00fcrlich nicht hei\u00dft, dass sie kein soziales Umfeld haben, aber es ist nat\u00fcrlich ein anderer Lebensstil. Wenn ich sage, ich will auch in WGs wohnen oder in Hausgemeinschaften oder Projekten, wo die Leute auch etwas miteinander zu tun haben, das ist noch mal etwas anderes als die Nachbarschaft, mit der man sich gut versteht und wo man sich auch mal Eier ausleihen kann.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nAls realistisch, bodenst\u00e4ndig, harmoniebed\u00fcrftig. Und ich w\u00fcrde sagen, dass ich ein ausgepr\u00e4gtes Sicherheitsbed\u00fcrfnis habe. Also dass mir eine finanzielle Sicherheit wichtig ist, dass mir ein Platz zum Wohnen, also etwas Stabiles und auf Dauer eingerichtetes Wohnen wichtig ist. Ich brauche so Konstanten im Leben.\n\n#### Hast Du Hobbys?\n\nMein Beruf ist mein Hobby. [Lacht.]\n\n#### War das jetzt ironisch oder ernst gemeint?\n\nBeides. Wenn man einen Beruf hat, wo man merkt, dass es einem Spa\u00df macht und man auch das Feedback bekommt, dass man es gut kann, dann f\u00e4llt mir das schwer, das so zu trennen. Ansonsten mache ich viel Sport, ich lese zu wenig, ich entspanne mich zu wenig \u2013 aber vom Prinzip her sind das alles Dinge, die ich unheimlich gerne mache. Ich lege gerne auf. Also Sport und Auflegen, das ist eigentlich das, was ich in meiner Freizeit so am liebsten mache. Und Menschen treffen, die ich sehr gerne mag, also mit Freunden etwas unternehmen.\n\n#### Bist du in irgendeinem Verein?\n\nIm Sportverein.\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nEine gro\u00dfe. Gar nicht, dass ich zu Hause so viel Musik h\u00f6re, aber dadurch, dass ich auflege, bin ich immer auf der Jagd nach Musik, die mich mitrei\u00dft. Aber Musik hat auf jeden Fall eine Bedeutung bei den schwierigen Lebenssituationen. Da gibt es immer verschiedene Lieder. Das m\u00fcssen gar nicht zwangsl\u00e4ufig traurige Lieder sein, aber wenn ich die dann h\u00f6re, dann sind die mit der Situation verkn\u00fcpft.\n\n#### Hast du eine bestimmte Lieblingsmusik?\n\nNee, ich h\u00f6re tats\u00e4chlich so jeden Schrott. Ich kann eher sagen, was ich nicht so h\u00f6re: Jazz finde ich total grauenvoll, mit Soul kann ich auch nicht so viel anfangen, Hardcoremusik ist auch nicht so meins. Und ich h\u00f6re keine Volksmusik. Aber mit so Volksliedern kann ich durchaus etwas anfangen. Ich habe ja mal im Chor gesungen. Ich h\u00f6re mittlerweile auch gerne mal klassische Musik \u2013 da dann auch nur die Burner, also klassische Musik, die jeder kennt: Mozart, Bach, also keine extravaganten klassischen Komponisten oder so. Das hat dann eher so einen beruhigenden, entspannenden Effekt.\n\n#### Gibt es sonstige Medien, die du konsumierst?\n\nIch guck zu viel Fernsehen. Definitiv. Also wenn ich das jetzt erz\u00e4hle \u2013 ist ja grauenvoll! [Lacht.] Ich gucke gerne Comedyserien, das ist was, wo ich oft dr\u00fcber lachen kann. Ich gucke total gerne Scrubs, das ist eine Krankenhaus-Comedyserie. Das liegt nat\u00fcrlich auch ein bisschen nahe bei mir, aber da kann ich auch noch beim zweiten oder dritten Mal sehen dr\u00fcber lachen. Total gerne sehe ich Oliver Kalkofe. Dieses ganze neue Comedyzeug, das es so massenhaft gibt mittlerweile, das finde ich nicht komisch. Ich kann stundenlang Dschungelcamp gucken \u2013 fantastisch! Von mir aus k\u00f6nnte das ganze Jahr \u00fcber Dschungelcamp sein. [Lacht.] Ich gucke aber auch Krimiserien. Ich gucke keine traurigen oder bedeutungsschweren Filme; das hat etwas damit zu tun, dass ich mich beruflich oder privat politisch mit so viel Schei\u00df auseinandersetzen muss, da muss ich nicht auch noch einen traurigen Film gucken. Da hab ich schon Drama genug. Aber dass ich irgendwelche Tageszeitungen regelm\u00e4\u00dfig lese oder Magazine, bestimmte Radiosendungen, k\u00f6nnte ich nicht so sagen, die Titanic wahrscheinlich am ehesten.\n\n#### Woher beziehst du deine Informationen zum aktuellen Zeitgeschehen?\n\nMeistens w\u00fcrde ich das lieber alles gar nicht so genau wissen. Also ich gucke Nachrichten im Fernsehen.\n\n#### Interessierst du dich f\u00fcr Mode?\n\nMode schon \u2013 ich bin eifrige Konsumentin von Klamotten. [Lacht.]\n\n#### Gibt es bestimmte Marken, die du bevorzugst?\n\nC & A. [Lacht.] Wenn man sich immer viel kauft, aber eigentlich nicht das Geld daf\u00fcr zur Verf\u00fcgung hat, dann geht man halt dahin, wo man es sich leisten kann. Ich bin keine Markenfetischistin.\n\n#### Wie verbringst du deine Freizeit? Und mit wem?\n\nZunehmend mit mir alleine zu Hause. Ansonsten treffe ich mich mit Freunden, gehe essen oder in die Kneipe oder auf Partys. Aber das dann auch gezielt, also ich laufe nicht \u00fcber den Kiez und guck mal, was der Abend so bringt. Ich bin viel beim Sport und da auch individualsportlerisch, also Laufen, Schwimmen \u2013 wof\u00fcr ich mich dann auch nicht extra verabrede. Mal in die Sauna, aber auch durchaus alleine. Und nat\u00fcrlich vorm Computer.\n\n#### Computerspiele oder Internet?\n\nEher Internet. Also Facebook. Wobei ich auch immer nicht so richtig wei\u00df, was ich eigentlich im Internet soll. Also ich sitz da nicht stundenlang vor, aber ich besch\u00e4ftige mich mehrmals t\u00e4glich damit.\n\n#### Wie w\u00fcrdest du \"autonome Szene\" definieren oder jemandem beschreiben, der noch nie etwas davon geh\u00f6rt hat?\n\nEhrlich gesagt habe ich mich auf theoretischer Ebene damit bisher nicht gro\u00df auseinandergesetzt und habe es perspektivisch auch nicht vor. Ich glaube, dass damit oftmals eine ganz grunds\u00e4tzliche Unzufriedenheit mit den herrschenden Lebensverh\u00e4ltnissen einhergeht. Also gar nicht so speziell Rassismus oder dass man findet, dass Schindluder mit dem Umweltschutz getrieben wird oder so, sondern dass es eher eine generelle Unzufriedenheit ist, die relativ viele Bereiche umfasst, und der Wunsch danach, das anders zu machen. Und es f\u00e4ngt damit an, dass man im zwischenmenschlichen Miteinander versucht, Sachen anders zu machen: dass man R\u00e4ume gemeinschaftlich gestaltet, dass man Projekte gemeinschaftlich angeht \u2013 auch Projekte, die gar nicht per se einen politischen Hintergrund haben. Wohnen auf eine andere Art und Weise, gemeinsam Veranstaltungen machen, gemeinsam Feiern auf eine bestimmte Art und Weise. Mit Leuten, wo deine Meinung genauso viel z\u00e4hlt wie meine Meinung und dass wir uns dar\u00fcber austauschen und versuchen, ein Konsensprinzip einzuhalten, und auch versuchen, in diesem gemeinsamen Austausch, in diesem gemeinsamen Gestalten, etwas anders zu machen, was mehr den Vorstellungen entspricht, die wir davon haben, wie so gemeinschaftliches Leben funktionieren sollte.\n\n#### Was hat das mit dem Begriff \"autonom\" zu tun oder woran w\u00fcrdest du das festmachen?\n\nDas ist ja letztendlich grundlegend anders als wie es in der Gesellschaft funktioniert: Wir halten uns an Gesetze, die wer aufgestellt hat, wir halten uns an Regeln, es gibt \u00fcberall Hierarchien, sei es in der Schule oder im Beruf \u2013 da gibt es Menschen, die sagen, was wir zu tun haben oder was ich zu lassen habe oder was richtig und was falsch ist, und es geht nicht darum, mit denen zu diskutieren oder auszuhandeln, sondern das dann umzusetzen. Oder gesamtgesellschaftlich: Es gibt eine gew\u00e4hlte Regierung, die nach dem Mehrheitsprinzip gew\u00e4hlt wurde und die sagt, \"wo es langgeht\". Deshalb ist es schon grunds\u00e4tzlich, wie wir versuchen, miteinander zu reden und etwas zu gestalten, etwas Anderes und letztlich damit schon kontr\u00e4r zum gesamtgesellschaftlichen Funktionieren. Also das macht es dann vielleicht \"autonom\", indem man versucht, die Dinge selber zu regeln, anders zu gestalten, ohne auch abh\u00e4ngig zu sein von Geld von anderen zum Beispiel und von den Strukturen, die wir halt so gelernt haben, wie es zu laufen hat. Wenn man versucht, sich abzugrenzen von den Notwendigkeiten, die einem eingetrichtert werden, von Regeln, von Hierarchien, und sich auch absichtlich au\u00dferhalb der Gesellschaft definiert \u2013 was man nat\u00fcrlich nie ist.\n\n#### Bist du Mitglied in einer konkreten Gruppe?\n\nN\u00f6. Fr\u00fcher durchaus. Ich habe jetzt gerade \u00fcberlegt, ob ich so etwas wie die Gestaltung eines Projektes als Gruppe definieren w\u00fcrde, weil ich das die letzten Jahre haupts\u00e4chlich gemacht habe. Also ich bin schon mehrere Jahre nicht mehr in einer festen Polit-Gruppe. Bei so einer Gruppe ist klar, wer dazugeh\u00f6rt und wer nicht dazugeh\u00f6rt und was man gemeinsam f\u00fcr Projekte macht. Und die Gestaltung eines autonomen Projektes, Raumes ist dann doch etwas Offeneres.\n\n#### Was habt ihr da gemacht fr\u00fcher?\n\nSelbststudium, gelesen, Veranstaltungen organisiert: Diskussionsveranstaltungen oftmals anlassbezogen zum Jugoslawienkrieg, zu Drogenpolitik, zu Brechmitteleins\u00e4tzen, zu Gentrification, zu Rassismus, zu Antisemitismus; also es gab Gruppen, die sich mit unterschiedlichen Sachen besch\u00e4ftigt haben, und es gab Gruppen, die sich extra zu bestimmten Themen zusammengefunden haben.\n\n#### Kannst du dich noch an den ersten Kontakt mit der linken Szene erinnern?\n\nSicherlich, also ganz zu Anfang war ich Teil von so einer Punker-Szenerie, am Rande aufgrund des jungen Alters.\n\n#### Wie alt warst du da?\n\n13, 14, als ich zum ersten Mal auf einer Demo war, eine Hafenstra\u00dfendemo. Das war so ein Alter, wo man das noch gar nicht \u00fcberblicken konnte, was das jetzt irgendwie hei\u00dft, was f\u00fcr Stilrichtungen es auch politisch in so einer linken Szene gibt.\n\n#### Gibt es deiner Meinung nach Ankn\u00fcpfungspunkte zwischen der linken Szene und anderen Jugendkulturen?\n\nAm ehesten wahrscheinlich im HipHop und Punk, Hardcore, Fu\u00dfball sicherlich auch. Ich glaube, dass die autonome Szene letztendlich so eine Schnittstelle ist f\u00fcr Leute, die entweder noch nicht sich irgendwo eingeordnet haben oder das auch bewusst nicht wollen, weil in der autonomen Szene Praxis immer eine gro\u00dfe Rolle spielt und das attraktiv ist f\u00fcr Leute, die sich neu orientieren, an Dingen teilzuhaben, z. B. der konkreten Gestaltung von einem Autonomen Zentrum oder einer Demo. Und es gibt viele Leute, die einfach keinen Bock haben, sich einzusortieren: Ich bin jetzt antideutsch, ich bin jetzt Punk oder Anti-Atom oder so. In der autonomen Szene kommt vieles zusammen. Aber die autonome Szene ist nicht nur Freiheit und Spa\u00df und DIY [Do it yourself], sondern auch sehr anstrengend \u2013 das ist die Kehrseite der Medaille. Was sich in der Theorie sch\u00f6n anh\u00f6rt: \"Wir diskutieren Sachen aus, deine Meinung ist genauso wichtig wie meine\" und so, ist in der Praxis auch anstrengend. Wir haben au\u00dferdem auch bestimmte Strukturen, die es einzuhalten gilt, und die sind f\u00fcr Au\u00dfenstehende nicht immer leicht zu \u00fcberblicken und nachzuvollziehen.\n\n#### Was sind das f\u00fcr Strukturen?\n\nNa ja, dass Entscheidungen nicht einfach nach Lust und Laune und im Zweifelsfall auch nicht zeitnah getroffen werden, sondern dass alle Meinungen geh\u00f6rt werden m\u00fcssen. Das macht es oftmals schwierig und unpraktisch. Dass jeder so sein kann, wie er m\u00f6chte, also auch so aussehen kann, das ist f\u00fcr viele auch schwierig. Das hei\u00dft nicht, dass er oder sie alles machen darf, es gibt bestimmte No-Gos, die eher noch definiert sind als bestimmte Strukturen.\n\n#### K\u00f6nntest du welche benennen?\n\nDie klassischen Antis: Dass sexistisches Verhalten nicht geduldet wird, dass rassistisches Verhalten nicht geduldet wird, dass antisemitisches Verhalten und Spr\u00fcche nicht geduldet werden, dass homophobes Verhalten nicht geduldet wird \u2013 und das ist f\u00fcr viele aktuelle Jugendkulturen, mit Spr\u00fcchen wie \"Das ist ja voll schwul\" oder wo so eine Sexualisierung nach au\u00dfen hin zumindest aus meiner Perspektive zunimmt, dann auch schwierig nachzuvollziehen. Da kannst du dir dann den Mund fusselig reden, und das ist vielen zu anstrengend und dann sagen sie: \"Ach nee, lass mal, hab ich keinen Bock drauf.\" Also die autonome Szene arbeitet sich eher daran ab, was nicht geht und was nicht geduldet wird, als an der Definition von dem, was geht oder was m\u00f6glich ist.\n\n#### W\u00fcrdest du die autonome Szene als Jugendkultur bezeichnen?\n\nDadurch, dass die meisten in der Jugend oder im jungen Erwachsenenalter mehr zeitlichen Spielraum haben als sp\u00e4ter, hat es etwas davon. Aber ich glaube, dass es bei vielen \u00c4lteren eher Sachzw\u00e4nge sind, die sie dazu veranlassen, sich zur\u00fcckzuziehen. Sicherlich auch eine bestimmte Abgegessenheit von bestimmten Diskussionen, aber es gibt ja auch Leute, die immer noch dabei sind. Also ich w\u00fcrde den Gro\u00dfteil der Szene zwischen 17 und 30 einordnen.\n\n#### Gibt es einen eigenst\u00e4ndigen Autonomen-Stil?\n\nIch w\u00fcrde sagen nein. Andere w\u00fcrden vielleicht etwas anderes sagen, aber was ich schon meinte mit der Schnittstelle: Es treffen sich da ja auch bestimmte Gruppierungen mit ihren bestimmten Stilen. Sei das so eine Antifa-Fraktion, die in Funktionskleidung unterwegs ist, in Carhartt-Klamotten oder schwarzer Allwetter-Jacke, die Hippie-Fraktion mit Dreadlocks und bunter, oder die Punker und Hardcore-Fraktion bis unter die Haarspitzen t\u00e4towiert. Also ich habe in ganz jungen Jahren auch Wert drauf gelegt auf eine gewisse Uniformierung und auf ein gewisses Nach-Au\u00dfen-Tragen einer Meinung. So ein Kleidungsstil, von dem ich damals der Meinung war, der w\u00fcrde auch meine Meinung transportieren. Aber davon habe ich mich schon vor langer Zeit verabschiedet. Die J\u00fcngeren sind da schon in ihrem jeweiligen Stil etwas konformer oder uniformierter. Auf Demonstrationen ziehen sich in so einem \"Schwarzen Block\" auch die Hippies schwarz an.\n\n#### Gibt es andere popkulturelle Produkte, die zu so einem Stil dazu geh\u00f6ren w\u00fcrden?\n\nErstaunlich ist ja, dass in so einer Szene Musik sehr abgefeiert wird, von der Au\u00dfenstehende das nicht denken w\u00fcrden, wie 80er, 90er Trash-Musik, Scooter wurde wild abgefeiert die letzten Jahre. Jetzt geht es eher so in den Elektro-Pop oder Elektro-Punk. Da machen viele gro\u00dfe Augen, wenn sie zu einer Party ins Autonome Zentrum kommen und dann l\u00e4uft da Scooter oder Bl\u00fcmchen. Das ist dann schon ein bisschen skurril.\n\n#### Welche Themen sind dir f\u00fcr dein Engagement besonders wichtig?\n\nAntisemitismus auf jeden Fall, das war und ist immer ein Thema, wo ich sensibel bin. Sexismus ist ein Thema, wo die Szene ja auch wahrlich nicht frei von ist und es doch an den Frauen meistens h\u00e4ngenbleibt, das zu thematisieren oder einzugreifen in bestimmten Situationen. Das waren auf jeden Fall immer Punkte. Drogenpolitik... das ist schon 90er, das ist v\u00f6llig out mittlerweile, aber findet man in so Stadtteilentwicklungsgeschichten nat\u00fcrlich immer wieder.\n\n#### Zu dem Sexismus-Thema, das du ansprichst: Ist die Szene m\u00e4nnlich dominiert?\n\nM\u00e4nnlich dominiert w\u00fcrde ich nicht unbedingt sagen, aber prozentual gesehen sind es sicherlich mehr M\u00e4nner als Frauen. Und auch in den Machenden-Positionen \u2013 Positionen h\u00f6rt sich jetzt so nach Hierarchie an, das meine ich damit nicht, aber was so Organisierung von Geschichten angeht \u2013 w\u00fcrde ich schon sagen, dass es mehr M\u00e4nner gibt, die in Verantwortung gehen, als Frauen.\n\n#### Du hast eben gesagt, Sexismus geh\u00f6re zu den No-Gos der Szene, und trotzdem ist die Szene nicht frei von Sexismus?\n\nLetztendlich ist das so ein Punkt, wo gerne automatisch reproduziert wird, was man gesellschaftlich gelernt hat und was schnell auch als \"nicht so schlimm\" abgetan wird wie wenn man etwas Rassistisches sagt. Aber es gibt nat\u00fcrlich sexuelle \u00dcbergriffe in der Szene, es gibt Vergewaltigungen in der Szene und den Versuch, damit einen Umgang zu finden. Es ist auf jeden Fall immer wieder Thema und betrifft nicht nur Leute, die zuf\u00e4llig mal auf Partys gelandet sind oder die sich am Rande der Szene aufhalten, sondern auch Leute, die seit Jahren mit dabei sind.\n\n#### Was ist ein \"Umgang mit einer Vergewaltigung\"?\n\nIch hab schon an relativ vielen gescheiterten Versuchen teilgenommen, einen \"Umgang\" zu finden. Davon ausgehend, dass der \"rechtsstaatliche\" Umgang damit schwierig ist, also Anzeige erstatten, Beweise sammeln und im Zweifel f\u00fcr den Angeklagten. Abgesehen davon, dass das ein Umgang ist, der nicht derjenigen oder demjenigen, dem das widerf\u00e4hrt, gerecht wird, geht es nat\u00fcrlich auch darum, dass jemand aus Zusammenh\u00e4ngen raus muss, der eine Vergewaltigung begangen hat, es geht auch darum, Gruppenbildung zu vermeiden (was meistens nicht funktioniert), also dass sich Leute um den Mann formieren und Leute um die Frau formieren und dann gegenseitige Schuldzuweisungen passieren. Aber es reicht eigentlich auch nicht zu sagen: \"Du bist jetzt raus, wir wollen dich nicht mehr\", das w\u00e4re ja in derselben Logik wie Wegsperren, sondern wichtig ist auch, eine Auseinandersetzung einzufordern mit dem, was jemand getan hat \u2013 in der Hoffnung, daran m\u00f6ge sich dann etwas \u00e4ndern. Nicht einfach zu sagen: \"Das war schei\u00dfe, jetzt bist du raus aus den Strukturen und was du dann tust, ist uns egal.\" Aber da gibt es unterschiedlichste Ansichten und das ist schon extrem schwierig. Schon die Definition, was ist eine Vergewaltigung gewesen, was war ein \u00dcbergriff, das ist einfach schon wahnsinnig schwer und da gibt es auch immer wiederkehrende Diskussionen \u00fcber die Definitionsmacht, das Definitionsrecht: Hei\u00dft Nein! wirklich nein oder hei\u00dft es erst beim dritten Mal nein... Also letztendlich ganz viele Diskussionen, die meistens unleidlich sind und oft auch nicht wirklich zu etwas f\u00fchren. Also das geh\u00f6rt zu den schwierigsten Herausforderungen in so einer linken Szene, der Umgang mit sexuellen \u00dcbergriffen, mit Vergewaltigungen. Mir f\u00e4llt nichts ein, was so schwierig ist und so viel Sprengkraft besitzt.\n\n#### Wie ist das Verh\u00e4ltnis der autonomen Szene zum Staat und zu Gewalt \u2013 oder pers\u00f6nlicher: Wie ist dein Verh\u00e4ltnis zum Staat und zur Polizei?\n\nGekennzeichnet von Misstrauen, Resignation, Desinteresse fast auch an einem bestimmten Punkt. Ich hab auf jeden Fall nicht das Gef\u00fchl, dass ich mich irgendwie besch\u00fctzt oder aufgehoben f\u00fchle oder dass sich um mich gek\u00fcmmert wird. Das ist nicht die Funktion, die der Staat in meinem Leben erf\u00fcllt \u2013 also das tut er sicherlich objektiv: Wenn ich jetzt arbeitslos werden w\u00fcrde, w\u00fcrde ich auch in den Genuss bestimmter Leistungen kommen, und das ist nat\u00fcrlich von Vorteil. Aber dass mich staatliche Politik in meinem Alltag so wahnsinnig ber\u00fchren w\u00fcrde oder Dinge, die ich in der Tagesschau sehe, das l\u00f6st eher so ein Kopfsch\u00fctteln aus bei mir. Aber nat\u00fcrlich hat es direkten Einfluss auf mich und ich \u00e4rgere mich \u00fcber vieles. Ich finde vieles furchtbar.\n\n#### Kannst du Beispiele nennen, was du furchtbar findest?\n\nNa ja, Versch\u00e4rfung von Ausl\u00e4nderpolitik, Integrationsdebatten \u2013 grauenvoll. Oder Innere-Sicherheit-Diskurse jagen mir Schauer \u00fcber den R\u00fccken. Ja, mein Verh\u00e4ltnis zum Staat ist eher ablehnend. Und das trifft generell f\u00fcr die Szene zu. Ein ablehnendes, misstrauisches Verh\u00e4ltnis. Und zur Polizei: Wenn man gen\u00fcgend schwierige Erfahrungen gemacht hat, dann ist auch eher nicht das \"Freund-und-Helfer\"-Bild da. Und damit meinte ich nicht, dass ich geblitzt wurde oder abgeschleppt.\n\n#### Sondern?\n\nWahrzunehmen, wie bei Aktionen, bei Demonstrationen die Meinungsfreiheit massiv eingeschr\u00e4nkt wird, wie Leute drangsaliert werden, wie systematisch versucht wird zu verhindern, dass die Meinung ausgedr\u00fcckt werden kann, dass die auch kundgetan werden kann, bis hin zum Drangsalieren von Obdachlosen oder Drogenverk\u00e4ufern oder DrogenkonsumentInnen oder wem auch immer. Wenn man das \u00f6fter miterlebt hat, kann man kein positives Verh\u00e4ltnis zur Polizei entwickeln, das geht einfach nicht.\n\n#### Die autonome Szene wird ja als sehr gewaltbereite Szene dargestellt in den Medien und vom Verfassungsschutz. Ist das so? Oder was hast du f\u00fcr ein Verh\u00e4ltnis zu Militanz?\n\nIch kann das gut nachvollziehen und mich erfreuen auch durchaus bestimmte Nachrichten \u00fcber militante Aktionen. Ich kann gut verstehen, dass die Wut \u00fcber bestimmte Verh\u00e4ltnisse auch einen militanten Ausdruck finden muss, weil nur dann findet sie n\u00e4mlich auch Geh\u00f6r. Du kannst Pamphlete schreiben, B\u00fccher, Kundgebungen veranstalten, Flugbl\u00e4tter verteilen und wirst ignoriert. Damit will ich nicht sagen, dass diese Dinge unn\u00f6tig sind, aber nat\u00fcrlich hast du die gr\u00f6\u00dfte Aufmerksamkeit \u00fcber militante Aktionen. Ich wei\u00df nicht, ob man damit den gew\u00fcnschten Effekt erzielt, aber mit etwas anderem erzielt man ihn eben auch nicht, und es gibt zumindest eine gewisse Aufmerksamkeit, und ich kann das Bed\u00fcrfnis, dass die Wut und die Verzweiflung \u00fcber Verh\u00e4ltnisse auch ein Ventil braucht, gut nachvollziehen.\n\n#### W\u00fcrdest du sagen, Gewalt geh\u00f6rt automatisch zur Szene?\n\nEs geh\u00f6rt automatisch zur Szene, weil sie staatlicherseits mit sehr viel Gewalt und Repression belegt wird. Wenn Demonstrationen nur noch stattfinden im dreifachen Polizeispalier oder bei jeder schwierigen Bewegung sofort reingekn\u00fcppelt wird oder Leute massiv mit \u00dcberwachung und Repression im Sinne von Androhung auf der Stra\u00dfe oder auch juristisch \u00fcberzogen werden... \u2013 man ist sozusagen st\u00e4ndig mit Gewalt staatlicherseits konfrontiert und deswegen geh\u00f6rt Gewalt zur autonomen Szene. Aber dass die autonome Szene von sich aus per se gewaltt\u00e4tig ist oder sagt, \"das ist das Mittel der Wahl, um Dinge zu \u00e4ndern\", das w\u00fcrde ich so nicht sehen.\n\n#### Man k\u00f6nnte ja auch einen Zusammenhang herstellen zwischen der Infragestellung des Staates und der Infragestellung des Gewaltmonopols?\n\nSicherlich wird das Gewaltmonopol des Staates in Frage gestellt. Aber dass das zwangsl\u00e4ufig nach sich zieht, dass wir jetzt deswegen Gewalt anwenden k\u00f6nnen wie der Staat, das geht so nicht. Ich w\u00fcrde eher sagen, wir nehmen ganz viel Gewalt wahr, die entweder uns widerf\u00e4hrt oder anderen, vornehmlich halt denjenigen, die sich nicht wehren k\u00f6nnen, und deswegen beantwortet man bestimmte Dinge auch mit Gewalt gegen Sachen \u2013 nicht gegen Menschen, da ist ja auch noch mal ein deutlicher Unterschied. Aber dass das zwangsl\u00e4ufig dazugeh\u00f6rt, dass man sich da irgendwie so ein Recht nimmt, das sehe ich so nicht.\n\n#### Wie siehst du den Konflikt zum Thema Antideutsche?\n\nDas ist nat\u00fcrlich ein j\u00e4hre-, wenn nicht jahrzehntelang schwelender Konflikt und ich glaube eine k\u00fcnstliche Zuspitzung, die an der Realit\u00e4t vorbei geht. Das ist sicherlich einer der Konflikte, der am meisten Sprengkraft in den letzten Jahren entwickelt hat, der die Szene grunds\u00e4tzlich ver\u00e4ndert hat. Ich glaube, dass die Wahrheit oder die L\u00f6sung irgendwo in der Mitte liegen m\u00fcsste. Ich selbst werde von Antiimp-Seite aus als Antideutsche oder zumindest als antideutsch-freundlich eingestuft, weil f\u00fcr mich einfach das Existenzrecht Israels total wichtig ist. Und das ist ja schon in bestimmten Zusammenh\u00e4ngen nicht Konsens, wo man da schon grunds\u00e4tzlich dr\u00fcber diskutiert in antiimperialistischen Zusammenh\u00e4ngen, und deswegen ist mir das auch relativ fern.\n\n#### Antideutsch, Antiimp, Antifa, Autonome \u2013 ist das alles autonome Szene oder w\u00fcrdest du das trennen?\n\nAls \u00dcberbegriff vielleicht linke Szene, wobei ja die Antiimp-Antideutsch-Debatte davon gepr\u00e4gt ist, dass man sich gegenseitig das Links-Sein abspricht und zum Teil auch schon sich selber exkludiert, weil mit den anderen will man ja nix zu tun haben und schon gar nicht in einen Topf geschmissen werden. Ich glaube, dass das Bed\u00fcrfnis einzuteilen, zu kategorisieren, dem ganzen Kind einen Namen zu geben, von au\u00dfen gr\u00f6\u00dfer ist als von innen. F\u00fcr mich pers\u00f6nlich ist dieses ganze Labeling nicht so wichtig. Ich glaube, dass die autonome Szene tats\u00e4chlich so eine Schnittstelle ist, wo sich Leute vielleicht auch nur f\u00fcr einen bestimmten Zeitraum zugeh\u00f6rig f\u00fchlen. Wobei es ja in den letzten Jahren Bestrebungen gegeben hat, dem Ganzen mehr eine Richtung zu geben, wie zum Beispiel Autonome Vollversammlung in Hamburg oder Kongresse zu dem Thema \"Was ist eigentlich autonome Politik und was wollen wir eigentlich?\". Aber das ist nat\u00fcrlich auch alles sehr langwierig und ich w\u00fcrde l\u00fcgen, wenn ich sagen w\u00fcrde, da bin ich voll dabei. Also es gibt Bestrebungen, dem Ganzen mehr theoretischen Background und F\u00fclle zu geben. Ich finde das immer ein bisschen unwichtig, weil es gibt Menschen, mit denen mag ich einfach Sachen zusammen machen, und es gibt Menschen, mit denen mag ich keine Sachen zusammen machen, oder Leute, die sich vielleicht zur Szene zugeh\u00f6rig f\u00fchlen, deren Meinung ich aber vielleicht doch schwierig finde. Da muss man halt sehen, wie man sich so zusammengeschustert bekommt.\n\nDas Interview f\u00fchrte Melanie Gro\u00df.\n\n## \"Ich m\u00f6chte, dass hier Menschen gleich behandelt werden.\"\n\nW., 41 Jahre, lebt in Hamburg.\n\n#### Was ist dein Familienstand?\n\nLedig.\n\n#### Lebst du in einer festen Partnerschaft?\n\nJa.\n\n#### Welche Ausbildung haben deine Eltern?\n\nMein Vater ist Lehrer auf dem zweiten Bildungsweg. Meine Mutter ist ungelernte Altenpflegerin. Beide sind in Rente.\n\n#### Was ist deine gegenw\u00e4rtige Situation? Was machst du beruflich?\n\nIch arbeite Teilzeit als Sozialarbeiterin.\n\n#### Wie bewertest du deine aktuelle Situation?\n\nJetzt gerade f\u00fchle ich mich gut, aber wenn ich mir mehr Gedanken dar\u00fcber mache, denke ich nat\u00fcrlich, dass meine Situation prek\u00e4r ist. Mein Arbeitsplatz wird alle drei Jahre neu verhandelt mit der Beh\u00f6rde und man wei\u00df nie, wie das dann laufen wird. Wenn ich das global sehe, geht es mir ganz sch\u00f6n gut.\n\n#### Woran w\u00fcrdest du das festmachen?\n\nDass ich nicht hungere und eine Wohnung habe und die auch noch Hartz-IV-f\u00f6rderungsf\u00e4hig ist, falls ich meinen Arbeitsplatz verliere, und dass ich Kompetenzen habe, mit Beh\u00f6rden zu verhandeln, ob ich Arbeitslosengeld bekomme.\n\n#### Und dein Beruf \u2013 w\u00fcrdest du sagen, das ist ein Traumberuf?\n\nNein. [Lacht.] Ich wollte es immer werden, bin es dann geworden mit 27, und seitdem denke ich, ich h\u00e4tte es nicht werden sollen.\n\n#### Warum?\n\nAls Sozialarbeiterin arbeitest du mit Leuten zusammen, die der Leistungsgesellschaft nicht entsprechen, also du reparierst die ganze Zeit. Gedacht hab ich immer, dass das auch gut ist, wenn da gute Leute arbeiten und Menschen unterst\u00fctzen, die an den Rand gedr\u00e4ngt worden sind. Das denke ich immer noch, aber das ist so anstrengend. Es ist wahnsinnig anstrengend auf Dauer. Und deshalb w\u00e4re es gut, wenn man das eine gewisse Zeit macht und dann etwas anderes machen d\u00fcrfte.\n\n#### W\u00fcrdest du etwas anders machen als deine Eltern?\n\nJa, auf jeden Fall. [Lacht.] Mit meinem Vater hab ich gar keinen Kontakt schon seit Jahren, aber ich erinnere mich noch, dass er BILD-Zeitung gelesen hat und CDU-W\u00e4hler war, aber ich kann mich nicht erinnern, dass er sich dar\u00fcber hinaus in irgendeiner Art und Weise politisch interessiert oder engagiert hat. Ich wei\u00df nicht genau, ob meine Eltern nicht analytisch denken k\u00f6nnen oder ob sie es einfach nur nicht getan haben, und ich glaube, das tue ich schon.\n\n#### Haben deine Eltern eine Rolle gespielt f\u00fcr deine politische Entwicklung?\n\nEine tragende nicht \u2013 vielleicht als Abgrenzung, da m\u00fcsste ich mal dr\u00fcber nachdenken.\n\n#### Hast du Kinder oder m\u00f6chtest du Kinder haben?\n\nIch habe keine Kinder und m\u00f6chte auch keine haben.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nIch w\u00fcrde mich beschreiben als aufmerksamer Mensch und als sensitiver Mensch, und deshalb hab ich Depressionen. [Lacht.] Ich nehme viel wahr und ich glaube, dass ich mehr wahrnehmen kann als viele andere und mich das wahnsinnig anstrengt. Also dass das eine F\u00e4higkeit und ein Fluch ist.\n\n#### Hast du Hobbys?\n\nMeine Katzen. Ich besch\u00e4ftige mich sehr viel mit meinen Katzen. Ich boxe. Ich hab auch noch ziemlich viel Antifa-Sachen, die ich mache, aber das w\u00fcrde ich nicht als Hobby bezeichnen. Aber das nimmt Zeit ein und ist auch mein Interesse. Aber als Hobby w\u00fcrde ich sagen: meine Katzen, meine Blumen auf dem Balkon und mein kleiner Garten und dass ich viele soziale Kontakte habe. Und ich lese gerne skandinavische Krimis.\n\n#### Bist du in irgendeinem Verein?\n\nIm Boxverein und in einem Verein, der sich drogenpolitisch engagiert.\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nEindeutig eine gro\u00dfe Bedeutung. Rocksteady und Ska.\n\n#### Warum hat das so eine gro\u00dfe Bedeutung?\n\nWeil ich total gerne Musik h\u00f6re. Ich mag den Sound gerne, ich mag die B\u00e4sse gerne. Ich mag das gerne, wenn ich die B\u00e4sse im Bauch f\u00fchlen kann. Und Rocksteady und Ska ist auch Black Music \u2013 in ihren Urspr\u00fcngen zumindest, da wird textlich viel \u00fcber Missst\u00e4nde und Ungerechtigkeiten gesungen und das ist weniger religi\u00f6s als Reggae halt oder gar nicht. Und da gibt's ganz viele Liebeslieder \u2013 find ich toll. [Lacht.]\n\n#### Interessierst du dich f\u00fcr Mode, K\u00f6rperschmuck?\n\nIch mag T\u00e4towierungen. Nat\u00fcrlich nicht jedes Motiv, da achte ich schon drauf. Und doch, so ein bisschen Mode schon. Rocksteady und Ska haben auch anh\u00e4ngig Modelabels, Ben Sherman und Fred Perry vor allem, und gerade jetzt hab ich mir ein Polohemd gekauft.\n\n#### Wie verbringst du deine Freizeit?\n\nIch bin ziemlich viel zu Hause in meiner eigenen Wohnung. Gerne auch allein. Das klappt meistens nicht, weil meine ganzen Nachbarinnen da sind, insofern besch\u00e4ftige ich mich ziemlich viel mit meinen Nachbarinnen und deren Problemen, ehrlich gesagt. Und mit meinen Katzen und mit meinen Pflanzen. Und mit meinem Freund.\n\n#### Nach welchen Kriterien schlie\u00dft du Freundschaften?\n\nIch schlie\u00df keine Freundschaften mehr [lacht], ich kenn gen\u00fcgend Leute. Nach Sympathie. Und mit Rechten bin ich nicht befreundet, das kann ich mir auch nicht vorstellen \u2013 die treffe ich aber auch nicht. So wie ich wohne und so, das hat schon etwas Inseliges.\n\n#### Wenn du die linke Szene jemandem beschreiben solltest, der noch nie davon geh\u00f6rt hat, was w\u00fcrdest du dann sagen?\n\nIch w\u00fcrde beschreiben, dass f\u00fcr mich das Besondere daran ist, dass jeder eine Stimme hat und dass die auch erst mal geh\u00f6rt wird. Es ist irgendwie eine Vorstellung von etwas Solidarischem, etwas Sozialem, etwas Nicht-Ausgrenzenden.\n\n#### Gibt es bestimmte zentrale Inhalte, die die Szene hat?\n\nDa ich mich ja der Antifa-Szene zugeh\u00f6rig f\u00fchle, gibt es da nat\u00fcrlich Inhalte. Also ich arbeite ja, wenn ich politisch arbeite, inzwischen nicht mehr in Gruppen organisiert, sondern eher so Stra\u00dfen-Antifa, und da ist der minimale Konsens, dass eine Nazi-Demo sich m\u00f6glichst nicht bewegen kann.\n\n#### Was ist Stra\u00dfen-Antifa?\n\nF\u00fcr mich, dass man guckt, wo die sich wie bewegen, wo die versuchen, Fu\u00df zu fassen, Info-Tische haben, Leute verpr\u00fcgeln, irgendwelche Veranstaltungen von denen nicht stattfinden lassen und protestieren, demonstrieren. Vor Ort zu sein und mindestens den Einsatz von Bullen teuer zu machen. Am besten ist nat\u00fcrlich, wenn die nicht gehen d\u00fcrfen, aber ansonsten soll es teuer werden, damit das nicht normal ist, was die verbreiten. Aber die Inhalte, da w\u00fcrde ich schon sagen, dass es immer wieder grob auf den Punkt kommt: Ich m\u00f6chte, dass hier Menschen gleich behandelt werden und nicht diskriminiert werden. Bei Nazis kommt diese Haltung nat\u00fcrlich am meisten raus \u2013 also mir ist schon klar, dass dieses Denken aus der b\u00fcrgerlichen Mitte und aus neoliberalen politischen Statements letztendlich entspringt, aber da habe ich keinen Ansatzpunkt und auch keine Lust, mir irgendetwas zu suchen, mich zu engagieren. Bei Nazis ist das so einfach. Als ich noch organisiert war in Gruppen, da hatten wir auch schon sehr so AntiRa- [Antirassismus-]Sachen gemacht, aber da hab ich nicht mehr die Geduld und die Kraft zu.\n\n#### Du hast eben die Szene als linke Polit-Szene bezeichnet \u2013 w\u00fcrdest du \u00fcberhaupt den Begriff \"Autonome\" verwenden?\n\nNein, ich glaube nicht. Vielleicht, weil mir das ein bisschen zu abstrakt ist. Die organisierten Gruppen, in denen ich war, die hie\u00dfen nie Autonome. Also ich kenne einzelne Personen, die sagen: Ich bin Autonome. Aber ich glaube auch in Abgrenzung zu Antiimps vor allen Dingen. Aber ich f\u00fchle mich nicht so als Autonome, weil ich gar nicht so richtig wei\u00df...\n\n#### Wo ist denn der Unterschied zwischen Autonomen und Antiimps?\n\nDie Autonomen sind mehr so in der Flora und die Antiimps im Hafen. [Lacht.] Die Antiimps sind ein bisschen \u00e4lter und haben sich auch mehr mit den politischen Gefangenen aus der RAF befasst und da viel ihrer Zeit und Kraft reingesteckt. Und Autonome haben das nicht als Schwerpunkt. Die Autonomen waren die Coolen und sind auch mehr unter sich. Ich kann mich auch nicht daran erinnern, dass es so richtig Autonomen-Nachwuchs gegeben hat. Was ich auch einen gro\u00dfen Fehler finde \u00fcberhaupt von der Linken, dass die sich nicht um den Nachwuchs gek\u00fcmmert haben. Die waren immer so konspirativ.\n\n#### Und wo w\u00fcrdest du da die Antifas einsortieren?\n\nDa sind aus allen Bereichen welche, inklusive sogar Sch\u00fclerinnen.\n\n#### Gibt es von dieser Szene denn Ankn\u00fcpfungspunkte an Jugendkulturen?\n\nZu HipHop, Rap schon, weil es da so ein minimal politisches Verst\u00e4ndnis gibt, das auch \u00fcber Musik transportiert wird.\n\n#### Und zu Rocksteady und Ska, gibt es da Verkn\u00fcpfungen?\n\nDa ist das so selbstverst\u00e4ndlich, dass da keine Rassisten sind. Aber da sind ja ganz viele Skinheads, und das ist ja auch wieder eine Jugendkultur, die aber darauf bedacht sind zu sagen: \"Wir sind nicht politisch.\"\n\n#### Gibt es einen linken oder einen Antifa-Kleidungsstil, Codes etc.?\n\nIch glaube schon. So Elektro oder Techno, immer wieder Musik der 80er Jahre, so Partys kommen gut an. Es gibt in der Szene von hippiesk bis schicke Markenkapuzenpullis alles. Es ist nicht mehr so eng, wie es mal war. Man muss nicht schwarz tragen und kann trotzdem eine dazugeh\u00f6rige Kleidung tragen. Es gibt ja immer auch so Hippie-Leute, junge Leute, die Dreadlocks haben und Kleidung, die nicht teuer war, und 'ne Schlaghose mit Blumen, und die machen nat\u00fcrlich genauso VoK\u00fc oder Antifa wie die, die in schicken Markenjeans und -turnschuhen und Pullovern daherkommen.\n\n#### Was f\u00fcr Themen, neben denen, die du schon genannt hast, Ungerechtigkeit, Antifa, sind dir f\u00fcr dein Engagement besonders wichtig?\n\nAngefangen hat mein politisches Engagement im drogenpolitischen Bereich. Da ging es um Entkriminalisierung und um Verbesserung von Lebensumst\u00e4nden \u2013 das hat sich dann nat\u00fcrlich mit meinem Beruf vermischt und ich w\u00fcrde sagen, dass ich immer noch meine Lohnarbeit so gestalte, f\u00fcr Menschen, die von struktureller Ungleichheit betroffen sind in ihrem Leben, im Wohnen, im Arbeiten, die keine Bildung haben, kein Geld haben. Das ist mir schon auch wichtig. Mir ist auch wichtig, dass es weiterhin bezahlbaren Wohnraum gibt. Ich glaube, zum Schluss komme ich immer wieder darauf, womit ich angefangen habe bei dem Interview: Ich m\u00f6chte gerne, dass hier alle Menschen gleich behandelt werden und dieselben Chancen habe, und wenn ich dann eine Illegalisierte habe und wir \u00fcberlegen, wie bekommen wir sie jetzt legal, dann kann ich sagen, da mache ich AntiRa-Arbeit oder Sozialarbeit oder AntiRa-Sozialarbeit, aber letztendlich geht es mir um einen gerechten Umgang mit Menschen.\n\n#### Was denkst du \u00fcber den Konflikt zwischen Antiimps und Antideutschen?\n\nMan kann sich dem ja leider nicht entziehen, aber dadurch, dass ich nicht mehr aktiv in einer Gruppe bin, hat mir niemand die Pistole auf die Brust gelegt, eine Position haben zu m\u00fcssen, und dar\u00fcber bin ich sehr gl\u00fccklich. Ich habe das Gef\u00fchl, dass sich das alle paar Jahre wiederholt. Irgend so ein Riesenkonflikt in der Szene, jetzt ist es halt das: Antideutsche, Antiimps, Vegan-Geschichten, das war auch mal, T\u00e4tersch\u00fctzer \u2013 also Sexismus in der linken Szene. Ja, es betrifft mich tats\u00e4chlich immer mal wieder, dass ich dazu gefragt werde, wie ich noch mit einzelnen Leuten, die mal in der linken Szene waren, befreundet sein kann, weil die doch die und die Extremposition haben, meistens aus dem Antiimp-Bereich, und das finde ich dann immer ganz furchtbar. Ich w\u00fcrde mich selber als antideutsch definieren, aber nicht wie die Antideutschen. Ich finde das eine Frechheit, dass die diese Begrifflichkeit besetzt und \u00fcbernommen haben, weil ich eigentlich denke, ich bin antinationalistisch und deshalb bin ich antideutsch. Aber meine Position ist, dass ich \u00fcberhaupt keine Staaten mag und letztendlich auch \u2013 und von mir aus auch als letztes \u2013 Israel nicht, auch das ist nicht meins.\n\n#### Du sagst, solche Konflikte wiederholen sich \u2013 welche Bedeutung haben die dann f\u00fcr die Szene?\n\nIch habe das Gef\u00fchl, dass solche Konflikte immer kommen, wenn der \u00e4u\u00dfere Feind zu gro\u00df ist, dass sich die Szene dann untereinander zerfleischt. Das ist schrecklich, denn dann machen wir mit unseren eigentlichen Themen nicht weiter, wenn wir uns intern die ganze Zeit streiten und nur noch dar\u00fcber reden m\u00fcssen, wer aus welchen Gr\u00fcnden in welchen politischen Institutionen und H\u00e4usern Hausverbot bekommt, weil er dies und das und jenes hat, weil auf der Demo jetzt die und die nicht mehr mitgehen k\u00f6nnen, die sollen eine eigene machen und so.\n\n#### Setzt du dich eigentlich mit Theorien auseinander?\n\nN\u00f6. [Lacht.] Sonst w\u00fcsste ich wahrscheinlich, was Autonome sind.\n\n#### Wie kommst du eigentlich an Infos aus der Szene? Wie informierst du dich?\n\nInternet. Szenezeitschriften und Mundpropaganda. Also Indymedia, antifa. info und ZECK.\n\n#### Was h\u00e4ltst du von B\u00fcndnissen mit anderen linken Bewegungen, wie zum Beispiel der neuen Anti-AKW-Bewegung?\n\nGrunds\u00e4tzlich finde ich das gut, B\u00fcndnisse zu machen, wenn das nicht zu weit geht.\n\n#### Wo w\u00fcrdest du Grenzen ziehen?\n\nAlso, ich wei\u00df ja, dass bei den Antifa-Geschichten auch Jusos sind und da nehme ich das auch in Kauf. Oder teilweise bei Tierschutz-Sachen, ich bin da nicht so engagiert, finde ich aber total gut, aber da sind auch Leute dabei, die nicht wirklich zur linken Szene geh\u00f6ren.\n\n#### Was h\u00e4ltst du von Parteipolitik?\n\nNichts.\n\n#### Gehst du w\u00e4hlen?\n\nJa.\n\n#### Was hast denn du f\u00fcr ein Verh\u00e4ltnis zum Staat, zur Polizei?\n\nEin schlechtes. [Lacht.] Da ich ja f\u00fcr Gerechtigkeit bin, das widerspricht sich. Ich bekomme alle m\u00f6glichen \u00dcbergriffe von Seiten der Polizei mit, ich finde den Staat schei\u00dfe, weil da Positionen festgelegt sind, die ich als ungerecht empfinde.\n\n#### Fast alle Jugendkulturen sind m\u00e4nnlich dominiert \u2013 gilt das auch f\u00fcr die linke Szene?\n\nJa.\n\n#### Woran machst du das fest?\n\nWeil die zahlenm\u00e4\u00dfig \u00fcberlegen sind, und wenn man sich das so anguckt, an wen die Presse sich in Hamburg zum Beispiel wendet, wenn sie mit dem Macher der Flora reden will, ist das ein Typ seit Jahr und Tag, nie eine Frau. Das hat vielleicht auch was mit dem Alter zu tun. Frauen in meinem Alter finde ich selten in der Szene. Die bekommen Kinder und sind dann einfach nicht mehr so pr\u00e4sent wie die Typen, die irgendwie immer bleiben. Gef\u00fchlt sind das immer mehr M\u00e4nner \u2013 vielleicht auch, weil sie bei den Vollversammlungen mehr reden.\n\n#### Was hast du f\u00fcr eine Einstellung zu militanten Aktionen?\n\nDer Zweck heiligt die Mittel. Es kann schon sein, dass so was auch mal vorkommen muss, also dass in einer Aktion auch Militanz ist. Ich habe aber gleichzeitig auch Angst davor, dass jemandem etwas passiert. Das finde ich nat\u00fcrlich auch grunds\u00e4tzlich nicht gut. Aber ich denke, dass das manchmal legitim ist.\n\n#### Hast du eine Idee, wie gro\u00df die Szene ist?\n\nNein. In Hamburg w\u00fcrde ich mich dann auf die Zahlen vom Verfassungsschutz beziehen [lacht], der was von 1.000 Militanten erz\u00e4hlt. Aber da sind glaube ich die Dazugereisten auf Demos und so schon mitgez\u00e4hlt. Hamburg sch\u00e4tze ich so auf 400 \"Gewaltbereite\", aber wenn man die gesamte linksautonome oder linksradikale Szene z\u00e4hlt, komme ich so auf 2.500.\n\n#### Wenn du die verschiedenen Bezeichnungen benutzt: linke Szene, linksradikal, Antiimp, linksautonom, Antifa, Antideutsche \u2013 geh\u00f6rt das alles zusammen und ist linke Szene, ist radikale Linke dann der \u00dcberbegriff?\n\nJa, w\u00fcrde ich schon sagen. Aber die Antideutschen w\u00fcrde ich nicht mehr da rein tun, jedenfalls nicht die in Hamburg. Die sind nicht links, das sind Rassisten gegen alles Islamische. Rassisten k\u00f6nnen f\u00fcr mich nicht links sein.\n\nDas Interview f\u00fchrte Melanie Gro\u00df.\n\n## \"Meine Zukunft? Definitiv keine schei\u00df-b\u00fcrgerliche Kleinfamilie!\"\n\nMattias, 24 Jahre, lebt in Berlin.\n\n#### Lebst du in einer festen Partnerschaft?\n\nJa, seit drei Jahren.\n\n#### Was machen deine Eltern beruflich?\n\nMein Vater ist gelernter Baufacharbeiter und arbeitet heute bei einer Baufirma und verl\u00e4dt dort Betonfertigteile oder ist auch mal auf Montage unterwegs. Meine Mutter ist gelernte G\u00e4rtnerin, hat sp\u00e4ter als Telefonistin gearbeitet und ist seit ein paar Jahren Hausfrau.\n\n#### Und was ist deine gegenw\u00e4rtige Situation?\n\nIch lebe in einer WG mit zwei meiner ehemaligen Mitstudenten. Mit denen habe ich auch w\u00e4hrend des Studiums zusammengewohnt. Beruflich bin ich seit circa einem Jahr selbst\u00e4ndig im Bereich Online-Marketing.\n\n#### Wie bewertest du deine aktuelle Situation? Und deine Zukunft?\n\nIch bin momentan ziemlich gl\u00fccklich und dankbar f\u00fcr mein Leben. Ich f\u00fchle mich in Berlin wohl und zu Hause, habe eine tolle Freundin und viele tolle Menschen als Freunde. Die einzelnen Dinge funktionieren eigentlich grad gut. Trotzdem passt das gro\u00dfe Ganze nicht zusammen, was mich nach wie vor unzufrieden stimmt. Ich investiere zum Beispiel zu viel Zeit in meine Lohnarbeit, die an und f\u00fcr sich okay ist, aber ich m\u00f6chte nicht mehr im Online-Marketing arbeiten. Also im Alltag ist es okay, aber es passt nicht in meine Lebenspl\u00e4ne, es kostet halt viel Zeit. Und f\u00fcr das Geld, was ich bekomme, kann ich mir nun mal nicht die Zeit zur\u00fcckkaufen, andere Dinge zu tun, die mich interessieren oder mir wichtig sind.\n\n#### Was sind das f\u00fcr Dinge?\n\nPolitische Arbeit, Lesen, Sport, solche Sachen. Und konstruktiv an meinem zuk\u00fcnftigen Leben arbeiten. Momentan f\u00fchlt es sich so ein bisschen Murmeltier-m\u00e4\u00dfig an, also irgendwie zu viel alltagsgetrieben und immer dasselbe: Arbeit \u2013 Essen \u2013 Schlafen. Dieses Hamsterradleben, das ich nie wollte. [Lacht.]\n\n#### Wie sieht denn dann das Leben aus, das du gern h\u00e4ttest? Hast du bereits einen Plan, etwas zu \u00e4ndern? Tr\u00e4ume und Ziele f\u00fcr die Zukunft?\n\nJa, schon. Ich mache mir, seitdem ich 12, 13 bin, sehr viele Gedanken zum Leben an sich, meinem Leben, stelle mir Sinnfragen und versuche zu reflektieren. Manchmal sicherlich zu viele Fragen und zu viel Gr\u00fcbelei. Momentan plane ich meinen Ausbruch. Ich will noch mal weg. Der Plan ist ein Jahr sogenanntes \"Work and Travel\". Ich hoffe, mir w\u00e4hrend der Zeit klar zu werden, wie die einzelnen Dinge, die mir wichtig sind, in einem gro\u00dfen Ganzen besser zusammenpassen, als das jetzt der Fall ist.\n\nIrgendwann m\u00f6chte ich mit Menschen, die mir wichtig sind, zusammenleben. In einem Kollektiv. Also definitiv keine schei\u00df-b\u00fcrgerliche Kleinfamilie! Ob in der Stadt oder auf dem Land, keine Ahnung, aber gemeinsam in einem R\u00fcckzugsraum, den ich mir mit Menschen teile, die eine \u00e4hnliche Sicht aufs Leben haben und mit denen es sich halt einfach leben l\u00e4sst. Eigentlich ja das klassische linke Klischee: Kollektiv, Selbstverwaltung. Gern auch so voll \u00f6kologisch mit eigenem Garten, wo Gem\u00fcse und Obst angebaut wird, und halt M\u00f6glichkeiten zur freien Entfaltung. Also dass jede und jeder sich so einbringen kann mit dem, was er oder sie gerne macht und kann. Vielleicht ist das in meinem Kopf auch schon recht konkret an ein Bild gekoppelt. Ein paar Menschen, mit denen das funktionieren k\u00f6nnte, hab ich bereits im Sinn.\n\n#### Ist dir das Land wichtig, in dem du leben willst?\n\nIrgendwie nicht, weil es von den Menschen abh\u00e4ngt, mit denen ich lebe, das habe ich bereits gelernt: Es kann \u00fcberall dein Zuhause sein, wenn da Menschen sind, mit denen du gl\u00fccklich bist und gut auskommst. Andererseits wollte ich nie in Deutschland leben bzw. alt werden. Ich glaube, so sp\u00e4testens mit 15 habe ich mir das erste Mal gesagt, dass ich mit 25 hier weg sein muss. Ich mag Deutschland nicht: Die Menschen, wie sie denken, das ganze Gesellschaftliche drum herum. Aber inzwischen frage ich mich auch, wie sich etwas \u00e4ndern soll, wenn ich einfach woanders hingehe, wie ich mein Leben sinnvoll einbringen kann und etwas erreichen kann. Ich habe so ein bisschen die Hoffnung, dass ich durch mein Jahr im Ausland etwas mehr Klarheit bekomme.\n\n#### Wovor hast du Angst?\n\nEs mag aufgesetzt oder abgehoben klingen, aber irgendwie gibt es nichts mehr, vor was ich so richtig Angst habe. Klar, im Alltag gibt's schon Sachen. In Berlin Fahrrad fahren ist Adrenalin pur, du musst st\u00e4ndig Schiss haben, \u00fcber'n Haufen gefahren zu werden. Da hab ich irgendwie Angst davor. Ansonsten gibt's Situationen wie Nazigruppen auf meinem Weg, Bullen auf Demos in Kampfmontur, das ist auch h\u00e4sslich. Da hab ich auch jedes Mal Angst, dass es gleich knallen k\u00f6nnte. Auch aus der Erfahrung, dass es durchaus passiert. Ansonsten lebe ich sehr Karma-getrieben. Tue Gutes und dir passiert Gutes. [Lacht.]\n\nVielleicht ist da doch noch etwas, was mir Angst machen w\u00fcrde: Wenn ich nicht mehr die M\u00f6glichkeit h\u00e4tte, frei zu entscheiden. Also durch einen Unfall zum Beispiel, 'ne L\u00e4hmung, eine Behinderung oder so was... das w\u00e4re sicher hart. Aber ich habe mir nie ernsthaft dr\u00fcber Gedanken gemacht. Ich versuche insgesamt, \u00c4ngste abzubauen. Ich bin \"nat\u00fcrlich\" auch sehr sicherheitsbezogen erzogen worden, also nur kein Risiko eingehen, wenn die entsprechenden Sicherheiten nicht vorhanden sind. Das ist so 'ne Angst, die echt schei\u00dfe ist und dich aufh\u00e4lt. Ich versuche, mich nicht von solchen \u00c4ngsten beeindrucken zu lassen... Lieber mit Mut an das Leben rangehen. Okay, vielleicht zur n\u00e4chsten Frage, bevor ich albern oder zu philosophisch werde. [Lacht.]\n\n#### Was \u00e4rgert dich am meisten?\n\nIch versuche, mich nicht zu \u00e4rgern. Bringt nix. Also entweder, ich kann etwas an den Dingen, die mich \u00e4rgern, \u00e4ndern \u2013 dann mache ich das auch oder versuche es zumindest. Oder ich kann nichts \u00e4ndern, dann bringt es auch nichts, Energie f\u00fcr \u00c4rger aufzubringen. Im Zustand st\u00e4ndigen \u00c4rgerns und Verbitterung will ich nicht enden, ich w\u00fcrde immer versuchen, etwas zu \u00e4ndern.\n\n#### Was macht dich w\u00fctend?\n\nBei Wut ist das anders, Wut ist ein wichtiger Motor, um Dinge zu \u00e4ndern, \u00fcber die ich mich \u00e4rgere. Wut bringt dich dazu, Dinge zu \u00e4ndern. Also meine ganzen politischen Aktivit\u00e4ten sind irgendwie getrieben von Wut. Wut habe ich zum Beispiel \u00fcber Engstirnigkeit und Verschlossenheit, Rassismus und Fremdenfeindlichkeit nat\u00fcrlich, Dogmatismus, Elitarismus. Wenn ich Menschen begegne, die entsprechende Statements fallen lassen, werde ich immer erst mal kurz w\u00fctend, weil ich vielleicht nicht ganz so schlagfertig bin, wie ich gern sein w\u00fcrde. Dann sag ich da auch was.\n\n#### Kannst du ein Beispiel nennen?\n\nNa ja, die Klassiker so, ob am Elterntisch zu Hause oder im erweiterten Freundeskreis irgendwo auf 'ner Party: \"Wenn die Ausl\u00e4nder aber kriminell sind...\"; \"Ich bin kein Rassist, aber...\"; \"Manche T\u00fcrken sind aber auch einfach nur...\" Das endet immer in 'nem Statement, wo ich gern ausrasten w\u00fcrde, dann aber doch zu oft die Contenance bewahre. Dar\u00fcber \u00e4rger ich mich dann auch manchmal. [Lacht.] Besonders schlimm sind auch elit\u00e4re Yuppies, die mit Anfang zwanzig an nichts anderes denken als ihre Karriere und Kohle. Ich meine, nicht jeder Mensch braucht der gro\u00dfe Revolution\u00e4r zu sein, aber wer mit Anfang zwanzig sein Leben schon auf erwachsen sein, Geld verdienen und Karriere machen gepolt hat und keine Tr\u00e4ume unabh\u00e4ngig von Geld und Karriere hat... Diese Menschen machen mich echt w\u00fctend. Sie sind in meinen Augen daran schuld, dass immer wieder so viel Schei\u00dfe passiert. Es ist wirklich schwierig, nicht permanent von diesem Klischee zu sprechen, aber leider best\u00e4tigt es sich so oft. Diese ganzen jungen Elite-Studenten, vor allem die BWL-Heinis, die glauben alle, sie seien erwachsen und h\u00e4tten die Welt verstanden. Und in Wirklichkeit funktionieren sie nur und machen sich zum Affen f\u00fcr Konzerne und Unternehmen, die irgendeinen Schei\u00df produzieren. Hach, ich glaube, ich \u00e4rgere mich gerade...\n\n#### Deine Eltern scheinen ja nicht dieses Karriereleben zu haben. W\u00fcrdest du trotzdem etwas anders machen als sie?\n\nDie Frage ist schwierig. Logo, ich lebe anders als meine Eltern, alleine, weil ich in anderen Verh\u00e4ltnissen lebe. Ich habe zum Beispiel studiert. Ich habe f\u00fcr mein Verst\u00e4ndnis schon alles anders gemacht als sie: Ich bin in die Gro\u00dfstadt gegangen und arbeite zumindest formell freiberuflich und nicht in einem Lohnverh\u00e4ltnis. Ich wei\u00df, dass ich auch dieses aufgeben und ins Ausland gehen werde. Was dann kommt, ist ungewiss, aber ich werde nicht wie mein Vater in einem Job tagein, tagaus zehn bis zw\u00f6lf Stunden verbringen und mich davon physisch wie psychisch kaputt machen lassen. Nein, das ist definitiv nicht mein Leben.\n\n#### Haben deine Eltern eine Rolle f\u00fcr deine politische Entwicklung gespielt?\n\nDefinitiv, meine Mutter. Sie wird das nie so wahrhaben wollen, aber sie hat mich genau dahin gebracht, wo ich jetzt bin. Vor allem politisch. Und das, obwohl es eine Zeit gab, so mit 16, nach meiner ersten Festnahme, als sie alles getan hat, um mich von meiner \"radikalen, linken Sicht\" abzubringen. Da war auch alles dabei: Musik eingezogen, Poster abgenommen, Klamotten weggenommen. Sie hat in der Musik immer den Ausl\u00f6ser meiner geistigen Haltung gesehen. Was nat\u00fcrlich so nicht stimmte, weil sich das gegenseitig beeinflusst hat. Am Anfang waren da die Dinge, die mich angekotzt haben: Zw\u00e4nge, Polizei, Nazis, Arbeits-Tristesse. Und dann kam die Musik, die genau gegen diese Dinge Stellung bezog. Also ich habe im Alter von zw\u00f6lf Jahren meine musikalische Findungsphase gehabt, hat aber glaub ich jeder Jugendliche. Bei mir kamen da Die \u00c4rzte, aber direkt danach, \u00fcber Freunde, auch WIZO, Terrorgruppe und Co. Also schon etwas \"h\u00e4rter\" in Text und Inhalten. [Lacht.] Bewirkt hat das Eingreifen meiner Mutter damals nat\u00fcrlich genau das Gegenteil: Meine politische Anti-Haltung, die zum gro\u00dfen Teil auch typische Jugendrebellion und Provokation war, hat sich nur noch verst\u00e4rkt. Auf der anderen Seite hat mich meine Mutter sehr moderat erzogen, also nicht laissez faire, aber ich hatte schon viele Freiheiten und sie hat mich vor allem immer darin best\u00e4rkt, neue Dinge auszuprobieren. Sie ist au\u00dferdem sehr DDR-gepr\u00e4gt. In einem positiven Sinne. Also Themen wie Solidarit\u00e4t und Zusammenhalt, gegenseitige Hilfe, Sparsamkeit. Das sind Sachen, die habe ich von ihr gelernt und die haben vor allem meine politische Entwicklung beeinflusst.\n\n#### M\u00f6chtest du mal Kinder haben?\n\nIch m\u00f6chte definitiv mal Kinder haben, bin mir aber nicht sicher, ob auch eigene. Das ist halt auch die Idee, in 'nem Kollektiv zu leben. Da sollen auch Kinder sein, um die ich mich gerne k\u00fcmmere, ob es meine eigenen, also leiblichen, sind, ist mir nicht so wichtig.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nIch bin ein von Grund auf ehrlicher und emotionaler Mensch, sehr hilfsbereit und offen f\u00fcr neue Dinge. Deswegen auch immer etwas rastlos und besch\u00e4ftigt. Ich nehme mir aber immer Zeit f\u00fcr Freunde, weil ich wei\u00df, wie wichtig es ist, Menschen zu haben, die auch f\u00fcr mich da sind. Und ich bin ziemlich gut organisiert und organisierend, das ist glaube ich so das Spie\u00dfigste, was mich ausmacht. Meinen Hang zum Perfektionismus habe ich inzwischen ganz gut im Griff. Ich bin sehr bestimmend, weil ich meistens genau wei\u00df, was ich will. Kann aber auch andere gut \u00fcberzeugen, da ich sehr argumentativ und analytisch bin. Ansonsten bin ich gern kindisch oder kindlich, ich wei\u00df nicht, was es besser trifft, aber ich treibe gern Schabernack oder albere herum. Humor ist wichtig f\u00fcr mich und ich mag vor allem den schwarzen. Das merken andere h\u00e4ufig auch an meiner leicht zynischen Art. Manchmal f\u00fchle ich mich etwas schizophren, da ich zur Melancholie neige. Dann hasse ich alle Menschen und w\u00fcrde am liebsten f\u00fcr mich alleine leben.\n\n#### Was, meinst du, denken andere von dir?\n\nIch wei\u00df, dass ich manchmal etwas einsch\u00fcchternd r\u00fcberkomme, da ich sehr bestimmt bin. Meine aufgeschlossene und ehrliche Art sorgt immer daf\u00fcr, dass mich andere Menschen entweder sehr schnell annehmen oder ablehnen. Manchmal habe ich den Eindruck, als halten mich andere f\u00fcr elit\u00e4r und arrogant. Auf der anderen Seite halten mich viele f\u00fcr sehr vertrauenerweckend und engagiert. Menschen, die mich nicht kennen, halten mich auch f\u00fcr \"freakig\" oder sehen in mir einen Spinner, weil ich mich h\u00e4ufig nonkonform kleide. Grad was \u00c4u\u00dferlichkeiten angeht, bin ich halt Punk, also nicht so klischeehaft mit Iro und Lederjacke, sondern ich zieh mich einfach an, wie ich mag. [Lacht.]\n\n#### Welche Hobbys hast du?\n\nSport. Sport kommt definitiv an erster Stelle. Also ich mache Krav Maga und Parkour. Das ist einfach wichtig f\u00fcr meinen pers\u00f6nlichen Ausgleich, auch zur Arbeit, den ganzen Tag am Rechner sitzen. Danach kommen Kochen und Freunde, wobei sich die beiden Dinge perfekt kombinieren lassen. Essen ist mega-wichtig f\u00fcr mich und Kochen macht mir echt Spa\u00df. Backen genauso. Mit Freunden reden, sich einfach nur besuchen, sehen und \u00fcber die Dinge, das Leben oder konkrete Themen zu philosophieren, das ist super. Und Konzerte besuchen. Ganz wichtig. Also Musik generell ist mehr als ein Hobby. Ich h\u00f6re sehr viel Musik und habe auch mal in einer Punk-Band gesungen. Irgendwann m\u00f6chte ich das noch mal machen. F\u00fcr Lesen und Filme oder Serien schauen nehme ich mir nicht so viel Zeit, wie ich gerne m\u00f6chte, aber sehe ich auch als Hobby.\n\n#### Bist du in einem Verein aktiv?\n\nIch bin seit circa acht Jahren in verschiedenen gemeinn\u00fctzigen Vereinen aktiv, die sich f\u00fcr die F\u00f6rderung alternativer Jugendkultur einsetzen. Da habe ich schon einige Konzerte organisiert, das war so der \u00fcberwiegende Teil der Arbeit da. Und durchs Studium bin ich noch in einem Mediennachwuchsverein. Da werden unterschiedliche Medienprojekte, zum Beispiel Kurzfilmdrehs, organisiert.\n\n#### Bist du anderweitig gesellschaftlich\/politisch aktiv?\n\nIst nicht jede und jeder irgendwie gesellschaftlich aktiv? Also ich sag mal so, ich besch\u00e4ftige mich gern mit unterschiedlichen gesellschaftlich relevanten, auch politischen Themen und rede dar\u00fcber und teile meine Erfahrungen, Meinungen, Eindr\u00fccke gern mit anderen. Momentan zum Beispiel besch\u00e4ftige ich mich viel zum Thema Subkulturen unter der Einflussnahme und Assimilierung durch \"Rechte\", also Nazis. Dazu habe ich auch schon im \u00f6ffentlichen Rahmen referiert. Politisch aktiv bin ich definitiv, wobei ich auch hier denke, dass das jeder Mensch ist, der umgeben von anderen Menschen bzw. einer Gesellschaft ist. F\u00fcr mich bedeutet politisch aktiv zu sein vor allem, sich einzubringen, wo es geht, sei es auf Demos, Infoveranstaltungen, direkten Aktionen. Ich bin in keiner festen Organisation eingebunden oder habe spezielle Themen, denen ich mich widme, sondern suche mir die Sachen so raus, wie sie grad passen und wie ich zeitlich und \u00f6rtlich in der Lage bin, sie zu unterst\u00fctzen. Ansonsten lebe ich seit 1,5 Jahren vegan und seit 2,5 Jahren Straight Edge. Dar\u00fcber fragen mich echt viele Leute aus, immer wieder. Und ich gebe auch gerne Antworten. Es ist nicht so, dass ich missioniere, aber ich habe schon einen klaren Standpunkt, den ich vertrete, und f\u00fcr mich ist das auch ganz selbstverst\u00e4ndlich immer gesellschaftliches bzw. politisches Handeln.\n\n#### Wie genau sieht dein Standpunkt aus? Kannst du das kurz zusammenfassen?\n\nJa, eigentlich in einem Satz: Jeder Mensch muss herausfinden, was f\u00fcr ihn richtig ist, und sollte dabei stets offen f\u00fcr Diskurse sein. Was zum Beispiel Veganismus angeht, wo ja momentan eine Art Trend entsteht, k\u00f6nnen sich die Menschen nicht allem gesellschaftlichen Fortschritt und neuen Denkweisen verschlie\u00dfen. Viele sind tats\u00e4chlich interessiert, etwas in ihrem Leben zu \u00e4ndern, und jede oder jeder muss den besten Weg daf\u00fcr finden. Ich denke, dass ein Gespr\u00e4ch mit mir da einiges bewirken kann, bzw. wei\u00df auch, dass es so ist. Auch was mein \"drogenfreies\" Leben angeht, sehe ich das \u00e4hnlich. Also ich habe auch meine Laster und Sehns\u00fcchte und Gewohnheiten, aber ich m\u00f6chte mein Leben nicht von irgendwelchen Substanzen abh\u00e4ngig machen. Wenn du mit offenen Augen durchs Leben l\u00e4ufst, dann gibt es da viele Dinge, die dich berauschen und gl\u00fccklich machen k\u00f6nnen. Ist meine Meinung. Die sage ich, und viele finden es auch irgendwie beeindruckend, keinen Alkohol zu trinken, nicht zu rauchen usw., k\u00f6nnen es sich f\u00fcr sich aber nicht vorstellen, so zu leben. Ist okay, ich habe auch eine Weile gebraucht, um auf den Trichter zu kommen. [Lacht.]\n\n#### Welche Bedeutung hat Musik heute f\u00fcr dich?\n\nMusik ist Leben. Das ist alles. Seit ich zw\u00f6lf bin, h\u00f6re ich bewusst und intensiv Musik, in fast jeder Lebenslage, wahrscheinlich acht Stunden Minimum am Tag.\n\n#### Hast du eine Lieblingsmusik?\n\nIch hab bef\u00fcrchtet, dass das jetzt kommt... Ja und nein. Ich mag Abwechslung und bin auch da sehr offen f\u00fcr Ver\u00e4nderungen. Letzten Endes lande ich doch immer wieder beim Punk in all seinen Formen. Es gibt Bands, die h\u00f6re ich regelm\u00e4\u00dfig und wie zuf\u00e4llig und selbstverst\u00e4ndlich einmal im Monat. Weil die Musik einfach so stark ist und die Texte viel kompensieren. Wut, Gewalt, Ohnmacht, Kampf, Revolution, Ver\u00e4nderungsw\u00fcnsche, Widerstand usw. Oftmals kannst du nicht alles \u00e4ndern, aber es ist geil, jemand anderen zu kennen, der das so sagt, wie du denkst. Punk schafft das immer wieder am besten.\n\n#### Interessierst du dich f\u00fcr Mode oder auch K\u00f6rperschmuck?\n\nMode ist wohl das \u00fcberbewertetste Ding, was es gibt. Also ich nehme verschiedene Modetrends sicher wahr, wei\u00df aber nicht, inwiefern eine Ablehnung von Mode jetzt als Interesse gesehen werden kann. Desinteresse trifft es wohl eher. Bei K\u00f6rperschmuck sieht's dagegen anders aus. Ich liebe T\u00e4towierungen. Das hat aber bei mir nix mit Mode zu tun, ich meine, das sind Bilder f\u00fcr die Ewigkeit, die mich mein Leben lang an pers\u00f6nliche Dinge erinnern sollen.\n\n#### Hast du Vorbilder, Leitbilder, Idole?\n\nNein. Generell nicht. Es gibt sicherlich viele Menschen, die ich irgendwie bewundernswert finde, aufgrund ihrer Leistungen oder ihrer Lebensgeschichte. Einstein zum Beispiel oder Karl Marx, um mal Beispiele zu nennen, die jeder und jede irgendwie kennt. Allerdings habe ich bereits so viele interessante Menschen pers\u00f6nlich kennengelernt und auch in meinem Freundeskreis, dass es mir leichter f\u00e4llt, mich an diese Personen zu halten. Au\u00dferdem ist kein Mensch unfehlbar, also halte ich mich lieber an konkrete Dinge und lasse mich inspirieren. Da bin ich irgendwie zu sehr Anarchist: No gods, no masters, no idols.\n\n#### Wie verbringst du deine Freizeit?\n\nZusammenfassend mit Sport, Kochen, Musik und Freunden. Ich gehe zwei bis vier Mal die Woche zum Krav Maga. Kochen meistens am Wochenende oder unter der Woche abends, mit meiner Freundin oder Freunden oder einfach f\u00fcr mich alleine. Musik h\u00f6ren eh immer, aber oft auch mit Freunden gemeinsam zu Hause oder bei Konzerten nat\u00fcrlich. Und generell bin ich gern bei Freunden. Ich besuche sie auch gern, wenn sie in anderen St\u00e4dten oder meiner ehemaligen Heimat wohnen. Besuchen und Reisen ist eine echte Leidenschaft von mir. Ansonsten gibt's viel Kleinkram, der meine rare Freizeit neben den bereits erw\u00e4hnten Aktivit\u00e4ten auff\u00fcllt. Internet ist ganz schlimm. Da verbringe ich auch einige Freizeit mit. Meistens alleine vor meinem Bildschirm in einem Meer aus Texten, durch die ich mich lese, oder Bilder, die ich durchklicke. Furchtbare Zeitverschwendung oft. Aber meistens auch wichtig und f\u00f6rderlich f\u00fcr soziale Kontakte, Bildung usw.\n\n#### Nach welchen Kriterien schlie\u00dft du Freundschaften?\n\nNach gar keinen. Freundschaften schlie\u00dfen sich von alleine. Wenn es passt. Ich habe ein sehr sicheres Bauchgef\u00fchl und eine sehr gute Menschenkenntnis. Manchmal habe ich den Eindruck, ich lasse mich zu schnell von neuen Menschen beeindrucken und mitrei\u00dfen. Aber fast immer entstanden engere Freundschaften, wenn das Bauchgef\u00fchl in den ersten Sekunden passt. Das ist ja auch wissenschaftlich erwiesen, dass die meisten Menschen nicht mehr als drei Sekunden brauchen, um jemensch anderen einordnen und sympathisch zu finden.\n\n#### Geh\u00f6rst du einer Jugendkultur an?\n\nNein, definitiv nicht. Ich bewege mich seit circa zehn Jahren inmitten verschiedener Szenen und f\u00fchle mich der Punk-Szene auch irgendwie verb\u00fcndet. Aber ein richtiger Teil war ich nie. Ich mag Zusammengeh\u00f6rigkeitsgef\u00fchl, aber nur, wenn es nicht zu gro\u00df ist. Bei mir funktioniert das alles \u00fcber Freundschaften. Das ist eher meine Szene. Subkulturen frustrieren mich irgendwie oder stimmen mich zumindest traurig. Punk, Hardcore sind Subkulturen, die ich echt liebe wegen ihrer Euphorie, Entstehungsgeschichte und der Musik und dem Drumherum, aber es gibt keine Subkultur, die nicht ausverkauft ist. Das, was die meisten als Subkultur sehen, ist nichts als tote H\u00fclle, die h\u00e4ufig nur noch durch die Musik getragen wird. Eine Ausnahme gibt es sicherlich, und das ist auch am ehesten etwas, wo ich mich hingezogen f\u00fchle, die DIY-[Do-It-Yourself-]Szene. Wobei ich da wenig Zugang habe und auch denke, dass da in Deutschland nicht viel zu holen ist. Daf\u00fcr geht's uns einfach zu gut. [Lacht.] Mir f\u00e4llt grad das Tocotronic-Lied \"Ich m\u00f6chte Teil einer Jugendbewegung ein\". Das ist gro\u00dfartig, wie \u00fcberhaupt die ganze Band.\n\n#### Welche Jugendkulturen findest du absolut unsympathisch?\n\nNazis, Disco, HipHop. Zu Nazis muss ich nicht so viel erkl\u00e4ren, oder? Disco ist genau das Gegenteil einer Subkultur, Mainstream halt. Chartmusik, Fashion before Passion, sinnlos volllaufen lassen, Haufen Kohle raushauen, um in 'nem dunklen Raum mit hunderten anderen Zombies zu sein und das Ganze noch als Spa\u00df zu bezeichnen, nee, darauf komm ich nicht klar. Das hat ja auch null Charme, von Anfang bis Ende. Alle schminken sich bis zur Unkenntlichkeit, obwohl sie in 'nen dunklen Raum gehen, ziehen sich feinste Hemden und wei\u00dfe Jeans an, obwohl sie den ganzen Abend schwitzen... Und mit HipHop meine ich eigentlich deutschen HipHop und das ganze Proll-Ding. Also nicht klassischen Rap, so wirklich von der Stra\u00dfe, revolution\u00e4r, rebellisch, kritisch. Aber diese ganze Bling-Bling- und Ich-hab-den-Megaschwanz-und-fick-dichso-krass-Schei\u00dfe ist so bescheuert, meistens total aufgesetzt und zugleich menschenverachtend. Da kommt mir das kalte Kotzen. Dabei mag ich Rap an sich und h\u00f6re auch selber gern HipHop. Vorausgesetzt, es geht halt um richtige, wichtige Inhalte. Nicht ewig derselbe Schei\u00df.\n\n#### Welche Rolle spielen deine Freunde f\u00fcr dich?\n\nFreunde sind mega-wichtig. Das ist meine zweite Familie bzw. inzwischen meine erste. Wenn du Freunde hast, brauchst du sonst nichts auf der Welt. Richtige Freunde kannst du dir nirgends und f\u00fcr nix in der Welt kaufen. Freunde bzw. vor allem meine Freundin sind so wichtige Anker im Leben, R\u00fcckzugsr\u00e4ume oder Bezugspunkte, je nachdem, wie mensch das sehen m\u00f6chte.\n\n#### Und welche Rolle spielt die autonome Szene f\u00fcr dich?\n\nAh, wir kommen langsam zum Thema. [Lacht.] Ich sage mal so: Wie mit allem anderen bereits Erw\u00e4hnten ist es hier \u00e4hnlich: Ich bin die Szene und das, was ich daraus mache. Ich sehe mich schon als Teil dieser Szene, selbst wenn sich das vom Gef\u00fchl her nur auf Demos so anf\u00fchlt. Ansonsten sind es eben eher die Freunde, die machen meine Szene aus, was nicht hei\u00dft, dass wir nicht autonom sind.\n\n#### Woher beziehst du deine Informationen zum aktuellen Zeitgeschehen?\n\nWenn ich ehrlich bin, beziehe ich seit Jahren keinerlei aktuelle Informationen mehr auf aktivem Weg, sondern sie finden mich. Es ist sehr schwer dem g\u00e4nzlich aus dem Weg zu gehen. Vor allem, wenn mensch wie ich tagt\u00e4glich im Internet zu tun hat. Da kommen die News automatisch zu dir. Meistens \u00fcber Freunde, soziale Netzwerke, Arbeit und Kollegen. Dann lese ich nat\u00fcrlich auch mal was nach, meistens im Internet. Fernsehen schaue ich schon circa zwei Jahre nicht mehr, Radio noch l\u00e4nger nicht mehr geh\u00f6rt, wobei das ja ab und an nebenbei l\u00e4uft. Bei Freunden, in der Mitfahrgelegenheit, bei den Eltern zu Hause. Ich kaufe mir vielleicht einmal im Jahr eine Wochenzeitung, die ich dann nie komplett schaffe zu lesen. Nicht mal die Artikel, die mich interessieren. Das Hauptproblem f\u00fcr mich ist, dass es immer dieselben Nachrichten sind: Krieg, Tote, Ungl\u00fcck, Katastrophe, Verletzte. Nur der Ort, der Ausl\u00f6ser und die Anzahl der Opfer sind variabel. Warum sollte ich mir das jeden Tag geben? Au\u00dferdem folgen Medien einem, sehr perfiden System: Sie verkaufen jeden Tag Informationen und Geschichten, die aufs Neue spannend und interessant sein m\u00fcssen. Hier verliert sich der Blick aber schnell in Kleinigkeiten und Alltagsgeschehen. Die meisten Menschen verstehen nichts vom gro\u00dfen Ganzen und den Hintergr\u00fcnden. Diese sind auch nicht im Alltagsgesch\u00e4ft eines Journalisten mal nebenher recherchiert, das dauert, und Tagesnachrichten sind nicht der richtige Ort zur Verwertung. Deswegen lese ich auch lieber Brosch\u00fcren oder Hintergrundberichte oder ggf. ein Buch zu einer Thematik, weil jedes Ereignis eine lange Vorgeschichte hat, und das Ganze zu verstehen, ist f\u00fcr mich wesentlich wichtiger. Dann ist auch die Manipulation durch Einzelinformationen ohne Kontextzusammenhang schwieriger. Und wenn ich mich dann mal zu einem aktuellen Ereignis informiere, nutze ich so viele Quellen wie m\u00f6glich. Vor allem im Internet. Aber da f\u00e4llt dann auch auf, dass zumindest alle gr\u00f6\u00dferen Zeitungen und Portale, wie SPON.de, S\u00fcddeutsche, Frankfurter Allgemeine und wie sie alle hei\u00dfen, h\u00e4ufig \u00e4hnliche Meldungen und Texte ver\u00f6ffentlichen.\n\n#### Geh\u00f6rst du einer Religion oder Kirche an?\n\nNein. Ich bin getauft und konfirmiert, also protestantisch. Bin aber vor drei Jahren auch formell aus der Kirche ausgetreten. So richtig war ich da auch nie drin. Ich wollte nur diese komische Jugendweihe nicht mitmachen, wo du in die Gesellschaft aufgenommen wirst. Ich wollte nicht in die Gesellschaft.\n\n#### Welche Bedeutung hat Religion f\u00fcr dich?\n\nReligion ist aus meiner Sicht in erster Linie ein Gemeinschaftsding. Ich denke, jeder Mensch braucht etwas, an das er glaubt und an das er sich halten kann. Das ist auch wichtig und richtig so. Allerdings finde ich Religionen wie zum Beispiel das Christentum, den Islam oder auch den Hinduismus sehr verantwortungslos den Menschen gegen\u00fcber. Religion hat f\u00fcr mich viel mit Schicksal und dem Erdulden des Lebens zu tun, in blindem Gottvertrauen. Amen. Ich verstehe nicht, warum Menschen so geil drauf sind, sich institutionalisieren zu lassen. Glauben kann ich auch alleine, an Gott, mich selbst, Karma oder von allem etwas. Aber Religion versucht das Ganze wieder zu ordnen und zu interpretieren. Die Menschen brauchen halt die Ordnung, sonst w\u00e4ren sowohl Staat als auch Religion \u00fcberfl\u00fcssig.\n\n#### Wie w\u00fcrdest du \"autonom\", \"autonome Szene\" eigentlich definieren oder jemandem beschreiben, der noch nie etwas davon geh\u00f6rt hat?\n\nAutonom hei\u00dft ja erst einmal nicht mehr als selbst\u00e4ndig oder selbstbestimmt. Autonome sind m. E. freiheitsstrebende Menschen, die sich meist in kleinen, unabh\u00e4ngigen Gruppen organisieren, welche untereinander lose miteinander vernetzt sind. Die einzelnen Gruppen k\u00f6nnen dabei durchaus unterschiedliche Ziele verfolgen, geh\u00f6ren aber fast immer zur linksradikalen Bewegung. In der Praxis eint fast alle Autonomen die Ablehnung von Autorit\u00e4ten wie Staat und Kirche und das Streben nach Freir\u00e4umen. Die Summe der Gruppen und Einzelindividuen kann als \"autonome Szene\" bezeichnet werden, wobei ich auch hier gerne zum Szene-Begriff Abstand halten m\u00f6chte.\n\n#### Bist du Mitglied einer konkreten Gruppe?\n\nNein, keine konkrete Gruppe. Das ist nicht so mein Ding, da sich Gruppen meistens \u00fcber gemeinsame Ideale und Vorstellungen identifizieren, die nicht selten in Dogmen enden. Ich sehe Freunde als wichtigere Bezugsgruppe an und mache dann mit unterschiedlichen Leuten unterschiedliche Dinge, je nach Interesse. Mit den einen also eher theoretisch arbeiten und zum Beispiel an Infoveranstaltungen arbeiten. Mit anderen eher direkte Aktionen wie Graffiti, Streetart, Plakat-Aktionen, solche Dinge. Ich finde es wichtig, dass ich mit Menschen befreundet sein kann und mich auf sie verlassen kann, unabh\u00e4ngig von einem konkreten politischen Anspruch. Wenn dann daraus gemeinsame politische Aktionen entstehen, ist das voll cool, aber das sollte nicht der Selbstzweck der Freundschaft sein.\n\n#### Was bedeutet dann \"autonom\" f\u00fcr dich genau?\n\nF\u00fcr mich bedeutet es, sich so frei wie m\u00f6glich zu machen hinsichtlich des eigenen Handelns. Oder einfacher: in Bewegung bleiben und dabei niemals das Denken aufzuh\u00f6ren. F\u00fcr mich ist wichtig, dass ich das Gef\u00fchl habe, in der Welt etwas bewegt zu haben. Und wenn es am Ende nur ein Mensch ist, der durch mich ein besseres Leben hatte oder einen Tag in seinem Leben gl\u00fccklich war. Dabei will ich selbstbestimmt in der Wahl meiner Mittel und Wege sein, zumindest so weit wie m\u00f6glich. Eingeschr\u00e4nkt sind wir ja eh alle durch irgendwas. Aber ich denke dann auch immer an meinen Traum des unabh\u00e4ngigen Lebens im Kollektiv in einer libert\u00e4ren Gesellschaft, zu der ich einen konkreten Beitrag geleistet habe. Grunds\u00e4tzlich bin ich der Meinung, dass Informationen und Austausch sehr viel bewirken k\u00f6nnen. Also angefangen von Gespr\u00e4chen in kleiner Runde als auch Vortr\u00e4ge, Workshops und so weiter. Aus jedem Gespr\u00e4ch nimmst du ja irgendetwas mit und bist danach ein anderer Mensch. Viele brauchen halt ein bisschen Anschub zu begreifen, dass wir alle nicht frei sind, aber uns frei machen k\u00f6nnen.\n\n#### Wie war dein erster Kontakt zur autonomen Szene?\n\nAn einen konkreten ersten Kontakt kann ich mich nicht erinnern. Aber seitdem ich ungef\u00e4hr sechzehn bin, habe ich mich schon verst\u00e4rkt f\u00fcr linke Politik und deren Geschichte interessiert. Das wurde dann im Abitur noch mal intensiver, und durch Behandlung von Themen wie '68er Bewegung, APO, RAF usw. habe ich angefangen, mich konkret zur autonomen Szene, RAF und zum Anarchismus zu informieren. Das hat mich echt voll interessiert und war irgendwie ziemlich magisch. Ich habe damals gedacht, dass ich gern eher gelebt hatte und fr\u00fcher politisch mehr Bewegung war, wenngleich viele Entwicklungen anders verlaufen sind als urspr\u00fcnglich gedacht. Also ich war nie verliebt in Ulrike Meinhof oder wollte RAF-Terrorist sein, weil ich schon der Meinung bin, dass es Teilen der RAF irgendwann eher ums \"Gangsterleben\" anstatt politischer Ideale und Ziele ging. Na ja, und dann hat sich das so weiterentwickelt, mal hier und da Kontakt zu neuen Leuten gehabt, \u00fcbers Internet von Gruppen und Aktionen erfahren. Allerdings muss ich zugeben, dass ich da echt im Tal der Ahnungslosen lebte, halt auf 'nem Dorf in der osts\u00e4chsischen Provinz, da biste selten richtig rausgekommen und dann auch nicht wirklich im Sinne autonomer Aktionen. So richtige Kontakte haben sich erst sp\u00e4ter entwickelt, nach dem Abitur. Im Freundeskreis waren wir halt paar Punks und Chaoten, die sich oft zum Biertrinken getroffen haben, philosophiert haben und uns aber auch immer als politisch und linksradikal verstanden haben. Von so \"klassischen\" Autonomen als eigene Bewegung wusste ich zwar, da gab's aber keinerlei Ber\u00fchrungspunkte f\u00fcr mich pers\u00f6nlich.\n\n#### Ist die autonome Szene auch eine Jugendkultur?\n\nIch habe zumindest das Gef\u00fchl, dass sie sich ein bisschen dazu entwickelt hat. Allerdings ist die autonome Szene ja seit Beginn sehr erwachsen gewesen irgendwie. Also was so politischen Anspruch, konkrete Forderungen nach Ver\u00e4nderungen angeht, gab es da ja zumindest ungef\u00e4hre Vorstellungen, in welche Richtung eine Ver\u00e4nderung passieren sollte, wenngleich die Auffassungen sich im Detail unterschieden. Diese klare linksradikale, libert\u00e4re Ader und die N\u00e4he zu revolution\u00e4ren Ideen \u2013 Anarchismus, Kommunismus... \u2013, das ist schon etwas mehr als nur Jugendkultur. Also das subkulturelle Element ist sehr stark ausgepr\u00e4gt, aber so richtige eigene Elemente wie Musik oder Kleidung w\u00fcrde ich nicht f\u00fcr die autonome Szene festmachen wollen. Ich sehe die autonome Szene eher als Sammelbecken verschiedener Jugendkulturen. Die Leute, die Bock auf linke Politik haben, landen sehr wahrscheinlich in der autonomen Szene. Da ist der Szenehintergrund gar nicht so wichtig. Also ob Punk, Skinhead, HipHopper, Skater, Raver, was auch immer. Ich kenne inzwischen unterschiedliche Menschen aus unterschiedlichen Subkulturen, die gemeinsam linke Politik zum Beispiel in einer gemeinsamen Antifa-Gruppe machen. Aber die Politik und gemeinsame Ideale oder Ziele stehen da so \u00fcber dem Rest.\n\n#### Aber gibt es nicht einen eigenst\u00e4ndigen \"autonomen\" Stil oder ist auch das komplett unterschiedlich und individuell?\n\nMit zunehmender Popularit\u00e4t der Autonomen zum Beispiel durch gemeinsames Auftreten im sogenannten \"Schwarzen Block\" auf Demos ist da schon ein bisschen schwarze Uniformierung forciert wurden. Ich pers\u00f6nlich trage auch viel dunkle Sachen, schwarz zum Beispiel, aber auch nicht immer. Ist halt praktisch. Ob das jetzt als einheitlicher Stil durchgeht, wei\u00df ich nicht. Individualisierung findet dann sp\u00e4testens durch unterschiedliche Aufn\u00e4her statt. [Lacht.] So richtige Codes gibt's f\u00fcr mich zumindest nicht. Klar, das Anarchie-\"A\" ist super popul\u00e4r.\n\n#### Welche Musik spielt eine Rolle? Andere (pop)kulturelle Medien?\n\nPunk, Hardcore, HipHop und zunehmend auch Elektro spielen eine wesentliche Rolle. Ich denke allerdings, dass es an der Thematik liegt, mit denen sich die Songs besch\u00e4ftigen. Ich finde Filme ganz wichtig, sowohl Spielfilme als auch Dokumentationen, weil sie ein wunderbares Medium sind, sich mit Thematiken auseinanderzusetzen und neue Gesichtspunkte zu zeigen. Videokollektive oder so Guerilla-Geschichten entstehen oft aus dem autonomen Umfeld heraus und sprechen dann sowohl die eigene Szene als auch Au\u00dfenstehende gezielt f\u00fcr ein Thema an. Und Zines spielen eine gro\u00dfe Rolle. Eigene Magazine, sei es mit Hintergrundinfos, Geschichten, Tipps und Hilfe oder Veranstaltungsinformationen, sind auch wieder st\u00e4rker im Kommen, trotz Internet.\n\n#### Welche Themen sind dir f\u00fcr dein Engagement wichtig?\n\nIch bin grunds\u00e4tzlich erst mal Anarchist. Und als dieser glaube ich an eine freie Gesellschaft ohne \u00e4u\u00dfere Zw\u00e4nge. Somit sind Themen wie Freir\u00e4ume, Staatsgewalt und Unterdr\u00fcckung und Antikapitalismus sicherlich wegf\u00fchrend. Antifaschistisches Engagement ist f\u00fcr mich ehrlich gesagt normal. Also im Sinne von dar\u00fcber muss ich nicht reden, mich labeln etc. Das ist mir sowohl im Alltag wichtig und auch durch Teilnahme an Demos gegen Nazis, Rechtspopulismus oder \u00c4hnliches.\n\n#### Und welche politischen Themen sind dir besonders wichtig?\n\nIch lebe vegan und hasse Mackertum. An entsprechenden Stellen weise ich Mitmenschen dann auch gern direkt darauf hin, wenn sie diesbez\u00fcglich etwas Dummes sagen oder sich daneben benehmen. Ich geb mir auch M\u00fche zu gendern und das zur Gewohnheit werden zu lassen, nicht spezifisch zu sprechen. Aber ich bin ehrlich gesagt nicht \u00fcberm\u00e4\u00dfig im Thema Antisexismus, Gender oder so aktiv. Auch nicht im Bereich Tierrechte oder Veganismus. Ich sehe halt die pers\u00f6nliche Aufkl\u00e4rung im direkten Gespr\u00e4ch als meins, auch wenn ich da schon mal auf 'ner Demo war. Umweltschutz und Castor sind auch so Dinge, sind wichtig, logo. Aber praktiziere ich erst mal f\u00fcr mich selber und geh mal auf 'ne Gro\u00dfveranstaltung. Aber nicht so in der allt\u00e4glichen politischen Arbeit, nee.\n\n#### Wie stehst du zum Konflikt zwischen Antiimps und Antideutschen?\n\nOh je! Irgendwo in der Mitte vielleicht. Das Ganze steht und f\u00e4llt ja leider immer mit der Israel-Pal\u00e4stina-Frage. Das ist halt ein mega-komplizierter Konflikt, in dem ich mich immer noch nicht positionieren kann und will. Vielleicht werde ich das auch nie, egal, wie viel ich mich noch damit besch\u00e4ftige. Ich sehe mich grunds\u00e4tzlich eher auf der Seite der Antideutschen, wobei ich mich selbst gern als anti-national verstehe. Ich halte nichts von s\u00e4mtlichen k\u00fcnstlich geschaffenen Konstrukten, die Menschen eingrenzen alleine dadurch, dass sie vorhanden sind. Dieses Denken in Nationen widerspricht meiner anarchistischen Grundhaltung. Im Falle Israels ist das nat\u00fcrlich sehr schwierig. Ich denke, die Juden und die sich als Israelis f\u00fchlen, brauchen ihren Staat. Alleine als Schutzraum und aufgrund ihrer historisch einmaligen Situation. Da stehe ich auch klar dahinter. Nur wird bei Israel leider immer mit doppeltem Ma\u00df gemessen, wenn es zum Beispiel um die Frage nach der Legitimation milit\u00e4rischer Operationen geht. Da wird halt einem Land das Selbstverwaltungsrecht abgesprochen, welches bei jedem anderen selbstverst\u00e4ndlich ist. Ein konkretes Problem habe ich aber sowohl mit verk\u00fcrzten Kapitalismus- und auch Globalisierungskritiken als auch mit dogmatischem Pro-Israelismus.\n\n#### Setzt du dich auch mit Theorie auseinander?\n\nZu selten. Zumindest w\u00fcrde ich gern mehr Zeit auf politische Theorie verwenden. Es interessiert mich einfach, grundlegende Zusammenh\u00e4nge zu verstehen und andere Ideen zu reflektieren. Ich finde das wichtig f\u00fcr mein pers\u00f6nliches Weiterkommen, Horizonterweiterung eben. Ganz aktuell wieder verst\u00e4rkt mit Anarchismus und Kommunismus und Straight Edge und linke Politik. Au\u00dferdem interessieren mich alte Philosophen, das sind so die Basics, die immer noch passen. Also angefangen von Aristoteles und Sokrates bis Hegel und Marx. Das \u00e4ndert sich auch immer mal und momentan hab ich auch viel zu wenig Zeit f\u00fcrs Selbststudium.\n\n#### Wie informierst du dich \u00fcber szeneinterne Aspekte?\n\nZu neunzig Prozent \u00fcber's Internet, gemessen alleine an der H\u00e4ufigkeit. Dabei sind Foren sehr wenig, Blogs und Websites und Soziale Netzwerke, Facebook regelm\u00e4\u00dfig. Ansonsten Filme und Videos, Magazine, Flyer, was es so an Veranstaltungen gerade gibt und mich interessiert. Was da an interessant scheinenden Infos herumliegt, nehm ich mit und schau's durch. Irgendwas Spannendes gibt's eigentlich immer und \u00fcberall. Die Stra\u00dfen aus Zucker finde ich zum Beispiel sehr gut. Das ist ja quasi auch 'ne kollektiv verbreitete Zeitschrift, die quartalsweise erscheint und anl\u00e4sslich der Fu\u00dfball-WM gegr\u00fcndet wurde. Da stehen sehr inspirierende Artikel drin, die mich immer auf gewisse Themen sto\u00dfen, mit denen ich mich dann vertiefend besch\u00e4ftige.\n\n#### Wie stehst du zu B\u00fcndnissen zu anderen linken Bewegungen, zu sozialen Bewegungen?\n\nIch finde solche Dinge \u00e4u\u00dferst wichtig. Nat\u00fcrlich muss mensch schauen und beachten, dass gerade im B\u00fcndnis mit b\u00fcrgerlichen Gruppen und Bewegungen die eigene Schlagkraft und Aussage zur\u00fcckbleibt. Also wenn du als linksradikale Gruppe nur noch Castor blockieren im Wendland und Nazis blockieren am 13. Februar in Dresden als Aktionsfeld hast, dann ist die Frage nach dem Selbstzweck schon recht offensichtlich. Linke Politik sollte in meinen Augen in Bewegung bleiben und tonangebend sein, neue Entwicklungen und Denkprozesse forcieren und unterst\u00fctzen. Eigentlich ist es logisch, sich auf Gro\u00dfdemos gegen \u00dcberwachung und Atomkraft und Nazis zusammenzuschlie\u00dfen, aber das kann eben nicht alles sein. W\u00e4hrend viele der Teilnehmer von Gro\u00dfevents oder gr\u00f6\u00dferen Bewegungen sich auf ein konkretes Thema mit einem konkreten Ziel spezialisieren, m\u00fcssen linke Gruppen immer einen Schritt weitergehen, weiterdenken und \u00dcberwachung zum Beispiel aus einem anderen Blickwinkel kritisieren als zum Beispiel die Piraten das machen. Das ist auch gar nicht schwierig. Kontakte und Verbindungen schaffen immer Synergien und m\u00fcssen genutzt werden. Allerdings geht zu viel \u00d6ffnung auch oft mit der wachsenden Gefahr der Assimilation Hand in Hand.\n\n#### Wie stehst du zu Parteipolitik? Gehst du w\u00e4hlen?\n\nIch glaube nicht an Parteienpolitik. Wir m\u00fcssen uns alle selbst um unsere Belange k\u00fcmmern, gemeinsam. Daraus kann viel mehr entstehen und verwirklicht werden als durch Abgeordneten-Politik und Repr\u00e4sentative. Ich sehe das Ganze als hohles Konstrukt, ein nettes Schauspiel. Ich wei\u00df, dass sich nur etwas \u00e4ndern wird, wenn ich es mache. Nicht ich alleine, eben gemeinsam mit anderen. Auch da bin ich durch und durch Anarchist. [Lacht.]\n\n#### Wie ist dein Verh\u00e4ltnis zum Staat, zur Polizei?\n\nDer Ort, wo ich lebe, regional willk\u00fcrlich abgesteckt, hat nichts mit mir, meinem Leben, meiner Person zu tun. Staaten als Organisationssysteme funktionieren nie so gut wie Kollektive. Ich bin gegen Staaten. Punkt. Polizei, das sind die Bullen, die Staatsgewalt. Ich kenn auch gute Bullen, ja. Einer hat uns direkt im Dorf gegen\u00fcber gewohnt. Der war aber auch nur so lange gut, wie er keine Uniform trug. Wobei er uns auch ein, zwei Mal in der Verkehrskontrolle hat weiterfahren lassen, weil er uns kannte. [Lacht.] Aber generell ist es immer dasselbe: Der Mensch geht auf Arbeit, streift sich seine Uniform \u00fcber und dazu muss er seine Individualit\u00e4t, sein Mensch-Sein, seinen Charakter ablegen, sonst passt er nicht rein. Egal ob Bulle, Soldat oder Politesse. Wie soll das auch anders gehen. Wenn du anf\u00e4ngst, individuell, moralisch, menschlich zu denken und zu handeln, kannst du ja nicht nach Regeln und Gesetzen funktionieren. Da gibt's immer klare Vorgaben und die Ermessensgrenzen werden immer enger, auch durch immer mehr Gerichtsurteile, die zum Beispiel als Referenz dienen k\u00f6nnen. Ein Polizist im Alltag, der vielleicht an der chinesischen Botschaft steht, in seiner alten olivgr\u00fcnen Uniform, ist nat\u00fcrlich etwas komplett Anderes als der schwarze Kampfroboter vom USK [Unterst\u00fctzungskommando] auf 'ner Antinazi-Demo. Das unwohle Gef\u00fchl, wenn ich die Polizei auf der Stra\u00dfe sehe, egal wann und wo, bleibt bzw. ist immer da. Kein Staat, keine Staatsgewalt.\n\n#### Gibt es Ver\u00e4nderungen beim Umgang des Staates mit Autonomen?\n\nOh ja, sp\u00e4testens seit dem ganzen Aufkochen der Extremismusdiskussion und dem damit nun offenen Kampf gegen sogenannte Linksextremisten haben sich Ton und Gangart um Einiges versch\u00e4rft. Vor allem die Gleichsetzung von Links- und Rechtsextremen durch Staat und Medien schafft mich. Ich kann so viel Blindheit und Dummheit nicht verstehen. Das ist, als wenn ich sage, alle Obstb\u00e4ume sind Blumen, nur weil sie auch bunte Bl\u00fcten tragen. Also du schaust auf ein Ereignis und suchst nach etwas Vergleichbarem, und pl\u00f6tzlich ist alles eins, trotz komplett unterschiedlicher Ausgangspunkte, Absichten, Handlungsweisen. Die aktuelle Entwicklung l\u00f6st ganz ehrlich gro\u00dfes Unbehagen bei mir aus, da ich auch am eigenen Leib auf Demos gemerkt habe, wie krass die Bullen durchgreifen. Auch durch die breitere zivilgesellschaftliche Beteiligung an Massenblockaden von Nazi-Demos scheinen einige Leute kalte F\u00fc\u00dfe zu haben. Vor allem, wenn ich nach Sachsen schaue. Da gibt die Politik eine sehr harte Gangart vor, die setzt die Polizei dann um. Momentan bleibt noch abzuwarten, wie seitens der Judikative dann tats\u00e4chlich gegen die Leute vorgegangen wird. Momentan scheint vor allem Verunsicherung und Zerm\u00fcrbung eine Strategie seitens des Staates zu sein gegen\u00fcber autonomen Gruppen.\n\n#### Spielen internationale Vernetzungen f\u00fcr dich eine Rolle?\n\nUm ehrlich zu sein, bin ich daf\u00fcr nicht die ideale Ansprechperson. Ich bin froh, inzwischen in Berlin etwas Anschluss gefunden zu haben. Klar, ich habe auch ein paar internationale Kontakte, aber eher direkt zu Einzelpersonen und weniger Gruppen. F\u00fcr mich spielen die momentan keine wirkliche Rolle, ich kann mir aber sehr gut vorstellen, dass das in den n\u00e4chsten Jahren zunehmen wird. Ich hoffe, durch meinen Auslandsaufenthalt auch weitere Menschen kennenzulernen und Kontakte zu kn\u00fcpfen, die ich auch f\u00fcr zuk\u00fcnftige politische Arbeit nutzen kann. So Gruppen wie ABC \u2013 also Anarchist Black Cross \u2013 kenn ich nat\u00fcrlich, auch teilweise \u00fcber pers\u00f6nliche Kontakte. Das ist es dann aber auch.\n\n#### Wie hat sich die autonome Szene selbst im Laufe der Jahre ver\u00e4ndert? Was wei\u00dft du \u00fcber die Geschichte der Szene?\n\nIch wei\u00df, dass die italienische Arbeiterbewegung und der Operaismus Ende der 1960er Jahre wichtige Ausgangspunkte waren. Und in Deutschland die APO und '68er Bewegung. Aber so richtig die gesamte Historie k\u00f6nnte ich jetzt nicht erz\u00e4hlen. Seit Anfang der 2000er Jahre wuchs die antideutsche Str\u00f6mung ziemlich stark an. Vor allem nach dem 11. September und durch die zweite Intifada in Israel habe ich auch eine zunehmende Spaltung innerhalb \"der Linken\" wahrgenommen. An dieser Stelle sei auch mal betont, dass ich nicht wei\u00df, was genau immer mit \"der Linken\" gemeint ist. F\u00fcr mich ist linke Politik sehr heterogen, da ist nichts einheitlich Linkes, was irgendwie kritisiert werden k\u00f6nnte. Das ist so die Entwicklung, die ich als ziemlich bestimmend in den letzten Jahren wahrgenommen habe. Im Moment geht's wieder in eine konstruktive Richtung, weil ziemlich viele junge Menschen sich f\u00fcr linke Politik interessieren, w\u00e4hrend noch viel mehr komplett gar nichts mehr mit Politik zu tun haben wollen. Aber da geht noch Einiges. Auch \"Der kommende Aufstand\" blieb ja nicht unbeachtet, auch \u00fcber die autonome Szene hinaus. Da haben einige Leute auch aus meinem Bekanntenkreis noch mal ihren Blickwinkel gewechselt und wieder angefangen, sich st\u00e4rker einzubringen. Aber eine wirklich klare Richtung sticht f\u00fcr mich zurzeit noch nicht heraus.\n\n#### Wo siehst du Kritikpunkte, Probleme in der autonomen Szene? Gibt es gruppen\u00fcbergreifende Szenediskussionen?\n\nEigentlich steckt die Antwort schon in der Frage. Aus meiner Wahrnehmung heraus finden zu wenige konstruktive gruppen\u00fcbergreifende Szenediskussionen statt. Jeder kocht irgendwie so vor sich hin, seine eigene kleine Suppe. Das nervt tierisch und das sehe ich als ernsthaftes Problem. Vor allem in Berlin erscheint jedes Kollektiv, jede Gruppe immer super elit\u00e4r und auf sich bezogen. Da wird einfach zu wenig genetzwerkt und sich ge\u00f6ffnet gegen\u00fcber Neuen und Neuem. In der Kommunikation und Vernetzung muss die autonome Szene sich meiner Meinung nach besser aufstellen und mehr zusammenwachsen. Ich kann schlecht beurteilen, wie es fr\u00fcher mal war, aber habe schon von einigen geh\u00f6rt, dass auch das Internet viel Zusammenhalt gefressen hat. Du musst pl\u00f6tzlich keine Treffen und Plena und Lager mehr irgendwo im Wald organisieren, um dich auszutauschen und dich zu sehen. Du kannst es theoretisch jederzeit \u00fcber das Internet machen. Aber das ist nicht dasselbe. Und viele nutzen die Chance gar nicht. Bist du einmal richtig irgendwo in einem Kollektiv, zum Beispiel einem Hausprojekt in Berlin, hast du kaum noch die Notwendigkeit, mit Leuten aus anderen H\u00e4usern zusammenzuarbeiten.\n\n#### Fast alle Jugendkulturen sind m\u00e4nnlich dominiert. Gilt das auch f\u00fcr die autonome Szene?\n\nEs gibt viele Frauen in der autonomen Szene. Trotzdem ist auch sie m\u00e4nnlich dominiert. Das ist leider eine nat\u00fcrliche Konsequenz einer m\u00e4nnlich dominierten Welt. Das wirkt sich auch auf Subkulturen aus. Auch auf Autonome. Aber alleine durch Themen wie Antisexismus und Antispezismus ist die Szene insgesamt eher offen f\u00fcr \"Weibsvolk\". [Lacht.] Im Verh\u00e4ltnis kommen in meinem Bekanntenkreis auf jedes weibliche Wesen vielleicht drei m\u00e4nnliche, die ich zur autonomen Szene z\u00e4hlen kann.\n\n#### Wie bewertest du die \u00f6ffentliche Wahrnehmung der autonomen Szene?\n\nDie Medien freuen sich \u00fcber jedes brennende Auto und jedes Foto von 'nem Steinewerfer am 1. Mai. Das verkauft sich gut und pr\u00e4gt gleichzeitig auch die \u00f6ffentliche Wahrnehmung. Die Gewaltbereitschaft wird sehr h\u00e4ufig besprochen und gern diskutiert. Da geht es viel darum, Angst zu erzeugen, um Politik zu rechtfertigen, \u00dcberwachung zum Beispiel. Menschen, die sich aktiv in zivilgesellschaftlichen Prozessen beteiligen und direkte Kontakte zu Autonomen haben, haben halt auch die Chance, Vorbehalte abzubauen. Menschen, die zum Beispiel im direkten Umfeld von Hausprojekten leben, erz\u00e4hlen ja auch immer von positiven Erfahrungen mit Autonomen. Zumindest erinnere ich mich noch an die Liebig-14-R\u00e4umung, wo das der Fall war. Da w\u00e4ren wir wieder bei wichtigen Aufgaben der autonomen Szene f\u00fcr die n\u00e4chsten Jahre: Kommunikation, Verbindungen schaffen. Und der Isolation offen entgegenwirken.\n\n#### Geh\u00f6rt Gewalt nicht zur autonomen Szene dazu?\n\nNein, keinesfalls. Klar, je revolution\u00e4rer die Einstellung einer Gruppe, je radikaler ihr Ver\u00e4nderungswunsch, je wahrscheinlicher geh\u00f6rt Gewalt oder Militanz dazu. Wobei auch das jetzt recht unreflektiert ist. Grunds\u00e4tzlich geht es vielen ja wirklich \"nur\" um ein selbstbestimmtes Leben. Aber weil das so schwierig ist und h\u00e4ufig verwehrt bleibt, wird der Einsatz erh\u00f6ht. Das ist aber keine spezifische Eigenschaft der autonomen Szene, sondern so funktioniert ja leider unsere Welt. F\u00fcr mich ist es zum Beispiel logisch und konsequent, dass Leute, wenn ein Haus wie die Liebig ger\u00e4umt wird, auf die Stra\u00dfe gehen und auch Schaufenster einwerfen. Es ist leider sehr schwierig geworden, sich in so einer lauten Zeit ausreichend Geh\u00f6r zu verschaffen. Die oft angemahnte Verh\u00e4ltnism\u00e4\u00dfigkeit spielt in meinen Augen tats\u00e4chlich auch eine Rolle. Also Gegengewalt zum Schutz immer, aktiv gegen Nazis vorzugehen schlie\u00dft leider oft auch die Gewalt nicht aus, Gewalt gegen Sachen ist auch okay. Menschen ernsthaft oder gar vors\u00e4tzlich zu verletzen, stellt in meinen Augen schon eine Grenz\u00fcberschreitung dar. Und ich glaube, genau so legt sich jeder seine eigenen Werte und Grenzen fest.\n\n#### Wie sch\u00e4tzt du die gesamtgesellschaftliche Bedeutung der Autonomen ein? Welche M\u00f6glichkeiten hat die autonome Szene, etwas zu ver\u00e4ndern?\n\nGenau da scheiden sich momentan Realit\u00e4t und Utopie. Ganz klar muss es auch darum gehen, einen Einfluss zu nehmen und Ver\u00e4nderungen in der Gesellschaft mitzubestimmen. Die M\u00f6glichkeit besteht, weil jeder Mensch etwas \u00e4ndern kann und sich ja alles permanent im Wandel befindet. Zu sagen, ich kann doch nichts \u00e4ndern, ist einfach Quatsch. Klar ist der Anspruch, die Welt zu ver\u00e4ndern, und eine gerechtere Gesellschaft zu erschaffen, Utopie. Aber es geht ja gar nicht darum, das alles selber zu machen. Auch hier ist wieder das Gemeinsame gefragt. Also gemeinsam f\u00fcr eine Ver\u00e4nderung eintreten ist wichtig, und da kommt der autonomen Szene eine nicht zu untersch\u00e4tzende Bedeutung zu. Gerade in Verbindung zu anderen Bewegungen und in B\u00fcndnissen ist es sehr einfach, Einfluss zu nehmen, durch direkte Mitarbeit.\n\n#### Besteht denn \u00fcberhaupt das Interesse, die Gesellschaft als Ganzes zu \u00e4ndern, oder geht es nur darum, die eigenen Freir\u00e4ume zu bewahren?\n\nWenn es sich nur noch um die eigenen Freir\u00e4ume dreht, dann ist das Ziel linker Politik verfehlt. Freir\u00e4ume sollten als R\u00fcckzugs- und Schutzgebiet dienen und aus ihnen heraus m\u00fcssen die Ver\u00e4nderungen geschehen. Also Veranstaltungen geplant werden, Aktionen vorbereitet etc. Da ist jeder als Individuum verantwortlich, gar nicht mal abh\u00e4ngig von der Szene. Ich kann da nur pers\u00f6nlich f\u00fcr mein Interesse sprechen, und das ist ganz klar: einen Eindruck zu hinterlassen und viele Spuren in meinem Leben.\n\n#### Wie sch\u00e4tzt du die Zukunft der autonomen Bewegung ein?\n\nIch gehe davon aus, dass die Bewegung in den n\u00e4chsten Jahren weiter wachsen wird, aber vielleicht auch st\u00e4rker fluktuiert. Viele junge Menschen, die momentan in der autonomen Bewegung aktiv sind, erscheinen f\u00fcr mich spa\u00df-und erlebnisorientiert. Also da bleibt abzuwarten, ob sich da neue Strukturen entwickeln k\u00f6nnen. Ich kenne einige Beispiele, in denen die Leute erst voll dabei waren und nach zwei Jahren in ihrem Studium oder 'ner neuen Freundin interessantere Schaffensbereiche gefunden haben. [Lacht.] Nichtsdestotrotz bleibt der Einfluss erhalten und wird auch in die Zivilgesellschaft weitergetragen. Die Perspektive muss definitiv die Vernetzung sein und Einbindung von neuen Leuten. Eine gr\u00f6\u00dfere Herausforderung werden auch die staatlichen Repressionen. Aber ist grad schwierig, da eine Prognose abzugeben.\n\n#### Wie gro\u00df ist die autonome Szene in deiner Stadt, wie gro\u00df insgesamt in Deutschland? Gibt es da Ver\u00e4nderungen?\n\nIn Berlin ist sie ziemlich gro\u00df und umfasst sicherlich zwei- bis dreitausend Personen. In meiner Heimat vielleicht zwanzig. In ganz Deutschland vielleicht achttausend. Das wei\u00df der Verfassungsschutz aber sicher besser einzusch\u00e4tzen. [Lacht.] Ganz ehrlich: Ich wei\u00df es nicht genau, aber Tendenz steigend.\n\n#### Was w\u00fcrdest du selbst gern noch sagen? Gibt es etwas Wichtiges, was bisher nicht gefragt wurde?\n\nIch hoffe, dass die Mehrheit der Leute sich wieder mehr der Utopie widmen wird und aktiv an der Verwirklichung mitarbeitet. Und immer nur im eigenen Saft zu schmoren, tut auch niemandem gut. Also eine st\u00e4rkere Vernetzung und mehr Miteinander, Solidarit\u00e4t ist eine Waffe und so. Nicht nur die Aufn\u00e4her auf den Kapu n\u00e4hen, sondern auch aktiv daran arbeiten. \u00dcber s\u00e4mtliche Grenzen hinweg. Das ist so mein Wunsch.\n\n## Tierbefreierin\n\nW., 21 Jahre, stammt aus dem Saarland und lebt in Berlin.\n\n#### Erz\u00e4hl bitte erst mal etwas \u00fcber dich!\n\nIch bin 21 Jahre alt und mache zurzeit eine Ausbildung als Ergotherapeutin. Davor habe ich ein Studium angefangen, dass ich aber nach einem Semester abgebrochen habe. Ich studierte soziale Arbeit. Ich komme aus Saarbr\u00fccken. Meine Familie ist eher aus dem Mittelstand. Mein Vater hat beim Rundfunk gearbeitet. Meine Mutter arbeitet im B\u00fcro. Es ist also alles recht b\u00fcrgerlich.\n\n#### Hast du auch vor, Ergotherapeutin zu werden?\n\nEs ist eher ein Basic und dann mal gucken.\n\n#### Also hast du vielleicht noch vor zu studieren?\n\nStudieren ist eigentlich nicht so mein Ding, mal gucken, vielleicht irgendwann mal.\n\n#### Wie bewertest du deine aktuelle Situation? Und wie planst du dein Leben?\n\nMeine momentane Situation ist sehr festgefahren, weil ich die Ausbildung mache, und danach wei\u00df ich noch gar nicht, was kommen soll. Ich plane, ins Ausland zu gehen, um den Beginn des Arbeitslebens weiter nach hinten zu schieben. Nicht weil ich keine Lust hab, sondern einfach weil ich riesigen Schiss davor hab, und deswegen \u00fcberlege ich, was ich alles machen kann, damit ich nicht arbeiten muss. Ich w\u00fcrde mich gern selbstst\u00e4ndig machen, aber das erfordert ja sehr viel. Teilweise auch Dinge, von denen ich keine Ahnung hab. Also ich wei\u00df nicht, wie man sich selbstst\u00e4ndig macht. Ich habe auch Angst davor, irgendwo Angestellte zu sein, weil ich keine Lust habe, st\u00e4ndig einen auf den Deckel zu kriegen. Das ist eben genau das Bild, das man so vermittelt kriegt, auch in den Praktika, in denen ich bin. Also dieses Angestellten-Chef-Verh\u00e4ltnis, das hierarchische Prinzip bei der Arbeit entspricht nicht meinen Vorstellungen und davor hab ich Angst. Ansonsten hab ich eigentlich vor nicht so viel Angst. Ich mag Herausforderungen gerne, bis auf das mit der Arbeit. Ansonsten politisch gesehen denk ich \u00fcberhaupt nicht dar\u00fcber nach, weil das tats\u00e4chlich eher einen niedrigeren Stellenwert hat. Ich muss erst mal gucken, dass ich pers\u00f6nlich zurechtkomme, nur dann kann man auch Energie in solche Sachen stecken.\n\n#### Hast du einen Traumberuf?\n\nIch w\u00fcrde gern Reittherapie irgendwann mal anbieten, aber das wird leider nicht finanziert. Da m\u00fcsste ich mir irgendwelche reichen Leute rauspicken, die Lust haben, das zu bezahlen f\u00fcr ihre Kinder.\n\n#### Was macht dich w\u00fctend, was \u00e4rgert dich?\n\nFleischfresser. Auf jeden Fall Leute, die keine Lust haben zuzuh\u00f6ren, warum Veganismus eine sehr, sehr gute Alternative ist. W\u00fctend macht mich, dass wir keinen Mindestlohn in Deutschland haben. Und w\u00fctend machen mich M\u00e4nner, manchmal.\n\n#### Was w\u00fcrdest du anders machen als deine Eltern?\n\nIch w\u00fcrde nicht so fr\u00fch heiraten und Kinder kriegen, wobei der Zug ist eh schon abgefahren, weil ich jetzt schon \u00e4lter bin als meine Mutter damals. Und ich w\u00fcrde versuchen, offener zu sein f\u00fcr andere Lebensstile und nicht dieses festgefahrene.\n\n#### Haben deine Eltern, dein Umfeld in deiner Kindheit\/Jugend mit deiner politischen Entwicklung zu tun?\n\nNein, \u00fcberhaupt nicht. Also wenn du meinst, dass sie mich irgendwie dahin gebracht haben oder so, auf keinen Fall. Ich wollte eigentlich immer raus aus dem, wie es da ist, und habe nach Alternativen gesucht. Deswegen bin ich nach Berlin gekommen und dadurch da reingekommen. Aber sie haben mich nie mit Gedankengut gef\u00fcttert, was dazu gef\u00fchrt h\u00e4tte, dass ich ein kritisch denkender Mensch werde. Eher im Gegenteil.\n\n#### Was hei\u00dft im Gegenteil?\n\nNa ja, sie sind halt ein bisschen einfacher gestrickt, nicht so politisch, mittlerweile zwar schon, aber vor zehn Jahren noch gar nicht. Da waren andere Themen wichtiger, wie die Finanzierung, wie f\u00fcttern wir die Kinder durch oder so. Das kann ich aber auch verstehen. Man hat sich nicht ums Weltgeschehen gek\u00fcmmert oder um irgendwelche Missst\u00e4nde. Uns geht es gut \u2013 und das war das Ende des Horizonts.\n\n#### Warum bist du nach Berlin gekommen und wann?\n\nVor gut zwei Jahren. Ich wollte eigentlich in Koblenz studieren, um bei meiner Familie zu bleiben, aber dann gab es einen schweren Schicksalsschlag und meine Familie ist nicht damit zurechtgekommen oder ich hab mich einfach nicht gut aufgehoben gef\u00fchlt und dachte, ich muss weg, ich halte es dort nicht mehr aus. Und mein damaliger Freund wollte immer schon einmal nach Berlin. Er war voll der Punker. So haben wir unsere Sachen gepackt und uns eine Wohnung und einen Job besorgt und uns hier irgendwie eingenistet. Also ohne dar\u00fcber nachzudenken bin ich hierhergekommen.\n\n#### M\u00f6chtest du Kinder haben?\n\nIch glaube schon, ja. Aber ich finde es schwierig. Manchmal denke ich, es ist voll das Verbrechen, ein Kind in diese Welt zu setzen, und dann denke ich wieder, es wird schon gehen. F\u00fcr mich pers\u00f6nlich w\u00e4re es sch\u00f6n und erf\u00fcllend, aber ich habe nat\u00fcrlich auch Angst, dass ich dem Kind nicht das bieten kann, was es braucht. Ich bin ja noch jung, deshalb kann ich diese Entscheidung guten Gewissens noch herausschieben.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nIch glaube, man kann mich oft nicht nachvollziehen, weil ich oft hin und her springe, aber ansonsten wei\u00df ich auch nicht. Manchmal bin ich verschlossen, manchmal total offen. Ich bin auf jeden Fall engagiert und interessiert. Ich will immer wissen, warum die Dinge so sind, wie sie sind, und wenn ich es nicht rauskriegen kann, dann frustriert mich das.\n\n#### Welche Hobbys, Leidenschaften, Interessen hast du?\n\nIch hab gar keine Hobbys, eigentlich. Ich bin mal geritten, aber das ist auch \u00fcbelste Tierqu\u00e4lerei, hab ich irgendwann rausgefunden. Es gibt gewaltfreies Reiten, aber nicht wirklich hier, und au\u00dferdem, finde ich es nicht wirklich korrekt, sich auf ein Pferd zu setzen. Ich glaube, das kann man nur mit dem Kontakt zum Tier machen. Ansonsten lese ich, aber nur wenn ich eine Lesephase hab, das ist auch nicht so oft. Ich h\u00f6re viel Musik und treffe mich gern mit meinen Freunden und bin in der Tierbefreiungsgruppe. Aber das ist auch nicht wirklich ein Hobby, glaube ich.\n\n#### Welche Bedeutung hat Musik f\u00fcr dich? Welche Musik h\u00f6rst du?\n\nKeine Ahnung. Wenn man nach Hause kommt, macht man erst mal Musik an. Aber eine gro\u00dfe Bedeutung hat sie f\u00fcr mich nicht wirklich. Ich h\u00f6re eigentlich querbeet alles. Ich kenn mich \u00fcberhaupt nicht aus mit irgendwelchen Namen von Bands oder Musikrichtungen, weil es mich nie wirklich interessiert hat. Ich h\u00f6re Rock, aber das hei\u00dft ja auch schon wieder fast alles. Also ich h\u00f6re auf keinen Fall Punk. Ich war mal in der Punker-Szene, aber Punk-Musik finde ich richtig schei\u00dfe, weil die meisten strohdumme Texte haben. Oder solche Oi!-Bands, das geht gar nicht. Ich h\u00f6re alternative Musik. So Sachen, von denen keiner wirklich den Namen kennt.\n\n#### Hast du irgendwelche anderen Vorlieben, also Literatur, Film, Fernsehserien oder vielleicht im Bereich von Kunst?\n\nIch gucke gerne Filme und auch diverse Fernsehserien. Ich glaub, die sind ein bisschen peinlich. Ich gucke gerne King of Queens, Scrubs, Gilmore Girls.\n\n#### Interessierst du dich f\u00fcr Mode?\n\nIch interessiere mich nicht f\u00fcr die Mode auf den Laufstegen. Es ist schon interessant, aber auch total absurd, dieses Geld, was dahintersteckt. Ansonsten w\u00fcrde ich mich jetzt nicht anpassen den Leuten, mit denen ich zusammen rumh\u00e4nge, sondern ich trag das, was ich will.\n\n#### Nach welchen Kriterien suchst du Klamotten aus?\n\nSie m\u00fcssen mir gefallen. Also mein Kleiderschrank gibt alles her. Ich k\u00f6nnte auf eine Psychobilly-Party gehen und auf eine Gothic-Party, ich kann als Tussi rumlaufen, vielleicht jetzt nicht gerade mit pinken Haaren, genauso aber auch als Hippie, ich hab alles querbeet und zieh mich dann nach Tageslaune an.\n\n#### Spielt so etwas wie umweltgerechte Produktion oder menschengerechte Produktion eine Rolle? Wahrscheinlich kaufst du keine Ledersachen?\n\nJa richtig, ich hab noch welche, die ich halt auch noch trage, aber ich kaufe nichts Neues mehr aus Leder. Ich hab mich schon damit ein bisschen auseinandergesetzt, so von wegen Produktionsverh\u00e4ltnisse und so, aber es ist halt auch eine finanzielle Frage, weil oft sind diese Produkte sehr teuer und ich hab echt fast kein Geld und kaufe dann halt auch bei Zehmann oder Kik ein, obwohl ich wei\u00df, dass es eigentlich richtig mies ist, aber ich kann es mir halt nicht leisten. Ich kann eigentlich auch nicht zu H & M gehen. H & M ist eigentlich auch schei\u00dfe, aber ist ja eher noch billiger, aber da kann ich eigentlich nur selten einkaufen. Also ich mach's halt eher vom Geldbeutel abh\u00e4ngig, wo ich einkaufe.\n\n#### Wie verbringst du deine freie Zeit?\n\nIch lerne f\u00fcr die Schule, also das ist eine schulische Ausbildung, die ich mache. Ich muss Hausaufgaben machen und Vortr\u00e4ge vorbereiten und so. Ansonsten treffe ich Freunde oder zurzeit mache ich ein Archiv, also unsere Gruppe hat auch Zeitschriften, Zeitungsartikel und alles M\u00f6gliche zum Thema Tier, Tierbefreiung und so gesammelt und ich archiviere die jetzt. Und das nimmt schon Zeit in Anspruch.\n\n#### Wo macht ihr das Archiv?\n\nDas wei\u00df ich nicht, ob ich das sagen darf.\n\n#### Hast du Vorbilder, Leitbilder, Idole und so etwas?\n\nNein, eigentlich nicht, aus pers\u00f6nlicher und politischer Sicht habe ich einen Kumpel, den ich als Leitbild sehe, weil er so viel Ahnung hat, das imponiert mir halt. Ich w\u00e4re gern an so einem Punkt, wo ich wirklich Ahnung habe, was in der Szene so los ist, wie das mit den Strukturen ist und so und wie man was macht und organisiert... Ich habe das Gef\u00fchl, ich habe einfach nicht den Durchblick.\n\n#### Nach welchen Kriterien schlie\u00dft du Freundschaften?\n\nMan lernt jemanden kennen. Wenn ich mich verstanden f\u00fchle und das Gef\u00fchl habe, ich kann der Person was zur\u00fcckgeben, und das Ganze dann irgendwann verbindlich bleibt, dann entsteht eine Freundschaft. Also Kriterien wie Essverhalten oder so ber\u00fccksichtige ich nicht. Also richtig gute Freunde von mir sind auch omnivor [Allesfresser] und das kratzt mich dann nicht, es geht halt eher um die... Werte kann ich ja nicht sagen, weil sie eigentlich Werte vertreten, die ich nicht vertrete, aber es geht so um das Pers\u00f6nliche, ob man klarkommt und sich versteht. Es gibt ja mehr Themen als Essen, \u00fcber die man reden kann. Verl\u00e4sslichkeit ist auch wichtig.\n\n#### W\u00fcrdest du sagen, dass du einer Jugendkultur angeh\u00f6rst?\n\nEigentlich nicht. Ich bewege mich in einem gewissen Kreis und kenne gewisse Leute, die sich als Autonome bezeichnen, aber ich pers\u00f6nlich mache es einfach nicht. Ich glaube, daf\u00fcr habe ich nicht genug Selbstbewusstsein, mich hinzustellen und zu sagen, ich geh\u00f6re dazu. Ich f\u00fchle mich eher wie jemand, der so mitgezogen wird und auch seinen Teil beitr\u00e4gt und Energie reinsteckt, aber ich bin mir sicher, dass ich irgendwann mal wieder rausgehe, und deswegen w\u00fcrde ich diese Frage mit nein beantworten.\n\n#### Gibt es irgendwelche Szenen, die du sympathisch findest?\n\nDie Tierbefreiungsbewegung. Ansonsten die Punker. Ich bin schon als kleines Kind immer als Punker zum Fasching gegangen. Ich habe meine Mutter damit auf die Palme gebracht, aber ich fand es immer schon toll. Nur als ich dann drin war, fand ich es nicht mehr so toll.\n\n#### Und unsympathisch?\n\nDie Autonomen Nationalisten oder \u00fcberhaupt alle Rechten. Wobei das sich jetzt nicht nur auf Jugend bezieht, aber jugendlich und rechts geht gar nicht. Und ansonsten, manche sind skurril, aber so lange sie politisch korrekt sind, ist das cool.\n\n#### Und mit welchen Leuten kommst du \u00fcberhaupt nicht klar?\n\nIch komm nicht klar mit diesen Machotypen, also M\u00e4nnern, die frauenfeindlich sind, aber genauso auch diese \u00fcberheblichen Tussis, die voll raush\u00e4ngen lassen, dass sie Geld haben, oder sich lustig machen \u00fcber einen. Aber das ist unabh\u00e4ngig vom Geld, alle Menschen, die sich \u00fcber andere lustig machen nur, weil sie anders sind, gehen mir gar nicht ab. Ich habe schon eine gewisse Ablehnung gegen\u00fcber reichen Leuten. Es ist eigentlich Quatsch so pauschal, aber man wei\u00df ja immer nicht, wo der Reichtum herkommt... Aber so Leute, die diese V\u00f6llerei... die in die XXL-Restaurants gehen oder sonst wo leben und morgens schon Prosecco trinken und meinen, es sei ganz toll, und dann noch im Fernseher auftreten und Werbung daf\u00fcr machen, wie geil ihr Leben ist und dass es f\u00fcr sie total normal ist, an jedem Tag auf irgendwelchen Modeschauen rumzuh\u00e4ngen, die kotzen mich echt richtig an. Also ich habe keinen Fernseher mehr, aber ich habe echt viele Jahre ferngesehen und bei Vox und so gibt es ja diese Serien, wo so Leute ihr Leben vorstellen. Ganz, ganz schrecklich finde ich auch Familien oder Eltern, die sich nicht gen\u00fcgend um ihre Kinder k\u00fcmmern. Wir reden gerade in Soziologie \u00fcber solche Sachen, \u00fcber Schichten und so Hartz-IV-Leute und Kinder, die in die Arche zum Essen gehen, so etwas finde ich halt richtig schei\u00dfe, wenn die Eltern, nur vorm Fernseher rumhocken und nicht den Arsch hoch kriegen. Also ich kann verstehen, wenn es irgendwelche psychischen Probleme gibt, aber dann soll man sich Hilfe holen. Also das finde ich richtig ein No-Go.\n\n#### Und so direkt im Umfeld, in der Szene, gibt es da Leute, die gar nicht gehen?\n\nDie, die total krass drauf sind, die diesen Veganismus zu krass vertreten. Ein Beispiel: Da ist eine, die hat eine Zeit lang mit niemandem gesprochen, der nicht vegan war. Leute, die mich nerven, sind so Antifa-Leute, und da gibt es ganz viele dabei, die nicht wirklich Ahnung haben, aber das Maul weit aufrei\u00dfen. Und Hauptsache Stress machen und auch Demos f\u00fcr gute Sachen immer missbrauchen und auf die Polizei schimpfen, Bullen und so. Auf jeden Fall gibt es viele Gr\u00fcnde daf\u00fcr, aber wenn ich eine Demo gegen sonst was mache, dann kann ich das Thema rauslassen. Dann sollen sie eine Anti-Bullen-Demo machen und da k\u00f6nnen sie so viele Flaschen schmei\u00dfen wie sie wollen. Und dann treffen sie ja auch oft eigene Leute, was mich auch immer sehr sauer macht. Der schwarze Block kann gerne raus, manchmal, wenn sie es \u00fcbertreiben.\n\n#### Welche Rolle spielt die autonome Szene oder die linke Szene f\u00fcr dich?\n\nF\u00fcr mich pers\u00f6nlich verbinde ich damit Leute, die zumindest bestrebt sind, irgendwie ein aufgekl\u00e4rteres Leben zu f\u00fchren. Oder eins, wo man mehr aufeinander achtet. Wo man so den sozialen Gedanken verst\u00e4rkt hat, dass man Sachen teilt und f\u00fcr alle zug\u00e4nglich macht. Das geh\u00f6rt f\u00fcr mich dazu. Aber auch Aufmerksamkeit der \u00d6ffentlichkeit zu erregen. Das ist so ein bisschen das, was wir ja machen, unsere Gruppe. Die \u00d6ffentlichkeit aufmerksam machen auf Missst\u00e4nde und zeigen, was halt alles schief l\u00e4uft und warum manche Dinge, die in Deutschland ablaufen, einfach falsch sind, so wie sie sind, und dass das anderen schadet.\n\n#### Woher beziehst du Informationen zum aktuellen Zeitgeschehen? Und inwiefern interessiert es dich \u00fcberhaupt?\n\nAlso interessieren tut es mich stark. Oft verstehe ich, wenn es um irgendwelche globalen Sachen geht, die Zusammenh\u00e4nge einfach nicht. Also ich lese die taz und frage meine Freunde. Wenn ich irgendetwas nicht verstehe, dann frage ich die Leute um mich rum, von denen ich wei\u00df, die haben Ahnung. Weil, man kriegt irgendwo eine Info her, und dann sagt jemand anders, ja, aber eigentlich ist es ganz anders... Wei\u00dft du, was ich meine? Ich hab immer das Gef\u00fchl, okay, ich hab jetzt von irgendwo eine Info, aber das ist nicht die korrekte Insider-Info. Ich bin mir dann nicht sicher, ob ich die richtige Meinung zu einem Thema hab, weil ich nicht wei\u00df, ob ich alle Informationen zu dem Thema hab. Aber wenn es um spezielle Themen geht, wie die Befreiung oder so, da gibt es ja szeneinterne Sachen und die lese ich halt auch.\n\n#### Und andere eher linke Zeitungen neben der taz, wie die Jungle World oder so?\n\nNein, davon habe ich noch nicht mal geh\u00f6rt.\n\n#### Die haben sich mal abgespalten von der jungen welt.\n\nAch ja, das kenne ich.\n\n#### Die Jungle World ist schon eine sehr gute Zeitung, die zwar auch immer tendenziell so gem\u00e4\u00dfigt antideutsch ausgerichtet ist \u2013 kennst du das: die Antideutschen, Antiimperialisten? Dieser Konflikt?\n\nJa, da reden Leute in meinem Umfeld viel dr\u00fcber und ich hab es mir schon erkl\u00e4ren lassen, aber ich habe es nicht wirklich gerafft. Ganz ehrlich. Hab es einfach nicht gerafft.\n\n#### Wie w\u00fcrdest du autonom \u00fcberhaupt definieren?\n\nIch habe ehrlich gesagt nicht wirklich die Ahnung, wie ich das definieren w\u00fcrde. Nein, keine Ahnung.\n\n#### Wie w\u00fcrdest du die autonome Szene beschreiben? Falls irgendjemand keine Ahnung davon hat, was w\u00fcrdest du ihm erz\u00e4hlen?\n\nDass es eher so ein Oberbegriff ist f\u00fcr deutsche Gruppen, die sich als links verstehen. Es sind Leute, die sich f\u00fcr politische Freir\u00e4ume einsetzen. R\u00e4ume f\u00fcr Gruppen, wo man sich treffen kann, f\u00fcr au\u00dferparlamentarische Gruppen.\n\n#### Was macht ihr in eurer Gruppe?\n\nWir produzieren Flyer, wo \u00fcber das Thema Veganismus aufgekl\u00e4rt wird. Wir machen einen Brunch einmal im Monat, wo Leute vegan essen k\u00f6nnen und dann auch eben Flyer verteilt werden. Wir vernetzen uns auch mit anderen Gruppen, die dieselben Themen haben, um irgendwelche Kampagnen zu starten, wie zum Beispiel \"Gegen Pelz\". Es werden auch Demonstrationen organisiert, angemeldet und durchgef\u00fchrt.\n\n#### Wie war dein erster Kontakt zur autonomen Szene?\n\nAuf Demos. In Saarbr\u00fccken trifft sich ja immer alles, was irgendwie alternativ ist, ob Punk oder politisch, ist alles ein Haufen. Und da war einer, der war in der Antifa, und den habe ich gekannt. Aber ich habe mich nie getraut, ihn zu fragen, was er eigentlich macht, weil alle sagten, er w\u00e4re bescheuert und so. Das war alles so Kleinkinderkacke und deswegen ist es nie zum Gespr\u00e4ch gekommen. Bei uns galt immer: Antifa = doof. Irgendwann hatte ich der Antifa mal eine E-Mail geschrieben, weil ich mir \u00fcberlegt hatte, es w\u00e4re doch cool, da mitzumachen, weil die scheinen ja politisch irgendwie was zu machen, und die haben mir zur\u00fcckgeschrieben, dass ich einfach vorbeikommen soll, weil in der E-Mail wollten sie mir nat\u00fcrlich nicht erkl\u00e4ren, was sie machen. Ich bin dann aber nie hin, weil ich mich einfach nicht getraut hab, weil ich niemanden gekannt habe.\n\n#### W\u00fcrdest du die Punk-Szene in Saarbr\u00fccken als linke Szene bezeichnen?\n\nSie bezeichnen sich sicherlich als links, rechts sind sie auch nicht, aber ich w\u00fcrde sie als unpolitisch bezeichnen, weil sie sich gar keine politischen Gedanken machen. Die kaufen sich wirklich alle ihre Klamotten bei Nixgut [ein Punkversand] und schimpfen dar\u00fcber, dass in der heutigen Gesellschaft alle gleich aussehen blabla und sitzen den ganzen Tag auf der Saarwiese und saufen. Also die sind einfach gar nichts au\u00dfer hohl in der Birne.\n\n#### Hast du noch mit anderen Gruppen zu tun, die sich politisch engagieren, au\u00dfer Tierschutzgruppen?\n\nWir sind keine Tierschutzgruppe!!!\n\n#### Entschuldigung. Was ist denn der Unterschied zwischen Tiersch\u00fctzern und euch?\n\nDie Tiersch\u00fctzer sind f\u00fcr gr\u00f6\u00dfere K\u00e4fige und weniger schlimme T\u00f6tungsverfahren; wir sind daf\u00fcr, dass die Gesellschaft aufh\u00f6rt zu denken, dass Tiere dazu da sind, um sie zu benutzen. Tiere sind kein Gut, keine Ware. Wir wollen, dass es gar keine Tiere mehr in K\u00e4figen gibt.\n\n#### Seid ihr auch eine Art Jugendkultur?\n\nSchon, weil in letzter Zeit ist es auch in, vegan zu sein. Es gibt auch noch die sogenannte Fleischlinke, das sind alle Leute, die links sind, aber nicht vegan, sondern auch Fleisch essen. Aber viele junge Leute, die in irgendwelchen politischen Gruppen sind, sto\u00dfen jetzt an das Thema Vegan, und um dem Radikalismus noch eins draufzusetzen, werden viele selbst vegan. Nicht alle sind in der Tierbefreiung aktiv, sondern auch in der Antifa oder sonst was, aber sind eben auch vegan.\n\n#### Gibt es viele Auseinandersetzungen zwischen der Fleischlinken und den Veganen?\n\nJa, es gibt schon Querelen. Aber in unserem Bereich gibt es mehr Zoff zwischen den linken Tierbefreiern und den rechten Tierbefreiern, weil wir mit denen nat\u00fcrlich nicht zusammenarbeiten wollen.\n\n#### Was sind denn rechte Tierbefreier?\n\nDas sind Leute, die Tiere befreien, aber sich politisch rechts positionieren. Die stellen auch andere Vergleiche an und beziehen sich immer wieder positiv auf Hitler.\n\n#### Das sind dann die, die von \"H\u00fchner-KZs\" sprechen?\n\nNee, das ist ja PETA. \"Der Holocaust auf deinem Teller\", das ist PETA.\n\n#### Wie eng ist die Verbindung zu anderen Gruppen der autonomen Szene? Gibt es da klare Grenzen oder ist das eher ein flie\u00dfender \u00dcbergang?\n\nWenn wir zum Beispiel Geld sammeln bei diesem Brunch, dann ist das nicht nur f\u00fcr uns, sondern auch f\u00fcr andere linke Projekte, also da sind schon \u00dcberschneidungen. Wir sind nicht irgendwie isoliert, sondern versuchen, uns da so zu integrieren, damit das wirklich ein komplettes linkes Thema wird. Noch vor ein paar Jahren waren die Tierleute eher isoliert, aber das hat sich schon etwas ge\u00e4ndert. Aber ich bin erst seit einem Jahr in diesen Strukturen, und es ist nicht so viel Zeit, um durchzublicken. Das f\u00e4ngt ja alles langsam an: Man lernt Leute kennen, geht auf das Plenum, dann wird irgendwann geguckt, ist die Person cool, wollen wir die haben, ja, und dann muss man erst mal reinkommen in die Gruppe, also das Pers\u00f6nliche spielt da voll die Rolle. Und dann lernt man so viele neue Dinge kennen, von denen man keine Ahnung hatte, da ist ein Jahr ist nicht viel.\n\n#### Gibt es irgendwelche anderen Sachen, die f\u00fcr die Gruppe wichtig sind? Abseits der Politik?\n\nDie Form, wie man eine Beziehung f\u00fchrt, ist ein Thema. Man trifft viel seltener dort diese stinknormalen Beziehungsmuster. Da wird viel mehr Wert gelegt auf das Individuelle. Wer will was, jeder soll irgendwie berechtigt sein, seine Interessen zu verfolgen. So etwas wie Polygamie oder Polyamour\u00f6sit\u00e4t ist da eher ein Thema als in anderen Szenen oder Wirkungsschichten.\n\n#### Wirklich?\n\nAuf jeden Fall. Wenn wir gerade in der Schule in Soziologie \u00fcber Familienmodelle reden, dann kriegen die anderen, die ich eher so als normal oder unpolitisch bezeichnen w\u00fcrde, ganz gro\u00dfe Augen, wenn sie h\u00f6ren, dass es noch was anderes gibt als Vater, Mutter, Kind. Das ist f\u00fcr sie unvorstellbar, dass es irgendwie anders laufen k\u00f6nnte. Allein Fernbeziehung ist f\u00fcr die ein rotes Tuch: \"Was, es gibt Paare, die sehen sich mal zwei Wochen nicht?! Warum habe ich dann einen Partner?\" Ich bin erst in der linken Szene damit konfrontiert worden, dass man sich schon mal Gedanken macht und alternative Beziehungsstile f\u00fchrt. Klar, modernere Modelle von Familien gibt es \u00fcberall, aber gerade so was wie Polyamour\u00f6sit\u00e4t... man besch\u00e4ftigt sich mehr damit und mehr Leute in dieser Szene trauen sich das auszusprechen: Ich m\u00f6chte das aber so, ich will mich nicht einschr\u00e4nken. Weil man da ja immer sehr viel auf Akzeptanz und Toleranz pocht.\n\n#### Sind Geschlechterrollen nicht so sehr festgeschrieben?\n\nNein. Und das wird auch sehr stark akzeptiert. Da wird man ge\u00e4chtet, wenn man mal ein b\u00f6ses Wort fallen l\u00e4sst \u00fcber jemanden, der transsexuell ist oder so.\n\n#### In den meisten Jugendkulturen ist es ja so, dass die M\u00e4nner schon lauter sind. Jugendkulturen sind ganz oft m\u00e4nnlich gepr\u00e4gt und orientieren sich stark an m\u00e4nnlichen Werten usw. Und da haben es Frauen sehr schwer, da sie sich oftmals nur an diesen m\u00e4nnlichen Werten messen m\u00fcssen.\n\nIch glaub, das ist in der Antifa stark so. Bei uns ist es so, dass immer darauf geachtet wird, dass Frauen- und M\u00e4nneranteil zumindest ausgeglichen ist. Sollte ein Anteil mal geringer sein, gucken wir, woran kann das liegen, wie k\u00f6nnen wir das \u00e4ndern. Und wenn wir eine Diskussion haben, wird immer darauf geachtet, wer den gr\u00f6\u00dften Redeanteil hat, und dann wird die Person darauf aufmerksam gemacht nach dem Motto, halt dich mal etwas zur\u00fcck.\n\n#### Wie stehst du zu B\u00fcndnissen mit z. B. Tierschutzorganisationen oder PETA? Das kommt jetzt gar nicht in Frage, oder?\n\nNein, \u00fcberhaupt nicht. Also f\u00fcr mich pers\u00f6nlich nicht. Wenn man wirklich dieselben \u00dcberzeugungen hat und dieselben Methoden, dann ist es nat\u00fcrlich immer gut, sich zu vernetzen. Weil man sich gegenseitig unterst\u00fctzen kann und auch mit Infos versorgen kann.\n\n#### Gehst du w\u00e4hlen?\n\nIch hatte erst einmal die Gelegenheit zu w\u00e4hlen, und da habe ich es gemacht. Weil das nun mal das politische System ist, das wir haben, und wenn ich wenigstens dadurch die Chance habe, irgendwie mitzuwirken, Einfluss darauf zu nehmen, dann sollte ich die ergreifen. Es gibt ja Gegenpositionen dazu, und das kann ich auch verstehen, aber ich will mein Recht zu w\u00e4hlen in Anspruch nehmen. Allein schon, und das klingt jetzt vielleicht total naiv, aber allein schon weil ich eine Frau bin. Ich finde, dem sollte man irgendwie Tribut zeigen, dass andere Frauen sich daf\u00fcr eingesetzt haben, dass ich w\u00e4hlen gehen darf. Von mir aus kann man auch irgendeine Spa\u00dfpartei w\u00e4hlen. Einmal war das so, wo kaum einer w\u00e4hlen gegangen ist, und dadurch hat dann die NPD viel mehr Prozente erworben. Allein um dem entgegenzusteuern, muss man schon w\u00e4hlen gehen, irgendeine coolere Partei.\n\n#### Wie ist dein Verh\u00e4ltnis zum Staat, zur Polizei usw.?\n\nEs ist ja schon eine gute Sache, dass es so etwas gibt wie die Polizei, eine Instanz, die eingreift in Sachen wie Verbrechen, Mord, Diebstahl oder so. Es ist gut, dass es eine Instanz gibt, die sich darum k\u00fcmmert. Soweit ich wei\u00df, gibt es im anarchistischen Staat keine Polizei. Ich glaube nicht, dass man dann alles l\u00f6sen k\u00f6nnte, was da auf einen zukommt an Konflikten. Aber wie die Polizei bzw. manche Polizisten sich so verhalten, ist schon sehr kritisch zu sehen. Und dass sie Gewalt einsetzen k\u00f6nnen, ohne daf\u00fcr bestraft zu werden, was schon oft passiert ist. Also ich w\u00fcsste keine L\u00f6sung. Man kann die Polizei nicht abschaffen. Aber es gibt auf jeden Fall kritische Punkte bei der Polizei.\n\n#### Und wie siehst du den deutschen Staat?\n\nDa habe ich einfach keinen Bezug zu. Also ich frag mich, was soll das sein? Wenn das die Politiker sind, die sind sowieso alle nur am Schei\u00dfe labern. Mein Bild ist, dass die da ihr eigenes gro\u00dfes Spiel treiben und gar nicht auf den Einzelnen gucken, der von ihrer Politik betroffen ist. Ich glaube, die leben in einer ganz anderen Welt. Die kriegen das gar nicht mehr mit, wie der B\u00fcrger lebt und was das hei\u00dft, mit nur 300 \u20ac im Monat auskommen zu m\u00fcssen. Das k\u00f6nnen die sich doch gar nicht vorstellen. Die haben da eine gro\u00dfe Spielwiese und machen, was sie wollen, aber das ist nicht wirklich Politik f\u00fcr den Menschen, der hier lebt.\n\n#### Wie w\u00fcrdest du die \u00f6ffentliche Wahrnehmung eurer Arbeit einsch\u00e4tzen?\n\nWenn man das Wort Tierbefreiung erw\u00e4hnt, da denken die Leute, man w\u00fcrde irgendwie illegale Sachen machen wie Tiere aus K\u00e4figen rausklauen. Dies ist aber eine irrt\u00fcmliche Annahme, denn diese Tiere m\u00fcssen ja irgendwo hin. Wir w\u00fcrden niemals so etwas machen, erstens, weil es illegal ist, und wie gesagt diese Tiere irgendwo hin m\u00fcssen. Wenn es nicht die M\u00f6glichkeit gibt, sie entsprechend zu versorgen, warum sollte man das machen? Wenn man auf einer H\u00fchnerfarm alle H\u00fchner befreien w\u00fcrde, w\u00fcrden die verrecken, und das ist nun mal nicht unsere Intention. Das \u00f6ffentliche Bild ist wahrscheinlich eher die militanten Tierbefreier, die die Tiere aus den K\u00e4figen klauen. Das finde ich sehr schade.\n\n#### Wie stehst du zu militanten Aktionen?\n\nDa bin ich dagegen. Ich find es auch schwierig, \u00fcber \"die Autonomen\" zu reden, aber ich muss leider gestehen, dass auf Demonstrationen auch Flaschen fliegen, die wirklich nicht fliegen m\u00fcssten, dass ich das schon als hohe Gewaltbereitschaft sehe. F\u00fcr mich ist das absolut unvorstellbar, so etwas zu machen. Gerade auch, weil man die eigenen Leute treffen kann. Ich habe schon sehr oft Kommentare geh\u00f6rt wie: \"Geil, Demo, gehen wir uns mal mit Bullen schlagen.\" Wobei man da unterscheiden muss, es ist oft auch nur ein Machogehabe. Es kommt auch eher von M\u00e4nnern, von Frauen weniger, aber auch. Aber wie viel Ernst steckt dahinter? Ich glaube, in den meisten F\u00e4llen ist es nur Gerede, weil man sich dadurch auch profiliert. Man ist ja nicht nur politisch, man ist ja auch Mensch. Da gibt es Gruppen, Prozesse, wo man sich in den Vordergrund dr\u00e4ngen will. Man will irgendetwas beweisen. Ich glaube, dass es halt eher an solchen Dingen liegt.\n\n#### Wie gro\u00df ist die Tierbefreier-Szene in Berlin im Vergleich zu anderen Teilen der linken Szene?\n\nDie Tierbefreiungsleute sind ein kleiner Teil in Berlin. Man kennt sich. Ich glaube, das sagt es schon aus. Es gibt verschiedene Gruppen in Berlin und man kennt sich einfach. Es ist einfach nicht gro\u00df. Man sieht es auch an der Gr\u00f6\u00dfe der Demonstrationen, also bei einer Demonstration, die was mit Tierrechten oder Tierbefreiern zu tun hat, da ist man bei einer Zahl von 500 schon recht stolz auf die Demo. Andere linke Sachen sind einfach gr\u00f6\u00dfer. Aber in Berlin ist schon tierbefreiungsm\u00e4\u00dfig recht viel los. Auch in Hamburg. Aber in St\u00e4dten wie Saarbr\u00fccken oder so, da gibt es den Tierversuchsgegner, aber das war es auch schon.\n\nDas Interview f\u00fchrte Daniel Schneider.\n\n## \"Privat kann man die Musik h\u00f6ren, die man will, es sei denn, sie ist frauenfeindlich, homophob, antisemitisch oder rassistisch.\"\n\nM., 18 Jahre, lebt in Niedersachsen.\n\nEmm, mittelgro\u00df, halblange Haare, macht einen gelassenen Eindruck, schaut eher herausfordernd, wirkt aber charmant und l\u00e4chelt gerne. Wenn ich ein Phantombild\/Klischeebild eines Autonomen entwickeln sollte, w\u00e4re Emm ganz bestimmt nicht darauf wiederzuerkennen. Wir treffen uns in der Wohnung seiner Mutter und setzen uns in die K\u00fcche. Es ist eine sehr kleine, aber gem\u00fctliche K\u00fcche. Emm wohnt in einer Kleinstadt mit gut 30.000 EinwohnerInnen. Ich sehe die Stadt zum ersten Mal, sie ist nicht wirklich sch\u00f6n und einladend. Aber Emm wohnt hier gerne.\n\n#### Welche Ausbildung haben deine Eltern und was machen sie beruflich?\n\nEinzelhandelskaufmann und keine. Sie arbeiten beide in einem Supermarkt.\n\n#### Was ist deine gegenw\u00e4rtige Situation?\n\nIch mach zurzeit eine berufsvorbereitende Bildungsma\u00dfnahme, bin in einer Elektrowerkstatt t\u00e4tig und erlerne die Grundlagen der Elektrotechnik. Nebenher, um halt meine Miete zusammenzubekommen, arbeite ich auch in einem Supermarkt an der Kasse.\n\n#### Wie bewertest du denn deine aktuelle Situation? Zufriedenstellend oder eher ausbauf\u00e4hig?\n\nDoch, ich finde schon zufriedenstellend. Ich habe jetzt meine eigene Wohnung, steh auf eigenen F\u00fc\u00dfen. Klar, ich arbeite sehr viel, aber wer auf eigenen Beinen stehen will, muss auch daf\u00fcr arbeiten, finde ich. Geht halt nicht anders. Mir gef\u00e4llt's und ich bin gl\u00fccklich so.\n\n#### Das ist sch\u00f6n. Wie siehst du denn deine Zukunft?\n\nDa mache ich mir nicht so sehr Gedanken drum. Ich lebe eigentlich mehr in den Tag hinein.\n\n#### Hast du bestimmte Tr\u00e4ume, die du dir erf\u00fcllen m\u00f6chtest?\n\nNein, ich nehme es, wie es kommt. Allerdings pass ich auch auf, dass es so kommt, wie ich es nehmen m\u00f6chte.\n\n#### Hast du einen bestimmten Traumberuf?\n\nJa, ich w\u00fcrde gerne in der IT-Branche arbeiten, am liebsten als Programmierer, aber auch auf der technischen Seite. Bisher haben mir jedoch die Firmen einen Strich durch die Rechnung gemacht wegen meines schlechten Abschlusses. Ich habe einen Realschulabschluss von 3,0.\n\n#### Aber das ist doch gar nicht schlecht!\n\nJa, nein, es gibt halt Interessierte in der Branche, die einen viel besseren Abschluss hingelegt haben. Dass ich jedoch schon selbst Programme schreibe, wird meist gar nicht beachtet.\n\n#### Gibt es etwas, wovor du Angst hast?\n\n[Atmet langsam tief ein und denkt nach.] Das ist eine ungew\u00f6hnliche Frage. Wei\u00df ich gar nicht, wie ich darauf antworten soll. Konkret f\u00e4llt mir da nichts ein.\n\n#### Okay, auch keine Zukunfts\u00e4ngste?\n\nNein.\n\n#### Das ist sch\u00f6n. Was \u00e4rgert dich denn?\n\nOh, da gibt es so viel, aber das alles aufzuz\u00e4hlen, br\u00e4uchte den gesamten Tag.\n\n#### Na gut, ich formuliere mal um, wenn du von deiner pers\u00f6nlichen Situation ausgehst, was macht dich w\u00fctend?\n\nEs gibt immer wieder Leute, zum Beispiel auch in meiner Werkstatt, die bringen irgendwie scherzhaft beil\u00e4ufig so menschenverachtende Sachen, wo ich dann am liebsten sagen w\u00fcrde: \"Ey Leute, Schluss jetzt, so nicht!\" Manchmal m\u00f6chte ich ihnen eine auf den Kopf geben, wobei ich mir wieder denke, ich muss mit ihnen auskommen, und versuche deshalb, es ihnen vern\u00fcnftig zu erkl\u00e4ren. Nur es geht einfach nicht in ihren Kopf. Die wollen es nicht verstehen. Und genau das macht mich w\u00fctend.\n\n#### Verstehe ich gut, du musst dich halt arrangieren.\n\nJa, ich muss mit den Leuten arbeiten, verbringe also viel Zeit mit ihnen, muss mit ihnen klarkommen und versuche deshalb, meinen Standpunkt zu erkl\u00e4ren, warum mich das sauer macht. Aber es wird halt immer schlimmer, denn dann macht es mich sauer, dass sie es einfach nicht verstehen. Erst sagen sie: Ja ja, du hast ja Recht, und im n\u00e4chsten Satz kommt schon wieder so ein Ding. Sie raffen es einfach nicht!\n\n#### Geht es dabei um Intoleranz im Allgemeinen?\n\nEs sind teilweise ausl\u00e4nderfeindliche, menschenverachtende, eher sexistische Sachen, was ja mittlerweile leider \u00fcberall vorkommt in der Jugendsprache, und wenn man ihnen es dann erkl\u00e4rt, bringt es einfach nichts. Wenn man gegen eine Wand redet, f\u00fchlt sich das wohl genauso an. Aber es sind gr\u00f6\u00dftenteils nette Leute. Ich muss es halt immer wieder sagen, ihnen immer wieder meinen Standpunkt erkl\u00e4ren.\n\n#### Was w\u00fcrdest du anders machen als deine Eltern?\n\n[Z\u00f6gert, trinkt einen Schluck Kaffee und \u00fcberlegt, schweigt circa eine Minute, ich m\u00f6chte ihn nicht unter Druck setzen und schweige ebenfalls.] Ich muss ganz ehrlich sagen, dass ich an meiner Mutter wirklich nichts bem\u00e4ngeln kann. Ich habe ein super Verh\u00e4ltnis zu ihr und es ist fast schon wie eine sehr gute Freundin, wir k\u00f6nnen \u00fcber alles reden, k\u00f6nnen etwas miteinander unternehmen. Wenn ich dann allerdings die Beziehung zu meinem Vater betrachte, zu ihm hatte ich zwei oder drei Jahre \u00fcberhaupt keinen Kontakt oder immer nur sporadisch, oder so super Rei\u00dfer, dass dann zu Weihnachten die Ostersachen geschickt wurden, also sprich Schoko-Osterhase etc., oder drei Wochen nach dem Geburtstag ruft er an und br\u00fcllt in den H\u00f6rer: \"Hey! Herzlichen Gl\u00fcckwunsch!!!\" Das deprimiert mich. Also wenn ich irgendwann Kinder haben sollte, dann werde ich auf jeden Fall versuchen, das zu vermeiden, was mein Vater da veranstaltet hat. Wenn ich mit der Frau, also der Mutter der Kinder nicht mehr klarkommen sollte, so werde ich doch versuchen, mit den Kindern weiter Kontakt zu haben. Ich m\u00f6chte es nicht so machen wie mein Vater. Gegen meinen Vater hege ich einen ziemlich gro\u00dfen Groll, ich spreche ihn auch nur noch mit seinem Vornamen an, in der Hoffnung, dass er etwas merkt, aber er tat es nicht.\n\n#### Du w\u00fcrdest also Kinder haben wollen?\n\nJa, schon. Das klingt jetzt vielleicht klischeehaft, aber es m\u00fcsste schon mit der Richtigen sein. Ich m\u00fcsste mir schon vorstellen k\u00f6nnen, dass ich mit ihr zusammenleben will. Und wenn, dann erst in weiter, weiter Zukunft!\n\n#### Also f\u00fchlst du dich auch der Verantwortung gewachsen?\n\nJa klar f\u00fchl ich mich dem gewachsen! Also wenn es jetzt passieren w\u00fcrde, m\u00fcsste ich halt zusehen, wie ich damit klarkomme. Aber ich setze mich schon jetzt damit auseinander und habe f\u00fcr mich befunden, dass ich mich dem gewachsen f\u00fchle.\n\n#### Haben die Beziehungen zu deinen Eltern eine Rolle gespielt bei deiner politischen Entwicklung?\n\nNein, eigentlich nicht. Ich bin in die Szene gekommen, als ich von meiner Mutter weggezogen bin. Da hatte sie \u00fcberhaupt keinen Einfluss drauf.\n\n#### Wie alt warst du, als du von deiner Mutter weggezogen bist?\n\nIch war in der achten Klasse und hatte keinen Bock mehr auf die Lehrer und verklemmten Bewohner meines kleinen norddeutschen \u00d6rtchens. Meine Lehrer haben mir ein katastrophales Abschlusszeugnis vorhergesagt und meinten noch zu meiner Mutter: \"Er kann sich anstrengen, wie er will, das wird nicht besser.\" Damit war f\u00fcr mich klar, dass ich da weg muss, also habe ich mich selbst in eine psychiatrische Klinik eingewiesen. Ich habe zwar nur das Wochenende dort verbracht, aber danach hatte ich die M\u00f6glichkeit, in einem betreuten Wohnprojekt unterzukommen. Dort traf ich einige meiner jetzigen Freunde zum ersten Mal, sie pr\u00e4gten mich wohl politisch. Bis dahin hatte ich mich \u00fcberhaupt nicht f\u00fcr Politik oder dergleichen interessiert. Hab mich immer nur gewundert, was die f\u00fcr einen Schei\u00df labern. Im Erz\u00e4hlen und dann Diskutieren bin ich wohl entflammt, wenn du mir das Wortspiel gestattest.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nDas ist fast so wie die Frage: Wer bist'n du? Wenn du zu einer dir fremden Gruppe kommst und dich zu ihnen stellst, kommt zuerst die absch\u00e4tzende Frage: \"Ja sag mal, wer bist du denn?\" So eine Frage kann ich nicht beantworten. Jeder sollt sich selbst von mir ein Bild machen. Ich bin zufrieden mit mir und das reicht!\n\n#### Was andere von dir denken, ist dann sicher auch egal?\n\nJa, total egal. Wer mich mag, mag mich halt, und vielleicht mag ich ihn dann auch, wer nicht, der dann nicht, das ist mir so was von egal.\n\n#### Hast du Hobbys?\n\nIn meiner Freizeit sitze ich sehr gerne am Computer und programmiere Spiele, spiele diese Spiele nat\u00fcrlich sehr gerne oder helfe meiner Mutter beim Forum pflegen. Oder ich helfe Freunden und Bekannten, Homepages zu erstellen, bin auch in mehreren Communities aktiv, wo man sich gegenseitig hilft beim Programmieren und so weiter. Nat\u00fcrlich open source, alles was geschrieben wird, steht jedem zur Verf\u00fcgung. Ansonsten mit Freunden treffen.\n\n#### Bist du in einem Verein aktiv?\n\nNein.\n\n#### Und anderweitig gesellschaftlich oder politisch aktiv?\n\nPolitisch aktiv kann man nicht so direkt sagen. Ich war auf vielen Demos, wo das Thema mich interessiert hat. Hab mir die Kundgebungen angeh\u00f6rt und dann auch welche mitorganisiert, hab auch eine VoK\u00fc [Volksk\u00fcche] mit organisiert und gekocht, aber ein politisches Statement an sich habe ich nicht abgegeben.\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nDie Musik, die ich gerne h\u00f6re, hat eine sehr gro\u00dfe Bedeutung f\u00fcr mich. \u00dcber die Musik habe ich auch einige Freunde kennengelernt. Bis dahin hatte ich Probleme, Freunde zu finden, obwohl ich ein sehr aufgeschlossener Typ bin und auch auf Leute zugehe. Aber durch die Musik findet man schneller Freunde, die dieselben Interessen und Einstellungen pflegen. Mittlerweile h\u00f6re ich viele verschiedene Richtungen, wichtig ist nur, dass sie mich anspricht. Anfangs war es viel Punk-\/Partymusik.\n\n#### Ist Punk gleichbedeutend f\u00fcr dich mit Partymusik?\n\nNein, aber mit Punk verbinde ich immer Party. Es gab bei mir eine Zeit, da war nur Party angesagt, da hab ich nur Punk geh\u00f6rt; sofern ich die Texte verstand, waren sie immer lustig. Bis ich mich mit dem speziellen Thema besch\u00e4ftigt habe. Ich habe in einer Kneipe einen Typen getroffen, der mir was erz\u00e4hlt hat \u00fcber die autonome Szene, wir sind \u00fcber Musik ins Gespr\u00e4ch gekommen. Er hat sich \u00fcber die hirnlose Musik aufgeregt, die ich gefeiert habe. Er hat mir geraten, mal \"richtige\" Musik zu h\u00f6ren, und hat mich mitgenommen. So kam ich in den Genuss, zum ersten Mal Holger Burner zu h\u00f6ren. Er macht nicht nur Antifa-Musik, sondern auch antikapitalistische, antimilitaristische, halt sehr politische Songs. Als ich die Texte zum ersten Mal h\u00f6rte, dachte ich nur: Das stimmt ja irgendwie. So kam ich zum Nachdenken und in die Szene.\n\n#### Hat dich das gepr\u00e4gt?\n\nJa, ich denke schon. Selbst meine Mutter hat die Musik gerne geh\u00f6rt und h\u00f6rt sie immer noch gerne.\n\n#### Hast du auch eine Lieblingsliteratur, Filme, Kunst etc.?\n\nNee, gar nicht. Also ich schau gerne mal einen Film, auch gerne mal was Anspruchsvolles, bei dem man nachdenken muss, auch mal mit politischem Hintergrund, zum Beispiel den Film \u00fcber die RAF, den habe ich verschlungen und, was ich sonst nicht mache, ich habe ihn sogar mehrmals gesehen. Ansonsten w\u00fcrde ich mich nicht festlegen wollen.\n\n#### Also eher zum Konsumieren bzw. Abschalten?\n\nJa, genau.\n\n#### Interessierst du dich f\u00fcr Mode?\n\nDas erkl\u00e4rt sich glaube ich von selbst, wenn du mich anschaust, oder? [Lacht, er hat einen Blaumann an und tr\u00e4gt Arbeitsschuhe.] Ich interessiere mich \u00fcberhaupt nicht daf\u00fcr. Na ja, nicht mehr. Eine Zeit lang habe ich Klamotten f\u00fcr Kumpels und Freunde geschneidert. Irgendwann fing ich an, meine Kutte selbst zu schneidern und die Leute sagten dann: Boah, das sieht aber Hammer aus, kannst du mir das nicht auch mal schneidern? Also hab ich angefangen, Klamotten zu schneidern.\n\n#### Was f\u00fcr Klamotten? Hast du was da?\n\nJa, warte. [Er holt aus dem Nebenzimmer einen Styroportorso, auf dem eine Klamotte drapiert ist, die eindeutig dem Punk zuzuordnen ist, das typische Karomuster, Nieten, Schnallen, L\u00f6cher, N\u00e4hte, alles per Handarbeit angebracht, nicht schick, aber originell.] Das kann man jetzt nicht Mode nennen, Mode ist, was irgendwie alle gerade toll finden oder was gerade in ist. Ich hatte halt versucht, meinen eigenen Style zu finden. Ich mochte Metal und Punk gerne und hatte versucht, beides miteinander zu verkn\u00fcpfen. Das ist mir zwar so nicht gelungen, aber mir hat es gefallen. Mittlerweile gehe ich ins Gesch\u00e4ft und schaue nur, ob die Hose sitzt und bequem ist und ob ich mich darin wohlf\u00fchle.\n\n#### Hast du Leitbilder, Idole, Vorbilder?\n\nDa muss ich mal nachdenken, ob ich fr\u00fcher welche hatte. Hatte ich mal Vorbilder? Also zurzeit nicht.\n\n#### Und der Holger Burner, ist das f\u00fcr dich ein Leitbild?\n\nIdol oder Leitbild kann man es nicht nennen, sonst w\u00fcrde ich ihm nacheifern wollen, will hei\u00dfen, dass ich auch Musik machen wollen w\u00fcrde, und das mache ich nicht, mal abgesehen davon, dass ich es auch nicht k\u00f6nnte. Er macht HipHop, das k\u00f6nnte ich gar nicht. Vielleicht hab ich versucht, seiner Einstellung nachzueifern, obwohl es das auch nicht richtig trifft. Ich kenne ihn nicht pers\u00f6nlich, kann also seine Einstellungen nicht als solche erleben. Ich habe nur versucht, mich mit seinen Texten zu identifizieren.\n\n#### Also m\u00fcsstest du denjenigen als ganzen Menschen wahrnehmen k\u00f6nnen, um ihn als Vorbild einstufen zu k\u00f6nnen?\n\nIch m\u00fcsste ihn auf jeden Fall mal getroffen haben und mich mit ihm unterhalten haben. Dann m\u00fcsste ich noch denken, toll, so m\u00f6chte ich sein! Aber wenn ich jetzt so dar\u00fcber nachdenke, m\u00f6chte ich gar kein Idol haben, ich bin mit mir zufrieden und m\u00f6chte kein anderer Mensch sein.\n\n#### Wie verbringst du deine Freizeit?\n\nMit meinem Computer zu Hause und mit Freunden auf der Stra\u00dfe oder in der Kneipe.\n\n#### Nach welchen Kriterien schlie\u00dft du Freundschaften?\n\nWenn ich mich mit denen ganz gut unterhalten kann, sie mir sympathisch sind und es Spa\u00df macht. Ich muss mich in deren Gegenwart wohl und aufgehoben f\u00fchlen. Wenn wir unterwegs sind, muss ich mich sicher f\u00fchlen, das sind so meine Kriterien. Ob nun klein, gro\u00df, dick, d\u00fcnn, das ist mir wumpe.\n\n#### Geh\u00f6rst du einer bestimmten Jugendkultur oder Szene an?\n\n[Z\u00f6gern und Schweigen.]\n\n#### Oder siehst du dich eher als Einzelk\u00e4mpfer?\n\nMittlerweile bin ich schon der Einzelg\u00e4nger, ich war mal in einer Subkultur unterwegs, der Metal-Punk-Szene, dort habe ich viele Freunde gefunden, aber ich bin da rausgewachsen. Es war halt wenig Inhalt, es ging nur um Party machen und Spa\u00df. Mein kleiner Bruder h\u00e4ngt jetzt auch mit solchen Typen ab, dann geh ich ab und an mit, aber nicht mehr so, dass ich dazu geh\u00f6re.\n\n#### Welche Jugendkultur ist dir absolut unsympathisch?\n\nIch wei\u00df nicht, ob man es so als Jugendszene bezeichnen kann, aber es f\u00fchlen sich halt doch viele junge Leute da hingezogen; hier in meiner Stadt, einer der Hochburgen der Autonomen Nationalisten, ist diese Szene immer pr\u00e4sent. Das ist eine Gruppe, Szene, mit denen ich \u00fcberhaupt nicht sympathisieren kann. In einer anderen Stadt, in der ich lebte, gab es eine Gruppe, die nannte sich Anti-Antifa. Wer f\u00e4hig ist, solch eine Gruppe zu gr\u00fcnden, der muss doch einfach nur bescheuert sein. Kann ich \u00fcberhaupt nicht verstehen, alleine schon der Gedanke einer Anti-Antifa, der ist so hirnrissig! Ich meine, wenn du Faschist bist, das ist doch Schei\u00dfe genug, aber dann noch diese Gruppe, also bitte! Aber ich glaube doch nicht, dass das eine Jugendkultur ist, es sind viele \u00e4ltere Leute aktiv dabei.\n\n#### Gibt es noch mehr Leute, mit denen du nicht klarkommst?\n\nEs sind die aktiven Faschisten, die mich w\u00fctend werden lassen. Ansonsten komme ich mit jedem klar, so lange er bei Diskussionen auch mal Einsicht zeigt und nicht stur auf seiner eigenen Meinung beharrt. Zum Beispiel meine Kollegen, sie sind so anders als ich, aber ich komme mit ihnen klar.\n\n#### Warum, was ist mit denen?\n\nWir sind eine ziemlich bunt gemischte Truppe, es sind viele Ausl\u00e4nder dabei, die sich in ihren eigenen Gruppen treffen und auch Pause machen. In der Gesamtgruppe bilden sie aber auch eine Dominanz im Zusammenhalt. Dann gibt es noch die Leute, meist Deutsche, die bl\u00f6de Spr\u00fcche rei\u00dfen, aber an sich auch nette Menschen sein k\u00f6nnen. Die denken einfach nicht nach, bevor sie sprechen, erlebt man ja sehr h\u00e4ufig. Und diese Leute halte ich immer wieder an, \u00fcber ihr Gesagtes nachzudenken. Obwohl mich so manches Gerede wirklich w\u00fctend macht, aber wir sind so eine super Truppe.\n\n#### Wenn davon jetzt einer ein aktiver Faschist w\u00e4re, w\u00fcrdest du dann auch versuchen, mit ihm klarzukommen?\n\nSo lange er diskussionsbereit ist oder vielleicht Einsicht zeigt. Befreunden w\u00fcrde ich mich mit ihm h\u00f6chstwahrscheinlich nicht, aber ich k\u00f6nnte mit ihm arbeiten, sagen wir mal so. Ich habe sogar gerade einen in meiner Gruppe, er ist neu reingekommen, der ist genau so drauf, ob er nun aktiv ist, ist noch nicht absehbar, aber man sieht und merkt ihm seine Gesinnung gleich an. Ich wei\u00df es, kann mit ihm arbeiten und mich auch mit ihm unterhalten, aber wir werden keine Freunde. Mir ist auch mal auf einer Party passiert, dass ich mich super mit jemandem unterhalten habe und hinterher erst gemerkt habe, halt-stopp, da stimmt was nicht. Es k\u00f6nnen sehr nette und umg\u00e4ngliche Menschen sein, aber sobald sie radikal werden und ihre Ideologie verbreiten wollen, da werde ich vorsichtig. Dann w\u00fcrde ich auch nicht versuchen, damit klarzukommen, geht ja auch nicht. W\u00fcrde er ja andersrum auch nicht.\n\n#### Welche Rolle spielen Freunde f\u00fcr dich?\n\nEine sehr gro\u00dfe Rolle. Ich hatte eine Zeit lang keine oder keine richtigen Freunde. Da bin ich fast eingegangen, wurde schon fast depressiv. Wenn ich jetzt keine Freunde mehr h\u00e4tte, w\u00fcrde ich kaputtgehen. Ich w\u00fcrde nur noch arbeiten gehen, nach Hause gehen und am Computer sitzen, schlafen gehen, Schicht im Schacht. Nee, das w\u00e4r Schei\u00dfe!\n\n#### Und welche Rolle spielt die autonome Szene?\n\nSie spielt eine sehr gro\u00dfe Rolle f\u00fcr mich dadurch, dass ich viele Newsletter abonniert habe, um immer auf dem neuesten Stand und \u00fcber die aktuellen Themen informiert zu sein.\n\n#### Welche Seiten bevorzugst du?\n\nIndymedia und \u00e4hnliche Seiten, ich m\u00f6chte nicht zu viel preisgeben.\n\n#### Welche Rolle spielt die autonome Szene in deinem momentanen Leben?\n\nAls ich hier hergezogen bin, hab ich mir erst mal eine Gruppe gesucht und wollte schauen, wie die so drauf sind. Man muss bedenken, dass ich hier \u00fcberhaupt keinen kannte und ein Niemand in der Szene war. Wie zu erwarten habe ich nichts gefunden, logischerweise h\u00e4ngen keine Zettel aus: Suche Mitstreiter! Also hab ich erst mal einen Job gesucht und gefunden. Habe dann \u00fcber meinen \u00e4lteren Bruder eine Gruppe gefunden, allerdings in einer Nachbarstadt. Diese Gruppe ist politisch sehr aktiv und f\u00fcr mich sehr wichtig; ich bin zwar nicht direkt involviert, also kein Gruppenmitglied, nehme nur als geduldeter Gast am Plenum teil. Und wenn ich im Moment nicht so viel arbeiten m\u00fcsste, w\u00fcrde ich auch, um jetzt mal ein aktuelles Thema anzuschneiden, auf die Anti-Atom-Demo gehen und sogar einige organisieren, aber durch meine aktuelle pers\u00f6nliche Lage bin ich gezwungen, viel zu arbeiten und Geld zu verdienen. Mir ist die Atom-Lobby schon eine ganze Weile ein Dorn im Auge, wenn man das sagen kann, wir wissen doch alle, es geht anders! Warum m\u00fcssen die da weiter machen? Strom geht auch ohne Atomenergie, da muss man doch nicht so gef\u00e4hrliche Sachen machen. Und dann diese Zerm\u00fcrbungstaktik der Regierung, mal wollen sie die Atomwerke abschalten, dann wieder dran lassen; man gewinnt den Eindruck, dass sie sich selber alle nicht \"gr\u00fcn\" sind, entschuldige das Wortspiel. Dieses hin und her ist unglaublich. Einfach abschalten und fertig.\n\n#### Deine Infos zum aktuellen Zeitgeschehen beziehst du also vornehmlich aus dem Internet. Gibt es noch andere Medien, die du pr\u00e4ferierst?\n\nDie Infos, die durch Mundpropaganda zu mir gelangen, sind meist jedoch die interessantesten, weil da die zeitnahen Infos zum Geschehen weitergereicht werden.\n\n#### Geh\u00f6rst du einer Religion oder Kirche an?\n\nNein. Also ich sag mal so, irgendwie muss da oben im Himmel schon irgendwas sein, sonst w\u00e4re ich mindestens schon hundertmal gestorben. Irgendwie muss schon eine h\u00f6here Macht existieren, daran glaube ich schon, aber ohne mich einem bestimmten Glauben anzuschlie\u00dfen. Bei den Regeln, die die haben, w\u00e4re das auch gar nicht m\u00f6glich. Jede Religion hat so ihre eigenen Regeln, du darfst dies nicht essen, das nicht trinken, musst das tragen und was wei\u00df ich. Das ist mir pers\u00f6nlich zu doof.\n\n#### Sag mir doch mal bitte, wie du \"autonom\" oder die \"autonome Szene\" definieren w\u00fcrdest. Stell dir vor, ich h\u00e4tte keinen blassen Schimmer davon, wie w\u00fcrdest du es mir erkl\u00e4ren?\n\nDas ist so die Horrorfrage f\u00fcr mich, weil ich \u00fcberhaupt nicht w\u00fcsste, wie ich es dir erkl\u00e4ren sollte. Ich kann dir nur von mir erz\u00e4hlen und wie ich meine Erfahrungen gemacht habe. Ich w\u00fcrde dir erz\u00e4hlen, was ich gemacht habe, wie ich reingekommen bin. Dann m\u00fcsstest du noch wissen, dass es verschiedene politische Richtungen bei den Autonomen gibt.\n\n#### Geh doch einfach vom Begriff aus.\n\nDer Begriff an sich bedeutet ja, selbstbestimmt und auch selbstverwaltend leben. Es geht darum, dass man sich die Welt so macht, wie man sich die w\u00fcnscht, und zwar so, dass es f\u00fcr die gesamte Gesellschaft durchf\u00fchrbar ist. Also nicht f\u00fcr die einzelne Person, sondern f\u00fcr alle. Es sollte keine Ober- oder Mittelschicht geben, sondern alle sollten die M\u00f6glichkeit haben, dasselbe zu verdienen. Es sollte m\u00f6glich sein, dass jeder Zugang zur selben Bildungsart hat zum Beispiel. Es muss einfach das Recht existieren, f\u00fcr alle alles zug\u00e4nglich zu machen. Es wird zwar immer wieder gesagt: \"Wir machen gar keinen Unterschied, alle sind gleich!\", aber dem ist ja nicht so. Besser k\u00f6nnte ich es jetzt nicht erkl\u00e4ren.\n\n#### Denkst du, dass es der autonomen Szene wirklich darum geht, um Freir\u00e4ume f\u00fcr die Allgemeinheit zu schaffen, oder ob sie nur eigene Partikularinteressen sch\u00fctzen oder durchsetzen wollen?\n\nIch denke mal, der Grundgedanke war schon f\u00fcr die Allgemeinheit gedacht. Mittlerweile hat es sich beschr\u00e4nkt, da es f\u00fcr die Allgemeinheit doch sehr schwer zu verwirklichen ist. Man m\u00fcsste es in kleineren Schritten machen. Nun ist es aber eher so, dass in erster Linie echt f\u00fcr die eigenen Freir\u00e4ume gek\u00e4mpft wird. Und wenn das geschafft sein sollte, ist der Weg geebnet f\u00fcr die Allgemeinheit.\n\n#### Du bist in keiner konkreten Gruppe Mitglied?\n\nNein, mittlerweile habe ich Kontakt zu verschiedenen Gruppen und je nachdem wie ich Zeit habe, klinke ich mich ein und mache mit. Das Thema sollte allerdings auch zu meiner \u00dcberzeugung passen, denn wenn mir das Thema zu hoch gegriffen w\u00e4re, w\u00e4re mir mittlerweile die Zeit zu schade.\n\n#### Magst du mir erz\u00e4hlen, wie dein erster Kontakt zur autonomen Szene war?\n\nJa, ich war mit meinem Bruder und ein paar Kumpels unterwegs, auf Kneipentour. Wir trafen dann ein paar Leute, mit denen wir uns unterhalten haben. Schade, dass ich mich nicht mehr genau an das Gespr\u00e4ch erinnern kann, die haben mir ganz genau erkl\u00e4rt, was autonom bedeutet. Dann h\u00e4tte ich es dir besser erkl\u00e4ren k\u00f6nnen. [Lacht.] Ich habe schnell gemerkt, dass ich mich sehr gut damit identifizieren konnte, und wollte genau so leben.\n\n#### War es f\u00fcr dich relevant, dass die Autonomen von der Gesellschaft, also der breiten Masse kritisiert wurden, dass sie als Au\u00dfenseiter wahrgenommen werden?\n\nDas kann sein, ist mir so nicht bewusst, aber kann durchaus sein. Mir und meinem Bruder war daran gelegen, noch mehr Informationen von den Autonomen und \u00fcber sie zu bekommen. \u00dcber diesen einen Kontakt hat sich f\u00fcr uns ein ganzes Netzwerk aufgetan.\n\n#### Wie alt warst du denn, als du die ersten Kontakte gekn\u00fcpft hast?\n\nMit 15 habe ich angefangen, mich f\u00fcr Politik zu interessieren, habe begonnen, Nachrichten bewusst zu lesen und zu h\u00f6ren, und kurz vor meinem 16. Geburtstag habe ich die ersten Kontakte zur autonomen Szene gekn\u00fcpft.\n\n#### Hast du den Kontakt gezielt gesucht? Konntest du die Person gleich einordnen und hast daraufhin das Gespr\u00e4ch gesucht oder ist er auf dich zugekommen?\n\nNein, wir sind \u00fcber Musik ins Gespr\u00e4ch gekommen. Wir sa\u00dfen zuf\u00e4llig in der gleichen Kneipe am gleichen Tisch und haben uns zuerst \u00fcber Musik unterhalten und sind dann auf die aktuelle Politik zu sprechen gekommen. Ich war von seiner Einstellung begeistert und habe erst relativ sp\u00e4t mitbekommen, dass er der autonomen Szene angeh\u00f6rt; er ist nicht offensiv damit umgegangen.\n\n#### Gibt es Ankn\u00fcpfungspunkte an andere Jugendkulturen?\n\nDie gibt's schon, ja klar. Zum Beispiel die Antifa und Punks, wenn da ein Fascho vorbeikommt, ist Ende im Gel\u00e4nde, der wird platt gemacht. Es sind also eher die Jugendkulturen der linken Szene, die da Ankn\u00fcpfungspunkte aufweisen. Ich habe auch mitbekommen, dass immer mehr Metaler links eingestellt sind. Sicher, da gibt es auch viele, denen Politik am Arsch vorbei geht, aber vermehrt ist ein Linksruck zu verzeichnen, ist mir jedenfalls hier aufgefallen.\n\n#### Denkst du, dass die autonome Szene eine Jugendkultur ist?\n\nNein, eine Jugendkultur nicht unbedingt. Ich kenne viele Leute, die sind \u00fcber 50 und seit etlichen Jahren aktiv. Zwar sind sie nicht mehr auf Demos dabei, aber sie organisieren und veranstalten sie oder fahren den Lauti [Lautsprecherwagen] und machen, was gemacht werden muss. Also eine Jugendkultur w\u00fcrde ich es nicht nennen, denn ich kenne echt viele alte S\u00e4cke aus der Szene.\n\n#### Gibt es einen eigenst\u00e4ndigen autonomen Stil?\n\nAuf einer Demo auf jeden Fall. Der \"Schwarze Block\" sagt eigentlich alles. Ja sicher, einige laufen auch in der Freizeit so rum, nicht vermummt, aber halt ganz in schwarz, denen gef\u00e4llt es dann aber auch. Ich kenne aber auch Leute, die laufen in ihrer Freizeit kunterbunt rum mit Hawaiihemd und gelber Hose und so.\n\n#### Ich hab geh\u00f6rt, dass auf Demos ein Wort als Code benutzt wird, wenn zum Beispiel Gefahr von Seiten der Polizei droht und der R\u00fcckzug, die Flucht angeraten ist?\n\nDas ist unterschiedlich von Gruppe zu Gruppe; ich hatte das mal auf einer Demo, da war ich einer gr\u00f6\u00dferen Gruppe zugeteilt, da wurde so etwas \u00c4hnliches veranlasst. Viele Leute hatten ein Funkger\u00e4t dabei und wenn jemand gemerkt hat, das wird hier langsam gef\u00e4hrlich, die bauen einen Kessel \u2013 dann wurde \u00fcber Funk ein bestimmtes Wort gesagt, dann haben wir uns verzogen. Aber das ist sicher in jeder Gruppe anders, nur die Struktur \u00e4hnelt sich. Es ist ja nicht eine Gruppe, sondern viele kleine, so muss keine lange Kette der Nachrichten\u00fcbermittlung absolviert werden. Es gibt auch nicht ein Wort, was alle benutzen, sondern es variiert von Gruppe zu Gruppe. Es geh\u00f6rt halt zu den Vorsichtsma\u00dfnahmen, die man treffen muss, um sich und die anderen zu sch\u00fctzen in Anbetracht der Willk\u00fcr der \"aufpassenden Polizei\". Wir hatten auch immer Nummern auf den Armen oder Beinen f\u00fcr den EA [Ermittlungsausschuss]. Wenn du gekascht wirst, steht dir ein Anruf zu, und wenn du die notierte Nummer anrufst, deinen Namen sagst und wo du dich gerade befindest, holt ein Anwalt dich innerhalb von 24 Stunden raus.\n\n#### Wo hast du die Nummer her?\n\nDie gibt's immer vom Veranstalter der Demo, der organisiert vorher den EA. Das sind halt Leute, die sitzen in irgendeinem Haus in irgendeiner Stadt und warten auf den Anruf, den sie dann umgehend weiterleiten. Die Nummer ist immer eine Handynummer, die danach deaktiviert ist.\n\n#### Was ist der Ermittlungsausschuss?\n\nDas ist der Stab Leute, die von den Handys aus angerufen werden und sich darum k\u00fcmmern, dass man wieder rauskommt, wenn man gekascht wird. Es ist eine Gruppe von Anw\u00e4lten, die aus der linken Szene kommen. Die Mittelsm\u00e4nner und -frauen k\u00fcmmern sich darum, dass die richtigen Handynummern verteilt werden, und geben uns Ratschl\u00e4ge, wo wir uns die Nummer hinschreiben sollten. Das Problem ist nur, dass die Polizei schon mal zu rigorosen Mitteln greift. Als ich gekascht wurde, hatten wir uns die Nummern mit einem Edding auf den Arm geschrieben. Die gesamte Gruppe wurde eingekesselt, beiseite genommen und gekascht. Dann wurde Mann f\u00fcr Mann, Frau f\u00fcr Frau durchgegangen, \u00c4rmel hochkrempeln, Hosenbeine hochkrempeln, die wussten ganz genau, wonach und vor allem wo sie zu suchen hatten. Sobald sie Nummern gefunden haben, wurden diese Nummern abgewaschen. Das geschah abseits von der gro\u00dfen Menschenmenge. Hinterher, als wir uns beschweren wollten, hatten wir keinen Beweis, also ist es nicht geschehen. Ich kann mit Fug und Recht sagen, die Sicherheit der Demo-Teilnehmer spielt eine sehr gro\u00dfe Rolle.\n\n#### Welche Musik spielt denn eine Rolle? Sei es auf Demos oder privat?\n\nPrivat kann man h\u00f6ren, was man will, es sei denn, sie ist frauenfeindlich, homophob, antisemitisch oder rassistisch. Viele wissen gar nicht, was sie da f\u00fcr Musik h\u00f6ren, oder sie h\u00f6ren gar nicht richtig zu. Eine Rolle auf Demos spielt sie wegen der Texte, da fahren die Lautis rum, die die Musik sehr laut abspielen. Nat\u00fcrlich Texte, die zum Thema passen, die polarisieren, zum Beispiel von Holger Burner. Die Musik macht es auch einfacher, die Leute auf der Demo, die nicht zu den Autonomen geh\u00f6ren, zu bewegen und zu kontrollieren. Sonst hast du nur einen Haufen Menschen, die durcheinander br\u00fcllen, und es kann nichts transportiert werden. So spielt die Musik, das erheitert einerseits und bringt die Massen dazu, die Texte durch Sprechgesang wiederzugeben. So haben sie Spa\u00df daran weiterzulaufen, Demo-Z\u00fcge sind ja doch sehr lang manchmal, Marathon auch gerne genannt. Wichtig ist, dass die Musik zum Thema passt.\n\n#### Welche Themen sind dir f\u00fcr dein Engagement besonders wichtig?\n\nAls Erstes und ganz schnell f\u00e4llt mir die Homophobie ein. Passend, weil meine Mutter eine Lesbe ist und mein Bruder schwul ist und ich auch viele schwule Freunde habe. Mir ist wichtig, dass die Leute diese Homophobie endlich mal ablegen. Ich hab damals schon in der Schule versucht zu vermitteln, dass es nicht gef\u00e4hrlich ist, aber mir scheint, der Weg ist noch lang. Dieses Thema ist mir eben sehr nah. Dann als n\u00e4chster Punkt der Faschismus und Rassismus, da komm ich gar nicht drauf klar. Und dann ist mir in der letzten Zeit etwas aufgefallen, was ich zum Teil \u00fcbertrieben finde, das Thema Sexismus. Da steigern sich Leute doch sehr rein. In Rassismus, Homophobie und Faschismus kann man sich nicht reinsteigern, das ist einfach so, aber beim Sexismus... muss man schon unterscheiden, differenzieren k\u00f6nnen. Klar, wenn es frauenfeindlich wird: \"Frauen geh\u00f6ren...\", da geht mir der Hut hoch, ich kann einfach nicht verstehen, wenn ein Mensch frauenfeindlich ist, ich kann es schlicht und einfach nicht verstehen, wie kommt man auf so was? Wieso verdient eine Frau im selben Job weniger als ein Mann? Wieso muss eine Frau mehr Geld in die Krankenversicherung einzahlen? Eine Kollegin von mir wollte so gerne Kfz-Mechatronikerin werden, aber sie kommt gar nicht erst in den Beruf rein, keiner will sie als Lehrling haben. Versteh ich alles nicht. Aber frauenfeindlich ist f\u00fcr mich nicht gleich sexistisch.\n\n#### Die autonome Szene entwickelt mitunter bestimmte Theorien, setzt du dich mit diesen auseinander?\n\nAnfangs schon, ich wollte mich schlau lesen, weil ich noch nicht so viel dar\u00fcber wusste, aber mittlerweile nicht mehr so, ich bin zeitlich zu sehr eingeschr\u00e4nkt. Wenn wir uns aber mal zum Bier treffen oder ich Zeit f\u00fcr ein Plenum habe, dann schon. Womit ich mich anfangs tiefergehend besch\u00e4ftigt habe, war die Anarchie-Theorie.\n\n#### Wie hast du dich damit auseinander gesetzt?\n\nIch habe mir gedacht, dass sie funktionieren kann, aber eigentlich nicht zu verwirklichen ist. Die Theorie kann einfach nicht funktionieren. Weil meines Erachtens der Mensch daf\u00fcr nicht geschaffen ist. Und die Theorie vom Sozialismus ist auch eine sch\u00f6ne Theorie, aber wurde noch nicht verwirklicht.\n\n#### Also bem\u00e4ngelst du die Praxisferne der Theorien?\n\nJa, sie erscheinen mir manchmal sogar weltfremd. Die stalinistischen oder maoistischen Theorien finde ich sogar grenzwertig, denn wenn man sich mit den Namensgebern dieser Theorien besch\u00e4ftigt, so merkt man schnell, dass es menschenfeindliche und gef\u00e4hrliche Machthaber waren. Wieso sollten dann die entwickelten Theorien sozial und gesund sein? Da gilt es noch viel zu lesen f\u00fcr mich.\n\n#### Wie informierst du dich \u00fcber szeneinterne Aspekte? Welche internen Medien nutzt du?\n\nIndymedia, anschlie\u00dfende Foren und \u00e4hnliche Seiten, die nicht nur das aktuelle Geschehen abdecken, sondern auch szeneinterne Informationen zeitnah weitergeben. Ich m\u00f6chte gar nicht so viel davon preisgeben.\n\n#### Das Plakat als Medium wird auch gerne genutzt, inwieweit ist es wichtig zur Informationsvermittlung?\n\nEs vermittelt Termine f\u00fcr Demos und auch einen Standpunkt, aber es ist eher ein Aufh\u00e4nger, kleiner Scherz am Rande und kein Medium f\u00fcr aktuelle gruppeninterne Informationen. Da greift die Mundpropaganda eher und auch schneller.\n\n#### Du darfst dieses Jahr zum ersten Mal w\u00e4hlen, wirst du dein Recht wahrnehmen?\n\nIch werde auf jeden Fall w\u00e4hlen gehen, aber ich werde nicht sagen, wen ich w\u00e4hle. Ich schwanke noch zwischen zwei Parteien, sie haben beide sehr gute Argumente und Ideen, aber ich wei\u00df es halt noch nicht.\n\n#### Setzt du dich mit den Programmen der jeweiligen Parteien auseinander?\n\nBevor ich w\u00e4hle auf jeden Fall, ich finde es sogar sehr wichtig, sonst k\u00f6nnte ich die falschen w\u00e4hlen. Ich muss ja wissen, was die vorhaben!\n\n#### Wie ist dein Verh\u00e4ltnis zum Staat und zur Polizei?\n\n\u00dcber die Bullen k\u00f6nnte ich stundenlang erz\u00e4hlen, aber das w\u00fcrde sich irgendwann wiederholen. Vorab muss ich sagen, dass die Allgemeinheit davon ausgeht, dass die linke Szene sich immer und andauernd im Konflikt mit der Polizei befindet. Dem ist nicht so. Ich kenne Polizisten, Freunde von meinem Onkel zum Beispiel, mit denen hab ich Party gemacht, mit denen kann man sich gut unterhalten, das sind ganz normale Menschen, die machen halt ihre Arbeit, gehen auf Streife, schauen, dass keiner Schei\u00dfe baut. Damit habe ich gar kein Problem. Womit ich ein Problem habe, das sind die sogenannten \"Pr\u00fcgelbullen\", die auf Demos \"aufpassen\". Was die an Willk\u00fcr an den Tag legen, ist unfassbar. Ein Beispiel: Gewerkschafts-Demo zum 1. Mai in Hannover, es war eine sehr lange Demo und wir wollten uns danach ausruhen. Wir sind zum Hauptbahnhof gegangen und wollten da mit anderen Leuten zusammentreffen. Also setzten wir uns vor den Hauptbahnhof auf den Bordstein, weil wir total kaputt waren. Das Problem war, dass es so nach und nach immer mehr Leute wurden, die kannten wir nat\u00fcrlich nicht, aber es wurden immer mehr. Und mit der Anzahl der Leute stieg die Anzahl der Polizei, die die Situation total falsch eingesch\u00e4tzt hatte und auf eine Spontandemo setzte. Letzten Endes wurde der gesamte Bahnhof von Hannover mit Sechserautos eingekesselt. Vor den Autos standen zweireihig die Pr\u00fcgelbullen in voller Montur. Hinterher haben wir erfahren, die haben aus mehreren St\u00e4dten Pr\u00fcgelbullen einfahren lassen, um den Hauptbahnhof komplett abzuschotten, da ist keiner mehr rausgekommen. Und alles nur, weil wir dort gesessen haben! Aber das Beste kommt noch, da habe ich n\u00e4mlich meinen Zahn verloren. Als wir da also sa\u00dfen, haben wir so langsam gemerkt, dass es eng wird und die immer n\u00e4her kommen, uns einkesseln wollen. Also hatten wir beschlossen, lieber abzuhauen. Dann ging es los. Auf einmal wurden zwei Kessel gebildet. In dem einen Kessel wurden den Anwesenden die Nummern abgewaschen, das kannten wir ja schon, und in dem anderen Kessel wurde wild drauflos gepr\u00fcgelt. Ich bin vorher zu einem Bullen gegangen und wollte fragen, warum sie uns denn einkesseln, wir sa\u00dfen ja nur, aber so weit kam ich gar nicht. Ich konnte nur meinen Mund \u00f6ffnen und hatte schon einen Kn\u00fcppel im Gesicht. Ein Bulle hielt mich fest und der andere hat draufgekloppt. Zur selben Zeit wollte ein Passant mit seinem Fahrrad vorbei, er hatte wohl von der anderen Seite des Kessels Fotos gemacht, da sind zwei Bullen hingerannt und haben ihn vom Fahrrad gezogen, einer hat ihn festgehalten, der andere hat auf ihn eingetreten. Bl\u00f6d nur f\u00fcr die Bullen, dass ein Video davon am n\u00e4chsten Tag im Internet zu sehen war. So was, genau das macht uns aggressiv. Mich besonders!\n\n[Er macht sich ein Bier auf, trinkt einen Schluck und entspannt sich wieder ein wenig.] Wenn so etwas passiert, fangen wir nat\u00fcrlich auch an und verteidigen uns gegen diese Bullen, soweit es m\u00f6glich ist nat\u00fcrlich. Das besonders Unfaire ist, dass die Bullen herumlaufen wie Robocop, total gepanzert mit f\u00fcnf, sechs Flaschen Reizgas am G\u00fcrtel, und bei uns alle waffen\u00e4hnlichen Sachen zu einer Verhaftung und Anklage f\u00fchren: Strafgesetzbuch, besonderer Teil (\u00a7\u00a7 80\u2013358), 7. Abschnitt \u2013 Straftaten gegen die \u00f6ffentliche Ordnung (\u00a7\u00a7 123\u2013145d), \u00a7 125 Landfriedensbruch. Ein eleganteres Wort f\u00fcr Terrorismus. Daf\u00fcr gehst du richtig lange in den Bau! Und nur mit diesen Bullen haben wir Probleme, die Streifenpolizisten machen nur ihren Job, aber diese Pr\u00fcgelbullen werden richtig darauf gedrillt. Die stehen schon und warten regelrecht darauf, endlich losrennen zu k\u00f6nnen. Meistens haben sie sich ihre Leute schon vorher ausgesucht und gehen gezielt auf diese los, und egal, was du machst, du kriegst auf die Fresse. Wenn du dich auf die Erde schmei\u00dft, wirste getreten, wenn du dich hinstellst und die Arme hochrei\u00dft, kriegste den Kn\u00fcppel, und wenn du dich wegdrehst oder gar wegrennst, kriegste Reizgas. Ich hab gesehen, wie sie zu f\u00fcnft auf einem Typen knieten und an seinen Haaren zerrten und ihn mit Reizgas bearbeiteten, obwohl er am Boden lag. Er hatte sich nicht gewehrt, aber letzten Endes bekam er eine Anklage wegen Widerstands gegen die Staatsgewalt mit 2.500 Euro Strafe. Was ist das f\u00fcr eine Staatsgewalt, wenn der Fokus auf der Gewalt liegt? Siehste, ich hab ja gesagt, dar\u00fcber k\u00f6nnte ich ewig quatschen, ich hab mich wohl ein wenig reingesteigert.\n\n#### Und dein Verh\u00e4ltnis zum Staat?\n\nIch m\u00f6chte nicht in manch einem anderen Land leben, muss ich echt zugeben. Deutschland hat Probleme, das wei\u00df ich, der Staat hat Probleme, der Staat als solches baut Schei\u00dfe, aber er macht auch ganz viele gute Sachen. Das muss ich auch zugeben. Die Politiker erz\u00e4hlen zwar, dass sie was \u00e4ndern wollen, aber sie machen es nicht. Ich sehe es als meine Pflicht und auch mein Recht an, den Staat an seine Pflichten und Versprechen zu erinnern.\n\n#### Wie hat sich denn die autonome Szene im Laufe der letzten Jahre ver\u00e4ndert?\n\nDas kann ich dir nicht so pauschal beantworten, aber was ich so mitbekomme, ist, dass sich immer mehr und kleinere Gr\u00fcppchen bilden und jeder macht sein eigenes Ding, also vertreten ihre eigenen Theorien. Das finde ich pers\u00f6nlich nicht so gut, weil man als gro\u00dfe Gruppe schneller und besser wahrgenommen wird.\n\n#### Wo siehst du weitere Kritikpunkte, Probleme in der autonomen Szene?\n\nEs gibt viel Streit untereinander. Viele haben andere Ansichten zu einem oder mehreren Themen, was an sich ja eine super Sache ist, aber manche schaukeln sich daran ziemlich hoch, was mitunter zu Feindschaft zwischen verschiedenen Gruppen f\u00fchren kann. Da das Grundprinzip doch eher dem Zusammenhalt gilt, arbeiten sie damit gegen sich selbst und gegen die Bewegung an sich.\n\n#### Wie bewertest du die \u00f6ffentliche Wahrnehmung der autonomen Szene?\n\nDie \u00f6ffentliche Wahrnehmung der autonomen Szene ist dank der Medien ziemlich schlecht. Wenn es IRGENDWO in IRGENDEINER Stadt IRGENDWIE Krawalle gab, hei\u00dft es sofort, die b\u00f6sen, b\u00f6sen Autonomen waren das! Jedoch gibt es aktuell Hoffnung, auch wenn sie eher tragischer Natur ist. Die Anti-Atom-Bewegung bekommt ziemlich gro\u00dfen Zuwachs auch von \"Otto-Normal-B\u00fcrgern\", denen es langsam reicht mit dem Hin und Her bei der Atomkraft... und dadurch hoffe ich, bekommen die Menschen einen besseren Einblick in die Szene, in das, was wir machen, und ich hoffe, sie verstehen uns dann besser.\n\n#### Geh\u00f6rt Gewalt zur autonomen Szene dazu?\n\nNat\u00fcrlich geh\u00f6rt Gewalt dazu, ohne ein bisschen Gewalt kann man heute nichts Gro\u00dfes mehr bewegen. Allerdings muss man dabei bedenken, in welche Richtung diese Gewalt gehen darf und soll. Wenn es w\u00e4hrend oder nach einer Demo zu \"Krawallen\" kommen sollte, dann sollte es nicht gegen das Auto des normalen Menschen gehen, der dann am n\u00e4chsten Tag nicht zur Arbeit kommt, sondern gegen Gegenst\u00e4nde des Staates. Denn es geht ja gegen den Staat. Denkst du, dass in \u00c4gypten oder den umliegenden L\u00e4ndern, wo aktuell ein Umbruch in der Politik stattfindet, etwas erreicht worden w\u00e4re, wenn man friedlich demonstriert und vielleicht ein paar Sitzblockaden errichtet h\u00e4tte? Das tun die Menschen dort schon lange und nichts hat es gebracht. Ich will Gewalt nicht verherrlichen, auch m\u00f6chte ich nicht deutlich machen, dass das Ziel der Autonomen eine Revolution in Deutschland ist, dennoch geh\u00f6rt Gewalt dazu, aber haupts\u00e4chlich gegen Gegenst\u00e4nde und nicht gegen Menschen. Obwohl wir uns gegen die Gewalt der Bullen und der Faschos wehren m\u00fcssen.\n\n#### Welche M\u00f6glichkeiten hat die autonome Szene, etwas zu ver\u00e4ndern?\n\nWenn wir als geschlossene Einheit auftreten, sind uns schon viele M\u00f6glichkeiten gegeben. Wir m\u00fcssen das Interesse der Gesellschaft wecken und Ankn\u00fcpfungspunkte finden.\n\n#### Okay, letzte Frage, was w\u00fcrdest du noch gerne sagen?\n\nDas war ein sehr langes Gespr\u00e4ch, du hast mir Fragen gestellt, \u00fcber die ich vorher so noch nicht nachgedacht hatte, das war interessant und aufschlussreich. Meine letzte Frage ins Mikro: Willste auch ein Bier?\n\n#### Was hab ich wohl geantwortet?\n\nDas Interview f\u00fchrte Bettina Thiele.\n\n## \"Extreme Aktionen wie gegen die sogenannte Grauzone finde ich daneben. Bands verbieten, obwohl man sie nicht wirklich kennt und sich nicht mit ihnen unterhalten hat, ist schei\u00dfe.\"\n\nViktor, 25 Jahre, lebt in Berlin, geboren in der DDR ein Jahr vor dem Mauerfall.\n\n#### Lebst du in einer festen Partnerschaft?\n\nMomentan nicht.\n\n#### Welche Ausbildung haben deine Eltern? Wie ist ihre berufliche Situation heute?\n\nMein Vater hat Schlosser gelernt und wurde sp\u00e4ter aus dem Betrieb rausrationalisiert. War zwei Wochen arbeitslos und ist jetzt beim Objektschutz, als Sicherheitsmensch. Und meine Mutter war im Bankwesen t\u00e4tig und wurde auch aus dem Betrieb rausgeworfen und ist jetzt am Flughafen, auch als Sicherheitsangestellte.\n\n#### Was ist deine gegenw\u00e4rtige Situation?\n\nIch mache gerade mein Abitur nach. Gelernt hab ich Maurer, das macht aber einfach keinen Sinn gerade.\n\n#### Wie bewertest du deine aktuelle Situation?\n\nEs k\u00f6nnte finanziell alles etwas besser sein, so mit Baf\u00f6g ist nicht so doll, aber eigentlich bin ich ein gut gen\u00e4hrter Mitteleurop\u00e4er. In Zukunft hoffentlich besser. Vorw\u00e4rts immer! [Lacht.]\n\n#### Wie sehen deine Tr\u00e4ume und Pl\u00e4ne f\u00fcr die Zukunft aus?\n\nAlso das Abitur ist so ein bisschen daf\u00fcr da, mich zu orientieren, ich habe noch keine richtigen Pl\u00e4ne. Ich will sp\u00e4ter auf jeden Fall mal eine vern\u00fcnftige Arbeit haben, irgendwas Sinnvolles in der Gesellschaft. Keine Ausbeutung halt. Vielleicht gibt es bis dahin ja einen Mindestlohn.\n\n#### Wovor hast du Angst?\n\nGesellschaftlich kann mich eigentlich nichts mehr erschrecken... Pers\u00f6nlich eigentlich nur, mal Opfer eines Angriffs von Nazis zu werden. Gerade bei uns, wo grad wieder ein neuer Naziladen er\u00f6ffnet hat...\n\n#### Was \u00e4rgert dich am meisten?\n\nDiese ganze gesellschaftliche Moral, so von oben beherrscht zu werden. Die Leute, die Geld haben, die einen von oben herab behandeln. Und die Politiker, die keinen Bezug mehr zur Gesellschaft haben und sich da irgendwas zurechtlegen, was nicht wirklich zu den Menschen passt.\n\n#### Was w\u00fcrdest du anders machen als deine Eltern?\n\nAn Stelle meines Alten w\u00fcrde ich meine Kinder nicht schlagen. Ansonsten... sie sind arbeitsame Menschen... Das bin ich eigentlich auch, also da w\u00fcrde ich nichts anders machen.\n\n#### Haben deine Eltern oder andere Verwandte eine Rolle f\u00fcr deine politische Entwicklung gespielt?\n\nSchon irgendwie... Also meine Family ist aus dem \"roten Politgewerbe\", also DDR noch, mein Opa war Auslandskader und war dadurch in Kuba und verschiedenen afrikanischen L\u00e4ndern und dadurch habe ich schon einiges mitbekommen. Mein Uropa war im KZ und ist kurz nach dem Krieg gestorben.\n\n#### Hast du Kinder?\n\nIch habe keine Kinder, aber irgendwann m\u00f6chte ich mal welche haben. Aber erst mal muss ich mich selbst orientieren und mein Leben ordnen.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nEin sehr ernster Mensch. Bei richtiger Motivation auch arbeitsam.\n\n#### Was, meinst du, denken andere von dir?\n\nAlso ich denke, einige halten mehr auf mich, als mir zusteht. Aber andere untersch\u00e4tzen mich. Weil ich ein schweigsamer Typ bin.\n\n#### Welche Hobbys hast du?\n\nFicken \u2013 Saufen \u2013 Oi!... Punk-Konzerte, Pogen, ich interessiere mich f\u00fcr Politik...\n\n#### Bist du in einem Verein aktiv?\n\nIch geh zum Fu\u00dfball, also als Unterst\u00fctzer, aber ansonsten habe ich mich noch an keine Gruppe gebunden. Ich wei\u00df zwar, die Mehrheit macht uns stark, aber ich kann einfach nicht jeder Idee hinterherlaufen. Also Gruppen sind schwierig, eine komplizierte Sache. Zum Beispiel, wenn ich sagen w\u00fcrde, ich bin in der Antifa, und die lehnen es ab, mit einer bestimmten Gruppe von Menschen zu reden, funktioniert das f\u00fcr mich nicht, man kommt einfach zu keinem Ergebnis, wenn man nicht miteinander redet.\n\n#### Bist du anderweitig gesellschaftlich\/politisch aktiv?\n\nIm allt\u00e4glichen Leben immer. Gerade jetzt in der Schule entstehen viele Diskussionen mit anders eingestellten Personen.\n\n#### Sind die interessiert oder eher ablehnend?\n\nEher interessiert. Also es gibt nat\u00fcrlich Ausnahmen... ein paar Konservative, die der Meinung sind, ohne Kapitalismus funktioniert die Welt nicht...\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nEine gro\u00dfe! Punk ist ja schon eine philosophische Kunst, teilweise, was so in den Texten drin steckt. Sich damit auseinanderzusetzen, das macht man ja schon, seit man 12, 13, 14 Jahre ist.\n\n#### Hast du eine Lieblingsmusik?\n\nOi!, Punk, Ska, Reggae, Hardcore, so die Richtung. Ab und zu auch mal eine Trash-Party.\n\n#### Interessierst du dich f\u00fcr Mode?\n\nHat ja in der autonomen Szene wohl alles mit der Farbe schwarz zu tun. Zur Solidarisierung trage ich auch schwarz, gerade auf Demos st\u00e4rkt es den Zusammenhalt. Aber so richtig \"Mode\"-m\u00e4\u00dfig bin ich eigentlich nicht interessiert.\n\n#### Hast du Vorbilder?\n\nRichtige Vorbilder nicht, aber beeindruckend waren so Pers\u00f6nlichkeiten wie Karl Marx, Mahatma Gandhi, Leute, die geistig sich bet\u00e4tigt haben. Aber eher als Einfluss, nicht als Vorbild. Mit vierzehn Jahren ungef\u00e4hr hatte ich mal einen T\u00fcrsteher als Vorbild. Ein starker Kerl, der sich durchsetzen konnte, aber auch geistig was auf dem Kasten hatte.\n\n#### Wie verbringst du deine Freizeit?\n\nMit einer gro\u00dfen Clique in einem Jugendklub. Wir kochen, wir trinken, wir quatschen oder planen. Man regt sich gemeinsam auf. F\u00fcrs Abi lernen. Feiern gehen.\n\n#### Nach welchen Kriterien schlie\u00dft du Freundschaften?\n\nAlso man muss mit einem Menschen reden k\u00f6nnen. Und derjenige muss nicht gegen andere hetzen. Und dann ergibt sich das so... Meistens lernt man Freunde ja in der \"Szene\" kennen, also hat man schon die gleichen Interessen. Also man lernt auch Leute in der Schule kennen, aber da ist es schwieriger. Zum Beispiel ist bei uns ein M\u00e4del in der Klasse, \"aus gutem Elternhaus\" mit ihrem Gucci oder Louis-Vuitton-T\u00e4schchen... und die hat halt kaum Ahnung, was sie auch selbst mal in Politikwissenschaften meinte, sie hat gesagt: \"Manchmal denke ich wirklich, ich bin dumm.\" Mit jemandem wie ihr, mit dem man sich auch nicht unterhalten kann, ist es halt schwieriger, eine Freundschaft zu schlie\u00dfen.\n\n#### Geh\u00f6rst du einer Jugendkultur an?\n\nAuf jeden Fall, Punk-Szene, linke Szene, alles in der Richtung so.\n\n#### Welche Jugendkulturen findest du absolut unsympathisch?\n\nSo Rock 'n' Roller\/Psychos, die so \"rummackern\", das ist nicht mein Ding. Oder so Popper, die oberfl\u00e4chlich sind. Nazis, CDU-Jugend, alles so rechts au\u00dfen halt.\n\n#### Und welche Rolle spielt die autonome Szene f\u00fcr dich?\n\nEine wichtige Rolle in meinem Leben. Allein schon immer neue Spr\u00fcche auf Aufn\u00e4hern, die einen zum Denken anregen. Ansonsten gibt es auch in der Szene einige Gruppen, die zu elit\u00e4r sind, was einen ankotzt, dass die Leute mit Gewalt sich \u00fcber andere hinwegsetzen und \u00fcber diese urteilen. Wodurch einige doch fast wieder wie Faschos sind.\n\n#### Woher beziehst du deine Informationen zum aktuellen Zeitgeschehen?\n\nEigentlich alles, was es so gibt, also Zeitung, Fernsehen, Internet. Und man tauscht sich \u00fcber Freunde aus.\n\n#### Welche Medien nutzt du generell?\n\nIch bin wohl ein guter Deutscher und gucke jeden Abend die Abendschau. Manchmal lese ich die junge welt, weil die auch Hintergrund- und Geschichtswissen mit vermitteln.\n\n#### Geh\u00f6rst du einer Religion oder Kirche an?\n\nNein. Religion ist f\u00fcr mich Opium f\u00fcrs Volk. Es bringt die Leute in eine gewisse Abh\u00e4ngigkeit. Schafft aber auch Zusammengeh\u00f6rigkeit. Allerdings bei der christlichen Kirche die Historie mit Hexenverbrennung usw. oder in der neueren Zeit die Missbrauchsskandale, das macht sie auf jeden Fall nicht sympathischer. Dann eher Buddhismus oder \u00c4hnliches. Also eher was Freies, wo keine so strengen Regeln vorgegeben werden.\n\n#### Wie w\u00fcrdest du \"autonom\", \"autonome Szene\" eigentlich definieren?\n\nSich von dem gesellschaftlichen Denken zu l\u00f6sen und selbst etwas zu machen und selbst\u00e4ndig zu denken. Dass man versucht, sich immer frei zu entscheiden, sich nicht unterdr\u00fccken zu lassen und auch keine anderen zu unterdr\u00fccken. Und f\u00fcr mich geh\u00f6ren die Grunds\u00e4tze von Antifaschismus, Anarchismus, Antikapitalismus dazu, so als Voraussetzung.\n\n#### Bist du Mitglied einer konkreten Gruppe?\n\nNur was Fu\u00dfball angeht. Wir sind mittlerweile so 20\u201325 Leute und versuchen auch, das rechte Gesocks von unserem Verein fern zu halten. Steht auch in der Vereinssatzung richtig drin, allerdings wurde es bisher nicht durchgesetzt. Aber das wollen wir jetzt machen, weil momentan l\u00e4uft noch so 'n M\u00e4del durchs Vereinsheim, und da wollen wir unseren Einfluss st\u00e4rken, dass die nichts mehr dort zu suchen hat. Wir sind jedoch keine Hooligans, eher Ultras.\n\n#### Wie war dein erster Kontakt zur autonomen Szene?\n\nDas fing halt irgendwie an \u00fcber so Sachen wie RAF, \"die b\u00f6sen Terroristen\", was man halt so mitbekommen hat \u00fcber die Medien. \u00dcber eine Ex-Freundin habe ich damals einige Leute und L\u00e4den der linken Szene kennengelernt und bin ins Gespr\u00e4ch mit verschiedenen Menschen gekommen, und dann hat sich ein Interesse entwickelt.\n\n#### Gibt es einen eigenst\u00e4ndigen \"autonomen\" Stil?\n\nDer Trend, schwarz herumzulaufen, dann die ganzen DIY-Geschichten, also sich selbst Accessoires, Taschen oder so zu n\u00e4hen. Tunnels, Piercings, so was, daran erkennt man die Leute schon. Allerdings werden die Sachen leider auch von anderer Seite \u00fcbernommen, so dass es langsam schwieriger wird. Wenn man 'nen Antifa und einen Anti-Antifa nebeneinander stellt ohne Aufn\u00e4her und Buttons, hat man kaum eine Chance, die zu unterscheiden. Das ist so die Gefahr bei den Leuten mit der neuen autonomen rechten Bewegung, dass sie sich versuchen einzuschleichen.\n\n#### Setzt du dich auch mit Theorie auseinander?\n\nJa, mit Marxismus. Eine Zeitlang habe ich den Kommunismus gehasst aufgrund von Stalinismus und DDR. Dann habe ich mich aber erst mit Lenin befasst, der auf dem Sterbebett schon vor Stalin gewarnt hat, und anschlie\u00dfend mit Trotzki, der ein gro\u00dfer Feind von Stalin war. Also wurde ja alles irgendwie versaut durch Stalin.\n\n#### Wie informierst du dich \u00fcber szeneinterne Aspekte? Welche szeneinternen Medien nutzt du?\n\nDie Clique als Netzwerk, Internetseiten, so was wie Stressfaktor und Pukemusic f\u00fcr Konzerte, Indymedia...\n\n#### Wie stehst du zu Parteipolitik? Gehst du w\u00e4hlen?\n\nIch gehe w\u00e4hlen, auch wenn ich mir denke, dass es nichts bringt. Ich w\u00e4hle halt die Linke, oder die Piraten sind seit kurzem auch eine interessante Partei...\n\n#### Wie ist dein Verh\u00e4ltnis zum Staat, zur Polizei etc.?\n\nPolizei, Staat, Staat, Polizei... Polizeistaat... Also es muss eine Art Organisation geben, sonst w\u00fcrde alles schiefgehen. Und es muss auch eine Art Ordnungskraft geben, so wegen Diebstahl und Gewalt, aber das ist auch wieder ein soziales Problem. Wenn alle gleichberechtigt w\u00e4ren und dasselbe h\u00e4tten, d\u00fcrfte es zumindest keinen Neid geben. Wenn man also die M\u00f6glichkeit h\u00e4tte, mit ehrlicher Arbeit alles zu erreichen, dann w\u00e4re das nicht so schlimm. Ansonsten, die Polizei ist halt auch so ein elit\u00e4rer Club, die denken, sie sind was Besseres, sollten sie auch eigentlich sein, also so ein St\u00fcck weit eine Vorbildfunktion \u00fcbernehmen. Benehmen sich aber oftmals unter aller Sau.\n\n#### Wo siehst du Kritikpunkte, Probleme in der autonomen Szene?\n\nManchmal an einzelnen Leuten, die mehr reden als machen... Gut finde ich, dass die autonome Szene eigentlich f\u00fcr alle offen ist und einen weiten Kreis an Leuten zul\u00e4sst. Also eigentlich gibt es nicht so viele Kritikpunkte bis auf die Spaltungsgeschichte mit den Anti-Ds und Antiimps. Und einige, die meinen, sie haben die absolute Meinung und davon darf es keine Abweichung geben. Also es gibt halt schwarze Schafe.\n\n#### Geh\u00f6rt Gewalt zur autonomen Szene dazu?\n\nEs ist einerseits ein Schutz, dass nicht jeder dumme Mitl\u00e4ufer mit reinkommt, aber es hindert auch am Vorankommen der Leute. Ich denke, es geh\u00f6rt dazu und man wird anders nicht mehr wahrgenommen. Es ist momentan notwendig, um etwas zu \u00e4ndern. Das merkst du sp\u00e4testens, wenn du \u00fcber Jahre auf Demos gehst ohne extremere Aktionen und es \u00e4ndert sich gar nichts!\n\n#### Welche Einstellung hast du zu militanten Aktionen?\n\nAlso extreme Aktionen wie momentan gegen die sogenannte \"Grauzone\" finde ich daneben; ich kann nicht Menschen ausgrenzen und blo\u00dfstellen und Sachen verbieten, nur weil jemand nicht so ein gro\u00dfes politisches Interesse hat, das ist daneben. Bands verbieten, obwohl man sie nicht wirklich kennt und sich nicht mit ihnen unterhalten hat, ist schei\u00dfe. F\u00fcr solche Aktionen wie am Ostkreuz, wo Stromkabel gekappt wurden, bin ich eher zu haben... Ist ja irgendwie schon antikapitalistisch und einfach mal die Aufmerksamkeit darauf zu lenken, dass die DB ein Ausbeuterverein ist, bei denen der Service immer schlechter wird.\n\n#### Welche M\u00f6glichkeiten hat die autonome Szene, etwas zu ver\u00e4ndern? Besteht \u00fcberhaupt das Interesse, die Gesellschaft als Ganzes zu \u00e4ndern, oder geht es nur darum, die eigenen Freir\u00e4ume zu bewahren?\n\nSie haben schon ein gro\u00dfes Ziel, also auch ein gesamtgesellschaftliches Interesse. Sicher auch die eigenen Freir\u00e4ume, aber auch die ganze Gesellschaft, weil was jetzt ge\u00e4ndert wird, ist ja auch f\u00fcr unsere Kinder ausschlaggebend. Autonome Menschen sind global denkende Menschen... Wenn man in Amerika die Leute sieht, die mit 500 kg auf den H\u00fcften rumrennen, und dagegen die afrikanischen Kinder verhungern sieht, will man schon mehr ver\u00e4ndern.\n\n#### Wie sch\u00e4tzt du die Zukunft der autonomen Bewegung ein?\n\nSie kann immer nur Nadelstiche setzen... Ist ja wie die Durchsetzung einer annehmbaren Arbeitswoche und einer Rente, Arbeitslosengeld, das zu erreichen, hat einige Zeit gedauert...\n\n#### Wie gro\u00df ist die autonome Szene in deiner Stadt, wie gro\u00df insgesamt in Deutschland? Gibt es da Ver\u00e4nderungen?\n\nSo circa 3.000 Leute in Berlin. Aber ich kann das schlecht einsch\u00e4tzen, da sie in manchen St\u00e4dten doch gr\u00f6\u00dfer ist und in anderen fast gar nicht existiert. Statistisch ist es auch schwieriger, weil: Wen z\u00e4hlt man wirklich zu den Autonomen?\n\n## \"Die Arroganz und Naivit\u00e4t der Menschen heute macht mir schon Angst.\"\n\nHenry, 29 Jahre, lebt in Berlin.\n\n#### Was ist dein Familienstand? Lebst du in einer festen Partnerschaft?\n\nIch bin ledig.\n\n#### Welche Ausbildung haben deine Eltern? Wie ist ihre berufliche Situation heute?\n\nMein Vater ist im \u00d6ffentlichen Dienst und meine Mutter hat diverse Ausbildungen und Umschulungen hinter sich, derzeit arbeitet sie als Laborantin.\n\n#### Was ist deine gegenw\u00e4rtige Situation?\n\nIch studiere und arbeite nebenbei, die Branche lasse ich mal lieber ungenannt.\n\n#### Wie bewertest du deine aktuelle Situation? Und deine Zukunft?\n\nAn sich hab ich es noch recht gut. Durch mein Studium lerne ich viele interessante Dinge und das in einem Bereich, der noch viele Jahrzehnte ben\u00f6tigt wird. Durch meinen Nebenjob bin ich unabh\u00e4ngig von irgendwelchen Staatsgeldern und mache mir an sich f\u00fcr die Zukunft nicht wirklich viele Sorgen, weil ich auch nicht der Mensch bin, der weit voraus denkt, lieber nur in Wochen, nicht Jahren. Mir schwebt eher vor, das Land irgendwann zu verlassen und mein Gl\u00fcck woanders zu suchen (wenn ich mal weit voraus denken m\u00fcsste), da ich von diesem Staat nicht viel halte, was nicht bedeutet, dass alle anderen Staaten in irgendeiner Weise \"besser\" w\u00e4ren. Ich sehe nur einfach f\u00fcr die Zukunft der Menschen in diesem Staat schwarz, da dieser auf dem besten Wege ist, sich zu einem \u00dcberwachungsstaat, wie es die USA oder Gro\u00dfbritannien schon sind, zu verwandeln, und ich will auf gewisse Freiheiten einfach nicht verzichten m\u00fcssen.\n\n#### Wovor hast du Angst?\n\nMeine Freunde zu verlieren, aber auch, dass die jetzige Entwicklung weitergeht. Mit Ausbeutung von armen Menschen und der generellen Umverteilung von unten nach oben. In diesem Zusammenhang hab ich die meiste Angst davor, dass es den Menschen weiterhin so gleichg\u00fcltig ist, wie es derzeit der Fall ist, da sie medial auf andere \"Probleme\" gelenkt werden, wie die \"b\u00f6sen Hartz-IV-Empf\u00e4nger\" oder \"ausl\u00e4ndische Banden\" und dieser ganze Mist. Wenn sich alles weiter in eine \u00fcberwachte Gesellschaft mit gl\u00e4sernen Menschen verwandelt oder gar wieder den Weg in eine Diktatur geht, ohne dass Widerstand geleistet wird... Das macht mir schon arg Angst, da die Arroganz und Naivit\u00e4t der Menschen schon heute zeigt, dass das nicht abwegig ist.\n\n#### Was \u00e4rgert dich am meisten? Was macht dich w\u00fctend?\n\nGenau dieser Egoismus des Menschen oder noch viel schlimmer, die Naivit\u00e4t: Denn viele suchen die Probleme oder Fehler eben nicht dort, wo sie entstehen bzw. herkommen, also sozusagen ganz oben, bei den Regierungen, sondern lassen sich ganz einfach lenken und schimpfen dann auf die \"Ausl\u00e4nder\" und Hartz-IV-Empf\u00e4nger und \u00c4hnliche. Lassen sich generell von Medien verbl\u00f6den und leben in ihrer kleinen Auto-TV-Balkon- und Was-geht-mich-fremdes-Elend-an-Welt. Das ist keineswegs so arrogant gemeint, wie es klingt, aber in diversen Situationen bin ich auf genau diese Haltung getroffen. Dass fremde Probleme eben nicht die eigenen sind, also \"einen auch nix angehen\", und Dinge, gegen oder f\u00fcr die andere k\u00e4mpfen (f\u00fcr eine bessere Welt, gegen Rassismus etc.), ja eh nur utopischer Schwachsinn seien. Und genau diese Arroganz und Naivit\u00e4t macht mich w\u00fctend...\n\n#### Was w\u00fcrdest du anders machen als deine Eltern?\n\nNun ja, meine hatten mich nach der Schilderung meiner ersten Demo-Erfahrung extrem unter Druck gesetzt und mir den Besuch von Demos und \u00c4hnlichem verboten. Das hat nat\u00fcrlich nichts gebracht und mein Engagement nicht gemindert, ich musste mir nur immer neue Ausreden ausdenken, wieso ich mal wieder zwei Tage nicht zu Hause sein werde. Sie hatten einfach Angst um mich, sowohl aus gesundheitlichen als auch strafrechtlichen Gr\u00fcnden, was ich ihnen ja nicht vorwerfen kann und werde, aber etwas mehr Zuspruch oder generell ein Gespr\u00e4ch dar\u00fcber h\u00e4tte nicht geschadet.\n\n#### Wie war denn deine erste Demo-Erfahrung?\n\nDas war Ende 2000, beim sogenannten \"Aufstand der Anst\u00e4ndigen\", als rund 1.500 Faschos in Berlin vom Ostbahnhof \u00fcber'n Alexanderplatz bis zur Friedrichstra\u00dfe marschieren wollten. Ich hatte bis dahin nur bei Schulstreiks oder \u00c4hnlichem mitgemacht, aber nie aus politischer \u00dcberzeugung eine Demo besucht. Ein Schulkumpel hatte mich dann einfach gefragt, ob ich nicht mal dabei sein m\u00f6chte, und wir sind zu zweit recht fr\u00fch los zum Alexanderplatz, um mal die Lage zu checken, und es waren schon massig Leute, vor allem Punks und Antifas unterwegs, viele sogar bewaffnet (mit Holzlatten und \u00c4hnlichem), was mich anfangs etwas abgeschreckt hatte. Aber wir sind dann der Nazi-Demo entgegen gelaufen, mussten uns vor den Polizeiketten zur\u00fcckziehen, die die Stra\u00dfe vor der Demo r\u00e4umen sollten, und hatten uns mit gut 150 Antifas zur\u00fcckgezogen, um dann, als die Nazi-Demo vorbeilief, durch die Bullenketten durchzubrechen und die Nazi-Demo anzugehen. Es flog einiges, aber es kam nicht zu gro\u00dfer Gewalt, bis aus anderen Seitenstra\u00dfen immer mehr vermummte Gruppen erschienen. Wir sind dann Richtung Alexanderplatz weiter und haben uns dort den Blockierern angeschlossen, die kurze Zeit sp\u00e4ter zusammengekn\u00fcppelt wurden, aber wegen der Masse an Leuten sind die Bullen nicht durchgekommen und die Nazi-Demo wurde umgeleitet in eine Seitenstra\u00dfe. Ab diesem Moment gab es einen erheblichen Gewaltausbruch und es flog alles in die Nazi-Demo; es kam zu heftigen K\u00e4mpfen und die Nazi-Demo wurde beendet und die Faschos in Z\u00fcge gestopft. In der Seitenstra\u00dfe ging es dann noch einige Zeit weiter nur gegen die Polizei.\n\nWas mich an diesem Tag so faszinierte, war nicht die Gewalt, sondern die Entschlossenheit der Gegendemonstranten (der aktiven, nicht die komische Kundgebung am Roten Rathaus damals) und der Zusammenhalt bei Verletzungen oder \u00c4hnlichem. Das war mir in der Form komplett neu und sorgte daf\u00fcr, dass ich von nun an \u2013 gegen das Verbot meiner Eltern \u2013 an immer mehr solchen Aktionen, Demos usw. teilgenommen habe. Ich meine, da haben sich Leute, die sich nicht kannten, gegenseitig geholfen, bei Verletzungen versorgt, sich gegenseitig verteidigt und gegen die Staatsmacht aufbegehrt, weil sie die gleichen Interessen bzw. Feinde hatten, und das war ein gro\u00dfartiges Gef\u00fchl, Teil einer solchen Masse zu sein. Die Bereitschaft, f\u00fcr etwas einzustehen, obwohl Repression und Verletzungen die Folgen sein k\u00f6nnen, fand ich unheimlich beeindruckend.\n\n#### Haben deine Eltern oder andere Verwandte eine Rolle f\u00fcr deine politische Entwicklung gespielt?\n\nFamili\u00e4r gab es keinerlei politischen Einfluss. H\u00f6chstens von meinem Gro\u00dfvater, welcher im Dritten Reich einiges an Gr\u00e4ueltaten der Nazis miterlebte und immer auch stolz auf mein antifaschistisches Engagement war. Er hat es nie direkt ge\u00e4u\u00dfert, aber so etwas ist schon erkennbar.\n\n#### Hast du Kinder?\n\nIch hab noch keine und plane auch noch keine, aber in Zukunft sicher. Am liebsten jedoch nicht unbeabsichtigt, sondern schon mit der richtigen Person.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nSo etwas fiel mir schon immer sehr schwer. Vielleicht sehr \u00fcberm\u00fctig, was das Politische angeht, auf der anderen Seite aber auch schon mal faul. Mitunter rechthaberisch, aber auch hilfsbereit und sozial engagiert.\n\n#### Was, meinst du, denken andere von dir?\n\nAndere w\u00fcrden mich wohl als ziemlich nerdig sehen. Au\u00dferdem kann ich auch schon mal etwas nervig sein, gelte aber auch als sehr entertainend. Im Arbeitsleben werde ich als \u00e4u\u00dferst stressresistent und humorvoll wahrgenommen.\n\n#### Welche Hobbys hast du?\n\nIch schaue sehr gerne Filme und Serien, h\u00f6re viel Musik, spiele selbst in einer Band und besuche sehr gerne Konzerte. Ansonsten noch weggehen und ab und an Kampfsport.\n\n#### Bist du in einem Verein aktiv?\n\nJa, ich bin ehrenamtlich t\u00e4tig in einem Jugendverein.\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nMusik hat f\u00fcr mich eine sehr gro\u00dfe Bedeutung. Sie ist ein sehr guter Zeitvertreib, aber auch ein klasse Weg, um politische Statements zu verbreiten, Menschen zu verbinden und kennenzulernen sowie Kontakte zu kn\u00fcpfen. Au\u00dferdem auch, um Erfahrung zu verarbeiten oder sich abzulenken. Musik zu machen ist auch ein guter Ausgleich, um Stress abzubauen.\n\n#### Hast du eine Lieblingsmusik?\n\nNein, ich bin vielem gegen\u00fcber aufgeschlossen, was handgemacht ist, wie Hardcore, Punk und \u00c4hnliches. Musik muss f\u00fcr mich handgemacht sein, zusammengeklickte Musik hat mir noch nie gefallen. Als Ausgleich ist aber auch Klassik super, als Musiker hat man eh ein anderes Verh\u00e4ltnis zur Musik als der g\u00e4ngige Konsument.\n\n#### Welche sonstigen (pop)kulturellen Medien konsumierst du (zum Beispiel Literatur, Film, Serien, Kunst etc.)?\n\nInternet, insbesondere Internet-Meme (deren Entwicklung und Nutzung, also Dinge wie Lolcats, Trollface usw.). Filme, sehr gerne Kino, dort gibt es kaum Filmgenres, die ich nicht gucke. Von Kom\u00f6dien bis Horror mag ich eigentlich fast alles. Serien mag ich auch besonders gerne, aber am liebsten in Originalsprache und da am meisten Kom\u00f6dien. Bei B\u00fcchern interessieren mich vor allem Fantasy- und Sachb\u00fccher. Auch Dokumentationen schaue ich sehr gerne, insbesondere historische Dokus.\n\n#### Hast du Vorbilder, Leitbilder, Idole?\n\n\"Kill your idol!\" Nein. Ich halte absolut nichts davon, irgendwelchen Personenkulten zu folgen oder Menschen zu verg\u00f6ttern oder auch nur deren Gesicht auf einem Shirt zu tragen. Es gibt und gab Menschen, deren Engagement ich beeindruckend finde, deren Taten ich bewundere usw., aber das hei\u00dft nicht, dass ich nun mit einem Subcommandante-Marcos-Shirt herumlaufe oder unbedingt wie Gandhi leben m\u00f6chte; ich versuche, mein Leben lieber in die eigene Hand zu nehmen. Aber gewisse Ideen oder Anreize kann man sich da ja schon holen.\n\n#### Wie verbringst du deine Freizeit? Mit wem und wo?\n\nEs gibt keinen festen Ort und Zeit, mit wem ist auch nicht festgefahren. Es ist ein weiter Freundes- und Bekanntenkreis, viele Veranstaltungen, Freunde und Plena (auch deutschlandweit). Ansonsten noch viel Internet, Filme, Serien, Bandproben und -auftritte.\n\n#### Nach welchen Kriterien schlie\u00dft du Freundschaften?\n\nPolitische Gesichtspunkte spielen schon eine Rolle. Da Freundschaften \u00fcber l\u00e4ngere Zeit entstehen, merkt man schnell, wie Menschen zu Sachen und Situationen stehen. Deshalb w\u00fcrde ich einem Fascho keinen Raum lassen f\u00fcr eine Freundschaft. Es gibt Leute, mit denen ich nichts zu tun haben m\u00f6chte aufgrund ihres Verhaltens anderen Menschen gegen\u00fcber: sexistisches Verhalten, Bestehlen, Hintergehen von Menschen etc., in meinen Augen unmenschliches und respektloses Verhalten. Freundschaften entwickeln sich, da habe ich keine Schablone.\n\n#### Geh\u00f6rst du einer Jugendkultur an?\n\nIch war mal Punk, na ja, eigentlich nicht so richtig, wenn ich ehrlich sein soll. Ich hab mir aus verschiedenen Szenen das f\u00fcr mich Praktische rausgesucht oder was mir gefallen hat (aus Gothic, Punk, Hardcore). Generell mag ich dieses Schubladendenken nicht so.\n\n#### Gibt es Jugendkulturen, die du sympathisch findest?\n\nSympathisch sind f\u00fcr mich Gothic, Punk, Hardcore, sonst finde ich nur noch Sachen wie zum Beispiel Extremsport interessant, beispielsweise Skaten, Klettern etc. Klar, Skins auch \u2013 \"Skins & Punks united\", ihr generelles antirassistisches Grundverst\u00e4ndnis etc. \u2013, dennoch gef\u00e4llt mir ihr Au\u00dfenbild und Machogehabe nicht. Auch in anderen Jugendkulturen gibt es Sachen, die unheimlich interessant sind.\n\n#### Welche Jugendkulturen findest du absolut unsympathisch?\n\nAbsolut unsympathisch gibt es eigentlich nicht, solange sie kein d\u00e4mliches Machogehabe an den Tag legen oder irgendwelchen rechten Mist tolerieren oder gar aus\u00fcben.\n\n#### Mit welchen Leuten kommst du \u00fcberhaupt nicht klar?\n\nMit Nazis, Menschen, die sich anderen gegen\u00fcber nicht respektvoll benehmen. Auch mit v\u00f6llig starren \"Sekten\" komme ich gar nicht klar \u2013 so Sachen wie \"Du bist nicht f\u00fcr mich, also bist du gegen mich\" finde ich absolut bescheuert. Ich habe mit diesen Personen keine Basis und kann auch ihr Verhalten nicht tolerieren, weshalb ich es nicht einsehe, mich mit diesen Menschen auseinanderzusetzen.\n\n#### Welche Rolle spielt die autonome Gruppe oder Szene f\u00fcr dich?\n\nIch selbst bin in keiner Gruppe, deswegen spielt eine bestimmte Gruppe keine gro\u00dfe Rolle f\u00fcr mich, die Szene allerdings schon. Diese ist eine Ansammlung von Menschen mit \u00e4hnlichen Zielen, mit denen man viel zusammen machen kann, die zu einigen Dingen bereit sind und auch gewisse Wege mit bestreiten, zu denen andere nicht bereit sind.\n\n#### Woher beziehst du deine Informationen zum aktuellen Zeitgeschehen?\n\nInternet, gruppenspezifische Sachen durch Freunde aus spezifischen Gruppen. Ab und zu mal Radio und Zeitung, aber generell kein TV.\n\n#### Welche Medien nutzt du generell?\n\nInternet, sehr selten Radio, TV traue ich nicht und f\u00fcr t\u00e4gliches Zeitunglesen fehlt mir das Geld. Internet ist sehr individuell, man kann viel Verschiedenes lesen und sich so seine eigene Meinung bilden. Viel lesen tue ich Spiegel Online, gulli.com, heise.de und diverse Blogs, Radio nichts Festes, Zeitungen ab und zu mal Jungle World, junge welt, Neues Deutschland und taz.\n\n#### Geh\u00f6rst du einer Religion oder Kirche an?\n\nNein, ich bin Agnostiker. Ich g\u00f6nne es den Menschen, wenn sie sich dort gut aufgehoben f\u00fchlen und ihre Religion als Halt und R\u00fcckzug ben\u00f6tigen. Wenn aber Menschen versuchen, Leuten ihren Glauben aufzudr\u00e4ngen, oder andere Menschen wegen dem Glauben ausschlie\u00dfen, dann passt es mir nicht.\n\n#### Wie w\u00fcrdest du \"autonom\", \"autonome Szene\" eigentlich definieren?\n\nDas ist eine Gruppe von Menschen, die unabh\u00e4ngig von Parteien und Gewerkschaften politische Arbeit macht und dabei auch nicht immer staatlich legitimierte Mittel nutzt. Diese Gruppen sind meist lose Zusammenh\u00e4nge und hierarchielos aufgebaut. Was sie vor allem ausmacht, ist die Bereitschaft, andere, eher un\u00fcbliche Wege zu beschreiten, sei es durch Militanz oder Spa\u00dfguerilla-Aktionen. Es ist vor allem eine inhomogene Masse, was es schwerer macht, sie zu fassen bzw. zu beschreiben und genau zu definieren.\n\n#### Warum bist du nicht Mitglied einer konkreten Gruppe?\n\nWeil ich mich nicht einem Gruppenkonsens unterordnen m\u00f6chte. Ich gehe auf Veranstaltungen und Aktionen, die mich interessieren. Ich kenne viele Leute, die in Gruppen sind, und unterst\u00fctze diese auch sehr gerne bei Aktionen und Kampagnen, aber Teil einer festen Gruppe, wie der AAB damals oder der ALB heute, wollte ich nie sein.\n\n#### Was bedeutet \"autonom\" f\u00fcr dich genau?\n\nDass ich v\u00f6llig unabh\u00e4ngig mit den Entscheidungen, die ich treffe, nur mir selber gegen\u00fcber verantwortlich bin und selbst \u00fcber die Mittel entscheide. Nat\u00fcrlich kann ich nicht bei Aktionen sinnlos Leute gef\u00e4hrden oder andere angreifen, ohne Konsequenzen f\u00fcrchten zu m\u00fcssen, denn ein generelles respektvolles Verhalten sollte immer gegeben sein, und ich hab auch kein Problem damit, wenn andere, die sich entsprechend falsch verhalten, also Leute durch Unsinn gef\u00e4hrden oder auch hilflose Leute verletzen oder \u00c4hnliches, entsprechend behandelt werden.\n\n#### Was hei\u00dft das genau?\n\nNun, Leute, die bei einem Riot aus der hintersten Reihe Steine werfen, werden nie die Richtigen treffen, sondern immer nur die eigenen Leute im R\u00fccken verletzen. Da gibt es dann von Ansage und Schubsen bis richtig auf die Fresse eigentlich alles, wie mit diesen Personen umgegangen werden sollte und auch wird. Ich kann ja verstehen, wenn Leute Angst haben, weiter vorne zu stehen, und erste Schritte in den militanten Widerstand sind erf\u00fcllt von Angst, viel Adrenalin und irrationalem Verhalten, da die Situation v\u00f6llig neu, total un\u00fcbersichtlich und auch voller Gefahren ist, aber wie sollen die das anders lernen? Klar werden dadurch \"Neulinge\" verprellt, aber lieber weniger Leute, daf\u00fcr aber entschlossener und vor allem mit dem Kopf bei der Sache als sinnloses Zerst\u00f6ren und Leute gef\u00e4hrden.\n\n#### Seit wann wei\u00dft du von der Existenz einer autonomen Szene?\n\nIch sch\u00e4tze mal Mitte der 90er Jahre d\u00fcrfte ich erfahren haben, dass so etwas wie \"Autonome\" existiert, hatte es aber nicht wirklich begriffen, da ich zu dieser Zeit keinerlei Interesse an Politik hatte.\n\n#### Wie kam es denn, dass du dich pl\u00f6tzlich f\u00fcr Politik interessiert hast?\n\nPl\u00f6tzlich nicht wirklich. Wo ich aufgewachsen bin, gab es viele Faschos in den 90er Jahren und die haben auch unpolitische Menschen zusammengeschlagen und bei Langeweile durch die Kieze gejagt. Aber nicht einmal das war f\u00fcr mich so wirklich ein Grund, gegen die zu sein. Ich hab mit der Zeit einfach im Freundeskreis und in der Schule eine zunehmende Politisierung beobachtet, allerdings mitunter in beide Richtungen, also zerbrachen da auch Kontakte und es kam zu offenen Anfeindungen und ich wollte mich da nirgends mitrei\u00dfen lassen. Nach einigen Monaten oder fast zwei Jahren habe ich mir dann aber gedacht, ich mache mir selber ein Bild von beiden Richtungen und habe mir diverse Infomaterialien geben lassen (ich kannte ja sowohl Faschos als auch Antifas) und mich da einige Zeit mit belesen. Ich hab da auch viele Geschichtsb\u00fccher gelesen und kam dann irgendwann zum eigentlich offensichtlichen Schluss, dass es bei den Faschos keinerlei Argumentation f\u00fcr deren Taten und \"Ansichten\" gibt. Da hat in meinen Augen gar nichts zusammengepasst oder gestimmt, da ich nie Probleme mit Ausl\u00e4ndern hatte, sondern immer mit den Faschos, weil ich Menschen nicht wegen ihrer Religion oder Hautfarbe einordne oder angehe, sondern wegen ihres Charakters. Mir war das Geschwafel viel zu stumpf, kurz gedacht und allgemein unsinnig. Ich hab mich dann keineswegs als links bezeichnet oder bin mit Iro und \"Gegen-Nazis\"-Badge durch die Gegend gerannt. Ich habe nur f\u00fcr mich den Schluss gezogen, dass die rechte Richtung keineswegs meinem Denken entsprach, und mich eher als Antifaschist gesehen und meine Bekannten und Freunde von da an eher in diesen Kreisen gefunden. Eine weitergehende Politisierung fand dann aber nicht durch diese Leute statt, da hatte ich auch keinen Bock zu, mich in irgendeine Richtung dr\u00e4ngen oder belabern zu lassen, sondern ich hab mich bei Interesse einfach selber mit Infomaterialien versorgt und in der Bibliothek Entsprechendes ausgeliehen und gelesen. Wobei, ohne Einfl\u00fcsse sind Freunde und Bekannte ja nie, das ist mir schon klar.\n\n#### Gibt es Ankn\u00fcpfungspunkte an (andere) Jugendkulturen?\n\nSicherlich zu einigen Punks, da auch sie gepr\u00e4gt sind von der Ablehnung von Hierarchien und eigentlich alle (auch wenn ich ungern verallgemeinere) eine antifaschistische Grundeinstellung haben. Auch zum Hardcore, durch das Gerechtigkeitsempfinden und die Ablehnung von hierarchischen Strukturen. Das hei\u00dft, zu einigen Jugendkulturen gibt es Ber\u00fchrungspunkte und auch personelle \u00dcberschneidungen.\n\n#### Ist die autonome Szene auch eine Jugendkultur?\n\nBin ich eher nicht der Meinung, da es sich zwar vorwiegend um Jugendliche handelt, aber auch Erwachsene in der autonomen Szene unterwegs sind. Au\u00dferdem w\u00fcrde ich die autonome Szene als viel zu inhomogen bezeichnen, als dass so etwas wie eine allgemeing\u00fcltige Jugendkultur daraus entstehen k\u00f6nnte oder entstanden w\u00e4re, da es in so vielen Punkten keinerlei Gemeinsamkeiten gibt \u2013 beispielsweise Musik, Kleidungsstil. Klar, ich bin mir auch bewusst, dass es bei den Punks, Skins, HipHoppern usw. diverse politische Ansichten, unterschiedliche Kleidungsstile, Musikgeschm\u00e4cker etc. gibt, aber es w\u00e4re bei weitem einfacher, eine Gruppe von Personen diesen Jugendkulturen zuzuordnen als eine Gruppe von Autonomen. Bei Letzteren bezeichnen sich ja viele auch gar nicht als solche. Auch denke ich, dass eine Kommerzialisierung, wie sie mit eigentlich allen Jugendkulturen irgendwann einhergeht, bei der autonomen Szene nie der Fall sein d\u00fcrfte. Okay, beim Punk h\u00e4tte das vor drei\u00dfig Jahren auch jeder gesagt. Also mal schauen. Gruselig w\u00e4re es trotzdem.\n\n#### Gibt es einen eigenst\u00e4ndigen \"autonomen\" Stil, welche Codes etc. gibt es?\n\nStumpf gesagt: schwarze Kleidung und Kapuzenpullis, aber das w\u00e4r zu einfach. Da gibt es gl\u00fccklicherweise keine generellen \"Vorgaben\". Ich finde es eigentlich ganz gut, dass nicht alle Personen, die ich als autonom bezeichnen w\u00fcrde, auch als solche erkennbar sind. Ist ja auch immer ganz gut, so einer gesteigerten Repression vorzubeugen. Es gibt schon ein paar erkennbare Codes, aber nicht um andere auszugrenzen. Symboliken wie das Hausbesetzersymbol \u2013 Kreis mit dem blitzf\u00f6rmigen N \u2013, 1312 etc.\n\n#### Was bedeutet 1312?\n\nNun, einfach mal wie bei der 88 bei den Faschos das Alphabet benutzen. [Grinst.]\n\n#### Welche Musik spielt eine Rolle?\n\nDie Leute h\u00f6ren alles, von Elektro bis Punk. Es gibt viele Elektro- und Trash-Partys, aber auch Punk- und Hardcore-Konzerte. Alle suchen sich das raus, was sie m\u00f6gen, das ist f\u00fcr mich auch einer der Gr\u00fcnde, wieso ich es nicht als Jugendkultur bezeichnen w\u00fcrde, da diese sich ja meist \u00fcber Musik definieren.\n\n#### Welche Themen sind dir f\u00fcr dein Engagement besonders wichtig?\n\nMir selber vor allem Antifaschismus und Anti-Atom, aber auch Antirassismus, Freir\u00e4ume und Gentrifizierung. Antifaschismus und -Atom sind mir die wichtigsten, da Antifaschismus mich in die Szene gebracht hat. Anti-Atom, weil es von oben diktiert ist und da Menschen \u00fcber die Gesundheit anderer entscheiden, da wird \u00fcber Leichen gegangen, das hei\u00dft, es ist ein Wirtschaftszweig, der derma\u00dfen gegen menschliche Interessen agiert, dass dessen Weiterf\u00fchrung f\u00fcr mich ein Verbrechen darstellt. Ein kleiner Schritt dabei ist, dass in meiner WG \u00d6ko-Strom flie\u00dft, kostet nicht viel mehr und \u00e4ndert ein kleines bisschen was, vor allem in unseren K\u00f6pfen, und ansonsten Teilnahmen an den Gegenaktivit\u00e4ten bei den Castor-Transporten und \u00e4hnlichen Protesten. Auch Antisexismus ist mir wichtig, da ich mir dar\u00fcber nie wirklich Gedanken gemacht hatte, bevor ich mit dieser Problematik zu tun hatte. Nat\u00fcrlich wurde mir gleichberechtigtes Handeln von meinen Eltern beigebracht und kein Rollenleben vorgelebt, weil der Haushalt gleichberechtigt unter allen aufgeteilt war und beide Elternteile arbeiten waren, aber tiefer gehend hatte ich mich damit nie besch\u00e4ftigt, wahrscheinlich auch, weil ich es als selbstverst\u00e4ndlich erachtete, keinen Unterschied zwischen Mann und Frau in diesem Sinne zu sehen, also einen der beiden zu bevorzugen. Ich achte zwar auch in meiner Sprache darauf zu transgendern, aber das W\u00f6rtchen \"man\" benutze ich noch das ein oder andere Mal, bin aber meist darauf bedacht, Studierende zu sagen, wenn andere Studenten meinen und \u00c4hnliches. Das halten viele f\u00fcr Schwachsinn, aber f\u00fcr mich f\u00e4ngt das schon bei so kleinen Schritten wie der Sprache an.\n\nTierrecht ist mir schon als Thema wichtig, aber da hatte ich leider kaum die Zeit und Energie, um mich reinzuh\u00e4ngen, da alleine Antifaschismus schon sehr viel Kraft kostet. Und ich bin kein Vegetarier. Zum veganen Lebensstil, den ich vorziehen w\u00fcrde, denn wenn schon, dann doch richtig, fehlt mir einfach das letzte bisschen Kraft und auch das Geld, was vielleicht wie eine Ausrede klingt, aber ich investiere da mehr Energie in andere Themengebiete.\n\nAntimilitarismus ist als Thema genauso wichtig und Teilnahme an Protesten da auch mein Weg, dies zu unterst\u00fctzen, aber organisatorisch war ich bisher meist nur bei Antifa-Arbeit dabei. Das hei\u00dft, Demos organisieren, Kampagnen aufbauen, Plakatieren, Soli-Partys organisieren, Infoabende und was da alles dazu geh\u00f6rt.\n\nAnarchismus war mir als Thema nie wirklich wichtig, da ich mich nicht als Anarchist bezeichnen w\u00fcrde. F\u00fcr mich ist der Mensch viel zu arrogant und selbstverliebt, als dass eine solche Utopie funktionieren k\u00f6nnte. Da bin ich Pessimist.\n\n#### Als was w\u00fcrdest du dich dann bezeichnen? Kommunist, Sozialist?\n\nWeder noch. Am ehesten noch als Basisdemokraten. Ich w\u00fcrde mir weitaus mehr Mitsprache w\u00fcnschen, viel mehr basisdemokratische Z\u00fcge, da ich schon glaube, dass die Menschen teilnehmen w\u00fcrden, wenn sich auch was ver\u00e4ndern w\u00fcrde.\n\n#### Und wenn die Masse Atomkraftwerke m\u00f6chte...\n\nOkay, dann ist das erst einmal so, aber durch gute Kampagnenarbeit, die nicht immer viel Geld braucht, sondern viele engagierte Menschen und vor allem gute Argumente, kann auch das zum Guten ver\u00e4ndert werden, wenn auch der Gegner dort mit der dickeren Brieftasche ein schwerer Brocken ist. Rein von der Utopie w\u00fcrde ich mir Kommunismus w\u00fcnschen, wenn alle Menschen genug zum Leben h\u00e4tten und es nicht bei jedem Menschen den Wunsch nach f\u00fcnf Flatscreens und drei Autos g\u00e4be... Aber ich halte es leider f\u00fcr nicht mehr als eine Utopie.\n\n#### Setzt du dich auch mit Theorie auseinander?\n\nEher weniger, nun, Adorno und Marx habe ich schon gelesen, aber ich \u00fcberlasse es lieber den Menschen, die dazu auch die Zeit und Kraft haben.\n\n#### Wie informierst du dich \u00fcber szeneinterne Aspekte? Welche szeneinternen Medien nutzt du?\n\nIndymedia, Linksunten, Interim sind meine Hauptquellen. Freundes- und Bekanntenkreise spielen nat\u00fcrlich auch eine Rolle. Antifa.de und Umbruch spielen eine eher nachgesetzte Rolle, ansonsten diverse aktuelle Blogs bei gewissen Kampagnen und \u00c4hnliches.\n\n#### Wie stehst du zu B\u00fcndnissen zu anderen linken Bewegungen, zu sozialen Bewegungen?\n\nSehr positiv, weil man dort Leute mit anderen Ideen erreicht, da auch unterst\u00fctzen und voneinander lernen kann, insbesondere bei Anti-Atom. Die Fingertaktik ging ja damals beim G8-Gipfel 2007 bei Rostock auch sehr gut auf, da hat die Anti-Globalisierungsbewegung hervorragend von der Anti-Atom-Bewegung gelernt und es gab dort reichlich personelle \u00dcberschneidungen. Ich fand es auch richtig, der sozialen Bewegung bei den Anti-Hartz-IV-Protesten antikapitalistische Z\u00fcge zu geben, das hei\u00dft, mit entsprechenden Bl\u00f6cken aufzutreten und dem Ganzen eine radikalere Note zu geben, denn Rechte werden nun mal nicht erbettelt, sondern erk\u00e4mpft.\n\n#### Wie stehst du zu Parteipolitik? Gehst du w\u00e4hlen?\n\nJa, ich gehe w\u00e4hlen, lehne die hiesigen Parteien allerdings fast alle ab; w\u00e4hle aber die Piraten aufgrund ihres Engagements im Datenschutz, wobei ich mittlerweile bei einigen Parteientscheidungen auch schon am \u00dcberlegen bin, ob das die richtige Wahl ist oder ung\u00fcltig w\u00e4hlen doch wieder meine Option wird (ist im Vergleich zu gar nicht w\u00e4hlen die klar bessere Alternative).\n\n#### Wie ist dein Verh\u00e4ltnis zum Staat, zur Polizei etc.?\n\nIch lehne jeden Staat ab, da ich diese nur als Konstrukte der Unterdr\u00fcckung und Ausbeutung erachte, nicht als Strukturen, um Menschen zu helfen! Nicht jeder Bulle ist ein Staatsb\u00fcttel oder Schl\u00e4ger in Uniform, das will ich nur mal klarstellen, aber wenn ich das x-te Mal gesehen habe, dass kleine M\u00e4dchen und alte Menschen geschlagen und regelrecht gequ\u00e4lt werden, dann glaube ich einfach nicht mehr an die Menschlichkeit dieser Personen in Uniform. Ich habe oft genug gesehen, wie Polizisten beim Aus\u00fcben ihrer T\u00e4tigkeit, damit meine ich Gewaltaus\u00fcbung, gegrinst haben und es genossen haben, Menschen zu qu\u00e4len. Da die in Aus\u00fcbung ihres Berufes niemals mit Konsequenzen rechnen m\u00fcssen und ihre Macht ausnutzen, habe ich s\u00e4mtlichen Respekt vor diesen Personen verloren. Ich sehe nicht mehr den Menschen, sondern Roboter, die f\u00fcr den Staat agieren. Deshalb ist es mir egal, was ihnen passiert, und hei\u00dfe Gewalt gegen die Riot-Cops generell gut, da diese sich ihren Job und ihre Eins\u00e4tze ausgesucht haben. Die k\u00f6nnten sich doch immer noch krank melden oder k\u00fcndigen, aber sie sind bereit, f\u00fcr Staat und Kapital ihren Kopf hinzuhalten, dann sollen sie aber auch nicht \u00fcber ihre Gewerkschaft heulen, wenn sie auch mal eine Beule bekommen, oder gleich nach Datenschutz schreien, wenn sie Nummern tragen sollen, damit ihre Opfer wenigstens eine kleine Chance haben, sich zu wehren. Aber das zeigt nur erneut, wie diese Personen denken. W\u00fcrden sie sich immer an Recht und Gesetz halten, br\u00e4uchten sie doch vor Nummern an ihrer Uniform keine Angst haben.\n\n#### Gibt es Ver\u00e4nderungen beim Umgang des Staates\/der Polizei mit Autonomen?\n\nJa, die Polizeitaktik in Berlin ist gerade sehr erfolgreich, da sie in kleinen Gruppen infiltrieren und dann zerstreuen, es kommt immer seltener zu gro\u00dffl\u00e4chigen Barrikadenk\u00e4mpfen, sondern eher zu Kleingruppenaktionen. Generell werden schwarz gekleidete Menschen anders behandelt als bunt gekleidete. Das Strafma\u00df ist subjektiv in einigen F\u00e4llen gestiegen. Ich muss sagen, Staat und Polizei haben schon sehr gut gelernt, mit dem Auftreten von Black Blocks und generell der Gewalt bei Aktionen umzugehen, darum verlagert sich vieles auch in unangemeldete Aktionen oder kleinere Aktionen und es kommt immer seltener zu gro\u00dfen Events, wie es derzeit noch Dresden im Februar ist. Da wei\u00df jede\/r, dass was geht!\n\n#### \"Dass da was geht\": Meinst du, da fahren auch viele hin, weil es Gewalt geben wird, und deren eigentlicher politischer Anspruch ist entweder nicht vorhanden oder schon l\u00e4ngst vergessen?\n\nNat\u00fcrlich gibt es auch solche Leute bei diesen Events. Das will ich gar nicht bezweifeln oder kleinreden, denn es gibt immer Leute, die nur den Nervenkitzel suchen, aber auf eine Gesinnungspolizei kann nun wirklich kein Mensch Bock haben.\n\n#### Welche internationalen Vernetzungen gibt es, spielen f\u00fcr dich eine Rolle?\n\nIn Sachen Hausbesetzung und Antifaschismus gibt es viele Vernetzungen. Im Thema Anti-Globalisierung vor allem seit dem G8-Gipfel in Genua 2001. Vernetzung spielt f\u00fcr mich eine gro\u00dfe Rolle, ich war schon oft im Ausland, um zu unterst\u00fctzen und voneinander zu lernen. Sich in sozialen K\u00e4mpfen unterst\u00fctzen wie Griechenland und Italien, aber auch wenn ausl\u00e4ndische Antifas kommen, um uns zu unterst\u00fctzen, wie zum Beispiel in Dresden. Oder der CSD in Warschau. Was ich sagen will, es gibt reichlich Netzwerke, aber ich bin da bei keinem ein direkter Teil, sondern z\u00e4hle zu den Leuten, die dann zu Aktionen mitfahren oder mal Busse oder Autos organisieren.\n\n#### Wie hat sich die autonome Szene im Laufe der Jahre ver\u00e4ndert?\n\nWas ich von fr\u00fcher wei\u00df, wei\u00df ich auch nur aus sekund\u00e4ren Quellen und mitunter auch von Zeitzeugen, da ich es ja nicht selbst erlebt habe. Aber in den mehr als zehn Jahren, die ich \"dabei\" bin, hat sich auch schon einiges ge\u00e4ndert. Sowohl personell als auch was die Repression angeht, die Polizeitaktiken, aber auch die Events als solche. Fr\u00fcher ging bei Demos meist mehr und es gab mehr Gro\u00dfevents, aber gerade gro\u00dfe Fascho-Demos wurden mit den Jahren oft genug zerschlagen und erfolgreich blockiert, so dass deren Szene schon geschw\u00e4cht wurde, habe ich den Eindruck. Die Leute haben aber auch gelernt, sich besser zu vernetzen, nat\u00fcrlich auch bedingt durch bessere Kommunikationswege. Internationale Kommunikation ist schlicht g\u00fcnstiger und einfacher geworden. Ich habe auch den Eindruck, dass sich die Leute vor Aktionen und Events besser vorbereiten und sich auch mehr einen Kopf um die Konsequenzen, sowohl rechtlich als auch was die \u00f6ffentliche Wahrnehmung angeht, machen.\n\n#### Wo siehst du Kritikpunkte, Probleme in der autonomen Szene? Gibt es gruppen\u00fcbergreifende Szenediskussionen?\n\nNa ja, die Militanzdebatte \u2013 kurz gesagt: Darf Gewalt angewendet werden und wenn ja, in welchem Rahmen und wie wird dies am besten vermittelt? \u2013 gibt es immer noch und wird wohl auch nie aufh\u00f6ren, denn da gibt es zu viele Positionen und Ansichten zu. Auch taucht oft genug der Begriff des Mackertums bzw. Mackergehabes auf, das hei\u00dft, der Vorwurf der Gewaltaus\u00fcbung, um anderen etwas zu beweisen etc., aber da halte ich nicht viel von. Ein Problem ist, einhergehend mit der Militanz, die Au\u00dfenwirkung der autonomen Szene, das hei\u00dft, wie nimmt ein \"Normalb\u00fcrger\" diese wahr, was halten diese von den Autonomen. Aber ich gebe da nicht so viel drauf. Auch wenn Ver\u00e4nderungen ohne eine Basis fast unm\u00f6glich sein d\u00fcrften, so bin ich nicht bereit, Faschos marschieren zu lassen, weil ein Anwohner vielleicht wegen brennender M\u00fclltonnen weinen k\u00f6nnte.\n\n#### Wie bewertest du generell die \u00f6ffentliche Wahrnehmung der autonomen Szene?\n\nLege ich selber nicht so viel Wert drauf. Es gibt viele, die der Meinung sind, dass das wichtig ist, aber ich denke, dass jeder, der sich f\u00fcr die Szene interessiert, auch mit den Gruppen in Kontakt kommt. Au\u00dferdem kann Sachbesch\u00e4digung nicht sch\u00f6n geredet werden. Und auch ein brennender Golf in einer Barrikade ist mitunter schwer zu vermitteln, aber f\u00fcr mich z\u00e4hlt da mehr der Erfolg einer Aktion als nur der gew\u00e4hlte Weg. Klarstellen m\u00f6chte ich nur, dass ich sinnloses Zerst\u00f6ren generell ablehne, es muss schon ein Ziel haben (eine zerst\u00f6rte Bank, eine Barrikade, um sich besser zur\u00fcckziehen zu k\u00f6nnen, usw.). Aber ich bekomme schon mit, dass einige \"Normalb\u00fcrger\" Militanz nicht mehr generell ablehnen bzw. langsam lernen, dass mitunter auch mal andere Wege als das Stammtischgemeckere oder brennende Kerzen notwendig sind.\n\n#### Also lehnst du friedlichen Protest ab?\n\nNein, keineswegs! Ich habe auch gro\u00dfen Respekt vor Sitzblockierern und \u00c4hnlichen, aber sehe dies nur als einen Weg, etwas zu ver\u00e4ndern oder zu verhindern. Erfolgreicher Protest muss vielschichtig sein und die Aktionsformen m\u00fcssen sich respektieren. Aber brennende Kerzen? Nun ja, da halte ich es mit dem Liedtext \"Lichterketten\" von der Punk-Band Sluts'N: \"So lange Lichter nur in H\u00e4nden brennen \/ k\u00f6nnen Millionen auf die Stra\u00dfe gehen. \/ So lange Lichter nicht in K\u00f6pfen brennen \/ ist damit \u00fcberhaupt noch nichts geschehen.\"\n\n#### In der \u00f6ffentlichen Wahrnehmung werden Autonome immer mit Gewalt in Verbindung gebracht. Wie siehst du das: Geh\u00f6rt Gewalt zur autonomen Szene dazu?\n\nEs ist eines der Mittel, die genutzt werden. Es existiert eine gewisse Grundhaltung, was die Gewalt angeht, au\u00dferdem ist es schlicht medienwirksamer, um so auf Aktionen hinzuweisen. Wie w\u00e4re die Anti-Globalisierungsbewegung wohl aufgestellt, wenn es Seattle 1999 oder Genua 2001 nicht gegeben h\u00e4tte? Wie viele Menschen h\u00e4tten vom Naziaufmarsch j\u00e4hrlich im Februar in Dresden erfahren, wenn dieser nicht auch durch militante Mittel 2010 angegangen worden w\u00e4re? Das ist jetzt kein \"auf die Schulter klopfen\", aber das ist eine der vielen Konsequenzen von Militanz. Es gibt aber auch genug M\u00f6glichkeiten f\u00fcr Personen, die Gewalt generell ablehnen, beispielsweise Spa\u00df-Guerilla.\n\n#### Welche Einstellung hast du zu militanten Aktionen?\n\nF\u00fcr mich es ein legitimes Mittel, definitiv, um sich zu wehren bei Polizei- und\/ oder Nazi\u00fcbergriffen, oder um Blockaden erfolgreicher durchzuf\u00fchren. Des Weiteren, um Widerstand erfolgreich darzubieten. Gewalt gegen Sachen finde ich absolut in Ordnung, so lange der Kopf benutzt wird, das hei\u00dft, kein Familienwagen umgedreht oder der Tante-Emma-Laden entglast wird. Gewalt gegen Menschen hei\u00dfe ich nur im Falle von Gegenwehr f\u00fcr sinnvoll.\n\n#### Wurdest du schon auf deine Gewaltbereitschaft angesprochen oder daf\u00fcr kritisiert?\n\nEs gibt Dinge, \u00fcber die rede ich nicht mit jeder Person. Ich werde ja auch nicht direkt von Arbeitskollegen oder der Familie angesprochen, ob ich auf Demos Gewalt anwende. Es kam aber schon im Verlauf von Gespr\u00e4chen zu diesem Thema und ich halte es da eher offen, ob ich dazu bereit bin oder nicht, das hei\u00dft, ich sage nicht direkt: \"Ja, dann hab ich dem Fascho ins Gesicht geschlagen\" oder: \"Dann hab ich da eine M\u00fclltonne umgeworfen\" oder \u00c4hnliches, ich rede da halt ungern dar\u00fcber, weil ich ja auch nicht wei\u00df, wer genau zuh\u00f6rt und wie diese Person damit umgeht, und habe einfach keinen Bock auf Repression. Leute, die mich n\u00e4her kennen, wissen schon, ob sie auf mich z\u00e4hlen k\u00f6nnen in gewissen Situationen, um es mal so auszudr\u00fccken, und es kommt auch schon mal vor, dass \u00fcber gro\u00dfe Ereignisse sp\u00e4ter geredet wird, auch mit Menschen, die nicht zu Militanz bereit sind, die diese aber nicht ablehnen. Ich passe da schon etwas auf, wem ich was erz\u00e4hle. Kritik gab es auch schon einige Male, aber ich rede lieber mit den Leuten direkt, als es abzutun und \"mein Ding durchzuziehen\", denn in meinem direkten Umfeld ist mir schon wichtig, wie die Leute denken, und da bin ich Diskussionen auch nicht abgeneigt.\n\n#### Und deine Eltern?\n\nIch halte sie nicht f\u00fcr dumm und denke, sie wissen Bescheid, aber ich habe es nie zugegeben oder gro\u00dfartig davon erz\u00e4hlt. Sie wissen, dass ich viel unterwegs bin und gewisse Aktionsformen nicht verteufele, aber ob ich daran teilnehme, habe ich nie offen erkl\u00e4rt. Sie werden es sich aber denken k\u00f6nnen.\n\n#### Wie sch\u00e4tzt du die gesamtgesellschaftliche Bedeutung der Autonomen ein? Welche M\u00f6glichkeiten hat die autonome Szene, etwas zu ver\u00e4ndern?\n\nIch denke, dass die Gesellschaft als solche diese eher ablehnt, da der \"gemeine Deutsche\" halt mit seinem Leben zufrieden ist und lieber meckert, aber zu Taten eben eher nicht bereit ist, um mal grob zu verallgemeinern. Den Glauben, dass die gro\u00dfe Revolution bald kommt und das System schon erkennt, wie der richtige Weg w\u00e4re, habe ich schon lange verloren. Ich sehe mich auch nicht als Teil eines Besseren oder Mitglied in einer Gruppe, die jetzt die bessere Welt herbeiackert, aber ich denke schon, dass wir in kleinen Schritten etwas ver\u00e4ndern. Sei es den Castor-Transport so teuer wie m\u00f6glich zu machen; gro\u00dfe Fascho-Events zu st\u00f6ren und zu blockieren; dem einen oder anderen Fascho klarzumachen, dass er\/sie da den falschen Weg geht; unsere Missbilligung kundzutun, indem Regierende nicht seelenruhig am Tisch sitzen und labern k\u00f6nnen, sondern sich zur\u00fcckziehen m\u00fcssen bei ihren Gipfeln; Freir\u00e4ume zu erk\u00e4mpfen oder deren R\u00e4umungen zu erschweren usw. usf. Auch kleine Erfolge sind nun mal Erfolge!\n\n#### Hei\u00dft das, dir ist die Ver\u00e4nderung der Gesellschaft als solches nicht wichtig?\n\nDas hab ich nicht gesagt. Wenn es m\u00f6glich w\u00e4re, diese Welt zu einer besseren zu machen, w\u00e4re ich der Letzte, der etwas dagegen t\u00e4te, aber ich bin einfach zu pessimistisch, um noch daran zu glauben, die gro\u00dfe Revolution steht bevor.\n\n#### Wie sch\u00e4tzt du die Zukunftsperspektiven der autonomen Bewegung ein?\n\nWiderstand wird niemals aussterben, aber wir steuern auf immer gr\u00f6\u00dfere \u00dcberwachungsmaschinerien zu. Sich zu wehren wird schwerer werden, und ich glaube eher an Dystopien wie 1984, bin aber Revolutionsromantiker genug, um auch an so etwas wie V wie Vendetta zu glauben, das hei\u00dft, dass sich die Gesellschaft auch wehren w\u00fcrde, wenn es irgendwann zu weit geht. Passend zum Thema finde ich derzeitige Widerstandskonzepte wie die von Anonymous \u00e4u\u00dferst reizvoll und auch ausbauf\u00e4hig, da sie weltweit operieren und auch Widerstandsbewegungen wie die in Nordafrika unterst\u00fctzen. Welche Rolle die autonome Szene zuk\u00fcnftig spielen wird, ist nat\u00fcrlich schwer zu sagen, ich hab ja auch keine Glaskugel, aber ich nehme an, dass gesteigerte Repression und \u00dcberwachung es uns schon schwerer machen wird, allerdings entstehen so auch immer wieder neue Widerstandskonzepte, und etwas vorsichtiger zu sein ist ja auch nicht unbedingt schlecht. Ich kann mir die Autonomen einfach nicht wegdenken, das w\u00e4re selbst mir zu pessimistisch.\n\n#### Was w\u00fcrdest du selbst gern noch sagen?\n\nIch will keine \u00dcberzeugungsarbeit leisten oder irgendwelche Aktionen verteidigen oder so, mir w\u00e4re wichtig, dass Menschen sich mehr um andere Menschen sorgen, einfach mal \u00fcber den eigenen Tellerrand blicken, gewisse Grund\u00e4ngste ablegen und bereit sind, Widerstand zu leisten gegen Ungerechtigkeiten. Das f\u00e4ngt schon dabei an, wenn Eltern ihre Kinder schlagen, Menschen andere Mitarbeiter sexuell bel\u00e4stigen und auch bei kleinen Geld- oder Essensspenden f\u00fcr Menschen, die weniger haben. Ich erwarte von keinem Menschen, sich pr\u00fcgelnden Bullen in den Weg zu stellen oder einen Nazi-Laden zu entglasen. Ich erwarte von den Menschen nur, dass sie endlich mal die Augen aufmachen und nicht jeder Zeitung gleich alles glauben, sich alternativ informieren und trotz m\u00f6glicher R\u00fcckschl\u00e4ge vielleicht auch mal bereit sind, gegen unbequeme Dinge aufzubegehren. Sich einfach mal trauen, \"Nein!\" zu sagen, und ihre Kr\u00e4fte nutzen, die sie haben, und nicht nur sinnlos auf der Couch sitzen oder in der Kneipe meckern. Sich aufregen ist einfach, andere beschuldigen genauso, aber mal einfach selbst den Arsch hochzubekommen, ist es genauso, es muss nur der erste Schritt getan werden!\n\n## \"Antifaschismus ist als Grundlage einer linksradikalen Organisierung nicht geeignet.\"\n\nBj\u00f6rn, 35 Jahre alt, lebt in Berlin.\n\nIch treffe Bj\u00f6rn bei uns im Archiv der Jugendkulturen. Wir sind durch zahlreiche Ver\u00f6ffentlichungen von ihm auf ihn aufmerksam geworden. Seine klare, differenzierte Sprache in diesen Artikeln gefiel mir auf Anhieb. Ich freute mich darauf ihn kennenzulernen. Bj\u00f6rn ist nicht viel gr\u00f6\u00dfer als ich, aber extrem durchtrainiert. Er ist w\u00e4hrend des Interviews sehr ernsthaft und nachdenklich.\n\n#### Kannst du dich ganz kurz in Stichpunkten beschreiben.\n\nGeboren und aufgewachsen bin ich im alten Westberlin. Meine Eltern geh\u00f6rten nicht gerade zum Bildungsb\u00fcrgertum und besonders wohlhabend waren sie auch nicht. Ganz grob kann meine soziale Herkunft als untere Mittelschicht ohne Migrationshintergrund zusammengefasst werden. Heute wohne ich in einer kleinen Wohngemeinschaft in Kreuzberg. Ich arbeite gleichzeitig in drei verschiedenen Jobs, von denen ich trotzdem nicht so richtig leben kann. Vielleicht kann man mich als \"working poor\" oder prek\u00e4r Besch\u00e4ftigten bezeichnen. Nach b\u00fcrgerlichen Standards ist meine Biografie als Karriereverweigerer sicherlich keine Erfolgsgeschichte. Aber das ist nichts, was mich gro\u00df bek\u00fcmmern w\u00fcrde.\n\n#### Was sind das f\u00fcr Jobs, in denen du arbeitest?\n\nIch arbeite in mehreren sehr, sehr unterschiedlichen Bereichen mit abwechselnder Intensit\u00e4t. Die T\u00e4tigkeit, die mich gerade am meisten Zeit und Nerven kostet, weil sie mir am wenigsten Spa\u00df macht, ist mein Wochenendjob als K\u00fcnstler- und G\u00e4stebetreuer.\n\n#### Was ist denn ein K\u00fcnstler- und G\u00e4stebetreuer?\n\nDas ist in meinem Fall nur die umst\u00e4ndliche und tats\u00e4chlich eher schmeichelhafte Bezeichnung f\u00fcr T\u00fcrsteher. Das ist keine Sache, auf die ich besonders stolz bin, aber das hat sich halt so ergeben.\n\n#### Ist dir der Job unangenehm?\n\nMir ist die Rolle, die ich da einnehmen muss, oft unangenehm. Der Job als T\u00fcrsteher ist es, anderen Menschen Grenzen zu setzen. Die Funktion ist also zumeist repressiv. Ganz objektiv erf\u00fcllt man im Mikrokosmos des Clubs die gleiche Rolle, die die Polizei in der Gesellschaft erf\u00fcllt. Man sorgt daf\u00fcr, dass bestimmte Regeln eingehalten werden. Klar, oft macht man das alles ja auch im eigenen Interesse der feiernden Leute. Wenn jemand zum Beispiel zu viel getrunken oder irgendwelche Drogen eingeworfen hat, dann passt man eben auf, dass die anderen Leute nicht die Hilflosigkeit der Person ausnutzen. Man sorgt auch daf\u00fcr, dass gegebenenfalls eine medizinische Betreuung gew\u00e4hrleistet wird. Aber manchmal muss man eben auch verhindern, dass Leute, die auf Droge oder sonst wie schr\u00e4g drauf sind, anderen Leuten den Abend versauen. Hin und wieder muss man dann auch die eigene Autorit\u00e4t gegen andere durchsetzen. Manchmal erlebt man da sehr unsch\u00f6ne Situationen, die sehr belastend sein k\u00f6nnen.\n\n#### Gewaltt\u00e4tigkeiten?\n\nK\u00f6rperliche Gewaltt\u00e4tigkeiten kommen vor, sind aber die Ausnahme. Oft kann ich das besser verarbeiten als das ganz normale Elend des Partyalltags. Ich meine damit nicht nur dieses erb\u00e4rmliche, machohafte Balzverhalten von Sp\u00e4tpubertierenden, sondern da sind diese kleinen Respekt- und R\u00fccksichtslosigkeiten der Leute. Mich nervt eher, wie die Leute miteinander umgehen. Ich w\u00fcrde behaupten, dass ich den Job recht gut mache. Ich habe eben eine gewisse Lebenserfahrung, mir bestimmte Fach- und Sozialkompetenz angeeignet, die mir da weiterhelfen. Mittlerweile habe ich daher auch den Luxus, dass ich mir meinen Arbeitgeber selbst aussuchen kann. Zum Beispiel arbeite ich grunds\u00e4tzlich nicht f\u00fcr Clubs und Veranstalter, die nicht mal den Mindestlohn zahlen oder eine rassistische Einlasspolitik von mir verlangen.\n\n#### Was sind deine anderen T\u00e4tigkeiten?\n\nDas ist eher kreatives Kopfarbeiten im weitesten Sinne. Einer der Jobs findet im Rahmen eines explizit linken, selbstverwalteten Kollektiv-Betriebes statt.\n\n#### Was bedeutet das, in einem Kollektiv zu arbeiten?\n\nDas bedeutet, dass die Firma tats\u00e4chlich in der Hand der Leute ist, die da arbeiten. Wichtige Entscheidungen werden basisdemokratisch getroffen. Was alle betrifft, wird also auch von allen gemeinsam diskutiert und gleichberechtigt entschieden. Klingt alles in der Theorie ganz toll, ist aber in der Praxis oft nervenaufreibend und anstrengend. Andauernd endlose Diskussionen! Trotzdem m\u00f6chte ich nicht mehr in einem anderen Rahmen arbeiten. Also entweder arbeite ich weiter als Selbstst\u00e4ndiger oder gemeinsam mit anderen im Kollektiv.\n\n#### Geh\u00f6rst du einer Jugendkultur an?\n\nNa ja, ich bin jetzt 35 Jahre alt und, um ehrlich zu sein, gar nicht mehr so jugendlich, wie ich es selbst gerne h\u00e4tte.\n\n#### Geh\u00f6rtest du denn einer Jugendkultur an?\n\nNein, es sei denn \"couchpotato\" w\u00fcrde als Jugendkultur von dir gewertet. Ich war nie Punker, nie Gruftie oder so etwas, sondern ein ziemlich langweiliger Einzelg\u00e4nger und Nerd, der Raumschiff Enterprise toll fand.\n\n#### Gibt es denn eine Jugendkultur, die du sympathisch findest?\n\nVielleicht arbeite ich schon zu lange als T\u00fcrsteher, vielleicht bin ich auch einfach in einem Alter, in dem man Jugendliche ganz pauschal als laut, obsz\u00f6n und anstrengend empfindet. Es gibt aber ein paar Jugendszenen, die ich sympathisch finde. Wenn ich mir vorstelle, ich w\u00e4re noch mal 16 oder 18 Jahre alt, dann w\u00fcrde ich mich vermutlich f\u00fcr diese vegane Hardcore-Straight-Edge-Szene begeistern k\u00f6nnen.\n\n#### Gibt es eine Jugendszene, die du besonders unsympathisch findest?\n\nAlles, wo viel Alkohol flie\u00dft, ist mir per se eher unsympathisch. Ich kann zum Beispiel gar nichts mit dieser Fu\u00dfball-, Oi!-Skin- oder Bierdosenpunker-Kultur anfangen. Aber nat\u00fcrlich verurteile ich niemanden daf\u00fcr, wenn er oder sie ein FC-Union-Fan ist und am liebsten in Horden mit einer Bierdose in der Hand um die H\u00e4user zieht. Ich habe auch kein Problem damit, wenn Leute ihr Wochenende auf Electro-Partys mit Drogen verbringen. Wer gerne Fasching im eigenen Kopf feiern oder mal Urlaub von sich selbst machen will, dem w\u00fcnsche ich viel Spa\u00df. Aber es ist nicht mein Ding. Ich kann damit nichts anfangen. Ich denke, das liegt daran, dass es in meiner Familie eine lange Tradition von Drogen- und Alkoholmissbrauch gibt.\n\n#### Hat Religion eine Bedeutung f\u00fcr dich?\n\nF\u00fcr mich ist die Auseinandersetzung mit dem Tod sehr wichtig. Das Sterben geh\u00f6rt eben zum Leben dazu, auch wenn es oft verdr\u00e4ngt oder tabuisiert wird. In meinem Umfeld sind schon sehr fr\u00fch Menschen gestorben und ich habe bereits als Kind einen Umgang mit dem Tod finden m\u00fcssen. Ich glaube nicht an den lieben Gott, nicht an den Teufel, an Himmel und H\u00f6lle \u2013 trotzdem bin ich kein Atheist. Ich w\u00fcrde mich selbst als Agnostiker bezeichnen.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nVielleicht w\u00fcrde ich mich als Grenzg\u00e4nger zwischen verschiedenen Bereichen beschreiben. Ich sitze immer irgendwo zwischen allen St\u00fchlen. Ich habe keinen festen Platz, wo ich hingeh\u00f6re. Darunter leide ich aber nicht.\n\n#### Bist du ein Autonomer?\n\nWenn in den Medien von \"Autonomen\" gesprochen wird, f\u00fchle ich mich zwar irgendwie angesprochen, aber so ganz vorbehaltlos w\u00fcrde ich mich nicht so bezeichnen. Ich w\u00fcrde mich ja auch nicht Kommunist nennen, obwohl ich mich stark auf eine links- beziehungsweise r\u00e4tekommunistische Denktradition beziehe. Aber mit Lenin, Stalin, Mao und diesem ganzen Arbeiter- und Bauern-Begl\u00fcckungsmumpitz habe ich nichts, aber auch gar nichts zu tun. Genauso wenig bin ich ein klassischer Anarchist, auch wenn f\u00fcr mich der Anarchosyndikalismus ein ganz wichtiger Bezugspunkt ist. Ich m\u00f6chte einfach nicht mit den falschen Leuten und Ideen in Verbindung gebracht werden.\n\n#### Wie definierst du \"autonom\"? Was sind die Autonomen? Stell dir vor, du m\u00fcsstest es jemandem erkl\u00e4ren, der gar keine Ahnung davon hat.\n\nEine ad-hoc-Definition ist sehr schwer. Die Autonomen sind nichts Statisches, keine Partei, keine Denkschule und trotzdem eine politische Fraktion. Die Autonomen sind etwas sehr Nebul\u00f6ses, das sich zudem noch laufend ver\u00e4ndert. Sie sind ein politisches Milieu, ein Sammelbecken, eine Str\u00f6mung oder ein Spektrum. Es gibt darin verschiedene Verdichtungen, die sich manchmal sogar widersprechen. Am einfachsten ist es, wenn man die Autonomen historisch herleitet. Dann m\u00fcsste ich dir jetzt von der italienischen \"Autonomia\"-Bewegung von 1977, den Operaisten um die Zeitschrift Autonomie \u2013 Neue Folgen aus Frankfurt, der Anti-AKW-Bewegung und dem Westberliner H\u00e4userkampf von 1981 erz\u00e4hlen. Ich w\u00fcrde erz\u00e4hlen, dass die Autonomen immer so etwas wie die linksradikale und militante Au\u00dfenboje der Neuen Sozialen Bewegungen der 80er Jahre waren und sich dann irgendwann mal selbstst\u00e4ndig gemacht haben. Das alles ist aber lange vor meiner aktiven Zeit passiert. Ich kenne es auch nur aus Geschichtsb\u00fcchern und Erz\u00e4hlungen. Also komme ich mal lieber zum Wesentlichen: Autonom bedeutet nicht nur unabh\u00e4ngig, sondern selbstbestimmt. Gemeint ist damit, dass Autonome jeder paternalistischen Stellvertreterpolitik durch Parteien und Gewerkschaften eine Absage erteilen und sich lieber selbst und eigenverantwortlich organisieren. Die Autonomen sind also eine linksradikale, sozialrevolution\u00e4re und au\u00dferparlamentarische Bewegung.\n\n#### Was bedeutet es f\u00fcr dich pers\u00f6nlich, ein Autonomer zu sein?\n\nAuch auf die Gefahr hin, jetzt ganz pathetisch zu klingen \u2013 das bedeutet f\u00fcr mich, die Welt, so wie sie eingerichtet ist, nicht einfach hinzunehmen und von der Ver\u00e4nderung her zu denken. Das bedeutet, mal frei nach dem autonomen Theoretiker Karl Heinz Roth, die bestehenden Verh\u00e4ltnisse zu negieren, nicht nur mental, nicht nur theoretisch, sondern im gesamten Lebensvollzug. Also ein Leben gegen diese Verh\u00e4ltnisse zu f\u00fchren.\n\n#### Was bedeutet das im Alltag, dieses Leben gegen die Verh\u00e4ltnisse?\n\nDass ich mich in Fundamentalopposition zu den Verh\u00e4ltnissen, wie die Herrschaftsverh\u00e4ltnisse sind, begreife und gegen sie aktiv werde. Also an der Abschaffung eben dieser Verh\u00e4ltnisse aktiv mitwirke, so gut ich eben kann. Das bedeutet, dass man sich ins Handgemenge der sozialen K\u00e4mpfe begibt. Mit dazu geh\u00f6rt auch, dass man selbst diese Macht- und Herrschaftsverh\u00e4ltnisse nicht reproduziert und die eigenen Verstrickungen darin reflektiert. Ganz banal bedeutet das dann auch, nicht rassistisch, sexistisch oder homophob zu handeln.\n\n#### Wie war dein erster Kontakt zur autonomen Szene?\n\nIch bin im Berlin der 80er Jahre gro\u00df geworden, da hatten die Autonomen eine innenpolitische Gr\u00f6\u00dfe, die sich hin und wieder bemerkbar gemacht hat. Es gab viele besetzte H\u00e4user, Parolen an den W\u00e4nden und Demonstrationen. Als Kind fand ich vor allem Hausbesetzungen sehr interessant. Die Idee, das Miteinander anders organisieren zu wollen, fand ich total spannend und faszinierend. Dann gab es bestimmte Schl\u00fcsselmomente, wie das Einwerfen einer Deutschen Bank, was ich hautnah miterlebt habe. Das muss Mitte der 80er gewesen sein, da bin ich beim Einkaufen pl\u00f6tzlich einer linksradikalen Demo begegnet. Aus der Demonstration raus haben dann Vermummte die Scheiben einer Bank eingeworfen und \"Freiheit f\u00fcr S\u00fcdafrika!\" rangeschrieben. Das ging alles ganz schnell und war gut und arbeitsteilig organisiert. Ich bin dann zu meiner Mutter gelaufen und habe gefragt, was es denn mit S\u00fcdafrika auf sich hat. Sie hat mir dann von der Apartheid erz\u00e4hlt, was ich ziemlich skandal\u00f6s fand. Dass die Vermummten die Fensterscheiben einer Bank eingeworfen haben, die mit dem Apartheidregime Gesch\u00e4fte machte, empfand ich als gerechtfertigt. Ich fand das sogar richtig gut! Das war das erste Mal, dass ich einen Gesetzesbruch als v\u00f6llig richtig empfunden habe.\n\nEin weiterer Schl\u00fcsselmoment war, als 1987 die Polizei ganz Kreuzberg mit Stra\u00dfensperren abriegelte, weil bef\u00fcrchtet wurde, dass es zu Demonstrationen gegen den Besuch vom damaligen US-Pr\u00e4sidenten Ronald Reagan kommen k\u00f6nnte. Auch die U-Bahn fuhr f\u00fcr Stunden nicht. 1988 gab es dann das IWF-Treffen in Berlin, das die ganze City mit Demonstranten und Polizisten f\u00fcllte. Ich wollte damals genau w\u00e4hrend der Aktionstage ein Geschenk f\u00fcr meine Freundin im KaDeWe [Kaufhaus des Westens] kaufen. Ich fuhr mit meinem hart ersparten Taschengeld zum Kaufhaus. Als ich am Tauentzien ankam, war das KaDeWe geschlossen und die Schaufenster waren mit Spanplatten vernagelt, davor standen Polizisten mit Schildern. Dann kam die Demo mit einigen Vermummten, es gab einen Polizeieinsatz, dann Steinw\u00fcrfe und ich war dazwischen. Alles ein wenig be\u00e4ngstigend, aber auch sehr spannend, ich konnte es aber noch nicht ganz einordnen. Das waren f\u00fcr mich die ersten Kontakte, die mir klar machten, es gibt offensichtlich ein subversives Milieu.\n\nJahre sp\u00e4ter bin ich dann selbst aktiv geworden. Als am 1. Januar 1994 die Zapatistas im s\u00fcdmexikanischen Chiapas rebellierten, war ich total begeistert. Ich schrieb auf meine Federtasche und auf das Schulklo \"EZLN-Viva Zapata!\". Ich war damals ein ziemlich nerdiger Einzelg\u00e4nger, der in der Schule nicht mehr klarkam. Anstatt zur Schule zu gehen, habe ich lieber alleine im Park gesessen und Zeitung gelesen. Im Sommer 1994 stand da, dass der Ministerpr\u00e4sident der Volksrepublik China, Li Peng, zu Besuch kommen w\u00fcrde. Li Peng ist einer der Verantwortlichen f\u00fcr das Massaker vom Tian'anmen Platz. Also bin ich alleine zum Schloss Charlottenburg gefahren, um mich mal pers\u00f6nlich bei Herrn Peng und seinen deutschen Gastgebern \u00fcber die Menschenrechtsverletzungen zu beschweren. Ich bin da hin, hab gewartet und bin im richtigen Moment \u00fcber die Absperrung gesprungen und fast von der Peng-Limousine \u00fcberfahren worden.\n\nEbenfalls im v\u00f6lligen Alleingang habe ich 1995 beschlossen, mich w\u00e4hrend der UN-Klimakonferenz an die T\u00fcren des ICC [Internationales Congress Centrum] zu kleben, um gegen die v\u00f6llig verfehlte Klimapolitik zu demonstrieren. Die ersten Kampagnen, an denen ich mich dann aktiv beteiligt habe, waren gegen die Versenkung der \"Brent Spar\", diesem schwimmenden \u00d6ltank von Shell\/Esso in der Nordsee, und gegen die Atombombentests auf dem Mururoa-Atoll. Dabei habe ich auch mehr und mehr eine gewisse jugendliche Alltagsmilitanz praktiziert. Ich habe massenhaft Wahlplakate zerst\u00f6rt, \"Viva EZLN\" an Hausw\u00e4nde gespr\u00fcht und irgendwann auch mal einem Naziskin auf die Nase gehauen und ihm seinen \"Deutschland den Deutschen!\"-Aufn\u00e4her von der Jacke gerissen. Im gleichen Zeitraum habe ich Kontakt in die autonome Szene gekn\u00fcpft. Das wiederum war sehr entt\u00e4uschend.\n\n#### Warum?\n\nIch kam aus dem Nirgendwo, niemand kannte mich. Ich kam ja nicht \u00fcber Freunde, Partys oder subkulturelle Bekanntschaften zu diesem autonomen Milieu, sondern ich kam da hin, weil ich eine gewisse Kritik an der Gesellschaft hatte. Ich sah anders aus als die \u00fcblichen linken Jugendlichen. So etwas weckte Misstrauen. Es war also sehr schwer, Anschluss zu finden. Ich habe festgestellt, dass es keine offene Szene ist. Sie ist nicht offen f\u00fcr Neue, nicht offen f\u00fcr Interessierte. Das fand ich sehr \u00e4rgerlich und kontraproduktiv, denn wenn man gesellschaftliche Relevanz haben m\u00f6chte, dann muss man auch Leute einbinden k\u00f6nnen, die anders ticken. Also hab ich selbst eine Gruppe gegr\u00fcndet und anschlie\u00dfend erfolgreich mit einer anderen Gruppe fusioniert.\n\n#### Bist du heute noch aktiv?\n\nKlar! Ich bin nach wie vor im Rahmen eines Pools von Aktivisten aktiv, die sich zum Teil schon sehr lange kennen. Wir finden uns immer wieder anlassbezogen in wechselnder Konstellation der immer Gleichen zusammen. Phasenweise bilden wir auch dass, was gemeinhin als anonyme autonome Gruppe bezeichnet wird. In meinem Fall sind das zumeist eher Diskussionszusammenh\u00e4nge als eine feste Gruppe.\n\n#### Was macht ihr, wenn ihr euch trefft? Zu welchen Anl\u00e4ssen werdet ihr aktiv?\n\nManchmal treffen wir uns, weil es ein spannendes neues Buch oder einen neuen Text gibt. Dann lesen wir gemeinsam den Text und diskutieren dar\u00fcber. Manchmal ist es auch eine Demonstration, die wir dann inhaltlich oder auch mal ganz praktisch begleiten. Eigentlich mache ich heute genau das, was ich an den Autonomen immer kritisiert habe, n\u00e4mlich diese Form der weitgehend unverbindlichen Szene- und Freundschaftspolitik. Ich mache das jetzt einfach deswegen, weil ich nicht mehr die Zeit und wahrscheinlich nicht mehr die Energie habe, mich wie fr\u00fcher in festen Strukturen einzufinden.\n\n#### Was sind das f\u00fcr feste Strukturen gewesen?\n\nDer Verfassungsschutz hat uns fr\u00fcher immer sehr schmeichelhaft als die \"organisierten Autonomen\" bezeichnet. Wir haben in den 90er Jahren die Autonomen und insbesondere die autonome Antifa stark ver\u00e4ndert und nachhaltig gepr\u00e4gt. Wir sind eine Art Reformationsbewegung innerhalb der Autonomen in den 90er Jahren gewesen.\n\n#### Magst du darauf genauer eingehen?\n\nDazu muss ich sehr weit ausholen. Als ich Mitte der 90er Jahre angefangen habe, mich f\u00fcr linksradikale, au\u00dferparlamentarische Politik zu begeistern, setzten sich die \"organisierten Autonomen\" gegen das \u00e4ltere autonome Milieu der 80er und fr\u00fchen 90er Jahre durch. Die klassischen Autonomen waren als Szenen mit diversen halbklandestinen, nach au\u00dfen sehr abgeschotteten Gr\u00fcppchen organisiert. Zusammengehalten wurde das Ganze durch ein recht diffuses \"Wir\"-Gef\u00fchl, das nicht \u00fcber konkrete politische Positionen gebildet wurde, sondern vor allem \u00fcber einen gemeinsamen subkulturellen Habitus. Viele Leute waren zwar irgendwie dabei, zu einem einheitlichen politischen Handeln jenseits der typischen Einpunktpolitik war die Szene aber nicht f\u00e4hig. Es gab weder eine vern\u00fcnftige \u00d6ffentlichkeitsarbeit noch eine kontinuierliche Theoriebildung. An diesen offensichtlichen Schw\u00e4chen gab es eine immer wiederkehrende Selbstkritik. Anfang der 90er Jahre kam es dann in der autonomen Szene zu einer Organisierungsdebatte. Weil zeitgleich die neofaschistische Offensive mit den Pogromen von Hoyerswerda und Rostock tobte, war Antifaschismus damals das dringendste Themenfeld. Hier ging es zum Teil ganz handfest um den Schutz von linken Strukturen und Menschen, die von Nazis bedroht wurden. Es war auch das Thema, an dem wir die autonome Linke sich sammeln, organisieren und neu konstituieren wollten. Alle konnten sich darauf einigen, dass Antifaschismus ein ganz wichtiges und essentielles Aktionsfeld ist.\n\nUnter dem Motto \"Organisiert die autonome Antifa\" wurden dann feste verbindliche Strukturen geschaffen. Eigentlich machten wir da nichts anderes, als die totalen Basisstandards von politischer Arbeit zu etablieren. Wir gaben uns einen Namen, eine Adresse, mieteten ein B\u00fcro, verschickten Infopakete und gingen auf Interessierte zu. Ganz wichtig war uns noch Jugend- und B\u00fcndnisarbeit. Das war zu dem Zeitpunkt Neuland f\u00fcr die Szene. Genauso neu war, dass wir T-Shirts verkauften und Konzerte mit angesagten Bands veranstalteten, um unsere Gruppenkasse aufzubessern. Demonstrationen wurden von uns nicht nur als politische Selbstdarstellung, sondern als Event geplant. W\u00e4hrend am 1. Mai andere Gruppen mit einem kleinen Lautsprecherwagen herumfuhren und langweilige Redebeitr\u00e4ge verlasen, kamen wir mit einem Sattelschlepper, wie man sie in Berlin von der Love Parade her kannte. Wir hatten schlie\u00dflich Leute mit BWL-Abschluss und Erfahrungen in der Werbung und Eventmanagement bei uns. Die haben uns beigebracht, wie man eine politische Kampagne aufzieht, wie man bestimmte Momente ausnutzen kann, um Aufmerksamkeit auf sich zu ziehen. Wir lernten, dass man so eine Gruppe wie einen Betrieb aufziehen und mit einer Corporate Identity ausstatten muss. Wir wurden zur professionellen Pop-Antifa. Die \u00c4sthetik, die wir damals geschaffen haben, hat sich bis weit \u00fcber die autonome Antifa hinaus etabliert und wird bis heute europaweit kopiert.\n\n#### Kannst du etwas dar\u00fcber sagen, wie die Gruppe aufgebaut und strukturiert war?\n\nEs gibt nicht so viele Wege, vern\u00fcnftig eine solche Gruppe basisdemokratisch zu organisieren. Bei uns gab es eine monatliche Vollversammlung, ein w\u00f6chentliches Delegiertentreffen und diverse Arbeitsgruppen, die themenspezifisch gearbeitet haben. Von den traditionellen Autonomen wurden wir immer als partei\u00e4hnlich, sektenhaft oder als Kaderorganisation kritisiert. Mit dieser Kritik lagen sie zwar nicht richtig, aber eben auch nicht v\u00f6llig falsch. So ein Projekt kann man nur mit hoch motivierten, sehr engagierten Leuten stemmen. Wir waren so ein Kern von drei\u00dfig oder vierzig Leuten, wir wohnten oft zusammen in Wohngemeinschaften und wir hatten die Regel, dass wir zehn Prozent unseres Einkommens in die Gruppenkasse steckten. Wir haben das gemacht, weil uns allen das Projekt sehr wichtig war. Nat\u00fcrlich betrieben wir auch eine Art Kaderpolitik. In anderen Politgruppen \u00fcbernahmen die Kleptomanen stets die Verwaltung der Gruppenkasse, Legastheniker schrieben die Presseerkl\u00e4rung, und wer nichts zu sagen hatte, aber sich gerne reden h\u00f6rte, durfte auf jedem Podium sprechen. So etwas gab es bei uns nicht! Wir hatten zeitweise neben einer unglaublichen Motivation und Energie einige interne Mechanismen, die an die K-Gruppen der 70er Jahre erinnern. Ein ganz finsteres Beispiel daf\u00fcr: Es gibt Arbeiten, die keiner gerne macht, die aber eben gemacht werden m\u00fcssen. Bei uns war das plakatieren gehen, das hei\u00dft, man zieht um die H\u00e4user und klebt politische Plakate an die W\u00e4nde. Das wollte niemand machen. Nun hatten wir immer so ein gemeinsames w\u00f6chentliches Treffen. Auf diesen Treffen wurde gefragt, wer plakatieren gehen m\u00f6chte. Da meldete sich dann niemand freiwillig. Weil die Aufgabenverteilung meistens am Anfang des Treffens stand, kamen dann einige Leute mit Absicht zu sp\u00e4t. Das war damals sowieso ein gro\u00dfes Problem, die Leute p\u00fcnktlich zu den Treffen zu bekommen. Es war wirklich ein Riesenproblem, Studenten und Punker p\u00fcnktlich irgendwo hin zu bekommen. Also f\u00fchrten wir die gef\u00fcrchteten Plakatierpunkte ein: Die Leute, die zu sp\u00e4t kamen, bekamen f\u00fcr jeweils f\u00fcnf Minuten Versp\u00e4tung einen Plakatierpunkt. Wer zehn Minuten zu sp\u00e4t war, bekam zwei Punkte, wer f\u00fcnfzehn Minuten zu sp\u00e4t war, drei Punkte. Wer die meisten Plakatierpunkte hatte, musste plakatieren gehen. Wir hatten also interne Repressionsmechanismen, die dazu dienten, die Leute zu disziplinieren und gleichzeitig die unbeliebten Arbeiten zu erledigen. Das sind genau die Sachen, bei denen ich heute die H\u00e4nde \u00fcber dem Kopf zusammenschlage und mir denke, was wir uns damals f\u00fcr einen Unsinn ausgedacht haben. Aber ich lernte in diesen Gruppenzusammenh\u00e4ngen, effektiv zu arbeiten.\n\n#### Gibt es diese Gruppe noch?\n\nNein. Heute existieren aber diverse andere Gruppen, die aus unserer hervorgegangen sind. Es gibt so etwas wie Nachfolgeorganisationen, Spaltungsprodukte und einige politische Erbschleicher.\n\n#### Was hat zur Aufl\u00f6sung der Gruppe gef\u00fchrt?\n\nEntscheidend war eine sehr komplexe Diskussion um die Perspektiven und Grundlagen revolution\u00e4rer Politik. Wir merkten, dass es einige offene Fragen gab, worauf wir keine Antwort wussten. Wir konnten irgendwann nicht mehr so tun, als ob wir welche h\u00e4tten. Die wachsende inhaltliche Unsicherheit und die damit einhergehende Fraktionierung nahmen dann der Gruppe viel von ihrer Dynamik. Mit dem Wegfall der gemeinsamen Gruppenidentit\u00e4t setzte der rapide Zerfall der Strukturen ein. Als wir als Gruppe nicht mehr arbeitsf\u00e4hig waren, l\u00f6sten wir uns auf. R\u00fcckblickend muss ich aber sagen, dass dieser Aufl\u00f6sungsprozess die spannendste und lehrreichste Zeit war, die ich je in einer Politgruppe erlebt habe.\n\n#### Was waren das f\u00fcr Diskussionen, an denen sich die Gruppe gespalten hatte?\n\nUnser grundlegendes Problem war immer gewesen, dass wir eine strukturelle Antwort auf eine inhaltliche Krise gegeben haben. Dass es der radikalen Linken in den 90er Jahren so schlecht ging, lag nicht nur daran, dass sie so schlecht organisiert war. Wir wollten die Perspektivlosigkeit und gesellschaftliche Isolation der Autonomen \u00fcberwinden, dazu wollten wir mit der str\u00f6mungs\u00fcbergreifenden Organisierung eine Art melting pot schaffen. Antifaschismus als kleinster gemeinsamer Nenner der radikalen Linken sollte uns zusammenbringen. Im praktischen Anti-Nazi-Kampf waren wir schlie\u00dflich auch recht kompetent. Aber sobald es zu einer ernsthaften theoretischen Diskussion dar\u00fcber kam, was denn nun genau der Faschismus ist, taten sich gro\u00dfe Gr\u00e4ben und manchmal auch Abgr\u00fcnde auf. Die antideutsche Kritik, die damals vor allem Bezug auf die Goldhagen-Debatte [Autor von Hitlers willige Vollstrecker] nahm, sorgte dann daf\u00fcr, dass auch in der autonomen Antifa die naive und letztendlich auch verniedlichende Vorstellung vom Nationalsozialismus als einer Veranstaltung von Staat und Kapital mal kritisch hinterfragt wurde. Die Erkenntnis, dass der deutsche Faschismus eine weit ins Proletariat hineinreichende Massenbewegung war, war bis dato nur partiell zu uns durchgedrungen. Entsprechend arbeiteten wir mit so platten Parolen wie: \"Hinter dem Faschismus steht das Kapital\". Das war zwar nicht falsch, aber eben nur die halbe Wahrheit.\n\nDie Grundlage unserer Politik war der \"revolution\u00e4re Antifaschismus\". Das Konzept impliziert, dass die Motivation, gegen die bestehende Ordnung vorzugehen, eben nicht die Verh\u00e4ltnisse selbst sind, sondern das ihnen innewohnende Potenzial eines viel gr\u00f6\u00dferen \u00dcbels, also des Faschismus. Wir versuchten da anhand eines konkreten Aufh\u00e4ngers, einer augenscheinlichen Ungerechtigkeit, mit der Kritik aufs Ganze zu gehen und behaupteten einfach: \"Antifa ist der Kampf ums Ganze.\" Diese Vorgehensweise zieht sich wie ein roter Faden durch die Geschichte der Bewegungslinken. Entweder man tritt eine Protestbewegung los oder man springt auf und versucht den Protest zu radikalisieren. Das Dilemma, an dem die Linke dabei gedanklich scheitert, ist, dann zu belegen, warum nun gerade der Castortransport, die Nazis, der Rassismus, die heteronormative Matrix oder \u00c4hnliches jetzt die gesellschaftlichen Widerspr\u00fcche sind, die derart bedeutend f\u00fcr die Verh\u00e4ltnisse sind, dass sie nicht in ihr aufgel\u00f6st werden k\u00f6nnen, sondern nur mit den Verh\u00e4ltnissen. Ganz simpel formuliert war unser \"revolution\u00e4rer Antifaschismus\" gar nicht revolution\u00e4r. Wir haben es einfach nur behauptet. Antifaschismus ist f\u00fcr sich selbst eine sehr vern\u00fcnftige Angelegenheit, aber eben nichts Revolution\u00e4res. Als Dreh- und Angelpunkt oder Grundlage einer linksradikalen Organisierung ist er nicht geeignet. Mit dieser Erkenntnis ist Ende der 90er die linksradikale Antifa-Bewegung mit ihrem v\u00f6llig unzul\u00e4nglichen Konzept des \"revolution\u00e4ren Antifaschismus\" an ihren Endpunkt geraten. Dass wir uns organisiert den Nazis entgegengestellt haben, dass wir zusammengekommen sind und wichtige Debatten angesto\u00dfen haben \u2013 das bereut aber niemand.\n\n#### Was ist aus den Aktivisten der Gruppe geworden?\n\nWie zu erwarten, haben viele Karriere gemacht. Einige sind nun Anw\u00e4lte, \u00c4rzte, Psychologen, Journalisten oder Unternehmensberater. Manche von denen haben mit ihrer Vergangenheit gebrochen. Einige sind in NGOs, Gewerkschaften und Stiftungen untergekommen und machen da jetzt ganz vern\u00fcnftige Sachen. Viele sind immer noch politisch aktiv. Einige haben die Nachfolgeprojekte aufgebaut, andere haben sich in diverse Lesekreise zur\u00fcckgezogen. Ich glaube allen gemein ist, dass sie ihre Zeit in der Gruppe als sehr lehrreich und wichtig bezeichnen w\u00fcrden.\n\n#### Wie beurteilst du den Streit zwischen Antideutschen und Antiimperialisten?\n\nDie Begriffe \"antideutsch\" und \"Antiimperialist\" sind heute innerlinke Fremdzuschreibungen, die nichts erkl\u00e4ren. Beispielsweise tauchen in Berlin periodisch diese ultranervigen zwanzigj\u00e4hrigen Politaktivisten auf, die mich wahlweise mit der Forderung nach uneingeschr\u00e4nkter Solidarit\u00e4t mit Israel oder der ebenfalls uneingeschr\u00e4nkten antiimperialistischen Solidarit\u00e4t mit Pal\u00e4stina nerven. Beide gehen mir auf den Keks. Beiden geht es nicht um die Sache, sondern um linke Identit\u00e4tsbildung, und beide nutzen daf\u00fcr den Nahostkonflikt. Was soll man denn noch dazu sagen? Wenn Staatsterror auf Staatsgr\u00fcndungsterror trifft, also gewaltsam f\u00fcr eine falsche Idee gestritten wird, gibt es da f\u00fcr mich als sozialrevolution\u00e4ren Staatsfeind keine Parteinahme.\n\nMeine Position als Autonomer war immer ganz klar antinational. Mit dem linksnationalistischen Antiimperialismusquatsch konnte ich nie etwas anfangen. Das ganze antiimperialistische Denkmuster, dieses manich\u00e4ische Weltbild, das den Globus in zwei Sph\u00e4ren aufteilt, ist mir fremd. F\u00fcr die klassischen Antiimps gibt es nur die Wei\u00dfen, die ausbeutenden Imperialisten der globalen Metropole, und die ausgebeuteten schwarzen, armen V\u00f6lker auf der anderen Seite. Dieses im kolonialistischen Befreiungskampf verhaftete Denken ebnet viele Widerspr\u00fcche v\u00f6llig ein. Der postimperialistischen Weltordnung mit einer antiimperialistischen Befreiungsstrategie entgegentreten zu wollen, funktioniert einfach nicht.\n\nDie antideutsche Linke ist komplizierter. Ein Teil hat sich in der Auseinandersetzung um die zweite Intifada und den 11. September im Namen eines sehr fragw\u00fcrdigen Verst\u00e4ndnisses von Fortschritt und Moderne von der Linken verabschiedet. Sie glauben, dass die Alternative zum Elend des Kapitalismus nur noch die v\u00f6llige Barbarei sei. Mit dieser Position haben sie vor den Verh\u00e4ltnissen feige kapituliert und sind mit fliehenden Fahnen ins Lager der Liberalen und Neokonservativen gewechselt. Ein anderer Teil geh\u00f6rt nach wie vor zur radikalen Linken und widmet sich vor allem der Ideologiekritik. Das finde ich schon viel spannender! Dass es in der radikalen Linken heute \u00fcberhaupt eine so eine intensive Auseinandersetzung mit Antisemitismus, Antizionismus und Antiamerikanismus gibt, ist eben dieser Str\u00f6mung zu verdanken.\n\n#### Wie ist dein Verh\u00e4ltnis zur Polizei?\n\nIch habe zum Teil sehr r\u00fcde Sachen mit der Polizei, insbesondere mit der Polizei im europ\u00e4ischen Ausland erlebt. Ich habe Narben von Kn\u00fcppelschl\u00e4gen, Gummigeschossen und Schockgranaten. Ich habe erlebt, wie unbewaffnete Demonstranten von der Polizei niedergeschossen wurden und Gefangene misshandelt worden sind. Vor diesem Hintergrund ist es schwer, die Polizei nicht verzerrt wahrzunehmen. Trotzdem ist mein Verh\u00e4ltnis zur Polizei bis heute relativ n\u00fcchtern. Ich versp\u00fcre keinen Hass. Die Polizei ist blo\u00df ein Instrument zur Durchsetzung der staatlichen Ordnung. Sie sorgt daf\u00fcr, dass die Gesetze eingehalten und \u00dcbertretungen verfolgt werden. Dass ich als Staatsfeind mit der Polizei in Konflikt gerate, ist klar. Kein Staat darf es sich schlie\u00dflich erlauben, dass seine Feinde ungehindert an seiner \u00dcberwindung arbeiten k\u00f6nnen. Ich denke, dass es eines der gro\u00dfen Probleme der Autonomen ist, sich so auf die Polizei zu fixieren. Leider sind viele schlicht unf\u00e4hig, gesellschaftliche Verh\u00e4ltnisse analytisch zu erfassen. Die \"Bullen\" werden zur Projektionsfl\u00e4che f\u00fcr den Zwangscharakter der Verh\u00e4ltnisse gemacht.\n\n#### Hat sich dein Verh\u00e4ltnis zur Polizei ge\u00e4ndert?\n\nIch habe nach wie vor ein antagonistisches Verh\u00e4ltnis zum Staat, zur Nation, zum Kapital und daher nat\u00fcrlich auch zur Polizei. Aber ich sehe heute vieles differenzierter als noch vor einigen Jahren. Das liegt aber nicht an einer irgendwie erfolgreich verlaufenden Befriedung oder Resozialisierung durch das Gef\u00e4ngnis, sondern an einer Ver\u00e4nderung in meinem Verst\u00e4ndnis von Gesellschaft und Revolution.\n\n#### Revolution?\n\nWenn man von Revolution spricht, wird man in der Regel ausgelacht, dabei ist nichts l\u00e4cherlicher, als die \u00e4tzenden Verh\u00e4ltnisse als unumst\u00f6\u00dflich zu begreifen. Die Abschaffung des Kapitalismus ist ein ganz notwendiger Schritt in Richtung einer menschenw\u00fcrdigen Existenz aller Menschen. Leider hebt sich der Kapitalismus nicht automatisch auf, auch wenn das einige linke Esoteriker behaupten. Die historische \u00dcberwindung muss gewollt und planm\u00e4\u00dfig ins Werk gesetzt werden. Aber ich habe ein anderes Revolutionsmodell als die klassischen Anarchisten oder Marxisten-Leninisten, Maoisten, Trotzkisten. Meine Politik zielt nicht auf die \u00dcbernahme der Macht oder die Eroberung eines Staatsapparates ab. Mir geht es um einen radikalen, allumfassenden Umsturz des Bestehenden und eine grunds\u00e4tzliche Ver\u00e4nderung der Daseinswelt. Ich will viel mehr als nur das Privateigentum an den Produktionsmitteln abschaffen. Man kann deshalb nicht einfach den Staat, das System, die Verh\u00e4ltnisse milit\u00e4risch bek\u00e4mpfen und seine Beamten, Soldaten und Funktion\u00e4re durch eine bewaffnete Gegenmacht besiegen. Die Revolution ist in erster Linie ein Erkenntnisprozess, der sich in ver\u00e4nderten zwischenmenschlichen Verkehrsformen der Menschen und einer neuen g\u00e4nzlich anderen Organisation der Gesellschaft ausdr\u00fcckt. Die gewaltsame Erst\u00fcrmung des Bundeskanzleramtes und der Polizeikaserne in Ruhleben k\u00f6nnen dabei bestenfalls eine ganz kleine Rolle spielen, m\u00fcssen es aber nicht.\n\nIch meine, eigentlich ist es doch ganz banal: Wir sitzen alle auf diesem Planeten fest und machen uns selbst das Leben zur H\u00f6lle. Es gibt genug zu essen und trotzdem verhungern jede Sekunde Menschen. Selbst hier auf der globalen Gewinnerseite, in den reichen Metropolen, ist das Leben ungem\u00fctlich eingerichtet, weil diese Gesellschaft auf Konkurrenz basiert. Die Welt wird aufgeteilt in ganz wenige Gewinner und viele Verlierer. Manager und Bauarbeiter, Hausfrauen wie Supermodels, Polizisten wie Junkies leiden unter dem permanenten Konkurrenzkampf, der Degradierung zum blo\u00dfen Warensubjekt und dem Zwang zur Lohnarbeit, auch wenn die einen ihre Haut etwas teurer verkaufen k\u00f6nnen als die anderen. Niemand findet das so richtig toll, und doch beteiligen sich alle aus Mangel an Alternativen am kollektiven Amoklauf. Am Ende verlieren wir alle dabei. Ich glaube, es ist daher das Vern\u00fcnftigste, die Welt ganz anders einzurichten.\n\n#### Welche Einstellung hast du zu militanten Aktionen?\n\nIch bin kein Pazifist und war wegen der Beteiligung an militanten Auseinandersetzungen lange im Gef\u00e4ngnis. Grunds\u00e4tzlich halte ich die Sprengung eines Gef\u00e4ngnisses f\u00fcr kein Verbrechen, den Bau dagegen schon. Wenn eine vern\u00fcnftige, reflektierte Gewaltanwendung helfen kann, gewaltt\u00e4tige Verh\u00e4ltnisse zu \u00fcberwinden, dann bef\u00fcrworte ich ganz klar diese Gewaltanwendung. Mir geht es um die radikale Beendigung von Herrschafts-, also Gewaltverh\u00e4ltnissen. Alle Handlungsformen, die diesen Zweck erf\u00fcllen, sind diskutabel.\n\n#### Was ist denn deiner Meinung nach das geeignetste politische Mittel zur Ver\u00e4nderung der Gesellschaft?\n\nDie Bildung von kritischem Bewusstsein! Die gesellschaftlichen Verh\u00e4ltnisse werden heute von den meisten Menschen so selbstverst\u00e4ndlich wie ein Naturgesetz, wie die Schwerkraft hingenommen. Damit stirbt auch der Gedanke daran, dass es anders werden kann und dass Menschen um ihre Befreiung k\u00e4mpfen k\u00f6nnen. Ich glaube, dass es heute vor allem darum geht, die Idee der menschlichen Emanzipation am Leben zu halten. Etwas anderes als die Anziehungs- und \u00dcberzeugungskraft ihrer Ideen kann die radikale Linke eh nicht bieten. Die Autonomen k\u00f6nnen nur so etwas wie Impulsgeber sein. Sie sind ein Laboratorium subversiver Ideen und Praktiken. In bestimmten gesellschaftlichen Situationen, wie zum Beispiel in Griechenland vor drei Jahren, k\u00f6nnen sie zeitweise als Avantgarde von sozialen Bewegungen agieren. Aber gerade das Beispiel Griechenland zeigt die Begrenztheit der Autonomen. Jenseits vom Aufstand bekommen sie nichts hin. Es mangelt ihnen schlicht an einer langfristigen Strategie und einer Perspektive jenseits der Revolte. Johannes Agnoli hat mal \u00fcber die Autonomen gesagt, dass sie die Richtigen sind, die alles falsch machen. Ich glaube, das stimmt.\n\nDas Interview f\u00fchrte Bettina Thiele.\n\n## \"Wenn am 1. Mai sogar die Schaufenster linker Kollektive in Kreuzberg eingeschlagen werden, l\u00e4uft irgendwas falsch.\"\n\nLena, 36 Jahre, lebt in Berlin, stammt aus Kiel.\n\nLena hat \"irgendwas Sozialwissenschaftliches\" studiert und wohnt seit \u00fcber zehn Jahren in Berlin-Friedrichshain. Wir treffen uns in ihrer Wohnung. Wir begegnen einer sehr freundlichen jungen Frau mit langen blonden Haaren, die etwas wirr durcheinanderh\u00e4ngen. Sie ist sehr schlank, mittelgro\u00df und tr\u00e4gt schwarze Kleidung: eine schwarze Dachdeckerhose, ein schwarzes T-Shirt mit dem Aufdruck einer Metal- oder Crust-Band, der Schriftzug ist kaum zu lesen, und eine schwarze Kapuzenjacke. Wir sitzen in ihrem Wohnzimmer, h\u00f6ren den St.-Pauli-Radiosender, den sie erst k\u00fcrzlich f\u00fcr sich entdeckt hat, trinken dunkle Kr\u00e4utergetr\u00e4nke und Kaffee mit Sojamilch. Sie raucht viele selbst gedrehte Zigaretten, w\u00e4hrend sie mit uns spricht, sie lacht viel, schaut aber immer wieder auch ernst und nachdenklich vor sich hin. An den W\u00e4nden h\u00e4ngen Plakate ihrer Lieblingsbands (von Depeche Mode \u00fcber Mot\u00f6rhead bis... But Alive ist alles dabei), Party-Flyer, Plakate von Hardcore-Konzerten in Hausprojekten. Das Sofa, auf dem wir sitzen, ist schwarz, die Tischdecke ist schwarz, und irgendwo entdecke ich sogar eine schwarze Kerze. Sie besitzt sehr viele Schallplatten und Maxi-Singles aus den 80ern und, ja, viele alte Musikkassetten \u2013 ein Kind der 80er, eindeutig! Im B\u00fccherregal stehen Werke von Marx, Engels und Lenin, Max Weber und Adorno, aber auch Romane von Jean-Paul Sartre und Max Frisch. Irgendwo klebt ein Aufkleber aus den 90ern: \"Nazis raus: aus den K\u00f6pfen. PDS\" und an anderer Stelle: \"animal liberation front\" oder auch \"fight sexism\".\n\nIm Vorfeld hatte sie uns erz\u00e4hlt, dass sie lange in der autonomen Szene war, heute immer noch mit ihr sympathisiert, aber \"nicht mehr so aktiv\" sei. Sie wirkt sehr reflektiert und manchmal geradezu versunken in ihre Gedanken.\n\n#### W\u00fcrdest du etwas \u00fcber deine Schulzeit erz\u00e4hlen?\n\nDie war nicht so toll. Ich habe st\u00e4ndig die Schule gewechselt. Bis 15, 16 war ich eher sch\u00fcchtern, aber dadurch bin ich komisch aufgefallen. Ich war schon so ein bisschen autistisch und... verschlossen, und dann habe ich mich manchmal gemeldet und irgendetwas ganz Merkw\u00fcrdiges gesagt, was meiner Ansicht nach von hoher Intelligenz zeugte, so Sachen, auf die andere Kinder nicht unbedingt gekommen w\u00e4ren, was aber dann als merkw\u00fcrdig eingestuft wurde. Das Drama des begabten Kindes... [Lacht.]\n\n#### Was waren deine au\u00dferschulischen Aktivit\u00e4ten?\n\nEhrlich gesagt, in der Teenie-Zeit war ich relativ unpolitisch, also ich war ziemlich durch Punk beeinflusst. Ich habe mich selber ziemlich punkig gesehen oder eben gruftiem\u00e4\u00dfig, schon jugendkulturell, aber politisch war ich \u00fcberhaupt nicht. Es gab bei uns Leute, die so Antifa-Kram gemacht haben, die hab ich ausgelacht. Wie man als Punk halt so ist, eher anarchom\u00e4\u00dfig, mit Politik wollte ich da gar nichts zu tun haben. Das kam eigentlich alles sp\u00e4ter.\n\n#### Den Abschluss hast du aber in der \u00fcblichen Zeit gemacht? Abitur mit 19?\n\nJa. Ich fand Schule zwar immer schlimm, ich fand die Atmosph\u00e4re bedr\u00fcckend, und dieser Zwang, unter dem habe ich schon gelitten. Ich hab trotzdem immer gute Noten gehabt.\n\n#### Und die Familie, mit der du aufgewachsen bist?\n\nIch hatte ein sehr schlechtes Verh\u00e4ltnis zu meiner Mutter und ein ganz gutes Verh\u00e4ltnis zu meinem Vater. Dann haben sich meine Eltern getrennt, ich bin bei meiner Mutter geblieben und... wenn man aufw\u00e4chst mit Leuten, mit denen man sich \u00fcberhaupt nicht versteht... Meine Mutter hatte dann einen neuen Freund und den habe ich gehasst. Ich h\u00e4tte lieber bei meinem Vater gelebt, aber das ging nicht. Was kann es Schlimmeres geben f\u00fcr ein Kind oder einen Jugendlichen, wenn man sich da, wo man lebt, nicht wohl f\u00fchlt und das Gef\u00fchl hat, hier geh\u00f6re ich nicht hin und hier ist immer Streit und Stress.\n\n#### Schreckt dich das davon ab, eine eigene Familie zu gr\u00fcnden?\n\nAbsolut ja, total. Weil ich erlebt habe, was da abgehen kann, das will ich nat\u00fcrlich \u00fcberhaupt nicht wiederholen. Da bin ich relativ pessimistisch, weil ich sehe, dass man doch viel von den Eltern \u00fcbernimmt und das auch so schnell nicht loswird, auch wenn man das vielleicht gerne w\u00fcrde, auch wenn man sich einbildet, man k\u00f6nnte das. Aber das erfordert doch harte Arbeit an sich selber. Und mir ist die Verantwortung zu gro\u00df.\n\n#### Und Kinder willst du gar nicht haben? Oder vielleicht in irgendwelchen anderen Konstellationen?\n\nIch glaube, dass die Konstellation nicht entscheidend ist; das Entscheidende f\u00fcr mich ist, ob ich selber in der Lage bin, einem Kind zu geben, was es braucht, und die entsprechende Verantwortung zu tragen, und da ich nicht mal mit mir selbst klarkomme... Ich w\u00fcrde das nicht kategorisch ausschlie\u00dfen und sagen, auf keinen Fall, aber die Frage hat sich bisher nicht gestellt. Ich w\u00fcrde das nicht alleine machen wollen, und bisher hab ich noch niemanden getroffen, mit dem ich das h\u00e4tte zusammen machen k\u00f6nnen.\n\n#### Und wie findest du die Lebensweise, die du jetzt hast?\n\nIch w\u00fcrde nicht sagen, dass ich zufrieden bin in meinem Leben, aber es ist schon besser geworden. Ich denke, dass ich mich weiterentwickelt habe, auch in der Auseinandersetzung mit meinen Eltern, und das ist auch notwendig f\u00fcr die Weiterentwicklung. Da ver\u00e4ndern sich bei mir auch bestimmte Denkweisen und Perspektiven f\u00fcr die Zukunft.\n\n#### Du hast gesagt, du warst vom Punk beeinflusst. Warst du selbst Punk oder geh\u00f6rtest du einer anderen Jugendkultur an?\n\nIch f\u00fchlte mich schon mehreren Jugendkulturen zugeh\u00f6rig, also in verschiedenen Phasen. Ich war mal Gruftie, so in den 80ern\/90ern, sp\u00e4ter kam Punk dazu, dann kam diese autonome Sache, im weitesten Sinne die linksradikale Szene.\n\n#### W\u00fcrdest du die autonome Szene auch unter Jugendkultur einordnen?\n\nEigentlich nicht, weil der Impuls ein anderer ist; gleichzeitig gibt es aber auch Aspekte, die jugendkulturell daran sind und \u00c4hnlichkeiten mit Jugendkulturen haben. Jugendkultur klingt so ein bisschen verniedlichend, so als ob das nur eine Jugendphase w\u00e4re, denn eigentlich geht es ja um was Politisches. Andererseits sind es ja \u00fcberwiegend j\u00fcngere Leute und viele ziehen sich auch zur\u00fcck, wenn sie \u00e4lter werden. Und es gibt auf jeden Fall jugendkulturelle Elemente: die Musik, der Style etc. \u2013 das \u00fcberschneidet sich ja oft mit der Punk- und Hardcore-Szene.\n\n#### Was war dir wichtig in den Jugendkulturen, in denen du warst? Eher Kleidung oder eher Musik, T\u00e4towierungen, K\u00f6rperschmuck?\n\nMusik, grunds\u00e4tzlich immer Musik, weil ich eigentlich sehr \u00fcber Musik funktioniere. [Lacht.] Weil mich Musik immer am meisten \u00fcberzeugt bei irgendwas, die Musik transportiert auch am ehesten das Lebensgef\u00fchl. Die Musik ist f\u00fcr mich immer das gewesen, wodurch mich etwas angezogen hat. Und dann im weiteren Verlauf, wenn ich die Szenen kennengelernt habe, passten auch andere Sachen dazu, die mir gefallen haben. Also meinetwegen in der Gruftie-Szene diese Traurigkeit, die in der Musik zum Ausdruck kommt, auch dieses D\u00fcstere, Morbide, und dazu passt nat\u00fcrlich dann auch, dass man Schwarz tr\u00e4gt, dass man dementsprechend sich die Haare f\u00e4rbt und so weiter. Und im Punk genauso, im Punk ist f\u00fcr mich die Wut, die Aggression und das Zerst\u00f6rerische interessant, und das kommt in der Musik zum Ausdruck. Mode aber auch, doch, Style, wenn man das so nennen kann, obwohl die ja immer so tun, als ob sie keinen Style haben, aber nat\u00fcrlich gibt es da einen Style, und das hat f\u00fcr mich schon auch eine Rolle gespielt. [Sie grinst und zeigt mir eine \"Schwarztrinker\"-Tasse mit einem dunklen Getr\u00e4nk.]\n\n#### Und warum bist du nicht in der Gothic-Szene geblieben?\n\nWeil die mir zu unpolitisch waren, so einfach ist das. Ich hab sp\u00e4ter auch in der autonomen Szene ehemalige Gothics kennengelernt, da gibt es manchmal durchaus personelle \u00dcberschneidungen, wahrscheinlich wegen der schwarzen Outfits, ja, wirklich! Und die haben mir auch alle gesagt: Die Gothics waren mir zu unpolitisch. Und wieder an bestimmten Stellen waren mir die Autonomen zu oberpolitisch oder zu dogmatisch. Da gibt es nicht viel Sinn f\u00fcr \u00c4sthetik oder Style und Humor und so. Gothic ist auch nicht unbedingt eine Spa\u00dfkultur, aber das ist ja auff\u00e4llig, dass da wirklich der Style und die Mode eine riesige Rolle einnimmt. Die stehen zwei Stunden vor dem Spiegel und machen sich sch\u00f6n. Und das ist so was, was mir immer viel Spa\u00df gemacht hat als Kind schon, Verkleiden, und das hat in der autonomen Szene vordergr\u00fcndig \u00fcberhaupt keinen Raum.\n\n#### Und wie \u00e4u\u00dfert sich deine Begeisterung f\u00fcr die entsprechende subkulturelle Musik?\n\nMusik ist sehr wichtig f\u00fcr mich, ich w\u00fcrde fast sagen, das Wichtigste. Ich habe sogar schon Aussagen getroffen wie, dass Musik wichtiger f\u00fcr mich ist als Menschen! Weil Musik immer da ist.\n\n#### Ist deine Freizeitgestaltung dementsprechend? Hast du mal in einer Band gespielt?\n\nIn einer Band habe ich leider nie gespielt, wollte ich mal, aber ich habe nie die richtigen Leute gefunden oder vielleicht bin ich nicht leidenschaftlich genug oder kreativ, sondern eher Konsumentin. Aber ich wollte schon fr\u00fcher Musik machen, ich hab hier auch 'ne Bass-Gitarre stehen, und ich wollte singen, aber das habe ich nicht realisiert.\n\n#### Hast du sonst noch Hobbys... oder peinliche Vorlieben? So was wie Lindenstra\u00dfe gucken? [Hier grinsen alle, weil wir uns schon im Vorgespr\u00e4ch als Lindenstra\u00dfen-Fans geoutet haben.]\n\nDas sage ich laut! Lindenstra\u00dfe ist 'ne totale Punk-Sendung! Da gab's sogar eine Platte, Wir warten auf die Lindenstra\u00dfe mit den Goldenen Zitronen und anderen Punk-Bands. Also wenn man das jetzt als Hobby bezeichnen kann? Also Fernsehen guck ich leider wirklich ziemlich viel im Moment, hab da so meine Lieblingsserien... Das ist \u00fcbrigens auch ein wichtiges Stichwort in der autonomen Szene. Wenn man mich vor zehn Jahren gefragt h\u00e4tte, da hatte ich keinen Fernseher, da hab ich \u00fcberhaupt nie ferngesehen, h\u00f6chstens Filme. Und wenn ich Filme gesehen habe, dann im Kino oder auf Video. Da gab es schon eine starke Zensur, also man hat nicht alles angeguckt, nur, was auch politisch wertvoll war. Man redet ja auch \u00fcber das, was man guckt, und so was wie Desperate Housewives war da nicht angesagt. Ich glaube auch nicht, dass ich jemandem aus der Szene erz\u00e4hlt habe, dass ich Lindenstra\u00dfe geguckt habe. Bei anderen Freunden ging das, aber in der autonomen Szene hatte ich das Gef\u00fchl, dass ich aufpassen muss, was ich da sage, was ich zugebe, wie ich drauf bin. Lindenstra\u00dfe w\u00e4re etwas gewesen, wo die die Augen verdreht h\u00e4tten, so was kann man doch nicht gucken. Ist doch alles Konsumindustrie gewesen, Fernsehen generell. Man hat einfach politisch korrekte Filme geguckt.\n\n#### Gab es da auch Bands, die man besser nicht zugeben durfte?\n\nStimmt, bei mir ist das dann auch kollidiert mit der Gruft-Geschichte. Weil das alles andere als p.c. ist, da ist vieles sexistisch, aber auch politisch v\u00f6llig daneben, und da musste ich mir auch harte Sachen anh\u00f6ren von meinen autonomen Freunden, richtige moralische Standpauken, dass ich \"solche\" Musik h\u00f6re. Es war wirklich teilweise so, dass ich diese Musik dann heimlich geh\u00f6rt habe. Und ich hatte auch eine Phase, wo ich drauf und dran war, die ganze Musik richtig auszusortieren und zu verschenken oder zu verkaufen. Aber irgendwann habe ich dann gedacht, was mach ich hier eigentlich, das kann ja wohl nicht wahr sein, ich h\u00f6re die Musik doch gerne! Vielleicht ist das der Punkt, wo ich gemerkt habe, dass ich im Grunde meines Herzens doch einiges davon nicht vertreten kann. Also dieses Dogmatische, dieses Strenge, dass alles ausgew\u00e4hlt sein muss \u2013 nee, ich m\u00f6chte, wenn ich dazu Bock habe, auch etwas h\u00f6ren, was v\u00f6llig un-p.c. ist, und mir den \u00fcbelsten Schei\u00df reinziehen, der politisch-moralisch gar nicht vertretbar ist. Aber man kann auch nicht generell sagen, dass da alle Leute super-dogmatisch sind, nicht umsonst gab es dann auch viele Leute, die sich als \"undogmatische Linke\" bezeichnet haben. Andererseits gibt es halt so bestimmte Dinge, wie Rassismus, Sexismus; Nationalismus, die absolut abgelehnt werden, was ja eigentlich auch gut ist. Und das wird dann eben auf alles bezogen bzw. alles wird quasi darauf gepr\u00fcft: Kunst, Kultur etc. Insgesamt vielleicht etwas zu streng oder humorlos. Aber das waren halt auch die Leute, die sehr ernsthaft politisch unterwegs waren, nicht gerade Anarchos oder Spa\u00dffraktion, da gab es auch Leute, die ganz anders drauf waren...\n\n#### Kann man Angeh\u00f6rige der autonomen Szene an einer bestimmten Musik, Kleidung oder anderen Merkmalen erkennen?\n\nFinde ich grunds\u00e4tzlich erst mal ganz schwierig, weil ich pers\u00f6nlich ein ganz anderes Verst\u00e4ndnis von Autonomen habe, also der Begriff \"autonom\" bedeutet f\u00fcr mich etwas anderes als das, was klischeehaft darunter verstanden wird. Deswegen habe ich ein Problem damit zu sagen, die sehen alle \"so\" aus und h\u00f6ren \"die und die\" Musik. Das sehe ich grunds\u00e4tzlich nicht so, weil ich in dieser Szene alle m\u00f6glichen Leute kennengelernt habe, die ganz unterschiedliche Musik geh\u00f6rt haben, die sich teilweise ganz unterschiedlich gekleidet haben, die sich aber trotzdem alle als Autonome verstanden haben. Nat\u00fcrlich kann ich hier auch die ganzen Klischees reproduzieren und sagen, die tragen alle schwarze Kapuzenpullis und so, das ist aber nicht meine Auffassung davon.\n\n#### Wie kam es denn zu deinem Interesse an der autonomen Szene?\n\nMich hat einfach beeindruckt, das muss in der Pubert\u00e4t gewesen sein, in den 80ern, als ich zum ersten Mal etwas dar\u00fcber gelesen oder geh\u00f6rt habe, wie die leben. Dass sie in besetzten H\u00e4usern wohnen und fernab von jeglicher Restgesellschaft ihr eigenes Leben auf die Beine stellen. Das hat mich schon angezogen.\n\n#### Was war denn konkret die erste Kontaktaufnahme? Hausprojekt, Party, Freunde? Oder geht man da hin und sagt, so, ich will jetzt autonom sein?\n\nJa, ein bisschen war es bei mir so. Als ich 15 oder 16 war, da gab es mal einen Artikel im Stern oder Spiegel, was meine Eltern eben so gelesen haben, \u00fcber Hausbesetzer in Potsdam, und das war schon einer der Gr\u00fcnde, warum ich nach Berlin gegangen bin, weil ich wusste, da gibt es eben diese Szene. Dann habe ich Leute kennengelernt, auch \u00fcber die Uni, und dann hat man mal einen Flyer gefunden und ist zu Veranstaltungen gegangen in einem von diesen Hausprojekten. Das war so der Erstkontakt, wo das Interesse anfing. Es gab damals auch \u00dcberschneidungen, zum Beispiel Gruftie-Partys in besetzten H\u00e4usern. So kam man eben von einer Szene in die andere. Aber ich habe auch oft versucht, mich genau an diesen Schnittstellen zu bewegen. Ich bin am liebsten auf Gruftie-Partys gegangen, die auch ein bisschen punkig waren, zum Beispiel im Drugstore, da gab es eine Party, die hie\u00df \"Darkstore\", einmal im Monat, da bin ich am liebsten hingegangen, weil da liefen auch Punks rum, da war das so ein bisschen gemischter.\n\n#### Was waren denn deine ersten Aktivit\u00e4ten in der autonomen Szene jenseits von Partys?\n\nDemos, der Erstkontakt sind doch meistens Demos. Mal eine Tierrechts-Demo, weil ich war auch Vegetarierin damals schon, und hatte auch schon von Veganern geh\u00f6rt, dann waren wir mal auf einer Anti-Pelz-Demo, und da hat man auch wieder Leute kennengelernt. Meine erste Demo war allerdings die Luxemburg-Liebknecht-Demo im Januar! Da war ich gerade ein halbes Jahr in Berlin. Und da war ein Autonomen-Block. Ich war damals mit Punks unterwegs, die kannte ich aus dem Haus, wo ich gewohnt habe, und mit denen bin ich da mal hingefahren. Ich wusste damals gar nicht, worum es da eigentlich ging, ich war noch so unpolitisiert, war nicht einmal informiert, wer Rosa Luxemburg ist. [Lacht.] Ich hatte \u00fcberhaupt keine Ahnung! Ich bin da einfach mitgefahren, weil ich mit den Leuten da abhing. Und so l\u00e4uft das bei vielen. Ich w\u00fcrde sagen, dass der h\u00e4ufigste Einstieg in Szenen \u00fcber andere Leute ist. Und die Punks sind dann straight in den autonomen Block, und das war tats\u00e4chlich meine erste Begegnung. Das war beeindruckend. Das war eine gro\u00dfe Demo mit allen m\u00f6glichen Bl\u00f6cken, und da l\u00e4uft alles M\u00f6gliche nebeneinander her, und dann gab's da den autonomen Block und die sind genau da rein. Und dann war ich dabei, sozusagen. So f\u00e4ngt so was an.\n\n#### Hast du konkret in bestimmten Gruppen mitgemacht?\n\nJa.\n\n#### Machst du noch mit?\n\nNein. Das ist auch alles, was ich dazu sagen will. [Lacht.]\n\n#### Dann beschreib doch mal die autonome Szene aus deiner Sicht?\n\nEin loser Zusammenschluss von verschiedenen Gruppen und Gr\u00fcppchen, die sich sehr weit links positionieren, also radikal links. Kennzeichnend ist, dass die versuchen, ein Leben in Freir\u00e4umen jenseits der Restgesellschaft \u2013 mehr oder weniger \u2013 aufzubauen, wo es nat\u00fcrlich Durchl\u00e4ssigkeiten gibt; aber sie versuchen schon relativ stark, ihre eigenen R\u00e4ume und Freir\u00e4ume zu schaffen, ihre eigenen Vorstellungen von einem guten Leben umzusetzen.\n\n#### Und das gilt auch f\u00fcr dich?\n\nF\u00fcr mich pers\u00f6nlich ist das vielleicht noch mal ein bisschen anders. Ich verstehe unter autonom erst mal nur selbstbestimmt. Und ich verstehe das Wort fast im psychologischen Sinn, autonom bedeutet ja auch von der Pers\u00f6nlichkeit her oder auch in Beziehungen Unabh\u00e4ngigkeit. Also bin ich in der Lage, selber zu bestimmen, was ich will und wie ich leben will, und mich nicht danach zu riehten, was andere m\u00f6glicherweise von mir erwarten und was die gesellschaftlichen Konventionen sind, was ich von meinen Eltern mitbekommen habe usw.\n\n#### Und geht es dir eher um Freir\u00e4ume oder um gesamtgesellschaftliche Ver\u00e4nderungen?\n\nUm beides. Um die Freir\u00e4ume zu erk\u00e4mpfen geht es definitiv, und vor allem geht es den Leuten in der autonomen Szene sehr viel darum, diese hier schon zu leben. Es gibt ja auch ganz andere politische Fraktionen, auch linksradikale politische Fraktionen, die das v\u00f6llig l\u00e4cherlich finden, zu meinen, man k\u00f6nnte hier, so wie die Gesellschaft strukturiert ist, ein eigenst\u00e4ndiges Leben oder ein unabh\u00e4ngiges Leben f\u00fchren, und dass man erst mal den Kampf ums Ganze f\u00fchren muss. Die autonome Szene ist schon immer auch sehr auf sich bezogen und wie gestalte ich mein eigenes Leben im Hier und Jetzt schon so unabh\u00e4ngig wie m\u00f6glich, und wie mache ich in meinen kleinen R\u00e4umen und in meinem kleinen Mikrokosmos das politisch korrekte Essen, oder wie diskutiere ich irgendwelches sexistisches Redeverhalten am K\u00fcchentisch aus. Das habe ich immer schon so gesehen als Versuch, im Hier und Jetzt in bestimmten Freir\u00e4umen ein besseres Leben aufzubauen. Aber dieser Kampf in die Gesellschaft hinein findet nat\u00fcrlich auch statt, mal mehr, mal weniger. Es gibt unterschiedliche Gr\u00fcppchen, es gibt eben welche, die begn\u00fcgen sich mehr damit, ihre Landkommune zu gr\u00fcnden oder auf ihrem Wagenplatz zu sitzen und ansonsten nicht viel zu machen, und dann gibt es welche, die eher ihre Energie auf den politischen Kampf konzentrieren und sagen, ich k\u00fcmmere mich jetzt nicht so darum, ob ich Veganer bin oder ob ich meinen M\u00fcll trenne oder so. Es gibt in der autonomen Szene Leute, die sich als Anarchisten bezeichnen, und es gibt Leute, die w\u00fcrden sich als Kommunisten sch\u00e4rfster Couleur bezeichnen.\n\n#### Wie sieht es aus mit Feminismus in der autonomen Szene?\n\nOh, das ist ein wichtiges Thema. Nat\u00fcrlich gibt es auch in der autonomen Szene eine M\u00e4nnerdominanz. Aber das gibt es auch in allen anderen Szenen, und ich fand teilweise die M\u00e4nnerdominanz in der Punk-Szene viel schlimmer. Gerade weil Punks sich oft nicht als politisch verstehen, wird das nicht reflektiert, und unter diesem Deckmantel des Anarchismus, wo dann gesagt wird, wir k\u00f6nnen ja alle machen, was wir wollen, wird dann auch sich \u00fcbelst verhalten und benommen, weil immer davon ausgegangen wird, die Frauen sind ja auch alle tough und k\u00f6nnen machen, was sie wollen. Da habe ich in der autonomen Szene wesentlich positivere Erfahrungen gemacht, weil die Leute sich als politisch verstehen und einen anderen moralischen Anspruch haben. In der autonomen Szene habe ich das immer so erlebt, dass alles ausdiskutiert wurde, alles, in jeder Gruppe. Wenn ich von autonomer Szene spreche, meine ich nat\u00fcrlich nicht die ultraprolligen Steineschmei\u00dfer, die hier zum 1. Mai anreisen und einen auf Touri machen. Die w\u00fcrde ich nicht dazurechnen, das sind oft ganz unpolitische Vorstadt-Kids.\n\nGrunds\u00e4tzlich gibt es ja Sexismus \u00fcberall in allen Szenen so wie in der Restgesellschaft auch, das ist klar. Also auch in der autonomen Szene. Und nat\u00fcrlich waren es auch meistens die Frauen, die diese Themen eingebracht hatten. Wenn man eine Gruppe hatte, gemischtgeschlechtlich, waren es nat\u00fcrlich die Frauen, die gesagt haben, hier gibt es aber sexistisches Redeverhalten, weil die Typen reden schon wieder zwei Stunden, das ist ja auch Sexismus, nicht nur Militanz und so. F\u00fcr mich gibt es auch ganz viele Intellektuelle, die sind dann verbale Macker, die reden drei Stunden und h\u00f6ren sich selber gerne reden. Unter Militanz wird ja gerade auch dieses Martialische, dieses Stra\u00dfenk\u00e4mpferm\u00e4\u00dfige verstanden. Aber was ich teilweise viel schlimmer fand, ist eher intern, also bei politischen Diskussionen oder so, ein bestimmter Gestus von Typen, der immer so selbstverst\u00e4ndlich daherkommt, dass sie superschlaue Sachen sagen, und die Frauen trauen sich dann nicht. Aber es gibt auch eine autonome Queer-Szene zum Beispiel, und die sehen das zum Beispiel als Hauptagitationsfeld. Aber nat\u00fcrlich gibt es Sexismus in der Szene. F\u00fcr mich gibt es \u00fcberall Sexismus, wie soll es dann in der autonomen Szene keinen Sexismus geben?! Sexismus, das sind doch Strukturen, die in allem drinstecken, also Machtstrukturen...\n\nNat\u00fcrlich gibt es dieselben Dinge, die es woanders gibt, auch in der autonomen Szene, nur dass die Leute einen anderen Anspruch an sich selber haben und sagen, wir verstehen uns als politisch links, linksradikal, das hei\u00dft, wir versuchen, genau diese Strukturen aufzubrechen und zu hinterfragen und zu reflektieren, auch, dass wir selber Teil dessen sind. Das habe ich oft in der autonomen Szene kennengelernt, da gab es eine hohe Selbstreflektion. Diese ganze Besch\u00e4ftigung mit sich selbst, das ist fast schon Psycho-Seminar, was die teilweise machen. Kennzeichnend f\u00fcr die autonome Szene ist, dass alles und jedes immer stundenlang ausdiskutiert wird, weil selten etwas eindeutig ist. Nat\u00fcrlich gibt es Sachen, die allgemein als sexistisch empfunden werden, aber da sind eben auch die Grauzonen. Wenn die Betreiber des Kneipenkollektivs X sagen, der Typ hat einen sexistischen Spruch gebracht, die Frau hat sich bel\u00e4stigt gef\u00fchlt, wird das normalerweise erst mal mit dem Typen ausdiskutiert und er muss Stellung dazu beziehen. Wenn er das aber wiederholt tut, dann fliegt er raus. Und so wurden Konflikte in dem Sinne gel\u00f6st. In die Frauenr\u00e4ume kommen eh keine Typen rein, da wird das dann gleich so gel\u00f6st. In den anderen R\u00e4umen, auf Konzerten, auf Partys, auf Veranstaltungen werden Leute bestimmt, die daf\u00fcr zust\u00e4ndig sind, auf so was zu achten, so das Antisexismus-Team, die achten darauf und beobachten so ein bisschen. Und wenn die sehen, da gibt es einen Konflikt zwischen einem. Typen und einer Frau oder der Typ macht die Frau dumm an und sie f\u00fchlt sich bel\u00e4stigt, dann gehen die hin und k\u00fcmmern sich darum. Also insgesamt hat Sexismus und Anti-Sexismus einen hohen Stellenwert in der autonomen Szene. Es wird sich viel damit auseinandergesetzt, das fand ich immer positiv.\n\n#### Wie sieht es aus mit der Gewalt in der autonomen Szene? Wo w\u00fcrdest du mitmachen, was kannst du noch guthei\u00dfen und wann ist bei dir Schluss?\n\nIch habe definitiv immer ein Problem gehabt mit Gewalt gegen Personen, andererseits habe ich auch nicht unbedingt ein ganz schlimmes moralisches Problem damit, wenn ein Nazi auf die Fresse kriegt. Die pr\u00fcgeln ja auch andere Menschen tot oder z\u00fcnden sie an. Als Verteidigung sowieso, es gibt ja viele Leute, die Selbstverteidigung machen, gerade als Frau, das finde ich okay. Was Gewalt gegen Sachen angeht, damit habe ich wesentlich weniger Probleme. Ich finde diese Art von Gewalt aber nicht unbedingt ein ad\u00e4quates Mittel von ernsthafter Politik. Was nutzt mir das, wenn ich ein Auto anz\u00fcnde? Ich finde nicht, dass man mit v\u00f6llig friedlichem Protest besonders viel erreicht, aber trotzdem sehe ich, wenn jetzt zuf\u00e4llig irgendein unschuldiger Gr\u00fcner [lacht] oder so sein Auto in Kreuzberg geparkt hat und das ist dann am n\u00e4chsten Tag kaputt, nicht richtig viel Sinn darin. Ich halte auch nicht viel davon, irgendeiner Bank die Schaufenster einzuschlagen, \u00fcberhaupt gar nicht. Ich hab kein moralisches Problem damit, wenn das Leute machen, von wegen Sachbesch\u00e4digung oder so, aber ich denke, das lenkt eher von den eigentlichen Zielen ab, n\u00e4mlich die bestehenden Verh\u00e4ltnisse ernsthaft zu ver\u00e4ndern. Und ob sich dadurch was ver\u00e4ndert, wenn man das Medienklischee reproduziert, dass alle Autonomen \u00fcble Gewaltt\u00e4ter sind... ich wei\u00df nicht. Ich halte diese Art von symbolischem Protest f\u00fcr kein besonders ad\u00e4quates Mittel von Politik. Als ich jung und neu in dieser Szene war, hatte das noch einen gewissen Reiz und eine gewisse Faszination, dieses Gewaltsame und Zerst\u00f6rerische. Man will ja auch mal seine Wut loswerden. Aber wenn am 1. Mai z. B. sogar die Schaufenster linker Kollektive in Kreuzberg eingeschlagen werden, kann ich nur sagen: Hier l\u00e4uft irgendwas falsch. Das ist einfach nur dumm. Aber ich vermute, dass das eben oft die eher unpolitischen Demo-Touristen sind, von denen so was ausgeht. Von den Leuten, die ich so kenne, w\u00fcrde niemand im Traum auf so was kommen.\n\n#### Gibt es andere linke Bewegungen, mit denen du dich anfreunden kannst? Anti-Atom, Umweltschutz?\n\nIch sehe das nicht so getrennt, weil die autonome Szene auch immer mit all diesen Bewegungen zu tun hatte. Ich w\u00fcrde das gar nicht so abgrenzen. Es gibt Autonome, die auf Anti-Atom-Proteste fahren, es gibt Autonome, die sich stark antirassistisch engagieren. Ich habe auch themenm\u00e4\u00dfig vieles ausprobiert, und bei Anti-Atom-Protesten war ich auch schon. Aber das war bei mir nie so ein Schwerpunkt. Meine pers\u00f6nlichen thematischen Schwerpunkte waren doch mehr Gender, also Feminismus im Sinne der heutigen Gender-Studies, und Antifa.\n\n#### Wirst du weiter politisch aktiv sein?\n\nIch bin jetzt schon Mitte drei\u00dfig und ich habe immer ein Leben gef\u00fchrt, was anders ist als die meisten anderen Leben, und ich werde mich nie im b\u00fcrgerlichen Leben zu Hause f\u00fchlen, wirklich, auch wenn es da nat\u00fcrlich Ber\u00fchrungspunkte gibt, und so wird Politik immer ein Teil meines Lebens bleiben, und ich werde auch wahrscheinlich immer meine linksradikale Einstellung behalten. Aber ich werde auch nicht meine gesamte Energie darauf verwenden. Es wird immer wichtig f\u00fcr mich bleiben, ich werde mich immer daf\u00fcr interessieren und mich damit besch\u00e4ftigen, aber ich werde dem nicht mein ganzes Leben verschreiben.\n\n#### Und hast du sonst noch Ziele?\n\nEben nicht, das ist auch mein Problem. Keine konkreten Pl\u00e4ne. Es gibt einfach Sachen und Themen, die mich interessieren, gesellschaftliche Bereiche und Berufsfelder, da werde ich mal weiter gucken und suchen und mehr und mehr meine Richtung finden. Aber ich w\u00fcrde jetzt nicht sagen, dass ich genau wei\u00df, was ich will, endg\u00fcltig.\n\nDas Interview f\u00fchrte Anna Ziegler.\n\n## Live dabei am 1. Mai \u2013 Die \"Revolution\u00e4re 1. Mai Demo\" in Kreuzk\u00f6lln\n\nVon Martin Gegenheimer\n\nDer 1. Mai begann entspannt. Die Sonne schien und wir legten den Anfahrttermin auf etwa 16 Uhr, um die angek\u00fcndigte und in den Medien herbeibeschworene St\u00f6raktion der \"Linksextremisten\" auf dem MyFest miterleben zu k\u00f6nnen. Ab 18 Uhr wollten wir die traditionelle \"Revolution\u00e4re 1. Mai Demonstration\" dokumentierend begleiten, auf der ich auch einen Kollegen erwartete, der noch auf dem Fahrrad zur Beobachtung der Lage in Kreuzberg unterwegs war.\n\nIch packte meine Fotoausr\u00fcstung und meinen \"Demohelm\" ein und machte mich auf den Weg nach \"Kreuzberg 36\". Bei der Anreise \u00fcber den Tempelhofer Damm fehlte \u00fcberraschenderweise der \u00fcbliche Truppenauflauf der Polizei am Platz der Luftbr\u00fccke. Stattdessen waren nur zwei verlorene Wagen vor dem Eingang zum Polizeipr\u00e4sidenten zu sehen, die sich auf die Abfahrt Richtung Kreuzberg vorzubereiten schienen. Auf der Skalitzer Stra\u00dfe begegneten uns die ersten abgestellten Konvois mit Polizisten in Schlagschutzausr\u00fcstung, die alle entspannt mit ihren Handys hantierten.\n\nMeine Freundin suchte einen m\u00f6glichst ruhigen Platz zum Abstellen unseres Motorrollers, um nah am Geschehen sein zu k\u00f6nnen. Im Notfall sollte es aber auch m\u00f6glich sein, dieses schnell verlassen zu k\u00f6nnen. Eine knifflige Aufgabe, wenn man das Risiko minimieren wollte, den Roller eventuellen Barrikadenbauten zu opfern. Der perfekte Parkplatz schien sich nahe eines Hauseinganges auf der Skalitzer Stra\u00dfe zu befinden. Nach dem Abstellen hielt dort jedoch auch eine komplette Polizeihundertschaft, was den Platz minder optimal erscheinen lie\u00df. Trotzdem entschlossen wir uns, das Risiko einzugehen, und bewegten uns \u00fcber den Platz des Kottbusser Tors Richtung MyFest. Kleinere Trupps gelangweilter Einsatzpolizisten standen in voller Montur herum, neben einzelnen \"Sp\u00e4hern\", die, R\u00fccken an R\u00fccken, das Geschehen zu \u00fcberblicken versuchten.\n\nAm Anfang der Adalbertstra\u00dfe fielen mir weitere Polizisten in Zweierteams auf, die offenbar neben allem, was man zerst\u00f6ren oder anz\u00fcnden k\u00f6nnte, postiert waren. Es sah so aus, als h\u00e4tte jedes Bushaltestellenh\u00e4uschen und jeder M\u00fclleimer einen Bodyguard erhalten. Ein St\u00fcckchen weiter tauchten die ersten Zivilkr\u00e4fte auf: schlecht getarnt mit schwarzen Sonnenbrillen, Softshelljacken, Bauchtaschen und alle mit verr\u00e4terischen Schallschl\u00e4uchen der Funkger\u00e4te im Ohr. Der krampfhafte Versuch, sich \"szenetypisch\" zu kleiden und in der Masse unterzutauchen, scheiterte schon allein an der Gr\u00fcppchenbildung. Ein St\u00fcck weiter sa\u00dfen die ersten Punks und linken Skins mit ihren Bierchen, die gut gelaunt den kommenden Herausforderungen entgegenfeierten und von den Einsatzkr\u00e4ften gemustert wurden.\n\nNachdem der Hochhausquerriegel passiert war, waberte eine Rauchwolke auf uns zu und Grill \u00fcber Grill pumpte die Ger\u00fcche von K\u00f6fte, Nackensteak und Rostbratwurst in die Luft. Es schien, als ob jeder, der einen Grill besa\u00df, sich vor seinem Haus am MyFest beteiligen wollte, um sein Taschen- oder Haushaltsgeld aufzubessern. Dazwischen befanden sich besser ausgestattete und professionelle St\u00e4nde von Bars und Restaurants. Aber die Masse an Menschen, die sich \u00fcber die Oranienstra\u00dfe quetschte, war dichter, als ich es mir in meinen schlimmsten Weihnachtsmarkt-Alptr\u00e4umen h\u00e4tte vorstellen k\u00f6nnen. Wir versuchten, m\u00f6glichst schnell wieder aus dem \"Fest\" zu gelangen. Selbst wenn man gewollt h\u00e4tte, erschien es unm\u00f6glich, etwas zu trinken zu kaufen, da sich vor jedem Stand eine Warteschlange befand, die mit den Schlangen der vorgelagerten St\u00e4nde zu verschmelzen drohte. Wir dr\u00e4ngelten uns an den Wartenden und Feiernden vor den B\u00fchnen durch den L\u00e4rm. An jeder B\u00fchne war es m\u00f6glich, mindestens noch eine weitere zu h\u00f6ren, was ein unangenehmes Musikwirrwarr erzeugte. Am Heinrichplatz stoppte mich kurz ein Plakat mit der Aufschrift \"Willkommen zum polizeilich organisierten MyFest \u2013 Saufen, Fressen, Dr\u00e4ngeln\". \u00dcber das \"polizeilich organisiert\" lie\u00dfe sich bestimmt streiten, \u00fcber den Rest eher nicht.\n\nAuf einmal stand ich \u00fcberrascht vor einer Frau. Zwischen uns befand sich ein surreal gro\u00dfer Freiraum von ca. 2-3 Metern. Neben ihr stand ein Bollerwagen voll Kinder. Sie hob den Arm, in der Hand einen gro\u00dfen Pflasterstein, und warf ihn in meine Richtung. Nach einer Schrecksekunde schnellte er in ihre Hand zur\u00fcck. Beim N\u00e4herkommen enttarnte ich die handwerklich geschickt fabrizierte Bastelarbeit als Styroporkubus, der mit grauer Farbe auf Stein getrimmt und mit einer Schnur befestigt worden war. Die Kinder spielten mit ihren Exemplaren. \"Lustige Idee!\" dachte ich im Vorbeidr\u00e4ngeln und w\u00fcnschte ihnen, an diesem 1. Mai nur Polizisten mit starken Nerven zu treffen.\n\n\u00dcber die Waldemarstra\u00dfe versuchten wir entlang der t\u00fcrkischen Folkloreb\u00fchne dem Get\u00fcmmel Richtung Manteuffelstra\u00dfe zu entkommen. Mein Mitleid mit Hunden und Kindern, die von ihren Herrchen und Eltern hinter sich hergeschleift wurden, stieg. Es fiel auf, dass im Gegensatz zu den letzten Jahren die Dachhoheit der Polizei anscheinend nicht gegeben war. Die Jahre zuvor waren auf fast allen Flachd\u00e4chern der Umgebung Kamerateams zu sehen gewesen. Heute war davon nichts zu erkennen. Auch die Hinterh\u00f6fe wurden offensichtlich nicht massiv als Gruppenkraftwagen-Parkplatz und Polizei-Grillplatz genutzt wie in den Jahren davor.\n\nIn der Manteuffelstra\u00dfe am G\u00f6rlitzer Bahnhof machten wir eine Pause und suchten zwischen \u00fcberquellenden M\u00fclleimern Erholung. Als erstes Res\u00fcmee war festzuhalten, dass das MyFest sehr weit davon entfernt war, unterwandert und gesprengt zu werden. Doch die an uns vorbeiziehenden und gut gelaunten Fahnentr\u00e4ger erinnerten uns rasch an den baldig anstehenden Beginn der \"Revolution\u00e4ren 1. Mai Demonstration\". So schlugen wir uns durch diverse Seitenstra\u00dfen, vorbei an Polizeiwagen aus aller Herren Bundesl\u00e4nder, Richtung Kottbusser Br\u00fccke, die den Startpunkt der diesj\u00e4hrigen Demo darstellte.\n\nDoch vor dem Erreichen gab es noch ein Hindernis: der erste Polizeiriegel des Tages. Taschenkontrolle! Im Vorjahr w\u00e4re an diesem Punkt meine Teilnahme schon fast gescheitert, da die Herren einer Einsatzhundertschaft aus Niedersachsen mich, aufgrund eines mitgef\u00fchrten und deutlich mit \"PRESSE\" beschrifteten Helmes, nicht passieren lassen wollten. Auch das Argument, dass ich rund 2.000 \u20ac in Fotoausr\u00fcstung bei mir f\u00fchren w\u00fcrde, konnte weder meinen Ansprechpartner in der Kette noch seinen Vorgesetzten \u00fcberzeugen. Erst der hinzugezogene Berliner Einsatzleiter entschied das Ganze unb\u00fcrokratisch mit einem Augenzwinkern und einem Kopfnicken, begleitet von einem gemurmelten: \"Lass ihn durch\", zu meinen Gunsten. Doch in diesem Jahr wurde die Kette von karnevalserprobten Polizisten aus NRW organisiert. Man begr\u00fc\u00dfte mich mit einem entspannten \"Hallo. In dem Beutel haben Sie bestimmt einen Helm. Das ist kein Problem, aber ich m\u00fcsste doch noch einmal kurz hineinsehen\", und Sekunden sp\u00e4ter passierte ich die Sperre.\n\nDas Warten begann. Entspannt wirkende Polizisten vorm lokalen Bio-Markt hatten noch genug Zeit, mit interessierten, Bier trinkenden und schwarz gekleideten Demog\u00e4ngern zu plaudern. W\u00e4hrend sich auf der Kottbusser Br\u00fccke die Masse der Demonstrationsteilnehmer sammelte, standen wir vor dem Zug an der Ecke Maybachufer. Der Termin zum Demobeginn um 18 Uhr verstrich und es war noch immer nichts anderes als Musik vom Lautsprecherwagen zu h\u00f6ren. Doch pl\u00f6tzlich tauchten auf dem Dach des Hauses Kottbusser Damm\/ Ecke Mariannenstra\u00dfe f\u00fcnf Vermummte auf, die ein Plakat entrollten. Darauf zu sehen war ein riesiger arabischer Schriftzug: \"Yalla\" (\"Los geht's\"), mit dem Untertitel \"Klasse gegen Klasse\". Damit wurde Bezug auf die revolution\u00e4ren Vorg\u00e4nge im arabischen Raum genommen, was im Vorfeld in weiten Teilen der Presse schon berichtet und diskutiert wurde. Viele, inklusive der Polizei, vermuteten dahinter auch den Versuch, arabische Jugendliche f\u00fcr Ausschreitungen am 1. Mai gewinnen zu k\u00f6nnen. Unter Freude und Gr\u00f6len der Wartenden wurden anschlie\u00dfend Flugbl\u00e4tter vom Dach geworfen und bengalische Feuer entz\u00fcndet.\n\nKurz darauf fing der Demonstrationszug unter dem Motto \"F\u00fcr die soziale Revolution weltweit!\" mit reichlich Versp\u00e4tung endlich an, sich Richtung Hermannplatz zu bewegen \u2013 der Anfang einer ungew\u00f6hnlich und verwirrend anmutenden Demoroute, die sich konstant vom \u00fcblichen \"1. Mai Kreuzberg\" rund um die Oranienstra\u00dfe entfernte und sich zu gro\u00dfen Teilen durch Neuk\u00f6lln schl\u00e4ngelte.\n\nDas vorgegebene Tempo war rasanter als gedacht. Noch kurz zuvor diskutierten ein paar Demoteilnehmer neben mir \u00fcber die zu sehenden Fahnen: \"Wei\u00dft du, wer das ist?\" \u2013 \"Keine Ahnung, nie gesehen! Ich glaub, die \u00f6kologische Linke... ?!\". Ansonsten fielen die Transparente eher durch Abwesenheit auf.\n\nIch begab mich weit nach vorn, um die schnell marschierende Menge mit dem gro\u00dfen Frontplakat knipsen zu k\u00f6nnen. Kurz bevor sie bei mir war, l\u00f6ste sich eine kleine Gruppe von drei bis vier Vermummten, die auf einen anderen Fotografen losging und auf ihn einschlug. Kritik folgte aus dem Demozug: \"Eyyyy! Lasst das!\". Das half nicht wirklich. Ein Kameramann bekam seine Kamera durch einen Schlag direkt auf die Linse ins Gesicht gedr\u00fcckt, und ich beschloss, etwas mehr Abstand beim Fotografieren einzuhalten. W\u00e4hrenddessen bewegte sich mein Kollege, den ich bis dahin noch nicht getroffen hatte, risikofreudig weiter mit dem Fotografen-Block an der Spitze fort.\n\nSprechch\u00f6re wechselten sich ab und es ging schnurstracks den Kottbusser Damm hinunter. Der Lautsprecherwagen ging in den folgenden und nach hinten immer bunter werdenden Teilnehmermengen unter. Das Qu\u00e4ken der HipHop-Musik, die in diesem Jahr die politischen Parolen im Zug ersetzte, war an der Spitze nicht mehr zu vernehmen. Dort gab der \"Schwarze Block\" das Tempo vor und die Massen folgten. Sp\u00e4tere Sch\u00e4tzungen schwankten zwischen 9.000 bis 15.000 Teilnehmern in diesem Jahr.\n\nEs ging vorbei an unz\u00e4hligen Zuschauern, die an diesem Tag ihrem Fenster den Vorzug vor dem Fernseher gaben. Ein fr\u00f6hliches P\u00e4rchen mit rosa \"Hassmaske\" (Sturmhaube) beobachtete das Treiben genauso wie einige offensichtlich mit alpiner Ausr\u00fcstung ausgestattete Polizisten, die das Dach des Hauses Kottbusser Damm\/Ecke Urbanstra\u00dfe in Besitz genommen hatten, von dem im Vorjahr ein Feuerl\u00f6scher auf Polizisten geworfen worden war. Die Polizei hielt sich ansonsten scheinbar im Hintergrund. In den parallel verlaufenden Stra\u00dfen waren jedoch das Blaulichtmeer und die \"Truppenbewegungen\" kaum zu \u00fcbersehen. Am Stra\u00dfenrand h\u00f6rte mein Kollege noch Jugendliche angesichts der Demo diskutieren: \"Sind das die Nazis? Oder die gegen Nazis?\"\n\nAm Hermannplatz standen die ersten Polizeiwagen zum Abriegeln und Leiten der Demo bereit. Die ersten B\u00f6ller explodierten zur Freude der Teilnehmer. Der Aufzug bog rasch Richtung Sonnenallee ab. Zwischen den B\u00fcschen auf der Sonnenallee kam es immer wieder zu kleineren Tempoversch\u00e4rfungen. Die Kunden der diversen Spielcasinos, die es rund um den Hermannplatz wie Sand am Meer zu geben scheint, lugten belustigt aus den T\u00fcren.\n\nNichts schien darauf hinzudeuten, dass sich irgendjemand an der Route auf die \"Randale-Demo\" vorbereitet h\u00e4tte. Und dies, obwohl ich von Neuk\u00f6llner Bekannten geh\u00f6rt hatte, dass in den Tagen vor dem 1. Mai gezielt Casinos unter antikapitalistischer Zielsetzung angegriffen worden waren. W\u00e4hrend in Kreuzberg zum 1. Mai in der Regel alles mehr oder weniger vernagelt wird, was als Feindbild (Banken etc.) gelten oder zum kostenlosen Versorgen des leiblichen Wohles (Superm\u00e4rkte etc.) dienen k\u00f6nnte, schienen an der Neuk\u00f6llner Demoroute so gut wie gar keine Vorkehrungen getroffen worden zu sein. Obwohl weiterhin kaum Polizei zu sehen war, schien die Grundstimmung aggressiver zu sein als in den Jahren davor.\n\nDie Demo bog ab und machte sich flink durch die Fuldastra\u00dfe den Berg hinauf auf der Flughafenstra\u00dfe Richtung Boddinstra\u00dfe. Die Route f\u00fchrte an zwei ungesicherten Banken vorbei. Keine Absperrung, keine Sicherung, kein Holz. Die dumpfen Einschl\u00e4ge von Steinen, die die Schaufenster durchschlugen, wurden h\u00f6rbar, w\u00e4hrend sich der Zug konstant und rasch weiterbewegte. Auf dem kurzen St\u00fcck, das uns wieder \u00fcber die Hermannstra\u00dfe f\u00fchrte, trafen wir meinen Kollegen endlich pers\u00f6nlich.\n\nNach wenigen hundert Metern sollte die Demo gleich wieder nach links in die Werbellinstra\u00dfe abbiegen. Doch dort stand ein einsamer ziviler Kamerawagen der Polizei. W\u00e4hrend ein Polizist aus dem Verdeck des T4 Busses filmte, standen zwei andere in Kampfmontur vor dem Bulli und beobachteten, wie die Demonstration auf sie zukam.\n\nDunkle Erinnerungen erwachten in mir: Drei Jahre zuvor gab es eine \u00e4hnliche Situation, bei der die \"Revolution\u00e4re 1. Mai Demonstration\" in der Wiener Stra\u00dfe vor der Feuerwache au\u00dfer Kontrolle geriet. Dort war von polizeilicher Seite ein baugleicher Wagen in der Mitte der Stra\u00dfe platziert worden, um die Demonstration offensichtlich zu beobachten. Die Teilnehmer wussten jedoch ganz genau, dass aus den Bussen mit den get\u00f6nten Scheiben oft gefilmt und fotografiert wird, und griffen den Bus sofort an. Damals wurde ein Trupp Polizisten eingekesselt und es kam im sp\u00e4teren Verlauf zu massiven Steinw\u00fcrfen. \u00c4hnlich unbeliebt war der baugleiche Bus auch dieses Jahr. Deshalb musste er die Szenerie fluchtartig verlassen und fuhr die Werbellinstra\u00dfe hinunter. Auf dem Weg Richtung Karl-Marx-Stra\u00dfe wurde mir erst bewusst, dass die Route direkt an einer Neuk\u00f6llner Polizeiwache vorbeif\u00fchrte. Wir bewegten uns etwas schneller vorw\u00e4rts, um nicht zwischen die Fronten zu geraten, und verloren den gerade erst getroffenen Kollegen auf dem Rad dabei wieder.\n\nAus der Ferne bekamen wir ein kleineres Scharm\u00fctzel an der \u00f6rtlichen Einkaufsmeile nur am Rande mit und bewegten uns rasch Richtung Rollbergstra\u00dfe, um gute Bilder von der vor\u00fcberziehenden Demo zu bekommen. An der Polizeiwache angekommen, mussten wir zu unserem Erstaunen feststellen, dass auch hier keinerlei Ma\u00dfnahmen getroffen worden waren. Polizisten standen vereinzelt an den ge\u00f6ffneten Fenstern und sp\u00e4hten hinaus. Erst als die Demo sich n\u00e4herte, schlossen sich die Fenster pl\u00f6tzlich. Zwei unbeteiligte Schaulustige standen vor der Wache und bestaunten den Demozug. Aus diesem l\u00f6ste sich \u00fcberraschenderweise nur eine Person, die einen Stein auf das Geb\u00e4ude warf. Dieser war erschreckend schlecht geworfen und verfehlte alle Fenster des fast komplett verglasten Geb\u00e4udes, traf daf\u00fcr um Haaresbreite einen der Schaulustigen, der den Fehler gemacht hatte, sich nicht wie sein Freund rasch in die Rollbergstra\u00dfe zur\u00fcckzuziehen. Wir k\u00fcrzten durch eben diese Stra\u00dfe ab, um den Zug in der Karl-Marx-Stra\u00dfe wieder zu begleiten.\n\nWir gingen auf der Karl-Marx-Stra\u00dfe der Demo entgegen und mussten feststellen, dass diese am Ende der Werbellinstra\u00dfe gestoppt worden war. Die Polizei begann, einzelne Leute aus dem Zug zu ziehen und festzunehmen. Wie wir sp\u00e4ter erfuhren, waren es Leute, die verd\u00e4chtigt wurden, an den vorherigen K\u00e4mpfen teilgenommen zu haben. Anwohner sammelten sich auf den Balkonen und fingen die Szenerie mit ihren Kamerahandys ein. Die Festgenommenen wurden den bereitstehenden Gefangenentransportern rasch zugef\u00fchrt, aber nicht rasch genug. W\u00e4hrend ein Verd\u00e4chtiger noch verfrachtet wurde, wurde die Demo zwar von Veranstalterseite um 20:17 Uhr f\u00fcr beendet erkl\u00e4rt, setzte sich aber wieder in Bewegung. Schnell wurden die T\u00fcren zugeknallt und der Zuruf an die Fahrerin \"FAHR!\" ert\u00f6nte. Die Demo kam am ehemaligen Standplatz des Wagens nur wenige Sekunden sp\u00e4ter an. Dieser war Richtung Hermannplatz davongezischt.\n\nW\u00e4hrend versucht wurde, einen neuen Veranstalter zu finden, der die Demo weiter Richtung Hasenheide leitete, warteten wir auf der Karl-Marx-Stra\u00dfe. Ein Teil der Einsatzhundertschaft schlenderte in Kettenformation \u00fcber die leere Stra\u00dfe.\n\nIch wollte dem Aufzug wieder entgegengehen, um zu sehen, was geschehen war, und versuchte, die Kette zu kreuzen. Der Plan, normal durch die anr\u00fcckende Kette zu laufen, wurde jedoch unterbrochen. Da die Polizisten sich r\u00fcckw\u00e4rts bewegten und nur der Truppf\u00fchrer in meine Richtung sah, entschied ich mich, bei ihm zu passieren, um nicht in Konflikte zu geraten. Gerade dies erwies sich jedoch als die falsche Entscheidung. Er schubste mich vor sich her. \"Sie behindern unsere Ma\u00dfnahmen!\" Mein Hinweis, dass ich nur queren wollte, um Bilder zu machen, fruchtete nicht. Von den Seiten ert\u00f6nten Zurufe wie: \"Lass den Fotografen in Ruhe\" und einfach nur: \"EYYYYYYYY!\". Ich entschloss mich zum taktischen R\u00fcckzug in einen Hauseingang.\n\nEin Gr\u00fcppchen von drei Leuten blieb jedoch auf dem Gehweg stehen und wurde angerempelt. Zwei von ihnen steckten die Erfahrung wacker weg, w\u00e4hrend der Dritte anfing, sich mit dem Rempler zu streiten. Dieser entschloss sich aber, weiter Teil der Kette zu bleiben, und lief nach einem kurzen Wortgefecht weiter, w\u00e4hrend der streitlustige Passant sich von hinten an den Abr\u00fcckenden anpirschte und ihn zu treten versuchte. Er verfehlte sein Ziel nur knapp. Weder der Beamte noch einer seiner Kollegen bemerkten den Vorfall.\n\nUnmittelbar danach kam der Demozug wieder in Schwung und bewegte sich Richtung Hermannplatz. Der \"Schwarze Block\" wurde mittlerweile von einer rennenden .Polizeikette begleitet. Ein beeindruckendes Bild! Ab und zu explodierten B\u00f6ller.\n\nDirekt am Hermannplatz ging es jedoch schon wieder nicht mehr weiter. Polizeiketten hatten den Platz zur rechten Seite abgesperrt. Weitere versperrten den geplanten Weg \u00fcber die Hasenheide zum S\u00fcdstern und auch der Weg nach links zur\u00fcck auf die Hermannstra\u00dfe war gesperrt. Die Demo schien eingekesselt. Karstadt sah aus wie eine bewachte Festung, gesichert von Polizisten, deren Fahrzeugen und Absperrgittern. Verunsicherung machte sich breit. Niemand wurde durchgelassen, egal ob er nach Hause wollte oder nur einmal austreten. Aggressionen folgten unmittelbar. Die ersten Scharm\u00fctzel nahmen ihren Lauf. Immer wieder flogen kleinere Hagel von Wurfgeschossen auf die nun omnipr\u00e4senten Polizisten. B\u00f6ller knallten.\n\nUm einen \u00dcberblick zu gewinnen, stellte ich mich auf die Fahrbahn direkt an der Ecke Karl-Marx-Stra\u00dfe\/Ecke Hermannstra\u00dfe. Eine Frau blieb mit ihrem Rad direkt vor mir stehen und wies mich auf meinen schlecht gew\u00e4hlten Standpunkt hin: \"Geh lieber weg!\" Sie zeigte auf die Filiale einer Bank direkt in meinem R\u00fccken. Dutzende Demonstranten verhakten ihre Arme und standen den Polizeitrupps wie ein Bollwerk gegen\u00fcber. Ich entschloss mich dazu, meinen Helm aufzusetzen. Das 1.-Mai-typische Hin und Her nahm zum ersten Mal seinen Lauf. Erstaunlicherweise flog nichts Richtung Bank.\n\nAuf der Karl-Marx-Stra\u00dfe wurden die Teilnehmer der nun endg\u00fcltig beendeten Demonstration tr\u00f6pfchenweise rausgelassen. Auf dem Herrmannplatz rauchte und knallte es hingegen weiter. Wasserwerfer standen schon bereit. Wer es aus den Polizeiabsperrungen heraus schaffte, zog, mangels Abschlusskundgebung, parallel in verschiedenen Seitenstra\u00dfen Richtung Kottbusser Tor ab.\n\nNach einiger Zeit und Beruhigung der Lage wurden auch bei uns in der Spitze die .Polizeiketten gel\u00f6st. Jedoch nur Richtung S\u00fcdstern. Der Weg \u00fcber den Hermannplatz blieb weiter verwehrt. Vor der Karstadt-Filiale sammelte sich eine kleine Gruppe vermeintlicher Autonomer im jugendlichen Alter, die b\u00f6se Blicke mit vorbeiziehenden Einsatzhundertschaft-Polizisten austauschten. Letztere achteten peinlich genau darauf, der Gruppe nicht den R\u00fccken zuzudrehen. Die Jugendlichen versuchten offensichtlich noch, sich per Handy umzuorganisieren: \"Wir stehen vorm Karstadt. Wo seid ihr? Ist bei euch eine Bank oder ein Netto in der N\u00e4he?\"\n\nTrotz offizieller Beendigung der Demo wurden die Abr\u00fcckenden weiter von der Polizei geleitet. Der Weg f\u00fchrte \u00fcber Jahnstra\u00dfe Richtung Graefe-Kiez. An einer Baustelle kam es erneut zum Tumult. Jemand hatte sich durch die offenstehende Absperrung einer Baustelle gezw\u00e4ngt und sich auf ein Dixi-Klo zur\u00fcckgezogen. Begleitend mitlaufende Polizisten vermuteten offenbar eine bevorstehende Bedrohung und enterten die Baustelle. Taschenlampen durchschnitten die dunkle Seitenstra\u00dfe und Gebr\u00fcll und Geschubse setzte ein, bis der Verd\u00e4chtige mit offener Hose wieder aus dem Klo gezogen werden konnte.\n\nW\u00e4hrenddessen begab sich mein Kollege an den urspr\u00fcnglich vorgesehenen Zielort, den S\u00fcdstern. Dort herrschte g\u00e4hnende Leere: keine Polizei, keine Demonstranten und sonst auch nicht viel. Hier wusste man von der Demo und die Anwohner hatten schon im Vorfeld ihre Autos woanders geparkt. Trotz beendeter Demo schienen die Stra\u00dfen immer noch weitl\u00e4ufig gesperrt zu sein. Ein Polizei-Hubschrauber kreiste weiterhin \u00fcber dem Gebiet.\n\nDie Masse der Demoteilnehmer setzte hingegen ihren Weg mit uns \u00fcber die Sch\u00f6nleinstra\u00dfe und den Kottbusser Damm fort. Neues Ziel wurde das MyFest. Dies bef\u00fcrchteten anscheinend auch die Polizeikr\u00e4fte, die die Kottbusser Br\u00fccke kurzerhand sperrten. W\u00e4hrend Ausweichbewegungen \u00fcber das Planufer und das Maybachufer in Richtung der n\u00e4chsten Br\u00fccken einsetzten, entschieden wir uns abzuwarten.\n\nNach etwa einer halben Stunde, die Masse ehemaliger Demonstranten war nach links und rechts abgeflossen, fing die Polizei an, Leute Richtung Kottbusser Tor durchzuschleusen. Zwischen zwei Polizeitransportern war eine L\u00fccke, durch die GENAU ein Mensch passte. Durch diese fingen sich die Wartenden an zu zw\u00e4ngen. Konzertatmosph\u00e4re kam auf. Alle standen dicht gedr\u00e4ngt und gut gelaunt wie vor einem Club und harrten der Dinge, die sie erwarten w\u00fcrden. Keiner wusste, nach welchen Vorgaben die polizeilichen T\u00fcrsteher die Leute durchlassen w\u00fcrden. Endlich durchgedr\u00e4ngelt, wurde jeder von begutachtenden Beamte erwartet, die \"Verd\u00e4chtige\" und mitgef\u00fchrte Taschen nach ihrem eigenen System aussonderten und zus\u00e4tzlich durchsuchten. Nach dieser Prozedur war der Weg endlich wieder frei.\n\nDas Wetter verschlechterte sich. Es wurde windig und immer k\u00e4lter. Am Kottbusser Tor angekommen, sammelten sich vermehrt Gr\u00fcppchen, die es auch \u00fcber die Hobrecht- und die Admiralbr\u00fccke zur\u00fcck nach Kreuzberg geschafft hatten. Die \u00fcblichen 1.-Mai-Gefechte setzten wieder ein, wobei es in diesem Jahr auf Seiten der Polizei mehrere Kollateralsch\u00e4den gab. Vereinzelte Flaschenw\u00fcrfe wurden mit sofortigen Festnahmen geahndet, auf die Sprechch\u00f6re folgten und wieder W\u00fcrfe. Der gro\u00dfe Knall blieb, mangels motivierter Randalierer, aus.\n\nDas Ritual kam nur schwer gegen die vorherrschende Feierstimmung an, die von Techno-Musik gesch\u00fcrt wurde. Autonome Gruppen waren nirgends zu identifizieren. Gelegentliche .B\u00f6llerexplosionen und Gejohle, gepaart mit \"Haut ab, haut ab!\"-Sprechch\u00f6ren, stellten die Highlights des Geschehens dar. Mit Politik oder der autonomen Szene hatte das nichts zu tun. Lediglich ein paar Betrunkene suchten so noch den Nervenkitzel. Ein Jugendlicher beschimpfte ein paar Polizisten ohne Konsequenzen aufs \u00dcbelste und wurde daraufhin von seinen Kumpels ausgiebig gefeiert. Die Polizei drehte ansonsten in kleinen Gruppen ihre Runden, durch die Menge und griff bei Zwischenf\u00e4llen Einzelpersonen mit Pfefferspray an. Das sorgte f\u00fcr steten Nachschub bei den wartenden Sanit\u00e4tern. Ansonsten galt auch f\u00fcr alle anderen Beteiligten: Man wartete.\n\nDie Situation blieb ein wenig angespannt, aber weitgehend harmlos, und transformierte sich zusehends in Langeweile. Abseits vom Kotti hatte die Polizei ohnehin nichts mehr zu tun. Einzelne Polizisten hingen bei McDonalds rum oder sa\u00dfen am Stra\u00dfenrand. Um ca. 1 Uhr wurde der Bereich um das Kottbusser Tor endg\u00fcltig ger\u00e4umt. Wir hatten uns jedoch schon zwei Stunden vorher gegen diesen, bis in die fr\u00fche Nacht andauernden Kreislauf entschlossen.\n\nDer Motorroller stand noch immer an seinem Platz, auch er hatte einen extrem ruhig verlaufenen 1. Mai in Kreuzberg heil \u00fcberstanden und transportierte uns zuverl\u00e4ssig wieder zur\u00fcck.\n\n## \"Die Szene definiert sich ganz klar dar\u00fcber, dass sie sich als was Besseres f\u00fchlen als andere Leute.\"\n\nLeonie, 34 Jahre, lebt in Berlin.\n\nIch treffe Leonie im Archiv der Jugendkulturen. Es war ihr lieber, sich an einem neutralen Ort zu treffen. Da sie wenig Zeit hat, treffen wir uns an einem Montagabend nach der Arbeit. Leonie ist eine sehr h\u00fcbsche junge Frau, sie ist sehr klein, hat kurze, schwarz gef\u00e4rbte Haare und tr\u00e4gt einen Nasenring sowie mehrere Ohrringe. Sie wirkt recht nat\u00fcrlich, ist ungeschminkt. Sie tr\u00e4gt eine blaue Jeans, einen schwarzen Kapuzenpullover, dar\u00fcber eine schwarze Weste und Turnschuhe. Sie ist sehr freundlich und eher zur\u00fcckhaltend, bl\u00fcht aber im Laufe des Gespr\u00e4ches zunehmend auf. Ich biete ihr Tee an, aber sie trinkt ausschlie\u00dflich Leitungswasser aus einer kleinen Plastikflasche. Sie wirkt recht humorvoll und lacht \u00f6fter, spricht aber meistens sehr ernst und bestimmt. Sie betonte vorab mehrfach, dass sie Probleme mit der Bezeichnung \"autonom\" hat, weil sie Schubladendenken und Kategorien f\u00fcr Menschen ablehnt. Sie m\u00f6chte daher nicht unbedingt als \"Autonome\" bezeichnet werden.\n\n#### Kommst du aus Berlin, urspr\u00fcnglich...?\n\nNee, ich bin ganz viel umgezogen in meinem Leben.\n\n#### Und wo bist du geboren?\n\nMuss ich das alles so genau sagen?\n\n#### Nein, musst du nicht. Was ist dein Familienstand? Lebst du in einer festen Partnerschaft?\n\nJa.\n\n#### Und was machst du derzeit?\n\nIch habe gerade mein Studium abgeschlossen, Sozialarbeit, und fange nun an zu arbeiten in dem Bereich.\n\n#### Wie bewertest du deine aktuelle Situation, bist du zufrieden?\n\nAlso f\u00fcr den Moment bin ich zufrieden, aber in der Zukunft w\u00fcrde ich schon lieber mit Fl\u00fcchtlingen arbeiten oder mehr in 'nem politischen Rahmen arbeiten. Also mir gef\u00e4llt die Arbeit schon ganz gut und das liegt quasi auf dem Weg, wo ich hin m\u00f6chte, aber in der Zukunft w\u00fcrde ich das lieber verbinden.\n\n#### Also du hast schon konkrete Vorstellungen und Pl\u00e4ne in Richtung Zukunft?\n\nIch stelle mir schon meine Zukunft so vor, dass ich gerne mit Leuten zusammenleben w\u00fcrde in 'nem Projekt, dass man was politisch zusammen macht, sich nicht abkapselt oder so, vielleicht die Nachbarschaft einbezieht... Also ich habe noch keine konkreten Vorstellungen, aber definitiv etwas mit anderen Leuten, zusammenleben, und ich hab schon die Vorstellung, dass meine berufliche Arbeit und was ich in meinem Leben privat mache, zusammenkommt, nicht so weit auseinander liegt.\n\n#### Jetzt lebst du nicht mit anderen zusammen?\n\nDoch, ich wohne in 'ner kleineren WG mit vier Leuten, aber ich stell mir schon vor, in der Zukunft mit mehr Leuten zusammenzuwohnen, also vielleicht so 8\u201310 Leute, in einem richtig gr\u00f6\u00dferen Projekt und eventuell auch auf dem Land, aber es k\u00f6nnte auch in der N\u00e4he von der Stadt sein, so ganz klar ist das noch nicht.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\n[Lacht.] Oh Gott, das ist die Frage, die ich in Einstellungstests immer am schlimmsten finde... Da f\u00e4llt mir echt gar nicht viel ein. Ich bin relativ zielstrebig, w\u00fcrde ich mal so sagen. Ich bin jemand, der ziemlich alles immer relativ weit voraus plant, und ich hab eigentlich immer genaue Vorstellungen, was ich gut finde und was ich machen will in meinem Leben, also ich hab selten Schwierigkeiten, dass ich nicht so richtig wei\u00df oder so, sondern ich bin eher so, dass ich so eine Vorstellung hab, was ich die n\u00e4chsten f\u00fcnf Jahre grob machen will, und das dann auch durchziehe.\n\n#### Cool. Was meinst du, denken andere von dir? Die Frage geh\u00f6rt dazu...\n\n[Lacht.] Das ist ja noch viel schlimmer. Ich wei\u00df nicht, was andere Leute von mir denken... Ich glaub, wenn ich jetzt mal einfach von meiner WG ausgehe, dass Leute mich so wahrnehmen, dass ich gerne Zeit mit Leuten verbringe, aber auch Zeit f\u00fcr mich selber brauche, also dass ich mich manchmal zur\u00fcckziehe, mich ein bisschen abgrenze und mein Ding so allein mache manchmal.\n\n#### Wovor hast du Angst?\n\n[Lacht.] Jetzt ehrlich? Hm... Am meisten habe ich immer echt Angst, dass Sachen nicht so klappen, wie ich mir die vorstelle. Ich hab manchmal schon ziemlich genaue Vorstellungen und werde immer so 'n bisschen fuchsig, wenn die dann nicht so richtig hinhauen, dass einem dann irgendwas in die Quere kommt. Aber das ist eher was, was mich nervt, wenn Sachen nicht so hinhauen oder wenn ich mich nicht so richtig vorbereiten kann, also wenn ich mich irgendwie unvorbereitet f\u00fchle in Situationen...\n\n#### Bei was zum Beispiel?\n\nAlles von Arbeit \u00fcber so 'ne Interviewsituation bis besonders nat\u00fcrlich Situationen, wo man sich irgendwie produzieren muss, wo man bewertet wird.\n\n#### Was \u00e4rgert dich am meisten? Was macht dich w\u00fctend?\n\nEine Sache, die mich auf jeden Fall immer richtig w\u00fctend macht, ist, wenn man das Gef\u00fchl hat, dass Leute sich in einer bestimmten Weise darstellen und das so gar nicht der Realit\u00e4t entspricht, so was finde ich immer richtig \u00e4rgerlich; und besonders \u00e4rgerlich finde ich es, wenn andere Leute das dann nicht wahrnehmen.\n\n#### Kannst du was \u00fcber deine Eltern erz\u00e4hlen?\n\nMeine Eltern haben beide Biologie studiert und meine Mutter ist dann Hausfrau geworden, mein Vater arbeitete, ist aber gerade in Rente gegangen. Ich hab ein relativ gutes Verh\u00e4ltnis zu meinen Eltern; wir sind jetzt nicht so 'ne ganz enge Family, ich h\u00e4ng nicht jeden Tag mit denen rum oder so, aber wenn man Zeit miteinander verbringt, ist es gut und dann verstehen wir uns auch gut. Meine Eltern sind sehr offen und die haben auf keinen Fall solche Erwartungen, dass man f\u00fcr sie da sein m\u00fcsste oder so und so oft da aufkreuzen m\u00fcsste. Ich hab das Gef\u00fchl, ich hab ziemlich viel Freiheit, wenn ich das im Vergleich zu anderen manchmal sehe.\n\n#### Hast du Geschwister?\n\nJa, ich habe einen j\u00fcngeren Bruder.\n\n#### Gibt es irgendwas, was du anders machen w\u00fcrdest als deine Eltern?\n\nWenn ich Kinder h\u00e4tte? Ich glaub, ich w\u00fcrde relativ \u00e4hnlich rangehen, erstaunlich, ne? So in der Beziehung zu Kindern bin ich schon auch ein Fan davon, Freiheiten zu lassen und zu vertrauen, dass sie die richtige Entscheidung machen, und die auch als eigenst\u00e4ndige Person wahrzunehmen und nicht mich \u00fcber die zu definieren zum Beispiel. Aber auch da zu sein, wenn Hilfe gebraucht wird, das ist auch ganz wichtig, also da habe ich schon das Gef\u00fchl bei meinen Eltern, dass sie halt da sind, wenn ich Hilfe brauche. Aber man muss auch sagen, das ist eine Entwicklung bei meinen Eltern, das war nicht immer. Meine Mutter hatte sehr unterschiedliche Erziehungsmethoden durch meine Kindheit und Jugend; die hatte immer so Phasen, wo sie so ein Buch gelesen hat oder sich auf irgendeinen bestimmten Erziehungsstil eingefahren hat, und das hat dann wieder komplett gewechselt. Und das ist total verwirrend als Kind und als Jugendlicher sp\u00e4ter auch, und das fand ich nicht besonders hilfreich, das w\u00fcrde ich auf keinen Fall machen.\n\n#### M\u00f6chtest du denn selber Kinder haben?\n\nBis jetzt bin ich mir da noch nicht so richtig sicher. Ich hab nie gesagt, ich muss unbedingt Kinder haben sp\u00e4ter, und ich konnte es mir fr\u00fcher auch gar nicht vorstellen. Das ist im Augenblick f\u00fcr mich nicht Thema, das ist nichts, wor\u00fcber ich gerade nachdenke, aber ich kann nicht ausschlie\u00dfen, dass ich in f\u00fcnf Jahren pl\u00f6tzlich dann doch Bock drauf habe. Aber es ist nicht etwas, was ich fest geplant h\u00e4tte.\n\n#### Haben deine Eltern oder andere Verwandte eine Rolle f\u00fcr deine politische Entwicklung gespielt?\n\nNee. So gar nicht, weil die einfach alle komplett unpolitisch sind.\n\n#### Vielleicht gerade deswegen.\n\nJa, in dem Sinne vielleicht, dass sie einfach so unpolitisch sind. Aber doch, ein bisschen hat wahrscheinlich der neue Partner von meiner Mutter eine Rolle gespielt, weil der einfach so 'n Arschloch war und ich dadurch ganz krass doch irgendwie politisiert wurde, sozusagen in der Auseinandersetzung mit diesem Typ.\n\n#### Inwiefern?\n\nIch w\u00fcrde mal sagen, dass meine Politisierung unter anderem auch dar\u00fcber lief, dass der sich halt als so 'n ganz krasser Macker entpuppt hat und ich schon, als ich ziemlich jung war, das ganz klar als Sexismus enttarnt hab und dachte, das geht ja gar nicht so. Das war also, bevor ich irgendwas dr\u00fcber gelesen habe. Ich wei\u00df nicht, wo das herkam, aber dann hab ich mich ein bisschen mehr damit besch\u00e4ftigt, und das war ausschlaggebend auch f\u00fcr andere Sachen dann.\n\n#### Hast du Hobbies?\n\nDerzeit nicht. [Lacht.] Also im Moment ganz wenig, durch die Arbeit. Vorher habe ich immer wahnsinnig viele Hobbies gehabt und dies und das und jenes, aber gerade mache ich fast nichts au\u00dfer arbeiten. Ich versuche, manchmal ein bisschen Sport zu machen, aber da passiert gerade auch nicht mehr so richtig viel.\n\n#### Und Freizeitgestaltung im Allgemeinen, was machst du so?\n\nManchmal habe ich auch bisschen Freizeit, ich gehe schon mal aus nat\u00fcrlich, so mit Freunden, und gehe schon hier und da noch auf Punk-Konzerte zum Beispiel oder auch auf Polit-Veranstaltungen, wenn ich Zeit finde oder Leute hingehen.\n\n#### Was zum Beispiel?\n\nMich interessiert schon besonders die Fl\u00fcchtlingsproblematik, wenn da Soli-Partys sind oder irgendwelche Infoveranstaltungen, gehe ich hin, aber auch zu so Queer-Bewegungen, das interessiert mich, auch was da so neuere Entwicklungen in der Politik-Szene sind. Aber ich bin nicht irgendwie darin involviert, so was selbst zu organisieren oder so was.\n\n#### Das war fr\u00fcher anders?\n\nJa, fr\u00fcher h\u00e4tte ich da mitorganisiert oder mitgeholfen... und wenn es die Cocktailbar gewesen w\u00e4re... nicht unbedingt Vortr\u00e4ge halten, aber so was halt.\n\n#### In was f\u00fcr Gruppen warst du fr\u00fcher?\n\nAls ich j\u00fcnger war, war das schon die Antifa-Szene, und dann hat sich das immer weiter bewegt zu so politischen Caf\u00e9s, also Vok\u00fc-[\"Volxk\u00fcche\"-]Caf\u00e9-Arbeit, aber nicht so sehr mit Schwerpunkt auf ein spezielles Thema, sondern eher so breit, wo man halt, was auch immer f\u00fcr Themen gerade aktuell sind, diskutiert oder dazu Infoveranstaltungen macht... Sp\u00e4ter bin ich dann in diese ganze Anti-Globalisierungs-Bewegung so rein, das war auf jeden Fall eine Zeit lang ein wichtiger Themenbereich f\u00fcr mich, und dann ging das halt los mit Antira-[Antirassismus-]Arbeit und in Fl\u00fcchtlingsgruppen arbeiten.\n\n#### Was habt ihr da gemacht in den Fl\u00fcchtlingsgruppen?\n\nIch hab in zwei verschiedenen Fl\u00fcchtlingsgruppen gearbeitet: Eine war eher, Leuten direkt zu helfen, die eine Aufenthaltsgenehmigung brauchten und in diesem ganzen Papierkram steckten, und einfach auch alles vermitteln, was an Unterst\u00fctzung gebraucht wurde, Termine bei \u00c4rzten oder was auch immer. Und als Letztes hab ich in einer Gruppe gearbeitet, die selbst von Fl\u00fcchtlingen organisiert ist, die selber Gelder kriegen \u00fcber den Staat und dann kucken m\u00fcssen, was f\u00fcr Projekte sie aufziehen wollen, und das ist dann alles von Schwimmunterricht \u00fcber mit den Kids Fu\u00dfballspielen gehen bis zu Sprachunterricht organisieren.\n\n#### Bist du in einem Verein aktiv?\n\nNein.\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nDas variiert. Ich hab tats\u00e4chlich Phasen, wo das nicht so gro\u00dfes Thema ist in meinem Leben, und dann habe ich so Phasen, wo das wieder gr\u00f6\u00dfere Bedeutung hat. Also ich gehe zum Beispiel total gerne tanzen, aber es gibt Phasen, da mach ich das einfach nicht. Es gibt Phasen, wo ich zu den Konzerten gehe und mich mit den Bands besch\u00e4ftige, und dann gibt's Phasen, wo das kein Thema ist.\n\n#### Punk oder auch was anderes?\n\nIn erster Linie schon Punk-Konzerte [lacht], ich h\u00f6r fast nix anderes, wenn ich ausgehe. Aber ich h\u00f6r Punk-Musik tats\u00e4chlich nur live, und zum Tanzen gehen finde ich Punk-Musik auch nicht ganz so toll, da w\u00fcrde ich mich f\u00fcr was anderes entscheiden, und zu Hause h\u00f6re ich sehr gemischte Musik, so alles durcheinander.\n\n#### Warum gerade Punk?\n\nPunk ist bei mir mit Jugend verbunden, das war die erste Subkultur, mit der ich mich identifiziert hab, als ich ganz jung war. Da war ich halt auch in 'ner Gegend, wo ich komplett von Punks umgeben war; alle meine Freunde waren Punks, und ich hab so das Gef\u00fchl, das sind meine Wurzeln und das ist meine Jugend gewesen.\n\n#### Hast du sp\u00e4ter noch einer anderen Jugendkultur angeh\u00f6rt als Punk?\n\nIch bin irgendwann umgezogen in eine Gegend, wo es nicht mehr viele Punks gab und ich ein bisschen vereinzelt war, und das war dann, wo ich so politisiert wurde und dann angefangen habe mit Antifa-Arbeit, und das w\u00fcrde ich auch als eine Art Subkultur, Jugendkultur fast bezeichnen.\n\n#### Du w\u00fcrdest die Antifa-Szene auch als Jugendkultur sehen?\n\nJa, zumindest da, wo ich war, definitiv war das 'ne Jugendkultur. Das war auch ganz interessant, weil das eine bestimmte Altersgruppe war an der Schule, die das haupts\u00e4chlich getragen hat, und das ist dann h\u00e4ufig auch damit gestorben, dass die Leute wieder weggezogen sind oder \u00e4lter geworden sind.\n\n#### Welche jugendkulturellen Elemente hat sie denn, einfach das Alter der Leute oder noch irgendwas anderes?\n\nDa gibt's eine ganze Menge, von bestimmten Dresscodes, so ein ganzer Habitus, das ganze Auftreten... Also du erkennst die Leute, die Antifa machen, rein \u00e4u\u00dferlich an ihren Klamotten, an der Art, wie sie reden...\n\n#### Zum Beispiel?\n\nBei uns in der Schule haben die Leute, die sich politisch engagiert haben, schon im Politikunterricht ganz klar Positionen bezogen, und nicht unbedingt Positionen, die konform waren mit dem, was andere Leute gedacht haben. Das war schon so 'ne Gruppierung, und das war auch anderen Leuten klar, dass das eine Gruppierung ist, die sich auch abgrenzen will.\n\n#### Wie ist denn das Verh\u00e4ltnis Antifa zu Autonomen?\n\nDa gibt's einen ganz flie\u00dfenden \u00dcbergang. Autonome haben \u00dcberschneidungen mit verschiedenen Str\u00f6mungen, aber auf jeden Fall ganz klar mit Antifa; also ich w\u00fcrde sagen, dass alle Leute, die sich der autonomen Szene zugeh\u00f6rig f\u00fchlen, Antifa-Arbeit unterst\u00fctzenswert finden, aber nicht unbedingt alle machen Antifa-Arbeit [\u00fcberlegt], aber auf jeden Fall ein guter Teil. Aber umgekehrt bezeichnen sich nicht alle Antifas als Autonome. Mit dieser ganzen Israel-Pal\u00e4stina-Debatte ist das nat\u00fcrlich ein St\u00fcck auseinandergebrochen, ganz klar, dass sich das teilweise sogar gegen\u00fcbersteht heute. Die Schnittmenge war in den 90er Jahren gr\u00f6\u00dfer, da war das sehr viel gemischter.\n\n#### Aber h\u00e4ttest du dich damals noch als Autonome bezeichnet, wenn dich jemand gefragt h\u00e4tte?\n\nNee, das w\u00e4r kein Label gewesen, was ich mir gegeben h\u00e4tte. Ich h\u00e4tte mich ganz klar als Antifa bezeichnet damals, aber je \u00e4lter ich geworden bin, desto mehr bin ich davon weggekommen, mich \u00fcberhaupt hinter einen Begriff zu stellen. Feminismus ist da eine Ausnahme, ich bin ganz klar Feministin, aber sonst bin ich immer mehr davon weggekommen, irgendwelche Labels zu benutzen f\u00fcr mich selber.\n\n#### Wie w\u00fcrdest du \"autonom\", \"autonome Szene\" eigentlich definieren oder jemandem beschreiben, der noch nie etwas davon geh\u00f6rt hat?\n\nIch w\u00fcrde in den 80er Jahren ansetzen und sagen, das fing damals an, hing mit der Hausbesetzer-Szene zusammen und hatte einen anarchistischen Hintergrund... Dieser Begriff \"autonom\" kommt von unabh\u00e4ngig sein und damit ist ganz klar die Abgrenzung zu so Herrschaftsverh\u00e4ltnissen gemeint und dieses ohne Hierarchien leben zu wollen. Es ist ja 'ne kleine Welt f\u00fcr sich, die ganze Besetzer-Szene, die Kneipen und so, zum Beispiel in der Rigaer Stra\u00dfe kann man im Prinzip nur in den Strukturen leben, ohne wirklich nach au\u00dferhalb zu gehen... Und Leute entwickeln dann eben auch so Ideen von \u00fcberhaupt keine Ber\u00fchrungspunkte mehr mit der Mainstream-Gesellschaft, also: Ich geh containern, ich geh nicht arbeiten, ich lebe in meinem besetzten Haus und ich geh abends in die besetzte Kneipe und ich beweg mich auch nur mit Leuten, die genau das Gleiche denken, und damit ist man dann frei von all diesen Fehlern oder all den Sachen, die man kritisiert, nicht haben will.\n\n#### Eine interessante Frage w\u00e4re, ob es eigentlich darum geht, in dieser autonomen Szene, gesamtgesellschaftliche Ver\u00e4nderungen zu erzielen oder nur die eigenen Partikularinteressen zu verteidigen, die eigenen Freir\u00e4ume zu bewahren?\n\nIn anderen L\u00e4ndern, in denen ich war, sind die autonomen Szenen sehr viel mehr daran interessiert, die Gesellschaft als Ganzes zu ver\u00e4ndern, sich nicht abzugrenzen und abzuschotten, sondern offen zu sein f\u00fcr Leute, und es wird auch ganz viel Wert darauf gelegt, dass man so r\u00fcberkommt, dass man offen ist, dass Leute sich auf einen zutrauen, dass man seine Werte quasi auch verbreiten kann. Das ist auch meine Hauptkritik an der autonomen Szene in Deutschland. Was ich am allerproblematischsten hier finde, ist, dass diese Szene ganz klar nur darauf konzentriert ist, ihre eigene kleine Szene zu haben und diese bessere Welt f\u00fcr sich zu haben, was aber dar\u00fcber l\u00e4uft, dass sich abgegrenzt wird von dem schlechten Anderen und das schlechte Andere darf da auch nicht reinkommen, und deswegen sind alle anderen Leute auch gleich richtig schei\u00dfe und denen werden dann auch die Scheiben eingeworfen, wenn da das Caf\u00e9 an der Ecke aufmacht, weil das dann ein Yuppie-Caf\u00e9 ist. Damit \u00fcberzeugt man nicht andere Leute von seinem Standpunkt, aber ich denke, dass man andere Leute immer \u00fcberzeugen muss, und wenn man was leben will, was besser sein sollte, sollte das doch zumindest offen f\u00fcr alle Leute sein.\n\nViele von diesen Parolen, die da geklopft werden, oder andere Sachen, werden dazu benutzt, andere Leute auszugrenzen, die noch nicht ganz kapieren, wie das funktioniert. Also die Begr\u00fcndung ist nat\u00fcrlich, dass man keinen Sexismus oder bestimmte andere Sachen nicht haben will in der Szene und deshalb Leute rausschmei\u00dft, die grenz\u00fcberschreitend sind, was ja zum gewissen Ma\u00df okay ist, aber nat\u00fcrlich gibt es Hierarchien, manche Leute genie\u00dfen bestimmte Privilegien und k\u00f6nnen sehr wohl Sachen machen, w\u00e4hrend quasi Fremde, die reinkommen in die Szene und sich mit diesen ganzen Verhaltensweisen noch nicht auskennen, knallhart ausgegrenzt werden. Die Szene definiert sich ganz klar dar\u00fcber, dass sie was Besseres sind als andere Leute, dass sie sich als was Besseres f\u00fchlen dadurch, dass sie sich so klar abgrenzen und dadurch einen bestimmten Status f\u00fcr sich innehaben, der \u00fcber ein innen\/ au\u00dfen, gut\/schlecht funktioniert und man sich so selbst dar\u00fcber erh\u00f6ht und sein Selbstbewusstsein aufpoliert. Das klingt jetzt ganz b\u00f6se, aber ich find das schon ungew\u00f6hnlich in Deutschland, wie das hier in der autonomen Szene abl\u00e4uft zum Teil und wie die Szene auch so auftritt in der Art, dass Leute in der Nachbarschaft nicht unbedingt das Gef\u00fchl haben, mit denen will ich mich jetzt befreunden oder das sind ganz nette Jungs und M\u00e4dels.\n\nIn anderen L\u00e4ndern ist der Schwerpunkt viel mehr darauf gerichtet, wie kommen wir r\u00fcber, wie k\u00f6nnen wir das vermitteln an Leute, dass die das verstehen, also viel mehr von der anderen Perspektive gedacht wird und nicht von der eigenen Perspektive, sondern gekuckt wird, wenn du jetzt die Oma von nebenan bist, wie soll die denn diese Parole verstehen und wie bereitet man die Sachen auf, dass da eine Kommunikation stattfindet. Deshalb frage ich mich auch manchmal bei der autonomen Szene, ist denn die Idee \u00fcberhaupt, die Gesellschaft zu ver\u00e4ndern, oder ist die Idee einfach nur, seinen eigenen Freiraum zu haben, aber auch nur f\u00fcr bestimmte Leute, die in erster Linie wei\u00df sind oder aus der Mittelklasse kommen, also das ist eine ganz klare demografische Tatsache. Diese ganze autonome Szene ist schon sehr privilegiert und ich finde, da fehlt manchmal so ein bisschen die Selbstkritik.\n\n#### Lassen sich die sozialen und biografischen Milieus beschreiben, aus denen sich die Angeh\u00f6rigen der autonomen Szene mehrheitlich rekrutieren?\n\nDefinitiv: Die meisten kommen aus der Mittelklasse und haben studiert oder studieren noch, kommen jedenfalls ganz klar aus privilegierten Verh\u00e4ltnissen, b\u00fcrgerlich und akademisch. Und altersm\u00e4\u00dfig ist es so ein Ding zwischen Ende Teenager-Alter bis drei\u00dfig, und dann h\u00f6rt das bei den meisten schon irgendwie wieder auf. Klar, es gibt vereinzelt Leute, die das l\u00e4nger machen, aber bei den meisten ist das schon auf 'nen gewissen Zeitraum begrenzt. Die Szene wechselt sich also alle paar Jahre wieder aus, Neue kommen rein und Alte gehen.\n\n#### Ist damit die autonome Szene auch eine Jugendkultur?\n\nSo \u00e4hnlich wie Antifa finde ich schon auch; es gibt da eine bestimmte Art, Sachen zu diskutieren, eine bestimmte Art zu reden, so diese ganzen Sachen.\n\n#### Gibt es denn einen eigenst\u00e4ndigen autonomen Stil, zum Beispiel Kleidung, Codes etc.?\n\nEs gibt schon so ein paar Merkmale, das kommt schon von den Anf\u00e4ngen in den 80er Jahren, so dieses schwarze Klamotten tragen, den schwarzen Kapuzi, meistens so BW-Hosen, dieser ganze Dresscode, aber auch so ein Habitus, also dass so ein bestimmtes, mackeriges Auftreten da schon h\u00e4ufig mit einhergeht. Es gibt auch Str\u00f6mungen in der Szene, die da gegenhalten, aber wenn man das so im Ganzen sieht, w\u00fcrde ich schon sagen, dass das noch stark dominiert. Es gibt Frauen in der Szene, okay, aber ich w\u00fcrde mal behaupten, dass der Anteil kleiner ist, so 60 : 40. Also im Vergleich zu anderen Jugendkulturen ist die autonome Szene nicht so krass m\u00e4nnlich dominiert. Die Punk-Szene zum Beispiel finde ich noch ein bisschen krasser und in der autonomen Szene gibt es zumindest inhaltlich eine Kritik auch, das ist den Leuten zumindest bewusst, dass es ein Problem ist, aber trotzdem bleibt das Problem bestehen. Und ich glaube auch, dass sich Frauen dem so anpassen im Verhalten, Auftreten generell, auf Demos, die Art, wie Aktionen geplant werden, was erwartet wird. Und umso mehr das in Aktionismus umschl\u00e4gt, wird das dann auch immer krasser so mackerig, immer st\u00e4rker sind es dann Typen, die Reden schwingen und so.\n\n#### Welche Rolle spielen denn Frauen in der autonomen Szene?\n\nIch denke schon, dass die Rolle von Frauen in der autonomen Szene ganz klar ist, diesen Antisexismus zu vertreten. Das sind ja meistens Frauen, die dann sagen, so geht's aber nicht, und das wird auch ein bisschen erwartet, dass Frauen diese Meinung haben und diese Sachen durchk\u00e4mpfen und diese Konfrontation von Frauen initiiert stattfindet. Das ist selten so, dass M\u00e4nner das einbringen. Die M\u00e4nner in der Szene haben viele Themen und Rollen, die Frauen haben vor allem eine Hauptrolle: Antisexismus.\n\n#### Wie war denn dein erster Kontakt zur autonomen Szene?\n\nDas lief \u00fcber die Antifa-Szene. Als ich mit Antifa-Arbeit angefangen hab, war ich noch relativ naiv und ziemlich jung, bin da so reingerutscht, weil ich nat\u00fcrlich Nazis schei\u00dfe fand und es auch Probleme mit Nazis im Alltag gab. Und dann \u00fcber die Zeit er\u00f6ffnet sich einem ja viel mehr, was alles linke Szene noch ist, dass es da ganz viele Facetten gibt. Man geht dann halt auf Partys in besetzten H\u00e4usern, da gibt es noch andere Leute, die machen nicht alle Antifa, das mischt sich dann immer mehr. Auch diese Begrifflichkeit \"Autonome\" ist da schon benutzt worden, man geht auf Demos und dann steht sp\u00e4ter immer in der Zeitung: die Autonomen. Man wird so bezeichnet, damit f\u00e4ngt es an. Also ich kann mich nicht entsinnen, dass wir uns selbst in der Zeit als Autonome bezeichnet h\u00e4tten explizit, aber das fing so langsam an, in einen Topf reingepackt zu werden. Und dann ging man zu Quetschenpaua-Konzerten, und er redet ganz klar von Autonomen: \"Kommt zu den Autonomen!\"\n\n#### Habt ihr euch damals auch mit Theorien besch\u00e4ftigt?\n\nWir haben uns schon eine ganze Zeit mit Anarchismus besch\u00e4ftigt, zu Theorien und Praktiken was gelesen und auch diskutiert am Anfang. Bei mir ging das ja dann immer so weiter, nat\u00fcrlich Marx sp\u00e4ter, dann ging es halt um Kommunismus und sich damit auseinanderzusetzen, dann Horkheimer, Adorno usw. Dann auch viel \u00fcber diese ganzen militanten Gruppen in verschiedenen L\u00e4ndern, \u00fcber Italien, Spanien, S\u00fcdamerika auch und so ab 2000 dann viel Zapatistas, deren Theorien. Aber das haben nicht alle gemacht, das hat dann schon mit der Uni zusammengehangen, so Lesezirkel und so was, das muss nicht unbedingt in der Szene notwendigerweise stattfinden. Da gibt es viele, die es nicht so machen.\n\n#### Spielten Filme eine Rolle?\n\n[Lacht.] Mit der Antifa haben wir echt Nazifilme geguckt, also um das bei Infoveranstaltungen zu diskutieren. Sp\u00e4ter haben wir ganz viel Filme \u00fcber S\u00fcdamerika geguckt und viel \u00fcber die '68er in Deutschland.\n\n#### Ist die Besch\u00e4ftigung mit Theorie wichtig in der Szene?\n\nDas spielt schon eine Rolle, also man muss mithalten k\u00f6nnen. Das sind ja Sachen, wor\u00fcber man sich unterh\u00e4lt, und dann muss man bestimmte Sachen auch schon mal geh\u00f6rt oder mal gelesen haben, dass man sich darauf beziehen kann, und auch so 'n bisschen halt mit Namen um sich werfen. Ein ganz gro\u00dfes Thema ist nat\u00fcrlich der Nationalsozialismus, das habe ich eben bei der Aufz\u00e4hlung der Literatur ganz vergessen, weil das so selbstverst\u00e4ndlich ist. Da kommt man gar nicht drum herum, es gibt niemanden in der Szene, der sich mit diesem Thema nicht viel besch\u00e4ftigt hat oder einen ziemlich guten Hintergrund und viel geschichtliches Wissen angeeignet hat.\n\n#### Wie stehst du zum Konflikt zwischen Antiimps und Antideutschen?\n\n[Lacht.] Diese Konflikt ist tats\u00e4chlich ein deutsches Ph\u00e4nomen, also das gibt es international nicht, diese Spaltung, das ist ganz klar deutsch, und so sehr ich auch verstehe, dass es wichtig ist, sich gerade in Deutschland mit diesem Thema auseinanderzusetzen, und dass es diese Kritik gibt innerhalb der Szene an diesen Pro-Pal\u00e4stina-Anh\u00e4ngerInnen, die es nirgendwo anders gibt auf der Welt in der linken Szene, finde ich doch die extremeren Ausl\u00e4ufer problematisch. Vor allem muss die Szene noch handlungsf\u00e4hig bleiben und sich nicht derma\u00dfen in die Haare kriegen, dass man keine Politik mehr machen kann. Ich glaube, das hat was damit zu tun, dass zum Beispiel in Berlin einfach die Szenen so gro\u00df sind, dass sie sich das leisten k\u00f6nnen, auseinanderzufallen, w\u00e4hrend es in kleineren Orten eher so homogenisiert ist, was nicht unbedingt nur was Positives ist, aber dazu f\u00fchrt, dass unabh\u00e4ngig von der inhaltlichen Diskussion immer so ein bisschen mehr bedacht wird, was ist denn der Effekt von dieser Diskussion, hilft es der Szene weiter? Das Thema Antideutsche spielt ja heute schon nicht mehr die gro\u00dfe Rolle wie noch vor ein, zwei Jahren, aber in Berlin gibt's eigentlich so alle paar Jahre eine extreme Debatte \u00fcber irgendwas, die die Szene krass spaltet und viele nur noch schwarz oder wei\u00df sehen l\u00e4sst.\n\n#### Wie stehst du zu B\u00fcndnissen mit anderen linken und Protestbewegungen?\n\nIch bin ein ganz gro\u00dfer Fan von B\u00fcndnissen, das ist total wichtig, um irgendwas \u00fcberhaupt auf die Reihe zu kriegen, \u00fcberhaupt Durchschlagkraft zu haben, geh\u00f6rt zu werden oder irgendwas ver\u00e4ndern zu k\u00f6nnen. Nat\u00fcrlich muss man vorsichtig sein, mit wem man B\u00fcndnisse macht, weil man nat\u00fcrlich nicht zu starke Kompromisse machen will, aber grunds\u00e4tzlich denke ich schon, dass es mehr passieren sollte.\n\n#### Gehst du w\u00e4hlen?\n\nUnterschiedlich. Ich hab schon Jahre nicht gew\u00e4hlt oder bewusst ung\u00fcltig gew\u00e4hlt und manchmal w\u00e4hl ich auch.\n\n#### Wovon h\u00e4ngt es ab?\n\nEs gibt Wahlen, wo ich das Gef\u00fchl hab, das macht keinen Unterschied, ob die Partei oder die Partei drankommt, und dann gibt's die Jahre, wo man das Gef\u00fchl hat, es gibt eine echte Gefahr, einen extremen Rechtsruck oder so, und da gehe ich schon w\u00e4hlen, das ist mein Kriterium.\n\n#### Wie ist dein Verh\u00e4ltnis zum Staat, zur Polizei?\n\nIch sehe die Polizei als Institution, die daf\u00fcr da ist, um Herrschaftsverh\u00e4ltnisse abzusichern und die bestehenden Verh\u00e4ltnisse so zu lassen, wie sie sind, und alles, was dem entgegensteht, niederzumachen in allen Formen, mit allen Mitteln.\n\n#### Nach dem Ende des linken Terrorismus durch zum Beispiel die RAF sind es vor allem Autonome, die als linke Gefahr f\u00fcr die \u00f6ffentliche Ordnung und Sicherheit wahrgenommen werden. Wie gef\u00e4hrlich sind Autonome tats\u00e4chlich?\n\nGanz ehrlich: Ich sehe da keine Gefahr durch die Autonomen, so wie die Autonomen heutzutage organisiert sind. [Lacht.] Im Vergleich gerade zur RAF, das ist ein extremer Unterschied, die RAF war einfach sehr gut organisiert und zu allem M\u00f6glichen bereit, und wenn ich da die Autonomen sehe, das ist wesentlich unstrukturierter. Das h\u00e4ngt nat\u00fcrlich auch mit dem Repressionsapparat zusammen, es ist heute ein anderes Klima als damals, als die RAF entstanden ist. Die waren ja der Meinung, kurz vor dem n\u00e4chsten faschistischen Staat zu stehen; nat\u00fcrlich geht man da mit einer ganz anderen Motivation ran, alles zu geben. Die sind in Trainingscamps gegangen, die haben Leute erschossen, Politiker entf\u00fchrt usw., und im Vergleich dazu sind die Autonomen so harmlos. Die Aktionen sind teilweise schon gezielt und gut durchdacht, aber nicht darauf ausgelegt, einen massiven Schaden anzurichten, die w\u00e4ren nicht bereit, Menschenleben aufs Spiel zu setzen, was auch positiv ist, und damit [lacht] sind die nat\u00fcrlich auch nicht so eine Gefahr. Die Autonomen sind dem Staat sicherlich ein Dorn im Auge, denn der Staat will alles kontrollieren und die hat er nicht unter Kontrolle.\n\nDas Interview f\u00fchrte Anna Ziegler.\n\n## \"Am 11. September hab ich mich total gefreut. Ich hab echt gejubelt.\"\n\nSuzana, 39 Jahre, lebt in Berlin.\n\nSuzana kommt urspr\u00fcnglich aus S\u00fcddeutschland und wohnt seit \u00fcber 15 Jahren in Berlin-Friedrichshain. Wir treffen uns in ihrer Wohnung. Ich treffe auf eine junge Frau, die auf den ersten Blick sehr lebenslustig wirkt. Sie redet viel und lacht oft dabei. Sie ist mittelgro\u00df, sehr d\u00fcnn und hat dunkelblonde lange Haare. Sie tr\u00e4gt eine schwarze Bundeswehr-Hose, ein wei\u00dfes M\u00e4nner-Unterhemd und dar\u00fcber ein schwarzes Strickj\u00e4ckchen. Sie ist ungeschminkt, wirkt sehr nat\u00fcrlich. Ihre Wohnung ist sehr ordentlich und ger\u00e4umig, recht dunkel, aber gediegen eingerichtet. Sie sitzt in ihrem Wohnzimmer auf dem Teppich, w\u00e4hrend sie mit mir spricht. Wir trinken gr\u00fcnen Tee, sie raucht, obwohl sie eigentlich aufgeh\u00f6rt hat, wie sie mir erz\u00e4hlt \u2013 aber weil sie ein bisschen nerv\u00f6s ist, hat sie sich heute doch mal wieder eine Schachtel Zigaretten gekauft. Ich schaue mich in ihrem Zimmer um, es ist recht spartanisch eingerichtet, viele B\u00fccher, alte M\u00f6bel, an den W\u00e4nden h\u00e4ngen fast keine Bilder oder \u00c4hnliches, und ich bin etwas erstaunt, als ich an der Wand eine Rosenkranzkette entdecke. In der Ecke stehen eine alte E-Gitarre, ein Bass, ein Verst\u00e4rker. Auf der Fensterbank steht eine kleine schwarze Engelsfigur.\n\n#### Dein Name Suzana ist kroatisch, oder? Bist du denn in Deutschland geboren?\n\nJa, aber ich habe nicht immer in Deutschland gelebt. Nur das erste Jahr, dann war ich dreieinhalb Jahre wieder in Jugoslawien und dann bin ich wieder zur\u00fcckgekommen \u2013 worden. [Lacht.] Also im Alter von f\u00fcnf Jahren.\n\n#### Und warum warst du wieder in Kroatien?\n\nWeil meine Eltern dachten, sie k\u00f6nnten mehr Geld verdienen, wenn die Kinderchen nicht dabei sind. [Lacht.] Ja, das war damals so \u00fcblich. Da hat man Kinder teilweise zu Gro\u00dfeltern geschickt, um sie erst zum Schulbeginn wieder zu holen.\n\n#### Hast du Geschwister?\n\nJa, eine Schwester. Die ist vier Jahre \u00e4lter.\n\n#### Und wie ist dein Verh\u00e4ltnis zu ihr?\n\nKaum Kontakt. Wir haben kein so 'n geschwisterliches Verh\u00e4ltnis. Vielleicht auch dem Umstand geschuldet, dass wir uns bis auf das erste Jahr nicht \u00fcber den Weg gelaufen sind und uns eigentlich gar nicht kannten und bei der R\u00fcckf\u00fchrung erst richtig kennengelernt haben.\n\n#### Deine Schwester war nicht in Jugoslawien?\n\nDoch, aber die war in 'ner anderen Familie. Wir haben nicht realisiert, dass wir Geschwister sind.\n\n#### Was, echt?\n\nHm.\n\n#### Was habt ihr f\u00fcr ein Verh\u00e4ltnis zu den Eltern?\n\nMeine Eltern sind geschieden, und ich habe keinen Kontakt zu meinem Vater, den hab ich circa dreimal im Leben gesehen. Aber mit meiner Mutter hab ich eigentlich ein ganz gutes Verh\u00e4ltnis, wenn wir uns denn mal besuchen, also jetzt nicht so regelm\u00e4\u00dfig, aber im Prinzip haben wir schon ein Verh\u00e4ltnis.\n\n#### Welche Ausbildung haben deine Eltern bzw. was macht deine Mutter beruflich?\n\nMeine Mutter ist gelernte Verk\u00e4uferin, hat aber, als sie nach Deutschland gekommen ist, nicht lange in dem Beruf gearbeitet und dann verschiedene Jobs gemacht, was man so macht als Gastarbeiter, im Hotelgewerbe, im Krankenhausgewerbe. Mein Vater ist gelernter Maurer, hat eine Woche vor seiner Abiturpr\u00fcfung die Schule geschmissen und war dann im Milit\u00e4r wohl lange Zeit und sp\u00e4ter... keine Ahnung. Dadurch, dass wir keinen Kontakt haben, wei\u00df ich das gar nicht so genau.\n\n#### Seit wann waren deine Eltern in Deutschland?\n\nDas muss so '68 herum gewesen sein.\n\n#### Was w\u00fcrdest du anders machen als deine Eltern?\n\n[Einen Moment erstaunt, lacht dann.] Alles! Ich w\u00fcrde meine Kinder nicht wegschicken im Kleinkindalter, ich w\u00fcrde sie statt zu Duckm\u00e4usertum eher zu Selbstbewusstsein erziehen... Ja, ich w\u00fcrde sozusagen zur Utopie aufrufen, ich w\u00fcrde meinem Kind, selbst wenn's sagen w\u00fcrde, es will den Mond neu erfinden, sagen, versuch das, mach das. Und ich w\u00fcrde auch nur dann Kinder in die Welt setzen, wenn ich sie auch finanzieren kann.\n\n#### M\u00f6chtest du denn selbst Kinder haben?\n\nNein! [Lacht.]\n\n#### Warum nicht?\n\nWeil die sozialen Umst\u00e4nde nicht daf\u00fcr gegeben sind, weder in meinem pers\u00f6nlichen Leben noch in der Gesellschaft. Aber das ist jetzt nicht was Schlimmes f\u00fcr mich, wenn man Kinder kriegt, ich find Kinder ganz toll. Ich freu mich immer, wenn Freunde Kinder kriegen, dann partizipiere ich daran.\n\n#### W\u00fcrdest du sagen, dass deine Eltern oder andere Verwandte eine Rolle f\u00fcr deine politische Entwicklung gespielt haben?\n\nJa. Mit Migrationshintergrund kommt man ganz schnell in Ber\u00fchrung mit Ungerechtigkeiten innerhalb der Gesellschaft, die strukturell verankert sind, und da hab ich sehr fr\u00fch schon mitgekriegt, wie beispielsweise meine Mutter beim Jugendamt damals Kindergeld beantragen wollte, irgendwann Anfang der 80er, als alleinerziehende Mutter mit schlechten Deutschkenntnissen. Und ich war dabei, da war ich vielleicht so acht Jahre alt, und hab mitgekriegt, wie sie ganz fies behandelt wurde von dieser Frau, die da gearbeitet und zu ihr gesagt hat: \"Gehen Sie doch zur\u00fcck in Ihr Heimatland, was wollen Sie denn hier \u00fcberhaupt!\" und so. Und meine Mutter wusste halt von Gesetzen nicht so viel, sie ist dann wirklich gegangen und hat in ihrem Leben nie wieder Kindergeld beantragt. Ansonsten ist meine Mutter eigentlich schon 'n ganz peaciger Mensch, der also weder Rassist noch Faschist ist, deswegen sind wir nicht totale Kontra-Gegner oder so, aber sie ist schon mehr politikfrustriert und sagt: Man kann ja eh nix machen... Und das ist ja nicht falsch.\n\n#### Warum?\n\nWeil wir immer noch die kroatischen P\u00e4sse haben und das als Nicht-EU-B\u00fcrger in Deutschland nicht geduldet wird, auch wenn du hier geboren bist.\n\n#### Also keine deutsche Staatsb\u00fcrgerschaft?\n\nUnd also auch kein Wahlrecht. Auch keine doppelte Staatsb\u00fcrgerschaft. Also eigentlich hat man alle Pflichten, aber keine Rechte. Was sich ja angeblich 2015 \u00e4ndern wird, aber ich hoffe mal nicht. Bin 'n ausgesprochener EU-Gegner.\n\n#### Warum?\n\nWeil ich es ganz schlimm, finde, was die EU an Politik macht. Siehe kriegspolitische Einmischung auf anderen Kontinenten, Asylgesetzgebung, Drittweltl\u00e4nder-Abkommen, Nahrungsmittelverordnungsgesetze, also diese ganzen wichtigen Dinge, die passieren ja inzwischen nur noch im EU-Parlament, gar nicht mehr auf nationaler Ebene. Und von daher halte ich das f\u00fcr 'ne ganz schlimme Bande. [Lacht.] Es ist nat\u00fcrlich nicht mehr r\u00fcckg\u00e4ngig zu machen, das ist mir schon klar.\n\n#### Und wenn du w\u00e4hlen k\u00f6nntest, w\u00fcrdest du w\u00e4hlen gehen?\n\nDefinitiv ja. Ich w\u00fcrde sogar Wahlwerbung machen. [Lacht.] Wahrscheinlich w\u00fcrde ich selbst kandidieren.\n\n#### F\u00fcr welche Partei?\n\nF\u00fcr die Linken wahrscheinlich oder f\u00fcr die Gr\u00fcnen, die Gr\u00fcnen mal wieder ein bisschen weiter nach links schubsen. Also wenn ich jetzt ganz auf Erfolgsfilm und ganz mutig w\u00e4re und ganz viel Zeit \u00fcbrig h\u00e4tte oder das hauptberuflich machen w\u00fcrde, dann w\u00fcrde ich mich sogar an die SPD rantrauen. Ich w\u00fcrde alles machen, um dieser liberalen kapitalistischen Schei\u00dfe mal wieder 'n bisschen aufkl\u00e4rerischen Touch zu verleihen. Also Politik ist ganz wichtig...\n\n#### Auch Parteipolitik?\n\nJa. Auch Stadtpolitik, jede Politik ist wichtig. Politik f\u00e4ngt im Kleinen an. Und das ist manchmal sogar effektiver, da hat man kurzfristig mehr Erfolgschancen. Siehe Baden-W\u00fcrttemberg, hat sich ja auch alles ge\u00e4ndert auf einmal.\n\n#### Wie ist deine gegenw\u00e4rtige Situation?\n\nSchlecht. [Lacht.] Ich bin momentan in der Schwebe, weil ich noch nicht genau wei\u00df, ob ich meinen Studienabschluss kriege, ansonsten h\u00e4tte ich halt ein Zwischenzeugnis... der Geisteswissenschaften... \u2013 muss ich das genauer erkl\u00e4ren? Ansonsten bin ich so Tagel\u00f6hnerin, wie man so sch\u00f6n sagt, also die meisten Jobs, in denen ich arbeite, hab ich nicht gelernt.\n\n#### Wie sehen deine Pl\u00e4ne und Tr\u00e4ume f\u00fcr die Zukunft aus?\n\nIch bin momentan dabei, mir zu \u00fcberlegen, wie ich den Fu\u00df in die soziologische T\u00fcr kriege. Tr\u00e4ume? Ich wei\u00df nicht, man nimmt, was man kriegt heutzutage. Ich w\u00fcrde schon ganz gerne irgendwas arbeiten, was ich auch studiert habe.\n\n#### Hast du einen Traumberuf?\n\nMehrere eigentlich, von Politik \u00fcber \u00d6ffentlichkeitsarbeit, Wissenschaftsjournalismus, Schriftstellerin bis hin zu Jugendarbeit kann ich mir eigentlich alles ganz gut vorstellen. Also traumhaft w\u00e4re, wenn man sich irgendwie selbstst\u00e4ndig machen k\u00f6nnte.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nIch bin ein Mensch, der sagt, was er denkt, auch wenn das nicht so gerne geh\u00f6rt wird und zu Konsequenzen f\u00fchren kann. Das ist Interpretationssache, ob man sagt, das ist besonders mutig oder besonders dumm, aber ich bin wenig anpassungsf\u00e4hig wahrscheinlich und... sehr selbstbewusst... bewusst \u00fcber das Selbst. Und ich bin auf jeden Fall unstet, ich will immer viele Sachen auf einmal machen, und dann mache ich eine Sache nicht ganz richtig. Also ich f\u00fchre Dinge, die ich anfange, selten zu Ende und bin auch relativ unzuverl\u00e4ssig.\n\n#### Wovor hast du Angst?\n\nVor Verarmung, vor Einsamkeit und vor Versagen. [Lacht.]\n\n#### In welcher Hinsicht Versagen?\n\nDass ich mein Selbstbild vielleicht falsch einsch\u00e4tze und meine gesellschaftlichen und privaten Ziele nicht erreiche und am Schluss so 'ne ungl\u00fcckliche Existenz f\u00fcr immer und ewig f\u00fchren muss. F\u00fcr immer Hartz-IV-Empf\u00e4nger sein oder so was.\n\n#### Was meinst du mit gesellschaftlichen Zielen?\n\nMeine Weltanschauungen, dass die sich quasi mein Leben lang nur als Tr\u00e4ume verwirklichen lassen, aber wenn ich dann am Schluss feststelle, ich konnte \u00fcberhaupt nichts zur Verbesserung der Gesellschaft beitragen, das w\u00e4re f\u00fcr mich Versagen. Auch wenn es in meinem pers\u00f6nlichen Leben gut laufen w\u00fcrde. Also ich brauche schon eine gewisse gesellschaftliche Evidenz, dass da irgendein Wirken war, was so K\u00fcnstler immer von sich sagen, dass sie irgendwas zur\u00fccklassen wollen auf der Welt, um sie ein St\u00fcck zu ver\u00e4ndern, diesen Anspruch hab ich im sozialpolitischen Sinne dann schon.\n\n#### Was w\u00fcrdest du denn gerne zur\u00fccklassen?\n\nIch w\u00fcrde ganz gerne B\u00fccher schreiben. Also wenn ich meine Magisterarbeit nicht schreiben darf, weil die keiner mehr annimmt, hab ich mir \u00fcberlegt, dass ich daraus vielleicht so eine Art Roman mache, 'nen gesellschaftskritischen Roman. Mit empirischen Zahlen [lacht] und Evidenzen, aber mit noch mehr Pfeffer und Salz in der Suppe.\n\n#### Was \u00e4rgert dich am meisten? Was macht dich w\u00fctend?\n\nUnsere gesellschaftliche Struktur, unsere Institutionen hier machen mich wahnsinnig w\u00fctend. Also ich meine alle Institutionen; ich glaube, Habermas hat mal gesagt, die Gr\u00fcnde, die zur Erschaffung von Institutionen f\u00fchren, sind dieselben, die beim Individuum zu Neurosen f\u00fchren. Bei einzelnen Menschen macht mich Meinungslosigkeit am allermeisten w\u00fctend, Menschen, die einfach kein Gesicht haben und das zeigen k\u00f6nnen, sondern bei jedem Menschen irgendwie anders sind. Und asoziales Verhalten.\n\n#### Lebst du eigentlich in einer festen Partnerschaft?\n\nNein, ich bin Single und gl\u00fccklich. Ich bin \u00fcberzeugter Single.\n\n#### Wie verbringst du deine Freizeit? Hast du Hobbys?\n\nIch habe momentan dadurch, dass ich keine feste Besch\u00e4ftigung habe, relativ viel Zeit, von daher sind manchmal die Grenzen flie\u00dfend. Wenn ich ein Buch lesen m\u00f6chte, lande ich doch bei wissenschaftlicher Literatur, nicht nur, weil ich denke, da k\u00f6nnte ich ja eventuell noch mal 'ne Pr\u00fcfung machen, sondern weil mich das wirklich total interessiert. Ansonsten h\u00f6re ich gerne Musik und versuche ja immer wieder, selber Musik zu machen, aber die Disziplin fehlt und das Geld f\u00fcr 'n gutes Instrument. Ansonsten muss ich ehrlich gestehen, bin ich ein bisschen computers\u00fcchtig geworden in den letzten zwei Jahren, also ich bin sehr auf Kommunikationsplattformen unterwegs.\n\n#### Auf welchen Plattformen bist du unterwegs?\n\nDas sag ich jetzt lieber nicht. Aber ich guck zehnmal am Tag, ob ich Post bekommen habe [lacht], da muss man echt aufpassen; also ich hoffe, das ist nur die Anfangseuphorie. Ansonsten schaue ich gerne Leute, ist das ein Hobby? Leute beobachten? Sich vor die T\u00fcr setzen und gucken...\n\n#### Das ist interessant.\n\nIn Berlin machen das wirklich viele Leute und ich finde, dass es nicht nur soziologisch interessant ist, das ist wirklich wie Fernsehen. Also ich schaue gerne Leute, ich unterhalte mich gerne mit fremden Leuten und lerne gerne Leute kennen. Ansonsten fehlt mir zu befriedigender Freizeitgestaltung schlicht das Geld. Sonst w\u00e4ren das sicherlich Reisen, Fotografieren, Konzerte, Theater.\n\n#### Was sind das f\u00fcr Leute, mit denen du dich unterh\u00e4ltst?\n\nGanz unterschiedliche. Je unterschiedlicher und mir fremder, desto interessanter. Ich unterhalte mich sogar gerne mit Touristen. [Lacht.]\n\n#### Nach welchen Kriterien schlie\u00dft du Freundschaften?\n\nDas ist eine ganz heikle Frage. Denn ich bin eine von denen, die tats\u00e4chlich so Dogmata haben, was Freundschaften betrifft, was ganz schwierig ist f\u00fcr manche Leute auch, dass ich unterm Strich schon sagen muss, dass ich so gut wie keine Freunde habe. F\u00fcr mich ist Freundschaft dann gegeben, wenn ich jemand vor mir habe, bei dem ich merke, da kann ich so sein, wie ich bin, vor allen Dingen auch mit den negativen Eigenschaften, ohne mich daf\u00fcr rechtfertigen zu m\u00fcssen. Ein Freund sollte einem sagen, was er an einem nicht gut findet, aber er sollte die Sache so gestalten, dass man sich nicht abgewertet f\u00fchlt, sondern dass man damit arbeiten kann, dass man sich selber weiterentwickeln kann und nicht abgebremst wird in seiner Entwicklung. Ich hatte mal eine Zeit, in der ich mit vielen anderen Migranten-Kids rumgehangen habe, und da war das oberste Gesetz, dass man f\u00fcreinander einsteht mit allem, was dir an Mitteln zur Verf\u00fcgung steht, auch wenn's f\u00fcr dich selber Nachteile bedeutet, und das hat mir damals so gut gefallen, das gef\u00e4llt mir heute noch sehr gut. Also dass ich f\u00fcr einen Freund vielleicht sogar mehr tun w\u00fcrde als f\u00fcr mich selbst. Aber den meisten bedeutet Freundschaft eher, dass man einfach Spa\u00df zusammen hat und was zusammen tut, was man sowieso tun w\u00fcrde, also betreutes Beisammensein oder so. [Lacht.] Und das versteh ich darunter halt nicht, weil das kann ich auch mit Bekannten oder Interessengruppen.\n\n#### Bist oder warst du mal in einem Verein aktiv?\n\nJa, ich war in mehreren eingetragenen Vereinen. Ich war in einigen sozialpolitisch motivierten Gruppen, um gewisse Interessen zu vertreten, und die waren einfach meistens Vereine, und da hat man sich entweder als f\u00f6rderndes Mitglied, aktives Mitglied oder unterst\u00fctzendes Mitglied sozusagen der Vereinsmeierei gewidmet. Derzeit bin ich in drei Vereinen eingetragen.\n\n#### Dein haupts\u00e4chliches politisches und gesellschaftliches Engagement liegt aber mehr in der Vergangenheit, oder?\n\nAlso das Erste, was mir einf\u00e4llt, ist der Arbeitskreis Asyl, das war damals in Baden-W\u00fcrttemberg Anfang der 90er. Das war diese ganz schlimme Zeit in Deutschland, wo \u00fcberall die Asylbewerberheime gebrannt hatten und ich mir dachte, jetzt machst du mal richtig was, und bin dann in den Arbeitskreis Asyl mit reingegangen, wo \u00e4ltere, oft sehr versierte politische B\u00fcrger der Stadt aktiv waren, von Feministinnen bis zu Sprechern des Studentenrates \u00fcber Gr\u00fcnen-Politiker, alle m\u00f6glichen Leute waren da vertreten, aber eben auch Jugendliche, und dort haben wir dann verschiedene Dinge gemacht wie Arbeitsloseninitiativen, also versucht, Leute, die langzeitarbeitslos oder unverschuldet arbeitslos geworden sind, an gute Beratungsstellen zu vermitteln usw., aber vor allem eben, auf selbstverwaltete Art und Weise mit den Menschen in den Asylbewerberheimen zu arbeiten. Ich war noch Sch\u00fclerin damals und dachte, die nehmen mich bestimmt gar nicht ernst, aber ich hatte Gl\u00fcck, weil es gab in der Zeit ganz viele Menschen aus dem ehemaligen Jugoslawien und ich war die Einzige, die die Sprache konnte, und so habe ich jeden Mittag in irgendwelchen Asylbewerberheimen verbracht und dort die Lebensbedingungen der Menschen aufgenommen und auch versucht, dort mit den Heimleitern zu vermitteln, bei Gespr\u00e4chen zu \u00fcbersetzen, was so die Probleme sind. Und nachts habe ich dann mit meiner Jugendgruppe auf unsere Asylbewerberheime aufgepasst, also wirklich patrouilliert vor denen, damit auch ja nichts passiert.\n\nDann war ich in der Antifa, die hie\u00df damals so, war aber sehr heterogen, also da waren auch ganz seri\u00f6se politische Menschen aktiv und nicht irgendwelche Steineschmei\u00dfer, wie man heutzutage oft glaubt. Und dann war ich noch bei einem autonomen Jugendzentrum, und da hatten wir auch ganz viel politische Arbeit zu tun, weil unser Jugendzentrum von der Schlie\u00dfung bedroht war, und dieser Kampf von der Kampfansage bis zur letztendlichen Schlie\u00dfung ging \u00fcber Jahre, also da hat man ganz viel Erfahrung sammeln d\u00fcrfen. Die Antifa selber hat sich dann auch gespalten in einzelne Subgruppen: Die Feminismus-Gruppe traf sich montags, mittwochabends hatte man dann die Tierrechts-Gruppe, donnerstags hatte man die Gruppe, die die Prober\u00e4ume verwaltete und Konzerte organisierte und am Wochenende hat man eben vor Asylbewerberheimen rumpatrouilliert.\n\nDann hat man aktive Anti-Vergewaltigungs-Arbeit gemacht, da haben wir versucht, in Dialog zu gehen mit Prostituierten. Die meisten Prostituierten kamen ja aus Russland und anderen Ostblockl\u00e4ndern und haben in ganz schlimmen Zust\u00e4nden da gearbeitet auf der Stra\u00dfe. Und daran hat sich auch irgendwann unsere Feminismus-Gruppe gespalten, weil viele Frauen gesagt haben: Prostitution unterst\u00fctzen, das geht gar nicht, und ich hab dann meine eigene Gruppe gegr\u00fcndet und gesagt: Wir brauchen ein Bordell f\u00fcr unsere russischen Frauen. Wir haben leider nicht soo viel erreicht, aber immerhin vielen Frauen zumindest mal die M\u00f6glichkeit gegeben, an uns ranzutreten, wenn sie ein Anliegen hatten.\n\nUnd die Tierrechts-Gruppe hat sich dann auch gespalten, da gab's Leute, die in die Kriminalit\u00e4t gegangen sind mit ihren Aktivit\u00e4ten, andere haben ganz normal Pamphlete aufgesetzt gegen irgendwelche Zirkusveranstalter oder Pharma-Konzerne, Demonstrationen organisiert, Flugbl\u00e4tter verteilt. Und als wir dann gemerkt haben, es verl\u00e4uft sich im Nichts, haben wir 'ne Zeitschrift gegr\u00fcndet und da unsere Aktionen dokumentiert, wo jede kleine Gruppe ihren Beitrag geschrieben hat, was hat jede gemacht in dem Monat, und dann kam so einmal im Monat diese Zeitschrift raus, durch die man auch den \u00dcberblick \u00fcber alle anderen Gruppen hatte, denn man konnte nicht in jede Gruppe gehen.\n\n#### In dieser Tierrechts-Gruppe warst du auch?\n\nJa, ich war Springer. [Lacht.] Kann man so sagen.\n\n#### Wie w\u00fcrdest du \"autonom\", \"autonome Szene\" eigentlich definieren oder jemandem beschreiben, der noch nie etwas davon geh\u00f6rt hat?\n\nF\u00fcr mich ist Autonomie Selbstverwaltung. In allererster Linie selbstverwaltete Struktur. Das bedeutet eben, dass es keine Hierarchien gibt, dass es sehr wohl Regeln gibt, die Regeln aber im Konsens immer wieder neu diskutiert und entschieden werden m\u00fcssen, und nicht per Mehrheitswahlrecht oder Abstimmung. Nat\u00fcrlich ist Mehrheit immer wichtig, aber wichtiger ist die Argumentation. Also wenn zwanzig Leute gegen was abstimmen und es hat aber nicht wirklich Sinn, dann haben drei Leute immer noch die M\u00f6glichkeit, in einem demokratischen Basisgespr\u00e4ch zu erkl\u00e4ren, warum sich vielleicht die anderen zwanzig t\u00e4uschen, und sie zu \u00fcberzeugen, so dass dann trotzdem noch mal anders entschieden wird.\n\n#### Das klingt nach Endlosdebatten ohne Entscheidungen...\n\nManchmal ist es schon so, dass man sich basisdemokratisch derma\u00dfen zerfetzt, dass letztlich nichts r\u00fcberkommt, dass man sich manchmal schon gew\u00fcnscht h\u00e4tte, man h\u00e4tte so eine gewisse autorit\u00e4re Stimme darin gehabt, die sagt: So Leute, wir machen das jetzt so und Schluss. Und ihr werdet sehen, es wird gut f\u00fcr euch sein. [Lacht.] Aber das ging ja nicht, es mussten immer alle einverstanden sein.\n\n#### Also quasi ein Konsensprinzip mit Veto-Option?\n\nGenau. Man braucht daf\u00fcr ganz viel Toleranz, so 'ne gewisse Frustrationstoleranz auch, denn es kann dir passieren, dass du glaubst, du hast 'ne ganz tolle Idee, und die ist vielleicht auch toll, aber alle anderen sagen: Nein, es passt jetzt einfach nicht. [Lacht.] Man braucht eben einen langen Atem und muss am Ball bleiben und dann eben immer wieder versuchen, noch besser zu argumentieren. Ich erinnere mich noch, als w\u00e4re es gestern gewesen: Wir hatten damals ein paar Jugendliche, die hatten massive Drogenprobleme, also heroinabh\u00e4ngige Jugendliche, aber wir hatten keine Fixerstuben in der Stadt, und ich hatte dann den Antrag gestellt, dass diese betreffenden drei, vier Personen, wenn ich um 19 Uhr den Laden aufmache, dahin kommen k\u00f6nnen, dort die \u00d6rtlichkeiten nutzen k\u00f6nnen, dass man ein Auge darauf hat, wenn was passiert, und ich in der Lage gewesen w\u00e4re, eventuell die Ambulanz zu rufen oder Erste Hilfe zu leisten. Es wurde einstimmig abgelehnt. Und das war so eine Situation, wo man dann wei\u00df: Es w\u00e4re eigentlich 'ne gute Idee und super-sozial und so was braucht es auch, aber wir k\u00f6nnen das nicht machen, weil, dann schlie\u00dfen die uns den Laden, weil es hie\u00dfe, wir w\u00e4ren 'ne Drogenkneipe, und dann haben wir gar nichts mehr. Also mussten wir sagen: Ey Leute, ihr k\u00f6nnt hier keine Drogen konsumieren, und wenn wir das feststellen, dann m\u00fcsst ihr leider Hausverbot bekommen. Aber ich hab... kann man ja jetzt vielleicht sagen, ich hab mich dr\u00fcber hinweggesetzt, ich hab alles riskiert und von zwei Leuten gesagt, sie k\u00f6nnen das bei mir machen, wenn ich da bin. Das nehme ich auf meine Kappe. Also so ganz autonom, ganz anarchisch hab ich das nicht gemacht. Da war ja noch ein Sozialarbeiter, eine ganz liebe Seele von einem Menschen, unser Papa eigentlich, und der wusste das, weil es w\u00e4re ja doch ein bisschen zu krass gewesen, das ohne sein Wissen zu machen. Aber er hat gemeint, okay, wir machen das, und wenn was schief geht, sagen wir halt, wir wussten das nicht.\n\n#### Du hattest schon Anarchismus erw\u00e4hnt und es klang so, als ob du dich mit anarchistischen Ideen besch\u00e4ftigt h\u00e4ttest?\n\nAlso in meinem pers\u00f6nlichen Leben damals hab ich in meiner Freizeit durchaus auch mit den Stra\u00dfenpunks und Obdachlosen abgehangen und ganz klar anarchische Strukturen kennengelernt im Sinne von: \"Ich nehme mir, was ich brauche, und brauch das auch niemandem zu erkl\u00e4ren, warum ich das jetzt mache.\" Aber ich hab dann tats\u00e4chlich durch Literatur erkannt... so 'n russischer Autor...\n\n#### Bakunin?\n\nJa, den nat\u00fcrlich auch, das ist die Grundliteratur. [Lacht.] Makarenko! Der hat in der Ukraine ein Kinderheim gegr\u00fcndet und versucht, das anarchistisch zu leiten. Doch er hat dann gemerkt, die Stra\u00dfenkids tanzen ihm auf der Nase rum, die haben gemacht, was sie wollten, das war ein Haufen von kleinen Banditen und Schl\u00e4gern und jeder war unversch\u00e4mt und gewaltt\u00e4tig ihm gegen\u00fcber. Und so hat er gemerkt, dass sein Haus letztlich so nicht aufrechterhalten werden kann. Er hat dann dieses R\u00e4tesystem eingef\u00fchrt, mit einem Funken Autorit\u00e4t, und das hat funktioniert. Man kann Freiheit im Grunde dann weiterentwickeln, wenn man sie auch z\u00fcgelt auf eine gewisse Art und Weise, gerade bei Jugendlichen und Kindern; wenn du denen eine gewisse Struktur gibst, werden sie sich innerhalb dieser Struktur freier bewegen k\u00f6nnen paradoxerweise, als wenn du sie v\u00f6llig frei l\u00e4sst. Und das war damals f\u00fcr mich ein ganz neuer Gedanke. Ich dachte davor immer, der Mensch ist gut, und wenn wir in der Anarchie leben w\u00fcrden, w\u00fcrde ja keiner auf die Idee kommen, jemand anders zu \u00fcberfallen, wenn doch jeder dasselbe h\u00e4tte. Aber es geht eben nicht, weil die Menschen nehmen, was sie kriegen k\u00f6nnen. Aber in den meisten Anarchie-B\u00fcchern ist es ja auch so erkl\u00e4rt, dass es nicht darum geht, dass alle alles d\u00fcrfen, sondern dass es auch innerhalb einer anarchischen Gesellschaft durchaus Instanzen gibt, die Recht und Unrecht organisieren. Aber ich glaube schon, dass es prinzipiell trotzdem m\u00f6glich ist; ich halte Anarchie durchaus f\u00fcr m\u00f6glich f\u00fcr Kleinstgruppen mit \u00e4hnlichen Charakterstrukturen und Interessen usw., dann funktioniert es.\n\n#### Wie war eigentlich dein allererster Kontakt zur autonomen Szene?\n\nDas war im Autonomen Jugendzentrum in Konstanz, da gab es sonntags eine Jugend-Disco, mehr so 'ne Nachmittags-Kinderdisco, in der es keinen Alkohol gab, in der man nicht rauchen durfte usw. Und da war ich zehn Jahre alt und hatte mit meiner Freundin einen Sch\u00fclerausweis gef\u00e4lscht, so dass wir reingekommen sind. Irgendwie hat's mir dann doch nicht so gefallen und ich hab mir gedacht, ach, jetzt gehst du mal unten rein und guckst dir die Punks an. Die waren alle ein bisschen \u00e4lter, so 15, 16, und fanden das ganz toll, dass so 'n Kind sich daf\u00fcr interessiert, und waren alle furchtbar nett zu mir. Innerhalb von k\u00fcrzester Zeit hatte ich dann auch kurze gr\u00fcne Stachelhaare, damit ich nicht mehr so auffalle und mich von den Poppern da oben abgrenzte.\n\n#### Kannst du sagen, was genau dich daran angezogen hat?\n\nIn erster Linie tats\u00e4chlich die Optik der Leute. Fand ich sehr interessant diese bunten V\u00f6gel, die ich vorher immer so ein bisschen als unfreundlich und schlecht gelaunt wahrgenommen habe. Oben in der Teenie-Disco waren mehr so die Migranten-Jugendlichen, das waren fast nur Italiener, T\u00fcrken und Jugoslawen, die dementsprechend auch optisch so ein bisschen anders aussahen. Und die Musik! \"Stairway to Heaven\" war mein erstes Lieblingslied damals, das hab ich hoch und runter geh\u00f6rt...\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nDamit steht und f\u00e4llt alles. [Lacht.] Das ist heute noch so. Musik ist das wichtigste! Ja, und ich w\u00fcrde mich selber als Textfetischistin bezeichnen, das hat weniger was mit Musik tats\u00e4chlich zu tun, aber Songtexte sind f\u00fcr mich die Versinnbildlichung von eigenen Gedanken, die andere zwar konstruiert haben, aber die trotzdem das Sprachrohr meiner eigenen Gedanken sind, die einen weiterentwickeln k\u00f6nnen oder Freiraum f\u00fcr neue Welten schaffen. Das ist so ein bisschen wie f\u00fcr andere Drogen, sie er\u00f6ffnet dir neue Welten, die quasi jederzeit abrufbar sind. In der Anfangszeit war f\u00fcr mich immer ganz wichtig, dass man sich dazu bewegen konnte, also pogen konnte, dass man Aggressionen ablassen konnte, obwohl ich ja nie so ein besonders gewaltt\u00e4tiger Pogo-Mensch war. Jemand hat mal gesagt, Musik und Fu\u00dfball verbindet die Leute, und das ist schon so: Wenn du Menschen triffst, die dieselbe Musik gut finden wie du, hast du sofort einen Zugang zu denen, was du sonst nicht so schnell hast, alles andere bedarf erst mal Diskussionen und gemeinsamer Erlebnisse usw. Wenn du auf einem Konzert bist und alle finden die Band toll, dann ist das so ein Gemeinschaftserlebnis... oh Gott, das klingt ja schon wie bei den Nazis. Kameradschaft! [Lacht.]\n\n#### Hast du eine Lieblingsmusik?\n\nDa bin ich variabel, aber letztlich habe ich auch meine Lieblingsbands, und wenn eine mal meine Lieblingsband war, dann h\u00f6r ich die auch heute noch gerne. Also es gibt nichts, was ich mal ganz toll fand und heute ganz schlimm finde, keine einzige. Noch nicht mal Iron Maiden, sogar die Toten Hosen! Aber so szene\u00fcbergreifend bin ich sehr variabel, ich h\u00f6r eigentlich fast alles, was von Herzen kommt, meinen Geist befl\u00fcgelt und meinen Gef\u00fchlen wie Wut und Melancholie Ausdruck verleiht \u2013 haupts\u00e4chlich Gruftiemusik, also Gothic, derzeit.\n\n#### Geh\u00f6rst oder geh\u00f6rtest du denn mal einer Jugendkultur an?\n\nJa, den Autonomen! [Lacht.] Also mehreren, ich habe mehrere Jugendszenen durchlaufen, angefangen bei... damals hat man \"Hippies\" gesagt, heute ist das ja mehr oder weniger ein Schimpfwort, dann habe ich ein wenig die Metal-Szene gestreift, bin dann aber relativ schnell bei den Punks gelandet, also bei den Deutsch-Punks, klar, allerdings bei den politischen Punks. Aber eigentlich so die meiste Zeit \u00fcber h\u00e4tte ich mich schon eher als autonom bezeichnet, da ich mich eigentlich gar nicht irgendwo zugeh\u00f6rig gef\u00fchlt hatte und dachte, bei den Autonomen kann man sich individueller entfalten, denn da sind die Grenzen nicht so eng gezogen wie bei allen anderen.\n\n#### Mit welchen Leuten kommst du \u00fcberhaupt nicht klar?\n\nMit Leuten, die sich hinter Szenen verstecken, also jemand, der so sagt: Ich bin nicht die Beate aus Leipzig, sondern ich bin die Gruftie-Beate aus Leipzig. So was mag ich nicht, denn letztlich stehe ich ja auf Individualismus und nicht auf Menschen, die sich von Gruppendynamiken leiten lassen. Mir ist ein politisch schr\u00e4g denkender Mensch, der argumentieren kann, lieber als jemand, der mit Phrasen um sich wirft, ohne sie selber begr\u00fcnden zu k\u00f6nnen.\n\n#### Woher beziehst du deine Informationen zum aktuellen Zeitgeschehen?\n\nHaupts\u00e4chlich durch's Internet, weil ich pers\u00f6nlich finde, dass die Aufkl\u00e4rung \u00fcber das, was in der Welt los ist, sich ganz schlecht aus den Massenmedien hierzulande rausziehen l\u00e4sst, und ich versuche halt auch, mir durch die Internetanalysen weitere Denkanst\u00f6\u00dfe zu holen. Ich guck aber auf jeden Fall jeden Tag Nachrichten und Tagesthemen, und politische Debatten zieh ich mir alle rein. Radio h\u00f6r ich auch total viel, Deutschlandradio.\n\n#### Geh\u00f6rst du einer Religion oder Kirche an?\n\nOffiziell geh\u00f6re ich tats\u00e4chlich der r\u00f6misch-katholischen Kirche an, bin nicht \u00fcberzeugte Katholikin, aber auch nicht so negativ betroffen, dass ich austreten w\u00fcrde. Aber ich w\u00fcrde mich selbst als multireligi\u00f6s bezeichnen, ich sehe durchaus den roten Faden zwischen allen Religionen und hol auch im Zuge meines Studiums, aber auch aufgrund meines pers\u00f6nlichen Interesses manchmal das eine oder andere Heilige Buch raus und lese dann mal darin. Meine Mutter w\u00fcrde sagen, ich bin eher spirituell als religi\u00f6s; ich bete jetzt nicht zu Gott oder so, es sei denn, mir geht es total schlecht, dann manchmal schon.\n\n#### Und welche Bedeutung hat das f\u00fcr dich in deinem Leben, diese religi\u00f6se Seite?\n\nDas hat auf jeden Fall eine meditative Bedeutung. Man sagt ja, Religion ist entstanden, weil die Menschen Angst vor dem Tod haben. Ich glaub schon, dass man sich, gerade, wenn man das Leben als besonders ungerecht empfindet, was bei mir oft der Fall ist, damit beruhigt. Ich halte mich zwar f\u00fcr multireligi\u00f6s, aber ich schlie\u00dfe auch nicht aus, dass tats\u00e4chlich nach dem Tod gar nichts mehr ist, und wenn es dann so ist, dann ist es halt so. Dann ist es halt bl\u00f6d gelaufen. [Lacht.]\n\n#### Du glaubst also schon an ein Leben nach dem Tod?\n\nIch glaub auch an ein Leben vor der Geburt, das w\u00fcrde die Reinkarnationstheorie mit einbringen, also das kann ich mir schon vorstellen.\n\n#### Okay, zur\u00fcck zur autonomen Szene: Welche Rolle hatte die autonome Szene f\u00fcr dich?\n\nH\u00e4tte ich diese Phase nicht gehabt, w\u00e4re ich ein komplett anderer Mensch geworden. Vieles bei meiner Pers\u00f6nlichkeitsentwicklung und der Abl\u00f6sung vom Elternhaus und der Abl\u00f6sung von Autorit\u00e4ts\u00e4ngsten oder \u00c4ngsten vor individualistischen Entscheidungen, prinzipiell eigentlich mein gesamtes Weltbild hat dort sozusagen seine Wiege gehabt.\n\n#### Denkst du, dass die autonome Szene auch eine Jugendkultur ist?\n\nIm weitesten Sinne w\u00fcrde ich das unterschreiben insofern, als ich denke, dass man in der Jugend mit der autonomen Szene anf\u00e4ngt. Ich halte es f\u00fcr eine Jugendkultur, weil es viele Aspekte birgt, die durchaus der Leichtigkeit einer Jugend angemessen sind und weniger im Erwachsenenalter vorhanden sind: die Risikobereitschaft, auch contra Gesetzgebung zu agieren, m\u00f6glicherweise auch das gr\u00f6\u00dfere Bed\u00fcrfnis nach einer Utopie, dass man sich gr\u00f6\u00dfere Chancen ausrechnet, was ver\u00e4ndern zu k\u00f6nnen...\n\n#### Gr\u00f6\u00dfere Naivit\u00e4t?\n\nJa, im guten Sinne naiv, dass man st\u00e4rker noch daran glaubt, dass man aktiv daran beteiligt sein kann, etwas ganz schnell ganz massiv zu ver\u00e4ndern. Das ist das Privileg der Jugend, nat\u00fcrlich nicht in aller Ausschlie\u00dflichkeit.\n\n#### Wie stehst du zu militanten Aktionen?\n\nIrgendwie w\u00e4re ich halt schon gerne auch Terrorist... [lacht] muss ich mal ehrlich so sagen, je \u00e4lter ich werde... Rein theoretisch bin ich schon ganz sch\u00f6n radikal, glaub ich. Also, dass ich das richtig gut finde, dass man f\u00fcr Gerechtigkeit t\u00f6ten w\u00fcrde. Am 11. September hab ich mich total gefreut, das wei\u00df ich noch genau, hier in dem Zimmer mit meinem kleinen Fernseher, morgens war das ja irgendwie, und da hab ich schon brav Fernsehen geguckt, einen Film mit Louis de Fun\u00e8s, und nach 'ner halben Stunde kommt auf einmal das Testbild. Und dann kamen Nachrichten mit World-Trade-Center und \u00fcberhaupt, und dann haben sie das erste Bild gezeigt, noch bevor der zweite Turm dran war, und es war noch nicht sicher, ist das jetzt ein Anschlag oder nur ein ganz schlimmer Unfall, aber ich dachte: Was es auch immer ist, es ist ein Segen Gottes! [Lacht.] Und ich dachte so: Jaaa! [Lacht.] Ich hab hier echt gejubelt. Und dann kam der zweite Anschlag und ich dachte: Super, super! Mir war nicht klar, dass das aus der arabischen Welt kommt, aber ich dachte, was ist das f\u00fcr eine gute Idee! Ich frag mich mein Leben lang, wie kann ich den Kapitalismus zerst\u00f6ren, und so einfach kann man denen echt mal Angst machen. Fand ich ganz toll. Nat\u00fcrlich hat's mir dann leid getan, wie die Menschen durch die Luft geflogen sind, das war schon schlimm, aber ich dachte mir halt: Anders geht's ja nicht, es geht ja nicht anders! Wie willst du denn Amerika sonst sagen, dass sie nicht der Nabel der Welt sind? Auf jeden Fall war an diesem Abend dann Gruftie-Disko, und da war ich verabredet mit ein paar Leuten und bin dann hin, total gute Laune, so: Leute, habt ihr auch schon mitgekriegt, ist das nicht geil! Ich hab gedacht, die steinigen mich. Ein paar Leute waren den Tr\u00e4nen nahe, weil sie so ersch\u00fcttert waren. Und auch die sogenannten Anti-Amerikanisten sa\u00dfen da alle und waren total traurig: \"Wie viele Menschen und dieses Leid...\", und ich meinte: Wie viele Kinder sterben jeden Tag in Afrika aufgrund der imperialistischen Weltanschauung! Und dann gab's so Grundsatzdiskussionen. Ich bin echt 'n Gandhi-Fanatiker, gewaltloser Widerstand ist das Ehrenwerteste, was es auf dieser Welt gibt, aber ich muss sagen, die Zeiten sind halt... vorbei. Die sind vorbei! Du kannst Menschenketten bilden und Kerzen anz\u00fcnden und zum Boykott gegen Nestl\u00e9 aufrufen, aber es wird sich an der Struktur dadurch nichts \u00e4ndern, es wird sich nichts \u00e4ndern! Fr\u00fcher konnte man noch losgehen und 'nen Bankraub und Post\u00fcberfall machen und einen Politiker umzubringen drohen oder dessen Kind entf\u00fchren, aber das w\u00fcrde ja heute gar nicht mehr gehen, weil die Politik ist entpersonalisierter und das Kapital ist in Aktien virtuell auf der Welt verteilt, also kannst du das Geld nicht mehr nehmen und an die Armen verteilen.\n\nDie Menschen trifft nur, was ihr pers\u00f6nliches Leben betrifft und ihre pers\u00f6nliche Sicherheit, wenn jeder Angst hat. Ist halt leider nach hinten losgegangen, jetzt haben sie alle Angst vor Islamisten. Aber da bin ich schon arrogant; die Leute, die wirklich islamophob sind, sind wirklich so die ungebildeten Menschen... Wie dumm sind denn die Menschen, dass sie heute noch glauben, dass Religion an irgendetwas eine Schuld tr\u00e4gt? Das ist pure Dummheit. Wenn man das im Mittelalter geglaubt hat, wenn man noch nicht die Bibel lesen konnte, okay, aber wenn man heute noch glaubt, dass der Koran die Aufforderung f\u00fcr den allgegenw\u00e4rtigen Dschihad ist, dann hat man den Koran nicht gelesen oder nicht verstanden. Und mit so Leuten unterhalte ich mich gar nicht dar\u00fcber. Das sind so Leute, die den Spiegel lesen und da steht drin, dass es deswegen Attent\u00e4ter gibt, weil die alle vierzig Jungfrauen haben wollen, wenn sie im Paradies sind. Das steht im Spiegel und dann muss es ja auch so sein. Und das sind die Leute, die dann sagen, neulich im Urlaub, da war so 'n Islamist, so 'n voll krasser mit schwarzer Kutte und Bart und ich hatte richtig Angst neben dem. Seit wann gibt es denn 'nen Terroranschlag von 'nem b\u00e4rtigen Gelehrten? Das ist genau so wie der sekund\u00e4re Antisemitismus einfach nur Geschw\u00e4tz von Leuten, die sich noch nie mit der Substanz auseinandergesetzt haben.\n\n#### Also du h\u00e4ltst terroristische Anschl\u00e4ge f\u00fcr sinnvoll oder f\u00fcr gerechtfertigt...\n\nF\u00fcr notwendig leider...\n\n#### Okay, in dem Sinne beinhaltet das ja schon die Antwort auf die Frage, welche Einstellung du zu militanten Aktionen hast...\n\nIch bin eigentlich von Hause aus gegen Gewalt und ich mag Gewalt nicht, wenn sie keine Berechtigung hat. Gewalt ist f\u00fcr mich dann verabscheuungsw\u00fcrdig, wenn jemand, der mehr Macht und mehr Mittel hat, auf jemanden losgeht, der mittellos dasteht, das ist f\u00fcr mich Gewalt. Aber wenn ein System so \u00fcberm\u00e4chtig ist wie die NATO-Gesellschaft, dann gibt es kein friedliches Veto-Mittel dagegen, das die tats\u00e4chlichen Ungerechtigkeiten auf dieser Welt jemals \u00e4ndern k\u00f6nnte. Und Fakt ist, dass es den meisten Menschen auf dieser Welt und damit im Kapitalismus schlecht geht. Dass jetzt Gewalt ausgerechnet aus der arabischen Ecke kommt, das spielt f\u00fcr mich nur eine zweitrangige Rolle; die Frage ist, gibt es \u00fcberhaupt irgendeine M\u00f6glichkeit f\u00fcr jemand anderen, dagegenzuhalten. Selbst wenn alle buddhistischen M\u00f6nche dieser Welt morgen einen Brief schreiben w\u00fcrden, man sieht doch beim Dalai Lama, wo es hinf\u00fchrt, der hat eine gro\u00dfe Fangemeinde, der hat gro\u00dfe Anerkennung \u00fcberall in der Welt, aber es \u00e4ndert nichts f\u00fcr sein Land und sein Volk, das wird nach wie vor getreten und die ganze Kultur zerst\u00f6rt. Als Pazifist ist man heutzutage nicht mehr realit\u00e4tskonform. Bei mir pers\u00f6nlich hat sich das nicht durch den 11. September erst entwickelt, sondern tats\u00e4chlich schon in dem Jugoslawien-Krieg, dass ich meinen Pazifismus und meine Peace-Kette im wahrsten Sinne abgelegt habe; die habe ich zwar noch, ich sch\u00e4tze diese Kette sehr und ich bin sehr f\u00fcr Peace und Woodstock f\u00fcr alle, aber leider ist die Gesellschaft nicht so, und wenn eine \u00dcbermacht losgeht und Frauen vergewaltigt und Kinder ermordet und der ganze Rest der Welt schaut zu, wie es heutzutage in ganz vielen L\u00e4ndern der Welt so ist, dann finde ich es notwendig, dass es Menschen gibt, die sagen: Okay, dann besorge ich mir jetzt auch 'ne Waffe und bringe den Aggressor um. Nat\u00fcrlich hab ich auch keine Lust, dass in Berlin Bomben hochgehen wie in London oder so, aber andererseits ist das vielleicht nicht die einzige M\u00f6glichkeit, aber die tempor\u00e4r aufmerksamkeitsst\u00e4rkste, die Menschen hier aufzur\u00fctteln und zu sagen: Wie ihr lebt, das ist einfach nicht richtig und man muss daran was \u00e4ndern.\n\nDas Interview f\u00fchrte Anna Ziegler.\n\n## \"Kopftuch oder Sturmmaske?\"\n\nJan, 34 Jahre, wohnt in Konstanz.\n\nDer Kontakt mit Jan wird mir von einem ehemaligen Uni-Kollegen vermittelt. Jan ist nach eigener Angabe 34 Jahre alt, sieht aber j\u00fcnger aus. Er tr\u00e4gt schlichte schwarze Kleidung und einen Haarschnitt, den man im weitesten Sinn als Nerd-Frisur bezeichnen kann. Er wohnt in einem Wohnblock in einem Konstanzer Vorort in einer Ein-Zimmer-Wohnung. Wir unterhalten uns in seiner rot angestrichenen K\u00fcche, da sein Zimmer zu unaufger\u00e4umt ist, wie er findet. Auf dem Fensterbrett steht eine Zimmerpflanze, die zu seiner gro\u00dfen Freude gerade eine rote Bl\u00fcte bekommen hat. Wie die Pflanze hei\u00dft, wei\u00df er aber nicht. Die Wohnungseinrichtung besteht ansonsten zum Gro\u00dfteil aus Kisten. Da die K\u00fcche mit dem Luxus eines K\u00fchlschranks und einer Geschirrsp\u00fclmaschine ausgestattet ist, bietet er mir wahlweise Bier oder regionalen Wein aus gl\u00e4nzend gesp\u00fclten Gl\u00e4sern an, die er mir stolz pr\u00e4sentiert.\n\n#### Trinkt der Konstanzer Autonome Wein?\n\nNein, Bier! Aber nat\u00fcrlich kein Tannenz\u00e4pfle.\n\n#### Ruppaner?\n\nNein, man hat schon Geschmack. Das Freiburger, das ist okay.\n\n#### Wo kommst du denn urspr\u00fcnglich her?\n\nAus Balingen, ich wei\u00df nicht, ob du das kennst. Zollern-Albkreis, 40.000 Einwohner. Das ist so ein Ausl\u00e4ufer von der schw\u00e4bischen Alb. Ein kleines Industriest\u00e4dtchen, Feinmechanik vor allem. Es ist ja hier \u00fcberall so, dass die St\u00e4dte relativ industriell sind. Schaffe-schaffe-H\u00e4usle-baue, wie sich des geh\u00f6rt hier.\n\n#### Und wie lange hast du da gelebt?\n\nIch bin da gleich nach dem Abitur weg.\n\n#### Wie war denn die Schulzeit?\n\nAch, lustig. Ich hatte da ja keine Schwierigkeiten. Schulzeit war eigentlich nett.\n\n#### Hast du dich mit deinen Lehrern gut verstanden?\n\nHervorragend! Mit den meisten schon, ja.\n\n#### Und wie war deine Familie?\n\nAch Gott, das ist kompliziert, Familienchaos, wie das halt so ist, also die meiste Zeit war ich mit meiner Mutter allein.\n\n#### Geschwister?\n\nKeine.\n\n#### Und wieso mit deiner Mutter allein?\n\nNa ja, die haben sich irgendwann getrennt und da kam auch nichts Neues mehr nach.\n\n#### War das schlimm mit der Trennung?\n\nN\u00f6, eigentlich nicht. Ich hatte dann mehr so die volle Aufmerksamkeit, so schlecht ist diese Situation f\u00fcr ein Kind nicht unbedingt. [Grinst.] Also in dem Sinn bin ich schon ein Mama-Kind.\n\n#### Wie alt warst du denn bei der Trennung?\n\nElf.\n\n#### Und was hast du so in deiner Freizeit gemacht? Gab es da irgendwas, Jugendzentren, Jugendorganisationenen?\n\nJa, ein Jugendhaus gab es da, die \"Insel\", war schon wichtig fr\u00fcher. Da hat man halt die Leute getroffen \u2013 so kam ich auch in die Szene rein. Das Jugendhaus selber ist so ein klassisches st\u00e4dtisches Ding gewesen mit Sozialarbeitern, ich glaube, die hatten zwei Hauptamtliche plus einen Zivi [Zivildienstleistenden]. Die Kundschaft, die kriminell wurde, hat ihre Sozialstunden da abgeleistet, das war so ein bisschen selbstverst\u00e4rkend. Da gab es h\u00e4ufig mal so kleinere Delikte, die dann in zwei Wochen Jugendhaus-Putzen m\u00fcndeten. So hat sich das dann im Prinzip auch gereinigt. [Lacht.] Ansonsten hatten die die \u00fcblichen Angebote, so Werkst\u00e4tten, den klassischen T\u00f6pferkurs und HipHop f\u00fcr Anf\u00e4nger gab es dann sp\u00e4ter auch. Aber im Prinzip hat sich alles eigentlich im Caf\u00e9 abgespielt. Und dann gab es Prober\u00e4ume, wo auch irgendwelche Bands geprobt haben. Da hing man halt rum.\n\n#### Was f\u00fcr Bands waren das?\n\nSchrammel-Punk. Ich wei\u00df nicht, ob jemals was aus Balingen ber\u00fchmt wurde... hat dann wahrscheinlich im Insel-Keller angefangen. Aber ich vermute, so richtig \u00fcber die Zollernalbkreis-Prominenz hat es da niemand geschafft.\n\n#### Und dann war die Schule fertig und du bist weg?\n\nJa, erst mal musste ich ja Zivildienst machen. Ich habe mir was in Berlin in einer Behinderteneinrichtung gesucht, weil ich halt in eine gro\u00dfe Stadt wollte. Und dann bin ich nach Hamburg weiter.\n\n#### Hast du eine Ausbildung gemacht nach dem Zivildienst?\n\nIch habe versucht, ein Studium anzufangen, zweimal umgesattelt und es dann doch irgendwann abgebrochen.\n\n#### Und was machst du jetzt?\n\nIch musste ja mal irgendwie Geld verdienen und habe halt angefangen, das, was ich sowieso kann, zu Geld zu machen und mit Webseiten programmieren und anderem Computerzeug angefangen, mein Geld zu verdienen, und das mache ich heute noch. Und inzwischen auch so, dass ich davon leben kann.\n\n#### Hast du schon Kinder?\n\nNein, und lege auch keinen gro\u00dfen Wert darauf. Im Moment lebe ich auch allein \u2013 bin damals ja hier wieder hergekommen wegen Frauengeschichten, was dann nat\u00fcrlich wieder nicht klappte, und bin doch hier gestrandet. Obwohl ich eigentlich nie selber auf die Idee k\u00e4me, nach Konstanz zu ziehen, aber wenn man mal da ist, dann bleibt man doch kleben.\n\n#### Und wie lange bist du jetzt wieder in Konstanz?\n\nAcht Jahre sind das jetzt. Man merkt halt, dass man dann doch tr\u00e4ge wird. Und hier unten ist ja schon sch\u00f6n zu leben, au\u00dferdem ging mir dieses Gro\u00dfst\u00e4dtische, gerade die Szene, extrem auf den Keks.\n\n#### Was hat dich da gest\u00f6rt?\n\nDie Selbstdarstellerdichte ist da extrem hoch. Die innerszenischen Grabenk\u00e4mpfe sind deutlich extremer als das hier in dem kleineren Kontext ist, weil man hier auch keine so dichte Szene in dem Sinne hat, das sind ja eher so vernetzte Einzelpersonen. Man hat zwar schon kleine Gruppen, in denen man auch politisch arbeitet, die sind aber zum Teil auch wirklich so regional gestreut. Das hei\u00dft, man muss sich da schon eher bewusst treffen, als dass man sich jeden Abend in irgendeinem Keller \u00fcber den Weg l\u00e4uft. Wobei das nicht f\u00fcr alle gilt, aber f\u00fcr die eher politisch Arbeitenden.\n\n#### Dann bist du nicht in einer konkreten Gruppe, sondern du triffst dich mit einzelnen Leuten?\n\nDas ist zum Teil schon strukturiert. Aber das h\u00e4ngt auch sehr themenspezifisch davon ab, was man macht.\n\n#### Und welche Themen sind aktuell?\n\nDer Dauerbrenner ist nat\u00fcrlich Antifa, die ist aber im Konstanzer Raum nicht so stark organisiert, da ist man relativ verbunden mit den umgebenden St\u00e4dten, Radolfzell, Singen und so. Man lebt ja, was das angeht, hier unten relativ idyllisch. Das Zweite ist die Freiburg-Connection, das klassische Freiraum-Thema. Freiburg hat ja eine lange Besetzer-Tradition und die haben im Moment mit dieser Wagenburg ein Thema, wo auch viele Demos sind und wo man mal mit kleinen Gr\u00fcppchen nach Freiburg f\u00e4hrt und da die Leute unterst\u00fctzt.\n\n#### Inwieweit hast du irgendwann mal Jugendkulturen angeh\u00f6rt oder geh\u00f6rst du noch an?\n\nF\u00fcr Jugendkultur f\u00fchle ich mich jetzt zu alt. Es gab immer so eine nat\u00fcrliche N\u00e4he zum Punk, aber ich war nie jemand, der mit gr\u00fcnen Haaren durch die Gegend laufen muss... Ich lauf jetzt auch nicht mit Anzug durch die Stra\u00dfe, aber diese Dresscodes waren schon immer nicht ganz so meins.\n\n#### Und dein Musikgeschmack?\n\nIst schon auch beeinflusst aus dem Punk, und zu meiner Zeit gab es da immer noch diese gro\u00dfe Indie-Rock-Welle Anfang der 90er und Grunge und was dazugeh\u00f6rt.\n\n#### Gibt es \u00dcberschneidungen oder Zusammenh\u00e4nge, dass dich die Musik in die Szene gebracht hat oder dass du durch die Szene auf eine bestimmte Musik gekommen bist?\n\nNee. Ich glaube nicht, dass Musik tats\u00e4chlich Leute zu irgendwas bringt. Nat\u00fcrlich hat man immer Slime geh\u00f6rt, und das war nat\u00fcrlich die Parolen-Dreschmaschine, wo man sehr viel gelernt hat f\u00fcr den t\u00e4glichen Parolen-Haushalt, aber das best\u00e4rkt ja meistens eher die Position, die man hat. Aber eigentlich hat man das auch nur phasenweise geh\u00f6rt und dann doch die unpolitische Musik, das war dann sp\u00e4ter wichtiger. Diese Codes, die man nat\u00fcrlich aufschnappt, die haben irgendwie so eine sich selbstbest\u00e4rkende Wirkung, wenn man st\u00e4ndig um die Ohren gehauen kriegt, wie schlimm der Kapitalismus ist. Aber dass der schlimm ist, das hat man ja schon immer gewusst.\n\n#### Also Musik st\u00e4rkt Positionen, die man schon hat? Die muss man doch auch irgendwo her haben? Oder bist du allein in deinem Kinderzimmer gesessen und irgendwann fiel dir ein, Kapitalismus ist eigentlich schlimm? Wie bist du da drauf gekommen?\n\nDas sind Fragen! Ich wei\u00df nicht, wann wird man denn politisch, mit f\u00fcnfzehn, sechzehn oder so, wenn man sich daf\u00fcr interessiert, und das lief dann eher \u00fcber Zeitungen, manchmal hat man auch Flugbl\u00e4tter irgendwo aufgeschnappt. Gerade im Jugendhaus liegt dann doch mal gern was rum, man liest das Zeug und kriegt so schon was mit, auch wenn man es meistens nicht versteht \u2013 es gibt ja nichts Unverst\u00e4ndlicheres als linke Flugbl\u00e4tter manchmal. [Lacht.] Man liest das, hat gar keine Ahnung, von was die reden, aber das beeindruckt einen nat\u00fcrlich noch viel mehr. Und man liest ja eher gerade so Zeitschriften, Internet gab es damals ja nicht, im Jugendhaus lag aber schon die Konkret aus und so was, wo man sich zumindest linke Basics aneignete, die sich dann nat\u00fcrlich verfeinerten, und die Berlin-Fahrer brachten dann sp\u00e4ter die hei\u00dfen Bl\u00e4tter mit. Das war damals auch die hei\u00dfe Zeit, wo heimlich die radikal unter der Schulbank weitergereicht wurde. Wir hatten damals einen \u00e4lteren... wir waren f\u00fcnfzehn, sechzehn, und der war siebzehn, achtzehn..., der immer nach N\u00fcrnberg, diese alte KOMM-Connection, fuhr und dann sp\u00e4ter vor allem nach Berlin. Und dann gab es auch die Entdeckung des wilden Ostens, damals, wo gerade die hei\u00dfe Zeit der Mainzer Stra\u00dfe [Anfang der 90er Jahre besetzte H\u00e4userzeile in Ostberlin] und \u00c4hnliches war, und der uns da immer viel besorgt hat.\n\n#### Das KOMM in N\u00fcrnberg war auch ein Jugendzentrum?\n\nDas war eigentlich ein Kulturzentrum, das aus einer Besetzung am Bahnhof hervorging und dann sp\u00e4ter quasi von der Stadt eingemeindet wurde. Das war damals auch die Hoch-Zeit, Anfang der 90er, der AABO, die ja heute kaum mehr existiert, die Antifaschistische Aktion\/Bundesweite Organisation, das kennt jeder aus dem Jahrgang, und die war auch nicht unumstritten. Damals war ja G\u00f6ttingen die Hauptstadt des autonomen Deutschlands, was heute auch wieder niemand mehr glauben kann. Ja, die G\u00f6ttinger waren damals schon die harte ML-[Marxismus-Leninismus-]Fraktion, die auch schon versucht haben, so eine kleine leninistische Kader-Organisation aufzubauen. Von G\u00f6ttingen aus wurden bundesweit Zellen gegr\u00fcndet in der ganzen Republik. In G\u00f6ttingen war der harte Kern und es gab dann \u00fcberall Basis-Organisationen. N\u00fcrnberg war auch sehr hart drauf. Und so '94, '95 ist dann alles in sich zusammengebrochen, die Gr\u00fcnde wei\u00df ich gar nicht mehr so genau. Aber man mochte diese arrogante G\u00f6ttinger Fraktion ja nicht, die einem die Welt erkl\u00e4ren wollte. Und das KOMM ging fast zur gleichen Zeit langsam zu Ende... Also es existiert heute immer noch, aber es ist nicht mehr wichtig.\n\n#### Noch mal zu deiner Person zur\u00fcck: Wie sieht es aus mit Hobbys, Freizeitgestaltung in irgendeiner Form, die autonomenkompatibel ist?\n\nWas sind autonomenkompatible Hobbys? Briefmarkensammeln mit schwarzen Bildern drauf? [Lacht.] Hobbys ist so ein Unwort... Ich glaube, Leute mit Leidenschaften haben keine Hobbys. Also nat\u00fcrlich tut man was in seiner Freizeit... Wir streben ja an, die Zeit zu befreien!\n\n#### Ich wollte wissen, ob du auf Konzerte gehst oder lieber Sport machst oder ob du dich in deiner ganzen Freizeit mit autonomen Sachen besch\u00e4ftigst?\n\nAuf Konzerte gehe ich immer noch gern, wobei das ja hier nicht so richtig viel ist. Von Konstanz aus f\u00e4hrt man gerne in die Schweiz r\u00fcber zu den Z\u00fcrichern, die \"Rote Fabrik\", und da gibt es auch noch so ein paar Rest-H\u00e4user in der Bahnhofsgegend, wo es dann auch diverse Keller gibt, und es gibt auch auf dem Weg nach Z\u00fcrich in Winterthur und Frauenfeld nette Jugendclubs, wo auch gute Konzerte sind.\n\n#### Wie ist es mit B\u00fcchern und Informationen \u00fcbers Internet?\n\nAutonome B\u00fccher sind immer ziemlich ungenie\u00dfbar, B\u00fccher nehme ich nicht so richtig wahr. Ich denke aber schon, dass der Zeitschriftenmarkt ganz wichtig ist, auch was die Selbstverst\u00e4ndigung ist. Nat\u00fcrlich gibt es das schon alles im Netz. Da schau ich mir schon die ganze Bandbreite an.\n\n#### Was h\u00e4ltst du von Indymedia?\n\nIch schau Indymedia nicht mehr so regelm\u00e4\u00dfig an, das ist inzwischen ziemlich uninformativ geworden. Es gab mal eine Zeit am Anfang, wo es auch von der Moderation her und die Beitr\u00e4ge qualitativ h\u00f6herwertiger waren. Inzwischen ist das mehr so ein Benachrichtigungsmedium: Oi!-Band so und so aus der Grauzone spielt heute in Potsdam, kommt alle und schreit laut. Finde ich nicht mehr so zielf\u00fchrend, schon gar nicht, was den Informationsgehalt betrifft. Zur Vernetzung ist es schon noch wichtig, aber auch nicht mehr so stark. Und wir haben ja hier unten auch noch unser eigenes Indymedia, das linksunten.indymedia.org. Aber im Regelfall erscheint alles, was bei Links-Unten erscheint, auch beim gro\u00dfen Indymedia. Nur ist es spezifisch hier f\u00fcr die Region Baden W\u00fcrttemberg, ein St\u00fcck Bayern, ein bisschen auch noch die Schweizer mit drin. Nachrichten aus Z\u00fcrich kommen noch ab und zu hier an, aber auch das funktioniert nicht so richtig mit der internationalen Solidarit\u00e4t. [Lacht.] Aber man hat dann wenigstens so eine Regionalversion, das sind zu achtzig Prozent Freiburg-Nachrichten. Die Infrastruktur von Freiburg ist schon traditionell gut, was die linke Szene angeht, mit Radio Dreyeckland und was es da alles gibt. Die hatten ja schon Piratenradios, wo die Gro\u00dfst\u00e4dte noch davon getr\u00e4umt haben.\n\n#### Und sind deine Freundschaften und Beziehungen auch szeneintern oder von au\u00dferhalb?\n\nBeides.\n\n#### Und gibt es Leute, die dich ablehnen?\n\nIch hoffe! [Lacht.] Sonst w\u00e4re der ganze Schei\u00df hier umsonst! Soll ich Namen nennen?\n\n#### N\u00f6, aber vielleicht wirst du ja von Nazis verhauen oder b\u00fcrgerliche Menschen finden dich ganz schrecklich, wenn du l\u00e4nger als drei S\u00e4tze redest.\n\nN\u00f6, ich kann mit b\u00fcrgerlichen Menschen ganz hervorragend, au\u00dferdem muss ich bei denen mein Geld verdienen. Insofern ist man da ja auch zu Kompromissen f\u00e4hig und, ich muss sagen, auch bereit. So was wie eine organisierte Anti-Antifa gibt es hier ja nicht. Das \"autonome Leben\" ist hier also schon ein St\u00fcck weit gem\u00fctlicher. Was ich auch sch\u00e4tze, weil ich ja den anderen Pol auch kenne, dass man sich in autonomen Kreisen gegenseitig verachtet und die K\u00f6pfe einschl\u00e4gt. Es geh\u00f6rt nat\u00fcrlich ein St\u00fcck weit auch dazu, dassman diese Kompromisslosigkeit hat in gewissen Dingen, ansonsten k\u00f6nnte man ja gleich zu den Gr\u00fcnen gehen. Aber klar, wie es sich dann zuspitzt und personalisiert wird, du kennst ja die klassischen Konflikte: Antideutsche gegen Antiimps, die Frauen gegen die coolen Macker, die Anti-Speziesisten gegen die Fleischfresser...\n\n#### Und wo stehst du da?\n\nAuf der richtigen Seite! [Lacht.] Ich genie\u00dfe, dass das hier unten zum gro\u00dfen Teil ein bisschen anti-ideologisierter ist, dass man wirklich an politischen Fragen sich lang hangelt und nicht an irgendwelchen zu eng verschnittenen Weltbildern, die zum Teil wirklich nicht mehr diskursf\u00e4hig sind. Ich kann nat\u00fcrlich schon viel von der antideutschen Kritik an der klassischen Linken gut nachvollziehen. Aber ob man das dann so zuspitzen muss, dass man gleich mit Amerika-F\u00e4hnchen durch die Gegend laufen muss...\n\n#### Und andere gr\u00f6\u00dfere Bewegungen, Gr\u00fcne, Anti-Atomkraft, Stuttgart 21, wie stehst du dazu?\n\nDas hat ja mit autonomer Politik wohl nichts mehr zu tun. Ich bin in Bahnhofsfragen ja eher Stalinist und sage: Alles, was die Industrie voranbringt, bringt die arbeitende Klasse voran! Kommunismus hei\u00dft: Sozialismus plus Industrialisierung! Oder so \u00e4hnlich. Insofern verstehe ich die Stuttgarter nicht, aber das sind auch keine Genossen, das sind ja eher gr\u00fcne Hausfrauen, denen langweilig ist, weil die Gatten zu viel verdienen. Anti-AKW ist nat\u00fcrlich eine ganz andere Baustelle, da sind wir auch dabei, wenn irgendwas ist. Das ist hier in Konstanz nat\u00fcrlich wieder eher die alberne Gr\u00fcnen-Fraktion, die da auf die Stra\u00dfe geht, aber man ist da schon dabei. Man f\u00e4hrt dann auch zu gr\u00f6\u00dferen Anl\u00e4ssen, gerade nach Stuttgart, da gab es die Umzingelung des Parlaments vor einem halben Jahr, da waren wir dann auch da. Sind allerdings mit der Bahn gefahren, insofern waren wir da nicht solidarisch mit der Bahn-Verweigerer-Fraktion. [Lacht.]\n\n#### Wie stehst du denn zum Staat und zur Polizei?\n\nIch habe da eine eher entspannte Position. Die machen ihren Job und ich meinen dagegen. Und jeder macht ja seine Sache ganz gut. Sie wahrscheinlich erfolgreicher als wir. Nat\u00fcrlich hat man da Konflikte und haut dann auch rein, aber manchmal sind die auch sehr nett. Und was \"der Staat\" ist, wei\u00df ich auch nicht so genau.\n\n#### Aber geht es denn eher darum, die Gesellschaft und den ganzen Staat zu ver\u00e4ndern, oder darum, Freir\u00e4ume zu schaffen, wo man sich nach seinen eigenen Regeln bewegen kann?\n\nDas ist eine Frage von Realismus und dem, was man jetzt erreichen kann und was nicht. Ich denke schon, dass am Anfang nat\u00fcrlich das gro\u00dfe Nein! steht und die gro\u00dfe Kritik an \u2013 und jetzt kommen wir wieder auf die gro\u00dfen Baustellen \u2013 dem Staat. Aber das ist ja nicht der Kern autonomer Politik, da hatten wir immer diese drei gro\u00dfen Antis: Anti-Kapital, Anti-Faschismus und Anti-Sexismus. Da kann man nat\u00fcrlich noch ein paar kleinere As drumrum setzen, aber da kommt der Staat erst mal nicht vor. Die Bullen ja, die symbolisieren nat\u00fcrlich vieles von dem, was ich gerade meinte, von diesen klassischen Ismen, die man bek\u00e4mpft, und sind nat\u00fcrlich der erste Konfrontationspol, dem man gegen\u00fcber tritt. Wenn man auf der Demo erst mal mit einer Taschendurchsuchung begr\u00fc\u00dft wird, ist das nat\u00fcrlich da. Aber klar, es ist auch von unserer Seite immer eine bewusst gew\u00e4hlte Konfrontation. Uns haben fr\u00fcher auch immer diese wilden Stra\u00dfenschlacht-Geschichten aus Berlin fasziniert und eigentlich dazu gebracht, uns mit dem Zeug n\u00e4her zu besch\u00e4ftigen. Insofern ist auch Militanz ein sinnvolles politisches Mittel, um Aufmerksamkeit zu kriegen. Die Anti-AKW-Bewegung w\u00e4re mit diesen gr\u00fcnen Langweilern nie dahin gekommen, wo sie heute ist! Wenn sie nicht massiv die Kosten in die H\u00f6he getrieben h\u00e4tte durch Konfrontation. Das ist auch die Strategie, die die Stuttgarter Nicht-Genossen gerade verfolgen, die Kosten durch Aktionen so in die H\u00f6he zu treiben, dass es auch f\u00fcr einen vern\u00fcnftig Denkenden der anderen Seite unrentabel wird, das zu machen. Insofern halte ich Militanz in einem gewissen Bereich einfach f\u00fcr ein wichtiges Mittel, um mal Positionen r\u00fcberzukriegen und zu mobilisieren und ja, auch ein bisschen kritische Masse zu erzeugen.\n\n#### Differenzierst du zwischen Gewalt gegen Sachen und Gewalt gegen Menschen?\n\nJa, musst du schon machen, das ist klar. Es ist aber die Frage, ob man jede Gewalt gegen Menschen ablehnen muss, das glaube ich nicht. Klar, gegen Sachen hat man weniger Bedenken. Allerdings, wenn man jetzt diese ganzen brennenden Autos anguckt, ob das irgendeinen Effekt hat au\u00dfer, dass es anregt zu Versicherungsbetrug? Ich glaube, dass inzwischen \u00fcber die H\u00e4lfte dieser Autos von ihren Eigent\u00fcmern abgefackelt wird.\n\n#### Wo w\u00e4ren denn deine Grenzen? Gibt es Sachen, wo du sagst, da mache ich nicht mit?\n\nIch muss vor allem hinter der politischen Haltung stehen, und da gibt es nat\u00fcrlich unsinnige Anl\u00e4sse, das ist ja ein Hauptgegenstand der ganzen Konflikte in der Szene, gegen welchen Krieg man sein darf und welcher Krieg angewandter Antifaschismus ist. Da habe ich bei manchen Aktionen schon gro\u00dfe Bauchschmerzen und werde mich nat\u00fcrlich nicht beteiligen. Genauso, wie mir manche Aktionsformen einfach fremd sind, zum Beispiel sinnlose Gewaltorgien gegen Personen, die nur in einer funktionalen Rolle sind. Wenn ein Arbeitsamtmitarbeiter erstochen wird, halte ich das weniger f\u00fcr einen politischen Akt. Es gibt gerade die gro\u00dfe Diskussion um einen, der eine Arbeitsamtmitarbeiterin abgestochen hat und jetzt im Hungerstreik ist, und dem gerade eine gro\u00dfe Solidarit\u00e4tsbewegung der Befreit-die-politischen-Gefangenen-Liga ankommt, weil der jetzt als politischer Held gegen Hartz IV, oder was auch immer er da gekriegt hat, antritt. Und da mache ich zum Beispiel nicht mit. Das f\u00fchrt aber auch \u2013 da habe ich bei Indymedia mal wieder mit Leidenschaft die Debatten gelesen \u2013 in der Szene zu herben Debatten, wo ich auch sage, das ist nicht mehr politisch motivierte Gewalt und da muss man auch nicht allzu solidarisch auftreten, weil man muss ja bei Solidarit\u00e4t auch immer im Auge halten, wenn wir es doch noch schaffen, uns selbstorganisiert zu bewegen, dann muss das auch mit Leuten sein, die man ertr\u00e4gt. Auch wenn man das Langzeitziel der befreiten Gemeinschaften nicht st\u00e4ndig vor Augen hat, sollte man doch das konkrete Handeln darauf ausrichten, dass das dann auch lebensf\u00e4hige Zusammenh\u00e4nge sind.\n\n#### Wie siehst du denn deine Zukunft? Wie realistisch sind Ver\u00e4nderungen?\n\nAlso ich bin jetzt nicht so dringend auf Ver\u00e4nderungen in meinem pers\u00f6nlichen Leben aus. Ich denke, dass ich da einen ganz guten Mittelweg fahre. Anders sieht es nat\u00fcrlich aus, wenn du Richtung Gesellschaft fragst. Da bin ich dann schon skeptischer, was gro\u00dfe Ver\u00e4nderungsperspektiven angeht. Man kann ganz gut Sachen verhindern, die v\u00f6llig fatal sind, aber ich glaube nicht mehr, dass wir noch auf eine in irgendeiner Form befreitere Gesellschaft zusteuern. Es sieht ja eher in vielen Dingen ganz anders aus. Ich sehe uns aber auch nicht in die absolute Katastrophe schlittern, sondern ich glaube, dass viele Bewegungen, gerade jetzt die Atomgeschichte in den letzten Jahren, doch einiges im Kleinen bewegt haben, ohne dass ich jetzt zu reformistisch bin, aber das gro\u00dfe Ganze werden wir mittelfristig nicht zerschlagen k\u00f6nnen.\n\n#### Ist die autonome Szene eigentlich eine M\u00e4nnerszene?\n\nIch glaube schon, dass sich die Szene den Vorwurf der M\u00e4nnerdominanz schon vom Ursprung her machen lassen muss. Hat vielleicht auch schon mit dieser Rekrutierung \u00fcber Gewalt zu tun. Wenn ich so an uns denke, uns haben immer diese wilden Stra\u00dfenschlacht-Geschichten fasziniert. Autonome Politik ist nat\u00fcrlich auch viel R\u00e4uber und Gendarm spielen auf st\u00e4dtischem Niveau. Und das ist, auch wenn das jetzt sexistisch klingen mag, halt ein Jungs-Ding. Auch das \u00e4u\u00dfere Erscheinungsbild von Autonomen, der ber\u00fchmte Schwarze Block und so, ist nat\u00fcrlich m\u00e4nnerdominiert, das sind auch M\u00e4nneraktionsformen. Ich bin nicht sicher, ob das f\u00fcr die Szene als Ganzes stimmt, die intern auch viele Frauen hat und die in anderen Aktionsformen als jetzt dem ber\u00fchmten Stra\u00dfenkampf auch da sind. Aber ich glaube schon, dass der Vorwurf ein St\u00fcck weit gelten muss. Gerade bei den J\u00fcngeren sind die M\u00e4dchen eher die, die, notgedrungen, das Ideologische mitbringen. Die bringen die anderen Themen rein, wo die Jungs nichts mit anfangen k\u00f6nnen. Zum Beispiel das Thema Frauen selbstverst\u00e4ndlich, das ist was, was Jungs gar nicht in den Kopf kommt, und dann auch diese Tierrechtsgeschichten und Veganismus. Das sind, wenn man ehrlich ist, schon eher die Frauenvarianten der autonomen Bewegung. Auch wenn die M\u00e4nner das im Regelfall nicht zugeben werden und dann sagen, nat\u00fcrlich, uns sind auch Tierrechte wichtig. Insofern haben die Frauenfraktionen da auch eine diskursive Macht bewiesen, indem sie solche Themen auf die Agenda gesetzt und durchgebracht haben, wo die Jungs nicht einmal auf die Idee gekommen w\u00e4ren, dass das Probleme sind.\n\n#### Ich hatte immer den Eindruck, diese richtigen Vegan-Leute, das waren diese Hardcore-Typen, und das war dann auch wieder M\u00e4nnersache. Frauen waren eher so vegetarisch, nett. Und dieses vegan, gar kein Fleisch und gar kein Spa\u00df und gar keine Drogen, das kam doch wieder mehr von den Jungs?\n\nJa, kann sein. Aber in meinem Umfeld waren das immer die Frauen, die damit ankamen. Du merkst schon, das ist ein Thema, das mir komplett fremd ist, das mit den Tieren und so. Aber das waren dann auch meine ersten Begegnungen, gerade in Berlin, und das waren halt einfach Frauen, die mir mit ihren bemitleidenswerten Hunden \u00fcber den Weg liefen, die irgendwelche Rasurstellen hatten, die aus Tierversuchsanlagen befreit worden waren. Ich hatte den Eindruck, die wurden auch von den Frauen selbst befreit. Und die waren nat\u00fcrlich strikt vegan. Das war so mein erstes Bild von diesem Teil der Szene, in dem ich mich selber aber auch h\u00f6chst ungern bewege. [Lacht.]\n\n#### Und wie stehst du zu Religion?\n\nIch halte das immer noch f\u00fcr Unsinn. Ich wei\u00df gar nicht, ob ich das so ein richtig schlimmes Thema finde. Also das war fr\u00fcher ein wichtiges Thema, auch in der Szene, diese Papst-Proteste und so was, das war mal mobilisierender als es heute ist. Mir war der polnische Papst immer sympathischer als dieser verkappte Nazi danach. Klar machte sich das immer an diesen \"Keine Kondome\"-Themen fest, aber der alte Karol Wojtyla... Wir wollten sogar eine Punk-Band namens Bloody Wojty\u0142as gr\u00fcnden, weil der zu dem Zeitpunkt in unserer Szene schon so ein kleines Feindbild war. Und damals fand man Religion schon b\u00f6se, und es gibt ja auch die ber\u00fchmte Hymne von Slime: \"Religion hat Millionen von Menschen get\u00f6tet\", und da wurde dann von den Kreuzz\u00fcgen bis zu sonst was alles der Kirche angelastet und die Kirche war der gr\u00f6\u00dfte Feind. Ich pers\u00f6nlich sehe die Kirchen im Moment nicht als eine Instanz weder des B\u00f6sen noch des Kapitalismus an. Die sind ja selber sehr auf dem absteigenden Ast. Ich glaube, ich k\u00e4mpfe lieber gegen andere Sachen. Aber das war schon mal ein Thema. Anfang der 90er Jahre hatten wir dann auf einmal das deutsche Wiedervereinigungs-Thema, das war dann die Geburtsstunde des Antideutschtums, auf einer entspannteren Basis, als das heute diskutiert wird, aber damit hat man sich nat\u00fcrlich identifiziert. Das kann ich heute auch noch. Das war diese klassische \"Nie-wieder-Deutschland\"-Geschichte, das war schon ein gro\u00dfes Ding, da ging dann auch die Papst-Kritik ein bisschen unter. Das waren doch eher die postmodernen 80er, wo man sich noch mit Religion besch\u00e4ftigen konnte, bevor die reale Welt wieder einbrach. Meines Wissens ist das in der autonomen Szene gerade kein wichtiges Thema, au\u00dfer, dass man sich dar\u00fcber streitet, wie man sich zum Islam verh\u00e4lt und ob das jetzt eine pseudofaschistische Ideologie ist, was nat\u00fcrlich die Bahamas-Fraktion verkauft und als eines der Hauptprobleme ansieht, oder ob man das doch eher entspannt sehen sollte und den Islam als eine Religion ansehen sollte, die genauso schei\u00dfe ist wie die anderen auch. Wobei ich eher der letzteren Fraktion angeh\u00f6re. Aber ansonsten ist Religion jenseits dieser Islamismus-Debatte kein wichtiges Thema. Ich kenne auch kaum Leute, die religi\u00f6s sind.\n\n#### Und wie stehst du zum Islamismus?\n\nAlso ich glaube schon, dass man die Bewegung von der Religion unterscheiden muss. Die Religion halte ich f\u00fcr genauso bescheuert wie die anderen auch, aber ich sehe nicht, dass das zwangsl\u00e4ufig in eine pseudofaschistische Ideologie f\u00fchrt, die der Islamismus in seiner harten Form darstellt und damit auch ein nat\u00fcrlicher Feind ist wie alle Bewegungen, die darauf abstellen, dass man \u00fcber religi\u00f6se Begr\u00fcndungen, \u00fcber irgendwelche Ziele im Jenseits Politik machen kann. Aber es ist schon in der Szene ein Riesen-Spalter, da ist dann immer die Frage: Kopftuch oder Sturmmaske?\n\nEr grinst und wir beenden das Interview.\n\n## \"St\u00e4ndig ein Plenum nach dem anderen und immer Stellung nehmen. Es h\u00e4tte nur noch gefehlt, dass wir Antr\u00e4ge f\u00fcrs Molliwerfen stellen sollten.\"\n\nGerd, 52 Jahre, lebt in einem Dorf in Sachsen-Anhalt.\n\nWilly, den ich zuerst interviewt habe, hat mir Gerd vorgestellt. Gerd ist ein Punk in den F\u00fcnfzigern. Er sieht aus, wie ein Punk aussehen sollte, zerrissene Hose und durchl\u00f6chertes T-Shirt. Wenn er lacht, legt sich sein gesamtes Gesicht in Falten, die Augen sind klar und er nimmt seine Umwelt aufmerksam wahr. Gerd ist ungef\u00e4hr 1,95 m gro\u00df und an den Beinen und Armen t\u00e4towiert. Wir trafen uns auf dem Bauwagenplatz in einer sehr gro\u00dfen Stadt, aber er war nur zu Besuch da. W\u00e4hrend des Interviews trank er sechs Bier.\n\n#### Hallo Gerd, wie geht's?\n\nGut und selbst?\n\n#### Prima! Fangen wir an? Wie bewertest du deine aktuelle Situation oder Zukunft?\n\nMeine aktuelle Situation ist die, dass ich hier auf dem Bauwagenplatz bin, weil ich zu einer Party eingeladen wurde. Meine Zukunft wird so aussehen, dass ich drei Tage durchsaufe und mich vielleicht noch mit Nazis pr\u00fcgeln werde und dann wieder nach Hause fahre.\n\n#### Gut, ich stelle die Frage anders; wie bewertest du dein momentanes Leben allgemein?\n\nIch hatte dich schon verstanden, wollte nur sehen, wie du reagierst. Tja, ich bin ganz zufrieden mit meinem Leben. Ich lebe im Osten in einem Dorf. Da brauchste gar nicht so \u00fcberrascht kieken. Wir haben uns ein Geh\u00f6ft gesucht und auch gleich eins gefunden, was wir mieten durften.\n\n#### Wen meinst du mit \"wir\"? Deine Frau und du?\n\n[Er lacht schallend.] Nee, meine Kumpels und ich. Wir sind so im Schnitt zehn oder zw\u00f6lf Punks und wir versuchen, uns selbst zu versorgen, nur das Bier k\u00f6nnen wir nicht selbst brauen, ansonsten sind wir fast unabh\u00e4ngig vom Supermarkt. Was wir nicht selbst anbauen k\u00f6nnen, bringen uns die Nachbarn, daf\u00fcr helfen wir beim Schlachten und auf'm Acker.\n\n#### Ist das dein Traum gewesen?\n\nN\u00f6, das hat sich so ergeben. Wir haben erst in einem besetzten Haus gewohnt, dann haben wir Mietvertr\u00e4ge bekommen und dann wurden die Mieten nach und nach erh\u00f6ht, so dass wir sie nicht mehr bezahlt bekommen haben. Dann kam noch der st\u00e4ndige Stress mit den Nazis, und irgendwie hatten wir die Schnauze voll, da haben wir uns umgesehen. Erst in der Stadt, und dann kam irgendwie der Gedanke mit dem Dorf. Wer da genau drauf kam, wei\u00df ich gar nicht mehr, ist aber auch egal.\n\n#### Du bist also nicht verheiratet?\n\nIch bin geschieden, zwei Mal.\n\n#### Lebst du momentan in einer festen Partnerschaft?\n\nNein.\n\n#### Welche Ausbildungen haben deine Eltern?\n\nWei\u00df ich nicht.\n\n#### Ist dir die Frage zu pers\u00f6nlich?\n\nFragen zu meinen Eltern beantworte ich grunds\u00e4tzlich nicht, das sollte dir reichen.\n\n#### Dann erz\u00e4hl mir doch bitte, was dich am meisten \u00e4rgert, was dich so richtig w\u00fctend macht?\n\nEindeutig Nazis!\n\n#### Magst du ein bisschen n\u00e4her darauf eingehen?\n\nDa gibt es gar nicht so viel zu erz\u00e4hlen, wir haben regelm\u00e4\u00dfig Schl\u00e4gereien mit denen. Als wir noch in der Stadt gewohnt haben, wollten die jeden Freitag unser Haus st\u00fcrmen, aber daraus wurde nie etwas. Immer wenn wir in unserem Club Konzerte hatten, kamen die bl\u00f6den Piepels angeschissen, als ob die um Arsch voll bettelten. Als wir auf's Land zogen, hatten wir anfangs Ruhe vor denen, aber irgendwann standen wieder welche vor der T\u00fcr. Es ist ja nicht so, dass ich sie mir suche und ihnen denn eine auf den Kopp haue, nein, die kommen zu uns und fangen \u00c4rger an, da muss ich mich wehren, n\u00fctzt ja nix.\n\n#### Und wovor hast du Angst? Gibt es \u00fcberhaupt etwas, wovor du Angst hast?\n\nMeine Fresse, davon hat mir Willy nichts gesagt, der Arsch. [Er schweigt eine ganze Weile.] Ich habe eigentlich vor nichts Angst, auch nicht vor dem Tod, aber ich bemerke, dass ich mit zunehmendem Alter gewisse \u00c4ngstlichkeiten an mir wahrnehme.\n\n#### Inwiefern?\n\nEs ist einfach so, dass ich Angst habe, meine Freunde durch Gevatter Tod zu verlieren. Wir sind in einem Alter, wo das durchaus m\u00f6glich ist, und unser Lebenswandel k\u00f6nnte da eine helfende Hand sein. [Er grinst jetzt frech und ich habe den Eindruck, dass er auch davor keine Angst hat.]\n\n#### Hast du eigentlich Kinder?\n\nJa, ich hab eine Tochter. [Er sagt es sehr stolz.] Sie studiert gerade in Italien.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nGar nicht, das ist mir zu seltsam. Wieso sollte ich mich selbst beschreiben? Ich will mich ja nicht bewerben, und ganz ehrlich, wenn ich mich selbst beschreibe und meine gl\u00e4nzenden Vorz\u00fcge hervorhebe, bin ich ja wohl identifizierbar. Verstehste?! Kleines Dorf voll mit Punkern und so ein alter Sack beschreibt sich erst mal flei\u00dfig...\n\n#### Gut, das ist ein Argument, obwohl ich eigentlich auf deine inneren Werte abzielte...\n\nNun, das musst du entweder selbst rausfinden oder Mutma\u00dfungen anstellen.\n\n#### Was denkst du, was andere \u00fcber dich denken?\n\nDas ist mir total egal, schau mich an, ich laufe wie der letzte Penner rum, krieg im Bus immer einen Fensterplatz und sitze alleine. Das geht mir alles am Arsch vorbei. Wenn allerdings Freunde von mir auf irgendwelchen Tratsch h\u00f6ren, ist es in erster Linie ihre Sache, so lange sie mir Gelegenheit geben, was klarzustellen. Ansonsten w\u00e4ren es ja Idioten.\n\n#### Sagst du immer, was du denkst, und gibst es ungefiltert weiter?\n\nJo, das ist f\u00fcr die anderen unbequem, aber sie haben sich daran gew\u00f6hnt und merken, dass es auf Dauer zeitsparend ist. Ich rede halt nicht um den hei\u00dfen Brei herum. Allerdings ist mir so auch fast jeder Fettnapf sicher.\n\n#### Hast du Hobbys?\n\nSch\u00f6ner Themensprung, da musste noch ein bisschen \u00fcben. Hobbys? Das ist doch das, was man betreibt, nachdem das Tagwerk getan ist und sich entspannen m\u00f6chte? Also in der Freizeit. Mein Leben besteht aus Freizeit! Wenn man so will, habe ich ein Hobby: Ich mach Musik und manage eine israelische Punk-Band. Das ist mir wichtig und macht Spa\u00df, auch wenn ich jetzt noch \u00f6fter stramm bin.\n\n#### Welche Bedeutung hat Musik denn f\u00fcr dich?\n\nWichtig schon, aber nicht \u00fcberlebenswichtig. Wenn ich Texte schreibe, so will ich auch, dass meine Worte geh\u00f6rt werden. Sie wurden mit dem Alter immer pers\u00f6nlicher, anfangs noch hoch politisch und zur Revolution aufrufend und jetzt handeln sie davon, dass ich die Sache vergeigt habe.\n\n#### Bist du in einem Verein aktiv?\n\n[Er lacht wieder.] In einem Fu\u00dfballverein, ich sag aber nicht, in welchem.\n\n#### Welche sonstigen popkulturellen Produkte konsumierst du au\u00dfer Musik?\n\nOh, wir dr\u00fccken uns heute gew\u00e4hlt aus, popkulturelle Produkte konsumieren. [Er \u00e4fft mich nach und lacht schallend.] Ich h\u00f6re Radio, guck in die Glotze und lese ab und an ein Schundblatt, geh auf Konzerte oder mache Konzerte, ist das popkulturell genug?\n\n#### Holst du dir so Informationen \u00fcber das Tagesgeschehen oder konsumierst du ohne Reflektion?\n\nAutsch, das hat gesessen! Wir wohnen in einem kleinen Dorf und sind auf die Informationen aus den einschl\u00e4gigen Medien angewiesen. Bei uns ist das Internet die Verbindung zur Welt da drau\u00dfen. Da k\u00f6nnen wir uns auch gezielter Infos zum aktuellen Geschehen holen und k\u00f6nnen im Vorfeld filtern. Du siehst, ich kann mich auch gew\u00e4hlt ausdr\u00fccken, wenn ich will.\n\n#### Welche Seiten pr\u00e4ferierst du?\n\nIn erster Linie Indymedia und Antifa-Seiten. Momentan steppt ja der B\u00e4r bei euch in Berlin. Die brennenden Autos finde ich aber schei\u00dfe. Ich glaube auch nicht, dass das die Autonomen oder Antifas waren. Ich vermute, dass die Nazis da drin h\u00e4ngen. Jetzt so kurz vor den Wahlen k\u00f6nnen sie so noch einmal richtig Stunk machen und alle gegen die andern aufbringen. Das ist echt eine kranke Schei\u00dfe.\n\n#### Hast du eigentlich Vor- oder Leitbilder? Oder jemals gehabt?\n\nNee, nicht mal als Kind. Vorbilder hatten f\u00fcr mich schon immer etwas mit Autorit\u00e4t zu tun, und mit Autorit\u00e4t habe ich m\u00e4chtige Probleme. Ich bin mein eigener Herr und denke und handle wie ich will.\n\n#### Schlie\u00dft du schnell Freundschaften?\n\nIch schlie\u00dfe wirklich schnell Freundschaften.\n\n#### Welche Kriterien sind dir da wichtig?\n\nMir ist Offenheit sehr wichtig. Manchmal wirke ich auf andere provozierend, aber nur so kann ich an den Reaktionen erkennen, wer mithalten kann und mir auch Kontra gibt. Wenn ich das merke, wei\u00df ich, dass sich die M\u00fche lohnt.\n\n#### Du geh\u00f6rst ja der Punk-Szene an, gibt es f\u00fcr dich eine Jugendszene, mit der du sympathisierst?\n\nJa, mittlerweile komm ich auf HipHop ganz gut klar. Ein paar meiner Kumpels haben Kinder, die jetzt Jugendliche sind. Die haben bei jedem Besuch die Musik gleich mitgeschleppt. Aber mit so manchen Texten von denen komm ich gar nicht klar. Muss ich ja auch nicht.\n\n#### Und welche Szene ist f\u00fcr dich ein No-Go?\n\nGanz logisch: die Nazis. Das sind solche Volldeppen, das geht gar nicht. [Eine Zornesfalte bildet sich auf seiner Stirn.] Vor drei Wochen haben wir auf unserem Geh\u00f6ft ein Open-Air-Konzert gegeben. Es waren f\u00fcnf Bands und etliche Leute aus allen Teilen Europas da, also der Hof war voll. Auf einmal rauschten die verschissenen Nazi-Biker an und wollten auf dem Hof Stunk machen. Die haben so die Jacke voll bekommen, dass sie genauso schnell wieder abhauen mussten. Unglaublich, die Penner.\n\n#### Welche Rolle spielen deine Freunde in deinem Leben?\n\nEine sehr wichtige. Es ist schon komisch, als Jugendlicher waren Kumpels auch wichtiger als alles andere. Dann ging es los mit Familie gr\u00fcnden und Freundschaften spielten eine untergeordnete Rolle. Aber jetzt, wo ich eine Familie abgeschrieben habe, sind mir Freunde \u00fcberaus wichtig. Zum Gl\u00fcck sind mir ein paar meiner \u00e4ltesten Freunde erhalten geblieben, mit denen bin ich sogar schon im Kindergarten gewesen. Wir sind halt Kinder des Ostens und haben so manche Schei\u00dfe erlebt, das schwei\u00dft zusammen.\n\n#### Wart ihr auch zusammen in der autonomen Szene?\n\nJa, auf jeden Fall, wir haben ganz vorne gestanden. In jedem besetzten Haus hatten wir ein Wohnung zusammen, und wir haben uns auf den \u00e4tzenden Plena immer rausgehauen, wenn wieder einer Mode war. Oft war ich es, der angeschissen wurde. Einmal wurde mir Schwulenfeindlichkeit vorgeworfen, nur weil ich offen ansprach, dass der eine Mitbewohner sich kleine Jungs von der Stra\u00dfe holte. Na ja, wir haben das dann anders gel\u00f6st.\n\n#### Welche Rolle spielte die autonome Szene f\u00fcr dich?\n\nEine sehr gro\u00dfe. Endlich hatte ich die M\u00f6glichkeit, wahre Partizipation zu praktizieren. Allerdings wurde es immer schwieriger. In der Gruppe, in der ich war, wurde so viel zerredet. St\u00e4ndig ein Plenum nach dem anderen und immer Stellung nehmen. Es h\u00e4tte nur noch gefehlt, dass wir Antr\u00e4ge f\u00fcrs Molliwerfen stellen sollten. Aber auch das habe ich geschafft, irgendwann mochten sie mich sogar und ich konnte ein wenig Einfluss nehmen.\n\n#### Inwiefern Einfluss?\n\nZum Beispiel beim Entwerfen der Plakatmotive oder Aktionen anf\u00fchren. Eine Aktion war: Nach der R\u00e4umung eines Hauses in Berlin haben wir Plakate geklebt mit der Ank\u00fcndigung, dass wir es zur\u00fcckerobern werden. Tja, wir waren schon l\u00e4ngst wieder drin, wollten die Bullen nur verarschen. Am n\u00e4chsten Morgen war die gesamte Stra\u00dfe verriegelt, \u00fcberall nur Bullen. Sie standen da einfach nur rum. Wir haben dann unsere Boxen auf die Fensterbretter gestellt und volles Brett Einst\u00fcrzende Neubauten gespielt. Mir haben denn sogar die Bullen leidgetan, denn die erste Platte von denen war echt schei\u00dfe. Das ging zwei Tage so, wir wussten, dass sie irgendwann angreifen, also sind wir zum Kacken nicht aufs Klo, sondern haben in s\u00e4mtliche Eimer und Sch\u00fcsseln gemacht. Das waren super Wurfgeschosse. In den zwei Tagen hatten wir auch Zeit, ein Gitter vor die T\u00fcr zu schwei\u00dfen, die T\u00fcr wurde bei der ersten R\u00e4umung entfernt. Kurz gesagt, sie kamen nicht rein, und als wir dann auch noch auf ihre Dienstwagen ballerten, sind sie abgehauen.\n\n#### Was meinst du mit geballert? Hattet ihr Schusswaffen?\n\nNein, wir haben mit einer Zwille oder Mollies geschossen, das hatte aber auch einen guten Effekt.\n\n#### Wie w\u00fcrdest du \"autonom\" oder \"autonome Szene\" definieren?\n\nF\u00fcr mich bestand die autonome Szene aus mehreren Gruppen, die verschiedene Teilziele, aber ein gro\u00dfes gesamtes Ziel hatten. Sie agierten in ihren Gruppen und schlossen B\u00fcndnisse mit anderen Gruppen, um ihre jeweiligen Ziele zu erreichen. Und die Mittel, die zum Erfolg f\u00fchrten, waren zahlreich und manchmal sogar nachhaltig. Anfangs war es uns egal, welche Mittel wir einsetzten und ob wir andere verletzten. Wir wurden gerade aus dem Ost-Knast freigekauft, wir wurden drangsaliert und gefoltert, da war genug Wut in uns. Es war die absolute Freiheit, endlich so leben zu k\u00f6nnen, wie man es wollte. Es war das perfekte Leben.\n\n#### Warst du Mitglied in einer konkreten Gruppe?\n\nJa, aber ich werde dir ganz bestimmt nicht sagen, wie die hie\u00df. Wir waren eine ziemlich gro\u00dfe Gruppe. Unser Gemeinschaftsraum im besetzten Haus war dementsprechend riesig und beim Plenum war es manchmal schwer, den Redner oder die Rednerin zu verstehen. Kurz vor meinem Ausstieg haben sie dann ja auch kleinere Gr\u00fcppchen gebildet, da konnten sich auf einmal nicht mehr alle so gut leiden.\n\n#### Mit welchem Thema hat sich deine Gruppe besch\u00e4ftigt?\n\nAntifa nat\u00fcrlich. Unsere Etage war f\u00fcr Materialbeschaffung f\u00fcr Demos und Transparente und Plakate zust\u00e4ndig. Mehr m\u00f6chte ich dazu nicht sagen.\n\n#### Wie war dein erster Kontakt mit der autonomen Szene?\n\nGanz klassisch in der Kneipe. Wir haben so rumphilosophiert und unsere Tischnachbarn haben sich irgendwann eingeklinkt und von ihren Projekten erz\u00e4hlt. Ich war total begeistert, ein selbstverwaltetes Haus mit Volksk\u00fcche und Proberaum. Das war f\u00fcr mich das Paradies, da wollten wir sofort mitmachen. Jetzt im Nachhinein denke ich, dass sie Leute f\u00fcrs Grobe brauchten, da kamen wir ihnen ganz recht. Na was soll's, ich fand die Zeit sehr bereichernd.\n\n#### Wusstest du schon vorher von der Existenz der Autonomen?\n\nJa, aber nur bruchst\u00fcckweise, was dahintersteckte, konnte ich nur ahnen. Wie gesagt, wir sa\u00dfen im Ostknast, die hatten es nicht so mit der Informationsvermittlung.\n\n#### Gibt es deines Erachtens bei der autonomen Szene Ankn\u00fcpfungspunkte an andere Jugendkulturen?\n\nDazu m\u00fcsste ich erst mal feststellen, dass die autonome Szene eine Jugendkultur ist, da bin ich mir aber noch nicht so ganz sicher. Jedoch bin ich mir sicher, dass ich nicht mehr dazu sagen kann oder will. Mir ist aufgefallen, dass viele Gruppen oder, wie du es nennst, Jugendkulturen kriminalisiert werden, sobald sie mit der autonomen Szene sympathisieren.\n\n#### Findest du nicht, dass die autonome Szene eine Jugendkultur ist?\n\nNein, es ist eine Bewegung, ein Lebensstil. Vielleicht mit jugendkulturellen Elementen. Sie besitzt ja gerade bei Jugendlichen einen gewissen Reiz, es setzt ein Prozess der politischen Reife ein, der selbstbewusst und stolz macht.\n\n#### Gibt es einen eigenst\u00e4ndigen autonomen Stil?\n\nIn der Kleidung schon, das wei\u00df ja inzwischen jeder. Aber ich habe festgestellt, dass es auch einen eigenst\u00e4ndigen Stil gibt, Probleme anzugehen. Das hat mich letzten Endes auch von ihnen entfernt. Wenn Probleme erkannt werden, so wird versucht, sie wegzudiskutieren. Da wird ein Plenum nach dem anderen abgehalten, aber nach zwei Wochen besteht das Problem immer noch.\n\n#### Was spielt die Musik in der autonomen Szene f\u00fcr eine Rolle?\n\nOh, eine gro\u00dfe. Die Musik ist so etwas wie ein Sprachrohr, in ihr b\u00fcndeln sich alle Meinungen und Anschauungen ihrer Macher und H\u00f6rer. Die Musik kann ber\u00fchren und fesseln, sie kann Einfluss auf deinen Gem\u00fctszustand haben, \u00fcber die Musik kannst du neue Leute kennenlernen. Es werden Botschaften f\u00fcr Insider versandt. Nat\u00fcrlich ist Musik ein sehr wichtiger Faktor in der autonomen Szene.\n\n#### Welche Themen waren dir im Engagement in der autonomen Szene wichtig?\n\nMir war wichtig, die Stadt Nazi-frei zu bekommen. Die Themen Menschenrechte und -w\u00fcrde sind mir ein ganz pers\u00f6nliches Anliegen gewesen, sind sie \u00fcbrigens immer noch.\n\n#### Und bei den Themen Antisexismus, Antirassismus, Antimilitarismus, Umweltschutz und Tierrechte, wie stehst du dazu?\n\nViele einzelne Gruppen beackern ein bestimmtes Thema, bei uns war es die Antifa. Das macht mich allerdings nicht blind und taub f\u00fcr die anderen The-men. Auf jeden Fall habe ich mich auch f\u00fcr die anderen engagiert, sie waren mir genauso wichtig. Jetzt nat\u00fcrlich auch noch, aber ich werde nicht mehr so oft mit ihnen konfrontiert, so dass ich weniger bis gar nicht dar\u00fcber nachdenke.\n\n#### Wie stehst du zum Konflikt zwischen den Antiimps und den Antideutschen?\n\nF\u00fcr mich sind das Vollidioten, sie denken alle, sie h\u00e4tten die Welt neu erfunden und sie stehen auf der einzig richtigen Seite. Was mich abschreckt, ist ihre Vehemenz, bei beiden Gruppen, sie vertreten ihre Meinung mit der Message: Wer nicht f\u00fcr uns ist, ist gegen uns, und wer gegen uns ist, ist ein Schwein! Es m\u00fcsste doch gar keinen Konflikt geben, es gibt diese beiden Gruppen und fertig. Was soll der ganze Schei\u00df. Die israelische Punk-Band, mit der ich gerade durch Deutschland toure, kann es \u00fcberhaupt nicht verstehen, dass wir in Deutschland solche Konflikte austragen. Ich finde beide Gruppen echt daneben und verstehe nicht, was sie wollen, deshalb besch\u00e4ftige ich mich auch nicht mehr mit ihren Themen.\n\n#### Aber sonst setzt du dich grunds\u00e4tzlich mit Theorien auseinander?\n\nNa ja, nicht grunds\u00e4tzlich und zurzeit sowieso nicht, aber als ich aktiv war, auf jeden Fall. Ich hab mich anfangs damit ein bisschen schwer getan. Ich komme nicht aus der intellektuellen Ecke, hab nicht das Abitur und war auch rhetorisch nicht der aufkommende Stern, aber ich blieb am Ball und habe mich mit den Themen auseinandergesetzt. Das muss man ja auch erst mal k\u00f6nnen, aufgestellte Theorien hinterfragen. Die Staatskritik und die vermeintliche Rechtfertigung militanter Eins\u00e4tze war meist Thema Nummer eins.\n\n#### Wie ist dein Verh\u00e4ltnis zum Staat und zur Polizei?\n\nEs ist besser geworden. [Er lacht sehr laut.] Neulich in Hamburg haben ein paar Bengels eine Party zerst\u00f6rt. Stell dir vor, da kommen die in eine Punker-Kneipe rein und wollen einen auf Macker machen. Pl\u00f6tzlich hauen die einem M\u00e4del die Nase ein und z\u00fccken ihre Messer. Wir haben nicht die Bullen gerufen, sie haben nicht damit gerechnet, dass wir in vielen Ecken Kn\u00fcppel zu stehen haben und meist mehr Leute sind als die, die zu sehen sind. Letzten Endes haben die die Bullen gerufen und was von \u00dcberfall gefaselt. Die St\u00f6cker haben wir nat\u00fcrlich entsorgt, Kohleofen sei Dank, aber die Bullen waren diesmal richtig nett zu uns. Sicherlich, weil sie die Schl\u00e4ger schon kannten und sich insgeheim freuten, dass sie selbst mal aufs Maul bekommen haben. Was wollte ich damit sagen? [F\u00fcnf Bier sind leer.] Ach ja, mein Verh\u00e4ltnis zu Bullen und Staat. Es ist entspannter; so lange sie mich in Ruhe lassen, lasse ich sie auch in Ruhe.\n\n#### Wie stellst du zu Parteipolitik? Gehst du w\u00e4hlen?\n\nJa, ich gehe w\u00e4hlen und w\u00e4hle meist die APPD, mir f\u00e4llt nichts anderes ein. Zu Parteien kann ich nur sagen, es ist so wie bei der Gewerkschaft, sie sind zu Firmen mutiert, die nur noch auf ihren Profit bedacht sind. Ihnen geht es nicht mehr um die Sache, na ja, vielleicht vereinzelt. Die gehen mir alle geh\u00f6rig auf den Sack mit ihrer Scheinheiligkeit, und wer mir erz\u00e4hlen will, dass Politik und Wirtschaft zwei verschiedene Sachen sind, der h\u00e4lt mich f\u00fcr bl\u00f6de.\n\n#### Informierst du dich immer noch \u00fcber szeneinterne Aspekte?\n\nN\u00f6, eigentlich nicht. Wenn ich mal wieder in Hamburg oder Berlin bin, treffe ich meist alte Bekannte wieder, dann bekomme ich das Neueste erz\u00e4hlt. In meinem Dorf muss ich nicht wissen, ob wieder etwas geplant ist. Au\u00dferdem bin ich raus, ich sympathisiere zwar noch, bin aber passiv.\n\n#### Wie hast du dich in deiner aktiven Zeit informiert? Welche Medien hast du genutzt?\n\nDamals lief alles nur \u00fcber Mundpropaganda, manchmal verschl\u00fcsselte Nachrichten, aber das war es dann aber auch schon. Handy und Internet waren noch nicht so angesagt. [Er grinst wieder.]\n\n#### Wie stehst du zu B\u00fcndnissen mit anderen linken Bewegungen oder sozialen Protestbewegungen?\n\nIch bef\u00fcrworte das, weil es die Masse macht. Als ich noch aktiv war, hat sich die autonome Szene viel zu sehr abgeschottet. Sie war eine geschlossene Gesellschaft, was zu der Zeit damals vielleicht richtig war, aber die Kommunikation nach drau\u00dfen erschwerte. Jede Aktion hatte ja eine zugrundeliegende Theorie, die wiederum zum B\u00fcrger transportiert werden musste, so dass er uns verstehen kann oder zumindest die Beweggr\u00fcnde nachvollziehen kann. In B\u00fcndnissen ist dies viel besser und schneller realisierbar, allerdings sind die Aktionen nicht mehr militant und bed\u00fcrfen keiner Kommunikation, die Aktionen sprechen f\u00fcr sich.\n\n#### Wie hat sich die Szene im Laufe der Jahre noch ver\u00e4ndert?\n\nDie Szene ist irgendwie abwartender geworden, kaum noch plumper Aktionismus.\n\n#### Wo siehst du Kritikpunkte und Probleme in der autonomen Szene?\n\nMeine Kritik hast du ja eigentlich schon geh\u00f6rt, es ist das st\u00e4ndige Diskutieren, das ist nervt\u00f6tend. Dann ist f\u00fcr mich der bestehende Widerspruch zu sehr sichtbar. Einerseits wird f\u00fcr gerechtere L\u00f6hne und ein gutes Miteinander gek\u00e4mpft, andererseits gehen die meisten Autonomen irgendwelchen Jobs nach, dessen Stundenlohn unter aller Sau ist. Und das gute Miteinander ist auch f\u00fcr den Arsch, irgendwie haben es Autonome schwerer, andere mit ihren Macken und Unzul\u00e4nglichkeiten zu akzeptieren, die streiten selbst in den eigenen WGs mit ihren Mitbewohnern rum.\n\n#### Fast alle Jugendkulturen sind m\u00e4nnlich dominiert. Gilt das auch f\u00fcr die autonome Szene?\n\nAlso ich denke schon, dass es mehr m\u00e4nnliche als weibliche Autonome gibt, jedoch ist der Umgang untereinander ein anderer. Die Frauen haben den Mut, Themen explizit anzusprechen, auch wenn sie Probleme mit dem Machismus einiger M\u00e4nner haben. Das finde ich sehr erhellend. In unserer Gruppe waren ungef\u00e4hr vierzig Prozent Frauen, sie waren die reinen Organisationstalente.\n\n#### Wie bewertest du die \u00f6ffentliche Wahrnehmung der autonomen Szene?\n\nDie Medien filtern ja schon vorher, welche Nachrichten sie in welcher Dosis verabreichen werden. Die meisten Leute sind passive Nachrichtenvernichter, sie lassen sie sich sozusagen bringen, wenig Eigeninitiative. F\u00fcr diese Leute sind die Autonomen genau die Auto-Anz\u00fcnder, die auch so gezeigt werden. Es gibt aber noch viele, die Nachrichten hinterfragen und so auch das Bild der Autonomen. Es wird von ihnen ja immer nur im negativen Zusammenhang berichtet. So negativ, wie die Medien \u00fcber die Autonomen berichten, so negativ ist auch das Bild in der \u00d6ffentlichkeit. Traurig, aber wahr.\n\n#### In der \u00f6ffentlichen Wahrnehmung werden Autonome immer mit Gewalt in Verbindung gebracht. Wie siehst du das, geh\u00f6rt Gewalt zur autonomen Szene dazu?\n\nIch glaube schon, aber nur gegen Sachen. Gezielte Gewalt gegen Menschen, die nicht dem Selbstschutz dient, lehne ich grunds\u00e4tzlich ab. So geht es \u00fcbrigens der Mehrheit der Autonomen, das glaubt nur kaum jemand.\n\n#### Wie ist deine Einstellung zu militanten Aktionen?\n\nDamals war ich grunds\u00e4tzlich daf\u00fcr; um Ziele zu erreichen, war mir jedes Mittel recht, meine Schwelle war sehr niedrig.\n\n#### Wie sch\u00e4tzt du die gesamtgesellschaftliche Bedeutung der Autonomen ein? Welche M\u00f6glichkeiten hat die autonome Szene, etwas zu ver\u00e4ndern?\n\nDiese Frage k\u00f6nntest du auf einem Plenum stellen, dann k\u00f6nntest du bis morgen diskutieren. Ich wei\u00df es nicht. Ja, ich finde sie f\u00fcr die Gesellschaft wichtig, sie geh\u00f6rt dazu und sollte nicht kriminalisiert werden. Ja, ich wei\u00df, sie machen ungesetzliche Dinge, aber sie sind nicht kriminell, sondern politisch. Leider hat die autonome Szene nicht die Macht, wirklich etwas entscheidend zu ver\u00e4ndern, aber die autonome Szene hat die Macht, etwas zu bewegen, wenn auch nur millimeterweise.\n\n#### Denkst du, dass sie wirklich die Gesellschaft als Ganzes \u00e4ndern m\u00f6chten, oder geht es nur um die Wahrung von Freir\u00e4umen?\n\nNein, ich denke, dass sie die Gesellschaft als Ganzes \u00e4ndern wollen. Klar gibt es auch dort Leute, die nur ihrer kriminellen Energie freien Lauf lassen wollen, die sogenannten Event-Orientierten, aber dem Gro\u00dfteil geht es darum, den Kapitalismus nicht weiter zu n\u00e4hren und gleiches Recht f\u00fcr alle zu schaffen.\n\n#### Wie sch\u00e4tzt du die Zukunft und Perspektiven der autonomen Bewegung ein?\n\nVor zwei Jahren noch h\u00e4tte ich gedacht, dass sie sich selbst demontiert, auf einmal sind so viele Leute ausgestiegen, haben sich sozusagen sozialisiert, sind aus den WGs in Appartements gezogen und haben B\u00fccher geschrieben, aber mittlerweile sehe ich das nicht mehr. Der harte Kern macht weiter, und ich glaube, so lange es Nazis gibt, gibt es auch Autonome.\n\n#### Vielen Dank Gerd, f\u00fcr deine Geduld. Gibt es vielleicht noch etwas, was du sagen m\u00f6chtest?\n\nDas klingt so, als wolltest du mich gleich abf\u00fchren. [Er lacht und trinkt den letzten Schluck Bier.]\n\nDas Interview f\u00fchrte Bettina Thiele.\n\n## \"Wenn Parteien mit Autonomen klark\u00e4men, w\u00e4re ich auch in einer Partei.\"\n\nSimon, 35, lebt in Berlin-Friedrichshain.\n\nEr ist in Berlin-West geboren und aufgewachsen. Seit etwa f\u00fcnf Jahren wohnt er nun in Friedrichshain, nachdem er lange Zeit in Neuk\u00f6lln gelebt hat. Er hat Soziologie studiert und ist gerade dabei, seine Dissertation zu schreiben. Ich begegne einem jung wirkenden Mann, mittelgro\u00df, dunkelblonde Haare, recht schlank und eher unauff\u00e4llig. Er tr\u00e4gt eine blaue Jeans und ein graues T-Shirt. Seine Wohnung ist f\u00fcr eine Ein-Zimmer-Wohnung recht gro\u00df und hell, sehr ordentlich und wohnlich, eher spartanisch eingerichtet. Das Auff\u00e4lligste an seinem Zimmer ist das riesige B\u00fccherregal, in dem wohl an die 500 B\u00fccher stehen. Viele kulturwissenschaftliche und philosophische B\u00e4nde, Foucault, Adorno, B\u00fccher \u00fcber den Nationalsozialismus. Er sammelt auch politische Zeitschriften, die alle s\u00e4uberlich archiviert sind.\n\nSimon bezeichnet sich als \"Post-Autonomer\", hatte lange Zeit mit der autonomen Szene zu tun und ist immer noch stark politisch engagiert. Ich merke schnell, dass er sich sehr gut mit allen m\u00f6glichen politischen Themen, Debatten und Theorien auskennt. Das Interview f\u00fchren wir in der K\u00fcche, die ebenso schlicht und spartanisch eingerichtet ist.\n\n#### Wie w\u00fcrdest du dich selbst beschreiben?\n\nALG-II-Empf\u00e4nger, ich mag Neuk\u00f6lln und Freundschaft ist mir wichtiger als Familie, zwangsl\u00e4ufig, weil ich keine hatte so richtig... Ich mag Spiele-Abende und hatte irgendwann auch mal ein bisschen was mit Politik zu tun. [Lacht.]\n\n#### Hast du jetzt auch noch mit Politik zu tun oder w\u00fcrdest du das auf die Vergangenheit beziehen?\n\nAlso bisher bestand das politisch aktiv sein darin, organisiert zu sein, sich mit anderen zusammenzutun, kontinuierlich irgendwelche Treffen zu haben, und insofern bin ich politisch interessiert, aber gerade nicht politisch aktiv. Ich bin gerade doch eher mit mir selbst besch\u00e4ftigt, insofern ich mir Sorgen um meine Zukunft mache.\n\n#### Wie ist denn deine gegenw\u00e4rtige Situation und wie stellst du dir deine Zukunft vor?\n\nMein Traum w\u00e4re, mir ein Stipendium ergattern zu k\u00f6nnen und eine Diss zu schreiben, damit ich endlich den finanziellen Background habe, um mich entspannt und ruhig zu artikulieren. Inhaltlich interessiert mich dieses Schnittfeld von Medizin und Politik: linke Medizinkritik, Psychiatrie, Medizingeschichte, Medizin im Nationalsozialismus, Gesundheitspolitik, Humangenetik, Biologie, Biologismus, Bev\u00f6lkerungspolitik, Rassismus.\n\n#### Wie war denn dein Verh\u00e4ltnis zu deinen Eltern?\n\nKompliziert. Ich bin ein Einzelkind, bei meiner Mutter aufgewachsen hier in Berlin. Meine Mutter war Verk\u00e4uferin bei einer gro\u00dfen Firma, vollzeitbesch\u00e4ftigt, hatte kaum Zeit, insofern habe ich meine Mutter, meine Kindheit so ein bisschen verpasst. Erst seitdem sie Rentnerin ist, ist das irgendwie spannender, also erst im sp\u00e4ten Alter habe ich meine Mutter so ein bisschen besser kennengelernt und bin jetzt halt damit besch\u00e4ftigt, Verst\u00e4ndnis daf\u00fcr aufzubringen, warum sie so geworden ist, wie sie ist. Und meinen Vater habe ich nicht gesehen, au\u00dfer einmal im Gerichtssaal, da ging's um Unterhaltszahlungen. 18 Jahre lang hat er halt kaum bezahlt, weil er noch zw\u00f6lf andere Kinder hat oder dreizehn, und dar\u00fcber hat sich meine Mutter sehr ge\u00e4rgert. Bis zum 18. Lebensjahr haben sich enorme Schulden angesammelt, also praktisch Kindergeld, Unterhalt, den er h\u00e4tte zahlen m\u00fcssen, aber nicht gezahlt hat, insofern gab es einen Schuldentitel, der nie umgesetzt wurde, weil's halt nichts zu pf\u00e4nden gab, und dann gab's halt noch mal, als ich im Studium war, eine Gerichtsverhandlung, wo es darum ging, noch was aus ihm herauszupressen.\n\n#### Wie alt warst du \u00fcberhaupt, als deine Eltern sich getrennt haben?\n\nDa gab's mich noch gar nicht, das war vor meiner Geburt. Ich bin ein ungewolltes Kind, w\u00e4re es medizinisch m\u00f6glich gewesen, w\u00e4re ich abgetrieben worden.\n\n#### Und hast du deinen Vater \u00fcberhaupt besser kennengelernt oder hattet ihr Kontakt?\n\nMeine Mutter hat so vor gut zehn Jahren versucht, einen Kontakt zwischen ihm und mir herzustellen; das hat sie auf eine recht dumme Art und Weise getan und mein Vater hat das noch getoppt, das sah n\u00e4mlich folgenderma\u00dfen aus [lacht]: Meine Mutter hat einfach dem zwei, drei Briefe geschickt, ihm Fotos von mir mitgeschickt, so erz\u00e4hlt, wer ich bin, so \u00fcber meinen Kopf hinweg, wie meine Mutter halt so ist, und mein Vater hat mir dann wiederum einen Brief geschrieben, den fand ich so bl\u00f6d, dass ich auf den nicht geantwortet hab, also in dem Brief war keine einzige Frage an mich drin, stattdessen nur blabla dar\u00fcber, dass Blut dicker ist als Wasser und was ich denn von ihm alles lernen k\u00f6nnte...\n\n#### Hast du Kontakt zu deinen Geschwistern?\n\nIch hab ein paar von denen kennengelernt, die haben mich mal ausgegraben und ich fand die sehr nett, war allerdings selbst gerade sehr mit irgendwelchem Beziehungsstress besch\u00e4ftigt und konnte mich dann nicht so drum k\u00fcmmern, und als ich dann irgendwie am Kontakt interessiert war, ist das leider von deren Seite aus nicht erwidert worden, insofern ist es halt wieder eingeschlafen.\n\n#### Schreckt dich deine eigene famili\u00e4re Erfahrung ab, selbst eine Familie zu gr\u00fcnden, oder wie ist deine Einstellung in der Hinsicht?\n\nIch habe selbst keinen Wunsch nach einer Familiengr\u00fcndung, w\u00fcrde das aber mitmachen, wenn ich mit einer Person zusammen w\u00e4re, die unbedingt auf family steht. Wenn sie sich das halt w\u00fcnscht, k\u00f6nnte ich mir vorstellen, wenn ich sie denn liebe, das einfach spa\u00dfeshalber mit ihr mitzumachen, und dann w\u00e4chst man da schon irgendwie hinein. Aber den Herzenswunsch, Vater zu werden, hab ich jetzt nicht.\n\n#### W\u00fcrdest du sagen, dass deine Mutter oder andere Verwandte eine Rolle f\u00fcr deine politische Entwicklung gespielt haben?\n\nMeine Mutter hat keine Geschwister, das einzige, was ich an Familie in meiner Kindheit mitbekommen hab, das war meine Tante, die ich irgendwie skurril und cool fand, und sonst hatte ich leider nur so 'ne Art Schrecken vor meiner eigenen Familie, insofern ich die alle ziemlich unterbelichtet fand, also das waren ziemliche Vollprolls eher [lacht], mit denen ich mich nicht gut unterhalten und identifizieren konnte. Die waren lebensfroh, aber irgendwie dumm [lacht], und da habe ich nichts f\u00fcr mich entdeckt. Meine Mutter hat sich schon f\u00fcr Nachrichten interessiert und f\u00fcr Pers\u00f6nlichkeiten, aber die hat sich halt die Leute angeschaut und dann gesagt, wen sie mochte oder nicht mochte; die ist sozusagen nicht in der Lage, \u00fcber Politik oder das Politische zu reflektieren. F\u00fcr sie ist Politik 'ne pers\u00f6nliche Geschichte, so dass sie halt Helmut Kohl als Pers\u00f6nlichkeit sympathisch fand, w\u00e4hrend sie Gerhard Schr\u00f6der oder so nicht mochte. Umso lustiger und umso skurriler die waren, umso mehr mochte sie die auch, also so jemanden wie Kohl oder Angela Merkel mag sie, und wenn die halt so geschliffen sind wie Gerhard Schr\u00f6der oder Westerwelle, dann mag sie die nicht.\n\n#### Ist das bei dir auch so? Man \u00fcbernimmt ja doch mehr von den Eltern als man glaubt...\n\nJa, ich bin auch nicht in der Lage, irgendwie Ideen nachzuh\u00e4ngen, so wie andere das k\u00f6nnen, ich mag das auch ziemlich lebensnah und lebenspraktisch. Bei mir m\u00fcssen Politiker politische Richtungen verk\u00f6rpern und dann kann ich mich davon abgrenzen oder werde auch ganz warm. In der Schule hab ich mal ein Referat gehalten \u00fcber Che Guevara und fand das ganz toll, dass er Asthma hatte und irgendeinen Tee trinken musste, also diese pers\u00f6nlichen Noten haben mich fasziniert, mit so was Abstraktem wie \"Kommunismus\" konnte ich nichts anfangen.\n\n#### Wer waren denn deine Vorbilder?\n\nAlso das Jahr 1989 war ein recht spannendes Jahr f\u00fcr mich, da einfach Berlin so der Nabel der Welt war und ganz viel passiert ist. Und da gab's das Rathaus Sch\u00f6neberg, vor dem sich ganz viele Politiker versammelt hatten, um die Nationalhymne zu singen, und im Hintergrund wurde von Leuten dagegen getr\u00e4llert und gebuht und gepfiffen, und das hat meine Mutter vor'm Fernseher so erregt, dass sie mit mir zum Kurf\u00fcrstendamm wollte, wo Helmut Kohl noch mal auftauchen sollte und Momper, der damals Regierender B\u00fcrgermeister von Berlin war. Ich war irritiert, warum pfeifen da Leute, warum freuen sie sich nicht, warum gibt es da welche, die sich nicht \u00fcber den Fall der Mauer freuen, und das fand ich spannend oder f\u00fchlte mich irgendwie lustig infiziert, und gleichzeitig bin ich immer noch mit meiner Mutter mit zum Kurf\u00fcrstendamm, weil meine Mutter so eine Art \u00dcber-Ich in mir pr\u00e4sentierte; wenn meine Mutter sich da dr\u00fcber \u00e4rgert, dann hat das seinen wichtigen Grund. Ich konnte mich emotional also von meiner Mutter \u00fcberhaupt nicht abgrenzen. Dann kam ich von der Realschule auf ein Gymnasium, und da habe ich dann halt Linke kennengelernt, die genau auf meiner Wellenl\u00e4nge waren: Die waren durchweg alle extravagant, spleenig und auch auf der Suche, was ich anscheinend irgendwie auch war, und das Tollste war, die waren ein Jahr \u00e4lter, also in der Jahrgangsstufe \u00fcber mir, so dass ich dann auch zu den Klassentreffen von der Klasse \u00fcber mir eingeladen wurde, und da hab ich wichtige Freunde kennengelernt, halt auch Linke, die auf dem Schulhof heimlich die radikal verkauften und sich \u00fcber die RAF unterhielten. Und 1990 hab ich ein Haus besetzt: Kaum drei Monate am Gymnasium, bin ich gefragt worden, hast du Lust, ein Haus zu besetzen, und da habe ich gesagt: ja. Da war ich 16. Ich hab das gar nicht als ein politisches Projekt verstanden, sondern ich hab mich mit meiner Mutter \u00fcberhaupt nicht verstanden und das war sozusagen n\u00fctzlich als Alternative zu dem, was mir praktisch meine Mutter bot.\n\n#### Das war dein Einstieg in die linke oder autonome Szene?\n\nJa genau, in die linksradikale Szene oder einfach die linke Szene... Heutzutage kommt mir das so vor, als ob man viel mehr aufpassen muss, als was man sich definiert, fr\u00fcher konnte man jeden Monat was anderes sein, mal war man Anarchist, im n\u00e4chsten Monat war man Kommunist und dann wieder Autonomer, das war okay.\n\n#### Geh\u00f6rtest du denn mal einer Jugendkultur an?\n\nEine sehr gute Freundin meinte irgendwann zu den Fotos, auf denen ich zu sehen war, ich war wohl mal Popper. [Wir lachen.] Das Erste, was ich versucht habe zu leben, war sicher mein Monty-Python-Helge-Schneider-Fan-Dasein, ich wollte Helge Schneider und Monty Python leben, nicht irgendwie wie andere mir Dreadlocks wachsen lassen oder irgendeine Musikrichtung mitmachen oder so, sondern meine Helden waren Helge Schneider und Monty Python. Nat\u00fcrlich hab ich HipHop-Mucke geh\u00f6rt wie alle und die Charts auf rs2.\n\n#### Und was wolltest du konkret...\n\nIch wollte auffallen...\n\n#### Indem du das nachahmst? Oder worum ging es?\n\nDas zu feiern und andere damit zu verst\u00f6ren, indem man seltsame Gangarten auf dem Flur macht usw., ja, und auch schockieren, selbst spleenig sein, das gefiel mir irgendwie. Ich fand Stinos [Stinknormale] nie toll, sondern ich fand alle sympathisch, die irgendwie dabei waren, sich abzugrenzen. Ein sehr wichtiger Freund damals, der war 'ne Klasse \u00fcber mir, ich war hellauf begeistert, als er mir aus dem Mathematikunterricht erz\u00e4hlte, dass er mal wieder nicht seine Hausaufgaben gemacht hatte und als er vom Lehrer gefragt wurde, was sind Graphen, nur sagte: Grafen fahren Kutschen. Seine gesamten Schulsachen hatte er in einem gro\u00dfen Seesack, unsortiert, das hie\u00df, er hat jeden Morgen seinen gro\u00dfen Sack auf den Tisch geknallt, wo irgendwie 15 Kilo lose Papierbl\u00e4tter umherschwirrten, und darin nie was gefunden oder erst mal 'ne Viertelstunde lang suchen m\u00fcssen. So 'ne Leute mochte ich, die habe ich geliebt und die haben mich adoptiert.\n\n#### Welche Bedeutung hat Musik f\u00fcr dich?\n\nDas schwankt. Wenn ich viel Zeit hab, mag ich auch mal wieder Musik h\u00f6ren oder auch Musik entdecken. Da mein Englisch so schlecht ist, bin ich dabei recht stark auf deutschsprachige Musik angewiesen. Also heute kann ich \u00fcberhaupt nichts mehr mit Gangster-Rap anfangen, umso intellektueller, umso besser, und nicht mehr so 'n Schrott. Ein paar Ikonen mag ich immer noch, so jemanden wie Ice-T, und ich lasse mich auch gerne von anderen beeinflussen.\n\n#### Hast du eine Lieblingsmusik?\n\n[\u00dcberlegt] Ja, ich h\u00f6r ziemlich gerne Elektro, aber auch so Elektro-Punk, also das muss nicht nur so artiger Elektro sein, das kann sich schon auch recht punkig anh\u00f6ren.\n\n#### Warum gerade die?\n\nWeil Punk in der Lage ist, recht gut so Herz-Schmerz zu artikulieren, und Elektro wiederum so was Gen\u00fcsslich-Lockeres hat, das ist so die Spannung f\u00fcr mich.\n\n#### Interessierst du dich f\u00fcr Mode oder auch K\u00f6rperschmuck?\n\nMeine Mutter hat bei C & A gearbeitet, zu Mode hab ich also ein sehr unentspanntes Verh\u00e4ltnis, ich bin sozusagen C & A-Kind, hab zwangsl\u00e4ufig immer C & A-Klamotten tragen m\u00fcssen und hab die gehasst. Ich trage Klamotten aus Notwehr, also unter der Perspektive, was muss man tragen, damit man in Ruhe gelassen wird. [Lacht.] Ich trage \u00fcberhaupt keine Marken, mach sozusagen bei diesem ganzen Zug mit von \"No Logo\" von Naomi Klein, keine Labels tragen, ziemlich einfarbige St\u00fccke nur zu tragen usw. Ich mag es, so ganz geschmacksneutral auszusehen. W\u00e4hrend andere es nicht auf Nazi-Demos schaffen, hab ich das ziemlich oft mit Bekannten und Freunden auf die Demos geschafft, weil ich so geschmacksneutral aussehe und nicht den \u00fcberkandidelten Autonomen raush\u00e4ngen lasse.\n\n#### Und was macht man dann da? Beobachten?\n\nWarten. Auf 'ne Gelegenheit, ob man da vielleicht am Rande dann zu zweit einen erwischt mal.\n\n#### Wozu?\n\n[Lacht.]... um ihm was auf die Nase zu geben.\n\n#### Wie verbringst du ansonsten deine Freizeit?\n\nAus irgendeinem Grund, den ich nicht wei\u00df, ist mein halber Freundeskreis schwul oder lesbisch und insofern verbringe ich halt ziemlich viel Zeit mit Leuten, die Tunten sind oder zur Kuchenfraktion geh\u00f6ren... Also ich bin nicht der typische Antifa, der lauter Antifa-Freunde hat und mit Jungs abh\u00e4ngt, die heterosexuell sind und Party machen, sondern ich bin von diesem Antifa-V\u00f6lkchen auf gewisse Art und Weise abgeschnitten und verbringe stattdessen ziemlich viel Zeit in so 'ner ganz komischen, randst\u00e4ndigen Ecke mit so anpolitisierten Leuten, die nicht so ganz in der Polit-Szene drin stecken, sondern eher am Rande stehen. Insofern wei\u00df ich, was \"queer\" bedeutet, kenne mich mit der Theoretisierung des Geschlechterverh\u00e4ltnisses aus, hab als Autonomer versucht, alles richtig zu machen... Also die Autonomen, wenn man denn so ein bisschen l\u00e4stern will, zeichnen sich dadurch aus, dass sie immer alles richtig machen wollen, bis zur eigenen \u00dcberforderung, bis das irgendwann ein Problem ist, so dass man daran kaputtgehen kann.\n\n#### W\u00fcrdest du dich denn als heterosexuell bezeichnen?\n\nJetzt ja.\n\n#### Fr\u00fcher nicht?\n\nIch hab halt auch meine Erlebnisse gehabt... [Lacht.] Ich hatte Kontakte mit M\u00e4nnern, aber sie sind immer so geendet, dass die mehr wollten als ich. Ich fand das halt unterhaltsam, zu flirten, aber im Endeffekt konnte ich mit bestimmten Sexualpraktiken nicht dienen und das hat dann halt entt\u00e4uscht. In meinen wilden Zwanzigern hatte ich \u00fcber vier Jahre eine lockere Beziehung zu 'nem Mann. Aber lieb hab ich den bis heute!\n\n#### Aber freundschaftlich?\n\nJa.\n\n#### Was hast du denn generell so f\u00fcr Beziehungskonzepte?\n\nDie Frage mag ich jetzt nicht beantworten. N\u00e4chste Frage.\n\n#### Dann kommen wir mal zur autonomen Szene. Wie w\u00fcrdest du denn ganz allgemein \"autonom\" definieren oder die autonome Szene jemandem beschreiben, der noch nie etwas davon geh\u00f6rt hat?\n\nEinmal gibt es ja die Parteien, also sozusagen eine offiziell etablierte politische Sph\u00e4re, die zum gro\u00dfen Teil von Berufspolitikern getragen wird, die bestimmen, was Gesetz werden soll und wie wir leben sollen. Und dann gibt es Leute, die k\u00f6nnen sich halt nicht so sehr mit dieser etablierten Sph\u00e4re identifizieren, finden diese delegierte Demokratie Murks und sind halt irgendwie [lacht] autonomer, nehmen das selbst in die Hand. Fr\u00fcher nannte man das Au\u00dferparlamentarische Opposition, sp\u00e4ter waren das dann die Neuen Sozialen Bewegungen und vielleicht sind die Autonomen auch eine soziale Bewegung, die sich dadurch auszeichnet, dass sie nicht auf irgendeine ferne Zukunft warten, sondern ziemlich unruhig sind, die wollen im Hier und Jetzt eine Ver\u00e4nderung, und das unterscheidet sie von kommunistischen Richtungen. Die Autonomen sind nicht die gro\u00dfen Zukunftsstrategen, die sich in gro\u00dfen Organisationen zusammentun wollen, weshalb sie auch als \"Spontis\" denunziert werden, sondern hier und jetzt soll das richtige Leben her.\n\n#### Aber waren die Spontis nicht Vorl\u00e4ufer der Autonomen?\n\nJa. Aber die Autonomen werden auch als Spontis denunziert, das hab ich selbst noch so mitbekommen, dass die AAB zu uns, den anderen Antifas, in der Uni immer gesagt hat: \"Spontis\", weil die halt organisierter waren als wir.\n\n#### AAB?\n\nAntifaschistische Aktion Berlin, was im Endeffekt ein Export ist aus Passau, weil die Passauer Antifa mit Hilfe des Paragraphen 129a kriminalisiert wurde, und deswegen ist fast die komplette Passauer Antifa in den 90ern nach Berlin umgezogen und hat hier die AAB aufgebaut.\n\n#### Okay, und du warst bei einer Uni-Antifa?\n\nGenau. Irgendwann wurde ich gefragt, ob ich diese Liste weitermache, weil die, die die Liste bis dahin gemacht hatten, da raus wollten und Nachwuchs gesucht haben, und dann hab ich die halt weitergef\u00fchrt. Das war eigentlich nie mein Wunsch, in 'nen Asta zu gehen, und ich hatte da auch eher eine randst\u00e4ndige Position. Antifa war nie mein Zuhause, mit Antifas konnte ich nichts anfangen, ich hab immer gesagt, Antifas sind die, die nicht tanzen k\u00f6nnen. Ich hab mich eher als Antira verstanden und mich noch so als Rest-Autonomer, sp\u00e4ter als Post-Autonomer und ganz individuell als bunter Schmetterling definiert... Auf jeden Fall bin ich zwar ins Antifa-Referat gegangen, hab aber nie das gemacht, was Antifas machen sollten, n\u00e4mlich mich mit Neonazis besch\u00e4ftigt. W\u00e4hrend das Antifa-Infoblatt so was hinbekommt wie eine Art \"Glatzen-Memory\", also Bildchen mit Namen drunter, und die werden dann von Antifas auswendig gelernt und die wollen die die ganze Zeit enttarnen und outen und so, besch\u00e4ftigen sich sozusagen in ihrem ganzen Alltag damit, rechte Strukturen ausfindig zu machen, ihre Recherchearbeit zu betreiben usw., hat mich das halt \u00fcberhaupt nicht interessiert. Ich hab mich viel mehr damit besch\u00e4ftigt, was im wissenschaftlichen Mainstream passiert, wo sind da die Linken, wo sind da die Rechten, also diese Intellektuellengeschichte. Ich wollte die Welt verstehen, mir begreifbar machen, anderen begreifbar machen und gegen wissenschaftlichen Rassismus vorgehen. Insofern konnte ich auch mit Antiras viel anfangen, wobei Antiras ja sehr damit besch\u00e4ftigt sind wiederum, so 'ne Art euphorischen Umgang mit schwarzen Menschen zu pflegen; also es gibt ja Autonome, die machen permanent diese Arbeit mit Migranten und waren dadurch in der Lage, mit total fremden Menschen ihren sozialen Umgang zu pflegen, und so war ich halt nicht, ich bin schon recht distanziert und sozio phob, und das schl\u00e4gt sich darin auch nieder, dass ich gar nicht so einen gro\u00dfen Umgang mit Migranten oder Illegalen gepflegt habe, sondern mich eher mit Theorien auseinandersetze, so 'ne Art B\u00fccherwurm geworden bin.\n\n#### Okay, das klingt so, als ob du dich sehr viel mit bestimmten Theorien auseinandersetzt. Welche Theorien beeinflussen dich da?\n\nMein Lieblingsphilosoph ist Michel Foucault [z\u00f6gert]... Aber ich versuch auch, den Blick f\u00fcr mich zu erweitern, versuche erstens zu verstehen, was denn Wolfgang Fritz Haug und der ganze Argument-Verlag, also diese ganzen Gramscianer, gegen Foucault haben, wie sie sich abgrenzen, aber auch adaptiert haben, also Gemeinsamkeiten und Unterschiede zwischen den beiden ber\u00fchmtesten linksradikalen Ideengebern. Fr\u00fcher konnte ich auch viel mit der Kritischen Theorie anfangen, hab sie zum Teil sogar auch verstanden, aber im Moment ist mein Interesse ein bisschen gestorben; stattdessen, interessiere ich mich jetzt sehr f\u00fcr die klassische Soziologie, f\u00fcr Max Weber usw. Fr\u00fcher war ich sehr lesefeindlich, da war ich so ein Bauch-Linksradikaler und hab eigentlich wenig gelesen, erst an der Uni brach das so richtig auf, lesen zu wollen, sich wo durchzuk\u00e4mpfen und nicht so sehr nachzuplappern. In meiner ersten Politikgruppe gab's eine kleine Autonomen-Bibel, das war Klaus Viehmann, die Triple-Oppression-Theorie [Dreifachunterdr\u00fcckung] mit den drei Widerspr\u00fcchen Rassismus, Sexismus und Kapitalismus, das war so die typische Lekt\u00fcre Anfang der 90er Jahre bei Autonomen. Dann hat man wiederum die irgendwann kritisiert. Also man war auf der Suche, meine erste Politikgruppe Spuk \u2013 Sch\u00fclerInnenplenum undogmatisch-kreativ \u2013 war schon ziemlich verkopft. Wir haben uns immer abgegrenzt von den anderen Antifa-Gruppen. Es gab drei gro\u00dfe Jugendgruppen in Berlin und wir waren weder die Epis, also Edelwei\u00dfpiraten, noch die Antifa-Jugendfront, sondern wir waren eher schon so die Mainstream... In unserer Gruppe waren so Lehrerkinder, da wurde halt immer erst gedacht und dann getan, also das ging da schon recht reflexiv zu.\n\n#### Und das war eine autonome Gruppe?\n\nJa, wir waren eine riesengro\u00dfe Gruppe, teils 60 Leute.\n\n#### Geh\u00f6rt Gewalt zur autonomen Szene dazu?\n\nJa, sie kann dazu geh\u00f6ren. Muss nicht. Sie ist halt ein Mittel. Es gibt auch nichtmilitante Autonome, aber wenn man sich als Autonomer versteht, geh\u00f6rt auf jeden Fall eine Solidarit\u00e4t mit denen dazu, die militant sind. Also sich selbst nicht zu trauen, das und das zu machen, aber klammheimliche Freude empfinden, wenn dann was kaputt geht.\n\n#### Wie stellst du zu Parteipolitik? Gehst du w\u00e4hlen?\n\nJa. Ich hab einmal in meinem Leben die SPD gew\u00e4hlt, [z\u00f6gert] als ich das erste Mal ALG II bekam, als ich endlich mal Kohle hatte, habe ich aus Begeisterung einmal aus Spa\u00df SPD gew\u00e4hlt. Und sonst w\u00e4hl ich einfach PDS. Auf jeden Fall muss man w\u00e4hlen gehen, da bin ich \u00fcberhaupt nicht dieser Anarcho.\n\n#### Und warum sollte man w\u00e4hlen gehen?\n\nWeil sonst gibt man die Stimme an die Rechten. Also muss man leider bei diesem Spiel mitmachen und das kleinste \u00dcbel w\u00e4hlen.\n\n#### Wie stehst du generell zu Parteipolitik?\n\n[Z\u00f6gert.] Wenn Parteien mit Autonomen klark\u00e4men, w\u00e4re ich auch in einer Partei. Ich k\u00f6nnte mir zum Beispiel vorstellen, bei den Gr\u00fcnen einen Job zu haben. Aber ich k\u00f6nnte es einfach nicht lassen, die zu provozieren, so 'ne Art Selbst-Outing, dass ich Autonomer bin, also praktisch so, als wenn die Sowjetunion der NATO beigetreten w\u00e4re, das w\u00e4r nat\u00fcrlich das Lustigste \u00fcberhaupt, wenn das ginge, wenn die das aushalten w\u00fcrden; aber das w\u00fcrden sie ja nicht aushalten, sie w\u00fcrden einen dann wieder aussto\u00dfen, Parteiausschlussverfahren. Also ich hab gar nichts gegen Parteien per se.\n\n#### Und wie ist dein Verh\u00e4ltnis zum Staat oder so staatlichen Institutionen wie Polizei etc.?\n\nDa ich kein MGler bin, nicht zur Marxistischen Gruppe usw. geh\u00f6re, sondern Foucaultianer bin, gibt es f\u00fcr mich den Staat gar nicht, sondern es gibt nur zig Millionen Menschen hier in Deutschland, die irgendwas machen. Ja, Polizei nervt, ich hab auch was gegen Polizisten und ihr Verhalten, gegen polizeiliche Arroganz. Wenn wir jedenfalls gute Polizisten h\u00e4tten, dann w\u00e4re das vielleicht schon ein Fortschritt. Also ich bin eigentlich daf\u00fcr, das, was Amnesty International macht, zu versuchen, also man muss die Polizei auf Menschenrechte trimmen, eigentlich auch auf den Rechtsstaat trimmen, weil die gehen zu weit, die haben den schon verlassen. Die d\u00fcrften nicht kaserniert ausgebildet werden, also da werde ich B\u00fcrgerrechtler, weil die eigentlich erst zu Arschl\u00f6chern gemacht werden, um dann Arschl\u00f6cher auch sein zu k\u00f6nnen. Insofern sollten Linksradikale auch so was wie Polizeisoziologie betreiben und so f\u00fcr \u00c4nderungsvorschl\u00e4ge sorgen. Nat\u00fcrlich w\u00e4re es nett, wenn man sie irgendwann abschafft, diese Einrichtung. Das Gewaltmonopol des Staates gibt es, es wird permanent durchgesetzt, bei jeder H\u00e4userr\u00e4umung, auf jeder Demo usw. Ja, aber nichtsdestotrotz gibt es den Staat nicht, sondern es gibt Menschen, die dieses und jenes machen. Da bin ich auch freundschaftlich diesen Gramscianern gesonnen, die einfach sagen, der Staat ist ein Teil der b\u00fcrgerlichen Gesellschaft. Das gef\u00e4llt mir schon mal besser, so als Aussage.\n\n#### W\u00fcrdest du sagen, dass das eine typische autonome Position ist?\n\nJa, es gibt dieses Sprechen unter Autonomen \u00fcber den Polizeistaat, aber das ist ein heuristischer und nicht weiter durchdrungener Begriff. So intellektuell sind die Autonomen gar nicht, die tun sich eher holprig und schwer mit Theorienbildung, das sind nur ganz wenige, die Autonomen scheinen nicht sehr sich in Sprache zu versuchen.\n\n#### Gibt es \u00c4nderungen im Umgang des Staates bzw. der Polizei mit Autonomen?\n\nJa, es ist immer schwieriger, man muss immer \u00fcberraschender sein, darum konzentriert sich das auch so um Kleinst- und Nachtaktionen. Auf jeden Fall gibt es weniger Autonome und die sind auch weniger militant als fr\u00fcher. Man ist immer weniger selbstbewusst und traut sich auch immer weniger und geht gro\u00dfen Polizeiansammlungen aus dem Weg, viel mehr als fr\u00fcher. Das war derma\u00dfen ein Ereignis, dass vor zwei Jahren mal ein Molli flog, und ich hab Demos mitbekommen, da flogen mehr als zwanzig Mollis. Also ich komme mir echt vor mittlerweile wie ein Dinosaurier. Ich m\u00f6chte jetzt nicht den Molli-Wurf verherrlichen, aber immerhin kann ich sagen, dass ich sozusagen die M\u00f6glichkeit sehr vermisse.\n\nDas Interview f\u00fchrte Anna Ziegler.\n\n## \"Gewalt sollte immer das Salz in der Suppe sein und nicht die Suppe selber!\"\n\nWolfgang, Ende 50, lebt in Berlin.\n\nWolfgang ist geschieden und ohne Partnerschaft. Tr\u00e4gt schwarze Kleidung und ist ein \"Ehemaliger\". Wir sind in seiner Wohnung im f\u00fcnften Stock in Berlin-Kreuzberg. Er fragt zur Begr\u00fc\u00dfung: \"M\u00f6chtest du Kaffee oder gleich einen Molotow-Cocktail?\" Wir trinken erst Kaffee. Im Hintergrund l\u00e4uft Musik, nicht zu laut, aber doch deutlich h\u00f6rbar.\n\n#### Macht Forschung \u00fcber die autonome Szene heute noch Sinn?  \nUnd wie sch\u00e4tzt du den momentanen Forschungsstand ein?\n\nAlso ich bin kein Theoretiker, ich war oder bin praktischer Autonomer. Forschen sollen die, die...\n\n####... wissenschaftlich arbeiten?\n\nJa. Aber was sollen die denn erforschen? Dass und wie viel die autonome Szene zerst\u00f6rt hat? Oder warum? Dazu sollte vielleicht auch gekl\u00e4rt werden, nach welchen Aspekten geforscht wird? Wird die autonome Szene nur negativ betrachtet? Einen Aufkl\u00e4rungsbedarf sehe ich. Gerade hier in Kreuzberg! St\u00e4ndig liest und h\u00f6rt man von Mietsteigerungen, die Hausbesitzer sahnen jetzt so richtig ab und gentrifizieren (\u00fcbrigens ein unheimlich sch\u00f6nes Wort f\u00fcr die Leute, die ihr Kapital vermehren wollen). Schau dich um, das Klientel unseres Viertels hat sich immens ver\u00e4ndert, es gibt keine Ans\u00e4tze, wo etwas dagegengesetzt werden kann, und die Leute begreifen das auch nicht, sondern machen wie die Lemminge alles mit. Ich finde, man sieht gerade in der Hinsicht, wie losgel\u00f6st die autonome Szene von den normalen Bev\u00f6lkerungsschichten inzwischen ist. Da ist nichts mehr, was vermittelt wird. Oder was an kreativen Aktionen versucht wurde, die Hausgemeinschaften, die alternativ Lebenden, Mietergruppen mit einzubinden und auf eine gemeinsame Linie gegen irgendwelche Hausbesitzer oder so zu kommen. Das klappt nicht, vielleicht sollte da mal geforscht werden.\n\n#### Dann setzt du also Forschung mit Aufkl\u00e4rung gleich?\n\nJa, denn geforscht haben damals die Altanarchisten, zum Beispiel Kropotkin [russischer Anarchist und Schriftsteller aus dem sp\u00e4ten 19. Jahrhundert, Die Eroberung des Brotes], um aufzukl\u00e4ren. Die haben den Begriff ja erst mal mit Leben gef\u00fcllt. Aber f\u00fcr mich gibt es keine Autonomie, die nur am Schreibtisch entsteht und besteht.\n\n#### Welche Ver\u00e4nderungen hast du bei den Autonomen seit ihrer Entstehung wahrgenommen und wie lassen sie sich mit den gesellschaftlichen Wandlungsprozessen in Verbindung bringen?\n\nFr\u00fcher waren die Gruppierungen der Antiimps, jetzt sind es wohl die Antideutschen, nicht so stark ausgepr\u00e4gt. Die autonome Szene war nicht so zersplittert. Es gab einen gemeinsamen Feind, den es zu bek\u00e4mpfen galt, Widerspr\u00fcche und Nebenwiderspr\u00fcche wurden erst mal hinten angestellt. Es gab Wichtigeres.\n\n#### Sind denn die Autonomen aus den 80er Jahren mit denen von heute zu vergleichen?\n\nDie Spr\u00fcche haben sich nicht ge\u00e4ndert, h\u00f6chstens der modebewusste Stil. Also die Hasskappen zum Beispiel werden jetzt anders gepflegt. Die Farben und die Machart haben sich ver\u00e4ndert, aber sonst? Nein, hat sich nichts ge\u00e4ndert. Es werden immer noch die gleichen stilistischen M\u00f6glichkeiten in Anspruch genommen.\n\n#### Du meinst die Parolen? Die Argumente?\n\nJa, die sind die gleichen geblieben.\n\n#### Lassen sich denn die sozialen und biografischen Milieus beschreiben, aus denen sich die Angeh\u00f6rigen der autonomen Szene zusammensetzen?\n\nDas Vorurteil w\u00e4re: schw\u00e4bische Kleinstadtkinder [er lacht], die bevorzugt aus der Enge ihres Elternhauses... \u2013 ach nein, das waren die von der RAF, die aus Pfarrhausfamilien kamen... Nee, ich denke mal, jeder, unabh\u00e4ngig von Herkunft und Geschlecht, der sich ungerecht behandelt f\u00fchlt und auch ein bisschen unabh\u00e4ngiger im Denken ist, findet sich in einer bestimmten Szene, also auch in den Autonomen, wieder.\n\n#### Welche Rolle spielen die Frauen in der autonomen Szene?\n\nWie fast \u00fcberall in dieser Gesellschaft eine untergeordnete... Ich kenne keine Statistiken, aber der Frauenanteil bei Demos und Hausbesetzungen d\u00fcrfte nicht \u00fcber f\u00fcnfzig Prozent liegen.\n\n#### Gibt's \u00fcberhaupt geschlechtsspezifische Unterschiede? Verhalten sich die Frauen in autonomen Gruppen zum Beispiel aggressiver oder kommunikativer?\n\nZwei meiner Freundinnen aus der autonomen Szene kamen aus Schwaben und waren sehr fordernd und sehr aggressiv, wenn es nicht so lief, wie sie es wollten. Eine Zeit lang war es so, dass Frauen in politischen Zusammenh\u00e4ngen Sachen fordern konnten, die nicht unbedingt mit der realen Lebenssituation in Einklang zu bringen waren. Dort mussten st\u00e4ndig Kompromisse gefunden werden, was wir auch taten, weil wir ja alle sehr bewusst waren und an uns arbeiten wollten.\n\n#### Was sind die zentralen Medien und Informationsvermittler f\u00fcr die Szene?\n\nDas ist in erster Linie das Internet. Es greift immer mehr und es ist sehr sch\u00f6n, dass da so gut wie nicht zensiert werden kann. Dass es also wirklich \"free speech\" ist, freie Meinung \u00fcberall. Ansonsten radikal und Interim. Es gibt immer wieder Wellen, bei denen versucht wird, diese Zeitschriften zu kriminalisieren und mundtot zu machen. Das ist einfach l\u00e4cherlich! Vorgestern sind in so einem Infoladen in einem kleinen Kaff circa drei\u00dfig B\u00fcttel eingeritten, um f\u00fcnf Interim, die da wochenlang herumlagen, zu beschlagnahmen. S\u00e4mtliche Personen wurden ID-behandelt und teilweise festgenommen, das steht dann in keinem Verh\u00e4ltnis mehr. Das ist l\u00e4cherlich. Kann ich kiffen oder ist das zu unpolitisch?\n\n#### Nach dem Ende des linken Terrorismus durch zum Beispiel die RAF sind es vor allem Autonome, die als linke Gefahr f\u00fcr die \u00f6ffentliche Ordnung und Sicherheit wahrgenommen werden. Wie gef\u00e4hrlich sind Autonome tats\u00e4chlich f\u00fcr die \u00f6ffentliche Sicherheit und Demokratie?\n\n[Er lacht laut.] Ist die Frage ernst gemeint?\n\n#### Nat\u00fcrlich!\n\nIn den momentan herrschenden gesellschaftlichen Zust\u00e4nden ist das alles (leider, leider, leider) Kinderkacke, was gespielt wird. Es gibt immer weniger Freir\u00e4ume, wo Leute wirklich sich ausleben k\u00f6nnen. Beispiel: Im Tommy-Haus [Tommy Weisbecker Haus] ist jahrelang mit \u00f6ffentlichen Geldern befriedet worden; pl\u00f6tzlich, als dort eine Informationsveranstaltung zu den politischen Gefangenen der RAF durchgef\u00fchrt wurde, wurde der Geldhahn abgedreht. Bestimmt nur ein Zufall... Wie war die Frage?\n\n#### Wie gef\u00e4hrlich sind sie nun wirklich f\u00fcr die \u00f6ffentliche Sicherheit und Demokratie?\n\n\u00dcberhaupt nicht! Leider nicht!\n\n#### Also verpuffen die Aktionen eher?\n\nEs ist nicht vermittelbar. Das, was an Aktionen passiert, ist nicht vermittelbar. Ich hab manchmal auch 'ne klammheimliche Freude, wenn ein sogenanntes Bonzenauto abgefackelt wird, aber es ist nicht vermittelbar, warum das passiert! Und der B\u00fcrger, der durch die Springerpresse manipuliert wird, hat daf\u00fcr einfach kein Verst\u00e4ndnis und der wei\u00df auch nicht, wieso die Leute, die so etwas tun, gar nicht offiziell darlegen, warum sie das machen. Kann nat\u00fcrlich auch sein, dass die Autos gar nicht von Autonomen abgefackelt wurden, sondern Versicherungsbetrug sind. Nein, ein ganz klares Nein: Sie sind der \u00f6ffentlichen Sicherheit und Demokratie nicht gef\u00e4hrlich! Sie werden jedoch so dargestellt und dementsprechend auch wahrgenommen, und das ist das Gef\u00e4hrliche. Das schnallt nur keiner.\n\n#### Weil da die Aufkl\u00e4rung an sich fehlt? Oder das \u00f6ffentliche Interesse?\n\nJeder ist mit sich selbst besch\u00e4ftigt. Die Leute, die noch Arbeit haben, sind froh dar\u00fcber und halten still. Jetzt streiken die Bahner und was h\u00f6rt man da f\u00fcr Meinungen? \"Was, die Leute kommen nicht zur Arbeit!\" Sie sind komplett emp\u00f6rt, dass die nicht zur Arbeit kommen. Aber wenn sie davon selber betroffen sind, wenn es um ihren Streik geht, um ihren Lohn und ihre sozialen Rechte, dann streiken sie. Ich war jahrelang Gewerkschafter und habe mich auch dar\u00fcber radikalisiert. Ich war manchmal entsetzt, dass die Leute nichts mit der Gewerkschaft am Hut hatten, nichts daf\u00fcr tun wollten, noch nicht mal in die Gewerkschaft eingetreten sind, aber die Hand aufgehalten haben, wenn es darum ging, die 1,9 Prozent Lohnerh\u00f6hung in Anspruch zu nehmen. Es bei\u00dft sich. Wenn Leute Anspr\u00fcche haben, m\u00fcssen sie auch was daf\u00fcr tun.\n\n#### Ist die Reduzierung der autonomen Szene auf Gewalt sinnvoll und richtig?\n\nN\u00f6. Da gibt es einen sch\u00f6nen Satz aus der radikal 53 aus dem Jahr 1973 (kann allerdings auch ein ganz anderer Jahrgang sein): \"Gewalt sollte immer das Salz in der Suppe sein und nicht die Suppe selber!\" Und das ist eigentlich der einzige Satz, der stimmt. Wenn Leute irgendwas abfackeln oder kaputt machen, siehe damals '68, als die Springerautos abgefackelt wurden, dann ist es ein klares politisches Zeichen, was ich vollkommen richtig finde. Wenn die RAF allerdings einen kleinen US-Soldaten erschossen hat, Pimental, nur um an den Ausweis von dem zu kommen, so ist das vollkommen daneben. Unabh\u00e4ngig davon hat die RAF grunds\u00e4tzlich meine Solidarit\u00e4t, das hei\u00dft aber nicht, mit jedem Schei\u00dfdreck einverstanden zu sein.\n\n#### Gibt es eine einheitliche Haltung innerhalb der autonomen Szene zur Gewalt?\n\nNein, nat\u00fcrlich nicht, da musst du nur mal auf s\u00e4mtliche Internet-Plattformen schauen, die Meinungen verlaufen meistens kontr\u00e4r. Dabei ist es vielleicht wichtig zu unterscheiden, ob die Gewalt gegen Sachen oder Personen gemeint ist. Ich denke, dass da eher eine positive Meinungseinheit bei Gewalt gegen Sachen besteht, besonders, wenn die begr\u00fcndet wird. Dann kommt es auch immer auf die Emotionen an, die dahinterstecken. Wenn ich zum Beispiel auf einen aggressiven Fascho treffe, so kann es sein, dass es zur Gewalt kommt, auch wenn ich sonst gegen Gewalt bin.\n\n#### Wie sch\u00e4tzt du denn die gesamtgesellschaftliche Bedeutung der Autonomen ein?\n\nGeh doch mal auf die Stra\u00dfe und frag Tante M\u00fcller von nebenan: \"Was halten Sie von den Autonomen?\" 80 Prozent denken, es ist 'ne unbekannte Automarke, 19,9 Prozent sagen, das sind alles Verbrecher. Und nur 0,1 Prozent der Gesamtbev\u00f6lkerung setzt sich tats\u00e4chlich mit den Zielen oder auch Idealen und Tr\u00e4umen von autonomen Gruppen auseinander. Ich denke mal, keine gesellschaftliche Verankerung.\n\n#### Und welche M\u00f6glichkeiten haben die Autonomen, da etwas zu ver\u00e4ndern?\n\nTja [lacht kurz auf], Liebe, Lachen, Glauben, K\u00e4mpfen. Vorleben! Praktisch vorleben und auf Leute zugehen.\n\n#### Also offener sein?\n\nViel offener sein, auf jeden Fall. Ich war damals viel offener. Nicht dass ich besser war als die anderen, aber ich hab mit jedem geplaudert. Sei es mit der Supermarkt- oder B\u00e4ckereifachverk\u00e4uferin oder dem Bankangestellten, wo ich heimlich mein Geld abgehoben habe. Ich war zu jedem freundlich und zu jedem nett, und mich konnte man auch fragen, und so konnte ich mich auch austauschen und mit ihnen diskutieren. Ohne meine Meinung verstecken zu m\u00fcssen. Aber man hat mich sehr oft als Menschen im Sinne von: Der ist ja gar nicht so! eben anders wahrgenommen, als wenn ich gr\u00f6lend und marodierend durch die Gegend gezogen w\u00e4re.\n\n#### Du findest, dass die autonome Szene sich zu sehr abschottet?\n\nZu klerikal sind, ja! Sie grenzen sich ja sogar gegeneinander noch mal ab. Ein Beispiel: Eine bekannte Band von mir gab in der K\u00f6pi ein Solikonzert f\u00fcr eine Tierrechtsorganisation. Allerdings benutzten sie farbige Plakate zur Ank\u00fcndigung und einen Werbetr\u00e4ger unterhalb des Plakates. Jetzt d\u00fcrfen sie in der K\u00f6pi nicht mehr spielen, haben verschissen. Das darf nicht sein: keine farbigen Plakate und keine Werbung. Und dementsprechend darf diese Band jetzt dort nicht mehr spielen. Und das verstehe ich einfach nicht, wie man wegen einem Popelfatz nicht mehr normal miteinander reden kann?\n\n#### Sie sind intolerant untereinander?\n\nParagraph 227 des ungeschriebenen Gesetzes, ja!\n\n#### Denkst du, dass sie \u00fcberhaupt ein Interesse haben, die Gesellschaft als Ganzes zu \u00e4ndern? Oder geht es nur darum, eigene Partikularinteressen und Freir\u00e4ume zu verteidigen?\n\nDie Gesellschaft ver\u00e4ndert sich jeden Tag, st\u00fcndlich. Von ganz allein. Und dieses Wort \"Partikularinteressen\" zeigt deutlich, dass es schon mit der Sprache beginnt. Sehr viele Leute k\u00f6nnen sich gar nicht miteinander austauschen, weil sie die Sprache oder Worte des Gegen\u00fcbers gar nicht verstehen. Ich bin ein einfacher Mensch des Volkes, ich hatte auch ein Leben vor der Autonomie, den Herrn Dutschke habe ich sehr oft nicht verstanden, weil ich nicht die Macht dieser Worte hatte. Das war aber auch nicht die Frage. Was ist ein Freiraum? Wenn Tante M\u00fcller in ihrem Kleingarten sitzt, dann ist das auch ein Freiraum. Und wenn Leute nicht gesellschaftskonform sind, dann sind sie einer st\u00e4ndigen Repression ausgesetzt. Deshalb m\u00fcssen sie f\u00fcr ihre Freir\u00e4ume k\u00e4mpfen, was allerdings der Ver\u00e4nderung der Gesellschaft als Ganzes bedarf.\n\n#### Wie beschreibst du \"autonom\"?\n\n\"Wir sind die wilden Horden, wir pl\u00fcndern und wir morden. Wir waschen uns nie, Anarchie!\" Reicht das?\n\n#### Wo ist das her? Aus einem Lied?\n\nNein, das ist eine Eigenkreation, ist in der Kneipe entstanden, und irgendwann haben wir es dann immer auf Demos gebr\u00fcllt.\n\n#### Wie sch\u00e4tzt du die Zukunft der autonomen Bewegung ein?\n\nNeue Generationen bringen immer auch neue Ans\u00e4tze und neue M\u00f6glichkeiten. Ich hoffe und w\u00fcnsche mir sehr, dass kommende Generationen derma\u00dfen unzufrieden sind, dass sie sich trauen, radikaler und freier zu denken und zu handeln. Gerade jetzt, wo alles nur noch um Kommerz geht, man muss die neuesten Nike-Schuhe tragen, sonst ist man nicht in, dass sich diese Zust\u00e4nde wieder gesellschaftlich ver\u00e4ndern k\u00f6nnen. Ich hoffe es sehr! F\u00fcr die autonome Szene w\u00fcrde es auch mehr Zulauf bedeuten; wie sich das jedoch auswirkt, bleibt abzuwarten.\n\n#### Kannst du ungef\u00e4hr absch\u00e4tzen, wie gro\u00df die autonome Szene in Deutschland ist?\n\nDa m\u00fcsste man erst mal definieren, wer sich im weitesten Sinne alles dazu z\u00e4hlt. Berlin und Hamburg sind eindeutig die Headquarters. Dann traditionell dort, wo Studenten sich zusammenfinden, zum Beispiel Heidelberg oder Freib\u00fcrg. Aber auch in jeder kleineren Stadt, in jedem Dorf, wenn dort autonome Jugendzentren sind, gibt es einige Leute, die sich anders begreifen, autonomer begreifen. Allerdings hat man auch den bitteren Beigeschmack, dass es fast immer vom Staat finanziert wird. Dementsprechend schwebt auch immer der leise Hauch dar\u00fcber, dass die Leute dadurch kanalisiert werden, dass es eine nicht selbst bestimmte Autonomie ist, sondern kritische Jugendliche werden durch die Aufsicht von Erziehern an der langen Leine gelassen. Jetzt bin ich schon wieder abgeschweift. Der Verfassungsschutz kann besser beurteilen, wer zum harten Kern oder zum weiten Umland geh\u00f6rt. Ich kann und will das auch nicht sch\u00e4tzen.\n\n#### Denkst du, dass es Ver\u00e4nderungen im Sinne von Fluktuation gibt?\n\nNat\u00fcrlich! Die Alten treten ab, die Familie ist gegr\u00fcndet, das Studium ist beendet, die erste Vorstrafe hat noch mal einen heiligenden Erfolg gekr\u00f6nt. Es fluktuiert sogar stark. Es gibt kaum Autonome, die \u00fcber drei\u00dfig Jahre alt sind.\n\n#### Wie sieht es mit der Solidarit\u00e4t der Autonomen untereinander aus?\n\nIch finde es sehr wichtig, dass Leute, die f\u00fcr l\u00e4ngere Zeit aufgrund von politischen Aktionen in den Knast kommen, irgendwann nicht feststellen m\u00fcssen: Huch, es ist ja niemand mehr da! Dass die dort vereinsamen und keine Chance haben, in die Szene zur\u00fcckzukehren, oder noch viel schlimmer, keine Solidarit\u00e4t mehr haben. Leute, die drin sind, k\u00f6nnen sehr leicht vergessen werden.\n\n#### Ist es so, dass die Autonomen untereinander wenig Solidarit\u00e4t aufzeigen?\n\nNein, das kann man nicht sagen. Aber bei der starken Fluktuation ist es halt zu bef\u00fcrchten. In den Zusammenh\u00e4ngen, die ich kenne, finden viele Soli-Aktionen statt. Da wird sehr viel getan. Es ist so, dass die private Solidarit\u00e4t eher fl\u00f6ten geht als die, wenn es darum geht, f\u00fcr jemanden eine Party zu organisieren, um eine Strafe zu bezahlen. Oder ihm im Knast B\u00fccher zukommen zu lassen. Das funktioniert. F\u00fcr mich wurde auch mal ein Soli-Konzert gegeben. Es war wunderbar, es ist unheimlich viel Geld rumgekommen, die Anw\u00e4lte tanzen wahrscheinlich immer noch im Dreieck vor Freude. Es war wirklich wunderbar. Aber trotzdem stand ich dann morgens um vier Uhr alleine in einem riesengro\u00dfen Raum und konnte zehn Stunden alleine sauber machen, weil alle gegangen waren.\n\n#### Wie war dein erster Hontakt zur autonomen Szene?\n\nIch war Supermarktleiter. Hab mir aber immer die taz gekauft, bin im Betriebsrat gewesen und ich fand sehr sympathisch, was die Hausbesetzer-Szene machte. Ich hatte mir meine erste Lederjacke zugelegt, was bei meinen Vorgesetzten nat\u00fcrlich \u00fcberhaupt nicht gut ankam. Und eines Abends, als ich mit meiner Assistentin den Laden abschloss, lief gerade eine Demo vorbei. Bullen standen vor dem Laden, nahmen mich und warfen mich in die Demo rein, \u00e4u\u00dferst r\u00fcde! Und meine Assistentin nat\u00fcrlich auch. Ab da war ich dabei!\n\n#### Wie alt warst du da?\n\nEtwas \u00fcber drei\u00dfig.\n\n#### Denkst du, dass es auch Ankn\u00fcpfungspunkte an andere Jugendkulturen gibt?\n\nAutonomie ist keine Jugendkultur! Autonomie ist ein \u00e4u\u00dferst schwammiger politischer Begriff. Jugendkultur definiert sich \u00fcber Punk, \u00fcber Skins, \u00fcber Popper, \u00fcber was wei\u00df ich nicht noch alles. Das l\u00e4sst sich nicht vergleichen, nein. Es gibt ja bei Punks, wo man annehmen sollte, dass sie automatisch links sind, auch Rechte. Jugendgruppen definieren sich ganz stark unpolitisch! Nein, Autonome sind keine Jugendkultur, obwohl sie von Jugendlichen gepr\u00e4gt sind.\n\n#### Einen eigenst\u00e4ndigen autonomen Stil, gibt es den?\n\nJa, der \u00e4u\u00dferst erfolgreich inzwischen von den autonomen Nationalisten kopiert wird, die \u00fcbrigens alles klauen. Die klauen die ganzen Spr\u00fcche und h\u00f6ren selbst Slime und Ton Steine Scherben, was f\u00fcr 'ne Schande!\n\n#### Spielt denn die Musik f\u00fcr die Autonomen eine Rolle!\n\nNa klar! Slime haben mich gepr\u00e4gt! Sie haben mich aufgew\u00fchlt, haben bei mir tiefe Emotionen erzeugt und es mit einfachen Texten auf den Punkt gebracht. Selbst heute, auf jeder Demo h\u00f6rst du immer noch Slime. Allein schon, um es \"Polizei, SA, SS\" den Bullen zu zeigen.\n\n#### Welche Themen waren denn f\u00fcr dein Engagement besonders wichtig?\n\nEigentlich alles, was gegen die herrschenden Normen ging. Ich war noch nie ein gro\u00dfer Theoretiker, ich war mehr ein praktischer Mensch. Ich hab noch nie gro\u00dfartig Schrifts\u00e4tze verfassen k\u00f6nnen oder Erkl\u00e4rungen schreiben wollen, sondern ich hab die Basis praktisch durch Konzerte, durch Teilnahme an Infol\u00e4den oder anderen politischen Gruppierungen unterst\u00fctzt.\n\n#### Wie stehst du zu Parteipolitik? Welche Partei w\u00fcrdest du denn w\u00e4hlen, wenn du w\u00e4hlen gehst?\n\nIch war noch nie w\u00e4hlen!\n\n#### Nein? Dein Verh\u00e4ltnis zu Staat und Polizei ist wie? Damals und heute, gibt es da Unterschiede?\n\nIch bin abgeschliffen. Ich hab mein stillschweigendes Agreement inzwischen gefunden, ich kann ja auch nicht mehr so schnell wegrennen. Polizei wird, so wie ich sie erlebt habe, f\u00fcr mich immer ein Feindbild sein, das steht einfach mal fest. Es gibt f\u00fcr mich keine gute Polizei mehr.\n\n#### Wo siehst du denn Kritikpunkte und Probleme in der autonomen Szene?\n\nDas w\u00fcrde den Zeitplan dieses Nachmittags sprengen! Das hatten wir auch schon: Intoleranz und die Unf\u00e4higkeit zur Selbstkritik. Es ist eine kleine klerikale Szene.\n\n#### Und die \u00f6ffentliche Wahrnehmung der autonomen Szene, wie wertest du die?\n\nFrau Mustermann bekommt von denen, au\u00dfer die Springerpresse br\u00fcllt mal wieder ganz aufgebracht, \u00fcberhaupt nichts mit. Das ist f\u00fcr sie ein nicht verst\u00e4ndliches Ph\u00e4nomen.\n\n#### Weil die Autonomen immer nur mit Gewalt in Verbindung gebracht werden?\n\nUnd das mit wenig Inhalten.\n\n#### Ich merke, du wirst immer einsilbiger, ich komme auch schon zur letzten Frage, sie ist vielleicht sehr pers\u00f6nlich: Was hat bei dir zum Ausstieg aus der autonomen Szene gef\u00fchrt?\n\nEin ganz schlimmer Vorfall und er macht mich immer noch w\u00fctend. Aber ich werde mich nicht dazu \u00e4u\u00dfern.\n\n#### Vielen Dank f\u00fcr das Gespr\u00e4ch, es war sehr interessant.\n\nDas Interview f\u00fchrte Bettina Thiele.\n\n## Von der Sponti-Schule zum Antifa-Shopping.\n\nWolf Wetzel, 58 Jahre, Publizist, lebt in Frankfurt.\n\nWolf Wetzel, Jahrgang 1956, war Mitglied der ehemaligen autonomen L. U.P. U.S.-Gruppe, in den 90er Jahren Mitglied im antirassistischen Plenum\/ Frankfurt, Mitinitiator des Aufrufes Die Brandstifter sitzen in Bonn (Bundestagblockade anl\u00e4sslich der Abschaffung des Asylrechtes 1993) und zwischen 2001 und 2007 Mitglied in der AntiNaziKoordination\/ANK in Frankfurt. Seit 2011 Vorstandsmitglied von Business Crime Control\/BCC Frankfurt.\n\nZur Geschichte der H\u00e4userk\u00e4mpfe in den 70er und 80er Jahren finden sich viele Texte in dem von Wolf Wetzel zusammengestellten Buch: H\u00e4userk\u00e4mpf I: Wir wollen alles \u2013 der Beginn einer Bewegung, Bibliothek des Widerstandes, Band 21, LAIKA-Verlag Hamburg, plus zahlreiche Videofilme zum Thema.\n\n#### Vielleicht kannst du zu Beginn noch mal kurz was zu deiner eigenen Person sagen, was du machst und wie du dich heutzutage einordnen w\u00fcrdest in Bezug auf die radikale Linke oder die autonome Szene?\n\nEin gutes St\u00fcck meiner Besch\u00e4ftigung ist die publizistische Aufarbeitung der Geschichte der radikalen Linken. Das ist ja gerade in Berlin im Moment ungewollt Thema. Die H\u00e4userkampf-Bewegung der 80er Jahre steht vor dem Punkt, die Erfolge verteidigen zu m\u00fcssen, da sehr viele Nutzungsvertr\u00e4ge auslaufen und die Stadt Berlin nat\u00fcrlich Interesse daran hat, das Thema \"H\u00e4userkampf\" ganz still abzuwickeln. Von daher gibt es gute Gr\u00fcnde, zur\u00fcckzuschauen und sich selbst zu fragen: Mit was ist man angetreten, was hat man davon (nicht) einl\u00f6sen k\u00f6nnen und woran lag es? Lag es an einem selbst, lag es an den anderen, an gl\u00fccklichen\/widrigen Umst\u00e4nden? Und dann bin ich nach wie vor politisch aktiv, wie zum Beispiel in der Anti-Nazi-Koordination, Dresden 2010 oder alles, was es politisch noch gibt und wo ich das Gef\u00fchl habe, das k\u00f6nnte auch aufgrund meiner Geschichte f\u00fcr mich interessant sein.\n\n#### Wann beginnt deine \"politische\" Geschichte?\n\nIm Prinzip beginnt meine Geschichte in den 70er Jahren in Frankfurt. Ein gl\u00fccklicher Umstand und gl\u00fcckliche Zeiten, weil in den 70er Jahren gab es einen recht massiven H\u00e4userkampf, der auch viele prominente Gr\u00f6\u00dfen hervorgebracht hat wie Joschka Fischer und Daniel Cohn-Bendit und viele andere. In der Zeit bin ich gro\u00df geworden. Aber da war ich noch nicht richtig politisch aktiv. Da war ich so 17 oder 18, bin auf die Demonstrationen vom H\u00e4userrat gegangen und habe geschaut, dass ich einigerma\u00dfen mit den Ereignissen Schritt halten konnte. Das hat aber wesentlich meine politische Geschichte gepr\u00e4gt, weil es in diesem H\u00e4userkampf und in dem, was damals Spontis genannt wurde, eine ganz eigene politische Antwort auf '68 gab. Und das war f\u00fcr uns als Jugendliche damals sehr interessant. Also der Marsch durch die Institutionen, den ja ein gro\u00dfer Teil der '68er gemacht hat, interessierte uns \u00fcberhaupt nicht. Die Resignation interessierte uns nat\u00fcrlich auch nicht. Von daher waren die Spontis f\u00fcr uns wichtig, weil zwischen Resignation und dem Versuch, mit dem Marsch durch die Institutionen die Verh\u00e4ltnisse zu \u00e4ndern, haben wir versucht, viel aufzuschnappen, was die sich dazu gedacht haben. Und von daher war das so \u00fcber Jahre meine zweite Schule oder die eigentliche Schule, in die ich gegangen bin.\n\n#### Hast du diese Szene gesucht oder hat die dich schon umgeben und du bist dadurch da reingewachsen? Oder gab es ein bestimmtes Ereignis, dass du angefangen hast, dich mehr zu engagieren?\n\nBeides. Die H\u00e4userkampf-Bewegung in Frankfurt war nat\u00fcrlich nur die letzte Klammer, es gab ganz viele Ausstrahlungspunkte und viele politische Orte, wo diese Idee virulent war. Es gab eine Jungarbeiterbewegung, es gab die Heimkinder, die revoltiert haben, es gab eine starke Jugendzentrumsbewegung, in der ich auch aktiv war. Damals ging es um selbstverwaltete Jugendzentren, also im Prinzip eine \u00e4hnliche Adaption: Du besetzt irgendein Geb\u00e4ude und forderst ein selbstverwaltetes Jugendzentrum. Und es gab nat\u00fcrlich auch schon die Frauenbewegung. Die wollten ja nicht nur was von den Herrschenden, sondern auch von den M\u00e4nnern, mit denen sie zusammen sein wollten \u2013 oder eben nicht mehr. Zur selben Zeit haben wir einen Schulstreik gemacht, da ging es um den Numerus clausus, wo wir versucht haben, ein bisschen die Ideen, die \u00fcberall um uns herum zirkulierten, aufzugreifen und umzusetzen. Andere Gruppen haben ein Haus besetzt in Offenbach, ich glaube, es war das erste besetzte Haus nach 1945, und da sind wir als Sch\u00fcler damals hingegangen, weil die auch ein autonomes Zentrum forderten.\n\n#### W\u00fcrdest du sagen, dass es eine Jugendbewegung war zu der damaligen Zeit?\n\nIch finde den Begriff viel zu begrenzend. Die Leute sind nicht auf die Stra\u00dfe gegangen oder dem Wunsch nach einem anderen Leben gefolgt, weil sie jung waren, sondern weil sie andere Ideen hatten. Weil sie mit dem, was sie vorfanden in ihrer Familie, in der Schule, auf der Stra\u00dfe, nicht zufrieden waren. Das Problem, dass diejenigen, die dann \u00e4lter werden, irgendwann resignieren oder sich zur\u00fcckziehen oder das Gegenteil machen von dem, von dem aus sie losgelaufen sind, das ist weniger ein Problem des Alterns als das Problem, damit umzugehen, dass nat\u00fcrlich die Vorstellungen, die man hat, nicht herzlich willkommen sind in der Gesellschaft. Auch wenn es in der Demokratie formal ja erw\u00fcnscht ist, andere Ideen zu haben. Aber wenn man auf diese Ideen besteht, bekommt man die Kehrseite mit, n\u00e4mlich, dass man daf\u00fcr ziemlich viel einstecken muss, mit Repression umgehen muss, nicht Karriere macht, nicht gef\u00f6rdert wird, nicht als belebendes Element der Demokratie begriffen wird, sondern eigentlich alles erf\u00e4hrt, was einem das Leben schwerer macht.\n\n#### Vielleicht kannst du noch etwas zur Rolle von Frauen in der Bewegung sagen?\n\nDu bist 17 oder 18, du hast ein spie\u00dfiges bis reaktion\u00e4res Elternhaus, was damals der Durchschnitt war, die alle noch im Wesentlichen von der Nazizeit gepr\u00e4gt waren, du hast also dieses Bed\u00fcrfnis, da auszubrechen und irgendetwas anderes zu erleben. Auch eine Beziehung und Freundinnen zu haben und nicht zu warten, bis man sie heiratet, also all das, was in den 70er Jahren ja noch massiv an Konventionen und Vorstellungen da war, wie so ein Ehegl\u00fcck aussehen soll. Die Bewegung, der H\u00e4userkampf oder der Kampf um die Jugendzentren, das war nat\u00fcrlich ein Traum, weil zum ersten Mal Platz war, wo man etwas ausprobieren konnte, etwas anders machen konnte als das, was die Eltern von einem erwartet haben. Und von daher war es erst mal f\u00fcr alle Jugendlichen, f\u00fcr Jungs und M\u00e4dchen, ein gro\u00dfartiger Ort, aufzuatmen und auszuprobieren, herumzuexperimentieren. Das Terrain zu erkunden, was gibt es jenseits der Ehe, was hei\u00dft eine \"lockere Beziehung\"? Wie soll man das machen, wenn es eigentlich so gut wie keine Vorbilder gibt? Irgendwie hat man nat\u00fcrlich ein bisschen was von der '68er Bewegung mitbekommen, aber im Wesentlichen schon durch den Filter der Nachrichtenmedien. Wir waren ja selbst nicht dabei. Da gibt es ja angeblich den ber\u00fchmten Spruch \"Wer zweimal mit der derselben pennt, geh\u00f6rt schon zum Establishment\". War das jetzt wirklich die Meinung der '68er Bewegung in der Mehrheit oder war das einfach ein medialer Plot, um irgendetwas f\u00fcr allgemeing\u00fcltig zu erkl\u00e4ren, was f\u00fcr die '68er Generation mit Sicherheit nicht galt? F\u00fcr uns \u2013 oder zumindest f\u00fcr mich, kann ich nur sagen \u2013 war es einfach erst mal toll, alles auszuprobieren oder erleben zu k\u00f6nnen, was jenseits dieser spie\u00dfigen, elterlichen Verh\u00e4ltnisse war.\n\nWie war unser Verh\u00e4ltnis zu den M\u00e4dchen? Also ich kann mich jetzt selten dran erinnern, dass es richtige Auseinandersetzungen gab: Wie k\u00f6nnte ein Mann sein, anders sein als soldatisch zu sein, herrschs\u00fcchtig und die Frau als Nebengesch\u00f6pf oder Beiwerk zu betrachten? Man hat nur alles versucht oder ausprobiert, was jenseits dieser autorit\u00e4ren oder reaktion\u00e4ren Zuschreibung stattfand. Und vieles hat auch nicht geklappt, weil wir ja alle in den Verh\u00e4ltnissen gro\u00df geworden sind, und mussten dann nat\u00fcrlich merken, wo man sie wieder reproduziert, wo man genau dasselbe macht, wo man Frauen ebenfalls abf\u00e4llig behandelt. Warum sollten wir vom Planeten Utopia gefallen sein? Das wird immer hin und her gegangen sein, und nicht ohne Grund entstanden dann ja auch aus der Bewegung heraus Frauengruppen, die nicht nur das Patriachat ganz oben in der Regierung oder in der Wirtschaft oder im Milit\u00e4r verortet haben, sondern es gab auch die Kritik, dass viele M\u00e4nner, mit denen sie zusammen waren, mit denen sie eine Beziehung hatten, ganz viel von dem mit ins Bett genommen haben, mit in die Beziehung, was man eigentlich bek\u00e4mpfen wollte. Und das war auch der Grund, warum es dann eine eigenst\u00e4ndige Frauenbewegung gab, warum sich Frauen selbst organisiert haben und es bei uns in den 80er Jahren dann auch Frauen- und M\u00e4nnerplenen gab.\n\n#### Also das mit der Gleichberechtigung hat nicht so unbedingt geklappt?\n\nWie sollte es auch? Man kann sich nicht zum neuen Mann deklarieren. Abgesehen davon haben sich auch die Frauen in der Frauenbewegung nicht sofort als neues Geschlecht etablieren k\u00f6nnen. Alle leben in Verh\u00e4ltnissen, die sie v\u00f6llig anders gepr\u00e4gt haben und die du nicht sofort absch\u00fcttelst. Der Unterschied zwischen M\u00e4nnern und Frauen ist nat\u00fcrlich, und da haben Frauen zu Recht drauf hingewiesen, dass das sich l\u00f6sen f\u00fcr M\u00e4nner schwieriger ist als f\u00fcr Frauen, weil M\u00e4nner in dieser Gesellschaft privilegiert sind, selbst wenn sie lange Haare haben und Linke sind. Und Privilegien abzugeben ist immer schwerer als Unterdr\u00fcckung abzuwerfen. Von daher war der \"Leidensdruck\" unter M\u00e4nnern nicht so gro\u00df. Die H\u00e4userkampf-Bewegung oder die Bewegungen, die es \u00fcberhaupt gab, waren weder nur ein Abbild dessen, was sie angeblich bek\u00e4mpft haben, noch verk\u00f6rperten sie bereits den Himmel auf Erden. Sie haben darum gerungen, und in manchen Bewegungen gelang das mehr oder war das mehr Thema.\n\n#### Gibt es die gleichen Probleme heute immer noch oder hat sich im Laufe der Zeit doch was ver\u00e4ndert?\n\nEine Bewegung, die die Verh\u00e4ltnisse, wie sie sind, nicht hinnimmt, ist nicht qua dieser Positionierung anders, sondern sie beschreibt nur einen Weg, auf dem sie entlanggehen will und auf dem sie sich ver\u00e4ndern wird. Das wird sich auch in den n\u00e4chsten hundert Jahren nicht \u00e4ndern, das ist immer ein Ringen um das, was man gegenw\u00e4rtig ablehnt und was man als erstrebenswert, als Ziel formuliert. Wir waren weder am Ziel noch werden andere Bewegungen am Ziel sein, sondern sie werden sich im besten Fall dem Ziel n\u00e4hern. Von daher sind viele Auseinandersetzungen verschwunden und kein Gegenstand politischer Auseinandersetzung mehr. Auseinandersetzungen, die es fr\u00fcher vehementer gab, weil man in der Bewegung gelebt hat, manchmal von morgens bis nachts. Ich kann mich kaum erinnern, dass es in letzter Zeit Diskussion um Frauenbilder, M\u00e4nnerrollen, autorit\u00e4res Verhalten und dominantes Verhalten gab. Wer fragt heute danach, welche Beziehungen reaktion\u00e4r, welche Beziehungen progressiv sind, sein k\u00f6nnten?\n\n#### Aus welchen Milieus, mit welchem Bildungshintergrund kamen die Leute der Bewegung damals?\n\nEs gab sehr viele in den 80er Jahren, die \"freiwillig\" in prek\u00e4ren Verh\u00e4ltnissen lebten, also nicht viel Geld verdienen wollten, ihr Studium abgebrochen hatten oder das so nebenher gemacht haben, mit wenig Geld auskamen, was damals nat\u00fcrlich auch zum Teil m\u00f6glich war, weil man zum einen relativ gut verdient hat und zum anderen auch das Sozialhilfesystem oder Baf\u00f6g hatte, was einem das Leben nicht besonders schwer machte. Wenn ich mich an die Leute erinnere, die ich kannte, dann lebten die meisten nicht \u00fcppig, aber man konnte damit leben. Also ein Gutteil der Autonomen, die ich kannte, kam familiengeschichtlich eher aus der Mittelschicht als aus der Unterklasse. Diejenigen, die wirklich in erb\u00e4rmlichen Verh\u00e4ltnissen lebten, als Jungarbeiter oder nat\u00fcrlich noch schlimmer in Obdachlosensiedlungen, waren kaum repr\u00e4sentiert.\n\n#### Und heute?\n\nHeute ist es \u00e4hnlich, nur dass ein gro\u00dfer Teil nicht mehr in \"freiwilligen\" prek\u00e4ren Lebensverh\u00e4ltnissen steckt, sondern in prek\u00e4ren Lebensverh\u00e4ltnissen, die aufgezwungen sind durch Hartz IV oder einen Job, der befristet und mies bezahlt ist. Ich kenne relativ viele, die in diesen Verh\u00e4ltnissen leben. Aber ich glaube auch, und das ist ein weiterer Unterschied zu fr\u00fcher, dass die Spreizung eine viel gr\u00f6\u00dfere ist. Fr\u00fcher verdienten wir alle zwischen Sozialhilfe 700 Mark und 1.100 Mark, wenn jemand einen Job hatte. Heute ist der Unterschied zwischen Hartz IV 365 Euro und 4.000 Euro. Weil es nat\u00fcrlich auch Leute in der Bewegung gibt, ob als Antifa oder als Autonome oder als Anti-AKW-Gegner, die arbeiten im Computerbereich, sozusagen in der Business Class, und verdienen ohne Probleme 3.000 bis 4.000 Euro. Diese irrsinnige Spreizung ist nicht der Bewegung geschuldet, aber sie spiegelt sich in der Bewegung wider, und das ist nat\u00fcrlich ein ganz gro\u00dfes Problem. Wie willst du in einer Bewegung soziale und \u00f6konomische Unterschiede ausbalancieren, die so irrsinnig sind?!\n\n#### Wart ihr eher eine deutsche oder eine internationale Bewegung in Frankfurt?\n\nDas ist oft ein berechtigter Vorwurf, dass die meisten Bewegungen in Deutschland sehr deutsch gepr\u00e4gt sind und im Prinzip den Ausschluss reproduzieren, den sie eigentlich abstrakt beklagen. Eine der wenigen Bewegungen, wo dies so nicht stattgefunden hat, war die H\u00e4userkampf-Bewegung in den 70er Jahren. Das lag daran, dass es sehr starke migrantische Organisationen gab. In den 70er Jahren kamen viele nicht nur hierher, um zu arbeiten. Es gab sehr starke politische Organisationen, wie Lotta Continua aus Italien, die italienische Mieterunion, die selbstst\u00e4ndig Forderungen aufgestellt hat und die K\u00e4mpfe aus ihrem Heimatland mitgenommen hat. Die erste Hausbesetzung zum Beispiel in Frankfurt war eine Mischung aus Migrantenfamilien und Obdachlosenfamilien. In den 70er Jahren spielte es eine gro\u00dfe Rolle, nicht nur sich selbst zu bewegen und f\u00fcr sich selbst Vorteile zu erk\u00e4mpfen, sondern es ging darum, die Arbeitsverh\u00e4ltnisse der Arbeiter, die als sogenannte Gastarbeiter in die gro\u00dfen Fabriken geholt wurden, zu thematisieren. Es ging auch um die Lebensverh\u00e4ltnisse in sozialen Brennpunkten, es ging auch um die Thematisierung von Heimkindern. R\u00fcckblickend finde ich das einen der ganz gro\u00dfen Anspr\u00fcche, nicht nur f\u00fcr sich selbst etwas zu erk\u00e4mpfen, sondern tats\u00e4chlich die Verh\u00e4ltnisse drumherum im Blick zu haben. Und dabei haben Migranten und Migrantinnen eine gr\u00f6\u00dfere Rolle gespielt, was nat\u00fcrlich auch daran lag, dass man sie nicht als Opfer wahrgenommen hat, sondern sie sich als Handelnde selbst in diese K\u00e4mpfe gestellt haben. In den 90er Jahren, als es dann noch einmal eine starke Bezugnahme auf Fl\u00fcchtlinge gab, in Folge der rassistischen Pogrome und der Asylpolitik der Regierung, da gab es auch viele aus autonomen Zusammenh\u00e4ngen, die \"Fl\u00fcchtlingsarbeit\" gemacht haben. Dort war es tats\u00e4chlich das Problem, dass es sich ganz oft um Opfer von rassistischen \u00dcbergriffen handelte und das Verh\u00e4ltnis zu diesen Fl\u00fcchtlingen oft ein sozialarbeiterisches war.\n\n#### Und heute? Hat sich nicht noch eine gr\u00f6\u00dfere Distanz zwischen zum Beispiel der migrantischen Jugend in den Gro\u00dfst\u00e4dten und der autonomen Szene entwickelt?\n\nZuvor noch etwas zur \"autonomen Szene\": Wenn es so viele Autonome g\u00e4be, wie in den Medien berichtet wird, m\u00fcssten das mehrere zehntausend sein. Das steht im krassen Verh\u00e4ltnis zu der Realit\u00e4t derer, die sich als Autonome begreifen. Es gibt immer kleinere Versuche, genau diesen Graben zwischen medialer \u00dcberpr\u00e4senz und realer Unterpr\u00e4senz aufzul\u00f6sen. Das, was ganz oft im Zusammenhang mit Antifa als \"autonom\" bezeichnet wird, da w\u00fcrde die Antifa in gro\u00dfer Mehrheit mit Sicherheit sagen, dass sie nicht Autonome sind, sondern im besten Fall sind sie autonome Antifa, was aber m\u00e4\u00dfig bis wenig mit dem Grundgedanken von autonomer Theorie und Praxis zu tun hat.\n\n#### Du w\u00fcrdest die Antifa nicht als Teil der autonomen Szene begreifen?\n\nSie verstehen sich selbst zum guten Teil nicht so. Die, die mit dem antideutschen Label unterwegs sind, und die, die dies ablehnen, sind da mehr als gespalten. Die Ersteren z\u00e4hlen sich definitiv nicht zu den Autonomen, sondern treten mit dem Nimbus auf, dass das, was die Autonomen gemacht haben, alles Schei\u00dfe war und dass sie jetzt endlich alles richtig machen. Ein anderer Teil der Antifa steht schon irgendwo in dieser Tradition autonomer Politik, aber es reduziert sich erst mal nur auf den Anti-Nazi-Kampf. Und das ist etwas, was wir immer versucht haben zu vermeiden, also Antifaschismus auf einen Anti-Nazi-Kampf oder \u00fcberhaupt politisches Engagement auf einen Teilbereich zu reduzieren. In den 80er Jahren gab es \u00fcberall autonome Plenen in vielen St\u00e4dten, es gab regionale Strukturen, wo sich autonome Gruppen organisiert haben, es gab bundesweite Strukturen, die sich organisiert haben. Auch wenn diese Versuche nicht massiv waren, gab es doch Bem\u00fchungen, das zu bestimmen, was \u00fcber eine Kritik der Verh\u00e4ltnisse hinausweist: Wo will man hin, was sollte eigentlich folgen, wenn der schlimmste sch\u00f6nste Fall eintr\u00e4fe, das kapitalistische System samt Regierung verschwunden w\u00e4re, was kommt dann? Es gab also zumindest Versuche in den 80er Jahren, so etwas wie eine Utopie zu formulieren.\n\nWeil du gefragt hast, was ist der Unterschied zu heute: Genau dies w\u00e4re ein elementarer Unterschied. Es gibt heute keine Utopie, im Gro\u00dfen und Ganzen gibt es nur eine mehr oder weniger starke Ablehnung der Verh\u00e4ltnisse. Manches artikuliert sich vorsichtig, im zivilen Ungehorsam, andere formulieren sich rabiat. Aber eine Vision von dem, was danach kommen k\u00f6nnte, das ist \u00fcberhaupt nicht Gegenstand der Bewegung oder der Initiativen oder der Auseinandersetzung. Und ich glaube, dass das auch ein gro\u00dfes Drama ist. Denn keine Bewegung wird sich lange halten, wenn sie keine Utopie hat. Wenn sie nicht eine Vorstellung hat, die \u00fcber das hinausgeht, was sie gerade bl\u00f6de findet, was sie gerade bek\u00e4mpfen will. Und das ist kein deutsches Problem, das ist kein europ\u00e4isches Problem, das ist ein weltweites Problem, weil man nat\u00fcrlich meint, das, was im Ostblock passiert ist, das hat sich als falsch, als unemanzipativ erwiesen, und der Kapitalismus ist zwar Schei\u00dfe, aber man bekommt wenigstens Geld und kann abends in Kino gehen und zweimal in Urlaub fahren. Dass Sozialismus wenig mit dem zu tun hat, was im Ostblock passiert ist, und noch weniger mit dem, was in der DDR passiert ist, ist das eine. Genauso wichtig w\u00e4re es, das zu umrei\u00dfen, was anders sein k\u00f6nnte und m\u00fcsste. Und da w\u00fcrde ich schon sagen, dass die autonome Bewegung von der Idee her nat\u00fcrlich eine Vision hatte, die bezog sich eben nicht auf Parteien, sondern es gab und gibt die Vorstellung von r\u00e4tedemokratischen Strukturen, also keine Parteiendemokratie mehr, sondern eine direkte Demokratie, in der die Leute selbst entscheiden, keine Delegation von Macht, sondern Selbsterm\u00e4chtigung, und nat\u00fcrlich geh\u00f6rt auch eine Wirtschaftsform dazu, die nicht kapitalistisch ist, sondern auf Kollektiveigentum beruht, was auch noch einmal etwas anderes w\u00e4re als Staatseigentum.\n\n#### Ich mache noch mal den Sprung zur\u00fcck zu Jugendkultur und Jugendbewegung: Gab es Beziehungen zu anderen Jugendkulturen? Gab es bestimmte Szenen, die wichtig waren, au\u00dferhalb der explizit politischen Szenen?\n\nJa, ich glaube schon. Der Anteil der Autonomen, die irgendwo in einem Sub-Milieu lebten, in besetzten H\u00e4usern, irgendwo zwischendrin, war relativ hoch. Es gab endlos viele \"illegale Orte\", Kneipen, Zentren, die sehr viele anzogen, aus sehr unterschiedlichen Gr\u00fcnden. Und die Alternativbewegung gab es damals ja auch schon, kleine Klitschen, Werkst\u00e4tten, Bio-L\u00e4den etc. Es gab Druckereien, es gab Kneipen, die sich selbst finanziert haben, oder Musikveranstaltungen, die gemacht wurden, und viele Ber\u00fchrungspunkte, die weit dar\u00fcber hinausgingen, was f\u00fcr \"autonom\" gehalten wurde. Das ist ja immer der entscheidende Punkt: Eine Bewegung ist dann stark und strahlt dann eine Faszination aus, wenn sie mehr erreicht als ihre Beteiligten, wenn sie ausstrahlt auf diejenigen, die eigentlich damit nichts zu tun haben, aber in den Bann geraten, was ja ganz oft bei der Hausbesetzer-Bewegung der Fall war. Dasselbe gilt f\u00fcr die (Underground-)Musik-Szene und die Drogen-Szene \u2013 wenn es keine harten Drogen waren.\n\n#### Gab es bestimmte Musik-Szenen, die besonders wichtig waren oder vielleicht auch Einfluss hatten auf die Szene?\n\nDa bin ich nicht so ganz der Richtige... Aber ich hatte f\u00fcr ein Buch alle Freunde, Freundinnen gefragt, was sie damals geh\u00f6rt haben, was f\u00fcr sie damals wichtig war an Musik. Und ich w\u00fcrde sagen, dass es eine bestimmte Musik gab. Das hei\u00dft nicht, dass die Gruppen Mitglieder der autonomen Bewegung waren, aber sie haben durch das, was sie an Musik gemacht haben, an Texten, da reingespielt. Und da ist in den 80er Jahren zum einen die Punk-Bewegung zu nennen, zum anderen die \"Neue Deutsche Welle\", Bands wie Fehlfarben und Extrabreit. In den 70er Jahren war Ton Steine Scherben \"konkurrenzlos\". Mit Sicherheit w\u00fcrden achtzig Prozent der Leute, die am H\u00e4userkampf beteiligt waren, wenn du sie fragst, was war eure Band, eure Musik, sofort sagen: Ton Steine Scherben, was sonst?\n\n#### Wei\u00dft du, wie es heute ist?\n\nEs gibt immer noch Bands, die bedeutend sind, weil sie die Themen aufgreifen, weil sie politische Texte haben. Aber die Stile sind sehr unterschiedlich geworden, haben sich so gespreizt wie die Einkommen. Die einen finden Punk noch klasse, die anderen finden House, Indie, das, was auch in den konventionellen Diskotheken und Clubs an verschiedenen Musikrichtungen auf den Plattenteller kommt, gut. Die Bandbreite ist gr\u00f6\u00dfer geworden.\n\n#### Und wie sieht es in au\u00dfermusikalischen Bereichen aus? Sport-Szenen sind wahrscheinlich eher nicht so relevant, aber bestimmte Szenen wie Veganer-Szene vermutlich schon. Gab es so was auch schon in den 80er Jahren, dass vegetarische Ern\u00e4hrung wichtig war?\n\n[Lacht.] Nein, also so etwas gab es in den 80ern nicht. Ich kann mich nicht dran erinnern, dass wir uns dar\u00fcber gestritten haben, wo das \"Verbrechen\" anf\u00e4ngt, ob beim Huhn oder dem Ei... Wir haben uns \u00fcber ganz viele Sachen zerstritten, wo man in dem System involviert ist, wo man eben auch dasselbe macht, wie Miete zahlen oder ein Bahnticket zieht, aber das bezog sich nie auf das Essen. Niemand von uns hat \u00fcber die Wahl des Essens, vegan oder vegetarisch oder fleischig, eine politische Aussage getroffen.\n\n#### Gibt es denn so etwas wie einen eigenen Stil? Hat Mode schon eine Rolle gespielt in den 80er Jahren? Heutzutage gibt es bei j\u00fcngeren Leuten eine gro\u00dfe Verbreitung von bestimmten Haarschnitten, und auch T\u00e4towierungen und Piercings spielen eine Rolle, bestimmte Klamotten...\n\n[Lacht.] Antifa zu sein ist heute ein teures Vergn\u00fcgen. Viele sprechen schon humorig von den ganz vielen Labels, die man haben muss, um angesagt zu sein. Und es gibt ja tats\u00e4chlich heute eine Menge Antifa-Shops... Diese Tendenz zur Vermarktung und selbst Marke zu werden in einem kommerziellen Sinne gab es in den 80er Jahren nicht. Es gab keine L\u00e4den, die autonome Klamotten oder autonome Schuhe und autonome Kopfbedeckungen produziert haben, sondern unser Ort in den 80er Jahren war der Flohmarkt. Heute gehen alle zum Piercing und dann in den Antifa-Shop \u2013 oder du brauchst nur noch im Internet anklicken \u2013 und schon bist du ein Antifa.\n\n#### Aber du willst jetzt nicht ernsthaft behaupten, dass euch in den 80ern Jahren euer Aussehen vollkommen egal war?\n\nNat\u00fcrlich waren wir genauso darum bem\u00fcht, uns irgendwo kenntlich zu machen, denn das hei\u00dft ja immer auch, sich abzusetzen von dem, was man nicht will. Von daher tauchte nicht umsonst der Schwarze Block auf, weil wir Schwarz f\u00fcr die richtige Farbe hielten, weil sie Anarchismus symbolisierte. Das Wesentliche war jedoch, dass die Klamotten praktisch und billig waren. Ganz viele von uns haben Bundeswehrstiefel gehabt, weil die lange halten und billig waren, oder Bundeswehrhosen, die dann schwarz gef\u00e4rbt wurden, was sein musste, da niemand in Tarnfarben der Bundeswehr herumlaufen wollte, und Lederjacken. Daf\u00fcr gab es bereits L\u00e4den, die auch Lederjacken n\u00e4hten und um Einiges teurer waren als das, was man auf dem Flohmarkt bekam. Das Bed\u00fcrfnis, sich abzusetzen, auch im \u00c4u\u00dferen sichtbar zu machen, dass man anders ist als die Mehrheit, ist nat\u00fcrlich in jeder Bewegung ein wichtiges Motiv. Das Problem ist, wenn man es verstetigt und meint, eine Bewegung erkennt man nur daran, dass man sofort auf die Leute zeigen kann.\n\n#### Ist vielleicht das Ganze ein St\u00fcck weit zu einer Art Mode geworden oder auch zu einer Eventkultur, wo man sich zu bestimmten Anl\u00e4ssen wie etwa zum 1. Mai trifft und ansonsten eigentlich nichts so gro\u00dfartig macht?\n\nNein, nicht wirklich. Ich w\u00fcrde schon sagen, dass das weniger den Leuten anzulasten ist, das ist einfach der bunte Kapitalismus. Die Kommerzindustrie schaut ja nicht, was die Leute im Kopf haben, sondern: Ist das ein Markt? Und nat\u00fcrlich f\u00f6rdern wir damit etwas, was unseren Interessen nicht unbedingt entspricht... Es werden alle eingekleidet, ob Lesben, Schwule, Queer-Leute oder Transvestiten. Der Kapitalismus hat kein Problem, die verschiedenen Geschm\u00e4cker und die verschiedenen sexuellen Vorlieben einzukleiden, auszustatten und zu melken. Im Gegenteil, nichts befeuert den Kapitalismus mehr, als dass man das Warenangebot vergr\u00f6\u00dfert, indem, man es diversifiziert. Wenn du nur normale Ehepaare hast, dann hat es sich mit dem Konsum schnell erledigt. Wenn du zus\u00e4tzlich Schwule hast, wenn du Lesben hast, dann hast du jedes Mal einen Markt. Das ist ein Unterschied zu den 70er, 80er Jahren: Ton Steine Scherben wurden nirgendwo gespielt, kein Radiosender brachte das, keine Fernsehsendung strahlte ein Ton-Steine-Scherben-Konzert aus. Und das lag nat\u00fcrlich daran, dass die Vermarktung ein Problem ist, wenn der Inhalt des Marktproduktes die Zerst\u00f6rung des Marktes zum Thema hat. Also die Abschaffung des Kapitalismus, das Ende der Lohnarbeit und so weiter. Da gibt es immanente Grenzen der Verwertung. Dass man trotzdem immer wieder aus einer Bewegung bestimmte Insignien herausl\u00f6st, wie das Che-Guevara-Symbol, das dann auch Banker tragen, weil sie das extrem witzig finden, wenn sie auf einer Party ein bisschen avantgardistisch auftreten, oder den Mao auf dem R\u00fccken tragen oder auf dem Oberarm. Das passiert nat\u00fcrlich immer. Aber eine Bewegung insgesamt zu vermarkten, die klar in ihren Botschaften und in dem, was sie tut, die Abschaffung dessen fordert, was daraus Geld machen m\u00f6chte... wenn diese Grenze existiert, ist die Vermarktung auch begrenzt. Doch bei vielem, was jetzt vermarktet wird, ob du in einen Frauenladen gehst oder in einen Schwulenladen oder in einen Punkerladen, wo du alles kriegst, was ein Punker haben muss, geht es es im Gro\u00dfen und Ganzen um Lebensstil.\n\n#### Du hast eben die queere Szene oder lesbische, schwule, Transgender-Szene erw\u00e4hnt. Inwiefern haben denn diese Szenen eine Rolle gespielt in der autonomen Bewegung?\n\nEs gab all dies am Rand, aber ohne besondere Aufmerksamkeit, ohne besondere Heraushebung. Weder gab es eine st\u00e4ndige Auseinandersetzung darum, noch kann ich mich erinnern, dass wir uns darum bem\u00fcht haben. Wir hatten mit uns genug zu tun und kamen gar nicht auf die Idee, uns zu fragen: Was will die Schwulenbewegung, was wollen wir von der Lesbenbewegung? Kurzum, es gab weder eine besondere Sympathie noch eine besondere Abneigung. Lediglich zur Lesben-Szene war das Verh\u00e4ltnis ganz leicht ein anderes. Es gab einige Frauengruppen innerhalb autonomer Zusammenh\u00e4nge, in denen das Lesbisch-Sein eine besondere Rolle bzw. Gegenstand von Auseinandersetzungen war. Ein Konflikt zwischen Hetero-Frauen und Lesben. Letztere warfen ersteren vor, sich um eine klare Auseinandersetzung zu dr\u00fccken, keinen klaren Schnitt zu machen. F\u00fcr einige von ihnen waren Patriarchatskritik und eine Partnerschaft mit einem Mann eine unglaubw\u00fcrdige, halbherzige Melange. Diese Auseinandersetzungen spielten schon in die Bewegung rein. Das hat auch damit etwas zu tun, dass lesbisch oder schwul sein keine politische Aussage ist.\n\n## Die Zeitschrift radikal als Sprachrohr der linken autonomen Szene \u2013 Eine Medienanalyse\n\nvon Rana Holsti\n\n### Z\u00e4hne zeigen\n\nEin Poster hing Mitte der 1990er Jahre \u00fcberall, wo die Partys lang und die Musik toll war, in WGs, in Kneipen mit menschenfreundlichen Preisen und in L\u00e4den, die Konzerte jenseits des Mainstream veranstalteten. Sch\u00f6n war es gerade nicht, dieses Plakat, das konnte nicht der Grund sein, warum es auf einmal allerorts auftauchte. Es zeigt eine wilde Fratze mit aufgerissenen Augen und gefletschten Hauern, es leuchtet in schwarz-rot und der Schriftzug darauf besagt: \"Die Z\u00e4hne zeigt, wer das Maul aufmacht \u2013 lebt und lest radikal\". Das Poster war Teil einer Solidarit\u00e4tskampagne, deren Unterst\u00fctzung leicht gemacht wurde, konnten doch die paar Mark Soli-Beitrag f\u00fcr Eintritt oder Getr\u00e4nkepreise selbst mit Lehrlingslohn oder Baf\u00f6g noch gestemmt werden. Schwieriger war es schon, an das Objekt heranzukommen, mit dem man da solidarisch war. Die Zeitschrift radikal, deren Titelbild von der Ausgabe 134\/2 sich auf dem Plakat befand, war seit 1984 verboten, der Handel damit seit 1986 illegal. Ohne ein gewisses Ma\u00df an subkulturellen Kontakten war sie damals kaum zu kriegen und darf bis heute nicht offen verkauft werden. Im Intro der Ausgabe 152 vom Dezember 1994 wird erkl\u00e4rt: \"Die radikal ist kriminalisiert, weil u. a. das Recht auf freie Meinungs\u00e4u\u00dferung ein Lacher ist, wenn es wirklich in Anspruch genommen wird. Genauso ist es mit den meisten Rechten. Es sind nur schriftlich fixierte Spielregeln der Herrschenden, die nach Bedarf erweitert oder gestrichen werden. Sie schicken ihre elenden Handlanger vor gegen alle, die au\u00dferhalb des erlauchten Spielkreises stehen, oder sich bewusst dorthin stellen. Gerne spielen wir mit der radikal ein anderes Spiel.\"\n\nBeim Durchbl\u00e4ttern des Heftes stellte sich aber schon das erste Stirnrunzeln ein: \"Internationaler Frauen- u. Lesbenkampftag\", \"Antirassismus in Deutschland\", \"Erziehung und Sexualit\u00e4t\", \"Von heute an kein Pornoprogramm\", \"Radio als Kommunikationsmittel \u2013 ein Aufruf zur Nutzung nichtkommerzieller Radios\", so lauteten die Artikel\u00fcberschriften, die auf Anhieb ins Auge fielen. Klar, dass das kontroverse Themen waren, die nicht unbedingt den konservativen Mainstream begeisterten. Aber kriminell? Da musste der Blick schon gezielter auf die Rubrik OLGA fallen \u2013 eine Ladenpl\u00fcnderung hier, eine Anschlagserkl\u00e4rung da \u2013 und tats\u00e4chlich auch eine Bauanleitung f\u00fcr einen \"Wurfanker\", geeignet zur Behinderung der verhassten Castor-Transporte der Bahn. Allerdings sieht diese technische Zeichnung \u00e4hnlich kryptisch aus wie die Bedienungsanleitung einer elektrischen Handoberfr\u00e4se \u2013 auch so ein Ger\u00e4t, an dem sich Laien nicht ohne Weiteres vergreifen w\u00fcrden. Beim Weiterbl\u00e4ttern findet sich eine Aufforderung zur Teilnahme an einem. \"Autonomie-Kongress\" zu Ostern 1995 an der Berliner Humboldt-Universit\u00e4t. Nicht gerade einladend, was die kleinen trockenen Textausz\u00fcge aus dem zugeh\u00f6rigen Reader dort erwarten lie\u00dfen. Aber damit sind wir beim Thema und beim Stein des Ansto\u00dfes \u00fcberhaupt \u2013 der autonomen Bewegung.\n\nWorum es hier gehen soll, ist, einen Blick auf die autonome Szene zu werfen, wie sie sich in der radikal darstellt \u2013 aber nicht darum, zentrale Themen wie die schier endlose Militanzdebatte abzuarbeiten. Zwar mag diese die \"Street-fighter\" ebenso brennend interessieren wie den Staatsschutz, und auch die ber\u00fcchtigten \"Bombenbauanleitungen\", die allesamt nicht gerade zum Nachbau inspirieren, auf deren Erscheinen aber immer wieder im Zusammenhang mit der radikal hingewiesen wird und auf die diese Zeitung zum Teil auch reduziert wird. Zudem pr\u00e4sentiert sich die radikal auch noch als an der RAF interessiert, aber diese, ihre Methoden und Politik nicht bedingungslos bef\u00fcrwortend oder gar als \"RAF-Blatt\" auftretend. Diese und andere \"gro\u00dfe\" Themen wie \"Hausbesetzungen, Knastkampf, Anti-Atomkraft, Antiimperialismus oder Antifa sollen hier zwar auch angerissen werden, aber dazu gibt es bereits einiges zu lesen, von wissenschaftlichen Abhandlungen \u00fcber die etablierte Presse bis hin zu Verfassungsschutzberichten.\n\nIm Folgenden sollen jedoch auch die Themen und Diskussionen aufgegriffen werden, die nicht immer wieder dieselben sind, sondern jene, die durch ihre Seltenheit oder Begrenztheit auffielen und damit auch die Leerstellen oder die Ignoranz aufzeigen, die in den szeneinternen Debatten vorhanden sind, bei aller, gerade dieser Szene eigenen, ausgepr\u00e4gten Bereitschaft zu Selbstkritik und Reflektion. Deutlich wird dies beispielsweise, wenn es um Ostdeutschland geht oder bei Gender- und auch Queer-Thematiken, deren phasenweise oder v\u00f6llig mangelnde Pr\u00e4senz sicher nicht anzeigt, dass es in der autonomen Bewegung keine \"Ossis\", Frauen oder nicht-heterosexuellen Menschen gab und gibt. Das ist das eine. Das andere ist, dass die radikal \u2013 im Gegensatz zu Publikationen aus dem politisch rechten Spektrum, die Menschen(-gruppen) verachtende Positionen vertreten \u2013 auf Werte verweist, die in der b\u00fcrgerlichen Gesellschaft entschieden mehr Beachtung finden k\u00f6nnten: Solidarit\u00e4t und Unterst\u00fctzung marginalisierter Gruppen ebenso wie die Kreativit\u00e4t und der Idealismus, sich hierarchiefreie und damit humanere Utopien oder zumindest Freir\u00e4ume auszudenken und damit ein St\u00fcck weit zu schaffen, dass der Wert von Menschen nicht nur an ihrer Leistungsf\u00e4higkeit und Kaufkraft gemessen wird. Auch wenn die Versuche zu deren Umsetzung selbst in der autonomen Bewegung offensichtlich nicht immer einfach, erfolgreich und ohne Konfliktpotential abliefen \u2013 neue Denkmuster und Verhaltensmaximen erwachsen schlie\u00dflich auch erst mal in \"alten\" K\u00f6pfen. Letztlich beinhaltet jede noch so widerst\u00e4ndige Subkultur und politische Bewegung auch Elemente dessen, gegen das sie sich abgrenzt. Aber genau das macht auch die differenzierte Auseinandersetzung mit ihnen viel spannender als zum x-ten Mal dieselben Klischees herunterzubeten. Also keine vermummten, schwarz gekleideten Steineschmei\u00dfer in der radikal? Doch, auch, aber nicht nur und dazu noch einiges mehr.\n\n### Das praktische Instrument\n\nVor mittlerweile fast 40 Jahren erschien die erste Ausgabe der radikal. Das Titelblatt vom 18.6.1976 tr\u00e4gt noch die Unterschrift: Sozialistische Zeitung f\u00fcr Westberlin. Darauf abgebildet war ein \u00fcberdimensionaler Fingerabdruck, dazu der Hinweis auf Generalbundesanwalt Buback mit dem Zitat \"nach SZ, 28.1.76\": \"Es sei w\u00fcnschenswert, wenn die Fingerabdr\u00fccke s\u00e4mtlicher Staatsb\u00fcrger der Bundesrepublik aufgenommen w\u00fcrden. Dies sei zurzeit aus politischen Gr\u00fcnden leider noch nicht erreichbar\" \u2013 heute gibt es Menschen, die sich freiwillig auf diese Form von Registrierung einlassen. Aber zur\u00fcck: Konzipiert war die radikal als Zeitung mit einem 14-t\u00e4gigen Rhythmus. Die erste Ausgabe erschien in einer Auflage von 3.000 St\u00fcck und wurde zum Gro\u00dfteil im Handverkauf in Buchl\u00e4den und in Kneipen angeboten. Herausgegeben wurde die erste Nummer von einer Redaktionsgruppe in einer Zeit, als in Westberlin eine zersplitterte linke Parteienlandschaft existierte, in der jede Einzelne der konkurrierenden K-Parteizeitungen unertr\u00e4glich war. Auf Seite 2 der radikal-Erstausgabe folgt daher die Erkl\u00e4rung, man wolle keine Parteizeitung sein, sondern ein \"Diskussionsforum und [...] praktisches Instrument im Dienst grunds\u00e4tzlich aller antikapitalistischen Gruppen in West-Berlin\". Und weiter hei\u00dft es: \"Wir m\u00f6chten mithelfen, die politische Praxis der vielen Parteien abzuklopfen auf das, was tr\u00e4gt, was die emanzipatorischen Bed\u00fcrfnisse der Menschen f\u00f6rdert, was die politische Schlagkraft st\u00e4rkt, Klarheit verbessert und was den Weg erhellt, den wir im Kampf um eine neue Gesellschaft gehen m\u00fcssen.\"\n\nHier wurden auch die weiteren \"praktischen Aufgaben\" aufgelistet, die man sich zum Ziel gesetzt hatte und die nachfolgend und bis ins Jahr 1980 in jeder weiteren Ausgabe an dieser Stelle zu finden waren: \"Zusammenfassung aller Termine der politischen Linken in einem zentralen Veranstaltungskalender; \u00dcberblick \u00fcber die Aktionen der Linken in West-Berlin durch Kurzberichte; Offenes Diskussionsforum \u00fcber wichtige politische Probleme; Unterst\u00fctzung von Anstrengungen, Aktionseinheiten der Linken gegen die zunehmende Faschisierung; Unterst\u00fctzung von Versuchen alternative Lebensformen zu entwickeln; Angebot von Diensten zu praktischen Problemen: Kleinanzeigen, Beratungen, Tips.\"\n\nWer heute in den fr\u00fchen radikal-Zeitungen nach den Urspr\u00fcngen der autonomen Bewegung sucht, wird entt\u00e4uscht. Inhaltlich war diese erste Ausgabe vergleichsweise zahm, um nicht zu sagen: langweilig. Sie bot Artikel zu einem Chor- und einem Mieterfest, einem Tutorenstreik an der Uni, Elternprotesten gegen Sparma\u00dfnahmen in der Schule und zur B\u00fcrgerinitiative gegen die geplante Westtangente. Berichtet wurde au\u00dferdem vom Antirepressionskongress des Sozialistischen B\u00fcros, und im \"Forum\", das der Auseinandersetzung zu bestimmten Inhalten aus unterschiedlichen Positionen dienen sollte, ging es um Vor- und Nachteile von Studierendengremien wie AStA und UStA. Zwar gab es praktische Tipps wie die \"Wichtigen Hinweise f\u00fcr Demonstranten\" zum Verhalten auf Demos und kritische Auseinandersetzungen mit Themen wie den Bestimmungen zum Schusswaffengebrauch oder \u2013 einer der Dauerbrenner der folgenden Jahrg\u00e4nge \u2013 Berufsverboten, so richtig in die Nesseln setzte sich diese erste radikal jedoch nur mit dem \"effe\"-Artikel und dessen Kritik am \"insularischen separatismus in der frauenbewegung\". Dieser Beitrag l\u00f6ste erste heftige Diskussionen und Reaktionen aus, denn die Meinungen zur Frauenbewegung waren gespalten. Prompt weigerten sich die damals f\u00fcr die Druckvorlagen zust\u00e4ndigen Mitarbeiterinnen von viva frauendruck, weitere Repro-Arbeiten f\u00fcr die radikal auszuf\u00fchren und argumentierten in der zweiten Ausgabe der radikal: \"Die Trennung Frau\/Mann ist keine von der Frauenbewegung konstruierte, sondern eine gesellschaftlich vorhandene. Gerade weil viele Frauen anfingen, dies zu begreifen, ist die Frauenbewegung entstanden. Sie ist der Versuch, sich der Verst\u00fcmmelung durch die Gesellschaft bewusst zu werden, und keine Anti-Mann-Bewegung, wie der Artikel suggeriert. Sie richtet sich solange gegen M\u00e4nner, solange diese ihre degenerierten barbarisch verformten Bed\u00fcrfnisstrukturen und Lustformen' [...] nicht aufbrechen und damit auch jeden notwendigen Ansatz von Frauen, die dies versuchen, unterdr\u00fccken, weil es f\u00fcr sie jetzt SELBST Ver\u00e4nderung bedeuten w\u00fcrde. Gerade durch die Polemiken wie sie in der radikal erscheinen [...] werden diese Ver\u00e4nderungsversuche, die wir unterst\u00fctzen wollen, boykottiert.\"\n\nDie zweite Ausgabe verweist bereits auf eine Diskussion, die lange Zeit zentral in und um die radikal gef\u00fchrt werden w\u00fcrde: Die Haltung zur Frage der Militanz innerhalb der linken Szene. Das Titelblatt vom 1. Juli 1976 ziert das gezeichnete Bild eines jungen Mannes mit einer Handgranate in einem Caf\u00e9. Am Rand findet sich die Erl\u00e4uterung: \"Emile Henry warf am. 12. Februar 1894 eine selbstgebastelte Bombe auf trinkende und prassende Bourgeois in einem Pariser Lokal\". Hinzugef\u00fcgt wurde das mahnende Zitat Mao Tse-tungs: \"Wenn das Bewusstsein der Massen noch nicht geweckt ist, und wir dennoch einen Angriff unternehmen, so ist das Abenteurertum\". In dieser Ausgabe werden aus Anlass des Todes von Ulrike Meinhof unter der \u00dcberschrift \"Zur Frage der revolution\u00e4ren Gegengewalt\" Statements aus vergangenen Jahren gegen\u00fcbergestellt. \u00dcberhaupt ist das Mitte bis Ende der 1970er Jahre hei\u00df umstrittene Thema RAF Dreh- und Angelpunkt sowohl szene- als auch redaktionsinterner Auseinandersetzungen und Positionierungen. Reflektiert werden hier nicht nur die unterschiedlichen Standpunkte, sondern auch die Folgen, die diese Spaltung bewirkt, und deren Auswirkungen auf den Zusammenhalt innerhalb der linken Szene.\n\nIntern wurde in den ersten Ausgaben noch der Name der Zeitung zur Diskussion gestellt \u2013 die Leser_innen wurden hier zur Mitbestimmung aufgefordert. Eine der Grundideen f\u00fcr die Beibehaltung des Namens radikal war es, der \"Radikalenhetze offensiver entgegen zu treten\". Denn es sei besser, den Begriff zu akzeptieren, als vor ihm davonzulaufen. Zudem wollte man sich zum einen von Gruppierungen wie den Jusos absetzen und zum anderen \"nicht an den Symptomen herumkratzen, sondern ihre wirklichen Ursachen angehen, das \u00dcbel an der Wurzel packen\". Die Ziele waren also hoch gesteckt. Und vorsorglich gab es in dieser Ausgabe als praktische Tipps noch Ratschl\u00e4ge zum Verhalten bei Hausdurchsuchungen.\n\nTats\u00e4chlich aber bleibt die Diskussion um die Frauenbewegung in den Ausgaben des ersten Jahres der radikal das kontroverseste Thema. In einer Frauenkneipe durfte die Zeitung nicht einmal mehr verkauft werden. Die f\u00fcnfte Ausgabe tr\u00e4gt dann auch den programmatischen Untertitel \"Wohin geht die Frauenbewegung?\" und besch\u00e4ftigt sich mit dem Vorwurf, dass die radikal frauenfeindlich sei, und dazu mit den \"Schwierigkeiten unseres Redaktionskollektivs, sich zur Frauenbewegung zu verhalten\". In diesem Kontext geben zwei M\u00e4nner aus der Redaktion reichlich verkopfte Statements zur \"Autonomie der Frauenbewegung\" und zur \"Haltung der M\u00e4nner zu Feministinnen\" ab, und auch KBW, KPD, GIM und KB \u00e4u\u00dfern sich zur Frauenbewegung, wobei die Gruppe Internationaler Marxisten immerhin die Frauenbewegung f\u00fcr notwendig zu erachten scheint, denn \"M\u00e4nner nehmen Frauen diesen Kampf nicht ab\". Das Thema Patriarchatskritik blieb auch weiterhin ein umstrittener Punkt, sowohl im Redaktionskollektiv der radikal, das mehrheitlich aus M\u00e4nnern bestand, als auch in den Beitr\u00e4gen zu sogenannten Frauen-Themen insgesamt.\n\nWeitere Hauptinhalte dieser Anfangszeit waren Haftbedingungen und Hungerstreiks politischer Gefangener, diverse B\u00fcrgerinitiativen und Protestbewegungen, die sich zumeist Umweltbelange wie den Widerstand gegen Atomanlagen auf ihre Fahnen geschrieben hatten. Auch andere Themen sind bis heute aktuell, so zum Beispiel der \"Wegweiser durch den Sozialamtsdschungel\" in der radikal Nr. 17, mit Ratschl\u00e4gen zum Umgang mit dieser ungeliebten, manchmal aber auch unvermeidbaren Institution, was eine historische Kontinuit\u00e4t menschenunw\u00fcrdiger Behandlung auf Beh\u00f6rden erkennen l\u00e4sst. Daneben gab es Mitteilungen aus diversen, zum Teil heute noch existierenden alternativen Projekten und dem Westberliner Stadtleben. So bietet eine kleine Meldung aus der Nr. 27 einen der seltenen heiteren Momente bei der Lekt\u00fcre der fr\u00fchen radikal-Ausgaben: \"Mit Gips und Zement ausgegossene Toilettenbecken und Schanktische fand die Polizei in einem Gro\u00dflokal vor, in dem zurzeit Frauenboxk\u00e4mpfe 'oben ohne' stattfinden (TS)\". Ansonsten waren Uni-Streiks in fast jeder Ausgabe pr\u00e4sent und gegen die chronisch in der Kritik stehenden zu hohen Fahrpreise der \u00f6ffentlichen Verkehrsmittel wurde auf der letzten Seite jeder radikal ein gro\u00dfer roter Punkt zum Ausschneiden angeboten. Diesen konnten sich Autofahrer_innen an die Scheibe kleben, wenn sie bereit waren, BVG-boykottierende Mitreisende einzuladen.\n\nSp\u00fcrbar bissiger wurde der Tonfall der radikal, als die Ereignisse im Kontext des \"Deutschen Herbstes\" 1977 die Seiten f\u00fcllten. In der Ausgabe Nr. 18 wird ausf\u00fchrlich zum Tod Bubacks Stellung bezogen, wobei deutlich wird, dass es \"in der radikal redaktion [...] keine einheitliche einsch\u00e4tzung \u00fcber die Sache mit buback [gibt]\". In einem Beitrag hei\u00dft es: \"Siegfried Buback ist tot. Die ihn erschossen haben, begreifen die Tat als einen Akt revolution\u00e4rer Kriegsf\u00fchrung. Doch die gro\u00dfen Worte klingen hohl. [...] Das 'Urteil', das die T\u00e4ter \u00fcber Buback gesprochen haben, mag richtig sein und dem entsprechen, was auch wir denken. Das 'Recht' es zu vollziehen hatten sie aber n i c h t. [...] Auch politisch war die Erschie\u00dfung Bubacks ein Fehler.\"\n\nAuch wenn man hier Gewalt nicht gerade bef\u00fcrwortete, wurde diese Ausgabe von der Staatsanwaltschaft umgehend sanktioniert, und so vermeldet die Redaktion in der Nr. 28: \"In einem Schreiben vom 19. Oktober teilt der Erste Staatsanwalt am Landgericht Berlin, Weber, dem presserechtlich Verantwortlichen von radikal mit, 'dass ich gegen Sie ein Ermittlungsverfahren wegen des Verdachts der Verunglimpfung des Andenkens eines Verstorbenen, \u00a7189 StGB, f\u00fchre.' [...] Wir werden uns gegen eine Kriminalisierung unserer Arbeit nach Kr\u00e4ften wehren und z\u00e4hlen auf Eure Solidarit\u00e4t.\"\n\nEin weiterer Stein des Ansto\u00dfes war der Abdruck des ber\u00fcchtigten G\u00f6ttinger \"Mescalero\"-Briefes \"Buback \u2013 Ein Nachruf\" in der radikal Nr. 22. Auch der \"Mescalero\" hatte sich keineswegs f\u00fcr uneingeschr\u00e4nkte Gewalt ausgesprochen, zitiert wurde in anderen Medien zumeist nur die \u00c4u\u00dferung der \"klammheimlichen Freude\" als \"unmittelbare Reaktion [...] nach dem Abschuss von Buback\" zu Beginn des Textes. Am Ende steht hier jedoch: \"Unsere Gewalt kann nicht die Al Capones sein, eine Kopie des offenen Stra\u00dfenterrors und des allt\u00e4glichen Terrors, nicht autorit\u00e4r, sondern antiautorit\u00e4r und deswegen umso wirksamer. Um der Machtfrage willen (O Gott!) d\u00fcrfen Linke keine Killer sein, keine Brutalos, keine Vergewaltiger, aber sicher auch keine Heiligen, keine Unschuldsl\u00e4mmer. Einen Begriff und eine Praxis zu entfalten von Gewalt\/Militanz, die fr\u00f6hlich sind und die den Segen der beteiligten Massen haben, das ist (zum praktischen Ende gewendet) unsere Tagesaufgabe.\"\n\nDass die kontroverse Auseinandersetzung mit der Militanzfrage in der radikal quasi Tradition hat, kann anhand zahlreicher Beispiele gezeigt werden. So findet sich in der Nr. 25 ein kritischer Beitrag unter der \u00dcberschrift \"Gewalt(re)aktionen\": \"Die Anschl\u00e4ge auf Drenckmann, Lorenz, Buback, Ponto und jetzt Schleyer wecken in mir ganz andere Gef\u00fchle als bei vielen Genossen. Ich f\u00fchle keinerlei Befriedigung, Freude dar\u00fcber, ob klammheimlich oder offen, sondern nur Wut und Ohnmacht angesichts einer Entwicklung, die uns noch weiter zur\u00fcckwirft, als wir schon sind. [...] Nur wenn sich das Ziel der Befreiung von Gewalt auch in unserem gegenw\u00e4rtigen Denken, F\u00fchlen und Handeln niederschl\u00e4gt, ist es ein lebendiges Ziel, ist es mehr als eine Phrase. Deshalb haben Linke, so meine ich, als eine Grundlinie ihrer Politik immer m \u00f6 g l i c h s t w e n i g Gewalt anzuwenden! Dieses 'm\u00f6glichst wenig' bedeutet nun nicht den Verzicht auf jegliche Gewaltanwendung [...]. Gegengewalt ist in der einen oder anderen Form immer wieder unumg\u00e4nglich, weil das herrschende System selbst gewaltt\u00e4tig agiert, weil wir uns gegen diese Gewaltt\u00e4tigkeit manchmal nur mit Gegengewalt wirksam wehren k\u00f6nnen (Hausbesetzungen als letztes und einzig wirksames Mittel f\u00fcr Mieter, ihren Wohnraum zu erhalten oder akzeptablen zu erhalten, und dergl.).\"\n\nAuch in der folgenden Ausgabe wird im \"Forum zur Gewaltfrage\" weiter diskutiert. Das Sozialistische B\u00fcro West-Berlin distanziert sich explizit von militanten Aktionen und kritisiert: \"Die Aktionen der RAF leisten [...] gerade dem Abbau b\u00fcrgerlicher Freiheitsrechte Vorschub, weil sie von Seiten der Herrschenden zum willkommenen Anlass genommen werden k\u00f6nnen, Terrorismus als reale Bedrohung propagandistisch auszuschlachten, und eine Legitimation der Durchsetzung autorit\u00e4rer Strukturver\u00e4nderung erleichtern.\" Die Rote Hilfe Westberlin entgegnet zu diesem Thema, sie sei \"nicht bereit, sich an dem staatlich verordneten Distanzierungs- und Abgrenzungswahn zu beteiligen, der innerhalb der Linken ausgebrochen ist\" und erkl\u00e4rt: \"Der Staat will die Linke auf eine totale Absage auf revolution\u00e4re Gewalt verpflichten. Letztlich also auch darauf, herrschende Verh\u00e4ltnisse zu akzeptieren. Wir brauchen nicht die Geschichte zu bem\u00fchen, um zu wissen, dass die Herrschenden nie ihren Sessel freiwillig r\u00e4umen werden, sondern im Gegenteil ihre Macht mit Brutalit\u00e4t und Gewalt aufrechterhalten wollen. [...] Ziel der Kampagne des Staatsapparates ist es, die Linke an der Frage der Gewalt weiter zu spalten in einen friedlichen, anst\u00e4ndigen Teil auf der einen Seite, einen gewaltt\u00e4tigen, terroristischen Teil auf der anderen Seite. [...] Der gemeinsame Gegner, Staat und Kapital, wird allm\u00e4hlich aus den Augen verloren, der Widerspruch zwischen uns und dem Staat wird verschleiert. Der Gegner ist nun nicht mehr der Staat, sondern die Guerilla, die Schuld an allem \u00dcbel ist [...]. So findet man in der Guerilla einen bequemen S\u00fcndenbock f\u00fcr die eigene Hilflosigkeit im Kampf gegen die Repression des Staates. [...] Es gibt eine breite Palette von M\u00f6glichkeiten des Widerstandes, angefangen von Reden mit dem Nachbarn \u00fcber Beteiligung an Demos, Veranstaltungen bis hin zu bewaffneten Angriffen auf Einrichtungen und Repr\u00e4sentanten des Systems. Alle diese M\u00f6glichkeiten k\u00f6nnen richtig sein und es ist falsch, eine gegen die andere auszuspielen. Genau dar\u00fcber, meinen wir, ist es wichtig, sich sowohl mit dem SB als auch der Stadtguerilla auseinanderzusetzen. Eine solche Diskussion ist aber nur dann m\u00f6glich, wenn nicht von vorneherein bestimmte Gruppen und ihre bewaffneten Kampfformen als au\u00dferhalb der Linken stehend diffamiert werden.\"\n\nEin Thema, das unmittelbar mit der autonomen Bewegung zu tun hat, ist der Widerstand gegen Atomkraftwerke und atomare Wiederaufbereitungsanlagen in Brokdorf, Wackersdorf und Gorleben. Bereits in einem Bericht in der Nr. 8 vom November 1976 wird geschildert, wie mit Gewalt, Wasserwerfern und Tr\u00e4nengas gegen Demonstranten vorgegangen wurde. Weiter berichtet radikal Nr. 10 \u00fcber den Widerstand gegen den geplanten Bau von Kernkraftwerken und warnt ausf\u00fchrlich vor dem \"unbegrenzten Wahnsinn\" der Kernenergie, kl\u00e4rt \u00fcber die Technik und Sicherheitsprobleme auf. Zum Ende der 1970er Jahre wird dieser Widerstand zunehmend aggressiver. Plakativ dazu zeigt der Titel der Nr. 55 das Bild eines Menschen mit Gewehr vor einem Atomkraftwerk und der Unterschrift: \"Bewaffnete Linke??\".\n\nIm selben Zeitraum tauchen die ersten Meldungen von Instandbesetzungen auf. Mit dieser Formulierung wurde ausdr\u00fccklich darauf hingewiesen, dass die Besetzer_innen sich Wohnungen nicht einfach widerrechtlich aneignen, sondern diese auch in Eigenleistung wieder bewohnbar machen w\u00fcrden. Aufgrund einer absurden Sanierungspolitik der Stadt standen zu dieser Zeit ganze Wohnh\u00e4user leer und wurden dem Verfall preisgegeben, um sie dann abrei\u00dfen, zu lassen und durch teure Neubauten zu ersetzen, w\u00e4hrend die Wohnungsknappheit im Westteil Berlins immer unertr\u00e4glicher wurde. So wie es immer wieder zu R\u00e4umungen der besetzten Wohnungen kam, wurde auch in sch\u00f6nster Regelm\u00e4\u00dfigkeit neu besetzt. Den Beginn der 1980er Jahre pr\u00e4gte dann die erste gro\u00dfe Phase des H\u00e4userkampfes in Westberlin, an dem die autonome Bewegung ma\u00dfgeblich beteiligt war (siehe auch das Interview mit Michael Wildenhain in diesem Buch).\n\nEin weiterer Eckpfeiler autonomer Bestrebungen k\u00fcndigt sich auch bereits in den ersten Jahrg\u00e4ngen der radikal an. Mit Aufrufen gegen Neonazi-Aufm\u00e4rsche und Beitr\u00e4gen \u00fcber die NPD und Kneipen, die als Treffpunkte von Alt- und Jung-Nazis dienten, wurde vor faschistischen Tendenzen gewarnt. Berichtet wurde von Antifa-Demos und \u00fcber eine antifaschistische und antirassistische Initiative unter dem Motto \"Rock ist wieder politisch\" nach dem englischem Vorbild des rock against racism. Das erste von der Naturfreundejugend und dem Spandauer Sch\u00fclerladen im Rahmen dieses Projekts organisierte Konzert stie\u00df bei der damaligen radikal-Leser_innenschaft jedoch nicht unbedingt auf Begeisterung, wie ein Leser_innenbrief aus der Nr. 62 illustriert, der zudem erkennen l\u00e4sst, wie weit die Linke von der subkulturellen Str\u00f6mung des Punk damals noch entfernt war: \"Als ich die Dachluke [eine Jugendclubdisco] betrat, dachte ich in einem Disco-Flipperladen gelandet zu sein, wo sich die Wochenendpunker zum Stelldichein trafen, und nicht auf einer politischen Veranstaltung. Das Konzert hatte noch nicht begonnen, es gab keinerlei Vorinformation \u00fcber den Zweck der Veranstaltung und das Publikum machte aus Langeweile seine eigene Show, mit Lederkluften angetan, rot-gr\u00fcnen Haaren im B\u00fcrstenschnitt, als Pomadeigel oder im Frack mit Schmetterlingsbrille, egal ob Mann oder Frau. [...] Die erste Gruppe fing langsam an. Sie nannte sich 'Breitmaulfrosch' und war eine Sch\u00fclergruppe aus Spandau. [...] Doch diese Gruppe ging bald unter in den Rufen des Publikums, aufzuh\u00f6ren, da sie wohl zu viel Jazz machten. Punk war die Parole des Abends. Ein engagierter Beitrag \u00fcber die Arbeit des Sch\u00fclerladens Spandau und seine Probleme mit Neonazis erzeugte Langeweile, noch mehr der Beitrag eines Vertreters der Naturfreundejugend. Jetzt spielte die Punkgruppe 'Auswurf', die ihr Debut am ersten Mai am Spreewaldplatz hatte. Ihre Mitglieder sind jugendliche Arbeiter. Ihre Musik war laut, um nicht zu sagen nur aggressiv, und von dem politischen Anspruch ihrer Texte war durch das Schlagzeugget\u00f6se leider nichts zu merken. Selbst die wenigen Wortfetzen, die \u00fcber die Anlage kamen, waren nicht mehr zu verstehen als 'Auswurf'. Doch das Publikum st\u00f6rte sich daran gar nicht, sondern tanzte in einem steifen abgehackten Stil, sich gegenseitig anrempelnd und kreischend. Wo bleibt da die Politik? Ist es genug, eine Musik zu machen, die sich vom Bisherigen abhebt, sei es, auf eine Art h\u00e4rter zu spielen als der Rock oder durch aggressives Losschreien in das Publikum. Ich fand auf jeden Fall das ganze Konzert fragw\u00fcrdig in seinem Anspruch, einen Beitrag im Widerstand gegen die Neonazis zu leisten.\"\n\nZum Ende des Jahres 1979 ist eine Umbruchstimmung in der Redaktion sp\u00fcrbar. Der Titel der Ausgabe Nr. 69 zeigt eine in die Toilette gestopfte radikal, dar\u00fcber steht etwas holprig: \"Wir wissen nicht, ob es besser wird, wenn es anders wird, aber es muss anders werden, damit es besser wird!\". Das Thema des Forums lautet \"radikal in der Krise\", und damit wurde das eigene Zeitungskonzept in Frage gestellt. In der nachfolgenden Ausgabe verk\u00fcndet die niederschmetternde \u00dcberschrift eines Berichts anl\u00e4sslich des \u00fcberregionalen Alternativzeitungstreffens: \"Alle Alternativzeitungen stellen Erscheinen ein\". Die Arbeitsgruppe kam zu dem Ergebnis: \"Unsere Zeitungen sind langweilig, trostlos, traurig. Unsere Zeitungen sind Zeitungen der Bewegung, und wo sich nichts bewegt, l\u00e4sst sich nichts beschreiben\". Und diese Diagnose wurde zweifellos zu Recht gestellt. Provokant schlug die Gruppe der radikal darauf vor: \"H\u00f6rt auf Zeitungen zu machen, h\u00f6rt auf zu reden, handelt!\"\n\nDie Ausgabe Nr. 72\/73 k\u00fcndigt dann ein \"Treffen Autonomer Gruppen in Hamburg\" an: \"W\u00e4hrend der Bl\u00fctezeit des Spontitums hat es in mehreren westdeutschen St\u00e4dten (z. B. Berlin, Frankfurt, Hamburg) sog. Spontiplenen gegeben. Ganz gegen jede Spontanit\u00e4t haben sich die Leute aus den unterschiedlichsten autonomen Projekten beinahe regelm\u00e4\u00dfig getroffen, um gemeinsame Probleme zu bequatschen. Als die einzelnen Plenen immer mehr zu Laber-Cl\u00fcbchen degenerierten, haben sie sich nach und nach aufgel\u00f6st. In Hamburg wird jetzt wieder ein Versuch gemacht, ein Forum, f\u00fcr die autonomen Gruppen ins Leben zu rufen. In der Einleitung dazu hei\u00dft es: 'wir halten eine verst\u00e4rkte Zusammenarbeit unter uns f\u00fcr existenziell, da sich zurzeit f\u00fcr uns gef\u00e4hrliche tendenzen entwickeln: einmal die integration vieler vorher unabh\u00e4ngiger gruppen in die herrschenden gesellschaftlichen strukturen (z. B. gr\u00fcne und bunte listen), zum anderen die isolation und kriminalisierung von gruppen, die an der notwendigkeit, auch praktisch widerstand zu leisten, festhalten.'\" Die Einladung zu diesem Treffen sei, so die radikal, \"sicher auch f\u00fcr einzelne Berliner interessant\" und man vermittle Mitfahrgelegenheiten, denn: \"wir fahren selber hin\".\n\n### Die Bewegung der Bewegung\n\nOffenbar hatte sich mit dem Beginn der 1980er Jahre in der radikal tats\u00e4chlich etwas getan \u2013 die Inhalte wurden auffallend offensiver. Die erste Ausgabe des Jahres 1980 \u00fcberschreibt einen Beitrag zu AKW-Gegnern mit \"Die Militanten und die Bewegung\". Die Redaktion war zu der \u00dcberzeugung gelangt, dass Demonstrationen nicht ausreichen w\u00fcrden, um das Atomprogramm zu blockieren. Dezent formuliert hei\u00dft es: \"Vielleicht finden wir eine M\u00f6glichkeit in der konkreten Auseinandersetzung mit Erfahrungen aus militanten Widerstandsformen. Konkrete Auseinandersetzung soll hei\u00dfen: Bereitschaft zum Lernen und Weiterentwickeln\". Einige Seiten weiter wird zudem eine Art Erlebnisbericht unter dem Titel \"und dann habe ich einen Molli in ein Kaufhaus geworfen\" geboten, der die Aktion politisch nicht mehr als mit einer diffusen Kapitalismuskritik begr\u00fcndet. Die Nr. 75 enth\u00e4lt ein Bekennerschreiben zu einer Brandstiftung bei einem \"Miterbauer\" eines neuen Hochsicherheitstrakts im Moabiter Knast, welcher in diesem Zusammenhang als \"neues KZ\" betitelt wird. Und schlie\u00dflich ist die Ausgabe Nr. 76 die letzte, die den Untertitel Sozialistische Zeitung f\u00fcr Westberlin tr\u00e4gt.\n\nDie Nr. 77, deren Cover sich mit einer gespannten Zwille und dem Satz: \"Wir haben keine Chance, aber wir nutzen sie...\" kampfbereit zeigt, f\u00e4llt dann auch durch die Titelunterschrift radikal sein ist die sache an der wurzel fassen auf, eine Zeile, die in Gedichtform bereits auf dem Deckblatt der Nr. 75 zu finden war. In der Hausmitteilung dieser Ausgabe wird dazu erkl\u00e4rt, dass die radikal nun immer zum Monatsanfang erscheinen werde. Inhaltliche Schwerpunkte sollen k\u00fcnftig auf einem monatlichen Plenum aller Interessierten bestimmt werden. Dank einer eigenen Druckmaschine k\u00f6nne man bei Bedarf aktuelle Extra- und Flugbl\u00e4tter drucken. In der Folge werde die radikal also mehr ein Diskussions- als ein Mitteilungsblatt von und f\u00fcr die Westberliner Szene sein.\n\nIm Inhalt finden sich in dieser und der folgenden Ausgabe, die sich als Zeitung f\u00fcr Freiheit und Abenteuer betitelt und einen \"Aufruf zum schwarzen bunten Block am 1. Mai\" enth\u00e4lt, neben Beitr\u00e4gen zum mehr oder weniger gewaltfreien Widerstand in Brokdorf und Gorleben auch Nachdrucke aus der Zeitschrift Autonomie \u2013 Materialien gegen die Fabrikgesellschaft, die den Versuch einer theoretischen Unterf\u00fctterung der neu bewegten radikal erkennen lassen. Auffallend ist eine H\u00e4ufung von Bekennerschreiben diverser militanter Gruppen, die wiederum nicht unwidersprochen bleiben. Im vorigen Jahr gab es lediglich einen Beitrag dieser Art, den Text \"Hunde wollt ihr ewig bellen\" der RZ [Revolution\u00e4re Zellen], in dem Erkl\u00e4rungen zu diversen Aktionen, von zerst\u00f6rten Fahrscheinautomaten (aus Protest gegen hohe Fahrpreise und schlechte Bef\u00f6rderungsbedingungen) bis hin zu einer Bombe im Gerichtsgeb\u00e4ude (wegen der Entscheidung gegen eine Aufhebung der Abtreibungsverbote trotz massiver Kampagne der Frauenbewegung) abgegeben wurden. Der Text wurde in radikal 55 in seinen Hauptteilen im \"Forum zur Gewaltdiskussion\" abgedruckt und von der Redaktion kommentiert: \"Durch die Umst\u00e4nde [die Frankfurter Alternativzeitung Pflasterstrand war aufgrund dieses Textes beschlagnahmt worden] sind wir nat\u00fcrlich gezwungen zu erkl\u00e4ren (was ohnehin nicht viel hilft), dass mit dem Abdruck nicht verbunden ist, dass sich die Zeitung mit dem Inhalt des Textes identifiziert. Uns geht es darum, die presserechtliche Freiheit, unterschiedliche und gerade auch radikale Positionen zu dokumentieren und in die Debatte zu werfen, nicht wegen des \u00fcberzogenen Schutzbed\u00fcrfnisses des Staates einfach freiwillig einzustellen und nur noch Regierungsmeinung zuzulassen.\"\n\nAnfangs war es in der radikal, wie allgemein in der Alternativpresse der 1970er Jahre, noch darum gegangen, eine Gegen\u00f6ffentlichkeit herzustellen und dazu Texte zu publizieren, die in der b\u00fcrgerlichen Presse nirgends gedruckt worden w\u00e4ren. Angesichts der allgemeinen Stagnation und Befriedung innerhalb der linken Szene wurde genau dieses Konzept in Frage gestellt, denn es schien nicht viel damit erreicht worden zu sein, und so begann sich auch die journalistische Grundhaltung in der Redaktion zu \u00e4ndern. Nun wurde zentral von der eigenen Welt und den eigenen Ansichten berichtet und begonnen, sich mit deren Schwerpunkten auseinanderzusetzen. Damit wurde auch Militanz in den eigenen Reihen diskutierbar.\n\nIn der Ausgabe 79 wird die Erkl\u00e4rung der Amerikahaus-Besetzer vom 17.5.1980 in Berlin von Seiten der radikal konkret kritisiert: \"Es ist politisch falsch und gef\u00e4hrlich, die hier auf der Erde vorhandenen Probleme auf dogmatische Formeln zu reduzieren, die Erkl\u00e4rung tut das in der Tradition der RAF-Politik und -Strategie. [...] Wir allerdings lehnen das autorit\u00e4re Prinzip des ML-Dogmatismus ab und wollen versuchen, unsere Kritik an der Erkl\u00e4rung darzustellen, ohne dabei in das b\u00fcrgerliche 'Anti-Terror-Geschrei' einzusteigen.\"\n\nIn derselben Ausgabe berichtet wiederum \"1 Autonomer\" ausf\u00fchrlich \u00fcber die \"fr\u00f6hliche Militanz\" der Auseinandersetzungen anl\u00e4sslich der Rekrutenvereidigung (\"Karneval in Bremen\") am 6. Mai. Prompt findet sich dann dazu wieder ein Leserbrief mit Kritik am \"b\u00fcrgerlichen Pathos der 'Fighter'\". Thematisiert wurden aber auch andere Formen von Widerstand und Verweigerung. Statt des bisherigen Klagens \u00fcber Berufsverbote und den Mangel an zumutbaren Ausbildungs- und Arbeitspl\u00e4tzen besch\u00e4ftigte man sich im Forum der Nr. 78 unter dem Motto \"Arbeit ist Verrat am Proletariat\" mit den Vorz\u00fcgen der Arbeitslosigkeit: \"Wir wissen alle, wozu der \u00fcberm\u00e4\u00dfige Genuss von Arbeit f\u00fchrt: Zum Tode \u2013 fr\u00fcher oder sp\u00e4ter, direkt oder indirekt, bewusst oder unbewusst. Auf keinen Fall aber in die Freiheit. Zumindest nicht in die, die wir meinen. Und ganz sicher nicht die Arbeit, die SIE uns geben.\" Dazu geliefert werden Ausz\u00fcge aus der \u2013 umgehend beschlagnahmten \u2013 Brosch\u00fcre Lieber Krankfeiern als gesund Schuften des Hamburger Gesundheitskollektivs, das effektive Tipps zum Simulieren von Krankheitssymptomen gibt.\n\nNach wie vor bietet die radikal, in der Juni-Ausgabe 1980 mit dem Untertitel Zeitung f\u00fcr unkontrollierte Bewegungen, noch Informationen zu aktuellen Terminen, internationalen Brennpunkten wie Nicaragua, El Salvador oder Iran und zum Westberliner Stadtgeschehen wie die Gr\u00fcndung des Mehringhofs in Kreuzberg als \"neue Heimat\" diverser alternativer Projekte. Ebenso gibt es Berichte \u00fcber vergleichsweise harmlose Aktionen. So hatte eine Gruppe Radfahrer eigenm\u00e4chtig die Martin-Luther-Stra\u00dfe an ihrer gef\u00e4hrlichsten Stelle mit einem Fahrradstreifen auf der rechten Fahrspur verziert, weil der Senat keine Fahrradwege baute. Aber auch in dieser Ausgabe wird kundgetan, dass es \"so wie bisher mit der Zeitung nicht mehr weitergehen k\u00f6nne\", man distanziert sich von den etablierten Alternativen und rechnet mit den \"Gesch\u00e4fts(ver) f\u00fchrern der Alternativbewegung\", so der thematische Schwerpunkt der Nr. 79, ab. Damit war f\u00fcr die radikal ein weiterer Schritt weg von der akzeptierten Linken getan.\n\nHauptthema der provokant Zeitung der Bewegung 1. Juli genannten Ausgabe Nr. 80 war die auf einem Flugblatt verk\u00fcndete \"offizielle Aufl\u00f6sung\" der Bewegung 2. Juni und deren \u00dcbertritt in die RAF. Zwei Wochen sp\u00e4ter detonierte im Kreuzberger Rathaus eine Bombe. Die Verantwortung daf\u00fcr \u00fcbernahm laut Bekennerschreiben das \"Kommando der Bewegung 2. Juni\". Die urspr\u00fcngliche, selbst so bezeichnete \"unklare Bestimmung\" des 2. Juni war es, \"spontane proletarische politik\" zu machen: \"wir haben die revolution\u00e4re theorie, die analyse der bedingungen \u2013 aus der heraus die strategie und taktik, die kontinuit\u00e4t und perspektive des kampfes erst entwickelt werden kann \u2013 f\u00fcr unwichtig gehalten und 'drauflosgek\u00e4mpft', mit dem ziel, die jugendlichen anzut\u00f6rnen.\" Dieses Konzept schien bei der autonomen Bewegung Anklang zu finden. Die Reaktion in Form eines in Ausz\u00fcgen abgedruckten l\u00e4ngeren Flugblattes, so die radikal-Redaktion, \"macht deutlich: 'Wir sind alle vom 2. Juni'\".\n\nDer Umbruch ist der Zeitung nun auch optisch anzumerken. Nicht nur ist das i-Punkt-Sternchen auf dem radikal-Schriftzug der Nr. 81 nicht einfach rot wie bisher, es enth\u00e4lt den Begriff \"Umbruch\" in einer designerischen Meisterleistung zudem w\u00f6rtlich. Offensichtlich hat man sich inzwischen auch mit der Idee des Punk angefreundet. Der Druck ist in m\u00fchsam zu entzifferndem Pink gehalten und \u00dcberschriften aus ausgeschnittenen Buchstaben erinnern eher an das Layout eines Punk-Fanzines als an das einer Zeitung. Ganzseitige Collagen mit entkontextualisierten Zitaten lassen Dada-Anleihen erkennen. Abgesehen davon enth\u00e4lt das Impressum keine praktischen Aufgaben mehr. Stattdessen erkl\u00e4rt sich die radikal selbst: \"radikal [...] ist \u2013 seit der Zerschlagung des INFOs und des BUG INFOs \u2013 die einzige linke Stadtzeitung in Restberlin. Sie wurde ins Leben gerufen als ein Diskussionsforum f\u00fcr die verschiedenen Fraktionen, Gruppierungen und Parteien der Linken. Dieses Konzept, die damals noch relevanten Gruppen in einen Dialog treten zu lassen, erwies sich allerdings als nicht realisierbar. Im Laufe der Zeit verloren die ML-Ans\u00e4tze an Bedeutung, Basisbewegungen \u00fcberwucherten die Ruinen der K-Gruppen. Diese Entwicklung spiegelt sich in der radikal wider [...]. Die 80er Jahre begannen f\u00fcr die radikal mit einigen zerm\u00fcrbenden Krisen, wodurch die Zeitung, entgegen dem gewohnten 14-t\u00e4gigen Rhythmus, nur noch einmal im Monat erscheinen konnte. Seitdem versuchen wir eine neue Konzeption an die Stelle der alten zu setzen. Doch dazu bedarf es noch l\u00e4ngerer politischer Diskussionen. [...] Zun\u00e4chst \u2013 wir sind alle Freizeitjournalist\/innen und somit auf Beitr\u00e4ge von dir und mir angewiesen. Es gibt Leute, die aufgrund ihrer beruflichen Situation nur wenige Stunden an unbezahlter Arbeit in die Zeitung einbringen k\u00f6nnen, f\u00fcr andere ist radikal der Mittelpunkt ihrer Arbeit. Bezahlt wird bisher nur der Buchladen- und Kioskvertrieb mit einem Stundenlohn von DM 5.-. Die technische Erstellung einer Zeitung von z. Z. 20-24 Seiten DIN A3 erfordert einen Haufen Maloche: Wir setzen radikal selber; Lay-out, das Legen und der Handverkauf sind weitere Arbeitsschritte, die bei jeder Nummer anfallen. Dabei k\u00f6nnen wir jede Form von Unterst\u00fctzung gut gebrauchen!!\"\n\nAuch der Tonfall der Beitr\u00e4ge hat sich ver\u00e4ndert, zum Teil sind sie ironischer geworden und haben den g\u00e4hnreizerregenden Betroffenheitspathos der fr\u00fcheren Jahre abgelegt, zum Teil merkt man ihnen aber auch an, dass da jemand etwas zu viel Foucault gelesen hat und wohl dachte, dies beim Verfassen der Beitr\u00e4ge zur Selbstdarstellung dringend umsetzen zu m\u00fcssen: \"Befriedung und Marginalisierung \u2013 das sind also die Herrschaftsstrategien des zuk\u00fcnftigen Sonnenstaates, denn der moderne Polizeiapparat wei\u00df, dass es inzwischen 'nicht mehr um die Verbrechensbek\u00e4mpfung, sondern um die Beseitigung der Bedingungen von Kriminalit\u00e4t geht'. F\u00fcr uns als autonome 'Bewegung' kann das nur bedeuten, uns nicht \u00fcber paranoische Maschinen zu definieren, nicht unseren verinnerlichten Grenzen aufzusitzen, uns keine Diskurse aufdr\u00e4ngen zu lassen, sei es die geordnete orgasmische Entladung der Bev\u00f6lkerungsw\u00fcnsche auf einen Herrschaftsk\u00f6rper im Ph\u00e4nomen der Kriegsangst, sei es die gro\u00dfe Therapie-Utopie.\" Dank Punk wird der verquaste Erguss am Ende doch noch zu einer greifbaren und kompakten Aussage heruntergebrochen, denn da hei\u00dft es: \"Oder die Punks und Hans-a-plast sagens so: 'Alles geht kaputt, alles geht in Schutt \u2013 und ich lache!'\". Mit dem Bezug zur italienischen Autonomia-Bewegung wird zudem ein Versuch zur theoretischen Selbstverortung unternommen.\n\nDie internen Mitteilungen in der Nr. 82 betreffen juristische Konsequenzen, sprich: eine Redaktionsdurchsuchung, angeblich wegen des der radikal regelm\u00e4\u00dfig beiliegenden Knastblatt, aber auch eine Durchsuchung der Druckerei Contrast, in der die Filmmontagen f\u00fcr die Druckplatten der radikal Nr. 80 beschlagnahmt wurden. Grund f\u00fcr die Kriminalisierung war die Betitelung dieser Ausgabe als Zeitung der Bewegung 1. Juli, von der Redaktion selbst als ein \"Gag [...] im Zusammenhang mit dem Erscheinungsdatum der Nr. 80\" betrachtet. Offenbar hatte das den Humor der Staatsanwaltschaft jedoch \u00fcberstrapaziert und so warf diese dem presserechtlich Verantwortlichen, den Druckern und der halben Berliner Szene (als \"Hersteller und Verteiler\") Unterst\u00fctzung einer kriminellen Vereinigung nach \u00a7 129 vor. Die \u00dcberschrift auf der 82. Ausgabe lautet daher konsequent: Zeitung der unbekannten Hersteller und Vertreiber.\n\nWeitere Inhalte dieser Ausgabe betrafen die laufenden \"Instand(be)setzungen\", die als Ma\u00dfnahme gegen Wohnraumzerst\u00f6rung und Wohnungsnot gerechtfertigt wurden. Die Kriminalisierung wurde hier schlicht umgedreht: \"Zerst\u00f6rung von brauchbarem Wohnraum durch Spekulanten und Politiker ist unverantwortlich und im Grunde 'kriminell'. Diese Sanierungspolitik l\u00e4sst sich nicht mit dem Polizeikn\u00fcppel vertuschen\", hei\u00dft es auf dem Titelblatt der Nr. 82 und es wird zu einer Demo am 10.10.1980 gegen drohende R\u00e4umungen aufgerufen. Im Heft wird zudem mit einem Brief an den Senator f\u00fcr Bau und Wohnungswesen (\"Lieber instandbesetzen als kaputt besitzen\") eine Aktionswoche der Hausbesetzer angek\u00fcndigt.\n\nEine Rarit\u00e4t im Kontext der radikal findet sich auch in dieser Ausgabe, und zwar wurde mit \"Schwuler Oktober \u2013 auch f\u00fcr Heteros \u2013 vom 10.-26.\" ein bundesweites \"Schwulentreffen\" in Berlin angek\u00fcndigt, das dem Erfahrungs- und Kulturaustausch dienen und verschiedene Bereiche von Theater, Film und Literatur bis hin zu Rockmusik und Cabaret miteinbeziehen sollte. Abgesehen von einer Auseinandersetzung in der radikal Nr. 47 \u00fcber die mangelnde Repr\u00e4sentation der Anliegen schwuler M\u00e4nner \u2013 Lesbenthemen wurden ohnehin direkt an die Frauenbewegung abgeschoben \u2013 in der im Entstehen begriffenen \"bunten alternativen Liste\" war das Thema Homosexualit\u00e4t in der linken Szene offenbar extrem unterrepr\u00e4sentiert. Auch in der Nr. 65 beklagte ein sich selbst als \"heterosexuell\" bezeichnender Mann die mangelnde Unterst\u00fctzung der Christopher-Street-Day-Demo, auf der er unter den 400 Teilnehmenden \"keinen Bekannten aus der Sponti-, AKW- und so Szene gesehen habe\", und das h\u00e4tte ihn \"recht frustig\" gemacht. Angesichts von etwa eintausend Unterst\u00fctzern auf einer Nicaragua-Demo fragt er: \"Wie entwickelt sich Betroffenheit und warum bei der einen Sache, bei der anderen wenig oder gar nichts? [...] Sexuelle Unterdr\u00fcckung, gibt es die? Frauen werden von M\u00e4nnern unterdr\u00fcckt. M\u00e4nner werden von M\u00e4nnern unterdr\u00fcckt. Heteros unterdr\u00fccken Homos usw. usw. Wen betrifft das? Mich, iihh bewahre. Aach, ich verga\u00df \u2013 nicht einmal die Frau-Mann Beziehungen werden in der 'Linken' konsequent angegangen. Wie k\u00f6nnte ich da verlangen, sich mit der 'Homosexuellenproblematik' auseinanderzusetzen.\" Trotz dieses berechtigten Vorwurfs bleibt es um queere Thematiken in der radikal aber auch weiterhin still.\n\nEine nachfolgende Sonderausgabe 83 ist das Ergebnis des Diskussionswochenendes zum Thema \"Wir sind alle vom 2.6.\" im September 1980. Ausf\u00fchrlich geschildert werden interne Auseinandersetzungen unter dem Titel \"Militanz ist wieder diskutierbar\", dazu bietet das Heft eine Art Geschichtsschreibung der Bewegung 2. Juni. Reklamiert wird, dass ein \"Mythos des bewaffneten Kampfes\" aufgebaut werde, der letztendlich zum Ziel habe, Teile der Widerstandsbewegung zu entsolidarisieren. So wurde den Entf\u00fchrern des Politikers Peter Lorenz von Seiten der Polizei Intelligenz, Phantasie und Sendungsbewusstsein zugesprochen, ein akademischer Hintergrund unterstellt und deren gelungene Strategie anerkannt. radikal spottet: \"Die Entf\u00fchrer m\u00fcssen demnach Menschen sein mit \u00fcbernat\u00fcrlichen Begabungen und F\u00e4higkeiten, die ein Durchschnittsmensch nicht besitzt.\" In ihrer Erkl\u00e4rung stellen die T\u00e4ter klar: \"Wir sind keine Phantome und auch nicht 'krankhaft genial', wie Parteien, Presse und Polizei sich und der Bev\u00f6lkerung einreden wollen, um ihre eigene Erb\u00e4rmlichkeit zu bem\u00e4nteln.\" Die an der Entf\u00fchrung Beteiligten seien vor allem Handwerker und Lehrlinge gewesen. Zur Erkl\u00e4rung dieses Widerspruchs hei\u00dft es, dass durch die Mystifizierung sowohl eine Identifikation mit militanten Aktionen als auch deren Sinnhaftigkeit zunichte gemacht werden solle.\n\nWeiter best\u00fcnde \"ein zweiter Entsolidarisierungsversuch der staatlichen Konterrevolution\u00e4re darin, revolution\u00e4ren Widerstand auf die Stufe der Barbarei zu dr\u00fccken\". Zu diesem Zweck werde versucht, \"das Bild wahllos um sich schie\u00dfender, bombender, ausgeflippter Desperados in die K\u00f6pfe zu dr\u00fccken, die vor der Gem\u00fcsefrau an der Ecke nicht halt machen, Kinder notfalls auch noch entf\u00fchren, das Leben Unschuldiger und Unbeteiligter leichtfertig aufs Spiel setzen, ganze Fu\u00dfballstadien bedrohen, Bahnhofs Schlie\u00dff\u00e4cher explodieren lassen und es nat\u00fcrlich auch auf Atombombenkraftwerke abgesehen haben.\" Etwas pathetisch wird schlie\u00dflich erkl\u00e4rt, dass der Mut zu radikalen Aktionen nicht von \u00fcbermenschlichen F\u00e4higkeiten abhinge, sondern bestimmt sei \"durch eine unbestechliche Haltung zur menschlichen Moral, eine unverdorbene Sensibilit\u00e4t f\u00fcr politisches Recht und Unrecht und die einfache F\u00e4higkeit praktisch Partei zu ergreifen\".\n\nAuf derselben Seite ist ein Bekennerschreiben einer \"autonomen Zelle aus der Bewegung 2. Juni\" abgedruckt, das einen etwas anderen Tonfall anschl\u00e4gt: \"Die Bullen haben heute im KaDeWe und in der deutschen Oper zwei 'hochgef\u00e4hrliche Sprengs\u00e4tze' entsch\u00e4rft. Beim \u00d6ffnen der Bomben fanden die Pigs neben einem brisanten Arsenal von Mohrenk\u00f6pfen und einem Wecker auch folgende Erkl\u00e4rung: Wir gratulieren euch zu der au\u00dfergew\u00f6hnlichen Leistung, diese Bombe gefunden und entsch\u00e4rft zu haben. Brav. Ihr habt euch um die FDGO sehr verdient gemacht. Als Anerkennung f\u00fcr diesen todesmutigen und staatserhaltenden Einsatz freuen wir uns, euch diese Negerk\u00fcsse \u00fcberreichen zu k\u00f6nnen.\" Abgesehen von den 1980 offenbar noch nicht antirassistisch hinterfragten Formulierungen k\u00f6nnte man meinen, dass hier dem \u00dcbermenschlichkeitsmythos eine Art Klamaukmythos entgegengesetzt werden sollte.\n\nMit der Nr. 84 wurde der Bruch von der sozialistischen Zeitung f\u00fcr Westberlin zur zeitung f\u00fcr unkontrollierte bewegungen (hier: f\u00fcr d'Bewegig \u2013 von de Bewegig), verbunden mit einem \"bruch mit gro\u00dfen teilen der szene und linken\", endg\u00fcltig vollzogen. Das zeigt, flankiert von entsprechenden Leser_innenbriefen, das Intro: \"Diese trennung schmerzt uns nicht, im gegenteil, sie kommt reichlich sp\u00e4t! Wir stellen in schlagworten fest: die alternativbewegung ist gescheitert, sie funktioniert als ein nebenzyklus des kapitalismus und ist damit keinerlei gefahr f\u00fcr das system. Doch genau mit diesen krisen und klassenk\u00e4mpfen wird eine radikale infragestellung von staat, gesellschaft und produktion einhergehen.\"\n\nDie Neuausrichtung der radikal wird w\u00f6rtlich als die \"f\u00fcr eine autonome anarchistische bewegung\" bezeichnet, und das aktuelle Selbstverst\u00e4ndnis liest sich wie folgt: \"Wir wollen nicht eine zeitung f\u00fcr, sondern von sch\u00fclern, punks, siemensarbeitern, jobbern, hausfrauen, rockern, massenarbeitern, arbeitsemigranten, \u2013 nen paar alternativen nichtstuern, arbeitsscheuen, aussteigern, kiffern, tr\u00e4umern, stadtguerillos... Wir wollen keine avantgarde sein, nur vermitteln, was wir denken und f\u00fcr richtig\/wichtig erachten; die zeitung nicht f\u00fcr jemand machen, sondern als ausdruck von bewegungen; wollen multiplikator von ans\u00e4tzen zu aufstandsbewegungen sein, ein faktor eines aufstandes, unkontrollierte bewegungen, auf das es brodelt und kocht... warum aufst\u00e4nde, das zu erkl\u00e4ren, ersparen wir uns... Kein blatt f\u00fcr leute, die dabei sind, sich einzurichten; den kreis der szene sprengen. Raus aus unserem gem\u00fctlichen angstvollen ghetto; anders gesagt: sabotage und subversion sind \u00fcberall, die szene ist nur ein teil davon: schule schw\u00e4nzen, krank feiern, wilde streiks, ladendiebstahl, h\u00e4userkampf, schwarzfahren...\" Unterschrieben wurde mit: \"ein kleiner teil der radikal-redaktion jacques mesrine, wurzelsepp, temrock\". Weiter geht es mit der Distanzierung von der alternativen Linken auch in der Nr. 85 vom Dezember 1980. Diese hat als Hauptthema die Kritik am Konsum, thematisiert explizit alternative Konsumenten und offeriert passend dazu am Anfang ein \"radikal-Geschenkpaket\" als \"unheimlich dufte Weihnachts\u00fcberraschung\", die \"auf keinem WG-Gabentisch fehlen\" sollte, angepriesen als \"f\u00fcr linke Verantwortungstr\u00e4ger unbedingt aus zeit\u00f6konomischen Gr\u00fcnden empfehlenswert\". Bei genauerer Betrachtung dieses h\u00fcbsch zusammengestellten und sicher zum Bedauern mancher Leser_innen wohl nicht ernst gemeinten Angebots ergibt sich eine Charakteristik des ehemaligen Zielpublikums, auf das nun weniger Wert gelegt wird:\n\n\"-1 Langspielplatte 'The harder they come' von Jimmy Cliff, dem bekannten Reggae Superstar, der Originalsound aus den rebel-slums der 3. Welt!\n\n-1 Paar Original 'Roots' (Bitte unbedingt Schuhgr\u00f6\u00dfe angeben!) mit leichten Farbmakeln\n\n-1 P\u00e4ckchen Henna (damit k\u00f6nnt ihr neben den verschiedenen K\u00f6rperhaaren auch eine einheitliche F\u00e4rbung der 'Roots' in Eigenproduktion bewerkstelligen).\n\n-1 Flasche 'Revolution\u00e4rer Wein aus Okzitanien' \u2013 Jeder Schluck aus dieser Flasche ist dein solidarischer Beitrag von praktisch-wein-nachtlichem Internationalismus!\n\n-1 schwarz-rotes Pal\u00e4stinensertuch \u2013 Endlich wird der Qual der Wahl zwischen rot und schwarz ein Ende bereitet!\n\n-1 Buch 'Marijuana-Anbau in der Wohnung' \u2013 ein beliebter Beitrag zum Sch\u00f6ner Wohnen, der in keiner Alternativ-WG fehlen sollte!\n\n-1 Jahresabonnement der radikalsten deutschsprachigen Zeitung (die mit den brandhei\u00dfen super-Power-Ermittlungsverfahren): 'radikal', die Zeitung f\u00fcr Freiheit und Abenteuer\"\n\nZum autonomen Kernthema H\u00e4userkampf bietet diese Ausgabe einen liebevoll gestalteten Stadtplan von Kreuzberg 36, auf dem besetzte H\u00e4user eingezeichnet sind, mit der Aufforderung, \"da laufend neue H\u00e4user besetzt werden, erg\u00e4nzt ihn selber...\". Dazu gibt es den Hinweis, \"Leute, die im falle einer r\u00e4umung nicht tatenlos zusehen wollen, treffen sich bei alarm (\u00fcber telefonkette) am kottbusser tor oder am moritz-platz\", und es folgen die praktischen Tipps: \"Lasst euch was einfallen: baut barricaden! besetzt br\u00fccken und kreuzungen! st\u00f6rt, wo ihr k\u00f6nnt! organisiert euch in gruppen! schafft \u00f6ffentlichkeit!\".\n\nEbenfalls im Dezember erschien kurz darauf ein radikal-Extrablatt, die Zeitung f\u00fcr eine sch\u00f6ne Bescherung. Die Schlagzeile der Sondernummer \"'INNERE UNRUHEN' Berliner Autonome schlagen zu (siehe Foto)\" verweist auf eine darunter befindliche Fotografie. Diese zeigt vor dem Hintergrund behelmter Polizisten eine \u00e4ltere Dame mit dunkler Sonnenbrille, die aus einer eingeschlagenen Glasvitrine auf dem Kudamm diverse Stehlampen befreit. Die Bildunterschrift zitiert Springers Morgenpost vom 14.12.1980: \"Unter den Pl\u00fcnderern viele junge Frauen\". In dieser Sondernummer ging es um die am 12.12.1980 anl\u00e4sslich der versuchten R\u00e4umung eines besetzten Hauses beginnenden Stra\u00dfenschlachten in Kreuzberg und am Kudamm. Eine Chronologie verzeichnet die Ereignisse, die Forderungen der Besetzer nach der Freilassung Inhaftierter und auch die zum Teil schwer Verletzten. Die R\u00fcckseite des Extrablatts beschw\u00f6rt den Zusammenhalt zwischen den gem\u00e4\u00dfigten Demonstranten und den, wie es hier selbstironisch hei\u00dft, \"Chaoten\". Die Aufforderung: \"Gewalt oder Gewaltlosigkeit, Militanz oder Witz, stehen sich dabei nicht als Alternativen gegen\u00fcber, sondern m\u00fcssen einander erg\u00e4nzen\", hat die radikal mit dieser Sondernummer jedenfalls bestens erf\u00fcllt.\n\nDie folgenden Ausgaben berichten von den \u00fcber 200 Neubesetzungen der n\u00e4chsten Monate, unter anderem auch \u00fcber das erste von der bereits l\u00e4nger existierenden autonomen Frauen- und Lesbenbewegung besetzte, m\u00e4nnerfreie \"Hexenhaus\" ebenso wie von den H\u00e4userk\u00e4mpfen in anderen westdeutschen St\u00e4dten \u2013 sogenannte \"Notizen aus der Provinz\" \u2013 und auch in der Schweiz und in den Niederlanden. Mit weiteren Bekennerschreiben zu diversen Anschl\u00e4gen und dem Titelbild der Nr. 86, darauf eine Horde muskelbepackter angriffslustiger Urmenschenwesen und der Schlachtruf \"Wir wollen alles\", schien f\u00fcr die radikal endg\u00fcltig die Anarchie ausgebrochen zu sein. Dennoch sind die Zeichen f\u00fcr Ver\u00e4nderungen und eine drohende Entzweiung innerhalb der Besetzer_innenbewegung schon wieder erkennbar: \"Von der Offensive der spontanen Aktion sind wir nun \u00fcber den Verhandlungs- und Spaltungsapparat der Bewegung selbst in die Defensive geraten\". Offenkundig bef\u00fcrwortete die Redaktion dabei weiterhin den Standpunkt der Nichtverhandler_innen.\n\nNach wie vor war die radikal auch immer wieder Kriminalisierungsversuchen seitens der Staatsanwaltschaft ausgesetzt, die jedoch zun\u00e4chst im Sande verliefen. So wird in der \"Hausmitteilung\" der Nr. 88 ironisch verk\u00fcndet: \"Wir bitten die zust\u00e4ndigen Staatsorgane hiermit \u00f6ffentlich von weiteren Zusendungen abzusehen, in denen uns mitgeteilt wird, dass wieder ein neues Ermittlungsverfahren gegen die radikal eingeleitet worden ist. Wir verlieren n\u00e4mlich langsam den \u00dcberblick bzw. haben aufgeh\u00f6rt, mitzuz\u00e4hlen. Auch die teilweise echt gut gemachte Public Relation in Springerpresse und im ZDF langweilt uns zunehmend.\"\n\nNach einem Demonstrationsaufruf in der radikal 86 vom Januar 1981, der von diversen Bekennerschreiben \u2013 \"Bombe f\u00fcr Reaktorsicherheit\", \"Brandanschlag\" auf einen Zulieferer der Atomindustrie \u2013 begleitet wird, ist Brokdorf schlie\u00dflich in der 88. Ausgabe das zentrale Thema. Laut des zitierten Anti-AKW-Telegramm war der Bau des AKW aufgrund der massiven Proteste vier Jahre lang auf Eis gelegt worden. In der Februarausgabe hei\u00dft es nun, dass \"seit dem 6.2.81 in Brokdorf weitergebaut [wird]\" und der k\u00e4mpferische Widerstand daher langfristig und mit verschiedenen Mitteln weitergef\u00fchrt werde und man sich auch weiterhin nicht nur auf Demos und Aktionen am Bauzaun beschr\u00e4nke. Parallel dazu geht es auch im Kontext der Hausbesetzungen um \"das revolution\u00e4re Schweigen des Steines\". Reflektiert wurde auch die \"Sprachlosigkeit der Bewegung\". Nicht in dem Sinne, dass den autonomen Autor_innen die Worte ausgehen w\u00fcrden, als vielmehr darauf zielend, dass nun die Zeit des Handelns und nicht des Ver-Handelns oder gar Rechtfertigens angebrochen sei. Und dennoch zeichnete sich immer deutlicher eine Spaltung der H\u00e4userbewegung in Verhandlungswillige und -unwillige ab.\n\nMit der vermeintlichen \"Sprachlosigkeit\" der Jugendunruhen, die sich (ebenso wie Graffiti, unkonventionelle Kleidung und provozierendes Verhalten) auch als eine neue Sprache interpretieren l\u00e4sst, gegen welche die Erwachsenengeneration weit sprachloser erscheint, besch\u00e4ftigte sich auch eine \"erste staatliche Studie zu den Unruhen\". Die Publikation stammte aus der Schweiz, und in dem in der radikal nachgedruckten Text hei\u00dft es: \"Unsere Zukunft ist nicht von randalierenden Jugendlichen gef\u00e4hrdet, sondern von Krankheitserscheinungen unserer Gesellschaft, die Jugendliche zu Auflehnung veranlassen. Ohne die Jugend hat unsere Gesellschaft keine Chance. Mit einer passiven, angepassten, resignierten Jugend hat sie eine nur krankere Zukunft. Im Dialog mit Jugendlichen, die anders denken, und die vielleicht auch nur laut sagen, was wir h\u00f6chstens leise zu denken wagen, k\u00f6nnen wir m\u00f6glicherweise an einer besseren Zukunft arbeiten.\" Die Studie wird in einer vorangestellten Einleitung zwar als \"differenzierte Analyse\", verbunden mit einer \"Gest\u00e4ndnisproduktion von gesellschaftlichen Vers\u00e4umnissen und Fehlern\" betrachtet, aber auch als Versuch, \"uns in ihre Gesellschaft zu integrieren\". Daher wird von Seiten der radikal auch hier schon im Vorfeld gefordert: \"Doch lassen wir uns nicht vereinnahmen, verweigern wir den Dialog mit der Macht, lasst uns in den L\u00f6chern, den Br\u00fccken, in den Betonmauern die Diktatur des Sinnes zersprengen, das Delirium in die Ordnung der Kommunikation einf\u00fchren, das Begehren, die Wut, die Verr\u00fccktheit, die Ungeduld, die Verweigerung sprechen lassen.\"\n\nAuch in sp\u00e4teren Ausgaben wurden Studien, die sich thematisch mit der autonomen Szene oder ihr nahestehenden Aspekten besch\u00e4ftigten, nicht gerade wohlwollend aufgenommen. Ob ihre Initianten nun Soziologen oder Kunsthochschulen waren, sie standen immer unter dem Generalvorwurf des Integrations- und Befriedungsbestrebens und machten sich grunds\u00e4tzlich unbeliebt. In der Ausgabe Nr. 89 konzentriert sich die radikal ausschlie\u00dflich auf eine Auseinandersetzung mit Faschismus und Nationalsozialismus und zeigt damit, dass sie neben den militanten Aufrufen, programmatischen Versuchen und der eher flachen Literatur-\"von uns unten\";-Sondernummer vom Januar inhaltlich und intellektuell immer noch eine Menge zu bieten hat.\n\n### Der Kampf geht weiter, doch wohin? \u2013 Das Dilemma der Unintegrierbarkeit\n\nMit einem traurigen Clown auf der Titelseite und der deprimierten Frage \"Bewegung kaputt?\" kehrte die Nr. 90\/91 wieder zur\u00fcck in das \"Get\u00fcmmel der Stra\u00dfe\", wie es im Vorwort hei\u00dft, \"um wieder dabei zu sein, wenn die Bewegung den Bach rauf oder runter treibt\". Erkl\u00e4rend dazu hei\u00dft es jedoch einige Zeilen weiter: \"Der Aufmacher der Seite und Titel des Schwerpunktthemas [...] soll nicht unserer Resignation und schon gar nicht irgendeiner verinnerlichten Niederlage Ausdruck verleihen, sondern auf die Zerfalls- und Spaltungslinien des H\u00e4userkampfs deuten und vielleicht helfen, uns alle dar\u00fcber hinweg zu katapultieren \u2013 zu neuen Ufern.\" Das Gef\u00fchl einer Niederlage manifestierte sich dennoch an der gewaltsamen endg\u00fcltigen R\u00e4umung eines besetzten Hauses am Berliner Fr\u00e4nkelufer. Dass es jedoch, gerade angesichts der immer n\u00e4her r\u00fcckenden Kriminalisierung der radikal, eine breite Solidarit\u00e4t mit der Zeitung und damit auch mit der autonomen Bewegung gab, zeigt der Erfolg der Unterst\u00fctzungsaufrufe unter dem Motto \"Wir sind keine kleine kriminelle Vereinigung, wir sind eine gro\u00dfe\" neben dem Impressum, das eine Liste der etwa zwanzig mitherausgebenden Projekte und Unternehmen beinhaltet.\n\nAktuell ist in dieser Ausgabe auch die immer wieder thematisierte Warnung vor \"Zivis\" \u2013 Spitzel der Polizei in Zivilkleidung \u2013 auf Veranstaltungen wie der Hungerstreik-Demo in Berlin-Neuk\u00f6lln im M\u00e4rz 1981, mit gleichzeitiger Ermahnung, trotz allem nicht in Paranoia zu verfallen (\"Nicht jeder unauff\u00e4llige Auff\u00e4llige im Alter zwischen 21 und 50 Jahren ist ein Zivi\"), und Tipps, woran diese zu erkennen w\u00e4ren und was zu tun sei (\"Wir habens auf der Demo erlebt, dass das blo\u00dfe Abtasten den Bullen verscheucht hat\"). Raumgreifender werden in diesen Ausgaben jedoch die kooperationsbereiten Alternativen als Gegner thematisiert, sei es das Netzwerk Selbsthilfe, die taz, UFA-Fabrik oder die Alternative Liste. Bereits in der vorigen Ausgabe war unter der \u00dcberschrift \"Die alternativen Techniken des sanften Staatsschutzes\" von \"Geheimverhandlungen zwischen Sozialdemokratie und Netzwerk\" die Rede gewesen, in denen es darum gehen sollte, \"wie man sich der staatlichen Kontrolle entziehende Teile der Jugend wieder in den Staat integrieren, oder sie wenigstens auf einem alternativen Nebenzyklus kaltstellen kann\". Dazu wieder einmal der Aufruf: \"Wir lassen uns nicht integrieren. Wir werden hier schon noch alles zur Wiese machen. Passt blo\u00df auf!!!\"\n\nEs scheint, als ob mit dieser gebetsm\u00fchlenartigen Wiederholung der Unintegrierbarkeit der autonomen Bewegung ein Zusammenhalt und eine Entschlossenheit herbeibeschworen werden sollte, die in der Realit\u00e4t schon l\u00e4ngst nicht mehr existierten. Dazu arbeitete man sich noch am Operaismus nach dem Vorbild der italienischen Autonomia-Bewegung ab, der als theoretisches Grundger\u00fcst dienen sollte. Im Artikel \"Sabotage\" wird erkl\u00e4rt: \"Autonomie ist l\u00e4ngst ein politischer Begriff geworden, der sich weit \u00fcber italienische Grenzen und die Organisation der 'Autonomia' verbreitet hat. 'Autonomie, aber subito' hei\u00dft eine Parole der Z\u00fcricher Bewegung. Autonomie hei\u00dft es auch in Berlin. \u00dcber die genaue Definition dieses Begriffes herrscht allerdings viel Unklarheit. [...] Autonomie ist jene Radikalit\u00e4t, die die sp\u00e4tkapitalistischen Normen bis in den letzten Winkel, am Arbeitsplatz, in den Schulen und Lernfabriken, im Freizeit und Konsumangebot, in den Wohnsilos der Zwangsehen, ablehnt, negiert. Diese Radikalit\u00e4t konnten die Herrschenden noch niemals verstehen, denn sonst m\u00fcssten sie sofort von der historischen B\u00fchne abtreten.\"\n\nMit der Nr. 93 wird deutlich, dass sich die autonome Bewegung nicht nur auf Westberlin und -Deutschland beschr\u00e4nkte. Der Nachdruck einer Darstellung der Jugendunruhen in der Schweiz beschreibt die \"Z\u00fcricher N\u00e4chte\" als ein \"Potlatch der Zerst\u00f6rung\". Die darauf folgende Sommerausgabe l\u00e4sst erkennen, dass auch diese radikal-Redaktion sich bereits dezimiert hat. Zum Teil urlaubsreif in Richtung S\u00fcden gereist, zum Teil entnervt ob der R\u00e4umungen und der st\u00e4ndigen Nachtwachen in den besetzten H\u00e4usern, war ein personeller Engpass entstanden und man suchte wieder nach \"neuen K\u00f6pfen\" und am besten gleich \"neuen Ideen\". Vorerst wurde aber auch in dieser Ausgabe noch einmal die Auseinandersetzung mit der Autonomia-Bewegung aufgegriffen. Ins Auge f\u00e4llt in der Nr. 94\/95, dass die ganze letzte Seite der autonomen Frauenbewegung und dem Anarcha-Feminismus gewidmet ist. So endet die Ausgabe mit dem expliziten Aufruf an Frauen, nicht nur vermehrt schriftlich, sondern auch aktionistisch in Erscheinung zu treten.\n\nDie nachfolgenden Ausgaben sind dem Personalnotstand entsprechend eher d\u00fcnn ausgefallen. Eine Extra-Ausgabe im August k\u00fcndigt mit dem Untertitel Zeitung f\u00fcr ein Spektakel des Grauens \u2013 F\u00fcr den Untergang des Abendlandes den TUWAT-Kongress ab dem 25. des Monats an, der ein vierw\u00f6chiges \"Festival, ein Ausdruck f\u00fcr den Kampf, der uns verbindet\" sein soll. Vollmundig hei\u00dft es im Aufruf: \"Wir werden Wehrd\u00f6rfer auf allen Pl\u00e4tzen Berlins errichten, die Stadt mit Chaoten, Terroristen, Punks, Hippies, Gammlern, Alkis, Schwulen, Anti, Lesben, Kraakern, AJZ-K\u00e4mpfern, Schwarzen, Ausgeflippten usw. \u00fcberfluten, die Stadt muss stinken und brodeln\". Das Ganze dekoriert mit der gelungenen Zeichnung einer Stra\u00dfenschlacht zwischen Punks und Bullen, beschwert man sich im selben Atemzug \u00fcber die Darstellung in der etablierten Presse: \"'Ekel, Abscheu und Chaosfurcht sch\u00fctteln das offizielle Berlin', so t\u00f6nt der SFB-Kommentator aus dem \u00c4ther, 'Terroristen \u00fcberfallen unsere Stadt', quillt aus Springers bunt bedrucktem Klopapier und die CDU f\u00fchlt sich von der 'Beendigung des demokratischen Lebens' bedroht. [...] Ein Ereignis wird kriminalisiert, bevor es stattfinden kann.\"\n\nDie Frage, was denn anderes von der b\u00fcrgerlichen Presse erwartet wurde, dr\u00e4ngt sich hier unwillk\u00fcrlich auf. Offenbar war den Schreibenden mittlerweile die fr\u00f6hliche Ironie, die noch in den Winterausgaben vorherrschte, verloren gegangen. Und auch sonst setzt man sich mit gebotenem Ernst damit auseinander, inwieweit eine Solidarit\u00e4t mit befreiungsnationalistischen Bewegungen wie der der IRA\/INLA in Nordirland sinnvoll ist.\n\nDie zweite schmale Extra-Ausgabe enth\u00e4lt neben TUWAT-Veranstaltungsterminen einen Versuch, zehn verh\u00e4ltnism\u00e4\u00dfig knappe Thesen zur theoretischen Verortung der Bewegung zu formulieren. Diese sagen im Kern nichts anderes aus, als was in der radikal im Grunde immer wieder postuliert wird: die Ablehnung von Herrschaft und Staat ebenso wie einer zentralen Ideologie und Organisationseinheit und einem Dialog mit der \"Macht\"; die Abgrenzung zu den \"Alternativen\" und ihren Strategien; Subkultur (und nicht Arbeitskampf) als Grundlage; ein diffuser, aber kein traditioneller Anarchismus und das Eintreten f\u00fcr einen \"Normalvollzug\" im Knast. In der darauf folgenden Ausgabe 98\/1 im September 1981 werden diese Thesen ausgiebig und wortreich in Frage gestellt, erg\u00e4nzt oder verworfen und reformuliert. Zudem geht es noch um Sinn und Unsinn von relativen Freir\u00e4umen, reformistischen B\u00fcndnispartnern und Antiimperialismus, um Bewusstsein und Fantasie. So richtig zufriedenstellend waren die Thesen wohl doch nicht \u2013 die Versuche zur Theorieformulierung erstrecken sich noch \u00fcber einige weitere Ausgaben und die Diskussion dazu erscheint endlos. Auch der \"Schwarze Block\" meldet sich ausf\u00fchrlich zu Wort.\n\nWeniger theorieorientiert klingt eine Gruppe aus M\u00fcnchen, die unter dem Namen Freizeit 81 agierte. Die radikal 98\/2-10\/81, Zeitung f\u00fcr Anarchie und Wohlstand, pr\u00e4sentiert ein Interview des M\u00fcnchner Blatt mit der Gruppe, in dem diese sich im Punk verorten. Dazu wurden Konzerte veranstaltet, Fanzines verfasst und Destruktion (wie Entglasung von Banken) sowohl als kreativ als auch als Racheakt am Staat und am Materialismus begriffen. Auf der R\u00fcckseite der Ausgabe finden sich noch einige pragmatische Statements von Freizeit 81:\n\n\"FREIZEIT '81 \u2013 die Bezeichnung f\u00fcr alles und nichts [...]\n\nFREIZEIT '81 ist kein 'Schwarzer Block' und keine feste Gruppe [...]\n\nFREIZEIT '81 sind Flugbl\u00e4tter, Plakate, Zeitungen, Konzerte, Spr\u00fchaktionen, Steine, Bankenschl\u00f6sser zukleben, Mollies \u2013 jeder auf seine Art\n\nFREIZEIT '81 ist gewaltlos oder militant, legal oder illegal, \u00e4ngstlich oder stark, auf jeden Fall: GEF\u00dcHL UND H\u00c4RTE! [...]\n\nFREIZEIT '81 ist Widerstand aus dem Bauch, eine unkontrollierte Reflexbewegung. Niemand kann mit jeder Aktion einverstanden sein, aber jeder sollte seine eigenen Sachen machen. Nur Mut, aber PASST BLOSS AUF! Bildet kleine Gruppen (nur mit bekannten Leuten) oder bleibt Einzelk\u00e4mpfer. [...]\n\nFREIZEIT '81 ist eine Festveranstaltung gegen die BRD!\n\nFREIZEIT '81 hat irgendwas mit Politik zu tun!\"\n\nVor allem dieses \"irgendwas mit Politik\" d\u00fcrfte den theoriesuchenden Berlinern ziemlich sauer aufgesto\u00dfen sein. Erstaunlich genug ist dann, dass der zur Diskussion auffordernde Artikel \"Thesen zur Verweigerung des Krieges durch die Bewegung\", der in der folgenden Ausgabe die Ver\u00e4nderungen innerhalb der Bewegung, die machtvolle Gegenoffensive des Staates und die Desertation des Punk als Bewegungsmusik thematisiert, neben Rock against Reality auch Freizeit 81 als Unterzeichnende auff\u00fchrt.\n\nMit \"R\u00fcckzugsgefecht\" betitelte die radikal im Winter den ersten Artikel ihrer 99. Ausgabe und als Zeitung gegen den freiwilligen R\u00fcckzug in die Reservate sich selbst. W\u00e4hrend im Vorwort noch s\u00fcffisant berichtet wird, mit welch schlichten Mitteln ein erneuter Prozess der Staatsanwaltschaft gegen die radikal zum Freispruch gef\u00fchrt wurde, zeigt dieser erste Artikel, wie erfolglos die Taktik zur Neubesetzung von H\u00e4usern \u2013 zehn ernsthafte Versuche und sechzig zum Schein \u2013 war. Ebenso sollten die in diesem und den letzten Heften umfassend dargestellten Protestaktionen gegen die Frankfurter Startbahn West langfristig nicht das erw\u00fcnschte Ergebnis erzielen.\n\n### Die Flaute des Sturms und die L\u00f6cher im Segel\n\nDas Gedicht \"Dark Star\" auf dem Cover der 100. Ausgabe der radikal \u2013 Zeitung f\u00fcr Jugend und Sch\u00f6nheit steht im krassen Gegensatz zum beigef\u00fcgten, die Bewegung symbolisierenden Bild eines schw\u00e4chlichen, milde l\u00e4chelnden Greises in einer Badewanne:\n\n\"nach dem besinnlichen reinigungsprozess\n\nwar es der jugendlichen bewegung gelungen\n\nunter auferbietung ungeahnter potenzen\n\nihren wohltrainierten\n\nfesten k\u00f6rper\n\nempor zu recken\n\nder intelligent-neugierige blick\n\nwar hoffnungsvoll und trotzig\n\nin die zukunft geworfen\n\nwas hatte man schon zu bef\u00fcrchten?\"\n\nAuf Seite drei hatte die radikal jedoch angesichts des gutb\u00fcrgerlich-weihnachtlichen Konsumrausches wieder zu der heiteren Bissigkeit des vergangenen Winters zur\u00fcckgefunden. Ein Aufruf zur massiven St\u00f6rung des Kudammer Weihnachtsmarktes und der Grunewalder Villenidylle am 19.12.1981 hatte offenbar f\u00fcr ein Polizeiaufgebot an diesen Orten gesorgt, welches nicht nur den Einkaufsw\u00fctigen die jahresendzeitliche Besinnlichkeit verg\u00e4llt haben mag. Der radikale Beobachter themroc schildert den Auftritt der alarmierten Ordnungsh\u00fcter: \"Ich war schon sehr gespannt, als ich mich zur angegebenen Zeit dem Kudamm n\u00e4herte. [...] Der Einladung, am letzten verkaufsoffenen Samstag vor dem Fest der Liebe noch ein bisschen Abwechslung im eint\u00f6nigen Konsumentendasein zu bieten, waren fast alle gefolgt. In unkontrollierbaren Horden bev\u00f6lkerten sie den Boulevard, bauten Wagenburgen vor besonders reich dekorierten Schaufenstern, liefen, einer Parade gleich, in 4er-Pulks und mit Festtagsuniformen immer wieder den beliebten Damm entlang. Nicht wenige hatten sich als B\u00fcrger getarnt, nur die st\u00e4ndig piepsenden Antennen wollten nicht so recht ins gewohnte Bild passen. [...] W\u00e4hrenddessen konnte sich die Genossin Butters\u00e4ure unbemerkt ins nahezu hermetisch abgeriegelte KaDeWe schleichen \u2013 um dort zu verdunsten. Das weihnachtlich-stimulierte Konsumentenvolk war gern bereit, der sich st\u00e4ndig weiter ausbreitenden Weltlichkeit im Tempel der Ware Platz zu machen und str\u00f6mte zum Ausgang. Doch dort wurden sie verst\u00e4ndnislos von den Agenten der Ordnung beschnuppert. [...] Alles in allem konnte dieser Tag durchaus als eine beeindruckende und gelungene Abwechslung bewertet werden, auch wenn nicht alle Lichter brannten, weil der Brandsatz unterm festlichen Baume wohl zu fr\u00fch in Ruprechts Sack verschwand.\"\n\nVon einer anderen Aktion zur vorweihnachtlichen Stimmungsaufhellung wird auf Seite vier berichtet. Offenbar lief auch die autonome Schenkfreudigkeit um diese Jahreszeit zur Hochform auf: \"Rechtzeitig vor dem Weihnachtsfest sind auf bisher ungekl\u00e4rte Weise etwa 100.000 (Abendschau) Sammelkarten der BVG in die Briefk\u00e4sten der Berliner Bev\u00f6lkerung gelangt. In einem Begleitschreiben der BVG wird darauf hingewiesen, die Freikarten zu einem Besuch im vom Autobahnbau bedrohten Tegeler Forst zu benutzen. Diese an sich sehr l\u00f6bliche Initiative der BVG bietet allerdings, bei aller Anerkennung, einen Grund zur Kritik. Im Zuge der \u00f6ffentlichen Sparma\u00dfnahmen wurde der Auftrag an die billigste Druckerei vergeben, so dass die miese Druckqualit\u00e4t selbst von den BVG-Kontrolleuren nicht immer anerkannt wurde, was zeitweilig zu Verz\u00f6gerungen im \u00f6ffentlichen Nahverkehrssystem f\u00fchrte.\"\n\nWeit weniger erheiternd spricht ein Beitrag einige Seiten weiter von der \"jetzigen Bewegungsflaute\" und sucht Gr\u00fcnde daf\u00fcr auch in der zunehmenden Ausbreitung der Heroinabh\u00e4ngigkeit in der Kreuzberger H\u00e4user-Szene. Die Auseinandersetzung mit dieser, sich den Z\u00fcricher Zust\u00e4nden ann\u00e4hernden Tendenz in Kombination mit einer auffallenden Zur\u00fcckhaltung des Rauschgiftdezernates l\u00e4dt zu Spekulationen \u00fcber eine absichtliche und forcierte Aush\u00f6hlung der Szene auf diesem Wege ein.\n\nWas nicht weiter \u00fcberrascht: Die unerm\u00fcdliche Theoriediskussion geht in eine neue Runde und erstreckt sich auch in dieser Ausgabe \u00fcber mehrere eng bedruckte Seiten mit flie\u00dfenden \u00dcberg\u00e4ngen in eine Nabelschau der H\u00e4user-Szene, die mit mal eher pragmatischen, mal eher theoretischen Ans\u00e4tzen verzweifelt festzuhalten versucht, was ihr zunehmend zu entgleiten droht: Der Zusammenhalt und die Kraft, sich ihrer Aufl\u00f6sung zu widersetzen. Ob damit in der Szene noch etwas zu retten war, l\u00e4sst sich angesichts dieses kurzen Kommentars zum Thema \"Autonomie\" in der Februarausgabe 1982 bezweifeln: \"Eine Sache, ein Lebensgef\u00fchl oder wie auch immer gewinnt durch st\u00e4ndiges Wiederk\u00e4uen nicht an Inhalt. Ganz im Gegenteil, liebe radikale, es w\u00fcrgt einen. Bei mir war es sch\u00e4tzungsweise beim. 1500. Mal, als mich \u00dcbelkeit befiel. Wie mag es Sensibleren ergehen? Magengeschw\u00fcre, Suff, vielleicht sogar Heroin. Aber k\u00e4ut ruhig weiter. Unsere Worte waren schon immer gewaltiger und witziger als unsere Taten.\"\n\nJenseits der eigenen Befindlichkeiten r\u00fcckte in diesem Winter neben der Auseinandersetzung mit den Unruhen in Irland zunehmend die mit der Solidarnosc-Bewegung und der Verh\u00e4ngung des Kriegsrechts in Polen in den Fokus der radikal. Aber auch die Konflikte in Mittelamerika im Zusammenhang mit der US-amerikanischen Expansionspolitik wurden immer wieder aufgegriffen und differenziert betrachtet, die eigene Position dazu reflektiert und in Frage gestellt, zum Teil auch die Solidarit\u00e4t mit Befreiungsbewegungen aufgek\u00fcndigt.\n\nDas titelgebende Thema der radikal 102 unter der \u00dcberschrift Zeitung der Computer-Guerilla allerdings ist die fortschreitende Entwicklung der Mikroelektronik und deren Auswirkungen auf die Arbeitswelt. Die radikal selbst stand dem entsprechenden herstellungstechnischen Fortschritt weit aufgeschlossener gegen\u00fcber als manch traditioneller Medienbetrieb, in dem die Rationalisierung von Arbeitspl\u00e4tzen durch computergest\u00fctzte Produktionsabl\u00e4ufe problematisch erschien. Im Gegensatz dazu reduzierte sich f\u00fcr alternative Kollektivbetriebe nur die Selbstausbeutung: \"Gro\u00dfe Zeitungskonzerne haben bei der Einf\u00fchrung der 'Neuen Technik' noch die sozialen Konsequenzen am Hals. Sozialpl\u00e4ne und Manteltarifvertr\u00e4ge verz\u00f6gern den hemmungslosen Einsatz der Mikroprozessoren, so dass Axel Caesar bis vor kurzem seine Facharbeiter mit dem Bleisatz bem\u00fchen musste, w\u00e4hrend die radikal schon auf der 2. Generation des Satzcomputers herumhaut.\"\n\nBemerkenswert ist, wie treffend damals, im Fr\u00fchling 1982, die Bedeutung der Computertechnik und -vernetzung eingesch\u00e4tzt wurde: \"Die Kapazit\u00e4t des Bildschirmcomputers ist unbegrenzt, Reisebuchungen, Abfragen des Kontostands, die neueste Ausgabe der BZ, Kaisers Sonderangebote und die 1.952 Seiten des Quelle-Katalogs kann man \u00fcber den Fernseher bestaunen. Der Bildschirmtextcomputer ist mit einem externen Rechner gekoppelt, z. B. mit dem vom Quelle-Versand; Frau Annemaria Baierlein findet auf Seite 253 ein entz\u00fcckendes rosa Negligee. Per Knopfdruck bestellt sie es sich und automatisch wird der Betrag von 94,52 DM von ihrem Konto abgebucht, der Computer bedankt sich namentlich bei der Dame und bl\u00e4ttert die n\u00e4chste Seite auf. Schmei\u00dft man nun den gesamten gespeicherten M\u00fcll von Quelle bis Springer weg und denkt sich die einzelnen Fernseher untereinander verbunden, so dass praktisch jeder mit jedem kommunizieren kann und sich aus dem Computer alle gew\u00fcnschten Informationen holen kann, hat das zwar mit Romantik nicht mehr viel gemein, aber weist einen kleinen Schritt in die Dimension einer erfreulicheren Zukunft.\"\n\nAber die radikal w\u00e4re nicht sie selbst, wenn nicht unvermeidlich der folgende Absatz wieder in einer der gewohnten Kampfansagen m\u00fcnden w\u00fcrde: \"Die Mystifikation der Technik aufzubrechen soll nun nicht hei\u00dfen, sie zu idealisieren. Technologie ist nicht herrschaftsfrei \u2013 Technologie an sich gibt's nicht. Wir sollten das, wie irgendwer mal sagte, dialektisch sehen. Solange wir die Technologie nicht f\u00fcr uns selbst nutzen k\u00f6nnen, als Teil unserer Revolte gegen das Kapital, sollten wir sie zerst\u00f6ren, wo wir uns ihr unbeobachtet n\u00e4hern k\u00f6nnen.\"\n\nSeit der Ausgabe 101 gibt es anstelle der \"Notizen aus der Provinz\" die Rubrik \"Herzschl\u00e4ge\", die \u00fcber die \"An- und R\u00fcckschl\u00e4ge des westdeutschen Widerstands (einschl. angrenzende Gebiete)\" berichtet. Zentral sind dabei jedoch \"klare inhaltliche Kriterien, sich von reaktion\u00e4ren oder faschistischen Anschl\u00e4gen abzugrenzen\". Weiter hei\u00dft es unter der \u00dcberschrift \"Soll es doch brennen...\" in der Ausgabe 104: \"Au\u00dferdem ist es f\u00fcr die Linke (genauso wie f\u00fcr uns) klar, dass Anschl\u00e4ge auf Objekte gerichtet sind und Menschen dabei nicht gef\u00e4hrdet werden d\u00fcrfen\".\n\nIn den folgenden Ausgaben wird die Sorge um die vermehrten R\u00e4umungen besetzter H\u00e4user sp\u00fcrbar und manifestiert sich in immer zahlreicher werdenden Aktionen und den dazu abgedruckten Meldungen und Bekennerschreiben. Dazu reflektiert der Artikel \"K\u00e4seklau & Bombenbau\": \"Im Fr\u00fchjahr konnten Aktionsformen, die vorher nur d\u00fcrftig im Schatten des Pflastersteins vegitierten, im Licht unserer hungrigen Aufmerksamkeit zu neuem Ansehen erbl\u00fchen. Anschl\u00e4ge etablierten sich zur mehr und mehr bestimmenden Aktionsform. [...] Ohnmacht \u00fcberwinden, M\u00f6glichkeiten und Ziele aufzeigen, waren Motivationen der neuen Welle. [...] Noch sind Anschl\u00e4ge meist nur symbolische Angriffe gegen das System. Sie zerst\u00f6ren zwar, gelten auch direkt dem Ziel, doch vor allem sind sie Werkzeuge im Krieg um die K\u00f6pfe, sind Teil der \u00d6ffentlichkeitsarbeit \u2013 nur mit anderen Mitteln.\" Kritisiert wurde aber auch, dass derartige nachtaktive Aktionen, \u00e4hnlich der Situation in den 1970er Jahren, eine kontraproduktive Wirkung haben k\u00f6nnen, wie die Mythenbildung um den \"\u00fcbernat\u00fcrlichen Revolution\u00e4r\" oder eine Spaltung der Bewegung \u2013 in auf der einen Seite \"K\u00e4seklau\", auf der anderen \"Bombenbau\" \u2013 durch die propagandistische Fokussierung auf einzelne militante Gruppen wie die RZ. Effektiver sei das \"massenhafte Leben und Erleben der allt\u00e4glichen Subversion\".\n\nAuch die Queer-Fraktion meldet sich zu Wort, um mobil zu machen, allerdings auf einer anderen Ebene: \"He(te)roische Arroganz und K\u00e4lte [...] wir haben diese coole, tolerante Hetero-Politmacker-scene satt, die uns genauso unterdr\u00fcckt wie der Rest der hetero-genormten Schweine-Welt. Kotz! Daher wollen wir die Zusammenrottung aller lebendigen und schwulen Anarchis, Radikalen, Spontis, Freaks, Hippies usw., um den Rest der Bewegung und der Stadt in ihrer Hetero-Idylle zu terrorisieren.\" Und dies zu Recht, denn so etwas wie das oben geforderte Leben allt\u00e4glicher Subversion schien zumindest auf der Gender-Ebene nicht stattzufinden. Neben den Politmackern und militanten Stra\u00dfenk\u00e4mpfern findet sich kaum eine Spur einer subversiven M\u00e4nnlichkeit. Ebenso ausgespart wurden seit \u00fcber einem Jahr die sogenannten Frauen-Themen, abgesehen von dem erw\u00e4hnten Appell an Anarchafeministinnen und einem vereinzelten Artikel zu Vergewaltigung und frauenfeindlicher Justiz in der radikal 99. Lediglich Frauenbilder im w\u00f6rtlichen Sinn, zumeist mit wallendem Haar und knapper Bekleidung, wurden noch als dekorative Elemente ins Layout eingef\u00fcgt.\n\nSp\u00e4testens die Nr. 105 l\u00e4sst erkennen, dass sich die autonome Szene schon seit geraumer Zeit f\u00fcr einen Staatsbesuch aus Amerika r\u00fcstete. Anl\u00e4sslich des Nato-Gipfels in Bonn war f\u00fcr den 11.6.1982 ein Besuch Ronald Reagans in Berlin angek\u00fcndigt. Grund genug f\u00fcr die radikal, sich auf ein umfassendes USA-Bashing einzulassen. Die Beitr\u00e4ge reichen von der Schilderung einer eigenartigen Amerikareise bis hin zur Abarbeitung an der Geschichte der Nato und bieten erwartungsgem\u00e4\u00df eine Ausarbeitung von Demo-Strategien jenseits eines \"plumpen Kr\u00e4ftemessen\" mit der Polizei, das angesichts des Gro\u00dfaufgebots ohnehin kaum zu gewinnen w\u00e4re. Nachfolgend reflektiert dann die Ausgabe 106 den Ablauf der nicht genehmigten Gegendemonstration \u2013 oder vielmehr deren Nicht-Ablauf, da sie am Nollendorfplatz eingekesselt wurde. Zwar hielt dieser Kessel nicht stand, aber das Gef\u00fchl einer Niederlage schien sich nachhaltig eingestellt zu haben. Das Layout dekoriert die entsprechenden Seiten mit Randstreifen, auf denen \"Waterloo\" \u2013 \"Stalingrad\" \u2013 \"Little Big Horn\" und \u00c4hnliches zu lesen ist.\n\nHighlights finden sich in dieser Zeit eher am Rande: So beschwert sich ein aufgebrachter Stuttgarter Skinhead in einem Leserbrief in der Nr. 105 bitterst dar\u00fcber, dass eine Meldung \u00fcber Auseinandersetzungen zwischen Berliner Punks und Skins in der Ausgabe 101 den Anschein erweckt h\u00e4tte, alle Skinheads w\u00e4ren Faschisten. Dies sei ein unhaltbares Vorurteil, und ihre \"Opfer\" seien doch nur \"Kapitalisten, Spie\u00dfer und Bonzen und ab und zu vielleicht mal 'n Ted, 'n Popper oder sonst 'n Wixer\".\n\nEine glorreiche Eigenleistung zum Thema Selbstironie bietet die radikal auf der vorletzten Seite in Form eines, einer Werbeanzeige nachempfundenen, Sonderangebots eines angeblichen \"Radikal-Versands\". Als \"Werte\" in \"preisg\u00fcnstigen A & P-Paketen\" werden hier unter anderem offeriert: \"Ausgewogenheit & Pluralismus \u2013 F\u00fcr jeden etwas!: Nur 1.50\", \"Anarchie & Pyromanie \u2013 So frei wie das Feuer im Wind: Nur 14.50\", \"Antiimperialismus & Pfadfinderturn \u2013 \"Einer f\u00fcr alle, alle f\u00fcr einen!' K\u00e4mpfen Sie gemeinsam mit uns gegen die Ami-Pest!: Nur 2.50\" oder auch \"Atomstaat & Pazifismus \u2013 Friedlich in der Katastrophe: geschenkt!\".\n\n\u00c4hnlich bissig geht es im Vorwort der Nr. 106 (Untertitel: Zeitung f\u00fcr den rei\u00dfenden Absatz) weiter. Offenbar waren Bekanntheitsgrad und Interesse an der radikal seit dem vorherigen Monat in mehrfacher Hinsicht massiv angestiegen: \"Der Finanzchef frohlockte. \u00dcberall erschienen Gratis-Anzeigen der radikal, mit denen Tagesspiegel, Morgenpost und sogar die 'Abendschau' beweisen wollten, wie sehr sich unsere verehrten Leser_innen von uns zur heftigen Anteilnahme am angek\u00fcndigten Besuchsspektakel verleiten lassen w\u00fcrden. Der Pressestelle des Innensenators ein herzliches 'God bless you' f\u00fcr diese PR-Aktion bei Zielgruppen, die wir sonst publizistisch eher streifen. Im Vorfeld des Besuchs hatte somit die Politisierung einiger eher unauff\u00e4llig uniformierter B\u00fcttel einen Grad erreicht, dass sie sich endlich mal ernsthaft f\u00fcr unsere kleine Zeitung zu interessieren begannen. So lie\u00dfen es sich die Sch\u00fctzer ' unserer' Verfassung nicht nehmen, die Aktivit\u00e4ten unserer Vertriebsabteilung zu beobachten. Dabei folgten sie ihrer Beute mit mindestens zehn Automobilen \u2013 sogar ein unauff\u00e4lliger Motorradfahrer stie\u00df uns sofort ins Auge \u2013 mit geb\u00fchrendem Abstand. Wo sich auch die Damen und Herren vom Vertrieb in Kneipen niederlie\u00dfen, immer war dort bald der Nebentisch von aufdringlich unauff\u00e4lligen Neugierigen besetzt. Warum so kontaktscheu? Wir sind keine Halbg\u00f6tter in Schwarz \u2013 und durchaus sehr b\u00fcrgernah. Ernsthaft denken wir daran, unseren F\u00e4nklub dermal einst zum Tee zu bitten, falls er uns nicht zuvor kommt, gell.\n\n\u00dcber den Sommer kommt die radikal in diesem Jahr ohne bemerkenswerte thematische Innovationen. Man sorgte sich weiter um Atomkraft und Hungerstreiks im Knast, um die (Nicht-)Verhandlungsbereitschaft in der H\u00e4userszene, sprengte hier und da mal etwas in die Luft und interessierte sich tats\u00e4chlich auch f\u00fcr die Rolle der Frau \u2013 allerdings nicht in den eigenen Reihen, sondern in der pal\u00e4stinensischen Befreiungsbewegung. Man kritisierte weiterhin vehement die USA, solidarisierte sich mit der polnischen Arbeiter_innenbewegung und dachte auch mal \u00fcber das eigene Verh\u00e4ltnis zu Arbeit\/Job versus Geldmangel\/Armut nach.\n\nDie eigene Szene schien in einer Art Stillstand angekommen zu sein. In der Oktoberausgabe Nr. 109 wurden auch die Beschwerden seitens der Leser_innenschaft zur Kenntnis genommen und abgedruckt. Eine Leserin kommentiert: \"was mich jedes mal ankotzt, ist die Sprache. Wessen Sprache sprechen wir? Wer erfindet immer neue, nur f\u00fcr Fachmenschen verst\u00e4ndliche W\u00f6rter?\". Ein anderer Leser w\u00fcnscht sich insgesamt mehr Unterhaltungswert: \"Liebe radischen, mir liegt schon lange etwas auf dem Herzen [...]: die radi ist eine langweilige und kopflastige Zeitung. Es gibt zwar auch einige Pluspunkte, welche ich euch anrechne (das lay-out z. B. ist geil), eure inhaltlichen Sachen aber hinterlassen bei mir einen faden\/trockenen Nachgeschmack. Was ich meine ist, dass die Beitr\u00e4ge (z. B. Libanon) mit mir nicht viel bzw. gar nichts zu tun haben. Auch wenn dies die autonomen Freiheitshelden nicht gerne h\u00f6ren. Ich w\u00fcrde es z. B. geil finden, wenn mal was \u00fcber Musik in der radi zu lesen w\u00e4re. Es gibt doch genug interessante, politische Gruppen, \u00fcber die man mal was schreiben k\u00f6nnte (Crass, Dead Kennedys, Black Flag, Slime etc.) Au\u00dferdem erscheint mir die radi auch etwas humorlos. [...] Ihr sollt kein zweites MAD werden und auch keine Alternative zum Pop-Teenie-Bolschwismus [...] des 'Sounds', ihr solltet aber ruhig mal etwas Auflockerndes und Kulturelles bringen.\" Ernsthaft umgesetzt wurde davon nicht viel.\n\nMit der Ausgabe 110, der Zeitung f\u00fcr unkonventionelle Bewegung, geht es wieder ans Eingemachte. In der \"Hausmitteilung\" hei\u00dft es: \"Seit geraumer Zeit erwarten wir eigentlich einen Besuch 'der Herren des Morgengrauens' in unseren gut abgesicherten Verliesen. Doch auf die Einladung, uns bei der Arbeit behilflich zu sein, bekamen wir nur vage Antworten \u2013 auch Ger\u00fcchte genannt. Dass ihr Interesse an einer konstruktiven Zusammenarbeit doch noch nicht ganz verschwunden ist, bewiesen sie jedenfalls wieder bei der letzten Produktion, wo sie unauff\u00e4llig um uns rumschlawenzelten und ihre N\u00e4chte in einem Wohnmobil absa\u00dfen. Doch selbst der aus den Redaktionsr\u00e4umen dringende Kaffeeduft konnte ihre Sch\u00fcchternheit nicht bezwingen, endlich auszusteigen, tief einzuatmen und mit einem freundlichen L\u00e4cheln auf dem Gesicht unseren Empfang mit W\u00fcrde entgegenzunehmen. Jedenfalls mussten wir nach zwei Tagen traurig feststellen, dass sie ihren Versuch aufgegeben hatten und einfach von dannen fuhren und uns mit unseren vielen \u00dcberraschungen alleine lie\u00dfen.\"\n\nInteressant ist hier auch der Artikel \"Die neuen Bunten\", der eine Art neue Generation der Subkultur ins Visier nimmt und sich mit einer ersten Form von Gentrifizierung infolge der Sanierungen in Kreuzberg befasst. Wurde der damals noch randst\u00e4ndige Bezirk selbst von seinen autonomen Bewohner_innen oft als \"Sumpf\" und \"Ghetto\" bezeichnet, schien er nun reizvoll f\u00fcr \"die etwas besser oder nur bunter gekleideten Bev\u00f6lkerungsschichten\" geworden zu sein. W\u00e4hrend der Er\u00f6ffnungsfeier eines \"exquisiten Lampengesch\u00e4fts in der Oranienstra\u00dfe\" wurde die Fensterscheibe eingeworfen, die Party ruiniert und sich gepr\u00fcgelt, denn so etwas d\u00fcrfe sich nicht \"bruchlos in die schwarz-rote Realit\u00e4t einf\u00fcgen\". Dazu gibt es aber auch selbstkritische Gedankeng\u00e4nge im Sinne von: Was wir da gemacht haben, ist das, was die sonst mit uns machen, blo\u00df umgekehrt. Nur das Aussehen der Leute hat nicht gefallen, man war angetrunken und schon gab es Randale. Der Artikel referiert weiter \u00fcber Punk und dessen \"Missgeburt\" New Wave mit schickerer Kleidung \u2013 eine richtige Schickeria wie in M\u00fcnchen sei das auf der Party aber nicht gewesen. Im Zuge einer Beschreibung der Entstehung verschiedener Subkulturen und Bewegungen wie den Hippies und Punks wird die Frage nach den \"kulturellen [...] Leitfiguren im H\u00e4userkampf\" gestellt: \"Was f\u00fcr eine Art Kultur hat er sich zugelegt, der 'typische' Autonome, wie sieht sie aus? [...] Man k\u00f6nnte sagen: Lederjacke, PLO-Tuch, Kampfstiefel, schwarzer Helm. Gut, dies w\u00e4re ein Beispiel f\u00fcr eine ideale Demo-Ausr\u00fcstung, aber im Alltag laufen doch viele Leute, auch noch so militante, ganz anders rum. Es l\u00e4sst sich einfach nicht sagen: So oder so sieht der durchschnittliche H\u00e4userk\u00e4mpfer aus, oder anders, diese oder andere Sachen wirst du bei den meisten Bewegten ganz bestimmt finden. Es l\u00e4sst sich weder in Bezug auf \u00c4u\u00dferlichkeiten sagen, noch in Bezug darauf, was die Leute im Kopf haben. Diese ' Vielfalt' der Bewegung ist h\u00e4ufig bejubelt, manchmal auch hart kritisiert worden. Der Staatsschutz wollte sich damit gar nicht anfreunden, und einige Politiker rennen immer noch rum auf der Suche nach den Hinterm\u00e4nnern (!) und R\u00e4delsf\u00fchrern. [...] Das einzig Gemeinsame liegt also in der Verschiedenheit? Es liegt, wie festgestellt, nicht in der Lebensweise und auch nicht in den teils sehr wirren, teils zu klaren Zielvorstellungen von dem, was mensch mal erreichen will. Es liegt aber sehr wohl in dem gemeinsamen Feind. Hier treffen sich alle wieder, die Lederjacken und Bunthaare, die Spiritualisten, Marxisten, Anarchisten und derweil auch wieder Kommunisten, die Randgruppen aus fast allen Bereichen: Ausl\u00e4nder, Schwule, Frauen, Jobber, Kiffer, Rentner usw. Wer ist der gemeinsame Feind? Es sind Ausbeutung und Unterdr\u00fcckung!\" Das klingt schon fast wieder r\u00fchrend, zeigt aber auch, dass zumindest manche sich in Kreuzberg nicht durch die etwas schickeren Klamotten der neuen Nachbar_innen von ihrer Toleranz und Offenheit abbringen lassen wollten: \"Was ist nun mit der Neuen Deutschen Buntheit (dies in geschickter Vermeidung der Bezeichnung 'Schickies')? Was uns an ihnen besonders auff\u00e4llt ist ihre Kleidung, die betonte Auff\u00e4lligkeit, ein anderen gegen\u00fcber etwas arrogantes Gehabe, eine neue Betonung von K\u00f6rperlichkeit, ein leichter Hauch von Dekadenz. Keine Vorliebe f\u00fcr versyphte, d\u00fcstere Anarchokneipen im Stil der 70er Jahre, sondern abfahren auf Neon, grelle Farben und \u2013 mit Verlaub gesagt \u2013 kitschige Einrichtungen, die irgendwas haben von den 50ern. Aber zweifellos machen sie sich durch diese \u00c4u\u00dferlichkeiten zu Outsidern, die von den Normalos garantiert nicht beim Trampen mitgenommen werden. Und sie sehen auch nicht gerade danach aus, als w\u00fcrden sie morgens um f\u00fcnf in die Fabrik rennen. Sie setzen sich bewusst ab von der Durchschnittlichkeit, die unsere Gesellschaft beherrscht. [...] Wahrscheinlich sind die neuen Bunten ziemlich unpolitisch. [...] Jedenfalls \u2013 es gibt sie, sp\u00e4testens in der n\u00e4chsten Wohnung, und f\u00fcr mich sind sie als gleichberechtigte Mitbewohner im Kiez anzusehen, denen man nicht wahllos aufs Maul haut.\"\n\nDass die Ansichten in diesem Artikel nicht auf ungeteilte Zustimmung stie\u00dfen, zeigen kritischere Repliken in Heft 111. Relevanter als jene ist in dieser Ausgabe allerdings der Bericht von einer umfassenden Razzia, in deren Zuge \"14 Privatwohnungen, 2 besetzte H\u00e4user, 3 Druckereien, 2 Buchl\u00e4den, 1 Verlag, 1 Buchvertrieb, 1 Photosetzerei sowie die Vereinsr\u00e4ume der Zeitungskooperative\" durchsucht worden waren. Zwar kam dies f\u00fcr die Redaktion nicht unerwartet und blieb zudem ohne nennenswerte Erfolge f\u00fcr den Staatsschutz, dennoch wurde die Aktion als \"harter Schlag\", \"Initialz\u00fcndung f\u00fcr eine h\u00e4rtere Gangart des Justiz- und Bullenapparats\" und massiver Einsch\u00fcchterungsversuch empfunden. Die radikal sieht sich selbst als \"Schnittpunkt verschiedenster Str\u00f6mungen\" und stelle daher \"ein bevorzugtes Objekt dar, um exemplarisch gegen die Szene vorzugehen\".\n\n### Die Repression startet durch\n\nDas neue Jahr 1983 begann mit einer Aufr\u00e4umaktion in der radikal-Redaktion, die unter anderem \"elektronisches Ungeziefer\" zutage f\u00f6rderte: Eine Wanze wurde gefunden und zur Strecke gebracht. Anl\u00e4sslich dieses Ereignisses wird in der mit Zeitung f\u00fcr animalische Reflexe betitelten Ausgabe 112 der Artikel \"Search und destroy\" ver\u00f6ffentlicht, der praktische Tipps gibt, wie diese und andere Abh\u00f6rger\u00e4tschaften unsch\u00e4dlich gemacht werden k\u00f6nnen.\n\nEine etwas anders geartete \"Tiergeschichte\" setzt sich kritisch mit Rudelhierarchien, M\u00e4nnerb\u00fcnden und Mackerverhalten, versinnbildlicht durch Paviane (mit den \"gr\u00f6\u00dften Muckis und dem rotesten Arsch\"), in der Kreuzberger autonomen Szene auseinander. Die Verfasserin (oder m\u00f6glicherweise auch der Verfasser) unter dem sch\u00f6nen Pseudonym. Anna Bolika stellt fest: \"Zum Gl\u00fcck f\u00fcr sie [die M\u00e4nner-Paviane] gibt es den Leoparden, der gr\u00f6\u00dfte Feind f\u00fcr die Paviane (so wie f\u00fcr uns zum Beispiel die Yanks): Taucht der Leopard auf, stellen die M\u00e4nnchen sofort alle Konkurrenzk\u00e4mpfe ein, klopfen einander auf die Schulter, lausen einander kurz mal und f\u00fchlen sich gut \u2013 bis der Kampf mit dem Feind vorbei ist. Sollte \u00fcber l\u00e4ngere Zeit kein Feind aufzutreiben sein, halluzinieren sie einfach einen. [...] So kommen auch M\u00e4nnchen zu dem, was sie brauchen. Die Verhaltensforschung nennt das 'Bildung von M\u00e4nnerb\u00fcnden auf der Basis von Aggression'. Die Zusammenh\u00e4nge werden klarer, wenn man Paviane und Leoparden nach K 36 verlegt. Die Skins marschieren z. B. mit einem Leopardenfell \u00fcber der Schulter zum Heinrichplatz. Danach wurde es bei den T\u00fcrken gesichtet, denen es die Borns [eine Rockertruppe] abnahmen, die verscheuerten es an die Schickis, die es mit Neon bestrahlt an der wei\u00dfen Wand aufh\u00e4ngten. Zwischendurch kriegten es auch die H-Dealer in die Finger, verdealten es aber immer gleich weiter. Und getreulich folgte jedem Auftauchen des Leoparden ein marschierender M\u00e4nnerbund, der sich, nach Kampf und Sieg, wieder in die allt\u00e4glichen Konkurrenzk\u00e4mpfe aufl\u00f6ste, bis die Frustration wieder einen neuen Leoparden auf den Plan rief.\"\n\nNicht weiter \u00fcberraschend erhielt dieser Artikel in der Nr. 113 ein ausf\u00fchrliches Feedback. Allerdings nicht von einem der angesprochenen Macker, sondern von einem schwulen Autonomen (\"bzw.\/und\/oder ein autonomer Schwuler\"), der aus der Perspektive auf einen \"rosaarschigen Pavian\" die Geschichte und die Mackerkritik ob der von ihm erlebten Ausgrenzung aus den homosozialen, dabei aber homophoben und (offiziell) heterosexuellen M\u00e4nnerb\u00fcnden weiterf\u00fchrt: \"noch viel schlimmer f\u00fcr unseren freund [der rosaarschige Pavian] war jedoch, dass seine kumpels \u00fcber seine neidlosen und liebevollen blicke, die er auf ihre \u00e4rsche warf, sehr ver\u00e4rgert war'n [...] und sie verbaten ihm derartige blicke, obwohl doch \u2013 wie der rosa pavian bei seinen intensiven studien inzwischen rausgefunden hatte \u2013 es viele paviane gab, deren roter arsch von zeit zu zeit auch pl\u00f6tzlich eine rosa f\u00e4rbung annahm. er hatte bemerkt, dass das den betroffenen genossen dann jedoch eher mit peinlichkeit erf\u00fcllte und er versuchte, seinen arsch zu verstecken oder mit hilfe roter farbe wieder den NORMALEN schein zu wahren. (...) unser pavian-freund wollte davon nichts mehr wissen, hatte er doch erkannt, dass sein hauptfeind nicht drau\u00dfen in der steppe in gestalt eines leoparden sitzt, sondern unter den pavianen selbst in form des konkurrenzverhaltens der roten selbst-ver-arscher.\"\n\nEin neues Ziel, auf dessen Bek\u00e4mpfung man sich einigen konnte, stand allerdings schon vor der T\u00fcr und wurde auch in den folgenden Ausgaben immer wieder aufgegriffen: Die f\u00fcr den April geplante \"Schafsz\u00e4hlung\", wie sie in der radikal im Februar 1983 genannt und in gewohnter Tierliebe auch mit entsprechendem Sch\u00e4fchenlayout dekoriert wurde. Es wird zum Boykott der Volksz\u00e4hlung aufgerufen, denn diese sei eine \"Potenzierung der Rasterfahndung\". In diesem Zusammenhang wird auch in der folgenden Ausgabe in einem Beitrag unter dem Titel \"Datenschatten. Zur Modernisierung der Herrschaft durch Informationstechnologie\" vor dem erwarteten Missbrauch der so gewonnenen Daten gewarnt.\n\nDie Nr. 114 setzt sich aber auch kritisch mit Anschl\u00e4gen und Bekennerschreiben auseinander, die nur scheinbar den RZ oder der Guerilla Diffusa zuzuordnen waren, in Machart (zu professionell), Ausrichtung (Gef\u00e4hrdung von Familien von US-Soldaten) oder Stil (w\u00f6rtliche Kopien alter Begleitschreiben) jedoch erkennen lie\u00dfen, dass sie entweder von faschistischen Gruppen oder dem Staatsschutz selbst inszeniert worden sein mussten. Diskutiert werden ebenso nicht wirklich gut durchdachte Aktionen aus den eigenen Reihen wie der \"z\u00fcndende Kloreiniger\", der nicht wie geplant im Supermarkt f\u00fcr Unruhe sorgte, sondern auf dem K\u00fcchentisch einer ahnungslosen Kreuzberger WG landete, die diesen todesmutig und erfolgreich selbst entsch\u00e4rfte. Kritisiert wird zudem die Abgehobenheit der RZ, die doch wieder \"in die Bewegung\" zur\u00fcckkehren solle. Die RZ antworteten und warnten zwei Ausgaben sp\u00e4ter, in der Zeitung f\u00fcr Entkrampfung davor, Antiimperialismus mit Antiamerikanismus zu verwechseln. Einzelne US-Amerikaner anzugreifen sei tats\u00e4chlich nicht ihr Ziel.\n\nDas Cover der Doppelnummer 115\/116, genannt Zeitung f\u00fcr Liebe und Tod, besticht durch rosafarbenes Design und die Abbildung zweier Frauen (wallendes Haar, sp\u00e4rliche Bekleidung), die eine mit Rehblick, die andere in Kampfpose, und tats\u00e4chlich finden sich im Inhalt zwei lobenswerte Artikel, einer \u00fcber die indische \"Banditenk\u00f6nigin Phoolan Devi, R\u00e4cherin indischer Frauen\", der andere zum Thema \"Die pal\u00e4stinensische Frau und der bewaffnete Kampf\". Au\u00dferdem enth\u00e4lt diese Ausgabe auffallend viele und ziemlich schlechte Gedichte. \u00dcber die Frauen in der eigenen Bewegung wurde jedoch wieder nichts geschrieben.\n\nIm Sp\u00e4tsommer 1983 erschien eine zweite Doppelnummer (118\/119), die zu mehr Szenebeteiligung aufruft \u2013 \"Hoffnungsvoll auch, weil unser Segel ein wenig unter L\u00f6chern leidet\". Zwei der als Herausgeber gef\u00fchrten Personen befanden sich in Haft und offenbar ging der Redaktion wieder die Puste aus. Zwar waren nach der Durchsuchung im vergangenen Jahr zahlreiche Solidarit\u00e4tsbekundungen eingegangen (einige Beispiel daf\u00fcr finden sich auf der R\u00fcckseite der Nr. 112) und im Impressum sind mittlerweile die Namen von um die 40 linken Gruppen und Unternehmen als Mitherausgeber abgedruckt. In der Nr. 120 zeichneten sogar \"500 Menschen und Kollektive\" als presserechtlich verantwortliche Herausgeber, dennoch scheint sich ein Gef\u00fchl der Desintegration eingestellt zu haben. Ziel war es daher jetzt, \"diese Isolation zu durchbrechen, die Zeitung wieder breiter und vielf\u00e4ltiger diskutieren zu wollen und mehr den Hin- und R\u00fcckfluss zu suchen: Der Traum aller Medienschaffenden, bei dem die Grenzen zwischen Konsumenten und Produzenten immer mehr ins Flie\u00dfen geraten, bis sie letztlich verschwinden\". Selbstkritik und Aufarbeitung waren angesagt. Im Artikel \"Ansichten zu Themen der Zeit\" hei\u00dft es treffend: \"Nix gegen die vielgeliebten entscheidenden Schl\u00e4ge in die Fresse des Schweinesystems, aber Herzschl\u00e4ge ohne Hirnstr\u00f6me sind just boring\".\n\nIm September verk\u00fcndete die radikal erfreut die Freilassung der beiden Inhaftierten bis zu deren Prozess. Ebenso bemerkenswert ist, dass das Berliner Kammergericht am 18.8.1983 \"h\u00f6chstrichterlich entschieden [hat]: Ein Bulle darf Bulle genannt werden! Das Wort 'Bulle' [...] sei 'im heutigen Sprachgebrauch nicht mehr unbedingt als beleidigend' anzusehen\". Ansonsten bietet die Nr. 120 ausf\u00fchrliche Auseinandersetzungen mit der Geschichte des Berliner H\u00e4userkampfs, einer Bush-Demo, Lateinamerika und der Startbahn West inklusive eines Textes der RZ. Und ein Hirnstrom scheint hinsichtlich feministischer Beitr\u00e4ge reanimiert worden zu sein: \u00dcber eine ganze Doppelseite erstrecken sich zwei Beitr\u00e4ge, die sich mit Gewalt und Sexismus gegen Frauen befassen. Der Artikel der Roten Zora erkl\u00e4rt zudem die Zusammenh\u00e4nge von Anschl\u00e4gen gegen Betreiber einer philippinische Frauen an deutsche M\u00e4nner verkuppelnden Partnervermittlung, die eine Form legalen Frauenhandels darstellte. Sogar auf dem Titelbild der Ausgabe 120 sind diesmal drei wehrhafte Frauen mit kampf- und alltagstauglicher Frisur und Kleidung abgebildet.\n\nDas Licht dieser Glanzleistung beginnt aber angesichts des Vorworts der n\u00e4chsten Ausgabe schon wieder zu verblassen. In diesem wird herablassend erkl\u00e4rt, man h\u00e4tte sich \"beim letzten Mal nicht zum Untertitel 'Mittelm\u00e4\u00dfige Zeitung f\u00fcr gem\u00e4\u00dfigte Bewegungen' durchringen [k\u00f6nnen] \u2013 was in Bezug auf die Redaktion, den Inhalt und die Bewegung einer Berechtigung nicht entbehrt h\u00e4tte\", habe aber nun zum \"radikalen \u00dcberma\u00df zur\u00fcckgefunden\" und beruft sich auf die mittlerweile \u00fcber 1.000 solidarisch Herausgebenden. Der zweite Teil des Vorworts berichtet von der polizeilichen Beschlagnahmung von 1.500 Exemplaren dieser Ausgabe, die aber nichtsdestotrotz nachgedruckt worden sei. Das Cover ziert die Randbemerkung: \"wegen rei\u00dfendem Absatz \u2013 2. Auflage\".\n\nDie Geschichte einer Bewegung zu schreiben und damit ihr Scheitern zu erkl\u00e4ren, setzt sich nach dem H\u00e4userkampf auch mit dem Thema Startbahn West fort. Inhaltlich neu ist die Darstellung der Ostberliner Szene aus der Perspektive eines ehemaligen DDR-Bewohners. Ein in der Juli-Ausgabe beschriebener Ausflug eines Westlers nach Ostberlin hatte lediglich die stereotypen Eindr\u00fccke eines tristen und bedr\u00fcckenden Stadt-Teils ohne Alternativszene reproduziert. Unter dem Titel \"Blick in' Osten\" wurde nun im Oktober der Prenzlauer Berg als Szenebezirk benannt und ebenfalls von Hausbesetzungen berichtet, wenn auch unter einer anderen methodischen Vorgehensweise. Der beigef\u00fcgte Artikel \"Punks in der Kirche\" rankt sich um eine der subversiven und stets \u00fcberf\u00fcllten Blues-Messen in Ostberlin, die ein West-Autor besucht hatte und dabei feststellte, dass er die empfundene Distanz zur Alternativ-Szene im Osten doch nicht h\u00e4tte \u00fcberwinden k\u00f6nnen. Das mag soziologisch zwar alles interessant sein, eine Steigerung zur vorhergegangenen radikal-Ausgabe ist jedoch nur in Form einer noch 'radikaleren' Arroganz sp\u00fcrbar.\n\nVergleichsweise erfrischend liest sich in dem Artikel \"Bremerhaven geilo!\" zur Vorbereitung auf eine antimilitaristische Aktion eine Passage, in der die autonom engagierte, norddeutsche Landjugend der latent hysterisch wirkenden Berliner Fraktion gegen\u00fcbergestellt wird: \"Die Bewegung als provinzielle hat sich ein Gutst\u00fcck an Homogenit\u00e4t bewahrt, der Hippie und der Punk sind kommunikationsf\u00e4hig, es gibt eine Subkultur, die eher noch expandiert, und was politische Arbeit betrifft \u2013 es gibt bestimmt Wichtigeres im Leben \u2013 die wird eben gemacht, ziemlich kontinuierlich, was gerade anliegt. AKW, ein Hauch von H\u00e4userkampf, na jetzt eben Bremerhaven, Nordenham, Muni-Z\u00fcge und so. Wer von Revolution spricht, bekommt ein mildes, ungl\u00e4ubiges L\u00e4cheln als Antwort, das zugleich fragt: Meinst du das ernst, willst du mich verarschen? Klar kann man sich nicht alles bieten lassen, von den Spie\u00dfern, dem Staat, den RD-Bullen, den Yankees und so, aber eine hardcore-Mentalit\u00e4t, die als Perspektive nur das Grab oder den Hochsicherheitstrakt hat, diese impotenzverbergende Straightheit Kreuzberger Finstergucker-Sekten, das f\u00e4llt hier nicht auf fruchtbaren Boden.\"\n\nDass diese Haltung der Landjugend in der radikal 121 nicht gerade mit Begeisterung kommentiert wird, verwundert nicht. Sich permanent verfolgt und bespitzelt sehend, arbeitete die Redaktion bereits an geheimen Orten. In Form einer Karibik-Piratenstory werden in der Nr. 122 auch interne Konflikte angesprochen: Einige haben die Redaktion verlassen, andere rauchen sich in sinnlosen Grabenk\u00e4mpfen auf: \"die Frauen gegen die Machos, die Revolution\u00e4re und 'Sozialbanditen' gegen diese aus Europa abgehauenen verkrachten Studies. Wir waren mal eine Bande, im Moment ist es wohl nur das Schiff, die 'Mama', die uns zusammenh\u00e4lt\".\n\nIn der Ausgabe vom Dezember 1983 erschien eine gepfefferte Antwort der RZ auf die Kritik der radikal an deren vorherigen Beitr\u00e4gen und traf damit einen neuralgischen Punkt: \"Militanz und Aktionen werden zwar noch f\u00fcr gut befunden, aber nur solange sie keine politischen Zusammenh\u00e4nge herstellen bzw. Kontinuit\u00e4t beweisen. 'Hingehen, abfackeln, abhauen.' Alles was dar\u00fcber hinausgeht, wird als potentielle Kaderpartei beschimpft [...]. Was z\u00e4hlt, ist die 'neue Subjektivit\u00e4t'. Wie f\u00fchl ich mich hier am wohlsten?\" Dabei, so argumentiert \"eine RZ\" weiter, h\u00e4tte doch gerade \"die Radi [...] eine wichtige Funktion als Diskussionsforum des militanten undogmatischen Widerstands\". Der Beitrag eines \"bazillus optimistiku\u00df\" schlie\u00dft sich an und fordert von der radikal: \"Anstatt \u00fcber andere zynisch und somit destruktiv herzuziehen, solltet ihr mal wieder eigene Standpunkte und Ideen positiv formulieren \u2013 weg von einer polemischen Plattheit hin zu einer spannenden Auseinandersetzung\". Dem gegen die RZ gerichteten Vorwurf der Abgehobenheit wird ebensolche auf Seiten der Redaktion entgegengestellt \u2013 die \"radis\" seien es doch, die wieder \"in die Bewegung\" zur\u00fcckkehren sollten. Und in einem Leserbrief steht ganz schlicht: \"Warum macht Ihr ne Zeitung, wenn Ihr nix im Kopf habt?\"\n\nMit der radikal schien es also zu Ende zu gehen \u2013 oder auch nicht. Im Intro der Ausgabe vom M\u00e4rz\/April 1984 hei\u00dft es hochphilosophisch: \"Am Anfang steht das Ende, sonst w\u00e4re das Neue das Alte.\" Zuletzt aber wurden die Texte eher langatmiger und unlesbarer, man k\u00f6nnte auch sagen, intellektueller, aber das hie\u00dfe, Verquastheit mit Intellektualit\u00e4t gleichzusetzen. Auch angesichts entnervter Leser_innenbriefe bleibt zu fragen, f\u00fcr wen eigentlich noch geschrieben wurde. Betitelte sich die Nr. 124\/125 vom Jahresanfang noch als Zeitung f\u00fcr die Neuordnung der Welten, nannte sich die Ausgabe 126\/127 Fachblatt f\u00fcr alles, was Terroristen Spa\u00df macht und pr\u00e4sentierte ein seitenlanges Interview mit Jean Baudrillard unter dem Titel \"Der Tod des politischen Subjekts\". Ein beigef\u00fcgter Artikel erkl\u00e4rt dazu: \"Wer mit Baudrillard nichts anfangen kann, betrachte ihn als Schmuck, den wir uns leisten. Wer sich der Qual des Lesens und dem Versuch des Verstehens unterzieht, wird neben einer Menge offensichtlichen Unsinns doch genau wieder die Begriffe entdecken, auf die sich die Diskussion zentriert: Die Revolte und die Macht. Die Macht der Revolte. Das Ziel der Geschichte. Die Massen, die dieses Ziel erk\u00e4mpfen oder sich ihm verweigern sollen. Die gesellschaftliche Entwicklung, die technologische Entwicklung, die Entwicklung im Medienbereich und die Entwicklung des revolution\u00e4ren Subjekts, \u00fcber dessen Existenz am Rande, \u00fcber dessen Verortung zentral gestritten wird. Was bei Baudrillard \u2013 und bei allen anderen \u2013 etwas zu kurz kommt, ist der Alltag, oder auch: das konkrete Leben. M\u00fcsste nicht eigentlich das am Anfang stehen?\"\n\nOffenbar war man von einem Neuanfang doch noch ein St\u00fcck weit entfernt. Die radikal tauchte erst mal ab, oder, wie es am Ende des Vorworts hei\u00dft: \"zumindest f\u00fcr eine Weile gehen die Lichter aus\".\n\n### Going underground\n\nIm Herbst 1984 erschien die 128. Ausgabe der radikal \u2013 illegal, versteckt und unerreichbar, au\u00dfer \u00fcber eine Postfachadresse in der Schweiz. Vorausgegangen waren der Prozess wegen \u00a7 129a \"Werbung und Unterst\u00fctzung einer terroristischen Vereinigung\" (aufgrund der abgedruckten RZ-Texte) und die Verurteilung der beiden Angeklagten zu 30-monatigen Haftstrafen, die diese jedoch nicht antreten mussten, da sie als Abgeordnete der Gr\u00fcnen ins Europaparlament gew\u00e4hlt worden waren. Die Repressionen gegen die radikal gingen jedoch weiter. Im Sommer hatte sich trotz allem eine neue Redaktion gebildet, die nun im Intro \"Frohe Botschaft \u2013 der Kampf geht weiter\" verk\u00fcndet: \"So Leute, habt ihr also gedacht, es w\u00e4re zu Ende mit der radikal... Habt ihr sie schon auf der Liste der von ' Verfassungsorganen' totgepr\u00fcgelten linken Initiativen angekreuzt und archiviert... Wir wollen nicht verhehlen, dass das Frohlocken dar\u00fcber, der Repression ein Schnippchen geschlagen zu haben, einen guten Teil der Motivation dieser Nummer ausgemacht hat.\" Und es wird alles aufgeboten, was an Provokation in den Sinn kommt, frei nach dem Motto: \"Wenn schon illegal, dann richtig. Dann wollen wir auch \u00fcber die Dinge diskutieren, die in anderen Zeitungen in der Form heute wohl nicht m\u00f6glich sind\".\n\nDas Cover ist in schwarz-rot gehalten, undatiert und mit dem Foto eines Friedhofs versehen, dazu der Untertitel: Anleitung f\u00fcr den Herzinfarkt von Staat und Staatsanwalt. Eine neue Zeitung sollte es sein, in einer Auflage von 6.000 St\u00fcck, mit altem Namen und auf \"Basis des Austausches und gegenseitiger Stellungnahmen\", wobei diese Nummer wenig eigene Kommentare enthalte und vor allem den Informationen diene, die als Diskussionsanreize zu betrachten seien. Und die Leser_innen wurden, bez\u00fcglich etwaiger Vermutungen, von wem und wo die Zeitung gemacht werde, auf \"Maul halten\" eingeschworen.\n\nDie Inhalte betreffen neben Internationalismus-Themen den militanten Widerstand im Wendland \u2013 der Tag X der ersten Castor-Transporte r\u00fcckt n\u00e4her \u2013 ebenso wie gegen Nato-Herbstman\u00f6ver und die Startbahn West. Dazu gibt es vermeintliche Enth\u00fcllungsgeschichten \u00fcber Justizbeamte, eine eher dekorative als technisch taugliche Anleitung zum Bombenbau, ein altes RZ-Interview von 1975 und, auf der letzten Seite, ein \"Frauen-Thema\": die Hintergrundgeschichte zu einem Anschlag gegen eine gyn\u00e4kologische Arztpraxis. Dies ist umso erstaunlicher angesichts der Tatsache, dass autonome Frauen mittlerweile eine eigene Zeitschrift, die bereits in der Ausgabe 126\/127 angek\u00fcndigte Anagan, herausgaben, in der dieser Anschlag bereits diskutiert wurde.\n\nAuch die zweite radikal der neuen Redaktion, fortlaufend weiter nummeriert als 129, stellt sich als reine Info-Ausgabe dar. Schwerpunktthema ist der Hungerstreik im Knast und die RAF. Die hier oberfl\u00e4chlich und schlicht nebeneinandergestellten Texte wurden erst in der nachfolgenden Ausgabe ausf\u00fchrlich diskutiert und kritisch aufgearbeitet. Der Forderung eines Leser_innenbriefes in der Februarausgabe von 1983, mehr \u00fcber Szene-Spitzel zu schreiben, wird hier verst\u00e4rkt nachgekommen, zwei enttarnte M\u00e4nner werden ausf\u00fchrlich und mit Fotos portr\u00e4tiert, ein Dritter verd\u00e4chtigt. Ein weiterer gr\u00f6\u00dferer Artikel besch\u00e4ftigt sich mit der Aufl\u00f6sung der H\u00e4userbewegung und der Bed\u00fcrfnisse der Bewohner_innen ignorierenden Stadtplanung und -erneuerung, ein anderer mit der paramilit\u00e4rischen und der neonazistischen Szene nahestehenden Wehrsportgruppe TeNo, die vom Verfassungsschutz auffallend wenig behelligt wurde. Den Abschluss der Nr. 129 bildet eine korrigierte und erweiterte Anleitung zum Bombenbau und ebenso zum F\u00e4lschen von Ausweispapieren. Dazu erfolgt die Aufforderung, Verbesserungsvorschl\u00e4ge einzusenden.\n\nAufgrund der erh\u00f6hten \"Sicherheitsma\u00dfnahmen\", finanzieller und interner Probleme erschien die radikal nun unregelm\u00e4\u00dfig und in der Nr. 130 wird eine Adresse in Wien als Kontaktm\u00f6glichkeit angegeben. Diese Ausgabe versteht sich als Diskussions-Nummer f\u00fcr \"unsere Diskussionen in der Gruppe, von denen wir annehmen, dass sie so oder \u00e4hnlich bei vielen von Euch gef\u00fchrt werden\". Au\u00dferdem gibt es praxisnahe Informationen und Tipps zu folgenden Themen: \"Sei es der Widerstand gegen deine 'Zwangsverpflichtung' zur Bundeswehr, der erneute Nachschlag als Erg\u00e4nzung zu den explosiven Basteleien der ersten Nummern, Erkl\u00e4rungen und Flugis, oder der Einstieg in das Thema Neue Technologien anhand einer realit\u00e4tsbezogenen Geschichte \u00fcber die M\u00f6glichkeiten des Hackens und der Computer-Sabotage. Der Artikel \u00fcber die Allianz soll was von den Verflechtungen dieses Schweinebetriebes vermitteln, damit nicht nur die Gegenseite unsere Strukturen durchleuchtet, sondern auch umgekehrt. Im Anschluss empfehlen ein paar Tips geeignete M\u00f6glichkeiten des Widerstands, in Form der Selbstbereicherung an diesen Wirtschaftsgiganten: den geplanten Versicherungsbetrug.\"\n\nDie Ausgabe 131 (\"Nach 7,5 Monaten Schweigen unsere 4. Nummer\") beginnt mit einer ausf\u00fchrlichen Darstellung der Konflikte innerhalb der Redaktionsgruppe, vereinfacht formuliert als Diskrepanz zwischen den \"Dominanten\" und den \"Dominierten\", was \u2013 nicht weiter \u00fcberraschend \u2013 in etwa dem Verh\u00e4ltnis der M\u00e4nner bzw. der \"Macker\" zu den Frauen der Gruppe entsprach und ziemlich ern\u00fcchtert klingt: \"An diesem Punkt haben wir bereits eine Phase erreicht, an der wir mit Reden nicht mehr weiterkommen, berechtigterweise haben die Dominierten keine Lust mehr, die Dominanten immer wieder auf ihre Fehler hinweisen zu m\u00fcssen, von den Dominanten sind die Fehler zwar erkannt, aber noch nicht beseitigt worden.\"\n\nNichts Neues an der Genderfront also, aber immerhin erschienen drei feministische Beitr\u00e4ge in dieser Ausgabe (allerdings auf nur 3 Seiten von insgesamt 87), eingef\u00fchrt mit den Worten: \"Im Folgenden bringen wir drei Papiere, die uns Berliner Frauen geschickt haben. Die Abgrenzung gegen\u00fcber m\u00e4nnerdominierten gemischten Zusammenh\u00e4ngen, die in allen drei Beitr\u00e4gen deutlich wird, sollte den M\u00e4nnern endlich mal zu denken geben. Und zwar \u00fcber den von ihnen kaum begriffenen und erst recht nicht gef\u00fchrten Kampf gegen patriarchale Strukturen in der Gesellschaft und in ihrem eigenen Verhalten. Solche Abgrenzungen machen die Begrenztheit auch linken m\u00e4nnlichen Denkens deutlich. Sie sind mit dem Wutgebr\u00fcll 'Spaltung!' nicht zu beantworten, sondern Herausforderung zu selbstkritischem Denken. Auf der Suche nach neuer m\u00e4nnlicher und weiblicher Identit\u00e4t \u2013 im Kampf f\u00fcr ein herrschaftsfreies Verh\u00e4ltnis zwischen den Geschlechtern.\"\n\nDie Texte geben einen Einblick in die Situation der Frauen und Lesben in der autonomen Szene, wie der Artikel zum Frauenstra\u00dfenfest in Berlin Ende September 1985 zeigt: \"Erschreckend und vielsagend waren die heftigen Diskussionen, die das Fest bei vielen 'angesagten' Szene-M\u00e4nnern ausgel\u00f6st hat. Sie f\u00fchlten sich ausgegrenzt und gekr\u00e4nkt in ihrem Glauben an die eigene Fortschrittlichkeit und f\u00fchrten sich auf, als w\u00fcrde ihnen Gewalt angetan. Ihr Denken gipfelte in \u00c4u\u00dferungen, die das Gewaltverh\u00e4ltnis zwischen M\u00e4nnern und Frauen auf den Kopf stellen, wie z. B.: Ein Frauenstra\u00dfenfest ohne M\u00e4nner zu machen, w\u00e4re das Gleiche, wenn Wei\u00dfe in S\u00fcdafrika Schwarze von einem Fest ausschlie\u00dfen! Sie h\u00e4tten nichts mit einem durchschnittlichen Vergewaltiger zu tun und suchten doch gleichzeitig nach M\u00f6glichkeiten, sich in ihrer alten M\u00e4nnerrolle zu best\u00e4tigen, indem sie sich z. B. als Besch\u00fctzer des Festes anboten.\"\n\nEin weiterer Knackpunkt innerhalb der Szene war die fortschreitende Zersplitterung, die in ihrer Form an die Situation der K-Gruppen Mitte der 1970er Jahre erinnert und in der radikal anl\u00e4sslich eines Treffens in der Hamburger Hafenstra\u00dfe bewusst gemacht wird: \"Wir nennen \/ nannten uns Autonome, unsere Gemeinsamkeit ist \/ war eine Bewegung und die Revolte, das Gesp\u00fcr f\u00fcr unsere Kraft, wenn wir zusammen an einer Stelle und nicht zersplittert an hundert Fronten k\u00e4mpfen. Aus dieser Gemeinsamkeit sind viele rausgefallen oder haben sich bewusst rausgezogen. Heute sind es autonome Antifaschisten, autonome Kommunisten, autonome Kollektive und Jobbergruppen. Das Hauptwort ist zum Eigenschaftswort geworden und dr\u00fcckt weniger Zusammenhalt als das Auseinanderdriften in verschiedene Ans\u00e4tze aus.\"\n\nZentral in der Diskussion \u2013 aber nicht in deren grunds\u00e4tzlicher Bef\u00fcrwortung \u2013 sind in dieser Ausgabe auch wieder Zul\u00e4ssigkeit und Ausma\u00df der Militanz, die von konspirativen Kleingruppen ausgeht. Wurde diese in der Anfangszeit der Zeitung noch als eine Art fast unumg\u00e4nglicher Notwehr gegen die Gewalt von staatlicher Seite aus betrachtet und erschien sie ab 1980 zuweilen als Spa\u00dffaktor, so wurde doch im allgemeinen eine Grenze gezogen, wenn es um Gewalt gegen Personen ging. Auch an dieser Frage zeigte sich nun die Spaltung der Redaktion. Ein hier abgedruckter Artikel \"Direkte Aktion\" pl\u00e4diert als Einzelmeinung explizit f\u00fcr Anschl\u00e4ge gegen verantwortliche Personen, also die \"Damen und Herren in den Chefetagen\": \"Derart k\u00f6nnen unsere Aktionen \u00fcber den reinen Symbolcharakter hinausgehen, weil weniger der behebbare Sachschaden als vielmehr die direkte Gef\u00e4hrdung der Herrschenden und Nutznie\u00dfer bewirkt wird. Am Ende st\u00fcnde ein Zustand, wo weniger die Person als vielmehr die Funktionen in diesem System mit einem Hauch von Gefahr umgeben w\u00fcrden, der nicht nur ihre Besetzung erschwert, sondern gleichzeitig ein breites Bewusstsein hierf\u00fcr schaffen w\u00fcrde.\" Im Anschluss folgt ein Kommentar, der gleich zu Beginn klarstellt: \"Nein! Mit dem vorangegangenen Text identifiziert sich die Redaktion nicht!\". Wie dem auch sei, diese und die folgende Ausgabe quellen nur so \u00fcber von Anschlagserkl\u00e4rungen und \u2013 auch innerhalb der Szene umstrittenen \u2013 Bastelanleitungen. Ebenso wird der Anschlag der RAF auf die milit\u00e4rische Air-Base in Frankfurt diskutiert.\n\nTeil 1 und 2 der Nr. 132 erschienen 1986 mit einer neuen Kontaktadresse in Kopenhagen. Die \"Dominierten\"-Fraktion in der Redaktion machte Fortschritte, einige Seiten besch\u00e4ftigen sich mit dem Standpunkt der Frauen zum Thema Militanz: \"schon l\u00e4nger schwirrte in unseren K\u00f6pfen der gedanke, platz zu schaffen f\u00fcr eine kontinuierliche, breitere auseinandersetzung mit militantem widerstand von frauen, da diskussionen dar\u00fcber oft abgeblockt oder abgeschottet gef\u00fchrt werden. viele frauen haben angefangen, ihren kampf gegen sexismus und patriarchat in praktische militanz umzuwandeln, indem sie personen oder institutionen, die f\u00fcr frauenverachtung stehen, angegriffen haben. zus\u00e4tzlich betroffen von allt\u00e4glicher sexistischer gewalt gehen frauen von anderen bedingungen als typen aus, entwickeln ihre eigenen aktionformen. wir glauben, dass es viele frauen gibt, die \u00e4hnliche erfahrungen machen\/gemacht haben wie wir. die genervt sind vom verhalten vieler typen auf demos oder bei aktionen, dass patriarchat im linken widerstand selten zum thema gemacht wird, dass hierarchie f\u00fcr viele immer noch ein fremdwort ist, oder h\u00f6chstens im zusammenhang mit den schweinen gesehen wird. [...] beim schreiben der folgenden seiten, wo wir \u00f6fters stecken geblieben sind, wurde uns bewusst, [...] dass unsere spr\u00e4che verst\u00fcmmelt ist und wir unsere eigene erst finden m\u00fcssen. [...] das h\u00e4lt uns aber nicht davon ab, dar\u00fcber eine diskussion einzul\u00e4uten. [...] schreibt uns, welche gedanken dazu in euren k\u00f6pfen brodeln, was ihr in euren diskussionen entwickelt habt, und schickt uns erkl\u00e4rungen von aktionen, die ihr in diesem zusammenhang gemacht bzw. mitgekriegt habt!!!\"\n\nSchwerpunkte bildeten neben der Dokumentation des Anstiegs diverser \"Direkter Aktionen\" gegen Atomanlagen auch Antiimperialismus-Themen. Als ein besonderer Stein des Ansto\u00dfes erwies sich jedoch das Gru\u00dfwort an die RAF auf der R\u00fcckseite von Teil 2: \"wir gr\u00fc\u00dfen die genossen\/innen aus der stadtguerilla, die siemens-vorstandsmitglied beckurts liquidiert haben. siemens steht f\u00fcr beteiligung an sdi, europ\u00e4isierung, mik, akw-forschung, umstrukturierung, ausbeutung weltweit. Hier hat der revolution\u00e4re kampf seine kraft entwickelt, und den verantwortlichen noch mal verdeutlicht, dass wir sie nicht vergessen werden!! kraft und liebe allen k\u00e4mpfenden menschen!\" Beigelegt war diesem Teil der radikal ein Kommunique der RAF vom 9. Juli 1986 mit einer Erkl\u00e4rung zum Attentat auf Karl Heinz Beckurts. Daraufhin gab es bis in den Fr\u00fchling 1987 wieder zahlreiche Durchsuchungen linker Buchhandlungen und Wohnungen mit nachfolgenden Anklagen und Prozessen. Nun sollte also nicht nur die Zeitung selbst, sondern auch deren \u00f6ffentlicher Verkauf kriminalisiert werden. Nachdem ein Gro\u00dfteil der Ausgabe 132 beschlagnahmt wurde, kam es endg\u00fcltig zum Bruch in der Redaktion.\n\n### Es brodelt im Untergrund\n\nNach einer weiteren monatelangen Pause erschien erst wieder im August 1987 eine neue Ausgabe der radikal. Die Nr. 133 nennt sich Zeitung f\u00fcr den brodelnden Untergrund und gibt eine Kontaktadresse in Amsterdam an. Sie dokumentiert ausf\u00fchrlich die Auseinandersetzungen unter den verschiedenen radikal-Gruppen und zeigte damit die m\u00f6glichen Wege auf, welche die Zeitung in Zukunft gehen k\u00f6nnte. Hier noch am\u00fcsant ist der Einfall, ganzseitig aus dem Vorwort von Lenins Was tun? zur Problematik des Zeitungmachens zu zitieren, kommentiert mit der Bemerkung, dass \"diese Gedanken im Zusammenhang mit einer bolschewistischen Organisations-Struktur stehen\", die \"in ihrer althergebrachten Weise f\u00fcr uns indiskutabel\" ist.\n\nAus Anlass der Reaktorkatastrophe in Tschernobyl werden praktische Tipps zum Ums\u00e4gen von Strommasten geboten. Neben Anschlagserkl\u00e4rungen, Berichten von antifaschistischen Aktionen und einer Zusammenstellung der Neuigkeiten zum Thema Repression linker Publikationen war der 1. Mai 1987 in Kreuzberg ein Schwerpunktthema. Die damals erstmalig nach Aufl\u00f6sung des Stra\u00dfenfestes am Lausitzer Platz stattfindenden Kiezkrawalle bildeten den Auftakt allj\u00e4hrlich wiederkehrender Stra\u00dfenschlachten. Hier wurde bereits die Frage gestellt, inwieweit die massiven Ausschreitungen und Pl\u00fcnderungen \u00fcberhaupt als politisch motiviert anzusehen seien, da die Beteiligung von Kiezbewohner_innen aller Art und Altersgruppen offenkundig war.\n\nMit dem Erscheinen der beiden Teilhefte der Ausgabe 134 im Jahr 1988 und wieder unter Schweizer Adresse begann eine neue Phase der radikal. Die Zeitung wurde nun von verschiedenen Gruppen gemacht. Das Kernthema Knast wird in Kombination mit Aussageverweigerung angesprochen und in den folgenden Ausgaben aufgrund der andauernden Hungerstreiks und Solidarit\u00e4tsaktionen immer wieder aufgegriffen. Zentral ist im ersten Teil auch die Diskussion um die Umstrukturierung und Gentrifizierung in Kreuzberg, dem Zentrum autonomen Lebens in Berlin, aufgeh\u00e4ngt an der skandaltr\u00e4chtigen \"K\u00fcbel\"-Aktion gegen das Restaurant Maxwell in der Oranienstra\u00dfe. Das Horrorzukunftsszenario wurde damals so antizipiert, wie es heute Realit\u00e4t im Kiez ist: \"Die Neuen brauchen zahlkr\u00e4ftige Kunden [...]. Also holt man sich die Kundschaft aus anderen Stadtteilen und der BRD. Und wie wird das gemacht? Kreuzberg wird zum Kneipenviertel, St\u00fcck f\u00fcr St\u00fcck, erst die belebten Stra\u00dfen, dann die Nebenstra\u00dfen, dann die etwas abgelegenen Ecken. Kreuzberger N\u00e4chte sind lang!! Das sieht dann so aus: Erst kommt ne Disco (sind ja schon welche da, sogar mit T\u00fcrsteher), daneben eine Boutique, dann ein Music-Cafe, dann ein Restaurant, dann noch ein Restaurant, dann ein Schnellimbiss usw. Immer donnerstags gehts dann los: Die Leute kommen, gut gelaunt und mit vollem Portemonnaie. Die Autos werden am liebsten in einer stillen Seitenstra\u00dfe geparkt, zur Not auch auf dem B\u00fcrgersteig, daf\u00fcr l\u00e4uft man ja auch mal 5 Minuten. Dann k\u00e4mmen unsere Ferieng\u00e4ste die Stra\u00dfen durch, meist ne kleine Gruppe, sch\u00f6n gestailt 'guck mal da, lass uns doch mal hier rein', usw. Dann endlich um 2 Uhr nachts str\u00f6men sie zu ihren Autos zur\u00fcck, und du denkst, wat wolln die blo\u00df alle? Aber uns kann das ja egal sein, weil wir dann eh nicht mehr hier wohnen. Weil die Wohnungen f\u00fcr uns schon l\u00e4ngst zu teuer sind [...]. Dieser Umbau von Kreuzberg richtet sich nicht nur gegen Leute, die politisch aktiv sind, sondern gegen die ganze soziale Struktur.\" Ebenso umstritten waren die, im zweiten Teil dieser Ausgabe thematisierten, \u00fcberfallartigen Attacken auf kleine alternative Programmkinos, die im Zuge der feministisch-autonomen PorNo-Aktionen stattfanden.\n\nInsgesamt beinhaltet dieses Teilheft auffallend viele \u2013 auch, aber nicht nur, von M\u00e4nnern verfasste \u2013 Texte, die sich mit der Kritik an patriarchalen Strukturen auseinandersetzen, selbst in den eigenen Reihen: \"Die einen (M\u00e4nner) sehen Diskussionen um das Patriarchat im Kampf als Behinderung an, st\u00e4ndig mit ihrer eigenen Rolle konfrontiert zu sein. Schlimmstenfalls kommt der Vorwurf der Spaltung, begr\u00fcndet \u00fcber Erfahrungen mit Frauen \u2013 die von ihrem Selbstverst\u00e4ndnis her mit Typen gar nichts mehr zu tun haben wollen und die dementsprechend kompromisslos vorgehen. Die anderen (M\u00e4nner) kapitulieren vor der Komplexit\u00e4t des Patriarchats, stellen die Erfahrungen der Kollektive entgegen, wollen also keine Pauschalurteile \u00fcber M\u00e4nner h\u00f6ren und sind erm\u00fcdet, weil keine Praxis den Diskussionen folgt. Vor diesem Hintergrund finden wir es wichtig, dass innerhalb dieser Zeitung die anti-patriarchale Debatte von M\u00e4nnern fortgesetzt wird, weitergehend die Gr\u00fcndung von M\u00e4nnergruppen und ein Standpunkt zu Vergewaltigung.\"\n\nAuch die nachfolgenden Ausgaben enthalten verst\u00e4rkt frauenspezifische Beitr\u00e4ge. Offenbar hat hier ein Um- oder zumindest Mit-Denken der Kritik am Sexismus in der radikal Fu\u00df gefasst. Zudem hatte sich ein Frauenplenum gegr\u00fcndet, das sich mit der Problematik des Internationalen W\u00e4hrungsfonds (IWF) und der Weltbank (WB) speziell f\u00fcr Frauen besch\u00e4ftigte. Angesichts der Erfolge der Aktionen der Roten Zora gegen einen deutschen Bekleidungsmarkt, die laut radikal zu verbesserten Arbeitsbedingungen koreanischer N\u00e4herinnen f\u00fchrten, wurde auch hier zum Widerstand aufgerufen.\n\nDen Gro\u00dfteil der im Oktober 1988 erschienenen Nr. 135, auf dem Titel Pflastersteine und ein hysterischer Polizist, f\u00fcllt dann auch eine Chronologie der Widerstandsaktionen zum IWF-Kongress. Die zahlreichen Attacken auf die \"Bonzen und Benzen\" lesen sich wie ein gut geschriebener Abenteuerroman und als voller Erfolg f\u00fcr die autonomen K\u00e4mpfer_innen. Angriffspunkte im Umfeld des Kongresses waren zudem einige Firmen und Gro\u00dfkonzerne, die sich aufgrund ihrer Profite mit R\u00fcstungs-, Atom- und Biotechnologie unbeliebt gemacht hatten.\n\nWeiter ging es im Jahr 1989 in der zweiteiligen Nr. 136 mit Informationen zum Thema Kriminalisierung und wie mit Spitzeln und Denunziation umzugehen sei. Schwerpunkt der Mai-Ausgabe ist der Antifaschismus, zu dem es auch in den vorherigen Ausgaben bereits einige Beitr\u00e4ge gab. Neben NPD und DVU stehen hier die relativ neue Partei der Republikaner und deren damalige Wahlerfolge im Fokus. Insgesamt werden die Ausgaben dieses Jahres durch Aufarbeitung von Basiswissen bestimmt. Es gibt Chronologien zum Hungerstreik der RAF- und anderer Inhaftierter in der Nr. 138\/1, zur Frauenbewegung in der Nr. 138\/2 und zu linker und widerst\u00e4ndiger Politik im Sinne einer \"Geschichte von unten\" unter dem Titel \"Gegen das Vergessen\" in der Nr. 139 ebenso wie Hintergrundinformationen zu einem boykott- und angriffsw\u00fcrdigen Tankstellenkonzern, ebenfalls in der Nr. 139. Zwar wird auch dazu eine Brandsatz-Bauanleitung geliefert, insgesamt steigt mittlerweile jedoch die Anzahl der qualitativ h\u00f6herwertigen Beitr\u00e4ge im Vergleich zur Quantit\u00e4t der Anschlagserkl\u00e4rungen. Einzelne Auseinandersetzungen wurden an konkreten Ereignissen festgemacht, wie die Frage nach dem Militanzpotential von Frauen angesichts der 1.-Mai-Krawalle oder nach den Intersektionen von Rassismus und Sexismus.\n\nInternes wurde kaum noch thematisiert, mit Ausnahme des Postverlustes, den die wechselnden Adressen (Nr. 135 und 136: Z\u00fcrich; Nr. 137 und 138: Amsterdam; danach wieder Z\u00fcrich) verursachten, auch wenn zun\u00e4chst eine polizeiliche \u00dcberwachung vermutet wurde. Aber dieses Beispiel zeigt, welche H\u00fcrden die klandestine Produktion der radikal in den Weg stellte. Das Intro der 138 berichtet: \"beim Versuch, eine schei\u00dfe nicht zu wiederholen, ist erst recht ein haufen hingelegt worden. in der letzten Nr. 137 wechselte die kontaktadresse von z\u00fcrich nach amsterdam. aus amsterdam haben wir ab mitte mai \u00fcberhaupt keine post bekommen, und im juni nur einen teil. \u00fcberst\u00fcrzt und entgegen eines anderslautenden beschlusses in der struktur wurde eine 'warnung' ver\u00f6ffentlicht. darin hie\u00df es sinngem\u00e4\u00df: die schweinebullen zocken uns kohle und briefe, blo\u00df nix mehr nach amsterdam schicken. w\u00e4re besser gewesen, ein bisschen zu warten, denn schon eine woche sp\u00e4ter gab's die 'entwarnung'. [...] f\u00fcr die Ver\u00f6ffentlichung gab's gute gr\u00fcnde. erstens wollten wir schnell reagieren und den briefefluss an die bullen \u2013 schien sooo klar \u2013 stoppen. und zweitens die allerh\u00f6chstwahrscheinliche ursache nennen \u2013 im nachhinein oberpeinlich, denn die bullen waren's nicht.\"\n\nNach einer Pause von sieben Monaten seit dem Erscheinen der 139. Ausgabe kam im Juni 1990 die Nr. 140 mit einer neuen Wiener Adresse heraus. Auf den zwischenzeitlichen Fall der Mauer reagierte die radikal mit einem Artikel unter dem Titel \"Schlagt die Faschisten wo ihr sie trefft\", beginnend mit einer ganzseitigen Chronologie faschistischer Aufm\u00e4rsche und Angriffe auf besetzte H\u00e4user, Linke und Migrant_innen. Die nachfolgende Analyse der Situation l\u00e4sst eine \u00dcberforderung angesichts der rasanten Ereignisse erkennen: \"Seit dem 9. November laufen die politischen Entwicklungen im Raketentempo ab. Die Wiedervereinigung steht vor der Haust\u00fcr \u2013 das BRD-Kapital str\u00f6mt geiernd in den Osten. [...] Der nationale Aufbruch kitzelt die Kleinb\u00fcrgerIn heraus \u2013 die Aussicht auf Konsum, die 'Heimkehr', die Aussicht, 'wer' zu sein, besser zumindest als die 'Anderen', Platz auch mal an der Sonne, her mit den Weibern \u2013 die Pornoindustrie nistet sich in der DDR ein. Die Linke in der BRD ist am Verarbeiten, versucht erste Kontakte zu kn\u00fcpfen und zieht gerade aus Wohnungsmangel von Westberlin nach Ostberlin H\u00e4user besetzen. Die Linke in der DDR ist in der Mehrzahl demoralisiert und genauso zersplittert wie wir. Die Ans\u00e4tze autonomer Organisierung sind gering, aber da! Und in der Gesellschaft bricht aus, was lange g\u00e4rte, aber am G\u00e4ngelband lag: die rassistische und sexistische Einstellung der Kleinb\u00fcrgerInnen und Prolls, die sich nur behaupten k\u00f6nnen, wenn sie anderen ihre Daseinsberechtigung absprechen. [...] Ob sie selber vom Kapital ausgenommen werden, spielt eher eine zweitrangige Rolle. Wichtig ist, dass sie immer noch 'besser' zu sein meinen und auch gesellschaftlich werden, wenn der eigene Machtanspruch nur lautstark und brutal genug untermauert wird. Wenn die Abgrenzung zum Angriff wird: der deutsche Bieder\/Saubermann als aktives Gegenbild zum faulen Gesocks, der spie\u00dfige Drecksack mit der 'gesunden' Fortpflanzung gegen die 'verluderten Homos' oder auf Selbstbestimmung bestehenden Frauen. [...] Eine 'radikale Linke' macht eine an der Situation vorbeigehende Demonstration mit der Parole 'Gegen die Wiedervereinigung'. Statt l\u00e4ngst abgefahrenen Z\u00fcgen hinterherzurennen, w\u00e4re es besser, den Tatsachen ins Auge zu sehen.\"\n\nAls erste Reaktion auf die Geschehnisse wurde gefordert, \"den Faschos [...] eine entsprechende militante Gegenwehr in den Weg zu stellen\", notwendig seien aber auch ein solidarischer Zusammenhalt und die Kombination verschiedenster Aktionsformen. Dazu wurden Listen mit Namen und Adressen von FAP- und DVU-Mitgliedern ver\u00f6ffentlicht. Dass Denunziation auch in dieser Form keine Glanzleistung war, wird im n\u00e4chsten Heft einger\u00e4umt: \"Der uralte Stand (1988) sowie die Durchmischung der Liste von F\u00fchrern der Faschistenszene und eher kleineren Lichtern sind sicherlich ein gro\u00dfes Problem dieser Liste, denn so k\u00f6nnen sich darauf Leute befinden, die der Faschistenszene l\u00e4ngst den R\u00fccken gekehrt haben. [...] Es ist ein \u00e4hnliches Problem wie beim Abdrucken der Anleitungen, da k\u00f6nnen wir auch immer wieder nur betonen 'alles in kleinen Mengen auszuprobieren' oder hier 'alles muss genauestens nachrecherchiert werden'.\" Diese Einsicht w\u00e4re bereits vor der Publikation solcher Listen w\u00fcnschenswert gewesen. Trotzdem sollte es sie weiter geben, sie w\u00fcrden jedoch nur auf Anfrage zugeschickt. Insgesamt bleibt der Eindruck, dass das Thema Wiedervereinigung angesichts des Ausma\u00dfes der politischen Entwicklungen und der damit einhergehenden Verunsicherung (oder vielleicht gerade deswegen) einen vergleichsweise geringen Raum in der radikal einnimmt.\n\nDie weiteren Seiten sind hier, wie auch in den n\u00e4chsten Heften wieder, mit RAF- und RZ-Papieren, sonstigen Anschlagserkl\u00e4rungen oder auch Bauanleitungen f\u00fcr Zeitz\u00fcnder gef\u00fcllt. Zudem wird die in der Nr. 139 begonnene und sehr lesenswerte Geschichtsserie \"Gegen das Vergessen\" fortlaufend weitergef\u00fchrt. Das Thema Mauerfall wird lediglich in einem Beitrag zum 1. Mai 1990 noch einmal aufgenommen, diesmal mit einer konkreten Beschreibung der negativen Ver\u00e4nderungen auf dem Arbeits- und Wohnungsmarkt und auf den (Kreuzberger) Stra\u00dfen. In einem \"Interview mit Ines und Emil aus Kreuzberg\", beide aus dem autonomen Spektrum, wird aber auch der Frage nach dem politischen Gehalt der Militanz zum 1. Mai differenziert nachgegangen.\n\nZudem meldet sich die autonome Frauengruppe der radikal explizit zu Wort und stellt ihre Angriffspunkte vor: \"Wie ihr euch vielleicht schon denken k\u00f6nnt, ist Frauenkampf mehr als das Thematisieren von 'frauenspezifischen' Themen. [...] Wir denken, es gibt nicht den Frauenkampf, den Weg f\u00fcr Frauen, nicht den Punkt, auf den sich jetzt alle Frauen st\u00fcrzen m\u00fcssen. Sicher gibt es Kristallisationspunkte wie z. B. die Gen- und Reproduktionstechnologien. Aber dies ist nur die eine Seite, die der objektiven Analyse der Schweinestrategien und daraus abgeleitet die Vorrangigkeit und Wichtigkeit unserer Angriffsziele. Die andere Ebene ist die, dass Frauenkampf sich entwickeln muss, wo die unterschiedlichen Frauen sich bewegen: im Stadtteil, in der Fabrik... da, wo wir gemeinsam als Frauen mit sexistischen Angriffen konfrontiert sind... gegen sexistische Gyn-\u00c4rzte, Krankenh\u00e4user mit schweinischen Gyn-Abteilungen, \u00a7 218, Pornographie in Werbung, Sexl\u00e4den. Da wo Frauen international ausgebeutet werden in einer weltweit verzahnten Struktur: Weltmarktfabriken [...], Sextourismus...\" Neu im Gegensatz zum verallgemeinernden Ansatz der Frauenbewegung der 1970er Jahre ist hier und im weiteren Verlauf des Artikels, dass kein biologistisches \"Wir gegen das Patriarchat\"-Denken vorgestellt wird, sondern unterschiedliche soziale und kulturelle Hintergr\u00fcnde und Ausgangspositionen von Frauen mitgedacht wurden. Und mit feministischen Positionen und Debatten ging es auch in der Nr. 141 vom November 1990 weiter.\n\nBemerkenswert ist, dass in der ganzen Zeit seit der Mauer\u00f6ffnung die neue Hausbesetzungswelle, die sich nun haupts\u00e4chlich in Ostberlin abspielt, so gut wie gar nicht thematisiert wurde. Von den zahlreichen Neubesetzungen \u2013 bis Juli 1990 sind es weit \u00fcber 100 im Ostteil der Stadt \u2013 erf\u00e4hrt man nur im Hintergrund anl\u00e4sslich der Attacken Rechtsradikaler oder nebenbei, wenn von Kreuzberg die Rede ist. Dort gab es zwar ab 1989 auch wieder Besetzungen, die meisten H\u00e4user hielten sich aber nicht allzu lange. Zudem hatte sich eine eigene BesetzerInnenzeitung (BZ) im Osten gegr\u00fcndet. Lediglich in dem \"Leipzisch\"-Interview in der Nr. 141\/2 vom November 1990 erf\u00e4hrt man \u00fcberhaupt etwas aus \"dem Osten\". Die Unzufriedenheit mit dem Verlauf und Inhalt des Interviews ist den \"Wessis\" aber deutlich anzumerken. So beschr\u00e4nkte sich die Berichterstattung der radikal weiter auf Westberlin und Westdeutschland, gelegentlich auch Wien oder Kopenhagen und die Antiimp-Themen im Rest der Welt.\n\nMit der radikal-Ausgabe 142 vom M\u00e4rz 1991 kam aus aktuellem Anlass das neue gro\u00dfe Thema Golfkrieg zur Sprache mit dem Motto: \"Deutsche Waffen, deutsches Geld, morden mit in aller Welt\". Neben einem in der Redaktion umstrittenen Artikel zur Politik der RAF er\u00f6rtert ein weiterer Beitrag \u2013 wieder einmal \u2013 Sinn und Unsinn der Bauanleitungen, die nach wie vor nicht ganz fehlerfrei zu sein scheinen und damit eine Gef\u00e4hrdung f\u00fcr alle Seiten beinhalten. Au\u00dferdem wurde ein neues Projekt unter dem etwas ungl\u00fccklichen Titel \"Piefke\" ins Leben gerufen, das sp\u00e4ter als \"OLGA\" weitergef\u00fchrt wird und dem autonomen Informations- und Erfahrungsaustausch im Sinne einer st\u00e4rkeren Einbeziehung des jugendlichen Nachwuchses und der Geschehnisse in der \"Provinz\" dienen sollte. Die Leser_innen wurden ausdr\u00fccklich dazu aufgefordert, eigene Beitr\u00e4ge einzusenden. In dieser Rubrik wird die R\u00e4umung der Mainzer Stra\u00dfe im November 1990 kurz angesprochen. Deutlich wird jedoch nur, dass die stets in der Kritik stehende Verhandlungsbereitschaft der Ostberliner Besetzer_innen nicht zu den erw\u00fcnschten Zielen f\u00fchrte und diese H\u00e4userbewegung noch schneller am Ende war als jene zehn Jahre zuvor.\n\nZwei Monate sp\u00e4ter nahm sich die radikal mit dem Artikel \"Go Trabi Go!\" doch noch der ehemaligen DDR an. Im Fokus steht der dort offen zutage tretende Rassismus, der auch im Westen existiere, da aber durch Wohlstand abgefedert werde. Von der schw\u00e4chelnden autonomen Bewegung wurde jedoch mehr Begeisterung f\u00fcr emanzipatorische und linksradikale Inhalte innerhalb einer \"Massenprotestbewegung\" wie der im Osten erwartet. Kreativ dagegen ist die Bastelanleitung in der Nr. 143. Waren es in der vorherigen Ausgabe noch technische Ratschl\u00e4ge zum Autoklau, wird diesmal unter dem Titel \"Gegen den Sozialabbau, organisiert den Kaufhausklau!\" eine N\u00e4hanleitung f\u00fcr die Herstellung einer Tasche vorgestellt, die durch ein integriertes Alugewebe die Sicherheitsanlagen in Kaufh\u00e4uern \u00fcberlisten soll. Die nachfolgende Ausgabe bietet zudem \"Tips und Trix zur kostenlosen Aneignung von Strom und Gas \u2013 Wir bohren eine Steigeleitung an\". In sp\u00e4teren Ausgaben folgten weitere Hinweise zur erfolgreichen \"Warenumverteilung\" und sogar ein Computerkurs.\n\nFast das gesamte erste Teilheft der Nr. 144 besch\u00e4ftigt sich mit Antifa-Aktionen und -Diskussionen und dem Versuch, Thesen zu einer \"Autonomen Organisierung\" vorzustellen. Im zweiten Teil setzt sich ein l\u00e4ngerer Artikel mit dem Titel \"1. Mai und kein Ende \u2013 Ein Nachschlag von Frauen \u2013 (Ex) Westberlin\" ein weiteres Mal mit der \"Mackermilitanz\" auseinander, allerdings auch mit der Kritik aus dem Ostteil, ausgerechnet von der BesetzerInnenzeitung, an der \"Randale\". Die Antwort aus dem Westen darauf: \"So lesen wir z. B., dass wir mit der abendlichen Randale die positiven Momente von Demo und Fest relativiert haben, bei wem? Vielleicht geht ihr zu sehr von euch selbst aus, wenn ihr behauptet, dass die Randale keine positiven Momente haben. Was ist mit den vielen hundert, die mitgemacht haben, was mit den 2000, die sich 89 beteiligten und all denen, die damit sympathisieren? Alles hirnlose Militante? [...] Das hei\u00dft im Gegenzug nicht, dass Randale das Einzigste und Beste ist, was es gibt. Aber massenmilitante Aktionen sind eine Form von Widerstand, an der sich auch solche beteiligen, die sonst nicht so gut organisiert sind [...]. Massenmilitante Aktionen k\u00f6nnen durchaus eine Perspektive aufzeigen. Zu sehen, dass es m\u00f6glich ist, die Bullen zu vertreiben oder z. B. einen Supermarkt auszur\u00e4umen, wenn sich viele daf\u00fcr zusammentun, kann auch eine sehr positive Erfahrung f\u00fcr viele bedeuten, kann die Ohnmacht f\u00fcr diesen Moment durchbrechen und Hoffnung machen. [...] Es ist schon eine ziemliche Unversch\u00e4mtheit jener Ostberliner BesetzerInnen, uns als heldenhafte H\u00fcnengestalten zu beschreiben. Heldenhaft, na ja, das w\u00fcrde vielleicht noch gehen, [...] aber wie H\u00fcnen sehen wir nun wirklich nicht aus. Da zeigt sich der schon oben kritisierte gesellschaftlich gepr\u00e4gte Blick auf die Realit\u00e4t. Wahrscheinlich habt ihr nur M\u00e4nnern zugesehen bzw. den 1.80 aufw\u00e4rts-M\u00e4nnern nachgeschaut. Au\u00dfer dieser st\u00e4ndigen Unsichtbarmachung regt uns aber der unsolidarische besserwisserische Ton auf. Da werden keine Fragen gestellt, da ist alles klar. Keine\/r hat Steinw\u00fcrfe auf Wannen zur ruhmreichen Schlacht gegen das 'Schweinesystem' hochstilisiert, wie ihr behauptet. [...] Wir erwarten von welchen, die sich als autonom bezeichnen, einen anderen Umgang damit.\"\n\nSo viel zum Ost-West-Verh\u00e4ltnis in der autonomen Szene. Es scheint, dass selbst im Herbst 1991 der Graben zwischen den beiden Stadt-Teil-Fraktionen noch tiefer ist als zuvor die Mauer hoch war.\n\nNach einer zweiteiligen Ausgabe 145 im Februar 1992 erschien die Nr. 146 erst im November \u2013 mit der Aufschrift \"Totgesagte leben I\u00e4nger!?\" und dekoriert mit dem Konterfei von Herman Munster.\n\nDie Nabelschau der Redaktion erstreckt sich \u00fcber dreimal zwei Seiten und erkl\u00e4rt aus verschiedenen Perspektiven interne Differenzen als Begr\u00fcndung f\u00fcr das lange Aussetzen. So, wie sich die Autonomen in einer Art Krise befanden, schien es auch innerhalb der Redaktion zu sein. Auf ein notd\u00fcrftig zusammengeschriebenes Flugblatt, das die radikal im Vorfeld zu diesem Thema herausgegeben hatte, waren zahlreiche Leser_innenbriefe erschienen, von denen einer treffend bemerkt: \"Was ihr als euer aktuelles Problem beim Zeitungmachen beschreibt, ist meiner Meinung nach gar kein spezielles Problem. Das ist die zugespitzte Form des Normalzustandes der allermeisten autonomen ,Zusammenh\u00e4nge', sozusagen in konjunkturell versch\u00e4rfter Form. Ihr sagt: 'Wir wollten und haben nie ein verbindendes Statut f\u00fcr uns verabschiedet. Deshalb basierte unsere Zusammenarbeit auf entwickelten Kriterien und dem Willen und dem Wirken aller Beteiligten.' Das ist in unseren Kreisen ja auch so \u00fcblich, und auch ganz praktisch, da mensch sich ja dann zusammen gleich in die Praxis st\u00fcrzen kann, aber hat eben auch Nachteile. Zum Beispiel dass es nicht m\u00f6glich ist, so eine Struktur \u00fcber einen sehr beschr\u00e4nkten Rahmen auszubauen, weil dann nie ausgesprochene oder auch nur bemerkte Differenzen pl\u00f6tzlich mitten in der Praxis auf dem Tisch liegen und einen blockieren, wenn das Zusammenspiel nicht mehr klappt. Weil n\u00e4mlich au\u00dferhalb eines sehr pers\u00f6nlichen Kreises so ein unausgesprochenes Einverst\u00e4ndnis nicht lange halten kann, da eine sonst die inhaltlichen Differenzen absorbierende Vertrautheit fehlt.\"\n\nEs gab einzelne Gruppen, die zwar keine konkrete Spaltung herbeif\u00fchrten, jedoch zwei nicht ganz klar getrennte \"Tendenzfraktionen\" ausgebildet hatten. Innerhalb der Redaktion benannten sie sich als \"die Einen\" und \"die Anderen\", ein eigenes Intro schrieb aber auch die Frauengruppe. Den Ausl\u00f6ser des Konfliktes bildete die Rubrik \"OLGA\" oder vielmehr deren. Umfang und die Relevanz, die diesem Teil des Heftes im Verh\u00e4ltnis zu politischen Diskussionsbeitr\u00e4gen einger\u00e4umt werden sollte und womit auch die redaktionelle Richtlinie beeinflusst wurde. Stand noch zu Beginn der klandestinen Produktion im Vordergrund, einfach irgendwie weiterzubestehen, ging es jetzt um die Bestimmung der Inhalte und, unausgesprochen, auch um hierarchische Tendenzen.\n\nDie Frauengruppe f\u00fchlte sich offenbar nach wie vor nicht angemessen repr\u00e4sentiert und stellte fest, dass der \"Dreh- und Angelpunkt unserer politischen Arbeit au\u00dferhalb der radikal [liegt]\". Sie war bereits dazu \u00fcbergegangen, manche Artikel auch oder nur in der mittlerweile als eigene autonome Frauen\/Lesben-Zeitung existierenden Publikation Amazora zu ver\u00f6ffentlichen, mit der Begr\u00fcndung: \"Einiges wollen wir nur f\u00fcr Frauen schreiben, nur mit Frauen diskutieren. Ist ja nichts Neues. Wir finden es richtig, die Begrenztheit der Diskussion f\u00fcr Frauen in einem gemischtgelesenen Medium \u00f6ffentlich aufzuzeigen und nicht so zu tun, als g\u00e4be es sie f\u00fcr uns nicht.\" In ihrem Intro fassen sie auch den Gruppenkonflikt am kompaktesten zusammen: \"Es gibt innerhalb der radikal zwei Tendenzen. Eine, die die radikal in erster Linie als Dienstleistungsunternehmen versteht, das Zugeschicktes dokumentiert, Regionalzeitungen auswertet und dadurch einen \u00dcberblick \u00fcber autonome Aktivit\u00e4ten und Diskussionen verschafft, redaktionelle inhaltliche Arbeit finden die VertreterInnen dieser Tendenz zweitrangig. Die andere Tendenz, zu der wir uns z\u00e4hlen, will \u00fcber Dokumente hinaus mit eigenen Positionen in laufende Diskussionen eingreifen oder auch Sachen anschieben. Selbstverst\u00e4ndlich sollen zugeschickte Beitr\u00e4ge, Erkl\u00e4rungen, Berichte ihren Platz in der radi haben. [...] Als Problem kommt noch unser seltenes, bestenfalls viertelj\u00e4hrliches Erscheinen dazu.\"\n\nDie Frauengruppe ver\u00f6ffentlichte in den folgenden Ausgaben neben einzelnen Artikeln noch eine dreiteilige Serie zu Prostitution und stieg mit deren letztem Teil in der Nr. 148 ebenso wie andere Gruppen im Herbst 1993 ganz aus der radikal-Redaktion aus.\n\n### Radikal in der Krise\n\nNachdem eine RZ bereits im Januar 1992 ihre Aufl\u00f6sung bekannt gegeben hatte, gibt es in der Nr. 147 vom M\u00e4rz 1993 ein Interview mit einer anderen zum Thema Fl\u00fcchtlingskampagne und militanter Widerstand. Deutlich wird, in welchem Dilemma militante Gruppen steckten, seit die Rechten verst\u00e4rkt auf deren Methoden zur\u00fcckgriffen. Bei Anschl\u00e4gen auf \"Ausl\u00e4nderbeh\u00f6rden\" sei nicht mehr klar, dass diese sich gegen die staatliche Fl\u00fcchtlingspolitik richteten, zum Teil w\u00fcrden diese in der Presse als migrantenfeindliche rechtsradikale Angriffe dargestellt. Eindeutig unterschieden sich die Aktionen der RZ jedoch nach wie vor in dem Anspruch, dass \"ein Risiko f\u00fcr unbeteiligte Menschen ausgeschlossen ist\".\n\nDie Auseinandersetzung mit patriarchalen Strukturen war auch weiterhin ein Knackpunkt innerhalb der Bewegung. Erkannt wurde zudem, \"wie schlecht es um die Wechselbeziehung zwischen militanten und nicht-militanten Gruppen steht\". Trotzdem wird behauptet: \"Auf jeden Fall sind wir nicht am Ende unserer Politik.\" Mit einer 200-seitigen Dokumentation von Texten zu RAF und RZ, die auf Anfrage zugeschickt w\u00fcrde, hatte sich die radikal-Redaktion zum Ziel gesetzt, dem nachlassenden Interesse linker Zeitungen an \"Guerilla und Stadtguerilla\" entgegenzuwirken.\n\nZu fr\u00fcher zentralen Themen wie dem H\u00e4userkampf gab es nur noch vereinzelte Meldungen, auch von der Anti-Atom-Bewegung lie\u00df sich kaum noch etwas in der radikal finden. Die mit einem Papier der G\u00f6ttinger Antifa (M) in der Nr. 144 losgetretene Diskussion um eine autonome Organisierung angesichts der konstatierten \"Krise der autonomen Bewegung\" zieht sich \u00fcber mehrere Ausgaben, ohne zu nennenswerten Ergebnissen oder einer Einigung zu kommen. Zur Debatte stand nach wie vor, ob vorrangig an internen Widerspr\u00e4chen wie Sexismus innerhalb der Szene gearbeitet werden m\u00fcsse, wie es die autonomen Kritiker_innen fordern, oder ob das Bestreben der Antifa (M) nach einer verst\u00e4rkten gemeinschaftlichen Organisierung und Strukturierung verschiedener Splittergruppen mit Orientierung nach au\u00dfen ma\u00dfgeblich voranzutreiben sei.\n\nNeben den unvermeidlich weiterlaufenden Auseinandersetzungen mit militanten Gruppen, der Antifa (M) und der RAF, aktualisiert um den Tod von Wolfgang Grams im Juni 1993, bietet die Nr. 148 nach acht Monaten Pause im November 1993 Ausarbeitungen zur Krise linker Medien oder auch zum Bild \"der Autonomen\" in den b\u00fcrgerlichen Medien, wie es die autonome a.f.r.i.k.a-gruppe\/mittlerer neckar darstellt: \"Und mit 'den Autonomen' haben sie [die b\u00fcrgerlichen Medien] eine politische Richtung gefunden, die sich selbst nur allzu gern symbolisch-visuell konstruiert und stilisiert. Von dem jeweiligen Selbstverst\u00e4ndnis profitieren beide. F\u00fcr 'die Autonomen' l\u00e4sst sich gar vermuten: Ohne derartige Medien w\u00fcrden sie in dieser Form nicht (mehr) existieren. Die Attraktivit\u00e4t der Autonomen ist nicht zuletzt das Ergebnis eines Mediendiskurses, der sie zu jenen entschlossenen militanten K\u00e4mpfern in schwarz stilisiert und die symbolische Repr\u00e4sentanz mit Lederjacke, Stein und Molli immer mehr und immer von Neuem zum unhinterfragten Selbstbild zahlreicher (m\u00e4nnlicher) junger Aktivisten werden l\u00e4sst.\"\n\nDaran h\u00e4ngt sich die Frage nach dem Stellenwert von Militanz im Kontext von Jugendkulturen an: \"Nun war es in der Vergangenheit wenigstens so, dass eine bestimmte Form (Militanz) einer bestimmten inhaltlichen Orientierung ('Links' \u2013 was auch immer das sein mochte) eindeutig zuordenbar war. Die Autonomen besa\u00dfen immerhin eine Art von Aktionsmonopol, auch wenn die (b\u00fcrgerlichen) Medien \u00fcber deren Deutungsmonopol verf\u00fcgten. Inzwischen ist aber auch hier einiges in Bewegung geraten. Es gibt keine Selbstverst\u00e4ndlichkeit mehr, mit der sich rebellierende Jugendkulturen als links, als 'alright', bewerten l\u00e4sst. Aktionen mit Steinen und Molotowcocktails sowie Auseinandersetzungen mit der bewaffneten Staatsgewalt zeichnen nunmehr auch die rassistischen Pogrome gegen Fl\u00fcchtlinge aus. [...] Wir sehen momentan die Notwendigkeit, dass das politische Projekt einer autonomen und radikalen Linken nicht mehr in erster Linie \u00fcber eine bestimmte Form ('Militanz') bestimmt werden darf. Besonders aufgefallen ist uns das in dem merkw\u00fcrdigen 'Internationalistischen Block' der Demonstration in Bonn [...]. Wenn der einzige Inhalt einer Manifestation darin besteht, wie gef\u00e4hrlich wir aussehen, dann brauchen wir keine eigene Abschlusskundgebung. Bereits im Demozug mutierte die Folklore autonomer Trachten zum Ersatz f\u00fcr fehlende Inhalte.\"\n\nIm Kontrast dazu steht der Bericht von der zweiw\u00f6chigen \"Tunten-Terror-Tour 1993\", deren Teilnehmer sich optisch nicht den g\u00e4ngigen \"autonomen Trachten\" unterordneten und sich auch im Tonfall ihrer Selbstdarstellung von der verbissenen Ernsthaftigkeit der laufenden Diskussionen abhoben. Ausgangspunkt war es, \"durch Fummel und Aktionen [...], als Schwule neben der b\u00fcrgerlichen Schwulenbewegung, \u00d6ffentlichkeit zu den Schwerpunkten Rassismus und Schwulenfeindlichkeit [zu schaffen und] damit den verkrusteten Christopher-Street-Day-Feierlichkeiten auf unsere Art [zu] begegnen\". Mit der Kritik, dass die urspr\u00fcnglichen Inhalte des CSD in den Hintergrund treten und immer mehr f\u00fcr ein \"karnevalistisches Zur-Schau-Gestelle\" herhalten w\u00fcrden, wurden Plakataktionen durchgef\u00fchrt, \"Tunten-Terror-Blocks\" innerhalb von Demos gebildet und in Fulda auf dem Domplatz ein \"Sex-in\" veranstaltet. Vor allem Plakate mit dem Text \"Am 27.6. wurde Wolfgang Grams von Bullen exekutiert. Trauer, Wut, Zorn\" f\u00fchrten immer wieder zu Durchsuchungen des Reisebusses und zu Beschlagnahmungen. Mit einer Kundgebung unter dem Motto: \"Rassismus t\u00f6tet \u2013 wehrt euch, greift ein, Tunten-Terror-Tour gegen Rassismus, Faschismus, Lesben- und Schwulenfeindlichkeit und Patriarchat!\" wurde zudem ein homosexueller Neonazi besucht. Das Fazit der Aktivisten lautete: \"Die teilweise \u00fcberaus heftigen Reaktionen unserer heterosexuell veranlagten Mitmenschen auf unser Auftauchen in Fummel zeigen uns die Notwendigkeit, diese Art von Tunten-Terror-Touren nicht immer, aber immer \u00f6fter zu machen. Es ist f\u00fcr uns wichtig im Fummel zu agieren, um das herrschende M\u00e4nnlichkeitsbild zu zerkratzen und um uns von der integrativen Politik der b\u00fcrgerlichen Schwulenbewegung abzugrenzen indem wir als offen tuntig-schwul auftreten. Obwohl wir auch einiges an Schwulenfeindlichkeit abbekommen haben (v. a. von Bullen), aber auch gerade deswegen hat es viel Spa\u00df gemacht und es gibt nicht nur bl\u00f6de Menschen... Wir kommen wieder!\"\n\nNeben diesem Hinweis, dass autonome Strukturen auch jenseits der heterosexuellen Matrix existierten, waren \u2013 trotz des Ausstiegs der radikal-Frauen \u2013 feministische Aspekte ebenfalls zun\u00e4chst nicht aus der Zeitung verschwunden. Die ehemaligen radikal-Frauen und andere ausgestiegene Gruppen beteiligten sich weiter mit Beitr\u00e4gen, insgesamt hatte sich das Zeitungskollektiv jedoch sp\u00fcrbar ausged\u00fcnnt.\n\nBei einer bundesweiten Gro\u00dfrazzia am 13. Juni 1995, vorgeblich im Zusammenhang mit den militanten Gruppen K.O.M.I.T.E.E. und AIZ, mit \u00fcber f\u00fcnfzig Hausdurchsuchungen fielen der Bundesanwaltschaft weitreichende Informationen zu Struktur und Vertrieb der radikal in die H\u00e4nde. Die beschlagnahmten Aufzeichnungen interner Kommunikation und Adressenlisten f\u00fchrten zu mehreren Verhaftungen und zur Flucht einiger Projektmitglieder und ein halbes Jahr sp\u00e4ter zu weiteren Durchsuchungen. Vorgeworfen wurden ihnen Mitgliedschaft in einer kriminellen Vereinigung (damit war die Zeitung selbst gemeint) und Werbung f\u00fcr terroristische Vereinigungen oder auch, wie es in einem drei Jahre sp\u00e4ter publizierten mehrseitigen Flugblatt hei\u00dft: \"ja, in deutscher Ordentlichkeit sogar Steuerhinterziehung\". Nach einem Jahr, genau am 13.6.1996, stellten sich drei der Abgetauchten, ein Vierter im November. Auch im Juni 1996 und danach kam es noch zu Hausdurchsuchungen aufgrund der Abo-Listen. Im Herbst wurde von Seiten der Bundesanwaltschaft sogar versucht, die ins Internet gestellten Seiten der radikal zu blockieren, was jedoch nur dazu f\u00fchrte, dass diese von Privatpersonen und dem Provider \"gespiegelt\" wurden und zeitweise zu den meistge\u00f6ffneten im Netz geh\u00f6rten.\n\nZudem erfuhr die radikal eine ganz gro\u00dfe Solidarit\u00e4tswelle. Neben den diversen und beliebten Spendenaktionen und Veranstaltungen erschien im Fr\u00fchjahr 1997 ein Soli-Heft Dokumentation kriminalisierter Hefte von einer Herausgeber_innengruppe, die prominente Namen von Rechtsanw\u00e4lt_innen, Wissenschaftler_innen, Journalist_innen und Abgeordneten verzeichnete, unter anderem Elmar Altvater, Peter Grottian, Jutta Ditfurth und Angela Mar-quardt, au\u00dferdem zahlreiche linke Verb\u00e4nde und Vereinigungen. Im Vorwort erkl\u00e4rten die Herausgeber_innen: \"Unabh\u00e4ngig davon, ob wir den Inhalt der Zeitschrift guthei\u00dfen oder ablehnen, wenden wir uns entschieden gegen den \u2013 wiederholten \u2013 Versuch, eine missliebige Publikation zum Schweigen zu bringen. [...] Die Auseinandersetzung um publizierte Thesen \u2013 auch, wenn sie diese Gesellschaftsform kritisieren \u2013 darf und soll nicht mit staatlicher Repression gef\u00fchrt werden. Der Versuch der Bundesanwaltschaft, die Redaktion einer Zeitschrift zu einer kriminellen Vereinigung zu erkl\u00e4ren, ist der Versuch, das Recht auf Meinungsfreiheit auszuh\u00f6hlen.\"\n\nDas Heft enth\u00e4lt Texte aus den letzten Ausgaben der radikal, die aufgrund der Themen (z. B. \"Wie erkenne ich eine Briefbombe?\") oder der beteiligten Gruppen (\"Anschlagserkl\u00e4rung der Rote Zora\"; \"Interview mit einer militanten gruppe\") kriminalisiert wurden. Au\u00dferdem wurde eine Unterst\u00fctzer_innen-Zeitung namens radikale zeiten von einer Solidarit\u00e4tsgruppe herausgegeben.\n\nDennoch war die erneute und massive Repressionswelle ein schwerer Schlag f\u00fcr die radikal. W\u00e4hrend die Justizverfahren liefen, erschienen bis Juni 1996 nur zwei weitere Ausgaben, ver\u00f6ffentlicht von \"alten\" Redaktionsmitgliedern, die nicht im Zugriff der Bundesanwaltschaft standen, um, wie es in dem Flugblatt sp\u00e4ter hei\u00dft, \"die Kontinuit\u00e4t aufrecht zu erhalten\", aber ob und wie es danach weitergehe, sei noch unklar. Auf keinen Fall solle zuk\u00fcnftig der Druck des regelm\u00e4\u00dfigen Erscheinens bestehen bleiben.\n\nBereits in der Nr. 150 hatte sich ein Beitrag mit dem \"Abtauchen\" und Leben in der Illegalit\u00e4t besch\u00e4ftigt \u2013 schon damals hatten einige radikal-Leute bemerkt, dass sie observiert wurden, ahnten aber noch nicht, dass die Mitarbeit an der Zeitung der Grund daf\u00fcr war. Im Heft 153\/2 besch\u00e4ftigt sich nun aus aktuellem Anlass der Artikel \"Tausend Augen\" mit \"Observation \u2013 und was du dagegen tun kannst\", ein weiterer mit dem Leben im Exil und ein anderer mit Aussageverweigerung. Ein Jahr nach dem gro\u00dfen Schlag erschien die Nr. 154 und berichtet im Intro: \"Als Ergebnis mussten sie schlie\u00dflich im Dezember bei der Aktenschlie\u00dfung kleinlaut feststellen, dass sie keinerlei Verbindungen zu anderen Organisationsans\u00e4tzen belegen k\u00f6nnen. Die BAW [Bundesanwaltschaft] entschlie\u00dft sich aufgrund der fehlenden Verbindung zu RAF, RZ etc. der radikal das G\u00fctesiegel 'Unterst\u00fctzung von terroristischen Gruppen' wieder zu entziehen und es bei profaner 'Werbung' zu belassen. Weiterhin bleibt aber der Vorwurf, wir seien eine 'kriminelle Vereinigung'\".\n\nEin neues Thema in dieser Ausgabe war der Internetauftritt der radikal, dazu lieferte sie auch gleich den passenden Computerkurs \"Sicheres Arbeiten mit dem Computer \u2013 2. Teil\" und zeigte sich damit, was die Nutzung des digitalen Technikfortschritts angeht, ihrer Zeit weit voraus.\n\nDann wurde es sehr lange Zeit still um die radikal. Zwar erschien das Soli-Heft im Fr\u00fchjahr 1997, mit der Herausgabe hatte die eigentliche radikal jedoch nichts zu tun. In der Zwischenzeit entstand auch das Buch 20 Jahre radikal \u2013 Geschichte und Perspektiven autonomer Medien, das verschiedene Aspekte zur Entwicklung der Zeitung aus der Sicht einiger Insider darstellt. Das oben bereits erw\u00e4hnte Flugblatt Ein Zusammenhang der radikal zum 13.6.1995, dem Davor und Danach wurde erst im M\u00e4rz 1998 ver\u00f6ffentlicht und gab konkrete Informationen zu Hintergr\u00fcnden und Fortgang der Repression von 1995, um Misstrauen, Spekulationen und Ger\u00fcchten entgegenzutreten und weiteren Schaden in der Szene zu vermeiden, der durch fortgesetztes Stillschweigen zu entstehen drohte. Die Verfahren gegen insgesamt neun Angeklagte waren mittlerweile gegen Zahlung von Geldbu\u00dfen eingestellt worden und, so hei\u00dft es: \"Der Versuch der BAW, die radikal juristisch als kriminelle Vereinigung dingfest zu machen, ist misslungen.\" Da seit der M\u00f6glichkeit zur Akteneinsicht klar geworden war, welches Ausma\u00df an Informationen der BAW zugegangen ist, k\u00f6nne nun nichts mehr verraten werden, was der Gegenseite nicht schon bekannt sei. Ein zweites, noch ausf\u00fchrlicheres Heft zu den Ereignissen und zur Geschichte der radikal erschien sp\u00e4ter unter dem Titel Ehemalige Mitarbeiter der radikal zum 13.6.1995, dem Davor und Danach und arbeitete die Geschehnisse detailliert auf.\n\n### Reset, Neustart \u2013 und Flop\n\nTote tragen keine karos! lautet der Untertitel einer trotz allem im Mai 1998 herausgegebenen radikal Nr. 155. Nachdem sich also die \"Alten\" mit der letzten Publikation verabschiedet hatten, trat nun eine neue Redaktion an, \"die Fu\u00dfstapfen dieses traditionsschweren Projekts weiterzuf\u00fchren\", wie die etwas verungl\u00fcckte Formulierung auf Seite drei verk\u00fcndet. Das Intro wird, schauderhaft um Witzigkeit und Ironie bem\u00fcht, so weitergef\u00fchrt: \"Wir sind noch relativ jung, aber nicht so richtig, denn wir haben eine Menge Erfahrung. Wir sind Danger-Seekers. Gefahrensucher. Echte Adrenalin-Junkies. Wir sind st\u00e4ndig auf der Suche nach dem ultimativen Kick, dem absoluten Nervenkitzel. Kennengelernt haben wir uns auf dem Hauptbahnhof. Wir waren unterwegs ins Dynamo-Stadion, die M\u00e4dels als Cheerleaders, die Jungs nat\u00fcrlich als Hools. Den M\u00e4dels fiel auf, dass ihre P\u00fcschel denselben rosaroten. Farbton hatten wie der Iro von Ratte, unserm Punky. Tja, und so lernten wir uns kennen und lieben. Seither sind wir ein unschlagbares Team. St\u00e4ndig suchen wir das neuste Abenteuer. Wir haben schon so ziemlich alles ausprobiert: Bungee-Jumping, Airbagging, Subway-Surfing, Skinhead-Bashing, Drive-by-Shooting, Herzklappen-Piercing... Und irgendwann nach dem 13.6.95 wurde unser Top-favorite-Adrenaline-Kick radikal. Das bringt volle Kanne fun, so was hatten wir noch nie! Zum Beispiel mussten wir uns schon oft mit Detektiven, Wachsch\u00fctzern oder Bullen rumschlagen, aber bei der radikal hat das alles andere Dimensionen. Mein lieber Scholli, das bringt unsere Herzen erst so richtig in Wallung...\"\n\nWer das Heft jetzt nicht entnervt zuklappt, erf\u00e4hrt auf der n\u00e4chsten Seite in einem ertr\u00e4glicheren Tonfall, dass sich die neue Redaktion aus \"Frauen und M\u00e4nnern aus unterschiedlichen politischen und sozialen Zusammenh\u00e4ngen\" zusammensetzt, die \"vor der Situation [stehen], ein Bewegungsblatt ohne Bewegung zu sein\": \"Vielleicht kann mensch bei der Antifa oder den Auseinandersetzungen um Gorleben ansatzweise von Bewegungen sprechen, aber von einer linksradikalen\/autonomen Bewegung mit einer Vielfalt an Debatten und einer nicht nur punktuellen Mobilisierungsf\u00e4higkeit merken wir herzlich wenig.\"\n\nTrotzdem, so die Redaktion, soll die radikal als Forum erhalten bleiben, um \"das Politische jenseits staatlicher Legalit\u00e4tsgrenzen zu formulieren, zu debattieren und militant zu erstreiten\". Die \"neuen Wege\", die diese Redaktion beschreiten wollte, klingen insgesamt nicht viel anders als die Ans\u00e4tze bisheriger Redaktionen. Daf\u00fcr wartet dann aber der erste Artikel mit einer \u00fcberraschenden und gelungenen Analyse von Science-Fiction-Filmen aus linker Perspektive auf. Das neue Heft zeigt sich auch sonst so vielf\u00e4ltig wie fr\u00fchere Ausgaben. \"Ein Beitrag von Frauen\" besch\u00e4ftigt sich mit Chipkarten und Daten\u00fcberwachung, diverse Befreiungskampfthemen (IRA, PKK, Kurdistan, Chiapas) werden fortgesetzt, die Tipps und Tricks betreffen Abh\u00f6rung \u00fcber Telefone, eine Bauanleitung erkl\u00e4rt das Herstellen von Hakenkrallen. F\u00fcr Antifa-Infos gibt es eine gesonderte Spalte und eine neue Rubrik namens \"Anschlagsbrett\" sollte etabliert werden und forderte zu Leser_innen-Beitr\u00e4gen auf. Ein gr\u00f6\u00dferer Artikel unter dem Titel \"Identit\u00e4t und Kampfkultur\" besch\u00e4ftigt sich mit grundlegenden Fragen zu Militanz. Dazu gibt es Anschlagserkl\u00e4rungen militanter Gruppen ebenso wie Gedichte und, wie fr\u00fcher auch, kleine Pointen in Form von Cartoons.\n\nEin Meilenstein wird ganz am Schluss pr\u00e4sentiert: Die Aufl\u00f6sungserkl\u00e4rung der RAF, eingef\u00fchrt mit einer nicht unkritischen Anmerkung einiger Redaktionsmitglieder: \"Nun ist sie also da, die Aufl\u00f6sungserkl\u00e4rung der RAF. Sie beinhaltet als einen wesentlichen Aspekt die Aussage, dass es nach 1977 innerhalb des Projekts Bem\u00fchungen gab, sich anderen linksradikalen Gruppen anzun\u00e4hern, um eine gemeinsame Handlungsbasis zu erreichen. Dass all diese Versuche, zun\u00e4chst im Frontkonzept umgesetzt, trotzdem relativ starr an den Vorgaben der Guerilla, am 'alles oder nichts' und an der milit\u00e4rischen Aktion festhielten und deshalb fehlschlugen, hatten auch diejenigen festgestellt, die sich zu der Generation nach 1984 und damit zu einem gro\u00dfen Teil zu den VerfasserInnen der Aufl\u00f6sungserkl\u00e4rung z\u00e4hlen. [...] Die RAF war in den Strudel des Zerfalls geraten, wie so viele andere Projekte und Gruppen auch. In dieser Situation ist die Erkl\u00e4rung der Aufl\u00f6sung konsequent und richtig. Nach dem langen Schweigen ist es sogar erfreulich, dass sie \u00fcberhaupt erscheint und das Verschwinden nicht ein sang- und klangloses ist, wie z. B. das der restlichen revolution\u00e4ren Zellen, die vor Jahren noch vorgaben, weitermachen zu wollen, als sich eine ihrer Gruppen verabschiedete. St\u00f6ren mag, dass die GenossInnen der nun ehemaligen RAF heute Erkenntnisse als ihre eigenen ausgeben, wof\u00fcr vor Jahren ihre solidarischen KritikerInnen als halbherzig, wenn nicht gar als Verr\u00e4terInnen beschimpft worden sind.\" Die Einf\u00fchrung endet mit dem kernigen Gru\u00df: \"Viel Gl\u00fcck im neuen Leben und lasst euch nicht erwischen!\". Und dann machte die radikal wieder \u00fcber ein Jahr lang Pause.\n\nDas Intro der vergleichsweise d\u00fcnnen Ausgabe 156 (31 Seiten im Kontrast zu den 99 der letzten Ausgabe) beklagt das Erstarken des Neoliberalismus und die damit einhergehende Tendenz zur Vereinzelung: \"Vorbei scheint die Zeit, in der mit Lust und Energie an kollektiven Strukturen und Zusammenh\u00e4ngen diskutiert wurde. Der Versuch, sich als Linksradikale eine lebendige Struktur zu erarbeiten, die sich im st\u00e4ndigen Diskussionsprozess befindet, unseren Alltag tr\u00e4gt und politisch zukunftsweisende Positionen erstreitet, scheint gegessen zu sein. Die Absicherung des privaten Gl\u00fccks durch die Lohnarbeit via 40-Stundenwoche (zuz\u00fcglich \u00dcberstunden) erklimmt rasant den obersten Stellenwert, und zwar gesellschaftsweit. Die Zeit f\u00fcr politisches Engagement begrenzt sich zunehmend auf die befriedeten 'Restnischen' sozialer \u00dcberbleibsel am WG-Tisch.\" Zudem werde das Redaktionskollektiv nicht gr\u00f6\u00dfer und komme nicht wirklich voran.\n\nUm dem Neoliberalismus etwas entgegenzusetzen, sollte nun jenseits der Aussicht auf eine Existenz in einer subkulturellen Nische erneut \u00fcber Kommunismus diskutiert und dieser \"neu begr\u00fcndet\" werden. W\u00f6rtlich hei\u00dft es: \"Dem Begriff des Kommunismus eine neue Grundlage zu geben, hei\u00dft, mit ihm, in dem Spannungsfeld von Autonomie und Kollektivit\u00e4t, eigener politischer Zusammenh\u00e4nge und Gesellschaft, eine lebbare Alltagsbasis zu entwickeln.\"\n\nAuch andere redaktionelle Beitr\u00e4ge, dekoriert mit einem tanzenden Pinup, lassen inhaltlich eine v\u00f6llige Planlosigkeit erkennen, und die Zusammenstellung von zwei Artikeln, die in kritischer Absicht Krieg und Militarismus thematisieren, mit der anschlie\u00dfend abgedruckten Anleitung zum Bau eines Sprengsatzes erscheint nahezu grotesk. Angesichts dieses qualitativen Absturzes verwundert es nicht weiter, dass die radikal ihr Erscheinen f\u00fcr die n\u00e4chsten f\u00fcnf Jahre einstellte.\n\n### Die schwarze Serie\n\nEine im Fr\u00fchjahr 2004 herausgegebene Nr. 157 er\u00f6ffnete die sogenannte schwarze Serie, die insgesamt vier Hefte umfasst. Das Cover des ersten ist ganz in schwarz gehalten mit leuchtend orangem Schriftzug und dem Untertitel: Mundorgel f\u00fcr Militante. Ein Praxisheft mit Tipps & Tricks soll es sein und tats\u00e4chlich besteht es neben einem f\u00fcnfseitigen Intro nur aus Bauanleitungen rund um den militanten Bastelbedarf. Dazu wird schlicht erkl\u00e4rt: \"Wir k\u00f6nnen hier keine politische Orientierung anbieten, weder f\u00fcr die, denen die Aktionsfelder abhanden gekommen sind, noch f\u00fcr die, die ihren Aktionsschwerpunkt noch nicht gefunden haben. Das Bed\u00fcrfnis, militant zu handeln, jedoch ist, wie die letzten Wochen und Monate gezeigt haben, so manches Mal weiter verbreitet, als wir bisher angenommen haben.\" Weiter angemerkt wird lediglich, dass militante Aktionen nur als \"Ergebnis einer konkreten politischen Auseinandersetzung\" sinnvoll seien. Das Heft erschien vor dem Hintergrund der seit 2001 in der Berliner Autonomen-Zeitschrift Interim gef\u00fchrten Militanzdebatte, auf die im Vorwort dieser radikal auch verwiesen wird, was die scheinbare Kopflosigkeit dieses \"Praxisheftes\" etwas relativiert.\n\nIm Intro werden im Weiteren dann doch ein paar \"politische Einsch\u00e4tzungen\" gegeben, die kurz die Entwicklungen innerhalb der Szene umrei\u00dfen. Erkennbar ist eine neue Orientierung der radikalen Linken an internationalem Austausch im Zuge von Gipfelmobilisierungen oder No-border-camps, auch bei Protesten gegen Sozialabbau sei die \"fr\u00fcher szenetypische Selbstbezogenheit\" aufgegeben worden und es wird Kooperationsbereitschaft gezeigt: \"So beteiligen sich Linksradikale in einigen St\u00e4dten an den Sozialforen, Umsonst-Kampagnen oder \u00e4hnlichen Initiativen. Selbst bei Attac soll es welche geben. Eine verallgemeinerbare Diskussion um die M\u00f6glichkeiten und Grenzen in solchen B\u00fcndnissen gibt es unseres Wissens noch nicht. Und auch das k\u00f6nnen wir gar nicht leisten. Wir blicken jedoch mit Interesse auf die neuen B\u00fcndnis-Ans\u00e4tze etwa bei feministischen, Crossover-, Gender- oder Queergruppen.\"\n\nAuch in den \"eher klassischen Bereichen der Autonomen\" sei einiges passiert, wie neue Versuche, H\u00e4user zu besetzen, zeigen w\u00fcrden, ebenso in der Antifa, beim Castor-Protest und auf weiteren Gebieten. Sinn der Aufz\u00e4hlung sei es, der verbreiteten Perspektivlosigkeit etwas entgegenzusetzen und nochmals zu betonen: \"Ein militantes Agieren [geh\u00f6rt] f\u00fcr viele Gruppen zu den Optionen politischer Intervention.\" Die kritisch betrachtete Globalisierung ist ein zentrales Thema und das Aufzeigen der zahlreichen damit zusammenh\u00e4ngenden negativen Entwicklungen nicht nur vor der eigenen Haust\u00fcr.\n\nDie gut ein Jahr sp\u00e4ter herausgegebene zweite schwarze Ausgabe \u2013 die Nr. 158, diesmal mit Star Wars-Design und dem Untertitel M\u00f6ge die Nacht mit Euch sein in Gr\u00fcn abgesetzt \u2013 vertieft die im letzten Heft angesprochenen Inhalte. Der angefugte \"Praxisblock\" fiel verh\u00e4ltnism\u00e4\u00dfig klein aus. Zeitgem\u00e4\u00df waren \"Einige n\u00fctzliche Informationen zum Umgang mit Mobiltelefonen\", eher \"historisch\" der angeblich simple Sprengsatz \"Ein Klassiker namens 'Nobelkarossentod'\". Pr\u00e4sentiert wurde ein Interview mit der seit 2001 unter diesem Namen existierenden militanten gruppe (mg), die den Ansto\u00df f\u00fcr die Militanz-debatte in der Interim gegeben hatte und sich weder bei den Autonomen noch bei den Antiimps verorten lassen wollte. In ihrer langatmigen Selbstdarstellung formuliert die mg ihre Positionierung als \"Synthese eines Sozialrevolution\u00e4ren und antiimperialistischen Ansatzes auf kommunistischer Grundlage\". Vorgestellt wurde zudem die \"haupts\u00e4chlich von Autonomen initiierte Agenturschlusskampagne\", die bundesweit als Gegenreaktion auf die Einf\u00fchrung von Hartz IV geplant war, aber nur m\u00e4\u00dfige Mobilisierung hervorrief. Ein weiterer Beitrag thematisiert den Fall eines der letzten Dinosaurier aus der Zeit des H\u00e4userkampfs in Form eines Interviews mit ehemaligen BewohnerInnen des im Sommer 2005 ger\u00e4umten Berliner Hausprojekts Yorckstra\u00dfe 59.\n\nAuch die im Fr\u00fchjahr 2006 erschienene Nr. 159 (schwarz mit \"Street-fighter\"-Design und Schriftzug in pink: make capitalism history) befasst sich schwerpunktm\u00e4\u00dfig mit Globalisierungskritik und Gipfelmobilisierungen \u2013 im Jahr 2007 k\u00fcndigte sich der G8-Gipfel in Heiligendamm an \u2013 und der Militanzdebatte. In diesem Rahmen wird eine Antwort der Autonomen Gruppen (AG) auf das Interview mit der mg abgedruckt und ein sich damit \u00fcberschneidender Text der mg, der Fragen an die AG formuliert. Im Intro wurde schon darauf hingewiesen, dass die radikal nicht \u00f6fter als im Schnitt einmal im Jahr erscheinen k\u00f6nne. Die damit verbundenen Aktualit\u00e4tsprobleme w\u00fcrden durch andere Medien wie der w\u00f6chentlich erscheinenden Interim oder das Internet aufgefangen, eine Rubrik wie die OLGA werde es daher nicht mehr in der radikal geben.\n\nDie Ausgabe vom Winter 2006 gl\u00e4nzt in schwarz und gold: 30 Jahre radikal. Neben den obligatorischen Themen Militanz und G8 besteht ein Gro\u00dfteil des Heftes aus gek\u00fcrzten Nachdrucken von Teilen des Buches 20 Jahre radikal. Ein weiterer Artikel befasst sich mit den Repressionen gegen\u00fcber dem Autonomen Zusammenschluss (AZ) in Magdeburg. Im historischen R\u00fcckblick wird das zwischenzeitlich geplante Vorhaben der radikal-Redaktion angesprochen, sich verst\u00e4rkt den \"sozialen K\u00e4mpfen\" und einer \"Wiederbelebung des guten alten Internationalismus\" zu widmen, anstatt an einer \"eingefahrenen Nischenpolitik\" mit der \"Welt der besetzten H\u00e4user, Wagenburgen und Infol\u00e4den\" als Bezugspunkt festzuhalten. Trotzdem sei die Zeitung noch szeneorientiert geblieben und auch manch andere Widerspr\u00fcche seien nicht aufgel\u00f6st worden. Mit dieser 160. Ausgabe wurde die schwarze Serie beendet.\n\n### Quo vadis, radikal?\n\nNach diesmal zweieinhalb Jahren Pause erschien im Sommer 2009 eine Nr. 161, \u00e4u\u00dferlich mit einem Schutzumschlag perfekt getarnt und innen mit dem Untertitel Publikation der revolution\u00e4ren Linken. Die Zeitung pr\u00e4sentiert sich noch textlastiger als jemals zuvor und wird wiederum von einem neuen Redaktionskollektiv herausgegeben, das sich selbst als Revolution\u00e4re Linke (RL) vorstellt: \"Wir sind ein Zusammenschluss von Gruppen und AktivistInnen, die aus den verschiedenen Str\u00f6mungen der revolution\u00e4ren Linken kommen und auch auf unterschiedlichen Ebenen aktiv sind. Als Zusammenhang entstanden sind wir durch langj\u00e4hrige Diskussionen um und \u00fcber die verschiedenen Beitr\u00e4ge der Militanzdebatte der letzten Jahre in Interim und der Radi. Was uns zusammengef\u00fchrt hat, ist der gemeinsame Wille, das in der Debatte Diskutierte in die Praxis umzusetzen und den widerstandsebenen\u00fcbergreifenden revolution\u00e4ren Aufbauprozess konkret voranzutreiben. Wir haben uns als Organisierung ,Revolution\u00e4re Linke (RL)' konstituiert. Als diese sind wir in erster Linie bestrebt, die Str\u00f6mungen der revolution\u00e4ren Linken mitzugestalten, voranzutreiben und eine kontinuierliche Weiterentwicklung abzusichern. [...] Unser strategischer Ansatzpunkt zielt auf die Schaffung und permanente Ausdehnung proletarisch-revolution\u00e4rer Autonomie auf allen Ebenen und in allen Lebensbereichen, das hei\u00dft auf politisch-ideologischem, kulturellem, organisatorischem usw. Gebiet. Es geht uns darum, durch die Etablierung eigener revolution\u00e4rer Werte, Normen und Strukturen die Gegenmacht von unten aufzubauen. Durch das St\u00e4rken der eigenen Seite soll das Kr\u00e4fteverh\u00e4ltnis zwischen den Klassen dahingehend verschoben werden, dass die herrschenden Ausbeutungs- und Unterdr\u00fcckungsstrukturen aufgehoben und der Aufbau einer klassenlosen Gesellschaft erm\u00f6glicht werden.\"\n\nStatt einer Auslandsanschrift gab es nun eine E-Mail-Adresse. Redaktionelle Beitr\u00e4ge sind mit dem K\u00fcrzel RL unterzeichnet, manche zudem mit den markigen Zeilen: \"F\u00fcr den revolution\u00e4ren Aufbauprozess! F\u00fcr den Kommunismus!\". Das Anliegen der Redaktion wird ansonsten so formuliert: \"Wie ihr euch sicher vor dem beschriebenen Hintergrund vorstellen k\u00f6nnt, wird sich das Gesicht der radikal ver\u00e4ndern. Wir wollen uns mit den neuen Ausgaben von der \"schwarzen Reihe\" vor allem dadurch unterscheiden, dass die radikal mehr als bisher einer politischen Linie folgen und aktiv eigene Ans\u00e4tze entwickeln soll, also mehr als ein 'blo\u00dfes' Zur-Verf\u00fcgung-stellen eines unzensierten Diskussionsraumes.\" \"Im besten Fall\", so hei\u00dft es auch, wird die radikal \"der kollektive Agitator und Organisator einer revolution\u00e4ren Linken\".\n\nGeplant ist, so das Intro, in Anlehnung an das Geschichtsprojekt der mg eine Reihe mit Beitr\u00e4gen zur Widerstandsgeschichte zu ver\u00f6ffentlichen, angefangen mit der US-amerikanischen Stadtguerilla Weather. Eine weitere Serie solle unter dem Titel \"F\u00fcr ein revolution\u00e4res Leben\" firmieren und sich \"mit den Kernfragen der subjektiven Rolle und Weiterentwicklung im revolution\u00e4ren Aufbauprozess besch\u00e4ftigen\", als zentrale Ansatzpunkte w\u00fcrden \"Kaderbildung, Schulung und Basisarbeit\" betrachtet, dazu notwendig seien \"Verantwortung und Disziplin\".\n\nEin weiterer Beitrag pr\u00e4sentiert die Aktionsform der \"Stencils\", anhand von Schablonen erstellte Bilder, die im Rahmen von \"klandestin-militanter Agitation und Propaganda\", kurz: \"Miliprop\", Verwendung finden k\u00f6nnten.\n\nDas Schwerpunktthema des Heftes bildet jedoch die mg. Deren Abschlussbeitrag zur Militanzdebatte ist bereits \u00e4lteren Datums, aber dazu gibt es noch ein neueres Interview mit der mg \u2013 und auch hier meint man am Ende des Textes die roten Fahnen f\u00f6rmlich schwingen zu sehen. Zun\u00e4chst gibt die mg im Interview jedoch Anschlagserkl\u00e4rungen ab (Ziele waren \"Einrichtungen der Sozialtechnokratie\" wie Geb\u00e4ude eines Sozialgerichts und eines Jobcenters ebenso wie \"imperialistisches Kriegsmaterial\"), aber auch ihre Aufl\u00f6sung bekannt \u2013 oder vielleicht auch nicht, die Ansagen sind diesbez\u00fcglich etwas nebul\u00f6s. Betont wird aber, \"dass der uns zugeschobene versuchte Brandanschlag auf BW-LKWs am 31.7.2007 in Brandenburg\/Havel nicht in unserem 'Planungsb\u00fcro' ersonnen und von dort ausgef\u00fchrt wurde\", mit der selbstbewussten Erkl\u00e4rung: \"Das, was uns beinahe kr\u00e4nkt, ist, dass man uns f\u00fcr so dusslig h\u00e4lt, alle Normen einer klandestinen Aktion zu verletzen, um schnurstracks in eine vorbereitete Bullenfalle zu tappen.\"\n\nDeutliche Kritik wird von Seiten der mg an der Praxis des innerst\u00e4dtischen \"Wagensports\" ge\u00e4u\u00dfert, die, obwohl der Widerstand gegen Gentrifizierung eines ihrer zentralen Anliegen sei, dann doch nicht als Mittel der Wahl angesehen werde: \"Au\u00dferhalb jedes Vermittlungsverh\u00e4ltnisses steht dabei die narzisstische Brandsatzlegerei des 'Nobelkarossentods'. Im 'Militantismus' ist zumindest noch eine Restspur von Politik zu entdecken, wenn auch als Ersatzhandlung, aber die allabendliche alternative Freizeitbesch\u00e4ftigung des automobilen Herumz\u00fcndelns tr\u00e4gt nur noch zur Diskreditierung von militanten und klandestinen Aktionsformen bei. \u00dcber eine gewisse Zeit haben wir diese Form des n\u00e4chtlichen Streifzuges noch als eine Lektion des zu erlernenden Umgangs mit Brandsatzmitteln f\u00fcr uns als annehmbar zu interpretieren versucht. Das gelingt uns seit geraumer Zeit nicht mehr; die negativen Begleitumst\u00e4nde dieses falschen Feuerzaubers sind f\u00fcr uns zu m\u00e4chtig (relative Wahllosigkeit der 'Objektauswahl', verkohlte 'Beif\u00e4nge' von daneben abgestellten PKWs, selbst ge\u00f6ffnetes Einfallstor f\u00fcr eine breite mediale Hetze etc.), als dass wir dieses sinn-lose Tun tolerieren wollen.\"\n\nDie Neuorientierung der radikal stie\u00df bei der Leser_innenschaft nicht nur auf Begeisterung. Die vehementesten Kritiken unterstellten gar, sie w\u00fcrde nun von BKA-Leuten verfasst werden. Ebenso schien Protest von \"alten\" radikal-Macher_innen gekommen sein, wie das Vorwort der n\u00e4chsten Ausgabe erkennen l\u00e4sst: \"Was uns wirklich \u00fcberrascht hat, waren die Reaktionen einiger ehemaliger GenossInnen, die sich nicht zu dumm waren, f\u00fcr die radikal ein Copyright zu beanspruchen und uns einen 'Retro'-Vorwurf unterzujubeln.\" Den Vorwurf, eine M\/L-Truppe oder \"Scheuklappen-Linke\" zu sein, weist die Redaktion weit von sich. Ihre Basis sei \"keine auswendig gelernte Lehrbuchmeinung, sondern orientiert sich an den Notwendigkeiten eben dieses revolution\u00e4ren Prozesses\". Wiederholt wird erkl\u00e4rt: \"Unsere inhaltliche Linie gr\u00fcndet sich auf der Dialektik eines sozialrevolution\u00e4r-klassenk\u00e4mpferischen (also antikapitalistisch, antirassistisch und antipatriarchal) und antiimperialistisch-internationalistischen Ansatzes mit der klar definierten Zentralperspektive des Kampfes f\u00fcr den (libert\u00e4ren) Kommunismus. Proletarischer Klassenstandpunkt und proletarischer Internationalismus sind vor dem Hintergrund einer kommunistischen Befreiungsperspektive untrennbar, ein unaufl\u00f6sliches Band der eigenen Politik als revolution\u00e4re Linke.\"\n\nNicht hinterfragt wurde dabei offenbar, ob reale Vertreter_innen des Proletariats willens und in der Lage sind, ein derartiges Geschwurbel zu lesen. Dennoch hei\u00dft es von Seiten der RL, sie verstehe die radikal \"als M\u00f6glichkeit, unsere Positionen und Ans\u00e4tze unter die Leute zu bringen\". Auch was das Layout angeht, entspricht die radikal mittlerweile dem Begriff der \"Bleiw\u00fcste\", wof\u00fcr fr\u00fchere Redaktionen sich bei einzelnen, etwas klobigeren Artikeln immer wieder entschuldigt haben. Es gibt kaum Bilder, keine Farbe oder etwas originelleren Textsatz, der das Lesen auflockern w\u00fcrde. Einzige Ausnahme ist ein Art Newsticker, der sich in Form eines fortlaufenden grauen Streifens am Rand der Heftseiten befindet und Informationen zu militanten Aktionen aus dem In- und Ausland enth\u00e4lt. Allerdings werden hier nur stark verk\u00fcrzt Datum, Ort, Mittel, Ziel und bestenfalls ein kurzer Slogan aufgelistet, Hintergr\u00fcnde und Anl\u00e4sse der Anschl\u00e4ge bleiben oft im Unklaren.\n\nEine spezielle Form von Design findet sich daf\u00fcr in der Ausgabe 162 bei den. Kommuniques der RL anl\u00e4sslich des Aufrufs zum Wahlboykott oder zum \"Streik als Mittel des Klassenkampfes\". Hier wurde das RL-Emblem in Form von Hammer und Sichel plus Stern gestylt. Insgesamt hinterlassen die Beitr\u00e4ge der RL den Eindruck, dass, w\u00fcrde man die Begriffe \"Klasse\", \"Proletariat\" und \"kommunistisch\" herausstreichen, h\u00f6chstens noch der halbe Text stehen bliebe. Dieser w\u00e4re dann aber vielleicht lesbarer. Und w\u00fcrde das leidige und endlose Thema Militanz etwas k\u00fcrzer gefasst werden, w\u00e4re m\u00f6glicherweise auch wieder Raum f\u00fcr die sogenannten Nebenwiderspr\u00fcche, zu denen erl\u00e4utert wird: \"Unser theoretischer Ausgangspunkt ist es nicht, die Existenz eines alles \u00fcberlagernden '.Hauptwiderspruches' (Kapitalismus) auszurufen, mit dessen. Aufl\u00f6sung dann alle anderen 'Nebenwiderspr\u00fcche' (Rassismus und Sexismus) automatisch verschwinden. Vielmehr gehen wir von der Verzahnung der verschiedenen eigenst\u00e4ndigen Widerspr\u00fcche aus, wobei wir den Kapitalismus\/Imperialismus aktuell als den dominantesten betrachten \u2013 in dem Sinne, dass dieser alle anderen aufs engste mitbedingt und beeinflusst.\" Ein schwacher Versuch, den Begriff der Dominanz aufzuweichen \u2013 und die Heftinhalte sprechen zudem eine deutliche und selbstherrliche Mackersprache, auch wenn ein bisschen \"mensch\"-statt-\"man\"-Makeup aufgetragen wird.\n\nZur Umsetzung z\u00fcndender Ideen, wie dem Legen eines Brandsatzes am Geb\u00e4ude der Agentur f\u00fcr Arbeit in Berlin-Wedding \u2013 eine entsprechende Erkl\u00e4rung der Revolution\u00e4ren Aktionszellen (RAZ) befindet sich in Heft 162 \u2013, wird eine Bastelanleitung f\u00fcr das \"Modell Gasaki\" abgedruckt, was wiederum f\u00fcr einen Beschlagnahmungsbeschluss gegen\u00fcber der radikal sorgte. Mit weiteren Anschlagserkl\u00e4rungen warten auch die Ausgaben 163 und 164 auf. Neben unendlichen Beitr\u00e4gen zum Thema Militanz gibt es ebenso ausufernde Selbstdarstellungstexte der RL. Aus Anlass eines aktuellen Prozesses wird in der Nr. 163 vom Sommer 2010 sogar das Thema RAF wieder aufgew\u00e4rmt und der wieder sehr kreative Tarnumschlag k\u00fcndigt \"Die roten R\u00e4cher\" an.\n\nBemerkenswert ist der Artikel \"Wir m\u00fcssen uns mehr einmischen!\" der Hamburger antimilitaristischen FrauenLesbenTransgender-Gruppe AMIGA, die sich als \"linke autonome radikale Feministinnen\" beschreibt und mit den Verkn\u00fcpfungen von Militarismus mit patriarchalen, rassistischen und global-kapitalistischen Strukturen auseinandersetzt. Der Text ist informativ, gut strukturiert, kompakt und lesbar und kommt dankenswerterweise ohne rote F\u00e4hnchen und inflation\u00e4re Verwendung der oben genannten Begriffe aus. Vermutlich verdankt er seinen Abdruck dem Umstand, dass er ein Zitat von Rosa Luxemburg enth\u00e4lt, lie\u00dfe sich zumindest boshaft spekulieren. Dieser Text, darauf wird im Vorfeld verwiesen, ist auch im Rahmen einer neuen autonomen Zeitung namens autonomes Bl\u00e4ttchen \u2013 gegen Repression und f\u00fcr antimilitaristische Praxis erschienen. Ohne weiteren Kommentar dagegen wird ein Beitrag aus der Interim ver\u00f6ffentlicht, der sich mit der jetzigen Ausrichtung der radikal kritisch auseinandersetzt.\n\nDie Ausgabe 164 aus dem Jahr 2011 r\u00fcckt das Thema Militanz zwar nicht g\u00e4nzlich in den Hintergrund, bietet aber immerhin inhaltlich etwas mehr Abwechslung, wenn auch \u2013 wie im Interim-Text der letzten Ausgabe treffend formuliert \u2013 nach wie vor in un\u00fcbersehbarer \"Rotf\u00e4rbung\". Die Beitr\u00e4ge sind zum Teil historisch orientiert, wie die Artikel zum Zeitungsprojekt FIZZ aus den 1970er Jahren, zur italienischen Organisation Lotta Continua oder zum Sozial-Rebellentum. Andere betreffen aktuelle Entwicklungen im Kontext von Stadtteilpolitik, Zust\u00e4nden in der \"Provinz\" oder den Libyen-Krieg.\n\nMit einem zweiten Teil zum Sozial-Rebellismus, Texten zu Argentinien und zur \"Occupy\"-Bewegung ebenso wie ''Tipps & Tricks im H\u00e4userkampf' zum Thema \"Enteignung und Vergesellschaftung von Wohnraum\" wurde 2012 eine 165. radikal herausgegeben. Auch hier bleibt das Thema Militanz in vielen Facetten pr\u00e4sent \u2013 angefangen von der \"Massenmilitanz und den Aufstandsperspektiven der revolution\u00e4ren Linken\", \u00fcber RAZ-Texte und bis hin zu der Frage, ob \"die Barrikade ein \u00fcberholtes Kampfmittel\" sei.\n\n### Lebt und lest radikal?\n\nDie radikal als Sprachrohr der autonomen Szene \u2013 ist sie das noch? Sie war es sicherlich in den 1980er Jahren, vor allem in der ersten H\u00e4lfte, als die Westberliner Hausbesetzungswelle auf ihrem H\u00f6hepunkt war. Auch in den 1990ern hatte sie als \u00fcberregionales Szene-Medium noch Relevanz und lie\u00df sich selbst durch Repression nicht unterkriegen. Die radikal diente jedoch nicht nur als Sprachrohr. Ebenso stie\u00df sie aktiv Diskussionen zu neuen oder sonst kaum angesprochenen Themen an, brachte informative und inhaltlich fundierte Beitr\u00e4ge wie die \"Gegen das Vergessen\"-Serie und reflektierte zentrale Debatten \u2013 nicht unparteiisch, aber auch nicht einstimmig oder unkritisch. Trotz inhaltlicher Schwerpunktwechsel begleitete sie die linke Szene und forderte immer wieder aufs Neue sowohl Widerspruch als auch Solidarit\u00e4t heraus. Phasenweise hatte sie sich Antiimperialismus oder Antifaschismus, Patriarchats- oder Rassismus-Kritik gr\u00f6\u00dfer auf ihre Fahnen geschrieben, zu anderen Zeiten fehlten genau diese Ans\u00e4tze wieder.\n\nKontinuit\u00e4t bewies und beweist die radikal bis heute an ihrem umstrittensten Punkt, der Haltung zur Militanz, welche sie zwar nicht uneingeschr\u00e4nkt bef\u00fcrwortet(e), aber auch nie ganz ablehnte. Vor allem war diese Zeitung ein Forum f\u00fcr Meinungen und Informationen, die sonst nirgends gedruckt wurden. Und auch wenn sp\u00e4ter durch die M\u00f6glichkeiten des Internet ein Vielfaches auch an brisanten Meldungen einstell- und abrufbar wurde, so blieb die radikal doch ein Kristallisationsort, an dem solche Nachrichten geb\u00fcndelt wurden und nicht in der virtuellen Datenmasse untergingen. \u00dcber den (Un-)Sinn des Abdruckens praktischer Anleitungen zu militanten Aktionen kann man sicherlich geteilter Meinung sein, zumal die h\u00e4ufige Fehlerhaftigkeit diese noch gef\u00e4hrlicher machte als urspr\u00fcnglich beabsichtigt. Dass \"die Vermittlung von Techniken und Know-how immer noch zum Selbstverst\u00e4ndnis des Projektes geh\u00f6rt\", wie es in der Jubil\u00e4umsausgabe hei\u00dft, erscheint als Rechtfertigung reichlich platt, die Annahme, dass kaum jemand durch diese Anleitungen zum Nachbau inspiriert w\u00fcrde, als Argument etwas lapidar. Die Zeitung aber aufgrund des Vorwurfs der Bildung einer kriminellen Vereinigung zu verbieten, hatte eher den Effekt, dass ein Mythos um sie entstand (ob ihr das nun recht war oder nicht), als dass sie damit zum Verschwinden gebracht wurde. Vielmehr steigerten der Reiz des Verbotenen und der Ruf der Klandestinit\u00e4t noch die Attraktivit\u00e4t der radikal und f\u00fchrten letztendlich zu der Z\u00e4higkeit, mit der immer wieder neue Redaktionsgruppen und stetige Leser_innenresonanzen sie auch nach Jahrzehnten noch am Leben erhalten.\n\nEin gr\u00f6\u00dferes Hindernis als die 1984 erfolgte Kriminalisierung scheint f\u00fcr die Weiterexistenz der Zeitung die Fragmentierung der autonomen Szene zur Mitte der 1990er Jahre hin gewesen zu sein. Zwar ergab sich dadurch mit szeneinternen Auseinandersetzungen um \"Widerspr\u00fcche\" und \"Nebenwiderspr\u00fcche\" zun\u00e4chst ein thematischer Dauerbrenner, eine Krise der autonomen Bewegung war dennoch \u2013 oder gerade darum \u2013 deutlich zu erkennen. Die zahlreichen Verweise in der radikal auf Artikel und ausf\u00fchrlichere Diskussionen in der Berliner Autonomen-Zeitung Interim, die am 1. Mai 1988 erstmalig erschien und auch \u00fcberregionale Verbreitung erfuhr, zeigen, dass die radikal nicht die einzige relevante Publikation der autonomen Bewegung war, wenn nicht gar, dass die Interim ihr nach und nach den Rang als zentrales Szene-Medium abgelaufen hat. Gerade die szenerelevante Militanzdebatte in den Nullerjahren wurde in ihrer ausf\u00fchrlichen Breite in der Interim gef\u00fchrt und in der radikal erst mit Versp\u00e4tung nach deren gro\u00dfer Flaute aufgegriffen und lediglich geb\u00fcndelt dargestellt.\n\nDie Ausrichtung schlie\u00dflich, die die radikal mit dem letzten Redaktionswechsel angenommen hat, l\u00e4sst bezweifeln, dass die Zeitung heute noch als Hauptmedium der autonomen Szene betrachtet werden kann. Vielmehr erscheint die radikal heute als eine Art \"RL-Organ\", dass gelegentlich noch Beitr\u00e4ge aus autonomen Zusammenh\u00e4ngen aufnimmt. Die RL selbst schrieb: \"Die radikal ist eine Publikation, die im Namen eines Projektes erscheint und zugleich als publizistisches Sprachrohr einer Str\u00f6mung funktionieren soll.\" Welche Str\u00f6mung auch immer damit gemeint ist, vordergr\u00fcndig gibt sich die Redaktion nach allen Seiten offen, aber der Bruch zu ihren Vorg\u00e4nger_innen liegt auf der Hand. Globalisierungskritische Ans\u00e4tze und Aktionen konnten sich der Unterst\u00fctzung von autonomer Seite aus noch relativ sicher sein, wie die Inhalte der schwarzen Serie zeigen. Dass jedoch autonome Zusammenh\u00e4nge mit ihrer oftmals selbst beklagten Theorie-, Organisations- und Sprachlosigkeit in der end- und distanzlos auf Lenin & Co rekurrierenden RL aufgehen, erscheint fraglich. Wer im Jahr 2014 nach DEM Sprachrohr der autonomen Szene sucht, sollte wohl eher die Augen nach anderen Medien offenhalten.\n\n## ExpertInneninterviews\n\n## \"Ein Maximum an Demokratie ist aus unserer Perspektive nicht erstrebenswert.\"\n\nProf. Uwe Backes, 54 Jahre, Mitbegr\u00fcnder der vergleichenden Extremismusforschung.\n\nDas Interview findet im Hannah-Arendt-Institut f\u00fcr Totalitarismusforschung an der Technischen Universit\u00e4t in Dresden statt. Mein Interviewpartner ist der Politikwissenschaftler Professor Uwe Backes, Mitbegr\u00fcnder der vergleichenden Extremismusforschung.\n\nProf Uwe Backes, Jahrgang 1960, studierte Politikwissenschaft, Geschichtswissenschaft und Germanistik an der Universit\u00e4t Trier. Seit 1999 ist er stellvertretender Direktor des Hannah-Arendt-Instituts f\u00fcr Totalitarismusforschung an der Technischen Universit\u00e4t Dresden. Zugleich ist er Inhaber einer au\u00dferplanm\u00e4\u00dfigen Professur f\u00fcr vergleichende Diktaturforschung am Institut f\u00fcr Politikwissenschaft der TU Dresden.\n\nUwe Backes forscht vor allem in den Bereichen Demokratietheorie, Ideologiegeschichte, vergleichende Extremismusforschung und Diktaturen. Gemeinsam mit Alexander Gallus und Eckhard Jesse gibt er das Jahrbuch Extremismus & Demokratie heraus, dessen 25. Band 2013 bei Nomos\/Baden-Baden erschienen ist.\n\n#### Zun\u00e4chst einmal interessiert mich Ihr Hintergrund. Was sind die inhaltlichen Schwerpunkte Ihrer Arbeit?\n\nDie Perspektive der von mir vertretenen Extremismusforschung unterscheidet sich von verbreiteten psychologischen und soziologischen Ans\u00e4tzen. Mit dem normativen Begriff \"Extremismus\" ist eine spezifische Aufmerksamkeitsrichtung verbunden, die unsere Arbeiten pr\u00e4gt. Bedeutsam in diesem Zusammenhang ist das Konzept der streitbaren Demokratie, wie es in der Verfassung der Bundesrepublik verankert worden ist; dieses Konzept ist aus der Erfahrung mit den beiden Diktaturen in Deutschland entstanden und will dazu beitragen, dass die Demokratie nicht erneut durch extremistische Bewegungen gef\u00e4hrdet wird. Unsere Aufmerksamkeit gilt allen Gef\u00e4hrdungen des demokratischen Verfassungsstaates. Wir sehen den demokratischen Verfassungsstaat als ein historisches Entwicklungsprodukt von hohem Wert an, das erhalten und weiterentwickelt werden soll, und von daher betrachten wir in unserer Arbeit alle Str\u00f6mungen aufmerksam, die grundlegende Werte und Spielregeln der demokratischen Verfassungsstaaten implizit oder explizit negieren. Dabei geraten zun\u00e4chst einmal Bestrebungen in den Blick, die im Zusammenhang mit dem historischen Nationalsozialismus stehen; dar\u00fcber hinaus aber auch andere rechtsextremistische, an autorit\u00e4rer und nationalistischer Staatlichkeit orientierte Konzepte. Auf der anderen Seite gilt unsere Aufmerksamkeit einer Linken, die in ihrem emphatischen Demokratie- und Gleichheitsverst\u00e4ndnis den demokratischen Verfassungsstaat in seiner Institutionenordnung nicht anerkennen kann, weil sie auf utopische, egalit\u00e4re Konzepte der Zukunft hingerichtet ist und dann zwar in ihren Zielsetzungen diametral entgegengesetzte Konzepte wie rechtsextreme Strukturen hat, aber in ihren Mitteln und in ihrer politischen Praxis, wenn sie an die Macht gelangt, sehr \u00e4hnliche Herrschaftsmethoden kultiviert, wie sie aus den rechtsextremen Kontexten bekannt sind.\n\n#### Sie nannten das in einem Ihrer Texte \"geistige Isomorphie im Kern\"...\n\nJa, Isomorphie im Sinne von Gleichf\u00f6rmigkeit, aber wir tun keineswegs das, was uns gelegentlich vorgeworfen wird, den Rechts- und Linksextremismus gleichsetzen. Das w\u00e4re eine Torheit. Dann m\u00fcssten wir ja auch nicht begrifflich zwischen rechts und links unterscheiden. Aber was manche Vertreter der Linken auf die Palme bringt, ist, dass sie durch die Perspektive, die wir einnehmen, auf eine \u00e4hnliche Untersuchungsebene kommen, n\u00e4mlich Bedrohung des demokratischen Verfassungsstaates, wie rechtsextreme Gruppen, obwohl nat\u00fcrlich in den Idealen, im Ethos, im Politikverst\u00e4ndnis grundlegende Differenzen festzustellen sind. Aber das Entscheidende f\u00fcr uns ist eben, dass der demokratische Verfassungsstaat das Geh\u00e4use ist, in dem die Freiheiten der Menschen wirksam gesch\u00fctzt und aufgehoben sind. Das wird oft falsch gesehen, gerade bei linken Kritikern, die sich emphatisch auf die Demokratie berufen, im Sinne einer Gleichheitsvorstellung von Volkssouver\u00e4nit\u00e4t, die aber nicht sehen, dass der demokratische Verfassungsstaat eine Balance darstellt aus demokratischen und verfassungsstaatlichen, rechtsstaatlichen Grundgedanken. Das sind zwei Elemente, die in einem Spannungsverh\u00e4ltnis zueinander stehen. Das hei\u00dft, ein Maximum an Demokratie ist aus unserer Perspektive nicht erstrebenswert, weil die Demokratie auch Grenzen haben muss. Sie findet ihre Grenze im Rechtsstaat, im Verfassungsstaat. Demokratie, verstanden als radikale Gleichheit oder als radikale Realisierung von Volkssouver\u00e4nit\u00e4t, w\u00fcrde ja auch hei\u00dfen, dass quasi alles zur Disposition von Mehrheitsentscheidungen steht, und das ist im Verfassungsstaat eben nicht der Fall. Es gibt einen Verfassungskern, die Idee der Menschenw\u00fcrde, die grundlegenden Rechte des Menschen, der gesch\u00fctzt ist, auch vor Mehrheitsentscheidungen. Der Deutsche Bundestag kann nicht \u00fcber alle Fragen eine souver\u00e4ne Entscheidung herbeif\u00fchren. Im Grundgesetz sind die Artikel 1 und 20 der Ver\u00e4nderung durch den Gesetzgeber entzogen worden, das ist die normative Grundlage des Staates. Die Menschenrechte sind der Kern dieser normativen Grundlage. Der Schutz der Menschenrechte durch Institutionen, Gewaltenteilung durch Institutionen, das ist im Artikel 20 GG normiert. Der als unab\u00e4nderlich geltende Verfassungskern beruht auf einer bewussten Entscheidung der Verfassungssch\u00f6pfer von 1948\/49, die sich damit absetzen wollten von der Weimarer Reichsverfassung, die in all ihren Bestimmungen durch den demokratischen Gesetzgeber abgeschafft, ver\u00e4ndert werden konnte.\n\nDie Interpretation von Teilen der Linken, die meinen, das Grundgesetz sei eine antifaschistische Grundordnung, ist unhaltbar. Als das Grundgesetz beraten wurde, wenn Sie sich die Protokolle des Parlamentarischen Rates ansehen, ist das klar, war da nat\u00fcrlich zun\u00e4chst einmal das Erleiden der nationalsozialistischen Diktatur bestimmend. Die Mitglieder des Parlamentarischen Rates sahen aber auch schon die neue Diktatur im \u00f6stlichen Deutschland heraufziehen: die Gr\u00fcndung der SED 1946, die beginnende Austreibung und Marginalisierung der anderen demokratischen Parteien, die Monopolisierung der politischen Macht, gest\u00fctzt auf die Macht der sowjetischen Milit\u00e4radministration. Denken Sie an die Berlin-Blockade 1948\/49, das war nat\u00fcrlich auch eine Erfahrung, und deshalb ist das Grundgesetz keine rein antifaschistische, sondern ebenso eine antikommunistische, man m\u00fcsste sagen antiextremistische oder antitotalit\u00e4re Verfassung, die sich gegen diese beiden Herausforderungen des demokratischen Verfassungsstaates richtet. Der Islamismus, die religi\u00f6sen Fundamentalismen, die dann sp\u00e4ter auch eine Rolle spielten, waren noch nicht am Horizont der Zeit, aber diese beiden Formen waren es, und unser Ansatz kn\u00fcpft an diese historischen Erfahrungen an und f\u00fchrt sie in die Gegenwart fort. Wir sind keine unkritischen Anbeter des demokratischen Verfassungsstaates, wir sind uns klar dar\u00fcber, dass man alle politischen Formen weiter entwickeln muss, wir m\u00fcssen reformf\u00e4hig sein. Aber es muss eben unter Bewahrung dessen geschehen, was sich als historischer Erfahrungsschatz herausgebildet hat. Wer sagt, wir wollen den Bundestag abschaffen und durch direkte Demokratie ersetzen, der muss zun\u00e4chst den Beweis erbringen, dass die Institutionen, die er vorschl\u00e4gt, auch funktionieren. Und daran hapert es. Diejenigen, die radikale Konzepte entwickeln, also in der radikalen Linken will ich mal sagen, bleiben uns meistens die Antwort darauf schuldig, wie das denn funktionieren soll. In der Kritik sind sie stark, aber auch ma\u00dflos. Diese Ma\u00dflosigkeit erkl\u00e4rt sich zum Teil durch das mangelnde Verantwortungsbewusstsein, was Alternativen angeht, denn ich kann einen bestehenden Zustand nur dann verantwortungsvoll kritisieren, wenn ich eine Vorstellung davon habe, wie ich etwas besser machen kann. Wir haben ja nirgends eine perfekte Ordnung, und wenn ich Kritik \u00fcbe, muss ich mich der Frage stellen, wie kann ich es verbessern? Dann muss ich konkrete Schritte vorschlagen und sagen, so und so machen wir das. Und wenn ich verantwortungsvoll bin, dann sag' ich, wir probieren das mal in einem kleinen Rahmen. Die extreme Linke neigt dazu, gewaltige Sozialexperimente auf ihre Agenda zu stellen, ohne verantwortungsethisch korrekt zu \u00fcberlegen, wie sind die Risiken dieser Experimente, wie sind denn die Aussichten auf Erfolg des Experiments. Die \u00fcben oft scharfsinnige Kritik an Missst\u00e4nden und Fehlentwicklungen in den demokratischen Verfassungsstaaten, aber sie bleiben uns die Antwort komplett schuldig, wie wir denn diese Probleme in den Griff kriegen k\u00f6nnten. Dahinter steckt etwas, was f\u00fcr extreme Linke charakteristisch ist: Die Menschheitsgeschichte wird als eine einzige gro\u00dfe Fehlentwicklung angesehen. Irgendwo ist quasi der Wagen entgleist, und nun wird eine L\u00f6sung, eine Aufl\u00f6sung der Geschichte gesucht. Es ist oft gesagt worden, dass dieses Interpretationsschema von Geschichte viel zu tun hat mit der radikal-christlichen Vorstellung der Erl\u00f6sung. Das kommt urspr\u00fcnglich aus dem Judentum: der Messias, der uns die Erl\u00f6sung bringt. Seit dem radikalen Teil der Aufkl\u00e4rung \u2013 es gibt auch einen gem\u00e4\u00dfigten Teil der Aufkl\u00e4rung, der f\u00fcr Gewaltenteilung eintritt, denken sie an Montesquieu \u2013 haben wir es im Grunde mit einer politisch-messianischen Konzeption zu tun, die auf eine endg\u00fcltige L\u00f6sung des Grundproblems der Menschheit zielt. Die Menschheit ist quasi verknotet und wir m\u00fcssen diesen Knoten aufl\u00f6sen, wir k\u00f6nnen den Knoten aufl\u00f6sen. Ein Christ traditioneller Art w\u00fcrde sagen, dass der Knoten gar nicht aufgel\u00f6st werden kann, weil der Mensch eben ein S\u00fcnder ist. Die Begrenztheit des Menschen verhindert es, dass wir Perfektion im Diesseits erreichen. Der Liberale dagegen ist mehr Empiriker, er wird sagen, dass das mit dem Knoten erst mal probiert werden sollte, dass etwas vorgeschlagen wird nach dem Prinzip \"Trial and Error\". Sie n\u00e4hern sich der L\u00f6sung schrittweise, dann k\u00f6nnen sie immer noch korrigieren. Wenn es dann in die falsche Richtung geht, dann ziehen sie daraus die Konsequenz und steuern um. Aber der Liberale geht nicht davon aus, dass er den Knoten aufl\u00f6sen kann. Diese Sehnsucht nach der Aufl\u00f6sung des Knotens, pr\u00e4gt einen Teil der Aufkl\u00e4rung und der Str\u00f6mungen, die aus ihr hervorgegangen sind. Karl Marx hatte unter seinen Vorfahren Rabbiner, und in der j\u00fcdisch-messianischen \u00dcberlieferung steckt ebenso wie im Christentum, das ja aus dem Judentum hervorgegangen ist, eine Vorstellung die auf Erl\u00f6sung\/Aufl\u00f6sung im Diesseits zielt. Sie wurde allerdings nur von kleinen Gruppen vertreten. Es gab Gruppen, wie etwa die Wiedert\u00e4ufer in M\u00fcnster, die versuchten, das \"neue Jerusalem\" aufzubauen. Aber die Mehrheit der Juden und Christen war immer der Meinung, das ist eine Sache, die wir nur im Jenseits erreichen k\u00f6nnen. Die sozialistisch-kommunistische Bewegung und der Anarchismus sind sich hingegen in ihrem diesseitsreligi\u00f6sen Endziel einig. Die \u00dcberzeugung von der Perfektibilit\u00e4t des Menschen auf der Grundlage einer sehr optimistischen Anthropologie ist hier wegweisend.\n\nUnd hier gibt es durchaus auch strukturelle \u00c4hnlichkeiten mit rechtsextremen Gruppierungen. Es gibt auch sehr viele wesentliche Unterschiede, die man nicht unterschlagen darf, das ist vollkommen klar, nur glaube ich, dass die linksextremen Varianten des politischen Messianismus auf lange Sicht eine gr\u00f6\u00dfere, verf\u00fchrerische Faszinationskraft auf Menschen aus\u00fcben werden, aber vielleicht bin ich zu optimistisch. Denn wenn ich der Meinung w\u00e4re, dass die Rechtsextremisten die gr\u00f6\u00dfere Faszinationskraft aus\u00fcbten, m\u00fcsste ich davon ausgehen, dass das B\u00f6se im Menschen dominant ist. Der Linksextremismus appelliert ja an das Gute im Menschen, er hat nur eben kein Konzept, wie die b\u00f6sen Seiten des Menschen unter Kontrolle gebracht werden k\u00f6nnen. Der liberale Ansatz hat immer auf der einen Seite ein gewisses Vertrauen in den Menschen, dass er anderen als Freien und Gleichen begegnen kann, dass er verhandeln kann, ein Miteinander-Reden und -Diskutieren m\u00f6glich ist. Auf der anderen Seite ist der Liberale aber davon \u00fcberzeugt, dass der Mensch Institutionen braucht, die die Menschen vor sich selber sch\u00fctzen, die zugleich sehr klug arrangiert sein m\u00fcssen, damit von den Institutionen selbst keine Gefahr ausgeht, dass rechtsstaatliche Sicherungen eingebaut werden, \"checks and balances\", wie die Angelsachsen sagen. Der freiheitssichernde Wert dieses komplizierten gewaltenkontrollierende Institutionengef\u00fcges, das manchmal auch \u00fcberkompliziert ist und seine M\u00e4ngel hat, wird von linksextremen K\u00f6pfen nicht gesehen. Man opfert sozusagen den Verfassungsstaat dem politisch-messianischen Heilsprojekt. Aber das ist jetzt die Ebene der gro\u00dfen ideologischen Auseinandersetzungen; wenn wir uns mit den Gruppen der sogenannten Autonomen besch\u00e4ftigen, m\u00fcssen wir in die Niederungen jugendsubkultureller Milieus eintauchen.\n\n#### Dann lassen Sie uns gleich eintauchen: Was wissen wir eigentlich \u00fcber die autonome Szene?\n\nWir wissen \u00fcber die autonome Szene viel weniger, als wir \u00fcber die rechtsextremen Szenen wissen, weil wir dar\u00fcber in den vergangenen zwanzig Jahren wesentlich weniger geforscht haben. Es gibt eine Masse an Forschungen \u00fcber rechtsextreme Gruppen, darunter sehr solide empirische Studien, aber wir haben viel weniger \u00fcber die Autonomen. Es ist aber nicht so, dass wir gar nichts wissen. Das Beste, was wir haben, sind inhaltsanalytische Studien, die sich auf die Auswertungen der Organe der autonomen Szene konzentrieren. Hier liegt ja ein reiches Informationsmaterial vor, weil die Szene eine sprechende Szene ist. Sie teilt uns mehr mit als zum Beispiel viele rechtsextreme Gruppen, die oft eher schweigsam, stumm oder wortkarg sind, w\u00e4hrend die autonome Szene Debatten f\u00fchrt; sie legt Wert darauf, sich zu entwickeln, es ist eine diskutierende Szene.\n\n#### Eher untereinander oder auch nach au\u00dfen?\n\nAuch nach au\u00dfen, denken sie an die Rechtfertigungsschreiben, die regelm\u00e4\u00dfig eingehen, wenn irgendwo ein Sprengstoff- oder Brandanschlag ver\u00fcbt wurde. Es gibt eine F\u00fclle an solchen Versuchen der Rechtfertigung, die dann in den Szeneorganen abgedruckt werden, die an Redaktionen von Zeitungen verschickt werden. In der Szene werden Militanz-Debatten seit vielen Jahren gef\u00fchrt, eigentlich seitdem die Szene entstanden ist Ende der 70er, Anfang der 80er Jahre. Es gibt eine kontroverse Diskussion zwischen Szeneangeh\u00f6rigen \u00fcber die Legitimit\u00e4t und Begrenzung von Gewaltanwendung. \u00dcber die legitimen und illegitimen Formen der Gewaltanwendung, was ist erlaubt, und diese Debatten kann man nat\u00fcrlich in Szeneorganen nachlesen und versuchen, sie in ideologische Kontexte und Theorietraditionen einzuordnen. Das ist in einigen Arbeiten auf \u00fcberzeugende Weise geschehen. Die Szene hat sich auch selbst in B\u00fcchern mitgeteilt, von der AG Grauwacke zum Beispiel entstanden zwei B\u00e4nde, die informativ sind, die auch eine gewisse selbstkritische Reflexion der Szene zeigen. Inzwischen gibt es auch Literatur von etwas \u00e4lter gewordenen Autonomen, die aus der Szene aussteigen und etwas \u00fcber ihre fr\u00fchere Heimat schreiben oder sich eben mit den B\u00f6sen auseinandersetzen, die Kritik an der eigenen Szene ge\u00fcbt haben, wie das Buch zur \"Gegnerbestimmung\", das vor einiger Zeit erschienen ist. Wir haben bei der Auswertung dieser Quellen allerdings auch Schwierigkeiten, weil wir nicht genau wissen, wie repr\u00e4sentativ die \u00c4u\u00dferungen sind, die in den Organen ver\u00f6ffentlicht werden. Sicherlich sind das Reflexionen, die eher von einem Kern ausgehen und nicht ohne weiteres repr\u00e4sentativ f\u00fcr diejenigen sind, die sich am Rande der Szene bewegen. Da k\u00f6nnen wir wie in den rechtsextremen Szenen beobachten, dass der harte Aktivistenkern eher von jungen M\u00e4nnern dominiert wird, aber auch der Frauenanteil ist nicht ganz irrelevant in der Szene. Dann haben wir da noch Mitl\u00e4ufer, dann haben wir sogenannte \"eventorientierte\" Jugendliche, die einfach mal auf den Zug aufspringen und ein St\u00fcckchen mitfahren und dann wieder abspringen, die nicht automatisch das teilen, was vom Kern der Gruppen in der \u00d6ffentlichkeit vertreten wird.\n\n#### Wie definieren Sie den Begriff \"autonome Szene\" und wer geh\u00f6rt Ihrer Meinung dazu?\n\nWenn man das definieren will, ist es sinnvoll, beim Selbstverst\u00e4ndnis der Szene anzusetzen, und nat\u00fcrlich ist das dann wieder die Selbstbestimmung derer, die \u00fcber einen l\u00e4ngeren Zeitraum hinweg in dieser Szene waren und \u00fcber ihre eigene Position reflektieren. Insofern w\u00fcrde es keinen Sinn machen, einen von der Szene losgel\u00f6sten Begriff von autonom zu beschreiben, da der Begriff an sich, \"Autonome\", von der Szene aufgebracht worden ist. Der Begriff stammt aus Norditalien, wo die Autonomia in den Auseinandersetzungen bestimmter Teile der gewerkschaftlich orientierten Linken eine Rolle gespielt hat. Nat\u00fcrlich, der Begriff der Autonomie, der Selbstgesetzlichkeit, also seine Normen selber zu setzen, bringt zun\u00e4chst einmal ein Bestreben nach Unabh\u00e4ngigkeit zum Ausdruck. Unabh\u00e4ngigkeit von fest gef\u00fcgten Traditions- und Organisationszusammenh\u00e4ngen der Linken. Es wird nicht infrage gestellt, dass man selbst links ist, das geh\u00f6rt zum Selbstverst\u00e4ndnis dazu. Jedoch distanziert man sich von einer Traditionslinken, wie sie in den realsozialistischen L\u00e4ndern verankert war und in entsprechenden kommunistischen Parteien zum Ausdruck kam, mit ihren Ablegern ja auch im westlichen Deutschland, der DKP und ihrem Umfeld. Wir m\u00fcssen im Grunde zur\u00fcckgehen in die Zeit, als noch die SED und ihre Ausl\u00e4ufer in Westdeutschland die linksextreme Szene bestimmt haben mit ihren finanziellen M\u00f6glichkeiten, indem Erich Honecker in regelm\u00e4\u00dfigen Abst\u00e4nden Schecks unterschrieben hat. Der letzte Scheck, den er noch unterzeichnet hatte, bevor er zur\u00fcckgetreten ist, belief sich nach meiner Erinnerung auf 87 Millionen Westmark. Sie sehen, das waren keine geringen finanziellen. Mittel, die da zur Verf\u00fcgung gestellt wurden. Das war die Tradition der Westpolitik der SED, die mit ihrer Gr\u00fcndung begonnen hat; man wollte den Sozialismus in ganz Deutschland aufbauen. Anfang der 50er Jahre sind Heerscharen von Instrukteuren in den Westen geschickt worden. Das waren in Schulen ausgebildete junge Leute, die in Schnellkursen ausgebildet wurden. Sie bekamen Anweisungen aus Ostberlin und sind zu den Westgenossen gefahren, um denen in den Kommunistischen Zellen zu sagen, wie sie ihre politische Arbeit, zum Beispiel in den Wahlk\u00e4mpfen, machen sollen. Den Leuten im Westen hat das nicht immer gut gefallen, aber sie mussten es sich gefallen lassen, weil sie ja vom. Osten finanziert wurden. Diese Kontinuit\u00e4t der Westpolitik der SED geht bis 1989, unmittelbar bis zum Sturz Honeckers ist das durchgehalten worden. Und die autonome Szene ist eine Szene, die davon nichts wissen will. Sie ist links, radikal links in ihrem Selbstverst\u00e4ndnis, sie brechen mit dem kapitalistischen System, sind genauso antiimperialistisch, lehnen aber instinktiv den Parteikommunismus ab, den Leninismus mit der Avantgarde, die auf dem Prinzip des demokratischen Zentralismus in einem Kaderapparat beruht. Die Autonomen gingen aus der Sponti-Bewegung hervor, in der die Gedanken der Spontaneit\u00e4t und Kreativit\u00e4t eine gro\u00dfe Rolle spielten, wo Organisationen wegen ihrer mangelnden demokratischen Mitwirkungsm\u00f6glichkeiten misstrauisch betrachtet wurden, weil das Individuum eben dadurch unterdr\u00fcckt werde und seine kreativen F\u00e4higkeiten nicht entfalten k\u00f6nne. Da sind sicherlich anarchistische Ideen eingeflossen; beim Anarchismus spielt die individuelle Entfaltung eine wesentlich gr\u00f6\u00dfere Rolle als im Kommunismus, weil dort die Bereitschaft, sich unterzuordnen in die Gemeinschaft \u2013 \"Wir schreiten Seit an Seit\" \u2013 viel st\u00e4rker ist. Also diese ganze Tradition wird von dieser Szene abgelehnt, das macht sie dann aber auch attraktiver, als die Tradition zusammenbricht. 1989\/90 kann die autonome Szene f\u00fcr sich mit Recht reklamieren, nicht belastet zu sein durch den autokratischen Sozialismus des Ostens. Das hat die Anziehungskraft der Szene f\u00fcr junge Leute erh\u00f6ht, das erkl\u00e4rt auch, warum die Szene in den fr\u00fchen 90er Jahren angewachsen ist, allerdings dann auch wieder etwas geschrumpft. Heute stabilisiert sie sich in ihrem Personenpotenzial; wenn wir den Verfassungsschutzzahlen glauben, und die kennen die Szene recht gut, weil sie sie eben auf Landesebene sorgf\u00e4ltig beobachten.\n\n#### Warum wird sie heute wieder attraktiver?\n\nDie zentralen Gr\u00fcnde daf\u00fcr sind die Anti-Globalisierungsbewegung und der Antifaschismus. Das h\u00e4ngt zusammen mit den Erfolgen der NPD, mit der relativen Vitalit\u00e4t militanter rechtsextremistischer Jugendkulturen, mehr im \u00f6stlichen als im westlichen Deutschland. In einer Untersuchung, die ich gerade auf dem Tisch liegen habe und die am Hannah-Arendt-Institut erarbeitet worden ist, geht es um die Interaktionsdynamik. Zun\u00e4chst einmal hatten wir vor zu untersuchen, wie die NPD-Wahlerfolge auf die rechtsextremen militanten Gruppen wirken; da gibt es widerstreitende Hypothesenbildungen in den Sozialwissenschaften. Dann haben wir aber auch gepr\u00fcft, ob in der Gewaltdynamik die linksextremen Gruppen eine Rolle spielen. Schlie\u00dflich hat sich das als ein Schl\u00fcsselfaktor erwiesen, dass im Grunde die NPD-Wahlmobilisierung und die Antifa-Gegenmobilisierung gegen die rechtsextremen Bestrebungen f\u00fcr die Stimulierung der Konfrontationsgewalt zwischen den militanten Kontrahenten verantwortlich waren. Man muss also die Gewalt, die von einer militanten Antifa ausgeht, in einem Interaktionsgef\u00fcge sehen mit Anti-Antifa-Gewalt und rechtsextremistischen Wahlmobilisierungen, mit ihren von Jugendlichen gepr\u00e4gten militanten, gewaltgeneigten R\u00e4ndern.\n\n#### W\u00fcrden Sie sagen, dass die autonome Szene eine Jugendkultur ist?\n\nJedenfalls besitzt sie jugendkulturelle Elemente, und es ist liegt auf der Hand, dass die Szene von jungen Leuten getragen wird. Man findet da nicht mehr sehr viele, die weit \u00fcber drei\u00dfig sind. Der Fluktuationsgrad ist sehr gro\u00df. Es ist wohl so, dass man einige Jahre in der Szene verbringt und dann wieder ausscheidet. Aber es scheint auch danach so zu sein, dass in vielen F\u00e4llen eine gewisse Grundsolidarit\u00e4t erhalten bleibt. Denn es ist nicht einfach, mit Ehemaligen der Szene in Kontakt zu kommen. Wir haben mal versucht, Interviews zu f\u00fchren mit Szeneaussteigern, also mit Leuten, die aus der Szene herausgewachsen sind, aber das ist nicht gelungen. Einen ehemaligen Rechtsextremisten oder sogar aktive Angeh\u00f6rige rechtsextremer Gruppen zu befragen ist viel einfacher.\n\n#### Gehen Rechtsextreme offensiver mit ihrer Einstellung um?\n\nDie Rechtsextremen sind sozial marginalisiert, sie werden in der deutschen Gesellschaft aufgrund unserer Geschichte, aufgrund der intensiven Auseinandersetzung, die wir mit dieser Geschichte gef\u00fchrt haben und noch f\u00fchren, von einer ganz gro\u00dfen Mehrheit der Gesellschaft, 80 % +, als eine Gefahr angesehen. Allensbach hat eine Studie ver\u00f6ffentlicht (Institut f\u00fcr Demoskopie Allensbach Zuwanderer in Deutschland, PDF-Dokument unter <http:\/\/www.bertelsmann-stiftung.de\/bst\/de\/media\/xcms_bst_dms_28825_28831_2.pdf> am 13.06.2011), in der eine Frage aus dem Fragenkatalog lautete: Welche Gruppen m\u00f6chten Sie nicht als Nachbarn haben? Die Rechtsextremisten waren ganz oben im Ranking. Diese Frage ist mittlerweile nur noch in der \u00f6sterreichischen Version vorhanden. \u00d6sterreich hat eine andere Erinnerungskultur als Deutschland, aber auch dort war das Ergebnis \u00e4hnlich.\n\nAlso auf der einen Seite haben wir in der Gesellschaft Resonanz und Anf\u00e4lligkeit f\u00fcr rechtsextreme Vorstellungen, Einstellungen und Denkweisen. Das streitet keiner ab, aber man muss auf der anderen Seite auch gleichzeitig konstatieren, welch gro\u00dfe Ablehnung diese Szene erf\u00e4hrt, und das von einer doch sehr gro\u00dfen Mehrheit der Gesellschaft. Vor allem auch in der historischen Perspektive, was nicht unbedingt den jungen Leuten, die sich jetzt in den Szenen herumtreiben, immer gerecht wird, also individuell betrachtet; wenn man sich einen jungen Rechtsextremisten von 18 Jahren einmal anschaut, fragt man sich, wie ist denn der gepr\u00e4gt worden? Klar, dass der mit Adolf nichts mehr zu tun haben kann, dass da ganz andere Hintergr\u00fcnde und motivationale Faktoren eine Rolle spielen, warum der junge Mann in die Szene reingekommen ist. Die Linksextremen f\u00fchlen sich nicht im gleichen Ma\u00dfe sozial marginalisiert. Sie sind es objektiv gesehen auch nicht in dem Ma\u00dfe, wie es bei den Rechtsextremen der Fall ist. Sie k\u00f6nnen sich eher als ein halbwegs akzeptierter Teil einer linken Protestkultur verstehen, die noch ein Bestandteil der demokratischen Gesellschaft der Bundesrepublik Deutschland ist. Dieses Gef\u00fchl, Bestandteil einer legitimen Protestkultur zu sein, das k\u00f6nnen die Rechtsextremisten kaum haben. Dieses Selbstverst\u00e4ndnis erkl\u00e4rt auch vieles, zum Beispiel die Schwierigkeiten, an einen Linksextremen heranzukommen, die Art und Weise, wie Journalisten dar\u00fcber schreiben, wie Wissenschaftler sich damit besch\u00e4ftigen; warum an der Freien Universit\u00e4t Berlin so wenig Forschungsprojekte zu dem Thema gemacht worden sind, w\u00e4re auch interessant. Das h\u00e4ngt alles zusammen mit der linken Pr\u00e4gung der Universit\u00e4t und den Karrieren, den Lebenswegen auch von Dozenten an der Freien Universit\u00e4t Berlin, die inzwischen meist vern\u00fcnftige Leute geworden sind, die mit dem Radikalismus ihrer Jugendzeit nicht mehr viel zu tun haben wollen, aber die gewisse instinktive Abwehrreflexe an den Tag legen; die Diskussionen h\u00e4tten sie eigentlich gar nicht n\u00f6tig, das ist intellektuell auch nicht sehr redlich, muss ich sagen. Redlicher w\u00e4re es, wenn man \u00fcber die eigene Biografie offen spricht und sagt, ich geh\u00f6rte damals zu einer Protestkultur, f\u00fcr uns war es normal, auch mal Steine zu werfen, oder wir haben Dinge gemacht, von denen muss man sich heute distanzieren, das darf man eigentlich nicht tun, da haben wir Grenzen \u00fcberschritten. Wir haben in unseren Lehrveranstaltungen junge Leute in einer Art und Weise \u00fcber diesen Staat informiert, da muss man heute sagen, das war nicht sehr verantwortungsvoll. So m\u00fcsste man eigentlich dar\u00fcber sprechen, \u00fcber dieses Kapitel. Aber sie werden kaum Jemanden finden, der diese kritische, selbstkritische Distanz aufbringt. Das ist auch ein Grund, warum nicht so viel geforscht worden ist. Es gibt einige Wissenschaftler, aber es sind nicht viele. Dieter Rucht ist einer der Wenigen, die sich in Berlin als Bewegungsforscher damit auseinandergesetzt haben, und wir haben ihm wichtige Einsichten zu verdanken.\n\n#### Wie sehen Sie die \u00f6ffentliche Wahrnehmung der autonomen Szene? Oder anders, ist die mediale Darstellung der autonomen Szene authentisch?\n\nEs ist ja doch so, dass die Szene am ehesten durch ihre Gewaltexzesse in der \u00d6ffentlichkeit wahrgenommen wird. Und das ist nicht in jeder Hinsicht repr\u00e4sentativ f\u00fcr die Szene. Aber die Szene wird durch eine Aversion gegen diesen unterdr\u00fcckenden Staat wahrgenommen, verbunden mit einem Wirtschaftssystem, das als kapitalistisch abgelehnt wird; also die politisch-\u00f6konomischen Strukturen werden hier in einer engen Verbindung miteinander gesehen; das ist ein klassischer Ansatz, wie wir ihn aus dem Marxismus kennen, da steht man in einer guten alten linken Tradition der Wirklichkeitswahrnehmung. Damit verbunden ist die Rechtfertigung einer dosierten Anwendung von Gewalt in bestimmten Situationen; kaum einer w\u00fcrde \"revolution\u00e4re Gewalt\" grunds\u00e4tzlich ausschlie\u00dfen, also der Status quo soll ja revolution\u00e4r \u00fcberwunden werden, das ist Szenekonsens. Aber es werden auch Gewaltbegrenzungsdiskurse gef\u00fchrt, und man diskutiert \u00fcber die Legitimit\u00e4t bestimmter Formen der Gewalt. Dann gibt es innerhalb der Szene viele lammfromme intellektuelle Schreiber, die nie einen Stein in die Hand nehmen w\u00fcrden, und auch nicht jeder, der sich in der Szene bewegt, ist schon ein potenzieller Gewaltt\u00e4ter. Insofern ist die Wahrnehmung in der \u00d6ffentlichkeit nicht repr\u00e4sentativ. Die Vielfalt der intellektuellen Diskussionen wird nat\u00fcrlich kaum wahrgenommen. Die Wahrnehmung ist im Wesentlichen die Perspektive der Bedrohung durch Gewalttaten im Demonstrationsgeschehen. Das spielt eine ganz wichtige Rolle, weil ein Gro\u00dfteil der Gewalttaten im Demonstrationsgeschehen ver\u00fcbt wird, dann nat\u00fcrlich spektakul\u00e4re Gewalttaten, die von kleinen geschlosseneren Gruppen wie der Marxistischen Gruppe im Berliner und Brandenburger Raum ver\u00fcbt worden sind. Hier in Dresden gab es einen Brandanschlag auf die Offiziersschule des Heeres, der von einem gr\u00f6\u00dferen planerischen Vorlauf zeugte und durch das Ausma\u00df des Schadens und die Art der Tatausf\u00fchrung gr\u00f6\u00dfere Medienaufmerksamkeit gefunden hat. Aber das ist eben immer die Gewaltperspektive. Insofern kann das \u00f6ffentliche Bild nicht das korrekte Bild der Vielschichtigkeit und Heterogenit\u00e4t der Szene abbilden.\n\n#### Ist eine Ver\u00e4nderung in der Szene zu erkennen? Im Zeitraum von den 80ern bis heute betrachtet?\n\nWas man ganz bestimmt sagen kann, ist, dass bestimmte Themen Konjunktur haben und dann wieder an Bedeutung verlieren. Die mobilisierungstr\u00e4chtigen Themen, die im Zentrum der Bewegung stehen, wechseln. Zum Beispiel spielte die Kernkraft in den 80er Jahren zeitweilig eine sehr gro\u00dfe Rolle, ist dann aber f\u00fcr viele Jahre von der Bildfl\u00e4che verschwunden und erst in neuester Zeit durch die japanischen Ereignisse und ihre Folgen wieder en vogue. Der Antifaschismus war von Anfang an ein Thema, aber nat\u00fcrlich hatte das Thema Konjunkturen, die gekn\u00fcpft waren an teils spektakul\u00e4re Ereignisse, zum Beispiel Anfang der 90er Jahre die Exzesse in Rostock, Solingen und Hoyerswerda, die zu einer antifaschistischen Gegenmobilisierung f\u00fchrten. Letztlich war die breite gesellschaftliche Mobilisierung im \"Aufstand der Anst\u00e4ndigen\" f\u00fcr die Szene aber gar nicht gut, denn das Thema wurde ihnen von den gem\u00e4\u00dfigten Gruppen genommen. Der Einzug der NPD in das s\u00e4chsische Landesparlament 2004 mit 9,2 Prozent der Stimmen war noch so ein Schl\u00fcsselereignis als mobilisierungstr\u00e4chtiges Thema f\u00fcr die Szene. Es gibt auch andere bedeutsame Themen. Urspr\u00fcnglich war es ja die sogenannte Hausbesetzer-Szene in den Gro\u00dfst\u00e4dten, die hier eine wichtige Rolle spielte. Die Frage nach preisg\u00fcnstigem Wohnraum haben wir heute wieder in der sogenannten Gentrifizierungsdebatte. Man sieht, es ist ein altes Thema, das mal ganz im Zentrum steht und dann wieder an Bedeutung verliert, weil andere Themen in den Mittelpunkt r\u00fccken.\n\n#### Liegt die Themenvielfalt an der Spezifikation als Fokus auf jeweils eigene Themen einzelner autonomer Gruppen?\n\nDie Szene hatte nie ein Zentrum und wollte es auch nicht. Man hat zwar diese kommunikative Netzwerkstruktur durch die Szenemedien, aber die Szene organisiert sich lokal, allenfalls regional, nicht national. Obwohl Versuche unternommen worden sind, diese dezentralisierte Struktur aufzubrechen und die Aktivit\u00e4ten st\u00e4rker zu b\u00fcndeln, wie in G\u00f6ttingen durch die Antifa \"M\". Aber sie sind gescheitert. Es gibt in der neuesten Zeit Versuche, sich mit einem kommunistischen ideologischen Hintergrund fester zu organisieren, aber das ist nicht repr\u00e4sentativ f\u00fcr die gesamte Szene, es ist nur ein kleiner Teil davon. Diese dezentralisierte Organisationsstruktur f\u00fchrt auch dazu, dass die Themenvielfalt relativ gro\u00df ist.\n\n#### Lassen sich soziale, biografische Milieus beschreiben, aus denen sich die autonome Szene rekrutiert?\n\nDar\u00fcber wissen wir viel zu wenig. Wir haben gar keine vergleichenden Studien \u00fcber Lebensl\u00e4ufe, da ist noch sehr viel zu leisten. Ob wir das \u00fcberhaupt tun k\u00f6nnen, ist fraglich, weil die Szene sich der Exploration so stark verweigert, und wir wissen dar\u00fcber sehr wenig, es liegen kaum Forschungsergebnisse zu dieser Frage vor. Man wei\u00df aus einigen T\u00e4terstudien, was allerdings nicht repr\u00e4sentativ f\u00fcr die gesamte Szene ist, dass das Bildungsniveau h\u00f6her ist als in den rechtsextremen Gruppen, dass Abiturienten und Studierende einen recht hohen Anteil an der Szene haben, aber wir wissen eigentlich nichts dar\u00fcber, ob es eine bestimmte Sozialstruktur oder andere Besonderheiten gibt. Wir wissen, dass es \u00fcberwiegend junge Leute sind und dass sie nach einer gewissen Zeit aus der Szene wieder aussteigen und sich relativ spurlos in die Gesellschaft reintegrieren.\n\n#### Wissen Sie, welche Rolle Frauen in der autonomen Szene spielen?\n\nFrauen spielen eine gr\u00f6\u00dfere Rolle in der autonomen Szene als in der rechtsextremen Szene. Wobei wir feststellen k\u00f6nnen, dass auch in der rechtsextremen Szene der Frauenanteil langsam ansteigt. Was nat\u00fcrlich mit gesamtgesellschaftlichen Entwicklungen zusammenh\u00e4ngt und dass sich einfach das Rollenverst\u00e4ndnis und die Verhaltensformen von jungen M\u00e4nnern und Frauen ann\u00e4hern. In den autonomen und linksextremen Gruppen ist der Frauenanteil schon immer h\u00f6her gewesen. Man m\u00fcsste jetzt schauen, ob das ein Spiegelbild der entsprechenden Abiturienten- und Studierendenjahrg\u00e4nge ist. Wahrscheinlich nicht ganz, denn das w\u00fcrde ja hei\u00dfen, dass wir eine 50-zu-50-Verteilung h\u00e4tten, und die haben wir nicht in der autonomen Szene. Die Gewaltt\u00e4tigkeit ist sicherlich ein wichtiger Grund, dies zu verhindern. Frauen haben im Schnitt gr\u00f6\u00dfere Schwierigkeiten als M\u00e4nner mit physischer Gewaltanwendung, auch in autonomen Zusammenh\u00e4ngen. Die Rechtsextremisten leben die physische Gewaltsamkeit geradezu lustvoll aus, indem sie oft mit unglaublicher Brutalit\u00e4t auf ihre Opfer eindreschen. Bei der linksextremen Szene ist es h\u00e4ufiger die Distanzwaffe. Fr\u00fcher waren es Zwillen, mit denen gerne geschossen wurde, jetzt sind es Steine und Flaschen. Hingegen agiert man weniger in der face-to-face-Situation, also in unmittelbarem k\u00f6rperlichem Kontakt zum Kontrahenten. Ein wesentlicher Punkt ist, dass in der rechtsextremen Szene Frauenbilder vermittelt werden, die f\u00fcr junge Frauen nicht attraktiv sind. Die Frauen dort sind meist die Freundinnen von rechtsextremistischen Aktivisten, die so in die Szene hineinkommen und dann ideologisch die entsprechenden Frauenbilder vertreten. Es ist klar, dass die egalit\u00e4re Geschlechterinterpretation, die in der autonomen Szene vertreten wird, dem heutigen, verbreiteten Verst\u00e4ndnis von Geschlechterbeziehungen weit eher entspricht als das Rollenverst\u00e4ndnis, das in rechtsextremen Szenen \u00fcberwiegt.\n\n#### Wie gef\u00e4hrlich ist die autonome Szene f\u00fcr die \u00f6ffentliche Sicherheit oder gar Demokratie?\n\nF\u00fcr die Demokratie ist die autonome Szene keine Gefahr. F\u00fcr die \u00f6ffentliche Sicherheit ist es schon ein Problem, weil bestimmte Gruppen der Gesellschaft in das Zielfernrohr der Szene geraten und dann entsprechend gef\u00e4hrdet sind. Allerdings muss man sagen, dass die gezielte T\u00f6tung von Menschen bisher keine Rolle spielt. Es gab immer wieder Todesf\u00e4lle, aber es ist nicht typisch f\u00fcr die Art der Gewaltanwendung der autonomen Szene, den Tod von Menschen anzustreben. Bedeutsam sind hingegen Sachbesch\u00e4digungen, da werden jedes Jahr gro\u00dfe Sch\u00e4den angerichtet, die die \u00d6ffentlichkeit zu tragen hat oder auch Privatpersonen zu tragen haben, wenn etwa Fahrzeuge \"abgefackelt\" werden. Da sagen sich manche Autonome vielleicht, das zahlt sowieso die Versicherung, aber sie bedenken nicht, dass Versicherungssch\u00e4den auch von der Allgemeinheit getragen werden. Besonders besorgniserregend ist die Gewalt, die Polizeibeamte erleiden. Da haben wir es mit einer zunehmenden Hemmungslosigkeit im Demonstrationsgeschehen zu tun. Dort sind die Polizisten in einer sehr undankbaren Situation; ihre Aufgabe ist es zum Beispiel, Demonstrationen von Rechtsextremisten zu sch\u00fctzen, und sie sehen sich dann sehr oft von gewaltt\u00e4tigen linken Demonstranten angegriffen. Schwere Verletzungen von Polizeibeamten haben in den letzten Jahren erheblich zugenommen, nat\u00fcrlich auch durch die gesteigerte Konfrontationsgewalt von rechts gegen links wie auch von links gegen rechts.\n\nVor drei Jahren gab es einen Angriff auf eine Polizeiwache im Schanzenviertel in Hamburg, der allerdings nicht typisch f\u00fcr die Szene war und hoffentlich nicht typisch f\u00fcr die Szene wird. Da wurden sogar noch die T\u00fcr verrammelt und Brands\u00e4tze ins Geb\u00e4ude geworfen. Wir k\u00f6nnen auch einen Anstieg schwerer k\u00f6rperlicher Gewalttaten gegen \"Faschisten\" feststellen. Im Vergleich aber ist die K\u00f6rperverletzung bei den rechtsmotivierten T\u00e4tern weit h\u00e4ufiger zu beobachten als bei linksmotivierten. Es gibt allerdings auch bei extremen Linken dehumanisierende Ideologien und menschenverachtende Einstellungen, die sich in entfesselter Gewalt gegen Personen \u00e4u\u00dfern. Es gibt linksextreme Hassmusik gegen \"Faschos\" oder Polizeibeamte wie den Spruch: \"All Cops are Bastards\", ACAB abgek\u00fcrzt, auch Wirtschaftsunternehmen oder Sozialbeh\u00f6rden sind Angriffsobjekte von Anschl\u00e4gen. Insofern besteht durchaus ein Problem f\u00fcr die innere Sicherheit. Allerdings ist das Gewaltpotenzial der rechtsextremen Szene h\u00f6her. Au\u00dferdem gibt es im Rechtsextremismus eine Reihe von Delikten, die keine Entsprechung auf der linksextremen Seite haben. Wenn sie irgendwo ein Hakenkreuz anpinseln, ist das ein kriminelles Delikt, Hammer und Sichel hingegen sind nicht strafrechtlich bewehrt. Die Verfolgung der rechtsmotivierten Propaganda-Delikte besch\u00e4ftigt die Polizei \u00fcber alle Ma\u00dfen. Dabei d\u00fcrften diese Delikte zum Teil deshalb ver\u00fcbt werden, weil sie strafbewehrt sind; Jugendliche reizt bekanntlich der Normbruch. Es w\u00e4re aus meiner Sicht empfehlenswert, die Aufmerksamkeit der Polizei auf Gewalttaten zu konzentrieren.\n\n#### Wie wichtig ist denn die Arbeit des Verfassungsschutzes und der verdeckten Ermittler in der autonomen Szene?\n\nWir wissen ja, dass die rechtsextreme Szene ziemlich durchsetzt ist mit V-Leuten; mein Eindruck ist, dass den Verfassungsschutzbeh\u00f6rden das bei den Autonomen viel weniger gelungen ist. Das h\u00e4ngt mit der gr\u00f6\u00dferen Solidarit\u00e4t und Geschlossenheit der Szene zusammen. Zumindest die Kerngruppen halten zusammen wie Pech und Schwefel. Es ist nicht einfach, in diese Struktur einzudringen. Folglich sprudeln die Quellen da weit sp\u00e4rlicher als in der rechtsextremen Szene.\n\n#### Aber Sie finden es wichtig, dass es geschieht?\n\nIch bin ein Anh\u00e4nger unseres Systems der Verfassungsschutzbeh\u00f6rden. Ich wei\u00df, dass das Kritik hervorruft, aber der Verfassungsschutz war ja auch historisch gesehen eine Reaktion auf die deutsche Geschichte. Die deutschen Verfassungsschutzbeh\u00f6rden geh\u00f6ren heute im internationalen Vergleich zu den demokratischsten und offensten Nachrichtendiensten, die relativ transparent sind und scharf kontrolliert werden, auch wenn hier Defizite bestehen m\u00f6gen. Ich beurteile das Wirken des Inlandsnachrichtendienstes \u00fcberwiegend positiv. Allerdings sehe ich bestimmte Formen der Berichtspraxis kritisch, weil sie in Freiheitsrechte eingreifen und aus meiner Sicht mehr Zur\u00fcckhaltung w\u00fcnschenswert w\u00e4re. Aber im Gro\u00dfen und Ganzen halte ich die Beobachtung durch Verfassungsschutzbeh\u00f6rden f\u00fcr wichtig, um die Demokratie gegen ihre Feinde zu sch\u00fctzen, und insofern halte ich es auch f\u00fcr legitim, dass Verfassungsschutzbeh\u00f6rden versuchen, Informanten in diesen Szenen zu gewinnen. Wir wissen, dass V-Leute keine James Bonds sind, die in diese Szenen eingeschleust werden, sondern die Verfassungsschutzbeh\u00f6rden treten an unsichere Kantonisten heran, von denen sie glauben, dass sie kooperationswillig sein k\u00f6nnten. Sie bleiben in der Szene, liefern aber Informationen. Es ist nicht einfach, diese V-Leute zu f\u00fchren, und es ist vor dem Hintergrund der NSU-Erfahrungen zu \u00fcberlegen, wie die Praxis der V-Mann-F\u00fchrung strenger zu regeln w\u00e4re.\n\n#### Wie sch\u00e4tzen Sie die gesamtgesellschaftliche Bedeutung der autonomen Szene ein? Hat sie die M\u00f6glichkeit, da etwas zu ver\u00e4ndern?\n\nSie ist ja eher ein Randph\u00e4nomen innerhalb der Neuen Sozialen Bewegungen, aber im negativen Sinne haben sie ein gewisses Gewicht, sie ruinieren den Ruf der Neuen Sozialen Bewegungen mit ihrer Gewaltt\u00e4tigkeit. Problematisch ist es, wenn Parteien wie die Linke Autonome als einen Teil ihrer Klientel betrachten und versuchen, Klientelpolitik zu betreiben. Dadurch kann den Autonomen ein Wirkungsfeld zuwachsen, das sie ansonsten gar nicht h\u00e4tten. Das ist eine gewisse Gefahr, die ich sehe, und wenn man so etwas erkennen kann, muss man auch kritisch in der \u00d6ffentlichkeit darauf hinweisen, damit da nicht durch die Hintert\u00fcr Wirkungen entstehen, die die Szene aufgrund ihres Potenzials eigentlich nicht haben kann. Aber ich sehe die Autonomen nicht als ein Ph\u00e4nomen an, durch das unsere politische Agenda erheblich bestimmt werden k\u00f6nnte.\n\n#### Besteht bei der autonomen Szene ein Interesse, die Gesellschaft als Ganzes zu \u00e4ndern, oder geht es darum, die eigenen Partikularinteressen bzw. Freir\u00e4ume zu bewahren?\n\nZumindest die ideologisch tonangebenden Kr\u00e4fte der Szene, die sich in entsprechenden Szeneorganen artikulieren, haben durchaus ein Interesse, die Gesellschaft als Ganzes revolution\u00e4r zu ver\u00e4ndern. Dieses Interesse w\u00fcrde ich sogar als zentral ansehen, es ist ein verbindendes Element. Jedoch entspricht das menschliche Verhalten nicht immer dem, was ideologisch propagiert wird, die propagierten Ziele des Handelns stimmen oft nicht mit den tats\u00e4chlichen Handlungsmotiven der Menschen \u00fcberein. Nat\u00fcrlich sind unsere zentralen Interessen und Bed\u00fcrfnisse, dass man Spa\u00df haben will, dass Gemeinschaftserlebnisse gesucht werden, dass man sich f\u00fcr etwas engagieren will, aber auch Zusammenhalt finden, sich in der Kneipe treffen und Anerkennung in der Gemeinschaft erfahren will. Diese Motive sind in allen Jugendkulturen zentral und \u00fcberlagern nicht selten im realen Verhalten die ideologisch propagierten Ziele und Motive. Aber das schreiben sie nicht in ihre Zeitschriften, da muss die Revolution im Mittelpunkt stehen.\n\n#### Wie sch\u00e4tzen Sie Zukunft und Perspektiven der autonomen Szene ein?\n\nIch glaube, dass hier Neigungen, Sehns\u00fcchte und Illusionen eine Rolle spielen, die schwer auszurotten sind. Die man vielleicht auch nicht ausrotten sollte. Wenn die Menschen dann \u00fcbers Ziel hinausschie\u00dfen, im Radikalismus des Jugendalters, das bekannterma\u00dfen einen \u00dcberschuss an Idealismus produziert, wenn man Tr\u00e4umen anh\u00e4ngt, sich Illusionen macht und gro\u00dfe Theorien entwickelt, so ist das im Jugendalter h\u00e4ufiger anzutreffen als bei den \"\u00e4lteren Semestern\", die schon einige Erfahrungen im Leben gemacht haben und wissen, dass nicht alles so l\u00e4uft, wie man es gerne h\u00e4tte, und es Grenzen f\u00fcr das eigene Handeln gibt. Der politische Illusionismus, der in diesen Szenen gepflegt wird, d\u00fcrfte kaum zu entzaubern sein, er wird immer ein Gegenstand politischer Bildung bleiben. Es geht darum, den jungen Menschen auf der einen Seite ihre Ideale nicht auszutreiben, aber auf der anderen Seite ein Gef\u00fchl f\u00fcr das Realisierbare und politisch Machbare, f\u00fcr die Grenzen des Machbaren zu vermitteln, was ganz zentral f\u00fcr die politische Urteilskraft ist: also Alternativen absch\u00e4tzen zu k\u00f6nnen, M\u00f6glichkeitsr\u00e4ume auszuloten, aber auch zu wissen, was v\u00f6llig unm\u00f6glich ist, was gef\u00e4hrlich ist, Abenteuer zu vermeiden, Risiken sorgf\u00e4ltig zu kalkulieren, verantwortlich zu handeln, indem man die Ziele des eigenen Handelns bedenkt. Daran mangelt es generell bei jungen Leuten, und das ist eine Herausforderung. Das sind wichtige Aufgaben politischer Bildung, aber da wird man sich wohl bei den Hartgesottenen, die bei Interim oder radikal ihre Beitr\u00e4ge schreiben, die Z\u00e4hne ausbei\u00dfen. Jedoch bei vielen anderen jungen Leuten, die eine gewisse Affinit\u00e4t zu diesen Szenen haben, kann man erfolgreich sein, wenn man sich auf die Leute einl\u00e4sst und versucht, ins Gespr\u00e4ch zu kommen, eine gewisse Offenheit an den Tag legt; wir wissen, dass sie nicht bildungsresistent sind, um mal dieses etwas zynische Wort zu benutzen. Wir kennen die Bildungsresistenz aus der rechtsextremistischen Szene, wo Box-Kurse oder Anti-Aggressions-Training wirksamer sind als die Auslotung politischer M\u00f6glichkeitsr\u00e4ume. Bei denen, die im Kielwasser der autonomen Szene anzutreffen sind, w\u00fcrde ich mir durchaus gr\u00f6\u00dfere Hoffnungen machen.\n\n#### Die Programme gegen Rechtsextremismus wurden erweitert, um gegen alle Formen des Extremismus \u2013 also auch gegen den Linksextremismus \u2013 vorgehen zu k\u00f6nnen. Diese Programmausweitung basiert auf Ihrer Extremismus-Theorie...\n\nDie Extremismus-Theorie gibt es ja schon l\u00e4nger, als ich an der Universit\u00e4t t\u00e4tig bin; ich bin weder der Erfinder der Theorie noch war ich Berater von Frau Schr\u00f6der.\n\n#### Aber das Konzept der Extremismus-Theorie ist doch aus Ihrer Feder?\n\nDas Extremismuskonzept ergibt sich doch aus der Verfassungsordnung der Bundesrepublik Deutschland (wie anderer demokratischer Verfassungsstaaten). Wenn man auf der Grundlage der Verfassung der Bundesrepublik Deutschland Konzepte gegen Bedrohungen des demokratischen Verfassungsstaates entwickelt, dann ist es naheliegend, nicht ausschlie\u00dflich Konzepte zur Auseinandersetzung mit dem Rechtsextremismus zu entwickeln. Es ist klar, dass die \u00dcbergriffe auf Minderheiten, die sich in Deutschland Anfang der 90er Jahre h\u00e4uften, einen ziemlichen Schub ausl\u00f6sten. Mit dem sogenannten Aufstand der Anst\u00e4ndigen ab 2000 ging es eigentlich richtig los mit den Pr\u00e4ventionsprogrammen. Damals war es eine rot-gr\u00fcne Bundesregierung, die diese Pr\u00e4ventionsprogramme auf den Weg gebracht hat. Damals h\u00e4tte man durchaus denken k\u00f6nnen, dass man Gewalt, die von linksextremen Strukturen ausgeht, nicht v\u00f6llig ausblendet. Es ist erkl\u00e4rungsbed\u00fcrftig, warum diese Programme immer nur gegen Rechtsextremismus entwickelt worden sind. Auf der Grundlage der Verfassung h\u00e4tte man es eigentlich von vornherein etwas breiter anlegen k\u00f6nnen, anlegen sollen. Die neue Bundesregierung hat dann die Akzente anders gesetzt. Dass die Linkspartei damit ein Problem hat, ist aus ihrer Geschichte heraus verst\u00e4ndlich, aber wenn die Linke auf dem Boden der Verfassung endlich ankommen will, darf sie diesem Problem nicht ausweichen. So lange sie sich wie eine Lobby f\u00fcr ehemalige Stasi-Agenten, fr\u00fchere SED-Gr\u00f6\u00dfen und militante Antifaschisten verh\u00e4lt, muss sie sich kritische Fragen gefallen lassen, ob sie denn auf dem Boden der Verfassung der Bundesrepublik Deutschland steht. Mit einer gewissen Besorgnis stelle ich fest, dass bei unserer gem\u00e4\u00dfigten Linken durch das Schielen auf m\u00f6gliche Koalitionen in der Zukunft die demokratische Streitbarkeit nach links erlahmt. Zugleich muss man anerkennen, dass die Auseinandersetzung mit dem Rechtsextremismus eine zentrale \u00f6ffentliche Aufgabe ist und bleibt.\n\n#### Literatur\n\nA.G. Grauwacke: Autonome in Bewegung. Aus den ersten 23 Jahren. Assoziation A., Berlin 2003.\n\nBackes, Uwe: Politische Extreme. Eine Wort- und Begriffsgeschichte von der Antike bis zur Gegenwart. Vandenhoeck & Ruprecht, G\u00f6ttingen 2006.\n\nBackes, Uwe\/Mletzko, Matthias \/Stoye, Jan: Rechtsextreme Wahlmobilisierung und politisch motivierte Gewalt. Wolter Kluwer\/Luchterhand, M\u00fcnchen 2010.\n\nBackes, Uwe\/Gallus, Alexander\/Jesse, Eckhard (Hrsg.): Jahrbuch Extremismus & Demokratie, Band 25. Nomos, Baden-Baden 2013.\n\nDustin Hoffmann, Karsten: \"Rote Flora\". Ziele, Mittel und Wirkungen eines linksautonomen Zentrums in Hamburg. Nomos, Baden-Baden 2011.\n\nMohr, Markus\/R\u00fcbner, Hartmut: Gegnerbestimmung. Sozialwissenschaft im Dienst der \"inneren Sicherheit\". Unrast, M\u00fcnster 2010.\n\n## \"Dass ihr deklariertes Ziel, den Kapitalismus und den Staat zu bek\u00e4mpfen, aussichtslos ist, ist offenkundig.\"\n\nProf. Dr. Albert Scherr, 56 Jahre, Soziologe und Sozialp\u00e4dagoge.\n\nProf. Dr. Albert Scherr, Jahrgang 1958, ist Soziologe und Sozialp\u00e4dagoge. Er studierte Soziologie und P\u00e4dagogik an der Universit\u00e4t Frankfurt, wo er 1985 promovierte. Von 1981 bis 1983 und von 1985 bis 1988 war er als Sozialarbeiter in der Jugendarbeit und Jugendsozialarbeit t\u00e4tig. Anschlie\u00dfend war er wissenschaftlicher Mitarbeiter an der Universit\u00e4t Bielefeld und von 1990 bis 2001 Professor f\u00fcr Soziologe und Jugendarbeit an der Fachhochschule Darmstadt. Er wurde 1998 an der Universit\u00e4t Karlsruhe als Soziologe f\u00fcr Allgemeine Soziologie habilitiert. Seit 2001 ist er Professor an der P\u00e4dagogischen Hochschule Freiburg und Leiter des Instituts f\u00fcr Soziologie. Albert Scherr ist Vorstandsmitglied im Komitee f\u00fcr Grundrechte und Demokratie (www.grundrechtekomitee.de) und Redaktionsbeirat der Zeitschriften Soziale Probleme und Sozial Extra.\n\n#### Was wissen wir eigentlich \u00fcber die autonome Szene? Wie sch\u00e4tzen Sie den Forschungsstand ein?\n\nEs gibt zwar eine Reihe von Ver\u00f6ffentlichungen zur Geschichte und zum Selbstverst\u00e4ndnis der Autonomen und auch vielf\u00e4ltige Selbstdarstellungen. Was aber bislang fehlt, ist eine seri\u00f6se empirische Forschung, die Einblicke in die Praktiken und Strukturen der Autonomen erm\u00f6glicht, also sich nicht einfach an den Selbstdarstellungen der Szene einerseits, den Beobachtungen des Verfassungsschutzes andererseits orientiert. Denn in beiden F\u00e4llen handelt es sich ja um Quellen, die in hohem Ma\u00dfe interessengeleitet sind, die also keine verl\u00e4ssliche Grundlage f\u00fcr eine wissenschaftliche Auseinandersetzung bieten. Eine angemessene Forschung sollte drei Bedingungen erf\u00fcllen: Sie sollte sich erstens der Gefahr ihrer Instrumentalisierung durch den Verfassungsschutz bewusst sein und sich konsequent einer solchen Instrumentalisierung verweigern. Zweitens kann sie sich aber selbstverst\u00e4ndlich auch nicht einfach auf die Seite der Autonomen begeben, sondern muss Distanz zu ihrem Gegenstand wahren und kann auf die Kritik problematischer Sachverhalte und Entwicklungen nicht verzichten. Drittens ist es erforderlich, nicht einfach nur \u00fcber die Autonomen zu forschen, sondern auch die politischen und medialen Konstellationen, in denen die Entstehung und Entwicklung der Autonomen verortet ist. Zum Beispiel kann man die Entwicklung der Autonomen nicht verstehen, ohne sich mit rechtsextremer und rassistischer Gewalt seit den 1990er Jahren und den staatlichen Reaktionen darauf zu befassen.\n\n#### Wer geh\u00f6rt zur autonomen Szene?\n\nDie autonome Szene kann als ein Netzwerk lokaler und \u00fcberregionaler Gruppen beschrieben werden, die sich \u00fcber szenetypische Publikationen, Internetforen und Aktionsb\u00fcndnisse zueinander in Beziehung setzen. Selbstverst\u00e4ndlich gibt es keine formellen Mitgliedschaften wie in Organisationen, aber bedeutsame informelle Zugeh\u00f6rigkeitsregulierungen. Zu unterscheiden sind dabei ein innerer Kern und ein weiteres Umfeld. Die Zugeh\u00f6rigkeit beruht meiner Einsch\u00e4tzung nach vor allem auf der Mitgliedschaft in Szenen und Gruppen, die als dichte Gesinnungsgemeinschaften zu charakterisieren sind. Das hei\u00dft: Zugeh\u00f6rigkeit setzt eine weitgehende ideologische \u00dcbereinstimmung voraus und schlie\u00dft die Bereitschaft ein, sich in der Gruppe wie in einer erweiterten Familie zu verhalten, also z. B. nicht nach au\u00dfen offen zu kommunizieren, was innen geschieht. Begr\u00fcndet ist das einerseits in dem Feindbild, das in Teilen der politischen \u00d6ffentlichkeit und durch den Verfassungsschutz gepflegt wird, anderseits in einer Selbstabschlie\u00dfung nach au\u00dfen, die Bestandteil des Selbstverst\u00e4ndnisses ist. In Teilen der Szene gibt es zudem eine Ideologie der Militanz, die als Rechtfertigung f\u00fcr die \u00dcberschreitung strafrechtlicher Normen verstanden wird. Auch dies tr\u00e4gt zu einer klaren Grenzziehung nach au\u00dfen bei, die manchmal zu einem Selbstverst\u00e4ndnis als klandestiner Geheimbund tendiert.\n\n#### Wie w\u00fcrden Sie einen \"typischen\" Autonomen, eine \"typische\" Autonome beschreiben?\n\nTypisch f\u00fcr die Szene sind ein hoher Anteil von Gymnasiasten, Studenten und Akademikern sowie ein relativ ausgewogenes Geschlechterverh\u00e4ltnis. Der bzw. die typische Autonome versteht sich als hoch politisierter Aktivist, der f\u00fcr sich in Anspruch nimmt, die Machtverh\u00e4ltnisse einer Gesellschaft, die als kapitalistische und nationalistische verstanden wird, zu durchschauen. Dies verbindet sich mit einer Militanz und einem politisch-moralischen Rigorismus, deren positive Seite ein starkes Engagement im Bereich des Antifaschismus ist. Dessen Kehrseite besteht in der Tradierung eines traditionellen Musters k\u00e4mpferischer M\u00e4nnlichkeit und ideologischen Vereinfachungen: Der und die typische Autonome betrachtet den Staat als kapitalistisches Machtorgan und die Polizei als Repressionsapparat, und entsprechend fungieren \"die Bullen\" als Feindbilder.\n\n#### Welche Ver\u00e4nderungen haben Sie bei den Autonomen seit ihrer Entstehung wahrgenommen? Sind die Autonomen aus den 80er und 90er Jahren mit denen der heutigen Zeit zu vergleichen?\n\nIn den 90er Jahren war die Auseinandersetzung mit dem damals neuen Rechtsextremismus der zentrale Fokus der Autonomen. Mit durchaus guten Gr\u00fcnden konnten sie sich als entschiedene Gegner der extremen Rechten beschreiben und dadurch Sympathien in einem breiteren Umfeld gewinnen. Dieser Gegner ist f\u00fcr die Autonomen in einigen Regionen inzwischen kaum mehr greifbar und die Dynamik der inhaltlichen Auseinandersetzung mit der extremen Rechten ist erlahmt. Mein Eindruck ist, dass sich die Autonomen in diesen Regionen deshalb gegenw\u00e4rtig in einem Such- und Orientierungsprozess befinden, wobei noch unklar ist, welche Themen sich perspektivisch dazu eignen, die eigene Militanz politisch zu artikulieren.\n\n#### Lassen sich die sozialen, biografischen Milieus beschreiben, aus denen sich die Angeh\u00f6rigen der autonomen Szene mehrheitlich rekrutieren?\n\nHierzu fehlt eine zureichende empirische Forschung. Im Sinne eines spekulatives R\u00fcckschlusses von den Artikulationsformen auf die sozialen Hintergr\u00fcnde w\u00e4re zu vermuten: Es handelt sich um Jugendliche und junge Erwachsene aus akademischen Milieus sowie um Bildungsaufsteiger, und es w\u00fcrde mich nicht \u00fcberraschen, wenn empirische Forschung einen deutlichen Bezug zum Protestantismus aufzeigen w\u00fcrde.\n\n#### Inwiefern?\n\nZu beobachten ist ein politischer und moralischer Rigorismus, f\u00fcr den eine protestantische Sozialisation immer schon eine f\u00f6rderliche Voraussetzung war. Denkbar sind aber auch andere biografische Hintergr\u00fcnde. So z. B. die Jugendtypische Radikalisierung politischer Haltungen aus dem eigenen Elternhaus.\n\n#### Ist die autonome Szene eine Jugendkultur?\n\nEs handelt sich einerseits um eine politisierte Sub- oder Gegenkultur, die, im Unterschied zu den klassischen Jugendkulturen, keine explizite Distanzierung von der 'Welt der Erwachsenen' vornimmt. Und in autonomen Szenen sind durchaus auch Erwachsene aktiv. Andererseits kann die autonome Szene insofern als eine Jugendkultur charakterisiert werden, wie sie f\u00fcr zahlreiche ihrer Mitglieder ein lebensphasentypisches \u00dcbergangsph\u00e4nomen darstellt; und der politisch-moralische Rigorismus ist zwar kein exklusives Merkmal von Jugendkulturen, in diesen aber in besonders starker Weise ausgepr\u00e4gt. Zudem leitet sich die Form der politischen Militanz nicht einfach aus den Inhalten ab, sondern weist typische Merkmale m\u00e4nnlicher Jugendlichkeit auf. Die Art und Weise, wie Konflikte zwischen Polizisten und Autonomen, etwa bei Demonstrationen, ausgetragen, werden, hat Aspekte, die f\u00fcr Konflikte zwischen Gruppen m\u00e4nnlicher Jugendlicher charakteristisch sind.\n\n#### Was macht die autonome Szene abseits von politischen Inhalten aus, z. B. in Bezug auf Kleidungs- und Lebensstil?\n\nAuff\u00e4llig ist die Uniformierung des Kleidungsstils. Der autonome Spruch \"Ich bin nichts, ich kann nichts, gebt mir eine Uniform!\" adressiert die Polizei, die Selbstuniformierung der Autonomen ist aber nicht zu \u00fcbersehen und Teil der Selbstinszenierung als verschworene Gesinnungsgemeinschaft. Meine Beobachtungen deuten aber nicht darauf hin, dass von einem einheitlichen Lebensstil gesprochen werden kann.\n\n#### Nach dem Ende des linken Terrorismus durch z, iL die R.A.F. sind es vor allem Autonome, die als (linke) Gefahr f\u00fcr die \u00f6ffentliche Ordnung und Sicherheit wahrgenommen werden. Wie gef\u00e4hrlich sind Autonome tats\u00e4chlich f\u00fcr die \u00f6ffentliche Sicherheit oder gar Demokratie?\n\nBeobachtbar sind gegenw\u00e4rtig vielf\u00e4ltige Bem\u00fchungen, insbesondere des Verfassungsschutzes und der Polizei, die Autonomen als Gefahr zu inszenieren. Diese Bem\u00fchungen basieren auf einer Verdachtslogik, die von lokalen Problemen ausgeht, die vor allem in Berlin und Hamburg zu finden sind. Als Urheber werden \"die Autonomen\" dargestellt und dann ganz generell postuliert, dass \"die Autonomen\" eine Gefahr f\u00fcr Sicherheit und Ordnung seien. Es spricht vieles daf\u00fcr, die Ursache dieses Konstrukts in den Interessen der konstruierenden Organisationen zu sehen: Die inszenierte Bedrohung dient der Legitimationsbeschaffung von Polizei und Verfassungsschutz. Gleichwohl ist nicht zu vernachl\u00e4ssigen, dass die politisch-moralischen \u00dcberzeugungen der Autonomen eine problematische Selbstlegitimation umfassen. Nach dem Muster: Im Kampf gegen das B\u00f6se ist den Guten vieles erlaubt. Es gibt in der autonomen Szene durchaus problematische Tendenzen, vor allem im Sinne der Durchf\u00fchrung und Rechtfertigung von Sachbesch\u00e4digungen und gelegentlich auch von physischen \u00dcbergriffen auf Polizisten, die als Bullen denunziert, und auf diejenigen, die von Autonomen als Nazis definiert werden. Das Rechtfertigungsmodell, dass der eigene gute Zweck auch schlechte Mittel rechtfertigt, ist innerhalb der autonomen Szene meiner Wahrnehmung nach nicht konsequent in Frage gestellt.\n\n#### Ist die Reduzierung der autonomen Szene auf Gewalt sinnvoll und richtig?\n\nZweifellos nicht. Der Versuch, die Autonomen zentral als gewaltbereit oder gewaltt\u00e4tig darzustellen, ist ersichtlich Bestandteil einer Kriminalisierungsstrategie und als solcher zu analysieren und zu kritisieren. Zudem verkennt dies die genuin politischen Motive, die f\u00fcr die autonome Szene grundlegend sind. An die Stelle einer politischen Auseinandersetzung mit den Autonomen tritt damit der Versuch, sie straf- und verfassungsrechtlich zu kontrollieren.\n\n#### Wie sch\u00e4tzen Sie die gesamtgesellschaftliche Bedeutung der Autonomen ein? Besteht \u00fcberhaupt das Interesse, die Gesellschaft als Ganzes zu \u00e4ndern, oder geht es nur darum, die eigenen Partikularinteressen zu verteidigen, die eigenen Freir\u00e4ume zu bewahren?\n\nBei den Autonomen handelt es sich um eine marginale politische Gegenkultur, deren gesellschaftspolitischer Einfluss gering ist. Politisch bedeutsam sind die Autonomen als politische und jugendkulturelle Opposition zu rechten Jugendszenen sowie als politisierendes Sozialisationsmilieu. Dass ihr deklariertes Ziel, den Kapitalismus und den Staat zu bek\u00e4mpfen, aussichtslos ist, ist offenkundig. Dagegen ist der Versuch, gegenkulturelle Freir\u00e4ume zu schaffen, aussichtsreicher. Und ich sehe darin mehr als ein Partikularinteresse. Denn es gibt durchaus ein dar\u00fcber hinausgehendes Interesse an einer Gesellschaft, die durch soziokulturelle Vielfalt gekennzeichnet ist und in der progressive politische Gegenkulturen M\u00f6glichkeiten der Entfaltung vorfinden.\n\n#### Welche Erfahrungen haben Sie pers\u00f6nlich im Umgang mit Autonomen gemacht?\n\nIch habe Autonome immer wieder als engagierte und kompetente politische Diskutanten und als freundliche Menschen erlebt. Grenzen der Verst\u00e4ndigung entstehen aber dann, wenn die ideologischen Legitimationen der Militanz gegen Infragestellungen abgeschottet werden.\n\n#### Viele Programme gegen \"Rechtsextremismus\" wurden von der letzten Bundesregierung erweitert, um gegen alle Formen des \"Extremismus\" \u2013 also auch den \"Linksextremismus\" \u2013 vorgehen zu k\u00f6nnen. Wie sehen Sie die Programmausweitung und die der zugrunde liegenden \"Extremismustheorie\"?\n\nDer Programmausweitung liegt keine Extremismustheorie zugrunde, sondern ein recht schlichtes politisches Denkmodell. Dieses missachtet die liberalen Kernannahmen des Grundgesetzes und zielt auf eine formierte Gesellschaft, in welcher der Spielraum dessen, was politisch gedacht und gesagt werden darf, erheblich eingeschr\u00e4nkt ist. Daraus resultiert die Vorstellung, dass politische Denkweisen, die sich jenseits des Spektrums der etablierten Parteien bewegen, extrem und insofern inakzeptabel oder gar gef\u00e4hrlich seien. Zudem erlaubte es die Programmausweitung der CDU, ein Problem zu l\u00f6sen, das sie seit den 90er Jahren mit den Programmen gegen Rechtsextremismus hatte: Das Problem, wie unsichtbar gemacht werden kann, dass es durchaus ein konservatives Sympathisantenumfeld der extremen Rechten gibt und die Grenzen zwischen der sog. Mitte der Gesellschaft und der extremen Rechten unscharf sind. Wenn dagegen generell \"der Extremismus\" zum Gegner erkl\u00e4rt wird, l\u00f6st sich dieses Abgrenzungsproblem auf einfache Weise. Denn Extremismus in dem von ihr verstandenen Sinn muss sich die CDU nicht vorwerfen lassen, und zweifellos auch keine ideologische N\u00e4he zum Antikapitalismus der Autonomen. Etwas theoretischer gesprochen: Extremismustheorien haben einen systematischen blinden Fleck: Sie \u00fcbersehen, dass die radikale Linke, ganz im Gegensatz zur extremen Rechten, nicht gegen Demokratie und Menschenrechte ist, sondern aktiv f\u00fcr die Ausweitung von Demokratie und Menschenrechten eintritt. Genau das ist aber die entscheidende Grundlage f\u00fcr eine politische Auseinandersetzung und eine politische Bildungsarbeit mit den Autonomen.\n\n#### Wo w\u00e4re noch ein gro\u00dfer Forschungsbedarf hinsichtlich der autonomen Szene?\n\nIch gehe davon aus, dass es sich bei der autonomen Szene um eine komplexe Gemengelage aus 1. ernst zu nehmendem politischen Engagement, 2. etwas zu einfach gefassten gesellschaftskritischen Sichtweisen und 3. Ritualen und Dynamiken politisch-ideologischer Vergemeinschaftung handelt. Interessant w\u00e4re meines Erachtens eine Forschung, die das Zusammenwirken dieser drei Elemente etwas n\u00e4her in den Blick nimmt und dadurch Einsichten in M\u00f6glichkeiten der politischen Auseinandersetzung und der politischen Bildungsarbeit gewinnt.\n\n#### Warum interessiert Sie pers\u00f6nlich dieses Thema?\n\nIch habe durchaus Sympathien f\u00fcr die antikapitalistischen und antirassistischen Motive der Autonomen, aber eine sehr kritische Haltung gegen\u00fcber Dogmatismen, Schlie\u00dfungstendenzen und den immanenten Dynamiken ihrer politisch-ideologischen Vergemeinschaftung. Ich halte es f\u00fcr wichtig, gegen die Versuche einer inneren Ausb\u00fcrgerung der Autonomen vorzugehen und mit ihnen in einen Dialog einzutreten.\n\n#### Literatur\n\nGeronimo: Autonome in Bewegung: Aus den ersten 23 Jahren. Assoziation A, Berlin 2003.\n\nScheerer, Sebastian: \"Deutschland: Die ausgeb\u00fcrgerte Linke\", in: Soziale Entwicklung und Terrorismus. Bd. 1. Suhrkamp, Frankfurt 1988, S. 193\u2013429.\n\nScherr, Albert: \"Ordnungsstiftende und illegitime Gewalt. Perspektiven reflexiver Gewaltforschung\", in: Soziale Passagen, H. 2\/2010, S. 169\u2013181.\n\nScherr, Albert: \"Jugendarbeit mit linksextremen Jugendlichen?\", in: deutsche Jugend, H. 6, 2010, S. 251\u2013255.\n\nScherr, Albert: Jugendsoziologie. 9. Auflage. VS Verlag f\u00fcr Sozialwissenschaften, Wiesbaden 2009.\n\nScfawarzmeier, Jan: Die Autonomen zwischen Subkultur und sozialer Bewegung. Books on Demand, Hamburg 2000.\n\n## Feindbeobachtung\n\nFelix Krautkr\u00e4mer, 35 Jahre, Redakteur der Junge Freiheit.\n\nFelix Krautkr\u00e4mer, 1979 in Freiburg im Breisgau geboren, studierte Geschichte, Politik und \u00d6ffentliches Recht und arbeitet seit November 2007 f\u00fcr die Junge Freiheit derzeit als Redakteur f\u00fcr Innenpolitik und verantwortlich f\u00fcr die Online-Redaktion.\n\n#### Wie sch\u00e4tzen Sie den Forschungsstand \u00fcber die autonome Szene ein?\n\nWie beim gesamten Thema Linksextremismus gibt es auch hier erheblichen Nachholbedarf, vor allem an wissenschaftlicher Literatur. Es gibt, mit einigen publizistischen Ausnahmen aus dem Umfeld der beiden Extremismusforscher Eckhard Jesse und Uwe Backes, so gut wie keine Untersuchungen \u00fcber die autonome Szene, ihre einzelnen Bet\u00e4tigungsfelder, regionale Unterschiede, Mobilisierungsmethoden, Informations- und Kommunikationswege, das Verh\u00e4ltnis zu anderen linksextremistischen Gruppen oder internationale beziehungsweise l\u00e4nder\u00fcbergreifende Zusammenarbeit. Und zum Thema Gewalt, Stichwort Militanzdebatte. Der Einsatz von Gewalt als legitimes Mittel in der politischen Auseinandersetzung spielt in der autonomen Szene eine wichtige Rolle. Auch w\u00e4re es interessant zu untersuchen, ob und wie sich die Themenfelder der autonomen Szene in den vergangenen 20 Jahren ver\u00e4ndert haben. Gibt es neue Priorit\u00e4ten, gibt es Kontinuit\u00e4ten, wie reagiert die autonome Szene auf den gesellschaftlichen Wandel und politische Ver\u00e4nderungen?\n\n#### Welche Ver\u00e4nderungen haben Sie bei den Autonomen seit ihrer Entstehung wahrgenommen und inwiefern lassen sie sich mit gesellschaftlichen Wandlungsprozessen in Verbindung setzen?\n\nEs gibt ja nicht die Autonomen, sondern Thematik und Vorgehensweise autonomer Gruppen sind immer auch von aktuellen politischen und gesellschaftlichen Entwicklungen gepr\u00e4gt. Hinzu kommen regionale Unterschiede. In Mitteldeutschland beispielsweise spielt der Kampf um sogenannte \"Freir\u00e4ume\", also die Besetzung leerstehender H\u00e4user und selbstverwaltete autonome Jugend- und Kulturzentren, nicht so eine bedeutende Rolle wie beispielsweise in Berlin oder Hamburg. Und selbst hier l\u00e4sst sich beobachten, dass dieses Kernanliegen der autonomen Bewegung mal mehr und mal weniger Konjunktur hat. In Berlin und Hamburg spielte das Thema zu Beginn des Jahrtausends l\u00e4ngst nicht mehr die Rolle wie noch in den beiden Jahrzehnten zuvor. Seit einigen Jahren gewinnt der Kampf um Freir\u00e4ume unter dem Schlagwort \"Gentrifizierung\" allerdings wieder an Bedeutung. Die R\u00e4umung der \"Liebig 14\" in Berlin oder der drohende Verkauf der \"Roten Flora\" in Hamburg mobilisierte zunehmend die Szene. Insgesamt l\u00e4sst sich aber feststellen, dass es gewisse Themen gibt, die f\u00fcr jede autonome Gruppe zu jeder Zeit wichtig waren und sind. Unterschiedlich ist nur die einzelne Gewichtung. Die bedeutendsten davon sind sicherlich der Kampf f\u00fcr \"Freir\u00e4ume\" und gegen Gentrifizierung, gegen Rassismus und Neonazismus, gegen Kapitalismus und Kommerz, gegen Polizei und Repression (\"\u00dcberwachung\") und Bundeswehr (\"Militarismus\"). Zus\u00e4tzlich gibt es Themen, die keine prim\u00e4ren Anliegen von Autonomen sind, dennoch aber f\u00fcr Aktionen oder zur Mobilisierung der eigenen Szene genutzt werden, zum Beispiel der Kampf gegen Atomkraft, gegen Sexismus sowie f\u00fcr Tierschutz\/Tierrechte.\n\n#### Sind die Autonomen aus den 80er Jahren mit denen aus der heutigen Zeit zu vergleichen?\n\nVom Habitus her schon. Formulierungen, Vorgehensweise und Feindbilder der heutigen Autonomen \u00e4hneln denen der 80er und 90er Jahre beziehungsweise sind mit diesen identisch. Selbst die Szene-Medien wie Interim sind die gleichen geblieben. Allerdings suchen die heutigen Autonomen teilweise, wenn auch anonym, gezielter die \u00d6ffentlichkeit, zum Beispiel \u00fcber Erkl\u00e4rungen im Internet \u2013 und das nicht nur, um irgendwelche Bekennerschreiben zu ver\u00f6ffentlichen. Erinnert sei beispielsweise an das auf Indymedia ver\u00f6ffentlichte Foto, auf dem sich im April 2010 etwa 50 Berliner Autonome in martialischer Pose mit Schlagst\u00f6cken und Baseballschl\u00e4gern unter der Parole \"Nazis auf's Maul\" ablichten lie\u00dfen. Das war mit Sicherheit mehr f\u00fcr die Presse als f\u00fcr die eigene Szene oder zur Abschreckung von Rechtsextremisten in Prenzlauer Berg gedacht. In den vergangenen zehn Jahren war zudem eine gewisse Zersplitterung der Szene zu beobachten. Zwar gab es zwischen autonomen Gruppen immer noch St\u00e4dte- und l\u00e4nder\u00fcbergreifende Zusammenarbeit, aber nicht mehr in dem Ausma\u00df wie noch bis zur Mitte der 90er Jahre. Auch hat die autonome Szene in einigen ihrer einstigen Hochburgen wie beispielsweise G\u00f6ttingen an Schlagkraft verloren. Seit einiger Zeit bem\u00fchen sich die autonomen Gruppen durch \"Autonome Vollversammlungen\" und \"Kongresse f\u00fcr autonome Politik\" jedoch wieder verst\u00e4rkt um eine engere Vernetzung und Absprache untereinander.\n\n#### Lassen sich die sozialen, biografischen Milieus beschreiben, aus denen sich die Angeh\u00f6rigen der autonomen Szene mehrheitlich rekrutieren?\n\nSt\u00e4dtisch, durchschnittlich bis gut gebildet, selten Arbeiterkinder, eher Mittelschicht. Autonome gehen weniger einer geregelten Arbeit nach. Es gibt aber auch einen hohen Anteil an Auszubildenden oder Studenten. Sie sind in der Regel kinderlos, unter 30 (laut einer Studie des Berliner Verfassungsschutzes waren die Tatverd\u00e4chtigen linker Gewalt im Durchschnitt 23 \u00bd Jahre alt) und der M\u00e4nneranteil \u00fcberwiegt deutlich.\n\n#### Ist die autonome Szene eine Jugendkultur?\n\nNur bedingt. Es gibt zwar mittlerweile so etwas wie einen autonomen \"Lifestyle\" mit eigener Szene-Bekleidung und eigenen Szene-Getr\u00e4nken. Es gibt extra Online-Shops, in denen man sich, nicht gerade kosteng\u00fcnstig, szenekonform einkleiden kann. Die autonome Szene bem\u00fcht sich durchaus, auf Jugendliche attraktiv zu wirken. Auch beim Thema Musik hat man sich weiterentwickelt und beschr\u00e4nkt sich l\u00e4ngst nicht mehr nur auf Punk. Seit einigen Jahren bietet die autonome Szene mit linksradikalem HipHop auch eine Musikrichtung an, die eher den Nerv heutiger Jugendlicher trifft. Gegen das Argument einer Jugendkultur spricht aber der hohe Grad der Politisierung der Autonomen. Zwar gibt es auch hier Mitl\u00e4ufer, der Anteil der \"\u00dcberzeugungst\u00e4ter\" \u00fcberwiegt aber. Der \"harte Kern\" steht \"Mode-Autonomen\" zudem ablehnend bis feindlich gegen\u00fcber, was der Entwicklung einer Jugendkultur entgegenwirkt.\n\n#### Welche Rolle spielen Frauen in der autonomen Szene?\n\nSicherlich sind Frauen in der autonomen Szene gemessen an ihrem Bev\u00f6lkerungsanteil unterrepr\u00e4sentiert, aber das ist bei allen extremistischen Gruppen so. Je radikaler, desto h\u00f6her der M\u00e4nneranteil. Es gibt aber durchaus auch Frauen im \"Schwarzen Block\" und in Gruppen, die Anschl\u00e4ge ver\u00fcben. Laut einer Untersuchung des Berliner Verfassungsschutzes waren von 864 Tatverd\u00e4chtigen linker Gewalt 81 Prozent m\u00e4nnlich und 19 Prozent weiblich. Dabei spielte es keine Rolle, ob sich die Gewalt gegen Personen oder gegen Sachen richtete. Der Anteil von Frauen bei politisch links motivierter Gewalt liegt damit deutlich h\u00f6her als ihr Anteil bei allgemeinkriminellen Gewaltdelikten. Dies k\u00f6nnte daher kommen, dass sich Frauen in der linksextremen Szene besonders gen\u00f6tigt f\u00fchlen, zu beweisen, dass sie den M\u00e4nnern in nichts nachstehen. Schlie\u00dflich wird hier ja die Gleichheit der Geschlechter in besonderem Ma\u00dfe propagiert.\n\n#### Was sind Ihrer Meinung nach die zentralen Medien und Informationsvermittler f\u00fcr die autonome Szene?\n\nZum einen das Internet, hier sind besonders Indymedia sowie zahlreiche regionale Antifa-Seiten zu nennen. Gerade f\u00fcr Aktionen wie das sogenannte \"Nazi-Outing\", bei dem Personen, die als politische Gegner ausgemacht sind, gezielt mit ihren privaten Daten wie Adresse, Arbeitgeber, Kfz-Kennzeichen, Freunden und so weiter in die \u00d6ffentlichkeit gezerrt werden, wird das Internet genutzt. Das Outing findet zuerst per Flugblatt in der Nachbarschaft der betroffenen Person statt und hinterher wird das Outing dann nochmals \u00fcber das Internet verbreitet, um so den sozialen Druck zu erh\u00f6hen. Daneben spielen aber vor allem die Szenepublikationen eine wichtige Rolle. Die bedeutendste ist hier mit Sicherheit Interim. Daneben dienen auch Infol\u00e4den und Szenetreffs dem Informationsaustausch und der Vernetzung. Insgesamt laufen Treffen von Autonomen aber sehr konspirativ ab. Die Szene ist sehr misstrauisch, regelm\u00e4\u00dfige Kontakte gibt es nur unter pers\u00f6nlich Bekannten. Die Kontaktaufnahme f\u00fcr Au\u00dfenstehende ist schwierig.\n\n#### Wie gef\u00e4hrlich sind Autonome tats\u00e4chlich f\u00fcr die \u00f6ffentliche Sicherheit oder gar Demokratie?\n\nEine wirklich ernstzunehmende Gefahr droht der Demokratie hierzulande weder von rechten noch von linken politischen Extremisten. Mit der \u00f6ffentlichen Sicherheit sieht das schon anders aus. Diese kann vor und w\u00e4hrend wichtiger Ereignisse f\u00fcr die Szene (R\u00e4umung von Szeneobjekten, Sicherheitskonferenzen, 1. Mai, Castor-Transporte, Demonstrationen von Rechtsextremisten) durchaus bedroht sein, wie Sabotageakte an Bahnanlagen, Brandstiftungen, Anschl\u00e4ge mit Sprengs\u00e4tzen, aber auch Vandalismus, Landfriedensbruch, K\u00f6rperverletzungen, N\u00f6tigungen und Bedrohungen zeigen. Ein \"eindrucksvolles\" Beispiel hierf\u00fcr lieferte kurz vor Weihnachten 2013 die autonome Szene in Hamburg.\n\n#### Ist die Reduzierung der autonomen Szene auf Gewalt sinnvoll und richtig?\n\nEs stellt sich weniger die Frage, ob es sinnvoll und richtig ist, sondern warum die autonome Szene h\u00e4ufig mit Gewalt in Verbindung gebracht wird. Und dies kommt nun mal daher, dass der \u00fcberwiegende Teil ein unkritisches bis positives Verh\u00e4ltnis zur Gewalt hat.\n\n#### Gibt es eine einheitliche Haltung innerhalb der autonomen Szene zur Gewalt?\n\nSowenig wie es die autonome Szene gibt, sowenig gibt es eine einheitliche Haltung zur Gewalt. Die Tendenz ist allerdings eindeutig positiv. Die Mehrheit der Autonomen h\u00e4lt Militanz f\u00fcr ein legitimes Mittel im Kampf f\u00fcr die eigenen politischen Vorstellungen. Unterschiedliche Ansichten gibt es aber zum Einsatz von Gewalt gegen Menschen. Ein Teil der Szene h\u00e4lt Gewalt gegen bestimmte Personengruppen, zum Beispiel Polizisten oder Rechtsextremisten, f\u00fcr richtig und notwendig. Andere lehnen dies ab.\n\n#### Wie sch\u00e4tzen Sie die gesamtgesellschaftliche Bedeutung der Autonomen ein?\n\nAutonome sind, wie alle radikalen oder extremistischen Gruppierungen, Randgruppen. Ihr gesamtgesellschaftlicher Einfluss ist daher eher gering. Allerdings kommt es immer wieder zu B\u00fcndnissen mit etablierten Parteien und gesellschaftlichen Institutionen, zum Beispiel bei Demonstrationen gegen Rechtsextremisten, was den Einfluss der autonomen Szene steigert und ihre Positionen gesellschaftsf\u00e4hig macht. Wirkliche Ver\u00e4nderungen sind nur dann m\u00f6glich, wenn sich ein Anliegen der Autonomen mit dem breiterer Gesellschaftsgruppen deckt wie bei den Anti-Atom-Protesten. Nur braucht es daf\u00fcr die Autonomen im Grunde genommen nicht. Ansonsten geht es vielen Autonomen meines Erachtens nach haupts\u00e4chlich um ihre pers\u00f6nlichen Anliegen, beispielsweise um den Erhalt eines besetzten Hauses. Sicherlich gibt es auch ein paar Idealisten, aber die Masse der Autonomen verh\u00e4lt sich \u2013 abgesehen von den regelm\u00e4\u00dfigen Demonstrationen gegen Rechtsextremismus, Abschiebung oder \"Polizeigewalt\" \u2013 solange verh\u00e4ltnism\u00e4\u00dfig ruhig, bis sie sich durch ein Ereignis oder bestimmte Entwicklungen beeintr\u00e4chtigt f\u00fchlt. Wobei dies nicht zwangsl\u00e4ufig die eigene Gruppe selbst betreffen muss, sondern auch anderswo in Deutschland, Europa und teilweise weltweit stattfinden kann. Als beispielsweise 2007 das Ungdomshuset in Kopenhagen ger\u00e4umt wurde, kam es in Deutschland gleich in mehreren St\u00e4dten zu Solidarit\u00e4tsdemonstrationen.\n\n#### Wie sch\u00e4tzen Sie die Zukunft der autonomen Bewegung ein?\n\nEs wird immer wieder Phasen geben, in denen die autonome Szene st\u00e4rkeren Zulauf hat und sich einer gr\u00f6\u00dferen Aufmerksamkeit erfreut. Das l\u00e4sst sich etwa auch seit ein paar Jahren wieder beobachten, vor allem in einigen westdeutschen Gro\u00dfst\u00e4dten. Dabei spielte neben den zahlreichen Wahlen 2013 auch die sich versch\u00e4rfende Fl\u00fcchtlingsproblematik (Antirassismus, Solidarit\u00e4t mit Fl\u00fcchtlingen) eine Rolle. Ebenso wie die \u00f6ffentlichere Pr\u00e4senz der Bundeswehr, die nach dem Wegfall der Wehrpflicht nun intensiver um Nachwuchs werben muss (Antimilitarismus). Langfristig werden die Autonomen, abgesehen vielleicht von letzten Bastionen in Berlin und Hamburg, in der Bedeutungslosigkeit verschwinden. Ganz einfach, weil immer weniger Menschen sich mit ihren Zielen identifizieren beziehungsweise Verst\u00e4ndnis f\u00fcr ihre Vorgehensweise aufbringen. Den Autonomen fehlt es an einem massentauglichen Thema. Es k\u00f6nnte aber auch sein, dass sich die Autonomen in dem Moment erledigt haben, in dem hierzulande kein Geld mehr zur Unterst\u00fctzung ihrer Anliegen vorhanden ist; wenn Druckerzeugnisse und Veranstaltungen nicht mehr aus F\u00f6rdermitteln querfinanziert werden, wenn Szenetreffs nicht mehr zu einem Gutteil von der \u00f6ffentlichen Hand getragen werden. Gerade letzteres ist ohnehin aus autonomer Sicht etwas inkonsequent. Die Treffs bezeichnen sich gerne als \"autonome Zentren\", in denen der Staat nach Ansicht der Betreiber nichts zu suchen hat. Doch dessen Geld nimmt man gerne, sei es, indem die Stadt oder die Kommune die Miete sowie die Nebenkosen bezahlt oder die R\u00e4umlichkeiten mietfrei zur Verf\u00fcgung stellt. Wenn solche Unterst\u00fctzungen wegfallen, werden zwar die Krawalle kurzzeitig wieder aufflammen und eventuell auch h\u00e4rter ausfallen. Sp\u00e4testens, wenn die Szene aber merkt, dass trotz allem Druck kein Geld mehr flie\u00dft \u2013 schlicht, weil es einfach nichts mehr zu verteilen gibt \u2013 und der Staat dagegenh\u00e4lt, wird sich auch das wieder legen.\n\n#### Viele Programme gegen \"Rechtsextremismus\" wurden von der letzten Bundesregierung erweitert, um gegen alle Formen des \"Extremismus\" \u2013 also auch den \"Linksextremismus\" \u2013 vorgehen zu k\u00f6nnen. Wie sehen Sie die Programmausweitung und die ihr zugrunde liegende \"Extremismustheorie\"?\n\nWenn schon von einem Extremismusbegriff ausgegangen wird, dann muss dieser konsequenterweise auch auf beiden Seiten des politischen Spektrums angewandt werden \u2013 links wie rechts. Insofern ist die Ausweitung der Anti-Extremismusprogramme richtig. Auf einem anderen Blatt steht allerdings, ob der Extremismusbegriff \u00fcberhaupt sinnvoll ist. Wer legt denn fest, was extremistisch ist und was nicht? Meistens doch der Staat beziehungsweise die regierenden Parteien. Ein Beispiel: In L\u00e4ndern, in denen die Linkspartei auf der Oppositionsbank sitzt, taucht sie meistens im Verfassungsschutzbericht auf. In den L\u00e4ndern, in denen sie an der Regierung beteiligt ist, handelt es sich auf einmal um unverd\u00e4chtige Demokraten. Was die F\u00f6rderprogramme angeht: Ich halte wenig von solchen Programmen. F\u00fcr die Bek\u00e4mpfung von politischen und religi\u00f6sen Extremisten sind die Sicherheitsbeh\u00f6rden zust\u00e4ndig \u2013 und nicht die regierenden Parteien oder die sogenannte \"Zivilgesellschaft\".\n\n#### Nicht erst seit dem gescheiterten NPD-Verbotsantrag wissen wir, dass der Verfassungsschutz ein wichtiger Aufbauhelfer f\u00fcr die rechtsextreme Szene ist Welche Bedeutung haben Ihrer Einsch\u00e4tzung nach Verfassungsschutz und verdeckte Polizeiermittler hinsichtlich der autonomen Szene?\n\nAuch wenn es in der Vergangenheit einige prominente F\u00e4lle von staatlicher Ein-flussnahme auf die autonome Szene gegeben hat (mg-Prozess, Mark Kennedy und \"Simon Brenner\"), d\u00fcrfte es sich dabei eher um Einzelf\u00e4lle handeln. Die ideologische Geschlossenheit und die gefestigte politische \u00dcberzeugung der autonomen Szene machen es f\u00fcr die Geheimdienste schwieriger, sie zu infiltrieren, als es bei der rechtsextremen Szene der Fall ist. Hinzu kommt, dass die linksextreme Szene den Verfassungsschutz als \"Aufbauhelfer\" nicht braucht, da sie \u00fcber ausreichende Unterst\u00fctzung bis hinein in die etablierten Parteien und die sogenannte \"Mitte der Gesellschaft\" verf\u00fcgt.\n\n## \"'Jugendkultur' ist ein Ausdruck wie ein Sack, in den man alles reinstecken kann, was man selbst, \u00e4lter geworden, nicht versteht.\"\n\nProf. Dr. Wolf-Dieter Narr, 77 Jahre, Menschenrechtsaktivist und Demonstrationsbeobachter.\n\nProf. Dr. Wolf-Dieter Narr, Jahrgang 1937, studierte von 1957 bis 1962 in W\u00fcrzburg, T\u00fcbingen und Erlangen. Seine Promotion und die Habilitation 1969 erfolgten in Konstanz. Er war Fellow an der John F. Kennedy School of Government der Harvard University und lehrte von 1971 bis 2002 als Professor f\u00fcr empirische Theorie der Politik am Otto-Suhr-Institut (OSI) der Freien Universit\u00e4t Berlin. Er verzichtete zusammen mit Peter Grottian auf ein Drittel seiner Professur, um damit eine Professur f\u00fcr Frauenforschung zu erm\u00f6glichen.\n\nEr beteiligte sich im Januar 1959 am Studentenkongress gegen Atomr\u00fcstung, war 1968 Mitbegr\u00fcnder der Bundesassistentenkonferenz, einer treibenden Kraft bei der sp\u00e4teren Hochschulreform. 1969 trat er aus Protest gegen die Gro\u00dfe Koalition aus der SPD aus. Mitte der 70er Jahre arbeitete an der Freien Universit\u00e4t Berlin unter seiner Leitung eine kleine Gruppe zum Themenkomplex \"Politik Innere Sicherheit\". Aus dieser \"AG B\u00fcrgerrechte\" entstand ein gemeinn\u00fctziger Verein mit dem Namen Institut f\u00fcr B\u00fcrgerrechte & \u00f6ffentliche Sicherheit e. V., der bis heute das Magazin B\u00fcrgerrechte und Polizei herausgibt, bei dem Wolf-Dieter Narr Mitherausgeber ist. 1978\/1979 war er Mitglied des Deutschen Beirats beim 3. Russell-Tribunal \u00fcber die Situation der Menschenrechte in der Bundesrepublik. Aus dessen Unterst\u00fctzerbewegung ging 1980 das Komitee f\u00fcr Grundrechte und Demokratie hervor, dessen Sprecherposten Narr zeitweise innehatte. Ebenso war er Mitorganisator des Foucault-Tribunals zur Lage der Psychiatrie, das 1998 in Berlin stattfand. Seit langer Zeit engagiert er sich in der Fl\u00fcchtlingspolitik, so lebte er beispielsweisef\u00fcr zwei Tage im Fl\u00fcchtlingslager Bramsche. Seit 2002 publiziert Wolf-Dieter Narr auch in der Zeitung Graswurzelrevolution.\n\nSeine zahlreichen (Buch-)Publikationen besch\u00e4ftigen sich mit Menschenrechten, Globalisierung und Demokratie.\n\n#### Was wissen wir \u00fcber die autonome Szene?\n\nDas ist sehr punktuell. Wenn von Autonomen gesprochen wird, dann mit der Verwendung des bestimmten Artikels, dann hei\u00dft es DIE Autonomen. Aus meiner eigenen Besch\u00e4ftigung, sowohl was die Lekt\u00fcre angeht als auch den Umgang mit Gruppierungen, die zu den Autonomen gerechnet werden, wei\u00df ich, dass der bestimmte Artikel nicht stimmt. Sie werden im Grunde wie ein gr\u00e4uliches Subjekt charakterisiert, vor dem man Angst haben muss. Sie schwanken angeblich zwischen Anarchie und Terrorismus. Beide Bezeichnungen und \u00c4u\u00dferungsformen werden in Deutschland zusammengeschmissen. Meine Erfahrungen beginnen mit der Mescalero-Aff\u00e4re. Damals hat sich eine Gruppe von Stadtindianern in G\u00f6ttingen 1977 nach dem Mord an Herrn Buback ge\u00e4u\u00dfert. Sie haben in einem Brief ihre gezielt missinterpretierte \"klammheimliche Freude\" an Bubacks Tod ge\u00e4u\u00dfert. Wenn man den Brief ohne die offiziell erzeugte Panik liest, ist nichts von einem Lob dieses Anschlags oder einer Freude zu erkennen. Die G\u00f6ttinger Gruppe hat vor allem ihre Distanz zur Aktion ge\u00e4u\u00dfert; sie haben aber nicht auf die Abscheudr\u00fcsen gedr\u00fcckt und Begriffe gebraucht, die erregte Reaktionen hervorriefen. In den Medien wurden nur Ausschnitte des Briefes ver\u00f6ffentlicht. Kollegen von mir haben sich damit auseinandergesetzt. Die bundesrepublikanische Fang-Frage lautete seinerzeit: Wie h\u00e4ltst du es mit der RAF und der von ihr propagierten Gewalt? Man konnte kaum noch vern\u00fcnftig differenzieren, weil jeder entweder polizeilich oder auch von den RAF-Anh\u00e4ngern zum Feind gestempelt worden ist. Entgegen solcher hysterischer Reaktionen haben Kollegen von mir den gesamten Brief herausgegeben, damit ihn jeder vollst\u00e4ndig lesen kann. Ich h\u00e4tte mitmachen k\u00f6nnen, mir war jedoch das Vorwort nicht ausreichend. Als hinterher eine geradezu staatsoffizielle Kampagne gegen die Kollegen entfesselt wurde, als sei die Publikation des Briefes fast schon ein terroristischer Akt, habe ich es sehr bedauert, nicht mitver\u00f6ffentlicht zu haben. W\u00e4re ich verfolgt worden, w\u00e4re ich nicht zu Kreuze gekrochen wie die meisten der Herausgeber aus Niedersachsen. Die mussten sich dem damaligen Wissenschaftsminister Prestel unters widerrufgekr\u00fcmmte Joch beugen. Sie f\u00fcrchteten \u2013 im Auge der panikerhitzten Situation begr\u00fcndet \u2013 um ihre Positionen. Der Einzige, der das nicht unterschrieben hat, war Peter Br\u00fcckner. Er wurde dann auch dienstlich suspendiert. Die gesamte \u00f6ffentliche Reaktion auf den publizierten Brief gegen\u00fcber denjenigen war ma\u00dflos, die ihn nur herausgegeben hatten, sich aber nicht damit identifizierten. So wie der Brief als Mescalero-Aff\u00e4re aufgeblasen worden ist, zeigt sie die ganze Perspektivlosigkeit und den Mangel an Augenma\u00df der verfolgenden Beh\u00f6rden wie der politisch-publizistischen Repr\u00e4sentanten. Das ist bei nahezu allen Fragen sog. \u00f6ffentlicher Sicherheit heute noch der Fall \u2013 auch darum ein in negativen Folgen anhaltendes St\u00fcck unbew\u00e4ltigter bundesdeutscher Vergangenheit!\n\nWir haben angesichts des notorischen Mangels einigerma\u00dfen verl\u00e4sslicher Informationen im Komitee f\u00fcr Grundrechte und Demokratie fr\u00fch beschlossen, Demonstrationsbeobachtung zu betreiben, damit eine unabh\u00e4ngige Gruppierung vor Ort ist. Mitunter beobachten wir mit zwanzig Leuten und mehr teilnehmend eine Demonstration und versuchen, verl\u00e4ssliche Informationen zu bieten. Die letzte Demonstration, an der ich teilnahm, fand rund um Heiligendamm statt. Eine Gruppe \u00e4u\u00dferlich erkenntlicher 'Autonomer' nahm daran teil. Die Art und Weise, wie Autonome verschw\u00f6rungspraktisch projektiv als allzeit \"gewaltbereit\" stilisiert werden, belegt die vorhin genannte Ma\u00dflosigkeit des konstruierenden politisch-polizeilichen Blicks. Aus eigenen Erfahrungen wei\u00df ich, wie sehr Projektionen verzerren k\u00f6nnen. Als bildeten \"die\" Autonomen eine teuflische Gefahr. Am zweiten oder dritten Tag der Demonstrationen in Heiligendamm, als wir wieder von einem Ort zum n\u00e4chsten marschierten, habe ich erlebt, wie geradezu diszipliniert sich der sogenannte Schwarze Block benommen hat. Die Polizei hat die Demonstrationsz\u00fcge immer wieder ohne erkennbaren Grund aufgehalten. Die Polizei hat nicht rational nachvollziehbare Ma\u00dfnahmen ergriffen und zum Schluss sogar die friedliche Demonstration aufgel\u00f6st. \"Die\" Autonomen haben sich verhalten, wie man sich verhalten w\u00fcrde, wenn man zu guten B\u00fcrgerleuten geladen wird, ein wei\u00dfes Tischtuch auf dem Tisch, auf das man keinen Fleck machen m\u00f6chte. Sie verhielten sich aggressionslos. Kein fernes Zeichen von Gewalt war erkenntlich, obwohl es zig Situationen gab, die h\u00e4tten, und sei's nur spielerisch, verlocken k\u00f6nnen. Beispielsweise haben neben der Demonstrationsstrecke lockere Steine gelegen, niemand hat einen Stein genommen und gegen die Polizei geworfen.\n\nAuch in den 80er Jahren, als in Wackersdorf gegen das geplante Kernkraftwerk demonstriert wurde, war ich als Demonstrationsbeobachter dabei. \"Die\" Autonomen waren junge Erwachsene, etwa Studierende, mit denen ich zu tun hatte. Sie \u00e4ngstigten sich angesichts der Polizei, die ihrerseits unachtsam und rabiat mit ihnen umgegangen ist. Als Beobachter hielt ich mich mitten im Block auf. Weil ich nicht schwarz gekleidet und schon \u00e4lter war, misstraute man mir. Ob ich etwa nicht \"Verfassungssch\u00fctzer\" sei. Ich sollte ihnen deshalb meinen Ausweis zeigen. Wir einigten uns darauf, dass niemand sich ausweisen m\u00fcsse. Es gibt Situationen, zum Beispiel manche 1.-Mai-Demonstration in (West-)Berlin, da haben manche, die gemeinsam eine sogenannte Spa\u00dfguerilla bildeten, auch Steine geworfen. Ich habe jedoch keine Gruppe erlebt, die bewusst auch \"nur\" Sachen zerst\u00f6rt h\u00e4tte oder Polizeileute oder andere in irgendeiner Weise absichtlich gef\u00e4hrdet h\u00e4tte. Wenn es darauf angekommen w\u00e4re, h\u00e4tte man die von ihr ausgehenden beschr\u00e4nkten Gefahren umgehen k\u00f6nnen. Ich halte es f\u00fcr ein wechselseitig produziertes Ger\u00fccht, Autonome seien grunds\u00e4tzlich, wie es in der pauschalen Diskriminierung hei\u00dft, \"gewaltbereit\". Sie k\u00f6nnen dann konsequenterweise polizeilich wie schon Gewaltt\u00e4tige behandelt werden. Diese nicht bewiesene pr\u00e4ventive Annahme erlaubt nur ihrerseits einen enthemmten polizeilichen Gewaltgebrauch. Provozierende gibt es \u00fcberall, bekanntlich auch in der Polizei. Ich war auf vielen Demos dabei. Soweit ich mit Autonomen zu tun hatte, war ich nie gef\u00e4hrdet. Ich habe keine Situation miterlebt, die es rechtfertigte, sie zum Kollektiv-Subjekt zu erheben, das voll terroristischer Energie sei, wie es im \u00a7 129 StGB, diesem einem grundrechtlich demokratischen Strafrecht entgegengesetzten Paragraphen, rechtsunsicher normiert worden ist. Ich habe nie etwas gesehen, dass man sagen k\u00f6nnte, da gibt es eine Gruppe, die man so verfolgen muss, dass ihr nicht die Chance einger\u00e4umt werden d\u00fcrfe, sich politisch zu artikulieren.\n\n#### Denken Sie, dass die autonome Szene eine Jugendkultur ist?\n\n\"Jugendkultur\" ist ein Ausdruck wie ein Sack, in den man alles reinstecken kann, was man selbst, \u00e4lter geworden, nicht versteht. Mein Eindruck ist, dass die Mehrheit der Menschen, die sich ab und an und zeitweise als \"autonom\" verstehen, junge Leute sind. Es sind in der Mehrheit Leute, die aus vielen Gr\u00fcnden im zeitweisen Abstand zur etablierten Bundesrepublik leben. Das tue ich \u00fcbrigens aus guten Gr\u00fcnden ein l\u00e4ngst alt gewordenes Leben lang auch. Es sind oft Leute, die noch nicht fest in Berufen stecken, die vielfach arbeitslos sind oder erst sich in der Ausbildung befinden. Deswegen sind auch so viele Studies dabei \u2013 oder waren es, bevor aus dem Studium ein prek\u00e4rer Lauf durch Notenspie\u00dfe und keuchende Konkurrenz geworden ist. Es sind meist Intellektuelle, die noch nicht vom 'System' kooptiert worden sind. Das soll allerdings nicht hei\u00dfen, dass jede und jeder, die oder der einen Beruf haben, sich brav verhalten und politisch nur noch wollen, was ohnehin der Fall ist. Da braucht man nur mich angucken. [Er lacht schelmisch.] Unter dem Begriff Autonome sind vierzig Jahre lang bunte Jugendgruppen verstanden worden. Vielleicht mit einem kontinuierlichen Element: wenigstens gelegentlich aufm\u00fcpfig zu sein. Es ist ein Element dessen, was man unter Jugendkultur versteht. Man kann indes nicht genauer sagen, welcher Art die Jugendkultur jeweils ist und in welche Richtung sie sich bewegen k\u00f6nnte.\n\n#### Hat sich die Szene innerhalb der Jahrzehnte ver\u00e4ndert?\n\nDie Studentenbewegung der endsechziger Jahre ist aus vielen Gr\u00fcnden zustande gekommen. Verstehen kann man sie nur, ber\u00fccksichtigt man, dass fast alle eine feste Arbeit schon inne oder in Aussicht hatten. Wenige mussten darum bangen, sie zu verlieren. Mir war nie bange darum, dass ich keinen Beruf erlernen k\u00f6nnte. Ich hatte immer die Wahl, den Beruf des Journalisten oder den des Lehrers erlernen zu k\u00f6nnen, weil es so viele Angebote gab. Dass ich an der Uni gelandet bin, war ein Zufall, wahrscheinlich, weil es so leicht m\u00f6glich war. Das hat sich erst Mitte der 70er Jahre ver\u00e4ndert. Aber in den 80er Jahren, als die sozialstaatlichen Regelungen reduziert worden sind, wurde es noch schwieriger. Die Bundesrepublik hat sich erheblich ver\u00e4ndert. Wie mittlerweile die Menschen behandelt werden, die Hartz IV beziehen, ist unglaublich und menschenverachtend. Die Integrit\u00e4t jedes Einzelnen ist doch eine Angelegenheit, den zu sch\u00fctzen ist.\n\n#### Also ging mit der Ver\u00e4nderung in der Bundesrepublik an sich auch eine Ver\u00e4nderung der Autonomen einher?\n\nDie Autonomen kann man nicht aus sich selber heraus erkl\u00e4ren. Man muss immer den Kontext ber\u00fccksichtigen, also die M\u00f6glichkeiten, die es in der Republik gab, und gleichzeitig die Repressalien, die es auch gab. Die Entstehung der Studentenbewegung ist nur zu erkl\u00e4ren, wenn der massive Anti-Kommunismus verstanden wird in Verbindung mit der Springer-Presse, die heute \u00fcberhaupt nicht mehr problematisiert wird.\n\n#### Die autonome Szene wird medial meist auf Gewalt reduziert, finden Sie das authentisch und richtig?\n\nDie Unterstellung, autonom hei\u00dft Gewalt, ist falsch. Autonom hei\u00dft, sich selber bestimmen, das ist eine wunderbare Sache. Autonom wollen wir alle sein. Autos hei\u00dft \"selbst\" und nomos ist das Gesetz, also hei\u00dft es Selbstbestimmung. Ich m\u00f6chte dazu sagen, dass die Mehrheit derjenigen, die mit dem Etikett \"autonom\" versehen werden oder die selber das Etikett \u00fcbernehmen, keine physische Gewalt gegen Andere anwendet. Gewalt ist kein Instrument der Autonomen. Genau das ist die Begrenzung zur RAF. Als die RAF 1972 nach dem Kaufhausbrand in diese Richtung ging, als sie wie beim V\u00f6lkerrecht darauf gedrungen hat, dass sie wie ein eigener Staat behandelt wird, als sie sich also vermeintlich \"im Krieg\" mit der BRD befand, da fiel die Gewalthemmung weg. Das genau ist bei den Autonomen nicht der Fall. Da ist eine Gewalthemmung da. Vielleicht ist sie ab und zu gering, aber ich w\u00fcrde auf jeden Fall sagen, dass physische und zerst\u00f6rerische Gewalt nicht etwas ist, was zum Ziel der Gruppierungen geh\u00f6rt. Die Medien spielen eine gro\u00dfe Rolle. Von seri\u00f6sen Medien bis zur BILD-Zeitung. Die FAZ hat schon immer kundig vom \"Terrorismussumpf\" geredet. Sie hat selbstredend Typen wie mich im Sumpf gesehen. Der SPD-Vorsitzende der Nachkriegszeit, bis zu seinem Tod 1952, Kurt Schumacher, der auch schon Reichstagsabgeordneter der SPD in der Weimarer Republik war, hat Goebbels vorgeworfen, die Nazis betrieben \"einen permanenten Appell an den inneren Schweinehund des Menschen\". Das machten BILD-Zeitung und FAZ jedenfalls zu den terroristischen Hatzzeiten auch. Es gibt Leute, die sofort Angst bekommen und von Gewalt reden, ohne je von den (Un-)Taten des staatlichen Monopols legitimer physischer Gewalt zu reden oder der vielen Verh\u00e4ltnissen und Aktionen zugrunde liegenden \"strukturellen Gewalt\", sobald \u00c4u\u00dferungen etwa Jugendlicher von ihrer Normalit\u00e4tsvorstellung abweichen. Mit Thomas Mann zu reden: Wenn die \"Wonnen der Gew\u00f6hnlichkeit\" gef\u00e4hrdet scheinen, heben die Spatzen an, im Sturzflug zu kreischen (ein treffliches Bild aus einem. Gedicht Heinrich B\u00f6lls).\n\n#### Denken Sie, dass unter solchen Umst\u00e4nden auch die Extremismustheorie entstanden ist?\n\nOhne Frage! Von meiner Warte aus ist der \"Verfassungsschutz\", ein antib\u00fcrgerlicher Geheimdienst im Innern, eine Institution, die nicht akzeptabel ist. Die innere Feinderkl\u00e4rung ist ungeheuerlich. Der Verfassungsschutz dr\u00e4ngt die Informationen in die Richtung des vermeintlich Extremen. Er verwechselt oft Verbalisierungen mit Taten. Die Art und Weise, wie der Verfassungsschutz Dokumente und Analysen anfertigt, sind durchschnittlich UAS, wie das im Schw\u00e4bischen so sch\u00f6n hei\u00dft: \"unter aller Sau!\" Und zwar im Sinne urteilsfundierter analytischer Arbeit und dem allemal zu f\u00fchrenden Nachweis der Verl\u00e4sslichkeit quellenkritisch gewonnener Informationen. Man sch\u00fctzt die Verfassung nicht, indem man innere Feindbildung betreibt und die Meinungsfreiheit reduziert.\n\n#### Wie sch\u00e4tzen Sie die gesamtgesellschaftliche Bedeutung der autonomen Szene ein?\n\nSehr marginal. Eine polizeilich politisch mit herrschenden Pausbacken je und je voll gepustete Gewaltblase, die trefflich zum Zwecke aktueller Restriktion nen, auch Repressionen pseudolegitim instrumentalisiert werden kann.\n\n#### Hat die autonome Szene die M\u00f6glichkeit, etwas zu ver\u00e4ndern?\n\nSie ist kein institutionalisiertes Subjekt, das in herrschaftlich unterstellter Weise verschw\u00f6rungspraktisch arbeitete.\n\n#### Unter keinen Umst\u00e4nden?\n\nIch w\u00fcsste nicht, wie sie es schaffen soll. Selbst die Hartz-IV-Betroffenen und andere mehr, ausl\u00e4ndische Mitb\u00fcrgerinnen und Mitb\u00fcrger vor allem, schaffen es nicht, sich gegen die menschenrechtlichen Verletzungen zureichend zu wehren. Wenn sie es schafften, wie in Kairo mehrere Wochen lang Hunderttausende zum Reichstag zu bringen, sozusagen die Revolution von innen, dann k\u00f6nnte der \"repr\u00e4sentative Absolutismus\" nicht so uneingeschr\u00e4nkt weiter einseitig Politik betreiben wie heute. \"Die\" Autonomen werden, wie schon bemerkt, viel mehr in ihrer Bedeutung \u00fcberh\u00f6ht, \"Schwarze Bl\u00f6cke\" und andere, meine Ururgro\u00dfmutter \u00e4ngstigenden Verschw\u00f6rungszirkel, um innere Aufr\u00fcstungen zu legitimieren. Daf\u00fcr k\u00f6nnen sie nichts.\n\n#### Besteht denn ein Interesse der Autonomen, die Gesellschaft als Ganzes zu \u00e4ndern?\n\nMir ist nicht bekannt, dass sie ein Konzept h\u00e4tten, die Bundesrepublik gleichsam umzuw\u00e4lzen. Die analytischen, nicht zu reden von den organisatorisch praktischen M\u00f6glichkeiten von \u00fcberwiegend eher losen Gruppierungen wie den Autonomen sind nicht derart, dass sie solches auch nur hoffen k\u00f6nnten.\n\n## \"Das Potential f\u00fcr linksradikales Denken sehe ich in Deutschland nicht.\"\n\nProf. Dr. Melanie Gro\u00df, 39 Jahre, Feministin.\n\nProf. Dr. Melanie Gro\u00df, Jahrgang 1975, forscht und lehrt zu Jugend, Jugend- und Protestkulturen vor dem Hintergrund sozialer Ungleichheiten und Differenz an der FH Kiel und im Feministischen Institut Hamburg.\n\n#### Was wissen wir eigentlich \u00fcber die autonome Szene? Wie sch\u00e4tzen Sie den Forschungsstand ein?\n\nDie autonome Szene als Einheit ist sehr d\u00fcrftig bis gar nicht sozialwissenschaftlich erforscht. Verschiedene Aspekte oder Teilkulturen der autonomen Szene sind allerdings immer mal wieder Gegenstand von wissenschaftlichen Publikationen. Sebastian Haunss beispielsweise publiziert zu dem Thema und hat in seiner Dissertation zum Thema \"Prozesse kollektiver Identit\u00e4t bei den Autonomen und in der Schwulenbewegung\" eine Diskursanalyse vorgelegt. Ich selber habe zu feministischem linksradikalem Widerstand und zu Ladyfesten, die sehr eng mit der autonomen Szene verbunden sind, geforscht. Es gibt nat\u00fcrlich auch Ver\u00f6ffentlichungen aus der Szene selbst, z. B. Autonome in Bewegung oder Gender und H\u00e4userkampf. M. E. gibt es durchaus eine ganze Reihe an Publikationen zu dieser Szene, aber eben nicht in der Form: Das ist die autonome Szene, denn diese Szene ist eigentlich im Singular nicht zu haben, sondern besteht aus vielen Teilszenen und heterogenen Gruppen, die als Gesamtheit schwer zu erfassen sind. Zudem sprechen wir hier von Personen, die sich gerade auch aus politischen Gr\u00fcnden einer Erforschung als Szene oder Milieu widersetzen. Dies w\u00fcrde nach Auffassung einiger Gruppen der 'Politik der ersten Person' widersprechen, da aus dieser Perspektive z. B. Aussagen einzelner Personen oder Gruppen nicht verallgemeinert werden d\u00fcrfen. Ein weiterer Grund, sich der Erforschung zu verweigern, ist nat\u00fcrlich auch die empfundene und tats\u00e4chliche Bedrohung durch staatliche \u00dcberwachungsorgane, die mit einem grunds\u00e4tzlichen Misstrauen gegen\u00fcber denjenigen einhergehen, die \u00fcber Gruppen und Personen sowie deren Ansichten Ausk\u00fcnfte haben wollen. Ein pers\u00f6nlicher, von Vertrauen gekennzeichneter Zugang zu dieser Szene ist absolut notwendig, um Ausk\u00fcnfte zu erhalten \u2013 das gilt forschungspraktisch aber im Prinzip f\u00fcr alle Gruppen oder Szenen.\n\n#### Wer geh\u00f6rt zur autonomen Szene?\n\nDie Szene setzt sich aus verschiedenen Einzelpersonen und Gruppen zusammen, die sich selbst politisch als radikal links bezeichnen und sich in der Regel in szenetypischen R\u00e4umen bewegen und diese gestalten. Solche R\u00e4ume sind zumeist Autonome Zentren, wie beispielsweise die Rote Flora in Hamburg. Der Grad der Einbindung und Aktivit\u00e4t innerhalb dieser Zentren variiert nat\u00fcrlich. Die Zugeh\u00f6rigkeit zur Szene erfolgt grunds\u00e4tzlich durch die eigene Definition. Das hei\u00dft, wenn ich mich zugeh\u00f6rig f\u00fchle, bin ich Teil der Szene. Da unterscheidet sie sich nicht von anderen Szenen.\n\n#### Wie w\u00fcrden Sie einen \"typischen\" Autonomen, eine \"typische\" Autonome beschreiben?\n\nGrunds\u00e4tzlich ein Mensch, der sich stark politisch engagiert und dessen Art der Lebensf\u00fchrung stark von seinen politischen Ansichten beeinflusst ist. Es ist jemand, der oder die eine herrschaftskritische Perspektive auf Gesellschaft einnimmt und Kapitalismus als Herrschaftsverh\u00e4ltnis betrachtet. Dar\u00fcber hinaus werden andere Herrschaftsverh\u00e4ltnisse benannt: (Hetero-)Sexismus und Rassismus sind neben Kapitalismus zentrale Themen der radikalen Linken. Wie diese Herrschaftsverh\u00e4ltnisse allerdings zusammenh\u00e4ngen und ob das eine Herrschaftsverh\u00e4ltnis z. B. Voraussetzung oder Wirkung der anderen ist, ist dabei immer wieder Gegenstand von zum Teil heftig gef\u00fchrten Diskussionen. Aktuell w\u00e4re ein Autonomer, eine Autonome jemand, der oder die sich zum Beispiel engagiert gegen Gentrification in den St\u00e4dten und zu diesem Zweck Vortragsreihen organisiert oder u. a. leerstehende H\u00e4user besetzt, der oder die sich in Antifa-Gruppen organisiert, Aktionstage zu Geschlechterverh\u00e4ltnissen plant oder sich in der Anti-AKW-Bewegung engagiert. Meistens sind die Personen gleichzeitig in mehreren Bereichen aktiv. Die Wahrscheinlichkeit, im Laufe der Biografie mindestens einmal Teil einer WG gewesen zu sein, ist recht hoch, ebenso die Wahrscheinlichkeit, Demonstrationen zu organisieren oder zu besuchen, die dann in meist dunkler szenetypischer Kleidung aufgesucht werden. Die Freizeit wird h\u00e4ufig in Autonomen Zentren und\/oder mit politischer Arbeit verbracht.\n\n#### Lassen sich die sozialen, biografischen Milieus beschreiben, aus denen sich die Angeh\u00f6rigen der autonomen Szene mehrheitlich rekrutieren?\n\nDas kann nicht sozialwissenschaftlich fundiert beantwortet werden, da es dar\u00fcber wenig Zahlen gibt. Menschen, die sich innerhalb der Autonomen Zentren bewegen, sind unterschiedlichen Alters, Geschlechts und unterschiedlicher sozialer Herkunft. Die Szene ist meiner Einsch\u00e4tzung nach allerdings durchaus stark akademisch gepr\u00e4gt \u2013 besteht aber keinesfalls nur aus Menschen mit akademischem Hintergrund. Starken Einfluss haben auch z. B. Jugendkulturen wie die Punk-Szene.\n\n#### Ist die autonome Szene selbst eine Jugendkultur?\n\nEs ist eine Szene, die starke jugendkulturelle Bez\u00fcge aufweist: Kultur, Stil und Musik sind durchaus als Ausdrucksformen politischer Inhalte wichtig und werden in Autonomen Zentren artikuliert und gestaltet. Nur die Sache mit dem Alter und der zeitlichen Begrenzung der Zugeh\u00f6rigkeit passt nicht so ganz: Menschen verbleiben durchaus auch bis ins Alter in dieser Szene. Das ist allerdings ein Ph\u00e4nomen, das auch f\u00fcr Jugendkulturen wie Punk G\u00fcltigkeit hat. Auch hier werden Protagonist_innen immer \u00e4lter.\n\n#### Was macht die autonome Szene abseits von politischen Inhalten aus, z. B. in Bezug auf Musikgeschmack, popkulturelle Vorlieben, Kleidungs-, Lebensstil etc.?\n\nStil und Politik sind meiner Meinung nach nicht zu trennen, da sie aufs Engste miteinander verwoben sind. So ist es eine politische Entscheidung, sich beispielsweise nicht dem Prinzip des Konsums zu unterwerfen, der als Teil des Herrschaftssystems Kapitalismus verstanden wird. Daraus entsteht dann ein Stil, der gebrauchte Kleidung und Bricolage kultig werden l\u00e4sst. Eine sexis-muskritische Perspektive sorgt zudem daf\u00fcr, dass die Kleidung h\u00e4ufig auch nicht bruchlos als typisch weiblich eingeordnet werden kann \u2013 m\u00e4nnlich schon eher. Es gibt auch eine starke Affinit\u00e4t zu Musik jenseits des Mainstreams, egal ob HipHop, Pop oder Hardcore \u2013 es sei denn, es ist eine ironische Aneignung: Take-That-Songs werden auf linksradikalen Partys zwischen Politsongs aufgelegt. Grunds\u00e4tzlich ist die Idee, dass Politik im Privaten anf\u00e4ngt, der Grund f\u00fcr die Auspr\u00e4gung der Lebensstile: Das Leben in Gemeinschaften, das gemeinsame Wirtschaften in H\u00e4usern (ob besetzt oder gemietet), das Infragestellen monogamer Zweierbeziehungen, die Hinterfragung eigener geschlechtlicher Identit\u00e4t etc. sind wichtige Themen. Es wird versucht, dem aktuellen Gesellschaftssystem eine andere Lebensweise entgegenzustellen und dadurch auch zu zeigen, dass es neben den gesellschaftlich anerkannten Normen auch andere M\u00f6glichkeiten der Vergemeinschaftung gibt \u2013 wenn auch immer in aktuell m\u00f6glichen Grenzen.\n\n#### Welche Rolle spielen Frauen in der autonomen Szene?\n\nFrauen sind in der Szene durchaus sehr pr\u00e4sent und haben h\u00e4ufig ein feministisches oder queer-feministisches Selbstverst\u00e4ndnis. Nur weil die Szene ihrem Selbstverst\u00e4ndnis nach antisexistisch ist, hei\u00dft das aber noch lange nicht, dass es hier keine sexistischen Strukturen gibt. Das zeigen die vielen szeneinternen Debatten zu diesem Thema. Allerdings ist es eine Szene, die sich diesem Problem zumindest nicht verweigert und die Auseinandersetzungen f\u00fchrt. Somit geh\u00f6ren zur autonomen Szene Debatten um Gender, Queer, Transgender und Feminismus genauso wie um sexualisierte Gewalt und \u00dcbergriffe innerhalb der Szene. Au\u00dferdem gibt es Kritik an \"m\u00e4nnlichem Mackerverhalten\", das es auch in dieser Szene gibt und das beispielsweise eine Selbststilisierung mancher Autonomer als 'm\u00e4nnliche Stra\u00dfenk\u00e4mpfer' angreift. Diskussionen zeigen auch immer wieder, dass durchaus auch Frauen f\u00fcr mackerhaftes Verhalten kritisiert werden.\n\n#### Was sind die zentralen Medien und Informationsvermittler f\u00fcr die autonome Szene?\n\nBesonders wichtig ist das Internet \u2013 hier gibt es eine F\u00fclle von Seiten, die zu Informationszwecken genutzt werden k\u00f6nnen. Am bekanntesten ist sicherlich Indymedia, interessant sind aber zum Beispiel auch Seiten wie der Bewegungsmelder in Hamburg. Neben dem Internet sind auch Zeitschriften wichtige Medien: die Interim zum Beispiel.\n\n#### Nach dem Ende des linken Terrorismus durch z, B. die R.A.F. sind es vor allem Autonome, die als (linke) Gefahr f\u00fcr die \u00f6ffentliche Ordnung und Sicherheit wahrgenommen werden. Wie gef\u00e4hrlich sind Autonome tats\u00e4chlich f\u00fcr die \u00f6ffentliche Sicherheit oder gar Demokratie?\n\nWir haben es ja hier nicht mit einer Massenbewegung zu tun, insofern halte ich den Umgang mit der autonomen Szene f\u00fcr deutlich \u00fcberzogen. Nat\u00fcrlich fordert sie den Staat heraus, weil sie seine Machtposition und das Gewaltmonopol in Frage stellt \u2013 dennoch zeigen uns gro\u00dfe Umst\u00fcrze und Revolutionen immer wieder, dass die dahinter stehenden Ideen von sehr vielen Menschen der Gesellschaft getragen werden m\u00fcssen. Dieses Potential f\u00fcr linksradikales Denken sehe ich in Deutschland nicht. Meines Erachtens sind rechte Gruppierungen hier sehr viel gef\u00e4hrlicher, da sie auf ein breiteres gesellschaftliches Echo zur\u00fcckgreifen k\u00f6nnen. Das zeigten zuletzt auch wieder Studien aus der Friedrich-Ebert-Stiftung (z. B. Decker\/Kiess\/Br\u00e4hler 2012), die nachweisen konnten, dass rassistisches Denken in der Mitte der Gesellschaft stattfindet und nicht nur an imaginierten 'R\u00e4ndern'.\n\n#### Ist die Reduzierung der autonomen Szene auf Gewalt richtig?\n\nDas ist meiner Meinung nach eine unzul\u00e4ssige Verk\u00fcrzung. Zwar ist Militanz ein Thema in der radikalen Linken, das immer wieder diskutiert wird, aber es ist eben nur ein Thema unter vielen. L\u00e4ngst nicht alle, die sich der Szene zugeh\u00f6rig f\u00fchlen, agieren mit militanten Aktionen. Die vielen Diskussionen, die beispielsweise auf linken Portalen gef\u00fchrt werden, zeigen, wie differenziert und bisweilen auch hochemotional hier mit der Gewaltfrage umgegangen wird. Fakt ist aber, dass die Szene deshalb in der \u00d6ffentlichkeit wahrgenommen wird, weil ihr Gewaltt\u00e4tigkeit zugeschrieben wird. Dabei werden in Presseberichterstattungen zu Demonstrationen beispielsweise die Inhalte immer wieder systematisch ausgeblendet und fast schon reflexartig von \"Randalierern\" gesprochen, wenn es zu Zusammenst\u00f6\u00dfen mit der Polizei gekommen ist. So geschah es auch zuletzt im Anschluss an eine Demonstration der radikalen Linken am 21. Dezember 2013 in Hamburg. Diese Demonstration wurde von der Polizei gestoppt, nachdem sie erst zehn Meter gelaufen war. In der Folge kam es zu Auseinandersetzungen mit der Polizei und einem breiten Presseecho. Dabei sind die Themen der Demonstration, n\u00e4mlich der Umgang des Hamburger Senats mit Fl\u00fcchtlingen, die aus Lampedusa in Hamburg gestrandet sind, die Forderung nach dem Erhalt des Autonomen Zentrums Rote Flora sowie der Umgang mit Gentrifizierungsprozessen auf St. Pauli und in anderen Hamburger Stadtteilen lediglich von einigen Medien aufgegriffen worden. Im Zentrum gerade der regionalen Presse standen vielmehr die gewaltsamen Auseinandersetzungen mit der Polizei und das im Anschluss eingerichtete gro\u00dffl\u00e4chige Gefahrengebiet.\n\n#### Wie sch\u00e4tzen Sie die gesamtgesellschaftliche Bedeutung der Autonomen ein? Besteht \u00fcberhaupt das Interesse, die Gesellschaft als Ganzes zu \u00e4ndern, oder geht es nur darum, die eigenen Freir\u00e4ume zu bewahren?\n\n\"Wir wollen kein St\u00fcck vom Kuchen \u2013 wir wollen die ganze B\u00e4ckerei!\" Das ist ein bekannter Ausdruck in der linken Szene, der durchaus auf den Versuch verweist, das Ganze \u00e4ndern zu wollen. Aber: Zur Macht geh\u00f6rt Widerstand \u2013 diese foucaultsche Formel ist auch f\u00fcr linken Protest g\u00fcltig. Das viel kritisierte System hat nicht nur Herrschaftsformen produziert, sondern auch diese Form des Widerstandes. Er ist also durchaus auch Teil oder Produkt des kritisierten Systems. Insofern steht die autonome Szene auch nicht au\u00dferhalb der gesellschaftlichen Strukturen und ist von ihnen gepr\u00e4gt. Zum einen k\u00f6nnte eine Deutung sein, dass Gesellschaften immer Gruppen brauchen, die radikal sind, um sich ihrer eigenen 'Normalit\u00e4t' immer wieder zu versichern und sie stets aufs Neue zu konstruieren. Zum anderen w\u00e4re eine Deutung, dass radikal linke Gruppen auch ein Korrektiv f\u00fcr Gesellschaften sind, weil sie immer wieder den Finger auf die Wunde soziale Ungleichheiten legen und damit den Mainstream n\u00f6tigen, sich zu solchen Themen zu verhalten.\n\n#### Wo w\u00e4re noch ein gro\u00dfer Forschungsbedarf hinsichtlich der autonomen Szene?\n\nInteressant w\u00e4re meines Erachtens, was Menschen dazu bringt, sich auf diese Weise politisch zu engagieren. Das k\u00f6nnte Hinweise daf\u00fcr liefern, wie politische Handlungsf\u00e4higkeit erlebt wird und wie sie unterst\u00fctzt werden kann.\n\n#### Warum interessiert Sie pers\u00f6nlich dieses Thema?\n\nDie aktuellen neoliberalen gesellschaftlichen Entwicklungen f\u00fchren zu einer Versch\u00e4rfung sozialer Ungleichheiten, die meines Erachtens demokratische Werte und die gesellschaftliche Sicherheit immer mehr gef\u00e4hrden. Insofern ist es f\u00fcr mich als Wissenschaftlerin mit emanzipatorischem Selbstverst\u00e4ndnis besonders interessant, emanzipatorische Protestbewegungen und deren Vorstellungen von Handlungsf\u00e4higkeit, deren Perspektiven auf die M\u00f6glichkeiten und Grenzen von Widerstand und Kritik zu untersuchen. Es geht auch mir um die Suche nach einer gerechteren Gesellschaft.\n\n#### Literatur\n\nA.G. Grauwacke: Autonome in Bewegung. Aus den ersten 23 Jahren. Assoziation A., Berlin 2003.\n\nAmantine: Gender und H\u00e4userkampf. Genderspezifische Aspekte und anti-patriarchale Kampfe in den H\u00e4userbewegungen in der BRD und Westberlin. Unrast, M\u00fcnster 2011.\n\nDecker, Oliver\/Kiess, Johannes\/Br\u00e4hler, Elmar: Die Mitte im Umbruch. Rechtsextreme Einstellungen in Deutschland 2012. J. H. W. Dietz Nachf., Bonn 2012.\n\nGro\u00df, Melanie: \"Internet als Plattform politischer Interventionen: Ladyfeste im Netz\", in: kommunikation@gesellschaft, Jg. 7, Beitrag 4, 2006. Online-Publikation: <http:\/\/www.soz.uni-frankfurt.de\/K.G\/B4_2006_gross.pdf>.\n\nGro\u00df, Melanie: \"riot grrrls und ladyfeste \u2013 Angriffe auf die heterosexuelle Matrix\", in: Rohmann, Gabriele (Hg): Krasse T\u00f6chter. M\u00e4dchen in Jugendkulturen. Archiv der Jugendkulturen, Berlin 2007, S. 71-81\n\nGro\u00df, Melanie: \"Feministischer Widerstand aus post-\/queer-\/linksradikal-feministischer Perspektive\", in: Gro\u00df, Melanie; Winker, Gabriele (Hg): Queer-| Feministische Kritiken neoliberaler Verh\u00e4ltnisse. Unrast, M\u00fcnster 2007, S. 169-189.\n\nGro\u00df, Melanie: Geschlecht und Widerstand, post... | queer... | linksradikal... Helmer, K\u00f6nigstein 2008.\n\nGro\u00df, Melanie: \"Gender Trouble und queere Interventionen \u2013 Angriffe auf Heterosexismus und Zweigeschlechtlichkeit\", 2010, in: Bois, Marcel\/H\u00fcttner, Bernd (Hg): Beitr\u00e4ge zur Geschichte einer pluralen Linken. Theorien und Bewegungen nach 1968, Heft 2: 48-51 (PDF).\n\nHaunss, Sebastian: Identit\u00e4t in Bewegung. Prozesse kollektiver Identit\u00e4t bei den Autonomen und in der Schwulenbewegung. VS Verlag f\u00fcr Sozialwissenschaften, Wiesbaden 2004.\n\n## Die Wut ist vorhersehbar.\n\nEin Kommentar zu den Entwicklungen rund um Hamburg am 21.12.2013 von Melanie Gro\u00df\n\nRund um eine Demonstration in Hamburg, die am 21.12.13 stattfinden sollte, aber nach wenigen Metern durch die Polizei vermutlich rechtswidrig gestoppt wurde, entspinnt sich eine bemerkenswerte Eskalationsspirale, die Anlass zum Innehalten und zur differenzierten Analyse geben sollte. Wir erleben eine Medienstrategie der verbalen Aufr\u00fcstung, wenn, noch bevor die Ursachen der Eskalation bearbeitet wurden, bereits medienwirksam nach mehr Befugnissen f\u00fcr die Polizei gerufen wird. So sind nun die Forderungen nach Gummigeschossen, der Einsatz von Teasern oder gar der Gebrauch von Schusswaffen in der Diskussion, die durch unterst\u00fctzende Zeitungsartikel flankiert werden. So wurde etwa in der Hamburger Morgenpost die Mahnwache f\u00fcr Polizeibeamtinnen dargestellt oder Interviews mit PolizistInnen ver\u00f6ffentlicht, in denen diese \u00fcber ihre \u00c4ngste w\u00e4hrend der Aus\u00fcbung ihres Berufes zu Wort kommen. Selbstverst\u00e4ndlich sind solche Berichte notwendig, es f\u00e4llt jedoch auf, dass die Gewalterfahrungen auf der Seite der Protestierenden deutlich weniger Aufmerksamkeit erfahren bzw. diese nur als \"Krawallmacher\", \"Chaoten\" oder wie durch den stellvertretenden Landessprecher der Deutschen Polizeigewerkschaft Werminghaus als \"Abschaum\" bezeichnet wurden. Diese Medienberichterstattung \u00fcberdeckt die Notwendigkeit, die Wut auf Polizei und staatliche Ordnungspolitik nachzuvollziehen und ihr mit angemessenen Mitteln zu begegnen.\n\nDie Anw\u00e4ltInnen der Roten Flora ver\u00f6ffentlichten inzwischen zudem eine Pressemitteilung, in der die polizeiliche Meldung, es habe am 29.12.2013 einen zweiten Angriff auf die Davidwache gegeben, bei der PolizistInnen schwer verletzt worden seien, als strategische \"Desinformation\" und damit als L\u00fcge bezeichnet wurde. Demnach k\u00f6nnte der Angriff auf PolizistInnen, der am selben Abend auf dem Kiez in der N\u00e4he der Reeperbahn stattgefunden haben soll, auch von KiezbesucherInnen ausgegangen sein \u2013 das r\u00e4umt die Polizei inzwischen selbst ein. Die BAG Kritische Polizisten, die mit einigen Mitgliedern auch auf der Demonstration in Hamburg am 21.12.13 eingesetzt war, spricht in ihrer Pressemitteilung von rechtswidrigen Polizeieins\u00e4tzen rund um die verhinderte Demonstration. Die Wortwahl ist drastisch: \"Skrupellosigkeit\", \"polizeiliche Eskalationsstrategie\", \"unwirkliche Brutalit\u00e4t durch Polizeibeamte\" sowie eine Pressearbeit, die einem \"Opernball\" \u00e4hneln w\u00fcrde.\n\nNoch bevor gekl\u00e4rt ist, was eigentlich genau der Grund f\u00fcr die gew\u00e4hlte Polizeistrategie der Einsatzleitung auf der Demonstration am 21.12.2013 war, legten bereits etliche digitale Zeugnisse des Tages nahe, dass das Aufhalten der Demonstration nach wenigen Metern sowie der darauf folgende Einsatz von Gewalt durch die Polizei ausgesprochen fragw\u00fcrdig sind. Von schwerer K\u00f6rperverletzung auf wehrlose und schlie\u00dflich regungslos am Boden liegende junge Menschen berichten schockierte AnwohnerInnen. Eine Landtagsabgeordnete aus NRW berichtet von willk\u00fcrlichen Jagdszenen, Videomaterial zeigt massiven Einsatz von Schlagst\u00f6cken und Pfefferspray aus dichter Distanz. Auch die via Twitter geschilderten Erfahrungen von AnwohnerInnen und BesucherInnen des neu eingerichteten und in dieser Form auch neuartigen Gefahrengebietes werfen ein ungutes Licht auf die Polizeistrategie.\n\nWir m\u00fcssen uns fragen, ob ein Gesellschaftsentwurf, der den Anschein erweckt, dass soziale Konflikte durch die Polizei gel\u00f6st werden sollen, und damit ordnungspolitisch statt politisch gel\u00f6st werden, mit den Grundprinzipien der Demokratie vereinbar ist. Auch ohne die aktuellen Vorkommnisse in Hamburg erleben wir seit einigen Jahren Prozesse, die Menschen in prek\u00e4ren Lebenslagen ordnungspolitisch entfernen und verdr\u00e4ngen, w\u00e4hrend gleichzeitig soziale Ungleichheiten steigen und Mittel der Sozialen Arbeit gek\u00fcrzt werden \u2013 Prozesse, auf die gerade die \"Recht-auf-Stadt\"-Bewegung, die MitveranstalterIn der Demonstration am 21.12.2013 gewesen ist, immer wieder hingewiesen hat. Ursachenforschung oder gar -bek\u00e4mpfung von sozialen Problemen und Konflikten werden immer seltener \u00f6ffentlich differenziert betrieben. Gerade Jugendliche sind \u00fcberdurchschnittlich h\u00e4ufig von ordnungspolitischen Ma\u00dfnahmen betroffen, weil sie eine ausgesprochen sichtbare Gruppe im \u00f6ffentlichen Raum bilden: Kinder und Jugendliche, die in armen Verh\u00e4ltnissen in St\u00e4dten aufwachsen, bev\u00f6lkern die Bushaltestellen, Shopping-Malls und Vorpl\u00e4tze von Imbissbuden, weil sie zu Hause in beengten Verh\u00e4ltnissen schlicht wenig M\u00f6glichkeiten haben, sich mit ihren FreundInnen zu treffen. Das Geld fehlt, um \u2013 sozial erw\u00fcnscht \u2013 die Getr\u00e4nke in einem Caf\u00e9 oder Bistro zu konsumieren. Der Aufenthalt im \u00f6ffentlichen Raum ist aber nicht nur Ausdruck eines Mangels, sondern auch ein Element der Teilhabe an der Gesellschaft. Machen Jugendliche oder Menschen in prek\u00e4ren Lebenslagen davon Gebrauch, werden sie von B\u00fcrgerInnen und Polizei oftmals aber diffus als bedrohlich empfunden. Hier wird nicht nach den Ursachen f\u00fcr m\u00f6glicherweise tats\u00e4chlich hin und wieder problematischem Verhalten etwa bei in Konflikt geratenen Jugendgruppen gesucht, sondern stattdessen werden sie mit einem Generalverdacht \u00fcberzogen, der dann auch rabiate und fragw\u00fcrdige Ausgrenzungs- und Verdr\u00e4ngungsprozesse legitimieren soll. Vergleichbares geschieht auf Demonstrationen: Gerade auf Demonstrationen, die der linken Szene zugeordnet werden, haben wir einen \u00fcberdurchschnittlichen Anteil an jungen Menschen, die diese Demonstrationen besuchen und auch mitplanen. Das ist beachtlich, denn gerade dieser Generation wird in der Regel politisches Desinteresse bescheinigt. Ihnen wird immer \u00f6fter schlicht unterstellt, dass sie einer \"gef\u00e4hrlichen\" Generation angeh\u00f6ren, die mit staatlichen Mitteln der Kontrolle und Repression bedacht werden m\u00fcsste. Zugleich wird der Begriff \"Jugendliche\" selbst daf\u00fcr eingesetzt, politische Proteste zu verharmlosen und ihnen Ernsthaftigkeit abzusprechen. Europaweit sehen wir in den letzten Jahren immer wieder Bilder von Demonstrationen, die ein Ausma\u00df von Gewalt und auch staatlicher Repression zeigen, die Anlass zum Innehalten geben sollte. Hiervon sind grunds\u00e4tzlich mehrere Altersgruppen betroffen, jedoch f\u00e4llt auf, dass gerade die junge Generation in einem hohen Ma\u00dfe von polizeilicher Gewalt betroffen ist. Die Art und Weise, wie dies derzeit in Hamburg geschieht und nun mit dem Durchsetzen polizeilicher Sonderrechte in einem eigens definierten \"Gefahrengebiet\" oder den \"Gefahreninseln\" noch verst\u00e4rkt wird, stellt demokratische Grundprinzipien in Frage. Wenn Menschen via Twitter \u00e4u\u00dfern, dass sie sich die Wahl ihrer Kleidung gut \u00fcberlegen m\u00fcssen, wenn sie derzeit in Altona oder auf St. Pauli einkaufen gehen, dann haben wir es mit massiven Einschnitten in die Pers\u00f6nlichkeitsrechte zu tun.\n\nEs ist ein hohes Ma\u00df an kulturellem Kapital vonn\u00f6ten, um gesellschaftliche Zusammenh\u00e4nge zu problematisieren und den eigenen Protestaktionen \u00fcberhaupt Geh\u00f6r zu verschaffen bzw. mit Hilfe eigener Publikationen und Artikulationen etwa via Twitter Gegen\u00f6ffentlichkeiten herzustellen. Die nachwachsende Generation hat es mit einer Gesellschaft zu tun, die sich ihr und ihren Anliegen gegen\u00fcber vor allem durch Ignoranz in den Weg stellt. Die zahlreichen sozialen Probleme und Konflikte bleiben bemerkenswert ungel\u00f6st und die direkt Betroffenen werden in der Regel nicht 'geh\u00f6rt': Das Ansteigen von Armut in Deutschland, die je nach Sch\u00e4tzung bereits jedes dritte bis f\u00fcnfte Kind betrifft. Drohende Abw\u00e4rtsspiralen der europ\u00e4ischen Wirtschaft, die Menschen in L\u00e4ndern wie Griechenland bereits erreicht haben und auch in Deutschland von einigen bef\u00fcrchtet werden. Zunehmende rassistische Haltungen und rechte \u00dcbergriffe sowie eine restriktive Fl\u00fcchtlingspolitik betreffen junge Menschen entweder direkt oder zeigen eine gesellschaftliche Entwicklung, die sie als \u00e4u\u00dferst problematisch einsch\u00e4tzen. Katastrophale Prognosen in Bezug auf die Entwicklung des Klimas und vieles mehr sind die Themen und wom\u00f6glich der k\u00fcnftige Alltag der jetzt jungen Generation. Gleichzeitig werden Proteste der jungen Generation immer \u00f6fter kriminalisiert und abgewertet. Die Wut ist vorhersehbar. Nicht zuf\u00e4llig erleben europ\u00e4ische Gro\u00dfst\u00e4dte so genannte Riots, an denen vor allem junge Menschen beteiligt sind. An dieser Stelle sei an die gewaltsamen Auseinandersetzungen in den franz\u00f6sischen Vorst\u00e4dten oder in London erinnert, die ebenfalls durch soziale Konflikte entstanden sind und schlie\u00dflich vor allem ordnungspolitisch 'gel\u00f6st' wurden. Es sei auch daran erinnert, welche Auswirkungen die Missachtung des Wunsches junger Menschen, ihr selbstverwaltetes Jugendzentrum in Kopenhagen zu erhalten, gehabt hat. Nach der R\u00e4umung gab es tagelange Auseinandersetzungen mit der Polizei und eine hohe Frustration bei den Betroffenen, die gesellschaftliche R\u00e4ume eigenverantwortlich mitgestalten wollten. Eine solche Politik f\u00fchrt nur kurzfristig zu geordneten Verh\u00e4ltnissen. Langfristig ist sie nicht geeignet zur Befriedung und Integration, sondern wird stattdessen die sozialen Konflikte versch\u00e4rfen und befeuern.\n\nDie aktuellen Ereignisse in Hamburg hinterlassen den Eindruck, dass hier mit gezielten Medienstrategien durch die Polizei und ihre Gewerkschaften eine Stimmung erzeugt werden soll, die eine Aufr\u00fcstung der Polizei und ihrer Befugnisse einleiten soll. Mehr noch \u2013 der \"Mangel an politischer F\u00fchrung\" in Hamburg, wie ihn der Bund der Kritischen Polizisten diagnostiziert, k\u00f6nnte auch gewollt sein, wenn staatliche Organe sich aufgrund k\u00fcnftig auch weiter zuspitzender sozialer Konflikte zunehmend auf soziale Unruhen und Aufst\u00e4nde einstellen und daf\u00fcr ihre Befugnisse weiter ausbauen sowie autorit\u00e4re Strukturen erproben.\n\nErstver\u00f6ffentlicht in: analyse & kritik. Zeitung f\u00fcr linke Debatte und Praxis. Nr. 590 vom 21. Januar 2014.\n\n## Bibliographie zum Buchprojekt Autonome\n\nDie folgende, sicher nicht vollst\u00e4ndige, aber vielleicht hilfreiche Literaturliste zur autonomen Szene enth\u00e4lt den Bibliotheksbestand des Archiv der Jugendkulturen e. V. (www.jugendkulturen.de), Stand September 2014.\n\nA.G. Grauwacke: Autonome in Bewegung. Aus den ersten 23 Jahren. Assoziation A, Berlin 2003.\n\nAG Spa\u00df muss sein (Hrsg.): Spa\u00dfguerilla. Unrast, M\u00fcnster 1994 (Reprint).\n\nAgentur BILWET: Bewegungslehre. Botschaften aus einer autonomen Wirklichkeit., Ed. ID-Archiv, Berlin\/Amsterdam 1991.\n\nAK Wantok (Hrsg.): Perspektiven autonomer Politik. Unrast, M\u00fcnster 2010.\n\nAkstinat, Simon: Wir sind die Guten. Edition Braus, Berlin 2012.\n\nAlternative Tagungsst\u00e4tte e. V. (Hrsg.): 30 Jahre Alternative! Das Buch. Eigenverlag, L\u00fcbeck 2008\n\nAmann, Marc (Hrsg.): go.stop.act! Die Kunst des kreativen Stra\u00dfenprotests. Geschichten \u2013 Aktionen \u2013 Ideen. 3. \u00fcberarb. Aufl. Trotzdem-Verlag, Frankfurt\/Main 2011.\n\namantine: Gender und H\u00e4userkampf. Genderspezifische Aspekte und anti-patriarchale Kampfe in den H\u00e4userbewegungen in der BRD und Westberlin. Unrast, M\u00fcnster 2011.\n\nAnders, Freia: \"Wohnraum, Freiraum, Widerstand. Die Formierung der Autonomen in den Konflikten um Hausbesetzungen Anfang der achtziger Jahre\", in: Reichardt, Sven\/Siegfried, Detlef (Hrsg.): Das alternative Milieu. Antib\u00fcrgerlicher Lebensstil und linke Politik in der Bundesrepublik Deutschland und Europa 1968-1983. (Hamburger Beitr\u00e4ge zur Sozial- und Zeitgeschichte; 47). Wallstein, G\u00f6ttingen 2010, S. 473-498.\n\nAnti a71\/73 \u2013 Protestreader 96\/97. (Brosch\u00fcre zu den Aktionstagen vom. 17.-24. November 1996 gegen die Waldautobahn in Th\u00fcringen), 1996.\n\nAntifa-Texte 2. (Brosch\u00fcre) Eigenverlag, Stuttgart 1986.\n\nAntifaschistische Aktion Berlin (Hrsg.): Das Konzept Antifa. Grundsatztexte und Konkretes. Informationsbrosch\u00fcre der Antifaschistischen Aktion Berlin 1995-98. Antifaschistische Aktion, Berlin 1998.\n\nAntifaschistische Aktion (Hrsg.): Tipps und Tricks f\u00fcr Antifas: reloaded, 1. Aufl. Unrast, M\u00fcnster 2009.\n\nAntifaschistische Linke Berlin (Hrsg.): Antifa... und wie!? Tips und Trix zur politischen Praxis. ALB, Berlin 2010.\n\nAntifaschistischer Linke Berlin (Hrsg.): Total Extrem. Die (neue) Funktion der Totalitarismus- und Extremismusideologien. Berlin 2010 [\u2192 Reader; pdf vorh.].\n\nArzt, Ingo: Militante Gruppe. Internetseite der Bundeszentrale f\u00fcr politische Bildung, Bonn: 2009 [\u2192 Reader]. <http:\/\/www.bpb.de\/politik\/extremismus\/linksextremismus\/33609\/militante-gruppe?p=all>, [31.7.2014).\n\nautonome (ex-)StipendiatInnen, einige: \"'Oops! We did it again.' Demo und Riots in Rostock am 2. Juni 2007 aus autonomer Sicht\", in: Rilling, Rainer (Hrsg.): Eine Frage der Gewalt \u2013 Antworten von links. Dietz, Berlin 2008 [pdf vorh.].\n\nAutonome L.U.P.U.S.-Gruppe: Die Hunde bellen... von A bis RZ. Eine Zeitreise durch die 68er Revolte und die militanten K\u00e4mpfe der 70er bis 90er Jahre. Unrast, M\u00fcnster 2001.\n\nAutonome L.U.P.U.S.-Gruppe: Lichterketten und andere Irrlichter. Texte gegen finstere Zeiten. Ed. ID-Archiv, Berlin\/Amsterdam 1994.\n\nBacia, J\u00fcrgen\/Scherer, Klaus-J\u00fcrgen: Passt blo\u00df auf! Was will die neue Jugendbewegung? Olle und Wolter, Berlin 1981.\n\nBackes, Uwe\/Gallus, Alexander\/ Jesse, Eckhard (Hrsg.): Jahrbuch Extremismus & Demokratie (E & D) \u2013 Sonderdruck, 22. Jg. Nomos, Baden-Baden 2010.\n\nBackes, Uwe (u. a.) (Hrsg.): Extremistenj\u00e4ger!? Der Extremismus-Begriff und der demokratische Verfassungsstaat. (Endstation Rechts; 1). Adebor, Schwerin 2011.\n\nBackes, Uwe\/Jesse, Eckhard: Vergleichende Extremismusforschung (Extremismus und Demokratie; 11). Nomos, Baden-Baden 2005.\n\nBergsdorf, Harald\/H\u00fcllen, Rudolf van: Linksextrem \u2013 Deutschlands untersch\u00e4tzte Gefahr? Ferdinand Sch\u00f6ningh, Paderborn 2012.\n\nBernd, Matthias\/Holm, Andrej: \"Wir bleiben alle?\", in: StadtRat (Hrsg.): Umk\u00e4mpfte R\u00e4ume. Verl. Libert\u00e4re Assoziation, Verl. der Buchl\u00e4den Schwarze \u2013 Rote Stra\u00dfe, Hamburg u. a. 1998, S. 155-167.\n\nBeyeler, Susanne (u. a.): Allein machen sie dich ein! Ein Buch zum Rauch-Haus-Film. Zusammengestellt vom Filmkollektiv. Filmkollektiv, Berlin 1974.\n\nBirke, Peter\/H\u00fcttner, Bernd\/Oy, Gottfried (Hrsg.): Alte Linke \u2013 Neue Linke? Die Protestbewegungen der 1968er Jahre in der Diskussion. Dietz, Berlin 2010 [online verf\u00fcgbar, pdf. vorh.].\n\nBlechschmidt, Andreas: \"Vom 'Gleichgewicht des Schreckens'\", in: StadtRat (Hrsg.): Umk\u00e4mpfte R\u00e4ume. Verlag Libert\u00e4re Assoziation, Verlag der Buchl\u00e4den Schwarze \u2013 Rote Stra\u00dfe, Hamburg u. a. 1998, S. 83-101.\n\nBock, Marlene\/Reimitz, Monika\/Richter, Horst-Eberhard: Zwischen Resignation und Gewalt \u2013 Jugendprotest in den achtziger Jahren. Leske + Budrich, Opladen 1989.\n\nBois, Marcel\/H\u00fcttner, Bernd (Hrsg.): Beitr\u00e4ge zur Geschichte einer pluralen Linken, Heft 1 und 2, rls-papers, Mai 2010 [online verf\u00fcgbar, pdf. vorh.\u2192 Reader].\n\nBrodkorb, Mathias (Hrsg.): Extremistenj\u00e4ger!? Der Extremismus-Begriff und der demokratische Verfassungsstaat. Adebor, Banzkow 2011.\n\nBrumlik, Micha: \"Autonome. Gewaltbereitschaft als verbindendes Lebensgef\u00fchl\", in: Heitmeyer, Wilhelm (Hrsg.): Jugend \u2013 Staat \u2013 Gewalt: Politische Sozialisation von Jugendlichen, Jugendpolitik und politische Bildung. Juventa, Weinheim\/M\u00fcnchen 1989, S. 175-190.\n\nBrumlik, Micha: \"Linksradikale Jugendkultur? Mutma\u00dfungen \u00fcber ein unbekanntes Terrain\", in: Richard, Birgit\/Kr\u00fcger, Heinz-Hermann (Hrsg.): inter-cool 3.0. Jugend \u2013 Bild \u2013 Medien. Ein Kompendium zur aktuellen Jugendkulturforschung. Finck, M\u00fcnchen 2010, S. 299-310.\n\nChe 2001: \"Konjunkturen der Solidarit\u00e4t oder vom Mitgef\u00fchl zum Miteinander. Zur Geschichte der autonomen Antirassismusgruppen in Deutschland\", in: Bois, Marcel\/H\u00fcttner, Bernd (Hrsg.): Beitr\u00e4ge zur Geschichte einer pluralen Linken, rls-papers, Heft 2: Theorien und Bewegungen nach 1968, Mai 2010, S. 45-60 [online verf\u00fcgbar, pdf. vorh. \u2192 Reader].\n\nDovermann, Ulrich (Hrsg.): Linksextremismus in der Bundesrepublik Deutschland. Bundeszentrale f\u00fcr politische Bildung, Bonn 2011.\n\nErmittlungsausschu\u00df (Hrsg.): Pa\u00dft blo\u00df auf \u2013 [Lieber instand-besetzen als kaputt-besitzen]. Dokumentation. Dezember 1980. Berlin 1981.\n\nHillenkamp, Sven: \"Glatzk\u00f6pfe und Betonk\u00f6pfe. Skinheads und Autonome\", in: Farin, Klaus (Hrsg.): Die Skins. Mythos und Realit\u00e4t. Tilsner, Bad T\u00f6lz 2001, S. 173-205.\n\nHSK 13 (Hrsg.): hoch die kampf dem. 20 Jahre Plakate autonomer Bewegungen. Verlag Libert\u00e4re Assoziation \u2013 Verlag der Buchl\u00e4den Schwarze Risse \u2013 Rote Stra\u00dfe, Hamburg u. a. 1999 [Beilage: CD-ROM Bilddatenbank].\n\nHSK 13 (Hrsg.): vorw\u00e4rts bis zum nieder mit. 30 Jahre Plakate unkontrollierter Bewegungen. Assoziation A, Berlin 2001 [Beilage: CD-ROM Bilddatenbank].\n\nFarin, Klaus: Jugendkulturen in Deutschland 1950\u20131989. (Kap. 8: Die Achtziger. Eine Generation steigt aus: Hausbesetzer II; Gef\u00fchl und H\u00e4rte \u2013 die Autonomen.) Bundeszentrale f\u00fcr politische Bildung, Bonn 2006, S. 145-151; S. 152-157.\n\nFarin, Klaus\/Seidel, Eberhard: Krieg in den St\u00e4dten. Jugendgangs in Deutschland. (\"Streetfighter gegen rechts\"; \"Anmache ist Anmache.\") Archiv der Jugendkulturen, Berlin 2012, S. 131-147; S. 158-172.\n\nForum f\u00fcr kritische Rechtsextremismusforschung (Hrsg.): Ordnung. Macht. Extremismus. Effekte und Alternativen des Extremismus-Modells. VS, Wiesbaden 2011.\n\nGeronimo: Feuer und Flamme. 1: Zur Geschichte und Gegenwart der Autonomen. ID-Verlag im IISG, Amsterdam 1990.\n\nGeronimo: Feuer und Flamme. 2: Kritiken, Reflexionen und Anmerkungen zur Lage der Autonomen. Ed. ID-Archiv, Berlin\/Amsterdam 1992.\n\nGeronimo: Glut & Asche. Reflexionen zur Politik der autonomen Bewegung. Unrast, M\u00fcnster 1997.\n\nGeronimo: Feuer und Flamme. Zur Geschichte der Autonomen (\u00fcberarbeitete Neuauflage des Titels von 1990). Ed. ID-Archiv, Berlin\/Amsterdam 1995.\n\nGriese, Hartmut M.: \"'Jugend(sub)kultur(en)' \u2013 Facetten, Probleme und Diskurse\", in: Roth, Roland\/Rucht, Dieter (Hrsg.): Jugendkulturen, Politik und Protest. Leske + Budrich, Opladen 2000, S. 37\u201347.\n\nGross, Almut\/Schultze, Thomas: Die Autonomen. Urspr\u00fcnge, Entwicklung und Profil der Autonomen. Konkret Literatur, Hamburg 1997.\n\nGro\u00df, Melanie: Geschlecht und Widerstand. post... | queer... | linksradikal... Helmer, K\u00f6nigstein\/Taunus 2008.\n\nH\u00e4rlin, Benny: \"Von Haus zu Haus \u2013 Berliner Bewegungsstudien\", in: Kursbuch 65: Der gro\u00dfe Bruch \u2013 Revolte 81. Rotbuch Verlag, Berlin 1981.\n\nHaller, Michael: Aussteigen oder rebellieren \u2013 Jugendliche gegen Staat und Gesellschaft. Rowohlt, Reinbek 1983.\n\nHanloser, Gerhard: \"Zwischen Klassenkampf und Autonomie. Die Neue Linke und die soziale Frage\", in: Birke, Peter\/H\u00fcttner, Bernd\/Oy, Gottfried (Hrsg.): Alte Linke \u2013 Neue Linke? Die Protestbewegungen der 1968er Jahre in der Diskussion. Dietz, Berlin 2010, S. 150-160 [online verf\u00fcgbar, pdf. vorh. \u2192 Reader].\n\nHartmann, Detlef: Die Alternative. Leben als Sabotage. Zur Krise der technologischen Gewalt. Junk, Berlin 1989.\n\nHaunss, Sebastian: Identit\u00e4t in Bewegung. Prozesse kollektiver Identit\u00e4t bei den Autonomen und in der Schwulenbewegung (B\u00fcrgergesellschaft und Demokratie Band 19). VS, Wiesbaden 2004.\n\nHaunss, Sebastian: \"Antiimperialismus und Autonomie \u2013 Linksradikalismus seit der Studentenbewegung\", in: Roth, Roland\/Rucht, Dieter (Hrsg.): Die Sozialen Bewegungen in Deutschland seit 1945. Ein Handbuch. Campus, Frankfurt\/Main 2008, S. 447-473.\n\nHitzler, Ronald\/Niederbacher, Arne: Leben in Szenen. Formen jugendlicher Vergemeinschaftung heute, 3., vollst. \u00fcberarb. Aufl. (Erlebniswelten; Bd. 3). Kap. 14: Antifa-Szene. VS, Wiesbaden 2010, S. 149-161.\n\nHoffmann, Martin (Hrsg.): SubversionsReader. Texte zu Kultur und Politik. Schl\u00fcsseltexte der militanten Linken und Diskussionsbeitr\u00e4ge zu Theorie und Praxis der Bewegungen nach '68. Ed. ID-Archiv, Berlin u. a. 1998.\n\nH\u00fcttner, Bernd: \"Weder Glut noch Asche. Probleme einer Geschichtsschreibung der Autonomen\", in: Z. Zeitschrift Marxistische Erneuerung, H. 32\/1997. Thema: EU und die Linke, S. 220ff. [\u2192 Reader; Volltext: <http:\/\/www.unrast-verlag.de\/unrast,6,1,26.html>].\n\nH\u00fcttner, Bernd: Archive von unten. Bibliotheken und Archive der neuen sozialen Bewegungen und ihre Best\u00e4nde. AG Spak, Neu-Ulm 2003.\n\nH\u00fcttner, Bernd: \"Ziviler Ungehorsam, Militanz und Gewalt\", in: Rilling, Rainer (Hrsg.): Eine Frage der Gewalt. Antworten von links. Dietz, Berlin 2008, S. 39-43 [pdf vorh. \u2192 Reader].\n\nH\u00fcttner, Bernd: \"'Geschichte von unten' und radikale Linke\", in: arranca! Nr. 29 (Fr\u00fchjahr 2004), S. 50-52 [online verf\u00fcgbar; \u2192 Reader].\n\nID-Archiv im IISG, Amsterdam (Hrsg.).: Die Fr\u00fcchte des Zorns. Texte und Materialien zur Geschichte der Revolution\u00e4ren Zellen und der Roten Zora. Ed. ID-Archiv, Berlin 1993, Bd. 1. und Bd. 2.\n\nJaschke, Hans-Gerd: Linksextremismus. Internetseite der Bundeszentrale f\u00fcr politische Bildung, Bonn 2008 [\u2192 Reader]. <http:\/\/www.bpb.de\/politik\/extremismus\/linksextremismus\/33598\/linksextremismus>, [31.7.2012]\n\nJaschke, Hans-Gerd: Politischer Extremismus. VS, Wiesbaden 2006.\n\nKailitz, Steffen: Politischer Extremismus in der Bundesrepublik Deutschland. Eine Einf\u00fchrung. VS, Wiesbaden 2004.\n\nKeller, Mirja\/K\u00f6gler, Lena\/Krawinkel, Moritz\/Schlemermeyer, Jan: Antifa. Geschichte und Organisierung. Schmetterling, Stuttgart 2011.\n\nKongresslesebuchgruppe: Der Stand der Bewegung \u2013 18 Gespr\u00e4che \u00fcber linksradikale Politik. Lesebuch zum Autonomie-Kongre\u00df 1995. Eigenverlag, Berlin 1995.\n\nKrautkr\u00e4mer, Felix: Das linke Netz. Das Zusammenspiel zwischen der Wochenzeitung \"Die Zeit\", dem \"Netz Gegen Nazis\", der Bundeszentrale f\u00fcr Politische Bildung und der linksextremen Szene (Studien Extremismus und Freiheit; H. 2). Ed. Junge Freiheit, Berlin 2010.\n\nKuhn, Gabriel: Jenseits von Staat und Individuum. Individualit\u00e4t und autonome Politik. Unrast, M\u00fcnster 2007.\n\nKremeyer, Norbert (Hrsg.)\/Roschek, Michael (Mitverf.): Heute schon gelebt? Alltag u. Utopie, Verlag 2000, Offenbach 1981.\n\nLanger, Bernd: Operation 1653. stay rude \u2013 stay rebel. Pl\u00e4ttners, Berlin 2004.\n\nLaurisch, Bernd: Kein Abri\u00df unter dieser Nummer. 2 Jahre Instandbesetzung in der Cuvrystra\u00dfe in Berlin-Kreuzberg (Werkbund-Archiv; 7). Anabas 1981, Gie\u00dfen.\n\nLecorte, Tomas: Wir tanzen bis zum Ende \u2013 Die Geschichte eines Autonomen. Galgenberg, Hamburg 1992.\n\nLeipziger Antifa (LEA): \"Militanz ohne Mythen. Wieso der Streit um das politische Element im Militanzbegriff so wichtig ist\", in: PHASE 2, 12\/2008 [\u2192 Reader].\n\nLindner, Werner: Jugendprotest seit den f\u00fcnfziger Jahren. Dissens und kultureller Eigensinn. Kap. 5.7: Die autonomen Szenen. Leske + Budrich, Opladen 1996, S. 380-387.\n\nMallet, Carl-Heinz: Die Leute von der Hafenstra\u00dfe. \u00dcber eine andere Art zu leben. Edition Nautilus, Hamburg 2000.\n\nMecklenburg, Jens (Hrsg.): Antifa Reader. Antifaschistisches Handbuch und Ratgeber (Antifa-Edition). Elefanten-Press, Berlin 1996.\n\nMletzko, Matthias: \"Merkmale politisch motivierter Gewalttaten bei militanten autonomen Gruppen\", in: Backes, Uwe\/Jesse, Eckhard (Hrsg.): Jahrbuch Extremismus & Demokratie. Band 11. Nomos, Baden-Baden 1999, S. 180-199.\n\nMletzko, Matthias: \"Gewaltdiskurse und Gewalthandeln militanter Szenen \u2013 Unterschiede am Beispiel 'Antifa' und 'Anti-Antifa' Teil 1, in: Kriminalistik Nr. 8\/9 2001, S. 543 \u2013 548; Teil 2, in: Kriminalistik Nr. 10 2001, S. 639 \u2013 644 [\u2192 Reader; pdf vorh.].\n\nMletzko, Matthias: Gewalthandeln linker und rechter militanter Szenen im Vergleich. Erw. Version eines gleichnamigen Aufsatzes, in: Aus Politik und Zeitgeschichte (APuZ) Nr. 44\/2010 [\u2192 Reader; word-Dok vorh.].\n\nM\u00f6ller, Nadine: Hausbesetzungen in Deutschland. Am Beispiel der Mainzer Strasse in Berlin. Grin, M\u00fcnchen 2011.\n\nM\u00fcller-M\u00fcnch, Ingrid\/Prosinger, Wolfgang\/Rodenbladt, Sabine\/Stibler, Lisa u. a.: Besetzung \u2013 weil das W\u00fcnschen nicht geholfen hat. Rowohlt, Reinbek 1981.\n\nNeugebauer, Gero: \"Extremismus \u2013 Linksextremismus \u2013 Rechtsextremismus. Einige Anmerkungen zu Begriffen, Forschungskonzepten, Forschungsfragen und Forschungsergebnissen\", in: Schubarth, Wilfried\/St\u00f6ss, Richard (Hrsg.): Rechtsextremismus in der Bundesrepublik Deutschland. Eine Bilanz (Schriftenreihe der Bundeszentrale f\u00fcr politische Bildung; 368). Bpb, Bonn 2000, S. 13-37 [\u2192 Reader].\n\nNeugebauer, Gero: Extremismus \u2013 Linksextremismus \u2013 Rechtsextremismus. Begriffsdefinitionen und Probleme. Internetseite der Bundeszentrale f\u00fcr politische Bildung, Bonn 2008 [\u2192 Reader; pdf vorh.]. <http:\/\/www.bpb.de\/politik\/extremismus\/linksextremismus\/33591\/definitionen-und-probleme>, [31.7.2014].\n\nNitsche, Rainer (Hrsg.): H\u00e4userk\u00e4mpfe 1872\/1920\/1945\/1982. Transit, Berlin 1981.\n\nNotz, Gisela: \"Selbstverwaltung und Alternativbewegungen\", in: Bois, Marcel\/H\u00fcttner, Bernd (Hrsg.): Beitr\u00e4ge zur Geschichte einer pluralen Linken, rls-papers, Heft 2: Theorien und Bewegungen nach 1968, Mai 2010, S. 11-15 [online verf\u00fcgbar, pdf. vorh. \u2192 Reader].\n\nOff, Jan: Happy Endstadium. Ventil, Mainz 2012.\n\nPfahl-Traughber, Armin: Die Autonomen zwischen Anarchie und Bewegung, Gewaltfixiertheit und Lebensgef\u00fchl. Zu den Besonderheiten einer linksextremistischen Subkultur. Internetseite der Bundeszentrale f\u00fcr politische Bildung, Bonn 2008 [\u2192 Reader], <http:\/\/www.bpb.de\/politik\/extremismus\/linksextremismus\/33632\/autonome>, [31.7.2014].\n\nPfahl-Traughber, Armin: Der Antifaschismus als Thema linksextremistischer Agitation, B\u00fcndnispolitik und Ideologie. Eine Analyse zu ideengeschichtlichen Hintergr\u00fcnden und strategischen Funktionen eines Kampf begriffs. Internetseite der Bundeszentrale f\u00fcr politische Bildung, Bonn 2008 [\u2192 Reader], <http:\/\/www.bpb.de\/politik\/extremismus\/linksextremismus\/33612\/antifaschism.us>, [31.7.2014]\n\nProjektgruppe Antifa (Hrsg.): Antifa: Diskussionen und Tips aus der antifaschistischen Praxis. Ed. ID-Archiv, Berlin\/Amsterdam 1994.\n\nReimitz, Monika: \"Niemand soll mich zu fassen kriegen. Gespr\u00e4che mit Vertretern der Autonomen\", in: Bock, Marlene u. a.: Zwischen Resignation und Gewalt. Jugendprotest in den achtziger Jahren. Leske + Budrich, Opladen 1989, S. 133-149.\n\nRichter, Horst-Eberhard: \"Gewalt bei Demonstrationen\", in: Bock, Marlene u. a.: Zwischen Resignation und Gewalt. Jugendprotest in den achtziger Jahren, Leske + Budrich, Opladen 1989, S. 121-132.\n\nRilling, Rainer (Hrsg.): Eine Frage der Gewalt \u2013 Antworten von links. Dietz, Berlin 2008 [pdf vorh.].\n\nRoth, Roland: \"Demonstrieren, Blockieren. Neue Qualit\u00e4ten in der politischen Auseinandersetzung?\", in: Heitmeyer, Wilhelm (Hrsg.): Jugend-Staat \u2013 Gewalt: Politische Sozialisation von Jugendlichen, Jugendpolitik und politische Bildung. Juventa, Weinheim\/M\u00fcnchen 1989, S. 175-190.\n\nRucht, Dieter (Hrsg.): Berlin, 1. Mai 2002. Politische Demonstrationsrituale (B\u00fcrgergesellschaft und Demokratie; 11). Leske + Budrich, Opladen 2003.\n\nRubner, Hartmut: \"Linker Radikalismus und Geschichtspolitik. Ein \u00dcberblick\", in: Bois, Marcel\/H\u00fcttner, Bernd (Hrsg.): Beitr\u00e4ge zur Geschichte einer pluralen Linken, rls-papers, Heft 1: Theorien und Bewegungen vor 1968, Mai 2010, S. 59-64 [online verf\u00fcgbar, pdf. vorh. \u2192 Reader].\n\nSch\u00f6nberger, Klaus\/Sutter, Ove (Hrsg.): Kommt herunter, reiht euch ein... Eine kleine Geschichte der Protestformen sozialer Bewegungen. Assoziation A, Berlin u. a. 2009.\n\nSch\u00fcnemann, Uwe: Die untersch\u00e4tzte Gefahr \u2013 Linksextremismus in Deutschland. Rede des Ministers f\u00fcr Inneres, Sport und Integration des Landes Niedersachsen, Uwe Sch\u00fcnemann MdL, am 29. September 2008 in Hannover, Sankt Augustin. Konrad-Adenauer-Stiftung, Berlin 2008 [\u2192 Reader].\n\nSchuhmacher, Nils: \"'Die Antifa'. Zwischen revolution\u00e4rer Imagination und b\u00fcrgerschaftlichem Engagement\", in: Bois, Marcel\/H\u00fcttner, Bernd (Hrsg.): Beitr\u00e4ge zur Geschichte einer pluralen Linken, rls-papers, Heft 2: Theorien und Bewegungen nach 1968, Mai 2010, S. 57-60 [online verf\u00fcgbar, pdf. vorh. \u2192 Reader].\n\nSchultens, Ren\u00e9\/Glaser, Michaela (Hrsg.): 'Linke' Militanz im Jugendalter. Befunde zu einem umstrittenen Ph\u00e4nomen. Deutsches Jugendinstitut e. V., Halle 2013.\n\nSchwarzmeier, Jan: Die Autonomen zwischen Subkultur und sozialer Bewegung. Books on Demand, Hamburg 2001.\n\nSedlmaier, Alexander: \"Konsumkritik und politische Gewalt in der linksalternativen Szene der siebziger Jahre\", in: Reichardt, Sven\/Siegfried, Detlef (Hrsg.): Das alternative Milieu. Antib\u00fcrgerlicher Lebensstil und linke Politik in der Bundesrepublik Deutschland und Europa 1968-1983 (Hamburger Beitr\u00e4ge zur Sozial- und Zeitgeschichte; 47). Wallstein, G\u00f6ttingen 2010, S. 185-205.\n\nSemler, Christian: Die radikale Linke und die RAF. Essay. Internetseite der Bundeszentrale f\u00fcr politische Bildung <http:\/\/www.bpb.de\/apuz\/30196\/die-radikale-linke-und-die-raf-essay>, [31.7.2014] und in: Aus Politik und Zeitgeschichte, Bd. 40-41\/2007), S. 3-5 [\u2192 Reader].\n\nSoneenberg, Uwe: \"Agitation und Aufkl\u00e4rung. Zur Geschichte linker Buchl\u00e4den nach 1968'\", in: Bois, Marcel\/H\u00fcttner, Bernd (Hrsg.): Beitr\u00e4ge zur Geschichte einer pluralen Linken, rls-papers, Heft 2: Theorien und Bewegungen nach 1968, Mai 2010, S. 16-19 [online verf\u00fcgbar, pdf. vorh. \u2192 Reader].\n\nSuttner, Andreas: \"Beton brennt\". Hausbesetzer und Selbstverwaltung im Berlin, Wien und Z\u00fcrich der 80er. Lit, M\u00fcnster 2011.\n\nThiel, Wolfgang\/Wirth, Hans-J\u00fcrgen: \"Mu\u00df denn Leben S\u00fcnde sein? Notizen, Assoziationen und Interpretationen zu Gespr\u00e4chen mit Hausbesetzern und Punks\", in: Bock, Marlene u. a.: Zwischen Resignation und Gewalt. Jugendprotest in den achtziger Jahren. Leske + Budrich, Opladen 1989, S. 11-42.\n\nWetzel, Wolf: T\u00f6dliche Sch\u00fcsse. Eine dokumentarische Erz\u00e4hlung. Unrast, M\u00fcnster 2008.\n\nWildenhain, Michael: Zum Beispiel K. Rotbuch, Berlin 1983.\n\nWilmer, Daniela: Autonome. Projektarbeit zum Thema \"Jugendkulturen und ihre Darstellungsformen\". 3. Semester des Ausbildungsgangs Modejournalismus\/Medienkommunikation. Akademie Mode & Design, Berlin 2009.\n\n### Brosch\u00fcren des Verfassungsschutzes\n\nBundesamt f\u00fcr Verfassungsschutz (Hrsg.): Feindbilder im politischen Extremismus. Gegens\u00e4tze, Gemeinsamkeiten und ihre Auswirkungen auf die Innere Sicherheit. K\u00f6ln 2003 [\u2192 Reader; pdf vorh.].\n\nBundesamt f\u00fcr Verfassungsschutz (Hrsg.): Extremistische Globalisierungskritik nach Heiligendamm. K\u00f6ln 2008 [\u2192 Reader; pdf vorh.].\n\nBundesministerium des Innern (Hrsg.): Extremismus in Deutschland. Erscheinungsformen und aktuelle Bestandsaufnahme (Texte zur Inneren Sicherheit). Berlin 2004.\n\nH\u00fcllen van, Rudolf: \"Antiamerikanismus im Rechts- und Linksextremismus\", in: Bundesamt f\u00fcr Verfassungsschutz (Hrsg.): Feindbilder im politischen Extremismus. Gegens\u00e4tze, Gemeinsamkeiten und ihre Auswirkungen auf die Innere Sicherheit. Ein Symposium des Bundesamtes f\u00fcr Verfassungsschutz am 1. Oktober 2003. K\u00f6ln 2004, S. 63-75.\n\nInnenministerium des Landes Nordrhein-Westfalen. Abteilung Verfassungsschutz (Hrsg.): Linksextremismus in NRW. D\u00fcsseldorf 2008 [\u2192 Reader].\n\nLandesamt f\u00fcr Verfassungsschutz Hessen (Hrsg.): \"Staat. Nation. Kapital. Schei\u00dfe.\" \u2013 Einblicke in die autonome Bewegung. Wiesbaden 2010 [\u2192 Reader; pdf vorh.].\n\nLandesamt f\u00fcr Verfassungsschutz Sachsen (Hrsg.): Autonome Szene im Freistaat Sachsen. Dresden 2010 [\u2192 Reader; pdf vorh.].\n\nMeyer, Thomas: \"Islamismus, Linksextremismus und der 'Westen'\", in: Bundesamt f\u00fcr Verfassungsschutz (Hrsg.): Feindbilder im politischen Extremismus. Gegens\u00e4tze, Gemeinsamkeiten und ihre Auswirkungen auf die Innere Sicherheit. Ein Symposium des Bundesamtes f\u00fcr Verfassungsschutz am 1. Oktober 2003. K\u00f6ln 2004, S. 76-93.\n\nNieders\u00e4chsisches Ministerium f\u00fcr Inneres, Sport und Integration. Verfassungsschutz (Hrsg.): Linksextremismus \u2013 die untersch\u00e4tzte Gefahr? Extremismus-Symposium des Nieders\u00e4chsischen Verfassungsschutzes am 28. Mai 2009 in Hannover. Tagungsdokumentation. Hannover 2009 [\u2192 Reader].\n\nSenatsverwaltung f\u00fcr Inneres und Sport. Abteilung Verfassungsschutz (Hrsg.): Linke Gewalt in Berlin. Berlin 2009.\n\nSenatsverwaltung f\u00fcr Inneres und Sport. Abteilung Verfassungsschutz (Hrsg.): Linksextremistische Protestvorbereitungen gegen den G 8-Gipfel 2007. Berlin 2007 [\u2192 Reader].\n\nTh\u00fcringer Landesamt f\u00fcr Verfassungsschutz (Hrsg.): Linksextremistische Gewalt \u2013 Gef\u00e4hrdungen, Ursachen und Pr\u00e4vention. 9. Symposium des Th\u00fcringer Landesamtes f\u00fcr Verfassungsschutz am 4. November 2010 in Erfurt, Augustinerkloster. Erfurt 2010.\n\nVerfassungsschutz Brandenburg (Hrsg.): Schwarze Bl\u00f6cke rechts und links. Autonome Extremisten auf Gewaltkurs. Eine Veranstaltung des Verfassungsschutzes am 24. Juni 2010 in Potsdam. Tagungsband, Potsdam 2010 [\u2192 Reader; pdf vorh.].\n\nVerfassungsschutz Nordrhein-Westfalen (Hrsg.): Linksextremismus in NRW. D\u00fcsseldorf 2008 [\u2192 Reader; pdf vorh.].\n\nAu\u00dferdem.: Zahlreiche Jahresberichte des Bundesamtes und der meisten Landes\u00e4mter f\u00fcr Verfassungsschutz aus den letzten vier Jahrzehnten.\n\n### Weitere Literaturhinweise\n\n(Monographien, Sammelb\u00e4nde, Zeitschriftenaufs\u00e4tze, graue Literatur)\n\nAltvater, Elmar u. a. (Hrsg.): Radikal. Dokumentation kriminalisierter Texte. Eigendruck, Berlin 1997.\n\nArndt, Susan u. a. (Hrsg.): Berlin, Mainzer Stra\u00dfe: \"Wohnen ist wichtiger als das Gesetz.\" Basis-Druck, Berlin 1992.\n\n\u00c4sthetik und Kommunikation: H. 99\/Jg. 28. Thema: Demos. R\u00fcckkehr zur Stra\u00dfe?\n\nAtzert, Thomas\/Fanizadeh, Andreas (Redaktion) \/ Wohlfahrtsaussch\u00fcsse (Hrsg.): Etwas Besseres als die Nation: [Texte und Materialien zur Abwehr des gegenrevolution\u00e4ren \u00dcbels]. Ed. ID-Archiv, Berlin\/Amsterdam 1994.\n\nAus Politik und Zeitgeschichte (APuZ): Extremismus. Beilage zur Wochenzeitung Das Parlament 44\/2010. Bundeszentrale f\u00fcr politische Bildung, Bonn 2010.\n\nAust, Stefan\/Rosenbladt, Sabine (Hrsg.): Hausbesetzer. Wof\u00fcr sie k\u00e4mpfen, wie sie leben und wie sie leben wollen. Hoffmann u. Campe, Hamburg 1981.\n\nAutonome: Zur autonomen Szene in G\u00f6ttingen. Rauchen ist ungesund \u2013 daf\u00fcr sind wir permanent vermummt. Autonome Rauchzeichen c\/o Buchladen Rote Strasse, G\u00f6ttingen 1993.\n\nAutonome Antifa (M): \"Antifaschismus auf antiimperialistischer Grundlage\", in: Projektgruppe (Hrsg.): Antifa, S. 49-68 (o. Jahresangabe).\n\nAutonome\/antiimperialistische Gruppe: Wo ist die authentische Darstellung unserer Politik geblieben...? Einige \u00dcberlegungen zur bundesweiten Situation nach dem Hungerstreik. Marburg 1989.\n\nautonome lupus-Gruppe: \"Autonome Bewegung \u2013 Langlauf oder Absturz\", 1986, abgedruckt als: \"Eigentlich Schnee von gestern?\", in: Atom Nr. 41, 18. Jg., 1994, S. 26-32.\n\nautonome lupus-Gruppe: Stand autonomer Bewegung \u2013 Langlauf oder Abfahrt im Sturz, Brosch\u00fcre im Archiv der Sozialen Bewegungen, Hamburg 1987, Kurzfassung in: Schwarzer Faden Nr. 25, 3\/87, S. 13\u201314.\n\nautonome lupus-Gruppe: \"Nicht mehr l\u00e4nger Bauern auf dem Schachbrett der Gener\u00e4le\", in: Schwarzer Faden Nr. 23, 1\/87, S. 22\u201330.\n\nautonome lupus-Gruppe: Geschichte, Rassismus und das Boot. Wessen Kampf gegen welche Verh\u00e4ltnisse? Ed. ID-Archiv, Berlin\/Amsterdam 1992.\n\nAutonome Studis (Bolschewiki): Mit den \u00fcberlieferten Vorstellungen radikal brechen. Ein Blick \u00fcber den Tellerrand autonomer Basisbanalit\u00e4ten. 2., erw. Auflage, \u00e7a ira, Freiburg 1990.\n\nAutonomie-Kongress: Auf den Weg ins 21. Jahrhundert. Vorbereitungsreader, Berlin 1994.\n\nBackes, Uwe: \"Rechts- und linksextremistische Gewalt in Deutschland \u2013 vergleichende Betrachtungen\", in: Extremismus in Deutschland \u2013 Schwerpunkte, Perspektiven, Vergleich. Politische Studien 7\/2007. S. 31-43.\n\nBalistier, Thomas: \"Stra\u00dfenprotest in der Bundesrepublik Deutschland. Einige Entwicklungen, Besonderheiten und Novit\u00e4ten in den Jahren 1979 bis 1983\", in: Warneken, Bernd J\u00fcrgen (Hrsg.): Massenmedium Stra\u00dfe. Zur Kulturgeschichte der Demonstration. Campus, Frankfurt\/Main u. a. 1991, S. 257-281.\n\nBalistier, Thomas: Stra\u00dfenprotest. Formen oppositioneller Politik in der Bundesrepublik Deutschland zwischen 1979 und 1989. Westf\u00e4lisches Dampfboot, M\u00fcnster 1996.\n\nBaron, Udo: \"'Fight the System. Fuck the law!' \u2013Autonome in Bewegung\", in: Deutschland Archiv \u2013 Zeitschrift f\u00fcr das vereinigte Deutschland (2) 2010, S. 197-203.\n\nBecker, Helmut\/Eigenbrodt, J\u00f6rg\/May, Michael: \"Cliquen und Raum. Zur Konstitution von Sozialr\u00e4umen bei unterschiedlichen sozialen Milieus von Jugendlichen\", in: Neidhardt, Friedhelm (Hrsg.): Gruppensoziologie. Sonderheft 25 der KZfSS. Opladen u. a. 1983.\n\nBlank, Bettina: \"Deutschland, einig Antifa\"? \"Antifaschismus\" als Agitationsfeld von Linksextremisten (Extremismus und Demokratie; 28). Nomos, Baden-Baden 2014.\n\nBreyvogel, Wilfried (Hrsg.): Autonomie und Widerstand. Zur Theorie und Geschichte des Jugendprotests. Rigodon, Essen 1983.\n\nBude, Eva: Verpisst Euch! Sex and drugs and hardcore-punk. Europa-Verlag, Hamburg u. a. 2005.\n\nBusch, Heiner: \"Die B\u00fcrgerinitiative in der Lederjacke. Anmerkungen zu den Autonomen\", in: Vorg\u00e4nge 101, 28(5) 1989, S. 62-67.\n\nButterwegge, Christoph: \"Eine fatale Gleichung. Rechtsextremismus und Linksradikalismus sind keine miteinander verwandten Extremismusformen\", in: Neue Praxis 6\/2009, S. 639-641.\n\nClarke, John: \"Stil\", in: Ders. u. a.: Jugendkultur als Widerstand. Milieus, Rituale, Provokationen. Syndikat, Frankfurt\/Main 1979, S. 133-157.\n\nDer Spiegel: \"'Das Wort Nation ist so tot.' Drei Autonome und der Berliner Innensenator J\u00f6rg Sch\u00f6nbohm (CDU) \u00fcber Gewalt, das Grundgesetz und die Zukunft der Hauptstadt\", in: Spiegel 21\/1998.\n\nDer Spiegel: \"'Der Traum, von N\u00e4chten ohne Gesetz.' Die bundesdeutschen Autonomen und ihr Verh\u00e4ltnis zur Gewalt\", in: Spiegel 19\/1989, S. 118.\n\ndie tageszeitung: \"Sachschaden. H\u00e4user und andere K\u00e4mpfe\", in: taz-Journal Nr. 3, Berlin 1981.\n\nDr\u00fccke, Bernd: Zwischen Schreibtisch und Stra\u00dfenschlacht? Anarchismus und libert\u00e4re Presse in Ost- und Westdeutschland. Klemm & Oelschl\u00e4ger, Ulm 1998.\n\nFarin, Klaus: \"Ein anderer Blick. Alternative und oppositionelle Jugendkulturen der 90er Jahre\", in: Breyvogel, Wilfried (Hrsg.): Stadt, Jugendkulturen und Kriminalit\u00e4t. Dietz, Bonn 1998, S. 198-208.\n\nFelS (Hrsg.): Texte zur Kritik an den Autonomen. Organisationsdebatte. Berlin 1992.\n\nFrankfurter, H: \"Das rebellische Kind, oder: 'Nie wieder erwachsen!'\", in: HKS 13 (Hrsg.): hoch die kampf dem. 20 Jahre Plakate autonomer Bewegungen. Verlag Libert\u00e4re Assoziation, Hamburg, Verlag der Buchl\u00e4den Schwarze Risse \u2013 Rote Stra\u00dfe, Hamburg u. a. 1999, S. 124-135.\n\nFriemann, Andrea: Theorien und Feindbilder der deutschen Autonomen. Grin, M\u00fcnchen 2002.\n\nFrombeloff, Carl-Maria (Hrsg.):... und es begann die Zeit der Autonomie. Politische Texte von Karl Heinz Roth. Libert\u00e4re Assoziation, Hamburg; Schwarze Risse, Berlin; Rote Strasse, G\u00f6ttingen 1993.\n\nGansner, Hans Peter: Desperado. Roman einer Bewegung. Rotpunkt, Z\u00fcrich 1981.\n\nHaunss, Sebastian: \"Das Innere sozialer Bewegungen. Strukturelle Konfliktlinien basisdemokratischer Bewegungsorganisationen\", in: Roth, Roland\/Rucht, Dieter (Hrsg.): Jugendkulturen, Politik und Protest. Vom Widerstand zum Kommerz? Leske + Budrich, Opladen 2000, S. 141-163.\n\nHegar, A.: \"Modell Kreuzberg\", in: Konkret 1\/1990, S. 80-84.\n\nHeiland, Silke\/Keupp, Heiner\/Straus, Florian: Gesellschaftliches bzw. politisches Engagement, Jugendprotest und die Wahl der Mittel. Eine Studie des Instituts f\u00fcr Praxisforschung und Projektberatung im Auftrag der Bayerischen Landeszentrale f\u00fcr politische Bildungsarbeit. M\u00fcnchen 2013.\n\nHillenkamp, Sven: \"Die Autonomen. Zwischen kultureller Wirklichkeit und politischer Wirksamkeit\", in: Forschungsjournal Neue Soziale Bewegungen 2\/1995, S. 54-66.\n\nHoffmann-Axthelm, Dieter: \"Stra\u00dfengewalt und Autonomiebewegung in Berlin-Kreuzberg\", in: Neue Gesellschaft\/Frankfurter Hefte 2\/1989, S. 148-157.\n\nHolm, Andrej: Die Restrukturierung des Raumes. Stadterneuerung der 90er Jahre in Ostberlin. Interessen und Machtverh\u00e4ltnisse. Transcript, Bielefeld 2006.\n\nH\u00fcllen, Rudolf van: Linksextremismus \u2013 eine vernachl\u00e4ssigte Gefahr (Aktuelle Fragen der Politik; 44). Konrad-Adenauer-Stiftung, Referat f\u00fcr Publ., Sankt Augustin 1997.\n\nH\u00fcttner, Bernd: \"Die Krise der autonomen Medien\", in: Contraste 10\/1993 (nachgedruckt in Interim Nr. 259, Swing Nov. 1993, radikal Nr. 148 und in der Brosch\u00fcre zum radikal-Verfahren 1995).\n\nH\u00fcttner, Bernd: \"Der (Still-)stand der Bewegung. Kongressbericht\", in: Z. Zeitschrift Marxistische Erneuerung 23\/1995.\n\nH\u00fcttner, Bernd: \"Die Autonomen\", in: Z. Zeitschrift Marxistische Erneuerung 24\/1995.\n\nH\u00fcttner, Bernd: \"Linke Beziehungsgeschichten\", in: discus 3\/2005 (ersch. Sept. 2006), Nachdruck in Interim Nr. 644 vom 26.10.2006, auch online verf\u00fcgbar.\n\nH\u00fcttner, Bernd: \"When I'm 44 \u2013 Kinder, \u00c4lterwerden und das linke Leben\", in: arranca! Nr. 33 (unter Pseudonym).\n\nIG Rote Fabrik (Hrsg.): Zwischenberichte. Zur Diskussion \u00fcber die Politik der bewaffneten und militanten Linken in der BRD, Italien und der Schweiz, [Red.: Alvaro Baragiola]. ID-Verlag, Berlin 1998.\n\nInstitut f\u00fcr Elbvertiefung, Bewegungslehre und Politikberatung (Hrsg.): Autonomie-Kongre\u00df der undogmatischen linken Bewegungen \u2013 Standpunkte \u2013 Provokationen \u2013 Thesen. Unrast, M\u00fcnster 1997.\n\nJenrich, Holger: Anarchistische Presse in Deutschland. 1945\u20131985. Trotzdem, Grafenau-D\u00f6ffingen 1988.\n\nKn\u00fctter, Hans-Helmutn\/Winckler, Stefan (Hrsg.): Handbuch des Linksextremismus. Die untersch\u00e4tzte Gefahr. Stocker, Graz\/Stuttgart 2002.\n\nKolling, Heide: Honig aus dem zweiten Stock. Berliner Hausprojekte erz\u00e4hlen. Assoziation A, Berlin u. a. 2008.\n\nKongress der Radikalen Linken: Deutschland? Nie wieder! Frankfurt\/Main 1990.\n\nKongressvorbereitungsgruppe (Hrsg.): Die radikale Linke. Reader zum Kongress vom 1.\u20133. Juni 1990 in K\u00f6ln. Konkret-Literatur-Verlag, Hamburg 1990.\n\nKritik und Praxis Berlin: \"Kommunismus und politische Praxis\", in: Demo-Punk\/Kritik und Praxis Berlin (Hrsg.): Indeterminate! Kommunismus. Texte zur \u00d6konomie, Politik und Kultur. Unrast, M\u00fcnster 2005, S. 313-322.\n\nLanger, Bernd: Kunst als Widerstand. Plakate, \u00d6lbilder, Aktionen, Texte der Initiative Kunst und Kampf. Pahl-Rugenstein, Bonn 1997.\n\nLauterbach, J\u00f6rg: Staats- und Politikverst\u00e4ndnis autonomer Gruppen in der BRD. Dissertation im Fachbereich Gesellschaftswissenschaften der Johann-Wolfgang-Goethe-Universit\u00e4t zu Frankfurt am Main 1999.\n\nMail\u00e4nder, Ulf\/Zander, Ulrich: Das kleine Westberlin-Lexikon \u2013 Von \"Autonome\" bis \"Zapf\" \u2013 die alternative Szene der siebziger und achtziger Jahre. Schwarzkopf, Berlin 2003.\n\nManns, Haide und Treusch, Wolf-S\u00f6ren: \"'Hau weg die Schei\u00dfe'. Autonomer Widerstand in der Bundesrepublik\", in: Vorg\u00e4nge 85, 26, 1987, S. 65-74.\n\nManrique, Matthias: Marginalisierung und Militanz. Jugendliche Bewegungsmilieus im Aufruhr (Campus Forschung; Bd. 677). Campus, Frankfurt am Main\/New York 1992.\n\nMedico International u. a.: 20 Jahre Radikal. Geschichte und Perspektiven autonomer Medien. Verlag Libert\u00e4re Assoziation, Hamburg, Unrast, M\u00fcnster, Verlag der Buchl\u00e4den Schwarze Risse\/Rote Stra\u00dfe, Hamburg u. a., Ed. ID-Archiv, Berlin 1996.\n\nNeu, Viola: Rechts- und Linksextremismus in Deutschland. Wahlverhalten und Einstellungen (Zukunftsforum Politik; 98). Konrad-Adenauer-Stiftung, Sankt Augustin\/Berlin 2009.\n\nO. A.: \"Ich sag wie es ist.\" [Papier (angeblich) autonom\/antiimperialistischer Gruppen mit stark marxistisch-leninistischem Einschlag], Hamburg 1988.\n\nO. A.: Vier freye Geister im Nebelschleyer: Die autonome Bewegung und die RAF. Eine v\u00f6llig unautorisierte Stellungnahme zum Eliteverein RAF, in taz-Journal: 20 Jahre Deutscher Herbst. Berlin 1997, S. 39.\n\nPfahl-Traughber, Armin: \"Die Autonomen. Portrait einer linksextremistischen Subkultur\", in: Aus Politik und Zeitgeschichte, B 9\u201310\/1998, S. 36-46.\n\nPorta, Donatella della: \"Gewalt und die neue Linke\", in: Heitmeyer, Wilhelm\/Hagan, John (Hrsg.): Internationales Handbuch der Gewaltforschung. Westdeutscher Verlag, Wiesbaden 2002, S. 479-500.\n\nRada, Uwe: \"Wie lange wird es die Autonomen noch geben?\", in: taz 18.04.1995, S. 3.\n\nRote Flora Dokumentation: Teil 1: Den Widerstand in den Vierteln organisieren! Hamburg 1988.\n\nRoth, Roland\/Rucht, Dieter (Hrsg.): Jugendkulturen, Politik und Protest. Vom Widerstand zum Kommerz? Leske + Budrich, Opladen 2000.\n\nRoth, Roland\/Rucht, Dieter (Hrsg.): Die sozialen Bewegungen in Deutschland seit 1945. Ein Handbuch, Campus, Frankfurt\/New York 2008.\n\nRucht, Dieter u. a.: Soziale Bewegungen auf dem Weg zur Institutionalisierung? Zum Strukturwandel \"alternativer\" Gruppen in beiden Teilen Deutschlands. Campus, Frankfurt\/Main u. a. 1997.\n\nRucht, Dieter: \"Gewalt und neue soziale Bewegungen.\" Heitmeyer, Wilhelm\/Hagan, John (Hrsg.): Internationales Handbuch der Gewaltforschung. Westdeutscher Verlag, Wiesbaden 2002, S. 461-478.\n\nScheer, Joseph\/Espert, Jan: \"Deutschland, Deutschland, alles ist vorbei.\" Alternatives Leben oder Anarchie? Die neue Jugendrevolte am Beispiel der Berliner 'Scene' Bernard & Graefe, M\u00fcnchen 1982.\n\nSchr\u00f6ter, Lorenz: \"Tod einer Freiheitsk\u00e4mpferin\", in: Zeitmagazin 8, 18.02.1999, S. 18-22.\n\nSchultze, Thomas: Die Autonomen. Profil einer radikalen Systemopposition. Diplomarbeit am Institut f\u00fcr Politische Wissenschaft, Hamburg 1991.\n\nSternsdorff, Hans-Wolfgang: \"Wir machen nicht aus Spa\u00df Krawall\". Spiegel-Interview mit den Kreuzberger Autonomen Alex und Ronnie \u00fcber ihre Strategie der Gewalt. Spiegel 19\/1989, S. 117-120.\n\nStrobl, Ingrid u. a.: Drei zu Eins. ID-Archiv, Berlin\/Amsterdam 1993. <http:\/\/www.idverlag.com\/BuchTexte\/DreiZuEins\/DreiZuEins.html>, [31.7.2014].\n\nTolmein, Oliver: \"Die Rote Zora z\u00fcndelt nicht\", in: Konkret 1\/1988.\n\nWildenhain, Michael: Die kalte Haut der Stadt. Roman. Fischer-Taschenbuch, Frankfurt am Main 1995.\n\nWolfschlag, Claus-M.: \"Antifa ist Pop. Zur popul\u00e4rkulturellen Konstituierung einer radikalen Linken\", in: Kn\u00fctter, Hans-Helmuth\/Winckler, Stefan (Hrsg.): Handbuch des Linksextremismus. Stocker, Graz 2002, S. 98-118.\n\nWolfschlag, Claus-M.: Das \"antifaschistische Milieu\": vom \"schwarzen Block\" zur \"Lichterkette\" \u2013 die politische Repression gegen \"Rechtsextremismus\" in der Bundesrepublik Deutschland. Gek\u00fcrzte, aktualisierte Sonderausgabe. Stocker, Graz\/Stuttgart 2002.\n\nZinkernagel, Moritz: Die Auseinandersetzung um das Autonome Zentrum Wuppertal 1986 \u2013 1990. Grin, M\u00fcnchen 2007.\n\nAu\u00dferdem verf\u00fcgt die Pr\u00e4senzbibliothek des Archiv der Jugendkulturen \u00fcber eine gro\u00dfe Sammlung autonomer u. a. Zeitschriften, Fanzines und weitere Szene-Publikationen.\n\nWer wichtige Literatur vermisst: Das Archiv der Jugendkulturen freut sich \u00fcber jede Spende. Kontakt: bibliothek@jugendkulturen.de.\n\n## Inhalt\n\n  1. Cover\n  2. Schmutztitel\n  3. Impressum\n  4. Titel\n  5. Widmung\n  6. Inhalt\n  7. Die Emporten\n    1. Geschichte\n    2. Organisationsstrukturen\n    3. Szene-Medien\n    4. Theorie und Aktionsfelder\n    5. Sexismus\n    6. Militanz\n    7. Antiimps und Antideutsche\n    8. Autonome - eine Jugendszene?\n    9. Literatur\n    10. Fazit\n    11. Literaturverzeichnis\n  8. \"Ruf niemals um Hilfe, es konnte die Polizei kommen!\"\n  9. \"Dieser Staat richtet sich gegen mich, und aus diesem Grund ist das Verhaltnis zwischen mir und dem Staat ein feindliches.\"\n  10. \"Israel ist halt das Mallorca der Antideutschen.\"\n  11. \"Naturlich macht es mich wutend zu sehen, wie wir hier leben und wie andere woanders leben mussen.\"\n  12. \"Ich mochte, dass hier Menschen gleich behandelt werden.\"\n  13. \"Meine Zukunft? Definitiv keine schei\u00df-burgerliche Kleinfamilie!\"\n  14. Tierbefreierin\n  15. \"Privat kann man die Musik horen, die man will, es sie denn, sie ist frauenfeindlich, homophob, antisemitisch oder rassistisch.\"\n  16. \"Extreme Aktionen wie gegen die sogenannte Grauzone finde ich daneben. Bands verbieten, obwohl man sie nicht wirklich kennt und sich nicht mit ihnen unterhalten hat, ist schei\u00dfe.\"\n  17. \"Die Arroganz und Naivitat der Menschen heute macht mir schon Angst.\"\n  18. \"Antifaschismus ist als Grundlage einer linksradikalen Organisierung nicht geeignet.\"\n  19. \"Wenn am 1. Mai sogar die Schaufenster linker Kollektive in Kreuzberg eingeschlagen werden, lauft irgendwas falsch.\"\n  20. Live dabei am 1. Mai - Die \"Revolutionare 1. Mai Demo\" in Kreuzkolln\n  21. \"Die Szene definiert sich ganz klar daruber, dass sie sich als was Besseres fuhlen als andere Leute.\"\n  22. \"Am 11. September hab ich mich total gefreut. Ich hab echt gejubelt.\"\n  23. \"Kopftuch oder Sturmmaske?\"\n  24. \"Standig ein Plenum nach dem anderen und immer Stellung nehmen. Es hatte nur noch gefehlt, dass wir Antrage furs Molliwerfen stellen sollten.\"\n  25. \"Wenn Parteien mit Autonomen klarkamen, ware ich auch in einer Partei.\"\n  26. \"Gewalt sollte immer das Salz in der Suppe sein und nicht die Suppe selber!\"\n  27. Von der Sponti-Schule zum Antifa-Shopping\n  28. Die Zeitschrift radikal als Sprachrohr der linken autonomen Szene - Eine Medienanalyse\n    1. Zahne zeigen\n    2. Das praktische Instrument\n    3. Die Bewegung der Bewegung\n    4. Der Kampf geht weiter, doch wohin? - Das Dilemma der Unintegrierbarkeit\n    5. Die Flaute des Sturms und die Locher im Segel\n    6. Die Repression startet durch\n    7. Going underground\n    8. Es brodelt im Untergrund\n    9. Radikal in der Krise\n    10. Reset, Neustart - und Flop\n    11. Die schwarze Serie\n    12. Quo vadis, radikal?\n    13. Lebt und lest radikal?\n  29. ExpertInneninterviews\n  30. \"Ein Maximum an Demokratie ist aus unserer Perspektive nicht erstrebenswert.\"\n    1. Prof. Uwe Backes, 54 Jahre, Mitbegrunder der vergleichenden Extremismusforschung.\n  31. \"Dass ihr deklariertes Ziel, den Kapitalismus und den Staat zu bekampfen, aussichtslos ist, ist offenkundig.\"\n    1. Prof. Dr. Albert Scherr, 56 Jahre, Soziologe und Sozialpadagoge.\n  32. Feindbeobachtung\n    1. Felix Krautkramer, 35 Jahre, Redakteur der Junge Freiheit.\n  33. \"'Jugendkultur' ist ein Ausdruck wie ein Sack, in den man alles reinstecken kann, was man selbst, alter geworden, nicht versteht.\"\n    1. Prof. Dr. Wolf-Dieter Narr, 77 Jahre, Menschenrechtsaktivist und Demonstrationsbeobachter.\n  34. \"Das Potential fur linksradikales Denken sehe ich in Deutschland nicht.\"\n    1. Prof. Dr. Melanie Gro\u00df, 39 Jahre, Feministin.\n  35. Die Wut ist vorhersehbar\n  36. Bibliographie zum Buchprojekt Autonome\n    1. Broschuren des Verfassungsschutzes\n    2. Weitere Literaturhinweise\n  37. Cover-Ruckseite\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# \u00cdNDICE\n\nPORTADA\n\nSINOPSIS\n\nPORTADILLA\n\nCITA\n\nPR\u00d3LOGO\n\nCAP\u00cdTULO PRIMERO\n\nCAP\u00cdTULO SEGUNDO\n\nCAP\u00cdTULO TERCERO\n\nCAP\u00cdTULO CUARTO\n\nCAP\u00cdTULO QUINTO\n\nCAP\u00cdTULO SEXTO\n\nCAP\u00cdTULO S\u00c9PTIMO\n\nCAP\u00cdTULO OCTAVO\n\nCAP\u00cdTULO NOVENO\n\nCAP\u00cdTULO D\u00c9CIMO\n\nCAP\u00cdTULO UND\u00c9CIMO\n\nEP\u00cdLOGO\n\nNOTAS\n\nCR\u00c9DITOS\nGracias por adquirir este eBook\n\nVisita Planetadelibros.com y descubre una  \nnueva forma de disfrutar de la lectura\n\n\u00a1Reg\u00edstrate y accede a contenidos exclusivos!\n\nPrimeros cap\u00edtulos\n\nFragmentos de pr\u00f3ximas publicaciones  \nClubs de lectura con los autores  \nConcursos, sorteos y promociones  \nParticipa en presentaciones de libros\n\nComparte tu opini\u00f3n en la ficha del libro  \ny en nuestras redes sociales:\n\nExplora Descubre Comparte\n\n# SINOPSIS\n\nBruselas es el coraz\u00f3n de la UE y tambi\u00e9n de esta novela. Esta obra es un viaje a las interioridades de la vida pol\u00edtica europea a trav\u00e9s de cinco historias conectadas que corren paralelas, cada una con un protagonista diferente, y que se centran en la vida diaria de la capital europea desde diversas perspectivas. La capital sigue de cerca el destino de sus personajes en sus frecuentes encuentros, ofreciendo una visi\u00f3n de sus vidas profesionales as\u00ed como de sus coloridas vidas privadas, alejadas de las oficinas y congresos. La mirada penetrante de Menasse, cargada de humor e iron\u00eda, hace de esta novela una lectura tan irresistible como importante en los tiempos actuales de nacionalismos, desde el Brexit hasta la independencia de Catalu\u00f1a.\n\n#\n\nRobert Menasse\n\nLa capital\n\nTraducci\u00f3n del alem\u00e1n por\n\nCarmen Gauger\n\n#\n\nR\u00eaver, c'est le bonheur;\n\nattendre, c'est la vie\n\nVICTOR HUGO\n\n# PR\u00d3LOGO\n#\n\n\u00a1Por ah\u00ed va un cerdo! David de Vriend lo vio al abrir una ventana del sal\u00f3n para pasear la mirada una \u00faltima vez por la plaza antes de abandonar para siempre el piso. No era un sentimental. Hab\u00eda vivido all\u00ed sesenta a\u00f1os, hab\u00eda visto esa plaza a lo largo de sesenta a\u00f1os, y ahora aquello hab\u00eda llegado a su fin. Eso era todo. Era su frase preferida: siempre que contaba, informaba, atestiguaba algo, dec\u00eda dos o tres frases y despu\u00e9s: \u00abEso era todo\u00bb. Para \u00e9l, esa frase era el \u00fanico resumen leg\u00edtimo de cada momento o cap\u00edtulo de su vida. La empresa de mudanzas hab\u00eda recogido los pocos enseres que se llevaba a la nueva direcci\u00f3n. Enseres: una palabra curiosa pero que a \u00e9l no le dec\u00eda nada. Luego los hombres de la empresa encargada de dejar completamente vac\u00eda la vivienda llegaron para llevarse todo lo dem\u00e1s, no s\u00f3lo lo que no estaba bien clavado y remachado, sino tambi\u00e9n los clavos y los remaches: arrancaron, desmontaron, transportaron, hasta que el piso qued\u00f3 vac\u00edo y como reci\u00e9n barrido. Mientras a\u00fan segu\u00edan all\u00ed la cocina y la cafetera expr\u00e9s, De Vriend se hab\u00eda hecho un caf\u00e9, hab\u00eda observado a los hombres con cuidado de no interponerse en su camino, durante mucho tiempo hab\u00eda sostenido la taza de caf\u00e9 vac\u00eda hasta que finalmente la hab\u00eda dejado caer en una bolsa de basura. Luego los hombres se marcharon y el piso qued\u00f3 vac\u00edo. \u00abComo reci\u00e9n barrido\u00bb, seg\u00fan estipulaba el contrato. Eso era todo. Una \u00faltima mirada por la ventana. Ah\u00ed abajo no hab\u00eda nada que \u00e9l no conociera, y ahora deb\u00eda marcharse porque los tiempos hab\u00edan cambiado; y ahora ve\u00eda..., s\u00ed, en efecto: \u00a1all\u00ed abajo hab\u00eda un cerdo! En pleno Bruselas, en Sainte-Catherine. Ten\u00eda que haber salido de la rue de la Braie, corr\u00eda junto a la valla que hab\u00edan puesto delante de la casa por las obras, De Vriend se asom\u00f3 a la ventana y vio que el cerdo, sorteando ahora a algunos transe\u00fantes en la esquina con la rue du Vieux March\u00e9 aux Grains, casi chocaba con un taxi.\n\nKai-Uwe Frigge, impulsado hacia delante por el frenazo, cay\u00f3 de nuevo en el asiento e hizo una mueca. Llegaba tarde. Estaba impaciente. \u00bfQu\u00e9 pasaba ahora? No llegaba realmente tarde, s\u00f3lo que cuando ten\u00eda una cita le gustaba llegar diez minutos antes de la hora, sobre todo en d\u00edas lluviosos, para asearse un momento en el ba\u00f1o, el pelo h\u00famedo, las gafas empa\u00f1adas, antes de encontrarse con la persona con la que hab\u00eda quedado.\n\n\u00a1Un cerdo! \u00bfLo ha visto, monsieur?, exclam\u00f3 el taxista. \u00a1Por poco se me mete en el coche! Se inclin\u00f3 m\u00e1s all\u00e1 del volante: \u00a1Por ah\u00ed! \u00a1Por ah\u00ed! \u00bfLo ve?\n\nAhora lo ve\u00eda Kai-Uwe Frigge. Pas\u00f3 el dorso de la mano por el cristal, el cerdo se marchaba corriendo hacia un lado, el cuerpo h\u00famedo del animal brillaba, sucio y rosado, a la luz de las farolas.\n\n\u00a1Hemos llegado, monsieur! No puedo acercarme m\u00e1s. \u00a1Bueno, qu\u00e9 cosas! \u00a1Por poco se me mete un cerdo en el coche! \u00a1Vaya suerte que he tenido, desde luego!\n\nFenia Xenopoulou estaba en el restaurante Menelas sentada en la primera mesa junto al ventanal que daba a la plaza. Estaba de mal humor por haber llegado tan pronto. No denotaba mucha seguridad en s\u00ed misma lo de estar all\u00ed sentada esperando a que \u00e9l llegara. Estaba nerviosa. Temiendo que hubiera atascos por la lluvia, hab\u00eda calculado mal la duraci\u00f3n del trayecto. Ya iba por el segundo ouzo. El camarero rondaba a su alrededor como una molesta avispa. Ella ten\u00eda la mirada fija en la copa y se oblig\u00f3 a no tocarla. El camarero trajo una jarra de agua fresca, luego un platito de aceitunas... y exclam\u00f3: \u00a1Un cerdo!\n\n\u00bfC\u00f3mo dice? Fenia levant\u00f3 la vista, vio que el camarero miraba hipnotizado hacia la plaza y entonces lo vio ella tambi\u00e9n: el cerdo corr\u00eda en direcci\u00f3n al restaurante en un rid\u00edculo galope, las cortas patitas movi\u00e9ndose r\u00edtmicamente hacia delante y hacia atr\u00e1s bajo el orondo y pesado cuerpo. Ella pens\u00f3 primero que era un perro, uno de esos repugnantes bichos cebados por viudas, pero no, era un cerdo, en efecto. Casi como sacado de un libro de cuentos; ve\u00eda el hocico y las orejas como l\u00edneas, como contornos, as\u00ed se dibuja un cerdo a los ni\u00f1os, pero \u00e9se parec\u00eda salido de un libro infantil de terror. No era un jabal\u00ed; aunque muy sucio, se trataba sin lugar a dudas de un rosado cerdo dom\u00e9stico que ten\u00eda algo de demencial, algo de amenazador. Por el ventanal resbalaba el agua de la lluvia, borrosamente Fenia Xenopoulou vio que el cerdo frenaba de pronto delante de unos transe\u00fantes; con las patas estiradas, resbalaba, se ladeaba, doblaba las patas, se incorporaba y retroced\u00eda al galope, ahora en direcci\u00f3n al hotel Atlas. En ese momento, Ryszard Oswiecki abandonaba el hotel. Ya al salir del ascensor, mientras atravesaba el hall, se hab\u00eda echado sobre la cabeza la capucha de su chaqueta, y ahora sal\u00eda a la lluvia, deprisa pero no demasiado, no quer\u00eda llamar la atenci\u00f3n. La lluvia era una suerte: capucha, paso r\u00e1pido, dadas las circunstancias eso era completamente normal y poco llamativo. Nadie deb\u00eda declarar despu\u00e9s que hab\u00eda visto huir a un hombre, de tal edad m\u00e1s o menos, de aproximadamente tal altura, y el color de la chaqueta... s\u00ed, s\u00ed, eso tambi\u00e9n lo recordaba. R\u00e1pidamente se volvi\u00f3 hacia la derecha, oy\u00f3 voces excitadas, un grito y un jadeo extra\u00f1amente chill\u00f3n. Se par\u00f3 un instante, mir\u00f3 hacia atr\u00e1s. Y entonces descubri\u00f3 al cerdo. No pod\u00eda creer lo que ve\u00eda. Hab\u00eda un cerdo entre dos de esos pilares de hierro forjado que bordeaban la explanada del hotel; la cabeza inclinada, en la postura de un toro al ataque, ten\u00eda un no s\u00e9 qu\u00e9 de rid\u00edculo, pero tambi\u00e9n de amenazador. Aquello era un completo misterio: \u00bfde d\u00f3nde hab\u00eda salido ese cerdo, por qu\u00e9 estaba all\u00ed? A Ryszard Oswiecki le daba la impresi\u00f3n de que en esa plaza, al menos en la medida en que \u00e9l la abarcaba con la vista, la vida se hab\u00eda quedado inm\u00f3vil y congelada, los peque\u00f1os ojos del animal reflejaban centelleantes la luz de ne\u00f3n de la fachada del hotel: entonces Ryszard Oswiecki empez\u00f3 a correr. Corri\u00f3 hacia la derecha, volvi\u00f3 de nuevo la vista atr\u00e1s, el cerdo, resollando, alz\u00f3 la cabeza, retrocedi\u00f3 varios pasos, se dio la vuelta y atraves\u00f3 corriendo la plaza en direcci\u00f3n a la hilera de \u00e1rboles que hab\u00eda delante del Centro Cultural Flamenco De Markten. Los transe\u00fantes que hab\u00edan observado la escena segu\u00edan con la vista al cerdo y no al hombre de la capucha, y ahora Martin Susman vio al animal. Viv\u00eda en la casa contigua al hotel Atlas, abr\u00eda justo en ese momento la ventana para ventilar, y no daba cr\u00e9dito a sus ojos: \u00a1aquello parec\u00eda un cerdo! Acababa de meditar sobre su vida, sobre las casualidades que hab\u00edan dado lugar a que \u00e9l, hijo de campesinos austriacos, viviera y trabajara ahora en Bruselas; por su estado de \u00e1nimo, todo le parec\u00eda absurdo y extra\u00f1o, pero un cerdo corriendo libremente por all\u00ed abajo, por la plaza, era demasiado absurdo, eso s\u00f3lo pod\u00eda ser una estratagema de su imaginaci\u00f3n, una proyecci\u00f3n de sus recuerdos. Miraba pero ya no ve\u00eda al cerdo.\n\nEl cerdo corr\u00eda en direcci\u00f3n a la iglesia de Sainte-Catherine, cruz\u00f3 la rue Sainte-Catherine, se mantuvo a la izquierda, sorteando a los turistas que sal\u00edan de la iglesia, corri\u00f3 dejando a un lado la iglesia, el quai aux Briques; los turistas re\u00edan, seguramente pensaban que el estresado y ya casi colapsado animal era parte del folclore, alg\u00fan fen\u00f3meno local. Muchos hojear\u00edan despu\u00e9s la gu\u00eda de viaje buscando una explicaci\u00f3n. \u00bfNo se hostigan tambi\u00e9n los toros en Espa\u00f1a, en ciertos d\u00edas de fiesta, por las calles de la ciudad de Pamplona? Quiz\u00e1 se hace lo mismo en Bruselas con los cerdos. Cuando se presencia lo inexplicable all\u00ed donde no se espera comprenderlo todo, qu\u00e9 amena es entonces la vida.\n\nEn ese momento, Gouda Mustafa dobl\u00f3 la esquina y casi se top\u00f3 con el cerdo. \u00bfCasi? \u00bfNo le hab\u00eda tocado, no le hab\u00eda rozado la pierna? \u00bfUn cerdo? Gouda Mustafa, presa del p\u00e1nico, salt\u00f3 a un lado, perdi\u00f3 el equilibrio y cay\u00f3 al suelo. Y ahora estaba tendido en un charco, se revolcaba, lo que empeoraba la situaci\u00f3n, pero no era el barro de la calle, era el contacto con el animal impuro, si lo hab\u00eda habido, lo que le hac\u00eda sentirse sucio.\n\nVio entonces una mano tendida hacia \u00e9l, vio el rostro de un se\u00f1or de edad, un rostro triste, preocupado; mojado por la lluvia, aquel hombre parec\u00eda llorar. Era el profesor Alois Erhart. Gouda Mustafa no entendi\u00f3 lo que dec\u00eda, s\u00f3lo la palabra okay.\n\n\u00a1Okay! \u00a1Okay!, dijo Mustafa.\n\nEl profesor Erhart sigui\u00f3 hablando en ingl\u00e9s, dijo que \u00e9l tambi\u00e9n se hab\u00eda ca\u00eddo ese mismo d\u00eda, pero estaba tan confuso que dijo \u00abfailed\u00bb en lugar de \u00abfell\u00bb. Gouda Mustafa no le entendi\u00f3 y dijo otra vez: \u00a1Okay!\n\nPero ya llegaba la sirena. La salvaci\u00f3n. Polic\u00eda. Toda la plaza daba vueltas, centelleaba, vibraba en la luz azul. Los veh\u00edculos de emergencia avanzaban velozmente hacia el hotel Atlas. El cielo de Bruselas cumpl\u00eda con su deber: llov\u00eda. Ahora parec\u00eda que llov\u00edan gotas en reluciente azul. Acompa\u00f1adas, adem\u00e1s, de una fuerte racha de viento que a no pocos transe\u00fantes les arranc\u00f3 y puso del rev\u00e9s el paraguas. Gouda Mustafa tom\u00f3 la mano del profesor Erhart, se dej\u00f3 ayudar. Su padre le hab\u00eda prevenido contra Europa.\n\n# CAP\u00cdTULO PRIMERO\n\nEN REALIDAD NO TIENE POR QU\u00c9 HABER CONEXIONES, PERO SIN ELLAS TODO SE DESCOMPONDR\u00cdA.\n\n#\n\n\u00bfQui\u00e9n invent\u00f3 la mostaza? No es un buen comienzo para una novela. Por otra parte, no puede haber un buen comienzo, porque, bueno o menos bueno, no hay comienzo. Porque toda primera frase imaginable es ya un final, aunque contin\u00fae despu\u00e9s. Est\u00e1 al final de miles y miles de p\u00e1ginas que nunca fueron escritas: de la prehistoria.\n\nCuando se empieza a leer una novela, en el fondo habr\u00eda que pasar hacia atr\u00e1s las hojas ya despu\u00e9s de la primera frase. Era el sue\u00f1o de Martin Susman, eso le habr\u00eda gustado ser: un narrador de prehistorias. Hab\u00eda interrumpido la carrera de Arqueolog\u00eda y s\u00f3lo entonces..., da igual, eso no viene al caso aqu\u00ed, forma parte de la prehistoria de la que ha de prescindir todo comienzo de novela porque, si no, al final nunca hay un comienzo.\n\nMartin Susman estaba sentado ante el escritorio, hab\u00eda apartado el port\u00e1til y sacaba de dos tubos, poni\u00e9ndolas en un plato, dos mostazas diferentes, una inglesa, fuerte, y una alemana, dulce, y se preguntaba qui\u00e9n habr\u00eda inventado la mostaza. \u00bfQui\u00e9n dio en la extravagante idea de producir una pasta que cubre por completo el sabor de un manjar, sin tener ella buen sabor? \u00bfY c\u00f3mo era posible que eso llegara a convertirse en art\u00edculo de masas? Es, pens\u00f3, un producto como la Coca-Cola. Un producto que nadie echar\u00eda de menos si no existiera. Cuando regresaba a casa, Martin Susman hab\u00eda comprado en un supermercado de la cadena Delhaize, en el boulevard Anspach, dos botellas de vino, un ramo de tulipanes, una salchicha y como complemento, evidentemente, mostaza, y adem\u00e1s dos tubos, porque no pod\u00eda decidirse entre dulce y fuerte.\n\nLa salchicha saltaba y crepitaba ahora en la sart\u00e9n, el fuego estaba muy alto, la grasa se quemaba, la salchicha se carbonizaba, pero Martin no prestaba atenci\u00f3n a eso. Sentado, miraba fijamente la rosquilla de mostaza algo m\u00e1s clara y amarillenta, y a su lado la otra marr\u00f3n oscura, puestas en el plato blanco, esculturas en miniatura de excrementos de perro. Contemplar mostaza en un plato mientras en la sart\u00e9n se quema una salchicha no est\u00e1 descrito a\u00fan en la literatura especializada como s\u00edntoma inequ\u00edvoco y t\u00edpico de la depresi\u00f3n: nosotros, sin embargo, podemos interpretarlo como tal.\n\nLa mostaza en el plato. La ventana abierta, la cortina de lluvia. El aire mohoso, el hedor a carne carbonizada, el crujido de la tripa reventada y de la grasa quemada, las esculturas de excrementos en el plato de porcelana: entonces Martin Susman oy\u00f3 el disparo.\n\nNo se asust\u00f3. Hab\u00eda sonado como si en el piso vecino hubieran descorchado una botella de champ\u00e1n. Sin embargo, detr\u00e1s de la pared de particular delgadez no hab\u00eda una vivienda sino una habitaci\u00f3n de hotel. Al lado estaba el hotel Atlas: qu\u00e9 nombre eufem\u00edstico para ese edificio estrecho en el que se alojaban sobre todo encorvados lobbistas que arrastraban tras de s\u00ed una maleta con ruedas. Sin que le importara gran cosa, Martin Susman o\u00eda a menudo a trav\u00e9s de la pared cosas que \u00e9l no pon\u00eda precisamente empe\u00f1o en o\u00edr. Reality show o, qui\u00e9n sabe, s\u00f3lo reality, ronquidos o gemidos.\n\nLa lluvia era m\u00e1s intensa. Martin quer\u00eda salir de casa. Estaba bien preparado para Bruselas. En su fiesta de despedida en Viena le hab\u00edan hecho regalos deliberadamente oportunos para Bruselas, entre ellos nueve paraguas, desde el cl\u00e1sico long brit\u00e1nico, pasando por el knirps alem\u00e1n y hasta el italiano mini en tres colores de Benetton, adem\u00e1s de dos ponchos de lluvia para ciclistas.\n\nEstaba sentado inm\u00f3vil delante de su plato, los ojos clavados en la mostaza. Si m\u00e1s tarde pudo decir a la polic\u00eda a qu\u00e9 hora hab\u00eda sonado el disparo, se debi\u00f3 al hecho de que el supuesto estallido de un tap\u00f3n de champ\u00e1n lo hab\u00eda animado a abrir una botella de vino. Cada d\u00eda aplazaba la bebida lo m\u00e1s posible, en cualquier caso no beb\u00eda nunca antes de las 19.00. Mir\u00f3 el reloj: eran las 19.35. Fue al frigor\u00edfico, sac\u00f3 el vino, apag\u00f3 el fog\u00f3n, ech\u00f3 la salchicha en el cubo de la basura, puso la sart\u00e9n en el fregadero, abri\u00f3 el grifo. El agua burbujeaba en la sart\u00e9n caliente. \u00a1Ya est\u00e1s otra vez pensando en las musara\u00f1as!, dec\u00eda su madre apretando los labios cuando \u00e9l, sentado ante un libro y con mirada ausente, clavaba la vista en el vac\u00edo en lugar de ayudar en la pocilga limpiando y dando de comer a los cerdos.\n\nEl doctor Martin Susman continu\u00f3 sentado, un plato con mostaza ante \u00e9l, se sirvi\u00f3 una copa de vino, luego otra; la ventana estaba abierta, de vez en cuando se levantaba, iba hacia la ventana, miraba un rato fuera, luego se sentaba de nuevo a la mesa. A la tercera copa, entr\u00f3 luz azul por la ventana y pas\u00f3 por las paredes de su habitaci\u00f3n. Los tulipanes emit\u00edan r\u00edtmicos reflejos azulados en el florero de la chimenea. Son\u00f3 el tel\u00e9fono. No lo cogi\u00f3. Que siguiera sonando. Martin Susman mir\u00f3 en la pantalla qui\u00e9n llamaba. No lo cogi\u00f3.\n\nPrehistoria. Es tan significativa y al mismo tiempo tan exigua y temblorosa como la l\u00e1mpara del sagrario de la iglesia de Sainte-Catherine, en la otra punta de la place du Vieux March\u00e9 aux Grains, donde viv\u00eda Martin Susman.\n\nAlgunos transe\u00fantes se hab\u00edan refugiado en la iglesia huyendo de la lluvia, estaban reunidos en grupos sin saber qu\u00e9 hacer o paseaban por la nave, los turistas hojeaban sus gu\u00edas y segu\u00edan la ruta de los puntos de inter\u00e9s art\u00edstico: \u00abVirgen negra, siglo XIV\u00bb, \u00abRetrato de santa Catalina\u00bb, \u00abP\u00falpito t\u00edpicamente flamenco, probablemente de Mechelen\u00bb, \u00abTumba de Gilles-Lambert Godecharle\u00bb...\n\nDe vez en cuando un rel\u00e1mpago.\n\nEl hombre sentado en un banco de la iglesia parec\u00eda rezar. Apoyado sobre los codos, la barbilla descansando sobre las manos cruzadas, la espalda encorvada. Llevaba una chaqueta negra con capucha, que se hab\u00eda echado sobre la cabeza, y si en el dorso de la chaqueta no pusiera GUINNESS, habr\u00eda podido pasar, a primera vista, por un monje en su h\u00e1bito.\n\nLa chaqueta de la capucha se deb\u00eda seguramente a la lluvia de Bruselas, pero la impresi\u00f3n que produc\u00eda con eso delataba, sin embargo, algo fundamental sobre aquel hombre. En efecto, a su modo era un monje: consideraba lo monacal, o la idea que \u00e9l ten\u00eda de ello \u2014\u200cascesis, meditaci\u00f3n y ejercicios\u2014, su salvaci\u00f3n en una vida incesantemente amenazada por el caos y la intemperancia. Para \u00e9l eso no estaba vinculado a una orden religiosa o a un monasterio, ni tampoco a una huida del mundo: independientemente de cu\u00e1l fuera su oficio o su funci\u00f3n, cada hombre pod\u00eda, deb\u00eda incluso, ser un monje en su \u00e1mbito, el siervo, concentrado en su misi\u00f3n, de una voluntad superior.\n\nAmaba contemplar al hombre torturado en la cruz y pensar en la muerte. Eso era para \u00e9l cada vez purificaci\u00f3n de los sentimientos, sujeci\u00f3n del pensamiento y refuerzo de su energ\u00eda.\n\nAs\u00ed era Mateusz Oswiecki. Su nombre de pila, que tambi\u00e9n figuraba en su pasaporte, era no obstante Ryszard. Oswiecki no se convirti\u00f3 en Mateusz hasta que empez\u00f3 a estudiar en el seminario de la academia de Lubranski, en Pozna\u0144, donde cada \u00abdisc\u00edpulo iluminado\u00bb recib\u00eda como sobrenombre el nombre de uno de once ap\u00f3stoles. Lo hab\u00edan bautizado de nuevo y ungido como \u00abMateo, el publicano\u00bb. Aunque dej\u00f3 el seminario, mantuvo el nombre como su nom de guerre. Las fronteras en las que ten\u00eda que presentar su pasaporte las pasaba como Ryszard. En el servicio secreto, seg\u00fan hab\u00edan declarado algunos antiguos contactos, se le conoc\u00eda como Matek, el diminutivo de Mateusz. As\u00ed se hac\u00eda llamar por sus compa\u00f1eros. Como Mateusz cumpl\u00eda su misi\u00f3n, como Matek se le buscaba, como Ryszard se les escurr\u00eda de las manos.\n\nOswiecki no estaba rezando. No formulaba en su interior frases que comenzaban con \u00abSe\u00f1or\u00bb y que siempre eran s\u00f3lo deseos, \u00abdame fuerzas...\u00bb para hacer esto o lo otro, \u00abbendice...\u00bb esto o lo otro... No hab\u00eda que pedir nada a un esp\u00edritu absoluto que guardaba silencio. \u00c9l contemplaba al hombre clavado en la cruz. La experiencia de ese hombre, de car\u00e1cter ejemplar para la humanidad y expresada al final tambi\u00e9n con palabras, era la de estar completamente abandonado en el momento de la confrontaci\u00f3n con lo absoluto: cuando la envoltura recibe un rasgu\u00f1o, cuando se rompe, se raja, se perfora y se rasga, cuando los gritos de dolor de la vida pasan a ser un gemido y, finalmente, silencio. S\u00f3lo en el silencio est\u00e1 la vida cerca del esp\u00edritu omnipotente, que, en inconcebible capricho, hizo salir de \u00e9l lo contrario de su ser: el tiempo. Desde el instante de su nacimiento el hombre puede recordar y recordar, llegar cada vez m\u00e1s atr\u00e1s, m\u00e1s y m\u00e1s atr\u00e1s, perpetuamente m\u00e1s atr\u00e1s; pero no llegar\u00e1 a ning\u00fan comienzo y con su rid\u00edculo concepto de tiempo s\u00f3lo comprender\u00e1 una cosa: que antes de ser, no fue eterno. Y ya puede pensar con anticipaci\u00f3n, desde el momento de su muerte hasta todo el porvenir; no llegar\u00e1 a un final, s\u00f3lo a esta evidencia: ya no ser\u00e1 eterno. Y el entreacto entre eternidad y eternidad es el tiempo: el ruido, el vocer\u00edo, el golpeteo de las m\u00e1quinas, el zumbido de los motores, el estallido y estampido de las armas, los gritos de dolor y el desesperado alarido de placer, las corales de las masas furiosas y de las alegremente enga\u00f1adas, el retumbar del trueno y el jadear de angustia en el terrario microsc\u00f3pico de la Tierra.\n\nMateusz Oswiecki contemplaba al hombre torturado.\n\nNo hab\u00eda plegado las manos. Con las manos entrelazadas se clavaba las u\u00f1as en los dorsos de las manos hasta que cruj\u00edan los nudillos y le escoc\u00eda la piel. Ese dolor era anterior a \u00e9l mismo. Pod\u00eda volver a sentirlo con desesperaci\u00f3n en cualquier momento. A principios de 1940, su abuelo Ryszard hab\u00eda pasado a la clandestinidad para luchar contra los alemanes en la resistencia polaca, a las \u00f3rdenes del general Stefan Rowecki. Ya en abril del mismo a\u00f1o fue traicionado, detenido, torturado y finalmente fusilado p\u00fablicamente, como partisano, en Lublin. En aquella fecha la abuela estaba embarazada de ocho meses, el ni\u00f1o naci\u00f3 en Kielce en mayo de 1940 y recibi\u00f3 el nombre de su padre. Para escapar a una eventual corresponsabilidad familiar lo llevaron a Pozna\u0144, a la familia de un t\u00edo abuelo, donde se crio y donde a los diecis\u00e9is a\u00f1os vivi\u00f3 la sublevaci\u00f3n. El joven bachiller se uni\u00f3 al grupo del comandante Franczak para luchar en la resistencia anticomunista. Oper\u00f3 en acciones de sabotaje, despu\u00e9s en secuestros de confidentes de la polic\u00eda estatal... y en 1964 un compa\u00f1ero le traicion\u00f3 por 6.000 zlotys. Fue detenido en un piso franco y torturado hasta la muerte en un s\u00f3tano de la SB, el Servicio de Seguridad. En aquel tiempo, su novia, Marija, ya estaba embarazada, el ni\u00f1o naci\u00f3 en febrero de 1965 en la aldea de Kocize Gorne y lo bautizaron con el nombre de su abuelo y de su padre. Otro hijo que no conocer\u00eda a su progenitor. La madre no le contaba mucho. En una ocasi\u00f3n: \u00abNos ve\u00edamos en los campos de labranza o en el bosque. Ven\u00eda a nuestras citas con una pistola y con granadas\u00bb.\n\nUn abuelo eternamente silencioso. Un padre eternamente silencioso. Los polacos, \u00e9sa era la doctrina de Matek, siempre hab\u00edan luchado por la libertad de Europa, quien entraba en la lucha se hab\u00eda criado en el silencio y luchaba hasta que entraba en el silencio.\n\nSu madre iba con \u00e9l a ver a los curas, buscaba intercesores, compraba cartas de recomendaci\u00f3n, confiaba en la protecci\u00f3n que pod\u00eda brindar la Iglesia. Finalmente lo dej\u00f3 con los hermanos maristas de Pozna\u0144. All\u00ed comprob\u00f3 \u00e9l mismo la vulnerabilidad del cuerpo humano: la sangre es un lubricante y lubrificante cuando penetra en la envoltura, la piel no es sino pergamino h\u00famedo sobre el que un cuchillo dibuja mapas, la boca y la garganta que grita, un agujero negro que se tapona hasta que se extingue el \u00faltimo grito y se limita a absorber lo que deber\u00eda dar vida. Y all\u00ed aprendi\u00f3 un concepto completamente nuevo de \u00absubterr\u00e1neo\u00bb. Cuando los alumnos recibieron sus nombres apost\u00f3licos, los llevaron a las catacumbas de la grandiosa catedral de Pozna\u0144, a las b\u00f3vedas y c\u00e1maras funerarias subterr\u00e1neas, bajando por escaleras que brillaban y fulguraban a la luz de las antorchas, hasta el s\u00f3tano m\u00e1s profundo y, a trav\u00e9s de una \u00faltima y tosca galer\u00eda, a una c\u00e1mara que result\u00f3 ser una capilla, hundida en la tierra, de la muerte y de la vida eterna: una b\u00f3veda de ca\u00f1\u00f3n empotrada en la piedra en el siglo X de la era cristiana, a cien pies bajo la tierra empapada de sangre de Polonia. En la cara frontal de aquel recinto se alzaba una cruz monumental con una figura de Cristo atrozmente realista, detr\u00e1s de ella relieves de \u00e1ngeles que sal\u00edan de la piedra o que parec\u00edan entrar en ella y atravesarla, horriblemente llenos de vida a la luz oscilante de las llamas. Delante una Virgen, como nunca la viera antes el joven Ryszard, en ninguna iglesia, en ninguna ilustraci\u00f3n de sus libros: llevaba un embozo que la enmascaraba por completo. La Virgen ten\u00eda un manto echado sobre frente, nariz y boca de forma que, a trav\u00e9s de una estrecha abertura en el manto, s\u00f3lo se ve\u00edan sus ojos, las cuencas de los ojos tan hondas y tan muertas como no pod\u00eda ser de otro modo al cabo de mil a\u00f1os de l\u00e1grimas. Todo ello, al igual que el altar, esculpido y configurado en la piedra y la arcilla fangosa del estrato geol\u00f3gico all\u00ed perforado. Bancos de roca fr\u00eda sobre los que estaban sentados, dando la espalda a Ryszard y a los otros alumnos que entraban, once monjes con h\u00e1bitos negros, las cabezas inclinadas cubiertas por sus capuchas.\n\nCondujeron a los alumnos por el pasillo central entre los monjes en oraci\u00f3n, hacia delante, hacia Cristo, donde se santiguaron y luego se les indic\u00f3 que se dieran la vuelta. Ryszard volvi\u00f3 la vista atr\u00e1s y entonces lo vio: bajo las capuchas asomaban calaveras, los rosarios que hab\u00eda en las manos de los monjes colgaban de dedos que eran huesos. Los monjes eran esqueletos.\n\nBajo la tierra se est\u00e1 m\u00e1s cerca de Dios que en las cimas de los montes.\n\nMateusz Oswiecki se golpe\u00f3 varias veces la frente con las yemas de los dedos. Sent\u00eda la carne pesada y mohosa. Y en su cavidad abdominal, a la izquierda y un poco por debajo del ombligo, sent\u00eda un escozor. Lo sab\u00eda: ah\u00ed escuece la muerte. No le daba miedo. Le quitaba el miedo.\n\nEsos esqueletos con h\u00e1bitos eran los restos mortales del obispo de la misi\u00f3n, Jordanes, y de los miembros del consejo fundador del obispado de Pozna\u0144. Desde hac\u00eda casi mil a\u00f1os permanec\u00edan all\u00ed en oraci\u00f3n eternamente silenciosa. Delante de esos once esqueletos le fue dado a cada alumno un nombre elegido entre once nombres de ap\u00f3stoles. \u00bfOnce? \u00bfNo estaba Judas? S\u00ed. Pero darle a un alumno el nombre de Pedro, del primer vicario de Cristo en la tierra, habr\u00eda sido una usurpaci\u00f3n. El elegido tambi\u00e9n se convierte en Pedro llam\u00e1ndose Juan o Pablo.\n\nMateusz Oswiecki apret\u00f3 las palmas de las manos contra sus o\u00eddos. Ten\u00eda tantas voces en la cabeza. Cerr\u00f3 los ojos. Demasiadas im\u00e1genes. Eso no era recuerdo, no era prehistoria. Eso estaba all\u00ed ahora, ahora, como \u00e9l estaba sentado delante del Crucificado. Y como el escozor en el vientre. No ten\u00eda miedo, s\u00f3lo la angustiosa sensaci\u00f3n que se tiene antes de un examen importante, de una tarea dif\u00edcil. Un examen al que uno s\u00f3lo puede presentarse una vez es el m\u00e1s dif\u00edcil. Volvi\u00f3 a abrir los ojos, levant\u00f3 la vista y mir\u00f3 la llaga del costado del ya liberado.\n\nEn el fondo, Mateusz Oswiecki envidiaba a sus v\u00edctimas. Ellos ya lo hab\u00edan dejado todo atr\u00e1s.\n\nSe levant\u00f3, sali\u00f3 de la iglesia, ech\u00f3 una breve ojeada a la luz azul que danzaba delante del hotel Atlas, y despacio, con la cabeza inclinada, la capucha hundida hasta los ojos, avanz\u00f3 a trav\u00e9s de la lluvia en direcci\u00f3n a la estaci\u00f3n de metro de Sainte-Catherine.\n\nCuando Alois Erhart regres\u00f3 al hotel Atlas, al principio le prohibieron el paso. Al menos interpret\u00f3 la mano extendida de un polic\u00eda ante la entrada del hotel como una intimaci\u00f3n a que se detuviera. Lo que dijo el polic\u00eda no lo entendi\u00f3. \u00c9l no sab\u00eda mucho franc\u00e9s.\n\nYa hab\u00eda visto de lejos la luz giratoria azul de la polic\u00eda y de la ambulancia y hab\u00eda pensado en un suicidio. Se hab\u00eda acercado despacio al hotel y enseguida hab\u00eda tenido la misma sensaci\u00f3n que lo hab\u00eda asaltado al mediod\u00eda: como si la nada en la que se precipita m\u00e1s pronto o m\u00e1s tarde cada ser humano se extendiera de s\u00fabito, a manera de anuncio o hasta de exigencia, por la caja tor\u00e1cica y la cavidad abdominal. R\u00edgido y sin aliento hab\u00eda sentido el milagro: que en la envoltura limitada del cuerpo pueda extenderse infinitamente un vac\u00edo creciente. El alma como agujero negro que absorb\u00eda y hac\u00eda desaparecer todas las experiencias de una vida entera, hasta que ya s\u00f3lo quedaba la nada, el vac\u00edo absoluto, completamente negro pero sin la suavidad de una noche sin estrellas.\n\nAhora estaba all\u00ed, ante las escaleras de entrada al hotel, con los huesos doloridos y los m\u00fasculos que ard\u00edan de cansancio, detr\u00e1s de \u00e9l unos cuantos mirones, y dec\u00eda en ingl\u00e9s que \u00e9l se alojaba en ese hotel, que ten\u00eda all\u00ed una habitaci\u00f3n; lo cual no produc\u00eda ning\u00fan cambio en aquel brazo extendido. La situaci\u00f3n le parec\u00eda tan surrealista que no se habr\u00eda extra\u00f1ado si le hubieran detenido. Pero \u00e9l no era s\u00f3lo el hombre de edad avanzada cuyo cuerpo empezaba a rebelarse definitivamente, era tambi\u00e9n el doctor Erhart, catedr\u00e1tico em\u00e9rito que durante media vida hab\u00eda significado autoridad. Turista, dijo con voz firme, \u00e9l era turista. \u00a1All\u00ed! En aquel hotel. Y deseaba ir a su habitaci\u00f3n. Tras lo cual el funcionario lo acompa\u00f1\u00f3 al hall y lo condujo ante un hombre de casi dos metros de altura, en la mitad de la cincuentena y con un traje gris muy estrecho, que le pidi\u00f3 la documentaci\u00f3n.\n\n\u00bfPor qu\u00e9 agachaba la cabeza el profesor? Ve\u00eda el abultado vientre lleno de gas de aquel hombre gigantesco y de pronto sinti\u00f3 compasi\u00f3n. Hay personas que en su masiva presencia f\u00edsica parecen eternamente fuertes, siempre en forma, nunca achacosos, hasta que de pronto, como fulminados por un rayo, yacen en el suelo, muertos a una edad de la que se dice: no es edad para morirse. Siempre orgullosos de su constituci\u00f3n, se tienen por inmortales mientras pueden alzar su cuerpo delante de otros, lucirlo delante de otros. Esas personas no se ve\u00edan nunca confrontadas con la cuesti\u00f3n de qu\u00e9 decisi\u00f3n tomar\u00edan cuando fueran viejos y enfermos cr\u00f3nicos, cuando en tiempo no lejano fueran incapaces de valerse por s\u00ed solos. Ese hombre estaba caduco y podrido en su interior, muy pronto se derrumbar\u00eda, s\u00f3lo que lo ignoraba.\n\nEl profesor Erhart le tendi\u00f3 el pasaporte.\n\n\u00bfCu\u00e1ndo hab\u00eda llegado? Parlez-vous fran\u00e7ais? \u00bfNo? English? Cu\u00e1ndo hab\u00eda salido del hotel. Si hab\u00eda estado en el hotel entre las diecinueve y las veinte horas.\n\n\u00bfPor qu\u00e9 esas preguntas?\n\nBrigada de homicidios. Hab\u00edan matado a tiros a un hombre en una habitaci\u00f3n del hotel.\n\nLe dol\u00eda el brazo derecho. El profesor Erhart pens\u00f3 que quiz\u00e1 estaba llamando la atenci\u00f3n por c\u00f3mo se pasaba una y otra vez la mano por el brazo, c\u00f3mo lo apretaba, lo masajeaba.\n\nSac\u00f3 del bolsillo de su impermeable su c\u00e1mara digital, la encendi\u00f3. Pod\u00eda mostrar d\u00f3nde hab\u00eda estado: en cada foto pon\u00eda la hora en que la hab\u00eda hecho.\n\nEl hombre sonri\u00f3. Mir\u00f3 las fotos una por una. Por la tarde en el barrio de Europa, plaza Schuman. El edificio Berlaymont, el edificio Justus Lipsius. El letrero con el nombre de una calle, RUE JOSEPH II. \u00bfPor qu\u00e9 este letrero?\n\n\u00a1Soy austriaco!\n\nAh, vale.\n\nLa escultura El sue\u00f1o de Europa en la rue de la Loi. La figura de bronce de un ciego (o son\u00e1mbulo) que desde un pedestal da un paso al vac\u00edo. \u00a1Qu\u00e9 no fotografiar\u00e1n los turistas! Aqu\u00ed est\u00e1. Diecinueve quince: Grand Place. Varias fotos all\u00ed hasta las diecinueve veintiocho. Luego la \u00faltima foto: veinte horas y cuatro, Sainte-Catherine, la nave de la iglesia. El hombre sigui\u00f3 pulsando el cursor, y ya ven\u00eda otra vez la primera foto. Entonces lo puls\u00f3 en direcci\u00f3n contraria. El Cristo, el altar, delante, en un banco, un hombre en cuyo dorso pon\u00eda GUINNESS.\n\nSonri\u00f3 y le devolvi\u00f3 la m\u00e1quina.\n\nCuando Alois Erhart entr\u00f3 en su habitaci\u00f3n se acerc\u00f3 a la ventana, mir\u00f3 la lluvia a trav\u00e9s del cristal, se pas\u00f3 la mano por el pelo h\u00famedo, escuch\u00f3 en su interior. No oy\u00f3 nada. Cuando hab\u00eda llegado hacia mediod\u00eda, enseguida hab\u00eda abierto la ventana, asom\u00e1ndose mucho finalmente para tener una mejor perspectiva de la plaza: se hab\u00eda asomado demasiado, casi hab\u00eda perdido el equilibrio, los pies no tocaban el suelo, ya ve\u00eda acercarse el asfalto, fue algo muy r\u00e1pido, tom\u00f3 impulso hacia atr\u00e1s, cay\u00f3 al suelo delante de la ventana d\u00e1ndose un golpe en el brazo contra el radiador, qued\u00f3 sentado en el suelo en una postura rid\u00edcula... y con la sensaci\u00f3n de encontrarse en la ca\u00edda libre que hab\u00eda evitado en el \u00faltimo momento, una sensaci\u00f3n que tal vez se tenga en el segundo anterior a la muerte. Luego se hab\u00eda levantado, se hab\u00eda sentado en la cama, jadeante, y de pronto sinti\u00f3 aquella euforia: era libre. A\u00fan. Pod\u00eda decidir con plena independencia. Y tomar\u00eda la decisi\u00f3n. A\u00fan no. Pero a tiempo. Suicidio, un concepto est\u00fapido. Autodeterminaci\u00f3n, hombre libre. Sab\u00eda que deb\u00eda hacerlo y, de pronto, supo tambi\u00e9n que pod\u00eda hacerlo. La muerte. Ahora lo ten\u00eda claro, era tan banal y f\u00fatil e inevitable como el punto \u00abruegos y preguntas\u00bb al final del orden del d\u00eda. Era el momento en que ya no hab\u00eda nada. Deb\u00eda superar la muerte de un salto. Deb\u00eda saltar.\n\nNo quer\u00eda morir como su mujer. Tan desvalida al final, dependiendo de que \u00e9l le...\n\nCogi\u00f3 el mando a distancia, encendi\u00f3 el televisor. Se quit\u00f3 la camisa, vio que ten\u00eda un cardenal en el brazo derecho. Puls\u00f3 en el mando a distancia: \u00a1adelante! Se quit\u00f3 el pantal\u00f3n: \u00a1adelante! Los calcetines: \u00a1adelante! Los calzoncillos: \u00a1adelante! Aterriz\u00f3 entonces en el canal Arte. Iba a empezar un largometraje, un cl\u00e1sico: De aqu\u00ed a la eternidad. Hac\u00eda d\u00e9cadas que hab\u00eda visto aquella pel\u00edcula. Se tendi\u00f3 en la cama. Una voz dijo: \u00abLes presenta este film parship.de, la acreditada agencia de citas\u00bb.\n\nNo era casualidad que Fenia Xenopoulou, justo en el momento en que la ambulancia entraba en la plaza y se o\u00eda la sirena, pensara en salvarse. Desde hac\u00eda d\u00edas no pensaba en otra cosa, hab\u00eda pasado a ser casi una idea fija y por eso tambi\u00e9n lo pensaba ahora: \u00a1Salvaci\u00f3n! \u00a1Tiene que salvarme!\n\nEstaba cenando en el restaurante Menelas, situado enfrente del hotel Atlas, junto con Kai-Uwe Frigge, a quien, desde la breve aventura amorosa que hab\u00edan tenido dos a\u00f1os atr\u00e1s, llamaba en privado Fridsch, sin explicar si alteraba su nombre convirti\u00e9ndolo en \u00abFritz\u00bb, porque era alem\u00e1n, o si alud\u00eda a \u00abFridge\u00bb, el frigor\u00edfico, porque \u00e9l, con su estilo objetivo y correcto, daba impresi\u00f3n de frialdad. Frigge, un hombre larguirucho y \u00e1gil en la cuarentena, natural de Hamburgo, desde hac\u00eda diez a\u00f1os en Bruselas, en las guerras internas, en las intrigas y trueques que preceden por naturaleza a la constituci\u00f3n de un nuevo gabinete de la Comisi\u00f3n Europea hab\u00eda tenido suerte (o m\u00e1s bien no se hab\u00eda fiado de su propia suerte) y hab\u00eda dado un salto espectacular en su carrera: ahora era jefe de gabinete en la Direcci\u00f3n General de Comercio y, con ello, el influyente jefe de gabinete de uno de los comisarios m\u00e1s poderosos de la Uni\u00f3n.\n\nSi en aquella ciudad llena de restaurantes de primer orden se citaron en uno griego, que adem\u00e1s result\u00f3 ser m\u00e1s bien mediocre, no fue por deseo de Fenia Xenopoulou; ella no ten\u00eda nostalgia ni a\u00f1oranza del sabor y los aromas de la cocina de su tierra. Lo hab\u00eda propuesto Kai-Uwe Frigge: quer\u00eda dar una muestra de solidaridad a su compa\u00f1era griega, ahora que, despu\u00e9s de la casi bancarrota estatal de Grecia y del cuarto paquete salvador de la UE, pecaminosamente caro, el prestigio de \u00ablos griegos\u00bb entre los compa\u00f1eros y en la opini\u00f3n p\u00fablica estaba totalmente por los suelos. \u00c9l se sent\u00eda seguro de contar con un punto a su favor cuando propuso por e-mail como lugar de encuentro \u00ab\u00bfMenelas? En el Vieux March\u00e9 aux Grains, Sainte-Catherine, griego y muy bueno al parecer\u00bb, y ella le hab\u00eda respondido \u00abOkay\u00bb. Eso a ella le daba igual. Llevaba ya mucho tiempo viviendo y trabajando en Bruselas para ocuparse de patriotismos. Lo que quer\u00eda era: salvaci\u00f3n. La suya propia.\n\nLlamar paraguas de salvamento al fondo que deb\u00eda impedir la bancarrota de Grecia era ya de una involuntaria comicidad, dijo Frigge. Bueno, s\u00ed, en esta casa nuestra lo de las met\u00e1foras es cuesti\u00f3n de suerte.\n\nA Fenia no le hac\u00eda ninguna gracia, no entend\u00eda lo que \u00e9l quer\u00eda decir, pero solt\u00f3 una risa radiante. Sinti\u00f3 que se pon\u00eda una m\u00e1scara, y no estaba segura de si se notaba la afectaci\u00f3n o si segu\u00eda funcionando lo que antes nunca le fallaba: que la magistral combinaci\u00f3n de m\u00fasculos faciales, sincronizaci\u00f3n, dientes blanqu\u00edsimos y una c\u00e1lida mirada diera una imagen de irresistible naturalidad. Tambi\u00e9n para lo artificial hay que tener un talento natural, pero Fenia, debido a su tropez\u00f3n en la carrera \u2014\u200c\u00a1a su edad!, \u00a1iba a cumplir cuarenta a\u00f1os!\u2014, estaba tan angustiada que ya no ten\u00eda claro si contaba con su talento natural, a saber: agradar premeditadamente. Dudar de s\u00ed misma, as\u00ed lo sent\u00eda ella, cubr\u00eda de una especie de sarna su figura.\n\nKai-Uwe s\u00f3lo hab\u00eda pedido una ensalada campera, el primer impulso de Fenia fue decir: Yo tambi\u00e9n. Pero luego se oy\u00f3 pedir yiuvetsi. Estaba tibio y nadaba en grasa. \u00bfPor qu\u00e9 no se controlaba m\u00e1s? Empezaba a perder la l\u00ednea. Deb\u00eda tener cuidado. El camarero llen\u00f3 otra vez la copa. Ella mir\u00f3 la copa de vino y pens\u00f3: ochenta calor\u00edas m\u00e1s. Bebi\u00f3 un poquito de agua, reuni\u00f3 todas sus fuerzas y mir\u00f3 a Kai-Uwe; apretando el vaso de agua con ambas manos contra su labio inferior trat\u00f3 de dirigirle una mirada c\u00f3mplice y al mismo tiempo seductora. En su interior maldec\u00eda. \u00bfQu\u00e9 le estaba pasando?\n\n\u00a1Paraguas de salvamento!, dijo Kai-Uwe. En alem\u00e1n se pueden formar esos neologismos y, con que aparezcan tres veces en el Frankfurter Allgemeine, cualquier persona culta los considera completamente normales. Y luego ya no hay quien los elimine. La jefa lo dec\u00eda delante de todas las c\u00e1maras. Los traductores lo pasaban fatal. El ingl\u00e9s y el franc\u00e9s conocen el salvavidas y el paraguas. Pero, por favor, nos preguntaban, \u00bfqu\u00e9 es un Rettungsschirm, un paraguas de salvamento? Los franceses lo tradujeron de entrada por \u00abparachute\u00bb. Entonces lleg\u00f3 la protesta del palacio del El\u00edseo: un paraca\u00eddas no impide la ca\u00edda, s\u00f3lo la ralentiza, eso era una se\u00f1al equivocada; que los alemanes tuvieran a bien...\n\nCuando \u00e9l com\u00eda una aceituna, poniendo el hueso en el plato, a Fenia le parec\u00eda que s\u00f3lo inger\u00eda el sabor de la aceituna y que devolv\u00eda las calor\u00edas a la cocina.\n\nEntonces empez\u00f3 el ulular de la sirena, luego la luz azul, azul, azul, azul...\n\n\u00bfFridsch?\n\n\u00bfS\u00ed?\n\nTienes que... ya iba a pronunciarlo: salvarme. Pero no pod\u00eda decir eso. Se corrigi\u00f3: ayudarme. No, ten\u00eda que dar la impresi\u00f3n de persona competente, no desvalida.\n\n\u00bfS\u00ed? Por la ventana del restaurante mir\u00f3 hacia el hotel Atlas. Vio que sacaban de la ambulancia una camilla, que unos hombres entraban con ella en el hotel. Por muy cerca del hotel que estuviera el Menelas, la distancia era todav\u00eda muy grande como para pensar en la muerte. Para \u00e9l era s\u00f3lo una coreograf\u00eda, las personas se mov\u00edan con luz y sonido.\n\nTienes que... \u2014\u200cya lo hab\u00eda dicho, ahora quer\u00eda desdecirse, pero ya no era posible\u2014 comprender..., \u00a1pero ya lo haces! Lo s\u00e9, t\u00fa comprendes que yo...\n\n\u00bfS\u00ed? \u00c9l la mir\u00f3.\n\nLas sirenas de los coches de la polic\u00eda.\n\nFenia Xenopoulou hab\u00eda trabajado primero en la Direcci\u00f3n General de Competencia. El comisario, un espa\u00f1ol, no ten\u00eda ni idea de nada. Pero todo comisario es tan bueno como su equipo, y ella hab\u00eda llamado la atenci\u00f3n como miembro excelente de un gabinete que funcionaba perfectamente. Se divorci\u00f3. No ten\u00eda tiempo ni ganas de que cada dos o, despu\u00e9s, cada tres o cuatro fines de semana, en su apartamento de Bruselas, o en Atenas cuando ella iba all\u00ed, hubiera un hombre que depart\u00eda sobre esta o aquella intimidad de la sociedad ateniense mientras fumaba puros como la caricatura de un nuevo rico. \u00a1Se hab\u00eda casado con un abogado estrella y ech\u00f3 de su piso a un abogadillo de provincias! Luego ascendi\u00f3 un pelda\u00f1o y entr\u00f3 en el gabinete del comisario de Comercio. All\u00ed se hacen m\u00e9ritos si se derriban restricciones comerciales. Para ella no hab\u00eda ya vida privada, no hab\u00eda trabas, s\u00f3lo exist\u00eda el libre comercio mundial. Cre\u00eda de verdad que la carrera que ve\u00eda ante ella ser\u00eda la recompensa por haber contribuido a mejorar el mundo. Fair trade era para ella una tautolog\u00eda. Pues trade era la condici\u00f3n previa del fairness global. El comisario, un holand\u00e9s, ten\u00eda escr\u00fapulos. Era tan incre\u00edblemente correcto. Fenia trabajaba duro para calcular cu\u00e1ntos florines costaban sus escr\u00fapulos. \u00a1Aquel hombre, en efecto, segu\u00eda operando en florines! \u00a1El laurel que recibi\u00f3 cuando ella lo convenci\u00f3 val\u00eda su peso en oro! Ahora ten\u00eda que llegar el siguiente salto. Despu\u00e9s de las elecciones europeas, Fenia esperaba seguir ascendiendo cuando se constituyera de nuevo la Comisi\u00f3n. Y en efecto: la ascendieron. Le asignaron un departamento. \u00bfD\u00f3nde estaba el problema? Ella tom\u00f3 aquel ascenso por un descenso, por un baj\u00f3n en su carrera, por una relegaci\u00f3n: ten\u00eda a su cargo la Direcci\u00f3n C (Comunicaci\u00f3n) en la Direcci\u00f3n General de Educaci\u00f3n y Cultura.\n\n\u00a1Cultura!\n\nElla hab\u00eda estudiado Econom\u00eda, London School of Economics, Postgraduate en la Stanford University, hab\u00eda entrado por concurso, y ahora estaba metida en la Cultura: \u00a1eso ten\u00eda incluso menos sentido que jugar al Monopoly! La Cultura era una secci\u00f3n insignificante, sin presupuesto, sin peso en la Comisi\u00f3n, sin influencia ni poder. Los compa\u00f1eros dec\u00edan de la Cultura que era una secci\u00f3n-pantalla. \u00a1Si al menos lo fuera! Las pantallas son importantes para el d\u00eda a d\u00eda. Pero la Cultura no era ni siquiera falacia porque no hab\u00eda nadie pendiente de lo que hac\u00eda la Cultura y a quien se pudiera enga\u00f1ar. Cuando el comisario de Comercio o de Energ\u00eda, y hasta cuando la comisaria de Pesca ten\u00eda que ir al ba\u00f1o durante una sesi\u00f3n de la Comisi\u00f3n, se dejaba de discutir y se esperaba hasta que \u00e9l o ella volviera. Pero cuando la comisaria de Cultura ten\u00eda que salir, se continuaba discutiendo con toda indiferencia; m\u00e1s a\u00fan, ni siquiera se advert\u00eda si estaba sentada a la mesa de negociaci\u00f3n o en el retrete.\n\nFenia Xenopoulou hab\u00eda tomado un ascensor que iba hacia arriba pero que despu\u00e9s, inesperadamente, se hab\u00eda quedado detenido entre dos pisos.\n\n\u00a1Tengo que salir!, dijo. Cuando regres\u00f3 del ba\u00f1o vio que \u00e9l hablaba por tel\u00e9fono. No hab\u00eda esperado.\n\nPor la gran ventana, Fridsch y Fenia miraban al hotel, silenciosos como un viejo matrimonio que se alegra de que pase algo sobre lo que poder decir unas frases.\n\n\u00bfQu\u00e9 ocurre ah\u00ed?\n\nNi idea. Quiz\u00e1 alguien haya tenido un infarto en el hotel, dijo Fridsch.\n\n\u00a1Pero por un ataque al coraz\u00f3n no llega al momento la polic\u00eda!\n\nCierto, dijo \u00e9l. Y tras una breve pausa casi habr\u00eda dicho: hablando del coraz\u00f3n, \u00bfc\u00f3mo va tu vida amorosa? Pero reprimi\u00f3 la pregunta.\n\nA ti te preocupa algo, dijo \u00e9l.\n\n\u00a1S\u00ed!\n\nPuedes cont\u00e1rmelo.\n\nEscuch\u00f3 asintiendo una y otra vez, de cuando en cuando dec\u00eda muy despacio okay para mostrarle que la segu\u00eda y al final dijo: \u00bfQu\u00e9 puedo hacer por ti?\n\nTienes que reclamarme. \u00bfPuedes..., s\u00ed, eso, reclamarme? Quiero volver a Comercio. \u00bfO puedes hablar con Queneau? T\u00fa te entiendes bien con \u00e9l. Y \u00e9l te hace mucho caso. A lo mejor puede hacer algo. Tengo que salir de la Cultura. \u00a1All\u00ed me ahogo!\n\nS\u00ed, dijo \u00e9l. De pronto tuvo miedo. Eso es tal vez mucho prometer. Sinti\u00f3 una ansiedad que no pod\u00eda explicarse. \u00c9l nunca reflexionaba sobre su vida. Hab\u00eda reflexionado sobre su vida antes, en alg\u00fan momento, mucho antes, cuando a\u00fan no ten\u00eda experiencia en la vida. Hab\u00edan sido fantas\u00edas, sue\u00f1os, hab\u00eda confundido los sue\u00f1os con la reflexi\u00f3n. No se pod\u00eda decir que hubiera ido detr\u00e1s de sus sue\u00f1os. Hab\u00eda ido, igual que cuando se va a un and\u00e9n determinado, all\u00ed donde empieza el viaje a un destino determinado. Desde entonces estaba sobre las v\u00edas. En su fuero interno sab\u00eda que a menudo hab\u00eda sido pura suerte que no descarrilara. Pero mientras se estaba sobre las v\u00edas no hab\u00eda nada sobre lo que reflexionar. La vida. Funciona o no funciona. Si funciona se sustituye \u00abla vida\u00bb por \u00abse\u00bb. Se funciona. \u00c9l no pensaba todo eso. Simplemente, lo ve\u00eda con claridad. Confund\u00eda esa claridad con un suelo seguro sobre el que caminaba sin tener que reflexionar a cada paso. Pero ahora hab\u00eda una ligera oscilaci\u00f3n sobre ese suelo. \u00bfPor qu\u00e9? No se lo preguntaba. S\u00f3lo sent\u00eda esa ligera ansiedad. \u00a1He de ir un momento al ba\u00f1o!\n\nSe lav\u00f3 las manos, se contempl\u00f3 en el espejo. No se ve\u00eda como un extra\u00f1o. No ser un extra\u00f1o tampoco equivale, por otra parte, a ser un \u00edntimo. Cogi\u00f3 de su monedero una pastilla de Viagra. Siempre llevaba una. La mastic\u00f3, tom\u00f3 un trago de agua, luego se lav\u00f3 otra vez las manos.\n\nSab\u00eda que Fenia deb\u00eda salir al d\u00eda siguiente por la ma\u00f1ana muy temprano, igual que \u00e9l. O sea, que ten\u00edan que irse pronto a la cama. Ten\u00edan que funcionar.\n\nTomaron un taxi a Ixelles, al apartamento de \u00e9l. \u00c9l fingi\u00f3 deseo, ella fingi\u00f3 un orgasmo. Hab\u00eda buena qu\u00edmica. A trav\u00e9s de la ventana parpadeaba la luz azul del anuncio luminoso del nightclub Le cerf bleu, en la acera de enfrente. Kai-Uwe Frigge se levant\u00f3 y corri\u00f3 la cortina.\n\n\u00bfHab\u00eda un hombre en la ventana? El vengador negro. El espectro. El hombre en la sombra. Parec\u00eda una figura de c\u00f3mic pintada en la pared de la casa abandonada: todas las ventanas de esa casa, casi justo enfrente del hotel Atlas, en la esquina de la rue de la Brai, estaban oscuras, el escaparate del local comercial cegado con planchas de madera, sobre los tablones aleteaban jirones de carteles medio arrancados. Al lado, en la pared de la casa, grafitis, palabras escritas con espray, ilegibles; \u00bfornamentos, escritura secreta, s\u00edmbolos? Delante de la casa, una valla de obras, sobre ella un letrero de la empresa de derribos De Meuter. El comisario Brunfaut sab\u00eda, evidentemente, que esa figura negra enmarcada por una ventana de la primera planta de la casa muerta no era un grafiti. Pero daba esa impresi\u00f3n. Por doquier en aquella ciudad, las paredes de las casas y los muros cortafuegos estaban pintados hasta el remate de los tejados con im\u00e1genes de c\u00f3mics, con copias y variaciones de los dibujos de Herg\u00e9 o Morris, con los animales de Bonom o con obras de los j\u00f3venes que se ten\u00edan por los sucesores de esos artistas. Si Bruselas era un libro abierto, entonces era un volumen de c\u00f3mics.\n\nEl comisario Brunfaut hab\u00eda salido del hotel Atlas para dar a los compa\u00f1eros del veh\u00edculo de servicio la orden de recorrer las casas vecinas y preguntar si a la hora en cuesti\u00f3n no habr\u00eda mirado quiz\u00e1 alguien por la ventana y visto algo.\n\n\u00a1Empieza bien el a\u00f1o, comisario!\n\nCada d\u00eda empieza bien, dijo Brunfaut. La lluvia hab\u00eda amainado, el comisario, con las piernas abiertas plantadas en el suelo, se subi\u00f3 la cinturilla del pantal\u00f3n y, mientras hablaba con los hombres, pase\u00f3 la mirada por las fachadas de las casas de enfrente. Y entonces la vio: la silueta enmarcada por la ventana.\n\nHab\u00eda, en efecto, un hombre en la ventana. De una casa en derribo. El comisario levant\u00f3 la vista, lo mir\u00f3 fijamente. El hombre no se movi\u00f3. \u00bfEra un hombre de verdad? \u00bfO un mu\u00f1eco? \u00bfPor qu\u00e9 iba a haber detr\u00e1s de esa ventana un mu\u00f1eco? \u00bfO era una sombra cuyo contorno le enga\u00f1aba? \u00bfO un grafiti? El comisario sonri\u00f3. No externamente, claro. Por dentro. No, all\u00ed hab\u00eda un hombre. \u00bfMiraba hacia abajo? \u00bfVe\u00eda que el comisario levantaba la vista hacia \u00e9l? \u00bfQu\u00e9 hab\u00eda visto?\n\n\u00a1Venga!, dijo el comisario Brunfaut. \u00a1A trabajar! \u00a1T\u00fa te encargas de esa casa, t\u00fa de aquella otra! Y t\u00fa...\n\n\u00bfTambi\u00e9n ese inmueble en ruinas? \u00a1Si est\u00e1 vac\u00edo!\n\n\u00a1S\u00ed, \u00e9se tambi\u00e9n! \u00a1Sube arriba y mira!\n\nPero la silueta hab\u00eda desaparecido.\n\nSe alej\u00f3 de la ventana. \u00bfD\u00f3nde ten\u00eda sus cigarrillos? Quiz\u00e1 en el abrigo. El abrigo estaba sobre la silla de la cocina, el \u00fanico mueble que quedaba en aquel piso. David de Vriend fue a la cocina, cogi\u00f3 el abrigo. \u00bfQu\u00e9 quer\u00eda? El abrigo. \u00bfPor qu\u00e9? Indeciso, segu\u00eda all\u00ed, mirando el abrigo. Era hora de marcharse. S\u00ed. All\u00ed ya no hab\u00eda nada. Que hacer. El piso estaba completamente vac\u00edo. Mir\u00f3 una mancha rectangular en la pared. All\u00ed hab\u00eda estado colgado un cuadro. Bosque junto a Boortmeerbeek, un paisaje id\u00edlico. Eso a\u00fan lo recordaba: c\u00f3mo lo hab\u00eda colgado en aquel sitio. Luego lo hab\u00eda tenido ante la vista a lo largo de toda una vida, hasta que ya no lo ve\u00eda. Y ahora: un hueco. S\u00f3lo se ve\u00eda que all\u00ed hab\u00eda habido algo que ya no estaba. Historia de una vida: un contorno vac\u00edo sobre un papel pintado pegado sobre otra historia anterior. Por debajo se ve\u00eda el contorno del armario que hab\u00eda estado all\u00ed. \u00bfQu\u00e9 hab\u00eda guardado dentro? Lo que se va juntando en una vida. \u00a1La suciedad de detr\u00e1s! Entonces sale a la vista. Polvo hecho grumos, estr\u00edas de suciedad grasienta, ennegrecida por el humo, cubierta de moho. Puedes limpiar durante toda tu vida, s\u00ed, limpia tu vida, pero al final, cuando se vac\u00eda la vivienda, lo que queda es suciedad. Detr\u00e1s de cada botella que limpias, detr\u00e1s de cada fachada que enluces. Eres joven y crees que todav\u00eda no hay nada corrompido, enmohecido y podrido, cuando de pronto quitan de en medio tu vida. \u00bfEres joven y crees que la vida te ha dado poco a\u00fan? Pero la suciedad de detr\u00e1s es siempre la suciedad de tu vida. La suciedad permanece porque t\u00fa eres suciedad y aterrizas en la suciedad. Pero si llegas a viejo: has tenido suerte. Sin embargo has estado enga\u00f1\u00e1ndote aunque hayas limpiado durante toda esa vida que te han regalado: al final lo vac\u00edan todo y \u00bfqu\u00e9 se ve? Suciedad. Est\u00e1 detr\u00e1s de todo, debajo de todo, es la base de todo lo que has limpiado. Una vida limpia. Es la que has tenido. Hasta que la suciedad aparece. All\u00ed estaba el fregadero. \u00c9l fregaba sin interrupci\u00f3n. Nunca tuvo lavavajillas. Fregaba cada plato, cada taza nada m\u00e1s utilizarlos. Cuando tomaba un caf\u00e9, solo, y desde luego estaba solo, casi siempre estuvo solo, se beb\u00eda el caf\u00e9 de pie, justo al lado del fregadero para poder fregar la taza al momento, tomar el \u00faltimo sorbo de caf\u00e9 y abrir el grifo, para \u00e9l eso siempre fue una unidad: enjuagar, secar hasta dejar la taza brillante y volver a colocarla en su sitio, para que todo est\u00e9 limpio, eso siempre fue importante para \u00e9l, una vida limpia, y luego: \u00bfqu\u00e9 se ve ahora all\u00ed donde estuvo el fregadero? Moho, verd\u00edn, viscosidad, porquer\u00eda. Hasta en la oscuridad o en la penumbra se ve\u00eda la ro\u00f1a. Ya no quedaba nada, todo se lo hab\u00edan llevado, pero eso segu\u00eda all\u00ed, eso se ve\u00eda: los manchones de suciedad detr\u00e1s de una vida de limpieza.\n\nVolvi\u00f3 a echar el abrigo sobre la silla. Quer\u00eda... \u00bfqu\u00e9? Mir\u00f3 alrededor. \u00bfPor qu\u00e9 no se iba? Deb\u00eda irse. Escapar. \u00c9se ya no era el piso en el que hab\u00eda vivido. Eran s\u00f3lo las habitaciones en las que hab\u00eda habido una vida anterior. Har\u00eda un \u00faltimo recorrido. \u00bfPara qu\u00e9? \u00bfMirar habitaciones vac\u00edas? Fue al dormitorio. Donde hab\u00eda estado la cama, el suelo de madera era m\u00e1s claro, el rect\u00e1ngulo que se marcaba all\u00ed en la penumbra parec\u00eda una gran escotilla. Rode\u00e1ndolo se dirigi\u00f3 a la ventana; \u00bfpor qu\u00e9 no pasaba por encima?, \u00bfpor qu\u00e9 daba un rodeo en esa habitaci\u00f3n vac\u00eda como si tuviera miedo de que ese rect\u00e1ngulo pudiera abrirse de verdad y devorarlo? \u00c9l no ten\u00eda miedo. All\u00ed hab\u00eda estado siempre la cama, y fue de la puerta a la ventana como hab\u00eda ido, rodeando la cama, toda su vida. Mir\u00f3 al exterior: casi al alcance de la mano, la escalera de incendios del edificio contiguo, un colegio. Una vez al a\u00f1o hab\u00eda un simulacro, sonaba una sirena y los estudiantes ensayaban c\u00f3mo se bajaba, deprisa y en orden, por la escalera de incendios. Cu\u00e1ntas veces hab\u00eda estado David de Vriend mirando por esa ventana. La huida. Un ensayo. Al alcance de la mano: qu\u00e9 f\u00e1cilmente se dice eso. La escalera estaba al alcance de la mano cuando se mud\u00f3 a esa casa. Fue tambi\u00e9n para \u00e9l en aquel entonces un argumento para quedarse con el piso. Est\u00e1 muy bien ubicado, dijo el vendedor, y De Vriend hab\u00eda contemplado por esa ventana la escalera de incendios y le hab\u00eda dado la raz\u00f3n: S\u00ed, la ubicaci\u00f3n es buena. Pens\u00f3 que, si fuera necesario, desde esa ventana estar\u00eda de un salto en esa escalera y podr\u00eda escapar mientras por delante seguir\u00edan golpeando en la puerta. De eso se crey\u00f3 capaz, sin duda, eso lo habr\u00eda conseguido. Pero hoy..., hoy era impensable. Ahora la escalera estaba fuera del alcance de la mano, inaccesible. Desde hac\u00eda medio siglo, los ni\u00f1os que all\u00ed ensayaban la huida ten\u00edan siempre la misma edad, siempre eran ni\u00f1os, s\u00f3lo \u00e9l hab\u00eda envejecido, demasiado al final, se hab\u00eda vuelto d\u00e9bil y achacoso y hab\u00eda perdido facultades. Mir\u00f3 por la ventana y vio... que no estaba a su alcance. Record\u00f3 que quer\u00eda fumar. Ten\u00eda que marcharse por fin, salir de all\u00ed; atraves\u00f3 el pasillo, no entr\u00f3 en la cocina donde estaba su abrigo con los cigarrillos, sino en el sal\u00f3n. Se detuvo indeciso, mir\u00f3 alrededor buscando. Una habitaci\u00f3n vac\u00eda. Quer\u00eda..., \u00bfqu\u00e9 quer\u00eda all\u00ed a\u00fan? Fue a la ventana, s\u00ed: tener otra vez esa vista, la vista de la plaza en la que hab\u00eda pasado toda una vida que hab\u00eda sido un regalo, y donde hab\u00eda tratado de encontrar su \u00ablugar en la vida\u00bb.\n\nMir\u00f3 abajo, a la luz azul. No pensaba en nada. Ten\u00eda fr\u00edo. Sab\u00eda por qu\u00e9. Ni siquiera pensaba que lo sab\u00eda y que eso no merec\u00eda ni un pensamiento m\u00e1s. Estaba dentro de \u00e9l, un saber antiguo. No ten\u00eda que formularse en la mente. Mir\u00f3 inm\u00f3vil los coches de la polic\u00eda de abajo, el coraz\u00f3n se le contrajo, se le dilat\u00f3 de nuevo, un encogerse de hombros del alma.\n\nCuando a\u00fan era profesor, siempre quiso acostumbrar a sus alumnos a prescindir en sus redacciones de eso: de las frases de blablabl\u00e1-coma-pens\u00f3.\n\nNo hab\u00eda manera de que lo dejaran. Los ni\u00f1os cre\u00edan de verdad que las personas, cuando est\u00e1n solas, tienen continuamente en la cabeza frases con \u00abpensaba \u00e9l\u00bb, \u00abpensaba ella\u00bb. La verdad es que, bajo aquel firmamento sin Dios, hab\u00eda un inconcebible silencio hasta dentro de la mente. Nuestra verborrea es s\u00f3lo el eco de ese silencio. Fr\u00edo, su coraz\u00f3n se contra\u00eda y se dilataba. Se contra\u00eda, se dilataba. Aspiraba el aire, lo expulsaba. \u00a1C\u00f3mo vibraba la luz azul!\n\nEntonces oy\u00f3 el timbre. Luego el golpear de un pu\u00f1o contra la puerta del piso. Fue a la cocina, se puso el abrigo. Fue al dormitorio. Alguien segu\u00eda aporreando la puerta. David de Vriend dio otra vez el peque\u00f1o rodeo cuando fue a la ventana. Mir\u00f3 fuera. No estaba al alcance de la mano. Se sent\u00f3 en el suelo, se encendi\u00f3 un pitillo. Ese golpear. Ese palpitar.\n\n# CAP\u00cdTULO SEGUNDO\n\nLAS IDEAS MOLESTAN, COSA QUE NO SUCEDER\u00cdA SIN ELLAS.\n\n#\n\nLa depresi\u00f3n es para quien puede permit\u00edrsela. Martin Susman pod\u00eda sobrevivir: trabajaba en el \u00abArca de No\u00e9\u00bb. Era funcionario de la Comisi\u00f3n Europea, Direcci\u00f3n General de Educaci\u00f3n y Cultura, adscrita a la Direcci\u00f3n C \u00abComunicaci\u00f3n\u00bb, y dirig\u00eda el departamento EAC-C-2 \u00abPrograma y disposiciones culturales\u00bb.\n\nEntre ellos, los compa\u00f1eros de trabajo llamaban a su departamento simplemente el \u00abArca de No\u00e9\u00bb o, m\u00e1s brevemente, \u00abel Arca\u00bb. \u00bfPor qu\u00e9? Un arca carece de destino. Se balancea con las corrientes, se columpia sobre las olas, afronta las borrascas y s\u00f3lo quiere una cosa: salvarse a s\u00ed misma y lo que lleva a bordo.\n\nMartin Susman no hab\u00eda tardado mucho en comprenderlo. Al principio estuvo feliz y orgulloso por haber conseguido tal empleo, sobre todo porque ning\u00fan partido ni autoridad administrativa alguna de Austria lo hab\u00eda enviado a Bruselas como END (Expert National D\u00e9tach\u00e9), sino que \u00e9l mismo hab\u00eda solicitado directamente el puesto en la Comisi\u00f3n y hab\u00eda ganado el concurso: \u00a1era, pues, realmente un funcionario europeo, y sin compromiso nacional! Y despu\u00e9s comprob\u00f3 que la Direcci\u00f3n General de Educaci\u00f3n y Cultura no ten\u00eda ning\u00fan prestigio dentro de la Comisi\u00f3n Europea y que nadie la tomaba en serio. En el aparato dec\u00edan simplemente \u00abla Cultura\u00bb cuando hablaban de esa direcci\u00f3n general, la \u00abEducaci\u00f3n\u00bb la suprim\u00edan, aunque en ese campo se hab\u00edan logrado notables \u00e9xitos, por ejemplo la elaboraci\u00f3n y realizaci\u00f3n del programa Erasmus. Y cuando dec\u00edan \u00abla Cultura\u00bb, era con un tonillo como si, por ejemplo, los brokers de Wall Street dijeran \u00abnumism\u00e1tica\u00bb, el hobby de un exc\u00e9ntrico pariente. Pero tambi\u00e9n en la opini\u00f3n p\u00fablica, en la medida en que \u00e9sta se interesaba por ello, la imagen de la \u00abCultura europea\u00bb era mala. Martin Susman llevaba poco tiempo en el cargo y a\u00fan le\u00eda los peri\u00f3dicos patrios \u2014\u200ct\u00edpico error de principiante\u2014, cuando en Austria estall\u00f3 la indignaci\u00f3n porque, tal como se le\u00eda en la prensa, hab\u00edan \u00abamenazado\u00bb a los austriacos con la Cultura: todo Estado miembro de la UE ten\u00eda derecho a un puesto de comisario, el gobierno nominaba a una persona, y el presidente de la Comisi\u00f3n le asignaba un departamento. Cuando despu\u00e9s de las elecciones europeas de entonces se cubrieron de nuevo los departamentos, se rumore\u00f3 que el comisario nominado por Austria recibir\u00eda la Cultura. El gobierno de coalici\u00f3n austriaco se enzarz\u00f3 en discusiones porque el partido del comisario designado barruntaba una intriga del compa\u00f1ero de coalici\u00f3n; protestaron, los peri\u00f3dicos austriacos caldearon el ambiente contando con la disposici\u00f3n a indignarse de sus lectores: \u00ab\u00a1Nos amenaza la Cultura!\u00bb. O bien: \u00ab\u00a1Austria ha de darse por satisfecha con la Cultura!\u00bb.\n\nEso, como reacci\u00f3n, resulta asombroso si se tiene en cuenta que ese pa\u00eds, bueno, quiz\u00e1 no se \u00abconceb\u00eda\u00bb pero s\u00ed se complac\u00eda en denominarse naci\u00f3n de cultura. Por otra parte, tal reacci\u00f3n correspond\u00eda a la imagen y a la importancia que la Cultura ten\u00eda justamente en las estructuras de poder europeas. Imagen e importancia depend\u00edan del nivel del presupuesto que pod\u00eda repartir un departamento y de la influencia en las \u00e9lites pol\u00edticas y econ\u00f3micas. Y en el caso de la Cultura ambos factores eran negativos. Al final al comisario austriaco no le fue asignado el Departamento de Cultura, sino la \u00abpol\u00edtica regional\u00bb, lo que caus\u00f3 j\u00fabilo en la naci\u00f3n de cultura. \u00abNosotros \u2014\u200cdijeron ahora los peri\u00f3dicos austriacos\u2014 tenemos un presupuesto de 337 mil millones.\u00bb\n\nLa Cultura le toc\u00f3 a Grecia. Esto parece del todo congruente cuando se piensa en la Antig\u00fcedad griega como fundamento de la cultura europea, o ingeniosamente c\u00ednico si se quiere relacionar la degradaci\u00f3n de la democracia en Europa con la sociedad esclavista de la Antig\u00fcedad griega, pero en realidad la cosa era de lo m\u00e1s simple: debido a su ya interminable crisis econ\u00f3mica y presupuestaria, Grecia estaba totalmente desacreditada y, por ello, indefensa, y ten\u00eda que aceptar lo que le dieran. El departamento menospreciado. No era una tarea, era un castigo: a quien no sabe manejar el dinero, m\u00e1s vale no ponerle dinero en la mano, a \u00e9se se le asigna, pues, el departamento sin presupuesto. La comisaria griega, una mujer en\u00e9rgica, luchaba por tener un equipo fuerte en el que poder confiar y que pudiera darle a ella alguna relevancia pol\u00edtica en la Comisi\u00f3n. Consigui\u00f3 reclamar a algunos compatriotas que ya dispon\u00edan de experiencia en el aparato de la Comisi\u00f3n, que estaban bien conectados con otras direcciones generales y ten\u00edan excelente fama, a fin de ocupar con ellos las posiciones clave de su direcci\u00f3n general. As\u00ed fue como Fenia Xenopoulou fue \u00abretirada\u00bb de Comercio y ascendida a jefa de la direcci\u00f3n en el Arca en la que trabajaba Martin Susman.\n\nFenia no hab\u00eda podido declinar ese ascenso. Quien quer\u00eda hacer carrera en el aparato de la Comisi\u00f3n Europea deb\u00eda mostrar mobilit\u00e9. Quien no estaba dispuesto a ello y rechazaba una oferta de cambio de esfera de acci\u00f3n quedaba fuera del juego. As\u00ed que se hab\u00eda trasladado al Arca con el plan de probar a\u00fan mejor su mobilit\u00e9: disponi\u00e9ndose al momento a aspirar al siguiente cambio con especial consideraci\u00f3n de la visibilit\u00e9. Para el ascenso en el aparato eso era igualmente decisivo: ser visible, trabajar de manera que se llamara constantemente la atenci\u00f3n.\n\nFenia sab\u00eda lo que era la miseria. La hab\u00eda conocido. Gozaba de esa ferviente energ\u00eda que poseen a menudo las personas que tienen grabada a fuego en el alma la miseria de su origen y que nunca logran distanciarse de ella, por muy lejos que lleguen, porque siempre llevan su alma consigo. Desde su primera oportunidad en la vida hab\u00eda demostrado de continuo que estaba dispuesta a aprovechar la ocasi\u00f3n. Cuando le ense\u00f1aban una puerta dici\u00e9ndole: Si encuentras la llave, sales al aire libre por esa puerta, ella buscaba minuciosamente la llave; estaba tambi\u00e9n dispuesta a limar durante mucho tiempo todas las llaves posibles hasta que por fin alguna encajara, pero de vez en cuando llegaba el momento en que cog\u00eda un hacha y destrozaba la puerta. El hacha se convirti\u00f3 finalmente en su llave maestra.\n\nMartin Susman no aguantaba a Fenia. Desde su incorporaci\u00f3n al Arca, el ambiente laboral hab\u00eda empeorado. Era evidente que ella despreciaba el trabajo que se efectuaba all\u00ed y, al mismo tiempo, presionaba de un modo insoportable para que resaltara y obtuviera mayor visibilidad.\n\nFenia Xenopoulou dorm\u00eda bien. Para ella, el sue\u00f1o era una parte del dominio del cuerpo, de la autodisciplina. Se acoplaba al sue\u00f1o como a un cargador de bater\u00edas. Encog\u00eda las piernas y los brazos, arqueaba la espalda, apretaba la barbilla contra el pecho. Y ya estaba recargando pilas para el combate del d\u00eda siguiente. Pero cuando dorm\u00eda no so\u00f1aba.\n\n\u00bfHe roncado?, le pregunt\u00f3 Fridsch por la ma\u00f1ana.\n\nNo, he dormido bien.\n\nComo un ni\u00f1o.\n\nS\u00ed.\n\nNo, en realidad como un embri\u00f3n.\n\n\u00bfEmbri\u00f3n?\n\nS\u00ed. Por la postura que ten\u00edas. Me recordaba fotos de embriones. \u00bfQuieres caf\u00e9?\n\nNo, gracias. Tengo que marcharme enseguida. Quer\u00eda darle un beso de despedida y decir \u00abPiensa en m\u00ed\u00bb, pero no lo hizo, s\u00f3lo salud\u00f3 con la cabeza y dijo: Tengo que...\n\nDe camino a la oficina, Martin Susman hab\u00eda recibido las \u00faltimas informaciones. Cuando el tiempo lo permit\u00eda, o sea, cuando no llov\u00eda, iba en bicicleta al trabajo. As\u00ed hac\u00eda un poco de ejercicio, pero \u00e9sa no era la raz\u00f3n principal. El metro le pon\u00eda triste. Esas caras cansadas, grises, ya a primera hora de la ma\u00f1ana. Esa afectada disposici\u00f3n de la gente a parecer siempre, con sus maletines y carteras con ruedas, din\u00e1mica y capacitada y competitiva, m\u00e1scaras mal ajustadas bajo las que se pudr\u00edan los verdaderos rostros. Esas miradas al vac\u00edo cuando sub\u00edan los mendigos con sus acordeones, tocaban una canci\u00f3n y ped\u00edan unas monedas con un envase de yogur. \u00bfQu\u00e9 canciones eran? Martin no habr\u00eda podido decirlo, quiz\u00e1 canciones de moda de los a\u00f1os veinte y treinta del siglo pasado, \u00e9poca de entreguerras. Esa oleada de gente que se mov\u00eda mec\u00e1nicamente, que sub\u00eda con fuertes pisadas las escaleras mec\u00e1nicas fuera de servicio, que avanzaba despu\u00e9s por los sucios pasillos revestidos de tablones de contrachapado de las obras eternas del metro, pasando por los cuchitriles de los trozos de pizza y los kebabs, por el olor a secreciones humanas y a putrefacci\u00f3n, y finalmente la bocanada de viento al subir a la calle, a la luz del d\u00eda que ya no penetra en el \u00e1nimo sombr\u00edo. Martin prefer\u00eda ir en bicicleta. Muy pronto se hab\u00eda hecho socio del EU-Cycling-Group. Esa asociaci\u00f3n pon\u00eda a disposici\u00f3n de cada funcionario de la UE que se hac\u00eda socio, primero un entrenador personal que le ense\u00f1aba, por ejemplo, a salir vivo de la traves\u00eda en bicicleta de la estaci\u00f3n de Montgomery; el entrenador averiguaba el camino m\u00e1s seguro para ir de casa al lugar de trabajo y despu\u00e9s recorr\u00edan juntos varios d\u00edas ese camino, y se aprend\u00eda tambi\u00e9n a pegar encima de los coches la pegatina \u00a1EST\u00c1 USTED ESTORBANDO! al pasar junto a los que estaban aparcados en un carril bici. Las pegatinas no deterioraban los coches, se las pod\u00eda despegar con facilidad. El EU-Cycling-Group era un gran \u00e9xito, la proporci\u00f3n de ciclistas en el tr\u00e1fico de Bruselas se hab\u00eda duplicado en pocos a\u00f1os gracias a los funcionarios de la UE. A Martin le gustaba sobre todo que se formaran espont\u00e1neamente grupos en el camino de casa a la oficina. Cuando sal\u00eda de casa por la ma\u00f1ana, se encontraba a m\u00e1s tardar en el boulevard Aspach con el primer compa\u00f1ero, luego con el segundo, hasta que al final eran a menudo un pelot\u00f3n de ocho o diez ciclistas. Los funcionarios alemanes adelantaban en bicicletas de carreras al pelot\u00f3n, pedaleaban camino del trabajo en ropa funcional, como si tuvieran que ganar un crit\u00e9rium, por eso eran casi \u00fanicamente alemanes los que antes de comenzar el trabajo utilizaban las duchas de las oficinas situadas en el s\u00f3tano del edificio. Los holandeses en sus omafietsen o los colegas de pa\u00edses mediterr\u00e1neos iban relajados, pedaleaban apaciblemente en traje completo, sin sudar, unos junto a otros, charlaban y as\u00ed se enteraban de muchas cosas mejor que en la cantina: todos los nuevos rumores, intrigas, avances en la carrera. Para estar informado, esas conversaciones en bicicleta eran m\u00e1s importantes que la lectura de la European Voice y por lo menos tanto como el estudio del Financial Times.\n\nEn la rue de l'\u00c9cuyer se hab\u00eda unido al pelot\u00f3n Bohumil Szmekal, su amigo y compa\u00f1ero de la C-I (Pol\u00edtica Cultural y Di\u00e1logo Intercultural), y menos de doscientos metros m\u00e1s adelante, en la rue d'Arenberg, oyeron los gritos de Kass\u00e1ndra Mercouri, la jefa de equipo de Fenia Xenopoulou. Bohumil y Martin frenaron y esperaron hasta que Kass\u00e1ndra se uni\u00f3 a ellos, dejaron pasar al pelot\u00f3n y continuaron su camino los tres juntos.\n\n\u00bfTienes ya una idea?, pregunt\u00f3 Bohumil. Luego exclam\u00f3 excitado \u00ab\u00a1Ten cuidado!\u00bb y se\u00f1al\u00f3 un coche aparcado delante de ellos en el carril bici. Sac\u00f3 a la velocidad del rayo una pegatina de la bolsa que llevaba en bandolera y, conduciendo sin manos, separ\u00f3 el papel y despu\u00e9s de dar un rodeo al coche le plant\u00f3 la pegatina en la ventanilla lateral. Hubo un concierto de bocinazos.\n\n\u00a1Toma ya! \u00a1\u00c9sa est\u00e1 bien pegada!, dijo triunfal.\n\nT\u00fa, con tus adhesivos, eres m\u00e1s peligroso que los coches, dijo Kass\u00e1ndra. Era una mujer regordeta en la treintena, de mirada siempre bondadosa o preocupada, junto a la que Bohumil, bajito y delgado, aunque unos a\u00f1os mayor que ella, parec\u00eda un chico travieso. \u00c9l sonri\u00f3. Ahora dime por fin: \u00bfhas tenido la idea salvadora? El trabajo de la direcci\u00f3n entera est\u00e1 completamente bloqueado porque todav\u00eda nadie...\n\n\u00bfQu\u00e9 clase de idea? No s\u00e9 a lo que te refieres.\n\n\u00a1El Big Jubilee Project! A\u00fan no has contestado al e-mail colectivo. Yo, por cierto, tampoco.\n\n\u00bfEl Big Jubilee Project? \u00a1Pensaba que no hab\u00eda que comentarlo!\n\nS\u00ed. Todos se hacen los tontos. No vendr\u00e1 nada. Nadie lo considera importante. No es de extra\u00f1ar cuando pienso en el fracaso de hace cinco a\u00f1os.\n\nEntonces yo no estaba a\u00fan aqu\u00ed.\n\n\u00bfPor qu\u00e9 fracaso? La ceremonia en el Parlamento con los embajadores infantiles fue muy conmovedora. \u00a1Ni\u00f1os de toda Europa! La manera en que expresaron sus deseos para el futuro, paz y...\n\n\u00a1S\u00e1ndra, por favor! \u00a1Embajadores infantiles! \u00a1Eso fue abusar de los ni\u00f1os! Por suerte la opini\u00f3n p\u00fablica no se dio cuenta de nada. As\u00ed que mi idea es... \u00a1Cuidado! Dio un giro s\u00fabito a la bici, empuj\u00f3 a Martin hasta el centro de la calle, ya ten\u00eda otra vez en la mano una pegatina, que dej\u00f3 caer, sin embargo, Martin lo empuj\u00f3 de vuelta al carril bici al tiempo que gritaba: \u00a1Est\u00e1s loco!\n\nS\u00ed. Bueno, yo opino lo siguiente: como es bien sabido, aprender de la historia significa: \u00a1Nunca m\u00e1s! Eso no puede repetirse. Se acabaron los aniversarios. Es caro y penoso. No puedo comprender por qu\u00e9 Xeno lo considera tan importante.\n\nTodas las direcciones generales est\u00e1n relacionadas entre ellas. Si se emplea a fondo en la suya puede destacar, dijo Kass\u00e1ndra.\n\nAhora est\u00e1 apretando las clavijas. Hoy, a las once, sesi\u00f3n. Quiere o\u00edr nuestras ideas.\n\nYo lo he entendido de un modo muy distinto, dijo Martin. Yo pensaba...\n\nQuiz\u00e1 se aplace la sesi\u00f3n. No est\u00e1 confirmado a\u00fan, pero la jefa quiere tener hoy una breve cita con el presidente. Por cierto: \u00bfsab\u00e9is lo que est\u00e1 leyendo?\n\n\u00a1No me interesa en absoluto!\n\n\u00bfQuieres decir un libro? \u00bfXeno leyendo? \u00a1Venga ya, S\u00e1ndra, est\u00e1s desvariando!\n\nUn libro, s\u00ed. Y no desvar\u00edo. Tuve que procur\u00e1rselo a toda prisa. \u00a1No vais a creerlo!\n\nDilo de una vez.\n\n\u00a1Atenci\u00f3n!\n\n\u00a1Escucha bien!\n\nVer\u00e9is: la jefa se est\u00e1 preparando desde hace d\u00edas, cual oficial del Estado Mayor, para la entrevista con el presidente. Quiere saberlo todo sobre \u00e9l, desde sus redes de contactos hasta su plato favorito, todo, incluso su libro favorito. Tal vez sea posible sacarlo a relucir en una charla informal. En eso es puntillosa en extremo.\n\n\u00bfEl presidente tiene un libro favorito?\n\nProbablemente El hombre sin atributos, dijo Martin.\n\n\u00bfEl hombre sin atributos? Ser\u00eda un buen t\u00edtulo para su autobiograf\u00eda.\n\n\u00a1Chicos, por favor! \u00a1Escuchad! Lo ha averiguado por canales privados. El presidente tiene realmente un libro favorito. Una novela. Eso no lo sabe el gran p\u00fablico. Y tiene por lo visto varios ejemplares, porque lo lee en todas partes. Uno est\u00e1 junto a su cama. Otro, sobre su escritorio, en su despacho. Probablemente habr\u00e1 alguno m\u00e1s en el apartamento de su pareja. El rostro de Kass\u00e1ndra brillaba. \u00bfUna ligera capa de sudor? \u00bfDiversi\u00f3n? En fin, dijo, lo cierto es que yo tuve que comprar el libro, y ahora la jefa lo est\u00e1 leyendo.\n\nXeno lee literatura, pens\u00f3 Martin con extra\u00f1eza, \u00a1una novela! Por su carrera incluso est\u00e1 dispuesta a leer una novela.\n\nFenia Xenopoulou estaba sentada ante su escritorio y le\u00eda. Lo que le\u00eda la dejaba perpleja. Sab\u00eda leer con gran rapidez, hab\u00eda aprendido a hacer una especie de escaneo de las p\u00e1ginas, a ordenar al momento las informaciones en compartimentos dentro de la cabeza, de donde, en caso de necesidad, pod\u00eda sacarlos en un abrir y cerrar de ojos. Pero aquello era una novela. Para eso no ten\u00eda ninguna estructura mental, \u00bfde qu\u00e9 iba aquello? \u00bfCu\u00e1les eran las informaciones que podr\u00edan acreditarse como \u00fatiles? \u00bfQu\u00e9 deb\u00eda, por todos los santos, retener en la memoria? All\u00ed se contaba la vida de un hombre, s\u00ed, muy bien, pero \u00bfqu\u00e9 le importaba a ella aquel hombre tan totalmente desconocido? Adem\u00e1s hab\u00eda vivido en una \u00e9poca por completo distinta, as\u00ed no piensa ni act\u00faa hoy nadie. Sobre todo: \u00bfhab\u00eda vivido realmente o era inventado? Seg\u00fan Google, el hombre hab\u00eda existido de verdad, por lo visto en su \u00e9poca hab\u00eda desempe\u00f1ado un papel sobresaliente, que influy\u00f3 en el orden pol\u00edtico del continente y, en \u00faltimo t\u00e9rmino, del mundo entero. Pero tan important\u00edsimo tampoco pudo ser, porque de lo contrario habr\u00eda o\u00eddo hablar de \u00e9l en el colegio. Seguramente era m\u00e1s bien un caso para especialistas, y hasta estos mismos no estaban de acuerdo en c\u00f3mo hab\u00eda que valorar, a fin de cuentas, el papel que hab\u00eda desempe\u00f1ado aquel hombre.\n\nSigui\u00f3 pasando hojas con impaciencia, se salt\u00f3 un cap\u00edtulo. No lo entend\u00eda: all\u00ed no hab\u00eda \u2014\u200cal menos hasta ahora\u2014 decisiones pol\u00edticas. Sino amor. Todo estaba escrito desde la perspectiva de una mujer que amaba a ese hombre. Pero en la entrada de Wikipedia sobre aquel hombre no ven\u00eda el nombre de la mujer. Y no quedaba claro si ella le quer\u00eda de verdad, es decir, hasta ahora no estaba claro a\u00fan. En cualquier caso, se sent\u00eda retada a acaparar su atenci\u00f3n, a lograr una influencia sobre \u00e9l. Pero si esa mujer era un invento de la novelista, \u00bfqu\u00e9 sentido ten\u00eda leer que ella, una ficci\u00f3n, trataba de adquirir poder sobre un hombre que en tiempos hist\u00f3ricos hab\u00eda sido realmente poderoso? Si la autora quiso mostrar de qu\u00e9 modo una mujer puede llegar a tener poder sobre hombres poderosos, \u00bfpor qu\u00e9 no escribi\u00f3 una gu\u00eda pr\u00e1ctica? En la novela hab\u00eda intrigas y graciosos enredos, luchas con competidores pol\u00edticos, pero en \u00faltimo t\u00e9rmino \u2014\u200cFenia hojeaba, le\u00eda cada vez con m\u00e1s impaciencia, le\u00eda una p\u00e1gina, hojeaba diez\u2014, en \u00faltimo t\u00e9rmino todo iba a parar al amor o, mejor, a la poca importancia que ten\u00eda el poder pol\u00edtico cuando se trataba del poder del amor. \u00bfSe pod\u00eda decir eso as\u00ed? Pues era absurdo. \u00a1Las novelas son puro absurdo!\n\nFenia se recost\u00f3 en la silla. \u00bf\u00c9se era el libro favorito del presidente? \u00a1El presidente estaba loco! \u00a1Todos esos pensamientos! Lo que pensaba ella, lo que pensaba \u00e9l: \u00bfde d\u00f3nde lo sacaba la autora? Si ese hombre hab\u00eda existido en la realidad, entonces, sin duda alguna, habr\u00eda alguna fuente en alg\u00fan archivo: documentos, contratos, credenciales, pero \u00bfpensamientos? Los pensamientos no se conservan ni se han conservado jam\u00e1s en documentos. Quien tiene la cabeza en su sitio evita cualquier cosa que pueda llevar a que se lean sus pensamientos.\n\nCerr\u00f3 los ojos, pens\u00f3 de pronto en la tarde anterior con Fridsch, en la noche. \u00bfHab\u00eda pensado ella de verdad que \u00e9l..., hab\u00eda pensado \u00e9l que ella...?\n\nEstaba sentada, pero crey\u00f3 tambalearse. Abri\u00f3 con fuerza los ojos, se dio \u00e1nimos... y en ese momento vio en la pantalla del ordenador: Nuevo mensaje de Kass\u00e1ndra Mercouri. \u00abLamentablemente, la cita con el presidente es imposible hoy. El gabinete del presidente propone fechas en los pr\u00f3ximos d\u00edas.\u00bb\n\nCerr\u00f3 el libro, lo apart\u00f3.\n\nA: B. Szmekal (Di\u00e1logo Intercultural), M. Susman (Disposiciones Cultura), H. Athanasiadis (Valoraci\u00f3n), C. Pinheiro da Silva (Diversidad Ling\u00fc\u00edstica), A. Klein (Competencia Medi\u00e1tica)\n\n\u2014Fenia hizo una breve pausa, luego borr\u00f3 Helene Athanasiadis\u2014\n\nAsunto: Jubilee Project\n\nConfirmaci\u00f3n de cita: 11.00 horas sala de reuniones. Espero propuestas.\n\nSon\u00f3 el tel\u00e9fono, Martin Susman mir\u00f3 la pantalla, era un n\u00famero local, de Bruselas, que no conoc\u00eda; lo cogi\u00f3 y se arrepinti\u00f3 al momento. Era su hermano.\n\n\u00a1Soy yo!\n\nS\u00ed. Hola, Florian.\n\nSab\u00edas que ven\u00eda a Bruselas.\n\nS\u00ed.\n\nHace d\u00edas que intento hablar contigo. No coges el tel\u00e9fono.\n\n...\n\nAnoche llam\u00e9 por lo menos diez veces. \u00bfPor qu\u00e9 no lo coges? \u00bfO me llamas despu\u00e9s t\u00fa mismo?\n\n\u00bfAnoche? Hab\u00eda un problema.\n\nT\u00fa siempre tienes problemas. Yo tambi\u00e9n tengo problemas, por eso...\n\nEra que...\n\nEn cualquier caso, he llegado. Ya estoy en el hotel. En el Marriott. Tengo mi primera cita ahora mismo. \u00bfNos vemos para cenar? \u00bfHasta qu\u00e9 hora trabajas?\n\nHasta las siete o las siete y media.\n\nVale. Pasa a recogerme a las ocho y media.\n\n\u00bfPor el hotel?\n\nClaro, por el hotel. Y luego me ense\u00f1as un restaurante donde se pueda fumar.\n\nNo se puede fumar en ning\u00fan sitio.\n\nImposible. Entonces hasta las ocho y media. Y s\u00e9 puntual, hermanito.\n\nEl Big Jubilee Project. En realidad hab\u00eda sido Mrs. Atkinson quien hab\u00eda tenido la idea. Era la nueva directora general de la DG KOMM, el Servicio de Comunicaci\u00f3n de la Comisi\u00f3n Europea, responsable tambi\u00e9n de su corporate image, y \u00e9sta, como hab\u00eda mostrado la \u00faltima encuesta del eurobar\u00f3metro, se hab\u00eda hundido estrepitosamente. Ella lo tuvo claro desde el primer momento: deb\u00eda llevar la direcci\u00f3n general de manera distinta a sus predecesores; un trabajo de prensa sin pena ni gloria, el rutinario spokesman's service y la coordinaci\u00f3n formal de las sopor\u00edferas oficinas de informaci\u00f3n en los Estados miembros no bastar\u00edan. No s\u00f3lo ten\u00edan los peores \u00edndices desde 1973, cuando comenzaron los sondeos sistem\u00e1ticos de la opini\u00f3n p\u00fablica en los pa\u00edses de la UE, sino que los resultados actuales de las encuestas hab\u00eda que calificarlos de desastre absoluto: seis meses antes, aproximadamente un 49 por ciento de los ciudadanos de la UE hab\u00edan valorado como b\u00e1sicamente positivo el trabajo de la Comisi\u00f3n, y ya entonces este resultado supon\u00eda \u00abun m\u00ednimo hist\u00f3rico\u00bb; inconcebible que fuera a\u00fan inferior. Ahora \u2014\u200cexplotando todas las posibilidades de la mistificaci\u00f3n\u2014 la cifra estaba en un escaso 40 por ciento, la mayor ca\u00edda en la historia del eurobar\u00f3metro, mayor a\u00fan que el hundimiento de la cuota de aceptaci\u00f3n en el a\u00f1o 1999, cuando la Comisi\u00f3n tuvo que dimitir en pleno por un esc\u00e1ndalo de corrupci\u00f3n. En aquel entonces, la bajada en picado del 67 por ciento al 59 por ciento se vivi\u00f3 como una cat\u00e1strofe: \u00bfahora qu\u00e9 era entonces? \u00bfY por qu\u00e9?\n\nMrs. Atkinson estudiaba papeles, listas, c\u00e1lculos porcentuales, gr\u00e1ficos, estad\u00edsticas, y se preguntaba c\u00f3mo se hab\u00eda llegado a tal dram\u00e1tica p\u00e9rdida de confianza en la instituci\u00f3n. Al nuevo presidente de la Comisi\u00f3n se le hab\u00edan dispensado grandes alabanzas anticipadas en los principales medios europeos, pero quien sac\u00f3 provecho de ello no fue la Comisi\u00f3n, sino el Parlamento Europeo, cuyo prestigio hab\u00eda subido casi cinco puntos porcentuales. Por primera vez en la historia, el presidente logr\u00f3 cumplir con la cuota femenina, y no s\u00f3lo en los miembros de la comisi\u00f3n \u2014\u200cdoce de veintiocho eran ahora mujeres\u2014, sino tambi\u00e9n en el nivel superior de las direcciones generales. La cuota femenina era all\u00ed ahora de un 40 por ciento escaso. Ella misma hab\u00eda sacado provecho de tal cuota, cosa que, como ella dec\u00eda, pod\u00eda admitir sin dudar por ello de su propia capacidad; al contrario, se deb\u00eda a la aplicaci\u00f3n consecuente de la cuota el hecho de que Mrs. Atkinson no quedara derrotada por el arribista, carente por completo de cualificaci\u00f3n, George Morland, ese ser despreciable de quien se habl\u00f3 en un primer momento para el puesto y que ahora iba por ah\u00ed dibujando una imagen caricaturesca de ella como t\u00edpico ejemplo de la estupidez de la cuota. Seg\u00fan hab\u00eda o\u00eddo decir, andaba contando que era fr\u00eda como el hielo, hasta tal punto que la frialdad de sus manos la atormentaba y que por eso siempre estaba sentada ante su escritorio con un enorme manguito... \u00a1En fin, mujeres!\n\nProbablemente, esa descabellada idea ya lo dec\u00eda todo sobre aquel intrigante: que la asociara con un enorme manguito mostraba de modo inequ\u00edvoco su horror a la vagina, t\u00edpico del hombre brit\u00e1nico de clase alta.\n\nMrs. Atkinson hab\u00eda estudiado Marketing y Management en la European Business School de Londres y hab\u00eda terminado la carrera con un meritorio trabajo sobre \u00abmarketing contrainductivo\u00bb. Reflexion\u00f3 si no deb\u00eda dar un giro ofensivo a la historia con el fin de desbaratar las intrigas de Mr. Morland y hacer del manguito su marca personal, un manguito enorme, gigantesco, con lo cual la caricatura de Morland resultar\u00eda ins\u00edpida y al mismo tiempo reforzar\u00eda la marca. Pero eso no era lo que la ocupaba ahora. Se preguntaba por qu\u00e9 ese \u00e9xito de la Comisi\u00f3n, la cuota femenina, esa clara se\u00f1al en favor de las posibilidades de las mujeres en este continente, no hab\u00eda mejorado la imagen de la Comisi\u00f3n. La proporci\u00f3n de mujeres en el Parlamento Europeo era s\u00f3lo de un 35 por ciento, pero el prestigio del Parlamento aumentaba, incluso en votantes femeninas de todas las edades, lo que estaba bien, pero el de la Comisi\u00f3n ca\u00eda en picado, y eso era enigm\u00e1tico; ah\u00ed estaba el problema, y su tarea consist\u00eda ahora en detener e invertir esa tendencia. Cu\u00e1les eran los aspectos criticables, a qu\u00e9 se deb\u00eda la mala imagen de la Comisi\u00f3n. Clich\u00e9s. Prejuicios. Siempre lo mismo. Falta de legitimaci\u00f3n democr\u00e1tica, proliferaci\u00f3n de la burocracia, man\u00eda reguladora. Le parec\u00eda significativo que no se criticaran las tareas propias de la Comisi\u00f3n, por lo visto la gente no las conoc\u00eda. \u00abSe inmiscuye en asuntos que mejor ser\u00eda regular a nivel nacional\u00bb, 59 por ciento; pero \u00abCumple sus tareas mal\u00bb o \u00abmuy mal\u00bb tuvo, en conjunto, s\u00f3lo un 5 por ciento escaso de respuestas afirmativas. Hab\u00eda que analizar esa contradicci\u00f3n. Se preguntaba por qu\u00e9 ninguno de sus predecesores hab\u00eda cuestionado el m\u00e9todo de la encuesta del eurobar\u00f3metro ni introducido un cambio. Si se propone a la gente marcar con una cruz la frase \u00abse inmiscuye en asuntos que mejor ser\u00eda regular a nivel nacional\u00bb, entonces lo har\u00e1 un determinado tanto por ciento. \u00a1Esos tipos del claro-que-es-verdad! \u00a1Esos idiotas del si-es-lo-que-yo-digo-siempre! Pero si se formulara que la Comisi\u00f3n protege a los ciudadanos de las injusticias que resultan de las diferencias entre los sistemas jur\u00eddicos nacionales, entonces el resultado ser\u00eda muy distinto.\n\nAhora lo ten\u00eda claro: su misi\u00f3n no pod\u00eda ser mejorar la imagen de la UE, sino que ten\u00eda que ocuparse directamente de la imagen de la Comisi\u00f3n Europea. Y la idea de c\u00f3mo lo conseguir\u00eda la tuvo una hora despu\u00e9s, animada por el champ\u00e1n Charlemagne Brut. Porque en ese momento se abri\u00f3 de golpe la puerta de su despacho, vio entrar a Catherine, su secretaria, con una tarta sobre la que ard\u00edan bengalas, y detr\u00e1s del humo y de las estrellas chispeantes vio: s\u00ed, en efecto, era el presidente, y detr\u00e1s de \u00e9l se met\u00eda a empellones cada vez m\u00e1s gente en la habitaci\u00f3n, su comisario, directores, ponentes, su departamento al completo, y cantaban Happy Birthday.\n\nTen\u00eda un cumplea\u00f1os terminado en cero. Ah, s\u00ed. Ella no le hab\u00eda dado importancia. Su marido estaba en Londres. Su hija, en Nueva York. Los dos hab\u00edan hecho una breve llamada telef\u00f3nica. Y amigos con los que habr\u00eda podido festejarlo..., a\u00fan no ten\u00eda all\u00ed, en Bruselas. Y ahora era el centro de atenci\u00f3n. Sorpresa. Habl\u00f3 el presidente. Unas palabras. Nada formal, muy personal, tambi\u00e9n una peque\u00f1a alusi\u00f3n a su imagen, que acab\u00f3 en risa general. Personas que ella s\u00f3lo conoc\u00eda de saludarlas, gente de la segunda tercera cuarta planta, re\u00edan abiertamente mir\u00e1ndola a la cara, las copas de champ\u00e1n espumeaban, tintineaban al chocar en el brindis, la besaron en las mejillas, le apretaban el brazo, le daban golpecitos en el hombro, gente que sab\u00eda poco o nada de ella le mostraba simpat\u00eda o disposici\u00f3n a la simpat\u00eda, el comisario levant\u00f3 la copa, dijo que cu\u00e1nto se alegraba de tener a esa competente y en todo magn\u00edfica colaboradora en su equipo, en esa importante funci\u00f3n, qu\u00e9 bien lo de la cuota, \u00e9l, personalmente, estaba a favor de una cuota femenina del 99 por ciento, claro que no quer\u00eda perder su trabajo, pero fuera de \u00e9l estar\u00eda encantado de rodearse s\u00f3lo de mujeres...; silbidos de los hombres, gritos de \u00a1machismo!, \u00a1machismo! de las mujeres, todo se disolvi\u00f3 en risa general y Mrs. Atkinson cort\u00f3 la tarta que ahora estaba sobre la carpeta del eurobar\u00f3metro en su escritorio, migajas y crema sobre las estad\u00edsticas, ceniza de las bengalas sobre la tumba de la opini\u00f3n europea.\n\nY luego se qued\u00f3 otra vez sola, todos hab\u00edan vuelto al trabajo, y ella, de pie junto al ventanal de su despacho, miraba a la rue de la Loi, a la cinta de coches oscuros que pasaban desliz\u00e1ndose y brillaban bajo la ligera llovizna; se frot\u00f3 las manos, pasando alternativamente una por el dorso de la otra, dando masajes y friccionando los dedos, que eran muy largos y finos y tend\u00edan a perder de pronto el color, a ponerse blancos e insensibles. Luego se sent\u00f3 de nuevo ante el escritorio; algo rumoreaba en su interior, esper\u00f3 a verlo m\u00e1s claro, hab\u00eda all\u00ed a\u00fan una copa medio llena de champ\u00e1n, tom\u00f3 un sorbito, reflexion\u00f3, se acab\u00f3 la copa. Se frot\u00f3 los dedos, luego busc\u00f3 en Google: \u00abComisi\u00f3n Europea fundaci\u00f3n\u00bb. \u00bfCu\u00e1ndo era el cumplea\u00f1os de la Comisi\u00f3n Europea? \u00bfHab\u00eda una especie de cumplea\u00f1os de la Comisi\u00f3n? \u00bfEl d\u00eda de la fundaci\u00f3n? \u00c9sa era la idea: no bastaba con vender lo mejor posible el trabajo diario de la Comisi\u00f3n, hab\u00eda que aplaudirla, hab\u00eda que inducir a la gente a felicitarla por su existencia, hab\u00eda que alabarla en lugar de mendigar aceptaci\u00f3n, hab\u00eda que rectificar estereotipos y desmentir rumores y leyendas. Hab\u00eda que poner en el centro la Comisi\u00f3n y no siempre, de manera abstracta y general, \u00abla UE\u00bb. Porque \u00bfqu\u00e9 es la UE? Diversas instituciones que hac\u00edan cada una rancho aparte y que defend\u00edan distintos intereses, pero si el conjunto ten\u00eda un sentido se deb\u00eda s\u00f3lo a que exist\u00eda la Comisi\u00f3n, que, ella s\u00ed, representaba a la totalidad. As\u00ed lo ve\u00eda ella. Hab\u00eda que generar una situaci\u00f3n en la que la Comisi\u00f3n estuviera alegremente en el centro, como alguien que cumple a\u00f1os y lo felicitan. \u00bfTen\u00eda, pues, la Comisi\u00f3n una fecha de nacimiento? Eso no era tan f\u00e1cil de determinar. \u00bfEra el d\u00eda en que se fund\u00f3 la Comisi\u00f3n de la CEE o la fecha en que se fund\u00f3 la Comisi\u00f3n Europea en su forma actual despu\u00e9s del acuerdo de fusi\u00f3n de la CE? En un caso, la Comisi\u00f3n cumplir\u00eda sesenta a\u00f1os dentro de tres, en el otro caso, dentro de dos a\u00f1os, cincuenta. Cincuenta le gustaba m\u00e1s. Medio siglo. Eso se vend\u00eda mejor. Y aplicado a la edad de los hombres: en pleno rendimiento, con experiencia, a\u00fan no en la transici\u00f3n a la inutilidad de la vejez. Adem\u00e1s, dos a\u00f1os era un periodo de tiempo razonable para una perfecta preparaci\u00f3n, mientras que tres a\u00f1os podr\u00edan ser quiz\u00e1 demasiado tiempo, muchas cosas podr\u00edan torcerse.\n\nSigui\u00f3 indagando. \u00bfHab\u00eda habido ya conmemoraciones? S\u00ed. Celebraciones pesadas y carentes de est\u00edmulo, con discursos solemnes, palabras amables para los predecesores, un poco de incienso para las etapas previas de la UE, cincuenta a\u00f1os de los Tratados de Roma, sesenta a\u00f1os de la fundaci\u00f3n de la CECA... \u00bfA qui\u00e9n hab\u00eda interesado eso? A nadie. \u00bfY para qu\u00e9 serv\u00eda decirles a los esc\u00e9pticos y adversarios de la UE lo estupenda que hab\u00eda sido la fundaci\u00f3n de la CECA? Como si se felicitara a un abuelo que sufre demencia senil porque hubo un tiempo en el que estaba cuerdo..., mientras que los nietos, con perfecta indiferencia, llevan mucho tiempo haci\u00e9ndolo todo de un modo completamente distinto.\n\nGrace Atkinson vio sobre la mesa de cristal, delante del tresillo, una botella abierta de champ\u00e1n. Quedaba un poquito, se lo sirvi\u00f3, bebi\u00f3. Estaba achispada cuando decidi\u00f3 enviar un e-mail a algunas secciones de las que, como ella cre\u00eda, pod\u00eda esperar inter\u00e9s por su plan, apoyo e ideas. Primero ten\u00eda que conseguir de un modo informal compa\u00f1eros de lucha antes de poder dar comienzo al procedimiento formal. Una gran fiesta de aniversario con ocasi\u00f3n del pr\u00f3ximo cincuentenario de la fundaci\u00f3n de la Comisi\u00f3n Europea, eso escribi\u00f3, le parec\u00eda una posibilidad de poner las tareas y los m\u00e9ritos de esa instituci\u00f3n en el centro de la atenci\u00f3n p\u00fablica, de robustecer su corporate identity, de mejorar su imagen, de celebrarlo alegremente y de ese modo dejar de estar a la defensiva.\n\nBorr\u00f3 la palabra alegremente, luego la escribi\u00f3 otra vez, asinti\u00f3, de eso se trataba en definitiva, se frot\u00f3 las manos y se lanz\u00f3 al toro. En la l\u00ednea de \u00abasunto\u00bb escribi\u00f3: \u00abBig Jubilee Project: se acab\u00f3 el lloriqueo\u00bb.\n\nHab\u00eda sido idea de Mrs. Atkinson. Fenia Xenopoulou fue la primera en responder... y enseguida hizo suyo el proyecto. Era cosa del Departamento de Cultura, opinaba Fenia, no cab\u00eda duda. Era la ocasi\u00f3n que ella hab\u00eda estado esperando para mostrar visibilit\u00e9. Y convirti\u00f3 a Martin Susman en su sherpa, que deb\u00eda cargar con el peso del proyecto.\n\nAl principio Grace Atkinson se alegr\u00f3 mucho de haber encontrado tan r\u00e1pidamente una compa\u00f1era de lucha tan entusiasta. Y al final estaba aliviada porque, debido a la abrumadora movilizaci\u00f3n de la desastrosa Cultura, se olvid\u00f3 de que hab\u00eda sido ella quien hab\u00eda tenido esa, al fin y a la postre, catastr\u00f3fica idea.\n\nEspero propuestas, hab\u00eda dicho Fenia Xenopoulou con tono excitado, es de enorme importancia y s\u00e9 que vosotros... Recorri\u00f3 con la vista el grupo y dijo en voz demasiado alta varias frases con ampulosos y dram\u00e1ticos adjetivos, seguramente pensaba que infund\u00eda \u00e1nimos, ese lenguaje corporal con la tropa, y Martin hab\u00eda bajado los ojos para esquivar su mirada, por lo que ahora ve\u00eda a Fenia sin cabeza, s\u00f3lo ve\u00eda su top ce\u00f1id\u00edsimo, la falda estrecha y ajustada, las piernas en los leotardos opacos y pens\u00f3: esta mujer est\u00e1 encorsetada, est\u00e1 metida en una armadura que la mantiene unida. La falda era de fin\u00edsimo pa\u00f1o, pero Martin ten\u00eda la impresi\u00f3n de que saltar\u00eda en pedazos si se daban golpes en ella. No era posible quitarse la prenda, habr\u00eda que romperla...\n\nBueno, \u00bfqu\u00e9 hacemos?\n\nUna vez m\u00e1s, Bohumil era ir\u00f3nico-destructivo. En primer lugar y ante todo, dijo, \u00bfqu\u00e9 no deber\u00edamos hacer? Evitar absolutamente todo lo que se ha hecho hasta ahora en aniversarios: cosas lamentables, suavizadas por ser casi todo a puerta cerrada. Folletos en papel satinado para el contenedor de papel. Discursos solemnes de d\u00edas de fiesta en d\u00edas de trabajo.\n\n\u00bfMartin?\n\nNo hab\u00eda visto la reacci\u00f3n de Fenia a las palabras de Bohumil, estaba mir\u00e1ndole los pies, los peque\u00f1os bultos que hab\u00eda m\u00e1s arriba de la abertura de sus estrechos zapatos.\n\n\u00bfMartin?\n\nA m\u00ed no me interesa el tema, es lo que habr\u00eda querido decir \u00e9l. Decidi\u00f3 dar simplemente la raz\u00f3n a todos, para no exponerse.\n\nDada la importancia del asunto, dijo en direcci\u00f3n a Fenia, estaba claro que \u2014\u200cy ahora en direcci\u00f3n a Bohumil\u2014 no se deb\u00edan repetir los errores del pasado. Bohumil ten\u00eda raz\u00f3n al recordar que..., pero Fenia ten\u00eda por supuesto toda la raz\u00f3n al esperar que. \u00bfCu\u00e1les hab\u00edan sido los errores cometidos en aniversarios anteriores? Continu\u00f3. No hab\u00eda habido m\u00e1s que la siguiente idea: celebrar un aniversario condicionado por la ocasi\u00f3n. Pero la ocasi\u00f3n por s\u00ed sola no era una idea. Que una instituci\u00f3n exista desde hace tantos y tantos a\u00f1os: muy bien, de acuerdo, pero \u00bfcu\u00e1l es la idea, qu\u00e9 idea est\u00e1 en el centro? Ha de ser convincente, ha de entusiasmar a la gente de manera que quiera realmente festejar el d\u00eda por ese motivo.\n\nAs\u00ed hab\u00eda ca\u00eddo Martin Susman en la trampa. Despu\u00e9s de discutir alg\u00fan tiempo esto y aquello, Fenia Xenopoulou dijo: Se acab\u00f3, el \u00fanico que al parecer hab\u00eda reflexionado al respecto era Martin. Era absolutamente l\u00f3gico lo que hab\u00eda dicho. Lo esencial era una idea central. Encomend\u00f3 a Martin que desarrollara la idea y que escribiera el correspondiente borrador. \u00bfCu\u00e1nto tiempo calculaba \u00e9l que necesitar\u00eda?\n\n\u00a1\u00bfDos meses?! Hab\u00eda que pensarlo muy a fondo, y hab\u00eda que discutirlo tambi\u00e9n con compa\u00f1eros de otras direcciones generales.\n\nUna semana, dijo Fenia.\n\nImposible. La semana siguiente ten\u00eda ese viaje oficial que tambi\u00e9n exig\u00eda bastante preparaci\u00f3n y...\n\nBueno, entonces dos semanas, algunos bullet points. \u00a1Eso s\u00ed que podr\u00e1s! Y con los compa\u00f1eros no discutiremos hasta que no hayamos presentado el paper. \u00bfEst\u00e1 claro? \u00a1Lo presentamos nosotros!\n\nMartin Susman estaba furioso y excitado cuando se march\u00f3 a las seis a casa, despu\u00e9s de haber liquidado los asuntos m\u00e1s urgentes del d\u00eda. A mitad del camino empez\u00f3 a llover, ten\u00eda el chubasquero en la bolsa de la bici, pero se la hab\u00eda dejado en la oficina. Lleg\u00f3 a casa calado hasta los huesos y temblando de fr\u00edo y se meti\u00f3 al momento bajo la ducha. Pero el agua no se puso caliente del todo, y la cortina de la ducha, como atra\u00edda por un im\u00e1n, se le pegaba, fr\u00eda, a la espalda. Furioso, la apart\u00f3 de un manotazo, con lo que casi la arranc\u00f3 de la barra. Lo primero que har\u00eda al d\u00eda siguiente ser\u00eda encargar que se cambiara la est\u00fapida cortina por una mampara de ducha, pero sab\u00eda que \u00e9sa era otra vez una de las ideas que nunca llevar\u00eda a la pr\u00e1ctica. Se envolvi\u00f3 en el albornoz, sac\u00f3 una botella de Jupiler de la nevera y se sent\u00f3 en la butaca frente a la chimenea. Ten\u00eda que tranquilizarse, respirar hondo, relajarse. Observ\u00f3 los libros de la chimenea.\n\nCuando Martin Susman se mud\u00f3 all\u00ed, al principio no daba cr\u00e9dito. La chimenea estaba fuera de uso desde que el piso dispon\u00eda de calefacci\u00f3n central. El casero hab\u00eda montado dos anaqueles en el hueco y colocado libros encima. Probablemente le pareci\u00f3 bonito y acogedor. M\u00e1s tarde, Martin lo hab\u00eda visto tambi\u00e9n en pisos de amigos y conocidos situados en inmuebles antiguos: libros en chimeneas fuera de uso.\n\nEn la chimenea de Martin hab\u00eda varias gu\u00edas de Bruselas, ediciones viejas y manoseadas, dejadas all\u00ed probablemente por anteriores inquilinos, algunos vol\u00famenes de un diccionario enciclop\u00e9dico del a\u00f1o 1914, tres atlas, uno de 1910, otro de 1943 y el tercero de 1955, y una docena larga de vol\u00famenes de la colecci\u00f3n \u00abCl\u00e1sicos de la literatura universal\u00bb del Club del Libro flamenco: \u00abEn cada tomo cuatro obras cl\u00e1sicas oportunamente abreviadas\u00bb, publicados en los a\u00f1os sesenta. Cuando Martin, ya instalado en el piso, pas\u00f3 revista una tarde a los libros, se qued\u00f3 escandalizado, no, esa palabra era demasiado fuerte; se qued\u00f3 desagradablemente impresionado: \u00bfeso era el progreso? \u00bfNo quemar libros, sino ponerlos en una chimenea fr\u00eda \u00aboportunamente abreviados\u00bb?\n\nAhora miraba los lomos de los libros, beb\u00eda su cerveza, fumaba un par de cigarrillos. Que hiciera el borrador para el proyecto del aniversario: qu\u00e9 desfachatez. Como si \u00e9l fuera un redactor de textos publicitarios que ha de vender el producto \u00abComisi\u00f3n UE\u00bb. Dirigi\u00f3 la mirada a su escritorio, all\u00ed estaba a\u00fan el plato con la mostaza reseca y cubierta de costra. \u00bfCu\u00e1l es la idea de la mostaza? Que la a\u00f1adimos. Genial. Convincente publicidad televisiva: apuestos j\u00f3venes exprimen gozosos y sonrientes mostaza sobre el plato gritando felices: \u00a1Viva, viva, ponla encima! Y no pueden contenerse de dichosos que son. Y las rosquillas de mostaza se siguen enroscando r\u00edtmicamente hacia lo alto en los platos, empiezan a bailar como al son de la flauta de un encantador de serpientes: Viva, viva, ponla encima. Eso era... Hizo un esfuerzo, se visti\u00f3 y se encamin\u00f3 hacia el Marriot. Se llev\u00f3 el cl\u00e1sico long, que bajo la lluvia ofrec\u00eda protecci\u00f3n para dos.\n\nHab\u00eda dejado de llover. El asfalto h\u00famedo, las fachadas de las casas y los transe\u00fantes brillaban a la luz de las farolas y de los tubos fluorescentes del puesto de patatas fritas, como si un pintor flamenco hubiera acabado de dar el barniz a ese cuadro. Entretanto, Martin hab\u00eda vivido tantas veces aquel ambiente crepuscular despu\u00e9s de d\u00edas de lluvia en Bruselas que ya casi le daba una sensaci\u00f3n de hogar. S\u00ed, all\u00ed estaba en casa. Compr\u00f3 cigarrillos al indio del night-shop de la esquina de la rue Sainte-Catherine. Despu\u00e9s de pagar, el indio dec\u00eda siempre \u00abDank u wel\u00bb si Martin hablaba franc\u00e9s y \u00abMerci, monsieur\u00bb si Martin hab\u00eda pedido en flamenco su marca de cigarrillos. Eso se podr\u00eda interpretar, pero quiz\u00e1 no hab\u00eda en ello nada que interpretar, era as\u00ed, y en alg\u00fan momento formaba parte simplemente, como tantas otras peque\u00f1eces, de la sensaci\u00f3n que Martin ten\u00eda de encontrarse a gusto viviendo, en cierto modo, entre muchos mundos.\n\nEl viento no era fuerte, pero s\u00ed fr\u00edo; Martin camin\u00f3 muy deprisa y lleg\u00f3, claro, con demasiada antelaci\u00f3n al Marriot. Su hermano, sin embargo, esperaba ya en el hall del hotel, con una expresi\u00f3n en el semblante... tan severa y arrogante, una cara que dec\u00eda: Siempre he cumplido los mandamientos divinos, entonces puedo esperar que.\n\nMartin conoc\u00eda bien esa cara. Siempre que se encontraba con su hermano, ve\u00eda en \u00e9l a su padre.\n\nSe saludaron con un abrazo, que result\u00f3 a\u00fan m\u00e1s inc\u00f3modo que otras veces porque Florian manten\u00eda una cartera apretada contra su cuerpo.\n\n\u00bfTomamos un taxi?\n\nNo. He reservado en el Belga Queen. Cinco minutos a pie.\n\nCaminaron en silencio. Finalmente, Martin pregunt\u00f3:\n\n\u00bfC\u00f3mo est\u00e1 Renate?\n\nBien.\n\n\u00bfY los ni\u00f1os?\n\nSon buenos estudiantes. \u00a1Gracias a Dios!\n\nNo es que Martin se avergonzara de su origen. Solamente no sab\u00eda si para \u00e9l era un problema que ese origen se le hubiera vuelto algo tan ajeno o que, por muy ajeno que ahora le resultara, le saliera una y otra vez al encuentro. Su padre hab\u00eda muerto dieciocho a\u00f1os atr\u00e1s, el 2 de noviembre, o sea, justo el D\u00eda de Difuntos. Muy prematuramente y de modo horriblemente tr\u00e1gico. Mientras Martin vivi\u00f3 en Austria, tuvo que revivir ese trauma cada 2 de noviembre. Cuando le\u00eda el peri\u00f3dico, cuando ve\u00eda la televisi\u00f3n o simplemente sal\u00eda de casa, ten\u00eda que recordarlo ya d\u00edas antes del 2 de noviembre: viene el D\u00eda de Difuntos. Y con ello: el aniversario de la muerte de su padre. Y estaba claro que ten\u00eda que viajar a su casa, para eso no hab\u00eda excusa porque era fiesta oficial, un general y morboso aniversario. En Bruselas, el 2 de noviembre no era d\u00eda festivo. All\u00ed, la historia propia y personal pod\u00eda o podr\u00eda desaparecer, pero cuando su hermano ven\u00eda, al instante era D\u00eda de Difuntos. Sin palabras. El padre se hab\u00eda metido en la m\u00e1quina. Era la frase repetida una y otra vez, que se hab\u00eda metido en la m\u00e1quina. Como si s\u00f3lo hubieran tenido una. Era la trituradora fina. Comoquiera que aquello ocurriese, se meti\u00f3 con el brazo en la maquinaria de triturar, la m\u00e1quina literalmente lo devor\u00f3 y \u00e9l se desangr\u00f3. Chill\u00f3 como un cerdo. \u00c9sa era la frase: chill\u00f3 como un cerdo. Despu\u00e9s hubo quien dijo que s\u00ed, que lo hab\u00eda o\u00eddo. Pero \u00bfpor qu\u00e9 no acudi\u00f3 nadie en su socorro? Porque era lo m\u00e1s natural, lo m\u00e1s normal, lo habitual en aquella granja: los alaridos de los cerdos. Con unos mil doscientos cerdos y matanzas diarias en aquella granja, ya no se sacan conclusiones de un solo chillido. Eso hab\u00eda dicho Felber, el maestro carnicero. \u00abSacar conclusiones\u00bb, dijo. Pero \u00bfc\u00f3mo se sabe entonces que chill\u00f3 como un cerdo? Tiene que haber gritado, dijeron todos. En eso estaban de acuerdo. Tiene que haber gritado de un modo incre\u00edble. Pero s\u00f3lo un momento. Porque se pierde muy pronto la conciencia. As\u00ed era. Ocurre muy deprisa. Por supuesto que los cerdos comprenden algo cuando... pero en un abrir y cerrar de ojos est\u00e1n anestesiados. Y ya los est\u00e1 devorando la m\u00e1quina. \u00a1Su padre era tan incansable! Quiso, en un corto intervalo, triturar unos desechos de animales que quedaban por el suelo. La empresa agr\u00edcola ya hab\u00eda crecido en aquel entonces incre\u00edblemente, pero a\u00fan no estaba tan superorganizada y con tan perfecta log\u00edstica. Su madre llam\u00f3 al m\u00e9dico, pero estaba, claro, completamente trastornada, y llam\u00f3 al doctor Schafzahl, el veterinario. Pero, en cualquier caso, ya todo llegaba tarde. Unos d\u00edas despu\u00e9s, Martin, que ten\u00eda diecis\u00e9is a\u00f1os, cont\u00f3 en el colegio riendo que su madre hab\u00eda llamado al doctor Schafzahl, y como nadie se re\u00eda repiti\u00f3: a Schafzahl, al \u00abovejero\u00bb, a una granja de cerdos. Luego estuvo mudo d\u00edas y d\u00edas y finalmente fue a confesarse para que el cura lo absolviera del pecado de haber hecho un chiste con la muerte de su padre.\n\nEl hermano cuatro a\u00f1os mayor tom\u00f3 entonces la granja a su cargo, el pr\u00edncipe heredero, de todos modos eso estaba concertado y proyectado desde siempre, s\u00f3lo que no tan pronto, y a \u00e9l, a Martin, el segund\u00f3n, el \u00abpayaso\u00bb, el desma\u00f1ado (\u00a1\u00abNo es de extra\u00f1ar, si siempre est\u00e1 leyendo!\u00bb), le permitieron estudiar, eso tambi\u00e9n estuvo claro siempre: \u00e9l pod\u00eda estudiar lo que quisiera, y \u00ablo que quisiera\u00bb significaba que a la familia le daba igual lo que hiciera mientras no viniera con exigencias y no fuera una carga. Arqueolog\u00eda.\n\nCuando los hermanos Susman entraron en el restaurante Belga Queen, Florian avanz\u00f3 despacio hasta el centro de la sala, ignor\u00f3 al camarero que se le interpuso en el camino y exclam\u00f3: \u00a1Eh! \u00bfEsto qu\u00e9 es? \u00bfUna catedral?\n\nMartin le dijo al camarero que ten\u00edan una reserva a nombre del doctor Susman, y a Florian: No, un antiguo banco. Bell\u00edsimo art d\u00e9co. Cenamos aqu\u00ed, en la antigua sala de las taquillas, y luego vamos al s\u00f3tano, a la c\u00e1mara acorazada, all\u00ed est\u00e1 ahora la sala de fumadores.\n\nCuando Florian tom\u00f3 completamente a su cargo la granja y su madre se jubil\u00f3, a Martin lo indemnizaron con una suma que hasta su mayor\u00eda de edad fue administrada fiduciariamente y que \u00e9l nunca cuestion\u00f3 y sobre la que nunca discuti\u00f3. Ese dinero le hab\u00eda permitido estudiar en la universidad c\u00f3modamente y pensar despu\u00e9s sin presi\u00f3n c\u00f3mo quer\u00eda ganarse la vida. Si se part\u00eda del valor de la empresa, sin duda la operaci\u00f3n no hab\u00eda sido justa, pero a Martin no le importaba, hab\u00eda bastado para abrir posibilidades y \u00e9l hab\u00eda sabido aprovecharlas. Ahora, sin embargo, hac\u00edan como si la familia hubiera permitido a Martin estudiar y le hubiera procurado ese estupendo trabajo en la Comisi\u00f3n Europea, para que \u00e9l, en esa posici\u00f3n, pudiera hacer lobbismo a favor de los intereses econ\u00f3micos de su hermano. \u00c9sa era la raz\u00f3n por la que Martin siempre ten\u00eda miedo cuando Florian se anunciaba y quer\u00eda verle en Bruselas. La granja, que ya en vida de su padre era muy considerable, Florian la hab\u00eda ampliado y convertido en la mayor empresa austriaca de producci\u00f3n de porcino, en una de las mayores de Europa; ya hac\u00eda mucho tiempo que no dec\u00eda, como su padre, \u00abgranja\u00bb, sino \u00abempresa\u00bb, y opinaba que no hab\u00eda nada m\u00e1s absurdo que la pol\u00edtica de la UE relativa a la producci\u00f3n y al comercio de cerdos. En su opini\u00f3n, aquello estaba en manos de ignorantes o de locos, sobornados o chantajeados o ideol\u00f3gicamente ofuscados por la mafia de los protectores de animales y por el lobby vegetariano. No ten\u00eda sentido alguno discut\u00edrselo, lo dec\u00eda en serio, pues \u00e9l ve\u00eda c\u00f3mo funcionaba aquello, \u00e9l conoc\u00eda la realidad pr\u00e1ctica. Hab\u00eda acumulado experiencia. Empez\u00f3 a comprometerse pol\u00edticamente, alcanz\u00f3 altos puestos en grupos de intereses y as\u00ed iba con cierta frecuencia a negociar a Bruselas. Hac\u00eda poco tiempo que lo hab\u00edan elegido presidente de The European Pig Producers, una red de los m\u00e1s destacados productores de cerdos del continente. En esa funci\u00f3n y al mismo tiempo como jefe de la corporaci\u00f3n de productores austriacos de cerdos hab\u00eda tenido aquel d\u00eda varias citas con diputados del Parlamento Europeo y con funcionarios de la Comisi\u00f3n Europea.\n\n\u00a1Oye, f\u00edjate!, dijo Florian cuando miraba la carta, gulasch de cerdo en cerveza de cerezas. Interesante. Si est\u00e1 bueno, les pido la receta. Y la pongo en la p\u00e1gina web.\n\nMartin pidi\u00f3 moules et frites. Y una botella de vino. Luego dijo: \u00bfQu\u00e9 tal tu jornada de hoy? Era una frase vac\u00eda y no intent\u00f3 en absoluto plantear la pregunta como si fuera de verdad interesante. Sab\u00eda que con ella desencadenaba una avalancha, pero de eso se trataba precisamente, y Martin quer\u00eda pasar el trago de una vez.\n\n\u00bfQue qu\u00e9 tal mi jornada? \u00bfT\u00fa qu\u00e9 crees? He tenido que tratar con idiotas. \u00a1As\u00ed ha sido mi d\u00eda! No entienden nada. \u00a1No son capaces de cambiar su pol\u00edtica, pero hoy me conminan a cambiar de nombre!\n\n\u00bfCambiar de nombre? \u00bfPor qu\u00e9 has de cambiar de nombre?\n\nYo no. Te lo explico enseguida. Primero tienes que saber lo siguiente: todo productor de cerdos quiere entrar, como es natural, en el mercado chino. China es el mayor importador de carne de cerdo a escala mundial. La demanda de China es enorme, ese pa\u00eds es el mercado en crecimiento.\n\nEso es bueno, \u00bfno?\n\nS\u00ed. Ser\u00eda bueno. Pero la UE es incapaz de sacar adelante un acuerdo comercial con China. Los chinos no negocian con la UE, sino con cada pa\u00eds por separado. Y cada Estado cree que puede ultimar un superconvenio bilateral para \u00e9l solo, sacar ventaja a los otros y obtener en solitario el mayor lucro, pero la realidad es que China se sirve de todos contra todos. Adem\u00e1s, ning\u00fan pa\u00eds puede asumir por s\u00ed solo las dimensiones que est\u00e1n all\u00ed en juego. Ni en a\u00f1os. Te doy un ejemplo: hace poco me llaman por tel\u00e9fono a la corporaci\u00f3n. \u00bfCu\u00e1ntas orejas de cerdo puede suministrar Austria?\n\n\u00bfOrejas de cerdo?\n\nS\u00ed, orejas de cerdo. Era uno del Ministerio de Comercio de China. Yo digo: En Austria sacrificamos cada a\u00f1o cinco millones de cerdos. O sea, diez millones de orejas. \u00c9l dice: Muy poco. Se despide muy cort\u00e9s y cuelga. \u00bfEntiendes? Si China necesita, digamos, cien millones de orejas de cerdos y si hubiera un convenio de la UE con China, entonces podr\u00edamos suministrar el 10 por ciento de esa cantidad. Pero \u00bfcu\u00e1l es la situaci\u00f3n? Austria a\u00fan no tiene un convenio bilateral con China, no se est\u00e1 tramitando ning\u00fan convenio com\u00fan de los Estados de la UE: y yo ya puedo tirar mis orejas de cerdo, en Austria son desechos de matanza. Y, sin embargo, en China las orejas de cerdo son una especialidad, hay una enorme demanda, y nosotros las tiramos o estamos encantados si alg\u00fan fabricante de comida para gatos viene a recogerlas gratuitamente.\n\nPero incluso si hubiera convenios, no se puede producir s\u00f3lo orejas de cerdo, para ello se necesita el cerdo entero. No se puede criar y alimentar tales cantidades de cerdos enteros s\u00f3lo por la demanda china de orejas de cerdo, porque \u00bfqu\u00e9 haces con el resto?\n\n\u00bfEres tonto o qu\u00e9? Entonces ya no hay resto. El resto lo tenemos ahora. Despojos de matanza. Las orejas de cerdo son s\u00f3lo un ejemplo. Los chinos no comen s\u00f3lo jam\u00f3n, solomillo, tocino, paletilla, eso por supuesto; sino que consumen tambi\u00e9n orejas, cabezas, rabos, se lo comen todo, se quedan con todo. Lo que aqu\u00ed es despojo de matanza ellos lo compran al precio del solomillo. Con otras palabras: un convenio sobre el comercio de cerdos con China significar\u00eda un incremento del 20 por ciento en la cifra de ventas, y a medio plazo, debido a la demanda, un crecimiento del cien por cien, o sea, que se duplicar\u00eda la producci\u00f3n europea de cerdos. Eso, compr\u00e9ndelo, eso ser\u00eda el mercado en crecimiento. No hay ning\u00fan ramo de la industria que tenga tales pron\u00f3sticos.\n\nComprendo, dijo Martin, y eso no fue bueno, aquel aburrido \u00abcomprendo\u00bb, esforzadamente ben\u00e9volo y de mal simulada cortes\u00eda. Su hermano le mir\u00f3 de una manera que le asust\u00f3. Dijo a toda prisa: No comprendo. Si existe esa posibilidad y hay tal demanda desde China, \u00bfpor qu\u00e9...?\n\nPorque tus colegas est\u00e1n locos. No tienen ni idea. En lugar de obligar a los Estados miembros a ceder la competencia a la Comisi\u00f3n a fin de hacer un convenio comercial de la UE con China, financiando al mismo tiempo mediante subvenciones el incremento de la producci\u00f3n de cerdos, ven c\u00f3mo China juega al divide y vencer\u00e1s y toman medidas para reducir la producci\u00f3n de cerdos en Europa. La Comisi\u00f3n opina que hay demasiados cerdos en Europa. Eso lleva a ca\u00edda de precios y etc. \u00bfQu\u00e9 hacen entonces? Menos subvenci\u00f3n. Incluso incentivos econ\u00f3micos para que las empresas cierren. S\u00ed, y ahora en Europa tenemos la siguiente situaci\u00f3n: un exceso de producci\u00f3n para el mercado interior que lleva a la ca\u00edda de los precios y, al mismo tiempo, un bloqueo frente a un mercado para el que producimos demasiado poco. Medidas que reducen a\u00fan m\u00e1s la producci\u00f3n, al mismo tiempo ninguna medida para que entremos en el mercado en el que podr\u00edamos vender el doble.\n\nEntretanto hab\u00eda llegado la comida.\n\n\u00bfQu\u00e9 tal est\u00e1 tu cerdo en cerveza de cerezas?\n\n\u00bfC\u00f3mo? Ah, s\u00ed. S\u00ed, no est\u00e1 mal. En cualquier caso: lo que ahora har\u00eda falta son inversiones en una dimensi\u00f3n que no puede permitirse ninguna empresa ella sola. Por tanto, fomento, no reducci\u00f3n. Promover, pol\u00edtica ofensiva de crecimiento. \u00bfComprendes? En lugar de eso, nos imponen condiciones. Protecci\u00f3n de animales. Se proh\u00edbe la cr\u00eda en jaulas, son obligatorias las cajas aireadas y con difusores. Sistema HDT...\n\nNo quiero preguntar lo que es eso.\n\nEs caro. Consume la ganancia. Mira, te voy a ense\u00f1ar una cosa. Abri\u00f3 la cartera que hab\u00eda tra\u00eddo, busc\u00f3 entre las hojas, sac\u00f3 una.\n\nMira: estad\u00edstica del precio del cerdo en la UE en el \u00faltimo semestre. 15 de julio: ca\u00edda de los precios en Europa; baj\u00f3n del 18 por ciento. 22 de julio: se ha tocado fondo. \u00a1Pues nada de eso! 19 de agosto: poco movimiento en los mercados. 9 de septiembre: el precio del cerdo cae un 21 por ciento. 16 de septiembre: gran retroceso de las cotizaciones. 21 de octubre: el precio del cerdo cae un 14 por ciento... \u00bfSigo leyendo?\n\nNo.\n\nRetroceso, ca\u00edda de precios, tocar fondo, nueva ca\u00edda de precios. Y de la UE no viene ninguna reacci\u00f3n. Desde comienzo de a\u00f1o... \u00a1Mira! Aqu\u00ed est\u00e1. Desde que empez\u00f3 el a\u00f1o, 48 criadores de porcino cierran cada d\u00eda la puerta de su granja para siempre. Y miles de ellos que han tratado de mantenerse a flote est\u00e1n metidos en un procedimiento judicial por dilaci\u00f3n de la quiebra. Y, sin embargo, podr\u00edamos producir el doble pagando el comprador un precio 20 por ciento m\u00e1s alto por el cerdo entero, s\u00f3lo habr\u00eda que invertir una vez en la infraestructura de modo coordinado y hablar con China. Pero expl\u00edcale esto al se\u00f1or Frigge. Me dice que la UE tiene por desgracia una agenda distinta en la producci\u00f3n de cerdos. Al mismo tiempo proh\u00edben que haya subvenciones a los Estados miembros, pues eso ser\u00eda alteraci\u00f3n de la competencia. \u00bfConoces t\u00fa a ese Frigge?\n\nNo.\n\nNo es posible. Es compa\u00f1ero tuyo. No entiendo lo que se trae entre manos. Esc\u00fachame: tienes que hablar con \u00e9l unas palabritas, debes hacerle ver claramente, as\u00ed, entre vosotros, que...\n\n\u00a1Florian! \u00a1La Comisi\u00f3n no funciona como la Asociaci\u00f3n de Agricultores Austriacos!\n\n\u00a1No me vengas con \u00e9sas! \u00bfPara qu\u00e9 te tenemos ah\u00ed, si no?\n\nAhora hay una cosa que no comprendo: antes has dicho no s\u00e9 qu\u00e9 de cambio de nombre. \u00bfQu\u00e9 quer\u00eda el se\u00f1or Frigge, qu\u00e9 nombre has de cambiar?\n\nNo, no era Frigge. Eran los se\u00f1ores diputados del Parlamento. All\u00ed no hab\u00eda ninguna se\u00f1ora. Si no, tal vez habr\u00eda podido yo flirtear un poco, pero eran s\u00f3lo hombres y duros como una piedra en su estupidez. Eran del grupo parlamentario del Partido Popular Europeo. \u00bfComprendes?\n\nNo.\n\nPartido Popular Europeo. Yo esperaba que, para m\u00ed, eso fuera pan comido, puesto que soy del Partido Popular Austriaco. Aqu\u00ed en el Parlamento Europeo el grupo se llama EPP, European People's Party.\n\n\u00bfY entonces?\n\nBueno, yo vengo aqu\u00ed como presidente de la European Pig Producers, es decir, tambi\u00e9n EPP, \u00bfcomprendes? Yo ten\u00eda la misi\u00f3n de negociar sobre dos puntos: subvenciones para el incremento de la producci\u00f3n de porcino y coordinaci\u00f3n de la exportaci\u00f3n europea de porcino. No hablamos de ello ni un minuto. Los diputados dijeron que ante todo ten\u00edamos que cambiar nuestro nombre y nuestro logotipo. No era de recibo que en Google, cuando se busca el Partido Popular Europeo, EPP, aparezcan al momento s\u00f3lo cerdos. \u00a1No te r\u00edas! Les dije que eso era dif\u00edcil. Somos una organizaci\u00f3n transnacional, registrada por la polic\u00eda de sociedades en cada Estado miembro. Ha sido un gran esfuerzo. \u00bfSabes lo que ellos propusieron? Como nos llamamos The European Pig Producers, que pongamos tambi\u00e9n el The en la abreviatura, y entonces nos llamar\u00edamos TEPP. Inconcebible, qu\u00e9 c\u00ednicos.\n\nPero \u00bfno hab\u00e9is discutido en alem\u00e1n?\n\nNo, ninguno era alem\u00e1n.\n\nEntonces no era cinismo. Ellos no saben lo que significa TEPP.[1]\n\nFlorian reba\u00f1\u00f3 con pan el \u00faltimo resto de la salsa del gulasch, as\u00ed lo hac\u00eda ya desde ni\u00f1o. El plato de Florian no hay que fregarlo despu\u00e9s de la comida, dec\u00eda siempre su madre.\n\nUn poco dulce esta salsa de cerveza de cerezas. \u00bfHas dicho que se puede fumar ah\u00ed en la c\u00e1mara acorazada? Ens\u00e9\u00f1amela. Necesito con urgencia un cigarrillo.\n\nMarcharon a casa como hermanos, cogidos del brazo, tambale\u00e1ndose y vacilando por el pavimento de Bruselas. Hab\u00edan bebido tambi\u00e9n gin tonics, seducidos por la oferta, hab\u00edan fumado puros. El efecto fue perceptible cuando se levantaron de las butacas y m\u00e1s a\u00fan cuando salieron al aire libre. Despu\u00e9s de dejar Martin a su hermano en el hotel, empez\u00f3 de nuevo a llover y \u00e9l se dio cuenta de que hab\u00eda olvidado el paraguas en el Belga Queen. Lleg\u00f3 empapado a su casa, se quit\u00f3 la chaqueta y el pantal\u00f3n, abri\u00f3 la nevera, vacil\u00f3 un momento, despu\u00e9s sac\u00f3 una Jupiler y se sent\u00f3 junto a la chimenea. Su hermano le hab\u00eda dado una revista (\u00abMira lo que te he tra\u00eddo: salgo en la portada.\u00bb), que ahora Martin no le\u00eda, s\u00f3lo miraba: \u00abTHINK PIG! El bolet\u00edn de informaci\u00f3n de la EPP\u00bb.\n\n# CAP\u00cdTULO TERCERO\n\nEN EL FONDO, LA MUERTE ES S\u00d3LO EL PRINCIPIO DE LAS SECUELAS.\n\n#\n\nDurante el trayecto de la Gare Centrale a la Comisar\u00eda General, situada en la rue March\u00e9 au Charbon, \u00c9mile Brunfaut se detuvo en repetidas ocasiones, miraba alrededor, paseaba la vista por las fachadas de las casas, observaba a las personas que ten\u00edan una tarea o una meta y que, por as\u00ed decirlo, pon\u00edan en marcha la ciudad. Le gustaba Bruselas a primera hora de la ma\u00f1ana, cuando la ciudad se despertaba. Respir\u00f3 hondo varias veces, suspir\u00f3, pero percibi\u00f3 cierta congoja, no eran suspiros de felicidad. Cuando atraves\u00f3 la Grand Place se detuvo de nuevo, mir\u00f3: \u00a1qu\u00e9 esplendor! En efecto, aquella plaza mostraba su belleza s\u00f3lo a esa hora temprana, antes de que la ocuparan las masas de turistas. Odiaba a los turistas, esos cazadores que s\u00f3lo buscaban la confirmaci\u00f3n de los clich\u00e9s que llevaban en la cabeza, gente que hab\u00eda sustituido sus ojos por tabletas y m\u00e1quinas de fotos, que obstru\u00edan el camino y convert\u00edan la ciudad llena de vida en un museo y a las personas que trabajaban all\u00ed en comparsas de la imagen de la ciudad, en empleados de museos y lacayos. Bruselas era ya una ciudad pol\u00edglota y multicultural antes de que esas masas, que all\u00ed sobraban, llegaran de todos los confines del mundo. Respir\u00f3 hondo, apret\u00f3 la cartera contra el vientre y trat\u00f3 de estirar el t\u00f3rax lo m\u00e1s posible. Miraba fijamente. Como un turista. \u00a1Qu\u00e9 bonito! \u00a1Qu\u00e9 hermosa era aquella plaza! No estaba feliz, sent\u00eda una nostalgia alarmante, una sensaci\u00f3n de tristeza. En el a\u00f1o 1914, hab\u00eda contado su abuelo, Bruselas era la ciudad m\u00e1s bella y rica del mundo; luego llegaron tres veces, dos veces con botas y fusiles, luego con zapatillas deportivas y m\u00e1quinas de fotos. Nos metieron en una prisi\u00f3n y nos pusieron en libertad como sirvientes. \u00c9mile Brunfaut nunca sinti\u00f3 cari\u00f1o por su abuelo, lo hab\u00eda respetado, eso s\u00ed, y en el fondo tambi\u00e9n admirado, pero durante su vida no pudo sentir cari\u00f1o por aquel hombre viejo y amargado. Ahora \u00e9l tambi\u00e9n envejec\u00eda. Muy prematuramente. Le gustaba Bruselas a primera hora de la ma\u00f1ana: nunca hab\u00eda tenido antes un pensamiento as\u00ed. Cruzaba simplemente esa plaza para ir al trabajo. Ahora ve\u00eda Bruselas..., s\u00ed, como si fuera a despedirse. \u00bfPor qu\u00e9? No ten\u00eda intenci\u00f3n... Sigui\u00f3 andando, con prisa, quer\u00eda tomar su caf\u00e9 antes del briefing de las ocho y prepararse. No sab\u00eda que aquello exist\u00eda realmente: los presentimientos. Era comisario. Poco dado a presentimientos, conjeturas, ensue\u00f1os. Su abuelo siempre dec\u00eda: So\u00f1ar con una cerveza no quita la sed. Y el comisario pensaba igual y tampoco habr\u00eda sido distinto si hubiera optado por otra profesi\u00f3n.\n\nIba a ser, en efecto, el d\u00eda en el que tendr\u00eda que despedirse. Pens\u00f3 que era el vientre. Su vientre grande y flatulento le oprim\u00eda el pulm\u00f3n y lo comprim\u00eda, as\u00ed lo percib\u00eda \u00e9l, y opinaba que a eso se deb\u00edan sus ahogos, que siempre sonaban como un suspiro.\n\nEra un g\u00e9lido d\u00eda de enero, bajo un cielo bajo y ceniciento. La tierra que hoy abrir\u00eda un enterrador estaba tan dura como el pavimento de aquella suntuosa plaza.\n\nEn el briefing de las ocho se vio obligado a informar de que no ten\u00edan nada sobre el caso \u00abasesinato en el Atlas\u00bb, absolutamente nada. Se pasaba continuamente la mano por el vientre, antes hab\u00eda tomado con el caf\u00e9 un croissant cuyas grasientas migajas se le hab\u00edan quedado pegadas en la mano; hablaba y se limpiaba, segu\u00eda hablando y segu\u00eda limpi\u00e1ndose, parec\u00eda un tic nervioso. Hab\u00eda un cad\u00e1ver, hombre, de identidad desconocida. Se hab\u00eda registrado en el hotel con nombre falso, supuestamente era un h\u00fangaro de Budapest, pero su pasaporte era falso. La recepcionista hab\u00eda declarado que hablaba ingl\u00e9s con un fuerte acento, pero que no pod\u00eda decir si ese acento era h\u00fangaro. Los del laboratorio hab\u00edan trabajado r\u00e1pidamente y a conciencia, pero ni la dactiloscopia ni la odontolog\u00eda y serolog\u00eda forense aportaron indicio alguno, no exist\u00edan correspondencias en el banco de datos de la Police F\u00e9d\u00e9rale. Igual de infructuoso hab\u00eda sido el an\u00e1lisis bal\u00edstico de la bala mortal. A lo mejor llegaba algo de la Europol. El informe de la autopsia confirmaba solamente la evidencia: era una ejecuci\u00f3n, un tiro en la nuca desde muy cerca. Aparentemente, el asesino no hab\u00eda buscado ni robado nada en la habitaci\u00f3n. Los objetos personales de la v\u00edctima no aportaban ning\u00fan indicio sobre su verdadera identidad ni, menos a\u00fan, sobre un posible m\u00f3vil. No hubo nada que llamara la atenci\u00f3n, salvo un cerdo. S\u00ed, un cerdo. Varias personas interrogadas que estuvieron a la hora aproximada del crimen en las inmediaciones del hotel Atlas y algunos vecinos de la zona declararon que les hab\u00eda llamado la atenci\u00f3n un cerdo que corr\u00eda suelto por delante del hotel. Completamente misterioso, dijo el comisario Brunfaut; despu\u00e9s de todas las indagaciones y los interrogatorios que hemos hecho hasta ahora sobre este caso, tenemos un \u00fanico punto de partida concreto: un cerdo..., y ni siquiera sabemos si ese cerdo tiene algo que ver con el caso. Se pas\u00f3 las manos de nuevo por el pecho, las puso despu\u00e9s sobre el vientre, lo oprimi\u00f3 y respir\u00f3 hondo. \u00a1Se\u00f1ores!\n\nNinguno de los oficiales dijo nada. \u00c9mile Brunfaut no pens\u00f3 que tal vez le ocultaran algo que \u00e9l no sab\u00eda a\u00fan, o que no mencionaran una idea que no se le hab\u00eda ocurrido a \u00e9l; se levant\u00f3 y pidi\u00f3 a su equipo de colaboradores que pasaran a la peque\u00f1a sala de reuniones.\n\nEn cuanto al estado de las cosas no podemos hacer nada, s\u00f3lo lo siguiente, dijo. Primero, esperar por si recibimos de la Europol una respuesta a los datos que hemos enviado. Segundo, el cerdo. No conocemos la identidad de la v\u00edctima, pero quiz\u00e1 podamos averiguar la identidad del cerdo. Compuso una risa forzada. Un cerdo as\u00ed no viene a Bruselas como turista, en avi\u00f3n, y se va despu\u00e9s al centro a pasear. Tiene que haber un due\u00f1o al que se le haya escapado o que lo haya abandonado. As\u00ed que interrogaremos a todos los granjeros de la regi\u00f3n de Bruselas dedicados a la cr\u00eda de cerdos. Y, sobre todo, tercero: quiero saber qui\u00e9n era el hombre que estaba en la ventana de la casa en derribo. Es posible que haya visto algo. Quiz\u00e1 era el due\u00f1o del piso, o el due\u00f1o del inmueble. Eso se puede averiguar enseguida. Quiero saberlo a las 13 horas, cuando est\u00e9 de regreso. Ahora he de ir al cementerio.\n\nS\u00f3lo los cementerios siguen teniendo educaci\u00f3n.\n\nHac\u00eda demasiado calor en la habitaci\u00f3n y David de Vriend se fue derecho a la ventana para abrirla. Comprob\u00f3 que s\u00f3lo era posible abatirla un poco, la abertura era tan peque\u00f1a que ni siquiera se pod\u00eda sacar por ella una mano. Mir\u00f3 abajo, a las l\u00e1pidas que se alzaban marcialmente, en fila india, bajo el cielo profundo y gris, y pregunt\u00f3 si se pod\u00eda cambiar, o mejor dicho, quitar ese bloqueo de las ventanas.\n\nLa se\u00f1ora Jos\u00e9phine explic\u00f3 que no se la pod\u00eda llamar enfermera, pues aquello no era un hospital, sino una residencia de mayores, \u00bfno es cierto, se\u00f1or De Vriend?\n\nHablaba muy alto, casi a gritos, esto se hab\u00eda convertido en su segunda naturaleza tras tantos a\u00f1os tratando con personas mayores, en su mayor\u00eda medio sordas. David de Vriend cerr\u00f3 los ojos como si as\u00ed pudiera tambi\u00e9n cerrar los o\u00eddos. La ventana... \u00ab... por su propia seguridad...\u00bb, la oy\u00f3 gritar o vociferar, \u00e9l s\u00f3lo quer\u00eda que esa mujer se marchara. No aguantaba ese tono de ordeno y mando, tampoco su forzada amabilidad, su boca perpetuamente dilatada en una sonrisa. Sab\u00eda que estaba siendo injusto, pero si hubiera justicia en la vida, entonces se habr\u00eda librado de esto. Ahora la mujer estaba a su lado, grit\u00e1ndole al o\u00eddo: qu\u00e9 estupendo todo ese verde delante de la ventana, \u00bfverdad? \u00c9l se dio media vuelta, se quit\u00f3 la chaqueta. La tir\u00f3 sobre la cama. Ella y su equipo estaban siempre a su disposici\u00f3n, dijo ella. Si necesitaba ayuda o le surg\u00eda alg\u00fan problema, s\u00f3lo ten\u00eda que llamar, all\u00ed por el tel\u00e9fono interior, o all\u00ed, junto a la cama, con el timbre, \u00bfsabe, se\u00f1or De Vriend? Mir\u00f3 alrededor entusiasmada, como si aquel diminuto apartamento fuera una suite de lujo, abri\u00f3 los brazos y exclam\u00f3: \u00a1\u00c9ste es, pues, desde ahora su peque\u00f1o reino! \u00a1Aqu\u00ed se sentir\u00e1 a gusto!\n\nEra una orden. Estupefacto vio que le tend\u00eda la mano. \u00c9l tard\u00f3 en reaccionar. Ella quer\u00eda ya retirar la mano cuando \u00e9l le tendi\u00f3 por fin la suya. Hubo un breve titubeo hasta que por fin consumaron el apret\u00f3n de manos. As\u00ed que le deseo lo mejor; vio entonces su n\u00famero tatuado en el antebrazo, \u00bfsabe?, dijo con voz queda. Luego se march\u00f3. Y De Vriend ech\u00f3 una ojeada a su peque\u00f1o reino y se pregunt\u00f3 por qu\u00e9 no le hab\u00eda llamado la atenci\u00f3n aquello cuando visit\u00f3 diversas residencias de mayores y se decidi\u00f3 por \u00e9sta: en aquella habitaci\u00f3n todo estaba fijado y atornillado. No hab\u00eda un solo mueble que se pudiera mover y cambiar de sitio. No s\u00f3lo la cama con la mesilla de noche, sino tambi\u00e9n el armario, mitad armario ropero con puertas lacadas en blanco, mitad vitrina con puertas de cristal; tambi\u00e9n eran fijos la peque\u00f1a mesa y el banco en forma de ele que la rodeaba, el televisor estaba atornillado a la pared, incluso el cuadro de encima de la cama \u2014\u200cVenecia bajo la lluvia, en estilo pseudoimpresionista\u2014 estaba colocado de tal manera que no era posible quitarlo. \u00bfPor qu\u00e9 Venecia? \u00bfY por qu\u00e9 bajo la lluvia? \u00bfHab\u00eda que consolar a los bruselenses que llegaban a la vejez mostr\u00e1ndoles que en los lugares m\u00e1s bellos del mundo tambi\u00e9n llov\u00eda? Un peque\u00f1o frente de cocina encastrado. No hab\u00eda nada que se pudiera mover, cambiar, poner de otra manera. Ni siquiera una silla. Todo era inalterable y definitivo. Fue al armario, detr\u00e1s de un cristal estaban los pocos libros que hab\u00eda tra\u00eddo, inmovilizados entre dos sujetalibros de cer\u00e1mica que representaban unos cerdos leyendo. Un regalo de su \u00faltima clase de bachilleres antes de la jubilaci\u00f3n. Iba a sacar libros, ponerlos aqu\u00ed y all\u00e1, sobre la mesa, sobre la cama, ser\u00edan lo \u00fanico que se mover\u00eda en aquella habitaci\u00f3n. Abri\u00f3 la puerta del armario, pase\u00f3 la mirada por los lomos de los libros, otra vez; dud\u00f3... \u00bfQu\u00e9 quer\u00eda? \u00bfLeer? \u00bfHab\u00eda querido leer? No. Estaba all\u00ed de pie mirando los lomos de los libros, cerr\u00f3 el armario. Quer\u00eda... \u00bfQu\u00e9? Salir. Quer\u00eda salir. Se acerc\u00f3 a la ventana. El cementerio municipal de Bruselas. No ten\u00eda nada al alcance de la mano. Pero s\u00ed al alcance de la vista. Se puso ropa de abrigo.\n\nHab\u00eda s\u00f3lo unos pasos de la residencia de mayores Maison Hanssens, en la rue de l'Arbre Unique, a la puerta principal del cementerio. El fr\u00edo g\u00e9lido. El cielo gris. La puerta de hierro forjado. Le tranquiliz\u00f3 ver p\u00e1jaros, cornejas y gorriones. Y tantas toperas entre las tumbas, no recordaba haber visto jam\u00e1s tantas toperas en un cementerio. Y por todas partes crec\u00edan all\u00ed setas entre la hiedra com\u00fan, un sinn\u00famero de setas, eran... eran... no le ven\u00eda el nombre. Lo sab\u00eda, daba igual, no eran comestibles. Eso era todo. Hab\u00eda all\u00ed una tumba literalmente vuelta del rev\u00e9s, volteada por las gruesas ra\u00edces de un \u00e1rbol inmenso. Al lado, l\u00e1pidas despedazadas por \u00e1rboles desplomados o ramas ca\u00eddas encima. Musgo sobre las piedras destrozadas. \u00c1rboles nuevos, reci\u00e9n plantados, junto a otros viejos, ca\u00eddos o talados, que se pudr\u00edan junto a las tumbas. En aquel campo de la muerte los \u00e1rboles tambi\u00e9n mor\u00edan y se hund\u00edan en la tierra. Sobre las l\u00e1pidas antiguas hab\u00eda colgadas peque\u00f1as coronas de yeso. A veces dos o tres, algunas de ellas tambi\u00e9n estaban en el suelo, delante de las l\u00e1pidas o junto a la tumbas. Como si ni\u00f1os morbosos hubieran jugado a tirarse aros.\n\nSe deten\u00eda delante de alguna tumba, le\u00eda los nombres, observaba las fotos esmaltadas. Le gustaba ir a los cementerios, le parec\u00eda hermoso que la gente tuviera tumbas en las que pon\u00eda su nombre. Gente que hab\u00eda muerto y a la que despu\u00e9s se pod\u00eda visitar. Vio tumbas de ni\u00f1os, de personas que hab\u00edan muerto muy j\u00f3venes, de enfermedades, por accidentes o asesinadas, tr\u00e1gicos destinos, pero ten\u00edan una tumba. Mientras hubiera cementerios hab\u00eda una promesa de civilizaci\u00f3n. Sus propios padres, su hermano, sus abuelos ten\u00edan tumbas en el aire. Ning\u00fan lugar que se pudiera visitar, cuidar, un lugar en el que se pudiera colocar una l\u00e1pida. Ning\u00fan lugar de descanso. S\u00f3lo un perpetuo desasosiego que no pod\u00eda encontrar un lugar de paz. En el recuerdo que morir\u00eda con \u00e9l s\u00f3lo hab\u00eda una \u00faltima imagen de su familia, captada con la \u00faltima mirada: y \u00e9sta era una simple afirmaci\u00f3n interior. No hab\u00eda visto el rostro de su madre sino s\u00f3lo su mano, que se agarraba a su brazo hasta que \u00e9l se solt\u00f3; no ten\u00eda una \u00faltima imagen de su padre sino s\u00f3lo el recuerdo de su grito: \u00ab\u00a1Qu\u00e9date! \u00a1Nos vas a traer la desgracia!\u00bb, y su hermano peque\u00f1o: sin rostro, s\u00f3lo la espalda de un ni\u00f1o que se apretaba contra su madre. \u00bfY qu\u00e9 m\u00e1s? Recuerdos que parec\u00edan robados del repertorio de recuerdos de otros: recuerdos padre-madre-hijo, recuerdos como los de todo el mundo, los m\u00e1s felices. Tan negros como la ceniza de fotos quemadas.\n\nA su padre le gustaba la tarte au riz. \u00c9se era un recuerdo. Y al mismo tiempo no lo era. No ten\u00eda una imagen de ello. C\u00f3mo la familia estaba sentada en torno a la mesa y el padre dec\u00eda, el rostro brill\u00e1ndole de satisfacci\u00f3n: \u00ab\u00a1Mmmm, hoy tenemos por fin otra vez tarte au riz!\u00bb, y c\u00f3mo la madre pon\u00eda la tarta sobre la mesa y el padre ped\u00eda moderaci\u00f3n a los ni\u00f1os: \u00ab\u00a1Esperad! \u00a1No os lanc\u00e9is de esa manera!\u00bb, y su madre dec\u00eda: \u00abPrimero un buen trozo para vuestro padre\u00bb y... \u00a1Todo mentira! No hab\u00eda imagen del recuerdo, no hab\u00eda pel\u00edcula del recuerdo, no se ve\u00eda sentado a la mesa con la familia, con la tarte au riz, s\u00f3lo esa frase: \u00abA pap\u00e1 le gustaba la tarte au riz\u00bb. Pero \u00bfpor qu\u00e9? \u00bfPor qu\u00e9 aquella frase? \u00bfY de d\u00f3nde ven\u00eda la frase? \u00bfPrecisamente esa frase? \u00bfComo recuerdo de una vida? Al mismo tiempo una frase muerta, enterrada en su cabeza. Vio entonces una l\u00e1pida en la que estaba grabado:\n\nTOUT PASSE\n\nTOUT S'EFFACE\n\nHORS DU SOUVENIR\n\nSe detuvo, contempl\u00f3 largo rato aquella inscripci\u00f3n, se inclin\u00f3, cogi\u00f3 un guijarro y lo puso sobre esa tumba.\n\nTantas tumbas destruidas. El vandalismo de la naturaleza. L\u00e1pidas levantadas por ra\u00edces de \u00e1rboles, panteones destrozados por ramas quebradas y \u00e1rboles ca\u00eddos, losas tragadas por plantas par\u00e1sitas. Los monumentos desintegrados de la competici\u00f3n humana, del ansia de representaci\u00f3n; mausoleos ruinosos, mohosos, que hab\u00edan de dar monumental testimonio del poder y la riqueza de una familia, ahora se desmoronaban y s\u00f3lo mostraban eso: caducidad. Delante, letreros puestos por la administraci\u00f3n del cementerio: el arriendo de este emplazamiento de tumba ha terminado a final del a\u00f1o.\n\nSin dinero mueren incluso las tumbas.\n\nEstaba cansado, reflexion\u00f3 un momento sobre si no ser\u00eda mejor volver. No, quer\u00eda acabar de inspeccionar el entorno en el que ahora viv\u00eda.\n\nTorci\u00f3 a la izquierda, sin prestar atenci\u00f3n a los indicadores, CEMENTERIO MILITAR ALEM\u00c1N, COMMONWEALTH WAR GRAVES, NEDERLANDSE OORLOGSGRAVEN, y ya empezaban las filas, correctamente alineadas, de tumbas siempre iguales, que en su infinita uniformidad, despu\u00e9s del caos vivo y casi vociferante de la parte civil del cementerio, transmit\u00edan dram\u00e1tica quietud y belleza, la perfecta liberaci\u00f3n del robo de la vida en una est\u00e9tica de dignidad.\n\nA la edad de 24 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 20 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 26 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 19 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 23 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 23 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 22 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 31 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 24 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 39 a\u00f1os \u2013 muerto por la patria.\n\nA la edad de 21 a\u00f1os \u2013 muerto por la patria.\n\nMort pour la patrie, for the glory of the nation, slachtoffers van den plicht.\n\nQuien hac\u00eda ese recorrido pasaba revista a las filas como un general a un ej\u00e9rcito de muertos, como un presidente a la formaci\u00f3n militar en una recepci\u00f3n oficial en el Hades. Cerr\u00f3 los ojos. Y justo en ese momento alguien le dirigi\u00f3 la palabra. Un se\u00f1or que le preguntaba si hablaba alem\u00e1n o ingl\u00e9s.\n\nUn poco de alem\u00e1n.\n\nSi sab\u00eda d\u00f3nde estaba el mausoleo del amor incondicional.\n\n\u00bfC\u00f3mo dice?\n\nEl hombre dijo que en la gu\u00eda de viajes hab\u00eda le\u00eddo eso, \u00bfle entend\u00eda? \u00bfS\u00ed? Bueno. As\u00ed que en la gu\u00eda. Ten\u00eda que estar all\u00ed, en alguna parte. Mausoleo del amor incondicional. \u00bfNo lo sabe usted?\n\nNo lo sab\u00eda, dijo De Vriend.\n\nEl profesor Erhart dio las gracias y sigui\u00f3 caminando. Al final de la avenida vio un edificio delante del cual hab\u00eda algunas personas; a lo mejor le informaban all\u00ed. A\u00fan ten\u00eda algo de tiempo. La mayor\u00eda de los participantes del reflection group New Pact for Europe llegar\u00edan en el transcurso de la ma\u00f1ana, por lo que la primera reuni\u00f3n de ese d\u00eda estaba fijada para las 13.00 horas. Pero \u00e9l hab\u00eda llegado dos d\u00edas antes porque, ya que le hab\u00edan invitado a ir a Bruselas, quer\u00eda ver algo de la ciudad y no estar metido todo el tiempo en un espacio cerrado y climatizado. En Viena no ten\u00eda ni obligaciones ni familia. En ese aspecto se hallaba en la situaci\u00f3n m\u00e1s horrible en la que se pod\u00eda estar a su edad: era libre. Gracias a su excelente prestigio acad\u00e9mico, a\u00fan recib\u00eda de vez en cuando invitaciones como \u00e9sta; las aceptaba siempre y se preparaba minuciosamente, aunque \u2014\u200co quiz\u00e1 porque\u2014 ten\u00eda la creciente sensaci\u00f3n de que ya no daba conferencias, sino de que, con sus papers, hac\u00eda una lectura p\u00fablica de su testamento. Pero entonces, que as\u00ed fuera: har\u00eda saber a los herederos que, m\u00e1s all\u00e1 del esp\u00edritu del siglo, hab\u00eda un legado que ellos estaban llamados a aceptar.\n\nAquel d\u00eda Alois Erhart hab\u00eda ido primero a visitar la tumba de Armand Moens, el economista, anta\u00f1o en boca de todos y hoy olvidado, que fue catedr\u00e1tico de la Universidad de Lovaina y que desarroll\u00f3 en los a\u00f1os sesenta del siglo pasado una teor\u00eda de la econom\u00eda pol\u00edtica posnacional, de la que dedujo la necesidad de fundar una Rep\u00fablica Unida Europea. La creciente interconexi\u00f3n de las econom\u00edas nacionales, las rec\u00edprocas dependencias resultantes de ello, el poder cada vez mayor de las empresas multinacionales y la importancia creciente de los mercados financieros internacionales no permitir\u00e1n ya a las democracias nacionales cumplir su misi\u00f3n de configurar las condiciones bajo las cuales han de vivir las personas y de garantizar la tendencia a una equidad distributiva. \u00ab\u00a1Cerrad los parlamentos nacionales!\u00bb: \u00e9se era el grito de guerra de un aut\u00e9ntico dem\u00f3crata que quer\u00eda reinventar la democracia teniendo en cuenta la situaci\u00f3n hist\u00f3rica. Que su tesis sobre la necesaria desaparici\u00f3n de las democracias nacionales no fuera tildada de escandalosa o de absurda utop\u00eda se debi\u00f3 a la libertad bufonesca de aquellos tiempos, y si finalmente Moens no pudo imponerse frente a los economistas pol\u00edticos nacionales, los \u00abrumiantes\u00bb (como \u00e9l los llamaba), se debi\u00f3 a la misma raz\u00f3n: \u00abLa libertad bufonesca en el \u00e1mbito acad\u00e9mico nos ayud\u00f3 al principio, pero al final consolid\u00f3 el poder de los verdaderos bufones\u00bb, escrib\u00eda en sus memorias.\n\nCuarenta y cinco a\u00f1os atr\u00e1s, siendo un joven estudiante, Erhart hab\u00eda asistido en Alpbach a una conferencia de Armand Moens, profesor invitado, y desde entonces se consider\u00f3 su disc\u00edpulo. Hab\u00eda le\u00eddo con fidelidad todas sus publicaciones. Cuando \u00e9l mismo public\u00f3 un trabajo por primera vez y lo envi\u00f3 a su maestro, \u00e9ste ya estaba enfermo de muerte. Moens a\u00fan respondi\u00f3 con una carta, pero ya no hubo m\u00e1s intercambios porque Moens muri\u00f3 pocos d\u00edas despu\u00e9s. Ahora, Erhart estaba, emocionado, ante la tumba:\n\nArmand Joseph Moens 1910-1972\n\nA un lado delante de la tumba hab\u00eda una peque\u00f1a placa esmaltada en la que pon\u00eda:\n\nTOEN HIJ HET MEEST NODIG WAS,\n\nWERD HIJ VERGETEN.\n\nSTUDENTEN WERKGROEP MOENS EED\n\nAAN DE KATHOLIEKE UNIVERSITEIT LEUVEN\n\n\u00abCuando m\u00e1s lo necesitaban, lo olvidaron.\u00bb Grupo estudiantil de trabajo Juramento de Moens de la Universidad Cat\u00f3lica de Lovaina.\n\nSobre la tumba hab\u00eda flores recientes y una botella de aguardiente. Y cerditos de la suerte. En diversos tama\u00f1os y de diversos materiales: pl\u00e1stico, peluche, madera, cer\u00e1mica; Alois Erhart no entend\u00eda lo de los cerdos. Hizo una foto. Luego otra, ahora de la l\u00e1pida y de la placa, sin los cerditos.\n\nCuando investigaba d\u00f3nde se encontraba la tumba del profesor Moens tropez\u00f3 con el dato de que el cementerio municipal de Bruselas tambi\u00e9n contaba con esta atracci\u00f3n tur\u00edstica: el mausoleo del amor incondicional. Es lo que ahora buscaba. Un bar\u00f3n bruselense, Alois Erhart hab\u00eda olvidado el nombre, que hab\u00eda hecho una fortuna con participaciones en minas del Congo belga, se hab\u00eda enamorado perdidamente en un viaje a la colonia de una mujer a la que se llev\u00f3 con \u00e9l a Bruselas para casarse con ella. \u00ab\u00a1Una negra!\u00bb Eso le acarre\u00f3 no s\u00f3lo la reprobaci\u00f3n de la alta sociedad de Bruselas, sino sobre todo varios problemas jur\u00eddicos que, tras larga lucha, finalmente super\u00f3, en parte con ayuda de los mejores abogados, en parte mediante el pago de sumas considerables. El amor del bar\u00f3n resisti\u00f3 todos los embates. \u00abPrefiero la reprobaci\u00f3n con esta mujer que la aprobaci\u00f3n sin ella.\u00bb A la boda, finalmente permitida, no acudi\u00f3 ninguno de los invitados, excepto la vieja y extravagante condesa Adolphine Marat, que despu\u00e9s de la ceremonia ofreci\u00f3 un t\u00e9 en su palacio. Los testigos de la boda fueron dos obreros que estaban reparando la tapa de un canal delante del registro civil y que, por cincuenta francos, estuvieron dispuestos a interrumpir su trabajo. La condesa de Marat, censurada por haber ofrecido una recepci\u00f3n en honor de aquella pareja, se justific\u00f3 con las legendarias palabras: \u00ab\u00a1Si \u00e9l est\u00e1 dispuesto a dar su nombre a esa mujer, yo podr\u00e9 darle un t\u00e9!\u00bb.\n\nEsa mujer, que se llamaba Libelulle (el profesor Erhart recordaba muy bien el nombre), muri\u00f3 poco despu\u00e9s, en el a\u00f1o 1910, tras el parto de su hijo, que naci\u00f3 muerto, estrangulado por el cord\u00f3n umbilical. El bar\u00f3n, ah, s\u00ed: se llamaba Caspers, Victor Caspers, loco de dolor, encarg\u00f3 a un arquitecto franc\u00e9s la construcci\u00f3n de un suntuoso mausoleo para su amor en el Cimeti\u00e8re de la Ville, un espacio para meditar en cuyo techo hab\u00eda un orificio, formado y calculado para que cada a\u00f1o, en el d\u00eda y a la hora de la muerte de su amada, cayera una mancha luminosa en forma de coraz\u00f3n sobre su sarc\u00f3fago.\n\nEso es lo que quer\u00eda ver el profesor Erhart. Hab\u00eda esperado que para una atracci\u00f3n as\u00ed habr\u00eda r\u00f3tulos e indicadores, pero nada de nada. \u00bfHabr\u00eda varios cementerios en Bruselas? \u00bfSe hab\u00eda equivocado de cementerio?\n\nAhora hab\u00eda llegado al edificio que ya hab\u00eda visto de lejos y delante del cual se hab\u00eda reunido entretanto un grupo considerable de gente.\n\nLe sorprendi\u00f3 verlo, inconfundible, entre los all\u00ed presentes. Ah\u00ed estaba, enorme y pesado: era el polic\u00eda que lo hab\u00eda interrogado en el hotel, no cab\u00eda duda, era aquel enorme comisario. Se detuvo, clav\u00f3 la vista en \u00e9l, sus miradas se cruzaron. Erhart no estaba seguro de si le hab\u00eda reconocido, pues ahora el comisario tuvo que concentrarse en otra cosa: dos hombres que se hab\u00edan acercado a \u00e9l a paso r\u00e1pido lo saludaron, intercambiaron unas palabras, luego entraron en el edificio que, como ahora ve\u00eda Erhart, era el crematorio.\n\nNo formaba parte de las tareas del comisario Brunfaut estar presente en la incineraci\u00f3n de la v\u00edctima de un crimen. Y tampoco hab\u00eda para ello ning\u00fan motivo t\u00e9cnico relacionado con la instrucci\u00f3n de la causa. Despu\u00e9s de un crimen, se retiene el cad\u00e1ver y se le hace la autopsia. Despu\u00e9s queda libre para su inhumaci\u00f3n. Si se conoce la identidad de la v\u00edctima y hay familiares, \u00e9stos organizan el entierro. Si no se conoce la identidad, antes de pasadas 48 horas despu\u00e9s de la autopsia se procede a la incineraci\u00f3n por orden municipal. Para ello se presenta un funcionario del ayuntamiento, que comprueba los papeles, confirma el n\u00famero de referencia del muerto, lee durante cinco minutos un texto sobre la caducidad de la vida y el descanso eterno, para garantizar, conforme a las directrices de la UE, un m\u00ednimo de dignidad humana al entierro, luego el ata\u00fad desciende a la c\u00e1mara de combusti\u00f3n. Posteriormente, las cenizas se esparcen, en realidad se vuelcan, sobre la zona verde contigua al crematorio y se pone en una estela una placa con el nombre o, si \u00e9ste se desconoce, con el n\u00famero de referencia policial del muerto. No era de esperar que alg\u00fan sospechoso, y menos a\u00fan el asesino, apareciera en esa ceremonia de la que nadie sab\u00eda el lugar ni la hora fuera de los funcionarios que se ocupaban del caso. Por otra parte, siempre hab\u00eda p\u00fablico, gente del entorno que iba con regularidad a pasear al cementerio, jubilados, viudas, madres empujando carritos infantiles, que se deten\u00edan por respeto o por curiosidad.\n\nPero el comisario Brunfaut no hab\u00eda ido hasta all\u00ed por su caso, sino porque ese d\u00eda era el aniversario de la muerte de su abuelo. Para esa ocasi\u00f3n, muchos a\u00f1os atr\u00e1s a\u00fan se reun\u00eda una impresionante y con el tiempo cada vez m\u00e1s reducida cantidad de gente junto a la tumba de su abuelo, h\u00e9roe de la resistencia belga. Se contaban historias, se beb\u00eda aguardiente, se entonaban canciones. Y al final, La Braban\u00e7onne.[2] Al llegar \u00abLes peuples libres sont amis!\u00bb, aquellos ancianos que cantaban con \u00e9nfasis, que vociferaban, parec\u00edan una cuadrilla de dementes. En el verso \u00abLe Roi, la Loi, la Libert\u00e9!\u00bb siempre hab\u00eda uno que, cual director, deten\u00eda de golpe al coro con un gesto de la mano y exclamaba: \u00a1No podemos tenerlo todo! \u00bfA qu\u00e9 podemos renunciar? Y todos: \u00a1Al rey! \u00bfY a qu\u00e9 no podemos renunciar? Y todos: \u00a1A la ley y la libertad!\n\nDe adolescente, a \u00c9mile Brunfaut le intimidaban bastante esos rituales, aquellos \u00e9xtasis junto a una tumba le parec\u00edan penosos, cre\u00eda que el olor a bolas de naftalina de los trajes de los viejos era olor a p\u00f3lvora. M\u00e1s tarde, tras la muerte de sus padres, empez\u00f3 a sentir respeto y admiraci\u00f3n por el hombre que le hab\u00eda dado tanto miedo de ni\u00f1o, y..., s\u00ed, tambi\u00e9n orgullo. Y m\u00e1s tarde, cuando las l\u00e1grimas estaban dispuestas a brotar de sus sacos lagrimales cada vez m\u00e1s abultados y habr\u00eda podido abrazar a las personas que se hab\u00edan reunido a\u00f1o tras a\u00f1o junto a esa tumba, ya no hab\u00eda nadie, ninguna persona viva, que se acordara de su abuelo y de sus haza\u00f1as. \u00c9l, sin embargo, cada a\u00f1o iba all\u00ed ese d\u00eda y pasaba una hora al pie de la tumba en solitaria contemplaci\u00f3n. Y como ya estaba ah\u00ed, despu\u00e9s fue al crematorio, donde incineraban en esos momentos a \u00absu caso\u00bb. No hab\u00eda esperado que le hiciera avanzar en sus pesquisas: por eso le sorprendi\u00f3 tanto ver all\u00ed a un hombre con el que, en el curso de las primeras tomas de declaraci\u00f3n, hab\u00eda hablado en el lugar de los hechos. Al principio, el hombre le result\u00f3 familiar, pero no acababa de localizarlo y tuvieron que pasar m\u00e1s de diez minutos hasta que supo con seguridad de qu\u00e9 lo conoc\u00eda. Sali\u00f3 inmediatamente de la sala del crematorio, pero el hombre se hab\u00eda ido. Brunfaut recorri\u00f3 todav\u00eda varias calles del cementerio, pero no lo encontr\u00f3.\n\nAbandon\u00f3 el recinto. Justo enfrente de la puerta estaba Le Rustique, un bar en el que entraba siempre a tomar algo cuando visitaba la tumba de su abuelo. Brunfaut se pregunt\u00f3 por qu\u00e9 estaban tapiadas las ventanas del piso de arriba. Era impensable que all\u00ed viviera alguien que no soportara la vista del cementerio. Nadie se amuralla porque le deprimen las vistas. Alguien as\u00ed no se habr\u00eda instalado all\u00ed. \u00bfQu\u00e9 enigma encerraban esas ventanas tapiadas?\n\nComo de costumbre, Brunfaut pidi\u00f3 un stoemp, el plato favorito de su abuelo y, para \u00e9l, un sabor sentimental de la infancia. Stoemp es stoemp, dec\u00eda siempre el abuelo, lo determinante en \u00e9l era naturalmente la calidad de la salchicha: ten\u00eda que reventar al pincharla con el tenedor. Para ello, la piel de la salchicha hab\u00eda de ser de tripa aut\u00e9ntica y no esa especie de media artificial de pl\u00e1stico que cada vez se usa m\u00e1s, s\u00edntoma dram\u00e1tico de la desaparici\u00f3n de la cultura del obrero belga. All\u00ed, en Le Rustique, el stoemp a\u00fan era aut\u00e9ntico. Simple, genuino, perfecto. Tom\u00f3 adem\u00e1s una sencilla Stella Artois de barril y para terminar una copita de Genever. \u00c9mile Brunfaut suspir\u00f3. Luego regres\u00f3 a la comisar\u00eda.\n\nCuando \u00c9mile Brunfaut volvi\u00f3 a la \u00abmine de charbon\u00bb, el agente que estaba de servicio le dijo que el comisario jefe le estaba esperando y le ped\u00eda que fuera inmediatamente a su despacho.\n\nBrunfaut hab\u00eda dicho que ir\u00eda al cementerio y que estar\u00eda de vuelta hacia las 13.00 horas. Y todos hab\u00edan hecho un gesto de asentimiento. Era la una y cinco. \u00bfEl jefe ten\u00eda otra vez que hacerse el importante? Brunfaut esperaba una reconvenci\u00f3n porque no hab\u00eda motivo fundado para que fuera a pasear al cementerio y adem\u00e1s volviera tarde. Se encogi\u00f3 de hombros, no de forma literal, claro, sino mentalmente, esper\u00f3 con paciencia el ascensor, camin\u00f3 despu\u00e9s pensativo por el pasillo hasta el despacho del jefe, llam\u00f3 a la puerta con los nudillos y entr\u00f3.\n\n\u00a1El mundo al rev\u00e9s!, pens\u00f3 al instante. Ven\u00eda del cementerio, pero el entierro, \u00e9sa era su impresi\u00f3n, ten\u00eda lugar all\u00ed. A la izquierda del comisario jefe estaba el juez instructor, a su derecha el fiscal, todos con semblantes extremadamente serios.\n\n\u00a1Si\u00e9ntese, Brunfaut!\n\nQue el juez instructor controlara de cerca al comisario jefe no le extra\u00f1aba demasiado: al fin y al cabo, era el verdadero jefe, que daba continuamente instrucciones y quer\u00eda que le informaran con regularidad de c\u00f3mo iban las diligencias. Pero la presencia del fiscal puso enseguida en alerta a Brunfaut. Porque eso significaba que ah\u00ed hab\u00eda una intervenci\u00f3n pol\u00edtica.\n\n\u00bfDe qu\u00e9 sirve ponerse en alerta cuando ya suena la alarma, cuando las consecuencias del peligro son ya un hecho irrevocable?\n\nS\u00ed, en aquella habitaci\u00f3n ten\u00eda lugar un entierro. El entierro del \u00abcaso Atlas\u00bb.\n\nBueno, dijo el comisario jefe Maigret. Brunfaut estaba convencido de que la carrera de aquel imb\u00e9cil s\u00f3lo se deb\u00eda a la casualidad de llevar ese apellido, una casualidad de lo m\u00e1s desgraciada para la ciudad. \u00c9l no dijo nada, miraba impasible c\u00f3mo aquel Maigret buscaba las palabras. Brunfaut mir\u00f3 en actitud expectante a Maigret, \u00e9ste miraba en busca de ayuda al juez instructor, y el juez instructor al fiscal, quien dijo finalmente: Muchas gracias, comisario, por su tiempo. Justo est\u00e1bamos hablando del asesinato en el hotel Atlas, en el que usted, si estoy bien informado...\n\nS\u00ed, dijo Brunfaut.\n\nBueno, dijo el comisario jefe Maigret.\n\nHay nuevos datos, dijo el juez instructor, monsieur De Rohan. Lo \u00fanico que Brunfaut encontraba interesante en aquel vanidoso Rohan era su mujer. La hab\u00eda conocido en una ocasi\u00f3n en una fiesta de Navidad, una joven dulce y delicada de grandes ojos perfilados en negro, a la que cada vez que iba a decir algo, De Rohan, con una sonrisa, callaba diciendo: \u00abY t\u00fa, ma biche, c\u00e1llate ahora\u00bb. Brunfaut hab\u00eda tenido al momento el deseo de acostarse con ella. No sab\u00eda si la deseaba de verdad o era s\u00f3lo af\u00e1n de humillar a su marido. Y bebi\u00f3 lo bastante como para dec\u00edrselo a ella al o\u00eddo: de modo muy directo y est\u00fapido. Ella lo mir\u00f3 con los ojos muy abiertos, \u00e9l se avergonz\u00f3 al instante, y ella respondi\u00f3: Hoy no puede ser. Ll\u00e1mame ma\u00f1ana.\n\nRohan se acarici\u00f3 con adem\u00e1n narcisista el pelo perfectamente peinado con secador y pidi\u00f3 al comisario jefe Maigret que explicara al comisario Brunfaut el estado de la investigaci\u00f3n.\n\nBrunfaut not\u00f3 que al fiscal le repel\u00eda la torpeza de aquellos polic\u00edas y que s\u00f3lo esperaba a que por fin se hablara sin rodeos y \u00e9l pudiera marcharse y dedicarse otra vez a cosas m\u00e1s importantes.\n\nBueno, dijo el comisario Maigret. Se trata de lo siguiente: hay razones bien fundadas para no seguir efectuando pesquisas en este caso.\n\n\u00bfComprende?\n\nNo, dijo Brunfaut, no comprendo. \u00bfQuiere decir: no seguimos investigando, yo no sigo investigando o, no se sigue investigando?\n\nEra la tercera vez en los \u00faltimos cinco a\u00f1os que le hab\u00edan llamado al lugar del crimen y que hab\u00eda estado delante de un cad\u00e1ver... que al d\u00eda siguiente ya no existir\u00eda. \u00bfSon \u00e9stas las razones bien fundadas: que Bruselas es la ciudad del juicio final? \u00bfDe la resurrecci\u00f3n de los muertos? \u00bfSe ha unido el alma del asesinado con su cuerpo y donde no hay muerto no hay caso? \u00bfLo ha confirmado la medicina forense?\n\nYa, dijo Maigret, comprendo...\n\nBrunfaut mir\u00f3 furioso a aquel imb\u00e9cil. Su necio peinado de erizo. Retorcido con gomina. Como si el apretad\u00edsimo nudo de corbata le pusiera los pelos de punta.\n\nComprendo que usted, bueno, que usted no comprenda esto ahora, pero...\n\nLa cosa era muy sencilla, intervino entonces Rohan, muy f\u00e1cil de entender. Ya no tenemos nada que ver con ese caso, ni usted, ni nosotros, nadie de aqu\u00ed. Y la explicaci\u00f3n que le doy ahora es estrictamente confidencial, usted la oye, pero jam\u00e1s la ha dicho nadie, \u00bfest\u00e1 claro? As\u00ed pues: hay una \u00fanica instituci\u00f3n que tiene el poder de quitarnos un caso as\u00ed, que lo hace desaparecer y que, a su manera, puede aclararlo. Y esa instituci\u00f3n es tan poderosa porque en realidad, es decir, oficialmente, no existe. No es accesible, ya me entiende usted, accede a todos esos casos pero no es accesible. Se trata de intereses que...\n\nIntereses, dijo Brunfaut.\n\nExacto. Nos entendemos.\n\nEl fiscal miraba en silencio al uno y al otro, hac\u00eda gestos de asentimiento.\n\nQueda entre nosotros, dijo Brunfaut, y de nuevo asinti\u00f3 el fiscal. S\u00ed, dijo Brunfaut, queda entre nosotros, como en una policiaca de la tele.\n\n\u00bfC\u00f3mo dice?\n\nOrden de la m\u00e1s alta autoridad, dijo Brunfaut, intervenci\u00f3n pol\u00edtica para impedir la instrucci\u00f3n de la causa, misteriosas alusiones, por lo dem\u00e1s, silencio: todo eso es insoportablemente estereotipado, pero el clich\u00e9, naturalmente, ha de ser completado: por un comisario que se ve obligado, por s\u00ed solo...\n\nUsted no va a...\n\nY que al final, como h\u00e9roe...\n\nUsted seguro que no va a acometer nada por cuenta propia, dijo el fiscal. Es una orden. Y seg\u00fan he sabido hace un momento, se ha dado curso a su solicitud de vacaciones.\n\n\u00a1Pero si yo no he solicitado vacaciones!\n\nEs un peque\u00f1o malentendido, dijo Maigret, yo he dicho que el comisario Brunfaut a\u00fan no ha agotado sus vacaciones, que le quedan muchos d\u00edas libres.\n\nBrunfaut sinti\u00f3 que se ahogaba, respir\u00f3 hondo.\n\nBueno, estupendo, dijo Rohan, entonces gaste ahora sus d\u00edas de vacaciones, rel\u00e1jese, seg\u00fan me han dicho, ha tenido usted un estr\u00e9s enorme y...\n\nEl fiscal se levant\u00f3, Maigret y Rohan saltaron de sus asientos, Brunfaut se puso de pie despacio, entonces not\u00f3 un pinchazo en el pecho y cay\u00f3 otra vez en la silla. El fiscal baj\u00f3 la vista hacia \u00e9l, dijo: \u00a1Se\u00f1ores!\n\n\u00c9mile Brunfaut fue a su despacho y comprob\u00f3 que la carpeta del \u00abAtlas\u00bb con el informe sobre la entrada en acci\u00f3n, con las primeras actas de los interrogatorios, con las fotos del lugar del crimen y el resultado de la autopsia hab\u00eda desaparecido de su mesa. Por otra parte, \u00e9l lo hab\u00eda almacenado todo en su ordenador. Introdujo su contrase\u00f1a: pero tambi\u00e9n en el escritorio virtual del monitor hab\u00eda desaparecido la correspondiente carpeta. Abri\u00f3 la papelera virtual: el expediente tampoco estaba entre los documentos borrados. El acta de las actividades, todo lo que concern\u00eda a aquel caso estaba borrado: cu\u00e1ndo hab\u00eda llegado la orden de intervenir y acudir al hotel Atlas, qu\u00e9 veh\u00edculos policiales y cu\u00e1ndo hab\u00edan llegado al lugar del crimen, qu\u00e9 funcionarios estaban de servicio, el primer informe sobre la recogida de las huellas, todo hab\u00eda desaparecido, el caso se hab\u00eda evaporado.\n\nResoll\u00f3, apret\u00f3 hacia abajo el vientre para aliviar el pulm\u00f3n, respir\u00f3 hondo, se desabroch\u00f3 el cintur\u00f3n y solt\u00f3 el bot\u00f3n de la pretina. Mir\u00f3 la pantalla. \u00bfCu\u00e1nto tiempo? \u00bfUn minuto? \u00bfDiez minutos? Not\u00f3 que ya no miraba la pantalla, sino que se observaba a s\u00ed mismo: \u00bfc\u00f3mo reaccionar\u00eda? No lo sab\u00eda. Se vio sentado all\u00ed como un cad\u00e1ver desplomado sobre una silla. Entonces sus dedos se lanzaron de nuevo al teclado; busc\u00f3 en Google: \u00bfqu\u00e9 informaci\u00f3n hab\u00eda aparecido en los medios sobre el asesinato en el hotel Atlas? Nada. Cualquiera que fuera el t\u00e9rmino que introduc\u00eda, nada, sin resultado. No hab\u00eda habido ning\u00fan art\u00edculo en ning\u00fan peri\u00f3dico. El crimen no hab\u00eda tenido lugar.\n\nLevant\u00f3 la vista y s\u00f3lo ahora not\u00f3 que tambi\u00e9n hab\u00edan limpiado su rotafolio: hab\u00edan arrancado la hoja en la que durante la \u00faltima reuni\u00f3n hab\u00eda escrito con grandes letras may\u00fasculas HOTEL ATLAS flecha CERDO y cinco signos de interrogaci\u00f3n.\n\nTuvo un extra\u00f1o pensamiento: \u00bfera \u00e9se el momento en el que por fin deb\u00eda convertirse en el nieto?\n\nEn el nieto del c\u00e9lebre combatiente de la resistencia.\n\nCogi\u00f3 el tel\u00e9fono y convoc\u00f3 a sus colaboradores. Estaba, eso lo percib\u00eda, furiosamente decidido.\n\nLlegaron el inspector jefe, el comisario auxiliar, los tres inspectores; el comisario Brunfaut apag\u00f3 el ordenador, levant\u00f3 la vista, mir\u00f3 atentamente los rostros de aquellos hombres y comprendi\u00f3 al instante: estaban al tanto y hac\u00eda tiempo que hab\u00edan aceptado la realidad. Era in\u00fatil intentar nada. Se levant\u00f3, dijo que quer\u00eda despedirse, porque sus vacaciones \u2014\u200cnot\u00f3 que se le escurr\u00eda el pantal\u00f3n y lo sujet\u00f3 r\u00e1pidamente\u2014, porque se iba de vacaciones y..., delante de los hombres no quer\u00eda abrocharse el pantal\u00f3n ni el cintur\u00f3n, as\u00ed que exclam\u00f3: \u00a1Hale, fuera de aqu\u00ed!\n\nY ahora esos conformistas, esos probos y educados oportunistas se despachar\u00e1n a gusto comentando el rid\u00edculo que he hecho. Se le humedecieron los ojos, fue al rotafolio, cogi\u00f3 un rotulador y escribi\u00f3: La Loi, la Libert\u00e9! Luego se acord\u00f3 de un epitafio que hab\u00eda visto de paso en el cementerio, y escribi\u00f3 debajo, en may\u00fasculas:\n\nTOUT PASSE\n\nTOUT S'EFFACE\n\nHORS DU SOUVENIR\n\nLuego tom\u00f3 su cartera \u2014\u200cestaba vac\u00eda\u2014 y se march\u00f3.\n\nEl algoritmo que filtra todo lo posible y que tambi\u00e9n ha ordenado lo narrado hasta ahora es absurdo, claro, pero sobre todo es tranquilizador: el mundo es confeti, pero a trav\u00e9s de \u00e9l lo vemos como un mosaico.\n\n\u00bfHa sido por la visita de Brunfaut al crematorio por lo que ahora ha surgido la siguiente asociaci\u00f3n?\n\nNuevo e-mail: \u00abAsunto: Auschwitz. Su visita\u00bb.\n\nMartin Susman ten\u00eda fr\u00edo. Llov\u00eda, y por eso no hab\u00eda ido al trabajo en bicicleta sino en metro. El viento subterr\u00e1neo en los t\u00faneles y galer\u00edas era distinto, m\u00e1s duro y agresivo, que el viento del viaje en bicicleta. Y el vaho caliente, de reba\u00f1o, en los vagones abarrotados no era un alivio, le daba miedo, miedo de enfermedades contagiosas, pero tem\u00eda especialmente que se le contagiara aquella apat\u00eda y resignaci\u00f3n que siempre hac\u00eda presa de la gente en los trenes.\n\n\u00abDistinguido se\u00f1or Susman, celebro poder saludarle pronto en Auschwitz.\u00bb\n\nHab\u00eda ido a buscar a la cantina una taza de t\u00e9 y ahora estaba sentado ante el ordenador para pasar revista a sus e-mails.\n\n\u00abPor supuesto, ir\u00e9 a recogerlo al aeropuerto de Cracovia, y le llevar\u00e9 personalmente en coche al campo. Me reconocer\u00e1 por el letrero con su nombre que mantendr\u00e9 en alto.\u00bb\n\nAsqueado, Susman dej\u00f3 el t\u00e9 sobre la mesa. Se sinti\u00f3 enfermo s\u00f3lo porque se estaba bebiendo ese t\u00e9 por miedo a ponerse enfermo.\n\nEl viaje en misi\u00f3n oficial. Ya estaban hechos todos los preparativos. El servicio cient\u00edfico y el museo del campo de exterminio Auschwitz-Birkenau estaban subvencionados por la UE, representantes de la Comisi\u00f3n Europea tomaban parte cada a\u00f1o el 27 de enero en el acto conmemorativo de la liberaci\u00f3n del campo. En nombre de la Direcci\u00f3n General de Cultura enviaban ese a\u00f1o a Martin Susman, que tambi\u00e9n deb\u00eda gestionar la activaci\u00f3n y el control de los recursos empleados.\n\n\u00abQuiero a\u00fan, si me lo permite, darle buen consejo para viaje. Importante ropa abrigada. En Auschwitz-Birkenau fr\u00edo helador en esta \u00e9poca. No queremos en absoluto se ponga enfermo en Auschwitz.\n\n\u00bbEn \u00faltima visita a Berl\u00edn compr\u00e9 en unos almacenes ropa interior, la mejor que he tenido nunca. No s\u00e9 c\u00f3mo se llama la marca, pero por favor, vaya a tienda, pida ropa interior alemana. Digo siempre ropa interior alemana porque comprada en Berl\u00edn y seguro es made in Germany. Eso ser\u00e1 conocido en Bruselas. \u00a1Ropa interior alemana! Le aconsejo comprar. \u00a1Ropa interior alemana es lo mejor para Auschwitz!\u00bb\n\nMartin Susman clic\u00f3 para responder, escribi\u00f3 tres frases amables, abri\u00f3 el e-mail siguiente, se levant\u00f3 y sali\u00f3 del despacho, mir\u00f3 hacia el despacho de Bohumil Szmekal, que tecleaba con prisa... y levant\u00f3 en alto un paquete de cigarrillos, Szmekal asinti\u00f3 y ambos salieron a la escalera de incendios para fumarse un cigarrillo.\n\nMrzne jak v rusk\u00fdm filmu, dijo Bohumil. Martin, evidentemente, no lo entendi\u00f3, pero le dio la raz\u00f3n: S\u00ed, necesitaremos ropa interior alemana.\n\nDavid de Vriend sali\u00f3 del cementerio. Ten\u00eda fr\u00edo. Lo toleraba bien, ya hab\u00eda pasado fr\u00edos m\u00e1s terribles, y sin un abrigo como el que ahora llevaba. Decidi\u00f3 entrar en Le Rustique, un bar que hab\u00eda enfrente, a tomar un bocado y beber algo que le calentara, un vaso de vino tinto. Entr\u00f3, encontr\u00f3 sitio all\u00ed mismo a la izquierda, junto a la ventana. La camarera trajo la carta, pregunt\u00f3: \u00bfEs usted de la Maison Hanssens, de la residencia de mayores? Ens\u00e9\u00f1eme sus sellos para la oferta.\n\n\u00bfSellos?\n\n\u00a1Para el descuento!\n\nNo, no, dijo De Vriend \u2014\u200c\u00e9l no sab\u00eda nada de tales sellos, al menos la se\u00f1ora Jos\u00e9phine no le hab\u00eda dicho nada de eso\u2014, yo soy normal, quiero decir, cliente normal.\n\nMuy bien, dijo ella poni\u00e9ndole delante la carta; \u00e9l pidi\u00f3 un vaso de vino tinto, s\u00ed, vino de la casa. Y pregunt\u00f3: De comer, alguna menudencia. \u00bfQu\u00e9 me recomienda?\n\nAqu\u00ed tenemos las cosas normales, dijo ella se\u00f1alando la carta, y cada d\u00eda un men\u00fa anticrisis.\n\n\u00bfMen\u00fa anticrisis?\n\nS\u00ed. Primero algo muy sustancioso y despu\u00e9s algo muy dulce. La gente lo pide mucho. Hoy choucroute \u00e0 l'ancienne, luego mousse au chocolat. Dieciocho euros sin sellos. Y si antes quiere un duo de fondue, fromages et crevettes, entonces veinticinco euros.\n\nMir\u00f3 a aquella risue\u00f1a mujer y se pregunt\u00f3 c\u00f3mo ser\u00eda tratar todos los d\u00edas con asistentes a un entierro, no con los muertos pero s\u00ed con los familiares.\n\nBueno, un men\u00fa anticrisis, dijo, sin fondue.\n\n\u00a1Y sin sellos! D'accord!\n\nEsper\u00f3 mirando por la ventana. M\u00e1s all\u00e1, la entrada del cementerio. S\u00f3lo ahora, a distancia, mir\u00e1ndolo desde all\u00ed, cay\u00f3 en la cuenta de que la puerta de entrada al cementerio ten\u00eda cierto parecido con la verja de entrada a Birkenau.\n\nEntonces lleg\u00f3 su vino tinto.\n\nUna puerta de hierro forjado siempre se parece a una puerta de hierro forjado. \u00bfY los postes a derecha e izquierda? \u00bfQu\u00e9 iba a haber, si no, a derecha e izquierda de una puerta de hierro forjado? Como la gente que hab\u00eda en el campo: eran seres humanos, \u00bfqu\u00e9 iban a ser, si no? Sin embargo, la impresi\u00f3n de que hay un parecido es absurda. No lo hay. Eso era todo.\n\n# CAP\u00cdTULO CUARTO\n\nSI PUDI\u00c9RAMOS VIAJAR AL FUTURO, ESTAR\u00cdAMOS A\u00daN M\u00c1S LEJOS.\n\n#\n\nMartin Susman quer\u00eda llevar a cabo ese viaje a Polonia a ser posible sin detrimento f\u00edsico ni ps\u00edquico. Nunca se habr\u00eda imaginado que precisamente ese viaje le dar\u00eda la idea, es m\u00e1s: la idea fija para el Big Jubilee Project, y que al final casi trastornar\u00eda por completo su vida.\n\nPero de momento sufr\u00eda con los preparativos.\n\nSe qued\u00f3 asombrado cuando la vendedora interrumpi\u00f3 su tartamudeo: Bien s\u00fbr, por supuesto que ella conoc\u00eda la ropa interior alemana; nombr\u00f3 una marca, y, naturalmente, ellos vend\u00edan ese \u2014\u200csonri\u00f3\u2014 producto alem\u00e1n de alta calidad.\n\nMartin le hab\u00eda preguntado a Kass\u00e1ndra Mercouri si sab\u00eda de alguna tienda especializada en ropa interior y ella le hab\u00eda aconsejado que fuera a Ixelles, a la Galerie Toison d'Or, all\u00ed hab\u00eda una tienda con un gran surtido, la tienda se llamaba Toll\u00e9, no, se llamaba Fronde, s\u00ed, seguro: Fronde. En cualquier caso, en el r\u00f3tulo de la tienda pon\u00eda en grandes caracteres ROPA INTERIOR, UNDERWEAR, dijo ella, adem\u00e1s reconocer\u00eda la tienda enseguida por el escaparate. \u00c9sos lo ten\u00edan todo. Ella compraba su ropa interior s\u00f3lo all\u00ed.\n\nUna vez que Martin hubo encontrado la tienda \u2014\u200cFronde Dessous\u2014 y mirado el escaparate, vio de pronto a la maternal Kass\u00e1ndra con otros ojos. \u00bfAll\u00ed compraba ella su ropa \u00edntima? \u00bfKass\u00e1ndra? Pens\u00f3 que por lo visto \u00e9l no se hab\u00eda explicado con suficiente claridad, que, ahora lo ve\u00eda, sin la menor duda era un malentendido. Vio..., s\u00ed, en efecto, ropa interior, maravillosos, refinad\u00edsimos, realmente seductores dessous, pero \u00bfpara \u00e9l? Y: \u00bfpara Auschwitz?\n\nMir\u00f3 alrededor, vio enfrente un Adventure Shop, all\u00ed uno pod\u00eda comprar todo lo que necesitaba para subir al Mount Everest..., tal vez deber\u00eda buscar ah\u00ed su equipo antihielo; \u00bfhab\u00eda pensado de verdad ahora en \u00abequipo antihielo\u00bb? Era tan rid\u00edculo. No pod\u00eda determinar qu\u00e9 le parec\u00eda m\u00e1s agobiante: con su cuerpo blando, fofo, cruzar enfrente donde los curtidos y viriles aventureros o..., no: Kass\u00e1ndra le hab\u00eda recomendado la tienda Fronde, y por eso entr\u00f3 decidido en ella.\n\nCuando trataba de explicarle a la vendedora lo que quer\u00eda se sinti\u00f3 como un provinciano de diecisiete a\u00f1os que dirige la palabra por primera vez a una chica en una discoteca de la capital. Cuando dijo \u00abropa interior alemana\u00bb: \u00abQuiero decir que hay una ropa interior muy abrigada, creo que de una marca alemana, no s\u00e9 si usted comprende lo que quiero decir, en cualquier caso muy abrigada...\u00bb, cerr\u00f3 los ojos como si temiera que aquella mujer leyera en su mirada que \u00e9l, en su imaginaci\u00f3n, la ve\u00eda con el mismo dessous que llevaban los maniqu\u00edes del escaparate.\n\nBien s\u00fbr! Hab\u00eda un armario con muchos cajones como los que \u00e9l hab\u00eda visto en las farmacias; abri\u00f3 uno, lo hizo deslizarse de nuevo, abri\u00f3 otro y sac\u00f3 varios envoltorios de celof\u00e1n que extendi\u00f3 ante \u00e9l sobre el mostrador. Aqu\u00ed est\u00e1n, \u00bfes esto lo que quiere? Camiseta, calzoncillos largos, calcetines, y esto son mu\u00f1equeras de lana. Cien por cien angora. Y mire, aqu\u00ed lo pone: calidad alemana. Cr\u00e9ame, con estas prendas tendr\u00e1 m\u00e1s calor que en el infierno.\n\nSe rio. O, digamos: que en una sauna. \u00bfSe va usted de viaje?\n\nS\u00ed, dijo \u00e9l. A... Polonia.\n\nOh. No conozco Polonia. Pero me imagino que esto viene muy bien all\u00ed, porque ya es casi Siberia. Rio, abri\u00f3 un paquete, extendi\u00f3 ante \u00e9l un calzoncillo largo, pas\u00f3 la mano por la tela, dijo: \u00a1Mire! T\u00f3quelo. \u00bfNota qu\u00e9 blando y c\u00e1lido es? Est\u00e1 hecho con el pelo de esos conejos, angora, \u00bfcomprende? Pero de Alemania, es decir: garant\u00eda de que no ha habido maltrato animal. Y aqu\u00ed est\u00e1 el certificado: esta prenda cumple tambi\u00e9n las nuevas normas de la UE para ropa interior.\n\n\u00bfC\u00f3mo dice?\n\nS\u00ed, monsieur. A m\u00ed tambi\u00e9n me extra\u00f1\u00f3. Hace poco ha estado aqu\u00ed el representante y nos lo ha explicado. Se trata de la reacci\u00f3n de la prenda ante el fuego, eso ya est\u00e1 regulado.\n\n\u00bfMe est\u00e1 diciendo \u2014\u200cMartin sonri\u00f3 ir\u00f3nico\u2014 que la prenda produce tanto calor que existe el peligro de que se inflame?\n\nLa chica sonri\u00f3. No, pero en cierto modo tiene que ver con eso: no puede ser inflamable. No s\u00e9 tampoco por qu\u00e9. Y la angora es propiamente..., es pelo de conejo. Y eso es, claro, extremadamente inflamable. Pero ya no. Ahora hay que impregnarla de alguna cosa. UE, \u00bfentiende? Quiz\u00e1 porque son sobre todo los fumadores quienes compran estas prendas, porque siempre tienen que estar al aire libre, en el fr\u00edo. Y por eso hay ahora esa norma de la UE: para que los fumadores no se prendan fuego. Se rio. O tambi\u00e9n en la cama.\n\n\u00bfEn la cama?\n\nS\u00ed, cuando los fumadores se van a la cama con un cigarrillo y se quedan dormidos...\n\nEntonces arde la cama...\n\nS\u00ed, pero esta ropa interior no. \u00a1Eso ya est\u00e1 regulado! Mire, aqu\u00ed: \u00abComportamiento de ropa interior ante el fuego conforme con la norma de la UE...\u00bb.\n\nNo me lo creo, mademoiselle.\n\nYo tampoco, dijo ella.\n\nLo primero que hizo Kai-Uwe Frigge ese lunes fue echar una ojeada a la lista \u00abValise Voyage \u00e0 Doha\u00bb, que Madeleine, su secretaria, le hab\u00eda dejado sobre la mesa para su firma. Frigge hab\u00eda introducido eso all\u00ed: cada lunes, Madeleine le presentaba una lista en la que figuraba su dresscode de martes a lunes, conforme a sus citas y sus obligaciones. Por lo general, Frigge firmaba la lista, que inmediatamente despu\u00e9s Madeleine enviaba por e-mail a Dubravka, su ama de llaves. \u00c9sta preparaba cada ma\u00f1ana su vestimenta tal como estaba en la lista, o, cuando \u00e9l iba de viaje, la met\u00eda en su maleta.\n\nEsto se sab\u00eda en la casa y hab\u00eda quien sonre\u00eda despectivamente o hac\u00eda observaciones ir\u00f3nicas, pero eso no compromet\u00eda el buen nombre de Frigge, al contrario: esa extravagancia demostraba que era un pragm\u00e1tico duro como el pedernal hasta en el menor detalle y que dispon\u00eda de la capacidad de encontrar soluciones originales para sudar menos cuando hab\u00eda que correr o para mojarse menos cuando hab\u00eda que nadar. En ambientes burocr\u00e1ticos, tal fama equivale al m\u00e1s alto t\u00edtulo nobiliario.\n\nHay una an\u00e9cdota significativa de la \u00e9poca estudiantil de Frigge, hab\u00eda contado Frauke Diestel, de la DG Energ\u00eda. Ella era en aquella \u00e9poca compa\u00f1era de estudios de Frigge, en la Universidad de Hamburgo, y durante alg\u00fan tiempo vivi\u00f3 con \u00e9l en un piso compartido. Cont\u00f3 que un d\u00eda Kai-Uwe regal\u00f3 todas sus camisas de color y de dibujos y compr\u00f3 en unas rebajas del Shopping Center Hamburger Meile diez camisas blancas id\u00e9nticas. \u00c9l lo explic\u00f3 del siguiente modo: ahora ahorraba tiempo cada ma\u00f1ana al no tener que pensar qu\u00e9 camisa pegaba con qu\u00e9 chaqueta o jersey; una camisa blanca pegaba siempre, dijo, y era igual lo que se pon\u00eda encima. Ahora, no ten\u00eda que pensar largo tiempo, pod\u00eda coger simplemente por la ma\u00f1ana la camisa superior de la pila del armario y pon\u00e9rsela, y cuando se pon\u00eda la n\u00famero ocho sab\u00eda que ten\u00eda que llevar las camisas sucias a la lavander\u00eda y recogerlas cuando se pon\u00eda la n\u00famero diez, de manera que al d\u00eda siguiente pod\u00eda empezar otra vez con la primera camisa. De alg\u00fan modo aquello era absurdo, pero ten\u00eda su l\u00f3gica. Hab\u00eda conseguido las camisas blancas a tan buen precio porque eran unas antiguallas invendibles pasadas de moda en las que todav\u00eda hab\u00eda que meter varillas en la punta de los cuellos. Pero \u00e9l estaba encantado: Esto es cultura, hab\u00eda dicho. Las mangas eran muy largas, pero Frigge encontr\u00f3 en un mercadillo antiguos brazaletes que se pon\u00eda para acortarlas. Para \u00e9l eso tambi\u00e9n formaba parte de la \u00abantigua cultura\u00bb. Sent\u00eda debilidad por los accesorios masculinos. En aquel entonces surgieron aquellas pel\u00edculas de g\u00e1nsteres y de mafiosos en las que todos los hombres llevaban esos ajustamangas, y se puso de moda: \u00a1de pronto Kai-Uwe, con su manera extravagante y pragm\u00e1tica y su total falta de inter\u00e9s por las necedades de la moda, pas\u00f3 a ser una especie de creador de tendencias! Aunque a Kai-Uwe se le interprete mal, dijo Frauke, estaba claro que eso beneficiaba su reputaci\u00f3n.\n\nDespu\u00e9s de repasar la lista, Kai-Uwe Frigge estaba de mal humor. Madeleine hab\u00eda olvidado otra vez lo que le hab\u00eda dicho ya tantas veces: en viajes a pa\u00edses c\u00e1lidos no se necesitaba ropa interior ligera y vaporosa, al contrario, justamente en pa\u00edses c\u00e1lidos hab\u00eda que llevar tambi\u00e9n ropa de abrigo, finos chalecos de cachemir, por ejemplo, y en cualquier caso, camisetas. En negociaciones y reuniones y durante las comidas est\u00e1 uno ininterrumpidamente en salas climatizadas, con temperaturas brutalmente bajas; en ning\u00fan sitio se pasa m\u00e1s fr\u00edo que entre esos jeques del desierto, donde el fr\u00edo se considera un lujo y el lujo es el objetivo de la vida. Si en Doha no se est\u00e1 precisamente paseando por la calle \u2014\u200c\u00bfy qui\u00e9n hace eso, y a santo de qu\u00e9 hacer eso?\u2014, se pasa m\u00e1s fr\u00edo que sentado en un banco de un parque del norte de Finlandia.\n\nLlam\u00f3 a Madeleine, le orden\u00f3 rehacer la lista. Olv\u00eddese de todos esos trapos de lino y de seda, eso vale para Estrasburgo en verano, pero no para Doha. Lana, cachemir, \u00bfde acuerdo? Chalecos y camisetas. Y pa\u00f1uelos, bufanda. Y ponga tambi\u00e9n en la lista, bajo \u00abvarios\u00bb, el cargador del tel\u00e9fono y la tableta, y bet\u00fan para los zapatos. Para que Dubra lo meta tambi\u00e9n en la maleta.\n\nMadeleine asinti\u00f3 y fue hacia la puerta.\n\n\u00a1Madeleine!\n\n\u00bfS\u00ed, monsieur?\n\nOtra cosa m\u00e1s. Por favor, ponga tambi\u00e9n en la lista el turbante azul.\n\nNo.\n\nS\u00ed. Nunca se sabe. Es posible que alguna vez tengamos que... \u2014\u200ctosi\u00f3 ligeramente\u2014 salir al aire libre.\n\nKai-Uwe Frigge mir\u00f3 el reloj. Ahora ten\u00eda que ocuparse de la \u00abcerdada\u00bb, como \u00e9l lo llamaba.\n\nMateusz Oswiecki quer\u00eda rezar antes de que despegara el avi\u00f3n. Ten\u00eda que recogerse. Le atormentaba haber ejecutado al hombre equivocado.\n\nAntes de pasar por el control de seguridad vio a activistas que repart\u00edan octavillas, una docena larga de hombres y mujeres j\u00f3venes que llevaban la misma camiseta amarilla, en el pecho un eslogan que \u00e9l no alcanzaba a leer. Tres polic\u00edas estaban indecisos a un lado, mientras otro hablaba con un activista, otro por radio.\n\nMateusz aminor\u00f3 el paso, para, de entrada, contemplar la escena, luego aceler\u00f3, pasajero con prisas que no quer\u00eda perder el vuelo, con manifiesta impaciencia intent\u00f3 escabullirse. Hab\u00eda llegado casi a la barrera cuando una activista se interpuso en su camino. Excuse me, sir, may I..., \u00e9l no reaccion\u00f3 y trat\u00f3 de pasar al lado. Do you speak english, sir? Sir? No la mir\u00f3, maniobr\u00f3 con su maleta de ruedas para pasar al lado. Parlez-vous fran\u00e7ais? Volez-vous vers la Pologne? Are you going to Poland? Sir? It is important, een vraag, mijnheer. \u00c9l baj\u00f3 la cabeza, vio de reojo que un polic\u00eda estaba mirando en su direcci\u00f3n y se sinti\u00f3 seguro. Era casi c\u00f3mico: se escapar\u00eda con ayuda de un polic\u00eda que ten\u00eda que intervenir cuando molestaban a un pasajero. Pero Mateusz no quer\u00eda llegar hasta ese extremo, no quer\u00eda verse envuelto en algo que pudiera atraer a la polic\u00eda. La mujer le tend\u00eda una hoja, vio que en ella hab\u00eda una foto de un hombre, ten\u00eda todo el aspecto de una foto de busca y captura. \u00bfEra una orden de captura? Mateusz puso el ticket sobre la pantalla de la barrera, se encendi\u00f3 en rojo, \u00bfqu\u00e9 ocurr\u00eda? Sir, please, \u00bfvuela usted con Polish Airlines? \u00bfVuelo LO 236? Tenemos una informaci\u00f3n importante. Sab\u00eda que era absurdo, s\u00ed, que todo se complicar\u00eda a\u00fan m\u00e1s si ahora dec\u00eda: Sorry, tengo prisa. Porque entonces dar\u00eda comienzo una conversaci\u00f3n, ella dir\u00eda que s\u00f3lo iba a reclamar su atenci\u00f3n un momentito, \u00e9l tendr\u00eda que dar tambi\u00e9n alguna respuesta; no: coloc\u00f3 sin palabras el ticket otra vez sobre la pantalla, de nuevo se ilumin\u00f3 en rojo, frot\u00f3 el billete varias veces en todas direcciones, \u00bfpor qu\u00e9 demonios no funcionaba aquello? Entonces se encendi\u00f3 la luz verde, se abrieron los batientes de cristal y pas\u00f3. Se puso en la cola que avanzaba despacio hacia el security check. Vio que algunos pasajeros le\u00edan la octavilla. Una vez pasado el esc\u00e1ner busc\u00f3 una indicaci\u00f3n que lo llevara a las salas de culto.\n\nQuedaba m\u00e1s de una hora hasta el embarque. Pas\u00f3 con su maleta por las tiendas, caminaba cada vez m\u00e1s deprisa, all\u00ed estaban ya las puertas de embarque: \u00bfd\u00f3nde quedaba la capilla del aeropuerto? Retrocedi\u00f3, no encontr\u00f3 ninguna indicaci\u00f3n. Quer\u00eda rezar. Hab\u00eda matado al hombre equivocado. Lo hab\u00eda visto con claridad tras las \u00faltimas \u00f3rdenes recibidas. Finalmente encontr\u00f3 una placa con un pictograma que pod\u00eda representar a una persona arrodillada rezando, a su lado una flecha que se\u00f1alaba un pasillo lateral. All\u00ed, de nuevo la figura arrodillada y la flecha que llevaba a una escalera hacia arriba.\n\nSigui\u00f3 las flechas, pens\u00f3 en san Sebasti\u00e1n, en su pecho asaeteado. Pocos d\u00edas antes, el 20 de enero, d\u00eda del santo, patrono de los soldados y combatientes contra los enemigos de la Iglesia, hab\u00eda implorado su protecci\u00f3n y rezado por el \u00e9xito de su misi\u00f3n en Bruselas, pero algo se hab\u00eda torcido y no se explicaba qu\u00e9. Las flechas llevaban a un pasillo vigilado por videoc\u00e1maras. Sigui\u00f3 caminando, con la cabeza gacha, se pas\u00f3 el pa\u00f1uelo por la frente como si se enjugara el sudor para que las c\u00e1maras no captaran su rostro; sab\u00eda que su prudencia era exagerada. Esas c\u00e1maras de vigilancia eran antediluvianas. \u00bfNevaba en aquel pasillo? Claro que no. Pero durante 48 horas esas c\u00e1maras almacenar\u00edan fotos de una resoluci\u00f3n tan tosca que s\u00f3lo podr\u00eda verse la borrosa imagen de un hombre que caminaba en medio de una nevisca. A derecha e izquierda macetas de flores. De pl\u00e1stico. C\u00e1\u00f1amo. No cab\u00eda duda, eran plantas de c\u00e1\u00f1amo de pl\u00e1stico. \u00bfQui\u00e9n hab\u00eda tenido la idea de colocar en el pasillo que llevaba a las capillas plantas de c\u00e1\u00f1amo de pl\u00e1stico? Por fin lleg\u00f3 a las salas de culto. Hab\u00eda una sala para cada gran comunidad religiosa. Cat\u00f3licos, protestantes, jud\u00edos, musulmanes y ortodoxos. Todas estaban vac\u00edas, m\u00e1s que eso: tan vac\u00edas como si en ellas nunca hubiera entrado nadie.\n\nMateusz sinti\u00f3 un intenso dolor cuando entr\u00f3 en la sala de oraci\u00f3n cat\u00f3lica. Aquel espacio era de una incre\u00edble fealdad. Incre\u00edble. Esa palabra era grotesca en un lugar para creyentes. Sinti\u00f3 un escozor punzante por debajo del ombligo, sudor fr\u00edo en la frente, avanz\u00f3 unos pasos, solt\u00f3 la maleta, sac\u00f3 el pa\u00f1uelo del bolsillo y se enjug\u00f3 el sudor mientras apretaba la otra mano contra el vientre. Entonces la maleta se volc\u00f3 con un estruendo cuando Mateusz estaba con el pa\u00f1uelo del sudor en la mano delante de Jesucristo. El frente de la sala estaba recubierto de varios tablones de madera, de ellos colgaba el Crucificado: pero sin cruz. Como si al Hijo de Dios no lo hubieran clavado en una cruz sino en una empalizada. Del techo colgaba un foco que proyectaba un duro y blanco rayo de luz sobre Jesucristo, como si, despu\u00e9s de haber sido clavado en la empalizada, debiera someterse a un \u00faltimo interrogatorio. Delante hab\u00eda un peque\u00f1o altar de madera que parec\u00eda m\u00e1s una de esas radiogramolas que al final de los setenta del siglo pasado hab\u00edan tra\u00eddo muchos polacos de sus viajes al oeste y que despu\u00e9s siguieron amueblando sus cuartos de estar como eterno memento de la anhelada modernidad. En la pared lateral colgaba un tr\u00edptico, \u00f3leo sobre lienzo, extra\u00f1amente indeciso entre abstracto y figurativo. En el cuadro de la izquierda se distingu\u00eda una puesta de sol, en cualquier caso, una bola roja que ca\u00eda sobre una muchedumbre o que planeaba sobre ella, las personas eran tal vez cardenales con su sotana escarlata, pero tal vez no fuesen tampoco cardenales, sino reflejos luminosos del sol que se pon\u00eda con colores rojos, o llamas o plantas. La parte central presentaba algo que parec\u00eda un platillo volante ensartado, pero quiz\u00e1 fuese una planta de incineraci\u00f3n de basura. Lo m\u00e1s claro era la imagen de la derecha: un charco de sangre bajo una luz blanca brillante de la que surg\u00eda una cruz blanca. Junto a la cruz estaba pintada una frase: UBI LUX IBI BLUT. \u00c9l sab\u00eda lat\u00edn, lo hab\u00eda estudiado, claro, en el seminario, pero eso no lo entend\u00eda: \u00bfqu\u00e9 significaba blut? \u00bfQu\u00e9 clase de palabra era \u00e9sa? \u00abDonde hay luz, hay...\u00bb Se acerc\u00f3 m\u00e1s, mir\u00f3 si hab\u00eda le\u00eddo mal, trat\u00f3 de descifrar aquella enigm\u00e1tica palabra, blut; no conoc\u00eda ese vocablo. Pero ahora vio que probablemente, no, con toda seguridad, la palabra era deus, pintada con trazos tan d\u00e9biles y confusos que parec\u00eda querer meterse en la sombra de la imprimaci\u00f3n del cuadro. Junto al tr\u00edptico hab\u00eda dos grandes tallas de madera, recordaban a los pastores de los belenes navide\u00f1os, pero m\u00e1s a\u00fan a seminaristas en camis\u00f3n.\n\nMateusz hab\u00eda estado de pie en camis\u00f3n, descalzo sobre el fr\u00edo suelo de piedra, recogimiento se llamaba eso, cuando un seminarista despu\u00e9s de la oraci\u00f3n nocturna recib\u00eda la orden de permanecer a pie firme en la galer\u00eda de las arcadas junto a la estatua del santo asignado, con vistas al patio por abajo y al cielo estrellado por arriba, para meditar sobre las \u00abtres preguntas\u00bb hasta que lo llamara el padre prior para que diera las respuestas, a veces despu\u00e9s de dos o tres horas, a veces al d\u00eda siguiente despu\u00e9s de la oraci\u00f3n matutina. \u00bfC\u00f3mo es de grande la duda sobre la fuerza de tu fe? \u00bfHasta qu\u00e9 punto est\u00e1s seguro de que vencer\u00e1s esa duda? \u00bfCon qu\u00e9 obras quieres probar la fuerza de tu fe?\n\nA Mateusz le acomet\u00eda una extra\u00f1a excitaci\u00f3n, no s\u00f3lo una agitaci\u00f3n o miedo general, sino literalmente excitaci\u00f3n, una energ\u00eda sexual o er\u00f3tica, cuando sent\u00eda bajo las plantas de los pies la lisura y el fr\u00edo de la piedra, el fr\u00edo que sub\u00eda en \u00e9l a trav\u00e9s de los pies y que endurec\u00eda y tensaba sus m\u00fasculos, su tejido, mientras percib\u00eda la lisa superficie de la piedra como la piel de un cuerpo, piel marm\u00f3rea, piel sagrada, piel-de-la-Madre-de-Dios, que \u00e9l tocaba, contra la que se estrechaba, con la que se fund\u00eda. Se hab\u00eda situado junto a la estatua de san Sebasti\u00e1n sin saber si era casualidad o una decisi\u00f3n cuidadosamente tomada por el prior cuando \u00e9ste le orden\u00f3 llevar a cabo all\u00ed su recogimiento.\n\nMateusz hab\u00eda buscado la conversaci\u00f3n con el Padre, no porque dudara de su fe, sino porque ten\u00eda dudas sobre c\u00f3mo vivirla. Estaba preparado para la lucha, pero quer\u00eda traer antes un hijo al mundo, como hab\u00edan hecho su padre y su abuelo antes de ir a la batalla.\n\n\u00bfQuieres que siga vivo tu apellido? \u00bfTu sangre? \u00bfAlgo tuyo? Vivir\u00e1s eternamente cuando mueras, pero \u00bfquieres seguir viviendo aqu\u00ed en la tierra?\n\nMateusz era otra vez Ryszard y no pudo dar una respuesta.\n\nRecogimiento. Antes de la oraci\u00f3n matutina lo hab\u00edan encontrado tendido sobre las losas de la galer\u00eda como si hubiera intentado poner la mayor cantidad posible de piel en contacto con la piedra. Estaba entumecido, luego tuvo fiebre varios d\u00edas. Despu\u00e9s respondi\u00f3 a las tres preguntas. De modo muy convincente y a entera satisfacci\u00f3n del prior. Pero no pod\u00eda quedarse en el seminario.\n\nAquel dolor punzante. Mateusz desvi\u00f3 la mirada de las figuras de bel\u00e9n, mir\u00f3 alrededor. Quer\u00eda rezar. All\u00ed no pod\u00eda. Apret\u00f3 una mano contra su diafragma, lanz\u00f3 un gemido, se sec\u00f3 el sudor de la frente. No le quedaba mucho tiempo.\n\nRespir\u00f3 hondo varias veces, sali\u00f3 de la capilla del aeropuerto y se dirigi\u00f3 a la puerta de embarque.\n\nEn un principio hab\u00eda querido volar de vuelta a Varsovia, una vez cumplida su misi\u00f3n. Pero aquella noche dejaron para \u00e9l en el hotel un sobre que por la ma\u00f1ana le entregaron en recepci\u00f3n. Encontr\u00f3 all\u00ed un billete de avi\u00f3n para Estambul, as\u00ed como la confirmaci\u00f3n de una reserva de hotel en esa misma ciudad. Mateusz sab\u00eda que no era una nueva orden, no pod\u00eda serlo. Cada nueva misi\u00f3n empezaba con un dosier sobre el objetivo y se planeaba y preparaba con todo detalle. Y una vez cumplida una misi\u00f3n, ning\u00fan soldado hab\u00eda recibido nunca otra nueva a la ma\u00f1ana siguiente. Para la seguridad de cada acci\u00f3n, la posterior retirada a la retaguardia ten\u00eda una importancia tan grande como la exacta planificaci\u00f3n anterior. \u00c9l s\u00f3lo pod\u00eda explic\u00e1rselo as\u00ed: el objetivo hab\u00eda huido a Estambul, y eso significaba tambi\u00e9n que \u00e9l hab\u00eda matado al hombre equivocado. Pero, adem\u00e1s, que aquello era una trampa. Si quer\u00edan deshacerse de \u00e9l, \u00e9se era el m\u00e9todo m\u00e1s sencillo: hab\u00eda jurado obediencia absoluta. A un animal hab\u00eda que atraerlo a la trampa. A un soldado s\u00f3lo hab\u00eda que darle la orden de marchar hacia la trampa.\n\nAh\u00ed hab\u00eda gato encerrado. Para operaciones fuera de la zona Schengen ten\u00edan especialistas propios. Mateusz confiaba en su pasaporte, que sin duda era perfecto, pero los controles en la frontera Schengen eran m\u00e1s minuciosos y no quer\u00eda verse obligado a depender de que su pasaporte resistiera tal control.\n\nHab\u00eda viajado hasta el aeropuerto y tratado de embarcar con su anterior billete a Varsovia. La mujer de la taquilla dijo que \u00e9l mismo hab\u00eda anulado el vuelo.\n\nNo.\n\nS\u00ed. Usted ya no est\u00e1 en la lista de pasajeros, monsieur. Lo anul\u00f3 anoche.\n\n\u00a1Es un malentendido! Quiero ese vuelo.\n\nLo siento, no puedo extenderle una tarjeta de embarque. Ya no tiene billete para ese vuelo.\n\n\u00a1Pero lo he pagado!\n\nLa mujer tecle\u00f3 en el ordenador, mir\u00f3, tecle\u00f3, mir\u00f3 y dijo: El precio del billete, deducidos los gastos de anulaci\u00f3n, ha sido reintegrado a su tarjeta de cr\u00e9dito.\n\n\u00bfMi tarjeta de cr\u00e9dito? Pero si no tengo... \u00a1En fin! Entonces quiero otro billete. Saco otro billete.\n\nLo siento mucho, monsieur, pero el vuelo est\u00e1 completo. No hay plazas.\n\nPero tengo que ir a Polonia. Hoy.\n\n\u00bfEs usted polaco, monsieur? \u00bfS\u00ed? Entonces podemos hablar en polaco, drogi Panie. Mi padre es polaco. Vino a Bruselas como hidr\u00e1ulico, fontanero. Conoci\u00f3 aqu\u00ed a mi madre. Encontraremos una soluci\u00f3n. Gdy zaleje woda, trzeba wymieni\u0107 rur\u0119.\n\nQuedaba un asiento libre para el vuelo a Cracovia dos horas m\u00e1s tarde. O una hora m\u00e1s tarde a Frankfurt con enlace a Varsovia.\n\nAs\u00ed result\u00f3 que al final viajaba en el mismo avi\u00f3n que Martin Susman. Pero \u00bfqu\u00e9 importancia pueden tener relaciones, enlaces y conexiones si los interesados no saben nada de ello?\n\nMartin Susman estaba irritado por su absurda idea de ponerse la ropa interior de abrigo ya en el viaje. Para no pasar fr\u00edo cuando llegara a Cracovia. Empez\u00f3 a sudar como un cerdo en el taxi al aeropuerto. El taxi, evidentemente, ten\u00eda calefacci\u00f3n, probablemente excesiva, y en su pelo de conejo se sent\u00eda como si tuviera mucha fiebre. \u00bfPor qu\u00e9 se dec\u00eda sudar como un cerdo? Como hijo de un granjero de cerdos sab\u00eda, claro, que los cerdos no sudan, no pueden transpirar por la piel. De ni\u00f1o emple\u00f3 una vez ese giro. \u00bfPor qu\u00e9? Porque se dice. Su padre le hab\u00eda reprendido. Los cerdos no sudan. Y no hay que hacer todo lo que hacen los dem\u00e1s; si otros dicen estupideces, t\u00fa no tienes que decirlas tambi\u00e9n.\n\nPero \u00bfpor qu\u00e9 se dice?\n\nPorque mucha gente tiene un problema con la sangre. Antes, cuando hac\u00edan la matanza en casa, ve\u00edan cu\u00e1nto sangraban los cerdos, y entonces llamaban sudor a la sangre. Un eufemismo, \u00bfcomprendes? As\u00ed no suena tan mal. Los cazadores llaman todav\u00eda hoy sudor a la sangre de los animales, y al sabueso que busca y captura un animal que sangra, herido por los disparos, en alem\u00e1n lo llaman perro del sudor.\n\nPero decimos morcilla de sangre y no morcilla de sudor.\n\nYa basta, hab\u00eda dicho su padre, entra en casa y ayuda a tu madre.\n\nDesde entonces no hab\u00eda empleado nunca m\u00e1s ese giro, pero ahora, en el taxi camino del aeropuerto, estaba de pronto otra vez all\u00ed, en su cabeza, con el recuerdo de que en realidad se trataba de sangre, derramamiento de sangre, r\u00edos de sangre, un ba\u00f1o de sangre.\n\nCuando lleg\u00f3 al aeropuerto, Martin Susman hab\u00eda gastado un paquete entero de pa\u00f1uelos de papel para secarse el sudor, llevaba en la mano una masa compacta de papel h\u00famedo cuando baj\u00f3 del taxi, ahora ya no le quedaban pa\u00f1uelos, se pasaba la manga por la cara, era in\u00fatil, sudaba y sudaba. Sangre. Quiso comprar m\u00e1s pa\u00f1uelos, fue de ac\u00e1 para all\u00e1, con lo que a\u00fan sudaba m\u00e1s. Finalmente decidi\u00f3 ir directamente a la puerta de embarque, lo m\u00e1s despacio posible, sentarse all\u00ed, quiz\u00e1 al no moverse dejaba de sudar. Estaba furioso consigo mismo, deber\u00eda haber sabido que era completamente absurdo ponerse esa ropa en situaciones en las que de todos modos no iba a pasar fr\u00edo. En Cracovia ir\u00edan a recogerlo al aeropuerto, lo llevar\u00edan en un taxi con calefacci\u00f3n a un hotel con calefacci\u00f3n, all\u00ed tendr\u00eda posibilidad de cambiarse, tendr\u00eda que haberse puesto esa ropa all\u00ed, antes de continuar viaje al campo de exterminio; y ahora dudaba que la ropa empapada de sudor se secara a tiempo en el hotel; probablemente se quedar\u00eda colgada all\u00ed sec\u00e1ndose mientras \u00e9l, sin esa ropa, estaba en el campo a merced del fr\u00edo mortal.\n\nEstaba fren\u00e9tico. Se odiaba a s\u00ed mismo. Ten\u00eda treinta y ocho a\u00f1os y a\u00fan era incapaz de vestirse solo de modo adecuado a una situaci\u00f3n, a lo que \u00e9sta requer\u00eda. Le vino a la cabeza desenvolverse en la vida; cu\u00e1ntas veces lo hab\u00eda o\u00eddo: Este ni\u00f1o no sabe desenvolverse en la vida. No es capaz de afrontar la vida. \u00a1Pero afortunadamente tenemos a Florian!\n\nDesenvolverse en la vida no estaba muy lejos de voluntad de vivir. Sab\u00eda o cre\u00eda saber c\u00f3mo era esa relaci\u00f3n. Inseparables. Alternativamente, o se remontan o se hunden. En individuos, familias, grupos sociales, sociedades enteras. \u00c9l hab\u00eda tenido suerte: su falta de capacidad para desenvolverse en la vida no llevaba a un r\u00e1pido final de la vida, su voluntad de vivir pod\u00eda quebrantarse, pero \u00e9l pod\u00eda as\u00ed, quebrantado, seguir caminando por la vida. Pero ten\u00eda miedo cuando en los medios surg\u00edan continuamente esas gu\u00edas pr\u00e1cticas para la vida y empezaban a fulgurar sus frases hueras cargadas de ideolog\u00eda: \u00abHay que poder desconectar\u00bb, \u00abHay que aprender a dejarse caer\u00bb... No sab\u00edan lo que dec\u00edan. Eso se pod\u00eda estudiar en los cuatro estratos arqueol\u00f3gicos de las excavaciones, all\u00ed estaba muy claramente datado cu\u00e1ndo hab\u00eda comenzado: el desconectar, el dejarse caer, la muerte que predicaban las gu\u00edas pr\u00e1cticas para la vida. El tercer estrato.\n\nMuy poco antes de pasar el security check se le ofreci\u00f3 un cuadro extra\u00f1o y confuso. Tuvo la impresi\u00f3n de que, en medio de la oleada de pasajeros, se enfrentaban dos equipos, uno en ropa amarilla, otro en ropa azul. \u00bfHab\u00eda un partido, una competici\u00f3n? Un partido no, pero una competici\u00f3n s\u00ed, en cierto sentido. Una joven en uniforme amarillo se dirigi\u00f3 a \u00e9l: Disculpe, se\u00f1or, \u00bfvuela a Polonia?\n\nS\u00ed, dijo. Ella le mir\u00f3, a \u00e9l le result\u00f3 violento; qu\u00e9 iba a pensar al ver su cara mojada, sus ojos enrojecidos. Ella sonri\u00f3, sigui\u00f3 hablando deprisa. Era activista de la organizaci\u00f3n pro derechos humanos Stop Deportation y...\n\n\u00bfC\u00f3mo dice?\n\nStop Deportation, dijo ella se\u00f1alando el eslogan de su camiseta:\n\nNO BORDER\n\nNO NATION\n\nSTOP DEPORTATION\n\nEs que van a expulsar a un hombre que...\n\nLleg\u00f3 entonces un integrante del equipo azul, un polic\u00eda, que dijo: \u00bfLe est\u00e1n molestando, se\u00f1or? Para su informaci\u00f3n: es una concentraci\u00f3n autorizada, pero podemos disolverla si hay pasajeros que se sienten importunados.\n\nNo, no, dijo Martin Susman, est\u00e1 okay, est\u00e1 okay. A m\u00ed no me importunan.\n\nCon la mano, se apart\u00f3 varias veces hacia el pelo el sudor de la frente. El polic\u00eda asinti\u00f3, se alej\u00f3, habl\u00f3 con otro pasajero al que un activista hab\u00eda enredado en una conversaci\u00f3n.\n\nSusman supo que iban a deportar a un checheno, perseguido y torturado por motivos pol\u00edticos. Hab\u00eda llegado a la UE a trav\u00e9s de Polonia. Ahora iban a deportarlo a Polonia y desde all\u00ed lo entregar\u00edan a Rusia. Rusia estaba catalogada por las autoridades competentes como pa\u00eds seguro para chechenos. Puro cinismo, le informaron. Se hab\u00eda demostrado en varias ocasiones que los chechenos entregados a Rusia desaparec\u00edan en s\u00f3tanos de tortura. La mujer le dio una octavilla. \u00c9ste es el hombre, dijo, Aslan Achmatow. Est\u00e1 traumatizado y le espera m\u00e1s tortura y la muerte. Es un esc\u00e1ndalo para los derechos humanos, monsieur. \u00bfEst\u00e1 usted de acuerdo conmigo? Aqu\u00ed tiene lo que puede hacer como pasajero para impedir la deportaci\u00f3n si ve a ese hombre en el avi\u00f3n. Exija hablar con el piloto y p\u00eddale que suspenda la deportaci\u00f3n, por razones humanitarias y de seguridad del vuelo. \u00c9l es el mando a bordo, puede rechazar el transporte de pasajeros que no viajan voluntariamente en su avi\u00f3n.\n\nLa chica hablaba cada vez m\u00e1s deprisa mientras \u00e9l le\u00eda la hoja.\n\n\u00a1Ah\u00ed lo tiene todo! Ni\u00e9guese a tomar asiento y a abrocharse el cintur\u00f3n, haga ver a los otros pasajeros que no se trata de un transporte normal, sino de un acto de violencia y...\n\nPerdone usted, dijo Martin, pero aqu\u00ed pone que se trata del vuelo LO 236 a Varsovia. \u00a1Pero yo vuelo a Cracovia!\n\nOh, perd\u00f3n. Habr\u00eda..., claro. Gracias. Le agradezco su paciencia, su comprensi\u00f3n. Qu\u00e9dese con la hoja, por favor. Por las informaciones generales. Porque hay cada vez m\u00e1s deportaciones y... Gracias. Y que tenga un buen d\u00eda.\n\nSe dio media vuelta, \u00e9l la sigui\u00f3 un momento con la vista, ahora hablaba con otro pasajero, en el dorso de su camiseta pon\u00eda RESISTANCE IS POSSIBLE.\n\nYa todos en sus asientos y concluido el embarque, una mujer se levant\u00f3 y avanz\u00f3 mirando las filas de pasajeros sentados a derecha e izquierda. Antes de pasar a la business class, una azafata se interpuso en su camino:\n\n\u00bfBusca el aseo, madame? Est\u00e1 al final del avi\u00f3n. Pero ahora no puede utilizarlo, madame. Tiene que sentarse y que ponerse el cintur\u00f3n.\n\nNo quiero ir al aseo, dijo ella, y luego, en voz alta: \u00a1Quiero hablar con el comandante! Por lo visto en este avi\u00f3n hay un pasajero que no vuela voluntariamente. Y quiero saber...\n\n\u00a1Por favor! Tiene usted que...\n\nHemos de saber si efectivamente est\u00e1 en este avi\u00f3n contra su voluntad. \u00a1Llame al comandante, por favor!\n\nSe dio media vuelta y retrocedi\u00f3 por el pasillo central. Mesdames, messieurs, en esta nave hay un hombre que va a ser deportado. Ay\u00fadenme, por favor, a que ese hombre...\n\n\u00a1Madame, se lo ruego! Tiene que sentarse y...\n\nLa mujer continu\u00f3 imperturbable su camino, lleg\u00f3 a la fila en la que estaba sentado Susman.\n\n... hemos de darle la posibilidad de salir del avi\u00f3n.\n\nEl vecino de asiento de Susman ten\u00eda la mirada fija en su peri\u00f3dico, la mujer que estaba en el asiento del pasillo cerr\u00f3 los ojos, el hombre de al lado, en el asiento de la ventana, no paraba de limpiar su smartphone.\n\nMartin Susman se levant\u00f3 para ver mejor lo que ocurr\u00eda. Al instante ten\u00eda a su lado a una azafata que le conmin\u00f3 a sentarse inmediatamente y a abrocharse el cintur\u00f3n.\n\nS\u00ed, dijo, \u00a1un momento! S\u00f3lo quer\u00eda... Abri\u00f3 el compartimento para equipajes y sac\u00f3 de su bolsa el paquete de chicles de nicotina. La mujer se detuvo entonces, se dirigi\u00f3 a un pasajero y pregunt\u00f3: \u00bfEs usted el se\u00f1or Achmatow?\n\nEl hombre no reaccion\u00f3. Se hab\u00eda echado una capucha sobre la cabeza, la barbilla descansaba en el pecho.\n\n\u00bfHabla usted ingl\u00e9s, sir? \u00bfEs usted Mr. Achmatow?\n\nMateusz Oswiecki levant\u00f3 la mirada y neg\u00f3 con la cabeza. La mujer vacil\u00f3, en un primer momento no estaba segura de si dec\u00eda que no sab\u00eda ingl\u00e9s o que no era la persona que buscaba. Ambos se miraron. Mateusz no hab\u00eda entendido bien de qu\u00e9 se trataba, pero comprend\u00eda que aquella mujer retrasaba la salida y la odiaba por ello. La mir\u00f3 a la cara, sus miradas se encontraron y...\n\nAlgo le pas\u00f3 en ese momento. En el diafragma, all\u00ed donde estaba localizado el dolor. Era como si hubiese reventado una vena y la sangre, caliente y dulce, se extendiera por su vientre. Mateusz no ten\u00eda pensamientos, en su cabeza no se formaba ninguna frase. De pronto le pesaban much\u00edsimo los p\u00e1rpados, luchaba por mantener abiertos los ojos, para ver c\u00f3mo lo miraba aquella mujer, quer\u00eda permanecer en esa mirada, saborear un anhelo que no hab\u00eda conocido, y el sentimiento de un cobijo que hab\u00eda conocido pero olvidado y que ahora estaba all\u00ed como la imagen de un recuerdo: de ni\u00f1o y con mucha fiebre, hab\u00eda visto, como a trav\u00e9s de la niebla, el rostro de su madre, que se inclinaba sobre su lecho de enfermo y sonre\u00eda. Esa imagen de su madre, como una aparici\u00f3n en la niebla, le hab\u00eda quitado todo el miedo, incluso el miedo a morir si se dejaba vencer y cerraba los ojos. Eso era de novela cursi. Ya no era un ni\u00f1o, hab\u00eda tenido que endurecerse y despreciaba el sentimentalismo. Lo que ahora sent\u00eda no estaba claro, era tan borroso como esa imagen de su recuerdo. El nost\u00e1lgico deseo de una infancia protegida, por haberla tenido o por no haberla tenido, lo comparten todos, terroristas lo mismo que pacifistas. \u00c9l s\u00f3lo quer\u00eda... su mirada... pero la mujer ya se hab\u00eda ido. Ped\u00eda disculpas por el retraso del despegue, ped\u00eda ayuda para evitar la deportaci\u00f3n. Martin Susman la sigui\u00f3 con la vista, los pasajeros guardaban silencio, continuaban inm\u00f3viles en sus asientos, crey\u00f3 reconocer por las miradas que algunos simpatizaban con la mujer, otros cerraban los ojos, bajaban la cabeza y, de pronto, un auxiliar de vuelo estaba a su lado: tiene usted que sentarse inmediatamente, si\u00e9ntese, por favor, y abr\u00f3chese el cintur\u00f3n. El hombre puso con suave presi\u00f3n una mano sobre el hombro de Martin; aument\u00f3 la presi\u00f3n. Cuando Martin se dej\u00f3 caer en el asiento oy\u00f3 una voz masculina: \u00a1Cierra el pico de una vez y si\u00e9ntate!\n\nOtra voz: \u00a1Deje usted de impedir el despegue! Se ha equivocado usted de avi\u00f3n. \u00a1Ese hombre est\u00e1 en el avi\u00f3n a Varsovia! \u00a1Lo pon\u00eda en la octavilla!\n\nLa mujer: Lo han cambiado a este vuelo. Por las protestas contra su deportaci\u00f3n. He recibido un SMS diciendo que ahora est\u00e1 en este avi\u00f3n. Para deportarlo a Polonia sin llamar la atenci\u00f3n.\n\nDe pronto un joven se puso de pie, inmediatamente delante de Martin Susman y grit\u00f3: No deportation! Y de m\u00e1s adelante lleg\u00f3 una voz de mujer: Solidarit\u00e9!\n\nMartin Susman se asom\u00f3 fuera del asiento y mir\u00f3 hacia atr\u00e1s, al pasillo; la mujer estaba de pie en la \u00faltima fila, y vio c\u00f3mo se inclinaba sobre un pasajero. Por su posici\u00f3n retorcida, Martin sinti\u00f3 una violenta y dolorosa tirantez en la espalda, desde las v\u00e9rtebras lumbares hasta la nuca, deber\u00eda levantarse, pens\u00f3, pero no quer\u00eda correr ning\u00fan riesgo, \u00bfqu\u00e9 riesgo? Se levant\u00f3, se estir\u00f3, apret\u00f3 las manos contra la espalda, el joven que hab\u00eda delante de \u00e9l volvi\u00f3 a sentarse, la azafata y el auxiliar de vuelo hab\u00edan desaparecido, y Martin oy\u00f3 que una mujer le dec\u00eda a un pasajero de la \u00faltima fila: \u00bfMr. Achmatow? \u00bfEs usted Mr. Achmatow?\n\nYes!\n\nEl hombre se levant\u00f3. \u00bfEra \u00e9l de verdad? No llevaba esposas ni le acompa\u00f1aba ning\u00fan polic\u00eda. Pero parec\u00eda aturdido, como bajo el efecto de tranquilizantes.\n\nLa mujer le ense\u00f1\u00f3 la hoja con su foto para estar segura, \u00e9l dijo: Yes!\n\nYa pas\u00f3, dijo la mujer. No tenga miedo, qu\u00e9dese de pie, qu\u00e9dese de pie, simplemente, y nos bajaremos del avi\u00f3n.\n\nEl hombre empez\u00f3 a llorar. Las manos en el rostro, con las mu\u00f1ecas una contra otra como si estuviera esposado.\n\nSubieron polic\u00edas, se llevaron a los dos. Los pasajeros aplaudieron. \u00bfPor qu\u00e9? \u00bfPor el valor c\u00edvico de la mujer? \u00bfO porque hab\u00eda intervenido la autoridad p\u00fablica? \u00bfO porque el avi\u00f3n pod\u00eda despegar por fin? Cada cual ten\u00eda sus motivos. La suma de ellos daba eso: \u00a1Aplausos!\n\nEl avi\u00f3n de Frigge sal\u00eda dentro de cuatro horas. Dubra preparaba la maleta. Y \u00e9l a\u00fan ten\u00eda una cita con su colega George Morland, de la DG AGRI. Entre la AGRI y la TRADE siempre hab\u00eda habido conflictos y discusiones por las competencias, era como una tradici\u00f3n, por no decir un antiguo juego. Pero ahora el conflicto se hab\u00eda agudizado y aquello ya no se pod\u00eda tratar con una sonrisa, ni pactar un acuerdo de caso en caso y luego irse juntos a tomar una cerveza o, cuando no hab\u00eda qu\u00edmica con los agricultores ecol\u00f3gicos, lamentar cort\u00e9smente no tener tiempo para tomar una cerveza. Ahora era guerra declarada, ahora hab\u00eda que rearmarse y buscar la batalla final. El punto de controversia que hab\u00eda llevado a la escalada eran precisamente los cerdos. Eso era lo que Frigge llamaba \u00abla cerdada\u00bb, en la Comisi\u00f3n otros llegaban a hablar, a prop\u00f3sito del conflicto entre la TRADE y la AGRI, de \u00abla guerra de los cerdos\u00bb. La AGRI quer\u00eda reducir las subvenciones y lograr as\u00ed que disminuyera la producci\u00f3n de cerdos a fin de detener la ca\u00edda de los precios de la carne porcina en el mercado europeo; la TRADE, sin embargo, quer\u00eda fomentar m\u00e1s a\u00fan la producci\u00f3n porque ve\u00eda en el comercio exterior, sobre todo con China, grandes posibilidades de crecimiento. Por eso quer\u00eda la TRADE competencias para negociar para toda Europa en terceros pa\u00edses la exportaci\u00f3n de productos de carne de cerdo y conseguir que en el continente se organizara la producci\u00f3n de porcino de acuerdo con la demanda de los mercados mundiales; por su parte, la AGRI s\u00f3lo quer\u00eda regular el mercado interior, conseguir est\u00e1ndares comunes y que los est\u00e1ndares veterinarios recayeran bajo la responsabilidad de la DG SANCO. Y ambos quer\u00edan que los tratados de comercio exterior permanecieran bajo la soberan\u00eda de los distintos Estados.\n\nEn cualquier caso, el resultado de esas luchas de competencias era que cada pa\u00eds europeo negociaba por s\u00ed solo con China, Europa quedaba desarticulada, por la competencia entre los Estados europeos la ca\u00edda de precios era a\u00fan mayor, tanto en el mercado interior como en el comercio exterior, mientras que ning\u00fan Estado pod\u00eda ya atender en solitario a la demanda internacional, porque al mismo tiempo los granjeros de cerdos se ve\u00edan empujados a abandonar su trabajo. A Frigge todo eso le parec\u00eda completamente absurdo. Y ese Morland le pon\u00eda fren\u00e9tico. Frigge se preguntaba... \u00bfEn el fondo por qu\u00e9? \u00bfPor qu\u00e9 se excitaba de esa manera? La Comisi\u00f3n, al fin y al cabo, no ten\u00eda de momento ninguna competencia para negociar en nombre de todos los Estados miembros; los Estados miembros estaban encantados de, entretanto, poder sacar provecho de la situaci\u00f3n y obtener cada uno el m\u00e1ximo para \u00e9l solo. Era, sin duda, una conclusi\u00f3n err\u00f3nea, en alg\u00fan momento se dar\u00edan cuenta, pero por ahora \u00e9l no pod\u00eda cambiarla, podr\u00eda dejar que las cosas pasaran por encima de su escritorio, mirar sin ninguna carga emocional, no fastidiar a nadie y alguna vez moverlas de nuevo hacia arriba... \u00a1Pero no! Esa situaci\u00f3n le parec\u00eda tan irracional que no era capaz de permanecer impasible. As\u00ed que, donde pod\u00eda, bloqueaba el business as usual para forzar una decisi\u00f3n.\n\nLa discusi\u00f3n por las competencias se basaba en que el cerdo era una materia transversal: el cerdo vivo, en la pocilga, \u00abpertenec\u00eda\u00bb a la DG AGRI; despu\u00e9s de la matanza, en forma de jam\u00f3n, tocino, filetes, embutido, lo que fuera, en cualquier caso como \u00abprocessed agricultural good\u00bb era competencia de la DG GROW, y s\u00f3lo cuando abandonaba Europa, por as\u00ed decirlo, en buque de carga o en cami\u00f3n, pertenec\u00eda a la DG TRADE. El problema era que no se pod\u00eda negociar sobre el cerdo en contenedor si no se pod\u00eda decidir sobre el cerdo en la pocilga del suelo patrio. La GROW era en ese caso inofensiva. All\u00ed se ocupaban de regular el listado del contenido de sustancias, de definir el empleo m\u00e1ximo de productos farmac\u00e9uticos y qu\u00edmicos, de criterios de calidad. A ellos el cerdo les importaba un comino, s\u00f3lo ten\u00eda que llevar la etiqueta correcta. El partido decisivo se jugaba entre la AGRI y la TRADE.\n\nDesde hac\u00eda semanas, George Morland evitaba tener una conversaci\u00f3n con Frigge. A los e-mails respond\u00eda con evasivas: hablemos m\u00e1s tarde sobre ello poniendo todos los factores sobre la mesa. Pero cuando Frigge propon\u00eda una cita concreta, respond\u00eda remitiendo a su agenda enormemente cargada. Los comisarios se manten\u00edan en segundo plano. Eran nuevos y quer\u00edan antes ponerse al d\u00eda. Pero el tiempo apremiaba. El gobierno holand\u00e9s, el alem\u00e1n y el austriaco eran los que m\u00e1s hab\u00edan avanzado en sus negociaciones con China. En el curso del \u00faltimo a\u00f1o natural, la canciller alemana hab\u00eda estado ocho veces en China. La semana siguiente viajar\u00eda a Pek\u00edn el presidente austriaco con el avi\u00f3n cargado de ministros, de representantes de intereses de la industria, del comercio y la agricultura; en primer\u00edsimo lugar de la agenda: el comercio porcino. Poco despu\u00e9s ya se hab\u00edan anunciado otra vez en Pek\u00edn los holandeses. Si alguno de esos pa\u00edses lograba ultimar un sustancial convenio bilateral con China, entonces, visto desde la perspectiva pol\u00edtica, era m\u00e1s bien improbable que la UE obtuviera competencias para negociar. Y entonces empezar\u00e1 el gran forcejeo, a lo bruto, el vender m\u00e1s barato, el intento de eliminar a los vecinos. En lugar de proceder en com\u00fan, se matar\u00e1n unos a otros y con su ansia de crecimiento nacional producir\u00e1n una crisis europea. Eso estaba m\u00e1s claro que el agua, como dec\u00eda Kai-Uwe. Morland sab\u00eda, evidentemente, que ese d\u00eda Kai-Uwe Frigge ten\u00eda que hacer un viaje oficial. Y era infame que por fin propusiera esa cita justo tres horas antes del embarque de Frigge.\n\nFrigge, sin perder la calma, hab\u00eda aceptado la cita. Y ahora estaba sentado enfrente de aquel cerdo. La asociaci\u00f3n era una vulgaridad, pero Frigge no pudo evitarla. No soportaba a ese Morland, lo ten\u00eda por insidioso, c\u00ednico e irresponsable. Eso por s\u00ed solo ya justificaba una expresi\u00f3n malsonante. Pero a ello se a\u00f1ad\u00eda la apariencia de Morland: su rostro redondo y rosado y en medio, como un enchufe de la pared, su peque\u00f1a y ancha nariz. Ten\u00eda treinta y tantos a\u00f1os, pero parec\u00eda mucho m\u00e1s joven, ese reto\u00f1o de la alta sociedad brit\u00e1nica, como si acabara de afeitarse, sus mejillas siempre brillaban rojizas e irritadas. Ten\u00eda pelo abundante y pelirrojo y lo llevaba cortado a cepillo, a Frigge le parec\u00edan cerdas.\n\nFrigge hab\u00eda nacido en Hamburgo, en el seno de una familia de profesores de secundaria. Internacionalismo hanse\u00e1tico, conocimiento de la deuda hist\u00f3rica alemana, una grande y abstracta exigencia de paz y justicia en el mundo, esfuerzo y honestidad personal, recelo frente a modas y mainstream: \u00e9sos eran los jalones que hab\u00edan colocado sus padres para delimitar el terreno en el que se hab\u00eda criado. Sab\u00eda que, en lo concerniente a Morland, era injusto. Pero sab\u00eda tambi\u00e9n que ten\u00eda motivos m\u00e1s que de sobra para serlo.\n\nMorland se miraba las u\u00f1as mientras \u00e9l explicaba su visi\u00f3n de las cosas. Frigge cerr\u00f3 los ojos, no quer\u00eda ver aquella indolente actitud. Morland ten\u00eda en todo raz\u00f3n. S\u00ed, as\u00ed era. \u00c9sa era la situaci\u00f3n. Pero la diferencia no consist\u00eda en que Frigge viera de un modo distinto la situaci\u00f3n, sino en que Morland la encontraba razonable y la defend\u00eda, mientras que Frigge quer\u00eda acabar con tal situaci\u00f3n.\n\nOkay, George, dijo Frigge, imag\u00ednate lo siguiente: \u00a1Eres un siervo de la gleba!\n\n\u2014\u00bfPor qu\u00e9 voy a hacer eso?\n\n\u2014Es s\u00f3lo un juego mental. As\u00ed que...\n\n\u2014\u00a1Con una idea as\u00ed no quiero jugar!\n\n\u2014Bueno, vale. En tiempos hubo siervos de la gleba. Right? Eso lo sabes. Y ahora imag\u00ednate: un siervo de la gleba va a su se\u00f1or y le dice que tiene que hablar con \u00e9l.\n\n\u2014\u00bfLos esclavos pod\u00edan hablar as\u00ed, sin m\u00e1s, con sus amos?\n\n\u2014No lo s\u00e9, da igual, ahora s\u00f3lo se trata de lo que dice el siervo de la gleba, no el esclavo, pero por m\u00ed, tambi\u00e9n el esclavo, da igual, \u00e9l va y dice: Se\u00f1or, la servidumbre no me parece bien, no es digna del ser humano, contradice a la Biblia...\n\n\u2014\u00bfEst\u00e1 esa historia en la Biblia? No la conoc\u00eda.\n\n\u2014En la Biblia est\u00e1 que ante Dios todos los hombres son iguales, y \u00e9se era un argumento del siervo, as\u00ed que...\n\n\u2014\u00bfSab\u00eda leer? \u00bfY lat\u00edn? Que yo sepa, en la Edad Media la Biblia s\u00f3lo estaba en lat\u00edn, y la mayor\u00eda de la gente era analfabeta.\n\n\u2014Okay. Sin Biblia. En cualquier caso, el siervo de la gleba no estaba de acuerdo con su servidumbre. Y propone a su amo, con algunas razones sensatas, que le d\u00e9 la libertad. \u00bfQu\u00e9 le responder\u00e1 el amo?\n\n\u2014D\u00edmelo t\u00fa.\n\n\u2014Explicar\u00e1 al campesino que es siervo de la gleba porque ya lo era su padre y lo era tambi\u00e9n su abuelo con el abuelo del amo, que el mundo es como es, y adem\u00e1s desde generaciones atr\u00e1s, desde siempre, y que eso ha de tener un sentido.\n\n\u2014Yo dir\u00eda que es un argumento razonable. \u00bfO no te parece?\n\n\u2014Okay, George, ahora dime: \u00bfhay todav\u00eda siervos de la gleba?\n\n\u2014No lo s\u00e9. \u00bfEn alguna parte del mundo?\n\n\u2014\u00a1George! \u00a1Otra vez! Un siervo de la gleba en alg\u00fan sitio de Europa se queja y...\n\n\u2014En la Edad Media, en lugar de alcanzar la libertad lo habr\u00edan descuartizado, supongo yo.\n\n\u2014Exacto. Y el se\u00f1or dice que eso ha sido siempre as\u00ed. Pero ahora te vuelvo a preguntar: \u00bfsiguen existiendo los siervos de la gleba? \u00bfLo ves? Ah\u00ed quer\u00eda yo ir a parar: todo lo que has dicho es verdad, completamente cierto... pero s\u00f3lo inside the box. Objetivamente, sin embargo, es absurdo y, en cualquier caso, visto en perspectiva, insostenible. Continuamente ha ido desapareciendo lo que parec\u00eda hecho para toda una eternidad y...\n\n\u2014\u00bfTe refieres a la UE?\n\n\u2014No, me refiero a los intereses nacionales. Porque es absurdo que los Estados europeos constituyan un mercado com\u00fan pero en el comercio exterior no establezcan una comunidad. Que cada cerdo que abandona Europa s\u00f3lo pueda entrar en el mercado mundial con un visado de su Estado nacional. Okay, es as\u00ed, pero alguna vez ser\u00e1 diferente porque las cosas cambian. O sea, podr\u00edamos organizarlo ya, enseguida, de un modo m\u00e1s sensato.\n\n\u2014Reflexionar\u00e9 sobre tu historia de la servidumbre de la gleba. Aunque no estoy seguro de si es exactamente lo que se dice un ejemplo oportuno.\n\nKai-Uwe Frigge sab\u00eda, como es natural, por qu\u00e9 Morland se resist\u00eda a que siguiera desarroll\u00e1ndose la pol\u00edtica comunitaria: no era europeo, sino ante todo s\u00fabdito brit\u00e1nico, y en la Comisi\u00f3n no era funcionario europeo, sino justamente eso, un s\u00fabdito brit\u00e1nico en el funcionariado europeo. Y era una f\u00e9rrea pol\u00edtica de la Gran Breta\u00f1a atajar cualquier m\u00ednimo traspaso de soberan\u00eda nacional a Bruselas. Han renovado con dinero de la UE su totalmente ruinoso Manchester, pero no lo agradecen, sino que esas fachadas de la ciudad, hoy tan acicaladas, son para ellos la prueba de que el capitalismo manchesteriano triunfar\u00e1 una y otra vez sobre todos sus competidores. Ese cerdo fofo y perfumado probablemente empezaba el d\u00eda cantando Rule, Britannia! mientras tomaba el early morning tea y... Frigge respir\u00f3 hondo. Luego se levant\u00f3, dijo:\n\nWell, tengo que marcharme al aeropuerto. \u00a1Seguimos hablando la semana que viene!\n\nCuando quieras, dijo Morland.\n\nFrigge se hab\u00eda preparado una despedida con golpe de efecto. Mientras se pon\u00eda el abrigo, dijo: Por cierto, supongo que estar\u00e1s informado. El gobierno alem\u00e1n firmar\u00e1 en las pr\u00f3ximas semanas un convenio bilateral de comercio con China. Okay, s\u00f3lo comercio porcino. Eso no es de gran inter\u00e9s para United Kingdom.\n\n\u00bfEso es seguro?\n\nS\u00ed. Es definitivo.\n\nFrigge se abroch\u00f3 el abrigo, meti\u00f3 documentos en su cartera.\n\nEs exclusivo, es pr\u00e1cticamente la puerta de entrada de la econom\u00eda alemana en el mercado chino. Y ah\u00ed no se trata s\u00f3lo de estad\u00edsticas de exportaci\u00f3n.\n\nTendi\u00f3 la mano a Morland.\n\nLos grandes inversores sabr\u00e1n interpretarlo, los mercados financieros acusar\u00e1n reacciones. El centro financiero de London City perder\u00e1 relevancia, la Bolsa de Frankfurt ganar\u00e1 de manera decisiva.\n\nFrigge dio a Morland unos golpecitos en la espalda.\n\n\u00bfNo es curioso? \u00bfTribulaci\u00f3n inglesa s\u00f3lo por cerdos alemanes? Okay, tengo que marcharme. Ll\u00e1mame la semana que viene, es preciso que continuemos hablando del tema. Estoy seguro de que encontraremos la manera de hacer esto de un modo m\u00e1s razonable, m\u00e1s justo. Pero en eso la Comisi\u00f3n ha de mostrar unanimidad.\n\nFrigge abri\u00f3 la puerta, se dio la vuelta otra vez hacia Morland, dijo sacudiendo la cabeza: \u00a1Cerdos!, y se ech\u00f3 a re\u00edr. A\u00fan sonre\u00eda en el taxi que le llevaba al aeropuerto.\n\nCuarenta y uno cuarenta y dos cuarenta y tres cuarenta y cuatro cinco seis siete ocho nueve, cincuenta. \u00a1Respirar hondo! Cincuenta y uno cincuenta y dos tres cuatro\n\niba por en medio de la calle, cada paso era una patada en el suelo, un jadear el n\u00famero del paso, cincuenta y siete ocho nueve \u00a1se-senta! \u00a1Respirar hondo! Sesenta y uno sesenta y dos tres... Por qu\u00e9 contaba los pasos, quer\u00eda saber cu\u00e1ntos pasos hab\u00eda desde la puerta de entrada hasta el final, desde la entrada en el final hasta la salida del final, quer\u00eda abarcar la dimensi\u00f3n de aquel lugar, de aquella calle que parec\u00eda infinitamente larga, la calle hacia el infinito. Ante \u00e9l estaba la calle con el blancor inocente de la nieve, de un blanco inocente toda aquella inmensa superficie, por qu\u00e9 se asocia el blanco con la inocencia, incluso all\u00ed, en aquel lugar, el color de un fr\u00edo mort\u00edfero bajo la luz muerta del sol de invierno. El vaho de la respiraci\u00f3n ante la boca, con cada n\u00famero, sesenta y seis siete ocho nueve...\u00a1se-tenta! Un viento glacial le soplaba en el rostro.\n\nMartin Susman not\u00f3 de pronto una suave presi\u00f3n en el hombro, setenta y uno setenta y dos setenta y... una mano sobre su hombro: \u00a1Tiene que abrocharse el cintur\u00f3n, por favor!\n\nSe sobresalt\u00f3, abri\u00f3 los ojos. S\u00ed, dijo, \u00a1por supuesto!\n\nEl vuelo de retorno de Cracovia a Bruselas. \u00bfJadeaba? Respiraba con dificultad. Se abroch\u00f3 el cintur\u00f3n de seguridad, alz\u00f3 la mano hasta el ventilador y cerr\u00f3 el dispositivo. Luego se le cerraban otra vez los ojos, notaba el sudor fr\u00edo en la frente, tiritaba. Se hab\u00eda resfriado, claro. Hab\u00eda tenido miedo de aquel viaje, se hab\u00eda preparado a disgusto, a la defensiva, para la visita del monumento conmemorativo y del museo, por miedo al shock, por miedo a ver lo indescriptible. Pero la musealizaci\u00f3n mata a la muerte y reconocer impide el shock del conocer. Los distribuidores autom\u00e1ticos donde los turistas pod\u00edan comprar en el campo por diez zlotys bebidas calientes o barras de chocolate le hab\u00eda producido un shock m\u00e1s fuerte que los montones de cabellos, de zapatos o de gafas, vistos ya tantas veces en fotograf\u00edas o en documentales. El fr\u00edo hab\u00eda sido lo m\u00e1s espantoso. Penetraba en todas partes, en la piel, en los huesos, el soplo helado en el largo corredor de la historia. En la carpa de Auschwitz para las celebraciones todav\u00eda fue relativamente soportable, pero en Birkenau no hubo perd\u00f3n, nunca en su vida hab\u00eda pasado un fr\u00edo tal. Su abuela siempre llevaba varias faldas y chaquetas unas sobre otras y siempre repet\u00eda: \u00ab\u00a1Quien se mantiene abrigado resiste!\u00bb. As\u00ed de abrigada, con varias capas, estaba incluso en el establo, en el calor del establo. Y cuando hac\u00eda un fr\u00edo glacial sol\u00eda decir: \u00abEsto trae la muerte\u00bb. Ese recuerdo, en el viaje de vuelta a su piso bien caldeado de Bruselas, le resultaba penoso, como si hubiera dicho a su vecina de asiento: Birkenau, eso ha sido para morirse. \u00a1Me refiero al fr\u00edo! \u00bfQu\u00e9, si no? \u00a1Era para morirse!\n\nJade\u00f3. Ten\u00eda la nariz taponada. Bostez\u00f3, en el fondo un ansioso respirar hondo, luego empez\u00f3 de nuevo a dormitar. Ten\u00eda el asiento del pasillo en una fila de tres. O\u00eda las voces de sus dos vecinas de asiento, como de lejos, como desde el recuerdo. Hablaban en alem\u00e1n, risue\u00f1as y animadas.\n\nVolvi\u00f3 a verse caminando por la calle del campo, respirando con dificultad, contando como un poseso los pasos, resist\u00eda al viento, marchaba muy inclinado hacia delante, las nubes cerraban el cielo como pesados p\u00e1rpados, la amplia y blanca superficie se volv\u00eda gris ceniza. Notaba que en \u00e9l todo ced\u00eda, no opuso resistencia a esa sensaci\u00f3n, su cabeza cay\u00f3 sobre el pecho. Entonces not\u00f3 un viento ascendente, se sinti\u00f3 llevado hacia lo alto, perdi\u00f3 el suelo bajo los pies, volaba. Se asombr\u00f3 de poder volar, al mismo tiempo ten\u00eda una extra\u00f1a confianza, era l\u00f3gico en cierto modo, ascender en el aire de manera tan leve e incorp\u00f3rea era lo m\u00e1s natural. \u00bfLo estaba viendo alguien? Deseaba que todo el mundo mirara y lo viera: c\u00f3mo se elevaba en el aire, c\u00f3mo sub\u00eda a las nubes girando y balance\u00e1ndose en las corrientes. O\u00eda las voces alemanas, tan cercanas, tan lejanas, hablaban de cosas muy distintas, de arte y literatura, de libros, y ve\u00eda libros abiertos que volaban como p\u00e1jaros, su canto llenaba el aire mientras \u00e9l miraba hacia abajo, a la amplitud del campo. Desde arriba \u2014\u200ceso se daba en primer curso de Arqueolog\u00eda\u2014 se pod\u00eda ver bajo la superficie de la tierra, se ve\u00eda una profundidad que no se percib\u00eda cuando se caminaba sobre ella. Cuando se camina y se mira alrededor, se ve una superficie cubierta de nieve. Si se vuela sobre ella se ven estructuras, espacios separados unos de otros, el campo se descompone en un ret\u00edculo de campos. Seg\u00fan lo que est\u00e9 debajo de la superficie, tierra inocente o despojos hundidos de civilizaci\u00f3n, cad\u00e1veres, piedras de edificios sumergidos, venas de agua o antiguos s\u00f3tanos y sistemas de canalizaci\u00f3n o fosas s\u00e9pticas y letrinas terraplenadas, la superficie reacciona de modo distinto, la vegetaci\u00f3n crece con m\u00e1s o menos fuerza, cuanta m\u00e1s historia tanto m\u00e1s detallado el campo a vista de p\u00e1jaro. Sobre la delgada capa de tierra que cubre las piedras de una civilizaci\u00f3n desaparecida, la vegetaci\u00f3n crece con menos exuberancia que sobre una fosa com\u00fan donde brota la hierba como se espera de la hierba: que crezca y que recubra enseguida. Pero tambi\u00e9n en una superficie enteramente cubierta de nieve hay diferencias visibles: la temperatura del suelo en una tierra inocente es distinta de la de una capa delgada sobre piedras o sobre madera podrida o sobre una fosa com\u00fan; todav\u00eda pasadas unas d\u00e9cadas el proceso de descomposici\u00f3n de los cad\u00e1veres calienta el suelo, y por eso all\u00ed la nieve est\u00e1 helada y ah\u00ed est\u00e1 endurecida y aqu\u00ed est\u00e1 mate y transparente y a punto de derretirse. Quien sobrevuela el terreno ve esa superficie reticulada y sabe d\u00f3nde hay que excavar.\n\nVe\u00eda delante al profesor Krinzinger, su antiguo maestro, que dec\u00eda: La arqueolog\u00eda moderna no empieza excavando, \u00a1empieza volando!\n\nDe pronto, el profesor volaba a su lado, le gritaba algo. \u00bfQu\u00e9? Hab\u00eda un estruendo tan grande en el aire que Martin no le entendi\u00f3 enseguida, ve\u00eda que el profesor se\u00f1alaba una y otra vez con el pulgar hacia el suelo y que gritaba algo al mismo tiempo.\n\n\u00bfQu\u00e9?\n\n\u00a1Baja, baja!\n\nAhora lo entend\u00eda: \u00a1Baja! Tenemos otra misi\u00f3n. Los arque\u00f3logos tenemos que desenterrar civilizaciones y no cr\u00edmenes.\n\nPero...\n\nCaminamos sobre una base insegura, pero pisamos fuerte, con nuestras botas, pateamos el suelo, incluso con pasos suaves es cada paso un pisot\u00f3n, lo importante es que los pies est\u00e9n calientes, Martin ve\u00eda botas, por doquier botas calientes, y ahora percib\u00eda m\u00e1s claramente esas voces femeninas que ya ten\u00eda todo el tiempo en el o\u00eddo.\n\n\u2014A m\u00ed me gust\u00f3 bastante la novela. Pero los sue\u00f1os me atacaban un poco los nervios.\n\n\u2014La novela es cl\u00e1sica.\n\n\u2014S\u00ed, por eso quer\u00eda leer por fin el libro, pero a m\u00ed no me gusta que haya sue\u00f1os en las novelas. Ella siempre est\u00e1 so\u00f1ando algo, la descripci\u00f3n es minuciosa, totalmente surrealista, eso por lo visto es po\u00e9tico. Quiero decir: lo que hace o vive un personaje puedo entenderlo, pero los sue\u00f1os...\n\n\u2014Pero la novela tiene lugar en el fascismo. Entonces uno puede tener pesadillas.\n\n\u2014No, no, lo que digo es que, si en un libro sale un sue\u00f1o, prefiero dormir yo.\n\nMartin ve\u00eda todas aquellas botas, c\u00e1lidas y c\u00f3modas, una clase alemana de colegio de excursi\u00f3n en Auschwitz. Una profesora: \u00a1Thorsten! \u00bfQu\u00e9 te pasa? \u00bfEst\u00e1s dormido? \u00a1\u00danete al grupo!\n\nDos adolescentes hablan en turco. Un profesor los conmina a no hablar all\u00ed turco, uno de ellos responde: \u00a1Aqu\u00ed precisamente no hablamos alem\u00e1n!\n\nMartin se mareaba. Era como si diera vueltas, cada vez m\u00e1s deprisa, a su alrededor todo se volv\u00eda nebuloso, s\u00f3lo a veces fulguraba de s\u00fabito una imagen, o\u00eda una frase, alguien dec\u00eda algo sobre carb\u00f3n, un estudiante preguntaba: Disculpe, \u00bfqu\u00e9 es carb\u00f3n?\n\nUn aviso por megafon\u00eda. Les habla el comandante. Abr\u00f3chense los cinturones. Turbulencias.\n\nMartin Susman est\u00e1 delante del crematorio del campo principal de Auschwitz. Ha visto la c\u00e1mara de gas, los hornos, es exactamente igual que en las fotos que conoc\u00eda, fotos en blanco y negro, y lo que ahora ve\u00eda en la realidad era efectivamente blanco y negro. Se sent\u00eda: \u00bfc\u00f3mo? No pod\u00eda decirlo, no ten\u00eda palabras porque consternado ya no era una palabra alemana, sino una suerte de ap\u00f3sito alem\u00e1n para el alma. Eso era un pensamiento, pero en el sue\u00f1o lo ten\u00eda ante los ojos. Estaba delante del edificio y se encend\u00eda un cigarrillo. De pronto aparecieron dos hombres uniformados: se acercaron a \u00e9l, uno le golpe\u00f3 la mano que sosten\u00eda el cigarrillo, el otro dijo algo en polaco, luego en ingl\u00e9s: No smoking here!\n\nSobre el pecho de Martin oscilaba una tarjeta: GUEST OF HONOUR \/ GOSC HONOROWY \/ EHRENGAST en Auschwitz. Le present\u00f3 la tarjeta al hombre, luego lleg\u00f3 corriendo el se\u00f1or Zeromski y exclam\u00f3: \u00a1Herr Doktor, Herr Doktor, tenemos que meternos en la carpa! Empieza el acto.\n\nSe despert\u00f3, porque el avi\u00f3n ten\u00eda sacudidas, vacilaba y temblaba. Un ni\u00f1o lloraba.\n\nAl d\u00eda siguiente cogi\u00f3 la baja. Se qued\u00f3 cinco d\u00edas en casa. Tres d\u00edas tuvo fiebre. El quinto d\u00eda le vino la idea y esboz\u00f3 un primer borrador para el Jubilee Project.\n\n# CAP\u00cdTULO QUINTO\n\nLOS RECUERDOS NO SON MENOS FIABLES QUE TODO LO DEM\u00c1S QUE NOS IMAGINAMOS.\n\n#\n\nEl amor es una ficci\u00f3n. Fenia Xenopoulou nunca hab\u00eda entendido toda esa parafernalia que hay montada en torno al amor. Ella ten\u00eda ese sentimiento por algo de otro mundo, algo cuya existencia no estaba demostrada, como el agua en el planeta Marte. La gente le\u00eda sobre eso en revistas multicolores, como Jrisi Kardia o Loip\u00f3n, que informan sobre las aventuras amorosas de los artistas de Hollywood y de las estrellas del pop y sobre las bodas de ensue\u00f1o de las princesas. Algunos consideraban posible el amor porque lo deseaban, pero todos los que conoc\u00eda Fenia hab\u00edan capitulado en alg\u00fan momento. Su madre dijo una vez en la peluquer\u00eda sobre la desventurada y enamorada Lady Diana: \u00abLo que ella nunca obtuvo yo nunca lo he obtenido mucho m\u00e1s barato\u00bb.\n\nEn la familia de Fenia, que ella supiera, nadie hab\u00eda amado nunca. En ese sentido enf\u00e1tico de que una exaltaci\u00f3n de sentimientos hubiera sido el motivo de una boda o que la frustraci\u00f3n de esos sentimientos hubiera sido el origen de una tragedia. A excepci\u00f3n de su t\u00edo Kostas, el hermano mayor de su padre, al que Fenia no conoci\u00f3 pero que viv\u00eda en los relatos familiares como el loco que se dio muerte por haber sentido un amor inmortal. Esa contradicci\u00f3n la hab\u00eda inquietado mucho de ni\u00f1a: darse muerte y amor inmortal. Probablemente no se hablaba tanto de \u00e9l como le parec\u00eda a ella retrospectivamente, y era s\u00f3lo que lo que ella o\u00eda excitaba su imaginaci\u00f3n y le produc\u00eda angustia. El t\u00edo Kostas hab\u00eda amado, fan\u00e1ticamente, como dec\u00edan, y como no pudo conseguir a su adorada se march\u00f3 y se uni\u00f3 a la resistencia. Con la palabra adorada, la peque\u00f1a Fenia pensaba en la Virgen Mar\u00eda, en \u00e9xtasis religioso, y eso tal vez no fuera tan equivocado. Pero m\u00e1s a\u00fan la hab\u00eda inquietado en aquel entonces el t\u00e9rmino resistencia. No sab\u00eda qu\u00e9 guerra o qu\u00e9 guerra civil hab\u00eda imperado all\u00ed, eso fue antes de que naciera y para ella estaba tan lejos como las guerras del Peloponeso, de las que oy\u00f3 hablar entonces o poco despu\u00e9s en el colegio. El t\u00edo Kostas, contaban, \u00abno hab\u00eda regresado\u00bb. En su imaginaci\u00f3n, \u00abel t\u00edo de la resistencia\u00bb se encontraba en un infierno en el que los muertos, pero eternamente enamorados, luchaban contra esa fatalidad que se llama amor y adoraci\u00f3n. Se hab\u00eda imaginado ese infierno oscuro, muy sofocante y h\u00famedo, peligroso de un modo poco claro y en cualquier caso no como el lugar al que uno quisiera ir absolutamente..., aunque ella no deseara otra cosa que marcharse de su aldea chipriota abrasada por el sol, de aquella tierra pedregosa, reseca, con los mezquinos olivos cuyo brillo plateado era s\u00f3lo una estafa, o sea, belleza para otros, para turistas embelesados cuyo dinero dejaba sobrevivir al pueblo que desde hac\u00eda mucho tiempo ya no pod\u00eda vivir de las aceitunas. Los turistas llegaban para ser conducidos al \u00abba\u00f1o de Afrodita\u00bb. El agua de ese manantial, dicen, confiere la eterna juventud a quien se ba\u00f1a en \u00e9l. All\u00ed hab\u00eda retozado la diosa del amor con Adonis. Por otra parte, aquella atracci\u00f3n tur\u00edstica era s\u00f3lo una insignificante poza natural en las rocas que hab\u00eda m\u00e1s arriba del pueblo, casi siempre seca, al lado ten\u00eda un gran r\u00f3tulo de madera en el que se le\u00eda:\n\nNON POTABLE WATER\n\nPLEASE DO NOT SWIM\n\nLos turistas fotografiaban la poza seca y el letrero y re\u00edan. \u00c9sos eran, pues, los disc\u00edpulos de la diosa del amor. Despu\u00e9s del colegio, Fenia les vend\u00eda agua mineral, que sub\u00eda al monte en dos bolsas nevera. Ahorraba. Quer\u00eda marcharse.\n\nTard\u00f3 a\u00f1os en comprender que el t\u00edo estaba realmente muerto desde hac\u00eda mucho tiempo, que hab\u00eda ca\u00eddo luchando como partisano y reposaba bajo tierra en alg\u00fan sitio. Los partisanos, pensaba ella ahora, eran gente que no aceptaba la realidad, por tanto ten\u00edan en efecto mucho en com\u00fan con los enamorados. A ella le parec\u00eda absurdo, totalmente absurdo, luchar contra generales griegos siendo griego, en lugar de defenderse contra los turcos, que hab\u00edan ocupado la mitad de la isla.\n\nFenia ten\u00eda una idea diferente de la felicidad y de la lucha a trav\u00e9s de la cual quer\u00eda alcanzar su felicidad. Quer\u00eda marcharse. Y remontar el vuelo. En su calidad de griega chipriota con las mejores notas pod\u00eda estudiar en Grecia. Quer\u00eda estudiar en Atenas. La madre apoyaba el plan de Fenia con sus modestos ahorros. \u00bfQuer\u00eda Fenia a su madre? Sab\u00eda que, en definitiva, se trataba de pagar con creces: se trataba del dinero que ella, una vez coronados con \u00e9xito los estudios, pudiera enviar a casa. Toda la familia hac\u00eda un esfuerzo supremo. \u00c9sa era la definici\u00f3n de amor que entend\u00eda Fenia. Con peque\u00f1os regalos y gran testarudez, el padre movilizaba a las personas que conoc\u00eda, las cuales a su vez conoc\u00edan a otras, hasta que organiz\u00f3 para Fenia un viaje en barco de Limassol a Lavrio. Era un buque de carga que no admit\u00eda pasajeros. El capit\u00e1n estuvo dispuesto a llevar a Fenia, por as\u00ed decirlo, como poliz\u00f3n tolerado. El ferry habr\u00eda sido muy caro y un vuelo, impagable. De Lavrio a Atenas ten\u00eda que arregl\u00e1rselas ella sola. No fue dif\u00edcil, por aquel trayecto pasaba un cami\u00f3n tras otro. Una amiga hab\u00eda profetizado: tendr\u00e1s que pagar con sexo. Fenia no pag\u00f3. Los camioneros dejaban subir a una guapa muchacha y luego ten\u00edan a su lado, en el asiento del copiloto, a una mujer de hielo que impon\u00eda respeto. En Atenas hall\u00f3 alojamiento en casa de unos parientes lejanos. Con la distancia aumenta el precio de la solidaridad familiar. Los parientes pidieron demasiada \u00abcontribuci\u00f3n a la pensi\u00f3n\u00bb, mucho m\u00e1s de lo acordado antes por carta. Su presupuesto, sus ahorros y los de su madre, se desvanecieron con rapidez. No le permit\u00edan coger nada de la nevera que no hubiera comprado ella: aunque pagaba una contribuci\u00f3n alimenticia. Cuando la familia com\u00eda carne por la noche, a ella s\u00f3lo le daban la verdura y las patatas, y al final los huesos de la pierna de cordero, si es que a\u00fan quedaba algo. Se sent\u00eda humillada, pero era demasiado orgullosa para escribirlo a casa. Ten\u00eda la mochila preparada y miraba alrededor. Una compa\u00f1era de la universidad la llev\u00f3 con ella al Spilia tou Platona, el local que marcaba tendencia entre la jrisi neolaia, la \u00abjuventud dorada\u00bb de Atenas.\n\n\u2014\u00bfNo es caro?\n\n\u2014Es caro. S\u00f3lo invertimos en una copa. Y luego seguro que los hombres nos invitan. Los hombres m\u00e1s interesantes frecuentan el Spilia.\n\nAll\u00ed conoci\u00f3 al doctor, el abogado Jorgos Chatzopoulos, al que pronto llam\u00f3 Chatz o Schatz; no estaba claro si eso era s\u00f3lo un diminutivo cari\u00f1oso de Chatzopoulos o una alusi\u00f3n a las amiguitas de los nazis que en su momento fueran estigmatizadas como germaniki Schatz:[3] Jorgos Chatzopoulos era el heredero de un bufete que su abuelo hab\u00eda tomado a su cargo durante la ocupaci\u00f3n alemana, despu\u00e9s de que el abogado jud\u00edo al que pertenec\u00eda fuera deportado. Pero eso Fenia no lo sab\u00eda. Y ten\u00eda un concepto exagerado de Jorgos. Cogi\u00f3 su mochila y se fue a vivir con \u00e9l. Para ella, fue el primer hombre de mundo. Quince a\u00f1os mayor, generoso, un hombre que sab\u00eda discutir de vinos franceses con los camareros en restaurantes caros. Casi habr\u00eda cre\u00eddo que aquello exist\u00eda: ese sentimiento de amor de las princesas. Se casaron. En la boda, Fenia tuvo que re\u00edrse cuando Schatz, en unas palabras que dirigi\u00f3 a los invitados, habl\u00f3 de \u00abamor eterno\u00bb. Sonaba como una historia dulzona de Jrisi Kardia, la revista del coraz\u00f3n. Y, en efecto, \u00e9l hab\u00eda vendido las fotos de la boda a la revista, pero s\u00f3lo apareci\u00f3 un breve art\u00edculo, media p\u00e1gina con dos fotos, y despu\u00e9s result\u00f3 que \u00abvendido\u00bb tampoco era del todo correcto: hab\u00eda pagado por ello.\n\nQu\u00e9 orgullosos estaban sus padres. Y se preocuparon cuando poco despu\u00e9s se enteraron de que Fenia no era feliz. No es que se preocuparan por ella, sino por su matrimonio. El proceso de desencanto de aquella pareja fue demasiado deprisa. Cuando se entregaba a Schatz en la piscina con jacuzzi de su casa, Fenia percib\u00eda con insoportable claridad qu\u00e9 ins\u00edpido era todo aquello: \u00e9l estaba orgullos\u00edsimo de su piscina, no saboreaba el lujo que hab\u00eda logrado en la vida, sino la sensaci\u00f3n de causar impresi\u00f3n con \u00e9l; saboreaba los s\u00edmbolos de una vida privilegiada, pero no la vida misma, estaba entusiasmado con que \u00e9l, \u00e9l, pudiera poseer a esa bella joven, estaba enamorado de s\u00ed mismo, pero ella tuvo enseguida la sensaci\u00f3n de ser intercambiable, \u00e9l cre\u00eda estar \u00abhaciendo el amor\u00bb \u2014\u200cuna formulaci\u00f3n que a ella le parec\u00eda m\u00e1s est\u00fapida que cualquier expresi\u00f3n soez\u2014 y s\u00f3lo estaba am\u00e1ndose a s\u00ed mismo.\n\nA trav\u00e9s de \u00e9l, Fenia accedi\u00f3 a otros c\u00edrculos y all\u00ed vio que su marido no era el gran personaje que fing\u00eda ser en Spilia, sino un nervioso peque\u00f1oburgu\u00e9s que halagaba y lisonjeaba a los verdaderamente ricos, en el fondo un picapleitos que con los peces podridos que tra\u00eda a tierra ganaba lo suficiente para creer que ya estaba en la antesala del dinero y el poder.\n\nCuando Fenia se retrajo y sigui\u00f3 de modo cada vez m\u00e1s consecuente sus propios derroteros, Schatz crey\u00f3 de pronto que la quer\u00eda. Lo demostr\u00f3 mediante reproches expresados de manera emocional, mediante un miedo neur\u00f3tico a perderla, que \u00e9l tomaba por prueba de amor, mediante una borrasca sentimental, tan violenta que se la pod\u00eda confundir con ansias asesinas. Especialmente escandaloso le pareci\u00f3 a Fenia que le exigiera agradecimiento. Era el colmo: pedir agradecimiento a otros cuando uno se ha dado satisfacci\u00f3n a s\u00ed mismo.\n\nLe hab\u00eda facilitado econ\u00f3micamente la vida estudiantil, okay, pero ella lo habr\u00eda conseguido tambi\u00e9n sin \u00e9l, mientras que \u00e9l, sin ella, se habr\u00eda divertido menos y, sobre todo, tal como la hab\u00eda engalanado y exhibido, habr\u00eda tenido menos notoriedad en sus c\u00edrculos. Fenia estudiaba Econom\u00eda, y el balance estaba por debajo del nivel. Y la beca para Inglaterra la hab\u00eda conseguido sin su ayuda y con ella pod\u00eda irse, quer\u00eda marcharse, remontar el vuelo.\n\nPas\u00f3 a ser un matrimonio de fin de semana, cada vez a m\u00e1s distancia, primero en Londres, despu\u00e9s en Bruselas. En la \u00faltima visita que \u00e9l le hizo, al despertarse ella y verlo en su cama, contempl\u00f3 sus sudorosos cabellos grises, su rostro abotargado por el alcohol y pens\u00f3: Hoy me resulta m\u00e1s ajeno que la primera vez.\n\nLe pareci\u00f3 una buena definici\u00f3n para el final.\n\nEse pensamiento la llen\u00f3 de alborozo. Durante el desayuno estuvo tan alegre y relajada como no lo estaba desde hac\u00eda mucho tiempo. Porque todo lo ve\u00eda con claridad. \u00c9se fue el momento en el que Schatz mostr\u00f3 verdadera grandeza. No entendi\u00f3 mal la situaci\u00f3n, pareci\u00f3 liberado tambi\u00e9n, estuvo de buen humor y cuando sali\u00f3 con la maleta del apartamento de ella dijo: El amor es una ficci\u00f3n.\n\nS\u00ed.\n\n\u00a1Que te vaya bien!\n\nS\u00ed. Y a ti tambi\u00e9n.\n\nY qu\u00e9 absurdo era, qu\u00e9 demencial, que Fenia estuviera sentada ahora ante su escritorio, incapaz de trabajar, porque esperaba con anhelo, con el anhelo de una enamorada... la llamada de Fridsch. \u00c9l hab\u00eda regresado la v\u00edspera de su viaje a Doha, hab\u00eda tenido aquella ma\u00f1ana una entrevista con Queneau, en la que tambi\u00e9n iba a mencionar de pasada el tema de Fenia, para sondear qu\u00e9 posibilidades hab\u00eda de que dejara Cultura. Hab\u00eda prometido llamarla en cuanto acabara. Ella, sentada all\u00ed, ten\u00eda la mirada clavada en el tel\u00e9fono. Cog\u00eda el auricular, volv\u00eda a colgarlo. No, ella no pod\u00eda llamarlo, ten\u00eda que hacerlo \u00e9l. Cogi\u00f3 su smartphone, mir\u00f3 si hab\u00eda una llamada perdida suya o si hab\u00eda enviado alg\u00fan mensaje, no, lo puso junto al teclado de su ordenador, pas\u00f3 revista a los e-mails, cuarenta y siete e-mails no le\u00eddos, pero ninguno suyo, volvi\u00f3 a coger el smartphone, s\u00ed, claro, la cobertura era buena, volvi\u00f3 a ponerlo sobre la mesa. Lo que la desconcertaba era que le daba completamente igual lo que Fridsch contara sobre la entrevista, si Queneau hab\u00eda hecho alguna insinuaci\u00f3n que pudiera interpretarse en el sentido de que estaba dispuesto a apoyar su deseo de cambio en el esp\u00edritu de la mobilit\u00e9: lo que ella quer\u00eda era o\u00edr la voz de Fridsch. Contara lo que contara. S\u00f3lo, simplemente, o\u00edr su voz. Se sent\u00eda como... s\u00ed, \u00bfc\u00f3mo? Aquello era absurdo: anhelaba o\u00edr su voz.\n\nMartin Susman entr\u00f3 en el Arca a las ocho de la ma\u00f1ana. De la cantina llegaba al foyer el olor de croissants reci\u00e9n horneados. El aroma, al que normalmente no era capaz de oponer resistencia, le recordaba hoy a una f\u00e1brica de productos qu\u00edmicos y lo tom\u00f3 como una se\u00f1al de que a\u00fan no estaba recuperado completamente. Delante del ascensor se encontr\u00f3 con dos hombres j\u00f3venes de la Task Force Ukraine que trabajaban en la sexta planta. Bohumil Szmekal los hab\u00eda denominado \u00ablos salamandras\u00bb, una expresi\u00f3n que hab\u00eda pasado a emplearse mucho en el Arca cuando hablaban del equipo de trabajo de esa \u00abtask force\u00bb. Por eso se pod\u00eda hablar sobre ellos, sobre \u00ablos salamandras\u00bb, con desprecio o con iron\u00eda aunque estuvieran en la mesa contigua de la cantina. \u00c9sta es la nueva generaci\u00f3n que est\u00e1 con nosotros, hab\u00eda explicado Bohumil, no europeos, sino arribistas de las instituciones europeas, son como las salamandras, uno puede echarlas al fuego pero no se queman, su principal caracter\u00edstica es su indestructibilidad.\n\nEran j\u00f3venes con correctos trajes ajustados, con grandes nudos de corbata y pelo engominado, ya visualmente el contraste m\u00e1s radical con el equipo de la Cultura; escurridizos y \u00e1giles, y con una cortes\u00eda formal que Kass\u00e1ndra calificaba como \u00abaplastante\u00bb: cinco minutos de charla con los salamandras y me deprimo.\n\n\u00bfCu\u00e1l es vuestra labor aqu\u00ed? Hab\u00eda preguntado Bohumil a un salamandra cuando la Task Force Ukraine se instal\u00f3 all\u00ed, m\u00e1s arriba que ellos. Se enter\u00f3 de que iban a desarrollar programas de ayuda para Ucrania a fin de apoyar el movimiento democr\u00e1tico despu\u00e9s de la revoluci\u00f3n del Maid\u00e1n. El desaf\u00edo consist\u00eda en repartir dinero que no ten\u00edan. No les hab\u00edan asignado ning\u00fan presupuesto nuevo propio. Por eso hac\u00edan el cl\u00e1sico repackaging: cuando no se tiene algo nuevo, se le da un nuevo envoltorio a lo viejo. As\u00ed pues, antiguos programas de ayuda, existentes desde hac\u00eda mucho tiempo, fueron empaquetados con nuevos t\u00edtulos y nuevas condiciones en nuevas combinaciones para hacer de ellos nuevos paquetes de ayuda, por lo que con antiguos presupuestos surgieron nuevos combates por la distribuci\u00f3n que llevaron a nuevas estad\u00edsticas en las que nuevos datos porcentuales y curvas gr\u00e1ficas ofrec\u00edan nueva din\u00e1mica. Esa tarea era el bautismo de fuego ideal para aquellos j\u00f3venes ambiciosos: al final no hab\u00eda nada que ganar salvo la propia supervivencia en las condiciones dadas o, mejor dicho, la persistencia de antiguas condiciones con mejores posibilidades personales de futuro.\n\nTener que esperar el ascensor con los dos salamandras no mejor\u00f3 el humor de Martin Susman.\n\n\u00bfC\u00f3mo est\u00e1? \u00abEstupendamente\u00bb habr\u00eda sido la respuesta adecuada, como es natural, pero Martin ten\u00eda ganas de fastidiar, dijo \u00ab\u00a1Estoy hecho una mierda!\u00bb, le encant\u00f3 la cara que pusieron los salamandras y a\u00f1adi\u00f3: \u00a1He cogido un resfriado horrible!\n\nLo siento mucho.\n\nMucho, dijo tambi\u00e9n el segundo salamandra.\n\nY ahora Martin fue a por todas: En Ucrania ha hecho un fr\u00edo del demonio.\n\n\u00a1Oh! \u00bfHa estado en Ucrania?\n\nYes, sir! Y no es de extra\u00f1ar que mi sistema inmunol\u00f3gico se haya desmoronado. Aquella gente est\u00e1 tan frustrada, tan desenga\u00f1ada de nosotros, de la UE. Se sienten abandonados y...\n\nLos salamandras resplandecieron: Oh, s\u00ed, conocemos el problema, tiene usted toda la raz\u00f3n. Nosotros...\n\n\u00a1Toda la raz\u00f3n!\n\nLo sabemos, ahora debemos...\n\nLleg\u00f3 el ascensor, la puerta se abri\u00f3.\n\n\u00bfAl tercero, no?\n\nS\u00ed, dijo Martin.\n\nEl salamandra apret\u00f3 el tres y el seis y dijo: Debemos mejorar nuestra comunicaci\u00f3n. \u00a1Tiene usted toda la raz\u00f3n! Por eso unimos ahora nuestras fuerzas en la comunicaci\u00f3n.\n\nLa Comisi\u00f3n tiene que venderse mejor, y nosotros...\n\nEl ascensor se detuvo, la puerta se abri\u00f3. \u00a1Venderse! \u00a1Si ellos supieran lo que han dicho!, pens\u00f3 Martin. \u00a1Adi\u00f3s!\n\n\u00a1Que tenga usted un buen d\u00eda!\n\n\u00a1Muy buenos d\u00edas! \u00a1Y que se mejore!\n\nLa puerta del ascensor se desliz\u00f3 hasta cerrarse, Martin respir\u00f3 fuerte, con la boca abierta, la nariz estaba taponada. Hab\u00eda vuelto demasiado pronto al trabajo, por otra parte: ten\u00eda que terminar ya su borrador para el Jubilee Project y envi\u00e1rselo por fin a Xeno por correo electr\u00f3nico. Podr\u00eda haberlo acabado tambi\u00e9n en su casa, pero conoci\u00e9ndola, le har\u00eda presentarse enseguida en la sala de reuniones para hablar sobre el borrador con \u00e9l y con algunos otros compa\u00f1eros del departamento. Por tanto, ten\u00eda que estar presente y mantenerse a la espera.\n\nPas\u00f3 junto al despacho de ella, la puerta estaba cerrada. Pas\u00f3 por el despacho de Bohumil, la puerta estaba abierta y Bohumil subido a una escalerilla de tijera. Cuando vio a Martin exclam\u00f3: Ahoj!\n\nAhoj!\n\nMartin ten\u00eda la cabeza tan embotada que s\u00f3lo cuando lleg\u00f3 a su despacho cay\u00f3 en la cuenta de que en realidad tendr\u00eda que haberse detenido y preguntarle a Bohumil qu\u00e9 hac\u00eda subido en esa escalera. Daba igual. Estuvo puliendo durante una hora que le pareci\u00f3 una eternidad sus bullet points, los envi\u00f3 a Xeno. Luego fue reduciendo poco a poco su atasco de e-mails. La mayor parte se respond\u00eda por s\u00ed solo o se hab\u00eda solucionado durante su baja. Hab\u00eda un e-mail de Florian. \u00abQuerido hermano descastado: la semana que viene voy a Par\u00eds, delegaci\u00f3n econ\u00f3mica con Nuestro Honorable Presidente y el pres. de la C\u00e1mara de Comercio. Seg\u00fan parece \u2014\u200ctengo informaci\u00f3n del delegado de comercio austr.\u2014 las pr\u00f3ximas negociaciones ser\u00e1n un \u00e9xito... que acabar\u00e1 siendo una cat\u00e1strofe. El pres. no tiene la menor idea, el convenio que se va a firmar nos expone al chantaje. Me pregunto qui\u00e9nes son aqu\u00ed los cerdos... T\u00fa tienes por fuerza que...\u00bb Martin Susman se levant\u00f3 y se estir\u00f3. Quer\u00eda fumar, necesitaba desesperadamente un cigarrillo. As\u00ed que tan enfermo no estaba. A\u00fan no hab\u00eda reacci\u00f3n de Xeno. Mir\u00f3 en direcci\u00f3n a Bohumil, pero hab\u00eda desaparecido junto con la escalerilla. Martin sali\u00f3 a la escalera de incendios, se fum\u00f3 temblando de fr\u00edo dos cigarrillos, luego regres\u00f3 a su despacho. Escribi\u00f3 el informe sobre el viaje, hizo la nota de los gastos, liquid\u00f3 alg\u00fan otro detalle administrativo, es decir, rellen\u00f3 tablas. Luego tramit\u00f3 peticiones de estudiantes, hab\u00eda otra vez dos que quer\u00edan hacer pr\u00e1cticas; transmiti\u00f3 la petici\u00f3n a otra oficina. Un estudiante estaba haciendo una tesis doctoral sobre European Studies en la Universidad de Passau, su tema era la pol\u00edtica cultural europea, partiendo de la cita de Jean Monnet \u00abSi pudiera volver a empezar, comenzar\u00eda por la cultura\u00bb. Martin Susman no sab\u00eda por qu\u00e9, pero esos e-mails llegaban en un promedio de dos por semana. El estudiante solicitaba un comentario sobre esa cita a la Direcci\u00f3n General de Cultura de la Comisi\u00f3n Europea. La respuesta se escrib\u00eda por s\u00ed sola. No hay ninguna prueba de que Monnet dijera realmente esa frase, a\u00fan menos de que la publicara en alg\u00fan sitio. E incluso si la hubiera dicho, sin m\u00e1s aclaraciones al respecto no estar\u00eda en absoluto claro lo que hab\u00eda querido decir con ese \u00abcomenzar\u00eda por la cultura\u00bb. \u00bfCantar el Himno de la Alegr\u00eda y fundar despu\u00e9s la CECA? La cultura era a priori universal, por tanto siempre hab\u00eda establecido entre los seres humanos una comunidad y compenetraci\u00f3n que tambi\u00e9n, por fin, hab\u00eda que realizar pol\u00edticamente. Y el intercambio de culturas regionales, que se hab\u00eda acreditado en la pr\u00e1ctica como de eminente importancia para la cohesi\u00f3n de Europa, s\u00f3lo gracias a las conquistas pol\u00edticas del proyecto europeo se hab\u00eda hecho posible de modo cada vez m\u00e1s intenso: con la desaparici\u00f3n de las fronteras, con la libertad para viajar y fijar residencia, con el libre comercio en un mercado com\u00fan.\n\nSe detuvo. \u00bfEran frases hueras? Por otra parte: \u00bfexiste una verdad que pueda repetirse cien veces sin que se convierta en frase huera? Le molestaba la nariz taponada, ten\u00eda miedo de que el resfriado acabara en sinusitis, notaba un martilleo alarmante en la frente. \u00bfPor qu\u00e9 se tomaba tanto tiempo con ese e-mail para el estudiante? Su borrador para el Jubilee Project: lo que hab\u00eda escrito all\u00ed no eran frases hueras. A\u00fan no ten\u00eda respuesta de Xeno. Le extra\u00f1aba. Mir\u00f3 el reloj. Era la una. Y Xeno no reaccionaba. \u00bfPor qu\u00e9 no reaccionaba?\n\nSe levant\u00f3, sali\u00f3 del despacho, quer\u00eda dejar aquella celda de trabajo. En el pasillo se tropez\u00f3 con Bohumil.\n\n\u2014\u00bfEst\u00e1s enfermo?\n\n\u2014S\u00ed.\n\n\u2014\u00bfEnfermo de amores?\n\n\u2014\u00bfC\u00f3mo se te ocurre eso?\n\n\u2014Tienes toda la pinta. Completamente confuso.\n\nDavid de Vriend estaba en medio de su habitaci\u00f3n y se preguntaba por qu\u00e9 estaba all\u00ed. Ten\u00eda la intenci\u00f3n de llevar algo a cabo, de hacer algo: pero \u00bfqu\u00e9? No, ahora no se lo preguntaba, miraba s\u00f3lo alrededor como si buscara una ocupaci\u00f3n o \u2014\u200csu mirada recay\u00f3 sobre el tel\u00e9fono\u2014 como si esperase algo, s\u00ed, como si esperase una llamada. Se sent\u00f3 en una butaca mirando al tel\u00e9fono. \u00a1Olvidado! Ten\u00eda la sensaci\u00f3n de estar olvidado, completamente olvidado por toda la gente y hasta por la muerte. \u00bfHab\u00eda a\u00fan alguien que se acordara de \u00e9l?\n\nLa luz de enero, una superficie gris plata en el marco de la ventana, como la puerta de un apartado de correos o de una caja fuerte. La llave, perdida; la combinaci\u00f3n, olvidada. O la puerta de hierro de un b\u00fanker, detr\u00e1s, la oscura galer\u00eda hacia la muerte.\n\nSe levant\u00f3 otra vez, se acerc\u00f3 a la ventana. Abajo estaba el cementerio. \u00bfQui\u00e9n iba a acordarse de \u00e9l? Si todos yac\u00edan all\u00ed, bajo las l\u00e1pidas y bajo la neblina gris. No, todos no.\n\nSe hab\u00eda vuelto hura\u00f1o desde que las personas m\u00e1s allegadas hab\u00edan ido muriendo una tras otra. Los hijos de esas personas segu\u00edan desde hac\u00eda tiempo su propio camino, que los llevaba muy lejos, a un mundo m\u00e1s venturoso o con una desventura muy diferente. En sus \u00faltimos tiempos de Sainte-Catherine a\u00fan le saludaban de vez en cuando por la calle... \u00bfQui\u00e9n ser\u00e1 \u00e9se? Alg\u00fan antiguo alumno, que ya va teniendo el pelo gris. Y \u00e9l, asombrado, devolv\u00eda el saludo. Eso era todo. Y ahora estaba solo en la residencia de ancianos Huis Hanssens, deb\u00eda compartir espacios comunes con personas que eran de su generaci\u00f3n pero que nunca hab\u00edan sido sus coet\u00e1neos porque no hab\u00edan tenido que compartir sus experiencias, la desdicha de ellos era la vejez, la de \u00e9l era la vida. No, all\u00ed no hab\u00eda nada que compartir salvo emanaciones de las bolas de naftalina de los trajes y vestidos, de orina, sudor, de las c\u00e9lulas corrompidas del cuerpo, s\u00f3lo las l\u00e1grimas no tienen olor. Hab\u00eda querido olvidar, pero eso hab\u00eda llevado a que lo olvidaran a \u00e9l.\n\nSe sent\u00f3 a la mesa. Hab\u00eda un bol\u00edgrafo. Se levant\u00f3 de nuevo, mir\u00f3 alrededor, en alg\u00fan sitio deb\u00eda de estar su bloc de notas. \u00bfD\u00f3nde estaba el bloc de notas? Pocos d\u00edas antes hab\u00eda estado all\u00ed una doctora, una psic\u00f3loga de alguna oficina municipal encargada de las residencias de la tercera edad, para tener con \u00e9l una, c\u00f3mo lo llamaban, entrevista de acomodaci\u00f3n. \u00bfUna qu\u00e9? Una entrevista de asesoramiento. Ella hab\u00eda tra\u00eddo aquel gran bloc de notas. Hab\u00eda venido para facilitarle, como ella dijo, la entrada en el ocaso de la vida y procurarle ayuda en la configuraci\u00f3n del ocaso de su vida, sobre todo para quitarle un eventual miedo al ocaso de la vida: dec\u00eda incesantemente \u00abocaso de la vida\u00bb, hasta que David de Vriend la interrumpi\u00f3: Ser\u00eda igual de falaz pero en cierta forma m\u00e1s agradable que, en lugar de \u00abocaso de la vida\u00bb, dijera usted simplemente \u00abetapa de la vida\u00bb. Que era la \u00faltima lo sab\u00eda \u00e9l de todos modos, pero incluso entonces pod\u00eda haber d\u00edas soleados y no s\u00f3lo un perpetuo ocaso. La doctora era tan afanosamente comprensiva. Lo que a De Vriend le pareci\u00f3 especialmente insoportable fue que aquella mujer tan delgada llevara el pelo cortado al rape: \u00bfpor qu\u00e9? \u00bfEso era moderno ahora? En los \u00faltimos tiempos hab\u00eda visto varias veces en las calles a j\u00f3venes con la cabeza rapada y con tatuajes. \u00bfSab\u00edan lo que hac\u00edan, lo que expresaban as\u00ed, qu\u00e9 asociaciones despertaban? \u00c9l hab\u00eda querido olvidar las cabezas rapadas y los esqueletos, y ahora le enviaban a esa mujer. Aquello le puso agresivo. \u00a1M\u00e1rchese! \u00a1Me est\u00e1 ofendiendo! Y se puso pat\u00e9tico: \u00a1Est\u00e1 ofendiendo la memoria del mundo!\n\nLa doctora era comprensiva. Hizo preguntas, al final le cont\u00f3 que hab\u00eda recibido quimioterapia. C\u00e1ncer de mama. Pero que para ella era importante seguir trabajando, porque...\n\nDe Vriend sinti\u00f3 verg\u00fcenza. Y guard\u00f3 silencio. Guard\u00f3 silencio y la dej\u00f3 hablar, ni una palabra equivocada otra vez, de vez en cuando asent\u00eda con la cabeza, y asinti\u00f3 cuando ella sac\u00f3 del bolsillo ese bloc de notas, lo coloc\u00f3 sobre la mesa y dijo: Le he tra\u00eddo esto. Un peque\u00f1o consejo: anote sus pensamientos e intenciones. Yo conozco eso, cr\u00e9ame: uno tiene un prop\u00f3sito y luego lo olvida. Pero si lo escribe enseguida, siempre puede usted comprobarlo: \u00bfhe hecho lo que quer\u00eda hacer? \u00bfMe he ocupado de lo que hab\u00eda estado pensando? Es un buen entrenamiento contra el olvido si uno se acostumbra a escribirlo todo.\n\n\u00bfD\u00f3nde estaba el bloc de notas? Ah\u00ed. Junto a la cama.\n\nSe sent\u00f3 a la mesa, cogi\u00f3 el bol\u00edgrafo. El bloc era de gran formato, por arriba ten\u00eda una tira de cart\u00f3n bajo la cual se pod\u00edan arrancar las hojas. Sobre esa tira, junto al escudo de Bruselas, pon\u00eda: BRUXELLES NE VOUS OUBLIE PAS! \/ BRUSSEL ZAL U NIET TE VERGETEN!\n\nQuer\u00eda confeccionar una lista, los nombres de todos aquellos que hab\u00edan sobrevivido con \u00e9l y que a\u00fan viv\u00edan o que tal vez a\u00fan viv\u00edan, porque \u00e9l no hab\u00eda tenido noticia de su muerte. \u00bfPor qu\u00e9? \u00c9l ten\u00eda recuerdos. Se abr\u00edan paso bruscamente. En su memoria de pronto relampagueaban nombres, De Vriend ve\u00eda rostros, o\u00eda tonos de voz, distingu\u00eda ojos negros, ve\u00eda gestos y ademanes, y sent\u00eda el hambre, esa trituradora de la vida que devora la grasa del cuerpo, despu\u00e9s aniquila los m\u00fasculos y luego el alma, que uno descubre, si acaso, cuando el hambre deviene met\u00e1fora: hambre de vivir. \u00c9l sent\u00eda esa hambre ahora, ya no tan fuerte, pero la sent\u00eda, y quer\u00eda hacer esa lista, escribir con qui\u00e9n hab\u00eda compartido esa hambre y... levant\u00f3 la vista. Hambre era el concepto equivocado, con hambre se describe la sensaci\u00f3n de quienes comen bien pero han suprimido una comida. Eso no ten\u00eda nada que ver con el hambre a la que \u00e9l hab\u00eda sobrevivido. Los vivientes y los supervivientes hablaban s\u00f3lo por casualidad la misma lengua y generaban un perpetuo malentendido al emplear los mismos conceptos.\n\nEmpez\u00f3 a escribir: \u00abSupervivientes\u00bb. As\u00ed quer\u00eda denominar su lista, \u00e9sa era su idea: qui\u00e9n exist\u00eda a\u00fan que hablara su lenguaje. De pronto son\u00f3 el tel\u00e9fono. Se detuvo un momento. Empez\u00f3 a escribir la palabra supervivientes, el tel\u00e9fono le importunaba, dej\u00f3 el bol\u00edgrafo sobre la mesa y fue a contestar.\n\nEra la se\u00f1ora Jos\u00e9phine. Que por qu\u00e9 no bajaba a comer. \u00a1No habr\u00e1 olvidado el almuerzo! Algo tenemos que comer, \u00bfno es cierto, se\u00f1or De Vriend? Vociferaba. No querremos pasar hambre, \u00bfverdad?\n\n\u00c9l era el \u00fanico de los que se val\u00edan que no hab\u00eda ido al comedor y...\n\n\u00bfEl \u00fanico de qu\u00e9?\n\nHay pescado, con arroz y verduras, es saludable y se digiere bien. Y...\n\nS\u00ed. S\u00ed. Se me ha pasado la hora. Ya voy.\n\nDavid de Vriend se anud\u00f3 una corbata, se puso una chaqueta y baj\u00f3 en ascensor al comedor. Mir\u00f3 si hab\u00eda una mesa en la que pudiera estar solo. No. La se\u00f1ora Jos\u00e9phine corri\u00f3 hacia \u00e9l, lo condujo a una mesa. Dijo que se alegraba de que hubiera bajado, que estaba preocupada. \u00bfNo queremos sufrir un mareo, verdad, se\u00f1or De Vriend?\n\nA la mesa hab\u00eda sentados dos hombres y una mujer que la se\u00f1ora Jos\u00e9phine le present\u00f3: un juez jubilado, un catedr\u00e1tico de universidad, de Historia, jubilado tambi\u00e9n y una antigua funcionaria del registro civil, viudos los tres. Eran muy amables y, al momento, De Vriend los encontr\u00f3 insoportables. Estaban tan \u2014\u200cDe Vriend buscaba la palabra\u2014 tan, \u00bfc\u00f3mo se dice \u00aben esta vida\u00bb? Llevaban bastante tiempo all\u00ed, conoc\u00edan el sistema, las estructuras, los h\u00e1bitos, ten\u00edan sus contactos con la direcci\u00f3n y el personal, lo conoc\u00edan todo bien y estaban bien adaptados, m\u00e1s a\u00fan, pod\u00edan ayudar a un reci\u00e9n llegado pero tambi\u00e9n hacerle dif\u00edcil la vida. Eso estaba claro al cabo de pocos minutos. Y luego vino la pregunta: \u00bfY qu\u00e9 ha hecho usted en la vida?\n\nDe Vriend entendi\u00f3, como es natural, que s\u00f3lo quer\u00edan saber qu\u00e9 profesi\u00f3n hab\u00eda ejercido, pero se atragant\u00f3 con la sopa, tosi\u00f3, llegaba entonces su plato de pescado, mientras sus compa\u00f1eros de mesa tomaban ya el postre, cr\u00e8me de lait. De Vriend apart\u00f3 el plato de sopa, empez\u00f3 a comerse el pescado, com\u00eda deprisa, no porque quisiera recuperar tiempo y alcanzar a los otros, sino porque quer\u00eda terminar lo antes posible con aquel almuerzo y marcharse; engull\u00eda el pescado y de pronto sinti\u00f3 que una espina se le met\u00eda en la tr\u00e1quea y se quedaba clavada, expuls\u00f3 aire varias veces con fuerza para desprender la espina, pero tuvo la impresi\u00f3n de que se clavaba a\u00fan m\u00e1s, se pon\u00eda de trav\u00e9s, le entr\u00f3 p\u00e1nico, jade\u00f3 carraspeando una y otra vez con fuerza y tragando y expulsando en\u00e9rgicamente aire. Se puso en pie de un salto, se inclin\u00f3, intent\u00f3 alternativamente tragar la espina y expulsarla, pero estaba clavada en la tr\u00e1quea, le imped\u00eda respirar. Se golpe\u00f3 el pecho, expuls\u00f3 aire con todas sus fuerzas, lo vio todo rojo y grit\u00f3. Primero un aaaaa como un graznido que iba en aumento, luego una maldici\u00f3n, el profesor y la funcionaria del registro se levantaron de un salto, desde otras mesas miraron asustados, la se\u00f1ora Jos\u00e9phine acudi\u00f3 a toda prisa. El profesor le dio golpes en la espalda y dijo: \u00a1Respire! Una y otra vez repet\u00eda: \u00a1Respire! \u00a1Respire! La funcionaria del registro trat\u00f3 de entregarle un vaso de agua, la se\u00f1ora Jos\u00e9phine se coloc\u00f3 detr\u00e1s de \u00e9l, le rode\u00f3 el pecho con los brazos, le apretaba y sacud\u00eda, \u00e9l la repel\u00eda con los codos, jadeaba.\n\nLa funcionaria trat\u00f3 de meterle los dedos en la boca, De Vriend la apart\u00f3 de un empuj\u00f3n, ella vacil\u00f3 y cay\u00f3 sobre una silla.\n\nHist\u00e9rico, gritaba que aquello no pod\u00eda ser verdad, que \u00e9l hab\u00eda sobrevivido a un campo de concentraci\u00f3n y ahora mor\u00eda por una espina, hasta que de pronto se detuvo porque not\u00f3 que s\u00f3lo notaba la espina como una ligera irritaci\u00f3n y que ya ni siquiera pod\u00eda decir si estaba realmente clavada en su garganta. Le sal\u00eda un poco de saliva de la boca, se sent\u00f3 jadeante y dijo finalmente: Est\u00e1 bien. Est\u00e1 bien.\n\n\u00bfTodo normal otra vez?\n\nS\u00ed.\n\n\u00bfSeguro?\n\nS\u00ed.\n\n\u00bfNecesita un m\u00e9dico?\n\nNo.\n\nDe Vriend respir\u00f3 varias veces profundamente, pidi\u00f3 disculpas y se fue a su apartamento.\n\nAll\u00ed se acost\u00f3, pero sent\u00eda una inquietud tan fuerte que no pudo seguir tumbado, se levant\u00f3 de nuevo, se sent\u00f3 a la mesa. All\u00ed estaba el bloc de notas. En \u00e9l, escrito a mano por \u00e9l, una palabra sin terminar: \u00abSuperviv\u00bb. Antes del almuerzo hab\u00eda querido escribir \u00abSupervivientes\u00bb y confeccionar una lista, pero el tel\u00e9fono le hab\u00eda interrumpido. Ahora s\u00f3lo pon\u00eda all\u00ed \u00absuperviv\u00bb, eso era lo que hab\u00eda escrito. \u00bfPor qu\u00e9? Se encendi\u00f3 un cigarrillo y cerr\u00f3 los ojos.\n\nFue justo un enterrador quien conoc\u00eda el camino al amor eterno. El profesor Erhart se hab\u00eda dirigido finalmente a \u00e9l, pregunt\u00e1ndole por el mausoleo del amor incondicional, y aquel hombre estaba enterado. Se llama del amor eterno, no del amor incondicional, dijo, apoyado en su pala, yo no s\u00e9 si existe el amor incondicional. El amor eterno, s\u00ed. Se refiere usted al mausoleo con el coraz\u00f3n iluminado sobre el sarc\u00f3fago, \u00bfno? S\u00ed, claro. Pues se ha equivocado de cementerio. El mausoleo del amor eterno est\u00e1 en el cementerio de Laeken.\n\n\u00bfD\u00f3nde?\n\nLaeken. En el norte de Bruselas.\n\nTom\u00f3 un taxi, dormit\u00f3 durante el viaje, que dur\u00f3 m\u00e1s de lo que hab\u00eda pensado, y se encontraba en un extra\u00f1o estado de trance hipn\u00f3tico cuando lleg\u00f3 a Laeken. Le dol\u00eda el brazo del hematoma, pero ahora, en su estado casi son\u00e1mbulo, sent\u00eda ese dolor s\u00f3lo como una ligera y agradable presi\u00f3n, como si su difunta mujer se le hubiera colgado del brazo, la sent\u00eda como si caminara a su lado, y a cada paso \u00e9l parec\u00eda adaptarse m\u00e1s a su ritmo y a su velocidad. Era una locura, claro. Sacudi\u00f3 la cabeza, literalmente, se llam\u00f3 a la raz\u00f3n. Ahora notaba m\u00e1s el dolor del brazo, y tambi\u00e9n una desagradable sensaci\u00f3n de entumecimiento en los pies hinchados, que mov\u00eda cuidadosamente uno despu\u00e9s de otro como si fueran pr\u00f3tesis a las que no estaba habituado.\n\nNada m\u00e1s cruzar la puerta de entrada hab\u00eda una oficina de la administraci\u00f3n del cementerio. All\u00ed le dieron, a petici\u00f3n suya, el plano en el que estaban consignadas las tumbas de celebridades y los monumentos y lugares conmemorativos de importancia hist\u00f3rica. El empleado marc\u00f3 adem\u00e1s con una X d\u00f3nde estaba el mausoleo del amor eterno. Al profesor le sorprendi\u00f3 que el empleado le diera esa informaci\u00f3n con una expresi\u00f3n de mortal tristeza, que le entregara el plano casi con cara de consternaci\u00f3n. \u00bfQu\u00e9 le ocurr\u00eda a aquel mausoleo para que preguntar por \u00e9l produjera tal reacci\u00f3n? Luego pens\u00f3 que se tratar\u00eda seguramente de una d\u00e9formation professionnelle. Aquel hombre trabajaba en un cementerio y en alg\u00fan momento su rostro se hab\u00eda convertido en una m\u00e1scara de condolencia. Y por eso hasta el amor eterno \u2014\u200cen forma de mausoleo\u2014 no era para \u00e9l sino un motivo de aflicci\u00f3n.\n\nEl profesor Erhart todav\u00eda no hab\u00eda superado la muerte de su mujer. Se preguntaba si a\u00fan dispondr\u00eda de ese tiempo que al parecer cura todas las heridas. Y, en caso afirmativo, si eso ser\u00eda deseable. El dolor que sent\u00eda desde la muerte, precedida de atroz agon\u00eda, de su mujer le recordaba de un modo tan..., s\u00ed, tan vivo la gran felicidad tard\u00eda que hab\u00eda disfrutado con ella, y estaba seguro de que ese recuerdo pasar\u00eda a ser una mera frase huera si la herida llegaba a cicatrizar de verdad.\n\nCaminaba sosteniendo el plano por el camino de grava y se extra\u00f1\u00f3 de no o\u00edr c\u00f3mo cruj\u00eda. Los caminos de grava cruj\u00edan en todas las pel\u00edculas y novelas. Se detuvo. Qu\u00e9 silencio hab\u00eda. Las ramas de los \u00e1rboles se inclinaban silenciosas ante el viento, sin sonido alguno bat\u00edan sus alas las cornejas. A lo lejos algunas personas cruzaban la calle, como sombras que se deslizaban en silencio, como las nubes grises en el cielo. Sigui\u00f3 caminando y... s\u00ed, en efecto: como si hubiera una capa de algod\u00f3n sobre la grava, o\u00eda apenas sus propios pasos.\n\nLuego se hall\u00f3 frente al monumento f\u00fanebre. Se cercior\u00f3 varias veces de que era efectivamente el mausoleo del amor eterno que estaba buscando, pero no cab\u00eda duda. Era triste. \u00bfQu\u00e9 esperaba? El Taj Mahal no, por supuesto, pero al menos algo que enorgulleciera, algo que, a la m\u00e1s hermosa escala humana, diera una expresi\u00f3n arquitect\u00f3nica a la idea y a la experiencia de un amor infinito, eternidad en el material de la eternidad, en la piedra. Pero aquello era una ruina. El tejado con el c\u00e9lebre orificio exactamente medido por el que deb\u00eda caer un coraz\u00f3n luminoso sobre el sarc\u00f3fago estaba quebrado, el costado izquierdo del mausoleo se hab\u00eda desprendido, por lo que se hab\u00edan formado grietas y desviaciones entre los sillares, de los que sal\u00edan malas hierbas, la puerta de hierro adornada con dos corazones en llamas estaba oxidada y cerrada con una cadena, uno de los batientes colgaba torcido de sus bisagras, de lo que resultaba un resquicio por el que se pod\u00eda mirar en el interior, pero no se ve\u00eda el sarc\u00f3fago, sino s\u00f3lo suciedad, incluso basura pl\u00e1stica, \u00bfc\u00f3mo hab\u00eda entrado aquello all\u00ed?\n\nA la izquierda, clavado en el suelo, hab\u00eda un letrero de madera, ya podrido y musgoso, que indicaba que el arriendo de ese terreno sepulcral hab\u00eda terminado en agosto de 1990 y que los eventuales descendientes deb\u00edan presentarse en la administraci\u00f3n del cementerio. Al lado hab\u00eda una placa esmaltada en un marco de hierro forjado que declaraba aquel mausoleo monumento cultural.\n\nLa idea de un amor eterno que toma la eternidad tan al pie de la letra que se asegura tambi\u00e9n su supervivencia en el recuerdo hab\u00eda fascinado a Alois Erhart. Pero mientras la eternidad s\u00f3lo sea algo hecho por los hombres, no un absoluto, sino una relaci\u00f3n entre los humanos, en el fondo un convenio, en alg\u00fan momento, como toda obra humana, se termina r\u00e1pida y despiadadamente.\n\nTen\u00eda que haberlo sabido. \u00c9l hab\u00eda necesitado que pasara una peque\u00f1a eternidad, a saber, hasta su sesenta cumplea\u00f1os, por tanto cuarenta a\u00f1os de matrimonio, para experimentar por primera vez en lo m\u00e1s hondo ese sentimiento: amor eterno. Y entonces hab\u00eda dicho: \u00a1Te querr\u00e9 eternamente!\n\nQu\u00e9 pat\u00e9tico. Y, en efecto, se hab\u00eda asombrado de s\u00ed mismo cuando hab\u00eda pronunciado aquella frase. Hab\u00eda tenido entonces la sensaci\u00f3n de haber llegado. Y m\u00e1s tarde se extra\u00f1\u00f3 de que no hubiera visto enseguida con claridad que no pod\u00eda haber una eternidad: \u00e9sta no es sino un breve descanso en el viaje de la historia. S\u00e9 que te querr\u00e9 eternamente, hab\u00eda dicho \u00e9l: y dos a\u00f1os despu\u00e9s muri\u00f3 su mujer. Y, haya o no haya una vida despu\u00e9s de la muerte, o sea, vida eterna, la frase del amor eterno es, exactamente igual que el sentimiento que hac\u00eda surgir la frase misma, mero recuerdo, o sea, historia.\n\n\u00a1Puro sentimentalismo! En el fondo hab\u00eda sido lo siguiente: Alois Erhart tuvo que llegar a los sesenta a\u00f1os para saber que hab\u00eda realmente eso: buen sexo.\n\nDurante toda su vida no hab\u00eda entendido las crecientes habladur\u00edas y chismorreos sobre \u00abbuen sexo\u00bb. \u00bfHab\u00eda pensado realmente \u00abtoda su vida\u00bb? Eso podr\u00eda haber sido de su padre, que empleaba tales expresiones. En cualquier caso \u00abbuen sexo\u00bb le parec\u00eda una necedad, la dudosa ideologizaci\u00f3n de un instinto humano que no ten\u00eda ni de lejos un fundamento y explicaci\u00f3n tan convincente como la cuesti\u00f3n de qu\u00e9 es la \u00abbuena cocina\u00bb con respecto al instintivo deseo humano de alimentaci\u00f3n. Alois Erhart era de los que opinaban que \u00abse come lo que en casa se pone sobre la mesa\u00bb. Se est\u00e1 agradecido y se hace sobre ello el signo de la cruz. Se hab\u00eda criado en la posguerra, en la reconstrucci\u00f3n, sab\u00eda qu\u00e9 era pasar necesidad, y pronto comprendi\u00f3 que con el incremento del bienestar aumentan las exigencias, pero no entend\u00eda por qu\u00e9 hab\u00eda de serlo el buen sexo, el sexo libre, algo que hab\u00eda que discutir y conquistar pol\u00edticamente, como si fuera una prestaci\u00f3n social a la que cada persona tiene derecho del mismo modo que el libre acceso a la universidad o a cobrar una pensi\u00f3n. Eso hab\u00eda sido en los a\u00f1os sesenta y setenta del siglo anterior, era su generaci\u00f3n la que hab\u00eda proclamado la \u00abrevoluci\u00f3n sexual\u00bb, pero \u00e9l no hab\u00eda formado parte de ella.\n\nSu padre pose\u00eda una tienda de art\u00edculos deportivos en la Mariahilfer Strasse, una de las grandes arterias comerciales de Viena, o sea, estaba bien situada, pero \u00bfde qu\u00e9 sirve la mejor ubicaci\u00f3n de una tienda en una \u00e9poca sin poder adquisitivo? Su padre, entusiasmado con aquella \u00abnueva \u00e9poca\u00bb, hab\u00eda inaugurado la tienda siendo joven, euf\u00f3rico y lleno de audacia, en el a\u00f1o 1937, justo todav\u00eda en \u00e9poca de entreguerras. \u00bfPor qu\u00e9 art\u00edculos de deporte? El padre era un gimnasta fan\u00e1tico, se llamaba a s\u00ed mismo \u00abhermano gimnasta\u00bb como afiliado a la asociaci\u00f3n Turnvater-Jahn de Viena; adem\u00e1s era futbolista, jugaba en el Wacker Viena, donde, tras la venta de Josef Mahal al Austria de Viena, accedi\u00f3 pronto a la primera divisi\u00f3n al convertirse en su sustituto. \u00abEl jud\u00edo Mahal, con su avaricia, me trajo la suerte \u2014\u200chab\u00eda contado su padre\u2014, se pas\u00f3 al Austria por diez chelines por partido, as\u00ed entr\u00e9 yo en el equipo y estuve m\u00e1s que satisfecho con cinco chelines.\u00bb\n\nInauguraci\u00f3n de la tienda. Pero el negocio marchaba mal. \u00bfQui\u00e9n compraba botas de f\u00fatbol en tiempos de desempleo masivo e hiperinflaci\u00f3n, si faltaba el dinero para calzado normal? En aquel entonces muchos ni\u00f1os iban descalzos al colegio. En la tienda, el padre sacaba brillo a las bicicletas, vend\u00eda de vez en cuando una \u00abcamiseta Jahn\u00bb y rozaba la quiebra. En el a\u00f1o 1939 concibi\u00f3 esperanzas cuando consigui\u00f3 por sus contactos vender una partida bastante grande de tiendas de campa\u00f1a y de \u00abvajilla de campamento\u00bb al Jungvolk y a las Juventudes Hitlerianas de Viena, un a\u00f1o despu\u00e9s cerr\u00f3. En 1944, la casa de la Zollerstrasse donde viv\u00edan sus padres qued\u00f3 destruida en un bombardeo, ellos salvaron la vida en un refugio antia\u00e9reo y se mudaron al almac\u00e9n, que a\u00fan exist\u00eda, de la tienda de la Mariahilfer Strasse. All\u00ed naci\u00f3 Alois Erhart. \u00abEres un ni\u00f1o de almac\u00e9n\u00bb, le gustaba decir a su madre, y esa frase a \u00e9l le parec\u00eda tan normal como \u00abQu\u00e9 malos eran los tiempos en aquel entonces\u00bb. Hasta que no fue a la universidad no entendi\u00f3 qu\u00e9 inconcebible era el cinismo de aquella frase y le prohibi\u00f3 a gritos a su madre decirla otra vez.[4] Pasaron a\u00f1os de nuevo hasta que comprendi\u00f3 que su madre era demasiado ingenua para ser culpable, o tambi\u00e9n que su culpa estaba en su ingenuidad y que por eso ella quedaba eximida de toda responsabilidad. Cuando llamaba \u00abni\u00f1o de almac\u00e9n\u00bb a su Loisl, a su Luisito, que hab\u00eda venido al mundo en el almac\u00e9n de su tienda, no era para ella sino un juego con palabras que le resultaban familiares porque en cierto modo estaban en el ambiente, era una broma desatinada en la desatinada miseria que hab\u00eda vivido. Era una \u00abmadre alemana\u00bb, de cuyo gran coraz\u00f3n y empat\u00eda con las personas pr\u00f3ximas a ella hab\u00edan abusado sin que ella misma llegara nunca a entenderlo. Los nazis hab\u00edan hecho un ideal de su imagen de la mujer y de la madre, y ese ideal, sin que ella hubiera tenido otro, no pod\u00eda quedar derogado por una derrota en una guerra. En tiempos malos no estuvo sujeto a modas, y en tiempos mejores estaba tanto m\u00e1s vigente. Abnegada era tambi\u00e9n una palabra as\u00ed, y ella lo era, m\u00e1s a\u00fan ahora que su hijo estudiaba en la universidad, y lloraba cuando el estudiante volv\u00eda a casa y la insultaba llam\u00e1ndola \u00abvieja bruja nazi\u00bb. En esa \u00e9poca le gustaba decir frases que empezaban con estas palabras: \u00abCuando yo falte...\u00bb, entonces \u00e9l la echar\u00e1 de menos. Entonces \u00e9l comprender\u00e1 lo que ella hab\u00eda hecho por \u00e9l. Entonces \u00e9l lamentar\u00e1 lo injusto que hab\u00eda sido con ella. Entonces ya ver\u00e1 que. Entonces ya ver\u00e1 qu\u00e9. Entonces ya ver\u00e1 c\u00f3mo. Cuando ella ya no est\u00e9. Ella, que en opini\u00f3n de su hijo estaba varada en un tiempo pasado, esperaba justicia despu\u00e9s de su muerte, en ella chocaban en su alma las eternidades, el eterno ayer y la vida eterna despu\u00e9s de la muerte. Alois evitaba cada vez m\u00e1s a su madre, su mirada, cuando \u00e9l estudiaba en la mesa de la cocina, las conversaciones con ella, la discusi\u00f3n, las l\u00e1grimas, y se iba corriendo a la Mariahilfer Strasse, a la tienda, se sentaba con sus apuntes en el almac\u00e9n. Pero no era una regresi\u00f3n, no era un retorno al \u00abni\u00f1o del almac\u00e9n\u00bb. Era una huida hacia delante. Hacia el futuro que estaba perfil\u00e1ndose all\u00ed. El auge econ\u00f3mico se percib\u00eda ahora claramente, los negocios de su padre iban cada vez mejor. Desde el campeonato mundial de 1954, las botas de f\u00fatbol con aquellos tacos modernos eran el m\u00e1ximo deseo de todos los chiquillos futbolistas, y ahora, a principios de los sesenta, la mayor\u00eda de los padres pod\u00edan permitirse satisfacer el deseo de sus hijos. Y aut\u00e9nticos balones de cuero. Y aut\u00e9ntico atuendo deportivo. Todo ten\u00eda que ser \u00abaut\u00e9ntico\u00bb, nada de suced\u00e1neos ni de como-si, nada de conformarse con lo que \u00abya hay\u00bb, s\u00f3lo porque durante la escasez se hab\u00eda tenido eso m\u00e1s o menos. Lo que hab\u00eda ahora estaba en los escaparates de las tiendas y en los estantes de los supermercados y uno pod\u00eda comprarlo y uno pod\u00eda permit\u00edrselo. Como su madre, que compraba ahora Fru Fru, en lugar de revolver, como antes, mermelada casera en un vaso de leche agria. Lo hecho en casa era suced\u00e1neo, lo comprado, aut\u00e9ntico. La tienda del padre marchaba bien ahora, y ten\u00eda un vendedor, el se\u00f1or Schramek, antiguo conocido de su \u00e9poca de la asociaci\u00f3n de gimnasia, y finalmente tambi\u00e9n una aprendiza, la Trude.\n\nLa Trudi. Ten\u00eda diecis\u00e9is a\u00f1os, un cuerpo musculoso con el que se mov\u00eda el\u00e1sticamente entre los estantes. Como un animal noble, pensaba Alois, inseguro de si esa comparaci\u00f3n no era muy est\u00fapida. Llevaba el pelo cortado a lo gar\u00e7on, muy corto, que estaba de moda entonces entre las chicas j\u00f3venes y que Alois consideraba de lo m\u00e1s chic. El fin\u00edsimo tejido de su bata de trabajo azul parec\u00eda casi transparente; cuando ella cruzaba las franjas de luz que entraban en la habitaci\u00f3n por la ventana, \u00e9l ve\u00eda los contornos de su cuerpo como a trav\u00e9s de rayos X. Era una chica muy seria, pero a veces, cuando \u00e9l dec\u00eda algo, se re\u00eda con una risa tan alegre e ingenua que Alois se quedaba extasiado y, en lugar de estudiar, reflexionaba sobre c\u00f3mo podr\u00eda hacerla re\u00edr. Le llamaba la atenci\u00f3n que ella, cada vez con m\u00e1s frecuencia, bajo cualquier pretexto miraba desde la tienda hacia atr\u00e1s, hacia el almac\u00e9n. Pero no se re\u00eda de los chistes que \u00e9l se hab\u00eda preparado.\n\nUn a\u00f1o despu\u00e9s se casaron. Alois necesit\u00f3 un permiso de su padre, Trudi era hu\u00e9rfana de guerra y ya estaba inscrita como mayor de edad.\n\nLa huida hacia delante: marcharse de casa. El padre de Alois Erhart conoc\u00eda de antes a un compa\u00f1ero de partido que ahora ten\u00eda influencia en la concesi\u00f3n de viviendas sociales del municipio de Viena. As\u00ed logr\u00f3 la joven pareja un piso barato en el Friedrich-Engels-Hof, del distrito II, justo en el a\u00f1o en que renovaban y cambiaban las letras rojas que hab\u00eda en la fachada de ese complejo de viviendas. Los nazis hab\u00edan arrancado \u00abFriedrich\u00bb y la \u00abs\u00bb de Engels, el bloque de viviendas tuvo que llamarse en la \u00e9poca del nacionalsocialismo \u00abEngel-Hof\u00bb, \u00abPatio del \u00e1ngel\u00bb.\n\nEn el Engels-Hof ya renovado, en su peque\u00f1o piso del ayuntamiento, para Alois Erhart no hab\u00eda nada m\u00e1s lejano que los pisos compartidos y las comunas en las que en aquel entonces se discut\u00eda la revoluci\u00f3n sexual.\n\nCuarenta a\u00f1os despu\u00e9s comprendi\u00f3 lo que era \u00abbuen sexo\u00bb. Que hab\u00eda de verdad esa sensaci\u00f3n.\n\nHab\u00edan seguido juntos despu\u00e9s de que el amor y el deseo se hubieran separado hac\u00eda mucho tiempo. Hab\u00edan seguido juntos despu\u00e9s de haber desaparecido ambos, el amor y el deseo. El respeto y la solidaridad se hab\u00edan instalado en su piso compartido. Entre sus amigos y conocidos, Alois Erhart era el \u00fanico que no estaba divorciado. \u00c9l dec\u00eda: es un buen matrimonio.\n\nFue un domingo, hab\u00edan dormido mucho, pero por el motivo que fuera no se levantaron enseguida, como de costumbre. Un d\u00eda soleado, por las dos ventanas del dormitorio ca\u00edan franjas de luz sobre la cama. \u00c9l la mir\u00f3. Le dol\u00eda la espalda. Ella le puso la mano sobre la espalda. \u00c9l gui\u00f1\u00f3 los ojos ante la luz y luego... \u00bfpor qu\u00e9 hizo eso de pronto? Se incorpor\u00f3 y apart\u00f3 la manta. Le subi\u00f3 el camis\u00f3n, sintiendo al hacerlo un breve y punzante dolor en las v\u00e9rtebras lumbares, como una sacudida el\u00e9ctrica. Gimi\u00f3, ella se quit\u00f3 el camis\u00f3n. Y sonri\u00f3. \u00bfAsombrada? \u00bfInquisitiva? \u00c9l contemplaba el cuerpo de ella, lo observaba cuidadosamente, le\u00eda cada arruga y cada raicilla venosa azul o roja y cada peque\u00f1a capa de grasa como un mapa en el que estaba dibujado un largo camino com\u00fan, un camino de vida con altibajos, y se apret\u00f3 contra ella excitado, llor\u00f3, se apret\u00f3, la luz, la mirada radiosc\u00f3pica y, de pronto, con m\u00e1xima excitaci\u00f3n, lo sinti\u00f3: una fusi\u00f3n en la que sus almas se tocaron.\n\nY ella re\u00eda. Trudi. Sus almas se tocaban. \u00c9se era el secreto, pens\u00f3 Alois Erhart, eso era, pues, el \u00abbuen sexo\u00bb, que le dio una satisfacci\u00f3n insospechada hasta entonces y que al mismo tiempo atizaba una y otra vez el deseo y la avidez: tocar un cuerpo de manera que las almas se tocaran.\n\nDos a\u00f1os despu\u00e9s, Trudi hab\u00eda muerto. El amor eterno. Tan breve era la eternidad.\n\n\u00bfPausa de cigarrillo?\n\n\u00a1Okay!\n\n\u00a1No, espera! Nada de escalera de incendios, dijo Bohumil. Hace mucho fr\u00edo, y t\u00fa sigues enfermo. Ven a mi despacho.\n\nPero el... Martin se\u00f1al\u00f3 con el dedo hacia arriba, no sab\u00eda c\u00f3mo se dec\u00eda detector de humo en ingl\u00e9s. Bohumil le entendi\u00f3:\n\nAntes le he sacado la bater\u00eda. No funciona.\n\nBohumil se sent\u00f3 ante su escritorio, se puso un cigarrillo en la boca y sonri\u00f3 como un pilluelo. Martin Susman se sent\u00f3 frente a \u00e9l en la silla de las visitas, mir\u00f3 al techo.\n\nY para m\u00e1s seguridad he pegado un esparadrapo al sensor. \u00bfFuego?\n\nMartin se encendi\u00f3 un cigarrillo.\n\nSoy funcionario, dijo Bohumil, estoy acostumbrado a trabajar con asiduidad. Adem\u00e1s, pegar una alarma que no funciona... \u00a1Si eso no es una met\u00e1fora de nuestro trabajo! Bueno, por lo menos no pasamos fr\u00edo. Pero dime: \u00bfqu\u00e9 has hecho en Ucrania?\n\n\u00bfYo? \u00bfEn Ucrania? \u00bfDe d\u00f3nde sacas eso?\n\nMe lo han dicho. Un salamandra ha contado que estuviste en Ucrania y le pareci\u00f3 muy interesante lo que comentaste...\n\n\u00a1Qu\u00e9 tonter\u00eda! \u00bfDe d\u00f3nde han sacado eso? He estado en Polonia. Auschwitz. \u00a1Pero si lo sab\u00edas!\n\nPor eso me ha extra\u00f1ado. \u00bfQu\u00e9 nos dice eso sobre nuestra Task Force de aqu\u00ed, de la casa? \u00bfCreen los salamandras que Auschwitz est\u00e1 en Ucrania?\n\n\u00bfY si tuvieran raz\u00f3n? Auschwitz est\u00e1 en todas partes.\n\nTienes fiebre.\n\nS\u00ed.\n\n\u00bfPor qu\u00e9 no te vas a casa y te metes en la cama?\n\nEstoy esperando a Xeno. He de hablar con ella.\n\nMartin sac\u00f3 su smartphone, al hacerlo se le enredaron los dedos en la tarjeta de Auschwitz que llevaba a\u00fan en el bolsillo de la chaqueta, mir\u00f3 si ten\u00eda alg\u00fan mensaje de Xeno; en ese mismo momento, Fenia Xenopoulou, dos despachos m\u00e1s adelante, miraba su Blackberry por si hab\u00eda llegado por fin un mensaje de Fridsch. Esa simultaneidad no era telepat\u00eda ni tampoco casualidad, sino s\u00f3lo probabilidad: porque entretanto Fenia consultaba su tel\u00e9fono cada minuto.\n\nMartin se sac\u00f3 la tarjeta del bolsillo y meti\u00f3 en \u00e9l su smartphone.\n\n\u00bfC\u00f3mo te ha ido en Auschwitz?\n\n\u00a1As\u00ed!, dijo Martin alarg\u00e1ndole la tarjeta de identificaci\u00f3n a Bohumil.\n\nGuest of Honour en Auschwitz, dijo Bohumil. \u00a1Qu\u00e9 guay!\n\nDale la vuelta. Y lee lo que pone.\n\nNo pierda esta tarjeta. En caso de p\u00e9rdida no tiene derecho a permanecer en el campo. \u00bfEs esto..., esto es \u2014\u200cBohumil daba la vuelta a la tarjeta en todas direcciones\u2014 aut\u00e9ntico? \u00bfTe dieron de verdad este distintivo en Auschwitz? \u00bfY lo luc\u00edas colgado del cuello? \u00bfEn serio?\n\nClaro, es muy serio. En el aniversario de la liberaci\u00f3n de Auschwitz el campo est\u00e1 cerrado para turistas, ese d\u00eda hay jefes de Estado, altos representantes y diplom\u00e1ticos de todos los pa\u00edses posibles, hay ciertas medidas de seguridad, bueno, yo lo entiendo, pero...\n\nPero ese emblema es como un chiste malo, como una parodia...\n\nS\u00ed. Todo. Cuando me encend\u00ed un cigarrillo en la calle del campo delante de las ruinas del crematorio, de pronto hab\u00eda un hombre en uniforme delante de m\u00ed que dijo: No smoking in Auschwitz.\n\nBohumil sacudi\u00f3 la cabeza, expuls\u00f3 humo y dijo: Hitler no fumaba.\n\nEra grotesco. Como tambi\u00e9n las m\u00e1quinas expendedoras del campo, de las que se sacaban bebidas calientes. La empresa de esas m\u00e1quinas se llama Enjoy! En Auschwitz hac\u00eda un fr\u00edo horrible y me alegr\u00e9 de conseguir un caf\u00e9 caliente. Pero quiz\u00e1 la normalidad s\u00f3lo nos escandaliza o asombra all\u00ed donde no la esperamos. Quiero decir, esta tarjeta no es una parodia c\u00ednica, es perfectamente normal. Que tenga un efecto grotesco, que deber\u00eda estar redactada de otro modo, configurada de otro modo, s\u00f3lo lo pensamos respecto a ese lugar. Que en cierta manera todo tendr\u00eda que ser distinto: eso s\u00f3lo lo pensamos respecto a ese lugar. Pero si ahora invertimos eso, si por todas partes vi\u00e9ramos lo normal, lo habitual, desde esa perspectiva... \u00bfEntiendes lo que quiero decir? Por eso he dicho antes: Auschwitz est\u00e1 en todas partes. S\u00f3lo que no lo vemos. Si pudi\u00e9ramos verlo, entonces comprender\u00edamos lo curioso, lo c\u00ednico de una normalidad que aqu\u00ed en Europa deber\u00eda ser una respuesta a Auschwitz, una lecci\u00f3n sacada de esa historia. No me entiendas mal, no se trata de una tarjeta redactada con m\u00e1s sensibilidad o de m\u00e1quinas de caf\u00e9 m\u00e1s respetuosas, hablo de algo fundamental...\n\nS\u00ed, okay. Bohumil apag\u00f3 su cigarrillo. La conversaci\u00f3n se le estaba volviendo filos\u00f3fica en exceso. \u00c9l ten\u00eda un car\u00e1cter risue\u00f1o, pensaba que un poco de iron\u00eda bastaba perfectamente para ser un ciudadano cr\u00edtico. No ten\u00eda planes de hacer carrera, pero tampoco le apetec\u00eda arriesgar lo que ten\u00eda o lo que tal vez podr\u00eda alcanzar. Apreciaba a Martin, pero a veces le cansaba su melancol\u00eda. Contempl\u00f3 pensativo el cenicero. Era de hierro colado negro y representaba la caricatura de un africano \u2014\u200clabios abultados, pelo rizado y faldita de rafia\u2014 que formaba con las manos un recipiente para recoger la ceniza. Estaba sentado sobre un taburete en el que pon\u00eda: LE CONGO RE\u00c7OIT LA CIVILISATION BELGE. Hab\u00eda comprado ese cenicero a\u00f1os atr\u00e1s en el mercadillo de Bruselas de la place du Jeu de Balle.\n\nOye, empez\u00f3 Martin.\n\n\u00bfS\u00ed?, dijo Bohumil.\n\nEn ese momento entr\u00f3 Kass\u00e1ndra, que se qued\u00f3 cortada cuando vio la nube de humo. Martin apag\u00f3 su cigarrillo en el cenicero, que s\u00f3lo le llam\u00f3 la atenci\u00f3n en ese momento, Bohumil grit\u00f3: \u00a1Fuego! \u00a1Socorro! \u00a1Los documentos! \u00a1Los documentos! \u00a1Llama a los bomberos! Se ech\u00f3 a re\u00edr, se levant\u00f3, abri\u00f3 la ventana. No te preocupes, dijo, he asesinado al detector de humos.\n\nSois como ni\u00f1os, dijo Kass\u00e1ndra. \u00a1Martin! \u00a1Te buscan! Xeno quiere hablar contigo.\n\nEl cerdo se convirti\u00f3 en poqu\u00edsimo tiempo en estrella medi\u00e1tica. Primero en el peri\u00f3dico gratuito Metro apareci\u00f3 un breve reportaje sobre el hecho de que algunos peatones dec\u00edan haber visto un cerdo que andaba suelto por Sainte-Catherine. El art\u00edculo estaba redactado en tono ir\u00f3nico, como si se hubiera avistado un platillo volante; estaba ilustrado con la foto de archivo de un delicioso cerdito y subtitulado con la frase: \u00ab\u00bfQui\u00e9n conoce a este cerdo?\u00bb. Entonces cada vez m\u00e1s gente llam\u00f3 a la redacci\u00f3n o escribi\u00f3 e-mails diciendo que ellos tambi\u00e9n se hab\u00edan tropezado con ese cerdo, y se quejaban de que tambi\u00e9n hab\u00edan informado a la polic\u00eda pero la polic\u00eda no hab\u00eda tomado en serio lo que ellos hab\u00edan visto, y de que el tono del art\u00edculo y la ilustraci\u00f3n eran una trivializaci\u00f3n y un enga\u00f1o de cara a la opini\u00f3n p\u00fablica, ya que se trataba de un animal mucho m\u00e1s grande y agresivo, tal vez un jabal\u00ed, en cualquier caso, un peligro p\u00fablico.\n\nMetro se dio cuenta del potencial de aquella historia y a\u00f1adi\u00f3 otra en la primera p\u00e1gina. Hab\u00edan preguntado a vecinos de Sainte-Catherine, \u00abciudadanos preocupados\u00bb que se sent\u00edan abandonados y no sab\u00edan si dejar que sus hijos fueran solos al colegio y si las mujeres pod\u00edan salir solas mientras hiciera estragos por las calles un jabal\u00ed posiblemente rabioso. Una tal madame Eloise Fourier preguntaba a la redacci\u00f3n de Metro si el gas pimienta era aconsejable contra los jabal\u00edes, cosa que el profesor Kurt van der Koot, profesor de la Frije Universiteit Brussel, neg\u00f3 en respuesta a las preguntas de Metro. El gas pimienta s\u00f3lo pod\u00eda aumentar el descontrol del Sus scrofa, \u00e9se era el t\u00e9rmino t\u00e9cnico. La pimienta, como por cierto tambi\u00e9n la sal y el comino, s\u00f3lo era recomendable, por tanto, para el asado de cerdo. Ese chiste malo del profesor, quien hasta entonces era un desconocido para el gran p\u00fablico y, como se supo despu\u00e9s, investigador del comportamiento y especialista en lobos, desencaden\u00f3 una shitstorm que llev\u00f3 a que la chispa saltara a otros peri\u00f3dicos. El peri\u00f3dico Le soir public\u00f3 una entrevista con el jefe de polic\u00eda de la comisar\u00eda Centre Ville, un flamenco que ya estaba desde hac\u00eda mucho tiempo en el punto de mira del peri\u00f3dico. Adem\u00e1s, el deseo del peri\u00f3dico de ejecutar a aquel hombre se un\u00eda a la ingenuidad con que \u00e9ste se hizo el harakiri (\u00ab\u00bfQu\u00e9 medidas ha tomado?\u00bb \u00abHe dado orden a los perreros municipales de capturar a ese cerdo cuando lo descubran en alg\u00fan sitio.\u00bb \u00ab\u00bfPor qu\u00e9 a los perreros?\u00bb \u00abTenemos muchos perros vagabundos. Por eso el ayuntamiento dispone de empleados que los recogen. Pero no tenemos laceros que recojan cerdos.\u00bb El peri\u00f3dico hizo el siguiente comentario: \u00abEl plan es tan perfecto como su franc\u00e9s\u00bb.) Cada vez interven\u00edan m\u00e1s testigos oculares, De Morgen tra\u00eda ahora cada d\u00eda un plano de la regi\u00f3n de Bruselas en el que estaba marcado gr\u00e1ficamente con banderines d\u00f3nde y cu\u00e1ndo hab\u00eda aparecido otra vez el cerdo. As\u00ed se dieron cuenta entonces de que entretanto el cerdo era omnipresente. Por ejemplo, un d\u00eda lo vieron en Anderlecht, poco despu\u00e9s en Uccla y luego lo avistaron otra vez en Molenbeek.\n\nEl profesor Kurt van der Koot, empe\u00f1ado en recuperar su buen nombre, public\u00f3 en De Morgen un art\u00edculo marcadamente objetivo en el que relacionaba la m\u00e1xima velocidad que pod\u00eda alcanzar un cerdo avanzando a la carrera con las distancias que ten\u00eda que haber recorrido, y as\u00ed demostr\u00f3 que emp\u00edricamente s\u00f3lo hab\u00eda dos posibilidades: o bien, tesis primera, no se trataba s\u00f3lo de un cerdo, sino que ten\u00eda que haber varios. Porque seg\u00fan el diagrama trayectoria-tiempo era completamente imposible que un \u00fanico cerdo hubiera podido estar donde los testigos oculares dec\u00edan haberlo visto. Pero tambi\u00e9n, tesis segunda, no hab\u00eda cerdo alguno, sino s\u00f3lo la ficci\u00f3n de un cerdo en las mentes de una poblaci\u00f3n irresponsablemente insegura, es decir, una proyecci\u00f3n hist\u00e9rica colectiva. Hab\u00eda habido en la historia, a\u00f1ad\u00eda, algunos casos documentados de tal histeria colectiva, por ejemplo en la cr\u00f3nica de la ciudad de N\u00faremberg se menciona que en el a\u00f1o 1221 hab\u00eda sido visto un unicornio. Aunque \u00e9l era esc\u00e9ptico en cuanto a que el cerdo de Bruselas fuera efectivamente un caso comparable: porque en todos los ejemplos hist\u00f3ricos se hab\u00eda tratado de seres fabulosos y no de animales domesticados, adem\u00e1s desde el final de la Edad Media no se hab\u00eda vuelto a ver ni a describir un animal fabuloso dotado de facultades sobrenaturales, como la omnipresencia. Pero de eso deduc\u00eda \u00e9l que en el caso actual no pod\u00eda tratarse ni de un cerdo de ficci\u00f3n ni de un \u00fanico cerdo, sino de una manada de cerdos observados por separado en distintos lugares de Bruselas.\n\n\u00a1Una manada! \u00bfY qu\u00e9 hac\u00eda el jefe de la polic\u00eda? \n\n# CAP\u00cdTULO SEXTO\n\n\u00bfSE PUEDE PLANEAR UN COMEBACK DEL FUTURO?\n\n#\n\nThe past forms the future, without regard to life.\n\nDif\u00edcil decir por qu\u00e9 esa frase hizo feliz a Fenia Xenopoulou o, si feliz era quiz\u00e1 una palabra excesiva, en cualquier caso la puso de buen humor. Fridsch hab\u00eda llamado, por fin hab\u00eda llamado y le hab\u00eda dicho que un cambio a corto plazo a otra direcci\u00f3n general era casi imposible. Pues la Comisi\u00f3n hab\u00eda sido renovada hac\u00eda poco y el presidente esperaba ahora, sobre todo de los funcionarios de niveles directivos, que primero se afianzase cada cual en su puesto. A\u00fan era muy pronto para cambios de posici\u00f3n y cargos. But... para poner mucho \u00e9nfasis en la confortante parte siguiente de esa informaci\u00f3n, Fridsch acentu\u00f3 en especial but, hizo una breve pausa, y Xeno pens\u00f3 en butter, mantequilla, \u00faltimo tango en Par\u00eds, luego en butterflies, mariposas, ella sent\u00eda butterflies en el est\u00f3mago, al menos ten\u00eda esa sensaci\u00f3n, y Fridsch dijo otra vez but, y: que Queneau la ten\u00eda muy presente y tambi\u00e9n otros funcionarios influyentes, todos estaban satisfechos con su trabajo hasta la fecha, su rendimiento hasta el momento gozaba realmente de m\u00e1xima aceptaci\u00f3n y ahora la cuesti\u00f3n no era qu\u00e9 deseaba ella, sino que permaneciera visible y siguiera llamando la atenci\u00f3n; Xeno escuchaba, no hab\u00eda sufrido un desenga\u00f1o, estaba okay, s\u00ed, s\u00ed, estaba okay, y luego ya no supo lo que \u00e9l hab\u00eda dicho entonces, c\u00f3mo fue la transici\u00f3n, el caso es que \u00e9l dijo de pronto esa frase: \u00abThe past forms the future, without regard to life\u00bb. Esa frase se le qued\u00f3 grabada, todav\u00eda un rato despu\u00e9s de la conversaci\u00f3n telef\u00f3nica segu\u00eda d\u00e1ndole vueltas en la cabeza, la tradujo para ella a su lengua materna y comprob\u00f3 que no s\u00f3lo en tratados internacionales y en leyes hab\u00eda que tener en cuenta los m\u00ednimos matices de cada palabra en las diferentes traducciones, sino tambi\u00e9n en una..., s\u00ed, \u00bfqu\u00e9?, en una frase tan extraordinariamente personal. Una frase, simplemente. Sobre la vida. Su vida. Una frase sobre la vida, tan clara como un par\u00e1grafo jur\u00eddico, que sin embargo en griego, como constat\u00f3 con asombro, necesitaba interpretaciones que complicaban la frase de un modo infernal... \u00bfCon qu\u00e9 concepto hab\u00eda que traducir the past? El pasado, parelth\u00f3n, y la historia, ist\u00f3ria, no son en griego tan acordes como en the past, que en cierto modo tambi\u00e9n incluye la history. \u00bfTodo lo sucedido? Sucedido \u00bfa qui\u00e9n? \u00bfHistoria individual? Por tanto, \u00bflo vivido, la biograf\u00eda? \u00bfO m\u00e1s bien en general, por as\u00ed decirlo la historia universal? En ingl\u00e9s eso se deja sin delimitar, sin embargo se tiene la sensaci\u00f3n de m\u00e1xima exactitud. En griego hay que aclarar en la traducci\u00f3n esas cuestiones..., y todo deviene as\u00ed menos claro y en cierto modo m\u00e1s limitado, algo que hay que interpretar. \u00bfTiene lo ya pasado un comienzo definido y un final definido, o no est\u00e1 claro cu\u00e1ndo comenz\u00f3 y si termin\u00f3? \u00bfSe repite o ha sido \u2014\u200co es\u2014 \u00fanico e irrepetible? De ello depend\u00eda la construcci\u00f3n del verbo griego, en ingl\u00e9s estaba en presente, pero en la traducci\u00f3n hab\u00eda que elegir quiz\u00e1 el aoristo o el imperfecto o el perfecto, seg\u00fan se definiera lo que hizo o hab\u00eda hecho el pasado. Y la divert\u00eda que, entonces, esa frase inglesa viniera a decir, en \u00faltimo t\u00e9rmino, exactamente esto: que el origen de ella se hallaba en contradicci\u00f3n con su vida; el llegar a esta convicci\u00f3n quiz\u00e1 fuese ya la traducci\u00f3n o al menos una interpretaci\u00f3n v\u00e1lida de \u00abThe past forms the future, without regard to life\u00bb.\n\nMand\u00f3 llamar a Martin Susman. \u00c9ste entr\u00f3 por fin en su despacho, permaneci\u00f3 de pie indeciso. Fenia sonri\u00f3. \u00c9l estaba asombrado, no la conoc\u00eda as\u00ed. Recibi\u00e9ndolo con una sonrisa. Con gesto amable. Martin s\u00f3lo pod\u00eda entenderlo mal. \u00bfTan buena impresi\u00f3n le hab\u00eda causado el borrador que le hab\u00eda enviado? No esperaba eso, ya se hab\u00eda arrepentido de haber escrito y enviado ese documento con fiebre, o sea, sin control, por otra parte...\n\nEra el traje, el traje gris de Martin, barato y con rodilleras. Un hombre con un m\u00ednimo de sentido de la elegancia, pensaba Fenia, nunca comprar\u00eda un traje as\u00ed. Pero un hombre a quien la idea o el sentido de la elegancia le daba completamente igual, tampoco. \u00c9ste llevar\u00eda puesto con indolente indiferencia algo funcional aunque c\u00f3modo, pero jam\u00e1s ese disfraz de rat\u00f3n. Fenia mir\u00f3 a Martin y se imagin\u00f3 c\u00f3mo en una casa de confecci\u00f3n, en la secci\u00f3n que se llamaba, de modo totalmente inapropiado para \u00e9l, \u00abde caballeros\u00bb, pasaba con la mano varios trajes colgados de la barra y de pronto se\u00f1alaba ese traje gris y dec\u00eda: Quiero probarme \u00e9se.\n\nToma asiento, Martin, por favor.\n\nLe parec\u00eda tan c\u00f3mico aquello. La idea de que en el probador se met\u00eda en ese traje, se miraba en el espejo y pensaba: \u00a1S\u00ed! \u00a1Me queda bien! Y c\u00f3mo se daba la vuelta un momento ante el espejo y le dec\u00eda al vendedor: Me lo dejo puesto.\n\nTuvo que contener la risa.\n\nMartin, aunque sin saber a qu\u00e9 atenerse, estaba encantado. Una desconcertante sensaci\u00f3n.\n\n\u00bfHas le\u00eddo mi borrador?, pregunt\u00f3.\n\nS\u00ed, claro, dijo ella. No pod\u00eda apartar la vista de su atuendo, sonriendo clavaba miradas en su traje como agujas en un mu\u00f1eco de vud\u00fa. Siempre llevaba puesto un traje gris as\u00ed, nunca lo hab\u00eda visto vestido de otro modo. Se imagin\u00f3 que necesitara un traje nuevo. El \u00fanico nuevo en el que se reconoc\u00eda en el espejo era otra vez exactamente el mismo traje gris. Con cualquier otro, \u00e9l pensar\u00eda: \u00e9se no soy yo. La costumbre no da seguridad, al contrario, uno se vuelve inseguro. Frente a todo lo dem\u00e1s. A rayas: demasiado formal. Azul, era quiz\u00e1 apropiado para la noche, pero no para el d\u00eda. Una tela m\u00e1s clara, demasiado dandi. Cualquier dibujo, cualquier corte moderno, no eran para el trabajo, la oficina no es una pasarela. Fenia se imaginaba c\u00f3mo se empe\u00f1aba el vendedor en presentarle alternativas; no, no. Martin empezar\u00eda a sudar, casi presa del p\u00e1nico, el traje gris est\u00e1 okay, dir\u00eda, me quedo con el gris, \u00e9se soy yo. El de gris.\n\nFenia Xenopoulou baj\u00f3 la cabeza, ten\u00eda el borrador de Martin colocado delante de ella sobre la mesa, pasaba muy suavemente la yema del dedo coraz\u00f3n sobre el papel, de ac\u00e1 para all\u00e1, de ac\u00e1 para all\u00e1, luego levant\u00f3 la vista, mir\u00f3 a Martin y dijo: \u00a1Auschwitz! \u00bfQu\u00e9 has pensado al proponerlo? He de admitir que me asust\u00e9 al leerlo. Pens\u00e9 que est\u00e1s... espera. Aqu\u00ed: Auschwitz como lugar de nacimiento de la Comisi\u00f3n Europea. Ah\u00ed est\u00e1. Pens\u00e9: esto es una locura. \u00bfQu\u00e9 te ocurre, Martin? \u00bfEst\u00e1s enfermo?\n\nSudaba, se pasaba con la mano el sudor de la frente al pelo, dijo: He estado enfermo varios d\u00edas, s\u00ed. Me enfri\u00e9 en... Me he enfriado durante el viaje. Pero. Ya estoy mejor.\n\nBueno. Pero \u00bfpuedes explicarme esto? Estamos buscando una idea que podamos poner, s\u00ed, que hemos de poner en el centro de nuestro aniversario. En eso est\u00e1bamos de acuerdo: un aniversario es un motivo pero no es una idea. Por tanto, c\u00f3mo podemos lograr que la gente note que la Comisi\u00f3n es necesaria, m\u00e1s a\u00fan, no s\u00e9 c\u00f3mo decirlo. Que somos sexis, que tenemos algo \u2014\u200ccarraspe\u00f3\u2014, s\u00ed, que la gente est\u00e1 contenta de que existamos. Que la gente espera cosas de nosotros. Que hay algo que nos une. \u00bfComprendes? \u00c9sa ser\u00eda la idea. Y t\u00fa vienes con Auschwitz.\n\nTodav\u00eda una hora antes, cuando fumaba con Bohumil en su despacho, Martin Susman habr\u00eda estado contento si Xeno le hubiera dicho que su propuesta era un perfecto absurdo, tiramos esto a la papelera y asunto olvidado. Hab\u00eda tenido miedo de eso... pero tambi\u00e9n esperanza. M\u00e1s vale ahora la breve humillaci\u00f3n, hab\u00eda pensado, que despu\u00e9s todo ese trabajo que sin duda alguna comportar\u00eda una serie de rechazos y de complicaciones en la casa. Pero tal como le ven\u00eda ahora Xeno, esa mujer blindada, con una sonrisa que a \u00e9l en principio le hab\u00eda sorprendido pero que en realidad parec\u00eda como puesta en el rostro con una pincelada de Photoshop, esa vulgar afectaci\u00f3n frente a la que \u00e9l estaba sentado sin parar de sudar, eso no pod\u00eda aceptarlo. Hab\u00eda...\n\nPero en el paper he explicado por qu\u00e9 tenemos que partir de Auschwitz. Okay, eran s\u00f3lo algunas frases breves, pens\u00e9...\n\nEntonces expl\u00edcamelo otra vez, Martin.\n\nSe levant\u00f3, llevaba una falda negra con una cremallera ribeteada de rojo trazada en diagonal. Martin pens\u00f3: como si su vientre estuviera tachado. Y sin embargo provisto de un mecanismo para poder abrirlo en un santiam\u00e9n.\n\n\u00bfCaf\u00e9? Sobre una mesita lateral ten\u00eda una m\u00e1quina de Nespresso propia. \u00bfLeche? \u00bfAz\u00facar? Martin neg\u00f3 con la cabeza. Ella volvi\u00f3 a sentarse ante el escritorio, sosteniendo su taza de caf\u00e9 con ambas manos. Martin pens\u00f3 que \u00e9l hab\u00eda sostenido la taza de caf\u00e9 en Auschwitz exactamente as\u00ed para calentar sus dedos entumecidos.\n\nMartin tosi\u00f3. Sorry, dijo y: \u00a1Es sin embargo la idea de la Comisi\u00f3n, as\u00ed lo pone en los documentos fundacionales, en aquellas declaraciones de intenciones y sideletters! Okay, suena bastante abstracto, pero tambi\u00e9n es clar\u00edsimo: la Comisi\u00f3n no es una instituci\u00f3n internacional sino supranacional, por tanto no sirve de mediadora entre naciones, sino que est\u00e1 por encima de las naciones y defiende los intereses comunes de la Uni\u00f3n y de sus ciudadanos. No busca compromisos entre naciones, quiere superar los cl\u00e1sicos conflictos y antagonismos nacionales en un desarrollo posnacional, es decir, en lo com\u00fan. Se trata de lo que une a los ciudadanos de este continente, y no de lo que los separa. Monnet escribi\u00f3...\n\n\u00bfQui\u00e9n?\n\nJean Monnet. Escribi\u00f3: Los intereses nacionales son abstractos, lo com\u00fan de los europeos es concreto.\n\nFenia vio que hab\u00eda llegado un e-mail. S\u00ed, \u00bfy qu\u00e9 m\u00e1s?, dijo. Nacional, supranacional: eso era para ella hilar demasiado fino, ella era chipriota, pero en cuanto a identidad nacional era griega. El e-mail era, como ve\u00eda, de Fridsch. Lo abri\u00f3 y dijo: \u00bfqu\u00e9 tiene que ver eso con Auschwitz?\n\nLo que es o ha de ser la Comisi\u00f3n, dijo Martin, s\u00f3lo se pudo pensar despu\u00e9s de Auschwitz. Una instituci\u00f3n que lleva a los Estados a renunciar poco a poco a derechos nacionales de soberan\u00eda y...\n\n\u00bfCu\u00e1ndo? \u00bfD\u00f3nde?, tecle\u00f3 Fenia. (Fridsch le hab\u00eda preguntado si tendr\u00eda tiempo y ganas de cenar con \u00e9l.)\n\n\u00a1Auschwitz!, dijo Martin. Las v\u00edctimas llegaron de todos los pa\u00edses de Europa, todos llevaban puestos la misma ropa a rayas, todos viv\u00edan a la sombra de la misma muerte, y todos ten\u00edan, en la medida en que sobrevivieron, el mismo deseo, a saber, la garant\u00eda, v\u00e1lida para todo el tiempo futuro, del reconocimiento de los derechos humanos. Nada en la historia ha unido las distintas identidades, mentalidades y culturas de Europa, las religiones, las denominadas razas, tan diversas, y las ideolog\u00edas, antes tan enfrentadas, nada ha creado una comunidad tan fundamental de todos los seres humanos como la experiencia de Auschwitz. Las naciones, las identidades nacionales, todo hab\u00eda perdido validez, espa\u00f1ol o polaco, italiano o checo, austriaco, alem\u00e1n o h\u00fangaro, todo hab\u00eda perdido validez, la religi\u00f3n, el origen, todo hab\u00eda quedado subsumido en un anhelo com\u00fan, el deseo de sobrevivir, el deseo de una vida en dignidad y libertad.\n\n\u00bfAl italiano? (Fridsch)\n\nO.K. (Fenia)\n\nEsa experiencia y la convicci\u00f3n general de que ese crimen no ten\u00eda que repetirse jam\u00e1s fue lo que hizo posible el proyecto de la uni\u00f3n de Europa. O sea, de que existamos nosotros. Y por eso Auschwitz es...\n\nFenia mir\u00f3 a Martin, dijo: But...\n\n\u00a1\u00c9sta es la idea! La superaci\u00f3n del sentimiento nacional. Nosotros somos los guardianes de esa idea. Nuestros testigos son los supervivientes de Auschwitz. Los supervivientes no son s\u00f3lo testigos de los cr\u00edmenes cometidos en los campos, son tambi\u00e9n los testigos de la idea surgida de ellos, es la idea de que realmente hay algo com\u00fan, y...\n\nPasta Divina, 16, rue de la Montagne. 8 pm? (Fridsch)\n\nO.K.! (Fenia)\n\nMartin ten\u00eda la impresi\u00f3n de que Xeno reflexionaba y a\u00f1adi\u00f3: La seguridad de una vida con dignidad, felicidad, derechos humanos, eso es desde Auschwitz una exigencia perpetua. \u00bfNo es cierto? Eso lo entiende cualquiera. Eso tenemos que hacerlo patente: que somos la instituci\u00f3n de esa exigencia. Los guardianes de ese acuerdo perpetuamente v\u00e1lido. Nunca m\u00e1s: eso es Europa. \u00a1Somos la moraleja de la historia!\n\nFenia lo mir\u00f3 asombrada. Qu\u00e9 vital era de pronto aquel sudoroso hombre gris.\n\nPor ello fueron seres humanos a la muerte, su muerte ha sido un crimen, para cada uno en particular absolutamente absurdo, pero queda sin embargo esta \u00faltima consecuencia: por ello fueron a la muerte, en \u00faltimo t\u00e9rmino, y eso permanece por toda la eternidad.\n\nAunque Xeno no ten\u00eda verdadera conciencia de ello, sonaba como un eco proveniente de la honda caverna oscura de su propia prehistoria, sonaba como: inmortal, ir a la muerte.\n\nMir\u00f3 a Martin. Ahora parec\u00eda muy seria, muy pensativa. \u00c9l se pregunt\u00f3 si a lo mejor la hab\u00eda convencido, aunque no hab\u00eda terminado a\u00fan con su argumentaci\u00f3n.\n\nFenia no reflexionaba mucho nunca sobre s\u00ed misma y, cuando lo hac\u00eda, era sobre posibilidades, sobre objetivos, pero no sobre estados de \u00e1nimo ni sentimientos. Buen estado general: \u00e9se era para ella el estado ideal de insensibilidad, en un sentido muy amplio, y eso significaba no sentirse importunada por estados de \u00e1nimo. Los sentimientos eran para ella estados de \u00e1nimo.\n\n\u00bfTienes cigarrillos?\n\nS\u00ed, claro, dijo Martin extra\u00f1ado.\n\nFenia se levant\u00f3, abri\u00f3 la ventana y dijo: \u00bfMe das uno?\n\nNo sab\u00eda que fumabas.\n\nA veces. Muy raras veces. Uno.\n\nEstaban de pie muy cerca uno del otro en el estrecho \u00e1ngulo del batiente abierto de la ventana y fumaban; Martin esperaba que ella dijera algo, ten\u00eda la impresi\u00f3n de que quer\u00eda decir algo, pero echaba bocanadas de humo con el gesto forzado del fumador aficionado, hac\u00eda un fr\u00edo g\u00e9lido, finalmente Martin dijo: Es la \u00faltima oportunidad.\n\nElla lo mir\u00f3 con asombro. Hac\u00eda un fr\u00edo horrible junto a la ventana abierta, Martin pens\u00f3 que ten\u00edan que acercarse m\u00e1s para darse calor mutuamente, se asust\u00f3, trat\u00f3 de ganar distancia, y ella dijo: \u00bfC\u00f3mo dices?\n\nSon cada vez menos, dijo Martin. Muy pronto ya no quedar\u00e1 nadie que haya sobrevivido a un campo de exterminio. \u00bfEntiendes? Debemos ponerlos en el centro de la conmemoraci\u00f3n. \u00c9sta es la idea: ellos dan testimonio de a qu\u00e9 monstruosos cr\u00edmenes llev\u00f3 el nacionalsocialismo en la vieja Europa y, al mismo tiempo, dan testimonio de todo lo com\u00fan que se hizo patente de modo tan radical mediante los campos de exterminio, a saber...\n\nHac\u00eda un fr\u00edo tan atroz junto a la ventana abierta.\n\n... y la Comisi\u00f3n es garante de todo lo com\u00fan en materia de dignidad y de justicia y por eso...\n\nMartin arroj\u00f3 el cigarrillo por la ventana, retrocedi\u00f3 un paso, Fenia lanz\u00f3 fuera asimismo el cigarrillo y cerr\u00f3 la ventana.\n\n\u00bfSe sabe cu\u00e1ntos viven a\u00fan?\n\nNo lo s\u00e9. S\u00f3lo s\u00e9 que en el aniversario de la liberaci\u00f3n de Auschwitz apenas estuvieron presentes una docena, y todos ten\u00edan entre ochenta y cinco y noventa y cinco a\u00f1os, calculo. Hace pocos a\u00f1os fueron por lo visto m\u00e1s de doscientos.\n\nVale. Entonces averigua eso: \u00bfcu\u00e1ntos viven a\u00fan? Y luego discutiremos qu\u00e9 hacemos concretamente, c\u00f3mo los ponemos en el centro de la celebraci\u00f3n. Todos, o... \u00bfsabes lo que veo ante m\u00ed? Miles...\n\n\u00a1Tantos ya no viven, seguro!\n\nNo, espera. Si los invitamos a todos con sus familias y descendientes, los hijos, los nietos, bisnietos, entonces ser\u00e1n a lo mejor miles, y luego, c\u00f3mo dir\u00eda yo \u2014\u200chizo un amplio movimiento con la mano\u2014, entonces nos declaramos todos simb\u00f3licamente hijos suyos. Y a nuestros hijos los declaramos nietos suyos y...\n\nNo lo s\u00e9 bien, pero creo que la mayor parte de la descendencia de los supervivientes de Auschwitz no vive en Europa.\n\nS\u00ed. Pero. \u00bfCambia eso algo? S\u00ed, quiz\u00e1. Bueno...\n\nReflexion\u00f3, luego dijo: Los otros puntos de tu paper est\u00e1n okay, eso lo dejamos de momento as\u00ed. Son las cosas habituales que se han de tener en cuenta en una celebraci\u00f3n de ese g\u00e9nero. Pero lo que ahora necesitamos con urgencia son hechos y n\u00fameros. \u00bfCu\u00e1ntos viven a\u00fan, sobre todo en Europa?\n\nReflexion\u00f3 de nuevo. Martin se pregunt\u00f3 si deb\u00eda sentarse otra vez. Pero ella no parec\u00eda dispuesta a sentarse, estaba de pie delante de la ventana mirando hacia fuera y finalmente dijo: Puede que baste con uno. En el fondo s\u00f3lo necesitamos una figura simb\u00f3lica, para la Europa unida, para lo com\u00fan, para la exigencia de este trabajo nuestro.\n\nPrimero quiere millares, luego s\u00f3lo uno: \u00bfen qu\u00e9 direcci\u00f3n deb\u00eda seguir trabajando \u00e9l? La mir\u00f3. Ella se miraba a s\u00ed misma hacia abajo, se quitaba ceniza de la blusa.\n\nCuando el profesor Erhart lleg\u00f3 al primer meeting del reflection group New Pact for Europe, \u00e9l era el \u00fanico que llevaba cartera. Era realmente c\u00f3mico: \u00e9l mismo se dio cuenta enseguida y tuvo la impresi\u00f3n de que los otros tambi\u00e9n se percataron de ello; con regocijo o s\u00f3lo con extra\u00f1eza, pero lo notaron.\n\nFue el \u00faltimo que entr\u00f3 porque al principio se hab\u00eda equivocado de camino. El encuentro ten\u00eda lugar en el R\u00e9sidence Palace, detr\u00e1s del ayuntamiento, en la rue de la Loi, una direcci\u00f3n que, en el fondo, era dif\u00edcil no encontrar; si se ven\u00eda de la estaci\u00f3n de metro Schuman, se sal\u00eda pr\u00e1cticamente delante. Pero junto al edificio hab\u00eda obras, delante una pasarela cortada, una verja que cerraba el paso, bloques de hormig\u00f3n. Alois Erhart hab\u00eda pensado que deb\u00eda rodear el terreno en obras para llegar a la trasera del edificio del ayuntamiento, as\u00ed que sigui\u00f3 por la rue de la Loi, pero no encontr\u00f3 posibilidad de doblar a la izquierda y llegar a una calle paralela que llevara otra vez a la parte de atr\u00e1s del edificio del ayuntamiento. Vio entonces la boca de metro de la estaci\u00f3n de Maelbeek, lo que significaba que ya se hab\u00eda alejado de Schuman un trayecto entero entre dos estaciones de metro. \u00a1C\u00f3mo pod\u00eda haberse alejado tanto! Por otra parte: no ve\u00eda otra posibilidad y avanz\u00f3 indeciso un poco m\u00e1s por la misma calle. Por fin vino una calle lateral a la izquierda. Torci\u00f3 por ella, rue de Tr\u00e8ves, luego otra vez a la izquierda, rue Jacques de Lalaing, le\u00eda los nombres de las calles, como si pudiera tranquilizarle que las calles por las que se extraviaba tuvieran nombre. Se detuvo y sac\u00f3 de su cartera el plano de Bruselas, busc\u00f3, comprob\u00f3 que si continuaba por Jacques de Lalaing llegar\u00eda a la Chauss\u00e9e d'Etterbeek, que llevaba por debajo de la rue de la Loi sin una posibilidad, o por lo menos no hab\u00eda ninguna marcada all\u00ed, de subir otra vez a la parte posterior del edificio del ayuntamiento. As\u00ed que se dio la vuelta y desanduvo el camino. De nuevo en el terreno de las obras descubri\u00f3 que entre las rejas y las planchas amarillas de madera contrachapada hab\u00eda un peque\u00f1o e insignificante pasadizo al R\u00e9sidence Palace.\n\nCuando entr\u00f3 en el edificio no sab\u00eda, naturalmente, ad\u00f3nde dirigirse. En el centro del hall hab\u00eda un infopoint, donde dos chicas informaron con gran amabilidad al profesor Erhart. No, no sab\u00edan d\u00f3nde estaba, all\u00ed en la casa, el European Policy Center. El think-tank New Pact for Europe era para ellas absolutamente desconocido. C\u00f3mo se llamaba, le preguntaron. El profesor Erhart dijo su apellido, una chica tecle\u00f3 en el ordenador, dijo con amable sonrisa que lo sent\u00eda, que lo sent\u00eda mucho, pero que all\u00ed no hab\u00eda nadie que se llamara as\u00ed. Pero soy yo quien se llama as\u00ed, dijo el profesor, yo pensaba que usted quer\u00eda decir... okay... que quer\u00eda ver a... \u00a1Espere! Abri\u00f3 la cartera, guardaba impreso el e-mail con los detalles de organizaci\u00f3n del primer meeting. Sac\u00f3 la hoja de la cartera. Mire, dijo, Mister Pinto, European Policy Center, primer encuentro del reflection group New Pact for Europe, vea usted. Max Kohnstamm Room, 4th floor.\n\nOh, dijo la chica. Aclarado. Cuarta planta. El ascensor est\u00e1 ah\u00ed detr\u00e1s, a la derecha.\n\nDe modo que entr\u00f3 el \u00faltimo. Pero no llegaba con excesivo retraso. Si no se hubiera extraviado, habr\u00eda llegado demasiado pronto. Sol\u00eda ser el primero, por miedo a llegar tarde.\n\nHab\u00eda llevado todo el tiempo la cartera con la mano izquierda por los dolores que segu\u00eda teniendo en el brazo derecho. Ahora sinti\u00f3 un dolor agudo tambi\u00e9n en el izquierdo. Levant\u00f3 la cartera y la apret\u00f3 con los brazos cruzados contra el pecho. Quer\u00eda aliviar los brazos, pero parec\u00eda que utilizaba la cartera como escudo, como si quisiera protegerse medrosamente. \u00c9sa era la imagen que ofrec\u00eda cuando entr\u00f3 en la sala.\n\nUn hombre se acerc\u00f3 a \u00e9l con una gran sonrisa.\n\n\u00bfMister Erhart?\n\nYes.\n\n\u00bfEl profesor de Austria?\n\nFrom Vienna, yes.\n\nSoy Ant\u00f3nio Oliveira Pinto, el director de nuestro reflection group. Encantado de verle aqu\u00ed, dijo el hombre. Hablaba perfectamente alem\u00e1n.\n\nLamento el retraso, esas obras...\n\nS\u00ed, dijo el hombre riendo alegremente, Europa es un complicado terreno en obras. Y por eso estamos aqu\u00ed, nuestra tarea es discutir: \u00bfqu\u00e9 estamos construyendo, propiamente?\n\nYo no soy arquitecto, y...\n\nJaja, humor vien\u00e9s, \u00bfno? Muy bien. As\u00ed pues, propongo que de momento cobre usted fuerzas y, dentro de veinte minutos, empezamos en el meeting-room con la presentaci\u00f3n de cada uno. No es arquitecto, ja, ja, muy bueno.\n\nAlois Erhart estaba all\u00ed de pie, abrazando su cartera y mirando alrededor. Sobre una mesa hab\u00edan montado un bufet; junto a una serie de mesas altas, hombres y mujeres de pie, los miembros del think-tank, com\u00edan con tenedores de cart\u00f3n en platos de cart\u00f3n y charlaban y miraban, o no hablaban, sonre\u00edan y miraban.\n\nAlois Erhart cogi\u00f3 ahora otra vez la cartera con la mano izquierda para dejar libre la derecha a fin de poder sostener un plato; pero \u00bfc\u00f3mo iba a poner ahora en el plato la ensalada de pasta o el rosbif? Sujet\u00f3 la cartera bajo la axila izquierda, puso el plato en la mano izquierda, trat\u00f3 de coger de la fuente, con la derecha, un poco de ensalada de pasta... y la cartera cay\u00f3 al suelo. Se agach\u00f3 para recogerla y entonces la ensalada de pasta que \u00e9l ya hab\u00eda amontonado en su plato se escurri\u00f3. Puso de pie en el suelo la cartera y \u00e9sta se volc\u00f3 a un lado. Curiosamente, eso le puso nervioso; que la cartera no estuviera de pie, sino tumbada. La cogi\u00f3 y la apoy\u00f3 en la pared. En cierto modo, aquello le intranquilizaba: que la cartera estuviera apoyada all\u00ed y \u00e9l, cuando se serv\u00eda en el bufet, se hallara tan alejado de ella. De modo que dej\u00f3 el plato, fue a buscar otra vez la cartera, la coloc\u00f3 entre los pies mientras se serv\u00eda en el bufet. Despu\u00e9s ten\u00eda que llegar hasta alguna de las mesas altas. Con el plato en la mano derecha, un vaso de zumo de manzana en la izquierda, avanzando a pasitos muy cortos, trat\u00f3 de llevar de alg\u00fan modo la cartera entre los pies, y casi hubiera dado un tropez\u00f3n, pero entonces peg\u00f3 un ligero puntapi\u00e9 a la cartera, avanz\u00f3 un paso y volvi\u00f3 a empujarla con el pie, para llevarla as\u00ed por delante de \u00e9l hasta la mesa, y a m\u00e1s tardar en ese momento \u00e9l, o mejor, su cartera estaba en el centro de la atenci\u00f3n general. Y el profesor Erhart vio que, aparte de \u00e9l, all\u00ed nadie ten\u00eda una cartera; algunos llevaban mochila, estaban all\u00ed de pie con sus chepas y sus manos libres, otros ten\u00edan al lado maletitas de ruedas en las que se apoyaban indolentemente con una mano. Y \u00e9l, el viejo de la cartera escolar.\n\nEra, en efecto, su cartera del colegio. No la consigui\u00f3 hasta muy tarde, en el grado superior del bachillerato. Antes no hubo dinero para eso. O su padre opinaba que la compra de una cartera escolar era un gasto superfluo, teniendo \u00e9l en su tienda bolsas de deporte. Eran de tela, una suerte de saco marinero que se cerraba con un cordel que luego formaba una lazada que serv\u00eda de asa. En el fondo era una bolsa de deporte pero m\u00e1s grande, y el joven Alois se avergonzaba de que su padre, al fin y al cabo due\u00f1o de una tienda, o sea, empresario, le obligara a ir al instituto de la Amerlingstrasse, para chicos de buenas familias, con aquella extra\u00f1a bolsa que no ten\u00eda ning\u00fan compa\u00f1ero. Cuando por fin consigui\u00f3 una aut\u00e9ntica cartera estaba rebosante de felicidad. Era de piel cosida a mano. Su padre la hab\u00eda comprado en Weinberger, a un \u00abproductor de accesorios de moda en piel\u00bb, un poco m\u00e1s abajo de la Mariahilfer Strasse, con un buen descuento, despu\u00e9s de haber dado \u00e9l muchas facilidades al guarnicionero cuando \u00e9ste compr\u00f3 un equipo de esquiar para su hijo.\n\nAlois estaba tan orgulloso de su cartera de piel que la pon\u00eda junto a la cama cuando se acostaba, para que estuviera en su campo visual en cuanto se despertaba. Le gustaba el ruido que hac\u00edan los cilindros del cierre de brillante n\u00edquel al encajar, con un sonoro clic, cuando preparaba la cartera para ir al instituto. De vez en cuando frotaba la cartera con una pasta grasienta para que la piel no se cuarteara. La cartera ten\u00eda una correa que se enganchaba con un nudo corredizo en el dorso cuando se la quer\u00eda llevar a la espalda, pero Alois no la us\u00f3 nunca, prefer\u00eda llevar la cartera en la mano como una persona mayor y, en alg\u00fan momento, acab\u00f3 perdiendo la correa.\n\nM\u00e1s tarde aparecieron las modernas mochilas escolares, de colorines, con dibujos llamativos y hechas de alg\u00fan material sint\u00e9tico, en el fondo cart\u00f3n plastificado, y Alois sent\u00eda una mezcla de repugnancia y de compasi\u00f3n cuando ve\u00eda a los ni\u00f1os que iban a la escuela con esas rid\u00edculas maletitas con Snoopy y Batman colgadas a la espalda. Su cartera de piel lo acompa\u00f1aba hasta hoy. La piel se hab\u00eda vuelto entretanto algo m\u00e1s blanda y ten\u00eda una hermosa p\u00e1tina de un brillo mate. Y en esa cartera met\u00eda todo lo que necesitaba para una ocasi\u00f3n como \u00e9sta. En una funda de pl\u00e1stico transparente, dos hojas con un breve guion para la intervenci\u00f3n personal de cinco minutos que, al igual que los otros, hab\u00eda de dar en el turno de apertura; una funda de pl\u00e1stico transparente con los e-mails impresos que hab\u00eda recibido del se\u00f1or Pinto en la preparaci\u00f3n del encuentro, una carpeta con su texto sobre la reforma de la Uni\u00f3n que, tan pronto hubiera ocasi\u00f3n, quer\u00eda presentar; un bloc de notas y un plumier. Se preguntaba qu\u00e9 llevaban los otros en sus repletas mochilas y en sus maletitas de ruedas.\n\nAnte una mesa alta, primer amable cambio de impresiones. Oh, \u00bfusted es el profesor Erhart? Me alegro mucho. Me alegro. Un placer. Yo soy, yo soy, s\u00ed, y yo soy. \u00c9se. \u00c9sa. \u00c9se. Mucho gusto en conocerle. Mucho gusto en conocerle. Un franc\u00e9s empez\u00f3 a contar algo, el franc\u00e9s escolar del profesor Erhart no bastaba para entender aquel franc\u00e9s dialectal... hasta que se dio cuenta de que el franc\u00e9s hablaba ingl\u00e9s y se concentr\u00f3 en su ensalada de pasta. Entonces Ant\u00f3nio Oliveira Pinto dio unas palmadas y exclam\u00f3: Se\u00f1oras y se\u00f1ores, por favor, vamos a empezar.\n\nEra todo muy r\u00e1pido: el profesor Erhart se dio cuenta enseguida de que aqu\u00e9l no era su sitio, o bien de que en aquel grupo \u00e9l no ten\u00eda la menor posibilidad de \u00e9xito con lo que se propon\u00eda. Todos eran parecid\u00edsimos entre s\u00ed. S\u00f3lo \u00e9l era distinto. Estaba informado de que ese nuevo think-tank se reunir\u00eda seis veces ese a\u00f1o, dos d\u00edas cada vez, para, al final, entregar al presidente de la Comisi\u00f3n un documento con los resultados de sus an\u00e1lisis y propuestas para salir de la crisis y para estabilizar la Uni\u00f3n. A Alois Erhart le hab\u00eda sorprendido que s\u00f3lo se contara con un periodo de doce d\u00edas, y repartidos a lo largo de todo un a\u00f1o, para elaborar un plan que ofreciera una soluci\u00f3n a la crisis europea. Pero tambi\u00e9n hab\u00eda visto la invitaci\u00f3n como una oportunidad de integrar sus ideas en ese sistema.\n\nAhora estaban sentados en c\u00edrculo en la Max Kohnstamm Room, Alois Erhart sac\u00f3 de la cartera sus hojas con los apuntes para su primer statement, todos los dem\u00e1s sacaron de sus mochilas o sus maletitas ordenadores port\u00e1tiles o tabletas, Ant\u00f3nio Oliveira Pinto dijo con la amplia y radiante sonrisa de un hombre que acaba de vivir el momento m\u00e1s dichoso de su vida, Once again welcome y, de pronto, hubo un ruido estruendoso, la vecina de Erhart agach\u00f3 la cabeza, un hombre se levant\u00f3 de un salto, a otro le resbal\u00f3 el port\u00e1til de las rodillas: \u00bfqu\u00e9 era aquello? Un p\u00e1jaro hab\u00eda chocado contra el cristal de la ventana, s\u00ed, eso ten\u00eda que haber sido, un p\u00e1jaro... Uno que afirmaba haberlo visto dijo: Un gran p\u00e1jaro negro... todos se levantaron, se acercaron a la ventana, se ve\u00eda en efecto una peque\u00f1a mancha de sangre y una pluma pegada al cristal.\n\nExtra\u00f1o que Alois Erhart, aquel hombre en el fondo de su alma tan felizmente conservador, pasara a convertirse en ese c\u00edrculo en triste revolucionario.\n\nSi al comisario \u00c9mile Brunfaut no le hubieran dado vacaciones, no habr\u00eda tenido tiempo de ir al m\u00e9dico. Y luego quiz\u00e1 tampoco habr\u00eda intentado resolver el enigma del caso \u00abAsesinato en el Atlas\u00bb.\n\nAhora, con el torso descubierto y el pantal\u00f3n desabrochado, estaba tendido en la camilla del m\u00e9dico y not\u00f3 angustiado que le asaltaba el miedo, un miedo silencioso, paralizante. \u00a1Respire hondo! Un miedo que le cortaba la respiraci\u00f3n. Extra\u00f1o que Brunfaut no hubiera pensado hasta entonces en que tambi\u00e9n era mortal, aunque de continuo estaba confrontado con cad\u00e1veres. Pero \u00e9l era precisamente el que viv\u00eda y ten\u00eda la misi\u00f3n de conducir al culpable de la muerte al justo castigo. Y \u00e9se se llamaba por lo general cadena perpetua, lo que incluso si se pon\u00eda en libertad prematuramente al autor del crimen sonaba a indescriptible eternidad de una vida cuyo final no conoce nadie.\n\nPeligrosas persecuciones, tiroteos y cosas semejantes, eso sal\u00eda por televisi\u00f3n, pero no pasaba en su trabajo, y si ocurr\u00eda alguna vez, hab\u00eda especialistas para ello, \u00e9l por su parte jam\u00e1s lo hab\u00eda vivido en todos sus a\u00f1os de servicio, nunca se hab\u00eda encontrado en una situaci\u00f3n en la que se viera forzado a tener miedo a la muerte. Pero ahora, con ese m\u00e9dico que no era ni experto en criminal\u00edstica ni forense, sino un normal\u00edsimo especialista en medicina general que le hab\u00eda examinado, apretando un poquito aqu\u00ed, golpeando un poquito all\u00e1...\n\nBrunfaut se aboton\u00f3 la camisa mientras el m\u00e9dico escrib\u00eda la orden de traslado a la cl\u00ednica para el exacto esclarecimiento de los s\u00edntomas, ya que...\n\nTuvo que pensar en la muerte. En la propia. Sin coqueter\u00eda. El m\u00e9dico ten\u00eda una sospecha. Sab\u00eda algo. Y en el hospital confirmar\u00edan lo que el m\u00e9dico sab\u00eda o sospechaba. La enfermedad mortal. De pronto Brunfaut no ten\u00eda duda alguna de que estaba presenciando c\u00f3mo elaboraban su sentencia de muerte. Viv\u00eda ese momento como irreal y a la vez se viv\u00eda a s\u00ed mismo, de un modo radical desconocido hasta entonces, como real. Nadie est\u00e1 tan fuera del mundo y al mismo tiempo tan en s\u00ed mismo como quien de pronto est\u00e1 perdido en una niebla impenetrable. P\u00e1nico y voluntad de supervivencia dilaceran el cuerpo, la cabeza est\u00e1 caliente, el pecho fr\u00edo y entumecido. El m\u00e9dico golpeaba las teclas, muy arr\u00edtmicamente, alzando una y otra vez las cejas y mirando al monitor, tip tip tip pausa clic pausa tip tip pausa, luego como un tamborileo y despu\u00e9s una parada como los sonidos de un coraz\u00f3n desesperado conectado a un amplificador. Y Brunfaut, como si hiciera ejercicios de traducci\u00f3n a otra lengua que estaba aprendiendo, formulaba mentalmente una tras otra, despacio e inseguro, las preguntas: \u00bfc\u00f3mo reaccionar? \u00bfC\u00f3mo reaccionar\u00e9 yo, cuando est\u00e9 el diagn\u00f3stico, cuando tenga el diagn\u00f3stico claro y seguro? \u00bfRebelarme y luchar? \u00bfQuerr\u00e9 luchar? \u00bfMe dejar\u00e9 ir, me dar\u00e9 por vencido? \u00bfEnga\u00f1arme a m\u00ed mismo, dejarme enga\u00f1ar, esperar contra toda evidencia? \u00bfSentir\u00e9 autocompasi\u00f3n, o placer, podr\u00e9 sentir a\u00fan placer, aprender\u00e9 a sentir placer en los \u00faltimos goces? \u00bfTendr\u00e9 ataques de furia o ser\u00e9 cari\u00f1oso? \u00bfCari\u00f1oso con qui\u00e9n?\n\nEl m\u00e9dico carraspe\u00f3 y Brunfaut se rio de pronto; qu\u00e9 tiempos aqu\u00e9llos, cuando estar enfermo era id\u00edlico y paradisiaco; durante poqu\u00edsimo tiempo, un segundo todo lo m\u00e1s, tuvo aquella imagen en la cabeza: \u00e9l, arrebujado en un blando edred\u00f3n de plumas, dispensado de asistir a clase, su madre tan cari\u00f1osa, con la mano sobre su frente caliente, tan sol\u00edcita, prepara un t\u00e9 y, para que se vaya recuperando, le prepara despu\u00e9s su manjar preferido. Dormitar, so\u00f1ar, leer. La dulce experiencia del amor en forma de compasi\u00f3n y solicitud. Y esta convicci\u00f3n: todo volver\u00e1 a estar bien. Todo estaba bien...\n\nEl doctor hablaba por tel\u00e9fono: \u00bfA primera hora de la ma\u00f1ana no es posible?... Comprendo... \u00bfAs\u00ed que a las 13.00 horas?... D'accord! \u00a1Muchas gracias!\n\nMa\u00f1ana a las 13.00 horas en el Europa-Hospital St. Michel, dijo el m\u00e9dico, mejor en ayunas. Con este volante, aqu\u00ed tiene. El esclarecimiento, o sea, los ex\u00e1menes necesarios durar\u00e1n unos tres d\u00edas. Si durasen m\u00e1s, puede usted en cualquier caso marchar a casa el fin de semana. Eso lo decidir\u00e1 el m\u00e9dico jefe, el doctor Drumont. Acabo de hablar con \u00e9l. Con \u00e9l est\u00e1 usted en las mejores manos.\n\nY entonces a \u00c9mile Brunfaut le ocurri\u00f3 algo extra\u00f1o: el miedo le liber\u00f3. Era \u00e9sa su sensaci\u00f3n y finalmente su pensamiento: liberaci\u00f3n.\n\nLa sentencia de muerte o, digamos, el comprender que era mortal lo entendi\u00f3 de pronto como una liberaci\u00f3n para actuar. Deb\u00eda hacer lo que hab\u00eda que hacer. Los polic\u00edas de permiso ten\u00edan prohibido efectuar pesquisas por su cuenta. Pero \u00bfqu\u00e9 castigo pod\u00eda temer \u00e9l ahora? Morir sabiendo que no hab\u00eda actuado era el \u00fanico castigo que tem\u00eda, ser\u00eda la muerte m\u00e1s atroz. \u00bfSentimental? La historia no es otra cosa que un movimiento de p\u00e9ndulo entre emoci\u00f3n y banalidad. Y el mortal se ve empujado una vez hacia un lado, otra vez hacia el otro.\n\nEl comisario Brunfaut se levant\u00f3, contempl\u00f3 desde arriba al m\u00e9dico con la mirada que ten\u00eda su abuelo. El famoso h\u00e9roe de la resistencia que hab\u00eda dado su nombre a una calle de Bruselas. La mirada del abuelo que \u00e9l tem\u00eda de ni\u00f1o. Cuando \u00e9l, el peque\u00f1o \u00c9mile, yac\u00eda en el lecho con ligera fiebre, con resfriado y dolor de garganta, con la infusi\u00f3n de hojas de salvia preparada por su madre, estar enfermo era id\u00edlico y paradisiaco, y un d\u00eda el abuelo, de pie ante \u00e9l, le mir\u00f3 desde arriba y dijo: No existe el estar enfermo. Uno s\u00f3lo est\u00e1 enfermo cuando cae al suelo. Y entonces uno est\u00e1 muerto. Y la madre, que entraba justamente con una infusi\u00f3n en el cuarto, grit\u00f3: \u00bfQu\u00e9 est\u00e1s diciendo? Deja al ni\u00f1o en paz. \u00bfPor qu\u00e9 le metes miedo?\n\nBrunfaut cogi\u00f3 el volante para la cl\u00ednica, dio las gracias al m\u00e9dico y se march\u00f3. Mentalmente baj\u00f3 la mirada hacia el ni\u00f1o que hab\u00eda sido; el ni\u00f1o estaba asustado, el ni\u00f1o ten\u00eda miedo. \u00c9l no.\n\nAhora estaba en la resistencia. Hasta caer al suelo. La Loi, la Libert\u00e9!\n\nCamin\u00f3 despacio en direcci\u00f3n al centro, ten\u00eda tiempo que llenar, hab\u00eda quedado con su amigo Philippe Gaultier una hora m\u00e1s tarde en el restaurante L'Ogenblik, en las galer\u00edas junto a la Grand Place.\n\nCompr\u00f3 bombones en la Neuhaus de la Grand Place.\n\nDe \u00e9stos de aqu\u00ed, por favor, en una cajita para nueve unidades.\n\n\u00a1Nueve bombones Le d\u00e9sir! D'accord. \u00bfSe la envuelvo para regalo?\n\nS\u00ed, por favor.\n\nLa se\u00f1ora se alegrar\u00e1. Para m\u00ed, Le d\u00e9sir es nuestro mejor bomb\u00f3n.\n\n\u00bfQu\u00e9 se\u00f1ora? Me lo regalo a m\u00ed mismo.\n\nOh.\n\nBrunfaut mir\u00f3 a la dependienta y de pronto sinti\u00f3 compasi\u00f3n. Y autocompasi\u00f3n. Hab\u00eda destruido una situaci\u00f3n id\u00edlica, aunque s\u00f3lo fuese la ficci\u00f3n de una situaci\u00f3n de venta. \u00bfPor qu\u00e9 era tan desconsiderado? Ya no pod\u00eda permit\u00edrselo. Desconsiderado. Pag\u00f3, cogi\u00f3 la cajita art\u00edsticamente empaquetada y dijo: He cambiado de opini\u00f3n. Quiero regalar estos bombones a una se\u00f1ora, a una se\u00f1ora cuya sonrisa hoy me ha hechizado.\n\nY entreg\u00f3 el paquete a la dependienta.\n\nSe march\u00f3.\n\nTodo est\u00e1 bien, pens\u00f3 con signos de exclamaci\u00f3n e interrogaci\u00f3n, siempre que la verg\u00fcenza abrase m\u00e1s que el miedo a la muerte.\n\nLleg\u00f3 un cuarto de hora antes a L'Ogenblik. Se tom\u00f3 una copa de champ\u00e1n mientras esperaba a Philippe.\n\nPhilippe era director del PED, el Centro de Procesamiento Electr\u00f3nico de Datos de la polic\u00eda de Bruselas, quince a\u00f1os m\u00e1s joven que Brunfaut y, pese a la diferencia de edad, su mejor amigo. Los un\u00eda en gran medida el hecho de que ambos fueran \u00abportadores de la bufanda h\u00fameda\u00bb, as\u00ed se denominaban los seguidores del club de f\u00fatbol RSC Anderlecht que no se perd\u00edan ni un partido en campo propio: hab\u00edan vertido tantas l\u00e1grimas en su bufanda de forofo que \u00e9sta ya no se secaba. Ambos opinaban, como comprobaron una vez mientras tomaban una cerveza despu\u00e9s del trabajo, que entonces, tras el incre\u00edble esc\u00e1ndalo de corrupci\u00f3n, cuando se supo que el club, antes del partido de vuelta de la semifinal de la copa de la UEFA contra el Nottingham Forest, hab\u00eda sobornado al \u00e1rbitro con 27.000 libras esterlinas, se ten\u00eda que haber dado un paso como se\u00f1al de un nuevo comienzo. Aunque hubiera sido una se\u00f1al de lo m\u00e1s simple, simb\u00f3lica, un peque\u00f1o cambio en el nombre del club, para dejar claro que a partir de ese momento aquel club despegaba de nuevo y ya no ten\u00eda nada que ver con corrupci\u00f3n y sobornos. RSC Anderlecht: \u00bfc\u00f3mo habr\u00eda podido ser ese cambio? Suprimir la erre, hab\u00eda dicho \u00c9mile Brunfaut, s\u00f3lo para marcar una se\u00f1al.\n\nPero \u00bfpor qu\u00e9 la erre?\n\nLe Roi, la Loi, la Libert\u00e9! \u00bfDe qu\u00e9 podemos prescindir? Le Roi!\n\nRieron. As\u00ed se encontraron tambi\u00e9n enseguida coincidiendo en pol\u00edtica en cuanto al sistema belga, a ese Estado desgarrado, que no deber\u00eda mantenerse precariamente unido mediante un rey, sino mediante el com\u00fan estatuto legal de una rep\u00fablica. Si bien la decisi\u00f3n del rey de no nombrar gobierno en la \u00e9poca en que B\u00e9lgica tuviera la presidencia del Consejo Europeo para no bloquear con disputas internas de coalici\u00f3n las necesarias decisiones de pol\u00edtica europea les pareci\u00f3 bien a los dos. Nunca, dijo Philippe, hab\u00eda funcionado B\u00e9lgica mejor que en aquella \u00e9poca sin gobierno.\n\nIban en peregrinaci\u00f3n al Constant-Vanden-Stadion de Anderlecht, lloraban en sus bufandas y se burlaban el uno del otro. Philippe contaba con entusiasmo que hab\u00eda visto jugar a Franky Vercauteren, uno como \u00e9se es lo que necesitaban ahora, un goleador genial. Bueno, no tienes ni idea, hab\u00eda dicho \u00c9mile, \u00e9l, de m\u00e1s edad, hab\u00eda visto jugar a Paul van Himst, comparado con \u00e9l, Vercauteren era m\u00e1s lento que una tortuga.\n\n\u00bfAntes era todo mejor? Nada era mejor, era simplemente muy distinto.\n\n\u00a1S\u00ed, claro! \u00a1Distinto! Pero \u00bfno era realmente mejor? Antes Anderlecht era un distrito jud\u00edo de Bruselas. Era el centro clandestino de Bruselas, por el club y por los caf\u00e9s y las tiendas. Ahora es un distrito musulm\u00e1n, los jud\u00edos se marcharon, y a nadie que yo conozca se le ocurrir\u00eda ir hasta all\u00ed y entrar en un caf\u00e9, y menos a\u00fan acompa\u00f1ado de una mujer, a quien los musulmanes no permitir\u00edan pisar un caf\u00e9.\n\nConoces a Gerrit Beers, de aseguramiento de pruebas, \u00bfverdad? Se ha mudado a Anderlecht, dice que all\u00ed los pisos son m\u00e1s baratos, todo es mucho m\u00e1s easy-going, y es fumador. All\u00ed les importa un bledo que est\u00e9 prohibido fumar. Le sirven un caf\u00e9 fant\u00e1stico y a los hombres, con sus narguiles, les da igual si se enciende un cigarrillo.\n\nComo en Molenbeek.\n\nS\u00ed. Los tiempos cambian. El club dejar\u00e1 pronto el estadio de all\u00ed y se trasladar\u00e1 al nuevo estadio Rey Balduino. Entonces el club seguir\u00e1 llam\u00e1ndose Anderlecht, pero no jugar\u00e1 en Anderlecht. Y t\u00fa dir\u00e1s que antes todo era mejor. Y hoy te quejas porque Anderlecht ya no es como era hace veinte a\u00f1os.\n\nBueno, hoy no han estado tan mal. 2-1 contra el Lovaina, ha sido aceptable.\n\nTres a\u00f1os atr\u00e1s, Philippe hab\u00eda pedido a \u00c9mile que fuera su testigo de boda. Un a\u00f1o m\u00e1s tarde, Philippe fue padre y \u00c9mile el padrino de la peque\u00f1a Jo\u00eblle. Ya era m\u00e1s que un amigo, era familia.\n\n\u00c9mile Brunfaut se bebi\u00f3 la copa de champ\u00e1n y pidi\u00f3 otra. Philippe era exactamente la persona que necesitaba en ese momento: un genial inform\u00e1tico, al mismo tiempo totalmente digno de confianza y solidario. Eso esperaba \u00e9l. No, no, estaba seguro.\n\nLe sirvieron la segunda copa, tom\u00f3 un sorbito y ya ten\u00eda a Philippe delante de \u00e9l: \u00a1El resto de la vida empieza con champ\u00e1n y termina con una tisana! \u00bfQu\u00e9 me cuentas? \u00bfQu\u00e9 tal en el m\u00e9dico?\n\nSe abrazaron, Philippe se sent\u00f3, dijo: Y lo que tambi\u00e9n quisiera saber: \u00bflo tienes ya localizado y detenido?\n\n\u00bfLo? \u00bfQu\u00e9? \u00bfA qui\u00e9n?\n\nHombre, al cerdo. \u00bfNo has le\u00eddo hoy el peri\u00f3dico?\n\nAh, el cerdo. Tengo una pista. Recogimos material gen\u00e9tico. Ma\u00f1ana tendr\u00e1s que comparar el ADN con el de todos los cerdos registrados en el banco de datos de la Europol.\n\nPhilippe se ech\u00f3 a re\u00edr. Ya sabes que siempre estoy a tu disposici\u00f3n.\n\nJusto de eso quer\u00eda hablar contigo.\n\nHablaron y comieron y bebieron. Antes la comida era mejor. \u00bfT\u00fa crees? S\u00ed. Pero aqu\u00ed no ha cambiado absolutamente nada. S\u00ed, excepto la comida. \u00bfPor qu\u00e9? Ya com\u00edamos aqu\u00ed asado de cordero hace diez a\u00f1os. S\u00ed, de acuerdo, pero el de antes era mejor. Bueno, quiz\u00e1, pero en lo dem\u00e1s... en cualquier caso en lo dem\u00e1s aqu\u00ed no ha cambiado nada. Quiz\u00e1 habr\u00eda sido preferible pedir la lubina a la parrilla con el risotto de esp\u00e1rragos. \u00bfEsp\u00e1rragos en invierno? Vienen de Tailandia, pon\u00eda en la carta. \u00a1Esp\u00e1rragos de Tailandia, vaya, no me digas! Aqu\u00ed siempre hemos comido cordero, que est\u00e1 muy bien. No s\u00e9, sabe a cad\u00e1ver, antes nunca pens\u00e9 en que un asado de cordero es un cad\u00e1ver. Oye, ya est\u00e1 bien, \u00bfqu\u00e9 es lo que te ocurre? No pasa nada, no, no pasa nada.\n\nBrunfaut cont\u00f3 que el m\u00e9dico lo hab\u00eda derivado al Europa-Hospital, y que al d\u00eda siguiente le har\u00edan un chequeo all\u00ed.\n\n\u00bfHa dicho si tiene alguna idea?\n\nNo. S\u00f3lo ha dicho que hay que hacer un reconocimiento m\u00e1s a fondo.\n\nBusca todas las posibilidades. Eso est\u00e1 bien. Y luego lo tendr\u00e1s claro. De modo que yo no me preocupar\u00eda por ahora.\n\nS\u00ed. Quiz\u00e1. Quiz\u00e1 tengas raz\u00f3n. Sea como sea: no estoy fuera de combate.\n\n\u00bfEs decir?\n\n\u00bfT\u00fa sabes que me han apartado del caso Atlas y me han dado vacaciones?\n\nS\u00ed.\n\n\u00bfSabes por qu\u00e9?\n\nPensaba que me lo contar\u00edas t\u00fa.\n\nPero yo no lo s\u00e9.\n\n\u00bfQue no lo sabes? \u00bfNo te han dicho los motivos?\n\nNo.\n\nNecesito otra copa de vino.\n\nEscucha, Philippe, han borrado todos los datos relativos al caso Atlas. Yo estuve en el lugar del crimen, all\u00ed se llev\u00f3 a cabo el aseguramiento de pruebas, yo he hecho los primeros interrogatorios: todo eso ha dejado de existir. Expedientes, actas, documentos: todo ha desaparecido sin dejar rastro, el asesinato se ha desvanecido como si no hubiera existido el cad\u00e1ver que yo vi. Cuando regres\u00e9 a mi ordenador, todo hab\u00eda desaparecido, como tragado por una aspiradora. Alguien lo hacke\u00f3. Probablemente no s\u00f3lo mi ordenador, sino el sistema entero. Y el fiscal est\u00e1 metido en el ajo. Me gustar\u00eda saber por qu\u00e9.\n\nLo comprendo.\n\nTienes que ayudarme.\n\nEl camarero recogi\u00f3 la mesa, y Philippe chasque\u00f3 los dedos, indic\u00f3 el lugar donde acababa de estar el plato de \u00c9mile y dijo: \u00a1El cad\u00e1ver ha desaparecido!\n\n\u00a1No bromees! Siento lo que he dicho antes. Completamente en serio: el caso ha desaparecido y si hay alguien que a lo mejor puede remontarse al inicio del proceso y saber c\u00f3mo ocurri\u00f3 y qui\u00e9n lo hizo, \u00e9se eres t\u00fa. T\u00fa eres el inform\u00e1tico jefe, t\u00fa controlas todo el sistema del PED de la polic\u00eda de Bruselas. T\u00fa tienes que encontrar ese agujero.\n\n\u00bfC\u00f3mo voy a explicarlo? No puedo iniciar una b\u00fasqueda en el departamento sin dar explicaciones. Y adem\u00e1s contra una orden del fiscal.\n\n\u00bfT\u00fa sabes que hay orden del fiscal? No. \u00bfLo ves? No tienes que dar explicaciones. Tienes que hacerlo y ya est\u00e1.\n\nEs muy complicado explicarte ahora c\u00f3mo funciona lo del acceso al almacenamiento central de datos, cu\u00e1ntos dispositivos de seguridad hay instalados ah\u00ed y cu\u00e1nta burocracia es necesaria para avanzar s\u00f3lo dos de quiz\u00e1 veinte pasos.\n\nNo tienes que hacerlo oficialmente, mi pregunta no es si crees que te dar\u00e1n autorizaci\u00f3n, sino si puedes hacerlo.\n\nSer\u00eda contrario a la ley.\n\nEscucha, Philippe, un asesinato es un delito oficial, un acto criminal que la fiscal\u00eda ha de perseguir de oficio. Pero si la fiscal\u00eda no lo hace, sino que, por el contrario, encubre el asesinato, entonces la ley ha sido quebrantada por el propio Estado y quienes entonces se sirven de m\u00e9todos ilegales para esclarecerlo son los defensores de la ley. Si me ayudas y tenemos \u00e9xito, habremos sido nosotros quienes hemos obrado conforme a la ley.\n\nBueno, vale. Primero lo intentar\u00e9 desde tu acceso. Dame tu contrase\u00f1a. Si descubren algo, es que t\u00fa has estado jugueteando con el ordenador durante tus vacaciones, \u00bfvale?\n\nVale.\n\n\u00bfUna mousse au chocolat?\n\nClaro. \u00bfPor qu\u00e9 vamos a cambiar nuestros h\u00e1bitos precisamente hoy? \u00bfC\u00f3mo est\u00e1 Jo\u00eblle?\n\nMatek sab\u00eda que no ten\u00eda posibilidad de desaparecer sin dejar rastro. Ya deb\u00edan de saber que no se hab\u00eda montado en el avi\u00f3n a Estambul. Seguramente tambi\u00e9n se habr\u00edan planteado si hab\u00eda volado a Polonia aunque hubieran anulado su billete a Varsovia. Para ellos no era dif\u00edcil averiguar en poqu\u00edsimo tiempo que estaba en la lista de pasajeros de un vuelo a Cracovia. As\u00ed pues, cuando lleg\u00f3 a Cracovia, pod\u00eda dar por seguro que le segu\u00edan a un paso de distancia.\n\nEso lo hab\u00eda aprendido ya en la instrucci\u00f3n b\u00e1sica de \u017bo\u0142nierz Chrystusa: ni siquiera intentes borrar tu rastro: es imposible. No trates de borrar tus huellas: nada estimula tanto a los perseguidores para continuar sobre la pista como tropezar con huellas que has tratado de borrar. Por tanto, si no puedes evitar dejarlas, entonces produce a\u00fan m\u00e1s. \u00a1Muchas huellas, huellas contradictorias! Mientras ellos las analizan, t\u00fa les tomas la delantera. Cuando regresen de rastrear pistas falsas, t\u00fa has aumentado la ventaja.\n\n\u00c9l sab\u00eda, por supuesto, que ellos sab\u00edan que \u00e9l sab\u00eda: pero eso nada cambiaba el hecho de que deb\u00edan seguir las huellas que \u00e9l produc\u00eda, y que lo hicieran con desconfianza o con total ingenuidad daba igual.\n\nCalculaba que necesitar\u00eda tres d\u00edas para averiguar qu\u00e9 hab\u00eda fallado en Bruselas y por qu\u00e9 entonces, contra el plan original, quisieron enviarlo a Estambul. Tres d\u00edas de ventaja, eso era factible, eso era rutina, y luego ya ver\u00eda.\n\nTras su llegada al aeropuerto de Cracovia fue al mostrador de informaci\u00f3n y all\u00ed se hizo llamar por el altavoz: \u00a1Se\u00f1or Mateusz Oswiecki, dir\u00edjase por favor a la ventanilla de la Krakow Express Shuttle Pastuszak! \u00a1Se\u00f1or Oswiecki, por favor! Su ch\u00f3fer le espera junto a la ventanilla de la Express Shuttle Pastuszak.\n\nSab\u00eda que los nombres de las personas a las que llamaban por altavoz quedaban guardados cuarenta y ocho horas. Luego fue a la taquilla del Shuttle-Service. En el aeropuerto de Bruselas hab\u00eda contratado por e-mail el transfer a la ciudad. Si entraban en su mailbox, ten\u00edan ya dos indicios. Pag\u00f3 con tarjeta de cr\u00e9dito. Tercer indicio. Se hizo llevar al hotel Europejski, ulica Lubicz.\n\nAl mediod\u00eda del d\u00eda siguiente sabr\u00edan lo que de todos modos no habr\u00eda podido ocultar: que hab\u00eda llegado a Cracovia. Un d\u00eda despu\u00e9s sabr\u00edan d\u00f3nde se hab\u00eda alojado. Al servirles en bandeja la direcci\u00f3n, pod\u00eda ponerlos sobre una pista falsa y dejarlos que buscaran en vano durante los tres d\u00edas que \u00e9l necesitaba: se registr\u00f3 en el hotel y pidi\u00f3 a la recepcionista que le buscase cu\u00e1ndo sal\u00eda al d\u00eda siguiente el primer tren para Varsovia. Ella tecle\u00f3 en el ordenador, mir\u00f3, mene\u00f3 la cabeza y dijo: \u00bfQuiere usted de verdad tomar el primer tren? Sale a las 4.52 y...\n\n\u00a1Eso es muy temprano!\n\nEl siguiente sale a las 5.41. Llega a las...\n\n\u00a1El siguiente, por favor!\n\nDespu\u00e9s hay trenes a las 6.31, luego a las 7.47 y...\n\nEl de las 6.31. \u00bfCu\u00e1ndo llega?\n\nA las 8.54 y el tren de las 7.47 llega a las 10.00.\n\n\u00c9se es muy tarde. El que llega a las 8.54 es perfecto. Diga otra vez, \u00bfa las seis...?\n\nSeis treinta y uno. De Krakow G\u0142\u00f3wny.\n\nMuy bien. Por favor, \u00bfpuede comprar el billete enseguida aqu\u00ed online e imprimirlo? Aqu\u00ed tiene mi tarjeta de cr\u00e9dito. Y pago tambi\u00e9n ya la habitaci\u00f3n. As\u00ed ahorro tiempo ma\u00f1ana por la ma\u00f1ana.\n\nBardzo zadowolony, panie Oswiecki.\n\nMatek llev\u00f3 su mochila a la habitaci\u00f3n, escribi\u00f3 una carta en el papel de cartas del hotel, que, junto con una de sus tarjetas de cr\u00e9dito, meti\u00f3 en un sobre que cerr\u00f3 despu\u00e9s de escribir la direcci\u00f3n. Luego se march\u00f3 del hotel. Ma\u00f1ana por la tarde tendr\u00edan seis pistas, que casaban bien l\u00f3gicamente, de que hab\u00eda llegado a Cracovia y de que a la ma\u00f1ana siguiente hab\u00eda continuado viaje a Varsovia. Pero \u00e9l se quedar\u00eda en Cracovia. Hasta que ellos lo comprendieran, ten\u00eda tiempo.\n\nPase\u00f3 hasta la Starowi\u015blna, sab\u00eda que all\u00ed hab\u00eda una de esas dudosas tiendas de tel\u00e9fonos m\u00f3viles usados. La tienda segu\u00eda existiendo, en efecto. Compr\u00f3 un Nokia antiguo y primitivo y una tarjeta prepaid de 100 zlotys. Matek mir\u00f3 al joven que abr\u00eda por la fuerza el tel\u00e9fono con una grapa doblada y colocaba la tarjeta, lo observ\u00f3 como si fuera un bicho en un terrario, repugnante y al mismo tiempo digno de compasi\u00f3n. En aquel muchacho, todo era un grito de socorro o de llamar la atenci\u00f3n y al mismo tiempo demostraci\u00f3n de porf\u00eda y desprecio. Su grotesco peinado, afeitado por los lados, el pelo de arriba largo y art\u00edsticamente revuelto, los gruesos mechones azulnegros fijados con gomina. Llevaba una camiseta roja en cuyo pecho se ve\u00eda un insultante dedo coraz\u00f3n. En su brazo derecho estaba tatuada una wolfsangel o trampa para lobos, debajo una mujer desnuda arrodillada y rodeada de cadenas. M\u00e1s interesante que esa pueril demostraci\u00f3n de fuerza era el antebrazo izquierdo: no cab\u00eda duda, el chico se autolesionaba con regularidad. Una serie de l\u00edneas rojas, cortes cubiertos de una costra m\u00e1s o menos reciente y hechos probablemente con una hoja de afeitar. Matek lo sab\u00eda por el seminario. Conoc\u00eda la sensaci\u00f3n que se tiene cuando se segregan las hormonas de la felicidad que alivian el dolor, pero que obran de una manera tan explosiva \u00fanicamente cuando uno se inflige a s\u00ed mismo dolor, cuando uno desv\u00eda con una hoja de afeitar el dolor del alma a la piel exterior. Endorfina y adrenalina, de eso se trataba. Matek hab\u00eda o\u00eddo decir que las mujeres viven esa sensaci\u00f3n en el estr\u00e9s y el dolor del parto. Dios lo ha dispuesto as\u00ed. Rasgu\u00f1os y cortes en brazos y vientre, eso estaba extendido en el seminario, a veces tambi\u00e9n, infligidos mutuamente, en la espalda, raras veces en los genitales.\n\nEl joven presion\u00f3 las dos partes del m\u00f3vil hasta que encajaron con un clic bien audible, puls\u00f3 varias teclas, mir\u00f3 la pantalla y dijo: Dopasowa\u0107!\n\nDzi\u0119kuj\u0119, dijo Matek, pag\u00f3 los ochenta zlotys por el m\u00f3vil y cien por la tarjeta, luego vacil\u00f3, fingi\u00f3 que de pronto ca\u00eda en la cuenta de algo, mir\u00f3 pensativamente en su cartera, dijo: Tengo una pregunta, a lo mejor podr\u00edas ayudarme. Sac\u00f3 un billete de cien euros, lo puso sobre el mostrador, la mano encima.\n\n\u00bfConoces por un casual a alguien que viaje a Varsovia?\n\nEl chico mir\u00f3 la mano de Matek, puesta sobre el billete.\n\nTendr\u00eda que preguntar un poco. \u00bfDe qu\u00e9 se trata? \u00bfViaje compartido?\n\nNo. Una carta. El que viajara podr\u00eda llev\u00e1rsela.\n\nMatek puso otro billete de cien euros sobre el mostrador.\n\n\u00bfPor qu\u00e9 no va a correos?\n\nLas estafetas est\u00e1n cerradas desde hace media hora. Y la carta corre prisa.\n\nCreo que mi hermano quer\u00eda viajar ma\u00f1ana a Varsovia. Tiene a su chica all\u00ed. Tendr\u00eda que preguntarle.\n\nMatek a\u00f1adi\u00f3 un billete de cincuenta.\n\nLa carta tendr\u00eda que estar all\u00ed como muy tarde a las diez.\n\nLe dar\u00e1 igual salir algo m\u00e1s pronto de lo que pensaba.\n\nTendr\u00e1 que salir muy pronto. Como muy tarde a las seis y media.\n\nQuerr\u00e1 tambi\u00e9n dinero para gasolina.\n\n\u00bfNo quer\u00eda viajar de todos modos para ver a su chica?\n\nMatek retir\u00f3 la mano de los billetes, sac\u00f3 la carta del bolsillo interior de su chaqueta, la puso sobre el dinero.\n\nMa\u00f1ana a las diez estar\u00e9 otra vez aqu\u00ed. Si para entonces he recibido una confirmaci\u00f3n por SMS \u2014\u200clevant\u00f3 en el aire el Nokia\u2014 de que ha llegado la carta, habr\u00e1 otra vez esa suma. Entonces tendr\u00e1 suficiente dinero para gasolina, entonces podr\u00e1 ir a ver a su chica y pasearse con ella veinte veces. Si es que el amor dura tanto tiempo.\n\nElla es fiel.\n\nEso est\u00e1 bien. Ser fiel siempre es algo bueno. La direcci\u00f3n est\u00e1 en el sobre.\n\nMatek se march\u00f3.\n\nVagabunde\u00f3 por la Starowi\u015blna hacia abajo en direcci\u00f3n al centro, al Rynek G\u0142\u00f3wny, el mercado central. La belleza y majestuosidad de la gran plaza medieval le emocionaba cada vez que iba a aquella ciudad. La inmensa plaza cuadrada estaba bordeada de palacios, s\u00f3lo la bas\u00edlica de Santa Mar\u00eda romp\u00eda la rigurosa simetr\u00eda. Con sus dos torres sobresal\u00eda de la fachada frontal de ese lado de la plaza como si hubiera dado un paso adelante, combativa, audaz, orgullosa, domin\u00e1ndolo todo, con las dos torres de diferente altura, cuya raz\u00f3n contaban antiguas leyendas, Matek las conoc\u00eda, claro, pero las tomaba casi por una usurpaci\u00f3n pagana. Para \u00e9l estaba clar\u00edsimo que s\u00f3lo pod\u00eda haber una raz\u00f3n de esa ruptura de la simetr\u00eda y de la armon\u00eda: ni siquiera al construir un templo sagrado le est\u00e1 permitido al hombre crear algo perfecto, porque perfecto es s\u00f3lo Dios y su plan de la creaci\u00f3n. De la mano del hombre no puede surgir una perfecci\u00f3n que pueda compararse con la perfecci\u00f3n de Dios, ni siquiera cuando el hombre cree que con esa pretensi\u00f3n le rinde el m\u00e1ximo honor. La bas\u00edlica de Santa Mar\u00eda, que se rebelaba contra el mercado pis\u00e1ndoles as\u00ed simb\u00f3licamente los dedos de los pies a los hombres que se dedicaban all\u00ed a sus negocios, mientras que ella al mismo tiempo se empinaba para asir las estrellas, con una torre se quedaba corta, con la otra ya estaba m\u00e1s cerca del cielo, expresi\u00f3n del af\u00e1n humano, que crece pero que fracasa en la perfecci\u00f3n: esa iglesia era para Matek la expresi\u00f3n m\u00e1s significativa de la relaci\u00f3n de los hombres con Dios. Muy distinta de Notre-Dame; un a\u00f1o antes Matek hab\u00eda tenido una misi\u00f3n en Par\u00eds. Como es natural, quiso ver la catedral de Notre-Dame y como es natural se qued\u00f3 impresionado en un primer momento cuando se encontr\u00f3 delante de ella. Pero... \u00bfqu\u00e9? Entonces lo hab\u00eda comprendido. Esa mente limitada, en el fondo endiosada, con la que se esperaba que unas reglas geom\u00e9tricas, aplicadas a pomposas dimensiones, pudieran reflejar la armon\u00eda divina del universo le hab\u00eda irritado, lo vio como una blasfemia. Y \u00e9sa era seguramente la raz\u00f3n por la que Dios hab\u00eda sido un fr\u00edo e indiferente espectador cuando el her\u00e9tico fil\u00f3sofo Abelardo fornic\u00f3 con la sobrina del can\u00f3nigo, Elo\u00edsa, en el altar de esa catedral. Matek estuvo escuchando cuando una gu\u00eda cont\u00f3 en esa iglesia, delante del altar, a un grupo de turistas ingleses que no paraban de re\u00edr, la siguiente historia: Y aqu\u00ed ocurri\u00f3, sobre este altar, ladies and gentlemen. Aqu\u00ed el joven profesor de filosof\u00eda Pedro Abelardo desflor\u00f3 a su gran amor, Elo\u00edsa, la sobrina del can\u00f3nigo de esta catedral. Abelardo y Elo\u00edsa, tan cantados y tan celebrados, \u00a1\u00e9ste es el altar de su amor! Matek encontr\u00f3 la decisi\u00f3n del can\u00f3nigo de hacer castrar al tal Abelardo completamente adecuada y justificada, y hasta ben\u00e9vola, pero incluso ese castigo que se llev\u00f3 efectivamente a cabo, como cont\u00f3 la gu\u00eda, no pudo anular, as\u00ed pensaba Matek, que ese orgulloso templo estuviera profanado para siempre. \u00c9l lo hab\u00eda sentido. Qu\u00e9 distinta la bas\u00edlica de Santa Mar\u00eda de Cracovia. La mir\u00f3 hacia arriba, eran las 19.00 horas, y como cada hora en punto el m\u00fasico de la torre empez\u00f3 a tocar el Przerwany Hejna\u0142: un toque de trompeta, para prevenir contra los enemigos que se aproximaban, que se interrumpe de pronto. En memoria del trompeta que durante el ataque de los t\u00e1rtaros en 1241 fue alcanzado por una flecha en la garganta, se tocaba s\u00f3lo hasta el \u00faltimo sonido que pudo sacar antes de caer muerto.\n\nMatek levant\u00f3 la vista buscando en la torre oriental al trompeta que deb\u00eda estar en una ventana, pero no pudo verlo, porque enseguida se cort\u00f3 el Hejna\u0142.\n\nNo entr\u00f3 en la iglesia. No pod\u00eda rezar entre los rel\u00e1mpagos de innumerables turistas haciendo fotos. Se dio la vuelta, cruz\u00f3 la plaza pasando por la lonja de los Pa\u00f1os, no se cansaba de mirar, pero tambi\u00e9n sab\u00eda que no deb\u00eda hacerlo muy de cerca. Las tiendas con las hermosas fachadas antiguas vend\u00edan tarjetas postales que mostraban las hermosas tiendas antiguas cuando a\u00fan no vend\u00edan tarjetas postales y recuerdos baratos. Los restaurantes se anunciaban con letreros que promet\u00edan \u00abcocina tradicional polaca\u00bb y no cultivaban otra tradici\u00f3n que la de despachar turistas a gran velocidad. Junto a la iglesia, donde antes estaba la gran librer\u00eda estatal, se encontraba ahora la flagshipstore, la tienda estrella de la cadena de tiendas de moda Zara. En las antiguas tiendas de pa\u00f1os, los turistas pod\u00edan comprar recuerdos de la antigua Cracovia jud\u00eda, postales con antiguas fotograf\u00edas y CD con m\u00fasica klezmer, pero tambi\u00e9n caricaturas de mal gusto de jud\u00edos al estilo del St\u00fcrmer, por ejemplo figuras en madera tallada del jud\u00edo avaricioso con la bolsa o la moneda de oro en la mano.\n\nSali\u00f3 de la plaza y dobl\u00f3 por la Grodzka, porque antes le gustaba comprar en la esquina el pan dulce de arroz polaco, ahora la tienda se llamaba Quality Burger. Continu\u00f3 por la Grodzka hasta el final, siempre adelante, caminaba y caminaba, a lo largo de la Stradomska, siempre adelante, los pasos r\u00edtmicos y la respiraci\u00f3n regular eran ahora su oraci\u00f3n, segu\u00eda y segu\u00eda hasta que lleg\u00f3 a la Pauli\u0144ska, donde hab\u00eda un peque\u00f1o restaurante que conoc\u00eda, el Kuchnia Adama, donde quer\u00eda comer algo. All\u00ed serv\u00edan el mejor bigos de la ciudad, y aunque para ese puchero hab\u00eda cien recetas distintas m\u00e1s o menos oficiales, para Matek s\u00f3lo all\u00ed, a dos vueltas de la esquina de la senda trillada para turistas, era aut\u00e9ntico. En modo alguno hab\u00eda que servirlo reci\u00e9n hecho, era realmente bueno cuando lo hab\u00edan recalentado varias veces durante varios d\u00edas. En Adam, la caldera de bigos estaba por lo menos una semana sobre el fog\u00f3n. As\u00ed el tocino daba toda su grasa al repollo, el piment\u00f3n picante desarrollaba todo su aroma, los tacos de carne se pon\u00edan maravillosamente suaves, pero: esas palabras son s\u00f3lo el susurro y la rima de los c\u00e1nticos que se cantaban al bigos de Adam, son azar, s\u00f3lo el est\u00f3mago comprender\u00e1 el bigos.\n\nMatek com\u00eda en silencio, claro, estaba solo, pero siempre com\u00eda, tambi\u00e9n cuando estaba solo, como si tuviera que atenerse a la prohibici\u00f3n de hablar durante las comidas. Una breve bendici\u00f3n de la mesa, casi susurrada sin sonido, con la cabeza inclinada, luego comer en silencio. Sin embargo, aquella noche ten\u00eda much\u00edsimos pensamientos en la cabeza, casi como un barullo de voces. O\u00eda a su madre, que hab\u00eda destruido su seguridad de estar bien guardado y protegido precisamente entreg\u00e1ndolo, para protegerlo, a las mazmorras de un subterr\u00e1neo, donde ya no hab\u00eda eso, ese venturoso olor de la cocina de una madre amorosa y sonriente. Ante \u00e9l humeaba el bigos y Matek pod\u00eda o\u00edrse a s\u00ed mismo charlando sin pausa cuando estaba sentado a la mesa con su madre, comiendo bigos y gol\u0105bki, fantas\u00edas heroicas, d\u00f3nde las habr\u00eda o\u00eddo \u00e9l, aquellas leyendas que contaba excitado mientras ella le escuchaba sonriente y dec\u00eda: \u00a1No olvides la comida! Y en aquel entonces \u00e9l ni siquiera sab\u00eda que ella llevaba un arma escondida debajo de la falda, la pistola del padre muerto. \u00bfD\u00f3nde estaba su padre? Eso no lo comprend\u00eda a\u00fan del todo mientras ella lo cog\u00eda en brazos, y luego se abrieron los brazos y lo entreg\u00f3, lo puso en manos de hombres santos, a los que se daba el nombre de padre, y de pronto tuvo hermanos, en un subterr\u00e1neo del que, despu\u00e9s de a\u00f1os de ascetismo, sali\u00f3 como \u017bo\u0142nierz Chrystusa para defender una patria en la que nunca hab\u00eda estado. \u00bfQui\u00e9n estuvo jam\u00e1s en ella? Su abuelo no, su padre no, y \u00e9l fue expulsado justo cuando quer\u00eda entrar en ella por una puerta trasera, por la puerta que su madre cerr\u00f3 de pronto de un portazo. Y o\u00eda la voz del padre prior, que con \u00edntima comprensi\u00f3n y una sonrisa que rezumaba grasa como aquel bigos, le explicaba que \u00e9l, Mateusz, el bueno de Matek, no estaba llamado al sacerdocio, sino a ser soldado de Cristo. \u00c9l era obediente, siempre lo hab\u00eda sido, primero porque confiaba en el mundo, y luego porque a \u00e9l lo hab\u00edan iniciado en el sentido y la raz\u00f3n de la obediencia, y ahora estaba ante una trampa y no sab\u00eda por qu\u00e9, pero no lo dudaba: le hab\u00edan tendido una trampa. O\u00eda a su madre, o\u00eda al padre prior, o\u00eda voces, poco claras, ininteligibles, de personas que no conoc\u00eda pero que hablaban de \u00e9l como de una figura sobre un tablero de ajedrez. Silentium!, exclam\u00f3, y luego otra vez: Silentium! Gritaba sin sonido, s\u00f3lo en la mente, quer\u00eda comer en silencio. Lanz\u00f3 un hondo suspiro enderezando el torso y mir\u00f3 en direcci\u00f3n a la camarera, que estaba fumando junto al letrero de NO FUMAR.\n\nRegres\u00f3 a pie al hotel, hizo en la habitaci\u00f3n sus ejercicios de gimnasia, luego se acost\u00f3.\n\nCuando a las seis de la ma\u00f1ana sali\u00f3 del hotel, los autocares tur\u00edsticos ya estaban delante del edificio, \u00abAuschwitz. Best price!\u00bb.\n\nFue al barrio de Kazimierz, tom\u00f3 en el Rubinstein un abundante desayuno, luego llam\u00f3 a Wojciech, su viejo amigo de los tiempos del seminario, que recibi\u00f3 entonces, cuando estudiaban con los hermanos del colegio, el sobrenombre apost\u00f3lico de Szymon, el alba\u00f1il. Ahora era padre del convento de los agustinos, que pertenec\u00eda a la iglesia de Santa Katarina de Cracovia. Matek conoc\u00eda su rutina diaria, la misa conventual ten\u00eda que haber terminado, ahora estar\u00eda disponible hasta tercias.\n\n\u00a1Mateusz, hermano m\u00edo! \u00bfEst\u00e1s en Cracovia? \u00bfC\u00f3mo te encuentras?\n\nS\u00ed, estoy en Cracovia. Me encuentro bien. Me gusta tanto recordar tiempos pasados, cuando pase\u00e1bamos por los jardines del convento y charl\u00e1bamos. Tenemos que hablar.\n\nAy, los jardines. Los hemos dado en arriendo, como aparcamiento. Triste, pero un buen negocio. La renovaci\u00f3n de la iglesia devora sumas horribles. S\u00ed, hablamos: \u00bfdespu\u00e9s de nonas?\n\nLlevo una mochila.\n\nS\u00e9 bienvenido.\n\nMatek ech\u00f3 una ojeada alrededor. Nadie lo miraba. Se arremang\u00f3 un poco la camisa, limpi\u00f3 el cuchillo con la servilleta y con suave presi\u00f3n se hizo un corte en el antebrazo izquierdo. El maldito cuchillo era romo, un t\u00edpico cuchillo de restaurante, lo lade\u00f3 ligeramente y pas\u00f3 la hoja por la piel esta vez presionando m\u00e1s; entonces, por fin, se form\u00f3 una peque\u00f1a raja en la piel y brot\u00f3 sangre, \u00e9l cerr\u00f3 los ojos y apart\u00f3 el cuchillo.\n\nA las nueve y media lleg\u00f3 el SMS: \u00abTransmitir\u00e9 con mucho gusto tus recuerdos\u00bb.\n\nAs\u00ed pues, en Varsovia el hermano Tomasz hab\u00eda recibido la carta. Tomasz ir\u00e1 a comer y pagar\u00e1 con la tarjeta de cr\u00e9dito de Matek. Luego ir\u00e1 a la Potockich, a la gran tienda de bolsos y maletas, comprar\u00e1 con la tarjeta de cr\u00e9dito una maleta y despu\u00e9s, en la estaci\u00f3n, sacar\u00e1 y pagar\u00e1 con la tarjeta un billete de tren a Budapest. Ellos averiguar\u00e1n eso. Tomasz cortar\u00e1 en trocitos la tarjeta y los tirar\u00e1. Matek calculaba que ten\u00eda 72 horas de ventaja hasta que ellos acabaran de examinar las pistas.\n\nFue al aseo, dej\u00f3 correr agua fr\u00eda sobre su antebrazo, mucho tiempo hasta que tuvo la sensaci\u00f3n de que el brazo estaba entumecido, luego se march\u00f3. Fue a la tienda de m\u00f3viles de la Starowi\u015blna, el chico llevaba la misma camiseta de la v\u00edspera, Matek puso el dinero sobre el mostrador.\n\nEra un d\u00eda inusitadamente c\u00e1lido y soleado para esa estaci\u00f3n del a\u00f1o.\n\nVagabunde\u00f3 por la ciudad, bajando por la ulica J\u00f3zefa, por la que iban grupos de turistas detr\u00e1s de letreros o banderines sostenidos en alto; en la ulica Bo\u017cego Cia\u0142a dobl\u00f3 a la izquierda, all\u00ed estaba la iglesia del Corpus Christi, la primera iglesia cat\u00f3lica pasado el barrio jud\u00edo, entr\u00f3, parec\u00eda que acababa de terminar la misa de la ma\u00f1ana, la gente se levantaba de los bancos, se dirig\u00eda a la salida, Matek permanec\u00eda de pie como una roca en el oleaje, a su izquierda y a su derecha la gente aflu\u00eda hacia el exterior, hasta que \u00e9l se dio la vuelta y abandon\u00f3 con los dem\u00e1s la iglesia, como si formara parte de un grupo; volvi\u00f3 a la J\u00f3zefa, una gran puerta cochera estaba abierta y permit\u00eda ver un bonito y escondido patio interior, al final de un pasillo ruinoso lleno de sacos de basura hab\u00eda un turista que fotografiaba con su smartphone, gritos de una gu\u00eda: \u00abThis way, please!\u00bb, una mujer dijo \u00ab... would be a perfect hideaway!\u00bb, un hombre re\u00eda, \u00abYou cannot escape\u00bb, el grupo segu\u00eda avanzando hacia la iglesia de Santa Katarina, jardines detr\u00e1s de puertas enrejadas, aparcamientos en los jardines, un joven empez\u00f3 a andar, corri\u00f3 hacia una mujer, se abrazaron, caminaron lentamente cogidos de la mano a lo largo de la ciega y silenciosa fachada del convento, pasando junto a la plaza con el altar del Milenio que constaba de un grupo de siete grandes figuras de bronce, santos, hombres de la Iglesia, de tama\u00f1o mayor que el natural, una alemana estaba delante y dec\u00eda: \u00ab\u00a1Anda, mira, \u00e9se es seguramente el papa polaco!\u00bb, un hombre dijo: \u00abS\u00ed, es Wojtyla\u00bb, otro: \u00abNo, ah\u00ed pone \u015aw. Stanis\u0142aw (1030-1079)\u00bb. Pasaban sacerdotes, se met\u00edan por la Augusti\u00e1nska, luego llegaron dos mujeres con pesadas bolsas, como si fueran detr\u00e1s de los sacerdotes, enseguida doblaron tambi\u00e9n la esquina, entretanto el grupo de turistas hab\u00eda seguido adelante, y las estatuas del milenio contemplaban con ojos ciegos una plaza vac\u00eda.\n\nLa Uni\u00f3n estaba a punto de desarticularse. Estaba en la mayor crisis desde su fundaci\u00f3n. Por honda convicci\u00f3n, Florian Susman hab\u00eda defendido desde hac\u00eda a\u00f1os ese proyecto, y tambi\u00e9n estaba dispuesto, naturalmente, a responsabilizarse. No se gru\u00f1e, se toman responsabilidades: \u00e9se hab\u00eda sido ya el credo de su padre. Quien monta una empresa acepta los riesgos. \u00bfC\u00f3mo se los puede evaluar y calcular de un modo responsable? Florian recordaba bien los tiempos en que sus padres se quedaban mucho tiempo despu\u00e9s de la cena sentados a la mesa y con serios semblantes sopesaban las probabilidades de \u00e9xito y los peligros que comportaba una inversi\u00f3n, financiada a cr\u00e9dito, en un matadero industrial concesionado. Las deudas podr\u00edan ser su hundimiento, pero la cobard\u00eda ante ese paso pod\u00eda significar, m\u00e1s definitivamente a\u00fan, el hundimiento de la granja. Hab\u00eda una arriesgada posibilidad de \u00e9xito, pero no exist\u00eda ninguna que excluyera todos los riesgos. Sus padres se sentaban all\u00ed y echaban cuentas, formulaban objeciones y tambi\u00e9n ten\u00edan enseguida argumentos para deshacerlas, en un platillo de la balanza pon\u00edan objeciones, en la otra esperanzas, no, objeciones contra objeciones. Florian escuchaba, era algo raro que sus padres no le hubiesen enviado a la cama, tal vez su padre opinaba que el pr\u00edncipe heredero deber\u00eda o\u00edr todo eso, si quer\u00eda, mientras que Martin, el menor, estaba tumbado en el sof\u00e1 leyendo hasta quedarse dormido y finalmente su madre lo llevaba en brazos a la cama \u2014\u200cno, nunca era algo tan cargado de ternura\u2014, lo empujaba a la cama.\n\n\u00abDioses, tumbas y sabios.\u00bb Florian estaba asombrado, hasta emocionado, por recordar ahora incluso el t\u00edtulo del libro que su hermano le\u00eda entonces una y otra vez, mientras que \u00e9l, Florian, se limitaba a estar all\u00ed sentado oyendo c\u00f3mo sus padres hablaban de lo que pod\u00edan y deb\u00edan hacerse cargo. En aquel entonces. En largas noches.\n\nFlorian conduc\u00eda despacio. Iba con tiempo. Hasta la noche no ten\u00eda que estar en Budapest, era primera hora de la tarde y se encontraba ya a veinte kil\u00f3metros de Nickelsdorf, la frontera entre Austria y Hungr\u00eda. Conduc\u00eda como en estado de trance, con tempomat, m\u00fasica suave de la radio del coche, programa regional, canciones populares continuamente interrumpidas por la publicidad: \u00abCu\u00e1nto me gustar\u00eda ser un cerdo trufero\u00bb, berreaba una voz, a lo que una sonora voz respond\u00eda: \u00abPero qu\u00e9 va, cerdito, \u00bfno crees que nuestras patatas saben mucho mejor? S\u00ed, claro, campesino, gru\u00f1e. Porque eres mi cerdito patatero. \u00bfSoy entonces tambi\u00e9n algo especial? S\u00ed, claro\u00bb.\n\nFlorian apag\u00f3 la radio.\n\nEn aquella \u00e9poca en que su padre, el modesto granjero de cerdos, convirti\u00f3 la granja apenas rentable en una empresa dedicada a criar y sacrificar cerdos, decidi\u00f3 tambi\u00e9n comprometerse en grupos de intereses. Pronto asumi\u00f3 funciones en asociaciones profesionales y en la Asociaci\u00f3n de Agricultores Austriacos. No hay que esperar a que hagan algo por nosotros, debe hacerlo uno mismo, hab\u00eda dicho. Sab\u00eda hablar, pero no pod\u00eda mejorar las condiciones del ramo y, menos a\u00fan, detener la ca\u00edda de los precios. As\u00ed que apost\u00f3 por la cantidad, para, con un margen de ganancia cada vez m\u00e1s reducido, tener a\u00fan beneficio. Las mayores inversiones elevaron el endeudamiento, pero tambi\u00e9n incrementaron el volumen de las transacciones. Y eso aument\u00f3 la relevancia del padre en los gremios. Florian se preguntaba si su padre, aquel hombre cada vez m\u00e1s nervioso e irritable, alguna vez, en un momento tranquilo, se hab\u00eda preguntado si exist\u00eda un camino de vuelta, de vuelta al punto en que necesidad y libertad quedaran en equilibrio, en que esfuerzo y trabajo se vieran premiados con satisfacci\u00f3n y seguridad. Probablemente no. Hay caminos que s\u00f3lo son de ida, sin posibilidad de retroceder. Como aquella autopista por la que se deslizaba, y si en esa calzada le ven\u00eda algo de frente, s\u00f3lo pod\u00edan ser kamikazes, un peligro.\n\nY de pronto Florian hab\u00eda tenido que ocupar el lugar de su padre. Asumir responsabilidades. Y comprob\u00f3 que todo se le quedaba peque\u00f1o. Eso es poco habitual en hijos de padres fuertes. Pero pronto lo vio con toda claridad: para salvar lo que el padre hab\u00eda construido necesitaba otros medios, medios bastante m\u00e1s amplios. Austria hab\u00eda entrado en la Uni\u00f3n Europea, y los grupos de intereses nacionales no comprendieron durante mucho tiempo que hab\u00edan ca\u00eddo en una trampa. Ellos defend\u00edan el mercado nacional que s\u00f3lo exist\u00eda ya en las mentes de viejos funcionarios, se sent\u00edan c\u00f3modos en un sistema de subvenciones que no llevaba a precios justos sino, con creciente derroche burocr\u00e1tico, a la dependencia de limosnas para las que a medio plazo ni siquiera hab\u00eda garant\u00edas, y no exist\u00eda ning\u00fan plan para el periodo posterior, cuando hubieran terminado las disposiciones de transici\u00f3n acordadas en las negociaciones para la entrada. Recordaba una sesi\u00f3n en la C\u00e1mara Econ\u00f3mica Federal de Viena cuyo tema eran las estrategias para la producci\u00f3n de cerdos. \u00c9l era joven entonces, muy inseguro a\u00fan, la sombra de su padre pesaba sobre \u00e9l. Le asombr\u00f3 la hostilidad con que reaccionaron los viejos funcionarios cuando \u00e9l hizo preguntas: como si en realidad lo cuestionara todo, especialmente a ellos, los se\u00f1ores de un mundo desaparecido, los pr\u00edncipes de la Atl\u00e1ntida.\n\nFue ingenuo, pero hab\u00eda comprendido lo m\u00e1s importante: \u00e9l necesitaba un campo de acci\u00f3n m\u00e1s amplio, con los grupos de intereses nacionales \u00e9l no avanzaba, dadas las nuevas condiciones que imperaban en Europa. Entonces hab\u00eda empezado a comprometerse en la Uni\u00f3n de Productores Europeos de Cerdos, la EPP. Y desde hac\u00eda un a\u00f1o era su presidente.\n\nLe adelant\u00f3 un coche de la polic\u00eda con luz azul y sirena, poco despu\u00e9s otro. Finalmente una ambulancia.\n\nUna vez al a\u00f1o, los representantes de la EPP se reun\u00edan en una ciudad europea para una asamblea general de tres d\u00edas. En ella eleg\u00edan al presidente o confirmaban al presidente en funciones. Hab\u00eda intercambio de experiencias, discusiones sobre las contradicciones entre directivas europeas y normativas especiales nacionales, confecci\u00f3n de cat\u00e1logos de demandas a los gobiernos europeos y a la Comisi\u00f3n, visita de empresas locales y cada a\u00f1o un tema principal: esta vez era \u00abComercio exterior porcino europeo\u00bb.\n\nEse a\u00f1o, la secci\u00f3n h\u00fangara hab\u00eda invitado a la reuni\u00f3n anual. Tras lo cual, en la Uni\u00f3n de Productores Europeos de Cerdos se produjeron tumultos y, a\u00fan en la fase de la preparaci\u00f3n organizativa de la asamblea, se lleg\u00f3 a la resistencia organizada. Eso ten\u00eda razones estatutarias y pol\u00edticas. Seg\u00fan los estatutos de la EPP, un representante del pa\u00eds anfitri\u00f3n entraba a formar parte de la presidencia de la EPP. Ahora bien, Hungr\u00eda estaba pol\u00edticamente proscrita porque el gobierno h\u00fangaro hab\u00eda expropiado a sangre fr\u00eda a los criadores europeos de porcino que, tras el cambio pol\u00edtico, hab\u00edan invertido en Hungr\u00eda y participado en empresas h\u00fangaras, y hasta ese momento segu\u00edan simplemente ignorando los escritos de la Comisi\u00f3n Europea en los que \u00e9sta reclamaba una toma de posici\u00f3n sobre esa ruptura del derecho europeo, y, finalmente, intimaba a revocar en un plazo determinado tales infracciones. Se form\u00f3 un grupo que reclamaba el boicot a Hungr\u00eda. Eran sobre todo los holandeses y los alemanes quienes exig\u00edan que se organizara la reuni\u00f3n anual en otra ciudad; se propuso Madrid, pues los cerdos del jam\u00f3n serrano e ib\u00e9rico estaban adquiriendo cada vez m\u00e1s relevancia. En cambio, sobre todo los austriacos, los italianos y los rumanos defend\u00edan la posici\u00f3n de que la asamblea hab\u00eda de celebrarse con mucha m\u00e1s raz\u00f3n a\u00fan en Hungr\u00eda para dejar bien claro que la EPP estaba dispuesta a defender incluso all\u00ed los intereses de sus miembros.\n\nEmpezaba a llover. Florian Susman mir\u00f3 la pantalla de su navegador: s\u00f3lo diez kil\u00f3metros hasta la frontera. Y otra vez ulular de sirenas, otra ambulancia pas\u00f3 a toda velocidad.\n\nFlorian hab\u00eda tenido como presidente un mont\u00f3n de trabajo para impedir la escisi\u00f3n de la EPP y elaborar un compromiso entre los bandos. El compromiso era fr\u00e1gil, consist\u00eda en el fondo en declaraciones de intenciones que luego habr\u00edan de discutirse en la asamblea. Pero era, al fin y a la postre, un compromiso, y la asamblea pod\u00eda celebrarse en Budapest como estaba proyectado. La secci\u00f3n h\u00fangara, como anfitriona, se hab\u00eda declarado dispuesta a firmar una nota de protesta ante el gobierno h\u00fangaro: faltaba por ver si lo har\u00edan de verdad. Porque los grandes productores h\u00fangaros de cerdos hab\u00edan sacado provecho de la renacionalizaci\u00f3n de las empresas, pero, por otra parte, ahora estaban subcapitalizados, y la exportaci\u00f3n de cerdos h\u00fangaros Mangalica hab\u00eda ca\u00eddo casi un 25 por ciento. Pero eso era el tema principal de la asamblea de aquel a\u00f1o.\n\nFlorian Susman no ten\u00eda miedo de que los votantes lo destituyeran como presidente. Al fin y al cabo, hab\u00eda alcanzado ese compromiso provisional, lo que en general hab\u00eda tenido un eco positivo, y hasta ese momento no se hab\u00eda presentado ning\u00fan candidato opositor.\n\nOtra vez sirenas y luces intermitentes azules. Parpadeaban luces en el espejo retrovisor, temblaban en el parabrisas ya algo empa\u00f1ado. Puso el ventilador, dos coches de la polic\u00eda le adelantaron a toda velocidad.\n\nEstaba seguro de que lo confirmar\u00edan como presidente, pero se preguntaba si \u00e9l quer\u00eda. Ya no era un ingenuo. Al contrario, corr\u00eda peligro de convertirse en el tipo de hombre pragm\u00e1tico que siempre hab\u00eda despreciado: uno que siempre hac\u00eda justo lo posible pero que no sab\u00eda llevar a cabo lo necesario. Se dirig\u00eda a un precipicio, pod\u00eda tratar de frenar, pero no pod\u00eda dar un volantazo.\n\nNo hab\u00eda, en verdad, forma de solucionar la escisi\u00f3n de la EPP; \u00e9l, al menos, no ve\u00eda c\u00f3mo: en esa asamblea de Budapest estar\u00edan unidos con los h\u00fangaros contra la Comisi\u00f3n Europea porque \u00e9sta no era capaz o no ten\u00eda la voluntad de negociar con China una cuota m\u00e1s alta de exportaci\u00f3n de cerdos y, al mismo tiempo, estar\u00edan unidos con la Comisi\u00f3n Europea contra los h\u00fangaros porque \u00e9stos violaban el derecho comunitario.\n\nSi la Uni\u00f3n de Productores Europeos de Cerdos deb\u00eda romperse, qu\u00e9 sentido ten\u00eda asumir la responsabilidad, qu\u00e9 absurdo era no querer nada de eso pero decir: asumo la responsabilidad. \u00bfPara qu\u00e9? Como una marioneta que sabe que hay personas con intereses comunes que se organizan para luego, en esa comunidad, dirimir tan inexorablemente los conflictos de intereses que al final ya no hay comunidad.\n\nVio entonces gente delante, en la autopista. \u00a1Gente andando! \u00a1En la autopista! Marchaban hacia \u00e9l. Peatones kamikazes. Hombres, mujeres, ni\u00f1os. Inclinados bajo capuchas de impermeables o con bolsas de pl\u00e1stico sobre la cabeza, algunos con mantas sobre los hombros o sobre la cabeza, algunos llevaban bolsas, otros arrastraban maletas, los limpiaparabrisas se mov\u00edan r\u00edtmicamente de un extremo a otro, como manos que quisieran borrar esa escena, hacerla desaparecer, entonces oy\u00f3 la voz del navegador: \u00abGire cuando sea posible. Gire cuando sea posible\u00bb. Era demencial. Estaba en una autopista, el navegador dec\u00eda que girase y ven\u00edan a su encuentro peatones. Puso la luz de emergencia, se acerc\u00f3 al paso, vio otra vez las luces azules, coches de polic\u00eda en el carril de emergencia, polic\u00edas que agitaban linternas tubulares. Par\u00f3 el coche. Cada vez m\u00e1s personas sal\u00edan de la cortina gris de lluvia a la luz de los faros. Eran muchas. Docenas. Cientos.\n\nDavid de Vriend no hab\u00eda pasado nunca en toda su vida, mejor dicho en su vida de superviviente, por la experiencia de que algo mejoraba, o le ayudaba o incluso le salvaba, cuando se esforzaba por ser amable. Y tampoco esperaba amabilidades. Cortes\u00edas, s\u00ed. La cortes\u00eda era civilizaci\u00f3n. Correcci\u00f3n. A eso quer\u00eda y ten\u00eda que atenerse. Pero cuando dec\u00eda \u00abencantado\u00bb, \u00bfpor qu\u00e9 ten\u00eda que fingir que de verdad estaba encantado?\n\nHab\u00eda podido mostrar sentimientos cuando los ten\u00eda de verdad. Amor, ese altruismo que pone de manifiesto lo m\u00e1s hermoso de uno mismo, y gratitud, una gratitud tan \u00edntima y existencial que hasta sustituye la perdida confianza en Dios. Y hab\u00eda aprendido a esconder sentimientos, miedo o la sensaci\u00f3n de vac\u00edo, de los cuales ya no pod\u00eda desembarazarse pero que al menos pod\u00eda empaquetar y apartar. Y hab\u00eda aprendido a ser desconfiado de una manera tan sensible que esa facultad funcionaba de forma discreta y esclarecedora, como un instrumento de visi\u00f3n nocturna. Pero amabilidad, sobre todo la amabilidad s\u00fabita para con la gente extra\u00f1a, era para \u00e9l pura far\u00e1ndula de comicastros, tan grotesca como un ojo de cristal que mira con forzada amabilidad.\n\nGoedemiddag, dijo con una cort\u00e9s inclinaci\u00f3n de cabeza cuando al salir de su apartamento vio al hombre que abr\u00eda en ese momento la puerta del apartamento vecino. Bonjour, dijo el hombre, que dio dos pasos en direcci\u00f3n a \u00e9l, al mismo tiempo que el chorro de luz de la l\u00e1mpara del techo ca\u00eda sobre su pelo, blanco como la nieve, haci\u00e9ndolo brillar como una aureola. Bonjour, monsieur.\n\nDavid de Vriend salud\u00f3 otra vez con la cabeza y quiso seguir deprisa su camino, pero mir\u00f3 con asombro s\u00f3lo un segundo, aunque ya demasiado tiempo, al hombre que brillaba a la luz del techo. Llevaba una gabardina con efecto moir\u00e9 que al menor movimiento pasaba del verde claro al beige, el rostro le brillaba como reci\u00e9n untado de crema.\n\nBonjour, monsieur, permettez-moi de me pr\u00e9senter, salud\u00f3 el hombre, que dijo llamarse Romain Boulanger, al tiempo que tend\u00eda la mano a De Vriend con una radiante sonrisa, como si aqu\u00e9l fuera el momento m\u00e1s feliz de su vida.\n\nDe Vriend le estrech\u00f3 literalmente la mano, dijo tambi\u00e9n su nombre y Aangenaam, enseguida se corrigi\u00f3: Enchant\u00e9! Todo aquello era a\u00fan de mucha cortes\u00eda, pero amenazaba convertirse en una conversaci\u00f3n acosadoramente amable.\n\nOh, as\u00ed que hablaba franc\u00e9s.\n\nTendr\u00eda que haber dicho por desgracia no muy bien, disculparse y seguir adelante, pero dijo: Oui, monsieur. Muchos flamencos hablaban un franc\u00e9s aceptable, y David de Vriend lo hablaba a la perfecci\u00f3n. En su d\u00eda, tras su huida del tren de la deportaci\u00f3n, hab\u00eda estado escondido dos a\u00f1os en casa de una familia valona, en Villers-la-Ville, de los catorce a los diecis\u00e9is a\u00f1os, hasta que poco antes de acabar la guerra lo denunciaron. El franc\u00e9s se convirti\u00f3 entonces en su segunda lengua materna, la lengua de sus segundos padres, en un sentido para \u00e9l existencial, la lengua del amor. Detest\u00f3 al instante la exageraci\u00f3n de aquella persona desconocida \u2014\u200cno recordaba su nombre\u2014 cuando dijo \u00abQuel bonheur\u00bb y otra vez \u00abQuel bonheur\u00bb, qu\u00e9 felicidad, y luego empez\u00f3 a hablar como un descosido: era el nuevo vecino, hab\u00eda llegado hoy mismo, qu\u00e9 bien haber conocido enseguida a su vecino, esperaba que fuera una buena vecindad, pero ya empezaba estupendamente, una suerte que monsieur De Vriend hablara franc\u00e9s, \u00e9l ya lo hab\u00eda comprobado, que en aquella casa algunos s\u00f3lo hablaban flamenco, incluso entre el personal, bueno, eso le hab\u00eda intranquilizado un poco nada m\u00e1s llegar, que en la Maison Hanssens hubiera empleados que no eran franc\u00f3fonos, o no muy versados, ya una cuidadora que quer\u00eda iniciarle en el reglamento de la casa, una tal madame Godelieve, apellido impronunciable...\n\nGodelieve.\n\nS\u00ed, monsieur, \u00bfla conoc\u00eda? \u00c9l, en cualquier caso, no la hab\u00eda entendido, pero por suerte fue posible arreglarlo, ahora se encargaba de \u00e9l madame Jos\u00e9phine...\n\n\u00a1Una suerte!\n\nLa gabardina de monsieur Boulanger cambiaba constantemente de color.\n\nS\u00ed, monsieur, muy simp\u00e1tica, muy complaciente, pero \u2014\u200cpuso una expresi\u00f3n p\u00edcara y alz\u00f3 el \u00edndice\u2014, no se la pod\u00eda tratar de enfermera, pero claro, era cierto, aquello no era un hospital, aunque ella llevara una cofia as\u00ed, \u00bfla conoc\u00eda \u00e9l?\n\nDavid de Vriend asinti\u00f3.\n\nEn cualquier caso, se alegraba infinito de tener un vecino tan simp\u00e1tico. \u00bfLlevaba mucho tiempo all\u00ed? Ten\u00eda que contarle sin falta, s\u00ed, sin falta, sus experiencias en la casa y darle consejos, quiz\u00e1 en la comida o, despu\u00e9s, tomando una copa de vino.\n\nDavid de Vriend no fue capaz de aceptar complacido esa propuesta y decir s\u00ed, por supuesto, con mucho gusto, buscaba una respuesta educada que no le obligara a nada, adem\u00e1s estaba distra\u00eddo porque el rostro de aquel hombre le recordaba a alguien, pero no sab\u00eda a qui\u00e9n. Monsieur Boulanger dio un corto paso, abandonando as\u00ed el directo cono luminoso del foco del techo, su pelo y su cara dejaron s\u00fabitamente de relucir, se pusieron grises, y \u00e9l dijo: \u00a1Le estoy entreteniendo! \u00bfHab\u00eda dicho de verdad arr\u00eater? \u00a1Disc\u00falpeme, por favor! \u00a1No le entretengo m\u00e1s! \u00a1Nos vemos!\n\nCuando De Vriend entr\u00f3 en el comedor comprob\u00f3 que no quedaba ninguna mesa en la que pudiera estar solo. Quiso dar media vuelta e ir a Le Rustique, entretanto ya le hab\u00edan dado vales de descuento para ese establecimiento, pero la se\u00f1ora Jos\u00e9phine ya lo hab\u00eda tomado bajo su tutela. Aqu\u00ed estamos por fin, dijo ella en voz tan alta que \u00e9l se sobresalt\u00f3, y le empuj\u00f3 en\u00e9rgicamente hacia una mesa en la que estaba el profesor al que ya \u00abhemos\u00bb, como dijo a voces la se\u00f1ora Jos\u00e9phine, conocido, \u00bfverdad, se\u00f1or De Vriend?, cuando aquel peque\u00f1o percance de la espina de pescado, \u00bfverdad?, pero hoy no hay peligro, tenemos un waterzooi riqu\u00edsimo. Profesor, \u00bfpuedo sentar con usted al heer de Vriend, que ya conoce?\n\nQue c\u00f3mo estaba, si se sent\u00eda a gusto en la casa, si ten\u00eda familiares que vinieran a verle: David de Vriend respond\u00eda con cortes\u00eda pero con mucha concisi\u00f3n a las preguntas con las que el profesor \u2014\u200chab\u00eda olvidado su nombre\u2014 quer\u00eda trabar conversaci\u00f3n. Luego vino un momento de silencio mientras tomaban el entrante, ensalada de hinojo con naranja, y De Vriend reflexionaba sobre si ser\u00eda de mala educaci\u00f3n preguntarle otra vez al profesor c\u00f3mo se llamaba, admitir por tanto que lo hab\u00eda olvidado, mientras que el profesor s\u00ed se dirig\u00eda a \u00e9l por su apellido, y pens\u00f3 que era m\u00e1s correcto hacerlo en lugar de disimular, con dificultad y, en \u00faltimo t\u00e9rmino, de modo precario, su peque\u00f1a falta de atenci\u00f3n.\n\nEl profesor no pareci\u00f3 molesto, le inform\u00f3 de buen grado. Se llamaba Gerrit Rensenbrink, dijo, sac\u00f3 su cartera, rebusc\u00f3 en ella y extrajo su tarjeta de visita, apart\u00f3 el plato y puso la tarjeta delante. Profesor de la Universidad de Lovaina, dijo, de pronto ten\u00eda en la mano un bol\u00edgrafo y tach\u00f3 KATHOLIEKE UNIVERSITEIT en la tarjeta. Porque estaba jubilado. Director del Centro de Investigaci\u00f3n de Historia Pol\u00edtica, dijo, y tach\u00f3 la correspondiente l\u00ednea en la tarjeta. Su campo especial de investigaci\u00f3n hab\u00eda sido la historia del nacionalismo y en ella sobre todo la historia de la colaboraci\u00f3n en B\u00e9lgica y Holanda durante la Segunda Guerra Mundial. \u00bfQu\u00e9 estaba tachando ahora? La direcci\u00f3n de correo electr\u00f3nico y el n\u00famero de tel\u00e9fono. Ya no los ten\u00eda.\n\nLuego dijo: Aqu\u00ed tiene, y le pas\u00f3 la tarjeta a De Vriend. En ese momento hubo un estruendo, era monsieur Boulanger que, entrando en el comedor, hab\u00eda cerrado la puerta lanz\u00e1ndola con demasiado \u00edmpetu. David de Vriend levant\u00f3 la vista, \u00c9mile Boulanger alz\u00f3 ambas manos en se\u00f1al de disculpa, dijo Pardon, messieursdames, mir\u00f3 en torno, descubri\u00f3 a De Vriend y corri\u00f3 alegre a su mesa.\n\nPuis-je me joindre \u00e0 vous?, dijo y: Estupendo que podamos proseguir tan r\u00e1pidamente nuestra charla.\n\nSe sent\u00f3, hizo un gesto de saludo al profesor Rensenbrink, era m\u00e1s que una inclinaci\u00f3n de cabeza, era casi una reverencia sentado, y dijo: Yo soy por as\u00ed decirlo el nuevo. Perm\u00edtame que me presente, me llamo...\n\nEmpez\u00f3 a hablar a chorros y De Vriend sinti\u00f3 de pronto un cansancio indecible. Retiraron los platos del entrante, tintineo, ya estaban all\u00ed los del waterzooi, m\u00e1s tintineo de platos, de pronto silencio: el profesor Rensenbrink hab\u00eda dicho que lamentaba no hablar franc\u00e9s.\n\n\u00a1Oh! Y monsieur Boulanger no hablaba neerland\u00e9s.\n\nA De Vriend siempre le hab\u00eda gustado el waterzooi, en cualquier caso nunca tuvo problemas con ese plato, a veces hab\u00eda waterzooi, en la cantina escolar hab\u00eda de vez en cuando, y \u00e9l siempre com\u00eda lo que hab\u00eda. Por supuesto, prefer\u00eda coq au vin cuando quer\u00eda pedir pollo en un restaurante, pero nunca se le habr\u00eda ocurrido hacer de aquello un problema; cuando hab\u00eda waterzooi, hab\u00eda waterzooi, y \u00e9l estaba agradecido. Mir\u00f3 los trozos de carne, levant\u00f3 la vista, el profesor Rensenbrink y monsieur Boulanger le miraron, \u00bfdesesperados? \u00bfCon cierto desamparo? Pero eso no ten\u00eda que ver con el waterzooi, que, en opini\u00f3n de De Vriend, ol\u00eda raro. \u00bfEra alg\u00fan ali\u00f1o que \u00e9l no conoc\u00eda o era olor de putrefacci\u00f3n?\n\nVous devez m'aider! \u00a1Tiene usted que ayudarme, monsieur De Vriend! Monsieur no habla franc\u00e9s, \u00bfser\u00eda tan amable de traducir?\n\nDe Vriend asinti\u00f3.\n\nBoulanger dijo, mientras volv\u00eda a saludar con la cabeza a Rensenbrink: Mon nom est Romain Boulanger...\n\nSon nom est Romain Boulanger...\n\nIk begrijp dat...\n\nJ'\u00e9tais journaliste jusqu'\u00e1 r\u00e9cemment chez Le Soir... Desde hac\u00eda diez a\u00f1os estaba jubilado, pero de vez en cuando escrib\u00eda art\u00edculos como aut\u00f3nomo, uno no pod\u00eda dejarlo, los se\u00f1ores sab\u00edan sin duda c\u00f3mo era, uno no pod\u00eda despedirse de su vida de un d\u00eda para otro, no era nada importante lo que a\u00fan escrib\u00eda, por supuesto, pero estaba agradecido de que le permitieran seguir escribiendo, y le gustaba, por ejemplo, la historia esa del cerdo fantasma, quiz\u00e1 se han enterado ustedes tambi\u00e9n, ese cerdo que... pero da igual; hizo una pausa y un movimiento de cabeza que significaba que De Vriend lo tradujera, por favor, al profesor Rensenbrink.\n\nAlors, dijo De Vriend, il a dit qu'il \u00e9tait un journaliste. Retrait\u00e9.\n\nPero que segu\u00eda escribiendo. Sobre un cerdo.\n\nBoulanger lo mir\u00f3 extra\u00f1ado, vacil\u00f3, De Vriend dijo, c'est tout, y Boulanger continu\u00f3: S\u00ed, si \u00e9l tuviera un vi\u00f1edo se ocupar\u00eda de \u00e9l con entusiasmo, o al menos una casa con jard\u00edn, quiz\u00e1 solamente cortar\u00eda rosas y leer\u00eda. Pero s\u00f3lo ten\u00eda un piso, un piso grande y bonito en Ixelles, pero qu\u00e9 hab\u00eda que hacer all\u00ed, y luego se hab\u00eda muerto su mujer y despu\u00e9s estaba deprimido, el piso, era un piso grande, pero le angustiaba estar all\u00ed, despu\u00e9s de morir su mujer ya no hab\u00eda sido posible all\u00ed una vida cotidiana, o sea, seguir con la vida cotidiana, s\u00f3lo un arrastrarse de all\u00e1 para ac\u00e1 entre las paredes y ya no hab\u00eda podido gestionar...\n\n\u00bfC\u00f3mo dice?\n\nGestionar, todo le resultaba demasiado y, al mismo tiempo, demasiado poco, \u00bflo entend\u00edan los se\u00f1ores?, en cualquier caso aquello ya no era su vida...\n\nWat heeft hij gezegd?\n\nDavid de Vriend respir\u00f3 hondo y reprodujo lo que hab\u00eda dicho Boulanger y, cuando vio el asombro del profesor Rensenbrink, a\u00f1adi\u00f3 que era comprensible. Que monsieur Boulanger tras la muerte de su mujer...\n\nOui, monsieur, dijo Boulanger, pero usted ha..., yo pensaba...\n\nEn ese momento, De Vriend sinti\u00f3 en el pecho una opresi\u00f3n que le cort\u00f3 la respiraci\u00f3n, al mismo tiempo sinti\u00f3 calor, era pura verg\u00fcenza, comprendi\u00f3 que...\n\nNo hab\u00eda traducido lo que monsieur Boulanger hab\u00eda dicho en franc\u00e9s, sino repetido todo el tiempo en franc\u00e9s.\n\nAgach\u00f3 la cabeza, mir\u00f3 los trozos de carne de su plato, se levant\u00f3 y sali\u00f3 corriendo del comedor, cuya puerta encaj\u00f3 de un portazo.\n\n# CAP\u00cdTULO S\u00c9PTIMO\n\n\u00bfC\u00d3MO SE PUEDE NO CREER EN EL FUTURO SI SE SABE QUE SE ES MORTAL?\n\n#\n\nLos d\u00edas eran cada vez m\u00e1s templados, hac\u00eda demasiado calor para esa estaci\u00f3n del a\u00f1o. En los encuentros en los pasillos, en la cantina o ante el ascensor se hac\u00edan humor\u00edsticos comentarios sobre el calentamiento de la Tierra.\n\n\u00a1En Bruselas sin lugar a dudas salimos ganando con el cambio!\n\nNos lo echar\u00e1n tambi\u00e9n en cara: un privilegio m\u00e1s para los funcionarios de Bruselas.\n\nEste tiempo c\u00e1lido me lo deb\u00e9is a m\u00ed, yo s\u00f3lo uso desodorantes con aerosol.\n\nCon las directivas sobre el clima tiramos piedras contra nuestro propio tejado.\n\nDe todos modos, nadie se atiene a ellas: ya ver\u00e9is, pronto tendremos palmeras en Bruselas.\n\nPero eso era el Arca y no la Direcci\u00f3n General de Pol\u00edtica Clim\u00e1tica, y en realidad nadie se re\u00eda por los chistecitos banales en charlas triviales, sino s\u00f3lo porque en esa ciudad lluviosa, en una estaci\u00f3n del a\u00f1o propiamente fr\u00eda, brillaba el sol desde hac\u00eda d\u00edas. El sol se reflejaba en los rostros resplandecientes de la gente, brillaba en sus ojos, reluc\u00eda en los cristales de las ventanas y centelleaba en la chapa met\u00e1lica del tr\u00e1fico urbano.\n\nDespu\u00e9s de la entrevista con Xeno, Martin Susman hab\u00eda redactado el documento para el Jubilee Project, hab\u00eda a\u00f1adido comentarios, ahora ten\u00eda que incluirlos en una nueva redacci\u00f3n para que, una vez finalizada, el documento sirviera de base a una inter-service-consultation. \u00c9se ser\u00eda el siguiente paso. Hab\u00eda prometido entregar el documento al final de la semana, pero a\u00fan quedaban varias cuestiones no resueltas, al menos una gran pregunta sin respuesta. Ten\u00eda que aclararla a toda prisa con Bohumil, porque el asunto era de su competencia. Fue a verle a su despacho, le pregunt\u00f3 si le apetec\u00eda comer algo al mediod\u00eda con \u00e9l.\n\nCon este tiempo podr\u00edamos caminar hasta la place Jourdan. Por ejemplo a la Brasserie L'Esprit. Creo que hasta es posible sentarse fuera.\n\nBuena idea. \u00bfLlamo y reservo una mesa?\n\nS\u00ed, por favor, entretanto voy a por mi chaqueta.\n\nPor la rue Joseph II pasaban tractores.\n\n\u00bfEs una manifestaci\u00f3n de agricultores?\n\n\u00bfC\u00f3mo?\n\nMartin grit\u00f3: \u00bfManifestaci\u00f3n de agricultores?\n\nBohumil se encogi\u00f3 de hombros.\n\nUna larga columna de tractores. Algunos ten\u00edan remolque, en los que hab\u00eda gente que gritaba algo que, sin embargo, con el ruido de motores, bocinas y silbatos, pasaba inadvertido.\n\nLa calles laterales estaban obstruidas por coches de la polic\u00eda colocados de trav\u00e9s.\n\nMartin y Bohumil marcharon en direcci\u00f3n al rond-point Schuman, era imposible conversar. Vieron que por la rue Archim\u00e9de y la avenue de Cortenbergh tambi\u00e9n llegaban, envueltos en el traqueteo del motor, tractores acarreando esti\u00e9rcol, entre medias marchaban grupos de gente con rastrillos y guada\u00f1as. Era un escenario amenazador y al mismo tiempo como de otra \u00e9poca, furia disfrazada de folclore. En la glorieta de Schuman, entre el edificio de la Comisi\u00f3n y el del ayuntamiento y hasta muy dentro de la rue de la Loi, hab\u00eda tractores parados, descargaban esti\u00e9rcol, desplegaban pancartas, apestaba a di\u00e9sel, negras nubecillas de gases de escape planeaban a la luz del sol, sobre un cami\u00f3n de carga estaba de pie una joven con los senos desnudos y enarbolando una tricolor; Martin se detuvo y mir\u00f3, los polic\u00edas le hicieron se\u00f1a de que continuara, continuez s'il vous pla\u00eet, doorlopen alstublieft, dirig\u00edan a los transe\u00fantes entre las vallas met\u00e1licas, llegaron a la rue Froissart, all\u00ed hab\u00eda m\u00e1s calma, pero ellos siguieron en silencio hasta la place Jourdan.\n\nEn la brasserie, o m\u00e1s exactamente, delante de la brasserie, pues se pod\u00eda estar fuera, en efecto, Martin y Bohumil se encendieron unos cigarrillos, echaron una ojeada a la carta, pidieron el men\u00fa del d\u00eda, waterzooi de la mer, y vino blanco y agua, Bohumil expuls\u00f3 al aire anillos de humo, y dijo: Es como en vacaciones, \u00bfverdad? Ya tengo miedo de volver a casa.\n\n\u00bfVolver a casa? \u00bfA qu\u00e9 te refieres?\n\nEl viernes he de viajar a Praga. El s\u00e1bado se casa mi hermana.\n\nLa camarera trajo el vino, Bohumil tom\u00f3 un sorbito, dijo: Y va a ser horrible. Se casa con Kv\u011btoslav Hanka; el nombre no te dice nada, pero en Praga es bastante conocido, m\u00e1s a\u00fan, es tristemente c\u00e9lebre. Es un, c\u00f3mo se dice en ingl\u00e9s, nosotros decimos k\u0159ikloun. S\u00ed, un camorrista. Un diputado bastante radical del \u00dasvit, eso es all\u00ed el partido nacionalista, claro, adversario radical de la UE. El absurdo total, \u00bfno? Yo trabajo en la Comisi\u00f3n Europea y mi cu\u00f1ado trabaja en la destrucci\u00f3n de la UE.\n\n\u00bfEn serio? Y no me digas que eres el padrino.\n\nNo, por supuesto que no. Tanto feeling s\u00ed tiene mi hermana. Hasta ahora. Estaba tan claro que ni siquiera pens\u00f3 en preguntarme. La puse verde cuando me habl\u00f3 de su enamoramiento. Me enter\u00e9 primero por la televisi\u00f3n. De vez en cuando veo por internet los telediarios checos. Y all\u00ed la vi, en un reportaje sobre un acto caritativo. \u00a1Caritativo! Esos asesinos organizan actos caritativos para criminales pobres. Y entonces lo vi, al se\u00f1or diputado, y una voz dec\u00eda: acompa\u00f1ado de su encantadora nueva pareja. \u00bfY qu\u00e9 veo? \u00a1A mi hermana! La llam\u00e9 al momento y le ped\u00ed explicaciones. Ella s\u00f3lo dijo: \u00a1Hombres!\n\n\u00bfHombres?\n\nS\u00ed, ella cree que las diferencias pol\u00edticas son un spleen masculino. Las mujeres entienden de amor, los hombres de est\u00fapidos combates.\n\n\u00bfAs\u00ed es tu hermana?\n\nSirvieron la comida, Bohumil meti\u00f3 su cuchara en el plato y la removi\u00f3 como si quisiera poner lo de abajo arriba, mene\u00f3 la cabeza, dijo: \u00bfTe imaginas la boda? \u00bfLa fiesta de la boda? Toda la algarada fascista de Praga estar\u00e1 presente, y Kv\u011btoslav ha vendido los derechos fotogr\u00e1ficos a Blesk.\n\n\u00bfA qui\u00e9n?\n\nA Blesk. Un peri\u00f3dico. Traducido: lightning. Prensa amarilla.\n\nLightning? Por lo visto, lo contrario de enlightenment.\n\nBohumil hizo un gesto de angustia.\n\nYo no ir\u00eda, dijo Martin.\n\nEs mi hermana. Y mi madre ha dicho que si yo no voy, se mata.\n\nYo no ir\u00eda, repiti\u00f3 Martin. Estaba extra\u00f1ado. Apreciaba a Bohumil y cre\u00eda conocerle. No habr\u00eda pensado que su alegre compa\u00f1ero, que acababa de parpadear tan risue\u00f1o al sol, pudiera tener un problema tan existencial. Pensaba que...\n\nBohumil dec\u00eda algo, Martin s\u00f3lo entendi\u00f3: tiempo de preguerra; \u00bfhab\u00eda dicho de verdad \u00abtiempo de preguerra\u00bb? Son\u00f3 entonces el m\u00f3vil de Martin, que lo cogi\u00f3 y dijo: Te llamo despu\u00e9s, estoy reunido, y pregunt\u00f3 a Bohumil: Perdona, \u00bfqu\u00e9 has dicho?\n\nBohumil inger\u00eda cucharadas de su waterzooi, apart\u00f3 de pronto el plato y dijo: En el fondo a m\u00ed esto no me gusta.\n\n\u00bfQu\u00e9?\n\nYo no soy historiador, dijo, pero para m\u00ed eso siempre ha sido historia, algo de antes, entiendes, edad de piedra, y ese cap\u00edtulo de la edad de piedra se llama \u00e9poca de preguerra: que radicales antagonismos pol\u00edticos dividan a las familias, uno se va con los fascistas, el otro con los comunistas, etc\u00e9tera. \u00bfMe apliqu\u00e9 demasiado poco en el colegio? Pero esto s\u00ed lo recuerdo, lo contaban as\u00ed: antes, en tiempos sombr\u00edos, el odio pol\u00edtico divid\u00eda a las familias. \u00bfQu\u00e9 pesadilla es \u00e9sta? \u00bfC\u00f3mo es que yo tengo hoy, hoy, esos tiempos sombr\u00edos en mi familia? Mi padre, por cierto, no va a la boda.\n\n\u00bfY eso no es motivo de suicidio para tu madre?\n\nNo, al contrario. A ella le gustar\u00eda que \u00e9l se suicidara. Se han separado y est\u00e1n pleiteando.\n\nMartin hab\u00eda querido hablar con Bohumil de algo importante, relacionado con el Jubilee Project, lo aplaz\u00f3 para despu\u00e9s, cuando estuvieran otra vez en la oficina, ahora ten\u00eda la sensaci\u00f3n de que \u00e9l, justamente \u00e9l, deb\u00eda animarle de alg\u00fan modo a \u00e9l, justamente a \u00e9l, a Bohumil. Levant\u00f3 la copa, dijo: Puedo consolarte. Piensa en Herman Van Rompuy.\n\nBohumil lo mir\u00f3 sin entender.\n\nEso hay que represent\u00e1rselo: Van Rompuy ha sido presidente del Consejo Europeo, o sea, presidente de la Uni\u00f3n Europea, su hermana es presidenta de los mao\u00edstas belgas, y su hermano es mandatario de los nacionalistas belgas, un separatista flamenco de la l\u00ednea dura. He le\u00eddo en el peri\u00f3dico que la familia se re\u00fane s\u00f3lo una vez al a\u00f1o: en Navidad.\n\nBohumil, que estaba tomando un trago de vino, se atragant\u00f3: \u00a1En Navidad! \u00a1El presidente de Europa, el nacionalista y la mao\u00edsta!\n\n\u00a1Y todos cantan Noche de paz!\n\n\u00a1Noche de paz! \u00a1Sopla! \u00bfEs verdad?\n\nS\u00ed. Por lo visto. Lo he le\u00eddo. Era una story en De Morgen.\n\nBohumil dijo riendo: Nos tomamos otra copa m\u00e1s.\n\nCuando regresaban a la oficina, la manifestaci\u00f3n se hab\u00eda disuelto, pasaron por la glorieta Schuman, entre vallas met\u00e1licas y junto a montones de esti\u00e9rcol que echaban a paletadas en los camiones de la limpieza municipales. Ol\u00eda mal. Re\u00eda el sol.\n\nDurante el camino de vuelta a la oficina, Bohumil estaba silencioso y pensativo. En el ascensor dijo: Anulo el vuelo del viernes. No voy a la boda. No quiero aparecer con Kv\u011btoslav Hanka en una foto que se publicar\u00e1 en Blesk.\n\n\u00bfY tu madre?\n\nLe digo que voy para Navidad.\n\nLuego golpe\u00f3 con el pu\u00f1o en el brazo a Martin y dijo sonriendo: \u00a1Noche de paz!\n\nMedia hora despu\u00e9s, Martin, Bohumil y Kass\u00e1ndra, sentados en la sala de reuniones, deliberaban sobre un update de los trabajos preparatorios del Jubilee Project. En el comentario al documento de Martin, Xeno hab\u00eda dicho que ten\u00edan que averiguar cu\u00e1ntos perseguidos y v\u00edctimas del Holocausto viv\u00edan a\u00fan. \u00bfHay un registro central de los supervivientes de campos de concentraci\u00f3n y de exterminio? \u00bfCu\u00e1ntos viven hoy en Europa, cu\u00e1ntos en Israel, en Estados Unidos o en otro lugar? \u00bfHay una instituci\u00f3n que pueda ser, como delegaci\u00f3n representativa de los supervivientes, un socio que coopere en la organizaci\u00f3n del evento?\n\nHab\u00eda que saberlo para decidir si era factible invitar a Bruselas a todos los supervivientes del Holocausto o al menos a un grupo realmente representativo.\n\nEso nos ha sorprendido mucho, dijo Bohumil. Hab\u00edamos esperado, claro, que hubiera un registro central de los supervivientes del Holocausto. Pero no hemos encontrado ninguno.\n\nKass\u00e1ndra: Ninguna de las instituciones a las que hemos pedido informaci\u00f3n ha respondido. Yad Vashem,[5] por ejemplo. Ninguna respuesta. Tras una nueva demanda lleg\u00f3 una, pero que tampoco lo es, aqu\u00ed la ten\u00e9is: que nuestro e-mail hab\u00eda sido trasmitido a un colaborador pertinente. Luego, durante d\u00edas, no pas\u00f3 nada otra vez. Volv\u00ed a escribir pidiendo que me comunicaran el nombre y la direcci\u00f3n de ese colaborador para ponerme directamente en contacto con \u00e9l. Ninguna respuesta. Hasta hoy. Luego el centro Wiesenthal[6] de Los \u00c1ngeles: ninguna respuesta. A nuestra nueva demanda nos respondieron que la documentaci\u00f3n de las v\u00edctimas de la Sho\u00e1 no era de la competencia del Wiesenthal Center. Ellos s\u00f3lo ten\u00edan una lista de los criminales de guerra del nacionalsocialismo a\u00fan vivos, esa lista estaba publicada en su p\u00e1gina web, pero no dispon\u00edan de un registro de las v\u00edctimas a\u00fan con vida de la Sho\u00e1. Nos indicaban que nos dirigi\u00e9ramos a Yad Vashem. Hemos enviado ese e-mail a Yad Vashem, pidiendo nuevamente informaci\u00f3n: no ha habido respuesta. Hemos escrito a todos los lugares conmemorativos: Auschwitz, Bergen-Belsen, Buchenwald, Mauthausen, etc., pero s\u00f3lo de Mauthausen ha llegado respuesta.\n\n\u00bfY qu\u00e9 ha escrito Mauthausen?\n\nAqu\u00ed est\u00e1: que ellos s\u00f3lo tienen una lista de los supervivientes de Mauthausen, que ni siquiera es completa, por el caos que hubo despu\u00e9s de la liberaci\u00f3n, en mayo del 45. Los supervivientes, en cuanto pudieron abandonar el campo, se dirigieron a diferentes autoridades e instituciones para obtener ayuda y papeles, y nada de eso se reuni\u00f3 en una oficina central. Y, adem\u00e1s, que del incompleto archivo de datos personales de que dispon\u00eda el servicio conmemorativo de Mauthausen s\u00f3lo una peque\u00f1a parte ten\u00eda los datos actualizados: y ni siquiera \u00e9sos eran del todo fiables. Cada a\u00f1o invitan al acto conmemorativo de la liberaci\u00f3n a las personas de las que tienen direcciones. Quien no reacciona durante a\u00f1os a esa invitaci\u00f3n puede haber muerto, pero tambi\u00e9n, simplemente, haber cambiado de direcci\u00f3n. El director del lugar conmemorativo de Mauthausen nos remiti\u00f3 \u2014\u200c\u00a1sorpresa!\u2014 a Yad Vashem, pero tambi\u00e9n a la Fundaci\u00f3n Sho\u00e1 de Steven Spielberg. \u00a1Una indicaci\u00f3n interesante! Y nos adjuntaban el texto del Juramento de Mauthausen, para recordarnos a nosotros, o sea, a la Comisi\u00f3n, que los Tratados de Roma se aten\u00edan a \u00e9l. El director escribi\u00f3 \u2014\u200cun momento, s\u00ed, aqu\u00ed lo tengo\u2014: el lema \u00abNunca m\u00e1s Auschwitz\u00bb era problem\u00e1tico porque pon\u00eda un campo por delante de todos los dem\u00e1s, es decir, en \u00faltimo t\u00e9rmino hac\u00eda un ranking de los campos, pero el Juramento de Mauthausen era universal y por eso estaba al comienzo del proyecto de unificaci\u00f3n europea, aunque hoy ya no se hable de eso.\n\nMartin asinti\u00f3. Es la raz\u00f3n por la que nosotros... Se interrumpi\u00f3, dijo: Empleamos Auschwitz como s\u00edmbolo, pero \u00e9l en el fondo ha entendido nuestra idea. \u00bfY has escrito a Spielberg?\n\nS\u00ed.\n\n\u00bfNo ha contestado?\n\nS\u00ed. Breve y conciso. Que s\u00f3lo hab\u00eda una lista de supervivientes que estuvieron dispuestos a contar, como testimonios, su historia delante de una c\u00e1mara. Pero que ellos no sab\u00edan cu\u00e1ntas v\u00edctimas de la Sho\u00e1 a\u00fan viv\u00edan, ni siquiera sab\u00edan cu\u00e1ntos testigos de aquello todav\u00eda segu\u00edan con vida. Las grabaciones se hicieron con personas que se declararon dispuestas a ello por iniciativa propia. El archivo es accesible a cualquiera. Para m\u00e1s detalles deb\u00edamos preguntar en...\n\nYad Vashem.\n\nExacto. Es decir, no sabemos nada.\n\nRealmente es algo raro, dijo Martin. Totalmente incomprensible. Los nazis consignaron en listas, con nombres y apellidos, datos personales, fecha de nacimiento, profesi\u00f3n, \u00faltima direcci\u00f3n, a todas las personas deportadas a un campo de concentraci\u00f3n, les pusieron n\u00fameros, los contaron y recontaron incesantemente, tacharon pulcramente de las listas a los asesinados: y despu\u00e9s de la liberaci\u00f3n todo se evapora en el aire.\n\n\u00a1La burocracia nazi!\n\nPero \u00bftoda la burocracia? Se los habr\u00eda tenido que consignar para...\n\nNo, dijo Bohumil. Muchos no quisieron o no pudieron regresar a los pa\u00edses de los que hab\u00edan sido expulsados o deportados. Nadie ten\u00eda inter\u00e9s en una lista m\u00e1s de \u00abdisplaced persons\u00bb. Se les prestaron los primeros auxilios y luego dejaron que los que pod\u00edan marcharse se marcharan.\n\nNo me lo creo, dijo Martin. \u00bfYad Vashem reconstruye todos los nombres de las personas que fueron asesinadas en los campos pero no tienen ning\u00fan inter\u00e9s por las que sobrevivieron? Yo no me lo creo. Esa lista tiene que existir, pero parece que hay inter\u00e9s en mantenerla oculta.\n\nCome on, Martin, dijo Kass\u00e1ndra, no hay una conjuraci\u00f3n. \u00bfQu\u00e9 raz\u00f3n habr\u00eda para ello? Hay muchas razones de por qu\u00e9 no conocemos el n\u00famero de supervivientes. Cuando despu\u00e9s de la liberaci\u00f3n pudieron abrirse paso hacia alguna parte, no dejaron direcci\u00f3n porque todav\u00eda no ten\u00edan ninguna. Y cuando despu\u00e9s empezaron a reorganizar su vida en alg\u00fan lugar, no escribieron a su antiguo campo de concentraci\u00f3n para comunicar d\u00f3nde se los pod\u00eda localizar: Martin, por favor, enti\u00e9ndelo, los supervivientes de los campos no son antiguos alumnos. Okay, algunos se dirigieron a los lugares conmemorativos, se pusieron a disposici\u00f3n como testigos de los hechos, para informar sobre lo que hab\u00edan vivido, algunos acudieron a los actos conmemorativos de la liberaci\u00f3n, algunos llegaron d\u00e9cadas despu\u00e9s con sus nietos, era su triunfo sobre Hitler, pero muchos ya no quisieron tener nada que ver con aquello, otros murieron muy pronto despu\u00e9s de la liberaci\u00f3n, eran supervivientes, s\u00ed, pero muy pronto tuvieron la muerte completamente normal de la posguerra, otros sent\u00edan verg\u00fcenza y no quer\u00edan quedar consignados nunca m\u00e1s en un fichero, otros guardaron silencio porque notaron que nadie quer\u00eda o\u00edr su historia, ni siquiera en Israel quer\u00edan escucharlos, a esos jud\u00edos embarazosos que proced\u00edan del matadero. \u00bfC\u00f3mo se habr\u00eda podido consignar y sistematizar todo eso?\n\nTenemos un problema, dijo Bohumil. La lista que quer\u00eda Xeno no existe. Carece de sentido averiguar la raz\u00f3n. Y hay una soluci\u00f3n sencilla al problema. \u00bfDe qu\u00e9 se trata en realidad? Del relato de la Comisi\u00f3n Europea. T\u00fa dices que la Comisi\u00f3n naci\u00f3 como respuesta al Holocausto, eso no ha de poder repetirse jam\u00e1s, nosotros garantizamos paz y justicia. Okay, pero para probar eso no necesitamos una lista completa de las v\u00edctimas que a\u00fan viven. \u00bfQuieres reunirlas en la rue de la Loi y pasar revista? \u00bfHacer un recuento?\n\n\u00a1No sigas por ah\u00ed! \u00a1C\u00e1llate, por favor!\n\nHay supervivientes de la Sho\u00e1 que son conocidos, dijo Kass\u00e1ndra, podr\u00edamos hacer una lista y ver qui\u00e9n de ellos, en nuestra celebraci\u00f3n...\n\n\u00bfHab\u00e9is preguntado en la Eurostat?\n\n\u00bfPor qu\u00e9 \u00edbamos a hacer eso?\n\nPor favor, Bohumil, dijo Martin. Tenemos una oficina europea de estad\u00edstica. Tenemos estad\u00edsticas de todo. Ellos lo saben todo. Saben cu\u00e1ntos huevos ponen las gallinas hoy en Europa. Tambi\u00e9n sabr\u00e1n cu\u00e1ntas v\u00edctimas del Holocausto viven hoy en Europa. Kass\u00e1ndra, preg\u00fantaselo y seguiremos hablando cuando tengamos la respuesta.\n\nKass\u00e1ndra escribi\u00f3 \u00abEurostat\u00bb en su bloc de notas, mir\u00f3 a Martin: Yo no quiero decir nada, pero \u00bfpor qu\u00e9 quieres precisamente ahora una informaci\u00f3n estad\u00edstica, un n\u00famero, sobre personas que fueron convertidas en n\u00fameros?\n\nAbri\u00f3 el bot\u00f3n del pu\u00f1o de su blusa, se arremang\u00f3. Escribi\u00f3 171185 con su bol\u00edgrafo en el antebrazo y se lo mostr\u00f3 a Martin.\n\n\u00bfQu\u00e9...? \u00bfQu\u00e9 es eso?\n\nMi fecha de nacimiento, dijo Kass\u00e1ndra.\n\nMartin Susman trabajaba a menudo hasta las siete o las siete y media. No le remord\u00eda la conciencia cuando ese d\u00eda sali\u00f3 de la oficina a las cuatro y media. No hab\u00eda pendiente nada de urgencia y el trabajo rutinario que tal vez viniera en la hora siguiente pod\u00eda despacharlo por la ma\u00f1ana. No ten\u00eda en casa nada de comida, pero no ten\u00eda hambre. Decidi\u00f3 tomar una cerveza de camino al metro, en el James Joyce Pub, en la rue Archim\u00e8de. Por all\u00ed circulaban veh\u00edculos blindados. Camin\u00f3 otro trecho, al Charlemagne, tambi\u00e9n all\u00ed y en la rue de la Loi hab\u00eda veh\u00edculos militares, cuyo acero lacado en verde y gris parec\u00eda tragar la luz del sol poniente. Patrullaban soldados, polic\u00edas desviaban los coches y a los transe\u00fantes los dirig\u00edan por vallas met\u00e1licas a estrechos pasadizos que llevaban a la estaci\u00f3n de metro, por estar cortado el acceso directo que hab\u00eda delante del edificio del ayuntamiento.\n\nLa situaci\u00f3n le record\u00f3 a Martin pel\u00edculas que hab\u00eda visto alguna vez, Z, o Desaparecido, o documentales de la televisi\u00f3n. Ve\u00eda muy poco la tele. Pero cuando en noches insomnes zapeaba por los canales, siempre se deten\u00eda en los documentales hist\u00f3ricos, la historia le interesaba m\u00e1s que las historias, sobre todo le fascinaban los documentales hist\u00f3ricos, los antiguos noticieros semanales que se proyectaban en los cines, as\u00ed como las grabaciones de aficionados que se desenterraban y utilizaban en documentales, mientras una voz sonora y grave hablaba de otra \u00e9poca. Ahora ten\u00eda esas im\u00e1genes en la cabeza, los tanques en la Wenzelsplatz despu\u00e9s de la represi\u00f3n de la Primavera de Praga, los carros blindados que pasaban por las calles de Santiago de Chile tras el golpe de Pinochet, la presencia militar en las calles de Atenas tras el golpe de los coroneles, temblorosas pel\u00edculas de aficionado en Super 8 y escenas en blanco y negro de antiguos telediarios. Martin ten\u00eda la impresi\u00f3n de que ahora, en la calle por la que caminaba, estaban proyectando ese material hist\u00f3rico creando una realidad virtual para la que le faltaba la consola de videojuegos. Los tanques se mov\u00edan como grandes escarabajos por la calle libre de coches, los pocos transe\u00fantes marchaban arrimados a las casas y a las vallas, y los tragaba la boca de metro.\n\nMartin no ten\u00eda miedo, recordaba que hab\u00eda una cumbre del Consejo Europeo con los jefes de Estado y de gobierno europeos. \u00c9sas eran las correspondientes medidas de seguridad. Se fue al James Joyce Pub, hombres con trajes y corbatas aflojadas estaban ante la barra charlando. Era la happy-hour.\n\nCamino de su casa compr\u00f3 un paquete de seis botellas de Jupiler en la tienda de la esquina de la rue Sainte-Catherine.\n\nGoedenavond.\n\nBonsoir, monsieur.\n\nAu revoir!\n\nTot ziens.\n\nEn casa se quit\u00f3 el pantal\u00f3n, le apretaba, estaba engordando, se despreciaba por ello, pero sin tomar una resoluci\u00f3n; en Bruselas se contaba el tiempo no en a\u00f1os, sino en kilos. Se fum\u00f3 un cigarrillo en camisa y calzoncillos junto a la ventana abierta, luego se sent\u00f3 en la butaca al lado de la chimenea en la que estaban los libros antiguos, encendi\u00f3 una vela, \u00bfpor qu\u00e9? Porque estaba all\u00ed. Bebi\u00f3 cerveza, mir\u00f3 c\u00f3mo entraban insectos por la ventana abierta y buscaban la luz de la vela, se met\u00edan en la llama y se quemaban.\n\nPara \u00e9l era la prueba de que Dios no exist\u00eda, de que no hab\u00eda ning\u00fan sentido en la creaci\u00f3n, o sea, de que no hab\u00eda creaci\u00f3n. Porque \u00bfqu\u00e9 sentido tiene crear una especie que se activa de noche, pero luego, en la oscuridad, busca la luz s\u00f3lo para perecer abrasada en ella? \u00bfQu\u00e9 utilidad tienen esos bichos, qu\u00e9 contribuci\u00f3n aportan a la postulada o esperada armon\u00eda de la naturaleza? Probablemente se han multiplicado ya antes de alg\u00fan modo, han puesto descendientes en el mundo que, al igual que ellos, se encuentran durante todo el largo y claro d\u00eda en un estado let\u00e1rgico para despu\u00e9s, al caer la noche, salir y buscar la luz que antes, sumidos en el sue\u00f1o, no hab\u00edan visto, s\u00f3lo para poner fin inmediatamente a su vida por una grotesca pulsi\u00f3n de muerte. En el crep\u00fasculo comienza el vuelo a la muerte. Se pegan a ventanas detr\u00e1s de las que hay luz, como si el cristal ofreciera alimento, revolotean en torno a l\u00e1mparas y farolas, como si tan cerca de la luz hubiera otra cosa aparte de la ceguera, y cuando descubren una vela o cualquier otro fuego abierto entonces encuentran su destino, la muerte inmediata en la que se precipitan, es decir, en las tinieblas de las que ven\u00edan.\n\nCon s\u00fabita decisi\u00f3n, el comisario Brunfaut se ape\u00f3 ya en Schuman, en lugar de viajar hasta Merode. Entre esas dos estaciones de metro estaba el Parc du Cinquantenaire, generalmente conocido como parque del Aniversario, por el que quer\u00eda dar un agradable paseo en ese d\u00eda de radiante sol. Se recet\u00f3 a s\u00ed mismo esa marcha a pie, oprimido por el miedo helador que se hab\u00eda apoderado de \u00e9l en el metro ante el tubo en el que lo meter\u00edan en el hospital. Ten\u00eda tiempo de sobra, con su nerviosismo hab\u00eda salido de casa demasiado pronto.\n\nLa salida a Justus Lipsius estaba cortada, la masa de gente lo empuj\u00f3 a la salida Berlaymont, donde se produjo un atasco porque la escalera mec\u00e1nica de ascenso no funcionaba. La gente se dirigi\u00f3 entonces a la escalera normal, en la que sin embargo se deten\u00eda continuamente y se echaba hacia un lado para dejar paso a los que descend\u00edan. Pero al mismo tiempo se les echaban encima los que llegaban despu\u00e9s, cargados con mochilas y maletas de ruedas. Brunfaut se apret\u00f3 contra el cuerpo su peque\u00f1a bolsa de viaje, o\u00eda gritos que proven\u00edan de arriba, de la salida, agudos silbatos, algunos transe\u00fantes que sub\u00edan la escalera se dieron la vuelta, cada vez bajaba m\u00e1s gente de arriba, Brunfaut no ten\u00eda idea de lo que ocurr\u00eda fuera, pero se vio obligado a retroceder y nad\u00f3 con la masa de vuelta al and\u00e9n. Llegaba ya un tren, Brunfaut se meti\u00f3 en \u00e9l y viaj\u00f3 una estaci\u00f3n m\u00e1s, hasta Merode.\n\nEn la avenue des Celtes, junto a la boca del metro, estaba la Brasserie La Terrasse. All\u00ed quer\u00eda esperar tomando una cerveza hasta la hora de la cita. La Terrasse estaba llena de gente pero quedaba una mesa libre y, aunque el establecimiento estaba situado directamente en una calle grande y de mucho tr\u00e1fico, \u00c9mile Brunfaut ten\u00eda la sensaci\u00f3n de encontrarse, detr\u00e1s de la pared de plantas verdes, en un oasis de paz. Paz. Reflexionar en paz. \u00bfQu\u00e9? \u00bfSobre qu\u00e9? Deb\u00eda tomar una decisi\u00f3n vital. Lo pensaba con esa pat\u00e9tica expresi\u00f3n: decisi\u00f3n vital. Y al momento fue demasiado para \u00e9l. Aunque ya viv\u00eda desde hac\u00eda alg\u00fan tiempo sabiendo que lo hab\u00edan despedido, no formalmente, pero s\u00ed: despedido de su vida, para \u00e9l aquello era \u00abs\u00fabito\u00bb, qu\u00e9 raro el tiempo que puede durar un \u00abs\u00fabito\u00bb.\n\nAl mismo tiempo se pregunt\u00f3 qu\u00e9 sentido iba a tener tomar una decisi\u00f3n vital s\u00f3lo porque ten\u00eda esa expresi\u00f3n en la cabeza, mientras que \u00e9l ni siquiera sab\u00eda...\n\nEl camarero. Brunfaut pidi\u00f3 una cerveza.\n\n\u00bfDeseaba tambi\u00e9n comer?\n\nDijo que no. S\u00f3lo quer\u00eda una cerveza.\n\n... mientras que \u00e9l ni siquiera sab\u00eda si a\u00fan le quedaba tiempo de vida.\n\nEl camarero trajo la cerveza, puso sobre la mesa al mismo tiempo el ticket de la cuenta y un papel en el que pon\u00eda: RESERVADO 12.30. Pidi\u00f3 cobrar enseguida. 12.30: era dentro de diez minutos. Era evidente que el camarero quer\u00eda tener libre la mesa enseguida por si llegaba alguien que s\u00ed quisiera comer.\n\nBrunfaut hab\u00eda sido siempre una persona que impon\u00eda respeto, ya s\u00f3lo por su presencia f\u00edsica, con un cuerpo grande que ocupaba terreno. Ahora estaba como aturdido, qu\u00e9 peque\u00f1o y confuso se sent\u00eda cuando levant\u00f3 la vista hacia el camarero.\n\nSe levant\u00f3. Respir\u00f3 hondo y se esponj\u00f3. Tendr\u00eda que haberme dicho que esta mesa est\u00e1 reservada. No tengo ganas de embucharme la cerveza a esa velocidad. Me planta usted delante el papel con el RESERVADO despu\u00e9s de haber pedido yo, me parece c\u00ednico y humillante. \u00a1Adi\u00f3s!\n\n\u00a1Pero, monsieur! No puede, \u00a1espere! \u00a1No puede usted marcharse sin m\u00e1s! Tiene que pagar la cerveza.\n\n\u00bfPor qu\u00e9? Si no me la bebo.\n\nEntonces tengo que llamar a la polic\u00eda.\n\nAqu\u00ed est\u00e1 mi placa. Vengo de lo m\u00e1s a prop\u00f3sito.\n\n\u00a1Oh! Perdone, se\u00f1or comisario. Por supuesto que puede quedarse en esta mesa todo el tiempo que quiera, cambiar\u00e9 la reserva, por supuesto, se\u00f1or comisario.\n\n\u00a1Ya no me apetece!\n\nEso fue s\u00f3lo una breve fantas\u00eda, que, por infantil que fuera, lo humill\u00f3 a\u00fan m\u00e1s. En la realidad pag\u00f3, dijo: No importa, de todos modos he de marcharme dentro de diez minutos. Tengo una cita y...\n\n\u00bfY qu\u00e9? Encima dej\u00f3 demasiada propina.\n\nMir\u00f3 unos minutos al vac\u00edo, mir\u00f3 la cerveza... \u00bfc\u00f3mo hab\u00eda podido olvidar que...? Se levant\u00f3 y se march\u00f3 sin haber bebido ni un sorbo.\n\n\u00c9mile Brunfaut cruz\u00f3 la avenue des Celtes y subi\u00f3 por la rue de Linthout. Hab\u00eda olvidado el n\u00famero, pero continu\u00f3 andando, pens\u00f3 que aunque no sab\u00eda el n\u00famero de la casa, la reconocer\u00eda.\n\nNo la reconoci\u00f3. Se pas\u00f3 de largo. En alg\u00fan momento lo tuvo claro y volvi\u00f3 sobre sus pasos. En lugar de llegar demasiado pronto logr\u00f3 casi llegar tarde. Estaba sudando. Ya en recepci\u00f3n y en la primera entrevista con el m\u00e9dico dar\u00eda la peor impresi\u00f3n posible.\n\n\u00a1Ah\u00ed! \u00a1Ahora lo ve\u00eda! El Europa-Hospital. Por fuera parec\u00eda una catedral neog\u00f3tica. Por eso se hab\u00eda pasado de largo. \u00bfQui\u00e9n va a esperar que un hospital parezca un templo hist\u00f3rico?\n\nEntr\u00f3: y se encontr\u00f3 de pronto en una estaci\u00f3n espacial. Blancas superficies de pl\u00e1stico, aluminio plateado, luz azul, luminosas cintas de colores en el suelo, como sistema de orientaci\u00f3n a las distintas especialidades; a Brunfaut le extra\u00f1\u00f3 que la gente que se mov\u00eda o estaba sentada all\u00ed no flotara ingr\u00e1vida en el espacio. Por otra parte: s\u00f3lo era, completamente banal, el hall de entrada de un hospital. Todo lavable, con brillo cl\u00ednico. Pero daba impresi\u00f3n de ser el escenario de una pel\u00edcula de ciencia ficci\u00f3n porque se entraba en aquella sala por la fachada de una catedral g\u00f3tica.\n\nBrunfaut estaba ante el panel del directorio. Lo primero que percibi\u00f3 de modo consciente fue: PSIQUIATR\u00cdA. Luego vio: MEDICINA INTERNA. Sigui\u00f3 la cinta luminosa azul del sistema de orientaci\u00f3n.\n\nInscripci\u00f3n, admisi\u00f3n, asignaci\u00f3n de la habitaci\u00f3n, primera entrevista m\u00e9dica con anamnesis. Luego el profesor Drumont explic\u00f3 qu\u00e9 ex\u00e1menes consideraba necesario hacer y que ser\u00eda posible llevarlos todos a cabo en dos d\u00edas. \u00c9l lo repartir\u00eda todo de manera adecuada. Despu\u00e9s, estaba seguro, se tendr\u00eda un diagn\u00f3stico de la afecci\u00f3n de Brunfaut. \u00bfEstaba en ayunas el se\u00f1or comisario? Brunfaut asinti\u00f3. Hoy no hab\u00eda comido ni bebido nada. Muy bien, dijo el m\u00e9dico jefe, entonces podemos liquidar enseguida la extracci\u00f3n de sangre. Eso lo har\u00e1 la enfermera se\u00f1orita Ana. Ella ir\u00e1 a su habitaci\u00f3n. Y me encargar\u00e9 de que despu\u00e9s le den a usted enseguida un peque\u00f1o refrigerio.\n\nLa enfermera que despu\u00e9s de sacarle sangre le llev\u00f3 t\u00e9 con gofres y unas fresas pregunt\u00f3 a Brunfaut enseguida lo que deseaba de cena.\n\nVeo en su historia cl\u00ednica que usted no \u2014\u200cle mir\u00f3\u2014, no est\u00e1 a\u00fan a dieta. As\u00ed que dieta normal. Puede elegir entre carne y vegetariano.\n\nBrunfaut mir\u00f3 el plato con los dos gofres y las tres fresas y dijo: Las dos, madame.\n\n\u00bfC\u00f3mo, las dos?\n\nSupongo que con la carne vendr\u00e1 una guarnici\u00f3n.\n\nHay boulettes sauce lapin.\n\n\u00bfY con eso?\n\nPur\u00e9 de patatas y zanahorias.\n\nPues lo que digo, eso es vegetariano. As\u00ed que me quedo con las boulettes, y tengo ambas cosas.\n\nEstaba asustado. Nunca antes hab\u00eda estado tan asustado. Pero algo en \u00e9l se rebelaba, le obligaba, por as\u00ed decirlo, a hacer como si no se tomara en serio nada de eso. Sobre la cama descansaba el pijama, como un cad\u00e1ver sin cuerpo. De un gancho junto a la cama colgaba, fl\u00e1ccido, su albornoz: \u00e9se era \u00e9l, despu\u00e9s de su desaparici\u00f3n. No se desvisti\u00f3, no se meti\u00f3 a\u00fan en la cama. La enfermera se march\u00f3. Brunfaut se comi\u00f3 un gofre, tom\u00f3 un trago de t\u00e9, sonri\u00f3 cuando se dio cuenta de que, conteniendo la respiraci\u00f3n y abriendo la puerta y mirando a derecha e izquierda, estaba comprobando si hab\u00eda moros en la costa. Sali\u00f3 de la habitaci\u00f3n, baj\u00f3 en el ascensor al hall para tomarse una cerveza en la cantina.\n\nNo hab\u00eda alcohol en la cantina del hospital. As\u00ed pues, sali\u00f3 del mundo de nave espacial y, a trav\u00e9s de la fachada, a la calle, dio unos pasos entre la oleada de gente que no pensaba en la muerte, encontr\u00f3 un caf\u00e9 con terraza y pidi\u00f3 una cerveza.\n\n\u00bfPeque\u00f1a, monsieur?\n\nGrande, por favor.\n\nEstaba sentado de manera que ten\u00eda ante la vista una farmacia.\n\nSudaba, se sec\u00f3 la frente con el pa\u00f1uelo. \u00bfTen\u00eda fiebre? No, simplemente era un d\u00eda caluroso. Por la abertura entre dos sombrillas el sol le quemaba la nuca y la espalda. Corri\u00f3 un poco la silla hacia un lado, se quit\u00f3 la chaqueta.\n\nEntonces son\u00f3 su m\u00f3vil. Era Philippe.\n\nOye, tengo algunas cosas que contarte. No por tel\u00e9fono. El cuadro no est\u00e1 claro a\u00fan, pero hay algunos \u2014\u200c\u00bfc\u00f3mo llamarlos\u2014 s\u00edntomas muy interesantes. No s\u00e9 si puedo continuar con esto, es muy arriesgado. Tenemos que hablarlo. \u00bfPodemos vernos ma\u00f1ana?\n\nEstoy en el hospital, dijo Brunfaut. Ya sabes, me hacen un chequeo. Ma\u00f1ana tengo una serie de reconocimientos, pero...\n\n\u00bfC\u00f3mo est\u00e1s? \u00bfQu\u00e9 dice el m\u00e9dico?\n\nComo t\u00fa dices: s\u00edntomas interesantes, pero el cuadro a\u00fan no est\u00e1 claro. \u00bfPuedes ma\u00f1ana a \u00faltima hora?\n\nAl caer la tarde. A las seis y media o las siete.\n\nVale. Entonces ven a verme al Europa-Hospital, rue de Linthout. Si vienes en metro, es la estaci\u00f3n de Merode.\n\nD'accord. Hasta ma\u00f1ana.\n\n\u00c9mile Brunfaut estaba en una habitaci\u00f3n de dos camas, pero, por suerte, la otra cama no estaba ocupada. As\u00ed pudo llevar a cabo por la noche algunas llamadas sin molestar a otro o sin verse obligado a salir al pasillo, pod\u00eda encender o apagar el televisor que estaba montado en la pared por encima de la mesa, frente a las camas, sin tener que ponerse de acuerdo con nadie, vio el informativo de la noche, hab\u00eda una entrevista con el presidente de la polic\u00eda, que rechazaba el reproche de inactividad, diciendo que, por desgracia, era muy dif\u00edcil capturar a un cerdo, si no se sab\u00eda cu\u00e1ndo y d\u00f3nde dar\u00eda el golpe la pr\u00f3xima vez. \u00bfHa dicho de verdad \u00abdar el golpe\u00bb?, se pregunt\u00f3 Brunfaut. Pero ya ven\u00eda la pregunta de la periodista: \u00bfQu\u00e9 quer\u00eda decir con \u00abdar el golpe\u00bb? Quer\u00eda decir, explic\u00f3, aparecer de pronto y provocar la inseguridad de los viandantes, aunque... Irritado, Brunfaut desconect\u00f3 el televisor, lo que precisamente pudo hacer por estar solo en la habitaci\u00f3n. As\u00ed tambi\u00e9n pudo estar desinhibidamente intranquilo en una noche muy intranquila, dar mil vueltas en la cama, levantarse una y otra vez, beber agua en el ba\u00f1o, ir al v\u00e1ter, tirar de la cadena, que hac\u00eda tanto ruido que se asust\u00f3, pod\u00eda maldecir cuando al volver a la cama choc\u00f3 con el borde, pudo roncar y tirarse pedos sin tener que poner todo el cuidado en guardar la decencia.\n\nPero esa suerte en la desgracia hab\u00eda desaparecido al d\u00eda siguiente. A primera hora de la ma\u00f1ana fueron a buscarlo para un ECG y cuando regres\u00f3 a la habitaci\u00f3n hab\u00eda un hombre en la otra cama. Estaba recostado en la cabecera doblada hacia arriba, era muy delicado, muy p\u00e1lido, casi transparente, su escaso pelo rubio estaba peinado con una raya muy marcada. \u00a1Llevaba un pijama a rayas! Seda azul oscura, las suaves rayas en color naranja. Las piernas dobladas, en el regazo un ordenador port\u00e1til.\n\nA Brunfaut a\u00fan le machacaba la cabeza el concepto extras\u00edstoles ventriculares, envuelto como en algod\u00f3n en las palabras tranquilizadoras del cardi\u00f3logo. Y ahora estaba en su habitaci\u00f3n aquel hombre, que saludaba tan alegre, como entusiasmado por la compa\u00f1\u00eda. Brunfaut contest\u00f3 al saludo, de pie entre ambas camas, salud\u00f3 otra vez con la cabeza al hombre y vio que, cosido en la pechera del pijama, llevaba un escudo, una serpiente azul celeste. \u00bfQu\u00e9...? El hombre tendi\u00f3 la mano a Brunfaut, dijo: Soy Maurice G\u00e9ronnez.\n\nEncantado. Brunfaut dijo su nombre y apellido e hizo una inclinaci\u00f3n, en el fondo se agach\u00f3 s\u00f3lo un poco para ver mejor el escudo, la serpiente era una estilizada ese, al lado pon\u00eda SOLVAY, debajo BRUSSELS SCHOOL OF ECONOMICS. Brunfaut se qued\u00f3 estupefacto. \u00c9l ten\u00eda una bufanda y una camiseta del RSC Anderlecht, a su ahijada Jo\u00eblle le hab\u00eda comprado en broma para el bautizo, en la fan-shop, un pelele con los colores del Anderlecht, pero nunca hab\u00eda visto ni o\u00eddo que alguien llevara el pijama de fan de una universidad.\n\nMonsieur G\u00e9ronnez quer\u00eda enseguida, como es natural, intercambiar historias cl\u00ednicas, Brunfaut dijo en pocas palabras que estaba all\u00ed s\u00f3lo para un chequeo, pura medida preventiva.\n\nS\u00ed, bueno, dijo G\u00e9ronnez, ya encontrar\u00e1n algo, siempre encuentran algo, a partir de los cincuenta puedes apostar la cabeza a que encuentran algo, y si los m\u00e9dicos no encuentran nada en un hombre de m\u00e1s de cincuenta, yo me pregunto qu\u00e9 han estudiado. Entonces hay que cambiar de hospital. Pero no tenga miedo, aqu\u00ed est\u00e1 usted en buenas manos, el Europa es el mejor, aqu\u00ed siempre encuentran algo. En mi caso es el bazo. \u00bfNo es curioso? Precisamente el bazo. Usted preguntar\u00e1 ahora por qu\u00e9 es curioso. D\u00edgame qu\u00e9 hace el bazo, cu\u00e1l es su funci\u00f3n. \u00bfLo ve? No lo sabe. Nadie lo sabe, pregunte a sus amigos, a sus conocidos, pregunte a la gente por la calle. El h\u00edgado, s\u00ed. El coraz\u00f3n, pues claro. El pulm\u00f3n, los ri\u00f1ones, no hay que haber estudiado medicina para saber lo que hacen esos \u00f3rganos, cu\u00e1l es su funci\u00f3n. Pero el bazo, venga, d\u00edgalo por fin: \u00bfcu\u00e1l es la funci\u00f3n del bazo? \u00bfLo ve?, es de lo m\u00e1s raro. El bazo lleva una vida miserable. Y, sin embargo, todos los dem\u00e1s \u00f3rganos sobre los que creemos estar enterados y que consideramos tan importantes no podr\u00edan trabajar a la larga si no estuviera el bazo. El bazo controla a todos los dem\u00e1s \u00f3rganos, lo sabe todo, los examina ininterrumpidamente. Previene enfermedades de los otros \u00f3rganos, aleja de la sangre part\u00edculas nocivas, acumula gl\u00f3bulos sangu\u00edneos, que expulsa cuando es necesario, se puede decir: los env\u00eda como un destacamento de intervenci\u00f3n. El coraz\u00f3n no nota cu\u00e1ndo tiene un problema el h\u00edgado, o al rev\u00e9s, los ri\u00f1ones intentan ejercer su oficio, sin distinguir si la funci\u00f3n pulmonar est\u00e1 limitada o no, pero el bazo, \u00e9se lo nota todo en todos y reacciona a todo. Y todo lo que hace el bazo lo perciben los otros \u00f3rganos. Es el gran comunicador, al mismo tiempo el servicio secreto que nadie tiene presente. \u00bfY por qu\u00e9 no tiene nadie presente el bazo? \u00bfPor qu\u00e9 no sabe nadie lo que hace el bazo? Justo por eso: porque por lo general no llama la atenci\u00f3n. El bazo es un \u00f3rgano que muy raras veces plantea problemas. Soluciona los problemas de otros \u00f3rganos, previene en lo posible sus enfermedades, pero \u00e9l no se pone enfermo pr\u00e1cticamente jam\u00e1s. \u00bfSabe lo que yo creo? Creo que hay algo de verdad en esa teor\u00eda de la psicosom\u00e1tica. \u00c9sa es mi sospecha. Usted puede alimentarse del modo m\u00e1s saludable, pero enferma del est\u00f3mago si siempre tiene que tragar algo en sentido figurado, \u00bfcomprende lo que quiero decir?\n\nS\u00ed, eso s\u00ed.\n\n\u00bfLo ve? Y en mi caso es el bazo. No es casualidad. Yo profesionalmente soy por as\u00ed decirlo un bazo, y hace alg\u00fan tiempo he notado que ya no puedo con ello, que no pod\u00eda aceptar lo que era mi misi\u00f3n, y...\n\nUsted, profesionalmente... \u00bfqu\u00e9 es? Quiero decir, el bazo no es una profesi\u00f3n.\n\nBrunfaut suspir\u00f3.\n\nTrabajo en la Comisi\u00f3n Europea, dijo, en la ECFIN, es la Direcci\u00f3n General de Asuntos Econ\u00f3micos y Financieros. Soy responsable de la comunicaci\u00f3n. Soy por as\u00ed decirlo, entre los distintos \u00f3rganos, el comunicador que act\u00faa en la sombra. Tengo que mantener junto y coordinar lo que cada cual trabaja a solas, prepararlo todo y, muy importante, escribir los discursos con los que el comisario defiende todo eso hacia fuera. Bien, y ahora imag\u00ednese usted un organismo, el pulm\u00f3n seriamente da\u00f1ado de fumar sin parar, el h\u00edgado por excesos de alcohol, el est\u00f3mago por qu\u00edmica alimentaria, y usted debe desintoxicar todo eso... y escribir los discursos en los que la boca anuncia que todo est\u00e1 magn\u00edficamente porque se hacen enormes esfuerzos para garantizar un mejor funcionamiento del organismo, por ejemplo amputando todos los dedos de las manos para ahorrarse cortar las u\u00f1as. Yo no pod\u00eda m\u00e1s con eso, monsieur Brunfaut. Hace tres a\u00f1os empec\u00e9 a tener dificultades porque ya no pod\u00eda funcionar. En aquel entonces me pusieron sobre la mesa el estudio que llev\u00f3 a cabo la Webster y la Portsmouth University junto con la Universidad de Econom\u00eda de Viena: \u00a1espere!\n\nTecle\u00f3 en su port\u00e1til. Aqu\u00ed est\u00e1. Lo he guardado. The Impact of Fiscal Austerity on Suicide Mortality. Es terrible, un estudio de larga duraci\u00f3n sobre la relaci\u00f3n entre los programas de austeridad para Grecia, Italia, Portugal y Espa\u00f1a y la evoluci\u00f3n de las tasas de suicidio en esos pa\u00edses. No quiero aburrirle ahora con estad\u00edsticas y n\u00fameros, pero s\u00f3lo esto: con el comienzo del programa de austeridad en Grecia, en el primer a\u00f1o la tasa de suicidio creci\u00f3 un 1,4 por ciento, suena a poco, un n\u00famero peque\u00f1o, pero, oiga, son seres humanos, y ahora viene esto: en el tercer a\u00f1o la curva sube dr\u00e1sticamente y tenemos entonces un n\u00famero que nos obliga a hablar de una epidemia, y el 91,2 por ciento de los casos de suicidio concierne a personas de m\u00e1s de sesenta a\u00f1os a las que hab\u00edan reducido o incluso suprimido la pensi\u00f3n y los seguros de asistencia sanitaria, en el cuarto a\u00f1o la proporci\u00f3n de los mayores de cuarenta a\u00f1os en la estad\u00edstica de suicidios crece, en su mayor\u00eda son desempleados de larga duraci\u00f3n y solteros. En el quinto a\u00f1o la disminuci\u00f3n del n\u00famero de desempleados corresponde, con una diferencia marginal del 0,8 por ciento, al n\u00famero de suicidios del a\u00f1o. Y ahora al rev\u00e9s, espere \u2014\u200ctecle\u00f3\u2014, aqu\u00ed est\u00e1: Irlanda. El ejemplo preferido de mi comisario. \u00a1Aqu\u00ed ha empezado de nuevo el crecimiento econ\u00f3mico! \u00a1Los alumnos modelo! Pero qu\u00e9 ense\u00f1a ese estudio: la tasa de suicidio que hab\u00eda subido antes dr\u00e1sticamente no ha retrocedido. El estudio muestra que el crecimiento coyuntural no ha llegado all\u00ed donde antes qued\u00f3 destruida la red social. \u00bfComprende?\n\nLos delicados l\u00f3bulos nasales del hombre vibraban.\n\nHe de confesar que yo me indign\u00e9 al leer eso. Escrib\u00ed un borrador para el comisario, para la reuni\u00f3n de los mi\u00e9rcoles de la Comisi\u00f3n, recuerdo que la primera frase era \u00e9sta: \u00abSomos asesinos\u00bb, y expuse algunos puntos sobre lo que \u00e9l, o sea, el comisario, ten\u00eda que proponer para cumplir con su misi\u00f3n: la protecci\u00f3n de los ciudadanos europeos. Envi\u00e9 una copia al director general, \u00e9l es al fin y al cabo el encargado de la econom\u00eda de los Estados miembros, en cualquier caso, desde entonces tengo molestias. Es el bazo, que ya no consigue desintoxicar y...\n\nEn ese momento entr\u00f3 la enfermera. \u00bfMonsieur Brunfaut? Le acompa\u00f1o a la sonograf\u00eda.\n\nBrunfaut pidi\u00f3 disculpas y sigui\u00f3 a la enfermera. Un redactor de discursos que soltaba discursos sin interrupci\u00f3n: eso era demasiado para \u00e9l. Aunque tuvo que admitirlo: en el fondo ese hombre era un compa\u00f1ero de armas.\n\nEl hematoma que el profesor Erhart se hab\u00eda hecho en el brazo al caer contra el radiador del hotel hab\u00eda pasado a ser una mancha azul oscura de considerable tama\u00f1o, que parec\u00eda m\u00e1s bien un tatuaje mal hecho de la superficie geogr\u00e1fica de Europa.\n\nDespu\u00e9s de la sesi\u00f3n del reflection group, el profesor Erhart se hab\u00eda negado a ir a una cena com\u00fan y hab\u00eda regresado enseguida en metro a Sainte-Catherine. Ahora estaba sentado en la terraza de la Brasserie Van Gogh, justo al lado de la iglesia; hab\u00eda visto al pasar, cuando caminaba de la estaci\u00f3n del metro al hotel, ostras, langostas y cangrejos colocados sobre un lecho de hielo, y sin pens\u00e1rselo m\u00e1s se hab\u00eda sentado ante una mesa para darse el gustazo. Fue literalmente lo que pens\u00f3: quer\u00eda darse el gustazo. Para consolarse. Pese a todo. Despu\u00e9s de la humillante escena de antes, en la sesi\u00f3n.\n\nHab\u00eda anochecido pero segu\u00eda haciendo tanto calor que el profesor Erhart se quit\u00f3 la americana y la colg\u00f3 en el respaldo de la silla. Vio entonces su involuntario tatuaje. Se asust\u00f3. Lo palp\u00f3 suavemente con las yemas de los dedos, suspir\u00f3 en voz baja, pero no era el dolor, en cualquier caso no el dolor local, era su desesperaci\u00f3n, su escozor interior.\n\nSe hab\u00eda comportado como uno de esos estudiantes antiautoritarios con los que hab\u00eda tenido que bregar como profesor, muchos a\u00f1os atr\u00e1s. Aunque se hab\u00eda entendido con ellos mejor que la mayor\u00eda de sus compa\u00f1eros, porque era capaz de distinguir a la gente capacitada y de tomar en serio las ideas por las que sent\u00eda entusiasmo, ve\u00eda con toda claridad que un comportamiento semejante no era decoroso en \u00e9l. Era profesor y no se hab\u00eda comportado como un profesor. \u00bfPod\u00eda decirse que era sencillamente un profesor que se saltaba las convenciones? No en esa \u00e9poca en la que lo no convencional s\u00f3lo se aceptaba si al mismo tiempo se presentaba como mainstream. Su comportamiento s\u00f3lo hab\u00eda sido est\u00fapido y escandaloso. Habr\u00eda sido mejor que hubiese guardado silencio lo m\u00e1s posible en la sesi\u00f3n del think-tank, haber pedido despu\u00e9s la palabra para hacer varios breves statements y poco a poco ir avanzando diplom\u00e1ticamente. Pero hab\u00eda sido tan inconcebiblemente est\u00fapido lo que hab\u00eda tenido que o\u00edr. Bueno, y qu\u00e9. A la estupidez tambi\u00e9n se le puede dar respuesta con sosiego y objetividad. Si por ejemplo un experto sostiene la tesis de que nuestro problema \u2014\u200chablando en met\u00e1fora\u2014 se llama \u00abexceso de peso\u00bb y que el mejor m\u00e9todo para combatir ese sobrepeso ser\u00eda comer m\u00e1s a\u00fan para obligar al cuerpo a excretar m\u00e1s y esa mayor excreci\u00f3n llevar\u00eda a una reducci\u00f3n del peso, entonces no se tendr\u00eda por qu\u00e9 vociferar y llamar idiota al experto. Ser\u00eda f\u00e1cil hacerlo de otra manera. \u00bfVerdad? Pues no. Eso fue lo fantasmag\u00f3rico, que en aquella reuni\u00f3n desde el principio reinaba el consenso de que la crisis de Europa s\u00f3lo pod\u00eda resolverse con los mismos m\u00e9todos que hab\u00edan llevado a la crisis. More of the same. \u00bfQue esta o aquella estrategia no hab\u00eda funcionado? \u00a1Entonces es que no se la hab\u00eda aplicado de modo consecuente! \u00a1A continuar con m\u00e1s consecuencia! More of the same! \u00bfQue con esta o aquella resoluci\u00f3n s\u00f3lo aumentaban los problemas? \u00a1S\u00f3lo de modo pasajero! \u00a1No cejar en los esfuerzos! More of the same! Aquello le hab\u00eda vuelto loco.\n\nPidi\u00f3 una docena de ostras, y luego media langosta. Y con ello un Chablis.\n\nChablis s\u00f3lo tenemos en botella, monsieur. En copa s\u00f3lo tenemos el vino de la casa, un sauvignon.\n\nEntonces tr\u00e1igame una botella de Chablis.\n\nSe pasaba una y otra vez las yemas de los dedos por su hematoma.\n\nLas ostras. Sorbi\u00f3 una tras otra y se pregunt\u00f3 por qu\u00e9 hab\u00eda cre\u00eddo que iba a disfrutarlo. Comer ostras. El sabor de las ostras no le recordaba ning\u00fan momento feliz anterior. Por eso no pod\u00edan hacerle feliz. Lo mejor de la langosta era que no ten\u00eda mucho que comer. Y \u00e9l no ten\u00eda paciencia para manejar las tenazas. Tampoco ten\u00eda hambre. S\u00f3lo hab\u00eda querido darse el gustazo. Y se hab\u00eda bebido media botella de vino. En la plaza, un hombre con acorde\u00f3n tocaba canciones de moda alemanas de los a\u00f1os treinta. Erhart las conoc\u00eda, sus padres ten\u00edan esos discos. Ahora estaba disfrutando algo: chuparse los dedos antes de meterlos en la taza de agua caliente con rodajas de lim\u00f3n.\n\nLo mejor fue que ese economista alem\u00e1n, en medio del acalorado debate que se llev\u00f3 a cabo en ingl\u00e9s, le dijo a Erhart en alem\u00e1n: \u00ab\u00a1Mod\u00e9rese usted!\u00bb. \u00a1Moderarse! \u00c9l precisamente ten\u00eda que moderarse en aquella desmesuradamente est\u00fapida discusi\u00f3n. Un experto en finanzas griego describi\u00f3 minuciosamente c\u00f3mo hab\u00eda surgido la deuda presupuestaria griega, y, con la autoridad de un hombre que se hab\u00eda puesto a salvo en Oxford, explic\u00f3 que sin m\u00e1s y mayores recortes en el sistema social griego no se pod\u00eda continuar. Un polit\u00f3logo precisamente italiano le apoy\u00f3 al momento e insisti\u00f3 en la necesidad de mantener los criterios de estabilidad. Al hacerlo gesticulaba moviendo ambas manos con los \u00edndices extendidos y describiendo monta\u00f1as rusas como si dirigiera un coro infantil. El fil\u00f3sofo franc\u00e9s \u2014\u200ca Erhart le pareci\u00f3 al principio interesant\u00edsimo que hubieran invitado al think-tank tambi\u00e9n a un fil\u00f3sofo\u2014 exhort\u00f3 a reforzar otra vez el eje alem\u00e1n-franc\u00e9s, una demanda a la que incluso la compa\u00f1era rumana dio su aprobaci\u00f3n. Solamente hubo una peque\u00f1a divergencia de opiniones entre dos alemanes que no pod\u00edan ponerse de acuerdo en si Alemania deber\u00eda ejercer su reivindicaci\u00f3n de liderazgo en Europa \u00abcon m\u00e1s conciencia de su val\u00eda\u00bb o \u00abcon m\u00e1s humildad\u00bb. Hasta ah\u00ed se hab\u00eda llegado y Erhart se hab\u00eda preguntado qu\u00e9 hab\u00eda ocurrido con aquellas personas para que, tras a\u00f1os de carrera universitaria y de lucha para conseguir c\u00e1tedras y posiciones llenas de responsabilidad, ahora no supieran otra cosa que eso: formular como desider\u00e1tum de la pol\u00edtica futura lo que se llevaba haciendo a\u00f1os y a\u00f1os. \u00a1Para eso no necesito un think-tank, hab\u00eda interrumpido Erhart, para eso s\u00f3lo necesito un peri\u00f3dico sensacionalista!\n\nEntonces empez\u00f3 el choque hasta que un alem\u00e1n, desconocido m\u00e1s all\u00e1 de los l\u00edmites del Departamento de Ciencias Econ\u00f3micas de la Universidad de Aquisgr\u00e1n, le espet\u00f3 a Erhart en alem\u00e1n: \u00ab\u00a1Mod\u00e9rese usted!\u00bb.\n\nUn profesor brit\u00e1nico de Ciencias de la Cultura, Universidad de Cambridge, dijo que el fundamento de la Europa com\u00fan es el cristianismo y que hoy estamos viendo c\u00f3mo se nos va ese \u00fanico denominador com\u00fan, tanto en lo pol\u00edtico-social en general como tambi\u00e9n en nuestro comportamiento individual.\n\nEntonces el profesor Erhart salt\u00f3 y...\n\nNo, dijo, no quer\u00eda postre. Vaci\u00f3 la botella de vino, pag\u00f3 y se march\u00f3. Lo hab\u00eda esperado todo, pero no la caricatura de todo. Conoc\u00eda a colegas de distintos pa\u00edses \u2014\u200cestaba en contacto con ellos\u2014 con los que se pod\u00eda discutir productivamente, hab\u00eda muchas iniciativas, fundaciones, ONG en las que se pod\u00eda dar por supuesto que sab\u00edan de lo que iba en Europa. Se carteaba con ellos, segu\u00eda sus blogs. Pero era muy poco lo que llegaba hasta el gran p\u00fablico. Por eso hab\u00eda puesto grandes esperanzas en ese think-tank New Pact, que estaba directamente unido al presidente de la Comisi\u00f3n Europea. Tan cerca del poder. Pero por lo visto tan cerca del poder s\u00f3lo hab\u00eda una burbuja, tan vacua como una burbuja de jab\u00f3n, pero sin embargo indestructible. Si se la pinchaba con la punta de una aguja, no estallaba, sino que segu\u00eda elev\u00e1ndose el\u00e1sticamente. Tropez\u00f3. Casi. Logr\u00f3 seguir en pie. El pavimento de Bruselas. La gente estaba sentada en las terrazas de los caf\u00e9s y parpadeaba ante el sol poniente. Un malabarista manten\u00eda cuatro, seis, ocho, \u00a1ocho! pelotas en el aire. El acordeonista. Erhart le ech\u00f3 una moneda en el sombrero, estaba tocando Hijo, vuelve pronto a casa. Los turistas se hac\u00edan selfies delante de la iglesia. Erhart atraves\u00f3 la plaza, pero no sigui\u00f3 hasta el hotel, sino que dobl\u00f3 por la rue Sainte-Catherine. Sin rumbo, de vez en cuando miraba los escaparates, pero s\u00f3lo ve\u00eda su p\u00e1lido rostro y las grandes gafas negras y el pelo blanco que se separaba de la cabeza como electrizado. Lleg\u00f3 a la rue des Poissoniers, vio all\u00ed en la esquina un caf\u00e9, el Caf\u00e9 Kafka, le pareci\u00f3 oportuno y entr\u00f3 a tomar una copa de vino. Pero ya estaba bastante achispado. Siempre le hab\u00eda gustado beber, pero por lo general en momentos solemnes, no por frustraci\u00f3n. Antes s\u00f3lo hab\u00eda pedido la botella de Chablis porque le hab\u00edan ense\u00f1ado que con ostras se bebe Chablis. Su mujer sab\u00eda esas cosas. Trudi. Si a\u00fan viviera, la llamar\u00eda y ella dir\u00eda: Ma\u00f1ana debes hacerlo mejor. T\u00fa tienes una visi\u00f3n. No ofendas a los otros. Lim\u00edtate a intentar explicarles tu visi\u00f3n.\n\nPag\u00f3 y sigui\u00f3 adelante. Cruz\u00f3 el boulevard Anspach, vio a mano izquierda la bonita y antigua fachada de una tienda, daba la impresi\u00f3n de ser una elegante joyer\u00eda, se dirigi\u00f3 a ella; \u00bfpor qu\u00e9? No necesitaba joyas. Trudi hab\u00eda muerto. Y a ella tampoco le hab\u00edan interesado nunca las alhajas. Era la fachada. En el letrero de la tienda pon\u00eda MYSTICAL BODIES. Mir\u00f3 el escaparate. Agujas y l\u00e1pices, con piedrecillas en un extremo, dibujos: \u00bfqu\u00e9 era eso? Por fin comprendi\u00f3: all\u00ed ofrec\u00edan piercings y tatuajes.\n\nEntr\u00f3. Un joven sentado ante un escritorio grande y vac\u00edo, tal como uno se lo imaginar\u00eda en el despacho del presidente de un Estado, levant\u00f3 la vista.\n\nErhart dijo que le gustar\u00eda tener un tatuaje. La situaci\u00f3n le parec\u00eda tan irreal y al mismo tiempo tan pl\u00e1stica como un intenso sue\u00f1o. \u00c9l hab\u00eda pensado que los tatuadores iban tatuados de pies a cabeza, pero aquel chico no llevaba ning\u00fan tatuaje, en cualquier caso ninguno visible.\n\nUsted desea...\n\nS\u00ed, dijo Erhart, y quit\u00e1ndose la americana present\u00f3 el brazo al chico: quiero 12 estrellas de cinco puntas aqu\u00ed encima, en torno a este... a este cardenal.\n\nEso es un hematoma.\n\nS\u00ed.\n\n\u00bfY yo tengo que tatuarle estrellas encima?\n\nS\u00ed, por favor.\n\nPero \u00bfpor qu\u00e9?\n\n\u00bfNo parece Europa?\n\n\u00bfC\u00f3mo dice?\n\n\u00a1Mire! Esto es la pen\u00ednsula ib\u00e9rica, y este peque\u00f1o saliente es, de modo inconfundible, la bota, \u00bfno?\n\n\u00bfItalia?\n\nS\u00ed. Y aqu\u00ed, donde salen flecos, est\u00e1 Grecia. \u00a1Si se ve muy bien!\n\nOkay, con mucha imaginaci\u00f3n. Pero las proporciones son err\u00f3neas, es, no, no es Europa, es una caricatura. Pero en todo caso se curar\u00e1: es lo que espero al menos para usted.\n\nYo veo a Europa en este cardenal. Y ahora quiero las estrellas correspondientes. \u00bfQu\u00e9 costar\u00eda?\n\nNo. Yo no hago eso. Hay vasos sangu\u00edneos da\u00f1ados, hay capilares reventados, ah\u00ed no pincho yo, eso no puedo controlarlo. Yo no lo tocar\u00eda. Y dentro de unas semanas habr\u00e1 desaparecido de todos modos. Entonces tendr\u00eda usted estrellas, pero la raz\u00f3n de ellas habr\u00e1 desaparecido, \u00bfpor qu\u00e9...?\n\n\u00bfAs\u00ed que nada de estrellas para una Europa en desaparici\u00f3n?\n\nSorry, caballero, yo eso no lo hago.\n\nEn el Arca nadie se habr\u00eda imaginado qu\u00e9 violenta tempestad desencadenar\u00eda en la Comisi\u00f3n el Jubilee Project. Sin embargo, la tormenta se hab\u00eda anunciado del mismo modo que se anuncian las grandes tempestades: con una calma casi fantasmag\u00f3rica.\n\nPrimero la Eurostat hab\u00eda enviado obedientemente datos. La respuesta fue extensa, cuajada de n\u00fameros, pero no \u00fatil.\n\n\u00a1Puro material estadista!, hab\u00eda dicho Bohumil en alem\u00e1n a Martin, encogi\u00e9ndose de hombros.\n\nQuieres decir, estad\u00edstico.\n\nS\u00ed.\n\nA partir de las tablas num\u00e9ricas, las f\u00f3rmulas y los gr\u00e1ficos, la Eurostat informaba de algo que dej\u00f3 a Martin tan estupefacto que ley\u00f3 tres veces el documento y luego lo contempl\u00f3 con incredulidad durante una hora. Lo que escrib\u00eda el estad\u00edstico de la Eurostat ven\u00eda a significar que el individuo, en todos los c\u00e1lculos basados en estad\u00edsticas, era un factor de perturbaci\u00f3n, pens\u00f3 Martin. Tambi\u00e9n podr\u00eda leerse la informaci\u00f3n as\u00ed: con su voluntad inescrutable, Dios convierte todos los datos estad\u00edsticos disponibles sobre los seres humanos en papel mojado.\n\nSe sabe, por ejemplo, cu\u00e1ntos nonagenarios, hombres y mujeres, viven hoy en Europa. Y se sabe que la brecha entre hombres y mujeres en la esperanza de vida se va cerrando cuanto m\u00e1s avanza la edad. Las mujeres de noventa a\u00f1os tienen a\u00fan, en promedio, una esperanza de vida de cuatro a\u00f1os; los hombres, de tres a\u00f1os y nueve meses. El n\u00famero de supervivientes de la Sho\u00e1 en 1945 s\u00f3lo pod\u00eda calcularse. Sobre la proporci\u00f3n de hombres y mujeres no hay datos. Pero si se parte de que la diferencia de esperanza de vida de hombres y mujeres se va igualando seg\u00fan aumenta la edad, y se extrapola a la esperanza de vida de los supervivientes de la Sho\u00e1 sin distinci\u00f3n de sexos para averiguar cu\u00e1ntos probablemente siguen vivos hoy, tal intento fracasa porque la esperanza de vida es muy distinta seg\u00fan los pa\u00edses y no se sabe c\u00f3mo est\u00e1n repartidos los supervivientes por pa\u00edses. Es diferente si el superviviente de la Sho\u00e1 est\u00e1 en Alemania, en Polonia, en Rusia, en Israel o en Estados Unidos. Y luego habr\u00eda tambi\u00e9n que tener en cuenta si alguien tiene dinero o vive por debajo del umbral de la pobreza. Hay un c\u00e1lculo de un dem\u00f3grafo israel\u00ed del a\u00f1o 2005, v\u00e9ase nota, seg\u00fan el cual el 40 por ciento de los supervivientes de la Sho\u00e1 viven en o bajo el umbral de la pobreza. Esas personas tienen indudablemente todas las de perder y de esto se deduce que de \u00e9stos ya no queda nadie con vida, lo cual por otra parte no puede comprobarse, ya que otra estad\u00edstica apunta en sentido contrario: las personas que en su juventud han vivido largos periodos de hambre tienen mayor esperanza de vida y en la vejez pueden adaptarse f\u00edsicamente a la penuria mejor que quienes nunca se han visto forzados a adaptarse fisiol\u00f3gicamente. Se sabe adem\u00e1s que no s\u00f3lo supervivientes de la Sho\u00e1, sino tambi\u00e9n gran parte de la poblaci\u00f3n civil de los territorios afectados por la guerra o de los que estuvieron ocupados han sufrido hambre epid\u00e9mica, por lo que no hay f\u00f3rmula con la que poder calcular exclusivamente la esperanza de vida y el n\u00famero probable de los actuales supervivientes de la Sho\u00e1.\n\nY despu\u00e9s, el estad\u00edstico de la Eurostat volv\u00eda a la esperanza de vida de los nonagenarios actuales mencionada al principio. Escrib\u00eda: \u00abSi partimos de que los m\u00e1s j\u00f3venes de los supervivientes actuales de la Sho\u00e1 nacieron en el a\u00f1o 1929 \u2014\u200cporque para entrar en un campo de concentraci\u00f3n hab\u00eda que tener al menos diecis\u00e9is a\u00f1os, todos los de menos edad eran gaseados inmediatamente\u2014, entonces s\u00f3lo sabemos, por las estad\u00edsticas de la esperanza de vida, que tiene que haber cierto n\u00famero de supervivientes. Pero incluso si supi\u00e9ramos su n\u00famero exacto no podr\u00edamos decir si la estad\u00edstica es aplicable a ellos, es decir, si cumplen con el promedio estad\u00edstico. Todos han de tener m\u00e1s de noventa a\u00f1os, por tanto todos tienen te\u00f3ricamente un promedio de esperanza de vida de tres a\u00f1os y nueve meses a cuatro a\u00f1os. Pero es posible que dentro de un a\u00f1o el cien por cien de ese n\u00famero desconocido para nosotros haya muerto o que el cien por cien siga con vida. Ambos casos est\u00e1n dentro de la l\u00ednea de fluctuaci\u00f3n\u00bb. Y luego ven\u00eda la frase que le bailaba a Martin ante los ojos como si estuviera escrita con may\u00fasculas: ESO YA NO ES ESTAD\u00cdSTICA; ESO ES DESTINO.\n\nMartin transmiti\u00f3 esta informaci\u00f3n, con un comentario, a Xeno. Propuso dejar de momento en suspenso la cuesti\u00f3n de si se deb\u00eda poner en el centro de la conmemoraci\u00f3n el mayor n\u00famero posible de supervivientes de la Sho\u00e1 (en la medida en que era posible abarcarlos) o un peque\u00f1o grupo representativo (representantes de distintos pa\u00edses) o s\u00f3lo un representante ejemplar. Lo decisivo era por de pronto obtener aprobaci\u00f3n general a esta idea: la celebraci\u00f3n deb\u00eda percibirse como posibilidad de mostrar al gran p\u00fablico europeo que la Comisi\u00f3n no es s\u00f3lo \u00abla guardiana de los Tratados de la Uni\u00f3n\u00bb (como pone en su p\u00e1gina web), sino sobre todo tambi\u00e9n la guardiana del juramento, m\u00e1s amplio y completo, de que una ruptura de la civilizaci\u00f3n europea como la de Auschwitz nunca se repetir\u00e1. Esa \u00abcl\u00e1usula de la perpetuidad\u00bb, escrib\u00eda Martin, hab\u00eda que presentarla como el verdadero n\u00facleo de la Comisi\u00f3n, pues la Comisi\u00f3n, de \u00abburocracia\u00bb abstracta, pasaba a ser una \u00abautoridad moral\u00bb, y con la presentaci\u00f3n de los \u00faltimos testimonials de la Sho\u00e1 pod\u00eda establecerse la necesaria relaci\u00f3n emocional de la opini\u00f3n p\u00fablica con el trabajo de la Comisi\u00f3n. La mala imagen de la Comisi\u00f3n se deb\u00eda en \u00faltimo t\u00e9rmino a que se la ve\u00eda como aparato de una comunidad exclusivamente econ\u00f3mica que defend\u00eda una pol\u00edtica econ\u00f3mica rechazada cada vez por m\u00e1s personas. Por eso hab\u00eda que recordar de modo consecuente la idea fundacional, con las palabras de Jean Monnet: \u00abTodos nuestros esfuerzos son la doctrina de nuestra experiencia hist\u00f3rica: el nacionalismo lleva al racismo y a la guerra, y como consecuencia radical, a Auschwitz\u00bb.\n\nPor esta raz\u00f3n, el primer presidente de la Comisi\u00f3n, el alem\u00e1n Walter Hallstein, pronunci\u00f3 su discurso inaugural en Auschwitz. M\u00e1s tarde esa idea fue recogida por los presidentes de la Comisi\u00f3n Jacques Delors y Romano Prodi. Y tambi\u00e9n el nuevo presidente declar\u00f3 el 27 de enero en la fiesta de la liberaci\u00f3n de Auschwitz que \u00abel entreverado econ\u00f3mico de las naciones\u00bb no era \u00abuna finalidad en s\u00ed misma s\u00f3lo para generar crecimiento econ\u00f3mico\u00bb, sino \u00abrequisito indispensable para el sentido m\u00e1s profundo del proyecto europeo: impedir en el futuro la obstinaci\u00f3n nacionalista y as\u00ed, finalmente, el nacionalismo que lleva a resentimientos y agresiones contra otros, a la divisi\u00f3n de Europa, al racismo y, en \u00faltimo t\u00e9rmino, a Auschwitz\u00bb.\n\nFinalmente Martin escrib\u00eda en su e-mail a Xeno que \u00e9l recomendaba encarecidamente financiar el proyecto no con el presupuesto de la UE, sino s\u00f3lo con el de la Comisi\u00f3n. De ese modo no habr\u00eda necesidad de consultarlo con el Consejo y con el Parlamento (con las previsiblemente largas negociaciones y al final improductivos compromisos), y la mejora de imagen ser\u00eda finalmente en su totalidad para la Comisi\u00f3n.\n\nXeno inform\u00f3 a Mrs. Atkinson y le pidi\u00f3 aprobaci\u00f3n para financiar el proyecto exclusivamente con recursos del presupuesto de la Comisi\u00f3n. Mrs. Atkinson, sin embargo, ten\u00eda entretanto otros problemas. Unos d\u00edas atr\u00e1s, en las redes sociales hab\u00eda surgido el rumor de que la Comisi\u00f3n, sobornada por lobbistas de las principales compa\u00f1\u00edas farmac\u00e9uticas, planeaba prohibir la homeopat\u00eda. En el espacio de una sola jornada hab\u00edan llegado un mill\u00f3n y medio de e-mails de protesta de toda Europa y casi hab\u00edan colapsado el servidor de la Comisi\u00f3n. El Bild-Zeitung alem\u00e1n trajo como titular la noticia falsa en caracteres gigantescos, si bien entre signos de interrogaci\u00f3n: \u00bfHAN PERDIDO EL JUICIO LOS FUNCIONARIOS DE BRUSELAS? Tambi\u00e9n informaban el Sun, el Kronenzeitung, el Blesk, el Hola y hasta El Pa\u00eds, el France Soir y, aunque no en portada, el Lib\u00e9ration. Y todas esas vociferantes noticias desembocaron en una convocatoria de protesta ante la Comisi\u00f3n contra las multinacionales y sus lobbistas. Mrs. Atkinson se retorc\u00eda las manos sentada ante su escritorio. Sus largos y delicados dedos estaban fr\u00edos y azulados. Los friccionaba, los masajeaba, los apretaba mientras reflexionaba de qu\u00e9 manera pod\u00eda contrarrestar ese disparate de manera efectiva. Una nota de prensa con un claro desmentido s\u00f3lo hab\u00eda hecho efecto en el Neue Z\u00fcrcher Zeitung, lo que no obstante s\u00f3lo hab\u00eda llevado a una nueva shitstorm en las redes sociales: todo el mundo conoc\u00eda a las multinacionales que tienen su sede en Suiza. Atkinson se preguntaba por qu\u00e9 unos medios a los que dif\u00edcilmente se pod\u00eda calificar como peri\u00f3dicos de lucha anticapitalista llamaban a un combate contra empresas multinacionales y con ello daban de palos sobre todo a la Comisi\u00f3n Europea, que a su vez, sin embargo, luchaba contra el poder incontrolado de las multinacionales. \u00bfNo hab\u00eda logrado recientemente la Comisi\u00f3n que se impusieran multas de miles de millones a Microsoft y Amazon?\n\nMrs. Atkinson era economista de profesi\u00f3n y no experta en comunicaci\u00f3n, aunque eso fuera ahora su campo de trabajo. Hab\u00eda entrado en funciones para mejorar la imagen de la Comisi\u00f3n, hab\u00eda planeado una ofensiva y desde entonces s\u00f3lo estaba a la defensiva. El presidente de la Comisi\u00f3n la hab\u00eda llamado a su despacho por esa historia de la homeopat\u00eda: que si ten\u00eda alg\u00fan plan de c\u00f3mo se pod\u00eda poner freno a ese menoscabo del buen nombre de la Comisi\u00f3n y comunicar mejor sus m\u00e9ritos y su rendimiento.\n\nS\u00ed. Naturalmente.\n\n\u00bfY cu\u00e1ndo se podr\u00e1 ver el resultado de ese plan?\n\nAhora mismo no sabr\u00eda decirle.\n\n\u00c9l s\u00f3lo llamar\u00eda \u2014\u200cexpresado con prudencia\u2014 plan a un plan cuando un resultado deseable fuera realista y tambi\u00e9n de pronta comprobaci\u00f3n.\n\nYes, sir.\n\nSe friccionaba las manos. La idea de Fenia Xenopoulou no la ayudaba ahora. Pero le agradec\u00eda su entrada en acci\u00f3n. A medio o largo plazo pod\u00eda ser una ayuda. Contest\u00f3 as\u00ed: \u00abConcedo financiaci\u00f3n del presupuesto de la Comisi\u00f3n, pero pido plan exacto de los gastos, listado de los recursos necesarios, tambi\u00e9n en lo personal. Go ahead!\u00bb.\n\nXeno dio entonces su okay a Martin. Por favor, ma\u00f1ana, a m\u00e1s tardar, una \u00abnota\u00bb para la inter-service consultation. Con: importe aproximado de los recursos econ\u00f3micos necesarios, plan cronol\u00f3gico, recursos necesarios, tambi\u00e9n en lo personal, contribuciones que se desean de las otras DG.\n\nDespu\u00e9s busc\u00f3 diligentemente entre todos los papeles de su mesa, en la mesa auxiliar y en la estanter\u00eda, la novela que hab\u00eda abandonado hac\u00eda m\u00e1s de tres semanas, la novela favorita del presidente. De la oficina de \u00e9ste le hab\u00edan dado por fin una fecha para la cita. Era una feliz coincidencia. Ahora ten\u00eda algo que pod\u00eda presentar: el proyecto con el que colocar\u00eda a la Cultura, el \u00faltimo mono de la Comisi\u00f3n, en el centro de la atenci\u00f3n general. Quien consiguiera eso, y eso tendr\u00eda que comprenderlo el presidente, deb\u00eda recibir un cargo m\u00e1s importante en la Comisi\u00f3n. Lo mejor ser\u00eda en la DG TRADE, donde podr\u00eda colaborar con Fridsch. Por otra parte: \u00bfera bueno colaborar tan \u00edntimamente con el hombre que ella...?, \u00bfqu\u00e9? Le horrorizaba pensar siquiera en la palabra amor. Y ten\u00eda la sensaci\u00f3n de que tambi\u00e9n \u00e9l deb\u00eda aprender a superar cierta distancia profesional. Durante la cena poco tiempo atr\u00e1s en el restaurante italiano, hab\u00eda sido educado y amable como se suele ser con buenos amigos o con estimados colaboradores, pero despu\u00e9s, cuando se fueron juntos a la cama, al final \u00e9l llor\u00f3. Es s\u00f3lo sudor, dijo cuando ella le enjug\u00f3 las l\u00e1grimas del rostro, pero Fenia estaba segura, eran l\u00e1grimas de felicidad y de emoci\u00f3n.\n\nEncontr\u00f3 la novela. Sab\u00eda de lo que quer\u00eda hablar con el presidente, pero pensaba que sin duda no le vendr\u00eda mal leer su libro preferido, de ese modo tendr\u00eda m\u00e1s sinton\u00eda con \u00e9l.\n\nHoje\u00f3 aqu\u00ed y all\u00e1, finalmente empez\u00f3 a leer en un pasaje cualquiera y se detuvo espantada cuando ley\u00f3 esta frase: \u00abEn una ocasi\u00f3n mand\u00f3 venir a una esteticista para probar distintos maquillajes para cuando yaciera en el ata\u00fad, llorada por su afligido amante\u00bb. Lo que la espant\u00f3 fue que al instante se vio ella en esa situaci\u00f3n: yaciendo en el ata\u00fad, perfectamente maquillada, con una sonrisa que s\u00f3lo puede hacer surgir m\u00e1gicamente en el rostro, al entrar en la eternidad, el pensamiento en el amante. Y Fridsch...\n\n# CAP\u00cdTULO OCTAVO\n\n\u00abGET INTO TROUBLE, GOOD TROUBLE.\u00bb\n\n#\n\nEn torno a la Piet\u00e0 giraba la luz azul, sobre la Piet\u00e0 daba vueltas el helic\u00f3ptero de salvamento. Cada vez m\u00e1s gente, hombres y mujeres, ancianos, j\u00f3venes y ni\u00f1os llegaban a esa escena, muchos se deten\u00edan asustados, miraban, pero la mayor\u00eda se acercaba deprisa a los polic\u00edas que con los brazos extendidos formaban una fila transversal en la autopista. \u00a1Stop! \u00a1Det\u00e9nganse! Los polic\u00edas intentaban alejar a la gente, bloquear la autopista para que el helic\u00f3ptero pudiera aterrizar, pero la muchedumbre iba en aumento y avanzaba hacia ellos y junto a ellos y a los veh\u00edculos policiales estacionados. Esas personas no comprend\u00edan la situaci\u00f3n, no ve\u00edan a los heridos, no atribu\u00edan importancia alguna a los veh\u00edculos destrozados, s\u00f3lo pensaban que all\u00ed iban a detenerlas y a echarlas otra vez para atr\u00e1s, tal vez tomaban el helic\u00f3ptero de salvamento por un helic\u00f3ptero militar o de la polic\u00eda, un torpe gesto amenazador de la polic\u00eda de fronteras austriaca, eso no pod\u00eda detenerlas, hab\u00edan pasado ya la frontera h\u00fangaro-austriaca, ya hab\u00edan llegado muy lejos y quer\u00edan continuar hacia Alemania, no dejaban que las detuvieran.\n\nHab\u00eda tambi\u00e9n all\u00ed periodistas que filmaban y fotografiaban y que estorbaban. Y la imagen de la Piet\u00e0 en medio de ese caos dar\u00eda la vuelta al mundo: la mujer vestida de negro, con pa\u00f1uelo en la cabeza y sentada en una maleta; atravesado sobre su regazo yac\u00eda un hombre con traje de ejecutivo. En el rostro de la mujer, la lluvia como l\u00e1grimas. Con su mano derecha sosten\u00eda la cabeza del hombre, la izquierda la hab\u00eda alzado a lo alto, la cabeza estaba echada hacia atr\u00e1s, miraba hacia arriba, en la foto daba la impresi\u00f3n de que la mujer del pa\u00f1uelo en la cabeza incriminaba desesperada al cielo. Levantaba la vista hacia el helic\u00f3ptero.\n\nEsa mujer se hab\u00eda dado cuenta antes que nadie de que hab\u00eda de alg\u00fan modo que atender al hombre.\n\nMientras tiraba de su maleta hab\u00eda o\u00eddo el estruendo, algo que parec\u00eda una explosi\u00f3n, hab\u00eda visto, sin comprender nada a\u00fan, que la gente que iba delante se dispersaba corriendo, se apartaba, gritaba y, de pronto, ella estaba delante del coche destrozado del que sobresal\u00eda un hombre que gem\u00eda.\n\nEra Florian Susman.\n\nLa gente que caminaba por la autopista hab\u00eda marchado hacia \u00e9l, de frente, coches de la polic\u00eda con luz azul y sirenas hab\u00edan pasado junto a \u00e9l y se hab\u00edan detenido un poco m\u00e1s adelante. \u00c9l ya iba lento, finalmente hab\u00eda frenado del todo y detenido el coche. Hab\u00eda puesto la luz de emergencia. Hab\u00eda visto a un polic\u00eda que, agitando una larga linterna tubular, se acercaba a \u00e9l. El polic\u00eda estaba a\u00fan quiz\u00e1 a unos veinte metros de distancia cuando de pronto grit\u00f3, grit\u00f3 de una manera que, durante una fracci\u00f3n de segundo que al mismo tiempo fue un momento eterno, Florian s\u00f3lo vio ese gritar, vio por el parabrisas mojado de lluvia la boca abierta del polic\u00eda como enfocada con un zoom y a la vez grotescamente desfigurada. Entonces el polic\u00eda dio un salto hacia un lado.\n\nEl estruendo, el enorme choque, el ruido agresivo de la carrocer\u00eda destrozada, la explosi\u00f3n de los neum\u00e1ticos: nada de eso recordaba Florian despu\u00e9s, s\u00f3lo el instante brev\u00edsimo en el que con un asombro mayor que el shock y el dolor se sinti\u00f3 prisionero en una angosta c\u00e1psula lanzada de un lado a otro con una inconcebible fuerza. Inmovilizado, vio im\u00e1genes borrosas por las que \u00e9l pasaba patinando, una confusa pel\u00edcula, curiosamente sin sonido.\n\nS\u00f3lo recobr\u00f3 por poco tiempo la conciencia en la cl\u00ednica de traumatolog\u00eda, cuando con una gran tijera le separaron la ropa del cuerpo. Abri\u00f3 los ojos, la tijera pasaba sobre su tronco, divid\u00eda su camisa polo, estaban como abri\u00e9ndole, vio un rostro, oy\u00f3: \u00bfMe comprende? \u00bfMe entiende usted?\n\n\u00c9l dijo algo sobre cerdos, los cerdos, no se entend\u00eda, luego perdi\u00f3 de nuevo la conciencia.\n\nUn taxista de Burgenland, que ese d\u00eda ya hab\u00eda hecho varios viajes al paso de frontera Nickelsdorf para llevar refugiados a Viena, a la Estaci\u00f3n del Oeste, desde donde pod\u00edan continuar en tren a M\u00fanich, regresaba a toda velocidad para el transporte siguiente, era un buen negocio, y r\u00e1pido, cada uno de esos pobres diablos pagaba sin rechistar tres veces el precio de la carrera. Con su precipitaci\u00f3n, su avaricia, su traj\u00edn, no vio que delante de \u00e9l el tr\u00e1fico se hab\u00eda detenido. Y as\u00ed, sin frenar, hab\u00eda ido a incrustarse a todo gas en el coche de Florian Susman.\n\nLa mujer que con ayuda de su hijo hab\u00eda sacado a Florian cuidadosamente del coche destrozado, que luego lo hab\u00eda tendido sobre su regazo y mantenido en alto la cabeza, fue su salvaci\u00f3n. Florian ten\u00eda rota una v\u00e9rtebra de la espalda, pero la cuidadosa salida del coche y la posterior estabilizaci\u00f3n hab\u00edan impedido que la m\u00e9dula espinal resultara da\u00f1ada, de lo contrario habr\u00eda quedado paral\u00edtico. Florian lo entendi\u00f3 cuando Martin le trajo a la cl\u00ednica los peri\u00f3dicos con la foto de la Piet\u00e0. \u00ab\u00a1Sales en portada!\u00bb\n\nA causa de esa foto, el Occidente cristiano que temblaba ante la invasi\u00f3n musulmana incurri\u00f3 en el sentimentalismo durante un hist\u00f3rico segundo. La musulmana que salv\u00f3 a Florian era una Virgen Madre.\n\n\u00bfQu\u00e9 habr\u00eda ocurrido si Florian no hubiera tenido ese accidente? Tal vez Martin Susman habr\u00eda podido impedir o al menos restringir las turbulencias que provoc\u00f3 su borrador para el aniversario, si se hubiera quedado en Bruselas en lugar de tomar de inmediato un avi\u00f3n a Viena para estar al lado de su hermano. Pero eso fue lo que ocurri\u00f3, mientras Martin se ocupaba en Viena de su hermano, en la Comisi\u00f3n Europea de Bruselas surgieron conflictos y discusiones que muy r\u00e1pidamente alcanzaron tal intensidad emocional que ya no pudo haber ninguna soluci\u00f3n racional y ni siquiera un compromiso. \u00bfY qui\u00e9n era el culpable de esa conmoci\u00f3n, qui\u00e9n hab\u00eda tenido aquella demencial idea? \u00bfMrs. Atkinson? \u00bfXeno? Martin.\n\nPor otra parte: \u00bfpuede haber culpables cuando cada cual cumple con su deber? \u00bfQu\u00e9 es deber? \u00bfEl cumplimiento de las reglas burocr\u00e1ticas, de los procedimientos fijados? \u00bfO la defensa de intereses con los que uno est\u00e1 comprometido o se siente comprometido? Todo queda triturado entre las grandes ruedas de arriba y las peque\u00f1as ruedecillas de abajo y al final no ha ocurrido nada, aunque el crujir y rechinar de la molienda al principio haya comportado nerviosismo y excitaci\u00f3n. Sin embargo, Xeno y \u00e9l, antes de que \u00e9l saliera para Viena, a\u00fan estaban completamente seguros de que el proyecto de la conmemoraci\u00f3n empezar\u00eda su andadura sin contratiempos. Hab\u00edan tomado la calma que precede a la tormenta por ausencia de objeciones, por aprobaci\u00f3n t\u00e1cita. Y se sent\u00edan confirmados y protegidos por el aliento y el apoyo de \u00abarriba del todo\u00bb.\n\nXeno hab\u00eda obtenido por fin la cita con el presidente de la Comisi\u00f3n, dos d\u00edas antes del inter-service-meeting que hab\u00eda convocado por el Jubilee Project. Es decir, en realidad no con el presidente en persona, sino con su jefe de gabinete. Pero ya eso era una distinci\u00f3n, un reconocimiento de su trabajo y un claro inter\u00e9s por su persona, porque en general los funcionarios en la posici\u00f3n de Xeno recib\u00edan una cita como mucho con un empleado del gabinete. \u00bfEra ese privilegio quiz\u00e1 la consecuencia de la intervenci\u00f3n de Fridsch, que la hab\u00eda recomendado con insistencia para m\u00e1s altas tareas? Por otra parte: \u00bfno hab\u00eda esperado ella m\u00e1s, a saber, una cita con el presidente en persona? \u00bfNo se hab\u00eda preparado minuciosamente para \u00e9l? \u00bfNo hab\u00eda estudiado por esa raz\u00f3n su biograf\u00eda, sus aficiones, sus man\u00edas, y hasta le\u00eddo su libro preferido? Pero ya empez\u00f3 a darse cuenta cuando le prometieron la cita (\u00ab\u00bfDe qu\u00e9 se trata?\u00bb, \u00abLo intentaremos\u00bb), y luego despu\u00e9s cuando le daban largas, hasta que Fridsch le dijo finalmente: una cita con el presidente es s\u00f3lo una cita con un empleado de su gabinete. Y m\u00e1s a\u00fan si se pertenece a la Cultura.\n\nSonre\u00eda.\n\nImag\u00ednate, dijo, que el presidente no existe en la realidad. Despu\u00e9s de Jacques Delors no ha habido presidente. Despu\u00e9s ya s\u00f3lo ha habido marionetas. El gabinete mueve los hilos. Cada palabra que dice el presidente la dicen sus ventr\u00edlocuos. Todo lo que decide est\u00e1 decidido hace tiempo, y si firma algo, le conducen la mano. \u00bfHas visto en la televisi\u00f3n c\u00f3mo el presidente, en una reuni\u00f3n con jefes de Estado, de pronto le tira de la corbata a uno y al otro le da un peque\u00f1o empuj\u00f3n? Eso es lo \u00fanico no preparado y aut\u00f3nomo que se puede permitir, por as\u00ed decirlo, su nota personal en esa mec\u00e1nica del poder, eso es su ir\u00f3nico juego: \u00e9l, que cuelga de tantos hilos, se burla de ello en pantomima, tirando y empujando, como si fuera \u00e9l quien maneja los hilos. As\u00ed que, hab\u00eda dicho Fridsch, obtendr\u00e1s tu cita con el presidente, pero no esperes una cita con la marioneta.\n\nY as\u00ed Xeno estaba sentada enfrente del jefe de gabinete del presidente de la Comisi\u00f3n, Romolo Strozzi, cuyo nombre completo, seg\u00fan hab\u00eda le\u00eddo ella en Wikipedia, era Romolo Augusto Massimo Strozzi, \u00faltimo reto\u00f1o sin sucesi\u00f3n de una antigua familia noble italiana. En las instituciones europeas corr\u00edan algunas an\u00e9cdotas sobre \u00e9l y sobre su poco convencional estilo, se dec\u00eda que era un personaje fuera de lo com\u00fan, y Xeno comprob\u00f3 asombrada que seguramente hab\u00eda que entender eso literalmente, por su apariencia exterior: Strozzi llevaba un traje azul, un pa\u00f1uelo amarillo, un chaleco rojo que realzaba su barriga y al mismo tiempo la manten\u00eda bien apretada. No estaba gordo, s\u00f3lo lo bastante relleno como para demostrar, muy especialmente tambi\u00e9n con el rojo llamativo de su chaleco, que desde luego no era un asceta. Eso era poco habitual a esa escala del poder, dominada por los \u00abenarcas\u00bb, procedentes de las canteras de ejecutivos como la \u00c9cole Nationale d'Administration, hombres muy esbeltos con trajes ni llamativos ni muy caros, asc\u00e9ticos en todos los aspectos: capaces de negociar durante horas y durante noches enteras. No parec\u00edan necesitar apenas comida ni pr\u00e1cticamente dormir, les bastaban pocas palabras y pocos gestos, evitaban el excesivo edulcoramiento de su alma por el dulzor de la empat\u00eda, no necesitaban p\u00fablico, les bastaba el metabolismo en el interior del poder, renunciaban al brillo exterior. En su vida y en su trabajo no hab\u00eda ornamentaci\u00f3n, todo era tan claro como invisible. Xeno apreciaba mucho profesionalmente ese tipo de hombre, as\u00ed lo hab\u00eda aprendido ella y para ellos se hab\u00eda preparado en sus escuelas elitistas, con ellos hab\u00eda tenido que ver en su carrera hasta ese momento: y hela aqu\u00ed ahora sentada frente a aquel conde barroco italiano que le presentaba su barriga colorada y que hablaba gesticulando como un director de opereta, de manera que al hacerlo a ella le bailaba ante los ojos el anillo de sello. No era rid\u00edculo, impon\u00eda sin duda alguna respeto y reverencia, en un hombre de su posici\u00f3n no era concebible otra cosa. Era s\u00f3lo que Xeno estaba desconcertada con aquel estilo y no sab\u00eda c\u00f3mo reaccionar. No s\u00f3lo hablaba italiano, alem\u00e1n, ingl\u00e9s y franc\u00e9s, sino que inici\u00f3 la entrevista, lami\u00e9ndose los labios de contento, en griego cl\u00e1sico. Cuando Xeno le mir\u00f3 totalmente perpleja, se disculp\u00f3: lamentablemente, su griego moderno era tan rudimentario que a ella le producir\u00eda dolores. Y adem\u00e1s olvidaba siempre que, para los griegos, el griego cl\u00e1sico era una lengua tan ajena como el kisuaheli.\n\n\u1fbf\u0395\u03bd \u1f00\u03f1\u03c7\u1fc7 \u1f26\u03bd \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2, dijo, y a\u00f1adi\u00f3: \u1fbf\u0391\u03bb\u03bb\u1fbf \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u1f26\u03bd \u1f01\u03bc\u03b1\u03c1\u03c4\u03bf\u03b5\u03c0\u03ae\u03c2. Al principio era la palabra. Pero la palabra era equivocada. Je suis d\u00e9sol\u00e9, dijo riendo.\n\nXeno estaba intimidada por su ataque de hilaridad. Justo antes de esa cita se hab\u00eda informado sobre el conte Strozzi, para calibrarlo mejor, para que no la pillara de sorpresa y, en las negociaciones con \u00e9l, poder reaccionar de modo adecuado y lo m\u00e1s r\u00e1pidamente posible. Pero s\u00f3lo ahora comprend\u00eda, demasiado tarde, lo que significaba en realidad todo lo que hab\u00eda o\u00eddo y le\u00eddo sobre \u00e9l: los Strozzi hab\u00edan recibido el t\u00edtulo de nobleza de manos del emperador del Sacro Imperio Romano Germ\u00e1nico Federico II, estaban emparentados por sangre o por matrimonio con la alta nobleza austriaca, alemana y checa. El abuelo de Romolo Strozzi fue criminal de guerra como comandante de una unidad del noveno ej\u00e9rcito italiano, que en 1941 y 1942 llev\u00f3 a cabo fusilamientos en masa en Montenegro, pero su padre, licenciado por la academia diplom\u00e1tica, fue en 1964 el miembro m\u00e1s joven del equipo negociador que prepar\u00f3 para el gobierno italiano el tratado de fusi\u00f3n de la Comunidad Europea, lo que llev\u00f3 a la fundaci\u00f3n del Consejo comunitario y de la Comisi\u00f3n. Su t\u00edo abuelo austriaco, el conde Nikolaus Khevenh\u00fcller, fue un nacionalsocialista fan\u00e1tico, que todav\u00eda en enero de 1945 recibi\u00f3 el nombramiento de Gauleiter de Carintia, pero ya a principios de mayo pas\u00f3 a la clandestinidad en Espa\u00f1a, donde hasta su muerte en 1967 vivi\u00f3 tranquilamente como \u00abconsejero\u00bb de la Direcci\u00f3n General de Seguridad, DGS, la polic\u00eda secreta espa\u00f1ola, con una pensi\u00f3n honor\u00edfica del general\u00edsimo Franco. Por su parte, su t\u00eda abuela Marion, nacida Von Tirpitz, se cas\u00f3 con el combatiente de la resistencia alemana Ulrich Hesse, fue pol\u00edtica comunal socialdem\u00f3crata de Hannover y secretaria de la Asociaci\u00f3n de V\u00edctimas del Nacionalsocialismo.\n\nEsa historia familiar era seguramente el fundamento de la cita m\u00e1s c\u00e9lebre atribuida a Romolo Strozzi: \u00abL'Europe, c'est moi\u00bb.\n\nUna historia familiar as\u00ed era, naturalmente, fascinante, pero para Xeno tambi\u00e9n en \u00faltimo t\u00e9rmino enigm\u00e1tica: era completamente ajeno a su naturaleza que todo aquello pudiera seguir influyendo y acu\u00f1ara la biograf\u00eda de una persona. Ella ten\u00eda una imagen de la familia en la que los antepasados eran algo sobre lo que s\u00f3lo se ten\u00eda conocimiento desde que exist\u00eda la fotograf\u00eda, y entonces tampoco se sab\u00eda mucho m\u00e1s que los nombres, en el fondo eran personas que probablemente no hab\u00edan vivido de modo muy distinto al de los padres, personas que se apoyaban y ayudaban mutuamente, cautivos de su situaci\u00f3n material, ten\u00eda que haber sido as\u00ed porque no hab\u00eda historias sobre ellos, ellos no hicieron historias, s\u00f3lo de vez en cuando hab\u00eda un caso especial como su t\u00edo Kostas, el del amor inmortal, y luego, al final, vino la ruptura radical: fue ella misma, que hab\u00eda dejado todo atr\u00e1s. Cuando Xeno ley\u00f3 el extenso art\u00edculo de Wikipedia sobre Romolo Strozzi no se sinti\u00f3 impresionada con todas las historias de los or\u00edgenes y la ascendencia de aquel hombre; para ella, era puro relumbr\u00f3n: Strozzi era jefe de gabinete del presidente de la Comisi\u00f3n, pero el art\u00edculo daba a entender que su profesi\u00f3n primera era la de descendiente, y eso a Xeno le parec\u00eda un desatino. Otra informaci\u00f3n s\u00ed le hab\u00eda causado respeto y asombro: en los Juegos Ol\u00edmpicos de verano de 1980, Romolo Strozzi hab\u00eda ganado una medalla en esgrima: bronce en sable individual.\n\n\u00bfLo sab\u00edas?, hab\u00eda preguntado a Fridsch.\n\nS\u00ed, hab\u00eda dicho \u00e9l, lo hab\u00eda o\u00eddo. Fue en los Juegos Ol\u00edmpicos de Mosc\u00fa. Se dice que a Strozzi le benefici\u00f3 que muchos pa\u00edses, no s\u00e9 cu\u00e1ntos, boicotearan los juegos por la entrada de las tropas sovi\u00e9ticas en Afganist\u00e1n. Debido a eso, algunos esgrimidores de nivel mundial no estuvieron presentes.\n\nPero \u00e9l se clasific\u00f3, luch\u00f3 y obtuvo una medalla.\n\nS\u00ed, es cierto. \u00bfY sabes lo m\u00e1s interesante? Esto me lo cont\u00f3 Queneau cuando hablamos una vez de Strozzi: los italianos no boicotearon los juegos, pero tampoco se presentaron bajo su bandera nacional, sino bajo la bandera ol\u00edmpica: los cinco anillos sobre fondo blanco. Y en los homenajes a los deportistas italianos ganadores no se toc\u00f3 su himno nacional, sino el Himno a la alegr\u00eda. Al parecer, la familia Strozzi tuvo una gran influencia en esa decisi\u00f3n del Comit\u00e9 Ol\u00edmpico italiano.\n\nXeno mir\u00f3 a Strozzi, el anillo de sello brincaba ante sus ojos, y lo que ella ten\u00eda en la mente sobre aquel hombre del vientre rojo era: medalla ol\u00edmpica en esgrima. Ella no conoc\u00eda ese deporte. \u00bfPor qu\u00e9 iba a conocerlo? Strozzi hab\u00eda ganado su medalla con el sable. No era florete. Si Xeno hubiera sabido la diferencia, ahora habr\u00eda podido evaluar mejor el desarrollo de la conversaci\u00f3n.\n\nElla esperaba que \u00e9l entrara en materia sin rodeos. Esos hombres tienen poco tiempo. \u00c9l le preguntar\u00eda sin m\u00e1s pre\u00e1mbulos qu\u00e9 pod\u00eda hacer por ella, luego mostrar\u00eda o fingir\u00eda inter\u00e9s, y ella tendr\u00eda que exponer con rapidez y precisi\u00f3n su demanda, de manera que la reacci\u00f3n de \u00e9l tendiera hacia \u00abtener inter\u00e9s\u00bb. Pero para su asombro dijo: \u00bfSabe lo que me interesa? Me gustar\u00eda saber qu\u00e9 opina. \u00bfQu\u00e9 le parece la prohibici\u00f3n del burkini? Usted, que es mujer. Me interesa de verdad. \u00bfOpina que hombres como el alcalde de Niza pueden determinar lo que han de ponerse las mujeres, mejor dicho, en este caso, determinar que deben desvestirse? Una mujer tiene que desvestirse, \u00bfes \u00e9sta nuestra civilizaci\u00f3n cristiana? \u00bfS\u00ed? \u00bfUsted qu\u00e9 dice? Si supiera cu\u00e1ntas demandas tenemos: que la Comisi\u00f3n ha de tomar cartas en el asunto.\n\nXeno se hab\u00eda quedado sin habla.\n\nStrozzi sonri\u00f3. Es mucho pedir, dijo. Bueno, yo, en particular, opino que el burkini protege a las mujeres del c\u00e1ncer de piel.\n\nXeno no sab\u00eda si Strozzi esperaba en serio que ella...\n\nPero la gente exige cada vez con m\u00e1s insistencia que se proh\u00edba. \u00bfEn qu\u00e9 podr\u00edamos basarnos para hacer eso? \u00bfLucha contra el fanatismo y la ortodoxia? No hay ninguna directiva que nos obligue a ello. Por suerte. Tendr\u00edas que echar el cierre. Porque entonces deber\u00edamos prohibir tambi\u00e9n el caft\u00e1n y el streimel, y...\n\n\u00bfY el qu\u00e9?\n\nEl streimel. Es ese enorme sombrero redondo de piel de los jud\u00edos ortodoxos.\n\nPero ah\u00ed hay una diferencia, dijo Xeno casi sin voz.\n\nPor supuesto que hay una diferencia. En todo lo que se asemeja hay diferencias. Y todo lo que se distingue de otra cosa, se asemeja. Mire lo que le digo: tendr\u00edamos que prohibir hasta los trajes de ejecutivo. En esta casa estoy rodeado de hombres con trajes de ejecutivo. Es como un uniforme. Todos tienen el mismo aspecto. Da miedo. Y cr\u00e9ame, todos esos hombres son a su manera ortodoxos y fan\u00e1ticos. \u00bfDir\u00eda usted ahora que han de quitarse sus trajes?\n\nXeno miraba perpleja a Strozzi, que se ech\u00f3 a re\u00edr, se recost\u00f3 en el asiento y abri\u00f3 los brazos. Luego se inclin\u00f3 hacia delante, sonriente a\u00fan, pero ahora cambiando agradablemente a una seria curiosidad, y dijo: Pero no quiero robarle su precioso tiempo. D\u00edgame sin pre\u00e1mbulos qu\u00e9 puedo hacer por usted.\n\nQue Strozzi dijera que no quer\u00eda robarle su tiempo no era s\u00f3lo la inversi\u00f3n ir\u00f3nica de la situaci\u00f3n, era una cl\u00e1sica finta circular, como dir\u00eda un esgrimidor. Xeno no sab\u00eda hacer una parada porque ni siquiera sab\u00eda lo que era. No ten\u00eda la menor idea de c\u00f3mo puede imprimir su sello en un hombre la esgrima. Por eso ella, que siempre se preparaba tan minuciosamente para cada situaci\u00f3n, no lo estaba en absoluto para Strozzi. Sortear la clara intenci\u00f3n del otro, esquivar y burlar, la formaci\u00f3n de las fintas, finta circular, finta de golpe, finta de pase, y luego el tocado despu\u00e9s del s\u00fabito e inesperado asalto. Y en un abrir y cerrar de ojos, todo ha pasado, se estrechan la mano con expresi\u00f3n de respeto y m\u00e1xima reverencia. Y ya estaba llevando un becario a Xeno a un ascensor y acompa\u00f1\u00e1ndola abajo, al foyer del Berlaymont, ella sali\u00f3 a la luz de un sol casi explosivo, regres\u00f3 como aturdida a la rue Joseph II, a su despacho. \u00bfY ahora qu\u00e9?\n\nCon el inesperado comienzo en griego cl\u00e1sico, Strozzi la hab\u00eda desconcertado y atacado despu\u00e9s por sorpresa cambiando al franc\u00e9s. Ella le sigui\u00f3 aunque en franc\u00e9s se sent\u00eda insegura, habr\u00eda preferido hablar en ingl\u00e9s, que dominaba a la perfecci\u00f3n, igual que \u00e9l. Strozzi ten\u00eda que haber sabido eso, estaba informado a fondo. En el transcurso de la conversaci\u00f3n, en franc\u00e9s, Strozzi pudo moverse con m\u00e1s soltura y elegancia que ella y dominar a su gusto el duelo. Y lo del burkini: \u00bflo hab\u00eda dicho realmente en serio? No hab\u00eda podido decir eso en serio: era la perfecta maniobra de diversi\u00f3n. Se qued\u00f3 tan desconcertada que ya no estuvo alerta, hab\u00eda perdido la concentraci\u00f3n. Y ahora, en el camino de vuelta a su despacho, no ten\u00eda claro a\u00fan qu\u00e9 resultados pr\u00e1cticos se derivar\u00edan de la entrevista. Al contrario: se repet\u00eda a s\u00ed misma que a fin de cuentas se hab\u00eda batido bien, interpretaba mentalmente una y otra vez los momentos esenciales de la conversaci\u00f3n, como si pasara un videoclip, lo hiciera retroceder, lo volviera a pasar hasta estar convencida: S\u00ed, ha sido un triunfo. Ella hab\u00eda presentado puntos d\u00e9biles pero, en definitiva, hab\u00eda sido una victoria.\n\nSab\u00eda que \u00e9l sab\u00eda que lo que a ella le importaba en el fondo era el cambio a otra direcci\u00f3n general. Porque por esa raz\u00f3n hab\u00eda solicitado originariamente la cita. \u00c9sa era tambi\u00e9n la raz\u00f3n de por qu\u00e9 hab\u00edan tardado tanto tiempo en conced\u00e9rsela. Porque as\u00ed no se hac\u00edan las cosas. Y sin intervenci\u00f3n tampoco habr\u00eda tenido nunca esa cita. Y he aqu\u00ed que ella no hab\u00eda tocado ese tema. Hab\u00eda presentado el Jubilee Project. Hab\u00eda pensado que era una jugada perfecta. Ella ser\u00eda quien mostrar\u00eda la importancia y los m\u00e9ritos de la Comisi\u00f3n y mejorar\u00eda la imagen de la Comisi\u00f3n en la opini\u00f3n p\u00fablica europea. Ella era la que hab\u00eda tenido la idea, la que ten\u00eda un programa y la que sab\u00eda ponerlo en pr\u00e1ctica. Despu\u00e9s ser\u00eda evidente que merec\u00eda una posici\u00f3n m\u00e1s importante en esa instituci\u00f3n. Ahora eso ya no ten\u00eda que formularse expl\u00edcitamente. Lo que ahora necesitaba era s\u00f3lo la aprobaci\u00f3n y el apoyo formal del presidente. Si \u00e9l declaraba que deseaba esa celebraci\u00f3n del aniversario, entonces ya no hab\u00eda marcha atr\u00e1s, entonces todos ten\u00edan que tirar de la misma soga. Xeno hab\u00eda entregado el borrador de Martin a Strozzi, le hab\u00eda explicado en l\u00edneas generales la idea, poniendo la mira en que quedase claro lo siguiente: se trataba de la Comisi\u00f3n, no de \u00abla UE\u00bb, se trataba de quitarle a la Comisi\u00f3n la imagen de una instituci\u00f3n de bur\u00f3cratas apartados de la realidad y de posicionarla como guardiana de los derechos humanos y de las ense\u00f1anzas de la historia. Por eso tambi\u00e9n era importante financiar el proyecto s\u00f3lo con el presupuesto de la Comisi\u00f3n, y sobre todo tambi\u00e9n necesitaba, como era l\u00f3gico, el pleno apoyo del presidente. Ese proyecto, dijo ella, interesar\u00eda sin duda alguna al presidente, justo en tiempos como \u00e9stos en que la Comisi\u00f3n ten\u00eda un aut\u00e9ntico problema de imagen. Ella se imaginaba que la celebraci\u00f3n del aniversario se inauguraba con un discurso de car\u00e1cter fundamental del presidente y...\n\nD'accord, hab\u00eda dicho Strozzi, d'accord. Creo que no exagero la elasticidad de mis atribuciones si...\n\nPardon?\n\nCreo, dijo sonriendo, que tengo el pouvoir de hacerle a usted enseguida, sin informarme a mi vez, esta promesa: el presidente apoya esa idea y pronunciar\u00e1 un discurso en la inauguraci\u00f3n. Enseguida har\u00e9 redactar el acta de nuestra reuni\u00f3n con esta promesa. La recibir\u00e1 usted hoy mismo.\n\n\u00c9se fue el triunfo de Xeno. Ten\u00eda lo que quer\u00eda. Eso se dec\u00eda a s\u00ed misma al llegar al 70 de la rue Joseph II y, nada m\u00e1s entrar, ir a la cantina a por un caf\u00e9. Con su caf\u00e9 se dirigi\u00f3 a una mesa del patio, a la que estaban sentados dos salamandras, se sent\u00f3 con ellos, sinti\u00f3 de pronto una c\u00e1lida simpat\u00eda hacia el conte Strozzi, s\u00ed, los trajes de ejecutivo deb\u00edan prohibirse, pregunt\u00f3 si alguien ten\u00eda un cigarrillo, ahora era el momento de un excepcional cigarrillo, los salamandras dieron un respingo hacia atr\u00e1s como si hubiera preguntado por opio o ars\u00e9nico. Entonces Martin y Bohumil llegaron al patio con tazas de caf\u00e9, Xeno los llam\u00f3 con la mano, dijo: Good news! El Jubilee Project es desde ahora deseo del presidente. \u00bfTiene alguien un cigarrillo para m\u00ed?\n\nTen\u00eda una sensaci\u00f3n de malestar. La reprimi\u00f3. Lo que reprim\u00eda era que Strozzi al final hab\u00eda dicho dos o tres frases sobre la planificaci\u00f3n: Ah, s\u00ed, de eso me encargar\u00e9 yo, de c\u00f3mo incluimos a los Estados miembros en el proyecto.\n\n\u00bfA los Estados miembros? \u00bfO sea, al Consejo?, hab\u00eda dicho Xeno. \u00bfPara qu\u00e9? Est\u00e1bamos de acuerdo en que el proyecto es cosa de la Comisi\u00f3n.\n\nS\u00ed, eso est\u00e1 claro. Pero los Estados miembros fundaron la Comisi\u00f3n.\n\nBien s\u00fbr.\n\nAh\u00ed no estuvo Xeno alerta. Ese \u00abbien s\u00fbr\u00bb la dej\u00f3 definitivamente al descubierto. No percibi\u00f3 el golpe de sable. Y ya hab\u00eda perdido. Con su \u00abbien s\u00fbr\u00bb hab\u00eda cargado con el lastre de las instituciones que, como Martin hab\u00eda propuesto con muy buen sentido, no deb\u00edan meter baza en aquello: el Consejo y el Parlamento. En lugar de tirar de una cuerda, se tiraba ahora de un ovillo de muchas hebras; ser\u00edan determinantes muchos intereses en lugar del inter\u00e9s com\u00fan. Y ya pocos d\u00edas despu\u00e9s, ella, que tanto deseaba mostrar visibilit\u00e9, s\u00f3lo ten\u00eda el deseo de ser invisible, de descargarlo todo en Martin, que ya estaba en la cl\u00ednica de traumatolog\u00eda Lorenz B\u00f6hler de Viena a la cabecera de la cama de su hermano herido.\n\nPero antes a\u00fan hubo el inter-service-meeting. All\u00ed tambi\u00e9n march\u00f3 todo a pedir de boca. La mayor parte de las direcciones generales ignoraron la sesi\u00f3n. Para cada uno de la Comisi\u00f3n que quer\u00eda sacar adelante un proyecto, el desinter\u00e9s general era un gran alivio. As\u00ed, sin tener que entend\u00e9rselas con innumerables opiniones y sus contrarias, con propuestas improductivas y cr\u00edtica estrecha de miras, se pod\u00eda dar enseguida grandes pasos y crear hechos consumados, despu\u00e9s de los cuales ya no era posible retroceder. Se hab\u00eda informado a todos.\n\nParticip\u00f3 naturalmente alguien de la DG COMM (Comunicaci\u00f3n), al fin y al cabo el proyecto hab\u00eda partido en su origen de Mrs. Atkinson, con quien Xeno estaba tambi\u00e9n en continuo contacto. Acudi\u00f3 una representante de la DG HOME (Migraci\u00f3n e Interior), lo que resultar\u00eda muy productivo porque la conmemoraci\u00f3n del Holocausto pertenec\u00eda a los \u00e1mbitos de trabajo de esa direcci\u00f3n general y \u00e9sta pod\u00eda aportar no poca competencia y no pocos contactos al efecto. Acudi\u00f3 un joven de la DG TRADE (Comercio), eso lo hab\u00eda dispuesto Fridsch, quien por lo visto quer\u00eda estar informado sobre el proyecto de Xeno, el joven s\u00f3lo tomaba algunos apuntes y de vez en cuando asent\u00eda con movimientos de cabeza. Fue sorprendente que acudiera uno de la DG JUST (Justicia y Consumidores). Como se supo, se deb\u00eda a que el funcionario de la JUST competente para la colaboraci\u00f3n con la EAC (Cultura) era nieto de supervivientes franceses del Holocausto. Eso le hab\u00eda interesado a Martin al momento: \u00bfviv\u00edan sus abuelos? No, por desgracia. No, desde hac\u00eda m\u00e1s de treinta a\u00f1os.\n\n\u00bfNo hay nadie de la AGRI?, pregunt\u00f3 Martin ir\u00f3nicamente al comienzo de la sesi\u00f3n.\n\nLa DG AGRI (Agricultura) era la secci\u00f3n con el mayor presupuesto, una suerte de Estado en el Estado, con dur\u00edsima pol\u00edtica de intereses pero, como era p\u00fablico y notorio, con poco compromiso para los intereses de otras direcciones generales. El representante de la COMM dijo: Los agricultores s\u00f3lo son competentes cuando ya ha crecido hierba sobre el asunto.\n\nEvidentemente, en esa reuni\u00f3n no s\u00f3lo no hubo objeciones contra una gran campa\u00f1a de imagen de la Comisi\u00f3n, sino que tampoco cuestion\u00f3 nadie la idea de poner a supervivientes de Auschwitz en el centro del aniversario de la Comisi\u00f3n. La informaci\u00f3n de que ese proyecto era deseo del presidente hab\u00eda sido confirmada entretanto, tambi\u00e9n ayud\u00f3 el hecho de que el borrador de Martin hubiese sido aceptado in toto y s\u00f3lo se discutieron algunos puntos pragm\u00e1ticos y relacionados con la organizaci\u00f3n: plan cronol\u00f3gico, medios de financiaci\u00f3n, recursos, tambi\u00e9n en lo personal. Despu\u00e9s de una hora y media escasa, el meeting hab\u00eda terminado y entonces pareci\u00f3 que definitivamente todo iba sobre ruedas.\n\nViernes por la tarde. En el camino a casa, Martin Susman hab\u00eda comprado en la tienda de los quesos del Vieux March\u00e9 una baguette, una botella de Sancerre Blanc y un peque\u00f1o surtido de quesos. El vendedor, un chico joven, a quien se le hac\u00eda la boca agua con todo lo que cortaba con deleite y empaquetaba con amor, le hab\u00eda recomendado encarecidamente que comprara tambi\u00e9n una mostaza de higos, del Tesino, nueva en la gama de productos de la tienda. No va a cre\u00e9rselo, dijo, pero es mejor que la moutarde aux figues de Borgo\u00f1a, y se bes\u00f3 sonoramente y con entusiasmo las yemas de los dedos. Y con el queso de cabra, dijo, es imprescindible la mostaza de higos, pero qu\u00e9 estoy dici\u00e9ndole, eso ya lo sabr\u00e1 usted, pero esta vez, dijo, es indispensable que se quede con la del Tesino.\n\nS\u00ed, entonces me quedo esta vez con la del Tesino, dijo Martin, que nunca hab\u00eda comprado all\u00ed mostaza de higos.\n\nEn casa puso en un plato el queso, la mostaza sobre la mesa. \u00bfQueso con mostaza? Parti\u00f3 un trozo de la baguette, sab\u00eda a algod\u00f3n. Hac\u00eda un calor asfixiante, Martin se quit\u00f3 el pantal\u00f3n y los zapatos, abri\u00f3 una ventana. El vino no estaba fr\u00edo. Puso la botella en el congelador, sac\u00f3 de la nevera una Jupiler, se asom\u00f3 a la ventana abierta y baj\u00f3 la vista a la plaza. Se bebi\u00f3 la cerveza en la botella, se fum\u00f3 adem\u00e1s un cigarrillo, mir\u00f3 por la ventana al hervidero de abajo, la ceniza ca\u00eda de su cigarrillo, en el plato se fund\u00eda y se deshac\u00eda el queso.\n\nLa imagen que se ofrec\u00eda a Martin desde la ventana le recordaba un libro infantil que le gustaba mucho y que repasaba continua y minuciosamente ya antes de aprender a leer. Se titulaba La ciudad y era un libro para buscar im\u00e1genes, en gran formato, con gruesas p\u00e1ginas de cart\u00f3n que mostraban escenas multicolores rebosantes de im\u00e1genes. Su madre nunca tuvo tiempo de mirar el libro con \u00e9l, tampoco recordaba qui\u00e9n se lo hab\u00eda regalado, pues sus padres no se lo habr\u00edan comprado. Pero Florian, su hermano mayor, a veces se sentaba en su cama por la noche y los dos miraban juntos el libro, como miraba \u00e9l ahora la plaza. \u00bfD\u00f3nde est\u00e1 la vendedora de flores?\n\n\u00a1Aqu\u00ed!\n\n\u00bfD\u00f3nde est\u00e1 el polic\u00eda?\n\n\u00a1Aqu\u00ed!\n\n\u00bfD\u00f3nde est\u00e1 el cartero?\n\n\u00a1Aqu\u00ed!\n\n\u00bfD\u00f3nde est\u00e1 el cami\u00f3n de los bomberos?\n\n\u00a1Aqu\u00ed!\n\n\u00bfD\u00f3nde est\u00e1 el surtidor?\n\n\u00a1Aqu\u00ed!\n\n\u00bfD\u00f3nde est\u00e1 el puesto de verduras?\n\n\u00a1Aqu\u00ed!\n\n\u00bfD\u00f3nde est\u00e1 el hombre de los pantalones cortos y la m\u00e1quina de fotos?\n\n\u00a1Aqu\u00ed!\n\n\u00bfD\u00f3nde est\u00e1 la mujer con la bolsa de la compra?\n\n\u00a1Aqu\u00ed! \u00bfD\u00f3nde est\u00e1n los soldados con las ametralladoras?\n\n\u00a1Aqu\u00ed, aqu\u00ed, aqu\u00ed y aqu\u00ed!\n\nDe pronto son\u00f3 su smartphone. Martin mir\u00f3 la pantalla, no conoc\u00eda el n\u00famero, cogi\u00f3 la llamada.\n\nAs\u00ed, en calzoncillos, con una botella de cerveza en la mano mirando fascinado \u00abla ciudad\u00bb, se enter\u00f3 de que su hermano estaba en una cl\u00ednica de traumatolog\u00eda.\n\nCuando Alois Erhart ten\u00eda doce a\u00f1os se hizo socio del MAC, el Mariahilfer Athletik Club, un peque\u00f1o y activo club deportivo del distrito. Por lo que recordaba Erhart, fue deseo de su padre y no suyo. No hubo discusi\u00f3n: Alois ten\u00eda que ser, evidentemente, socio del \u00abClub\u00bb. \u00bfQu\u00e9 iba a decir, si no, la gente? \u00bfEs que el hijo del comerciante de art\u00edculos deportivos no es deportista? El mundo era m\u00e1s peque\u00f1o entonces, se pensaba en la identidad del distrito. Si se viv\u00eda en el distrito 6 de Viena, se sab\u00eda enseguida todo, qui\u00e9n qu\u00e9 c\u00f3mo y por qu\u00e9, desde la Laimgrube arriba hasta el Magdalenengrund abajo, pasando por Gumpendorf y llegando hasta Wienzeile, a la izquierda. Alois recordaba que su padre hablaba entusiasmado de una boda que se hab\u00eda celebrado en la parroquia de San Egidio, en la Gumpendorfer Platz: \u00ab\u00a1Fue la boda m\u00e1s hermosa que se viera jam\u00e1s en Mariahilf!\u00bb. \u00a1Mariahilf! \u00a1No Viena! All\u00ed se era \u00abMariahilfer\u00bb y si se bajaba por la Mariahilfer Strasse, pasando por la Babenbergerstrasse, al primer distrito, entonces se iba \u00aba la ciudad\u00bb. En el Caf\u00e9 Kafka, de la Capistrangasse, se murmuraba que al \u00abchaval\u00edn\u00bb, al hijo de \u00abErhart, el de la tienda de deportes\u00bb, siempre se le ve\u00eda con libros, pero nunca con un bal\u00f3n. Y en un santiam\u00e9n, Alois era socio del club. Tuvo que elegir una \u00absecci\u00f3n\u00bb. Nada de gimnasia, eso era para mujeres. La gimnasia con aparatos era algo completamente ajeno a su ser, ya le ten\u00eda miedo en el colegio, en la clase de gimnasia no consigui\u00f3 en la barra ni el m\u00e1s elemental molinete. Por otra parte, el profesor de gimnasia del MAC era divertido y simp\u00e1tico: J\u00e1nos G\u00f6rgey, un refugiado h\u00fangaro del 56 que se llamaba a s\u00ed mismo \u00abpadre de la gimnasia J\u00e1nos\u00bb, le recibi\u00f3 con un fascinante acento h\u00fangaro: \u00abDonde se hace gimnasia, puedes quedarte sin miedo, porque las malas personas no tienen barras\u00bb. Pero no, ni barras paralelas, ni potro, ni barra fija. El MAC era famoso por su secci\u00f3n de boxeo. Presentaba a los campeones austriacos en las tres clases de peso. El entrenador de boxeo, Toni Marchardt, pellizc\u00f3 a Alois en el brazo superior, dijo con voz ronca algo ininteligible y le mir\u00f3 con tal desprecio que Alois se sinti\u00f3 confirmado en su opini\u00f3n de que el boxeo no era un deporte, sino un comportamiento exc\u00e9ntrico propio de locos. Estaba dispuesto a inscribirse en la secci\u00f3n de f\u00fatbol, ah\u00ed conoc\u00eda las reglas, de f\u00fatbol se discut\u00eda tambi\u00e9n en el colegio, por tanto \u00e9l podr\u00eda hablar con m\u00e1s conocimiento de causa, y pens\u00f3 que all\u00ed s\u00f3lo ten\u00eda que correr un poco como los dem\u00e1s y no llamar mucho la atenci\u00f3n, siempre hab\u00eda otros que quer\u00edan el bal\u00f3n a toda costa.\n\nEl bal\u00f3n.\n\nUn d\u00eda el entrenador, el se\u00f1or Horak, despu\u00e9s del entrenamiento, que hab\u00eda sido una batalla de barro porque llov\u00eda a c\u00e1ntaros en el prado de Denzel, dio a Alois el bal\u00f3n del club para que se lo llevara a casa. En aquel entonces a\u00fan se jugaba con balones de cuero cosidos a mano, los llamados \u00abaut\u00e9nticos\u00bb, era un objeto de valor con el que los socios del club se distingu\u00edan de los chicos de la calle, que jugaban en el parque con balones hechos de trapos viejos o con balones baratos de pl\u00e1stico, como globos pero un poco mejores.\n\nAlois ten\u00eda a su cargo esa vez el cuidado del bal\u00f3n, es decir, eliminar del tan maltratado bal\u00f3n barro, excrementos y humedad de la lluvia, frotar grasa en los peque\u00f1os desgarros y grietas y luego, cuando el cuero ya \u00abhab\u00eda recuperado sus grasas\u00bb, frotar y sacar brillo al bal\u00f3n con un pa\u00f1o suave, \u00abcomo si fuera el par de zapatos que uno se pone para una audiencia con el emperador\u00bb.\n\nAlois Erhart r\u00ede silenciosamente para sus adentros. En realidad, pensaba, hab\u00eda aprendido ya entonces algo que \u00e9l a\u00fan no pod\u00eda entender: con qu\u00e9 constancia sigue influyendo la historia hasta en lo banal.\n\nQuiz\u00e1 tuvo el se\u00f1or Horak un impulso pedag\u00f3gico y crey\u00f3 que pod\u00eda mover a Alois a comprometerse e identificarse m\u00e1s con el club si le encomendaba esa tarea. Quiz\u00e1 hab\u00eda notado el se\u00f1or Horek que Alois no daba muestras de que le gustara ir al club para sufrir lo indecible en los entrenamientos, para estar sentado durante el partido en el banco de los reservas, pero ser siempre el soporte publicitario de su padre, el \u00fanico con las botas de f\u00fatbol m\u00e1s modernas, provistas de tacos recambiables y de venta en Sport-Erhart.\n\nAs\u00ed que Alois se llev\u00f3 el bal\u00f3n a casa, el domingo deb\u00eda devolverlo para el partido contra los ottakringos. Uno de los m\u00e1s importantes partidos de la temporada, porque con Ottakring hab\u00eda una especial rivalidad: los mariahilfers llamaban entonces despectivamente a los ottakringos los b\u00e1varos, o incluso los germanos, eso ten\u00eda razones hist\u00f3ricas que nadie sab\u00eda ya bien. Al parecer, Ottakring fue fundada, como arrabal de Viena, por inmigrantes b\u00e1varos. Esa leyenda se fusion\u00f3 entonces de alg\u00fan modo con el odio generalizado a los piefkes, los alemanes, que, por supuesto, ten\u00edan la culpa de toda la desdicha de la guerra, de la posguerra y la ocupaci\u00f3n. Era grotesco, pero caldeaba a\u00fan m\u00e1s las emociones que ya hab\u00eda de todas maneras por la rivalidad tradicional de los distritos del centro de la ciudad con los exteriores, los que estaban al otro lado del cintur\u00f3n.\n\nAs\u00ed pues, llegaron los ottakringos. Y los mariahilfers no ten\u00edan bal\u00f3n.\n\n\u00c9ste se hallaba en la habitaci\u00f3n de Alois, en el oscuro rinc\u00f3n al lado del armario. Alois no se hab\u00eda presentado al partido. Cuando hubo decidido que ya no volver\u00eda al club, hab\u00eda olvidado el bal\u00f3n y por eso no lo hab\u00eda devuelto.\n\nEs f\u00e1cil imaginar lo que se murmur\u00f3 el lunes en el Caf\u00e9 Kafka de la Capistrangasse. Pap\u00e1 Erhart s\u00f3lo pudo subsanar el esc\u00e1ndalo donando al club un flamante \u00abaut\u00e9ntico\u00bb y un equipo completo de ropa deportiva. Y a su hijo le ley\u00f3 seriamente la cartilla.\n\nAlois Erhart estaba sentado en un banco del cementerio de Bruselas, hab\u00eda echado la cabeza para atr\u00e1s, cerrado los ojos y sonre\u00eda. \u00bfPor qu\u00e9 le ven\u00eda ahora todo aquello a la memoria?\n\nLa fiabilidad, hab\u00eda dicho su padre, es en la vida el alfa y la omega. Haz lo que quieras, pero en tu vida ha de haber una f\u00e9rrea ley: ser absolutamente de fiar con dos grupos de personas, con las personas a las que quieres y con las personas a las que necesitas.\n\nYo no quiero al se\u00f1or Horak, dijo Alois.\n\nSu padre lo mir\u00f3 en silencio.\n\nY tampoco lo necesito.\n\n\u00bfEst\u00e1s seguro? \u00bfEst\u00e1s seguro de que no vas a necesitarlo nunca? \u00bfNi a ninguno de tus compa\u00f1eros de equipo?\n\nAlois mir\u00f3 a su padre en silencio.\n\n\u00bfAs\u00ed que has comprendido? Repite lo que te he dicho.\n\nTengo que ser una persona de fiar.\n\n\u00bfCon qui\u00e9n?\n\nCon los que quiero y con los que necesito.\n\nNo, hijo m\u00edo. Ya hab\u00edamos avanzado. As\u00ed que: \u00bfcon qui\u00e9n?\n\nAlois mir\u00f3 a su padre en silencio.\n\nTienes que ser fiable siempre. Por principio. Con todos aquellos a los que quieres, eso de por s\u00ed. Pero tambi\u00e9n con todos los dem\u00e1s, porque nunca sabes a qui\u00e9n puedes necesitar y qui\u00e9n puede perjudicarte. \u00bfEst\u00e1 claro?\n\nTengo que ser fiable siempre.\n\nCuando prometes algo, \u00bfqu\u00e9 tienes que hacer?\n\nCumplirlo.\n\nCuando asumes una tarea, \u00bfqu\u00e9 tienes que hacer?\n\nEsa tarea, tengo que... tengo que...\n\nLlevarla a cabo.\n\nSi se espera algo de ti y no has dicho al momento con claridad que no puedes y si tampoco tienes buenas razones para no querer llevarlo a cabo, \u00bfqu\u00e9 tienes que hacer?\n\nAlois mir\u00f3 a su padre.\n\nPues eso: hacerlo. No quiero que en el Kafka vuelvan a echarme en cara que no s\u00e9 educar a mi hijo. \u00bfEst\u00e1 claro?\n\nS\u00ed, padre.\n\n\u00bfPor qu\u00e9 le ven\u00eda ahora a la memoria todo eso a Alois Erhart, medio enternecido, medio regocijado, mientras, sentado en un banco del cementerio de Bruselas, contemplaba una tumba y esperaba?\n\nEstaba disgustado por haber tomado de nuevo el avi\u00f3n a Bruselas, al segundo meeting del think-tank New Pact for Europe. Estaba disgustado cuando pag\u00f3 el vuelo, estaba disgustado cuando hizo la maleta, disgustado en el taxi que le llev\u00f3 al aeropuerto, en el avi\u00f3n rebosaba furia contra s\u00ed mismo, fue agresivo con la joven recepcionista de aflautada voz cuando se registr\u00f3 en el hotel Atlas, porque todo aquello le atacaba horriblemente los nervios, ese arrastre de maletitas de ruedas por Bruselas d\u00e1ndose importancia, ese trascendente participar en meetings, ese responder a frases hueras con frases hueras, la susurrante transformaci\u00f3n de cero ideas en un babil\u00f3nico galimat\u00edas, le parec\u00eda absurdo, completamente in\u00fatil, era tiempo perdido. Quer\u00eda llevar rodando el bal\u00f3n a la esquina y olvidar.\n\nPero hab\u00eda aceptado. Estaba en ese grupo. M\u00e1s a\u00fan, se hab\u00eda declarado dispuesto a pronunciar la conferencia de apertura de esa segunda ronda de deliberaciones. Se hab\u00eda comprometido a esa tarea. La pelota estaba con \u00e9l. Por eso hab\u00eda venido. \u00c9l era de fiar.\n\nSonri\u00f3.\n\nTen\u00eda que serlo. Formaba parte de \u00e9l y esa fiabilidad le hab\u00eda llevado lejos. Desde Mariahilf, por todo el mundo, hasta s\u00ed mismo. \u00bfQu\u00e9 era en cambio el desenga\u00f1o que hab\u00eda sentido en el primer encuentro del think-tank? \u00bfQu\u00e9 fue aquello comparado con el pobre desprecio \u2014\u200c\u00e9l, el altruista, ten\u00eda que confes\u00e1rselo\u2014, s\u00ed, el desprecio hacia los miembros del grupo?\n\n\u00bfPod\u00eda decir eso de un modo tan general? \u00bfQue todos eran despreciables? Hab\u00eda sin duda diferencias. En cualquier caso hab\u00eda grados de desprecio y grados de su efectividad. El profesor Erhart divid\u00eda a los miembros de think-tank en tres clases: estaban, primero, los vanidosos. S\u00ed, bueno, vanidosos eran todos, en el fondo, en cierto sentido tambi\u00e9n \u00e9l. Habr\u00eda que precisar: los nada-m\u00e1s-que-vanidosos. Para ellos el think-tank era de m\u00e1xima importancia: porque ellos estaban en \u00e9l. Pero en eso se agotaba ya su importancia, porque s\u00f3lo se trataba de notar la propia importancia dentro de s\u00ed mismos y emitirla al exterior. Erhart conoc\u00eda a esos tipos, sab\u00eda c\u00f3mo musitaban con aire trascendental, en casa, en sus institutos de la universidad o en otras instituciones en las que trabajaban: \u00abMa\u00f1ana, por cierto, tengo que viajar a Bruselas, se\u00f1or colega, ya sabe usted, estoy en el advisory group del presidente de la Comisi\u00f3n\u00bb. Era su elixir de vida: el efecto en su inmediato entorno profesional, el orgullo de haber llegado tan lejos que ya no ten\u00edan que escuchar, sino que se dignaban escuchar. Se entusiasmaban f\u00e1cilmente \u2014\u200cconsigo mismos\u2014 cuando hablaban, demostraciones ret\u00f3ricas de la pura dicha que era poder intervenir con su palabra. Jam\u00e1s ten\u00edan un pensamiento original y tampoco pod\u00edan entender ni apreciar ning\u00fan pensamiento que no hubiera sido ya citado alternativamente cien veces por gente como \u00e9l y asegurado con notas a pie de p\u00e1gina. En el fondo eran inofensivos. \u00bfLo eran realmente? Eran aquellos con los que en tales grupos, cuando se trataba de tomar decisiones y determinaciones, se pod\u00eda generar la mayor\u00eda.\n\nY luego estaban los idealistas. Aunque: \u00bfno eran todos en cierta medida idealistas? \u00c9l tambi\u00e9n. Solamente que sus ideales eran diferentes. Lo que a uno le parec\u00eda ideal, por ejemplo tener unos ingresos varias veces superiores a los del resto, porque en una sociedad de rendimiento \u00e9l se hab\u00eda impuesto con su propio rendimiento, estaba en pugna con el ideal de justicia distributiva de otro. Eso eran banalidades que Erhart ya hab\u00eda discutido en el primer semestre de Ciencias Econ\u00f3micas. En el fondo se daba el nombre de idealistas s\u00f3lo a quienes no sacaban ventaja de serlo. Los nada-m\u00e1s-que-idealistas. En un primer momento eran aliados contra los vanidosos, pero la alianza fracasaba muy deprisa porque siempre hab\u00eda alg\u00fan aspecto, alg\u00fan detalle, opuesto a sus abnegados ideales. Y entonces no pod\u00edan continuar. Eran tan altruistas que, \u00abpara poder mirar al espejo\u00bb y verse en \u00e9l ten\u00edan que poseer algo que fuera de ellos solos. Entonces eran ellos. Cuando hab\u00eda que votar y tomar decisiones, de pronto ya no estaban libres de toda claudicaci\u00f3n: entonces su preocupaci\u00f3n consist\u00eda en impedir un mal mayor con su aprobaci\u00f3n y convertirlo en un mal menor. Por otra parte, esos nada-m\u00e1s-que-idealistas no sol\u00edan ser determinantes para formar mayor\u00eda. Eran muy pocos. Por lo general bastaban para la mayor\u00eda los nada-m\u00e1s-que-vanidosos. Sin embargo, llamaba la atenci\u00f3n que los idealistas, por lo general, estaban de acuerdo con los vanidosos. Por lo visto, lo familiar, lo que se entend\u00eda por s\u00ed solo, les parec\u00eda menos peligroso, un mal menor en comparaci\u00f3n con lo que no se sab\u00eda a ciencia cierta y no pod\u00edan poner en consonancia con su conciencia. Est\u00fapido juego de palabras, pens\u00f3 Erhart, ciencia-conciencia, y se pidi\u00f3 disculpas a s\u00ed mismo. Por otra parte, tampoco era tan malo. Sonri\u00f3. En cualquier caso, ese enga\u00f1o funcionaba asombrosamente bien: lo seguro, lo realista que siempre aparec\u00eda con tablas y estad\u00edsticas, con casillas y flechas, y lo que se pod\u00eda hacer de modo realista, otra vez casillas y flechas, hoja tras hoja se llenaba el rotafolio con casillas y flechas, con rotuladores en distintos colores, ya s\u00f3lo el movimiento necesario para mover hacia atr\u00e1s un folio del rotafolio, por encima del marco del rotafolio, ten\u00eda algo grande, algo din\u00e1mico, y... \u00a1zisz\u00e1s! Ya est\u00e1. Y en otra hoja otras casillas, que se unen con flechas... Pero en fin: as\u00ed no funcionaba ni este mundo ni ning\u00fan otro, y mucho menos a\u00fan el mundo futuro. Pero hab\u00eda que hacer s\u00f3lo una casilla para los idealistas, escribir en una casilla uno de sus ideales, hacer unas cuantas flechas desde esa casilla hasta arriba, hasta el presidente, unas cuantas flechas desde abajo hasta esa casilla, exclamar al mismo tiempo: Demand-driven, bottom-up, no top-down, y ya se ten\u00eda en el revoltijo de las flechas y las l\u00edneas de uni\u00f3n una red en la que estaban pillados los idealistas.\n\nEntonces sonre\u00edan los del tercer grupo: sonre\u00edan sabiendo, como los vanidosos, pero sabiendo mejor y riendo los \u00faltimos, es decir, riendo mejor si los idealistas s\u00f3lo hab\u00edan impedido lo peor. \u00c9sos eran los lobbistas. Aunque ah\u00ed tambi\u00e9n hab\u00eda que distinguir: \u00bfno era \u00e9l mismo, el profesor Alois Erhart, tambi\u00e9n un lobbista? \u00bfLobbista de una idea? \u00bfLobbista de determinados intereses, aunque precisamente \u00e9stos, en su opini\u00f3n, eran para provecho de la generalidad? Una idea as\u00ed no la ten\u00edan los lobbistas, ni siquiera pod\u00edan imaginarse que la hubiera. La generalidad, el inter\u00e9s general, era para ellos algo a lo que ten\u00edan que vender lo que ellos vend\u00edan. Vender y comprar, \u00e9se era su mundo y quiz\u00e1 cre\u00edan incluso que en ello consist\u00eda el \u00fanico inter\u00e9s general. En tales advisory groups ellos eran representantes de multinacionales, eran representantes de fundaciones de las multinacionales. No hab\u00eda que menoscabar todo lo que patrocinaban, financiaban, subvencionaban, no se deb\u00eda siquiera poner pegas a lo que invest\u00edan en meras coartadas culturales, todo eso ten\u00eda en efecto de vez en cuando gran provecho social, y el profesor Erhart no quer\u00eda negarlo, \u00e9l estaba ya muy curtido no s\u00f3lo como economista, sino en lo relativo a la consecuci\u00f3n de recursos de terceros en su universidad. Pero lo que le volv\u00eda loco y tambi\u00e9n lo desesperaba tanto de ese think-tank era que ellos siempre zanjaban en alg\u00fan momento toda discusi\u00f3n y la neutralizaban siempre con el mismo mantra: necesitamos m\u00e1s crecimiento. Lo que quiera que se discutiese llevaba a la pregunta: \u00bfc\u00f3mo generamos m\u00e1s crecimiento? Un u\u00f1ero en el dedo del pie era un problema de crecimiento, objet\u00f3 una vez Erhart, y con ello s\u00f3lo cosech\u00f3 incomprensi\u00f3n, pero la general p\u00e9rdida de confianza en las instituciones europeas era consecuencia de la falta de crecimiento, el \u00e9xito amenazador del populismo de derechas..., clar\u00edsimo: si hubiera m\u00e1s crecimiento no habr\u00eda crecimiento del populismo de derechas. \u00bfY c\u00f3mo se pod\u00eda generar m\u00e1s crecimiento? Clar\u00edsimo: con m\u00e1s liberalizaci\u00f3n. En lugar de dar reglas comunitarias a la Uni\u00f3n, cada Estado miembro deber\u00eda reducir, cada uno de por s\u00ed, el mayor n\u00famero posible de reglas. As\u00ed nunca se lograr\u00eda nunca una uni\u00f3n real, pero s\u00ed crecimiento, y eso ser\u00eda lo mejor para la Uni\u00f3n. Al final, eso era clar\u00edsimo ya ahora, el grupo New Pact for Europe entregar\u00eda al presidente un documento en el que propon\u00eda lo siguiente: Debemos procurar que haya m\u00e1s crecimiento. El presidente dar\u00e1 amablemente las gracias, encomiar\u00e1 el importante trabajo del grupo y luego archivar\u00e1 el documento sin leerlo, porque no ten\u00eda que leer ese documento para poder decir en el pr\u00f3ximo discurso b\u00e1sico o ya en la pr\u00f3xima entrevista: Debemos procurar que haya m\u00e1s crecimiento.\n\nErhart sab\u00eda que esos lobbistas no eran b\u00e1sicamente unos c\u00ednicos, no todos. Cre\u00edan de verdad lo que dec\u00edan, primero porque no hab\u00edan aprendido otra cosa, y segundo porque hab\u00edan aprendido a ganarse la vida as\u00ed. Su mantra se pagaba bien, todo lo dem\u00e1s, menos o nada. Eso es al fin y al cabo una experiencia. No se pod\u00eda echar en cara a una persona el deseo de bienestar, tampoco el af\u00e1n de riqueza, pero s\u00ed esto: que sea sobornable. Y lo son. Objetivamente. Con su ignorancia en cuanto a las ideas que no se ajustan al esquema para cuya defensa han sido pagados. Cuando hablaban del futuro, hablaban de una prolongaci\u00f3n, con un m\u00ednimo de contratiempos, del presente, y no del futuro. Ellos no lo entend\u00edan porque cre\u00edan que el futuro consist\u00eda en tendencias que triunfan incesantemente. Un lobbista hab\u00eda dicho en la \u00faltima sesi\u00f3n: La tendencia va ahora claramente en direcci\u00f3n xy: tenemos que asegurarnos de que estamos preparados para esa evoluci\u00f3n. Erhart hab\u00eda dicho entonces: A finales de los a\u00f1os veinte, la tendencia iba claramente en direcci\u00f3n al fascismo en toda Europa. \u00bfEra correcto estar preparado para esa evoluci\u00f3n o no habr\u00eda sido correcto oponer resistencia?\n\nLos vanidosos estaban desconcertados, los lobbistas sonrieron y, est\u00fapidamente, los idealistas s\u00f3lo hicieron gestos de asentimiento, y despu\u00e9s de todos modos se apearon del carro porque en las siguientes consideraciones de Erhart hab\u00eda detalles que ellos no pod\u00edan aceptar.\n\nS\u00ed, Erhart hab\u00eda sido ingenuo. Sus publicaciones de los \u00faltimos a\u00f1os hab\u00edan dado origen a la invitaci\u00f3n a ese c\u00edrculo. Pero \u00e9l hab\u00eda atribuido demasiado valor a aquello. Crey\u00f3, en efecto, que colaborando de manera constante en aquel advisory group, por as\u00ed decirlo en la antesala del presidente de la Comisi\u00f3n, poco a poco podr\u00eda influir en las \u00e9lites pol\u00edticas y poner algo en movimiento. Colaborar en programas apropiados para salvar la Uni\u00f3n Europea. Y luego, la pelota estar\u00eda en el tejado de los l\u00edderes pol\u00edticos de Europa.\n\nPero as\u00ed no funcionaba ese juego. Lo hab\u00eda visto con claridad ya muy pronto.\n\nPero su keynote s\u00ed la pronunciar\u00eda. Hab\u00eda dicho que s\u00ed. Aunque le volviera loco ver qu\u00e9 in\u00fatil era todo. Se hab\u00eda comprometido, \u00e9l era de fiar. Y era tambi\u00e9n una deuda con su maestro Armand Moens, cuya tumba contemplaba ahora. Hab\u00eda llegado a Bruselas hacia el mediod\u00eda, el meeting con su keynote empezaba a las 18.00 horas. Para salvar ese lapso de tiempo hab\u00eda decidido ir otra vez al cementerio de Bruselas y visitar la tumba de su maestro, a quien citaba al comienzo de la conferencia: \u00abEl siglo XX habr\u00eda debido ser la transformaci\u00f3n de la econom\u00eda nacional del siglo XIX en la econom\u00eda de la humanidad del siglo XXI. Eso qued\u00f3 frustrado de forma tan atroz y criminal que despu\u00e9s resurgi\u00f3 ese anhelo y con m\u00e1s apremio. Pero s\u00f3lo en la conciencia de una peque\u00f1a \u00e9lite pol\u00edtica cuyos sucesores pronto dejaron de entender ambas cosas: la criminal energ\u00eda del nacionalsocialismo y las resoluciones, marcadas por esa experiencia, que ya hab\u00edan sido llevadas a la pr\u00e1ctica\u00bb.\n\nHab\u00eda reescrito su ponencia despu\u00e9s de haber decidido que ya no ir\u00eda m\u00e1s al club. No ve\u00eda por qu\u00e9 iba a intentar con gran paciencia a lo largo de un a\u00f1o cambiar del banco de reservas al campo de juego. Hab\u00eda sido su error: creer que podr\u00eda jugar y al mismo tiempo cambiar las reglas del juego. Eso era imposible. Jam\u00e1s podr\u00eda convencer a nadie de ese c\u00edrculo, del mismo modo que no se pod\u00eda parar una cadena de montaje si cada d\u00eda, mientras se hacen con paciencia los mismos movimientos, se dice al compa\u00f1ero que uno tiene otra idea de lo que es un trabajo razonable. As\u00ed pues, cumplir\u00eda con su deber y dar\u00eda su conferencia inaugural, pero de manera que quedara claro que con ello abandonaba el club. Hab\u00eda escrito un texto radical, totalmente absurdo para ese grupo. Ahora era \u00e9l quien ten\u00eda el bal\u00f3n. Y se hab\u00eda encargado con toda lealtad de que el bal\u00f3n estuviera engrasado.\n\n\u00bfHabla usted ya con los muertos?\n\nEl profesor Erhart levant\u00f3 la vista; ante \u00e9l hab\u00eda un hombre mayor cuyos ojos azul celeste formaban un raro contraste con las pobladas cejas negras: el hombre ten\u00eda as\u00ed algo radiante y sombr\u00edo a la vez. El pelo ralo pero negro a\u00fan parec\u00eda dibujado con tinta china sobre su cabeza inclinada. Llevaba un traje muy bueno que era un poco demasiado holgado y demasiado abrigado para ese d\u00eda c\u00e1lido. El hombre hab\u00eda dicho: Praat U ook al met de doden? El profesor Erhart no entendi\u00f3. No sab\u00eda flamenco, pero sab\u00eda que en esa lengua siempre se equivocaba uno cuando, siendo de habla alemana, cre\u00eda entender algo. \u00bfLe dec\u00eda en ingl\u00e9s que no le entend\u00eda? Le vino de pronto a la memoria \u00abKannitverstaan\u00bb,[7] pero antes de que pudiera decirlo, el anciano repiti\u00f3 la frase en franc\u00e9s. El franc\u00e9s de Erhart era malo, durante un a\u00f1o fue profesor visitante en la Panth\u00e9on-Sorbonne Paris 1, daba la clase en ingl\u00e9s, se esforz\u00f3 durante aquel tiempo en aprender franc\u00e9s, pero aprendi\u00f3 enseguida que era mejor decir que no se dominaba el idioma.\n\nPero s\u00ed pudo formar esta frase: \u00abLos muertos no responden\u00bb.\n\nEl problema de los idiomas, sab\u00eda Erhart, era que, si no los domina uno al menos medianamente, dice siempre s\u00f3lo lo que puede decir y no lo que quiere decir. La diferencia es la tierra de nadie entre las fronteras del mundo. En realidad \u00e9l hab\u00eda querido decir: \u00abLos muertos ya han dado sus respuestas antes de que los vivos tuvieran las preguntas\u00bb. Pero su franc\u00e9s no daba para tanto.\n\nEl viejo sonri\u00f3. \u00bfPod\u00eda sentarse?\n\nClaro. Por favor.\n\nDavid de Vriend tom\u00f3 asiento, dijo: Hay muy pocos bancos aqu\u00ed. \u00c9ste es el \u00fanico hasta \u2014\u200cextendi\u00f3 la mano muy hacia delante\u2014, hasta los h\u00e9roes de guerra.\n\nJade\u00f3, respir\u00f3 hondo varias veces. Le cansaba mucho caminar. En realidad De Vriend hab\u00eda querido pasar la tarde en su habitaci\u00f3n, con las persianas cerradas hasta que amainara lo m\u00e1s intenso del calor. Al poco tiempo hab\u00eda perdido en la habitaci\u00f3n oscura la sensaci\u00f3n del tiempo.\n\nNo sab\u00eda cu\u00e1nto tiempo se hab\u00eda quedado sentado all\u00ed metido en cavilaciones. Sinti\u00f3 sed.\n\nAbri\u00f3 la nevera, sac\u00f3 el bloc de notas. Era el bloc en el que hab\u00eda escrito los nombres de los supervivientes que le iban viniendo poco a poco a la memoria porque en el transcurso de los a\u00f1os hab\u00eda tenido contacto espor\u00e1dico con ellos o porque de vez en cuando hab\u00eda o\u00eddo o le\u00eddo algo sobre ellos. Eran nueve nombres. Cinco de ellos estaban tachados. Mir\u00f3 extra\u00f1ado la lista. Record\u00f3 que a\u00fan ten\u00eda que tachar otro nombre: Gustave Jakubowicz. Tras la liberaci\u00f3n de Auschwitz estudi\u00f3 Derecho en Bruselas y Par\u00eds y lleg\u00f3 a ser un relevante abogado defensor de los derechos humanos; en los \u00faltimos a\u00f1os \u2014\u200cjubilado hac\u00eda tiempo\u2014 representaba a refugiados que iban a ser deportados. De Vriend hab\u00eda le\u00eddo la noticia de su muerte en el peri\u00f3dico. Busc\u00f3 el bol\u00edgrafo. Subi\u00f3 la persiana, vio asombrado en qu\u00e9 luz radiante estaba inmerso el cementerio: el verde de las copas de los \u00e1rboles, el blanco del sendero de grava, el gris plata de las piedras, todo parec\u00eda brillar.\n\nEntonces hab\u00eda decidido salir.\n\nAlois Erhart pensaba que el viejo que se hab\u00eda sentado junto a \u00e9l en el banco necesitaba hablar, que quer\u00eda conversaci\u00f3n, y ahora le resultaba desagradable estar sentado junto a aquel hombre jadeante y no decir nada. \u00bfH\u00e9roes de guerra? \u00bfA qu\u00e9 se refer\u00eda? Probablemente en ese cementerio hab\u00eda m\u00e1s adelante una zona para ca\u00eddos en la guerra mundial. \u00bfQu\u00e9 iba a decir \u00e9l a eso? Buscaba palabras. S\u00ed, se\u00f1or, dijo por fin, muy pocos bancos. Y luego: \u00bfViene usted a visitar a familiares que en la guerra...?, y ahora no sab\u00eda la palabra francesa para ca\u00eddos, \u00bfqu\u00e9 significa caer? Bueno, pod\u00eda decir muertos, \u00e9l conoc\u00eda morir, pero el hombre ya dec\u00eda: No, vengo a pasear. Para nosotros este cementerio es el lugar por donde caminamos.\n\n\u00bfPara nosotros?\n\nVivo ah\u00ed, en la residencia de mayores. Maison Hannsens. Eso es todo.\n\nAhora pasaba un hombre a quien Erhart en un primer impulso quiso saludar porque cre\u00eda conocerle, le sonaba su cara, pero \u00bfde qu\u00e9? \u00bfQui\u00e9n era? S\u00ed, ahora lo recordaba, era el comisario de la enorme barriga que le hab\u00eda interrogado en el hotel, en su primera visita a Bruselas. Caminaba a paso r\u00e1pido, sin mirar, su barriga, pens\u00f3 Erhart, hab\u00eda disminuido.\n\nEl profesor Erhart mir\u00f3 el reloj. Era hora de ponerse en camino, cambiarse en el hotel y dirigirse al meeting.\n\nEl comisario Brunfaut moder\u00f3 el paso, jadeaba. Su camisa, h\u00fameda de sudor, se le pegaba al vientre y a la espalda; se quit\u00f3 la americana. No hab\u00eda tenido debidamente en cuenta lo larga que era esa avenida que llevaba a las tumbas de los soldados. Donde estaban las v\u00edctimas de la Segunda Guerra Mundial hab\u00eda un monumento conmemorativo, \u00abLe Mur des Fusill\u00e9s\u00bb, muy f\u00e1cil de encontrar, enfrente hab\u00eda un banco. All\u00ed le hab\u00eda citado Philippe. Brunfaut llegaba tarde, y Philippe le hab\u00eda pedido encarecidamente que fuera puntual. Porque ir\u00eda tambi\u00e9n otra persona que ten\u00eda muy poco tiempo.\n\n\u00bfQui\u00e9n?\n\nLo ver\u00e1s. No puedo dec\u00edrtelo por tel\u00e9fono.\n\n\u00bfRelacionado con...?\n\nS\u00ed, exacto.\n\n\u00bfPor qu\u00e9 all\u00ed?\n\nEso quiere... mi amigo. Y all\u00ed podemos hablar con toda tranquilidad. Los monumentos apenas los visita nadie, s\u00f3lo los pol\u00edticos en el aniversario del final de la guerra. Y \u00e9se ya ha pasado. S\u00f3lo estaremos nosotros y algunas coronas secas del aniversario.\n\nBrunfaut mir\u00f3 el reloj. Ya llegaba tarde casi quince minutos. Empez\u00f3 a correr. Se vio entonces desde fuera, opin\u00f3 que daba una impresi\u00f3n insoportablemente penosa, con su trote nervioso que no era ir al paso pero tampoco correr. Camin\u00f3 otra vez m\u00e1s despacio, con el pa\u00f1uelo h\u00famedo se sec\u00f3 otra vez el sudor de la cara. \u00bfPor qu\u00e9 hac\u00eda tanto calor? \u00a1Era Bruselas, no el Congo!\n\nPor fin vio delante los cuadril\u00e1teros con las cruces blancas. \u00a1Y all\u00ed! Aquello deb\u00eda de ser el monumento que dec\u00eda Philippe.\n\nLo vio claramente delante, caminaba y caminaba y sin embargo ten\u00eda la sensaci\u00f3n de que no avanzaba. Era una pesadilla.\n\nHac\u00eda semanas que Philippe fue a verlo al hospital para contarle lo que con sus posibilidades hab\u00eda podido averiguar sobre el caso del Atlas. Mejor dicho: sobre la desaparici\u00f3n del caso del Atlas.\n\nNuestro departamento de inform\u00e1tica, hab\u00eda explicado Philippe, no es malo, sabemos bastante, y yo he interpretado con gran libertad los l\u00edmites de la legalidad. Pero no debes olvidarlo: somos la polic\u00eda de Bruselas, o sea, nunca el \u00faltimo grito en tecnolog\u00eda. El conjunto se ve dificultado por una red de niveles de mantenimiento de secreto; \u00bfc\u00f3mo explic\u00e1rtelo? Es m\u00e1s o menos as\u00ed: cuando, por ejemplo, hay una informaci\u00f3n, mejor dicho la noticia de una informaci\u00f3n, que, digamos, el servicio secreto franc\u00e9s tiene especial inter\u00e9s en mantener secreta, entonces quiz\u00e1 tiene acceso a ella nuestra Suret\u00e9 de l'\u00c9tat, pero no nuestra polic\u00eda. Si se intenta hackearlo, tienen alarma, claro. Ahora imag\u00ednate que noten que el ataque de piratas inform\u00e1ticos viene de la polic\u00eda. Y luego est\u00e1 tambi\u00e9n la Europol. Los servicios de polic\u00eda de los Estados miembros europeos deber\u00edan colaborar entre ellos e intercambiar informaci\u00f3n. El problema es que ese intercambio no funciona. Cada Estado quiere, por supuesto, saberlo todo de los otros, pero ninguno quiere soltar prenda. Te vienen entonces con sus constituciones: todo lo que por desgracia, por desgracia, no les permite la constituci\u00f3n nacional. Es decir, ah\u00ed no se mueve nada, cada informaci\u00f3n se convierte en una aguja en el pajar. Siempre hay alguien que sabe d\u00f3nde est\u00e1 la aguja, pero \u00bfqui\u00e9n sabe d\u00f3nde est\u00e1 el que sabe eso? Es decir, tenemos dos montones de paja. No, tenemos cientos de montones de paja, y en dos de ellos hay una aguja que buscamos, una en cada uno de ellos. Pero si se consigue encontrarla, eso significa que hemos encontrado la caja fuerte en la que est\u00e1 guardado lo que nos interesa. Ahora tenemos que forzar la caja fuerte. Y si se consigue, lo primero que vemos al abrir la caja fuerte es otra c\u00e1mara acorazada con una combinaci\u00f3n a\u00fan m\u00e1s complicada. Entiendes, \u00bfverdad? Te doy ahora un ejemplo concreto de la experiencia pr\u00e1ctica: cuando ha tenido lugar un ataque terrorista, hab\u00eda en todos los grados de seguridad y en todos los niveles detr\u00e1s de muchas puertas cerradas de cajas fuertes todas las informaciones que habr\u00edan sido apropiadas para impedir ese ataque. Pero nadie las reuni\u00f3. De eso nos enteramos a veces por los peri\u00f3dicos. Y luego en alguna parte de Europa tiene que dimitir un ministro del Interior. Pero eso no cambia nada en el sistema. Al rev\u00e9s, si alguna vez, sobre la base de informaciones de los servicios secretos, ha de impedirse un ataque pero algo sale mal, los servicios secretos no tienen el menor inter\u00e9s en que eso aparezca en los peri\u00f3dicos y entonces el caso desaparece. Un muerto en la habitaci\u00f3n de un hotel no es como treinta muertos despu\u00e9s de una bomba en el aeropuerto. Eso puede encubrirse. Pues los servicios secretos no tienen ning\u00fan inter\u00e9s en que se proceda a indagar e investigar y se discuta p\u00fablicamente por qu\u00e9 un polic\u00eda mata de un tiro a un turista en la habitaci\u00f3n de un hotel. Y con eso estamos en el caso Atlas. No puedo probarlo, pero estoy cien por cien seguro de que es una historia de servicios secretos. La Suret\u00e9? No. Y tampoco el SGRS. La historia es mayor. Mucho mayor. Hemos empezado con la reconstrucci\u00f3n de tu disco duro. Todo lo almacenado en un ordenador y luego borrado puede reconstruirse, a no ser que los documentos no se hayan borrado en el ordenador personal, sino en la unidad central. Bueno, esto es basic. En cualquier caso, as\u00ed hemos procedido. No s\u00f3lo hay que encontrar puntos d\u00e9biles por los que poder penetrar en otros sistemas, hay que hacerlo tambi\u00e9n de manera que no se pueda averiguar de d\u00f3nde procede el ataque. Mientras nos mov\u00edamos en el sistema belga, fue relativamente f\u00e1cil. Me resulta bastante familiar, s\u00e9 c\u00f3mo act\u00faa nuestra gente y s\u00e9 tambi\u00e9n d\u00f3nde tienen que ahorrar, bajo qu\u00e9 limitaciones y obst\u00e1culos trabajan. Y esto es ahora t\u00edpicamente belga: la polic\u00eda judicial ha invertido realmente mucho en el cifrado de sus documentos, en medidas de seguridad y de defensa contra ataques externos. Pero lo que ha olvidado es la protecci\u00f3n de su papelera. Lo que se borra desde la unidad central va a parar a una papelera central, es l\u00f3gico. Quiz\u00e1 tienen alguna copia de seguridad en alg\u00fan sitio, a eso no tengo acceso, por supuesto. Pero est\u00e1 tambi\u00e9n en la papelera, para decirlo de un modo simplificado. Y en \u00e9sa puedo rebuscar. \u00bfNo es curioso? Han pensado en un agresor exterior que se interese por sus documentos secretos, pero no en que alguien rebusque en la papelera. En cualquier caso, por ah\u00ed hemos tratado de avanzar. En alguna parte ha de haber un punto d\u00e9bil por el que accedamos a m\u00e1s informaciones, no s\u00f3lo sobre qu\u00e9 fue borrado y encubierto, sino tambi\u00e9n sobre qui\u00e9n quer\u00eda eso y por qu\u00e9. No me mires as\u00ed. Te digo enseguida lo que yo creo, porque no puedo probar absolutamente nada. Hemos encontrado, en efecto, un punto d\u00e9bil. Nos es imposible hackear los ordenadores de los servicios secretos, eso para nosotros es como intentar abrir una caja fuerte con un palillo de dientes. Pero se puede reconocer la red que forman y, si yo interpreto bien todos los indicios, en medio de todo eso est\u00e1 la OTAN. S\u00ed. La OTAN: pero espera. Ahora viene: el sistema tiene, sin embargo, un punto d\u00e9bil. Y \u00e9se es el ordenador del arzobispado de Pozna\u0144. S\u00ed, Pozna\u0144. \u00bfQu\u00e9 significa, qu\u00e9 es eso? Es la di\u00f3cesis cat\u00f3lica romana m\u00e1s antigua de Polonia. All\u00ed se juntan algunas informaciones de servicios secretos, pero en mucha mayor medida salen de all\u00ed informaciones hacia la OTAN y servicios secretos cooperantes. \u00a1No me mires as\u00ed! Sabes bien que me ayuda Armin de Boor: cuando Armin y yo fuimos a parar all\u00ed, nos miramos desconcertados y entonces Armin tuvo que echarse a re\u00edr. Esto es de locos, dijo, deprisa, mete el c\u00f3digo de acceso. Es una palabra, s\u00f3lo una palabra. S\u00ed, dije, pero \u00bfcu\u00e1l? Debemos hacer saltar el manojo de llaves. Rio y dijo, \u00bfno lo ves? Tienen una mente de lo m\u00e1s simple, mete Judas. Tiene que ser una palabra cargada de significado para un cura cat\u00f3lico. Pero no era Judas. Armin dijo, un momento, tal vez Judas se escriba de un modo distinto en polaco. Abri\u00f3 un programa de traducci\u00f3n, nos enteramos de que en polaco Judas se escribe Judasz. Pero no era eso. Armin busc\u00f3 cerveza en la nevera, bebimos, de pronto dijo: \u00a1Claro! Claro que no es Judas. Ellos no quieren traicionar, quieren saberlo todo. Tecle\u00f3 algo en el traductor, luego lo puso como c\u00f3digo: y la puerta se abri\u00f3. El c\u00f3digo era Bozeoko: Ojo de Dios.\n\n\u00bfEl ojo de Dios?\n\nS\u00ed.\n\n\u00bfLa Iglesia cat\u00f3lica?\n\nArzobispado de Pozna\u0144. S\u00ed.\n\n\u00c9mile Brunfaut exhal\u00f3 un quejido.\n\n\u00bfQu\u00e9 pasa?, pregunt\u00f3 Philippe.\n\nMi bazo, dijo Brunfaut.\n\nQue el encubrimiento del asesinato del hotel Atlas no remit\u00eda s\u00f3lo a un fiscal belga, sino que, de alg\u00fan modo, la OTAN ten\u00eda las manos metidas en el asunto, para \u00c9mile era demasiado, en efecto. Lo olvidamos, hab\u00eda dicho a Philippe. Olvidarlo no puedo, hab\u00eda respondido Philippe, pero hacer, ya no har\u00e9 nada.\n\nYa no lo tocamos, dijo \u00c9mile.\n\nNo, ya no lo tocamos m\u00e1s. \u00bfCu\u00e1ndo sales del hospital? El domingo que viene, a las 15.00 horas, el club juega contra el Brujas.\n\nTenemos que ir.\n\n\u00a1Iremos!\n\nDurante las semanas siguientes, \u00c9mile Brunfaut se hab\u00eda ocupado sobre todo de su salud. Es decir, que cuando fumaba lo hac\u00eda con cargo de conciencia, que beb\u00eda su Duvel y luego su querido Ros\u00e9 de modo excepcional y s\u00f3lo vasito a vasito, que suprimi\u00f3 la cerveza Mort Subite, y que en todo lo que com\u00eda recortaba la grasa visible y la apartaba al borde del plato. Miraba sus frites largo tiempo y con desconfianza antes de \u00abprobarlas\u00bb y se com\u00eda \u00fanicamente dos terceras partes de la raci\u00f3n, \u00e9sa era su dieta, porque las moules eran pr\u00e1cticamente s\u00f3lo prote\u00ednas. En cualquier caso, daba paseos a pie con m\u00e1s frecuencia que antes. Pero a las tres semanas volvi\u00f3 totalmente a sus antiguos h\u00e1bitos y tom\u00f3 la liberaci\u00f3n y el placer que sinti\u00f3 por claros s\u00edntomas de restablecimiento. Acudi\u00f3 de nuevo a su trabajo, le devolvieron su placa, su ordenador y un mont\u00f3n de tareas burocr\u00e1ticas. Hab\u00eda m\u00e1s informes que muertos y, con alegre serenidad, el comisario Brunfaut pens\u00f3 que eso estaba muy bien. Maigret pas\u00f3 por su despacho para comprobar en una confusa conversaci\u00f3n trivial si Brunfaut hab\u00eda olvidado de verdad el asesinato del Atlas. Pero \u00bfc\u00f3mo se puede examinar si alguien ha olvidado algo sin record\u00e1rselo? Brunfaut estaba tan regocijado con la ingenuidad de Maigret que se sinti\u00f3 definitivamente confirmado en su sensaci\u00f3n de ser el mismo de antes. No, \u00e9l no tocaba m\u00e1s el caso.\n\nS\u00f3lo que no pod\u00eda dejarlo del todo.\n\nLa OTAN: eso era demasiado fuerte para \u00e9l. Tampoco habr\u00eda sabido c\u00f3mo efectuar pesquisas en esa direcci\u00f3n, por mucha que fuera su prudencia. Pero lo que s\u00ed ten\u00eda era el nombre de la v\u00edctima, mejor dicho, los tres nombres, porque en la habitaci\u00f3n del hotel se encontraron tres pasaportes diferentes. Inmediatamente despu\u00e9s de que le llamaran para aquel caso, Brunfaut hab\u00eda anotado esos nombres en su bloc y lo ten\u00eda a\u00fan en su poder, un bloc de espiral no lo borraba nadie. Y a lo que tambi\u00e9n le daba vueltas era a la cuesti\u00f3n de qu\u00e9 pod\u00eda tener que ver la Iglesia cat\u00f3lica, mejor dicho, una di\u00f3cesis, con aquello. Con los nombres no avanz\u00f3 nada, ninguno de los tres estaba en los archivos de la polic\u00eda, y ni siquiera registrado o empadronado en ning\u00fan lugar de Europa. Eso s\u00f3lo pod\u00eda significar que los tres pasaportes eran falsos. Para \u00e9l y para sus posibilidades era un callej\u00f3n sin salida. \u00bfY la participaci\u00f3n de la di\u00f3cesis de Pozna\u0144? En los apuntes que tomaba escrib\u00eda siempre VAT como abreviatura de Vaticano, porque no pod\u00eda imaginarse que una di\u00f3cesis cat\u00f3lica colaborase con servicios secretos sin que el Vaticano lo supiera. \u00c9l s\u00f3lo pod\u00eda especular. De modo que no hab\u00eda mentido cuando dio claramente a entender a Philippe y sobre todo a Maigret que abandonaba aquel caso. Pues lo \u00fanico que hac\u00eda era clavar la vista en casillas vac\u00edas como un complicado sudoku que no sab\u00eda resolver.\n\nTanto mayor fue su sorpresa cuando Philippe le cit\u00f3 de pronto en el cementerio para ese asunto. Por lo visto \u00e9l, sin decir palabra, tambi\u00e9n hab\u00eda seguido pendiente del caso y ten\u00eda alg\u00fan pez en el anzuelo.\n\nCuando Brunfaut lleg\u00f3 por fin sudando y jadeando al Mur des Fusill\u00e9s, busc\u00f3 el banco en el que Philippe y \u00absu amigo\u00bb deb\u00edan estar esper\u00e1ndole. Pero all\u00ed no hab\u00eda ning\u00fan banco. No delante de aquel enorme monumento AUX VICTIMES INNOCENTES DE LA FURIE TEUTONNE. \u00bfQuiz\u00e1 detr\u00e1s, en el lado opuesto? \u00bfO a un lado? \u00bfO se hab\u00eda referido Philippe a otro monumento? Contempl\u00f3 el campo con las innumerables cruces blancas. No es que nunca hubiera visto un cementerio militar, pero por primera vez estaba escandalizado... de que le pareciera hermoso. Estaba all\u00ed de pie, respiraba hondo y aquel gran cuadril\u00e1tero rodeado de setos, con las cruces blancas todas iguales, le pareci\u00f3 bello. Despu\u00e9s de todos los t\u00famulos, l\u00e1pidas, mausoleos, capillas con los que los muertos o sus descendientes quer\u00edan sobrepujar a los otros, despu\u00e9s de todas las esculturas de angelotes llorando, de \u00e1ngeles llorando, de madres llorando, en granito, en m\u00e1rmol, en bronce y en acero fino, despu\u00e9s de toda la proliferaci\u00f3n de plantas rastreras y trepadoras, despu\u00e9s de tanto desasosiego en aquel campo interminable del sosiego final, all\u00ed por fin se descansaba. El absoluto descanso \u00f3ptico. Le pareci\u00f3 bello en un sentido radicalmente est\u00e9tico, como si aquella parte del cementerio fuese una instalaci\u00f3n, el proyecto de un artista que se dedicaba al lenguaje formal del descanso, exento de todo sentido. Si daba un paso a la derecha o un paso a la izquierda, entonces en aquel campo con las cruces colocadas a exactamente la misma distancia y las mismas filas se abr\u00edan cada vez otras perspectivas, l\u00edneas, diagonales, l\u00edneas de fuga, y a \u00e9l le pareci\u00f3 lleno de sentido; l\u00edneas de fuga. L\u00edneas de fuga cambiantes, pero que todas, en cualquier perspectiva, se\u00f1alaban siempre en la misma direcci\u00f3n: hacia la eternidad. La eternidad estaba en todas partes, igual que al final la liberaci\u00f3n de sentido y significado. En honor de las trayectorias vitales hab\u00eda quedado eliminada toda trayectoria concreta, a la memoria de las v\u00edctimas se sacrific\u00f3 la memoria de que cada vida individual hab\u00eda sido \u00fanica e irrecuperable. S\u00f3lo hab\u00eda forma, simetr\u00eda, armon\u00eda. Integraci\u00f3n en una imagen est\u00e9tica. Y en modo alguno hab\u00eda resistencia en la muerte. Brunfaut estaba horrorizado de que a \u00e9l, a ese ser vivo sudoroso, jadeante, maloliente, aquello le pareciera hermoso. No bueno. Hermoso.\n\nPero \u00bfd\u00f3nde estaba Philippe? De pie delante del monumento, Brunfaut miraba a su alrededor. Entonces vio de repente que un cerdo atravesaba un seto y empezaba a hozar entre las blancas cruces. \u00a1El cerdo! Met\u00eda y volv\u00eda a meter el hocico en la tierra, hurgaba en ella, escarbaba la tierra con las pezu\u00f1as, golpeaba con el dorso una cruz que despu\u00e9s se quedaba torcida, el cerdo segu\u00eda escarbando y hozando, la cruz empez\u00f3 a caerse poco a poco. El comisario Brunfaut, que en el curso de su vida profesional nunca hab\u00eda estado frente a hombres armados pero s\u00ed hab\u00eda tenido que entrenarse para ello en ejercicios de simulaci\u00f3n, sinti\u00f3 frente a aquel animal un miedo y un desvalimiento desconocidos. No sab\u00eda qu\u00e9 hacer. Su idea fue acercarse al cerdo como si pudiera arrestarlo. Qu\u00e9 cosa m\u00e1s rid\u00edcula. Su impulso fue salir corriendo. \u00bfHuir \u00e9l de un cerdo? Independientemente de lo que Brunfaut hizo en aquel momento \u2014\u200c\u00e9l mismo no pudo decirlo m\u00e1s tarde\u2014, dar uno o dos pasos hacia delante, o retroceder unos pasos, o ambas cosas, un irresoluto hacia delante y hacia detr\u00e1s, el cerdo alz\u00f3 la cabeza, lanz\u00f3 un sonido horrible y se fue corriendo, con fuerza animal, en una diagonal recta atravesando el campo de la simetr\u00eda armoniosa, y Brunfaut constat\u00f3 con un quejido que estaba sentado. Estaba sentado sobre el camino de grava, en una mano su pa\u00f1uelo h\u00famedo, la otra aferrada a las piedras del camino. Con rasgu\u00f1os en las palmas de las manos y un dolor punzante desde la rabadilla hasta la espalda. Y sobre las tumbas soplaba el viento.\n\nDe regreso al hotel, el profesor Erhart se meti\u00f3 bajo la ducha, se puso una camisa limpia, luego el traje ligero azul; se mir\u00f3 en el espejo: azul Europa. Sonri\u00f3 en su interior. \u00a1Qu\u00e9 casualidad! Prescindi\u00f3 de la corbata.\n\nLuego sac\u00f3 de su cartera escolar la carpeta con su keynote. Las leng\u00fcetas de cuero del doble cierre de resorte se estaban resquebrajando. Pens\u00f3 que en casa tendr\u00eda que frotarla con un poco de grasa. Junto a la cama hab\u00eda una butaca, en realidad, un asiento en forma de concha, sin acolchar, cubierto de napa roja. Erhart se sent\u00f3 y coloc\u00f3 los pies sobre la cama. Era inc\u00f3moda y estrecha. Se incorpor\u00f3 trabajosamente de ese medio huevo y se sent\u00f3 en la cama. Quer\u00eda releer otra vez su ponencia antes de ir al meeting. La hab\u00eda escrito en ingl\u00e9s, su dominio del idioma era excelente, desde que fuera profesor visitante muchos a\u00f1os atr\u00e1s en la London School of Economics y en la University of Chicago, pese a ello la hab\u00eda revisado un profesor de ingl\u00e9s amigo.\n\n\u00bfDe verdad quieres dar esta ponencia?\n\nS\u00ed.\n\nMe gustar\u00eda estar presente.\n\nErhart memoriz\u00f3 a media voz su discurso a la velocidad con que iba a pronunciarlo. Hab\u00eda puesto en marcha el cron\u00f3metro de su smartphone. Diecisiete minutos. Dos minutos de m\u00e1s. Daba igual. No se trataba de dos minutos, sino de su vida. Eso era demasiado pat\u00e9tico. Se pregunt\u00f3 qu\u00e9 le ocurr\u00eda. Se sent\u00eda como perteneciente a otra \u00e9poca. Estaba sentado en la cama, con las hojas de su conferencia sobre las piernas y contemplaba el sombr\u00edo empapelado marr\u00f3n de la habitaci\u00f3n del hotel. Por qu\u00e9 recordaba eso ahora: palabras raras, palabras que le sonaban ajenas, que de ni\u00f1o ped\u00eda a su madre que le explicara cuando las le\u00eda en un libro y no las entend\u00eda: paroxismo, soliviantado, solaz, ungulado, urdir, doler...\n\nMam\u00e1, aqu\u00ed pone: le dol\u00edan los enflaquecidos caballos del coche. No lo comprendo.\n\nPero t\u00fa sabes lo que son caballos de coche. Caballos que tiran del coche.\n\nS\u00ed, eso lo s\u00e9. Pero: le dol\u00edan. \u00bfSignifica que los caballos le hac\u00edan da\u00f1o porque tiraban mal del coche?\n\nNo, significa que le daban pena.\n\n\u00c9l se hab\u00eda quedado sentado largo tiempo, con contenido asombro porque doler significara \u00absentir pena\u00bb.\n\nEl profesor Erhart hizo un esfuerzo y se puso en camino.\n\n# CAP\u00cdTULO NOVENO\n\nLA FIN, UN PROLONGEMENT DU PR\u00c9SENT, NOUS-M\u00caMES UNE CONDITION PR\u00c9ALABLE DU PASS\u00c9.\n\n#\n\nEl cerdo fue grabado por una c\u00e1mara de seguridad del hotel Sheraton, en la place Charles Rogier, una secuencia muy breve, el cerdo aparece a paso lento, con la cabeza levantada, como si diera un agradable paseo tomando el aire del incipiente verano, un transe\u00fante se aparta de un salto, otros se detienen at\u00f3nitos, algunos sacan sus tel\u00e9fonos m\u00f3viles para fotografiarlo y de pronto ha desaparecido del campo visual. Ese v\u00eddeo se subi\u00f3 a YouTube, con el t\u00edtulo \u00abAankomst van een afgevaardigde op de conferentie van de dieren\u00bb, por un usuario que se llamaba Zinneke. En el Sheraton dio comienzo una investigaci\u00f3n para saber qui\u00e9n del personal de seguridad, con acceso a los datos almacenados en las c\u00e1maras, era el tal Zinneke. El gerente del hotel tem\u00eda un deterioro de la imagen del establecimiento si circulaba p\u00fablicamente un v\u00eddeo que mostraba un cerdo corriendo suelto ante la entrada del Sheraton. Pero no hubo deterioro de imagen, al contrario. El v\u00eddeo fue compartido en Facebook y en poqu\u00edsimo tiempo tuvo m\u00e1s de treinta mil likes. El Metro s\u00f3lo pudo publicar una foto del cerdo, tras lo cual enviaron al peri\u00f3dico m\u00e1s fotos, procedentes de las c\u00e1maras de seguridad del Carrefour en la Chauss\u00e9e de Louvain, de la estafeta de correos en la avenue de la Braban\u00e7onne y de la embajada austriaca en la rue Kortenberg. Todas esas fotos estaban tan borrosas y tan movidas que el profesor Kurt van der Koot, que ahora ten\u00eda una columna fija en Metro, no pudo decir con seguridad absoluta si se trataba siempre del mismo cerdo o de distintos cerdos. Una manada intranquilizar\u00eda a la gente, pensaba ahora, pero un \u00fanico cerdo que paseaba por Bruselas la emocionar\u00eda, despertar\u00eda en ella un amor casi infantil a los animales, eso tendr\u00eda todos los ingredientes necesarios para la formaci\u00f3n de una leyenda. Kurt van der Koot no era enemigo de su popularidad y, por lo tanto, no quer\u00eda oponerse a necesidades colectivas. Por eso cinco d\u00edas despu\u00e9s de la publicaci\u00f3n del primer v\u00eddeo en YouTube puso en marcha esta acci\u00f3n: \u00abBruselas tiene un cerdo. \u00bfC\u00f3mo ha de llamarse?\u00bb.\n\nPropuestas de nombres a la redacci\u00f3n. Tres semanas hasta el cierre de admisi\u00f3n. El profesor Van der Koot salv\u00f3 el lapso de tiempo que quedaba hasta entonces con la serie de art\u00edculos \u00abEl cerdo como met\u00e1fora universal\u00bb. En nuevas entregas diarias, mostraba el ancho de banda del bien y del mal, de la felicidad y la desgracia, del amor sentimental, del desprecio y el odio profundo, del erotismo y la vileza, para todo lo cual el cerdo ten\u00eda que servir de s\u00edmbolo, era el \u00fanico animal que cubr\u00eda como met\u00e1fora toda la amplitud de los sentimientos humanos y de las im\u00e1genes ideol\u00f3gicas del mundo, desde el cerdito de la suerte hasta el cerdo asqueroso, \u00abser un cerdo\u00bb, osaba entrar incluso en terreno pol\u00edtico y peroraba sobre los conceptos de \u00abcerdo jud\u00edo\u00bb y \u00abcerdo nazi\u00bb, luego tambi\u00e9n sobre el cerdo prohibido en las religiones y sobre los simp\u00e1ticos Babe, Piggy y los tres cerditos. La serie fue un gran \u00e9xito de p\u00fablico, en gran parte por las ilustraciones: fotos de cerditos gracios\u00edsimos, facs\u00edmiles de antiguas caricaturas que presentaban a emperadores, generales y presidentes como cerdos, reproducciones de pinturas que mostraban al cerdo en el arte (much\u00edsimos likes tuvo un dibujo de Tomi Ungerer que mostraba a una cerda madre leyendo a sus lechoncillos un cuento: \u00ab\u00c9rase una vez un carnicero...\u00bb), figuras y adornos varios, desde el cerdito como hucha hasta el cerdo cocinero, desde el cazado al cazador, y fotos de objetos de la vida cotidiana; no hab\u00eda apenas, como el propio Van der Koot constat\u00f3 con el mayor asombro, objeto de uso diario que no hubiera tomado alguna vez la forma de un cerdo: jarras de cerveza, saleros, zapatillas, gorras, hasta tostadores...\n\nLa redacci\u00f3n convoc\u00f3 un jurado de famosos que de las propuestas enviadas deb\u00eda sacar primero una longlist, luego una shortlist, para elegir finalmente el nombre ganador. Del jurado formaban parte: el cantante de m\u00fasica popular Barthold Gabalier, la actriz Sandra Vall\u00e9e, el renombrado futbolista y m\u00e1ximo goleador de la Jupiler Pro League Jaap Mulder, la viuda del antiguo alcalde de Bruselas Daniela Collier, el caricaturista Roger Lafarge, que desde sus caricaturas sobre Mahoma estaba bajo protecci\u00f3n policial, el escritor y cronista de Bruselas Geert van Istendael, el cocinero con dos estrellas Kim King, ma\u00eetre de cuisine en Le cochon d'or, y el artista Wim Delvoye, conocido porque tatuaba sus cuadros en cerdos. El presidente del jurado y portavoz era, naturalmente, el catedr\u00e1tico de universidad Kurt van der Koot.\n\nRomolo Strozzi era un hombre que apenas se inmutaba. Lo que a otros pod\u00eda sorprender, en \u00e9l generaba como mucho una disposici\u00f3n a la iron\u00eda. Nada le era ajeno: \u00bfqu\u00e9 pod\u00eda entonces producirle asombro? Hab\u00eda visto mucho en su vida y lo que no hab\u00eda visto le hab\u00eda sido transmitido como c\u00famulo de experiencias de su familia y de sus antepasados. Adem\u00e1s hab\u00eda le\u00eddo mucho. Y en el campo que \u00e9l cultivaba conoc\u00eda cada grumo, cada piedra, cada mala hierba. Por eso tuvo que sonre\u00edr para sus adentros cuando esa Fenia Xenopoulou cit\u00f3 de pronto el libro favorito del presidente, de modo forzadamente casual pero obedeciendo inequ\u00edvocamente a un plan t\u00e1ctico. Demostraba que se hab\u00eda preparado con cierta energ\u00eda neur\u00f3tica. Pero eso a \u00e9l no pod\u00eda sorprenderle. Sab\u00eda que la gente hace todo tipo de cosas. La mujer hab\u00eda fallado la finta. \u00bfHab\u00eda pensado de verdad que \u00e9l dir\u00eda al presidente: Por cierto, esa se\u00f1ora Xenopoulou tiene la misma novela favorita que usted, monsieur le pr\u00e9sident? \u00bfHab\u00eda cre\u00eddo de verdad que con eso ganar\u00eda puntos?\n\nSe sent\u00f3 en una mesa delante del Caf\u00e9 Franklin, esquina rue Franklin y rue Archim\u00e8de, en el lado de Archim\u00e8de, o sea, a la sombra. Era un d\u00eda muy caluroso y quer\u00eda fumar un purito mientras esperaba a Attila Hidekuti, el jefe de protocolo del presidente del Consejo Europeo. Ten\u00eda que hablar con \u00e9l de modo informal sobre la se\u00f1ora Xenopoulou y su denominado Jubilee Project.\n\nDe pronto hab\u00eda un gran cerdo frente a \u00e9l. Un hombre en un grotesco disfraz de cerdo, un disfraz de peluche rosa que le cubr\u00eda todo el cuerpo. Sosten\u00eda un palo con un cartel en un extremo. Apoy\u00f3 el cartel en la fachada del bar, se sent\u00f3 en la mesa vecina, se quit\u00f3 la cabeza, es decir, su cabeza de cerdo; apareci\u00f3 un rostro masculino enrojecido, chorreando sudor, pelo rubio empapado en sudor. El hombre, que ten\u00eda aproximadamente la edad de Strozzi, se pas\u00f3 varias veces el brazo de peluche rosa por la cara y dijo a la camarera, que estaba sirviendo un caf\u00e9 a Strozzi: \u00a1Una cerveza, por favor!\n\n\u00bfLe sorprende? Lo entiendo, dijo volvi\u00e9ndose a Strozzi. No me desprecie, se lo ruego. Estoy en paro desde hace meses. A mi edad es dif\u00edcil. Al final acab\u00e9 plant\u00e1ndome con un letrero en el boulevard Anspach, delante de la bolsa: \u00abAcepto cualquier trabajo\u00bb. Y me han dado \u00e9ste. Llevar por ah\u00ed este cartel. Disfrazado de cerdo por el barrio europeo. Publicidad, dijo enjug\u00e1ndose de nuevo el sudor.\n\nStrozzi se volvi\u00f3 y ley\u00f3 el cartel:\n\nSLAGERIJ\n\nVAN KAMPEN\n\nFIJNSTE VLEES, BESTE WORST!\n\nVOOR BESTELLINGEN:\n\nLET OP! NIEUW TELEFOONNUMMER!\n\nMucha gente se r\u00ede. Algunos me preguntan c\u00f3mo soy capaz de hacer algo as\u00ed. \u00bfYa nadie puede imaginarse de lo que son capaces las personas que pasan necesidad? \u00bfCree usted que es un plato de gusto, con este calor y este disfraz?\n\nStrozzi sac\u00f3 su cartera, la camarera trajo una cerveza al hombre, sonri\u00f3 y pregunt\u00f3: \u00bfAlgo m\u00e1s? \u00bfUna mazorca, quiz\u00e1?\n\nStrozzi dej\u00f3 un billete de cinco euros sobre la mesa y se march\u00f3. En la otra acera tecle\u00f3 un SMS a Attila: \u00a1Cita no en Franklin! Estoy en el Kitty O'Shea's, blvd Charlemagne.\n\nEstaba en calzoncillos y calcetines en el balconcito y cepillaba cuidadosamente su traje. En esos d\u00edas c\u00e1lidos y secos, los caminos de grava del cementerio estaban muy polvorientos, cada paso entre las filas de muertos levantaba polvo que sub\u00eda por las perneras y tambi\u00e9n penetraba en el tejido de la chaqueta. David de Vriend era muy cuidadoso con su vestimenta. Desde su retorno a la vida, despu\u00e9s de la liberaci\u00f3n, daba mucha importancia a los buenos trajes de excelente pa\u00f1o. Como profesor de secundaria no hab\u00eda ganado mucho dinero, pero s\u00ed el suficiente para, al final, encargar trajes a medida y no llevar ninguno de confecci\u00f3n. Cepillaba y pensaba en el pan. \u00bfPor qu\u00e9 pensaba en el pan? Cepillaba cuidadosa y pacientemente, estaba encantado con su cepillo para la ropa que comprara cuarenta a\u00f1os atr\u00e1s en Walter Witte, la tienda de \u00abart\u00edculos de uso diario\u00bb del boulevard Anspach. El se\u00f1or Witte en persona le hab\u00eda recomendado ese cepillo, magn\u00edfica calidad, se\u00f1or De Vriend, este cepillo le sobrevivir\u00e1, el mejor cepillo para la ropa, crin de caballo alem\u00e1n, incrustado a mano en el cuerpo de madera de roble.\n\nDe Vriend se hab\u00eda quedado inm\u00f3vil un momento, \u00abAlem\u00e1n... \u00bfqu\u00e9? \u00bfCrin?\u00bb, y de pronto not\u00f3 que, sin resistencia interior, la calidad de sus necesidades diarias pod\u00eda ser para \u00e9l m\u00e1s importante que los fantasmas del pasado. Compr\u00f3 ese cepillo alem\u00e1n que iba a sobrevivirle, que era inocente, y quiz\u00e1 lo fueran tambi\u00e9n las manos que lo hab\u00edan fabricado. Cepill\u00f3 su traje, en la habitaci\u00f3n sonaba el tel\u00e9fono, lo o\u00eda pero no lo relacionaba con \u00e9l. El sonido no le era familiar y tampoco esperaba ninguna llamada. Se dice siempre que quien ha sobrevivido a un campo de concentraci\u00f3n o de exterminio es incapaz de tirar ni un trozo de pan durante el resto de su vida. Ahora eso ven\u00eda otra vez en el peri\u00f3dico. La hija de Gustave Jakubowicz lo hab\u00eda dicho en una entrevista en De Morgen, despu\u00e9s de morir su padre, el famoso abogado de los derechos humanos: Los ni\u00f1os ten\u00edamos que comer a menudo pan duro, no nos daban pan reciente hasta que hab\u00edamos consumido el antiguo, mi padre no pod\u00eda tirar el pan, simplemente le resultaba imposible. De Vriend cepillaba. \u00a1Gustave, oh, Gustave! El tel\u00e9fono volvi\u00f3 a sonar. Gustave apreciaba los trajes de primera calidad y la baguette reci\u00e9n hecha de las cestitas de los restaurantes. \u00a1Se acabaron los trajes ra\u00eddos, s\u00f3lo tejidos buenos y gruesos! Nada de confecci\u00f3n, y sobre todo nada con rayas, y ninguna gorra, la cabeza siempre descubierta. Quien hab\u00eda estado en un campo sab\u00eda lo que significaba \u00abninguna gorra\u00bb. Era la muerte. Por eso, despu\u00e9s, la consigna era \u00abvivir\u00bb. Libertad. El mejor tejido y la cabeza descubierta. De Vriend cepillaba con habilidad, estaba en calzoncillos en el balc\u00f3n, una pernera del traje sobre el brazo izquierdo, y pasaba r\u00edtmicamente el cepillo por la tela, inmerso en ese movimiento como un violinista. En alguna parte son\u00f3 otra vez un tel\u00e9fono. Ten\u00eda cuatro trajes a medida. Para el invierno dos de tweed grueso, un Harris de espiga y un Donegal algo m\u00e1s suave de Salt & Pepper. Para entretiempo uno azul oscuro de lana virgen y otro m\u00e1s ligero, pero sin embargo agradablemente abrigado, de mohair en gris marengo. No ten\u00eda trajes de verano. Hab\u00eda pasado demasiado fr\u00edo en su vida, el verano era para \u00e9l un periodo de transici\u00f3n. Un d\u00eda muy c\u00e1lido no le importaba, y el mohair gris que estaba cepillando era tan maravillosamente ligero. \u00bfCu\u00e1nto tiempo hac\u00eda que lo ten\u00eda? Muchos a\u00f1os, eran seguramente ya... muchos a\u00f1os.\n\nDe pronto not\u00f3 una mano que le agarraba con fuerza el brazo izquierdo, esa mano le tiraba hacia atr\u00e1s, casi se le hab\u00eda ca\u00eddo el cepillo. Pero, bueno, qu\u00e9 estamos haciendo ah\u00ed, grit\u00f3 la se\u00f1ora Jos\u00e9phine. No podemos estar desnudos en el balc\u00f3n, \u00bfverdad, se\u00f1or De Vriend?\n\n\u00c9l la mir\u00f3, y ella le segu\u00eda oprimiendo el brazo y dijo en voz demasiado alta: Ahora entramos y nos vestimos, \u00bfverdad?\n\nSi \u00e9l no era sordo. No la entendi\u00f3 enseguida s\u00f3lo porque ella gritaba tanto.\n\n\u00bfNo ha o\u00eddo el tel\u00e9fono?, grit\u00f3 ella. As\u00ed que ahora entramos tranquilamente en la habitaci\u00f3n, ande, entre, s\u00ed, y ah\u00ed, \u00bflo ve?, ah\u00ed est\u00e1 su camisa, ahora nos la ponemos y..., pero est\u00e1 h\u00fameda, c\u00f3mo habr\u00e1 sudado usted, \u00bfverdad?, entonces cogemos una nueva.\n\nAbri\u00f3 en\u00e9rgicamente un armario, mir\u00f3, meti\u00f3 la mano y De Vriend dijo: \u00a1No! No quer\u00eda, no quer\u00eda permitir que alguien abriera sin m\u00e1s su ropero y rebuscara en sus cosas... Pero ella ya estaba diciendo: \u00c9sta es una camisa preciosa, una bonita camisa blanca, \u00e9sta nos la ponemos ahora.\n\nLa se\u00f1ora Jos\u00e9phine le quit\u00f3 el cepillo que \u00e9l a\u00fan ten\u00eda en la mano, lo dej\u00f3 en la mesita; el pantal\u00f3n del traje se le hab\u00eda escurrido del brazo a De Vriend y estaba en el suelo. Ella le ayud\u00f3 a ponerse la camisa, al hacerlo vio otra vez el n\u00famero tatuado en su brazo, meti\u00f3 deprisa el brazo en la manga de la camisa, quiso decir \u00a1Bien!, pero no dijo nada.\n\nRecogi\u00f3 del suelo el pantal\u00f3n, se lo ofreci\u00f3. En silencio. \u00c9l se lo puso. Se abroch\u00f3 la camisa, y la correa del pantal\u00f3n. Ella mir\u00f3 alrededor, vio que los zapatos estaban junto a la cama, \u00e9l vio ad\u00f3nde miraba, fue a la cama, se sent\u00f3 y se calz\u00f3. La mir\u00f3, ella le mir\u00f3, luego \u00e9l se inclin\u00f3 y se at\u00f3 los zapatos. Se incorpor\u00f3, la mir\u00f3. Ella asinti\u00f3.\n\nLa se\u00f1ora Jos\u00e9phine era una experta cuidadora de ancianos. En sus casi veinte a\u00f1os de servicio hab\u00eda visto mucho. Y durante su formaci\u00f3n profesional tambi\u00e9n hab\u00eda hecho un curso de psicolog\u00eda y s\u00f3lo hac\u00eda dos a\u00f1os que hab\u00eda terminado su \u00faltimo curso de perfeccionamiento. Ella fue la m\u00e1s sorprendida cuando se oy\u00f3 decir de pronto: \u00bfAuschwitz?\n\n\u00c9l asinti\u00f3.\n\nQuiso levantarse. Pero no pudo. Se qued\u00f3 sentado en la cama.\n\nElla pens\u00f3 que hab\u00eda ido demasiado lejos. As\u00ed que dio un paso m\u00e1s: \u00bfC\u00f3mo era aquello? \u00bfQuiere contarme?\n\nSinti\u00f3 un horror que le imped\u00eda respirar. Por haber hecho esa pregunta.\n\nDe Vriend, sentado en la cama, la mir\u00f3, luego dijo: Form\u00e1bamos para el recuento. Form\u00e1bamos para el recuento. Eso era todo.\n\nCuando Jos\u00e9phine hubo abandonado la habitaci\u00f3n, De Vriend se qued\u00f3 un rato sentado en la cama, luego se levant\u00f3, pase\u00f3 por la habitaci\u00f3n, mir\u00f3 alrededor... y vio su cepillo.\n\nLentamente se desvisti\u00f3, cogi\u00f3 el cepillo, se ech\u00f3 una pernera del pantal\u00f3n sobre el brazo izquierdo, y desnudo en el balc\u00f3n empez\u00f3 a cepillar.\n\nEl jefe de gabinete Strozzi sab\u00eda, naturalmente, que el presidente de la Comisi\u00f3n no podr\u00eda en absoluto oponerse a una iniciativa que quer\u00eda dar brillo a la imagen de la Comisi\u00f3n y aumentar su prestigio. Por eso hab\u00eda garantizado al momento a Fenia Xenopoulou el apoyo del presidente. Una carte blanche. Pero Strozzi sab\u00eda tambi\u00e9n que ese extra\u00f1o proyecto iba a producir m\u00e1s problemas que ganancias. La idea del Jubilee Project era absurda, y aunque se la pod\u00eda justificar muy bien como hab\u00eda demostrado perfectamente la se\u00f1ora Xenopoulou, pol\u00edticamente era cualquier cosa menos oportuna. Por eso la carte blanche era una finta, un infalible truco de Strozzi, el viejo espadach\u00edn de la burocracia: cuando se quiere eliminar una idea, primero hay que aprobarla y prometer pleno apoyo. Despu\u00e9s de eso, cada cual se pone alegremente al descubierto. Lo mejor de ello es que, muchas veces, uno no tiene despu\u00e9s que asestar el golpe decisivo. Era un viejo chiste de esgrimidores: si consigues inducir a tu adversario a que se haga el harakiri, t\u00fa ya no tienes que atacar, sino s\u00f3lo tener cuidado de que no te caiga agonizante en los brazos. Y con Fenia Xenopoulou eso hab\u00eda funcionado una vez m\u00e1s: entusiasmada con su aprobaci\u00f3n, sin darse cuenta se hab\u00eda mostrado de acuerdo, naturalmente, con su propuesta de poner ese proyecto de la Comisi\u00f3n en conocimiento de los representantes de las naciones que hab\u00edan fundado la Comisi\u00f3n. \u00bfQu\u00e9 habr\u00eda podido aducir en contra? Al mismo tiempo, \u00e9l ya se hab\u00eda puesto de pie, d\u00e1ndole a entender as\u00ed que la entrevista hab\u00eda terminado. M\u00e1s tarde ella no podr\u00e1 decir que \u00e9l la atac\u00f3 por la espalda. Al contrario: fue cara a cara. Ahora \u00e9l s\u00f3lo ten\u00eda que ocuparse de que no cayera en sus brazos y le manchara de sangre el chaleco. Y para ello bastaba con una conversaci\u00f3n: con su amigo Attila, el jefe de protocolo del presidente del Consejo.\n\nEra una situaci\u00f3n absurda para una reuni\u00f3n: aquellos dos altos funcionarios en el Kitty O'Sheas's, el irish pub a espaldas del edificio Berlaymont, sentados con t\u00e9 helado ante una mesa pegajosa de cerveza derramada, rodeados de bebedores de Guinness y de jugadores de dardos que charlaban y vociferaban.\n\nEn cualquier caso, aqu\u00ed no pueden escucharnos, con este jaleo, dijo Attila en su simp\u00e1tico hungl\u00e9s, que es ingl\u00e9s con \u00e1-z\u00e1n-too h\u00fan-ga-rr\u00f3.\n\nStrozzi sonri\u00f3. Ten\u00eda desde hac\u00eda a\u00f1os una excelente base de conversaci\u00f3n con Attila, en exquisita sinton\u00eda hab\u00edan resuelto ya muchos problemas. Cuando surg\u00edan conflictos entre Comisi\u00f3n y Consejo, es decir, con mucha frecuencia, o cuando el presidente de la Comisi\u00f3n quer\u00eda algo del presidente del Consejo, es decir, no precisamente raras veces, entonces Strozzi prefer\u00eda hablar con Hidekuti y no con el jefe de gabinete del presidente del Consejo, Lars Ekel\u00f6f, ese sueco luterano de l\u00ednea dura para quien el conde barroco italiano era por definici\u00f3n un personaje inquietante. A la inversa, una vez hab\u00eda dicho Strozzi despectivamente sobre Ekel\u00f6f: En puntos conflictivos no puede uno entenderse con un hombre que en cada punto se siente moralmente superior y por eso entiende cada compromiso como traici\u00f3n a su moral. Y hab\u00eda a\u00f1adido con una sonrisa ir\u00f3nica: Nunca se puede atraer a Ekel\u00f6f a terreno descubierto porque \u00e9l s\u00f3lo consta de muro, \u00e9l es el muro en s\u00ed. Si se lo pudiera eludir, no habr\u00eda nada detr\u00e1s, s\u00f3lo un olor que se lleva el viento, los vapores del farise\u00edsmo.\n\nEl contraste entre el norte y el sur discurr\u00eda exactamente entre esos dos hombres que trabajaban en Bruselas, al norte y al sur de la rue de la Loi.\n\nY nosotros, los h\u00fangaros, nos v\u00e9-mos trri-turr-\u00e1dos \u00e9n-tre ambos. (Hidekuti.)\n\nHidekuti miraba ahora con preocupaci\u00f3n a los jugadores de dardos que estaban de pie a desagradable poca distancia de \u00e9l. Las flechas vuelan aqu\u00ed muy bajo, dijo.\n\nUn jugador de dardos le salud\u00f3, Hidekuti devolvi\u00f3 con la cabeza el saludo, corri\u00f3 la silla a un lado, luego otro de los jugadores levant\u00f3 el vaso de cerveza y brind\u00f3 a la salud de Hidekuti y de Strozzi.\n\nCome on, nos ponemos un poco m\u00e1s all\u00e1, dijo Strozzi, y luego: son ingleses. END brit\u00e1nicos. Desde el comienzo de las negociaciones para su salida, algunos se preparan aqu\u00ed con cerveza y dardos para el regreso a casa. Yo los prefiero a los ingleses que siguen trabajando mientras la salida no est\u00e9 definitivamente sellada pero que no hacen sino obstruir laboriosamente nuestro trabajo.\n\n\u00bfPor eso me has pedido que venga aqu\u00ed? \u00bfTienes problemas con funcionarios de mi casa?\n\nNo, dijo Strozzi, y le habl\u00f3 del Jubilee Project.\n\nHidekuti entendi\u00f3 al momento que ese proyecto ten\u00eda que granjearse mucho rechazo. No era tanto el hecho de que la Comisi\u00f3n planeara algo por su cuenta, contra las otras instituciones europeas o al menos sin incluirlas, aunque eso ya era desde luego muy problem\u00e1tico, no, era fundamentalmente la idea de hacer desfilar a testimonials que con sus biograf\u00edas y trayectorias vitales confirmaran que el nacionalismo hab\u00eda llevado a los mayores cr\u00edmenes de la historia de la humanidad, finalmente a Auschwitz, por lo que la obligaci\u00f3n moral de la Comisi\u00f3n ten\u00eda que ser trabajar para la superaci\u00f3n de las naciones. Que de la frase huera y repetida \u00abNunca m\u00e1s Auschwitz\u00bb se deduzca la necesidad de \u00absuperar el nacionalismo, en \u00faltimo t\u00e9rmino superar las naciones\u00bb y presentar eso a la opini\u00f3n p\u00fablica europea como exigencia moral y como misi\u00f3n pol\u00edtica de la Uni\u00f3n Europea: eso nunca lo aceptar\u00edan los jefes de Estado y de gobierno nacionales.\n\nTenemos expertos para todo, dijo Hidekuti. Podemos hacer llover y podemos hacerlo de forma que la Comisi\u00f3n se encuentre con el aguacero encima y sin paraguas.\n\nLo s\u00e9, dijo Strozzi. Por eso te lo cuento.\n\n\u00abNunca m\u00e1s Auschwitz\u00bb es bueno y correcto.\n\nS\u00ed.\n\nEso pod\u00e9is decirlo cada domingo en un discurso.\n\nS\u00ed. Para que no se olvide. No olvidar nunca, eso hay que repetirlo una y otra vez.\n\nExacto, pero eso no es un programa pol\u00edtico.\n\nLa moral no ha sido nunca un programa pol\u00edtico.\n\nSobre todo si la moral produce conflictos.\n\nExacto. El Consejo no podr\u00eda aceptarlo nunca: la superaci\u00f3n de las naciones. Significar\u00eda la guerra. Contra la Comisi\u00f3n. Y la revuelta popular contra Europa en todos los pa\u00edses.\n\nExacto.\n\n\u00bfEntonces?\n\nTe he comprendido. A-plas-ta-re-mos ese proyecto antes de que salga a la luz p\u00fablica.\n\nStrozzi sab\u00eda que pod\u00eda fiarse de su amigo Attila.\n\nY Attila Hidekuti trabaj\u00f3 muy bien. Tampoco le cost\u00f3 tanto. Una firma, una llamada telef\u00f3nica, en el fondo chasquear los dedos. Con eso empuj\u00f3 una bola que empuj\u00f3 a otra y as\u00ed sucesivamente. De ah\u00ed surgi\u00f3 una din\u00e1mica propia que pronto nadie supo qui\u00e9n hab\u00eda desencadenado, pero que segu\u00eda proporcionando energ\u00eda ininterrumpidamente hasta que la \u00faltima bola rod\u00f3 en la nada, en el out, en un agujero negro. De eso se trataba. \u00c9se era el trabajo de Hidekuti. Al final quien lo hab\u00eda provocado todo era s\u00f3lo una de esas bolas que empujaba a otra, en el fondo un bolindre o s\u00f3lo un grano peque\u00f1ito, finalmente algo invisible, un \u00e1tomo: el n\u00facleo fisionable de inconcebible energ\u00eda pol\u00edtica. Ya al d\u00eda siguiente el ministro de Exteriores h\u00fangaro llam\u00f3 a su \u00abapreciado colega y querido amigo\u00bb, el ministro de Exteriores austriaco, y le inform\u00f3 de que la Comisi\u00f3n, con el pretexto de celebrar un aniversario, quer\u00eda iniciar un proceso que abocar\u00eda en la supresi\u00f3n de las naciones europeas.\n\n\u00bfT\u00fa sabes lo que eso significa, querido amigo, que la UE decrete que Austria no es una naci\u00f3n?, le pregunt\u00f3. No se pod\u00eda decir: hip\u00f3crita, porque la naci\u00f3n era realmente su santuario. Pero el propio, el h\u00fangaro. Que Austria fuera una naci\u00f3n o un accidente de trabajo de la historia, reducido con raz\u00f3n, por su megaloman\u00eda, a un peque\u00f1o Estado de mestizos, eso le daba igual en \u00faltimo t\u00e9rmino, aunque \u00aben privado\u00bb, como le gustaba decir, se inclinaba m\u00e1s bien por lo segundo, por el accidente. Pero \u00e9l sab\u00eda que ten\u00eda un aliado si, como lo hab\u00eda formulado frente a su jefe de gobierno, \u00able rascaba un poco los huevos\u00bb al nacionalismo del vecino.\n\nUnos ochenta y seis mil millones de neuronas se comunicaron, en milisegundos tuvieron lugar en miles de c\u00e9lulas complejos procesos el\u00e9ctricos, sustancias qu\u00edmicas transmisoras cumplieron con su deber y las sinapsis funcionaron, en resumen: el ministro de Exteriores austriaco reflexion\u00f3. Y en un abrir y cerrar de ojos hab\u00eda sondeado las alternativas y tomado una decisi\u00f3n. Alternativa uno: permanecer inactivo de momento, esperar hasta que la Comisi\u00f3n hiciera p\u00fablico ese proyecto, y luego saltar a la palestra como defensor de la naci\u00f3n austriaca \u00abcontra la UE\u00bb. Las sinapsis se pusieron al rojo de puro placer, pero \u00bfqu\u00e9 era aquello? Empezaban a emitir una luz roja intermitente. A la clientela anti-EU ya le hab\u00eda prestado \u00e9l un buen servicio con sus intervenciones relacionadas con la pol\u00edtica europea con los refugiados; un paso m\u00e1s, entrar en el terreno del rechazo b\u00e1sico de la idea europea (y estaba bien que adem\u00e1s \u00e9sta fuese tan poco clara), no s\u00f3lo perturbar\u00eda la \u00abeconom\u00eda\u00bb, sino que lo situar\u00eda en las proximidades del partido de los bravucones de derechas que con su nacionalismo de \u00abAustria primero\u00bb ten\u00edan cada vez m\u00e1s partidarios. \u00c9l no quer\u00eda ser el marinero al lado del patr\u00f3n, quer\u00eda ser popular sin el descr\u00e9dito del populismo, por tanto estaba claro: si la naci\u00f3n y el nacionalismo se convert\u00edan en un gran tema fundamental y p\u00fablico, \u00e9l llevaba todas las de perder. Por eso, alternativa dos: ten\u00eda que impedir ese proyecto. Si pod\u00eda impedir una discusi\u00f3n b\u00e1sica sobre la naci\u00f3n y su defensa, entonces en cada cuesti\u00f3n concreta \u00e9l pod\u00eda aparecer como representante de los intereses austriacos, de los intereses de los votantes nacionales y, al mismo tiempo, tambi\u00e9n como europeo: entonces el patr\u00f3n ser\u00eda \u00e9l.\n\nDio las gracias a su querido amigo, el colega h\u00fangaro, prometi\u00f3 \u00abevidentemente\u00bb adecuada colaboraci\u00f3n, convoc\u00f3 a los miembros de su gabinete y reparti\u00f3 las tareas. Todos salieron corriendo de su despacho, s\u00f3lo el jefe de prensa se qued\u00f3 y se aclar\u00f3 la garganta. Record\u00f3 al ministro que a\u00fan ten\u00edan que rellenar el cuestionario.\n\n\u00bfQu\u00e9 cuestionario?\n\nPara Madonna, esa revista femenina. Donde tuvimos la semana pasada la sesi\u00f3n de fotos.\n\nAh, s\u00ed. Entonces rell\u00e9nalo.\n\nMe gustar\u00eda leerlo con usted, se\u00f1or ministro. Las preguntas de car\u00e1cter privado, por ejemplo: libro preferido.\n\n\u00bfQu\u00e9 propones t\u00fa?\n\nEs tradici\u00f3n en Austria que los pol\u00edticos escojan El hombre sin atributos. Por debajo de eso, en realidad, no es posible. Y tab\u00fa es, en cualquier caso, un autor vivo. La gente no quiere a ninguno que viva.\n\nBueno, vale, entonces somos muy austriacos. El hombre sin atributos. Por lo que s\u00e9, tambi\u00e9n le gustaba a Kreisky.\n\nY a Sinowatz, a Klima y a Gusenbauer.\n\n\u00bfS\u00f3lo los socialistas?\n\nNo, tambi\u00e9n a Mock, Khol y hasta a Molterer.\n\nBueno, no puedo quedar por debajo.\n\nY ahora: \u00bfel personaje preferido de la literatura?\n\nPero \u00bfqu\u00e9 pasa con esa revista femenina? \u00bfSon germanistas todas las que trabajan ah\u00ed?\n\nNo, se\u00f1or ministro. Son s\u00f3lo esas dos preguntas. Luego viene ya m\u00fasica y comida.\n\nAh, bueno. As\u00ed que personaje preferido. \u00bfC\u00f3mo se llama el de El hombre sin atributos?\n\nUlrich. Pero no lo aconsejar\u00eda. Lo dicho: sin atributos. Adem\u00e1s he buscado en Google: tiene un problema de incesto. Propongo: Arnheim.\n\n\u00bfQui\u00e9n es \u00e9se?\n\nLe pega, se\u00f1or ministro. Se le considera un \u00abgran hombre\u00bb, pol\u00edtico e intelectual. Y tiene una intensa relaci\u00f3n amorosa plat\u00f3nica.\n\n\u00bfEn serio?\n\nEn El hombre sin atributos.\n\n\u00a1Fen\u00f3meno!\n\nAl d\u00eda siguiente, el gobierno polaco dio instrucciones a los funcionarios polacos de los gabinetes para que truncaran esa \u00abcampa\u00f1a\u00bb de la Comisi\u00f3n Europea, dirigida contra el orgullo de la naci\u00f3n polaca. Sobre todo hab\u00eda que hacer observar a la DG COMM que el campo de exterminio de Auschwitz era un crimen alem\u00e1n y por eso un problema exclusivamente alem\u00e1n. La Rep\u00fablica Federal Alemana estaba cordialmente invitada a desmontar del suelo polaco el campo de exterminio alem\u00e1n y exponerlo en Alemania como museo. En cualquier caso, una cultura de memoria de cr\u00edmenes cometidos por fuerzas de ocupaci\u00f3n en suelo polaco no serv\u00eda para hacer de baldaqu\u00edn moral sobre una comunidad econ\u00f3mica.\n\nDel ministro de Exteriores austriaco lleg\u00f3 una nota para el presidente del Consejo Europeo que dejaba claro, sin lugar a equ\u00edvocos, que la Rep\u00fablica de Austria estaba a favor y en contra: apoyaba la iniciativa de la Comisi\u00f3n Europea, pero no pod\u00eda darle su aprobaci\u00f3n en la forma proyectada. La Oficina de Exteriores, en nombre del gobierno federal austriaco, apoyaba sin reservas la iniciativa de la Comisi\u00f3n Europea, \u00abComunicar mejor Europa a los ciudadanos\u00bb, pero en Austria no era comunicable que un campo polaco en el que hab\u00eda habido miles de v\u00edctimas austriacas promoviera el cuestionamiento de la naci\u00f3n austriaca.\n\nEl embajador de la representaci\u00f3n permanente de la Rep\u00fablica Checa en la Uni\u00f3n Europea hizo llegar una nota diplom\u00e1tica de protesta formulada con m\u00e1s acritud: el gobierno checo no permitir\u00eda que la Uni\u00f3n Europea proyectara una campa\u00f1a de la denominada \u00abconfrontaci\u00f3n cr\u00edtica con la historia\u00bb, en la que, una vez m\u00e1s, se tachar\u00eda a Chequia del mapa. Para ello no hab\u00eda ni pod\u00eda haber ning\u00fan mandato.\n\nPocas horas m\u00e1s tarde lleg\u00f3 de la representaci\u00f3n permanente de Eslovaquia un escrito de contenido similar.\n\nAttila Hidekuti sonri\u00f3. Los pa\u00edses peque\u00f1os, como era de esperar, eran los primeros en saltar a la palestra cuando se cuestionaba a su naci\u00f3n... \u00bfqu\u00e9?, \u00bfla identidad?, \u00bfel honor?, \u00bfo incluso el derecho a existir? En eso se pod\u00eda confiar. Con eso se pod\u00eda trabajar. La gran pregunta, la definitiva, era c\u00f3mo reaccionar\u00eda Alemania. Y Francia. Inglaterra estaba fuera de juego aunque todav\u00eda anduvieran movi\u00e9ndose por el terreno de juego. Hidekuti consideraba posible que el Reino Unido diera instrucciones a sus funcionarios de la casa para que apoyaran el proyecto y conminaran a hacerlo p\u00fablico para despu\u00e9s poderlo explotar en pol\u00edtica interior como una prueba m\u00e1s de la necesidad del Brexit. Inglaterra, pens\u00f3 Hidekuti, se pod\u00eda utilizar como otro medio de coacci\u00f3n contra el Arca y la DG COMM para detener enseguida el proyecto, antes de que se hiciera p\u00fablico.\n\nLars Ekel\u00f6f se estaba conteniendo cuando entr\u00f3 en el despacho de Hidekuti. Hab\u00eda interiorizado hasta tal punto la exigencia de ser siempre y en todo lugar absolutamente correcto que s\u00f3lo durante un momento tuvo el impulso de entrar como una tromba en la habitaci\u00f3n y gritarle: \u00ab\u00bfQu\u00e9 co\u00f1o es esto?\u00bb. Pero nunca se permit\u00eda emociones incontroladas y lenguaje soez que pudieran insultar u ofender. Nunca. Ten\u00eda, naturalmente, la sospecha de que en esas extra\u00f1as protestas que, procedentes de los ministerios de Asuntos Exteriores y de las embajadas de algunos Estados miembros, llegaban al gabinete del presidente, Hidekuti ten\u00eda metidas las manos de una manera u otra. Ese h\u00fasar h\u00fangaro, de ojos que relampagueaban siempre llenos de picard\u00eda y esa sonrisa fofa sobre la papada, siempre ten\u00eda las manos metidas en todas partes. No pod\u00eda probarlo pero ten\u00eda la sospecha de que Hidekuti siempre inventaba problemas con cuya soluci\u00f3n se daba luego importancia ante el presidente. Y todo ello, cada vez, sin que \u00e9l, Ekel\u00f6f, el jefe de gabinete, se enterase. Respir\u00f3 hondo, entr\u00f3 y dijo: Tengo un peque\u00f1o problema, estoy seguro de que t\u00fa puedes ayudarme.\n\nHidekuti pod\u00eda.\n\nUna persona especialmente activa se est\u00e1 dando importancia en la Comisi\u00f3n, explic\u00f3. Pero ya he hablado con el presidente. De momento estamos a la espera. El asunto se estrangular\u00e1 por s\u00ed solo.\n\nLars Ekel\u00f6f no era el tipo que esperaba a ver c\u00f3mo algo \u00abse estrangulaba\u00bb. \u00bfQu\u00e9 palabras insoportables eleg\u00eda otra vez el jefe de protocolo? Trat\u00f3 de aclarar el asunto, y eso acarre\u00f3 que por lo pronto la se\u00f1ora Atkinson tuviera problemas.\n\nHidekuti sonri\u00f3. Todo marchaba como \u00e9l lo hab\u00eda previsto.\n\nQuien ama la libertad y quien ama la verdad ya no sabe amar. Lo hab\u00eda dicho un d\u00eda su abuelo, y \u00c9mile Brunfaut, colegial en aquella \u00e9poca, se qued\u00f3 impresionado sin haber comprendido exactamente por qu\u00e9. Hab\u00eda reflexionado largo tiempo sobre esa frase, como sobre un enigma que le inquietaba enormemente, y por eso sin duda se le hab\u00eda quedado grabada. Brunfaut ve\u00eda delante de \u00e9l al abuelo, c\u00f3mo contaba algo y finalmente dec\u00eda aquella frase, su rostro surcado de arrugas y malhumorado que el peque\u00f1o \u00c9mile entonces entend\u00eda tan mal, como expresi\u00f3n de una intimidante arrogancia y de falta de empat\u00eda: si entonces hubiera conocido esos t\u00e9rminos. Probablemente hab\u00eda hablado el abuelo del tiempo de la resistencia, de qu\u00e9 si no, y de que la desconfianza, radical desconfianza, era un seguro de vida, no bueno, pero el \u00fanico. Uno pod\u00eda protegerse relativamente a s\u00ed mismo y a las personas m\u00e1s allegadas si se compart\u00eda con ellos lo menos posible y no se confiaba ni en aquellos a los que uno amaba. Maravillosos y valientes hombres y mujeres se vieron traicionados por amigos, hermanos, padres y hasta por los propios hijos, por personas que los quer\u00edan. El amor no era un espacio de libertad y no ofrec\u00eda protecci\u00f3n.\n\nM\u00e1s tarde, su abuelo hab\u00eda muerto hac\u00eda ya tiempo, Brunfaut hab\u00eda empezado a comprender poco a poco la frase: cuando se hizo polic\u00eda. Cuando aprendi\u00f3 a ser desconfiado por principio, a no creer nada de lo que le contaban, a considerar toda apariencia como encubrimiento intencionado y cada declaraci\u00f3n r\u00e1pida y sin rodeos, de entrada, como un intento de enmascarar algo. Pero se hab\u00eda jurado no aceptar esa deformaci\u00f3n profesional, no permitir en modo alguno que influyera en su vida privada, en su relaci\u00f3n con las personas que \u00e9l quer\u00eda.\n\nComo es natural, no se piensa cada d\u00eda de la vida en un prop\u00f3sito de ese g\u00e9nero. Pero ahora Brunfaut ten\u00eda motivos para pensar en \u00e9l, y se ufanaba de haber obrado, en el fondo, bastante bien: amaba tiernamente y sin desconfianza a las personas vinculadas a \u00e9l, amaba sin temor la libertad, y amaba con confianza b\u00e1sica e inconmovible la verdad, ya fuera en forma de sinceridad frente a las personas queridas, ya fuera como resultado de pesquisas e indagaciones, y, s\u00ed, tambi\u00e9n, incluso como exigencia de la prensa liberal.\n\nAl mismo tiempo ten\u00eda que confesarse \u2014\u200cy ahora ese pensamiento le escandalizaba y confund\u00eda\u2014 que todo eso quiz\u00e1 ya no era cierto. \u00bfQue \u00e9l amaba? \u00bfDe verdad? \u00bfNo ten\u00eda que confesarse que ahora tendr\u00eda que decir: hab\u00eda amado?\n\nYa no pod\u00eda amar sin reservas. De un golpe hab\u00eda olvidado c\u00f3mo se hac\u00eda. \u00bfPod\u00eda ser eso verdad?\n\nLo ocurrido en el cementerio. Le hab\u00eda conmocionado. Y no fue s\u00f3lo el cerdo, que al principio lo puso en tal estado de confusi\u00f3n y de temor, no, era m\u00e1s bien el hecho de que despu\u00e9s, con el pantal\u00f3n roto, con dolor de espalda y rasgu\u00f1os en la mano, a\u00fan vag\u00f3 media hora larga sin encontrar a Philippe, menos a\u00fan al \u00abamigo\u00bb de \u00e9ste, el cual, sin embargo, hab\u00eda sido la raz\u00f3n de la cita en aquel lugar. Luego hab\u00eda encontrado un banco, se hab\u00eda sentado y llamado por tel\u00e9fono varias veces a Philippe, pero s\u00f3lo oy\u00f3 el contestador autom\u00e1tico. Y luego pas\u00f3 adem\u00e1s un viejo a su lado, se sent\u00f3 junto a \u00e9l en el banco y dijo: \u00bfHabla usted ya con los muertos?\n\nTodo eso fue horrible, Brunfaut huy\u00f3, ahora realmente a la carrera, por toda la avenida, pasando tambi\u00e9n junto a la tumba de su abuelo, corriendo y jadeando hasta la salida y hasta su coche. Con infernales pinchazos en el costado, como si un gran signo de interrogaci\u00f3n se clavara, a manera de hoz, en el entramado de su alma, un dolor situado m\u00e1s en lo hondo que los rasgu\u00f1os, un dolor al que pudo darle finalmente un nombre cuando estuvo de nuevo en su casa y metido en la ba\u00f1era. Lo que le dol\u00eda era la s\u00fabita y honda desconfianza, mejor dicho: la p\u00e9rdida de confianza.\n\nIncluso su desconfianza profesional como polic\u00eda se hab\u00eda basado en una confianza b\u00e1sica: la confianza en el Estado de derecho. Bueno, siempre hab\u00eda habido intervenciones pol\u00edticas, cuando hombres influyentes estaban comprometidos en asuntos delicados, pero en el fondo eso era infantil, pod\u00eda obstruir los molinos de la justicia pero no socavar a la larga el derecho, y mucho menos a\u00fan trat\u00e1ndose de delitos oficiales como el asesinato. Pero el encubierto crimen del Atlas ya hab\u00eda quebrantado su confianza m\u00e1s de lo que \u00e9l quer\u00eda confesarse a s\u00ed mismo. La cuesti\u00f3n era c\u00f3mo se manejaba aquello: \u00bfcomo el abuelo? \u00bfO como Philippe? Y eso era lo que ahora le dol\u00eda tanto: que de pronto ya no se fiaba de Philippe. De su mejor amigo, del padre de su ahijada Jo\u00eblle. De pronto lo vio actuando de modo que daba lugar a sospechas; todo lo que le hab\u00eda contado era difuso, sobre la OTAN y el Vaticano, cuentos de miedo, apropiados para abandonar inmediatamente el caso, y de pronto viene con nuevas informaciones, cu\u00e1les, eso lo aclarar\u00eda un informante en el cementerio: y luego no aparecieron ni \u00e9l ni el misterioso informante, y de pronto tampoco lo localizaba por tel\u00e9fono.\n\nBrunfaut empuj\u00f3 el pato de pl\u00e1stico que se balanceaba entre sus rodillas en el agua, y se pregunt\u00f3 si Philippe no tendr\u00eda quiz\u00e1 la misi\u00f3n, en primer lugar, de convencerle de que proseguir con la investigaci\u00f3n no ten\u00eda sentido y, todo lo m\u00e1s, le pon\u00eda en peligro, y luego, de comprobar con el cuento chino del informante si realmente se hab\u00eda apartado del caso o si, lleno de curiosidad, segu\u00eda ocup\u00e1ndose de \u00e9l.\n\nEl ba\u00f1o le vino bien. No le aliviaba el dolor pero le relajaba. Ten\u00eda la sensaci\u00f3n de pensar ahora con claridad, pero precisamente lo que estaba pensando le intranquilizaba. Hizo ondas, el pato bailaba estoicamente sobre el agua, chocaba contra su vientre, se daba la vuelta y se balanceaba entre sus rodillas, le dio una patada, el pato dio un brinco para mecerse despu\u00e9s otra vez en el agua.\n\nA Brunfaut nunca le hab\u00eda gustado el fiscal. Lo respetaba, s\u00ed. Pero, al mismo tiempo, lo despreciaba. Un hombre que se identificaba con el Estado tan ciegamente que confund\u00eda a los m\u00e1s poderosos e influyentes del Estado con el Estado y que por eso, s\u00f3lo en casos excepcionales, claro, estaba incluso dispuesto, en inter\u00e9s del Estado, a violar el derecho que el Estado deb\u00eda garantizar. Pero \u00bften\u00eda Brunfaut que apreciarlo para entenderlo? No. Siempre que aparec\u00eda, estaba claro, hab\u00eda de por medio determinados intereses. Y esos intereses estaban claros. En el fondo aquello era siempre verdadero y esa verdad no necesitaba ni una relaci\u00f3n de confianza ni amor. \u00a1Oh, Philippe! Brunfaut dio un golpe en el agua con la palma de la mano. Yo confiaba en ti. \u00bfMe has enga\u00f1ado?\n\nEl agua se enfriaba y Brunfaut se pregunt\u00f3 si no era v\u00edctima de un gran espejismo a causa de un desgraciado azar. Quiz\u00e1 no era atinada su sospecha y Philippe segu\u00eda siendo el fiel amigo al que pod\u00eda querer y en el que pod\u00eda confiar.\n\nPero la desconfianza se hab\u00eda instalado en su coraz\u00f3n, estaba all\u00ed y no pod\u00eda desterrarla mediante una decisi\u00f3n.\n\nEl pato hab\u00eda sido el recept\u00e1culo de un champ\u00fa, de un champ\u00fa infantil, \u00abgarantizado sin l\u00e1grimas\u00bb, de ni\u00f1o le hab\u00eda gustado ese pato-champ\u00fa y siempre lo guardaba cuando se acababa, m\u00e1s tarde incluso super\u00f3 todos los traslados y cambios en su vida. El pato ten\u00eda en la cola un cierre de rosca por el que sal\u00eda el champ\u00fa.\n\nBrunfaut apret\u00f3 el pato con los dos pies dentro del agua. Cuando retiraba los pies, el pato daba un salto hacia arriba, se balanceaba y flotaba.\n\nNo pod\u00eda hundirse. Siempre se quedar\u00eda arriba del todo. De eso pod\u00eda uno fiarse. Brunfaut desenrosc\u00f3 el tap\u00f3n, apret\u00f3 el pato hacia abajo, el agua empez\u00f3 a meterse dentro, \u00e9l coloc\u00f3 los brazos sobre el borde de la ba\u00f1era, estir\u00f3 las piernas y vio c\u00f3mo el pato se hund\u00eda poco a poco.\n\nEl profesor Erhart casi hab\u00eda llegado tarde otra vez. Tom\u00f3 el metro como siempre, se baj\u00f3 en Schuman, pero la salida a Justus Lipsius estaba cerrada. As\u00ed pues, tom\u00f3 la salida a Berlaymont. Se encontr\u00f3 entonces no s\u00f3lo en el lado equivocado de la rue de la Loi, sino tambi\u00e9n por debajo de la calle, en esa extra\u00f1a hondonada en la que estaba el edificio Berlaymont. Cuando hab\u00eda rodeado el vallado y subido otra vez a la calle, comprob\u00f3 que no era posible cruzar la rue de la Loi. Hab\u00edan colocado verjas de contenci\u00f3n a lo largo de la acera, detr\u00e1s hab\u00eda veh\u00edculos militares. Polic\u00edas militares hac\u00edan con impaciencia se\u00f1ales de continuar a la gente que sal\u00eda del metro. \u00a1Sigan! \u00a1No se detengan!\n\nYo tengo que ir ah\u00ed enfrente, dijo Erhart, tengo que ir a...\n\n\u00a1Siga usted! \u00a1Siga hacia delante!\n\n\u00c9l habr\u00eda preferido subir a la glorieta Schuman para llegar desde all\u00ed a la acera de Justus Lipsius, pero Erhart entendi\u00f3 los gestos que hac\u00eda el polic\u00eda como que deb\u00eda ir en la otra direcci\u00f3n, justo all\u00ed abajo, donde ya se extraviara la \u00faltima vez. Avanz\u00f3 r\u00e1pidamente, balanceando los brazos, en la mano derecha llevaba su vieja cartera escolar, que con ese paso apresurado le golpeaba continuamente la rodilla o la corva. Ten\u00eda que ir hasta Maalbeek, all\u00ed podr\u00eda por fin cambiar de acera. La pr\u00f3xima vez, pens\u00f3, no viajar\u00eda en el metro hasta Schuman, sino que se bajar\u00eda ya en la estaci\u00f3n de Maalbeek. Si es que hab\u00eda una pr\u00f3xima vez. Despu\u00e9s de la keynote que pronunciar\u00eda dentro de diez minutos. Retrocedi\u00f3 todo el camino recorrido, hasta el terreno en obras junto al edificio Justus Lipsius, donde busc\u00f3 el pasadizo entre las rejas de la obra y las placas de contrachapado, el corredor que llevaba al R\u00e9sidence Palace, donde ten\u00eda lugar el meeting. Todo hab\u00eda cambiado desde entonces, como es natural, solamente no hab\u00eda cambios en el caos general. Se volvi\u00f3 hacia la izquierda, dio despu\u00e9s unos pasos a la derecha, s\u00f3lo ve\u00eda alambradas, a su espalda los veh\u00edculos militares, delante de \u00e9l la alambrada que imped\u00eda el paso, se sent\u00eda como un animal prisionero o acosado. Jadeaba, apretaba su cartera contra el pecho, la cartera con la ponencia que en el fondo era un discurso sobre la libertad. Sobre liberaci\u00f3n. Cuando menos, un discurso de la autoliberaci\u00f3n.\n\nErhart lleg\u00f3 el \u00faltimo, naturalmente. No realmente demasiado tarde, pero el \u00faltimo. Ahora estamos todos, dijo gozoso Mr. Pinto. \u00bfQuiere un caf\u00e9 antes de empezar? \u00bfAgua?\n\nS\u00ed, muchas gracias, dijo Erhart. Mir\u00f3 alrededor, salud\u00f3 a unos y a otros, le devolvieron el saludo. Qu\u00e9 perfectos eran. Ni la menor part\u00edcula de polvo de la calle en sus zapatos: \u00bfconocer\u00edan otro camino? \u00bfNo hab\u00edan tenido que cruzar la obra? Sin arrugas en pantalones y chaquetas, en ninguna camisa la menor mancha de sudor. \u00bfC\u00f3mo hab\u00edan llegado hasta all\u00ed? Hac\u00eda un calor tan horrible fuera que no hac\u00eda falta rodear las vallas, como \u00e9l, uno sudaba ya caminando despacio.\n\nMister Pinto pregunt\u00f3: \u00bfEstar\u00eda listo ahora, profesor?\n\nEl profesor Erhart estaba listo. Su vida era un perpetuo estar listo. Los tiempos cambian, pero en el fondo s\u00f3lo cambia lo que puede cambiar. Termin\u00f3 de beberse el caf\u00e9, hizo un gesto de asentimiento.\n\nCuando le invitaron por primera vez a un congreso, siendo un jovenc\u00edsimo ayudante de c\u00e1tedra, se hab\u00eda comprado especialmente para esa ocasi\u00f3n un traje nuevo. Hab\u00eda logrado participar, con una ponencia, en el Foro Cient\u00edfico de Alpbach, el pueblo alpino del Tirol donde cada a\u00f1o se reun\u00edan \u00e9lites de la econom\u00eda, famosos cient\u00edficos de diversas disciplinas y artistas de \u00e9xito para intercambiar ideas. Su profesor, el doctor Schneider, le hab\u00eda procurado esa invitaci\u00f3n para hacerle un favor o al menos para mantenerle en buena disposici\u00f3n, al fin y al cabo Erhart ya hab\u00eda escrito algunos art\u00edculos que luego publicaba el profesor Schneider con su nombre. Erhart se hab\u00eda sentido honrado y lisonjeado, y s\u00f3lo m\u00e1s tarde tom\u00f3 conciencia de a qu\u00e9 rid\u00edcula docilidad le hab\u00eda conducido ese horizonte de honor: no iba a dar una conferencia p\u00fablica, sino s\u00f3lo una breve ponencia en un grupo de trabajo; pero, de todos modos, estar\u00eda presente en Alpbach y, si se mostraba dispuesto, tomar\u00eda contacto con personas famosas e influyentes. As\u00ed pues, quer\u00eda dar la mejor impresi\u00f3n posible. Por eso el traje nuevo, su primer traje con chaleco, y zapatos nuevos. Embetun\u00f3 adem\u00e1s esos zapatos que no se hab\u00eda puesto nunca y les sac\u00f3 brillo. Y luego se encontr\u00f3 en una sala donde hab\u00eda caf\u00e9 y bollos de reques\u00f3n, los zapatos nuevos le hac\u00edan da\u00f1o, y en su traje nuevo se sent\u00eda como disfrazado, no era \u00e9l quien, literalmente, lo llevaba.\n\nVio c\u00f3mo sir Karl Popper bajaba la vista a las inclinadas espaldas de pol\u00edticos y funcionarios austriacos... que de pronto se fueron en desbandada hacia el secretario de Estado estadounidense para, inclin\u00e1ndose m\u00e1s a\u00fan, recoger en el hueco de la mano la ceniza de su puro.\n\nY luego Erhart lo vio: Armand Moens.\n\nEl primer congreso de Erhart. Y la \u00faltima aparici\u00f3n en p\u00fablico de Armand Moens antes de su muerte, pocas semanas despu\u00e9s. El \u00fanico encuentro entre maestro y disc\u00edpulo, Erhart quiz\u00e1 habr\u00eda dicho entonces incluso: entre Dios y ap\u00f3stol... Y de lo que hablaron fue de vestimenta.\n\nErhart se qued\u00f3 asombrado de la dejadez con que aquel hombre famoso iba vestido. Llevaba un ra\u00eddo pantal\u00f3n de pana, un jersey gris, con manchas (\u00bfde caf\u00e9?) en la pechera, encima una barata chaqueta azul de nailon.\n\nErhart se acerc\u00f3 para presentarse y hacer patente su reverencia al venerado cient\u00edfico.\n\nMoens era viejo y estaba enfermo. En las \u00faltimas. Erhart se arrepinti\u00f3 al momento de haberle hablado. Le habr\u00eda gustado discutir con \u00e9l sobre su libro, El final de la econom\u00eda nacional y el sistema econ\u00f3mico de una rep\u00fablica posnacional, pero cuando Erhart estuvo frente a \u00e9l, al momento vio con claridad que eso ya no era posible. Aquel cutis amarillo con manchas marrones, los ojos acuosos, los labios h\u00famedos de saliva..., lleg\u00f3 un estudiante con un libro de Moens y le pidi\u00f3 que lo firmara. Fue insoportable para Erhart ver cu\u00e1nto tiempo necesit\u00f3 Moens para escribir con letra temblorosa su nombre. Erhart no recordaba lo que \u00e9l dijo despu\u00e9s, s\u00f3lo sab\u00eda que Moens no contest\u00f3 a eso, sino que dijo: Todos parecen aqu\u00ed tan disfrazados.\n\nErhart: \u00bfC\u00f3mo dice?\n\n\u00bfNo lo ve usted? Toda esta gente con los trajes que llevan en Viena, en Par\u00eds y en Oxford \u2014\u200cten\u00eda dificultades para hablar\u2014, \u00e9stos, estos disfraces, aqu\u00ed, ante la madera de cembro y ante toda esta est\u00e9tica alpina... \u00a1disfrazados! \u00a1Parecen disfrazados! Y los otros, los que han venido con loden y traje regional, porque Tirol, pensaban, chaqueta tirolesa, porque Tirol... parecen verdaderamente disfrazados. \u00a1M\u00edrelo usted! S\u00f3lo gente disfrazada. Un carnaval cient\u00edfico.\n\nErhart no sab\u00eda qu\u00e9 responder, finalmente dijo: No deber\u00edamos disfrazarnos nunca.\n\nY Armand Moens dijo con voz asombrosamente alta y brusca: \u00a1No!\n\nDe regreso en Viena, en el instituto, Alois Erhart escribi\u00f3 en un papel:\n\n\u00ab\u00a1NO!\u00bb\n\nArmand Moens\n\n... y lo clav\u00f3 con una chincheta en la pared delante de su escritorio. Sab\u00eda que eso era infantil, pero que tampoco lo era, al mismo tiempo. Era una peque\u00f1a sacudida el\u00e9ctrica. \u00ab\u00a1No!\u00bb nunca era equivocado. \u00bfNunca? \u00a1Nunca!\n\nSe aboton\u00f3 su arrugada chaqueta, para tapar las manchas de sudor de la camisa, y sigui\u00f3 a Mr. Pinto a la sala donde iba a dar su keynote.\n\nCuando Kass\u00e1ndra Mercouri iba ese d\u00eda a la oficina en bicicleta, se encontr\u00f3, como casi siempre, con Bohumil en la rue d'Arenberg. Kass\u00e1ndra estaba excitada, impaciente, quer\u00eda soltarlo todo enseguida, contar su fin de semana, estaba tan orgullosa de lo que hab\u00eda averiguado, era asombroso, era tan importante..., pero en lugar de eso dijo: \u00bfQu\u00e9 tienes? \u00bfQu\u00e9 pasa?\n\nEl siempre tan alegre, turbulento, infantilmente temerario ciclista Bohumil pedaleaba en silencio, con rostro amargado, no sacaba del bolsillo ninguna de sus pegatinas \u00a1EST\u00c1 USTED ESTORBANDO! cuando ve\u00eda un coche aparcado en el carril bici. Ella siempre se inquietaba cuando \u00e9l hac\u00eda esas maniobras arriesgadas, pero ahora estaba acongojada porque no las hac\u00eda.\n\n\u00a1Venga, cuenta de una vez! \u00bfQu\u00e9 pasa?\n\nHe estado el fin de semana en casa. En Praga.\n\nKass\u00e1ndra tuvo que colocarse detr\u00e1s de Bohumil cuando ambos rodearon un coche que estaba en el segundo carril, mientras un autob\u00fas pasaba a toda mecha a su izquierda. Luego se puso de nuevo a su altura, Bohumil guardaba silencio.\n\nAs\u00ed que has estado en Praga. \u00bfVisita a la familia? Allons! \u00bfQu\u00e9 ha pasado?\n\nLa famille est la mort de la raison!\n\n\u00a1Bohumil!\n\nEn el fondo nada especial. No me ha sorprendido. O digamos m\u00e1s bien: ahora me asombro de que me haya sorprendido. He estado con mis padres. Eh bien! Los padres son los padres. Luego quise ver a mi hermana, comer con ella en el U Zav\u00ea\u0161enyho, pato con lombarda, como siempre. \u00a1No quiso!\n\n\u00bfTu hermana no quer\u00eda verte?\n\nNo en el restaurante y s\u00f3lo ella y yo. Quer\u00eda que fuera a su casa.\n\nPero eso es muy amable.\n\nNo. Ella sabe cu\u00e1nto me gusta el pato del U Zav\u00ea\u0161enyho. Y siempre ha sido as\u00ed. All\u00ed nos cit\u00e1bamos, all\u00ed com\u00edamos y nos cont\u00e1bamos todo, todas las novedades, todos los secretos, todos los cotilleos. No, yo no quer\u00eda ir a su casa. Se ha casado hace poco y...\n\n\u00bfConoces a su marido? \u00bfEntonces era una invitaci\u00f3n para estar con los dos?\n\nElla dijo: No viniste a nuestra boda. Dijo: S\u00e9 por qu\u00e9, claro. Ahora vienes a casa y le das la mano a mi marido. Har\u00e9 pato. Pero t\u00fa le das la mano a mi marido. En nuestra casa.\n\n\u00bfY d\u00f3nde est\u00e1 el problema?\n\nEn un coche aparcado delante de ellos se abri\u00f3 de golpe la puerta.\n\nBohumil fren\u00f3 de modo tan brusco que casi fue a dar con la cabeza en el suelo. Kass\u00e1ndra desvi\u00f3 de golpe su bicicleta hacia la izquierda y luego enseguida hacia la derecha, casi la hab\u00eda pillado una camioneta. Se detuvo y se ape\u00f3. El coraz\u00f3n le golpeaba en el pecho, le palpitaban las sienes. Bohumil se baj\u00f3, vocifer\u00f3 al conductor que hab\u00eda abierto la puerta sin mirar. El hombre se disculp\u00f3 varias veces, Bohumil pas\u00f3 junto al coche arrastrando su bicicleta hasta ponerse al lado de Kass\u00e1ndra, dej\u00f3 caer la bici, se sent\u00f3 sobre el cap\u00f3 de un coche aparcado y empez\u00f3 a llorar.\n\nKass\u00e1ndra se sent\u00f3 a su lado, le pas\u00f3 el brazo por los hombros, dijo: No ha pasado nada. No ha pasado nada. Ha habido suerte.\n\n\u00a1No ha habido suerte!\n\nEl conductor estaba all\u00ed de pie, p\u00e1lido como un muerto, Kass\u00e1ndra hizo un gesto con la mano indicando que se quitara de en medio.\n\nNada de ha habido suerte, repiti\u00f3 Bohumil, y se sec\u00f3 los ojos con el dorso de la mano. As\u00ed que fui all\u00ed, a casa de mi hermana. Ella quer\u00eda que le diera la mano a su marido. Y entonces, \u00e9l se neg\u00f3 a darme la mano a m\u00ed. \u00c9l, oye bien, \u00e9l no acept\u00f3 mi mano tendida. La ignor\u00f3. Me mir\u00f3 con su cara grasienta, pagada de s\u00ed misma, las manos en los bolsillos del pantal\u00f3n, y me dijo: Y smrade zasranej!\n\n\u00bfC\u00f3mo?\n\nTu es un cr\u00e9tin d'idiot!\n\nNon! Ce n'est pas vrai!\n\n\u00a1S\u00ed! Que yo estaba comprado por las multinacionales, que en Bruselas, por un opulento sueldo, estoy traicionando los intereses nacionales de la Rep\u00fablica Checa, que yo era un par\u00e1sito del pueblo, etc. Todo eso en el vest\u00edbulo de su casa. Junto al perchero.\n\n\u00bfY t\u00fa qu\u00e9 dijiste, qu\u00e9 hiciste?\n\nBohumil se ech\u00f3 a re\u00edr, respir\u00f3 con fuerza por la nariz y dijo: \u00bfQue qu\u00e9 hice? Retirar la mano. Y luego le dije a mi hermana: Si seguimos mucho tiempo aqu\u00ed discutiendo se va a quemar el pato. Y ella: No hay pato. S\u00f3lo hab\u00eda que poner las cosas en su sitio.\n\nKass\u00e1ndra le acarici\u00f3, apret\u00f3 su cabeza contra su pecho, le pas\u00f3 la mano por la cabeza. Era rid\u00edculo: estaba acariciando su casco de ciclista.\n\nDe pronto hab\u00eda un hombre de pie delante de ellos, grit\u00e1ndoles. El due\u00f1o del coche en cuyo cap\u00f3 estaban sentados. Bohumil levant\u00f3 la cabeza, sac\u00f3 de su bolsa en bandolera una pegatina, separ\u00f3 cuidadosamente el papel, se levant\u00f3 y le plant\u00f3 al hombre la pegatina en la frente. El hombre se tambale\u00f3 hacia atr\u00e1s, Bohumil cogi\u00f3 su bicicleta y dijo a Kass\u00e1ndra: \u00a1Venga! \u00a1Tenemos que llegar al trabajo!\n\nKass\u00e1ndra estaba asombrada de la rapidez con que se hab\u00eda sentado en la bicicleta; pedalearon con fuerza, en silencio, s\u00f3lo en la avenue des Arts dijo Bohumil: Mi hermana es cinco a\u00f1os menor que yo. Cuando iba al colegio, yo le hac\u00eda los deberes. Nadie dec\u00eda que fuera tonta o vaga. Era la princesa. Ahora va a tener un hijo de un fascista. Y nadie de la familia se altera. El tipo es amable con ellos, canta con bonita voz viejas canciones populares, tiene bastante buena pinta, gana bien y no es comunista. Esto es lo que hoy cuenta entre nosotros.\n\nKass\u00e1ndra no sab\u00eda qu\u00e9 decir. S\u00f3lo cuando llegaron, cuando pusieron el candado a las bicicletas y se dirig\u00edan al ascensor, dijo ella: Yo tambi\u00e9n tengo algunas cosas que contar de mi fin de semana.\n\nDelante del ascensor hab\u00eda dos salamandras. Saludaron con exagerada urbanidad, la puerta del ascensor se abri\u00f3, un salamandra pregunt\u00f3: Al cuarto, \u00bfverdad?\n\nKass\u00e1ndra asinti\u00f3, el salamandra oprimi\u00f3 tres y cuatro, pregunt\u00f3 amablemente: \u00bfHan tenido un buen fin de semana?\n\nEl fin de semana ha sido una mierda, dijo Bohumil.\n\nY Kass\u00e1ndra sinti\u00f3 un impulso, un placer insolente y desatinado, y dijo para su propia sorpresa: S\u00ed, y hasta ahora tambi\u00e9n el lunes es una mierda.\n\n\u00a1Oh!\n\nEl ascensor se balanceaba hacia lo alto, muy despacio, dada la situaci\u00f3n, deprimentemente despacio, y Bohumil dijo: Y el ascensor es tambi\u00e9n una mierda.\n\nKass\u00e1ndra se rio. Por dentro.\n\nEn la tercera planta los dos salamandras saltaron fuera del ascensor.\n\nBonne journ\u00e9e!\n\nBonne journ\u00e9e!\n\ny Bohumil se ech\u00f3 a re\u00edr. Kass\u00e1ndra dijo: Cu\u00e1nto me alegro de que vuelvas a re\u00edrte. Y ahora te cuento de mi fin de semana. A ti y a Xeno. Es importante. Y va a sorprenderte.\n\nFenia Xenopoulou ya estaba sentada ante su mesa de trabajo, con una taza de caf\u00e9 de la cantina. Iba a hacer otra vez un calor agobiante, la ventana estaba abierta; ya ahora, a las ocho de la ma\u00f1ana, el aire era caliente, pero Fenia Xenopoulou parec\u00eda tiritar. Abarcaba la taza con las dos manos, como si quisiera calentarse con ella. Pero eso era tal vez s\u00f3lo un h\u00e1bito. No ten\u00eda fr\u00edo. Como mucho en la psique. Ten\u00eda resaca. No f\u00edsica, sino moral. Hab\u00eda pasado la noche en casa de Fridsch y en un principio no hab\u00eda logrado decirle que... pero luego, s\u00ed, ya muy tarde, cuando el momento en verdad ya no era favorable, hab\u00eda propuesto que... y \u00e9l... se duerme... y ella... sosten\u00eda la taza de caf\u00e9 con las dos manos y sent\u00eda verg\u00fcenza de que ella entonces... hab\u00eda apretado la almohada sobre la cara de \u00e9l y... ella s\u00f3lo quer\u00eda ver si \u00e9l todav\u00eda era capaz de sentir algo, \u00bfo los hombres, cuando ya no hay alb\u00famina, son incapaces de sentir nada? \u00c9l se liber\u00f3 pataleando, la apart\u00f3 empuj\u00e1ndola, vocifer\u00f3, y ella rompi\u00f3 a llorar..., bueno, entonces \u00e9l la tom\u00f3 en sus brazos y...\n\nKass\u00e1ndra entr\u00f3 en la habitaci\u00f3n, \u00bfpor qu\u00e9 estaba tan excitada? Tenemos que hablar, tienes tiempo, es importante, para el Jubilee Project, ah, tienes caf\u00e9, buena idea, voy yo tambi\u00e9n a por uno, y d\u00edselo a Bohumil, \u00bfest\u00e1s de acuerdo, dentro de diez minutos aqu\u00ed, en tu despacho?\n\n\u00bfTienes un cigarrillo?\n\nNo, no fumo. Pero si quieres fumar, entonces hablamos mejor en el despacho de Bohumil, el cacharro ese de arriba... \u00e9l lo ha..., c\u00f3mo decirlo, bueno, en su despacho se puede fumar sin alarma.\n\nUn cuarto de hora despu\u00e9s estaban sentadas en el despacho de Bohumil, Kass\u00e1ndra hab\u00eda tra\u00eddo caf\u00e9 para todos, Xeno fumaba por primera vez tres cigarrillos seguidos ante testigos y Kass\u00e1ndra habl\u00f3 del cuartel de Dossin en Mechelen.\n\nA Kass\u00e1ndra le gustaba hacer excursiones en tren los fines de semana. Disfrutaba que \u00abdesde Bruselas estuviera todo tan cerca\u00bb, su Europa, como sol\u00eda decir: se estaba en menos de una hora y media en Par\u00eds, en dos horas y media en Londres, en \u00c1msterdam o Colonia en menos de dos horas. A veces se marchaba el domingo por la ma\u00f1ana y volv\u00eda por la noche, a veces viajaba ya el s\u00e1bado y pasaba una noche fuera. Visitaba entonces museos y galer\u00edas, se reun\u00eda con amigos en bistr\u00f3s y se permit\u00eda de vez en cuando alguna bonita prenda de una boutique. Pero ese fin de semana no hab\u00eda tomado el Thalys, sino un tren regional. A Mechelen, a s\u00f3lo treinta kil\u00f3metros de Bruselas, menos de media hora.\n\nHab\u00eda le\u00eddo en Le Soir la necrol\u00f3gica de Gustave Jakubowicz, el c\u00e9lebre abogado bruselense, que, como ella sab\u00eda, hab\u00eda desempe\u00f1ado un importante papel en la historia del Tribunal Europeo de Derechos Humanos; el hombre era una leyenda, en plena actividad hasta el final, hasta que ahora, a los noventa a\u00f1os escasos, acababa de morir. Pero lo que hab\u00eda atra\u00eddo en especial la atenci\u00f3n de Kass\u00e1ndra era la l\u00ednea sobre el autor del art\u00edculo necrol\u00f3gico: \u00abJean Nebenzahl, colaborador cient\u00edfico del Centro de Documentaci\u00f3n para el Holocausto y los Derechos Humanos del cuartel de Dossin de Mechelen\u00bb. El cuartel de Dossin hab\u00eda sido durante la ocupaci\u00f3n el campo de concentraci\u00f3n belga de las SS, desde el que fueron deportados a Auschwitz jud\u00edos, roman\u00edes y combatientes de la resistencia. Kass\u00e1ndra quiz\u00e1 hab\u00eda o\u00eddo alguna vez que ese cuartel era ahora un museo, pero no sab\u00eda que hubiera all\u00ed un centro de investigaci\u00f3n, una instituci\u00f3n cient\u00edfica que renovaba sistem\u00e1ticamente la historia de las deportaciones a Auschwitz. Escribi\u00f3 un e-mail a Jean Nebenzahl, que respondi\u00f3 enseguida: con mucho gusto estaba dispuesto a verla el domingo, a recorrer con ella la exposici\u00f3n y a contestar, en la medida de lo posible, a todas sus preguntas.\n\nKass\u00e1ndra era una funcionaria responsable. Viaj\u00f3 a Mechelen y se vio all\u00ed con Jean Nebenzahl, porque pensaba que eso pod\u00eda ser interesante para el Jubilee Project. Pero nunca se le habr\u00eda ocurrido apunt\u00e1rselo como horas extraordinarias o no viajar a Mechelen hasta que se le \u00abconcediera como viaje de servicio\u00bb y despu\u00e9s se le tuviera en cuenta en el balance final. A ella le interesaba aquello, era una excursi\u00f3n de domingo en la que pod\u00eda ver algo nuevo, aprender algo, y si resultaba en efecto \u00fatil para el proyecto, tanto mejor.\n\nY Jean Nebenzahl era un cient\u00edfico responsable, que se pon\u00eda a disposici\u00f3n tambi\u00e9n en domingo, \u00abfuera de las horas de servicio\u00bb cuando alguien de la Comisi\u00f3n Europea se interesaba por su trabajo y viajaba por eso a Mechelen. Era cada vez m\u00e1s dif\u00edcil llamar la atenci\u00f3n de la gente sobre el trabajo de ese centro de investigaci\u00f3n y reunir los fondos necesarios para mantenerlo. Por eso estaba casi conmovido por el inter\u00e9s de aquella funcionaria europea..., sobre la que \u00e9l tambi\u00e9n se hab\u00eda informado enseguida en Google: campo de actividades, fotograf\u00edas.\n\nNo tiene que darme las gracias, dijo, no soy un bur\u00f3crata sin alma, aunque en esta casa est\u00e9 sentado ante el escritorio de Eggert Reeder. \u00bfQue qui\u00e9n era? Era el jefe de la administraci\u00f3n alemana militar en B\u00e9lgica, organiz\u00f3 la deportaci\u00f3n de m\u00e1s de treinta mil jud\u00edos a Auschwitz, lo condenaron despu\u00e9s de la guerra a doce a\u00f1os de presidio... y despu\u00e9s Konrad Adenauer lo indult\u00f3. Porque \u00e9l se hab\u00eda limitado a estar sentado ante un escritorio. \u00c9l no era responsable del asesinato de los jud\u00edos en Auschwitz. S\u00f3lo los consignaba en listas durante sus horas de servicio, para que fueran enviados en orden al matadero. No era un fan\u00e1tico, eso es indudable, nunca hizo horas extraordinarias. Despu\u00e9s del indulto cobr\u00f3, de la Rep\u00fablica Federal de Alemania, una pensi\u00f3n de funcionario. Ten\u00eda suficientes a\u00f1os de servicio. Y yo estoy sentado hoy ante su mesa de despacho y trabajo con esas listas.\n\nJean Nebenzahl era un hombre bien parecido, de la misma edad aproximadamente que Kass\u00e1ndra, y muy semejante en el tipo: no delgado \u2014\u200cKass\u00e1ndra desconfiaba de los hombres enjutos de carnes, opinaba que tend\u00edan al ascetismo, es decir, a la rigidez y a la melancol\u00eda, pero Jean tampoco estaba gordo\u2014, los gordos a Kass\u00e1ndra le parec\u00edan amorfos, sin atractivo y sin autocontrol, pero no hab\u00eda que generalizar, o sea, la mayor\u00eda de los gordos, o en cualquier caso muchos de ellos, resultaban sospechosos, para Kass\u00e1ndra, de desali\u00f1o y descuido. Jean era simplemente un hombre, alto, fuerte, y sin embargo blando, as\u00ed ve\u00eda ella lo que \u00e9l, en su autoapreciaci\u00f3n, habr\u00eda calificado de \u00abun poquito demasiado relleno\u00bb. Y ella se qued\u00f3 extasiada con sus ojos casta\u00f1os y su rizado pelo negro.\n\nY por qu\u00e9 crees t\u00fa que a nosotros nos va a interesar que te hayas enamorado, pregunt\u00f3 Xeno.\n\nEn el despacho de Bohumil s\u00f3lo hab\u00eda dos sillas, la del escritorio y una para las visitas. Bohumil hab\u00eda ofrecido a Xeno la del escritorio, pero ella prefer\u00eda estar de pie. Xeno baj\u00f3 la vista con gesto de impaciencia hacia Kass\u00e1ndra, que estaba sentada en la silla de las visitas. Kass\u00e1ndra se levant\u00f3 de un salto: \u00bfNo comprend\u00e9is? \u00a1Lo he dicho con claridad! \u00a1Ellos tie-nen las lis-tas! \u00a1En Mechelen! Ah\u00ed est\u00e1 conservado en su totalidad el archivo del servicio de seguridad de las SS, que era el competente en deportaciones. Hemos mantenido correspondencia con el mundo entero y lo ten\u00edamos aqu\u00ed, ante nuestras propias narices: con el tren carreta a treinta minutos de nosotros. Tambi\u00e9n s\u00e9 cu\u00e1ntos supervivientes de Auschwitz viven a\u00fan, y tengo sus nombres.\n\n\u00bfCu\u00e1ntos son?\n\nDiecis\u00e9is, dijo Kass\u00e1ndra.\n\n\u00bfCu\u00e1ntos dices?\n\n\u00a1Diecis\u00e9is!\n\n\u00bfS\u00f3lo diecis\u00e9is? \u00bfA escala mundial?\n\nEn la medida en que estaban en listas de deportaci\u00f3n y luego registrados como supervivientes, en resumen: en la medida en que estuvieron registrados de alguna forma y alguien los conoce, s\u00ed.\n\n\u00bfY hay direcciones de contacto?\n\nJean dijo: No est\u00e1n totalmente al d\u00eda. Puede ser que en algunos sean err\u00f3neas porque no hay contacto regular con cada uno de ellos. Pero en principio, s\u00ed.\n\n\u00bfY en qu\u00e9..., c\u00f3mo dir\u00eda yo, en qu\u00e9 estado? Quiero decir, c\u00f3mo est\u00e1n de salud..., quiero decir, \u00bfpodr\u00edan viajar y presentarse en p\u00fablico?\n\nDe cinco se sabe que acuden con regularidad a colegios o a programas de testigos de la \u00e9poca.\n\n\u00bfCinco?\n\nS\u00ed. Y uno es un caso especial. Un tal David de Vriend. Y vive aqu\u00ed, en Bruselas. Jean opina que, si \u00e9l ha entendido bien nuestro proyecto, ese De Vriend ser\u00eda nuestro testigo ideal.\n\n\u00bfPor qu\u00e9?\n\nNo es s\u00f3lo uno de los \u00faltimos supervivientes de Auschwitz, es el \u00faltimo jud\u00edo a\u00fan vivo del legendario vig\u00e9simo tren de deportaci\u00f3n a Auschwitz. Fue el \u00fanico tren de deportados asaltado por combatientes de la resistencia y detenido en pleno trayecto. Cortaron con tenazas el alambre con el que estaban asegurados los cerrojos de las puertas de los vagones de ganado, abrieron las puertas y gritaron a los jud\u00edos que saltaran fuera y huyeran. A quien saltaba le daban cincuenta francos y una direcci\u00f3n segura. La mayor\u00eda tuvo miedo, tem\u00edan que los alemanes los mataran a tiros si trataban de huir. Se quedaron en el tren, que tras un breve tiroteo del cuerpo de guardia de las SS con los de la resistencia, continu\u00f3 el viaje. Todos los que no hab\u00edan saltado entraron en la c\u00e1mara de gas inmediatamente despu\u00e9s de la llegada a Auschwitz. Pero De Vriend fue uno de los que saltaron del tren.\n\nPero t\u00fa has dicho que estuvo en Auschwitz.\n\nEsa huida del vig\u00e9simo tren de deportados fue en abril de 1943. \u00c9l fue a parar a una familia de un pueblo, ya no recuerdo el nombre, que lo hizo pasar por un sobrino de Bruselas. Era jovenc\u00edsimo entonces y estaba gravemente traumatizado: sus padres se hab\u00edan quedado en el tren. Podr\u00eda haber esperado al final de la guerra viviendo con la familia que lo hab\u00eda acogido, pero quiso luchar, \u00bfquiz\u00e1 liberar a sus padres? \u00bfLiberar a Europa? En junio de 1944 se uni\u00f3, como el miembro m\u00e1s joven, al grupo de resistencia Europe Libre, \u00e9se era el grupo en torno a Jean-Richard Brunfaut, de quien quiz\u00e1 hay\u00e1is o\u00eddo hablar, bueno, como m\u00ednimo conoc\u00e9is la rue Brunfaut. Ese grupo se volvi\u00f3 legendario por sus temerarias acciones, pero tambi\u00e9n porque se distingu\u00eda pol\u00edticamente de todos los otros grupos de la resistencia. Era el \u00fanico que con su nombre no luchaba por una B\u00e9lgica libre, sino por una Europa libre. Quer\u00edan tambi\u00e9n enseguida para despu\u00e9s de la guerra, para despu\u00e9s de la victoria sobre los nazis, la abolici\u00f3n de la monarqu\u00eda belga y la fundaci\u00f3n de una rep\u00fablica europea. Brunfaut y sus compa\u00f1eros trabajaron tambi\u00e9n hasta el final de sus vidas contra el r\u00e9gimen fascista de Espa\u00f1a y Portugal, contra Franco y Salazar, de quienes las potencias vencedoras curiosamente se olvidaron cuando liberaron Europa. En cualquier caso: en agosto de 1944, David de Vriend fue traicionado, detenido y deportado a Auschwitz. No lo enviaron a la c\u00e1mara de gas. Era joven y fuerte. Sobrevivi\u00f3 aquellos meses hasta la liberaci\u00f3n. Despu\u00e9s de la guerra se hizo profesor. No quer\u00eda aparecer por los centros de ense\u00f1anza de vez en cuando, como muchos otros testigos de su \u00e9poca, \u00e9l quer\u00eda ser profesor para poder ocuparse cada d\u00eda de las generaciones siguientes. No quer\u00eda ser testigo, quer\u00eda ser pedagogo. As\u00ed pues, \u00bfqu\u00e9 dec\u00eds? Si sacamos adelante la idea de Martin y contamos para ello con la aprobaci\u00f3n del presidente, entonces tenemos que poner a ese hombre en el centro de la celebraci\u00f3n. En \u00e9l lo tenemos todo: a una v\u00edctima del racismo, a un combatiente de la resistencia, a una v\u00edctima de la colaboraci\u00f3n y la traici\u00f3n, a un testigo del campo de exterminio, a un visionario de la Europa posnacional sobre la base de los derechos humanos. La historia y la lecci\u00f3n extra\u00edda de la historia: todo en una persona, en la persona de ese profesor.\n\nMuy bien, dijo Xeno. \u00a1Qu\u00e9 \u00e9nfasis arrebatador! S\u00f3lo que hay un peque\u00f1o problema.\n\nLa se\u00f1ora Jos\u00e9phine estaba preocupada por De Vriend. Era una mujer imparcial, que procuraba tratar con la mayor equidad posible a todos sus \u00abprotegidos\u00bb, como ella los llamaba, ya le fueran simp\u00e1ticos, desagradables o incluso odiosos, ya se mostraran comunicativos o reservados, amables o agresivos. Jos\u00e9phine pensaba que todos ellos ten\u00edan sus razones para ser como eran all\u00ed, razones biogr\u00e1ficas que en aquella casa sal\u00edan claramente a la superficie cuando comprend\u00edan que en la Maison Hanssens no ten\u00edan otra cosa que hacer que esperar medio aletargados el final de sus vidas, mientras segu\u00edan fingiendo que eran los hu\u00e9spedes de un balneario.\n\nTodos los que ella cuidaba estaban al final de sus d\u00edas, pero a\u00fan no hab\u00edan hecho las paces con su vida. \u00c9sa era la experiencia de Jos\u00e9phine, su convicci\u00f3n. Cada d\u00eda se imaginaba lo que eso significaba. Para cada uno de ellos. Y en eso eran todos iguales, y en esa igualdad ella no distingu\u00eda entre protegidos f\u00e1ciles de cuidar y molestos, simp\u00e1ticos y antip\u00e1ticos. David de Vriend no hab\u00eda mostrado nunca la necesidad de comunicarse con ella m\u00e1s de lo necesario. Y cuando daba las gracias por algo sonaba m\u00e1s a una despedida que a una muestra de agradecimiento. Por tanto, no se pod\u00eda decir que De Vriend fuera un protegido al que hab\u00eda que querer y al que por eso se prestaba atenci\u00f3n de modo especialmente afectuoso. Y, sin embargo, Jos\u00e9phine pensaba que ten\u00eda una especial responsabilidad frente al se\u00f1or De Vriend. \u00bfEra por el n\u00famero que llevaba en el brazo? Se preguntaba eso y al mismo tiempo se prohib\u00eda tal pensamiento. Ella era correcta, igual de atenta con todos. A cada uno de ellos la vida le hab\u00eda jugado alguna mala pasada.\n\nAs\u00ed, se precipit\u00f3 con las mejores intenciones en la habitaci\u00f3n de De Vriend, con dos peri\u00f3dicos, y gritando: Nunca viene usted...\n\nDe Vriend estaba sentado en su butaca, vestido s\u00f3lo con un calzoncillo.\n\nJos\u00e9phine grit\u00f3: Hace ya d\u00edas que no le veo en la sala com\u00fan, donde est\u00e1n los peri\u00f3dicos. Pero tenemos que leer peri\u00f3dicos, \u00bfverdad, se\u00f1or De Vriend? \u00bfO ya no queremos saber lo que pasa en el mundo? No, no, queremos estar enterados, queremos mantener la cu-rio-si-dad, \u00bfverdad, se\u00f1or De Vriend? \u00bfQu\u00e9 prefiere usted leer, Le Soir o De Morgen? Creo que usted es lector de De Morgen, \u00bfverdad? Ahora vamos a entrenar un poquito nuestras peque\u00f1as c\u00e9lulas grises, y... A Jos\u00e9phine, evidentemente, le atacaba los nervios la apat\u00eda de aquel De Vriend, pero a pesar de todo trataba de animarlo a que siguiera alerta, con curiosidad, comunicativo, antes de hundirse en un letargo completo.\n\nDavid de Vriend tom\u00f3 el peri\u00f3dico, lo mir\u00f3 fijamente, luego empez\u00f3 a pasar despacio las hojas, hasta que de pronto se inclin\u00f3 sobre el peri\u00f3dico y clav\u00f3 la vista en \u00e9l.\n\n\u00bfQuiere que leamos juntos un art\u00edculo? Le interesa...\n\nDe Vriend se levant\u00f3, pase\u00f3 por la habitaci\u00f3n, fue de un lado a otro, miraba, buscaba, la se\u00f1ora Jos\u00e9phine lo observ\u00f3 con extra\u00f1eza: \u00bfQu\u00e9 busca usted?\n\nMi bloc de notas. \u00bfNo lo ha le\u00eddo? Las esquelas. Tengo que tachar un nombre, otro nombre m\u00e1s, de mi lista.\n\n# CAP\u00cdTULO D\u00c9CIMO\n\nGDY WSZYSTKO BY\u0141O NA PR\u00d3\u017bNO, NAWET NAJPI\u0118KNIEJSZE WSPOMNIENIE NAS NIE POCIESZY. I JAK TU SZUKA\u0106 USPRAWIEDLIWIENIA?\n\n#\n\n\u00c9mile Brunfaut estaba en el cuarto de ba\u00f1o, de pie y desnudo de espaldas al espejo, y trataba de mirarse por encima del hombro para comprobar si hab\u00eda un hematoma o una rozadura a la altura del sacro o del coxis. Al principio el ba\u00f1o lo hab\u00eda relajado pero, cuanto m\u00e1s tiempo segu\u00eda metido en la ba\u00f1era, tanto m\u00e1s fuerte era el dolor que ten\u00eda por encima de las nalgas, sin duda consecuencia de la ca\u00edda.\n\nLas v\u00e9rtebras cervicales crujieron y rechinaron, pero no consigui\u00f3 volver la cabeza lo suficiente para ver en el espejo el final de la espalda. Ahora se a\u00f1adieron a los dolores en la rabadilla intensos dolores cervicales. Brunfaut sab\u00eda, evidentemente, que su cuerpo no era tan flexible, m\u00f3vil y el\u00e1stico como el de una gimnasta rusa, pero le deprim\u00eda estar tan r\u00edgido. En las pausas en el trabajo de la comisar\u00eda, su colega Jules Meunier hac\u00eda \u00abpara no oxidarse\u00bb ejercicios de yoga y, cuando hab\u00eda largas reuniones, incluso el pino. \u00a1Qu\u00e9 rid\u00edculo le parec\u00eda eso a Brunfaut! Sin embargo, por otra parte era tan extravagante que ya casi resultaba simp\u00e1tico. Pero Brunfaut nunca lo habr\u00eda admitido. Probablemente ten\u00eda raz\u00f3n Jules. Brunfaut estaba convencido de que Jules pod\u00eda volver la cabeza y contemplarse en el espejo, sin el menor problema y con toda tranquilidad, la espalda y la rabadilla, sin contracciones musculares ni dolores.\n\nBrunfaut trat\u00f3 ahora de masajearse la nuca, all\u00ed detr\u00e1s estaba todo duro como un madero: y ahora sonaba el tel\u00e9fono. Sali\u00f3 corriendo del ba\u00f1o al dormitorio, donde se hab\u00eda desvestido, pero su tel\u00e9fono no estaba all\u00ed; corri\u00f3 al sal\u00f3n, all\u00ed estaba, sobre el escritorio. Descolg\u00f3 y se qued\u00f3 inm\u00f3vil. Era Philippe.\n\nEscucha, dijo Brunfaut, por tel\u00e9fono no hablamos sobre eso. S\u00ed, quiero que me lo expliques. Claro. \u00bfNos vemos? \u00bfD\u00f3nde? \u00bfCaf\u00e9 Kafka? \u00bfD\u00f3nde est\u00e1? \u00bfRue des Poissonniers? Esquina rue Antoine Dansaert. Entendido. \u00bfDentro de hora y media? D'accord.\n\nCuando lleg\u00f3 al caf\u00e9, Philippe a\u00fan no estaba. Brunfaut hab\u00eda llegado m\u00e1s de quince minutos antes, por tanto eso no pod\u00eda significar nada, sin embargo tuvo al momento la desagradable sensaci\u00f3n de que Philipe andaba jugando con \u00e9l de alguna manera y que otra vez iba a dejarlo plantado.\n\nDejarlo plantado: pero \u00bfc\u00f3mo? \u00c9mile Brunfaut apenas pod\u00eda estar sentado. Le dol\u00eda la rabadilla de un modo insoportable. S\u00f3lo si cargaba el peso sobre una nalga resultaba relativamente soportable estar sentado. Pero \u00bfcu\u00e1nto tiempo conseguir\u00eda aguantar? Se levant\u00f3 y fue a la barra. Atormentado, cambi\u00f3 una y otra vez el peso de una pierna a la otra, se bebi\u00f3 de un trago la cerveza, pidi\u00f3 otra, adem\u00e1s un Genever. Mir\u00f3 el reloj. \u00c9l desde luego no esperar\u00eda media hora o tres cuartos a un Philippe que luego no llegar\u00eda de todos modos. Desde luego que no. Diez minutos como m\u00e1ximo. Se bebi\u00f3 la ginebra, cogi\u00f3 su vaso de cerveza y se march\u00f3 fuera, a la puerta del caf\u00e9. Qu\u00e9 calor hac\u00eda. No recordaba ninguna primavera ni ning\u00fan comienzo de verano en Bruselas tan caluroso, tan aplastante, tan brutal. El asfalto, el empedrado, las paredes de las casas almacenaban el calor, lo irradiaban y ni siquiera un soplo de aire era un alivio, sino que le sacud\u00eda a uno el calor en la cara como un mazazo. Se a\u00f1ad\u00eda a eso ahora, en ese momento, una luz de lo m\u00e1s extra\u00f1a, que no parec\u00eda natural, era muy poco antes de la puesta de sol, pero all\u00ed, en el desfiladero de la calle, no se ve\u00eda el sol, naturalmente, s\u00f3lo se ve\u00edan bandas de luz de un rosa amarillento, que \u2014\u200cBrunfaut levant\u00f3 la vista\u2014 revest\u00edan el cielo como de un barniz venenoso.\n\n\u00c9mile Brunfaut era una persona po\u00e9tica. S\u00f3lo que no lo sab\u00eda porque le\u00eda poco. Y poes\u00eda, nunca. De todos los poemas que en su d\u00eda aprendi\u00f3 en el colegio \u2014\u200cmuchos no eran\u2014 s\u00f3lo recordaba uno: \u00ab\u00c0 une passante\u00bb, porque el verso \u00abUn \u00e9clair... puis la nuit! - Fugitive beaut\u00e9\u00bb le hab\u00eda emocionado de un modo especial. M\u00e1s tarde, cuando ya era comisario y su equipo tanteaba a ciegas, \u00e9l los animaba con la par\u00e1frasis: La nuit... puis l'\u00e9clair! - Le fugitif est visible. En su opini\u00f3n, \u00e9se hab\u00eda sido el \u00fanico m\u00e9rito po\u00e9tico de su vida. Pero ah\u00ed se minusvaloraba. Ahora estaba dolorosamente afectado por esa luz y la sent\u00eda como met\u00e1fora: y eso era sin duda un acto po\u00e9tico. La luz oblicua. De pronto todo estaba inmerso en luz oblicua. Lo familiar adquir\u00eda una apariencia venenosa, y tambi\u00e9n \u2014\u200cmir\u00f3 la casa de la esquina, enfrente del letrero de la calle: POISSONNIERS\u2014 irisaba como un pez.\n\nLe habr\u00eda gustado permanecer m\u00e1s tiempo en esa luz, en ese ambiente: no es que fuera agradable para \u00e9l, pero no, s\u00ed: le parec\u00eda acorde, que concordaba, s\u00ed, era acorde con su estado. Era eso, era la luz de sus dolores ps\u00edquicos, pero los dolores f\u00edsicos no los soportaba. Acab\u00f3 de beberse la cerveza, quiso entrar en el caf\u00e9 y pagar, marcharse a casa, cuando de pronto estaba Philippe delante de \u00e9l y le abrazaba. \u00bfPor qu\u00e9 tan alegre, y por qu\u00e9 le abrazaba con tanta fuerza?\n\nBrunfaut solt\u00f3 un breve gemido y se deshizo del abrazo. Philippe puso cara de preocupaci\u00f3n, pregunt\u00f3: \u00bfQu\u00e9 pasa? \u00bfTe duele algo?\n\n\u00bfPor qu\u00e9 a Brunfaut la cara de preocupaci\u00f3n de su amigo le pareci\u00f3 exagerada? \u00bfC\u00f3mo pod\u00eda creer Philippe que \u00e9l se iba a tragar esa comedia? Y si no era comedia, \u00bfc\u00f3mo pod\u00eda \u00e9l mismo creer que su mejor amigo era capaz de algo as\u00ed?\n\nTan furioso como si pisoteara el suelo para cerciorarse de que no se mov\u00eda y no le desaparec\u00eda debajo de los pies, dijo: S\u00ed, ten\u00eda dolores, se hab\u00eda ca\u00eddo en el cementerio. En el cementerio. \u00bfNo le sonaba aquello?\n\nRespir\u00f3 hondo: Est\u00e1bamos citados all\u00ed, \u00bfno es cierto? T\u00fa no estabas. Seguro que puedes explic\u00e1rmelo.\n\nDios m\u00edo, \u00bfpor qu\u00e9 te ca\u00edste? \u00bfTe has lastimado?\n\n\u00bfY si yo digo ahora: Porque no te vi all\u00ed a ti, sino a un fantasma?\n\nPhilippe, evidentemente, quiso decir algo pero no lo dijo, mene\u00f3 la cabeza, luego se\u00f1al\u00f3 el vaso de cerveza vac\u00edo de Brunfaut: Entremos, necesitamos algo de beber.\n\nEra la hora en la que el caf\u00e9 se llenaba deprisa. Ya no quedaba ninguna mesa libre, pero \u00c9mile Brunfaut dijo que \u00e9l de todos modos no pod\u00eda estar sentado.\n\nMe ca\u00ed sobre el trasero. En nuestro punto de encuentro en el cementerio. Es la rabadilla. Duele de un modo atroz.\n\nHizo un gesto al hombre de detr\u00e1s de la barra: \u00a1Dos cervezas!\n\nTampoco podr\u00e9 estar mucho tiempo de pie. As\u00ed que no vamos a andar con rodeos. \u00bfQu\u00e9 ha pasado? \u00bfPor qu\u00e9 no acudiste a la cita? \u00bfQu\u00e9 es eso del misterioso amigo al que \u00edbamos a ver all\u00ed? \u00bfEra tu amigo ese se\u00f1or mayor que me pregunt\u00f3 si hablo con los muertos? \u00bfQuieres explicarme si era una consigna para que lo reconociera? \u00bfY por qu\u00e9 no se te pod\u00eda localizar despu\u00e9s por tel\u00e9fono? Philippe, expl\u00edcamelo, te lo ruego. Y te lo pido de todo coraz\u00f3n: expl\u00edcamelo de manera que yo pueda entenderlo.\n\nNo te lo vas a creer, dijo Philippe, pero...\n\nEntonces lleg\u00f3 la cerveza.\n\n\u00c9mile Brunfaut levant\u00f3 su vaso, dijo: Sant\u00e9! As\u00ed que no me lo voy a creer. \u00bfY qu\u00e9 m\u00e1s?\n\nEscucha, dijo Philippe. Todo esto se puede explicar de un modo muy sencillo. El problema es s\u00f3lo que, siendo muy l\u00f3gico, suena de lo m\u00e1s inveros\u00edmil.\n\nT\u00fa conseguir\u00e1s que me lo crea.\n\nNo me lo parece. No te he visto nunca tan desconfiado, te est\u00e1s convirtiendo en tu abuelo, ten cuidado, es esa desconfianza que destruye la confianza. Da igual, ahora te cuento la historia, muy brevemente para que puedas meterte en la cama enseguida. Por cierto, Jo\u00eblle te env\u00eda saludos y pregunta cu\u00e1ndo vendr\u00e1s por fin a vernos. Le dir\u00e9 que tenga paciencia porque est\u00e1s enfermo. Bueno: empez\u00f3 con que recib\u00ed una carta. Esa historia, t\u00fa sabes a lo que me refiero, ya la hab\u00edamos dado por concluida. Y entonces lleg\u00f3 la carta. Insisto: carta. No un e-mail, no un mensaje electr\u00f3nico. Casi no la veo, porque en casa cuando vac\u00edo el buz\u00f3n es para echarlo todo enseguida a la basura. Siempre hay s\u00f3lo anuncios. En cualquier caso, en esa carta me escrib\u00eda una persona, que se llamaba a s\u00ed misma tambi\u00e9n \u00abPersonne\u00bb, que hab\u00eda ido desandando mis pasos.\n\n\u00bfDesandando mis pasos?\n\nS\u00ed. Por lo visto, cuando yo trataba de avanzar en nuestro asunto y averiguar c\u00f3mo hab\u00edan podido borrar de tu ordenador el caso Atlas, llegu\u00e9, de alg\u00fan modo al menos, a la antesala de un sistema que \u2014\u200cdig\u00e1moslo con prudencia\u2014 estaba implicado en ello. No lo s\u00e9 con seguridad. Sea como sea, all\u00ed alguien not\u00f3 que yo trataba de piratearlo. Y si ahora eso es un asunto de gran importancia, entonces ese Personne sabe en brev\u00edsimo tiempo que era yo. C\u00f3mo me llamo, d\u00f3nde vivo, todo. Ellos tienen esas posibilidades. Personne me escribi\u00f3, pues, una carta, y explic\u00f3 esa manera de tomar contacto diciendo que la carta de toda la vida, enviada por correo normal y superlento, era la \u00fanica forma de comunicaci\u00f3n que no se pod\u00eda almacenar, leer, evaluar y utilizar en tu contra en ninguna parte. Lo que antes se llamaba buz\u00f3n secreto, el escondrijo para mensajes secretos, es hoy el buz\u00f3n completamente normal de una casa. Pero bueno. T\u00fa conoces a L\u00e9o Aubry, del laboratorio. Un buen chico. Siempre dispuesto a ayudar. Y de absoluta confianza, \u00bfverdad? Eso. Le di la carta a \u00e9l. El papel: papel normal\u00edsimo, el m\u00e1s vendido, en cualquier gran superficie lo tienes a cuatro euros los quinientos folios. La impresora, en la medida en que se puede concluir algo a partir de la tinta, un simple Canon, la impresora m\u00e1s vendida en B\u00e9lgica. En el papel, ni una m\u00ednima huella de ADN o de lo que sea que pudiera dar una pista sobre el remitente.\n\nBien. Pero \u00bfqu\u00e9 pon\u00eda en la carta?\n\nQue me hab\u00eda atrevido a llegar lejos. Que era imposible que mis superiores estuvieran de acuerdo. Que por lo visto yo era alguien que luchaba en solitario, fuera del trabajo. Que \u00e9l tambi\u00e9n lo era.\n\n\u00bf\u00c9l? \u00bfEn qu\u00e9 notabas que Personne era un hombre?\n\nBuena pregunta. Lo he supuesto.\n\n\u00bfAh, s\u00ed? \u00bfY qu\u00e9 m\u00e1s?\n\n\u00c9l... Estoy seguro de que es un hombre. Dec\u00eda que no era como un whistleblower, uno que est\u00e1 dispuesto a destruir su propia vida, pero que simpatizaba con todos los que buscan grietas por las que pueda filtrarse la verdad.\n\n\u00bfAs\u00ed lo formulaba?\n\nS\u00ed. Y me ofrec\u00eda su ayuda. Si yo estaba interesado en continuar comunic\u00e1ndome con \u00e9l, deb\u00eda abstenerme de seguir intentando penetrar en el sistema, porque \u00e9l ya no pod\u00eda garantizar por m\u00e1s tiempo que se reprimiera la alerta que yo desencadenaba. \u00c9l me procurar\u00eda las informaciones que necesito. Que si estaba de acuerdo, a una hora determinada del d\u00eda siguiente deb\u00eda introducir en Google la consulta: Danza de la lluvia de los indios hopi.\n\n\u00bfHopi qu\u00e9? \u00bfQu\u00e9 me est\u00e1s contando? \u00a1Es todo absurdo!\n\nNo, no es absurdo. Por lo visto ese Personne tiene la posibilidad de ver lo que yo hago en mi ordenador. Y si introduzco lo que \u00e9l pide y entro en una de las p\u00e1ginas que entonces se ofrecen, sabe que acepto su propuesta: sin que eso llame la atenci\u00f3n de alguna forma en el sistema.\n\n\u00bfY t\u00fa lo has hecho?\n\nS\u00ed.\n\nNecesito otra cerveza.\n\nYo tambi\u00e9n. \u00bfY sabes lo que lleg\u00f3? Yo introduje, pues, \u00abdanza de la lluvia, indios hopi\u00bb y Google me recomend\u00f3 enseguida: Teor\u00eda sist\u00e9mica y nuevos movimientos sociales. Problemas de identidad en la sociedad del riesgo.\n\nNo comprendo.\n\nNo hay nada que comprender. Es el t\u00edtulo de un libro, y en ese libro hay por lo visto un cap\u00edtulo sobre los indios hopi y las danzas de la lluvia. Por el motivo que sea. Entr\u00e9 en eso.\n\n\u00bfY entonces?\n\nDos d\u00edas despu\u00e9s recib\u00ed la segunda carta.\n\n\u00bfY c\u00f3mo has respondido?\n\nHaciendo preguntas en Google en mi ordenador a las horas que \u00e9l dec\u00eda. Las palabras buscadas eran mi respuesta o mis preguntas. Al parecer, est\u00e1 metido en un sitio donde puede controlar qui\u00e9n busca y qu\u00e9 busca en Google.\n\n\u00bfCu\u00e1ntas veces hab\u00e9is..., quiero decir, cu\u00e1nto tiempo ha funcionado eso?\n\n\u00bfTres semanas? Tal vez cuatro...\n\n\u00bfY no me has contado nada? Hemos estado en el partido del Anderlecht contra el Mechelen, hemos llorado en nuestras bufandas. \u00bfC\u00f3mo fue posible que el Mechelen nos ganara por 2 a 0? Luego nos tomamos cinco cervezas, por lo menos, y hablamos de toda clase de cosas, pero t\u00fa no me contaste nada, ni mencionaste a ese Personne. Tiene que haber sido entonces... tiene que haber sido en esos d\u00edas.\n\nS\u00ed, pero antes quer\u00eda saber si era serio. Habr\u00eda podido ser un chiflado.\n\nPero \u00bfno era un chiflado?\n\nNo. Mejor dicho, no lo s\u00e9. Hay datos interesantes y dignos de cr\u00e9dito. De \u00e9l lleg\u00f3 el dosier sobre la colaboraci\u00f3n del Vaticano con los servicios secretos occidentales. Lo he le\u00eddo, fue asombroso, casi fant\u00e1stico, pero al mismo tiempo tambi\u00e9n totalmente l\u00f3gico y comprensible, eran piezas de un puzle que encajaban perfectamente. Mira, ning\u00fan servicio secreto del mundo tiene los recursos, los medios econ\u00f3micos y personales para formar una red de agentes que se extienda por el globo de una manera que corresponda, ni siquiera aproximadamente, al estado de la globalizaci\u00f3n. As\u00ed que ahora tienen a sus agentes en los focos de m\u00e1ximo inter\u00e9s. Pero \u00bfqui\u00e9n conf\u00eda en ellos, qui\u00e9n les da informaci\u00f3n? S\u00f3lo los que de todos modos colaboran con los gobiernos de esos servicios secretos, es decir: las informaciones de esos agentes no se distinguen en lo esencial de los informes que env\u00eda un embajador a su pa\u00eds. Y adem\u00e1s: \u00bfD\u00f3nde est\u00e1 el siguiente foco de inter\u00e9s? \u00bfQu\u00e9 empezar\u00e1 a arder ma\u00f1ana mientras se invierten millones en el trabajo de quiz\u00e1 treinta agentes que est\u00e1n metidos en los territorios en crisis, en un par de hoteles que a\u00fan funcionan, con zona de wellness? Y veinte de treinta son de la CIA, en unos lugares se estorban unos a otros, pero hay sitios en los que no tienen a nadie. Y esto es el poderos\u00edsimo servicio secreto. Pero, bueno, ahora una sencilla pregunta: \u00bfqui\u00e9n tiene metido un agente en cada m\u00edsero rinc\u00f3n? El Vaticano. \u00bfPor qu\u00e9? En cada pueblo perdido hay un sacerdote. \u00bfQui\u00e9n se entera en cada pueblo perdido de los secretos m\u00e1s secretos? El cura, en muy buena parte a trav\u00e9s de la confesi\u00f3n. Aunque esto quiz\u00e1 no lo cubra todo, es sin duda mucho m\u00e1s de lo que incluso los servicios secretos mejor provistos pueden organizar en cuanto a informaci\u00f3n. Y por eso, amigo m\u00edo, los servicios secretos compiten entre ellos con todos los medios por el favor del Vaticano, por la colaboraci\u00f3n y el trato con la Iglesia. Eso era as\u00ed en la guerra fr\u00eda, y ya ni siquiera es un secreto. Ahora hay otro enemigo. Ya no es el comunismo ateo, hoy el enemigo se llama islam.\n\nPero... Espera. Un musulm\u00e1n no va a ir a un cura a confesar que ha cometido o tiene planeado cometer un atentado. Eso es absurdo.\n\nNo, claro que no. Pero los buenos cristianos le cuentan al cura lo que les ha llamado la atenci\u00f3n por sospechoso, por ejemplo, en los nuevos inquilinos del piso vecino o de la casa vecina o de la casa de enfrente, ellos est\u00e1n detr\u00e1s de sus ventanas con prism\u00e1ticos y observan las ventanas de las casas de la acera de enfrente. \u00bfEs la curiosidad un pecado? La nuestra seguro que no. Pero lo mismo que nosotros solemos informar del resultado de nuestras pesquisas, el cristiano lo dice en la confesi\u00f3n. Y por eso a\u00fan existe el eje construido en la guerra fr\u00eda entre los servicios secretos y el Vaticano.\n\n\u00bfT\u00fa te crees eso?, pregunt\u00f3 Brunfaut.\n\nPhilippe se qued\u00f3 perplejo un momento, luego se ech\u00f3 a re\u00edr. Yo no soy religioso. Yo no creo. Y t\u00fa puedes creer lo que quieras. Yo te doy los hechos. \u00bfC\u00f3mo est\u00e1, por cierto, tu rabadilla?\n\nOtra cerveza y una Genever y estar\u00e1 mejor.\n\nBueno. En eso estoy a tu altura. En fin, que Personne apunta a que la Iglesia dispone de una especie de escuadr\u00f3n de la muerte, que con la aprobaci\u00f3n de los servicios secretos liquida simplemente a presuntos terroristas o a los llamados predicadores del odio. Es decir, a personas de las que se esperan actos terroristas y contra las que el Estado de derecho no tiene suficiente margen de maniobra para quitarlas de en medio legalmente. Y ah\u00ed entra el caso Atlas. Guerreros de Dios hacen el trabajo y los servicios secretos los respaldan reduciendo a continuaci\u00f3n a la nada el caso correspondiente. Personne me ha enviado una lista con catorce casos de asesinato que hubo en Europa el a\u00f1o pasado; sobre ninguno de ellos sali\u00f3 nada en los peri\u00f3dicos.\n\n\u00bfLo has comprobado?\n\nS\u00ed. No he encontrado la m\u00ednima alusi\u00f3n a ninguno de los asesinatos de la lista. Eso significa: no los ha habido, o el encubrimiento fue tan perfecto que no existe el menor indicio.\n\nPero entonces nos metemos en el terreno de las teor\u00edas conspirativas.\n\nNo, no nos metemos. Porque si buscas hoy datos sobre el asesinato del Atlas, no encontrar\u00e1s nada. Nada. Absolutamente nada. Pero nosotros sabemos que lo hubo. Y lo que necesitamos no es una prueba para los catorce asesinatos de la lista, sino una explicaci\u00f3n del asesinato en el hotel Atlas. Y la explicaci\u00f3n de Personne es condenadamente l\u00f3gica. Sant\u00e9!\n\nAlgo le resultaba discordante a Brunfaut. Y su experiencia de polic\u00eda le dec\u00eda: cuando en una leyenda algo te resulta raro, entonces es muy probable que realmente haya gato encerrado.\n\nNo comprendo por qu\u00e9 no me iniciaste en el asunto ni me mantuviste al corriente, dijo.\n\nS\u00ed que lo hice, dijo Philippe. Es decir, yo te conozco. Sab\u00eda que ten\u00eda que presentarte algo m\u00e1s que una historia as\u00ed. Hechos. Por eso quise encontrarme con Personne. Por eso a las horas convenidas s\u00f3lo hice en mi ordenador b\u00fasquedas con variaciones del lema \u00abencuentro\u00bb. Tres d\u00edas despu\u00e9s una carta con esta propuesta: Encuentro en el cementerio, tal como te lo dije.\n\n\u00bfY entonces tienes por fin tu cita con ese fantasma y no vas?\n\nPero \u00bfqu\u00e9 dices? Estuve all\u00ed. Por supuesto. No tengo ni idea de d\u00f3nde estabas t\u00fa. Quiz\u00e1 en el monumento equivocado, quiz\u00e1 a la hora equivocada, qu\u00e9 s\u00e9 yo. Yo, en cualquier caso, estuve all\u00ed. Y estuve sentado en el banco. Y os esper\u00e9 a Personne y a ti. Entonces son\u00f3 mi m\u00f3vil. Lo cog\u00ed y una voz dijo: Mister Philippe Gaultier? Yo dije s\u00ed. \u00c9l pregunt\u00f3: \u00bfEst\u00e1 sentado en el banco que hemos acordado como punto de encuentro? Comprend\u00ed qui\u00e9n era y dije s\u00ed. \u00c9l: Lev\u00e1ntese, por favor. Yo: \u00bfC\u00f3mo dice? \u00bfPor qu\u00e9? \u00c9l: Lev\u00e1ntese, por favor. Me levant\u00e9, \u00e9l: Dese media vuelta y d\u00edgame lo que ve. Me pareci\u00f3 grotesco, dije: Oiga usted, yo no quiero jueguecitos, y \u00e9l: No son jueguecitos. \u00bfQu\u00e9 ve? Yo dije: Un \u00e1rbol. Y pens\u00e9: qu\u00e9 rid\u00edculo todo, \u00bfad\u00f3nde va esto a parar? \u00c9l dijo: \u00bfDetr\u00e1s? Yo: Tumbas. Tumbas de soldados. Cruces blancas. \u00c9l: Muy bien. \u00bfY detr\u00e1s? Yo: Nada. S\u00f3lo un campo inmenso de cruces blancas. \u00c9l: Entonces levante la vista. \u00bfQu\u00e9 ve ahora? Yo: Nada. No s\u00e9 lo que quiere usted o\u00edr. \u00c9l: Quiero o\u00edr lo que usted ve. Yo: Nada, \u00e1rboles, cielo. \u00c9l: \u00bfY entre los \u00e1rboles y el cielo? \u00bfDetr\u00e1s del cementerio? Yo: S\u00ed, dos grandes edificios, como dos inmensos bloques de queso emmental. \u00c9l: Exacto. \u00bfSabe usted qu\u00e9 es eso? Yo: \u00bfLa OTAN? \u00c9l: Correcto. Ahora ya tiene usted la informaci\u00f3n que puedo darle. Trabaje con ella o d\u00e9jelo. \u00a1Ciao, se\u00f1or polic\u00eda!\n\n\u00bfEstuviste en el cementerio y recibiste esa llamada?\n\nS\u00ed. Entonces te esper\u00e9 tres cuartos de hora, luego me march\u00e9.\n\n\u00bfY por qu\u00e9 desconectaste el tel\u00e9fono? Te llam\u00e9 varias veces porque no estabas y...\n\nMi tel\u00e9fono no funcionaba. De pronto yo no pod\u00eda ni llamar ni recibir llamadas. Y cuando funcion\u00f3 otra vez, te llam\u00e9 inmediatamente. Por eso estamos ahora aqu\u00ed.\n\nA Brunfaut le encant\u00f3 la historia. Realmente sugestiva. Nunca habr\u00eda considerado a Philippe capaz de algo as\u00ed. Pero no le cre\u00eda ni una sola palabra. C\u00f3mo le dol\u00eda eso.\n\nTengo muchos dolores, dijo. No te enfades, he de irme a casa. Vio que Philippe no hab\u00eda tocado su ginebra. \u00c9mile cogi\u00f3 el vaso, se lo bebi\u00f3 de un trago, dijo \u00c0 bient\u00f4t, mon ami! Luego sali\u00f3 cojeando. Se dio cuenta de que cojeaba, no quer\u00eda, trat\u00f3 de caminar derecho y sin da\u00f1o visible, pero no lo consigui\u00f3, sali\u00f3 del Kafka cojeando y habr\u00eda querido llorar a gritos.\n\nQue Matek no hab\u00eda volado a Estambul, sino a Cracovia lo supieron ya el mismo d\u00eda. Y que los indicios de que hab\u00eda continuado viaje a Varsovia al d\u00eda siguiente eran una pista falsa puesta por \u00e9l lo supieron tres d\u00edas despu\u00e9s. Matek no tuvo confirmaci\u00f3n de ello, pero lo daba por seguro. Y sab\u00eda tambi\u00e9n que, si no part\u00eda al cabo de tres d\u00edas, causar\u00eda un conflicto de conciencia al padre Szymon, su amigo \u00edntimo de la \u00e9poca del seminario. Szymon le hab\u00eda dado refugio en el convento de los agustinos creyendo que Matek necesitaba otra vez un tiempo de recogimiento y contemplaci\u00f3n. Szymon era absolutamente leal, Matek sab\u00eda que pod\u00eda confiar en \u00e9l, pero tambi\u00e9n que Szymon nunca podr\u00eda entender que \u00e9l, Matek, all\u00ed, en el convento, se escond\u00eda en realidad de autoridades eclesi\u00e1sticas. Conoc\u00edan sus contactos, por tanto estaba claro que, a partir del cuarto d\u00eda, Szymon estaba en su punto de mira. E igual de claro estaba c\u00f3mo resolver\u00eda Szymon el conflicto entre la fiel amistad y el voto de obediencia que hab\u00eda hecho como sacerdote. Matek hab\u00eda aprovechado los tres d\u00edas para meditar, para reflexionar sobre su situaci\u00f3n, para cargar pilas. Pero ahora deb\u00eda abandonar el convento. Ten\u00eda dos posibilidades: seguir viajando, alojarse en hoteles baratos en los que no se tomaban tan al pie de la letra el registro y los documentos de identidad, no utilizar tarjetas bancarias de ning\u00fan tipo, evitar en lo posible c\u00e1maras de vigilancia en lugares p\u00fablicos, no usar nunca el port\u00e1til. Matek ser\u00eda un submarino, invisible, ilocalizable. Por otra parte: no tendr\u00eda posibilidad alguna de averiguar qu\u00e9 hab\u00eda fallado en Bruselas, en el hotel Atlas, y lo que ahora pensaban hacer con \u00e9l. Y su dinero en met\u00e1lico le alcanzar\u00eda como mucho para otra semana. Esa semana que pasar\u00eda como submarino no mejorar\u00eda en nada su situaci\u00f3n ni aportar\u00eda nuevos datos. Segunda posibilidad: meterse en la guarida del le\u00f3n. Ten\u00eda que averiguar lo que hab\u00eda ocurrido y en qu\u00e9 situaci\u00f3n estaba \u00e9l. Y s\u00f3lo hab\u00eda un lugar en el que podr\u00eda enterarse de todo eso: Pozna\u0144. Ellos no contar\u00edan con que \u00e9l, despu\u00e9s de haber desaparecido, fuera directamente a la central. Era peligroso. Por otra parte: si la cosa sal\u00eda mal, podr\u00eda mostrarse humilde y alegar que hab\u00eda regresado voluntariamente.\n\nLe dio un abrazo de despedida a Szymon, dijo: Gracias, hermano, le tom\u00f3 ambas manos. \u00a1Dios te proteja!\n\nSzymon sonri\u00f3: \u00a1Dios te proteja a ti! Y: \u00a1Buen viaje a Pozna\u0144!\n\nEra dif\u00edcil que Matek perdiese la serenidad. Estaba constantemente en guardia, calculaba todas las posibilidades y, seg\u00fan cre\u00eda \u00e9l, estaba preparado en cada situaci\u00f3n para cada eventualidad. Ten\u00eda la sangre fr\u00eda de un hombre que era ya soldado de cuarta generaci\u00f3n. Pero eso no se lo esperaba. \u00ab\u00a1Buen viaje a Pozna\u0144!\u00bb: fue como una sacudida el\u00e9ctrica que lo aturdi\u00f3 brevemente. Respir\u00f3 hondo, dej\u00f3 su mochila y dijo: T\u00fa sabes...\n\nSzymon asinti\u00f3.\n\n... que voy a Pozna\u0144. Pero yo no te lo he dicho.\n\nTe esperan all\u00ed. Y no tienes nada que temer.\n\n\u00bfQu\u00e9 sabes t\u00fa, hermano Szymon? \u00bfY por qu\u00e9 no me has dicho nada?\n\nNo me has preguntado. Aparec\u00edas para los ejercicios, para la oraci\u00f3n en com\u00fan y para las horas de silencio, y ven\u00edas a las comidas, excepto por la noche, y guardabas silencio, no s\u00f3lo durante la sopa. Fuera de eso pasabas horas arrodillado en la capilla ante la Madre de Dios del Consuelo. Si un hermano me pregunta, yo doy informaci\u00f3n, pero t\u00fa no preguntaste.\n\nPero \u00bft\u00fa diste informaci\u00f3n?\n\nS\u00ed.\n\n\u00bfTe preguntaron por m\u00ed?\n\nSzymon asinti\u00f3.\n\nMatek mir\u00f3 al suelo, levant\u00f3 despacio la cabeza. Vio el h\u00e1bito negro de Szymon, el cintur\u00f3n de cuero negro, la esclavina negra, de la que sobresal\u00eda un cuello gris sobre el que, bajo la negra capucha, estaba el rostro gris de Szymon. Matek baj\u00f3 la mirada otra vez, se observ\u00f3 las manos, tambi\u00e9n eran grises, las dej\u00f3 caer, desaparecieron en el negro gris\u00e1ceo, por encima de la piedra negra del suelo de aquel sombr\u00edo vest\u00edbulo. Ahora Matek mir\u00f3 a Szymon directamente a la cara. Los labios de Szymon estaban rojos. Como si se los hubiera mordido hasta hacerse sangre. Ahora te pregunto yo, dijo Matek. \u00bfQu\u00e9 sabes? \u00bfQu\u00e9 puedes decirme?\n\nTen\u00edas una misi\u00f3n. No s\u00e9 cu\u00e1l. Algo fue mal. No s\u00e9 qu\u00e9. No fue culpa tuya. Te esperan. No tienes nada que temer. Es lo que deb\u00eda decirte si me preguntabas.\n\nMatek mir\u00f3 a Szymon, asinti\u00f3, tom\u00f3 su cabeza entre las manos, lo atrajo hacia \u00e9l y apret\u00f3 su boca contra los labios rojo sangre de Szymon. El rojo sangre, lo \u00fanico brillante en aquel espacio que en ese momento era un espacio c\u00f3smico y al mismo tiempo una esclusa cerrada al mundo.\n\nLuego sali\u00f3 del convento al aire libre, a la amenazadora y amenazada libertad.\n\nDespu\u00e9s de los d\u00edas en silenciosa penumbra tras los gruesos muros, la deslumbrante luz del d\u00eda cay\u00f3 sobre \u00e9l como un rayo.\n\nLa DG AGRI no hab\u00eda reaccionado a la inter-service consultation en el asunto del Jubilee Project ni enviado a nadie al meeting. La organizaci\u00f3n de aniversarios y solemnidades no le interesaba a nadie de esa direcci\u00f3n general y, menos a\u00fan, si el centro de la celebraci\u00f3n no era un escaparate de los \u00e9xitos de la pol\u00edtica agraria europea. Y a la AGRI le interesaba menos todav\u00eda si la DG COMM encargaba la preparaci\u00f3n del aniversario ni m\u00e1s ni menos que a la Cultura, esa \u00abarca en dique seco\u00bb, como la hab\u00eda llamado una vez George Morland. El elefante sab\u00eda que no se puede convertir a un mosquito en elefante.\n\nY ahora era precisamente ese George Morland de la AGRI quien en la Comisi\u00f3n, tras los primeros ataques del Consejo, empez\u00f3 tambi\u00e9n a anudar hilos que hab\u00edan de convertirse en una red para que cayera en ella el proyecto.\n\nGeorge Morland, como la mayor\u00eda de los funcionarios ingleses, no gozaba de mucha estima en la casa. Los brit\u00e1nicos, eso lo hab\u00eda dicho incluso el presidente en una ocasi\u00f3n, aceptan all\u00ed una \u00fanica regla vinculante: que ellos son fundamentalmente una excepci\u00f3n. En efecto, los ingleses siempre estaban bajo la sospecha de postergar los intereses de la Comunidad frente a los intereses de Londres. En muchos casos, la sospecha era fundada. Pero en otros casos la cosa era m\u00e1s complicada: Reino Unido era en efecto por principio, se quisiera o no, un caso especial. La Corona inglesa ten\u00eda posesiones que, sin embargo, jur\u00eddicamente no formaban parte del Reino Unido, como la Isla de Man o las islas del Canal, lo que constitu\u00eda un problema insoluble en t\u00e9rminos de desarrollo de una pol\u00edtica fiscal europea: los para\u00edsos fiscales de un miembro a los que no se ten\u00eda acceso desde el punto de vista legal. La reina era formalmente jefa del Estado de los Estados de la Commonwealth, lo que implicaba sutilidades jur\u00eddicas, por ejemplo en todos los tratados comerciales que la UE firmaba con Estados no pertenecientes a la UE. Si esa situaci\u00f3n especial no se hubiera tenido en cuenta en las normativas de excepci\u00f3n, Australia, por ejemplo, de pronto habr\u00eda formado parte del mercado interior europeo. La relaci\u00f3n con Inglaterra no hab\u00eda sido sencilla desde el principio, pero hubo desde luego ingleses que en Bruselas se hicieron europeos. Y habr\u00eda que decir en favor de George Morland que en sus a\u00f1os en Bruselas no s\u00f3lo aprendi\u00f3 algo de franc\u00e9s, sino que llev\u00f3 a cabo un importante trabajo de pol\u00edtica europea. En su funci\u00f3n en la AGRI siempre hab\u00eda sido un ardiente defensor y promotor de la agricultura de peque\u00f1a escala, y aunque lo hab\u00eda hecho porque quer\u00eda ver el paisaje ingl\u00e9s cuidado en el sentido tradicional y no destruido por inmensos complejos agrarios y monocultivos, eso fue, sin embargo, tambi\u00e9n de inter\u00e9s general europeo. Y ah\u00ed Morland, ese reto\u00f1o de la upper class, tampoco se dej\u00f3 sobornar por la industria agraria, por los consorcios productores de semillas y por sus lobbies. \u00c9l, o su familia, ten\u00eda considerables propiedades rurales en el Yorkshire oriental, que estaban arrendadas a varios peque\u00f1os granjeros. Morland conoc\u00eda sus esfuerzos y sus apuros. Defender sus intereses contra la intensificaci\u00f3n radical de la agricultura era un caso cl\u00e1sico de inter\u00e9s personal que serv\u00eda al inter\u00e9s general. El \u00fanico monocultivo que \u00e9l aceptaba era el campo de golf.\n\nMorland era pues un caso muy ambivalente. \u00c9l sab\u00eda que no era popular, pero en principio esto ten\u00eda poco que ver con su trabajo en la Comisi\u00f3n. Ya lo hab\u00eda sufrido en su juventud, primero en el colegio y luego en la universidad, en Oxford. Era una persona de aspecto poco afortunado y, a primera vista, algo estrafalario, y por mucho que se esforzara, su modo de ser no atra\u00eda. Su rostro redondo y ros\u00e1ceo, su nariz achatada, el fuerte pelo rojizo que s\u00f3lo pod\u00eda domesticar si lo cortaba a cepillo, el cuerpo robusto y peque\u00f1o. De ni\u00f1o hab\u00eda pasado muchas noches llorando en la almohada, por los apodos que le pon\u00edan. De cosas peores que la burla lo hab\u00eda salvaguardado su origen, que al final \u2014\u200cen una especie de autodefensa ps\u00edquica\u2014 acab\u00f3 convirti\u00e9ndolo en una persona arrogante, pero al mismo tiempo enormemente ambiciosa. Aprendi\u00f3 a hacerse respetar a trav\u00e9s de su carrera y sus diversos cargos, y era, con ir\u00f3nica sonrisa, un hombre de la vieja escuela: en caso de duda, que le temiera quien de ning\u00fan modo quer\u00eda apreciarlo.\n\nNow is the winter of our discontent \/ Made glorious summer by this sun of Brussels.\n\nPero el sol se oscureci\u00f3. \u00c9l era un END, Expert National D\u00e9tach\u00e9, su periodo de Bruselas se terminaba. Y en todo el caos de las negociaciones por la salida de Gran Breta\u00f1a de la Uni\u00f3n hab\u00eda cometido sin darse cuenta un grave error que en su pa\u00eds hab\u00eda da\u00f1ado considerablemente su reputaci\u00f3n. Los alemanes hab\u00edan firmado un convenio bilateral con China que les abr\u00eda el mercado chino para su producci\u00f3n de cerdos. \u00a1De cerdos! \u00c9l no se lo hab\u00eda tomado en serio, hab\u00eda estado muy implicado en el boicot de cualquier intento de llevar a cabo un convenio comunitario de la Uni\u00f3n con China, hab\u00eda querido defender privilegios para el Reino Unido y no pudo prever las consecuencias. \u00a1Ese Kai-Uwe Frigge ten\u00eda raz\u00f3n! Las turbulencias en el mercado financiero de London City hab\u00edan sido considerables, y hab\u00edan acelerado el traspaso de importantes fondos a Frankfurt. \u00a1Por los cerdos! Morland estaba consternado. Para \u00e9l era del todo incomprensible que el hecho de que China quisiera importar tambi\u00e9n los despojos de los cerdos tuviera una importancia econ\u00f3mica tan enorme. En los tiempos del hambre, los irlandeses compraban por unos peniques pies de cerdo y los herv\u00edan despu\u00e9s durante varias horas, eso era comida de miseria de quien viv\u00eda en la pobreza extrema. Y en Londres los carniceros regalaban orejas de cerdo a los clientes fijos. Y la cabeza de cerdo..., qu\u00e9 decir de eso. En la boca de un cerdo muerto hab\u00eda metido \u00e9l su pene como rito de iniciaci\u00f3n para ser socio del Bullingdon Club de Oxford, la exclusiva asociaci\u00f3n estudiantil para alumnos de las mejores familias. Tener que hacer aquello para poder formar parte del club fue su \u00faltima humillaci\u00f3n, suavizada por la borrachera y la algarab\u00eda. Despu\u00e9s ya s\u00f3lo hubo aceptaci\u00f3n y elogio. El cerdo puede contener trazas de tories. S\u00ed. \u00a1Ja, ja! C\u00f3mo se re\u00edan ahora los alemanes. Venden despojos a precio de solomillo, pero Inglaterra no participa de ello, y dentro de poco Reino Unido estar\u00e1 fuera.\n\nEs absurdo, completamente irracional, pero esa historia de los cerdos fue una raz\u00f3n fundamental para que George Morland pasara ahora a la obstrucci\u00f3n radical. Si Inglaterra hab\u00eda salido perjudicada, que al menos se burlara de los causantes del perjuicio. Y todo lo que le sal\u00eda mal a la Comisi\u00f3n fortalec\u00eda la posici\u00f3n brit\u00e1nica en las pr\u00f3ximas negociaciones. Y si la Comisi\u00f3n, por lo visto bajo los auspicios del presidente, preparaba una campa\u00f1a de imagen, entonces que fracasara. Una mala imagen de la Comisi\u00f3n era algo bueno. Para Inglaterra.\n\nMorland se recost\u00f3 en su sill\u00f3n de oficina y empez\u00f3 a limarse las u\u00f1as. \u00bfA qu\u00e9 se deb\u00eda que sus u\u00f1as fueran de pronto tan quebradizas, se astillaran y se rompieran? Limaba y pensaba. De vez en cuando soplaba y se quitaba del pecho las limaduras.\n\n\u00a1Y la buena de Mrs. Atkinson! Morland sonri\u00f3. No ten\u00eda importancia a nivel nacional y menos a\u00fan en pol\u00edtica europea, pero ser\u00eda una bonita nota a pie de p\u00e1gina en la historia de sus esfuerzos pol\u00edticos si con el fracaso del Jubilee Project tambi\u00e9n quedara da\u00f1ado ese fr\u00edgido personaje con su manguito. S\u00f3lo por la cuota femenina hab\u00eda obtenido ese puesto al que \u00e9l aspiraba y para el que en un principio hab\u00eda sido barajado como favorito. George Morland no lo admitir\u00eda nunca, no ser\u00eda exactamente eso que \u00e9l denominar\u00eda \u00abnecesidad objetiva\u00bb, pero la mera idea de hacer caer a Mrs. Atkinson le gustaba mucho.\n\nSi reflexionaba bien sobre todo ello, ten\u00eda una idea clara de lo que hab\u00eda que hacer ahora. Unas cuantas citas para almorzar con compa\u00f1eros competentes de otras direcciones generales, mejor en Martin's, que ten\u00eda un bonito jard\u00edn, all\u00ed los colegas que fumaban estaban felices y mucho m\u00e1s relajados, m\u00e1s abiertos, all\u00ed \u00e9l tendr\u00eda que servirles argumentos, cortados a medida, que los intranquilizasen y los pusieran en contra del proyecto.\n\nMorland cambi\u00f3 de lima. Despu\u00e9s de la gruesa, ahora la fina.\n\nEso originar\u00eda de entrada cierta din\u00e1mica propia, comentarios, rumores, y ese desasosiego habr\u00eda que encaminarlo con prudencia en una direcci\u00f3n, de forma que surgiera la necesidad de un grupo de trabajo en el Consejo para discutir a fondo el problema y solucionarlo.\n\n\u00abSolucionar el problema.\u00bb Tambi\u00e9n en cuanto a esa expresi\u00f3n era conservador George Morland. En los \u00faltimos a\u00f1os hab\u00eda tenido lugar en la casa una asombrosa desviaci\u00f3n del lenguaje, y a nadie le llamaba la atenci\u00f3n, o al menos nadie lo hab\u00eda comentado ni cuestionado. Si antes se dec\u00eda resolver un problema, ahora eso era llevar el problema hacia una soluci\u00f3n. Si antes se hab\u00eda dicho tomar una decisi\u00f3n, ahora se dec\u00eda provocar una decisi\u00f3n. En lugar de analizar algo, ahora hab\u00eda que someter a an\u00e1lisis. Si antes era se toman medidas, ahora se ponen en marcha medidas. Se podr\u00eda hacer todo un diccionario del nuevo \u00abcomitology language\u00bb, y era asombroso c\u00f3mo en aquella babilonia determinadas tendencias ling\u00fc\u00edsticas pasaban de inmediato a formar parte de todas las lenguas. George Morland era lo bastante sensible para percatarse de eso. No era semi\u00f3logo, ni hermen\u00e9utico, ni ling\u00fcista, pero ten\u00eda la clara sensaci\u00f3n de que esa evoluci\u00f3n era una se\u00f1al, ten\u00eda un significado sintom\u00e1tico del estado de la Comisi\u00f3n, de su desvalimiento, de su entumecimiento. Poner algo en marcha era inequ\u00edvocamente algo distinto, algo m\u00e1s defensivo que hacer algo. Esas expresiones delataban que ya no se trataba de una meta, sino s\u00f3lo del camino. As\u00ed m\u00e1s o menos lo ve\u00eda \u00e9l. Pero no lo aceptaba. \u00c9l se quedaba con el cl\u00e1sico solucionar un problema, y en aquel caso eso significaba, sin rodeos: Kill the project, kill Mrs. Atkinson.\n\nAhora cogi\u00f3 el suave cepillo de u\u00f1as para eliminar los diminutos residuos de polvo que pudiera haber dejado la lima, luego sac\u00f3 del caj\u00f3n del escritorio el esmalte incoloro y se pint\u00f3 con entusiasmo las u\u00f1as, pensando con un discreto asomo de sarcasmo en Mrs. Atkinson, que escond\u00eda sus fr\u00edos dedos con las u\u00f1as mordidas en un manguito.\n\nY ya dos semanas despu\u00e9s pudo adherirse, sin inspirar la menor sospecha, al deseo general de que se formara un grupo de trabajo del Consejo, dependiente del CAC (\u00abCultural Affairs Committee\u00bb).\n\nMrs. Atkinson lo supo enseguida: eso era el final del proyecto, que ella misma tampoco hab\u00eda deseado realmente. Hab\u00eda sido s\u00f3lo una iniciativa de la Cultura. Hacia fuera el proyecto estaba vinculado totalmente a esa Xenopoulou, que se hab\u00eda dado con aquello una enorme importancia. Xeno por su parte no estaba tan segura, pensaba que si todav\u00eda hab\u00eda necesidad de discutirlo, Martin deb\u00eda ocuparse de eso. Puesto que el proyecto hab\u00eda sido idea de Martin Susman. Y ella le hab\u00eda encomendado todo el trabajo de organizaci\u00f3n.\n\nY Martin no estaba all\u00ed.\n\nEn los terrenos de la residencia de mayores Maison Hanssens hab\u00eda habido originariamente una f\u00e1brica de l\u00e1pidas. Piet Hanssens, cuarta generaci\u00f3n de tallistas de la piedra, no tuvo descendencia y no encontr\u00f3 a nadie que se hiciera cargo de la f\u00e1brica y siguiera con ella. Cuando a la edad de setenta y tres a\u00f1os pas\u00f3 una humillante odisea por hospitales y asilos debido a su silicosis y ya no pudo trabajar, dej\u00f3 en testamento su casa, la nave del taller y los terrenos a la Ville de Bruxelles, con la condici\u00f3n de que la ciudad, o la regi\u00f3n de Bruselas, construyera en ese lugar una residencia digna para mayores. Luego cerr\u00f3 los ojos. La ciudad, en apuros econ\u00f3micos, acept\u00f3 la herencia, pero pasaron a\u00f1os hasta que por fin, con ayuda de la UE, del Fondo Europeo de Desarrollo Regional y del Fondo Social Europeo, la antigua f\u00e1brica de l\u00e1pidas qued\u00f3 convertida y ampliada en un moderno \u00abcentro de atenci\u00f3n a la tercera edad\u00bb. En la antigua nave del taller se encontraba ahora el comedor, en la antigua sala de exposici\u00f3n, la biblioteca y la sala de estar de la residencia, fuera de eso no quedaba nada de la sustancia de la construcci\u00f3n originaria, nada recordaba la historia de aquel lugar.\n\nCasi nada. Detr\u00e1s de la salida lateral, junto a la biblioteca, que en realidad era una puerta de emergencia, estaban colocadas verticalmente sobre una franja de hierba una docena escasa de l\u00e1pidas lisas, sin nada grabado, restos de la antigua f\u00e1brica. No estaba claro si esas piedras simplemente hab\u00edan quedado olvidadas o las hab\u00edan dejado intencionadamente all\u00ed como reminiscencia de la historia del lugar. Normalmente, aparte del portero, monsieur Hugo, que tambi\u00e9n cortaba el c\u00e9sped que hab\u00eda en torno a la casa, nadie ten\u00eda ocasi\u00f3n de verlas.\n\nY entonces David de Vriend las descubri\u00f3. Hab\u00eda querido salir de la casa, no recordaba por qu\u00e9, durante un momento se qued\u00f3 indeciso cuando sali\u00f3 del ascensor en la planta baja; qu\u00e9 quer\u00eda, ad\u00f3nde quer\u00eda ir, busc\u00f3 la salida hacia la izquierda en lugar de hacia la derecha, se encontr\u00f3 de pronto delante de la salida de emergencia, oprimi\u00f3 el gran travesa\u00f1o rojo con el que abr\u00eda y se encontr\u00f3 delante de las l\u00e1pidas, que contempl\u00f3 asombrado: \u00e9l no quer\u00eda ir al cementerio, s\u00f3lo comer algo. Vio que en esas l\u00e1pidas no hab\u00eda nombres: \u00bfun cementerio de gente an\u00f3nima? Pero \u00bfpor qu\u00e9 tan pocas l\u00e1pidas? \u00bfPor qu\u00e9 un cementerio tan peque\u00f1o? Miles, cientos de miles de personas carec\u00edan ya de nombre cuando murieron, los nombres de millones de personas fueron eliminados antes de ser enviados a la muerte, hab\u00edan sido convertidos en n\u00fameros, aunque eran innumerables, y all\u00ed \u2014\u200cmir\u00f3 y empez\u00f3 a contar\u2014 eran s\u00f3lo: dos, tres, cuatro, cinco... y un cuidador le cogi\u00f3 del brazo; al abrir la puerta de emergencia, De Vriend hab\u00eda disparado una alarma.\n\n\u00bfQu\u00e9 hace usted ah\u00ed? \u00bfQuer\u00eda salir al exterior? \u00bfS\u00ed? \u00c9sta no es la puerta. Venga, yo le llevo: \u00bfad\u00f3nde quer\u00eda ir?\n\nDavid de Vriend dijo entonces con voz firme que quer\u00eda ir a comer.\n\n\u00bfAl comedor?\n\n\u00a1No! Fuera, al restaurante, al... se\u00f1al\u00f3 con el dedo \u00edndice: Al de all\u00ed, all\u00ed. Ah\u00ed al lado.\n\nPoco despu\u00e9s estaba sentado en Le Rustique, la camarera le trajo una copa de vino tinto y \u00e9l estaba avergonzado. Fue otro momento de lucidez. Y la lucidez se llamaba verg\u00fcenza. Se pregunt\u00f3 por qu\u00e9 \u00e9l...\n\nClaro que sab\u00eda por qu\u00e9 \u00e9l...\n\nY se puso furioso. As\u00ed no quer\u00eda...\n\nHac\u00eda un calor insoportable. De Vriend se quit\u00f3 la americana, se remang\u00f3 las mangas de la camisa, se enjug\u00f3 con el pa\u00f1uelo el sudor de la frente. No pod\u00eda pensar. Hab\u00eda demasiado ruido. En la mesa de al lado, el parloteo de una familia numerosa, los gritos de los ni\u00f1os. Irritado, mir\u00f3 en esa direcci\u00f3n. Y sonri\u00f3. Era un reflejo. Siempre sonre\u00eda cuando ve\u00eda ni\u00f1os. Feliz o comprensivo, o simplemente por educaci\u00f3n.\n\nVio entonces que una ni\u00f1a lo miraba con curiosidad. \u00bfQu\u00e9 edad tendr\u00eda? Unos ocho a\u00f1os. Sus miradas se encontraron. Entonces ella se acerc\u00f3 a su mesa.\n\n\u00a1No, por Dios!, pens\u00f3.\n\nCool!, dijo la ni\u00f1a se\u00f1alando el n\u00famero tatuado en el brazo de De Vriend.\n\n\u00bfEs de verdad?\n\nS\u00ed, dijo \u00e9l poni\u00e9ndose la chaqueta.\n\nCool!, dijo ella, ense\u00f1\u00e1ndole el tatuaje adhesivo de su brazo.\n\nCuatro caracteres de escritura china.\n\nPero no es de verdad, dijo ella. De verdad no me dejan todav\u00eda.\n\nSabes lo que significa, pregunt\u00f3 De Vriend. \u00bfNo? Pero \u00bfte gusta? \u00bfS\u00ed?\n\nFue tocando los signos con el dedo.\n\nEl primero: Todos\n\nEl segundo: los hombres\n\nEl tercero: son\n\nEl cuarto: cerdos\n\n...\n\nHe le\u00eddo mal, dijo De Vriend y toc\u00f3\n\nel segundo: los viejos\n\nel cuarto: callados.\n\nEl profesor Alois Erhart sigui\u00f3 a Ant\u00f3nio Oliveira Pinto al meeting room. Vio a los miembros del reflection group sentados en semic\u00edrculo en torno a la silla que \u00e9l iba a ocupar: un semic\u00edrculo de port\u00e1tiles y tabletas, detr\u00e1s miradas dirigidas hacia abajo, hacia los monitores, o\u00eda el silencioso y r\u00e1pido tectec de los teclados.\n\nErhart se sent\u00f3. Poco a poco las miradas se dirigieron a \u00e9l.\n\n\u00bfAll\u00ed s\u00f3lo iba a tener lugar una discusi\u00f3n? Era enga\u00f1oso. Se trataba de su ejecuci\u00f3n, del final de su vida en el mundo de los expertos. Pero \u00bfno era eso a lo que Erhart quer\u00eda llegar? \u00bfQu\u00e9 se dice en espera de una ejecuci\u00f3n? \u00daltimas palabras. Ahora hab\u00eda llegado el momento, pens\u00f3, a eso justamente quer\u00eda llegar desde hac\u00eda mucho tiempo: \u00faltimas palabras.\n\n\u00a1Con qu\u00e9 alegr\u00eda salud\u00f3 el se\u00f1or Pinto a todos los presentes! S\u00f3lo el profesor griego que ense\u00f1aba en Oxford escribi\u00f3 a\u00fan un momento algo en su port\u00e1til, ten\u00eda que ser muy importante y urgente, en cualquier caso, una demostraci\u00f3n de importancia y urgencia. Erhart sonri\u00f3, dijo: \u00bfHa terminado, se\u00f1or colega? \u00bfPodemos empezar?\n\n\u00daltimas palabras. Era una historia que se remontaba a la primera publicaci\u00f3n cient\u00edfica de Erhart, que hab\u00eda aparecido en la Revista Trimestral sobre Investigaci\u00f3n Econ\u00f3mica de la Universidad de Viena. En aquella \u00e9poca era a\u00fan ayudante de c\u00e1tedra. En esa publicaci\u00f3n expon\u00eda la teor\u00eda de la econom\u00eda pol\u00edtica posnacional de Armand Moens y la cimentaba con alg\u00fan nuevo material estad\u00edstico sobre la evoluci\u00f3n del comercio mundial. Henchido de orgullo, Erhart envi\u00f3 un ejemplar de su ensayo a Armand Moens, quien, ante la estupefacci\u00f3n de Erhart, respondi\u00f3 a vuelta de correo. Ahora Erhart llevaba consigo esa carta de respuesta, y un extracto de ella era parte de la breve ponencia que daba ahora.\n\nErhart empez\u00f3 con una cita de Armand Moens: \u00abEl siglo XX habr\u00eda debido ser la transformaci\u00f3n de la econom\u00eda nacional del siglo XIX en la econom\u00eda de la humanidad del siglo XXI. Eso qued\u00f3 frustrado de forma tan atroz y criminal que despu\u00e9s resurgi\u00f3 ese anhelo y con m\u00e1s apremio. Pero s\u00f3lo en la conciencia de una peque\u00f1a \u00e9lite pol\u00edtica cuyos sucesores pronto dejaron de entender ambas cosas: la criminal energ\u00eda del nacionalsocialismo y las resoluciones, marcadas por esa experiencia, que ya hab\u00edan sido llevadas a la pr\u00e1ctica\u00bb.\n\nAlgunos tecleaban en sus port\u00e1tiles. Erhart no sab\u00eda si tomaban apuntes o contestaban e-mails. Le daba igual. Ten\u00eda a\u00fan trece o quince minutos, ten\u00eda tiempo, su momento llegar\u00eda.\n\nErhart repas\u00f3 muy brevemente la evoluci\u00f3n de la econom\u00eda global hasta la Primera Guerra Mundial, y con algunas cifras explic\u00f3 el retroceso radical con el nacionalismo y el fascismo, y vio que ya ahora, en el minuto cinco de su conferencia, algunos se aburr\u00edan. Nada les aburr\u00eda tanto como el recuerdo del fascismo y del nacionalsocialismo. Era un cap\u00edtulo oscuro, el libro que contiene ese cap\u00edtulo est\u00e1 cerrado, otro libro est\u00e1 abierto hace tiempo, esa contabilidad es ahora estupenda, excepto en algunos pa\u00edses remolones, all\u00ed hay que obrar con energ\u00eda, \u00e9sa es nuestra misi\u00f3n, nosotros somos los nuevos contables.\n\nS\u00f3lo un ejemplo, dijo Erhart, para el par\u00e9ntesis de 1914 a 1945: si el comercio mundial sigue desarroll\u00e1ndose linealmente en los pr\u00f3ximos a\u00f1os como en los pasados veinte a\u00f1os \u2014\u200caunque ni siquiera podemos estar seguros de que sea as\u00ed\u2014, entonces en 2020 se habr\u00e1 alcanzado el volumen del comercio mundial de 1913. O sea, nos acercamos lentamente al estado de la globalizaci\u00f3n de la preguerra.\n\n\u00a1Eso es absurdo! \u00a1Eso no puede ser!\n\n\u00a1Se despertaban! \u00a1Ay, si supieran que todav\u00eda faltaba mucho para que estuvieran del todo despiertos!\n\n\u00bfPor qu\u00e9 dice usted \u00ababsurdo\u00bb, se\u00f1or colega? Es material estad\u00edstico comprobado, dijo Erhart. Yo s\u00f3lo quer\u00eda record\u00e1rselo, no pensaba que no lo conocieran.\n\nLuego Erhart ofreci\u00f3 otras tres citas de Moens con las que del desarrollo de la econom\u00eda transnacional deduc\u00eda la necesidad de nuevas instituciones democr\u00e1ticas que deb\u00edan reemplazar a los parlamentos nacionales. Bueno, el rodeo se hab\u00eda quedado muy reducido, pero Erhart ya no ten\u00eda mucho tiempo y quer\u00eda llegar al shock.\n\nRespir\u00f3 hondo, luego dijo: Y ahora quiero contarles algo. He citado varias veces a Armand Moens. Ustedes lo han aceptado m\u00e1s o menos. Han pensado quiz\u00e1, okay, Moens no es mainstream, son citas de un conocido economista, y ustedes, se\u00f1oras y se\u00f1ores, en sus trabajos y en sus intervenciones de palabra, citan a otros, citan los nombres que ahora son mainstream. No buscan la verdad, porque ven en el mainstream el \u00faltimo estadio de la verdad. \u00a1Esperen! \u00a1Esperen! No digo que yo sepa qu\u00e9 es la verdad. Digo s\u00f3lo que tenemos que pregunt\u00e1rnoslo. Y digo que de ninguna manera nos acercamos m\u00e1s a ella si nos orientamos por el esp\u00edritu del siglo, o sea, por los intereses actuales de pocos, para los que la mayor\u00eda de las personas s\u00f3lo son un asiento a deducir en su contabilidad. Es igual. Lo que quiero contar es esto: en mi primera publicaci\u00f3n cient\u00edfica analic\u00e9 la teor\u00eda de Armand Moens. Lleno de orgullo, le envi\u00e9 el art\u00edculo. Yo no lo esperaba, pero me respondi\u00f3. Quiero leerles un pasaje de su carta: Querido se\u00f1or Erhart, etc\u00e9tera, etc\u00e9tera, s\u00ed, aqu\u00ed: Lo que ha hecho usted es lisonjero para m\u00ed y le hace merecedor de elogios. Me ha citado d\u00e1ndome su aprobaci\u00f3n y manteniendo al mismo tiempo todas las reglas de la citaci\u00f3n. Lo que usted ha aportado es una perfecta primera publicaci\u00f3n, conforme a las reglas del juego de nuestro oficio. Pero imag\u00ednese que ahora usted se muere y esta publicaci\u00f3n fuera lo \u00fanico que quedara de usted. \u00bfSeguir\u00eda estando entonces contento con ella? \u00bfNo tiene usted pensamientos, visiones que sobrepasen con mucho lo que ha citado? \u00bfEs este art\u00edculo realmente lo que usted quiere comunicar al mundo, lo que s\u00f3lo usted puede decir, lo que ha de seguir surtiendo efecto caso de que usted ya no tenga posibilidad de decir nada? Yo digo: \u00a1NO!\n\nNO escrito con may\u00fasculas, dijo Erhart.\n\nY ahora le digo algo m\u00e1s. Si usted se considera disc\u00edpulo m\u00edo, como escribe en su carta, entonces debe aprender ante todo lo siguiente: en cualquier cosa que diga p\u00fablicamente, en cualquier cosa que publique, ha de partir de la idea de que podr\u00edan ser sus \u00faltimas palabras. Imag\u00ednese que cuando d\u00e9 su pr\u00f3xima conferencia sabe que inmediatamente despu\u00e9s va a morir: \u00bfqu\u00e9 dir\u00eda en ese caso? Todav\u00eda puede decir algo una vez, una \u00faltima vez, entre la vida y muerte, \u00bfqu\u00e9 ser\u00eda? Estoy seguro de que dir\u00eda algo distinto de lo que ha escrito en este art\u00edculo. Y si no es as\u00ed, entonces tampoco habr\u00eda tenido que escribirlo. \u00bfMe comprende? Hay un sinn\u00famero de frases con las que se puede sustentar la propia vida, conquistar un cargo y defenderlo, frases que al final entran en obras completas y en escritos conmemorativos, yo no digo que todas sean equivocadas o innecesarias, pero lo que necesitamos con urgencia son frases con la aspiraci\u00f3n existencial de las \u00faltimas palabras, palabras que luego no dormitan en un archivo sino que despiertan a los hombres, quiz\u00e1 incluso a gente que hoy no ha nacido a\u00fan. As\u00ed pues, querido se\u00f1or Erhart, env\u00edeme otro texto. Me gustar\u00eda saber lo que usted escribe bajo esta premisa: \u00e9sta es mi \u00faltima oportunidad de decir algo. Y luego le dir\u00e9 si tiene sentido que siga publicando.\n\nErhart levant\u00f3 la vista. No cont\u00f3 que despu\u00e9s de esa carta fue incapaz de escribir nada durante semanas, hasta que se enter\u00f3 de que Armand Moens hab\u00eda muerto. Vio que en la sala hab\u00eda un extra\u00f1o ambiente que \u00e9l no sab\u00eda valorar. Ant\u00f3nio Pinto exclam\u00f3: Muchas gracias por esa interesante... mmm, sugerencia, profesor Erhart, \u00bfalguien quiere...?\n\nUn momento, por favor, dijo Erhart, todav\u00eda no he terminado.\n\nPerd\u00f3n, dijo Pinto, as\u00ed que a\u00fan vienen last words, por as\u00ed decirlo. \u00a1Adelante, profesor!\n\nHe tratado de mostrar, dijo Erhart, que necesitamos algo completamente nuevo, una democracia posnacional, para poder configurar un mundo en el que ya no haya econom\u00eda nacional. En esta tesis, que yo defender\u00e9 hasta mi muerte, hay dos problemas. Primero: ni siquiera ustedes, las \u00e9lites de las ciencias econ\u00f3micas internacionales, miembros de innumerables think-tanks y de gremios asesores de Estados de la UE pueden imaginar algo as\u00ed, pueden aceptar esa idea. Todos ustedes siguen pensando con los criterios de presupuestos y democracias nacionales. Como si no hubiera un mercado y un sistema monetario comunes, como si no hubiera libertad de residencia para los flujos financieros y las cadenas de creaci\u00f3n de riqueza. Ustedes creen en serio que algo mejora en Europa si se sanea el presupuesto griego, o sea, un presupuesto nacional, de una manera que hace derrumbarse en Grecia los sistemas de sanidad, de educaci\u00f3n y de pensiones. Entonces para ustedes todo est\u00e1 en regla. \u00bfSaben cu\u00e1l es su problema? Son gatos en una jaula y ni siquiera es seguro que existan. Como realidad, ustedes y sus teor\u00edas s\u00f3lo se presuponen. Esa suposici\u00f3n previa hace posible que se puedan hacer facturas, y como esas facturas son posibles, eso vale enseguida como prueba de que esas cuentas reflejan la realidad y no puede ser de otra manera. \u00a1Esperen, esperen! Podr\u00e1n irritarse enseguida, s\u00f3lo quisiera a\u00f1adir unas frases m\u00e1s. Okay, lo reconozco: ustedes son expertos del statu quo. Nadie lo conoce mejor que ustedes, nadie tiene m\u00e1s saber de insider que ustedes. Pero no tienen ni idea de historia ni representaci\u00f3n alguna del porvenir. \u00bfQue no? Espere, profesor Stephanides, una pregunta: si usted hubiera vivido en la \u00e9poca de la sociedad griega esclavista y le hubieran preguntado si se imaginaba un mundo sin esclavos, usted habr\u00eda dicho: No. En absoluto. \u00a1Usted habr\u00eda dicho que la sociedad esclavista es la condici\u00f3n previa de la democracia! \u00bfNo es cierto? No, no, profesor Matthews, espere. Por favor. Yo me lo imagino a usted en Manchester, en la \u00e9poca del capitalismo manchesteriano. Si le hubieran preguntado entonces qu\u00e9 hab\u00eda que hacer para asegurar a Manchester como emplazamiento industrial usted habr\u00eda dicho: De ning\u00fan modo se debe ceder ante esos sindicatos que en lugar de una jornada de catorce horas exigen la jornada de ocho horas, la prohibici\u00f3n del trabajo infantil y que quieren hasta una renta de vejez e invalidez, porque eso har\u00eda peligrar totalmente lo atractivo de ese emplazamiento... y, profesor Matthews, \u00bfahora qu\u00e9? \u00bfExiste a\u00fan Manchester? Y prescinda de esa sonrisa desde\u00f1osa, se\u00f1or Mosebach. Con esa radicalidad con la que usted defiende hoy intereses alemanes habr\u00eda ido a parar, si hubiera nacido antes, entre los acusados de los juicios de N\u00faremberg. Y ni siquiera ve usted eso con claridad. Pero no tiemble, querido Mosebach, a las personas como usted siempre se las indulta, porque eso s\u00ed que lo ve cualquier perito: no lo hizo usted con mala intenci\u00f3n, usted s\u00f3lo estaba ofuscado. Usted es un mero seguidor. Y \u00e9se es el problema de todos ustedes. Todos ustedes no hacen sino marchar al ritmo que les marcan. Se indignan si hoy alguien les dice esto, pero son exactamente los mismos que ma\u00f1ana, si hay una cat\u00e1strofe y despu\u00e9s un proceso, dir\u00e1n para disculparse que s\u00f3lo han sido seguidores, peque\u00f1as ruedecillas en el engranaje. Y ahora les pregunto: \u00bfsaben siquiera qu\u00e9 discutimos? Discutimos sobre el desarrollo de la Uni\u00f3n Europea, una comunidad posnacional nacida del reconocimiento del error hist\u00f3rico que ustedes consideran tambi\u00e9n \u00abnormal\u00bb: as\u00ed es el mundo, as\u00ed son los hombres, quieren definirse por la pertenencia a una naci\u00f3n, quieren definir qui\u00e9n pertenece a ella y qui\u00e9nes son los otros, y quieren sentirse mejores que otros y quieren, si tienen miedo de otros, partirles el cr\u00e1neo, eso es completamente normal, as\u00ed son los hombres, lo importante es que el presupuesto nacional est\u00e9 en el marco de los criterios acordados.\n\nGracias, muchas gracias, profesor Erhart, dijo Ant\u00f3nio Pinto. \u00bfHay preguntas de...?\n\nPor favor, Mr. Pinto, no he terminado. Dos minutos m\u00e1s, se lo ruego.\n\nLa cartera escolar se le hab\u00eda escurrido del regazo y hab\u00eda ca\u00eddo al suelo, asimismo las hojas con su ponencia, llevaba ya mucho tiempo hablando sin manuscrito, su ponencia se le hab\u00eda ido por otros caminos, pero la meta a la que quer\u00eda llegar, el punto principal de su radical intervenci\u00f3n iba a decirlo a toda costa. Dos minutos m\u00e1s, por favor, para mi resumen. No, para mi visi\u00f3n. Realmente last words. Okay? Okay! As\u00ed pues, resumo primero: Estados nacionales en competencia no son una uni\u00f3n, aunque tengan un mercado com\u00fan. Estados nacionales que compiten entre s\u00ed en una uni\u00f3n bloquean ambas cosas, la pol\u00edtica europea y estatal. \u00bfQu\u00e9 har\u00eda falta ahora? El ulterior desarrollo hacia una uni\u00f3n social, hacia una uni\u00f3n fiscal: o sea, la elaboraci\u00f3n de condiciones b\u00e1sicas que conviertan la Europa de colectivos en competencia en una Europa de ciudadanos soberanos y de iguales derechos. \u00c9sa era la idea, eso era aquello con lo que hab\u00edan so\u00f1ado los fundadores del proyecto de la Uni\u00f3n Europea, porque ellos ten\u00edan sus experiencias. Pero todo esto no se puede alcanzar mientras se siga atizando la conciencia nacional contra todas las experiencias hist\u00f3ricas y mientras el nacionalismo siga siendo, sin alternativa alguna, la oferta de identificaci\u00f3n que se hace a los ciudadanos. \u00bfC\u00f3mo se puede, entonces, fomentar la conciencia de que los hombres de este continente son ciudadanos europeos? Habr\u00eda diversas peque\u00f1as iniciativas. Por ejemplo, se podr\u00edan sustituir todos los pasaportes nacionales por un pasaporte europeo. Un pasaporte de la Uni\u00f3n Europea en el que se consigne el lugar de nacimiento, pero no la nacionalidad. Creo que ya s\u00f3lo esto tendr\u00eda alg\u00fan efecto en la conciencia de la generaci\u00f3n que crezca con un pasaporte as\u00ed. Y eso ni siquiera comportar\u00eda gastos.\n\nErhart vio que los idealistas del grupo meneaban dudosos la cabeza pero estaban dispuestos a reflexionar sobre tal idea.\n\nEso, sin embargo, no es suficiente, continu\u00f3. Necesitamos tambi\u00e9n, y sobre todo, un fuerte s\u00edmbolo de concordia, tiene que ser un proyecto concreto com\u00fan, que, como esfuerzo com\u00fan, ponga en el mundo lo com\u00fan, necesitamos algo que sea de todos y que nos una como ciudadanos de la Uni\u00f3n Europea, porque fueron los ciudadanos de la Europa com\u00fan a todos quienes la quisieron y la gestaron y no la recibieron simplemente en herencia. Una primera, audaz, grande y consciente obra cultural de la historia posnacional, y ha de ser al mismo tiempo de importancia pol\u00edtica y de fuerza simb\u00f3lica psicol\u00f3gica. \u00bfAd\u00f3nde quiero ir a parar?\n\nErhart vio que por lo menos ahora algunos daban la impresi\u00f3n de estar esperando con atenci\u00f3n lo que vendr\u00eda despu\u00e9s. Respir\u00f3 hondo y dijo: La Uni\u00f3n Europea debe construir una capital, una capital nueva, una capital ideal, planificada.\n\nEl profesor Stephanides sonri\u00f3: La discusi\u00f3n sobre qu\u00e9 ciudad de Europa obtendr\u00eda la categor\u00eda de capital de la Uni\u00f3n est\u00e1 muerta y enterrada. Eso es agua pasada. Fue una decisi\u00f3n sensata no dar ese t\u00edtulo a ninguna ciudad, tampoco a Bruselas, sino repartir las diferentes instituciones europeas por diferentes ciudades en diferentes pa\u00edses.\n\nNo me ha entendido usted, colega Stephanides. Yo no he dicho que una ciudad haya de recibir el t\u00edtulo de capital. Tengo muy claro que eso s\u00f3lo seguir\u00eda fomentando el nacionalismo en los pa\u00edses cuyos ciudadanos se sentir\u00edan dirigidos por una capital ajena que al mismo tiempo es la capital de otra naci\u00f3n. \u00c9se es tambi\u00e9n el problema de Bruselas. Aunque yo al principio consider\u00e9 razonable a Bruselas como capital de la UE: la capital de un Estado nacional fracasado, la capital de un pa\u00eds con tres lenguas oficiales. Pero no, pens\u00e9: Europa debe construir una nueva capital. Una nueva ciudad cuya construcci\u00f3n sea obra de la Uni\u00f3n y no una antigua capital imperial o nacional en la que la Uni\u00f3n s\u00f3lo estuviera en realquiler.\n\n\u00bfY d\u00f3nde quiere usted construir esa ciudad? \u00bfEn qu\u00e9 tierra de nadie? \u00bfEn el centro geogr\u00e1fico del continente? \u00bfLa naci\u00f3n m\u00e1s rica y poderosa de Europa no logra siquiera construir un aeropuerto para una capital y usted sue\u00f1a sin m\u00e1s con construir una ciudad entera? Mosebach sacudi\u00f3 la cabeza con una sonrisa indulgente.\n\n\u00bfO sea, una suerte de Brasilia europea? Como experimento mental me parece interesante, dijo Dana Dinescu, la polit\u00f3loga rumana que ense\u00f1aba en Bolonia.\n\nPor supuesto, dijo Erhart, que esa ciudad no se puede construir en una tierra de nadie. En Europa ya no hay tierra de nadie, no hay ni un metro cuadrado de suelo que no tenga su historia. Y por eso la capital europea ha de construirse, evidentemente, en un lugar cuya historia haya sido determinante para la idea de la uni\u00f3n de Europa, una historia que quiera superar a nuestra Europa pero que al mismo tiempo no pueda caer jam\u00e1s en el olvido. Tiene que ser un lugar en el que la historia se sienta y se viva incluso cuando haya muerto el \u00faltimo que la haya vivido o sobrevivido. Un lugar como antorcha perpetua para la futura pol\u00edtica en Europa.\n\nErhart mir\u00f3 a los asistentes. \u00bfHab\u00eda alguien que ya sospechara lo que iba a venir? Dana sonre\u00eda y lo observaba con curiosidad. Stephanides miraba a la ventana con afectado aburrimiento. Mosebach tecleaba algo en su port\u00e1til. Pinto miraba el reloj. Pero diez segundos despu\u00e9s todos miraban a Erhart con la boca abierta. Desconcertados. Trece segundos despu\u00e9s, Erhart, el prestigioso profesor em\u00e9rito, era historia como miembro del think-tank New Pact for Europe.\n\nDijo: Y por eso la Uni\u00f3n tiene que construir su capital en Auschwitz. En Auschwitz ha de surgir la nueva capital europea, proyectada y levantada como ciudad del futuro, al mismo tiempo la ciudad que nunca puede olvidar. \u00abNunca m\u00e1s Auschwitz\u00bb es la base sobre la que se levant\u00f3 el edificio de la Uni\u00f3n Europea. Al mismo tiempo es una promesa para el futuro. Debemos construir ese futuro como centro tangible y funcional. \u00bfTienen ustedes el valor de reflexionar sobre esta idea? \u00c9se ser\u00eda un resultado de nuestro reflection group: una recomendaci\u00f3n al presidente de la Comisi\u00f3n para que se convoque un concurso de arquitectura para el proyecto y la construcci\u00f3n de una capital europea en Auschwitz.\n\nAlois Erhart puso su maleta sobre la cama del hotel Atlas para hacer el equipaje. Le ard\u00eda la cara, pensaba que ten\u00eda fiebre. Lo que acababa de vivir le quemaba. Descorri\u00f3 la cortina y mir\u00f3 por la ventana, hacia abajo, a la plaza. C\u00e1mara lenta, pens\u00f3. All\u00e1 abajo hab\u00eda un hervidero de gente a c\u00e1mara lenta. Todo se mov\u00eda despac\u00edsimo en un calor opresivo, como si los movimientos de todos fueran un movimiento com\u00fan con un objetivo com\u00fan... que deb\u00eda posponerse durante el mayor tiempo posible.\n\nErhart hab\u00eda aprendido que el proyecto de uni\u00f3n europea se basaba en este consenso: el nacionalismo y el racismo hab\u00edan llevado a Auschwitz y nunca deb\u00edan repetirse. Ese \u00ab\u00a1Nunca m\u00e1s!\u00bb era la base de todo lo dem\u00e1s, la cesi\u00f3n de la soberan\u00eda de los Estados miembros a instituciones supranacionales y la configuraci\u00f3n consciente de una entreverada econom\u00eda transnacional. \u00c9sa era tambi\u00e9n la base de la obra principal de Armand Moens, que en cuanto economista empez\u00f3 a reflexionar sobre c\u00f3mo se pod\u00eda organizar pol\u00edticamente una econom\u00eda posnacional. A esa cuesti\u00f3n hab\u00eda dedicado tambi\u00e9n su vida como cient\u00edfico el profesor Erhart. Su vida, la vida de su maestro, la historia contempor\u00e1nea vivida, la conservaci\u00f3n de la paz social, el futuro del continente, todo eso descansaba sobre dos palabras: \u00ab\u00a1Nunca m\u00e1s!\u00bb. As\u00ed lo ve\u00eda Erhart. \u00ab\u00a1Nunca m\u00e1s!\u00bb es una promesa de eternidad, es una exigencia que aspira a tener validez perpetua. Ahora mor\u00edan los \u00faltimos que hab\u00edan sobrevivido a lo que nunca deber\u00eda volver a ocurrir. \u00bfY despu\u00e9s? \u00bfTen\u00eda tambi\u00e9n la eternidad una fecha de caducidad? Ahora hab\u00eda asumido esa responsabilidad una generaci\u00f3n que al menos en discursos oficiales a\u00fan se sent\u00eda obligada a pronunciar con voz contenida y apremiante ese \u00ab\u00a1Nunca m\u00e1s!\u00bb. Pero \u00bfy despu\u00e9s? Cuando haya muerto el \u00faltimo que pueda testimoniar a partir de qu\u00e9 shock quiso Europa reinventarse: entonces, para los vivos, Auschwitz estar\u00e1 tan hundido en el tiempo como las guerras p\u00fanicas.\n\nCuando Alois Erhart necesitaba una raz\u00f3n poderosa, objetiva, para explicarse sus dolores y entregarse inerme a ellos, entonces pensaba en esas grandes categor\u00edas pol\u00edticas e hist\u00f3rico-filos\u00f3ficas. Eso era realmente el Weltschmerz, la honda melancol\u00eda contra la que no existe remedio alguno.\n\nLos pragm\u00e1ticos s\u00ed sab\u00edan de remedios. Como su padre. En 1942, el padre de Erhart fue llamado a filas con la orden de incorporarse a la OrPo, Batall\u00f3n 316 de la Ordnungspolizei, que fue trasladada a Pozna\u0144 para llevar a cabo all\u00ed, con el t\u00edtulo de \u00ablucha contra partisanos\u00bb, fusilamientos de jud\u00edos. Alois Erhart hab\u00eda encontrado esa orden de alistamiento despu\u00e9s de la muerte de su padre, en una carpeta con papeles que hab\u00eda en su escritorio. Ya antes de la anexi\u00f3n de Austria, el padre se hab\u00eda hecho miembro del Partido Nacionalsocialista, despu\u00e9s pas\u00f3 a ser el proveedor del equipamiento deportivo y militar de la Liga de Muchachas Alemanas, de las Juventudes Hitlerianas y de la Turnerbund, la federaci\u00f3n de gimnasia. As\u00ed, en su calidad de \u00abimportante para la guerra\u00bb pudo sustraerse largo tiempo al alistamiento. Cuando tuvo que cerrar la tienda fue inevitable que lo llamaran a filas. Pero debido a sus buenos contactos y a sus m\u00e9ritos no lo enviaron al frente, sino a la retaguardia con un batall\u00f3n de la polic\u00eda.\n\n\u00bfSu padre estuvo durante la guerra en Pozna\u0144? \u00bf\u00c9l, Alois Erhart, vino al mundo en el almac\u00e9n de la tienda mientras su padre, como \u00abpolic\u00eda\u00bb, ejecutaba en Polonia a jud\u00edos? \u00bfY posteriormente nunca habl\u00f3 de eso? Erhart hab\u00eda estudiado aquellos documentos con incredulidad durante mucho tiempo y preguntado finalmente a su madre. Cuando \u00e9l muri\u00f3, ella ya estaba demente, pocos meses despu\u00e9s sigui\u00f3 el camino del padre. Pero entonces a\u00fan viv\u00eda y Erhart intent\u00f3 reavivar en ella alg\u00fan recuerdo, pero ella s\u00f3lo miraba y, de pronto, rio y dijo: \u00bfPolonia? Y empez\u00f3 a cantar. Sto lat, sto lat, cantaba enf\u00e1ticamente y con rostro feliz. Alois no entend\u00eda ni una palabra. Le sacudi\u00f3 los hombros y grit\u00f3 \u00a1Madre! \u00a1Madre! \u00bfQu\u00e9 est\u00e1s cantando? Erhart trat\u00f3 de guardar esas palabras en la memoria, aunque no las entendiera; logr\u00f3 retener sto lat y Jeszcze raz porque su madre repet\u00eda cantando esas palabras, corri\u00f3 al ba\u00f1o y las escribi\u00f3, fon\u00e9ticamente, aproximadamente como sonaban. Luego regres\u00f3 junto a su madre, que estaba sentada, silenciosa, ensimismada, y que ya no dijo nada m\u00e1s.\n\nAl d\u00eda siguiente Erhart pregunt\u00f3 a una compa\u00f1era de filolog\u00eda eslava, ella dijo que las palabras que \u00e9l hab\u00eda apuntado significaban \u00abcien a\u00f1os\u00bb y \u00abotra vez\u00bb. Y ella cre\u00eda que la madre de Erhart hab\u00eda cantado una vieja canci\u00f3n popular polaca, pero que \u00absto lat\u00bb era tambi\u00e9n un brindis. \u00bfLe ayudaba eso?\n\nNo.\n\n\u00bfC\u00f3mo se sab\u00eda su madre una canci\u00f3n popular polaca? \u00bfQu\u00e9 hab\u00eda hecho su padre en Pozna\u0144? \u00bfY por qu\u00e9 su madre, que ya no tiene recuerdos, canta en polaco \u00ab\u00a1Otra vez! \u00a1Otra vez! \u00a1Otra vez!\u00bb?\n\nAlois Erhart hizo la maleta sumido en cavilaciones y recuerdos. Se detuvo. \u00bfPor qu\u00e9 preparaba la maleta? Su vuelo de regreso no era hasta pasado ma\u00f1ana, la habitaci\u00f3n del hotel Atlas estaba reservada y pagada hasta pasado ma\u00f1ana. Porque ma\u00f1ana a\u00fan se celebraba una sesi\u00f3n del think-tank New Pact. Y s\u00f3lo porque \u00e9l ya no quisiera participar, ya no quisiera aparecer por all\u00ed, no ten\u00eda que partir inmediatamente. Tampoco ten\u00eda un billete de avi\u00f3n que permitiera un cambio de fechas. As\u00ed pues: un d\u00eda m\u00e1s en Bruselas.\n\nSe sent\u00f3 a la mesa, abri\u00f3 su port\u00e1til, quer\u00eda redactar de memoria un acta de la sesi\u00f3n, resumir las reacciones de los miembros de ese grupo. Por orden, conforme a las categor\u00edas en las que \u00e9l los hab\u00eda dividido. Empez\u00f3 con \u00abDie Eitlen\u00bb, los vanidosos, pero antes de que pudiera seguir escribiendo vio que el programa autom\u00e1tico de correcci\u00f3n lo hab\u00eda cambiado en \u00abDie Eliten\u00bb, las \u00e9lites.\n\nYa est\u00e1 bien, pens\u00f3, y cerr\u00f3 de golpe el port\u00e1til.\n\nMatek cogi\u00f3 a las 11.04 el intercity de Cracovia G\u0142\u00f3wny a Pozna\u0144 G\u0142. El viaje deb\u00eda durar escasamente cinco horas y veinte minutos. Para \u00e9l hab\u00eda terminado despu\u00e9s de tres horas escasas. Porque poco despu\u00e9s de \u0141\u00f3d\u017a, el maquinista accion\u00f3 el freno de seguridad, por lo que Matek, que justo en ese momento se hab\u00eda levantado para ir al aseo, se vio lanzado por el pasillo central del vag\u00f3n, choc\u00f3 contra el respaldo de un asiento, luego contra la puerta y qued\u00f3 tendido en el suelo. Trat\u00f3 de levantarse, pero no pod\u00eda apoyarse, su brazo derecho estaba separado del cuerpo de un modo poco natural, no le obedec\u00edan las piernas, no pod\u00eda encogerlas para ponerse de rodillas, algo le pasaba en el vientre, como si detr\u00e1s del ombligo hubiera reventado algo, como si se hubiera liberado una poderosa energ\u00eda que ahora flu\u00eda ardiente por su vientre, o\u00eda gemidos de gente, ten\u00eda que haber tambi\u00e9n otros heridos, de nuevo trat\u00f3 de incorporarse, pero s\u00f3lo pudo levantar un poco la cabeza, que dej\u00f3 caer de nuevo con un quejido. Alguien se inclinaba sobre \u00e9l, dec\u00eda algo, era una voz de mujer que inspir\u00f3 confianza a Matek, casi una especie de sensaci\u00f3n de seguridad, cerr\u00f3 los ojos. Vio a un ni\u00f1o que corr\u00eda por un campo con una cometa. Otros ni\u00f1os corr\u00edan detr\u00e1s del chiquillo, quer\u00edan quitarle la cometa, pero el muchacho era m\u00e1s r\u00e1pido y, cuanto m\u00e1s deprisa corr\u00eda, m\u00e1s sub\u00eda la cometa, la cuerda se desenrollaba tan r\u00e1pidamente que le ara\u00f1aba y le cortaba literalmente las palmas de las manos, ahora surgieron hombres con pistolas y fusiles que disparaban contra la cometa, pero la gran cruz, revestida de pa\u00f1o rojo y blanco, ya volaba tan alto que las balas no la alcanzaban, sus manos sangraban, la sangre goteaba en la tierra, la cometa sub\u00eda cada vez m\u00e1s alto, al cielo, entonces vio a su lado a su madre, que aplaud\u00eda riendo, y el ni\u00f1o solt\u00f3 la cometa, que se remont\u00f3 hasta el sol, all\u00ed donde el sol ya no cegaba, sino que se tornaba rojo oscuro y, finalmente, negro.\n\nAl d\u00eda siguiente, los peri\u00f3dicos informaron en toda Europa sobre el accidente de ferrocarril. En el trayecto de \u0141\u00f3d\u017a a Zgierz, un suicida se hab\u00eda arrojado al intercity con destino a Pozna\u0144, lo que tuvo bloqueado en ese trayecto el tr\u00e1fico de trenes durante m\u00e1s de tres horas.\n\nEsa noticia era poco com\u00fan. Se trataba de un accidente relativamente peque\u00f1o, local, y hab\u00eda un consenso entre los medios para no informar de tales sucesos, a fin de evitar las imitaciones. Que ese caso, sin embargo, se abriera camino hasta los medios, incluso hasta los europeos, ten\u00eda una raz\u00f3n sencilla: el muerto era, al menos en cuanto muerto, de inter\u00e9s general. El hombre que se hab\u00eda arrojado al tren era el octogenario Adam Goldfarb.\n\nDesde 1942 hab\u00eda habido tambi\u00e9n, junto al gueto de \u0141\u00f3d\u017a, un campo de concentraci\u00f3n juvenil en el que encerraban a ni\u00f1os jud\u00edos a partir de los dos a\u00f1os de edad. Y Adam Goldfarb era el \u00faltimo superviviente de ese campo de concentraci\u00f3n juvenil de \u0141\u00f3d\u017a. Hab\u00eda sido el \u00faltimo superviviente. El motivo de ese \u00abavivador del recuerdo\u00bb, as\u00ed ven\u00eda en el peri\u00f3dico, no se conoc\u00eda.\n\n# CAP\u00cdTULO UND\u00c9CIMO\n\nCUANDO ALGO SE DESCOMPONE, TIENE QUE HABER ESTADO CONECTADO.\n\n#\n\nLa primera sesi\u00f3n del grupo de trabajo del Consejo, asunto: Jubilee Project de la Comisi\u00f3n Europea, tuvo lugar precisamente en la tarde del d\u00eda en que los peri\u00f3dicos belgas y tambi\u00e9n algunos alemanes y franceses comentaban el esc\u00e1ndalo que hab\u00eda provocado la nueva exposici\u00f3n de los Mus\u00e9es Royaux des Beaux-Arts de Bruselas. Como alg\u00fan otro gran esc\u00e1ndalo, tambi\u00e9n \u00e9ste empez\u00f3 en peque\u00f1o. Primero, despu\u00e9s del vernissage de la exposici\u00f3n \u00abArte en v\u00eda muerta - Modernidad olvidada\u00bb aparecieron en medios locales breves reportajes m\u00e1s bien aburridos, banales y poco inspirados. Cuando una exposici\u00f3n colectiva muestra obras de artistas olvidados, entonces a los cr\u00edticos de arte, habitualmente m\u00e1s exigentes, les resulta dif\u00edcil criticar la selecci\u00f3n y quejarse por ejemplo de que ha sido olvidado este o aquel artista al que deber\u00edan haber expuesto. Pues la exposici\u00f3n presentaba s\u00f3lo artistas olvidados, y cada cr\u00edtico que entre esos olvidados hubiera echado de menos a uno que qued\u00f3 olvidado por el comisario de la exposici\u00f3n caer\u00eda en una trampa: recordar\u00eda a un olvidado s\u00f3lo para incluirlo en la lista de los olvidados. Surgi\u00f3 entonces una pregunta de inmensa complejidad te\u00f3rico-art\u00edstica: \u00bfhay arte significativo en una determinada \u00e9poca pero que despu\u00e9s se ha olvidado con raz\u00f3n? Parece evidente. Pero \u00bfpor qu\u00e9? Nosotros no olvidamos la \u00e9poca hist\u00f3rica, entonces \u00bfpor qu\u00e9 olvidamos ejemplos de su arte? \u00bfExiste el arte ejemplarmente olvidado, existe el artista paradigm\u00e1tico olvidado? \u00bfY hasta qu\u00e9 punto un artista olvidado merece el veredicto de \u00abolvidado\u00bb si un cr\u00edtico lo trae a la memoria, y hasta qu\u00e9 punto no est\u00e1 entonces olvidado o, al contrario, olvidado del todo si el cr\u00edtico s\u00f3lo advierte de que no habr\u00eda que olvidarlo en la lista de los olvidados?\n\nPor ese motivo la exposici\u00f3n no fue un gran \u00e9xito de cr\u00edtica; imperaba el siguiente tenor: en el fondo se trata de arte que, al fin y a la postre, no pudo triunfar en el mercado. Pero tampoco fue un fracaso: pues todas las obras expuestas fueron compradas por los Museos Reales en alguna ocasi\u00f3n, despu\u00e9s de 1945, por tanto, en determinado momento se valoraron de manera diferente, en el contexto de sus coet\u00e1neos se las consider\u00f3 excelentes, o al menos se las tuvo por trabajos de j\u00f3venes artistas muy prometedores. Y algunos cr\u00edticos reflexionaron por eso de modo m\u00e1s o menos original sobre esa misma cuesti\u00f3n: \u00bfc\u00f3mo puede ocurrir que algo se considere importante y poco despu\u00e9s caiga en el olvido?\n\nThomas Hebbelinck, el comisario de la exposici\u00f3n, revel\u00f3 en una entrevista en De Standard el motivo asombrosamente banal que hab\u00eda dado origen a esa exposici\u00f3n: los Museos Reales preparaban una gran exposici\u00f3n sobre Francis Bacon; s\u00f3lo la prima de seguro que exig\u00edan otras galer\u00edas para los pr\u00e9stamos engull\u00eda una parte tan grande del presupuesto que surgi\u00f3 la necesidad de hacer una exposici\u00f3n intermedia que no costara nada. Con obras, por tanto, de los fondos del museo. As\u00ed naci\u00f3 la idea de mostrar creaciones, adquiridas en su momento, de artistas olvidados, una idea que a \u00e9l le parec\u00eda realmente de gran inter\u00e9s y digna de discutirse. Pues la cuesti\u00f3n de qu\u00e9 olvidamos y por qu\u00e9 olvidamos y si existe quiz\u00e1 incluso en las obras expuestas un deseo colectivo de represi\u00f3n era, desde luego, de importancia general.\n\nCon esto parec\u00eda que para los medios el tema de la exposici\u00f3n quedaba zanjado.\n\nSin embargo, luego De Morgen public\u00f3 el gran art\u00edculo, ingenioso y brillante, de Geert van Istendael, el afamado intelectual bruselense, que recientemente tambi\u00e9n hab\u00eda aparecido en los medios como miembro del jurado de \u00abBruselas busca un nombre para su cerdo\u00bb. \u00c9l abri\u00f3 un frente completamente nuevo en una discusi\u00f3n sostenida con escaso entusiasmo durante un d\u00eda y abandonada por completo despu\u00e9s: el tema en el que se concentraba no era el arte olvidado, sino s\u00f3lo en la forma en que el comisario Hebbelink lo hab\u00eda puesto en escena. La exposici\u00f3n llevaba por t\u00edtulo: \u00abArte en v\u00eda muerta\u00bb. A trav\u00e9s de la gran sala de la exposici\u00f3n se hab\u00edan colocado v\u00edas de ferrocarril, sobre el final de la v\u00eda hab\u00eda un tope fijo, lo que, como escrib\u00eda Van Istendael, deb\u00eda expresar seguramente que \u00abesto es la estaci\u00f3n final\u00bb. A los visitantes de la exposici\u00f3n se los situaba al lado izquierdo de las v\u00edas, mientras que las obras de arte, esculturas, pinturas y dibujos, colgadas y colocadas muy pegadas unas a otras, se hallaban en el lado derecho.\n\nGeert van Istendael empezaba su art\u00edculo con esta frase: \u00abEn esta exposici\u00f3n, que le da a uno mucho que pensar, s\u00f3lo falta un peque\u00f1o pero importante detalle: encima de la puerta de entrada a la exposici\u00f3n, la frase EL ARTE LIBERA\u00bb.[8]\n\nY planteaba la pregunta de si el museo, o bien el comisario de la exposici\u00f3n, opinaba que se pod\u00eda comparar la relaci\u00f3n entre arte fracasado y exitoso con la selecci\u00f3n que ten\u00eda lugar en la rampa de Auschwitz. A la izquierda la vida, a la derecha la muerte. La presentaci\u00f3n de un arte que no ha logrado imponerse en el mercado como un amasijo de obras que al final de un trayecto en tren es enviado a la muerte \u2014\u200cporque c\u00f3mo, si no, hab\u00eda que interpretar las v\u00edas y el tope\u2014, mientras que en el lado izquierdo se les dice a los visitantes que ellos est\u00e1n entre los supervivientes, no era s\u00f3lo una trivializaci\u00f3n de Auschwitz, sino que mostraba tambi\u00e9n la necedad y la inconveniencia de la idea de tener que remitir siempre a Auschwitz. Y se planteaba la cuesti\u00f3n \u2014\u200cseg\u00fan Van Istendael\u2014 de \u00abcu\u00e1l es el mayor esc\u00e1ndalo: equiparar el arte malo con los jud\u00edos o ver el mercado del arte como una especie de doctor Mengele. De una manera u otra, esta exposici\u00f3n es un esc\u00e1ndalo, pero al mismo tiempo, eso esperamos, el \u00faltimo de su g\u00e9nero: pues a partir de ahora la amenazadora maza del fascismo es un arma de cart\u00f3n piedra hecha con el papel del cat\u00e1logo de una mala exposici\u00f3n de camaradas de cart\u00f3n que se llaman a s\u00ed mismos artistas\u00bb.\n\nFue un golpe bien asestado. Y de pronto la exposici\u00f3n, comentada en las secciones culturales muy cansinamente, era un esc\u00e1ndalo tratado con todo detalle en comentarios pol\u00edticos y art\u00edculos de fondo.\n\nHasta el m\u00e1ximo representante de la prensa conservadora belga, el redactor jefe, jubilado desde hac\u00eda m\u00e1s de diez a\u00f1os, del De Financieel-Economische Tijd, Tom Koorman, reapareci\u00f3 en De Tijd con un comentario: Esa exposici\u00f3n era un crimen, porque no equiparaba ning\u00fan crimen con el mayor crimen. El mundo libre ten\u00eda tambi\u00e9n la libertad de olvidar, y el mercado libre, incluido el mercado del arte, no se defin\u00eda en modo alguno a trav\u00e9s de la adoraci\u00f3n de cenizas.\n\nEsa ins\u00edpida o, cuando menos, desafortunada formulaci\u00f3n \u2014\u00abadoraci\u00f3n de cenizas\u00bb\u2014 con referencia a Auschwitz acarre\u00f3 m\u00e1s irritadas reacciones, aunque sin duda Koorman no hab\u00eda querido decir lo que se interpretaba. Pero tampoco el comisario Hebbelinck hab\u00eda querido nada de todo lo que le atribu\u00edan en las reacciones y comentarios. En cualquier caso, el d\u00eda en que el grupo de trabajo del Consejo se reun\u00eda no hubo peri\u00f3dico en el que no fuera tema el \u00abuso indebido\u00bb de Auschwitz.\n\nY George Morland dijo, nada m\u00e1s comenzar el meeting \u2014\u200cpero que no conste en acta, por favor\u2014: Si esta exposici\u00f3n, que sin duda tiene cierta afinidad con la idea de la COMM, hubiera sido ya una parte del proyectado aniversario..., well, eso no habr\u00eda sido exactamente lo que yo llamar\u00eda una gran mejora de imagen para la Comisi\u00f3n.\n\nCuando Mrs. Atkinson ley\u00f3 el acta de la sesi\u00f3n, supo que el proyecto, en esa forma, s\u00ed, supo que pod\u00eda olvidarlo. Ahora hab\u00eda dos posibilidades: endosarle el proyecto definitivamente al Arca, y as\u00ed hacerlo fracasar. Eso apenas acarrear\u00eda rumores en la casa, porque nadie esperaba del Arca nada verdaderamente luminoso. \u00bfQu\u00e9 hab\u00eda dicho hac\u00eda poco su colega Jean-Philippe Dupont sobre el Arca? \u00abJ'adore les lucioles, vraiment, elles sont magnifiques. Mais quand je veux travailler, elles ne me donnent simplement pas assez de lumi\u00e8re!\u00bb\n\nO bien ella insist\u00eda en la idea b\u00e1sica del Jubilee Project para mejorar la imagen de la Comisi\u00f3n pero se distanciaba de la idea en cuanto al contenido, que proven\u00eda del Arca. Pues una propuesta del grupo de trabajo era: \u00ab\u00bfPor qu\u00e9 los jud\u00edos? \u00bfPor qu\u00e9 no deporte?\u00bb.\n\nS\u00ed, pens\u00f3. \u00bfPor qu\u00e9 no? La idea del deporte que une a los pueblos, con eso se pod\u00eda trabajar, en el sentido del art\u00edculo 165, p\u00e1rrafo 1, Tratado-TFUE, como se consign\u00f3 en acta. La secci\u00f3n de deporte estaba tambi\u00e9n adjunta a la Direcci\u00f3n General de Educaci\u00f3n y Cultura, as\u00ed podr\u00eda entonces seguir trabajando con la se\u00f1ora Xenopoulou, y continuar remiti\u00e9ndose al presidente, que hab\u00eda dado su apoyo al Jubilee Project en general. Eso tambi\u00e9n constaba en acta. Sin embargo, se rechaz\u00f3 en\u00e9rgicamente que la Comisi\u00f3n actuara en solitario, con lo que el sentido del proyecto, lograr una mejora de imagen de la Comisi\u00f3n, quedaba anulado. Lo \u00fanico que se aceptaba era que el proyecto se financiara exclusivamente con el presupuesto de la Comisi\u00f3n, lo cual era dif\u00edcil de aceptar si se a\u00f1ad\u00edan el Consejo y el Parlamento y \u00e9stos, con todas sus continuas objeciones, interven\u00edan en el proyecto. \u00bfY la Cultura iba a aceptar que se rechazara su idea y se la obligara al mismo tiempo a cambiarla por una idea completamente distinta, adem\u00e1s sin la perspectiva de lograr una mejora exclusiva de imagen?\n\nGrace Atkinson se estrujaba los dedos. La cocina de Bruselas le sentaba bien. Ya hab\u00eda engordado cuatro kilos y estaba asombrada de que tambi\u00e9n la circulaci\u00f3n de sus manos y pies aparentemente funcionara mejor. Y ni rastro ya de palidez, de cutis blanco como el papel. Ahora sus mejillas estaban coloradas, como en los retratos de sir Thomas Lawrence, el pintor favorito de la reina. Eso pod\u00eda deberse a la copita de champ\u00e1n o, no quer\u00eda exagerar, al prosecco que beb\u00eda de vez en cuando. Hab\u00eda comprobado que una copita, s\u00f3lo una peque\u00f1a copita, o dos todo lo m\u00e1s, fomentaba su imaginaci\u00f3n, su mente estaba m\u00e1s abierta, al mismo tiempo ella m\u00e1s decidida, s\u00f3lo segu\u00eda estruj\u00e1ndose los dedos por la costumbre.\n\nMasajeaba y reflexionaba. En primer lugar ten\u00eda que averiguar c\u00f3mo reaccionaba Fenia Xenopoulou al acta de la sesi\u00f3n del Consejo.\n\n\u00bfLe escrib\u00eda un e-mail proponiendo un encuentro para hablar de c\u00f3mo se podr\u00edan adaptar las objeciones que all\u00ed les hac\u00edan?\n\nAbsurdo. No hab\u00eda nada que adaptar. Y un e-mail as\u00ed equivaldr\u00eda ya a un distanciamiento definitivo de la idea aportada por el Arca.\n\nGrace Atkinson se sent\u00eda mal. Ella era una persona leal. Hab\u00eda valorado muy positivamente y con toda sinceridad el compromiso de Fenia Xenopoulou. Lealtad y honestidad no eran para ella palabras vac\u00edas, sino principios hondamente arraigados en su alma, las herramientas humanas para seguir su camino con dignidad y tambi\u00e9n, sin duda, con el deseo de tener \u00e9xito. Se hab\u00eda metido en algo donde sobrevivir en lo profesional y en lo humano depend\u00eda tal vez de par\u00e1metros totalmente distintos, y no sab\u00eda si eso ten\u00eda que ver con el hecho de que all\u00ed trabajaran juntas personas de horizontes culturales muy distintos o se deb\u00eda a que los grandes sistemas burocr\u00e1ticos llevaban en principio a esas contradicciones. Hab\u00eda trabajado primero en comit\u00e9s de la University of London, luego en el gabinete del ministro de Exteriores ingl\u00e9s. En ambos casos se trataba de estructuras reducidas, aunque no transparentes. En el fondo todo se hab\u00eda desarrollado a puerta cerrada; las famosas puertas acolchadas eran met\u00e1fora y realidad a la vez. Pero ahora, ahora la observaban ininterrumpidamente, y todos los e-mails quedaban almacenados y agregados a un expediente que despu\u00e9s, al cabo de alg\u00fan tiempo, iba a Florencia, al archivo de la Uni\u00f3n Europea, donde hab\u00eda historiadores que rebuscaban en \u00e9l. Cuando se iba a tomar una decisi\u00f3n en el gabinete del ministro, en Londres, el debate duraba un m\u00e1ximo de treinta minutos, incluidos los rituales y la ret\u00f3rica del principio y del final. All\u00ed se reun\u00edan personas que ten\u00edan el mismo background, or\u00edgenes similares, y por eso tambi\u00e9n hab\u00edan estudiado en los mismos colegios, hablaban la misma lengua con el mismo acento, por el que se reconoc\u00edan entre ellos, sus c\u00f3nyuges eran de la misma clase social, ten\u00edan en un ochenta o noventa por ciento biograf\u00edas an\u00e1logas y en gran parte id\u00e9nticas experiencias. \u00bfQue hab\u00eda un problema? Esos antiguos alumnos de piel blanca, protestantes, de colegios de \u00e9lite, se hab\u00edan puesto de acuerdo en veinte minutos. Lo que cualquiera dec\u00eda en aquel c\u00edrculo sonaba como quien recita un mon\u00f3logo. \u00a1Pero all\u00ed, en Bruselas! All\u00ed se reun\u00edan constantemente personas de distintas lenguas e improntas culturales; sobre todo de los Estados del Este llegaban muchos que proven\u00edan de familias de obreros y artesanos y hab\u00edan pasado por experiencias muy distintas, y todo lo que Grace Atkinson sol\u00eda aclarar en veinte minutos duraba all\u00ed horas, d\u00edas, semanas. Ella lo encontraba fascinante. Ten\u00eda que confesarse a s\u00ed misma que las decisiones que se pod\u00edan tomar tan r\u00e1pidamente en el c\u00edrculo de las \u00e9lites de Inglaterra por lo general no correspond\u00edan a los intereses de la mayor\u00eda de la poblaci\u00f3n brit\u00e1nica, independientemente de qui\u00e9n gobernara. En Bruselas era al rev\u00e9s: hab\u00eda tantos y tan laborios\u00edsimos compromisos que por eso ya nadie, daba lo mismo d\u00f3nde, entend\u00eda que sus propios intereses estuvieran contenidos de alg\u00fan modo en ese compromiso. Era m\u00e1s complicado pero tambi\u00e9n m\u00e1s interesante; sin embargo, a veces pensaba: habr\u00eda que poder intervenir autoritariamente, con derecho a impartir \u00f3rdenes y a obrar con energ\u00eda y...\n\nMrs. Atkinson trag\u00f3 saliva. Esa idea la escandalizaba. En cualquier caso, nada de e-mails. No le habr\u00eda parecido juego limpio distanciarse de la se\u00f1ora Xenopoulou de manera que constara en archivos. Absolutamente poco limpio. Se sirvi\u00f3 otra copa de prosecco y determin\u00f3 llamar por tel\u00e9fono a Fenia Xenopoulou.\n\nCuando llam\u00f3 Fridsch y pregunt\u00f3 si ten\u00eda tiempo en el descanso de mediod\u00eda, Xeno pens\u00f3 que se trataba de las reprobaciones que hab\u00eda suscitado el Jubilee Project. Era muy importante para ella, hab\u00eda dicho \u00e9l, era una informaci\u00f3n que quer\u00eda comunicarle con urgencia, y le hab\u00eda propuesto un lunch en la Rosticceria Fiorentina, rue Archim\u00e8de. Okay, hab\u00eda dicho ella, dentro de una hora en la Rosti.\n\nXeno no era una ingenua. Pero ahora, al leer el acta del grupo de trabajo, se preguntaba, sin embargo, c\u00f3mo pod\u00eda estar tan sorprendida por din\u00e1micas que ella, con toda su experiencia, habr\u00eda tenido que prever y que esperar. Y por qu\u00e9 esos jueguecitos que all\u00ed se jugaban le parec\u00edan de pronto repugnantes, por habituales que fueran. Aceptaci\u00f3n general de la idea y luego tantas objeciones por separado y tantas propuestas de cambio que de la idea no quedaba nada.\n\nEn la novela que Xeno hab\u00eda le\u00eddo, la favorita del presidente, hab\u00eda un pasaje en el que el emperador promet\u00eda a su amante que, con todo el poder que le hab\u00edan concedido los dioses, har\u00eda realidad el viejo sue\u00f1o de los hombres: volar. Si fuera capaz de realizar ese milagro, no s\u00f3lo consolidar\u00eda su dominio, sino que tambi\u00e9n promover\u00eda la fe de los hombres en sus posibilidades y por eso fomentar\u00eda la dicha y la prosperidad de su imperio. Convocaba a los m\u00e1s importantes fil\u00f3sofos, sacerdotes y sabios de su \u00e9poca para trabajar en la soluci\u00f3n de esa tarea: lo que fracasaba muy pronto porque todos esos sabios ni siquiera pod\u00edan ponerse de acuerdo en qu\u00e9 p\u00e1jaro era el adecuado para arrancarle el secreto del vuelo. No ve\u00edan el volar, ve\u00edan s\u00f3lo las diferencias entre los p\u00e1jaros.\n\nLo que m\u00e1s desconcert\u00f3 a Xeno fue la reacci\u00f3n de los alemanes. El acta comenzaba, de modo totalmente rutinario, con \u00abGeneral aprobaci\u00f3n de la propuesta de COO y EAC de conmemorar solemnemente el cincuentenario de la fundaci\u00f3n de la Comisi\u00f3n Europea con la finalidad de mejorar la imagen de la Comisi\u00f3n (PT, IT, DE, FR, HU, BG, SI, AT, FR, UK, NL, HR, LV, SE, DK, EE, CR, EL, ES, LU). BG ha subrayado un inter\u00e9s especial por esta iniciativa, que tendr\u00e1 lugar en el periodo de su presidencia del Consejo\u00bb.\n\nY as\u00ed continuaba al principio, educadas aprobaciones hasta que llegaron las primeras objeciones: \u00abAprobaci\u00f3n de la propuesta de presupuesto, pero MS (IT, DE, FI, EE, CR, HU, SI, HR, FR) han exigido promesa vinculante de que tambi\u00e9n en caso de que se sobrepasen los costes, la financiaci\u00f3n del proyecto s\u00f3lo se podr\u00e1 llevar a cabo con el presupuesto administrativo de la Comisi\u00f3n, sin cargar el presupuesto general. El Consejo y el Parlamento no dar\u00edan su aprobaci\u00f3n. A pesar de ello, MS (DE, IT, FR, HU, PL) han insistido en incluir al Consejo y al Parlamento en la orientaci\u00f3n del contenido del proyecto\u00bb.\n\nEso ya era muy fuerte. Pero la consternaci\u00f3n de Xeno creci\u00f3 cuando ley\u00f3 los reparos en cuanto al contenido, sobre todo por parte de Alemania: \u00abDE ha cuestionado la idea de Auschwitz como base de la unificaci\u00f3n europea y ha subrayado que los musulmanes de Europa no pod\u00edan quedar excluidos de la obra de unificaci\u00f3n europea. (Conformes: UK, HUN, PL, AT, HR, CR)\u00bb.\n\nXeno cre\u00eda estar curada de espantos. En el curso de su carrera hab\u00eda vivido suficientes resistencias, bloqueos y obst\u00e1culos burocr\u00e1ticos. Y aunque en los \u00faltimos tiempos hab\u00eda perdido algo de seguridad en lo concerniente a c\u00f3mo continuar\u00eda su carrera, siempre hab\u00eda confiado en que sab\u00eda prever resistencias y, llegado el momento, estar convenientemente preparada. Pero esta objeci\u00f3n de los alemanes y la lista de los Estados que estaban conformes con ella, esa objeci\u00f3n en particular la dejaba sin palabras. Eso no lo hab\u00eda visto venir: que los alemanes se preocuparan por los musulmanes y que precisamente los pa\u00edses que en pol\u00edtica interior defend\u00edan de forma m\u00e1s radical el \u00aboccidente cristiano\u00bb apoyaran a los alemanes. Y que justamente los h\u00fangaros expresaran su inquietud por que no hubiera aprobaci\u00f3n por parte de una amplia opini\u00f3n p\u00fablica europea si el crimen cometido contra los jud\u00edos se colocaba en el centro de las celebraciones supuestamente identitarias y no se mencionaba que los jud\u00edos hac\u00edan ahora con los palestinos exactamente lo mismo que se hab\u00eda hecho antes con ellos. Por esta objeci\u00f3n obtuvieron el aplauso de los parlamentarios de izquierdas (de DE, GR, ES, PT, IT). Y \u00e9stos, los h\u00fangaros, recordaban tambi\u00e9n que el a\u00f1o siguiente ellos ocupar\u00edan la presidencia de la Alianza Internacional para el Recuerdo del Holocausto (IHRA) y que por eso estaban preparando de todos modos una serie de actos conmemorativos. Y luego los italianos: \u00abPropuesta de IT de celebrar los actos del aniversario en Roma, en recuerdo de los Tratados de Roma. Acto solemne en el palazzo Montecitorio, con el presidente del Parlamento, del Consejo y de la Comisi\u00f3n, del Comit\u00e9 Econ\u00f3mico y Social, del Banco Central y del Comit\u00e9 de las Regiones\u00bb..., y el siguiente aditamento le pareci\u00f3 a Xeno particularmente p\u00e9rfido: \u00ab... en la medida en que \u00e9stos puedan acordar una solemne declaraci\u00f3n com\u00fan (aprobaci\u00f3n: UK, DE, HUN, CR, LV, AT)\u00bb. Xeno se pregunt\u00f3 por primera vez por qu\u00e9 cuando hab\u00eda que tomar una resoluci\u00f3n interven\u00edan continuamente personas que ni siquiera conoc\u00edan los basics. \u00abEn recuerdo de los Tratados de Roma\u00bb: la Comisi\u00f3n no ten\u00eda su origen en los Tratados de Roma, sino en el Tratado de Par\u00eds y, en su forma actual, en la cumbre de La Haya. \u00bfY nadie del grupo de trabajo del Consejo dec\u00eda nada contra la propuesta de los italianos de celebrar el aniversario de la Comisi\u00f3n en Roma? \u00bfNi siquiera los franceses, que ten\u00edan que saberlo mejor? Nadie sab\u00eda ya nada de nada. \u00a1C\u00f3mo se pod\u00eda olvidar tanto y, sin embargo, hablar tanto! Visto as\u00ed, la propuesta adicional de los italianos era, por otra parte, conmovedora: \u00abA continuaci\u00f3n, verbena en el centro de Roma\u00bb.\n\nLa propuesta presentada por los polacos con las palabras \u00ab\u00bfPor qu\u00e9 los jud\u00edos? \u00bfPor qu\u00e9 no el deporte?\u00bb la consider\u00f3 tan escandalosa que, no s\u00f3lo en su interior, sino literalmente, sacudi\u00f3 asombrada la cabeza. Si se obraba de acuerdo con la masiva aceptaci\u00f3n de esa idea, todo quedar\u00eda en su secci\u00f3n, ya que al Arca tambi\u00e9n le correspond\u00eda el deporte europeo, pero all\u00ed ten\u00eda menos competencias y posibilidades que en la cultura. Los partidos nacionales populistas eran una nimiedad en comparaci\u00f3n con el nacionalismo de las federaciones deportivas de los Estados miembros.\n\nEn ese momento entr\u00f3 Kass\u00e1ndra en el despacho para informar de que en la oficina de empadronamiento hab\u00edan averiguado la \u00faltima direcci\u00f3n de David de Vriend: en la place du Vieux March\u00e9 aux Grains, en Sainte-Catherine. Pero el edificio hab\u00eda sido derribado hac\u00eda poco.\n\n\u00bfQui\u00e9n es David...? \u00bfQu\u00e9?\n\nLo hab\u00edamos comentado. Ser\u00eda ideal para nuestro proyecto. Y no hay confirmaci\u00f3n de su muerte. Probablemente est\u00e9 en una residencia de la tercera edad. Lo averiguaremos tambi\u00e9n.\n\n\u00bfMuerte?, dijo Xeno. Ahora estaba muy cansada. \u00bfQue no hay confirmaci\u00f3n? \u00a1Muchas gracias!\n\nEch\u00f3 una ojeada al reloj. Tengo que marcharme, dijo. Un almuerzo de trabajo.\n\nFridsch ya estaba all\u00ed cuando Xeno lleg\u00f3 a la Rosti. Sentado a una mesa delante del establecimiento, a pleno sol, como si la calle fuera un escenario y el sol un reflector dirigido s\u00f3lo hacia \u00e9l. Ella ya ten\u00eda esa imagen al verlo de lejos y acercarse y, al mismo tiempo, se asombr\u00f3 por primera vez de la extra\u00f1a palabra alemana para reflector: Schein-werfer, \u00ablanzador de luz\u00bb.\n\nNo pod\u00eda distinguir si tambi\u00e9n \u00e9l la hab\u00eda visto ya. Fridsch llevaba unas gafas de sol espejadas. A Xeno le pareci\u00f3 espantoso. Ella detestaba las gafas de espejo. Los ojos de la gente no se ve\u00edan. Eso era para Xeno el disfraz m\u00e1s horrible, era peor que el nicab y el burka, en los que al menos se ve\u00edan los ojos, la ventana al alma de las personas. Adem\u00e1s esas gafas le recordaban a hombres de los que hab\u00eda tenido miedo en su infancia. De los que la hab\u00eda prevenido su padre: Quien lleva esas gafas, quien no ense\u00f1a sus ojos, tiene un tenebroso secreto. \u00bfY qui\u00e9n tiene un secreto? Claro, la polic\u00eda secreta. Por eso se llama as\u00ed. Delata a personas que luego van a prisi\u00f3n, o las asesina enseguida, hab\u00eda dicho su padre, ech\u00e1ndole protectoramente el brazo por el hombro y atray\u00e9ndola hacia \u00e9l.\n\nConociendo a Fridsch, las habr\u00eda comprado en alg\u00fan mercadillo callejero, pero si las llevaba puestas ahora, hab\u00eda que contar posiblemente con que las gafas de espejo se estaban poniendo de moda otra vez.\n\nSe levant\u00f3 de un salto para saludarla. Como ella no le ve\u00eda los ojos, ve\u00eda ahora por primera vez con claridad meridiana que Fridsch ten\u00eda pelos en la nariz. Le sal\u00edan de las fosas nasales como patitas de ara\u00f1a. Vio al mismo tiempo su propia mirada en los cristales de las gafas. Detestaba los pelos en la nariz. Ella se depilaba las axilas, las piernas, se recortaba el vello del pubis, y ese hombre ni siquiera era capaz de cortarse esos est\u00fapidos pelitos que le sal\u00edan por la nariz.\n\n\u00bfQu\u00e9 le pasaba? Eso le preguntaba tambi\u00e9n ahora Fridsch: \u00bfQu\u00e9 te ocurre?\n\nProblemas...\n\n\u00bfTe deslumbra...\n\n... con el proyecto.\n\n... el sol? Podemos...\n\n... S\u00ed.\n\n... ir tambi\u00e9n al interior. He...\n\n... \u00bfS\u00ed?\n\n... reservado una mesa dentro y otra fuera.\n\nEra tan previsor. Y dentro adem\u00e1s se quitar\u00eda las gafas, pens\u00f3 Xeno.\n\nEl proyecto, no pienses en \u00e9l. Enseguida hablamos de eso, dijo Fridsch, mientras manten\u00eda abierta la puerta del restaurante, dej\u00e1ndola pasar primero, sigui\u00e9ndola despu\u00e9s con la mirada y palp\u00e1ndola \u00f3pticamente al hacerlo: con el orgullo del hombre que hab\u00eda conquistado a esa mujer y que al mismo tiempo se enternec\u00eda consigo mismo porque ese orgullo lo llenaba de sentimientos de ternura. De tierna ternura. \u00bfEra una tautolog\u00eda? Hab\u00eda sin duda en ella diversos grados. \u00a1La m\u00e1s tierna ternura! Como si se pusiera la mano sobre el vientre de una mujer embarazada: \u00bfqu\u00e9 estaba pensando \u00e9l? No pensaba absolutamente nada, no en palabras, pero si sus sentimientos se pudieran introducir en un programa que los transformara en lenguaje, entonces saldr\u00edan aproximadamente esas frases.\n\nFridsch se hab\u00eda peinado con una raya perfecta. Esa se\u00f1al de exactitud escrupulosa y de correcci\u00f3n irrit\u00f3 a Xeno. \u00bfQu\u00e9 no la irritar\u00eda ahora? Cuando hubieron tomado asiento y Fridsch, quit\u00e1ndose las gafas, se inclin\u00f3 por encima de la mesa hacia ella, Xeno le toc\u00f3 el pelo con la mano hasta que la raya qued\u00f3 deshecha, luego se ech\u00f3 a re\u00edr, quiz\u00e1 de un modo un poquito demasiado artificial, y dijo: \u00a1Mejor! As\u00ed pareces cinco a\u00f1os m\u00e1s joven.\n\n\u00bfQuiero yo eso? Yo hace cinco a\u00f1os no era tan feliz como ahora.\n\nEsa reacci\u00f3n la dej\u00f3 sin palabras. Lleg\u00f3 entonces la due\u00f1a, trajo las cartas, apunt\u00f3 las bebidas: Fridsch pidi\u00f3 agua; Xeno, vino.\n\nAh\u00ed ten\u00e9is todo lo que se necesita para sermonear y para beber, dijo la hostelera, Xeno asinti\u00f3 amablemente, no lo hab\u00eda entendido, la mujer hablaba en b\u00e1varo. Era italiana, de Mil\u00e1n, pero antes de instalarse en Bruselas hab\u00eda regentado un restaurante en M\u00fanich y all\u00ed aprendi\u00f3 alem\u00e1n. Y conoc\u00eda a Fridsch, sab\u00eda que era alem\u00e1n.\n\nHab\u00eda venido a Bruselas por un hombre, ella dice \u00abtodo un hombre\u00bb cuando cuenta que era muy bien parecido, \u00abun buen mozo\u00bb, pero luego result\u00f3 que \u00abno era lo que promet\u00eda\u00bb, en resumen: \u00abun timo\u00bb. A Fridsch le gustaba ese establecimiento, conoc\u00eda todas las historias.\n\nHace poco, a la hora del cierre, puso en el equipo de m\u00fasica La Internacional, cont\u00f3 Fridsch. Algunos clientes se quedaron asombrad\u00edsimos. \u00bfSabes por qu\u00e9 lo hizo? Por nostalgia de Mil\u00e1n, dijo.\n\nXeno lo mir\u00f3 sin entender.\n\nFridsch se rio. Su padre, le cont\u00f3, era un forofo del Inter de Mil\u00e1n, el club de f\u00fatbol m\u00e1s famoso de Mil\u00e1n. Y cuando el club lleg\u00f3 a la final de la Copa de Europa, contra el Real Madrid, \u00e9l viaj\u00f3 a Viena.\n\n\u00bfPor qu\u00e9 a Viena?\n\nPorque all\u00ed se jugaba la final. O sea, el Inter de Mil\u00e1n contra el Real Madrid. Antes del partido la banda militar austriaca deb\u00eda tocar los himnos de los clubs.\n\n\u00bfPor qu\u00e9 la banda militar?\n\nNo lo s\u00e9. Era as\u00ed. \u00bfT\u00fa te crees que en el campo de f\u00fatbol toca la filarm\u00f3nica de Viena? En cualquier caso: la banda toc\u00f3 primero el himno del Real Madrid. Luego deb\u00eda venir el de Mil\u00e1n. Pero por equivocaci\u00f3n hab\u00edan dado a la banda la partitura de La Internacional en lugar del himno del Inter de Mil\u00e1n. As\u00ed pues, de pronto son\u00f3 La Internacional comunista. Y algunos jugadores italianos cantaron tambi\u00e9n al comp\u00e1s: \u00a1Arriba, parias de la tierra!, bueno, no s\u00e9 c\u00f3mo es en italiano. Con el Real Madrid jugaba Ferenc Pusk\u00e1s, entonces probablemente el mejor jugador del mundo. Un h\u00fangaro que en 1956 huy\u00f3 de los tanques sovi\u00e9ticos y abandon\u00f3 Budapest. Estaba tan trastornado cuando oy\u00f3 antes del partido el himno comunista que despu\u00e9s se limit\u00f3 a errar por el campo en estado de shock, por lo que el Inter de Mil\u00e1n gan\u00f3 por 3 a 1 contra el favorito Real Madrid. Y por eso, en memoria de aquel triunfo, su padre tocaba en casa siempre La Internacional y por eso ella...\n\nFridsch vio que a Xeno no le interesaba nada lo que contaba. Pero \u00e9l estaba tan contento, tan feliz; de la abundancia de su coraz\u00f3n hablaba su boca. Lleg\u00f3 entonces la hostelera con las bebidas. A\u00fan no hab\u00edan mirado la carta, pidieron simplemente el men\u00fa del d\u00eda.\n\nAnyway, dijo Fridsch, lo que quer\u00eda decirte es muy importante. Escucha, tu Jubilee Project, mira...\n\n\u00bfHas le\u00eddo el acta del grupo de trabajo?\n\nClaro.\n\n\u00bfY qu\u00e9 es tan importante ah\u00ed?\n\nNada...\n\nElla le cort\u00f3, con voz un poco demasiado alta, de manera que los de las mesas vecinas los miraron: \u00bfQu\u00e9 est\u00e1s diciendo? \u00bfNada es importante, y es tan importante que me pides que venga para decirme eso?\n\n\u00a1No, escucha, por favor! Lo que quer\u00eda decirte es esto: no hay nada que puedas hacer por el proyecto; est\u00e1 muerto. Durante alg\u00fan tiempo deambular\u00e1 como t\u00edpico zombi de la Comisi\u00f3n por algunos departamentos e instancias y luego lo enterrar\u00e1n definitivamente. Lo que ahora debes hacer es mantenerte al margen. Deja que los sherpas lo lleven a los enterradores. No puedes defender la idea. No lograr\u00edas nada. T\u00fa ah\u00ed ya no pintas nada. La COMM quer\u00eda una conmemoraci\u00f3n, el presidente dice que \u00e9l apoyar\u00eda una buena idea, el grupo de trabajo del Consejo dice que no hay ninguna buena idea, o bien hace otras malas propuestas, todas sin perspectiva ninguna, porque todas son propuestas para cubrir el expediente, \u00bfcomprendes? Si alguien cree a\u00fan que con eso puede coronarse de laureles, d\u00e9jale entonces. Pero si alguien fracasa as\u00ed penosamente, no eres t\u00fa. \u00bfOkay? T\u00fa ya est\u00e1s fuera de esto porque t\u00fa \u2014\u200cy ahora viene...\n\nFridsch quer\u00eda imitar un toque de trompeta para introducir lo inesperado, pero la hostelera trajo las ensaladas y les dese\u00f3 buen provecho, lo dijo en b\u00e1varo, apenas inteligible.\n\nPorque t\u00fa, prosigui\u00f3 Fridsch, estar\u00e1s entonces en un sitio completamente distinto. Con un hermoso paso adelante en tu carrera, estar\u00e1s por ejemplo en la TRADE o en la HOME.\n\n\u00bfQu\u00e9 est\u00e1s diciendo?\n\n\u00bfNo era eso lo que quer\u00edas? Y yo he descubierto la triqui\u00f1uela para lograrlo. \u00a1Escucha! T\u00fa sigues siendo chipriota, \u00bfno?\n\nS\u00ed, ya lo sabes.\n\nPero \u00bfChipre era ya miembro de la UE cuando t\u00fa viniste a Bruselas?\n\nNo. En aquella \u00e9poca yo...\n\nT\u00fa viniste entonces con un ticket griego.\n\nS\u00ed. Porque soy griega.\n\nBueno, entonces qu\u00e9: \u00bfGriega o chipriota?\n\n\u00bfPor qu\u00e9 te r\u00edes as\u00ed? \u00bfEst\u00e1s burl\u00e1ndote de m\u00ed? Qu\u00e9 es tan divertido: soy una chipriota griega.\n\nBueno, vayamos despacio, dijo Fridsch: Grecia es miembro de la Uni\u00f3n Europea. Entretanto, la Rep\u00fablica de Chipre tambi\u00e9n es miembro de la Uni\u00f3n. Pero entonces, cuando Chipre a\u00fan no era miembro, t\u00fa, una chipriota, viniste aqu\u00ed en tu condici\u00f3n de griega.\n\nS\u00ed, eso fue entonces mi oportunidad. Siendo chipriota griega pod\u00eda obtener un pasaporte griego y...\n\nY ahora tienes otra oportunidad muy distinta. Por ser la Rep\u00fablica de Chipre desde hace alg\u00fan tiempo miembro de la Uni\u00f3n. Una peque\u00f1a isla. Y por si fuera poco, media peque\u00f1a isla. Con menos de un mill\u00f3n de habitantes, un Estado con aproximadamente el mismo n\u00famero de habitantes que Frankfurt. Es curioso, \u00bfno? \u00bfY qu\u00e9 hace all\u00ed la gente? \u00bfSon gu\u00edas tur\u00edsticos, profesores de buceo, olivareros? No lo s\u00e9. S\u00f3lo s\u00e9 una cosa...\n\nXeno lo mir\u00f3, sus ojos alegres, estaba preparando un efecto sorpresa, ella a\u00fan no entend\u00eda cu\u00e1l, algo le resultaba desagradable, como un ultraje muy sutil, que sin embargo no entend\u00eda a\u00fan, ahora no le habr\u00eda importado que se hubiera calado otra vez sus gafas de espejo.\n\nEsa diminuta Rep\u00fablica de Chipre no logra cumplir con la cuota de funcionarios que le corresponde aqu\u00ed en todos los niveles de la jerarqu\u00eda, no logra ocupar todos los puestos a que tienen derecho los chipriotas. Tienen poqu\u00edsimas personas cualificadas. \u00bfComprendes ahora ad\u00f3nde quiero ir a parar?\n\n\u00bfEso era la cosa importante que quer\u00edas decirme?\n\nS\u00ed. \u00bfNo es estupendo? Tan l\u00f3gico. Tan simple. T\u00fa te procuras un pasaporte de la Rep\u00fablica de Chipre y con tu CV obtienes de inmediato una direcci\u00f3n.\n\nPero entonces alguien tendr\u00eda que marcharse.\n\nLos ingleses se marchan. Otros se jubilan. Dentro de un mes en la TRADE ha de cubrirse de nuevo una direcci\u00f3n. Y luego otra en HOME. Y si los chipriotas, que s\u00f3lo han cubierto el cincuenta por ciento de los puestos que les corresponden, pueden proponer a alguien...\n\nPero yo he ganado mi plaza por concurso, hace mucho tiempo que no dependo de un programa nacional.\n\n\u00a1Tanto mejor! La Rep\u00fablica de Chipre se sentir\u00e1 muy honrada si una compatriota tan experimentada y con puesto fijo ocupa un lugar de gran responsabilidad en la Comisi\u00f3n.\n\n\u00bfY yo s\u00f3lo tengo que... otro pasaporte?\n\nS\u00ed. Que t\u00fa, evidentemente, obtendr\u00e1s enseguida.\n\nFridsch estaba radiante. Le sorprend\u00eda que Fenia no diera se\u00f1ales de entusiasmo.\n\n\u00a1Venga, ponte en marcha!\n\nAhora se com\u00edan los raviolis, hablaban poco. Fridsch pensaba que Fenia ten\u00eda que digerir la informaci\u00f3n. Lo otro importante que tambi\u00e9n quer\u00eda decirle, el asunto privado, lo aplaz\u00f3. Los sentimientos eran muy dif\u00edciles de entender, y apenas se los hab\u00eda formulado con palabras, se volv\u00edan inciertos. Pens\u00f3 que de momento ser\u00eda mejor que ella le estuviera agradecida.\n\nDespu\u00e9s del lunch, Fenia Xenopoulou, de nuevo ante su escritorio, empez\u00f3 a contestar e-mails, tecle\u00f3 rutinaria y autom\u00e1ticamente frases vac\u00edas, y pronto se detuvo. \u00bfC\u00f3mo manejar\u00eda la propuesta de Fridsch? Al poco rato ya no ve\u00eda el monitor, sino im\u00e1genes de sus recuerdos, y sus dedos, inm\u00f3viles, descansaban sobre el teclado. Se recost\u00f3 en el asiento. Ese asunto del pasaporte era, desde luego... se levant\u00f3 de un salto, abri\u00f3 la ventana. El aire espeso y caliente por el sol que penetr\u00f3 en la habitaci\u00f3n climatizada le record\u00f3 veranos de su infancia en Chipre. Si el cielo tambi\u00e9n estaba sin nubes entonces, para ella, sin embargo, no era un tiempo sin nubes como el que disfrutaban en prados soleados, entre caricias y juegos, los ni\u00f1os de familias pudientes. Se vio reflejada en el batiente abierto de la ventana, pero s\u00f3lo de manera muy borrosa, como si esa imagen fuera la proyecci\u00f3n de un tiempo lejano, no, no era as\u00ed: ve\u00eda cu\u00e1n dura se le hab\u00eda puesto la boca, las arrugas a derecha e izquierda, que en la imagen reflejada parec\u00edan como pintadas con ligeros toques por el pincel del aire. Era ella, pero no obstante era otra, era... Se acerc\u00f3 corriendo a la mesa, cogi\u00f3 el tel\u00e9fono y llam\u00f3 a Bohumil: \u00bfPuedes venir un momento a mi despacho?\n\nVino enseguida, y Xeno le pidi\u00f3 un cigarrillo.\n\nLos tengo en mi despacho, voy a por ellos, dijo, luego levant\u00f3 la vista hacia el detector de humo: \u00bfTraigo tambi\u00e9n la escalerilla y pego el cacharro ese de ah\u00ed arriba?\n\nNo hace falta, dijo ella, fumo en la ventana.\n\nBohumil volvi\u00f3, le dio la cajetilla, dijo: Puedes quedarte con ella. S\u00f3lo hay cinco. Yo tengo otra en el despacho.\n\nGracias. Muy amable. \u00bfTienes fuego?\n\nSe qued\u00f3 de pie fumando junto a la ventana, mir\u00f3 a Bohumil de una manera que a \u00e9l le result\u00f3 desagradable. Como si ella estuviera junto a ella misma. Y como si mirase a trav\u00e9s de \u00e9l. \u00bfEra por el Jubilee Project? \u00c9l sab\u00eda, por supuesto, que hab\u00eda problemas, y pensaba que por eso lo hab\u00eda llamado. Pero no toc\u00f3 el tema. Era fantasmag\u00f3rico. Ella era pragm\u00e1tica cien por cien, \u00e9l nunca la hab\u00eda visto tan trastornada. Okay, dijo \u00e9l, dio un paso atr\u00e1s y ya quer\u00eda salir cuando Xeno dijo: \u00bfT\u00fa tienes pasaporte?\n\nBohumil la mir\u00f3 sorprendido.\n\nQuiero decir, dijo ella, \u00bfqu\u00e9 clase de pasaporte tienes?\n\nEsperaba que \u00e9l respondiera: un pasaporte checo, claro. Y ella habr\u00eda hecho un gesto de asentimiento y le habr\u00eda envidiado por ese \u00abclaro\u00bb. Pero entonces se qued\u00f3 cortada. Porque \u00e9l dijo: Tengo pasaporte austriaco. \u00bfPor qu\u00e9 lo preguntas?\n\nElla lo mir\u00f3, mantuvo el cigarrillo delante de la boca, sin darle una calada, gui\u00f1\u00f3 los ojos, sac\u00f3 despu\u00e9s por la ventana la mano con el cigarrillo, mene\u00f3 la cabeza y dijo: \u00bfAcabas de decir pasaporte austriaco?\n\nS\u00ed. Tengo pasaporte austriaco. \u00bfPor qu\u00e9?\n\nEso quiero saber yo. \u00bfPor qu\u00e9? \u00a1Si eres checo!\n\nS\u00ed, pero nac\u00ed en Viena. En 1968, mis abuelos, cuando los tanques rusos aplastaron la Primavera de Praga... lo sabes, \u00bfno? Primavera de Praga.\n\nXeno asinti\u00f3.\n\nMis abuelos huyeron a Austria. Con mi padre. Que ten\u00eda entonces diecis\u00e9is a\u00f1os. Diez a\u00f1os despu\u00e9s mi padre se cas\u00f3 con mi madre, que era tambi\u00e9n hija de checos refugiados en Viena. Pero para entonces, los dos ten\u00edan ya la nacionalidad austriaca. En diciembre de 1989, o sea, inmediatamente despu\u00e9s de la revoluci\u00f3n, regresamos a Praga. Fue el triunfo de mis padres, el vuelco pol\u00edtico. Yo ten\u00eda diez a\u00f1os entonces. En 2002 me present\u00e9 al concurso en Bruselas. Yo hab\u00eda estudiado Ciencias Pol\u00edticas en Praga, pero quer\u00eda marcharme, y entonces me vino bien tener a\u00fan el pasaporte austriaco, porque Austria ya era miembro de la UE, pero Chequia todav\u00eda no. Por eso estoy aqu\u00ed, y \u2014\u200csonri\u00f3\u2014 por eso soy fumador y siempre tengo varias cajetillas de reserva.\n\nXeno le dirigi\u00f3 una mirada interrogante.\n\nWell, de ni\u00f1o pasaba cada tarde con mis padres en la taberna m\u00e1s llena de humo de Viena, en el Azyl, era el punto de encuentro de los checos disidentes huidos y exiliados. Mis padres iban all\u00ed cada d\u00eda, al anochecer, y no ten\u00edan dinero para un canguro, as\u00ed que, simplemente, me llevaban con ellos. All\u00ed discut\u00edan horas y horas con V\u00e1clav Havel, cuando estaba en Viena, con Pavel Kohout, Karel Schwarzenberg, Jaroslav Hutka y como se llamara toda aquella gente. Fumaban sin interrupci\u00f3n, todos. Yo estaba sentado o dormido a su lado, ya era adicto a la nicotina antes de fumarme el primer cigarrillo.\n\nSe ech\u00f3 a re\u00edr. Se le cort\u00f3 la risa cuando vio el rostro de Xeno.\n\n\u00bfY...?, pregunt\u00f3 ella.\n\nEso no fue el final, dijo \u00e9l \u2014\u200cque no hab\u00eda entendido \u00aband?\u00bb, sino \u00abend?\u00bb\u2014, o tal vez s\u00ed: para mi padre. Havel fue despu\u00e9s presidente; Schwarzenberg, ministro de Exteriores; Kohout casi recibi\u00f3 el Premio Nobel de Literatura, al menos es lo que \u00e9l contaba, y Hutka, que se hab\u00eda convertido en un divo de Radio Free Europe, recorr\u00eda el pa\u00eds con sus canciones de protesta hasta que lleg\u00f3 una generaci\u00f3n que ya no entend\u00eda esas canciones. Entonces se jubil\u00f3 con el t\u00edtulo de \u00ableyenda viva\u00bb. Mi padre se convirti\u00f3 en ministro de Educaci\u00f3n: y el d\u00eda de la jura del cargo le dio un infarto. Pas\u00f3 a la historia checa como el \u00abministro de los diez minutos\u00bb.\n\nLo siento mucho.\n\nGracias. Yo tambi\u00e9n.\n\nBueno. Entonces hablas alem\u00e1n, dijo Xeno.\n\nMiserablemente, dijo Bohumil en alem\u00e1n: \u00ablausig\u00bb.\n\n\u00bfLousy?\n\nYes.\n\nPero \u00bfpor qu\u00e9? Si t\u00fa...\n\nPorque desde mi regreso a Praga no he vuelto a hablarlo, es decir, desde que ten\u00eda diez a\u00f1os. Y en la \u00e9poca de Viena aprend\u00ed alem\u00e1n en la escuela primaria, claro, pero en casa siempre hablaba checo. En el fondo, s\u00f3lo me ha quedado una cosa: que no puedo evitar re\u00edrme de los germanismos del checo. Por ejemplo, pinktlich. Es una palabra checa que proviene del alem\u00e1n, donde p\u00fcnktlich significa puntual. Pero en checo significa todo lo que es antip\u00e1tico y repugnante y t\u00edpicamente alem\u00e1n: significa puntilloso, inflexible, insensible, radical sin concesiones, pagado de s\u00ed mismo, con disciplina prusiana... Cuando alguien es as\u00ed, en checo se dice que es pinktlich.\n\nSe rio. Y se interrumpi\u00f3 de inmediato. Aquel semblante r\u00edgido de Xeno.\n\nI see, dijo ella. \u00bfY... el pasaporte? \u00bfNunca has tenido problemas con \u00e9l?\n\nNo, \u00bfpor qu\u00e9? \u00bfQu\u00e9 clase de problemas? Da igual qu\u00e9 pasaporte tenga, es un pasaporte europeo.\n\nXeno hab\u00eda tirado por la ventana el cigarrillo, ahora sac\u00f3 otro de la cajetilla, se lo puso entre los labios y present\u00f3 su boca a Bohumil; si se eliminaba el cigarrillo, parec\u00eda una boca que invitaba a que la besaran.\n\n\u00c9l le dio fuego, ella murmur\u00f3 gracias y mir\u00f3 por la ventana, lo que Bohumil entendi\u00f3 como discreta exhortaci\u00f3n a que se marchara.\n\n\u00c9l dijo okay, ella no dijo nada, segu\u00eda mirando por la ventana. De modo que \u00e9l se march\u00f3. Ten\u00eda la sensaci\u00f3n de abandonar un dep\u00f3sito de cad\u00e1veres. \u00bfPodr\u00eda identificar a la muerta? La conoc\u00eda pero no estaba seguro.\n\n\u00bfCu\u00e1l era el problema de Xeno? La historia de Bohumil la hab\u00eda dejado tan sorprendida que estaba como petrificada. Ese Bohumil tan alegre. Pero tan f\u00e1cil no era la cosa. Ella estaba dividida. Ella era dos. No entend\u00eda por qu\u00e9 ella ten\u00eda que ser eso. La historia de \u00e9l era en cierto modo la historia de ella. Y sin embargo la de ella era completamente distinta. Eso la desconcertaba. De momento.\n\nSu pasaporte siempre hab\u00eda sido su pasaporte europeo, y no un documento de identidad nacional o \u00e9tnica. Para ella era la entrada al reino de la libertad, de la libertad de domiciliarse y de establecerse en Europa, era su salvoconducto para seguir su camino en Europa. \u03a7\u03b1\u1fd6\u03c1\u03b5, \u1f66 \u03c7\u03b1\u1fd6\u03c1\u03b5, \u1f18\u03bb\u03b5\u03c5\u03b8\u03b5\u03c1\u03af\u03b1, cantaba en Chipre con entusiasmo en la escuela cuando en determinadas ocasiones tocaban el himno nacional: \u00a1Salve, libertad! Pero que, por ser chipriota griega, tuviera que convertirse en nacionalista chipriota: eso jam\u00e1s se le habr\u00eda ocurrido, le era totalmente ajeno. \u00bfPor qu\u00e9 el lugar de nacimiento iba a tener m\u00e1s importancia que los derechos que se pod\u00edan, m\u00e1s a\u00fan, que deb\u00edan tenerse como ser humano? Libertad, eso lo entend\u00eda, pero \u00abChipre ante todo\u00bb[9] jam\u00e1s se le habr\u00eda pasado por la cabeza. Por esa raz\u00f3n no le sorprendi\u00f3 en absoluto cuando al llegar a Grecia para estudiar en la universidad comprob\u00f3 que all\u00ed cantaban el mismo himno. \u03a7\u03b1\u1fd6\u03c1\u03b5, \u1f66 \u03c7\u03b1\u1fd6\u03c1\u03b5, \u1f18\u03bb\u03b5\u03c5\u03b8\u03b5\u03c1\u03af\u03b1. Para ella, por tanto, no se trataba de profesi\u00f3n de fe nacional, y tampoco le result\u00f3 en absoluto extra\u00f1o que dos pa\u00edses tuvieran el mismo himno, para Xeno ese himno era simplemente un canto a la libertad; y qu\u00e9 bien le cuadraba a ella: \u00a1T\u00fa, que surgiste, inflamada de ira \/ de huesos de los sagrados griegos!\n\nSu ira la llevar\u00eda lejos. La ira era energ\u00eda, una fuerza productiva. La promesa de libertad no pod\u00eda significar: Atr\u00f3fiate en tu limitado espacio, pero tus pensamientos son libres. Mira bien los olivos del \u00e1rido terreno que hay delante de tu casa. Qu\u00e9 poco necesitan, y sin embargo sus hojas brillan al sol como la plata.\n\nSi tus pensamientos son libres, tambi\u00e9n deben serlo tus posibilidades, tus acciones, tus obras. Ya ten\u00eda claro eso a los doce a\u00f1os, cuando transportaba para turistas de todos los confines del mundo botellas de agua mineral al seco ba\u00f1o de Afrodita. De todos los confines del mundo \u2014\u200cesto lo hab\u00eda aprendido en la escuela\u2014 llegaba gente a Chipre, porque Chipre est\u00e1 muy cerca de Turqu\u00eda, de Grecia, de Siria y Egipto; siempre ha sido, por tanto, un punto de intersecci\u00f3n entre Europa, Asia y \u00c1frica. Chipre no era una naci\u00f3n, esa isla era una peque\u00f1a embarcaci\u00f3n, se balanceaba sobre las olas de la historia y sobre las mareas de las naciones y los imperios que emerg\u00edan y volv\u00edan a sumergirse.\n\nCuando recibi\u00f3 su pasaporte griego nunca pens\u00f3 que con \u00e9l abandonaba y traicionaba a su pa\u00eds de origen. El pasaporte griego era para ella un documento de viaje desde la isla que ten\u00eda una paloma en su escudo hasta el continente que dec\u00eda ser un proyecto de paz y que le ofrec\u00eda posibilidades de hacer carrera. Y ahora le parec\u00eda completamente absurdo renunciar a ese pasaporte y cambiarlo por otro que ni ten\u00eda otro valor ni promet\u00eda otra cosa que el antiguo; pero que a ella, chipriota griega, la obligaba a decidirse entre ser griega o chipriota. Deb\u00eda cambiar su pasaporte, en el que ella hab\u00eda visto un documento europeo, por otro que s\u00f3lo era una profesi\u00f3n de fe nacional... para poder hacer carrera en Europa. S\u00ed, era absurdo. Hab\u00eda trabajado suficiente tiempo en la Comisi\u00f3n para saber por experiencia que los nacionalistas atacaban cada vez con m\u00e1s brutalidad a esa Europa en la que ella quer\u00eda seguir su camino en libertad, con toda su ira resultante de la estrechez pasada, y que, como ahora pensaba Xeno de pronto, quiz\u00e1 era inconscientemente una ira contra las limitaciones que exig\u00edan que dijera: Yo soy... chipriota. O griega. O lo que sea. Quien dice: T\u00fa eres..., quiere decir: Qu\u00e9date donde est\u00e1s.\n\nLa propuesta de Fridsch trastornaba toda su vida. La identidad era s\u00f3lo un papel. \u00bfIba a ser ella otra persona distinta si cambiaba de papel? \u00bfSer\u00eda una persona distinta si en lugar de \u00ab\u00a1Salve, libertad!\u00bb ahora cantaba \u00ab\u00a1Salve, libertad!\u00bb, el himno del nuevo pasaporte, que era id\u00e9ntico al himno del antiguo pasaporte? S\u00ed: porque hab\u00eda cambiado un himno a la libertad por un himno nacional, y la misma letra y la misma melod\u00eda ten\u00edan por ello un significado muy distinto. Hab\u00eda nacido en Chipre, como griega, y en Grecia era griega nacida en Chipre. Era absurdo que se le exigiera ver esa identidad como una identidad doble que le exig\u00eda tomar una decisi\u00f3n: \u00a1Eres una esquizofr\u00e9nica, decide qui\u00e9n eres!\n\nLo terrible era que en lo m\u00e1s \u00edntimo sab\u00eda que con sus cavilaciones se estaba mintiendo a s\u00ed misma. Por supuesto que aprovechar\u00eda la oportunidad y cambiar\u00eda de pasaporte. Necesit\u00f3 dos horas para admitirlo. Era pragm\u00e1tica. Y eso no era sino una decisi\u00f3n pragm\u00e1tica. \u00bfPor qu\u00e9 ten\u00eda tales escr\u00fapulos? Porque percib\u00eda de alguna manera que, al hacerlo, algo mor\u00eda en ella. \u00bfY a qui\u00e9n le gusta morirse? La perspectiva de una vida mejor despu\u00e9s, ya se llame Dios o carrera, es s\u00f3lo un consuelo desesperado.\n\nEmpez\u00f3 a escribir un e-mail para Mrs. Atkinson, par\u00f3 y cerr\u00f3 el documento. Apareci\u00f3 la pregunta: \u00ab\u00bfGuardar como borrador?\u00bb.\n\nLe habr\u00eda gustado que en la vida fuera posible lo que ofrece el ordenador, guardar borradores. Hizo clic en \u00abNo guardar\u00bb, se recost\u00f3 en el asiento y pens\u00f3: okay.\n\nEran casi las 17.00 horas. Escribi\u00f3 a\u00fan un e-mail colectivo a sus colaboradores: \u00abReuni\u00f3n ma\u00f1ana 11.00 horas por entierro Jubilee Project\u00bb.\n\nEntonces ya habr\u00e1 regresado Martin Susman.\n\nBorr\u00f3 entierro e hizo clic en \u00abEnviar\u00bb.\n\nApag\u00f3 el ordenador y sali\u00f3. No ten\u00eda ganas de \u00abirse a casa\u00bb, a su peque\u00f1o apartamento funcional, en el fondo un lugar donde dormir con vestidor. Pero tampoco quer\u00eda quedarse all\u00ed, en un puesto de trabajo que en realidad ya hab\u00eda abandonado con su decisi\u00f3n de aquel d\u00eda. \u00bfBeber algo en cualquier sitio? Estaba indecisa. Si acaso, en el Het Lachende Varken, el caf\u00e9 de la calle donde viv\u00eda.\n\nSubi\u00f3 por la rue Joseph II a la estaci\u00f3n de metro Maelbeek. En el and\u00e9n mir\u00f3 el panel de informaci\u00f3n: pr\u00f3ximo tren dentro de seis minutos.\n\nDe modo que ocurr\u00eda en la vida real. Que un hombre se pueda convertir en un escarabajo.\n\nEse pensamiento era s\u00f3lo un peque\u00f1o pero t\u00edpico s\u00edntoma de que el en\u00e9rgico y robusto Florian Susman era de pronto otro: trastornado, desvalido, desesperado. No era persona de lecturas. El de los libros siempre fue su hermano peque\u00f1o, Martin.\n\n\u00bfQu\u00e9 est\u00e1s leyendo otra vez? \u00bfAventuras de indios?\n\nNo. Es un hombre que se transforma en un insecto, en un escarabajo.\n\n\u00bfPor obra de un mago?\n\nNo. As\u00ed, sin m\u00e1s. De golpe y porrazo. Se despierta y es un escarabajo.\n\nQu\u00e9 loco le pareci\u00f3 entonces su hermano peque\u00f1o. \u00bfC\u00f3mo pod\u00eda uno leer algo as\u00ed, perder el tiempo con esos libros raros? \u00c9l hab\u00eda sido el hijo-de-su-padre, el designado heredero del trono, idolatrado por su progenitor, aunque tratado sin ternura. \u00c9l, que alg\u00fan d\u00eda se har\u00eda cargo de la granja, no pod\u00eda ser un blandengue, nunca d\u00e9bil ni so\u00f1ador. El padre no manifestaba sentimientos. Si los mostraba alguna vez, era con una mirada de reconocimiento, una leve inclinaci\u00f3n de cabeza o pas\u00e1ndole a Florian torpemente el brazo por los hombros, con una leve presi\u00f3n que significaba: \u00a1Hijo m\u00edo!\n\nMartin fue el hijo-de-mam\u00e1, un ni\u00f1o abstra\u00eddo, que lloraba mucho, le\u00eda mucho y a menudo ten\u00eda miedo. Entonces corr\u00eda junto a su mutti, que lo proteg\u00eda pero a quien al mismo tiempo le costaba ser cari\u00f1osa: se hab\u00eda endurecido en la lucha por la vida, pasaba noches en blanco por las deudas que hab\u00edan contra\u00eddo para ampliar la granja y hacer de ella una empresa de cr\u00eda y sacrificio de cerdos. Todos los m\u00fasculos estaban tensos. Quien tiene que levantar un peso no puede acariciar. Eso no significaba que ella lo rechazara, aunque a veces se preguntaba irritada por qu\u00e9 su hijo era como era, y por eso pensaba que ten\u00eda que curtirse, que necesitaba una coraza, y tambi\u00e9n, desde luego, que, aunque fuera tan inepto, deb\u00eda mostrar disposici\u00f3n a asumir la parte de trabajo que le tocaba. Entonces, cuando lo sorprend\u00eda leyendo, lo enviaba al establo. Era absurdo, porque a los animales de cebo ya entonces los alimentaban de manera mec\u00e1nica y dos empleados limpiaban los establos tambi\u00e9n por procedimientos mec\u00e1nicos, Martin, por tanto, no hac\u00eda sino estorbar. Finalmente volv\u00eda a la cocina. All\u00ed pod\u00eda ayudar a guisar... o leer en la mesa. Hasta que ten\u00eda que ponerla cuando los hombres ven\u00edan a comer, el padre, el hermano mayor, los dos empleados de olor tan penetrante, hombres.\n\n\u00bfTransformado en un escarabajo? \u00bfAs\u00ed, sin m\u00e1s? \u00bfSin magia? \u00a1Vaya estupidez!\n\n\u00bfRecuerdas, pregunt\u00f3 Florian, qu\u00e9 edad ten\u00edamos entonces? \u00bfCatorce y diecis\u00e9is? Y ahora \u00e9l estaba tumbado all\u00ed como un escarabajo ca\u00eddo boca arriba. Ahora estaba transformado en un escarabajo desvalido. De pronto. As\u00ed, sin m\u00e1s. Y esperaba a que se ocuparan de \u00e9l. Esperaba calmantes en el gotero, esperaba la comida, esperaba ayuda. Cuando pod\u00eda, le\u00eda, primero s\u00f3lo peri\u00f3dicos, luego tambi\u00e9n libros que le tra\u00eda Martin. Cuando la lectura lo fatigaba, cuando se le cansaban los ojos y los brazos le pesaban, entonces dormitaba, cavilaba, so\u00f1aba. Entretanto el hermano peque\u00f1o se ocupaba de una serie de cosas que se presentaban y que hab\u00eda que resolver, mientras Florian, impotente, permanec\u00eda tendido boca arriba. Hablaba con el m\u00e9dico de planta y hablaba por tel\u00e9fono con la compa\u00f1\u00eda de seguros en la que Florian ten\u00eda un seguro suplementario privado. Averiguaba qu\u00e9 cirujano ten\u00eda m\u00e1s renombre, para conseguir que llevara a cabo la complicada y peligrosa operaci\u00f3n en la espalda de Florian, ten\u00eda que ser un maestro de la especialidad.\n\n\u00bfUn mago?\n\nNo, de un modo completamente pragm\u00e1tico, un maestro, dijo Martin.\n\nMartin inform\u00f3 al gremio, a la Uni\u00f3n Econ\u00f3mica, a los socios comerciales de Florian, al presidente de la European Pig Producers, pidi\u00f3 a la EPP, por encargo de Florian, un informe sobre la asamblea de Budapest, estaba en contacto constante con Marlene, la mujer de Florian, que ten\u00eda que ocuparse de la empresa, encontr\u00f3 un abogado especializado en delitos de tr\u00e1fico y en da\u00f1os por accidente, le encarg\u00f3 que representara a su hermano ante la compa\u00f1\u00eda de seguros del taxista causante del horrendo accidente, lo que dar\u00eda lugar a un proceso civil para reclamar indemnizaciones por da\u00f1os y perjuicios y por da\u00f1os personales.\n\nFlorian, entretanto, le\u00eda o clavaba la mirada en el techo. Un asombroso cambio de papeles, as\u00ed, sin m\u00e1s, de pronto, sin mago alguno.\n\nFlorian ten\u00eda ahora una placa de titanio y doce tornillos en la espalda, la espina dorsal estaba estabilizada, la m\u00e9dula espinal no estaba afectada, el peligro de par\u00e1lisis quedaba eliminado. Todos felicitaron a Florian por su suerte en la desgracia.\n\nEstaba tumbado boca arriba, so\u00f1aba, a veces suspiraba o gem\u00eda, y sonre\u00eda cuando su hermano susurraba algo, o le enjugaba el sudor de la frente, o le cog\u00eda la mano.\n\nCuando padre muri\u00f3, ten\u00eda la edad que tengo yo ahora, dijo Florian. Yo era joven entonces, pero pod\u00eda... \u2014\u200cmis hijos, si yo hubiera muerto ahora... Elisabeth tiene siete a\u00f1os, Paul, cinco\u2014 ser\u00eda...\n\n\u00bfNo es curioso que me haya pasado esto ahora, a la edad que ten\u00eda padre cuando...? \u00bfSabes lo que es extra\u00f1o? Yo nunca hab\u00eda reflexionado sobre la muerte. Ni siquiera ante la tumba abierta de padre. Echar una paletada de tierra y... s\u00ed, estaba en estado de shock. Pero no pensaba en la muerte, sino en m\u00ed. Para una persona viva, la muerte es siempre la muerte de otros.\n\nCavilaba.\n\nSi me hubiera muerto ahora, no habr\u00eda podido despedirme, dijo. Lo mismo que padre no pudo despedirse.\n\nGuard\u00f3 silencio. Luego dijo: \u00bfEs mejor poder despedirse? \u00bfO es s\u00f3lo m\u00e1s doloroso a\u00fan?\n\nReflexionaba.\n\nSi ahora estuviera impedido, \u00bfme habr\u00edas ayudado a poner fin a esto? Yo no habr\u00eda querido seguir viviendo en tal caso. \u00bfHabr\u00eda podido contar contigo? Ahora creo que puedo contar contigo.\n\nNo, dijo Martin.\n\nMartin apuraba los d\u00edas de vacaciones no gastados, el derecho a vacaciones para cuidar a familiares y finalmente tambi\u00e9n la posibilidad de d\u00edas libres sin sueldo. Lleg\u00f3 la primavera, el aire tibio entraba por las ventanas abiertas, los primeros p\u00f3lenes, la habitaci\u00f3n de la cl\u00ednica estaba caldeada en exceso, porque seg\u00fan el calendario ten\u00eda que seguir haciendo fr\u00edo y caldeaban seg\u00fan el calendario y no seg\u00fan la realidad; Florian se apartaba el edred\u00f3n, gritaba \u00ab\u00a1Ay!\u00bb cuando ten\u00eda que estornudar, la sacudida a\u00fan le produc\u00eda dolor en la espalda, sudaba, luego tiritaba por la corriente de aire de las ventanas abiertas, Martin ten\u00eda que taparlo de nuevo, hasta que volv\u00eda a apartar el edred\u00f3n, furioso, lo \u00fanico que pod\u00eda hacer con energ\u00eda el escarabajo tendido boca arriba.\n\nMartin hab\u00eda conservado un peque\u00f1o apartamento en Viena, en el segundo distrito, para tener donde meterse cuando de vez en cuando fuera a casa un par de d\u00edas, pero no hab\u00eda sido nunca un hogar, era un apeadero, una cocina americana en la que nunca hab\u00eda hecho m\u00e1s que caf\u00e9, ni abierto otro caj\u00f3n que el que conten\u00eda el abrebotellas, donde de una vez para otra la mermelada criaba moho y se pasaba la fecha de caducidad de la mantequilla. Una habitaci\u00f3n con cama y mesa. Y cajas de cart\u00f3n. Ocho grandes cajas de mudanza. Las hab\u00eda guardado all\u00ed cuando dej\u00f3 su antiguo piso porque se mudaba a Bruselas. Con el tiempo hab\u00eda olvidado lo que hab\u00eda en esas cajas. Su hogar. A\u00fan ten\u00eda tambi\u00e9n una habitaci\u00f3n en la empresa, en casa de sus padres, junto a los cerdos, a tres horas de distancia de Viena, eso tampoco era un hogar, \u00bfqu\u00e9 pintaba \u00e9l all\u00ed?\n\nA veces, cuando volv\u00eda de la cl\u00ednica, entraba en el restaurante Zum Sieg, a la vuelta de la esquina. All\u00ed serv\u00edan un gulasch muy aceptable, los viernes un pescado muy bueno. Un d\u00eda fue testigo de c\u00f3mo un alem\u00e1n que hab\u00eda entrado en el establecimiento acompa\u00f1ando a un vien\u00e9s pregunt\u00f3 con una irritaci\u00f3n de verdadero p\u00e1nico: Zum Sieg? Espero que esto no sea un restaurante nazi\n\nEl camarero, que pasaba por all\u00ed y que lo oy\u00f3, apoy\u00f3 los brazos en la mesa, se inclin\u00f3 hacia delante y dijo: \u00a1S\u00ed, t\u00edo! \u00a1Victoria de la clase obrera! \u00bfTa claro?\n\nMartin sonri\u00f3. Era como una se\u00f1al de los fantasmas de la historia, como un trozo de vasija sacado a la luz en una excavaci\u00f3n arqueol\u00f3gica. Despu\u00e9s el camarero pas\u00f3 a su lado y dijo: \u00a1Es mentira! Ya sabes. Nos llamamos Zum Sieg porque este local existe desde la victoria de Aspern, la victoria, hace mucho tiempo, de los austriacos contra Napole\u00f3n.\n\nUn nivel m\u00e1s profundo, otro trozo de vasija.\n\nUn s\u00e1bado desayunaba en el Karmelitermarkt, all\u00ed se encontr\u00f3 con Felix, un antiguo compa\u00f1ero de la universidad. No lo hab\u00eda reconocido. A \u00e9l su colega s\u00ed lo reconoci\u00f3. Minti\u00f3: Qu\u00e9 bien volver a verse. Tomaron caf\u00e9, charlaron, y Martin se dispuso a intercambiar sentimentalismos. Eso funcionaba. \u00a1Antes, s\u00ed, antes! \u00bfY te acuerdas de cuando...? S\u00ed, entonces. Gui\u00f1aban los ojos al sol, tomaron caf\u00e9, pasaron al vino. Y de pronto el sentimentalismo se transform\u00f3 en lamento lloroso. Martin cont\u00f3 \u2014\u200c\u00bfpor qu\u00e9 precisamente a \u00e9l? \u00bfPor qu\u00e9 a ese extra\u00f1o, con el pretexto biogr\u00e1fico de ser un viejo amigo? \u00a1Quiz\u00e1 justamente por eso!\u2014, Martin le cont\u00f3 que estaba mal, que era depresivo, que sufr\u00eda depresiones y que...\n\n\u00bfDepresiones? Qu\u00e9 me est\u00e1s contando, dijo Felix con morboso buen humor. Dime una cosa: \u00bfte lavas los dientes antes de acostarte?\n\nMartin le mir\u00f3 desconcertado. S\u00ed, claro, dijo.\n\nFelix se ech\u00f3 a re\u00edr. Entonces no tienes depresi\u00f3n. Mientras uno se lave los dientes, no se est\u00e1 depresivo. A lo sumo deprimido, dijo. S\u00e9 de qu\u00e9 hablo. Y se recogi\u00f3 hacia arriba las mangas y le mostr\u00f3 las cicatrices de las mu\u00f1ecas.\n\n\u00bfCu\u00e1ndo ha sido eso?\n\nQu\u00e9 m\u00e1s da, dijo Felix. En cualquier caso: yo, para entonces, ya no me lavaba los dientes.\n\nEntretanto, Florian se recuperaba, despacio, pero se recuperaba. Ya no quer\u00eda leer. Volv\u00eda a la vida. Y, esto era extra\u00f1o: al mismo tiempo empez\u00f3 de alg\u00fan modo a perder inter\u00e9s por la vida.\n\nSupo que en Budapest, en la asamblea anual de la Uni\u00f3n de Productores Europeos de Cerdos, hab\u00edan elegido un nuevo presidente. Era de esperar. Debido a su accidente en el viaje de ida, no hab\u00eda podido presentarse ni tampoco informar a la presidencia de la EPP sobre la causa de su ausencia. Y estaba claro que en ese momento eso s\u00f3lo pod\u00eda interpretarse err\u00f3neamente. Como si ya no le interesara su cargo, y ni siquiera que se llevara a cabo un relevo oficial. Por tanto, que hubieran elegido un nuevo presidente, eso pod\u00eda comprenderlo, eso no le ofend\u00eda, pero lo que le preocupaba sobremanera, m\u00e1s a\u00fan, lo que le pon\u00eda realmente fren\u00e9tico era que hubieran elegido presidente a un h\u00fangaro, al impresentable Bal\u00e1zs Gy\u00f6ngy\u00f6si, un nacionalista radical que hasta entonces hab\u00eda aprovechado su colaboraci\u00f3n y su compromiso en esa organizaci\u00f3n europea s\u00f3lo para procurarse ventajas para su gran empresa dedicada a la cr\u00eda de cerdos mangalicas. Trataba de utilizar a los European Pig Producers como lobbistas para registrar el \u00abcerdo mangalica h\u00fangaro\u00bb como denominaci\u00f3n de origen con marca registrada y de esa manera eliminar a los criadores de mangalica austriacos y alemanes. Adem\u00e1s, Gy\u00f6ngy\u00f6si hab\u00eda llamado la atenci\u00f3n repetidas veces por sus opiniones antisemitas. Para \u00e9l, la UE era una conjuraci\u00f3n del juda\u00edsmo internacional para destruir a las naciones europeas, que \u00e9l denominaba \u00abpueblos hospedadores\u00bb. Todas esas contradicciones \u2014\u200cpedir a la UE garant\u00eda legal para sus cerdos h\u00fangaros de raza, al mismo tiempo rechazar la UE, dedicarse a la cr\u00eda de cerdos, pero a sus enemigos mortales, los jud\u00edos, calificarlos de cerdos\u2014 no s\u00f3lo eran grotescas, eran, a juicio de Florian, infamatorias y peligrosas para la Uni\u00f3n. Por eso hab\u00eda tenido la intenci\u00f3n de proponer la exclusi\u00f3n de la EPP de Bal\u00e1sz Gy\u00f6ngy\u00f6si. Y he aqu\u00ed que justo ese hombre era el nuevo presidente de la EPP. \u00bfC\u00f3mo era posible?\n\nA esa asamblea de Budapest hab\u00edan acudido, l\u00f3gicamente, un gran n\u00famero de cebadores y matarifes h\u00fangaros. Al parecer, Gy\u00f6ngy\u00f6si hab\u00eda transportado en autobuses a la asamblea a docenas de ellos. Su candidato rival era un espa\u00f1ol, un tal Juan Ram\u00f3n Jim\u00e9nez, a quien Florian no conoc\u00eda. El problema fue, al parecer, que los alemanes y los holandeses se abstuvieron, mientras que los delegados de los pa\u00edses peque\u00f1os se reunieron en torno a los h\u00fangaros, lo que bast\u00f3 para vencer por mayor\u00eda de votos a franceses, italianos y espa\u00f1oles.\n\nM\u00e1s tarde, Florian supo por qu\u00e9: entretanto, los alemanes hab\u00edan cerrado, y ya estaba preparado para su firma, un convenio comercial bilateral con China, y asimismo los holandeses. La Uni\u00f3n de Productores Europeos de Cerdos y la cuesti\u00f3n de qui\u00e9n ser\u00eda ahora su presidente les importaba...\n\n... \u00a1una mierda!, exclam\u00f3 Florian. Eso les importa ahora, con permiso sea dicho, \u00a1una mierda!\n\nClav\u00f3 la mirada en el techo, estaba tumbado e inm\u00f3vil, pero Martin ten\u00eda la sensaci\u00f3n de que muy dentro de \u00e9l un animal saltaba rugiendo contra las barras de la jaula.\n\nD\u00edas despu\u00e9s. E-mail de Gabor Szab\u00f3, el \u00fanico colega h\u00fangaro que a\u00fan ten\u00eda contacto con Florian. Martin lo ley\u00f3 en voz alta. Productores h\u00fangaros de porcino negocian bilateralmente con China. Delegaci\u00f3n capitaneada por Bal\u00e1zs Gy\u00f6ngy\u00f6si estuvo en Pek\u00edn. \u00abImag\u00ednate: ha habido una primera recepci\u00f3n, con comida y sobre todo con brindis, y Bal\u00e1zs levant\u00f3 la copa y dijo cu\u00e1nto se alegraba y qu\u00e9 honrado se sent\u00eda y que brindaba por relaciones amistosas, etc\u00e9tera. Y luego: que el gobierno chino era un modelo para nosotros, h\u00fangaros, por la claridad y determinaci\u00f3n con que defend\u00eda los intereses del pueblo, para bien del pueblo, y que merec\u00eda especial admiraci\u00f3n, por ejemplo, la contundencia y energ\u00eda con que se procedi\u00f3 en su momento en la plaza de Tian'anmen contra los enemigos del Estado. Los chinos se irritaron much\u00edsimo. No estaban ni preparados para eso ni interesados en que se mencionara la masacre de Tiananmen. En las negociaciones que siguieron lo mismo se habr\u00edan podido leer alternativamente los listines telef\u00f3nicos de Pek\u00edn y de Budapest. Ya en el vuelo de regreso, Bal\u00e1zs hab\u00eda sido depuesto como jefe de delegaci\u00f3n y como presidente de la comunidad de intereses de los productores de porcino h\u00fangaros.\u00bb\n\nFlorian sonri\u00f3. Luego clav\u00f3 de nuevo la vista en el techo. Cavilaba. Martin le apret\u00f3 la mano. Florian la retir\u00f3.\n\nEn alg\u00fan momento, Martin se sinti\u00f3 absorbido por su hermano como por un vampiro. \u00bfEra una se\u00f1al de que ya todo era otra vez como antes, o casi como antes? Florian ya pod\u00eda acostarse de vez en cuando de lado, levantarse alg\u00fan rato y dar unos pasos.\n\nTengo que volver a Bruselas.\n\nNunca olvidar\u00e9 lo que has hecho por m\u00ed.\n\nTomar\u00e9 el avi\u00f3n el lunes pr\u00f3ximo. El fin de semana te ayudar\u00e9 a trasladarte a la cl\u00ednica de rehabilitaci\u00f3n.\n\nGracias.\n\n\u00bfQu\u00e9 har\u00e1s cuando salgas?\n\nPues ya ves.\n\n\u00bfQu\u00e9?\n\n\u00bfNo ves lo que hago? Estar tumbado sin moverme.\n\nMe refiero a cuando te den el alta.\n\nYa te lo he dicho. La UE paga indemnizaciones de cierre de empresa a los criadores de cerdos. Dan dinero por cada cerdo que ya no cebas. Despedir\u00e9 a todos los empleados. Contemplar\u00e9 desde mi sala de estar c\u00f3mo se va desmoronando la empresa. En alg\u00fan momento tus sucesores la desenterrar\u00e1n y sacar\u00e1n sus conclusiones. Entretanto yo cobrar\u00e9 la indemnizaci\u00f3n.\n\n\u00a1No lo dir\u00e1s en serio!\n\nS\u00ed. Invertir\u00e9 mi capital en Alemania, participar\u00e9 en alg\u00fan gran negocio de engorde, probablemente en T\u00f6nnies Fleisch, ah\u00ed tengo buenos contactos a trav\u00e9s de la EPP, y me introducir\u00e9 all\u00ed con mi experiencia y mis conocimientos. O no. En cualquier caso: cierro la empresa. \u00bfPuedes t\u00fa ver el futuro?\n\nNo.\n\n\u00bfNo ves nada?\n\nNo.\n\nYo tampoco. Ya no veo nada.\n\nEl denominado \u00abvuelo en pijama\u00bb a Bruselas (lunes a las 7.00 de la ma\u00f1ana) l\u00f3gicamente estaba completo. En \u00e9l iban todos los funcionarios y eurodiputados que hab\u00edan pasado el fin de semana en Viena y regresaban al trabajo, lobbistas austriacos y representantes de grupos de intereses que ya ten\u00edan citas por la ma\u00f1ana y que regresaban por la noche o al d\u00eda siguiente. Y probablemente iba tambi\u00e9n a bordo, como tantas veces, un comprometido profesor de secundaria acompa\u00f1ando a una clase, en el marco del programa, fomentado por los gobiernos, \u00abJ\u00f3venes europeos visitan el Parlamento Europeo\u00bb. Martin hab\u00eda conseguido billete para el vuelo de por la tarde, lo que era una suerte porque no se habr\u00eda despertado para el de la ma\u00f1ana ni probablemente para el del mediod\u00eda. No se hab\u00eda dormido hasta casi las cuatro de la madrugada, no pod\u00eda desconectar la mente. Al atardecer hab\u00eda llevado a su hermano a Klosterneuburg, a la cl\u00ednica de rehabilitaci\u00f3n, luego hab\u00eda comprado en la tienda griega de la Taborstrasse tres botellas de cerveza Mythos, un poco de queso y una botella de vino blanco Drama, en el turco de la esquina un pan de pita.\n\nCom\u00eda y beb\u00eda mirando las cajas de mudanza, trataba de imaginarse c\u00f3mo ser\u00eda cuando hiciera la pr\u00f3xima vez una visita a la casa paterna, a su hermano y a la familia y no hubiera all\u00ed un solo cerdo, los establos, la gran nave de engorde, el matadero, todo vac\u00edo, paralizado, los azulejos blancos sin manchas de sangre y tampoco de un blanco brillante despu\u00e9s de haberlos rociado y lavado con la manguera el se\u00f1or Hofer, sino de un gris ceniciento, de una sequedad cenicienta, el se\u00f1or Hofer prejubilado, todos los empleados despedidos, la naturaleza penetrar\u00eda en las naves desmanteladas; hiedra, helechos, plantas trepadoras, malas hierbas empezar\u00edan a crecer sobre el esti\u00e9rcol que hab\u00edan dejado los cerdos, la \u00faltima generaci\u00f3n de cerdos antes de que todo se paralizara... Las ventanas se romp\u00edan, con las heladas reventaban las ca\u00f1er\u00edas en los establos fr\u00edos, en las paredes aparec\u00edan grietas, semillas voladoras tocaban tierra, germinaban y echaban ra\u00edces, plantas de todo tipo devoraban el revoque y cuarteaban las paredes, creaban un biotopo para ratones, ratas, erizos, hormigas, ara\u00f1as, vencejos, avispones, gatos asilvestrados, y Martin se beb\u00eda la tercera Mythos y ya ve\u00eda c\u00f3mo se derrumbaba el tejado del matadero que estaba delante de la casa familiar, la parte dedicada a vivienda de la antigua granja, que ahora ten\u00eda dos pisos m\u00e1s y era mucho m\u00e1s amplia, y Martin abri\u00f3 la botella de vino y se pregunt\u00f3 si se pondr\u00edan de pie frente a las ventanas o estar\u00edan sentados en el banco que hab\u00eda delante de la casa, mirando c\u00f3mo ra\u00edces de malas hierbas y plantas silvestres y garras de toda clase de animales penetraban en la declinante historia familiar. Y cuando la empresa se convierta en polvo y desaparezca, \u00bfcu\u00e1nto tiempo podr\u00e1 cobrar su hermano la prima por cierre?\n\nTen\u00eda que irse a dormir. Se lav\u00f3 los dientes. Sonri\u00f3 para sus adentros: era una buena se\u00f1al. Una se\u00f1al menos buena era que despu\u00e9s volvi\u00f3 a sentarse a la mesa para fumarse otro cigarrillo y tomar otra copa de vino. Reflexion\u00f3 sobre lo que le esperaba en Bruselas. Por e-mails colectivos se hab\u00eda enterado, por supuesto, de que hab\u00eda problemas con el Jubilee Project. Hab\u00eda recibido tambi\u00e9n, claro, el acta del grupo de trabajo del Consejo. La hab\u00eda le\u00eddo por encima y no se la hab\u00eda tomado muy en serio. Para \u00e9l era determinante que, al parecer, Xeno quer\u00eda seguir con el proyecto, al menos no hab\u00eda llegado ning\u00fan stop! por su parte. Algunas noches Martin se hab\u00eda sentado delante del ordenador y hab\u00eda completado sus apuntes y anotado ideas que hicieran avanzar el proyecto. Aunque estaba de permiso quer\u00eda poder presentar algo cuando regresara. Al menos algunas noches, despu\u00e9s de pasar las tardes con su hermano en la cl\u00ednica, no sab\u00eda qu\u00e9 otra cosa pod\u00eda hacer.\n\nSobre todo hab\u00eda estado d\u00e1ndole vueltas a una idea: si se presenta a supervivientes de Auschwitz como testigos de la \u00e9poca para la idea del proyecto europeo de paz y para la misi\u00f3n hist\u00f3rica de la Comisi\u00f3n Europea, entonces ser\u00eda l\u00f3gico y razonable incluir tambi\u00e9n a funcionarios de la \u00e9poca en que se fund\u00f3 la Comisi\u00f3n, para que contaran con qu\u00e9 ideas, intenciones y esperanzas empezaron a trabajar entonces. Martin estaba convencido de que los funcionarios de la primera generaci\u00f3n sab\u00edan de qu\u00e9 iba aquello con mucha m\u00e1s precisi\u00f3n que las actuales \u00e9lites de la burocracia. Eso ser\u00eda, pensaba Martin Susman, como el otro brazo de la tenaza. Por un lado, los supervivientes de los campos de exterminio, que recordaban el juramento: nunca m\u00e1s nacionalismo y racismo. Por el otro, los representantes de la generaci\u00f3n fundadora de la Comisi\u00f3n Europea, que recordaban que se trataba exactamente de eso: de desarrollar una instituci\u00f3n supranacional para superar el nacionalismo y, en \u00faltimo t\u00e9rmino, las naciones.\n\nHab\u00eda escrito un e-mail a Kass\u00e1ndra: \u00bfQu\u00e9 opinas sobre esto?\n\nKass\u00e1ndra: Me ocupar\u00e9 de ello.\n\nUna semana despu\u00e9s. Kass\u00e1ndra: Primera generaci\u00f3n Comisi\u00f3n: a) muertos b) dementes c) no dementes, no capacitados para viajar. \u00bfQuieres seguir trabajando sobre esa idea? \u00bfEv. mensajes por v\u00eddeo de c)?\n\nMartin se hab\u00eda terminado el vino Drama pero a\u00fan no se sent\u00eda capaz de acostarse y dormir. Encontr\u00f3 una botella de grappa en la cocina. No lo hagas, pens\u00f3, y abri\u00f3 la botella. Se tambale\u00f3 ligeramente cuando retrocedi\u00f3 los tres pasos de distancia entre la cocina y la mesa.\n\nTal vez, pensaba, habr\u00eda que organizar el Jubilee Project de manera muy distinta. No andarse con rodeos. Sin medias tintas. Si la demencia y la muerte imped\u00edan que alguien pudiera informar y recordar, que era de lo que se trataba y se segu\u00eda tratando propiamente, entonces ten\u00edan que comparecer y responder dementes y muertos. \u00bfNo provocar\u00edan pavor y compasi\u00f3n y tal vez clarificaci\u00f3n? Incluso comprensi\u00f3n. De pronto una sociedad demente comprende lo que hab\u00eda querido ser, de pronto un continente enfermo de muerte recuerda la medicina que le hab\u00eda prometido la curaci\u00f3n, pero que \u00e9l hab\u00eda dejado de tomar y olvidado despu\u00e9s. \u00bfC\u00f3mo? \u00bfC\u00f3mo se pod\u00eda poner eso en pr\u00e1ctica? \u00bfActores? Habr\u00eda que contratar a actores que interpreten a funcionarios de la \u00e9poca en que se fund\u00f3 la Comisi\u00f3n, no a actores famosos, encomiados por representar toda clase de papeles, \u00e9sos ser\u00edan otra vez s\u00f3lo ellos aunque en otro papel, estrellas del pluralismo al que todo le es indiferente, no, se necesitar\u00edan actores de edad avanzada que en su d\u00eda fueron grandes idealistas, que nunca llegaron a ser famosos, que nunca pudieron consolidarse completamente, aunque dominaban su oficio y hab\u00edan pasado por experiencias que imprimieron su sello en su trabajo pero que para las generaciones siguientes ya no significaban nada, cuando lo importante era m\u00e1s la fama que la verdad y las frases vac\u00edas de la verdad base de la fama, la fama como base para hacer negocio y no como fanal de sentido y significado. Los actores fracasados no tendr\u00edan que interpretar nada, ser\u00edan lo que los difuntos padres fundadores presentar\u00edan si se los llevara ma\u00f1ana a un escenario: el respeto inalienable a los ideales de su juventud, la desesperaci\u00f3n por su fracaso y su ca\u00edda en el olvido, el intenso deseo de verse redescubiertos y recordados, y la dignidad de una idea m\u00e1s hermosa que toda la rocalla bajo la que fue enterrada. \u00bfNo hab\u00eda actores fracasados de ochenta o noventa a\u00f1os que no estaban chalados y que a\u00fan pod\u00edan aprenderse los textos? Ser\u00edan los representantes aut\u00e9nticos de la \u00e9poca fundacional europea.\n\nMartin bebi\u00f3 grappa en el vaso de lavarse los dientes.\n\nLo ve\u00eda ante \u00e9l como una pel\u00edcula: la marcha de los muertos, en la gran pantalla, marchaban de manera radial por todas las calles y avenidas hacia el edificio Berlaymont, una manifestaci\u00f3n de la historia reprimida, un fanal de los fundadores del proyecto de uni\u00f3n europea, y luego llegaba el ata\u00fad. \u00bfQu\u00e9 clase de ata\u00fad? \u00bfQui\u00e9n yac\u00eda dentro? El \u00faltimo jud\u00edo, pues claro, el \u00faltimo jud\u00edo que hab\u00eda sobrevivido a un campo de exterminio. \u00a1Muerto en fatal coincidencia justo en aquel se\u00f1alado aniversario de la Comisi\u00f3n! En el marco del aniversario ten\u00eda lugar entonces una pomposa procesi\u00f3n, un entierro solemne, m\u00e1s que un entierro oficial, el primer entierro supranacional, europeo, de la Uni\u00f3n, el presidente de la Comisi\u00f3n renueva ante el ata\u00fad este juramento: \u00a1Nunca m\u00e1s nacionalismo, racismo, Auschwitz! Y la eternidad se prolongar\u00eda tras la muerte del \u00faltimo testigo, el punto final quedar\u00eda sobrepasado y la historia ser\u00eda otra vez m\u00e1s que un p\u00e9ndulo cuyas oscilaciones ponen a los hombres en estado de ins\u00edpido trance. En la pel\u00edcula de Martin aparec\u00edan nubes negras, en un dram\u00e1tico espect\u00e1culo del cielo tan radical como un eclipse solar, las nubes se pon\u00edan delante del sol, delante de toda luz, con asombrosa rapidez, a c\u00e1mara r\u00e1pida; el cine se deten\u00eda ahora un breve instante porque Martin se quedaba indeciso ante las palabras c\u00e1mara r\u00e1pida, fumaba y miraba al vac\u00edo y pensaba: c\u00e1mara r\u00e1pida. Luego las nubes segu\u00edan abalanz\u00e1ndose, la oscuridad iba en aumento, llegaba una tormenta que arrancaba los sombreros de las cabezas de la gente, ve\u00eda sombreros revoloteando en el aire, la oscuridad era cada vez mayor y...\n\nDesvanecimiento. No era sue\u00f1o. En alg\u00fan momento, hacia las cuatro de la ma\u00f1ana, Martin se desvaneci\u00f3.\n\nTom\u00f3 un taxi al aeropuerto, durante el viaje casi se qued\u00f3 dormido. Dormit\u00f3 durante el vuelo. Tomaba aspirinas como si fueran Smarties. En el aeropuerto de Bruselas cogi\u00f3 en el nivel 0 el autob\u00fas al barrio de Europa. Desde all\u00ed camin\u00f3 unos pasos a la estaci\u00f3n de metro Maelbeek, ya que el acceso Berlaymont estaba otra vez cerrado. Ya s\u00f3lo quer\u00eda ir a casa. Nunca hasta entonces hab\u00eda tenido una tan \u00edntima sensaci\u00f3n de que el piso de Bruselas era su hogar. En el and\u00e9n mir\u00f3 el panel indicador: cuatro minutos a\u00fan.\n\nEl profesor Erhart ten\u00eda que dejar el hotel Atlas a las 11.00, era muy pronto para ir directamente al aeropuerto. Camin\u00f3 despacio por el Vieux March\u00e9 aux Grains, tirando de su maleta, que saltaba y brincaba sobre el adoquinado como si Bruselas quisiera sacud\u00edrselo de encima. \u00bfQu\u00e9 pod\u00eda hacer durante el tiempo de espera? \u00bfIr a comer? S\u00ed, pero hab\u00eda desayunado muy tarde, no ten\u00eda hambre. Camin\u00f3 hacia la estaci\u00f3n de metro Sainte-Catherine. \u00bfQu\u00e9 hacer? El calor era insoportable, estaba empezando a sudar. En el peri\u00f3dico hab\u00eda le\u00eddo sobre la exposici\u00f3n \u00abModernidad olvidada\u00bb y las violentas discusiones que hab\u00eda provocado. \u00bfIr\u00eda quiz\u00e1 a ver esa exposici\u00f3n? Estaba indeciso. Cuando lleg\u00f3 a la iglesia de Sainte-Catherine entr\u00f3 en ella resueltamente. Ten\u00eda tiempo. En la iglesia har\u00eda m\u00e1s fresco. Hab\u00eda pasado muchas veces delante de esa iglesia, pero s\u00f3lo hab\u00eda entrado en ella una vez, en su primera tarde en Bruselas. Entonces hab\u00eda buscado refugio ante un s\u00fabito chubasco. La iglesia ten\u00eda realmente toda la apariencia de una catedral. Quiz\u00e1 fuera art\u00edstica y culturalmente interesante.\n\nApenas hubo entrado se pregunt\u00f3 qu\u00e9 buscaba all\u00ed. En las filas de bancos hab\u00eda algunas personas que rezaban, los turistas levantaban los smartphones o las tabletas y hac\u00edan fotos, continuamente fulguraban los flashes, mientras en los altares laterales temblaban las llamas de los cirios votivos. En Viena tampoco pisaba jam\u00e1s una iglesia. \u00bfPor qu\u00e9 entonces iba a visitar una iglesia en Bruselas? Cuando ten\u00eda doce a\u00f1os lo llevaron con su clase del colegio a la catedral de San Esteban. Ni siquiera fue por motivos religiosos, sino en el marco de la asignatura de historia regional. Y a los quince a\u00f1os hab\u00eda acompa\u00f1ado una vez a la misa del gallo a su abuela, que, cuando oy\u00f3 a la muerte llamar a su puerta, se volvi\u00f3 creyente last-minute. Pero \u00e9l s\u00f3lo lo hizo despu\u00e9s que ella le pasara sin que nadie se enterara un billete de veinte schillings. Desde entonces no hab\u00eda entrado en ninguna otra iglesia. Estaba contento de no haber recibido educaci\u00f3n religiosa, comulgaba con el ate\u00edsmo fundamental de sus padres, aunque s\u00f3lo fue m\u00e1s tarde, mucho m\u00e1s tarde, cuando comprendi\u00f3 que hab\u00edan sido nacionalsocialistas de rigurosa observancia y por eso anticlericales.\n\nAvanz\u00f3 por la nave lateral izquierda y entonces un hombre que llevaba un traje negro y alzacuello le habl\u00f3:\n\nEst-ce que vous l'aimez aussi?\n\nPardon?\n\n\u00a1A la Virgen negra!\n\nErhart sigui\u00f3 la mirada del hombre, vio la estatua de la Virgen.\n\n\u00a1Un milagro! Lo ve usted, \u00bfverdad?\n\n\u00bfA qu\u00e9 se refiere? \u00bfA la cara? \u00bfPorque es negra?\n\nNo. F\u00edjese en la mano. \u00bfLa ve? El pulgar est\u00e1 roto. En la \u00e9poca de la Reforma, los protestantes devastaron la iglesia y arrojaron al canal esa imagen de ah\u00ed delante, y al caer se rompi\u00f3 el pulgar. \u00bfVe usted el sitio de la fractura? Y ahora cuente los dedos. \u00bfQu\u00e9 me dice? \u00bfLo ve? \u00a1Cinco dedos! Los cat\u00f3licos rescataron a la Virgen, la devolvieron a la iglesia y la colocaron de nuevo en su lugar. Y aunque le hab\u00edan cortado un dedo, ten\u00eda otra vez cinco dedos. \u00a1Un milagro! \u00bfLo ve usted?\n\nSe santigu\u00f3 con radiante sonrisa.\n\n\u00bfPuede ser, dijo Erhart, que antes tuviera seis dedos?\n\nEl hombre del traje negro lo mir\u00f3, se dio media vuelta y se march\u00f3.\n\nEl profesor Erhart sali\u00f3 de la iglesia, se dirigi\u00f3 a la estaci\u00f3n del metro. Ten\u00eda la intenci\u00f3n de viajar hasta la Gare Centrale y all\u00ed tomar un tren al aeropuerto. Pero habr\u00eda llegado demasiado pronto, para matar el tiempo habr\u00eda paseado ap\u00e1tico por las taxfree shops, finalmente se habr\u00eda comido un s\u00e1ndwich malo y bebido una cerveza, de puro aburrimiento otra cerveza m\u00e1s, despu\u00e9s otra vez dar una vuelta, tomar un caf\u00e9, luego sentarse en alg\u00fan sitio y esperar. Finalmente, como el tiempo no pasaba y no pasaba, comprar\u00eda chocolate belga, porque de B\u00e9lgica se trae chocolate como recuerdo, pero no ten\u00eda a nadie a quien pudiera o quisiera llevar algo, a Trudi le gustaba el chocolate, de vez en cuando \u00e9l le tra\u00eda uno de esos rulos de chocolate con la borla azul, de Milka, al principio cuando se citaba con ella, m\u00e1s tarde simplemente como peque\u00f1o detalle cuando Erhart volv\u00eda de la universidad a casa y cuando a la vuelta de la esquina del instituto, en la Grillparzerstrasse, todav\u00eda estaba aquella confiter\u00eda, el Bon-bon Kaiser, regentado por el anciano se\u00f1or Kaiser, que dec\u00eda frases como \u00abPresente mis respetos a su se\u00f1ora esposa, profesor\u00bb, cuando \u00e9l era simple auxiliar de c\u00e1tedra, y se alegraba cuando Trudi se alegraba, pero a \u00e9l no le seduc\u00eda el chocolate, \u00bfpor qu\u00e9 iba a comprarlo ahora? La \u00faltima vez, s\u00f3lo para que pasara el tiempo, hab\u00eda comprado en el aeropuerto de Bruselas una bombonera de Neuhaus y luego, en casa, estuvo rodando semanas por la cocina. Y todav\u00eda segu\u00eda all\u00ed en alg\u00fan rinc\u00f3n. No se ape\u00f3 en Gare Centrale, sino que sigui\u00f3 hasta Maelbeek, all\u00ed, muy cerca de la estaci\u00f3n, conoc\u00eda un restaurante italiano donde hab\u00eda almorzado una vez despu\u00e9s de un meeting del New Pact. Era agradable y sin complicaciones y la comida era tan buena que aun sin hambre uno pod\u00eda saborearla. Encontr\u00f3, en efecto, la Osteria Agricola Toscana. Cuando esperaba la comida y, mientras com\u00eda y tomaba vino, reflexion\u00f3 sobre su porvenir. Al menos se lo propuso y lo intent\u00f3. No era tan f\u00e1cil. Lo \u00fanico que pod\u00eda saber con seguridad en cuanto al futuro inmediato era que todo lo que com\u00eda y beb\u00eda lo metabolizar\u00eda y, de regreso a Viena, lo eliminar\u00eda. Se exhort\u00f3 a no tener pensamientos tan banales. No era tan f\u00e1cil. La comida le gustaba. Pero le parec\u00eda un derroche: una comida tan buena s\u00f3lo para \u00e9l, que no pod\u00eda compartirla con nadie. El vino era muy bueno. Reflexion\u00f3 sobre su porvenir. Pensaba que lo mismo pod\u00eda reflexionar sobre si hab\u00eda una vida despu\u00e9s de la muerte. S\u00ed, s\u00ed, pens\u00f3, eso era vivir en la memoria de otros, dejar huella. \u00bfPod\u00eda \u00e9l dejar algo que quedara en la memoria de la gente? \u00bfTen\u00eda alg\u00fan legado que dejar a la posteridad? \u00bfUn testamento? Pens\u00f3 que tal vez tendr\u00eda tiempo de escribir un libro. \u00bfSe pod\u00eda planear y escribir un libro como un testamento que estableciera una herencia para la posteridad? \u00bfUna autobiograf\u00eda tal vez? Quiz\u00e1 deber\u00eda escribir una autobiograf\u00eda, sus experiencias y reflexiones, para que en alg\u00fan momento, una vez al menos, fuera posible el recuerdo de lo que habr\u00eda podido ser y segu\u00eda agit\u00e1ndose en el fondo, sin hallar la liberaci\u00f3n. En la autobiograf\u00eda de Armand Moens hab\u00eda le\u00eddo: \u00abLa historia no es s\u00f3lo el relato de lo que fue, sino tambi\u00e9n la continua elaboraci\u00f3n de las razones por las que no pudo ser algo m\u00e1s razonable\u00bb. \u00c9se deb\u00eda ser el lema de su autobiograf\u00eda, pens\u00f3, y pidi\u00f3 la cuenta y un caf\u00e9 solo. Quer\u00eda escribir una autobiograf\u00eda que no contara su modesta vida, sino lo no vivido. Lo no vivido de su tiempo de vida. Ahora el tiempo se estaba acabando. Ten\u00eda que ir al aeropuerto. Pag\u00f3 toda la botella de vino.\n\nSe puso nervioso, se le hab\u00eda hecho tarde.\n\n\u00bfSer\u00eda mejor ir a la glorieta Schuman y coger el autob\u00fas al aeropuerto? \u00bfO volver al metro, tres estaciones hasta la Gare Centrale e ir en tren? Pens\u00f3 que el tren era m\u00e1s r\u00e1pido que el autob\u00fas. Corri\u00f3, con su maleta dando tumbos, a la estaci\u00f3n de Maelbeek, baj\u00f3 a saltos por la escalera mec\u00e1nica, not\u00f3 demasiado tarde que estaba fuera de servicio, en el and\u00e9n mir\u00f3 nervioso el panel indicador: dos minutos.\n\nDavid de Vriend oy\u00f3 el grito \u00ab\u00a1Qu\u00e9date!\u00bb, se tap\u00f3 los o\u00eddos, pero entonces le reson\u00f3 a\u00fan con m\u00e1s fuerza en la cabeza, como si ese \u00ab\u00a1Qu\u00e9date!\u00bb se moviera continuamente de un lado a otro entre las sienes, el grito se repet\u00eda y se repet\u00eda, \u00ab\u00a1Qu\u00e9date!\u00bb, y supo que ahora ten\u00eda que marcharse. Ahora mismo. Ya no hab\u00eda lugar a reflexi\u00f3n, s\u00f3lo la resoluci\u00f3n. Ahora mismo me voy de aqu\u00ed, me marcho.\n\nNi siquiera cerr\u00f3 la puerta despu\u00e9s de salir. No se encontr\u00f3 con nadie. En la escalera, abajo, en la recepci\u00f3n, m\u00e1s all\u00e1, en el comedor, delante, en la biblioteca, por todas partes hab\u00eda silencio, no se ve\u00eda a nadie. Despu\u00e9s del almuerzo, la mayor parte de los residentes dorm\u00edan o daban un paseo, o bien bajando por la rue de l'Arbre Unique hasta el riachuelo con el sauce llor\u00f3n y all\u00ed daban de comer a los p\u00e1jaros, o bien por el cementerio hasta el banco, se sentaban un rato, luego volv\u00edan para el t\u00e9. Las cuidadoras tomaban caf\u00e9 en la salita del personal y comentaban sus casos dif\u00edciles.\n\nDe Vriend abandon\u00f3 la Maison Hanssens como si \u00e9sta fuera un mundo sin seres humanos. O como un vag\u00f3n de tren lleno de muertos. \u00ab\u00a1Nos vas a traer la desgracia!\u00bb, \u00e9sas hab\u00edan sido las \u00faltimas palabras. Ten\u00eda que marcharse, marcharse lo antes posible. \u00bfAd\u00f3nde?\n\nHab\u00eda sido una decisi\u00f3n que no le hab\u00eda dejado tiempo para sopesar los pros y los contras. \u00a1Salir! \u00a1Soltarse de un tir\u00f3n y salir!\n\nSe fue a la puerta del cementerio, pero no entr\u00f3 en \u00e9l, ten\u00eda una direcci\u00f3n, a \u00e9sa deb\u00eda ir.\n\nCuando salt\u00f3 del tren, un joven le entreg\u00f3 a toda prisa un sobre en el que hab\u00eda un papel con una direcci\u00f3n segura y veinte francos. Todo ocurri\u00f3 con tanta rapidez. Despu\u00e9s de un tiroteo, el tren se puso otra vez en marcha, pero \u00e9l lo vio muy despacio, c\u00f3mo arrancaba el tren, la puerta corredera abierta del vag\u00f3n de ganado como un agujero negro, detr\u00e1s sus padres y su hermano peque\u00f1o, era como si esa imagen se moviera un cent\u00edmetro tras otro, disparos y un golpear y resollar, el estr\u00e9pito cada vez m\u00e1s r\u00e1pido del golpe de un hierro contra otro hierro, un empuj\u00f3n, el hombre le empuj\u00f3 otra vez y vocifer\u00f3: \u00a1Corre! \u00a1Busca la direcci\u00f3n que tienes \u2014\u200cy se\u00f1al\u00f3 el sobre que acababa de ponerle en la mano\u2014 ah\u00ed, ah\u00ed dentro! Y el tren tom\u00f3 velocidad, el agujero negro detr\u00e1s del que su familia se apretujaba ya no estaba, entonces pas\u00f3 otro agujero negro y otro m\u00e1s, y \u00e9l se dio la vuelta y vio gente que corr\u00eda por los sembrados. \u00bfCu\u00e1ntos eran? \u00bfCien? Vio c\u00f3mo aqu\u00ed y all\u00e1 algunos se tiraban al suelo o se desplomaban, heridos por balazos en la espalda, y \u00e9l se arroj\u00f3 al suelo, rod\u00f3 por el terrapl\u00e9n de las v\u00edas y se qued\u00f3 tendido en el suelo hasta que pas\u00f3 el tren desde el cual el cuerpo de guardia de las SS disparaba a los fugitivos. S\u00f3lo entonces empez\u00f3 a correr.\n\nVio delante, tendidas en el campo, a personas que se hab\u00edan tirado al suelo y que ahora se levantaban. Pas\u00f3 junto a personas que estaban tendidas y ya no se levantaban. Corri\u00f3 en la noche. Ten\u00eda una direcci\u00f3n.\n\nNo conoc\u00eda el camino. Entonces lleg\u00f3 un autob\u00fas, par\u00f3 ante la puerta del cementerio.\n\nAutob\u00fas n.\u00b0 4. A De Vriend eso no le dec\u00eda nada. Subi\u00f3. El autob\u00fas arranc\u00f3. Le llevaba fuera. Lo dejaba todo atr\u00e1s. Nada m\u00e1s llegar a Auschwitz, sus padres y su hermano peque\u00f1o fueron directamente a la c\u00e1mara de gas. No habr\u00eda podido salvarlos si no hubiera saltado del tren, si se hubiera quedado con ellos. Tampoco hubo tiempo para discutir: \u00bfsaltamos o no saltamos? \u00bfQu\u00e9 nos espera en un caso y qu\u00e9 nos espera en el otro? \u00c9l hab\u00eda saltado. \u00c9l hab\u00eda sobrevivido. Su padre, ese modesto contable, ese hombre fr\u00e1gil y cari\u00f1oso, de tristes ojos oscuros que no pod\u00eda aportar al funcionamiento del mundo otra cosa que su implacable correcci\u00f3n, su confianza en el control del debe y el haber, con un marcado orgullo que en el fondo era resistencia frente a los tiempos que corr\u00edan y frente a la sonrisa ir\u00f3nica y despectiva de los m\u00e1s grandes y m\u00e1s acomodaticios. Incluso en casa, entre las cuatro paredes propias, completamente en privado, sin que nadie lo viera, representaba el espect\u00e1culo de la correcci\u00f3n absoluta, como si el rey y el gobierno le mirasen y asintieran dando su aprobaci\u00f3n. Y su madre, tambi\u00e9n la ve\u00eda cuando la recordaba, siempre con aquella mirada triste y devota, los dos ten\u00edan esos ojos tristes, no porque vieran venir lo que iba a suceder, sino porque cre\u00edan que todo seguir\u00eda siendo siempre como era. No se hab\u00edan preocupado, s\u00f3lo se hab\u00edan hecho a las preocupaciones que ellos consideraban que eran su vida... y no que eran las piedras del camino hacia su muerte. De Vriend s\u00f3lo la hab\u00eda o\u00eddo gritar, s\u00ed, chillar, una \u00fanica vez: \u00a1Qu\u00e9date! Si se hubiera quedado, habr\u00eda ido a la c\u00e1mara de gas como ella. No hab\u00eda podido ni habr\u00eda podido salvarla. \u00bfEso es culpa?\n\nTen\u00eda una direcci\u00f3n.\n\nPersonas desconocidas le hab\u00edan ense\u00f1ado a tener orgullo y capacidad de resistencia. Le hab\u00edan querido como a un hijo propio. Cuando finalmente fue traicionado, no hubo tiempo suficiente para asesinar mediante el trabajo a un hombre joven y fuerte. Hab\u00eda tenido suerte. Desgracia, suerte en la desgracia, desgracia, otra vez suerte en la desgracia.\n\nNo encontraba la direcci\u00f3n. Sentado en el autob\u00fas, not\u00f3 que sus bolsillos estaban vac\u00edos. Ten\u00eda que recordar. Ten\u00eda que encontrar el camino, reconocerlo. Suspir\u00f3. Ten\u00eda que recordar. Pero all\u00ed s\u00f3lo hab\u00eda un agujero negro. Mir\u00f3 por la ventana. Lo que por all\u00ed pasaba no era un recuerdo. Ning\u00fan poste indicador, nada vinculado a alguna experiencia. Fachadas.\n\nAhora no hab\u00eda nada. Las puertas del autob\u00fas se abrieron y se cerraron. Luego el autob\u00fas pasaba otra vez, tambale\u00e1ndose, junto a las fachadas. Las puertas se abr\u00edan y se cerraban. Eso era todo.\n\nAlguien abri\u00f3 de un tir\u00f3n la puerta del vag\u00f3n. Una voz grit\u00f3: \u00a1Afuera! \u00a1Saltad fuera!\n\nLas puertas del autob\u00fas se abrieron. \u00a1Qu\u00e9date! \u00a1Nos vas a traer la desgracia!\n\nDe Vriend baj\u00f3 del autob\u00fas de un salto. Casi se cay\u00f3. Un hombre que esperaba en la parada del autob\u00fas le agarr\u00f3.\n\n\u00a1Corre! A esta direcci\u00f3n.\n\nDe Vriend mir\u00f3 alrededor, vio a gente que corr\u00eda calle abajo. \u00bfD\u00f3nde estaba? Delante de un agujero negro. Durante un breve instante lo reconoci\u00f3: estaci\u00f3n de metro de Maelbeek. Eso le dec\u00eda algo. \u00bfQu\u00e9? Se meti\u00f3 en ella, bajando la escalera. Ten\u00eda que reconocer el camino. Se dirigi\u00f3 al and\u00e9n y pens\u00f3 que \u00e9se era el camino.\n\nFaltaba un minuto.\n\nUn hombre con una bolsa. Una mujer que tecleaba algo en su smartphone. Un hombre con una maleta. El tren lleg\u00f3, se detuvo. Se abrieron las puertas. En la puerta abierta delante de \u00e9l vio a un ni\u00f1o de la mano de su madre. El ni\u00f1o se solt\u00f3 de un tir\u00f3n cuando sali\u00f3 del tren.\n\nEn ese momento explot\u00f3 la bomba.\n\nCuando la se\u00f1ora Jos\u00e9phine junto con monsieur Hugo, el portero de la Maison Hanssens, vaciaba la habitaci\u00f3n de David de Vriend, encontr\u00f3 un papel con una lista de nombres.\n\nMonsieur Hugo ech\u00f3 tres camisas en una caja de mudanza, dijo: Mucho no ten\u00eda.\n\nLa se\u00f1ora Jos\u00e9phine asinti\u00f3. Todos los nombres de esa lista estaban tachados.\n\nSon poqu\u00edsimos los que tienen mucho, dijo Hugo. Yo llevo trabajando ya ocho a\u00f1os aqu\u00ed en la casa, dijo, y sigo asombr\u00e1ndome de lo poco que queda de una persona al final.\n\nS\u00ed, dijo Jos\u00e9phine. Se sent\u00f3, mir\u00f3 extra\u00f1ada la hoja. Al final de la lista de los nombres tachados, David de Vriend hab\u00eda escrito su propio nombre.\n\nTen\u00eda unos pa\u00f1uelos muy bonitos, con monograma, dijo Hugo, y ech\u00f3 los pa\u00f1uelos en la caja.\n\nS\u00f3lo el propio nombre de David de Vriend no estaba tachado.\n\n\u00a1Ten\u00eda unos trajes fant\u00e1sticos! Realmente de inmejorable calidad. Los de la ayuda a los indigentes se alegrar\u00e1n. Pero si alguien pide limosna con este traje, no le dar\u00e1n ni un c\u00e9ntimo. A un hombre vestido con este traje \u2014\u200clevant\u00f3 el tweed de De Vriend\u2014 no le ayuda nadie.\n\nA Jos\u00e9phine le habr\u00eda gustado que se callara. No dijo nada. All\u00ed, en una mesita auxiliar delante de ella, hab\u00eda un bol\u00edgrafo. Lo cogi\u00f3, lo empu\u00f1\u00f3 como un cuchillo.\n\nY a todo esto, \u00bfa qu\u00e9 se dedicaba en la vida?, pregunt\u00f3 monsieur Hugo. \u00bfEra relevante en alg\u00fan aspecto? \u00bfPol\u00edtico o alto funcionario? Quiero decir..., porque la Comisi\u00f3n organiza su entierro.\n\nEl silencioso entierro de una \u00e9poca, pens\u00f3 Jos\u00e9phine.\n\nLo que echo de menos son las cosas de siempre: \u00e1lbumes de fotos, agendas de bolsillo, diarios. Muy poco com\u00fan. No ten\u00eda nada, ni siquiera un \u00e1lbum de fotos, y eso lo tiene normalmente todo el mundo, dijo, echando los tensores de zapatos en la caja.\n\nJos\u00e9phine se pregunt\u00f3 qu\u00e9 deb\u00eda hacer con aquella lista de nombres. \u00bfEcharla en la caja? \u00bfO en la papelera? \u00bfTachaba tambi\u00e9n el nombre David de Vriend? \u00bfHab\u00eda querido eso \u00e9l? \u00bfDej\u00f3 por eso all\u00ed en la mesita la hoja, junto con el bol\u00edgrafo? Para que ella despu\u00e9s...\n\nEl se\u00f1or Hugo ech\u00f3 el cepillo de dientes, la pasta, la tijera de u\u00f1as, el desodorante y la m\u00e1quina de afeitar en una bolsa de pl\u00e1stico, la bolsa en la caja de cart\u00f3n. La caja no se llenar\u00e1, dijo.\n\nEsa muerte atroz, pens\u00f3 Jos\u00e9phine. Que precisamente De Vriend en ese atentado... Por otra parte: \u00bfqu\u00e9 quiere decir \u00abprecisamente\u00bb \u00e9l? Para todos. Para cada uno de los que estaban... en el momento que no deb\u00edan... para todos los... veinte muertos y ciento treinta heridos graves.\n\nDobl\u00f3 la lista de los nombres, la meti\u00f3 en el bolsillo de su bata blanca, se dio unos golpecitos con la mano en el bolsillo y pens\u00f3: mientras su nombre no est\u00e9 tachado...\n\nEso era todo, dijo monsieur Hugo.\n\n# EP\u00cdLOGO\n#\n\nLos editores del peri\u00f3dico Metro sab\u00edan que los activistas por los derechos de los animales protestar\u00edan. Antes de empezar con su serie de art\u00edculos, Kurt van der Koot se lo advirti\u00f3 a la redacci\u00f3n. El director se hab\u00eda limitado a echarse a re\u00edr: la protesta de los radicales s\u00f3lo fortalecer\u00eda el apego de los lectores al peri\u00f3dico.\n\nLo extra\u00f1o fue lo tarde que lleg\u00f3 esa protesta. Fue al cabo de varias semanas, tras un art\u00edculo en Le Soir que era un ataque a Metro y al periodismo sensacionalista de la publicaci\u00f3n gratuita.\n\nUn c\u00ednico art\u00edculo en el que Le Soir somet\u00eda a discusi\u00f3n la sospecha de que el cerdo que corr\u00eda por Bruselas probablemente no exist\u00eda y que las fotos borrosas de las c\u00e1maras de seguridad eran falsificaciones. Esa serie de Metro \u2014\u200ccontinuaba el peri\u00f3dico\u2014 s\u00f3lo era sin duda un nuevo ejemplo de c\u00f3mo trabajaban los peri\u00f3dicos gratuitos: generando desasosiego con historias inventadas. El art\u00edculo iba ilustrado con una foto, tomada en el matadero de Van Kampen, que mostraba dos mitades de cerdo colgadas de ganchos. Al pie de la foto pon\u00eda: \u00ab\u00bfEl final del cerdo de Bruselas?\u00bb.\n\nAdjunta a ese art\u00edculo hab\u00eda una entrevista con Michel Moreau, presidente de Socorro Animal B\u00e9lgica, que calificaba la campa\u00f1a de Metro como \u00abel mayor esc\u00e1ndalo desde Marc Dutroux\u00bb.[11] Era escandaloso, dec\u00eda, utilizar para una campa\u00f1a publicitaria de un peri\u00f3dico a un cerdo que corr\u00eda por la ciudad, en lugar de salvar a ese cerdo, en el supuesto de que existiera. Las calles de la ciudad, a\u00f1ad\u00eda, no eran un espacio vital natural para un cerdo que, confrontado con los desaf\u00edos del asfalto, del gent\u00edo por las calles y del tr\u00e1fico rodado, deb\u00eda de encontrarse en un continuo estr\u00e9s que seguramente era m\u00e1s torturante para el animal que la explotaci\u00f3n en las jaulas de las f\u00e1bricas de animales. Y conminaba a \u00ablas autoridades competentes\u00bb a que aclarasen por fin si se trataba de un cerdo dotado de \u00abexistencia real\u00bb y, si \u00e9se era efectivamente el caso, a que dieran orden de capturar al animal, que un veterinario le hiciera un reconocimiento y lo llevaran a una granja donde se le criase de modo adecuado a su especie. \u00abEn tanto que protector de animales, soy prudente con las met\u00e1foras de animales, pero lo que aqu\u00ed est\u00e1 ocurriendo s\u00f3lo puede calificarse de cerdada\u00bb, dec\u00eda Moreau.\n\nAhora tambi\u00e9n Le Soir ten\u00eda su shitstorm. Docenas de lectores protestaron en cartas al director y en postings por comparar el maltrato animal con el abuso de menores y con los cr\u00edmenes de Marc Dutroux. En Facebook, la entrevista con Michel Moreau recibi\u00f3 en el espacio de pocas horas cientos de furiosos emoticonos.\n\nEl ataque a Metro result\u00f3 contraproducente y se convirti\u00f3 en el problema de Le Soir a corto plazo. Sin embargo, la redacci\u00f3n de Metro ten\u00eda un problema mayor a\u00fan que deb\u00eda resolver sin falta antes de que la opini\u00f3n p\u00fablica supiera siquiera de su existencia. La campa\u00f1a \u00abBruselas busca un nombre para su cerdo\u00bb se les hab\u00eda ido por completo de las manos. Los lectores pod\u00edan hacer sus propuestas online o dar su like a las propuestas de otros, al mismo tiempo, con cada nombre y cada clic se actualizaba el ranking de las propuestas, en correspondencia con el n\u00famero de nombres presentados y el n\u00famero de likes. Ese ranking estaba pensado como base sobre la cual el jurado formar\u00eda la longlist. Al principio llegaron propuestas f\u00e1ciles de explicar: miss Piggy, madame Cochon, Cerdito astuto.\n\nS\u00f3lo ten\u00eda relaci\u00f3n con Bruselas Varkentje Pis (17 likes), quiz\u00e1 tambi\u00e9n Catherine, porque al cerdo lo hab\u00edan visto por primera vez en Sainte-Catherine (21 likes). Pero luego ocurri\u00f3 algo inconcebible. Cientos de veces se indic\u00f3 un nombre que, con miles de likes, se puso a la cabeza del ranking: Mohamed. Eso s\u00f3lo pod\u00eda ser una campa\u00f1a organizada. Cuando la redacci\u00f3n se percat\u00f3 de ello elimin\u00f3 la p\u00e1gina. Varios miembros del jurado se retiraron. Ya no quer\u00edan participar en una acci\u00f3n que generaba un acto de agresi\u00f3n contra conciudadanos musulmanes.\n\nParamos esto, decidi\u00f3 el director. Sin decir nada. Dentro de poco tiempo estar\u00e1 olvidado. Por cierto, Kurt, dijo a Van der Koot, \u00bfse ha dado cuenta de que desde hace dos semanas ya no llegan fotos del cerdo? Y ning\u00fan aviso de que lo hayan visto en alg\u00fan sitio. Ha desaparecido. Desaparecido sin dejar rastro.\n\n\u00c0 suivre.\n\n# NOTAS\n\n[]. Los austriacos pronuncian TEPP como Depp, palabra que significa \u00abtonto\u00bb, \u00abmajadero\u00bb. (N. de la t.)\n\n[]. La Braban\u00e7onne es el himno nacional belga, compuesto en 1830. El texto tiene tres versiones: francesa, flamenca y alemana. (N. de la t.)\n\n[]. El sustantivo alem\u00e1n Schatz significa \u00abtesoro\u00bb y se usa en sentido figurado, aqu\u00ed despectivo: \u00abla queridita alemana\u00bb.\n\n[]. El t\u00e9rmino alem\u00e1n Lager significa \u00abalmac\u00e9n\u00bb, pero, entre otros significados, tiene tambi\u00e9n el de \u00abcampo de concentraci\u00f3n\u00bb.\n\n[]. Yad Vashem es el gran museo y centro de conmemoraci\u00f3n, documentaci\u00f3n e investigaci\u00f3n del Holocausto, situado en Jerusal\u00e9n. (N. de la t.)\n\n[]. Simon Wiesenthal (1908-2005), superviviente del Holocausto, dedic\u00f3 la mayor parte de su vida a localizar e identificar criminales de guerra nazis (fue \u00e9l quien localiz\u00f3 a Eichmann). El centro Simon Wiesenthal de Los \u00c1ngeles est\u00e1 dedicado a documentar v\u00edctimas del Holocausto y lleva un registro de criminales de guerra nazis. (N. de la t.)\n\n[]. Breve pero c\u00e9lebre relato del escritor alem\u00e1n Johann Peter Hebel, publicado en 1808. Un joven artesano alem\u00e1n, de paso por \u00c1msterdam, pregunta repetidas veces lleno de admiraci\u00f3n qui\u00e9n es el due\u00f1o de los hermosos edificios de esa ciudad y le responden \u00abKannitverstaan\u00bb (\u00abNo comprendo\u00bb). Cuando poco despu\u00e9s ve pasar un entierro, pregunta qui\u00e9n es el muerto y recibe la misma contestaci\u00f3n. El chico, convencido de que \u00abKannitverstaan\u00bb es un se\u00f1or riqu\u00edsimo que acaba de morir, llega as\u00ed a la conclusi\u00f3n de la caducidad de todo lo terrenal y se conforma con su humilde destino. (N. de la t.)\n\n[]. Kunst macht frei. Alusi\u00f3n al lema emplazado en la verja de acceso al campo de concentraci\u00f3n de Auschwitz y de otros campos de concentraci\u00f3n: Arbeit macht frei. \u00abEl trabajo libera.\u00bb (N. de la t.)\n\n[]. Alusi\u00f3n a la c\u00e9lebre primera estrofa (hoy prohibida en Alemania) del himno nacional alem\u00e1n: \u00abDeutschland, Deutschland \u00fcber alles\u00bb. (N. de la t.)\n\n[]. Sieg significa \u00abvictoria\u00bb. (N. de la t.)\n\n[]. El caso del delincuente sexual y asesino en serie belga Marc Dutroux (n. 1956) conmocion\u00f3 a la opini\u00f3n p\u00fablica por la monstruosidad de sus delitos (secuestro, tortura, abuso sexual y asesinato de varias ni\u00f1as y adolescentes) y por los numerosos errores en la investigaci\u00f3n. Fue condenado a cadena perpetua en 2004. (N. de la t.)\n\n#\n\nLa capital\n\nRobert Menasse\n\nNo se permite la reproducci\u00f3n total o parcial de este libro, ni su incorporaci\u00f3n a un sistema inform\u00e1tico, ni su transmisi\u00f3n en cualquier forma o por cualquier medio, sea \u00e9ste electr\u00f3nico, mec\u00e1nico, por fotocopia, por grabaci\u00f3n u otros m\u00e9todos, sin el permiso previo y por escrito del editor. La infracci\u00f3n de los derechos mencionados puede ser constitutiva de delito contra la propiedad intelectual (Art. 270 y siguientes del C\u00f3digo Penal)\n\nDir\u00edjase a CEDRO (Centro Espa\u00f1ol de Derechos Reprogr\u00e1ficos) si necesita reproducir alg\u00fan fragmento de esta obra.\n\nPuede contactar con CEDRO a trav\u00e9s de la web www.conlicencia.com o por tel\u00e9fono en el 91 702 19 70 \/ 93 272 04 47\n\nT\u00edtulo original: Die Hauptstadt\n\nDise\u00f1o de la portada, Planeta Arte & Dise\u00f1o\n\n\u00a9 de la fotograf\u00eda de la portada, Julien Pacaud\n\n\u00a9 Suhrkamp Verlag Berl\u00edn, 2017. Publicado de acuerdo con Suhrkamp Verlag Berl\u00edn\n\n\u00a9 por la traducci\u00f3n, Carmen Gauger, 2018\n\nDise\u00f1o original de la colecci\u00f3n: Josep Bag\u00e0 Associats\n\n\u00a9 Editorial Planeta, S. A., 2018\n\nSeix Barral, un sello editorial de Editorial Planeta, S. A.\n\nAvda. Diagonal, 662-664, 08034 Barcelona (Espa\u00f1a)\n\nwww.seix-barral.es\n\nwww.planetadelibros.com\n\nPrimera edici\u00f3n en libro electr\u00f3nico (epub): septiembre de 2018\n\nISBN: 978-84-322-3419-4 (epub)\n\nConversi\u00f3n a libro electr\u00f3nico: El Taller del Llibre, S. L.\n\nwww.eltallerdelllibre.com\n\u00a1Encuentra aqu\u00ed tu pr\u00f3xima lectura!\n\n---\n\n\u00a1S\u00edguenos en redes sociales!\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Donald Lainson\n\n\n\n\n\nTHE WOLVES AND THE LAMB\n\n\nBy William Makepeace Thackeray\n\n\n\n\nDRAMATIS PERSONAE.\n\n     MR. HORACE MILLIKEN, a Widower, a wealthy City Merchant.\n     GEORGE MILLIKEN, a Child, his Son.\n     CAPTAIN TOUCHIT, his Friend.\n     CLARENCE KICKLEBURY, brother to Milliken's late Wife.\n     JOHN HOWELL, M's Butler and confidential Servant.\n     CHARLES PAGE, Foot-boy.\n     BULKELEY, Lady Kicklebury's Servant.\n     MR. BONNINGTON.\n     Coachman, Cabman; a Bluecoat Boy, another Boy (Mrs. Prior's Sons).\n\n     LADY KICKLEBURY, Mother-in-law to Milliken.\n     MRS. BONNINGTON, Milliken's Mother (married again).\n     MRS. PRIOR.\n     MISS PRIOR, her Daughter, Governess to Milliken's Children.\n     ARABELLA MILLIKEN, a Child.\n     MARY BARLOW, School-room Maid.\n     A grown-up Girl and Child of Mrs. Prior's, Lady K.'s Maid, Cook.\n\n\n\n\nTHE WOLVES AND THE LAMB.\n\n\n\n\nACT I.\n\n\nScene.--MILLIKEN'S villa at Richmond; two drawing-rooms opening into\none another. The late MRS. MILLIKEN'S portrait over the mantel-piece;\nbookcases, writing-tables, piano, newspapers, a handsomely furnished\nsaloon. The back-room opens, with very large windows, on the lawn and\npleasure-ground; gate, and wall--over which the heads of a cab and a\ncarriage are seen, as persons arrive. Fruit, and a ladder on the walls.\nA door to the dining-room, another to the sleeping-apartments, &c.\n\n\nJOHN.--Everybody out; governor in the city; governess (heigh-ho!)\nwalking in the Park with the children; ladyship gone out in the\ncarriage. Let's sit down and have a look at the papers. Buttons fetch\nthe Morning Post out of Lady Kicklebury's room. Where's the Daily News,\nsir?\n\nPAGE.--Think it's in Milliken's room.\n\nJOHN.--Milliken! you scoundrel! What do you mean by Milliken? Speak of\nyour employer as your governor if you like; but not as simple Milliken.\nConfound your impudence! you'll be calling me Howell next.\n\nPAGE.--Well! I didn't know. YOU call him Milliken.\n\nJOHN.--Because I know him, because I'm intimate with him, because\nthere's not a secret he has but I may have it for the asking; because\nthe letters addressed to Horace Milliken, Esq., might as well be\naddressed John Howell, Esq., for I read 'em, I put 'em away and docket\n'em, and remember 'em. I know his affairs better than he does: his\nincome to a shilling, pay his tradesmen, wear his coats if I like. I\nmay call Mr. Milliken what I please; but not YOU, you little scamp of a\nclod-hopping ploughboy. Know your station and do your business, or you\ndon't wear THEM buttons long, I promise you. [Exit Page.]\n\nLet me go on with the paper [reads]. How brilliant this writing is!\nTimes, Chronicle, Daily News, they're all good, blest if they ain't. How\nmuch better the nine leaders in them three daily papers is, than nine\nspeeches in the House of Commons! Take a very best speech in the 'Ouse\nnow, and compare it with an article in The Times! I say, the newspaper\nhas the best of it for philosophy, for wit, novelty, good sense too. And\nthe party that writes the leading article is nobody, and the chap that\nspeaks in the House of Commons is a hero. Lord, Lord, how the world\nis 'umbugged! Pop'lar representation! what IS pop'lar representation?\nDammy, it's a farce. Hallo! this article is stole! I remember a passage\nin Montesquieu uncommonly like it. [Goes and gets the book. As he is\nstanding upon sofa to get it, and sitting down to read it, MISS PRIOR\nand the Children have come in at the garden. Children pass across stage.\nMISS PRIOR enters by open window, bringing flowers into the room.]\n\nJOHN.--It IS like it. [He slaps the book, and seeing MISS PRIOR who\nenters, then jumps up from sofa, saying very respectfully,]\n\nJOHN.--I beg your pardon, Miss.\n\nMISS P.--[sarcastically.] Do I disturb you, Howell?\n\nJOHN.--Disturb! I have no right to say--a servant has no right to be\ndisturbed, but I hope I may be pardoned for venturing to look at\na volume in the libery, Miss, just in reference to a newspaper\nharticle--that's all, Miss.\n\nMISS P.--You are very fortunate in finding anything to interest you in\nthe paper, I'm sure.\n\nJOHN.--Perhaps, Miss, you are not accustomed to political discussion,\nand ignorant of--ah--I beg your pardon: a servant, I know, has no right\nto speak. [Exit into dining-room, making a low bow.]\n\nMISS PRIOR.--The coolness of some people is really quite extraordinary!\nthe airs they give themselves, the way in which they answer one, the\nbooks they read! Montesquieu: \"Esprit des Lois!\" [takes book up which\nJ. has left on sofa.] I believe the man has actually taken this from the\nshelf. I am sure Mr. Milliken, or her ladyship, never would. The other\nday \"Helvetius\" was found in Mr. Howell's pantry, forsooth! It is\nwonderful how he picked up French whilst we were abroad. \"Esprit des\nLois!\" what is it? it must be dreadfully stupid. And as for reading\n\"Helvetius\" (who, I suppose, was a Roman general), I really can't\nunderstand how--Dear, dear! what airs these persons give themselves!\nWhat will come next? A footman--I beg Mr. Howell's pardon--a butler\nand confidential valet lolls on the drawing-room sofa, and reads\nMontesquieu! Impudence! And add to this, he follows me for the last two\nor three months with eyes that are quite horrid. What can the creature\nmean? But I forgot--I am only a governess. A governess is not a lady--a\ngoverness is but a servant--a governess is to work and walk all day with\nthe children, dine in the school-room, and come to the drawing-room to\nplay the man of the house to sleep. A governess is a domestic, only her\nplace is not the servants' hall, and she is paid not quite so well as\nthe butler who serves her her glass of wine. Odious! George! Arabella!\nthere are those little wretches quarrelling again! [Exit. Children are\nheard calling out, and seen quarrelling in garden.]\n\nJOHN [re-entering].--See where she moves! grace is in all her steps.\n'Eaven in her high--no--a-heaven in her heye, in every gesture dignity\nand love--ah, I wish I could say it! I wish you may procure it, poor\nfool! She passes by me--she tr-r-amples on me. Here's the chair she sets\nin [kisses it.] Here's the piano she plays on. Pretty keys, them fingers\nout-hivories you! When she plays on it, I stand and listen at the\ndrawing-room door, and my heart thr-obs in time! Fool, fool, fool! why\ndid you look on her, John Howell! why did you beat for her, busy heart!\nYou were tranquil till you knew her! I thought I could have been a-happy\nwith Mary till then. That girl's affection soothed me. Her conversation\ndidn't amuse me much, her ideers ain't exactly elevated, but they are\njust and proper. Her attentions pleased me. She ever kep' the best cup\nof tea for me. She crisped my buttered toast, or mixed my quiet tumbler\nfor me, as I sat of hevenings and read my newspaper in the kitching. She\nrespected the sanctaty of my pantry. When I was a-studying there, she\nnever interrupted me. She darned my stockings for me, she starched and\nfolded my chokers, and she sowed on the habsent buttons of which time\nand chance had bereft my linning. She has a good heart, Mary has. I know\nshe'd get up and black the boots for me of the coldest winter mornings.\nShe did when we was in humbler life, she did.\n\nEnter MARY.\n\nYou have a good heart, Mary!\n\nMARY.--Have I, dear John? [sadly.]\n\nJOHN.--Yes, child--yes. I think a better never beat in woman's bosom.\nYou're good to everybody--good to your parents whom you send half your\nwages to: good to your employers whom you never robbed of a halfpenny.\n\nMARY [whimpering].--Yes, I did, John. I took the jelly when you were in\nbed with the influenza; and brought you the pork-wine negus.\n\nJOHN.--Port, not pork, child. Pork is the hanimal which Jews ab'or. Port\nis from Oporto in Portugal.\n\nMARY [still crying].--Yes, John; you know everything a'most, John.\n\nJOHN.--And you, poor child, but little! It's not heart you want, you\nlittle trump, it's education, Mary: it's information: it's head, head,\nhead! You can't learn. You never can learn. Your ideers ain't no good.\nYou never can hinterchange em with mine. Conversation between us is\nimpossible. It's not your fault. Some people are born clever; some are\nborn tall, I ain't tall.\n\nMARY.--Ho! you're big enough for me, John. [Offers to take his hand.]\n\nJOHN.--Let go my 'and--my a-hand, Mary! I say, some people are born with\nbrains, and some with big figures. Look at that great ass, Bulkeley,\nLady K.'s man--the besotted, stupid beast! He's as big as a\nlife-guardsman, but he ain't no more education nor ideers than the ox he\nfeeds on.\n\nMARY.--Law, John, whatever do you mean?\n\nJOHN.--Hm! you know not, little one! you never can know. Have YOU ever\nfelt the pangs of imprisoned genius? have YOU ever felt what 'tis to be\na slave?\n\nMARY.--Not in a free country, I should hope, John Howell--no such a\nthing. A place is a place, and I know mine, and am content with the\nspear of life in which it pleases heaven to place me, John: and I wish\nyou were, and remembered what we learned from our parson when we went\nto school together in dear old Pigeoncot, John--when you used to help\nlittle Mary with her lessons, John, and fought Bob Brown, the big\nbutcher's boy, because he was rude to me, John, and he gave you that\nblack hi.\n\nJOHN.--Say eye, Mary, not heye [gently].\n\nMARY.--Eye; and I thought you never looked better in all your life\nthan you did then: and we both took service at Squire Milliken's--me as\ndairy-girl, and you as knife-boy; and good masters have they been to\nus from our youth hup: both old Squire Milliken and Mr. Charles as is\nmaster now, and poor Mrs. as is dead, though she had her tantrums--and I\nthought we should save up and take the \"Milliken Arms\"--and now we have\nsaved up--and now, now, now--oh, you are a stone, a stone, a stone!\nand I wish you were hung round my neck, and I were put down the well!\nThere's the hup-stairs bell. [She starts, changing her manner as she\nhears the bell, and exit.]\n\nJOHN [looking after her].--It's all true. Gospel-true. We were children\nin the same village--sat on the same form at school. And it was for her\nsake that Bob Brown the butcher's boy whopped me. A black eye! I'm not\nhandsome. But if I were ugly, ugly as the Saracen's 'Ead, ugly as that\nbeast Bulkeley, I know it would be all the same to Mary. SHE has never\nforgot the boy she loved, that brought birds'-nests for her, and\nspent his halfpenny on cherries, and bought a fairing with his first\nhalf-crown--a brooch it was, I remember, of two billing doves a-hopping\non one twig, and brought it home for little yellow-haired, blue-eyed,\nred-cheeked Mary. Lord, Lord! I don't like to think how I've kissed 'em,\nthe pretty cheeks! they've got quite pale now with crying--and she has\nnever once reproached me, not once, the trump, the little tr-rump!\n\nIs it my fault [stamping] that Fate has separated us? Why did my young\nmaster take me up to Oxford, and give me the run of his libery and the\nsociety of the best scouts in the University? Why did he take me abroad?\nWhy have I been to Italy, France, Jummany with him--their manners noted\nand their realms surveyed, by jingo! I've improved myself, and Mary has\nremained as you was. I try a conversation, and she can't respond. She's\nnever got a word of poetry beyond Watt's Ims, and if I talk of Byron or\nMoore to her, I'm blest if she knows anything more about 'em than the\ncook, who is as hignorant as a pig, or that beast Bulkeley, Lady Kick's\nfootman. Above all, why, why did I see the woman upon whom my\nwretched heart is fixed for ever, and who carries away my soul with\nher--prostrate, I say, prostrate, through the mud at the skirts of her\ngownd! Enslaver! why did I ever come near you? O enchantress Kelipso!\nhow you have got hold of me! It was Fate, Fate, Fate. When Mrs. Milliken\nfell ill of scarlet fever at Naples, Milliken was away at Petersborough,\nRooshia, looking after his property. Her foring woman fled. Me and the\ngoverness remained and nursed her and the children. We nursed the little\nones out of the fever. We buried their mother. We brought the children\nhome over Halp and Happenine. I nursed 'em all three. I tended 'em all\nthree, the orphans, and the lovely gu-gu-governess. At Rome, where she\ntook ill, I waited on her; as we went to Florence, had we been attacked\nby twenty thousand brigands, this little arm had courage for them all!\nAnd if I loved thee, Julia, was I wrong? and if I basked in thy beauty\nday and night, Julia, am I not a man? and if, before this Peri, this\nenchantress, this gazelle, I forgot poor little Mary Barlow, how could I\nhelp it? I say, how the doose could I help it?\n\nEnter Lady KICKLEBURY, BULKELEY following with parcels and a spaniel.\n\nLADY K.--Are the children and the governess come home?\n\nJOHN.--Yes, my lady [in a perfectly altered tone].\n\nLADY K.--Bulkeley, take those parcels to my sitting-room.\n\nJOHN.--Get up, old stoopid. Push along, old daddylonglegs [aside to\nBULKELEY].\n\nLADY K.--Does any one dine here to-day, Howell?\n\nJOHN.--Captain Touchit, my lady.\n\nLADY K.--He's always dining here.\n\nJOHN.--My master's oldest friend.\n\nLADY K.--Don't tell me. He comes from his club. He smells of smoke; he\nis a low, vulgar person. Send Pinhorn up to me when you go down stairs.\n[Exit Lady K.]\n\nJOHN.--I know. Send Pinhorn to me, means, Send my bonny brown hair, and\nsend my beautiful complexion, and send my figure--and, O Lord! O Lord!\nwhat an old tigress that is! What an old Hector! How she do twist\nMilliken round her thumb! He's born to be bullied by women: and I\nremember him henpecked--let's see, ever since--ever since the time of\nthat little gloveress at Woodstock, whose picter poor Mrs. M. made such\na noise about when she found it in the lumber-room. Heh! HER picture\nwill be going into the lumber-room some day. M. must marry to get rid\nof his mother-in-law and mother over him: no man can stand it, not M.\nhimself, who's a Job of a man. Isn't he, look at him! [As he has been\nspeaking, the bell has rung, the Page has run to the garden-door, and\nMILLIKEN enters through the garden, laden with a hamper, band-box, and\ncricket-bat.]\n\nMILLIKEN.--Why was the carriage not sent for me, Howell? There was no\ncab at the station, and I have had to toil all the way up the hill with\nthese confounded parcels of my lady's.\n\nJOHN.--I suppose the shower took off all the cabs, sir. When DID a man\never git a cab in a shower?--or a policeman at a pinch--or a friend when\nyou wanted him--or anything at the right time, sir?\n\nMILLIKEN.--But, sir, why didn't the carriage come, I say?\n\nJOHN.--YOU know.\n\nMILLIKEN.--How do you mean I know? confound your impudence!\n\nJOHN.--Lady Kicklebury took it--your mother-in-law took it--went out\na-visiting--Ham Common, Petersham, Twick'nam--doose knows where. She,\nand her footman, and her span'l dog.\n\nMILLIKEN.--Well, sir, suppose her ladyship DID take the carriage? Hasn't\nshe a perfect right? And if the carriage was gone, I want to know, John,\nwhy the devil the pony-chaise wasn't sent with the groom? Am I to bring\na bonnet-box and a hamper of fish in my own hands, I should like to\nknow?\n\nJOHN.--Heh! [laughs.]\n\nMILLIKEN.--Why do you grin, you Cheshire cat?\n\nJOHN.--Your mother-in-law had the carriage; and your mother sent for\nthe pony-chaise. Your Pa wanted to go and see the Wicar of Putney. Mr.\nBonnington don't like walking when he can ride.\n\nMILLIKEN.--And why shouldn't Mr. Bonnington ride, sir, as long as\nthere's a carriage in my stable? Mr. Bonnington has had the gout, sir!\nMr. Bonnington is a clergyman, and married to my mother. He has EVERY\ntitle to my respect.\n\nJOHN.--And to your pony-chaise--yes, sir.\n\nMILLIKEN.--And to everything he likes in this house, sir.\n\nJOHN.--What a good fellow you are, sir! You'd give your head off your\nshoulders, that you would. Is the fish for dinner to-day? Band-box\nfor my lady, I suppose, sir? [Looks in]--Turban, feathers, bugles,\nmarabouts, spangles--doose knows what. Yes, it's for her ladyship.\n[To Page.] Charles, take this band-box to her ladyship's maid. [To his\nmaster.] What sauce would you like with the turbot? Lobster sauce\nor Hollandaise? Hollandaise is best--most wholesome for you. Anybody\nbesides Captain Touchit coming to dinner?\n\nMILLIKEN.--No one that I know of.\n\nJOHN.--Very good. Bring up a bottle of the brown hock? He likes the\nbrown hock, Touchit does. [Exit JOHN.]\n\nEnter Children. They run to MILLIKEN.\n\nBOTH.--How d'you do, Papa! How do you do, Papa!\n\nMILLIKEN.--Kiss your old father, Arabella. Come here, George--What?\n\nGEORGE.--Don't care for kissing--kissing's for gals. Have you brought me\nthat bat from London?\n\nMILLIKEN.--Yes. Here's the bat; and here's the ball [takes one from\npocket]--and--\n\nGEORGE.--Where's the wickets, Papa. O-o-o--where's the wickets? [howls.]\n\nMILLIKEN.--My dear, darling boy! I left them at the office. What a silly\npapa I was to forget them! Parkins forgot them.\n\nGEORGE.--Then turn him away, I say! Turn him away! [He stamps.]\n\nMILLIKEN.--What! an old, faithful clerk and servant of your father and\ngrandfather for thirty years past? An old man, who loves us all, and has\nnothing but our pay to live on?\n\nARABELLA.--Oh, you naughty boy!\n\nGEORGE.--I ain't a naughty boy.\n\nARABELLA.--You are a naughty boy.\n\nGEORGE.--He! he! he! he! [Grins at her.]\n\nMILLIKEN.--Hush, children! Here, Arabella darling, here is a book for\nyou. Look--aren't they pretty pictures?\n\nARABELLA.--Is it a story, Papa? I don't care for stories in general.\nI like something instructive and serious. Grandmamma Bonnington and\ngrandpapa say--\n\nGEORGE.--He's NOT your grandpapa.\n\nARABELLA.--He IS my grandpapa.\n\nGEORGE.--Oh, you great story! Look! look! there's a cab. [Runs out.\nThe head of a Hansom cab is seen over the garden-gate. Bell rings. Page\ncomes. Altercation between Cabman and Captain TOUCHIT appears to go on,\nduring which]\n\nMILLIKEN.--Come and kiss your old father, Arabella. He's hungry for\nkisses.\n\nARABELLA.--Don't. I want to go and look at the cab; and to tell Captain\nTouchit that he mustn't use naughty words. [Runs towards garden. Page is\nseen carrying a carpet-bag.]\n\nEnter TOUCHIT through the open window smoking a cigar.\n\nTOUCHIT.--How d'ye do, Milliken? How are tallows, hey, my noble\nmerchant? I have brought my bag, and intend to sleep--\n\nGEORGE.--I say, godpapa--\n\nTOUCHIT.--Well, godson!\n\nGEORGE.--Give us a cigar!\n\nTOUCHIT.--Oh, you enfant terrible!\n\nMILLIKEN [wheezily].--Ah--ahem--George Touchit! you wouldn't\nmind--a--smoking that cigar in the garden, would you? Ah--ah!\n\nTOUCHIT.--Hullo! What's in the wind now? You used to be a most\ninveterate smoker, Horace.\n\nMILLIKEN.--The fact is--my mother-in-law--Lady Kicklebury--doesn't like\nit, and while she's with us, you know--\n\nTOUCHIT.--Of course, of course [throws away cigar]. I beg her ladyship's\npardon. I remember when you were courting her daughter she used not to\nmind it.\n\nMILLIKEN.--Don't--don't allude to those times. [He looks up at his\nwife's picture.]\n\nGEORGE.--My mamma was a Kicklebury. The Kickleburys are the oldest\nfamily in all the world. My name is George Kicklebury Milliken, of\nPigeoncot, Hants; the Grove, Richmond, Surrey; and Portland Place,\nLondon, Esquire--my name is.\n\nTOUCHIT.--You have forgotten Billiter Street, hemp and tallow merchant.\n\nGEORGE.--Oh, bother! I don't care about that. I shall leave that when\nI'm a man: when I'm a man and come into my property.\n\nMILLIKEN.--You come into your property?\n\nGEORGE.--I shall, you know, when you're dead, Papa. I shall have this\nhouse, and Pigeoncot; and the house in town--no, I don't mind about the\nhouse in town--and I shan't let Bella live with me--no, I won't.\n\nBELLA.--No; I won't live with YOU. And I'LL have Pigeoncot.\n\nGEORGE.--You shan't have Pigeoncot. I'll have it: and the ponies: and I\nwon't let you ride them--and the dogs, and you shan't have even a\npuppy to play with and the dairy and won't I have as much cream as I\nlike--that's all!\n\nTOUCHIT.--What a darling boy! Your children are brought up beautifully,\nMilliken. It's quite delightful to see them together.\n\nGEORGE.--And I shall sink the name of Milliken, I shall.\n\nMILLIKEN.--Sink the name? why, George?\n\nGEORGE.--Because the Millikens are nobodies--grandmamma says they are\nnobodies. The Kickleburys are gentlemen, and came over with William the\nConqueror.\n\nBELLA.--I know when that was. One thousand one hundred and one thousand\none hundred and onety-one!\n\nGEORGE.--Bother when they came over! But I know this, when I come into\nthe property I shall sink the name of Milliken.\n\nMILLIKEN.--So you are ashamed of your father's name, are you, George, my\nboy?\n\nGEORGE.--Ashamed! No, I ain't ashamed. Only Kicklebury is sweller. I\nknow it is. Grandmamma says so.\n\nBELLA.--MY grandmamma does not say so. MY dear grandmamma says that\nfamily pride is sinful, and all belongs to this wicked world; and that\nin a very few years what our names are will not matter.\n\nGEORGE.--Yes, she says so because her father kept a shop; and so did\nPa's father keep a sort of shop--only Pa's a gentleman now.\n\nTOUCHIT.--Darling child! How I wish I were married! If I had such a dear\nboy as you, George, do you know what I would give him?\n\nGEORGE [quite pleased].--What would you give him, god-papa?\n\nTOUCHIT.--I would give him as sound a flogging as ever boy had, my\ndarling. I would whip this nonsense out of him. I would send him to\nschool, where I would pray that he might be well thrashed: and if\nwhen he came home he was still ashamed of his father, I would put him\napprentice to a chimney-sweep--that's what I would do.\n\nGEORGE.--I'm glad you're not my father, that's all.\n\nBELLA.--And I'M glad you're not my father, because you are a wicked man!\n\nMILLIKEN.--Arabella!\n\nBELLA.--Grandmamma says so. He is a worldly man, and the world is\nwicked. And he goes to the play: and he smokes, and he says--\n\nTOUCHIT.--Bella, what do I say?\n\nBELLA.--Oh, something dreadful! You know you do! I heard you say it to\nthe cabman.\n\nTOUCHIT.--So I did, so I did! He asked me fifteen shillings from\nPiccadilly, and I told him to go to--to somebody whose name begins with\na D.\n\nCHILDREN.--Here's another carriage passing.\n\nBELLA.--The Lady Rumble's carriage.\n\nGEORGE.--No, it ain't: it's Captain Boxer's carriage [they run into the\ngarden].\n\nTOUCHIT.--And this is the pass to which you have brought yourself,\nHorace Milliken! Why, in your wife's time, it was better than this, my\npoor fellow!\n\nMILLIKEN.--Don't speak of her in THAT way, George Touchit!\n\nTOUCHIT.--What have I said? I am only regretting her loss for our sake.\nShe tyrannized over you; turned your friends out of doors; took your\nname out of your clubs; dragged you about from party to party, though\nyou can no more dance than a bear, and from opera to opera, though you\ndon't know \"God Save the Queen\" from \"Rule Britannia.\" You don't, sir;\nyou know you don't. But Arabella was better than her mother, who has\ntaken possession of you since your widowhood.\n\nMILLIKEN.--My dear fellow! no, she hasn't. There's MY mother.\n\nTOUCHIT.--Yes, to be sure, there's Mrs. Bonnington, and they quarrel\nover you like the two ladies over the baby before King Solomon.\n\nMILLIKEN.--Play the satirist, my good friend! laugh at my weakness!\n\nTOUCHIT.--I know you to be as plucky a fellow as ever stepped, Milliken,\nwhen a man's in the case. I know you and I stood up to each other for an\nhour and a half at Westminster.\n\nMILLIKEN.--Thank you! We were both dragons of war! tremendous champions!\nPerhaps I am a little soft as regards women. I know my weakness well\nenough; but in my case what is my remedy? Put yourself in my position.\nBe a widower with two young children. What is more natural than that\nthe mother of my poor wife should come and superintend my family? My own\nmother can't. She has a half-dozen of little half brothers and sisters,\nand a husband of her own to attend to. I dare say Mr. Bonnington and my\nmother will come to dinner to-day.\n\nTOUCHIT.--Of course they will, my poor old Milliken, you don't dare to\ndine without them.\n\nMILLIKEN.--Don't go on in that manner, George Touchit! Why should not my\nstep-father and my mother dine with me? I can afford it. I am a domestic\nman and like to see my relations about me. I am in the city all day.\n\nTOUCHIT.--Luckily for you.\n\nMILLIKEN.--And my pleasure of an evening is to sit under my own vine and\nunder my own fig-tree with my own olive-branches round about me; to sit\nby my fire with my children at my knees: to coze over a snug bottle of\nclaret after dinner with a friend like you to share it; to see the young\nfolks at the breakfast-table of a morning, and to kiss them and so off\nto business with a cheerful heart. This was my scheme in marrying, had\nit pleased heaven to prosper my plan. When I was a boy and came from\nschool and college, I used to see Mr. Bonnington, my father-in-law, with\nHIS young ones clustering round about him, so happy to be with him! so\neager to wait on him! all down on their little knees round my mother\nbefore breakfast or jumping up on his after dinner. It was who should\nreach his hat, and who should bring his coat, and who should fetch his\numbrella, and who should get the last kiss.\n\nTOUCHIT.--What? didn't he kiss YOU? Oh, the hard-hearted old ogre!\n\nMILLIKEN.--DON'T, Touchit! Don't laugh at Mr. Bonnington! he is as good\na fellow as ever breathed. Between you and me, as my half brothers and\nsisters increased and multiplied year after year, I used to feel rather\nlonely, rather bowled out, you understand. But I saw them so happy that\nI longed to have a home of my own. When my mother proposed Arabella for\nme (for she and Lady Kicklebury were immense friends at one time), I was\nglad enough to give up clubs and bachelorhood, and to settle down as a\nmarried man. My mother acted for the best. My poor wife's character,\nmy mother used to say, changed after marriage. I was not as happy as I\nhoped to be; but I tried for it. George, I am not so comfortable now as\nI might be. A house without a mistress, with two mothers-in-law reigning\nover it--one worldly and aristocratic, another what you call serious,\nthough she don't mind a rubber of whist: I give you my honor my mother\nplays a game at whist, and an uncommonly good game too--each woman\ndragging over a child to her side: of course such a family cannot be\ncomfortable. [Bell rings.] There's the first dinner-bell. Go and dress,\nfor heaven's sake.\n\nTOUCHIT.--Why dress? There is no company!\n\nMILLIKEN.--Why? ah! her ladyship likes it, you see. And it costs nothing\nto humor her. Quick, for she don't like to be kept waiting.\n\nTOUCHIT.--Horace Milliken! what a pity it is the law declares a widower\nshall not marry his wife's mother! She would marry you else,--she would,\non my word.\n\nEnter JOHN.\n\nJOHN.--I have took the Captain's things in the blue room, sir. [Exeunt\ngentlemen, JOHN arranges tables, &c.]\n\nHa! Mrs. Prior! I ain't partial to Mrs. Prior. I think she's an artful\nold dodger, Mrs. Prior. I think there's mystery in her unfathomable\npockets, and schemes in the folds of her umbrella. But--but she's\nJulia's mother, and for the beloved one's sake I am civil to her.\n\nMRS. PRIOR.--Thank you Charles [to the Page, who has been seen to let\nher in at the garden-gate], I am so much obliged to you! Good afternoon,\nMr. Howell. Is my daughter--are the darling children well? Oh, I am\nquite tired and weary! Three horrid omnibuses were full, and I have had\nto walk the whole weary long way. Ah, times are changed with me, Mr.\nHowell. Once when I was young and strong, I had my husband's carriage to\nride in.\n\nJOHN [aside].--His carriage! his coal-wagon! I know well enough who old\nPrior was. A merchant? yes, a pretty merchant! kep' a lodging-house,\nshare in a barge, touting for orders, and at last a snug little place in\nthe Gazette.\n\nMRS. PRIOR.--How is your cough, Mr. Howell? I have brought you some\nlozenges for it [takes numberless articles from her pocket], and if\nyou would take them of a night and morning--oh, indeed, you would get\nbetter! The late Sir Henry Halford recommended them to Mr. Prior. He\nwas his late Majesty's physician and ours. You know we have seen happier\ntimes, Mr. Howell. Oh, I am quite tired and faint.\n\nJOHN.--Will you take anything before the school-room tea, ma'am? You\nwill stop to tea, I hope, with Miss Prior, and our young folks?\n\nMRS. PRIOR.--Thank you: a little glass of wine when one is so faint--a\nlittle crumb of biscuit when one is so old and tired! I have not been\naccustomed to want, you know; and in my poor dear Mr. Prior's time--\n\nJOHN.--I'll fetch some wine, ma'am. [Exit to the dining-room.]\n\nMRS. PRIOR.--Bless the man, how abrupt he is in his manner! He quite\nshocks a poor lady who has been used to better days. What's here?\nInvitations--ho! Bills for Lady Kicklebury! THEY are not paid. Where is\nMr. M. going to dine, I wonder? Captain and Mrs. Hopkinson, Sir John and\nLady Tomkinson, request the pleasure. Request the pleasure! Of course\nthey do. They are always asking Mr. M. to dinner. They have daughters\nto marry, and Mr. M. is a widower with three thousand a year, every\nshilling of it. I must tell Lady Kicklebury. He must never go to these\nplaces--never, never--mustn't be allowed. [While talking, she opens all\nthe letters on the table, rummages the portfolio and writing-box, looks\nat cards on mantelpiece, work in work-basket, tries tea-box, and shows\nthe greatest activity and curiosity.]\n\nRe-enter John, bearing a tray with cakes, a decanter, &c.\n\nThank you, thank you, Mr. Howell! Oh, oh, dear me, not so much as that!\nHalf a glass, and ONE biscuit, please. What elegant sherry! [sips a\nlittle, and puts down glass on tray]. Do you know, I remember in better\ndays, Mr. Howell, when my poor dear husband--\n\nJOHN.--Beg your pardon. There's Milliken's bell, going like mad. [Exit\nJohn.]\n\nMRS. PRIOR.--What an abrupt person! Oh, but it's comfortable, this wine\nis! And--and I think how my poor Charlotte would like a little--she so\nweak, and ordered wine by the medical man! And when dear Adolphus comes\nhome from Christ's Hospital, quite tired, poor boy, and hungry, wouldn't\na bit of nice cake do him good! Adolphus is so fond of plum-cake, the\ndarling child! And so is Frederick, little saucy rogue; and I'll give\nthem MY piece, and keep my glass of wine for my dear delicate angel\nShatty! [Takes bottle and paper out of her pocket, cuts off a great\nslice of cake, and pours wine from wine-glass and decanter into bottle.]\n\nEnter PAGE.\n\nPAGE.--Master George and Miss Bella is going to have their teas down\nhere with Miss Prior, Mrs. Prior, and she's up in the school-room, and\nmy lady says you may stay to tea.\n\nMRS. PRIOR.--Thank you, Charles! How tall you grow! Those trousers would\nfit my darling Frederick to a nicety. Thank you, Charles. I know the way\nto the nursery. [Exit Mrs. P.]\n\nPAGE.--Know the way! I believe she DO know the way. Been a having cake\nand wine. Howell always gives her cake and wine--jolly cake, ain't it!\nand wine, oh, my!\n\nRe-enter John.\n\nJOHN.--You young gormandizing cormorant! What! five meals a day ain't\nenough for you! What? beer ain't good enough for you, hey? [Pulls boy's\nears.]\n\nPAGE [crying].--Oh, oh, do-o-n't, Mr. Howell. I only took half a glass,\nupon my honor.\n\nJOHN.--Your a-honor, you lying young vagabond! I wonder the ground don't\nopen and swallow you. Half a glass! [holds up decanter.] You've took\nhalf a bottle, you young Ananias! Mark this, sir! When I was a boy,\na boy on my promotion, a child kindly took in from charity-school, a\nhorphan in buttons like you, I never lied; no, nor never stole, and\nyou've done both, you little scoundrel. Don't tell ME, sir! there's\nplums on your coat, crumbs on your cheek, and you smell sherry, sir! I\nain't time to whop you now, but come to my pantry to-night after you've\ntook the tray down. Come without your jacket on, sir, and then I'll\nteach you what it is to lie and steal. There's the outer bell. Scud, you\nvagabond!\n\nEnter LADY K.\n\nLADY K.--What was that noise, pray?\n\nJOHN.--A difference between me and young Page, my lady. I was\ninstructing him to keep his hands from picking and stealing. I was\nlearning him his lesson, my lady, and he was a-crying it out.\n\nLADY K.--It seems to me you are most unkind to that boy, Howell. He is\nmy boy, sir. He comes from my estate. I will not have him ill-used. I\nthink you presume on your long services. I shall speak to my son-in-law\nabout you. [\"Yes, my lady; no, my lady; very good, my lady.\" John has\nanswered each sentence as she is speaking, and exit gravely bowing.]\nThat man must quit the house. Horace says he can't do without him, but\nhe must do without him. My poor dear Arabella was fond of him, but he\npresumes on that defunct angel's partiality. Horace says this person\nkeeps all his accounts, sorts all his letters, manages all his affairs,\nmay be trusted with untold gold, and rescued little George out of\nthe fire. Now I have come to live with my son-in-law, I will keep his\naccounts, sort his letters, and take charge of his money: and if little\nGeorgy gets into the grate, I will take him out of the fire. What is\nhere? Invitation from Captain and Mrs. Hopkinson. Invitation from Sir\nJohn and Lady Tomkinson, who don't even ask me! Monstrous! he never\nshall go--he shall not go! [MRS. PRIOR has re-entered, she drops a very\nlow curtsy to Lady K., as the latter, perceiving her, lays the cards\ndown.]\n\nMRS. PRIOR.--Ah, dear madam! how kind your ladyship's message was to the\npoor lonely widow woman! Oh, how thoughtful it was of your ladyship to\nask me to stay to tea!\n\nLADY K.--With your daughter and the children? Indeed, my good Mrs.\nPrior, you are very welcome!\n\nMRS. PRIOR.--Ah! but isn't it a cause of thankfulness to be MADE\nwelcome? Oughtn't I to be grateful for these blessings?--yes, I say\nBLESSINGS. And I am--I am, Lady Kicklebury--to the mother--of--that\nangel who is gone [points to the picture]. It was your sainted daughter\nleft us--left my child to the care of Mr. Milliken, and--and you, who\nare now his guardian angel I may say. You ARE, Lady Kicklebury--you\nare. I say to my girl, Julia, Lady Kicklebury is Mr. Milliken's guardian\nangel, is YOUR guardian angel--for without you could she keep her place\nas governess to these darling children? It would tear her heart in two\nto leave them, and yet she would be forced to do so. You know that some\none--shall I hesitate to say whom I MEAN--that Mr. Milliken's mother,\nexcellent lady though she is, does not love my child because YOU love\nher. You DO love her, Lady Kicklebury, and oh! a mother's fond heart\npays you back! But for you, my poor Julia must go--go, and leave the\nchildren whom a dying angel confided to her!\n\nLADY K.--Go! no, never! not whilst I am in this house, Mrs. Prior. Your\ndaughter is a well-behaved young woman: you have confided to me her long\nengagement to Lieutenant--Lieutenant What-d'you-call'im, in the Indian\nservice. She has been very, very good to my grandchildren--she brought\nthem over from Naples when my--my angel of an Arabella died there, and I\nwill protect Miss Prior.\n\nMRS. PRIOR.--Bless you, bless you, noble, admirable woman! Don't take it\naway! I must, I WILL kiss your dear, generous hand! Take a mother's, a\nwidow's blessings, Lady Kicklebury--the blessings of one who has known\nmisfortune and seen better days, and thanks heaven--yes, heaven!--for\nthe protectors she has found!\n\nLADY K.--You said--you had--several children, I think, my good Mrs.\nPrior?\n\nMRS. PRIOR.--Three boys--one, my eldest blessing, is in a\nwine-merchant's office--ah, if Mr. Milliken WOULD but give him an order!\nan order from THIS house! an order from Lady Kicklebury's son-in-law!--\n\nLADY K.--It shall be done, my good Prior--we will see.\n\nMRS. PRIOR.--Another, Adolphus, dear fellow! is in Christ's Hospital.\nIt was dear, good Mr. Milliken's nomination. Frederick is at Merchant\nTaylor's: my darling Julia pays his schooling. Besides, I have two\ngirls--Amelia, quite a little toddles, just the size, though not so\nbeautiful--but in a mother's eyes all children are lovely, dear Lady\nKicklebury--just the size of your dear granddaughter, whose clothes\nwould fit her, I am sure. And my second, Charlotte, a girl as tall as\nyour ladyship, though not with so fine a figure. \"Ah, no, Shatty!\" I say\nto her, \"you are as tall as our dear patroness, Lady Kicklebury, whom\nyou long so to see; but you have not got her ladyship's carriage and\nfigure, child.\" Five children have I, left fatherless and penniless by\nmy poor dear husband--but heaven takes care of the widow and orphan,\nmadam--and heaven's BEST CREATURES feed them!--YOU know whom I mean.\n\nLADY K.--Should you not like, would you object to take--a frock or two\nof little Arabella's to your child? and if Pinhorn, my maid, will let\nme, Mrs. Prior, I will see if I cannot find something against winter for\nyour second daughter, as you say we are of a size.\n\nMRS. PRIOR.--The widow's and orphans' blessings upon you! I said\nmy Charlotte was as tall, but I never said she had such a figure as\nyours--who has?\n\nCHARLES announces--\n\nCHARLES.--Mrs. Bonnington! [Enter MRS. BONNINGTON.]\n\nMRS. B.--How do you do, Lady Kicklebury?\n\nLADY K.--My dear Mrs. Bonnington! and you come to dinner of course?\n\nMRS. B.--To dine with my own son, I may take the liberty. How are my\ngrandchildren? my darling little Emily, is she well, Mrs. Prior?\n\nLADY K. [aside].--Emily? why does she not call the child by her blessed\nmother's name of Arabella? [To MRS. B.] ARABELLA is quite well, Mrs.\nBonnington. Mr. Squillings said it was nothing; only her grandmamma\nBonnington spoiling her, as usual. Mr. Bonnington and all your numerous\nyoung folk are well, I hope?\n\nMRS. B.--My family are all in perfect health, I thank you. Is Horace\ncome home from the city?\n\nLADY K.--Goodness! there's the dinner-bell,--I must run to dress.\n\nMRS. PRIOR.--Shall I come with you, dear Lady Kicklebury?\n\nLADY K.--Not for worlds, my good Mrs. Prior. [Exit Lady K.]\n\nMRS. PRIOR.--How do you do, my DEAR madam? Is dear Mr. Bonnington QUITE\nwell? What a sweet, sweet sermon he gave us last Sunday. I often say\nto my girl, I must not go to hear Mr. Bonnington, I really must not, he\nmakes me cry so. Oh! he is a great and gifted man, and shall I not have\none glimpse of him?\n\nMRS. B.--Saturday evening, my good Mrs. Prior. Don't you know that my\nhusband never goes out on Saturday, having his sermon to compose?\n\nMRS. P.--Oh, those dear, dear sermons! Do you know, madam, that my\nlittle Adolphus, for whom your son's bounty procured his place at\nChrist's Hospital, was very much touched indeed, the dear child, with\nMr. Bonnington's discourse last Sunday three weeks, and refused to play\nmarbles afterwards at school? The wicked, naughty boys beat the poor\nchild; but Adolphus has his consolation! Is Master Edward well, ma'am,\nand Master Robert, and Master Frederick, and dear little funny Master\nWilliam?\n\nMRS. B.--Thank you, Mrs. Prior; you have a good heart, indeed!\n\nMRS. P.--Ah, what blessings those dears are to you! I wish your dearest\nlittle GRANDSON---\n\nMRS. B.--The little naughty wretch! Do you know, Mrs. Prior, my\ngrandson, George Milliken, spilt the ink over my dear husband's bands,\nwhich he keeps in his great dictionary; and fought with my child,\nFrederick, who is three years older than George--actually beat his own\nuncle!\n\nMRS. P.--Gracious mercy! Master Frederick was not hurt, I hope?\n\nMRS. B.--No; he cried a great deal; and then Robert came up, and that\ngraceless little George took a stick; and then my husband came out, and\ndo you know George Milliken actually kicked Mr. Bonnington on his shins,\nand butted him like a little naughty ram?\n\nMRS. P.--Mercy! mercy! what a little rebel! He is spoiled, dear madam,\nand you know by WHOM.\n\nMRS. B.--By his grandmamma Kicklebury. I know it. I want my son to whip\nthat child, but he refuses. He will come to no good; that child.\n\nMRS. P.--Ah, madam, don't say so! Let us hope for the best. Master\nGeorge's high temper will subside when certain persons who pet him are\ngone away.\n\nMRS. B.--Gone away! they never will go away! No, mark my words, Mrs.\nPrior, that woman will never go away. She has made the house her own:\nshe commands everything and everybody in it. She has driven me--me--Mr.\nMilliken's own mother--almost out of it. She has so annoyed my dear\nhusband, that Mr. Bonnington will scarcely come here. Is she not always\nsneering at private tutors, because Mr. Bonnington was my son's private\ntutor, and greatly valued by the late Mr. Milliken? Is she not making\nconstant allusions to old women marrying young men, because Mr.\nBonnington happens to be younger than me? I have no words to express my\nindignation respecting Lady Kicklebury. She never pays any one, and\nruns up debts in the whole town. Her man Bulkeley's conduct in the\nneighborhood is quite--quite--\n\nMRS. P.--Gracious goodness, ma'am, you don't say so! And then what an\nappetite the gormandizing monster has! Mary tells me that what he eats\nin the servants' hall is something perfectly frightful.\n\nMRS. B.--Everybody feeds on my poor son! You are looking at my cap, Mrs.\nPrior? [During this time MRS. PRIOR has been peering into a parcel which\nMRS. BONNINGTON brought in her hand.] I brought it with me across the\nPark. I could not walk through the Park in my cap. Isn't it a pretty\nribbon, Mrs. Prior?\n\nMRS. P.--Beautiful! beautiful? How blue becomes you! Who would think you\nwere the mother of Mr. Milliken and seven other darling children? You\ncan afford what Lady Kicklebury cannot.\n\nMRS. B.--And what is that, Prior? A poor clergyman's wife, with a large\nfamily, cannot afford much.\n\nMRS. P.--He! he! You can afford to be seen as you are, which Lady K.\ncannot. Did you not remark how afraid she seemed lest I should enter her\ndressing-room? Only Pinhorn, her maid, goes there, to arrange the\nroses, and the lilies, and the figure--he! he! Oh, what a sweet, sweet\ncap-ribbon! When you have worn it, and are tired of it, you will give it\nme, won't you? It will be good enough for poor old Martha Prior!\n\nMRS. B.--Do you really like it? Call at Greenwood Place, Mrs. Prior, the\nnext time you pay Richmond a visit, and bring your little girl with you,\nand we will see.\n\nMRS. P.--Oh, thank you! thank you! Nay, don't be offended! I must! I\nmust! [Kisses MRS. BONNINGTON.]\n\nMRS. B.--There, there! We must not stay chattering! The bell has rung. I\nmust go and put the cap on, Mrs. Prior.\n\nMRS. P.--And I may come too? YOU are not afraid of my seeing your hair,\ndear Mrs. Bonnington! Mr. Bonnington too young for YOU! Why, you don't\nlook twenty!\n\nMRS. B.--Oh, Mrs. Prior!\n\nMRS. P.--Well, five-and-twenty, upon my word--not more than\nfive-and-twenty--and that is the very prime of life. [Exeunt Mrs. B. and\nMrs. P., hand in hand. As Captain TOUCHIT enters, dressed for dinner, he\nbows and passes on.]\n\nTOUCHIT.--So, we are to wear our white cravats, and our varnished boots,\nand dine in ceremony. What is the use of a man being a widower, if he\ncan't dine in his shooting-jacket? Poor Mill! He has the slavery now\nwithout the wife. [He speaks sarcastically to the picture.] Well, well!\nMrs. Milliken! YOU, at any rate, are gone; and with the utmost respect\nfor you, I like your picture even better than the original. Miss Prior!\n\nEnter Miss PRIOR.\n\nMISS PRIOR.--I beg pardon. I thought you were gone to dinner. I heard\nthe second bell some time since. [She is drawing back.]\n\nTOUCHIT.--Stop! I say, Julia! [She returns, he looks at her, takes her\nhand.] Why do you dress yourself in this odd poky way? You used to be\na very smartly dressed girl. Why do you hide your hair, and wear such a\ndowdy, high gown, Julia?\n\nJULIA.--You mustn't call me Julia, Captain Touchit.\n\nTOUCHIT.--Why? when I lived in your mother's lodging, I called you\nJulia. When you brought up the tea, you didn't mind being called Julia.\nWhen we used to go to the play with the tickets the Editor gave us, who\nlived on the second floor--\n\nJULIA.--The wretch!--don't speak of him!\n\nTOUCHIT.--Ah! I am afraid he was a sad deceiver, that Editor. He was\na very clever fellow. What droll songs he used to sing! What a heap of\nplay-tickets, diorama-tickets, concert-tickets, he used to give you! Did\nhe touch your heart, Julia?\n\nJULIA.--Fiddlededee! No man ever touched my heart, Captain Touchit.\n\nTOUCHIT.--What! not even Tom Flight, who had the second floor after the\nEditor left it--and who cried so bitterly at the idea of going out to\nIndia without you? You had a tendre for him--a little passion--you know\nyou had. Why, even the ladies here know it. Mrs. Bonnington told me that\nyou were waiting for a sweetheart in India to whom you were engaged; and\nLady Kicklebury thinks you are dying in love for the absent swain.\n\nJULIA.--I hope--I hope--you did not contradict them, Captain Touchit.\n\nTOUCHIT.--Why not, my dear?\n\nJULIA.--May I be frank with you? You were a kind, very kind friend to\nus--to me, in my youth.\n\nTOUCHIT.--I paid my lodgings regularly, and my bills without asking\nquestions. I never weighed the tea in the caddy, or counted the lumps of\nsugar, or heeded the rapid consumption of my liqueur--\n\nJULIA.--Hush, hush! I know they were taken. I know you were very good to\nus. You helped my poor papa out of many a difficulty.\n\nTOUCHIT [aside].--Tipsy old coal-merchant! I did, and he helped himself\ntoo.\n\nJULIA.--And you were always our best friend, Captain Touchit. When our\nmisfortunes came, you got me this situation with Mrs. Milliken--and,\nand--don't you see?--\n\nTOUCHIT.--Well--what?\n\nJULIA [laughing].--I think it is best, under the circumstances, that the\nladies here should suppose I am engaged to be married--or or, they might\nbe--might be jealous, you understand. Women are sometimes jealous of\nothers,--especially mothers and mothers-in-law.\n\nTOUCHIT.--Oh, you arch schemer! And it is for that you cover up that\nbeautiful hair of yours, and wear that demure cap?\n\nJULIA [slyly].--I am subject to rheumatism in the head, Captain Touchit.\n\nTOUCHIT.--It is for that you put on the spectacles, and make yourself\nlook a hundred years old?\n\nJULIA.--My eyes are weak, Captain Touchit.\n\nTOUCHIT.--Weak with weeping for Tom Flight. You hypocrite! Show me your\neyes!\n\nMISS P.--Nonsense!\n\nTOUCHIT.--Show me your eyes, I say, or I'll tell about Tom Flight and\nthat he has been married at Madras these two years.\n\nMISS P.--Oh, you horrid man! [takes glasses off.] There.\n\nTOUCHIT.--Translucent orbs! beams of flashing light! lovely lashes\nveiling celestial brightness! No, they haven't cried much for Tom\nFlight, that faithless captain! nor for Lawrence O'Reilly, that killing\nEditor. It is lucky you keep the glasses on them, or they would transfix\nHorace Milliken, my friend the widower here. DO you always wear them\nwhen you are alone with him?\n\nMISS P.--I never AM alone with him. Bless me! If Lady Kicklebury thought\nmy eyes were--well, well--you know what I mean,--if she thought her\nson-in-law looked at me, I should be turned out of doors the next day,\nI am sure I should. And then, poor Mr. Milliken! he never looks at\nME--heaven help him! Why, he can't see me for her ladyship's nose and\nawful caps and ribbons! He sits and looks at the portrait yonder, and\nsighs so. He thinks that he is lost in grief for his wife at this very\nmoment.\n\nTOUCHIT.--What a woman that was--eh, Julia--that departed angel! What a\ntemper she had before her departure!\n\nMISS P.--But the wind was tempered to the lamb. If she was angry--the\nlamb was so very lamblike, and meek, and fleecy.\n\nTOUCHIT.--And what a desperate flirt the departed angel was! I knew\nhalf a dozen fellows, before her marriage, whom she threw over, because\nMilliken was so rich.\n\nMISS P.--She was consistent at least, and did not change after marriage,\nas some ladies do; but flirted, as you call it, just as much as before.\nAt Paris, young Mr. Verney, the attache, was never out of the house:\nat Rome, Mr. Beard, the artist, was always drawing pictures of her:\nat Naples, when poor Mr. M. went away to look after his affairs at St.\nPetersburg, little Count Posilippo was for ever coming to learn English\nand practise duets. She scarcely ever saw the poor children--[changing\nher manner as Lady KICKLEBURY enters] Hush--my lady!\n\nTOUCHIT.--You may well say, \"poor children,\" deprived of such a woman!\nMiss Prior, whom I knew in very early days--as your ladyship knows--was\nspeaking--was speaking of the loss our poor friend sustained.\n\nLADY K.--Ah, sir, what a loss! [looking at the picture.]\n\nTOUCHIT.--What a woman she was--what a superior creature!\n\nLADY K.--A creature--an angel!\n\nTOUCHIT.--Mercy upon us! how she and my lady used to quarrel! [aside.]\nWhat a temper!\n\nLADY K.--Hm--oh, yes--what a temper [rather doubtfully at first].\n\nTOUCHIT.--What a loss to Milliken and the darling children!\n\nMISS PRIOR.--Luckily they have YOU with them madam.\n\nLADY K.--And I will stay with them, Miss Prior; I will stay with them! I\nwill never part from Horace, I am determined.\n\nMISS P.--Ah! I am very glad you stay, for if I had not YOU for a\nprotector, I think you know I must go, Lady Kicklebury. I think you\nknow there are those who would forget my attachment to these darling\nchildren, my services to--to her--and dismiss the poor governess. But\nwhile you stay I can stay, dear Lady Kicklebury! With you to defend me\nfrom jealousy I need not QUITE be afraid.\n\nLADY K.--Of Mrs. Bonnington? Of Mr. Milliken's mother; of the parson's\nwife who writes out his stupid sermons, and has half a dozen children of\nher own? I should think NOT indeed! I am the natural protector of these\nchildren. I am their mother. I have no husband! You STAY in this house,\nMiss Prior. You are a faithful, attached creature--though you were sent\nin by somebody I don't like very much [pointing to TOUCHIT, who went off\nlaughing when JULIA began her speech, and is now looking at prints, &c.,\nin next room].\n\nMISS P.--Captain Touchit may not be in all things what one could wish.\nBut his kindness has formed the happiness of my life in making me\nacquainted with YOU, ma'am: and I am sure you would not have me be\nungrateful to him.\n\nLADY K.--A most highly principled young woman. [Goes out in garden and\nwalks up and down with Captain TOUCHIT.]\n\nEnter Mrs. BONNINGTON.\n\nMISS P.--Oh, how glad I am you are come, Mrs. Bonnington. Have you\nbrought me that pretty hymn you promised me? You always keep your\npromises, even to poor governesses. I read dear Mr. Bonnington's sermon!\nIt was so interesting that I really could not think of going to sleep\nuntil I had read it all through; it was delightful, but oh! it's still\nbetter when he preaches it! I hope I did not do wrong in copying a part\nof it? I wish to impress it on the children. There are some worldly\ninfluences at work with them, dear madam [looking at Lady K. in the\ngarden], which I do my feeble effort to--to modify. I wish YOU could\ncome oftener.\n\nMRS. B.--I will try, my dear--I will try. Emily has sweet dispositions.\n\nMISS P.--Ah, she takes after her grandmamma Bonnington!\n\nMRS. B.--But George was sadly fractious just now in the school-room\nbecause I tried him with a tract.\n\nMISS P.--Let us hope for better times! Do be with your children, dear\nMrs. Bonnington, as constantly as ever you can, for MY sake as well as\ntheirs! I want protection and advice as well as they do. The GOVERNESS,\ndear lady, looks up to you as well as the pupils; SHE wants the teaching\nwhich you and dear Mr. Bonnington can give her! Ah, why could not Mr.\nand Mrs. Bonnington come and live here, I often think? The children\nwould have companions in their dear young uncles and aunts; so pleasant\nit would be. The house is quite large enough; that is, if her ladyship\ndid not occupy the three south rooms in the left wing. Ah, why, WHY\ncouldn't you come?\n\nMRS. B.--You are a kind, affectionate creature, Miss Prior. I do not\nvery much like the gentleman who recommended you to Arabella, you know.\nBut I do think he sent my son a good governess for his children.\n\nTwo Ladies walk up and down in front garden.\n\nTOUCHIT enters.\n\nTOUCHIT.--Miss Julia Prior, you are a wonder! I watch you with respect\nand surprise.\n\nMISS P.--Me! what have I done? a poor friendless governess--respect ME?\n\nTOUCHIT.--I have a mind to tell those two ladies what I think of Miss\nJulia Prior. If they knew you as I know you, O Julia Prior, what a short\nreign yours would be!\n\nMISS P.--I have to manage them a little. Each separately it is not so\ndifficult. But when they are together, oh, it is very hard sometimes.\n\nEnter MILLIKEN dressed, shakes hands with Miss P.\n\nMILLIKEN.--Miss Prior! are you well? Have the children been good? and\nlearned all their lessons?\n\nMISS P.--The children are pretty good, sir.\n\nMILLIKEN.--Well, that's a great deal as times go. Do not bother them\nwith too much learning, Miss Prior. Let them have an easy life. Time\nenough for trouble when age comes.\n\nEnter John.\n\nJOHN.--Dinner, sir. [And exit.]\n\nMILLIKEN.--Dinner, ladies. My Lady Kicklebury (gives arm to Lady K).\n\nLADY K.--My dear Horace, you SHOULDN'T shake hands with Miss Prior. You\nshould keep people of that class at a distance, my dear creature. [They\ngo in to dinner, Captain TOUCHIT following with Mrs. BONNINGTON. As they\ngo out, enter MARY with children's tea-tray, &c., children following,\nand after them Mrs. PRIOR. MARY gives her tea.]\n\nMRS. PRIOR.--Thank you, Mary! You are so very kind! Oh, what delicious\ntea!\n\nGEORGY.--I say, Mrs. Prior, I dare say you would like to dine best,\nwouldn't you?\n\nMRS. P.--Bless you, my darling love, I had my dinner at one o'clock with\nmy children at home.\n\nGEORGY.--So had we: but we go in to dessert very often; and then don't\nwe have cakes and oranges and candied-peel and macaroons and things! We\nare not to go in to-day; because Bella ate so many strawberries she made\nherself ill.\n\nBELLA.--So did you.\n\nGEORGY.--I'm a man, and men eat more than women, twice as much as women.\nWhen I'm a man I'll eat as much cake as ever I like. I say, Mary, give\nus the marmalade.\n\nMRS. P.--Oh, what nice marmalade! I know of some poor children--\n\nMISS P.--Mamma! don't, mamma [in an imploring tone].\n\nMRS. P.--I know of two poor children at home, who have very seldom nice\nmarmalade and cake, young people.\n\nGEORGE.--You mean Adolphus and Frederick and Amelia, your children.\nWell, they shall have marmalade and cake.\n\nBELLA.--Oh, yes! I'll give them mine.\n\nMRS. P.--Darling, dearest child!\n\nGEORGE [his mouth full].--I won't give 'em mine: but they can have\nanother pot, you know. You have always got a basket with you, Mrs.\nPrior. I know you have. You had it that day you took the cold fowl.\n\nMRS. P.--For the poor blind black man! oh, how thankful he was!\n\nGEORGE.--I don't know whether it was for a black man. Mary, get us\nanother pot of marmalade.\n\nMARY.--I don't know, Master George.\n\nGEORGE.--I WILL have another pot of marmalade. If you don't, I'll--I'll\nsmash everything--I will.\n\nBELLA.--Oh, you naughty, rude boy!\n\nGEORGE.--Hold YOUR tongue! I WILL have it. Mary shall go and get it.\n\nMRS. P.--Do humor him, Mary; and I'm sure my poor children at home will\nbe the better for it.\n\nGEORGE.--There's your basket! now put this cake in, and this pat\nof butter, and this sugar. Hurray, hurray! Oh, what jolly fun! Tell\nAdolphus and Amelia I sent it to them--tell 'em they shall never want\nfor anything as long as George Kicklebury Milliken, Esq., can give it\n'em. Did Adolphus like my gray coat that I didn't want?\n\nMISS P.--You did not give him your new gray coat?\n\nGEORGE.--Don't you speak to me; I'm going to school--I'm not going to\nhave no more governesses soon.\n\nMRS. P.--Oh, my dear Master George, what a nice coat it is, and how well\nmy poor boy looked in it!\n\nMISS P.--Don't, mamma! I pray and entreat you not to take the things!\n\nEnter JOHN from dining-room with a tray.\n\nJOHN.--Some cream, some jelly, a little champagne, Miss Prior; I thought\nyou might like some.\n\nGEORGE.--Oh, jolly! give us hold of the jelly! give us a glass of\nchampagne.\n\nJOHN.--I will not give you any.\n\nGEORGE.--I'll smash every glass in the room if you don't; I'll cut my\nfingers; I'll poison myself--there! I'll eat all this sealing-wax if you\ndon't, and it's rank poison, you know it is.\n\nMRS. P.--My dear Master George! [Exit JOHN.]\n\nGEORGE.--Ha, ha! I knew you'd give it me; another boy taught me that.\n\nBELLA.--And a very naughty, rude boy.\n\nGEORGE.--He, he, he! hold your tongue Miss! And said he always got wine\nso; and so I used to do it to my poor mamma, Mrs. Prior. Usedn't to like\nmamma much.\n\nBELLA.--Oh, you wicked boy!\n\nGEORGY.--She usedn't to see us much. She used to say I tried her nerves:\nwhat's nerves, Mrs. Prior? Give us some more champagne! Will have\nit. Ha, ha, ha! ain't it jolly? Now I'll go out and have a run in the\ngarden. [Runs into garden].\n\nMRS. P.--And you, my dear?\n\nBELLA.--I shall go and resume the perusal of the \"Pilgrim's Progress,\"\nwhich my grandpapa, Mr. Bonnington, sent me. [Exit ARABELLA.]\n\nMISS P.--How those children are spoilt! Goodness; what can I do? If I\ncorrect one, he flies to grandmamma Kicklebury; if I speak to another,\nshe appeals to grandmamma Bonnington. When I was alone with them, I had\nthem in something like order. Now, between the one grandmother and the\nother, the children are going to ruin, and so would the house too, but\nthat Howell--that odd, rude, but honest and intelligent creature, I\nmust say--keeps it up. It is wonderful how a person in his rank of life\nshould have instructed himself so. He really knows--I really think he\nknows more than I do myself.\n\nMRS. P.--Julia dear!\n\nMISS P.--What is it, mamma?\n\nMRS. P.--Your little sister wants some underclothing sadly, Julia dear,\nand poor Adolphus's shoes are quite worn out.\n\nMISS P.--I thought so; I have given you all I could, mamma.\n\nMRS. P.--Yes, my love! you are a good love, and generous, heaven knows,\nto your poor old mother who has seen better days. If we had not wanted,\nwould I have ever allowed you to be a governess--a poor degraded\ngoverness? If that brute O'Reilly who lived on our second floor had not\nbehaved so shamefully wicked to you, and married Miss Flack, the singer,\nmight you not have been Editress of the Champion of Liberty at this very\nmoment, and had your Opera box every night? [She drinks champagne while\ntalking, and excites herself.]\n\nMISS P.--Don't take that, mamma.\n\nMRS. P.--Don't take it? why, it costs nothing; Milliken can afford it.\nDo you suppose I get champagne every day? I might have had it as a girl\nwhen I first married your father, and we kep' our gig and horse, and\nlived at Clapham, and had the best of everything. But the coal-trade is\nnot what it was, Julia. We met with misfortunes, Julia, and we went\ninto poverty: and your poor father went into the Bench for twenty-three\nmonths--two year all but a month he did--and my poor girl was obliged to\ndance at the \"Coburg Theatre\"--yes you were, at ten shillings a week,\nin the Oriental ballet of \"The Bulbul and the Rose:\" you were, my poor\ndarling child.\n\nMISS P.--Hush, hush, mamma!\n\nMRS. P.--And we kep' a lodging-house in Bury Street, St. James's,\nwhich your father's brother furnished for us, who was an extensive\noil-merchant. He brought you up; and afterwards he quarrelled with my\npoor James, Robert Prior did, and he died, not leaving us a shilling.\nAnd my dear eldest boy went into a wine-merchant's office: and my poor\ndarling Julia became a governess, when you had had the best of education\nat Clapham; you had, Julia. And to think that you were obliged, my\nblessed thing, to go on in the Oriental ballet of \"The Rose and the\nBul--\"\n\nMISS P.--Mamma, hush, hush! forget that story.\n\nEnter Page from dining-room.\n\nPAGE.--Miss Prior! please, the ladies are coming from the dining-room.\nMrs. B. have had her two glasses of port, and her ladyship is now\na-telling the story about the Prince of Wales when she danced with him\nat Canton House. [Exit Page.]\n\nMISS P.--Quick, quick! There, take your basket! Put on your bonnet, and\ngood-night, mamma. Here, here is a half sovereign and three shillings;\nit is all the money I have in the world; take it, and buy the shoes for\nAdolphus.\n\nMRS. P.--And the underclothing, my love--little Amelia's underclothing?\n\nMISS P.--We will see about it. Good-night [kisses her]. Don't be seen\nhere,--Lady K. doesn't like it.\n\nEnter Gentlemen and Ladies from dining-room.\n\nLADY K.--We follow the Continental fashion. We don't sit after dinner,\nCaptain Touchit.\n\nCAPTAIN T.--Confound the Continental fashion! I like to sit a little\nwhile after dinner [aside].\n\nMRS. B.--So does my dear Mr. Bonnington, Captain Touchit. He likes a\nlittle port-wine after dinner.\n\nTOUCHIT.--I'm not surprised at it, ma am.\n\nMRS. B.--When did you say your son was coming, Lady Kicklebury?\n\nLADY K.--My Clarence! He will be here immediately, I hope, the dear boy.\nYou know my Clarence?\n\nTOUCHIT.--Yes, ma'am.\n\nLADY K.--And like him, I'm sure, Captain Touchit! Everybody does like\nClarence Kicklebury.\n\nTOUCHIT.--The confounded young scamp! I say, Horace, do you like your\nbrother-in-law?\n\nMILLIKEN.--Well--I--I can't say--I--like him--in fact, I don't. But\nthat's no reason why his mother shouldn't. [During this, HOWELL,\npreceded by BULKELEY, hands round coffee. The garden without has\ndarkened, as if evening. BULKELEY is going away without offering coffee\nto Miss PRIOR. JOHN stamps on his foot, and points to her. Captain\nTOUCHIT, laughing, goes up and talks to her now the servants are gone.]\n\nMRS. B.--Horace! I must tell you that the waste at your table is\nshocking. What is the need of opening all this wine? You and Lady\nKicklebury were the only persons who took champagne.\n\nTOUCHIT.--I never drink it--never touch the rubbish! Too old a stager!\n\nLADY K.--Port, I think, is your favorite, Mrs. Bonnington?\n\nMRS. B.--My dear lady, I do not mean that you should not have champagne,\nif you like. Pray, pray, don't be angry! But why on earth, for you,\nwho take so little, and Horace, who only drinks it to keep you company,\nshould not Howell open a pint instead of a great large bottle?\n\nLADY K.--Oh, Howell! Howell! We must not mention Howell, my dear Mrs.\nBonnington. Howell is faultless! Howell has the keys of everything!\nHowell is not to be controlled in anything! Howell is to be at liberty\nto be rude to my servant!\n\nMILLIKEN.--Is that all? I am sure I should have thought your man was big\nenough to resent any rudeness from poor little Howell.\n\nLADY K.--Horace! Excuse me for saying that you don't know--the--the\nclass of servant to whom Bulkeley belongs. I had him, as a great favor,\nfrom Lord Toddleby. That class of servant is accustomed generally not to\ngo out single.\n\nMILLIKEN.--Unless they are two behind a carriage-perch they pine away,\nas one love-bird does without his mate!\n\nLADY K.--No doubt! no doubt! I only say you are not accustomed here--in\nthis kind of establishment, you understand--to that class of--\n\nMRS. B.--Lady Kicklebury! is my son's establishment not good enough for\nany powdered monster in England? Is the house of a British merchant--?\n\nLADY K.--My dear creature! my dear creature! it IS the house of a\nBritish merchant, and a very comfortable house.\n\nMRS. B.--Yes, as you find it.\n\nLADY K.--Yes, as I find it, when I come to take care of my departed,\nangel's children, Mrs. Bonnington--[pointing to picture]--of THAT\ndear seraph's orphans, Mrs. Bonnington. YOU cannot. You have other\nduties--other children--a husband at home in delicate health, who--\n\nMRS. B.--Lady Kicklebury, no one shall say I don't take care of my dear\nhusband!\n\nMILLIKEN.--My dear mother! My dear Lady Kicklebury! [To T., who has come\nforward.] They spar so every night they meet, Touchit. Ain't it hard?\n\nLADY K.--I say you DO take care of Mr. Bonnington, Mrs. Bonnington, my\ndear creature! and that is why you can't attend to Horace. And as he\nis of a very easy temper--except sometimes with his poor Arabella's\nmother--he allows all his tradesmen to cheat him, all his servants to\ncheat him, Howell to be rude to everybody--to me amongst other people,\nand why not to my servant Bulkeley, with whom Lord Toddleby's groom of\nthe chambers gave me the very highest character.\n\nMRS. B.--I'm surprised that noblemen HAVE grooms in their chambers. I\nshould think they were much better in the stables. I am sure I always\nthink so when we dine with Doctor Clinker. His man does bring such a\nsmell of the stable with him.\n\nLADY K.--He! he! you mistake, my dearest creature! Your poor mother\nmistakes, my good Horace. You have lived in a quiet and most respectable\nsphere--but not--not--\n\nMRS. B.--Not what, Lady Kicklebury? We have lived at Richmond twenty\nyears--in my late husband's time--when we saw a great deal of company,\nand when this dear Horace was a dear boy at Westminster School. And we\nhave PAID for everything we have had for twenty years, and we have owed\nnot a penny to any TRADESMAN, though we mayn't have had POWDERED\nFOOTMEN SIX FEET HIGH, who were impertinent to all the maids in the\nplace--Don't! I WILL speak, Horace--but servants who loved us, and who\nlived in our families.\n\nMILLIKEN.--Mamma, now, my dear, good old mother! I am sure Lady\nKicklebury meant no harm.\n\nLADY K.--Me! my dear Horace! harm! What harm could I mean?\n\nMILLIKEN.--Come! let us have a game at whist. Touchit, will you make a\nfourth? They go on so every night almost. Ain't it a pity, now?\n\nTOUCHIT.--Miss Prior generally plays, doesn't she?\n\nMILLIKEN.--And a very good player, too. But I thought you might like it.\n\nTOUCHIT.--Well, not exactly. I don't like sixpenny points, Horace, or\nquarrelling with old dragons about the odd trick. I will go and smoke\na cigar on the terrace, and contemplate the silver Thames, the darkling\nwoods, the starry hosts of heaven. I--I like smoking better than playing\nwhist. [MILLIKEN rings bell.]\n\nMILLIKEN.--Ah, George! you're not fit for domestic felicity.\n\nTOUCHIT.--No, not exactly.\n\nHOWELL enters.\n\nMILLIKEN.--Lights and a whist-table. Oh, I see you bring 'em. You know\neverything I want. He knows everything I want, Howell does. Let us cut.\nMiss Prior, you and I are partners!\n\n\n\n\nACT II.\n\n\nSCENE.--As before.\n\n\nLADY K.--Don't smoke, you naughty boy. I don't like it. Besides, it will\nencourage your brother-in-law to smoke.\n\nCLARENCE K.--Anything to oblige you, I'm sure. But can't do without it,\nmother; it's good for my health. When I was in the Plungers, our doctor\nused to say, \"You ought never to smoke more than eight cigars a day\"--an\norder, you know, to do it--don't you see?\n\nLADY K.--Ah, my child! I am very glad you are not with those unfortunate\npeople in the East.\n\nK.--So am I. Sold out just in time. Much better fun being here, than\nhaving the cholera at Scutari. Nice house, Milliken's. Snob, but\ngood fellow--good cellar, doosid good cook. Really, that salmi\nyesterday,--couldn't have it better done at the \"Rag\" now. You have got\ninto good quarters here, mother.\n\nLADY K.--The meals are very good, and the house is very good; the\nmanners are not of the first order. But what can you expect of city\npeople? I always told your poor dear sister, when she married Mr.\nMilliken, that she might look for everything substantial,--but not\nmanners. Poor dear Arabella WOULD marry him.\n\nK.--Would! that is a good one, mamma! Why, you made her! It's a dozen\nyears ago. But I recollect, when I came home from Eton, seeing her\ncrying because Charley Tufton--\n\nLADY K.--Mr. Tufton had not a shilling to bless himself with. The\nmarriage was absurd and impossible.\n\nK.--He hadn't a shilling then. I guess he has plenty now. Elder brother\nkilled, out hunting. Father dead. Tuf a baronet, with four thousand a\nyear if he's a shilling.\n\nLADY K.--Not so much.\n\nK.--Four thousand if it's a shilling. Why, the property adjoins\nKicklebury's--I ought to know. I've shot over it a thousand times. Heh!\nI remember, when I was quite a young 'un, how Arabella used to go out\ninto Tufton Park to meet Charley--and he is a doosid good fellow, and a\ngentlemanlike fellow, and a doosid deal better than this city fellow.\n\nLADY K.--If you don't like this city fellow, Clarence, why do you come\nhere? why didn't you stop with your elder brother at Kicklebury?\n\nK.--Why didn't I? Why didn't YOU stop at Kicklebury, mamma? Because you\nhad notice to quit. Serious daughter-in-law, quarrels about management\nof the house--row in the building. My brother interferes, and politely\nrequests mamma to shorten her visit. So it is with your other two\ndaughters; so it was with Arabella when she was alive. What shindies you\nused to have with her, Lady Kicklebury! Heh! I had a row with my brother\nand sister about a confounded little nursery-maid.\n\nLADY K.--Clarence!\n\nK.--And so I had notice to quit too. And I'm in very good quarters here,\nand I intend to stay in 'em, mamma. I say--\n\nLADY K.--What do you say?\n\nK.--Since I sold out, you know, and the regiment went abroad, confound\nme, the brutes at the \"Rag\" will hardly speak to me! I was so ill, I\ncouldn't go. Who the doose can live the life I've led and keep health\nenough for that infernal Crimea? Besides, how could I help it? I was\nso cursedly in debt that I was OBLIGED to have the money, you know. YOU\nhadn't got any.\n\nLADY K.--Not a halfpenny, my darling. I am dreadfully in debt myself.\n\nK.--I know you are. So am I. My brother wouldn't give me any, not a\ndump. Hang him! Said he had his children to look to. Milliken wouldn't\nadvance me any more--said I did him in that horse transaction. He! he!\nhe! so I did! What had I to do but to sell out? And the fellows cut\nme, by Jove. Ain't it too bad? I'll take my name off the \"Rag,\" I will,\nthough.\n\nLADY K.--We must sow our wild oats, and we must sober down; and we must\nlive here, where the living is very good and very cheap, Clarence, you\nnaughty boy! And we must get you a rich wife. Did you see at church\nyesterday that young woman in light green, with rather red hair and a\npink bonnet?\n\nK.--I was asleep, ma'am, most of the time, or I was bookin' up the\nodds for the Chester Cup. When I'm bookin' up, I think of nothin' else,\nma'am,--nothin'.\n\nLADY K.--That was Miss Brocksopp--Briggs, Brown and Brocksopp, the great\nsugar-bakers. They say she will have eighty thousand pound. We will ask\nher to dinner here.\n\nK.--I say--why the doose do you have such old women to dinner here? Why\ndon't you get some pretty girls? Such a set of confounded old frumps as\neat Milliken's mutton I never saw. There's you, and his old mother Mrs.\nBonnington, and old Mrs. Fogram, and old Miss What's-her-name, the woman\nwith the squint eye, and that immense Mrs. Crowder. It's so stoopid,\nthat if it weren't for Touchit coming down sometimes, and the billiards\nand boatin', I should die here--expire, by gad! Why don't you have some\npretty women into the house, Lady Kicklebury?\n\nLADY K.--Why! Do you think I want that picture taken down: and another\nMrs. Milliken? Wisehead! If Horace married again, would he be your\nbanker, and keep this house, now that ungrateful son of mine has turned\nme out of his? No pretty woman shall come into the house whilst I am\nhere.\n\nK.--Governess seems a pretty woman: weak eyes, bad figure, poky, badly\ndressed, but doosid pretty woman.\n\nLADY K.--Bah! There is no danger from HER. She is a most faithful\ncreature, attached to me beyond everything. And her eyes--her eyes\nare weak with crying for some young man who is in India. She has his\nminiature in her room, locked up in one of her drawers.\n\nK.--Then how the doose did you come to see it?\n\nLADY K.--We see a number of things, Clarence. Will you drive with me?\n\nK.--Not as I knows on, thank you. No, Ma; drivin's TOO slow: and you're\ngoin' to call on two or three old dowagers in the Park? Thank your\nladyship for the delightful offer.\n\nEnter JOHN.\n\nJOHN.--Please, sir, here's the man with the bill for the boats; two\npound three.\n\nK.--Damn it, pay it--don't bother ME!\n\nJOHN.--Haven't got the money, sir.\n\nLADY K.--Howell! I saw Mr. Milliken give you a cheque for twenty-five\npounds before he went into town this morning. Look sir [runs, opens\ndrawer, takes out cheque-book]. There it is, marked, \"Howell, 25L.\"\n\nJOHN.--Would your ladyship like to step down into my pantry and see what\nI've paid with the twenty-five pounds? Did my master leave any orders\nthat your ladyship was to inspect my accounts?\n\nLADY K.--Step down into the pantry! inspect your accounts? I never heard\nsuch impertinence. What do you mean, sir?\n\nK.--Dammy, sir, what do you mean?\n\nJOHN.--I thought as her ladyship kept a heye over my master's private\nbook, she might like to look at mine too.\n\nLADY K.--Upon my word, this insolence is too much.\n\nJOHN.--I beg your ladyship's pardon. I am sure I have said nothing.\n\nK.--Said, sir! your manner is mutinous, by Jove, sir! if I had you in\nthe regiment!--\n\nJOHN.--I understood that you had left the regiment, sir, just before it\nwent on the campaign, sir.\n\nK.--Confound you, sir! [Starts up.]\n\nLADY K.--Clarence, my child, my child!\n\nJOHN.--Your ladyship needn't be alarmed; I'm a little man, my lady,\nbut I don't think Mr. Clarence was a-goin' for to hit me, my lady; not\nbefore a lady, I'm sure. I suppose, sir, that you WON'T pay the boatman?\n\nK.--No, sir, I won't pay him, nor any man who uses this sort of damned\nimpertinence!\n\nJOHN.--I told Rullocks, sir, I thought it was JEST possible you\nwouldn't. [Exit.]\n\nK.--That's a nice man, that is--an impudent villain!\n\nLADY K.--Ruined by Horace's weakness. He ruins everybody, poor\ngood-natured Horace!\n\nK.--Why don't you get rid of the blackguard?\n\nLADY K.--There is a time for all things, my dear. This man is very\nconvenient to Horace. Mr. Milliken is exceedingly lazy, and Howell\nspares him a great deal of trouble. Some day or other I shall take\nall this domestic trouble off his hands. But not yet: your poor\nbrother-in-law is restive, like many weak men. He is subjected to other\ninfluences: his odious mother thwarts me a great deal.\n\nK.--Why, you used to be the dearest friends in the world. I recollect\nwhen I was at Eton--\n\nLADY K.--Were; but friendship don't last for ever. Mrs. Bonnington and\nI have had serious differences since I came to live here: she has a\nnatural jealousy, perhaps, at my superintending her son's affairs. When\nshe ceases to visit at the house, as she very possibly will, things will\ngo more easily; and Mr. Howell will go too, you may depend upon it. I am\nalways sorry when my temper breaks out, as it will sometimes.\n\nK.--Won't it, that's all!\n\nLADY K.--At his insolence, my temper is high; so is yours, my dear. Calm\nit for the present, especially as regards Howell.\n\nK.--Gad! d'you know I was very nearly pitching into him? But once,\none night in the Haymarket, at a lobster-shop, where I was with some\nfellows, we chaffed some other fellows, and there was one fellah--quite\na little fellah--and I pitched into him, and he gave me the most\nconfounded lickin' I ever had in my life, since my brother Kicklebury\nlicked me when we were at Eton; and that, you see, was a lesson to me,\nma'am. Never trust those little fellows, never chaff 'em: dammy, they\nmay be boxers.\n\nLADY K.--You quarrelsome boy! I remember you coming home with your\nnaughty head SO bruised. [Looks at watch.] I must go now to take my\ndrive. [Exit LADY K.]\n\nK.--I owe a doose of a tick at that billiard-room; I shall have that\nboatman dunnin' me. Why hasn't Milliken got any horses to ride? Hang\nhim! suppose he can't ride--suppose he's a tailor. He ain't MY tailor,\nthough, though I owe him a doosid deal of money. There goes mamma with\nthat darling nephew and niece of mine. [Enter BULKELEY]. Why haven't you\ngone with my lady, you, sir? [to Bulkeley.]\n\nBULKELEY.--My lady have a-took the pony-carriage, sir; Mrs. Bonnington\nhave a-took the hopen carriage and 'orses, sir, this mornin', which the\nBishop of London is 'olding a confirmation at Teddington, sir, and Mr.\nBonnington is attending the serimony. And I have told Mr. 'Owell, sir,\nthat my lady would prefer the hopen carriage, sir, which I like the\nhexercise myself, sir, and that the pony-carriage was good enough for\nMrs. Bonnington, sir; and Mr. 'Owell was very hinsolent to me, sir; and\nI don't think I can stay in the 'ouse with him.\n\nK.--Hold your jaw, sir.\n\nBULKELEY.--Yes, sir. [Exit BULKELEY.]\n\nK.--I wonder who that governess is?--sang rather prettily last\nnight--wish she'd come and sing now--wish she'd come and amuse me--I've\nseen her face before--where have I seen her face?--it ain't at all a bad\none. What shall I do? dammy, I'll read a book: I've not read a book this\never so long. What's here? [looks amongst books, selects one, sinks down\nin easy-chair so as quite to be lost.]\n\nEnter Miss PRIOR.\n\nMISS PRIOR.--There's peace in the house! those noisy children are away\nwith their grandmamma. The weather is beautiful, and I hope they will\ntake a long drive. Now I can have a quiet half-hour, and finish that\ndear pretty \"Ruth\"--oh, how it makes me cry, that pretty story.\n[Lays down her bonnet on table--goes to glass--takes off cap and\nspectacles--arranges her hair--Clarence has got on chair looking at\nher.]\n\nK.--By Jove! I know who it is now! Remember her as well as possible.\nFour years ago, when little Foxbury used to dance in the ballet over\nthe water. DON'T I remember her! She boxed my ears behind the scenes,\nby jingo. [Coming forward]. Miss Pemberton! Star of the ballet! Light of\nthe harem! Don't you remember the grand Oriental ballet of the \"Bulbul\nand the Peri?\"\n\nMISS P.--Oh! [screams.] No, n--no, sir. You are mistaken: my name is\nPrior. I--never was at the \"Coburg Theatre.\" I--\n\nK. [seizing her hand].--No, you don't, though! What! don't you remember\nwell that little hand slapping this face? which nature hadn't then\nadorned with whiskers, by gad! You pretend you have forgotten little\nFoxbury, whom Charley Calverley used to come after, and who used to\ndrive to the \"Coburg\" every night in her brougham. How did you know it\nwas the \"Coburg?\" That IS a good one! HAD you there, I think.\n\nMISS P.--Sir, in the name of heaven, pity me! I have to keep my mother\nand my sisters and my brothers. When--when you saw me, we were in great\npoverty; and almost all the wretched earnings I made at that time were\ngiven to my poor father then lying in the Queen's Bench hard by. You\nknow there was nothing against my character--you know there was not. Ask\nCaptain Touchit whether I was not a good girl. It was he who brought me\nto this house.\n\nK.--Touchit! the old villain!\n\nMISS P.--I had your sister's confidence. I tended her abroad on her\ndeath-bed. I have brought up your nephew and niece. Ask any one if I\nhave not been honest? As a man, as a gentleman, I entreat you to keep my\nsecret! I implore you for the sake of my poor mother and her children!\n[kneeling.]\n\nK.--By Jove! how handsome you are! How crying becomes your eyes! Get up;\nget up. Of course I'll keep your secret, but--\n\nMISS P.--Ah! ah! [She screams as he tries to embrace her. HOWELL rushes\nin.]\n\nHOWELL.--Hands off, you little villain! Stir a step and I'll kill you,\nif you were a regiment of captains! What! insult this lady who kept\nwatch at your sister's death-bed and has took charge of her children!\nDon't be frightened, Miss Prior. Julia--dear, dear Julia--I'm by you.\nIf the scoundrel touches you, I'll kill him. I--I love you--there--it's\nhere--love you madly--with all my 'art--my a-heart!\n\nMISS P.--Howell--for heaven's sake, Howell!\n\nK.--Pooh--ooh! [bursting with laughter]. Here's a novel, by\njingo! Here's John in love with the governess. Fond of plush, Miss\nPemberton--ey? Gad, it's the best thing I ever knew. Saved a good bit,\ney, Jeames? Take a public-house? By Jove! I'll buy my beer there.\n\nJOHN.--Owe for it, you mean. I don't think your tradesmen profit much by\nyour custom, ex-Cornet Kicklebury.\n\nK.--By Jove! I'll do for you, you villain!\n\nJOHN.--No, not that way, Captain. [Struggles with and throws him.]\n\nK. [screams.]--Hallo, Bulkeley! [Bulkeley is seen strolling in the\ngarden.]\n\nEnter BULKELEY.\n\nBULKELEY.--What is it, sir?\n\nK.--Take this confounded villain off me, and pitch him into the\nThames--do you hear?\n\nJOHN.--Come here, and I'll break every bone in your hulking body. [To\nBULKELEY.]\n\nBULKELEY.--Come, come! whathever his hall this year row about?\n\nMISS P.--For heaven's sake don't strike that poor man.\n\nBULKELEY.--YOU be quiet. What's he a-hittin' about my master for?\n\nJOHN.--Take off your hat, sir, when you speak to a lady. [Takes up a\npoker.] And now come on, both of you, cowards! [Rushes at BULKELEY and\nknocks his hat off his head.]\n\nBULKELEY [stepping back].--If you'll put down that there poker, you\nknow, then I'll pitch into you fast enough. But that there poker ain't\nfair, you know.\n\nK.--You villain! of course you will leave this house. And, Miss Prior, I\nthink you understand that you will go too. I don't think my niece wants\nto learn DANCIN', you understand. Good-by. Here, Bulkeley! [Gets behind\nfootman and exit.]\n\nMISS P.--Do you know the meaning of that threat, Mr. Howell?\n\nJOHN.--Yes, Miss Prior.\n\nMISS P.--I was a dancer once, for three months, four years ago, when my\npoor father was in prison.\n\nJOHN.--Yes, Miss Prior, I knew it. And I saw you a many times.\n\nMISS P.--And you kept my secret?\n\nJOHN.--Yes, Ju--Jul--Miss Prior.\n\nMISS P.--Thank you, and God bless you, John Howell. There, there. You\nmustn't! indeed you mustn't!\n\nJOHN.--You don't remember the printer's boy who used to come to Mr.\nO'Reilly, and sit in your 'all in Bury Street, Miss Prior? I was that\nboy. I was a country-bred boy--that is if you call Putney country, and\nWimbledon Common and that. I served the Milliken family seven year. I\nwent with Master Horace to college, and then I revolted against service,\nand I thought I'd be a man and turn printer like Doctor Frankling. And I\ngot in an office: and I went with proofs to Mr. O'Reilly, and I saw\nyou. And though I might have been in love with somebody else before I\ndid--yet it was all hup when I saw you.\n\nMISS P. [kindly.]--YOU must not talk to me in that way, John Howell.\n\nJOHN.--Let's tell the tale out. I couldn't stand the newspaper\nnight-work. I had a mother and brothers and sisters to keep, as you had.\nI went back to Horace Milliken and said, Sir, I've lost my work. I and\nmine want bread. Will you take me back again? And he did. He's a kind,\nkind soul is my master.\n\nMISS P.--He IS a kind, kind soul.\n\nJOHN.--He's good to all the poor. His hand's in his pocket for\neverybody. Everybody takes advantage of him. His mother-in-lor rides\nover him. So does his Ma. So do I, I may say; but that's over now; and\nyou and I have had our notice to quit. Miss, I should say.\n\nMISS P.--Yes.\n\nJOHN.--I have saved a bit of money--not much--a hundred pound. Miss\nPrior--Julia--here I am--look--I'm a poor feller--a poor servant--but\nI've the heart of a man--and--I love you--oh! I love you!\n\nMARY.--Oh ho--ho! [Mary has entered from garden, and bursts out crying.]\n\nMISS P.--It can't be, John Howell--my dear, brave, kind John Howell.\nIt can't be. I have watched this for some time past, and poor Mary's\ndespair here. [Kisses Mary, who cries plentifully.] You have the heart\nof a true, brave man, and must show it and prove it now. I am not--am\nnot of your pardon me for saying so--of your class in life. I was bred\nby my uncle, away from my poor parents, though I came back to them after\nhis sudden death; and to poverty, and to this dependent life I am now\nleading. I am a servant, like you, John, but in another sphere--have\nto seek another place now; and heaven knows if I shall procure one, now\nthat that unlucky passage in my life is known. Oh, the coward to recall\nit! the coward!\n\nMARY.--But John whopped him, Miss! that he did. He gave it him well,\nJohn did. [Crying.]\n\nMISS P.--You can't--you ought not to forego an attachment like that,\nJohn Howell. A more honest and true-hearted creature never breathed than\nMary Barlow.\n\nJOHN.--No, indeed.\n\nMISS P.--She has loved you since she was a little child. And you loved\nher once, and do now, John.\n\nMARY.--Oh, Miss! you hare a hangel,--I hallways said you were a hangel.\n\nMISS P.--You are better than I am, my dear much, much better than I am,\nJohn. The curse of my poverty has been that I have had to flatter and to\ndissemble, and hide the faults of those I wanted to help, and to smile\nwhen I was hurt, and laugh when I was sad, and to coax, and to tack, and\nto bide my time,--not with Mr. Milliken: he is all honor, and kindness,\nand simplicity. Who did HE ever injure, or what unkind word did HE ever\nsay? But do you think, with the jealousy of those poor ladies over his\nhouse, I could have stayed here without being a hypocrite to both of\nthem? Go, John. My good, dear friend, John Howell, marry Mary. You'll be\nhappier with her than with me. There! There! [They embrace.]\n\nMARY.--O--o--o! I think I'll go and hiron hout Miss Harabella's frocks\nnow. [Exit MARY.]\n\nEnter MILLIKEN with CLARENCE--who is explaining things to him.\n\nCLARENCE.--Here they are, I give you my word of honor. Ask 'em, damn em.\n\nMILLIKEN.--What is this I hear? You, John Howell, have dared to strike a\ngentleman under my roof! Your master's brother-in-law?\n\nJOHN.--Yes, by Jove! and I'd do it again.\n\nMILLIKEN.--Are you drunk or mad, Howell?\n\nJOHN.--I'm as sober and as sensible as ever I was in my life, sir--I not\nonly struck the master, but I struck the man, who's twice as big, only\nnot quite as big a coward, I think.\n\nMILLIKEN.--Hold your scurrilous tongues sir! My good nature ruins\neverybody about me. Make up your accounts. Pack your trunks--and never\nlet me see your face again.\n\nJOHN.--Very good, sir.\n\nMILLIKEN.--I suppose, Miss Prior, you will also be disposed to--to\nfollow Mr. Howell?\n\nMISS P.--To quit you, now you know what has passed? I never supposed\nit could be otherwise--I deceived you, Mr. Milliken--as I kept a secret\nfrom you, and must pay the penalty. It is a relief to me, the sword has\nbeen hanging over me. I wish I had told your poor wife, as I was often\nminded to do.\n\nMILLIKEN.--Oh, you were minded to do it in Italy, were you?\n\nMISS P.--Captain Touchit knew it, sir, all along: and that my motives\nand, thank God, my life were honorable.\n\nMILLIKEN.--Oh, Touchit knew it, did he? and thought it\nhonorable--honorable. Ha! ha! to marry a footman--and keep a\npublic-house? I--I beg your pardon, John Howell--I mean nothing against\nyou, you know. You're an honorable man enough, except that you have been\ndamned insolent to my brother-in-law.\n\nJOHN.--Oh, heaven! [JOHN strikes his forehead, and walks away.]\n\nMISS P.--You mistake me, sir. What I wished to speak of was the fact\nwhich this gentleman has no doubt communicated to you--that I danced on\nthe stage for three months.\n\nMILLIKEN.--Oh, yes. Oh, damme, yes. I forgot. I wasn't thinking of that.\n\nKICKLEBURY.--You see she owns it.\n\nMISS P.--We were in the depths of poverty. Our furniture and\nlodging-house under execution--from which Captain Touchit, when he came\nto know of our difficulties, nobly afterwards released us. My father was\nin prison, and wanted shillings for medicine, and I--I went and danced\non the stage.\n\nMILLIKEN.--Well?\n\nMISS P.--And I kept the secret afterwards; knowing that I could never\nhope as governess to obtain a place after having been a stage-dancer.\n\nMILLIKEN.--Of course you couldn't,--it's out of the question; and may I\nask, are you going to resume that delightful profession when you enter\nthe married state with Mr. Howell?\n\nMISS P.--Poor John! it is not I who am going to--that is, it's Mary, the\nschool-room maid.\n\nMILLIKEN.--Eternal blazes! Have you turned Mormon, John Howell, and are\nyou going to marry the whole house?\n\nJOHN.--I made a hass of myself about Miss Prior. I couldn't help her\nbeing l--l--lovely.\n\nKICK.--Gad, he proposed to her in my presence.\n\nJOHN.--What I proposed to her, Cornet Clarence Kicklebury, was my heart\nand my honor, and my best, and my everything--and you--you wanted to\ntake advantage of her secret, and you offered her indignities, and you\nlaid a cowardly hand on her--a cowardly hand!--and I struck you, and I'd\ndo it again.\n\nMILLIKEN.--What? Is this true? [Turning round very fiercely to K.]\n\nKICK.--Gad! Well--I only--\n\nMILLIKEN.--You only what? You only insulted a lady under my roof--the\nfriend and nurse of your dead sister--the guardian of my children. You\nonly took advantage of a defenceless girl, and would have extorted your\ninfernal pay out of her fear. You miserable sneak and coward!\n\nKICK.--Hallo! Come, come! I say I won't stand this sort of chaff. Dammy,\nI'll send a friend to you!\n\nMILLIKEN.--Go out of that window, sir. March! or I will tell my servant,\nJohn Howell, to kick you out, you wretched little scamp! Tell that big\nbrute,--what's-his-name?--Lady Kicklebury's man, to pack this young\nman's portmanteau and bear's-grease pots; and if ever you enter these\ndoors again, Clarence Kicklebury, by the heaven that made me!--by your\nsister who is dead!--I will cane your life out of your bones. Angel in\nheaven! Shade of my Arabella--to think that your brother in your house\nshould be found to insult the guardian of your children!\n\nJOHN.--By jingo, you're a good-plucked one! I knew he was, Miss,--I told\nyou he was. [Exit, shaking hands with his master, and with Miss P., and\ndancing for joy. Exit CLARENCE, scared, out of window.]\n\nJOHN [without].--Bulkeley! pack up the Capting's luggage!\n\nMILLIKEN.--How can I ask your pardon, Miss Prior? In my wife's name\nI ask it--in the name of that angel whose dying-bed you watched and\nsoothed--of the innocent children whom you have faithfully tended since.\n\nMISS P.--Ah, sir! it is granted when you speak so to me.\n\nMILLIKEN.--Eh, eh--d--don't call me sir!\n\nMISS P.--It is for me to ask pardon for hiding what you know now: but if\nI had told you--you--you never would have taken me into your house--your\nwife never would.\n\nMILLIKEN.--No, no. [Weeping.]\n\nMISS P.--My dear, kind Captain Touchit knows it all. It was by his\ncounsel I acted. He it was who relieved our distress. Ask him whether my\nconduct was not honorable--ask him whether my life was not devoted to my\nparents--ask him when--when I am gone.\n\nMILLIKEN.--When you are gone, Julia! Why are you going? Why should you\ngo, my love--that is--why need you go, in the devil's name?\n\nMISS P.--Because, when your mother--when your mother-in-law come to hear\nthat your children's governess has been a dancer on the stage, they will\nsend me away, and you will not have the power to resist them. They ought\nto send me away, sir; but I have acted honestly by the children and\ntheir poor mother, and you'll think of me kindly when--I--am--gone?\n\nMILLIKEN.--Julia, my dearest--dear--noble--dar--the devil! here's old\nKicklebury.\n\nEnter Lady K., Children, and CLARENCE.\n\nLADY K.--So, Miss Prior! this is what I hear, is it? A dancer in my\nhouse! a serpent in my bosom--poisoning--yes, poisoning those blessed\nchildren! occasioning quarrels between my own son and my dearest\nson-in-law; flirting with the footman! When do you intend to leave,\nmadam, the house which you have po--poll--luted?\n\nMISS P.--I need no hard language, Lady Kicklebury: and I will reply to\nnone. I have signified to Mr. Milliken my wish to leave his house.\n\nMILLIKEN.--Not, not, if you will stay. [To Miss P.]\n\nLADY K.--Stay, Horace! she shall NEVER stay as governess in this house!\n\nMILLIKEN.--Julia! will you stay as mistress? You have known me for a\nyear alone--before, not so well--when the house had a mistress that is\ngone. You know what my temper is, and that my tastes are simple, and\nmy heart not unkind. I have watched you, and have never seen you out\nof temper, though you have been tried. I have long thought you good and\nbeautiful, but I never thought to ask the question which I put to you\nnow:--come in, sir! [to CLARENCE at door]:--now that you have been\npersecuted by those who ought to have upheld you, and insulted by those\nwho owed you gratitude and respect. I am tired of their domination, and\nas weary of a man's cowardly impertinence [to CLARENCE] as of a woman's\njealous tyranny. They have made what was my Arabella's home miserable\nby their oppression and their quarrels. Julia! my wife's friend, my\nchildren's friend! be mine, and make me happy! Don't leave me, Julia!\nsay you won't--say you won't--dearest--dearest girl!\n\nMISS P.--I won't--leave--you.\n\nGEORGE [without].--Oh, I say! Arabella, look here: here's papa a-kissing\nMiss Prior!\n\nLADY K.--Horace--Clarence my son! Shade of my Arabella! can you behold\nthis horrible scene, and not shudder in heaven! Bulkeley! Clarence! go\nfor a doctor--go to Doctor Straitwaist at the Asylum--Horace Milliken,\nwho has married the descendant of the Kickleburys of the Conqueror,\nmarry a dancing-girl off the stage! Horace Milliken! do you wish to\nsee me die in convulsions at your feet? I writhe there, I grovel there.\nLook! look at me on my knees! your own mother-in-law! drive away this\nfiend!\n\nMILLIKEN.--Hem! I ought to thank you, Lady Kicklebury, for it is you\nthat have given her to me.\n\nLADY K.--He won't listen! he turns away and kisses her horrible hand.\nThis will never do: help me up, Clarence, I must go and fetch his\nmother. Ah, ah! there she is, there she is! [Lady K. rushes out, as the\ntop of a barouche, with Mr. and Mrs. BONNINGTON and Coachman, is seen\nover the gate.]\n\nMRS. B.--What is this I hear, my son, my son? You are going to marry\na--a stage-dancer? you are driving me mad, Horace!\n\nMILLIKEN.--Give me my second chance, mother, to be happy. You have had\nyourself two chances.\n\nMRS. B.--Speak to him, Mr. Bonnington. [BONNINGTON makes dumb show.]\n\nLADY K.--Implore him, Mr. Bonnington.\n\nMRS. B.--Pray, pray for him, Mr. Bonnington, my love--my lost, abandoned\nboy!\n\nLADY K.--Oh, my poor dear Mrs. Bonnington!\n\nMRS. B.--Oh, my poor dear Lady Kicklebury. [They embrace each other.]\n\nLADY K.--I have been down on my knees to him, dearest Mrs. Bonnington.\n\nMRS. B.--Let us both--both go down on our knees--I WILL [to her\nhusband]. Edward, I will! [Both ladies on their knees. BONNINGTON with\noutstretched hands behind them.] Look, unhappy boy! look, Horace! two\nmothers on their wretched knees before you, imploring you to send away\nthis monster! Speak to him, Mr. Bonnington. Edward! use authority with\nhim, if he will not listen to his mother--\n\nLADY K.--To his mothers!\n\nEnter TOUCHIT.\n\nTOUCHIT.--What is this comedy going on, ladies and gentlemen? The ladies\non their elderly knees--Miss Prior with her hair down her back. Is it\ntragedy or comedy--is it a rehearsal for a charade, or are we acting\nfor Horace's birthday? or, oh!--I beg your Reverence's pardon--you were\nperhaps going to a professional duty?\n\nMR. B.--It's WE who are praying this child, Touchit. This child, with\nwhom you used to come home from Westminster when you were boys. You\nhave influence with him; he listens to you. Entreat him to pause in his\nmadness.\n\nTOUCHIT.--What madness?\n\nMRS. B.--That--that woman--that serpent yonder--that--that\ndancing-woman, whom you introduced to Arabella Milliken,--ah! and I rue\nthe day:--Horace is going to mum--mum--marry her!\n\nTOUCHIT.--Well! I always thought he would. Ever since I saw him and her\nplaying at whist together, when I came down here a month ago, I thought\nhe would do it.\n\nMRS. B.--Oh, it's the whist, the whist! Why did I ever play at whist,\nEdward? My poor Mr. Milliken used to like his rubber.\n\nTOUCHIT.--Since he has been a widower--\n\nLADY K.--A widower of that angel! [Points to picture.]\n\nTOUCHIT.--Pooh, pooh, angel! You two ladies have never given the\npoor fellow any peace. You were always quarrelling over him. You took\npossession of his house, bullied his servants, spoiled his children; you\ndid, Lady Kicklebury.\n\nLADY K.--Sir, you are a rude, low, presuming, vulgar man. Clarence! beat\nthis rude man!\n\nTOUCHIT.--From what I have heard of your amiable son, he is not in the\nwarlike line, I think. My dear Julia, I am delighted with all my heart\nthat my old friend should have found a woman of sense, good conduct,\ngood temper--a woman who has had many trials, and borne them with great\npatience--to take charge of him and make him happy. Horace, give me your\nhand! I knew Miss Prior in great poverty. I am sure she will bear as\nnobly her present good fortune; for good fortune it is to any woman to\nbecome the wife of such a loyal, honest, kindly gentleman as you are!\n\nEnter JOHN.\n\nJOHN.--If you please, my lady--if you please, sir--Bulkeley--\n\nLADY K.--What of Bulkeley, sir?\n\nJOHN.--He has packed his things, and Cornet Kicklebury's things, my\nlady.\n\nMILLIKEN.--Let the fellow go.\n\nJOHN.--He won't go, sir, till my lady have paid him his book and wages.\nHere's the book, sir.\n\nLADY K.--Insolence! quit my presence! And I, Mr. Milliken, will quit a\nhouse--\n\nJOHN.--Shall I call your ladyship a carriage?\n\nLADY K.--Where I have met with rudeness, cruelty, and fiendish [to Miss\nP., who smiles and curtsies]--yes, fiendish ingratitude. I will go, I\nsay, as soon as I have made arrangements for taking other lodgings. You\ncannot expect a lady of fashion to turn out like a servant.\n\nJOHN.--Hire the \"Star and Garter\" for her, sir. Send down to the\n\"Castle;\" anything to get rid of her. I'll tell her maid to pack her\ntraps. Pinhorn! [Beckons maid and gives orders.]\n\nTOUCHIT.--You had better go at once, my dear Lady Kicklebury.\n\nLADY K.--Sir!\n\nTOUCHIT.--THE OTHER MOTHER-IN-LAW IS COMING! I met her on the road with\nall her family. He! he! he! [Screams.]\n\nEnter Mrs. PRIOR and Children.\n\nMRS. P.--My lady! I hope your ladyship is quite well! Dear, kind Mrs.\nBonnington! I came to pay my duty to you, ma'am. This is Charlotte, my\nlady--the great girl whom your ladyship so kindly promised the gown for;\nand this is my little girl, Mrs. Bonnington, ma'am, please; and this\nis my Bluecoat boy. Go and speak to dear, kind Mr. Milliken--our best\nfriend and protector--the son and son-in-law of these dear ladies. Look,\nsir! He has brought his copy to show you. [Boy shows copy.] Ain't it\ncreditable to a boy of his age, Captain Touchit? And my best and most\ngrateful services to you, sir. Julia, Julia, my dear, where's your cap\nand spectacles, you stupid thing? You've let your hair drop down. What!\nwhat!--[Begins to be puzzled.]\n\nMRS. B.--Is this collusion, madam?\n\nMRS. P.--Collusion, dear Mrs. Bonnington!\n\nLADY K.--Or insolence, Mrs. Prior!\n\nMRS. P.--Insolence, your ladyship! What--what is it? what has happened?\nWhat's Julia's hair down for? Ah! you've not sent the poor girl away?\nthe poor, poor child, and the poor, poor children!\n\nTOUCHIT.--That dancing at the \"Coburg\" has come out, Mrs. Prior.\n\nMRS. P.--Not the darling's fault. It was to help her poor father in\nprison. It was I who forced her to do it. Oh! don't, don't, dear Lady\nKicklebury, take the bread out of the mouths of these poor orphans!\n[Crying.]\n\nMILLIKEN.--Enough of this, Mrs. Prior: your daughter is not going away.\nJulia has promised to stay with me--and--never to leave me--as governess\nno longer, but as wife to me.\n\nMRS. P.--Is it--is it true, Julia?\n\nMISS P.--Yes, mamma.\n\nMRS. P.--Oh! oh! oh! [Flings down her umbrella, kisses JULIA, and\nrunning to MILLIKEN,] My son, my son! Come here, children. Come,\nAdolphus, Amelia, Charlotte--kiss your dear brother, children. What, my\ndears! How do you do, dears? [to MILLIKEN'S children]. Have they heard\nthe news? And do you know that my daughter is going to be your mamma?\nThere--there--go and play with your little uncles and aunts, that's good\nchildren! [She motions off the Children, who retire towards garden. Her\nmanner changes to one of great patronage and intense satisfaction.] Most\nhot weather, your ladyship, I'm sure. Mr. Bonnington, you must find\nit hot weather for preachin'! Lor'! there's that little wretch beatin'\nAdolphus! George, sir! have done, sir! [Runs to separate them.] How ever\nshall we make those children agree, Julia?\n\nMISS P.--They have been a little spoiled, and I think Mr. Milliken will\nsend George and Arabella to school, mamma: will you not, Horace?\n\nMR. MILLIKEN.--I think school will be the very best thing for them.\n\nMRS. P.--And [Mrs. P. whispers, pointing to her own children] the blue\nroom, the green room, the rooms old Lady Kick has--plenty of room for\nus, my dear!\n\nMISS P.--No, mamma, I think it will be too large a party,--Mr. Milliken\nhas often said that he would like to go abroad, and I hope that now he\nwill be able to make his tour.\n\nMRS. P.--Oh, then! we can live in the house, you know: what's the use of\npayin' lodgin', my dear?\n\nMISS P.--The house is going to be painted. You had best live in your own\nhouse, mamma; and if you want anything, Horace, Mr. Milliken, I am sure,\nwill make it comfortable for you. He has had too many visitors of late,\nand will like a more quiet life, I think. Will you not?\n\nMILLIKEN.--I shall like a life with YOU, Julia.\n\nJOHN.--Cab, sir, for her ladyship!\n\nLADY K.--This instant let me go! Call my people. Clarence, your arm!\nBulkeley, Pinhorn! Mrs. Bonnington, I wish you good-morning! Arabella,\nangel! [looks at picture] I leave you. I shall come to you ere long.\n[Exit, refusing MILLIKEN's hand, passes up garden, with her servants\nfollowing her. MARY and other servants of the house are collected\ntogether, whom Lady K. waves off. Bluecoat boy on wall eating plums.\nPage, as she goes, cries, Hurray, hurray! Bluecoat boy cries, Hurray!\nWhen Lady K. is gone, JOHN advances.]\n\nJOHN.--I think I heard you say, sir, that it was your intention to go\nabroad?\n\nMILLIKEN.--Yes; oh, yes! Are we going abroad, my Julia?\n\nMISS P.--To settle matters, to have the house painted, and clear\n[pointing to children, mother, &c.] Don't you think it is the best thing\nthat we can do?\n\nMILLIKEN.--Surely, surely: we are going abroad. Howell, you will come\nwith us of course, and with your experiences you will make a capital\ncourier. Won't Howell make a capital courier, Julia? Good honest fellow,\nJohn Howell. Beg your pardon for being so rude to you just now. But my\ntemper is very hot, very.\n\nJOHN [laughing].--You are a Tartar, sir. Such a tyrant! isn't he, ma'am?\n\nMISS P.--Well, no; I don't think you have a very bad temper, Mr.\nMilliken, a--Horace.\n\nJOHN.--You must--take care of him--alone, Miss Prior--Julia--I mean Mrs.\nMilliken. Man and boy I've waited on him this fifteen year: with the\nexception of that trial at the printing-office, which--which I won't\ntalk of NOW, madam. I never knew him angry; though many a time I have\nknown him provoked. I never knew him say a hard word, though sometimes\nperhaps we've deserved it. Not often--such a good master as that is\npretty sure of getting a good servant--that is, if a man has a heart in\nhis bosom; and these things are found both in and out of livery. Yes, I\nhave been a honest servant to him,--haven't I, Mr. Milliken?\n\nMILLIKEN.--Indeed, yes, John.\n\nJOHN.--And so has Mary Barlow. Mary, my dear! [Mary comes forward.] Will\nyou allow me to introduce you, sir, to the futur' Mrs. Howell?--if Mr.\nBonnington does YOUR little business for you, as I dare say [turning to\nMr. B.], hold gov'nor, you will!--Make it up with your poor son, Mrs.\nBonnington, ma'am. You have took a second 'elpmate, why shouldn't Master\nHorace? [to Mrs. B.] He--he wants somebody to help him, and take care of\nhim, more than you do.\n\nTOUCHIT.--You never spoke a truer word in your life, Howell.\n\nJOHN.--It's my general 'abit, Capting, to indulge in them sort of\nstatements. A true friend I have been to my master, and a true friend\nI'll remain when he's my master no more.\n\nMILLIKEN.--Why, John, you are not going to leave me?\n\nJOHN.--It's best, sir, I should go. I--I'm not fit to be a servant in\nthis house any longer. I wish to sit in my own little home, with my own\nlittle wife by my side. Poor dear! you've no conversation, Mary, but\nyou're a good little soul. We've saved a hundred pound apiece, and if\nwe want more, I know who won't grudge it us, a good fellow--a good\nmaster--for whom I've saved many a hundred pound myself, and will take\nthe \"Milliken Arms\" at old Pigeoncot--and once a year or so, at this\nhanniversary, we will pay our respects to you, sir, and madam. Perhaps\nwe will bring some children with us, perhaps we will find some more in\nthis villa. Bless 'em beforehand! Good-by, sir, and madam--come away,\nMary! [going].\n\nMRS. P. [entering with clothes, &c.]--She has not left a single thing\nin her room. Amelia, come here! this cloak will do capital for you, and\nthis--this garment is the very thing for Adolphus. Oh, John! eh,\nHowell! will you please to see that my children have something to eat,\nimmediately! The Milliken children, I suppose, have dined already?\n\nJOHN.--Yes, ma'am; certainly, ma'am.\n\nMRS. P.--I see he is inclined to be civil to me NOW!\n\nMISS P.--John Howell is about to leave us, mamma. He is engaged to Mary\nBarlow, and when we go away, he is going to set up housekeeping for\nhimself. Good-by, and thank you, John Howell [gives her hand to JOHN,\nbut with great reserve of manner]. You have been a kind and true\nfriend to us--if ever we can serve you, count upon us--may he not, Mr.\nMilliken?\n\nMILLIKEN.--Always, always.\n\nMISS P.--But you will still wait upon us--upon Mr. Milliken, for a day\nor two, won't you, John, until we--until Mr. Milliken has found some\none to replace you. He will never find any one more honest than you, and\ngood, kind little Mary. Thank you, Mary, for your goodness to the poor\ngoverness.\n\nMARY.--Oh miss! oh mum! [Miss P. kisses Mary patronizingly].\n\nMISS P. [to JOHN].--And after they have had some refreshment, get a cab\nfor my brothers and sister, if you please, John. Don't you think that\nwill be best, my--my dear?\n\nMILLIKEN.--Of course, of course, dear Julia!\n\nMISS P.--And, Captain Touchit, you will stay, I hope, and dine with Mr.\nMilliken? And, Mrs. Bonnington, if you will receive as a daughter one\nwho has always had a sincere regard for you, I think you will aid in\nmaking your son happy, as I promise you with all my heart and all my\nlife to endeavor to do. [Miss P. and M. go up to Mrs. BONNINGTON.]\n\nMRS. BONNINGTON.--Well, there, then, since it must be so, bless you, my\nchildren.\n\nTOUCHIT.--Spoken like a sensible woman! And now, as I do not wish to\ninterrupt this felicity, I will go and dine at the \"Star and Garter.\"\n\nMISS P.--My dear Captain Touchit, not for worlds! Don't you know I\nmustn't be alone with Mr. Milliken until--until--?\n\nMILLIKEN.--Until I am made the happiest man alive! and you will come\ndown and see us often, Touchit, won't you? And we hope to see our\nfriends here often. And we will have a little life and spirit and gayety\nin the place. Oh, mother! oh, George! oh, Julia! what a comfort it is\nto me to think that I am released from the tyranny of that terrible\nmother-in-law!\n\nMRS. PRIOR.--Come in to your teas, children. Come this moment, I\nsay. [The Children pass quarrelling behind the characters, Mrs. PRIOR\nsummoning them; JOHN and MARY standing on each side of the dining-room\ndoor, as the curtain falls.]\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Wolves and the Lamb, by \nWilliam Makepeace Thackeray\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nAlso by Josh Weil\n\n_The New Valley_\n\n_The Great Glass Sea_\n\n# **THE AGE OF  \nPERPETUAL  \nLIGHT**\n\nStories\n\n## JOSH WEIL\n\nCopyright \u00a9 2017 by Josh Weil\n\nCover art and design by Nick Misani\n\nStories in this collection originally appeared in the following publications: \"No Flies, No Folly\" in _One Story_ ; \"Long Bright Line\" in _Virginia Quarterly Review, New Stories from the Midwest 2016_ , and the _Pushcart Prize XXXVIV: Best of the Small Presses_ ; \"The Essential Constituent of Modern Living Standards\" in _Ploughshares_ ; \"Angle of Reflection\" in _Narrative_ (as \"Mirza\"); \"The Point of Roughness\" in _Tin House_ ; \"Beautiful Ground\" in _Agni_ ; \"The First Bad Thing\" in _American Short Fiction_.\n\nAll rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher, except by a reviewer, who may quote brief passages in a review. Scanning, uploading, and electronic distribution of this book or the facilitation of such without the permission of the publisher is prohibited. Please purchase only authorized electronic editions, and do not participate in or encourage electronic piracy of copyrighted materials. Your support of the author's rights is appreciated. Any member of educational institutions wishing to photocopy part or all of the work for classroom use, or anthology, should send inquiries to Grove Atlantic, 154 West 14th Street, New York, NY 10011 or permissions@groveatlantic.com.\n\nFIRST EDITION\n\n_Published simultaneously in Canada_\n\n_Printed in the United States of America_\n\nFirst Grove Atlantic hardcover edition: September 2017\n\nLibrary of Congress Cataloging-in-Publication data available for this title.\n\nISBN 978-0-8021-2701-3\n\neISBN 978-0-8021-8877-9\n\nGrove Press\n\nan imprint of Grove Atlantic\n\n154 West 14th Street\n\nNew York, NY 10011\n\nDistributed by Publishers Group West\n\ngroveatlantic.com\n\n17 18 19 20 10 9 8 7 6 5 4 3 2 1\nNow I am here\n\nwhere street lamps renounce stars\n\nand keys tap like stuttering criers\n\ncalling the faithful to their solitary stations.\n\n\u2014from _From African Dust_ by Rema Boscov,  \nmy mother, to whom this book is dedicated\n\n# **CONTENTS**\n\nCover\n\nAlso by Josh Weil\n\nTitle Page\n\nCopyright\n\nEpigraph\n\nNo Flies, No Folly\n\nLong Bright Line\n\nThe Essential Constituent of Modern Living Standards\n\nAngle of Reflection\n\nThe Point of Roughness\n\nBeautiful Ground\n\nThe First Bad Thing\n\nHello From Here\n\nAcknowledgments\n\nBack Cover\n\n# [**NO FLIES,  \nNO FOLLY**](toc.html#ch1-R)\n\nOne by one the windows come alight. From up the hill, I watch: the Hartzlers' old stone house so dark, so still, it might be the new-turned soil of a garden bed\u2014huge, square, black\u2014and in it the orange lamplight blooming. Bloom, bloom, bloom. Mrs. Hartzler lighting the wicks. There: I can see her shape. It goes window to window, a bee drifting, till it reaches the first floor, again, and goes straight to\u2014where else?\u2014the kitchen. My stomach moans. I suck in my gut, tug the rucksack's belt more tight. On my shoulders I shrug the straps a little higher. Down I start towards the farm.\n\nSuch clanking! I am a houseful of noise, kindes playing tin can train, mamme scrubbing dishes by the well, tatte hammering a bent scythe blade flat. Even the dog with its tail\u2014whap! whap! Even the mice in the walls squeaking. It is all in the rucksack, and the rucksack is always on my back, and wherever I go, I take it.\n\nMy name is Yankel Yushrov, and I was not always a peddler. I was once, too, a lighter of lamps. Street lamps, in the city of Providence. I was once a seller of lemons in Baltimore. I was a greenhorn seeing from the deck of a ship for the first time the lights of New York. I was a beggar. I was a deserter. Once upon a time I absconded from the army of the tsar. Yes, and hidden in a hay cart, too. Once upon a time, I was a soldier. A draftee. The small sound you cannot hear in the dark on this road beneath the clanking of my pack is my spit landing. The other one you cannot hear is my sigh. I was a Russian, a Jew. A brother. A son. I will tell you something: there was a time when I was not even me.\n\nBy the time I reach the gate, the children have come out.\n\n\"Feter Yankel!\" they shout.\n\nAnd when I call them by their names\u2014\"Yonnie! Samuel! Rachael!\"\u2014I feel like a feter. Okay, okay, of course, how would I know how it feels to be an uncle? Sometimes there comes the thought: Probably, I am one. Or my brother is dead. Or both. With my fingers I bush up my mustache. Behind my glasses I make my eyes wide. I come for the kindes with a big stride, booming and clanking, making noises in my nose and throat from when I was a kinde myself\u2014my mamme snoring, my tatte gathering up the phlegm\u2014monster noises that send the children scattering among the panicked geese, the darting dogs. I chase their giggles in the dusk.\n\nUntil, from behind me, a light washes over the yard, paints the geese fiery, lifts from the dark the faces of the kindes. I stop, hands on my knees. The air is gone from me. I can feel her watching from the door. But when I look, I cannot see her face, just her shape broad and solid as the side of a milch cow, her ankles below her dress like the widest part of fence posts before they bury into the ground, breasts so heavy their weight makes them known even in the darkness of her body, shoulders even heavier, sleeves bunched to her elbows. Such forearms! Through the lace of her bonnet there glows the lamplight from inside the house. Behind her, in a chair shoved back from the table, scraping at something on his foot with the blade of a knife: Mr. Hartzler.\n\n\"Feter.\" It is the oldest, Samuel\u2014he must by now be the age of a bar mitzvah\u2014who is the first to ask, \"May I see it?\"\n\nI glance back at the doorway, his mother moving more into it to block his father with her skirts. \"Come,\" I say.\n\nOne by one, into the light thrown by the doorway, they do. Then the light is gone. I know she has shut the door, just as I know she is standing outside it, on the top stone step, in the cold, watching.\n\nI unbutton my coat, the top of my shirt. Into the collar goes my hand. Out it comes with a necklace, a simple copper haldzband that fifteen years ago my mamme gave to me. And attached to it? The flasch. A little glass bottle, no bigger than the poz of a newborn boy. A tin cap screwed on very tight. On the inside of the cap, in Hebrew letters so tiny it is a miracle a hand could make them: C. E. YUSHROV, APOTHECARY. But that nobody sees. The cap never comes off. What they could see, if they could read Hebrew, and had very good eyes, and were leaning so close to make their breath puff on my neck, is what my tatte stamped onto the top: NO FLIES, NO FOLLY.\n\nIt is by now more dark than dusk. Venus, yes. A moon yet, no. From the almost-gone light of the sky the haldzband is a faint thread stitched from my wrist to my neck. In the palm of my hand the glass of the flasch barely shows. With my other hand I take off my fox-hair cap.\n\n\"Here,\" I say, \"rub it on your head.\"\n\nBut Samuel is already doing it, rubbing as if his scalp is on fire, until his fine blond hair sticks out like a dandelion puff. Then he stops, and, holding his breath, leans his head down\u2014slow, slow\u2014towards my palm. The zap, when it comes, makes my shoulders jerk. As if attached to them by fishing lines, the other kindes step closer. Beneath his fuzz of hair: my hand. In my hand: the flasch glowing. A faint green glow like the eyes of night animals in the last of the light. For a moment, before it fades, the green light is on all the kindes' faces. It is in the teeth of their smiles. Watching their wonder, I have a feeling so odd: I hope Mrs. Hartzler sees it, that she will see _me_ giving it to them, that in exchange she might give to me a word, a look, a moment of her praise.\n\nInside, the brightness of the lamplight is a spark in the eye. Up from the table comes Ura Hartzler and\u2014one boot off, one boot on\u2014he clomps over. In one of his hands there is still the knife. With his other, he grasps my fingers.\n\n\"Still?\" he says, motioning with the knife at my mustache.\n\nEvery time I come he asks me, _Why don't you shave this?_ And every time, I ask him back, _Why don't you grow this?_ Three times a year for four years I have been coming.\n\n\"Even,\" he says, bringing the tip of his knife under my nose, \"I could do it for you now, g\u00fct?\"\n\n\"Schwindler,\" I say, pushing the knife away with one finger. \"You only want to steal my wonze from me so you can have it for yourself and put it there.\" He is a tall man and I am not, but with the same finger I reach up and poke the bit of shaved skin above his lip.\n\nHe makes a noise, his mouth jerking downwards into his beard. \"Kalt!\" he says. And I feel, for a second, how warm his skin is against my frozen fingertip.\n\nOver supper, we talk of beards. Not because of Ura's and my joking. Or even because Samuel shows the nick he gave himself below his chin, swears he has stubble on his cheek, wants me to feel.\n\n\"You better find a wife quick,\" I tell him, unable to keep from grinning at the thought of such a young kinde obliged to grow a beard.\n\n\"Or,\" his father says, grinning beneath his own, \"get better at shaving.\"\n\nNo, the reason we talk of beards is Mrs. Hartzler. She wants, she says, to learn of Russia. It used to be when I would mention something of that part of the world, she would sit, her fork held still, her mouth a little open as if through her lips she might taste the words I made, but always silent as the kindes.\n\nBut, maybe a year ago, she began to ask the questions. They started very simple: \"What is this in Yiddish?\" Scooping a ladleful of smashed potatoes onto my plate, or holding up the ladle, or pulling out from her bonnet a strand of hair, or drawing with her fingertip above her upper lip so I could feel it as clear as if it was brushing through the hairs above my own. \"Kartofel,\" I would tell her. \"Eslefl, har, wonze.\" Her face showing nothing with each word until there came one that was nothing like the word she knew. \"Wonze?\" she would ask, something in her eyes more bright, as if the strangeness of the word rubbed her the way her children did my flasch. \"Wonze,\" I would say. And she would tell me, \"In Pennsylvania Deutsch it is schnurbaertli,\" and I would say, a little too loud, a little too hard, so Ura would hear, \"I know.\"\n\nUntil, one visit, she asked, \"And in Russian?\"\n\nIt was after supper, in the sitting room, by the woodstove. We were standing over the rug on which I had spread my wares. In her hand, she held a box of matches.\n\n\"Schwebelen,\" I told her, in Yiddish.\n\nShe frowned. \"Nein,\" she said. \"In Russian.\"\n\nI glanced around the room. Ura was still at the table next door: I could see his stocking feet scratching one of the dogs behind its ears.\n\n\"Surely,\" she said, \"they have matches in Russia.\"\n\nI nodded. She waited, her eyebrows lifted just a little, her lips parted.\n\n\"Speechki,\" I said.\n\nHow strange, after so many years, the word felt in my mouth. How strange, watching her face hear it, to imagine for the first time what she would look like in the act of making love.\n\nShe shut her eyes, opened them. \"And,\" she said, \"in French?\"\n\nI told her I didn't speak French.\n\n\"What else do you speak?\"\n\n\"Nothing,\" I said. But I added, \"Well.\"\n\nNothing well: a stopper followed by a corkscrew. She slid open the matchbox, reached in, drew one out. Through the doorway to the dining room, I could see just one stocking foot now. It remained on the dog's head. But it had gone still.\n\n\"A little Polish,\" I whispered.\n\nThe whisper: a wineglass lifted.\n\n\"In Polish?\" she asked.\n\nAnd when I said nothing, she touched the match to the strip at the side of the box and struck it. The flame burst. The dog's head lifted. Did he hear it from so far? Did her husband?\n\nIn my surprise, I said, \"Mecze.\" I watched Ura's foot.\n\nUntil his wife said, \"What else?\"\n\nThen I looked at her. She was holding the match between us, letting the flame burn slowly down.\n\n\"What if he sees?\" I said.\n\n\"What else do you speak?\"\n\n\"What will he think?\"\n\n\"That I'm testing them.\"\n\nThe air just above the flame was wavery with heat. Through it, she looked at me. Mrs. Hartzler has eyes that are like lying on your back in the forest and looking up at all the flecks of green that are the leaves of the trees and in a moment of breeze seeing through them so many slivers of sky. She rattled the box. I could feel inside my chest all five hundred matchsticks knocking together.\n\n\"What else?\" she said.\n\nI watched the flame burn down towards her fingers. \"French,\" I said. She smiled. \"Allumettes,\" I told her.\n\nAnd she shut her eyes, the smile still on her lips, the flame still crawling towards her fingers, until it almost reached them. Then, her eyes opened. And, as I watched, she opened her mouth. And, watching me, she slid the burning match between her lips and shut them so they pressed around her fingers that still held the end, sealing the flame inside. She did not blink. I did not blink. Slowly, she opened her lips. Out of them, around her fingers, there came a curling cloud of smoke.\n\n\"Allumettes,\" she said through it. And, then, in her own tongue, \"Danke.\" She cleared her throat, shook the matches again, slipped them into her apron. \"How much?\" she said.\n\nThrough the thin screen of smoke that remained, I saw Ura in his socks padding towards us.\n\nAfter that, her questions became more difficult. She wanted to know what kinds of pies my mother made, what I ate on my journey by foot across Russia to Riga, on the steamship to New York. What was my favorite thing to eat on Delancey Street? And did I ever sell it from my cart? And how many other carts sold it too? And how many Jews did I think lived there? And Russians? And Irish? And Chinese? And all together, everyone? And what, she would ask, her eyes wide, her lips a little open, what was such a world of people like?\n\nDifficult? Yes. Because with each one, each time I came to show my wares, each time I again sat at their supper table, I could feel Ura's disapproval thickening. I could see on his face that his mind had begun to ask questions of its own.\n\nNow, reaching with the plate of chicken parts across the table to me, she is asking about beards. \"Surely in Russia,\" she says, \"you had one.\"\n\n\"I was a kinde,\" I say, hoping with my grin and shrug and taking of the best piece left\u2014the last breast\u2014to put an end to it.\n\nBut she only sets down the chicken plate and lifts her hands to her cheeks. \"Wasn't your face cold?\"\n\nWhen she takes her hands away they leave a gleam of grease on her skin. I want to take a rag and dip it in the kettle warming on the stove, press it there and rub. And so, I do not look at her. I look at the kindes. \"Kelt?\" I say to them. \"You want to know about kelt?\"\n\nThe youngest boy stops trying to push a chunk of chicken cartilage up his nose. He got it in pretty far; it sticks out while he stares.\n\n\"So kelt,\" I say, \"that at night, in winter, on guard duty outside the barracks, at our post far in the north of Karelia...\" I know what a name like that will do to their mother; I steal a look at her; her eyes are bright as the grease on her cheeks. \"... Near Petroplavilsk,\" I say, just to watch her lips open that little bit, \"it was so cold we were afraid to pischn. One button of our pants we would...\" I pull my fingertips apart and with my mouth make a popping sound. \"Only one. Enough to let out just the very tips of our pozen.\" Over their giggling I say, \"Like the nose of a little mouse sniffing.\" With my palms I cover my face. Through the crack between my hands I poke out the pink tip of my nose as if it is a poz poking through an unbuttoned fly. I wiggle it. My ears wiggle, too; I cannot do one without the other. When I take away my hands there they are, all of them, laughing. Ura is so loud the others, beneath his booming, are just open mouths and happy eyes. I wait until his laugh has died down enough to hear the others again before I go on: \"Through that tiny opening, we would pischn into bags. Ya! Bags! Little bags. Why?\" They are quiet now. The chunk of cartilage falls out of the boy's nose. It goes plink on the plate. \"Because,\" I say, \"our piss was warm. So warm! In the bags it was so steamy! And what do you think we did with these steamy warm bags?\" With my hands I make a bowl. Gently, I press it to my face. I roll my eyes to the ceiling. I let out a sound such as I would make at the first touch of my toches lowering into a hot bath. \"Oh!\" I say. I take off my glasses and press the imaginary steamy bag to my eyes, my lips, the back of my neck, saying, \"Oh!\" over and over, \"Oh!\"\n\nUra, struggling to break apart two leg bones, says through a grunt, \"Until the bag leaks.\"\n\n\"Leaks?\" I lift my brows so high they lift my glasses. And keeping my face like that so everyone will see, I reach to my coat on the back of my chair. In the pocket I fish around. Ura, over his twisting of the bones, watches. And just as I bring the bag out, the bones break with a snap. His elbows jerk.\n\n\"I hope,\" he says, \"that's not what I think it is.\"\n\n\"To remember\"\u2014I pat the pocket\u2014\"I keep one right here.\"\n\n\"Piss?\" Yonnie, the youngest, says.\n\nI look at him as if to say of course, but speak instead to his mother. \"Not one spill in fifteen years.\"\n\nLittle Rachael looks like she is thinking not so much about the spilling as about the length of time the piss has been in the bag. She crinkles up her nose.\n\nI reach across the table and hold the bag before her face. \"What do you smell?\" She refuses to breathe. I hold it in front of her little brother's nose. Then her older brother's. Then, reaching all the way across the table, under her mother's nose. I can feel her breath on my knuckles.\n\n\"Nothing!\" I say, and, over her plate, turn the bag upside down. I shake it. Not a drop comes out.\n\n\"What kind of bags?\" she says.\n\nAs if such a bag is too valuable to be away from my body for long, I pull it back. \"Canvas,\" I tell her. \"Soaked in turpentine and wax.\"\n\nUra chews the end of a bone. \"From Russia?\"\n\n\"Where can you get them?\" his wife says.\n\n\"This,\" Ura says, \"is how they make them in Russia?\"\n\n\"Do you have more?\" she asks.\n\n\"I think...\" I answer, \"I believe... I might... No, I am almost certain I must have, even right now, one or two still in my sack.\"\n\nBut once we are in the sitting room standing beside my rucksack, it is not the bags she wants. It is colder in there, the fire unlit\u2014the Amish, in all my years of knowing them, are never ones to waste wood\u2014and when I bend down to open the straps of my rucksack, she bends down behind me. I feel, in the pocket of my coat, her hand. On the straps, my own hands go still. From the kitchen: the sound of the kindes cleaning up the plates, of Ura talking to them, just on the other side of the wall. But she is right there. So close. Her soft, round face with its sun-hardened skin, dark beneath the white of her bonnet, her mouth a hard line. Is the line shaking? Through the lining of my coat pocket, I feel the back of her fingers against my thigh. I feel them find the bag, close around it. Yes, her mouth is shaking, just a little. I see it when her lips open, hear it on the murmur of disappointment she makes.\n\n\"What is it?\" I whisper.\n\n\"Dry,\" she says. \"But not still warm.\"\n\n\"The piss?\" A small laugh comes from my throat and I try to tamp it out before it gets past her ears. \"After fifteen years?\"\n\nShe smiles. \"Why do you lie, Yankel?\"\n\nIt is the first time she has ever called me by my given name. I slip my hand into the pocket. Her wrists are so big there is hardly room for mine. I can feel my blood beating against the bones at the back of her hand, her knuckles beneath my fingers, and, as I curl mine around hers, the warmth passing through our skin.\n\nI whisper, \"What is it that you want?\"\n\nSuch is the world through which I walk: Heavy stone homes more knolls than houses, their chimneys going through them like veins of rock. Even the barns are made of stone. Their slate roofs dare the wind to try to blow them anywhere. So gray, so solid, and in winter the round, red hexes painted on their walls look like rusty washers bolting them to the sky. Even the dirt roads are hard as pounded metal, beaten by the hooves of plow horses, by the boots of a million soldiers who marched them not even forty years ago. How fertile the fields are with bones! Yet, when the sun breaks through, when it sweeps them in patches, they look alive, rolling gentle as clouds between the ridgelines. The ridgelines are black straps holding them down. In their shadows: the hunkered milking barns, silos like turrets. Defending what? Against whom? The windows of the houses give no answer. Mostly they are dark, still as if sleeping, uninterested in the passing of days, decades, the century. It is 1901. March. I have been walking these roads for almost four years.\n\nThere I am, cresting a hill! First comes my voice, babbling to itself like a whole family at supper. Next, the smoke from my pipe, drifting into the air as if just on the other side of the hill there is, instead of me, the chimney of a small house. But wait. It is moving! Yes, there is its roof! No, that is my rucksack! So large it rises first atop the hill, above my head, as if the road itself is growing a hump. And then? Bit by bit? Comes me: the pale orange fuzz at the top of my fox-hair hat, the hairy earflaps swinging, my glasses foggy, my mustache dark with sweat. The rucksack's straps cut into my shoulders, the rope with which I tie it to my waist bunching my long coat like a bathrobe. If it is sunny, my flasch is out. It swings\u2014tock, tock\u2014against my chest, soaking in the light, so it will be ready for the kindes of the next farm. And there I go: the nails in the soles of my boots clacking against the pebbles on the road, the ash I knock from my pipe leaving behind me a trail, almost invisible, then gone, until as I make my way down towards the last farm for the day, there is left only the sound of my singing:\n\n_A bisl libe un a bisele glik_ ,\n\n_Di zun zol shaynen!_\n\n_Nor oyf eyn oygnblik_\n\n_Ven ikh zol kenen in mayn Hartz_\n\n_arayn brengen zonnenshayn_\n\n_oyf eyn minit!_\n\n_A little love and a little luck_ ,\n\n_The sun shall shine!_\n\n_Just a little bit_\n\n_When I'll know in my heart_\n\n_to bring the sunshine around_\n\n_in a minute!_\n\nDown there they will serve to me my supper. The Klopfenstiens, or Virklers, or Oesches, or Yordys. They will make for me a bed in the barn. Waglers, Beilers, Lapps, Knepps. My hundred families, two hundred, maybe more. Three or four I make it to in a day. Three or four times I return to each within in a year. They like me. I speak their language, or close enough. I like them. They remind me\u2014their plain clothes, their big families, their devotion to God, their faces ready for suffering, yes, even the beards\u2014of home.\n\nNever have I cheated them. Never have I sold a faulty good. Never have I duped or swindled. Never do I come to their doors without a smile. Usually also with a tune. Of their ways I do not make my judgments known. I do not speak against their God. I speak no Hebrew in their homes. Nor do I place in their hands the weight of my heart. Never do I eat a meal without bringing a gift, or sleep in a bed without leaving a nickel on the sheets, or spend a night in the barn only to wake in the morning and steal my breakfast from a milch cow's teat. Never do I tempt their sons, flirt with their daughters. Never have I slept with their wives.\n\nHow many times have I told that story of the kelt and the piss? How many leak-proof bags have I sold? How many children have I made laugh with my nose poking through the crack between my hands like the tip of a poz? How many times have I wiggled my nose for them, my ears wiggling along with it? How many times, then, in the moment before I split my hands apart, have I remembered: we could all twitch our noses and our ears, but it was only my breuder who could do one without the other. I remember the way he would move his face part by part\u2014one ear, one eyebrow, one nostril\u2014and how the rest of us would laugh. Tatte's laugh like low humming from the back of his throat. Mamme's rolling, hiccupping, a bell tossed down the side of a hill. My breuder's squeezed snorts as he tried to keep quiet beneath the twitching of his face. My own, unrecognizable. Then, my hands stay sealed in front of me. My nose: still. My ears: still.\n\n\"Mister Yushrov?\" the farmer, after some time, will say.\n\nHis wife: \"Is something wrong?\"\n\nTheir kindes: silent, watching.\n\nThis is what they do not understand. This is why they do not like my wonze. Not only Ura Hartzler. All of them. They tell me that it makes me look like a soldier.\n\n\"A soldier?\" I say.\n\n\"In the army of Germany,\" they tell me, \"every soldier wears beneath his nose a schnurbaertli.\" They say, \"We are a peaceful people here.\"\n\nI tell them this I know, but I remind them that here is not Germany, anymore. \"We are in America,\" I say. \"A new country. A country far away from our pasts. Here,\" I tell them, \"the soldiers wear beards.\"\n\nThey nod, they shrug. \"Still,\" they say, \"it makes you look like a German soldier.\"\n\nThis is why there is no Amish peddler to take my trade from me. This is why in the first spring of the first year of the new century, I go all the way to the capital of the county, I brave the busy streets, I push through the doors of the Reading Hardware and Dry Goods Company and spread out on the counter all the money I have saved. This is why I tell the clerk, \"I want to buy a cart.\"\n\nBy May I am back at the Hartzlers' farm again. Normally, it would be June, even July, but I have changed my route. I don't want to carry with me too long the thing that Esther Hartzler desired. I have wrapped it in crumpled pages of the Reading paper, and slid it inside one of a pair of mittens I hope to sell. They are lined with the thick, soft fleece of a sheep, but, still, taking extra care, I have rolled them into a blanket and tied it shut and wedged it, snug, into the corner of my cart.\n\nHow before my cart did I make it up these hills? How did I carry enough to keep from starving? It has been barely more than a month since I gave up my rucksack and, yet, I cannot conceive of how I once did it. My shoulders have forgotten the weight of the straps. My belly cannot remember the belt. I tell my buttocks, Forget the weight pressed down on you! Back, I say, forget the sweat of the pack! I brush my hands of it. Is it possible that for three years I trudged all these miles beneath that rucksack? That I thought us inseparable as a tortoise and its shell? That once I loved it as if it was, all wrapped in one, my companion and work and home? Such is the cuckolding delivered at the hands of progress.\n\nMy new cart has three wheels. On either side, attached to the axle, there are two large ones rimmed with iron and spoked with wood, set close enough to the back that if I wished I could reach out with a hand and feel, brushing against my palm, their rough-pocked hoops. With each step I take they grunt alongside my grunting. Sometimes I speak to them. Just a little farther, I say. I tell them, We can rest at the top. The third wheel speaks back. It is a little thing, hiding under the front of the cart like a kitten beneath a couch. Thin iron hoop, thin iron spokes, it squeaks. I have, attached to it, a rod: it runs beneath the cart and up through the back to the handlebars from which I can turn it. This way or that. Coming down a hill, running with the pull of the weight, I like to play with the little wheel, making it swoop back and forth across the road, back and forth, until the cart almost topples and, laughing, we have to rein ourselves in.\n\nAfter all, the cart is not truly a new one. The money I had was only enough to pick through the hardware store's back lot. So? Neither am I such a new one, anymore. Besides, I like the layers of old paint, the black scraped down to the red scraped down to the yellow scraped down to the wood. It is as if I can see on its sides the passage of time, the world rolling forwards. How long ago was the first coat painted? Who brushed it on? What did he carry? Who did he carry it to?\n\nBy the time that I begin the long, rattling trundle down the hill towards the Hartzler farm, it is approaching the hour for supper. The smoke wisping out of the kitchen chimney does to my stomach the same as the clanging of a bell. But my stomach's rumble is buried beneath the cart. And I know that the Hartzlers are not yet inside at the table. Below, the rich light of the end of the day spreads thick as butter over pastures still giddy with spring. Such green! Such plushness! It makes me want to reach down\u2014Yankel the Giant! Yankel the Golem!\u2014and plunge my fingers in.\n\nDown there, in the milking parlor, there is, I know, the whole family. Though the only one I can see is the girl. She is outside, in the small corral, moving among the cows that have already been milked, dumping grain into the troughs, the bucket flashing in the light, the feed a little explosion of gold, the cows shoving to smother it. On the other side of the barn, the ones still waiting to be relieved of their milk crowd the shut door, knocking, wanting in.\n\nInside the parlor I can still hear their dull thudding, but it is walled off and distant as my own heartbeat, buried beneath the sounds in the room: hoof clomps, shit spatter, the muzzles of four cows battering the wood bins to shlingn every last grain of feed, the slap of a wet cloth, the high, thin, shushing of the milk streaming into the pails: shush, shush, shush.\n\nI stand in the entryway reserved for humans, listening, watching. The only light is what streams in through the small, square windows that line the walls. It looks more red than it did even just outside. Maybe it is only my eyes adjusting to the dimness of the barn, or maybe it is only how this time of day the light changes with each breath, fast as butter browning in a pan, but it seems to me more than that: strange, wondrous, even frightening, as if the light is not what is changing but the change is in the world on which it falls, as if the two have come untethered, out of alignment, the barn not the barn, the cows not the cows, the man not the man, the woman not the woman, the boys not the boys. In and out of the beams of red light, pieces of them\u2014hands, arms, faces\u2014flicker like moths. I can feel them swirling and I stare hard at them, at the rest of their shapes in the shadows, to make them people again, four people: tatte, mamme, sohn, sohn.\n\nSlowly, I become aware that the mamme is talking to me. They'll be done soon she says. Into the light she raises an arm\u2014Esther Hartzler's forearm, her thick wrist, her fingers spattered white with milk\u2014and gestures for me to wait outside. Instead, I unbutton the cuffs of my sleeves. I crouch down. At the basin, I soap my hands, I wash my forearms. And, fingers dripping, I enter the parlor to join them.\n\nOver supper there is the good freundschaft of having worked together. Do they feel it, this family that every evening eats like this after milking? How can I know? I only know that the green onion shoots taste like the wild ones I would pick beside the wagon ruts in the dirt road I used to walk to school; that the slices of warm mutton are slathered in a sauce that smells like my mother's hands after she had stood on the stool in the sun of the window hanging to dry the sheaves of mint; that little Yonnie is not too shy to, in the middle of my chewing, reach over and, with his napkin, wipe at my mustache: green jelly on the cloth, the boy's green teeth grinning. Is it true that Ura talks more, and more easily, that he tells a little slower the story of a shunned man, an excommunicated woman, with a little more detail of a sickness among the hens and trouble with two bull calves, unfolding tales without rush, without worry, the way he would to a friend?\n\nEven Esther Hartzler doesn't make any tension. All supper, she asks not one question. Is she tired from the work? No. Her eyes are alive as ever. Even\u2014is it possible?\u2014more. Is her face flushed simply from the heat of the stove, of the steam coming off her plate? Are her cheeks more full, the creases around her eyes less deep, her lips a younger woman's? All supper I try not to look at her, and fail. There is something new in her that makes me grip my fork and knife to keep the tines and tip from rattling against my plate.\n\nI want to ask her, but even after supper, when Ura tells her that he and the children will clear the table while she goes out to my cart to see my wares, she says only that it is too dark. In the morning, she says. Even when I try with a look to tell her that I have brought for her her wish. Even when, following her into the kitchen, I whisper that I have already brought it in, that it is wrapped up in my bedroll _right here beneath my arm_ , she does not even glance behind her to see.\n\nFor an hour before bedtime we all sit in the kitchen, on the dining room chairs drawn into a circle, the whole family and me shelling peas\u2014plink, plink\u2014into a bucket in the middle, my eyes taking the whole of her in. Are her breasts even heavier? Is her belly more round? My heart rises like dough in the moist heat of my chest. But when we're done and Ura announces his departure for the outhouse and there is just her and me in the kitchen with overhead the footsteps of the children drumming to their beds, before I can ask her anything, she tells me, \"Bring it to your room tonight.\"\n\nLater\u2014so late it is maybe already morning\u2014she comes to me. She brings no lamp, no candle. There is just the creaking of her walking the upstairs hall\u2014I can feel each footstep on the bridge of bone in the center of my chest\u2014and the door opens, a blackness in the wall peeled away to show the pale light of the moon coming through the hallway window and in it her black shape. She is listening. I listen, too. From the bed against the other wall comes no rustle of sheets, no creak of slats beneath the mattress, only the same steady breathing that I have listened to for hours: Samuel, her older boy. Yonnie is next door in his sister's room, the two smallest of the family sharing a bed, her younger son's narrow cot left for me.\n\nSlowly, the moonlit doorway disappears, her dark shape disappearing with it into the darkness of the wall. For a moment, after the click of the door, we both wait, holding our breath. Then she is there. As soon as I feel her weight settle onto the mattress beside me, I know, somehow, that she is with child. And I want\u2014suddenly, overwhelmingly\u2014to touch her like I never have before, slide my palm over her belly, up the center of her chest along the same bone so thudded at in mine, rest my hand there between her breasts. But instead I reach down. Beneath the bed, my hand finds the bag, the cinch, begins to open it.\n\n\"No,\" she whispers.\n\nShe has bent close and the feeling of her breath on the skin of my ear mutes for a moment the sound.\n\n\"Don't you want to see it?\" I whisper back.\n\n\"We can't,\" she says.\n\n\"Yes,\" I tell her, \"I have it here.\"\n\nI can barely feel through the mattress the faint ripple in her body from her shaking her head. \"The light,\" she whispers.\n\n\"The light?\"\n\n\"It will wake him.\"\n\nI hush a laugh and keep it to a smile that I can feel spread over my whole face. And so that she can know it, I reach out and take her hand and press her palm to my mouth. I have been growing my mustache and her little finger pushes the hairs against my nose. It tickles. I can feel my breath wet her skin and I say into her moist palm, \"There is no light.\"\n\n\"Nein?\" she whispers, and I wish she would take my palm and put it to her mouth.\n\n\"It doesn't work like that,\" I say. \"It can't make light on its own.\" I let go of her hand. In the dark, I think I can just make out that she lifts it, that, perhaps\u2014I stare at the darkness where I know her face must be\u2014she presses her hand to her cheek.\n\nBut no. It must be pressed to her own mouth\u2014in eagerness, in worry?\u2014because her words are muffled when she says, \"Then get it.\"\n\nThrough the opening I have made in my bag, I feel the bottom of the wooden base, rounded and solid as a hoof, the sharp edge of one of its brass knobs, the short neck barely thin enough to let my fingertips meet when they wrap around it, the soft edge of the fleece that still covers the glass. I slip the mitten off. I lift it out.\n\nSitting up in the bed, I hold the Edison lamp in my lap. I can feel her searching for it, her gaze probing the darker part of the dark that is me, my shape that she knows must contain the bulb somewhere in it. I reach towards the vague shape of her, searching with my hand for hers. She is closer than I think; my fingertips touch her: such soft fabric curved over such softness beneath. Her breast. Barely touching, there my fingers hover, waiting for her to jerk away. Instead, touching the back of my hand: her fingertips. The fronts of her fingers lay themselves against the backs of mine. She presses.\n\nFor a long time we stay like that. I can hear, from somewhere outside, a whip-poor-will let loose its call. Above us, in the ceiling, a faint scrabbling: some small night creature's tiny nails. Across the room, her boy breathes on. Then I stop listening. Her other hand is on my knee. Brushing over the blanket, it drifts onto my thigh, up my thigh. I hold my breath. Her fingers find my hip, crawl up the soft rolls of my belly. Until with my free hand I find them and guide them.\n\nBeneath my fingers I feel hers curve around the bulb. She slides them first one way along the glass, then back the other, discovering its smoothness, its roundness, with as much care and awe as the first stroking of a baby's head. My own hand on her breast moves to match hers, my fingers feeling the nipple hard between them, hard as the spike of glass that rises off the top of the bulb. When she touches that, her hand stops. Mine does too.\n\n\"The glass blower,\" I whisper. \"Where they pull the globe off the pipe.\"\n\n\"I wish I could see it,\" she says.\n\n\"It needs another part,\" I tell her.\n\n\"To glow?\"\n\n\"To make the electricity. A generator.\"\n\n\"Electricity?\"\n\n\"A dynamo.\"\n\n\"Like this?\" And her hand that had been cupped over mine leaves my fingers alone on her breast. A touch on my chest. I let her fingers search. When they find the flasch, they stop. Holding it, she says, \"Isn't that what you told the children? That it is the electricity inside them, in their hair when they rub it, that makes it glow?\"\n\nI nod. She must feel it in the necklace. Because the next thing I know her breast has slipped away from my hand and her head is in the blanket at my lap, the hard, round top of her head rubbing back and forth against the wool, and then her face is pressed to me, her hair feathery all over my chest, tickling, tingling. I can see it floaty, wild, in the faint green light that is the phosphorous in the flasch at my neck beginning to glow. In the pale green I watch her turn to look at the lamp on my lap. For a moment it is there, curved and pointed like the domes on the churches back home in Russia, the glass of the globe showing the wires inside like a casement displaying some rare, tiny plant\u2014such delicate branches, such strange straight roots\u2014and I can feel through my nightshirt her cheek pressed against my chest. I can feel her smile. Slowly, the reflection of the flasch\u2014small green sliver curved by the curve of the glass\u2014fades. The lamp disappears back into darkness. There is the scratching of the animal from the ceiling. From across the room the breathing of the boy. I wait for the night bird's call.\n\n\"Where did you get this?\" she whispers.\n\n\"In Reading,\" I say. \"I found one last month\u2014\"\n\n\"No,\" she says. \" _This_.\" A slight tug on my necklace.\n\n\"My tatte,\" I tell her. \"He gave it to me when I left.\"\n\n\"Your home?\"\n\n\"Russia.\" I shut my eyes. Though it was already dark it seems I can feel each tug she makes even better, each movement of her fingers over the flasch.\n\n\"Did he write this?\" she asks.\n\n\"Ja.\"\n\n\"For you?\"\n\n\"He had it engraved on the cap of every bottle.\"\n\n\"What does it say?\"\n\nI tell her.\n\nThere it is: _whip-poor-will, whip-poor-will_.\n\nWhen it is done, she asks, \"He was an apothecary?\"\n\nThe bird lets loose its call again\u2014 _whip-poor-will_ \u2014and I want to say _is_ , I want to tell her he _is_ an apothecary, but I find I cannot make a sound.\n\n\"Ecclesiastes,\" she says.\n\nI nod.\n\n\"Just as a fly in the ointment...\"\n\n\"Dead flies,\" I correct her. \"Dead flies cause the ointment of the apothecary to send forth a stinking savor.\"\n\n\"So doth...\" she says.\n\n\"A little folly...\" I say.\n\n\"Him,\" she says, \"that is in reputation for honor and wisdom.\"\n\n\"Wisdom,\" I say, \"and honor.\"\n\nHer face leaves my chest and rubs at the blanket again. When it returns her hair is all static, again making the flasch glow. But when I look down she is not looking at the lamp. She is looking up at my face.\n\n\"Why did you leave?\" she whispers.\n\n\"My tatte?\"\n\n\"Your family.\"\n\nAbove, the animal scrabbles. Somewhere outside a cow lows. I let the glow die down.\n\n\"That story,\" I say. \"About the piss in the bags. It wasn't true.\"\n\n\"I know,\" she whispers.\n\nI tell her, \"There was a soldier, a boy, Avrom, who... It was so cold. On the coldest nights it was always us put on guard duty.\"\n\n\"You and Avrom?\"\n\n\"Any of us,\" I say. \"Us Jews. It was so cold the snot in our noses froze into tiny knives. They pricked our nostrils. Our noses were always bleeding. And the blut if it froze on your lips would take the skin away when you peeled it off. This is what happened to this Avrom. And the next night, his mouth was so raw, he asked the ofizir to please not make him go out. The ofizir said to him he would give him something to make him warm.\"\n\n\"Nein,\" she whispers.\n\n\"All over him. Some of the other ofizirim too. Until he was soaked. It was warm. He was steaming. For a moment. Then they made him go out.\"\n\n\"Don't tell me,\" she says.\n\n\"I found him.\"\n\n\"Don't.\"\n\nBeside us, across the room, there is the slow, steady breathing of her boy. I can tell she is listening to it because I feel her breathing slow again to match it. I try to make mine match hers, too.\n\n\"You didn't think you would live,\" she asks, \"till your time was done?\"\n\n\"In the army?\"\n\n\"Till you got out?\"\n\n\"I am a Jew,\" I tell her. \"There is no getting out. We are drafted for life.\" I try a quiet laugh. It comes out a loud breath. Her head shakes with it against my chest. \"That's okay,\" I say. \"We weren't expected to live long.\"\n\n\"So you fled?\"\n\n\"Ja.\"\n\n\"How?\"\n\n\"With the help of my family.\"\n\n\"How?\" she whispers, again.\n\n\"You know,\" I tell her, \"my name is not really Yankel.\" The cow has quit its lowing. The bird must have flown somewhere else. The ceiling above us is silent.\n\n\"What is it?\" she says.\n\nAnd, in the quiet, before I can say anything, we both hear: \"Mamm?\"\n\nAcross the room, against the dark wall, I can just make out his shape. Is he sitting up? Is he still even in his bed?\n\nThen she is off of mine, her weight gone from the mattress, her hand gone from mine, her hair from my chest, the touch of her face, of her breath, gone. Her shape disappears into the blackness at the other side of the room. Listening to her whisper to her son, I try to find her figure as if my eyes are connected to my ears. But I cannot. So I can only listen to her talk to him, until he is quiet again and I can hear the soft rustling of her moving around his bed, tucking his blankets back in.\n\nI say nothing until again there comes in the darkness across the room the strip of moonlit hallway. Then, before she can shut the door, I say her name: \"Esther.\"\n\nShe pauses.\n\nI wait.\n\nFinally: \"Ya, Mr. Yushrov?\"\n\n\"Do you have another?\" I say.\n\nHer silence, her waiting.\n\n\"Another blanket?\" I ask. \"Do you perhaps have a spare one for me?\"\n\nThis is how it was when I found him: He was pissing on his own hands. It was a clear day, windy. I wasn't supposed to relieve him from his shift till night, but as soon as I could gather enough warm clothes I snuck out. It wasn't long before I saw him. A dark figure against the glaring snow. He was standing away from the edge of the pine forest, out in the sun. In cold like that, the sun was useless\u2014a white sore in the hide of the sky\u2014but it showed the steam rising off his knuckles, off the spattered snow. By the time I got to him the steam was gone. The piss had become a thin sheen of ice on his hands. He couldn't move them. He couldn't put his poz back in his pants. He didn't know who I was. Below him, his mittens were two black holes in the snow. _Here_ , I told him, _I have dry mittens for you_. He didn't want them. He said his hands felt much better now. When I tried to make him put them on, he swung at me. The movement toppled him into the snow and the noise was terrible: all his clothes were stiff with ice; they crackled as he crushed them beneath him. He lay there, in the snow, his hands bare, his beard clumped with ice, the skin of his face the bluish gray of overcooked meat left out to chill. It was puffy as if the ofizir had beaten every inch of it with his fists. I tried to lift him. He scrambled away. And, as I watched, he began to take off his clothes. That was the worst thing: watching him, unable to move his fingers, unable to make his arms work, struggle out of his coat, flail and tear at his belt, shake his hat off, attacking his clothes with his teeth like some rabid animal turning on its own flesh. I tried to stop him. I failed. The last I saw of him he was staggering away, naked, a furious dark shape of a man, and then just a dark shape, and then nothing. What was he charging after? What did he think was after him?\n\n_Why did you leave?_ she asked. I could say because my mamme told me to go. That summer, when for the first time in a year she came to the camp to see me, she heard it all. And she wanted me gone from there. I could say because my tatte made me, or simply because my breuder gave his life to me so that I could. But how in return can I answer anything but why? Why would I do such a thing if there was any answer but that I had no choice at all?\n\nSo, yes, there is the story of my tatte and the cart full of hay. Yes, of me buried in it, too. The dogs at the checkpoint, the guards at the gate, the sweat of my tatte, the bayonet stabbed into the piles of grass, into my thigh, of my clenched teeth and no sound, of it not mattering\u2014the blood on the blade gave me away\u2014of blackmail and bribes, and finally, at the border, my breuder's papers, my breuder's name, yes, there is all of that, but why would I tell it? No. Such tales of the past are better left to the places where they happened, where they matter, where people need them, lands where men walk in the light of lanterns held by those behind them, unable to see farther ahead than where their shaking shadows meet the dark. Here, in America, what does it matter if behind all is black? Ahead, there is always at least a crack of light. Here, what matters is to head always towards it.\n\nThese are the long days of summer. The wineberries line the road. The sun lights them up, bright red buttons, and they do taste like wine, like drops of red wine warmed in the sun. Who else is so lucky as to lunch on them every day? Sometimes I even have half a loaf of fresh-baked bread I traded for that morning. Sometimes a jar of milk from the night before. Resting in the shade of a stone wall, or beneath a butternut tree, I can sing to myself if I wish, or slip silent into my thoughts, or simply sleep, and there is no one there to mind. The corn rises on either side of the road so high that sometimes, pushing my cart along, I cannot see above it. Then, it is as if I am an Israelite on a path opened up through a green sea, always opening before me, mile after mile. The steepest hills I'm grateful for: they keep my legs strong. The hottest days only make me look forward to the night. The nights I sleep where I want: on my own beneath the cart if I'm feeling solitary; if I want company, there is always a family happy to give me a bed. A dozen to choose from. A hundred if I plan ahead. A thousand in a year. And they wonder why I whistle and sing. Even, sometimes, I do it while in the act of making love. Yes, I have a lover, too. More than one. They like me. They like the songs I sing, the gifts I bring them. They give me deals: half off, two for one. And when I'm in town to restock, they let me lease a room above their rooms. And when I'm on the road, they even collect my mail.\n\nWhich is how, one day, almost five months since I last saw her, I get the letter I have been waiting for. _Monday the 19th_ , it says. It says: _Ura_ and _overnight in town_ and _dynamo_ and _come after eleven when the children will be asleep_.\n\nSo here I am again, on the hill overlooking the Hartzler farm. It is raining so hard that when the lights begin to go out in the windows of the house they seem doused by the downpour, like campfires, drowned to blackness. But, of course, it is Esther blowing out the lamps. One by one, the windows go dark\u2014puff, puff, puff\u2014until the whole house is gone, the darkness risen around it like the waters of a lake. All around me it sounds as if the waters are rushing in, too: from here to the horizon, the drumming on the leaves of the corn is deafening. Closer, beneath it: the gurgling of the gullies dug from wheel ruts in the road. The water sheeting off the brim of my hat, banging away at my shoulders beneath the slicker, splashed by the stomp of a hoof in the mud. A hoof? A hoof. My mule. I have one now, an old mare with bucked shins and worn-out teeth and large twitchy ears that when, on the most steep hills, I hop off my new wagon to walk beside her and urge her on, talking into them, these ears turn to me and seem to grow larger and sometimes I reach out and stroke them. When I nap too long in the sun it is her soft, old muzzle that wakes me. I have named her Reba. \"Reba,\" I call to her over the noise of the rain, \"let's go down.\"\n\nAdd to the sound of the rain a clanging and a creaking, the splash of the wheels. It is a small wagon, but neatly organized as my discarded cart, as my rucksack before it, small drawers of hardware at its front, and, along its sides, wider ones of neatly folded clothes, shelves to store the iron rods and stew pots too heavy to bring before, slats at the back to pin upright ax handles and shovel heads, feather dusters and brooms. Now, I can stay out on the road for twice as long, take requests, special orders. Beneath the board on which I sit I keep a spot free just for that. Jolting down at Reba's careful clop I can feel, knocking there, the great weight of the generator.\n\nWhat a pity it is dark and everything is under the tarp; how I would have liked for her to see the cart, the mule, and\u2014on the teamster's seat? Become the teamster?\u2014me. With a mustache to fit the part. All the last months I have grown it, from the mustache of a minor man to one of a giant\u2014a Mr. Tesla's small dark patch to the huge walrus wonze of Mr. Westinghouse\u2014until, by the end of the summer, it surpassed even the mustache of the man who next will be the president.\n\nAnd the day before I bought the cart, I took what had become of my face to the barber. I said to him, _Cut just a smidgen_. I told him, _Shape for me the mustache of a millionaire!_\n\n_Mr. Yushrov_ , he said to me, _if there's one man I know made of the stuff to make it come true..._\n\nThe next morning, riding out of Reading on my wagon, behind my mule, I felt it already had. It was so early the new electric lights along the street were still glowing in the dark, and as I passed between them I thought how young I was when I first came to this country. A kinde! I couldn't even grow a mustache! So long ago: from the ship the city glowed, still flickery with all the thousands of flames in all those thousands of lamps, but in the torch of the new lighthouse\u2014huge, metal, shaped like a woman\u2014the glow was so steady. It was the first electric light that I had ever seen. How fast time goes! One day I am a lighter of lamps, the next the lamps light themselves. Now, even the smaller cities\u2014Reading, Allentown, Lancaster, Harrisburg\u2014are electrified. At night, wandering between them on my wagon, I can tell where they are by the lighter patches they make of the sky. Between them it is still dark, all blackness of hills and fields and houses gone to sleep. But soon that will change too. Town by town, house by house, until the whole country will be lit. I have seen the way the light spreads from city to city along the coast, inland road by road, like moonlit foam frothing on top of a rushing flood. It is the places between, the patches of darkness, that will be drowned. It is the people like the ones who buy my goods, who hate my mustache because they cannot see that it no longer means what it once did in a world that no longer even is, it is them\u2014the Virklers and Waglers, the Klopfenstiens and Hartzlers\u2014who will go under with it. But I, I will do the only thing there is to do: I will swim to the top, ride the foam, get out in front of the tide.\n\nFor now, though, the places between are still dark, and this night I am in one of them. Maybe one day we will carry our electric lights with us even on our wagons, but not yet. I lead Reba through the rainy blackness towards the dark hulk of the barn, letting her gauge the route in her mule's way, until we are on the side that is farthest from the house. Luckily, it is leeward, too. The rain slashes at the corn\u2014it sounds like a sea swelling and churning\u2014but leaves us quietly dripping where we've stopped. I get down, feel for the front corner of the tarp, untie it, reach in, heave out the dynamo. Holding it like a baby to my chest, I leave the new cart and old Reba at the edge of the rain, and carry the generator into the barn.\n\nIn there, it is even blacker. I listen to the cows shoving each other to get out of the way of whatever has come through the door. I set the dynamo on the ground, reach to my chest, and pull from my shirt the flasch. Any daylight the phosphorus might have stored has long ago leaked out. I try rubbing it on my head, but my hair is wet. It gives me nothing. In the darkness, I turn, feel for the door, slide it shut. I drop to a crouch. Shuffling, I pat the ground for a place thickest with hay, least churned up by hooves. By the time I have reached it, I have lost all sense of how far I am from the dynamo. I sit. I wait.\n\nNot long. At first, I think it is some lightning bug, flicking on and off, on and off, the only speck of light in the dark. But when it appears again it is right outside the walls\u2014a quick glimpse of fire between the slats, flickering, disappearing, flaring again\u2014and I know it is her coming around to the door.\n\nThere is the moan of wood, the groan of the wheel on its track. There: her lantern held up as if to light me. But it shows her. From the rain her nightdress clings to her body. Beneath it, her breasts hang heavy as if they, too, have soaked up the wet. Her belly is huge: the fabric sticks to it so that the lamplight gleams off the point pushed up by the press of her navel.\n\nI rise to meet her. \"Esther,\" I say.\n\nShe looks as if she is about to say my name, but instead she turns her face to the dynamo on the ground. \"Is that it?\" she says. \"It looks heavy. You didn't rent the wagon just to bring it out to me?\"\n\n\"The wagon,\" I tell her, \"is not rented. I own it.\"\n\n\"Oh?\"\n\nI cannot hide my smile. \"And the mule.\"\n\n\"But only this spring,\" she says, \"you had barely enough to buy a cart.\"\n\n\"That,\" I tell her, \"was before farmers' wives began paying me to bring them dynamos.\" And I walk in my wet boots across the muddied hay to the one I have brought for her. I know she follows because the lantern light comes behind me as if dragged by my shadow. The cows jostle and shove to get out of our way, pushing against the wall until they surround us in a circle. The heavy scent of their wet coats, their warm shit. The sounds of their shifting. From the darkness the lantern draws their large and wondering eyes.\n\nDown I crouch on one side of the dynamo. Standing on the other, she reaches into her stretched-out housecoat, and for a moment I imagine her drawing from it a newborn child, bloody and bawling. But it is only a blanket wrapped around what I know must be the bulb. Lowering herself to squat, she sets her lantern on the barn floor, and hands to me the Edison lamp.\n\n\"Good,\" I tell her, tell her to watch while I connect the generator, so later she can do it on her own. Slowly, I uncoil the cord, I separate the wires, I twist them together, all of it slowly, to let her eyes take in each part of each movement of my hands. But when I look at her, her eyes are on my face. I feel, just below my skin, the blood tingling.\n\n\"Are there so many?\" she asks.\n\n\"Dynamos?\"\n\n\"Farmer's wives,\" she says, \"who want them.\"\n\nBy the time I have finished shaking my head\u2014a silent no\u2014my fingers are touching her lips, though I can't imagine reaching to her like that, or that she would let me. But there they are. And\u2014is it possible?\u2014there is the moist softness of her lips touching the ends of my fingers, untouching them again.\n\n\"I borrowed the money,\" I say.\n\n\"Debt?\" Her lips make the word against my fingertips.\n\n\"To move ahead,\" I tell her. \"Sometimes it is necessary\u2014\"\n\n\"From what?\"\n\nShe draws her lips away. I draw my hand back.\n\n\"Sometimes it is necessary,\" I say again, \"to take such a risk.\"\n\n\"Towards what?\"\n\n\"The future,\" I tell her. \"From the past.\"\n\n\"What about the present?\" she says. \"Isn't it the present that counts?\"\n\n\"Nein,\" I say, but before I can say more, she leans forward, across the generator, putting down both her hands to steady her against the awkward swinging of her belly's weight, and kisses me. Her breath is sour from sleep. I close my lips around it, and hold it in my mouth, letting it roll over my tongue like smoke.\n\nMaybe she thinks I am not kissing her back, because she takes her lips away, moves her face away again. She says, \"Are you sure?\"\n\n\"Ja,\" I say, and her eyebrows rise, her eyes seem to laugh. I go on, \"Because the present is like the dawn, or the dusk\u2014\" But before I can explain, her laughter leaves her eyes and finds its sound and comes out on her breath. The cows startle, mill about. In the quiet after, there is only her smile in the oil lamp's glow as she says, \"Make the light. Go on, you strange man. Show me how the electricity makes it work.\"\n\nAnd so? I place my hands on the handles. I begin to crank.\n\nInside the blown glass globe something stirs to life. Not a movement, not anything we can even see, but a feeling, a life, that just a second ago in the barn did not exist. Now, it is there, and with each crank I give the handles, it is growing. Even when the curled wire inside the bulb begins to glow it feels as if that is not the thing itself, but just the part that we can sense with our eyes. I glance across at her. Her eyes are on it, wide, unblinking. And as it brightens and brightens I watch her pupils dwindle and dwindle until it seems they must entirely disappear and leave her eyes only the blue-green irises filled with the yellow blinding light. She raises them to me. They are gleaming with tears. From the light? It cannot be from sadness: she is smiling. Smiling in such a way so that I see in her face, stronger than ever before, her daughter. She leans forward and lifts the glass of the lantern she had brought and blows the flame out. The brightness of the electric bulb seems to leap another notch. And in it, her smile leaps, too.\n\n\"Faster,\" she says, and even her voice sounds like a little girl.\n\nI speed up my cranking. The bulb glows even brighter.\n\n\"Faster,\" she says again, and, struggling to her feet, she stands above me, above the glowing lamp, her skirts lit and the bulge of her belly lit, and her top in its shadow, and above that her face, lit from below, smiling that strange kinde's smile. Then she begins to clap.\n\nThe sound shakes through the barn, panics the cows, knocks about the rafters, startles the birds that flutter in the far reaches of the light\u2014clap, clap, clap\u2014slow as if it comes from my own cranking hands. Until she begins to speed it up\u2014clap, clap, clap\u2014and I speed up to try to match her.\n\nMy arms are already beginning to burn by the time she starts to dance. I have seen the steps before\u2014at Amish weddings, at celebrations when the boys come home from their year away from the farm\u2014but never like this: her belly thrusting and swinging, her hips jerking with its strange sway, her breasts flung this way then that. She dances with the unguarded joy of a kinde, but with all of her woman's body to make it known, and, watching her, I begin to sing:\n\n_Bay mir bistu sheyn_ ,\n\n_Bay mir hostu heyn_ ,\n\n_Bay mir bistu eyner oyf der velt_.\n\n_To me, you are lovely_ ,\n\n_To me, you are charming_ ,\n\n_To me, you are the only one in the world_.\n\nI do not know where the old words come from: I have not heard them since I was a boy, in the kitchen with my breuder playing the flute he carved, me clacking with spoons, while our mamme, dancing from woodstove to basin to chopping block, kicked out her feet, threw back her head, and sang.\n\n_Bay mir bistu sheyn_ ,\n\n_Bay mir hostu heyn_ ,\n\n_Bay mir bistu tayerer fun gelt_.\n\n_To me, you are lovely_ ,\n\n_To me, you are charming_ ,\n\n_To me, you are more precious than money_.\n\nOnly when my arms are shaking, when my shoulders feel as if they will begin to glow like the wires in the bulb, only when my breath is too shallow to get out the sound, do I stop. The handles roll forward a moment on their own. Her clapping echoes, echoes. Her body slows. In the quickly fading glow of the bulb I stand to go to her, but she is already lowering herself to me and we meet strangely crouched beside the lamp, balanced for a moment, it seems, by our mouths together. Then the light is gone and we are toppling to the ground.\n\nIn the dark, close to my face, she whispers, \"Why like the dusk or dawn?\"\n\n\"Because,\" I say into her mouth, \"the present is always about to change.\" Against her neck, I say, \"Even every second, it is actually already changing.\" Into the soft skin between her breasts, I tell her, \"But the difference, the beauty, the beautiful difference is that we get to choose.\" I whisper into her neck, my hands lifting her skirt. \"Whether it becomes darker\"\u2014her hands unclasping my belt\u2014\"or more light.\"\n\nMaybe it is the word that makes us notice, maybe it is the way her belly blocks my hips from reaching hers, or the fact that when her hand reaches down and holds me, I am not ready, but, whatever it is, her other hand reaches to my flasch and touches it and we both stare: it is glowing. Around us is the quiet of the milling cows, and the last fluttering in the rafters of the birds, and on the roof the drumming of the rain.\n\nWith her free hand, she gives the flasch a little push. From her chest to mine it knocks. In the faint green glow, her lips look like they will say something, they look like they will say my name, and I think even for a moment I hear them say it. But I don't; they don't. Instead, they say, \"What is it?\"\n\n\"I don't know,\" I tell her. \"I'm sorry.\"\n\n\"Nein,\" she says, \"I mean I want to know what is your name.\"\n\nBut I have hidden my face in her breasts. I back my hips away from her hand. Her hand lets me go. I want to think that my breath simply feels excited against her skin. I want to think that what she feels between my face and her breasts is just the wetness of our sweat. But when her hand touches the back of my head, it is not a lover's touch. It is not even a woman's. It is a parent's, the way she might put her fingers in the hair of her young son, and I press my nose into her flesh, press my wet cheeks into her, and I am glad for the loudness of the rain.\n\nHer fingers stroke my hair, the backs of my ears, softly, back and forth. \"Hey, hey,\" she says, cooing. \"Tell me. Tell me what she was like.\"\n\n\"She,\" I say into her chest.\n\n\"Ja,\" she says. \"Tell me.\"\n\n\"She had eyes,\" I tell her, \"that were like lying on your back in the forest. Like lying on your back and looking up at all the leaves. And then, a breeze. And through them, the sky.\"\n\nBy the time I lift my face from her flesh, unstick my cheek from her breast, unpeel myself from her, the flasch has lost the last of its glow. I slide off her into the muddy hay. We lie there, listening to the rain. Only when I sense her moving, that she is about to rise, do I speak. \"My name,\" I tell her, into the dark, \"is Shimel.\"\n\n\"Shimel,\" the dark tells me back.\n\n\"Shimel,\" my mamme said, \"we must.\"\n\nI was seventeen. She was... She was... Can it be, she was not much older than I am now? My tatte, forty? My breuder? Two years younger, always two years younger, as he will always be, until either he or I is dead. Unless he is already.\n\n\"We must, Shimel,\" my tatte said.\n\nMy breuder said nothing. Usually he was the one full of talk so late at night. Midnight, it must have been, because that time of summer the sky over Petroplavilsk stayed lit by the low strange sun almost all the way through the night. And, yet, I remember clear as I remember the heat of the tea glass burning the skin of my fingertips, the feverish clicking of my mamme's needles as she tried to knit from fox-hair yarn a hat to keep me warm, the ceaseless shaking of my breuder's leg, the worse trembling of my tatte's beard, just as clear as that I remember how, outside the windows of our home, it was dark. The panes, though, refused to let it in, refused so long as the lantern on the table between us was lit, so long as the window glass shivered with its flickering, refused just as I refuse to forget the eyes of my mamme, the fierceness of her short-legged waddling walk, the downy dark hint of sideburns I would feel on her cheek when she pressed the side of her face to mine. My tatte's beard grew high on his cheeks, and low on his throat, all the way down into his shirt, where it tangled with the hair of his chest, and he would bush it up with the backs of his hands and huff like a bear to make us laugh. His red lower lip was always cracked. His skin was splotchy, smelled of the chemicals of his trade. He had wanted me to follow him into it. He had wanted my breuder not to have to. My breuder, whose nose was like mine but straighter, whose mouth was as big lipped and chapped as our tatte's, whose eyes were like our mamme's but with the leaves brown like fall and the sky a wet gray, my breuder whose name was Yankel.\n\n\"Yankel,\" our tatte told him, \"go and get your papers.\"\n\nMy breuder who sat there with his fists on his thighs, his leg shaking, shaking. My breuder whose name, whose papers, whose whole being I took.\n\nFifteen years ago. Nine since the last letter I got. Four since the last one I sent. Such a long time now since I have known I will never see any of them again.\n\nOh, but such a beautiful world! Such a beautiful country. When next I return to the Hartzler farm the sun is a red flame on a wick so low that the road is wavery, oiled with its sheen, and to either side the walls of corn make a chimney glass of the sky. The fields are wizening, the ears drying on the stalks, the leaves already papery and brown. All across the land they are lit by that sun, glowing orange like embers blown upon by the day's last breath. Ahead, my mule's ears glow just the same. And down in the farmyard the barn is painted hot. Starlings explode above it, a flurry of sparks.\n\nShe does not know I'm coming. Or didn't. But there is the barking of the dogs; there, a small figure hurrying across the yard, grain bucket flashing among the flashing backs of the scurrying ducks, chicken, geese; there, the flare of the front door opening into the sun, the clap of it shutting. Will she wonder what I have brought? Will she guess? Would she even know what a battery is? Rattling down towards the house in my wagon I think, tonight, after I chase the children, after Ura hounds me about my mustache, after supper and dark, tonight we will go out to my wagon as if to see my wares. And I will show her. _What is it for?_ she'll ask. _For when I'm gone_ , I'll tell her. For when I cannot crank the generator for her. So she does not have to. So she can go out to the barn any midnight that she wants, connect the lamp. _It holds the current_ , I'll say. _In this_ , she'll wonder. _It stores the light?_\n\nBut when I pull up at the gate, no children run out to meet me. Just the dogs, whining, yapping, making Reba shift and snort. I am standing up from the teamster's seat, about to jump down and quiet them with scratches around the ears, when I hear through their noise the sounds from the house. Shouting. Ura's voice breaking through the dog barks, Esther's breaking through the boom and roar of his. Never have I heard such sounds from inside their home, never from him or her. It stops me in midrise. Crouched like that, my thighs begin to shake. Beneath all the shouting another new sound breaks: the wailing of the baby.\n\nThen, sudden as shoving off a hood, everything is twice as loud. The door is open. The shouting, the cries spill out. With them she comes running. Out into the red blast of sunlight, the white of her bonnet flaring like a match head, her white apron a pile of kindling caught fire in the dark pit of her skirt, her shoes flashing as she stumbles down the steps and hits the ground and comes at a rush through the mad flapping birds, the bolting dogs, her hair loosed and her face red in the sun, redder even than her apron or her bonnet, burning up. Behind her, I catch a glimpse inside the house\u2014one of the kindes crouched on the floor, arms covering his head; another pressed flat against a wall, a package held to her chest: the baby?\u2014before the view is blocked by the shape of Ura bursting through.\n\n\"Go!\" she shouts to me. \"Go!\"\n\nBut I see the redness now\u2014it is blood, spread down from her nose, or up from her mouth, or anyway smeared over half her face\u2014and how then can I sit back down and whip the mule away?\n\n\"Go!\" she says, but I am already leaning towards her, my hand reaching for her shoulder. She tries to pull away but I catch the fabric of her sleeve and, clenching it, I feel with my other hand for the whip. \"Go!\" she is shouting, and then I am shouting it, too, and there is the crack of the whip marking the word, and the wagon jerks forward. Behind her, her husband charges across the yard, his beard a bush aflame, his big hands whipping in and out of the red light as he runs. Her hand is on mine, then\u2014trying to unclamp my fingers from her sleeve? Trying to grasp onto them with her own? And I let go the fabric and grip her flesh instead, grip so hard I can feel the bones, and as she's shouting, as he's nearing, I lean back and haul on her arm with all the strength I have.\n\nStill, surely, it would not be enough if she didn't help. I could not with one arm lift her above the wheels onto the seat. But there are her shoes scrambling at the wood side, and her other hand must be clamped somewhere, because she doesn't fall. She hangs, struggling, as I shout for the mule to go, as I struggle to pull her the rest of the way up, as Ura reaches the wagon. I do not know if he has a hold of it with one hand or both, or if he is trying to clamber up, or plant his boots and pit his legs against the mule's, but the wagon jerks, there is a crack, and then the mule pounds forward again. When I look back, Ura is standing in the dirt path that leads down the hill to his farm, a shard of wood in his hand, bellowing at us something I do not hear. Could it be he is shouting \"Go!\", that we are all shouting the same word, and all meaning something different? The only meaning that matters is the one the mule hears. Her hooves hammer the path, her sides beginning to heave as the wagon clatters and shakes its way up the hill.\n\nBy the time we are at the top, Esther has stopped shouting. She has stopped making any sound at all. Behind us, there is the fading barking of the dogs. Beneath us, the shuddering of the wagon, the slowing creak of its wheels. Before us, the drumming of the mule's hooves, drumming and then clopping, and then just the steady tock, tock, tock of Reba's slow walk on the road. The sun is down. The cornfields shiver, crackle. Dusk slowly fills the spaces between the stalks. We ride beside each other until the fields are full with it up almost to the flowers at the tips. There is the evening star. Sometime after it appears, she reaches behind her back and unties her apron. She eases it, gingerly, over her head. She folds it in her hand and presses it to her face. It covers her mouth so that her words when she speaks are muffled and I have to ask her to speak again.\n\n\"Where are you taking me?\" she says.\n\nIn the last light, the cloth at her face is so white it is nearly glowing. It flaps behind her like a strange sail unfurled alongside her cheek.\n\n\"Away from there,\" I tell her.\n\n\"Shimel,\" she says.\n\nI wait for her to go on. But that is all.\n\nThe crickets are relentless in the roadsides. The fireflies are so thick\u2014all across the fields, above the high weeds, filling the open road around the plodding mule\u2014that it seems as if there is a light for every cricket, that their blinks are timed to each throbbing rasp, as if the sound is the cranking of a thousand tiny dynamos and the blips are the electricity leaping to the tiny bulbs. But what then lights the stars? There they come.\n\nThere they are.\n\n\"Shimel,\" she says again, a long time later, \"you know I have to go back.\"\n\n\"I brought you a gift,\" I tell her. I can tell she looks at me by the change in the sound of the cloth flapping at her face.\n\n\"I want to go back,\" she says.\n\n\"A battery,\" I tell her. \"For when I'm not there. To crank the dynamo. So you don't have to. It stores\u2014\"\n\n\"Ura broke it,\" she says.\n\nThe fireflies flash and disappear, flash and disappear.\n\n\"We can get a new bulb,\" I tell her.\n\n\"The generator, too,\" she says. \"He took a splitter shaft to it. The bulb and the lamp and the whole thing.\"\n\nAnd I can hear his voice _\u2014Covetous_ and _greed_ and _blasphemy_ and _modernity_ and _displeasure of God_ \u2014like a lapping in the darkness that appears each time a firefly's blip is snuffed out. I watch the spots where the lights had been and look for the next and watch that spot, and it is only when my eyes hold on a spot that doesn't disappear, a blip that holds, that I realize where I am taking us. There is another, nearby the first, and another and another, as still and permanent and glimmering as water's reflection of stars, the darkness unable to smother them any more than ripples could on the surface of a lake.\n\nI point them out to her as if she has never seen a town. \"Reading,\" I say. She does not say anything, does not even nod. She does not look away from my face to them. I point again. Again, I say the name of the town.\n\n\"Yankel,\" she says.\n\n\"Shimel.\"\n\n\"Take me back.\"\n\n\"Do you see them?\" I say.\n\n\"Yankel,\" she says, \"I want to go back.\"\n\n\"Isn't it beautiful?\"\n\n\"My baby is there. My children.\"\n\n\"They stay just like that,\" I tell her. \"Lit all night.\"\n\n\"My husband is there,\" she says. And then she says, \"Mr. Yushrov\u2014\"\n\nI make a sound in my mouth that quiets her. Or maybe it is the fact that the click of my tongue catches the mule's ear: the clopping quickens, the wagon jerks forward, the wheels speed up their groans.\n\n\"Mr. Yushrov,\" she says again.\n\nAnd this time I use the whip.\n\nOld Reba breaks into a trot. We sit there, jolting on the board, Esther's free hand gripping the seat, her other holding the starlit cloth where it flutters at her cheek, and I try to tell her over the noise what it will be like to gallop out of the dark fields we have ridden through and into the first light of the city's streets, under the first lamps, the sudden brightness that hurts the eyes, that makes them water and ache, and still how you can't help but gaze at the bulbs burning away, steady, unflickering, unwavering, through the night, but she only shouts at me that she wants to go back.\n\nOn her face there is a thing I have never seen before, not from her or any other. It is as if she is afraid of me, or of the answer to her question that she has somehow seen, that I have always known, that has all these years frightened me. What man who takes a chance simply because it is held out to him has ever made it farther than the next gate in the road? Who among us ever has no choice? In her face I see she knows it now, what my breuder must have seen when he gave to me his chance at life, when I took it from him: the wanting, the ruthlessness of wanting that in this world is the only kind of wanting that survives.\n\n\"I want to go home!\" she shouts.\n\nI whip the mule into a gallop.\n\n\"I want to go home!\"\n\nAnd I shout it at the mule, too\u2014\"Home! Home!\"\u2014until the sound of the hooves is like a team of horses gone wild in fright and the night behind us is filled with the crashing of my wares shaken loose and falling to the road and my flasch is smacking against my chest, over and over, so hard that I know either it or my breastbone will break.\n\nBeside me, she is crying.\n\n\"Look!\" I demand. \"Look!\" And I try with my shaking finger, with all that I can put into my eyes, to make her see how it was coming off of the ocean at night, the wind in your face and the waves rolling the ship beneath you and the salt crusting in your eyes and the first glimpse of the lights\u2014there!\u2014the island\u2014there!\u2014rising out of the darkness of the water, breaking the horizon, so many lights, so wondrous, bright, new.\n\n# [**LONG BRIGHT  \nLINE**](toc.html#ch2-R)\n\nThrough the window Clara could see the men: dark still hats huddled together. The only thing moving was their pipe smoke. It curled in lamp-lit clouds. Then\u2014a whoop!\u2014the clouds blew, the huddle burst, the hats were flying.\n\nOut in the street the gaslights seemed to feel her father's cheer; on her mother's face she watched them gutter.\n\n\"Look at him.\" The woman's grip was strong as any man's. \"How happy!\" But the fingers were bonier, worn to hooks. \"Look,\" she commanded. \"And tell me where he sets his heart.\" Then the grip became a shove. Her mother's _fetch your father_ and _that damn club_.\n\nThe Society for Aeronautical Enthusiasm. Sometimes, when she was sad, or scared, or simply felt the overwhelming weight of her own being, she would intone the strange words like an incantation: _Aeronautical enthusiasm, aeronautical enthusiasm, aeronautical..._ She said it now... _enthusiasm_... starting up the station steps... _aeronautical..._ shoe-clacking through the empty lobby... _enthusiasm..._ to the shut door... _aeronautical enthusiasm_. She knocked.\n\nInside it was all smoke and suit backs, elbows at her head level, her father bending down, face flush as drunk, but eyes clear, grin pure, whoop a straight shot of glee. He scooped her up.\n\n\"Fifty-nine seconds!\"\n\nHow long had it been since her father had held her like that?\n\n\"Eight hundred and fifty feet!\"\n\nLifted her so high? With each hoist and drop she felt her years shake off, seven, six, five\u2014her brother's age\u2014Larry in the corner watching, _this is what it's like to be him_.\n\nBefore her face: a piece of paper, some smiling stranger lifting and lowering it for her to read. At the top, the station master's name. At the bottom, that of the man her father called _their father_ : Bishop M. Wright.\n\n\"The _Flyer_!\" Her father raised her high again. Near the ceiling the air made her eyes water. \"The _Flyer_!\" He lifted her into the pipe-smoke clouds.\n\nBut she wasn't, wouldn't be. The balloon ride he'd won\u2014best guess at time and distance of the first flight\u2014was a prize he unwrapped on the cold walk home: how they would scale the sunset, skim beneath the stars, a Christmas present more miracle than gift. Just not for her. _Why?_ The basket size, the limits on weight. _Besides_ , he said, _ascending so high would surely swell that head of yours_. He tugged her braid. _No doubt big as the balloon itself_. Laughed. While around them little Larry ran in circles, whooping.\n\nOn Christmas Eve all she wanted was to stay up late enough to watch them float by above. But if she did, her mother told her, putting her heavy shoulders into the rolling pin, how could Saint Nick bring her her gifts? She spoke in sentences choppy with work: What did Clara think they were doing up there, her father, her brother, in that balloon? Airborne beside the sleigh, pointing out good children's houses, steering the reindeer towards the right roofs. Why else would they have had to do it on Christmas Eve? The last word pressed out by a hard push. Why else leave their women alone this one night a year she liked to share a little brandy with her husband, squeeze beside him on the chair, sing carols, hear _that sweetness in your father's voice_ , let her own loose just a little...\n\n\"But what about Larry? _He_ won't be asleep.\"\n\nHer mother set the pin down, crouched: her face suddenly level with her daughter's, her eyes strangely soft, her brow smooth as the dough she'd rolled, her hand ice-cold on Clara's cheek. From her forearm flour fell like snow. \"You know,\" she said, \"they won't see anything. Going up at night. Sweetie, out there it'll just be cold and dark and not one damn thing to see.\"\n\nThere was the whole world. Edge to edge. Lit by the stark stare of a full Yule moon. And, out in the white-bright yard, at the verge of the snow-glowing fields, a seven-year-old girl illumined, looking up. Behind her: the sleeping house. Above: the starry sky. In it: nothing moving. She watched until her eyes stung. No teardrop silhouette slipped across the luminescent globe. Would the men up there have lit a lantern? Would she see its wink? The stars had fled the moon for the rest of the sky, piled so thick upon each other she had no hope of picking out the one she wanted.... _But there!_ Could it move so fast? Careen across like that? Disappear in a blink before... and she was running, running in the direction the light had shot, running for the place on the horizon where she was sure she'd seen it come to earth.\n\nSometime in the night her father found her. She woke to his hands unclamping her huddled curl, hauling her up. In his arms she shook so much the stars seemed to rattle. The moon was down. His stumbling, his breath: he was drunk.\n\n\"Did you see us?\" His quaking grin. \"We went right over.\"\n\n\"No you didn't.\"\n\n\"Yes!\" His teeth a pale tremulous strip. \"Your brother waved! I showed the house to Saint Nick. His sleigh landed right there!\"\n\n\"No it didn't.\"\n\n\"You missed it? All that commotion? The hooves on the roof? Us caroling while we circled above? You truly missed it, truly?\"\n\n_Had she?_ His breath fluttered against her face, or her face shuddered beneath it. _Had she?_ Inside they waited for her: the presents, lurking beneath the tree, irrefutable.\n\nPapa, what did it look like?\n\nOh, magnificent! The white balloon, the light of the\n\nmoon, the stars so close!\n\nNo, Papa, what did it look like down here?\n\nOh, so vast, so small, so strange to think we all live out\n\nour lives down there!\n\nNo, Papa.\n\nThe world! Astounding!\n\nWhat did _I_ look like? Papa, what was it like to look down\n\nand see me?\n\nLike this, the way it might have looked to him had he been ballooning again the following Christmas Eve: a small girl sneaking out after supper, away from the dwindling singing, the laughter\u2014mother, father, brother, home. His eight-year-old daughter slipping past the last light of the candle-lit windows, beyond the hay barn, the snow-blown fields, disappearing in the blackness beneath the sky. And then: a spark. A golden bloom. Like this, Clara: a little fiery face looking up out of all the darkness of the fields, your little face flickering with the light of the oil lamp you brought, your breath a sunset's clouds, your eyes two glittering stars.\n\n_Aeronautical enthusiasm, aeronautical enthusiasm..._\n\nAll the winter of '05 she chanted it, silently, to herself, a spell to melt the snow, a wish for the rush of spring, and had he looked for her that March, that April, May, on the days when she skipped school, shirked chores, left her brother to play their well-house echo game alone, her mother on her knees wringing out the wash, had he looked, her father would have found her here: in the Scotch broom behind the split rail fence that bordered the cattle paddock they called Huffman Prairie, here, on her belly, behind a scrim of reeds, watching him.\n\nWatching him watch them. Two men in tweed coats and flat caps obsessed with some giant machine. Day after day, they worked on it, trained it like a horse, except\u2014when one at last would mount it, and the other, joined by a helper, would give it a mighty running push\u2014it flew. Its muslin wings stock still, its engine roaring like a mud-stuck truck, its driver clinging to the controls, and yet it flew surely as any bird. A bird aloft with a man on its back. Its shadow swept the stampeding cows, the whooping men. Her father too. ( _Tell me where he sets his heart_.) He seemed to shiver in his clothes. _Here_ , Clara thought. Rose on tiptoe, raised a hand to his eyes. _Here_.\n\nSometimes he would join the others running down the field after the roaring bird. Sometimes he would help push it off the earth. If they were far away, she might not manage to tell him from the rest. She might imagine it was him up there guiding that flying machine. How she wanted to ask him! What did it feel like? What _would_ it: that wind in _her_ hair, that sudden lift below _her_ belly? But all June she lay flat on the ground, behind the weeds, keeping quiet.\n\nUntil, one day in mid-July, she screamed. How he heard her over the crash, she didn't know. But when she peeled her palms from her eyes, there was the wreckage slammed against the ground, and two far figures running: one towards the crumpled machine, the other towards her.\n\n\"No, no, no, no,\" her father said, and, \"can't\" and \"daughter\" and \"not how the world works. Don't you ever think of anyone but yourself? How could you make her do your work, your mother worry, poor sick woman!\"\n\nShe was? How had Clara not noticed it till now? She would have crouched beside the bed and shut her eyes and said the incantation, but learning what the words meant had leeched the magic from them. _Enthusiasm_ : merely eagerness. _Aeronautical_ : only relevant to things that lifted off the earth, took to the air, soared for the heavens.\n\nThis was the last thing her mother said to her, one word: _selfish_ , or _selfless_. She wasn't sure. She only knew her mother had turned in bed, clutched her daughter's face, held her gaze, said the word with such vehemence Clara flinched at the flecks of spit. Her mother's fingers gripped her skull, eyes bore into hers. But did she mean herself or Clara? Was it an apology or reprimand? Warning or wish?\n\nShe was at Huffman Prairie the day her mother died. September, windless. No figures in the field, no _Flyer_ for them to push. She walked out there alone. At the end of the narrow track, a launching dolly lay overturned. Behind it, in the wet green grass, the bleached board looked white as bone. She lay down on it, aligned her spine with the rail. Stared up at the sky. Spread her arms.\n\nThat evening, returning to her father's stricken silence, her brother's sobs, a home whose walls had become more thin, she would wonder if it was something you could see: the soul ascending. If that day she had simply been watching the wrong place in the firmament.\n\nFall had come by the time her father brought her back to the field. He told her it was her mother's dying wish.\n\n\"To see the _Flyer_?\" she asked.\n\n\"For you to see it,\" he said.\n\nHuffman's was crowded. Farmers and friends, the entire Society, even the old man her father called _their father_ , come out to watch the flight. That day, the machine made it above the windbreak and kept on, became a hawk, a kite, a sparrow, a spec, was gone. But she could hear the sound somewhere, coming back around, circling her, homing in.\n\nHer father seemed to have forgotten how to steer. She would hear his foot scuffs wandering the house, floor to floor, gliding room to room, his tail rudder busted, blown by winds only he could feel. Her brother: a brooding boy, so serious\u2014seven years old and up first to fry the bacon, last in from the barns\u2014so careful cleaning the plow horse's hooves, so watchful over the hung tobacco for the slightest sign of rot. He spent his eighth birthday on the back of a cart, alone in the cold, forking fresh hay to the shivering Jerseys, worry frozen on his face.\n\nWhile Clara spent her tenth in her room writing a letter to a woman she'd never met: _Dear Mrs. Miller, Can you please describe your ride in the dirigible? How long before you will go up again? Would you consider taking along a girl? I'm still quite small, and very light_. By the time she was a teen the walls around her were plastered with clippings, posters, photographs: Lieutenant Lahm alighting from his balloon in the field at Flyingdales Moor, the Wright brothers on the racetrack at Le Mans, the note she got back from Hart O. Berg ( _You never know... I'll ask Orville... Maybe one day you'll pilot your own!_ ), the first woman ever lifted off the earth inside an aeroplane signing her name with an _O_ bold as a daredevil's loop. And the daredevils themselves! Glenn Curtiss of that lush mustache, sly-eyed Arch Hoxsey with his wry smile. Her first pack of cigarettes she purchased solely for the Ralph Johnstone card inside: goggles raised, chin strap sharpening his jaw, that slight swerve she so loved in the cock of an eyebrow above his steady gaze. At school, the girls talked of which boys they'd like to date. She smoked her Meccas, kept quiet. The boys discussed their options for careers, mocked her when she mentioned Raymonde de Laroche. ( _Her license!_ she shouted. _That's France!_ they jeered.) _Why do you even bother?_ her few friends asked, when, already fourteen, she fought for a seat in the science class. _You know this is your last year, anyway_.\n\nStill, the day her father sat her down it was a shock. She had tried her best to bring him back, curled beside him on the couch reading aloud the news (Bl\u00e9riot across the English Channel! Latham breaks a thousand meters!). She brought him to her room to show him each poster she put up\u2014the Los Angeles expo, the Grande Faime d'Aviation de la Champagne: skies aswirl with aviators thick as bees\u2014hoping to rekindle the heart her mother had heard in his whoop that long-ago first flight. All spring she'd tutored boys in science class, saved just enough: two tickets to the aero meet in Indianapolis that June!\n\nBut here was her father shaking his head. \"You're a woman now,\" he told her.\n\nHer brother, across the table: \"We need you here.\"\n\n\"And what,\" she demanded, \"if I go anyway?\" To Indianapolis that June, or back to school that fall, or away forever.\n\n\"This isn't New York,\" her brother said.\n\n\"This is your home,\" her father told her.\n\n\"Sis\"\u2014Larry reached over, touched her arm\u2014\"can't you see this is your job?\"\n\nIn the last year, he had become taller than her.\n\nWhen summer came she stole away to Asbury Park. Walked to the train station, bought a seat east, saw the ocean for the first time, the boardwalk, the beach, slept on sand, discovered the effect her shoulders, her smile had on men, snapped up a ride, a ticket, was there on the field when Brookins crashed into the crowd, when Prince plummeted six thousand feet, watched with all the others the failure of his parachute, felt the communal shudder when he hit, would sometimes feel it again, back home, alone in the kitchen cracking the back of a bird she'd serve to her brother or sewing a split in her father's yellowed long johns or stepping off the train onto the platform of her small Ohio town the day that she returned. But for one August night, near the air show's close, standing at the edge of the Atlantic, looking up, she had been struck by a sudden sureness that it would be alright. The moon. The Milky Way. The Stardust Twins swooping through. That was what the papers called them after that first night flight, Johnstone and Hoxsey circling each other in the lunar glow, their pale-winged biplanes soaring smooth as owls. And her, beneath them, swept by the peace of certainty. Neck stretched back, face flat to the sky, she knew it: she was not meant to be up there; she was meant to be down here, here like a cairn seen from above, a landmark, her.\n\nLet Bessica Raiche climb behind the controls that October.\n\nLet Harriet Quimby claim her license from the Aero Club.\n\nHere is Clara Purdy, standing far out in her father's field, surrounded by electric lamps. All these months she has collected them, repaired ones given to her for free, purchased others with earnings made from sewing piecemeal, stolen the rest from her own home. Here is the cord she's spliced, snaking away to the windmill her brother installed just this past September. It is November now, cold and clear and night. No snow on the ground, the fields crisp and brown with a fall drought. She stands in the high grass, waiting for wind. Behind her the turbine is still, the generator asleep. Above, the sky is breathtaking. What is the chance someone flies over tonight? What does it matter? For the first time she understands what she felt in August: it doesn't. The wind will gust, the turbine will whir, the charge will shoot through the cord, the lamps will all light up, and she will know how it would look from high above\u2014the concentric circles, bright bulbs swirling inwards to her here at the center\u2014and that will be enough. The grass heads stir against her shins, dry as tinder. The filaments await the spark. Here it comes.\n\nAfter the fire, they lived in a fourth-floor flat in Dayton, the three of them cramped close, her father and brother away all day in the old bicycle building, assembling engines for the new Model B. The intricacy of understanding, the advanced industry, the demands of such a task: the Wrights wouldn't hire her. Instead, she spent her sixty-hour weeks on the National Cash Register line, setting small round buttons\u2014red number five, red number five, red number five\u2014in their place on the machine. At home, her father ate whatever she cooked in his usual silence. Her brother, chewing, wouldn't even look her way, ever unforgiving of conflagration she had caused. That spring, when flames consumed a shirtwaist factory in New York, he slid the _Daily News_ across to her ( _a hundred and a half dead_ \u2014jabbing his finger at the newsprint\u2014 _most of them women_ ). Beside it, her eyes took in a different headline: in India, a Sommer biplane had delivered the first airmail. _What_ , she thought, flushed with excitement. _What would come next?_ Her father, reading aloud one night, his voice aghast: a ship, an iceberg... He couldn't stand it, handed her the paper. She drew in breath. Was it true? Quimby had piloted across the Channel? Alone, a woman! _Eight thousand suffragists_ , her brother announced, slapping the paper, shaking his head, _marching on the capital like a bunch of Albanians trying to overthrow the Turks_.\n\nNearly a thousand days went by before the flood of 1913 gave her a whole week off. Alone with herself for the first time in years, she climbed the stairs to the roof, thunked down a satchel bulging with buttons. Every shift she'd swiped one round red number five and now, crouched high above streets deep with water, she lay them out with a typesetter's care. Above, surveying aeroplanes circled. She wrote them notes in big red letters: _COME CLOSER! LOWER A ROPE!_\n\nIn Austria-Hungary an airship floated, testing photographic tools. An army aviator tried out his loop-de-loops. Biplane, dirigible: ball of fire erupting in the summer sky. A few days later Archduke Ferdinand took a bullet to the throat. A few weeks after that, half of Europe was at war. In a few years, her brother too.\n\nMid-September, 1918: two thousand planes aswarm over Saint-Mihiel, blasting thick as scattershot across the sky. It was the biggest air battle the world had ever seen, and the next night, while her new husband finalized what that afternoon had wrought, Clara stared up past the strange shoulders (later, she couldn't recall if he'd removed his undershirt, if he'd still worn his glasses), at the ceiling of water-stain clouds and watched the dogfight in her mind: a welkin of dark specs swirling, the opposite of stars. If Abner Lowell noticed (later she would know of course he had) he said nothing (of course he wouldn't), just as the few guests\u2014his family, his friends\u2014had looked at her drained face and silently assumed it must be grief (her fallen brother, her heart-felled dad). A few whispered she must have married out of desperation (her new husband's bony chest, his paltry schoolteacher's wage), but they were wrong. Clara had married the man for his location.\n\n\"Near Toledo,\" he'd told her, shelling a hot peanut, slipping it into her palm. \"A little town you wouldn't know.\"\n\nShe'd chewed, flashed him a look of _try me_.\n\n\"Maumee,\" he'd said.\n\n\"On the Maumee River?\"\n\nLaughing, he'd coughed out, \"You!\" It was what he would later say was their first date. \"You've got to be a bargeman's daughter!\"\n\nSmiling, she'd swallowed, held out her empty hand.\n\n\"A little land,\" she told him, a few months later, after they'd moved up north. \"A little place of our own out in the country.\"\n\n\"But dear\"\u2014he called her _dear_ and _sweets_ and _Mrs. Lowell_ \u2014\"we have a house already. Right here, right around the corner from the school.\"\n\n\"Abner,\" she said, \"I grew up on a farm.\"\n\n\"But why so far?\"\n\n\"I want our children\u2014\"\n\n\"Why pick a place near precisely nothing?\"\n\nIt was true: the plot she'd found sat equally far from even the tiniest of towns. Grand Rapids, Whitehouse, Waterville. It wasn't even on the routes between them. But it was beneath the one she wanted, right below the one up there.\n\nMornings, she made him hot cornmeal muffins, liver and onions the way he loved, helped carry his schoolbooks out to the car, stood in the dirt drive waving. She kept the bills in order, the mousetraps empty, the dust down, herself up, a wife he'd want to come home to, a home he'd be happy to find her in. Except he wouldn't.\n\nAfternoons, after everything else was done, she'd change into her coveralls, head for the tractor shed. Alone in all the horse-worked county, the Allis-Chalmers had cost her every cent of life savings her father had left, and every day before dusk she would crank its engine over, hitch its thresher on, rumble out into her field. Or, as Abner called it, her canvas. The tractor he called her Big Bad Brush. In the summer, she painted with it, mowed her pictures into high grass. In the fall, she ploughed pen lines, the overturned topsoil dark as ink. Winter found her bundled in the aviator's coat and hat her husband gave her, long leather earflaps whipping in the wind while she made her etchings on the earth, her shovel a chisel, snow peeling away like curls of wood. She planted daffodils. Dug up the bulbs each fall, stitched them back into the dirt like needlework, watched her embroidery bloom sun bright each spring.\n\nFrom the sky, the airmail pilots watched it too. Twice a day they flew the route, eastward in morning, westward late afternoon. Perched atop the Allis-Chalmers, or kneeling in the new-turned dirt, or simply standing still in a swirl of snow, she would listen to the hum, scan the sky, smile up, wait for the dip of a wing, the tiny stick arm flung back and forth in a far-off wave. She would watch them dwindle away to Cleveland, morning awhirl in their propeller blades, watch them disappear towards Chicago, sunset on their struts.\n\nAnd home from his day at the Maumee Secondary School, hunched over the kitchen table doing his preparatory work, her husband would catch a flutter in the corner of his eye, look out the window: his wife in that distant field of daffodils, her breeze-swept hair all auburn fire in the late light, her cap lifted into the last of the sun, waving, waving. She always left his supper warming in the oven. He always let it warm till she was done, would come clomping in, shuck her boots, sit down to eat beside him. Sometimes, he'd go out to her. In winter storms, if visibility was bad, the pilots, searching for landmarks along the route, might fly so low their wheels seemed close enough to grab, the silver belly suddenly there tearing through the all-white sky, the aviator's face a flash of goggles, the airplane roaring by. He'd stand behind her, arms around her, feel the gust, the rush, the thrumming of the engine in the air.\n\n\"What's it this time?\" Abner would ask.\n\nShe'd tell him: President Roosevelt on his first flight, von Richthofen shot down over the Somme, the new airmail stamp. \"See there's the biplane, there's the '24', over there the 'CENTS.'\"\n\nAnd Abner would gaze at the indecipherable arcs in the grass, the random squiggles in the snow, the mystifying daffodils, and fill his face with what he hoped conveyed belief in her, faith that from above it would all be clear. Until she began digging portraiture in dirt. Eddie Rickenbacker. Bert Acosta. Jack Knight. Airmail pilots that might _right then_ fly over on the Chicago-Cleveland route. \"Do you think,\" Abner said, his voice very level, his eyes somewhere off in the field, \"that they might... I mean, that they really... That they could actually really recognize themselves?\"\n\nShe meant to nod, but instead found herself starting to shake inside his arms. Against her back, his chest shook too. Their laughter filled the field.\n\nOnce, after a fight:\n\nHim: Weren't kids the entire reason they'd moved out there?\n\nHer: That's just the way people without meaning in their lives try to make some.\n\nHim: And didn't she want that, too?\n\nHer: What did he think she did all afternoon?\n\nShe'd asked him, \"Why did you marry me?\" Because, he said, he loved her. \"What does that mean?\" He'd told her then he'd never seen someone so consumed by what most moved them, never been that close to such a burning need, wanted to assuage it or be burned up in it, to feel even a little of it in the warmth of what he felt for her.\n\nAbner was the one who brought her art books\u2014Klee, Kandinsky, Mondrian\u2014who suggested they drive four hours to Cleveland just to see a room bedeviled by Kazimir Malevich's strange blocks and bars. She didn't understand what stirred behind it, no more than she understood what stirred in her. But dragging the thresher in wild swaths, plowing scattered squares of earth, planting bulbs in shapes that seemed to suggest themselves, she was sure of this: it was a style that far better fit her tools. A tool itself that let her grasp at last at what she had begun to conceive of as a gift. A gift Abner had wrapped for her. The way he wrapped himself around her the night the postal service flew its first transcontinental flight.\n\nThat night in February of '21 the snow spilled down as if to douse their fire. They had lit it in the middle of her field, at dusk, no telling when\u2014or _if_ \u2014the plane would hurtle overhead. All day she'd waited for news, pulled open the Ford's door to retrieve the evening paper from her husband's outstretched hand: the last heard from the airmail pilot, he was headed straight into a storm above Cheyenne. Cheyenne to North Platte to Omaha to Iowa City to Chicago to right overhead: at night the only thing to guide him would be a few post office workers' flares, nothing to mark the path between but what bonfires a few farmers might keep alight. Abner helped her haul out the half-rotted boards, pour on the gasoline. They brought blankets, a bottle of bootleg, sat close to the fire. All around her: Abner's enfolding coat, his enveloping arms, the warmth of his breath on her cheek, of his cheek against her ear, of him waiting all the long night with her. Sometime after midnight, they lay down together between the blankets. Sometime before dawn, he fell asleep. Sometime after first light she woke her husband, straddling him with her heat, tenting him with her body, the bottom blanket rough on her knees, the top blanket blocking his view of the sky, her own eyes focused only on his face. \"I hear it,\" he said. \"Darling, I hear it.\" She shook her head, rocked on his hips, kissed him quiet.\n\nSo why did she still daydream pilots down? That they would see her wave, circle round, land in her field, take her up. In a de Havilland with a scooped-out second seat? Or curled on her knees in the bin behind the engine, the mail hatch sprung? A gloved hand on her cheek (she could smell the leather), the twist of her neck as she turned to see (she could feel her whipping hair) his goggled eyes, his chapped lips, her first kiss at five hundred feet, six, seven, a thousand.... Why, when she first heard of the postal service beacons\u2014fifty-foot towers erected all across the country, a trail of landlocked lighthouses flashing their specific signal (me, me, me) to pilots plying a sea of stars\u2014did she feel betrayed? They built one five miles away. At night she could see its blinking flare (here, here, here). Why when all those women (only one aviatrix shy of a hundred) came together in Curtiss Field to make their mark in the history of flight did Clara turn with even more determination to her own canvas?\n\nNow, she worked at night. She spent all the allowance Abner gave her on a single headlight for her Big Bad Brush, ate her suppers sitting on the tractor's seat, stayed out past the last window gone dark in her house. What sent her to the shed to sharpen thresher blades at the news of Earhart's first Atlantic flight? What about word of the woman's solo crossing kept Clara up till dawn mapping out her next work of art? Each one, each season, outdid the last, pushed her abilities to new feats of skill, scaled the atmosphere of her imagination. She clipped grass by hand, cut staggered banks in sweeping slopes, accomplished tricks of shading by varying stalk heights. She incorporated color in the spring, in winter watered carefully considered ditches to show the sky fleeting paintings made of glinting ice.\n\nAnd when, one summer morning in '37, the kitchen radio reported that the Queen of the Air had gone down over the ocean, was feared drowned, she found she couldn't breathe. She stood up from the table. It was a school day. The house was empty. The news announcer's voice seemed to cinch her throat. And, even before she heard the airplane's purring approach, she was fleeing into the field. Maybe it was her frantic waving, maybe the desperation in her face: this time the pilot swung around, returned, roared down to her.\n\nA Boeing Monomail, army drab, no room to sit anywhere but in his lap. The leather of his jumpsuit creaked. He had to reach between her legs to take hold of the controls. The noise was deafening, the tree line rushing. The earth dropped.\n\nThere was her world: the house, the empty driveway, the field where she did her work. The flesh on the back of her neck urged her _look at the sky_ , the giddy slide of her stomach told her _you're flying_. But she couldn't take her eyes off the small, and smaller, square of landscape that was her canvas below.\n\nAt her cheek, the aviator was saying something. His chapped lips brushed her ear. For a second, she pulled her stare away, glanced at him: his goggles were so smeared with engine grease she couldn't see his eyes, just the rawness of his sunburned nose, the wetness of his grin.\n\n\"Take me back,\" she said.\n\n\"Down?\" he shouted.\n\nShe shook her head. \"Back around, back over, I want to see it again.\"\n\nIt was the first time that she ever had. Till then, it had all existed solely in her mind. There, _there_ , if she concentrated hard enough she could forget the feeling of his hand creeping across her chest, if she fought the wind in her eyes and focused hard enough, she could imagine that there was nothing around her but air, that she was up there, flying, looking down, alone.\n\nMaybe it was the Depression. Early on she'd offered to plow her canvas under, grow vegetables instead, but Abner insisted no: even when parents lost their jobs, students still needed teachers. Though when, that autumn after her first flight, Lucas County consolidated its schools and proved him wrong, her homebound husband spent his days gazing at her field, his nights commenting on her progress, his energy in coming up with ways that he could help _\u2014Observe you from the roof, shout when you'reabout to lose the line. I know: a business in balloon rides! Listen, I'll write a letter to_ Life _magazine, to ARTnews, get you noticed!_ \u2014his whole self seeming to clutch at her work as if it could become in some way his. Maybe it was the fact that all his attempts to garner her attention finally did. Late in '41 rumors began to go around: her flowers sprouted in secret patterns; her tractor furrowed code; Mrs. Lowell was planting messages for Japs. That winter, at Abner's urging, she undertook a radical revision of her aesthetic. He brought home images of Far Eastern art, read her haikus. And in the fresh snow of the new year's first storm, helped her shovel a field full of brushstrokes:\n\nThere it was, black shovel lines in white, giant characters carved beneath the January sky. Two days later an army corporal showed up at their door, watched them while, with the Big Bad Brush, Clara plowed their work away. Afterwards, Abner admitted it was probably time to stop. He looked so sad, sitting there in his coat, his pants wet to the knees, his head hanging forward, his hands hiding his face. His fingers were all knuckles and loose skin. He was going bald. How had her husband become a man of fifty? How had she become a wife of forty-five?\n\nAnd maybe it was simply that: so much time together, so many years gone by.\n\nAt first, she didn't think of the separation as something that might last. Just a few months away from each other, Abner working at his new job in Bowling Green, she in Toledo, working thirty miles from him, doing her part to keep the country stocked in B-17s, apart only until the war was over, maybe a year at most. But it was four. Four years living in her own room in a single-sex boardinghouse, four years in which she found she liked working alongside other women, liked earning enough on her own, liked the feeling of finishing the nose cone of a behemoth bomber, assembling the canopy of something that would one day soar over Hamburg, Dresden, Mainz. She was the oldest woman working at Libbey-Owens-Ford. Gran Gunner the others called her, smoking cigarettes, snapping gum. On their lunch breaks they laughed about messages for airmen slipped into secret cracks, read aloud the _Blade_ 's dispatches from the fronts, passed around pamphlets by the old BCFA, debated its new milquetoast moniker\u2014Planned Parenthood\u2014and whether General Spaatz was right to bomb Jerry's oil before his rails, and if it made sense to join a dying WTUL, shared home-canned pickles, packs of cigarettes, wondered what they would do after all this was done.\n\nOn Fridays, after work, she would wonder the same. Abner would pull into the Libbey-Owens lot, take her back to a home that all weekend they would pretend still felt like theirs. The wall calendar Allis-Chalmers always sent her, now swapped for one his students made. The sink corner that had once held her hand cream, now crowded by his shaving mug. He'd move it over, turn the month. Sometimes by Sunday they could almost feel like them again. Though more and more she worked the weekend shifts\u2014overtime, extra pay, Saturday night out with the girls\u2014Sunday coming every other week, then once a month, then not. On the phone, Abner would speak things he'd never said before\u2014how much he'd wanted children; how he used to lose himself in his guitar; all he'd given up for her, would, still wanted to\u2014as if the distance between them made him brave. But it was just distance.\n\nWeeds, scrub willows, the driveway buried somewhere beneath the grass. She stood in the sun, seed heads scratching at her stockings, looking at the house. He'd written her: moved out, a simple flat nearer the school he worked at now, she could stay in the house, or sell it, _it's up to you_ , he'd transferred the deed to her. But not the car. She'd walked the last two miles. Such flat land: the whole way she'd watched the house grow near, the road to town slip out of sight, its tree line dwindling to distant shakings, far-off Toledo disappearing from her life. The air grew thin. Her stomach dropped. Her juddering heart: she might have been taking off, climbing up, seeing the earth fall away below. Except she wasn't looking down. Instead, on every side it seemed the world was drawing itself away from her. Once, long ago, in a dim Dayton stairwell, her arms beneath her father's arms, dragging him down, flight after flight, fast as she could, she had felt it\u2014in his eyes on her, in the thud, thud, thud of his heels on the steps, in the desperate heartbeats of his departing life\u2014had felt the world withdraw like that. There, on the landing where they'd stopped, he had watched her with such hurt, such hope, so much understanding ( _selfish? selfless?_ ), such loneliness suddenly inside her.\n\nStanding in the hot sun in front of the abandoned house, she set her suitcase down. Hiked up her skirt. Peeled away her stockings. They ringed her shins like thick black shackles. But God the breeze on her legs felt good.\n\nIt gusted all the time. Flat fields like runways for the rushing wind, windbreaks bent by its launch against them, the stolid brick house huddled close to the ground, Clara leaning forward, dress and coat and hair afloat behind her, her whole body seeming about to lift off into the sky. These days no more mail planes flew by. Just clouds and birds and the bellies of DC-4s, their fuselages perforated with passenger windows, their cargo holds carrying the mail alongside luggage now. How high up they flew! How far away they seemed! How fast they grew\u2014Comets and Constellations and Stratocruisers\u2014big as blue whales swimming through the sea above. Sometimes it struck her as strange: the way their shapes\u2014so much larger than the biplanes of before, but so much higher, too\u2014seemed from below to stay the same size to her. She hung tobacco in the barn to dry, stuffed advertisers' envelopes all winter, barely scraped by. Even with the checks Abner still sent her. She knew he couldn't afford to give so much, knew she should tear them up, just as the one time she'd seen him in Maumee she'd known she should leave him alone. But she had crossed the street to the lunch counter window, watched his shape stiffen at sensing her, his face furrow with the effort of staring into his shake. She'd turned away, gone farther down the street, glanced back: there on the sidewalk her still husband stood, his hat pushed up from his eyes as if to keep the brim from blocking even a sliver of her in his sight.\n\nAnd the next time an envelope came from Abner it contained not just the usual check but a story clipped from _Life. Is he the greatest living painter in the United States?_ Photographs of splattered paint, scattered color. To Clara they looked like her field when she'd first come home: a bird's-eye view of what nature could do without her. Sometimes at night lying alone in her bed she could hear the airliners droning overhead. Sometimes, midday, sun-bright rooms would dim for a second, go bright again. Their shadows passed over her there in her field, and she watched them sweep away, disappear, didn't even look up.\n\nBut look down. Out that oval window. There, on the ground\u2014what is that?\n\n\"I'd like to buy it,\" the gentleman said. He stood on her doorstep, pinstriped pants aflutter in the breeze, voice like the news on CBS. News of a collector out in California who'd sent this man to hunt out art.\n\nShe looked past him to the field, the barn, the shadow of the plane. She'd stitched the tar paper together scrap by scrap, covered it with black painted muslin that wavered, rippled, gave the sense of the shape moving. Though it was nailed down, glued, painted midpass atop the barn, the yard, the plowed-under field, its wingspan nine hundred feet long, its fuselage distorted as a real shadow's would have been by the slant of the sun. It had taken her three years.\n\n\"The barn?\" she asked.\n\n\"That too,\" he said, in his Edmund Chester voice.\n\nNow Chester was off the radio and everyone was watching television instead and her barn was gone to some hangar outside LA, and she was in a magazine. Some writer spurred by word of the sale had done some digging, discovered a defense department file, photographs of her early '40s shovelings, revealed the message she and Abner had sent the Japanese:\n\n_Clouds drift back and forth_\n\n_Over my fields\u2014I wonder:_\n\n_Can you see them too?_\n\nClara Lowell. There on the page her name seemed like another person's. She read the story of herself as if from far away, from before she'd taken Abner's name, a Purdy girl again who might tear out a page, pin it to her bedroom wall. She read it all until, halfway, she hit a thing she'd never heard: _soldier, bullet_ , how her brother had died. Facedown in a ditch it said. Shot from the air by a strafing ace. Her eyes kept moving along the page, her mind making out the words, but she was seeing Larry again: running, running, engulfed by the onrushing shadow of the plane.\n\nAnd there went her phone again, ringing, ringing. The Garner Agency, the Fineman Gallery, funding from an arts foundation in San Francisco where she spent the first year of the new decade peering down from the Golden Gate or up from a boat beneath it, devising a way to make the bay look as if the shadow of fuselage and wings had been painted on its waves. Across the ocean, over Pyongyang, jet fighters screamed into the sky. She couldn't hear them. No more than Jackie Cochran, three years later, could have heard the sonic boom she left behind along with all the other aviatrixes still shackled to sound. Clara was in New York City, affixing the faux reflection of an onrushing airliner to the steel and glass of the Empire State\u2014a tragic trompe l'oeil. Haunting, critics called it, heart-stopping. They said her work rang of the grim reaper, contained a sense of the moment made permanent, and yet seemed fleeting, too, as if to offer a possibility of reprieve. And so was also hopeful. And so when, high above the Colorado Plateau a DC-7 struck a Constellation, she was commissioned to commemorate the lost souls, spent the fall of '57 marking the Grand Canyon with two immense shadows facing each other across the chasm, their shapes distended exactly as the sun had stretched them the morning of that last day.\n\nIt would be known as Clara Lowell's final shadow piece. Even in the moment, hovering above in the helicopter the Park Service pilot flew, she knew it: the silhouettes were old shapes cut from the woman she used to be, not who she had become. Down in the station everyone else had moved on, too. They were crowded around a shortwave radio, listening to a faint, steady beep. _That's it_ , one of the rangers said. _It must be passing over us right now_.\n\nBack in Ohio she stood in the spot where her barn had once been and watched the tiny glint arc along its orbit. She had seen pictures of the Soviet's sphere. She wondered what, from that height, it could possibly see down here.\n\nOld ovals found in dusty bureau drawers, age-spotted hand-me-downs, rearviews salvaged from crashed cars, castoff skyscraper panes from construction lots in Cleveland: she collected shards of any size, from all over, carted them back to her small square of earth, slowly, piece by piece, resurfacing her old canvas in glass. Ten entire acres. Half as many years. Hundreds of thousands of seamless fits found from a million broken sides. By the time John Glenn radioed down Oh, that view is tremendous! he might have meant the flash of glinting land she'd covered.\n\nOr the sight of all the others who'd come to help. From the beginning she had watched in wonder\u2014these young seekers fleeing their old lives, stopping by on their way to wherever, stepping out of dusty cars, parking in her driveway for a day, a couple, crashing in her guestroom, on her couch, men on motorcycles with their plaid shirts unbuttoned, wind-wild hair, women wearing jeans, scarves in colors more vibrant than anything for miles\u2014but as they crouched by her side, helping find a fit for a piece, holding a glued edge together, she had begun to think of them as somehow akin to her. These kids who were a third her age! Who drove up blaring bands with names like the Del-Tones, the Animals, the Stones. \"Can't you hear my heartbeat?\" The girls sang it while they worked. Girls who said things like _out there it's trying to bury us alive_ , and _can't let it stifle your voice within_ , who laughed when she insisted she was still married. _Well_ , she said, _it's true I haven't seen him in, let's see, oh jeeze_.... And they told her she couldn't continue like that, it was a new era\u2014 _all that matters now is what will make you happy, what's the point in living if your life isn't true to you_ \u2014an age of self-fulfillment, of our own happiness not just pursued but caught, kept, held perpetually near all our hearts. They could have been her children, her grandchildren, but standing amid a group who'd helped her put the last broken piece of glass in place, she felt as if she had at last found her generation, kindred spirits, a moment in time in which she fit.\n\nOnly the babies gave her pause. The ones brought into her home on the hips of girls younger than Clara had been when she'd first entered the house herself. In their sounds she would hear her long-gone husband's late-night voice, his telephone pleas. And, watching the stare the babies settled on their mothers, she would wonder if Abner\u2014another's husband now?\u2014had felt enlarged by that enamored gaze. She hoped he had. Though whenever an infant was handed to her, she felt the opposite: the child's need tight as its fist around her finger, squeezing her down to fit its purpose, herself made small as the reflection in its unblinking eyes.\n\nInstead, she kept her sights on the work before her. There, in the field of mirrors, the sun shone up out of a sky in the ground. Clouds crept through the grass along the edge, floated into twin squares of blue, followed themselves. Soon, she knew, a contrail would cut across like a line of chalk. A 727 on its way to Chicago, or coming east, a hundred and more passengers peering out their windows. Staring down at the sky beneath herself, she tried to imagine what they would see.\n\nA blinding glare, according to the FAA. Clear the mirror off the ground: the agency's order that at last turned Clara Lowell into a household name. The destruction on the evening news, the documentary about the flood of youth answering the artist's call, the image censored around the world: all that mirror-cleared ground blanketed now by a thousand bodies stripped bare, a ten-acre square of naked flesh flashed upwards in an act of irreverence so communal it seemed to capture the entire decade's mood. By the '60s end she had lit an entire rural county's roads, spidering bright veins across the nighttime dark; she made a color photograph of one square mile of the earth, shot from a mile high, then blew it up to actual size, printed it in pieces, put them back together over the spot, so from the air it almost looked like life, but not.\n\nStill, it was her _Long Bright Line_ that Clara meant to be her masterpiece. She had convinced the postal service to loan her the antiquated airmail beacons for one night. Coast to coast, every twenty miles, they stood rusting, signals extinguished long ago, last remnants of an idea once pioneering, now obsolete. Until, for sixty seconds on the night of July 20, 1969, she would bring it back to life. All her funding had gone into the purchase and installation of a hundred and forty first-order Fresnel lenses, powerful as any lighthouse beam, mounted atop the towers, aimed straight up. The volunteers who manned the stations wanted only to share in what would happen at her signal: starting in New York the first would flare on, followed by a second to its west, and the one that was next, and the one after, a constellation untangled across the country into a single strand of terrestrial stars, a gleaming necklace laid atop the earth's dark breast. Seen only by the moon. And the astronauts on it.\n\nSometime early that afternoon the TV would show the lander touch down. Around sunset it would show Armstrong or Aldrin stepping onto the surface. By dark they were supposed to be done. And she would call the beacon in New York and start her signal to them.\n\nBut in between the landing and the moonwalk, her phone rang instead. She picked it up, heard breathing.\n\n\"Wasn't that incredible?\" the voice on the other end said.\n\nEven after all these years she knew him. While he talked all out of breath about the surface of the moon seen coming close, and closer, the shadow of the lander growing ( _that beep, beep, beep_ , he said, _can you still hear it?_ and, for some reason, laughed), she slid open the deck doors, let the scent of the ocean in. She stood there trying to smell it, trying not let him hear her inhale. Her old nose. Her old mind. Him? He must have been approaching eighty. He must have been becoming senile: how else could he have just now asked her to come see him?\n\n\"I'm sorry,\" she told him.\n\n\"Tonight,\" he said.\n\n\"I live in Los Angeles now.\"\n\n\"Clara...\"\n\n\"I'm busy, Abner.\"\n\n\"I'm dying,\" he said.\n\nShe sat on the couch, in the salt-sticky breeze, feeling it on the loose skin of her neck, her scalp beneath her thinned-out hair, remembering the scent of peanuts, engine oil, the soft fleece of the aviator cap, the warmth of the earflaps\u2014on winter days his fingers had brushed her chin, buckling the straps, just so she wouldn't have to shuck her gloves\u2014how he'd gazed up at her astride him that winter night, his face full of bonfire light, the way he'd looked at his new wife in the picture he'd sent so many years ago to share the birth of his first child. A daughter? A son? She sat in the breeze, trying to remember ( _what do you think I do all afternoon?_ ), listening to Armstrong speak from the moon.\n\n_There seems to be no difficulty... Definitely no trouble..._\n\n_Gee_ , Cronkite said, _that's good news_.\n\nAnd she leaned forward, turned the volume down. She sat in silence, staring. A gray, grainy picture. A pale blur she knew was the shape of a man but might have been anything that moved. The longer she watched, the more strange and beautiful and unworldly, unreachable, it seemed. The longer she looked, the more it broke into its parts\u2014stillness, shadow, something stirring\u2014the more she felt her own shape blurring too. If anyone had looked away from their TVs, glanced up at her window, aglow with the light broadcast off the moon... but who on earth would?\n\nShe was still watching the fuzz of the screen when, a long time later, the broadcast done, the window for her beacons passed, the phone rang again.\n\n\"What happened?\" The volunteer's voice came all the way across the country. Some young woman high up in her tower, finger on the switch. \"We're waiting,\" the voice said, crackly with distance, beginning to doubt.\n\nIn the hospital she meant to tell him it hadn't been his fault, but stepping into his room, she found herself unable to utter the words _work_ , or _self_ , or _matter_. Instead, she sat gazing again, this time at her once-husband's face. Someone had turned his sound off, too. When the nurse told her he could no longer speak, Clara asked what his last words had been. The nurse didn't know. She wondered: would they have even been meant for her to hear? She wondered: would she have even been able to? There was just the beeping of the EKG. The blinking of his eyes. Even when they were open, she could tell they didn't see her. But she watched them: blink, blink, blink.\n\nFlying home, she could not stop feeling her own lids opening and closing, even as she leaned towards the window, looked down. The lights of Dayton dwindling. The lights of its outskirts spread as far as she could see. Somewhere down there it struck her was what used to be her father's farm. It was smothered by suburbs now, buried beneath the unrelenting burn of each house's separate star, but once it would have been unlit, all of it, house and barns and trees and field and little girl looking up, all indistinguishable, dark as sleep. When the moon went behind the clouds the balloon must have seemed suspended in pure blackness. They must have held the basket tight, peered over its edge, thought, _How beautiful!_ So strange, she thought now, what we have done to the surface of the world. She shut her eyes\u2014simply paused her lids, stopped them from opening. And, taking in the emptiness before her, wondered how many seconds of each minute she'd spent like that, how many minutes of each day, how many hours, how many years.\n\n# [**THE ESSENTIAL  \nCONSTITUENT OF  \nMODERN LIVING  \nSTANDARDS**](toc.html#ch3-R)\n\nWe woke to a world opened up in holes, a dawn still stained with spots of night. Stoking the stove, we stood, went to the window, blew out the lamp to better see: out there, beneath the barely brightening sky, the snow-smooth pasture somehow gone guttate while we'd slept. Letting out the dog, we lingered in the doorway, the cold cutting through our nightshirts, down our gowns, gazed at what we gradually made out to be a line of fresh-dug pits. Big and black as tractor tires, startling as exit wounds ringed by bursts of dirt. Outside the milking parlor, amid the lowing, we squinted past the path they made through nearer fields, discerned a distant movement: men on the far-off slope, shovel blades flashing first light, pickaxes swinging in glinting arcs. We called for our husbands then, went to our wives, stood side by side in the slam and slam of steel bars driving down. In the corners of our mouths the egg yolk had gone clammy. On the stove, the bacon began to burn.\n\nDid we feel it even then? The spark in our frayed ends? The places where our casings had been chewed through? The air\u2014alive like the second before a flash of lightning\u2014touching our wires where they had been stripped bare?\n\n_Watt, socket, flashover, volt:_ for most of our lives the words had seemed as far from us as the city from our farms. But ever since Penn Power and Light had graced the surrounding counties with places on their grid we'd watched their electric glow creep closer to the horizons of our fields, each evening towns to all points blooming bright as moonflowers, each year\u2014'37, '38, '39\u2014hoping that our time was coming near.\n\nFor three decades we had dreamed of it. Trying to follow a last fly ball falling palely through a dying dusk. Lying beside first sweethearts in a deep barn dark. How we yearned to see his face when he made that sound, to glimpse the sweat slick on her belly. At dances we fled the heat and smoke, quit the lamp-lit grange hall before our legs were ready, our hearts still thumping for another set, wishing for wires that would let us whirl and whirl. In a handsaw's slow rasp we heard a blade buzz through in seconds. In whispered rumors of the hobo who'd slit the throat of so-and-so we saw bulbs brightening our barnyards all night. But even when we left in our mule carts, rode off to war, we lived barely a month beneath the hum of filaments before we shipped for dark-ditched France. And back again, driving our own cars, our headlights showed the unlit roads, our still-dark homes. Still giving birth by firelight. Still fumbling to light the lamp to tell what the hell the baby was bawling about. Sitting in the rocker, we made our kids get up to crank the radio, told them someday wires would arrive, bring all the music we could take, effortless entertainment, a pump in the well, an indoor privy, electric milkers to spare our hands from ever wringing out another heifer's teat.\n\nYet for all our hope, there passed another decade of depression, came another war, our children grown old enough to cross the Atlantic in our stead, and still each evening we sank into a century the cities had long left behind. Still PP&L refused to help us into the decade the rest of our country had begun. 1940. And not a single power line. Profits, they said. People per square mile. Amperes and installation costs and, _We're sorry, we can't_.\n\nBut _we_ could. That was what Bell told us. _You can. Yourselves_. Though he, himself, what did we know of him? That he had showed up in the grange one night? There in the dark rectangle of that open door? Said _union_ , said _come from Harrisburg, Philadelphia before_. And, we sensed, from somewhere even stranger, further back: that trace of Slavic on his tongue, that hint in his eye of unimagined hardships seen. He was a Jew. A Red. A man of must be thirty without a family, a wife. And yet we listened to him. Watched his spindly frame unfold from the creaking folding chair, so rod armed and slack trousered he fooled you into thinking he was tall. He introduced himself _\u2014Abe Bell_ \u2014said he was there to help if we would let him.\n\nBy the bonfire after, passing the bottle around, he spoke of loans and power plants with a heat that seemed to stoke the blaze. That week he showed up at barn doors, chicken coops, helped peel potatoes, stook the corn, talked to us of the Steamburg Association, the Rural Electrification Administration, made us feel like kids again listening to tales: how the REA had fought to pave the way for us, for _this_ , for now, the law passed a mere two years ago so we, county by county, could come together, vote to form our own cooperative, control our own electricity, hold a little power in our hands. Crouched by our woodstoves, bony fingers outstretched, shoulders crowded as if squeezed by a clamp, his booming voice made it all seem something proud, our struggle with PP&L a bold new front in an old war on our own shores.\n\nIn churches, barns, grange halls, we gathered, bickered, shouted down the company hacks sent to dissuade us, lifted up our hands, counted, began. The Tri-County Co-operative's own electric generation: by the winter of '41 we'd gotten all but the final go-ahead for the federal funds, contracted for construction of the plant, routed our lines, could feel in our fingers the hardwood handles of the shovels we'd use to dig the holes for our first poles. Until that January dawn we woke to find PP&L putting in their own.\n\n_Spite line_ , Bell told us, the dog flinching at the anger in his voice, Bell's head shaking away our offer of a chair, no time to stay a second longer than it took to make things clear: That law the government had passed, pushed through by Roosevelt and the REA? Turned out we could cooperatize, could get the grant, only if we weren't already served by someone else. Those diggers out in our fresh-snowed fields: they'd come to undercut us, snatch away the claim we'd staked to our own light. In our milking parlors, Bell paced the concrete floors that we washed down, explained how the poles PP&L were putting in were solely meant to hold us back, a thing he called _cream skimming_ , the bastards running a single line along a road, or through a couple farms, serving just enough of the easiest to reach to claim PP&L had brought a county electricity, while in reality they'd left the rest of it _\u2014the rest of us_ \u2014still in the dark. _Except_ , he said that morning, going farm to farm, _we won't let them_.\n\nHe owned no auto, no animal to ride, but we lent him our mule, our workhorse, watched our school-bound children step aside into the snowbanks as he hoof churned by. Heaving the bucket up the well, we caught his silhouette come shaking into that circle of reflected sky, felt the urgent fury in his eyes. And who didn't pause, mid\u2013pitchfork lift, at the sight of Abe Bell peddling like hell atop a bike? His clothes mud flecked, his fists red, his face raw, his bushy hair matted beneath his dark knit hat. He swept it off, steam rising from his head.\n\n_Listen_ , he said, spoke of a such-and-such _preemption clause_ , the power company legally constrained: we had six months to get our lines up first. _Listen_ , he said, and it was as if his words gave voice to what we'd so long felt. True, he was a different kind\u2014that beakish nose, those accordion-moaned tunes that floated from his rented room\u2014but in the last few months he had become a part of who we had become together. We farmers. We kinfolk and friends. We men who had known each other since we were boys, who had shared bulls, helped slaughter hogs, celebrated births, set legs, lent a hand, a blessing, a buck. We came in trucks, packed three to a cab, on carts, legs dangling off the back, rode in alone from far-off farms to the big stone barn below Berks Hill where, that evening, we all converged.\n\nTwo dozen of us. Maybe more. Slamming truck doors, shaking hands, saying little amid the scraping clangor of all the shovels dragged across the boards. It was sunset. The red light caught the steel spade heads, sparked where stones had sharpened clean the blades. We held them, squinting, while Abe Bell spoke. Then\u2014mad scatter of refracted flashing\u2014raised them to our shoulders, swung them across our necks, and, amid the din of half a hundred boots shoving through snow, followed him into the field.\n\nThrough the sun's last slant the workers watched us come. They were all across the pasture, gathered in teams around each hole. Behind them, the holes already dug dotted the snow; ahead, the unmarred surface spread up the hill towards a chink of sky cut from the windbreak. There, others stood silhouetted against the fulvous clouds, axes gone still. All down the hillside shovels hung at the end of a toss, black bars of rock breakers stuck motionless in earth, the heavy heads of pickaxes eased to a rest in front of frozen boots.\n\nTo them, the spray of snow kicked up by us must have seemed a breaker rolling up the hill. If we could catch their troubled faces as we closed, they must have seen the fire in our own, known how hard this thing could go: the nearest workers\u2014their patched coats and ragged trousers and unkempt beards catching our deepest fears of what could come from being kept in backward blackness while the world sped past\u2014stepped away, drew up their tools, seemed prepared to leave us to the work we would have begun if, from up near the broken windbreak, a different figure hadn't come striding down.\n\nHe wore a quilted coat, its collar up, a wool cap's flaps over his ears and while he walked he lit a smoke, hands cupped before his mustache, eyes focused on the flame all the way until he nearly smacked into the wall of us. He dropped his hands, looked up. _Gentlemen?_ he said. _What is it I can do for you?_\n\nBut he was talking to Abe Bell. If we could spark, he had discovered which of us would flip the switch. Bell stepped out of our ranks. He'd laid his shovel across his shoulders, wrists hooked over handle, arms hanging loose, but as they talked, their words blown to us on the cold wind _\u2014court ordered_ , and _eminent domain_ and _in violation of the law_ \u2014we saw Bell's fingers slip to grip the handle, turn to fists. And gripping our own a little harder, pushing a little closer behind, we wondered what might make someone like Bell come on his own out among the likes of us. What had he left? What had he run from?\n\nWhen he swung\u2014the sudden slash of blade through air\u2014we jerked as one. The mustached man lurched back as if foreseeing the spade connecting with his head. But Bell checked its swing, leveled it instead at the man's chest. _Go on_ , he said, and, with its tip, pressed just enough to dent the padding between the man's coat buttons. _Go on get back to work_. He drew his shovel away again. _While we get on with ours_.\n\nThat night the moon rose gibbous, waxing, a scant few days from full, and in its light we danced an eerie choreography, us and them. All the snow's surface strung with lamp-lit circles, each one illuminating a crew of three or four digging a hole. Beyond the light, we watched. Until the moment when they would pick up their lamps, shoulder their tools, move on. And we would step up, unshoulder ours, fill the hole back in. While ahead they trudged past another cluster of us doing the same, another and another, until at last they reached the last hole dug and set to work digging one more. We followed, gathered again, waited outside their lantern's reach for our next turn to undo another hour of their work. You would have thought we'd come to blows, angry exchanges, but we settled instead into a sort of understanding, stayed out of each other's way, spoke low among ourselves as if to keep a separation of sound as well, hung back when they hit a rock, gave them space while they took turns slamming down the spud bar's iron wedge. And when, each hour on the hour, word came along the digger line\u2014 _take five!_ \u2014we accompanied their thermos-pouring sounds with ours from what our wives had packed us, each finding a spot to sit, a split rail, a pasture rock, until the diggers shook out their coffee grounds, stood, went back to work unearthing what we would come along behind them to replace. A strange sight. The lamp-lit them, the moonlit us, our ceaseless weaving around each other in that long line all through that wide white pasture.\n\nSometimes, we would glimpse our instigator. He flickered in and out of lantern light\u2014Bell's breath a cloud, his teeth a glint\u2014taking our shovels for a spell or speaking to the diggers as we stood our silent watch. He was the only one who broke the pact, braved the broad backs and swinging picks, tried to make them see our side, broke a little of the border we kept between us so that as the night wore on they stayed a little after they were done, long enough to loose a couple words on us as, stepping up to work, we sent back a little laughter, a little later offered them a slice of ham, maybe at the next hole a swig of whiskey, a sympathetic grunt at the grind of shovel hitting rock. Even, sometimes, when one of their spud bars would finally break through, a cheer.\n\nMaybe it was that. Maybe the way we sat close enough on the next break to see them hold their warm tin cups against their tired eyes. Maybe it was how they stretched out the silent minutes, their necks craned back, their throats smokestacks of breath. They spoke of cities, of someone left back home, the way the moon was so much brighter here. The stars. They seemed never to have seen such a sky of them. We tilted our own heads back, tried to imagine what it would be like to not see it every night. And maybe the foreman, observing all his men, looked up too, lost himself for a moment in the firmament. The distant bells of town began to clang. We all stood silent through all twelve knells. Maybe it was the sight of a far-off lantern slowly separating from the flickers in the sky, splitting in two, becoming headlights. A neighbor's cart, a truck from the next county. All night they trickled in, and maybe it was simply that\u2014too many of us, too few of them. Sometime in the early hours the company trucks rumbled to life, rolled down the line collecting their laborers, and left.\n\nAt daybreak, when they came back, they found us waiting: a wall of farmers built across the entrance to the field. We stood wrapped in our coats, holding our shovels. Though those of us newly arrived, not knowing how far the company might go, had brought our timber axes, two-handed saws arced high over a shoulder, even, jutting here and there among the tools, a rifle barrel white with frost.\n\nWatching the trucks emerge out of the distant gray, it came clear: the ones who had brought saws and axes had been right. On the flatbeds, poles were piled, strapped down in hulking stacks: the company would redig the holes, keep us at bay in whatever way they had to until they had planted and set the poles. Ahead of the flatbeds, more trucks hauled workmen huddled beneath canvas flaps, last night's camaraderie stripped from their faces by the knowledge of what they'd been sent back to do. Standing within the headlights' burn, the shuddering of so many engines, the sheer tonnage of all those poles, made the more than a hundred of us seem suddenly small. And in that moment we knew the rifle bearers had been right too: each truck cab held the heat-blurred shape of someone in the passenger seat holding a gun.\n\nThe closest passenger window slid down. A huff of heat fumed out. All along the line of trucks the side windows did the same, a silent cannonade of steam blown around the long black barrels that jutted from each cab. Just as, from each of us, there rose our own wisp of warmth, amidst the breath of all our beasts\u2014mules and horses and dogs come down\u2014mixed with the exhaust plumes of the entire caravan. For a moment the air above us all seemed the only thing that moved.\n\nAnd then Abe Bell shouldered through. He held an ax, its handle gripped hard in his bony fist, the double-bladed head swinging like a clock weight at his side. A few feet from the front of the first truck, he stopped. The sun was still buried behind the eastward ridge and in the shadowed road last vestiges of night remained, and standing in it, lit by the headlights, he seemed almost to glow, as if he alone among us had been plugged in, his switch already flicked. But what could he do but stand there and burn? What was there to say he hadn't already said all night? Instead, he simply stepped to the driver's side, swung the ax. A low soft thud. At first we weren't even sure we'd heard it beneath the engine's mutter. Though somewhere above it, was that the high-pitched hiss of leaking air? The second ax blow boomed so loud it snapped our stares to the shotgun barrel, then back to Bell stumbling away from the truck's slumped front. For a second, watching him stagger, we thought he had been shot, then saw him regain his footing, hitch his grip, and start for the tire at the rear. From us, at that: a roar. Another, louder, as he swung the ax. And when that tire blew the sound that burst from us might have been what knocked him back. And split his face with a grin so wide we could feel it in our own teeth. Around the other side he went, the workers in the truck bed pushing at the canvas flaps to better see, all of us shifting to catch a glimpse of ax head swinging back, jacket flung out, that bush of hair blown wild with each exploding blast.\n\nBy the time he reached the final tire, on the front passenger side, our cheering had grown so loud we couldn't tell what the man with the shotgun might have said. But we could see Bell hear it. He stood with the ax gone still in his hands, his eyes on the tire, his whole self attuned to the barrel aimed at his head. Later, some of us would claim the man said: _assault_ or _self-defense_. Others insisted on: _I guess you've gotten close enough_. Whatever it was, we countered it with our own calls. And when Bell lifted the ax again, squared himself to swing it at the sidewall, our urging swelled until he must have felt plunged into a sea of sound. It must have filled his ears, drowned the voice that surely spoke inside him: _Don't_.\n\nAll our breath from all our shouting. All the lung heat of all the laborers erupting all at once. The sudden nostril steam of all the startled horses' snorts, puffs released from the sprung jaws of all the dogs. The thicker gust poured from each truck's exhaust. From above it would have seemed an aberrant cloud descended from the sky, dropped down upon the place where we had gathered, a strange mist making itself out of the clear cold air. Within it, the separate strand of smoke blown upwards from the fired gun would have barely been discernable. Until the fog shifted, thinned, broke apart to show a momentary picture\u2014the shadow of a figure twisted on his back: Abe Bell burst open, his lifeblood bright as sunrise on the frozen road\u2014before closing shut back over us again.\n\nSo goes our memory. In the years after, we would come to disagree on what we'd seen. Whether the blast had killed him on the spot or he'd expired elsewhere, whether some of us had wrenched the door open, dragged the shooter out, or, to a man, each shrunk back in shock. Whatever happened that morning we all agree on this: that the whole convoy of company trucks shifted into reverse, backed away, wound down the road; the way the commissioner of utilities came to our defense at the hearings that came after; how, after eight days of argument were over, we had won. That spring the director of the REA himself stood among all our families come from all three counties, and, in the seconds before he sent power coursing through the lines we'd built, spoke: _Because you live on farms_ and _denied the essential_ and _modern living standards_. He bemoaned our inability till then to pump water, refrigerate fruit, brood our chicks, save our children's eyes. _When you banded together_ , he said, _your efforts were opposed by..._\n\nWell, that we knew. That was over. By the time a year had passed there would be other co-ops among other long-neglected farms, thousands of miles of power lines, poles planted all across the state. The entire country. In those next decades it was as if the stars we had always seen above showered down upon the darkest swatches of our land, lit our once-black nightscape up. Our towns grew galaxies, subdivisions burning bright, our farms bespecked with the window light of renters' doublewides, new brick homes built for our children, the all-night glow of our security lights.\n\nBut still, generations gone by, we couldn't quite quit thinking about Abe Bell. As those of us who'd known him grew fewer, our minds more feeble, we would tell our children things that seemed each year a little more uncertain, a little less alike. Some of us insisted he had been taken to the hospital, seen at the hearings, was the one who had accused PP&L of contravening law. Some said he'd fled before then, fully recovered, some swore he'd never even chopped a tire, that the whole thing was a fable, there'd never been a man named Bell, it was we, alone\u2014 _your grandpa, your old uncle_ \u2014who'd risen against PP&L. _Wouldn't_ , we their children whispered among ourselves, _there have been an investigation?_ And we spouses: _Wouldn't PP &L have wanted to keep it quiet?_ While we teens, free from reprimand by great-grandparents no longer living, joked of Bell's golden years back in the country from which he'd come, the tales he must have told his comrades of socialism seeded in our very heartland, here.\n\nWhere, with each passing year, a little more of what he'd planted was lost with the land our siblings sold, his story leaked away with friends settled in the suburbs, the cities soon to steal our daughters, sons. Gone were the heavy horses we used to walk behind, rest our cheeks against their steaming sides to whisper things we couldn't tell our husbands, mothers, wives, the plowing now done by a team of one. Gone was the milk cow, dairy now a word meant for farms made for nothing else, except, perhaps, a trip to a neighbor's parlor, a cup of coffee at the kitchen table before the trip back home. Seldom and more seldom and soon replaced by men bringing bottles in trucks, then not even them to wave at us when we creaked open our front doors. Chicken coops collapsed, our last hens cooked back in, oh, when was it: '70, '71, '72? Later for some, though eventually we all wound up at the grocer's, paying our visits from behind passing carts, until\u2014the IGA, the Kroger, the Walmart\u2014even the chance of seeing someone we knew was lost in those enormous warehouses where we bought our plastic gallons now. Only the silence stayed. We tried to fill the holes the rest had left with the whooping of _The Price Is Right_ , wailing from _Days of Our Lives_ , the seconds ticking by on _Jeopardy!_ echoing around our empty living rooms until we replaced that, too, with the bleeps and blats of dial-up, news of one-time neighbors whose voices had once filled our phones, and then simply the silence of high-speed lines strung where we'd long ago fought for a single one to carry a little electricity into our homes.\n\nFrom which we disappeared for hours now, sunk into a screen, filled with the sense that somehow we had failed, that the world still hurtled past, and we were simply clinging to its speeding side. In far-off Africa it seemed everyone spoke on cell phones, on the Mongolian steppe herders lit their yurts with solar panes, in China they churned electricity from dams that did to villages in a few moments what had taken us a century. And some days, seeing the news of oceans climbing, watching the cities flooded on TV, a fear crept on us, touched us where the fingers of long-gone friends no longer did, filled the hollows left by dead husbands and wives, pushed at our chests the winter that a thousand Lithuanians froze in a blackout of their country's grid, pressed a little harder when we heard the Russians had devised a way to reflect the sun off the wings of satellites, solar mirrors they said would send down an unceasing luminescence, light their patch of earth all night. That sent us out into our fields, stilled us beneath the stars. Until the fields were no longer ours, the night sky no longer there.\n\nNow we live surrounded by vast swaths of corn and soybeans, unimaginable acres of apple trees, each county assigned a single crop. From our windows, we watch the mammoth machines, the migrant workers moving through in thousands, out there all night beneath the mirrored satellites that years ago our own country launched into the thermosphere, those moon-sized spots of gleam that crowd the sky, the Milky Way, dimmed all these years by our earthbound lights, having finally disappeared. The way our few remaining homes, standing alone atop the land that we have sold, will one day, too. When we die off and they at last are bulldozed.\n\nSometimes, on sleepless nights, wandering the empty rooms, wondering about some old acquaintance or once-close cousin, we go out into the eerie mirror light, halfway between dusk and day, and search the fields for suggestions of what our great-grandparents had done. The single wire they'd once strung has long since given way to a 400-kilovolt transmission line, a dozen conductors swooping between steel towers hundreds of feet high. But beneath them we can still make out, hidden in the grass like a toppled headstone, the remains of a rotted pole. Shuffling in the weeds we can sometimes even find the dip in the ground, a sunken place where once there must have been a hole. Times like that we feel the pressing worst, as if it is a curse brought on by murdered Bell, or by our own great-grandparents, or by ourselves, this charge in all our hearts, this flash that fires in us even now, this spark that drives us ever forward, still.\n\n# [**ANGLE OF  \nREFLECTION**](toc.html#ch4-R)\n\n_I. The Space Mirror_\n\nThey said it was the end of darkness as we knew it, an age of perpetual light: morning and day and dusk and dusk and dusk, and morning again. Launched into orbit, it would spin, big as a house, moving through the sky at twenty times the speed of sound, its reflective wings unfurling like the petals of some massive morning glory seeking sun. Something like that should have had a better name. It should have been baptized by some blind singer in an ancient Dane's mead hall. Moon Slayer, he would have called it. Star Demolisher. The Mohicans, whose burial mounds we still came upon in the woods, would have made it The One Who Keeps the Light Warm Until the Dawn Awakes. Hell, even NASA would have come up with something passable. But it was the Soviets who dreamed it up and so it was the Soviets\u2014who named their own country the Union of Soviet Socialist Republics, for God's sake\u2014who got to name it: Banner. Maybe it sounds better in Russian.\n\nIt was built to crash. By the time they actually got it up there the Soviet Union had eaten itself and was cracking its own bones for the marrow and there was a whiff of death about the whole thing. At a signal from Moscow, it would detach from the space station, find the sun, reflect it earthward, and slip a beam of daylight along the surface of the world, slicing through night, across the continents, the seas, racing towards Russia, hunting for home. It would circle the globe a few times, then\u2014foom!\u2014hit the atmosphere. Self-immolation.\n\nMaybe it just seems murderous now, looking back at it all. Then, there was a thrill to it. Junior high was conquered. High school was begun. We were all fifteen. Except Mirza. Nobody knew how old Mirza was. You'd have guessed maybe a couple years older than us. Unless you'd spent the time, like we had, those fall nights up in the old concrete plant, smoking weed, him as quiet as ever, and you'd look over and get just the bit of moonlight on those creases by his mouth, just the _feel_ of him, and think, _Fuck if he's not thirty_. Mirza Kojic. We called him Kojak, like the TV show. Then just Jack. Jack with his mouth like an old man's. He was obsessed with that space mirror. Never told us why. Never told us much of anything. Never really even talked.\n\nAfterwards, we called him Mirza again. Not to him. We didn't talk to him anymore. But when we talked _about_ him. To each other. Which, mostly, we didn't.\n\n_II. The Concrete Plant_\n\nWe were fifty feet up in the abandoned plant, the usual four of us smoking our usual weed in our usual spot. The funnel was wide enough that if you looked up you'd see a patch of night sky the size of a garage door. If you looked down, you'd see your knees (you're squatting with your back against the rusty wall), your shoes on the slant of metal angling towards the hole in the middle. The hole dropped straight down fifty feet into the mixing tank. It sat there between us, black as a gun barrel, just wide enough to fall through.\n\nIn the distance, a train pushed its warning noise down the tracks.\n\n\"We should jump it,\" Eli said.\n\n\"Goes too fast,\" Greg said. \"Get your leg chopped off.\" He sucked in, looked at us with bulging eyes.\n\n\"You wait till it slows, genius,\" Eli said.\n\nHolding the smoke in, Greg made a death-by-index-finger motion across his throat. He was bigger than the rest of us\u2014small black eyes, pale skin, gelled hair\u2014and he would have been a bully if he wasn't always stoned. He had two fake front teeth from when his hockey-obsessed dad, who dug up their entire backyard to make a training pond, checked him into a tree. \"Get your fucking hands sliced off,\" he croaked on his exhale. \"Slip under the wheels...\"\n\n\"Jockhead,\" Eli said. \"It's curved down.\"\n\nThis was asked: \"What the hell is curved down?\"\n\n\"It's what they call it when the train comes to a bend in the\u2014\"\n\n\"Who's they?\"\n\n\"I'm telling you, Pete. Take your hit and listen, eh?\" Eli had lived stateside since he was four, but ever since junior high he'd started turning \"out\" into \"aoot\" and wearing patches depicting the Canadian flag. He was as Jewish looking as you could get, except for his hair. It was what he called a killer ponytail, which meant it reached to his ass, or would have if he hadn't been forced to wind it in a turban. His parents were Sikhs. Dharamjot (formerly Bernard Riffkin) and Guru Rajni (was Shirley Goldberg) smelled of tea tree oil and did yoga behind the bleachers during soccer games. He hated them perhaps even more than Greg hated his dad. \"You walk the steel till you get to a good sharp bend,\" he said. \"And then dig a grave where the engineer can't see you. Grave digging is when you get down beside the tracks and lie flat and wait for a hotel. A hotel's what they call a boxcar. A cattle car's a barn. The whole thing is called riding the rooster. They have a very sophisticated lexicon.\" He stared at us. \"Hoboes, geniuses. Jesus, you all've gotta lay off the stuff. Except Jack. Jack needs another hit. Give him the thing, Pete. I mean, look at him.\"\n\nWhen you passed a joint to Mirza, he took it with the obeisance of a gentle-mouthed dog easing a biscuit from your fingers. His hands were shrunken and bony as the rest of him; he couldn't have been much over five feet, a hundred pounds. When he wasn't around, the three of us dreamed up horrors about what had happened to him back where he was from. He had a dent in his left brow like someone had cracked him with a pipe. He smiled a lot. You wanted to look away from his teeth.\n\n\"What's hoboes?\" he asked in what we assumed was just the way Serbians sounded.\n\nHe and his mother had showed up in Culver that spring. We saw her at the Stop and Shop or the school parking lot, smoking, returning our parents' greetings with silent nods; none of us had ever heard her speak a word of English, just the strange language she poured over Mirza while she fidgeted with his palm or collar or hair. We had seen him in class, of course, but we'd only started hanging out that summer. We were facing our first year out of the town school, about to start at the regional high, and he was one more Culverite in a world growing thin on Culverites, or we probably wouldn't have hung out at all. The only time we did was at the concrete plant. None of us invited him to our homes. None of us had been to his.\n\nWe explained the concept of hoboes.\n\n\"No,\" Eli corrected us. \"They didn't just _used_ to. They still do. They still _are_. I've been reading this book.\" If you started talking about something, Eli was reading a book on it. \" _We_ could be hoboes. Freighters, in the lingo. There's nothing to it. When the rooster shows, you run alongside, sync up, take a jump, and you're on.\" He took the spliff from Mirza, looked at it. \"You ate it,\" he said. \"You frigging mawed it. Somebody give him like a frigging Tums.\" Through his exhale, he said, \"Picture it. The four of us, eh? Chiggered to a car. Riding north, like off to Canada, kiss Culver good-bye, frigging Vermont, frigging school, 'rents, like fuck off Dad, I mean nothing, nothing but the rooster and us and woods and like, like...\" Holding the smoke in, he made a circling motion with his free hand, as if urging us to finish his sentence, then dropped the dead end in the chute. It drifted down the hole between us, a campfire spark in reverse. \"Peaceful,\" Eli croaked. \"Like totally frigging peaceful.\"\n\nThe train roared by. It rattled the old plant all the way up to our funnel, and we leaned back against the metal, feeling it. The funnel was at the bottom of an old silo, the whole thing way up on stilts, and beside it a second tower stood on its own spindly legs, the mixer gone, the long thin line of the conveyor angling from the distant ground into the sky, and we watched it lit up in a sudden sweep of the train's beam, shaking against the stars. After a while, it was hard to tell which was shivering and which was still.\n\nIn the quiet afterwards, Greg said, \"I'm gonna leave him a note. Dear Dad: Fuck off.\"\n\n\"I'm not even gonna leave a note,\" Eli said.\n\n\"Dear Dad: Shove a puck up your ass.\"\n\n\"You know what I'm going to do?\" Eli said.\n\n\"And pack it in there with your fucking stick.\"\n\n\"Take a dump on his yoga mat.\"\n\nWe all laughed except Mirza. He had shut his eyes. If he had looked even a little relaxed you would have thought he was asleep.\n\nIt was suggested that before meeting at the tracks to grave dig for the rooster we should all make lists of everything we hated about our dads, leave them on our kitchen tables.\n\n\"Like whatever, Pete,\" Eli said. \"Your dad isn't around enough for you to know what to hate. You probably don't even hate him.\"\n\n\"Sure.\"\n\n\"That's not hate,\" Eli said. \"'Sure' is like, like discontent.\"\n\nHe was assured it was like hate.\n\n\"So say hate. Jack hates his frigging dad.\" Eli switched on his flashlight. \"Frigging look at him. You can see it.\"\n\nHe didn't move, didn't open his eyes. In the beam, you could see his forehead dent pulse where the blood flowed under the skin.\n\n\"Hey,\" Greg said. \"How about this: hand the spliff around and when you get it you have to list all the things you hate about your dad, okay? Give me it.\"\n\nFrom when he started to when we were done was long enough to go through two more joints. Mirza took his tokes in silence. The rest of us moved on to the things we'd do when we came back from our year on the rails: Eli was going to dump gasoline on his dad's head, strike a match to the turban; Greg was going to make his dad swallow his front teeth. We even painted the scene of our escape: we'd tell them all to meet us at the railroad crossing and, as we passed, the signal flashing and chiming its theme, we'd shoot out our parents' windshields with Greg's pellet gun. We were so caught up that we didn't notice Mirza until Eli said, \"Jack?\"\n\nMirza had shoved himself to the edge of the center chute. Dangling his feet down the hole, knocking his heels against its throat, he bent over, peering down.\n\n\"He's looking for spent butts,\" Eli said.\n\nWe laughed.\n\n\"Get away from there,\" Greg said.\n\nMirza flicked on his flashlight, sent its light down the hole.\n\n\"Hey,\" Eli said. \"We're talking about the four of us.\"\n\n\"You can shoot the gun,\" Greg said. \"Take out the windshields.\"\n\n\"You can clip my frigging dad,\" Eli said.\n\n\"Shit,\" Greg said. \"You can blast mine in the fucking face.\"\n\nThe word \"fugitives\" was said like it couldn't help but make everyone grin and \"outlaws\" followed.\n\n\"Think of it, Jack,\" Eli said. \"All badass on the run? It'd be like frigging rape and pillage. It'd be like the frigging Vikings. C'mon, get back from the hole. Help us plan it, man.\"\n\nMirza nodded. But instead of scooting back, he just held out his flashlight and dropped it. A flickering beam shot up out of the chute, hit Mirza's face, flung itself around the funnel rim. There was a distant bang. The hole went black. Still hunched over it, Mirza said, \"In Vi\u00e4egrad I once saw man they catch on train. When they catch, they take his arms and hold off side so his legs hit ground and like this till his feet have no more skin, all his bones break. Then they drop. I saw man like this. Crawl all way from track. He has no pants.\" He looked up from the hole. That terrible smile.\n\n\"Jesus,\" Greg said.\n\nEli said, \"We were just making shit up.\"\n\nSometimes Mirza's smile looked built over his real face like a fence made to block a yard from a stranger's view.\n\nAfter that, we smoked in silence. Then we dropped the last one down the chute and climbed up to the funnel's rim. Bolted to the outside of the tower, a ladder dropped. Greg climbed over the lip of the funnel onto the rungs. Then Eli. Mirza waited at the rear. When you first stepped over the edge and turned your back to the earth it felt as if the ladder was tilting backwards, a sensation less like you were closing in on the ground than it was closing in on you. If you looked down, which you shouldn't, you saw the abandoned shack made miniature by distance, the crane-like conveyor belt that once carried the mix up past our funnel to another that had fallen long ago; it now lay far below, rusting like the wreck of a ship.\n\nWe had just started down from the funnel tower, our footsteps shaking through the ladder's rungs, when Mirza said, \"From Canada, is possible see Banner two point five exact.\"\n\n\"What's he say, Pete?\" Eli called up.\n\n\"He's talking about the space thing again.\" And then, to Mirza's shoes, directly above: \"You said we could see it from here.\"\n\n\"Yes. But in Canada we can be _in_ it. It is to go over Ontario. Aim light beam at Montreal, Toronto. If we chigger to train on day before, we are in this beam when it reflect down. Ten times more bright than moon, Pete. Half so big. Surface area size of sport field.\"\n\n\"It's going to look like half a moon up there?\"\n\n\"From Canada. This is what I'm saying. From here is very minor size only, such as this.\" He held one hand out against the dark sky, as if we could see his fingers.\n\nBelow, Eli was saying to Greg, \"Look, I never claimed it was safe. Safe's for like, like, safe's for our _'rents_. I mean nowadays the cars are full of drug gangs and violent frigging like very violent gangs.\"\n\n\"Like Jack,\" Greg said.\n\n\"No shit,\" Eli said. \"Hey, Jack,\" he shouted, \"what would you do if some drug gang, like, we jumped the rooster and some frigging gang was in the hotel. And they were like, like they had knives, or baseball bats or something? What would you do, eh?\"\n\nOn the rungs above, Mirza flashed a grin through the gap between his arms. Too quiet for the others to hear, he said, \"Is made of Kevlar.\"\n\n\"We're counting on you, Jack,\" Eli called. \"We're just frigging American teenagers. You're, you've _been_ there. What would you do, eh? Like back in Serbia, on one of those trains where they hang the...\"\n\nMirza started a noise that if you hadn't been close enough to see his face you might have thought was laughing.\n\n\"This is serious,\" Eli shouted up. He had just started laughing, too, when the ladder began to shake. \"Whoa fuck,\" Eli said.\n\nWe all joined him in pretty much the same. The tank clanged back at the ladder and the rungs vibrated like they wanted us off.\n\n\"Fuck off,\" someone said and \"Jack, Jack,\" and then, \"Grab his fucking ankles, Pete, or something\" and then, suddenly, he was done. The ladder went still again. Somewhere inside the tank something came loose and dropped all the way down, pinging at the walls: ping, ping, ping.\n\nUp above, we could see the soles of his shoes, pale and floating. He was hanging from a rung, all his weight pulling on thin stretched arms. We listened as much as watched, as if expecting to hear the creaks of his knuckles losing their grip.\n\nIt would have been easier for us if he had fallen. Oh, there still would have been the guilt, but it would have made sense. We had assumed his parents were divorced, but afterwards we began hearing rumors: that his dad was stuck in Canada ironing out immigration; was in Serbia, dead in a mass grave; was in jail awaiting trial for war crimes. Dead dad, us talking, him throwing himself off the tank; it would have been bad, but it would have been understandable and understandable would have eventually become forgivable. But it wasn't.\n\nBecause there was still that time, so late in the year that only a few beech trees clung to their leaves, when Eli christened us the Culver Liberation Front. We agreed to meet in the funnel, bring whatever the CLF might require: a topo map of Franklin County, a set of walkie-talkies, MREs from the Gulf War. Mirza took out a plastic bag full of combs. Each was black\u2014the cheap drugstore counter kind\u2014and he had attached them in pairs, hinged at one end, one half's teeth cut off to make a thin plastic handle, the other half superglued to a line of gleaming razor blades. He showed us how to flick one open. We sat in the new fall cold, practicing, excited not so much by the weapons as by the sense that for the first time Mirza was opening up to us, showing us what we had always known was in him.\n\nAnd there was the time we got in a fight. The truth is it wasn't really a group fight. Eli mostly watched, scared. Greg just shouted like a coach. Mirza wasn't there.\n\nAfterwards, they told Mirza, \"Pete should have used one of your combs on him\" and \"You've got to show Pete how to punch\" but mostly there was just the four of us plotting how to get the ones who'd done it out to our concrete plant, where we could take them on our turf. It was decided that in case anything went wrong, we'd use Mirza as backup.\n\nThis was said: \"He could take Greg's pellet gun and\n\nOne of us said, \"He could take Greg's pellet gun and go up there on the\n\nI said, \"He could take Greg's pellet gun and go up there on the conveyor belt.\" \"Jack,\" I said. I said, \"Jack, you'd see everything from up there. They couldn't touch you. You could just pick them off. You could shoot them all in the fucking face,\" I told him. I said, \"You could shoot them all in the fucking face.\" I said that.\n\n_III. The Railroad Tracks_\n\nI was fifteen, too short to call average, straight brown hair my mother still cut. When I looked in the mirror I thought there was something feminine about my face. Once, I trimmed my eyelashes. I was an only child and my mother was going back to school and I did all the cooking. I read spy novels and historical thrillers and there seemed only two possible careers for me: writer or CIA agent. My father lived abroad in the kinds of countries in which the novels took place and each letter I got I opened with a small prayer that he would have bought a ticket for me. I was as obsessed with the idea as Mirza was with his space mirror. That winter, just before I turned sixteen, I stopped caring. I gave up the idea of becoming an agency man. I quit reading spy novels. What I'm about to tell, here, I never told anyone, not my parents, not the shrink, not even Eli or Greg. And Mirza, well, Mirza already knew.\n\nIt was the second to last time I saw him, and the last time I would see him alone. We had taken to walking back from the plant together, just the two of us\u2014Greg and Eli lived on the other side of Culver\u2014following the railroad tracks until the bridge over Bull Hill Road where I'd leave him to walk the rest of the way alone.\n\nThat night we walked in silence, as usual. Just the whispering of our jackets, occasionally the small clatter of one us slipping off the rail. He was in front, balancing like a gymnast. Every now and then he would waver. His arms would lift like wings. We moved along, quiet and even paced, until he suddenly said, \"I have not thought of snow.\"\n\nNovember had come that weekend and brought the first snow with it. Just a few inches, but enough to change the appearance of the world: the usually black depths of the woods glowed with moonlight on branches; the dark ties between the rails were gone as if pried up and carted away over night; brown fields shone silver.\n\n\"What do you mean?\" I asked.\n\n\"Banner two point five.\"\n\nI groaned. Ever since they launched it, Mirza had talked about nothing else. From what he said, it was up there now, attached to the space station, waiting for the signal.\n\n\"I have not thought how much more bright,\" he said. \"Ten times moonlight, but look here at moonlight on snow. Is _already_ ten times more bright. Ten times plus ten times, Pete.\" He craned his neck to look at me, then walked on.\n\n\"Hey,\" I said to his back. \"What is it about it, anyway? I mean, it's such a crackpot idea. I mean, is there anybody who actually thinks it's going to work?\" He looked over his shoulder again, gave me his smile. The only smile I've ever seen that never made me want to smile back. \"Even if it does,\" I said, \"I mean great, it's just like a big flashlight.\" I sped up until I was on his heels. \"Jack.\" Sometimes he forgot to answer to our nickname for him and I touched his shoulder. \"I'm really asking.\"\n\n\"This is only test flight,\" he said. \"They have plan put one hundred, two hundred, maybe more. Imagine this. Would be possible to make on ground lighting enough for whole city. No more streetlight. No more headlight. In Siberia, even in north part of Canada, in winter is so much dark, yes? No more. Now daylight one hundred percent.\"\n\n\"Okay,\" I said. \"But\u2014\"\n\n\"One hundred percent productivity.\"\n\n\"You're talking efficiency?\"\n\n\"Pete! This is what I'm talking. They target one city, another city. Okay? Whole side of world dark, yes? Whole half of world sleeping. But one city, in all of whole side of world: daylight. So beautiful, Pete. You can't see?\"\n\n\"No.\"\n\nHe shook his head. \"You have not live there.\"\n\n\"Neither have you,\" I said. He turned away. \"Serbia's not north. Its not even as north as here.\"\n\nWe were almost to the bridge when the train's headlight winked through the trees. I stepped off the track and started down the snow-covered bank. Mirza just stood there.\n\n\"Hey,\" I called. \"Get off.\" I could hear the rumble. The headlight flickered through the trees, then broke around the corner, suddenly steady, coming fast. \"Jack,\" I shouted, \"don't be a dick.\"\n\nThe whistle ripped a hole in the air. The headlight slung his shadow on the tracks. Slowly, he jogged towards me, slid down the snowy gravel. For a moment, I could hear his breathing beside me, and then the engine blasted all nontrain noise out of the night. The cars banged by, wind roaring off their sides, thundering towards the border.\n\n\"You're crazy,\" I mouthed.\n\nI don't know if he understood me, but he smiled like he did\u2014that awful smile\u2014and then was gone, rushing up the bank towards the train. For a moment I just stood there, watching him reach the cars, run, picking up speed. By the time I scrambled after him, he was racing at full sprint. The train was still passing him. It was way too fast. Beside it, I couldn't even hear my own shout, couldn't hear anything but the bdoom-bdoom, bdoom-bdoom.\n\nI was almost to him when he jumped. His legs flailed, thin arms struggling. Then he was on it. Just like Eli said. I ran full out beside him, an arm's reach away. Clinging to the side, he watched me. In the blur of the dark, I saw him start to slide away, felt my legs begin to go, the sharp jab of my breath, and reached for him. I grabbed his ankle. I pulled.\n\nFor a strange weightless moment all of his mass was attached to all of mine, my wrist an extension of his foot, my arm a continuation of his leg. He was guiding me through the air and I was following, easy, as if gravity, for that floating space of time, had freed us from its grip. Then we hit.\n\nMy feet slammed down, still running, stumbled, crashed, and I was on all fours, my only thought to stay clear of the wheels, then rolling away down the embankment to a stop. Above me, the train boomed on. Then was gone. Just its noise draining out of the night. Then that was gone, too.\n\nLimping up the bank, I saw a large dark lump a dozen yards down the track. It wasn't moving. He was on his face. I dropped next to him, shoved my hands under his chest\u2014felt him breathe, felt the warm slick blood\u2014and rolled him over. His jacket was soaked. His hair was pasted to his head, blood smeared over his chin. He was missing teeth. His lower lip was torn and hanging.\n\n\"I'll get you home,\" I told him.\n\nLooking up at me, he mouthed something that might have been in Serbian, or might have just been swallowed in his throat.\n\nI leaned forward. \"I was trying to help,\" I said. \"I didn't want you to...\"\n\nAll these years later, I can still see his face full of blood and moonlight, his eyes on me, his horrible failure to smile.\n\nMirza lived down a long slope of oiled gravel twisting its way through coniferous woods. Most of the snow had gotten stuck in the high boughs and they were matted thick. The moonlight had gotten stuck in them too. It was dark. Limping along, my ankle throbbing, I clung to his jacket, convinced I was helping him stay on his feet. We passed no lights of neighbors' houses, no TV sounds. Occasionally, a mailbox showed itself, hunched on its post, lifeless as its salted drive.\n\nIt must have taken us an hour to get to his home. We didn't talk. I didn't think he was able. Until, just before his house, he stopped, leaned over, drooled a splatter of blood onto the snow. Slowly, painfully, he said something through his ripped mouth. I couldn't make it out. He had to say it again: \"We were just moved to Vi\u00e4egrad.\" He formed the words with torturous deliberation, eased them through his mouth. \"Bosniaks. Moslems. They come to us at home. With knife\u2014\" He drooled into the snow again, made a fist, lifted it to his mouth. As slowly and purposefully as he had talked, he mimed a blade across his face, carving a picture into his forehead, his chin bone, his cheeks. \"Ubica,\" he said. \"In face of my tata.\" Even in the dark I could tell he could see my eyes searching his, could feel him willing me to see in his face what they had done to his father's. Then he reached over and touched my cheek. His fingertip was cold and so gentle that if we hadn't been alone I would have blushed. He traced the same carving on my face. \"Ubica,\" he said. And I realized he was making the shape of a u.\n\nWhen we got to his home it was as dark as all the others, a garage-less, small cape with unfinished dormers and a front door that hovered three feet off the ground. No deck at its foot. Not even any steps.\n\nHe started up the short dirt drive. When I tried to go with him, he made a noise in his throat, unpeeled my fingers from his jacket. Still, I followed him until the light over the side entrance snapped on. He shoved me back then, so hard my heels slid in the snow. I could see his face, now: the blood thickened around his mouth, clumps of his hair frozen in red-stained spikes. His stare was harder than his shove. I backed out of the light, watched him walk around to the unfinished steps, open the storm door. The unlit inside swallowed him.\n\nI would have left then if a window hadn't cut its shape into the black. It was upstairs, a dormer. Another one went on next to it. I caught the figure of a woman, blurred, moving fast. In a third explosion of window light, she stopped. The edge of the glass sliced her down the center, showed only part of an oversized T-shirt, bony shoulders, piled gray-streaked hair; I could tell by her movements that she was upset and, by her tilt, that she was looking down the stairs.\n\nI kept expecting her to go to him. Instead, she took a step back, and he rose into view\u2014hair, face, chest\u2014as if coming up through the floor. He hadn't washed off the blood and he was trying to talk to her through his mangled mouth, to soothe her, but I could see she was furious. When he reached for her she smacked his hand away. Her shouting was loud enough now that I could hear. Her fury seemed unhinged from anything I could understand, unraveled from reason, not just the Serbian words, but the simple fact of it with her son looking how he did. When he tried again to touch her, she raised a hand as if to smack his cheek\u2014my throat shut\u2014but hit him between his shoulder and his neck instead, then slapped at his chest, then was beating him with fists. He tried to get ahold of her. She jerked away, was suddenly gone from the window frame. A door slammed. I could see her in the window\u2014a bedroom\u2014pacing.\n\nThe first-floor windows flung their light onto the snow. Mirza went straight through the kitchen into the living room, picked up the phone. There was the slight movement of his dialing, and then he went perfectly still, motionless as the dried flowers behind him hanging off a nail in the wall. On the floor above, his mother tore in and out the window frame\u2014jerking gestures, she was talking to herself\u2014before she was gone again. For a moment, the entire house was still. Then she burst into the kitchen.\n\nMirza took one look at her and hung up. They stood watching each other, both motionless, him with one hand pressed to his face, one reaching towards her, her with the arm I couldn't see raising a hand I couldn't see to what looked like the side of her head. He took a step. She stepped back. Between the dark lump of her fist and the dark mass of her hair I could just make out the shape of a gun.\n\nI don't know how long they stood like that, only that the forest quiet seemed to reclaim the night and I became aware of a highway nearby. Once in a while a truck would moan through the trees. Mirza was trying to talk to her. She was talking, too. It went on. The house gradually drifted back to sleep, as if grown used to the lights and the chattering and too tired to care. The cold bit my face. My feet were freezing. I thought about going home.\n\nWhen she finally put the gun on the counter, he went to her as if it was a signal they had arranged beforehand, and I had a strange feeling, as I watched him pull her to him, as she let him, that this was something they had done so many times before they had grown good at it.\n\n_IV. The Concrete Plant_\n\nI told myself I'd talk to someone about what I'd seen. My parents, a teacher. But in my head, it always sounded like a confession; I could hear the questions\u2014they were always aimed at me\u2014but never my answers. In my silence, I tried not to let my mind imagine what it was like to live in that house. I told no one. I did nothing.\n\nEven in going to the library there was something that seemed wrong. Not twenty-four hours after it happened, I was on the public bus to UVM. I went in before dark and when I got out again it was still light. That\u2014the easiness of it, the tiny bit of time it took\u2014seemed wrong, too. Ubica. It meant murderer. That night I lay on the floor of my room feeling his fingertip on my face. I told nobody that, either.\n\nA week went by and none of us had seen Mirza since he fell on the tracks. That's what I told Eli and Greg and everyone: he had slipped in the snow, fallen face-first on a rail. He didn't return our calls. He hadn't been in school. But one morning when we heard on the news that they had cut the mirror loose from its station, that one hour before midnight it was to streak across the sky, we knew where we would find him.\n\nIt was a quarter past eleven when we heard him come through the hole in the chain link. From inside the funnel we listened to his footsteps on the frozen gravel, the faint thumps of him climbing. But the longer we listened the more it didn't sound like he was on the ladder. It was as if he was climbing some other part of the plant. The sound drew closer, then seemed almost level with us, then kept going, the clanks and thuds coming from above our heads.\n\n\"Holy fuck,\" Greg said, \"he's on the belt.\"\n\nThrough the funnel's mouth, in the big square of sky above us, we saw him. He'd wedged a flashlight in one underarm and we watched the beam creep along the defunct conveyor belt as it rose past the funnel, his dark shape covering constellations and crawling on and freeing them behind.\n\n\"Jack!\" Eli called.\n\nThe shape stopped.\n\n\"Get the fuck off there!\" Greg shouted.\n\nMirza's flashlight showed down on us. \"Guys,\" he said. His voice sounded like his tongue was swollen.\n\n\"You can see fine from here,\" I called. \"You can see great.\"\n\n\"Pete.\"\n\n\"Hey,\" I said. \"How's your mouth?\"\n\nHe turned his flashlight on his face: a lumpy mess of stitched lip.\n\n\"Looks good,\" Eli said.\n\n\"Looks fucking badass,\" Greg said. \"Get the fuck off that thing and come show us.\"\n\nHe did something unreadable with his mashed face, then put the flashlight back under his arm. We followed him with our own beams as he crawled along the rubber belt, higher and higher on the crane-like arm. By the time he reached the end, he was beyond the range of our flashlights. He stopped, shut his off. A small black lump at the very tip. Each time a gust came, the whole high conveyor belt swayed.\n\nThe three of us talked about how to get him off. We talked about how freezing it was. Every now and then, one of us would call to him to ask if the mirror had flown by. He didn't answer. At midnight, we climbed down. Halfway to the ground, I stopped on the ladder.\n\n\"It's not gonna happen,\" I shouted up to him. \"There must have been a problem. They probably aborted, Jack. They'll probably try again tomorrow night. Come on. Come down. We'll walk home.\" I couldn't even tell if he was looking at me.\n\nDown on the ground, the others wanted to leave him.\n\n\"He'll come down eventually,\" Eli said.\n\n\"Yeah,\" I said. \"When he drops.\"\n\n\"Well we can't just stand here all night,\" Greg said.\n\nEli nodded. \"We'll freeze. We'll get like hypothermia.\"\n\n\" _He'll_ get hypothermia,\" I said.\n\n\"He's Eastern European,\" Eli said. \"Eastern Europeans are genetically capable of withstanding lower temperatures.\"\n\n\"Fuck off,\" I told him. Then: \"Maybe we should call someone.\"\n\n\"I could ride my bike to the nearest house,\" Greg said. \"Call his mom.\"\n\nI wouldn't let him. They argued with me until it was clear there was something they didn't know and then Greg said, \"Well what the fuck then?\"\n\n\"I don't know,\" I said. \"Maybe we should call the cops.\"\n\n\"Are you frigging crazy?\" Eli said.\n\n\"Okay, what about the fire department?\"\n\nBy the time Greg returned on his bike, we could see the flashers coming. Through the trees, the red and yellow lights pulsed nearer until they were out in the open and we could see the truck. It was a pickup. They didn't have the siren on.\n\nTwo of Culver's volunteers came out slamming doors. One wore a lumpy knit hat. The other was bald to the cold. We let them know what was going on. They let us know just how deep in shit we were. The one with the hat walked back to the pickup and got the floodlight blasting. It hit the concrete plant all at once, grabbed it out of the night like it had yanked it by a cord, then crawled up the conveyor belt to the top.\n\nThe bald one shouted to him. Mirza didn't say anything back. There were long minutes of one-sided conversation. After a while, it started to seem like the fireman was a fool, shouting things up at a night sky that didn't seem to care, and he stopped. He and the other volunteer talked too low for me to hear. When the bald one turned back to the conveyor belt, he shouted, \"Alright. I'm going to come get you, then.\" He was watching the ground as he walked, heading for the hole in the fence, so he didn't see Mirza take out the gun.\n\nThe other one shouted.\n\n\"What?\" The bald one said, then looked up. He had reached the fence and his hands were on it and it shook a little. \"Kid,\" he called. \"That better be a toy.\"\n\n\"Where did he get a gun?\" Eli whispered.\n\n\"Is that a toy?\" the other volunteer said. He was talking to Eli.\n\n\"I don't know,\" Eli said.\n\n\"Let it go,\" the bald one shouted up at Mirza. \"Just drop it and we'll forget it ever happened.\"\n\nThe flashers lit the fence yellow then red, yellow then red. The spotlight stayed on Mirza steady as if it was sewn into the sky.\n\n\"I'm coming in,\" the fireman shouted. \"If you shoot me, I'm going to be pissed.\"\n\n\"If you shoot...\" the other one started.\n\nThe bald one held up a hand. \"He's not going to shoot,\" he said. \"Kid? If you're going to shoot, tell me, okay? That only seems fair.\" Puffed in his parka, he was having trouble getting through the hole in the fence. \"I'm not armed,\" he shouted. \"So, how about you give me fair warning. Say, fire off a warning shot. How's that sound?\"\n\nHe was through to the other side and pulling his coat free from a snag when Mirza shouted something. All of us looked at him.\n\nIn the floodlight, he held the pistol pressed to his temple. Eli and Greg both said things they wouldn't normally say within hearing distance of adults. I was silent.\n\n\"Okay,\" the man in the fence called. \"That's not what I meant.\"\n\n\"Don't come close,\" Mirza shouted.\n\n\"You bet,\" the volunteer called. \"See. I'm stopped.\" He stood a few feet inside the fence. \"What do you want?\"\n\n\"Turn off light,\" Mirza shouted.\n\n\"Sure,\" he called back. \"We turn off the light, you drop the gun, we're all happy, right?\"\n\n\"Turn off first,\" Mirza said.\n\n\"Go ahead, Lyle,\" the bald one said. He didn't take his eyes of Mirza. \"Lyle, go ahead.\"\n\nAs soon as it was off, the flashers seemed brighter. They pulsed their colors over the trees, the fence, touched pieces of the concrete plant, reached partway up the conveyor belt. After about fifty feet, though, it was just a black solid strip stabbing into the dark bowl of the sky. The stars shivered and swam. When the light appeared, half the size of the moon and twenty times as bright, I felt my breath leave my body. I watched it, waiting for it to move, to slip uncannily through the web of the stars. It was only when it didn't that I realized it was Mirza's flashlight. He was pointing it straight at us. Then the beam slid off us and landed on the fireman inside the fence. That was when he started shooting.\n\n_V. The Space Mirror_\n\nMoon Slayer. Star Demolisher. We never saw it. That night, we thought it must have swept by too far to the north. Or maybe they just aimed it somewhere else. They were steering it from their control center in Moscow; this is what Mirza told us one of those nights that long-ago fall. He said the mirrors had to be angled just right to throw the reflection where they wanted. Maybe, we decided, of all the moments over all the miles of the earth, the few seconds over an abandoned concrete plant in northern Vermont were just the wrong few seconds. Maybe in those seconds whoever had been at the controls had just decided to give the nighttime a moment to itself, angled the sunlit wings away from the planet, let darkness catch its breath.\n\nI thought he would kill the men that night. They were both shouting\u2014we all were\u2014the bald one tearing at the fence, struggling to loosen his parka from the barbs, his shout suddenly pinched to a scream. But it was the wire that gashed his cheek, ripped a hole in his palm; the bullet just nicked his neck. One inch closer, the paper said. Fortunate, it said. They put Mirza away.\n\nBy the time we were finished with tenth grade, his mother was gone, too. It was assumed she moved down near Rutland to be closer to the juvie prison. But when they let him out, he came back alone. We were all off to college by then\u2014Greg and Eli and me\u2014and when I heard he had returned to Culver I swore I'd stay away. I did, for a few years.\n\nHe's a line repairman now. On call for the electric company. I don't know much more about the work other than it's dangerous. He lives alone in a trailer way up in the pine woods along Rattlesnake Gutter. Our postmistress tells me he has a lot of dogs. She doesn't know much more; nobody does. He doesn't talk with people. Or they don't talk with him. I've seen him twice. The first time was in the co-op. He was buying bulk. I only glimpsed his back\u2014greasy hat-mashed hair, beard bushing at his jaw, shoulders thin and brittle as when we were boys\u2014before I turned and slipped out of the store.\n\nEvery time the first snows come, I watch the slow pileup on the deck chairs, or lose myself in the swirl before my headlights, and decide to hire him to plow my drive. People say he does a good job, shows up before they're awake, sends them a bill. They don't know his phone number. It's not listed. Send him a letter they tell me, a check. Each winter I find myself on the north side of Rattlesnake Gutter, stalled in my car, looking through the passenger side at his house. Black trash bags are duct-taped over two windows. Through a third, I have seen the yellow inside of a microwave and, above it, a sagged line of gray socks hanging. Once the dogs start they keep on until I drive away again.\n\nThe second, and last, time I saw him was almost a year ago. The night before it had snowed big and I was halfway up Rattlesnake Gutter when I met him coming down. The gutter road is one car's width, cliff on one side and gorge on the other. We idled in each other's lights. Then he backed up, lowered his plow blade, and cleared a turnout for me. As he drove by I rolled down my window. He stopped just past. I had to lean out into the cold and crane my neck around. He watched me do it. After a moment, he raised his hand good-bye, began to roll his window back up. His brake lights paled.\n\nI said the first thing that came to me: \"You know it never deployed.\" He didn't look like he knew what I was talking about. \"Banner,\" I said. \"Znamya. The reflector never opened. Something with the controls, I guess. They had to jettison it.\" And then, in his silence, as if it was some kind of excuse: \"It was on the news.\" And then, of all things: \"I'm sorry.\"\n\n\"For what?\" he said.\n\n\"For everything. What happened here, then, your father, everything Jack.\" I had not thought of him as that in years.\n\nHe looked at me as if he had not known till right then who I was. \"My tata,\" he said, \"used to stand on new bridge over river Drina and throw women off just to show how he can shoot them before they hit water. Women he has raped.\"\n\nSnow was mashed against the curve of his plow blade and I watched a chunk fall off, silently, in the snow beside it.\n\n\"In Rutland,\" he said, \"I live two years uncomplicated life.\" He lifted his hand. \"There is no place for turn up there,\" he said. \"You must back whole way down.\" Then there was just the sound of his window glass squealing upward, his tires creaking away over the snow.\n\nMy wife says I can't know he's unhappy, tells me he would have ended up like that anyway, that long before I knew him things stronger than me shoved him on a trajectory I couldn't have changed. But on nights of bad weather I still can't sleep. Somewhere out there a limb is cracking, a tree is down, a line snaps at the rain. On those nights, awake beside the sleep breathing of my wife, it seems the whole of the natural world is waiting to punish me with what it has in store for him.\n\nI wonder if he thinks about that night I pulled him off the train. If he imagines how it would have been to ride on, to cross the border in a thickness of north woods, to unbuckle from the ladder and leap free of the cars. I wonder if he dreams of that artificial moon rising above the trees, bathing the forest in its beam, chasing the night off him into the depths of the woods. I wonder if that's what he saw each night after lights-out those two years he spent in his cell.\n\nThe first cold snap has already come and gone. We're in the last of a late Indian summer. One night soon I'll go to the concrete plant. The town tore it down long ago, turned it into a gravel pit. Each fall I walk the dirt road in and come out of the woods into the sudden openness of gouged earth. I try to imagine the pit lit with an unearthly glow, alive with the rumble of backhoes and cranes holding the dark at bay until the natural dawn. But there are just a few machines left alone for the night, the railroad tracks waiting atop their ties, the cold expanse of the sky. The air is quiet as only a treeless place can be.\n\n# [**THE POINT  \nOF ROUGHNESS**](toc.html#ch5-R)\n\nI'm behind the barn, splitting burn wood, when I see the bear coming for our daughter. It's December, dusk. At my back: high piles of cut rounds. Out in the field: the bucked trees stacked, their drag marks dark in all the snow, the pines looking almost black beyond. And between their trunks: a patch of true black moving. Everything else is still\u2014the stone wall, the glass greenhouse, the sledding hill behind our home, packed hard by the weight of my wife and daughter gone down run after run\u2014except a spot of orange: Orly in her snowsuit. Rolling snow boulders. Down by the old stone wall at the edge of the woods. Beneath the splitter's rumble, the shaking of the pine boughs is a silent ripple washing steadily towards her.\n\nFor a second I can feel her in my hands\u2014the heft of her when I first pick her up, my arms strained with her struggling\u2014and then it's just the log again and Orly is out there, suddenly standing straight up, staring into the trees. Her hands are bare\u2014she will not suffer gloves, shucks mittens as soon as she thinks she's out of sight\u2014her fingers stained so bright by markers I can see them slowly curling towards her palms. She takes a snowsuit-stiffened step. Another. The first time we zipped her into the hunter's camouflage, I crouched down, winked. _Hey bub_ , I said, _get me a beer, eh?_ Bess laughed. But Orly only asked, _Who's Bub?_ And when I poked her bright orange belly with a wriggly finger, my wife said _Ev_ , the way I knew meant _stop_.\n\nI shout it now\u2014 _Stop!_ \u2014I must\u2014 _Orly!_ \u2014because she goes still at the wall, small hands on the stones, standing on tiptoe, peering over. But in my ears there is no echo of my voice, only the thudding from the hog stall, the chugging of the splitter waiting for me to load another round. _Move_ , I tell myself, _run_ , can feel her grabbed away into my hug, her warmth close as I shush her, Bess glancing up, seeing us through the kitchen window... And I am grateful for the near dark that hides me still standing here. Loading the log. Cranking down the handle. There is something hypnotic about watching hydraulics work, the steel ram pushing the round, the first touch of the wedge, the slow cracking open of such a solid thing, and, anyway, it's only a black bear, probably already gone.\n\nThey show up every spring, shaggy from sleep, coats loose on shoulder bones. Our first year here in these New England hills, one night at the almost end of winter, we stood at the bedroom window, watching them. Two cubs digging at the compost pile, jerking back from pawed puffs of heat, shaking their heads, pouncing on the fleeting steam. I looked at Bess\u2014just that fall we'd faced it: my surety that I didn't want to threaten what we had by having kids; her struggle with what it meant then to stay in love with me\u2014but she was smiling, her laugh its own puff on the window. Wrapped in the comforter, we watched them play and play, as if they'd discovered a game that could keep them happy indefinitely. By summer we'd built a better compost bin but there were the blueberry bushes, the beehives, the scent of Bess and me so fresh from sex the grass was still imprinted on our skin. Once, in the orchard, lying beneath last apples beneath a late October sky, I looked at Bess above me\u2014her cloud of curly hair barely held back, the loosed strands swinging to her rocking hips, the lighter brown of a face impossible to look at without thinking of the sun, those breasts she mostly hid from its brightness, but not from me, paler still, still perfect\u2014and saw, not more than twenty feet behind her, that unmistakable shuffling shape. We leapt up, ready with hand claps, shouts, but it was already wheeling, as if shocked into flight by the sight of our naked selves. Winter would find us naked again, out in the greenhouse, beneath our marriage lights. Nine years ago we'd strung them up, three thousand tiny bulbs glittering above us as our guests watched us exchange our vows. Now, each Yule season, we light them again. Each solstice night we step back in, spread out our sheepskins beneath a galaxy of our own stars. Then, sweaty, steaming, we burst back out, whooping at the wildness of running naked through thigh-high snow, at the nearing woodstove warmth, the feeling that we've escaped the darkness for another year. Of course, by then, the bears are gone, curled up in caves, safely away.\n\nExcept for this one: a large black bear standing in the twilight, hardly ten feet from Orly. They stand so still, staring at each other, that when I move to lift away the log's split halves it seems a breaking of some understanding. The pieces clatter onto the cordwood. The bear doesn't flinch. Orly doesn't look at me. And I can feel it: the fear that she will. If I could see myself I would be shouting at me to do what anyone would do, what, as I reach for another round, I am aware I am not doing, am wondering, instead, what the bear will do, what Orly... For a moment, I think she's singing to it. Which could be true. Kids like her can have a way with animals the rest of us won't ever understand. I listen for her voice, a hum in my ears.\n\nOrly. It was Bess who chose her name\u2014Oralee, Hebrew for \"my light\"\u2014but it was me who shortened it. And Bess? I don't know what her name means. Nor Evan, not more than that it's Welsh. Once, we considered renaming each other, trading in our parents' choices for the language of this land where we'd become ourselves. I chose for her _Wanee-mbee-shkwa_ : Good Water Woman. And she for me: _Nawasnaneekan_. It means \"my light,\" too.\n\nAll the seasons grown together, all the water, the sun, everything taken from the soil over all those years, all starting to rip apart: that is what a piece of wood sounds like when it splits.\n\nSomehow through it I hear the screaming, see the lunge, the blackness blurred, the spot of orange scrambling backwards in a burst of snow exploding with the back door's blast of yellow light, Bess flashing through it, the frantic form of my wife running. And before the hatchet is in my hand I know that hers are empty, know, as I start running, too, that this animal\u2014which should not be here, which should be hibernating beneath the earth somewhere\u2014will turn from its small prey to the larger threat, charge my wife instead.\n\nThen I am there, our daughter screaming somewhere behind me, Bess holding her, and in front of me only the bear. The gleam of its lips, the wetness of its nose, the flecks of snow like spittle stuck in its muzzle hairs. I can see its breath, sense the nostrils flaring behind the steam, feel in its eyes\u2014too dark, too small, too buried in the blackness of too huge a head for me to make them out\u2014its fear. Its confusion. Or maybe that is my own. Because the world is coming back around the bear, the snow and the woods and the nearly nighttime sky, and I realize its shape is receding, though it takes me another second to understand that means it's backing away. Another few steps and it turns\u2014how huge its shoulders, how beautiful the rippling fur like water over a boulder in a creek\u2014and leaps the stone wall, crashes into the pines, disappears.\n\nIt's only then that I can look behind me. Already Bess is halfway up the hill. Orly in her arms. My wife's back, her thin shirt: she hadn't even put on a jacket. Above her shoulder there is the pale spot of our daughter's face. Below, something I think at first must be a flash of snow kicked up by Bess's boots. But it is her flesh, her feet. She's barefoot. Something about that makes me want to cry.\n\nWhat is it about a sick animal? What is it in us that can sense it? That knows when something has come uncoupled from its nature, gone somehow wrong? When I turn back to the woods, night is waiting beneath the trees, too dark to see what might be watching back. But I can feel it. I can feel its stare.\n\nThese are things that I have done: forgotten to buckle her into her car seat, left her alone in the bath, let her get between the hooves of our nanny goat and suckle from the unwashed teat. Last summer solstice I jumped the Kupala fire with our daughter in my arms. Bess used to jump the flames with me, our hands locked to bless the strength of our bond. Where was she then, when, crouching in that crackling light, I urged Orly to attempt the leap herself? Or when, a month ago, I stood at the edge of Howland Pond, watching our girl, her mouth-wet blond hair brittle from cold, her four-year-old face furrowed with focus, go farther and farther onto the ice? It was a cold day, our first true feel of winter, and the pond was singing. That has always been one of Bess's favorite things\u2014the low booms and long echoing cries ice makes when it is freezing\u2014and I stood there listening, wishing she was with me, while Orly wandered out beyond the shoreside ice, towards the crying coming from the ever thinner center.\n\nI can only imagine what it sounded like to her. There is no way for me to know. Our Orly hears things differently. We knew this from the start. I think it was what made Bess want her: four thousand miles from our home, in a Far North orphanage filled with the strange sounds of a language neither of us understood, Bess cooed a note and the six-month-old in her arms gurgled the same note back. Bess sang another; the baby replied, perfectly on pitch. Now they spend whole afternoons, Bess with her flute, Orly singing along. A free-form warbling, a swooping through the house, her small feet thumping time around and around the living room rug, her body a loose-flopping ungainly thing, but her singing beautiful. Magical. No matter what my wife plays, Orly matches it, mates it with a wholly intuited harmony, makes something so spectacular that, even after all this time, I can't help but stop, leave whatever task aside, shut my eyes.\n\nAnd there are times I have walked in on our four-year-old and found her alone humming a single note. One long note, broken only for breath. It was Bess who figured it out: our daughter imitating the refrigerator, or the fan, or the washing machine downstairs, turning their tones into a chord. The way the whistles of a kettle, or a spatula scraping a pan, must, in her ears, become something else. She pounds her little fists against her head, pulls at her hair, runs from me, screeching, leaving me standing with the shopping cart on its offending wheels, or in the blinking lights of some construction vehicle backing up, struggling against my own urge to shout. If I do, it doesn't matter, anyway. She doesn't understand angry. She doesn't recognize relief, or pain, or joy, or me, or Bess, or anyone. How can she without empathy? Oh, she _feels_ things. She _knows_ us. But she won't ever understand us, who we are, beyond the woman she calls _Mom_ , the man she comprehends as _Dad_. Which I am not. That is something _I_ can't feel, hard as I've tried.\n\nWhen we first brought her home I would try to make her smile. I'd make the faces I always did with other kids\u2014the children of couples who always said _you'd be so great_ and _got such a way_ \u2014and she wouldn't even watch. I'd try pointing out things that made me smile; she seemed unaware of what my finger, my eyes, my eager face might mean. _Honey_ , Bess says, _you just have to find your way of connecting to her, her ways of connecting to you_. But how can I when she won't so much as wave good-bye? Or say hello? Or, on some days, even answer to her name?\n\nEven now there are days when Orly hardly says a word. There are days when it seems all she does is scream. And hit. I have seen her beat at another child so hard she dislocated her own wrist. I have seen her bite her mother, witnessed Bess's bleeding neck, ear, lip, scratches on her eyelid that made my fingers shake too badly to unpeel the Band-Aid. Last year, when Bess's mother died, our daughter sat there singing, singing to her heart's content, as if my wife's heart wasn't broken, as if the child's own mother wasn't weeping while she held her.\n\nBut God she's beautiful. Even more than her long blond hair, fine as if spun from sunlight, more than her small face pale and heart shaped as a barn owl's, more than the Bering Sea blue of her eyes, it is her expression, the seeming serenity of her very detachment, the sense of something that can't quite be grasped. True too: she doesn't like to be touched, won't let me hug her. Though Bess has found a way, can stand perfectly still, arms by her sides, and through some connection that they've made communicate to Orly that she's available for hugging, and it is, yes, a beautiful thing to see the girl approach, tentatively extend her arms, wrap them around my barely breathing wife, squeeze ever so softly. _Each day_ , Bess says, _I understand her a little more_. She says Orly understands her, too, in ways she never knew she could be understood. She says one day our daughter could become a brilliant composer, a prodigy on any instrument, one of the many with her condition who find in it a singular unmatchable grace. That may be true. But what I do not tell Bess is that I am not worried about what our daughter might grow to be. She never will grow up, won't ever live away from home, won't, for all her life or all the rest of ours, get through a single day without some help from us. No, I would like to tell my wife, I am not worried about what will happen to her.\n\nThe day after the bear, I find its prints. The sun is splintering through the trees, painting the forest floor in streaks, and they are there, in the snow, sunk deep. Earlier, I'd come in from the goats and found Bess stirring oatmeal on the stove. Usually, I fix Orly's breakfast, but Bess took the fresh milk from me, said she'd pasteurize it herself. And after, when I started to take our daughter to feed the hog\u2014one of the few pleasures Orly and I share\u2014Bess held her back with a strange seriousness, her face stricken as if she'd suddenly seen the sow as some new threat.\n\nBut she was staring at me. Sometimes, stepping into the truck, or triggering the security light outside our home, I catch my reflection in the rearview mirror, the glass pane of the storm door, and see that same stare back. Last time I trimmed my beard, I could not quit thinking how strange: that hair grew from my cheeks, around my mouth, as if my face wanted itself to disappear. And how is that different from any man's? How, enlarged by Bess's makeup mirror, could it look so inhuman: hairs creeping from my pores, skin swallowed by its own pelt. I pressed it with the scissors' tip. Soft buckling, a bead of red. I touched my fingers to the cut, then to my tongue. That metallic tang: same as anybody else's blood. Except mine is in me. I have tried to find it\u2014this thing that must have lurked since long before Orly, since I was small as her\u2014but had I ever harmed a single kid who came through my grandmother's home? Of all the hundreds in all the afternoons of helping with her day care? Had I ever even raised my voice? Let alone a fist against another? Never. Not a fight. Not even one. Not even when, grown far too big for whipping, I'd still be sent out to the yard, forced to fetch a bamboo switch to replace the one my grandfather's beatings had begun to fray. Which I know is no defense for this. Nor is the fact that I had few friends, that before Bess I'd never had anything that felt worth keeping. Now, most days, I keep to myself, spend from dark to dark in silence, working alone on land my wife and I once worked side by side. Most nights I offer her a break from Orly, or do the dishes beneath the sounds of them upstairs, wait for the half hour we might have together if Bess doesn't fall asleep putting Orly to bed. Maybe in the morning I'll wake beside her before our daughter starts calling for her mother; maybe my wife will kiss me when I come in from letting the animals out; maybe for a few moments then we'll talk of something other than Orly before I start the hours of my own sounds amid the silence that has become my life.\n\nThough it is the sense of something else out there that brings me back. The unwatched woods, the emptiness between the trees: I look up from the paw print searching for the bear. Holding the heavy chain saw, I scan the shadows. As if I might sense it, hear a sound other than my own breathing. As if it, watching back, might recall my sound, recognize my scent, know me.\n\nBut once in the woodlot there is so much to do\u2014this year's cords all sold, next fall's still needing cutting, splitting, setting aside to season\u2014that it drives away everything but work. These are the weeks of darkness spreading sudden and fast. The day is there, then done. It's almost night by the time I shut the saw off, make my way out of the woods. The goats bang at the shed door as I pass, the chickens quiet in the coop. Home, I can hear Bess putting Orly in the bath. In the skillet: what's left of a supper they've already eaten, the iron handle still hot.\n\nNormally this is when we're closest, this season of long nights and late mornings, a few more moments managed together beneath warm covers, small celebrations we make together with each last turn of the Wheel of the Year. When we were young, still giddy with our newness, our shared escape from PhDs and years of folklore studies into this place where we could find our own fable, what had grown between us had seemed so wondrous it was almost overwhelming. We'd turned to the ways others had long tried to fathom the mysteries of their lives. Each Wren Day, we would stuff our jackets full of straw, dangle a chicken from a pole, drive door to door handing out plucked feathers to perplexed neighbors, as if to share with them a little of our luck. Home, we'd strip, light candles around the bathtub rim, slip in. Bess would lift a ladle over my head, shut her eyes so I could do the same.\n\nBut the splashing scares our daughter. She tries to touch the candle flames. And, anyway, these days our solstice rites are meant to shine a light on Orly. We dress up in Koliada costumes in lieu of straw-boy suits, drive neighbor to neighbor singing her birth country's songs, the hen exchanged for a plate of cookies cut in reindeer shapes. Orly holds them out the way we once did feathers. The neighbors seem to like this better. Maybe us better, too. Though this year Bess goes without me. Back from the woodlot, a few days before the solstice, I find a note _\u2014safer while it's still light_ \u2014a broken-antlered cookie. And though Bess spends the whole next day making Orly's Rozhanitsa dress, half the night embroidering it with the image of the branch-horned goddess, when I offer to take our daughter to the equipment shed, hunt through the box of buck racks, make an antler crown, I get only a shake of my wife's head.\n\nStill, I remind myself, all of it was always about more than only us. The first time that we heard the singing of the ice, Bess whispered, _This_. We were sitting on the abandoned beaver dam, my not-yet wife leaned back into my chest, the pond spread out before us. _I know_ , I told her. And, together, we listened: a sound as if the earth, conversing with the heavens, had decided to let us hear. I think it was that moment that we decided there must be something. Such safety: to know there is a hand to steer us, to help us shape our lives more caringly than fate. Such sweetness: to have found our faith in that together.\n\nEver since then we have gone to the pond when life seems worst, listened, tried to open our hearts to guidance. The year that Bess, at forty-one, was stricken by unbearable desire for a child. The night that I gave in. The day, after a year of trying, then of searching, that we brought our daughter home. The November, almost exactly one year later, that Orly was diagnosed. The autumn, so long ago, back when we had barely brushed our thirties, when I had tried to tell the woman I loved why I couldn't stand to share her with a baby, couldn't face the thought of losing the world we'd made between us to another life. Weeks of talking had solidified into an ice-hard choice: whether she would hold tight to what we had or leave it\u2014leave me\u2014for the chance to have a child with someone else. Scared, I'd been unable to stop talking, to let us listen to the sounds of the ice, until she pressed her fingers to my lips, said, _Shh_. And half an hour later: _Don't you know?_ I'd risen, then, as if the ice had finished her words for her, because I did.\n\nNow there is this: a week ago I carried the chain saw out to the woodlot and found a tree half-cut, a wedge sawed from its trunk, the felling hinge already made. A dangerous mistake, to leave a tree like that. But, standing there that morning, I couldn't remember having done it. And worse, the end of that day I had the urge to do it again. The chain saw idled. Alone in the empty woods, too close to dark to start another tree, I told myself to shut it off. Then stepped through the snow to a marked trunk, tugged the trigger, revved the blade. I cannot explain the way it felt to rip through to the point of the hinge, cut free the notch, and then shut off the chain saw, turn in the silence, and head home.\n\nThe next morning I couldn't take the tree down. I left it that way, prepared for felling, and worked all day felling others and, when it came time to quit, did it again. Except this time I cut a little deeper. Went an inch beyond where the hinge should be. Removed a little larger wedge. Left that tree standing too. Seeing that notch I knew was too wide, amateurish, edging on dangerous, I felt a surge of relief. Enough\u2014that tang of oil smoke, shiver of sweat freezing on my neck, the sense that I could breathe\u2014to make me need it again the next day. And the next. Over that week, I must have left two dozen trees throughout that section of our woods, each notched a little deeper, weakened a little more, a forest of half-felled deadfall traps waiting for the next wet snow, the first wild wind, to bring them down.\n\nOn solstice eve I make a cut so deep I keep expecting to hear the crack, to see the sudden tearing rush of canopy above, but when I stop almost two-thirds through, there is nothing but the stillness of all the trees and a sense of dread. I look up from the blade, scan the woods: just my breath gusting white through the visor's mesh, my veins squeezing my blood.\n\nAt home, tonight, we will prepare our marriage lights. It is the one remaining ritual we still keep for just the two of us, draping the greenhouse each winter solstice with the same strands, setting the same celestial cocoon aglow. And each year, the night before our anniversary, we sit together on our bedroom floor replacing all the bulbs that have burned out. For every one that we remove we tell each other something too hard to have brought up before, whatever we need to exorcise before the unborn year. And for each new bulb that we set in the old one's place we give each other an assuaging gift, maybe a memory, a touch, anything that commits us to each other for another year, and the year after that, and the one after that, and the one after.\n\nAlways, we start small\u2014last solstice eve, Bess told me how sad she was I'd shaved my beard even though I'd known she loved to feel it when we kissed; I talked of times, reading Orly to sleep, that Bess had drifted off without so much as wishing me good night\u2014and, reaching over, she smoothed my eyebrows; I massaged the soft spot on her palms; we helped each other towards the harder ones. For Bess, that year, it had been what I'd done to our new tradition: her, dressed as a goddess, emerging from the cellar to tell the fortunes of good girls, while, at her side, to keep bad ones in line, there stood the demon Krampus\u2014me in branching-antler headdress, sheepskin mask with carved-out eyes. Bess's hands shook as she recounted how far I'd taken it, how, through her shouts and Orly's screams, over our daughter's cringing shape, I'd kept on roaring, roaring, unable to stop.\n\nBy now, the branches above me have turned black, the sky nearly dark enough to swallow them. I stand, listening to their silence. The way I sat, silent, listening to Bess until she'd finished. Until her shaking hand threw the bad bulb out. And, in its place, she fit a new one. I don't remember what sweet thing she said. But I know she must have. We always do. It's why it works. In the darkness there is the small clatter of a twig knocked loose. Tonight, I think, Bess will sing our girl to sleep. There comes the stick's long falling to the ground. I'll do the dishes. Its faint landing in the accepting snow. We'll meet in our bedroom and begin. This one thing that even Orly has not altered. Yet.\n\nIn the space after the thought I can hear my own fear. I know I should go home, that supper must already be on the table, Orly buckled into her high chair, Bess blowing on her soup, but I do not know what I might do if this last thing goes, too. I set the chain saw down. There are the first pricks of stars.\n\nLong after dark, I come in through the kitchen door to find a tray of fresh-baked oat bars\u2014still warm, smelling of molasses\u2014waiting on the stove and, nestled beneath one corner, a note: _Shhhh. Put her to bed early_. And when, cupping a bit of oat bar in my palm, I pad up the stairs to our bedroom, Bess is there, cross-legged on the floor, surrounded by a forest of electric wires, the glow of a thousand bulbs. In the clearing where she sits: just enough space for me.\n\n\"You made it,\" she whispers.\n\nOur room abuts Orly's, the house was built two centuries ago, its walls are thin.\n\n\"I was in the woodlot,\" I whisper back.\n\n\"After dark?\"\n\nTaking a bite of oat bar, I mumble \"afterwards\" and, through my chewing, \"out to the pond.\"\n\nShe always seems to know when I am lying. So when she says, \"I was out there,\" it stops my jaw. But she goes on: \"A week ago. Couldn't hear anything, of course. This time of year. All frozen over. Though I tried to break it\"\u2014her face winces the way it does when she's embarrassed\u2014\"by throwing rocks.\"\n\nMy eyebrows rise, my jaw unsticks, but before I can say anything she tells me how she'd stood there hurling the biggest she could find, the way they'd clattered and skittered and in the end stayed silent on the surface of the still-solid ice.\n\n\"Oh,\" I say, taking another bite, \" _that's_ what those were.\" And, stepping across the lights, I settle on the floor. \"I thought they were some kind of sign.\"\n\nThis time the slight wince is for me.\n\n\"God,\" I say, \"I feel so stupid. I mean, I _prayed_...\"\n\n\"To the rocks?\"\n\nI look down at the lights around us. Our knees are almost touching.\n\n\"For what?\" she says.\n\n\"That we...\" Glancing back up, I find her eyes. \"That you might hear my need.\" I make my face go slack with wonder. \"Might bake us a pan of these.\" My wife has always been unable to keep down a smile. \"And that Orly might leave me one to eat.\"\n\nHer laugh is little more than a wider smile, but I press my finger to my lips, give Orly's door a sideways glance. When I look back, even the smile's gone. She reaches for my hand, whispers, \"Take a dead one.\" And we start.\n\nIt's never easy, but tonight I watch her with a different kind of worry, can't shake the feeling that each small admission of hurt or trouble is simply her preparing me for worse. Still, in between, we replace the bad bulbs with the good: the way that she surprises me at night with herbal teas she's dried herself; the fact that the day after she held my hands like this last year I started growing back my beard.\n\n\"Actually,\" I tell her, \"I started right then, that very night. I could feel it prickling my face.\"\n\n\"Me too,\" she says. And in the softened line of her mouth, the ease that settles over our hands, I can tell she's remembering those early hours of last winter's shortest day, how, after finishing the lights, we'd made love on the floor in the midst of all those bulbs, how it had felt like sneaking something slightly forbidden, how, later that same day, the thought of it had covered our lovemaking inside the greenhouse with a second sweetness, our bodies finding their new moment within the memory of the one before.\n\n\"It's not scratchy anymore,\" I whisper.\n\n\"Let me see,\" she says. And, leaning over, kisses me.\n\nMy beard is a cloud compressed between us, and, for a second, I wonder if it's going to be okay, no worse than last year, no harder than the year before that, but, by the time I've thought it, her mouth has left mine, the pressure gone from my lips, my beard back to my beard, my face, my face. On hers there is what I've been worrying would be there.\n\n\"Ev,\" she says, \"how could you not have seen that bear?\"\n\nMy face, my beard, but it is as if the air has changed. I say, \"What happened to taking a bulb?\"\n\nHer hands twitch. I tighten my fingers around them.\n\nShe says, \"You wouldn't have let it\u2014\"\n\n\"I didn't.\"\n\n\"But you took so long.\"\n\n\"It was just a black bear. We see them all the time. Remember that first spring? Those cubs?\"\n\n\"This wasn't a cub.\"\n\n\"Remember how we watched them? Just the two of us wrapped together in that blanket? We were standing right over there.\"\n\nWhen she turns to look, the skin below her jaw goes taut and I can see it\u2014our home in that first year, Orly's bedroom still a meditation studio, the lights now on the floor still glowing on the greenhouse, and it is becoming spring, the fields beginning to show their soil, and behind us the sheets are stained, and who cares with no one but us to see them, and soon we will return to the bed to stain them more\u2014and part of me doesn't want her to turn back.\n\nBut she does. Her eyes seem wet. \"I remember afterwards,\" she says. \"The way you were so straight with me that night.\" Easing one hand free from mine, she slips a new bulb from the case. \"How you told me you were so worried that you had pushed your own resolve on me, how you would understand if I still wanted to be a mother, give birth. How much that made me love you, your honesty, that I could trust you, that you would face something like that and talk it through and at the end we'd bring what we had come to with us to bed and how amazing it always was, then, like that, like that night.\"\n\nFor a second, the spot where we had stood seems so close, the window we gazed out of so near, that I almost think she's right: we still can, could, one night of talking, of taking it to bed... And then\u2014a rustle of sheets, the sound of a sigh, my eyes adjusting to the dimness beyond the lights\u2014I see the lump in the comforter, the depression at a pillow's edge, the hint of golden hair.\n\n\"Honesty?\" On my face I can feel the new bulb's heat. \"Straight with you?\" Out of the corner of my eye I can see it, suddenly too bright. \"Bess,\" I say, \"I _was_ out in the woodlot. I just didn't want you to know what I was doing. Because\"\u2014I look at the bed, make sure Orly's still sleeping\u2014\"it's been a long time since we've been like that. That way that you remember. Since we were _together_ like that. Part of each other's lives, of each other, _the_ other part, the _only_ other part. The way we were before\u2014\"\n\nHer voice is a whisper: \"Before what?\"\n\nMy voice is not: \"Her.\"\n\nThe word sounds worse after I've said it. I wish it would disappear from the room. But even after it has lost its sound it seems to stay.\n\n\"Evan,\" Bess says, \"can I still trust you?\"\n\nWhen I glance at the bed again Orly's eyes are open. I start to tell Bess.\n\nBut she says, \"Can she?\"\n\n\"Of course,\" I say.\n\n\"I want her to hear you say it.\"\n\n\"Of course you can, Orly,\" I tell the staring face. \"I'm your father.\" It feels strange saying it like that, such a statement, and afterwards I'm relieved I didn't add _practically_.\n\nBut Bess only shuts her eyes, shakes her head. \"Then what,\" she says, \"is wrong with you? What the _fuck_ \"\u2014with the child in the room the word is ten times worse\u2014\"are you doing? Who _are_ you?\"\n\n\"Your husband,\" I tell her. \"Who loves you.\"\n\nHer eyes snap open. And beneath her stare, I shut mine. In the silence between us I can feel Orly waiting. Or maybe she has rolled over, gone back to sleep, left us to do what people like us do.\n\n\"Evan.\" When I open my eyes something in Bess's face has softened. \"Can't you see what that's like for me? For me to know that the man I love, the only one I ever have, loves me so much he'd do anything to have me back?\"\n\n\"Not anything,\" I say, but my throat closes on the words. It is the worst thing I have ever known: the knowledge that she is right.\n\nReaching to her sides, she draws the masses of bulbs into a denser bunch around her. A tinkling hailstorm.\n\nIn the quiet after it stops, I say, \"Us. To have us back.\"\n\nShe mounds the lights over her legs until her lap is buried in them. \"Don't you think,\" she says, \"that I miss that just as much? That there are times I want it all gone too?\" She leans closer. \"Do you think it matters?\" All those bulbs: they flood her face with a light from underneath, make her features almost unknown to me. \"It _is_ ,\" she says. \"This is what it is, Ev. What _I_ am, now. Can you understand that? We _were_ each other's only other part. But now there's her. Now _this_ is me. Is _us_. Us _three_. And we will never be the same as how we were, _never_ , not even if somehow we could go back to the Eden of your memory.\"\n\n\"The one that I remember,\" I say.\n\n\"I know,\" she tells me.\n\n\"There's a difference.\"\n\n\"I remember it too. That's why I can still say I was always straight with you. Back then it _was_ true, you _were_ everything. I needed nothing else. And then I did.\"\n\n\"But we couldn't know\u2014\"\n\n\"That she wouldn't turn out perfect?\" Between us, Bess's fingers search the tangle of bulbs, as if still looking for the burnt-out ones. \"Evan,\" she says, \"that day, last week, I went out to the pond, it was the day after the bear. After you...\" I cannot seem to look at anything but her fingers. \"I _needed_ to hear those sounds. I know it should have been impossible, I thought it was, but, in the quiet after I'd thrown the rocks I _did_. Just not coming from the pond.\" Beneath her stare my fingers search the wires too. \"They were coming from the woods.\" And I want only to find the bad bulb first, hold it up, make her stop. \"Where Orly stood.\" But Bess's lap is too bright, and Orly's breathing too loud, and when I look away my eyes are too mottled from the light to make out our daughter's face.\n\nBess reaches out and turns my face to her. The flaring fades. And she is simply there, the sight of her as right as the feel of her hand on my cheek. \"I _had_ to let her sleep in here,\" she says. \"She was too scared.\"\n\n\"Of what?\"\n\n\"This night last year, last solstice eve...\"\n\n\"Of Krampus?\"\n\nShe takes the bulb out of my fingers. \"Of you.\"\n\nOur woods have always been most beautiful in new snow. The granite ledges distilled to wet black shapes beneath unbroken white, the boulders pared back to their bare forms rising from a forest floor become a sheet of smoothness, even the trees refined into elemental lines of darkness and light. There is a sense, while watching the snow sift through the canopy, drift down its slowly floating fall, that everything else has stopped. Such silence. Except the sound of my own breath.\n\nWe never did finish replenishing our marriage lights. Orly stirred; Bess went to her. After she had put our daughter to sleep again, she led me out to whisper what she wanted us to do. The Festival of Wild Women, the ancient maenads chasing their bull: this solstice day I'd be the beast, Orly would chase me, Bess beside her so she wouldn't be too scared. _She'll strip away the mask_ , Bess said, _see nothing terrible there_. And listening to her hope that it would help me, too, her _it will be all right_ and _pretty good again_ , I had said nothing of what I knew had always been the ending of the hunt.\n\nThis morning we were supposed to wrap up the rest of the lights, replace the last bad bulbs, string them over the greenhouse before the promised chase. But now on this late hour of this fastest-passing day I am here, instead, standing below a half-felled tree, watching as flake by flake its branches gather weight. This is the thing: if it was Eden, it was one we made ourselves. We already knew the world, chose to leave it for our own. Then chose\u2014her sudden need, my acquiescence, no one but ourselves to undermine us\u2014to leave that too. Watching my breath rise into the ceaselessly descending snow, I try to remember that. That it was us, not Orly. That she is simply the result of a decision that we made. That there are no decisions left, nothing to do but live with it.\n\nAbove, the branches seem a little thicker, the twigs at the end to bend a little more. What snow has fallen through has stuck in a long trail down the trunk. Somewhere below my shins I know the notch is a dark crossing cut through it. _This is what it is_. Hiding the hinge. _This is what I am now_. I listen for a creak.\n\nBut when it comes it's from another tree. I see it: a stirring in the canopy, snow dusting down. Stepping into the fall path, I check how deep I sawed the trunk, tilt my neck, let my head fall back, watch the wind pick up. This morning the forecast said it would get worse, the storm dangerous by dark. I'd meant to make it to the pond before it hit, but passing through the grove of groaning trees I'd seen the tracks, fresh paw prints sunk in the fresh snow, wending from trunk to wounded trunk, as if the bear had sniffed each wedge of inner wood and instead of sap, smelled blood. Walking its path, I'd stopped beneath each weakened tree, waiting, waiting, unable to leave. And how long have I been here now? Long enough for Bess to have dressed Orly in her maenad clothes? Begun to wonder if I will show, while with each minute it grows too cold, too close to dark, too late?\n\nIn the growing wind, all the wounded trees sound like a timber frame settling, then, straining with ever greater gusts, like the cracking open all at once of an old house's beams. Listening, I know what it is telling me, and feeling the fibers in each trunk struggling against the swaying weight, I move into the places where I know that each will fall, wait for the hinge to snap.\n\n_It is_ , Bess says, _this is what it is_ , and she is right.\n\nAround me the storm grows strong enough to shake the heavy limbs, send smaller branches clattering through the boughs, fling them into their own dark shapes cut through the snow, but not to bring a single widow maker down. By the time I finally start back, the gusts have given up swirling flakes for slicing streaks of sleet, and when I break into the open field I pull my hood low, lean forwards, stare at the snow spray before my boots, but I can feel our half-buried tractor, the goats milling in their shed, the dark mass of the greenhouse still unlit beneath its thickening shroud.\n\nAhead, the kitchen windows glow. Upstairs there is another window lit, but on the first floor they seem twice as bright. I know before I'm close enough to see inside: our marriage lights, all three thousand glimmering, amassed across the floor. Bess must have finished them herself, dragged them down the stairs, is somewhere in there preparing the strands for us to bring them out, wondering where I am. But when I'm nearly there, I stop. Through the long clear pane of the back door there is just Orly, hunched on the boards, her back almost against the glass, her shape silhouetted by all the lights before her. Still, I can see her wrapped white sheet, her belted waist, the wreath around her head, her hair so golden it seems about to burst into flame. She reaches to her lap, brings out a bulb. Carefully, purposefully, she puts it in an empty socket. Selects another. Does the same. Again, and again, and I know that this is how it will be, that they will glow out on the greenhouse night after night, as if things were no different than before, just another year a little further from the truth, that it would be like this until Orly turned eighteen if she was a child for whom eighteen would matter, if we were a couple who could hope to then reclaim our second lives, but when she is nineteen we will put up the greenhouse lights again, and when she is twenty, and twenty-one, and the next year, and the one after that, and the one after that, and the one after.\n\nFrom somewhere on the second floor there comes a long, low sound: Bess, blowing a single note on her recorder. When she's done Orly sings the same one back. Standing in the dark behind our house, listening to them pass their music between each other, it seems an entire language I'll never know.\n\nAnd, once inside the equipment shed, I can't hear anything but the thudding of my blood, the rustling as I hunt for last year's Krampus mask. When I lift it from a bag of rotted canvas, its reek is thick as the sifting dust. Something has eaten at the eyeholes, but the curling fleece still hangs from the woolen hat, the horns are still attached, and, outside again\u2014the sleet smaller, the air colder, the light nearly snuffed by the low churning clouds\u2014my face is warm beneath the mask. My breath makes it wet. With each step, the loosened antlers shake, their sawed-off ends a shiver on my skull.\n\nBack at the house the sounds my wife and daughter make seem nothing like the singing of the ice, and the freezing pond is nothing like the trees, and the trees are nothing like a voice, and for a second, standing just outside the windowed door, watching Orly turn and see me, I know we do not matter to the world\u2014which does not care, which will not know\u2014but only to each other, husband, wife, mother, daughter, and I am suddenly terrified that somewhere outside my fleece-blurred circles of sight my wife is watching, that on the other side of the glass door my daughter will be scared by what she sees, and before Orly can scream, I flip the sheepskin up\u2014a slap of cold against my sweating face\u2014and show her it's okay, it's me, _I'm your father_.\n\nFrom upstairs: a long low note again.\n\nBut Orly doesn't answer. She is alone, frozen still, her face caught between fright and... What? Who can tell? Who can ever know what she is feeling? If she is scared or excited? If she remembers the game we've promised her or thinks that this is something just to do with me? I let the mask fall into place, start to back away, motion for her to follow. Still, I'm almost beyond the window's light before she stands.\n\nRunning, I turn back once to see: the bright kitchen, the pale shape bursting out the door, the flapping sheet, her hair blown wild beneath a shaking halo of evergreen, her breath steam that catches for a second in the window's light before she is beyond it, into the night, after me.\n\nAhead there is the sledding hill, its sudden drop, the darker swath of snow down there, the stone wall darker still, the even deeper dark behind it. Beneath the trees it will be nearly black. The night will crowd my eyes. I'll see branches lunge into sight, hear the roaring wind, the clatter of crashing limbs. The crack of a hinge. And they will fall like buildings detonated in the dark, like avalanches crumbling out of the air, like all the logs I've ever loaded into the splitter ripping all at once, and through them I will listen for her footsteps.\n\n# [**BEAUTIFUL  \nGROUND**](toc.html#ch6-R)\n\n_For the first time yesterday we saw shadows_.\n\n\u2014Mike Horn and Borge Ousland, North Pole\n\nExpedition, March 23, 2006\n\nNow they were setting out together, the two greatest adventurers alive trying to reach the North Pole in the dead of winter, in the Arctic's twenty-four-hour dark, on foot, day after day no different from night after night, no motors, no dogs, no nothing but the cold and the ice and them hauling sledges through the ceaseless black.\n\n_Why_ , the interviewer asked, _would you do a thing like that?_\n\nOn the radio: laughter.\n\nIt seemed to Claire to fill the studio apartment, take up what little space there was for her and Todd. Standing in the steam above the stove, she shivered. Behind her, her husband stopped chopping. She knew\u2014the way she knew this dinner they'd made together some hundred times over the years\u2014that in the breath between last laughter and next question he would say something. Still stirring, she reached back with her other hand, touched her fingers to his lips.\n\n_What is it that scares you most?_ the interviewer asked.\n\nMoving around each other in the cramped corner kitchen, they listened to the voices of the explorers\u2014 _our personal limits_ \u2014squeezing close to make room for a swinging fridge door, reaching behind for a saucepan\u2014 _the psychological effects_ \u2014occasionally brushing hips.\n\n_So much time with Borge!_\n\nLaughter.\n\n_With Mike!_\n\nLaughter.\n\nThere was a time, she thought, when he might have kissed her fingertips. When she might have left them there long enough to let him.\n\n_Data Log. Day 55. Latest position: N 88\u00b0 34' 10\" E 83\u00b0 41' 32\". Distance to go: 160 km. Temperature: \u201315\u00b0 C. Days of food left: 13. Average daily distance required: 13 km_.\n\nEvery few days the men sent word by sat phone, someone updated the blog, and each evening she sat staring at the screen while Todd slow walked preparing for sleep, finally said, _Hitting the sack_ , then shut off the lamp. She knew he didn't understand\u2014equipment failure, polar bears, their breathless _Pitch-dark 24 hours a day!! You cannot see a thing!!!_ \u2014but to her it seemed miraculous: a message from that desperate corner of the earth sent into space, caught by a satellite, directed down to her. She would read it over until the screen went dim, then stare into the blackness as if it were a patch of night sky brought back from their life before the city, the stars now replaced by a reflection of the lamp, her desk, the three-paneled divider, the rest of the room that was the rest of their home, her husband unfolding their futon bed, willing her to join him. Once, she knew, a director had forbade him to speak, ordered him to show what he wanted with just his eyes. _Make her feel it_ , the director had said. _Make her do it_.\n\n_Stand up_ , Todd's look said now. _Come to me_.\n\nFor the past quarter hour she had been staring out from her sweatshirt hood at the words: _FIRST GLIMPSE OF THE SUN!_ Fifty-four days walking in darkness, thirty in such relentless lack of light all either of the men had seen was the other's headlamp, all they'd heard was the other's breath, steps, until, far off towards the rest of the world, the blackness had cracked, a red line aglow like a chink in a cast-iron stove, day by day a dim light dawning at noon, for an hour, then two\u2014 _The light is fantastic up here! It's bluish violet with a trace of red_ \u2014the still-hidden sun pushing at the curve of the earth, trying to rise.\n\nYears ago, in a cave in Mexico, Todd had shut off his headlamp. He had reached to her forehead and shut hers off too. It was so black he told her, _I can't see your hand in front of my face_. She breathed out, _That's because it's not_. And felt his fingers, faint as breath, tracing her brow, brushing her eyelid, hovering over her cheek, her mouth. _I can see you_ , he had said. _I can see your face in front of my hand_.\n\nBehind her, the screened divider shook. In a second, he would be there, peering over the blind. She tapped the space bar; the laptop's brightness bloomed: _FIRST GLIMPSE OF THE SUN!_\n\n\"Anything happen today?\" His voice was so close it made her jerk.\n\n_We looked at the temperature gauge at the same time, but there was no change_.\n\n\"No,\" she told him, shutting the browser's window, and then her hood was off\u2014he'd tugged it\u2014and, looking up, she couldn't help but grin. \"What in the world are you wearing?\"\n\n\"Baby,\" he said, in his best Barry White, his arms spreading so the cashmere sweater's neck revealed an even wider V of bushy chest. \"Hugh Hefner\u2014\"\n\n\"Oh no.\"\n\n\"\u2014wears sweaters just like this.\"\n\nLaughing, she stood, stepped around the screen. Below his waist he wore nothing but briefs, skimpy and light, bought for the backpacking trips they used to take. \"Wow,\" she said. He looked at her as if that meant she liked them. \"No,\" she told him. And when he asked, \"Well, what are _you_ wearing?\" she looked down\u2014baggy sweatpants beneath a sweatshirt stained from that morning's co-op shift\u2014reached back, replaced her hood, stood there with a sly smile.\n\n\"Come on,\" he said.\n\n\"You don't know what I'm wearing under this.\"\n\n\"My boxers.\"\n\n\"Maybe not.\"\n\n\"And my T-shirt.\"\n\n\"Maybe I'm not wearing any underwear. Maybe I _won't_ wear\u2014\"\n\n\"Claire, you _have_ to. It buys you time. It gives you something to take off _after_ you've taken off\u2014\"\n\n\"Todd\"\u2014she put her hands on his chest, slid them to his shoulders\u2014\"we aren't really going to do this, are we?\" He had small shoulders, as if made to fit her hands. She used to like that.\n\n\"Baby,\" he said, his voice back to his voice this time, \"we could just stay home, stay in together.\" Open some wine, take it to bed. Even after these fallow weeks, she could feel how his fingers would stroke her spine, his lips would brush her eyelids, the wetness of his mouth switching from breast to breast. When had his touch begun to feel like it was wearing out her vertebrae? When had the rhythm of his breathing begun to bother her? She wanted to keep her eyes open, tell him, _Stay on one nipple_ , wished his thighs weren't so hairy, his shoulders so slim, that the worry would leave his touch, the doubt disappear from his breath. When had his lovemaking become so desperate? When had it begun to make her feel desperate too?\n\n_Data log. Day Some Thousand Some Hundred and Something. Latest position: Brooklyn, Park Slope, btwn 8th and 9th Aves. Distance to go: God knows. The rest of her life? Another year? Another night? And in the morning?_\n\nSometimes, coming back from work, climbing up the subway stairs, she thought she could feel the ice slipping away beneath her. When they first left their families, their homes, flew north for Cape Artichesky, they had felt the wind with them. But by the time they stepped onto the frozen sea, just the two alone, the gusts were blowing against them, the island of ice beneath their feet drifting backwards, pushed so steadily by the sea chop that they would slog a dozen hours northwards only to wind up standing farther south.\n\nFor a year after college she and Todd had traveled, just backpacks and boots and a different place to sleep each night: a hammock-hung porch beneath a quiet volcano, the two of them swinging in their own cocoon; a sheetless hotel where they rolled on the floor, dirt pasted to their sweat; even a mist-thick mountain pasture, him in a saddle, her on his lap, the horse drifting through the fog, the feel of it moving beneath them.\n\nThey'd come to New York for Todd's career\u2014 _I'm not getting any younger_ , he'd said at twenty-four\u2014but she had wanted it, too: the dim sum Sundays and Senegalese in Clinton Hill; mariachis on the subway home; the old Puerto Rican men blasting Juan Tizol from boom boxes in the baskets of their bicycles; the documentary festival that left them feeling they had explored the farthest corners of the world. How had it gone from that to this? Their reliable wine store, neighborhood Thai; the tailor who hemmed all his pants; the stylist who'd done her hair since '98, '97... For six years they'd been members of the massive food co-op in Park Slope, triweekly work shifts wheeling out wet boxes of mysterious greens, smashing the cardboard in the crushing machine, manning the register alongside men in turbans, dreads, tailored suits, making change for women wearing hijabs, or with tattoos on their shaved heads, too many to ever learn more than a smattering of their names. At first it had been almost an adventure. Now it was just what they did some Saturdays if they couldn't find another member willing to swap: a fellow actor who understood Todd's callbacks, a lesbian whose crush on Claire made it easy to ask. He used to tease her about that. She used to ask how his auditions went. But now every time he came back giddy Claire knew it was from some connection, some buzz, something he'd felt with someone else.\n\nSometimes, in the black of night, fissures would open, cracks in the ice. They were called leads, and they struck straight down into the sea\u2014the water dark, the depths unfathomable. Sometimes they would appear in the headlamp beams. Sometimes they would start to split right between your feet. Snow obscured thin patches of ice. Darkness hid the telltale blue. _A potential minefield. We have to test the ice cap with our ski poles, tapping the tips in front before we place our feet_. And, last week, after her last class, after she'd come home to the cracks opening all around her and told her husband it wasn't working, they were breaking apart, she didn't want them to, after he had told her, _I know, I just don't know what to do_ , that night she had not been able to get the words out of her head: _It tests our personal limits_ , Horn had written, _the drive to go beyond, it tests our personal limits, mental, spiritual, physical, it tests our personal limits_... She had fallen asleep to the sound of his voice, the image of the two men held in her mind. They were putting on their survival suits, zipping them shut over their heavy clothes, sealing their faces behind clear plastic, peering through the steam of their breath at the freezing water below, jumping in.\n\n\"You don't touch someone if she doesn't want to be touched. You don't ask again, you don't say please, you leave her alone. And you? You don't have to say yes. You don't have to say anything. You don't have to take your clothes off. But you can. If you go into a playroom, you probably should. Even if you just want to watch. You can do that. But you can't take pictures, you can't shoot video, you keep pulling out your phone we're going to ask you to leave. We catch you doing drugs, we're going to ask you to leave. We see you engaged in intercourse, we're going to ask you to take it to a playroom.\"\n\nClaire was sure she'd seen the woman before: the buzzed hair at her temples, the face red as if her thick throat was squeezed by her turtleneck; over it, she wore one of the stained quilted coats that hung outside the co-op freezer door. In one hand she held a clipboard like shift leaders used. In the other, a Sharpie tied to a string.\n\n\"Say you go into a playroom\"\u2014the woman tapped the pen against the board\u2014\"and you see someone else engaged? Say you want to engage with them? Do you just jump in?\" She looked at Todd. \"No.\" Her jowls shook. \"You wait, you watch, you ask. At an appropriate time. The middle of an orgasm is not an appropriate time.\"\n\nWhen exactly would be? Claire wanted to ask. And what might be an appropriate method of asking? From the way Todd glanced at her, she could tell he was thinking the same thing. _My name's Todd, this is my wife, nice to meet you too, why yes she is_. Watching him fish through his wallet for the suggested contribution, she half expected him to ask, _And what, when your wife wakes you in the middle of the night to suggest something like this, is the appropriate response?_\n\n\"A little advice?\" The woman was looking at them with unsettling concern, as if she had lain there in the sheets between them. \"Before you go up, take a second together.\" Her voice was almost kind. \"Figure out just what's okay, what's not okay.\" She smiled at them.\n\nOn the corner, a Town Car pulled up. The clipboard woman's face swung towards it\u2014another couple stepping out\u2014and Claire heard the vestibule door buzz behind her; she could feel the thrum straight through Todd's hand on her back. It was in his eyes, too: he looked so determined, so intent on going through with what she'd said she wanted. She imagined him looking like that at some woman in there. She imagined having to tell him it was okay.\n\n\"I want you to know,\" he said, \"that no matter how hard this is...\" She had an urge to pull him towards the cab, drag him in, but he went on: \"No matter how hard it would be for me\"\u2014he bit his bottom lip\u2014\"if you want me to 'engage' with some, say, sex-crazed supermodel...\" He stared straight into her eyes, tried for earnest: \"I want you to know it's okay. It's okay with me.\"\n\nShe fought down her smile, furrowed her brow, nodded her head. \"I appreciate that.\"\n\n\"Even if there are two of them.\"\n\n\"Even twins?\"\n\nAnd they were laughing.\n\nInside, the apartment smelled like an attempt to smother old dog scent with candles. A tiny East Indian man with a giant black beard was taking coats. He'd rung her up at the register once or twice\u2014she remembered his difficulty reaching over the counter, the eye-widening odor when he had\u2014and for a moment she wondered if they would all be co-op workers here.\n\n_Parties_ , one of the women at the cheese-wrapping table had called them. _Salons_ , the LISTSERV had said. And, riding the freight elevator up to the main floor, Claire had asked how it worked: _By invitation only_ and _no unaccompanied men_ and _sure you can bring your husband, that's what it's for_.\n\nNow, the East Indian man was handing them a small stack of cards, varicolored as the flyers people thumbtacked to the message board. She turned one over: _Kiss me_ , in cursive. Claire looked at Todd.\n\n\"Okay,\" he said.\n\nShe flipped another: _Touch me where you want to_.\n\n\"Not okay,\" he said.\n\nAnd, together, they went through the next few in the stack: _Got a tattoo?_ Okay. _Tell me a story_. Whatever. _Lick my neck_. Why not.\n\nBut the pump-top bottle of hand sanitizer beside the cut-crystal bowl filled with condoms? A little creepy. The bottle of lube? Claire shook her head. Especially not there, in the hall, on the sideboard with the family photos and Chag Purim! holiday cards.\n\nVotives flickered on the living room windowsills; a few lamps glowed in corners. She could just make out the figures dancing, others come together on couches, the floor. They searched for a place to sit that wouldn't land them on someone else's lap and, having navigated that\u2014the bench by the baby grand was free\u2014discussed, above the din, what to do next.\n\nGoing to the kitchen for a drink seemed fine, despite the man jerking cocktail shakers without a shirt, bells bouncing on his nipple rings. Even over the music, Claire could hear them jangle. The music\u2014some new age mix of sitar and synth\u2014was turned so low it left the partyers dancing in the living room unhinged from any beat. Watching them, they could make out jiggling flesh in the low light, pale flashes flapping in and out of sight. Most of the men had lost their shirts, though some still had ties swinging from their necks. Around them swung the breasts\u2014barely bumps or stretch marked with weight, browned beneath tanning booths or tub-soak pale\u2014and Todd was saying, \"Not okay, not okay, it's going to ruin them for me,\" but, to Claire, it was fine, fascinating, even beautiful. She wondered why the women with implants had done it, what they had been trying to fix, whether it had worked. She heard Todd's \"Gin and tonic?\" his \"Be right back\" but she couldn't seem to answer, couldn't stop trying to picture herself out there.\n\nIt was only once he was gone that she grew aware of those right around her. Over against a window, two men stood, naked but for a single yarmulke pinned to the back of one's head. Arms crossed behind them, hands cupping each other's rears, they peered out through a lifted blind as if pretending they were somewhere people could see stars. For a moment, the image came to her: Horn and Ousland standing on the ice outside their tent like that. She smiled. They never mentioned how they felt about each other, the loneliness, the needs. She couldn't remember them even once mentioning the stars. That seemed so strange it knocked her back into the room: the men, now kissing; the cluster of women laughing together on the couch; the dancing crowd. She wished she had a headlamp; she would look at each person, light up her face, find his eyes, move on to the next.\n\nBehind her, the piano shook: a woman leaning against the lacquered wood, her back to Claire, her arm a piston. The man pressed to her making chimp sounds.\n\nClaire picked up the icebreaker cards, tried to focus on flipping through them: _Love your lips_. The creak of weight lifted onto the piano lid. _Taste me_. A thud.\n\n\"Check it,\" she heard behind her.\n\nThe woman was on the baby grand now, the chimp man's grin sinking out of sight between her legs, his eyes on Claire. Great, she thought. No matter how long it was before Todd went down on her again, she'd wind up seeing that. The next time she unzipped his jeans, there'd be that piston hand. It struck her then: what was taking her husband so long? Standing, she looked out at the milling crowd, tried to find him coming back from the bar. She would catch his eye: _This?_ her look would say. _This is not okay_. And he would send her back his own: _No, no it's not, now let's go home_.\n\nBut her eyes never made it to him. They caught on the woman's stillness: she stood at the edge of the roiling crowd, motionless as the wall behind her, alone but for the men and women who approached to ask her questions, hold out drinks, hands, even a few cards. Some tried to touch her; her body showed no sign of feeling it; they drew their fingers back. And not once did she so much as look at them. She might have been thirty, forty, it didn't matter: swan necked, dusk skinned, black hair done by someone who knew to get out of the way of those cheekbones, she looked like the kind of woman you saw behind the windows of Tribeca wine bars, on billboards, movie screens, but not alone in a swinger's club, letting her glance slip across you.\n\nClaire turned: Todd coming towards her. And maybe it was simply that. That she knew what he was going to say when he reached her _\u2014at least we gave it a_... and _maybe it's just not_...\u2014and tomorrow the adventurers near the top of the earth would add an entry to their blog, and it would say the world had changed, the sun had hovered on the horizon all the past day and all the night, and would for the next, and the next, the darkness become a ceaseless light, and on Monday, home, they would cook another supper same as a hundred others, and say the same things to each other they always said, and not say the same things they never did, and, sometime later, go to bed, to sleep, still themselves. In her hand the cards clicked and clicked at her fingernails. In her ears, the crunch and crunch of crampons on ice. And she was reaching into her purse, taking out a pen, scribbling on the back of a card, already starting to push through the crowd, moving away from her husband before he could stop her.\n\nThey had just finished their drinks and were rising to head for their coats when the woman came up to them, held out a card, and said straight to Claire, \"What does this mean?\"\n\n_We've walked right into a compression zone. Mountains of ice as far as the eye can see_.\n\n\"A compression zone,\" Claire tried to explain, \"is where the shifting of the ice\"\u2014she could feel the woman's stare\u2014\"it causes these huge mounds, like fifty feet.\"\n\n\"What in the world?\" Reaching over, Todd took the card from the woman's hand.\n\nClaire reached out and took the woman's hand in hers. \"I'm Claire,\" she said. Shaking her hand seemed wrong. Just holding it seemed wrong, too.\n\n\"Suvi,\" the woman said. \"We were just leaving.\"\n\n\"Oh.\" Claire let go. She was with a man. He stood a little apart. \"So were we.\"\n\n\"But.\" The woman flicked her eyes at the card.\n\n\"I know,\" Claire said. \"I'm kind of obsessed.\" That sounded as wrong as holding her hand had felt. \"I mean with the Arctic, with... Do you want to sit?\"\n\nUp close, Claire thought she must be part Inuit, maybe forty-five. Her tapering eyes, her wide cheeks: just sitting on the bench with her made Claire feel beautiful. \"So,\" she heard, and out of the corner of her eye saw the man extend a hand towards her staring husband, wiggle his fingers. \"Did you intend to give me this?\" She couldn't place his accent\u2014Russian? Swedish?\u2014but watching Todd turn over the card, there was no mistaking the man's grin. _Taste me_ , Claire thought, but she was already telling the woman about the expedition. \"It's the first time anyone's ever even tried to do it. But that's not _why_. They've both done harder things.\"\n\n\"Really?\" the woman said.\n\nIt didn't mean _tell me more_ , Claire knew that, but the woman's eyes stayed locked on her, and she couldn't help going on. \"You know the Arctic Circle?\"\n\nThe woman smiled. \"I'm from Karelia,\" she said. \"Near Finland.\"\n\n\"One of them circumnavigated\u2014\"\n\n\"I'm from Long Island.\" The man had reappeared with a pouf in his arms. \"One hour and a half from here.\" He set down the ottoman. \"At least that's where we're from now. Belle Terre.\" Squatting next to it, he raised his eyes to Todd, gave the leather top two pats. \"Actually, a little over an hour, this time of night, the way I drive.\" He made a fist, jerked his arm like he was shifting gears. There was so much muscle on him that his smile seemed a pushup for his jaw. \"My name's Oleg,\" he told them, nose flexing. He must have been sixty. At least six feet tall. He said he used to work in astronomy, and Claire pictured him retiring to learn ice climbing, trading days in the office for days at the gym. His head looked waxed, his shoulders huge.\n\nWatching Claire, Suvi smiled too. \"So, if _why_ is not that they're the first...\"\n\n\"It's everything,\" Claire told her. \"The bears, the cracks in the ice, the endless night\u2014\"\n\n\"You have beautiful lips,\" Suvi said.\n\n\"Thank you.\"\n\n\"She does,\" Oleg told Todd.\n\n\"I know,\" Todd said.\n\nClaire reached up, took his hand. \"He's been kissing them for almost ten years,\" she said and, gently, led him to sit on the ottoman. It huffed beneath him. Beneath her hand she could feel their rings, the tension in his fingers. \"You must know how it is,\" she said to Suvi. \"Finland, in the middle of winter. But they're so far north there's no sun at all, ever, for months. That was one of the two things that scared them most.\"\n\n\"What was the other?\" Oleg asked. He was still squatting, his legs so big they might as well have been a chair, and before she could answer he sprang a grin. \"So many months without their wives!\"\n\nFrom nearby in the crowd there came the sound of a woman ramping up to shrieking.\n\n\"Or\"\u2014Oleg swung his grin to Todd, shot out a little elbow jab\u2014\"maybe not.\"\n\nWhen Todd spoke, he had to shout over the ecstatic noise. \"The fact that they're together,\" he said. \" _That's_ the other thing.\" Maybe he said it louder than he had to. All three looked at him. \"The Amazon, the Arctic Circle, they've always done it solo. But now they're going together. Just the two of them. For months.\"\n\nClaire squeezed his hand.\n\n\"And you know what?\" Todd said. \"They're doing okay.\" He glanced at Claire. \"What was it they wrote?\"\n\n\"Todd\u2014\"\n\n\"A pretty good team,\" he prompted.\n\nWhat, Claire thought, were they doing to that woman?\n\n\"They wrote: 'We make a pretty good team.'\"\n\nNobody, she thought, needs to make that kind of noise. That kind of noise was never real. Suvi's eyes found hers as if to show she was thinking the same thing.\n\n\"Still seems crazy to me,\" Oleg said.\n\n\"But,\" Suvi said, \"beautiful, too.\"\n\n\"Sometimes,\" Claire told her, \"the ice is like a mosaic. They have to jump from floe to floe.\" She could feel both men looking at her, but she kept her eyes on Suvi's. \"Sometimes in the compression zones they can hear the sound of it under their feet, crushing and grinding.\"\n\n\"Well\"\u2014Oleg stood up\u2014\"I can hardly hear _anything_ in here. But it's quiet at home. And I can get us there in an hour.\" He buttoned his blazer. \"It's a Cayenne,\" he said, talking now to Todd. \"Twin turbocharged. Five hundred horses. Tell you what, in the morning, or the afternoon, or whenever we take you back, I'm going to let you drive.\"\n\nWhen Suvi stood, Claire stood too. On Todd's face, she could see _This, this is not okay_ , and she tried to show him _It will be_ , but he was putting it all in his eyes in a way that would have made that director proud: _Sit down_ and _Don't do this_ and _Please_. That was when Suvi leaned down and kissed him.\n\nAt last they are moving. The wind has died down, the helicopter motors have been heated, they have left their lodgings at Golomiyanniy, the ice at Cape Artichesky has frozen firm enough to land, and, after so much waiting, they are on their way. What a relief to take the first step, put words into action. _There is nothing but possibility ahead!_\n\nTurning left onto Atlantic, they head west towards the ramp, and by the time they are on the Brooklyn-Queens Expressway it is midnight. _The light is fantastic from here! It's orangish brown with a trace of ochre near the city's horizon_. The bulbs on the bridge cables could almost be stars; the headlights of oncoming cars, comets; the hand on Claire's thigh so unlike her husband's. Oleg is driving fast: beneath her, the thump, thump, thump of the expansion joints in the road. Her whole body clenches in expectation of the potholes. In the visor mirror, Todd is sitting next to the most beautiful woman Claire has ever seen. Certainly, the most beautiful woman she has ever seen beside her husband in the backseat of a Porsche SUV. Without a doubt, the most beautiful woman she has ever seen snorting coke.\n\n\"Eight point six miles,\" Oleg says as they complete their leg of the BQE and begin the Long Island Expressway. \"Seven minutes, fifty-four seconds.\" He has set the trip odometer on the GPS. \"But,\" he assures them, his hand slipping farther into her lap, \"now we are going to see real road.\"\n\nIn the mirror: Suvi's long fingers handing Todd a tight-rolled bill, an eye shadow case, white stripe across its top. They must be at mile nine by the time Claire hears her husband say, \"Sure.\"\n\nAt 9.7 Oleg snorts his line. Claire knows this because he asks her to hold the wheel. Leaning over, she can see the map of where they are, the blue dot moving away from where they were; then he is bent beneath her arm, his head ducked close to her lap, and she wonders if he can smell her nervousness, feel her heat. When it's her turn, she doesn't look at the miles. She watches the little blue dot sucked along the yellow line and feels it slip straight up between her eyes into her brain.\n\n\"Hey,\" she says to Oleg, \"how fast are we going?\"\n\nAs if in response, he turns on music. Frenetic techno stuff, like road trip music made not for them but for the blue dot, for its adventure on the GPS screen, music so bad she has to look back at Todd to see the expression on his face. But he is staring fiercely out the window at the night blowing by. Suvi stares the same way into Claire's eyes. And Claire thinks, No, the music is for them. It is the soundtrack for the trip they are on. The theme for this. And she likes it. She knows she shouldn't, but her body is bouncing just a little in her seat. That's when she realizes she hasn't felt the potholes at all. \"This _is_ a nice car,\" she says.\n\n\"Wait till you see the house,\" Oleg tells her.\n\nSuvi nods. About the car, the house, or just to the music?\n\n\"We must be making good time,\" Todd says, still staring out the window.\n\nThe look on his face is a picture of the hardening in his voice, and watching her husband Claire tries to soften it. \"Rapid progress,\" she says. \"A new record!\" There, at last, is his smile. \"An impressive distance covered today!\"\n\nTodd puffs a laugh out of his nose like a snort in reverse.\n\n\"Your husband,\" Suvi says, smiling too, \"is afraid to look at me.\"\n\n\"Eighty-five,\" Oleg tells them. \"Ninety. Ninety-five.\"\n\nWhen Claire turns back, the road is coming faster than she has ever seen it come before.\n\n\"Aaand,\" Oleg says, the word like a ramp for the volume of his voice, \"Et hundre!\"\n\nThe blue dot is a pinball struck.\n\n\"Et hundre og ti! Et hundre og tuje! Et hundre og tretti!\"\n\nIs a BB bursting down the barrel of a gun.\n\n\"Et hundre og femti!\"\n\nIs them.\n\nOn the GPS screen something is flashing. A turn approaching, a new route found, a course on the map waiting to be blazed, a signal sent from her into space and back down to her again, and maybe the satellite that catches it is the same that catches the voices of the men\u2014 _Life is short! Each minute is important_!\u2014maybe the wavelengths simply get mixed up, but, for a moment, feeling the night roar at her, rocketing towards that beautiful land\u2014 _Live while you can!_ \u2014she can hear them.\n\nBut which one was inside her? For a moment, she couldn't tell. It was so dark she couldn't even see the wall her palms were pressed against. They had gone down to a room in the basement and sat on a carpet of some animal's fur, around a glass-top table, watched Suvi cut the lines. _But why now?_ Oleg had wondered, aloud, without warning, his pale eyes on Claire. And she had started with _global warming, the world's attention_ \u2014but Oleg shook his head. _No_ , he said, _it's because of the global brightening_. And holding her eyes with his, he had leaned in, told her in a voice low and hushed as the start of a bedtime story how, far to the north, beside the Bering Sea, there glowed a circle of light illumined on the tundra all day, all night. _This_ , he opened his hands, _is only the start_. Which was when Todd had broken in. _Okay_ , her husband had said, _all right already_. They had all looked at him, and Claire could see on his face that he had glimpsed what had been growing on hers. _But we turn off the lights_ , he said. _All of them, understand?_ He held the rolled bill so tight his fingers dented it. _I don't want to see anything_.\n\nBut she could feel it. Either Oleg was the kind of older man whose entire body balded as it aged, or he had waxed, his hairless thighs so unlike her husband's, the muscles stiff with the same strain that in Todd's legs always showed as shaking. In her ear, the astronomer was whispering how good she felt. In the minutes after the coke was gone and the lights were off and she had discovered the softness of a woman's lips brushing her belly, she _had_ felt good, had wanted, then, to believe what Oleg said: that circle of ceaseless glow, that nighttime sun reflected onto that patch of earth. In the darkness, she had wondered if the caromed light would look a little different, if it might feel somehow strange against her skin. And, for a moment, beneath the strangers' touches, in the newness of Suvi's search and find, the woman learning her down there, she _had_ felt it, she _had_ believed in it, for a moment had not wondered where her husband was.\n\nWhere was he now? She tried to hear, but Oleg, behind her, was grunting the way he must at the gym when doing squats, gripping her breasts like climbing-wall holds, and, imagining him\u2014neck bulged, jaw clenched, face red\u2014she saw, for a second, the two men instead, hauling their heavy sledges through true dark, pushing their bodies with real need, and it suddenly seemed so foolish. Staring hard into the blackness, she tried to listen harder: Oleg's wife moaning, Claire's husband breathing. Reaching back, she pushed herself free, heard her name, heard it again, but she was walking away from the sound of it then, hands held out before her sightless eyes.\n\nSoon, the men would reach the Pole, take one last step, and be standing on it. And then? Any step they took from there could only take them back. To the south there would be the sun, slowly circling the horizon. No mere experiment, no mirrored wings, but a natural never-ending light. All day, all night, they would watch its circumnavigation, knowing it wasn't really moving at all, that it was them, the earth beneath them, making another turn. They would stand there until the sun was back where it had started. Then they would do the only thing there was to do: begin walking again.\n\nCrouching down, she felt the cold, hard floor. How many other women had Oleg pressed down on it? How many other men had Suvi let shove her spine against these boards? And how long would they keep doing it? Till his legs could no longer hold him? Till her back grew too weak? The only difference night after night the change that time would have worked in them anyway. And when the men on the Pole came home again, their skin windburned away, their lungs scraped clean by Arctic air, the life in them spent and replaced, would even they be truly different from when they'd left? To a lover? A wife? Crawling through the darkness, Claire felt for the curve of her husband's calf, his forearm's hair, the hand that she would know anywhere.\n\nBut it was a woman's that brushed her. Claire jerked her hip away. In the blackness, she could barely make them out: pale glint of teeth, dark ledge of back. Almost invisible, it rose and fell as if the air itself was undulating. For a moment, she could feel the movement just beneath her reached-out palm. Then she touched him. His shoulder stopped. From the darkness under him, the woman said something in a language Claire couldn't understand. Nearer, Todd's breathing gusted. Through his back she could feel his lungs fill, feel her own empty, hear the _what_ and _we_ and _doing_ in their air.\n\nAnd then he was rising, as if pushed by the darkness beneath him, and in the scuff of skin on floorboards, the space where the woman had been seemed to pull Todd down to it, to draw Claire over him, her hands reading the give of his skin, the bone she could feel beneath her palms when she lowered herself onto him. She pressed with all her weight to keep him still. And, her hands on his face, she traced him with her fingers\u2014the ripples raised above his brow, the blink of a lash, the slightly open mouth\u2014until she felt his fingers, his touch, his hands see her.\n\nWe wake together, the way our bodies after all this time have learned to do, and, silently, side by side, in the dark, we dress: long underwear, soft shell, overlayer, down layer, wind shell, neck gaiter, face mask. The rasping of our nylon mummy bags being stuffed into their sacks. The heavy, sour scent of our bodies. From somewhere deep beneath the tent floor: the groaning ice. Wind batters the sides, cold coming through. We breathe one last warm breath onto our frostbitten fingers, pull on our gloves, turn on our headlamps, reach for the zippered door.\n\nClimbing the stairs, we leave behind the sounds of the others still sleeping, and with each step discern a little more: vague shadows in the hall, faint shapes in unfamiliar rooms, windows like giant frames of scenes unknown. The glass in them is aglow with dawn. By the time we are out in it, the sun is up, a splinter of red.\n\n_How hard to get from here to home?_\n\nA spark.\n\n_How long a walk?_\n\nA coal.\n\n_How far?_\n\nBlown brighter and brighter.\n\n_Data log_.\n\nUntil.\n\n_Latest position_.\n\nIt catches.\n\n_Distance to go_.\n\nAnd we watch it burn up the sky.\n\n_The spring equinox has arrived. We can see now that what we were seeing before was only a reflection. It wasn't the sun at all. Now we are looking at the real sun. And it is something different altogether_.\n\n# [**THE FIRST  \nBAD THING**](toc.html#ch7-R)\n\nThere was one rule: he didn't ask what was in her past to make them hunt her so, and she didn't ask what was in his, period. Lying on the sticky linoleum in what space was left beside the four burner they had overturned, they passed a filterless lips to lips and promised each other that.\n\nSome hours ago she had shown up in the rear lot, announced by the gravel. He'd just begun his shift and was in the spray shed standing before the night's first pane of glass, adjusting the nozzle, when he heard the truck. Through the open garage door he watched it roll to a stop. He didn't like that the pickup had come around the back and his eyes flicked over the garage for what he might use if he had to. Then the engine quit and his eyes flicked back. She popped the door. The cab light flared. In its yellow glow she sat behind the wheel doing her makeup in a way he could tell was meant to show she knew someone was watching.\n\nBetween them the outside lot was aglow with an eerie luminescence barely distinguishable from the cab light glowing within it. Behind her truck, one of the glass carriers had been left stacked and on its back the panes reflected the sky, glimmering like day. But it was full into night. And what dim semblance of dusk hung over Rundgren's lot was just the closest thing to darkness there was left. Most nights it was even brighter\u2014the big sign out front readable from the far curve of the road, the garage's clear ceiling letting in plenty of light to work by for all his night-spanning shift\u2014but that night the garage was bulb lit, the sign turned on and flickering: RUNDGREN'S ILLUMINATION VARIATION. ADJUSTABLE OPACITY GLASS, REFLECTIVE RECOGNITION TECHNOLOGY.\n\nHearkening, the locals called it, nights like this when clouds left the air thick with the closeness of rain, the hills swaddled in dimness for a few short hours. People would get wistful. They gathered on porches, talked of how night used to be: _pitch black_ and _couldn't see your hand in front of your face_ and _stars_. They waited for the rain, and watched the sky. Up there, the pale lights of the space mirrors drifted. The coat of clouds was spotted with them: a thousand smothered moons glowing faintly through.\n\nIn their pallid light the entire hillside behind her truck was grayed over and still, as if in wait for the downpour, and the gravel lot was bluely waiting, and while he stood there waiting the sky let loose and the rain came down in one solid sheet.\n\nShe came walking through it. She had pulled on a poncho, yellow plastic down past her knees. Beneath it he could just make out her shape.\n\nHe pried his eyes away\u2014pulled the respirator over his mouth and flicked the pump switch and began laying on a thin coat of spray\u2014and when she came in his eyes were on her again. She shoved off her hood and stood there dripping rain on the rubber mat: low tocking drops that beaded blackly around her black rain-beaded boots.\n\nSliding his mask off, he said a hello. She said something back. But he was looking at her face, a face so pale her skin almost looked blue with the blood beneath her cheeks. Until he looked at her eyes and knew what blue was.\n\n\"You brought the rain,\" he said.\n\n\"Do I look like I like rain?\" she said.\n\n\"No, I guess not.\"\n\n\"Alright then.\"\n\nThe spray pump hummed between them.\n\nHe told her, \"If you're looking for the manager, Mr. Rundgren's\u2014\"\n\n\"I'm looking to stay away from the manager.\"\n\n\"There stands a wise woman,\" he said.\n\nShe smiled at him. She was missing a tooth at the side of her mouth and it did a black thing that was the opposite of wink. After a moment she said, \"You gonna spray the whole van?\"\n\n\"Shit.\" He jerked his look back to the nozzle, already thumbing the spray gun off: it had drifted from the glass and was pointed at the van he'd unloaded earlier that night. A murky slick of In-Solar window coating spread across the open cargo door. \"Fuck,\" he said and wiping at the smear with his coverall sleeve, thinking of how much the spray was worth, how little he made, what a hardass Mr. Rundgren was, said a few things worse.\n\nIn one move, she reached over, lifted the respirator mask from around his neck, and pressed it over his mouth. She stood there grinning. Then let go. The mask dropped. He could feel the cool air hit the sweat around his lips.\n\n\"I take it back,\" he said.\n\n\"What?\"\n\n\"All that about the wisdom bottled up in those blue, blue eyes.\"\n\n\"Don't even try,\" she said.\n\n\"I just did.\"\n\n\"Don't try again.\"\n\n\"About the wisdom bottled up in those blue, blue, _blue_ eyes.\"\n\nShe tried not to laugh and then he could see her think screw it and she laughed.\n\nShe said, \"You can call my eyes purple and I'm still not gonna fuck you.\"\n\nHe leaned back against the van, his shoulder blades pushing the coverall against the wet patch, let the spray do what damage it would. \"Yes ma'am,\" he said. \"I guess that saves me asking.\"\n\nShe laughed again. \"What can I do for you,\" she prompted.\n\n\"What can I do for you?\" he said.\n\n\"That.\" She was pointing at him.\n\nHe stuck his own gloved finger at his chest but she shook her head and he cranked his face to look behind him at the drying smear of spray. \"On what?\"\n\n\"My truck.\"\n\n\"You put that on the windshield and you're not gonna be able to\u2014\"\n\n\"No. My truck.\"\n\n\"The whole truck?\"\n\n\"Think like a paint job,\" she said.\n\n\"You have any idea how much that would cost?\"\n\n\"Why do you think I'm talking to you instead of the manager?\"\n\n\"You could be rich,\" he said.\n\n\"You could be handsome.\"\n\nHe grinned. He had not trimmed his beard in a long time and he could feel the whiskers tickle the corners of his mouth and wished he had. \"I'd lose my job,\" he said.\n\nShe stood looking at him. Between the tops of her lumberjack boots and the bottom of the yellow slicker there was a stretch of shinflesh, white. Beneath the plastic there was a dress whose color he couldn't tell other than that it was dark, cut in a shape he didn't know how to describe except that it came nary lower than her upper thigh and was loose around everywhere and then suddenly tighter just below her breasts and cut low, too, and it didn't matter because she was looking at him in a way that made him unable to keep his eyes from hers.\n\n\"Can I say something?\" he asked her.\n\n\"If you've got a tongue,\" she said.\n\nAnd he leaned against the van, letting the wet glaze soak into his back and wondering if she was working him in a way that he would find out later was just the way she worked men, or if she felt, too, their pulses passing back and forth like jump lightning making its cauterizing leap from vein to vein.\n\nShe drove the Ram into the garage while he stood inside and waved her forwards, his gloved hands motioning her a little more, a little more. He spread his palm for her to stop. She pushed the clutch and revved the gas and waited for him to scurry. But he just stood there, giving her a hard eye, and she felt it again, something in the way he looked at her that made her sure he had survived something worse than anything she'd have to up ahead. It eased her nerves. She thought on how it would feel to have that beside her where she was going. She shut the engine off. He already had a slab of cardboard pressed to the long window of the camper shell that humped the truckbed behind her, and in the sideview she watched a boxcutter appear in his hand, trace the line of her window, something in his quick ease with the blade, its sudden appearance and disappearance when he was done, that cranked her nerves back up again. She watched him through the windows of the cab while one by one he blacked them out. His hair was a tangled self-cut mess as greasy looking as his mechanic suit and his beard was so long it was tangled up with it and when he was done with all but the driver's side, he leaned in.\n\nIn-Solar or refracted glaze, he asked her. She told him whatever would change the color most. He told her he had no clue. On glass In-Solar let the daylight in and kept out the glow sent down from the sky all night. Refracted did the opposite. But neither was meant for metal. She told him _whatever_ and _take a chance_ and rolled her window up.\n\nHe taped it over. \"Get out,\" he told her.\n\nShe pressed the lock down. Sitting in the darkness inside the cab, listening to the thunder of the spray on the metal around her, she could imagine his face rippled with the shadows of the rain, and the wan light seeping through the clear glass roof and above that the clouds with the reflected light leaking through and above them the mirrors. There were a thousand of them, or more. The government kept sending new ones up. They drifted ceaselessly across the sky in their ceaseless drifting orbit, their giant reflectors spread like parachutes meant to catch the sun instead of wind. They took its brightness and refracted it and sent it down upon the land, and for years now they had taken night away from the world and replaced it with their constant gloaming. Even in the first days of her life, even in the moment when she must have opened her eyes for the first time and gazed upwards, that was all she had ever known of night.\n\nHe was still at work when she shoved open the door and burst out, gasping. She grabbed at the cardboard on a window of the camper shell and wrenched it off. He quit spraying. In the sudden quiet beneath the rain there was just the sound of her ripping tape loose and the cardboard skittering across the floor and then she was yanking the airgun from his hands and had let loose all over the shell, plastic and glass and all of it.\n\nWhen she handed him the nozzle back, he took it without looking at it, his eyes never leaving her. \"You know you won't be able to see out the back in the day.\"\n\n\"I'm not driving in the day,\" she said.\n\nThen she tore off the rest of the cardboard, left it scattered around the garage floor like something dropped from a great height and exploded, and got back in the truck. He stood behind it. In the passenger side mirror he could see her hand reach over and unlock the passenger door.\n\nThey drove to the place that was the closest thing he could claim as his, a trailer on Catawba Creek that belonged to a friend who had one day disappeared to no one knew where. On the road, she pulled over and got out and squatted above the mud at the edge of the wet scrub while in the cab he rifled through the dross of her life that she'd left on the seat, his eyes glancing out the window at her eyes staring back. The whole walk around the front of the truck her stare didn't quit. It stilled his searching. She climbed in and before she had even shut the door they started at it. She drove\u2014barely\u2014and told him where and how, and they barely made it across the yard of sogged grass and wetter mud to the pine steps that rose before the door, and they didn't make it past the door. It was a backroad off a backroad but at least two vehicles passed while they were at it on the mud-slick steps and she didn't care and he didn't either. The rain came down on the tin roof. Lower, heavier, in the pooling yard. He ran around naked and tanned except where he was white as a peeled frog, looking behind each cinder block piling for the key while she laughed at him. When he found it and got the door open, she grabbed him by the flesh at his waist and dug in her nails and she was stronger than he would have guessed but should have known.\n\nInside, they lay into each other like mountain cats tied tail to tail. That was how his grandfather used to say the hill people did it and when they were done he told it to her. She laughed.\n\n\"Tails tied nothing,\" she said.\n\nThe ceiling above them was made of laminate strips painted to look like wood. Some had come loose and bowed down. He couldn't remember if they had hung like that before. It wouldn't have surprised him if they'd been tight and flush an hour ago, two, whatever. In the living room a busted couch arm hung by its upholstery. In the bathroom the ceramic toothbrush holder once attached to the wall lay scattered in its many ceramic parts. In the bedroom: the bamboo blinds in a heap below the window, the mattress shucked of sheets, pillows strewn. In the kitchen: pieces of plates littered beneath the table and the ones that had survived lying there dumbfounded and whitefaced as if traumatized at how they had been used. And mud simply everywhere.\n\nThey lay there smoking. The rain drummed on the roof and then quit and there was just the rush of the creek roiling by. A breath of air blew in through the still-open door and drew a shiver across them.\n\nHe said, \"I know it's what you're supposed to say, but\u2014\"\n\n\"Then don't say it.\"\n\n\"But you know what.\"\n\n\"Yeah. Me never either.\"\n\n\"You said it,\" he said.\n\nShe nodded. \"How come we didn't go to your place?\"\n\n\"Look around you,\" he said.\n\nHer smile did something to him, already, that made him want to see it again as if it was something he needed or would get the shakes, and he hadn't known her half a day.\n\n\"You didn't know how it would be,\" she said.\n\n\"I knew. But also there's that I don't so much really have a place.\"\n\n\"Where do you sleep?\"\n\n\"Rundgren's,\" he said. \"In the cargo vans.\"\n\nShe looked at him.\n\n\"Yes ma'am,\" he said. \"I have my pick of the seats.\"\n\n\"They don't pay you enough to get yourself a room?\"\n\nHe took one last drag and blew three smoke rings and rubbed the butt out between his fingers and flicked it against the wall and it bounced back and almost landed on her. One of the things he'd done made her smile again. He wished he knew which it was. He was old enough to have learned there was a kind of woman that did it for him and knew it was the allure of something dark in them they hid or tried to hide, and knew too that learning what it was always made it leave. But watching her so close he could see the dark grains of makeup below her eyes he thought she was not just his kind of woman, but the woman the idea of his kind was based on, and if the feeling was going to leave he wanted it gone.\n\n\"What are you running from?\" he said.\n\n\"I'd like to see you try and shrink my head.\"\n\n\"I meant it literal.\"\n\n\"I'm not,\" she said.\n\n\"You keep a Chief's Special .45 just to the terrorize the squirrels?\"\n\nShe looked at him and she wasn't smiling and in her eyes was what he'd thought he'd glimpsed while they went at it but it was rarified and stripped clean now and in all truth it scared him.\n\nHe said, \"Well, if you didn't want me to find it, you shoulda hid it a better place than the glove box.\"\n\n\"Who says all the other hiding places aren't already full?\"\n\n\"I wouldn't doubt it,\" he said. \"I've never seen hollow tips for anything but a rifle shell before, at least not in a box in the truck of a woman who claimed not to be running from someone.\"\n\n\"What if I'm hunting someone instead.\"\n\n\"Then I'd pity them. But you're not.\"\n\n\"Okay,\" she said, \"I guess you want to tell me why you're living in the back of a van and covered up in that fucking beard and have got right there what I'm guessing was once a tattoo of a butterfly or a little bird or something, which I mean whatever, but why the fuck'd you try to hide it with that nasty hand pricked shit that looks just fucking nasty and, while you're at it, how about touching on why you took the boxcutter with you and how come you have spare blades sewn into pockets on the inside of your belt?\"\n\n\"You don't like the beard?\" he said.\n\n\"I didn't say that. And you wouldn't show me your driver's license picture.\"\n\n\"I told you I don't have one.\"\n\n\"My point,\" she said. \"And won't even tell me your fucking name.\"\n\n\"You didn't tell me yours,\" he said.\n\n\"There you go.\" She said it like he'd won her argument for her.\n\nHe got up and walked to the open door and stood there, naked, leaning against the doorjamb, looking out at the empty dirt road.\n\n\"I'm guessing you're gonna move on soon,\" he said. When she didn't say anything he looked back at her and she was lying there sprawled out still in her boots and nothing else and she rose onto her elbows and looked at him and he saw the way her breasts remade themselves into a new and equally perfect and still unexpected shape and knew that would be in his mind for a long, long time and he looked back at the road and said, \"Shit.\"\n\nHe heard her come up behind him. She didn't bring a towel or a sheet or anything but the pack of smokes, bright green in her hand, just stood there naked beside him naked in the door.\n\n\"You're gonna attract attention to yourself,\" he said.\n\nShe smacked the pack and drew one out and realized she had forgotten to bring the lighter. She stood there holding the long white useless thing. The rain had quit and the clouds had piled onto the horizon and the rest of the sky was clear. The whole valley was lit up like a work site, the pale trunks of the beech trees and the needles of the pines bright in the floodlit forest, the dirt road clear of darkness as a city street, the flat roofs of the trailers in the trailer park around them gleaming, and on them the pooled rainwater and in the water the reflections of the sky: all the mirrors, like a thousand full moons swarming, shining down their reflected shards of day.\n\nAcross the road there was a dog house in the full blast, in front of it a metal bowl turned over and on fire with gleam. There was a dog standing on its roof. There was no owner's house to go with it, or trailer, or anything. Just woods and a dirt patch and the dog house and then more woods. In the woods, he knew, there were still some animals left, but he had not seen a dog since what seemed forever. As a boy he'd had one, but after the mirrors were launched they had to put it down. It had gone mad at the constant light. All the animals did. They seemed to have lost all sense of what was natural to them. Some had stopped mating and died off. Some had migrated south, confused and desperate, and had not been able to outrun the light that always followed and had turned violent in a way that people talked of with a sense of shame. It was as if they no longer knew what they themselves were and they tried to get into the houses and garages and anywhere that was sometimes blessed with a manmade dark, and once there they did things to each other that people didn't want to look at it, so most of them were killed off, too. Others had gone north and it was rumored they had found the line where true night began again. People talked of the possibility of pets up there.\n\n\"I wonder whose dog that is,\" he said.\n\nShe reached over and wedged the unused cigarette behind his ear and left it there. She looked at him. \"Cute,\" she said.\n\n\"It's not even guarding anything,\" he said.\n\n\"Maybe it's guarding this trailer. Maybe it's your friend's.\"\n\n\"It'd be nothing but the bones of a dog.\"\n\n\"You should come with me,\" she said.\n\nA car sent its noise around the bend and came after it and the dog leapt off the roof of its house and hit the ground running and ran for no more than a second before it was jerked into a backflip and landed in the dirt. The car passed through the barking of the dog, its small yellow dusklights glowing and the rainwater spraying from its tires and in the thin spatter of its wake the dog did the same thing again.\n\n\"Would you look at that,\" he said.\n\n\"You heard me,\" she said.\n\nThe road puddles returned to rocking the reflections of the mirrors above and the dog stood there barking at the dwindling sound of the tires and stopped and then slowly climbed back onto its roof again. He knew it was just the end of the chain but he liked to think it was some invisible something that the dog had come up against and could not pass, he liked to think that, and he thought she would like to think it too and he told it to her.\n\nShe reached across to his head exactly as she had done with the cigarette and found with her fingers a grease-and-sweat-bound curl and played with it.\n\n\"Do I get to ask where you're going?\" he said.\n\n\"If you're coming,\" she said. She looked to the east and watched the top of the ridge where the new mirrors slid into view, the whole sky creeping at the speed of their orbit. He watched it too. \"Away from that,\" she said.\n\n\"North or south?\"\n\n\"Is that you asking?\"\n\nHe reached up and caught her hand at his neck and pulled her to him and said, \"You fucking bet.\"\n\nThey wound up in the exact same spot on the same patch of floor staring up at the same ceiling and none of it had been similar at all. The cigarette was still behind his ear. She took it and they both looked at it as if there was no way of accounting for the fact, as if she held between her fingers evidence of God's approval, and he lit it and they smoked it down while the light thrown by the mirrors probed the kitchen around them and they promised their one rule to each other. Then she reached over with the cigarette and ground it out on his chest. He shouted and leapt up and stood there swearing.\n\n\"You know you like it,\" she said.\n\nOutside, the dog stood on its roof, barking at them, or at the mirrors sweeping the sky above, or just at a world it had been pinned to and left in and given no way of understanding what for or even ever learning why.\n\nThey drove during the long dusk, which was what people used to call night but could not be called that now, and they kept to back roads, wending their way northwards along the spine of the Appalachians, up past Winchester and Stroudsburg and into Pennsylvania. The land bellied out and the horizons flattened and the barns changed to stone and the houses, too. The sky stayed the same. The mirrors filled it and moved on and it was filled by more. They were aimed by signal to throw their reflective beams down on cities as they passed, or prison yards, or factories, or forests where men worked cutting timber through all hours, or construction sites where buildings went up without pause for dark, or ballparks with their floodlights defunct, and anywhere where costs could be cut and productivity doubled by the simple continuation of light. Driving, they could see the bright patches of near daylight where the beams landed and they avoided them if they could. Elsewhere it was just the peripheral glow, a cold luminescence like the landscape under a full moon with its brightness cranked up to double. The mirrors floated where the stars used to be, winking as they shifted to meet a need, larger than stars, and smaller than moons, and like nothing the sky had seen before. The moon, when it was out, hung dully in the hidden darkness beyond them, the breathfog of a moonshape, resigned.\n\n\"Alright,\" she said out of nowhere, \"let's talk about the future then.\"\n\n\"Alright,\" he said.\n\n\"What are you gonna do up there?\"\n\n\"Try to get across the border.\"\n\n\"I mean after.\"\n\n\"How are we gonna get across it?\"\n\n\"I want to talk about the future future,\" she said.\n\n\"In the future I'm gonna ask you again how we're gonna get across the border.\"\n\nShe picked up something on the seat next to her and threw it at him. It hit the window like it would have done more than just hurt.\n\n\"Jesus,\" he said. He felt at the chip in the glass and said, \"Jesus you're a bitch.\" She was smiling sweetly at him and he forgave her so fast he forgot if he had ever held it against her. \"What are you gonna do?\" he said.\n\nThat was when she told him she had never seen the night. Not the real night, the way it was when he was a growing up. She had seen a planetarium. She spoke of it as if it was almost the real thing and he decided not to tell her how far from real it was. She drove and daydreamed aloud on how the real night must be and all her ideas were wrong in small ways\u2014he thought on how sounds crystallized in dark and how the moon broke the black horizon in a different place each night and sometimes it was orange and squashed by the sky it was trying to rise through as if the stars had amassed their weight against it and meant to push it back\u2014and in the middle of it he told her that planetariums weren't anything like the real thing at all.\n\n\"You've really never seen it?\" he said, and in her silence tried to count back to the years when the mirrors first began to fill the sky. \"How young are you anyway?\"\n\n\"How old are you?\" she said. And then: \"Old enough to have done something bad.\"\n\n\"You or me?\"\n\n\"Both.\"\n\n\"Why do you think I've done something bad?\" he asked her.\n\n\"Why do you think I have?\"\n\n\"Because you told me. You said I wasn't to ask what you'd done to run from, which implies there was something you did that warranted the running. Besides, a woman like you doesn't get past puberty without doing something to make her parents cry and moan and gnash their teeth.\"\n\n\"That's what you like about me,\" she said.\n\n\"I don't mean sexual.\"\n\n\"I don't either.\"\n\n\"Yeah,\" he said. \"It probably is.\"\n\n\"It's what I like about you,\" she said.\n\n\"You don't know anything about me.\"\n\n\"I know you won't let me ask you about anything, which if you think my one thing is something not to ask about, what does that say about your life full?\"\n\n\"It doesn't say I've done anything bad. Could be something bad was done to me. Could be there was just a general badness in the world and I've lived through it.\"\n\n\"Could be,\" she said. \"But it's not.\"\n\n\"Well you don't know, or will,\" he said.\n\n\"Yeah,\" she said. \"That's what I like in you.\"\n\n\"Not knowing me?\"\n\n\"Knowing there's something to know and not knowing it and knowing you won't let me know it.\"\n\n\"You're how old?\" he said.\n\n\"Twenty,\" she said.\n\nHe sat there listening to the bad music and watching the hotspots in the distance glowing bright where the mirrors were focusing their beams.\n\n\"What?\" she said.\n\n\"That's older than I'd thought.\"\n\nIt was nearly true dawn when they pulled onto an old farm road and by the time it dead-ended in a thickness of woods, the sunrise was slicing between the eastwards trees. Westwards a blinding doppelganger glinted: the last of the mirrors setting. They got out and stood watching the truck change color. It had been red with a white camper shell when she had brought it to him, and under the mirrors the spray had turned it the color of bruised cherries topped by the yellow of coffee-stained teeth and now in the full sun it was all deep brown. They had driven it wet and bugs and grit had stuck in it as it dried. With his fingers he peeled away some insect's wing, held it up into the sunlight, watched it darken to black.\n\nShe climbed into the truck bed and he tossed the wing and climbed in after her. Inside there was a pileup of suitcases in the corner by one of the wheel humps, all of a rolling set, and all unzipped upon wild spillages of female wear, and wedged in beside the other wheel hump was a plastic fishing tackle box unlatched and the plastic trays risen out of it and all the colors of all her makeup strewn and jumbled as if someone had shaken the box like a die, and behind the suitcases and the makeup sat a large plywood trunk, fit flush from bedwall to bedwall along the back of the cab, unpainted and lidded and locked. On it someone had wood burned the word HOWELL. Before it was another word that had been scratched and gouged till it was unreadable now. She was flapping out a wool army blanket and as she let it settle he saw it again, stenciled in black: HOWELL. And again the antecedent word defaced.\n\nShe shut the gate and pulled the rear windshield shut above it. The windows were all black and it would have been true dark if the light hadn't come pouring in behind them from the windshield of the truck cab. She swore at it, wanted to block it out, hang the blanket over the window to the cab, but he told her he was too beat to do anything but sleep and she lay down next to him and they lay there.\n\n\"Aren't you forgetting something?\" he said. Then he put two fingers of his other hand to the back of her head and puffed an explosion with his lips against her ear. He got up and slid open the rear window, reached into the cab and popped the glove box and brought back the .45. She took it from him and he gave her the cartridge box and watched her slip the clip out of the handle, load it full, slap it back in. She checked the chamber and cocked the hammer and locked the safety and the whole time he watched her trigger finger and it didn't once touch the guard. He had seen on the barrel, engraved on its steel, the same word written and before Howell was James.\n\n\"What you did,\" he said. \"Is it bad enough they'll come after you now that we're out of Virginia?\"\n\n\"They'll hunt me till I get across the border,\" she said.\n\n\"I figured.\"\n\nShe turned and looked at his face so close she could feel his breath on her eyes. The orange sun came through the window from the cab and lit his beard like a torch blazing around his cheeks, his cheekbones like hot stones, his eyelashes like sparks escaped. The weight of his head lay on her arm and numbed it and she felt the feeling drain out of it and loved to feel it go. His head, heavy as a boulder. His ear, jutting out from beneath the blanket, dark and folded in such mysterious patterns of flesh. She had spent her life so far in moments a whole lot less peaceful than this.\n\nShe said into the scent of his neck, \"What's it like? The real thing. At night.\"\n\n\"It's limitless,\" he told her. \"That's the difference. They try to re-create it but there's only so much they can create and you feel that, but the real thing you don't, you feel the opposite.\" He was at the edge of sleep and his eyes were closed and she could feel his beard brush her face as he talked. \"And it's always different,\" he said. \"You can't know it. You can't even know all that's in it, and even if you could, you can't know what is going to cover it one day, maybe there's some clouds, or maybe there's just some light nearby, or maybe there's dust in the air, or anything, it all changes it, and you can't ever know exactly how it's going to be, or even what it really is.\"\n\nOutside the birds were sending their birdsounds through the trees.\n\n\"I thought they mapped it,\" she said.\n\n\"Sure,\" he said. \"They mapped what they can know, but there's stuff out there too far for them to even guess at.\"\n\n\"I thought they had theories.\"\n\n\"Theories are just guesses,\" he said. \"And there are some things way beyond what even they have theories for.\"\n\nThey were quiet for a while.\n\n\"It's not like that in a planetarium,\" she said.\n\n\"No,\" he said.\n\nThen he was asleep. She nestled tighter to him and felt his heat warm her almost till she couldn't stand it and she held herself there and tried to sleep, too. A while later she heard him say, \"The mystery of it you feel it's dark you feel so dark you're...\" But that was all he said and when she lifted her head and looked at him his breath was even and heavy and his eyes were tremoring beneath their lids. He had an Adam's apple like a knuckle in his throat. She reached to it, so lightly, and with her fingertip rode his swallow. He slept on. When she let her hands hover over the place his heart would be, palms down, she could feel his chest hair rise to meet her lifelines, draw away again. He had a sunken chest, like someone had taken a hammer and chisel to it and banged along his breastbone. The hair on it was gray as an old man's, and beneath it there were the strange tattoos buried, self-done and lumpy with scars. She tried to make them out. They were as inscrutable as wormwood grown over with moss.\n\nOutside, the birds that had once been night birds and the ones that had once been day birds filled the forest with their confused cacophony. Below it, other animals made noises in the leaves. She knew they were small, probably, but they sounded large. She knew too that there were still some of the larger ones left in places like this, small patches of woods that no one went in anymore. A few last stragglers hanging on. They were the worst. It seemed impossible to drive them out or kill them off. They seemed to already have in them what the change had required, as if it had been hulking there in their genes all along.\n\nWhen she was nine her father stuck his handgun in his belt and took his shotgun in one hand and her hand in his other and led her out into the woods. It was the long dusk and the trees slivered the mirror light around them. Back then the sky was less full of them and in the patches of land between where the light was directed the night was dimmed down to almost dark. He made her stand there with him till the first one came through the underbrush. He gave her the handgun and told her, _Shoot it_. She got it in the gut and it came at them, scrabbling and falling and trying to stay on its feet, a trail of itself left steaming behind it, until it was just a few feet away and her father shot it in the skull. Then he said, _Wait_. It was not five minutes before the others came, slinking through the scrub and scampering over the dry leaves until they lunged upon the dead thing leaking its scent and turned on each other, a rolling churn of them amassed over the carcass, tearing apart anything close enough to tear, eating and fighting and howling in pain and more coming through the brush. See, her father told her. _There's no mystery to it. It's just bad_. He ran his thumbnail down the part of her pigtailed hair. _That's better isn't it? To know?_ He was the kind of man who thought the mirrors beautiful.\n\nThere was a time in the hottest part of the day when he woke to the press and ease of her breasts against his back and lifted her arm and drew himself away from her and crouched quiet and bent under the low ceiling of the camper shell, his spine missing the feel of her breathing. She had pulled the blanket over her eyes and left the rest of her to breathe. He lifted it off and draped it, working slowly and silently, until it covered the rear window of the cab and it was dark but for the places on the side windows where she had missed. Then he opened the tailgate and climbed out. In the full blast of the sun he circled the truck, starting at one long window of the plastic shell and moving to the rear window and to the one on the other side. He was barefoot, in his underwear. In his hand a small stone glinted. Against the glass it made a rough scratching sound.\n\nInside the truckbed again, he shut the tailgate behind him, and on his knees turned to look. She lay still sleeping where he had left her, but the draped blanket had settled darkness over her and the hundreds of holes he'd scratched in the window coating let in the daylight. All around her a small new galaxy floated. He had meant to wake her and show her, but from the west the sun came through the holes\u2014a hundred luminous threads all slanting\u2014and found her skin and turned her naked body into a lake reflecting. He could not bring himself to stir it. And when he finally did crawl back beside her, he lay for a long time looking at a small dot of light that moved across her neck, slow and steady as the turning of the world they were on. When it got to her collarbone he placed his hand over it. There it was on him, now, there between the tendons behind his knuckles, and he could imagine it going right through like a bullet, marking them forever with two halves of the same wound.\n\nShe woke just before mirror-rise. He lay watching her as he had lain the last few hours. She took in what he'd done. Nothing of her moved but her eyes. All around them the camper shell was dark and the stars he'd made were white and they watched them burn down slowly to orange and red and almost gone. The long side windows grew more and more clear, the spray coating losing its opacity as the sunlight drained, and the mirrors came, until through them they could see the dark shapes of the trees and the pale orbs of the mirrors filling the space where the hot stars had been. Out there the sleepless birds sang their ceaseless songs.\n\n\"What's your name, anyway?\" she said.\n\n\"Minor,\" he said. \"Juhle.\"\n\n\"Minor Juhle,\" she said.\n\n\"Mitchel,\" he added. \"Minor Mitchel Juhle.\" He had been calling himself that for such a long time he considered it true.\n\n\"You know what's funny?\" she said.\n\n\"What?\"\n\n\"My name is Pearl. Well my middle name. My first name is Drema.\"\n\n\"Drema Pearl.\"\n\n\"Isn't that funny? Pearl and Juhle?\"\n\n\"Not particu\u2014\"\n\nHer elbow caught him below the rib, hard, and he took a deep breath to ease the pain. \"Drema Pearl Bitch,\" he said.\n\n\"My last name's Howell,\" she said. \"Not that you asked.\"\n\n\"Drema Pearl Howell Bi\u2014\"\n\n\"Don't say it.\"\n\nA little while later they were on the road driving north again with the windows down and the mirror-light blowing through them on a roaring wind.\n\n\"Mmmm Juhle,\" she said.\n\n\"What?\" he said.\n\nShe said it again, slower: \"Mm-Mm Juhle. Can I call you that?\"\n\n\"You can call me baby,\" he said. Then he grinned at her and sang it. She joined him, but after the chorus neither of them knew the words so they just sat there howling into the wind a wordless semblance of the tune.\n\nThey drove on north through long miles of agri-industry, huge fields staggered in their planting, the ceaseless sunshine quickening their rush to yields so in these months farming was a constant race to meet the rolling harvest. They passed wide swaths of new soybeans, their small leaves glinting in the gloam, and stretches of paled plants ready for cutting, and vast strips already being cut, so many white workers' hats flashing in the mirror-light, as if a flock of migratory birds had alighted to feed on swarms of something heavenly and rare, the way they used to before the mirrors changed the migrations. He had seen it long ago and he commented on it. She only knew of it from nature shows she'd seen in school. He was older than she'd thought. He asked if it mattered. She said it sure didn't seem to did it. And it was midnight. They passed wide-built single-story factories with their clear glass roofs letting the light down onto floors full of figures working through the electricless post meridian hours and forklifts scuttling across mirror-lit lots and they passed through the lumberfarms of west New Jersey where forestfuls of lumberjacks worked the quick-grown giant pines and there was no sound but the rumbling of the truck engine and the constant sawing from all directions as if the woods had been infested by some horde of saw-jawed pests. And they saw New York City rising in the distance, the mirror-light glancing off the glass of all the roofs, and all around it the oceanfields of floating solar rafts gleaming with slabs of the reflected sun, and they drove on.\n\nThey didn't eat anything until it was well past midnight and he had taken the wheel. She turned in her seat and slid the rear window open and, reaching behind, unlocked the chest in the truckbed and lifted out a cookie tin. It was old and rusted around the lid. On the lid was an embossed scene of a cabin in winter some long ago night, its chimney smoke lost in the starry sky above the trees and its windows aglow in the blackness between the pines and in each pane a Christmas scene\u2014fire, children; roast bird, dog. She put it in her lap and pried the lid off. It was filled with small pellets of puffed grain so sugared its smell made the air in the cab feel sticky. She offered it to him. He looked at it and asked her what it was. She took a pellet and shoved it between his lips. He didn't want more. She shrugged and sat there eating handful after handful, lifting her palm to her chin and touching her tongue to a puff and snapping it back in. He said it was disgusting. When she was done, she licked her palm. \"Look at me,\" she told him, and slowly sucked the sugar off her fingers.\n\n\"Whatever,\" he said, as if it was too much.\n\n\"Whatever what?\" she said, and, staring him down to let him know she knew him already and how well, made him lick them for her all over again.\n\nThat was it as far as food. By the time they were in Connecticut he was starving, and when they passed the Indian reservation, he turned in without asking.\n\n\"Not with my money,\" she said.\n\n\"You don't have any money.\"\n\n\"I could have shitloads, for all you know. I could be\u2014\"\n\n\"If you had it, you would've paid for the spray job and left me staring.\"\n\n\"You think that's why?\"\n\n\"No. But you don't have it.\"\n\nShe reached over and gripped his wallet where it pressed the jeans atop his thigh. \"Feels pretty skinny to me,\" she said.\n\n\"There's enough in it for a steak and a coffee.\" They pulled into the casino's lot. \"Two steaks if we get them here.\"\n\n\"I don't eat meat,\" she said.\n\n\"Who said one of them's for you?\"\n\nThe main building was shaped like a tepee, a hundred feet high and made of Plexiglas and the mirror-light flooding in to show the gaming going on and all around it like a wagon circle sat the outbuildings\u2014from darkness vaults with their beds and black silence to the bulbous planetarium room grown beside the parking lot like a helium balloon tethered to earth. It was their busy time, near the end of the long dusk, when the sleepless gave up trying and wandered out to twenty-four-hour drugstores, or all-night coffee bars, or gambling joints that could help them pass the time till sunrise.\n\nHe parked and got out and shut the door and waited for her. Then he came around to her side and leaned down to the window and said, \"It's as safe here as anywhere. It's not even under federal jurisdiction.\" She sat. \"These are tribal cops,\" he said. \"What do you think they're gonna do, wander out to the woods and do a bad woman dance and the Great Spirit comes down and slips them a vision of your truck? What? Is this the silent treatment?\" He stood up and looked around and leaned back down again. \"Come on,\" he said, \"I'll buy you a squash.\" Then he shrugged and reached in past her and popped the glove compartment and took out the Chief's Special and unzipped his coverall and slid the pistol in his pantswaist and zipped back up and left her there.\n\nShe watched him through the windshield until he was almost at the door and then she burst out and ran across the lot and caught him in the lobby and jumped on his back and yanked his hair like she meant to scalp him. She was laughing or he couldn't have wrestled her off so easy.\n\n\"That was smart,\" he said when they were seated and most everyone had stopped looking at them.\n\n\"They just think we're in love,\" she said.\n\n\"Is that what this is?\"\n\n\"If you think this is love, you've got a worse past than I thought.\"\n\nHe ordered and she ordered and he told the waiter to bring her three of what she'd asked for. They were silent until it came and then they ate in silence. Halfway through he said, \"Yeah, that's what this is.\"\n\nAfterwards they pooled their cash and tried their luck with the slots. Their luck ran to two buckets of nickels. She thought it might run further. He thought it might run out. And when they went to swap the coins for bills and saw a man in uniform with a gun chatting at the cashier's window, neither one wanted to find out. On the way back to the lot, he left her for the washroom and took the buckets with him.\n\nShe watched him walk off. His figure had become familiar to her already. The way he strode with one sneaker turned in a little, hips swinging, elbows stuck out like lug bolts, his crazy unwashed curled-up wildness of hair. His neck was thick as her own thighs and when she lifted the curls that covered it it was white as them, too, vulnerable like the underbelly of a pet, and she had fallen asleep and risen awake with the smell of it close to her, and it did something to her that she had not known could be done, and she felt it even now. Just before he went inside the door, he turned to her and winked and held up the buckets and she could see where the fabric of his coverall beneath his arm tightened and at his side the bulge of the handle of her gun.\n\nWalking back to the truck, she wondered if what he'd done was as bad as he seemed to think. She decided it probably wasn't. She thought he was a good man deep down at heart and she liked that he was not one on the surface and she thought that his past probably lay somewhere in between, imagined for him a drug-dealing sentence, a stint in jail, a failed attempt to rob something, maybe a fight, a life of fights, and liked the idea of that, liked even the idea of someone shot, hurt bad, killed?\n\nShe was halfway across the lot when the revolving casino door whirred so fast the noise stopped her. She turned in time to see it spit him out. Between the giant plastic buffalo mounted on either side, he came running.\n\n\"Get the fuck!\" He shouted. \"Get the truck in the fuck!\" He was running too fast to keep the buckets still and coins were flying loose around him. They came down clattering all over his feet, a twinkling rain of mirror-glint, and bounced off his thighs and struck his shins and he ran through them. \"Get the fuck in the truck!\" He had almost reached her and she was turning to run with him when she saw a man coming out of the revolving doorway behind him. He was an Indian and he was dressed like a cop and in his hands was a rifle.\n\nThe shot smothered the rattling of the buckets. She had the cab of the truck open and she looked back. Minor had stopped running and he was trying to hold the buckets in one hand. His other was scrabbling at the zipper down his chest. Behind him she saw a flash of something arc in the air. Saw the reservation man thumbing another bullet into the chamber. Minor saw it too. He was running again when the second gunshot boomed. Her scream seemed to come directly out of the jerk his body gave. Then he was on the ground, dropped to half his height. His legs were bent under him. He was looking at his legs like he had just discovered them there. She was still screaming when he struggled up and she didn't stop till she heard him shouting at her.\n\n\"Take it take it take it!\" he was saying.\n\nShe ran back from the truck and reached for the buckets and he said, \"The fucking gun! The gun!\" He'd gotten his chest zipper down and his coverall was flapping at his front and she reached in and drew the .45 from his belt as he stumbled past. The Indian was jerking the bolt back. She raised the handgun and sighted at him and he saw her and froze. He shouted something. She held the gun in both hands. Behind her Minor was banging around. In a second the truck would start. She had never wanted anything more she thought, than to hear the truck start. The Indian stood there with his hand on the bolt and the rifle in his other hand and another brass casing in his teeth. He shook his head at her.\n\n\"Shoot him!\" she heard.\n\nBut she just held the gun and in the moment after Minor's shout it was as if the reservation man and she entered into some secret language neither of them had understood till just then. \"Start the truck!\" she shouted.\n\n\"I can't drive,\" he said. \"Shoot the fucker! Shoot him, shoot the fucker! I can't fucking drive!\"\n\nBut the Indian had stopped shaking his head, and stopped shouting. He reached up and took the bullet from his teeth. She ran. She had just yanked open the truckdoor when the shot banged. She threw herself belly-down onto the seat. Minor was there with her. The sound of the bullet-struck metal all over them. Minor shouted at her, something about keys, and she got herself in and shut the door and tried to start the truck but she couldn't find the ignition slot, and then the keys were out of her hand and Minor was saying \"Push the clutch!\" The truck roared. \"Drive!\" he said.\n\nThe gravel thundered against the parked cars. She pulled the truck around. The Indian had come out into the open lane and was following the path of the truck with his gun and Minor was leaning out the window with the pistol in his hand and there were four or five shots. The truck lurched and Minor's back was slammed against the window and she heard the blast of the air and could feel the rear drop at an angle and she shouted, \"Minor?\"\n\n\"I oughta kill the fucker,\" he said.\n\n\"Kill him,\" she said.\n\nHe fired off the rest of the round\u2014one long rattling blast. In the rearview she could see the ground smoking where the bullets had hit and the reservation cop standing with his gun and the wide space in all its wide quiet between the two.\n\nThey drove with a flapping in the rear and then dragging and then just the rim grinding away at the road and they knew they'd have to stop and they didn't.\n\nHe had been shot in the leg. He was bleeding all over the seat and he got a rag out of the back and tied it around his thigh and tried to bend it so he could get it past the dash and up high where it would not bleed so bad but it was already stiffening. He had to slide around on the seat to get it out the window.\n\nShe said, \"Did it hit the bone?\"\n\n\"No.\"\n\n\"You okay?\"\n\n\"Fucking great. Fucking never been better.\"\n\n\"You gonna be okay in an hour?\"\n\n\"I'll tell you in an hour.\"\n\nShe let her eyes leave the road long enough to take him in. He had shut his own against the pain. \"What happened in there?\" she said. He shrugged and she told him \"fuck that\" and said, \"He just started shooting.\"\n\n\"I took a piss.\"\n\n\"What happened to make him\u2014\"\n\n\"I unzipped is what happened.\"\n\n\"And he saw the gun?\"\n\n\"I guess you'd rather I'd have just showed him my ID. I guess you'd rather I'd have brought him out to the truck, out to you\u2014\"\n\n\"You said you didn't have an ID.\"\n\n\"So he could ask you for the license for the gun.\" He had shut his eyes against the pain and he opened them and looked at her and shut them again. \"Of course, I guess you'd have just shot him.\"\n\nThey drove on, too fast for the road and wrecking the wheel until a smaller road punched the side of theirs and she turned onto it skidding. When they had driven far enough, she pulled to the side and got out. He stayed half-prone with his leg out the window while she clanked around in the truckbed for the jack. He could feel all his blood rushing to his leg trying to refill it and could feel it leaking out just as fast and filling his pant leg instead. Back there the heavy spare wheel dragged on the metal and dropped to the ground with a thud. He tried to get his heart as low as possible. He tightened the rag. The truck jerked, began to rise. He couldn't see her where she was crouched but he could feel every crank of the jack.\n\n\"I'm gonna tell you something,\" she said. She waited, then she went on. \"Just because I've never shot somebody doesn't mean shit. You think that means something?\"\n\n\"Just that you haven't shot somebody,\" he said.\n\n\"Listen asshole, either you're the fucking worst shot I've ever seen or\u2014\"\n\n\"No, I meant to miss him.\"\n\n\"So don't fucking say anything.\"\n\n\"I didn't.\"\n\nThe truck stopped its rise and he heard her grunting at the lugs and then her breathing as she hauled the tire off. He listened to the spare go over the bolts and to her wrench the lugs tight again and then he heard her boots on the road. Then he saw her mirror-thrown shadow and then she was in the window looking at him.\n\n\"You didn't do anything, did you?\" she said.\n\n\"I never said I did.\"\n\n\"Huh,\" he said. She shook her head. She took a cigarette and lit it and stood there smoking and when he asked for a drag she said, \"You implied it.\"\n\n\"Give me a drag,\" he said. \"I'm fucking shot.\"\n\n\"It's probably the first time.\"\n\n\"Does it matter?\"\n\n\"Why won't you talk about your past then?\"\n\n\"Because I'm fucking shot. Give me a drag. I haven't lied to you and I won't.\"\n\n\"You haven't lied,\" she said, as if he had.\n\n\"Or ever will,\" he said. \"You can shoot me if I do.\"\n\nWhen she looked at him he felt a grin come up in him and she looked away, grinning, and said, \"Asshole.\"\n\nShe took another drag on the cigarette and turned and handed it to him. Leaning in the window, filling the space left by his leg, she looked at him. \"Does it hurt?\" she said.\n\n\"What do you think?\"\n\n\"What do you want to do?\"\n\n\"Get to the fucking border.\"\n\n\"Do you love me?\"\n\nHe watched her and smoked and watched her. \"I think so.\"\n\nShe took the cigarette from his fingers. \"If you knew what I'd done,\" she said. \"You wouldn't. You would leave me. That's why I won't tell you, okay?\"\n\nThey did up his leg as best as they could with what she had in her first aid box, him grinding his teeth while she poured disinfectant over both holes of the wound and her trying to kiss him quiet afterwards while she wrapped it. He had passed out and rewoke and gone through his hours of faintness by the time they started looking for a place to bed for the day. The old border couldn't be more than a half hour's drive north. He commented on how if they had still been citizens in good standing and the Northeast Kingdom had been what it once was, they would be home free by sunrise. She said \"citizens in good standing\" like the whole idea was funny, and then pointed out how no matter where the mirror-light stopped they would still have to pay the exit tax citizens in good standing were made to pay. They had lived their lives with the benefits of endless light and if they took the resources of themselves away the government would demand its reimbursement, its mirror tax.\n\n\"I've never been worth that much in my life,\" he said.\n\n\"And now you're crippled.\"\n\n\"Or had near to that much on me.\" He looked his question at her.\n\n\"Good thing we're not citizens in good standing,\" she said.\n\nHe told her, \"I said I'd ask it again.\"\n\n\"And I told you I had it covered,\" she said.\n\nThey drove through the dim landscape of inland Maine, glinting rocks and stunted, wind-bent pines and bushy fields striated with dry grasses sheened in the reflected light, and they talked about what it would be like to get out from under the mirrors. They had heard rumors of the life up there\u2014communes and villages and tribes eking out an existence on what they could grow in such barren land and what they could hunt and what life they could make heating peat for warmth and moving by firelight through long seasons of dark.\n\nHe told her he was going to farm. She laughed at him, and then saw he was serious. There was little that would grow there she said, maybe tubers, roots. He said that sounded like farming to him. \"Okay,\" she told him. \"Me too.\"\n\nThere was a strangeness over everything that was the tension of the real dawn coming into a brightness that was the exact brightness the mirrors had held over the world all night and for a moment it was as if the earth had ceased its forwards spin and had paused in the half light, waiting to see if the glow of the mirrors would hold sway or if the aura of the coming sun would outdo them. The aura outdid them. Just before the sun itself rose they hit a small town. All the houses were still sleeping, windows blacked out with their heavy curtains and lightproof shades. They took the first small road that looked like it would lead them away. Instead, it dead-ended at a library. It was all rust and molding bricks, half swallowed by scrub woods, long forsaken by the town.\n\nShe spoke of sinks and couches and the many weeks she'd spent sleeping in the truck. He wasn't sure how well he could walk so he let her figure out how to get in. She went off in the growing light. He watched her go. He listened to her legs brushing through the high grass. When she was gone there was still the sound. It wasn't her legs. From the edge of the woods came the noises of animals moving. Their eyes were lit in the gleam of the mirrors and their pelts shone in the dimness beneath the trees. He watched them, quick glimpses of oil-sleek fur like flashes of fishbacks breaking dark surface, and their eyes like the white buds of water lilies floating. He had heard they had gathered here, near the end of the mirrors' reach, the ones that chose not to go across into the natural night, the ones that chose to stay and hunt the others who came up from the south to try. For a moment he worried about her and he looked to see if there were any others besides the ones that were gathered in the brush surrounding him but all were drawn to him. He didn't know why and then the dizziness came and he shut his eyes against it and felt his leg throbbing and he thought for a moment that he had hallucinated them\u2014them and the eyes of the child, that girl, rolled back in their sockets, shivering white eyes, pupils gone beneath her lids, irises gone, just white and the veins shaking in it, and the pond of blood growing itself across the bright green plastic of the fake lawn her grandparents had tacked to the screen porch floor, and the grandmother in the woodshed like a pile of rags thrown in a corner and the log that had crushed the pile dropped onto her shins with a crack and left there rolling, and the grandfather rolling back and forth on the kitchen floor, holding the spilled insides of his belly inside the bag of his shirt and crying, and as Minor looked up from the trembling whites of the girl's skull-rolled eyes he could see every tiny square of the porch screen lit in the porch light and through them all the mirror-lit tulip trees with their pale blooms hanging\u2014and then he opened his eyes and the eyes of the animals were still there watching him and he knew why. His thigh felt like it was eating itself. It had stopped bleeding so much. His jeans had been so sodden beneath his coveralls that she had stripped them from him and dumped them in their wet mess on the rubber mat beneath his feet. He reached down now and balled them up. He had grown so used to the smell of his blood he could not tell anymore what it did to the air. Out there, they could tell.\n\nHe was glad when she came out through the front door and surprised by how quickly she had managed it. He called to her and warned her of the animals and she held up the gun to show she knew. He swung the door and got out. Standing on his good leg, with his bad one hurting just from touching the ground, he hurled the bloody jeans at the eyes in the woods. They crashed into the brush. The eyes scattered. He stood for a moment, watching them slink back, until he heard them snarling at each other over his offering. Then he limped as fast as he could to her coming through the scrub to meet him.\n\nInside, he found a couch and hauled his leg up onto the arm and lay there and told her he was not moving again until mirror-rise. She explored. He dozed off and woke to the sound of her rummaging in a desk and dozed and woke again to her talking excitedly to him while she held up a brick-sized bottle of white glue. When he woke next it was full morning and she was sitting naked on his chest. She had drawn all the shades and they were glowing green. Slivers of sunlight sliced in and found the books and dust motes and one lay slantwise across her body. She had filled her hair with the glue. It was pasted to the sides of her head and in the middle spiked up in a two-foot mohawk so thick with the stuff it had only dried crusty at he tips; the clammy rest flopped to a side. She reached up and straightened it while she talked, and each time she did her breast rose and tightened against her chest and then relaxed again and he smiled.\n\n\"I've been shot,\" he said.\n\n\"Poor baby,\" she said.\n\n\"I can't move enough.\"\n\n\"For what?\"\n\n\"For anything.\"\n\n\"Then don't.\" She got up off him.\n\n\"I'm sorry,\" he said.\n\n\"For what?\" she said again, and came back holding an old lime green tape dispenser full of yellowed tape in one hand. In the other she held heavy metal-handled scissors with blades like hedge shears. He lay there while she taped his wrists together and then his ankles, wondering the whole time if he was actually going to let her, and a little awed by the fact that he did.\n\nWhen she was done she put the tape aside and opened the scissors and, quick, so fast he couldn't do anything but say her name, she slashed them across the top of her thigh. She did it three times, pressing hard. He lay there, not knowing what to say, watching the blood well up on her in the gashed lines.\n\n\"You don't know what I'm capable of,\" she said.\n\nThey did things with each other then that they had not known existed before they did them and yet once done seemed like things they had been wanting to do since they had first met and when they were finished it was midday and they were both clammy and spent and more tired than they had ever been and neither of them could sleep.\n\nHe lay there and watched her while she browsed the books. Her mohawk had mostly dried and her hair was strangely translucent in the light. She pulled books and looked at them and put them back and between each one she picked at the dried glue on her palms.\n\n\"You're pretty messed up,\" she said.\n\n\"You're one to shrink my head.\"\n\nShe shrugged, pulled a book. \"You ever been to one? A psychiatrist?\" She looked up from it in her hands. \"Oh, I forgot, I can't ask you, um, anything.\"\n\n\"No,\" he said. \"I've never been.\"\n\n\"You should,\" she said.\n\n\"You should.\"\n\n\"I have.\"\n\n\"If you're gonna confess to me,\" he said. \"Don't.\"\n\n\"It's not like you think.\"\n\n\"I stopped thinking I could outthink it a while ago.\"\n\n\"I mean going to a psychiatrist. They don't put you on a couch and ask you about your fucking mother or what. They know everything now. About the mind. They know\u2014\"\n\n\"They don't know everything.\"\n\n\"Sorry, but...\"\n\n\"You mean the brain.\"\n\n\"... they do. I mean the mind. They can tell you what chemical makes you like french fries.\"\n\n\"I don't care to know.\"\n\n\"Or me.\"\n\n\"Good,\" he said. \"Then we're\u2014\"\n\n\"I mean or me. They can tell you what makes you like me. Or me like you.\"\n\n\"No they can't,\" he said.\n\nShe shrugged. \"They know what sadness is.\"\n\n\"Don't we all.\"\n\n\"And happiness, and hope and de-fucking-spair. They know what it _is_. They can point right to the part of your mind.\"\n\n\"You mean\u2014\"\n\n\"I mean the fucking soul,\" she said. \"They can tell you what brain transmitter fires what node to what receptor cell in what lobe, which you know what all of it means?\"\n\n\"That you're crazy about french fries.\" He smiled. \"With a fuckload of ketchup.\"\n\n\"Love. That's what love is. They know it. They can tell you exactly how it happens and why and they can\u2014\"\n\n\"You know what?\" he said. \"Keep that shit to yourself.\"\n\n\"The fucking soul,\" she said.\n\n\"No, that's not the\u2014\"\n\n\"Then what the fuck is?\"\n\nHe could feel the pain throbbing up from his leg, and it seemed to him he could feel the very nerves, each one, and the blood running through him and each vein that carried it and he said, \"You talk like we're fucking machines.\"\n\n\"I'm just telling it how it is.\"\n\nHe watched her reach the end of a row and disappear. He could hear her running the back of her nails across the spines of the books in another row. \"Do you believe that?\" he said.\n\n\"I don't think it's believe or not believe.\" She came into view and gave him a smile that was the first time she had smiled at him that he wished she hadn't. \"They can tell you what belief is too.\"\n\n\"Well, fuck it,\" he said. \"I don't want it.\"\n\n\"Me neither,\" she said.\n\n\"Come here,\" he told her.\n\nShe came and lay down on top of him, careful of his leg, just lay there without him having to say a word.\n\nAfter a while he said, \"I've been thinking about why I couldn't shoot that man.\" He could feel her breath on his chest and he knew she was listening. \"It's this clean slate.\" She shifted on him so she could look at him and he felt her breath on the underside of his neck. \"You don't know shit about me except whatever's been since we got together. And I don't know hardly shit about you. And fuck if I'm gonna fuck that up, you know?\"\n\n\"Yeah,\" she said.\n\n\"Fuck that up? No way. If there was a fucking rat over there\"\u2014he swept a hand at the whole place\u2014\"I wouldn't kill it. I wouldn't even fucking bother it. Not with you here. For all you know I'm a sweet, kind, good fucking man.\"\n\n\"Except for what you've done.\"\n\n\"Which you don't know. That's the beauty. That's the beauty, Drema. For you, that's here,\" he touched her temple so gently he wasn't even sure if his finger had actually made contact with her skin. \"That's exactly where it's best.\"\n\n\"What about for you?\" she said.\n\n\"The same.\"\n\n\"I couldn't shoot that man either.\"\n\n\"I know. That doesn't change anything.\"\n\n\"I couldn't leave you,\" she said.\n\n\"That's what I mean. Neither of us could. It'd be the first bad thing.\"\n\nShe kissed the underside of his neck. \"Sounds pretty good.\"\n\n\"Doesn't it?\" After a while he said to himself or to her or just plain said because it wanted saying, \"It does.\" He said it again, \"It does.\" And then they were asleep.\n\nTwo days later they were at the border. Or close enough Minor wanted to stop. They had long ago passed out of the realm of cities and they had driven through forests of dark jack pine and black spruce and white slashes of birch trees with mirror-light on their bark, and the woods had thinned and dropped off and in places rose again, and road by road their options for pathways north had dwindled to just the one.\n\nIt was well into dawn and almost full-out morning and Minor sat there looking at her, waiting for her to stop.\n\n\"It's gonna be light when we get to the checkpoint,\" he told her.\n\nShe shrugged. Her hair had turned to plastic. It had pulled at her scalp and he'd shaved it off as best he could, which was patchy. The mohawk was too solid with the glue to do anything but hack off the top six inches so she could fit in the truck cab. While he was looking at her the first sunlight struck her face and lit it all and she looked beautiful. The sun came up fast. All around them the land was flat and wider than any land either of them had seen and on the stubbled moraine the tiny stunted spruce and twisted tamarack were for a moment aflame and they could see for miles and they knew they could be seen too.\n\n\"Okay,\" he said. \"You can tell me. What your plan is. Go on. Lay it out.\"\n\nThere were no cigarettes left and she took a match and stuck it in her mouth and held it in her teeth and said something.\n\n\"What?\" he said.\n\nShe took out the match and said, \"That's the bay.\"\n\nHe reached down and grabbed the emergency break and would have pulled it if she hadn't taken her foot off the accelerator. They lurched and rolled to a stop. The bay was across the border; he didn't know by how much. There was no river or anything. The only way you could tell, he'd heard, was at night when you could see the last verge of mirror-light, or in the day when you could see the change in growth, beyond the range of the reflected beams, where the land hardened again into tundra formed under long winters of darkness that it was rumored still remained. That, and the guardtower.\n\nThey sat stopped in the middle of the black asphalt strip. No one came from anywhere.\n\n\"You know something I don't know?\" he said.\n\n\"Depends what you know.\"\n\n\"Let's say I know nothing.\"\n\nShe rolled down the window and spat out the match. The air that came in was cold. She rolled it up again.\n\n\"You were gonna just drive right up there,\" he said.\n\n\"Well.\"\n\n\"Well, fuck.\"\n\n\"Well, I've got ID.\" He stared at her. \"I wasn't just gonna leave you. They'd let you through.\"\n\n\"Me?\"\n\n\"They don't care about ID so much for people going out...\"\n\n\"Yeah.\"\n\n\"... just coming in.\"\n\n\"Unless you're wanted.\"\n\nA wind came out of nowhere and hit the truck and shook it and moved on.\n\n\"Just saying,\" he told her. \"Just throwing that out there, thought I'd, you know, just fucking you know just mention it.\"\n\n\"What are you so worried about? You're not the one that has to worry.\"\n\n\"I'm not worried about me,\" he said, and shoved open the door and got out and slammed it.\n\nHe stood slouched off his bad leg beside the truck looking at the sun coming up over the ocean somewhere out there too far to see, his hands wedged into his pockets and his neck drawn into his shoulders and the wind clawing at his hair. For a moment, with him turned away, she let her face do what it needed. She had known this was coming, this moment when he would find out the truth of her, and feared it, and now that it was here she knew it was going to be worse than she had thought. She ducked down so he wouldn't see and unlaced her boot and took it off and reached into her sock and took out the money. Bent down beneath the dash she knew this was the end and her face worked until she got it quiet and she straightened up again. She held the wad of bills into the bright sunlight for him to see. He was still turned away. She meant to throw the boot at the door with a bang that would bring him around but it hit the window and went through. The glass crashed onto the road. He turned, his hair glittering with the chips of glass, and she started laughing. There was nothing funny about his face, but she couldn't quit. She sat there laughing and holding the money up for him to see. Slowly, her laughing died down. He stood there. The wind was in his eyes.\n\n\"It's enough,\" she said. \"For both of us. It doesn't look like much but it's all hundreds.\"\n\nBehind him, the wind shook the tamarack.\n\n\"What?\" she said. \"I couldn't tell you, you know I couldn't tell you. What, Minor, what the fuck? I'm sorry about the window but I mean, what?\"\n\nHe said something in the wind that she couldn't hear and then he came into the truck and shut the door and the wind came in the busted window and he looked at the window, and said, \"There wasn't anyone, was there.\"\n\n\"Any who?\"\n\n\"Coming after you.\"\n\nShe peeled off a sliver of bills. \"Why do you think I have extra then?\" She said. \"You think some fucking kid stuck up here on watch duty won't turn an eye?\"\n\nHe looked at the money, then at her. \"There are some things you can bribe away and those are the small things, the things only laws care about and people don't, the things that don't matter.\"\n\n\"Minor\u2014\"\n\n\"You didn't do anything.\"\n\n\"That's not true.\"\n\nHe reached over and took the bribe money that she'd peeled off and sat there counting it. It didn't take long. \"You didn't do anything,\" he said again. Slowly he fanned the money and held it into the light.\n\n\"Minor,\" she said. \"I wasn't lying to you. You still don't know what\u2014\"\n\n\"I know it's nothing.\" He was looking at the light coming through the bills. He spoke very slowly. \"You said it was strange how that Indian cop just started shooting. Not a word first. Just saw me and started shooting. You said it.\"\n\nShe watched him and said just as slowly. \"It's bad, isn't it.\"\n\nHe didn't move anything, not his jaw, not his eyes.\n\n\"How bad?\" she said.\n\n\"I won't tell you.\"\n\n\"What did you do, Minor?\"\n\nHe gave the bills back to her. \"I thought you had a plan,\" he said. \"I thought we were going to get across.\"\n\n\"Minor\u2014\"\n\n\"You don't want to know.\"\n\n\"I'll tell you. I stole\u2014\"\n\n\"Don't,\" he said.\n\n\"I stole my dad's truck.\"\n\nHe shut his eyes. \"That's nothing.\"\n\n\"It's something.\"\n\n\"No, it's nothing.\" He was sitting there squeezing his eyes shut as if that could keep him from hearing. \"It's what you didn't.\"\n\nShe watched him and she could feel the desperation climbing up her and she said, \"Did you kill someone? Did you murder someone, Minor? What could you have done that was so bad? You have to tell me now. I've told you and you have to tell me or it won't work.\"\n\n\"It won't work,\" he said.\n\nShe grabbed him by the hair and yanked his head to her and he yanked it free and she brought her fist down as hard as she could on his thigh. He threw his head back and gritted his teeth. She hit him again.\n\nHe sat there with his head thrown back and said through his teeth, \"It won't work.\"\n\n\"I'll forgive you,\" she said. \"Whatever it is\u2014\"\n\n\"You'll know me. Like I know you now. Not knowing, Drema. It was always the not knowing that\u2014\"\n\nThat was when she climbed on him and all the pain of her on his leg shot through his entire face and she could see it pulsing in his strained neck and she kissed him. She kissed him hard and for a long time. When she was done, she stayed with her face right on top of his and said with her lips touching his, \"How bad was it, Minor?\"\n\nHe shook his head, his beard, against her face.\n\n\"I'm going to think of the worst thing I can think of,\" she said. \"And I'm going to tell you and you're\u2014\"\n\n\"Double it,\" he said.\n\nShe sat there on him looking at his face so close it almost didn't look like a face. She could see the creases in the landscape of his skin and the dirt and the oil of him and the hairs of his beard came from holes in his face as if his body was trying to rid itself of them, pushing them forever through and them forever coming, black and enduring so long as he endured, and even after, when all other parts of him had gone still, they would be the last thing to move.\n\nShe sat up away from him. His eyes stayed shut. Slowly, she climbed off and picked her way carefully across the road to where her boot lay like something they'd hit. She picked it up and put it on and laced it and came around to the driver's side and got in and he had not opened his eyes.\n\n\"Get out,\" she said.\n\nShe had left his door open and he just swung his good leg out. He put his weight on it and stood up into the sun and the wind and the huge sky stretching over it all.\n\n\"Tell me,\" she said. \"Or I'll leave you here. I'll just drive to the border and give them the money and go across and never see you again and what good is it then? Not knowing? What's the point in it then?\"\n\nHe had opened his eyes and he was looking at her. She thought he was looking at her as if he was trying to figure out who she was now that he knew she was not who he had imagined her to be, as if to see what she would become to him now.\n\nShe could not know that he was thinking of how young she sounded. Younger than she had told him, maybe as young as he had been when he'd slapped out through the screen door into the summer's long dusk and left them all lying in their blood behind him and run beneath the thick-leafed tulip trees and out into the open where the mirrors threw his shadow scrambling ahead as if in chase of his own life, of all the unknown rest of it that what he'd done that night had left for him. Nor could she know this: that through it all the image of her backing her father's truck down a long steep drive kept coming at him. And all that that one thing had led to. And how unfathomable were the hours and days and years to come that would shape her into what he could not even guess. And him too. The past unraveling behind them and the future as unknowable as any dreamed-of dark.\n\nShe could not know all that, but she knew that while she watched him back something came into his eyes that surprised her. It was a kind of realization, a kind of relief even, something that looked to her almost like happiness.\n\n\"I'll hunt you,\" he said. \"If you leave me...\"\n\n\"Why?\"\n\n\"... I'll get across and I'll hunt you.\"\n\n\"You would do that?\"\n\n\"I will hunt you down.\"\n\n\"Still?\"\n\nAll day clouds amassed to the west, piling on themselves and fusing together until they were a bank as solid as if the land itself had bent and risen up. In the afternoon they began to move. By dusk they had smothered the entire sky. Through them, the light of the mirrors filtered down, dim, and in them Minor could see the place where the light stopped: low in the sky, a seam, as if the firmament was made of two different bodies that met there. He pointed it out to her. She had already seen it.\n\nThey moved through the tundra in the cloud-dimmed dusk, her wrapped in the army blanket against the cold, and him shivering in his one-piece and limping badly, and between them they carried the plywood box that had been fitted to the truckbed, a dark slab long and narrow and heavy as a corpse. The permafrost had long ago melted and it was bog. They sloshed through it, their pants below their knees wet and heavy, feeling like ice. They talked in whispers that barely made it over the sucking sound of their steps, talked first of night-vision equipment and the guards and then of all that lay ahead, the Inuit tribes, their dogs, of the roving bands of ones like them and what they killed to live on and who they knew who had once said they were going up and how cold it would get and after ten minutes they had exhausted everything they knew about what was to come. Behind them, so far back it was no more than just a glint of glass and metal, they had left the truck. To their east, and not far enough, the light of the guard tower hung in the dim gray gloam. Ahead, so distant it was almost impossible to make out: the periphery seam in the clouds.\n\nThey didn't realize it was growing darker until they could hardly see the bushes in front of them anymore and they went on farther, and then they stopped. They set down the box in a dark so dark that they didn't know the box was near the ground until they felt it hit. They stood in that darkness and breathed it and felt it around them and she waved her hands in it and told him she was waving her hands and he could hear her sloshing about, playing like a child in first snow. From somewhere in the distant dark came the sounds of coyotes. He stood listening to the barks and yips. It was a joyous sound like he had not heard an animal make since he was a very small child. He looked towards the place from where he thought they came, what he hoped was north, and could see nothing but the blackness. After a while, she picked up her side of the box again. They stood there looking for the part that looked most black, and it all seemed as unknowable as the rest, and they began to walk.\n\n# [**HELLO  \nFROM HERE**](toc.html#ch8-R)\n\nOne by one the streetlamps come alight. From on my stoop, I watch: the strip of sky between the buildings, the seagulls swirling, the cobblestones below so wet from the past rain they could be rocks along a coast, on top of them a row of beacons burning. Burn, burn, burn. The lamplighter with his long pole turning back the dark. One hundred nights now I have sat here and, Mamme, each time he lights the flames I see your eyes behind the Sabbath candles. Tatte, when he sets his ladder against the cross rest and climbs up, I watch you reach to the display case in your apothecary's shop. Yankel, my breuder, every time I smell the lamp gas escape I think of you. And start to smile. And want to cry.\n\nHow can a son say good-bye forever? How can a brother?\n\nI sit on this stoop and the lamplighter stops to ask what I am writing and I tell him, \"A letter.\" And he asks to whom. And I say, \"My family.\"\n\n\"All of it?\" he says, as if he thinks I mean feters and tantes, zeydes and bubbes, everybody who is of my blood and from whose blood I came.\n\n\"Yes,\" I say. But I do not tell him I also mean neighbors and friends and everyone I knew in my world up until I left it.\n\n\"You'll be here all night,\" he says.\n\nI shrug. He knows I would be anyway.\n\n\"They live far away?\" he asks.\n\n\"A month's walk,\" I say.\n\nIf he finds that strange, he does not show it. Maybe he walked as long to get here. Maybe he jumped trains and begged farm wives for rides and hopped the carts of gypsies and was chased by thieves, beaten by thugs. Maybe he kept to back roads, too. Walked at night. Maybe he came from a town not so different from ours. Though I do not think he is a Jew.\n\n\"But in Russia?\" he says.\n\nI reassure him.\n\n\"Well,\" he says, \"tell them hello from here.\" And starts away. Only to turn back. \"Tell them,\" he calls, lifting his pole in a salute, \"I wish for them always a light in the dark!\"\n\nDown the rest of the unlit street he does his best to make it true. I watch him till he disappears.\n\nHello from here. Hello from 214 Talevu Street. It has been ninety-eight days since I sent you my first letter letting you know I had found lodging, a way to make a little money, had not just left on the next boat, would not until I had enough to buy at least one other ticket, that I would wait to hear from you, or till one of you\u2014sometimes I dreamed it could be all three\u2014might arrive at Stepashin's Photographic Studio and Supply, peer in past the cameras on display, their brass lenses and mahogany plate holders and black bellows, and ask for me.\n\nA month ago I sent a second letter. That one I addressed to Tatte's shop. Sometimes, sitting here, eating my supper of cold herring and old bread, I cannot help but think on what could have kept my words from reaching you. Then I cannot swallow. And when I think what if they _did_ reach you, and still you have not written back, I stand up from the stoop and go inside, or onto the street, anywhere away from where the thought had found me.\n\nStill, I would wait here a year. Another. If I did not now know that to remain even one more day would be too late.\n\nSo, a little after dawn I will fold up the blankets Stepashin loaned to me for sleeping. I will put on the hat that you, Mamme, knit for me the night I left and which, for each night since, has been my pillow. Inside Tatte's valise I will shut all that I have come to own: a change of clothes, a sailor's sweater, a coachman's gloves, a frayed bowtie, a pocketknife with a broken blade, three pencils, and whatever paper I have left after I have finished writing this. Then I will leave the shop. I will lock the door, slide the key beneath it. I will walk down to the docks wearing the too-large clothes I took from Tatte, Yankel's too-small shoes I used to replace my soldier's boots, the short coat with the pocket Mamme stitched into the inside in which I'll slip my steerage ticket along with the papers to prove to anyone who asks\u2014port police, inspectors from the Baltic-Atlantic Line, whoever Stepashin will have alerted\u2014that I am not an army deserter, not your eldest son, your older brother, not someone who had left his family to face the consequence of what he'd done, not me.\n\nMy name is Yankel. So many times have I said it to myself. Trying to breathe beneath the hay of the cart in which I fled\u2014Your name is Yankel\u2014walking a moonlit road through unmown fields\u2014Your name is Yankel\u2014wandering these winding streets besides the banks of the Daugarva\u2014Your name\u2014over and over, with each step, each puff of dust\u2014is Yankel. And still, inside my ears, my mind hears Shimel, my name for all my years till now. Shimel, my mind tells me, your name is Yankel. But behind my eyes, what can this name show me but you? My breuder's eyes, my breuder's face.\n\nYankel, if you knew how I study you! Like we were boys again, holding candles, standing shirtless before our reflections in the windows at night. We'd lean close, inspecting our upper lips for hints of fuzz, wondering would we be able to grow a wonze thick as our father's, would it be, on both our faces, the same brownish red? How you tried to match my muscles! How you mimicked the poses that I made! How, when you would fail to magically be bigger than your big brother, you would do what you always did to make me laugh. A little wiggle in your ears. Then your nose. Then your whole face while you tried to stop from laughing, too.\n\nLittle breuder, you should be here with me! We would walk streets lined with houses painted all the colors of Tatte's powders, drink kvas on the docks beneath the hulls of steamships so huge they could fit inside them even your farts. There are whole trolley cars pulled by teams of horses, a constant crush of droshkies clattering back and forth from stores to bars to caf\u00e9s to theaters. In summer, when I got here, the city bustled even after midnight. I have seen gentlemen placing bets on hedgehog races, ladies pedaling tricycles along the street. In the square before the National Theater I watched a demonstration of Yablochkov's candles and I have seen their bulbs that burn without a flame, squinted up at their electric light. And shut my eyes. And seen your face. And stood there, in that awestruck crowd, keeping my eyes closed as long as I could. If you could see it through your brother's eyes, you would understand. If I could tell it to our mother, she would show you.\n\nMamme, do you remember the way that we would draw? When I was seven, eight. You would try to teach me to take what my eyes saw, make my hand re-create it on a page. I would try. And fail. And fall into such fits, squeezing my fingers to nearly breaking, beating my fists against my forehead, until at last you uncovered where my talent lay. Go find a picture, you would say. And I would rush around searching for something I thought you'd like. Mittens stuck on fence slats to dry. A birch fallen across the brook. Which, then, I would sit and memorize. Every line, shape, shadow. Until I could bring it in my memory back to my mamme. I would sit on your lap, holding the pad, and you would reach around me with your pencil, bent so I could feel your heart against my back, your cheek against my own, while I described what I had seen in such detail that, as if your fingers contained the same anointment as Elisha's bones, you would bring it back to life on our page. Perfect as if drawn from your own sight.\n\nWhy did we never try a face? I do them now. Not through drawing, but with words: descriptions of strangers' faces scribbled down so that a loved one, somewhere far away, might unfold a page and, with the aid of what I'd written, draw themselves a picture in their own mind.\n\nEach day, I sit on my valise outside Stepashin's studio, waiting for the leavers to pass by. Sailors with sweethearts, soldiers shipping out, friends sentimental from a night of drinking, mothers and fathers spending last hours with a daughter or a son. People who stop at the shop's portrait display, cup their hands around their faces, peer inside: the painted backdrop, the velvet drapes, the chairs where someone else, someone with money, might sit before Stepashin's lens. Sometimes I'll see the magnesium flash caught in the face pressed to the glass. I'll wait until the yearner steps away. Then\u2014\"Excuse me, friend\"\u2014offer him a chance to take his loved one with him for a few coins.\n\nOn the cobblestones before me I place a crate, drape it with my jacket, invite the subject to sit. For a minute or two I simply watch them. Then I begin to write. Not only what I see\u2014bend of a nose, shape of an eye\u2014but what I watch happen before me: the way a lady's nostril flares at something her lover says, a child's eyes brighten at the approach of a horse and dray. Once a sailor leaned forward to whisper he'd prefer that I not mention his bad breath. But I did. Because I glimpsed his sister roll her eyes at the request, a fond exasperation I wanted her to feel again when, in a month or two, she might wish to remember. Another time, I watched an old man comb with careful fingers a part onto his completely hairless head. Saw his wife fight an urge to, with her own fingers, adjust the styling. So I adjusted: the hair that was still there only in the memory they shared became a way that she, over their years, would touch him. Each letter picture I pushed like this. Just a little bit. And for this they paid ten kopeks each. For this they came all day.\n\nAt the end of that first one, Stepashin stepped out of his shop, locked up, walked straight to me. Would I, he asked, do one of him?\n\n\"For who?\" I asked.\n\n\"For you,\" he said. \"So you can remember me tomorrow.\" Reaching over, he put ten kopeks in my pocket. \"When you will be gone from before my store.\"\n\nBut while he sat there in his trim beard and beaver hat, fierce eyes blinking out from the black frame of facial hair and pelt, I wrote, instead, a contract: to him I would pay a ruble each week, only to use the street outside his studio. To me, he would give a place to sleep, inside, just behind the front door. Where anyone trying to break in would have to wake me. I watched him take in, the way men have all of my life, my unimpressive height. \"I was a soldier,\" I told him.\n\n\"Was?\" he said. And I watched him calculate the age in my still-boyish face. I should have seen it then. But, at the time, he only asked my name.\n\nAt least I knew enough not to say Yankel, not to say Shimel. Instead, I told him Shura. A nickname that I knew would not mark me as a Jew.\n\nAleksander Aleksandrevitch Aleksandrei: the full formal version that, in a moment of panicked inspiration, I gave him, that, from that day on, he always used.\n\n\"Aleksander Aleksandrevitch,\" he would say, \"you may keep your things inside the storage closet.\"\n\n\"Aleksander Aleksandrevitch, please do not bring your supper in; I cannot have the studio stinking of fish.\"\n\n\"Aleksander Aleksandrevitch\"\u2014even after I had been opening the door for him, greeting him good morning every day for months\u2014\"if you must light the lamp before retiring to the floor, please take care, upon your waking, to leave the chimney clean.\"\n\nHis father had been an officer in the last tsar's army, an inventor of military machines, and Yefin Eduardovitch Stepashin paid close attention to any manner of device, though none more than his cameras. To me, he hardly ever turned an eye, rarely seemed to notice I was there. Which is, surely, why\u2014despite sometimes a slipup of mumbled Yiddish, despite this snitch of a nose here on my face\u2014it was an arrangement that, for a while, worked. Only during downpours, when he would rap on the window, beckon to me, would what was between us seem something more.\n\nWhile I shook out my hat, stood trying not to drip a puddle, he would speak of things I still struggle to understand. Sometimes, I was not even sure he was talking to me. Fingers fluttering among his cameras, he'd reattach a tiny rubber band, mumbling about a \"rebound shutter\", \"aperture\", \"exposure\", as if explaining concepts to the device itself. Sliding a wood frame into a slot, he spoke of _bromides, emulsion_ , as if to the sheet of glass it held. But for all the unfamiliar words he used, I began to see that he was talking, always, about only one thing: light. Not what his contraptions might do with it, but what it is.\n\nThe Luminiferous Aether. This Stepashin calls it. Light, he says, travels from a source\u2014the gas flame in this lamp above me, the sun that will tomorrow brighten the sky\u2014and hits whatever stands in its path\u2014this page, this pencil, my face\u2014and bounces off of it in waves. Waves that ripple through the aether waiting in the air. And, moving it, make it something we can see. The way water in a glass is clear till stirred. This is what enters our eyes. These luminiferous waves, this aether that is stirred in such a pattern that the things off which the light has bounced appear. Without the waves, the aether is invisible. Which is what we call dark. But even in the darkest blackness, the aether is still there. Waiting to be stirred. So we might see.\n\nI wish my mind was better made for this. I wish, Tatte, you were here to sharpen it. You would ask me your questions, the raised eyebrows and downward-dipping mouth that Yankel and I always feared. Because we knew that what was coming would make us question what we had till then assumed was true. Sometimes, I try to do it for myself\u2014I have attempted to see your silence from some angle I have not yet considered\u2014but it is not the same. You cannot say, the way you always would when we were stymied, \"Let's see if we can't think up an experiment.\" And, with your powders and potions, you would find a way to test even Stepashin's aether. When I was a kinde, I could watch for hours while you worked. Melting, measuring, filling bottles with mixtures for sick strangers, neighbors. Even, sometimes, for your son. Meyn zun, you would say, and invite me over to smell an unknown scent, ignite shavings into a crackling of colors, watch as you mixed into some oil a powder to make it glow. Phosphorus, you said that day before I left. And sealed the tiny vial, the glass flasch that I wear still on the chain you placed around my neck.\n\nSometimes, listening to Stepashin speak of waves and light I place a hand over my chest and feel it\u2014the flasch\u2014and, watching him work, am struck by a sudden sense of peace. An inexplicable closeness. Which is the only reason I can give for my mistake.\n\nIt had been raining. He had been talking. I had been thinking\u2014if only the waves of light could pass through solid things, travel in an uninterrupted line from ricochet to our eyes, you could see me from five hundred miles away, I could see you\u2014when he turned to me and asked, \"Aleksander Aleksandrevitch, do you know anyone by the name of Yushrov?\"\n\nSo strange, after all these months, to hear our name. From him. Still the answer should have been simple had my mouth not refused to make it, had it not, instead, as if needing to feel the word, said the name back: \"Yushrov?\"\n\n\"A Hebrew name,\" he said. He had finished closing up. We were stopped just inside the door, barely a foot apart, my hand on the knob, ready to open it for him, his on the handle of an umbrella still in the stand.\n\nI managed to make my head shake no. Though out of my mouth there came, unbidden, \"Why?\"\n\nOutside, the rain drummed at the door, ran down the windows, made the shadows on that side of his face appear to move. \"I received,\" he started, \"a letter\u2014\"\n\n\"From...?\" I watched him wonder: hadn't he just told me? Or did I think the name had been who it was addressed to?\n\n\"Yankel,\" he said.\n\nI tried to make my face a picture about which, had I been peering at it, I could not have found anything to write.\n\n\"Shimel,\" he said.\n\nAnd I could see on his face that I was failing.\n\n\"Do either of these names\u2014\"\n\n\"Two?\" I said. \"You received two? To each?\" He released the umbrella handle. I released the knob. My hand was shaking. \"Show them to me,\" I pleaded. \"Fima\"\u2014it was the only time I ever used the familiar form of his first name\u2014\"will you?\"\n\n\"That depends,\" he said, \"on who I'm showing them to.\"\n\nAnd so I told him, watching his eyes narrow, as if he was trying to bring something out of focus into sharpness, until, by the time that I was done, his lids had shut almost entirely, and I knew that I had been wrong to.\n\n\"So,\" he said, \"you _are_ a deserter.\"\n\nI should have caught the strange lack of surprise, as if I'd simply confirmed something he'd heard, but I was too intent on trying to paint for him a picture of why: the death sentence that is conscription for a Jew, the friends I'd seen succumb to officers' demands, suicide posts, unsurvivable conditions, the impossibility of staying alive for the mandated twenty-five years.\n\nTo which he said, \"You _are_ a Jew.\"\n\n\"I am,\" I told him, \"the same man who, for a quarter of a year, has greeted you each morning, who guards your shop at night.\"\n\nHe lifted the umbrella out of the stand. \"Who _did_.\"\n\nFrom under his other reaching hand, I grabbed the knob, held the door. \"At least,\" I begged, \"give me the letters.\"\n\nHe set his fingers on top of mine. Such an unexpected touch: my hand, without his on it, would have slipped off the brass. \"The only letter I received,\" he said, \"was from the military police. Telling me they had come across another from your family. Addressed to here.\" Where, he said, his voice soft as his artists' fingers felt, he would, out of respect for the time we had spent in each other's company, allow me to remain till morning, till the usual hour that he would open the store, when he would return with the police.\n\nAlready, it is almost tomorrow. Long ago, the last droshkies passed by Talevu Street, left the avenue that is just visible at its far end empty of all but the gas lamps' glow, gone still but for the shifting of the few remaining leaves.\n\nI know that I should try to sleep. But I cannot stop thinking about the police. Not the ones Stepashin will bring. But the ones who already must have come to you. Mamme, did they wake you at night, break in, destroy your drawings looking for letters from me? Or by then were you already gone?\n\nNo, there will be no sleep for me tonight. This last night beneath the sky we still share. Where I am going, your night will be my day, my day your night. And with each passing one I will become less your brother, less your son. Until I am someone so far from you in a life so different from this that these words will be the last not just to you, but from me. Mamme, you will say I will always be your son. Tatte, you will tell me it is only an ocean, another country, a few more miles. But I have already traveled more than a few from you. And I know that this is different. By this time tomorrow I will be far out on the sea, deep in a shaking hull, surrounded by the sounds of sleeping others, mouthing into the steerage dark a single syllable\u2014'new'\u2014trying to get it right\u2014'new'\u2014the first sound of the name of the place in which I will never again be called meyn breuder or meyn zun.\n\nOutside, the street seems to have already begun to disappear. There are only burning lamps, cobblestones fading away beneath their glow, gaps of blackness between. And my lantern in this window. Lighting me. I will let it burn all night. I will let the chimney blacken. I will wander among the cameras on their stands, touching all the knobs, turning the rings, making their rubber bellows creak. I will set a match to the magnesium, watch the flash explode. And in the darker darkness after, the acrid scent of smoke, I will push open the velvet curtains, step into the backdrop, sit on the chair, stare into the lens.\n\nWell, in fact, other than keeping the flame lit, I have passed these last hours tossing in my blankets, trying to imagine what I could write here at the end. Now, the blankets are folded. My valise is packed. Outside, the night is growing thin. And through the shop windows I can see it: the first faint waves of daylight reaching down into the street, ricocheting off wheel spokes, shutter slats, showing a flutter of wings, first slight stirrings in the aether. I like to think of it, there all night, waiting for a ripple of light to come and give the things of the world shape, abiding, always, everywhere. Even when there is no one to see it.\n\nFor the past hour, watching the window light lift out of the lantern's flicker all the devices that fill this room, I have been thinking about this century's beginning. How back then no one had so much as conceived a camera. How a mere fifty years ago the journey I am about to make would have taken twice as many days. How barely fifteen years before today not one soul had ever seen an electric light. And fifteen years from now? In the new century? Maybe by then we will have cut the ocean crossing in half again. Maybe we will have found a way to send light around the earth, and I will once again be able to see your face. And you mine. But until then, all the time that we are hidden from each other, I will know that you are there, in the dark, as there as the spokes, the shutter, the street outside\u2014its cobblestones, abandoned carts, the tops of hitching posts painted white by perching gulls\u2014that, lost to me all night, has returned.\n\nAnd here he comes again: the lamplighter, crossing from one side of the street to the other, unlatching the panes, reaching up, snuffing the flames. Watching him it seems as if he might have just kept walking. Followed the night around the globe, lighting lanterns until the dark crossed into day and, coming upon the flames already lit, he flipped his pole to its snuffing end, simply kept on. Maybe, some night in another city in another country I will see him coming down another street. I will call him over. I will ask him to bring a message back to you. \"Tell them,\" I will say, \"hello from here.\"\n\nHe nears. A nod. I nod back. Reaching his pole to the lamp above me, he says, \"They'll be up all night reading it, too,\" and I wish I had another hundred pages. I wish I had something more than this to send. If I had learned to use the contraptions that all these nights have kept me company, paid more attention to practical things Stepashin said\u2014how to load the plates, train a lens on where I'd sit, focus it on myself, somehow trigger the shutter\u2014if I knew the way to free the captured light and imprint it on a piece of paper, I would do it for you. But all I know to do is this:\n\nHis skin is puffy from lack of sleep. His hair sticks out from beneath the hat his mother knit, wild, unwashed. The same color as his brother's. But his mustache has grown in a little darker, a little more full. Almost, now, as thick as his tatte's. His tatte whose beard he used to like to squeeze in his small child's fists, just to see his fingers disappear. He wishes now he had such a beard to hide his own face. The puffiness of his lower lip, raw from biting it. The shaking of his upper, delicate\u2014always, he thought, too like a girl's, too like his mother's\u2014its shape as deeply pinched as the pie crusts that she makes. He can feel her fingers on his mouth. Pressing it. He shuts his eyes. With them closed he can see yours better. If he could he would stay like that forever, remembering his father's, his brother's, his mother's. But when he opens his eyes again, there is just him, sitting by a still-burning lamp, watching, in the window glass, his own reflection. Your oldest son, your only sibling, your once was Shimel saying good-bye. Good-bye, Yankel. Good-bye, Tatte. Good-bye, Mamme. I wish for you a light in the dark.\n\n# **ACKNOWLEDGMENTS**\n\nThe eight stories in this collection were written over an entire decade of my life. In them, I see not only earlier iterations of myself, but the mark of so many others who helped me along the way:\n\nMy first readers and longtime friends: Mike Harvkey, Johanna Lane, Elizabeth Kadetsky, Meghan Kenny, Robin Kirman, Suzanne Rivecca, Laura van den Berg, Jennifer Sheffield (now Weil), and my brother, Ben.\n\nThe talented and generous editors who first brought these stories to print: Hannah Tinti at _One Story_ ; Ralph Eubanks at _Virginia Quarterly Review_ ; Ladette Randolph at _Ploughshares_ ; Tom Jenks at _Narrative_ ; Cheston Knapp at _Tin House_ ; Bill Pierce at _Agni_ ; and Stacey Swann at _American Short Fiction_.\n\nThe team at Grove, who, over these years and three books has become so important to me: Elisabeth Schmitz, Morgan Entrekin, Katie Raissian, and all the rest.\n\nPJ Mark: agent, friend, indefatigable guide, and guardian of my work.\n\nOthers, too: Henia Lewen helped me with Yiddish in \"No Flies, No Folly\"; Naomi Williams (and her mother) translated the Japanese in \"Long Bright Line\"; Jeremy Mohawk translated the Mohican in \"The Point of Roughness\"; my sister-in-law, Lisa Rasco, helped me understand autism for the same story; Adam Day, through the Baltic Writing Residency, helped get me to Riga, in which \"Hello from Here\" is set; quotes from Mike Horn's North Pole winter expedition log appear in \"Beautiful Ground\"; and the acts of resistance by residents of four rural counties in Pennsylvania\u2014Adams, York, Franklin, and Cumberland\u2014almost eighty years ago inspired the events in \"The Essential Constituents of Modern Living Standards.\"\n\nTo all of them\u2014and to my always supportive parents, inlaws, and grandmother; to my children, Sadie and Cody, who fill my days with light; to the love of my life, Jen, who carries me through the darkness\u2014thank you.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Jonathan Ingram and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\nTHE POETICAL WORKS OF WILLIAM WORDSWORTH\n\nVOL. VIII\n\n[Illustration: _William Wordsworth_\n\n_after Thomas Woolner_\n\n_Printed by Ch Wittmann Paris_]\n\n\n\n\n                          THE POETICAL WORKS\n                                  OF\n                          WILLIAM WORDSWORTH\n\n                               EDITED BY\n                            WILLIAM KNIGHT\n\n                               VOL. VIII\n\n                     [Illustration: _Gallow Hill_\n\n                             _Yorkshire_]\n\n                                London\n                        MACMILLAN AND CO., LTD.\n                       New York: Macmillan & Co.\n                                 1896\n\n                        _All rights reserved._\n\n\n\n\nCONTENTS\n\n\n                                                                    PAGE\n                                 1834\n\n  Lines suggested by a Portrait from the Pencil of F. Stone            1\n\n  The foregoing Subject resumed                                        6\n\n  To a Child                                                           7\n\n  Lines written in the Album of the Countess of Lonsdale,\n      Nov. 5, 1834                                                     8\n\n                                 1835\n\n  \"Why art thou silent? Is thy love a plant\"                          12\n\n  To the Moon                                                         13\n\n  To the Moon                                                         15\n\n  Written after the Death of Charles Lamb                             17\n\n  Extempore Effusion upon the Death of James Hogg                     24\n\n  Upon seeing a  Drawing of the Bird of Paradise\n      in an Album                                                     29\n\n  \"Desponding Father! mark this altered bough\"                        31\n\n  \"Four fiery steeds impatient of the rein\"                           31\n\n  To ----                                                             32\n\n  Roman Antiquities discovered at Bishopstone, Herefordshire          33\n\n  St. Catherine of Ledbury                                            34\n\n  \"By a blest Husband guided, Mary came\"                              35\n\n  \"Oh what a Wreck! how changed in mien and speech!\"                  36\n\n                                 1836\n\n  November 1836                                                       37\n\n  To a Redbreast--(In Sickness)                                       38\n\n                                 1837\n\n  \"Six months to six years added he remained\"                         39\n\n  Memorials of a Tour in Italy, 1837--To Henry Crabb Robinson         41\n\n      I. Musings near Aquapendente, April, 1837                       42\n\n     II. The Pine of Monte Mario at Rome                              58\n\n    III. At Rome                                                      59\n\n     IV. At Rome--Regrets--in Allusion to Niebuhr and other\n             Modern Historians                                        60\n\n      V. Continued                                                    61\n\n     VI. Plea for the Historian                                       61\n\n    VII. At Rome                                                      62\n\n   VIII. Near Rome, in Sight of St. Peter's                           63\n\n     IX. At Albano                                                    64\n\n      X. \"Near Anio's stream, I spied a gentle Dove\"                  65\n\n     XI. From the Alban Hills, looking towards Rome                   65\n\n    XII. Near the Lake of Thrasymene                                  66\n\n   XIII. Near the same Lake                                           67\n\n    XIV. The Cuckoo at Laverna                                        67\n\n     XV. At the Convent of Camaldoli                                  72\n\n    XVI. Continued                                                    73\n\n   XVII. At the Eremite or Upper Convent of Camaldoli                 74\n\n  XVIII. At Vallombrosa                                               75\n\n    XIX. At Florence                                                  78\n\n    XX. Before the Picture of the Baptist, by Raphael,\n            in the Gallery at Florence                                79\n\n    XXI. At Florence--From Michael Angelo                             80\n\n   XXII. At Florence--From Michael Angelo                             81\n\n  XXIII. Among the Ruins of a Convent in the Apennines                82\n\n   XXIV. In Lombardy                                                  83\n\n    XXV. After leaving Italy                                          84\n\n   XXVI. Continued                                                    85\n\n  At Bologna, in Remembrance of the late Insurrections,\n      1837.--I.                                                       86\n\n      II. Continued                                                   86\n\n      III. Concluded                                                  87\n\n  \"What if our numbers barely could defy\"                             87\n\n  A Night Thought                                                     88\n\n  The Widow on Windermere Side                                        89\n\n                                 1838\n\n  To the Planet Venus                                                 92\n\n  \"Hark! 'tis the Thrush, undaunted, undeprest\"                       93\n\n  \"'Tis He whose yester-evening's high disdain\"                       94\n\n  Composed at Rydal on May Morning, 1838                              94\n\n  Composed on a May Morning, 1838                                     97\n\n  A Plea for Authors, May 1838                                        99\n\n  \"Blest Statesman He, whose Mind's unselfish will\"                  101\n\n  Valedictory Sonnet                                                 102\n\n                                 1839\n\n  Sonnets upon the Punishment of Death--\n\n     I. Suggested by the View of Lancaster Castle (on the\n           Road from the South)                                      103\n\n    II. \"Tenderly do we feel by Nature's law\"                        104\n\n   III. \"The Roman Consul doomed his sons to die\"                    105\n\n    IV. \"Is _Death_, when evil against good has fought\"              106\n\n     V. \"Not to the object specially designed\"                       106\n\n    VI. \"Ye brood of conscience--Spectres! that frequent\"            107\n\n   VII. \"Before the world had past her time of youth\"                107\n\n  VIII. \"Fit retribution, by the moral code\"                         108\n\n    IX. \"Though to give timely warning and deter\"                    109\n\n     X. \"Our bodily life, some plead, that life the shrine\"          109\n\n    XI. \"Ah, think how one compelled for life to abide\"              110\n\n   XII. \"See the Condemned alone within his cell\"                    110\n\n  XIII. Conclusion                                                   111\n\n   XIV. Apology                                                      112\n\n  \"Men of the Western World! in Fate's dark book\"                    112\n\n                                 1840\n\n  To a Painter                                                       114\n\n  On the same Subject                                                115\n\n  Poor Robin                                                         116\n\n  On a Portrait of the Duke of Wellington upon the Field\n      of Waterloo, by Haydon                                         118\n\n                                 1841\n\n  Epitaph in the Chapel-Yard of Langdale, Westmoreland               120\n\n                                 1842\n\n  \"Intent on gathering wool from hedge and brake\"                    122\n\n  Prelude, prefixed to the Volume entitled \"Poems chiefly\n      of Early and Late Years\"                                       123\n\n  Floating Island                                                    125\n\n  \"The Crescent-moon, the Star of Love\"                              127\n\n  \"_A Poet!_--He hath put his heart to school\"                       127\n\n  \"The most alluring clouds that mount the sky\"                      128\n\n  \"Feel for the wrongs to universal ken\"                             129\n\n  In Allusion to various Recent Histories and Notices of\n      the French Revolution                                          130\n\n      Continued                                                      131\n\n      Concluded                                                      131\n\n  \"Lo! where she stands fixed in a saint-like trance\"                132\n\n  The Norman Boy                                                     132\n\n  The Poet's Dream                                                   135\n\n  Suggested by a Picture of the Bird of Paradise                     140\n\n  To the Clouds                                                      142\n\n  Airey-Force Valley                                                 146\n\n  \"Lyre! though such power do in thy magic live\"                     147\n\n  Love lies Bleeding                                                 148\n\n  \"They call it Love lies bleeding! rather say\"                      150\n\n  Companion to the Foregoing                                         150\n\n  The Cuckoo-Clock                                                   151\n\n  \"Wansfell! this Household has a favoured lot\"                      153\n\n  \"Though the bold wings of Poesy affect\"                            154\n\n  \"Glad sight wherever new with old\"                                 154\n\n                                 1843\n\n  \"While beams of orient light shoot wide and high\"                  156\n\n  Inscription for a Monument in Crosthwaite Church, in\n      the Vale of Keswick                                            157\n\n  To the Rev. Christopher Wordsworth, D.D., Master of\n      Harrow School                                                  162\n\n                                 1844\n\n  \"So fair, so sweet, withal so sensitive\"                           164\n\n  On the projected Kendal and Windermere Railway                     166\n\n  \"Proud were ye, Mountains, when, in times of old\"                  167\n\n  At Furness Abbey                                                   168\n\n                                 1845\n\n  \"Forth from a jutting ridge, around whose base\"                    170\n\n  The Westmoreland Girl                                              172\n\n  At Furness Abbey                                                   176\n\n  \"Yes! thou art fair, yet be not moved\"                             176\n\n  \"What heavenly smiles! O Lady mine\"                                177\n\n  To a Lady                                                          177\n\n  To the Pennsylvanians                                              179\n\n  \"Young England--what is then become of Old\"                        180\n\n                                 1846\n\n  Sonnet                                                             181\n\n  \"Where lies the truth? has Man, in wisdom's creed\"                 182\n\n  To Lucca Giordano                                                  183\n\n  \"Who but is pleased to watch the moon on high\"                     184\n\n  Illustrated Books and Newspapers                                   184\n\n  Sonnet. To an Octogenarian                                         185\n\n  \"I know an aged Man constrained to dwell\"                          186\n\n  \"The unremitting voice of nightly streams\"                         187\n\n  \"How beautiful the Queen of Night, on high\"                        188\n\n  On the Banks of a Rocky Stream                                     188\n\n  Ode. Intimations of Immortality from Recollections of\n      Early Childhood                                                189\n\n                                 POEMS\n                                  BY\n                           WILLIAM WORDSWORTH\n                                AND BY\n                           DOROTHY WORDSWORTH\n                  NOT INCLUDED IN THE EDITION OF 1849-50\n\n                                 1787\n\n  Sonnet, on seeing Miss Helen Maria Williams weep at a\n      Tale of Distress                                               209\n\n  Lines written by William Wordsworth as a School Exercise\n      at Hawkshead, Anno \u00c6tatis 14                                   211\n\n                           1792 (or earlier)\n\n  \"Sweet was the walk along the narrow lane\"                         214\n\n  \"When Love was born of heavenly line\"                              215\n\n  The Convict                                                        217\n\n                                 1798\n\n  \"The snow-tracks of my friends I see\"                              219\n\n  The Old Cumberland Beggar (MS. Variants, not inserted\n      in Vol. I.)                                                    220\n\n                                 1800\n\n  Andrew Jones                                                       221\n\n  \"There is a shapeless crowd of unhewn stones\"                      223\n\n                                 1802\n\n  \"Among all lovely things my Love had been\"                         231\n\n  \"Along the mazes of this song I go\"                                233\n\n  \"The rains at length have ceas'd, the winds are still'd\"           233\n\n  \"Witness thou\"                                                     234\n\n  Wild-Fowl                                                          234\n\n  Written in a Grotto                                                234\n\n  Home at Grasmere                                                   235\n\n  \"Shall he who gives his days to low pursuits\"                      257\n\n                                 1803\n\n  \"I find it written of Simonides\"                                   258\n\n                                 1804\n\n  \"No whimsey of the purse is here\"                                  258\n\n                                 1805\n\n  \"Peaceful our valley, fair and green\"                              259\n\n  \"Ah! if I were a lady gay\"                                         262\n\n                                 1806\n\n  To the Evening Star over Grasmere Water, July 1806                 263\n\n  Michael Angelo in Reply to the Passage upon his Statue\n      of Night sleeping                                              263\n\n  \"Come, gentle Sleep, Death's image tho' thou art\"                  264\n\n  \"Brook, that hast been my solace days and week\"                    265\n\n  Translation from Michael Angelo                                    265\n\n                                 1808\n\n  George and Sarah Green                                             266\n\n                                 1818\n\n  \"The Scottish Broom on Bird-nest brae\"                             270\n\n  Placard for a Poll bearing an old Shirt                            271\n\n  \"Critics, right honourable Bard, decree\"                           271\n\n                                 1819\n\n  \"Through Cumbrian wilds, in many a mountain cove\"                  272\n\n  \"My Son! behold the tide already spent\"                            273\n\n                                 1820\n\n  Author's Voyage down the Rhine                                     273\n\n                                 1822\n\n  \"These vales were saddened with no common gloom\"                   275\n\n  Translation of Part of the First Book of the _\u00c6neid_               276\n\n                                 1823\n\n  \"Arms and the Man I sing, the first who bore\"                      281\n\n                                 1826\n\n  Lines addressed to Joanna H. from Gwerndwffnant in June 1826       282\n\n  Holiday at Gwerndwffnant, May 1826                                 284\n\n  Composed when a Probability existed of our being obliged\n      to quit Rydal Mount as a Residence                             289\n\n  \"I, whose pretty Voice you hear\"                                   295\n\n                                 1827\n\n  To my Niece Dora                                                   297\n\n                                 1829\n\n  \"My Lord and Lady Darlington\"                                      298\n\n                                 1833\n\n  To the Utilitarians                                                299\n\n                                 1835\n\n  \"Throned in the Sun's descending car\"                              300\n\n  \"And oh! dear soother of the pensive breast\"                       301\n\n                                 1836\n\n  \"Said red-ribboned Evans\"                                          301\n\n                                 1837\n\n  On an Event in Col. Evans's Redoubted Performances in Spain        303\n\n                                 1838\n\n  \"Wouldst thou be gathered to Christ's chosen flock\"                303\n\n  Protest against the Ballot, 1838                                   304\n\n  \"Said Secrecy to Cowardice and Fraud\"                              304\n\n  A Poet to his Grandchild                                           305\n\n                                 1840\n\n  On a Portrait of I.F., painted by Margaret Gillies                 306\n\n  To I.F.                                                            307\n\n  \"Oh Bounty without measure, while the Grace\"                       308\n\n                                 1842\n\n  The Eagle and the Dove                                             309\n\n  Grace Darling                                                      310\n\n  \"When Severn's sweeping flood had overthrown\"                      314\n\n  The Pillar of Trajan                                               314\n\n                                 1846\n\n  \"Deign, Sovereign Mistress! to accept a lay\"                       319\n\n                                 1847\n\n  Ode, performed in the Senate-House, Cambridge, on the 6th of\n      July 1847, at the First Commencement after the Installation\n      of His Royal Highness the Prince Albert, Chancellor of the\n      University                                                     320\n\n  To Miss Sellon                                                     325\n\n  \"The worship of this Sabbath morn\"                                 325\n\n  BIBLIOGRAPHIES--\n\n        I. Great Britain                                             329\n\n       II. America                                                   380\n\n      III. France                                                    421\n\n  ERRATA AND ADDENDA LIST                                            431\n\n  INDEX TO THE POEMS                                                 433\n\n  INDEX TO THE FIRST LINES                                           451\n\n\n\n\nPREFATORY NOTE\n\n\nThe American Bibliography is almost entirely the work of Mrs. St. John\nof Ithaca, and is the result of laborious and careful critical research\non her part. The French Bibliography is not so full. I have been\nassisted in it mainly by M. Legouis at Lyons, and by workers at the\nBritish Museum. I have also collected a German Bibliography, but it is\nin too incomplete a state for publication in its present form.\n\nThe English Bibliography is fuller than any of its predecessors; but\nthere is no such thing as finality in such work, especially when an\naddition to the literature of the subject is made nearly every week.\nMany kind friends, and coadjutors, have assisted me in it, amongst whom\nI may mention Dr. Garnett of the British Museum, and _very specially_\nMr. Tutin, of Hull, and also Mr. John J. Smith, St. Andrews, and Mr.\nMaclauchlan, Dundee. If I omit, either here or elsewhere, to record the\nassistance which I have received from any one, in my efforts to make\nthis edition of Wordsworth as perfect as is possible at this stage of\nliterary criticism and editorship, I sincerely regret it; but many of\nmy correspondents have specially requested that no mention should be\nmade of their names or their services.\n\nIn the Preface to the first volume of this edition there was an\nunfortunate omission. In returning the final proofs to press, I\naccidentally transmitted an uncorrected one, in which two names did\nnot appear. They were those of Mr. Thomas Hutchinson, Dublin, and\nMr. S. C. Hill, of Hughli College, Bengal. The former kindly revised\nmost of the sheets of Volumes I. and II., and corrected errors,\nbesides making other valuable suggestions and additions. When his own\nClarendon Press edition of Wordsworth was being prepared for press,\nMr. Hutchinson asked permission to incorporate in it materials which\nwere not afterwards inserted. This I granted cordially, as a similar\npermission had been given to Professor Dowden for his Aldine edition.\nThe unfortunate omission of Mr. Hutchinson's name was not discovered\nby me till after the issue of volumes I. and II. (which appeared\nsimultaneously), and it was first brought under my notice by Mr.\nHutchinson's own letters to the newspapers. My debt to Mr. Hutchinson\nis great; and, although I have already thanked him for the services\nwhich he has rendered to the world in connection with Wordsworthian\nliterature, I may perhaps be allowed to repeat the acknowledgment now.\nThe revised sheets of Vols. I. and II. of this edition were, however,\nsubmitted to others at the same time that they were sent to Mr.\nHutchinson; more especially to the late Mr. <DW18>s Campbell, and on his\ndeath to Mr. Belinfante, and then to the late Mr. Kinghorn, all of whom\nwere engaged by my publishers to assist in the work entrusted to me.\nThey \"turned on the microscope\" on my own work, and Mr. Hutchinson's;\nand to them I have been indebted in many ways.\n\nMr. Hill's services, in tracing the sources of numerous quotations from\nother poets which occur in Wordsworth's text, have been great. He sent\nme his discoveries, unsolicited, and I wish to express very cordially\nmy indebtedness to him. To discover some of these quotations--there\nare several hundreds of them--cost me much labour, before I had the\npleasure of hearing from, or knowing, Mr. Hill; and his assistance\nin this matter has been greater than that of any other person. It\nwill be seen that I have failed--after much study and extensive\ncorrespondence--to discover them all.\n\nIn addition to actual quotations--indicated by Wordsworth by inverted\ncommas in his poems--to trace parallel passages from other poets, or\nphrases which may have suggested to him what he recast and glorified,\nhas seemed to me work not unworthy of accomplishment. At the same time,\nand in the same connection, to discover the somewhat similar debts\nof later poets to Wordsworth, and to indicate this here and there in\nfootnotes, may not be wholly useless to posterity.\n\nMy obligations to my friend, Mr. <DW18>s Campbell, are greater than I can\nadequately express. He supplied me with much material, drawn from many\nquarters; and, although he did not always mention his sources, I had\nimplicit confidence in him, both as a literary man and a friend. After\nhis death, through the kindness of Mrs. Campbell, I examined some MS.\nvolumes of _Wordsworthiana_ written by him, which were of much use to\nme.\n\nSome of these were from unknown sources, which I should perhaps have\ntraced out before making use of them, but, in all my Wordsworth work, I\nhave acted from first to last on the legal opinion of a distinguished\nJudge, that the heir of the writer of literary work could alone\nauthorise its subsequent publication; and, since the heirs of the Poet\nhad kindly given me permission to collect and publish his works, I did\nso, with a view to the benefit of posterity.\n\nSome of Mr. Campbell's material was derived from MSS. now in the\npossession of Mr. T. Norton Longman, and I have to express my sincere\nregret that in the earlier volumes I copied from Mr. Campbell's\ntranscripts of these MSS.--which were lent to him on the condition\nthat no public use should be made of them without Mr. Longman's\npermission--some variations of the text, without mentioning the source\nwhence they were derived.\n\nI was unaware that these MSS. were lent to Mr. Campbell with the\ncondition attached, and regret very much that I am unable to trust my\nmemory to indicate now what variations of text I have quoted from them.\nBut I may add that Mr. Longman is about to publish a work which will\nenable Wordsworth students to become practically acquainted with the\ncontents of his MSS.\n\nIn reference to the poems not published by Wordsworth or his sister\nduring their lifetime, I have included in this volume not only fugitive\npieces printed in Magazines and elsewhere, but also those which have\nbeen since recovered from numerous manuscript sources. They are of\nvarying merit. It would be interesting to know, and to record in every\ninstance, where these manuscripts now are; but this is impossible. In\nmany cases the manuscripts have recently changed ownership. I have\nobtained a sight of many of them, and have been granted permission to\ntranscribe them, from the fortunate possessors of large autograph\ncollections, and also from dealers in autographs; but, after the sale\nof manuscripts at public auction-rooms, it is, as a rule, impossible to\ntrace them.\n\nIn many cases the MS. variants which have been published in previous\nvolumes occur in copies of the poems, transcribed by the Wordsworth\nhousehold in private letters to friends. I have occasionally indicated\nthis in footnotes; but, to have done so always would have disfigured\nthe pages, and frequently the notes would have been longer than the\ntext. To trace the present possessors of the MSS. would be well-nigh\nimpossible. It is perhaps worth mentioning that in several cases\nWordsworth entered as \"misprints\" in future editions, what some of his\neditors have considered \"new readings.\" _E.g._ in _The Excursion_, book\nix. l. 679, \"wild\" demeanour, instead of \"mild\" demeanour.\n\nOn Nov. 4, 1893, Mr. Aubrey de Vere wrote to me--\n\n    \"I earnestly hope that, in your 'monumental edition,' you will\n    restore the _Ode, Intimations of Immortality_, to the place\n    which Wordsworth always assigned to it, that of the High Altar\n    of his poetic Cathedral; remitting Quillinan's laureate Ode\n    on an unworthy, because 'occasional,' subject to an Appendix,\n    as a work that at the time of publication was attributed to\n    Wordsworth, but was written by another, though it probably\n    was seen by him, and had a line or two of his in it, and\n    corrections by him.\n\n    \"This is certainly the truth; and I should think that he\n    probably himself told all that truth to the officials, when\n    transmitting the Ode; but that they concealed the circumstance;\n    and that Wordsworth, then profoundly depressed in spirits, gave\n    no more thought to the subject, and soon forgot all about it.\u2026\n\n                        \"Yours very sincerely,\n\n                                                  \"AUBREY DE VERE.\"\n\nIt was in compliance with Mr. Aubrey de Vere's request that, in this\nedition, I departed, in a single instance, from the chronological\narrangement of the poems.\n\nIt may not be too trivial a detail to mention that I gladly gave\npermission to other editors of Wordsworth to make use of any of the\nmaterial which I discovered, and brought together, in former editions;\n_e.g._ to Mr. George, in Boston, for his edition of _The Prelude_ (in\nwhich, if the reader, or critic, compares my original edition with his\nnotes, he will see what Mr. George has done); and to Professor Dowden,\nTrinity College, Dublin, for his most admirable Aldine edition. For the\nlatter--which will always hold a high place in Wordsworth literature--I\nplaced everything asked from me at the disposal of Mr. Dowden.\n\nWhile these sheets are passing through the press, Dr. Garnett, of the\nBritish Museum--one of the kindest and ablest of bibliographers--has\nforwarded to me a contribution, previously sent by him to _The\nAcademy_, and printed in its issue of January 2, 1897.\n\nI have no means of knowing--or of ultimately discovering--whether that\nsonnet, printed as Wordsworth's, is really his. Dr. Garnett says, in\nhis letter to me, \"The verses were undoubtedly in Wordsworth's hand\";\nand, he adds, \"I think they should be preserved, because they are\nWordsworth's, and as an additional proof of his regard for Camoens,\nwhom he enumerates elsewhere among great sonnet-writers. I have added\na version of the quatrains, that the piece may be complete. From the\ncharacter of the handwriting, the lines would seem to have been written\ndown in old age; and I am not quite certain of the word which I have\ntranscribed as 'Austral.'\"\n\n      Vasco, whose bold and happy mainyard spread\n      Sunward thy sails where dawning glory dyed\n      Heaven's Orient gate; whose westering prow the tide\n    Clove, where the day star bows him to his bed:\n    Not sterner toil than thine, or strife more dread,\n      Or nobler laud to nobler lyre allied,\n      His, who did baffled Polypheme deride;\n    Or his, whose scaring shaft the Harpy fled.\n    Camoens, he the accomplished and the good,\n      Gave to thy fame a more illustrious flight\n      Than that brave vessel, though she sailed so far.\n    Through him her course along the Austral flood\n      Is known to all beneath the polar star,\n      Through him the Antipodes in thy name delight.\n\n                                         WILLIAM KNIGHT.\n\n\n\n\nWORDSWORTH'S POETICAL WORKS\n\n\n\n\n1834\n\n\nLINES\n\nSUGGESTED BY A PORTRAIT FROM THE PENCIL OF F. STONE\n\nComposed 1834.--Published 1835\n\n[This Portrait has hung for many years in our principal sitting-room,\nand represents J. Q.[1] as she was when a girl. The picture, though it\nis somewhat thinly painted, has much merit in tone and general effect:\nit is chiefly valuable, however, from the sentiment that pervades\nit. The anecdote of the saying of the monk in sight of Titian's\npicture was told in this house by Mr. Wilkie, and was, I believe,\nfirst communicated to the public in this poem, the former portion of\nwhich I was composing at the time. Southey heard the story from Miss\nHutchinson, and transferred it to the _Doctor_; but it is not easy to\nexplain how my friend Mr. Rogers, in a note subsequently added to his\n_Italy_, was led to speak of the same remarkable words having many\nyears before been spoken in his hearing by a monk or priest in front\nof a picture of the Last Supper, placed over a Refectory-table in a\nconvent at Padua.--I.F.]\n\nOne of the \"Poems of Sentiment and Reflection.\"--ED.\n\n    Beguiled into forgetfulness of care\n    Due to the day's unfinished task; of pen\n    Or book regardless, and of that fair scene\n    In Nature's prodigality displayed\n    Before my window, oftentimes and long                              5\n    I gaze upon a Portrait whose mild gleam\n    Of beauty never ceases to enrich\n    The common light; whose stillness charms the air,\n    Or seems to charm it, into like repose;\n    Whose silence, for the pleasure of the ear,                       10\n    Surpasses sweetest music. There she sits\n    With emblematic purity attired\n    In a white vest, white as her marble neck\n    Is, and the pillar of the throat would be\n    But for the shadow by the drooping chin                           15\n    Cast into that recess--the tender shade,\n    The shade and light, both there and every where,\n    And through the very atmosphere she breathes,\n    Broad, clear, and toned harmoniously, with skill\n    That might from nature have been learnt in the hour               20\n    When the lone shepherd sees the morning spread\n    Upon the mountains. Look at her, whoe'er\n    Thou be that, kindling with a poet's soul,\n    Hast loved the painter's true Promethean craft\n    Intensely--from Imagination take                                  25\n    The treasure,--what mine eyes behold see thou,\n    Even though the Atlantic ocean roll between.\n\n      A silver line, that runs from brow to crown\n    And in the middle parts the braided hair,\n    Just serves to show how delicate a soil                           30\n    The golden harvest grows in; and those eyes,\n    Soft and capacious as a cloudless sky\n    Whose azure depth their colour emulates,\n    Must needs be conversant with upward looks,\n    Prayer's voiceless service; but now, seeking nought               35\n    And shunning nought, their own peculiar life\n    Of motion they renounce, and with the head\n    Partake its inclination towards earth\n    In humble grace, and quiet pensiveness\n    Caught at the point where it stops short of sadness.              40\n\n      Offspring of soul-bewitching Art, make me\n    Thy confidant! say, whence derived that air\n    Of calm abstraction? Can the ruling thought\n    Be with some lover far away, or one\n    Crossed by misfortune, or of doubted faith?                       45\n    Inapt conjecture! Childhood here, a moon\n    Crescent in simple loveliness serene,\n    Has but approached the gates of womanhood,\n    Not entered them; her heart is yet unpierced\n    By the blind Archer-god; her fancy free:                          50\n    The fount of feeling, if unsought elsewhere,\n    Will not be found.\n\n                        Her right hand, as it lies\n    Across the slender wrist of the left arm\n    Upon her lap reposing, holds--but mark\n    How slackly, for the absent mind permits                          55\n    No firmer grasp--a little wild-flower, joined\n    As in a posy, with a few pale ears\n    Of yellowing corn, the same that overtopped\n    And in their common birthplace sheltered it\n    'Till they were plucked together; a blue flower                   60\n    Called by the thrifty husbandman a weed;\n    But Ceres, in her garland, might have worn\n    That ornament, unblamed. The floweret, held\n    In scarcely conscious fingers, was, she knows,\n    (Her Father told her so) in youth's gay dawn                      65\n    Her Mother's favourite; and the orphan Girl,\n    In her own dawn--a dawn less gay and bright,\n    Loves it, while there in solitary peace\n    She sits, for that departed Mother's sake.\n    --Not from a source less sacred is derived                        70\n    (Surely I do not err) that pensive air\n    Of calm abstraction through the face diffused\n    And the whole person.\n                          Words have something told\n    More than the pencil can, and verily\n    More than is needed, but the precious Art                         75\n    Forgives their interference--Art divine,\n    That both creates and fixes, in despite\n    Of Death and Time, the marvels it hath wrought.\n\n      Strange contrasts have we in this world of ours!\n    That posture, and the look of filial love                         80\n    Thinking of past and gone, with what is left\n    Dearly united, might be swept away\n    From this fair Portrait's fleshly Archetype,\n    Even by an innocent fancy's slightest freak\n    Banished, nor ever, haply, be restored                            85\n    To their lost place, or meet in harmony\n    So exquisite; but _here_ do they abide,\n    Enshrined for ages. Is not then the Art\n    Godlike, a humble branch of the divine,\n    In visible quest of immortality,                                  90\n    Stretched forth with trembling hope?--In every realm,\n    From high Gibraltar to Siberian plains,\n    Thousands, in each variety of tongue\n    That Europe knows, would echo this appeal;\n    One above all, a Monk who waits on God                            95\n    In the magnific Convent built of yore\n    To sanctify the Escurial palace. He--\n    Guiding, from cell to cell and room to room,\n    A British Painter (eminent for truth\n    In character,[2] and depth of feeling, shown                     100\n    By labours that have touched the hearts of kings,\n    And are endeared to simple cottagers)--\n    Came, in that service, to a glorious work,[3]\n    Our Lord's Last Supper, beautiful as when first\n    The appropriate Picture, fresh from Titian's hand,               105\n    Graced the Refectory: and there, while both\n    Stood with eyes fixed upon that masterpiece,\n    The hoary Father in the Stranger's ear\n    Breathed out these words:--\"Here daily do we sit,\n    Thanks given to God for daily bread, and here                    110\n    Pondering the mischiefs of these restless times,\n    And thinking of my Brethren, dead, dispersed,\n    Or changed and changing, I not seldom gaze\n    Upon this solemn Company unmoved\n    By shock of circumstance, or lapse of years,                     115\n    Until I cannot but believe that they--\n    They are in truth the Substance, we\n    the Shadows.\"[4]\n\n      So spake the mild Jeronymite, his griefs\n    Melting away within him like a dream\n    Ere he had ceased to gaze, perhaps to speak:                     120\n    And I, grown old, but in a happier land,\n    Domestic Portrait! have to verse consigned\n    In thy calm presence those heart-moving words:\n    Words that can soothe, more than they agitate;\n    Whose spirit, like the angel that went down                      125\n    Into Bethesda's pool, with healing virtue\n    Informs the fountain in the human breast\n    Which[5] by the visitation was disturbed.\n    ----But why this stealing tear? Companion mute,\n    On thee I look, not sorrowing; fare thee well,                   130\n    My Song's Inspirer, once again farewell![6]\n\n[1] Jemima Quillinan, the eldest daughter of Edward Quillinan,\nWordsworth's future son-in-law. The portrait was taken when she was a\nschool-girl, and while her father resided at Oporto.--ED.\n\n[2] Wilkie. See the Fenwick note.--ED.\n\n[3] 1837.\n\n    Left not unvisited a glorious work,\n\n    1835.\n\n[4] \"When Wilkie was in the Escurial, looking at Titian's famous\npicture of the Last Supper, in the Refectory there, an old Jeronymite\nsaid to him: 'I have sate daily in sight of that picture for now nearly\nthree score years; during that time my companions have dropt off, one\nafter another--all who were my seniors, all who were my contemporaries,\nand many, or most of those who were younger than myself; more than one\ngeneration has passed away, and there the figures in the picture have\nremained unchanged! I look at them till I sometimes think that they are\nthe realities, and we but shadows!'\n\nI wish I could record the name of the monk by whom that natural feeling\nwas so feelingly and strikingly expressed.\n\n    The shows of things are better than themselves,\n\nsays the author of the tragedy of Nero, whose name also I could wish\nhad been forthcoming; and the classical reader will remember the lines\nof Sophocles:\n\n    \u1f41\u03c1\u1ff6 \u03b3\u1f70\u03c1 \u1f21\u03bc\u1fb6\u03c2 \u03bf\u1f50\u03b4\u1f72\u03bd \u1f44\u03bd\u03c4\u03b1\u03c2 \u1f04\u03bb\u03bb\u03bf, \u03c0\u03bb\u1f74\u03bd\n    \u03b5\u1f34\u03b4\u03c9\u03bb', \u1f45\u03c3\u03bf\u03b9\u03c0\u03b5\u03c1 \u03b6\u1ff6\u03bc\u03b5\u03bd, \u1f74 \u03ba\u03bf\u03cd\u03c6\u03b7\u03bd \u03c3\u03ba\u03b9\u03ac\u03bd.\n\nThese are reflections which should make us think\n\n    Of that same time when no more change shall be\n    But steadfast rest of all things, firmly stayd\n    Upon the pillars of Eternity,\n    That is contrain to mutability;\n    For all that moveth doth in change delight:\n    But henceforth all shall rest eternally\n    With Him that is the God of Sabaoth hight,\n    O that great Sabaoth God grant me that Sabbath's sight.\n\n    SPENSER.\"\n\n(Southey, _The Doctor_, vol. iii. p. 235.)--ED.\n\n[5] 1837.\n\n    That \u2026\n\n    1835.\n\n[6] The pile of buildings, composing the palace and convent of San\nLorenzo, has, in common usage, lost its proper name in that of the\n_Escurial_, a village at the foot of the hill upon which the splendid\nedifice, built by Philip the Second, stands. It need scarcely be added,\nthat Wilkie is the painter alluded to.--W.W. 1835.\n\n\nTHE FOREGOING SUBJECT RESUMED\n\nComposed 1834.--Published 1835.\n\nOne of the \"Poems of Sentiment and Reflection.\"--ED.\n\n    Among a grave fraternity of Monks,\n    For One, but surely not for One alone,\n    Triumphs, in that great work, the Painter's skill,\n    Humbling the body, to exalt the soul;\n    Yet representing, amid wreck and wrong                             5\n    And dissolution and decay, the warm\n    And breathing life of flesh, as if already\n    Clothed with impassive majesty, and graced\n    With no mean earnest of a heritage\n    Assigned to it in future worlds. Thou, too,                       10\n    With thy memorial flower, meek Portraiture!\n    From whose serene companionship I passed\n    Pursued by thoughts that haunt me still; thou also--\n    Though but a simple object, into light\n    Called forth by those affections that endear                      15\n    The private hearth; though keeping thy sole seat\n    In singleness, and little tried by time,\n    Creation, as it were, of yesterday--\n    With a congenial function art endued\n    For each and all of us, together joined                           20\n    In course of nature under a low roof\n    By charities and duties that proceed\n    Out of the bosom of a wiser vow.\n    To a like salutary sense of awe\n    Or sacred wonder, growing with the power                          25\n    Of meditation that attempts to weigh,\n    In faithful scales, things and their opposites,\n    Can thy enduring quiet gently raise\n    A household small and sensitive,--whose love,\n    Dependent as in part its blessings are                            30\n    Upon frail ties dissolving or dissolved\n    On earth, will be revived, we trust, in heaven.[7]\n\n[7] In the class entitled \"Musings,\" in Mr. Southey's Minor Poems, is\none upon his own miniature picture, taken in childhood, and another\nupon a landscape painted by Gaspar Poussin. It is possible that every\nword of the above verses, though similar in subject, might have been\nwritten had the author been unacquainted with those beautiful effusions\nof poetic sentiment. But, for his own satisfaction, he must be allowed\nthus publicly to acknowledge the pleasure those two poems of his Friend\nhave given him, and the grateful influence they have upon his mind as\noften as he reads them, or thinks of them.--W.W. 1835.\n\n\nTO A CHILD\n\nWRITTEN IN HER ALBUM[8]\n\nComposed 1834.--Published 1835\n\n[This quatrain was extempore on observing this image, as I had often\ndone, on the lawn of Rydal Mount. It was first written down in the\nAlbum of my God-daughter, Rotha Quillinan.--I.F.]\n\nIn 1837 this was one of the \"Inscriptions.\" In 1845 it was transferred\nto the \"Miscellaneous Poems.\"--ED.\n\n    Small service is true service while it lasts:\n    Of humblest Friends, bright Creature! scorn not one![9]\n    The Daisy, by the shadow that it casts,\n    Protects the lingering dew-drop from the Sun.[10]\n\n[8] The original title (1835) was \"Written in an Album.\" In 1837 it was\n\"Written in the Album of a Child.\" In 1845 the title was reconstructed\nas above.\n\n[9] 1845.\n\n    Of Friends, however humble, scorn not one:\n\n    1835.\n\n[10] Compare the lines, written in 1845, beginning--\n\n        So fair, so sweet, withal so sensitive.\n\n    ED.\n\n\nLINES\n\nWRITTEN IN THE ALBUM OF THE COUNTESS OF LONSDALE,[11] NOV. 5, 1834\n\nComposed 1834.--Published 1835\n\n[This is a faithful picture of that amiable Lady, as she then was. The\nyouthfulness of figure and demeanour and habits, which she retained in\nalmost unprecedented degree, departed a very few years after, and she\ndied without violent disease by gradual decay before she reached the\nperiod of old age.--I.F.]\n\nThis was placed, in 1845, among the \"Miscellaneous Poems.\"--ED.\n\n    Lady! a Pen (perhaps with thy regard,\n    Among the Favoured, favoured not the least)\n    Left, 'mid the Records of this Book inscribed,\n    Deliberate traces, registers of thought\n    And feeling, suited to the place and time                          5\n    That gave them birth:--months passed, and still this hand,\n    That had not been too timid to imprint\n    Words which the virtues of thy Lord inspired,\n    Was yet not bold enough to write of Thee.\n    And why that scrupulous reserve? In sooth                         10\n    The blameless cause lay in the Theme itself.\n    Flowers are there many that delight to strive\n    With the sharp wind, and seem to court the shower,\n    Yet are by nature careless of the sun\n    Whether he shine on them or not; and some,                        15\n    Where'er he moves along the unclouded sky,\n    Turn a broad front full on his flattering beams:\n    Others do rather from their notice shrink,\n    Loving the dewy shade,--a humble band,\n    Modest and sweet, a progeny of earth,                             20\n    Congenial with thy mind and character,\n    High-born Augusta!\n                        Witness Towers, and Groves!\n    And Thou, wild Stream, that giv'st the honoured name[12]\n    Of Lowther to this ancient Line, bear witness[13]\n    From thy most secret haunts; and ye Parterres,                    25\n    Which She is pleased and proud to call her own,\n    Witness how oft upon my noble Friend\n    _Mute_ offerings, tribute from an inward sense\n    Of admiration and respectful love,\n    Have waited--till the affections could no more                    30\n    Endure that silence, and broke out in song,\n    Snatches of music taken up and dropt\n    Like those self-solacing, those under, notes\n    Trilled by the redbreast, when autumnal leaves\n    Are thin upon the bough. Mine, only mine,                         35\n    The pleasure was, and no one heard the praise,\n    Checked, in the moment of its issue, checked\n    And reprehended, by a fancied blush\n    From the pure qualities that called it forth.\n\n      Thus Virtue lives debarred from Virtue's meed;                  40\n    Thus, Lady, is retiredness a veil\n    That, while it only spreads a softening charm\n    O'er features looked at by discerning eyes,\n    Hides half their beauty from the common gaze;\n    And thus,[14] even on the exposed and breezy hill                 45\n    Of lofty station, female goodness walks,\n    When side by side with lunar gentleness,\n    As in a cloister. Yet the grateful Poor\n    (Such the immunities of low estate,\n    Plain Nature's enviable privilege,                                50\n    Her sacred recompense for many wants)\n    Open their hearts before Thee, pouring out\n    All that they think and feel, with tears of joy;\n    And benedictions not unheard in heaven:\n    And friend in the ear of friend, where speech is free             55\n    To follow truth, is eloquent as they.\n\n      Then let the Book receive in these prompt lines\n    A just memorial; and thine eyes consent\n    To read that they, who mark thy course, behold\n    A life declining with the golden light                            60\n    Of summer, in the season of sere leaves;[15]\n    See cheerfulness undamped by stealing Time;\n    See studied kindness flow with easy stream,\n    Illustrated with inborn courtesy;\n    And an habitual disregard of self                                 65\n    Balanced by vigilance for others' weal.\n\n      And shall the Verse not tell of lighter gifts\n    With these ennobling attributes conjoined\n    And blended, in peculiar harmony,\n    By Youth's surviving spirit? What agile grace!                    70\n    A nymph-like liberty, in nymph-like form,\n    Beheld with wonder; whether floor or path\n    Thou tread; or sweep--borne on the managed steed--[16]\n    Fleet as the shadows, over down or field,\n    Driven by strong winds at play among the clouds.                  75\n\n      Yet one word more--one farewell word--a wish\n    Which came, but it has passed into a prayer--\n    That, as thy sun in brightness is declining,\n    So--at an hour yet distant for _their_ sakes\n    Whose tender love, here faltering on the way                      80\n    Of a diviner love, will be forgiven--\n    So may it set in peace, to rise again\n    For everlasting glory won by faith.\n\n[11] 1837.\n\n    Countess of ----\n\n    1835.\n\n[12] The Lowther stream passes the Castle, and joins the Eamont below\nBrougham Hall, near Penrith.--ED.\n\n[13] 1837.\n\n                      Towers, and stately Groves,\n    Bear witness for me; thou, too, Mountain-stream!\n\n    1835.\n\n[14]\n\n    When hence \u2026\n\n    C.\n\n[15] Compare _September, 1819_, and _Upon the Same Occasion_, vol. vi.\npp. 201, 202, especially the lines in the latter--\n\n    Me, conscious that my leaf is sere,\n    And yellow on the bough, etc.\n\nED.\n\n[16] 1837.\n\n    Thou tread, or on the managed steed art borne,\n\n    1835.\n\n\n\n\n1835\n\nTwo Evening Voluntaries, two Elegies (on the deaths of Charles Lamb and\nJames Hogg), the lines on the Bird of Paradise, and a few sonnets, make\nup the poems belonging to the year 1835.--ED.\n\n\n\"WHY ART THOU SILENT? IS THY LOVE A PLANT\"\n\nComposed 1835 (or earlier).--Published 1835\n\n[In the month of January,--when Dora and I were walking from Town-end,\nGrasmere, across the Vale, snow being on the ground, she espied, in\nthe thick though leafless hedge, a bird's nest half-filled with snow.\nOut of this comfortless appearance arose this Sonnet, which was, in\nfact, written without the least reference to any individual object,\nbut merely to prove to myself that I could, if I thought fit, write in\na strain that Poets have been fond of. On the 14th of February in the\nsame year, my daughter, in a sportive mood, sent it as a Valentine,\nunder a fictitious name, to her cousin C.W.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Why art thou silent? Is thy love a plant\n    Of such weak fibre that the treacherous air\n    Of absence withers what was once so fair?\n    Is there no debt to pay, no boon to grant?\n    Yet have my thoughts for thee been vigilant--                      5\n    Bound to thy service with unceasing care,[17]\n    The mind's least generous wish a mendicant\n    For nought but what thy happiness could spare.\n    Speak--though this soft warm heart, once free to hold\n    A thousand tender pleasures, thine and mine,                      10\n    Be left more desolate, more dreary cold\n    Than a forsaken bird's-nest filled with snow\n    'Mid its own bush of leafless eglantine--\n    Speak, that my torturing doubts their end may know!\n\n[17] 1845.\n\n    \u2026 with incessant care,\n\n    C.\n\n    (As would my deeds have been) with hourly care,\n\n    1835.\n\n\nTO THE MOON\n\n(COMPOSED BY THE SEA-SIDE,--ON THE COAST OF CUMBERLAND)\n\nComposed 1835.--Published 1837\n\nOne of the \"Evening Voluntaries.\"--ED.\n\n    Wanderer! that stoop'st so low, and com'st so near\n    To human life's unsettled atmosphere;\n    Who lov'st with Night and Silence to partake,\n    So might it seem, the cares of them that wake;\n    And, through the cottage-lattice softly peeping,                   5\n    Dost shield from harm the humblest of the sleeping;\n    What pleasure once encompassed those sweet names\n    Which yet in thy behalf the Poet claims,\n    An idolizing dreamer as of yore!--\n    I slight them all; and, on this sea-beat shore                    10\n    Sole-sitting, only can to thoughts attend\n    That bid me hail thee as the SAILOR'S FRIEND;\n    So call thee for heaven's grace through thee made known\n    By confidence supplied and mercy shown,\n    When not a twinkling star or beacon's light                       15\n    Abates the perils of a stormy night;\n    And for less obvious benefits, that find\n    Their way, with thy pure help, to heart and mind;\n    Both for the adventurer starting in life's prime;\n    And veteran ranging round from clime to clime,                    20\n    Long-baffled hope's slow fever in his veins,\n    And wounds and weakness oft his labour's sole remains.\n\n      The aspiring Mountains and the winding Streams,\n    Empress of Night! are gladdened by thy beams;\n    A look of thine the wilderness pervades,                          25\n    And penetrates the forest's inmost shades;\n    Thou, chequering peaceably the minster's gloom,\n    Guid'st the pale Mourner to the lost one's tomb;\n    Canst reach the Prisoner--to his grated cell\n    Welcome, though silent and intangible!--                          30\n    And lives there one, of all that come and go\n    On the great waters toiling to and fro,\n    One, who has watched thee at some quiet hour\n    Enthroned aloft in undisputed power,\n    Or crossed by vapoury streaks and clouds that move                35\n    Catching the lustre they in part reprove--\n    Nor sometimes felt a fitness in thy sway\n    To call up thoughts that shun the glare of day,\n    And make the serious happier than the gay?\n\n      Yes, lovely Moon! if thou so mildly bright                      40\n    Dost rouse, yet surely in thy own despite,\n    To fiercer mood the phrenzy-stricken brain,\n    Let me a compensating faith maintain;\n    That there's a sensitive, a tender, part\n    Which thou canst touch in every human heart,                      45\n    For healing and composure.--But, as least\n    And mightiest billows ever have confessed\n    Thy domination; as the whole vast Sea\n    Feels through her lowest depths thy sovereignty;\n    So shines that countenance with especial grace                    50\n    On them who urge the keel her _plains_ to trace\n    Furrowing its way right onward. The most rude,\n    Cut off from home and country, may have stood--\n    Even till long gazing hath bedimmed his eye,\n    Or the mute rapture ended in a sigh--                             55\n    Touched by accordance of thy placid cheer,\n    With some internal lights to memory dear,\n    Or fancies stealing forth to soothe the breast\n    Tired with its daily share of earth's unrest,--\n    Gentle awakenings, visitations meek;                              60\n    A kindly influence whereof few will speak,\n    Though it can wet with tears the hardiest cheek.\n\n      And when thy beauty in the shadowy cave\n    Is hidden, buried in its monthly grave;[18]\n    Then, while the Sailor, 'mid an open sea                          65\n    Swept by a favouring wind that leaves thought free,\n    Paces the deck--no star perhaps in sight,\n    And nothing save the moving ship's own light\n    To cheer the long dark hours of vacant night--\n    Oft with his musings does thy image blend,                        70\n    In his mind's eye thy crescent horns ascend,\n    And thou art still, O Moon, that SAILOR'S FRIEND!\n\n[18] Compare--\n\n    When thou wert hidden in thy monthly grave,\n\nin the lines _Written in a Grotto_, p. 235.--ED.\n\n\nTO THE MOON\n\n(RYDAL)\n\nComposed 1835.--Published 1837\n\nOne of the \"Evening Voluntaries.\"--ED.\n\n    Queen of the stars!--so gentle, so benign,\n    That ancient Fable did to thee assign,\n    When darkness creeping o'er thy silver brow\n    Warned thee these upper regions to forego,\n    Alternate empire in the shades below--                             5\n    A Bard, who, lately near the wide-spread sea\n    Traversed by gleaming ships, looked up to thee\n    With grateful thoughts, doth now thy rising hail\n    From the close confines of a shadowy vale.\n    Glory of night, conspicuous yet serene,                           10\n    Nor less attractive when by glimpses seen\n    Through cloudy umbrage,[19] well might that fair face,\n    And all those attributes of modest grace,\n    In days when Fancy wrought unchecked by fear,\n    Down to the green earth fetch thee from thy sphere,               15\n    To sit in leafy woods by fountains clear!\n\n      O still belov'd (for thine, meek Power, are charms\n    That fascinate the very Babe in arms,\n    While he, uplifted towards thee, laughs outright,\n    Spreading his little palms in his glad Mother's sight)            20\n    O still belov'd, once worshipped! Time, that frowns\n    In his destructive flight on earthly crowns,\n    Spares thy mild splendour; still those far-shot beams\n    Tremble on dancing waves and rippling streams\n    With stainless touch, as chaste as when thy praise                25\n    Was sung by Virgin-choirs in festal lays;\n    And through dark trials still dost thou explore\n    Thy way for increase punctual as of yore,\n    When teeming Matrons--yielding to rude faith\n    In mysteries of birth and life and death                          30\n    And painful struggle and deliverance--prayed\n    Of thee to visit them with lenient aid.\n    What though the rites be swept away, the fanes\n    Extinct that echoed to the votive strains;\n    Yet thy mild aspect does not, cannot, cease                       35\n    Love to promote and purity and peace;\n    And Fancy, unreproved, even yet may trace\n    Faint types of suffering in thy beamless face.\n\n      Then, silent Monitress! let us--not blind\n    To worlds unthought of till the searching mind                    40\n    Of Science laid them open to mankind--\n    Told, also, how the voiceless heavens declare\n    God's glory; and acknowledging thy share\n    In that blest charge; let us--without offence\n    To aught of highest, holiest, influence--                         45\n    Receive whatever good 'tis given thee to dispense.\n    May sage and simple, catching with one eye\n    The moral intimations of the sky,\n    Learn from thy course, where'er their own be taken,\n    \"To look on tempests, and be never shaken\";[20]                   50\n    To keep with faithful step the appointed way\n    Eclipsing or eclipsed, by night or day,\n    And from example of thy monthly range\n    Gently to brook decline and fatal change;\n    Meek, patient, stedfast, and with loftier scope,                  55\n    Than thy revival yields, for gladsome hope![21]\n\n[19] Compare _The Triad_, vol. vii. p. 181.--ED.\n\n[20] Compare l. 6 of Shakespeare's sonnet, beginning--\n\n    Let me not to the marriage of true minds.\n\nED.\n\n[21] See a fragment of ten lines, which was written by Wordsworth in\nMS. after the above, in a copy of his poems. They are printed in the\nAppendix to this volume.--ED.\n\n\nWRITTEN AFTER THE DEATH OF CHARLES LAMB\n\n[Light will be thrown upon the tragic circumstance alluded to in this\npoem when, after the death of Charles Lamb's Sister, his biographer,\nMr. Sergeant Talfourd, shall be at liberty to relate particulars which\ncould not, at the time his Memoir was written, be given to the public.\nMary Lamb was ten years older than her brother, and has survived him as\nlong a time. Were I to give way to my own feelings, I should dwell not\nonly on her genius and intellectual powers, but upon the delicacy and\nrefinement of manner which she maintained inviolable under most trying\ncircumstances. She was loved and honoured by all her brother's friends;\nand others, some of them strange characters, whom his philanthropic\npeculiarities induced him to countenance. The death of C. Lamb himself\nwas doubtless hastened by his sorrow for that of Coleridge, to whom\nhe had been attached from the time of their being school-fellows at\nChrist's Hospital. Lamb was a good Latin scholar, and probably would\nhave gone to college upon one of the school foundations but for the\nimpediment in his speech. Had such been his lot, he would most likely\nhave been preserved from the indulgences of social humours and fancies\nwhich were often injurious to himself, and causes of severe regret to\nhis friends, without really benefiting the object of his misapplied\nkindness.--I.F.]\n\nIn the edition of 1837, these lines had no title. They were printed\nprivately,--before their first appearance in that edition,--as a small\npamphlet of seven pages without title or heading. A copy will be found\nin the fifth volume of the collection of pamphlets, forming part of\nthe library bequeathed by the late Mr. John Forster to the South\nKensington Museum. There are several readings to be found only in this\nprivately-printed edition. The poem was placed among the \"Epitaphs and\nElegiac Pieces.\"--ED.\n\nComposed November 19, 1835.--Published 1837\n\n    To a good Man of most dear memory[22]\n    This Stone is sacred.[23] Here he lies apart\n    From the great city where he first drew breath,\n    Was reared and taught; and humbly earned his bread,\n    To the strict labours of the merchant's desk                       5\n    By duty chained. Not seldom did those tasks\n    Tease, and the thought of time so spent depress,\n    His spirit, but the recompense was high;\n    Firm Independence, Bounty's rightful sire;\n    Affections, warm as sunshine, free as air;                        10\n    And when the precious hours of leisure came,\n    Knowledge and wisdom, gained from converse sweet\n    With books, or while he ranged the crowded streets\n    With a keen eye, and overflowing heart:\n    So genius triumphed over seeming wrong,                           15\n    And poured out truth in works by thoughtful love\n    Inspired--works potent over smiles and tears.\n    And as round mountain-tops the lightning plays,\n    Thus innocently sported, breaking forth\n    As from a cloud of some grave sympathy,                           20\n    Humour and wild instinctive wit, and all\n    The vivid flashes of his spoken words.\n    From the most gentle creature nursed in fields[24]\n    Had been derived the name he bore--a name,\n    Wherever christian altars have been raised,                       25\n    Hallowed to meekness and to innocence;\n    And if in him meekness at times gave way,\n    Provoked out of herself by troubles strange,\n    Many and strange, that hung about his life;[25]\n    Still, at the centre of his being, lodged                         30\n    A soul by resignation sanctified:\n    And if too often, self-reproached, he felt\n    That innocence belongs not to our kind,\n    A power that never ceased to abide in him,\n    Charity, 'mid the multitude of sins[26]                           35\n    That she can cover, left not his exposed\n    To an unforgiving judgment from just Heaven.\n    O, he was good, if e'er a good Man lived!\n\n    From a reflecting mind and sorrowing heart\n    Those simple lines flowed with an earnest wish,                   40\n    Though but a doubting hope, that they might serve\n    Fitly to guard the precious dust of him\n    Whose virtues called them forth. That aim is missed;\n    For much that truth most urgently required\n    Had from a faltering pen been asked in vain:                      45\n    Yet, haply, on the printed page received,\n    The imperfect record, there, may stand unblamed\n    As long as verse of mine shall breathe the air\n    Of memory, or see the light of love.[27]\n\n      Thou wert a scorner of the fields, my Friend,                   50\n    But more in show than truth;[28] and from the fields,\n    And from the mountains, to thy rural grave\n    Transported, my soothed spirit hovers o'er\n    Its green untrodden turf, and blowing flowers;\n    And taking up a voice shall speak (tho' still                     55\n    Awed by the theme's peculiar sanctity\n    Which words less free presumed not even to touch)\n    Of that fraternal love, whose heaven-lit lamp\n    From infancy, through manhood, to the last\n    Of threescore years, and to thy latest hour,                      60\n    Burnt on with ever-strengthening light, enshrined[29]\n    Within thy bosom.\n                              \"Wonderful\" hath been\n    The love established between man and man,\n    \"Passing the love of women;\" and between\n    Man and his help-mate in fast wedlock joined                      65\n    Through God,[30] is raised a spirit and soul of love\n    Without whose blissful influence Paradise\n    Had been no Paradise; and earth were now\n    A waste where creatures bearing human form,\n    Direst of savage beasts, would roam in fear,                      70\n    Joyless and comfortless. Our days glide on;[31]\n    And let him grieve who cannot choose but grieve\n    That he hath been an Elm without his Vine,\n    And her bright dower of clustering charities,\n    That, round his trunk and branches, might have clung              75\n    Enriching and adorning. Unto thee,\n    Not so enriched, not so adorned, to thee\n    Was given (say rather thou of later birth\n    Wert given to her) a Sister--'tis a word\n    Timidly uttered, for she _lives_, the meek,                       80\n    The self-restraining, and the ever-kind;\n    In whom thy reason and intelligent heart\n    Found--for all interests, hopes, and tender cares,\n    All softening, humanising, hallowing powers,\n    Whether withheld, or for her sake unsought--                      85\n    More than sufficient recompense!\n                                      Her love\n    (What weakness prompts the voice to tell it here?)\n    Was as the love of mothers; and when years,\n    Lifting the boy to man's estate, had called\n    The long-protected to assume the part                             90\n    Of a protector, the first filial tie\n    Was undissolved; and, in or out of sight,\n    Remained imperishably interwoven\n    With life itself. Thus, 'mid a shifting world,\n    Did they together testify of time[32]                             95\n    And season's difference--a double tree\n    With two collateral stems sprung from one root;\n    Such were they--such thro' life they _might_ have been\n    In union, in partition only such;\n    Otherwise wrought the will of the Most High;                     100\n    Yet, thro' all visitations and all trials,\n    Still they were faithful; like two vessels launched\n    From the same beach one ocean to explore[33]\n    With mutual help, and sailing--to their league\n    True, as inexorable winds, or bars                               105\n    Floating or fixed of polar ice, allow.[34]\n\n      But turn we rather, let my spirit turn\n    With thine, O silent and invisible Friend!\n    To those dear intervals, nor rare nor brief,\n    When reunited, and by choice withdrawn                           110\n    From miscellaneous converse, ye were taught\n    That the remembrance of foregone distress,\n    And the worse fear of future ill (which oft\n    Doth hang around it, as a sickly child\n    Upon its mother) may be both alike                               115\n    Disarmed of power to unsettle present good\n    So prized, and things inward and outward held\n    In such an even balance, that the heart\n    Acknowledges God's grace, his mercy feels,\n    And in its depth of gratitude is still.                          120\n\n      O gift divine of quiet sequestration!\n    The hermit, exercised in prayer and praise,\n    And feeding daily on the hope of heaven,\n    Is happy in his vow, and fondly cleaves\n    To life-long singleness; but happier far                         125\n    Was to your souls, and, to the thoughts of others,\n    A thousand times more beautiful appeared,\n    Your _dual_ loneliness. The sacred tie\n    Is broken; yet why grieve? for Time but holds\n    His moiety in trust, till Joy shall lead                         130\n    To the blest world where parting is unknown.[35]\n\n[22] 1837.\n\n    _To the dear memory of a frail good Man_\n\n    In privately printed edition.\n\n[23] Charles Lamb died December 27, 1834, and was buried in Edmonton\nChurchyard, in a spot selected by himself.--ED.\n\n[24] This way of indicating the _name_ of my lamented friend has been\nfound fault with, perhaps rightly so; but I may say in justification of\nthe double sense of the word, that similar allusions are not uncommon\nin epitaphs. One of the best in our language in verse, I ever read,\nwas upon a person who bore the name of Palmer\u2020; and the course of the\nthought, throughout, turned upon the Life of the Departed, considered\nas a pilgrimage. Nor can I think that the objection in the present\ncase will have much force with any one who remembers Charles Lamb's\nbeautiful sonnet addressed to his own name, and ending--\n\n    No deed of mine shall shame thee, gentle name!\n\nW. W. 1837.\n\n    \u2020 1840.\n\n    Pilgrim;\n\n    1837.\n\nProfessor Henry Reed, in his edition of 1837, added the following note\nto Wordsworth's. \"In _Hierologus_, a Church Tour through England and\nWales, I have met with an epitaph which is probably the one alluded to\nabove \u2026 a Kentish epitaph on one Palmer:\n\n    Palmers all our fathers were;\n    I, a Palmer lived here,\n    And traveyled sore, till worn with age,\n    I ended this world's pilgrimage,\n    On the blest Ascension Day\n    In the cheerful month of May.\"\n\nThe above is Professor Reed's note. The following is an exact copy of\nthe epitaph:--\n\n    _Palmers_ all our faders were;\n    I, a _Palmer_ livyd here\n    And travyld still till worne wyth age,\n    I endyd this world's pylgramage,\n    On the blyst assention day\n    In the cherful month of May;\n    A thowsand wyth fowre hundryd seven,\n    And took my jorney hense to heven.\n\n    (Printed by Weever.)\n\nED.\n\n[25] Compare Talfourd's _Final Memorials of Charles Lamb_,\n_passim_.--ED.\n\n[26] 1837.\n\n    _He had a constant friend--in Charity_;\n    HER _who, among_ a multitude of sins,\n\n    In privately printed edition.\n\n[27] 1837.\n\n    From a reflecting mind and sorrowing heart\n    This tribute flow'd, with hope that it might guard\n    The dust of him whose virtues call'd it forth;\n    But 'tis a little space of earth that man,\n    Stretch'd out in death, is doom'd to occupy;\n    Still smaller space doth modest custom yield,\n    On sculptured tomb or tablet, to the claims\n    Of the deceased, or rights of the bereft.\n    'Tis well; and tho', the record overstepped\n    Those narrow bounds, yet on the printed page\n    Received, there may it stand, I trust, unblamed\n    As long as verse of mine shall steal from tears\n    Their bitterness, or live to shed a gleam\n    Of solace over one dejected thought.\n\n    In privately printed edition.\n\nProfessor Dowden quotes, from \"a slip of MS. in the poet's\nhand-writing,\" the following variation of these lines--\n\n    'Tis well, and if the Record in the strength\n    And earnestness of feeling, overpass'd\n    Those narrow limits and so miss'd its aim,\n    Yet will I trust that on the printed page\n    Received, it there may keep a place unblamed.\n\nED.\n\n[28] Lamb's indifference to the country \"was a sort of 'mock apparel,'\nin which it was his humour at times to invest himself.\" (H. N.\nColeridge, Supplement to the _Biographia Literaria_, p. 333.)--ED.\n\n[29] 1837.\n\n    Burned, and with ever-strengthening light, enshrined\n\n    In privately printed edition.\n\n[30] 1837.\n\n    By God, \u2026\n\n    In privately printed edition.\n\n[31] 1837.\n\n    \u2026 Our days pass on;\n\n    In privately printed edition.\n\n[32] 1837.\n\n    Together stood they witnessing of time\n\n    In privately printed edition.\n\n[33] 1837.\n\n    Yet, in all visitations, through all trials\n    Still they were faithful, like two goodly ships\n    Launch'd from the beach, \u2026\n\n    In privately printed edition.\n\n[34] Compare the testimony borne to Mary Lamb by Mr. Procter (Barry\nCornwall), and by Henry Crabb Robinson.--ED.\n\n[35] 1837.\n\n                            \u2026 The sacred tie\n    Is broken, to become more sacred still.\n\n    In privately printed edition.\n\nWordsworth originally meant to write an epitaph on Charles Lamb,\nbut his verse grew into an elegy of some length. A reference to the\ncircumstance of its \"composition\" will be found in one of his letters,\nin a later volume.--ED.\n\n\nEXTEMPORE EFFUSION UPON THE DEATH OF JAMES HOGG\n\nComposed 1835.--Published 1835\n\n[These verses were written extempore, immediately after reading a\nnotice of the Ettrick Shepherd's death, in the Newcastle paper, to the\nEditor of which I sent a copy for publication. The persons lamented\nin these verses were all either of my friends or acquaintance. In\nLockhart's _Life of Sir Walter Scott_, an account is given of my\nfirst meeting with him in 1803. How the Ettrick Shepherd and I became\nknown to each other has already been mentioned in these notes. He was\nundoubtedly a man of original genius, but of coarse manners and low\nand offensive opinions. Of Coleridge and Lamb I need not speak here.\nCrabbe I have met in London at Mr. Rogers's, but more frequently and\nfavourably at Mr. Hoare's upon Hampstead Heath. Every spring he used to\npay that family a visit of some length, and was upon terms of intimate\nfriendship with Mrs. Hoare, and still more with her daughter-in-law,\nwho has a large collection of his letters addressed to herself. After\nthe Poet's decease, application was made to her to give up these\nletters to his biographer, that they, or at least part of them, might\nbe given to the public. She hesitated to comply, and asked my opinion\non the subject. \"By no means,\" was my answer, grounded not upon any\nobjection there might be to publishing a selection from these letters,\nbut from an aversion I have always felt to meet idle curiosity by\ncalling back the recently departed to become the object of trivial\nand familiar gossip. Crabbe obviously for the most part preferred the\ncompany of women to that of men, for this among other reasons, that\nhe did not like to be put upon the stretch in general conversation:\naccordingly in miscellaneous society his _talk_ was so much below what\nmight have been expected from a man so deservedly celebrated, that to\nme it seemed trifling. It must upon other occasions have been of a\ndifferent character, as I found in our rambles together on Hampstead\nHeath, and not so much from a readiness to communicate his knowledge\nof life and manners as of natural history in all its branches. His\nmind was inquisitive, and he seems to have taken refuge from the\nremembrance of the distresses he had gone through, in these studies\nand the employments to which they led. Moreover, such contemplations\nmight tend profitably to counterbalance the painful truths which he had\ncollected from his intercourse with mankind. Had I been more intimate\nwith him, I should have ventured to touch upon his office as a minister\nof the Gospel, and how far his heart and soul were in it so as to make\nhim a zealous and diligent labourer: in poetry, though he wrote much\nas we all know, he assuredly was not so. I happened once to speak of\npains as necessary to produce merit of a certain kind which I highly\nvalued: his observation was--\"It is not worth while.\" You are quite\nright, thought I, if the labour encroaches upon the time due to teach\ntruth as a steward of the mysteries of God: if there be cause to fear\n_that_, write less: but, if poetry is to be produced at all, make\nwhat you do produce as good as you can. Mr. Rogers once told me that\nhe expressed his regret to Crabbe that he wrote in his later works so\nmuch less correctly than in his earlier. \"Yes,\" replied he, \"but then\nI had a reputation to make; now I can afford to relax.\" Whether it\nwas from a modest estimate of his own qualifications, or from causes\nless creditable, his motives for writing verse and his hopes and aims\nwere not so high as is to be desired. After being silent for more than\ntwenty years, he again applied himself to poetry, upon the spur of\napplause he received from the periodical publications of the day, as he\nhimself tells us in one of his prefaces. Is it not to be lamented that\na man who was so conversant with permanent truth, and whose writings\nare so valuable an acquisition to our country's literature, should have\n_required_ an impulse from such a quarter? Mrs. Hemans was unfortunate\nas a poetess in being obliged by circumstances to write for money, and\nthat so frequently and so much, that she was compelled to look out for\nsubjects wherever she could find them, and to write as expeditiously as\npossible. As a woman, she was to a considerable degree a spoilt child\nof the world. She had been early in life distinguished for talent, and\npoems of hers were published while she was a girl. She had also been\nhandsome in her youth, but her education had been most unfortunate.\nShe was totally ignorant of housewifery, and could as easily have\nmanaged the spear of Minerva as her needle. It was from observing these\ndeficiencies, that, one day while she was under my roof, I _purposely_\ndirected her attention to household economy, and told her I had\npurchased _Scales_ which I intended to present to a young lady as a\nwedding present; pointed out their utility (for her especial benefit)\nand said that no m\u00e9nage ought to be without them. Mrs. Hemans, not in\nthe least suspecting my drift, reported this saying, in a letter to a\nfriend at the time, as a proof of my simplicity. Being disposed to make\nlarge allowances for the faults of her education and the circumstances\nin which she was placed, I felt most kindly disposed towards her, and\ntook her part upon all occasions, and I was not a little affected\nby learning that after she withdrew to Ireland, a long and severe\nsickness raised her spirit as it depressed her body. This I heard from\nher most intimate friends, and there is striking evidence of it in a\npoem written and published not long before her death. These notices\nof Mrs. Hemans would be very unsatisfactory to her intimate friends,\nas indeed they are to myself, not so much for what is said, but what\nfor brevity's sake is left unsaid. Let it suffice to add, there was\nmuch sympathy between us, and, if opportunity had been allowed me to\nsee more of her, I should have loved and valued her accordingly; as it\nis, I remember her with true affection for her amiable qualities, and,\nabove all, for her delicate and irreproachable conduct during her long\nseparation from an unfeeling husband, whom she had been led to marry\nfrom the romantic notions of inexperienced youth. Upon this husband I\nnever heard her cast the least reproach, nor did I ever hear her even\nname him, though she did not wholly forbear to touch upon her domestic\nposition; but never so that any fault could be found with her manner of\nadverting to it. --I.F.]\n\nThis first appeared in _The Athen\u00e6um_, December 12, 1835, and in\nthe edition of 1837 it was included among the \"Epitaphs and Elegiac\nPieces.\"--ED.\n\n    When first, descending from the moorlands,\n    I saw the Stream of Yarrow glide\n    Along a bare and open valley,\n    The Ettrick Shepherd was my guide.[36]\n\n    When last along its banks I wandered,                              5\n    Through groves that had begun to shed\n    Their golden leaves upon the pathways,\n    My steps the Border-minstrel led.\n\n    The mighty Minstrel breathes no longer,[37]\n    'Mid mouldering ruins low he lies;[38]                            10\n    And death upon the braes of Yarrow,\n    Has closed the Shepherd-poet's eyes:[39]\n\n    Nor has the rolling year twice measured,\n    From sign to sign, its stedfast course,\n    Since every mortal power of Coleridge                             15\n    Was frozen at its marvellous source;[40]\n\n    The rapt One, of the godlike forehead,[41]\n    The heaven-eyed creature sleeps in earth:\n    And Lamb, the frolic and the gentle,\n    Has vanished from his lonely hearth.[42]                          20\n\n    Like clouds that rake the mountain-summits,[43]\n    Or waves that own no curbing hand,\n    How fast has brother followed brother,\n    From sunshine to the sunless land!\n\n    Yet I, whose lids from infant slumber[44]                         25\n    Were earlier raised, remain to hear\n    A timid voice, that asks in whispers,\n    \"Who next will drop and disappear?\"\n\n    Our haughty life is crowned with darkness,\n    Like London with its own black wreath,                            30\n    On which with thee, O Crabbe! forth-looking,\n    I gazed from Hampstead's breezy heath.\n\n    As if but yesterday departed,\n    Thou too art gone before;[45] but why,\n    O'er ripe fruit, seasonably gathered,                             35\n    Should frail survivors heave a sigh?\n\n    Mourn rather for that holy Spirit,\n    Sweet as the spring, as ocean deep;\n    For Her who, ere her summer faded,\n    Has sunk into a breathless sleep.[46]                             40\n\n    No more of old romantic sorrows,\n    For slaughtered Youth or love-lorn Maid!\n    With sharper grief is Yarrow smitten,\n    And Ettrick mourns with her their Poet dead.[47]\n\n[36] Compare _Yarrow Visited_ (September, 1814), vol. vi. p. 35.--ED.\n\n[37] Compare _Yarrow Revisited_ (1831), vol. vii. p. 278.--ED.\n\n[38] Scott died at Abbotsford, on the 21st September 1832, and was\nburied in Dryburgh Abbey.--ED.\n\n[39] Hogg died at Altrive, on the 21st November 1835.--ED.\n\n[40] Coleridge died at Highgate, on the 25th July 1834.--ED.\n\n[41] Compare the _Stanzas written in my Pocket Copy of Thomson's\n\"Castle of Indolence\"_ (vol. ii. p. 307)--\n\n    Profound his forehead was, though not severe.\n\nED.\n\n[42] Lamb died in London, on the 27th December 1834.--ED.\n\n[43] \"This expression is borrowed from a sonnet by Mr. G. Bell, the\nauthor of a small volume of poems lately printed at Penrith. Speaking\nof Skiddaw he says--\n\n    Yon dark cloud 'rakes,' and shrouds its noble brow.\"\n\n(Henry Reed, 1837.)--ED.\n\n[44] 1845.\n\n    \u2026 slumbers\n\n    1837.\n\n[45] George Crabbe died at Trowbridge, Wiltshire, on the 3rd of\nFebruary 1832.--ED.\n\n[46] Felicia Hemans died 16th May 1835.--ED.\n\n[47]\n\n    Grieve rather for that holy Spirit\n    Pure as the sky, as ocean deep;\n    For her who ere the summer faded\n    Has sunk into a breathless sleep.\n\n    No more of old romantic sorrows\n    For slaughtered Youth or love-lorn Maid!\n    With sharper grief is Yarrow smitten,\n    And Ettrick mourns her Shepherd Poet dead.\n\n    C.\n\n\nUPON SEEING A  DRAWING OF THE BIRD OF PARADISE IN AN ALBUM\n\nComposed 1835.--Published 1836\n\n[I cannot forbear to record that the last seven lines of this Poem were\ncomposed in bed during the night of the day on which my sister Sara\nHutchinson died about 6 P.M., and it was the thought of her innocent\nand beautiful life that, through faith, prompted the words----\n\n    On wings that fear no glance of God's pure sight,\n    No tempest from his breath.\n\nThe reader will find two poems on pictures of this bird among my\nPoems. I will here observe that in a far greater number of instances\nthan have been mentioned in these notes one poem has, as in this\ncase, grown out of another, either because I felt the subject had\nbeen inadequately treated, or that the thoughts and images suggested\nin course of composition have been such as I found interfered with\nthe unity indispensable to every work of art, however humble in\ncharacter.--I.F.]\n\nOne of the \"Poems of Sentiment and Reflection.\"--ED.\n\n    Who rashly strove thy Image to portray?\n    Thou buoyant minion of the tropic air;\n    How could he think of the live creature----gay\n    With a divinity of colours, drest\n    In all her brightness, from the dancing crest                      5\n    Far as the last gleam of the filmy train\n    Extended and extending to sustain\n    The motions that it graces----and forbear\n    To drop his pencil! Flowers of every clime\n    Depicted on these pages smile at time;                            10\n    And gorgeous insects copied with nice care\n    Are here, and likenesses of many a shell\n    Tossed ashore by restless waves,\n    Or in the diver's grasp fetched up from caves\n    Where sea-nymphs might be proud to dwell:                         15\n    But whose rash hand (again I ask) could dare,\n    'Mid casual tokens and promiscuous shows,\n    To circumscribe this Shape in fixed repose;\n    Could imitate for indolent survey,\n    Perhaps for touch profane,                                        20\n    Plumes that might catch, but cannot keep, a stain;\n    And, with cloud-streaks lightest and loftiest, share\n    The sun's first greeting, his last farewell ray!\n\n      Resplendent Wanderer! followed with glad eyes\n    Where'er her course; mysterious Bird!                             25\n    To whom, by wondering Fancy stirred,\n    Eastern Islanders have given\n    A holy name----the Bird of Heaven!\n    And even a title higher still,\n    The Bird of God![48] whose blessed will                           30\n    She seems performing as she flies\n    Over the earth and through the skies\n    In never-wearied search of Paradise----\n    Region that crowns her beauty with the name\n    She bears for _us_----for us how blest,                           35\n    How happy at all seasons, could like aim\n    Uphold our Spirits urged to kindred flight\n    On wings that fear no glance of God's pure sight,\n    No tempest from his breath, their promised rest\n    Seeking with indefatigable quest                                  40\n    Above a world that deems itself most wise\n    When most enslaved by gross realities!\n\n[48] Compare, in Robert Browning's poem on Guercino's picture of _The\nGuardian-Angel at Fano_----\n\n    Thou bird of God.\n\nED.\n\n\n\"DESPONDING FATHER! MARK THIS ALTERED BOUGH\"\n\nComposed 1835.--Published 1835\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Desponding Father! mark this altered bough,[49]\n    So beautiful of late, with sunshine warmed,\n    Or moist with dews; what more unsightly now,\n    Its blossoms shrivelled, and its fruit, if formed,\n    Invisible? yet Spring her genial brow                              5\n    Knits not o'er that discolouring and decay\n    As false to expectation. Nor fret thou\n    At like unlovely process in the May\n    Of human life: a Stripling's graces blow,\n    Fade and are shed, that from their timely fall                    10\n    (Misdeem it not a cankerous change) may grow\n    Rich mellow bearings, that for thanks shall call:\n    In all men, sinful is it to be slow\n    To hope----in Parents, sinful above all.\n\n[49] Compare _The Excursion_ (book iii. l. 649), and the sonnet (vol.\nvi. p. 72) beginning----\n\n    Surprised by joy----impatient as the Wind.\n\nED.\n\n\n\"FOUR FIERY STEEDS IMPATIENT OF THE REIN\"\n\nComposed 1835.--Published 1835\n\n[Suggested on the road between Preston and Lancaster where it first\ngives a view of the Lake country, and composed on the same day, on the\nroof of the coach.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Four fiery steeds impatient of the rein\n    Whirled us o'er sunless ground beneath a sky\n    As void of sunshine, when, from that wide plain,\n    Clear tops of far-off mountains we descry,\n    Like a Sierra of cerulean Spain,                                   5\n    All light and lustre. Did no heart reply?\n    Yes, there was One;--for One, asunder fly\n    The thousand links of that ethereal chain;\n    And green vales open out, with grove and field,\n    And the fair front of many a happy Home;                          10\n    Such tempting spots as into vision come\n    While Soldiers, weary of the arms they wield\n    And sick at heart[50] of strifeful Christendom,\n    Gaze on the moon by parting clouds revealed.\n\n[50] 1837.\n\n    While Soldiers, of the weapons that they wield\n    Weary, and sick of strifeful \u2026\n\n    1835.\n\n\nTO ----\n\nComposed 1835.--Published 1835\n\n[The fate of this poor Dove, as described, was told to me at Brinsop\nCourt, by the young lady to whom I have given the name of Lesbia.--I.F.]\n\n    [Miss not the occasion: by the forelock take\n    That subtle Power, the never-halting Time,\n    Lest a mere moment's putting-off should make\n    Mischance almost as heavy as a crime.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    \"Wait, prithee, wait!\" this answer Lesbia[51] threw\n    Forth to her Dove, and took no further heed.\n    Her eye was busy, while her fingers flew\n    Across the harp, with soul-engrossing speed;\n    But from that bondage when her thoughts were freed                 5\n    She rose, and toward the close-shut casement drew,\n    Whence the poor unregarded Favourite, true\n    To old affections, had been heard to plead\n    With flapping wing for entrance. What a shriek\n    Forced from that voice so lately tuned to a strain                10\n    Of harmony!----a shriek of terror, pain,\n    And self-reproach! for, from aloft, a Kite\n    Pounced,----and the Dove, which from its ruthless beak\n    She could not rescue, perished in her sight!\n\n[51] Miss Loveday Walker, daughter of the Rector of Brinsop. See the\nFenwick note to the next sonnet.--ED.\n\n\nROMAN ANTIQUITIES DISCOVERED AT BISHOPSTONE, HEREFORDSHIRE\n\nComposed 1835.--Published 1835\n\n[My attention to these antiquities was directed by Mr. Walker, son\nto the itinerant Eidouranian Philosopher. The beautiful pavement was\ndiscovered within a few yards of the front door of his parsonage, and\nappeared from the site (in full view of several hills upon which there\nhad formerly been Roman encampments) as if it might have been the\nvilla of the commander of the forces, at least such was Mr. Walker's\nconjecture.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    While poring Antiquarians search the ground\n    Upturned with curious pains, the Bard, a Seer,\n    Takes fire:----The men that have been reappear;\n    Romans for travel girt, for business gowned;\n    And some recline on couches, myrtle-crowned,                       5\n    In festal glee: why not? For fresh and clear,\n    As if its hues were of the passing year,\n    Dawns this time-buried pavement. From that mound\n    Hoards may come forth of Trajans, Maximins,\n    Shrunk into coins with all their warlike toil:                    10\n    Or a fierce impress issues with its foil\n    Of tenderness--the Wolf, whose suckling Twins\n    The unlettered ploughboy pities when he wins\n    The casual treasure from the furrowed soil.\n\n\nST. CATHERINE OF LEDBURY\n\nComposed 1835.--Published 1835\n\n[Written on a journey from Brinsop Court, Herefordshire.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    When human touch (as monkish books attest)\n    Nor was applied nor could be, Ledbury bells\n    Broke forth in concert flung adown the dells,\n    And upward, high as Malvern's cloudy crest;[52]\n    Sweet tones, and caught by a noble Lady blest                      5\n    To rapture! Mabel listened at the side\n    Of her loved mistress: soon the music died,\n    And Catherine said, Here I set up my rest.\n    Warned in a dream, the Wanderer long had sought\n    A home that by such miracle of sound                              10\n    Must be revealed:--she heard it now, or felt\n    The deep, deep joy of a confiding thought;\n    And there, a saintly Anchoress, she dwelt\n    Till she exchanged for heaven that happy ground.\n\n[52] The Ledbury bells are easily audible on the Malvern hills.--ED.\n\n\n\"BY A BLEST HUSBAND GUIDED, MARY CAME\"[53]\n\nPublished 1835\n\n[This lady was named Carleton; she, along with a sister, was brought\nup in the neighbourhood of Ambleside. The epitaph, a part of it at\nleast, is in the church at Bromsgrove, where she resided after her\nmarriage.--I.F.]\n\nOne of the \"Epitaphs and Elegiac Pieces.\"--ED.\n\n    By a blest Husband guided, Mary came\n    From nearest kindred, Vernon[54] her new name;\n    She came, though meek of soul, in seemly pride\n    Of happiness and hope, a youthful Bride.\n    O dread reverse! if aught _be_ so, which proves                    5\n    That God will chasten whom he dearly loves.\n    Faith bore her up through pains in mercy given,\n    And troubles that were each a step to Heaven:\n    Two Babes were laid in earth before she died;\n    A third now slumbers at the Mother's side;                        10\n    Its Sister-twin survives, whose smiles afford\n    A trembling solace to her widowed Lord.\n\n      Reader! if to thy bosom cling the pain\n    Of recent sorrow combated in vain;\n    Or if thy cherished grief have failed to thwart                   15\n    Time still intent on his insidious part,\n    Lulling the mourner's best good thoughts asleep,\n    Pilfering regrets we would, but cannot, keep;\n    Bear with Him--judge _Him_ gently who makes known\n    His bitter loss by this memorial Stone;                           20\n    And pray that in his faithful breast the grace\n    Of resignation find a hallowed place.\n\n[53] 1837.\n\nIn the edition of 1835 the title was \"Epitaph.\"\n\n[54] 1837.\n\n    From nearest kindred, \u2026\n\n    1835.\n\n\n\"OH WHAT A WRECK! HOW CHANGED IN MIEN AND SPEECH!\"\n\nComposed 1835.--Published 1838\n\n[The sad condition of poor Mrs. Southey[55] put me upon writing this.\nIt has afforded comfort to many persons whose friends have been\nsimilarly affected.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Oh what a Wreck! how changed in mien and speech!\n    Yet--though dread Powers, that work in mystery, spin\n    Entanglings of[56] the brain; though shadows stretch\n    O'er the chilled heart--reflect; far, far within\n    Hers is a holy Being, freed from Sin.                              5\n    She is not what she seems, a forlorn wretch,\n    But delegated Spirits comfort fetch\n    To Her from heights that Reason may not win.\n    Like Children, She is privileged to hold\n    Divine communion;[57] both do live and move,                      10\n    Whate'er to shallow Faith their ways unfold,\n    Inly illumined by Heaven's pitying love;\n    Love pitying innocence not long to last,\n    In them--in Her our sins and sorrows past.\n\n[55] Mrs. Southey died 16th November 1837. She had long been an\ninvalid. See Southey's _Life and Correspondence_, vol. vi. p. 347.--ED.\n\n[56] 1842.\n\n    \u2026 for \u2026\n\n    1838.\n\n[57] Compare a remark of Wordsworth's that he never saw those with\nmind unhinged, but he thought of the words, \"Life hid in God.\" It is a\ncurious oriental belief that idiots are in closer communion with the\nInfinite than the sane are.--ED.\n\n\n\n\n1836\n\nSo far as can be ascertained, only one sonnet was written by Wordsworth\nin 1836. The verses _To a Redbreast_, by his sister-in-law, Sarah\nHutchinson, may however be placed alongside of the sonnet addressed to\nher.--ED.\n\n\nNOVEMBER 1836\n\nComposed 1836.--Published 1837.\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Even so for me a Vision sanctified\n    The sway of Death; long ere mine eyes had seen\n    Thy countenance--the still rapture of thy mien--\n    When thou, dear Sister![58] wert become Death's Bride:\n    No trace of pain or languor could abide                            5\n    That change:--age on thy brow was smoothed--thy cold\n    Wan cheek at once was privileged to unfold\n    A loveliness to living youth denied.\n    Oh! if within me hope should e'er decline,\n    The lamp of faith, lost Friend! too faintly burn;                 10\n    Then may that heaven-revealing smile of thine,\n    The bright assurance, visibly return:\n    And let my spirit in that power divine\n    Rejoice, as, through that power, it ceased to mourn.\n\n[58] Sarah Hutchinson--Mrs. Wordsworth's sister--died at Rydal on the\n23rd June 1836. It was after her that the poet named one of the two\n\"heath-clad rocks\" referred to in the \"Poems on the naming of Places,\"\nand which he called respectively \"Mary-Point\" and \"Sarah-Point.\" In\n1827 he inscribed to her the sonnet beginning--\n\n    Excuse is needless when with love sincere,\n\nand the lines she wrote _To a Redbreast_, beginning--\n\n    Stay, little cheerful Robin! stay,\n\nwere published among Wordsworth's own poems.\n\nThe sonnet written in 1806, beginning--\n\n    Methought I saw the footsteps of a throne,\n\nwas, Wordsworth tells us, a great favourite with S. H. He adds, \"When\nI saw her lying in death I could not resist the impulse to compose the\nsonnet that follows it.\" (See vol. iv. p. 46.)\n\nIn a letter to Southey (unpublished), Wordsworth refers to her death,\nand adds: \"I saw her within an hour after her decease, in the silence\nand peace of death, with as heavenly an expression on her countenance\nas ever human creature had. Surely there is food for faith in these\nappearances: for myself, I can say that I have passed a wakeful night,\nmore in joy than in sorrow, with that blessed face before my eyes\nperpetually as I lay in bed.\"\n\n\nTO A REDBREAST--(IN SICKNESS)\n\nPublished 1842\n\n[Almost the only verses by our lamented sister Sara Hutchinson.--I.F.]\n\nOne of the \"Miscellaneous Poems.\"--ED.\n\n    Stay, little cheerful Robin! stay,\n      And at my casement sing,\n    Though it should prove a farewell lay\n      And this our parting spring.\n\n    Though I, alas! may ne'er enjoy                                    5\n      The promise in thy song;\n    A charm, _that_ thought can not destroy,\n      Doth to thy strain belong.\n\n    Methinks that in my dying hour\n      Thy song would still be dear,                                   10\n    And with a more than earthly power\n      My passing Spirit cheer.\n\n    Then, little Bird, this boon confer,\n      Come, and my requiem sing,\n    Nor fail to be the harbinger                                      15\n      Of everlasting Spring.\n\n                                     S.H.\n\n\n\n\n1837\n\nThe poems belonging to the year 1837 include the \"Memorials of a Tour\nin Italy\" with Henry Crabb Robinson in that year, and one or two\nadditional sonnets.--ED.\n\n\n\"SIX MONTHS TO SIX YEARS ADDED HE REMAINED\"\n\nPublished 1837\n\nOne of the \"Epitaphs and Elegiac Pieces.\"--ED.\n\n    Six months to six years added he remained\n    Upon this sinful earth, by sin unstained:\n    O blessed Lord! whose mercy then removed\n    A Child whom every eye that looked on loved;\n    Support us, teach us calmly to resign                              5\n    What we possessed, and now is wholly thine![59]\n\n[59] This refers to the poet's son Thomas, who died December 1, 1812.\nHe was buried in Grasmere churchyard, beside his sister Catherine; and\nWordsworth placed these lines upon his tombstone. They may have been\nwritten much earlier than 1836, probably in 1813, but it is impossible\nto ascertain the date, and they were not published till 1837.--ED.\n\n\nMEMORIALS OF A TOUR IN ITALY 1837\n\nComposed 1837.--Published 1842\n\n[During my whole life I had felt a strong desire to visit Rome and the\nother celebrated cities and regions of Italy, but did not think myself\njustified in incurring the necessary expense till I received from Mr.\nMoxon, the publisher of a large edition of my poems, a sum sufficient\nto enable me to gratify my wish without encroaching upon what I\nconsidered due to my family. My excellent friend H.C. Robinson readily\nconsented to accompany me, and in March 1837, we set off from London,\nto which we returned in August, earlier than my companion wished or\nI should myself have desired had I been, like him, a bachelor. These\nMemorials of that tour touch upon but a very few of the places and\nobjects that interested me, and, in what they do advert to, are for\nthe most part much slighter than I could wish. More particularly do I\nregret that there is no notice in them of the South of France, nor of\nthe Roman antiquities abounding in that district, especially of the\nPont de Degard, which, together with its situation, impressed me full\nas much as any remains of Roman architecture to be found in Italy.\nThen there was Vaucluse, with its Fountain, its Petrarch, its rocks of\nall seasons, its small plots of lawn in their first vernal freshness,\nand the blossoms of the peach and other trees embellishing the scene\non every side. The beauty of the stream also called forcibly for the\nexpression of sympathy from one who, from his childhood, had studied\nthe brooks and torrents of his native mountains. Between two and three\nhours did I run about climbing the steep and rugged crags from whose\nbase the water of Vaucluse breaks forth. \"Has Laura's Lover,\" often\nsaid I to myself, \"ever sat down upon this stone? or has his foot ever\npressed that turf?\" Some, especially of the female sex, would have felt\nsure of it: my answer was (impute it to my years) \"I fear, not.\" Is it\nnot in fact obvious that many of his love verses must have flowed, I\ndo not say from a wish to display his own talent, but from a habit of\nexercising his intellect in that way rather than from an impulse of his\nheart? It is otherwise with his Lyrical poems, and particularly with\nthe one upon the degradation of his country: there he pours out his\nreproaches, lamentations, and aspirations like an ardent and sincere\npatriot. But enough: it is time to turn to my own effusions such as\nthey are.--I.F.]\n\n\nTO HENRY CRABB ROBINSON[60]\n\n    Companion! by whose buoyant Spirit cheered,\n    In[61] whose experience trusting, day by day\n    Treasures I gained with zeal that neither feared\n    The toils nor felt the crosses of the way,\n    These records take, and happy should I be                          5\n    Were but the Gift a meet Return to thee\n    For kindnesses that never ceased to flow,\n    And prompt self-sacrifice to which I owe\n    Far more than any heart but mine can know.\n\n                                W. WORDSWORTH.\n\nRYDAL MOUNT, _Feb. 14th, 1842._\n\n[60] The following is the Itinerary of the Italian Tour of 1837,\nsupplied by Mr. Henry Crabb Robinson. (See _Memoirs of Wordsworth_,\nvol. ii. pp. 316, 317.) The spelling of the names of places is\nRobinson's.\n\n    March, 1837.\n\n    19. By steam to Calais.\n    20. Posting to Samer.\n    21. Posting to Granvilliers.\n    22. Through Beauvais to Paris.\n    26. To Fontainbleau.\n    27. Through Nemours to Cosne.\n    28. To Moulins.\n    29. To Tarare.\n    30. To Lyons.\n    31. Through Vienne to Tain.\n\n    April.\n\n    1. Through Valence to Orange.\n    2. To Avignon; to Vaucluse and back.\n    3, 4. By Pont du Gard to Nismes.\n    5, 6. By St. Remi to Marseilles.\n    7. To Toulon.\n    8. To Luc.\n    9. By Frejus to Cannes.\n    10, 11. To Nice.\n    12. Through Mentone to St. Remo.\n    13. Through Finale to Savone.\n    14-16. To Genoa.\n    17. To Chiaveri.\n    18. To Spezia.\n    19. By Carrara to Massa.\n    20. To Lucca.\n    21. To Pisa.\n    22. To Volterra.\n    23. By Castiglonacco and Sienna.\n    24. To Radicofani.\n    25. By Aquapendente to Viterbo.\n    26. To Rome.\n\n    May.\n\n    13. Excursion to Tivoli with Dr. Carlyle.\n    17-21. Excursion to Albano, etc., etc., with Miss Mackenzie.\n    23. To Terni.\n    24. After seeing the Falls, to Spoleto.\n    25. To Cortona and Perugia.\n    26. To Arezzo.\n    27. To Bibiena and Laverna.\n    28. To Camaldoli.\n    29. From Muselea to Ponte Sieve.\n    30. From Ponte Sieve to Val Ombrosa and Florence.\n\n    June.\n\n    6, 7. To Bologna.\n    8. Parma.\n    9. Through Piacenza to Milan.\n    11. To the Certosa and back.\n    12. To the Lake of Como and back.\n    13. To Bergamo.\n    14. To Pallazuola and Isco.\n    15. Excursion to Riveri and back.\n    16. To Brescia and Desinzano.\n    17. On Lake of Garda to Riva.\n    19. To Verona.\n    20. Vicenza.\n    21. Padua.\n    22. Venice.\n    28. To Logerone.\n    29. To Sillian.\n    30. Spittal (in Carinthia).\n\n    July.\n\n    1. Over Kazenberg to Tweng.\n    2. Through Werfen to Hallein.\n    3. Excursion to Konigsee.\n    4, 5. To Saltzburg.\n    6. To Ischl. A week's stay in the Salzkammer Gut, viz.--\n    8. Gmund.\n    9. Travenfalls and back.\n    10. Aussee.\n    11. Excursion to lakes, then to Hallstadt.\n    13. Through Ischl to St. Gilgin.\n    14. Through Salzburg to Trauenstein.\n    15. To Miesbach.\n    16. To Tegernsee and Holzkirken.\n    17. To Munich.\n    21. To Augsburg.\n    22. To Ulm.\n    23. To Stuttgard.\n    24. To Besigham.\n    25. To Heidelberg.\n    28. Through Worms to Mayence.\n    29. To Coblenz.\n    30. To Bonn.\n    31. Through Cologne to Aix-la-Chapelle.\n\n    August.\n\n    1. To Louvain\n    2. To Brussels.\n    3. To Antwerp.\n    4. To Liege.\n    5. Through Lille to Cassell.\n    6. Calais.\n    7. London.\n\n[61] 1845.\n\n    To \u2026\n\n    1842.\n\n\nThe Tour of which the following Poems are very inadequate remembrances\nwas shortened by report, too well founded, of the prevalence of Cholera\nat Naples. To make some amends for what was reluctantly left unseen\nin the South of Italy, we visited the Tuscan Sanctuaries among the\nApennines, and the principal Italian Lakes among the Alps. Neither\nof those lakes, nor of Venice, is there any notice in these Poems,\nchiefly because I have touched upon them elsewhere. See, in particular,\n_Descriptive Sketches_, \"Memorials of a Tour on the Continent in 1820,\"\nand a Sonnet upon the extinction of the Venetian Republic.--W.W.\n\n\nI\n\nMUSINGS NEAR AQUAPENDENTE\n\nAPRIL, 1837\n\n                        [Not the less\n    Had his sunk eye kindled at those dear words\n    That spake of bards and minstrels.\n\nHis, Sir Walter Scott's, eye, _did_ in fact kindle at them, for the\nlines, \"Places forsaken now\" and the two that follow, were adopted from\na poem of mine which nearly forty years ago was _in part_ read to him,\nand he never forgot them.\n\n                        Old Helvellyn's brow\n    Where once together, in his day of strength,\n    We stood rejoicing.\n\nSir Humphry Davy was with us at the time. We had ascended from\nPatterdale, and I could not but admire the vigour with which Scott\nscrambled along that horn of the mountain called \"Striding Edge.\" Our\nprogress was necessarily slow, and was beguiled by Scott's telling many\nstories and amusing anecdotes, as was his custom. Sir H. Davy would\nhave probably been better pleased if other topics had occasionally been\ninterspersed, and some discussion entered upon: at all events he did\nnot remain with us long at the top of the mountain, but left us to find\nour way down its steep side together into the Vale of Grasmere, where,\nat my cottage, Mrs. Scott was to meet us at dinner.\n\n                        With faint smile\n    \u2026\n    He said, \"When I am there, although 'tis fair,\n    'Twill be another Yarrow.\"\n\nSee among these notes the one on _Yarrow Revisited_.\n\n    A few short steps (painful they were) apart\n    From Tasso's Convent-haven, and retired grave.\n\nThis, though introduced here, I did not know till it was told me at\nRome by Miss Mackenzie of Seaforth, a lady whose friendly attentions\nduring my residence at Rome I have gratefully acknowledged with\nexpressions of sincere regret that she is no more. Miss M. told me\nthat she accompanied Sir Walter to the Janicular Mount, and, after\nshowing him the grave of Tasso in the church upon the top, and a mural\nmonument, there erected to his memory, they left the church and stood\ntogether on the brow of the hill overlooking the City of Rome: his\ndaughter Anne was with them, and she, naturally desirous, for the sake\nof Miss Mackenzie especially, to have some expression of pleasure from\nher father, half reproached him for showing nothing of that kind either\nby his looks or voice: \"How can I,\" replied he, \"having only one leg\nto stand upon, and that in extreme pain!\" so that the prophecy was more\nthan fulfilled.\n\n    Over waves rough and deep.\n\nWe took boat near the lighthouse at the point of the right horn of\nthe bay which makes a sort of natural port for Genoa; but the wind\nwas high, and the waves long and rough, so that I did not feel quite\nrecompensed by the view of the city, splendid as it was, for the danger\napparently incurred. The boatman (I had only one) encouraged me saying\nwe were quite safe, but I was not a little glad when we gained the\nshore, though Shelley and Byron--one of them at least, who seemed to\nhave courted agitation from any quarter--would have probably rejoiced\nin such a situation: more than once I believe were they both in extreme\ndanger even on the lake of Geneva. Every man, however, has his fears\nof some kind or other; and no doubt they had theirs: of all men whom I\nhave ever known, Coleridge had the most of passive courage in bodily\nperil, but no one was so easily cowed when moral firmness was required\nin miscellaneous conversation or in the daily intercourse of social\nlife.\n\n    How lovely robed in forenoon light and shade,\n    Each ministering to each, didst thou appear,\n    Savona.\n\nThere is not a single bay along this beautiful coast that might not\nraise in a traveller a wish to take up his abode there, each as it\nsucceeds seems more inviting than the other; but the desolated convent\non the cliff in the bay of Savona struck my fancy most; and had I, for\nthe sake of my own health or that of a dear friend, or any other cause,\nbeen desirous of a residence abroad, I should have let my thoughts\nloose upon a scheme of turning some part of this building into a\nhabitation provided as far as might be with English comforts. There is\nclose by it a row or avenue, I forget which, of tall cypresses. I could\nnot forbear saying to myself--\"What a sweet family walk, or one for\nlonely musings, would be found under the shade!\" but there, probably,\nthe trees remained little noticed and seldom enjoyed.\n\n    This flowering broom's dear neighbourhood.\n\nThe broom is a great ornament through the months of March and April to\nthe vales and hills of the Apennines, in the wild parts of which it\nblows in the utmost profusion, and of course successively at different\nelevations as the season advances. It surpasses ours in beauty and\nfragrance,[62] but, speaking from my own limited observations only,\nI cannot affirm the same of several of their wild spring flowers,\nthe primroses in particular, which I saw not unfrequently but thinly\nscattered and languishing compared to ours.\n\nThe note at the end of this poem, upon the Oxford movement, was\nentrusted to my friend, Mr. Frederick Faber.[63] I told him what I\nwished to be said, and begged that, as he was intimately acquainted\nwith several of the Leaders of it, he would express my thought in the\nway least likely to be taken amiss by them. Much of the work they are\nundertaking was grievously wanted, and God grant their endeavours may\ncontinue to prosper as they have done.--I.F.]\n\n    Ye Apennines! with all your fertile vales\n    Deeply embosomed, and your winding shores\n    Of either sea, an Islander by birth,\n    A Mountaineer by habit, would resound\n    Your praise, in meet accordance with your claims                   5\n    Bestowed by Nature, or from man's great deeds\n    Inherited:--presumptuous thought!--it fled\n    Like vapour, like a towering cloud, dissolved.\n    Not, therefore, shall my mind give way to sadness;--\n    Yon snow-white torrent-fall, plumb down it drops                  10\n    Yet ever hangs or seems to hang in air,\n    Lulling the leisure of that high perched town,\n    AQUAPENDENTE, in her lofty site\n    Its neighbour and its namesake--town, and flood\n    Forth flashing out of its own gloomy chasm                        15\n    Bright sunbeams--the fresh verdure of this lawn\n    Strewn with grey rocks, and on the horizon's verge,\n    O'er intervenient waste, through glimmering haze,\n    Unquestionably kenned, that cone-shaped hill\n    With fractured summit,[64] no indifferent sight                   20\n    To travellers, from such comforts as are thine,\n    Bleak Radicofani![65] escaped with joy--\n    These are before me; and the varied scene\n    May well suffice, till noon-tide's sultry heat\n    Relax, to fix and satisfy the mind                                25\n    Passive yet pleased. What! with this Broom in flower\n    Close at my side! She bids me fly to greet\n    Her sisters, soon like her to be attired\n    With golden blossoms opening at the feet\n    Of my own Fairfield.[66] The glad greeting given,                 30\n    Given with a voice and by a look returned\n    Of old companionship, Time counts not minutes\n    Ere, from accustomed paths, familiar fields,\n    The local Genius hurries me aloft,\n    Transported over that cloud-wooing hill,                          35\n    Seat Sandal, a fond suitor of the clouds,[67]\n    With dream-like smoothness, to Helvellyn's top,[68]\n    There to alight upon crisp moss and range,\n    Obtaining ampler boon, at every step,\n    Of visual sovereignty--hills multitudinous,                       40\n    (Not Apennine can boast of fairer) hills\n    Pride of two nations, wood and lake and plains,\n    And prospect right below of deep coves shaped[69]\n    By skeleton arms, that, from the mountain's trunk\n    Extended, clasp the winds, with mutual moan                       45\n    Struggling for liberty, while undismayed\n    The shepherd struggles with them. Onward thence\n    And downward by the skirt of Greenside fell,[70]\n    And by Glenridding-screes,[71] and low Glencoign,[72]\n    Places forsaken now, though[73] loving still                      50\n    The muses, as they loved them in the days\n    Of the old minstrels and the border bards.--\n    But here am I fast bound; and let it pass,\n    The simple rapture;--who that travels far\n    To feed his mind with watchful eyes could share                   55\n    Or wish to share it?--One there surely was,\n    \"The Wizard of the North,\" with anxious hope\n    Brought to this genial climate, when disease\n    Preyed upon body and mind--yet not the less\n    Had his sunk eye kindled at those dear words                      60\n    That spake of bards and minstrels; and his spirit\n    Had flown with mine to old Helvellyn's brow,\n    Where once together, in his day of strength,\n    We stood rejoicing,[74] as if earth were free\n    From sorrow, like the sky above our heads.                        65\n\n      Years followed years, and when, upon the eve\n    Of his last going from Tweed-side, thought turned,\n    Or by another's sympathy was led,\n    To this bright land, Hope was for him no friend,\n    Knowledge no help; Imagination shaped                             70\n    No promise. Still, in more than ear-deep seats,\n    Survives for me, and cannot but survive\n    The tone of voice which wedded borrowed words\n    To sadness not their own, when, with faint smile\n    Forced by intent to take from speech its edge,                    75\n    He said, \"When I am there, although 'tis fair,\n    'Twill be another Yarrow.\"[75] Prophecy\n    More than fulfilled, as gay Campania's shores\n    Soon witnessed, and the city of seven hills,\n    Her sparkling fountains, and her mouldering tombs;                80\n    And more than all, that Eminence[76] which showed\n    Her splendours, seen, not felt, the while he stood\n    A few short steps (painful they were) apart\n    From Tasso's Convent-haven, and retired grave.[77]\n\n      Peace to their Spirits! why should Poesy                        85\n    Yield to the lure of vain regret, and hover\n    In gloom on wings with confidence outspread\n    To move in sunshine?--Utter thanks, my Soul!\n    Tempered with awe, and sweetened by compassion\n    For them who in the shades of sorrow dwell,                       90\n    That I--so near the term to human life\n    Appointed by man's common heritage,[78]\n    Frail as the frailest, one withal (if that\n    Deserve a thought) but little known to fame--\n    Am free to rove where Nature's loveliest looks,                   95\n    Art's noblest relics, history's rich bequests,\n    Failed to reanimate and but feebly cheered\n    The whole world's Darling--free to rove at will\n    O'er high and low, and if requiring rest,\n    Rest from enjoyment only.\n                              Thanks poured forth                    100\n    For what thus far hath blessed my wanderings, thanks\n    Fervent but humble as the lips can breathe\n    Where gladness seems a duty--let me guard\n    Those seeds of expectation which the fruit\n    Already gathered in this favoured Land                           105\n    Enfolds within its core. The faith be mine,\n    That He who guides and governs all, approves\n    When gratitude, though disciplined to look\n    Beyond these transient spheres, doth wear a crown\n    Of earthly hope put on with trembling hand;                      110\n    Nor is least pleased, we trust, when golden beams,\n    Reflected through the mists of age, from hours\n    Of innocent delight, remote or recent,\n    Shoot but a little way--'tis all they can--\n    Into the doubtful future. Who would keep                         115\n    Power must resolve to cleave to it through life,\n    Else it deserts him, surely as he lives.\n    Saints would not grieve nor guardian angels frown\n    If one--while tossed, as was my lot to be,\n    In a frail bark urged by two slender oars                        120\n    Over waves rough and deep,[79] that, when they broke,\n    Dashed their white foam against the palace walls\n    Of Genoa the superb--should there be led\n    To meditate upon his own appointed tasks,\n    However humble in themselves, with thoughts                      125\n    Raised and sustained by memory of Him\n    Who oftentimes within those narrow bounds\n    Rocked on the surge, there tried his spirit's strength\n    And grasp of purpose, long ere sailed his ship\n    To lay a new world open.\n                              Nor less prized                        130\n    Be those impressions which incline the heart\n    To mild, to lowly, and to seeming weak,\n    Bend that way her desires. The dew, the storm--\n    The dew whose moisture fell in gentle drops\n    On the small hyssop destined to become,                          135\n    By Hebrew ordinance devoutly kept,\n    A purifying instrument--the storm\n    That shook on Lebanon the cedar's top,\n    And as it shook, enabling the blind roots\n    Further to force their way, endowed its trunk                    140\n    With magnitude and strength fit to uphold\n    The glorious temple--did alike proceed\n    From the same gracious will, were both an offspring\n    Of bounty infinite.\n                          Between Powers that aim\n    Higher to lift their lofty heads, impelled                       145\n    By no profane ambition, Powers that thrive\n    By conflict, and their opposites, that trust\n    In lowliness--a mid-way tract there lies\n    Of thoughtful sentiment for every mind\n    Pregnant with good. Young, Middle-aged, and Old,                 150\n    From century on to century, must have known\n    The emotion--nay, more fitly were it said--\n    The blest tranquillity that sunk so deep\n    Into my spirit, when I paced, enclosed\n    In Pisa's Campo Santo,[80] the smooth floor                      155\n    Of its Arcades paved with sepulchral slabs,[81]\n    And through each window's open fret-work looked\n    O'er the blank Area of sacred earth\n    Fetched from Mount Calvary,[82] or haply delved\n    In precincts nearer to the Saviour's tomb,                       160\n    By hands of men, humble as brave, who fought\n    For its deliverance--a capacious field\n    That to descendants of the dead it holds\n    And to all living mute memento breathes,\n    More touching far than aught which on the walls                  165\n    Is pictured, or their epitaphs can speak,\n    Of the changed City's long-departed power,\n    Glory, and wealth, which, perilous as they are,\n    Here did not kill, but nourished, Piety.\n    And, high above that length of cloistral roof,                   170\n    Peering in air and backed by azure sky,\n    To kindred contemplations ministers\n    The Baptistery's dome,[83] and that which swells\n    From the Cathedral pile;[84] and with the twain\n    Conjoined in prospect mutable or fixed                           175\n    (As hurry on in eagerness the feet,\n    Or pause) the summit of the Leaning-tower.[85]\n    Nor[86] less remuneration waits on him\n    Who having left the Cemetery stands\n    In the Tower's shadow, of decline and fall                       180\n    Admonished not without some sense of fear,\n    Fear that soon vanishes before the sight\n    Of splendour unextinguished, pomp unscathed,\n    And beauty unimpaired. Grand in itself,\n    And for itself, the assemblage, grand and fair                   185\n    To view, and for the mind's consenting eye\n    A type of age in man, upon its front\n    Bearing the world-acknowledged evidence\n    Of past exploits, nor fondly after more\n    Struggling against the stream of destiny,                        190\n    But with its peaceful majesty content.\n    --Oh what a spectacle at every turn\n    The Place unfolds, from pavement skinned with moss,\n    Or grass-grown spaces, where the heaviest foot\n    Provokes no echoes, but must softly tread;                       195\n    Where Solitude with Silence paired stops short\n    Of Desolation, and to Ruin's scythe\n    Decay submits not.\n                          But where'er my steps\n    Shall wander, chiefly let me cull with care\n    Those images of genial beauty, oft                               200\n    Too lovely to be pensive in themselves\n    But by reflection made so, which do best\n    And fitliest serve to crown with fragrant wreaths\n    Life's cup when almost filled with years, like mine.\n    --How lovely robed in forenoon light and shade,                  205\n    Each ministering to each, didst thou appear\n    Savona,[87] Queen of territory fair\n    As aught that marvellous coast thro' all its length\n    Yields to the Stranger's eye. Remembrance holds\n    As a selected treasure thy one cliff,                            210\n    That, while it wore for melancholy crest\n    A shattered Convent, yet rose proud to have\n    Clinging to its steep sides a thousand herbs\n    And shrubs, whose pleasant looks gave proof how kind\n    The breath of air can be where earth had else                    215\n    Seemed churlish. And behold, both far and near,\n    Garden and field all decked with orange bloom,\n    And peach and citron, in Spring's mildest breeze\n    Expanding; and, along the smooth shore curved\n    Into a natural port, a tideless sea,                             220\n    To that mild breeze with motion and with voice\n    Softly responsive; and, attuned to all\n    Those vernal charms of sight and sound, appeared\n    Smooth space of turf which from the guardian fort\n    Sloped seaward, turf whose tender April green,                   225\n    In coolest climes too fugitive, might even here\n    Plead with the sovereign Sun for longer stay\n    Than his unmitigated beams allow,\n    Nor plead in vain, if beauty could preserve,\n    From mortal change, aught that is born on earth                  230\n    Or doth on time depend.\n                            While on the brink\n    Of that high Convent-crested cliff I stood,\n    Modest Savona! over all did brood\n    A pure poetic Spirit--as the breeze,\n    Mild--as the verdure, fresh--the sunshine, bright--              235\n    Thy gentle Chiabrera![88]--not a stone,\n    Mural or level with the trodden floor,\n    In Church or Chapel, if my curious quest\n    Missed not the truth, retains a single name\n    Of young or old, warrior, or saint, or sage,                     240\n    To whose dear memories his sepulchral verse[89]\n    Paid simple tribute, such as might have flowed\n    From the clear spring of a plain English heart,\n    Say rather, one in native fellowship\n    With all who want not skill to couple grief                      245\n    With praise, as genuine admiration prompts.\n    The grief, the praise, are severed from their dust,\n    Yet in his page the records of that worth\n    Survive, uninjured;--glory then to words,\n    Honour to word-preserving Arts, and hail                         250\n    Ye kindred local influences that still,\n    If Hope's familiar whispers merit faith,\n    Await my steps when they the breezy height\n    Shall range of philosophic Tusculum;[90]\n    Or Sabine vales[91] explored inspire a wish                      255\n    To meet the shade of Horace by the side\n    Of his Bandusian fount;[92]--or I invoke\n    His presence to point out the spot where once\n    He sate, and eulogized with earnest pen\n    Peace, leisure, freedom, moderate desires;                       260\n    And all the immunities of rural life\n    Extolled, behind Vacuna's crumbling fane.[93]\n    Or let me loiter, soothed with what is given\n    Nor asking more, on that delicious Bay,[94]\n    Parthenope's Domain--Virgilian haunt,                            265\n    Illustrated with never-dying verse,[95]\n    And, by the Poet's laurel-shaded tomb,[96]\n    Age after age to Pilgrims from all lands\n    Endeared.\n                And who--if not a man as cold\n    In heart as dull in brain--while pacing ground                   270\n    Chosen by Rome's legendary Bards, high minds\n    Out of her early struggles well inspired\n    To localize heroic acts--could look\n    Upon the spots with undelighted eye,\n    Though even to their last syllable the Lays                      275\n    And very names of those who gave them birth\n    Have perished?--Verily, to her utmost depth,\n    Imagination feels what Reason fears not\n    To recognize, the lasting virtue lodged\n    In those bold fictions that, by deeds assigned                   280\n    To the Valerian, Fabian, Curian Race,\n    And others like in fame, created Powers\n    With attributes from History derived,\n    By Poesy irradiate, and yet graced,\n    Through marvellous felicity of skill,                            285\n    With something more propitious to high aims\n    Than either, pent within her separate sphere,\n    Can oft with justice claim.\n                                And not disdaining\n    Union with those primeval energies\n    To virtue consecrate, stoop ye from your height                  290\n    Christian Traditions! at my Spirit's call\n    Descend, and, on the brow of ancient Rome\n    As she survives in ruin, manifest\n    Your glories mingled with the brightest hues\n    Of her memorial halo, fading, fading,                            295\n    But never to be extinct while Earth endures.\n    O come, if undishonoured by the prayer,\n    From all her Sanctuaries!--Open for my feet\n    Ye Catacombs, give to mine eyes a glimpse\n    Of the Devout, as, 'mid your glooms convened                     300\n    For safety, they of yore enclasped the Cross[97]\n    On knees that ceased from trembling, or intoned\n    Their orisons with voices half-suppressed,\n    But sometimes heard, or fancied to be heard,\n    Even at this hour.\n                      And thou Mamertine prison,[98]                 305\n    Into that vault receive me from whose depth\n    Issues, revealed in no presumptuous vision,\n    Albeit lifting human to divine,\n    A saint, the Church's Rock, the mystic Keys\n    Grasped in his hand;[99] and lo! with upright sword              310\n    Prefiguring his own impendent doom,\n    The Apostle of the Gentiles; both prepared\n    To suffer pains with heathen scorn and hate\n    Inflicted;--blessed Men, for so to Heaven\n    They follow their dear Lord!\n                                  Time flows--nor winds,             315\n    Nor stagnates, nor precipitates his course,\n    But many a benefit borne upon his breast\n    For human-kind sinks out of sight, is gone,\n    No one knows how; nor seldom is put forth\n    An angry arm that snatches good away,                            320\n    Never perhaps to reappear. The Stream\n    Has to our generation brought and brings\n    Innumerable gains; yet we, who now\n    Walk in the light of day, pertain full surely\n    To a chilled age, most pitiably shut out                         325\n    From that which _is_ and actuates, by forms,\n    Abstractions, and by lifeless fact to fact\n    Minutely linked with diligence uninspired,\n    Unrectified, unguided, unsustained,\n    By godlike insight. To this fate is doomed                       330\n    Science, wide-spread and spreading still as be\n    Her conquests, in the world of sense made known.\n    So with the internal mind it fares; and so\n    With morals, trusting, in contempt or fear\n    Of vital principle's controlling law,                            335\n    To her purblind guide Expediency; and so\n    Suffers religious faith. Elate with view\n    Of what is won, we overlook or scorn\n    The best that should keep pace with it, and must,\n    Else more and more the general mind will droop,                  340\n    Even as if bent on perishing. There lives\n    No faculty within us which the Soul\n    Can spare,[100] and humblest earthly Weal demands,\n    For dignity not placed beyond her reach,\n    Zealous co-operation of all means                                345\n    Given or acquired, to raise us from the mire,\n    And liberate our hearts from low pursuits.\n    By gross Utilities enslaved we need\n    More of ennobling impulse from the past,\n    If to the future aught of good must come                         350\n    Sounder and therefore holier than the ends\n    Which, in the giddiness of self-applause,\n    We covet as supreme. O grant the crown\n    That Wisdom wears, or take his treacherous staff\n    From Knowledge!--If the Muse, whom I have served                 355\n    This day, be mistress of a single pearl\n    Fit to be placed in that pure diadem;\n    Then, not in vain, under these chesnut boughs\n    Reclined, shall I have yielded up my soul\n    To transports from the secondary founts                          360\n    Flowing of time and place, and paid to both\n    Due homage; nor shall fruitlessly have striven,\n    By love of beauty moved, to enshrine in verse\n    Accordant meditations, which in times\n    Vexed and disordered, as our own, may shed                       365\n    Influence, at least among a scattered few,\n    To soberness of mind and peace of heart\n    Friendly; as here to my repose hath been\n    This flowering broom's dear neighbourhood,[101] the light\n    And murmur issuing from yon pendent flood,                       370\n    And all the varied landscape. Let us now\n    Rise, and to-morrow greet magnificent Rome.[102]\n\n[62] Wordsworth himself, his nephew tells us, had no sense of smell\n(see the _Memoirs_, by his nephew Christopher, vol. ii. p. 322).--ED.\n\n[63] Afterwards Father Faber, priest of the Oratory of St. Philip\nNeri.--ED.\n\n[64] Monte Amiata,--ED.\n\n[65] On the old high road from Siena to Rome.--ED.\n\n[66] The mountain between Rydal Head and Helvellyn.--ED.\n\n[67] Seat Sandal is the mountain between Tongue Ghyll and Grisedale\nTarn on the south and east, and the Dunmail Raise road on the west.--ED.\n\n[68] Compare _The Eclipse of the Sun_, l. 78, in \"Memorials of a Tour\non the Continent in 1820\" (vol. vi. p. 345).--ED.\n\n[69] Keppelcove, Nethermost cove, and the cove in which Red Tarn lies\nbounded by the \"skeleton arms\" of Striding Edge and Swirrel Edge.\nCompare _Fidelity_, l. 17, vol. iii. p. 45--\n\n    It was a cove, a huge recess,\n    That keeps, till June, December's snow.\n\nED.\n\n[70] Descending to Ullswater from Helvellyn, Greenside Fell and Mines\nare passed.--ED.\n\n[71] The Glenridding Screes are bold rocks on the left as you descend\nHelvellyn to Patterdale.--ED.\n\n[72] Glencoign is an offshoot of the Patterdale valley between\nGlenridding and Goldbarrow.--ED.\n\n[73] 1845.\n\n    \u2026 but \u2026\n\n    1842.\n\n[74] See the Fenwick note.--ED.\n\n[75] These words were quoted to me from _Yarrow Unvisited_, by Sir\nWalter Scott, when I visited him at Abbotsford, a day or two before his\ndeparture for Italy: and the affecting condition in which he was when\nhe looked upon Rome from the Janicular Mount, was reported to me by a\nlady who had the honour of conducting him thither.--W.W. 1842. See also\nthe Fenwick note to this poem, and compare Lockhart's _Memoirs of the\nLife of Sir Walter Scott_ (chapter lxxx. vol. x. p. 104).--ED.\n\n[76] The Janicular Mount.--ED.\n\n[77] See the Fenwick note prefixed to this poem.--ED.\n\n[78] He was then sixty-seven years of age.--ED.\n\n[79] See the Fenwick note.--ED.\n\n[80] The Campo Santo, or Burial Ground, founded by Archbishop Ubaldo\n(1188-1200).--ED.\n\n[81] \"There are forty-three flat arcades, resting on forty-four\npilasters.\u2026 In the interior there is a spacious hall, the open\nround-arched windows of which, with their beautiful tracery, sixty-two\nin number, look out upon a green quadrangle.\u2026 The walls are covered\nwith frescoes by the Tuscan School of the fourteenth and fifteenth\ncenturies, below which is a collection of Roman, Etruscan, and\nmediaeval sculptures.\u2026 The tombstones of persons interred here form the\npavement.\" (Baedeker's _Northern Italy_, p. 324.)--ED.\n\n[82] Ubaldo conveyed hither fifty-three ship-loads of earth from Mount\nCalvary, in the Holy Land, in order that the dead might repose in holy\nground.--ED.\n\n[83] The Baptistery in Pisa was begun in 1153 by Diotisalvi, and\ncompleted in 1278. It is a circular structure, covered by a conical\ndome, 190 feet high.--ED.\n\n[84] The Cathedral of Pisa is a basilica, built in 1063, in the Tuscan\nstyle, and has an elliptical dome.--ED.\n\n[85] The Campanile, or Clock-Tower, rises in eight stories to the\nheight of 179 feet, and (from its oblique position) is known as the\nLeaning-Tower.--ED.\n\n[86] 1845.\n\n    Not \u2026\n\n    1842.\n\n[87] See the Fenwick note to this poem. Savona is a town on the Gulf of\nGenoa, capital of the Montenotte Department under Napoleon.--ED.\n\n[88] The theatre in Savona is dedicated to Chiabrera, who was a native\nof the place.--ED.\n\n[89] If any English reader should be desirous of knowing how far I\nam justified in thus describing the epitaphs of Chiabrera, he will\nfind translated specimens of them in this Volume, under the head of\n\"Epitaphs and Elegiac Pieces.\"--W.W. 1842.\n\n[90] Tusculum was the birthplace of the elder Cato, and the residence\nof Cicero.--ED.\n\n[91] \"Satis beatus unicis Sabinis.\" _Odes_, ii. 18, 14.--ED.\n\n[92] See Horace, _Odes_, iii. 13.--ED.\n\n[93] See Horace, _Epistles_, i. 10, 49--\n\n    Haec tibi dictabam post fanum putre Vacunae.\n\nVacuna was a Sabine divinity. She had a sanctuary near Horace's Villa.\n(Compare Pliny, _Nat. Hist._ iii. 42, 47.) A traveller in Italy writes:\n\"Following a path along the brink of the torrent Digentia, we passed\na towering rock, on which once stood Vacuna's shrine.\" See also Ovid,\n_Fasti_, vi. 307.--ED.\n\n[94] The Bay of Naples. Neapolis (the new city) received its ancient\nname of Parthenope from one of the Sirens, whose body was said to have\nbeen washed ashore in that bay. Sil. 12, 33.--ED.\n\n[95] See _Georgics_, iv. 564.--ED.\n\n[96] Virgil died at Brundusium, but his remains were carried to his\nfavourite residence, Naples, and were buried by the side of the road\nleading to Puteoli--the Via Puteolana. His tomb is still pointed out\nnear Posilipo,--close to the sea, and about half way from Naples to\nPuteoli, the _Scuola di Virgilio_.\n\n\"The monument, now called the tomb of Virgil, is not on the road\nwhich passes through the tunnel of Posilipo; but if the Via Puteolana\nascended the hill of Posilipo, as it may have done, the situation of\nthe monument would agree very well with the description of Donatus.\"\n(George Long, in Smith's _Dictionary of Greek and Roman Biography_.)\n\nThe inscription said to have been placed on the tomb was as follows:--\n\n    Mantua me genuit, Calabri rapuere, tenet nunc\n      Parthenope. Cecini pascua, rura, duces.\n\nED.\n\n[97] The catacombs were subterranean chambers and passages, usually\ncut out of the solid rock, and used as places of burial, or of refuge.\nThe early Christians made use of the catacombs in the Appian Way for\nworship, as well as for sepulture.--ED.\n\n[98] The Carcer Mamertinus,--one of the most ancient Roman\nstructures,--overhung the Forum, as Livy tells us, \"imminens foro,\"\nunderneath the Capitoline hill. It still exists, and is entered from\nthe sacristy of the church of S. Giuseppe de Falagnami, to the left\nof the arch of Severus. It was originally a well (the _Tullianum_ of\nLivy), and afterwards a prison, in which Jugurtha was starved to death,\nand Catiline's accomplices perished. There are two chambers in the\nprison, one beneath the other; the lower-most containing, in its rock\nfloor, a spring, which rises nearly to the surface. For the legend\nconnected with it see the next note.--ED.\n\n[99] According to the legend, St. Peter, who was imprisoned in the\n_Carcer Mamertinus_ under Nero, caused this spring to flow miraculously\nin order to baptize his jailors. Hence the building is called _S.\nPietro in Carcere._--ED.\n\n[100] Compare \"Despondency Corrected,\" _The Excursion_, book iv. l.\n1058--\n\n    Within the soul a faculty abides, etc.\n\nED.\n\n[101] See the Fenwick note.--ED.\n\n[102] It would be ungenerous not to advert to the religious movement\nthat, since the composition of these verses in 1837, has made itself\nfelt, more or less strongly, throughout the English Church;--a movement\nthat takes, for its first principle, a devout deference to the voice of\nChristian antiquity. It is not my office to pass judgment on questions\nof theological detail; but my own repugnance to the spirit and system\nof Romanism has been so repeatedly and, I trust, feelingly expressed,\nthat I shall not be suspected of a leaning that way, if I do not join\nin the grave charge, thrown out, perhaps in the heat of controversy,\nagainst the learned and pious men to whose labours I allude. I speak\napart from controversy; but, with strong faith in the moral temper\nwhich would elevate the present by doing reverence to the past, I would\ndraw cheerful auguries for the English Church from this movement, as\nlikely to restore among us a tone of piety more earnest and real than\nthat produced by the mere formalities of the understanding, refusing,\nin a degree, which I cannot but lament, that its own temper and\njudgment shall be controlled by those of antiquity.--W.W. 1842.\n\n\nII\n\nTHE PINE OF MONTE MARIO[103] AT ROME\n\n[Sir George Beaumont told me that, when he first visited Italy,\npine-trees of this species abounded, but that on his return thither,\nwhich was more than thirty years after, they had disappeared from many\nplaces where he had been accustomed to admire them, and had become\nrare all over the country, especially in and about Rome. Several Roman\nvillas have within these few years passed into the hands of foreigners,\nwho, I observed with pleasure, have taken care to plant this tree,\nwhich in course of years will become a great ornament to the city\nand to the general landscape. May I venture to add here, that having\nascended the Monte Mario, I could not resist embracing the trunk of\nthis interesting monument of my departed friend's feelings for the\nbeauties of nature, and the power of that art which he loved so much,\nand in the practice of which he was so distinguished?--I.F.]\n\n    I saw far off the dark top of a Pine\n    Look like a cloud--a slender stem the tie\n    That bound it to its native earth--poised high\n    'Mid evening hues, along the horizon line,\n    Striving in peace each other to outshine.                          5\n    But when I learned the Tree was living there,\n    Saved from the sordid axe by Beaumont's care,[104]\n    Oh, what a gush of tenderness was mine!\n    The rescued Pine-tree, with its sky so bright\n    And cloud-like beauty, rich in thoughts of home,                  10\n    Death-parted friends, and days too swift in flight,\n    Supplanted the whole majesty of Rome\n    (Then first apparent from the Pincian Height)[105]\n    Crowned with St. Peter's everlasting dome.[106]\n\n[103] The Monte Mario is to the north-west of Rome, beyond the\nJaniculus and the Vatican. The view from the summit embraces Rome, the\nCampagna, and the sea. It is capped by the villa Millini, in which the\n\"magnificent solitary pine-tree\" of this sonnet still stands, amidst\nits cypress plantations.--ED.\n\n[104] \"It was Mr. Theed, the sculptor, who informed us of the pine-tree\nbeing the gift of Sir George Beaumont.\" H.C. Robinson. (See _Memoirs of\nWordsworth_, by his nephew, vol. ii. p. 330.)--ED.\n\n[105] From the _Mons Pincius_, \"collis hortorum,\" where were the\ngardens of Lucullus, there is a remarkable view of modern Rome.--ED.\n\n[106] Within a couple of hours of my arrival at Rome, I saw from\nMonte Pincio, the Pine tree as described in the sonnet; and, while\nexpressing admiration at the beauty of its appearance, I was told by\nan acquaintance of my fellow-traveller, who happened to join us at the\nmoment, that a price had been paid for it by the late Sir G. Beaumont,\nupon condition that the proprietor should not act upon his known\nintention of cutting it down.--W.W. 1842.\n\n\nIII\n\nAT ROME\n\n[Sight is at first sight a sad enemy to imagination and to those\npleasures belonging to old times with which some exertions of that\npower will always mingle: nothing perhaps brings this truth home to\nthe feelings more than the city of Rome; not so much in respect to the\nimpression made at the moment when it is first seen and looked at as\na whole, for then the imagination may be invigorated and the mind's\neye quickened; but when particular spots or objects are sought out,\ndisappointment is I believe invariably felt. Ability to recover from\nthis disappointment will exist in proportion to knowledge, and the\npower of the mind to reconstruct out of fragments and parts, and to\nmake details in the present subservient to more adequate comprehension\nof the past.--I.F.]\n\n    Is this, ye Gods, the Capitolian Hill?\n    Yon petty Steep in truth the fearful Rock,\n    Tarpeian named of yore,[107] and keeping still\n    That name, a local Phantom proud to mock\n    The Traveller's expectation?--Could our Will\n    Destroy the ideal Power within, 'twere done\n    Thro' what men see and touch,--slaves wandering on,\n    Impelled by thirst of all but Heaven-taught skill.\n    Full oft, our wish obtained, deeply we sigh;\n    Yet not unrecompensed are they who learn,                         10\n    From that depression raised, to mount on high\n    With stronger wing, more clearly to discern\n    Eternal things; and, if need be, defy\n    Change, with a brow not insolent, though stern.\n\n[107] The Tarpeian rock, from which those condemned to death were\nhurled, is not now precipitous, as it used to be: the ground having\nbeen much raised by successive heaps of ruin.--ED.\n\n\nIV\n\nAT ROME--REGRETS--IN ALLUSION TO NIEBUHR AND OTHER MODERN HISTORIANS\n\n    Those old credulities, to nature dear,\n    Shall they no longer bloom upon the stock\n    Of History, stript naked as a rock\n    'Mid a dry desert? What is it we hear?\n    The glory of Infant Rome must disappear,[108]                      5\n    Her morning splendours vanish, and their place\n    Know them no more. If Truth, who veiled her face\n    With those bright beams yet hid it not, must steer\n    Henceforth a humbler course perplexed and slow;\n    One solace yet remains for us who came                            10\n    Into this world in days when story lacked\n    Severe research, that in our hearts we know\n    How, for exciting youth's heroic flame,\n    Assent is power, belief the soul of fact.\n\n[108] Niebuhr, in his Lectures on Roman History (1826-29), was one of\nthe first to point out the legendary character of much of the earlier\nhistory, and its \"historical impossibility.\" He explained the way\nin which much of it had originated in family and national vanity,\netc.--ED.\n\n\nV\n\nCONTINUED\n\n    Complacent Fictions were they, yet the same\n    Involved a history of no doubtful sense,\n    History that proves by inward evidence\n    From what a precious source of truth it came.\n    Ne'er could the boldest Eulogist have dared                        5\n    Such deeds to paint, such characters to frame,\n    But for coeval sympathy prepared\n    To greet with instant faith their loftiest claim.\n    None but a noble people could have loved\n    Flattery in Ancient Rome's pure-minded style:                     10\n    Not in like sort the Runic Scald was moved;\n    He, nursed 'mid savage passions that defile\n    Humanity, sang feats that well might call\n    For the blood-thirsty mead of Odin's riotous Hall.\n\n\nVI\n\nPLEA FOR THE HISTORIAN\n\n    Forbear to deem the Chronicler unwise,\n    Ungentle, or untouched by seemly ruth,\n    Who, gathering up all that Time's envious tooth\n    Has spared of sound and grave realities,\n    Firmly rejects those dazzling flatteries,                          5\n    Dear as they are to unsuspecting Youth,\n    That might have drawn down Clio from the skies\n    To vindicate the majesty of truth.\n    Such was her office while she walked with men,[109]\n    A Muse, who,[110] not unmindful of her Sire                       10\n    All-ruling Jove, whate'er the[111] theme might be\n    Revered her Mother, sage Mnemosyne,\n    And taught her faithful servants how the lyre\n    Should[112] animate, but not mislead, the pen.[113]\n\n[109] Clio, daughter of Zeus and Mnemosyne, the first-born of the\nMuses, presided over History. It was her office to record the actions\nof illustrious heroes.--ED.\n\n[110] 1845.\n\n    Her rights to claim, and vindicate the truth.\n    Her faithful Servants while she walked with men\n    Were they who, \u2026\n\n    1842.\n\n[111] 1845.\n\n    \u2026 their \u2026\n\n    1842.\n\n[112] 1845.\n\n    And, at the Muse's will, invoked the lyre\n    To animate, \u2026\n\n    1842.\n\n[113]\n\n    Quem virum--lyra--\n    --sumes celebrare Clio?\n\n    W. W. 1842.\n\n\nVII\n\nAT ROME\n\n[I have a private interest in this Sonnet, for I doubt whether it\nwould ever have been written but for the lively picture given me by\nAnna Ricketts of what she had witnessed of the indignation and sorrow\nexpressed by some Italian noblemen of their acquaintance upon the\nsurrender, which circumstances had obliged them to make, of the best\nportion of their family mansions to strangers.--I.F.]\n\n    They--who have seen the noble Roman's scorn\n    Break forth at thought of laying down his head,\n    When the blank day is over, garreted\n    In his ancestral palace, where, from morn\n    To night, the desecrated floors are worn                           5\n    By feet of purse-proud strangers; they--who have read\n    In one meek smile, beneath a peasant's shed,\n    How patiently the weight of wrong is borne;\n    They--who have heard some learned Patriot treat[114]\n    Of freedom, with mind grasping the whole theme                    10\n    From ancient Rome, downwards through that bright dream\n    Of Commonwealths, each city a starlike seat\n    Of rival glory; they--fallen Italy--\n    Nor must, nor will, nor can, despair of Thee!\n\n\nVIII\n\nNEAR ROME, IN SIGHT OF ST. PETER'S\n\n    Long has the dew been dried on tree and lawn;\n    O'er man and beast a not unwelcome boon\n    Is shed, the languor of approaching noon;\n    To shady rest withdrawing or withdrawn\n    Mute are all creatures, as this couchant fawn,                     5\n    Save insect-swarms that hum in air afloat,\n    Save that the Cock is crowing, a shrill note,\n    Startling and shrill as that which roused the dawn.\n    --Heard in that hour, or when, as now, the nerve\n    Shrinks from the note[115] as from a mis-timed thing,             10\n    Oft for a holy warning may it serve,\n    Charged with remembrance of _his_ sudden sting,\n    His bitter tears, whose name the Papal Chair\n    And yon resplendent Church are proud to bear.\n\n[114] 1845.\n\n    They--who have heard thy lettered sages treat\n\n    1842.\n\n[115] 1845.\n\n      \u2026 voice \u2026\n\n    1842.\n\n\nIX\n\nAT ALBANO[116]\n\n[This Sonnet is founded on simple fact, and was written to enlarge,\nif possible, the views of those who can see nothing but evil in the\nintercessions countenanced by the Church of Rome. That they are in\nmany respects lamentably pernicious must be acknowledged; but, on the\nother hand, they who reflect, while they see and observe, cannot but\nbe struck with instances which will prove that it is a great error to\ncondemn in all cases such mediation as purely idolatrous. This remark\nbears with especial force upon addresses to the Virgin.--I.F.]\n\n    Days passed--and Monte Calvo would not clear\n    His head from mist; and, as the wind sobbed through\n    Albano's dripping Ilex avenue,[117]\n    My dull forebodings in a Peasant's ear\n    Found casual vent. She said, \"Be of good cheer;                    5\n    Our yesterday's procession did not sue\n    In vain; the sky will change to sunny blue,\n    Thanks to our Lady's grace.\" I smiled to hear,\n    But not in scorn:--the Matron's Faith may lack\n    The heavenly sanction needed to ensure                            10\n    Fulfilment; but, we trust, her upward track[118]\n    Stops not at this low point, nor wants the lure\n    Of flowers the Virgin without fear may own,\n    For by her Son's blest hand the seed was sown.\n\n[116] Albano, 10 miles south-east of Rome, is a small town and\nepiscopal residence, a favourite autumnal resort of Roman citizens. It\nis on the site of the ruins of the villa of Pompey. Monte Carlo (the\nMonte Calvo of this sonnet) is the ancient _Mons Latialis_, 3127 feet\nhigh. At its summit a convent of Passionist Monks occupies the site of\nthe ancient temple of Jupiter.--ED.\n\n[117] The ilex-grove of the Villa Doria is one of the most marked\nfeatures of Albano.--ED.\n\n[118] 1845.\n\n    Its own fulfilment; but her upward track\n\n    1842.\n\n\nX\n\n\"NEAR ANIO'S STREAM, I SPIED A GENTLE DOVE\"\n\n    Near Anio's stream,[119] I spied a gentle Dove\n    Perched on an olive branch, and heard her cooing\n    'Mid new-born blossoms that soft airs were wooing,\n    While all things present told of joy and love.\n    But restless Fancy left that olive grove                           5\n    To hail the exploratory Bird renewing\n    Hope for the few, who, at the world's undoing,\n    On the great flood were spared to live and move.\n    O bounteous Heaven! signs true as dove and bough\n    Brought to the ark are coming evermore,                           10\n    Given though we seek them not, but, while we plough[120]\n    This sea of life without a visible shore,\n    Do neither promise ask nor grace implore\n    In what alone is ours, the living Now.[121]\n\n[119] The Anio joins the Tiber north of Rome, flowing from the\nnorth-east past Tivoli.--ED.\n\n[120] 1845.\n\n    Even though men seek them not, but, while they plough\n\n    1842.\n\n[121] 1845.\n\n    \u2026 the vouchsafed Now.\n\n    1842.\n\n\nXI\n\nFROM THE ALBAN HILLS, LOOKING TOWARDS ROME\n\n    Forgive, illustrious Country! these deep sighs,\n    Heaved less for thy bright plains and hills bestrown\n    With monuments decayed or overthrown,\n    For all that tottering stands or prostrate lies,\n    Than for like scenes in moral vision shown,                        5\n    Ruin perceived for keener sympathies;\n    Faith crushed, yet proud of weeds, her gaudy crown\n    Virtues laid low, and mouldering energies.\n    Yet why prolong this mournful strain?--Fallen Power,\n    Thy fortunes, twice exalted,[122] might provoke                   10\n    Verse to glad notes prophetic of the hour\n    When thou, uprisen, shalt break thy double yoke,\n    And enter, with prompt aid from the Most High,\n    On the third stage of thy great destiny.[123]\n\n[122] The ancient Classic period, and that of the Renaissance.--ED.\n\n[123] This period seems to have been already entered. Compare Mrs.\nBrowning's \"Poems before Congress,\" _passim_.--ED.\n\n\nXII\n\nNEAR THE LAKE OF THRASYMENE\n\n    When here with Carthage Rome to conflict came,[124]\n    An earthquake, mingling with the battle's shock,\n    Checked not its rage;[125] unfelt the ground did rock,\n    Sword dropped not, javelin kept its deadly aim.--\n    Now all is sun-bright peace. Of that day's shame,                  5\n    Or glory, not a vestige seems to endure,\n    Save in this Rill that took from blood the name[126]\n    Which yet it bears, sweet Stream! as crystal pure.\n    So may all trace and sign of deeds aloof\n    From the true guidance of humanity,                               10\n    Thro' Time and Nature's influence, purify\n    Their spirit; or, unless they for reproof\n    Or warning serve, thus let them all, on ground\n    That gave them being, vanish to a sound.\n\n[124] The Carthaginian general Hannibal defeated the Roman Consul C.\nFlaminius, near the lacus Trasimenus, 217 B.C., with a loss of 15,000\nmen. (See Livy, book xxii. 4, etc.)--ED.\n\n[125] Compare _Hannibal, A Historical Drama_, by the late Professor\nJohn Nichol, act II. scene vi. p. 107--\n\n                            Here shall shepherds tell\n    To passing travellers, when we are dust,\n    How, by the shores of reedy Thrasymene,\n    We fought and conquered, while the earthquake shook\n    The walls of Rome.\n\nED.\n\n[126] Sanguinetto.--W.W. 1845.\n\n\nXIII\n\nNEAR THE SAME LAKE\n\n    For action born, existing to be tried,\n    Powers manifold we have that intervene\n    To stir the heart that would too closely screen\n    Her peace from images to pain allied.\n    What wonder if at midnight, by the side                            5\n    Of Sanguinetto or broad Thrasymene,[127]\n    The clang of arms is heard, and phantoms glide,\n    Unhappy ghosts in troops by moonlight seen;\n    And singly thine, O vanquished Chief![128] whose corse,\n    Unburied, lay hid under heaps of slain:                           10\n    But who is He?--the Conqueror. Would he force\n    His way to Rome? Ah, no,--round hill and plain\n    Wandering, he haunts, at fancy's strong command,\n    This spot--his shadowy death-cup in his hand.[129]\n\n[127] Lake Thrasymene is the largest of the Etrurian lakes, being ten\nmiles in length and three in breadth.--ED.\n\n[128] C. Flaminius.--ED.\n\n[129] After the battle of Lake Thrasymene, Hannibal did not push on to\nRome, but turned through the Apennines to Apulia, just as subsequently\nafter the battle of Cannas he remained inactive.--ED.\n\n\nXIV\n\nTHE CUCKOO AT LAVERNA[130]\n\nMAY 25TH 1837\n\n[Among a thousand delightful feelings connected in my mind with\nthe voice of the cuckoo, there is a personal one which is rather\nmelancholy. I was first convinced that age had rather dulled my\nhearing, by not being able to catch the sound at the same distance as\nthe younger companions of my walks; and of this failure I had a proof\nupon the occasion that suggested these verses. I did not hear the sound\ntill Mr. Robinson had twice or thrice directed my attention to it.]\n\n    List--'twas the Cuckoo.--O with what delight\n    Heard I that voice! and catch it now, though faint,[131]\n    Far off and faint, and melting into air,\n    Yet not to be mistaken. Hark again!\n    Those louder cries give notice that the Bird,                      5\n    Although invisible as Echo's self,[132]\n    Is wheeling hitherward. Thanks, happy Creature,\n    For this unthought-of greeting!\n                                      While allured\n    From vale to hill, from hill to vale led on,\n    We have pursued, through various lands, a long                    10\n    And pleasant course; flower after flower has blown,\n    Embellishing the ground that gave them birth\n    With aspects novel to my sight; but still\n    Most fair, most welcome, when they drank the dew\n    In a sweet fellowship with kinds beloved,                         15\n    For old remembrance sake. And oft--where Spring\n    Display'd her richest blossoms among files\n    Of orange-trees bedecked with glowing fruit\n    Ripe for the hand, or under a thick shade\n    Of Ilex, or, if better suited to the hour,                        20\n    The lightsome Olive's twinkling canopy--[133]\n    Oft have I heard the Nightingale and Thrush\n    Blending as in a common English grove\n    Their love-songs; but, where'er my feet might roam,\n    Whate'er assemblages of new and old,                              25\n    Strange and familiar, might beguile the way,\n    A gratulation from that vagrant Voice\n    Was wanting;--and most happily till now.\n\n    For see, Laverna! mark the far-famed Pile,\n    High on the brink of that precipitous rock,[134]                  30\n    Implanted like a Fortress, as in truth\n    It is, a Christian Fortress, garrisoned\n    In faith and hope, and dutiful obedience,\n    By a few Monks, a stern society,\n    Dead to the world and scorning earth-born joys.                   35\n    Nay--though the hopes that drew, the fears that drove,\n    St. Francis, far from Man's resort, to abide\n    Among these sterile heights of Apennine, [135]\n    Bound him, nor, since he raised yon House, have ceased\n    To bind his spiritual Progeny, with rules                         40\n    Stringent as flesh can tolerate and live;[136]\n    His milder Genius (thanks to the good God\n    That made us) over those severe restraints\n    Of mind, that dread heart-freezing discipline,\n    Doth sometimes here predominate, and works                        45\n    By unsought means for gracious purposes;\n    For earth through heaven, for heaven, by changeful earth,\n    Illustrated, and mutually endeared.\n\n      Rapt though He were above the power of sense,\n    Familiarly, yet out of the cleansed heart                         50\n    Of that once sinful Being overflowed\n    On sun, moon, stars, the nether elements,\n    And every shape of creature they sustain,\n    Divine affections; and with beast and bird\n    (Stilled from afar--such marvel story tells--                     55\n    By casual outbreak of his passionate words,\n    And from their own pursuits in field or grove\n    Drawn to his side by look or act of love\n    Humane, and virtue of his innocent life)\n    He wont to hold companionship so free,                            60\n    So pure, so fraught with knowledge and delight,\n    As to be likened in his Followers' minds\n    To that which our first Parents, ere the fall\n    From their high state darkened the Earth with fear,\n    Held with all Kinds in Eden's blissful bowers.                    65\n\n      Then question not that, 'mid the austere Band,\n    Who breathe the air he breathed, tread where he trod,\n    Some true Partakers of his loving spirit\n    Do still survive,[137] and, with those gentle hearts\n    Consorted, Others, in the power, the faith,                       70\n    Of a baptized imagination, prompt\n    To catch from Nature's humblest monitors\n    Whate'er they bring of impulses sublime.\n\n      Thus sensitive must be the Monk, though pale\n    With fasts, with vigils worn, depressed by years,                 75\n    Whom in a sunny glade I chanced to see,\n    Upon a pine-tree's storm-uprooted trunk,\n    Seated alone, with forehead sky-ward raised,\n    Hands clasped above the crucifix he wore\n    Appended to his bosom, and lips closed                            80\n    By the joint pressure of his musing mood\n    And habit of his vow. That ancient Man--\n    Nor haply less the Brother whom I marked,\n    As we approached the Convent gate, aloft\n    Looking far forth from his aerial cell,                           85\n    A young Ascetic--Poet, Hero, Sage,\n    He might have been, Lover belike he was--\n    If they received into a conscious ear\n    The notes whose first faint greeting startled me,\n    Whose sedulous iteration thrilled with joy                        90\n    My heart--may have been moved like me to think,\n    Ah! not like me who walk in the world's ways,\n    On the great Prophet, styled _the Voice of One_\n    _Crying amid the wilderness_, and given,\n    Now that their snows must melt, their herbs and flowers           95\n    Revive, their obstinate winter pass away,\n    That awful name to Thee, thee, simple Cuckoo,\n    Wandering in solitude, and evermore\n    Foretelling and proclaiming, ere thou leave\n    This thy last haunt beneath Italian skies                        100\n    To carry thy glad tidings over heights\n    Still loftier, and to climes more near the Pole.\n\n      Voice of the Desert, fare-thee-well; sweet Bird!\n    If that substantial title please thee more,\n    Farewell!--but go thy way, no need hast thou                     105\n    Of a good wish sent after thee; from bower\n    To bower as green, from sky to sky as clear,\n    Thee gentle breezes waft--or airs that meet\n    Thy course and sport around thee softly fan--\n    Till Night, descending upon hill and vale,                       110\n    Grants to thy mission a brief term of silence,\n    And folds thy pinions up in blest repose.\n\n[130] Laverna is a corruption of _Alverna_ (now called Alverniac). It\nis about five or six hours' walk from Camaldoli, on a height of the\nApennines, not far from the sources of the Anio. To reach it, \"the\nsouthern height of the Monte Valterona is ascended as far as the chapel\nof St. Romaiald; then a descent is made to Moggiona, beyond which the\npath turns to the left, traversing a long and fatiguing succession of\ngorges and <DW72>s; the path at the base of the mountain is therefore\npreferable. The market town of Soci in the valley of the Archiano\nis first reached, then the profound valley of the Corsaline; beyond\nit rises a blunted cone, on which the path ascends in windings to a\nstony plain with marshy meadows. Above this rises the abrupt sandstone\nmass of the _Vernia_, to the height of 850 feet. On its S.W. <DW72>,\none-third of the way up, and 3906 feet above the sea-level, is seen a\nwall with small windows, the oldest part of the monastery, built in\n1218 by St. Francis of Assisi. The church dates from 1284.\u2026 One of the\ngrandest points is the _Penna della Vernia_ (4796 feet), the ridge of\nthe Vernia, also known as _l'Apennino_, the 'rugged rock between the\nsources of the Tiber and Anio,' as it is called by Dante (_Paradiso_,\nii. 106).\u2026 Near the monastery are the _Luoghi Santi_, a number of\ngrottos and rock-hewn chambers in which St. Francis once lived.\" (See\nBaedeker's _Northern Italy_, 1886, p. 463.)\n\n\"The Monte Alverno, or Monte della Verni is situated on the border\nof Tuscany, near the sources of the Tiber and Anio, not far from the\nCastle of Chiusi, where Orlando lived.\" (Mrs. Oliphant's _Francis of\nAssisi_, chap. xvi. p. 248.)\n\nSee also Herzog's _Real-Encyklop\u00e4die f\u00fcr protestantische Theologie und\nKirche_, vol. iv. p. 655.--ED.\n\n[131] Compare _To the Cuckoo_, II. 3, 4 (vol. ii. p. 289)--\n\n                      \u2026 Bird,\n    Or but a wandering Voice?\n\nED.\n\n[132] Compare _To the Cuckoo_, l. 15 (vol. ii. p. 290)--\n\n    No bird, but an invisible thing.\n\nED.\n\n[133] From the difference in the colour of each side of the leaf,\na grove of olives when _wind-tossed_ is pre-eminently a \"twinkling\ncanopy.\"--ED.\n\n[134] See note, p. 67.--ED.\n\n[135] St. Francis of Assisi, founder of the order of Friars Minors,\nafter establishing numerous monasteries in Italy, Spain, and France,\nresigned his office and retired to this, one of the highest of the\nApennine heights. See note, p. 67. He was canonised in 1230. Henry\nCrabb Robinson tells us, \"It was at Laverna that he\" [W.W.] \"led me to\nexpect that he had found a subject on which he could write, and that\nwas the love which birds bore to St. Francis. He repeated to me a short\ntime afterwards a few lines, which I do not recollect amongst those\nhe has written on St. Francis in this poem. On the journey, one night\nonly I heard him in bed composing verses, and on the following day I\noffered to be his amanuensis; but I was not patient enough, I fear, and\nhe did not employ me a second time. He made inquiries for St. Francis's\nbiography, as if he would dub him his Leibheiliger (body-saint), as\nGoethe (saying that every one must have one) declared St. Philip Neri\nto be his.\" (See the _Memoirs of William Wordsworth_, by his nephew,\nvol. ii. p. 331)--ED.\n\n[136] The characteristic feature of the Franciscan order was its vow\nof Poverty, and Francis desired that it should be taken in the most\nrigorous sense, viz. that no individual member of the fraternity,\nnor the fraternity itself, should be allowed to possess any property\nwhatsoever, even in things necessary to human use.--ED.\n\n[137] The members of the Franciscan order were the Stoics of\nChristendom. The order has been powerful, and of great service to\nthe Roman Church--alike in literature, and in practical action and\nenterprise.--ED.\n\n\nXV\n\nAT THE CONVENT OF CAMALDOLI\n\nThis famous sanctuary was the original establishment of Saint Romualdo\n(or Rumwald, as our ancestors saxonised the name) in the 11th century,\nthe ground (campo) being given by a Count Maldo. The Camaldolensi,\nhowever, have spread wide as a branch of Benedictines, and may\ntherefore be classed among the _gentlemen_ of the monastic orders. The\nsociety comprehends two orders, monks and hermits; symbolised by their\narms, two doves drinking out of the same cup. The monastery in which\nthe monks here reside is beautifully situated, but a large unattractive\nedifice, not unlike a factory. The hermitage is placed in a loftier and\nwilder region of the forest. It comprehends between 20 and 30 distinct\nresidences, each including for its single hermit an inclosed piece of\nground and three very small apartments. There are days of indulgence\nwhen the hermit may quit his cell, and when old age arrives, he\ndescends from the mountain and takes his abode among the monks.\n\nMy companion had, in the year 1831, fallen in with the monk, the\nsubject of these two sonnets, who showed him his abode among the\nhermits. It is from him that I received the following[138] particulars.\nHe was then about 40 years of age, but his appearance was that of an\nolder man. He had been a painter by profession, but on taking orders\nchanged his name from Santi to Raffaello, perhaps with an unconscious\nreference as well to the great Sanzio d'Urbino as to the archangel.\nHe assured my friend that he had been 13 years in the hermitage and\nhad never known melancholy or ennui. In the little recess for study\nand prayer, there was a small collection of books. \"I read only,\" said\nhe, \"books of asceticism and mystical theology.\" On being asked the\nnames of the most famous[139] mystics, he enumerated _Scaramelli_, _San\nGiovanni della Croce_, _St. Dionysius the Areopagite_ (supposing the\nwork which bears his name to be really his),[140] and with peculiar\nemphasis _Ricardo di San Vittori_. The works of _Saint Theresa_ are\nalso in high repute among ascetics.[141] These names may interest some\nof my readers.\n\nWe heard that Raffaello was then living in the convent; my friend\nsought in vain to renew his acquaintance with him. It was probably a\nday of seclusion. The reader will perceive that these sonnets were\nsupposed to be written when he was a young man.--W.W. 1842.\n\nThe monastery of Camaldoli is on the highest point of the hills near\nNaples (1476 feet), and commands one of the finest views in Italy.--ED.\n\n    Grieve for the Man who hither came bereft,\n    And seeking consolation from above;\n    Nor grieve the less that skill to him was left\n    To paint this picture of his lady-love:\n    Can she, a blessed saint, the work approve?                        5\n    And O, good Brethren of the cowl, a thing\n    So fair, to which with peril he must cling,\n    Destroy in pity, or with care remove.\n    That bloom--those eyes--can they assist to bind\n    Thoughts that would stray from Heaven? The dream must cease       10\n    To be; by Faith, not sight, his soul must live;\n    Else will the enamoured Monk too surely find\n    How wide a space can part from inward peace\n    The most profound repose his cell can give.\n\n[138] 1845.\n\n    received these particulars.\n\n1842.\n\n[139] 1845.\n\n    famous Italian mystics,\n\n1842.\n\n[140] 1845.\n\n    _San Dionysia_, _Areopagitica_, and with\n\n1842.\n\n[141] 1845.\n\n    are among ascetics in high repute, but she was a Spaniard.\n\n1842.\n\n\nXVI\n\nCONTINUED\n\n    The world forsaken, all its busy cares\n    And stirring interests shunned with desperate flight,\n    All trust abandoned in the healing might\n    Of virtuous action; all that courage dares,\n    Labour accomplishes, or patience bears--                           5\n    Those helps rejected, they, whose minds perceive\n    How subtly works man's weakness, sighs may heave\n    For such a One beset with cloistral snares.\n    Father of Mercy! rectify his view,\n    If with his vows this object ill agree;                           10\n    Shed over it thy grace, and thus subdue[142]\n    Imperious passion in a heart set free:--\n    That earthly love may to herself be true,\n    Give him a soul that cleaveth unto thee.\n\n[142] 1845.\n\n    \u2026 and so subdue\n\n    1842.\n\n\nXVII\n\nAT THE EREMITE OR UPPER CONVENT OF CAMALDOLI\n\n    What aim had they, the Pair of Monks, in size[143]\n    Enormous, dragged, while side by side they sate,\n    By panting steers up to this convent gate?\n    How, with empurpled cheeks and pampered eyes,\n    Dare they confront the lean austerities                            5\n    Of Brethren, who, here fixed, on Jesu wait\n    In sackcloth, and God's anger deprecate\n    Through all that humbles flesh and mortifies?\n    Strange contrast!--verily the world of dreams,\n    Where mingle, as for mockery combined,                            10\n    Things in their very essences at strife,\n    Shows not a sight incongruous as the extremes\n    That everywhere, before the thoughtful mind,\n    Meet on the solid ground of waking life.[144]\n\n[143] In justice to the Benedictines of Camaldoli, by whom strangers\nare so hospitably entertained, I feel obliged to notice, that I saw\namong them no other figures at all resembling, in size and complexion,\nthe two Monks described in this Sonnet. What was their office, or the\nmotive which brought them to this place of mortification, which they\ncould not have approached without being carried in this or some other\nway, a feeling of delicacy prevented me from inquiring. An account has\nbefore been given of the hermitage they were about to enter. It was\nvisited by us towards the end of the month of May; yet snow was lying\nthick under the pine-trees, within a few yards of the gate.--W.W. 1842.\n\n[144] See note, pp. 72, 73.--ED.\n\n\nXVIII\n\nAT VALLOMBROSA[145]\n\n[I must confess, though of course I did not acknowledge it in the few\nlines I wrote in the Strangers' book kept at the convent, that I was\nsomewhat disappointed at Vallombrosa. I had expected, as the name\nimplies, a deep and narrow valley overshadowed by enclosing hills; but\nthe spot where the convent stands is in fact not a valley at all, but\na cove or crescent open to an extensive prospect. In the book before\nmentioned, I read the notice in the English language that if anyone\nwould ascend the steep ground above the convent, and wander over it, he\nwould be abundantly rewarded by magnificent views. I had not time to\nact upon this recommendation, and only went with my young guide to a\npoint, nearly on a level with the site of the convent, that overlooks\nthe Vale of Arno for some leagues. To praise great and good men has\never been deemed one of the worthiest employments of poetry, but the\nobjects of admiration vary so much with time and circumstances, and\nthe noblest of mankind have been found, when intimately known, to be\nof characters so imperfect, that no eulogist can find a subject which\nhe will venture upon with the animation necessary to create sympathy,\nunless he confines himself to a particular part or he takes something\nof a one-sided view of the person he is disposed to celebrate. This\nis a melancholy truth, and affords a strong reason for the poetic\nmind being chiefly exercised in works of fiction: the poet can then\nfollow wherever the spirit of admiration leads him, unchecked by such\nsuggestions as will be too apt to cross his way if all that he is\nprompted to utter is to be tested by fact. Something in this spirit I\nhave written in the note attached to the Sonnet on the King of Sweden;\nand many will think that in this poem and elsewhere I have spoken\nof the author of _Paradise Lost_ in a strain of panegyric scarcely\njustifiable by the tenor of some of his opinions, whether theological\nor political, and by the temper he carried into public affairs, in\nwhich, unfortunately for his genius, he was so much concerned.--I.F.]\n\n    Thick as autumnal leaves that strew the brooks\n    In Vallombrosa, where Etrurian shades\n    High over-arch'd embower.\n\n                                    PARADISE LOST.[146]\n\n    \"Vallombrosa--I longed in thy shadiest wood\n    To slumber, reclined on the moss-covered floor!\"[147]\n    Fond wish that was granted at last, and the Flood,\n    That lulled me asleep, bids me listen once more.\n    Its murmur how soft! as it falls down the steep,                   5\n    Near that Cell--yon sequestered Retreat high in air--[148]\n    Where our Milton was wont lonely vigils to keep\n    For converse with God, sought through study and prayer.\n    The Monks still repeat the tradition with pride,\n    And its truth who shall doubt? for his Spirit is here;[149]       10\n    In the cloud-piercing rocks doth her grandeur abide,\n    In the pines pointing heavenward her beauty austere;\n    In the flower-besprent meadows his genius we trace\n    Turned to humbler delights, in which youth might confide,\n    That would yield him fit help while prefiguring that Place        15\n    Where, if Sin had not entered, Love never had died.\n\n    When with life lengthened out came a desolate time,\n    And darkness and danger had compassed him round,\n    With a thought he would[150] flee to these haunts of his prime,\n    And here once again a kind shelter be found.                      20\n    And let me believe that when nightly the Muse\n    Did[151] waft him to Sion, the glorified hill,[152]\n    Here also, on some favoured height, he[153] would choose\n    To wander, and drink inspiration at will.\n\n    Vallombrosa! of thee I first heard in the page                    25\n    Of that holiest of Bards, and the name for my mind\n    Had a musical charm, which the winter of age\n    And the changes it brings had no power to unbind.\n    And now, ye Miltonian shades! under you\n    I repose, nor am forced from sweet fancy to part,                 30\n    While your leaves I behold and the brooks they will strew,\n    And the realised vision is clasped to my heart.\n\n    Even so, and unblamed, we rejoice as we may\n    In Forms that must perish, frail objects of sense;\n    Unblamed--if the Soul be intent on the day                        35\n    When the Being of Beings shall summon her hence.\n    For he and he only with wisdom is blest\n    Who, gathering true pleasures wherever they grow,\n    Looks up in all places, for joy or for rest,\n    To the Fountain whence Time and Eternity flow.                    40\n\n[145] The name of Milton is pleasingly connected with Vallombrosa in\nmany ways. The pride with which the Monk, without any previous question\nfrom me, pointed out his residence, I shall not readily forget. It may\nbe proper here to defend the Poet from a charge which has been brought\nagainst him, in respect to the passage in _Paradise Lost_, where this\nplace is mentioned. It is said, that he has erred in speaking of the\ntrees there being deciduous, whereas they are, in fact, pines. The\nfault-finders are themselves mistaken; the _natural_ woods of the\nregion of Vallombrosa _are_ deciduous, and spread to a great extent;\nthose near the convent are, indeed, mostly pines; but they are avenues\nof trees _planted_ within a few steps of each other, and thus composing\nlarge tracts of wood; plots of which are periodically cut down. The\nappearance of those narrow avenues, upon steep <DW72>s open to the sky,\non account of the height which the trees attain by being _forced_\nto grow upwards, is often very impressive. My guide, a boy of about\nfourteen years old, pointed this out to me in several places.--W.W.\n1842.\n\n[146] Compare _Paradise Lost_, book i. l. 302. Vallombrosa--the shady\nvalley--is 18 miles distant from Florence. Wordsworth's quotation from\nMilton was from memory. It is not quite accurate.--ED.\n\n[147] See for the two _first lines_, _Stanzas composed in the Simplon\nPass_.--W.W. 1842. (See vol. vi. p. 357.)--ED.\n\n[148] The monastery of Vallombrosa was founded about 1050, by S.\nGiovanni Gnalberto. It was suppressed in 1869, and is now converted\ninto the R. Instituto Forestale, or forest school. The \"cell,\" the\n\"sequestered retreat\" referred to by Wordsworth, is doubtless _Il\nParadisino_, or _Le Celle_, a small hermitage 266 feet above the\nmonastery, which is itself 2980 feet above the sea.--ED.\n\n[149] Compare Milton's letter to Benedetto Bonmattei of Florence,\nwritten during his stay in the city, September 10, 1638.--ED.\n\n[150] 1845.\n\n    \u2026 might \u2026\n\n    1842.\n\n[151] 1845.\n\n    Would \u2026\n\n    1842.\n\n[152] Compare _Paradise Lost_, book iii. l. 29--\n\n                                \u2026 but chief\n    Thee, Sion, and the flourie Brooks beneath,\n    That wash thy hallowed feet, and warbling flow,\n    Nightly I visit.\n\nED.\n\n[153] 1845.\n\n    \u2026 they \u2026\n\n    1842.\n\n\nXIX\n\nAT FLORENCE\n\n[Upon what evidence the belief rests that this stone was a favourite\nseat of Dante, I do not know; but a man would little consult his own\ninterest as a traveller, if he should busy himself with doubts as\nto the fact. The readiness with which traditions of this character\nare received, and the fidelity with which they are preserved from\ngeneration to generation, are an evidence of feelings honourable to\nour nature. I remember how, during one of my rambles in the course\nof a college vacation, I was pleased on being shown a seat near a\nkind of rocky cell at the source of the river, on which it was said\nthat Congreve wrote his _Old Bachelor_. One can scarcely hit on any\nperformance less in harmony with the scene; but it was a local tribute\npaid to intellect by those who had not troubled themselves to estimate\nthe moral worth of that author's comedies; and why should they? He\nwas a man distinguished in his day; and the sequestered neighbourhood\nin which he often resided was perhaps as proud of him as Florence of\nher Dante: it is the same feeling, though proceeding from persons one\ncannot bring together in this way without offering some apology to the\nShade of the great Visionary.--I.F.]\n\n    Under the shadow of a stately Pile,\n    The dome of Florence, pensive and alone,\n    Nor giving heed to aught that passed the while,\n    I stood, and gazed upon a marble stone,\n    The laurelled Dante's favourite seat.[154] A throne,               5\n    In just esteem, it rivals; though no style\n    Be there of decoration to beguile\n    The mind, depressed by thought of greatness flown.\n    As a true man, who long had served the lyre,\n    I gazed with earnestness, and dared no more.                      10\n    But in his breast the mighty Poet bore\n    A Patriot's heart, warm with undying fire.\n    Bold with the thought, in reverence I sate down,\n    And, for a moment, filled that empty Throne.\n\n[154] The _Sasso di Dante_ is built into the wall of the house, No. 29\nCasa dei Canonici, close to the Duomo.--ED.\n\n\nXX\n\nBEFORE THE PICTURE OF THE BAPTIST, BY RAPHAEL, IN THE GALLERY AT\nFLORENCE[155]\n\n[It was very hot weather during the week we stayed at Florence; and,\nnever having been there before, I went through much hard service, and\nam not therefore _ashamed_ to confess I fell asleep before this picture\nand sitting with my back towards the Venus de Medicis. Buonaparte--in\nanswer to one who had spoken of his being in a sound sleep up to the\nmoment when one of his great battles was to be fought, as a proof\nof the calmness of his mind and command over anxious thoughts--said\nfrankly, that he slept because from bodily exhaustion he could not help\nit. In like manner it is noticed that criminals on the night previous\nto their execution seldom awake before they are called, a proof that\nthe body is the master of us far more than we need be willing to allow.\nShould this note by any possible chance be seen by any of my countrymen\nwho might have been in the gallery at the time (and several persons\nwere there) and witnessed such an indecorum, I hope he will give up the\nopinion which he might naturally have formed to my prejudice.--I.F.]\n\n    The Baptist might have been ordain'd to cry\n    Forth from the towers of that huge Pile, wherein\n    His Father served Jehovah; but how win\n    Due audience, how for aught but scorn defy\n    The obstinate pride and wanton revelry                             5\n    Of the Jerusalem below, her sin\n    And folly, if they with united din\n    Drown not at once mandate and prophecy?\n    Therefore the Voice spake from the Desert, thence\n    To Her, as to her opposite in peace,                              10\n    Silence, and holiness, and innocence,\n    To Her and to all Lands its warning sent,\n    Crying with earnestness that might not cease,\n    \"Make straight a highway for the Lord--repent!\"\n\n[155] This sonnet refers to the picture of the young St. John the\nBaptist, now in the Tribuna, Florence, designed about the same time as\nthe Madonna di San Sisto, for Cardinal Colonna, who is said to have\npresented it to his doctor, Jacopo da Carpi. It has been much admired,\nand often copied; but it is inferior, both in drawing and in colouring,\nto the great works of Raphael. How much of it was actually from his\nhand is uncertain; and Baptist is painted rather like a Bacchus than a\nSaint.--ED.\n\n\nXXI\n\nAT FLORENCE--FROM MICHAEL ANGELO\n\n[However at first these two sonnets from Michael Angelo may seem in\ntheir spirit somewhat inconsistent with each other, I have not scrupled\nto place them side by side as characteristic of their great author,\nand others with whom he lived. I feel, nevertheless, a wish to know\nat what periods of his life they were respectively composed.[156] The\nlatter, as it expresses, was written in his advanced years, when it\nwas natural that the Platonism that pervades the one should give way to\nthe Christian feeling that inspired the other: between both there is\nmore than poetic affinity.--I.F.]\n\n    Rapt above earth by power of one fair face,\n    Hers in whose sway alone my heart delights,\n    I mingle with the blest on those pure heights\n    Where Man, yet mortal, rarely finds a place.\n    With Him who made the Work that Work accords                       5\n    So well, that by its help and through his grace\n    I raise my thoughts, inform my deeds and words,\n    Clasping her beauty in my soul's embrace.\n    Thus, if from two fair eyes mine cannot turn,\n    I feel how in their presence doth abide                           10\n    Light which to God is both the way and guide;\n    And, kindling at their lustre, if I burn,\n    My noble fire emits the joyful ray\n    That through the realms of glory shines for aye.\n\n[156] The second of the two sonnets translated by Wordsworth is No.\nlxxiii. in Signor Cesare Guast\u00ee's edition of Michael Angelo (1863).\n\nAT THE FOOT OF THE CROSS.\n\n_Scaro d'un' importuna._\n\nIt was evidently written in old age. The following is Mr. John\nAddington Symond's translation of the same sonnet.\n\n    Freed from a burden sore and grievous band,\n      Dear Lord, and from this wearying world untied,\n      Like a frail bark I turn me to Thy side,\n      As from a fierce storm to a tranquil land.\n    Thy thorns, Thy nails, and either bleeding hand,\n      With Thy mild gentle piteous face, provide\n      Promise of help and mercies multiplied,\n      And hope that yet my soul secure may stand.\n    Let not Thy holy eyes be just to see\n      My evil part, Thy chastened ears to hear,\n      And stretch the arm of judgment to my crime:\n    Let Thy blood only love and succour me,\n      Yielding more perfect pardon, better cheer,\n      As older still I grow with lengthening time.\n\n_The Sonnets of Michael Angelo Buonarroti and Tomaso Campanella_, by\nJohn Addington Symonds, p. 110.\n\nCompare Wordsworth's translation of other three sonnets by Michael\nAngelo (vol. iii. pp. 380-384).--ED.\n\n\nXXII\n\nAT FLORENCE--FROM M. ANGELO\n\n    Eternal Lord! eased of a cumbrous load,\n    And loosened from the world, I turn to Thee;\n    Shun, like a shattered bark, the storm, and flee\n    To thy protection for a safe abode.\n    The crown of thorns, hands pierced upon the tree,                  5\n    The meek, benign, and lacerated face,\n    To a sincere repentance promise grace,\n    To the sad soul give hope of pardon free.\n    With justice mark not Thou, O Light divine,\n    My fault, nor hear it with thy sacred ear;                        10\n    Neither put forth that way thy arm severe;\n    Wash with thy blood my sins; thereto incline\n    More readily the more my years require\n    Help, and forgiveness speedy and entire.\n\n\nXXIII\n\nAMONG THE RUINS OF A CONVENT IN THE APENNINES\n\n[The political revolutions of our time have multiplied, on the\nContinent, objects that unavoidably call forth reflections such as are\nexpressed in these verses, but the Ruins in those countries are too\nrecent to exhibit, in anything like an equal degree, the beauty with\nwhich time and nature have invested the remains of our Convents and\nAbbeys. These verses, it will be observed, take up the beauty long\nbefore it is matured, as one cannot but wish it may be among some of\nthe desolations of Italy, France, and Germany.--I.F.]\n\n    Ye Trees! whose slender roots entwine\n      Altars that piety neglects;\n    Whose infant arms enclasp the shrine\n      Which no devotion now respects;\n    If not a straggler from the herd                                   5\n    Here ruminate, nor shrouded bird,\n    Chanting her low-voiced hymn, take pride\n    In aught that ye would grace or hide--\n    How sadly is your love misplaced,\n    Fair Trees, your bounty run to waste!                             10\n\n    Ye, too,[157] wild Flowers! that no one heeds,\n    And ye--full often spurned as weeds--\n    In beauty clothed, or breathing sweetness\n    From fractured arch and mouldering wall--\n    Do but more touchingly recal                                      15\n    Man's headstrong violence and Time's fleetness,\n    Making[158] the precincts ye adorn\n    Appear to sight still more forlorn.\n\n[157] 1845.\n\n    And ye, \u2026\n\n    1842.\n\n[158] 1845.\n\n    And make \u2026\n\n    1842.\n\n\nXXIV\n\nIN LOMBARDY\n\n    See, where his difficult way that Old Man wins\n    Bent by a load of Mulberry leaves!--most hard\n    Appears _his_ lot, to the small Worm's compared,\n    For whom his toil with early day begins.\n    Acknowledging no task-master, at will                              5\n    (As if her labour and her ease were twins)\n    _She_ seems to work, at pleasure to lie still;--\n    And softly sleeps within the thread she spins.\n    So fare they--the Man serving as her Slave.\n    Ere long their fates do each to each conform:                     10\n    Both pass into new being,--but the Worm,\n    Transfigured, sinks into a hopeless grave;\n    _His_ volant Spirit will, he trusts, ascend\n    To bliss unbounded, glory without end.\n\n\nXXV\n\nAFTER LEAVING ITALY\n\n[I had proof in several instances that the Carbonari, if I may still\ncall them so, and their favourers, are opening their eyes to the\nnecessity of patience, and are intent upon spreading knowledge actively\nbut quietly as they can. May they have resolution to continue in this\ncourse! for it is the only one by which they can truly benefit their\ncountry. We left Italy by the way which is called the \"Nuova Strada de\nAllmagna,\" to the east of the high passes of the Alps, which take you\nat once from Italy into Switzerland. This road leads across several\nsmaller heights, and winds down different vales in succession, so that\nit was only by the accidental sound of a few German words that I was\naware we had quitted Italy, and hence the unwelcome shock alluded to in\nthe two or three last lines of the latter sonnet.--I.F.]\n\n    Fair Land! Thee all men greet with joy; how few,\n    Whose souls take pride in freedom, virtue, fame,\n    Part from thee without pity dyed in shame:\n    I could not--while from Venice we withdrew,\n    Led on till an Alpine strait confined our view[159]                5\n    Within its depths, and to the shore we came\n    Of Lago Morto, dreary sight and name,\n    Which o'er sad thoughts a sadder colouring threw.\n    Italia! on the surface of thy spirit,\n    (Too aptly emblemed by that torpid lake)                          10\n    Shall a few partial breezes only creep?--\n    Be its depths quickened; what thou dost inherit\n    Of the world's hopes, dare to fulfil; awake,\n    Mother of Heroes, from thy death-like sleep!\n\n[159] They left Venice by the Nuova Strada de Allmagna, resting\nat Logerone, Sillian, Spittal (in Carinthia), and thence on to\nSalzburg.--ED.\n\n\nXXVI\n\nCONTINUED\n\n    As indignation mastered grief, my tongue\n    Spake bitter words; words that did ill agree\n    With those rich stores of Nature's imagery,\n    And divine Art, that fast to memory clung--\n    Thy gifts, magnificent Region, ever young                          5\n    In the sun's eye, and in his sister's sight\n    How beautiful! how worthy to be sung\n    In strains of rapture, or subdued delight!\n    I feign not; witness that unwelcome shock\n    That followed the first sound of German speech,                   10\n    Caught the far-winding barrier Alps among.\n    In that announcement, greeting seemed to mock[160]\n    Parting; the casual word had power to reach\n    My heart, and filled that heart with conflict strong.\n\n[160] See the Fenwick note to the last sonnet.--ED.\n\n\nAT BOLOGNA, IN REMEMBRANCE OF THE LATE INSURRECTIONS, 1837[161][162]\n\nComposed 1837.--Published 1842\n\nThis was originally (1842) included in the \"Memorials of a Tour in\nItaly,\" but, in 1845, it was transferred, along with the two which\nfollow it, to the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n\nI\n\n    Ah why deceive ourselves! by no mere fit\n    Of sudden passion roused shall men attain\n    True freedom where for ages they have lain\n    Bound in a dark abominable pit,\n    With life's best sinews more and more unknit.                      5\n    Here, there, a banded few who loathe the chain\n    May rise to break it: effort worse than vain\n    For thee, O great Italian nation, split\n    Into those jarring fractions.--Let thy scope\n    Be one fixed mind for all; thy rights approve                     10\n    To thy own conscience gradually renewed;\n    Learn to make Time the father of wise Hope;\n    Then trust thy cause to the arm of Fortitude,\n    The light of Knowledge, and the warmth of Love.\n\n\nII\n\nCONTINUED\n\nComposed 1837.--Published 1842\n\n    Hard task! exclaim the undisciplined, to lean\n    On Patience coupled with such slow endeavour,\n    That long-lived servitude must last for ever.\n    Perish the grovelling few, who, prest between\n    Wrongs and the terror of redress, would wean                       5\n    Millions from glorious aims. Our chains to sever\n    Let us break forth in tempest now or never!--\n    What, is there then no space for golden mean\n    And gradual progress?--Twilight leads to day,\n    And, even within the burning zones of earth,                      10\n    The hastiest sunrise yields a temperate ray;\n    The softest breeze to fairest flowers gives birth:\n    Think not that Prudence dwells in dark abodes,\n    She scans the future with the eye of gods.\n\n\nIII\n\nCONCLUDED\n\nComposed 1837.--Published 1842\n\n    As leaves are to the tree whereon they grow\n    And wither, every human generation\n    Is to the Being of a mighty nation,\n    Locked in our world's embrace through weal and woe;\n    Thought that should teach the zealot to forego                     5\n    Rash schemes, to abjure all selfish agitation,\n    And seek through noiseless pains and moderation\n    The unblemished good they only can bestow.\n    Alas! with most, who weigh futurity\n    Against time present, passion holds the scales:                   10\n    Hence equal ignorance of both prevails,\n    And nations sink; or, struggling to be free,\n    Are doomed to flounder on, like wounded whales\n    Tossed on the bosom of a stormy sea.\n\n[161] This date was omitted in the edition of 1842.\n\n[162] The three sonnets, _At Bologna, in remembrance of the late\nInsurrections_, 1837, are printed as a sequel to the Italian Tour of\nthat year.--ED.\n\n\n\"WHAT IF OUR NUMBERS BARELY COULD DEFY\"\n\nComposed 1837.--Published 1837\n\nOne of the \"Poems dedicated to National Independence and Liberty.\"--ED.\n\n    What if our numbers barely could defy\n    The arithmetic of babes, must foreign hordes,\n    Slaves, vile as ever were befooled by words,\n    Striking through English breasts the anarchy\n    Of Terror, bear us to the ground, and tie                          5\n    Our hands behind our backs with felon cords?\n    Yields every thing to discipline of swords?\n    Is man as good as man, none low, none high?--\n    Nor discipline nor valour can withstand\n    The shock, nor quell[163] the inevitable rout,                    10\n    When in some great extremity breaks out\n    A people, on their own beloved Land\n    Risen, like one man, to combat in the sight\n    Of a just God for liberty and right.\n\n[163] 1837.\n\n    \u2026 nor stem \u2026\n\n    C.\n\n\nA NIGHT THOUGHT\n\nComposed 1837.--Published 1837\n\n[These verses were thrown off extempore upon leaving Mrs. Luff's\nhouse at Fox Ghyll one evening. The good woman is not disposed to\nlook at the bright side of things, and there happened to be present\ncertain ladies who had reached the point of life where _youth_ is\nended, and who seemed to contend with each other in expressing their\ndislike of the country and climate. One of them had been heard to say\nshe could not endure a country where there was \"neither sunshine nor\ncavaliers.\"--I.F.]\n\nThis poem was first published in _The Tribute, a Collection of\nMiscellaneous unpublished Poems by various Authors, edited by Lord\nNorthampton_, in 1837, \"for the benefit of the widow and family of the\nRev. Edward Smedley.\" (The same volume contained a poem by Southey on\nBrough Bells.) It next found a place in \"Poems chiefly of Early and\nLate Years\" (1842). A stanza given in _The Tribute_, No. 2 (see below),\nwas omitted afterwards.--ED.\n\n    Lo! where the Moon along the sky\n    Sails with her happy destiny;[164]\n    Oft is she hid from mortal eye\n          Or dimly seen,\n    But when the clouds asunder fly                                    5\n          How bright her mien![165]\n\n    Far different we--a froward race,[166]\n    Thousands though rich in Fortune's grace\n    With cherished sullenness of pace\n          Their way pursue,                                           10\n    Ingrates who wear a smileless face\n          The whole year through.\n\n    If kindred humours e'er would make[167]\n    My spirit droop for drooping's sake,\n    From Fancy following in thy wake,                                 15\n          Bright ship of heaven!\n    A counter impulse let me take\n          And be forgiven.[168]\n\n[164] 1842.\n\n    The moon that sails along the sky\n    Moves with a happy destiny,\n\n    1837.\n\n[165] 1837.\n\n    Not flagging when the winds all sleep,\n    Not hurried onward, when they sweep\n    The bosom of th' ethereal deep,\n        Not turned aside,\n    She knows an even course to keep,\n        Whate'er betide.\n\n    In the text of 1837 only.\n\n[166] 1842.\n\n    Perverse are we--a froward race;\n\n    1837.\n\n[167] 1842.\n\n    If kindred humour e'er should make\n\n    1837.\n\n[168] Compare the poem _To the Daisy_ (1802), beginning--\n\n    Bright Flower! whose home is everywhere.\n\nED.\n\n\nTHE WIDOW ON WINDERMERE SIDE\n\nPublished 1842\n\n[The facts recorded in this Poem were given me, and the character of\nthe person described, by my friend the Rev. R. P. Graves,[169] who\nhas long officiated as curate at Bowness, to the great benefit of the\nparish and neighbourhood. The individual was well known to him. She\ndied before these verses were composed. It is scarcely worth while\nto notice that the stanzas are written in the sonnet form, which was\nadopted when I thought the matter might be included in twenty-eight\nlines.--I.F.]\n\nOne of the \"Poems founded on the Affections.\"--ED.\n\n                         I\n\n    How beautiful when up a lofty height\n    Honour ascends among the humblest poor,\n    And feeling sinks as deep! See there the door\n    Of One, a Widow, left beneath a weight\n    Of blameless debt. On evil Fortune's spite                         5\n    She wasted no complaint, but strove to make\n    A just repayment, both for conscience-sake\n    And that herself and hers should stand upright\n    In the world's eye. Her work when daylight failed\n    Paused not, and through the depth of night she kept               10\n    Such earnest vigils, that belief prevailed\n    With some, the noble Creature never slept;\n    But, one by one, the hand of death assailed\n    Her children from her inmost heart bewept.\n\n                         II\n\n    The Mother mourned, nor ceased her tears to flow,                 15\n    Till a winter's noon-day placed her buried Son\n    Before her eyes, last child of many gone--\n    His raiment of angelic white, and lo!\n    His very feet bright as the dazzling snow\n    Which they are touching; yea far brighter, even                   20\n    As that which comes, or seems to come, from heaven,\n    Surpasses aught these elements can show.\n    Much she rejoiced, trusting that from that hour\n    Whate'er befel she could not grieve or pine;\n    But the Transfigured, in and out of season,                       25\n    Appeared, and spiritual presence gained a power\n    Over material forms that mastered reason.\n    Oh, gracious Heaven, in pity make her thine!\n\n                         III\n\n    But why that prayer? as if to her could come\n    No good but by the way that leads to bliss                        30\n    Through Death,--so judging we should judge amiss.\n    Since reason failed want is her threatened doom,\n    Yet frequent transports mitigate the gloom:\n    Nor of those maniacs is she one that kiss\n    The air or laugh upon a precipice;                                35\n    No, passing through strange sufferings toward the tomb,\n    She smiles as if a martyr's crown were won:\n    Oft, when light breaks through clouds or waving trees,\n    With outspread arms and fallen upon her knees\n    The Mother hails in her descending Son                            40\n    An Angel, and in earthly ecstasies\n    Her own angelic glory seems begun.\n\n[169] The late Archdeacon of Dublin, author of _Life of Sir William\nRowan Hamilton_, etc. He gives the date of the composition of the poem\nas 1837.--ED.\n\n\n\n\n1838\n\nIn 1838 Wordsworth wrote ten sonnets. These were published (along with\nthe one suggested by Mrs. Southey) for the first time in the volume of\ncollected Sonnets, several being inserted out of their intended place,\nwhile the book was passing through the press.\n\nThe _Protest against the Ballot_, which appeared in 1838, was never\nrepublished.--ED.\n\n\nTO THE PLANET VENUS\n\nUPON ITS APPROXIMATION (AS AN EVENING STAR) TO THE EARTH, JANUARY 1838\n\nComposed 1838.--Published 1838[170]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    What strong allurement draws, what spirit guides,\n    Thee, Vesper! brightening still, as if the nearer\n    Thou com'st to man's abode the spot grew dearer\n    Night after night? True is it Nature hides\n    Her treasures less and less.--Man now presides                     5\n    In power, where once he trembled in his weakness;\n    Science[171] advances with gigantic strides;\n    But are we aught enriched in love and meekness?[172]\n    Aught dost thou see, bright Star! of pure and wise\n    More than in humbler times graced human story;                    10\n    That makes our hearts more apt to sympathise\n    With heaven, our souls more fit for future glory,\n    When earth shall vanish from our closing eyes,\n    Ere we lie down in our last dormitory?[173]\n\n[170] It was afterwards printed in the _Saturday Magazine_, Oct. 24,\n1840.--ED.\n\n[171] 1845.\n\n    Knowledge\n\n    1838.\n\n[172] Compare Tennyson's _In Memoriam_, stanza cxx.--\n\n    Let Science prove we are, and then\n    What matters Science unto men, etc.\n\nED.\n\n[173] Compare the poem in vol. vii. p. 299, _To the Planet Venus, an\nEvening Star_.--ED.\n\n\n\"HARK! 'TIS THE THRUSH, UNDAUNTED, UNDEPREST\"\n\nComposed 1838.--Published 1838\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Hark! 'tis the Thrush, undaunted, undeprest,\n    By twilight premature of cloud and rain;\n    Nor does that roaring wind deaden his strain[174]\n    Who carols thinking of his Love and nest,\n    And seems, as more incited, still more blest.                      5\n    Thanks; thou hast snapped a fire-side Prisoner's chain,\n    Exulting Warbler! eased a fretted brain,\n    And in a moment charmed my cares to rest.\n    Yes, I will forth, bold Bird! and front the blast,\n    That we may sing together, if thou wilt,                          10\n    So loud, so clear, my Partner through life's day,\n    Mute in her nest love-chosen, if not love-built\n    Like thine, shall gladden, as in seasons past,\n    Thrilled by loose snatches of the social Lay.\n\nRYDAL MOUNT, 1838.\n\n[174] 1838.\n\n                      \u2026 undaunted, unopprest,\n    Struggling with twilight premature and rain.\n    Loud roars the wind, but smothers not his strain\n\n    MS.\n\n\n\"'TIS HE WHOSE YESTER-EVENING'S HIGH DISDAIN\"\n\nComposed 1838.--Published 1838\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    'Tis He whose yester-evening's high disdain\n    Beat back the roaring storm--but how subdued\n    His day-break note, a sad vicissitude!\n    Does the hour's drowsy weight his glee restrain?\n    Or, like the nightingale, her joyous vein                          5\n    Pleased to renounce, does this dear Thrush attune\n    His voice to suit the temper of yon Moon\n    Doubly depressed, setting, and in her wane?\n    Rise, tardy Sun! and let the Songster prove\n    (The balance trembling between night and morn                     10\n    No longer) with what ecstasy upborne\n    He can pour forth his spirit. In heaven above,\n    And earth below, they best can serve true gladness\n    Who meet most feelingly the calls of sadness.\n\n\nCOMPOSED AT RYDAL ON MAY MORNING, 1838[175]\n\nComposed 1st May 1838.--Published 1838\n\n[This and the following sonnet were composed on what we call the \"Far\nTerrace\" at Rydal Mount, where I have murmured out many thousands of\nverses.--I.F.]\n\nThis sonnet was first published in the Volume of Collected Sonnets\nin 1838. In 1842 it was classed among the \"Miscellaneous Sonnets\";\nbut in 1845 it was transferred to the \"Memorials of a Tour in Italy,\n1837.\"--ED.\n\n    If with old love of you, dear Hills! I share\n    New love of many a rival image brought\n    From far, forgive the wanderings of my thought:\n    Nor art thou wronged, sweet May! when I compare[176]\n    Thy present birth-morn with thy last,[177][178] so fair,           5\n    So rich to me in favours. For my lot\n    Then was, within the famed Egerian Grot\n    To sit and muse, fanned by its dewy air\n    Mingling with thy soft breath! That morning too,\n    Warblers I heard their joy unbosoming                             10\n    Amid the sunny, shadowy, Coliseum;[179]\n    Heard them, unchecked by aught of saddening hue,[180]\n    For victories there won by flower-crowned Spring,[181]\n    Chant in full choir their innocent Te Deum.\n\n[175] 1845.\n\nThe title in 1838 was \"COMPOSED ON MAY-MORNING, 1838\"; and \"RYDAL\nMOUNT\" was written at the foot of the sonnet.\n\n[176] 1838.\n\n    May, if from these thy northern haunts I share\n    Fond looks of mind for images remote\n    Fetched out of milder climates, blame me not,\n    Nor that, upris'n thus early, I compare\n\n    MS.\n\n    Let those who will or can, dear May, forbear\n    To rise and hail thy coming, I could not.\n    The vivid images of scenes remote\n    Rushing on memory urge me to compare\n\n    MS.\n\n    Dear native Hills, the love of you I share\n    With \u2026\n\n    MS.\n\n    Dear fields and native mountains, if I share\n    My love of youth with love of objects brought\n    {From far, by faithful memory, blame me not. }\n    {Fetched from a milder climate, blame me not.}\n    {From a distant land by memory, blame me not.}\n    {Nor that, upris'n thus early,   }\n    {Nor be displeased, sweet May, if} I compare\n    {May,}\n    {Thy } present \u2026\n\n    MS.\n\n[177] 1838.\n\n    \u2026 past,\n\n    MS.\n\n[178] On May morning, 1837, Wordsworth was in Rome with Henry Crabb\nRobinson.--ED.\n\n[179] The Flavian Amphitheatre, begun by Vespasian, A.D. 72, and\ncontinued by his son Titus, one of the noblest structures in Rome, now\na ruin. --ED.\n\n[180] 1845.\n\n    \u2026 of sombre hue,\n\n    1838.\n\n    \u2026 by thoughts of sombre hue,\n\n    MS.\n\n[181] 1838.\n\n                                    \u2026 too,\n    How my heart swelled when in the mighty ring,\n    The mouldering, shadowy, sunny Collosseum,\n    I heard with some sad thoughts of local hue\n    Warblers there lodged, for victories won by spring\n\n    MS.\n\n                                    \u2026 too,\n    Here did I a deathless joy embosoming,\n    {Mid   } the shadowy Collosseum,\n    {Within}\n    Hear not without sad thoughts of local hue\n\n    MS.\n\n                                    \u2026 too,\n    Heard I, a deathless joy embosoming,\n    Tho' not without sad thoughts of local hue,\n    Amid the shadowy, sunny, Collosseum,\n    Warblers there lodged, for victories won by Spring\n\n    MS.\n\n\nCOMPOSED ON A MAY MORNING, 1838[182]\n\nComposed 1838.--Published 1838[183]\n\nThis was one of the \"Miscellaneous Sonnets.\"--ED.\n\n    Life with yon Lambs, like day, is just begun,\n    Yet Nature seems to them a heavenly guide.[184]\n    Does joy approach? they meet the coming tide;\n    And sullenness avoid, as now they shun[185]\n    Pale twilight's lingering glooms,--and in the sun                  5\n    Couch near their dams, with quiet satisfied;[186]\n    Or gambol--each with his shadow at his side,[187]\n    Varying its shape wherever he may run.\n    As they from turf yet hoar with sleepy dew\n    All turn, and court the shining and the green,                    10\n    Where herbs look up, and opening flowers are seen;\n    Why to God's goodness cannot We be true,\n    And so, His[188] gifts and promises between,\n    Feed to the last on pleasures ever new?\n\n[182] 1845.\n\nThe title, in 1838, was \"COMPOSED ON THE SAME MORNING\"; referring to\nthe previous sonnet in that edition, beginning--\n\n    If with old love of you, dear Hills! I share.\n\n[183] There were so many tentative efforts in the construction of this\nsonnet, and the one which follows it, that I feel justified in printing\nthem from MS. sources.--ED.\n\n[184] 1838.\n\n    Life with yon mountain lambs is just begun,\n\n    MS.\n\n    Yon mountain Lambs whose life is just begun\n    Some guidance know to Man's grave years denied.\n\n    MS.\n\n    Your lives, ye mountain lambs, tho' just begun\n    A guidance know to our best years denied.\n\n    MS. sent to Mr. Clarkson.\n\n[185] 1838.\n\n    O that by Nature we were prompt the tide\n    Of joy to meet, as {they} are who {now } shun\n                       {ye  }         {there}\n\n    MS. sent to Mr. Clarkson.\n\n[186] 1838.\n\n    The lingering glooms of twilight, in the sun\n    To couch, with sober quiet satisfied.\n\n    MS. sent to Mr. Clarkson.\n\n                                \u2026 shun\n    Hollows unbrightened by the {rising} sun\n                                {morning}\n    On <DW72>s to couch with quiet satisfied.\n\n    MS.\n\n    To couch on <DW72>s where he his beams has tried,\n    Sporting and running wheresoe'er ye run.\n\n    MS.\n\n[187] 1838.\n\n    Couch near their dams; or frisk in sportive pride\n    Each with his playful shadow at his side,\n\n    MS.\n\n[188] 1838.\n\n    As they from turf hoary with unsunned dew\n    Turn and do one and all prefer the green\n    To chilly nooks, knolls cheered with glistening sheen,\n    Why may not we a kindred course pursue\n    And so, God's \u2026\n\n    MS.\n\n                                \u2026 shun\n    Hollows {enlivened   } by the rising sun\n            {unbrightened}\n    On <DW72>s to couch with quiet satisfied,\n    Or gambol each, his shadow at his side,\n    Running in sport wherever he may run.\n    As from dull turf hoary with unsunned dew\n    They turn, and one and all prefer the green\n    To chilly nooks, knolls {warmed} with glistening sheen,\n                            {cheered}\n    Why may not we a kindred course pursue\n    And so, Heaven's \u2026\n\n    MS.\n\n                                \u2026 shun\n    The lingering gloom of twilight in the sun,\n    To couch with sober quiet satisfied,\n    Or gambol each, his shadow at his side,\n    Varying its shape wherever he may run.\n\n    MS.\n\n    As they from turf with thick and sleepy dew\n    {{Yet} whitened o'er, turn and}\n    {{All}                       } prefer the green\n    {Turn, and do one and all    }\n    To chilly nooks, {<DW72>s} warm with glistening sheen,\n                     {knolls}\n    Why may not we thro' life such course pursue\n    And so, God's \u2026\n\n    MS.\n\n    As they from turf with thick and sleepy dew\n    Yet whitened o'er, turn and prefer the green;\n    To chilly nooks, <DW72>s warm with glistering sheen,\n    Why may not we such course through life pursue,\n    And so, God's gifts and promises between,\n    Feed \u2026\n\n    MS.\n\n\nA PLEA FOR AUTHORS, MAY 1838\n\n    Failing impartial measure to dispense\n    To every suitor, Equity is lame;\n    And social Justice, stript of reverence\n    For natural rights, a mockery and a shame;\n    Law but a servile dupe of false pretence,                          5\n    If, guarding grossest things from common claim\n    Now and for ever, She, to works that came[189]\n    From mind and spirit, grudge a short-lived fence.\n    \"What! lengthened privilege, a lineal tie,\n    For _Books_!\" Yes, heartless Ones, or be it proved                10\n    That 'tis a fault in Us to have lived and loved\n    Like others, with like temporal hopes to die;\n    No public harm that Genius from her course\n    Be turned; and streams of truth dried up, even at their source![190]\n\n[189] 1838.\n\n    {If} failing one strict measure to dispense\n    {When}\n    To all her suitors Equity is lame,\n    And social justice by fit reverence\n    Of natural right unswayed is but a name,\n\n    MS.\n\n    {Law but} the servile dupe of false pretence,\n    {And Law}\n\n    MS.\n\n    {When} guarding grossest things from common claim\n    {If}\n    Now, and for ever, She for work that came\n\n    MS.\n\n                                  \u2026 lame,\n    Justice unswayed, unmoved by reverence\n    For natural right {what is she but a name?}\n                      {is but an empty name, }\n\n    MS.\n\n[190] 1838.\n\n                            \u2026 from its course\n    Be turned, and streams of truth dried at their source.\n\n    MS.\n\n    From mind and spirit grudge a short-lived fence.\n    But no--{our} sages join in banded force\n            {the}\n    {That} books by right or wrong {may} glad the isle\n    {With}                         {to}\n    Say, {would} this serve the {future should our} course\n         {can }                {people if the   }\n    {Of pure domestic hopes be checked the while}\n    {Of prejudice be less opposed the while    }\n    {Should} toil-worn Genius want a cheering smile\n    {If   }\n    And streams of truth be dried up at their source?\n\n    MS.\n\n    Out of the mind grudges a short-lived fence.\n    {But no--the Sages join in banded force   }\n    {And how preposterous Sages is your course}\n    Who cry give books free passage thro' the isle.\n    {Say can this serve the people of our isle, }\n    {By right or wrong, for better or for worse,}\n    Friends to the people, what care ye the while\n    Tho' toil-worn genius want a cheering smile\n    And far-fetched truth be dried up at her source?\n\n    MS.\n\n\n\"BLEST STATESMAN HE, WHOSE MIND'S UNSELFISH WILL\"\n\nComposed 1838.--Published 1838\n\nOne of the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n    Blest Statesman He, whose Mind's unselfish will\n    Leaves him[191] at ease among grand thoughts: whose eye\n    Sees that, apart from magnanimity,\n    Wisdom exists not; nor the humbler skill\n    Of Prudence, disentangling good and ill                            5\n    With patient care. What tho'[192] assaults run high,\n    They daunt not him who holds his ministry,\n    Resolute, at all hazards, to fulfil\n    Its[193] duties;--prompt to move, but firm to wait,--\n    Knowing, things rashly sought are rarely found;                   10\n    That, for[194] the functions of an ancient State--\n    Strong by her charters, free because imbound,\n    Servant of Providence, not slave of Fate--\n    Perilous is sweeping change, all chance unsound.[195]\n\n[191] 1842.\n\n    \u2026 her\n\n    C. and 1838.\n\n[192] 1838.\n\n    \u2026 if\n\n    C.\n\n[193] 1838.\n\n    His\n\n    C.\n\n[194] 1838.\n\n    \u2026 in\n\n    C.\n\n[195]\n\n    All change is perilous, and all chance unsound.\n\n    SPENSER.--W.W. 1838.\n\nThe passage will be found in _The Fa\u00ebrie Queene_, book v. canto xii.\nstanza 36.--ED.\n\n\nVALEDICTORY SONNET[196]\n\nComposed 1838.--Published 1838\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Serving no haughty Muse, my hands have here\n    Disposed some cultured Flowerets (drawn from spots\n    Where they bloomed singly, or in scattered knots),\n    Each kind in several beds of one parterre;\n    Both to allure the casual Loiterer,                                5\n    And that, so placed, my Nurslings may requite\n    Studious regard with opportune delight,\n    Nor be unthanked, unless I fondly err.\n    But metaphor dismissed, and thanks apart,\n    Reader, farewell! My last words let them be--                     10\n    If in this book Fancy and Truth agree;\n    If simple Nature trained by careful Art\n    Through It have won a passage to thy heart;\n    Grant me thy love, I crave no other fee!\n\n[196] This closed the volume of sonnets published in 1838.--ED.\n\n\n\n\n1839\n\nThe fourteen \"Sonnets upon the Punishment of Death\" were originally\npublished in the _Quarterly Review_ (in December 1841), in an article\non the \"Sonnets of William Wordsworth\" by the late Sir Henry Taylor,\nauthor of _Philip van Artevelde_, and other poems. Towards the close of\nthis article (of 1841), after reviewing the volume of Sonnets published\nin 1838, Sir Henry adds, \"There is a short series _written two years\nago_, which we have been favoured with permission to present to the\npublic for the first time. It was suggested by the recent discussions\nin Parliament, and elsewhere, on the subject of the 'Punishment of\nDeath.'\"\n\nWhen republishing this and other critical Essays on Poetry, in\nthe collected edition of his works in 1878, Sir Henry omitted the\nparagraphs relating to these particular sonnets. Wordsworth published\nthe sonnets in his volume of \"Poems chiefly of Early and Late Years,\"\nin 1842.--ED.\n\n\nSONNETS UPON THE PUNISHMENT OF DEATH\n\nIN SERIES\n\nComposed 1839.--Published 1841\n\n\"In the session of 1836, a report by the Commissioners on Criminal\nLaw--of which the second part was on this subject (the Punishment of\nDeath)--was laid before Parliament. In the ensuing session this was\nfollowed by papers presented to Parliament by her Majesty's command,\nand consisting of a correspondence between the Commissioners, Lord\nJohn Russell, and Lord Denman. Upon the foundation afforded by these\ndocuments, the bills of the 17th July 1837--(7th Gul. IV. and 1st\nVict. cap. 84 to 89 and 91)--were brought in and passed. These acts\nremoved the punishment of death from about 200 offences, and left it\napplicable to high treason,--murder and attempts at murder--rape--arson\nwith danger to life--and to piracies, burglaries, and robberies, when\naggravated by cruelty and violence.\" (Sir Henry Taylor, _Quarterly\nReview_, Dec. 1841, p. 39.) Some members of the House of Commons--Mr.\nFitzroy Kelly, Mr. Ewart, and others--desired a further limitation\nof the punishment of death to the crimes of murder and treason only:\nand the question of the entire abolition of capital punishment being\nvirtually before the country, Wordsworth dealt with it in the following\nseries of sonnets.--ED.\n\n\nI\n\nSUGGESTED BY THE VIEW OF LANCASTER CASTLE (ON THE ROAD FROM THE SOUTH)\n\n    This Spot--at once unfolding sight so fair\n    Of sea and land, with yon grey towers that still\n    Rise up as if to lord it over air--\n    Might soothe in human breasts the sense of ill,\n    Or charm it out of memory; yea, might fill                         5\n    The heart with joy and gratitude to God\n    For all his bounties upon man bestowed:\n    Why bears it then the name of \"Weeping Hill\"?[197]\n    Thousands, as toward yon old Lancastrian Towers,\n    A prison's crown, along this way they past                        10\n    For lingering durance or quick death with shame,\n    From this bare eminence thereon have cast\n    Their first look--blinded as tears fell in showers\n    Shed on their chains; and hence that doleful name.\n\n[197] The name given to the spot from which criminals on their way to\nthe Castle of Lancaster first see it.--ED.\n\n\nII[198]\n\n\"TENDERLY DO WE FEEL BY NATURE'S LAW\"\n\n    Tenderly do we feel by Nature's law\n    For worst offenders: though the heart will heave\n    With indignation, deeply moved we grieve,\n    In after thought, for Him who stood in awe\n    Neither of God nor man, and only saw,                              5\n    Lost wretch, a horrible device enthroned\n    On proud temptations, till the victim groaned\n    Under the steel his hand had dared to draw.\n    But O, restrain compassion, if its course,\n    As oft befalls, prevent or turn aside                             10\n    Judgments and aims and acts whose higher source\n    Is sympathy with the unforewarned, who died[199]\n    Blameless--with them that shuddered o'er his grave,\n    And all who from the law firm safety crave.\n\n[198] \"The first sonnet prepares the reader to sympathise with the\nsufferings of the culprits. The next cautions him as to the limits\nwithin which his sympathies are to be restrained.\" (Sir Henry\nTaylor.)--ED.\n\n[199] 1842.\n\n    \u2026 that died\n\n    1841.\n\n\nIII[200]\n\n\"THE ROMAN CONSUL DOOMED HIS SONS TO DIE\"\n\n    The Roman Consul doomed his sons to die\n    Who had betrayed their country.[201] The stern word\n    Afforded (may it through all time afford)\n    A theme for praise and admiration high.\n    Upon the surface of humanity                                       5\n    He rested not; its depths his mind explored;\n    He felt; but his parental bosom's lord\n    Was Duty,--Duty calmed his agony.\n    And some, we know, when they by wilful act\n    A single human life have wrongly taken,                           10\n    Pass sentence on themselves, confess the fact,\n    And, to atone for it, with soul unshaken\n    Kneel at the feet of Justice, and, for faith\n    Broken with all mankind, solicit death.\n\n[200] \"In the third and fourth sonnets the reader is prepared to\nregard as low and effeminate the views which would estimate life and\ndeath as the most important of all sublunary conditions.\" (Sir Henry\nTaylor.)--ED.\n\n[201] Lucius Junius Brutus, who condemned his sons to die for the part\nthey took in the conspiracy to restore the Tarquins. (See Livy, book\nii.)--ED.\n\n\nIV\n\n\"IS _DEATH_, WHEN EVIL AGAINST GOOD HAS FOUGHT\"\n\n    Is _Death_, when evil against good has fought\n    With such fell mastery that a man may dare\n    By deeds the blackest purpose to lay bare?\n    Is Death, for one to that condition brought,\n    For him, or any one, the thing that ought                          5\n    To be _most_ dreaded? Lawgivers, beware,\n    Lest, capital pains remitting till ye spare\n    The murderer, ye, by sanction to that thought\n    Seemingly given, debase the general mind;\n    Tempt the vague will tried standards to disown,                   10\n    Nor only palpable restraints unbind,\n    But upon Honour's head disturb the crown,\n    Whose absolute rule permits not to withstand\n    In the weak love of life his least command.\n\n\nV\n\n\"NOT TO THE OBJECT SPECIALLY DESIGNED\"\n\n    Not to the object specially designed,\n    Howe'er momentous in itself it be,\n    Good to promote or curb depravity,\n    Is the wise Legislator's view confined.\n    His Spirit, when most severe, is oft most kind;                    5\n    As all Authority in earth depends\n    On Love and Fear, their several powers he blends,\n    Copying with awe the one Paternal mind.\n    Uncaught by processes in show humane,\n    He feels how far the act would derogate                           10\n    From even the humblest functions of the State;\n    If she, self-shorn of Majesty, ordain\n    That never more shall hang upon her breath\n    The last alternative of Life or Death.\n\n\nVI[202]\n\n\"YE BROOD OF CONSCIENCE--SPECTRES! THAT FREQUENT\"\n\n    Ye brood of conscience--Spectres! that frequent\n    The bad man's restless walk, and haunt his bed--\n    Fiends in your aspect, yet beneficent\n    In act, as hovering Angels when they spread\n    Their wings to guard the unconscious Innocent--                    5\n    Slow be the Statutes of the land to share\n    A laxity that could not but impair\n    _Your_ power to punish crime, and so prevent.\n    And ye, Beliefs! coiled serpent-like about\n    The adage on all tongues, \"Murder will out,\"[203]                 10\n    How shall your ancient warnings work for good\n    In the full might they hitherto have shown,\n    If for deliberate shedder of man's blood\n    Survive not Judgment that requires his own?\n\n[202] \"The sixth sonnet adverts to the effect of the law in preventing\nthe crime of murder, not merely by fear, but by horror, by investing\nthe crime itself with the colouring of dark and terrible imaginations.\"\n(Sir Henry Taylor.)--ED.\n\n[203] See Chaucer, _The Nonnes Priestes Tale_, l. 232.--ED.\n\n\nVII\n\n\"BEFORE THE WORLD HAD PAST HER TIME OF YOUTH\"\n\n    Before the world had past her time of youth\n    While polity and discipline were weak,\n    The precept eye for eye, and tooth for tooth,\n    Came forth--a light, though but as of day-break,\n    Strong as could then be borne. A Master meek                       5\n    Proscribed the spirit fostered by that rule,\n    Patience _his_ law, long-suffering _his_ school,\n    And love the end, which all through peace must seek.\n    But lamentably do they err who strain\n    His mandates, given rash impulse to controul                      10\n    And keep vindictive thirstings from the soul,\n    So far that, if consistent in their scheme,\n    They must forbid the State to inflict a pain,\n    Making of social order a mere dream.\n\n\nVIII[204]\n\n\"FIT RETRIBUTION, BY THE MORAL CODE\"\n\n    Fit retribution, by the moral code\n    Determined, lies beyond the State's embrace,\n    Yet, as she may, for each peculiar case\n    She plants well-measured terrors in the road\n    Of wrongful acts. Downward it is and broad,                        5\n    And, the main fear once doomed to banishment,\n    Far oftener then, bad ushering worse event,\n    Blood would be spilt that in his dark abode\n    Crime might lie better hid. And, should the change\n    Take from the horror due to a foul deed,                          10\n    Pursuit and evidence so far must fail,\n    And, guilt escaping, passion then might plead\n    In angry spirits for her old free range,\n    And the \"wild justice of revenge\"[205] prevail.\n\n[204] \"In the eighth sonnet the doctrine, which would strive to measure\nout the punishments awarded by the law in proportion to the degrees of\nmoral turpitude, is disavowed.\" (Sir Henry Taylor.)--ED.\n\n[205] See Bacon's Essay _Of Revenge_, beginning, \"Revenge is a sort of\nwild justice.\"--ED.\n\n\nIX\n\n\"THOUGH TO GIVE TIMELY WARNING AND DETER\"\n\n    Though to give timely warning and deter\n    Is one great aim of penalty, extend\n    Thy mental vision further and ascend\n    Far higher, else full surely shalt thou err.[206]\n    What is a State? The wise behold in her                            5\n    A creature born of time, that keeps one eye\n    Fixed on the statutes of Eternity,\n    To which her judgments reverently defer.\n    Speaking through Law's dispassionate voice the State\n    Endues her conscience with external life                          10\n    And being, to preclude or quell the strife\n    Of individual will, to elevate\n    The grovelling mind, the erring to recal,\n    And fortify the moral sense of all.\n\n[206] 1845.\n\n    \u2026 thou shalt err.\n\n    1842.\n\n\nX\n\n\"OUR BODILY LIFE, SOME PLEAD, THAT LIFE THE SHRINE\"\n\n    Our bodily life, some plead, that life the shrine\n    Of an immortal spirit, is a gift\n    So sacred, so informed with light divine,\n    That no tribunal, though most wise to sift\n    Deed and intent, should turn the Being adrift                      5\n    Into that world where penitential tear\n    May not avail, nor prayer have for God's ear\n    A voice--that world whose veil no hand can lift\n    For earthly sight. \"Eternity and Time\"\n    _They_ urge, \"have interwoven claims and rights                   10\n    Not to be jeopardised through foulest crime:\n    The sentence rule by mercy's heaven-born lights.\"\n    Even so; but measuring not by finite sense\n    Infinite Power, perfect Intelligence.\n\n\nXI[207]\n\n\"AH, THINK HOW ONE COMPELLED FOR LIFE TO ABIDE\"\n\n    Ah, think how one compelled for life to abide\n    Locked in a dungeon needs must eat the heart\n    Out of his own humanity, and part\n    With every hope that mutual cares provide;\n    And, should a less unnatural doom confide                          5\n    In life-long exile on a savage coast,\n    Soon the relapsing penitent may boast\n    Of yet more heinous guilt, with fiercer pride.\n    Hence thoughtful Mercy, Mercy sage and pure,\n    Sanctions the forfeiture that Law demands,                        10\n    Leaving the final issue in _His_ hands\n    Whose goodness knows no change, whose love is sure,\n    Who sees, foresees; who cannot judge amiss,\n    And wafts at will the contrite soul to bliss.\n\n[207] \"In the eleventh and twelfth sonnets the alternatives of\nsecondary punishment,--solitary imprisonment, and transportation,--are\nadverted to.\" (Sir Henry Taylor.)--ED.\n\n\nXII\n\n\"SEE THE CONDEMNED ALONE WITHIN HIS CELL\"\n\n    See the Condemned alone within his cell\n    And prostrate at some moment when remorse\n    Stings to the quick, and, with resistless force,\n    Assaults the pride she strove in vain to quell.\n    Then mark him, him who could so long rebel,                        5\n    The crime confessed, a kneeling Penitent\n    Before the Altar, where the Sacrament\n    Softens his heart, till from his eyes outwell\n    Tears of salvation. Welcome death! while Heaven\n    Does in this change exceedingly rejoice;                          10\n    While yet the solemn heed the State hath given\n    Helps him to meet the last Tribunal's voice\n    In faith, which fresh offences, were he cast\n    On old temptations, might for ever blast.\n\n\nXIII[208]\n\nCONCLUSION\n\n    Yes, though He well may tremble at the sound\n    Of his own voice, who from the judgment-seat\n    Sends the pale Convict to his last retreat\n    In death; though Listeners shudder all around,\n    They know the dread requital's source profound;                    5\n    Nor is, they feel, its wisdom obsolete--\n    (Would that it were!) the sacrifice unmeet\n    For Christian Faith. But hopeful signs abound;\n    The social rights of man breathe purer air;\n    Religion deepens her preventive care;                             10\n    Then, moved by needless fear of past abuse,\n    Strike not from Law's firm hand that awful rod,\n    But leave it thence to drop for lack of use:\n    Oh, speed the blessed hour, Almighty God!\n\n[208] \"In the thirteenth sonnet he anticipates that a time may come\nwhen the punishment of death will be needed no longer; but he wishes\nthat the disuse of it should grow out of the absence of the need, not\nbe imposed by legislation.\" (Sir Henry Taylor.)--ED.\n\n\nXIV\n\nAPOLOGY\n\n    The formal World relaxes her cold chain\n    For One who speaks in numbers; ampler scope\n    His utterance finds; and, conscious of the gain,\n    Imagination works with bolder hope\n    The cause of grateful reason to sustain;                           5\n    And, serving Truth, the heart more strongly beats\n    Against all barriers which his labour meets\n    In lofty place, or humble Life's domain.\n    Enough;--before us lay a painful road,\n    And guidance have I sought in duteous love                        10\n    From Wisdom's heavenly Father. Hence hath flowed\n    Patience, with trust that, whatsoe'er the way\n    Each takes in this high matter, all may move\n    Cheered with the prospect of a brighter day.\n\n    1840.[209]\n\n[209] In the editions of 1842, 1845, and 1850 the date \"1840\" follows\nthis poem. It may have been written in that year.--ED.\n\n\n\"MEN OF THE WESTERN WORLD! IN FATE'S DARK BOOK\"\n\nPublished 1842\n\nOne of the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n    Men of the Western World! in Fate's dark book\n    Whence these opprobrious leaves of dire portent?\n    Think ye your British Ancestors forsook\n    Their native Land, for outrage provident;\n    From unsubmissive necks the bridle shook                           5\n    To give, in their Descendants, freer vent\n    And wider range to passions turbulent,\n    To mutual tyranny a deadlier look?\n    Nay, said a voice, soft as the south wind's breath,\n    Dive through the stormy surface of the flood                      10\n    To the great current flowing underneath;\n    Explore the countless springs of silent good;\n    So shall the truth be better understood,\n    And thy grieved Spirit brighten strong in faith.[210]\n\n[210] These lines were written several years ago, when reports\nprevailed of cruelties committed in many parts of America, by men\nmaking a law of their own passions. A far more formidable, as being a\nmore deliberate mischief, has appeared among those States, which have\nlately broken faith with the public creditor in a manner so infamous.\nI cannot, however, but look at both evils under a similar relation to\ninherent good, and hope that the time is not distant when our brethren\nof the West will wipe off this stain from their name and nation.\n\nADDITIONAL NOTE.\n\nI am happy to add that this anticipation is already partly realised;\nand that the reproach addressed to the Pennsylvanians is no longer\napplicable to them. I trust that those other states to which it may yet\napply will soon follow the example now set them by Philadelphia, and\nredeem their credit with the world.--W.W. 1850.\n\n\"This editorial note is on a fly-leaf at the end of the fifth volume of\nthe edition, which was completed only a short time before the Poet's\ndeath. It contains probably the last sentences composed by him for the\npress. It was promptly added by him in consequence of a suggestion\nfrom me, that the sonnet addressed \"_To Pennsylvanians_\" was no longer\njust--a fact which is mentioned to shew that the fine sense of truth\nand justice which distinguish his writings was active to the last.\"\n(Note to Professor Reed's American Edition of 1851.)--ED.\n\n\n\n\n1840\n\nOnly four poems, viz. _Poor Robin_, two sonnets referring to Miss\nGillies, and one on Haydon's portrait of the Duke of Wellington, belong\nto 1840.--ED.\n\n\nTO A PAINTER\n\nComposed 1840.--Published 1842\n\n[The picture which gave occasion to this and the following sonnet was\nfrom the pencil of Miss M. Gillies, who resided for several weeks under\nour roof at Rydal Mount.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    All praise the Likeness by thy skill portrayed;[211]\n    But 'tis a fruitless task to paint for me,\n    Who, yielding not to changes Time has made,\n    By the habitual light of memory see\n    Eyes unbedimmed, see bloom that cannot fade,                       5\n    And smiles that from their birth-place ne'er shall flee\n    Into the land where ghosts and phantoms be;\n    And, seeing this, own nothing in its stead.\n    Couldst thou go back into far-distant years,\n    Or share with me, fond thought! that inward eye,[212]             10\n    Then, and then only, Painter! could thy Art\n    The visual powers of Nature satisfy,\n    Which hold, whate'er to common sight appears,\n    Their sovereign empire in a faithful heart.\n\n[211] Miss Gillies told me that she visited Rydal Mount in 1841, at the\ninvitation of the Wordsworths, to make a miniature portrait of the poet\non ivory, which had been commissioned by Mr. Moon, the publisher, for\nthe purpose of engraving. An engraving of this portrait was published\non the 6th of August 1841. The original is now in America. I think she\nmust have been wrong in her memory of the year, which was 1840. Miss\nGillies also told me that the Wordsworths were so pleased with what she\nhad done for Mr. Moon that they wished a replica for themselves, with\nMrs. Wordsworth added. She painted this; and a copy of it, subsequently\ntaken for Miss Quillinan, was long in her possession at Loughrigg\nHolme. It now belongs to Mr. Gordon Wordsworth. It is to the portrait\nof Mrs. Wordsworth that this sonnet and the next refer.--ED.\n\n[212] Compare the lines in vol. iii. p. 5--\n\n    They flash upon that inward eye\n    Which is the bliss of solitude.\n\nThe fact that these two lines had been added by Mrs. Wordsworth (see\nnote to the poem, p. 7) was doubtless remembered by the poet, when he\nwrote this sonnet suggested by her portrait.--ED.\n\n\nON THE SAME SUBJECT\n\nComposed 1840.--Published 1842\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Though I beheld at first with blank surprise\n    This Work, I now have gazed on it so long\n    I see its truth with unreluctant eyes;\n    O, my Belov\u00e8d! I have done thee wrong,\n    Conscious of blessedness, but, whence it sprung,                   5\n    Ever too heedless, as I now perceive:\n    Morn into noon did pass, noon into eve,\n    And the old day was welcome as the young,\n    As welcome, and as beautiful--in sooth\n    More beautiful, as being a thing more holy:                       10\n    Thanks to thy virtues, to the eternal youth\n    Of all thy goodness, never melancholy;\n    To thy large heart and humble mind, that cast\n    Into one vision, future, present, past.[213]\n\n[213] Compare--\n\n    O dearer far than light and life are dear (1824).\n    Let other bards of angels sing (1824).\n    Such age how beautiful! O Lady bright (1827).\n    What heavenly smiles! O Lady mine (1845).\n\nED.\n\n\nPOOR ROBIN[214]\n\nComposed March 1840.--Published 1842\n\n[I often ask myself what will become of Rydal Mount after our day.\nWill the old walls and steps remain in front of the house and about\nthe grounds, or will they be swept away with all the beautiful mosses\nand ferns and wild geraniums and other flowers which their rude\nconstruction suffered and encouraged to grow among them?[215]--This\nlittle wild flower--\"Poor Robin\"--is here constantly courting my\nattention, and exciting what may be called a domestic interest with the\nvarying aspects of its stalks and leaves and flowers.[216] Strangely do\nthe tastes of men differ according to their employment and habits of\nlife. \"What a nice well would that be,\" said a labouring man to me one\nday, \"if all that rubbish was cleared off.\" The \"_rubbish_\" was some of\nthe most beautiful mosses and lichens and ferns and other wild growths\nthat could possibly be seen. Defend us from the tyranny of trimness and\nneatness showing itself in this way! Chatterton says of freedom--\"Upon\nher head wild weeds were spread,\" and depend upon it if \"the marvellous\nboy\" had undertaken to give Flora a garland, he would have preferred\nwhat we are apt to call weeds to garden flowers. True taste has an eye\nfor both. Weeds have been called flowers out of place. I fear the place\nmost people would assign to them is too limited. Let them come near to\nour abodes, as surely they may, without impropriety or disorder.--I.F.]\n\nOne of the \"Miscellaneous Poems.\"--ED.\n\n    Now when the primrose makes a splendid show,\n    And lilies face the March-winds in full blow,\n    And humbler growths as moved with one desire\n    Put on, to welcome spring, their best attire,\n    Poor Robin is yet flowerless; but how gay                          5\n    With his red stalks upon this sunny day!\n    And, as his tufts[217] of leaves he spreads, content\n    With a hard bed and scanty nourishment,\n    Mixed with the green, some shine not lacking power\n    To rival summer's brightest scarlet flower;                       10\n    And flowers they well might seem to passers-by\n    If looked at only with a careless eye;\n    Flowers--or a richer produce (did it suit\n    The season) sprinklings of ripe strawberry fruit.\n    But while a thousand pleasures come unsought,                     15\n    Why fix upon his wealth or want[218] a thought?\n    Is the string touched in prelude to a lay\n    Of pretty fancies that would round him play\n    When all the world acknowledged elfin sway?\n    Or does it suit our humour to commend                             20\n    Poor Robin as a sure and crafty friend,\n    Whose practice teaches, spite of names to show\n    Bright colours whether they deceive or no?--\n    Nay, we would simply praise the free good-will\n    With which, though slighted, he, on naked hill                    25\n    Or in warm valley, seeks his part to fill;\n    Cheerful alike if bare of flowers as now,\n    Or when his tiny gems shall deck his brow:\n    Yet more, we wish that men by men despised,\n    And such as lift their foreheads overprized,                      30\n    Should sometimes think, where'er they chance to spy\n    This child of Nature's own humility,\n    What recompense is kept in store or left\n    For all that seem neglected or bereft;\n    With what nice care equivalents are given,                        35\n    How just, how bountiful, the hand of Heaven.\n\n    _March, 1840._\n\n[214] The small wild Geranium known by that name.--W.W. 1842.\n\n[215] These things remain comparatively unaltered. Rydal Mount has\nsuffered little in picturesqueness since Wordsworth's death; while the\nhouse, and the grounds, have gained in many ways by what the present\ntenant has done for them. It is impossible to keep such a place exactly\nas it was left by its greatest tenant; and Mr. Crewdson has certainly\nnot injured, but wisely improved the place.--ED.\n\n[216] Compare what is said of it in the _Memoirs of Wordsworth_, by his\nnephew, vol. i. p. 20.--ED.\n\n[217] 1849.\n\n    \u2026 tuft\n\n    1842.\n\n[218] 1845.\n\n    \u2026 want or wealth\n\n    1842.\n\n\nON A PORTRAIT OF THE DUKE OF WELLINGTON UPON THE FIELD OF WATERLOO, BY\nHAYDON[219]\n\nComposed August 31, 1840.--Published 1842\n\n[This was composed while I was ascending Helvellyn in company with my\ndaughter and her husband. She was on horseback, and rode to the top\nof the hill without once dismounting, a feat which it was scarcely\npossible to perform except during a season of dry weather; and a guide,\nwith whom we fell in on the mountain, told us he believed it had never\nbeen accomplished before by any one.--I.F.]\n\nOne of the \"Miscellaneous Sonnets\"; but first published in the \"Poems\nchiefly of Early and Late Years.\"--ED.\n\n    By Art's bold privilege Warrior and War-horse stand\n    On ground yet strewn with their last battle's wreck;\n    Let the Steed glory while his Master's hand\n    Lies fixed for ages on his conscious neck;\n    But by the Chieftain's look, though at his side                    5\n    Hangs that day's treasured sword, how firm a check\n    Is given to triumph and all human pride!\n    Yon trophied Mound shrinks to a shadowy speck\n    In his calm presence! Him the mighty deed\n    Elates not, brought far nearer the grave's rest,                  10\n    As shows that time-worn face, for he such seed\n    Has sown as yields, we trust, the fruit of fame\n    In Heaven;[220] hence no one blushes for thy name,\n    Conqueror, 'mid some sad thoughts, divinely blest!\n\n[219] Haydon worked at this picture of Wellington from June to\nNovember, 1839. (See his Autobiography, vol. iii. pp. 108-131.) He\nwrites under date, Sept. 4, 1840:--\"Hard at work. I heard from dear\nWordsworth, with a glorious sonnet on the Duke, and Copenhagen.\u2020 It is\nvery fine, and I began a new journal directly, and put in the sonnet.\nGod bless him.\" The following is part of Wordsworth's letter:--\n\n\"MY DEAR HAYDON,--We are all charmed with your etching. It is both\npoetically and pictorially conceived, and finely executed. I should\nhave written immediately to thank you for it, and for your letter\nand the enclosed one, which is interesting, but I wished to gratify\nyou by writing a sonnet. I now send it, but with an earnest request\nthat it may not be put into circulation for some little time, as it\nis warm from the brain, and may require, in consequence, some little\nretouching. It has this, at least, remarkable attached to it, which\nwill add to its value in your eyes, that it was actually composed while\nI was climbing Helvellyn last Monday.\"--ED.\n\n    \u2020 Wellington's war-horse.--ED.\n\n[220] 1842.\n\n                            \u2026 Since the mighty deed\n    Him years have brought far nearer the grave's rest,\n    He shows that face time-worn. But he such seed\n    Has sowed that bears, we trust, the fruit of fame\n    In Heaven.\u2026\n\n    From a copy sent to Haydon.\n\n\n\n\n1841\n\n\nEPITAPH\n\nIN THE CHAPEL-YARD OF LANGDALE, WESTMORELAND\n\nComposed 1841.--Published 1842\n\n[OWEN LLOYD, the subject of this epitaph, was born at Old Brathay,\nnear Ambleside, and was the son of Charles Lloyd and his wife Sophia\n(_n\u00e9e_ Pemberton), both of Birmingham, who came to reside in this part\nof the country, soon after their marriage. They had many children,\nboth sons and daughters, of whom the most remarkable was the subject\nof this epitaph. He was educated under Mr. Dawes, at Ambleside, Dr.\nButler, of Shrewsbury, and lastly at Trinity College, Cambridge, where\nhe would have been greatly distinguished as a scholar but for inherited\ninfirmities of bodily constitution, which, from early childhood,\naffected his mind. His love for the neighbourhood in which he was\nborn, and his sympathy with the habits and characters of the mountain\nyeomanry, in conjunction with irregular spirits, that unfitted him for\nfacing duties in situations to which he was unaccustomed, induced him\nto accept the retired curacy of Langdale. How much he was beloved and\nhonoured there, and with what feelings he discharged his duty under the\noppression of severe malady, is set forth, though imperfectly, in the\nepitaph.--I.F.]\n\nOne of the \"Epitaphs and Elegiac Pieces.\"--ED.\n\n    By playful smiles, (alas! too oft\n    A sad heart's sunshine) by a soft\n    And gentle nature, and a free\n    Yet modest hand of charity,\n    Through life was OWEN LLOYD endeared                               5\n    To young and old; and how revered\n    Had been that pious spirit, a tide\n    Of humble mourners testified,\n    When, after pains dispensed to prove\n    The measure of God's chastening love,                             10\n    Here, brought from far, his corse found rest,--\n    Fulfilment of his own request;--\n    Urged less for this Yew's shade, though he\n    Planted with such fond hope the tree;\n    Less for the love of stream and rock,                             15\n    Dear as they were, than that his Flock,\n    When they no more their Pastor's voice\n    Could hear to guide them in their choice\n    Through good and evil, help might have,\n    Admonished, from his silent grave,                                20\n    Of righteousness, of sins forgiven,\n    For peace on earth and bliss in heaven.\n\nThis commemorative epitaph to the Rev. Owen Lloyd--the friend of\nHartley Coleridge and of Faber--is carved on the headstone over his\ngrave in the churchyard at the small hamlet of Chapel Stile, Great\nLangdale, Westmoreland. The stone also carries the inscription, \"To\nthe memory of Owen Lloyd, M.A., nearly twelve years incumbent of this\nchapel. Born at Old Brathay, March 31, 1803, died at Manchester, April\n18, 1841, aged 38.\" See a letter of Wordsworth's referring to Lloyd\namongst his letters in a subsequent volume. In a previous edition I\nerred by giving this poem an earlier date. Professor Dowden has shown\nthe true one conclusively.\n\nWriting from Rydal on 11th August 1841, to his brother Christopher,\nWordsworth said, \"I send you with the last corrections an epitaph which\nI have just written for poor Owen Lloyd. His brother Edward forwarded\nfor my perusal some verses which he had composed with a view to that\nobject; but he expressed a wish that I would compose something myself.\nNot approving Edward's lines altogether, though the sentiments were\nsufficiently appropriate, I sent him what I now forward to you, or\nrather the substance of it, for something has been added, and some\nchange of expression introduced. I hope you will approve of it. I find\nno fault with it myself, the circumstances considered, except that it\nis too long for an Epitaph, but this was inevitable if the memorial was\nto be as conspicuous as the subject required, at least according to the\nlight in which it offered itself to my mind.\"--ED.\n\n\n\n\n1842\n\nThe poems of 1842 include _The Floating Island_, _The Norman Boy_, _The\nPoet's Dream_, _Airey-Force Valley_, the lines _To the Clouds_, and a\nnumber of miscellaneous sonnets.--ED.\n\n\n\"INTENT ON GATHERING WOOL FROM HEDGE AND BRAKE\"\n\nComposed 8th March 1842.--Published 1842\n\n[Suggested by a conversation with Miss Fenwick, who along with her\nsister had, during their childhood, found much delight in such\ngatherings for the purposes here alluded to.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Intent on gathering wool from hedge and brake\n    Yon busy Little-ones rejoice that soon\n    A poor old Dame will bless them for the boon:\n    Great is their glee while flake they add to flake\n    With rival earnestness; far other strife                           5\n    Than will hereafter move them, if they make\n    Pastime their idol, give their day of life\n    To pleasure snatched for reckless pleasure's sake.\n    Can pomp and show allay one heart-born grief?\n    Pains which the World inflicts can she requite?                   10\n    Not for an interval however brief;\n    The silent thoughts that search for stedfast light,\n    Love from her depths,[221] and Duty in her might,\n    And Faith--these only yield secure relief.\n\n    _March 8th, 1842._\n\n[221] 1845.\n\n    Love from on high, \u2026\n\n    1842.\n\n\nPRELUDE,\n\nPREFIXED TO THE VOLUME ENTITLED \"POEMS CHIEFLY OF EARLY AND LATE YEARS\"\n\nComposed March 26, 1842.--Published 1842\n\n[These verses were begun while I was on a visit to my son John at\nBrigham, and were finished at Rydal. As the contents of the volume,\nto which they are now prefixed, will be assigned to their respective\nclasses when my poems shall be collected in one volume, I should be at\na loss where with propriety to place this prelude, being too restricted\nin its bearing to serve for a preface for the whole. The lines towards\nthe conclusion allude to the discontents then fomented through the\ncountry by the agitators of the Anti-Corn-Law League: the particular\ncauses of such troubles are transitory, but disposition to excite and\nliability to be excited are nevertheless permanent, and therefore\nproper objects for the poet's regard.--I.F.]\n\nOne of the \"Miscellaneous Poems.\"--ED.\n\n    In desultory walk through orchard grounds,\n    Or some deep chestnut grove, oft have I paused\n    The while a Thrush, urged rather than restrained\n    By gusts of vernal storm, attuned his song\n    To his own genial instincts; and was heard                         5\n    (Though not without some plaintive tones between)\n    To utter, above showers of blossom swept\n    From tossing boughs, the promise of a calm,\n    Which the unsheltered traveller might receive\n    With thankful spirit. The descant, and the wind                   10\n    That seemed to play with it in love or scorn,\n    Encouraged and endeared the strain of words\n    That haply flowed from me, by fits of silence\n    Impelled to livelier pace. But now, my Book!\n    Charged with those lays, and others of like mood,                 15\n    Or loftier pitch if higher rose the theme,\n    Go, single--yet aspiring to be joined\n    With thy Forerunners that through many a year\n    Have faithfully prepared each other's way--\n    Go forth upon a mission best fulfilled                            20\n    When and wherever, in this changeful world,\n    Power hath been given to please for higher ends\n    Than pleasure only; gladdening to prepare\n    For wholesome sadness, troubling to refine,\n    Calming to raise; and, by a sapient Art                           25\n    Diffused through all the mysteries of our Being,\n    Softening the toils and pains that have not ceased\n    To cast their shadows on our mother Earth\n    Since the primeval doom. Such is the grace\n    Which, though unsued for, fails not to descend                    30\n    With heavenly inspiration; such the aim\n    That Reason dictates; and, as even the wish\n    Has virtue in it, why should hope to me\n    Be wanting that sometimes, where fancied ills\n    Harass the mind and strip from off the bowers                     35\n    Of private life their natural pleasantness,\n    A Voice--devoted to the love whose seeds\n    Are sown in every human breast, to beauty\n    Lodged within compass of the humblest sight,\n    To cheerful intercourse with wood and field,                      40\n    And sympathy with man's substantial griefs--\n    Will not be heard in vain? And in those days\n    When unforeseen distress spreads far and wide\n    Among a People mournfully cast down,\n    Or into anger roused by venal words                               45\n    In recklessness flung out to overturn\n    The judgment, and divert the general heart\n    From mutual good--some strain of thine, my Book!\n    Caught at propitious intervals, may win\n    Listeners who not unwillingly admit                               50\n    Kindly emotion tending to console\n    And reconcile; and both with young and old\n    Exalt the sense of thoughtful gratitude\n    For benefits that still survive, by faith\n    In progress, under laws divine, maintained.                       55\n\nRYDAL MOUNT, _March 26, 1842_.\n\n\nFLOATING ISLAND\n\nPublished 1842\n\nThese lines are by the Author of the _Address to the Wind_, etc.,\npublished heretofore along with my Poems. Those to a Redbreast are by a\ndeceased female Relative.--W.W. 1842.\n\n[My poor sister takes a pleasure in repeating these verses, which she\ncomposed not long before the beginning of her sad illness.--I.F.]\n\nOne of the \"Miscellaneous Poems.\"--ED.\n\n    Harmonious Powers with Nature work\n    On sky, earth, river, lake, and sea;\n    Sunshine and cloud, whirlwind and breeze,\n    All in one duteous task agree.\n\n    Once did I see a slip of earth                                     5\n    (By throbbing waves long undermined)\n    Loosed from its hold; how, no one knew,\n    But all might see it float, obedient to the wind;\n\n    Might see it, from the mossy shore\n    Dissevered, float upon the Lake,                                  10\n    Float with its crest of trees adorned\n    On which the warbling birds their pastime take.\n\n    Food, shelter, safety, there they find;\n    There berries ripen, flowerets bloom;\n    There insects live their lives, and die;                          15\n    A peopled world it is; in size a tiny room.\n\n    And thus through many seasons' space\n    This little Island may survive;\n    But Nature, though we mark her not,\n    Will take away, may cease to give.                                20\n\n    Perchance when you are wandering forth\n    Upon some vacant sunny day,\n    Without an object, hope, or fear,\n    Thither your eyes may turn--the Isle is passed away;\n\n    Buried beneath the glittering Lake,                               25\n    Its place no longer to be found;\n    Yet the lost fragments shall remain\n    To fertilize some other ground.\n\n                                                   D. W.\n\nThere is one of these floating islands in Loch Lomond in Argyll,\nanother in Loch Dochart in Perthshire, and another in Loch Treig\nin Inverness. Their origin is probably due to a mass of peat being\ndetached from the shore, and floated out into the lake. A mass of\nvegetable matter, however, has sometimes risen from the bottom of the\nwater, and assumed for a time all the appearance of an island. This\nhas been probably due to an accumulation of gas, within or under the\ndetached portion, produced by the decay of vegetation in extremely hot\nweather.\n\nSouthey, in an unpublished letter to Sir George Beaumont (10th July\n1824), thus describes the Island at Derwentwater: \"You will have seen\nby the papers that the Floating Island has made its appearance. It\nsank again last week, when some heavy rains had raised the lake four\nfeet. By good fortune Professor Sedgewick happened to be in Keswick,\nand examined it in time. Where he probed it a thin layer of mud lies\nupon a bed of peat, which is six feet thick, and this rests upon a\nstratum of fine white clay,--the same I believe which Miss Barker\nfound in Borrowdale when building her unlucky house. Where the gas is\ngenerated remains yet to be discovered, but when the peat is filled\nwith this gas, it separates from the clay and becomes buoyant. There\nmust have been a considerable convulsion when this took place, for a\nrent was made in the bottom of the lake, several feet in depth, and\nnot less than fifty yards long, on each side of which the bottom rose\nand floated. It was a pretty sight to see the small fry exploring this\nnew made strait and darting at the bubbles which rose as the Professor\nwas probing the bank. The discharge of air was considerable here, when\na pole was thrust down. But at some distance where the rent did not\nextend, the bottom had been heaved up in a slight convexity, sloping\nequally in an inclined plane all round: and there, when the pole was\nintroduced, a rush like a jet followed, as it was withdrawn. The thing\nis the more curious, because as yet no example of it is known to have\nbeen observed in any other place.\"\n\nAnother of these detached islands used to float about in Esthwaite\nWater, and was carried from side to side of the pool at the north end\nof the lake--the same pool which the swans, described in _The Prelude_,\nused to frequent. This island had a few bushes on it: but it became\nstranded some time ago. One of the old natives of Hawkeshead described\nthe process of trying to float it off again, by tying ropes to the\nbushes on its surface,--an experiment which was unsuccessful. Compare\nthe reference to the Floating or \"Buoyant\" Island of Derwentwater, and\nto the \"mossy islet\" of Esthwaite, in Wordsworth's _Guide through the\nDistrict of the Lakes_.--ED.\n\n\n\"THE CRESCENT-MOON, THE STAR OF LOVE\"\n\nPublished 1842\n\nOne of the \"Evening Voluntaries.\"--Ed.\n\n    The Crescent-moon, the Star of Love,\n      Glories of evening, as ye there are seen\n      With but a span of sky between--\n      Speak one of you, my doubts remove,\n    Which is the attendant Page and which the Queen?\n\n\n\"_A POET!_--HE HATH PUT HIS HEART TO SCHOOL\"\n\nPublished 1842\n\n[I was impelled to write this Sonnet by the disgusting frequency with\nwhich the word _artistical_, imported with other impertinences from the\nGermans, is employed by writers of the present day: for artistical\nlet them substitute artificial, and the poetry written on this system,\nboth at home and abroad, will be for the most part much better\ncharacterised.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    _A Poet!_--He hath put his heart to school,\n    Nor dares to move unpropped upon the staff\n    Which Art hath lodged within his hand--must laugh\n    By precept only, and shed tears by rule.\n    Thy Art be Nature; the live current quaff,                         5\n    And let the groveller sip his stagnant pool,\n    In fear that else, when Critics grave and cool\n    Have killed him, Scorn should write his epitaph.[222]\n    How does the Meadow-flower its bloom unfold?\n    Because the lovely little flower is free                          10\n    Down to its root, and, in that freedom, bold;\n    And so the grandeur of the Forest-tree\n    Comes not by casting in a formal mould,\n    But from its _own_ divine vitality.\n\n[222] Compare _A Poet's Epitaph_ (vol. ii. p. 75).--ED.\n\n\n\"THE MOST ALLURING CLOUDS THAT MOUNT THE SKY\"\n\nPublished 1842\n\n[Hundreds of times have I seen, hanging about and above the vale\nof Rydal, clouds that might have given birth to this sonnet, which\nwas thrown off on the impulse of the moment one evening when I was\nreturning from the favourite walk of ours, along the Rotha, under\nLoughrigg.--I.F.]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    The most alluring clouds that mount the sky\n    Owe to a troubled element their forms,\n    Their hues to sunset. If with raptured eye\n    We watch their splendour, shall we covet storms,\n    And wish the Lord of day his slow decline                          5\n    Would hasten, that such pomp may float on high?\n    Behold, already they forget to shine,\n    Dissolve--and leave to him who gazed a sigh.\n    Not loth to thank each moment for its boon\n    Of pure delight, come whensoe'er[223] it may,                     10\n    Peace let us seek,--to stedfast things attune\n    Calm expectations, leaving to the gay\n    And volatile their love of transient bowers,\n    The house that cannot pass away be ours.[224]\n\n[223] 1849\n\n    \u2026 whencesoe'er \u2026\n\n    1842.\n\n[224] Compare _To the Clouds_, I. 94, p. 145.--ED.\n\n\n\"FEEL FOR THE WRONGS TO UNIVERSAL KEN\"\n\nPublished 1842\n\n[This Sonnet is recommended to the perusal of those who consider that\nthe evils under which we groan are to be removed or palliated by\nmeasures ungoverned by moral and religious principles.--I.F.]\n\nOne of the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n    Feel for the wrongs to universal ken\n    Daily exposed, woe that unshrouded lies;\n    And seek the Sufferer in his darkest den,\n    Whether conducted to the spot by sighs\n    And moanings, or he dwells (as if the wren                         5\n    Taught him concealment) hidden from all eyes\n    In silence and the awful modesties\n    Of sorrow;--feel for all, as brother Men!\n    Rest not in hope want's icy chain to thaw\n    By casual boons and formal charities;[225]                        10\n    Learn to be just, just through impartial law;\n    Far as ye may, erect and equalise;\n    And, what ye cannot reach by statute, draw\n    Each from his fountain of self-sacrifice!\n\n[225] 1845.\n\n                                    \u2026 Men!--\n    Feel for the Poor,--but not to still your qualms\n    By formal charity or dole of alms;\n    Learn \u2026\n\n    1842.\n\n\nIN ALLUSION TO VARIOUS RECENT HISTORIES AND NOTICES OF THE FRENCH\nREVOLUTION\n\nPublished 1842\n\nOne of the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n    Portentous change when History can appear\n    As the cool Advocate of foul device;[226]\n    Reckless audacity extol, and jeer\n    At consciences perplexed with scruples nice!\n    They who bewail not, must abhor, the sneer                         5\n    Born of Conceit, Power's blind Idolater;\n    Or haply sprung from vaunting Cowardice\n    Betrayed by mockery of holy fear.\n    Hath it not long been said the wrath of Man\n    Works not the righteousness of God? Oh bend,                      10\n    Bend, ye Perverse! to judgments from on High,\n    Laws that lay under Heaven's perpetual ban\n    All principles of action that transcend\n    The sacred limits of humanity.\n\n[226] Wordsworth wrote this sonnet against Carlyle's _French\nRevolution_ in particular. Carlyle knew it, and this may in\npart--although only in part--account for Carlyle's indifference to\nWordsworth.--ED.\n\n\nCONTINUED\n\nPublished 1842\n\nOne of the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n    Who ponders National events shall find\n    An awful balancing of loss and gain,\n    Joy based on sorrow, good with ill combined,\n    And proud deliverance issuing out of pain\n    And direful throes; as if the All-ruling Mind,                     5\n    With whose perfection it consists to ordain\n    Volcanic burst, earthquake, and hurricane,\n    Dealt in like sort with feeble human kind\n    By laws immutable. But woe for him\n    Who thus deceived shall lend an eager hand                        10\n    To social havoc. Is not Conscience ours,\n    And Truth, whose eye guilt only can make dim;\n    And Will, whose office, by divine command,\n    Is to control and check disordered Powers?\n\n\nCONCLUDED\n\nPublished 1842\n\nOne of the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n    Long-favoured England! be not thou misled\n    By monstrous theories of alien growth,\n    Lest alien frenzy seize thee, waxing wroth,\n    Self-smitten till thy garments reek dyed red\n    With thy own blood, which tears in torrents shed                   5\n    Fail to wash out, tears flowing ere thy troth\n    Be plighted, not to ease but sullen sloth,\n    Or wan despair--the ghost of false hope fled\n    Into a shameful grave. Among thy youth,\n    My Country! if such warning be held dear,                         10\n    Then shall a Veteran's heart be thrilled with joy,\n    One who would gather from eternal truth,\n    For time and season, rules that work to cheer--\n    Not scourge, to save the People--not destroy.\n\n\n\"LO! WHERE SHE STANDS FIXED IN A SAINT-LIKE TRANCE\"\n\nPublished 1842\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Lo! where she stands fixed in a saint-like trance,\n    One upward hand, as if she needed rest\n    From rapture, lying softly on her breast!\n    Nor wants her eyeball an ethereal glance;\n    But not the less--nay more--that countenance,                      5\n    While thus illumined, tells of painful strife\n    For a sick heart made weary of this life\n    By love, long crossed with adverse circumstance.\n    --Would She were now as when she hoped to pass\n    At God's appointed hour to them who tread                         10\n    Heaven's sapphire pavement, yet breathed well content,\n    Well pleased, her foot should print earth's common grass,\n    Lived thankful for day's light, for daily bread,\n    For health, and time in obvious duty spent.\n\n\nTHE NORMAN BOY\n\nPublished 1842\n\n[The subject of this poem was sent me by Mrs. Ogle, to whom I was\npersonally unknown, with a hope on her part that I might be induced\nto relate the incident in verse; and I do not regret that I took the\ntrouble, for not improbably the fact is illustrative of the boy's\nearly piety, and may concur with my other little pieces on children\nto produce profitable reflection among my youthful readers. This is\nsaid, however, with an absolute conviction that children will derive\nmost benefit from books which are not unworthy the perusal of persons\nof any age. I protest with all my heart against those productions, so\nabundant in the present day, in which the doings of children are dwelt\nupon as if they were incapable of being interested in anything else. On\nthis subject I have dwelt at length in the poem on the growth of my own\nmind.--I.F.]\n\nOne of the \"Poems referring to the Period of Childhood.\"--ED.\n\n  High on a broad unfertile tract of forest-skirted Down,\n  Nor kept by Nature for herself, nor made by man his own,\n  From home and company remote and every playful joy,\n  Served, tending a few sheep and goats, a ragged Norman boy.\n\n  Him never saw I, nor the spot; but from an English Dame,             5\n  Stranger to me and yet my friend, a simple notice came,\n  With suit that I would speak in verse of that sequestered child\n  Whom, one bleak winter's day, she met upon the dreary Wild.\n\n  His flock, along the woodland's edge with relics sprinkled o'er\n  Of last night's snow, beneath a sky threatening the fall of more,   10\n  Where tufts of herbage tempted each, were busy at their feed,\n  And the poor Boy was busier still, with work of anxious heed.\n\n  There _was_ he, where of branches rent and withered and decayed,\n  For covert from the keen north wind, his hands a hut had made.\n  A tiny tenement, forsooth, and frail, as needs must be              15\n  A thing of such materials framed, by a builder such as he.\n\n  The hut stood finished by his pains, nor seemingly lacked aught\n  That skill or means of his could add, but the architect had wrought\n  Some limber twigs into a Cross, well-shaped with fingers nice,\n  To be engrafted on the top of his small edifice.                    20\n\n  That Cross he now was fastening there, as the surest power and best\n  For supplying all deficiencies, all wants of the rude nest\n  In which, from burning heat, or tempest driving far and wide,\n  The innocent Boy, else shelterless, his lonely head must hide.\n\n  That Cross belike he also raised as a standard for the true         25\n  And faithful service of his heart in the worst that might ensue\n  Of hardship and distressful fear, amid the houseless waste\n  Where he, in his poor self so weak, by Providence was placed.\n\n  ----Here, Lady! might I cease; but nay, let _us_ before we part\n  With this dear holy shepherd-boy breathe a prayer of earnest heart, 30\n  That unto him, where'er shall lie his life's appointed way,\n  The Cross, fixed in his soul, may prove an all-sufficing stay.\n\n\nTHE POET'S DREAM[227]\n\nSEQUEL TO THE NORMAN BOY\n\nPublished 1842\n\nOne of the \"Poems referring to the Period of Childhood.\"--ED.\n\n  Just as those final words were penned, the sun broke out in power,\n  And gladdened all things; but, as chanced, within that very hour,\n  Air blackened, thunder growled, fire flashed from clouds that hid\n     the sky,\n  And, for the Subject of my Verse, I heaved a pensive sigh.\n\n  Nor could my heart by second thoughts from heaviness be cleared,     5\n  For bodied forth before my eyes the cross-crowned hut appeared;\n  And, while around it storm as fierce seemed troubling earth\n     and air,\n  I saw, within, the Norman Boy kneeling alone in prayer.\n\n  The Child, as if the thunder's voice spake with articulate call,\n  Bowed meekly in submissive fear, before the Lord of All;            10\n  His lips were moving; and his eyes, upraised to sue for grace,\n  With soft illumination cheered the dimness of that place.\n\n  How beautiful is holiness!--what wonder if the sight,\n  Almost as vivid as a dream, produced a dream at night?\n  It came with sleep and showed the Boy, no cherub, not transformed,  15\n  But the poor ragged Thing whose ways my human heart had warmed.\n\n  Me had the dream equipped with wings, so I took him in my arms,\n  And lifted from the grassy floor, stilling his faint alarms,\n  And bore him high through yielding air my debt of love to pay,\n  By giving him, for both our sakes, an hour of holiday.              20\n\n  I whispered, \"Yet a little while, dear Child! thou art my own,\n  To show thee some delightful thing, in country or in town.\n  What shall it be? a mirthful throng? or that holy place and calm\n  St. Denis, filled with royal tombs,[228] or the Church of Notre\n     Dame?[229]\n\n  \"St. Ouen's golden Shrine?[230] Or choose what else would please\n     thee most                                                        25\n  Of any wonder Normandy, or all proud France, can boast!\"\n  \"My Mother,\" said the Boy, \"was born near to a bless\u00e8d Tree,\n  The Chapel Oak of Allonville;[231] good Angel, show it me!\"\n\n  On wings, from broad and stedfast poise let loose by this reply,\n  For Allonville, o'er down and dale, away then did we fly;           30\n  O'er town and tower we flew, and fields in May's fresh verdure\n     drest;\n  The wings they did not flag; the Child, though grave, was not\n     deprest.\n\n  But who shall show, to waking sense, the gleam of light that\n     broke\n  Forth from his eyes, when first the Boy looked down on that\n     huge oak,\n  For length of days so much revered, so famous where it stands       35\n  For twofold hallowing--Nature's care, and work of human hands?\n\n  Strong as an Eagle with my charge I glided round and round\n  The wide-spread boughs, for view of door, window, and stair that\n     wound\n  Gracefully up the gnarled trunk; nor left we unsurveyed\n  The pointed steeple peering forth from the centre of the shade.     40\n\n  I lighted--opened with soft touch the chapel's iron door,[232]\n  Past softly, leading in the Boy; and, while from roof to floor\n  From floor to roof all round his eyes the Child with wonder\n     cast,[233]\n  Pleasure on pleasure crowded in, each livelier than the last.\n\n  For, deftly framed within the trunk, the[234] sanctuary showed,     45\n  By light of lamp and precious stones, that glimmered here, there\n     glowed,\n  Shrine, Altar, Image, Offerings hung in sign of gratitude;\n  Sight that inspired accordant thoughts; and speech[235] I thus\n     renewed:\n\n  \"Hither the Afflicted come, as thou hast heard thy Mother say,\n  And, kneeling, supplication make to our Lady de la Paix;[236]       50\n  What mournful sighs have here been heard, and, when the voice was\n     stopt\n  By sudden pangs; what bitter tears have on this pavement dropt!\n\n  \"Poor Shepherd of the naked Down, a favoured lot is thine,\n  Far happier lot, dear Boy, than brings full many to this shrine;\n  From body pains and pains of soul thou needest no release,          55\n  Thy hours as they flow on are spent, if not in joy, in peace.\n\n  \"Then offer up thy heart to God in thankfulness and praise,\n  Give to Him prayers, and many thoughts, in thy most busy days;\n  And in His sight the fragile Cross, on thy small hut, will be\n  Holy as that which long hath crowned the Chapel of this Tree;       60\n\n  \"Holy as that far seen which crowns the sumptuous Church in Rome\n  Where thousands meet to worship God under a mighty Dome;[237]\n  He sees the bending multitude, He hears the choral rites,\n  Yet not the less, in children's hymns and lonely prayer, delights.\n\n  \"God for His service needeth not proud work of human skill;         65\n  They please Him best who labour most to do in peace His will:\n  So let us strive to live, and to our Spirits will be given\n  Such wings as, when our Saviour calls, shall bear us up to heaven.\"\n\n  The Boy no answer made by words, but, so earnest was his look,\n  Sleep fled, and with it fled the dream--recorded in this book,      70\n  Lest all that passed should melt away in silence from my mind,\n  As visions still more bright have done, and left no trace behind.\n\n  But oh! that Country-man of thine, whose eye, loved Child, can see\n  A pledge of endless bliss in acts of early piety,\n  In verse, which to thy ear might come, would treat this simple\n     theme,                                                           75\n  Nor leave untold our happy flight in that adventurous dream.[238]\n\n  Alas the dream,[239] to thee, poor Boy! to thee from whom it flowed,\n  Was nothing, scarcely can be aught, yet 'twas[240] bounteously\n     bestowed,\n  If I may dare to cherish hope that gentle eyes will read\n  Not loth, and listening Little-ones, heart-touched, their fancies\n     feed.                                                            80\n\n[227] 1845.\n\nThe title in 1842 was \"SEQUEL TO THE NORMAN BOY.\"\n\n[228] The Abbey Church of St. Denis, to the north of Paris,--one of the\nfinest specimens of French Gothic,--was the burial-place of the French\nKings for many generations.--ED.\n\n[229] In Paris.--ED.\n\n[230] The Church of St. Ouen, in Rouen, is the most perfect edifice of\nits kind in Europe.--ED.\n\n[231] \"Among ancient Trees there are few, I believe, at least in\nFrance, so worthy of attention as an Oak which may be seen in the 'Pays\nde Caux,' about a league from Yvetot, close to the church, and in the\nburial-ground of Allonville.\n\nThe height of this Tree does not answer to its girth; the trunk, from\nthe roots to the summit, forms a complete cone; and the inside of this\ncone is hollow throughout the whole of its height.\n\nSuch is the Oak of Allonville, in its state of nature. The hand of Man,\nhowever, has endeavoured to impress upon it a character still more\ninteresting, by adding a religious feeling to the respect which its age\nnaturally inspires.\n\nThe lower part of its hollow trunk has been transformed into a Chapel\nof six or seven feet in diameter, carefully wainscotted and paved, and\nan open iron gate guards the humble Sanctuary.\n\nLeading to it there is a staircase, which twists round the body of the\nTree. At certain seasons of the year divine service is performed in\nthis Chapel.\n\nThe summit has been broken off many years, but there is a surface at\nthe top of the trunk, of the diameter of a very large tree, and from it\nrises a pointed roof, covered with slates, in the form of a steeple,\nwhich is surmounted with an iron Cross, that rises in a picturesque\nmanner from the middle of the leaves, like an ancient Hermitage above\nthe surrounding Wood.\n\nOver the entrance to the Chapel an Inscription appears, which informs\nus it was erected by the Abb\u00e9 du D\u00e9troit, Curate of Allonville, in the\nyear 1696; and over a door is another, dedicating it 'To Our Lady of\nPeace.'\"--Vide 14 _No. Saturday Magazine_.--W.W. 1842.\n\n[232] 1845.\n\n    \u2026 touch a grated iron door,\n\n    1842.\n\n[233] 1845.\n\n    \u2026 his eyes the wondering creature cast,\n\n    1842.\n\n[234] 1845.\n\n    \u2026 a \u2026\n\n    1842.\n\n[235] 1845.\n\n    And swift as lightning went the time, ere speech\n\n    1842.\n\n[236] See note, p. 137.--ED.\n\n[237] St. Peter's Church.--ED.\n\n[238] This stanza was added in the edition of 1845.\n\n[239] 1845.\n\n    And though the dream, \u2026\n\n    1842.\n\n[240] 1845.\n\n    Was nothing, nor e'er can be aught, 'twas \u2026\n\n    1842.\n\n\nSUGGESTED BY A PICTURE OF THE BIRD OF PARADISE\n\nPublished 1842\n\n[This subject has been treated of in another note. I will here only, by\nway of comment, direct attention to the fact, that pictures of animals\nand other productions of Nature, as seen in conservatories, menageries,\nand museums, etc., would do little for the national mind, nay, they\nwould be rather injurious to it, if the imagination were excluded by\nthe presence of the object, more or less out of a state of Nature. If\nit were not that we learn to talk and think of the lion and the eagle,\nthe palm-tree, and even the cedar, from the impassioned introduction of\nthem so frequently into Holy Scripture, and by great poets, and divines\nwho wrote as poets, the spiritual part of our nature, and therefore\nthe higher part of it, would derive no benefit from such intercourse\nwith such subjects.--I.F.]\n\nOne of the \"Poems of the Imagination.\"--ED.\n\n    The gentlest Poet, with free thoughts endowed,\n    And a true master of the glowing strain,\n    Might scan the narrow province with disdain\n    That to the Painter's skill is here allowed.\n    This, this the Bird of Paradise! disclaim                          5\n    The daring thought, forget the name;\n    This the Sun's Bird, whom Glendoveers might own\n    As no unworthy Partner in their flight\n    Through seas of ether, where the ruffling sway\n    Of nether air's rude billows is unknown;                          10\n    Whom Sylphs, if e'er for casual pastime they\n    Through India's spicy regions wing their way,\n    Might bow to as their Lord. What character,\n    O sovereign Nature! I appeal to thee,\n    Of all thy feathered progeny                                      15\n    Is so unearthly, and what shape so fair?\n    So richly decked in variegated down,\n    Green, sable, shining yellow, shadowy brown,\n    Tints softly with each other blended,\n    Hues doubtfully begun and ended;                                  20\n    Or intershooting, and to sight\n    Lost and recovered, as the rays of light\n    Glance on the conscious plumes touched here and there?\n    Full surely, when with such proud gifts of life\n    Began the pencil's strife,                                        25\n    O'erweening Art was caught as in a snare.\n\n      A sense of seemingly presumptuous wrong\n    Gave the first impulse to the Poet's song;\n    But, of his scorn repenting soon, he drew\n    A juster judgment from a calmer view;                             30\n    And, with a spirit freed from discontent,\n    Thankfully took an effort that was meant\n    Not with God's bounty, Nature's love, to vie,\n    Or made with hope to please that inward eye\n    Which ever strives in vain itself to satisfy,                     35\n    But to recal the truth by some faint trace\n    Of power ethereal and celestial grace,\n    That in the living Creature find on earth a place.\n\n\nTO THE CLOUDS[241]\n\nPublished 1842\n\n[These verses were suggested while I was walking on the foot-road\nbetween Rydal Mount and Grasmere. The clouds were driving over the top\nof Nab-Scar across the vale: they set my thoughts a-going, and the rest\nfollowed almost immediately.--I.F.]\n\nFirst published (1842) in \"Poems chiefly of Early and Late Years,\"\nafterwards included in the \"Poems of the Imagination.\"--ED.\n\n    Army of Clouds! ye wing\u00e8d Host in troops\n    Ascending from behind the motionless brow\n    Of that tall rock,[242] as from a hidden world,\n    O whither with[243] such eagerness of speed?\n    What seek ye, or what shun ye? of the gale[244]                    5\n    Companions, fear ye to be left behind,\n    Or racing o'er[245] your blue ethereal field\n    Contend ye with each other? of the sea\n    Children, thus post ye over vale and height[246]\n    To sink upon your mother's lap--and rest?[247]                    10\n    Or were ye rightlier hailed, when first mine eyes\n    Beheld in your impetuous march the likeness\n    Of a wide army pressing on to meet\n    Or overtake some unknown enemy?--\n    But your smooth motions suit a peaceful aim;                      15\n    And Fancy, not less aptly pleased, compares\n    Your squadrons to an endless flight of birds\n    Aerial, upon due migration bound\n    To milder climes; or rather do ye urge\n    In caravan your hasty pilgrimage                                  20\n    To pause at last on more aspiring heights\n    Than these,[248] and utter your devotion there\n    With thunderous voice? Or are ye jubilant,\n    And would ye, tracking your proud lord the Sun,\n    Be present at his setting; or the pomp                            25\n    Of Persian mornings would ye fill, and stand\n    Poising your splendours high above the heads\n    Of worshippers kneeling to their up-risen God?\n    Whence, whence, ye Clouds! this eagerness of speed?\n    Speak, silent creatures.--They are gone, are fled,                30\n    Buried together in yon gloomy mass\n    That loads the middle heaven; and clear and bright\n    And vacant doth the region which they thronged\n    Appear; a calm descent of sky conducting\n    Down to the unapproachable abyss,                                 35\n    Down to that hidden gulf from which they rose\n    To vanish--fleet as days and months and years,\n    Fleet as the generations of mankind,\n    Power, glory, empire, as the world itself,\n    The lingering world, when time hath ceased to be.                 40\n    But the winds roar, shaking the rooted trees,\n    And see! a bright precursor to a train\n    Perchance as numerous, overpeers the rock\n    That sullenly refuses to partake\n    Of the wild impulse. From a fount of life                         45\n    Invisible, the long procession moves\n    Luminous or gloomy, welcome to the vale\n    Which they are entering, welcome to mine eye\n    That sees them, to my soul that owns in them,\n    And in the bosom of the firmament                                 50\n    O'er which they move, wherein they are contained,\n    A type of her capacious self and all\n    Her restless progeny.\n\n                          A humble walk\n    Here is my body doomed to tread, this path,\n    A little hoary line and faintly traced,[249]                      55\n    Work, shall we call it, of the shepherd's foot\n    Or of his flock?--joint vestige of them both.\n    I pace it unrepining, for my thoughts\n    Admit no bondage and my words have wings.\n    Where is the Orphean lyre, or Druid harp,                         60\n    To accompany the verse? The mountain blast\n    Shall be our _hand_ of music; he shall sweep\n    The rocks, and quivering trees, and billowy lake,\n    And search the fibres of the caves, and they\n    Shall answer, for our song is of the Clouds                       65\n    And the wind loves them; and the gentle gales--\n    Which by their aid re-clothe the naked lawn\n    With annual verdure, and revive the woods,\n    And moisten the parched lips of thirsty flowers--\n    Love them; and every idle breeze of air                           70\n    Bends to the favourite burthen. Moon and stars\n    Keep their most solemn vigils when the Clouds\n    Watch also, shifting peaceably their place\n    Like bands of ministering Spirits, or when they lie,\n    As if some Protean art the change had wrought,                    75\n    In listless quiet o'er the ethereal deep\n    Scattered, a Cyclades[250] of various shapes\n    And all degrees of beauty. O ye Lightnings!\n    Ye are their perilous offspring;[251] and the Sun--\n    Source inexhaustible of life and joy,                             80\n    And type of man's far-darting reason, therefore\n    In old time worshipped as the god of verse,[252]\n    A blazing intellectual deity--\n    Loves his own glory in their looks, and showers\n    Upon that unsubstantial brotherhood                               85\n    Visions with all but beatific light\n    Enriched--too transient were they not renewed\n    From age to age, and did not, while we gaze\n    In silent rapture, credulous desire\n    Nourish the hope that memory lacks not power                      90\n    To keep the treasure unimpaired. Vain thought!\n    Yet why repine, created as we are\n    For joy and rest, albeit to find them only\n    Lodged in the bosom of eternal things?\n\n[241] The title in the edition of 1842 was _Address to the Clouds_.--ED.\n\n[242] See the Fenwick note and compare Dorothy Wordsworth's Grasmere\nJournal, 31st January 1802.--ED.\n\n[243] 1842.\n\n    \u2026 in \u2026\n\n    MS.\n\n[244] 1842.\n\n    \u2026 wind\n\n    MS.\n\n[245] 1842.\n\n    \u2026 on \u2026\n\n    MS.\n\n[246] 1842.\n\n    \u2026 over dale and mountain height\n\n    MS.\n\n[247] 1842.\n\n    \u2026 mother's joyous lap?\n\n    MS.\n\n[248] 1842.\n\n    Or come ye as I hailed you first, a Flight\n    Aerial, on a due migration bound,\n    Embodied travellers not blindly led\n    To milder climes; or rather do ye urge\n    Your Caravan, your hasty pilgrimage\n    With hope to pause at last upon the top\n    Of some remoter mountains more beloved\n    Than these, \u2026\n\n    MS.\n\n[249] Compare, in the \"Poems on the Naming of Places\" (1805), the lines\nbeginning, \"When, to the attractions of the busy world,\" l. 48--\n\n    A hoary pathway traced between the trees.\n\nED.\n\n[250] The fifty-three small islands in the \u00c6gean surrounding Delos, as\nwith a circle (\u03ba\u03cd\u03ba\u03bb\u03bf\u03c2)--hence the name.--ED.\n\n[251] Compare Coleridge's _Hymn before Sunrise in the Vale of\nChamouni_--\n\n    Ye lightnings, the dread arrows of the clouds!\n\nED.\n\n[252] Sol = Phoebus = Apollo.--ED.\n\n\nAIREY-FORCE VALLEY\n\nPublished 1842\n\nFirst published (1842) in \"Poems, chiefly of Early and Late Years.\"\nAfterwards one of the \"Poems of the Imagination.\"--ED.\n\n                  ----Not a breath of air\n    Ruffles the bosom of this leafy glen.\n    From the brook's margin, wide around, the trees\n    Are stedfast as the rocks; the brook itself,\n    Old as the hills that feed it from afar,                           5\n    Doth rather deepen than disturb the calm\n    Where all things else are still and motionless.\n    And yet, even now, a little breeze, perchance\n    Escaped from boisterous winds that rage without,\n    Has entered, by the sturdy oaks unfelt,                           10\n    But to its gentle touch how sensitive\n    Is the light ash! that, pendent from the brow\n    Of yon dim cave,[253] in seeming silence makes\n    A soft eye-music of slow-waving boughs,\n    Powerful almost as vocal harmony                                  15\n    To stay the wanderer's steps and soothe his thoughts.\n\nThe Aira beck rises on the <DW72>s of Great Dodd, passes Dockray, and\nenters Ullswater between Glencoin Park and Gowbarrow Park, about two\nmiles from the head of the lake. The Force is quite near to _Lyulph's\nTower_, where the stream has a fall of about eighty feet. Compare the\nreference to it in _The Somnambulist_ (1833), and Wordsworth's account\nof \"Aira-Force,\" in his _Guide through the District of the Lakes_,\n\"Here is a powerful Brook, which dashes among rocks through a deep\nglen, hung on every side with a rich and happy intermixture of native\nwood; here are beds of luxuriant fern, aged hawthorns and hollies\ndecked with honeysuckles; and fallow deer glancing and bounding over\nthe lawns and through the thickets.\"--ED.\n\n[253] An ash-tree may still be seen at Aira-Force.--ED.\n\n\n\"LYRE! THOUGH SUCH POWER DO IN THY MAGIC LIVE\"\n\nComposed 1842 (or earlier).--Published 1842\n\nOne of the \"Poems of the Imagination.\"--ED.\n\n    Lyre! though such power do in thy magic live\n      As might from India's farthest plain\n      Recal the not unwilling Maid,\n          Assist me to detain\n          The lovely Fugitive:                                         5\n    Check with thy notes the impulse which, betrayed\n    By her sweet farewell looks, I longed to aid.\n    Here let me gaze enrapt upon that eye,\n    The impregnable and awe-inspiring fort\n    Of contemplation, the calm port                                   10\n    By reason fenced from winds that sigh\n    Among the restless sails of vanity.\n    But if no wish be hers that we should part,\n    A humbler bliss would satisfy my heart.\n          Where all things are so fair,                               15\n    Enough by her dear side to breathe the air\n          Of this Elysian weather;\n    And, on or in, or near, the brook, espy\n      Shade upon the sunshine lying\n          Faint and somewhat pensively;                               20\n      And downward Image gaily vying\n          With its upright living tree\n    'Mid silver clouds, and openings of blue sky\n    As soft almost and deep as her cerulean eye.\n\n    Nor less the joy with many a glance                               25\n    Cast up the Stream or down at her beseeching,\n    To mark its eddying foam-balls prettily distrest\n    By ever-changing shape and want of rest;\n          Or watch, with mutual teaching,\n          The current as it plays                                     30\n          In flashing leaps and stealthy creeps\n          Adown a rocky maze;\n    Or note (translucent summer's happiest chance!)\n    In the <DW72>-channel floored with pebbles bright,\n    Stones of all hues, gem emulous of gem,                           35\n    So vivid that they take from keenest sight\n    The liquid veil that seeks not to hide them.[254]\n\n[254] Compare Wordsworth's description of the Duddon as \"diaphanous,\nbecause it travels slowly,\"--ED.\n\n\nLOVE LIES BLEEDING\n\nComposed 1842.--Published 1842\n\n[It has been said that the English, though their country has produced\nso many great poets, is now the most unpoetical nation in Europe. It\nis probably true; for they have more temptation to become so than any\nother European people. Trade, commerce, and manufactures, physical\nscience, and mechanic arts, out of which so much wealth has arisen,\nhave made our countrymen infinitely less sensible to movements of\nimagination and fancy than were our forefathers in their simple state\nof society. How touching and beautiful were, in most instances, the\nnames they gave to our indigenous flowers, or any other they were\nfamiliarly acquainted with!--Every month for many years have we been\nimporting plants and flowers from all quarters of the globe, many of\nwhich are spread through our gardens, and some perhaps likely to be met\nwith on the few Commons which we have left. Will their botanical names\never be displaced by plain English appellations, which will bring them\nhome to our hearts by connexion with our joys and sorrows? It can never\nbe, unless society treads back her steps towards those simplicities\nwhich have been banished by the undue influence of towns spreading and\nspreading in every direction, so that city-life with every generation\ntakes more and more the lead of rural. Among the ancients, villages\nwere reckoned the seats of barbarism. Refinement, for the most part\nfalse, increases the desire to accumulate wealth; and while theories\nof political economy are boastfully pleading for the practice,\ninhumanity pervades all our dealings in buying and selling. This\nselfishness wars against disinterested imagination in all directions,\nand, evils coming round in a circle, barbarism spreads in every quarter\nof our island. Oh for the reign of justice, and then the humblest man\namong us would have more power and dignity in and about him than the\nhighest have now!--I.F.]\n\nOne of the \"Poems of the Fancy.\"--ED.\n\n    You call it, \"Love lies bleeding,\"--so you may,[255]\n    Though the red Flower, not prostrate, only droops,\n    As we have seen it here from day to day,\n    From month to month, life passing not away:\n    A flower how rich in sadness! Even thus stoops,                    5\n    (Sentient by Grecian sculpture's marvellous power)\n    Thus leans, with hanging brow and body bent\n    Earthward in uncomplaining languishment,\n    The dying Gladiator. So, sad Flower!\n    ('Tis Fancy guides me willing to be led,                          10\n    Though by a slender thread,)\n    So drooped Adonis bathed in sanguine dew\n    Of his death-wound, when he from innocent air\n    The gentlest breath of resignation drew;\n    While Venus in a passion of despair                               15\n    Rent, weeping over him, her golden hair\n    Spangled with drops of that celestial shower.\n    She suffered, as Immortals sometimes do;\n    But pangs more lasting far, _that_ Lover knew\n    Who first, weighed down by scorn, in some lone bower              20\n    Did press this semblance of unpitied smart\n    Into the service of his constant heart,\n    His own dejection, downcast Flower! could share\n    With thine, and gave the mournful name which thou wilt ever bear.\n\n[255] Compare _Midsummer Night's Dream_, act II. scene i. ll.\n165-168.--ED.\n\n\n\"THEY CALL IT LOVE LIES BLEEDING! RATHER SAY\"\n\nThe previous poem was originally composed in sonnet form; and it\nbelongs, in that form, to the year 1833. It occurs in a MS. copy of\nthe sonnets which record the Tour of 1833 to the Isle of Man and to\nScotland.--ED.\n\n    They call it Love lies bleeding! rather say\n    That in this crimson Flower Love bleeding _droops_,\n    A Flower how sick in sadness! Thus it stoops\n    With languid head unpropped from day to day\n    From month to month, life passing not away.                        5\n    Even so the dying Gladiator leans\n    On mother earth, and from his patience gleans\n    Relics of tender thoughts, regrets that stay\n    A moment and are gone. O fate-bowed flower!\n    Fair as Adonis bathed in sanguine dew,                            10\n    Of his death-wound, _that_ Lover's heart was true\n    As heaven, who pierced by scorn in some lone bower\n    Could press thy semblance of unpitied smart\n    Into the service of his constant heart.\n\n\nCOMPANION TO THE FOREGOING\n\nComposed (?)[256]--Published 1845\n\n    Never enlivened with the liveliest ray\n    That fosters growth or checks or cheers decay,\n    Nor by the heaviest rain-drops more deprest,\n    This Flower, that first appeared as summer's guest,\n    Preserves her beauty 'mid autumnal leaves                          5\n    And to her mournful habits fondly cleaves.\n\n    When files of stateliest plants have ceased to bloom,\n    One after one submitting to their doom,\n    When her coevals each and all are fled,\n    What keeps her thus reclined upon her lonesome bed?               10\n\n      The old mythologists, more impress'd than we\n    Of this late day by character in tree\n    Or herb, that claimed peculiar sympathy,\n    Or by the silent lapse of fountain clear,\n    Or with the language of the viewless air                          15\n    By bird or beast made vocal, sought a cause\n    To solve the mystery, not in Nature's laws\n    But in Man's fortunes. Hence a thousand tales\n    Sung to the plaintive lyre in Grecian vales.\n    Nor doubt that something of their spirit swayed                   20\n    The fancy-stricken Youth or heart-sick Maid,\n    Who, while each stood companionless and eyed\n    This undeparting Flower in crimson dyed,\n    Thought of a wound which death is slow to cure,\n    A fate that has endured and will endure,                          25\n    And, patience coveting yet passion feeding,\n    Called the dejected Lingerer, _Love lies bleeding_.\n\n[256] The date of the composition of this poem is uncertain, but, as\n\"companion\" to _Love lies Bleeding_, it must be placed in immediate\nsuccession to it.--ED.\n\n\nTHE CUCKOO-CLOCK\n\nComposed 1842.--Published 1842\n\n[Of this clock I have nothing further to say than what the poem\nexpresses, except that it must be here recorded that it was a\npresent from the dear friend for whose sake these notes were chiefly\nundertaken, and who has written them from my dictation.--I.F.]\n\nOne of the \"Poems of the Imagination.\"--ED.\n\n    Wouldst thou be taught, when sleep has taken flight,\n    By a sure voice that can most sweetly tell,\n    How far-off yet a glimpse of morning light,\n    And if to lure the truant back be well,\n    Forbear to covet a Repeater's stroke,                              5\n    That, answering to thy touch, will sound the hour;\n    Better provide thee with a Cuckoo-clock\n    For service hung behind thy chamber-door;\n    And in due time the soft spontaneous shock,\n    The double note, as if with living power,                         10\n    Will to composure lead--or make thee blithe as bird in bower.\n\n    List, Cuckoo--Cuckoo!--oft tho' tempests howl,\n    Or nipping frost remind thee trees are bare,\n    How cattle pine, and droop the shivering fowl,\n    Thy spirits will seem to feed on balmy air:                       15\n    I speak with knowledge,--by that Voice beguiled,\n    Thou wilt salute old memories as they throng\n    Into thy heart; and fancies, running wild\n    Through fresh green fields, and budding groves among,\n    Will make thee happy, happy as a child;                           20\n    Of sunshine wilt thou think, and flowers, and song,\n    And breathe as in a world where nothing can go wrong.\n\n    And know--that, even for him who shuns the day\n    And nightly tosses on a bed of pain;\n    Whose joys, from all but memory swept away,                       25\n    Must come unhoped for, if they come again;\n    Know--that, for him whose waking thoughts, severe\n    As his distress is sharp, would scorn my theme,\n    The mimic notes, striking upon his ear\n    In sleep, and intermingling with his dream,                       30\n    Could from sad regions send him to a dear\n    Delightful land of verdure, shower and gleam,\n    To mock the _wandering_ Voice[257] beside some haunted\n     stream.[258]\n\n    O bounty without measure! while the grace\n    Of Heaven doth in such wise, from humblest springs,               35\n    Pour pleasure forth, and solaces that trace\n    A mazy course along familiar things,\n    Well may our hearts have faith that blessings come,\n    Streaming from founts above the starry sky,\n    With angels when their own untroubled home                        40\n    They leave, and speed on nightly embassy\n    To visit earthly chambers,--and for whom?\n    Yea, both for souls who God's forbearance try,\n    And those that seek his help, and for his mercy sigh.\n\n[257] Compare _To the Cuckoo_ (vol. ii. p. 289)--\n\n    O Cuckoo! shall I call thee Bird,\n    Or but a wandering Voice?\n\nED.\n\n[258] Professor Dowden has appropriately called attention to the\nfact that the cuckoo-clock at Rydal Mount was not stopped during\nWordsworth's last illness.--ED.\n\n\n\"WANSFELL! THIS HOUSEHOLD HAS A FAVOURED LOT\"\n\nComposed 1842.--Published 1845\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Wansfell![259] this Household has a favoured lot,\n    Living with liberty on thee to gaze,\n    To watch while Morn first crowns thee with her rays,\n    Or when along thy breast serenely float\n    Evening's angelic clouds. Yet ne'er a note                         5\n    Hath sounded (shame upon the Bard!) thy praise\n    For all that thou, as if from heaven, hast brought\n    Of glory lavished on our quiet days.\n    Bountiful Son of Earth! when we are gone\n    From every object dear to mortal sight,                           10\n    As soon we shall be, may these words attest\n    How oft, to elevate our spirits, shone\n    Thy visionary majesties of light,\n    How in thy pensive glooms our hearts found rest.\n\n    _Dec. 24, 1842._\n\n[259] The Hill that rises to the south-east, above Ambleside.--W.W.\n1842.\n\n\n\"THOUGH THE BOLD WINGS OF POESY AFFECT\"\n\nComposed (?)--Published 1842\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Though the bold wings of Poesy affect\n    The clouds, and wheel around the mountain tops\n    Rejoicing, from her loftiest height she drops\n    Well pleased to skim the plain with wild flowers deckt,\n    Or muse in solemn grove whose shades protect                       5\n    The lingering dew--there steals along, or stops\n    Watching the least small bird that round her hops,\n    Or creeping worm, with sensitive respect.\n    Her functions are they therefore less divine,\n    Her thoughts less deep, or void of grave intent                   10\n    Her simplest fancies? Should that fear be thine,\n    Aspiring Votary, ere thy hand present\n    One offering, kneel before her modest shrine,\n    With brow in penitential sorrow bent!\n\n\n\"GLAD SIGHT WHEREVER NEW WITH OLD\"\n\nComposed 1842.[260]--Published 1845\n\nOne of the \"Poems of the Fancy.\"--ED.\n\n    Glad sight wherever new with old[261]\n    Is joined through some dear homeborn tie;\n\n    The life[262] of all that we behold\n    Depends upon that mystery.\n    Vain is the glory of the sky,[263]                                 5\n    The beauty vain of field and grove,\n    Unless, while with admiring eye[264]\n    We gaze, we also learn to love.[265]\n\n[260] A MS. copy of this fragment in Wordsworth's handwriting, 31st\nDecember 1842, fixes the date approximately.--ED.\n\n[261] 1845.\n\n    Look up, look round, let things unfold\n    Far as they may, their mysteries;\n    What profits it if new with old\n    Unites not with some homeborn ties.\n\n    MS. 31st Dec. 1842.\n\n    Welcome the sight when new with old\n\n    C.\n\n    Glad sight it is when new with old\n\n    MS. 1843.\n\n[262] 1845.\n\n    The good \u2026\n\n    C.\n\n[263] 1845.\n\n    \u2026 skies,\n\n    MS. 1843.\n\n[264] 1845.\n\n    \u2026 eyes\n\n    MS. 1843.\n\n[265] Compare the lines addressed to Mrs. Wordsworth in 1824,\nbeginning--\n\n    True beauty dwells in deep retreats.\n\nED.\n\n\n\n\n1843\n\nTwo sonnets, and an _Inscription_ for a monument to Southey, were\nwritten in 1843.--ED.\n\n\n\"WHILE BEAMS OF ORIENT LIGHT SHOOT WIDE AND HIGH\"\n\nComposed 1st January 1843.--Published 1845\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    While beams of orient light shoot wide and high,\n    Deep in the vale a little rural Town[266]\n    Breathes forth a cloud-like creature of its own,\n    That mounts not toward the radiant morning sky,\n    But, with a less ambitious sympathy,                               5\n    Hangs o'er its Parent waking to the cares\n    Troubles and toils that every day prepares.\n    So Fancy, to the musing Poet's eye,\n    Endears that Lingerer. And how blest her sway[267]\n    (Like influence never may my soul reject)[268]                    10\n    If the calm Heaven, now to its zenith decked[269]\n    With glorious forms in numberless array,\n    To the lone shepherd on the hills disclose\n    Gleams from[270] a world in which the saints repose.\n\n    _Jan. 1, 1843._\n\n[266] Ambleside.--W.W. 1845.\n\n[267] 1845.\n\n    \u2026 And blessed be her sway\n\n    MS.\n\n    So Fancy charms the musing Poet's eye\n    Fixed on that Lingerer \u2026\n\n    C.\n\n[268] 1845.\n\n    Ne'er may my soul like influence reject.\n\n    MS.\u2020\n\n[269] 1845.\n\n    Endear that Lingerer. And how blest her sway,\n    The faith how pure and holy in effect,\n    If the calm Heavens, now to their summit decked\n\n    MS.\u2020\n\n[270]\n\n    \u2026 of \u2026\n\n    MS.\u2020\n\n\n\u2020 These MS. variants occur in a copy of the sonnet written by\nWordsworth for Mrs. Arnold at Foxhowe.\n\n\nINSCRIPTION\n\nFOR A MONUMENT IN CROSTHWAITE CHURCH, IN THE VALE OF KESWICK\n\nComposed 1843.--Published 1845\n\nOne of the \"Epitaphs and Elegiac Pieces.\"--ED.\n\n    Ye vales and hills whose beauty hither drew\n    The poet's steps, and fixed him here, on you,\n    His eyes have closed! And ye, lov'd books, no more\n    Shall Southey feed upon your precious lore,\n    To works that ne'er shall forfeit their renown                     5\n    Adding immortal labours of his own--\n    Whether he traced historic truth, with zeal\n    For the State's guidance, or the Church's weal,\n    Or Fancy, disciplined by studious art,\n    Inform'd his pen, or wisdom of the heart,                         10\n    Or judgments sanctioned in the Patriot's mind\n    By reverence for the rights of all mankind.\n    Wide were his aims, yet in no human breast\n    Could private feelings meet for holier rest.\n    His joys, his griefs, have vanished like a cloud                  15\n    From Skiddaw's top; but he to heaven was vowed\n    Through his industrious life, and Christian faith\n    Calmed in his soul the fear of change and death.\n\nI received, from the late Lord Coleridge, the following extracts\nfrom letters written by Wordsworth to his father, the Hon. Justice\nColeridge, in reference to the Southey Inscription in Crosthwaite\nChurch. Wordsworth seems to have submitted the proposed Inscription to\nMr. Coleridge's judgment, and the changes he made upon it, in deference\nto the opinions he received, shew, as Lord Coleridge says, \"the extreme\ncare Wordsworth took to have the substance, and the expression also, as\nperfect as he could make it.\" The original draft of the \"Inscription\"\nwas as follows:--\n\n    SACRED TO THE MEMORY OF ROBERT SOUTHEY, WHOSE MORTAL REMAINS\n    ARE INTERRED IN THE ADJOINING CHURCHYARD. HE WAS BORN AT\n    BRISTOL, OCTOBER YE 4TH, 1774, AND DIED, AFTER A RESIDENCE OF\n    NEARLY FORTY YEARS, AT GRETA HALL IN THIS PARISH. MARCH 21ST,\n    1843.\n\n    Ye Vales and Hills, whose beauty hither drew\n    The Poet's steps, and fixed him here, on you\n    His eyes have closed; and ye, loved Books, no more\n    Shall Southey feed upon your precious lore,\n    To Works that ne'er shall forfeit their renown\n    Adding immortal labours of his own,\n    As Fancy, disciplined by studious Art\n    Informed his pen, or Wisdom of the heart,\n    Or judgments rooted in a Patriot's mind\n    Taught to revere the rights of all mankind.\n    Friends, Family--ah wherefore touch that string,\n    To them _so_ fondly did the good man cling!\n    His joys, his griefs, have vanished like a cloud\n    From Skiddaw's top; but He to Heaven was vowed\n    Through a long life; and calmed by Christian faith,\n    In his pure soul, the fear of change and death.\n\n    This Memorial was erected by friends of Robert Southey.\n\nAlteration in the Epitaph--\n\n                          \u2026 He to Heaven was vowed\n    Through a life long and pure; and Christian faith\n    Calmed in his soul the fear of change and death.--W.W.\n\n                                                   December the 6th.\n\n    MY DEAR MR. JUSTICE COLERIDGE,\n\n    Notwithstanding what I have written before, I could not but\n    wish to meet _your wishes_ upon the points which you mentioned,\n    and, accordingly, have added and altered as on the other side\n    of this paper. If you approve don't trouble yourself to answer.\n\n                        Ever faithfully yours,\n\n                                                     W. WORDSWORTH.\n\n        Ye torrents, foaming down the rocky steeps,\n        Ye lakes, wherein the spirit of water sleeps,\n        Ye vales and hills, etc.\n        Or judgments sanctioned in the Patriot's mind\n        By reverence for the rights of all mankind.\n        Friends, Family--within no human breast\n        Could private feelings need a holier nest.\n        His joys, his griefs, have vanished.\n\n    These alterations are approved of by friends here, and I hope\n    will please you.\n\n       *       *       *       *       *\n\n    MY DEAR MR. JUSTICE COLERIDGE,\n\n    Pray accept my thanks for the pains you have taken with the\n    Inscription, and excuse the few words I shall have to say upon\n    your remarks. There are two lakes in the Vale of Keswick; both\n    which, along with the lateral Vale of Newlands immediately\n    opposite Southey's study window, will be included in the words\n    \"Ye _Vales_ and Hills\" by everyone who is familiar with the\n    neighbourhood.\n\n    I quite agree with you that the construction of the lines\n    that particularize his writings is rendered awkward by so\n    many participles passive, and the more so on account of the\n    transitive verb _informed_. One of these participles may be\n    got rid of, and, I think, a better couplet produced by this\n    alteration--\n\n        Or judgments sanctioned in the Patriot's mind\n        By reverence for the rights of all mankind.\n\n    As I have entered into particulars as to the character of S.'s\n    writings, and they are so various, I thought his historic\n    works ought by no means to be omitted, and therefore, though\n    unwilling to lengthen the Epitaph, I added the two following--\n\n                              \u2026 Labours of his own,\n        Whether he traced historic truth with zeal\n        For the State's guidance, or the Church's weal,\n        Or Fancy, disciplined by studious Art,\n        Informed his pen, or wisdom of the heart,\n        Or judgments sanctioned in the Patriot's mind\n        By reverence for the rights of all mankind.\n\n    I do not feel with you in respect to the word \"so\"; it refers,\n    of course, to the preceding line, and as the reference is to\n    fireside feelings and intimate friends, there appears to me\n    a propriety in an expression inclining to the colloquial.\n    The couplet was the dictate of my own feelings, and the\n    construction is accordingly broken and rather dramatic,--but\n    too much of this. If you have any objection to the couplet\n    as altered, be so kind as let me know; if not, on no account\n    trouble yourself to answer this letter.\n\n    _Prematurely_ I object to as you do. I used the word with\n    reference to that decay of faculties which is not uncommon in\n    advanced life, and which often leads to dotage,--but the word\n    must not be retained.\n\n    We regret much to hear that Lady Coleridge is unwell, pray\n    present to her our best wishes.\n\n    What could induce the Bishop of London to forbid the choral\n    service at St. Mark's? It was in execution, I understand, above\n    all praise.\n\n                      Ever most faithfully yours,\n\n                                                     W. WORDSWORTH.\n\n       *       *       *       *       *\n\n                                               _December 2nd, '43._\n\n    MY DEAR MR. JUSTICE COLERIDGE,\n\n    The first line would certainly have more spirit by reading\n    \"your\" as you suggest. I had previously considered _that_;\n    but decided in favour of \"the,\" as \"your,\" I thought, would\n    clog the sentence in sound, there being \"ye\" thrice repeated,\n    and followed by \"_you_\" at the close of the 4th line. I also\n    thought that \"_your_\" would interfere with the application of\n    \"you\" at the end of the fourth line, to the _whole_ of the\n    particular previous images as I intended it to do. But I don't\n    trouble you with this Letter on that account, but merely to ask\n    you whether the couplet now standing:--\n\n        Large were his aims, yet in no human breast\n        Could private feelings find a holier nest,\n\n    would not be better thus\n\n        Could private feelings meet in holier rest.\n\n    This alteration does not quite satisfy me, but I can do no\n    better. The word \"_nest_\" both in itself and in conjunction\n    with \"_holier_\" seems to me somewhat bold and rather startling\n    for marble, particularly in a Church. I should not have thought\n    of any alteration in a merely printed poem, but this makes a\n    difference. If you think the proposed alteration better, don't\n    trouble yourself to answer this; if not, pray be so kind as to\n    tell me so by a single line. I would not on any account have\n    trespassed on your time but for this public occasion. We are\n    sorry to hear of Lady Coleridge's indisposition; pray present\n    to her our kind regards and best wishes for her recovery,\n    united with the greetings of the season both for her and\n    yourself, and believe me faithfully,\n\n                             Your obliged,\n\n                                                    WM. WORDSWORTH.\n\n    RYDAL MOUNT, _December 23rd, '43_.\n\n       *       *       *       *       *\n\n    TO THE MEMORY OF ROBERT SOUTHEY, A MAN EMINENT FOR GENIUS,\n    VERSATILE TALENTS, EXTENSIVE AND ACCURATE KNOWLEDGE, AND\n    HABITS OF THE MOST CONSCIENTIOUS INDUSTRY. NOR WAS HE LESS\n    DISTINGUISHED FOR STRICT TEMPERANCE, PURE BENEVOLENCE, AND WARM\n    AFFECTIONS; BUT HIS MIND, SUCH ARE THE AWFUL DISPENSATIONS OF\n    PROVIDENCE, WAS PREMATURELY AND ALMOST TOTALLY OBSCURED BY A\n    SLOWLY-WORKING AND INSCRUTABLE MALADY UNDER WHICH HE LANGUISHED\n    UNTIL RELEASED BY DEATH IN THE 69TH YEAR OF HIS AGE.\n\n    READER! PONDER THE CONDITION TO WHICH THIS GREAT AND GOOD\n    MAN, NOT WITHOUT MERCIFUL ALLEVIATIONS, WAS DOOMED, AND LEARN\n    FROM HIS EXAMPLE TO MAKE TIMELY USE OF THY ENDOWMENTS AND\n    OPPORTUNITIES, AND TO WALK HUMBLY WITH THY GOD.\n\n       *       *       *       *       *\n\n                    COPY OF THE PRINTED INSCRIPTION\n\n    SACRED TO THE MEMORY OF ROBERT SOUTHEY, WHOSE MORTAL REMAINS\n    ARE INTERRED IN THE ADJOINING CHURCHYARD. HE WAS BORN AT\n    BRISTOL, OCTOBER 4TH, 1774, AND DIED AFTER A RESIDENCE OF\n    NEARLY 40 YEARS AT GRETA HALL, IN THIS PARISH, MARCH 21ST, 1843.\n\n        Ye torrents, foaming down the rocky steeps,\n        Ye lakes, wherein the spirit of water sleeps,\n        Ye vales and hills, whose beauty hither drew\n        The Poet's steps and fixed him here, on you\n        His eyes have closed! and ye, loved books, no more\n        Shall Southey feed upon your precious lore,\n        To works that ne'er shall forfeit their renown\n        Adding immortal labours of his own--\n        Whether he traced historic truth, with zeal\n        For the State's guidance or the Church's weal,\n        Or Fancy, disciplined by studious art,\n        Informed his pen, or wisdom of the heart,\n        Or judgments sanctioned in the Patriot's mind\n        By reverence for the rights of all mankind.\n        Wide were his aims, yet in no human breast\n        Could private feelings find a holier nest.\n        His joys, his griefs, have vanished like a cloud\n        From Skiddaw's top; but he to Heaven was vowed\n        Through a long life, and calmed by Christian faith,\n        In his pure soul, the fear of change and death.\n\n    This Memorial was erected by friends of Robert Southey.\n\nEdward Quillinan wrote, 25th March 1843, \"Yesterday I drove Mr.\nWordsworth early over to Keswick, that he and I might attend the\nfuneral of Mr. Southey, who was buried in Crosthwaite churchyard there\nat eleven A.M. It was very affecting to see Kate Southey with her\nbrother Cuthbert, and brother-in-law Herbert Hill, at her father's\ngrave as the coffin was lowered into it. She looked as if she yearned\nto be there too. She says she has now got her father back again.\"--ED.\n\n\nTO THE REV. CHRISTOPHER WORDSWORTH, D.D., MASTER OF HARROW SCHOOL[271]\n\nAfter the perusal of his _Theophilus Anglicanus_, recently published.\n\nComposed 1843.--Published 1845\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Enlightened Teacher, gladly from thy hand\n    Have I received this proof of pains bestowed\n    By Thee to guide thy Pupils on the road\n    That, in our native isle, and every land,\n    The Church, when trusting in divine command                        5\n    And in her Catholic attributes, hath trod:\n    O may these lessons be with profit scanned\n    To thy heart's wish, thy labour blest by God!\n    So the bright faces of the young and gay\n    Shall look more bright--the happy, happier still;                 10\n    Catch, in the pauses of their keenest play,\n    Motions of thought which elevate the will\n    And, like the Spire that from your classic Hill\n    Points heavenward, indicate the end and way.\n\n    RYDAL MOUNT, _Dec. 11, 1843_.\n\n[271] The poet's nephew, afterwards Canon of Westminster, and Bishop of\nLincoln, and the biographer of his uncle.--ED.\n\n\n\n\n1844\n\nOnly four poems were written in 1844.--ED.\n\n\n\"SO FAIR, SO SWEET, WITHAL SO SENSITIVE\"\n\nComposed July 1844.--Published 1845\n\nOne of the \"Poems of Sentiment and Reflection.\"--ED.\n\n    So fair, so sweet, withal so sensitive,\n    Would that the little Flowers were born to live,\n    Conscious of half the pleasure which they give;\n\n    That to this mountain-daisy's self were known[272]\n    The beauty of its star-shaped shadow, thrown                       5\n    On the smooth surface of this[273] naked stone!\n\n    And what if hence a bold desire should mount\n    High as the Sun, that he could take account\n    Of all that issues from his glorious fount!\n\n    So might he ken how by his sovereign aid                          10\n    These delicate companionships are made;\n    And how he rules the pomp of light and shade;\n\n    And were the Sister-power that shines by night\n    So privileged, what a countenance of delight\n    Would through the clouds break forth on human sight!              15\n\n    Fond fancies! wheresoe'er shall turn thine eye\n    On earth, air, ocean, or the starry sky,\n    Converse with Nature in pure sympathy;[274]\n\n    All vain desires, all lawless wishes quelled,\n    Be Thou to love and praise alike impelled,                        20\n    Whatever boon is granted or withheld.[275][276]\n\n[272] Compare the lines _To a Child, written in her Album_, in\n1834.--ED.\n\n[273] 1844.\n\n    Its sole companion on this\n\n    C.\n\n[274] 1845.\n\n    Fond fancies' bond, between a smile and sigh,\n    Do thou more wise, where'er thou turn'st thine eye\n    Converse with Nature in pure sympathy.\n\n    C.\n\n                          \u2026 be taught to fix an eye\n    On holy Nature in pure sympathy.\n\n    C.\n\n    Fond fancies, wheresoe'er shall range thine eye\n    Among the forms and powers of earth or sky,\n    Converse with Nature in pure sympathy.\n\n    C.\n\n[275] 1845.\n\n    A thankful heart all lawless wishes quelled,\n    To joy, to praise, to love alike compelled,\n    Whatever boon be granted or withheld.\n\n    C.\n\nThe following variation of the two last stanzas is from a MS. copy by\nWordsworth.\n\n    Fond fancies! wheresoe'er shall range thine eye\n    Among the forms and powers of earth and sky,\n    Converse with nature in pure sympathy.\n    A thankful heart, all lawless wishes quell'd,\n    To joy, to praise, to love alike compell'd,\n    Whatever boon be granted or withheld.\n\n_August, 1844._--ED.\n\n[276] The following account of the circumstance which gave rise to the\npreceding poem is from the _Memoir_ of Professor Archer Butler, by Mr.\nWoodward, prefixed to the \"First Series\" of his Sermons. The late Rev.\nArchdeacon Graves, of Dublin (in 1849 of Windermere), in writing to Mr.\nWoodward, gives an interesting account of a walk, in July 1844, from\nWindermere, by Rydal and Grasmere, to Loughrigg Tarn, etc., in which\nButler was accompanied by Wordsworth, Julius Charles Hare, Sir William\nHamilton, etc. He says, \"The day was additionally memorable as giving\nbirth to an interesting minor poem of Mr. Wordsworth's. When we reached\nthe side of Loughrigg Tarn (which you may remember he notes for its\nsimilarity, in the peculiar character of its beauty, to the Lago di\nNemi--Dianae Speculum), the loveliness of the scene arrested our steps\nand fixed our gaze. The splendour of a July noon surrounded us and\nlit up the landscape, with the Langdale Pikes soaring above, and the\nbright tarn shining beneath; and when the poet's eyes were satisfied\nwith their feast on the beauties familiar to them, they sought relief\nin the search, to them a happy vital habit, for new beauty in the\nflower-enamelled turf at his feet. There his attention was arrested\nby a fair smooth stone, of the size of an ostrich's egg, seeming to\nimbed at its centre, and at the same time to display a dark star-shaped\nfossil of most distinct outline. Upon closer inspection this proved\nto be the shadow of a daisy projected upon it with extraordinary\nprecision by the intense light of an almost vertical sun. The poet drew\nthe attention of the rest of the party to the minute but beautiful\nphenomenon, and gave expression at the time to thoughts suggested by\nit, which so interested our friend Professor Butler, that he plucked\nthe tiny flower, and, saying that \"it should be not only the theme but\nthe memorial of the thought they had heard,\" bestowed it somewhere\ncarefully for preservation. The little poem, in which some of these\nthoughts were afterwards crystallised, commences with the stanza--\n\n    So fair, so sweet, withal so sensitive,\n    Would that the little flowers were born to live,\n    Conscious of half the pleasure that they give.\"\n\n_Memoir_, pp. 27, 28.--ED.\n\n\nON THE PROJECTED KENDAL AND WINDERMERE RAILWAY\n\nComposed October 12, 1844.--Published 1844[277]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Is then no nook of English ground secure\n    From rash assault?[278] Schemes of retirement sown\n    In youth, and 'mid the busy world kept pure\n    As when their earliest flowers of hope were blown,\n    Must perish;--how can they this blight endure?                     5\n    And must he too the ruthless change bemoan\n    Who scorns a false utilitarian lure\n    'Mid his paternal fields at random thrown?\n    Baffle the threat, bright Scene, from Orrest-head[279]\n    Given to the pausing traveller's rapturous glance:                10\n    Plead for thy peace, thou beautiful romance\n    Of nature; and, if human hearts be dead,\n    Speak, passing winds; ye torrents, with your strong\n    And constant voice, protest against the wrong.\n\n    _October 12th, 1844._\n\n[277] In the first edition of his pamphlet \"On the projected Kendal and\nWindermere Railway.\"--ED.\n\n[278] The degree and kind of attachment which many of the yeomanry\nfeel to their small inheritances can scarcely be over-rated. Near the\nhouse of one of them stands a magnificent tree, which a neighbour of\nthe owner advised him to fell for profit's sake. \"Fell it!\" exclaimed\nthe yeoman, \"I had rather fall on my knees and worship it.\" It happens,\nI believe, that the intended railway would pass through this little\nproperty, and I hope that an apology for the answer will not be thought\nnecessary by one who enters into the strength of the feeling.--W.W.\n1845.\n\nCompare the two letters on the Kendal and Windermere Railway,\ncontributed by Wordsworth to _The Morning Post_ in 1844, at Kendal,\nrevised and reprinted in the same year. See _The Prose Works of\nWordsworth_, vol. ii. pp. 383-405.--ED.\n\n[279] Orresthead is the height close to Windermere, to the north of the\ntown.--ED.\n\n\n\"PROUD WERE YE, MOUNTAINS, WHEN, IN TIMES OF OLD\"\n\nComposed 1844.--Published 1845[280]\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Proud were ye, Mountains, when, in times of old,\n    Your patriot sons, to stem invasive war,\n    Intrenched your brows; ye gloried in each scar:\n    Now, for your shame, a Power, the Thirst of Gold,\n    That rules o'er Britain like a baneful star,                       5\n    Wills that your peace, your beauty, shall be sold,\n    And clear way made for her triumphal car\n    Through the beloved retreats your arms enfold!\n    Heard YE that Whistle? As her long-linked Train\n    Swept onwards, did the vision cross your view?                    10\n    Yes, ye were startled;--and, in balance true,\n    Weighing the mischief with the promised gain,\n    Mountains, and Vales, and Floods, I call on you\n    To share the passion of a just disdain.\n\nThe following by Canon Rawnsley--suggested by an attempt to introduce\na mineral railway into Borrowdale--may be read in connection with\nWordsworth's two sonnets.--ED.\n\nA CRY FROM DERWENTWATER\n\n    Shall then the stream of ruinous Lodore\n      Not fill the valley with its changeful sound\n      Unchallenged! shall grey Derwent's sacred bound\n    Hear the harsh brawl and intermittent roar\n    Of mocking waves upon an iron shore,\n      Whereby nor health nor happiness is found!--\n      While steam-wains drag from Honister's heart wound\n    The long cooled ashes of its fiery core!\n\n    Burst forth ye sulphurous fountains, as ye broke\n      On Skiddaw, lick the waters, blast the trees,\n    And let men have the earth they would desire,--\n    As well go pass our children through the fire\n    With shrieks, Cath-Belus, round thine altar's smoke,\n      As let old Derwent hear such sounds as these.\n\n                                          H.D. RAWNSLEY.\n\n    WRAY VICARAGE, AMBLESIDE.\n\n[280] This sonnet was first published in _The Morning Post_, December\n17, 1844.--ED.\n\n\nAT FURNESS ABBEY\n\nComposed 1844.--Published 1845\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Here, where, of havoc tired and rash undoing,\n    Man left this Structure to become Time's prey\n    A soothing spirit follows in the way\n    That Nature takes, her counter-work pursuing.\n    See how her Ivy clasps the sacred Ruin[281]                        5\n    Fall to prevent or beautify decay;\n    And, on the mouldered walls, how bright, how gay,\n    The flowers in pearly dews their bloom renewing!\n    Thanks to the place, blessings upon the hour;\n    Even as I speak the rising Sun's first smile                      10\n    Gleams on the grass-crowned top of yon tall Tower[282]\n    Whose cawing occupants with joy proclaim\n    Prescriptive title to the shattered pile\n    Where, Cavendish,[283] _thine_ seems nothing but a name!\n\n[281] In the chancel of the church at Furness Abbey, ivy almost covers\nthe north wall. In the Belfry and in the Chapter House, it is the same.\nThe \"tower,\" referred to in the sonnet, is evidently the belfry tower\nto the west. It is still \"grass-crowned.\" The sonnet was doubtless\ncomposed on the spot, and if Wordsworth ascended to the top of the\nbelfry tower, he might have seen the morning sunlight strike the small\nremaining fragment of the central tower. But it is more likely that he\nlooked up from the nave, or choir, of the church to the belfry, when he\nspoke of the sun's first smile gleaming from the top of the tall tower.\n\"Flowers\"--crowfoot, campanulas, etc.--still luxuriate on the mouldered\nwalls. With the line,\n\n    Fall to prevent or beautify decay;\n\ncompare,\n\n    Nature softening and concealing,\n    And busy with a hand of healing,\n\nin the description of Bolton Abbey in _The White Doe of Rylstone_,\ncanto i. I. 118. Compare also the _Address from the Spirit of\nCockermouth Castle_, vol. vii. p. 347.--ED.\n\n[282] See preceding note.\n\n[283] Furness Abbey is the property of the Duke of Devonshire, whose\nfamily name is Cavendish.--ED.\n\n\n\n\n1845\n\nThe Poems of 1845 include one of the group \"On the Naming of Places,\"\n_The Westmoreland Girl_ (addressed to the Poet's grandchildren),\nseveral fragments addressed to Mrs. Wordsworth, and to friends, with\none or two Sonnets.--ED.\n\n\n\"FORTH FROM A JUTTING RIDGE, AROUND WHOSE BASE\"\n\nComposed 1845.--Published 1845\n\nOne of the \"Poems upon the Naming of Places.\"--ED.\n\n    Forth from a jutting ridge, around whose base\n    Winds our deep Vale, two heath-clad Rocks ascend[284][285]\n    In fellowship, the loftiest of the pair\n    Rising to no ambitious height; yet both,\n    O'er lake[286] and stream, mountain and flowery mead,              5\n    Unfolding prospects fair as human eyes[287]\n    Ever beheld. Up-led with mutual help,\n    To one or other brow of those twin Peaks\n    Were two adventurous Sisters wont to climb,\n    And took no note of the hour while thence they gazed,             10\n    The blooming heath their couch, gazed, side by side,\n    In speechless admiration. I, a witness\n    And frequent sharer of their calm[288] delight\n    With thankful heart, to either Eminence\n    Gave the baptismal name each Sister bore.                         15\n    Now are they parted,[289] far as Death's cold hand\n    Hath power to part the Spirits of those who love\n    As they did love. Ye kindred Pinnacles--\n    That, while the generations of mankind\n    Follow each other to their hiding-place                           20\n    In time's abyss, are privileged to endure\n    Beautiful in yourselves, and richly graced\n    With like command of beauty--grant your aid\n    For MARY'S humble, SARAH'S silent, claim,\n    That their pure joy in nature may survive                         25\n    From age to age in blended memory.\n\n[284] 1845.\n\n    Winds our sequestered vale, two rocks ascend\n\n    MS.\n\n[285] These two rocks rise to the left of the lower high-road from\nGrasmere to Rydal, after it leaves the former lake and turns eastwards\ntowards the latter. They are still \"heath-clad,\" and covered with the\ncoppice of the old Bane Riggs Wood, so named because the shortest\nroad from Ambleside to Grasmere used to pass through it; \"bain\" or\n\"bane\" signifying, in the Westmoreland dialect, a short cut. Dr.\nCradock wrote of them thus:--\"They are now difficult of approach,\nbeing enclosed in a wood, with dense undergrowth, and surrounded by\na high, well-built wall. They can be well seen from the lower road,\nfrom a spot close to the three-mile stone from Ambleside. They are\nsome fifty or sixty feet above the road, about twenty yards apart, and\nseparated by a slight depression of, say, ten feet. The view from the\neasterly one is now much preferable, as it is less encumbered with\nshrubs; and for that reason also is more heath-clad. The twin rocks\nare also well seen, though at a farther distance, from the hill in\nWhite Moss Common between the roads, which Dr. Arnold used to call 'Old\nCorruption,' and 'Bit-by-bit Reform.' Doubtless the rocks were far more\neasily approached fifty years ago, when walls, if any, were low and\nill-built. It is probable, however, that even then they were enclosed\nand protected; for heath will not grow on the Grasmere hills, on places\nmuch frequented by sheep.\" The best view of these \"heath-clad\" rocks\nfrom the lower carriage road is at a spot two or three yards to the\nwest of a large rock on the road-side near the milestone. The view\nof them from the Loughrigg Terrace walks is also interesting. The\ntwo sisters were Mary and Sarah Hutchinson (Mrs. Wordsworth and her\nSister); and, in the Rydal household, the rocks were respectively named\n\"Mary-Point,\" and \"Sarah-Point.\"--ED.\n\n[286] 1845.\n\n    O'er wood \u2026\n\n    MS.\n\n[287] 1845.\n\n    \u2026 eye\n\n    MS.\n\n[288] 1845.\n\n    \u2026 that deep \u2026\n\n    MS.\n\n[289] 1845.\n\n    Gone to a common home, their duty done,\n    In this dear vale the Sisters lived, but long\n    Have they been parted \u2026\n\n    C.\n\n    True to a common love, their early choice\n    In this dear Vale, the sisters lived, but long\n    Have they been parted-- \u2026\n\n    C.\n\n\nTHE WESTMORELAND GIRL[290]\n\nTO MY GRANDCHILDREN\n\nComposed June 6, 1845.--Published 1845\n\nOne of the \"Poems referring to the Period of Childhood.\"--ED.\n\n               PART I\n\n    Seek who will delight in fable\n    I shall tell you truth. A Lamb\n    Leapt from this steep bank to follow\n    'Cross the brook its thoughtless dam.[291]\n\n    Far and wide on hill and valley                                    5\n    Rain had fallen, unceasing rain,\n    And the bleating mother's Young-one\n    Struggled with the flood in vain:\n\n    But, as chanced, a Cottage-maiden\n    (Ten years scarcely had she told)                                 10\n    Seeing, plunged into the torrent,\n    Clasped the Lamb and kept her hold.\n    Whirled adown the rocky channel,\n    Sinking, rising, on they go,\n    Peace and rest, as seems, before them                             15\n    Only in the lake below.\n\n    Oh! it was a frightful current\n    Whose fierce wrath the Girl had braved;\n    Clap your hands with joy my Hearers,\n    Shout in triumph, both are saved;                                 20\n\n    Saved by courage that with danger\n    Grew, by strength the gift of love,\n    And belike a guardian angel\n    Came with succour from above.\n\n               PART II\n\n    Now, to a maturer Audience,                                       25\n    Let me speak of this brave Child\n    Left among her native mountains\n    With wild Nature to run wild.\n\n    So, unwatched by love maternal,\n    Mother's care no more her guide,                                  30\n    Fared this little bright-eyed Orphan\n    Even while at her father's side.\n\n    Spare your blame,--remembrance makes him\n    Loth to rule by strict command;\n    Still upon his cheek are living                                   35\n    Touches of her infant hand,\n\n    Dear caresses given in pity,\n    Sympathy that soothed his grief,\n    As the dying mother witnessed\n    To her thankful mind's relief.                                    40\n\n    Time passed on; the Child was happy,\n    Like a Spirit of air she moved,\n    Wayward, yet by all who knew her\n    For her tender heart beloved.\n\n    Scarcely less than sacred passions,                               45\n    Bred in house, in grove, and field,\n    Link her with the inferior creatures,\n    Urge her powers their rights to shield.\n\n    Anglers, bent on reckless pastime,\n    Learn how she can feel alike                                      50\n    Both for tiny harmless minnow\n    And the fierce and sharp-toothed pike.\n\n    Merciful protectress, kindling\n    Into anger or disdain;\n    Many a captive hath she rescued,                                  55\n    Others saved from lingering pain.\n\n    Listen yet awhile;--with patience\n    Hear the homely truths I tell,\n    She in Grasmere's old church-steeple\n    Tolled this day the passing-bell.                                 60\n\n    Yes, the wild Girl of the mountains\n    To their echoes gave the sound,\n    Notice punctual as the minute,\n    Warning solemn and profound.\n\n    She, fulfilling her sire's office,                                65\n    Rang alone the far-heard knell,\n    Tribute, by her hand, in sorrow,\n    Paid to One who loved her well.\n\n    When his spirit was departed\n    On that service she went forth;                                   70\n    Nor will fail the like to render\n    When his corse is laid in earth.[292]\n\n    What then wants the Child to temper,\n    In her breast, unruly fire,\n    To control the froward impulse                                    75\n    And restrain the vague desire?\n\n    Easily a pious training\n    And a stedfast outward power\n    Would supplant the weeds and cherish,\n    In their stead, each opening flower.                              80\n\n    Thus the fearless Lamb-deliv'rer,\n    Woman-grown, meek-hearted, sage,\n    May become a blest example\n    For her sex, of every age.[293]\n\n    Watchful as a wheeling eagle,                                     85\n    Constant as a soaring lark,\n    Should the country need a heroine,\n    She might prove our Maid of Arc.\n\n    Leave that thought; and here be uttered\n    Prayer that Grace divine may raise                                90\n    Her humane courageous spirit\n    Up to heaven, thro' peaceful ways.[294]\n\n[290] This Westmoreland Girl was Sarah Mackereth of Wyke Cottage,\nGrasmere. She married a man named Davis, and died in 1872 at Broughton\nin Furness. The swollen \"flood\" from which she rescued the lamb,\nwas Wyke Gill beck, which descends from the centre of Silver Howe.\nThe picturesque cottage, with round chimney,--a yew tree and Scotch\nfir behind it,--is on the western side of the road from Grasmere\nover to Langdale by Red Bank. The Mackereths have been a well-known\nWestmoreland family for some hundred years. They belong to the \"gentry\nof the soil,\" and have been parish clerks in Grasmere for generations.\nOne of them was the tenant of the Swan Inn referred to in _The\nWaggoner_--the host who painted, with his own hand, the \"famous swan,\"\nused as a sign. (See vol. iii. p. 81.)\n\nThe story of _The Blind Highland Boy_, which gave rise to the poem\nbearing that name, was told to Wordsworth by one of these Mackereths\nof Grasmere. (See the Fenwick note, vol. ii. p. 420.) In a letter to\nProfessor Henry Reed (31st July 1845) Wordsworth said this poem might\ninterest him \"as exhibiting what sort of characters our mountains\nbreed. It is truth to the letter.\"--ED.\n\n[291] 1845.\n\n    \u2026 its simple dam.\n\n    MS.\n\n[292] 1845.\n\n    \u2026 must lie in earth.\n\n    MS.\n\n[293] Compare _Grace Darling_, p. 311 in this volume.--ED.\n\n[294] 1845.\n\n    Leave that word--and here be offered\n    Prayer that Grace divine would raise\n    This humane courageous spirit\n    Up to Heaven through peaceful ways.\n\n    In a letter to Henry Reed, July 1845.\n\n\nAT FURNESS ABBEY\n\nComposed 1845.--Published 1845\n\nOne of the \"Miscellaneous Sonnets.\"--ED.\n\n    Well have yon Railway Labourers to THIS ground\n    Withdrawn for noontide rest. They sit, they walk\n    Among the Ruins, but no idle talk\n    Is heard; to grave demeanour all are bound;\n    And from one voice a Hymn with tuneful sound                       5\n    Hallows once more the long-deserted Quire[295]\n    And thrills the old sepulchral earth, around.\n    Others look up, and with fixed eyes admire\n    That wide-spanned arch, wondering how it was raised,\n    To keep, so high in air, its strength and grace:                  10\n    All seem to feel the spirit of the place,\n    And by the general reverence God is praised:\n    Profane Despoilers, stand ye not reproved,\n    While thus these simple-hearted men are moved?\n\n    _June 21st, 1845._\n\n[295] See the note to the previous sonnet on Furness Abbey, p. 168.--ED.\n\n\n\"YES! THOU ART FAIR, YET BE NOT MOVED\"\n\nComposed possibly in 1845.--Published 1845\n\nOne of the \"Poems founded on the Affections.\"--ED.\n\n    Yes! thou art fair, yet be not moved\n      To scorn the declaration,\n    That sometimes I in thee have loved\n      My fancy's own creation.\n\n    Imagination needs must stir;                                       5\n      Dear Maid, this truth believe,\n    Minds that have nothing to confer\n      Find little to perceive.\n\n    Be pleased that nature made thee fit\n      To feed my heart's devotion,                                    10\n    By laws to which all Forms submit\n      In sky, air, earth, and ocean.\n\n\n\"WHAT HEAVENLY SMILES! O LADY MINE\"\n\nComposed 1845.--Published 1845\n\nOne of the \"Poems founded on the Affections.\"--ED.\n\n    What heavenly smiles! O Lady mine\n    Through my[296] very heart they shine;\n    And, if my brow gives back their light,\n    Do thou look gladly on the sight;\n    As the clear Moon with modest pride\n      Beholds her own bright beams\n    Reflected from the mountain's side\n      And from the headlong streams.\n\n[296] 1845.\n\n    \u2026 this \u2026\n\n    MS.\n\n\nTO A LADY,\n\nIN ANSWER TO A REQUEST THAT I WOULD WRITE HER A POEM UPON SOME DRAWINGS\nTHAT SHE HAD MADE OF FLOWERS IN THE ISLAND OF MADEIRA\n\nComposed 1845.--Published 1845\n\nOne of the \"Poems of the Fancy.\"--ED.\n\n    Fair Lady! can I sing of flowers\n      That in Madeira bloom and fade,\n    I who ne'er sate within their bowers,\n      Nor through their sunny lawns have strayed?\n    How they in sprightly dance are worn                               5\n      By Shepherd-groom or May-day queen,\n    Or holy festal pomps adorn,\n      These eyes have never seen.\n\n    Yet tho' to me the pencil's art\n      No like remembrances can give,                                  10\n    Your portraits still may reach the heart\n      And there for gentle pleasure live;\n    While Fancy ranging with free scope\n      Shall on some lovely Alien set\n    A name with us endeared to hope,                                  15\n      To peace, or fond regret.[297]\n\n    Still as we look with nicer care,\n      Some new resemblance we may trace:\n    A _Heart's-ease_ will perhaps be there,\n      A _Speedwell_ may not want its place.                           20\n    And so may we, with charm\u00e8d mind\n      Beholding what your skill has wrought,\n    Another _Star-of-Bethlehem_ find,\n      A new[298] _Forget-me-not_.\n\n    From earth to heaven with motion fleet                            25\n      From heaven to earth our thoughts will pass,\n    A _Holy-thistle_ here we meet\n      And there a _Shepherd's weather-glass_;\n    And haply some familiar name\n      Shall grace the fairest, sweetest, plant                        30\n    Whose presence cheers the drooping frame\n      Of English Emigrant.\n\n    Gazing she feels its power beguile\n      Sad thoughts, and breathes with easier breath;\n    Alas! that meek that tender smile                                 35\n      Is but a harbinger of death:\n    And pointing with a feeble hand\n      She says, in faint words by sighs broken,\n    Bear for me to my native land\n      This precious Flower, true love's last token.                   40\n\n[297] 1845.\n\n      And there in sweet communion live:\n    Yet those loved most, in which we own\n      A touching likeness which they bear\n    To flower or herb, by Nature sown,\n      To breathe our English air.\n\n    MS.\n\n      And there in sweet communion live\n    Admired for beauty of their own,\n      Loved for the likeness some may bear\n    To flower \u2026\n\n    MS.\n\n    Thus tempted Fancy with free scope\n      Will range, and on these aliens set\n    Names among us endeared to none,\n      To hearts a fond regret.\n\n    MS.\n\n    So tempted \u2026\n      May range, \u2026\n\n    MS.\n\n[298]\n\n    Nor miss \u2026\n\n    MS.\n\n\nTO THE PENNSYLVANIANS\n\nComposed 1845.--Published 1845\n\nOne of the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n    Days undefiled by luxury or sloth,\n    Firm self-denial, manners grave and staid,\n    Rights equal, laws with cheerfulness obeyed,\n    Words that require no sanction from an oath,\n    And simple honesty a common growth--                               5\n    This high repute, with bounteous Nature's aid,\n    Won confidence, now ruthlessly betrayed\n    At will, your power the measure of your troth!--\n    All who revere the memory of Penn\n    Grieve for the land on whose wild woods his name[299]             10\n    Was fondly grafted with a virtuous aim,\n    Renounced, abandoned by degenerate Men\n    For state-dishonour black as ever came\n    To upper air from Mammon's loathsome den.[300]\n\n[299] To William Penn, son of Admiral Sir W. Penn, a printer and\nQuaker, Charles II. granted lands in America, to which he gave the name\nof Pennsylvania.--ED.\n\n[300] Mr. Ellis Yarnall wrote to me, April 27, 1885: \"The three last\nlines of the Sonnet _To the Pennsylvanians_, in regard to which you\ninquire, I think refer to what at the time Wordsworth wrote was known\nas the _repudiation_ by Pennsylvania of her State debt. The language,\nhowever, is too strong, inasmuch as there was _no_ repudiation. For a\nyear or two the _interest_ on the debt was unpaid, then payment was\nresumed. Members of Wordsworth's family, or his near friends, held, I\nbelieve, some of the Pennsylvania bonds. They held also, as appears\nfrom the _Memoirs_, Mississippi bonds, and these _were_ repudiated, or\nat least five million dollars of a certain class of Mississippi bonds.\nNo such wrong-doing is chargeable to Pennsylvania. I remember the\ndelight with which Professor Reed showed me the note on the fly-leaf at\nthe end of the fifth volume of the edition of 1850--words written at\nhis request, and the last sentences ever composed by the Poet for the\npress.\"--ED.\n\n\n\"YOUNG ENGLAND--WHAT IS THEN BECOME OF OLD\"\n\nComposed 1845.--Published 1845\n\nOne of the \"Sonnets dedicated to Liberty and Order.\"--ED.\n\n    Young England--what is then become of Old\n    Of dear Old England? Think they she is dead,\n    Dead to the very name? Presumption fed\n    On empty air! That name will keep its hold\n    In the true filial bosom's inmost fold                             5\n    For ever.--The Spirit of Alfred, at the head\n    Of all who for her rights watch'd, toil'd and bled,\n    Knows that this prophecy is not too bold.\n    What--how! shall she submit in will and deed\n    To Beardless Boys--an imitative race,                             10\n    The _servum pecus_ of a Gallic breed?\n    Dear Mother! if thou _must_ thy steps retrace,\n    Go where at least meek Innocency dwells;\n    Let Babes and Sucklings be thy oracles.\n\n\n\n\n1846\n\nThe poems written in 1846 were six sonnets, the lines beginning, \"I\nknow an aged man constrained to dwell,\" an \"Evening Voluntary,\" and\nother two short pieces.--ED.\n\n\nSONNET[301]\n\nComposed 1846.--Published 1850\n\nThis was placed among the \"Epitaphs and Elegiac Poems.\"--ED.\n\n    Why should we weep or mourn, Angelic boy,\n    For such thou wert ere from our sight removed,\n    Holy, and ever dutiful--beloved\n    From day to day with never-ceasing joy,\n    And hopes as dear as could the heart employ                        5\n    In aught to earth pertaining? Death has proved\n    His might, nor less his mercy, as behoved--\n    Death conscious that he only could destroy\n    The bodily frame. That beauty is laid low\n    To moulder in a far-off field of Rome;                            10\n    But Heaven is now, blest Child, thy Spirit's home:\n    When such divine communion, which we know,\n    Is felt, thy Roman-burial place will be\n    Surely a sweet remembrancer of Thee.\n\n[301] This sonnet refers to the poet's grandchild, who died at Rome\nin the beginning of 1846. Wordsworth wrote of it thus to Professor\nHenry Reed, \"_Jan. 23, 1846._ \u2026 Our daughter-in-law fell into bad\nhealth between three and four years ago. She went with her husband to\nMadeira, where they remained nearly a year; she was then advised to go\nto Italy. After a prolonged residence there, her six children (whom her\nhusband returned to England for), went, at her earnest request, to that\ncountry, under their father's guidance; then he was obliged, on account\nof his duty as a clergyman, to leave them. Four of the number resided\nwith their mother at Rome, three of whom took a fever there, of which\nthe youngest--as noble a boy of five years as ever was seen--died,\nbeing seized with convulsions when the fever was somewhat subdued.\"--ED.\n\n\n\"WHERE LIES THE TRUTH? HAS MAN, IN WISDOM'S CREED\"\n\nComposed 1846.--Published 1850\n\nOne of the \"Evening Voluntaries.\"--ED.\n\n    Where lies the truth? has Man, in wisdom's creed,\n    A pitiable doom; for respite brief\n    A care more anxious, or a heavier grief?\n    Is he ungrateful, and doth little heed\n    God's bounty, soon forgotten; or indeed,                           5\n    Must Man, with labour born, awake to sorrow[302]\n    When Flowers rejoice and Larks with rival speed\n    Spring from their nests to bid the Sun good morrow?\n    They mount for rapture as their[303] songs proclaim\n    Warbled in hearing both of earth and sky;                         10\n    But o'er the contrast wherefore heave a sigh?\n    Like those aspirants let us soar--our aim,\n    Through life's worst trials, whether shocks or snares,\n    A happier, brighter, purer Heaven than theirs.[304]\n\n[302] 1850.\n\n    Who that lies down and may not wake to sorrow\n\n    MS.\n\n[303] 1850.\n\n    They mount for rapture; this their \u2026\n\n    MS.\n\n[304] This sonnet was suggested by the death of Wordsworth's grandson\ncommemorated in the previous sonnet, and by the alarming illness of his\nbrother, the Master of Trinity College, Cambridge, and the expected\ndeath of a nephew (John Wordsworth), at Ambleside, the only son of his\neldest brother, Richard.--ED.\n\n\nTO LUCCA GIORDANO[305]\n\nComposed 1846.--Published 1850\n\nOne of the \"Evening Voluntaries.\"--ED.\n\n    Giordano, verily thy Pencil's skill\n    Hath here portrayed with Nature's happiest grace\n    The fair Endymion couched on Latmos-hill;\n    And Dian gazing on the Shepherd's face\n    In rapture,--yet suspending her embrace,                           5\n    As not unconscious with what power the thrill\n    Of her most timid touch his sleep would chase,\n    And, with his sleep, that beauty calm and still.\n    O may this work have found its last retreat\n    Here in a Mountain-bard's secure abode,                           10\n    One to whom, yet a School-boy, Cynthia showed\n    A face of love which he in love would greet,\n    Fixed, by her smile, upon some rocky seat;\n    Or lured along where green-wood paths he trod.\n\n    RYDAL MOUNT, 1846.\n\n[305] Lucca Giordano was born at Naples, in 1629. He was at first a\ndisciple of Spagnaletto, next of Pietro da Cortona; but after coming\nunder the influence of Correggio, he went to Venice, where Titian was\nhis inspiring master. In his own work the influence of all of these\npredecessors may be traced, but chiefly that of Titian, whose style\nof colouring and composition he followed so closely that many of his\nworks might be mistaken for those of his greatest master. The picture\nreferred to in this sonnet was brought from Italy by the poet's eldest\nson.--ED.\n\n\n\"WHO BUT IS PLEASED TO WATCH THE MOON ON HIGH\"\n\nComposed 1846.--Published 1850\n\nOne of the \"Evening Voluntaries.\"--ED.\n\n    Who but is pleased to watch the moon on high\n    Travelling where she from time to time enshrouds\n    Her head, and nothing loth her Majesty\n    Renounces, till among the scattered clouds\n    One with its kindling edge declares that soon                      5\n    Will reappear before the uplifted eye\n    A Form as bright, as beautiful a moon,\n    To glide in open prospect through clear sky.\n    Pity that such a promise e'er should prove\n    False in the issue, that yon seeming space                        10\n    Of sky should be in truth the stedfast face\n    Of a cloud flat and dense, through which must move\n    (By transit not unlike man's frequent doom)\n    The Wanderer lost in more determined gloom.\n\n\nILLUSTRATED BOOKS AND NEWSPAPERS\n\nComposed 1846.--Published 1850\n\nOne of the \"Poems of Sentiment and Reflection.\"--ED.\n\n    Discourse was deemed Man's noblest attribute,\n    And written words the glory of his hand;\n    Then followed Printing with enlarged command\n    For thought--dominion vast and absolute\n    For spreading truth, and making love expand.                       5\n    Now prose and verse sunk into disrepute\n    Must lacquey a dumb Art that best can suit\n    The taste of this once-intellectual Land.\n    A backward movement surely have we here,[306]\n    From manhood--back to childhood; for the age--                    10\n    Back towards caverned life's first rude career.\n    Avaunt this vile abuse of pictured page!\n    Must eyes be all in all, the tongue and ear\n    Nothing? Heaven keep us from a lower stage!\n\n[306] The _Illustrated London News_--the pioneer of illustrated\nnewspapers--was first issued on 14th May 1842. The painter and artist\nmay differ from the poet, in the judgment here pronounced; but had\nWordsworth known the degradation to which many newspapers would sink in\nthis direction, his censure would have been more severe.--ED.\n\n\nSONNET\n\nTO AN OCTOGENARIAN\n\nComposed 1846.--Published 1850\n\n    Affections lose their object; Time brings forth\n    No successors; and, lodged in memory,\n    If love exist no longer, it must die,--\n    Wanting accustomed food must pass from earth,\n    Or never hope to reach a second birth.[307]                        5\n    This sad belief, the happiest that is left\n    To thousands, share not Thou; howe'er bereft,\n    Scorned, or neglected, fear not such a dearth.\n    Though poor and destitute of friends thou art,\n    Perhaps the sole survivor of thy race,                            10\n    One to whom Heaven assigns that mournful part\n    The utmost solitude of age to face,\n    Still shall be left some corner of the heart\n    Where Love for living Thing can find a place.\n\n[307] Compare Tennyson's _Lines to J.S._--\n\n    God gives us love. Something to love\n      He lends us; but, when love is grown\n    To ripeness, that on which it throve\n      Falls off, and love is left alone.\n\nED.\n\n\n\"I KNOW AN AGED MAN CONSTRAINED TO DWELL\"\n\nComposed 1846.--Published 1850\n\nOne of the \"Miscellaneous Poems.\"--ED.\n\n    I know an aged Man constrained to dwell\n    In a large house of public charity,\n    Where he abides, as in a Prisoner's cell,\n    With numbers near, alas! no company.\n\n    When he could creep about, at will, though poor                    5\n    And forced to live on alms, this old Man fed\n    A Redbreast, one that to his cottage door\n    Came not, but in a lane partook his bread.\n\n    There, at the root of one particular tree,\n    An easy seat this worn-out Labourer found                         10\n    While Robin pecked the crumbs upon his knee\n    Laid one by one, or scattered on the ground.\n\n    Dear intercourse was theirs, day after day;\n    What signs of mutual gladness when they met!\n    Think of their common peace, their simple play,                   15\n    The parting moment and its fond regret.\n\n    Months passed in love that failed not to fulfil,\n    In spite of season's change, its own demand,\n    By fluttering pinions here and busy bill;\n    There by caresses from a tremulous hand.                          20\n\n    Thus in the chosen spot a tie so strong\n    Was formed between the solitary pair,\n    That when his fate had housed him 'mid a throng\n    The Captive shunned all converse proffered there.\n\n    Wife, children, kindred, they were dead and gone;                 25\n    But, if no evil hap his wishes crossed,\n    One living Stay was left, and on[308] that one\n    Some recompense for all that he had lost.\n\n    O that the good old Man had power to prove,\n    By message sent through air or visible token,                     30\n    That still he loves the Bird, and still must love;\n    That friendship lasts though fellowship is broken!\n\n[308] So all the editions have it; but, as Principal Greenwood\nsuggested to me, the true reading should be \"in that one.\"--ED.\n\n\n\"THE UNREMITTING VOICE OF NIGHTLY STREAMS\"\n\nComposed 1846.--Published 1850\n\nOne of the \"Poems of Sentiment and Reflection.\"--ED.\n\n    The unremitting voice of nightly streams\n    That wastes so oft, we think, its tuneful powers,\n    If neither soothing to the worm that gleams\n    Through dewy grass, nor small birds hushed in bowers,\n    Nor unto silent leaves and drowsy flowers,--                       5\n    That voice of unpretending harmony\n    (For who what is shall measure by what seems\n    To be, or not to be,[309]\n    Or tax high Heaven with prodigality?)\n    Wants not a healing influence that can creep                      10\n    Into the human breast, and mix with sleep\n    To regulate the motion of our dreams\n    For kindly issues--as through every clime\n    Was felt near murmuring brooks in earliest time;\n    As at this day, the rudest swains who dwell                       15\n    Where torrents roar, or hear the tinkling knell\n    Of water-breaks, with grateful heart could tell.\n\n[309] _Hamlet_, act III. scene i. l. 56.--ED.\n\n\n\"HOW BEAUTIFUL THE QUEEN OF NIGHT, ON HIGH\"\n\nComposed 1846.--Published 1850\n\nOne of the \"Miscellaneous Poems.\"--ED.\n\n    How beautiful the Queen of Night, on high\n    Her way pursuing among scattered clouds,\n    Where, ever and anon, her head she shrouds\n    Hidden from view in dense obscurity.\n    But look, and to the watchful eye\n    A brightening edge will indicate that soon\n    We shall behold the struggling Moon\n    Break forth,--again to walk the clear blue sky.\n\n\nON THE BANKS OF A ROCKY STREAM\n\nComposed 1846.--Published 1850\n\n    Behold an emblem of our human mind\n    Crowded with thoughts that need a settled home\n    Yet, like to eddying balls of foam\n    Within this whirlpool, they each other chase\n    Round and round, and neither find\n    An outlet nor a resting-place!\n    Stranger, if such disquietude be thine,\n    Fall on thy knees and sue for help divine.\n\n\nODE\n\nINTIMATIONS OF IMMORTALITY FROM RECOLLECTIONS OF EARLY CHILDHOOD\n\nComposed 1803-6.--Published 1807\n\n[This was composed during my residence at Town-end, Grasmere. Two years\nat least passed between the writing of the four first stanzas and\nthe remaining part. To the attentive and competent reader the whole\nsufficiently explains itself; but there may be no harm in adverting\nhere to particular feelings or _experiences_ of my own mind on which\nthe structure of the poem partly rests. Nothing was more difficult for\nme in childhood than to admit the notion of death as a state applicable\nto my own being. I have said elsewhere--\n\n                  A simple child,\n    That lightly draws its breath,\n    And feels its life in every limb,\n    What should it know of death!--\n\nBut it was not so much from feelings of animal vivacity that my\ndifficulty came as from a sense of the indomitableness of the Spirit\nwithin me. I used to brood over the stories of Enoch and Elijah, and\nalmost to persuade myself that, whatever might become of others, I\nshould be translated, in something of the same way, to heaven. With\na feeling congenial to this, I was often unable to think of external\nthings as having external existence, and I communed with all that I saw\nas something not apart from, but inherent in, my own immaterial nature.\nMany times while going to school have I grasped at a wall or tree to\nrecall myself from this abyss of idealism to the reality. At that\ntime I was afraid of such processes. In later periods of life I have\ndeplored, as we have all reason to do, a subjugation of an opposite\ncharacter, and have rejoiced over the remembrances, as is expressed in\nthe lines--\n\n              Obstinate questionings\n    Of sense and outward things,\n    Fallings from us, vanishings, etc.\n\nTo that dream-like vividness and splendour which invest objects of\nsight in childhood, every one, I believe, if he would look back, could\nbear testimony, and I need not dwell upon it here; but having in the\npoem regarded it as presumptive evidence of a prior state of existence,\nI think it right to protest against a conclusion, which has given\npain to some good and pious persons, that I meant to inculcate such a\nbelief. It is far too shadowy a notion to be recommended to faith, as\nmore than an element in our instincts of immortality. But let us bear\nin mind that, though the idea is not advanced in revelation, there is\nnothing there to contradict it, and the fall of man presents an analogy\nin its favour. Accordingly, a pre-existent state has entered into the\npopular creeds of many nations; and, among all persons acquainted with\nclassic literature, is known as an ingredient in Platonic philosophy.\nArchimedes said that he could move the world if he had a point whereon\nto rest his machine. Who has not felt the same aspirations as regards\nthe world of his own mind?[310] Having to wield some of its elements\nwhen I was impelled to write this poem on the \"Immortality of the\nSoul,\" I took hold of the notion of pre-existence as having sufficient\nfoundation in humanity for authorizing me to make for my purpose the\nbest use of it I could as a poet.--I.F.]\n\n    The Child is Father of the Man;\n    And I could wish my days to be\n    Bound each to each by natural piety.[311]\n\n                          I\n\n    There was a time when meadow, grove, and stream,\n    The earth, and every common sight,\n            To me did seem\n          Apparelled in celestial light,\n    The glory and the freshness of a dream.                            5\n    It is not now as it hath[312] been of yore;--\n          Turn wheresoe'er I may,\n            By night or day,\n    The things which I have seen I now can see no more.\n\n                          II\n\n            The Rainbow comes and goes,                               10\n            And lovely is the Rose,\n            The Moon doth with delight\n      Look round her when the heavens are bare,\n            Waters on a starry night\n            Are beautiful and fair;                                   15\n        The sunshine is a glorious birth;\n        But yet I know, where'er I go,\n    That there hath passed away a glory from the earth.\n\n                         III\n\n    Now, while the birds thus sing a joyous song,\n        And while the young lambs bound                               20\n            As to the tabor's sound,\n    To me alone there came a thought of grief:\n    A timely utterance gave that thought relief,\n            And I again am strong:\n    The cataracts blow their trumpets from the steep;                 25\n    No more shall grief of mine the season wrong;\n    I hear the Echoes through the mountains throng,\n    The Winds come to me from the fields of sleep,\n            And all the earth is gay;\n                Land and sea                                          30\n          Give themselves up to jollity,\n            And with the heart of May\n          Doth every Beast keep holiday;--\n            Thou Child of Joy,\n    Shout round me, let me hear thy shouts, thou happy Shepherd-boy!  35\n\n                          IV\n\n    Ye blessed Creatures, I have heard the call\n        Ye to each other make; I see\n    The heavens laugh with you in your jubilee;\n        My heart is at your festival,\n          My head hath its coronal,[313]                              40\n    The fulness of your bliss, I feel--I feel it all.[314]\n          Oh evil day! if I were sullen\n          While Earth herself is adorning,[315]\n              This sweet May-morning,\n          And the Children are culling[316]                           45\n              On every side,\n          In a thousand valleys far and wide,\n          Fresh flowers; while the sun shines warm,\n    And the Babe leaps up on his Mother's arm:--\n          I hear, I hear, with joy I hear!                            50\n          --But there's a Tree, of many, one,\n    A single Field which I have looked upon,\n    Both of them speak of something that is gone:\n          The <DW29> at my feet\n          Doth the same tale repeat:                                  55\n    Whither is fled the visionary gleam?\n    Where is it now,[317] the glory and the dream?\n\n                          V\n\n    Our birth is but a sleep and a forgetting:\n    The Soul that rises with us, our life's Star,\n          Hath had elsewhere its setting,                             60\n            And cometh from afar:\n          Not in entire forgetfulness,\n          And not in utter nakedness,\n    But trailing clouds of glory do we come\n          From God, who is our home:                                  65\n    Heaven lies about us in our infancy!\n    Shades of the prison-house begin to close\n          Upon the growing Boy,\n    But He beholds the light, and whence it flows\n          He sees it in his joy;                                      70\n    The Youth, who daily farther from the east\n          Must travel, still is Nature's Priest,\n          And by the vision splendid\n          Is on his way attended;\n    At length the Man perceives it[318] die away,                     75\n    And fade into the light of common day.[319]\n\n                          VI\n\n    Earth fills her lap with pleasures[320] of her own;\n    Yearnings she hath in her own natural kind,\n    And, even with something of a Mother's mind,\n          And no unworthy aim,                                        80\n          The homely Nurse doth all she can\n    To make her Foster-child, her Inmate Man,\n          Forget the glories he hath known,\n    And that imperial palace whence he came.\n\n                         VII\n\n    Behold the Child among his new-born blisses,                      85\n    A six years' Darling[321] of a pigmy size!\n    See, where 'mid work of his own hand he lies,\n    Fretted by sallies of his mother's kisses,\n    With light upon him from his father's eyes!\n    See, at his feet, some little plan or chart,                      90\n    Some fragment from his dream of human life,\n    Shaped by himself with newly-learned art;\n        A wedding or a festival,\n        A mourning or a funeral;\n          And this hath now his heart,                                95\n        And unto this he frames his song:\n          Then will he fit his tongue\n    To dialogues of business, love, or strife;\n        But it will not be long\n        Ere this be thrown aside,                                    100\n        And with new joy and pride\n    The little Actor cons another part;\n    Filling from time to time his \"humorous stage\"[322]\n    With all the Persons, down to palsied Age,\n    That Life brings with her in her equipage;                       105\n        As if his whole vocation\n        Were endless imitation.\n\n                         VIII\n\n    Thou, whose exterior semblance[323] doth belie\n        Thy Soul's immensity;\n    Thou best Philosopher, who yet dost keep                         110\n    Thy heritage, thou Eye among the blind,\n    That, deaf and silent, read'st the eternal deep,\n    Haunted for ever by the eternal mind,--\n          Mighty Prophet! Seer blest!\n          On whom those truths do rest,                              115\n    Which we are toiling all our lives to find,\n    In darkness lost, the darkness of the grave;[324]\n    Thou, over whom thy Immortality\n    Broods like the Day, a Master o'er a Slave,\n    A Presence which is not to be put by;[325]                       120\n    Thou little Child, yet glorious in the might\n    Of heaven-born freedom on thy being's height,[326]\n    Why with such earnest pains dost thou provoke\n    The years to bring the inevitable yoke,\n    Thus blindly with thy blessedness at strife?                     125\n    Full soon thy Soul shall have her earthly freight,\n    And custom[327] lie upon thee with a weight,[328]\n    Heavy as frost, and deep almost as life!\n\n                          IX\n\n            O joy! that in our embers\n            Is something that doth live,                             130\n            That nature yet remembers\n            What was so fugitive!\n    The thought of our past years in me doth breed\n    Perpetual benediction;[329] not indeed\n    For that which is most worthy to be blest;                       135\n    Delight and liberty, the simple creed\n    Of Childhood, whether busy or at rest,\n    With new-fledged hope still fluttering in his breast:--[330]\n            Not for these I raise\n            The song of thanks and praise;                           140\n          But for those obstinate questionings\n          Of sense and outward things,\n          Fallings from us, vanishings;\n          Blank misgivings of a Creature\n    Moving about in worlds not realised,                             145\n    High instincts before which our mortal Nature\n    Did tremble like a guilty Thing surprised:\n            But for those first affections,\n            Those shadowy recollections,\n          Which, be they what they may,                              150\n    Are yet the fountain light of all our day,\n    Are yet a master light of all our seeing;\n        Uphold us, cherish, and have power to make[331]\n    Our noisy years seem moments in the being\n    Of the eternal Silence: truths that wake,                        155\n          To perish never;\n    Which neither listlessness, nor mad endeavour,\n            Nor Man nor Boy,\n    Nor all that is at enmity with joy,\n    Can utterly abolish or destroy!                                  160\n          Hence in a season of calm weather,\n            Though inland far we be,\n    Our Souls have sight of that immortal sea\n            Which brought us hither,\n          Can in a moment travel thither,                            165\n    And see the Children sport upon the shore,\n    And hear the mighty waters rolling evermore.\n\n                          X\n\n    Then sing, ye Birds, sing, sing a joyous song!\n          And let the young Lambs bound\n          As to the tabor's sound!                                   170\n    We in thought will join your throng,\n          Ye that pipe and ye that play,\n          Ye that through your hearts to-day\n          Feel the gladness of the May!\n    What though the radiance which was once so bright                175\n    Be now for ever taken from my sight,\n        Though nothing can bring back the hour\n    Of splendour in the grass, of glory in the flower;\n          We will grieve not, rather find\n          Strength in what remains behind;                           180\n          In the primal sympathy\n          Which having been must ever be;\n          In the soothing thoughts that spring\n          Out of human suffering;\n          In the faith that looks through death,                     185\n    In years that bring the philosophic mind.\n\n                          XI\n\n    And O, ye Fountains, Meadows, Hills, and Groves,\n    Forebode not any severing[332] of our loves!\n    Yet in my heart of hearts I feel your might;\n    I only have relinquished one delight                             190\n    To live beneath your more habitual sway.\n    I love the Brooks which down their channels fret,\n    Even more than when I tripped lightly as they;\n    The innocent brightness of a new-born Day\n                  Is lovely yet;                                     195\n    The Clouds that gather round the setting sun\n    Do take a sober colouring from an eye\n    That hath kept watch o'er man's mortality;\n    Another race hath been, and other palms are won.[333]\n    Thanks to the human heart by which we live,                      200\n    Thanks to its tenderness, its joys, and fears,\n    To me the meanest flower that blows[334] can give\n    Thoughts that do often lie too deep for tears.[335]\n\nThis great _Ode_ was first printed as the last poem in the second\nvolume of the edition of 1807. At that date Wordsworth gave it the\nsimple title _Ode_, prefixing to it the motto, \"Paul\u00f2 majora canamus.\"\nIn 1815, when he revised the poem throughout, he named it--in\nthe characteristic manner of many of his titles--diffuse and yet\nprecise, _Ode. Intimations of Immortality from Recollections of Early\nChildhood_; and he then prefixed to it the lines of his own earlier\npoem on the Rainbow (March 1802):--\n\n    The Child is Father of the Man;\n    And I could wish my days to be\n    Bound each to each by natural piety.\n\nIt retained this longer title and motto in all subsequent editions. In\nthe editions 1807 to 1820, it was placed by itself at the end of the\npoems, and formed their natural conclusion and climax. In the editions\n1827 and 1832, it was inappropriately put amongst \"Epitaphs and Elegiac\nPoems.\" The evident mistake of placing it amongst these seems to have\nsuggested to Wordsworth, in 1836, its having a place by itself,--which\nhe gave it then and retained in the subsequent editions of 1842 and\n1849,--when it closed the series of minor poems in Volume V., and\npreceded the _Excursion_ in Volume VI. The same arrangement was adopted\nin the double-columned single volume edition of 1845.\n\nMr. Aubrey de Vere has urged me to take it out of its chronological\nplace, and let it conclude the whole series of Wordsworth's poems, as\nthe greatest, and that to which all others lead up. Mr. De Vere's wish\nis based on conversations which he had with the poet himself.\n\nThe _Ode, Intimations of Immortality_, was written at intervals,\nbetween the years 1803 and 1806; and it was subjected to frequent and\ncareful revision. No poem of Wordsworth's bears more evident traces\nin its structure at once of inspiration and elaboration; of original\nflight of thought and _afflatus_ on the one hand, and on the other of\ncareful sculpture and fastidious choice of phrase. But it is remarkable\nthat there are very few changes of text in the successive editions.\nMost of the alterations were made before 1815, and the omission of some\nfeeble lines which originally stood in stanza viii. in the editions\nof 1807 and 1815, was a great advantage in disencumbering the poem.\nThe main revision and elaboration of this Ode, however--an elaboration\nwhich suggests the passage of the glacier ice over the rocks of White\nMoss Common, where the poem was murmured out stanza by stanza--was all\nfinished before it first saw the light in 1807. In form it is irregular\nand original. And perhaps the most remarkable thing in its structure,\nis the frequent change of the keynote, and the skill and delicacy with\nwhich the transitions are made. \"The feet throughout are iambic. The\nlines vary in length from the Alexandrine to the line with two accents.\nThere is a constant ebb and flow in the full tide of song, but scarce\ntwo waves are alike.\" (Hawes Turner, _Selections from Wordsworth_.)\n\nIn the \"notes\" to the _Selections_ just referred to on Immortality,\nthere is an excellent commentary on this _Ode_, almost every line of\nwhich is worthy of minute analysis and study. Some of the following are\nsuggested by Mr. Turner's notes.\n\n    (1) _The Winds come to me from the fields of sleep._\n\nThe morning breeze blowing from the fields that were dark during the\nhours of sleep.\n\n    (2) --_But there's a Tree, of many, one._\n\nCompare Browning's _May and Death_--\n\n    Only one little sight, one plant\n    Woods have in May, etc.\n\n    (3) _The <DW29> at my feet_\n    _Doth the same tale repeat._\n\nFrench \"Pens\u00e9e.\" \"<DW29>s, that's for thoughts.\" Ophelia in _Hamlet_.\n\n    (4) _Our birth is but a sleep and a forgetting._\n\nThis thought Wordsworth owed, consciously or unconsciously, to Plato.\nThough he tells us in the Fenwick note that he did not mean to\n_inculcate_ the belief, there is no doubt that he clung to the notion\nof a life pre-existing the present, on grounds similar to those on\nwhich he believed in a life to come. But there are some differences in\nthe way in which the idea commended itself to Plato and to Wordsworth.\nThe stress was laid by Wordsworth on the effect of terrestrial life\nin putting the higher faculties to sleep, and making us \"forget the\nglories we have known.\" Plato, on the other hand, looked upon the\nmingled experiences of mundane life as inducing a gradual but slow\nremembrance (\u1f00\u03bd\u03ac\u03bc\u03bd\u03b5\u03c3\u03b9\u03c2) of the past. Compare Tennyson's _Two Voices_,\nand Wordsworth's sonnet, beginning--\n\n    Man's life is like a sparrow, mighty king.\n\n    (5) _Filling from time to time his \"humorous stage\"_\n    _With all the Persons,_\n\n_i.e._ with the _dramatis person\u00e6_.\n\n    (6) \u2026 _thou Eye among the blind,_\n    _That, deaf and silent, read'st the eternal deep._\n\nThere is an admirable parallel illustration of Wordsworth's use of this\nfigure (describing one sense in terms of another), in the lines in\n_Airey-Force Valley_--\n\n    A soft eye-music of slow-waving boughs.\n\n    (7) _Full soon thy Soul shall have her earthly freight,_\n    _And custom lie upon thee with a weight,_\n    _Heavy as frost, and deep almost as life!_\n\nCompare with this, the lines in the fourth book of _The Excursion_,\nbeginning--\n\n    Alas! the endowment of immortal power\n    Is matched unequally with custom, time.\n\n    (8) _Fallings from us, vanishings._\n\nThe outward sensible universe, visible and tangible, seeming to\nfall away from us, as unreal, to vanish in unsubstantially. See the\nexplanation of this youthful experience in the Fenwick note. That\nconfession of his boyish days at Hawkshead, \"many times, while going to\nschool, have I grasped at a wall or tree, _to recall myself from this\nabyss of idealism to the reality_\" (by which he explains those--\n\n    Fallings from us, vanishings, etc.),\n\nsuggests a similar experience and confession of Cardinal Newman's in\nhis _Apologia_ (see p. 67).\n\nThe late Rev. Robert Perceval Graves, of Windermere, and afterwards of\nDublin, wrote to me in 1850:--\"I remember Mr. Wordsworth saying, that\nat a particular stage of his mental progress, he used to be frequently\nso rapt into an unreal transcendental world of ideas that the external\nworld seemed no longer to exist in relation to him, and he _had to\nreconvince himself of its existence by clasping a tree, or something\nthat happened to be near him_. I could not help connecting this fact\nwith that obscure passage in his great _Ode on the Intimations of\nImmortality_, in which he speaks of--\n\n        Those obstinate questionings\n    Of sense and outward things;\n    Fallings from us, vanishings; etc.\"\n\nProfessor Bonamy Price further confirms the explanation which\nWordsworth gave of the passage, in a letter written to me in 1881,\ngiving an account of a conversation he had with the poet, as follows:--\n\n                                         \"OXFORD, _April 21, 1881_.\n\n    \"MY DEAR SIR,--You will be glad, I am sure, to receive an\n    interpretation, which chance enabled me to obtain from\n    Wordsworth himself of a passage in the immortal _Ode on\n    Immortality_.\u2026\n\n    \"It happened one day that the poet, my wife, and I were taking\n    a walk together by the side of Rydal Water. We were then by the\n    sycamores under Nab Scar. The aged poet was in a most genial\n    mood, and it suddenly occurred to me that I might, without\n    unwarrantable presumption, seize the golden opportunity thus\n    offered, and ask him to explain these mysterious words. So\n    I addressed him with an apology, and begged him to explain,\n    what my own feeble mother-wit was unable to unravel, and for\n    which I had in vain sought the assistance of others, what\n    were those 'fallings from us, vanishings,' for which, above\n    all other things, he gave God thanks. The venerable old man\n    raised his aged form erect; he was walking in the middle,\n    and passed across me to a five-barred gate in the wall which\n    bounded the road on the side of the lake. He clenched the top\n    bar firmly with his right hand, pushed strongly against it,\n    and then uttered these ever-memorable words: 'There was a time\n    in my life when I had to push against something that resisted,\n    to be sure that there was anything outside of me. I was sure\n    of my own mind; everything else fell away, and vanished into\n    thought.' Thought, he was sure of; matter for him, at the\n    moment, was an unreality--nothing but a thought. Such natural\n    spontaneous idealism has probably never been felt by any other\n    man.\n\n                                                    \"BONAMY PRICE.\"\n\nThis, however, was not an experience peculiar to Wordsworth, as\nProfessor Price imagined--and its value would be much lessened if it\nhad been so--but was one to which (as the poet said to Miss Fenwick)\n\"every one, if he would look back, could bear testimony.\"\n\nThe following is from S.T. Coleridge's _Biographia Literaria_ (chap.\nxxii. p. 29, edition 1817)--\n\n\"To the _Ode on the Intimations of Immortality from Recollections of\nEarly Childhood_, the poet might have prefixed the lines which Dante\naddresses to one of his own Canzoni--\n\n    Canzone, i' credo, che saranno radi\n    Color che tua ragione intendan bene:\n    Tanto lor sei faticoso ed alto.\n\n    O lyric song, there will be few, think I,\n    Who may thy import understand aright:\n    Thou art for them so arduous and so high!\n\nBut the Ode was intended for such readers only as had been accustomed\nto watch the flux and reflux of their inmost nature, to venture\nat times into the twilight realms of consciousness, and to feel a\ndeep interest in modes of inmost being, to which they know that the\nattributes of time and space are inapplicable and alien, but which yet\ncannot be conveyed, save in symbols of time and space. For such readers\nthe sense is sufficiently plain, and they will be as little disposed to\ncharge Mr. Wordsworth with believing the Platonic pre-existence in the\nordinary interpretation of the words, as I am to believe, that Plato\nhimself ever meant or taught it.\n\n    \u03c0\u03bf\u03bb\u03bb\u03ac \u03bc\u03bf\u03b9 \u1f51\u03c0' \u1f00\u03b3\u03ba\u1ff6\u03bd\u03bf\u03c2 \u1f60\u03ba\u03ad\u03b1 \u03b2\u03ad\u03bb\u03b7\n    \u1f14\u03bd\u03b4\u03bf\u03bd \u1f10\u03bd\u03c4\u1f76 \u03c6\u03b1\u03c1\u03ad\u03c4\u03c1\u03b1\u03c2\n    \u03c6\u03c9\u03bd\u1fb6\u03bd\u03c4\u03b1 \u03c3\u03c5\u03bd\u03b5\u03c4\u03bf\u1fd6\u03c3\u03b9\u03bd \u1f10\u03c2\n    \u03b4\u1f72 \u03c4\u1f78 \u03c0\u1fb6\u03bd \u1f11\u03c1\u03bc\u03b7\u03bd\u03ad\u03c9\u03bd\n    \u03c7\u03b1\u03c4\u03af\u03b6\u03b5\u03b9. \u03c3\u03bf\u03c6\u1f78\u03c2 \u1f41 \u03c0\u03bf\u03bb\u03bb\u1f70 \u03b5\u1f30\u03b4\u1f7c\u03c2 \u03c6\u03c5\u1fb7.\n    \u03bc\u03b1\u03b8\u03cc\u03bd\u03c4\u03b5\u03c2 \u03b4\u1f72 \u03bb\u03ac\u03b2\u03c1\u03bf\u03b9\n    \u03c0\u03b1\u03b3\u03b3\u03bb\u03c9\u03c3\u03c3\u03af\u1fb3, \u03ba\u03cc\u03c1\u03b1\u03ba\u03b5\u03c2 \u1f65\u03c2,\n    \u1f04\u03ba\u03c1\u03b1\u03bd\u03c4\u03b1 \u03b3\u03b1\u03c1\u03cd\u03b5\u03c4\u03bf\u03bd\n    \u0394\u03b9\u1f78\u03c2 \u03c0\u03c1\u1f78\u03c2 \u1f44\u03c1\u03bd\u03b9\u03c7\u03b1 \u03b8\u03b5\u1fd6\u03bf\u03bd.\n\n                 PINDAR, OLYMP. ii.\"[336]\n\nThe following parallel passages from _The Excursion_, _The Prelude_,\nRuskin's _Modern Painters_, Keble's _Praelectiones de Poeticae vi\nMedica_ (p. 788, Prael. xxxix.), and the _Silex Scintillans_ of\nHenry Vaughan, are quoted, in an interesting note to the _Ode_ on\nImmortality, in Professor Henry Reed's American edition of the Poems\n(1851).\n\n    I\n\n                      Ah! why in age\n    Do we revert so fondly to the walks\n    Of childhood--but that there the Soul discerns\n    The dear memorial footsteps unimpaired\n    Of her own native vigour--thence can hear\n    Reverberations; and a choral song,\n    Commingling with the incense that ascends,\n    Undaunted, toward the imperishable heavens,\n    From her own lonely altar?\n\n               _The Excursion_, book ix. ll. 36-44.\n\n    II\n\n                        Our childhood sits,\n    Our simple childhood, sits upon a throne\n    That hath more power than all the elements.\n    I guess not what this tells of Being past,\n    Nor what it augurs of the life to come; etc.\n\n             _The Prelude_, book v. ll. 507-511.\n\nIII\n\n\" \u2026 There was never yet the child of any promise (so far as the\ntheoretic faculties are concerned) but awaked to the sense of beauty\nwith the first gleam of reason; and I suppose there are few, among\nthose who love Nature otherwise than by profession and at second-hand,\nwho look not back to their youngest and least learned days as those of\nthe most intense, superstitious, insatiable, and beatific perception of\nher splendours. And the bitter decline of this glorious feeling, though\nmany note it not, partly owing to the cares and weight of manhood,\nwhich leave them not the time nor the liberty to look for their lost\ntreasure, and partly to the human and divine affections which are\nappointed to take its place, yet have formed the subject, not indeed of\nlamentation, but of holy thankfulness for the witness it bears to the\nimmortal origin and end of our nature, to one whose authority is almost\nwithout appeal in all questions relating to the influence of external\nthings upon the pure human soul.\n\n      Not for these I raise\n      The song of thanks and praise\n    But for those obstinate questionings, etc. etc.\n\nAnd if it were possible for us to recollect all the unaccountable and\nhappy instincts of the careless time, and to reason upon them with the\nmaturer judgment, we might arrive at more right results than either\nthe philosophy or the sophisticated practice of art has yet attained.\nBut we love the perceptions before we are capable of methodising or\ncomparing them.\" (Ruskin's _Modern Painters_, vol. ii. p. 36, part iii.\nch. v. sec. i.)\n\n\" \u2026 Etenim qui velit acutius indagare causas propensae in antiqua\nsaecula voluntatis, mirum ni conjectura incidat aliquando in commentum\nillud Pythagorae, docentis, animarum nostrarum non tum fieri initium,\ncum in hoc mundo nascimur; immo ex ignota quadam regione venire eas,\nin sua quamque corpora; neque tam penitus Lethaeo potu imbui, quin\npermanet quasi quidam anteactae aetatis sapor; hunc autem excitari\nidentidem, et nescio quo sensu percipi, tacito quidem illo et obscuro,\nsed percipi tamen. Atque hac ferme sententia extat summi hac memoria\nPoetae nobilissimum carmen; nempe non aliam ob causam tangi pueritiae\nrecordationem exquisita illa ac pervagata dulcedine, quam propter\ndebilem quendam prioris aevi Deique propioris sensum.\n\nQuamvis autem hanc opinionem vix ferat divinae philosophiae ratio,\nfatemur tamen eam eatenus ad verum accedere, quo sanctum aliquod\net grave tribuit memoriae et caritati puerilium annorum. Nosmet\ncerte infantes novimus quam prope tetigerit Divina benignitas; quis\nporro scit, an omnis illa temporis anteacti dulcedo habeat quandam\nsignificationem Illius Praesentiae?\" (Keble, _Praelectiones de Poeticae\nvi Medica_, p. 788, Prael. xxxix.)\n\n\"CORRUPTION\n\n    Sure, it was so. Man in those early days\n            Was not all stone and earth;\n    He shined a little, and by those weak rays,\n            Had some glimpse of his birth.\n    He saw Heaven o'er his head, and knew from whence\n            He came condemned hither,\n    And, as first Love draws strongest, so from hence\n            His mind sure progressed thither.\"\n\n                    Henry Vaughan, _Silex Scintillans_.\n\nMr. Reed also quotes a passage from Vaughan's poem _Childehood_; but\na more apposite passage may be found in _The Retreate_, in _Silex\nScintillans_.\n\n    Happy those early dayes, when I\n    Shined in my Angell-infancy!\n    Before I understood this place\n    Appointed for my second race,\n    Or taught my soul to fancy ought\n    But a white celestiall thought;\n    When yet I had not walkt above\n    A mile or two from my first Love,\n    And looking back, at that short space,\n    Could see a glimpse of his bright face;\n    When on some _gilded Cloud or Flowre_\n    My gazing soul would dwell an houre,\n    And in those weaker glories spy\n    Some shadows of eternity;\n    \u2026\n    But felt through all this fleshly dresse\n    Bright _shootes_ of everlastingnesse.\n\nThe extent of Wordsworth's debt to Vaughan has been discussed a good\ndeal. There was no copy of the _Silex Scintillans_ in the Rydal\nMount sale-catalogue. I believe that he had read _The Retreate_, and\nforgotten it more completely perhaps than Coleridge forgot Sir John\nDavies' _Orchestra, a Poem on Dancing_, when he wrote _The Ancient\nMariner_.\n\nThe following may be added from _The Friend_ (the edition of 1818),\nvol. i. p. 183:--\"To find no contradiction in the union of old and new\nto contemplate the Ancient of Days with feelings as fresh as if they\nthen sprang forth at his own fiat, this characterizes the minds that\nfeel the riddle of the world, and may help to unravel it! To carry on\nthe feelings of childhood into the powers of manhood, to combine the\nchild's sense of wonder and novelty with the appearances which every\nday, for perhaps 40 years, had rendered familiar,\n\n    With sun and moon and stars throughout the year\n    And man and woman----\n\nThis is the character and privilege of genius, and one of the marks\nwhich distinguish genius from talent.\"--ED.\n\n[310] Compare the Atman of the Vedanta Philosophy.--ED.\n\n[311] See vol. ii. p. 292.--ED.\n\n[312] 1820.\n\n    \u2026 has \u2026\n\n    1807.\n\n[313] Compare _The Idle Shepherd Boys_, ll. 28-30 (vol. ii. p.\n138).--ED.\n\n[314] 1807.\n\n    Even yet more gladness, I can hold it all.\n\n    MS.\n\n[315] 1836.\n\n    While the Earth herself \u2026\n\n    1807.\n\n    \u2026 itself \u2026\n\n    1827.\n\nThe text of 1832 returns to that of 1807.\n\n[316] 1836.\n\n    \u2026 pulling\n\n    1807.\n\n[317]\n\n    Where is it gone, \u2026\n\n    MS.\n\n[318] 1807.\n\n    \u2026 beholds it \u2026\n\n    MS.\n\n[319] Compare, in Bacon's Essay _Of Youth and Age_, \"A certaine Rabbine\nupon the Text, _Your Young Men shall see visions, and your Old Men\nshall dream dreames_, inferreth that Young Men are admitted nearer to\nGod than Old, because _Vision_ is a clearer Revelation than a Dreame.\"\n\nSee Professor Max M\u00fcller's note to his translation of the Upanishads\n(_Sacred Books of the East_, vol. xv. p. 164), beginning \"Drivudagomga\nuses a curious argument in support of the existence of another\nworld.\"--ED.\n\n[320] 1807.\n\n    \u2026 pleasure \u2026\n\n    MS.\n\n[321] 1815.\n\n    A four years' Darling \u2026\n\n    1807.\n\n[322] See, in Daniel's _Musophilus_, the introductory sonnet to Fulke\nGreville, l. 1.--ED.\n\n[323] 1807.\n\n    \u2026 presence \u2026\n\n    MS.\n\n[324] This line is not in the editions of 1807 and 1815.\n\n[325] The editions of 1807 and 1815 have, after \"put by\":\n\n          To whom the grave\n    Is but a lowly bed without the sense or sight\n          Of day or the warm light,\n    A place of thought where we in waiting lie;\n\n    MS.\n\nThe subsequent omission of these lines was due to Coleridge's\ndisapproval of them, expressed in _Biographia Literaria_.--ED.\n\n[326] 1815.\n\n    Of untamed pleasures, on thy Being's height,\n\n    1807.\n\n[327] 1807.\n\n    The world upon thy noble nature seize\n            With all its vanities,\n    And custom \u2026\n\n    MS.\n\n[328] Compare _The Excursion_, book iv. ll. 205, 206--\n\n    Alas! the endowment of immortal power\n    Is matched unequally with custom, time.\n\nED.\n\n[329] 1827.\n\n    Perpetual benedictions: \u2026\n\n    1807.\n\n[330] 1815.\n\n    Of Childhood, whether fluttering or at rest,\n    With new-born hope for ever in his breast:\n\n    1807.\n\n[331] 1815.\n\n    Uphold us, cherish us, and make\n\n    1807.\n\n[332] 1836.\n\n    Think not of any severing \u2026\n\n    1807.\n\n[333] Professor Dowden writes of this line: \"It is a sunset reflection,\nnatural to one who has 'kept watch o'er man's mortality': the day is\nclosing, as human lives have closed; the sun went forth out of his\nchamber as a strong man to run a race, and now the race is over and the\npalm has been won: all things have their hour of fulfilment.\" (See vol.\nv. p. 365, of his edition of Wordsworth's Poems.)--ED.\n\n[334] Compare the introduction to the first canto of _Marmion_--\n\n    The vernal sun new life bestows\n    Upon the meanest flower that blows,\n\nED.\n\n[335] Compare Wither's _The Shepherds Hunting_, the fourth eclogue, ll.\n368-380.--ED.\n\n[336] The text of Pindar, as given by S.T.C., is corrected in the above\nquotation.--ED.\n\n\n\n\nPOEMS BY WILLIAM WORDSWORTH AND BY DOROTHY WORDSWORTH NOT INCLUDED IN\nTHE EDITION OF 1849-50\n\n\n\n\n1787\n\n\nSONNET, ON SEEING MISS HELEN MARIA WILLIAMS WEEP AT A TALE OF\nDISTRESS[337]\n\n    She wept.--Life's purple tide began to flow\n    In languid streams through every thrilling vein;\n    Dim were my swimming eyes--my pulse beat slow,\n    And my full heart was swell'd to dear delicious pain.\n\n    Life left my loaded heart, and closing eye;                        5\n    A sigh recall'd the wanderer to my breast;\n    Dear was the pause of life, and dear the sigh\n    That call'd the wanderer home, and home to rest.\n\n    That tear proclaims--in thee each virtue dwells,\n    And bright will shine in misery's midnight hour;                  10\n    As the soft star of dewy evening tells\n    What radiant fires were drown'd by day's malignant pow'r,\n    That only wait the darkness of the night\n    To chear the wand'ring wretch with hospitable light.\n\n                                                   AXIOLOGUS.\n\n[European Magazine, 1787, vol. xi. p. 302.]\n\nS.T.C. addressed some lines to Wordsworth under the name Axiologus. The\nfollowing is a sample, sent to me by the late Mr. <DW18>s Campbell, _Ad\nVilmum Axiologum_.--ED.[338]\n\nAD VILMUM AXIOLOGUM\n\n    This be the meed, that thy song creates a thousand-fold echo!\n    Sweet as the warble of woods, that awakes at the gale of the morning!\n    List! the Hearts of the Pure, like caves in the ancient mountains\n    Deep, deep _in_ the Bosom, and _from_ the Bosom resound it,\n    Each with a different tone, complete or in musical fragments--\n    All have welcomed thy Voice, and receive and retain and prolong it!\n\n    This is the word of the Lord! it is spoken and Beings Eternal\n    Live and are borne as an Infant, the Eternal begets the Immortal--\n    Love is the Spirit of Life, and Music the Life of the Spirit!\n\n[337] The only justification for republishing this sonnet is that it is\nthe earliest authoritative record of Wordsworth's attempts in Verse. It\nis a much more authentic one than the _Extract from the conclusion of\na Poem, composed in anticipation of leaving School_, or than the lines\n_Written in very early Youth_, and beginning\n\n    Calm is all nature as a resting wheel.\n\nWordsworth dated the former of these poems 1786, but I do not believe\nthat he wrote that poem, and still less that he wrote \"Calm is all\nnature,\" etc., _as we now have it_, in that year. Doubtless he wrote\nverses on these two subjects; but the best evidence against the notion\nthat the text, as we now have it, was written in 1786, is this 1787\nsonnet on Miss Maria Williams. It is not only dated authoritatively,\nbut it was _published_ in 1787; and therefore serves (as nothing else\ncan until we come to 1793) as evidence in regard to the development of\nhis poetic power. The translation of Francis Wrangham's lines--which\nhe called _The Birth of Love_--in 1795, is further evidence in the\nsame direction. No doubt there were many poor poetic utterances by\nWordsworth later in life--failures in his manhood, as dismal as the\n\"Walford Tragedy\" was in his youth--but I think that the _Lines written\nin very early Youth_, and the _Extract from the Poem composed in\nanticipation of leaving School_, were rehandled by him, and the text\ngreatly improved before they were first published. The late Mr. J.\n<DW18>s Campbell wrote to me in 1892: \"Poets tell dreadful fibs about\ntheir early verses--as witness S.T.C. who declared he wrote _The Advent\nof Love_ at fifteen! I _know_ he didn't, and am going to print one or\ntwo of his prize school verses of that age, which I have found in his\nown fifteen-year-old fist.\"--ED.\n\n[338] I should add, in a footnote, that I have no knowledge of the\nsource whence Mr. Campbell derived this; but I am sure that it must\nhave reached him from an authentic one.--ED.\n\n\nLINES WRITTEN BY WILLIAM WORDSWORTH AS A SCHOOL EXERCISE AT HAWKSHEAD,\nANNO \u00c6TATIS 14\n\nIn the \"Autobiographical Memoranda\"--dictated at Rydal Mount in\n1847--Wordsworth said, \"The first verses which I wrote were a task\nimposed by my master: the subject _The Summer Vacation_, and of my\nown accord I added others upon _Return to School_. There was nothing\nremarkable in either poem; but I was called upon, among other scholars,\nto write verses upon the completion of the second century from the\nfoundation of the school in 1585, by Archbishop Sandys. These verses\nwere much admired, far more than they deserved, for they were but a\ntame imitation of Pope's versification, and a little in his style.\nThis exercise, however, put it into my head to compose verses from the\nimpulse of my own mind; and I wrote, while yet a schoolboy, a long poem\nrunning upon my own adventures, and the scenery of the county in which\nI was brought up.\"\n\nThe _Summer Vacation_, and the _Return to School_, were destroyed by\nWordsworth.--ED.\n\n    And has the Sun his flaming chariot driven\n    Two hundred times around the ring of heaven,\n    Since Science first, with all her sacred train,\n    Beneath yon roof began her heavenly reign?\n    While thus I mused, methought, before mine eyes,                   5\n    The Power of EDUCATION seemed to rise;\n    Not she whose rigid precepts trained the boy\n    Dead to the sense of every finer joy;\n    Nor that vile wretch who bade the tender age\n    Spurn Reason's law and humour Passion's rage;                     10\n    But she who trains the generous British youth\n    In the bright paths of fair majestic Truth:\n    Emerging slow from Academus' grove\n    In heavenly majesty she seem'd to move.\n    Stern was her forehead, but a smile serene                        15\n    \"Soften'd the terrors of her awful mien.\"[339]\n    Close at her side were all the powers, design'd\n    To curb, exalt, reform the tender mind:\n    With panting breast, now pale as winter snows,\n    Now flushed as Hebe, Emulation rose;                              20\n    Shame follow'd after with reverted eye,\n    And hue far deeper than the Tyrian dye;\n    Last Industry appear'd with steady pace,\n    A smile sat beaming on her pensive face.\n    I gazed upon the visionary train,                                 25\n    Threw back my eyes, return'd, and gazed again.\n    When lo! the heavenly goddess thus began,\n    Through all my frame the pleasing accents ran.\n\n      When Superstition left the golden light\n    And fled indignant to the shades of night;                        30\n    When pure Religion rear'd the peaceful breast\n    And lull'd the warring passions into rest,\n    Drove far away the savage thoughts that roll\n    In the dark mansions of the bigot's soul,\n    Enlivening Hope display'd her cheerful ray,                       35\n    And beam'd on Britain's sons a brighter day,\n    So when on Ocean's face the storm subsides,\n    Hush'd are the winds and silent are the tides;\n    The God of day, in all the pomp of light,\n    Moves through the vault of heaven, and dissipates the night;      40\n    Wide o'er the main a trembling lustre plays,\n    The glittering waves reflect the dazzling blaze;\n    Science with joy saw Superstition fly\n    Before the lustre of Religion's eye;\n    With rapture she beheld Britannia smile,                          45\n    Clapp'd her strong wings, and sought the cheerful isle.\n    The shades of night no more the soul involve,\n    She sheds her beam, and, lo! the shades dissolve;\n    No jarring monks, to gloomy cell confined,\n    With mazy rules perplex the weary mind;                           50\n    No shadowy forms entice the soul aside,\n    Secure she walks, Philosophy her guide.\n    Britain, who long her warriors had adored,\n    And deemed all merit centred in the sword;\n    Britain, who thought to stain the field was fame,                 55\n    Now honour'd Edward's less than Bacon's name.\n    Her sons no more in listed fields advance\n    To ride the ring, or toss the beamy lance;\n    No longer steel their indurated hearts\n    To the mild influence of the finer arts;                          60\n    Quick to the secret grotto they retire\n    To court majestic truth, or wake the golden lyre;\n    By generous Emulation taught to rise,\n    The seats of learning brave the distant skies.\n    Then noble Sandys, inspir'd with great design,                    65\n    Rear'd Hawkshead's happy roof, and call'd it mine;\n    There have I loved to show the tender age\n    The golden precepts of the classic page;\n    To lead the mind to those Elysian plains\n    Where, throned in gold, immortal Science reigns;                  70\n    Fair to the view is sacred Truth display'd,\n    In all the majesty of light array'd,\n    To teach, on rapid wings, the curious soul\n    To roam from heaven to heaven, from pole to pole,\n    From thence to search the mystic cause of things                  75\n    And follow Nature to her secret springs;\n    Nor less to guide the fluctuating youth\n    Firm in the sacred paths of moral truth,\n    To regulate the mind's disorder'd frame,\n    And quench the passions kindling into flame;                      80\n    The glimmering fires of Virtue to enlarge,\n    And purge from Vice's dross my tender charge.\n    Oft have I said, the paths of Fame pursue,\n    And all that virtue dictates, dare to do;\n    Go to the world, peruse the book of man,                          85\n    And learn from thence thy own defects to scan;\n    Severely honest, break no plighted trust,\n    But coldly rest not here--be more than just;\n    Join to the rigours of the sires of Rome\n    The gentler manners of the private dome;                          90\n    When Virtue weeps in agony of woe,\n    Teach from the heart the tender tear to flow;\n    If Pleasure's soothing song thy soul entice,\n    Or all the gaudy pomp of splendid Vice,\n    Arise superior to the Siren's power,                              95\n    The wretch, the short-lived vision of an hour;\n    Soon fades her cheek, her blushing beauties fly,\n    As fades the chequer'd bow that paints the sky,\n      So shall thy sire, whilst hope his breast inspires,\n    And wakes anew life's glimmering trembling fires,                100\n    Hear Britain's sons rehearse thy praise with joy,\n    Look up to heaven, and bless his darling boy.\n    If e'er these precepts quell'd the passions' strife,\n    If e'er they smooth'd the rugged walks of life,\n    If e'er they pointed forth the blissful way                      105\n    That guides the spirit to eternal day,\n    Do thou, if gratitude inspire thy breast,\n    Spurn the soft fetters of lethargic rest.\n    Awake, awake! and snatch the slumbering lyre,\n    Let this bright morn and Sandys the song inspire.                110\n\n        I look'd obedience: the celestial Fair\n        Smiled like the morn, and vanished into air.\n\n[339] This quotation I am unable to trace--ED.\n\n\n\n\n1792 (or earlier)\n\n\n\"SWEET WAS THE WALK ALONG THE NARROW LANE\"\n\nThis sonnet is found in one of Dorothy Wordsworth's letters to her\nfriend Miss Jane Polland, written from Forncett Rectory, on 6th May\n1792. She wrote:--\n\n\"I promised to transcribe some of William's compositions. As I made\nthe promise I will give you a little sonnet, but all the same I charge\nyou, as you value our friendship, not to read it, or to show it to\nany one--to your sister, or any other person.\u2026 I take the first that\noffers. It is only valuable to me because the lane which gave birth to\nit was the favourite evening walk of my dear William and me.\" \u2026 \"I have\nnot chosen this sonnet because of any particular beauty it has; it was\nthe first I laid my hands upon.\"--ED.\n\n    Sweet was the walk along the narrow lane\n    At noon, the bank and hedgerows all the way\n    Shagged with wild pale green tufts of fragrant hay,\n    Caught by the hawthorns from the loaded wain\n    Which Age, with many a slow stoop, strove to gain;                 5\n    And Childhood seeming still more busy, took\n    His little rake with cunning sidelong look,\n    Sauntering to pluck the strawberries wild unseen.\n    _Now_ too, on melancholy's idle dream\n    Musing, the lone spot with my soul agrees                         10\n    Quiet and dark; for through the thick-wove trees\n    Scarce peeps the curious star till solemn gleams\n    The clouded moon, and calls me forth to stray\n    Through tall green silent woods and ruins grey.\n\n\n\"WHEN LOVE WAS BORN OF HEAVENLY LINE\"\n\nComposed 1795 (or earlier).--Published 1795\n\nTranslated from some French stanzas by Francis Wrangham, and Printed\nin _Poems by Francis Wrangham_, M.A., Member of Trinity College,\nCambridge, London (1795), Sold by J. Mawman, 22 Poultry, pp. 106-111.\nIn the edition of 1795, the original French lines are printed side by\nside with Wordsworth's translation, which closes the volume.--ED.\n\n    When Love was born of heavenly line,\n      What dire intrigues disturb'd Cythera's joy!\n    Till Venus cried, \"A mother's heart is mine;\n      None but myself shall nurse my boy.\"\n\n    But, infant as he was, the child                                   5\n      In that divine embrace enchanted lay;\n    And, by the beauty of the vase beguiled,\n      Forgot the beverage--and pined away.\n\n    \"And must my offspring languish in my sight?\"\n      (Alive to all a mother's pain,                                  10\n    The Queen of Beauty thus her court address'd)\n      \"No: Let the most discreet of all my train\n    Receive him to her breast:\n    Think all, he is the God of young delight.\"\n\n    Then TENDERNESS with CANDOUR join'd,                              15\n      And GAIETY the charming office sought;\n    Nor even DELICACY stay'd behind:\n      But none of those fair Graces brought\n    Wherewith to nurse the child--and still he pined.\n    Some fond hearts to COMPLIANCE seem'd inclined;                   20\n      But she had surely spoil'd the boy:\n        And sad experience forbade a thought\n    On the wild Goddess of VOLUPTUOUS JOY.\n\n    Long undecided lay th' important choice,\n    Till of the beauteous court, at length, a voice                   25\n    Pronounced the name of HOPE:--The conscious child\n    Stretch'd forth his little arms, and smiled.[340]\n\n    'Tis said ENJOYMENT (who averr'd\n      The charge belong'd to her alone)\n    Jealous that HOPE had been preferr'd                              30\n      Laid snares to make the babe her own.\n\n    Of INNOCENCE the garb she took,\n    The blushing mien and downcast look;\n      And came her services to proffer:\n    And HOPE (what has not Hope believed!)                            35\n    By that seducing air deceived,\n      Accepted of the offer.\n\n    It happen'd that, to sleep inclined,\n      Deluded HOPE for one short hour\n      To that false INNOCENCE'S power                                 40\n    Her little charge consign'd.\n\n    The Goddess then her lap with sweetmeats fill'd\n      And gave, in handfuls gave, the treacherous store:\n    A wild delirium first the infant thrill'd;\n      But soon upon her breast he sunk--to wake no more.              45\n\n[340] Compare Gray's _Progress of Poesy_, iii. I. 87--\n\n                      The dauntless child\n    Stretch'd forth his little arms, and smiled.\n\nED.\n\n\nTHE CONVICT\n\nComposed (?).--Published 1798\n\n    The glory of evening was spread through the west;\n      --On the <DW72> of a mountain I stood,\n    While the joy that precedes the calm season of rest\n      Rang loud through the meadow and wood.\n\n    \"And must we then part from a dwelling so fair?\"                   5\n      In the pain of my spirit I said,\n    And with a deep sadness I turned, to repair\n      To the cell where the convict is laid.\n\n    The thick-ribbed walls that o'ershadow the gate\n      Resound; and the dungeons unfold:                               10\n    I pause; and at length, through the glimmering grate,\n      That outcast of pity behold.\n\n    His black matted hair on his shoulder is bent,\n      And deep is the sigh of his breath,\n    And with stedfast dejection his eyes are intent                   15\n      On the fetters that link him to death.\n\n    'Tis sorrow enough on that visage to gaze,\n      That body dismiss'd from his care;\n    Yet my fancy has pierced to his heart, and pourtrays\n      More terrible images there.                                     20\n\n    His bones are consumed, and his life-blood is dried,\n      With wishes the past to undo;\n    And his crime, through the pains that o'erwhelm him, descried,\n      Still blackens and grows on his view.\n\n    When from the dark synod, or blood-reeking field,                 25\n      To his chamber the monarch is led,\n    All soothers of sense their soft virtue shall yield,\n      And quietness pillow his head.\n\n    But if grief, self-consumed, in oblivion would doze,\n      And conscience her tortures appease,                            30\n    'Mid tumult and uproar this man must repose,\n      In the comfortless vault of disease.\n\n    When his fetters at night have so press'd on his limbs,\n      That the weight can no longer be borne,\n    If, while a half-slumber his memory bedims,                       35\n      The wretch on his pallet should turn,\n\n    While the jail-mastiff howls at the dull clanking chain,\n      From the roots of his hair there shall start\n    A thousand sharp punctures of cold-sweating pain,\n      And terror shall leap at his heart.                             40\n\n    But now he half-raises his deep-sunken eye,\n      And the motion unsettles a tear;\n    The silence of sorrow it seems to supply,\n      And asks of me why I am here.\n\n    \"Poor victim! no idle intruder has stood                          45\n      With o'erweening complacence our state to compare,\n    But one, whose first wish is the wish to be good,\n      Is come as a brother thy sorrows to share.\n\n    \"At thy name though compassion her nature resign,\n      Though in virtue's proud mouth thy report be a stain,           50\n    My care, if the arm of the mighty were mine,\n      Would plant thee where yet thou might'st blossom again.\"\n\n\n\n\n1798\n\n\n\"THE SNOW-TRACKS OF MY FRIENDS I SEE\"\n\nThe following incomplete stanzas were evidently written when _The\nComplaint of a Forsaken Indian Woman_ was being composed. They were all\ndiscarded, but have a biographical interest. I assign them to the year\n1798.--ED.\n\n    The snow-tracks of my friends I see,\n    Their foot-marks do not trouble me,\n    For ever left alone am I.\n    Then wherefore should I fear to die?\n    They to the last my friends did cherish                            5\n    And to the last were good and kind,\n    Methinks 'tis strange I did not perish\n    The moment I was left behind.\n\n    Why do I watch those running deer?\n    And wherefore, wherefore come they here?                          10\n    And wherefore do I seem to love\n    The things that live, the things that move?\n    Why do I look upon the sky?\n    I do not live for what I see.\n    Why open thus mine eyes? To die                                   15\n    Is all that now is left for me,\n    If I could smother up my heart\n    My life would then at once depart.\n    My friends, you live, and yet you seem\n    To me the people of a dream;                                      20\n    A dream in which there is no love,\n    And yet, my friends, you live and move.\n\n    When I could live without a pain,\n    And feel no wish to be alive,\n    In quiet hopelessness I sleep,                                    25\n    Alas! how quiet, and how deep!\n\n    Oh no! I do not, cannot rue,\n    I did not strive to follow you.\n    I might have dropp'd, and died alone\n    On unknown snows, a spot unknown.                                 30\n    This spot to me must needs be dear,\n    Of my dear friends I see the trace.\n    You saw me, friends, you laid me here,\n    You know where my poor bones shall be,\n    Then wherefore should I fear to die?                              35\n    Alas that one beloved, forlorn,\n    Should lie beneath the cold starlight!\n    With them I think I could have borne\n    The journey of another night,\n    And with my friends now far away                                  40\n    I could have lived another day.\n\n\nTHE OLD CUMBERLAND BEGGAR\n\nMS. Variants, not inserted in Vol. I.\n\n    (l. 3) On a small pile of humble masonry\n    Placed at the foot of \u2026\n\n    (l. 24) He travels on, a solitary man.\n    His age has no companion. He is weak,\n    So helpless in appearance that, for him\n    The sauntering horseman-traveller does not throw\n    With careless hand his pence upon the ground\n    But stops that he may lodge the coin\n    Safe in the old man's hat: nor quits him so,\n    But as he goes towards him turns a look\n    Sidelong and half-reverted.\u2026\n\n\n\n\n1800\n\n\nANDREW JONES\n\nComposed 1800.--Published 1800\n\n_Andrew Jones_ was included in the \"Lyrical Ballads\" of 1800, 1802,\n1805, and in the Poems of 1815. It was also printed in _The Morning\nPost_, February 10, 1801. It was not republished after 1815. With this\npoem compare _The Old Cumberland Beggar_.--ED.\n\n    I hate that Andrew Jones; he'll breed\n    His children up to waste and pillage.\n    I wish the press-gang or the drum\n    Would with its rattling music come,[341]\n    And sweep him from the village!                                    5\n\n    I said not this, because he loves\n    Through the long day to swear and tipple;\n    But for the poor dear sake of one\n    To whom a foul deed he had done,\n    A friendless man, a travelling <DW36>!                           10\n    For this poor crawling helpless wretch\n    Some horseman who was passing by,[342]\n    A penny on the ground had thrown;\n    But the poor <DW36> was alone\n    And could not stoop--no help was nigh.                            15\n\n    Inch-thick the dust lay on the ground\n    For it had long been droughty weather;\n    So with his staff the <DW36> wrought\n    Among the dust till he had brought\n    The half-pennies together.                                        20\n\n    It chanced that Andrew passed that way\n    Just at the time; and there he found\n    The <DW36> in the mid-day heat\n    Standing alone, and at his feet\n    He saw the penny on the ground.                                   25\n\n    He stooped and took the penny up:[343]\n    And when the <DW36> nearer drew,\n    Quoth Andrew, \"Under half-a-crown,\n    What a man finds is all his own,\n    And so, my friend, good-day to you.\"                              30\n\n    And _hence_ I said, that Andrew's boys\n    Will all be trained to waste and pillage:\n    And wished the press-gang, or the drum\n    Would with its rattling music come,[344]\n    And sweep him from the village!                                   35\n\n[341] 1815.\n\n    With its tantara sound would come,\n\n    1800.\n\n[342]\n\n    It chanc'd some Traveller passing by,\n\n    MS.\n\n[343] In the text of 1800, this line is, \"He stopped and took the\npenny up,\" but in the list of _errata_, \"stooped\" is substituted for\n\"stopped.\"--ED.\n\n[344] 1815.\n\n    With its tantara sound would come\n\n    1800.\n\n\n\"THERE IS A SHAPELESS CROWD OF UNHEWN STONES\"\n\nNumerous fragments of verse, more or less unfinished, occur in the\nGrasmere Journals, written by Dorothy Wordsworth. One of these--which\nis broken up into irregular fragments, and very incomplete--is\nevidently part of the material which was written about the old Cumbrian\nshepherd Michael. The successive alterations of the text of the poem\n_Michael_ are in the Grasmere Journal. These fragments have a special\ntopographical interest, from their description of Helvellyn, and its\nspring, the fountain of the mists, and the stones on the summit. On the\noutside leather cover of the MS. book there is written, \"May to Dec.\n1802.\"\n\nThe following lines come first:--\n\n    There is a shapeless crowd of unhewn stones[345]\n    That lie together, some in heaps, and some\n    In lines, that seem to keep themselves alive\n    In the last dotage of a dying form.\n    At least so seems it to a man who stands\n    In such a lonely place.\n\nThese are followed by a few lines, some of which were afterwards used\nin _The Prelude_ (see vol. iii. p. 269):--\n\n    Shall he who gives his days to low pursuits,\n    Amid the undistinguishable crowd\n    Of cities, 'mid the same eternal flow\n    Of the same objects, melted and reduced\n    To one identity, by differences\n    That have no law, no meaning, and no end,\n    Shall he feel yearning to those lifeless forms,\n    And shall we think that Nature is less kind\n    To those, who all day long, through a long life,\n    Have walked within her sight? It cannot be.\n\n        Mary Wordsworth, Dorothy Wordsworth,\n                William Wordsworth.\n           Sat. Eve., 20 past 6, May 29.\n\nOther fragments follow, less worthy of preservation. Then the passage,\nwhich occurs in book xiii. of _The Prelude_, beginning--\n\n    There are who think that strong affection, love,\n\n(see vol. iii. p. 361), with one or two variations from the final text,\nwhich were not improvements.\n\nFive lines on Helvellyn, afterwards included in the _Musings near\nAquapendente_ (see vol. viii. p. 47, ll. 61-65), come next.\n\nThe fragments referring to _Michael_ are written down, probably just\nas the brother dictated them to his sister, and would be--if not\nunintelligible--certainly without any literary connection or unity,\nwere they printed in the order in which they occur. I therefore\ntranspose them slightly, to give something like continuity to the\nwhole; which remains, of course, a torso.\n\n    I will relate a tale for those who love\n    To lie beside the lonely mountain brooks,\n    And hear the voices of the winds and flowers.\n    \u2026\n                                   \u2026 It befell\n    At the first falling of the autumnal snows,\n    Old Michael and his son one day went forth\n    In search of a stray sheep. It was the time\n    When from the heights our shepherds drive their flocks\n    To gather all their mountain family\n    Into the homestalls, ere they send them back\n    There to defend themselves the winter long.\n    Old Michael for this purpose had driven down\n    His flock into the vale, but as it chanced,\n    A single sheep was wanting. They had sought\n    The straggler during all the previous day\n    All over their own pastures, and beyond.\n    And now at sunrise, sallying forth again\n    Far did they go that morning: with their search\n    Beginning towards the south, where from Dove Crag\n    (Ill home for bird so gentle), they looked down\n    On Deep-dale-head, and Brothers water (named\n    From those two Brothers that were drowned therein);\n    Thence northward did they pass by Arthur's seat,[346]\n    And Fairfield's highest summit, on the right\n    Leaving St. Sunday's Crag, to Grisdale tarn\n    They shot, and over that cloud-loving hill,\n    Seat-Sandal, a fond lover of the clouds;\n    Thence up Helvellyn, a superior mount,\n    With prospect underneath of Striding edge,\n    And Grisdale's houseless vale, along the brink\n    Of Sheep-cot-cove, and those two other coves,\n    Huge skeletons of crags which from the coast\n    Of old Helvellyn spread their arms abroad\n    And make a stormy harbour for the winds.\n    Far went these shepherds in their devious quest,\n    From mountain ridges peeping as they passed\n    Down into every nook; \u2026\n                            \u2026 and many a sheep\n    On height or bottom[347] did they see, in flocks\n    Or single. And although it needs must seem\n    Hard to believe, yet could they well discern\n    Even at the utmost distance of two miles\n    (Such strength of vision to the shepherd's eye\n    Doth practice give) that neither in the flocks\n    Nor in the single sheep was what they sought.\n    So to Helvellyn's eastern side they went,\n    Down looking on that hollow, where the pool\n    Of Thirlmere flashes like a warrior's shield\n    His light high up among the gloomy rocks,\n    With sight of now and then a straggling gleam\n    On Armath's[348] pleasant fields. And now they came,\n    To that high spring which bears no human name,\n    As one unknown by others, aptly called\n    The fountain of the mists. The father stooped\n    To drink of the clear water, laid himself\n    Flat on the ground, even as a boy might do,\n    To drink of the cold well. When in like sort\n    His son had drunk, the old man said to him\n    That now he might be proud, for he that day\n    Had slaked his thirst out of a famous well,\n    The highest fountain known on British land.\n    Thence, journeying on a second time, they passed\n    Those small flat stones, which, ranged by traveller's hands\n    In cyphers on Helvellyn's highest ridge,\n    Lie loose on the bare turf, some half-o'ergrown\n    By the grey moss, but not a single stone\n    Unsettled by a wanton blow from foot\n    Of shepherd, man or boy. They have respect\n    For strangers who have travelled far perhaps,\n    For men who in such places, feeling there\n    The grandeur of the earth, have left inscribed\n    Their epitaph, which rain and snow\n    And the strong wind have reverenced.\n    \u2026\n      But soon as Luke, full ten years old, could stand\n    Against the mountain blasts, and to the heights\n    Not fearing toil, nor length of weary ways,\n    He with his Father daily went, and they\n    Were as companions, why should I relate\n    That objects which the shepherd lov'd before\n    Were dearer now? that from the Boy there came\n    Feelings and emanations, things which were\n    Light to the sun and music to the wind;\n    And that the old man's heart seem'd born again?\n      Thus in his Father's sight the Boy grew up;\n    And now when he had reached his eighteenth year,\n    He was his comfort and his daily hope.\n    \u2026\n    Though often thus industriously they passed[349]\n    Whole hours with but small interchange of speech,\n    Yet were there times in which they did not want\n    Discourse both wise and pleasant,[350] shrewd remarks\n    Of moral prudence,[351] clothed in images\n    Lively and beautiful, in rural forms,\n    That made their conversation fresh and fair\n    As is a landscape; and the shepherd oft\n    Would draw out of his heart the mysteries[352]\n    And admirations that were there, of God\n    And of his works: or, yielding to the bent\n    Of his peculiar humour, would let loose\n    His tongue, and give it the wind's freedom; then,\n    Discoursing on remote imaginations, strong\n    Conceits, devices, plans, and schemes,[353]\n    Of alterations human hands might make\n    Among the mountains, fens which might be drained,\n    Mines opened, forests planted, and rocks split,\n    The fancies of a solitary man.[354]\n    Not with a waste of words, but for the sake\n    Of pleasure which I know that I shall give\n    To many living now, have I described\n    Old Michael's manners and discourse, and thus\n    Minutely spoken of that aged Lamp\n    Round which the Shepherd and his household sate\n    --The light was famous in the neighbourhood\n    And was a public symbol \u2026\n\nThen follow four pages of Dorothy Wordsworth's Journal (May 4th and\n5th, 1802); and then, irregularly written, and with numerous erasures,\nthe remainder of these unpublished lines.\n\n                            \u2026 At length the boy\n    Said, \"Father, 'tis lost labour; with your leave\n    I will go back and range a second time\n    The grounds which we have hunted through before.\"\n    So saying, homeward, down the hill the boy\n    Sprang like a gust of wind: [and with a heart\n    Brimful of glory said within himself,\n    \"I know where I shall find him, though the storm\n    Have driven him twenty miles.\"\n    For ye must know][355] that though the storm\n    Drive one of those poor creatures miles and miles,\n    If he can crawl, he will return again\n    To his own hills, the spots where when a lamb\n    He learned to pasture at his mother's side.\n    Bethinking him of this, again the boy\n    Pursued his way toward a brook, whose course\n    Was through that unfenced tract of mountain ground\n    Which to his father's little farm belonged,\n    The home and ancient birthright of their flock.\n    Down the deep channel of the stream he went,\n    Prying through every nook. Meanwhile the rain\n    Began to fall upon the mountain tops,\n    Thick storm, and heavy, which for three hours' space\n    Abated not; and all that time the boy\n    Was busy in his search, until at length\n    He spied the sheep upon a plot of grass,\n    An island in the brook. It was a place\n    Remote and deep, piled round with rocks, where foot\n    Of man or beast was seldom used to tread.\n    But now, when everywhere the summer grass\n    Began to fail, this sheep by hunger pressed\n    Had left his fellows, made his way alone\n    To the green plot of pasture in the brook.\n    Before the boy knew well what he had seen\n    He leapt upon the island, with proud heart,\n    And with a shepherd's joy. Immediately\n    The sheep sprang forward to the further shore,\n    And was borne headlong by the roaring flood.\n    At this the boy looked round him, and his heart\n    Fainted with fear. Thrice did he turn his face\n    To either bank, nor could he summon up\n    The courage that was needful to leap back\n    'Cross the tempestuous torrent; so he stood\n    A prisoner on the island, not without\n    More than one thought of death, and his last hour.\n    Meantime the father had returned alone\n    To his own home, and now at the approach\n    Of evening he went forth to meet his son,\n    Nor could he guess the cause for which the boy\n    Had stayed so long. The shepherd took his way\n    Up his own mountain grounds, where, as he walked\n    Along the steep that overhung the brook,\n    He seemed to hear a voice, which was again\n    Repeated, like the whistling of a kite.\n    At this, not knowing why--as often-times\n    The old man afterwards was heard to say--\n    Down to the brook he went, and tracked its course\n    Upwards among the o'erhanging rocks; nor\n    Had he gone far ere he espied the boy\n    Right in the middle of the roaring stream.\n    Without distress or fear the shepherd heard\n    The outcry of his son: he stretched his staff\n    Towards him, bade him leap, which word scarce said\n    The boy was safe.\u2026\n    \u2026\n\nOf Michael it is said--\n\n    No doubt if you in terms direct had asked\n    Whether he loved the mountains, true it is\n    That with blunt repetition of your words\n    He might have stared at you, and said that they\n    Were frightful to behold, but had you then\n    Discoursed with him \u2026\n    Of his own business, and the goings on\n    Of earth and sky, then truly had you seen\n    That in his thoughts there were obscurities,\n    Wonder, and admiration, things that wrought\n    Not less than a religion in his heart.\n    And if it was his fortune to converse\n    With any who could talk of common things\n    In an unusual way, and give to them\n    Unusual aspects, or by questions apt\n    Wake sudden recognitions, that were like\n    Creations in the mind (and were indeed\n    Creations often), then when he discoursed\n    Of mountain sights, this untaught shepherd stood\n    Before the man with whom he so conversed\n    And looked at him as with a poet's eye.\n    But speaking of the vale in which he dwelt,\n    And those bare rocks, if you had asked if he\n    For other pastures would exchange the same\n    And dwell elsewhere, \u2026\n                          \u2026 you then had seen\n    At once what spirit of love was in his heart.\n    \u2026\n    I have related that this Shepherd loved\n    The fields and mountains, not alone for this\n    That from his very childhood he had lived\n    Among them, with a body hale and stout,\n    And with a vigorous mind \u2026\n                              \u2026 But exclude\n    Such reasons, and he had less cause to love\n    His native vale and patrimonial fields\n    Than others have, for Michael had liv'd on\n    Childless, until the time when he began\n    To look towards the shutting in of life.\n\nIn this MS. book there are also some of the original stanzas of _Ruth_,\nwith a few variations of text.--ED.\n\n[345] Compare the first line of those _Written with a Slate Pencil upon\na Stone, the largest of a Heap lying near a deserted Quarry, upon one\nof the Islands at Rydal_, vol. ii. p. 63.--ED.\n\n[346] Stone Arthur. See, in the \"Poems on the Naming of Places,\" the\none beginning--\n\n    There is an Eminence,\n\nED.\n\n[347] Bottom is a common Cumbrian word for valley.--ED.\n\n[348] Armboth, on the western side of Thirlmere.--ED.\n\n[349] Though in these occupations they would pass\u2020\n\n[350] \u2026 prudent, \u2026\u2020\n\n[351] Of daily Providence \u2026\u2020\n\n[352] \u2026 obscurities\u2020\n\n[353] Day-dreams, thoughts, and schemes.\u2020\n\n\u2020 These variants occur in a letter of Dorothy Wordsworth to Thomas\nPoole.--ED.\n\n[354] All doubt as to these fragments being originally intended to form\npart of _Michael_ is set at rest by a letter from Wordsworth to Thomas\nPoole, of Nether Stowey, written from Grasmere on the 9th of April\n1801, in which he gives first some new lines to be added to _Michael_,\nat pp. 210 and 211 of vol. ii. of the \"Lyrical Ballads\" (ed. 1800); to\nwhich letter Dorothy Wordsworth added the postscript, \"My brother has\nwritten the following lines, to be inserted page 206, after the ninth\nline--\n\n    Murmur as with the sound of summer flies;\"\n\nand then follow--\n\n    Though in these occupations they would pass\n    Whole hours, etc.\n\nas printed above.\n\nDorothy Wordsworth adds, \"Tell whether you think the insertion of these\nlines an improvement.\"--ED.\n\n[355] An erased version.--ED.\n\n\n\n\n1802\n\n\n\"AMONG ALL LOVELY THINGS MY LOVE HAD BEEN\"\n\nComposed April 12, 1802.--Published 1807\n\nThis poem--known in the Wordsworth household as _The Glowworm_--was\nwritten on the 12th of April 1802, during a ride from Middleham to\nBarnard Castle, and was published in the edition of 1807. It was never\nreproduced. The \"Lucy\" of this and other poems was his sister Dorothy.\nIn a letter to Coleridge, written in April 1802, he thus refers to\nthe poem, and to the incident which gave rise to it:--\"I parted from\nM---- on Monday afternoon, about six o'clock, a little on this side\nRushyford. Soon after I missed my road in the midst of the storm.\u2026\nBetween the beginning of Lord Darlington's park at Raby, and two or\nthree miles beyond Staindrop, I composed the poem the opposite page. I\nreached Barnard Castle about half-past ten.\u2026 The incident of this poem\ntook place about seven years ago between my sister and me.\"\n\nI think it probable that the \"incident\" occurred near Racedown,\nDorsetshire, where, in the autumn of 1795 Wordsworth settled with his\nsister. The following is Dorothy's account of the composition of the\npoem:--\"Tuesday, April 20, 1802.--We sate in the orchard and repeated\n_The Glowworm_, and other poems. Just when William came to a well, or\ntrough, which there is in Lord Darlington's park, he began to write\nthat poem of _The Glowworm_; interrupted in going through the town of\nStaindrop, finished it about two miles and a-half beyond Staindrop. He\ndid not feel the jogging of the horse while he was writing; but, when\nhe had done, he felt the effect of it.\u2026 So much for _The Glowworm_. It\nwas written coming from Middleham, on Monday, April 12, 1802.\"--ED.\n\n    Among all lovely things my Love had been;\n    Had noted well the stars, all flowers that grew\n    About her home; but she had never seen\n    A glow-worm, never one, and this I knew.\n\n    While riding near her home one stormy night                        5\n    A single glow-worm did I chance to espy;\n    I gave a fervent welcome to the sight,\n    And from my horse I leapt; great joy had I.\n\n    Upon a leaf the glow-worm did I lay,\n    To bear it with me through the stormy night:                      10\n    And, as before, it shone without dismay;\n    Albeit putting forth a fainter light.\n\n    When to the dwelling of my Love I came,\n    I went into the orchard quietly;\n    And left the glow-worm, blessing it by name,                      15\n    Laid safely by itself, beneath a tree.\n\n    The whole next day I hoped, and hoped with fear;\n    At night the glow-worm shone beneath the tree;\n    I led my Lucy to the spot, \"Look here,\"\n    Oh! joy it was for her, and joy for me!                           20\n\n\n\"ALONG THE MAZES OF THIS SONG I GO\"\n\nThis, and the next two fragments, by Wordsworth, are extracted from his\nsister's Grasmere Journal.--ED.\n\n    Along the mazes of this song I go\n    As inward motions of the wandering thought\n    Lead me, or outward circumstance impels.\n    Thus do I urge a never-ending way\n    Year after year, with many a sleep between,\n    Through joy and sorrow; if my lot be joy\n    More joyful if it be with sorrow sooth'd.\n\n\n\"THE RAINS AT LENGTH HAVE CEAS'D, THE WINDS ARE STILL'D\"\n\n    The rains at length have ceas'd, the winds are still'd,\n    The stars shine brightly between clouds at rest,\n    And as a cavern is with darkness fill'd,\n    The vale is by a mighty sound possess'd\n\n\n\"WITNESS THOU\"\n\n                                Witness thou\n    The dear companion of my lonely walk,\n    My hope, my joy, my sister, and my friend,\n    Or something dearer still, if reason knows\n    A dearer thought, or in the heart of love\n    There be a dearer name.[356]\n\n[356] Compare Byron's _Epistle to Augusta_--\n\n    My sister! my sweet sister! if a name\n    Dearer and purer were, it should be thine.\n\nIt is a mere coincidence, as Byron could not have seen the Wordsworth\nMS.--ED.\n\n\nWILD-FOWL\n\n                              The order'd troops\n    In spiral circles mount aloft, and soar\n    In prospect far above the denser air\n    That hangs o'er the moist plain. Again they view\n    The glorious sun, and while the light of day\n    Still gleams upon their polish'd plumes--the bright\n    Sonorous squadrons sing their evening hymn.\n\n\nWRITTEN IN A GROTTO\n\nPublished in _The Morning Post_, March 9, 1802\n\nI cannot affirm, with any certainty, that these lines were written by\nWordsworth; but I agree with Mr. Ernest Coleridge in thinking that they\nwere. He showed them to his relative--the late Chief Justice--who said\nthat he did not know who else _could_ have written them, at that time.\nLord Coleridge said the same to myself.--ED.\n\n    O moon! if e'er I joyed when thy soft light\n      Danc'd to the murmuring rill on Lomond's wave,\n    Or sighed for thy sweet presence some dark night\n      When thou wert hidden in thy monthly grave,[357]\n    If e'er on wings which active fancy gave                           5\n      I sought thy golden vale with dancing flight\n    Then stretcht at ease in some sequestered cave\n      Gaz'd on thy lovely Nymphs with fond delight,\n    Thy Nymphs with more than earthly beauty bright,\n    If e'er thy beam, as Smyrna's shepherds tell,                     10\n      Soft as the gentle kiss of amorous maid\n    On the closed eye of young Endymion fell[358]\n      That he might wake to clasp thee in the shade,\n    Each night while I recline within this cell\n    Guide hither, O sweet Moon, the maid I love so well.              15\n\nThe shepherds of Smyrna show a cave, where, as they say, Luna descended\nto Endymion, laid on a bed under a large oak which was the scene of\ntheir loves. See Chandler's _Travels in Asia Minor_.\n\n[357] Compare _To the Moon_, vol. viii. p. 15, l. 64.--ED.\n\n[358] Compare, in the \"Evening Voluntaries,\" _To Lucca Giordano_\n(1846), p. 183.--ED.\n\n\nHOME AT GRASMERE\n\nThe canto of Wordsworth's autobiographical poem, unpublished in _The\nPrelude_ (1851), and first given to the world in 1888, is appropriately\nentitled \"Home at Grasmere.\"\n\nThe introduction to _The Recluse_ was not only kept back by him during\nhis lifetime, but was omitted by his representatives--with what must be\nregarded as true critical insight--when _The Prelude_ was published in\n1850. As a whole, it is not equal to _The Prelude_. Certain passages\nare very inferior, but there are others that posterity must cherish,\nand \"not willingly let die.\" It was probably a conviction of its\ninequality and inferiority that led Wordsworth to give only one or two\nselected extracts from this canto to the world, in his own lifetime.\nTwo passages were printed in his _Guide to the District of the Lakes_;\nanother--a description of the flight and movement of birds--was\npublished in 1827, and subsequent editions, under the title of\n_Water-Fowl_; while the Bishop of Lincoln published other two passages\nin the _Memoirs_ of his uncle, beginning respectively--\n\n    On Nature's invitation do I come,\n\nand\n\n    Bleak season was it, turbulent and bleak.\n\nInternal evidence (see the numerous allusions to Dorothy, and the\nreference to John Wordsworth) shows that this canto of _The Recluse_\nwas written at Grasmere, not long after Wordsworth's arrival there,\nand certainly before his marriage. The text, as now printed, has been\ncarefully compared with the original MS. by Mr. Gordon Wordsworth. The\nMS. heading is--THE RECLUSE. BOOK FIRST, PART FIRST.\n\nHOME AT GRASMERE\n\n    Once to the verge of yon steep barrier came\n    A roving school-boy; what the Adventurer's age\n    Hath now escaped his memory--but the hour,\n    One of a golden summer holiday,\n    He well remembers, though the year be gone.                        5\n    Alone and devious from afar he came;\n    And, with a sudden influx overpowered\n    At sight of this seclusion, he forgot\n    His haste, for hasty had his footsteps been\n    As boyish his pursuits; and, sighing said,                        10\n    \"What happy fortune were it here to live!\n    And, (if a thought of dying, if a thought\n    Of mortal separation, could intrude\n    With paradise before him), here to die!\"\n    No prophet was he, had not even a hope,                           15\n    Scarcely a wish, but one bright pleasing thought,\n    A fancy in the heart of what might be\n    The lot of others, never could be his.\n      The station whence he looked was soft and green,\n    Not giddy yet aerial, with a depth                                20\n    Of vale below, a height of hills above.\n    For rest of body, perfect was the spot,\n    All that luxurious nature could desire,\n    But stirring to the spirit. Who could gaze\n    And not feel motions there? He thought of clouds                  25\n    That sail on winds, of breezes that delight\n    To play on water, or in endless chase\n    Pursue each other through the yielding plain\n    Of grass or corn, over and through and through,\n    In billow after billow, evermore                                  30\n    Disporting. Nor unmindful was the Boy\n    Of sunbeams, shadows, butterflies and birds,\n    Of fluttering Sylphs, and softly-gliding Fays,\n    Genii, and winged Angels that are Lords\n    Without restraint of all which they behold.                       35\n    The illusion strengthening as he gazed, he felt\n    That such unfettered liberty was his,\n    Such power and joy; but only for this end,\n    To flit from field to rock, from rock to field,\n    From shore to island, and from isle to shore,                     40\n    From open ground to covert, from a bed\n    Of meadow-flowers into a tuft of wood,\n    From high to low, from low to high, yet still\n    Within the bound of this high concave; here\n    Must be his home, this Valley be his world.                       45\n      Since that day forth the place to him--_to me_\n    (For I who live to register the truth\n    Was that same young and happy being) became\n    As beautiful to thought, as it had been,\n    When present, to the bodily sense; a haunt                        50\n    Of pure affections, shedding upon joy\n    A brighter joy; and through such damp and gloom\n    Of the gay mind, as ofttimes splenetic youth\n    Mistakes for sorrow darting beams of light\n    That no self-cherished sadness could withstand:                   55\n    And now 'tis mine, perchance for life, dear Vale,\n    Beloved Grasmere (let the Wandering Streams\n    Take up, the cloud-capped hills repeat, the Name),\n    One of thy lowly dwellings is my Home.\n      And was the cost so great? and could it seem                    60\n    An act of courage, and the thing itself\n    A conquest? who must bear the blame? sage man\n    Thy prudence, thy experience--thy desires;\n    Thy apprehensions--blush thou for them all.\n      Yes, the realities of life so cold,                             65\n    So cowardly, so ready to betray,\n    So stinted in the measure of their grace\n    As we pronounce them, doing them much wrong,\n    Have been to me more bountiful than hope,\n    Less timid than desire--but that is passed.                       70\n      On Nature's invitation do I come,[359]\n    By reason sanctioned--Can the choice mislead,\n    That made the calmest, fairest spot of earth,\n    With all its unappropriated good,\n    My own; and not mine only, for with me                            75\n    Entrenched, say rather peacefully embowered,\n    Under yon orchard, in yon humble cot,\n    A younger orphan of a home extinct,\n    The only daughter of my parents, dwells.\n      Aye, think on that, my heart, and cease to stir,                80\n    Pause upon that, and let the breathing frame\n    No longer breathe, but all be satisfied.\n    --Oh if such silence be not thanks to God\n    For what hath been bestowed, then where, where then\n    Shall gratitude find rest? Mine eyes did ne'er                    85\n    Fix on a lovely object, nor my mind\n    Take pleasure in the midst of happy thoughts,\n    But either She whom now I have, who now\n    Divides with me this loved abode, was there,\n    Or not far off. Where'er my footsteps turned,                     90\n    Her Voice was like a hidden Bird that sang,\n    The thought of her was like a flash of light,\n    Or an _unseen_ companionship, a breath,\n    Or fragrance independent of the wind.\n    In all my goings, in the new and old                              95\n    Of all my meditations, and in this\n    Favourite of all, in this the most of all.\n    --What Being, therefore, since the birth of man\n    Had ever more abundant cause to speak\n    Thanks, and if favours of the heavenly Muse                      100\n    Make him more thankful, then to call on verse\n    To aid him, and in Song resound his joy.\n    The boon is absolute; surpassing grace\n    To me hath been vouchsafed; among the bowers\n    Of blissful Eden this was neither given,                         105\n    Nor could be given, possession of the good\n    Which had been sighed for, ancient thought fulfilled\n    And dear Imaginations realized\n    Up to their highest measure, yea and more.\n      Embrace me then, ye Hills, and close me in,                    110\n    Now in the clear and open day I feel\n    Your guardianship; I take it to my heart;\n    'Tis like the solemn shelter of the night.\n    But I would call thee beautiful, for mild\n    And soft, and gay, and beautiful thou art,                       115\n    Dear Valley, having in thy face a smile\n    Though peaceful, full of gladness. Thou art pleased,\n    Pleased with thy crags, and woody steeps, thy Lake,\n    Its one green Island and its winding shores;\n    The multitude of little rocky hills,                             120\n    Thy Church and cottages of mountain stone\n    Clustered like stars some few, but single most,\n    And lurking dimly in their shy retreats,\n    Or glancing at[360] each other cheerful looks,\n    Like separated stars with clouds between.                        125\n    What want we? have we not perpetual streams,\n    Warm woods, and sunny hills, and fresh green fields,\n    And mountains not less green, and flocks, and herds,\n    And thickets full of songsters, and the voice\n    Of lordly birds, an unexpected sound                             130\n    Heard now and then from morn till latest eve,\n    Admonishing the man who walks below\n    Of solitude, and silence in the sky?\n    These have we, and a thousand nooks of earth\n    Have also these, but _no_ where else is found,                   135\n    No where (or is it fancy?) _can_ be found\n    The one sensation that is here; 'tis here,\n    Here as it found its way into my heart\n    In childhood, here as it abides by day,\n    By night, here only; or in chosen minds                          140\n    That take it with them hence, where'er they go.\n    'Tis, but I cannot name it, 'tis the sense\n    Of majesty, and beauty, and repose,\n    A blended holiness of earth and sky,\n    Something that makes this individual Spot,                       145\n    This small abiding-place of many men,\n    A termination, and a last retreat,\n    A centre, come from wheresoe'er you will,\n    A whole without dependence or defect,\n    Made for itself; and happy in itself,                            150\n    Perfect Contentment, Unity entire.\n      Bleak season was it, turbulent and bleak,[361]\n    When hitherward we journeyed, side by side,\n    Through bursts of sunshine and through flying showers,\n    Paced the long Vales--how long they were--and yet                155\n    How fast that length of way was left behind,\n    Wensley's rich Vale and Sedbergh's naked heights.\n    The frosty wind, as if to make amends\n    For its keen breath, was aiding to our steps,\n    And drove us onward like two ships at sea,                       160\n    Or like two birds, companions in mid air,\n    Parted and re-united by the blast.\n    Stern was the face of Nature. We rejoiced\n    In that stern countenance, for our souls thence drew\n    A feeling of their strength. The naked trees,                    165\n    The icy brooks, as on we passed, appeared\n    To question us. \"Whence come ye? to what end?\"\n    They seemed to say; \"What would ye,\" said the shower,\n    \"Wild wanderers, whither through my dark domain?\"\n    The sunbeam said, \"Be happy.\" When this Vale                     170\n    We entered, bright and solemn was the sky\n    That faced us with a passionate welcoming,\n    And led us to our threshold. Daylight failed\n    Insensibly, and round us gently fell\n    Composing darkness, with a quiet load                            175\n    Of full contentment, in a little shed\n    Disturbed, uneasy in itself as seemed,\n    And wondering at its new inhabitants.\n    It loves us now, this Vale so beautiful\n    Begins to love us! By a sullen storm,                            180\n    Two months unwearied of severest storm,\n    It put the temper of our minds to proof,\n    And found us faithful through the gloom, and heard\n    The Poet mutter his prelusive songs\n    With cheerful heart, an unknown voice of joy,                    185\n    Among the silence of the woods and hills;\n    Silent to any gladsomeness of sound\n    With all their Shepherds.\n                            But the gates of Spring\n    Are opened. Churlish Winter hath given leave\n    That she should entertain for this one day,                      190\n    Perhaps for many genial days to come,\n    His guests, and make them jocund. They are pleased,\n    But most of all the Birds that haunt the flood\n    With the mild summons; inmates though they be\n    Of winter's household, they keep festival                        195\n    This day, who drooped, or seemed to droop, so long;\n    They shew their pleasure, and shall I do less?\n    Happiest of happy though I be, like them\n    I cannot take possession of the sky,\n    Mount with a thoughtless impulse, and wheel there,               200\n    One of a mighty multitude, whose way\n    Is a perpetual harmony, and dance\n    Magnificent. Behold, how with a grace\n    Of ceaseless motion,[362] that might scarcely seem\n    Inferior to angelical, they prolong                              205\n    Their curious pastime, shaping in mid air,\n    And sometimes with ambitious wing that soars\n    High as the level of the mountain tops,\n    A circuit ampler than the lake beneath,\n    Their own domain;--but ever, while intent                        210\n    On tracing and retracing that large round,\n    Their jubilant activity evolves\n    Hundreds of curves and circlets, to and fro,\n    Upwards and downwards, progress intricate\n    Yet unperplexed, as if one spirit swayed                         215\n    Their indefatigable flight. 'Tis done--\n    Ten times and more, I fancied it had ceased;\n    But lo! the vanished company again\n    Ascending, they approach--I hear their wings\n    Faint, faint at first; and then an eager sound                   220\n    Passed in a moment--and as faint again!\n    They tempt the sun to sport among[363] their plumes;\n    Tempt the smooth water,[364] or the gleaming ice,\n    To show them a fair image; 'tis themselves,\n    Their own fair forms, upon the glimmering plain,                 225\n    Painted more soft and fair as they descend,\n    Almost to touch;--then up again aloft,\n    Up with a sally, and a flash of speed,\n    As if they scorned both resting-place and rest![365]\n    This day is a thanksgiving, 'tis a day                           230\n    Of glad emotion and deep quietness;\n    Not upon me alone hath been bestowed,\n    Me rich in many onward-looking thoughts,\n    The penetrating bliss; oh surely these\n    Have felt it, not the happy Quires of Spring,                    235\n    Her own peculiar family of love\n    That sport among green leaves, a blither train.\n      But two are missing--two, a lonely pair\n    Of milk-white Swans, wherefore are _they_ not seen\n    Partaking this day's pleasure? From afar                         240\n    They came, to sojourn here in solitude,\n    Choosing this Valley, they who had the choice\n    Of the whole world.[366] We saw them day by day,\n    Through these two months of unrelenting storm,\n    Conspicuous at the centre of the Lake,                           245\n    Their safe retreat. We knew them well, I guess\n    That the whole Valley knew them; but to us\n    They were more dear than may be well believed,\n    Not only for their beauty, and their still\n    And placid way of life, and constant love                        250\n    Inseparable, not for these alone,\n    But that _their_ state so much resembled ours,\n    They having also chosen this abode;\n    They strangers, and we strangers; they a pair,\n    And we a solitary pair like them.                                255\n    They should not have departed; many days\n    Did I look forth in vain, nor on the wing\n    Could see them, nor in that small open space\n    Of blue unfrozen water, where they lodged,\n    And lived so long in quiet, side by side.                        260\n    Shall we behold them, consecrated friends,\n    Faithful companions, yet another year\n    Surviving--they for us, and we for them--\n    And neither pair be broken? Nay perchance\n    It is too late already for such hope,                            265\n    The Dalesmen may have aimed the deadly tube,\n    And parted them; or haply both are gone\n    One death, and that were mercy given to both.\n    Recal my song the ungenerous thought; forgive,\n    Thrice favoured Region, the conjecture harsh                     270\n    Of such inhospitable penalty,\n    Inflicted upon confidence so pure.\n    Ah, if I wished to follow where the sight\n    Of all that is before mine eyes, the voice\n    Which speaks from a presiding Spirit here,                       275\n    Would lead me, I should whisper to myself;\n    They who are dwellers in this holy place\n    Must needs themselves be hallowed, they require\n    No benediction from the stranger's lips,\n    For they are blest already. None would give                      280\n    The greeting \"peace be with you\" unto them,\n    For peace they have, it cannot but be theirs,\n    And mercy, and forbearance. Nay--not these,\n    Their healing offices a pure goodwill\n    Precludes, and charity beyond the bounds                         285\n    Of charity--an overflowing love,\n    Not for the creature only, but for all\n    That is around them, love for every thing\n    Which in this happy region they behold!\n      Thus do we soothe ourselves, and when the thought              290\n    Is past we blame it not for having come.\n    What, if I floated down a pleasant Stream\n    And now am landed, and the motion gone,\n    Shall I reprove myself? Ah no, the stream\n    Is flowing, and will never cease to flow,[367]                   295\n    And I shall float upon that stream again.\n    By such forgetfulness the soul becomes,\n    Words cannot say, how beautiful. Then hail,\n    Hail to the visible Presence, hail to thee,\n    Delightful Valley, habitation fair!                              300\n    And to whatever else of outward form\n    Can give us inward help, can purify,\n    And elevate, and harmonise, and soothe,\n    And steal away, and for a while deceive\n    And lap in pleasing rest, and bear us on                         305\n    Without desire in full complacency,\n    Contemplating perfection absolute\n    And entertained as in a placid sleep.\n      But not betrayed by tenderness of mind\n    That feared, or wholly overlooked the truth,                     310\n    Did we come hither, with romantic hope\n    To find, in midst of so much loveliness,\n    Love, perfect love; of so much majesty\n    A like majestic frame of mind in those\n    Who here abide, the persons like the place.                      315\n    Not from such hope, or aught of such belief\n    Hath issued any portion of the joy\n    Which I have felt this day. An awful voice,\n    'Tis true, hath in my walks been often heard,\n    Sent from the mountains or the sheltered fields;                 320\n    Shout after shout--reiterated whoop\n    In manner of a bird that takes delight\n    In answering to itself; or like a hound\n    Single at chase among the lonely woods,\n    His yell repeating;[368] yet it was in truth                     325\n    A human voice--a Spirit of coming night,\n    How solemn when the sky is dark, and earth\n    Not dark, nor yet enlightened, but by snow\n    Made visible, amid a noise of winds\n    And bleatings manifold of mountain sheep,                        330\n    Which in that iteration recognise\n    Their summons, and are gathering round for food,\n    Devoured with keenness ere to grove or bank\n    Or rocky _bield_ with patience they retire.\n      That very voice, which, in some timid mood                     335\n    Of superstitious fancy, might have seemed\n    Awful as ever stray Demoniac uttered,\n    His steps to govern in the Wilderness;\n    Or as the Norman Curfew's regular beat,\n    To hearths when first they darkened at the knell:                340\n    That Shepherd's voice, it may have reached mine ear\n    Debased and under profanation, made\n    The ready Organ of articulate sounds\n    From ribaldry, impiety, or wrath\n    Issuing when shame hath ceased to check the brawls               345\n    Of some abused Festivity--so be it.\n    I came not dreaming of unruffled life,\n    Untainted manners; born among the hills,\n    Bred also there, I wanted not a scale\n    To regulate my hopes. Pleased with the good,                     350\n    I shrink not from the evil with disgust,\n    Or with immoderate pain. I look for Man,\n    The common creature of the brotherhood,\n    Differing but little from the Man elsewhere,\n    For selfishness, and envy, and revenge,                          355\n    Ill neighbourhood--pity that this should be--\n    Flattery and double-dealing, strife and wrong.\n      Yet is it something gained, it is in truth\n    A mighty gain, that Labour here preserves\n    His rosy face, a servant only here                               360\n    Of the fire-side, or of the open field,\n    A freeman, therefore, sound and unimpaired;\n    That extreme penury is here unknown,\n    And cold and hunger's abject wretchedness,\n    Mortal to body, and the heaven-born mind;                        365\n    That they who want, are not too great a weight\n    For those who can relieve. Here may the heart\n    Breathe in the air of fellow-suffering\n    Dreadless, as in a kind of fresher breeze\n    Of her own native element, the hand                              370\n    Be ready and unwearied without plea\n    From tasks too frequent, or beyond its power\n    For languor, or indifference, or despair.\n    And as these lofty barriers break the force\n    Of winds, this deep Vale,--as it doth in part                    375\n    Conceal us from the storm,--so here abides\n    A power and a protection for the mind,\n    Dispensed indeed to other solitudes,\n    Favoured by noble privilege like this,\n    Where kindred independence of estate                             380\n    Is prevalent, where he who tills the field,\n    He, happy man! is master of the field,[369]\n    And treads the mountains which his fathers trod.\n      Not less than half-way up yon Mountain's side\n    Behold a dusky spot, a grove of Firs,                            385\n    That seems still smaller than it is. This grove\n    Is haunted--by what ghost? a gentle spirit\n    Of memory faithful to the call of love;\n    For, as reports the dame, whose fire sends up\n    Yon curling smoke from the grey cot below,                       390\n    The trees (her first-born child being then a babe)\n    Were planted by her husband and herself,\n    That ranging o'er the high and houseless ground\n    Their sheep might neither want (from perilous storms\n    Of winter, nor from summer's sultry heat)                        395\n    A friendly covert. \"And they knew it well,\"\n    Said she, \"for thither as the trees grew up,\n    We to the patient creatures carried food\n    In times of heavy snow.\" She then began\n    In fond obedience to her private thoughts                        400\n    To speak of her dead husband. Is there not\n    An art, a music, and a strain of words\n    That shall be like the acknowledged voice of life,\n    Shall speak of what is done among the fields,\n    Done truly there, or felt, of solid good                         405\n    And real evil, yet be sweet withal,\n    More grateful, more harmonious than the breath,\n    The idle breath of softest pipe attuned\n    To pastoral fancies? Is there such a stream,\n    Pure and unsullied, flowing from the heart                       410\n    With motions of true dignity and grace?\n    Or must we seek that stream where Man is not?\n    Methinks I could repeat in tuneful verse,\n    Delicious as the gentlest breeze that sounds\n    Through that aerial fir-grove, could preserve                    415\n    Some portion of its human history\n    As gathered from the Matron's lips, and tell\n    Of tears that have been shed at sight of it,\n    And moving dialogues between this pair,\n    Who in their prime of wedlock, with joint hands                  420\n    Did plant the grove, now flourishing, while they\n    No longer flourish, he entirely gone,\n    She withering in her loneliness. Be this\n    A task above my skill; the silent mind\n    Has her own treasures, and I think of these,                     425\n    Love what I see, and honour humankind.\n      No, we are not alone, we do not stand,\n    My Sister, here misplaced and desolate,\n    Loving what no one cares for but ourselves;\n    We shall not scatter through the plains and rocks                430\n    Of this fair Vale, and o'er its spacious heights\n    Unprofitable kindliness, bestowed\n    On objects unaccustomed to the gifts\n    Of feeling, which were cheerless and forlorn\n    But few weeks past, and would be so again                        435\n    Were we not here; we do not tend a lamp\n    Whose lustre we alone participate,\n    Which shines dependent upon us alone,\n    Mortal though bright, a dying, dying flame.\n    Look where we will, some human hand has been                     440\n    Before us with its offering; not a tree\n    Sprinkles these little pastures but the same\n    Hath furnished matter for a thought; perchance,\n    For some one, serves as a familiar friend.\n    Joy spreads, and sorrow spreads; and this whole Vale,            445\n    Home of untutored shepherds as it is,\n    Swarms with sensation, as with gleams of sunshine,\n    Shadows or breezes, scents or sounds. Nor deem\n    These feelings, though subservient more than ours\n    To every day's demand for daily bread,                           450\n    And borrowing more their spirit, and their shape\n    From self-respecting interests, deem them not\n    Unworthy therefore, and unhallowed: no,\n    They lift the animal being, do themselves\n    By Nature's kind and ever-present aid                            455\n    Refine the selfishness from which they spring,\n    Redeem by love the individual sense\n    Of anxiousness with which they are combined.\n    And thus it is that fitly they become\n    Associates in the joy of purest minds,                           460\n    They blend therewith congenially: meanwhile,\n    Calmly they breathe their own undying life\n    Through this their mountain sanctuary. Long,\n    Oh long may it remain inviolate,\n    Diffusing health and sober cheerfulness,                         465\n    And giving to the moments as they pass\n    Their little boons of animating thought\n    That sweeten labour, make it seen and felt\n    To be no arbitrary weight imposed,\n    But a glad function natural to man.                              470\n      Fair proof of this, newcomer though I be,\n    Already have I gained. The inward frame\n    Though slowly opening, opens every day\n    With process not unlike to that which cheers\n    A pensive stranger, journeying at his leisure                    475\n    Through some Helvetian dell, when low-hung mists\n    Break up, and are beginning to recede;\n    How pleased he is where thin and thinner grows\n    The veil, or where it parts at once, to spy\n    The dark pines thrusting forth their spiky heads;                480\n    To watch the spreading lawns with cattle grazed,\n    Then to be greeted by the scattered huts,\n    As they shine out; and _see_ the streams whose murmur\n    Had soothed his ear while _they_ were hidden: how pleased\n    To have about him, which way e'er he goes,                       485\n    Something on every side concealed from view,\n    In every quarter something visible,\n    Half-seen or wholly, lost and found again,\n    Alternate progress and impediment,\n    And yet a growing prospect in the main.                          490\n      Such pleasure now is mine, albeit forced,\n    Herein less happy than the Traveller\n    To cast from time to time a painful look\n    Upon unwelcome things, which unawares\n    Reveal themselves; not therefore is my heart                     495\n    Depressed, nor does it fear what is to come,\n    But confident, enriched at every glance.\n    The more I see the more delight my mind\n    Receives, or by reflexion can create.\n    Truth justifies herself, and as she dwells                       500\n    With Hope, who would not follow where she leads?\n      Nor let me pass unheeded other loves\n    Where no fear is, and humbler sympathies.\n    Already hath sprung up within my heart\n    A liking for the small grey horse that bears                     505\n    The paralytic man, and for the brute--\n    In Scripture sanctified--the patient brute,\n    On which the <DW36>, in the quarry maimed,\n    Rides to and fro: I know them and their ways.[370]\n    The famous sheep-dog, first in all the Vale,                     510\n    Though yet to me a stranger, will not be\n    A stranger long; nor will the blind man's guide,\n    Meek and neglected thing, of no renown!\n    Soon will peep forth the primrose; ere it fades\n    Friends shall I have at dawn, blackbird and thrush               515\n    To rouse me, and a hundred warblers more;\n    And if those eagles to their ancient hold\n    Return, Helvellyn's eagles! with the pair\n    From my own door I shall be free to claim\n    Acquaintance as they sweep from cloud to cloud.                  520\n    The owl that gives the name to Owlet-Crag\n    Have I heard whooping, and he soon will be\n    A chosen one of my regards. See there\n    The heifer in yon little croft belongs\n    To one who holds it dear; with duteous care                      525\n    She reared it, and in speaking of her charge\n    I heard her scatter some endearing words\n    Domestic, and in spirit motherly\n    She being herself a Mother, happy Beast\n    If the caresses of a human voice                                 530\n    Can make it so, and care of human hands.\n      And ye as happy under Nature's care,\n    Strangers to me, and all men, or at least\n    Strangers to all particular amity,\n    All intercourse of knowledge or of love                          535\n    That parts the individual from his kind,\n    Whether in large communities ye keep\n    From year to year, not shunning Man's abode,\n    A settled residence, or be from far,\n    Wild creatures, and of many homes, that come                     540\n    The gift of winds, and whom the winds again\n    Take from us at your pleasure--yet shall ye\n    Not want, for this, your own subordinate place\n    In my affections. Witness the delight\n    With which erewhile I saw that multitude                         545\n    Wheel through the sky, and see them now at rest,\n    Yet not at rest, upon the glassy lake.\n    They _cannot_ rest, they gambol like young whelps;\n    Active as lambs, and overcome with joy.\n    They try all frolic motions; flutter, plunge,                    550\n    And beat the passive water with their wings.\n    Too distant are they for plain view, but lo!\n    Those little fountains, sparkling in the sun,\n    Betray their occupation, rising up,\n    First one and then another silver spout,                         555\n    As one or other takes the fit of glee,\n    Fountains and spouts, yet somewhat in the guise\n    Of play-thing fire-works, that on festal nights\n    Sparkle about the feet of wanton boys.\n    --How vast the compass of this theatre,                          560\n    Yet nothing to be seen but lovely pomp\n    And silent majesty; the birch-tree woods\n    Are hung with thousand thousand diamond drops\n    Of melted hoar-frost, every tiny knot\n    In the bare twigs, each little budding place                     565\n    Cased with its several beads, what myriads there\n    Upon one tree, while all the distant grove\n    That rises to the summit of the steep\n    Shows like a mountain built of silver light.\n    See yonder the same pageant, and again                           570\n    Behold the universal imagery\n    Inverted, all its sun-bright features touched\n    As with the varnish, and the gloss of dreams;\n    Dreamlike the blending also of the whole\n    Harmonious landscape; all along the shore                        575\n    The boundary lost, the line invisible\n    That parts the image from reality;\n    And the clear hills, as high as they ascend\n    Heavenward, so piercing deep the lake below.\n    Admonished of the days of love to come                           580\n    The raven croaks, and fills the upper air\n    With a strange sound of genial harmony;[371]\n    And in and all about that playful band,\n    Incapable although they be of rest,\n    And in their fashion very rioters,                               585\n    There is a stillness, and they seem to make\n    Calm revelry in that their calm abode.\n    Them leaving to their joyous hours I pass,\n    Pass with a thought the life of the whole year\n    That is to come, the throng of woodland flowers,                 590\n    And lilies that will dance upon the waves.\n      Say boldly then that solitude is not\n    Where these things are. He truly is alone,\n    He of the multitude whose eyes are doomed\n    To hold a vacant commerce day by day                             595\n    With objects wanting life, repelling love;\n    He by the vast Metropolis immured,\n    Where pity shrinks from unremitting calls,\n    Where numbers overwhelm humanity,\n    And neighbourhood serves rather to divide                        600\n    Than to unite. What sighs more deep than his,\n    Whose nobler will hath long been sacrificed;\n    Who must inhabit, under a black sky,\n    A City where, if indifference to disgust\n    Yield not, to scorn, or sorrow, living men                       605\n    Are ofttimes to their fellow-men no more\n    Than to the forest hermit are the leaves\n    That hang aloft in myriads--nay, far less,\n    For they protect his walk from sun and shower,\n    Swell his devotion with their voice in storms,                   610\n    And whisper while the stars twinkle among them\n    His lullaby. From crowded streets remote,\n    Far from the living and dead wilderness\n    Of the thronged world, Society is here[372]\n    A true Community, a genuine frame                                615\n    Of many into one incorporate.\n    _That_ must be looked for here, paternal sway,\n    One household under God for high and low,\n    One family, and one mansion; to themselves\n    Appropriate, and divided from the world                          620\n    As if it were a cave, a multitude\n    Human and brute, possessors undisturbed\n    Of this recess, their legislative hall,\n    Their Temple, and their glorious dwelling-place.\n      Dismissing therefore, all Arcadian dreams,                     625\n    All golden fancies of the golden age,\n    The bright array of shadowy thoughts from times\n    That were before all time, or is to be\n    Ere time expire, the pageantry that stirs\n    And will be stirring when our eyes are fixed                     630\n    On lovely objects, and we wish to part\n    With all remembrance of a jarring world,\n    --Take we at once this one sufficient hope,\n    What need of more? that we shall neither droop,\n    Nor pine for want of pleasure in the life                        635\n    Scattered about us, nor through dearth of aught\n    That keeps in health the insatiable mind;\n    That we shall have for knowledge and for love\n    Abundance; and that, feeling as we do\n    How goodly, how exceeding fair, how pure                         640\n    From all reproach is yon ethereal vault,\n    And this deep vale its earthly counterpart,\n    By which, and under which, we are enclosed\n    To breathe in peace, we shall moreover find\n    (If sound, and what we ought to be ourselves,                    645\n    If rightly we observe and justly weigh)\n    The inmates not unworthy of their home\n    The dwellers of their dwelling.\n                                  And if this\n    Were otherwise, we have within ourselves\n    Enough to fill the present day with joy,                         650\n    And overspread the future years with hope,\n    Our beautiful and quiet home, enriched\n    Already with a stranger whom we love\n    Deeply, a stranger of our father's house,\n    A never-resting Pilgrim of the Sea,[373]                         655\n    Who finds at last an hour to his content\n    Beneath our roof. And others whom we love\n    Will seek us also, sisters of our hearts,[374]\n    And one, like them, a brother of our hearts,\n    Philosopher and Poet,[375] in whose sight                        660\n    These mountains will rejoice with open joy.\n    --Such is our wealth; O Vale of Peace, we are\n    And must be, with God's will, a happy band.\n      Yet 'tis not to enjoy that we exist,\n    For that end only; something must be done.                       665\n    I must not walk in unreproved delight\n    These narrow bounds, and think of nothing more,\n    No duty that looks further, and no care.\n    Each being has his office, lowly some\n    And common, yet all worthy if fulfilled                          670\n    With zeal, acknowledgment that with the gift\n    Keeps pace, a harvest answering to the seed--\n    Of ill-advised Ambition and of Pride\n    I would stand clear, but yet to me I feel\n    That an internal brightness is vouchsafed                        675\n    That must not die, that must not pass away.\n    Why does this inward lustre fondly seek,\n    And gladly blend with outward fellowship?\n    Why do _they_ shine around me whom I love?\n    Why do they teach me whom I thus revere?                         680\n    Strange question, yet it answers not itself.\n    That humble roof embowered among the trees,\n    That calm fire-side, it is not even in them,\n    --Blest as they are--to furnish a reply,\n    That satisfies and ends in perfect rest.                         685\n    Possessions have I that are solely mine,\n    Something within which yet is shared by none,\n    Not even the nearest to me and most dear,\n    Something which power and effort may impart,\n    I would impart it, I would spread it wide,                       690\n    Immortal in the world which is to come.\n    Forgive me if I add another claim,\n    And would not wholly perish even in this,\n    Lie down and be forgotten in the dust,\n    I and the modest partners of my days                             695\n    Making a silent company in death;\n    Love, knowledge, all my manifold delights\n    All buried with me without monument\n    Or profit unto any but ourselves.\n    It must not be, if I, divinely taught,                           700\n    Be privileged to speak as I have felt\n    Of what in man is human or divine.\n      While yet an innocent little-one, with a heart\n    That doubtless wanted not its tender moods,\n    I breathed (for this I better recollect)                         705\n    Among wild appetites and blind desires,\n    Motions of savage instinct, my delight\n    And exaltation. Nothing at that time\n    So welcome, no temptation half so dear\n    As that which urged me to a daring feat.                         710\n    Deep pools, tall trees, black chasms, and dizzy crags,\n    And tottering towers; I loved to stand and read\n    Their looks forbidding, read and disobey,\n    Sometimes in act, and evermore in thought.\n    With impulses that scarcely were by these                        715\n    Surpassed in strength, I heard of danger, met\n    Or sought with courage; enterprize forlorn\n    By one, sole keeper of his own intent,\n    Or by a resolute few who for the sake\n    Of glory, fronted multitudes in arms.                            720\n    Yea to this hour I cannot read a tale\n    Of two brave vessels matched in deadly fight,\n    And fighting to the death, but I am pleased\n    More than a wise man ought to be. I wish,\n    Fret, burn, and struggle, and in soul am there;                  725\n    But me hath Nature tamed, and bade to seek\n    For other agitations, or be calm;\n    Hath dealt with me as with a turbulent stream,\n    Some nursling of the mountains, which she leads\n    Through quiet meadows, after he has learnt                       730\n    His strength, and had his triumph and his joy,\n    His desperate course of tumult and of glee.\n    That which in stealth by Nature was performed\n    Hath Reason sanctioned. Her deliberate voice\n    Hath said, \"Be mild and cleave to gentle things,                 735\n    Thy glory and thy happiness be there.\n    Nor fear, though thou confide in me, a want\n    Of aspirations that _have_ been, of foes\n    To wrestle with, and victory to complete,\n    Bounds to be leapt, darkness to be explored,                     740\n    All that inflamed thy infant heart, the love,\n    The longing, the contempt, the undaunted quest,\n    All shall survive--though changed their office, all\n    Shall live,--it is not in their power to die.\"\n      Then farewell to the Warrior's schemes, farewell               745\n    The forwardness of soul which looks that way\n    Upon a less incitement than the cause\n    Of Liberty endangered, and farewell\n    That other hope, long mine, the hope to fill\n    The heroic trumpet with the Muse's breath!                       750\n    Yet in this peaceful Vale we will not spend\n    Unheard-of days, though loving peaceful thoughts.\n    A voice shall speak, and what will be the theme?[18]\n\n[359] The following lines, 71-97, and 110-125, were first published in\nthe _Memoirs of Wordsworth_, in 1851.--ED.\n\n[360]\n\n    \u2026 on \u2026\n\n    1851.\n\n[361] The lines 152-167 were first published in the _Memoirs of\nWordsworth_ in 1851.--ED.\n\n[362]\n\n    Mark how the feathered tenants of the flood\n    With grace of motion \u2026\n\n    MS.\n\n[363]\n\n    \u2026 amid \u2026\n\n    MS.\n\n[364]\n\n    They tempt the water, or \u2026\n\n    MS.\n\n[365] The foregoing twenty-seven lines were published under the title\n_Water-Fowl_, in the 1827 edition of Wordsworth's \"Poetical Works.\"\nThey are also printed in the fifth edition of the _Guide through the\nDistrict of the Lakes in the North of England_ (section first).--ED.\n\n[366] Compare _Paradise Lost_, book xii. l. 646.--ED.\n\n[367] Compare, in the _After-Thought_ to \"The Duddon Sonnets\"--\n\n    Still glides the Stream, and shall for ever glide.\n\nED.\n\n[368] Compare, in _An Evening Walk_, l. 378--\n\n    Or yell, in the deep woods, of lonely hound.\n\nED.\n\n[369] Compare Wordsworth's numerous references to the Cumbrian and\nWestmoreland \"Statesmen,\" in his Prose Works, and elsewhere.--ED.\n\n[370] Compare _Peter Bell_.--ED.\n\n[371] Compare _The Excursion_, book iv. ll. 1175-1187.--ED.\n\n[372] Wordsworth says elsewhere that\n\n    Solitude is blithe Society.\n\nED.\n\n[373] John Wordsworth.--ED.\n\n[374] The Hutchinsons.--ED.\n\n[375] Coleridge.--ED.\n\n\n\"SHALL HE WHO GIVES HIS DAYS TO LOW PURSUITS\"\n\nThe following lines occur in the experimental efforts made by\nWordsworth to write an autobiographical poem. They occur in one of his\nsister's Journals, entitled \"May to December, 1802\"; and were probably\neither dictated to her in that year, or were copied by her from some\nearlier fragment. They stand related to passages in _The Prelude_. (See\nvol. iii. p. 269.)--ED.\n\n    Shall he who gives his days to low pursuits\n    Amid the undistinguishable crowd\n    Of cities, 'mid the same eternal flow\n    Of the same objects, melted and reduced\n    To one identity, by differences                                    5\n    That have no law, no meaning, and no end,\n    Shall he feel yearning to those lifeless forms,\n    And shall we think that Nature is less kind\n    To those, who all day long, through a busy life,\n    Have walked within her sight? It cannot be.                       10\n\n\n\n\n1803\n\n\n\"I FIND IT WRITTEN OF SIMONIDES\"\n\nPublished in _The Morning Post_, October 10, 1803\n\nS.T.C. writing to Tom Poole, October 14, 1803, said that Wordsworth\nwrote to _The Morning Post_ \"as W. L. D., and sometimes with no\nsignature.\" There is ample evidence that the following sonnet was\nwritten by Wordsworth. He had contributed five sonnets to _The Morning\nPost_ before the month of September 1803; and on the 10th of October in\nthat year the following appeared.--ED.\n\n    I find it written of Simonides,\n    That, travelling in strange countries, once he found\n    A corpse that lay expos'd upon the ground,\n    For which, with palms, he caus'd due obsequies\n    To be perform'd, and paid all holy fees.                           5\n    Soon after this man's ghost unto him came,\n    And told him not to sail, as was his aim,\n    On board a ship then ready for the seas.\n    Simonides, admonish'd by the ghost,\n    Remain'd behind: the ship the following day                       10\n    Set sail, was wreck'd, and all on board were lost.\n    Thus was the tenderest Poet that could be,\n    Who sang in antient Greece his loving lay,\n    Sav'd out of many by his piety.\n\n\n\n\n1804\n\n\n\"NO WHIMSEY OF THE PURSE IS HERE\"\n\nWriting to Sir George Beaumont, on Christmas Day, 1804, Wordsworth\nsaid: \"We have lately built in our little rocky orchard a circular\nhut, lined with moss, like a wren's nest, and coated on the outside\nwith heath, that stands most charmingly, with several views from the\ndifferent sides of it, of the Lake, the Valley, and the Church.\u2026 I will\ncopy a dwarf inscription which I wrote for it\" (_i.e._ the circular\nhut, in his Orchard-Garden) \"the other day before the building was\nentirely finished, which indeed it is not yet.\"[376]--ED.\n\n    No whimsey of the purse is here,\n    No pleasure-house forlorn;\n    Use, comfort, do this roof endear;\n    A tributary shed to cheer\n    The little cottage that is near,\n    To help it and adorn.\n\n[376] See the _Memorials of Coleorton_, vol. i. p. 81; and Wordsworth's\nletter on the subject in a later volume of this edition.--ED.\n\n\n\n\n1805\n\n\n\"PEACEFUL OUR VALLEY, FAIR AND GREEN\"\n\nThis is extracted from a copy of an appendix to _Recollections of a\nTour in Scotland_ by Dorothy Wordsworth, written by Mrs. Clarkson,\nSeptember-November 1805. It was composed by the poet's sister. In\nFebruary 1892 it was published in _The Monthly Packet_ under the title\n\"Grasmere: a Fragment,\" and with the signature \"Rydal Mount, September\n26, 1829.\" It is now printed from the MS. of 1805.--ED.\n\n    Peaceful our valley, fair and green;\n      And beautiful the cottages\n    Each in its nook, its sheltered hold,\n      Or underneath its tuft of trees.\n\n    Many and beautiful they are;                                       5\n      But there is one that I love best,\n    A lowly roof in truth it is,\n      A brother of the rest.\n\n    Yet when I sit on rock or hill\n      Down-looking on the valley fair,                                10\n    That cottage with its grove of trees\n      Summons my heart; it settles there.\n\n    Others there are whose small domain\n      Of fertile fields with hedgerows green\n    Might more seduce the traveller's mind                            15\n      To wish that there his home had been.\n\n    Such wish be his! I blame him not,\n      My fancies they, perchance, are wild;\n    I love that house because it is\n      The very mountain's child.                                      20\n\n    Fields hath it of its own, green fields;\n      But they are craggy, steep, and bare;\n    Their fence is of the mountain stone,\n      And moss and lichen flourish there.\n\n    And when the storm comes from the North                           25\n      It lingers near that pastoral spot,\n    And piping through the mossy walls,\n      It seems delighted with its lot.\n\n    And let it take its own delight,\n      And let it range the pastures bare                              30\n    Until it reach that grove of trees\n      ----It may not enter there!\n\n    A green unfading grove it is,\n      Skirted with many a lesser tree,\n    Hazel and holly, beech and oak,                                   35\n      A fair and flourishing company!\n\n    Precious the shelter of those trees!\n      They screen the cottage that I love;\n    The sunshine pierces to the roof\n      And the tall pine trees tower above.                            40\n\n    When first I saw that dear abode\n      It was a lovely winter's day:\n    After a night of perilous storm\n      The West wind ruled with gentle sway;\n\n    A day so mild, it might have been                                 45\n      The first day of the gladsome spring;\n    The robins warbled; and I heard\n      One solitary throstle sing:\n\n    A stranger in the neighbourhood,\n      All faces then to me unknown,                                    50\n    I left my sole companion-friend\n      To wander out alone.\n\n    Lur'd by a little winding path,\n      I quitted soon the public road,\n    A smooth and tempting path it was                                 55\n      By sheep and shepherds trod.\n\n    Eastward, toward the mighty hills\n      This pathway led me on,\n    Until I reach'd a lofty Rock\n      With velvet moss o'ergrown.                                      60\n\n    With russet Oak and tufts of Fern\n      Its top was richly garlanded;\n    Its sides adorn'd with Eglantine\n      Bedropp'd with hips of glossy red.\n\n    There too in many a shelter'd chink                               65\n      The foxglove's broad leaves flourish'd fair,\n    And silver birch whose purple twigs\n      Bend to the softest breathing air.\n\n    Beneath that rock my course I stay'd\n      And, looking to its summit high,                                 70\n    \"Thou wear'st,\" said I, \"a splendid garb,\n      Here winter keeps his revelry.\n\n    \"I've been a dweller on the plains,\n      Have sigh'd when summer days were gone;\n    No more I'll sigh; for winter here                                75\n      Hath gladsome gardens of his own.\n\n    \"What need of flowers? The splendid moss\n      Is gayer than an April mead;\n    More rich its hues of various green,\n      Orange and gold and glowing red.\"                               80\n\n    ----Beside that gay and lovely rock\n      There came with merry voice\n    A foaming streamlet glancing by,\n      It seem'd to say \"Rejoice!\"\n\n    My youthful wishes all fulfill'd,                                 85\n      Wishes matured by thoughtful choice,\n    I stood an Inmate of this vale,\n      How could I but rejoice?\n\n\n\"AH! IF I WERE A LADY GAY\"\n\nThe following two stanzas were added by Wordsworth to his sister's\npoem, entitled _The Cottager to her Infant_--composed in 1805, and\nissued in 1815 (see vol. iii. pp. 74, 75); but they were never\npublished in Wordsworth's lifetime.--ED.\n\n    Ah! if I were a lady gay\n    I should not grieve with thee to play;\n    Right gladly would I lie awake\n    Thy lively spirits to partake,\n      And ask no better cheer.                                         5\n\n    But, Babe! there's none to work for me,\n    And I must rise to industry;\n    Soon as the cock begins to crow\n    Thy mother to the fold must go\n      To tend the sheep and kine.                                     10\n\n\n\n\n1806\n\n\nTO THE EVENING STAR OVER GRASMERE WATER, JULY 1806\n\n                                The Lake is thine,\n    The mountains too are thine, some clouds there are,\n    Some little feeble stars, but all is thine,\n    Thou, thou art king, and sole proprietor.\n\n    A moon among her stars, a mighty vale,                             5\n    Fresh as the freshest field, scoop'd out, and green\n    As is the greenest billow of the sea.\n\n    The multitude of little rocky hills,\n    Rocky or green, that do like islands rise\n    From the flat meadow lonely there.                                10\n    \u2026\n    Embowering mountains, and the dome of Heaven\n    And waters in the midst, a Second Heaven.\n\n\nMICHAEL ANGELO IN REPLY TO THE PASSAGE UPON HIS STATUE OF NIGHT SLEEPING\n\nIn the first volume of a copy of the edition of 1836,--long kept by\nWordsworth at Rydal Mount, and afterwards the property of the late Lord\nColeridge--which has been referred to in the Preface to Vol. 1., and\nvery often in the footnotes to all the volumes, signed C.--Wordsworth\nwrote in MS. two translations of a fragment of Michael Angelo's on\nSleep, and a translation of some Latin verses by Thomas Warton on the\nsame subject. These fragments were never included in any edition of his\npublished works, and it is impossible to say to what year they belong.\nFrom their close relation to other translations from Michael Angelo,\nmade by Wordsworth in 1806, I assign them, conjecturally, to the same\nyear. The title is from Wordsworth's own MS.--ED.\n\n                        I\n\n    Grateful is Sleep, my life, in stone bound fast,\n    More grateful still: while wrong and shame shall last,\n    On me can Time no happier state bestow\n    Than to be left unconscious of the woe.\n    Ah then, lest you awaken me, speak low.                            5\n\n                       II\n\n    Grateful is Sleep, more grateful still to be\n    Of marble; for while shameless wrong and woe\n    Prevail, 'tis best to neither hear nor see.\n    Then wake me not, I pray you. Hush, speak low.\n\n\n\"COME, GENTLE SLEEP, DEATH'S IMAGE THO' THOU ART\"\n\n    Come, gentle Sleep, Death's image tho' thou art,\n    Come share my couch, nor speedily depart;\n    How sweet thus living without life to lie,\n    Thus without death how sweet it is to die.\n\nThe Latin verse by Thomas Warton, of which these lines are a\ntranslation, is as follows:--\n\n    Somne veni! quamvis placidissima Mortis imago es,\n      Consortem cupio te tamen esse tori;\n    Hue ades, haud abiture cit\u00f2! nam sic sine vita\n      Vivere quam suave est, sic sine morte mori!\n\nThomas Warton, Fellow of Trinity College, Oxford, and Professor of\nPoetry in that University, is chiefly known by his _History of English\nPoetry_ (1774-1781).--ED.\n\n\n\"BROOK, THAT HAST BEEN MY SOLACE DAYS AND WEEKS\"\n\nThe following version of the sonnet beginning \"Brook! whose society the\nPoet seeks,\" probably written in 1806 and first published in 1815 (see\nvol. iv. p. 52), has come to light since that volume was issued. The\nvariants throughout are sufficient to warrant its publication here. Had\nI received it earlier they would have appeared in vol. iv.--ED.\n\n    Brook, that hast been my solace days and weeks,\n    And months, and let me add the long year through,\n    I come to thee, thou dost my heart renew;\n    O happy Thing! among thy flowery creeks,\n    And happy, dancing down thy water-breaks:                          5\n    If I some type of thee did wish to view,\n    Thee, and not thee thyself, I would not do\n    Like Grecian Poets, give thee human cheeks,\n    Channels for tears! No Naiad should'st thou be;\n    Have neither wings, feet, feathers, joints, nor hairs.            10\n    It seems the Eternal Soul is clothed in thee\n    With purer robes than those of flesh and blood,\n    And hath bestowed on thee a better good;\n    The joy of fleshly life without its cares.\n\n\nTRANSLATION FROM MICHAEL ANGELO\n\nThe date of this is unknown, and the original MS. is difficult to\ndecipher. It is here and there illegible. It may belong to the year\nof the \"Ecclesiastical Sonnets,\" but I place it beside the other\ntranslation from Michael Angelo.--ED.\n\n    Rid of a vexing and a heavy load,\n    Eternal Lord! and from the world set free,\n    Like a frail Bark, weary I turn to Thee,--\n    From frightful storms into a quiet road.\n    On much repentance Grace will be bestow'd.                         5\n    The nails, the thorns, and thy two hands, thy face\n    Benign, meek, \u2026, offers grace\n    To sinners whom their sins oppress and goad.\n    Let not thy justice view, O Light Divine,\n    My fault, and keep it from thy sacred ear.                        10\n    \u2026\n    Cleanse with thy blood my sins, to this incline\n    More readily, the more my years require\n    Prompt aid, forgiveness speedy and entire.\n\n\n\n\n1808\n\n\nGEORGE AND SARAH GREEN\n\nComposed 1808.--Published 1839\n\nThis poem was first printed in De Quincey's \"Recollections of\nGrasmere,\" which appeared in _Tait's Edinburgh Magazine_, September\n1839, p. 573, and afterwards in his _Recollections of the Lakes_\n(1853), p. 23.\n\nThe text is printed as it is found in De Quincey's article. Doubtless\nWordsworth, or some member of the family, had supplied him with a\ncopy of these verses. Wordsworth himself seemed to have thought them\nunworthy of publication. A copy of the poem was transcribed at Grasmere\nby Dorothy Wordsworth for Lady Beaumont on the 20th April 1808. In\nthis copy there are numerous variations from the text as published by\nDe Quincey, and these are indicated in the footnotes. In the letter to\nLady Beaumont, Dorothy Wordsworth says, \"I am going to transcribe a\npoem composed by my brother a few days after his return. It was begun\nin the churchyard when he was looking at the grave of the Husband and\nWife, and is in fact supposed to be entirely composed there.\"\n\nWordsworth returned to his old home at Dove Cottage, Grasmere, after a\nshort visit to London, on the 6th April 1808; and there he remained,\ntill Allan Bank was ready for occupation. I therefore conclude that\nthis poem was written in April 1808.\n\nCompare De Quincey's account of the disaster that befell the Greens, as\nreported in his _Early Recollections of Grasmere_. The Wordsworths had\nevidently taken part in the effort to raise subscriptions in behalf of\nthe orphan children. They issued a printed appeal on the subject. The\nfollowing is an extract from a letter of Dorothy Wordsworth's to Lady\nBeaumont on the subject:--\n\n                                    \"GRASMERE, _April 20th, 1808_.\n\n    \"We received your letter this morning, enclosing the half of a\n    \u00a35 note. I am happy to inform you that the orphans have been\n    fixed under the care of very respectable people. The baby is\n    with its sister--she who filled the Mother's place in the house\n    during their two days of fearless solitude. It has clung to her\n    ever since, and she has been its sole nurse. I went with two\n    ladies of the Committee (in my sister's place, who was then\n    confined to poor John's bedside) to conduct the family to their\n    separate homes. The two Girls are together, as I have said; two\n    Boys at another Home; and the third Boy by himself at the house\n    of an elderly man who had a particular friendship for their\n    father. The kind reception that the children met with was very\n    affecting.\"\n\nSee the letters from Wordsworth to Richard Sharpe, Esq., Mark Lane,\nLondon, in a subsequent volume, referring to the catastrophe.--ED.\n\n    Who weeps for strangers? Many wept\n      For George and Sarah Green;\n    Wept for that pair's unhappy fate,\n      Whose grave may here be seen.[377]\n\n    By night, upon these stormy fells,[378]                            5\n      Did wife and husband roam;\n    Six little ones at home had left,\n      And could not find that home.[379]\n\n    For _any_ dwelling-place of man\n      As vainly did they seek.                                        10\n    He perish'd; and a voice was heard--\n      The widow's lonely shriek.[380]\n\n    Not many steps, and she was left[381]\n      A body without life--\n    A few short steps were the chain that bound[382]                  15\n      The husband to the wife.[383]\n\n    Now do those[384] sternly-featured hills\n      Look gently on this grave;\n    And quiet now are the depths[385] of air,\n      As a sea without a wave.                                        20\n\n    But deeper lies the heart of peace\n      In quiet more profound;[386]\n    The heart of quietness is here\n      Within this churchyard bound.[387]\n\n    And from all agony of mind                                        25\n      It keeps them safe, and far\n    From fear and grief, and from all need\n      Of sun or guiding star.[388]\n\n    O darkness of the grave! how deep,[389]\n      After that living night--                                       30\n    That last and dreary living one\n      Of sorrow and affright!\n\n    O sacred marriage-bed of death,\n      That keeps[390] them side by side\n    In bond of peace, in bond of love,[391]                           35\n      That may not be untied!\n\n[377] 1839.\n\n    Wept for that Pair's unhappy end,\n    Whose Grave may here be seen.\n\n    MS. letter of Dorothy Wordsworth's.\n\n[378] 1839.\n\n    \u2026 these stormy Heights,\n\n    MS.\n\n[379] 1839.\n\n    Six little ones the Pair had left,\n    And could not find their home.\n\n    MS.\n\n[380] 1839.\n\n    Down the dark precipice he fell,\n      And she was left alone,\n    Not long to think of her children dear,\n      Not long to pray, or groan.\n\n    Added in MS.\n\n[381] 1839.\n\n    A few wild steps--she too was left,\n\n    MS.\n\n[382] 1839.\n\n    The chain of but a few wild steps.\n\n    MS. in Dorothy Wordsworth's handwriting--sent to Lady Beaumont.\n\n[383] 1839.\n\nFour stanzas are here added in MS., only one of which need be given--\n\n    Our peace is of the immortal soul,\n      Our anguish is of clay;\n    Such bounty is in Heaven: so pass\n      The bitterest pangs away.\n\n[384] 1839.\n\n    Now do the \u2026\n\n    MS.\n\n[385] 1839.\n\n    \u2026 is the depth \u2026\n\n    MS.\n\n[386] 1839.\n\n    In shelter more profound.\n\n    MS.\n\n[387] 1839.\n\n    \u2026 ground.\n\n    MS.\n\n[388] 1839.\n\n    From fear, and from all need of hope\n      From sun or guiding star.\n\n    MS.\n\n[389] 1839.\n\n    \u2026 how calm,\n\n    MS.\n\n[390] 1839.\n\n    That holds \u2026\n\n    MS.\n\n[391] 1839.\n\n    In bond of love, in bond of God,\n\n    MS.\n\n\n\n\n1818\n\n\n\"THE SCOTTISH BROOM ON BIRD-NEST BRAE\"[392]\n\n    The Scottish Broom on Bird-nest brae[393]\n    Twelve tedious years ago,\n    When many plants strange blossoms bore\n    That puzzled high and low,\n    A not unnatural longing felt,                                      5\n    What longing would ye know?\n    Why, friend, to deck her supple twigs\n    With _yellow_ in full blow.\n\n    To Lowther Castle she addressed\n    A prayer both bold and sly,                                       10\n    (For all the Brooms on Bird-nest brae\n    Can talk and speechify)\n    That flattering breezes blowing thence\n    Their succour would supply,\n    Then she would instantly put forth                                15\n    A flag of _yellow_ dye.\n\n    But from the Castle turret blew\n    A chill forbidding blast,\n    Which the poor Broom no sooner felt\n    Than she shrank up so fast;                                       20\n    Her _wished-for_ yellow she forswore,\n    And since that time has cast\n    Fond looks on colours three or four\n    And put forth _Blue_ at last.\n    And now, my lads, the Election comes                              25\n    In June's sunshining hours,\n    When every field and bank and brae\n    Is clad with yellow flowers.\n    While faction Blue from shops and booths\n    Tricks out her blustering powers,                                 30\n    Lo! smiling Nature's lavish hand\n    Has furnished wreaths for ours.\n\n[392] \"Written, in my opinion, at the General Election of 1818.\"--(The\nRev. Thomas Hutchinson of Kimbolton.)\n\n[393] \"Bird-nest\" was the old name of Brougham Hall.--ED.\n\n\nPLACARD FOR A POLL BEARING AN OLD SHIRT\n\nWordsworth was deeply interested in the successive parliamentary\nelections for Westmoreland (see his \"Addresses to the Freeholders of\nWestmorland, 1818,\" in the Prose Works.) He particularly disliked\nLord Brougham's candidature. The following squib is in MS. at Lowther\nCastle. He wrote on the MS.--\"For a version of part of B.'s famous\nLondon Tower Speech see opposite page.\"--ED.\n\n    If money's slack,\n    The shirt on my back\n    Shall off, and go to the hammer:\n    Though I sell shirt and skin\n    By Jove I'll be in,\n    And raise up a radical clamor!\n\n\n\"CRITICS, RIGHT HONOURABLE BARD, DECREE\"\n\nI have found this in a catalogue of Autograph Letters, and have no\nknowledge of its date, or of the Bard referred to. Solomon Gesner wrote\na poem on _The Death of Abel_, which was translated into English. See\nfootnote to _The Prelude_, book vii. l. 564.--ED.\n\n    \"Critics, right honourable Bard, decree\n    Laurels to some, a night-shade wreath to thee,\n    Whose muse a sure though late revenge hath ta'en\n    Of harmless Abel's death, by murdering Cain.\"\n\nOn Cain, a Mystery, dedicated to Sir Walter Scott:--\n\n    \"A German Haggis from receipt\n    Of him who cooked the death of Abel,\n    And sent 'warm-reeking, rich and sweet,'\n    From Venice to Sir Walter's table.\"\n\n\n\n\n1819\n\n\n\"THROUGH CUMBRIAN WILDS, IN MANY A MOUNTAIN COVE\"\n\nIn 1819 Wordsworth wrote the sonnet beginning, \"Grief, thou hast lost\nan ever ready friend.\" In the note to that sonnet (vol. vi. p. 196)\nI have given a different version of its last six lines, from a MS.\nsonnet. But as these six lines also form the conclusion of another\nunpublished sonnet, it may be given in full by itself, in this\nAppendix.--ED.\n\n    Through Cumbrian wilds, in many a mountain cove,\n    The pastoral Muse laments the Wheel--no more\n    Engaged, near blazing hearth on clean-swept floor,\n    In tasks which guardian Angels might approve,\n    Friendly the weight of leisure to remove,                          5\n    And to beguile the lassitude of ease;\n    Gracious to all the dear dependencies\n    Of house and field,--to plenty, peace, and love.\n    There too did _Fancy_ prize the murmuring wheel;\n    For sympathies, inexplicably fine,                                10\n    Instilled a confidence--how sweet to feel!\n    That ever in the night-calm, when the Sheep\n    Upon their grassy beds lay couch'd in sleep,\n    The quickening spindle drew a trustier line.\n\n\n\"MY SON! BEHOLD THE TIDE ALREADY SPENT\"\n\nThe following sonnet occurs after the above in the same MS. whence both\nare extracted.--ED.\n\n    My Son! behold the tide already spent\n    That rose, and steadily advanced to fill\n    The shores and channels, working Nature's will\n    Among the mazy streams that backward went,\n    And in the sluggish Ports where ships were pent.                   5\n    And now, its task performed, the flood stands still\n    At the green base of many an inland hill,\n    In placid beauty and entire content.\n    Such the repose that Sage and Hero find,\n    Such measured rest the diligent and good                          10\n    Of humbler name, whose souls do like the flood\n    Of ocean press right on, or gently wind,\n    Neither to be diverted nor withstood\n    Until they reach the bounds by Heaven assigned.\n\n\n\n\n1820\n\n\nAUTHOR'S VOYAGE DOWN THE RHINE\n\n(THIRTY YEARS AGO)\n\n    The confidence of Youth our only Art,\n    And Hope gay Pilot of the bold design,\n    We saw the living Landscapes of the Rhine,\n    Reach after reach, salute us and depart;\n    Slow sink the Spires,--and up again they start!                    5\n    But who shall count the Towers as they recline\n    O'er the dark steeps, or in the horizon line\n    Striding, with shattered crests, the eye athwart?\n    More touching still, more perfect was the pleasure,\n    When hurrying forward till the slack'ning stream                  10\n    Spread like a spacious Mere, we there could measure\n    A smooth free course along the watery gleam,\n    Think calmly on the past, and mark at leisure\n    Features which else had vanished like a dream.\n\nThis sonnet was published in the first edition of the Memorials of\nthis Tour (1822), but was struck out of the next edition, and never\nrepublished. Its rejection by Wordsworth is curious.\n\nIt refers to the pedestrian tour which the Poet took, with his\nfriend Jones, in 1790, which he afterwards recorded in full in his\n_Descriptive Sketches_.\n\nDorothy Wordsworth, in her Journal of the Tour in 1820, refers to it\nthus:--\"Our journey through the narrower and most romantic passages\nof the Vale of the Rhine was connected with times long past, when my\nbrother and his Friend (it was thirty years ago) floated down the\nstream in their little Bark. Often did my fancy place them with a\nfreight of happiness in the centre of some bending reach, overlooked\nby tower or castle, or (when expectation would be most eager) at the\nturning of a promontory, which had concealed from their view some\ndelicious winding which we had left behind; but no more of my own\nfeelings, a record of his will be more interesting.\"\n\nShe then quotes the sonnet, beginning\n\n    The confidence of Youth our only Art.\n\nThere are also numerous allusions in Mrs. Wordsworth's Journal to\nthis early tour; _e.g._ under date August 13. \"We left Meyringen;\nsoon reached a sort of Hotel, which Wm. pointed out to us with great\ninterest, as being the only spot where he and his friend Jones were ill\nused, during the course of their adventurous journey--a wild looking\nbuilding, a little removed from the road, where the vale of Hasli\nends.\" Again, in describing the sunset from the woody hill Colline de\nGibet, overlooking the two lakes of Brienz and Thun, at Interlaken,\n\"with the loveliest of green vallies between us and Jungfrau,\" \"Surely\nWilliam must have had this Paradise in his thoughts when he began his\n_Descriptive Sketches_--\n\n    Were there, below, a spot of holy ground,\n    By Pain and her sad family unfound, etc.\n\nBut no habitation was there among these rocky knolls, and tiny\npastures. One fragment, something like a ruined convent, lurked under\na steep, woody-fringed crag. What a Refuge for a pious Sisterhood!\"\nCompare also the note to _Stanzas composed in the Simplon Pass_, vol.\nvi. p. 359.--ED.\n\n\n\n\n1822\n\n\n\"THESE VALES WERE SADDENED WITH NO COMMON GLOOM\"\n\nIn the _Memoirs of William Wordsworth_ by his nephew (the late Bishop\nof Lincoln) vol. i. chap. xxx. the following occurs as an addendum\ntransferred to the footnotes:--\n\n\"The first six lines of an epitaph in Grasmere Church were also his\ncomposition. The elegant marble tablet on which they were engraved was\ndesigned by Sir Francis Chantry, and prepared by Allan Cunningham,\n1822. It is over the chancel door.\"\n\nThe following is the Inscription:--\n\n                         In the Burial Ground\n              of this Church are deposited the remains of\n                         JEMIMA ANNE DEBORAH,\n                          second daughter of\n           Sir Egerton Brydges, of Denton Court, Kent, Bart.\n           She departed this life at the Ivy Cottage, Rydal,\n                     May 25th 1822, aged 28 years.\n                This memorial is erected by her husband\n\n                           EDWARD QUILLINAN.\n\nThe entire sonnet, of which Wordsworth wrote the \"first six lines,\" is\nas follows:--\n\n    These vales were saddened with no common gloom\n    When good Jemima perished in her bloom;\n    When, such the awful will of heaven, she died\n    By flames breathed on her from her own fireside.\n    On earth we dimly see, and but in part                             5\n    We know, yet faith sustains the sorrowing heart;\n    And she, the pure, the patient and the meek,\n    Might have fit epitaph could feelings speak;\n    If words could tell and monuments record,\n    How Treasures lost are inwardly deplored,                         10\n    No name by grief's fond eloquence adorned\n    More than Jemima's would be praised and mourned.\n    The tender virtues of her blameless life,\n    Bright in the daughter, brighter in the wife,\n    And in the cheerful mother brightest shone,--                     15\n    That light hath past away--the will of God[394] be done.\n\n[394]\n\n    \u2026 of Heaven \u2026\n\n    MS.\n\n\nTRANSLATION OF PART OF THE FIRST BOOK OF THE \u00c6NEID\n\nComposed 1823 (?).--Published 1836\n\nThis translation was included in the _Philological Museum_, edited\nby Julius Charles Hare, and published at Cambridge in 1832 (vol. i.\np. 382, etc.). Three Books were translated by Wordsworth, but the\ngreater portion is still in MS., unpublished. What is now reproduced\nappeared in the _Museum_. As it was never included by Wordsworth\nhimself in any edition of his Works, his own estimate of its literary\nvalue was slight. It was published by Professor Henry Reed in his\nAmerican reprint of 1851. Writing to Lord Lonsdale on 9th Nov. 1823,\nWordsworth says, \"I have just finished a Translation into English rhyme\nof the First _\u00c6neid_. Would you allow me to send it to you? I would\nbe much gratified if you would take the trouble of comparing some\npassages with the original. I have endeavoured to be much more literal\nthan Dryden, or Pitt--who keeps more close to the original than his\npredecessor.\"--ED.\n\n            TO THE EDITORS OF THE \"PHILOLOGICAL MUSEUM\"\n\n    Your letter, reminding me of an expectation I some time since\n    held out to you of allowing some specimens of my translation\n    from the _\u00c6neid_ to be printed in the _Philological Museum_\n    was not very acceptable; for I had abandoned the thought of\n    ever sending into the world any part of that experiment,--for\n    it was nothing more,--an experiment begun for amusement, and I\n    now think a less fortunate one than when I first named it to\n    you. Having been displeased in modern translations with the\n    additions of incongruous matter, I began to translate with a\n    resolve to keep clear of that fault, by adding nothing; but\n    I became convinced that a spirited translation can scarcely\n    be accomplished in the English language without admitting a\n    principle of compensation. On this point, however, I do not\n    wish to insist, and merely send the following passage, taken at\n    random, from a wish to comply with your request.--W.W.\n\n      But Cytherea, studious to invent\n    Arts yet untried, upon new counsels bent,\n    Resolves that Cupid, chang'd in form and face\n    To young Ascanius, should assume his place;\n    Present the maddening gifts, and kindle heat                       5\n    Of passion at the bosom's inmost seat.\n    She dreads the treacherous house, the double tongue;\n    She burns, she frets--by Juno's rancour stung;\n    The calm of night is powerless to remove\n    These cares, and thus she speaks to wing\u00e8d Love:                  10\n\n      \"O son, my strength, my power! who dost despise\n    (What, save thyself, none dares through earth and skies)\n    The giant-quelling bolts of Jove, I flee,\n    O son, a suppliant to thy deity!\n    What perils meet \u00c6neas in his course,                             15\n    How Juno's hate with unrelenting force\n    Pursues thy brother--this to thee is known;\n    And oft-times hast thou made my griefs thine own.\n    Him now the generous Dido by soft chains\n    Of bland entreaty at her court detains;                           20\n    Junonian hospitalities prepare\n    Such apt occasion that I dread a snare.\n    Hence, ere some hostile God can intervene,\n    Would I, by previous wiles, inflame the queen\n    With passion for \u00c6neas, such strong love                          25\n    That at my beck, mine only, she shall move.\n    Hear, and assist;--the father's mandate calls\n    His young Ascanius to the Tyrian walls;\n    He comes, my dear delight,--and costliest things\n    Preserv'd from fire and flood for presents brings.                30\n    Him will I take, and in close covert keep,\n    'Mid groves Idalian, lull'd to gentle sleep,\n    Or on Cythera's far-sequestered steep,\n    That he may neither know what hope is mine,\n    Nor by his presence traverse the design.                          35\n    Do thou, but for a single night's brief space,\n    Dissemble; be that boy in form and face!\n    And when enraptured Dido shall receive\n    Thee to her arms, and kisses interweave\n    With many a fond embrace, while joy runs high,                    40\n    And goblets crown the proud festivity,\n    Instil thy subtle poison, and inspire,\n    At every touch, an unsuspected fire.\"\n\n      Love, at the word, before his mother's sight\n    Puts off his wings, and walks, with proud delight,                45\n    Like young Iulus; but the gentlest dews\n    Of slumber Venus sheds, to circumfuse\n    The true Ascanius steep'd in placid rest;\n    Then wafts him, cherish'd on her careful breast,\n    Through upper air to an Idalian glade,                            50\n    Where he on soft _amaracas_ is laid,\n    With breathing flowers embraced, and fragrant shade.\n    But Cupid, following cheerily his guide\n    Achates, with the gifts to Carthage hied;\n    And, as the hall he entered, there, between                       55\n    The sharers of her golden couch, was seen\n    Reclin'd in festal pomp the Tyrian queen.\n    The Trojans, too (\u00c6neas at their head),\n    On couches lie, with purple overspread:\n    Meantime in canisters is heap'd the bread,                        60\n    Pellucid water for the hands is borne,\n    And napkins of smooth texture, finely shorn.\n    Within are fifty handmaids, who prepare,\n    As they in order stand, the dainty fare;\n    And fume the household deities with store                         65\n    Of odorous incense; while a hundred more\n    Match'd with an equal number of like age,\n    But each of manly sex, a docile page,\n    Marshal the banquet, giving with due grace\n    To cup or viand its appointed place.                              70\n    The Tyrians rushing in, an eager band,\n    Their painted couches seek, obedient to command.\n    They look with wonder on the gifts--they gaze\n    Upon Iulus, dazzled with the rays\n    That from his ardent countenance are flung,                       75\n    And charm'd to hear his simulating tongue;\n    Nor pass unprais'd the robe and veil divine,\n    Round which the yellow flowers and wandering foliage twine.\n\n      But chiefly Dido, to the coming ill\n    Devoted, strives in vain her vast desires to fill;                80\n    She views the gifts; upon the child then turns\n    Insatiable looks, and gazing burns.\n    To ease a father's cheated love he hung\n    Upon \u00c6neas, and around him clung;\n    Then seeks the queen; with her his arts he tries;                 85\n    She fastens on the boy enamour'd eyes,\n    Clasps in her arms, nor weens (O lot unblest!)\n    How great a God, incumbent o'er her breast,\n    Would fill it with his spirit. He, to please\n    His Acidalian mother, by degrees                                  90\n    Blots out Sichaeus, studious to remove\n    The dead, by influx of a living love,\n    By stealthy entrance of a perilous guest.\n    Troubling a heart that had been long at rest.\n\n      Now when the viands were withdrawn, and ceas'd                  95\n    The first division of the splendid feast,\n    While round a vacant board the chiefs recline,\n    Huge goblets are brought forth; they crown the wine;\n    Voices of gladness roll the walls around;\n    Those gladsome voices from the courts rebound;                   100\n    From gilded rafters many a blazing light\n    Depends, and torches overcome the night.\n    The minutes fly--till, at the queen's command,\n    A bowl of state is offered to her hand:\n    Then she, as Belus wont, and all the line                        105\n    From Belus, filled it to the brim with wine;\n    Silence ensued. \"O Jupiter, whose care\n    Is hospitable dealing, grant my prayer!\n    Productive day be this of lasting joy\n    To Tyrians, and these exiles driven from Troy;                   110\n    A day to future generations dear!\n    Let Bacchus, donor of soul-quick'ning cheer,\n    Be present; kindly Juno, be thou near!\n    And, Tyrians, may your choicest favours wait\n    Upon this hour, the bond to celebrate!\"                          115\n    She spake and shed an offering on the board;\n    Then sipp'd the bowl whence she the wine had pour'd\n    And gave to Bitias, urging the prompt lord;\n    He rais'd the bowl, and took a long deep draught;\n    Then every chief in turn the beverage quaff'd.                   120\n\n      Graced with redundant hair, Iopas sings\n    The lore of Atlas, to resounding strings,\n    The labours of the Sun, the lunar wanderings;\n    Whence human kind, and brute; what natural powers\n    Engender lightning, whence are falling showers.                  125\n    He haunts Arcturus,--that fraternal twain\n    The glittering Bears,--the Pleiads fraught with rain;\n    --Why suns in winter, shunning heaven's steep heights\n    Post seaward,--what impedes the tardy nights.\n    The learned song from Tyrian hearers draws                       130\n    Loud shouts,--the Trojans echo the applause.\n    --But, lengthening out the night with converse new,\n    Large draughts of love unhappy Dido drew;\n    Of Priam ask'd, of Hector--o'er and o'er--\n    What arms the son of bright Aurora wore;--                       135\n    What steeds the car of Diomed could boast;\n    Among the leaders of the Grecian host\n    How look'd Achilles, their dread paramount--\n    \"But nay--the fatal wiles, O guest, recount,\n    Retrace the Grecian cunning from its source,                     140\n    Your own grief and your friends'--your wandering course;\n    For now, till this seventh summer have ye rang'd\n    The sea, or trod the earth, to peace estrang'd.\"\n\n\n\n\n1823\n\n\n\"ARMS AND THE MAN I SING, THE FIRST WHO BORE\"\n\nThe following version of the first few lines of the _\u00c6neid_ were copied\nby Professor Reed of Philadelphia, with Mrs. Wordsworth's permission,\nduring a visit to Rydal Mount in 1854, four years after the poet's\ndeath. Mrs. Reed kindly sent them to me.--ED.\n\n    Arms and the Man I sing, the first who bore\n    His course to Latium from the Trojan shore,\n    A fugitive of fate. Long time was he\n    By powers celestial tossed on land and sea\n    Thro' wrathful Juno's far-famed enmity;\n    Much too from war endured till new abodes\n    He planted, and in Latium fixed his Gods,\n    Whence flows the Latin people, whence have come\n    The Alban Sites and walls of lofty Rome.\n\n\n\n\n1826\n\n\nLINES ADDRESSED TO JOANNA H. FROM GWERNDWFFNANT IN JUNE 1826\n\nBY DOROTHY WORDSWORTH[395]\n\n    A twofold harmony is here;\n    I listen with the bodily ear,\n    But dull and cheerless is the sound\n    Contrasted with the heart's rebound.\n\n    Now at the close of fervid June,                                   5\n    Upon this breathless hazy noon,\n    I seek the deepest darkest shade\n    Within the covert of that glade,\n\n    Which you and I first named our own\n    When primroses were fully blown,                                  10\n    Oaks just were budding, and the grove\n    Rang with the gladdest songs of love.\n\n    Then did the Leader of the Band,\n    A gallant thrush, maintain his stand\n    Unshrouded from the eye of day                                    15\n    Upon yon Beech's topmost spray.\n\n    Within the selfsame lofty tree\n    A thrush sings now--perchance 'tis he--\n    The lusty joyous gallant bird,\n    Which on that April morn we heard.                                20\n\n    But oh! how different that voice\n    Which bade the very hills rejoice.\n    Through languid air, through leafy boughs\n    It falls, and can no echo rouse.\n\n    But on the workings of my heart                                   25\n    Doth memory act a busy part;\n    That jocund April morn lives there,\n    Its cheering sounds, its hues so fair.\n\n    Why mixes with remembrance blithe\n    What nothing but the restless scythe                              30\n    Of Death can utterly destroy,\n    A heaviness, a dull alloy?\n\n    Ah Friend! thy heart can answer why.\n    Even then I heaved a bitter sigh,\n    No word of sorrow did'st thou speak,                              35\n    But tears stole down thy tremulous cheek.\n\n    The wished for hour at length was come,\n    And thou had'st housed me in thy home,\n    On fair Gwerndwffnant's billowy hill,\n    Had'st led me to its crystal rill,                                40\n\n    And led me through the dingle deep\n    Up to the highest grassy steep,\n    The sheep walk where the snow-white lambs\n    Sported beside their quiet dams.\n\n    But thou wert destined to remove                                  45\n    From all these objects of thy love,\n    In this thy later day to roam\n    Far off, and seek another home.\n\n    _Now_ thou art gone--belike 'tis best--\n    And I remain a passing guest,                                     50\n    Yet for thy sake, beloved Friend,\n    When from this spot my way shall tend,\n\n    And if my timid soul might dare\n    To shape the future in its prayer,\n    Then fervently would I entreat                                    55\n    Our gracious God to guide thy feet\n    Back to the peaceful sunny cot,\n    Where thou so oft hast blessed thy lot.\n\n[395] I owe my knowledge of this and the following poem to the nephew\nof Mrs. Wordsworth, the Reverend Thomas Hutchinson of Kimbolton,\nHerefordshire, who wrote: \"The two following poems were found among his\npapers on the demise of Mr. Monkhouse--a first cousin of Wordsworth;\nthe first in the hand-writing of Wordsworth's wife, and the second of\nher daughter.\"--ED.\n\n\nHOLIDAY AT GWERNDWFFNANT, MAY 1826\n\nIRREGULAR STANZAS\n\nBY DOROTHY WORDSWORTH\n\n    You're here for one long vernal day;\n    We'll give it all to social play,\n    Though forty years have rolled away\n        Since we were young as you.\n\n    Then welcome to our spacious Hall!                                 5\n    Tom, Bessy, Mary, welcome all!\n    Though removed from busy men,\n    Yea lonesome as the foxes' den,\n    'Tis a place for joyance fit,\n    For frolic games and inborn wit.                                  10\n\n    'Twas nature built this hall of ours;\n    She shap'd the bank; she framed the bowers\n        That close it all around;\n    From her we hold our precious right,\n    And here, thro' live-long day and night,                          15\n        She rules with modest sway.\n\n    Our carpet is our verdant sod;\n    A richer one was never trod\n        In prince's proud saloon.\n    Purple, and gold, and spotless white,                             20\n    And quivering shade, and sunny light,\n        Blend with the emerald green.\n\n    She opened for the mountain brook\n        A gentle winding pebbly way\n    Into this placid secret nook.                                     25\n    Its bell-like tinkling--list, you hear--\n    'Tis never loud, yet always clear\n        As linnet's song in May.\n\n    And we have other music here:\n    A thousand songsters through the year                             30\n    Dwell in these happy groves,\n    And in this season of their loves\n    They join their voices with the doves\n        To raise a perfect harmony.\n\n    Thus spake I while with sober pace                                35\n    We slipped into that chosen place\n    And from the centre of our Hall\n        The young ones played around,\n    Then, like a flock of vigorous lambs,\n    That quit their grave and slow-paced dams                         40\n        To frolic o'er the mead,\n\n    That innocent fraternal troop\n    Erewhile a steady listening group\n        Off starting--Girl and Boy\n    In gamesome race with agile bound                                 45\n    Beat o'er and o'er the grassy ground\n        As if in motion--perfect joy.\n\n    So vanishes my idle scheme\n    That we through this long vernal day,\n        Associates in their youthful play,                            50\n    With them might travel in one stream.\n        Ah! how should we whose heads are grey?\n    Light was my heart, my spirits gay,\n        And fondly did I dream.\n\n    But now, recalled to consciousness,                               55\n    With weight of years, of changed estate,\n    Thought is not needed to repress\n    Those shapeless fancies of delight\n    That flash before my dazzled sight\n        Upon this joy-devoted morn.                                   60\n\n    Gladly we seek the stillest nook\n    Whence we may read, as in a book,\n    A history of years gone by,\n    Recalled to faded memory's eye\n    By bright reflection from the mirth                               65\n    Of youthful hearts--a transient second-birth\n        Of our own childish days.\n\n    Pleasure unbidden is their guide\n    Their leader--faithful to their side\n    Prompting each wayward feat of strength:                          70\n    The ambitious leap, the emulous race,\n    The startling shout, the mimic chase,\n    The simple half-disguis\u00e8d wile\n    Detected through the flattering smile.\n\n    A truce to this unbridled course                                  75\n    Doth intervene--no need of force.\n    We spread upon the flowery grass\n    The noontide meal--each lad and lass\n    Obeys the call--we form a Round,\n    And all are seated on the ground.                                 80\n\n    The sun's meridian hour is passed,\n    Again begins the emulous race,\n    Again succeeds the sportive chase.\n    And thus was spent that vernal day,\n    Till twilight checked the noisy play;                             85\n    Then did they feel a languor spread\n    Over their limbs, the beating tread\n    Was stilled--the busy throbbing heart--\n        And silently we all depart.\n\n    The shelter of our rustic cot                                     90\n    Receives us, and we envy not\n    The palace, or the stately dome;\n    But wish that _all_ had such a home.\n    Each child repeats his nightly prayer\n    That God may bless their parents' care                            95\n    To guide them in the way of truth\n    Through helpless childhood, giddy youth.\n\n    The closing hymn of cheerful praise\n    Doth yet again their spirits raise;\n    But 'tis not now a thoughtless joy.                              100\n    For tender parents, loving friends,\n    And all the gifts God's blessing sends,\n    Feelingly do they bless his name.\n\n    That homage paid, the young retire\n    With no unsatisfied desire;                                      105\n    Theirs is one long, one steady sleep,\n    Till the sun, tip-toe on the steep\n    In front of our beloved cot,\n    Casts on the walls her brightest beams.\n    Within, a startling lustre streams.                              110\n    They all awaken suddenly;\n    As at the touch of magic skill,\n    Or, as the pilgrim, at the bell\n        That summons him to matin-prayer.\n\n    And is it sorrow that they feel?                                 115\n    Nay! call it not by such a name,\n    The stroke of sadness that doth steal\n    With rapid motion through their hearts,\n    When comes the thought that yesterday\n    With all its joys is passed away,                                120\n        The long expected happy day.\n\n    An instant--and all sadness goes;\n    Nor brighter looks the half-blown rose\n    Than does the countenance of each child\n    Whether of ardent soul or mild.                                  125\n    The hour was fixed--they are prepared--\n    And homeward now they must depart,\n    And after many a brisk adieu,\n    On pony trim, and fleet of limb,\n    Their bustling journey they pursue.                              130\n\n    The fair-hair'd gentle quiet maid,\n    And she who is of daring mood,\n    The valiant and the timid Boy\n    Alike are ranged to hardihood;\n    And wheresoe'er the troop appear                                 135\n    They scatter smiles, a hearty cheer\n    Comes from both old and young,\n    And blessings fall from many a tongue.\n\n    They reach the dear paternal roof,\n    Nor dread a cold or stern reproof,                               140\n    While they pour forth the history\n    Of three days' mirth and revelry.\n    Ah! Children, happy is your lot,\n    Still bound together in one knot\n    Beneath your tender mother's eye!                                145\n    Too soon these blessed days shall fly,\n    And brothers shall from sisters part;\n    And, trust me, whatsoe'er your doom,\n    Whate'er betide through years to come,\n    The punctual pleasures of your home                              150\n    Shall linger in your thoughts,\n    More clear than any future hope\n    Though fancy take her freest scope.\n    For oh! too soon your hearts shall own\n    The past is all that is your own.                                155\n\n    And every day of _festival_\n    Gratefully shall ye then recal,\n    Less for their own sakes than for this,\n    That each shall be a resting-place\n    For memory, and divide the race                                  160\n    Of childhood's smooth and happy years,\n    Thus lengthening out that term of life\n    Which governed by your parents' care\n    Is free from sorrow and from strife.\n\n\nCOMPOSED WHEN A PROBABILITY EXISTED OF OUR BEING OBLIGED TO QUIT RYDAL\nMOUNT AS A RESIDENCE\n\nThe following lines were written by Wordsworth in 1826. He never\npublished them. They were the result of a slight disagreement between\nthe Wordsworth family and the Le Flemings, which led the former to fear\nthat they might have to \"quit Rydal Mount as a residence.\" It was an\ninsignificant difference, and the Wordsworths did not leave their home.\nThe only thing worthy of record, in connection with the matter, is that\nthe fear of being dispossessed led the poet to write what follows.--ED.\n\n    The doubt to which a wavering hope had clung\n    Is fled; we must depart, willing or not;\n    Sky-piercing Hills! must bid farewell to you\n    And all that ye look down upon with pride,\n    With tenderness, embosom; to your paths,                           5\n    And pleasant dwellings, to familiar trees\n    And wild-flowers known as well as if our hands\n    Had tended them: and O pellucid Spring!\n    Unheard of, save in one small hamlet, here\n    Not undistinguished, for of wells that ooze                       10\n    Or founts that gurgle from yon craggy steep,\n    Their common sire, thou only bear'st his name.\n    Insensibly the foretaste of this parting\n    Hath ruled my steps, and seals me to thy side,\n    Mindful that thou (ah! wherefore by my Muse                       15\n    So long unthanked) hast cheered a simple board\n    With beverage pure as ever fixed the choice\n    Of hermit, dubious where to scoop his cell;\n    Which Persian kings might envy; and thy meek\n    And gentle aspect oft has ministered                              20\n    To finer uses. They for me must cease;\n    Days will pass on, the year, if years be given,\n    Fade,--and the moralising mind derive\n    No lessons from the presence of a Power\n    By the inconstant nature we inherit                               25\n    Unmatched in delicate beneficence;\n    For neither unremitting rains avail\n    To swell thee into voice; nor longest drought\n    Thy bounty stints, nor can thy beauty mar,\n    Beauty not therefore wanting change to stir                       30\n    The fancy pleased by spectacles unlooked for.\n      Nor yet, perchance, translucent Spring, had tolled\n    The Norman curfew bell when human hands\n    First offered help that the deficient rock\n    Might overarch thee, from pernicious heat                         35\n    Defended, and appropriate to man's need.\n    Such ties will not be severed: but, when we\n    Are gone, what summer loiterer will regard,\n    Inquisitive, thy countenance, will peruse,\n    Pleased to detect the dimpling stir of life,                      40\n    The breathing faculty with which thou yield'st\n    (Tho' a mere goblet to the careless eye)\n    Boons inexhaustible? Who, hurrying on\n    With a step quickened by November's cold,\n    Shall pause, the skill admiring that can work                     45\n    Upon thy chance-defilements--withered twigs\n    That, lodged within thy crystal depths, seem bright,\n    As if they from a silver tree had fallen--\n    And oaken leaves that, driven by whirling blasts,\n    Sunk down, and lay immersed in dead repose                        50\n    For Time's invisible tooth to prey upon\n    Unsightly objects and uncoveted,\n    Till thou with crystal bead-drops didst encrust\n    Their skeletons, turned to brilliant ornaments.\n    But, from thy bosom, should some venturous[396] hand              55\n    Abstract those gleaming relics, and uplift them,\n    However gently, toward the vulgar air,\n    At once their tender brightness disappears,\n    Leaving the intermeddler to upbraid\n    His folly. Thus (I feel it while I speak),                        60\n    Thus, with the fibres of these thoughts it fares;\n    And oh! how much, of all that love creates\n    Or beautifies, like changes undergo,\n    Suffers like loss when drawn out of the soul,\n    Its silent laboratory! Words should say                           65\n    (Could they depict the marvels of thy cell)\n    How often I have marked a plumy fern\n    From the live rock with grace inimitable\n    Bending its apex toward a paler self\n    Reflected all in perfect lineaments--                             70\n    Shadow and substance kissing point to point\n    In mutual stillness; or, if some faint breeze\n    Entering the cell gave restlessness to one,\n    The other, glassed in thy unruffled breast,\n    Partook of every motion, met, retired,                            75\n    And met again. Such playful sympathy,\n    Such delicate caress as in the shape\n    Of this green plant had aptly recompensed\n    For baffled lips and disappointed arms\n    And hopeless pangs, the spirit of that youth,                     80\n    The fair Narcissus by some pitying God\n    Changed to a crimson flower; when he, whose pride\n    Provoked a retribution too severe,\n    Had pined; upon his watery duplicate\n    Wasting that love the nymphs implored in vain.                    85\n      Thus while my Fancy wanders, thou, clear Spring,\n    Moved (shall I say?) like a dear friend who meets\n    A parting moment with her loveliest look,\n    And seemingly her happiest, look so fair\n    It frustrates its own purpose, and recalls                        90\n    The grieved one whom it meant to send away--\n    Dost tempt me by disclosures exquisite\n    To linger, bending over thee: for now,\n    What witchcraft, mild enchantress, may with thee\n    Compare! thy earthly bed a moment past                            95\n    Palpable to sight as the dry ground,\n    Eludes perception, not by rippling air\n    Concealed, nor through effect of some impure\n    Upstirring; but, abstracted by a charm\n    Of my own cunning, earth mysteriously                            100\n    From under thee hath vanished, and slant beams\n    The silent inquest of a western sun,\n    Assisting, lucid well-spring! Thou revealest\n    Communion without check of herbs and flowers,\n    And the vault's hoary sides to which they cling,                 105\n    Imaged in downward show; the flower, the Herbs,[397]\n    _These_ not of earthly texture, and the vault\n    Not _there_ diminutive, but through a scale\n    Of vision less and less distinct, descending\n    To gloom imperishable. So (if truths                             110\n    The highest condescend to be set forth\n    By processes minute), even so--when thought\n    Wins help from something greater than herself--\n    Is the firm basis of habitual sense\n    Supplanted, not for treacherous vacancy                          115\n    And blank dissociation from a world\n    We love, but that the residues of flesh,\n    Mirrored, yet not too strictly, may refine\n    To Spirit; for the idealising Soul\n    Time wears the features of Eternity;                             120\n    And Nature deepens into Nature's God.\n      Millions of kneeling Hindoos at this day\n    Bow to the watery element, adored\n    In their vast stream, and if an age hath been\n    (As books and haply votive altars vouch)                         125\n    When British floods were worshipped, some faint trace\n    Of that idolatry, through monkish rites\n    Transmitted far as living memory,\n    Might wait on thee, a silent monitor,\n    On thee, bright Spring, a bashful little one,                    130\n    Yet to the measure of thy promises\n    True, as the mightiest; upon thee, sequestered\n    For meditation, nor inopportune\n    For social interest such as I have shared.\n    Peace to the sober matron who shall dip                          135\n    Her pitcher here at early dawn, by me\n    No longer greeted--to the tottering sire,\n    For whom like service, now and then his choice,\n    Relieves the tedious holiday of age--\n    Thoughts raised above the Earth while here he sits               140\n    Feeding on sunshine--to the blushing girl\n    Who here forgets her errand, nothing loth\n    To be waylaid by her betrothed, peace\n    And pleasure sobered down to happiness!\n      But should these hills be ranged by one whose soul             145\n    Scorning love-whispers shrinks from love itself\n    As Fancy's snare for female vanity,\n    Here may the aspirant find a trysting-place\n    For loftier intercourse. The Muses crowned\n    With wreaths that have not faded to this hour                    150\n    Sprung from high Jove, of sage Mnemosyne\n    Enamoured, so the fable runs; but they\n    Certes were self-taught damsels, scattered births\n    Of many a Grecian vale, who sought not praise,\n    And, heedless even of listeners, warbled out                     155\n    Their own emotions given to mountain air\n    In notes which mountain echoes would take up\n    Boldly and bear away to softer life;\n    Hence deified as sisters they were bound\n    Together in a never-dying choir;                                 160\n    Who with their Hippocrene and grottoed fount\n    Of Castaly, attest that Woman's heart\n    Was in the limpid age of this stained world\n    The most assured seat of [        ]\n    And new-born waters, deemed the happiest source                  165\n    Of inspiration for the conscious lyre.\n      Lured by the crystal element in times\n    Stormy and fierce, the Maid of Arc withdrew\n    From human converse to frequent alone\n    The Fountain of the Fairies. What to her,                        170\n    Smooth summer dreams, old favours of the place.\n    Pageant and revels of blithe elves--to her\n    Whose country groan'd under a foreign scourge?\n    She pondered murmurs that attuned her ear\n    For the reception of far other sounds                            175\n    Than their too happy minstrelsy,--a Voice\n    Reached her with supernatural mandate charged\n    More awful than the chambers of dark earth\n    Have virtue to send forth. Upon the marge\n    Of the benignant fountain, while she stood                       180\n    Gazing intensely, the translucent lymph\n    Darkened beneath the shadow of her thoughts\n    As if swift clouds swept o'er it, or caught\n    War's tincture, 'mid the forest green and still,\n    Turned into blood before her heart-sick eye.                     185\n    Erelong, forsaking all her natural haunts,\n    All her accustomed offices and cares\n    Relinquishing, but treasuring every law\n    And grace of feminine humanity,\n    The chosen Rustic urged a warlike steed                          190\n    Toward the beleaguered city, in the might\n    Of prophecy, accoutred to fulfil,\n    At the sword's point, visions conceived in love.\n      The cloud of rooks descending thro' mid air\n    Softens its evening uproar towards a close[398]                  195\n    Near and more near; for this protracted strain\n    A warning not unwelcome. Fare thee well!\n    Emblem of equanimity and truth,\n    Farewell!--if thy composure be not ours,\n    Yet as thou still, when we are gone, wilt keep                   200\n    Thy living chaplet of fresh flowers and fern,\n    Cherished in shade tho' peeped at[399] by the sun;\n    So shall our bosoms feel a covert growth\n    Of grateful recollections, tribute due\n    To thy obscure and modest attributes                             205\n    To thee, dear Spring,[400] and all-sustaining Heaven!\n\n[396] The MS. has a second reading, \"covetous hand.\"--ED.\n\n[397] In MS. also \"its herbs.\"--ED.\n\n[398]\n\n    \u2026 to a close\n\n    From a MS. copied at Rydal by Professor Reed in 1854.\n\n[399]\n\n    \u2026 pecked at \u2026\n\n    From a MS. copied at Rydal by Professor Reed in 1854.\n\n[400]\n\n    \u2026 clear Spring \u2026\n\n    From a MS. copied at Rydal by Professor Reed in 1854.\n\n\n\"I, WHOSE PRETTY VOICE YOU HEAR\"\n\nThese lines were written for Miss Fanny Barlow of Middlethorpe Hall,\nYork. She was first married to the Rev. E. Trafford Leigh, and\nafterwards to Dr. Eason Wilkinson of Manchester.--ED.\n\n    I, whose pretty Voice you hear,\n    Lady (you will think it queer),\n    Have a Mother, once a Statue,\n    I, thus boldly looking at you,\n    Do the name of Paphus bear,                                        5\n    Fam'd Pygmalion's Son and Heir,\n    By that wondrous marble wife\n    That from Venus took her life.\n    Cupid's Nephew then am I,\n    Nor unskill'd his darts to ply;                                   10\n    But from Him I crav'd no warrant,\n    Coming thus to seek my Parent;\n    Not equipp'd with bow and quiver\n    Her by menace to deliver,\n    But resolv'd with filial care                                     15\n    Her captivity to share.\n    Hence, while on your toilet, She\n    Is doom'd a Pincushion to be,\n    By her side I'll take my place,\n    As a humble Needle-case;                                          20\n    Furnish'd too with dainty thread,\n    For a Sempstress thorough-bred.\n    Then let both be kindly treated,\n    Till the Term, for which She's fated\n    Durance to sustain, be over;                                      25\n    So will I ensure a Lover\n    Lady! to your heart's content;\n    But on harshness are you bent\n    Bitterly shall you repent,\n    When to Cyprus back I go                                          30\n    And take up my Uncle's bow.\n\n    _Composed_, and in part transcribed, for Fanny Barlow, by her\n                         affectionate Friend\n\n                                                  WM. WORDSWORTH.\n\n    RYDAL MOUNT,\n        _Shortest Day, 1826_.\n\n\n\n\n1827\n\n\nTO MY NIECE DORA\n\nBY DOROTHY WORDSWORTH\n\nThe following lines were written in Dora Wordsworth's \"Album,\" in which\nSir Walter Scott also wrote some verses.--ED.\n\n    Confiding hopes of youthful hearts,\n    And each bright visionary scheme,\n    Shall here remain in vivid hues\n    The hues of a celestial dream.\n\n    The farewell of the laurelled Knight                               5\n    Traced by a brave but tremulous hand,\n    Pledge of his truth and loyalty\n    Thro' changeful years, unchanged shall stand.\n\n    But why should I inscribe my name,\n    No Poet I--no longer young?                                       10\n    The ambition of a loving heart\n    Makes garrulous the tongue.\n\n    Memorials of thy aged Friend\n    Dora thou dost not need;\n    And when the cold earth covers her                                15\n    No flattery shall she heed.\n\n    Yet still a lurking wish prevails\n    That when from life we all have passed\n    The friends who loved thy Father's name\n    On her's a thought may cast.                                      20\n\n                             DOROTHY WORDSWORTH.\n\n    _January 1827._\n\n\n\n\n1829\n\n\n\"MY LORD AND LADY DARLINGTON\"\n\nThese lines were written by Wordsworth, after reading a sentence in\nthe Stranger's Book at \"The Station,\"--not a railway station!--on\nthe western side of Windermere lake, opposite Bowness. Their poetic\nmerit is slight, but they illustrate the honesty and directness of\nthe writer's mind. The Stranger's Book at \"The Station\" contained the\nfollowing:--\n\n    \"Lord and Lady Darlington, Lady Vane, Miss Taylor, and Captain\n    Stamp pronounce this Lake superior to Lac de Gen\u00e8ve, Lago\n    de Como, Lago Maggiore, L'Eau de Zurich, Loch Lomond, Loch\n    Katerine, or the Lakes of Killarney.\"-ED.\n\n    My Lord and Lady Darlington,\n    I would not speak in snarling-tone;\n    Nor, to you, good Lady Vane,\n    Would I give one moment's pain;\n    Nor Miss Taylor, Captain Stamp,                                    5\n    Would I your flights of _memory_ cramp.\n    Yet, having spent a summer's day\n    On the green margin of Loch Tay,\n    And doubled (prospect ever bettering)\n    The mazy reaches of Loch Katerine,                                10\n    And more than once been free at Luss,\n    Loch Lomond's beauties to discuss,\n    And wished, at least, to hear the blarney\n    Of the sly boatmen of Killarney,\n    And dipped my hand in dancing wave                                15\n    Of Eau de Zurich, Lac Gen\u00e8ve,\n    And bowed to many a major domo\n    On stately terraces of Como,\n    And seen the Simplon's forehead hoary,\n    Reclined on Lago Maggiore                                         20\n    At breathless eventide at rest\n    On the broad water's placid breast,\n    I, not insensible, Heaven knows,\n    To all the charms this Station shows,\n    Must tell you, Captain, Lord, and Ladies--                        25\n    For honest worth one poet's trade is--\n    That your praise appears to me\n    Folly's own hyperbole.\n\n\n\n\n1833\n\n\nTO THE UTILITARIANS\n\nThese lines were written and sent in a letter to Henry Crabb Robinson,\ndated 5th May 1833.--ED.\n\n    Avaunt this \u0153conomic rage!\n    What would it bring?--an iron age,\n    Where Fact with heartless search explored\n    Shall be Imagination's Lord,\n    And sway with absolute controul                                    5\n    The god-like Functions of the Soul.\n    Not _thus_ can knowledge elevate\n    Our Nature from her fallen state.\n    With sober Reason Faith unites\n    To vindicate the ideal rights                                     10\n    Of human-kind--the tone agreeing\n    Of objects with internal seeing,\n    Of effort with the end of Being.\n\nWordsworth added, in the letter to Robinson, \"Is the above\nintelligible? I fear not! I know, however, my own meaning, and that's\nenough for Manuscripts.\"--ED.\n\n\n\n\n1835\n\n\n\"THRONED IN THE SUN'S DESCENDING CAR\"\n\nThese lines were placed by Wordsworth amongst the \"Evening Voluntaries\"\nin the two editions of _Yarrow Revisited and other Poems_ (1835, 1836);\nbut they were never afterwards reprinted in his life-time.--ED.\n\nFor printing the following Piece, some reason should be given, as not\na word of it is original: it is simply a fine stanza of Akenside,[401]\nconnected with a still finer from Beattie[402]by a couplet of\nThomson.[403] This practice, in which the author sometimes indulges, of\nlinking together, in his own mind, favourite passages from different\nauthors, seemed in itself unobjectionable; but, as the _publishing_\nsuch compilations might lead to confusion in literature, he should\ndeem himself inexcusable in giving this specimen, were it not from\na hope that it might open to others a harmless source of _private_\ngratification.--W. W. 1835.\n\n    Throned in the Sun's descending car,\n    What Power unseen diffuses far\n    This tenderness of mind?\n    What Genius smiles on yonder flood?\n    What God in whispers from the wood                                 5\n    Bids every thought be kind?\n\n    O ever-pleasing solitude,\n    Companion of the wise and good.\n\n    Thy shades, thy silence, now be mine\n      Thy charms my only theme;                                       10\n\n    Why haunt the hollow cliff whose Pine\n      Waves o'er the gloomy stream;\n    Whence the scared Owl on pinions grey\n      Breaks from the rustling boughs,\n    And down the lone vale sails away                                 15\n      To more profound repose!\n\n[401] See his Ode V., _Against Suspicion_, stanza viii.--ED.\n\n[402] See his poem, _Retirement_, 1758.--ED.\n\n[403] See his _Hymn on Solitude_, which begins, \"Hail, ever-pleasing\nSolitude!\"--ED.\n\n\n\"AND OH! DEAR SOOTHER OF THE PENSIVE BREAST\"\n\nThe following ten lines were written by Wordsworth in a copy of his\nworks, after the lines _To the Moon_ (Rydal) 1835. They may have been\nintended as a possible sequel to them, or to the lines _To the Moon,\ncomposed by the Seaside--on the coast of Cumberland_ (1835).--ED.\n\n    And oh! dear soother of the pensive breast,\n    Let homelier words without offence attest\n    How where on random topics as they hit\n    The moments' humour, rough Tars spend their wit.\n    Thy changes, which to wiser Spirits seem                           5\n    Dark as a riddle, prove a favourite theme;\n    Thy motions, intricate and manifold,\n    Oft help to make bold fancy's flight more bold;\n    Beget strange themes; and to freaks give birth\n    Of speech as wild as ever heightened mirth.                       10\n\n\n\n\n1836\n\n\n\"SAID RED-RIBBONED EVANS\"\n\nOn the 26th of March 1836, Wordsworth sent the following lines to Henry\nCrabb Robinson; written, he tells him, \"immediately on reading Evans's\nmodest self-defence speech the other day.\" George de Lacy Evans was\nradical member of Parliament for Westminster. \"In 1835, he took command\nof the British Legion raised for the service of the Queen Regent of\nSpain against Don Carlos.\" (Professor Dowden.)--ED.\n\n    Said red-ribboned Evans:\n    \"My legions in Spain\n    Were at sixes and sevens;\n    Now they're famished or slain:\n    But no fault of mine,                                              5\n    For, like brave Philip Sidney,\n    In campaigning I shine,\n    A true knight of his kidney.\n    Sound flogging and fighting\n    No chief, on my troth,                                            10\n    E'er took such delight in\n    As I in them both.\n    Fontarabbia can tell\n    How my eyes watched the foe,\n    Hernani knows well                                                15\n    That our feet were not slow;\n    Our hospitals, too,\n    They are matchless in story;\n    Where her thousands Fate slew,\n    All panting for glory.\"                                           20\n    Alas for this Hero!\n    His fame touched the skies,\n    Then fell below zero,\n    Never, never to rise!\n    For him to Westminster                                            25\n    Did Prudence convey,\n    There safe as a Spinster\n    The Patriot to play.\n    But why be so glad on\n    His feats or his fall?                                            30\n    He's got his red ribbon,\n    And laughs at us all.\n\n\n\n\n1837\n\n\nON AN EVENT IN COL. EVANS'S REDOUBTED PERFORMANCES IN SPAIN\n\nMrs. Wordsworth sent this to Henry Crabb Robinson in 1837, \"to show\nyou that _we_ can write an Epigram--we _do not say_ a good one.\" She\nthen quoted it, and added, \"The Producer thinks it not amiss, as being\nmurmured between sleep and awake over the fire, while thinking of you\nlast night!\"--Ed.\n\n    The Ball whizzed by,--it grazed his ear,\n      And whispered as it flew,\n    \"I only touch--not take--don't fear,\n    For both, my honest Buccaneer!\n      Are to the Pillory due.\"\n\n\n\n\n1838\n\n\n\"WOULDST THOU BE GATHERED TO CHRIST'S CHOSEN FLOCK\"\n\nThe following lines were cut on the face of a rock at Rydal Mount in\n1838. There, they still remain.--ED.\n\n    Wouldst thou be gathered to Christ's chosen flock,\n    Shun the broad way too easily explored,\n    And let thy path be hewn out of the Rock,\n    The living Rock of God's eternal Word.\n\n\nPROTEST AGAINST THE BALLOT, 1838[404]\n\nComposed 1838.--Published 1838\n\n    Forth rushed, from Envy sprung and Self-conceit,\n    A Power misnamed the SPIRIT OF REFORM,\n    And through the astonished Island swept in storm,\n    Threatening to lay all Orders at her feet\n    That crossed her way. Now stoops she to entreat                    5\n    Licence to hide at intervals her head,\n    Where she may work, safe, undisquieted,\n    In a close Box, covert for Justice meet.\n    St. George of England! keep a watchful eye\n    Fixed on the Suitor; frustrate her request--                      10\n    Stifle her hope; for, if the State comply,\n    From such Pandorian gift may come a Pest\n    Worse than the Dragon that bowed low his crest,\n    Pierced by thy spear in glorious victory.\n\n[404] In his notes to the volume of Collected Sonnets (1838),\nWordsworth writes:--\"'_Protest against the Ballot._' Having in this\nnotice alluded only in general terms to the mischief which, in my\nopinion, the Ballot would bring along with it, without especially\nbranding its immoral and antisocial tendency (for which no political\nadvantages, were they a thousand times greater than those presumed\nupon, could be a compensation), I have been impelled to subjoin a\nreprobation of it upon that score. In no part of my writings have\nI mentioned the name of any contemporary, that of Buonaparte only\nexcepted, but for the purpose of eulogy; and therefore, as in the\nconcluding verse of what follows, there is a deviation from this rule\n(for the blank will be easily filled up) I have excluded the sonnet\nfrom the body of the collection, and placed it here as a public record\nof my detestation, both as a man and a citizen, of the proposed\ncontrivance.\"\n\nThen follows the sonnet beginning--\n\n    Said Secrecy to Cowardice and Fraud.\n\nED.\n\n\n\"SAID SECRECY TO COWARDICE AND FRAUD\"\n\nComposed, probably, in 1838.--Published 1838[405]\n\n    Said Secrecy to Cowardice and Fraud,\n    Falsehood and Treachery, in close council met,\n    Deep under ground, in Pluto's cabinet,\n    \"The frost of England's pride will soon be thawed;\n    Hooded the open brow that overawed                                 5\n    Our schemes; the faith and honour, never yet\n    By us with hope encountered, be upset;--\n    For once I burst my bands, and cry, applaud!\"\n    Then whispered she, \"The Bill is carrying out!\"\n    They heard, and, starting up, the Brood of Night                  10\n    Clapped hands, and shook with glee their matted locks;\n    All Powers and Places that abhor the light\n    Joined in the transport, echoed back their shout,\n    Hurrah for ----, hugging his Ballot-box![406]\n\n[405] This was first published in a note to the sonnet entitled\n_Protest against the Ballot_, in the volume of 1838. It was never\nrepublished by Wordsworth.\n\n[406] See the note to the previous sonnet. George Grote was the\nperson satirised. \"Since that time,\" adds Mr. Reed, in a note to his\nAmerican edition, \"Mr. Grote's political notoriety, as an advocate of\nthe ballot, has been merged in the high reputation he has acquired as\nprobably the most eminent modern historian of ancient Greece\"--ED.\n\n\nA POET TO HIS GRANDCHILD\n\n(SEQUEL TO THE FOREGOING)[407]\n\nPublished 1838\n\n    \"Son of my buried Son, while thus thy hand\n    Is clasping mine, it saddens me to think\n    How Want may press thee down, and with thee sink\n    Thy Children left unfit, through vain demand\n    Of culture, even to feel or understand                             5\n    My simplest Lay that to their memory\n    May cling;--hard fate! which haply need not be\n    Did Justice mould the Statutes of the Land.\n    A Book time-cherished and an honoured name\n    Are high rewards; but bound they Nature's claim                   10\n    Or Reason's? No--hopes spun in timid line\n    From out the bosom of a modest home\n    Extend through unambitious years to come,\n    My careless Little-one, for thee and thine!\"[408][409]\n\n[407] \"The foregoing\" was the Sonnet named _A Plea for Authors, May\n1838_.--ED.\n\n[408] 1836.\n\n    Son of my buried Son, whose tiny hand\n    Thus clings to mine, it {saddens} me to think\n                            {troubles}\n    That thou pressed down by poverty mayst sink\n    Even till thy children shall in vain demand\n    {Culture and neither feel nor} understand\n    {Culture required to feel and}\n    {My simplest lay that to their memory}\n    {My least recondite lay, which memory}\n    {Perchance may cleave}; hard fate, which need not be\n    {May keep in trust   }\n    Did justice mould the statutes of the land.\n    {A book time-cherished} and an honoured name\n    {A cherished volume   }\n    Are high rewards, but bound not {Reason's} claim.\n                                    {Nature's}\n    No--hopes {in fond hereditary line    }\n              {and wishes in a living line}\n    Spun from the bosom of a modest home\n    Extend thro' unambitious years to come,\n    My careless Little-one, for thee and thine!\n\n    MS.\n\n[409] The author of an animated article, printed in the _Law Magazine_,\nin favour of the principle of Serjeant Talfourd's Copyright Bill,\nprecedes me in the public expression of this feeling; which had been\nforced too often upon my own mind, by remembering how few descendants\nof men eminent in literature are even known to exist.--W.W. 1838.\n\nThis sonnet was not addressed to any grandson of the Poet's.--ED.\n\n\n\n\n1840\n\n\nON A PORTRAIT OF I.F., PAINTED BY MARGARET GILLIES[410]\n\nComposed 1840.--Published 1850\n\n    We gaze--nor grieve to think that we must die,\n    But that the precious love this friend hath sown\n    Within our hearts, the love whose flower hath blown\n    Bright as if heaven were ever in its eye,\n    Will pass so soon from human memory;                               5\n    And not by strangers to our blood alone,\n    But by our best descendants be unknown,\n    Unthought of--this may surely claim a sigh.\n    Yet, bless\u00e8d Art, we yield not to dejection:\n    Thou against Time so feelingly dost strive;                       10\n    Where'er, preserved in this most true reflection,\n    An image of her soul is kept alive,\n    Some lingering fragrance of the pure affection,\n    Whose flower with us will vanish, must survive.\n\n                                     WILLIAM WORDSWORTH.\n\n    RYDAL MOUNT,\n        _New Year's Day, 1840_.\n\n[410] See the note to the next sonnet.--ED.\n\n\nTO I.F.[411]\n\nComposed 1840.--Published 1850\n\n    The star which comes at close of day to shine\n    More heavenly bright than when it leads the morn,\n    Is friendship's emblem,[412] whether the forlorn\n    She visiteth, or, shedding light benign\n    Through shades that solemnize Life's calm decline,                 5\n    Doth make the happy happier. This have we\n    Learnt, Isabel, from thy society,\n    Which now we too unwillingly resign\n    Though for brief absence. But farewell! the page\n    Glimmers before my sight through thankful tears,                  10\n    Such as start forth, not seldom, to approve\n    Our truth, when we, old yet unchill'd by age,\n    Call thee, though known but for a few fleet years,\n    The heart-affianced sister of our love!\n\n                                   WILLIAM WORDSWORTH.\n\n    RYDAL MOUNT,\n        _Feb. 1840_.\n\n[411] This and the preceding sonnet, beginning \"We gaze--nor grieve\nto think that we must die,\" were addressed to Miss Fenwick, to whom\nwe owe the invaluable \"Fenwick Notes.\" Were it not that the date is\nvery minutely given, I would believe that they belong to 1841, as Miss\nGillies told me she resided at Rydal Mount in that year, when she\npainted Mrs. Wordsworth's portrait.--ED.\n\n[412] 1850.\n\n    Bright is the star which comes at eve to shine\n    More heavenly bright than when it leads the morn,\n    And such is Friendship, whether the forlorn, etc.\n\n    1840.\n\n\n\"OH BOUNTY WITHOUT MEASURE, WHILE THE GRACE\"\n\nIn his copy of the edition of 1845 at the close of the poem, _Animal\nTranquillity and Decay_ (1798) (see the \"Poem referring to the Period\nof Old Age,\" vol. i. p. 307), Henry Crabb Robinson wrote the following\nlines, sent to him by Wordsworth.--ED.\n\n    Oh Bounty without measure, while the Grace\n    Of Heaven doth in such wise from humblest springs\n    Pour pleasures forth, and solaces that trace\n    A mazy course along familiar things,\n    Well may our hearts have faith that blessings come                 5\n    Streaming from points above the starry sky,\n    With angels, when their own untroubled home\n    They leave, and speed on mighty embassy\n    To visit earthly chambers,--and for whom?\n    Yea, both for souls who God's forbearance try,                    10\n    And those that seek his help and for his mercy sigh.\n\n    _7th April 1840. My 70th Birthday._\n\n                                                    W.W.\n\n\n\n\n1842\n\n\nTHE EAGLE AND THE DOVE[413]\n\nThe following poem was contributed to, and printed in, a volume\nentitled \"_La Petite Chouannerie, ou Histoire d'un Coll\u00e8ge Breton sous\nl'Empire_. Par A. F. Rio. Londres: Moxon, Dover Street, 1842,\" pp. 62,\n63. The Hon. Mrs. Norton, Walter Savage Landor, and Monckton Milnes\n(Lord Houghton), were among the other English contributors to the\nvolume, the bulk of which is in French. It was printed at Paris, and\nnumbered 398 pages, including the title. It was a narrative of \"the\nromantic revolt of the royalist students of the college of Vannes in\n1815, and of their battles with the soldiers of the French Empire.\" (H.\nREED.)--ED.\n\nComposed (?).--Published 1842\n\n    Shade of Caractacus, if spirits love\n    The cause they fought for in their earthly home,\n    To see the Eagle ruffled by the Dove\n    May soothe thy memory of the chains of Rome.\n\n    These children claim thee for their sire; the breath               5\n    Of thy renown, from Cambrian mountains, fans\n    A flame within them that despises death,\n    And glorifies the truant youth of Vannes.\n\n    With thy own scorn of tyrants they advance,\n    But truth divine has sanctified their rage,                       10\n    A silver cross enchased with flowers of France\n    Their badge, attests the holy fight they wage.\n\n    The shrill defiance of the young crusade\n    Their veteran foes mock as an idle noise;\n    But unto Faith and Loyalty comes aid                              15\n    From Heaven, gigantic force to beardless boys.\n\n[413] In the volume from which the above is copied, the original French\nlines (commencing at p. 106) are printed side by side with Wordsworth's\ntranslation, which ends on p. 111, and closes the volume.--ED.\n\n\nGRACE DARLING[414]\n\nComposed 1842.--Published 1845\n\nWordsworth's lines on Grace Darling were printed privately, and\nanonymously, at Carlisle, before they were included in the 1845 edition\nof his works. A copy was sent to Mr. Dyce, and is preserved in the Dyce\nLibrary at South Kensington. Another was sent to Professor Reed (March\n27, 1843), with a letter, in which the following occurs: \"I threw it\noff two or three weeks ago, being in a great measure impelled to it\nby the desire I felt to do justice to the memory of a heroine, whose\nconduct presented, some time ago, a striking contrast to the inhumanity\nwith which our countrymen, shipwrecked lately upon the French coast,\nhave been treated.\"\n\nEdward Quillinan, writing on 25th March 1843, enclosed a copy, adding,\n\"Mr. Wordsworth desires me to send you the enclosed eulogy on Grace\nDarling, recently composed. He begs me to say that he wishes it kept\nout of the newspapers, as he has printed it only for some of his\nfriends, and his friends' friends more peculiarly interested in the\nsubject, for the present. Do not therefore give a copy to any one.\"\n\n\"Almost immediately after I had composed my tribute to the memory of\nGrace Darling, I learnt that the Queen and Queen Dowager had both just\nsubscribed towards the erection of a monument to record her heroism,\nupon the spot that witnessed it.\" (Wordsworth to Sir W. Gomm, March 24,\n1843.)--ED.\n\n    Among the dwellers in the silent fields\n    The natural heart is touched, and public way\n    And crowded streets resound with ballad strains,\n    Inspired by ONE whose very name bespeaks\n    Favour divine, exalting human love;                                5\n    Whom, since her birth on bleak Northumbria's coast,\n    Known unto few but prized as far as known,\n    A single Act endears to high and low\n    Through the whole land--to Manhood, moved in spite\n    Of the world's freezing cares--to generous Youth--                10\n    To Infancy, that lisps her praise--to Age\n    Whose eye reflects it, glistening through a tear\n    Of tremulous admiration. Such true fame\n    Awaits her _now_; but, verily, good deeds\n    Do no imperishable record find                                    15\n    Save in the rolls of heaven, where hers may live\n    A theme for angels, when they celebrate\n    The high-souled virtues which forgetful earth\n    Has witness'd. Oh! that winds and waves could speak\n    Of things which their united power called forth                   20\n    From the pure depths of her humanity!\n    A Maiden gentle, yet, at duty's call,\n    Firm and unflinching, as the Lighthouse reared\n    On the Island-rock, her lonely dwelling-place;\n    Or like the invincible Rock itself that braves,                   25\n    Age after age, the hostile elements,\n    As when it guarded holy Cuthbert's cell.[415]\n\n      All night the storm had raged, nor ceased, nor paused,\n    When, as day broke, the Maid, through misty air,\n    Espies far off a Wreck, amid the surf,                            30\n    Beating on one of those disastrous isles--\n    Half of a Vessel, half--no more; the rest\n    Had vanished, swallowed up with all that there\n    Had for the common safety striven in vain,\n    Or thither thronged for refuge.[416] With quick glance            35\n    Daughter and Sire through optic-glass discern,\n    Clinging about the remnant of this Ship,\n    Creatures--how precious in the Maiden's sight!\n    For whom, belike, the old Man grieves still more\n    Than for their fellow-sufferers engulfed                          40\n    Where every parting agony is hushed,\n    And hope and fear mix not in further strife.\n    \"But courage, Father! let us out to sea--\n    A few may yet be saved.\" The Daughter's words,\n    Her earnest tone, and look beaming with faith,                    45\n    Dispel the Father's doubts: nor do they lack\n    The noble-minded Mother's helping hand\n    To launch the boat; and with her blessing cheered,\n    And inwardly sustained by silent prayer,\n    Together they put forth, Father and Child!                        50\n    Each grasps an oar, and struggling on they go--\n    Rivals in effort; and, alike intent\n    Here to elude and there surmount, they watch\n    The billows lengthening, mutually crossed\n    And shattered, and re-gathering their might;                      55\n    As if the tumult, by the Almighty's will\n    Were, in the conscious sea, roused and prolonged,[417]\n    That woman's fortitude--so tried, so proved--\n    May brighten more and more!\n                                      True to the mark,\n    They stem the current of that perilous gorge,                     60\n    Their arms still strengthening with the strengthening heart,\n    Though danger, as the Wreck is near'd, becomes\n    More imminent. Not unseen do they approach;\n    And rapture, with varieties of fear\n    Incessantly conflicting, thrills the frames                       65\n    Of those who, in that dauntless energy,\n    Foretaste deliverance; but the least perturbed\n    Can scarcely trust his eyes, when he perceives\n    That of the pair--tossed on the waves to bring\n    Hope to the hopeless, to the dying, life--                        70\n    One is a Woman, a poor earthly sister,\n    Or, be the Visitant other than she seems,\n    A guardian Spirit sent from pitying Heaven,\n    In woman's shape. But why prolong the tale,\n    Casting weak words amid a host of thoughts                        75\n    Armed to repel them? Every hazard faced\n    And difficulty mastered, with resolve\n    That no one breathing should be left to perish,\n    This last remainder of the crew are all\n    Placed in the little boat, then o'er the deep                     80\n    Are safely borne, landed upon the beach,\n    And, in fulfilment of God's mercy, lodged\n    Within the sheltering Lighthouse.--Shout, ye Waves!\n    Send forth a song of triumph. Waves and Winds,\n    Exult in this deliverance wrought through faith                   85\n    In Him whose Providence your rage hath served![418]\n    Ye screaming Sea-mews, in the concert join!\n    And would that some immortal Voice--a Voice\n    Fitly attuned to all that gratitude\n    Breathes out from floor or couch, through pallid lips             90\n    Of the survivors--to the clouds might bear--\n    Blended with praise of that parental love,\n    Beneath whose watchful eye the Maiden grew\n    Pious and pure, modest and yet so brave,\n    Though young so wise, though meek so resolute--                   95\n    Might carry to the clouds and to the stars,\n    Yea, to celestial Choirs, GRACE DARLING'S name!\n\n[414] Grace Darling was the daughter of William Darling, the lighthouse\nkeeper on Longstone, one of the Farne Islands on the Northumbrian\ncoast. On the 7th of September 1838, the Forfarshire steamship was\nwrecked on these islands. At the instigation of his daughter, and\naccompanied by her, Darling went out in his lifeboat through the surf,\nto the wreck, and --by their united strength and daring--rescued the\nnine survivors.--ED.\n\n[415] St. Cuthbert of Durham, born about 635, was first a shepherd boy,\nthen a monk in the monastery of Melrose, and afterwards its prior. He\nleft Melrose for the island monastery of Lindisfarne; but desiring\nan austerer life than the monastic, he left Lindisfarne, and became\nan anchorite, in a hut which he built with his own hands, on one of\nthe Farne Islands. He was afterwards induced to accept the bishopric\nof Hexham, but soon exchanged it for the see in his old island home\nat Lindisfarne, and after two years there resigned his bishopric,\nreturning to his cell in Farne Island, where he died in 687. His\nremains were carried to Durham, and placed within a costly shrine.--ED.\n\n[416] Fifty-four persons had perished, before Grace Darling's lifeboat\nreached the wreck.--ED.\n\n[417] 1845.\n\n    As if the wrath and trouble of the sea\n    Were by the Almighty's sufferance prolonged,\n\n    In privately printed edition.\n\n[418] 1845.\n\nFor the last three lines, the privately printed edition has the single\none--\n\n    Pipe a glad song of triumph, ye fierce Winds.\n\n\n\"WHEN SEVERN'S SWEEPING FLOOD HAD OVERTHROWN\"\n\nComposed 23rd January 1842.--Published 1842\n\nIn 1842 a bazaar was held in Cardiff Castle to aid in the erection of\na Church, on the site of one which had been washed away by a flood in\nthe river Severn (and a consequent influx of waters into the estuary\nof the British Channel) two hundred years before. Wordsworth and James\nMontgomery were asked to write some verses, which might be printed and\nsold to assist the cause. They did so. The following was Wordsworth's\ncontribution.--ED.\n\n    When Severn's sweeping flood had overthrown\n    St. Mary's Church, the preacher then would cry:--\n    \"Thus, Christian people, God his might hath shown\n    That ye to him your love may testify;\n    Haste, and rebuild the pile.\"--But not a stone                     5\n    Resumed its place. Age after age went by,\n    And Heaven still lacked its due, though piety\n    In secret did, we trust, her loss bemoan.\n    But now her Spirit hath put forth its claim\n    In Power, and Poesy would lend her voice;                         10\n    Let the new Church be worthy of its aim,\n    That in its beauty Cardiff may rejoice!\n    Oh! in the past if cause there was for shame,\n    Let not our times halt in their better choice.\n\n    RYDAL MOUNT, _23rd Jan. 1842_.\n\n\nTHE PILLAR OF TRAJAN\n\nThe Fenwick note to _The Pillar of Trajan_ mentions that the author's\nson having declined to attempt to compete for the Oxford prize poem on\n\"The Pillar of Trajan,\" his father wrote it, to show him how the thing\nmight be done. This son--the Rev. John Wordsworth of Brigham--wrote\nLatin verse with considerable success; and as specimens of the poetic\nwork of Dorothy Wordsworth and of Sarah Hutchinson are included in\nthese volumes, the following _Epistola ad Patrem suum_, written at\nMadeira by John Wordsworth in 1844, may be reproduced.--ED.\n\n    I pete longinquas, non segnis Epistola, terras,\n      I pete, Rydaliae conscia saxa lyrae:\n    I pete qu\u00e0 valles rident, sylvaeque lacusque,\n      Quamvis Arctoo paen\u00e8 sub axe jacent.\n    Parvos quaere Lares, non aurea Tecta, poetae,                      5\n      Qui tamen ingenii sceptraque mentis habet.\n    Quid faciat genitor? valeatne, an cura senilis\n      Opprimat? Ista refer, filius ista rogat.\n    Scire velit, quare venias tu scripta _latine_?\n      Dic \"fugio linguam, magne poeta, tuam!                          10\n    Quem Regina jubet circumdare tempora lauro,\n      Quem ver\u00e8 vatem saecula nostra vocant.\"\n    Inde refer gressus responsaque tradita curae\n      Fida tuae, numeris in loca digna senis,\n    Haec ego tradiderim, majoribus ire per altum                      15\n      Nunc velis miserum me mea musa rapit.\n    Solvimus \u00e8 portu, navisque per aequora currit\n      Neptuni auxilio fluctifragisque rotis.\n    Neptunus videt attonitus, Neptunia conjux,\n      Omnis et aequorei nympha comata chori.                          20\n    Radimus Hispanum litus, loca saxea crebris\n      Gallorum belli nobilitata malis.\n    Haud mora, sunt visae Gades,[419] urbs fabula quondam,\n      Claraque ab Herculeo nomine, clara suo.\n    Hanc magnam cognovit Arabs, Romanus candem,                       25\n      Utraque gens illi vimque decusque tulit.\n    Hora brevis, fragilisque viris! similisque ruina\n      Viribus humanis omnia facta manet\n    Pulchra jaces, olim Carthaginis aemula magnae,\n      Nataque famosae non inhonesta Tyri!                             30\n    En! ratibus navale caret, nautis caret alnus,\n      Mercatorque fugit dives inane Forum.\n    Templa vacant pomp\u00e2, nitidisque theatra catervis,\n      Tristis et it foed\u00e2 foemina virque via.\n    Segnis in officiis, nec rectus ad aethera miles                   35\n      Pauperis et vestes, armaque juris habet.\n    Sic gens quaeque perit,[420] quando civilia bella\n      Viscera divellunt, jusque fidesque fugit.\n    Auspiciis laetam nostris lux proxima pandit\n      Te, Calpe[421] celsis imperiosa jugis.                          40\n    Urbs munimen habet nullo quassabile bello,\n      Claustrum Tyrrhenis, claustrum et Atlantis, aquis.\n    Undique nam vastae sustentant moenia rupes,\n      Quae torv\u00e8 in terras inque tuentur aquas.\n    Arteque sunt mir\u00e2 sectae per saxa cavernae,                       45\n      Atria sanguineo saeva sacrata Deo.\n    Urbs invicta tamen populis commercia tuta\n      Praebet, et in portus illicit inque Forum.\n    Hic Mercator adest Maurus cui rebus agendis.\n      Ah! nimis est cordi Punica prisca fides;                        50\n    Afer et \u00e8 mediis Libyae sitientis arenis,\n      Suetus in immund\u00e2 vivere barbarie;\n    Multus et aequoreis, ut quondam, Graius in undis,\n      Degener, antiquum sic probat ille genus;\n    Niliacae potator aquae, Judaeus, et omne                          55\n      Litus Tyrrhenum quos, et Atlantis, alit.\n    Hos qu\u00e0m dissimiles (linguae sive ora notentur)\n      Hos qu\u00e0m felices pace Britannus habet!\n    Anglia! dum pietas et honos, dum nota per orbem\n      Sit tibi in intacto pectore prisca fides;                       60\n    Dum pia cura tibi, magnos meruisse triumphos,\n      Justaque per populos jura tulisse feros;\n    Longinquas teneat tua vasta potentia terras,\n      Et maneat Calpe gloria magna Tibi!\n    Insula Atlanteis assurgit ab aequoris undis,                      65\n      Insula flammigero semper amata Deo,\n    Seu teneat celsi flagrantia signa Leonis,\n      Seu gyro Pisces interiore petat.\n    \"Hic ver assiduum atque alienis mensibus aestas,\"\n      Flavus et autumnus frugibus usque tumet.                        70\n    Non jacet Ionio felicior Insula ponto\n      Ulla, nec Eoi fluctibus oceani.\n    Vix, Madeira! tuum nunc refert dicere nomen,\n      Floribus, et Bacchi munere pingue solum.\n    Te vetus haud vanis cumulavit laudibus aetas,                     75\n      O fortunato conspicienda choro!\n    Haec nunc terra sinu nos detinet alma, proculque\n      A Patriae curis, anxietate domi.\n    Sic cepisse ferunt humanae oblivia curae\n      Quisquis Lethaeae pocula sumpsit aquae:                         80\n    Sic semota sequi studiisque odiisque docebas\n      Otia discipulos, docte Epicure, tuos.\n    Sed non ulla dies grato sine sole, nec ullo\n      Fruge carens hortus tempore,[422] fronde nemus;[423]\n    Nec levis ignotis oneratus odoribus aer,                          85\n      Quales doctus equum flectere novit Arabs;\n    Nec caecae quaecumque jacent sub rupe cavernae,[424]\n      Que\u00ees nunquam radiis Phoebus adire potest;\n    Nec currentis aquae strepitus,[425] nec saxa, petensque\n      Mons[426] excelsa suis sidera culminibus;                       90\n    Nec tranquilla quies, rerumque oblivia, ponti\n      Suadebunt iterum solicitare vias!\n    Rideat at quamvis haec vultu terra sereno,\n      Tabescit pravo gens malefida jugo:\n    Dum sedet heu! tristis morborum pallor in ore,                    95\n      Crebraque anhelanti pectore tussis inest.\n    Ambitus et luxus, totoque accersita mundo,\n      Que\u00ees omnis populus quoque sub axe peril;\n    Famae dira sitis, rerumque onerosa cupido,\n      Raptaque ab irato templa diesque Deo,                          100\n    Supplicium non lene suum, poenasque tulerunt;\n      Saep\u00e8 petis proprio, vir miser, ense latus!\n    Uxor adhuc aegros dilecta resuscitat artus;\n      Anxia cura suis, anxia cura mihi.\n    Altera quodque dies jam roboris attulit, illud                   105\n      Altera dura suis febribus abstulerit.\n    Aurea mens illi, mollique in pectore corda,\n      Et clarum long\u00e2 nobilitate genus.\n    Quanqu\u00e0m saepe trahunt Libycum non[427] aera sanum\n      (Gratia magna Dei), pignora nostra vigent.                     110\n    Iamque vale grandaeve Pater, grandaevaque Mater,\n      Tuque O dilecto conjuge laeta soror!\n    Quaeque pias nobis partes cognata ferebas,\n      Nomina vana cadunt, Tu mihi Mater eras;\n    Ingenioque mari, pietate ornata fideque,                         115\n      Sanguine nulla dom\u00fbs, semper amore, soror;\n    Tu quoque, care, vale, Frater, quamvis procul absis,\n      Per virides campos, qu\u00e0 petit aequor Eden.\n    Denique tota domus, cunctique valete propinqui,\n      Carmina plura mihi musa manusque negat.                        120\n\n    MADEIRAE, _MARTIIS CALENDIS_, 1844.\n\n[419] Cadiz.\n\n[420] Hispania hoc tempore bello civili divulsa fuit.\n\n[421] Gibraltar.\n\n[422] Sunt hibernis mensibus aurea mala.\n\n[423] Laureae sylvae sunt.\n\n[424] Antris abundat Insula.\n\n[425] Multos rivos natur\u00e2, mir\u00e2que humani ingenii arte constructos\ncontinet Madeira.\n\n[426] Pace Lusitanorum Insula nil nisi mons est, rectis culminibus mari\nconspicua.\n\n[427] Ventus ex Africa.--_Leste._\n\nSee also the _Carmen Maiis calendis compositum_, the _Carmen ad Maium\nmensem_, and the _Somnivaga_,--evidently by the same writer,--in the\nappendix to the second edition of _Yarrow Revisited_, 1836.--ED.\n\n\n\n\n1846\n\n\n\"DEIGN, SOVEREIGN MISTRESS! TO ACCEPT A LAY\"\n\nIn January 1846 Wordsworth sent a copy of his Poems to the Queen, for\nthe Royal Library at Windsor, and inscribed the following lines upon\nthe fly-leaf. For their republication I am indebted to the gracious\npermission of Her Majesty.--ED.\n\n    Deign, Sovereign Mistress![428] to accept a lay,\n      No Laureate offering of elaborate art;\n    But salutation taking its glad way\n      From deep recesses of a loyal heart.\n\n    Queen, Wife, and Mother! may All-judging Heaven                    5\n      Shower with a bounteous hand on Thee and Thine\n    Felicity that only can be given\n      On earth to goodness blest by grace divine.\n\n    Lady! devoutly honoured and beloved\n      Through every realm confided to thy sway;                       10\n    Mayst thou pursue thy course by God approved,\n      And He will teach thy people to obey.\n\n    As thou art wont, thy sovereignty adorn\n      With woman's gentleness, yet firm and staid;\n    So shall that earthly crown thy brows have worn                   15\n      Be changed for one whose glory cannot fade.\n\n    And now, by duty urged, I lay this Book\n      Before thy Majesty, in humble trust\n    That on its simplest pages thou wilt look\n      With a benign indulgence more than just.                        20\n\n    Nor wilt thou blame an aged Poet's prayer,\n    That issuing hence may steal into thy mind\n    Some solace under weight of royal care,\n    Or grief--the inheritance of humankind.\n\n    For know we not that from celestial spheres,                      25\n    When Time was young, an inspiration came\n    (Oh, were it mine!) to hallow saddest tears,\n    And help life onward in its noblest aim.\n\n                                              W.W.\n\n    _9th January 1846._\n\n[428] Compare the address presented by the Deputies of the Kingdom of\nItaly to Buonaparte, on Oct. 27, 1808, beginning, \"Deign, Sovereign\nMaster of all Things.\"--ED.\n\n\n\n\n1847\n\n\nODE, PERFORMED IN THE SENATE-HOUSE, CAMBRIDGE, ON THE 6TH OF JULY 1847,\nAT THE FIRST COMMENCEMENT AFTER THE INSTALLATION OF HIS ROYAL HIGHNESS\nTHE PRINCE ALBERT, CHANCELLOR OF THE UNIVERSITY.[429]\n\nINSTALLATION ODE\n\nComposed 1847.--Published 1847.\n\n             INTRODUCTION AND CHORUS\n\n    For thirst of power that Heaven disowns,\n    For temples, towers, and thrones,\n    Too long insulted by the Spoiler's shock,\n            Indignant Europe cast\n            Her stormy foe at last\n    To reap the whirlwind on a Libyan rock.\n\n                 SOLO.--TENOR\n\n            War is passion's basest game\n            Madly played to win a name;\n    Up starts some tyrant, Earth and Heaven to dare;\n            The servile million bow;\n    But will the lightning glance aside to spare\n            The Despot's laurelled brow?\n\n                   CHORUS\n\n    War is mercy, glory, fame,\n    Waged in Freedom's holy cause;\n    Freedom, such as Man may claim\n    Under God's restraining laws.\n    Such is Albion's fame and glory:\n    Let rescued Europe tell the story.\n\n       RECIT. (_accompanied_).--CONTRALTO\n\n    But lo, what sudden cloud has darkened all\n            The land as with a funeral pall?\n    The Rose of England suffers blight,\n    The flower has drooped, the Isle's delight,\n            Flower and bud together fall--\n    A Nation's hopes lie crushed in Claremont's desolate hall.\n\n                AIR.--SOPRANO\n\n    Time a chequered mantle wears;--\n          Earth awakes from wintry sleep;\n    Again the Tree a blossom bears,--\n        Cease, Britannia, cease to weep!\n    Hark to the peals on this bright May-morn!\n    They tell that your future Queen is born!\n\n            SOPRANO SOLO AND CHORUS\n\n        A Guardian Angel fluttered\n        Above the Babe, unseen;\n        One word he softly uttered--\n        It named the future Queen:\n    And a joyful cry through the Island rang,\n    As clear and bold as the trumpet's clang,\n      As bland as the reed of peace--\n        \"VICTORIA be her name!\"\n      For righteous triumphs are the base\n    Whereon Britannia rests her peaceful fame.\n\n                  QUARTETT\n\n    Time, in his mantle's sunniest fold,\n    Uplifted in his arms the child;\n    And, while the fearless Infant smiled,\n    Her happier destiny foretold:--\n      \"Infancy, by Wisdom mild,\n      Trained to health and artless beauty;\n      Youth, by Pleasure unbeguiled\n      From the lore of lofty duty;\n      Womanhood in pure renown,\n      Seated on her lineal throne:\n      Leaves of myrtle in her Crown,\n      Fresh with lustre all their own.\n      Love, the treasure worth possessing\n      More than all the world beside,\n      This shall be her choicest blessing,\n      Oft to royal hearts denied.\"\n\n         RECIT. (_accompanied_).--BASS\n\n    That eve, the Star of Brunswick shone\n        With stedfast ray benign\n    On Gotha's ducal roof, and on\n        The softly flowing Leine;\n    Nor failed to gild the spires of Bonn,\n        And glittered on the Rhine.--\n    Old Camus too on that prophetic night\n        Was conscious of the ray;\n    And his willows whispered in its light,\n        Not to the Zephyr's sway,\n    But with a Delphic life, in sight\n        Of this auspicious day:\n\n                  CHORUS\n\n    This day, when Granta hails her chosen Lord,\n      And proud of her award,\n      Confiding in the Star serene\n    Welcomes the Consort of a happy Queen.\n\n              AIR.--CONTRALTO\n\n    Prince, in these Collegiate bowers,\n    Where Science, leagued with holier truth,\n    Guards the sacred heart of youth,\n    Solemn monitors are ours.\n    These reverend aisles, these hallowed towers,\n    Raised by many a hand august,\n    Are haunted by majestic Powers,\n    The memories of the Wise and Just,\n    Who, faithful to a pious trust,\n    Here, in the Founder's spirit sought\n    To mould and stamp the ore of thought\n    In that bold form and impress high\n    That best betoken patriot loyalty.\n    Not in vain those Sages taught.--\n    True disciples, good as great,\n    Have pondered here their country's weal,\n    Weighed the Future by the Past,\n    Learned how social frames may last,\n    And how a Land may rule its fate\n    By constancy inviolate,\n      Though worlds to their foundations reel,\n    The sport of factious Hate or godless Zeal.\n\n                AIR.--BASS\n\n      Albert, in thy race we cherish\n      A Nation's strength that will not perish\n      While England's sceptered Line\n      True to the King of Kings is found;\n      Like that Wise[430] Ancestor of thine\n    Who threw the Saxon shield o'er Luther's life,\n    When first, above the yells of bigot strife,\n      The trumpet of the Living Word\n    Assumed a voice of deep portentous sound\n    From gladdened Elbe to startled Tiber heard.\n\n                 CHORUS\n\n    What shield more sublime\n    E'er was blazoned or sung?\n    And the PRINCE whom we greet\n    From its Hero is sprung.\n      Resound, resound the strain\n      That hails him for our own!\n    Again, again, and yet again;\n    For the Church, the State, the Throne!--\n    And that Presence fair and bright,\n    Ever blest wherever seen,\n    Who deigns to grace our festal rite,\n    The pride of the Islands, VICTORIA THE QUEEN!\n\n[429] This \"Ode\" was printed and sung at Cambridge on the occasion of\nthe installation of His Royal Highness Prince Albert as Chancellor of\nthe University. It was published in the newspapers of the following\nday, as \"written for the occasion by the Poet Laureate, by royal\ncommand.\"\n\nThere is no evidence, however, that Wordsworth wrote a single line\nof it. Dr. Cradock used to attribute the authorship to the poet's\nnephew, the late Bishop of Lincoln. It is much more likely that Edward\nQuillinan was the author of the whole, although Christopher Wordsworth\nmay have revised it. Mr. Aubrey de Vere wrote to me, November 12,\n1893, \"It was from Miss Fenwick that I heard that the Laureate poem\n(_Ode, etc._), was written by Quillinan, at Wordsworth's request, he\nhaving himself wholly failed in a reluctant attempt to write one. If\nhe _had_ written it, I doubt much whether he would ever have admitted\nit to a place among his works, for he did not hold 'Laureate Odes' in\nhonour, and had only taken the Laureateship on the condition that he\nwas to write none. Tennyson made the same condition: which could not,\nof course, interfere with either poet addressing lines to the Queen, if\nthey felt specially moved from within to do so.\"\n\nMiss Frances Arnold writes, \"Miss Quillinan was my authority for saying\nthat the Cambridge Ode had been written by her father, owing to the\ndeep depression in which Wordsworth then was.\"--ED.\n\n[430] Frederic the Wise, Elector of Saxony (1847).\n\n\nTO MISS SELLON\n\nThis sonnet exists, _in Wordsworth's handwriting_; but it is doubtful\nwhether it was written by him, or not. Possibly Mr. Quillinan wrote it.\nThe place, and the date of composition--given in MS.--are, \"Ambleside,\n22nd February, 1849.\" Miss Sellon was a relation of the late Count\nCavour.--ED.\n\n    The vestal priestess of a sisterhood who knows\n    No self, and whom the selfish scorn--\n    She seeks a wilderness of weed and thorn,\n    And, undiverted from the blessed mood\n    By keen reproach or blind ingratitude,                             5\n    A wreath she twines of blossoms lowly born--\n    An amaranthine crown of flowers forlorn--\n    And hangs her garland on the Holy Rood.\n    Sister of Mercy, bravely hast thou won\n    From men who winnow charity from Faith                            10\n    The Pharasaic sneer that treats as dross\n    The works by faith ordained. Pursue thy path,\n    Till, at the last, thou hear the voice--\"Well done,\n    Thou good and faithful servant of the Cross.\"\n\n\n\"THE WORSHIP OF THIS SABBATH MORN\"\n\nBY DOROTHY WORDSWORTH\n\nThese lines were published in _The Monthly Packet_, in July 1891, where\nthe following note is appended by Miss Christabel Coleridge:--\"Written\n_circa_ 1852-3, and given to Mrs. Derwent Coleridge.\" But Miss Edith\nColeridge, and Mr. E. H. Coleridge, tell me that they think they\n\"belong to an earlier period.\" Mr. Coleridge writes, \"I have heard Miss\nWordsworth repeat the lines now printed, seated in her arm-chair, on\nthe terrace at Rydal Mount.\"--ED.\n\n    The worship of this Sabbath morn,\n    How sweetly it begins!\n    With the full choral hymn of birds\n    Mingles no sad lament for sins.\n\n    Alas! my feet no more may join                                     5\n    The cheerful Sabbath train;\n    But if I inwardly lament,\n    Oh! may a will subdued all grief restrain.\n\n    No prisoner am I on this couch,\n    My mind is free to roam,                                          10\n    And leisure, peace, and loving friends,\n    Are the best treasures of an earthly home.\n\n    Such gifts are mine, then why deplore\n    The body's slow decay?\n    A warning mercifully sent                                         15\n    To fix my hopes upon a surer stay.\n\n\n\n\nA WORDSWORTH BIBLIOGRAPHY\n\n\n\n\nI.--_GREAT BRITAIN_\n\n\nI\n\nEDITIONS PUBLISHED DURING WORDSWORTH'S LIFETIME\n\nIn the Bibliographies by Mr. Tutin and Professor Dowden there are\nnumerous and valuable details as to these editions, which it is\nunnecessary to reproduce here.--ED.\n\n1\n\n1793. AN EVENING WALK. An Epistle; in verse. Addressed to a Young\nLady, from the Lakes of the North of England. By W. Wordsworth, B. A.,\nof St. John's, Cambridge. London: printed for J. Johnson, St. Paul's\nChurch-yard. 4to.\n\n2\n\n1793. DESCRIPTIVE SKETCHES. In verse. Taken during a pedestrian tour\nin the Italian, Grison, Swiss, and Savoyard Alps. By W. Wordsworth,\nB. A., of St. John's, Cambridge. Loca pastorum deserta atque otia\ndia.--_Lucret._ Castella in tumulis--Et longe saltus lateque\nvacantes.--_Virgil._ London: printed for J. Johnson, St. Paul's\nChurchyard. 4to.\n\n3\n\n1798. LYRICAL BALLADS, with a few other Poems. Bristol: printed by\nBiggs and Cottle; for T. N. Longman, Paternoster-Row, London. 8vo.\n\n1798. LYRICAL BALLADS, with a few other Poems. London: printed for J. &\nA. Arch, Gracechurch Street. 8vo.[431]\n\n4\n\n1800. LYRICAL BALLADS, with other Poems. In two volumes. By W.\nWordsworth. Quam nihil ad genium. Papiniane, tuum! Vol. I. Second\nEdition. [Vol. II.] London: printed for T. N. Longman and O. Rees,\nPaternoster-Row, by Biggs and Co., Bristol. 8vo.[432]\n\n5\n\n1802. LYRICAL BALLADS, with Pastoral and other Poems. In two volumes.\nBy W. Wordsworth. Quam nihil ad genium, Papiniane, tuum! Third Edition.\nLondon: printed for T. N. Longman & O. Rees, Paternoster-Row, by Biggs\nand Cottle, Crane-Court, Fleet-Street. 8vo.[433]\n\n6\n\n1805. LYRICAL BALLADS, with Pastoral and other Poems. In two volumes.\nBy W. Wordsworth. Quam nihil ad genium, Papiniane, tuum! Fourth\nEdition. London: printed for Longman, Hurst, Rees, & Orme, by R.\nTaylor and Co., 38 Shoe Lane. 8vo.[434]\n\n7\n\n1807. POEMS, in two volumes, By William Wordsworth, Author of the\nLyrical Ballads. _Posterius graviore sono tibi Musa loquetur Nostra,\ndabunt cum securos mihi tempora fructus._ Vol. I. [Vol. II.] London:\nprinted for Longman, Hurst, Rees, and Orme, Paternoster-Row. 12mo.\n\n8\n\n1809. CONCERNING THE RELATIONS OF GREAT BRITAIN, SPAIN, AND\nPORTUGAL, TO EACH OTHER, AND TO THE COMMON ENEMY, AT THIS CRISIS;\nand specifically as affected by the Convention of Cintra: _The whole\nbrought to the test of those principles by which alone the Independence\nand Freedom of Nations can be Preserved or Recovered_. Qui didicit\npatriae quid debeat;--Quod sit conscripti, quod judicis officium; quae\nPartes in bellum missi ducis. By William Wordsworth. London: printed\nfor Longman, Hurst, Rees, and Orme, Paternoster-Row. 8vo.\n\n9\n\n1814. THE EXCURSION, being a portion of The Recluse, a Poem. By William\nWordsworth. London: printed for Longman, Hurst, Rees, Orme, and Brown,\nPaternoster-Row. 4to.[435]\n\n10\n\n1815. POEMS BY WILLIAM WORDSWORTH: including Lyrical Ballads, and\nthe Miscellaneous Pieces of the Author. With additional Poems, a new\nPreface, and a Supplementary Essay. In two volumes. Vol. I. [Vol.\nII.] London: printed for Longman, Hurst, Rees, Orme, and Brown,\nPaternoster-Row. 8vo.[436]\n\n11\n\n1815. THE WHITE DOE OF RYLSTONE; OR, THE FATE OF THE NORTONS. A Poem.\nBy William Wordsworth. London: Printed for Longman, Hurst, Rees, Orme,\nand Brown, Paternoster-Row, by James Ballantyne and Co., Edinburgh.\n4to.[437]\n\n12\n\n1816. A LETTER TO A FRIEND OF ROBERT BURNS: occasioned by an intended\nrepublication of the account of the Life of Burns, by Dr. Currie;\nand of the Selection made by him from his Letters. By William\nWordsworth. London: Printed for Longman, Hurst, Rees, Orme, and Brown,\nPaternoster-Row. 8vo.[438]\n\n13\n\n1816. THANKSGIVING ODE, January 18, 1816. With other short Pieces,\nchiefly referring to Recent Public Events. By William Wordsworth.\nLondon: Printed by Thomas Davison, Whitefriars; for Longman, Hurst,\nRees, Orme, and Brown, Paternoster-Row. 8vo.\n\n14\n\n1818. TWO ADDRESSES TO THE FREEHOLDERS OF WESTMORELAND. Kendal: Printed\nby Airy and Bellingham. 8vo.\n\n15\n\n1819. PETER BELL, a Tale in Verse, by William Wordsworth. London:\nPrinted by Strahan and Spottiswoode. Printers-Street; for Longman,\nHurst, Rees, Orme, and Brown, Paternoster-Row. 8vo.[439]\n\n16\n\n1819. PETER BELL, A Tale in Verse, by William Wordsworth. Second\nEdition. London: Printed by Strahan and Spottiswoode, Printers-Street;\nfor Longman, Hurst, Rees, Orme, and Brown, Paternoster-Row. 8vo.\n\n17\n\n1819. THE WAGGONER, a Poem, to which are added, Sonnets. By William\nWordsworth. \"What's in a NAME?\" \"Brutus will start a Spirit as soon as\nC\u00e6sar,\" London: Printed by Strahan & Spottiswoode, Printers-Street; for\nLongman, Hurst, Rees, Orme, & Brown, Paternoster-Row. 8vo.[440]\n\n18\n\n1820. THE RIVER DUDDON, a Series of Sonnets; Vaudracour and Julia:\nand other Poems. To which is annexed, a Topographical Description\nof the Country of the Lakes, in the North of England. By William\nWordsworth. London: Printed for Longman, Hurst, Rees, Orme, and Brown,\nPaternoster-Row. 8vo.[441]\n\n19\n\n1820. THE MISCELLANEOUS POEMS OF WILLIAM WORDSWORTH. In four\nvolumes. London: Printed for Longman, Hurst, Rees, Orme, and Brown,\nPaternoster-Row. 12mo.[442]\n\n20\n\n1820. THE EXCURSION, being a portion of The Recluse, A Poem. By William\nWordsworth. Second Edition. London: Printed for Longman, Hurst, Rees,\nOrme, and Brown, Paternoster-Row. 8vo.\n\n21\n\n1822. MEMORIALS OF A TOUR ON THE CONTINENT, 1820. By William\nWordsworth. London: Printed for Longman, Hurst, Rees, Orme, and Brown,\nPaternoster-Row. 8vo.\n\n22\n\n1822. ECCLESIASTICAL SKETCHES. By William Wordsworth. London: Printed\nfor Longman, Hurst, Rees, Orme, and Brown, Paternoster-Row. 8vo.[443]\n\n23\n\n1822. A DESCRIPTION OF THE SCENERY OF THE LAKES IN THE NORTH OF\nENGLAND. Third Edition (now first published separately), with\nadditions, and illustrative remarks upon the Scenery of the Alps. By\nWilliam Wordsworth. London: printed for Longman, Hurst, Rees, Orme,\nand Brown, Paternoster-Row. 12mo.[444]\n\n24\n\n1827. THE POETICAL WORKS OF WILLIAM WORDSWORTH. In five volumes.\nLondon: Printed for Longman, Rees, Orme, Brown, and Green,\nPaternoster-Row. 12mo.[445]\n\n25\n\n1828. THE POETICAL WORKS OF WILLIAM WORDSWORTH. Complete in one volume.\nParis: Published by A. and W. Galignani, No. 18, Rue Vivienne. 8vo.[446]\n\n26\n\n1831. SELECTIONS FROM THE POEMS OF WILLIAM WORDSWORTH, ESQ., chiefly\nfor the use of Schools and Young Persons. London: Edward Moxon, 64 New\nBond Street. 12mo.[447]\n\n27\n\n1832. THE POETICAL WORKS OF WILLIAM WORDSWORTH. A new Edition. In four\nvolumes. London: Printed for Longman, Rees, Orme, Brown, Green, &\nLongman, Paternoster-Row. 8vo.[448]\n\n28\n\nSELECTIONS FROM THE POEMS OF WILLIAM WORDSWORTH, ESQ., chiefly for the\nuse of Schools and young persons. A New Edition. London: Edward Moxon,\nDover Street. MDCCCXXXIV.\n\n29\n\nThe Memorial Lines \"Written after the Death of Charles Lamb\" were\nissued privately, without title or date, probably late in 1835, or\nearly in 1836. 8vo. pp. 7.\n\n30\n\n1835. YARROW REVISITED, AND OTHER POEMS. By William Wordsworth.\n\n    Poets \u2026 dwell on earth\n    To clothe whate'er the soul admires and loves;\n    With language and with numbers.--AKENSIDE.\n\nLondon: printed for Longman, Rees, Orme, Brown, Green, & Longman,\nPaternoster-Row; and Edward Moxon, Dover Street. 12mo.\n\n31\n\n1835. A GUIDE THROUGH THE DISTRICT OF THE LAKES IN THE NORTH OF\nENGLAND, with a Description of the Scenery, &c. For the use of Tourists\nand Residents. Fifth Edition, with considerable additions. By William\nWordsworth. Kendal: published by Hudson and Nicholson; and in London by\nLongman & Co., Moxon, and Whittaker and Co. 12mo.\n\n32\n\n1836. YARROW REVISITED, AND OTHER POEMS. By William Wordsworth.\n\n    Poets \u2026 dwell on earth\n    To clothe whate'er the soul admires and loves;\n    With language and with numbers.--AKENSIDE.\n\nSecond Edition. London: printed for Longman, Rees, Orme, Brown, Green,\n& Longman, Paternoster-Row; and Edward Moxon, Dover Street. 8vo.[449]\n\n33\n\nTHE EXCURSION. A Poem. By William Wordsworth. A New Edition. London:\nEdward Moxon, Dover Street. MDCCCXXXVI. 8vo.[450]\n\n34\n\nTHE POETICAL WORKS OF WILLIAM WORDSWORTH. A New Edition. In six\nvolumes. Vol. I. (Vol. II.-VI.) London: Edward Moxon, Dover Street.\nMDCCCXXXVI.-MDCCCXXXVII. Fcap. 8vo.[451]\n\n35\n\nTHE SONNETS OF WILLIAM WORDSWORTH. Collected in one volume, with a\nfew additional ones, now first published. London: Edward Moxon, Dover\nStreet. MDCCCXXXVIII. 8vo.[452]\n\n36\n\nYARROW REVISITED; AND OTHER POEMS. By William Wordsworth. London:\nEdward Moxon, Dover Street. MDCCCXXXIX. 18mo.[453]\n\n37\n\nPOEMS, CHIEFLY OF EARLY AND LATE YEARS; including The Borderers, a\nTragedy. By William Wordsworth. London: Edward Moxon, Dover Street.\nMDCCCXLII. 8vo.[454]\n\n38\n\n1843. SELECT PIECES FROM THE POEMS OF WILLIAM WORDSWORTH. London: James\nBurns. Sq. 12mo.[455]\n\n39\n\n1844. KENDAL AND WINDERMERE RAILWAY. Two Letters, re-printed from\nthe Morning Post. Revised, with additions. Kendal: printed by R.\nBranthwaite and Son.\n\n40\n\n1845. THE POEMS OF WILLIAM WORDSWORTH, D.C.L., Poet Laureate, etc. etc.\nA New Edition. London: Edward Moxon, Dover Street. MDCCCXLV. Royal\n8vo.[456]\n\n41\n\n1847. ODE, performed in the Senate-House, Cambridge, on the sixth of\nJuly, M.DCCC.XLVII. At the first commencement after the Installation\nof his Royal Highness the Prince Albert, Chancellor of the University.\nCambridge: printed at the University Press. 4to.\n\n42\n\n1847. ODE on the installation of His Royal Highness Prince Albert as\nChancellor of the University of Cambridge. By William Wordsworth, Poet\nLaureate. London: Printed, by permission, by Vizetelley Brothers & Co.\nPublished by George Bell, Fleet Street. 4to.\n\n43\n\nTHE POETICAL WORKS OF WILLIAM WORDSWORTH, D.C.L., Poet Laureate, etc.\netc. In six volumes. A New Edition. London: Edward Moxon, Dover Street.\nMDCCCXLIX.-MDCCCL. 12mo.[457]\n\n[431] These two editions of 1798 are the same; but as Cottle sold to\nArch most of the copies printed, the majority bear the name of Arch as\npublisher.\n\nFour of the poems were by S.T. Coleridge, viz. _The Rime of the\nAncyent Marinere_; _The Foster-Mother's Tale_; _The Nightingale, a\nConversational Poem_; and _The Dungeon_.--ED.\n\n[432] The first volume of this edition is a reprint of the editions\nof 1798, _The Convict_ being left out. In it there is one poem by\nColeridge entitled _Love_, which was not in the edition of 1798. The\npoems in the second volume are new. The preface to Volume 1. contains\nWordsworth's poetical theory in its original form. This preface was\nincluded in the 1802 and 1805 editions of Lyrical Ballads, and also--in\nan expanded form--in almost every subsequent edition of his poems.--ED.\n\n[433] This was almost a reproduction of the two volumes of 1800, with\na few variations of text. The preface, however, was much enlarged.\nThe poem _A Character in the Antithetical Manner_ was left out, also\nColeridge's poem _The Dungeon_.--ED.\n\n[434] A reprint of the edition of 1802, with slight variations of\ntext.--ED.\n\n[435] The _Essay on Epitaphs_ inserted in the notes to this volume was\noriginally published in _The Friend_, February 22, 1810.--ED.\n\n[436] This was the first edition of Wordsworth's Poems arranged by\nhim under distinctive headings, viz. \"Poems referring to the Period\nof Childhood,\" \"Juvenile Pieces,\" \"Poems founded on the Affections,\"\n\"Poems of the Fancy,\" \"Poems of the Imagination,\" \"Poems proceeding\nfrom Sentiment and Reflection,\" \"Miscellaneous Sonnets,\" \"Sonnets,\netc., dedicated to Liberty,\" \"Poems on the Naming of Places,\"\n\"Inscriptions,\" \"Poems referring to the Period of Old Age,\" \"Epitaphs\nand Elegiac Poems,\" \"Ode, Intimations of Immortality from Recollections\nof Childhood.\" In it, he gave _dates_ to his poems.\n\nIn Volume I. is an engraving by Mr. Bromley from a picture by Sir\nGeorge Beaumont; Volume II. has an engraving by Mr. Reynolds from Sir\nGeorge's picture of Peele Castle in a storm.--ED.\n\n[437] The poem _The Force of Prayer; or, the Founding of Bolton Priory_\nfollows the _White Doe of Rylstone_; and the volume contains an\nengraving by Mr. Bromley from a painting of Bolton Abbey by Sir George\nBeaumont.--ED.\n\n[438] The \"Friend\" was Mr. James Gray, Edinburgh.--ED.\n\n[439] The volume contains an engraving by Mr. Bromley from a painting\nby Sir George Beaumont. In addition to _Peter Bell_, this volume\ncontained four sonnets.--ED.\n\n[440] This volume was dedicated to Charles Lamb.--ED.\n\n[441] In 1820 the four separate publications, _The Waggoner_, etc.,\n_Thanksgiving Ode_, etc., _Peter Bell_, etc., and _The River Duddon,\nVaudracour and Julia_, etc., were bound up together with their separate\ntitle-pages, and issued under the title, _Poems by William Wordsworth_,\nmaking Volume III. of the _Miscellaneous Poems_.--ED.\n\n[442] Each of these volumes contained an engraving from a picture by\nSir George Beaumont. They were \"Lucy Gray,\" \"Peter Bell,\" \"The White\nDoe of Rylstone,\" and \"Peele Castle.\" All had appeared in previous\neditions. The \"Advertisement\" states that this edition contains the\nwhole of the published poems of the Author, with the exception of _The\nExcursion_, and that a few Sonnets \"are now first published.\"\n\nIt is worthy of note that, in this edition, Wordsworth for the first\ntime abandoned the practice of putting in an apostrophe, instead of\na vowel letter, in words ending with \"ed,\" and in similar cases of\ncontraction.--ED.\n\n[443] Wordsworth added to this series of Sonnets, in the one-volume\nedition of 1845 which contained 132. In the first edition, there were\n102 sonnets.--ED.\n\n[444] This originally appeared as an Introduction to Wilkinson's\n_Select Views in Cumberland, Westmoreland, and Lancashire_, which was\npublished in 1810. In 1820 it was included (see No. 18) in _The River\nDuddon: A Series of Sonnets_. In 1823 a fourth edition appeared which\nwas a reprint of that of 1822.--ED.\n\n[445] To this edition Wordsworth prefixed the following\n\"Advertisement\":--\"In these volumes will be found the whole of the\nAuthor's published poems, for the first time collected in a uniform\nedition, with several new pieces interspersed.\"--ED.\n\n[446] In this edition--copied without authority, from the poet or\nhis publishers, and with many errata, from the issue of 1827--there\nis an engraving of Wordsworth by Mr. Wedgewood, after the portrait\nby Carruthers, now in the possession of Mr. Hutchinson at Kimbolton.\nThe Galignani edition of Southey is even worse; three poems, not by\nSouthey, being included in it.--ED.\n\n[447] The editor of these selections was Joseph Hine.--ED.\n\n[448] The \"Advertisement\" to this edition is as follows:--\"The contents\nof the last edition in five volumes are compressed into the present\nof four, with some additional pieces reprinted from miscellaneous\npublications.\"--ED.\n\n[449] As this volume (No. 32 in the list) was the last printed for the\nMessrs. Longman, and issued by that firm and by Mr. Moxon jointly,\nit is desirable to mention here, in a footnote, that, with the\nexception of _The Evening Walk_ and _Descriptive Sketches_ (which were\npublished by J. Johnson) every one of Wordsworth's works from 1798 to\n1836--thirty in number--were introduced to the world by the Messrs.\nLongman. It is questionable if any firm has ever had a similar \"record\"\nin connection with the works of any great poet.--ED.\n\n[450] A reprint of the sixth volume of the 1836-37 edition. It was\nagain reprinted in 1841, 1844, and 1847.--ED.\n\n[451] Volumes one and two are dated 1836; the remaining four 1837. This\nedition was stereotyped. It was reprinted in 1840, 1841, 1842, 1843,\n1846, 1849, etc.; and some of the reprints contain slight variations\nof text, etc. All the editions issued after 1841 include the volume,\n_Poems of Early and Late Years_ (see No. 37) as a seventh volume. After\n1850 _The Prelude_ was added as an eighth volume.\n\nIn the first volume of this edition there is a steel engraving by\nMr. Watt of a portrait of the Poet by W. Pickersgill, which is in\nSt. John's College, Cambridge. This engraving was reproduced in the\neditions of 1840, 1841, and following ones.--ED.\n\n[452] This edition includes (as its \"Advertisement\" tells us) \"twelve\nnew Sonnets which were composed while the sheets were going through the\npress.\"--ED.\n\n[453] Mr. Tutin writes in his Wordsworth Bibliography:--\"This Pocket\nedition of _Yarrow Revisited_, etc., is the third separate issue of the\nPoem. It seems to have been intended as a supplementary volume to the\nfour vol. edition of 1832, as the sheets of it are all imprinted 'Vol.\nv.,' but I have no direct proof that it was ever so issued.\"--ED.\n\n[454] In his \"Advertisement\" the Author states that about one-third of\nthe Poem _Guilt and Sorrow_ was written in 1794, and was published in\nthe year 1798 under the title of _The Female Vagrant_.--ED.\n\n[455] This volume is dedicated \"To her Most Sacred Majesty,\nVictoria.\"--ED.\n\n[456] Frequently republished. After 1851 _The Prelude_ was included.\nThe edition of 1869 has \"nine additional poems,\" dated 1846. All the\neditions which I have seen contain an engraving by Mr. Finden from the\nbust of Wordsworth by Chantrey--the original of which is at Coleorton\nHall--and a picture of Rydal Mount engraved by Mr. House after Finden.\nProfessor Dowden tells us that, in some later editions \"the Pickersgill\nportrait, engraved by J. Skelton, replaces Chantrey's bust.\" In this\nedition, as in that of 1815, Wordsworth gave dates to his poems.--ED.\n\n[457] Volumes I. and II. are dated 1849, and Volumes III.-VI. 1850.\n_The Excursion_ formed the sixth volume. It was reprinted separately in\n1851, 1853, and 1857.--ED.\n\n\nII\n\nEDITIONS OF THE POEMS, AND OF SELECTIONS FROM THEM, PUBLISHED AFTER THE\nPOET'S DEATH.\n\n1\n\n1850. THE PRELUDE, OR GROWTH OF A POET'S MIND; an Autobiographical\nPoem; by William Wordsworth. London: Edward Moxon, Dover Street. Demy\n8vo.\n\n2\n\n1851. THE PRELUDE, OR GROWTH OF A POET'S MIND; an Autobiographical\nPoem; By William Wordsworth. Second Edition. London: Edward Moxon,\nDover Street. Fcap. 8vo.\n\n3\n\n1855. SELECT PIECES FROM THE POEMS OF WILLIAM WORDSWORTH. London:\nEdward Moxon. Sq. 12mo.\n\n4\n\n1857. THE POETICAL WORKS OF WILLIAM WORDSWORTH. In six volumes. A new\nEdition. London: Edward Moxon, Dover Street. 8vo.[458]\n\n5\n\nTHE EARLIER POEMS OF WILLIAM WORDSWORTH. Corrected as in the latest\nEditions. With Preface, and Notes showing the text as it stood in 1815.\nBy William Johnston. London: Edward Moxon, Dover Street. Fcap. 8vo.\n\n6\n\n1859. THE DESERTED COTTAGE. By William Wordsworth. Illustrated with\ntwenty-one designs by Birket Foster, J. Wolf, and John Gilbert,\nengraved by the Brothers Dalziel. London: George Routledge and Co.,\nFarringdon Street. New York: 18 Beekman Street. Small 4to.[459]\n\n7\n\nPOEMS OF WILLIAM WORDSWORTH. Selected and Edited by Robert Aris\nWillmott, Incumbent of Bear Wood. Illustrated with one hundred designs\nby Birket Foster, J. Wolf, and John Gilbert, Engraved by the Brothers\nDalziel. London: George Routledge and Co., Farringdon Street. New York:\n18 Beekman Street, MDCCCLIX. Small 4to.\n\n8\n\nTHE WHITE DOE OF RYLSTONE; OR, THE FATE OF THE NORTONS. By William\nWordsworth. London: Longman, Brown, Green, Longmans, and Roberts. Small\n4to.[460]\n\n9\n\nPASSAGES FROM \"THE EXCURSION,\" by William Wordsworth, Illustrated\nwith Etchings on Steel by Agnes Fraser. London: published by Paul and\nDominic Colnaghi and Co., publishers to Her Majesty, 13 and 14 Pall\nMall East. Oblong 4to.[461]\n\n10\n\nTHE WHITE DOE OF RYLSTONE; OR, THE FATE OF THE NORTONS. With\nIllustrations by Birket Foster, and others. London: Longman, Brown,\nGreen, Longmans, and Roberts.\n\n11\n\nPASTORAL POEMS, by William Wordsworth. London: Sampson, Low, etc.\n\n12\n\n1864. THE SELECT POETICAL WORKS OF WILLIAM WORDSWORTH. Copyright\nEdition. In two volumes. Leipzig, Bernhard Tauchnitz.[462]\n\n13\n\n1865. A SELECTION FROM THE WORKS OF WILLIAM WORDSWORTH, Poet Laureate.\nMoxon's Miniature Poets. Selected and arranged by Francis Turner\nPalgrave. Published in London: Edward Moxon & Co., Dover Street. Sq.\n12mo.[463]\n\n14\n\nTHE POEMS OF WILLIAM WORDSWORTH. A new Edition. London: Edward Moxon &\nCo., Dover Street.\n\n15\n\n1867. THE WHITE DOE OF RYLSTONE; OR, THE FATE OF THE NORTONS. By\nWilliam Wordsworth. London: Bell and Daldy, 186 Fleet Street. 8vo.[464]\n\n16\n\n1869. THE POETICAL WORKS OF WILLIAM WORDSWORTH. A new Edition. London:\nEdward Moxon, Son, & Co., 44 Dover Street, Piccadilly.\n\n17\n\n1870. THE POETICAL WORKS OF WILLIAM WORDSWORTH. Edited, with a critical\nMemoir, by William Michael Rossetti. Illustrated by artistic etchings\nby Edwin Edwards. London: E. Moxon, Son, & Co., Dover Street. Small 4to.\n\n18\n\nTHE POETICAL WORKS OF WILLIAM WORDSWORTH. Edited, with a critical\nMemoir, by William Michael Rossetti. Illustrated by Henry Dell. London:\nE. Moxon, Son, & Co., Dover Street. 8vo.[465]\n\n19\n\n1876. THE PROSE WORKS OF WILLIAM WORDSWORTH. For the first time\ncollected, with additions from unpublished manuscripts. Edited, with\nPreface, Notes and Illustrations, by the Rev. Alexander B. Grosart, St.\nGeorge's, Blackburn, Lancashire. In three volumes. Volume I. Political\nand Ethical. Volume II. \u00c6sthetical and Literary. Volume III. Critical\nand Ethical. London: Edward Moxon, Son, and Co., 1 Amen Corner,\nPaternoster Row. 8vo.\n\n20\n\n1879. POEMS OF WORDSWORTH, chosen and edited by Matthew Arnold. London:\nMacmillan and Co. 18mo.[466]\n\n21\n\nTHE POETICAL WORKS OF WILLIAM WORDSWORTH. Edited by William Knight,\nLL.D., Professor of Moral Philosophy, St. Andrews. Edinburgh: William\nPaterson. MDCCCLXXXII. [MDCCCLXXXII.-- MDCCCLXXXVI.] 8 vols. Demy\n8vo.[467]\n\n22\n\nSELECTIONS FROM WORDSWORTH. Edited, with an Introductory Memoir, by J.\nS. Fletcher. London: Alex. Gardner, 12 Paternoster Row, and Paisley.\nMDCCCLXXXIII. Fcap. 8vo. Parchment.[468]\n\n23\n\n1883. WINNOWINGS FROM WORDSWORTH. Edited by J. Robertson. Simpkin & Co.\n1883.\n\n24\n\nTHE BROTHERS, AND OTHER POEMS FOUNDED ON THE AFFECTIONS. 18mo. Collins.\n\n25\n\n1884. THE RIVER DUDDON. A Series of Sonnets. By William Wordsworth.\nWith ten Etchings by R. S. Chattock, The Fine Art Society, 148 New Bond\nStreet, London. Folio.\n\n26\n\nTHE SONNETS OF WILLIAM WORDSWORTH. Collected in one volume, with an\nEssay on The History of the English Sonnet by Richard Chenevix Trench,\nD.D., Archbishop of Dublin, Chancellor of the Order of St. Patrick.\nLondon: Suttaby and Co., Amen Corner. MDCCCLXXXIV. 8vo.[469]\n\n27\n\nSELECTIONS FROM WORDSWORTH. By Misses Wordsworth. London: Kegan Paul, &\nCo. April 8, 1884.\n\n28\n\nTHE WORDSWORTH BIRTHDAY BOOK. Edited by Adelaide and Violet Wordsworth.\nLondon: Kegan Paul, Trench, & Co.\n\n29\n\nBIRTHDAY TEXTS FROM WORDSWORTH. Edinburgh: W. P. Nimmo. N. D.\n\n30\n\nTHE GOLDEN POETS. \"Wordsworth.\" London: Marcus Ward & Co. N. D.\n\n31\n\n1885. THE POETICAL WORKS OF WILLIAM WORDSWORTH, With a Prefatory\nNotice, Biographical and Critical. By Andrew James Symington. London:\nWalter Scott, 14 Paternoster Square and Newcastle-on-Tyne. 16mo.[470]\n\n32\n\nWORDSWORTH'S EXCURSION. THE WANDERER. Edited, with Notes, etc., by H.\nH. Turner. London: Rivingtons. N. D.\n\n33\n\nODE ON IMMORTALITY, AND LINES ON TINTERN ABBEY. Illustrated. Cassell.\n4to.\n\n34\n\nTINTERN ABBEY, ODES, AND THE HAPPY WARRIOR. 8vo. Chambers. (Republished\nin 1892.)\n\n35\n\n1887. THROUGH THE WORDSWORTH COUNTRY. By Harry Goodwin and Professor\nKnight. London: Swan Sonnenschein, Lowrey & Co., Paternoster Square.\nImperial 8vo.[471]\n\n36\n\nWORDSWORTH AND KEATS, Selections. In 16mo. M. Ward.\n\n37\n\n1888. THE COMPLETE POETICAL WORKS OF WILLIAM WORDSWORTH. With an\nIntroduction by John Morley. With a Portrait. London: Macmillan & Co.\nCrown 8vo.\n\n38\n\n1888. SELECTIONS FROM WORDSWORTH. By William Knight, and other Members\nof the Wordsworth Society. With Preface and Notes. London: Kegan\nPaul, Trench, & Co., 1 Paternoster Square. MDCCCLXXXVIII. Large Crown\n8vo.[472]\n\n39\n\n1888. THE RECLUSE. By William Wordsworth. London: Macmillan and Co.[473]\n\n40\n\n1888. THE POETICAL WORKS OF WORDSWORTH. With Memoir, Explanatory Notes,\netc. London: Griffith, Farren, & Co., Newbury House, Charing Cross Road.\n\n41\n\nPROSE WRITINGS OF WORDSWORTH: Selected and Edited, with an\nIntroduction, by William Knight. London: Walter Scott. No date.\n\n42\n\n1889. WE ARE SEVEN. Illustrated by Agnes Gardner King. 16mo.\n\n43\n\n1891. LYRICS AND SONNETS OF WORDSWORTH. With Introduction and\nBibliography. By Clement R. Shorter. Scott Library. 32mo.\n\n44\n\nTHE POETICAL WORKS OF WILLIAM WORDSWORTH. Edited, with Memoir, by\nEdward Dowden. London: George Bell & Sons. 1892-1893.[474]\n\n45\n\n1891. LYRICAL BALLADS, ETC. A reprint of the original edition of 1798.\nEdited by Edward Dowden. London: David Nutt. 16mo.\n\n46\n\n1891. THE WHITE DOE OF RYLSTONE, WITH THE SONG AT THE FEAST OF BROUGHAM\nCASTLE. Edited, with introduction and notes, by William Knight. Oxford:\nAt the Clarendon Press.\n\n47\n\nTHE POETICAL WORKS OF WILLIAM WORDSWORTH. Edinburgh: W.P. Nimmo, Hay,\nand Mitchell. 1892.\n\n48\n\nWORDSWORTH FOR THE YOUNG. With notes by J.C. Wright. 8vo. 1893.\n\n49\n\n1895. THE POETICAL WORKS OF WILLIAM WORDSWORTH, with introductions and\nnotes. Edited by Thomas Hutchinson, M.A. London: Henry Froude, Oxford\nUniversity Press Warehouse, Amen Corner, E.C.\n\n50\n\nTHE PENNY POETS, in \"The Masterpiece Library.\" Wordsworth. Nos. XXXII.\nand XXXVII.\n\n51\n\n1896. LYRIC POEMS. Edited by Ernest Rhys. 8vo. London: Dent & Co.\n\n52\n\nTHE PRELUDE; OR, GROWTH OF A POET'S MIND. 18mo. London: Dent & Co.\n\n53\n\n\"The Lansdowne Poets\" included one of Wordsworth. The \"Albion\" edition\nwas published by Messrs. Froude, Oxford University Press.[475]\n\n[458] In this edition--reprinted as \"The Centenary Edition\" in 1870,\n1881, and 1882--the Fenwick notes were printed, for the first time in\nfull, as prefatory notes to the poems.--ED.\n\n[459] Reproduced in 1864.--ED.\n\n[460] It contains illustrations by H. N. Humphreys and Birket\nFoster.--ED.\n\n[461] This volume contains eleven etchings of varying merit.--ED.\n\n[462] These are volumes 707 and 708 of Tauchnitz's \"Collection of\nBritish Authors.\"--ED.\n\n[463] It contains a steel engraving from Chantrey's bust of the Poet.\nThis selection was re-issued in 1866, and 1869; and, recently, in a\nsmall pocket edition.--ED.\n\n[464] This is a reprint, in a different form, of No. 8.--ED.\n\n[465] In this edition, which is a reprint, on smaller paper, of No. 19.\nthere is an engraving from one of the portraits of the Poet by Miss\nGillies. The engraving first appeared in Volume I. of _The New Spirit\nof the Age_, edited by R. H. Horne.--ED.\n\n[466] It contains an idealised engraving of one of Haydon's portraits\nof Wordsworth, after Lupton, by C. H. Jeens, and on the outside cover a\ndrawing of Dove Cottage.--ED.\n\n[467] In this edition the Poems were arranged for the first time\nin the chronological order of composition; the changes of text, in\nthe successive editions, were given in footnotes, with the dates of\nthese changes; many new readings, or suggested changes of text--which\nwere written by the Poet on the margins of a copy of the edition of\n1836-37, kept at Rydal Mount, and afterwards in the possession of Lord\nColeridge--were added; all the Fenwick notes were printed as Prefatory\nnotes; Topographical notes--containing allusions to localities in the\nEnglish Lake District, and elsewhere--were given; several Poems and\nFragments hitherto unpublished were printed; a Bibliography of the\nPoems, and of editions published in England and America from 1793 to\n1850 was added. Etchings of localities associated with the Poet, from\ndrawings by Mr. MacWhirter, were given as frontispieces to Volumes I.,\nII., III., IV., V., VI., and VII. The text adopted was Wordsworth's\nfinal text of 1849-50.--ED.\n\n[468] It contains an engraving of Rydal Mount on the fly-leaf.--ED.\n\n[469] This volume is a reprint of Wordsworth's own edition of his\nSonnets, published in 1838, with the addition of Archbishop Trench's\n_History of the English Sonnet_.--ED.\n\n[470] This is one of the volumes of _The Canterbury Poets_. It is only\na selection, though described on the title as \"The Poetical Works.\"--ED.\n\n[471] This volume contains fifty-five engravings from drawings by\nHarry Goodwin of scenes in the English Lake District associated with\nWordsworth, with the poems, or portions of poems, referring to the\nplaces.--ED.\n\n[472] The poems are arranged in chronological order of composition;\nand there is, as frontispiece, an etched portrait of the Poet from a\nminiature by Margaret Gillies in the possession of Sir Henry Doulton.\nAmongst those who contributed to it were Robert Browning, James\nRussell Lowell, the late Lord Selborne, Mr. R. H. Hutton, the Dean\nof Salisbury, the late Lord Coleridge, the Rev. Stopford Brooke, Mr.\nAubrey de Vere, the late Lord Houghton, Canon Rawnsley, the late\nPrincipals Shairp and Greenwood and Professor Veitch, Mr. Spence\nWatson, Mr. Rix, Mr. Heard, Mr. Cotterill, the late Bishop Wordsworth\nof St. Andrews, and the Editor.--ED.\n\n[473] In the prefatory advertisement to the first edition of _The\nPrelude_ 1850, it is stated that that poem was designed to be\nintroductory to _The Recluse_, and that _The Recluse_ if completed,\nwould have consisted of three parts. The second part is _The\nExcursion_. The third part was only planned. The first book of the\nfirst part was left in manuscript by Wordsworth. It was published for\nthe first time _in extenso_ in 1888.--ED.\n\n[474] This Aldine edition, by Professor Dowden, is one of great merit,\nand permanent value. Although it is not immaculate--as no literary work\never is--as a contribution to Wordsworthian Literature it will hold an\nhonoured place. Its \"critical apparatus\" is succinct and admirable.--ED.\n\n[475] Mr. Andrew Lang tells me that he is about to edit a _Selection_\nof the Poems, for the Messrs. Longman; which will, no doubt, be as\nuseful, and popular, as Matthew Arnold's Selection has been.--ED.\n\n\nIII\n\nESTIMATES OF WORDSWORTH IN VARIOUS BOOKS[476]\n\n1811. SEWARD, ANNA. Letters written between the Years 1784 and 1807.\nEdited by A. Constable, vol. vi. No. 66.[477] 8vo. Edinburgh.\n\n1817. COLERIDGE, S. T. Biographia Literaria; or, Biographical Sketches\nof my Literary Life and Opinions. 2 vols. 8vo. London: Rest Fenner.\nSecond Edition. London: William Pickering. 1847. Bohn's Standard\nLibrary. 1866.\n\nCOLERIDGE, S. T. In _The Friend, passim_. Second Edition. London: Rest\nFenner.\n\nHAZLITT, WILLIAM. The Round Table: a Collection of Essays on\nLiterature, Men, and Manners. Observations on Mr. Wordsworth's Poem,\n\"The Excursion.\" 12mo. London: Templeman. Also in Bohn's Standard\nLibrary. Edited by W. Carew Hazlitt. Pp. 158-176. London. 1871.\n\n1818. HAZLITT, WILLIAM. Lectures on the English Poets. 8vo. London:\nTaylor and Hessey. Also in Bohn's Standard Library. 1870.\n\n1819. HAZLITT, WILLIAM. Political Essays, with Sketches of Public\nCharacters. My First Acquaintance with Poets. 8vo. London: Templeman.\nAlso in Winterslow, pp. 255-277. Bohn's Standard Library. 1872.\n\n1823. SOLIGNY, VICTOIRE DE, COUNT, _pseud._ (_i.e._ Peter George\nPatmore, father of the late Coventry Patmore). Letters on England, vol.\nii. pp. 7-19. 8vo. London: Henry Colburn and Co.\n\n1824. LANDOR, W. S. Imaginary Conversations of Literary Men and\nStatesmen. Southey and Porson, i. 39. 8vo. London: Taylor and Hessey.\nNew Edition, i. 11, 68, 182. London: Edward Moxon. 1846. New Edition,\niv. 18. London: Chapman and Hall. 1876.\n\n1825. HAZLITT, WILLIAM. The Spirit of the Age; or, Contemporary\nPortraits. 8vo. London: Henry Colburn and Co.; Fourth Edition. George\nBell and Sons. 1886.\n\n1827. HONE, WILLIAM. The Table Book. Wordsworth, ii. 275. 8vo. London:\nHunt and Clarke.\n\nCOLERIDGE, S. T. Table Talk. July 21, 1832; July 31, 1832; February 16,\n1833.\n\n1833. MONTGOMERY, JAMES. Lectures on Poetry and General Literature,\ndelivered at the Royal Institution in 1830 and 1831. Wordsworth's\nTheory of Poetic Diction, pp. 134-141. 8vo. London: Longmans.\n\n1836. Conversations at Cambridge. The Poet Wordsworth and Professor\nSmythe, pp. 235-252. 8vo. London: John W. Parker.\n\n1837. COTTLE, JOSEPH. Early Recollections; chiefly relating to the late\nSamuel Taylor Coleridge, during his long Residence in Bristol. 2 vols.\n8vo. London: Longman, Rees and Co.\n\n1838. CHORLEY, H. F. The Authors of England. 4to. London. New Edition,\nrevised (by G. B.) London. 1861.\n\nHARE, JULIUS C. and AUGUSTUS W. Guesses at Truth, by Two Brothers.\nSecond Series. 8vo. London: Taylor and Walton. The Dedication of this\nedition is to William Wordsworth. New Edition, in one volume. Macmillan\nand Co. 1866.\n\n1840. HUNT, LEIGH. The Seer. \"Wordsworth and Milton,\" pp. 5-53. London:\nEdward Moxon.\n\nRUSKIN, JOHN. Modern Painters (1843-1860), _passim_ in all the five\nvolumes. London: George Allen.\n\n1843. CHAMBERS, ROBERT. Cyclop\u00e6dia of English Literature. Wordsworth,\nii. 322-333. Fourth Edition, revised by Robert Carruthers, LL.D. 1888.\n8vo. Edinburgh: William and Robert Chambers.\n\n1844. HORNE, R. H. A New Spirit of the Age. William Wordsworth and\nLeigh Hunt, vol. i. pp. 307-332. 12mo. London: Smith, Elder and Co.\n\nKEBLE, JOHN. Praelectiones Academicae Oxonii habitae, annis\nMDCCCXXXII.-MDCCCXLI., tom. ii. pp. 615, 789. 8vo. Oxonii: J. H. Parker.\n\n1845. GILFILLAN, GEORGE. A Gallery of Literary Portraits. 12mo.\nEdinburgh: Groombridge.\n\nCRAIK, E. L. Sketches of the History of Literature and Learning in\nEngland. Vol. vi., pp. 114-139. London: Charles Knight.\n\n1847. HOWITT, WILLIAM. Homes and Haunts of the most eminent British\nPoets, vol. ii. pp. 259-291. 8vo. London: Richard Bentley. Third\nEdition. Routledge and Sons. 1862.\n\nTUCKERMAN, HENRY T. Thoughts on the Poets. 8vo. London: J. Chapman.\n\n1849. GILFILLAN, GEORGE. A Second Gallery of Literary Portraits. 8vo.\nEdinburgh: Groombridge.\n\nSHAW, THOMAS B. Outlines of English Literature. Wordsworth, pp.\n518-526. 8vo. London: John Murray. Sixteenth Edition, edited by William\nSmith, D.C.L. 1887.\n\nTAYLOR, HENRY. Notes from Books. In four Essays. Wordsworth's Poetical\nWorks and Sonnets, pp. 1-186. 8vo. London: John Murray. Works: Author's\nEdition, vol. v. London: C. Kegan Paul and Co. 1878.\n\n1849-50. SOUTHEY, ROBERT. Life and Correspondence. Edited by the Rev.\nCharles Cuthbert Southey. 6 vols. Comments on Wordsworth in chaps,\nix.-xiii. xv. xix. xxvi. xxxii. and xxxvi. 8vo. London: Longman, Brown,\nGreen and Longmans.\n\n1851. GILLIES, R. P. Memoirs of a Literary Veteran; including Sketches\nand Anecdotes of the most distinguished Literary Characters from 1794\nto 1849. Wordsworth, vol. ii. pp. 136-173. 8vo. London: Richard Bentley.\n\nThe Poetic Companion, vol. i., pp. 168-173. A Biographical and Critical\nSketch of William Wordsworth.\n\nMOIR, D. M. Sketches of the Poetical Literature of the past\nHalf-Century, pp. 59-81; 120. Edinburgh: William Blackwood and Sons.\nThird Edition, 1856.\n\nWORDSWORTH, CHRISTOPHER. Memoirs of William Wordsworth, Poet-Laureate,\nD.C.L. 2 vols. 8vo. London: Edward Moxon. 1851.\n\n1852. JANUARY SEARLE (George S. Phillips). Memoirs of William\nWordsworth, compiled from Authentic Sources. 12mo. London: Partridge\nand Oakey.\n\nMITFORD, M. R. Recollections of a Literary Life; or, Books, Places, and\nPeople, vol. iii. chap. i. 8vo. London: Richard Bentley.\n\n1853. An Essay on the Poetry of Wordsworth, 72 pp. 8vo. Liverpool.\n\nAUSTIN, W. S., and JOHN RALPH. The Lives of the Poets-Laureate. With\nan Introductory Essay on the Title and Office. William Wordsworth, pp.\n396-428. 8vo. London: Richard Bentley.\n\nWRIGHT, JOHN. The Genius of Wordsworth harmonised with the Wisdom and\nIntegrity of his Reviewers. 8vo. London: Longman, Brown, Green and\nLongmans.\n\nSPALDING, WILLIAM. The History of English Literature. 8vo. Edinburgh:\nOliver & Boyd.\n\n1854. DE QUINCEY, THOMAS. Autobiographic Sketches. Early Memorials\nof Grasmere, vol. ii. pp. 104-141; William Wordsworth, pp. 227-314;\nWilliam Wordsworth and Robert Southey, pp. 315-345. 8vo. Edinburgh:\nJames Hogg. Also Collected Writings. New and Enlarged Edition. By David\nMasson. Edinburgh: Adam and Charles Black. 1889-90.\n\nSPALDING, WILLIAM. Wordsworth, pp. 849-851. Cyclop\u00e6dia of Biography,\nedited by Elihu Rich. 8vo. Glasgow: Richard Griffin and Co.\n\nMOORE, THOMAS. Memoirs, Journal, and Correspondence of. Edited by the\nRight Honourable Lord John Russell, vol. iii. pp. 161, 163; vol. iv.\npp. 48, 335; vol. vii pp. 72, 85, 197-8; vol. viii. pp. 69, 73, 291.\n\n1856. CARLYON, CLEMENT. Early Years and Late Reflections, vol. i. 8vo.\nLondon: Whittaker and Co.\n\nHOOD, E. P. William Wordsworth: a Biography. 8vo. London: W. and F. G.\nCash.\n\nMASSON, DAVID. Essays, Biographical and Critical: chiefly on English\nPoets. Wordsworth, pp. 346-390. 8vo. Cambridge: Macmillan and Co.\nReprinted from _The North British Review_, August 1850.\n\nROGERS, SAMUEL. Recollections of the Table Talk of Samuel Rogers. 8vo.\nLondon: Edward Moxon.\n\nWILSON, JOHN. Noctes Ambrosianae, vols. i.-iii. 8vo. Edinburgh: William\nBlackwood and Sons. New Edition, 1864.\n\nWILSON, JOHN. Essays, Critical and Imaginative. Wordsworth, vol. i. pp.\n387-408. 8vo. Edinburgh: William Blackwood and Sons.\n\n1857. DE QUINCEY, THOMAS. Sketches, Critical and Biographic. On\nWordsworth's Poetry, vol. v. pp. 234-268. 8vo. Edinburgh: James Hogg\nand Sons.\n\nREED, HENRY. Lectures on the British Poets. Wordsworth, Lecture XV.\n8vo. London.\n\nWILSON, JOHN. Recreations of Christopher North, vol. ii. Sacred Poetry.\nWordsworth, pp. 54-70. 8vo. Edinburgh: William Blackwood and Sons.\n\n1858. BRIMLEY, GEORGE. Essays. Edited by William George Clark, M.A.\nWordsworth's Poems, pp. 104-187. 8vo. Cambridge: Macmillan and Co.\nSecond Edition, 1860. Third Edition, 1882. Reprinted from _Fraser's\nMagazine_, 1851.\n\nROBERTSON, F. W. Lectures and Addresses on Literary and Social Topics.\nWordsworth, pp. 203-256. 8vo. London: Smith, Elder and Co.\n\nTHE ENGLISH CYCLOP\u00c6DIA. A New Dictionary of Universal Knowledge.\nConducted by Charles Knight. Wordsworth, vol. vi. pp. 808-812.\n\n1859. MILL, J. S. Dissertations and Discussions. Thoughts on Poetry and\nits Varieties, i. 63-94. 8vo. London: John W. Parker and Son. Second\nEdition. Longmans, Green, Reader and Dyer. 1867.\n\n1860. CARRUTHERS, R. William Wordsworth. The _Encyclop\u00e6dia Britannica_,\nEighth Edition, xxi. 929-932. 4to. Edinburgh: Adam and Charles Black.\n\n1861. CRAIK, GEORGE L. A Compendious History of English Literature,\nand of the English Language from the Norman Conquest. Wordsworth, ii.\n435-456; 463-467; 473. 8vo. London: Griffin, Bohn and Co.\n\n1862. GORDON, MRS. \"Christopher North.\" A Memoir of John Wilson,\ncompiled from Family Papers and other Sources. 2 vols. 8vo. Edinburgh:\nEdmonston and Douglas. New Edition, 1879.\n\nPATTERSON, A. S. Poets and Preachers of the Nineteenth Century: Four\nLectures, Biographical and Critical, on Wordsworth, Montgomery, Hall,\nand Chalmers. 8vo. Glasgow: A. Hall.\n\n1863. RUSHTON, WILLIAM. The Classical and Romantic Schools of English\nLiterature, as represented by Spenser, Dryden, Pope, Scott, and\nWordsworth. The Afternoon Lectures on English Literature, delivered in\nDublin, pp. 43-92. 8vo. London: Bell and Daldy.\n\n1864. COLQUHOUN, J. C. Scattered Leaves of Biography. IV.--Life of\nWilliam Wordsworth. 8vo. London: Macintosh.\n\nKNIGHT, CHARLES. Passages from a Working Life during half a century:\nwith a prelude of Early Reminiscences, vol. iii. chap. ii. pp. 27-29.\n\n1865. The Imperial Dictionary of Universal Biography. Edited by J. F.\nWALLER. Wordsworth, vol. vi. p. 1389. 8vo. London: W. Mackenzie.\n\n1865. DENNIS, JOHN. Evenings in Arcadia. Edited by John Dennis. 12mo.\nLondon.\n\n1868. BUCHANAN, ROBERT. David Gray, and Other Essays, chiefly on\nPoetry. Sampson Low.\n\nMACDONALD, GEORGE. England's Antiphon, pp. 303-7. 8vo. London.\n\nSHAIRP, J. C. Studies in Poetry and Philosophy. Wordsworth: the Man\nand the Poet, pp. 1-115. 8vo. Edinburgh: Edmonston and Douglas. Third\nEdition, 1876. Fourth Edition, 1886.\n\n_Chambers's Encyclop\u00e6dia._ A Dictionary of Universal Knowledge for the\nPeople. Wordsworth, vol. x. pp. 272-274. New Edition, pp. 737-740.\n1892. 8vo. London: W. and R. Chambers.\n\n1869. CLOUGH, A. H. Poems and Prose Remains. Lecture on the Poetry of\nWordsworth, vol. i. pp. 309-325. 8vo. London: Macmillan and Co.\n\nG., F. J. The Old College, being the Glasgow University Album for\nMDCCCLXIX. Edited by Students. William Wordsworth, pp. 243-259. 8vo.\nGlasgow: James Maclehose.\n\nGRAVES, R. P. Recollections of Wordsworth and the Lake Country. The\nAfternoon Lectures on Literature and Art, delivered in Dublin, pp.\n275-321. 8vo. Dublin: William M'Gee.\n\nMARTINEAU, HARRIET. Biographical Sketches. Mrs. Wordsworth, pp.\n402-408. 8vo. London: Macmillan and Co.\n\nROBINSON, HENRY CRABB. Diary, Reminiscences, and Correspondence.\nSelected and edited by Thomas Sadler. 3 vols. 8vo. London: Macmillan\nand Co.\n\n1870. EMERSON, R. W. English Traits, First Visit to England. Bohn's\nStandard Library; also Macmillan and Co. 1883.\n\n1871. HUTTON, R. H. Essays, Theological and Literary. Wordsworth and\nhis Genius, vol. ii. Literary Essays, pp. 101-146. 8vo. London: Strahan\nand Co. Second Edition, 1877.\n\nTAINE, H. A. History of English Literature. Translated by H. Van\nLaun. With a preface by the author. Vol. ii. pp. 248; 260-265. 8vo.\nEdinburgh: Edmonston and Douglas.\n\nHALL, S. C. A Book of Memories of Great Men and Women of the Age, from\nPersonal Acquaintance. London: Virtue and Co. Wordsworth, pp. 287-318.\n\n1872. COOPER, THOMAS, Life of: An Autobiography. Reminiscence of\nWordsworth (first published in _Cooper's Journal_, May 1850), pp.\n287-295.\n\nDE MORGAN, AUGUSTUS. A Budget of Paradoxes. Wordsworth and Byron, p.\n435. 8vo. London: Longmans, Green and Co.\n\nNEAVES, CHARLES (Lord Neaves). A Lecture on Cheap and Accessible\nPleasures. With a Comparative Sketch of the Poetry of Burns and\nWordsworth, etc. 8vo. Edinburgh.\n\nYONGE, CHARLES D. Three Centuries of English Literature. Wordsworth,\npp. 251-267. 8vo. London: Longmans, Green and Co.\n\n1873. COLERIDGE, SARA. Memoir and Letters. Edited by her Daughter. 2\nvols. 8vo. London: Henry S. King and Co.\n\nDEVEY, JOSEPH. A Comparative Estimate of Modern English Poets.\nWordsworth, pp. 87-103. 8vo. London: Moxon and Son.\n\nLONSDALE, HENRY. The Worthies of Cumberland. William Wordsworth, vol.\niv. pp. 1-40. 8vo. London: George Routledge and Sons.\n\nMORLEY, H. A First Sketch of English Literature. 8vo. London: Cassell,\nPetter, and Galpin.\n\nNICHOLS, W. L. The Quantocks and their Associations. A Paper read\nbefore the Members of the Bath Literary Club. 12mo. Bath. Printed for\nPrivate Circulation. Second Edition. London: Sampson Low, Marston and\nCo.\n\n1874. BROOKE, STOPFORD A. Theology in the English Poets. Wordsworth,\npp. 93-286. 8vo. London: Henry S. King and Co.\n\nMASSON, DAVID. Wordsworth, Shelley, Keats, and other Essays.\nWordsworth, pp. 3-74. 8vo. London: Macmillan and Co.\n\nWORDSWORTH, DOROTHY. Recollections of a Tour made in Scotland, A.D.\n1803. Edited by J. C. Shairp. 8vo. Edinburgh: Edmonston and Douglas.\n\n1875. FLETCHER, MRS. Autobiography. With Letters and other Family\nMemorials. 8vo. Edinburgh: Edmonston and Douglas.\n\n1876. FORSTER, JOHN. The Works and Life of Walter Savage Landor. Vol.\ni. The Life. 8vo. London: Chapman and Hall.\n\nLAMB, CHARLES. The Life, Letters, and Writings of Charles Lamb. Edited,\nwith Notes and Illustrations, by Percy Fitzgerald. References to, and\nCriticisms of Wordsworth in vols. i. ii. 8vo. London: E. Moxon and Co.\n\nLOWELL, J. RUSSELL. Among my Books. Second Series. Wordsworth, pp.\n201-251. 8vo. London: Sampson Low, Marston, Searle and Rivington.\n\nMORLEY, HENRY. Cassell's Library of English Literature. Vols. iii.,\niv., v. Wordsworth. 8vo. London: Cassell, Petter, and Galpin.\n\nSTEDMAN, E. C. Victorian Poets. 8vo. London: Chatto and Windus.\n\nTICKNOR, GEORGE. Life, Letters, and Journals. 2 vols. 8vo. London:\nSampson Low, Marston, Searle and Rivington.\n\n1877. DOYLE, SIR FRANCIS H. Lectures on Poetry delivered at Oxford.\nSecond Series. Wordsworth Lectures, i.-iii. pp. 1-77. 8vo. London:\nSmith, Elder and Co.\n\nSHAIRP, J. C. On Poetic Interpretation of Nature. Wordsworth as an\nInterpreter of Nature, pp. 225-270. 8vo. Edinburgh: David Douglas.\n\nADAMS (W. DAVENPORT). Dictionary of English Literature. Wordsworth, pp.\n700-701. 8vo. London: Cassell, Petter, and Galpin.\n\n1878. DOWDEN, E. Studies in Literature, 1789-1877. The Prose Works of\nWordsworth, pp. 122-158. 8vo. London: C. Kegan Paul and Co.\n\nKNIGHT, WILLIAM. The English Lake District as Interpreted in the Poems\nof Wordsworth. 12mo. Edinburgh: David Douglas. Second Edition, revised\nand enlarged 1891.\n\nROSSETTI, W. M. Lives of Various Poets. Wordsworth, pp. 203-218. 8vo.\nLondon: E. Moxon and Son.\n\nThe Treasury of Modern Biography. Edited by Robert Cochrane.\nWordsworth, pp. 98-116. 8vo. Edinburgh: W. P. Nimmo.\n\n1879. BAGEHOT, WALTER. Literary Studies. Edited by Richard Holt Hutton.\nWordsworth, Tennyson, and Browning; or, Pure, Ornate, and Grotesque Art\nin English Poetry, vol. ii. pp. 338-390. 8vo. London: Longmans, Green\nand Co.\n\nKNIGHT, WILLIAM. Studies in Philosophy and Literature. Wordsworth, pp.\n283-317. Nature as Interpreted by Wordsworth, pp. 405-426. 8vo. London:\nC. Kegan Paul and Co.\n\nSTEPHEN, LESLIE. Hours in a Library. Third Series. Wordsworth's Ethics,\npp. 178-229. 8vo. London: Smith, Elder and Co.\n\n1880. BAYNE, PETER. Two Great Englishwomen: Mrs. Browning and Charlotte\nBront\u00eb. With an Essay on Poetry, illustrated from Wordsworth, Burns,\nand Byron, pp. xi.-lxxviii. 8vo. London: James Clarke and Co.\n\nCHURCH, R. W. William Wordsworth. The English Poets. Edited by Thomas\nHumphry Ward, vol. iv. pp. 1-15. 8vo. London: Macmillan and Co.\n\nMAIN, DAVID M. A Treasury of English Sonnets. Edited from the Original\nSources, with Notes and Illustrations, pp. 365-390. 8vo. Manchester:\nAlexander Ireland and Co.\n\nMYERS, F. W. H. Wordsworth (English Men of Letters). 8vo. Macmillan and\nCo.\n\n1881. CARLYLE, THOMAS. Reminiscences. Edited by James Anthony Froude.\nVol. ii. pp. 330-341. 8vo. London: Longmans, Green and Co.\n\nDOWDEN, E. The Correspondence of Robert Southey with Caroline Bowles.\nEdited, with an Introduction, by Edward Dowden. 8vo. Dublin: Hodges,\nFiggis, and Co.\n\nMILNER, GEORGE. The Literature and Scenery of the English Lake\nDistrict. Reprinted from the Papers of the Manchester Literary Club,\nvol. vii. pp. 1-21. 8vo. Manchester.\n\nSHAIRP, J. C. Aspects of Poetry, being Lectures delivered at Oxford.\nThe Three Yarrows, pp. 316-344. The White Doe of Rylstone, pp. 345-376.\n8vo. Oxford: Clarendon Press.\n\nSHORTHOUSE, J. H. On the Platonism of Wordsworth. A Paper read to the\nWordsworth Society, 19th July 1881. 4to. Birmingham: Cornish Brothers.\n\nSYMINGTON, A. J. William Wordsworth: a Biographical Sketch, with\nSelections from his Writings in Poetry and Prose. 2 vols. 8vo. London:\nBlackie and Son.\n\n1882. BUCKLAND, ANNA. The Story of English Literature. 8vo. London:\nCassell and Co.\n\nCOTTERILL, H. B. An Introduction to the Study of Poetry. Wordsworth,\npp. 208-241. 8vo. London: Kegan Paul, Trench and Co.\n\nOLIPHANT, MRS. The Literary History of England in the end of the\nEighteenth and beginning of the Nineteenth Century. 3 vols. 8vo.\nLondon: Macmillan and Co.\n\nSCHERER, J. A History of English Literature. Translated from the German\nby M. V. 8vo. London: Sampson Low and Co.\n\nSEELEY, J. R. Natural Religion. By the Author of _Ecce Homo_, pp.\n94-111. 8vo. London: Macmillan and Co.\n\nIRELAND, ALEXANDER. Recollections of George Dawson, etc., pp. 22-25.\n\n1883. CAINE, T. HALL. Cobwebs of Criticism. A Review of the First\nReviewers of the \"Lake,\" \"Satanic,\" and \"Cockney\" Schools. Wordsworth,\npp. 1-29. 8vo. London: Elliot Stock.\n\nDENNIS, JOHN. Heroes of Literature: English Poets. William Wordsworth,\npp. 278-299. 8vo. London: S.P.C.K.\n\nHALL, S. C. Retrospect of a Long Life: from 1815 to 1883. Wordsworth,\nvol. ii. pp. 36-42. 8vo. London: Richard Bentley and Son.\n\nHAWTHORNE, N. English Note-Books, vol. ii. 8vo. London: Kegan Paul,\nTrench and Co.\n\nThe Lyme Parish Church Magazine. Lyme-Regis: Walton.\n\n1884. HOFFMANN, F. A. Poetry, its Origin, Nature, and History.\nWordsworth, chap. xxvi. pp. 359-375. 8vo. London: Thurgate and Sons.\n\nKERR, R. N. Our English Laureates and the Birds. Dundee: John Leng\nand Co. Pp. 29-51. (Originally published in the _Newcastle Weekly\nChronicle_.)\n\nNICHOLSON, ALBERT. The Literature of the English Lake District.\nManchester.\n\nSHORTER, C. K. William Wordsworth. The National Cyclop\u00e6dia: a\nDictionary of Universal Knowledge. New Edition. 8vo. London: W.\nMackenzie.\n\nTRAILL, H. D. Coleridge. English Men of Letters. 8vo. London: Macmillan\nand Co.\n\n1885. COURTHOPE, W. J. The Liberal Movement in English Literature.\nEssay III. Wordsworth's Theory of Poetry, pp. 71-108. 8vo. London: John\nMurray.\n\nELIOT, GEORGE. George Eliot's Life, as related in her Letters and\nJournals. By J. W. Cross. Vol. i. p. 61; iii. 388. 8vo. Edinburgh: W.\nBlackwood and Sons.\n\nHUTTON, LAWRENCE. Literary Landmarks, pp. 321-7. London: T. Fisher\nUnwin.\n\nCARNE, JOHN, Letters of, 1813-1837. Privately printed. Pp. 133-138.\n\nTAYLOR, SIR HENRY. Autobiography 1800-1875. 2 vols. 8vo. London:\nLongmans, Green and Co.\n\n1886. DAWSON, GEORGE. Biographical Lectures. Edited by George St.\nClair. The Poetry of Wordsworth, pp. 251-307. 8vo. London: Kegan Paul,\nTrench and Co.\n\nLAW, DAVID. Wordsworth's Country. A series of Five Etchings of the\nEnglish Lake District. 24mo. London: Robert Dunthorne.\n\nLEE, EDMUND. Dorothy Wordsworth. The Story of a Sister's Love. 8vo.\nLondon: James Clarke and Co. New and revised edition 1894.\n\nNICHOLSON, CORNELIUS. Wordsworth and Coleridge: Two Parallel Sketches.\nVentnor: R. Madley. 1886.\n\nNOEL, HON. RODEN B. W. Essays on Poetry and Poets. Wordsworth, pp.\n132-149. 8vo. London: Kegan Paul, Trench and Co.\n\nSWINBURNE, A. C. Miscellanies, Wordsworth and Byron, pp. 63-156. 8vo.\nLondon. 1886.\n\nLAUNCELOT CROSS (F. Carr). Thinkers of the World in relation to the\nNew Church. 1. Childhood as revealed in Wordsworth; 2. Wordsworth on\nInfancy and Youth. N.D.\n\n1887. DE VERE, AUBREY. Essays, chiefly on Poetry. The Genius and\nPassion of Wordsworth, vol. i. pp. 101-173; The Wisdom and Truth of\nWordsworth's Poetry, vol. i. pp. 174-264; Recollections of Wordsworth,\nvol. ii. pp. 275-295. 8vo. London: Macmillan and Co.\n\nGOODWIN, H., and WILLIAM KNIGHT. Through the Wordsworth Country. 8vo.\nLondon: Swan Sonnenschein, Lowrey and Co. Third Edition, 1892.\n\nLOWELL, J. RUSSELL. Democracy and other Addresses, pp. 137-156. 8vo.\nLondon: Macmillan and Co.\n\nMemorials of Coleorton: being Letters from Coleridge, Wordsworth and\nhis Sister, Southey, and Sir Walter Scott, to Sir George and Lady\nBeaumont of Coleorton, Leicestershire, 1803 to 1834. Edited, with\nIntroduction and Notes, by William Knight. 2 vols. 8vo. Edinburgh:\nDavid Douglas.\n\nSUTHERLAND, J. M. William Wordsworth: the Story of his Life, with\nCritical Remarks on his Writings. 8vo. London: Elliot Stock.\n\n1888. ARNOLD, MATTHEW. Essays in Criticism. Second Series. Wordsworth,\npp. 122-162. 8vo. London: Macmillan and Co.\n\nCHURCH, R. W. Dante and other Essays. William Wordsworth, pp. 193-219.\n8vo. London: Macmillan and Co.\n\nDOWDEN, E. Transcripts and Studies. The Text of Wordsworth's Poems, pp.\n112-152. 8vo. London: Kegan Paul, Trench and Co. Reprinted from _The\nContemporary Review_.\n\nINGLEBY, C. M. Essays. Edited by his Son. 8vo. Tr\u00fcbner and Co.\n\nMINTO, W. William Wordsworth. The _Encyclop\u00e6dia Britannica_, Ninth\nEdition, xxiv. pp. 668-676. 4to. Edinburgh: Adam and Charles Black.\n\nSANDFORD, MRS. HENRY. Thomas Poole and his Friends. 2 vols. 8vo.\nLondon: Macmillan and Co.\n\n1889. CLAYDEN, P. W. Rogers and his Contemporaries. 2 vols. 8vo.\nLondon: Smith, Elder and Co.\n\nHOWITT, MARY. Autobiography. Edited by her daughter Margaret Howitt. 2\nvols. 8vo. London: William Isbister.\n\nLetters from the Lake Poets, Samuel Taylor Coleridge, William\nWordsworth, Robert Southey, to Daniel Stuart. Printed for Private\nCirculation. Wordsworth, pp. 329-386. 8vo. London: West, Newman and Co.\n\nPATER, WALTER. Appreciations. With an Essay on Style. 8vo. London:\nMacmillan and Co.\n\nWORDSWORTHIANA. A Selection from Papers read to the Wordsworth Society.\nEdited by William Knight. 8vo. London: Macmillan and Co.\n\n1890. BOLAND, R. Yarrow, its Poets and Poetry, pp. 77-9. Dalbeattie.\n\nBROOKE, STOPFORD A. Dove Cottage, Wordsworth's Home from 1800-1808.\nDecember 21, 1799, to May 1808. 12mo. London: Macmillan and Co.\n\nDAVEY, SIR HORACE. Wordsworth. An Address read to the Stockton Literary\nand Philosophical Society. 8vo. Stockton-on-Tees. 1890.\n\nDAWSON, W. J. Makers of Modern English. Ch. x. William Wordsworth; ch.\nxi. The Connection between Wordsworth's Life and Poetry; ch. xii. Some\nCharacteristics of Wordsworth's Poetry; ch. xiii. Wordsworth's View of\nNature and Man; ch. xiv. Wordsworth's Patriotic and Political Poems;\nch. xv. Wordsworth's Personal Characteristics; ch. xvi. Concluding\nSurvey.\n\nMALLESON, F. A. Holiday Studies of Wordsworth, by Rivers, Woods, and\nAlps. The Wharfe, the Duddon, and the Stelvio Pass. 4to. Cassell and Co.\n\nM'WILLIAMS, R. Handbook of English Literature, pp. 456-466. London:\nLongmans, Green and Co.\n\nTUTIN, J. R. Birthday Texts. W. P. Nimmo.\n\n1891. DE QUINCEY, THOMAS. De Quincey Memorials. Being Letters and\nRecords here first published.\u2026 Edited, with Introduction, Notes,\nand Narrative, by Alexander H. Japp. 2 vols. 8vo. London: William\nHeinemann.\n\nGOSSE, E. Gossip in a Library. _Peter Bell_ and his Tormentors, pp.\n253-267. 8vo. London: W. Heinemann. Third Edition, 1893.\n\nGRAHAM, P. A. Nature in Books: some Studies in Biography. 8vo. London:\nMethuen and Co.\n\nMORLEY, JOHN. Studies in Literature. Wordsworth, pp. 1-53. 8vo. London:\nMacmillan and Co.\n\nSCHERER, EDMOND. Essays on English Literature, translated by George\nSaintsbury, with a Critical Introduction. 8vo. London: Sampson Low,\nMarston and Co.\n\nTUTIN, J. R. The Wordsworth Dictionary of Persons and Places, with\nthe Familiar Quotations from his Works (including full Index) and a\nchronologically-arranged List of his best Poems. 8vo. Hull: J. R. Tutin.\n\nWORDSWORTH, ELIZABETH. William Wordsworth. 8vo. London: Percival and Co.\n\n1892. CAIRD, EDWARD. Essays on Literature and Philosophy. Wordsworth,\nvol. i. pp. 147-189. 8vo. Glasgow: James Maclehose and Sons.\n\nDAWSON, W. J. Quest and Vision: essays in Life and Literature.\nWordsworth and his Message, pp. 41-72. 8vo. London: Hodder and\nStoughton.\n\nTUTIN, J. R. An Index to the Animal and Vegetable Kingdoms of\nWordsworth. Hull.\n\nTUTIN, J. R. Wordsworth in Yorkshire. First published in _Yorkshire\nNotes and Queries_. Part xix.\n\nWINTRINGHAM, W. H. The Birds of Wordsworth: Poetically, Mythologically,\nand Comparatively examined. 8vo. London: Hutchinson and Co.\n\n1894. CAMPBELL, J. <DW18>s. Samuel Taylor Coleridge. A Narrative of the\nEvents of his Life. 8vo. London: Macmillan and Co.\n\nMINTO, W. The Literature of the Georgian Era. Edited, with a\nBiographical Introduction, by William Knight, LL.D., pp. 140-177. 8vo.\nEdinburgh: William Blackwood and Sons.\n\nRAWNSLEY, H. D. Literary Associations of the English Lakes. 2 vols.\n8vo. Glasgow: James Maclehose and Sons.\n\n1895. COLERIDGE, S. T. Letters. Edited by Ernest Hartley Coleridge. 2\nvols. 8vo. London: William Heinemann.\n\nIn Lakeland, a Wordsworthic Pilgrimage, Easter 1895.\n\n1896. SAINTSBURY, GEORGE. A History of Nineteenth Century Literature\n(1780-1895). Wordsworth, pp. 49-56. 8vo. London: Macmillan and Co.\n\nA REMINISCENCE OF WORDSWORTH DAY. Cockermouth, April 7, 1896. Edited by\nthe Rev. H. D. Rawnsley, Hon. Canon of Carlisle. Cockermouth: A. Lang.\n\n[476] There are numerous notes and letters on Wordsworth in such\nJournals as _The Athen\u00e6um_, _The Academy_, _Notes and Queries_, the\nexamination of which will repay perusal. In _Notes and Queries_ there\nare at least twenty-four valuable ones which cannot be recorded\nhere.--ED.\n\n[477] A criticism of the \"dancing daffodils.\"--ED.\n\n\nIV\n\nCRITICAL ESTIMATES IN BOOKS, PAMPHLETS, MAGAZINES, AND REVIEWS\n\nIn the following section when the name of an author is placed within\nbrackets, it is to be understood that the name was not given on the\npublication of the Review, but that it is otherwise known.--ED.\n\n1793. \"Descriptive Sketches in Verse.\" _The Monthly Review_, xii. 216.\n\n\"An Evening Walk.\" _The Monthly Review_, xii. 218.\n\n1799. \"Lyrical Ballads, with a few other Poems.\" _The Monthly Review_,\nxxix. 202; _The British Critic_, xiv. 364.\n\n1801. \"Lyrical Ballads, with other Poems.\" In 2 vols. Second Edition.\n_The British Critic_, xvii. 125.\n\n1802. \"Lyrical Ballads, with other Poems.\" Vol. ii. _The Monthly\nReview_, xxxviii. 209.\n\n1807. \"Poems.\" In 2 vols. _The Edinburgh Review_, xi. 214. By Francis\nJeffrey. _Monthly Literary Recreations_, 65. (By Lord Byron.)\n\n1808. \"Poems.\" In 2 vols. _The Eclectic Review_, vii. 35.\n\n1809. \"Poems.\" In 2 vols. _The British Critic_, xxxiii. 298.\n\n1810. \"Concerning the relations of Great Britain, Spain, and Portugal,\nto each other, and to the Common Enemy, at this Crisis, etc.\" _The\nBritish Critic_, xxxiv. 305.\n\n1814. \"The Excursion; being a portion of The Recluse, a Poem.\" _The\nEdinburgh Review_, xxiv. 1. (By Francis Jeffrey); _The Quarterly\nReview_, xii. 100. (By Charles Lamb.)\n\n1815. \"Poems; including Lyrical Ballads, and the miscellaneous\npieces of the Author. With additional Poems, a new Preface, and\na supplementary Essay.\" _The Monthly Review_, lxxviii. 225; _The\nQuarterly Review_, xiv. 201. (By W. Gifford.)\n\n\"The Excursion; being a portion of The Recluse: a Poem.\" _The Eclectic\nReview_, xxi. 13; _The Monthly Review_, lxxvi. 123; _The British\nCritic_, iii. 449.\n\n\"The Excursion: being a portion of The Recluse: a Poem.\" _The British\nReview_, vi. 49.\n\n\"The White Doe of Rylstone.\" _The Quarterly Review_, xiv. 201. (By W.\nGifford.) _The Edinburgh Review_, xxv. 355. (By Francis Jeffrey.) _The\nMonthly Review_, lxxviii. 235.\n\n1816. \"The White Doe of Rylstone.\" _The Eclectic Review_, xxiii. 33.\n\n\"Thanksgiving Ode, with other short Pieces.\" _The Eclectic Review_,\nxxiv. 1.\n\n\"The White Doe of Rylstone.\" _The British Review_, vii. 370.\n\n1817. \"Thanksgiving Ode, with other short Pieces.\" _The Monthly\nReview_, lxxxii. 98.\n\n\"Observations on Mr. Wordsworth's Letter relative to a new Edition of\nBurns's Works.\" _Blackwood's Magazine_, i. 261.\n\n\"Vindication of Mr. Wordsworth's Letter to Mr. Gray on a new Edition of\nBurns.\" _Blackwood's Magazine_, ii. 65.\n\n\"Letter occasioned by N.'s Vindication of Mr. Wordsworth in last\nNumber.\" _Blackwood's Magazine_, ii. 201.\n\n1818. \"Essays on the Lake School of Poetry. I. Wordsworth's White Doe\nof Rylstone.\" _Blackwood's Magazine_, iii. 369.\n\n1819. \"Peter Bell: a Tale in Verse.\" _The Edinburgh Monthly Review_,\nii. 654; _Blackwood's Magazine_, v. 130; _The Eclectic Review_, xxx.\n62; _The Monthly Review_, lxxxix. 419; _The Literary Gazette_, 273.\n\n\"The Waggoner: a Poem, to which are added Sonnets.\" _The Monthly\nReview_, xc. 36; _The Edinburgh Monthly Review_, ii. 654; _Blackwood's\nMagazine_, v. 332; _The Eclectic Review_, xxx. 62.\n\n\"Benjamin the Waggoner, a ryghte merrie and conceitede Tale in Verse.\"\n_The Monthly Review_, xc. 41.\n\n\"Peter Bell: a Lyrical Ballad.\" _The Monthly Review_, lxxxix. 422; _The\nEclectic Review_, xxix. 473.\n\n\"Memoir of William Wordsworth, Esq.\" (with a portrait). _The New\nMonthly Magazine_, i. 48.\n\n1820. \"Lake School of Poetry--Mr. Wordsworth.\" _The New Monthly\nMagazine_, xiv. 361.\n\n\"Wordsworth.\" _The London Magazine_, i. 275, 435.\n\n\"Wordsworth's River Duddon, and other Poems.\" _The Gentleman's\nMagazine_, xc. 344; _The London Magazine_, i. 618; _The London Review\nand Literary Journal_, 523; _Blackwood's Magazine_, vii. 206; _The\nEclectic Review_, xxxii. 170; _The Monthly Review_, xciii. 132.\n\n\"The River Duddon, and other Poems.\" _The British Review_, xvi. 37.\n\n\"Essay on Poetry, with Observations on the Living Poets.\" _The London\nMagazine_, ii. 557.\n\n\"The Dead Asses: A Lyrical Ballad.\" _The Monthly Review_, xci. 322.\n\n\"Description of the Scenery of the Lakes.\" _Blackwood's Magazine_, xii.\n\n1822. \"Memorials of a Tour on the Continent.\" _The British Critic_,\nxviii. 522; _The Edinburgh Review_, xxxvii. 449. (By F. Jeffrey.)\n_Blackwood's Magazine_, xii. 175; _The British Review_, xx. 459; _The\nLiterary Gazette_, 192, 210; _The Museum_, i. 339.\n\n\"Ecclesiastical Sketches.\" _Blackwood's Magazine_, xii. 175; _The\nBritish Critic_, xviii. 522; _The Literary Gazette_, 123.\n\n1829. \"An Essay on the Theory and the Writings of Wordsworth.\"\n_Blackwood's Magazine_, xxvi. 453, 593, 774, 894.\n\n1831. \"Literary Characters--No. III. Mr. Wordsworth.\" _Fraser's\nMagazine_, iii. 557. By Pierce Pungent.\n\n\"Selections from the Poems of W. Wordsworth, chiefly for the use of\nSchools and Young Persons.\" _The New Monthly Magazine_, xxxiii. 304;\n_The Monthly Review_, ii. 602.\n\n1832. \"Gallery of Literary Characters--No. XXIX. William Wordsworth.\"\n_Frasers Magazine_, vi. 313.\n\n\"Poetical Works.\" New Edition. _Fraser's Magazine_, vi. 607.\n\n1833. \"What is Poetry? The two kinds of Poetry.\" _The Monthly\nRepository_, New Series, vii. 60, 714. By Antiquus (John Stuart Mill).\n\n1834. \"The Poetical Works of William Wordsworth.\" A New Edition. _The\nQuarterly Review_, lii. 317. (By Henry Taylor.)\n\n\"Selections from the Poems of William Wordsworth.\" _The Quarterly\nReview_, lii. 317. (By Henry Taylor.)\n\n1835. \"Yarrow Revisited, and other Poems.\" _The New Monthly Magazine_,\nxliv. 12; _Blackwood's Magazine_, xxxvii. 699; _Fraser's Magazine_,\nxi. 689; _The Quarterly Review_, liv. 181; _The Dublin University\nMagazine_, v. 680; _The Monthly Literary Gazette_, 257; _The Athen\u00e6um_,\n293; _The Monthly Review_, cxxxvii. 605; _The Monthly Repository_, New\nSeries, ix. 430.\n\n1838. \"Letter from Tomkins--Bagman _versus_ Pedlar.\" _Blackwood's\nMagazine_, xliv. 509.\n\n\"Our Pocket Companions.\" _Blackwood's Magazine_, xliv. 584.\n\n\"The Sonnets of William Wordsworth.\" _The Literary Gazette_, 540.\n\n1839. \"Lake Reminiscences, from 1807 to 1830--Nos. I.-III. William\nWordsworth; No. IV. William Wordsworth and Robert Southey.\" _Taits\nEdinburgh Magazine_, vi. I, 90, 246, 453. (By Thomas de Quincey.)\n\n1841. \"Wordsworth.\" _Blackwood's Magazine_, xlix. 359.\n\n\"The Sonnets of William Wordsworth.\" _The Quarterly Review_, lxix. 1.\n(By Henry Taylor.)\n\n1842. \"Poems, chiefly of Early and Late Years; including The\nBorderers.\" _The Monthly Review_, ii. 270; _The Eclectic Review_,\nlxxvi. 568; _The Christian Remembrancer_, iii. 655; _The Athen\u00e6um_, 757.\n\nCriticism in a Review of \"The Book of the Poets\" in _The Athen\u00e6um_. (By\nElizabeth Barrett Browning.)\n\n\"Poems of the Fancy,\" \"Poems of the Imagination.\" _The Gentleman's\nMagazine_, xvii. 3.\n\n\"Imaginary Conversation. Southey and Porson.\" _Blackwood's Magazine_,\nlii. 687. (By Walter Savage Landor.)\n\n1844. \"Oswald Herbst's Letters from England--No. II. Wordsworth and his\nPoetry.\" _Tait's Edinburgh Magazine_, xi. 641.\n\n1845. \"On Wordsworth's Poetry.\" _Tait's Edinburgh Magazine_, xii. 545.\n(By Thomas de Quincey.)\n\n\"Poems, chiefly of Early and Late Years; including The Borderers.\" _The\nGentleman's Magazine_, xxiv. 555.\n\n\"William Wordsworth.\" _Hogg's Weekly Instructor_, ii. 243.\n\n1850. \"William Wordsworth.\" _Chambers's Papers for the People_, v. I.\n\n\"William Wordsworth.\" _The Gentleman's Magazine_, New Series, xxxiii.\n668; _The Athen\u00e6um_, 447; _Sharpe's London Magazine_, xi. 349.\n\n\"Poetical Works.\" _The Eclectic Review_, xcii. 56; _The North British\nReview_, xiii. 473. (By David Masson.)\n\n\"The Prelude, or Growth of a Poet's Mind.\" _The Eclectic Review_,\nxcii. 550; _The Gentleman's Magazine_, xxxiv. 459; _Fraser's Magazine_,\nxlii. 119; _The Westminster Review_, liv. 271; _The British Quarterly\nReview_, xii. 549; _Tait's Edinburgh Magazine_, xvii. 521; _The Dublin\nUniversity Magazine_, xxxvi. 329; _The Literary Gazette_, 513; _The\nAthen\u00e6um_, 805; _Sharpe's London Journal_, xii. 185; _The London\nExaminer_, 478.\n\n\"William Wordsworth.\" _Household Words_, i. 210.\n\n\"Wordsworth and his Poetry.\" _Chambers's Journal_, xiii. 363. By C. R.\n\n\"Poetical Works.\" _The Christian Observer_, i. 307.\n\n\"Religious Character of Wordsworth's Poetry.\" _The Christian Observer_,\ni. 381.\n\n\"Death of Wordsworth.\" _The London Examiner_, 259, 265.\n\n\"The Poetry of Wordsworth.\" _The Wesleyan Methodist Magazine_, 27.\n\n1851. \"Memoirs of William Wordsworth.\" _Fraser's Magazine_, xliv.\n101, 186; _The Dublin University Magazine_, xxxviii. 77; _The Dublin\nReview_, xxxi. 313; _The Gentleman's Magazine_, New Series, xxxvi. 107;\n_The Athen\u00e6um_, 445.\n\n\"Poetical Works.\" _The Dublin Review_, xxxi. 313.\n\n\"The Prelude, or Growth of a Poet's Mind.\" _The Prospective Review_,\nvii. 94.\n\n1852. \"Memoirs of William Wordsworth.\" By Christopher Wordsworth. _The\nQuarterly Review_, xcii. 182.\n\n\"Memoirs of William Wordsworth, compiled from Authentic Sources.\" By\nJanuary Searle. _The Quarterly Review_, xcii. 182.\n\n\"Lives of the Illustrious. William Wordsworth.\" _The Biographical\nMagazine_, I.\n\n1853. \"William Wordsworth.\" _Sharpe's London Journal_, xvii. 148.\n\n\"The Genius of Wordsworth harmonised with the Wisdom and Integrity of\nhis Reviewers.\" By J. C. Wright. _The Athen\u00e6um_, 824.\n\n1855. \"William Wordsworth.\" _The Leisure Hour_, iv. 439.\n\n1856. \"Poems of William Wordsworth, D.C.L.\" _The Dublin Review_, xl.\n338.\n\n\"William Wordsworth.\" _Sharpe's London Journal_, xi. 349.\n\n1857. \"William Wordsworth. A Biography.\" By Edwin Paxton Hood. _The\nNational Review_, iv. 1.\n\n\"The Poetical Works of William Wordsworth.\" A New Edition. _The\nAthen\u00e6um_, 109.\n\n\"The Earlier Poems of William Wordsworth.\" Edited by William Johnston.\n_The Athen\u00e6um_, 109.\n\n\"Wordsworth's Sister.\" By E. P. Hood. _The Leisure Hour_.\n\n1859. \"Passages from Wordsworth's Excursion.\" Illustrated with Etchings\non Steel. By Agnes Fraser. _The Athen\u00e6um_, i, 361.\n\n\"William Wordsworth. A Biography.\" By Edwin Paxton Hood. _The Christian\nObserver_, lix. 156.\n\n\"A Talk about Rydal Mount.\" _Once a Week_, i. 107. (By Thomas\nBlackburne.)\n\n1860. \"Collected Works of William Wordsworth.\" A New and Revised\nEdition. _The British Quarterly Review_, xxxi. 79.\n\n\"The Prelude, or Growth of a Poet's Mind.\" _The British Quarterly\nReview_, xxxi. 79.\n\n\"Richard Baxter paraphrased by Wordsworth.\" Varieties in _The Leisure\nHour_.\n\n1863. \"The Poems of Hood and of Wordsworth.\" _The Christian Observer_,\nlxiii. 677.\n\n\"William Wordsworth.\" _The Leisure Hour_, xii. 628.\n\n1864. \"Wordsworth, Tennyson, and Browning; or, Pure, Ornate, and\nGrotesque Art in English Poetry.\" _The National Review_, xix. 27. W. B.\n(Walter Bagehot.)\n\n\"Wordsworth: the Man and the Poet.\" _The North British Review_, xli. 1.\n(By J. C. Shairp.)\n\n1865. \"Two Poets of England. Wordsworth and Landor.\" _Temple Bar_, xvi.\n106.\n\n\"Wordsworth at Rydal Mount in 1849.\" In _The Leisure Hour_.\n\n1866. \"Memories of the Authors of the Age.\" William Wordsworth. _The\nArt Journal_, xviii. 245, 273. S. C. Hall and Mrs. S. C. Hall.\n\n1868. \"Characteristic Letters\"; communicated by the author of Men I\nhave Known--W. Wordsworth.\n\n1870. \"Wordsworth at Work.\" _Chambers's Journal_, xlvii. 247.\n\n\"Personal Recollections of the Lake Poets.\" In _The Leisure Hour_, 651.\nThe Rev. Edward Whately.\n\n\"Wordsworth's Study,\" in _The Leisure Hour_.\n\n1871. \"A Century of Great Poets, from 1750 downwards--No. III. William\nWordsworth.\" _Blackwood's Magazine_, cx. 299.\n\n1872. \"Wordsworth impartially weighed.\" _Temple Bar_, xxxiv. 310.\n\n1873. \"Wordsworth.\" _Macmillan's Magazine_, xxviii. 289. Sir John Duke\nColeridge.\n\n\"Wordsworth's Three Yarrows.\" _Good Words_, xiv. 649. J. C. Shairp.\n\n1874. \"On Wordsworth.\" _The Fortnightly Review_, xxi. 455. Walter H.\nPater.\n\n\"William and Dorothy Wordsworth.\" _Chambers's Journal_, li. 513.\nWilliam Chambers.\n\n\"White Doe of Rylstone.\" _Good Words_, xv. 269. J. C. Shairp.\n\n\"The Cycle of English Song.\" _Temple Bar_, xl. 478.\n\n1875. \"The Prose Works of William Wordsworth.\" Edited by the Rev. A.\nB. Grosart. _The Fortnightly Review_, xxiv. 449. Edward Dowden. _The\nDublin University Magazine_, lxxxvi. 756.\n\n1876. \"Hours in a Library.\" Wordsworth's Ethics. _The Cornhill\nMagazine_, xxxiv. 206. Leslie Stephen.\n\n\"The Prose Works of William Wordsworth.\" Wordsworth and Gray. _The\nQuarterly Review_, cxli. 104.\n\n\"The Prose Works of William Wordsworth.\" Edited by the Rev. A. B.\nGrosart. _The London Quarterly Review_, xlvii. 102.\n\n1877. \"The Wordsworths at Brinsop Court.\" _Temple Bar_, xlix. 110.\n\n1878. \"The Text of Wordsworth's Poems.\" _The Contemporary Review_,\nxxxiii. 734. Edward Dowden.\n\n\"Wordsworth.\" _Transactions of the Cumberland Association for the\nAdvancement of Literature and Science_, Part III. William Knight.\n\n1879. \"Wordsworth.\" _Macmillan's Magazine_, xl. 193. Matthew Arnold.\n\n\"Matthew Arnold's Selections from Wordsworth.\" _The Fortnightly\nReview_, xxxii. 686. J. A. Symonds.\n\n1880. \"Milton and Wordsworth.\" _Temple Bar_, lx. 106.\n\n\"Wordsworth.\" _Frasers Magazine_, ci. 205. Edward Caird.\n\n\"Wordsworth's Poems.\" Selected and edited by Matthew Arnold. _The\nModern Review_, i, 235. William Knight.\n\n\"The Genius and Passion of Wordsworth.\" _The Month_, xxxviii. 465;\nxxxix. 1. Aubrey De Vere.\n\n1881. \"Carlyle's Reminiscences.\" Carlyle's Impressions of Wordsworth.\n_The Nineteenth Century_, lx. 1010. Henry Taylor.\n\n\"Wordsworth.\" _The Churchman_, March.\n\n1882. \"Wordsworth and Byron.\" _The Quarterly Review_, cliv. 53. Matthew\nArnold.\n\n\"My Rare Book.\" _The Gentleman's Magazine_, New Series, xxviii. 531.\nFrederick Wedmore.\n\n\"Wordsworth's Two Styles.\" _The Modern Review_, iii. 525. R. H. Hutton.\n\n\"A French Critic on Wordsworth--M. Sch\u00e9rer.\" _The Saturday Review_,\nliv. 565.\n\n\"Poetical Works.\" Edited by William Knight. _The Academy_, xxii. III.\nEdward Dowden. _The Spectator_, lv. 1141; _The Modern Review_, iii,\n861.\n\n\"Transactions of the Wordsworth Society--No. I. Bibliography of the\nPoems; No. II. On the Platonism of Wordsworth.\" J. H. Shorthouse. _The\nSpectator_, lv. 238.\n\n\"The Weak Side of Wordsworth.\" _The Spectator_, lv. 687.\n\n1883. \"Wordsworth and the Duddon.\" _Good Words_, xxiv. 573. F. A.\nMalleson.\n\n\"Address to the Wordsworth Society.\" _Macmillan's Magazine_, xlviii.\n154. Matthew Arnold.\n\n\"Poetical Works.\" Edited by William Knight. _The Spectator_, lvi. 614.\n\n\"In Wordsworth's Country.\" _The Yorkshire Illustrated Monthly_, 32. N.\nPaton.\n\n\"Poets' Pictures.\" _Temple Bar_, lxxx. 232.\n\n\"Old Age in Bath, to which are added a few unpublished remains of\nWordsworth.\" Henry Julian Hunter.\n\n1884. \"Wordsworth and Byron.\" _The Nineteenth Century_, xv. 583, 764.\nA. C. Swinburne.\n\n\"The Wisdom and Truth of Wordsworth's Poetry.\" _The Catholic World_.\nAubrey de Vere.\n\n\"Wordsworth and 'Natural Religion.'\" _Good Words_, xxv. 307. J. C.\nShairp.\n\n\"Wordsworth's Relations to Science.\" _Macmillan's Magazine_, l. 202. R.\nSpence Watson.\n\n\"Sonnets.\" Edited by the Archbishop of Dublin. _The Academy_, xxv. 108.\nSamuel Waddington.\n\n\"The Literature of the English Lake District.\" _The Manchester\nQuarterly_, No. xii. Albert Nicholson.\n\n\"A Stroll up the Brathay.\" _Good Words_, xxv. 392. Herbert Rix.\n\n\"The Liberal Movement in English Literature--III. Wordsworth's Theory\nof Poetry.\" _The National Review_, iv. 512. William John Courthope.\n\n1885. \"Wordsworth's Influence in Scotland.\" _The Spectator_, lviii.\n1292.\n\n\"Dorothy Wordsworth.\" _The Christian World Magazine_, 314, 360, 464,\n548.\n\n\"Archbishop Sandys' Endowed School, Hawkshead, near Ambleside.\nTercentenary Commemoration.\"\n\n1886. \"Wordsworth.\" _Temple Bar_, lxxvii. 336. Charles F. Johnson.\n\n\"Poetical Works.\" Edited by William Knight. _The Spectator_, lix. 355.\n\n1887. \"Memorials of Coleorton.\" Edited by William Knight. _The\nSpectator_, lx. 1656.\n\n\"Wordsworth, the Poet of Nature.\" _The Sunday Magazine_, xvi. 166.\nHenry C. Ewart.\n\n\"The Mystical Side of Wordsworth.\" _The National Review_, ix. 833. John\nHogben.\n\n1888. \"Mr. Morley on Wordsworth.\" _The Spectator_, lxi. 1807.\n\n\"The Recluse.\" _The Spectator_, lxi. 1852.\n\n\"Selections from Wordsworth.\" By William Knight, and other Members of\nthe Wordsworth Society. _The Spectator_, lxi. 1852.\n\n1889. \"Selections from Wordsworth.\" By William Knight, and other\nMembers of the Wordsworth Society. _The Athen\u00e6um_, i. 109.\n\n\"A Modern Poetic Seer.\" _The Christian World._\n\n\"The Recluse.\" _The Edinburgh Review_, clxix. 415. _The Academy_, xxxv.\n17. Edward Dowden. _The Saturday Review_, lxvii. 43; _The Athen\u00e6um_, i.\n109.\n\n\"Complete Poetical Works.\" With an Introduction by John Morley. _The\nEdinburgh Review_, clxix. 415. _The Academy_, xxxv. 17. Edward Dowden.\n_The Athen\u00e6um_, i. 109.\n\n\"Wordsworthiana.\" Edited by William Knight. _The Edinburgh Review_,\nclxix. 415; _The Academy_, xxxv. 229. Edward Dowden. _The Spectator_,\nlxii. 369.\n\n\"Wordsworth's Great Failure.\" _The Nineteenth Century_, xxvi. 435.\nWilliam Minto.\n\n\"The Life of William Wordsworth.\" By William Knight. _The Saturday\nReview_, lxvii. 732; _The Spectator_, lxiii. 143; _The Athen\u00e6um_, i.\n719.\n\n\"Wordsworth and the Quantock Hills.\" _The National Review_, xiv. 67.\nWilliam Greswell.\n\n1890. \"Lyrical Ballads.\" Edited by Edward Dowden. _The Spectator_,\nlxiv. 479.\n\n\"The Story of a Sonnet.\" _The Athen\u00e6um_, i. 641. James Bromley.\n\n\"Some Early Poems of Wordsworth.\" _The Athen\u00e6um_, ii. 320. J. D. C.\n(James <DW18>s Campbell).\n\n\"The Lyrical Ballads of 1800.\" _The Athen\u00e6um_, ii. 699. J. D. C.\n\n\"Wordsworth's Verses in his Guide to the Lake Country.\" _The Athen\u00e6um._\nJ. D. C.\n\n1891. \"Wordsworth's 'Immortal' Ode.\" _The Parent's Review_, i. 864,\n944; ii. 70.\n\n\"The Wordsworth Dictionary of Persons and Places,\" with the Familiar\nQuotations from his Works. (By J. R. Tutin.) _The Athen\u00e6um_, ii. 756,\n834.\n\n\"The College Days of William Wordsworth.\" _The Eagle_, xvi., No. 94. G.\nC. M. Smith.\n\n\"William Wordsworth.\" By Elizabeth Wordsworth. _The Athen\u00e6um_, ii. 516.\n\n1892. \"The Yarrow of Wordsworth and Scott.\" _Blackwood's Magazine_,\ncli. 638. John Veitch.\n\n\"The last Decade of the last Century.\" _The Contemporary Review_, lxii.\n422. J.W. Hales.\n\n\"The Influence of Burns on Wordsworth.\" _The Manchester Quarterly_, xi.\n285. George Milner.\n\n\"Wordsworth on Old Age.\" _Literary Opinion_, vii. 186, Sir Edward\nStrachey.\n\n\"The Birds of Wordsworth, practically, mythologically, and\ncomparatively examined.\" By William H. Wintringham. _The Athen\u00e6um_, i.\n594, 634, 666, 697.\n\n\"Dove Cottage,\" in _The Athen\u00e6um_, i. 727.\n\n\"The Poetical Works of William Wordsworth.\" Edited by Edward Dowden.\n_The Athen\u00e6um._ No. 3404.\n\n1893. \"Some Unpublished Letters of William Wordsworth.\" _The Cornhill\nMagazine_, New Series, xx. 257.\n\n\"Reminiscences of Scott, Campbell, Jeffrey, and Wordsworth.\" _The\nBookman_, iv. 47.\n\n\"Our Poet's Corner.\" _The Girls' Own Paper_, xiv. 772.\n\n\"Dove Cottage, Grasmere--Wordsworth's Home.\" _The Girls' Own Paper_,\nxiv. 772. Milward Wood.\n\n\"Down the Duddon with Wordsworth.\" _The Leisure Hour_, xlii. 532.\nHerbert Rix.\n\n\"Wordsworth's 'Grace Darling.'\" _The Athen\u00e6um_, No. 3440. Edward Dowden.\n\n\"Note by Wordsworth.\" _The Athen\u00e6um_, No. 3443. E. H. C. (Ernest H.\nColeridge).\n\n\"Wordsworth and the _Morning Post_.\" _The Athen\u00e6um_, No. 3445. E. H. C.\n\n1894. \"Wordsworth's 'Castle of Indolence' Stanzas.\" _The Fortnightly\nReview_, lxii. 685. T. Hutchinson.\n\n\"A Century of Wordsworth.\" _The Sunday at Home_, 641, 646. By E. S.\nCapper.\n\n1895. \"The Charm of Wordsworth.\" _Great Thoughts_, iv. 399.\n\n\"Wordsworth and Carlyle: a Literary Parallel.\" _Temple Bar_, cv. 261.\n\n\"Dorothy Wordsworth, 1771-1855.\" _Great Thoughts_, v. 56. Alexander\nSmall.\n\n1896. \"Wordsworth's Quantock Poems.\" _Temple Bar_, April 1896. William\nGreswell.\n\n\nV\n\nPARODIES ON WORDSWORTH\n\nTHE BATTERED TAR; OR, THE WAGGONER'S COMPANION. A Poem, with Sonnets,\netc. J. Johnston.\n\n1839. PETER BELL THE THIRD. By Miching Mallecho, Esq. (Percy B.\nShelley).\n\n1876. LITERARY REMAINS. By Catherine Maria Fanshawe. B. M. Pickering.\nLondon.\n\n1888. THE POETS AT TEA. _The Cambridge Fortnightly_ (Feb. 7).\n\n1819. THE DEAD ASSES. A Lyrical Ballad.\n\n1819. PETER BELL. a Lyrical Ballad. By John Hamilton Reynolds. London:\nTaylor and Hessey.\n\n1816. THE POETIC MIRROR; OR, THE LIVING BARDS OF BRITAIN, pp. 131-187.\n(By James Hogg.)\n\nThe Stranger; being a further portion of \"The Recluse,\" a poem.\n\nThe Flying Taylor; further extract from \"The Recluse,\" a poem.\n\nJames Rigg; still further extract from \"The Recluse,\" a poem. 12mo.\nLondon: Longmans. Second Edition. 1817.\n\n1888. HAMILTON, WALTER. Parodies of the Works of English and American\nAuthors, collected and annotated by Walter Hamilton. _William\nWordsworth_, pp. 88-106. 8vo. London: Reeves and Turner.\n\n\nVI\n\nPOEMS ADDRESSED TO WORDSWORTH, AND ALLUSIONS TO HIM BY CONTEMPORARY AND\nSUBSEQUENT POETS\n\n1. COLERIDGE, S. T. _To William Wordsworth, composed on the night\nafter his recitation of a poem on the growth of an individual mind._\nPublished in \"Sibylline Leaves.\"\n\n2. COLERIDGE, HARTLEY. _To William Wordsworth, on his seventy-fifth\nBirthday._\n\n3. WILSON, JOHN. In \"The Angler's Tent,\" p. 257 of the edition of 1858.\n\n4. KEATS, JOHN. In his Sonnets [the 2nd addressed to Haydon].\n\n5. SHELLEY, PERCY B. _To Wordsworth._ Another reference occurs in\n_Alastor_.\n\n6. MOIR, D. M. _To Wordsworth._ In _Blackwood's Magazine_, viii. 542;\nafterwards included amongst his \"Poems,\" vol. ii. p. 28. 1852.\n\n7, 8. BROWNING, MRS. _On a Portrait of Wordsworth by B. R. Haydon._\n(Sonnets.) 1866. Vol. ii. p. 264. Also in _Lady Geraldine's Courtship_,\nvol. ii. p. 109. 1866.\n\n9. ELLIOTT, EBENEZER. In _The Village Patriarch_. Book iv. 1840.\n\n10. TENNYSON, ALFRED LORD. In the Dedication of his _Poems_ \"To the\nQueen.\" March 1851.\n\n11, 12. ALFORD, HENRY. In _The School of the Heart_, pp. 66, 67; and\n_Recollections of Wordsworth's_ \"_Ruth_,\" p. 163. 1868.\n\n13. LOWELL, JAMES RUSSELL. In _A Fable for Critics_, p. 133. 1873.\n\n14, 15. BYRON, LORD. In _English Bards and Scotch Reviewers_. Also in\n_Don Juan_.\n\n16. HUNT, LEIGH. In _The Feast of the Poets_. This first appeared in\n_The Reflector_, which survived from 1810 to 1812.\n\n17. HEMANS, MRS. _To Wordsworth_, in her \"Miscellaneous Poems.\"\n\n18. Scenes and Hymns of Life. Dedicated to Wordsworth. p. 568. N. D.\n\n19. HALLAM, A. H. _Meditative Fragments._ No. vi. 1863.\n\n20, 21, 22. ARNOLD, MATTHEW. _Memorial Verses._ April 1850. Also in\n_Youth and Nature_, and in _Obermann Once More_. p. 203. 1869.\n\n23, 24, 25. DE VERE, SIR AUBREY. _In Rydal with Wordsworth_ (Sonnets).\np. 208. 1842. _Wordsworth._ Composed at Rydal, 1st Sept. 1860. p. 392.\n_Wordsworth, on Visiting the Duddon_, p. 393.\n\n26. TOLLEMACHE, The Hon. BEATRIX L. _Wordsworth_, in \"Safe Studies,\" p.\n409. 1884.\n\n27. TOLLEMACHE, The Hon. BEATRIX L. _To Wordsworth_, in \"Engleberg, and\nother Verses.\" 1890.\n\n28. BELL, GEORGE. _Rydal Mount_, in \"Descriptive and other\nMiscellaneous Pieces in Verse.\" Penrith, 1835.\n\n29. HOUGHTON, LORD. Sonnet beginning \"The hour may come,\" etc. Poetical\nWorks, vol. i. p. 267. 1876.\n\n30. WORSLEY, P. S. Stanzas to Wordsworth, in _Blackwood's Magazine_,\nxcii. pp. 92-93.\n\n31. AUSTIN, ALFRED. _Wordsworth at Dove Cottage._ 1890.\n\n32, 33. SCOTT, W. B. Poems (three Sonnets), pp. 180-182. 1875. Also in\n\"A Poet's Harvest Home,\" 1893. _Wordsworth_, p. 123.\n\n34, 35, 36. RAWNSLEY, H. D. In \"Sonnets at the English Lakes.\" IX.\n_Wordsworth's Seat, Rydal_; LI. _A Tree planted by William Wordsworth\nat Wray Castle_; LXII. _Wordsworth's Tomb._\n\n37. PAYNE, JAMES. _Wordsworth's Grave_, in \"Lakes in Sunshine.\" 1870.\n\n38. LANDOR, L. E. _On Wordsworth's Cottage, near Grasmere Lake_, in her\n\"Poetical Works,\" pp. 551-4. 1873.\n\n39. ALLINGHAM, WILLIAM. _On reading of the Funeral of the Poet\nWordsworth_, p. 258 of \"Poems.\" 1850.\n\n40. PALGRAVE, FRANCIS TURNER. _William Wordsworth_, in his \"Lyrical\nPoems.\" 1871.\n\n41. ANDERSON, G. F. R. _Wordsworth_, in \"The White Book of the Muses,\"\np. 67. 1895.\n\n42. DAWSON, JAMES, jun. _Wordsworth and Hartley Coleridge: in Grasmere\nChurchyard, Westmoreland._ In _Macmillan's Magazine_, xiii. 26.\n\n43. WATSON, WILLIAM. _Wordsworth's Grave._ Originally published in\nthe _National Review_, x. 40; afterwards included in the volume,\n\"Wordsworth's Grave, and other Poems.\" 1890.\n\n44. MATSURA (a Japanese poet). _Moonlight on Windermere_, translated by\nH. D. Rawnsley in _Murray's Magazine_, Oct. 1887.\n\n\n\n\nII.--_AMERICA_\n\nBIBLIOGRAPHY of the Various Editions of WORDSWORTH'S POETICAL WORKS,\nwhich have been printed and published in the United States of America,\nfrom 1801 to 1895, arranged in Chronological Order: also a BIBLIOGRAPHY\nOF CRITICAL ESSAYS, and BIOGRAPHICAL SKETCHES, of Wordsworth's Life and\nWorks in Books, Reviews, and Periodicals; with Notes, by Mrs. HENRY A.\nST. JOHN, Ithaca, New York.\n\n\nPREFATORY NOTE\n\nMy ideal in attempting to prepare a _Bibliography of Wordsworth in\nAmerica_ was high. I hoped to see each edition, or at least to identify\nthe editions hinted at in the various catalogues. I determined to\nread every article, in criticism, or review; and to know if the many\nreferences, given by Poole and other authorities, were correct. As\nis usually the case, the reality has fallen far short of the ideal.\nBut, while the results are not what were desired, there have been many\nfortunate discoveries.\n\nTwo things were learned to begin with. First, that astonishingly little\ncare had been taken to preserve the history of the early American\nEditions, or to preserve, even, the earlier American Periodicals.\nMost of our larger libraries are amazingly deficient in these works.\nSecond, it was found that existing Catalogues or Lists are not only\nfar from complete, but full of gross blunders. Roorbach (the Addenda,\nSupplements, etc.) was found to be a mere rehash of the old trade sales\nCatalogues, swarming with blunders. In the matter of dates, imprints,\nthe particular editions, the size of books, Roorbach is utterly\nuntrustworthy. Allibone (so far as Wordsworth is concerned) is also\nconfusing and incomplete. I did not find much in the various Public or\nCollege Library Catalogues.\n\nI wrote to the librarians of some of the older libraries, after I had\nmade out a preliminary list, to ascertain if they could add thereto any\neditions, from their cards or manuscript catalogues. From these sources\nI was enabled several times to solve seemingly insolvable problems.\n\nI had assistance from, and in some instances visited, the following\nlibraries: Cornell University, Boston Public Library, Boston Athen\u00e6um,\nHarvard College, Philadelphia Public Library, the Library College of\nPhiladelphia, Mercantile Library College, Philadelphia; the Public\nLibrary, St. Louis; that of Lennox and Astor, the University of\nVirginia, the State Library, Richmond, Va., and one or two other\nSouthern libraries. I have written more than one hundred letters\nto publishers, editors, authors, the descendants of early American\nWordsworthians, Professors of Literature, and professed Wordsworthians\nin Seminaries and Colleges. I have examined, or employed others to\nexamine, the following works for editions of Wordsworth: the _New York\nLiterary World_, _Norton's Literary Gazette_, _American Publishers'\nCircular_, _Publishers' Weekly_, _Catalogues of Congress Library_, _The\nPort Folio_, _American Quarterly Review_, _Knickerbocker Magazine_,\n_New York Quarterly Review_, _American Review_, _North American\nReview_. And this is but half of my story.\n\nPoole's \"Index,\" of course, was a great assistance. But I did not rely\naltogether on him, after I had discovered several mistakes in titles\nand numbering--mistakes which were confusing in the extreme. I have\nconsulted all other Indexes and Reference Lists that I could procure,\nand have carefully examined the periodicals in which it was possible\nthat such articles could be found.\n\nMy greatest light, however, came from responses to personal appeals,\nto those in the North, South, East, and West of the Country, who\nenlightened me in particular directions. I needed assistance, not only\nto discover the articles, but more particularly to secure the articles\nto read, or to procure proper persons to read the few articles that I\ncould not obtain. When valuable books were sent me, by express, from\ndistant College Libraries, that I might read for myself, I realised the\nbond there is between Wordsworthians.\n\nI cannot begin to speak of the delight that I have had in this work,\ndelight because of the response I have met with, and in opening\nup unknown and rich veins of criticism. I have learned too, that\nWordsworth has many enthusiastic followers in America.\n\nI have included in the Bibliography the accounts of visits paid to\nWordsworth by certain well-known Americans, a half-dozen poems on\nWordsworth, and three or four unpublished Lectures.\n\nI am exceedingly grateful to the many who (to my surprise) have\nanswered my questions, and have given me of their valuable time. I\nam especially indebted to Mr. George P. Philes, of Philadelphia, and\nalso to Mr. F. Saunders of the Astor Library, New York. Dean Murray of\nPrinceton rendered me exceedingly gracious service, and but for Mr.\nEdwin H. Woodruff of Stanford University, California, I should not have\nknown how or where to begin my investigations.\n\nIn all probability my work is not perfect. I would that it were. I only\nknow that I have been enabled, by enthusiasm alone, to lay a foundation\nfor Wordsworth Bibliography in America, that may be an assistance to\nfuture scholars, and will aid the next Wordsworthian who is brave\nenough to build enduringly.\n\nC. M. ST. JOHN.\n\n\nI\n\nAMERICAN EDITIONS OF WORDSWORTH\n\nINCLUDING A FEW WORKS WHICH ARE NOT STRICTLY EDITIONS OF WORDSWORTH\n\nI have endeavoured to include in this list every distinctive American\nedition of Wordsworth, published during the poet's lifetime, and\nsince his death. There are many others, issued with the imprints of\nhonourable publishers; which, upon investigation, were found to be\nEnglish reprints; to say nothing of those editions made from worn-out\nplates, and issued by houses of less reputation for honourableness.\nI was puzzled to account for so many editions of Matthew Arnold's\nSelections, some of them bearing the imprint of Harper Brothers, some\nof Macmillan, and several of Crowell. The Harpers wrote me that these\nvarious publications were possible in view of the fact that there\nwas no copyright of the work, and that all of them might properly be\ncalled American Editions. I have not placed those bearing the Macmillan\nimprint, of course, among purely American editions. Nor have I included\nthe several cheap ones of Crowell. The one of Crowell, given in the\nlist, is copyrighted by the Crowell Company.\n\nThe fact that the introduction of Wordsworth's poetry into America is\nso easily authenticated, and that the history of it is so concise,\nis my apology for deviating from ordinary bibliographical rule in\nincluding among the regular editions certain numbers of America's first\nLiterary Journal, and two or three other volumes.\n\nI have confined myself to a simple chronological arrangement of the\nEditions, with place of imprint, name of publisher, number, and size\nof volumes. This makes the most convenient list for easy reference,\nespecially as I have tried to mention technical points of difference.\n\nC. M. ST. JOHN.\n\n1\n\n1801. THE PORT FOLIO. (Edited by Joseph Dennie.) Philadelphia. 4to.\n\nThe following poems appeared in \"The Port Folio,\" vol. i., before the\npublication of the First American Edition of \"Lyrical Ballads\"--\n\n    (1) _Simon Lee_, p. 24.[478]\n    (2) _The Last of the Flock_, p. 48.\n    (3) _The Thorn_, p. 94.\n    (4) _The Mad Mother_, p. 232.\n    (5) _Anecdote for Fathers_, p. 232.\n    (6) _Ellen Irwin_, p. 391.\n    (7) _Strange Fits of Passion_, etc., p. 392.\n    (8) _The Waterfall and the Eglantine_, p. 408.\n    (9) _Lucy Gray_, p. 408.\n    (10) _Andrew Jones_, p. 408.\n\n2\n\n1801. INTRODUCTION TO THE ENGLISH READER. By Lindley Murray.\nPhiladelphia: Johnson and Warner. 12mo.[479]\n\n3\n\n1802. LYRICAL BALLADS, with Other Poems. In two volumes. By W.\nWordsworth.\n\n    Quam nihil ad genium, Papiniane, tuum!\n\nFrom the London second edition. Philadelphia: Printed and sold by James\nHumphreys. 2 vols. in one. 12mo.[480]\n\n4\n\n1823. THE AMERICAN FIRST CLASS BOOK. By John Pierpont. Boston: William\nB. Fowle. 1 vol. 12mo.[481]\n\n5\n\n1824. THE POETICAL WORKS OF WILLIAM WORDSWORTH. Boston: published by\nCummings, Hilliard and Co. 4 vols. 12mo.[482]\n\n6\n\n1833. SKETCH OF THE GENIUS AND CHARACTER OF WILLIAM WORDSWORTH. With\nSelections from his \"Lyrical Ballads.\"[483] Philadelphia: Greenbak's\nPeriodical Library. Vol. ii. pp. 181-202.\n\n7\n\n1835. YARROW REVISITED, and Other Poems. New York: R. Bartlett and S.\nRaynor. 16mo. pp. 17-244.\n\n1835. Same Title. Boston: R. Bartlett and S. Raynor. 16mo; also,\nBoston: James Munroe and Co. 16mo.\n\n1835. Same Title. Philadelphia. 12mo.\n\n8\n\n1836. YARROW REVISITED. Second Edition. Boston: William D. Ticknor.\n16mo.\n\n9\n\n1836. THE POETICAL WORKS OF WILLIAM WORDSWORTH. The first complete\nAmerican, from the last London, edition. New Haven: Peck and Newton. In\n1 vol. Royal 8vo.[484]\n\n10\n\n1836. THE COMPLETE POETICAL WORKS OF WILLIAM WORDSWORTH, together with\na Description of the Country of the Lakes, etc. Edited by Henry Reed.\nWith Portrait. Philadelphia: Kay and Brother. Royal 8vo; also, by James\nKay and Brother.[485]\n\n1839. Same Title. Philadelphia: Kay and Brother. Boston: Munroe and Co.\nPittsburg: Kay and Co.\n\n1844. Same Title. Philadelphia: James Kay jun.[486]\n\n11\n\n1842. WORDSWORTH'S POEMS. In \"The New World,\" vol. iv. No. 16.\nNew York: Park Benjamin, Editor. Sat. April 9, _Sonnet Written at\nFlorence_; April 16, _Address to the Clouds, Suggested by a Picture\nof the Bird of Paradise_; _Maternal Grief_ (\"New Poems, never before\npublished\"). May 7, _Guilt and Sorrow_ (\"From proof sheets received in\nadvance\").[487]\n\n12\n\n1843. POEMS FROM THE POETICAL WORKS OF WILLIAM WORDSWORTH. Selected by\nHenry Reed.\n\n    Go forth, my little Book; pursue thy way;\n    Go forth, and please the gentle and the good.\n\nPhiladelphia: John Locken. 32mo.\n\n(Entered according to the Act of Congress in 1841.)\n\n1846. Same Title. Philadelphia: Uriah Hunt and Son. 32mo.\n\nSame Title. New York: Leavitt and Co.[488]\n\n1853. Same Title. New York: Leavitt and Allen. 24mo.\n\n1856. Same Title.[489] New York: Leavitt and Allen.\n\n13\n\n1847. WORDSWORTH'S COMPLETE POETICAL AND PROSE WORKS.[490] In 5 vols.\n(In Press.) Philadelphia: Kay and Troutman. 12mo.\n\n14\n\n1849. POEMS OF WILLIAM WORDSWORTH: with an Introductory Essay on his\nLife and Writings. By H. T. Tuckerman. New York: C. S. Francis and Co.\n12mo. pp. 21-356; also, Boston: J. H. Francis.[491]\n\n15\n\n1849. THE EXCURSION: a Poem. New York: C. S. Francis and Co. 12mo.\n\n1850. THE EXCURSION, etc. New York: C. S. Francis and Co. 12mo.\n\n1852-55. The above was again republished.\n\n16\n\n1850. THE PRELUDE; or, Growth of a Poet's Mind. New York: Appleton and\nCo. 12mo.\n\n1850. THE PRELUDE, etc. Philadelphia: George S. Appleton and Co. 12mo.\n\n17\n\n1850. THE POETICAL WORKS OF WILLIAM WORDSWORTH. Boston: Phillips,\nSampson and Co. 12mo. Reprinted in 1857 and 1859.\n\n1859. Same Title. Boston: Phillips, Sampson and Co. 16mo.\n\n18\n\n1851. THE COMPLETE POETICAL WORKS OF WILLIAM WORDSWORTH. Edited by\nHenry Reed. Royal 8vo. Philadelphia: James Kay jun. and Brother. Also,\nKay and Troutman. Also, Troutman and Hayes. Also, Hayes and Zell. Also,\nPorter and Coates.[492]\n\n1852. THE COMPLETE POETICAL WORKS OF WILLIAM WORDSWORTH. Edited by\nHenry Reed. 8vo. Philadelphia: Troutman and Hayes.\n\n1860. THE COMPLETE POETICAL WORKS OF WILLIAM WORDSWORTH. Edited by\nHenry Reed. Royal 8vo. pp. 727.[493]\n\n19\n\n1854. THE POETICAL WORKS OF WILLIAM WORDSWORTH, with a Memoir.[494]\nBoston: Little, Brown and Co. Also, New York: Evans and Dickenson.\nAlso, Philadelphia: Lippincott, Grant and Co. 18mo. 7 vols.\n\n20\n\n1855. POETICAL WORKS OF W. WORDSWORTH. Portrait. Boston: Crosby and\nNichols(?) 12mo.\n\n21\n\n1855. THE PRELUDE. New York: Appleton and Co. 12mo. Second Edition.\n\n22\n\n1860. POETICAL WORKS OF WORDSWORTH.[495] 2 vols. New York: 12mo.\n\n23\n\n1863. SELECTIONS FROM WORDSWORTH, with an Essay by H. T. Tuckerman.\nPhiladelphia. 32mo.[496]\n\n1863. Same Title. Boston.\n\n24\n\n1865. POEMS OF NATURE AND SENTIMENT. By William Wordsworth. Elegantly\nillustrated. Philadelphia: E. H. Butler and Co.[497]\n\n25\n\nTHE POETICAL WORKS OF WILLIAM WORDSWORTH.[498] A new edition. Boston:\nCrosby and Nichols. 12mo.\n\n1867. THE POETICAL WORKS OF WILLIAM WORDSWORTH. A new edition. Boston:\nCrosby and Ainsworth. New York: Oliver S. Felt. 16mo. pp. 539.[499]\n\n26\n\n1870. THE EXCURSION: a Poem. A new edition. New York: J. Miller. 16mo.\n\n27\n\n1871-75. THE HOWE MEMORIAL PRIMER, in raised letters for the Blind.\nWORDSWORTH'S POETICAL WORKS, with a Memoir. Boston. 7 vols. 16mo.\nPortrait.\n\n28\n\n1876. WORDSWORTH'S POEMS. Selected and Prepared for Schools. Edited by\nH. N. Hudson. Boston: Ginn and Co. 12mo. \"Text-book of Prose and Poetry\nSeries.\"\n\n1882. Same Title. In paper. Hudson's Pamphlet Selections of Poetry.\n(No. VI. Wordsworth.)\n\n29\n\n1877. FAVORITE POEMS. Vest-pocket Series. Boston: Osgood. Illustrated.\n32mo.\n\n1877. FAVORITE POEMS. Illustrated. Boston, Massachusetts. (Printed at\nCambridge.) 16mo.\n\n30\n\n1877. THE POETICAL WORKS. New edition. Boston: Hurd and Houghton. 8vo.\n3 vols.\n\n31\n\n1878. THE POETICAL WORKS OF WILLIAM WORDSWORTH, with Memoir. 7 vols. in\n3. Boston: Houghton, Osgood and Co. Riverside Press. 8vo; also,\n\n1880. Same Title.[500]\n\n32\n\n1879. WORDSWORTH'S POEMS. Chosen and Edited by Matthew Arnold. Franklin\nSquare Library. New York: Harper and Brother. Paper 4to.\n\n1880. Another Edition.\n\n1891. Another Edition.\n\n33\n\n1881. THE EXCURSION, with a Biographical Sketch. English Classic\nSeries. New York: Clark and Maynard. 16mo.\n\n1889. Same Title. With Explanatory Notes. New York: Effingham, Maynard\nand Co.\n\n34\n\n1881-82. FAVORITE POEMS. By William Wordsworth. In Modern Classics, No.\nVII. Illustrated. Boston: Houghton, Mifflin and Co. 32mo.\n\n35\n\n1884. ODE, INTIMATIONS OF IMMORTALITY. By William Wordsworth.\nIllustrated. Boston: D. Lothrop Company. Small 4to. Copyright by D.\nLothrop.\n\n36\n\n1884. POEMS BY WILLIAM WORDSWORTH. Selected and Prepared for use in\nSchools. (From Hudson's _Text-Book of Poetry_.) Section I. Boston:\nGinn, Heath and Co. 12mo.\n\n37\n\n1888. PRELUDE; or, Growth of a Poet's Mind. With Notes by A. J. George.\nBoston: D. C. Heath and Co. 12mo.\n\n38\n\n1888. BITS OF BURNISHED GOLD, from William Wordsworth. Compiled by Rose\nPorter. New York: A. D. F. Randolph and Co. 12mo.\n\n39\n\n1889. SELECTIONS FROM WORDSWORTH. With Notes by A. J. George. Boston:\nD. C. Heath and Co. 12mo.\n\n40\n\n1889. MELODIES FROM NATURE. (From Wordsworth.) Illustrated. Boston: D.\nLothrop Company. 4to.\n\n41\n\n1889. SELECT POEMS OF WILLIAM WORDSWORTH.[501] Edited, with Notes, by\nW. J. Rolfe. With Engravings. New York: Harper Brothers. Square 16mo.\n\n42\n\n1889. POEMS BY WILLIAM WORDSWORTH. Selected and Prepared for use in\nSchool. Paper. (From Hudson's _Text-Book of Poetry_.) Section II. 12mo.\nBoston: Ginn and Co.\n\n43\n\n1890. SELECT POEMS FROM WORDSWORTH, with Explanatory Notes. Edited by\nJames H. Dillard. New York: Effingham, Maynard and Co. 12mo.\n\n44\n\n1890. PASTORALS, LYRICS AND SONNETS FROM THE POETIC WORKS OF WILLIAM\nWORDSWORTH. Boston and New York: Houghton, Mifflin and Co. 16mo. White\nand Gold Series.\n\n45\n\n1891. A SELECTION OF THE SONNETS OF WILLIAM WORDSWORTH.[502] With\nnumerous Illustrations. By A. Parsons. New York: Harper Brothers. 4to.\n\n46\n\n1891. WORDSWORTH FOR THE YOUNG. Selections. Illustrated. With an\nIntroduction for parents and teachers by Cynthia Morgan St. John.\nBoston: D. Lothrop Company. Small 4to. 153 pp.\n\n47\n\n1892. WORDSWORTH'S PREFACES AND ESSAYS ON POETRY. Edited by A. J.\nGeorge. (Heath's English Classics.) Boston: D. C. Heath and Co. 12mo.\n\n48\n\n1892. POEMS OF WORDSWORTH. Chosen and Edited by Matthew Arnold.\nIllustrated by Edmund H. Garrett. New York: Thomas Y. Crowell and Co.\n(Copyright 1892 by T. Y. Crowell.)\n\n[478] _Simon Lee_ was probably the first poem of Wordsworth's published\nin a Literary Journal in America, and is the beginning of Wordsworth's\nBibliography in U.S.A. A note in \"The Port Folio\" (vol. i. p. 24) is as\nfollows: \"The public may remember reading in some of the newspapers the\ninteresting little ballads, _We are Seven_, and _Goody Blake and Harry\nGill_. They were extracted from the 'Lyrical Ballads,' a collection\nremarkable for originality, simplicity, and nature.\u2026 The following,\n_Simon Lee_, is from the same work.\"\n\nIt is evident from this that two, at least, of Wordsworth's poems were\ncopied into American newspapers as early as 1800, and that Joseph\nDennie, the founder, as well as editor, of \"The Port Folio\"--the first\npurely Literary Journal established in this country--was the first\nAmerican champion of Wordsworth.\n\nC. M. ST. JOHN.\n\n[479] _The Pet Lamb_ appeared in this Book almost immediately after\nits publication in England. It was the first poem of Wordsworth's\npublished in a book in America. It was also the first instance of the\nintroduction of a poem of Wordsworth's into a School Book.\n\nC. M. ST. JOHN.\n\n[480] The first American edition, and the first work by Wordsworth,\nprinted in America. It looks as if the Poet found appreciative readers\nin America sooner than in England; the first edition of \"Lyrical\nBallads,\" which had fallen dead in his own country in 1798, being\npublished in Philadelphia in 1802. The American edition was delayed in\nthe press, in order to include certain pieces which first appeared in\nthe second (English) edition of 1802. See Humphreys' Preface.\n\nA copy of \"Lyrical Ballads,\" 1802, is in the possession of Judge Henry\nReed, with exactly the same title-page as the above, except that it\nreads--\n\n\"Printed by James Humphreys for Joseph Groff.\"\n\nIt is believed that the work was printed at the joint expense of\nHumphreys and Groff, each bookseller taking a certain number of copies\nupon which was placed his individual imprint. Both book-sellers\nadvertised the volumes almost simultaneously. I know of another copy\nof (1802) \"Lyrical Ballads,\" of which the first volume contains the\nimprint of Humphreys, and the second volume that of Groff. The two\nvolumes are bound together, and are _identical_ in type, paper, etc.\n\nC. M. ST. JOHN.\n\n[481] Amongst the contents there are four long extracts from _The\nExcursion_, with titles attributed to W.W. _Goody Blake and Harry Gill_\nis amongst the extracts from \"Lyrical Ballads,\" and there is a long\nnote to the former poem by Joseph Dennie.\n\nC. M. ST. JOHN.\n\n[482] The first collected edition of Wordsworth's Poems printed in\nAmerica.\n\nC. M. ST. JOHN.\n\n[483] The sketch is by R. H. Home. The poems are _The Last of the\nFlock_, _The Dungeon_, _The Mad Mother_, _Anecdote for Fathers_, _We\nare Seven_, _Lines Written in Early Spring_, _The Female Vagrant_,\n_Goody Blake and Harry Gill_, _The Waterfall and the Eglantine_, _The\nOak and the Broom_, _Lucy Gray_, _Hart-Leap Well_, _Lucy_, _Nutling_,\n_Ruth_.\n\nC. M. ST. JOHN.\n\n[484] Printed and published by Peck and Newton.\n\nC. M. ST. JOHN.\n\n[485] First double-column edition of the poems, adopted by Moxon in\n1845 edition.\n\nC. M. ST. JOHN.\n\n[486] The Boxall portrait was engraved for the above. I could not find\nthe 1844 imprint, but presume that it is the same as that of 1837 and\n1839.\n\nC. M. ST. JOHN.\n\n[487] In an editorial of April 16 of \"The New World\" is the following:\n\"We are enabled by the purchase of the printed sheets considerably in\nadvance of their publication in England to present the first and only\nAmerican Editions of new poems by William Wordsworth.\"\n\nC. M. ST. JOHN.\n\n[488] This is spoken of in Ellis Yarnall's Reminiscences as having no\ndate. When John Locken--the first publisher--failed, the plates passed\ninto the possession of Messrs. Uriah Hunt and Son. They retired from\nbusiness, and Messrs. Leavitt and Co. took the plates. It is possible\nthat there was an edition earlier than 1843.\n\nC. M. ST. JOHN.\n\n[489] The last two named are exactly as in 1843, except that they are\nprinted on larger paper. Why one is put down 32mo and the other 24mo is\na mystery!\n\nC. M. ST. JOHN.\n\n[490] If this edition was published, it seems to have disappeared. It\nis advertised in A. V. Blake's _American Booksellers' Complete Trade\nList_, published at Claremont, N.H., 1847.\n\nC. M. ST. JOHN.\n\n[491] Copyright in 1848. It contains about one-fifth of all\nWordsworth's poems. The Essay, which occupies ten pages, is taken \"by\npermission\" from Tuckerman's _Thoughts on Poets_.\n\nC. M. ST. JOHN.\n\n[492] In connection with this edition, I can vouch for the five firms\nof Publishers in Philadelphia, but I cannot explain it.\n\nC. M. ST. JOHN.\n\n[493] \"This edition contains some pieces omitted--inadvertently it is\nbelieved--from the latest London edition.\" Additional poems have been\nintroduced, and the arrangement changed since the 1839 edition.\n\nC. M. ST. JOHN.\n\n[494] This edition contains a remarkable \"Sketch of Wordsworth's\nLife,\" by James Russell Lowell, which was afterwards embodied, with\nadditions, in _Among my Books_. Mr. Ellis Yarnall believed that this\nedition was an English reprint. I doubt this from the fact that it is\n\"Entered according to the Act of Congress in 1854,\" and was \"Printed at\nCambridge by H.O. Houghton.\"\n\nC. M. ST. JOHN.\n\n[495] This edition is mentioned in some lists, but I am inclined to\ndoubt if it can be authenticated.\n\nC. M. ST. JOHN.\n\n[496] The size is given as 32mo. I have not seen the book.\n\nC. M. ST. JOHN.\n\n[497] Edited by Waldron J. Cheney, though not credited to him. C. M.\nST. JOHN.\n\n[498] No date is given to this edition. The firm-name and place of\nbusiness according to the Boston Directory would limit the date of\nthe title page at least to 1863-65. It is in the New Haven Library.\nAllibone notes a volume of \"Selections,\" Boston, 12mo, 1863, which may\nbe this.\n\nC. M. ST. JOHN.\n\n[499] I have placed the two works together, as they are closely\nrelated, if not identical. The edition contains _The Excursion_ and\nfifty-seven other poems.\n\nC. M. ST. JOHN.\n\n[500] From plates of the 1854 edition, with changes.\n\nC. M. ST. JOHN.\n\n[501] This excellent edition--as to selection, size, paper, binding,\nand illustrations--is the best handy edition of Wordsworth issued in\nAmerica.\n\nC. M. ST. JOHN.\n\n[502] Eighty-eight of the sonnets are here illustrated with rare skill\nand artistic effect. The illustrations first appeared in wood-cuts in\nHarper's _Monthly Magazine_.\n\nC. M. ST. JOHN.\n\n\nII\n\nREPRINTS, AND BOOKS, BOTH ENGLISH AND AMERICAN\n\nA Bibliography of Wordsworth in America is not complete without some\nreference to the many editions of Wordsworth, and of works pertaining\nto him, which have--for the most part--appeared simultaneously in\nEngland and America. These works cannot properly be termed American,\nbut they have been welcomed, and they have also supplied a want, on\nthis side of the Atlantic. The editions are confined, for the most\npart, to the last twenty years. I have endeavoured to select those\nwhich are of most value.\n\nC. M. ST. JOHN.\n\n1\n\n1859. WORDSWORTH'S PASTORAL POEMS. Illustrated. New York: D. Appleton\nand Co. 12mo.\n\n1875. Same Title. New York: Putnam. 12mo.\n\n2\n\n1859. POEMS BY WILLIAM WORDSWORTH. Selected and Edited by Robert Aris\nWillmott. Illustrated with 100 Designs by Birket Foster and others.\nLondon and New York: George Routledge and Co. 4to.\n\n1870. The above republished.\n\n3\n\n1869. THE POETICAL WORKS OF WILLIAM WORDSWORTH. Globe Edition. Square\n12mo. Philadelphia: Lippincott and Co.\n\n4\n\n1874. RECOLLECTIONS OF A TOUR MADE IN SCOTLAND. By Dorothy Wordsworth.\nEdited by J. C. Shairp. New York: G. P. Putnam's Sons. (Printed at the\nEdinburgh University Press.) 12mo.\n\n5\n\n1880. WORDSWORTH'S POEMS. Chosen and Edited by Matthew Arnold. Large\nPaper Edition. London and New York: Macmillan and Co. 8vo.\n\n1892. Same Title. With Steel Portrait. Printed on India paper. London\nand New York: Macmillan and Co. 8vo.\n\n6\n\n1881. WILLIAM WORDSWORTH: a Biography with Selections from Prose and\nPoetry. By A. J. Symington. Boston: Roberts Brothers. 2 vols. 16mo.\n\n7\n\n1885. ODE ON IMMORTALITY AND LINES ON TINTERN ABBEY. London and New\nYork: Cassell and Co. 12mo. (Popular Illustrated Series.)\n\n8\n\n1886. PASTORAL POEMS. London and New York: Cassell and Co. 4to.\n\n9\n\n1887. MEMORIALS OF COLEORTON. Edited, with Introduction and Notes, by\nWilliam Knight. Boston and New York: Houghton, Mifflin and Co. 12mo.\n(Printed at the Edinburgh University Press.)\n\n10\n\n1887. THROUGH THE WORDSWORTH COUNTRY. By William Knight. London and New\nYork: Scribner and Welford. Engraving. 8vo.\n\n11\n\n1888. THE COMPLETE POETICAL WORKS OF WILLIAM WORDSWORTH. With an\nIntroduction by John Morley. London and New York: Macmillan and Co.\nCrown 8vo.\n\n12\n\n1888. THE RECLUSE. London and New York: Macmillan and Co. 16mo.\n\n13\n\n1889. WORDSWORTHIANA. Edited by William Knight. London and New York:\nMacmillan and Co. 16mo.\n\n14\n\n1889. POETICAL WORKS, with Memoir. Illustrated. 8 vols. New York: A. C.\nArmstrong and Son. 16mo. (Printed at the University Press, Glasgow.)\n\n15\n\n1889. SELECTIONS FROM WORDSWORTH. By William Knight, and other Members\nof the Wordsworth Society. With Preface and Notes. New York: Scribner\nand Welford. 8vo.\n\n16\n\n1889. WORDSWORTH'S POETICAL WORKS. Edited by William Knight. New York:\nMacmillan and Co. 8 vols. 8vo. (First published in Edinburgh 1882-89.)\n\n17\n\n1889. LIFE OF WILLIAM WORDSWORTH. By William Knight. New York (and\nLondon): Macmillan and Co. 3 vols. 8vo. (First published in Edinburgh,\nin 1889.)\n\n18\n\n1891. WILLIAM WORDSWORTH. By Elizabeth Wordsworth. New York: Scribner.\n18mo. (Also London: Percival and Co.)\n\n19\n\n1889. EARLY POEMS BY WILLIAM WORDSWORTH. Edited by J. R. Tutin. London,\netc., and New York: George Routledge and Sons. (Routledge's Pocket\nLibrary.)\n\n20\n\n1890. DOVE COTTAGE, Wordsworth's Home from 1800 to 1808. By Stopford A.\nBrooke. Small paper. London and New York: Macmillan and Co.\n\n21\n\n1891. WORDSWORTH'S THE WHITE DOE OF RYLSTONE, etc. Edited with\nIntroduction and Notes by William Knight. (Clarendon Press Series.)\nLondon and New York: Macmillan and Co.\n\n22\n\n1892. WORDSWORTH'S LYRICS AND SONNETS. Selected and Edited by C. K.\nShorter. London: David Stott. New York: Macmillan and Co. 32mo.\n\n23\n\n1892. WORDSWORTH'S POETICAL WORKS. Edited with Memoir by E. Dowden. 7\nvols. 16mo. London: George Bell and Sons. New York: 112 Fourth Avenue.\n\n24\n\nGLEANINGS FROM WORDSWORTH. Edited by J. Robertson. Vest-pocket Edition.\nNew York: White, Stokes and Allen. (Printed at the University Press,\nGlasgow.)\n\n25\n\nWE ARE SEVEN. By William Wordsworth.[503] With Drawings by Mary L.\nGrow. Small 4to. New York: E. P. Dutton and Co.\n\n26\n\nODE. INTIMATIONS OF IMMORTALITY. With Biographical Sketch and Notes.\nBoston: Houghton, Mifflin and Co., \"Riverside Literature Series,\" No.\n76. March 1895.\n\n[503] This was lithographed and printed by Ernest Nister at Nuremberg.\n\nC. M. ST. JOHN.\n\n\nIII\n\nBOOKS CONTAINING BIOGRAPHICAL SKETCHES, AND CRITICAL ESSAYS\n\nTHE WRITERS ARE ARRANGED IN ALPHABETICAL ORDER\n\n1\n\n1867. ALGER, W. R. _The Genius of Solitude._ Boston: Roberts Brothers.\n16mo. _Wordsworth_, p. 277.\n\n2\n\n1859-71. ALLIBONE, S. A. _Critical Dictionary of English Literature,\nand British and American Authors._ Philadelphia: J. B. Lippincott Co. 3\nvols. Imperial 8vo. _Wordsworth_, vol. iii. pp. 2843-2849.\n\n3\n\n1884. BURROUGHS, J. \"Fresh Fields.\" Boston: Houghton, Mifflin and Co.\n16mo. _In the Wordsworth Country_, p. 161.[504]\n\n4\n\n1878. CALVERT, G. H. _Wordsworth; A Biographic, Aesthetic Study._\nBoston: Lee-Sheperd. 16mo.\n\n5\n\n1863. CALVERT, G. H. _Scenes and Thoughts in Europe._ Boston: 16mo.[505]\n\n6\n\n1873. CHANNING, W. ELLERY. Address before the Mercantile Library\nCompany of Philadelphia, May 11, 1841. Also in his \"Complete Works.\"\nBoston.[506]\n\n7\n\n1895. CHENEY, JOHN VANCE. _Thoughts on Poetry and the Poets._ Chicago.\nChapter X. is on Wordsworth.\n\n8\n\n1879. DESHLER, C. D. _Afternoons with the Poets._ New York: Harper and\nBrothers. 12mo. _Wordsworth._\n\n9\n\n1871. FIELDS, J. T. _Yesterdays with Authors._ Boston: Houghton,\nMifflin and Co.; also,\n\n1889. _Wordsworth, A Sketch_, p. 253.\n\n10\n\n1838. FROST, JOHN. _Select Works of the British Poets, with\nBiographical Sketches._ Philadelphia: Thomas Wardle. _Wordsworth._\n\n11\n\n1849. GRAHAM, G. F. _English Synonyms._ New York: D. Appleton and Co.\nEdited with an Introduction and Illustrative Authorities. By Henry\nReed.[507]\n\n12\n\n1854. GILES, H. T. _Illustrations of Genius._ Boston: Ticknor and\nFields. 16mo. _William Wordsworth_, pp. 239-266.\n\n13\n\n1886. GRISWOLD, H. T. _Home Life of Great Authors._ Chicago. 18mo.\n_William Wordsworth_, p. 43.\n\n14\n\n1849. GRISWOLD, R. W. _Sacred Poets of England and America._ New York.\n_Wordsworth._\n\n15\n\n1842. GRISWOLD, R. W. _Poets and Poetry of England._ Philadelphia:\nCarey and Hunt. A Review and Selections.\n\n16\n\nHODGKINS, LOUISE M. _Guide to Nineteenth Century Authors._ Boston.\n_Wordsworth Bibliography._\n\n17\n\n1884. HUDSON, H. N. _Studies in Wordsworth._ Boston: Little, Brown and\nCo.[508]\n\n18\n\n1886. JOHNSON, C. F. _Three Americans and Three Englishmen._ New York.\n_Wordsworth._\n\n19\n\n1864. LOWELL, J. R. _The Poetical Works of William Wordsworth._ Boston:\nLittle, Brown and Co. 4 vols. Vol. 1.--_A Sketch of Wordsworth's Life._\n\n20\n\n1876. LOWELL, J. R. _Among my Books._ Boston: Houghton, Mifflin and Co.\n_Wordsworth_,[509] pp. 201-251.\n\n21\n\n1887. LOWELL, J. R. _Democracy and other Addresses._ Boston: Houghton,\nMifflin and Co. _Wordsworth_,[510] 22 pp.\n\n22\n\n1885. MASON, E. T. _Personal Traits of British Authors._ New York:\nCharles Scribner's Sons. _William Wordsworth_, pp. 7-55.\n\nWhat follows is due to American Enterprise, but it is, of course, not\nstrictly American.\n\nC. M. ST. JOHN.\n\n23\n\n1883. MACDONALD, GEORGE. _The Imagination and other Essays_\n(\"Wordsworth's Poetry,\" pp. 245-263). Boston: D. Lothrop and Co.\n\n24\n\n1881. MYERS, F. W. H. _William Wordsworth._ (\"English Men of Letters\nSeries.\") New York: Harper and Brothers. 12mo.\n\n1884. Same Title. New York: J. W. Lovell. 12mo.\n\n1889. Same Title. New York. Harper and Brothers.\n\n25\n\n1838. OSBORN, LAUGHTON. _The Vision of Rubeta._[511] Boston: Weeks,\nJordan and Co. 8vo.\n\n26\n\n1846. OSSOLI, MARGARET FULLER. _Art, Literature, and the Drama._\nBoston. _Wordsworth._[512]\n\n27\n\n1885. PHILLIPS, MAUD GILLETTE. _A Popular Manual of English\nLiterature._ New York: Harper and Brothers. Vol. ii. pp. 217-264.\n\n28\n\n1851. REED, HENRY. _Memoirs of Wordsworth._ By C. Wordsworth. Edited by\nHenry Reed. Boston: Ticknor, Reed, and Fields.[513]\n\n29\n\n1857. REED, HENRY. _Lectures on the British Poets._ In two vols.\nPhiladelphia: Claxton, Remsen and Haffelfinger. Vol. ii. pp. 199-231.\nLecture XV.--_Wordsworth._\n\n30\n\n1870. REED, HENRY. _Lectures on the British Poets._ Philadelphia:\nClaxton, Reinsen and Haffelfinger. _Essay on the English Sonnet_, vol.\nii. pp. 235-272.[514]\n\n31\n\n1887. SAUNDERS, FREDERICK. _Story of some Famous Books._ New York:\nArmstrong and Son. _William Wordsworth_, p. 125.\n\n32\n\nSAUNDERS, FREDERICK. _Evenings with Sacred Poets._ New York: Randolph\nand Co. _Wordsworth._[515]\n\n33\n\n1894. SCUDDER, HORACE E. _Childhood in Literature and Art._ Boston:\nHoughton, Mifflin and Co. In the chapter entitled \"In English\nLiterature and Art,\" Wordsworth is dealt with (chap. vi. pp.\n145-157).[516]\n\n34\n\n1895. SCUDDER, VIDAD. _The Life of the Spirit in Modern English Poets._\nBoston: Houghton, Mifflin and Co. Crown 8vo.\n\n35\n\n1892. STEDMAN, C. E. _Nature and Elements of Poetry._ Boston: Houghton,\nMifflin and Co.[517]\n\n36\n\n1846. TUCKERMAN, H. T. _Thoughts on the Poets._ New York. _Genius and\nWritings of Wordsworth._\n\n37\n\n1882. WELSH, A. H. _Development of English Literature and Language._\nChicago. _Wordsworth_, vol. ii. pp. 330-339.\n\n38\n\n1850. WHIPPLE, E. P. _Essays and Reviews._ Boston: Houghton, Mifflin\nand Co. _Wordsworth_, vol. i. p. 222.[518]\n\n39\n\n1871. WHIPPLE, E. P. _Literature and Life._ Boston: Houghton, Mifflin\nand Co. _Wordsworth_, p. 253.[519]\n\n40\n\n1854. WILLIS, N. P. _Famous Persons and Places._ New York: Charles\nScribner.[520]\n\n[504] A reprint of the article was published in _The Century Magazine_,\n1884.\n\nC. M. ST. JOHN.\n\n[505] Not of much importance--the author praises Wordsworth and\ncriticises Jeffrey.\n\nC. M. ST. JOHN.\n\n[506] About the same in the \"Address\" as in the \"Complete Works.\"\n\nC. M. ST. JOHN.\n\n[507] Contains four hundred quotations from Wordsworth.\n\nC. M. ST. JOHN.\n\n[508] Contains 258 pages on Wordsworth.\n\nC. M. ST. JOHN.\n\n[509] The same as above with some corrections, and twenty-three new\npages added.\n\nC. M. ST. JOHN.\n\n[510] The above was first given as an address to \"The Wordsworth\nSociety,\" 1884, and appeared in _Wordsworthiana_ in 1889.\n\nC. M. ST. JOHN.\n\n[511] In the Appendix are about twenty pages containing a ferocious\ncriticism on \"Wordsworth, his Poetry and his Misrepresentations.\"\n\nC. M. ST. JOHN.\n\n[512] In the Memoirs of M. F. Ossoli (Boston, vol. iii. p. 84) there is\na short reference to Wordsworth.\n\nC. M. ST. JOHN.\n\n[513] Introduction and Editorial Notes by H. R., interesting and\nvaluable.\n\nC. M. ST. JOHN.\n\n[514] In the Lecture on the Sonnet, there are interesting allusions to\nWordsworth's Sonnets.\n\nC. M. ST. JOHN.\n\n[515] This book and the previous one have about half a dozen pages each\non Wordsworth.\n\nC. M. ST. JOHN.\n\n[516] The substance of this chapter on Wordsworth as a revealer of\nChildhood, first appeared in _The Atlantic Monthly_, October 1885.\n\nC. M. ST. JOHN.\n\n[517] In this volume there are many references to Wordsworth of\ninterest--especially at pp. 202, 206, 210 and 263--on _Subjective\nInterpretation, The Pathetic Fallacy_, etc.\n\nC. M. ST. JOHN.\n\n[518] This essay was also published in _The Complete Poetical Works_.\nPhiladelphia: James Kay jun. and Brothers, 1837. Also in _The North\nAmerican Review_, 1844.\n\nC. M. ST. JOHN.\n\n[519] The above appeared first in _The North American Review_. It was\n\"written when the news came of Wordsworth's death.\" It is not given\nelsewhere in this list.\n\nC. M. ST. JOHN.\n\n[520] Letter V. contains some characteristic remarks on Wordsworth\nby \"Christopher North,\" who gave Willis a note of introduction to\nWordsworth and Southey. Willis did _not_ write about Wordsworth in this\nbook. As it is inserted in some of the lists, I include it, with this\nexplanation.\n\nC. M. ST. JOHN.\n\n\nIV\n\nREVIEW AND MAGAZINE ARTICLES ON WORDSWORTH PUBLISHED IN AMERICA\n\nFROM 1801 TO 1840\n\nIn examining American Reviews and Magazines, for articles on\nWordsworth, I find--after much laborious search--only some\ninsignificant notices of his poems, of no critical or literary merit.\n\nI have carefully read each article which appears in this list, and I\nadd brief explanatory notes, indicative of the general tenor of the\narticles. It was disheartening to find that many of the references to\nWordsworth, in Poole's elaborate _Index to Periodical Literature_,\nwere inaccurate and misleading; and that nearly all the articles on\nWordsworth published in _Harper's Monthly Magazine_ for 1850 were\n\"conveyed\" from contemporary English journals.\n\n1\n\n1801. _The Port Folio._ Vol. i.\n\nMemoranda regarding the first publication of \"Lyrical Ballads\" in\nAmerica.\n\n1801. December, p. 407. The Original Prospectus of \"Lyrical\nBallads.\"[521] (James Humphreys publisher.)\n\n1801. P. 408.[522]\n\n1802. Vol. ii. p. 62.[523]\n\n1803. Vol. iii. p. 288.[524]\n\n1803. P. 320. Note on the poem beginning,\n\n\"A whirl-blast from behind the hill.\"\n\n1804. Vol. iv. p. 87. Announcement that the editor wishes to obtain a\ncopy of _Descriptive Sketches_ (1798) from some publisher or reader.\n\n1804. P. 96.[525]\n\n2\n\n1802. _The Philadelphia Gazette and Daily Advertiser._ (Published by\nSamuel Relf.) Friday, Jan. 15, \"Wordsworth's Lyrical Ballads.\" (The\npublisher's advertisement of the First American Edition.)\n\n3\n\n1819. DANA, R. H.[526] _North American Review._ Vol. xxiii. p. 276. In\nreview of Hazlitt's _English Poets_.\n\n4\n\n1824. _North American Review._ Vol. xviii. p. 356.[527]\n\n5\n\n1824. _United States Literary Gazette._ Vol. i. p. 245.[528]\n\n6\n\n1825. _The Atlantic Magazine_, vol. ii. pp. 334-348.\n\n7\n\n1827. _Christian Monthly Spectator._ Vol. ix. p. 244. (A short article\non Wordsworth.)\n\n8\n\n1832. PRESCOTT, W. H. _North American Review._ Vol. xxxv. pp. 171,\n173-176. (In a \"Review of English Literature of Nineteenth Century,\" is\nan important reference to Wordsworth.)\n\n9\n\n1836. EDWARDS, B. B. _American Biblical Repository._ Vol. vii. pp.\n187-204.[529]\n\n10\n\n1836. _American Quarterly Review._ Vol. xix. p. 66.[530]\n\n11\n\n1836. _American Quarterly Review._ Vol. xix. pp. 420-442.[531]\n\n12\n\n1836. FELTON, C. C. _The Christian Examiner._ Vol. xix. p. 375.[532]\n\n13\n\n1836. PORTER, NOAH. _Christian Quarterly Spectator._[533] Vol. viii.\npp. 127-151.\n\n14\n\n_Christian Monthly Spectator._ Vol xviii. p. 1.[534]\n\n15\n\n1837. _\"Waldie's\" Octavo Library._ (Edited by John J. Smith.)[535]\n\n16\n\n1837. _\"Waldie's\" Octavo Library._ March 21.[536]\n\n17\n\n1837. _Southern Literary Messenger._ Vol. iii. p. 705. \"By a\nVirginian.\"[537]\n\n18\n\n1837. WHIPPLE, E. P. _The Complete Poetical Works of William\nWordsworth_[538] (1837).\n\n19\n\n1839. _New York Review._ Vol. iv. pp. 1-71.[539]\n\n20\n\n1839. _American Biblical Repository._[540] Vol. i. pp. 206-239. (Second\nedition.)\n\n21\n\n1839. _Boston Quarterly Review._ Vol. ii. pp. 137-169. (A review of\n\"Wordsworth's Poetical Works,\" London, 1832.)\n\n22\n\n1839. _American Methodist Review._[541] Vol. xxi. p. 449.\n\n[521] An enthusiastic announcement.\n\nC. M. ST. JOHN.\n\n[522] An appreciatory and critical Introductory Note to _The Waterfall\nand the Eglantine_.\n\nC. M. ST. JOHN.\n\n[523] Editorial reporting the increasing popularity of \"Lyrical\nBallads,\" and further commendation of the poems.\n\nC. M. ST. JOHN.\n\n[524] Note on _The Fountain_.\n\nC. M. ST. JOHN.\n\n[525] An editorial announcement that \"Lyrical Ballads\" had reached\na third edition, and containing one of the most ardent tributes to\nWordsworth in the language.\n\nC. M. ST. JOHN.\n\n[526] Not long, but of much interest.\n\nC. M. ST. JOHN.\n\n[527] An unsigned and excellent review of the 1824 (Boston) edition\nof the poems. The writer remarks that not a volume of Wordsworth's\npoems has been published in America since 1802. Attributed to F.W.P.\nGreenwood.\n\nC. M. ST. JOHN.\n\n[528] Anonymous review of the 1824 (Boston) edition of the poems. One\nof the very best.\n\nC. M. ST. JOHN.\n\n[529] Sectarian in spirit, but on the whole fair to Wordsworth.\n\nC. M. ST. JOHN.\n\n[530] Anonymous. A well-written article of about twenty-four pages,\nreviewing _Yarrow Revisited_. It was one of the earliest reviews in an\nAmerican journal that claimed for Wordsworth a high order of genius. It\nwas probably written by Robert Walsh, the editor of the _Review_.\n\nC. M. ST. JOHN.\n\n[531] An article on Wordsworth's sonnets on Capital Punishment, in an\narticle on \"The English Sonnet.\" Judge Henry Reed found this to have\nbeen written by his father, Professor Henry Reed.\n\nC. M. ST. JOHN.\n\n[532] An appreciative criticism of eight pages.\n\nC. M. ST. JOHN.\n\n[533] Entitled \"Wordsworth and his Poetry.\" A review of the 1824\nedition and of _Yarrow Revisited_, Boston, 1835. An estimate of\nWordsworth's claims as a poet, and as a man. A more comprehensive,\nstronger, more inviting criticism (in appealing to those to whom the\npoetry is unknown) has not been written. It ranks, in my opinion, among\nthe best criticisms on Wordsworth written in America.\n\nC. M. ST. JOHN.\n\n[534] H. Tuckerman wrote an article on Wordsworth for his magazine.\nThis may be the article.\n\nC. M. ST. JOHN.\n\n[535] The number for 7th March contains a notice of Wordsworth, in a\nreview of Reed's _Complete Poetical Works of Wordsworth_ (1837).\n\nC. M. ST. JOHN.\n\n[536] Another mention of Reed's edition, and of the discovery that \"a\nfellow-townsman,\" Dr. T. C. James, anticipated the fact of Wordsworth's\npopularity. A quotation from \"Memoirs of Historical Society of\nPennsylvania\" to prove Dr. James' prophecy.\n\nC. M. ST. JOHN.\n\n[537] Writer unknown.\n\nC. M. ST. JOHN.\n\n[538] To class this review with others of an early date, I have placed\nit among Periodical Reviews. It appeared in _The North American\nReview_, 1844; and again, in 1850, in _Essays and Reviews_.\n\nC. M. ST. JOHN.\n\n[539] A review of Reed's 1837 edition of \"Wordsworth's Poetical Works.\"\nProfessor Henry Reed's son--Judge Henry Reed of Philadelphia--informs\nme that it was written by his father.\n\nC. M. ST. JOHN.\n\n[540] This article is entitled \"Modern English Poetry--Byron, Shelley,\nand Wordsworth.\"\n\nC. M. ST. JOHN.\n\n[541] By an unknown author.\n\nC. M. ST. JOHN.\n\n\nV\n\nCRITICISMS AND REVIEWS IN PERIODICALS FROM 1840 TO 1870\n\nArranged as far as possible according to merit. It is difficult\nto distinguish between the first twelve or fifteen. After them I\nhave placed the articles in the _Literary World_. Most of them have\nnot been noted in other lists, and are especially interesting, as\nbeing additional tributes of Wordsworth's intimate friend, Henry\nReed. I am indebted to Judge Henry Reed of Philadelphia, for more\ncarefully examining his father's papers, and to the _Literary World_\nfor ascertaining, as far as possible, all that his father wrote on\nWordsworth. The criticisms that immediately follow are not without\ninterest. The last half dozen are given, for the most part, because\nthey appear in _Poole's Index_, or in other lists. I have omitted two\nor three which are of no value whatever.\n\nC. M. ST. JOHN.\n\n1\n\n1844. WHIPPLE, E. P. _North American Review._[542] Vol. lix. pp.\n352-384.\n\n2\n\n1857. HAVEN, GILBERT. _Methodist Quarterly Review._ Vol. xxxix. p.\n362.[543]\n\n3\n\n1851. PASSMORE, J. C. _The Church Review._ Vol. iv. pp. 169-188.[544]\n\n4\n\n1866. ALGER, W. R. _Monthly Religious Magazine._ Vol. xxxvi. p. 294.\n\n5\n\n1850. MUZZEY, A. B. _The Christian Examiner._ Vol. xlix. p. 100. (The\ntitle of this article is \"Wordsworth, the Christian Poet.\")\n\n6\n\n1851. GOODWIN, H. M. _The New Englander._ Vol. xlvii. p. 309. (Title,\n\"Wordsworth as a Spiritual Teacher.\")\n\n7\n\n1851. _North American Review._ Vol. lxxiii. p. 473.[545]\n\n8\n\n1851. MOUNTFORD, W. _The Christian Examiner._ Vol. li. p. 275.[546]\n\n9\n\n1851. PORTER, NOAH. _The New Englander Magazine._ Vol. ix. p. 583.[547]\n\n10\n\n1851. WIGHT, ORLANDO WILLIAMS. _American Whig Review._ Vol. xiv. pp.\n68-81.[548]\n\n11\n\n1851. WIGHT, ORLANDO WILLIAMS. _American Whig Review._ Vol. xiii. pp.\n448-458.[549]\n\n12\n\n1854. _Presbyterian Quarterly Review._ Vol. ii. pp. 643-663.[550]\nArticle 1.\n\n13\n\n1854. _Presbyterian Quarterly Review._ Vol. iii. pp. 69-88.[551]\nArticle 2.\n\n14\n\n1841. TUCKERMAN, H. _Southern Literary Messenger._ Vol. vii. p. 105.\n\n15\n\n1850. _Literary World._ Vol. vi. p. 485. \"William Wordsworth.\"[552]\n\n16\n\n1850. REED, HENRY. _Literary World._ Vol. vi. pp. 581, 582. On\nWordsworth.\n\n17\n\n1850. REED, HENRY. _Literary World._ Vol. vii. pp. 205, 206. A second\nshort article.\n\n18\n\n1850. _Literary World._ \"The Prelude.\" Vol. vii. p. 167.[553]\n\n19\n\n1850. _Literary World._ \"Visit to Wordsworth's Grave.\" Vol. vii. p.\n225.[554]\n\n20\n\n1850. SPENCER, J. A. _Literary World._ \"Visit to Wordsworth.\" November\n23.[555]\n\n21\n\n1851. _Literary World._ Vols. viii. ix. (May 24, June 14, July 12,\nAugust 2.)[556] Reviews of Christopher Wordsworth's _Memoirs_ of his\nuncle.\n\n22\n\n1853. REED, HENRY. _Literary World._ Vol. xii. June 25.[557]\n\n23\n\n1850. _Southern Quarterly Review._ Vol. xviii. p. 1. Review of the\n_Poetical Works of Wordsworth_. London: Moxon, 1845.\n\n24\n\n1856. _United States Democratic Review._ Vol. vi. pp. 281-295. (New\nSeries.) Article 1. \"Of Wordsworth's life, beginning at Bristol.\"\n\n25\n\n1856. _United States Democratic Review._ Vol. vi. p. 363. (New Series.)\nArticle 2.\n\n26\n\n1850. _Graham Magazine._ Vol. i. pp. 105-116. Supposed to be written\nby Charles J. Peterson. (Signed P.) Review of the life and poetry\nof Wordsworth, written by one who confessed to an admiration for\nWordsworth's genius bordering on veneration.\n\nC. M. ST. JOHN.\n\n27\n\n1878. _American Journal of Education._ Wordsworth and Cambridge. Vol.\nxxviii. p. 426.[558]\n\n28\n\n1843. _United States Democratic Review._ Vol. xii. p. 158.[559]\n\n29\n\n1836-63. _Christian Review._ Vol. xvi. p. 434. \"Wordsworth as a\nReligious Poet.\"\n\n30\n\n1844. CUYLER, T. L. _Godey's Lady's Book._ Vol. xxviii. (January). \"On\nthe English Lakes and Wordsworth.\"\n\n31\n\n1850. _International Magazine._ Vol. i. p. 271. \"A Review of _The\nPrelude_, from _The Examiner_.\"\n\n32\n\n1855. _Brownson's Quarterly Review._ Vol. xii. p. 525. \"Wordsworth's\nPoetical Works.\"\n\n33\n\n1850. _Graham Magazine._ Vol. i. pp. 322, 323.[560]\n\n34\n\n1842. _United States Democratic Review._ Vol. x. pp. 272-288. (New\nSeries.)[561]\n\n35\n\n1865. _North American Review._ Vol. c. p. 508. Boston: Little, Brown\nand Co.\n\n36\n\n1850. _Southern Literary Messenger._ Vol. xvi. p. 474.[562]\n\n37\n\n1851. _Harper's Monthly Magazine._ Vol. iii. p. 502.[563]\n\n38\n\n1845. BOWEN, F. _North American Review._ Vol. lxi. p. 217.[564]\n\n39\n\n1863. ALGER, W. R. _North American Review._ Vol. xcvi. p. 141.[565]\n\n40\n\n1850. _Southern Literary Messenger._ Vol. xvi. p. 637.[566]\n\n41\n\n1863. WARD, J. H. _North American Review._ Vol. xcvii. p. 387.\n\n42\n\n1853. _The National Magazine._ Vol. iii. No. 7, \"An Estimate of\nWordsworth.\"\n\n43\n\n1853. _The Christian Observer._ Vol. 1. pp. 307-381.[567]\n\n44\n\n1858. \"The Genius of Wordsworth,\" in the \"Editor's Table\" of _Russell's\nMagazine_. Charleston, S.E. Vol. iii. pp. 271-274.\n\n[542] A review of the 1837 edition of Wordsworth's poems. Perhaps no\nabler or more comprehensive review of Wordsworth's life and writings\nhas been written than this, by America's foremost critic.\n\nC. M. ST. JOHN.\n\n[543] One of the best of the early American criticisms.\n\nC. M. ST. JOHN.\n\n[544] A review of the 1851 edition. Contains an earnest plea for the\nstudy of Wordsworth's poetry in America. One of the noblest criticisms\nwritten.\n\nC. M. ST. JOHN.\n\n[545] On the \"Life and Poetry of Wordsworth.\" A review of _The\nPrelude_. Unsigned; but the name is given elsewhere, as T. Chase.\n\nC. M. ST. JOHN.\n\n[546] A review of the _Memoirs of Wordsworth_, by his nephew, the\nBishop of Lincoln.\n\nC. M. ST. JOHN.\n\n[547] A review of Professor Reed's edition of the _Memoirs of\nWordsworth_, Boston, 1851.\n\nC. M. ST. JOHN.\n\n[548] A review of the _Memoirs_, signed O. W.W.\n\nC. M. ST. JOHN.\n\n[549] A review of _The Prelude_.\n\nC. M. ST. JOHN.\n\n[550] Anonymous. A short review of _The Prelude_, and, at greater\nlength, of _The Life_ (edited by Reed). An estimate of his work and\ninfluence.\n\nC. M. ST. JOHN.\n\n[551] Traces the literary life of the poet. Claims for Wordsworth the\nprecedence to Coleridge in the utterance of a spiritual Philosophy.\n\nC. M. ST. JOHN.\n\n[552] A notice of Wordsworth's death, unsigned; but Mr. Wilberforce\nEames--of the Lenox Library--informs me, that their library now owns\nMr. Evert A. Duyckinck's copy of the _Literary World_, and that\ngentleman's own initials are appended in pencil to this article. Mr.\nDuyckinck was editor of the _Literary World_.\n\nC. M. ST. JOHN.\n\n[553] Judge Reed, Professor Henry Reed's son, does not attribute this\narticle to his father. There is an impression that Professor Reed\npublished an article on _The Prelude_. His lecture on that poem was\nnever published.\n\nC. M. ST. JOHN.\n\n[554] Signed by R. F. Correspondence, _London Literary Gazette_, August\n31.\n\nC. M. ST. JOHN.\n\n[555] Possibly the same as in that scarce number of the _Southern\nLiterary Messenger_. Vol. xvi. p. 474.\n\nC. M. ST. JOHN.\n\n[556] These articles, in the opinion of Judge Henry Reed, are not by\nhis father, Professor Henry Reed.\n\nC. M. ST. JOHN.\n\n[557] Notice to those who wish to subscribe to the Memorial to\nWordsworth, signed.\n\nC. M. ST. JOHN.\n\n[558] An article on the University of Cambridge, and an account of\nWordsworth's residence at St. John's College, 1787-1791.\n\nC. M. ST. JOHN.\n\n[559] Six pages on Wordsworth's _Sonnet to Liberty_.\n\nC. M. ST. JOHN.\n\n[560] A brief review of _The Prelude_ and _Excursion_, and a comparison\nbetween the two poems.\n\nC. M. ST. JOHN.\n\n[561] On Wordsworth's sonnets in favour of Capital Punishment.\n\nC. M. ST. JOHN.\n\n[562] On the house at Rydal.\n\nC. M. ST. JOHN.\n\n[563] An unsigned, four paged article on Wordsworth, Byron Scott, and\nShelley.\n\nC. M. ST. JOHN.\n\n[564] In a \"Review of Longfellow's _Poets and Poetry of Europe_,\" a\npage on Wordsworth's influence.\n\nC. M. ST. JOHN.\n\n[565] In \"The Origin and Uses of Poetry,\" a few lines on Wordsworth.\n\nC. M. ST. JOHN.\n\n[566] A notice, with extracts from _The Prelude_.\n\nC. M. ST. JOHN.\n\n[567] \"The Religion of Wordsworth's Poetry.\"\n\nC. M. ST. JOHN.\n\n\nVI\n\nCRITICISMS AND REVIEWS IN PERIODICALS FROM 1870 TO 1895\n\nThese are not chronologically arranged by Mrs. St. John, but see her\nnote to Section V.--ED.\n\n1\n\n1882. DEWITT, DR. JOHN. _Presbyterian Review._ Vol. iii. p. 241.[568]\n\n2\n\n1884. BURROUGHS, JOHN. _The Century Magazine._ Vol. v. p. 418. This is\nentitled \"Wordsworth's Country.\"\n\n3\n\n1880. CRANCH, C. P. _The Atlantic Monthly._ Vol. xlv. p. 241. Entitled\n\"Wordsworth.\" A review of the 1880 Poetical Works (Boston). The writer\nnotes what he considers the chief excellency as well as defects of\nWordsworth's poetry.\n\n4\n\n1888. MURRAY, J. O. _The Homiletic Review._ Vol. xvi. pp. 295-304.\nTitle, \"The Study of Wordsworth's Poetry.\"\n\n5\n\n1890. PATTISON, T. H. _The Baptist Review._ Vol. xii. p. 265. \"The\nReligious Influence of Wordsworth.\"\n\n6\n\n1889. HUTTON, LAWRENCE. _Harper's Monthly Magazine._ Vol. lxxviii.[569]\n(in Literary Notes).\n\n7\n\n1880-1. CONWAY, MONCURE D. _Harper's Monthly Magazine._ \"The English\nLakes and their Genii.\" Vol. lxii. pp. 7, 161, 339.\n\n8\n\n1883. PEDDER, H. C. _The Manhattan._ Vol. ii. pp. 418-433.[570]\n\n9\n\n1876. YARNALL, ELLIS. _Lippincott's Magazine._ Vol. xviii. pp. 543-554,\n669-683. \"Walks and Visits in Wordsworth's Country.\" Written in the\nsummer of 1855 and 1857.\n\n10\n\n1871. FIELDS, J. T. _The Atlantic Monthly._ Vol. xxviii. p. 750. On\nWordsworth, in an article entitled \"Our Whispering Gallery.\" The same\narticle is cut down in _Yesterdays with Authors_.[571]\n\n11\n\n1892. PARSONS, EUGENE. _The Examiner._ Vol. lxx. p. 1. On \"Tennyson and\nWordsworth.\"\n\n12\n\n1888. WILLIAMS, T. C. _Andover Review._ Vol. ix. p. 30.\n\n13\n\n1889. NOBLE, FRED PERRY. _The Homiletic Review._ Vol. xviii. p. 306.\n\"The Value of Wordsworth to the Preacher.\"\n\n14\n\n1873. HIMES, JOHN A. _Lutheran Quarterly Review._ Vol. iii. p. 252.\n\"The Religious Faith of Wordsworth and Tennyson as shown in their\nPoems.\"\n\n15\n\n1881. JOHNSON, E. E. _American Church Review._ Vol. xxxiii. p. 139.\n\"Influence of Wordsworth's Poetry.\"\n\n16\n\n1886. COAN, T. M. _The New Princeton Review._ Vol. i. pp. 297-319.\n\"Wordsworth's Passion.\"\n\n17\n\n1889. VEDDER, H. C. _The New York Examiner_, August 28. \"The Decline of\nWordsworth.\"[572]\n\n18\n\n1877. COAN, T. M. _The Galaxy._ Vol. xxiii. pp. 322-336. \"Wordsworth's\nCorrections.\"[573]\n\n19\n\n1881. BOWEN, F. F. _The Dial._ Vol. i. p. 21. \"A Review of Myers'\nWordsworth.\"\n\n20\n\n1881. GERHART, R. L. _Reformed Quarterly Review._ Vol. xxviii. p. 344.\n\"Wordsworth and his Art.\"\n\n21\n\n1887. WOODBERRY, G. E. _The Nation._ Vol. xlv. p. 487. \"Wordsworth and\nthe Beaumonts.\"\n\n22\n\n1881. BROWNELL, W. C. _The Nation._ Vol. xxxii. p. 153. \"Myers'\nAccount of Wordsworth.\"\n\n23\n\n1872. CROFFUT, W. A. _Lakeside Monthly._ Vol. viii. pp. 418-425.\n\"Wordsworth.\"\n\n24\n\n1895. THORPE, F. W. _The Philadelphia Call._ \"The Home of Wordsworth.\"\nAutobiographic and critical.\n\n25\n\n1879. _Appleton's Journal._ Vol. xxii. p. 223. \"How to Popularise\nWordsworth.\"\n\n26\n\n1874. DE-VERE, A. _The Catholic World._ Vol. xix. p. 795.\n\"Recollections of Wordsworth.\"\n\n27\n\n1875. DE-VERE, A. _The Catholic World._ Vol. xxii. p. 329.\n\n28\n\n1891. PAGE, H. A. _The Century Magazine._ No. 1. pp. 453-864.\n\"Wordsworth and De Quincey. With hitherto unpublished letters.\"[574]\n\n29\n\n1853. _The National Magazine._ Vol. iii. pp. 36-40.\n\n30\n\n1853. _Brownson's Quarterly Review._ Vol. xii. 525.\n\n31\n\n1896. THEODORE W. HUNT in _Bibliotheca Sacra_. No. 66. \"William\nWordsworth.\"\n\n32\n\n1896. J. W. BRAY. _The Literary Democracy of Wordsworth_ in \"Poet\nLove.\" Vol. iii. No. 6.\n\n[568] On \"The Homiletic Value of Wordsworth's Poetry.\" One of the\nablest papers ever written on Wordsworth. It contains the best reply to\nMatthew Arnold's estimate of his poetry.\n\nC. M. ST. JOHN.\n\n[569] This is a review of Rolf's _Wordsworth's Selected Poems_.\nIt contains one of the most appreciative tributes to Wordsworth's\ninfluence which has appeared in America.\n\nC. M. ST. JOHN.\n\n[570] On \"Wordsworth and the Modern Age.\" Illustrated by W. St.\nJ. Harper, and other artists. It deals with the especial need of\nWordsworth's \"calming influence in the exacting competition for\nsuccess,\" and gives a comparison between Virgil and Wordsworth.\n\nC. M. ST. JOHN.\n\n[571] Of interest to Americans.\n\nC. M. ST. JOHN.\n\n[572] It aims to give some explanation of the lack of interest in\nWordsworth's poetry in later days.\n\nC. M. ST. JOHN.\n\n[573] An attempt, the writer says, to point out the corrections,\nleaving their interpretation to the reader.\n\nC. M. ST. JOHN.\n\n[574] Written by an Englishman, but published first in an American\nmagazine.\n\nC. M. ST. JOHN.\n\n\nVII\n\nVISITS TO WORDSWORTH BY EMINENT AMERICANS\n\nThe following books record visits made by eminent Americans to\nWordsworth.\n\nC. M. ST. JOHN.\n\n1\n\n1863. HAWTHORNE, N. _Our Old Home, and English Note-Books._ Vol. ii.\npp. 24-56, etc.; also,\n\n1883. Boston: Houghton, Mifflin and Co. \"A Visit to Wordsworth.\"\n\n2\n\n1856. EMERSON, R. W. _English Traits._ Boston: James Munroe and Co. pp.\n24-31; also,\n\n1881. Boston: Houghton, Mifflin and Co. Visit to Wordsworth, in chapter\nentitled \"First Visit to England.\"\n\n3\n\n1876. TICKNOR, GEORGE. _Life, Letters, and Journals._ Boston: James R.\nOsgood and Co. 2 vols. Vol. i. pp. 287, 288, etc. Vol. ii. p. 167, etc.\n\n4\n\n1836. DEWEY, ORVILLE. _The Old World and the New._ Boston: 2 vols. pp.\n89-96.\n\n5\n\n1884. BRYANT, W. C. Prose Works. In a chapter on \"Poets and Poetry of\nthe English Language\" (New York: D. Appleton and Co.) a few pages deal\nwith Wordsworth.\n\n\nVIII\n\nA FEW POEMS ON WORDSWORTH\n\n1\n\n1846. WALLACE, W. _Poem on Wordsworth._ New York: 12mo.\n\n2\n\n1850. FIELD, JAMES T. _Graham Magazine_ (October). \"Wordsworth.\"\n\n3\n\n1850. ALEXANDER, W. _Graham Magazine_ (November), p. 221. \"Wordsworth.\n(A Sonnet.)\"\n\n4\n\n1850. H. M. R. _Harpers Magazine._ \"Sonnet on the Death of Wordsworth.\"\nVol. i. p. 218.\n\n5\n\n1850. E. A. W. _Literary World._ \"Sonnet on Wordsworth.\" Vol. vii. p.\n255.\n\n6\n\n1874. WHITTIER, J. G. Whittier's Works. Boston: Houghton, Mifflin and\nCo. \"Poem on Wordsworth. Written on a blank leaf of _Wordsworth's\nMemoirs_, 1851.\" Vol. iv. p. 66.\n\n7\n\n1890. SCOLLARD, CLINTON (?) _Northern Christian Advocate._ \"The Poet's\nSeat. A Sonnet on Wordsworth. Written at Ambleside, 1890.\"\n\n8\n\n1893. \"To Wordsworth, after reading his XXX Ecclesiastical Sonnets\" in\n_The Echo and the Poet_, by William Cushing Bamburgh. N. Y. 1893.\n\n\nIX\n\nUNPUBLISHED LECTURES ON WORDSWORTH\n\nESSAYS OF SPECIAL INTEREST\n\n1\n\n1892. CORSON, HIRAM. \"The Divine Immanence in Nature, and the\nrelationship of the human spirit thereto, as presented in Wordsworth's\nPoetry.\"\n\n2\n\nWINCHESTER, C. T. \"The Lake District and Wordsworth.\"\n\n3\n\nPRENTISS, GEORGE L. \"Hurstmonceaux Rectory and Rydal Mount.\" (Personal\nRecollections.)\n\n4\n\nHOYT, A. S. \"Wordsworth, the Man and the Poet.\" (Imperfectly reported\nin _The Houghton Record_.)\n\n\n\n\nIII.--_FRANCE_\n\nWORDSWORTH IN FRANCE\n\nBy \u00c9MILE LEGOUIS, Professeur \u00e0 la Facult\u00e9 de Lettres, Universit\u00e9 de\nLyon, France\n\n\nI\n\nBIBLIOGRAPHY\n\nThere is no separate or whole book on Wordsworth that I know of.\n\nARTICLES IN MAGAZINES, OR CHAPTERS IN BOOKS\n\n_Voyage historique et litt\u00e9raire en Angleterre et en \u00c9cosse_, par\nAm\u00e9d\u00e9e Pichot (_passim_). 3 vols. in 8. Paris, 1829.[575] An English\ntranslation was published in London in 1825.\n\n_Revue Britannique._\n\nMai 1827. Wordsworth, Crabbe, and Campbell, pp. 61-79, a criticism\ntranslated from the _New Monthly Magazine_.\n\nF\u00e9vrier 1835. Po\u00e9sie domestique de la grande Bretagne, translated from\nthe _New Monthly Magazine_.\n\nJanvier 1836, p. 190. Compte-rendu de \"Yarrow Revisited and other\nPoems,\" translated from the _Repository of Knowledge_.\n\n_Revue des Deux Mondes._ 1er Ao\u00fbt 1835. William Wordsworth, par A.\nFontaney.[576]\n\n_Revue Contemporaine._ 15 D\u00e9cembre 1853. Po\u00e8tes contemporains de\nl'Angleterre: William Wordsworth et John Wilson, par L. \u00c9tienne.\n\n_Litt\u00e9rature anglaise_ de H. Taine.[577] 1864. Vol. iv. pp. 311-324.\n\n_\u00c9tudes sur la Litt\u00e9rature contemporaine_, par \u00c9d. Sch\u00e9rer.[578]\n\n_Revue critique d'histoire et de litt\u00e9rature._ 16 Janvier 1882. Article\nde James Darmesteter sur la Biographie de Wordsworth, par Myers.[579]\n\n_Essais de Litt\u00e9rature anglaise_, par James Darmesteter. Paris,\n1883.[580]\n\n_Histoire de la Litt\u00e9rature anglaise_, par M. L\u00e9on Boucher. Paris,\n1890. pp. 355-363.\n\n_La Renaissance de la Po\u00e9sie anglaise_, par Gabriel Sarrazin. 1887.\n\n_\u00c9tudes et Portraits_, par Paul Bourget. Vol. ii. \u00c9tudes\nanglaises.[581] 1888.\n\n_\u00c9tude sur la Vie et les \u0152uvres de Robert Burns_, par Auguste\nAngellier. Paris, 1892. (_Passim_, et surtout vol. ii. pp. 362-393,\n\u00c9tude sur le sentiment de la nature dans Wordsworth et autres po\u00e8tes\nanglais contemporains.)\n\n_Le g\u00e9n\u00e9ral Michel Beaupuy_, par Georges Bussi\u00e8re et \u00c9mile Legouis.\nParis, 1891.\n\n[575] Vol. ii. pp. 363-394.--ED.\n\n[576] This was signed Y, which was Fontaney's pseudonym.--E.L.\n\n[577] Wordsworth et la po\u00e9sie moderne de l'Angleterre.--_Histoire de la\nLitt\u00e9rature anglaise_, par H. Taine.--ED.\n\n[578] Vol. vi. pp. 127, 128, and vol. vii. pp. 1-59.--ED.\n\n[579] pp. 227-236.--ED.\n\n[580] pp. 227-236.--ED.\n\n[581] Vol. ii. pp. 83; 126-134.--ED.\n\n\nII\n\nTRANSLATIONS\n\nPas de traduction compl\u00e8te, ni de volume sp\u00e9cial de traductions de\nWordsworth.\n\nUne traduction par Fontaney annonc\u00e9e en 1837 comme devant para\u00eetre dans\nle _Biblioth\u00e8que anglo-fran\u00e7aise_, n'a pas paru.\n\nEn dehors des po\u00e8mes ou parties de po\u00e8mes traduit par les critiques\n\u00e9num\u00e9r\u00e9s plus haut, il n'y a gu\u00e8re de traduction en prose de quelque\nimportance.\n\nTRADUCTIONS EN VERS\n\nMADAME AMABLE TASTU. _We are Seven._\n\nSAINTE-BEUVE. _Joseph Delorme._ 1829.\n\n    \"Le plus long jour de l'ann\u00e9e,\" p. 88.\n    Sonnet, \"Personal Talk,\" p. 123.\n    \"Sonnet sur le Sonnet,\" p. 124.\n\n_Consolations._ 1830.\n\n    Sonnet, \"It is a beauteous evening,\" p. 234.\n    Sonnet, \"Not Love, nor War,\" p. 239.\n    Sonnet, \"Quand le po\u00e8te en pleurs,\" p. 236.\n\n_Pens\u00e9es d'Ao\u00fbt._ Trois sonnets imit\u00e9s de Wordsworth.\n\n      I. \"Reposez-vous et remerciez.\"\n     II. \"La Cabane du Highlander.\"\n    III. \"Le Ch\u00e2teau de Bothwell.\"\n\nSainte-Beuve cite en outre dans ses _Nouveaux Lundis_ des 21 et 22\nAvril 1862, trois sonnets de Wordsworth traduits en vers, par l'Abb\u00e9\nRoussel. Ces traductions assez pauvres de po\u00e9sie sont celles des\nsonnets suivants--\n\n    \"Nuns fret not.\u2026\"\n    \"Dark and more dark.\u2026\"\n    \"These words were uttered as in pensive mood.\"\n\nJEAN AICARD a traduit _We are Seven_ dans _La Chanson de l'Enfant_.\n\nPAUL BOURGET (_\u00c9tudes et Portraits_, vol. ii. _op. cit._) a traduit\nl'un des sonnets au Duddon.\n\n    \"What aspect bore the Man \u2026?\"\n\n\nIII\n\nINFLUENCE\n\nWordsworth's influence on French literature was altogether very slight,\nnor did it make itself felt till about 1830; when, after a very limited\nperiod, it silently died away.\n\nWordsworth was but little known by his contemporary Ch\u00e2teaubriand, who\nmerely names him among other poets in his _Essai sur la Litt\u00e9rature\nanglaise_. Byron, Walter Scott, and in a lesser degree Thomas\nMoore, were the only writers of Great Britain whose works told on\nour literature at that time. Villemain, in his criticism of Byron,\ncontemptuously dismisses all the so-called lake-poets to fix on his\nhero. He calls them: \"Des m\u00e9taphysiciens, raisonneurs sans invention,\nm\u00e9lancoliques sans passion, qui, dans l'\u00e9ternelle r\u00eaverie d'une vie\n\u00e9troite et peu agit\u00e9e, n'avaient produit que des singularit\u00e9s sans\npuissance sur l'imagination des autres hommes. Tel \u00e9tait Woodsworth\n(_sic_) et le subtil mais non touchant Col\u00e9ridge.\"\n\nTo Byron also, and to him alone (Ossian being excepted) among the\npoets of England, was Lamartine indebted. I am not sure that he names\nWordsworth once; but still the striking analogy between the ideas and\nimaginative style of both cannot fail to be noticed by the reader.\nWithout insisting on a parallel that might be drawn between many pages\nof _The Excursion_ and of _Jocelyn_, I will only point out two short\npieces of Lamartine that bear strong resemblance to two poems of\nWordsworth, so much so that they almost read like free imitations--\n\n          Lamartine                       Wordsworth's\n\n    \"A Augusta,\" _Recueillements  |\n    Po\u00e8tiques_, xx.               | _Nightingale and Stock-dove._\n                                  |\n    \"Le Fontaine du Foyard,\"      |\n    _Nouvelles Confidences_.      | _The Fountain._\n\nVictor Hugo, so far as I know, only names Wordsworth once, in _L'\u00c2ne_--\n\n        \u2026Young le pleureur des nuits,\n    Wordsworth l'esprit des lacs \u2026\n\nM. Sully Prudhomme when he wrote _A l'Hirondelle_ (stanzas, la vie\nint\u00e9rieure) appears to have borne in mind _To a Skylark_, \"Ethereal\nminstrel,\" etc.\n\nM. Copp\u00e9e has often been called a French Wordsworth, owing to his\npoetical collection called _Les Humbles_, wherein he shows the same\npartiality as the English Poet does for humble themes and characters,\ntogether with a bold attempt to naturalise trivial or ludicrous\ndetails in serious poetry; but there is no proof, as far as I know, of\nWordsworth's influence having been strong upon him.\n\nIf we except two or three disciples of Wordsworth, neither he, nor\nthe lake-poets taken as a whole, seem to have been much thought of, or\neven read, by our contemporary verse-writers. The word _Lakist_ has\ngenerally been used as a synonym for \"weak and doleful mysticism.\"\nEx.:--\n\n(_a_) _Revue Encyclop\u00e9dique._ 1831. Article de Pierre Leroux, sur la\n\"Po\u00e9sie de notre \u00c9poque.\" \"L'Angleterre a entendu autour de ses lacs\nbourdonner comme des ombres plaintives un essaim de po\u00e8tes ab\u00eem\u00e9s dans\nune mystique contemplation.\"\n\n(_b_) _Journal d'un Po\u00e8te_, par Alfred de Vigny. (Ed. Michel L\u00e9vy.\n1867. p. 80.) \"Barbier vient de publier _Il Pianto_. Les d\u00e9lices de\nCapone ont amolli son caract\u00e8re de po\u00e9sie et Brizeux a d\u00e9teint sur\nlui ses douces couleurs virgiliennes et laquistes (_sic_) d\u00e9rivant de\nSainte-Beuve.\"\n\n(_c_) TH\u00c9OPHILE GAUTIER (_Portraits Contemporains_, p. 174) almost\nseems to derive the word _Lakiste_ from Lamartine's poem called _Le\nLac_. He has just mentioned the poem and goes on: \"Il ne faut pas\ncroire que Lamartine, parce qu'il y a toujours chez lui une vibration\net une r\u00e9sonnance de harpe \u00e9olienne, ne soit qu'un m\u00e9lodieux _lakiste_\net ne sache que soupirer mollement la m\u00e9lancolie et l'amour. S'il a le\nsoupir, il a la parole et le cri \u2026\" (_Journal Officiel_, 8 Mars 1869.)\n\nI now come to the man who, first and foremost among our poets and\ncritics, paid due homage to Wordsworth, _i.e._ Sainte-Beuve. I have\nalready enumerated his several translations in verse from Wordsworth.\nStrange to say, the voluminous critic has no single article with\nWordsworth for its main subject; but, whoever will go through his many\nvolumes will find many judicious and admiring references to the poet.\n\nMoreover, as a poet, Sainte-Beuve has endeavoured to naturalise in\nFrance the poetic style that has been associated with the name of\nWordsworth. He expressly claims Wordsworth as one of his masters in his\n_Consolations_ xviii. \"A Antony Deschamps.\" Among his bosom-poets he\nreckons--\n\n        \u2026Wordsworth peu connu, qui des lacs solitaires\n    Sait t\u00f2us les bleus reflets, les bruits et les myst\u00e8res,\n    Et qui, depuis trente ans vivant au m\u00eame lieu,\n    En contemplation devant le m\u00eame Dieu,\n    A travers les soupirs de la mousse et de l'onde,\n    Distingue, au soir, des chants venus d'un meilleur monde.\n\nThe original attempt of Sainte-Beuve (for he was original in his very\nchoice of Wordsworth as a model at a time when Byron engrossed all\nthe admiration of the French poets) has been ably characterised by\nTh\u00e9ophile Gautier in his \"Portraits Contemporains\" (pp. 208, 209), an\narticle reprinted from _La Gazette de Paris_, 19 Novembre 1871:--\n\n    \"(Sainte-Beuve) avait \u00e9t\u00e9 en po\u00e9sie un inventeur. Il avait\n    donn\u00e9 une note nouvelle et toute moderne, et de tout le c\u00e9nacle\n    c'\u00e9tait \u00e0 coup s\u00fbr le plus r\u00e9ellement romantique. Dans cette\n    humble po\u00e9sie qui rappelle par la sinc\u00e9rit\u00e9 du sentiment et\n    la minutie du d\u00e9tail observ\u00e9 sur nature, les vers de Crabbe,\n    de Wordsworth, et de Cowper, Sainte-Beuve s'est fray\u00e9 de\n    petits sentiers \u00e0 mi-c\u00f4te, bord\u00e9s d'humbles fleurettes, o\u00f9 nul\n    en France n'a pass\u00e9 avant lui. Sa facture un peu laborieuse\n    et compliqu\u00e9e vient de la difficult\u00e9 de r\u00e9duire \u00e0 la forme\n    m\u00e9trique des id\u00e9es et des images non exprim\u00e9es encore ou\n    d\u00e9daign\u00e9es jusque-l\u00e0, mais que de morceaux merveilleusement\n    venus o\u00f9 l'effort n'est plus sensible!\"\n\nSainte-Beuve's admiration of Wordsworth is a well-known fact. Less\ngenerally known is the influence of this admiration on several poets\nof that time (_circa_ 1830-40), who, either through Sainte-Beuve's\nimitations, or with a direct knowledge of Wordsworth's poems, to the\nreading of which they had thus been stimulated, offer great marks of\nresemblance with Wordsworth. I have quoted a judgment of De Vigny that\nconsiders Brizeux and Barbier as having turned _laquistes_ through\nSainte-Beuve. I know no other immediate proof of this influence.\nPerhaps Barbier and Brizeux have consigned it somewhere. Anyhow Brizeux\nwith his glorification of his youthful years and school-time, with\nhis intense love of his native Brittany, his fond attachment to local\ncustoms and habits, his lamentations on the death of the poetical poet\nas embodied in his own province (_\u00c9l\u00e9gie de la Bretagne_), is to all\nextent and purposes the most thoroughly Wordsworthian of all our poets.\nThere may be more of Wordsworth's _philosophy_ in Lamartine, but there\nis more of his _poetry_ proper in Brizeux.\n\nThe influence of Wordsworth on Maurice de Gu\u00e9rin and Hippolyte de la\nMorvonnais, is more easily ascertained than the preceding. Here, again,\nSainte-Beuve appears to have been the intermediate agent.[582]\n\nIn 1832-33 Maurice de Gu\u00e9rin, fresh from the reading of the\n_Consolations_, and De la Morvonnais, who came to be a direct admirer\nof the Lake Poets, and chiefly of Wordsworth, set to write short\npoems which they aspired to make as little different from prose as\npossible, rejecting all traditional ornaments, and making little of\nthe rhythmical improvements of the _Romantiques_ proper. Some of those\npieces were inserted in a local paper as downright prose (no stop\nintervening at the end of the lines), whereas the said paper would\nnot have made room for verse.[583] This looks like trifling, but the\nearnestness of this attempted revolution is shown in the interesting\npoems of Maurice de Gu\u00e9rin. Another outcome of this was an intended\npublication on Wordsworth, of which it is impossible to say whether it\nwas to be a criticism, or a translation, of the English Poet. It is\nthus mentioned in a letter of Gu\u00e9rin to De la Morvonnais of June 30,\n1836: \"Nous avons adress\u00e9 des circulaires \u00e0 un grand nombre d'\u00e9diteurs\npour l'impression Wordsworth. Nous attendons la r\u00e9ponse d'un moment \u00e0\nl'autre.\" The answer must have been unfavourable, as nothing more was\nheard of the intended publication.\n\nThe early death of Gu\u00e9rin left it for De la Morvonnais alone to spread\nthe influence of Wordsworth's poetry in France. Of him we read in\nSainte-Beuve's _\u00c9tude sur Maurice de Gu\u00e9rin_:--\n\n    \"La Morvonnais, vers ce temps m\u00eame (1834), en \u00e9tait fort\n    pr\u00e9occup\u00e9 (des lakistes et de leur po\u00e9sie), au point d'aller\n    visiter Wordsworth \u00e0 sa r\u00e9sidence de Rydal Mount, pr\u00e8s des lacs\n    du Westmoreland, et de rester en correspondance avec ce grand\n    et pacifique esprit, avec ce patriarche de la Muse intime.\n    Gu\u00e9rin, sans tant y songer, ressemblait mieux aux Lakistes en\n    ne visant nullement \u00e0 les imiter.\"\n\nOf the supposed correspondence between Wordsworth and De la Morvonnais\nno trace remains. M. Hippolyte de la Blanchardi\u00e8re, De la Morvonnais'\ngrandson, has informed me that in the collection of his grandfather's\nletters there is no letter of Wordsworth to be found. That at least\na Study of Wordsworth existed at the time is proved by the following\npreface to his poem _La Th\u00e9ba\u00efde des Gr\u00e8ves_, written by his friend A.\nDuquesnel (ed. by Didier, Quai des Augustins. 1864. p. xxvii.)\n\n     \"Nous avons trouv\u00e9 dans les _Reliquiae_ du po\u00e8te de\n    l'Arguenon[584] de pr\u00e9cieuses \u00e9tudes sur les lakistes. Il\n    s'\u00e9tait passionn\u00e9 pour ces hommes dans les dix derni\u00e8res ann\u00e9es\n    de sa vie (1843-53).[585] Wordsworth lui semblait plus grand\n    que Byron, qu'il trouvait trop emphatique, trop solennel,\n    pas assez pr\u00e8s de la nature. L'auteur de _l'Excursion_ a\n    exerc\u00e9 une p\u00e9n\u00e9trante influence sur l'esprit et le c\u0153ur de la\n    Morvonnais, nous trouvons dans ses cahiers des traductions\n    en vers de Wordsworth, de Col\u00e9ridge, de Crabbe, qui, lui, ne\n    faisait pas partie de ce groupe. Nous les publierons peut-\u00eatre\n    un jour; elles ont d'autant plus d'int\u00e9r\u00eat que l'on ne conna\u00eet\n    gu\u00e8re les lakistes en France, que par de rares extraits. Il\n    s'\u00e9tait livr\u00e9, comme on le verra, \u00e0 une \u00e9tude approfondie de la\n    litt\u00e9rature anglaise. Son admiration pour Walter Scott \u00e9tait\n    inexprimable.\"\n\nThe study and translations above-mentioned have also been lost, many\nmanuscripts of De la Morvonnais having been destroyed.\n\nIt remains for me to point out some allusions to, or imitations of,\nWordsworth in the existing verse of De la Morvonnais.\n\nIn the _Th\u00e9ba\u00efde des Gr\u00e8ves_ (1838), \"Le Petit Patour\" is a close\nimitation of _We are Seven_, the conclusion being--\n\n    Cet enfant en sait plus que moi sur l'existence;\n    Savoir vivre est savoir souffrir avec constance.\n\n\"Le Vagabond,\" a story of a vagrant by whom the poet is taught\nresignation, is an imitation of _Resolution and Independence_.\n\nIn \"A Sainte-Beuve\" are found these two lines--\n\n    J'ai pos\u00e9 sous mon bras mon penseur solitaire,\n    Mon Wordsworth tant aim\u00e9 de l'amant du myst\u00e8re.\n\nIn \"Dispersion, \u00e0 Mistress Hemans,\" etc., we read this--\n\n    Nous primes un po\u00e8te, une femme ang\u00e9lique\n    Dont peu savent chez nous la voix m\u00e9lancolique,\n    Disciple de Wordsworth, le sublime penseur,\n    Des lakistes ch\u00e9ris je la nomme la s\u0153ur.\n\nIn \"Derni\u00e8res Paroles\" we find this praise of Wordsworth--\n\n    Or, ce soir-l\u00e0, je lus un homme de g\u00e9nie;\n    Celui dont la mystique et profonde harmonie\n    Sonne pour les \u00e9lus des po\u00e9tiques dons,\n    Et soul\u00e8ve notre \u00e2me en ses grands abandons \u2026\n                        \u2026Oh! ne pourrai-je voir\n    Ces lacs du Westmoreland, mon d\u00e9sir, mon espoir?\n    \u2026\n    Cet homme est honor\u00e9 des puissances secr\u00e8tes;\n    Lui mort, \u00e0 ses beaux lacs, romantiques retraites,\n    Des p\u00e8lerins viendront, penseurs religieux.\n    Le monde m\u00e9connut l'homme m\u00e9lodieux.\n\nI pass over many sonnets, and divers other poems, in which the\ninfluence of Wordsworth is unmistakable, and come to a last quotation\nwhich is useful to elucidate an allusion in Wordsworth's _The Poet's\nDream: Sequel to the Norman Boy_. In this poem, written in 1842,\nWordsworth says--\n\n    But oh! that Country-man of thine, whose eye, loved Child, can see\n    A pledge of endless bliss in acts of early piety,\n    In verse, which to thy ear might come, would treat this simple theme,\n    Nor leave untold our happy flight in that adventurous dream.\n\nAs Wordsworth read very little French poetry in his old age, I think he\nhere alludes to a poem of his admirer De la Morvonnais, who very likely\nsent him that _Th\u00e9ba\u00efde des Gr\u00e8ves_ (1838), in which Wordsworth was so\nhighly praised. The passage alluded to is taken from \"Solitude,\" and\nreads thus--\n\n    Enfant, Il (Dieu) te promet le domaine de l'ange\n      Si tu gardes l'amour et la foi des a\u00efeux,\n    Et sa m\u00e8re, aujourd'hui loin de l'humaine fange,\n      Que tu n'as pas connue et qui t'attend aux cieux.\n\nAs a whole, De la Morvonnais, though he imitates Wordsworth, is very\nunlike him. Of course I do not mean to compare the two, but even\nin like subjects he differs from Wordsworth, owing to a sort of\nconstitutional nervousness and brooding melancholy.[586]\n\n[582] Voir Maurice de Gu\u00e9rin, _Journal, Lettres et Po\u00e8mes_, publi\u00e9s par\nJ. S. Tr\u00e9butien avec Pr\u00e9face de Sainte-Beuve (1860).--E.L.\n\n[583] In the above work--_S\u00e9jour de M. de Gu\u00e9rin en Bretagne;\nImpressions et Souvenirs de M. Fran\u00e7ois du Breil de Marzan_, pp.\n434-441.--E.L.\n\n[584] H. de la Morvonnais.--E.L.\n\n[585] A mistake: his admiration of Wordsworth began before 1832.--E.L.\n\n[586] In _Voyage historique et litt\u00e9raire en Angleterre et en \u00c9cosse_,\npar Am\u00e9d\u00e9e Puchot, Lettre XXIV. there are numerous references to\nWordsworth. It begins with a quotation from _Tintern Abbey_. In\nLettre LXV. there is additional critical reference to Wordsworth and\nColeridge. In the _Album po\u00e9tique des jeunes personnes_, par Mme.\nTastu, there is a \"Sonnet imit\u00e9 de Wordsworth,\" by St. Beuve, pp. 101,\n102.\n\n    C'est un beau soir, un soir paisible et solennel,\n    A la fin du saint jour la nature en pri\u00e8re\n    Le tait, comme Marie \u00e0 genoux sur la pierre, etc.--ED.\n\nSee also the _Nouveaux Lundis_ of St. Beuve, 21 and 22 Avril 1862,\nwhere there are \"trois sonnets traduits en vers par l'Abb\u00e9 Roussel\"\nfrom Wordsworth.\n\n\n\n\nERRATA AND ADDENDA LIST\n\nREFERRING TO VOLUMES I. TO VIII.\n\n\n1. _Inistar omnium._--I wish to explain the accidental omission of Mr.\nT. Hutchinson's name amongst those who helped me in Volumes I. and II.\n(see the prefatory note to this volume), and also that of Mr. Hill. It\nwas due to my returning, \"for press,\" an uncorrected copy of my Preface.\n\n2. Vol. ii. p. 106, _Ruth_, l. 54--The following extract from Bartram's\n_Travels_, etc., illustrates Wordsworth's debt to him:--\n\n    Proceeding on our return to town in the cool of the evening\n    \u2026 we enjoyed a most enchanting view; \u2026 companies of young\n    innocent Cherokee virgins, some busy gathering the rich\n    fragrant fruit, others having already filled their baskets, lay\n    reclined under the shade of floriferous and fragrant native\n    bowers \u2026 disclosing their beauties to the fluttering breeze\n    \u2026 whilst other parties, more gay and libertine, were yet\n    collecting strawberries, or wantonly chasing their companions,\n    tantalising them, staining their lips and cheeks with the ripe\n    fruit.\n\n3. In vol. ii. p. 348, the date of publication should be Sept. 17,\n1802, not 1803.\n\n4. In _The Prelude_ (vol. iii. p. 202, book v. l. 26) the quotation\nwhich I could not trace is from Shakespeare, Sonnet No. 64--\n\n    This thought is as a death, which cannot choose\n    But weep to have that which it fears to lose.\n\n5. Vol. v. p. 113 (_The Excursion_, book iii. l. 187).--Mr. William\nE. Walcott--Laurence, Mass. U.S.A.--sends me the following variant\nreadings, which he has found in a copy of the edition of 1814--\n\n                      \u2026 crystal tube\n    Be lodged therein \u2026\n\nP. 151, book iv. l. 187--\n\n    Nor sleep, nor \u2026\n\n6. Vol. vii. p. 276.--This sonnet first appeared in the _New Monthly\nMagazine_, part ii. p. 26, under the title, _To B. R. Haydon. Composed\non seeing his Picture of Napoleon, musing at St. Helena_; and it is\ndated \"Saturday, June 11th, 1831.\"\n\n7. Vol. vii. p. 336.--This poem was published in the _Saturday\nMagazine_, May 18, 1844, in which the fifth line is--\n\n    Woe to the purblind men who fill.\n\n8. It may be worth mentioning (1) that the quotation (not noted,\nunfortunately, where it occurs)--\n\n    Some natural tears she drops, but wipes them soon,\n\nis from _Paradise Lost_, book xii. l. 645. See also _An Elegy delivered\nat the Hot Wells_, Bristol, July 1789. (2) That the phrase \"numerous\nverse\" is from _Paradise Lost_, book v. l. 150; and (3) that \"lenient\nhand of Time\" is from Bowles' sonnet--\n\n    O Time, who know'st a lenient hand to lay\n    Softest on sorrow's wound.\n\nAmongst those which I have failed to trace are the following:\n\n    _Ecclesiastical Sonnets_, II. xxxiv.--\n\n                \u2026 murtherer's chain partake,\n    Corded, and burning at the social stake.\n\n    xlv.--\n\n      \u2026 in the painful art of dying\n\n    _The Russian Fugitive_, Part II. l. 51--\n\n      \u2026 if house it be or bower.\n\n    _Elegiac Musings_, l. 41--\n\n    Let praise be mute where I am laid.\n\n    _Stanzas suggested in a Steamboat off Saint Bees' Heads_, l. 37--\n\n    Cruel of heart were they, bloody of hand.\n\n\n\n\nINDEX TO THE POEMS\n\n\n                                                            VOL.    PAGE\n\n  Aar, The Fall of the                                           vi  308\n\n  Abbeys, Old                                                   vii  100\n\n  Address from the Spirit of Cockermouth Castle                 vii  347\n\n  Address to a Child                                             iv   50\n\n  Address to Kilchurn Castle                                     ii  400\n\n  Address to my Infant Daughter, Dora                           iii   14\n\n  Address to the Scholars of the Village School of ----          ii   84\n\n  Admonition                                                     iv   34\n\n  \u00c6neid, Translation of Part of the First Book of the          viii  276\n\n  \"Aerial Rock--whose solitary brow\"                             vi  187\n\n  Affliction of Margaret--, The                                 iii    7\n\n  Afflictions of England                                        vii   72\n\n  After-Thought (Duddon)                                         vi  263\n\n  After-Thought (Tour on the Continent)                          vi  315\n\n  Airey-Force Valley                                           viii  146\n\n  Aix-la-Chapelle                                                vi  295\n\n  \"Alas! what boots the long laborious quest\"                    iv  216\n\n  Alban Hills, From the                                        viii   65\n\n  Albano, At                                                   viii   64\n\n  Alfred                                                        vii   24\n\n  Alfred, His Descendants                                       vii   25\n\n  Alice Fell; or, Poverty                                        ii  272\n\n  Aloys Reding                                                   vi  310\n\n  Ambleside                                                    viii  156\n\n  America, Aspects of Christianity in (Three Sonnets)           vii   84\n\n  American Episcopacy                                           vii   85\n\n  American Tradition                                             vi  246\n\n  Ancient History, On a celebrated Event in (Two Sonnets)        iv  242\n\n  Andrew Jones                                                 viii  221\n\n  Anecdote for Fathers                                            i  234\n\n  Animal Tranquillity and Decay                                   i  307\n\n  Anticipation (October 1803)                                    ii  436\n\n  Anticipation of leaving School, Composed in                     i    1\n\n  Apennines, Among the Ruins of a Convent in the               viii   82\n\n  Apology (Ecclesiastical Sonnets, 1st part)                    vii   18\n\n  Apology (Ecclesiastical Sonnets, 2nd part)                    vii   55\n\n  Apology (Sonnets upon the Punishment of Death)               viii  112\n\n  Apology (Yarrow Revisited)                                    vii  309\n\n  Applethwaite, At                                              iii   23\n\n  Aquapendente, Musings near                                   viii   42\n\n  Armenian Lady's Love, The                                     vii  232\n\n  Artegal and Elidure                                            vi   45\n\n  Authors, A plea for,                                         viii   99\n\n  Author's Portrait, To the                                     vii  318\n\n  Autumn (September)                                             vi   64\n\n  Autumn (Two Poems)                                             vi  201\n\n  Avarice, The last Stage of                                     ii   60\n\n  Avon, The (Annan)                                             vii  303\n\n  Bala-Sala, At                                                 vii  365\n\n  Balbi                                                          iv  237\n\n  Ballot, Protest against the                                  viii  304\n\n  Bangor, Monastery of Old                                      vii   13\n\n  Baptism                                                       vii   89\n\n  Barbara                                                        ii  178\n\n  Beaumont, Sir George, Epistle to                               iv  256\n\n  Beaumont, Sir George, Upon perusing the foregoing Epistle to   iv  267\n\n  Beaumont, Sir George, Picture of Peele Castle, painted by     iii   54\n\n  Beaumont, Sir George, Beautiful Picture, painted by            iv  271\n\n  Beaumont, Sir George, Elegiac Stanzas addressed to            vii  132\n\n  Beaumont, To Lady                                              iv   57\n\n  Beggar, The Old Cumberland                                      i  299\n\n  Beggars (Two Poems)                                            ii  276\n\n  \"'Beloved Vale!' I said, 'when I shall con'\"                   iv   35\n\n  Benefits, Other (Two Sonnets)                                 vii   40\n\n  Bible, Translation of the                                     vii   58\n\n  Binnorie, The Solitude of                                      ii  204\n\n  Bird of Paradise,  Drawing of the                    viii   29\n\n  Bird of Paradise, Suggested by a Picture of                  viii  140\n\n  Biscayan Rite (Two Sonnets)                                    iv  241\n\n  Bishops, Acquittal of the                                     vii   79\n\n  Bishops and Priests                                           vii   86\n\n  Black Comb, Inscription on a Stone on the side of              iv  281\n\n  Black Comb, View from the top of                               iv  279\n\n  \"Blest Statesman He, whose Mind's unselfish will\"            viii  101\n\n  Bologna, At (Three Sonnets)                                  viii   85\n\n  Bolton Priory, The Founding of                                 iv  204\n\n  Books and Newspapers, Illustrated                            viii  184\n\n  Borderers, The                                                  i  112\n\n  Bothwell Castle                                               vii  299\n\n  Boulogne, On being stranded near the Harbour of                vi  378\n\n  Bran, Effusion on the Banks of the                             vi   28\n\n  Breadalbane, Ruined Mansion of the Earl of                    vii  295\n\n  Brientz, Scene on the Lake of                                  vi  315\n\n  Brigham, Nun's Well                                           vii  347\n\n  Britons, Struggle of the                                      vii   11\n\n  Brothers, The                                                  ii  184\n\n  Brothers Water, Bridge at the foot of                          ii  293\n\n  Brougham Castle, Song at the Feast of                          iv   82\n\n  Brownie's Cell                                                 vi   16\n\n  Brownie, The                                                  vii  297\n\n  Brug\u00e8s (Two Poems)                                             vi  288\n\n  Brug\u00e8s, Incident at                                           vii  198\n\n  Buonapart\u00e9                                                     ii  323\n\n  Buonapart\u00e9                                                     ii  331\n\n  Buonapart\u00e9                                                     iv  228\n\n  Burial in the South of Scotland, A Place of                   vii  285\n\n  Burns, At the Grave of                                         ii  379\n\n  Burns, Thoughts suggested near the Residence of                ii  383\n\n  Burns, To the Sons of                                          ii  386\n\n  Butterfly, To a                                                ii  383\n\n  Butterfly, To a                                                ii  297\n\n  Calais, August 1802                                            ii  331\n\n  Calais, August 15, 1802                                        ii  334\n\n  Calais, Composed by the Seaside, near                          ii  330\n\n  Calais, Composed near                                          ii  332\n\n  Calais, Composed on the Beach, near                            ii  335\n\n  Calais, Fish-women at                                          vi  286\n\n  Calvert, Raisley                                               iv   44\n\n  Camaldoli, At the Convent of (Three Sonnets)                 viii   72\n\n  Canute                                                        vii   27\n\n  Canute and Alfred                                              vi  130\n\n  Castle, Composed at ----                                       ii  410\n\n  \"Castle of Indolence,\" Written in my Pocket Copy of\n      Thomson's                                                  ii  305\n\n  Casual Incitement                                             vii   14\n\n  Catechising                                                   vii   91\n\n  Cathedrals, etc.                                              vii  105\n\n  Catholic Cantons, Composed in one of the (Two Poems)           vi  312\n\n  Celandine, The Small                                          iii   21\n\n  Celandine, To the Small (Two Poems)                            ii  300\n\n  Cenotaph (Mrs. Fermor)                                        vii  135\n\n  Chamouny, Processions in the Vale of                           vi  363\n\n  Character, A                                                   ii  208\n\n  Charles the First, Troubles of                                vii   71\n\n  Charles the Second                                            vii   75\n\n  Chatsworth                                                    vii  272\n\n  Chaucer, Selections from (Three Poems)                         ii  238\n\n  Chiabrera, Epitaphs translated from                            iv  229\n\n  Chichely, Archbishop, to Henry V.                             vii   47\n\n  Child, Address to a                                            iv   50\n\n  Child, Characteristics of a, three years old                   iv  252\n\n  Child, To a (Written in her Album)                            viii   7\n\n  Childless Father, The                                          ii  181\n\n  Christianity in America, Aspects of (Three Sonnets)           vii   84\n\n  Churches, New                                                 vii  102\n\n  Church to be erected (Two Sonnets)                            vii  103\n\n  Churchyard, New                                               vii  104\n\n  Cintra, Convention of (Two Sonnets)                            iv  210\n\n  Cistertian Monastery                                          vii   37\n\n  Clarkson, Thomas, To                                           iv   62\n\n  Clergy, Corruptions of the Higher                             vii   49\n\n  Clergy, Emigrant French                                       vii  101\n\n  Clerical Integrity                                            vii   78\n\n  Clermont, The Council of                                      vii   30\n\n  Clifford, Lord                                                 iv   82\n\n  Clouds, To the                                               viii  142\n\n  Clyde, In the Frith of, Ailsa Crag                            vii  369\n\n  Clyde, On the Frith of                                        vii  370\n\n  Cockermouth Castle, Address from the Spirit of                vii  347\n\n  Cockermouth, In sight of                                      vii  346\n\n  Coleorton, Elegiac Musings in the grounds of                  vii  269\n\n  Coleorton, A Flower Garden at                                 vii  125\n\n  Coleorton, Inscription for an Urn in the grounds of            iv   78\n\n  Coleorton, Inscription for a Seat in the groves of             iv   80\n\n  Coleorton, Inscription in a garden of                          iv   76\n\n  Coleorton, Inscription in the grounds of                       iv   74\n\n  Coleridge, Hartley, To                                         ii  351\n\n  Collins, Remembrance of                                         i   33\n\n  Cologne, In the Cathedral at                                   vi  297\n\n  Commination Service                                           vii   96\n\n  Complaint, A                                                   iv   17\n\n  \"Complete Angler,\" Written on a blank leaf in the              vi  190\n\n  Conclusion (Duddon)                                            vi  262\n\n  Conclusion (Ecclesiastical Sonnets)                           vii  108\n\n  Conclusion (Miscellaneous Sonnets)                            vii  177\n\n  Conclusion (Prelude)                                          iii  367\n\n  Conclusion (Sonnets upon the Punishment of Death)            viii  111\n\n  Confirmation (Two Sonnets)                                    vii   92\n\n  Congratulation                                                vii  102\n\n  Conjectures                                                   vii    5\n\n  Contrast, The. The Parrot and the Wren                        vii  141\n\n  Convent in the Apennines                                     viii   82\n\n  Convention of Cintra, Composed while writing a Tract\n      occasioned by the (Two Sonnets)                            iv  210\n\n  Conversion                                                    vii   17\n\n  Convict, The                                                 viii  217\n\n  Cora Linn, Composed at                                         vi   26\n\n  Cordelia M----, To                                            vii  400\n\n  Cottage Girls, The Three                                       vi  351\n\n  Cottager to her Infant, The                                   iii   74\n\n  Council of Clermont, The                                      vii   30\n\n  Countess' Pillar                                              vii  307\n\n  Covenanters, Persecution of the Scottish                      vii   79\n\n  Cranmer                                                       vii   62\n\n  Crosthwaite Church                                           viii  157\n\n  Crusaders                                                     vii   41\n\n  Crusades                                                      vii   31\n\n  Cuckoo and the Nightingale, The                                ii  250\n\n  Cuckoo at Laverna, The                                       viii   67\n\n  Cuckoo Clock, The                                            viii  151\n\n  Cuckoo, To the                                                 ii  289\n\n  Cuckoo, To the                                                vii  169\n\n  Cumberland Beggar, The Old                                      i  299\n\n  Cumberland Beggar, The Old, MS. Variants                     viii  220\n\n  Cumberland, Coast of (In the Channel)                         vii  358\n\n  Cumberland, On a high part of the coast of                    vii  337\n\n  Daffodils, The                                                iii    4\n\n  Daisy, To the (Two Poems)                                      ii  353\n\n  Daisy, To the                                                  ii  360\n\n  Daisy, To the                                                 iii   51\n\n  Daniel, Picture of (Hamilton Palace)                          vii  303\n\n  Danish Boy, The                                                ii   96\n\n  Danish Conquests                                              vii   27\n\n  Danube, The Source of the                                      vi  303\n\n  Dati, Roberto                                                  iv  234\n\n  Dedication (Miscellaneous Sonnets)                            vii  159\n\n  Dedication (Tour on the Continent)                             vi  285\n\n  Dedication (White Doe of Rylstone)                             iv  102\n\n  Dedication (White Doe of Rylstone)                             vi   42\n\n  Departure from the Vale of Grasmere                            ii  377\n\n  \"Deplorable his lot who tills the ground\"                     vii   38\n\n  Derwent, To the River                                          vi  193\n\n  Derwent, To the River                                         vii  345\n\n  Descriptive Sketches                                            i   35\n\n  Descriptive Sketches                                            i  309\n\n  Desultory Stanzas                                              vi  382\n\n  Detraction which followed the Publication of a certain\n      Poem, On the                                               vi  212\n\n  Devil's Bridge, To the Torrent at the                         vii  129\n\n  Devotional Incitements                                        vii  314\n\n  Dion                                                           vi  116\n\n  Dissensions                                                   vii   10\n\n  Distractions                                                  vii   68\n\n  Dog, Incident characteristic of a favourite                   iii   48\n\n  Dog, Tribute to the Memory of the same                        iii   49\n\n  Donnerdale, The Plain of                                       vi  251\n\n  Dora, To (A little onward)                                     vi  132\n\n  Dora, To my Niece                                            viii  297\n\n  Douglas Bay, Isle of Man, On entering                         vii  360\n\n  Dover, Composed in the Valley near                             ii  341\n\n  Dover, Near                                                    ii  343\n\n  Dover, The Valley of (Two Sonnets)                             vi  380\n\n  Druidical Excommunication                                     vii    7\n\n  Druids, Trepidation of the                                    vii    6\n\n  Duddon, The River                                              vi  225\n\n  Dungeon-Ghyll Force                                            ii  138\n\n  Dunollie Castle (Eagles)                                      vii  292\n\n  Dunolly Castle, On Revisiting                                 vii  371\n\n  Dunolly Eagle, The                                            vii  372\n\n  Duty, Ode to                                                  iii   37\n\n  Dyer, To the Poet John                                         iv  273\n\n  Eagle and the Dove, The                                      viii  309\n\n  Eagles (Dunollie Castle)                                      vii  292\n\n  Eagle, The Dunolly                                            vii  372\n\n  Easter Sunday, Composed on                                     vi  194\n\n  Ecclesiastical Sonnets                                        vii    2\n\n  Echo, The Mountain                                             iv   25\n\n  Echo upon the Gemmi                                            vi  360\n\n  Eclipse of the Sun, The                                        vi  345\n\n  Eden, The River (Cumberland)                                  vii  385\n\n  Edward VI.                                                    vii   59\n\n  Edward VI. signing the Warrant                                vii   60\n\n  Egremont Castle, The Horn of                                   iv   12\n\n  Egyptian Maid, The                                            vii  252\n\n  Ejaculation                                                   vii  107\n\n  Elegiac Musings (Coleorton Hall)                              vii  269\n\n  Elegiac Stanzas (Goddard)                                      vi  371\n\n  Elegiac Stanzas (Mrs. Fermor)                                 vii  132\n\n  Elegiac Stanzas (Peele Castle)                                iii   54\n\n  Elegiac Verses (John Wordsworth)                              iii   58\n\n  Elizabeth                                                     vii   65\n\n  Ellen Irwin                                                    ii  124\n\n  Emigrant French Clergy                                        vii  101\n\n  Emigrant Mother, The                                           ii  284\n\n  Eminent Reformers (Two Sonnets)                               vii   66\n\n  Emma's Dell                                                    ii  153\n\n  Engelberg                                                      vi  316\n\n  Enghien, Duke d'                                               vi  114\n\n  \"England! the time is come when thou should'st wean\"           ii  432\n\n  England, Afflictions of                                       vii   72\n\n  Enterprise, To                                                 vi  218\n\n  Episcopacy, American                                          vii   85\n\n  Epistle to Sir George Beaumont                                 iv  256\n\n  Epistle to Sir George Beaumont, Upon perusing the foregoing    iv  267\n\n  Epitaph, A Poet's                                              ii   75\n\n  Epitaph in the Chapel-yard of Langdale                       viii  120\n\n  Epitaphs translated from Chiabrera                             iv  229\n\n  \"Ere with cold beads of midnight dew\"                         vii  145\n\n  \"Even as a dragon's eye that feels the stress\"                 vi   69\n\n  Evening of extraordinary splendour, Composed upon an           vi  176\n\n  Evening Star over Grasmere Water, To the                     viii  263\n\n  Evening Walk, An                                                i    4\n\n  Event in Ancient History, On a celebrated (Two Sonnets)        iv  242\n\n  Excursion, The                                                  v    1\n\n  Expostulation and Reply                                         i  272\n\n  Fact, A, and an Imagination                                    vi  130\n\n  Faery Chasm, The                                               vi  241\n\n  Fancy                                                          iv   36\n\n  Fancy and Tradition                                           vii  306\n\n  Fancy, Hints for the                                           vi  242\n\n  Farewell, A                                                    ii  324\n\n  Farewell Lines                                                vii  155\n\n  Farewell (Tour, 1833)                                         vii  341\n\n  Farmer of Tilsbury Vale, The                                   ii  147\n\n  Far-Terrace, The                                              vii  154\n\n  Father, The Childless                                          ii  181\n\n  Fathers, Anecdote for                                           i  234\n\n  Fermor, Mrs. (Cenotaph)                                       vii  135\n\n  Fermor, Mrs. (Elegiac Stanzas)                                vii  132\n\n  Fidelity                                                      iii   44\n\n  Filial Piety                                                  vii  231\n\n  Fir Grove (John Wordsworth)                                   iii   66\n\n  Fishes in a Vase, Gold and Silver                             vii  214\n\n  Fish-women                                                     vi  286\n\n  Flamininus, T. Quintius (Two Sonnets)                          iv  242\n\n  Fleming, To the Lady (Rydal Chapel), (Two Poems)              vii  109\n\n  Floating Island (D. W.)                                      viii  125\n\n  Florence (Four Sonnets)                                      viii   78\n\n  Flower Garden, A (Coleorton)                                  vii  125\n\n  Flowers                                                        vi  235\n\n  Flowers (Cave of Staffa)                                      vii  378\n\n  Flowers in the Island of Madeira                             viii  177\n\n  \"Fly, some kind Harbinger, to Grasmere-dale!\"                  ii  419\n\n  Foresight, or Children gathering Flowers                       ii  298\n\n  Forms of Prayer at Sea                                        vii   97\n\n  Forsaken Indian Woman, Complaint of a                           i  275\n\n  Forsaken, The                                                 iii   10\n\n  Fort Fuentes                                                   vi  328\n\n  Fountain, The                                                  ii   91\n\n  Fox, Mr., Lines composed on the expected death of              iv   47\n\n  France, Sky-prospect from the Plain of                         vi  377\n\n  Francesco Pozzobonnelli                                        iv  236\n\n  French Army in Russia (Two Poems)                              vi  107\n\n  French Clergy, Emigrant                                       vii  101\n\n  French Revolution                                              ii   34\n\n  French Revolution, In allusion to Histories of the\n      (Three Sonnets)                                          viii  130\n\n  French Royalist, Feelings of a                                 vi  114\n\n  Friend, To a (Banks of the Derwent)                           vii  348\n\n  Funeral Service                                                vi   97\n\n  Furness Abbey, At                                            viii  168\n\n  Furness Abbey, At                                            viii  176\n\n  Gemmi, Echo upon the                                           vi  360\n\n  General Fast, Upon the late (1832)                            vii  323\n\n  George the Third (November, 1813)                              iv  282\n\n  George the Third, On the death of                              vi  209\n\n  Germans on the Heights of Hockheim, The                        vi  216\n\n  Germany, Written in                                            ii   73\n\n  Gillies, Margaret, To (Two Poems)                            viii  114\n\n  Gillies, Margaret                                            viii  306\n\n  Gillies, Robert Pearce                                         vi   33\n\n  Gipsies                                                        iv   65\n\n  Glad Tidings                                                  vii   15\n\n  Gleaner, The                                                  vii  202\n\n  Glen-Almain, or, The Narrow Glen                               ii  393\n\n  Glencroe, At the Head of                                      vii  295\n\n  Glowworm, The                                                viii  231\n\n  Goddard, Elegiac Stanzas                                       vi  371\n\n  Gold and Silver Fishes in a Vase (Two Poems)                  vii  214\n\n  Goody Blake and Harry Gill                                      i  253\n\n  Gordale                                                        vi  185\n\n  Grace Darling                                                viii  310\n\n  Grasmere, Departure from the Vale of (August 1803)             ii  377\n\n  Grasmere, Home at                                            viii  235\n\n  Grasmere, Inscription on the Island at                         ii  213\n\n  Grasmere, Return to                                            ii  419\n\n  Grasmere Lake, Composed by the side of                         iv   73\n\n  Grave-stone, A (Worcester Cathedral)                          vii  201\n\n  \"Great men have been among us; hands that penned\"              ii  346\n\n  Green, George and Sarah                                      viii  266\n\n  Green Linnet, The                                              ii  367\n\n  Greenock                                                      vii  383\n\n  Greta, To the River                                           vii  344\n\n  \"Grief, thou hast lost an ever ready friend\"                   vi  195\n\n  Grotto, Written in a                                         viii  234\n\n  Guernica, Oak of                                               iv  245\n\n  Guilt and Sorrow                                                i   77\n\n  Gunpowder Plot                                                vii   69\n\n  Gustavus IV                                                    iv  227\n\n  Gwerndwffnant, Holiday at                                    viii  284\n\n  H. C., Six years old, To                                       ii  351\n\n  Hambleton Hills, After a journey across the                    ii  349\n\n  Happy Warrior, Character of the                                iv    7\n\n  Hart-Leap Well                                                 ii  128\n\n  Hart's-Horn Tree                                              vii  305\n\n  Haunted Tree, The                                              vi  199\n\n  Hawkshead, Written as a School Exercise at                   viii  211\n\n  Hawkshead School, In anticipation of leaving                    i    1\n\n  Hawkshead School, Address to the Scholars of                   ii   84\n\n  Haydon, To B. R.                                               vi   61\n\n  Haydon, To B. R. (Picture of Napoleon Buonaparte)             vii  276\n\n  Heidelberg, Castle of (Hymn for Boatmen)                       vi  301\n\n  Helvellyn, To ----, on her first ascent of                     vi  135\n\n  Henry Eighth, Portrait of                                     vii  166\n\n  Her eyes are wild                                               i  258\n\n  Hermitage (St. Herbert's Island)                               ii  210\n\n  Hermitage, Near the Spring of the                              vi  175\n\n  Hermit's Cell, Inscriptions in and near                        vi  170\n\n  Highland Boy, The Blind                                        ii  420\n\n  Highland Broach, The                                          vii  310\n\n  Highland Girl, To a                                            ii  389\n\n  Highland Hut                                                  vii  296\n\n  Hint from the Mountains                                        vi  156\n\n  Hints for the Fancy                                            vi  242\n\n  Historian, Plea for the                                      viii   61\n\n  Hoffer                                                         iv  213\n\n  Hogg, James, Extempore Effusion upon the death of            viii   24\n\n  Holiday at Gwerndwffnant                                     viii  284\n\n  Home at Grasmere                                             viii  235\n\n  Horn of Egremont Castle, The                                   iv   12\n\n  Howard, Mrs., Monument of (Wetheral), (Two Sonnets)           vii  386\n\n  Humanity                                                      vii  222\n\n  Hutchinson, Sarah, To                                         vii  162\n\n  Hymn for Boatmen (Heidelberg)                                  vi  301\n\n  Hymn, The Labourer's Noon-day                                 vii  408\n\n  I.F., To                                                     viii  307\n\n  Idiot Boy, The                                                  i  283\n\n  Illustrated Books and Newspapers                             viii  184\n\n  Illustration (The Jung-Frau)                                  vii   70\n\n  Imagination                                                    vi   67\n\n  Immortality, Ode, Intimations of                             viii  189\n\n  Indian Woman, Complaint of a Forsaken                           i  275\n\n  Infant Daughter, Address to my                                iii   14\n\n  Infant M---- M----, To the                                    vii  170\n\n  Infant, The Cottager to her                                   iii   74\n\n  Influence Abused                                              vii   26\n\n  Influence of Natural Objects                                   ii   66\n\n  Influences, Other                                             vii   19\n\n  Inglewood Forest, Suggested by a View in                      vii  304\n\n  Inscription for a Monument in Crosthwaite Church (Southey)   viii  157\n\n  Inscription for a Stone (Rydal Mount)                         vii  269\n\n  Inscriptions (Coleorton)                                       iv   74\n\n  Inscriptions (Hermit's Cell)                                   vi  170\n\n  Installation Ode                                             viii  320\n\n  Interdict, An                                                 vii   32\n\n  Introduction (Ecclesiastical Sonnets)                         vii    4\n\n  Introduction (Prelude)                                        iii  132\n\n  Invasion, Lines on the expected                                ii  437\n\n  Inversneyde                                                    ii  389\n\n  Invocation to the Earth                                        vi   95\n\n  Iona (Two Sonnets)                                            vii  379\n\n  Iona, The Black Stones of                                     vii  381\n\n  Isle of Man (Two Sonnets)                                     vii  362\n\n  Isle of Man, At Bala-Sala                                     vii  365\n\n  Isle of Man, At Sea off the                                   vii  359\n\n  Isle of Man, By the Sea-shore                                 vii  361\n\n  Isle of Man (Douglas Bay)                                     vii  360\n\n  Italian Itinerant, The                                         vi  338\n\n  Italy, After leaving (Two Sonnets)                           viii   84\n\n  \"It is no Spirit who from heaven hath flown\"                   ii  375\n\n  \"I watch, and long have watched, with calm regret\"             vi  197\n\n  Jedborough, The Matron of                                      ii  414\n\n  Jewish Family, A                                              vii  195\n\n  Joanna, To                                                     ii  157\n\n  Joanna H., Lines addressed to                                viii  282\n\n  Joan of Kent, Warrant for Execution of                        vii   60\n\n  Jones, Rev. Robert                                             vi  257\n\n  Journey Renewed                                                vi  257\n\n  June, 1820                                                     vi  214\n\n  Jung-Frau, The, and the Fall of the Rhine                     vii   70\n\n  Kendal, Upon hearing of the death of the Vicar of              vi   40\n\n  Kendal and Windermere Railway, On the projected              viii  166\n\n  Kent, To the Men of (October, 1803)                            ii  434\n\n  Kilchurn Castle, Address to                                    ii  400\n\n  Killicranky, In the Pass of                                    ii  435\n\n  King's College Chapel, Cambridge, Inside of (Three Sonnets)   vii  106\n\n  Kirkstone, The Pass of                                         vi  158\n\n  Kirtle, The Braes of                                           ii  124\n\n  Kitten and Falling Leaves, The                                iii   16\n\n  Laborer's Noon-day Hymn, The                                  vii  408\n\n  Lady, To a, upon Drawings she had made of Flowers in\n      Madeira                                                  viii  177\n\n  Lady E. B., and the Hon. Miss P., To the                      vii  128\n\n  Lamb, Charles, Written after the death of                    viii   17\n\n  Lancaster Castle, Suggested by the view of                   viii  103\n\n  Langdale, Epitaph in the Chapel-yard of                      viii  120\n\n  Laodamia                                                       vi    1\n\n  Last of the Flock, The                                          i  279\n\n  Last Supper, by Leonardo da Vinci, The                         vi  343\n\n  Latimer and Ridley                                            vii   61\n\n  Latitudinarianism                                             vii   76\n\n  Laud                                                          vii   71\n\n  Leonardo da Vinci, The Last Supper                             vi  343\n\n  Lesbia                                                       viii   32\n\n  Liberty (Gold and Silver Fishes)                              vii  216\n\n  Liberty (Tyrolese Sonnets)                                     iv  214\n\n  Liberty, Obligations of Civil to Religious                    vii   81\n\n  Liege, Between Namur and                                       vi  293\n\n  Lines, composed a few miles above Tintern Abbey                ii   51\n\n  Lines composed on the expected death of Mr. Fox                iv   47\n\n  Lines, Farewell                                               vii  155\n\n  Lines left upon a Seat in a Yew-tree                            i  108\n\n  Lines on the expected Invasion, 1803                           ii  437\n\n  Lines suggested by a Portrait from the Pencil of F. Stone\n     (Two Poems)                                               viii    1\n\n  Lines written as a School Exercise at Hawkshead              viii  211\n\n  Lines written in Early Spring                                   i  268\n\n  Lines written in the Album of the Countess of Lonsdale       viii    8\n\n  Lines written upon a Stone, upon one of the Islands at Rydal   ii   63\n\n  Lines written upon hearing of the death of the late Vicar\n      of Kendal                                                  vi   40\n\n  Lines written while sailing in a Boat at Evening                i   32\n\n  Liturgy, The                                                  vii   88\n\n  Loch Etive, Composed in the Glen of                           vii  291\n\n  Lombardy, In                                                 viii   83\n\n  London, Written in (1802), (Two Sonnets)                       ii  344\n\n  Longest Day, The                                               vi  153\n\n  Long Meg and her Daughters                                    vii  390\n\n  Lonsdale, The Countess of (Album)                            viii    8\n\n  Lonsdale, To the Earl of                                        v   20\n\n  Lonsdale, To the Earl of                                      vii  392\n\n  Louisa                                                         ii  362\n\n  Love, The Birth of                                           viii  215\n\n  Love lies bleeding (Two Poems)                               viii  148\n\n  Loving and Liking                                             vii  320\n\n  Lowther                                                       vii  391\n\n  Lowther, To the Lady Mary                                      vi  211\n\n  Lucca Giordano                                               viii  183\n\n  Lucy Gray; or, Solitude                                        ii   99\n\n  Lucy (Three Poems)                                             ii   78\n\n  Lucy (Three years she grew)                                    ii   81\n\n  Lycoris, Ode to (Two Poems)                                    vi  145\n\n  M. H., To                                                      ii  167\n\n  Madeira, Flowers in the Island of                            viii  177\n\n  Malham Cove                                                    vi  184\n\n  Manse, On the sight of a (Scotland)                           vii  286\n\n  March, Written in                                              ii  293\n\n  Margaret ----, The Affliction of                              iii    7\n\n  Mariner, By a Retired                                         vii  364\n\n  \"Mark the concentred hazels that enclose\"                      vi   71\n\n  Marriage Ceremony                                             vii   94\n\n  Marriage of a Friend, Composed on the Eve of the               iv  276\n\n  Marshall, To Cordelia                                         vii  400\n\n  Mary Queen of Scots, Captivity of                              vi  191\n\n  Mary Queen of Scots, Lament of                                 vi  162\n\n  Mary Queen of Scots (Workington)                              vii  349\n\n  Maternal Grief                                                 iv  248\n\n  Matron of Jedborough, The                                      ii  414\n\n  Matthew                                                        ii   87\n\n  May Morning, Composed on (1838)                              viii   97\n\n  May Morning, Ode composed on                                  vii  146\n\n  May, To                                                       vii  148\n\n  Meditation                                                    vii  401\n\n  Memory                                                        vii  117\n\n  \"Men of the Western World!\"                                  viii  112\n\n  Mental Affliction                                            viii   36\n\n  Merry England                                                 vii  343\n\n  Michael                                                        ii  215\n\n  Michael Angelo, From the Italian of (Three Sonnets)           iii  380\n\n  Michael Angelo, Translation from                             viii  265\n\n  \"Milton! thou should'st be living at this hour\"                ii  346\n\n  Missions and Travels                                          vii   23\n\n  Monasteries, Dissolution of the (Three Sonnets)               vii   52\n\n  Monasteries, Saxon                                            vii   22\n\n  Monastery, Cistertian                                         vii   37\n\n  Monastery of Old Bangor                                       vii   13\n\n  Monastic Power, Abuse of                                      vii   50\n\n  Monastic Voluptuousness                                       vii   51\n\n  Monkhouse, Mary                                               vii  170\n\n  Monks and Schoolmen                                           vii   39\n\n  Monument of Mrs. Howard (Two Sonnets)                         vii  386\n\n  Monument (Long Meg and her Daughters)                         vii  390\n\n  Moon, The (The Shepherd, looking eastward)                     vi   68\n\n  Moon, The (With how sad steps, O Moon)                         iv   38\n\n  Moon (The Crescent-moon, the Star of Love)                   viii  127\n\n  Moon, The (Sea-side)                                         viii   13\n\n  Moon, The (Rydal)                                            viii   15\n\n  Moon, The (Who but is pleased to watch)                      viii  184\n\n  Moon, The (How beautiful the Queen of Night)                 viii  188\n\n  Moon, The (Once I could hail)                                 vii  152\n\n  Morning Exercise, A                                           vii  178\n\n  Mosgiel Farm (Burns)                                          vii  383\n\n  Mother, The Mad                                                 i  258\n\n  Mother's Return, The                                           iv   63\n\n  Mountains, Hint from the                                       vi  156\n\n  Mull, In the Sound of                                         vii  293\n\n  Music, Power of                                                iv   20\n\n  Mutability                                                    vii  100\n\n  Naming of Places, Poems on the                                 ii  153\n\n  Namur and Liege, Between                                       vi  293\n\n  Natural Objects, Influence of                                  ii   66\n\n  \"Near Anio's stream, I spied a gentle Dove\"                  viii   65\n\n  Needlecase in the form of a Harp, On seeing a                 vii  157\n\n  <DW64> Woman                                                    ii  342\n\n  Newspaper, Composed after reading a                           vii  290\n\n  Nightingale, The                                               vi  214\n\n  Nightingale, The Cuckoo and the                                ii  250\n\n  Night Piece, A                                                  i  227\n\n  Night-thought, A                                             viii   88\n\n  Nith, On the Banks of                                          ii  383\n\n  Norman Boy, The                                              viii  132\n\n  Norman Conquest, The                                          vii   28\n\n  North Wales, Composed among the Ruins of a Castle in          vii  131\n\n  Nortons, The Fate of the                                       iv  100\n\n  November, 1806                                                 iv   49\n\n  November, 1813                                                 iv  282\n\n  November 1 (1815)                                              vi   63\n\n  Nunnery                                                       vii  388\n\n  Nun's Well, Brigham                                           vii  347\n\n  Nutting                                                        ii   70\n\n  Oak and the Broom, The                                         ii  174\n\n  Oak of Guernica                                                iv  245\n\n  Octogenarian, To an                                          viii  185\n\n  Ode, Installation                                            viii  320\n\n  Ode, Vernal                                                    vi  138\n\n  Ode (Who rises on the Banks of Seine)                          vi  104\n\n  Ode (1814) (When the soft hand)                                vi   96\n\n  Ode (1815) (Imagination--ne'er before content)                 vi   88\n\n  Ode, The Morning of the Day of Thanksgiving                    vi   74\n\n  Ode to Duty                                                   iii   37\n\n  Ode to Lycoris (Two Poems)                                     vi  145\n\n  Ode composed on May Morning                                   vii  146\n\n  Ode, Intimations of Immortality                              viii  189\n\n  Oker Hill in Darley Dale, A Tradition of                      vii  230\n\n  \"O Nightingale! thou surely art\"                               iv   67\n\n  \"On Nature's invitation do I come\"                             ii  118\n\n  Open Prospect                                                  vi  243\n\n  Ossian, Written in a blank leaf of Macpherson's               vii  373\n\n  Our Lady of the Snow                                           vi  318\n\n  Oxford, May 30, 1820 (Two Sonnets)                             vi  213\n\n  Painter, To a (Two Sonnets)                                  viii  114\n\n  Palafox                                                        iv  222\n\n  Palafox                                                        iv  228\n\n  Palafox                                                        iv  240\n\n  Papal Abuses                                                  vii   33\n\n  Papal Dominion                                                vii   34\n\n  Papal Power                                                   vii   36\n\n  Papal Unity                                                   vii   42\n\n  Parrot and the Wren, The                                      vii  141\n\n  Parsonage in Oxfordshire, A                                    vi  217\n\n  Pastoral Character                                            vii   87\n\n  Patriotic Sympathies                                          vii   74\n\n  Paulinus                                                      vii   15\n\n  Peele Castle, Suggested by a Picture of                       iii   54\n\n  Pelion and Ossa                                                ii  238\n\n  Pennsylvanians, To the                                       viii  179\n\n  Persecution                                                   vii    8\n\n  Personal Talk                                                  iv   30\n\n  Persuasion                                                    vii   16\n\n  Peter Bell                                                     ii    1\n\n  Peter Bell, On the detraction which followed                   vi  212\n\n  Pet-Lamb, The                                                  ii  142\n\n  Philoctetes                                                   vii  167\n\n  Picture, Upon the sight of a beautiful                         iv  271\n\n  Piety, Decay of                                               vii  163\n\n  Piety, Filial                                                 vii  231\n\n  Pilgrim Fathers (Two Sonnets)                                 vii   84\n\n  Pilgrim's Dream, The                                           vi  167\n\n  Pillar of Trajan, The                                         vii  137\n\n  Places of Worship                                             vii   87\n\n  Plea for Authors, A                                          viii   99\n\n  Plea for the Historian                                       viii   61\n\n  Poet and the Caged Turtledove, The                            vii  265\n\n  Poet's Dream, The                                            viii  135\n\n  Poet's Epitaph, A                                              ii   75\n\n  Poet to his Grandchild, A                                    viii  305\n\n  Point at issue, The                                           vii   58\n\n  Point Rash Judgment                                            ii  163\n\n  Poor Robin                                                   viii  116\n\n  Poor Susan, The Reverie of                                      i  226\n\n  Popery, Revival of                                            vii   61\n\n  Portrait, Lines suggested by a (Two Poems)                   viii    1\n\n  Portrait of I.F., On a                                       viii  306\n\n  Portrait of the Duke of Wellington, On a                     viii  118\n\n  Portrait, To the Author's                                     vii  318\n\n  Postscript (John Dyer)                                         vi  264\n\n  Power of Music                                                 iv   20\n\n  Power of Sound, On the                                        vii  203\n\n  Prayer at Sea, Forms of                                       vii   97\n\n  Prayer, The Force of                                           iv  204\n\n  Prelude, Prefixed to \"Poems of Early and Late Years\"         viii  123\n\n  Prelude, The                                                  iii  121\n\n  Presentiments                                                 vii  266\n\n  Primrose of the Rock, The                                     vii  274\n\n  Prioress' Tale, The                                            ii  240\n\n  Processions (Chamouny)                                         vi  363\n\n  Prophecy, A. February, 1807                                    iv   59\n\n  Punishment of Death, Sonnets upon the                        viii  103\n\n  Queen, To the                                                viii  319\n\n  Quillinan, To Rothay                                          vii  171\n\n  Railway, On the projected Kendal and Windermere              viii  166\n\n  Railways, etc.                                                vii  389\n\n  Rainbow, The                                                   ii  291\n\n  Ranz des Vaches, On hearing the                                vi  326\n\n  Recovery                                                      vii    9\n\n  Redbreast chasing the Butterfly, The                           ii  295\n\n  Redbreast, The                                                vii  410\n\n  Redbreast, To a                                              viii   38\n\n  Reflections                                                   vii   57\n\n  Reformation, General view of the Troubles of the              vii   64\n\n  Reformers, Eminent (Two Sonnets)                              vii   66\n\n  Reformers in Exile, English                                   vii   64\n\n  Regrets                                                       vii   99\n\n  Regrets, Imaginative                                          vii   56\n\n  Repentance                                                    iii   11\n\n  Reproof                                                       vii   21\n\n  Resolution and Independence                                    ii  312\n\n  Rest and be thankful                                          vii  295\n\n  Resting-place, The (Two Sonnets)                               vi  254\n\n  Retirement                                                    vii  165\n\n  Return                                                         vi  248\n\n  Return, The Mother's                                           iv   63\n\n  Reverie of Poor Susan                                           i  226\n\n  Rhine, Author's Voyage down the                              viii  273\n\n  Rhine, Upon the Banks of the                                   vi  299\n\n  Richard I                                                     vii   31\n\n  Richmond Hill (Thomson)                                        vi  214\n\n  Ridley, Latimer and                                           vii   61\n\n  Robinson, To Henry Crabb (Tour in Italy, 1837)               viii   41\n\n  Rob Roy's Grave                                                ii  403\n\n  Rock, Inscribed upon a                                         vi  173\n\n  Rocks, Two heath-clad                                        viii  170\n\n  Rocky Stream, Composed on the Banks of a                       vi  208\n\n  Rocky Stream, On the Banks of a                              viii  188\n\n  Rogers, Samuel, To                                            vii  280\n\n  Roman Antiquities                                            viii   33\n\n  Roman Antiquities (Old Penrith)                               vii  308\n\n  Roman Refinements, Temptations from                           vii   10\n\n  Romance of the Water Lily                                     vii  252\n\n  Rome (Two Sonnets)                                           viii   62\n\n  Rome, At (Three Sonnets)                                     viii   59\n\n  Rome, The Pine of Monte Mario at                             viii   58\n\n  Roslin Chapel, Composed in                                    vii  287\n\n  Rotha Q----, To                                               vii  171\n\n  Ruins of a Castle in North Wales                              vii  131\n\n  Rural Architecture                                             ii  206\n\n  Rural Ceremony                                                vii   98\n\n  Rural Illusions                                               vii  319\n\n  Russian Fugitive, The                                         vii  239\n\n  Ruth                                                           ii  104\n\n  Rydal, At, on May Morning (1838)                             viii   94\n\n  Rydal Chapel                                                  vii  109\n\n  Rydal, Written upon a Stone at                                 ii   63\n\n  Rydal, In the woods of                                        vii  176\n\n  Rydal Mere, By the side of                                    vii  403\n\n  Rydal Mount, Inscription for a Stone in the Grounds of        vii  269\n\n  S. H., To                                                     vii  162\n\n  Sacheverel                                                    vii   82\n\n  Sacrament                                                     vii   93\n\n  Sailor's Mother, The                                           ii  270\n\n  Saint Bees' Head, In a Steam-boat off                         vii  351\n\n  Saint Catherine of Ledbury                                   viii   34\n\n  Saint Gothard (Ranz des Vaches on the Pass of)                 vi  326\n\n  Saint Herbert's Island, Derwent-water (Hermitage)              ii  210\n\n  Saints                                                        vii   54\n\n  Salinero, Ambrosio                                             iv  233\n\n  Salisbury Plain, Incidents upon                                 i   77\n\n  San Salvador, The Church of                                    vi  332\n\n  Saxon Clergy, Primitive                                       vii   19\n\n  Saxon Conquest                                                vii   12\n\n  Saxon Monasteries                                             vii   22\n\n  Saxons                                                        vii   29\n\n  \"Say, what is Honour?--'Tis the finest sense\"                  iv  225\n\n  Schill                                                         iv  226\n\n  Scholars of the Village School of ----, Address to the         ii   84\n\n  School, Composed in anticipation of leaving                     i    1\n\n  School Exercise at Hawkshead, Written As a                   viii  211\n\n  Schwytz                                                        vi  324\n\n  Scottish Covenanters, Persecution of the                      vii   79\n\n  Scott, Sir Walter, Departure of                               vii  284\n\n  Sea-shore, Composed by the                                    vii  340\n\n  Sea-side, Composed by the                                      ii  330\n\n  Sea-side, By the                                              vii  338\n\n  Seasons, Thoughts on the                                      vii  229\n\n  Seathwaite Chapel                                              vi  249\n\n  Seclusion (Two Sonnets)                                       vii   20\n\n  Sellon, To Miss                                              viii  325\n\n  September 1, 1802                                              ii  342\n\n  September, 1815                                                vi   64\n\n  September, 1819                                                vi  201\n\n  Seven Sisters, The                                             ii  204\n\n  Sexton, To a                                                   ii   95\n\n  Sheep-washing                                                  vi  253\n\n  Shepherd-Boys, The Idle                                        ii  138\n\n  \"She was a Phantom of delight\"                                iii    1\n\n  Simon Lee                                                       i  262\n\n  Simplon Pass, Column lying in the                              vi  356\n\n  Simplon Pass, Stanza's composed in the                         vi  357\n\n  Simplon Pass, The                                              ii   69\n\n  Sister, To my                                                   i  270\n\n  Skiddaw                                                        ii  238\n\n  Sky-lark, To a                                                iii   42\n\n  Sky-lark, To a                                                vii  143\n\n  Sky-prospect--From the Plain of France                         vi  377\n\n  Sleep, To (Three Sonnets)                                      iv   42\n\n  Snow-drop, To a                                                vi  191\n\n  Sobieski, John                                                 vi  110\n\n  Solitary Reaper, The                                           ii  397\n\n  Solitude (The Duddon)                                          vi  245\n\n  Somnambulist, The                                             vii  393\n\n  Song at the Feast of Brougham Castle                           iv   82\n\n  Song for the Spinning Wheel                                    iv  275\n\n  Song for the Wandering Jew                                     ii  182\n\n  Sonnet, The                                                   vii  163\n\n  Sonnet, June, 1820 (Fame tells of groves)                      vi  214\n\n  Sonnet, September 1, 1802 (We had a female Passenger)          ii  342\n\n  Sonnet, September, 1802 (Inland, within a hollow vale)         ii  343\n\n  Sonnet, September, 1815 (While not a leaf seems faded)         vi   64\n\n  Sonnet, October, 1803 (One might believe)                      ii  430\n\n  Sonnet, October, 1803 (These times strike monied worldlings)   ii  432\n\n  Sonnet, October, 1803 (When, looking on the present face\n      of things)                                                 ii  433\n\n  Sonnet, November, 1806 (Another year!)                         iv   49\n\n  Sonnet, November, 1813 (Now that all hearts are glad)          iv  282\n\n  Sonnet, November 1, 1815 (How clear, how keen)                 vi   63\n\n  Sonnet, November, 1836 (Even so for me a Vision)             viii   37\n\n  Sound of Mull, In the                                         vii  293\n\n  Sound, The Power of                                           vii  203\n\n  Southey, Edith May                                            vii  157\n\n  Southey, (Inscription for monument)                          viii  157\n\n  Spade of a Friend, To the                                      iv    2\n\n  Spaniards (Three Sonnets)                                      iv  246\n\n  Spanish Guerillas, The French and the                          iv  248\n\n  Spanish Guerillas                                              iv  253\n\n  Sparrow's Nest, The                                            ii  236\n\n  Spinning Wheel, Song for the                                   iv  275\n\n  Sponsors                                                      vii   90\n\n  Spring, Lines written in Early                                  i  268\n\n  Staffa, Cave of (Four Sonnets)                                vii  376\n\n  Star and the Glow-worm, The                                    vi  167\n\n  Star-gazers                                                    iv   22\n\n  Staubbach, On approaching the                                  vi  306\n\n  Steamboats, Viaducts, and Railways                            vii  389\n\n  Stepping-stones, The (Two Sonnets)                             vi  239\n\n  Stepping Westward                                              ii  396\n\n  Stone, F., Lines suggested by a Portrait from the Pencil\n      of (Two Poems)                                           viii    1\n\n  Storm, Composed during a                                       vi  187\n\n  Stray Pleasures                                                iv   18\n\n  Stream, Composed on the Banks of a Rocky                       vi  208\n\n  Stream, On the Banks of a Rocky                              viii  188\n\n  Stream, Tributary                                              vi  250\n\n  Streams (The Duddon)                                           vi  255\n\n  Streams, The unremitting voice of nightly                    viii  187\n\n  Swan, The                                                      vi  198\n\n  Sweden, The King of                                            ii  338\n\n  Sweden, The King of                                            iv  227\n\n  Switzerland, Subjugation of                                    iv   60\n\n  Tables Turned, The                                              i  274\n\n  Tell, Effusion in presence of the Tower of                     vi  321\n\n  Temptations from Roman Refinements                            vii   10\n\n  Thanksgiving after Childbirth                                 vii   95\n\n  Thanksgiving Ode                                               vi   74\n\n  \"The leaves that rustled on this oak-crowned hill\"            vii  406\n\n  \"There is a bondage worse, far worse, to bear\"                 ii  431\n\n  \"There is a little unpretending Rill\"                          iv   53\n\n  There was a Boy                                                ii   57\n\n  \"The Stars are mansions built by Nature's hand\"                vi  210\n\n  \"This Lawn, a carpet all alive\"                               vii  228\n\n  Thomson's \"Castle of Indolence,\" Stanzas written in            ii  305\n\n  Thorn, The                                                      i  239\n\n  Thrasymene, Near the Lake of (Two Sonnets)                   viii   66\n\n  Thrush, The (Two Sonnets)                                    viii   93\n\n  Thun, Memorial near the Lake of                                vi  310\n\n  Tillbrook, Rev. Samuel                                         vi   65\n\n  Tilsbury Vale, The Farmer of                                   ii  147\n\n  Tintern Abbey, Lines, composed a few miles above               ii   51\n\n  To ---- in her seventieth year                                vii  172\n\n  To ---- Upon the birth of her First-born Child                vii  328\n\n  To ---- (Mrs. Wordsworth), (Two Poems)                        vii  121\n\n  To ---- (Look at the fate of summer flowers)                  vii  124\n\n  To ---- (Miscellaneous Sonnets--Dedication)                   vii  159\n\n  To ---- (Miscellaneous Sonnets--Conclusion)                   vii  177\n\n  To ---- (Wait, prithee, wait!)                               viii   32\n\n  To ---- on her First Ascent of Helvellyn                       vi  135\n\n  To ---- (The Haunted Tree)                                     vi  199\n\n  Torrent at Devil's Bridge                                     vii  129\n\n  Tour among the Alps (1791-2), (Descriptive Sketches)            i   35\n\n  Tour among the Alps (1791-2), (Descriptive Sketches)            i  309\n\n  Tour in Italy (1837), Memorials of a                         viii   39\n\n  Tour in Scotland (1803), Memorials of a                        ii  377\n\n  Tour in Scotland (1814), Memorials of a                        vi   15\n\n  Tour in Scotland (1831)                                       vii  278\n\n  Tour in the Summer of 1833                                    vii  341\n\n  Tour on the Continent (1820), Memorials of a                   vi  285\n\n  Toussaint L'Ouverture, To                                      ii  339\n\n  Tradition                                                      vi  253\n\n  Tradition, American                                            vi  246\n\n  Tradition, Fancy and                                          vii  306\n\n  Tradition of Oker Hill                                        vii  230\n\n  Trajan, The Pillar of                                         vii  137\n\n  Translation of the Bible                                      vii   58\n\n  Transubstantiation                                            vii   44\n\n  Triad, The                                                    vii  181\n\n  Tributary Stream                                               vi  250\n\n  Troilus and Cresida                                            ii  264\n\n  Trosachs, The                                                 vii  288\n\n  Turtledove, The Poet and the Caged                            vii  265\n\n  Twilight                                                       vi   67\n\n  Two April Mornings, The                                        ii   89\n\n  Two Thieves, The                                               ii   60\n\n  Tyndrum, Suggested at                                         vii  294\n\n  Tynwald Hill                                                  vii  366\n\n  Tyrolese, Feelings of the                                      iv  215\n\n  Tyrolese, On the final submission of the                       iv  217\n\n  Tyrolese Sonnets                                               iv  213\n\n  Ulpha, Kirk of                                                 vi  260\n\n  Uncertainty                                                   vii    7\n\n  Utilitarians, To the                                         viii  299\n\n  Valedictory Sonnet (Miscellaneous Sonnets)                   viii  102\n\n  Vallombrosa, At                                              viii   75\n\n  Vaudois, The (Two Sonnets)                                    vii   44\n\n  Vaudracour and Julia                                          iii   24\n\n  Venetian Republic, On the Extinction of                        ii  336\n\n  Venice, Scene in                                              vii   34\n\n  Venus, To the Planet (January 1838)                          viii   92\n\n  Venus, To the Planet (Loch Lomond)                            vii  299\n\n  Vernal Ode                                                     vi  138\n\n  Vienna, Siege of, raised by John Sobieski                      vi  110\n\n  Virgin, The                                                   vii   54\n\n  Visitation of the Sick                                        vii   96\n\n  Waggoner, The                                                 iii   76\n\n  Waldenses                                                     vii   46\n\n  Wallace's Tower                                                vi   26\n\n  Walton, Isaac                                                  vi  190\n\n  Walton's Book of Lives                                        vii   77\n\n  Wandering Jew, Song for the                                    ii  182\n\n  Wansfell                                                     viii  153\n\n  Warning, The                                                  vii  330\n\n  Wars of York and Lancaster                                    vii   48\n\n  Waterfall and the Eglantine, The                               ii  170\n\n  Water-fowl                                                     iv  277\n\n  Waterloo, After visiting the Field of                          vi  292\n\n  Waterloo, Occasioned by the Battle of (Three Sonnets)          vi  111\n\n  We are Seven                                                    i  228\n\n  Wellington, On a Portrait of the Duke of                     viii  118\n\n  Westall, Mr. W., Views of the Caves, etc., in Yorkshire, by\n      (Three Poems)                                              vi  183\n\n  Westminster Bridge, Composed upon                              ii  328\n\n  Westmoreland Girl, The                                       viii  172\n\n  \"Whence that low voice?--A whisper from the heart\"             vi  252\n\n  \"Where lies the truth? has Man, in wisdom's creed\"           viii  182\n\n  \"While Anna's peers and early playmates tread\"                vii  169\n\n  Whirl-blast, The                                                i  238\n\n  Whistlers, The Seven                                           iv   68\n\n  White Doe of Rylstone                                          iv  100\n\n  \"Who fancied what a pretty sight?\"                             ii  374\n\n  \"Why, Minstrel, these untuneful murmurings\"                   vii  161\n\n  Wicliffe                                                      vii   49\n\n  Widow on Windermere Side, The                                viii   89\n\n  Wild Duck's Nest, The                                          vi  189\n\n  Wild-Fowl                                                    viii  234\n\n  William the Third                                             vii   80\n\n  Winter (French Army), (Two Poems)                              vi  107\n\n  Wishing-gate, The                                             vii  189\n\n  Wishing-gate Destroyed, The                                   vii  192\n\n  Worcester Cathedral, A Grave-Stone in                         vii  201\n\n  Wordsworth, Catherine                                          vi   72\n\n  Wordsworth, Dora                                               vi  132\n\n  Wordsworth, John, Elegiac Verses in memory of                 iii   58\n\n  Wordsworth, John (Fir Grove)                                  iii   66\n\n  Wordsworth, To the Rev. Christopher                          viii  162\n\n  Wordsworth, To the Rev. Dr. (Duddon)                           vi  227\n\n  Wordsworth, Thomas                                           viii   39\n\n  Wren's Nest, A                                                vii  325\n\n  Yarrow Unvisited                                               ii  411\n\n  Yarrow Visited                                                 vi   35\n\n  Yarrow Revisited                                              vii  278\n\n  Yew-trees                                                      ii  369\n\n  Yew-tree Seat                                                   i  108\n\n  York and Lancaster, Wars of                                   vii   48\n\n  Young England                                                viii  180\n\n  Young Lady, To a                                               ii  365\n\n  Youth, Written in very early                                    i    3\n\n  Zaragoza                                                       iv  224\n\n\n\n\nINDEX TO FIRST LINES\n\n\n  VOL.  PAGE\n\n  A barking sound the Shepherd hears,                           iii   44\n\n  A Book came forth of late, called PETER BELL;                  vi  212\n\n  A bright-haired company of youthful slaves,                   vii   14\n\n  Abruptly paused the strife;--the field throughout              vi  216\n\n  A dark plume fetch me from yon blasted yew,                    vi  248\n\n  Adieu, Rydalian Laurels! that have grown                      vii  342\n\n  Advance--come forth from thy Tyrolean ground,                  iv  214\n\n  Aerial Rock--whose solitary brow                               vi  188\n\n  A famous man is Robin Hood,                                    ii  403\n\n  Affections lose their object; Time brings forth,             viii  185\n\n  A flock of sheep that leisurely pass by,                       iv   43\n\n  A genial hearth, a hospitable board,                          vii   87\n\n  A German Haggis from receipt                                 viii  272\n\n  Age! twine thy brows with fresh spring flowers,                ii  414\n\n  Ah! if I were a lady gay                                     viii  262\n\n  Ah, think how one compelled for life to abide,               viii  110\n\n  A humming Bee--a little tinkling rill--                         v  106\n\n  Ah, when the Body, round which in love we clung,              vii   19\n\n  Ah! where is Palafox? Nor tongue nor pen                       iv  240\n\n  Ah why deceive ourselves! by no mere fit,                    viii   86\n\n  Aid, glorious Martyrs, from your fields of light,             vii   64\n\n  Alas! what boots the long laborious quest                      iv  216\n\n  \"_A little onward lend thy guiding hand_\"                      vi  133\n\n  All praise the Likeness by thy skill portrayed,              viii  114\n\n  Along the mazes of this song I go,                           viii  233\n\n  A love-lorn Maid, at some far-distant time,                    vi  253\n\n  Ambition--following down this far-famed <DW72>                  vi  356\n\n  Amid a fertile region green with wood                         vii  301\n\n  Amid the smoke of cities did you pass                          ii  157\n\n  Amid this dance of objects sadness steals                      vi  299\n\n  Among a grave fraternity of Monks,                           viii    6\n\n  Among all lovely things my Love had been,                    viii  232\n\n  Among the dwellers in the silent fields,                     viii  310\n\n  Among the dwellings framed by birds                           vii  325\n\n  Among the mountains were we nursed, loved Stream!              vi  193\n\n  Among the mountains were we nursed, loved Stream!             vii  345\n\n  A month, sweet Little-ones, is past                            iv   63\n\n  An age hath been when Earth was proud                          vi  146\n\n  A narrow girdle of rough stones and crags,                     ii  164\n\n  And has the Sun his flaming chariot driven,                  viii  211\n\n  And is it among rude untutored Dales,                          iv  222\n\n  And is this--Yarrow?--_This_ the Stream                        vi   36\n\n  And, not in vain embodied to the sight,                       vii   40\n\n  \"And shall,\" the Pontiff asks, \"profaneness flow\"             vii   30\n\n  And what is Penance with her knotted thong;                   vii   50\n\n  And what melodious sounds at times prevail!                   vii   40\n\n  An Orpheus! an Orpheus! yes, Faith may grow bold,              iv   20\n\n  Another year!--another deadly blow!                            iv   49\n\n  A pen--to register; a key--                                   vii  117\n\n  A Pilgrim, when the summer day                                 vi  167\n\n  A plague on your languages, German and Norse!                  ii   73\n\n  A pleasant music floats along the Mere,                       vii   27\n\n  _A Poet!_--He hath put his heart to school,                  viii  128\n\n  A point of life between my Parents' dust,                     vii  346\n\n  Arms and the Man I sing, the first who bore                  viii  281\n\n  Army of Clouds! ye wing\u00e8d Host in troops,                    viii  142\n\n  A Rock there is whose homely front                            vii  274\n\n  A Roman Master stands on Grecian ground,                       iv  242\n\n  Around a wild and woody hill                                   vi  310\n\n  Arran! a single-crested Teneriffe,                            vii  370\n\n  Art thou a Statist in the van                                  ii   75\n\n  Art thou the bird whom Man loves best,                         ii  295\n\n  As faith thus sanctified the warrior's crest                  vii   42\n\n  A simple Child,                                                 i  231\n\n  As indignation mastered grief, my tongue,                    viii   85\n\n  As leaves are to the tree whereon they grow,                 viii   87\n\n  A slumber did my spirit seal;                                  ii   83\n\n  As often as I murmur here                                     vii  265\n\n  As star that shines dependent upon star                       vii   87\n\n  \"As the cold aspect of a sunless way\"                          vi  191\n\n  A Stream, to mingle with your favourite Dee,                  vii  129\n\n  A sudden conflict rises from the swell                        vii   82\n\n  As, when a storm hath ceased, the birds regain                vii    9\n\n  As with the Stream our voyage we pursue,                      vii   33\n\n  At early dawn, or rather when the air                          vi  185\n\n  A Traveller on the skirt of Sarum's Plain                       i   79\n\n  A trouble, not of clouds, or weeping rain,                    vii  284\n\n  At the corner of Wood Street, when daylight appears,            i  226\n\n  A twofold harmony is here                                    viii  282\n\n  Avaunt all specious pliancy of mind                            iv  247\n\n  Avaunt this \u0153conomic rage!                                   viii  299\n\n  A voice, from long-expecting thousands sent                   vii   79\n\n  A volant Tribe of Bards on earth are found,                   vii  119\n\n  Avon--a precious, an immortal name!                           vii  303\n\n  A weight of awe, not easy to be borne,                        vii  390\n\n  A whirl-blast from behind the hill                              i  238\n\n  A wing\u00e8d Goddess--clothed in vesture wrought                   vi  292\n\n  A Youth too certain of his power to wade                      vii  362\n\n  Bard of the Fleece, whose skilful genius made                  iv  273\n\n  Beaumont! it was thy wish that I should rear                  iii   23\n\n  Before I see another day,                                       i  276\n\n  Before the world had past her time of youth,                 viii  107\n\n  \"Begone, thou fond presumptuous Elf,\"                          ii  170\n\n  Beguiled into forgetfulness of care,                         viii    2\n\n  Behold an emblem of our human mind,                          viii  188\n\n  Behold a pupil of the monkish gown,                           vii   24\n\n  Behold her, single in the field,                               ii  397\n\n  Behold, within the leafy shade,                                ii  237\n\n  \"Beloved Vale!\" I said, \"when I shall con\"                     iv   35\n\n  Beneath the concave of an April sky,                           vi  138\n\n  Beneath these fruit-tree boughs that shed                      ii  367\n\n  Beneath yon eastern ridge, the craggy bound,                   iv   80\n\n  Be this the chosen site; the virgin sod,                      vii  103\n\n  Between two sister moorland rills                              ii   96\n\n  Bishops and Priests, blessed are ye, if deep                  vii   86\n\n  Black Demons hovering o'er his mitred head,                   vii   34\n\n  Bleak season was it, turbulent and bleak,                      ii  121\n\n  Blest is this Isle--our native Land;                          vii  109\n\n  Blest Statesman He, whose Mind's unselfish will,             viii  101\n\n  Bold words affirmed, in days when faith was strong            vii  359\n\n  Brave Schill! by death delivered, take thy flight              iv  226\n\n  Bright Flower! whose home is everywhere,                       ii  360\n\n  Bright was the summer's noon when quickening steps            iii  186\n\n  Broken in fortune, but in mind entire                         vii  365\n\n  Brook and road                                                 ii   69\n\n  Brook, that hast been my solace days and weeks,              viii  265\n\n  Brook! whose society the Poet seeks,                           iv   52\n\n  Brug\u00e8s I saw attired with golden light                         vi  288\n\n  But Cytherea, studious to invent,                            viii  277\n\n  But here no cannon thunders to the gale;                       vi  262\n\n  But liberty, and triumphs on the Main,                        vii  102\n\n  But, to outweigh all harm, the sacred Book,                   vii   58\n\n  But, to remote Northumbria's royal Hall,                      vii   15\n\n  But what if One, through grove or flowery mead,               vii   21\n\n  But whence came they who for the Saviour Lord                 vii   44\n\n  By a blest Husband guided, Mary came,                        viii   35\n\n  By antique Fancy trimmed--though lowly, bred                   vi  324\n\n  By Art's bold privilege Warrior and War-Horse stand,         viii  118\n\n  By chain yet stronger must the Soul be tied:                  vii   93\n\n  By playful smiles, (alas, too oft,                           viii  120\n\n  By such examples moved to unbought pains,                     vii   22\n\n  By their floating mill,                                        iv   18\n\n  By vain affections unenthralled,                              vii  135\n\n  Call not the royal Swede unfortunate,                          iv  227\n\n  Calm as an under-current, strong to draw,                     vii   80\n\n  Calm is all nature as a resting wheel                           i    4\n\n  Calm is the fragrant air, and loth to lose                    vii  317\n\n  Calvert! it must not be unheard by them                        iv   44\n\n  \"Change me, some God, into that breathing rose!\"               vi  237\n\n  Chatsworth! thy stately mansion, and the pride                vii  273\n\n  Child of loud-throated War! the mountain Stream                ii  401\n\n  Child of the clouds! remote from every taint                   vi  231\n\n  Clarkson! it was an obstinate hill to climb:                   iv   62\n\n  Closing the sacred Book which long has fed                    vii   98\n\n  Clouds, lingering yet, extend in solid bars                    iv   73\n\n  Coldly we spake. The Saxons, overpowered                      vii   29\n\n  Come, gentle Sleep, Death's image tho' thou art,             viii  264\n\n  Come ye--who, if (which Heaven avert!) the Land                ii  437\n\n  Companion! by whose buoyant Spirit cheered,                  viii   41\n\n  Complacent Fictions were they, yet the same,                 viii   61\n\n  Confiding hopes of youthful hearts,                          viii  297\n\n  Critics, right honourable Bard, decree                       viii  272\n\n  Dark and more dark the shades of evening fell;                 ii  349\n\n  Darkness surrounds us: seeking, we are lost                   vii    7\n\n  Days passed--and Monte Calvo would not clear,                viii   64\n\n  Days undefiled by luxury or sloth,                           viii  179\n\n  Dear be the Church, that, watching o'er the needs             vii   89\n\n  Dear Child of Nature, let them rail!                           ii  366\n\n  Dear Fellow-travellers! think not that the Muse,               vi  285\n\n  Dear native regions, I foretell,                                i    2\n\n  Dear Reliques! from a pit of vilest mould                      vi  114\n\n  Dear to the Loves, and to the Graces vowed,                   vii  350\n\n  Deep is the lamentation! Not alone                            vii   56\n\n  Degenerate Douglas! oh, the unworthy Lord!                     ii  410\n\n  Deign, Sovereign Mistress, to accept a lay,                  viii  319\n\n  Departed Child! I could forget thee once                       iv  249\n\n  Departing summer hath assumed                                  vi  202\n\n  Deplorable his lot who tills the ground,                      vii   38\n\n  Desire we past illusions to recal?                           vvii  360\n\n  Desponding Father! mark this altered bough                   viii   31\n\n  Despond who will--_I_ heard a voice exclaim,                  vii  368\n\n  Destined to war from very infancy                              iv  234\n\n  Did pangs of grief for lenient time too keen,                 vii  363\n\n  Discourse was deemed Man's noblest attribute,                viii  184\n\n  Dishonoured Rock and Ruin! that, by law,                      vii  292\n\n  Dogmatic Teachers, of the snow-white fur!                      vi  208\n\n  Doomed as we are our native dust                               vi  312\n\n  Doubling and doubling with laborious walk,                    vii  295\n\n  Down a swift Stream, thus far, a bold design                  vii   83\n\n  Dread hour! when, upheaved by war's sulphurous blast,          vi  329\n\n  Driven in by Autumn's sharpening air                          vii  410\n\n  Earth has not any thing to show more fair:                     ii  328\n\n  Eden! till now thy beauty had I viewed                        vii  385\n\n  Emperors and Kings, how oft have temples rung                  vi  113\n\n  England! the time is come when thou should'st wean             ii  433\n\n  Enlightened Teacher, gladly from thy hand                    viii  162\n\n  Enough! for see, with dim association                         vii   44\n\n  Enough of climbing toil!--Ambition treads                      vi  149\n\n  Enough of garlands, of the Arcadian crook,                    vii  294\n\n  Enough of rose-bud lips, and eyes                             vii  239\n\n  Ere the Brothers through the gateway                           iv   12\n\n  Erewhile to celebrate this glorious morn                       vi  195\n\n  Ere with cold beads of midnight dew                           vii  145\n\n  Ere yet our course was graced with social trees                vi  235\n\n  Eternal Lord! eased of a cumbrous load,                      viii   81\n\n  Ethereal minstrel! pilgrim of the sky!                        vii  143\n\n  Even as a dragon's eye that feels the stress                   vi   69\n\n  Even as a river,--partly (it might seem)                      iii  293\n\n  Even so for me a Vision sanctified                           viii   37\n\n  Even such the contrast that, where'er we move,                vii   71\n\n  Even while I speak, the sacred roofs of France                vii  101\n\n  Excuse is needless when with love sincere                     vii  162\n\n  Failing impartial measure to dispense                        viii   99\n\n  Fair Ellen Irwin, when she sate                                ii  124\n\n  Fair is the Swan, whose majesty, prevailing                    vi  116\n\n  Fair Lady! can I sing of flowers                             viii  177\n\n  Fair Land! Thee all men greet with joy; bow few,             viii   84\n\n  Fair Prime of life! were it enough to gild                    vii  165\n\n  Fair Star of evening, Splendour of the west,                   ii  330\n\n  Fallen, and diffused into a shapeless heap,                    vi  256\n\n  Fame tells of groves--from England far away--                  vi  214\n\n  Fancy, who leads the pastimes of the glad,                    vii  178\n\n  \"Farewell, deep Valley, with thy one rude House,\"               v  196\n\n  Farewell, thou little Nook of mountain-ground,                 ii  324\n\n  Far from my dearest Friend, 'tis mine to rove                   i    6\n\n  Far from our home by Grasmere's quiet Lake,                    iv  259\n\n  Father! to God himself we cannot give                         vii   90\n\n  Fear hath a hundred eyes that all agree                       vii   69\n\n  Feel for the wrongs to universal ken                         viii  129\n\n  Festivals have I seen that were not names:                     ii  334\n\n  Fit retribution, by the moral code                           viii  108\n\n  Five years have past; five summers, with the length            ii   51\n\n  Flattered with promise of escape                              vii  229\n\n  Fly, some kind Harbinger, to Grasmere-dale!                    ii  419\n\n  Fond words have oft been spoken to thee, Sleep!                iv   43\n\n  For action born, existing to be tried,                       viii   67\n\n  Forbear to deem the Chronicler unwise,                       viii   61\n\n  For ever hallowed be this morning fair,                       vii   15\n\n  For gentlest uses, oft-times Nature takes                      vi  316\n\n  Forgive, illustrious Country! these deep sighs,              viii   65\n\n  Forth from a jutting ridge, around whose base                viii  170\n\n  For thirst of power that Heaven disowns,                     viii  320\n\n  Forth rushed from Envy sprung and Self-conceit,              viii  304\n\n  For what contend the wise?--for nothing less                  vii   58\n\n  Four fiery steeds impatient of the rein                      viii   32\n\n  From Bolton's old monastic tower                               iv  106\n\n  From early youth I ploughed the restless Main,                vii  364\n\n  From false assumption rose, and fondly hail'd                 vii   36\n\n  From Little down to Least, in due degree,                     vii   91\n\n  From low to high doth dissolution climb,                      vii  100\n\n  From Nature doth emotion come, and moods                      iii  355\n\n  From Rite and Ordinance abused they fled                      vii   85\n\n  From Stirling castle we had seen                               ii  411\n\n  From that time forth, Authority in France                     iii  330\n\n  From the Baptismal hour, thro' weal and woe,                  vii   97\n\n  From the dark chambers of dejection freed,                     vi   34\n\n  From the fierce aspect of this River, throwing                 vi  308\n\n  From the Pier's head, musing, and with increase                vi  381\n\n  From this deep chasm, where quivering sunbeams play            vi  245\n\n  Frowns are on every Muse's face,                              vii  157\n\n  Furl we the sails, and pass with tardy oars                   vii   41\n\n  Genius of Raphael! if thy wings                               vii  195\n\n  Giordano, verily thy Pencil's skill                          viii  183\n\n  Glad sight wherever new with old                             viii  154\n\n  Glide gently, thus for ever glide,                              i   33\n\n  Glory to God! and to the Power who came                       vii  107\n\n  Go back to antique ages, if thine eyes                        vii  174\n\n  Go, faithful Portrait! and where long hath knelt              vii  318\n\n  Grant, that by this unsparing hurricane                       vii   57\n\n  Grateful is Sleep, my life, in stone bound fast,             viii  264\n\n  Great men have been among us; hands that penned                ii  346\n\n  Greta, what fearful listening! when huge stones               vii  344\n\n  Grief, thou hast lost an ever-ready friend                     vi  196\n\n  Grieve for the Man who hither came bereft,                   viii   72\n\n  Had this effulgence disappeared                                vi  177\n\n  Hail, orient Conqueror of gloomy Night!                        vi   78\n\n  Hail to the crown by Freedom shaped--to gird                    v  235\n\n  Hail to the fields--with Dwellings sprinkled o'er              vi  243\n\n  Hail, Twilight, sovereign of one peaceful hour!                vi   67\n\n  Hail, Virgin Queen! o'er many an envious bar                  vii   65\n\n  Hail, Zaragoza! If with unwet eye                              iv  224\n\n  Happy the feeling from the bosom thrown                       vii  159\n\n  Hard task! exclaim the undisciplined, to lean                viii   86\n\n  Hark! 'tis the Thrush, undaunted, undeprest,                 viii   93\n\n  Harmonious Powers with Nature work                           viii  125\n\n  Harp! could'st thou venture, on thy boldest string            vii   72\n\n  Hast thou seen, with flash incessant,                          vi  174\n\n  Hast thou then survived--                                     iii   14\n\n  Haydon! let worthier judges praise the skill                  vii  277\n\n  Here closed the Tenant of that lonely vale                      v  145\n\n  _Here Man more purely lives, less oft doth fall_,             vii   37\n\n  Here, on our native soil, we breathe once more                 ii  341\n\n  Here on their knees men swore; the stones were black,         vii  381\n\n  Here pause: the poet claims at least this praise,              iv  255\n\n  Here stood an Oak, that long had borne affixed                vii  305\n\n  Here, where, of havoc tired and rash undoing,                viii  168\n\n  Her eyes are wild, her head is bare,                            i  258\n\n  Her only pilot the soft breeze, the boat                      vii  160\n\n  \"High bliss is only for a higher state,\"                      vii  156\n\n  High deeds, O Germans, are to come from you!                   iv   59\n\n  High in the breathless Hall the Minstrel sate,                 iv   83\n\n  High is our calling, Friend!--Creative Art                     vi   61\n\n  High on a broad unfertile tract of forest-skirted Down,      viii  133\n\n  High on her speculative tower                                  vi  345\n\n  His simple truths did Andrew glean                             ii  174\n\n  Holy and heavenly Spirits as they are,                        vii   67\n\n  Homeward we turn. Isle of Columba's Cell,                     vii  382\n\n  Hope rules a land for ever green:                             vii  190\n\n  Hope smiled when your nativity was cast,                      vii  378\n\n  Hopes, what are they?--Beads of morning                        vi  170\n\n  How art thou named? In search of what strange land,           vii  129\n\n  How beautiful the Queen of Night, on high                    viii  188\n\n  How beautiful, when up a lofty height                        viii   90\n\n  How beautiful your presence, how benign,                      vii   19\n\n  How blest the Maid whose heart--yet free                       vi  351\n\n  How clear, how keen, how marvellously bright                   vi   63\n\n  \"How disappeared he?\" Ask the newt and toad;                  vii  297\n\n  How fast the Marian death-list is unrolled!                   vii   61\n\n  How profitless the relics that we cull,                       vii  308\n\n  How richly glows the water's breast                             i   32\n\n  How rich that forehead's calm expanse!                        vii  123\n\n  How sad a welcome! To each voyager                            vii  380\n\n  How shall I paint thee?--Be this naked stone,                  vi  232\n\n  How soon--alas! did Man, created pure--                       vii   35\n\n  How sweet it is, when mother Fancy rocks                       iv   36\n\n  Humanity, delighting to behold                                 vi  107\n\n  Hunger, and sultry heat, and nipping blast                     iv  248\n\n  I am not One who much or oft delight                           iv   31\n\n  I come, ye little noisy Crew,                                  ii   84\n\n  I dropped my pen; and listened to the Wind                     iv  211\n\n  I find it written of Simonides,                              viii  258\n\n  If from the public way you turn your steps                     ii  215\n\n  If Life were slumber on a bed of down,                        vii  351\n\n  If money's slack,                                            viii  271\n\n  If Nature, for a favourite child,                              ii   88\n\n  If there be prophets on whose spirits rest                    vii    5\n\n  If these brief Records, by the Muses' art                     vii  177\n\n  If the whole weight of what we think and feel,                vii  165\n\n  If this great world of joy and pain                           vii  336\n\n  If thou indeed derive thy light from Heaven,                  vii  175\n\n  If thou in the dear love of some one Friend                    ii  210\n\n  If to Tradition faith be due                                  vii  311\n\n  If with old love of you, dear Hills! I share                 viii   95\n\n  I grieved for Buonapart\u00e9, with a vain                          ii  323\n\n  I hate that Andrew Jones; he'll breed                        viii  221\n\n  I have a boy of five years old;                                 i  234\n\n  I heard (alas! 'twas only in a dream)                          vi  198\n\n  I heard a thousand blended notes,                               i  269\n\n  I know an aged Man constrained to dwell                      viii  186\n\n  I listen--but no faculty of mine,                              vi  326\n\n  Imagination--ne'er before content,                             vi   88\n\n  I marvel how Nature could ever find space                      ii  208\n\n  I met Louisa in the shade,                                     ii  362\n\n  Immured in Bothwell's Towers, at times the Brave              vii  299\n\n  In Brug\u00e8s town is many a street                               vii  198\n\n  In days of yore how fortunately fared                           v   67\n\n  In desultory walk through orchard grounds,                   viii  123\n\n  In distant countries have I been,                               i  279\n\n  In due observance of an ancient rite,                          iv  241\n\n  Inland, within a hollow vale, I stood;                         ii  343\n\n  Inmate of a mountain-dwelling,                                 vi  135\n\n  In my mind's eye a Temple, like a cloud                       vii  173\n\n  In one of those excursions (may they ne'er                    iii  367\n\n  Intent on gathering wool from hedge and brake                viii  122\n\n  In these fair vales hath many a Tree                          vii  269\n\n  In the sweet shire of Cardigan,                                 i  262\n\n  In this still place, remote from men,                          ii  393\n\n  In trellised shed with clustering roses gay,                   iv  102\n\n  Intrepid sons of Albion! not by you                            vi  111\n\n  In youth from rock to rock I went,                             ii  353\n\n  I rose while yet the cattle, heat-opprest,                     vi  257\n\n  I saw a Mother's eye intensely bent                           vii   92\n\n  I saw an aged Beggar in my walk;                                i  300\n\n  I saw far off the dark top of a Pine,                        viii   58\n\n  I saw the figure of a lovely Maid                             vii   74\n\n  Is _Death_, when evil against good has fought,               viii  106\n\n  I shiver, Spirit fierce and bold,                              ii  379\n\n  Is it a reed that's shaken by the wind,                        ii  331\n\n  Is then no nook of English ground secure,                    viii  166\n\n  Is then the final page before me spread,                       vi  382\n\n  Is there a power that can sustain and cheer                    iv  228\n\n  Is this, ye Gods, the Capitolian Hill,                       viii   59\n\n  _I thought of Thee, my partner and my guide_,                  vi  263\n\n  It is a beauteous evening, calm and free,                      ii  335\n\n  It is no Spirit who from heaven hath flown,                    ii  376\n\n  It is not to be thought of that the Flood                      ii  347\n\n  It is the first mild day of March:                              i  271\n\n  I travelled among unknown men,                                 ii   80\n\n  It seems a day                                                 ii   70\n\n  It was a beautiful and silent day                             iii  311\n\n  It was a dreary morning when the wheels                       iii  168\n\n  It was a _moral_ end for which they fought;                    iv  217\n\n  It was an April morning: fresh and clear                       ii  154\n\n  I've watched you now a full half-hour,                         ii  297\n\n  I wandered lonely as a cloud                                  iii    4\n\n  I was thy neighbour once, thou rugged Pile!                   iii   54\n\n  I watch, and long have watched, with calm regret               vi  197\n\n  I, who accompanied with faithful pace                         vii    4\n\n  I, whose pretty Voice you hear,                              viii  295\n\n  I will relate a tale for those who love                      viii  224\n\n  Jesu! bless our slender Boat,                                  vi  301\n\n  Jones! I as from Calais southward you and I                    ii  332\n\n  Just as those final words were penned, the sun broke out\n      in power,                                                viii  135\n\n  Keep for the Young the Impassioned smile                       vi  218\n\n  Lady! a Pen (perhaps with thy regard,                        viii    8\n\n  Lady! I rifled a Parnassian cave                               vi  211\n\n  Lady! the songs of Spring were in the grove                    iv   58\n\n  Lament! for Diocletian's fiery sword                          vii    8\n\n  Lance, shield, and sword relinquished--at his side            vii   20\n\n  Last night, without a voice, that Vision spake                vii   74\n\n  Let other bards of angels sing,                               vii  121\n\n  Let thy wheel-barrow alone                                     ii   95\n\n  Let us quit the leafy arbour,                                  vi  153\n\n  Lie here, without a record of thy worth,                      iii   50\n\n  Life with yon Lambs, like day, is just begun,                viii   97\n\n  Like a shipwreck'd Sailor tost                                vii  328\n\n  List, the winds of March are blowing;                         vii  331\n\n  List--'twas the Cuckoo.--O with what delight,                viii   68\n\n  List, ye who pass by Lyulph's Tower                           vii  394\n\n  Lo! in the burning west, the craggy nape                       vi  377\n\n  Lone Flower, hemmed in with snows and white as they            vi  191\n\n  Long-favoured England! be not thou misled,                   viii  131\n\n  Long has the dew been dried on tree and lawn,                viii   63\n\n  Long time have human ignorance and guilt                      iii  345\n\n  Lonsdale! it were unworthy of a Guest,                        vii  392\n\n  Look at the fate of summer flowers,                           vii  124\n\n  Look now on that Adventurer who hath paid                      iv  228\n\n  Lord of the vale! astounding Flood;                            vi   26\n\n  Loud is the Vale! the Voice is up                              iv   47\n\n  Loving she is, and tractable, though wild;                     iv  252\n\n  Lo! where she stands fixed in a saint-like trance,           viii  132\n\n  Lo! where the Moon along the sky,                            viii   88\n\n  Lowther! in thy majestic Pile are seen                        vii  392\n\n  Lulled by the sound of pastoral bells,                         vi  372\n\n  Lyre! though such power do in thy magic live,                viii  147\n\n  \"Man's life is like a Sparrow, mighty King!\"                  vii   16\n\n  Mark how the feathered tenants of the flood,                   iv  278\n\n  Mark the concentred hazels that enclose                        vi   71\n\n  Meek Virgin Mother, more benign                                vi  318\n\n  Men of the Western World! in Fate's dark book,               viii  112\n\n  Men, who have ceased to reverence, soon defy                  vii   68\n\n  Mercy and Love have met thee on thy road,                     vii    7\n\n  Methinks that I could trip o'er heaviest soil,                vii   66\n\n  Methinks that to some vacant hermitage                        vii   21\n\n  Methinks 'twere no unprecedented feat                          vi  255\n\n  Methought I saw the footsteps of a throne                      iv   46\n\n  'Mid crowded obelisks and urns                                 ii  387\n\n  Mid-noon is past;--upon the sultry mead                        vi  254\n\n  Milton! thou should'st be living at this hour:                 ii  346\n\n  Mine ear has wrung, my spirit sunk subdued,                   vii  104\n\n  \"_Miserrimus!_\" and neither name nor date,                    vii  201\n\n  Monastic Domes! following my downward way,                    vii  100\n\n  Most sweet it is with unuplifted eyes                         vii  401\n\n  Mother! whose virgin bosom was uncrost,                       vii   54\n\n  Motions and Means, on land and sea at war,                    vii  389\n\n  My frame hath often trembled with delight                      vi  250\n\n  My heart leaps up when I behold                                ii  292\n\n  My Lord and Lady Darlington                                  viii  298\n\n  My Son! behold the tide already spent,                       viii  273\n\n  Nay, Traveller! rest. This lonely Yew-tree stands               i  109\n\n  Near Anio's stream, I spied a gentle Dove,                   viii   65\n\n  Never enlivened with the liveliest ray,                      viii  150\n\n  Next morning Troilus began to clear                            ii  264\n\n  No fiction was it of the antique age:                          vi  241\n\n  No more: the end is sudden and abrupt,                        vii  309\n\n  No mortal object did these eyes behold                        iii  381\n\n  No record tells of lance opposed to lance,                     vi  258\n\n  Nor scorn the aid which Fancy oft doth lend                   vii   18\n\n  Nor shall the eternal roll of praise reject                   vii   78\n\n  Nor wants the cause the panic-striking aid                    vii   12\n\n  Not a breath of air,                                         viii  146\n\n  Not envying Latian shades--if yet they throw                   vi  230\n\n  Not hurled precipitous from steep to steep;                    vi  261\n\n  Not in the lucid intervals of life                            vii  402\n\n  Not in the mines beyond the western main,                     vii  400\n\n  Not, like his great Compeers, indignantly                      vi  303\n\n  Not Love, not War, nor the tumultuous swell                   vii  118\n\n  Not 'mid the World's vain objects that enslave                 iv  210\n\n  Not sedentary all: there are who roam                         vii   23\n\n  Not seldom, clad in radiant vest,                              vi  175\n\n  Not so that Pair whose youthful spirits dance                  vi  240\n\n  Not the whole warbling grove in concert heard                 vii  169\n\n  Not to the clouds, not to the cliff, he flew;                 vii  372\n\n  Not to the object specially designed,                        viii  106\n\n  Not utterly unworthy to endure                                vii   55\n\n  Not without heavy grief of heart did He                        iv  236\n\n  No whimsey of the purse is here,                             viii  259\n\n  Now that all hearts are glad, all faces bright,                iv  282\n\n  Now that the farewell tear is dried,                           vi  338\n\n  Now we are tired of boisterous joy,                            ii  420\n\n  Now when the primrose makes a splendid show,                 viii  116\n\n  Nuns fret not at their convent's narrow room;                  iv   28\n\n  Oak of Guernica! Tree of holier power                          iv  245\n\n  O blithe New-comer! I have heard,                              ii  289\n\n  O dearer far than light and life are dear,                    vii  122\n\n  O'er the wide earth, on mountain and on plain,                 iv  223\n\n  O'erweening Statesmen have full long relied                    iv  247\n\n  O Flower of all that springs from gentle blood,                iv  235\n\n  Of mortal parents is the Hero born                             iv  214\n\n  O for a dirge! But why complain?                              vii  132\n\n  O, for a kindling touch from that pure flame,                  vi  110\n\n  O for the help of Angels to complete                           vi  297\n\n  O Friend! I know not which way I must look                     ii  345\n\n  Oft have I caught, upon a fitful breeze,                      vii  373\n\n  Oft have I seen, ere Time had ploughed my cheek,              vii  163\n\n  Oft I had heard of Lucy Gray:                                  ii   99\n\n  Oft is the medal faithful to its trust                         iv   77\n\n  Oft, through thy fair domains, illustrious Peer!                v   20\n\n  O gentle Sleep! do they belong to thee,                        iv   42\n\n  O happy time of youthful lovers (thus                         iii   24\n\n  Oh Bounty without measure, while the Grace                   viii  308\n\n  Oh Life! without thy chequered scene                           vi  315\n\n  Oh! pleasant exercise of hope and joy!                        iii   35\n\n  Oh what a Wreck! how changed in mien and speech,             viii   36\n\n  Oh! what's the matter? what's the matter?                       i  254\n\n  \"O Lord, our Lord! how wondrously,\" (quoth she)                ii  240\n\n  O Moon! if e'er I joyed when thy soft light                  viii  235\n\n  O mountain Stream! the Shepherd and his Cot                    vi  245\n\n  Once did She hold the gorgeous east in fee;                    ii  336\n\n  Once I could hail (howe'er serene the sky)                    vii  152\n\n  Once in a lonely hamlet I sojourned                            ii  285\n\n  Once more the Church is seized with sudden fear,              vii   49\n\n  Once on the top of Tynwald's formal mound                     vii  366\n\n  Once to the verge of yon steep barrier came                  viii  236\n\n  One might believe that natural miseries                        ii  431\n\n  One morning (raw it was and wet--                              ii  270\n\n  One who was suffering tumult in his soul                       vi  187\n\n  On his morning rounds the Master                              iii   48\n\n  O Nightingale! thou surely art                                 iv   67\n\n  On, loitering Muse--the swift Stream chides us--on!            vi  242\n\n  \"On Man, on Nature, and on Human Life,\"                         v   23\n\n  On Nature's invitation do I come,                              ii  118\n\n  O now that the genius of Bewick were mine,                     ii   60\n\n  On to Iona!--What can she afford                              vii  379\n\n  Open your gates, ye everlasting Piles!                        vii  105\n\n  O there is blessing in this gentle breeze,                    iii  132\n\n  O thou who movest onward with a mind                           iv  231\n\n  O thou! whose fancies from afar are brought;                   ii  351\n\n  Our bodily life, some plead, that life the shrine,           viii  109\n\n  Our walk was far among the ancient trees:                      ii  167\n\n  Outstretching flame-ward his upbraided hand                   vii   62\n\n  <DW29>s, lilies, kingcups, daisies,                            ii  301\n\n  Part fenced by man, part by a rugged steep                    vii  286\n\n  Pastor and Patriot!--at whose bidding rise                    vii  349\n\n  Patriots informed with Apostolic light                        vii   85\n\n  Pause, courteous Spirit!--Balbi supplicates                    iv  237\n\n  Pause, Traveller! whosoe'er thou be                            vi  173\n\n  Peaceful our valley, fair and green;                         viii  259\n\n  Pelion and Ossa flourish side by side,                         ii  238\n\n  \"People! your chains are severing link by link;\"              vii  290\n\n  Perhaps some needful service of the State                      iv  230\n\n  Pleasures newly found are sweet                                ii  303\n\n  Portentous change when History can appear,                   viii  130\n\n  Praised be the Art whose subtle power could stay               iv  272\n\n  Praised be the Rivers, from their mountain springs            vii   45\n\n  Prejudged by foes determined not to spare,                    vii   71\n\n  Presentiments! they judge not right                           vii  266\n\n  Prompt transformation works the novel Lore;                   vii   17\n\n  Proud were ye, Mountains, when, in times of old,             viii  167\n\n  Pure element of waters! wheresoe'er                            vi  184\n\n  Queen of the Stars!--so gentle, so benign,                   viii   15\n\n  Ranging the heights of Scawfell or Black-Comb,                vii  358\n\n  Rapt above earth by power of one fair face,                  viii   81\n\n  Realms quake by turns: proud Arbitress of grace,              vii   32\n\n  Record we too, with just and faithful pen,                    vii   39\n\n  Redoubted King, of courage leonine,                           vii   31\n\n  Reluctant call it was; the rite delayed;                      vii  323\n\n  \"Rest, rest, perturb\u00e8d Earth!\"                                 vi   95\n\n  Return, Content! for fondly I pursued,                         vi  255\n\n  Rid of a vexing and a heavy load,                            viii  265\n\n  Rise!--they _have_ risen: of brave Aneurin ask                vii   11\n\n  Rotha, my Spiritual Child! this head was grey                 vii  171\n\n  Rude is this Edifice, and Thou hast seen                       ii  213\n\n  Sacred Religion! \"mother of form and fear,\"                    vi  249\n\n  Sad thoughts, avaunt!--partake we their blithe cheer           vi  253\n\n  Said red-ribboned Evans:                                     viii  302\n\n  Said Secrecy to Cowardice and Fraud,                         viii  304\n\n  Say, what is Honour?--'Tis the finest sense                    iv  225\n\n  Say, ye far-travelled clouds, far-seeing hills--              vii  287\n\n  Scattering, like birds escaped the fowler's net,              vii   64\n\n  Scorn not the Sonnet; Critic, you have frowned,               vii  163\n\n  Screams round the Arch-druid's brow the seamew--white         vii    6\n\n  Seek who will delight in fable,                              viii  172\n\n  See the Condemned alone within his cell,                     viii  110\n\n  See what gay wild flowers deck this earth-built Cot,          vii  296\n\n  See, where his difficult way that Old Man wins,              viii   83\n\n  Serene, and fitted to embrace,                                 vi  117\n\n  Serving no haughty Muse, my hands have here,                 viii  102\n\n  Seven Daughters had Lord Archibald,                            ii  204\n\n  Shade of Caractacus, if spirits love,                        viii  309\n\n  Shall he who gives his days to low pursuits                  viii  257\n\n  Shame on this faithless heart! that could allow                vi  214\n\n  She dwelt among the untrodden ways                             ii   79\n\n  She had a tall man's height or more;                           ii  278\n\n  She was a Phantom of delight                                  iii    2\n\n  She wept.--Life's purple tide began to flow                  viii  209\n\n  Shout, for a mighty Victory is won!                            ii  436\n\n  Show me the noblest Youth of present time,                    vii  181\n\n  Shun not this rite, neglected, yea abhorred,                  vii   96\n\n  Since risen from ocean, ocean to defy,                        vii  369\n\n  Six changeful years have vanished since I first               iii  247\n\n  Six months to six years added he remained,                   viii   39\n\n  Six thousand veterans practised in war's game,                 ii  435\n\n  Small service is true service while it lasts,                viii    8\n\n  Smile of the Moon!--for so I name                              vi  163\n\n  So fair, so sweet, withal so sensitive,                      viii  164\n\n  Soft as a cloud is yon blue Ridge--the Mere                   vii  405\n\n  Sole listener, Duddon! to the breeze that played               vi  234\n\n  Son of my buried Son, while thus thy hand,                   viii  305\n\n  Soon did the Almighty Giver of all rest                        iv  267\n\n  Spade! with which Wilkinson hath tilled his lands,             iv    3\n\n  Stay, bold Adventurer; rest awhile thy limbs                   iv  281\n\n  Stay, little cheerful Robin! stay,                           viii   38\n\n  Stay near me--do not take thy flight!                          ii  283\n\n  Stern Daughter of the Voice of God!                           iii   38\n\n  Strange fits of passion have I known:                          ii   78\n\n  Stranger! this hillock of mis-shapen stones                    ii   63\n\n  Stretched on the dying Mother's lap, lies dead                vii  387\n\n  Such age how beautiful! O Lady bright,                        vii  172\n\n  Such fruitless questions may not long beguile                  vi  246\n\n  Surprised by joy--impatient as the Wind                        vi   72\n\n  Sweet Flower, belike one day to have                          iii   51\n\n  Sweet Highland Girl, a very shower                             ii  390\n\n  \"Sweet is the holiness of Youth\"--so felt                     vii   59\n\n  Sweet was the walk along the narrow lane,                    viii  215\n\n  Swiftly turn the murmuring wheel!                              iv  275\n\n  Sylph was it? or a Bird more bright                           vii  319\n\n  Take, cradled Nursling of the mountain, take                   vi  233\n\n  Tax not the royal Saint with vain expense,                    vii  106\n\n  Tell me, ye Zephyrs! that unfold,                             vii  125\n\n  Tenderly do we feel by Nature's law,                         viii  104\n\n  Thanks for the lessons of this Spot--fit school               vii  377\n\n  That happy gleam of vernal eyes,                              vii  202\n\n  That heresies should strike (if truth be scanned              vii   10\n\n  That is work of waste and ruin--                               ii  298\n\n  That way look, my Infant, lo!                                 iii   16\n\n  The Baptist might have been ordained to cry,                 viii   80\n\n  The Bard--whose soul is meek as dawning day,                   vi  112\n\n  The captive Bird was gone;--to cliff or moor                  vii  371\n\n  The cattle crowding round this beverage clear                 vii  348\n\n  The Cock is crowing,                                           ii  293\n\n  The confidence of Youth our only Art,                        viii  273\n\n  The Crescent-moon, the Star of Love,                         viii  127\n\n  The Danish Conqueror, on his royal chair,                      vi  130\n\n  The days are cold, the nights are long,                       iii   74\n\n  The dew was falling fast, the stars began to blink;            ii  143\n\n  The doubt to which a wavering hope had clung                 viii  289\n\n  The embowering rose, the acacia, and the pine,                 iv   74\n\n  The encircling ground, in native turf arrayed,                vii  104\n\n  The fairest, brightest, hues of ether fade;                    vi   66\n\n  The feudal Keep, the bastions of Cohorn,                      vii  360\n\n  The fields which with covetous spirit we sold,                iii   12\n\n  The floods are roused, and will not soon be weary;            vii  388\n\n  The forest huge of ancient Caledon                            vii  304\n\n  The formal World relaxes her cold chain,                     viii  112\n\n  The gallant Youth, who may have gained,                       vii  281\n\n  The gentlest Poet, with free thoughts endowed,               viii  141\n\n  The gentlest Shade that walked Elysian plains                  ii  378\n\n  The glory of evening was spread through the west;            viii  217\n\n  The God of Love--_ah, benedicite!_                             ii  250\n\n  The imperial Consort of the Fairy-king                         vi  189\n\n  The imperial Stature, the colossal stride,                    vii  166\n\n  The Kirk of Ulpha to the pilgrim's eye                         vi  260\n\n  The Knight had ridden down from Wensley Moor                   ii  129\n\n  The Lake is thine,                                           viii  263\n\n  The Land we from our fathers had in trust,                     iv  215\n\n  The leaves that rustled on this oak-crowned hill,             vii  407\n\n  The leaves were fading when to Esthwaite's banks              iii  222\n\n  The linnet's warble, sinking towards a close,                 vii  403\n\n  The little hedgerow birds,                                      i  307\n\n  The lovely Nun (submissive, but more meek                     vii   52\n\n  The Lovers took within this ancient grove                     vii  306\n\n  The martial courage of a day is vain,                          iv  217\n\n  The massy Ways, carried across these heights                  vii  154\n\n  The Minstrels played their Christmas tune                      vi  227\n\n  The most alluring clouds that mount the sky,                 viii  128\n\n  The old inventive Poets, had they seen,                        vi  251\n\n  _The oppression of the tumult--wrath and scorn--_             vii   13\n\n  The order'd troops                                           viii  234\n\n  The peace which others seek they find;                        iii   11\n\n  The pensive Sceptic of the lonely vale                          v  327\n\n  The pibroch's note, discountenanced or mute;                  vii  290\n\n  The post-boy drove with fierce career,                         ii  273\n\n  The power of Armies is a visible thing,                        iv  254\n\n  The prayers I make will then be sweet indeed                  iii  382\n\n  The rains at length have ceas'd, the winds are still'd,      viii  233\n\n  There are no colours in the fairest sky                       vii   77\n\n  There is a bondage worse, far worse, to bear                   ii  431\n\n  There is a change--and I am poor;                              iv   17\n\n  There is a Flower, the lesser Celandine,                      iii   21\n\n  There is a little unpretending Rill                            iv   53\n\n  There is an Eminence,--of these our hills                      ii  162\n\n  _There is a pleasure in poetic pains_                         vii  166\n\n  There is a shapeless crowd of unhewn stones                  viii  223\n\n  There is a Thorn--it looks so old,                              i  242\n\n  There is a Yew-tree, pride of Lorton Vale,                     ii  370\n\n  There never breathed a man who, when his life                  iv  232\n\n  \"There!\" said a Stripling, pointing with meet pride           vii  384\n\n  There's George Fisher, Charles Fleming, and Reginald Shore,    ii  207\n\n  There's more in words than I can teach:                       vii  321\n\n  There's not a nook within this solemn Pass,                   vii  289\n\n  There's something in a flying horse,                           ii    3\n\n  There was a Boy; ye knew him well, ye cliffs                   ii   57\n\n  There was a roaring in the wind all night;                     ii  314\n\n  There was a time when meadow, grove, and stream,             viii  190\n\n  The Roman Consul doomed his sons to die,                     viii  105\n\n  The Sabbath bells renew the inviting peal;                    vii   96\n\n  The saintly Youth has ceased to rule, discrowned              vii   61\n\n  The Scottish Broom on Bird-nest brae                         viii  270\n\n  These times strike monied worldlings with dismay:              ii  432\n\n  These Tourists, heaven preserve us! needs must live            ii  184\n\n  These vales were saddened with no common gloom               viii  275\n\n  The Sheep-boy whistled loud, and lo!                          iii   58\n\n  The Shepherd, looking eastward, softly said,                   vi   68\n\n  The sky is overcast                                             i  227\n\n  The snow-tracks of my friends I see,                         viii  219\n\n  The soaring lark is blest as proud                            vii  214\n\n  The Spirit of Antiquity--enshrined                             vi  290\n\n  The stars are mansions built by Nature's hand,                 vi  210\n\n  The star which comes at close of day to shine,               viii  307\n\n  The struggling Rill insensibly is grown                        vi  239\n\n  The sun has long been set,                                     ii  327\n\n  The sun is couched, the sea-fowl gone to rest;                vii  338\n\n  The Sun, that seemed so mildly to retire,                     vii  337\n\n  The sylvan <DW72>s with corn-clad fields                        vi  201\n\n  The tears of man in various measure gush                      vii   60\n\n  The Troop will be impatient; let us hie                         i  114\n\n  The turbaned Race are poured in thickening swarms             vii   31\n\n  The unremitting voice of nightly streams,                    viii  187\n\n  The valley rings with mirth and joy;                           ii  138\n\n  The vestal priestess of a sisterhood who knows               viii  325\n\n  The Vested Priest before the Altar stands;                    vii   94\n\n  The Virgin Mountain, wearing like a Queen                     vii   70\n\n  The Voice of song from distant lands shall call                ii  338\n\n  The wind is now thy organist;--a clank                        vii  288\n\n  The woman-hearted Confessor prepares                          vii   28\n\n  The world forsaken, all its busy cares,                      viii   73\n\n  The world is too much with us; late and soon,                  iv   39\n\n  The worship of this Sabbath morn,                            viii  326\n\n  They called Thee MERRY ENGLAND, in old time;                  vii  343\n\n  They call it Love lies bleeding! rather say,                 viii  150\n\n  They dreamt not of a perishable home                          vii  107\n\n  The Young-ones gathered in from hill and dale,                vii   92\n\n  They seek, are sought; to daily battle led,                    iv  253\n\n  They--who have seen the noble Roman's scorn,                 viii   62\n\n  This Height a ministering Angel might select:                  iv  271\n\n  \"This Land of Rainbows spanning glens whose walls,\"           vii  299\n\n  This Lawn, a carpet all alive                                 vii  228\n\n  This Spot--at once unfolding sight so fair,                  viii  103\n\n  Those breathing Tokens of your kind regard,                   vii  217\n\n  Those had given earliest notice, as the lark                  vii   46\n\n  Those old credulities, to nature dear,                       viii   60\n\n  Those silver clouds collected round the sun                    vi  199\n\n  Those words were uttered as in pensive mood                    iv   37\n\n  Though I beheld at first with blank surprise                 viii  115\n\n  Though joy attend Thee orient at the birth                    vii  299\n\n  Though many suns have risen and set                           vii  148\n\n  Though narrow be that old Man's cares, and near,               iv   69\n\n  Tho' searching damps and many an envious flaw                  vi  343\n\n  Though the bold wings of Poesy affect                        viii  154\n\n  Though the torrents from their fountains                       ii  182\n\n  Though to give timely warning and deter                      viii  109\n\n  \"Thou look'st upon me, and dost fondly think,\"                vii  347\n\n  Thou sacred Pile! whose turrets rise                           vi  333\n\n  Threats come which no submission may assuage,                 vii   52\n\n  Three years she grew in sun and shower,                        ii   81\n\n  Throned in the Sun's descending Car                          viii  300\n\n  Through Cumbrian wilds, in many a mountain cove,             viii  272\n\n  Through shattered galleries, 'mid roofless halls,             vii  131\n\n  Thus all things lead to Charity, secured                      vii  102\n\n  Thus far, O Friend! have we, though leaving much              iii  153\n\n  Thus is the storm abated by the craft                         vii   48\n\n  Thy functions are ethereal,                                   vii  204\n\n  'Tis eight o'clock,--a clear March night,                       i  283\n\n  'Tis gone--with old belief and dream                          vii  192\n\n  'Tis He whose yester-evening's high disdain                  viii   94\n\n  'Tis not for the unfeeling, the falsely refined,               ii  147\n\n  'Tis said, fantastic ocean doth enfold                         vi  286\n\n  'Tis said, that some have died for love:                       ii  178\n\n  'Tis said that to the brow of yon fair hill                   vii  230\n\n  'Tis spent--this burning day of June!                         iii   76\n\n  To a good Man of most dear memory                            viii   18\n\n  To appease the Gods; or public thanks to yield;                vi  363\n\n  To barren heath, bleak moor, and quaking fen,                  vi   16\n\n  \"To every Form of being is assigned,\"                           v  353\n\n  To kneeling Worshippers no earthly floor                      vii   97\n\n  Too frail to keep the lofty vow                                ii  383\n\n  To public notice, with reluctance strong,                      vi   40\n\n  Toussaint, the most unhappy man of men!                        ii  339\n\n  Tradition, be thou mute! Oblivion, throw                      vii  293\n\n  Tranquillity! the sovereign aim wert thou                     vii  387\n\n  Troubled long with warring notions                             vi  175\n\n  True is it that Ambrosio Salinero                              iv  233\n\n  'Twas summer, and the sun had mounted high:                     v   26\n\n  Two Voices are there; one is of the sea,                       iv   61\n\n  Under the shadow of a stately Pile,                          viii   78\n\n  Ungrateful Country, if thou e'er forget                       vii   81\n\n  Unless to Peter's Chair the viewless wind                     vii   34\n\n  Unquiet Childhood here by special grace                       vii  170\n\n  Untouched through all severity of cold;                       vii  231\n\n  \"Up, Timothy, up with your staff and away!\"                    ii  181\n\n  Up to the throne of God is borne                              vii  408\n\n  Up! up! my Friend, and quit your books;                         i  274\n\n  Up with me! up with me into the clouds!                       iii   42\n\n  Urged by Ambition, who with subtlest skill                    vii   26\n\n  Uttered by whom, or how inspired--designed                     vi  306\n\n  Vallombrosa! I longed in thy shadiest wood                     vi  357\n\n  \"Vallombrosa--I longed in thy shadiest wood\"                 viii   76\n\n  Vanguard of Liberty, ye men of Kent,                           ii  434\n\n  \"Wait, prithee, wait!\" this answer Lesbia threw              viii   32\n\n  Wanderer! that stoop'st so low, and com'st so near           viii   13\n\n  Wansfell! this Household has a favoured lot,                 viii  153\n\n  Ward of the Law!--dread Shadow of a King!                      vi  209\n\n  Was it to disenchant, and to undo,                             vi  295\n\n  Was the aim frustrated by force or guile,                      vi  184\n\n  Watch, and be firm! for, soul-subduing vice,                  vii   10\n\n  \"Weak is the will of Man, his judgment blind;\"                 vi   67\n\n  We can endure that He should waste our lands,                  iv  246\n\n  Weep not, belov\u00e8d Friends! nor let the air                     iv  230\n\n  We gaze--nor grieve to think that we must die,               viii  306\n\n  We had a female Passenger who came                             ii  342\n\n  _We_ have not passed into a doleful City,                     vii  383\n\n  Well have yon Railway Labourers to THIS ground               viii  176\n\n  Well may'st thou halt--and gaze with brightening eye!          iv   34\n\n  Well sang the Bard who called the grave, in strains           vii  295\n\n  Well worthy to be magnified are they                          vii   84\n\n  Were there, below, a spot of holy ground                        i   37\n\n  Were there, below, a spot of holy ground,                       i  310\n\n  We saw, but surely, in the motley crowd,                      vii  376\n\n  We talked with open heart, and tongue                          ii   91\n\n  We walked along, while bright and red                          ii   89\n\n  What aim had they, the Pair of Monks, in size                viii   74\n\n  What aspect bore the Man who roved or fled,                    vi  237\n\n  What awful p\u00e9rspective! while from our sight                  vii  106\n\n  \"What beast in wilderness or cultured field\"                  vii   47\n\n  What beast of chase hath broken from the cover?                vi  360\n\n  What crowd is this? what have we here! we must not\n      pass it by                                                 iv   22\n\n  What heavenly smiles! O Lady mine                            viii  177\n\n  What He--who, mid the kindred throng                           vi   29\n\n  What if our numbers barely could defy                        viii   87\n\n  \"What is good for a bootless bene?\"                            iv  205\n\n  \"What know we of the Blest above\"                              vi  315\n\n  What lovelier home could gentle Fancy choose?                  vi  294\n\n  What mischief cleaves to unsubdued regret,                    vii  340\n\n  What need of clamorous bells, or ribands gay,                  iv  276\n\n  What sounds are those, Helvellyn, that are heard              iii  270\n\n  What strong allurement draws, what spirit guides,            viii   92\n\n  What though the Accused, upon his own appeal                  vii  223\n\n  What though the Italian pencil wrought not here,               vi  321\n\n  What way does the Wind come? What way does he go?              iv   50\n\n  \"_What, you are stepping westward?_\"--\"_Yea._\"                 ii  396\n\n  When Alpine Vales threw forth a suppliant cry,                vii   79\n\n  Whence that low voice?--A whisper from the heart,              vi  252\n\n  When Contemplation, like the night-calm felt                  iii  201\n\n  When, far and wide, swift as the beams of morn                 iv  244\n\n  When first descending from the moorlands,                    viii   27\n\n  When haughty expectations prostrate lie,                       vi  192\n\n  When here with Carthage Rome to conflict came,               viii   66\n\n  When human touch (as monkish books attest),                  viii   34\n\n  When I have borne in memory what has tamed                     ii  348\n\n  When in the antique age of bow and spear                      vii  115\n\n  When, looking on the present face of things,                   ii  433\n\n  When Love was born of heavenly line,                         viii  216\n\n  When Philoctetes in the Lemnian isle                          vii  167\n\n  When Ruth was left half desolate,                              ii  104\n\n  When Severn's sweeping flood had overthrown,                 viii  314\n\n  When the soft hand of sleep had closed the latch               vi   97\n\n  When thy great soul was freed from mortal chains,             vii   25\n\n  When, to the attractions of the busy world,                   iii   66\n\n  When years of wedded life were as a day                        vi   43\n\n  Where are they now, those wanton Boys?                         ii  281\n\n  Where art thou, my beloved Son,                               iii    7\n\n  Where be the noisy followers of the game                       vi  380\n\n  Where be the temples which, in Britain's Isle,                 vi   45\n\n  Where holy ground begins, unhallowed ends,                     vi  217\n\n  Where lies the Land to which yon Ship must go?                 iv   41\n\n  Where lies the truth? has Man, in wisdom's creed,            viii  182\n\n  Where long and deeply hath been fixed the root                vii   43\n\n  Where towers are crushed, and unforbidden weeds               vii  137\n\n  Where will they stop, those breathing Powers,                 vii  314\n\n  While Anna's peers and early playmates tread,                 vii  169\n\n  While beams of orient light shoot wide and high,             viii  156\n\n  While flowing rivers yield a blameless sport,                  vi  190\n\n  While from the purpling east departs                          vii  146\n\n  While Merlin paced the Cornish sands,                         vii  252\n\n  While not a leaf seems faded; while the fields,                vi   65\n\n  While poring Antiquarians search the ground,                 viii   33\n\n  While the Poor gather round, till the end of time             vii  307\n\n  While thus from theme to theme the Historian passed,            v  283\n\n  \"Who but hails the sight with pleasure\"                        vi  156\n\n  Who but is pleased to watch the moon on high,                viii  184\n\n  Who comes--with rapture greeted, and caress'd                 vii   75\n\n  Who fancied what a pretty sight                                ii  374\n\n  Who is the happy Warrior? Who is he                            iv    8\n\n  Who ponders National events shall find,                      viii  131\n\n  Who rashly strove thy Image to portray,                      viii   29\n\n  Who rises on the banks of Seine,                               vi  104\n\n  Who swerves from innocence, who makes divorce                  vi  260\n\n  Who weeps for strangers? Many wept,                          viii  267\n\n  Why art thou silent! Is thy love a plant,                    viii   12\n\n  Why cast ye back upon the Gallic shore,                        vi  378\n\n  \"Why, Minstrel, these untuneful murmurings--\"                 vii  161\n\n  Why should the Enthusiast, journeying through this Isle,      vii  343\n\n  Why should we weep or mourn, Angelic boy,                    viii  181\n\n  Why sleeps the future, as a snake enrolled,                   vii  108\n\n  Why stand we gazing on the sparkling Brine,                   vii  361\n\n  \"Why, William, on that old grey stone,\"                         i  272\n\n  Wild Redbreast! hadst thou at Jemima's lip                    vii  176\n\n  Wisdom and Spirit of the universe!                             ii   66\n\n  With copious eulogy in prose or rhyme                         vii  270\n\n  With each recurrence of this glorious morn                     vi  194\n\n  With how sad steps, O Moon, thou climb'st the sky,             iv   38\n\n  Within her gilded cage confined,                              vii  142\n\n  Within our happy Castle there dwelt One                        ii  306\n\n  Within the mind strong fancies work,                           vi  158\n\n  With little here to do or see                                  ii  358\n\n  \"With sacrifice before the rising morn\"                        vi    2\n\n  With Ships the sea was sprinkled far and nigh,                 iv   40\n\n  Witness thou,                                                viii  234\n\n  Woe to the Crown that doth the Cowl obey!                     vii   27\n\n  \"Woe to you, Prelates! rioting in ease\"                       vii   49\n\n  Woman! the Power who left his throne on high,                 vii   95\n\n  Wouldst thou be gathered to Christ's chosen flock,           viii  303\n\n  Wouldst thou be taught, when sleep has taken flight,         viii  151\n\n  Would that our scrupulous Sires had dared to leave            vii   99\n\n  Ye Apennines! with all your fertile vales,                   viii   45\n\n  Ye brood of conscience--Spectres! that frequent,             viii  107\n\n  Ye Lime-trees, ranged before this hallowed Urn,                iv   78\n\n  Ye sacred Nurseries of blooming Youth!                         vi  213\n\n  Ye shadowy Beings, that have rights and claims                vii  377\n\n  Yes! hope may with my strong desire keep pace,                iii  381\n\n  Yes, if the intensities of hope and fear                      vii   88\n\n  Yes, it was the mountain Echo,                                 iv   25\n\n  Yes! thou art fair, yet be not moved,                        viii  176\n\n  Yes, though He well may tremble at the sound,                viii  111\n\n  Ye Storms, resound the praises of your King!                   vi  109\n\n  Yet are they here the same unbroken knot                       iv   65\n\n  Yet many a Novice of the cloistral shade,                     vii   53\n\n  Yet more,--round many a Convent's blazing fire                vii   51\n\n  Ye, too, must fly before a chasing hand,                      vii   54\n\n  Ye torrents, foaming down the rocky steeps,                  viii  161\n\n  Ye Trees! whose slender roots entwine,                       viii   82\n\n  Yet Truth is keenly sought for, and the wind                  vii   76\n\n  Yet, yet, Biscayans! we must meet our Foes                     iv  242\n\n  Ye vales and hills whose beauty hither drew,                 viii  157\n\n  You call it, \"Love lies bleeding,\"--so you may,              viii  149\n\n  You have heard \"a Spanish Lady\"                               vii  232\n\n  YOUNG ENGLAND--what is then become of Old,                   viii  180\n\n  You're here for one long vernal day;                         viii  284\n\n                           END OF VOL. VIII\n\n           _Printed by_ R. & R. CLARK, LIMITED, _Edinburgh_.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Poetical Works of William\nWordsworth -- Volume 8 (of 8), by William Wordsworth\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Praise for Paul Lisicky's Lawnboy:\n\n\"The sexual awakening of a gay teenager leads to a peculiar, short-ranged rebellion in Paul Lisicky's moody, thoughtful first novel.... Lawnboy recalls standouts of the genre.\"\u2014SAN FRANCISCO CHRONICLE\n\n\"Lawnboy is, quite simply, the real thing, a novel of mystery and great beauty. The appearance of a writer like Paul Lisicky\u2014a writer who deeply respects the complexities of love and desire, who can find tragedy and transcendence almost everywhere he looks\u2014is a rare event. I read this book increasingly slowly, dreading the moment when I would have no more of it to read. Now that I've finished, all I can say is that I hope Paul Lisicky is hard at work on another one.\"\u2014MICHAEL CUNNINGHAM\n\n\"This novel is to young gay men growing up in the 1980s and 1990s what Edmund White's A Boy's Own Story was for gay men who grew up in the 1950s and 60s. It's the best gay coming-of-age novel I've read in a decade, written with a wonderfully crisp prose. This story concerns a young man trying to find a family anywhere he can; he drifts from situation to situation in South Florida until he finally makes peace with himself. This writer could be the next David Leavitt or Michael Cunningham.\"\u2014BOOKSENSE 76 NEWSLETTER\n\n\"Nobody writes about hilarious longing the way Paul Lisicky does. Some writers manage to be funny and sad in turn; in Lawnboy, Lisicky manages to be both at the same time. His characters are lovable and fallible; his prose is gorgeous. Lawnboy is a tribute to the endless series of life's first loves: your parents, your siblings, your best friends, your childhood fears. It's beautifully written, it's beautiful.\"\u2014ELIZABETH McCRACKEN\n\n\"Savvy enough to recognize the importance of buzz cuts and sleeveless shirts in gay identity formation, Lisicky is also smart enough not to rely on hackneyed consumer-culture signifiers, resulting in a lushly emotional, romantic and tragic pursuit.\"\u2014PAPER\n\n\"Lisicky's prose shines, at times hilarious, at others entrenched in sorrow and longing, but always gorgeous to read.... The reconciliations between the characters are moving and earned, graced with compassion and vitality.\"\u2014BRET ANTHONY JOHNSON, BOOK\n\n\"What distinguishes [Lawnboy] is the depth of humanity that flows into Lisicky's cascade of crisp, fresh sentences.... Lisicky never ceases to be funny, ironic, and surprising.... Lawnboy scythes a remarkably touching journey.\"\u2014LAMBDA BOOK REPORT\n\n\"This adventure-of-the-heart takes place in as evocative a landscape as any you'll find in fiction, its Floridian decay and lushness the perfect setting for a story dense with eroticism, disillusionment, and the surprising grace notes of renewal.\"\u2014BERNARD COOPER\n\n\"Lawnboy re-landscapes the front yard of American fiction.\"\u2014CAROL MUSKE DUKES\n\n\"The power of this book is its ability to touch you on so many different levels.\"\u2014HOUSTON VOICE\n\n\"Lisicky charts Evan's conflicting emotions deftly.... Humorous and moving... hitting musical notes of insight and wit.\"\u2014AUSTIN CHRONICLE\n\n\"Reading this often brilliant novel makes a critic want to apply the clich\u00e9 'promising,' which doesn't do justice to this accomplished and highly readable excursion into human emotions and the choices that we make, or have made for us.\"\u2014SUITE 101.COM\n\n# Lawnboy\n\nAlso by Paul Lisicky\n\nFamous Builder\n\nThe Burning House\n\nUnbuilt Projects\n\nThe Narrow Door\n\n# Lawnboy\n\n## a novel\n\n## Paul Lisicky\n\nGraywolf Press\nCopyright \u00a9 1998 by Paul Lisicky\n\nFirst published by Turtle Point Press\n\nPublication of this volume is made possible in part by a grant provided by the Minnesota State Arts Board, through an appropriation by the Minnesota State Legislature; a grant from the Wells Fargo Foundation Minnesota; and a grant from the National Endowment for the Arts, which believes that a great nation deserves great art. Significant support has also been provided by the Bush Foundation; the McKnight Foundation; and other generous contributions from foundations, corporations, and individuals. To these organizations and individuals we offer our heartfelt thanks.\n\nSupported by the Jerome Foundation in celebration of the Jerome Hill Centennial and in recognition of the valuable cultural contributions of artists to society\n\nPublished by Graywolf Press\n\n250 Third Avenue North, Suite 600\n\nMinneapolis, Minnesota 55401\n\nAll rights reserved.\n\nwww.graywolfpress.org\n\nPublished in the United States of America\n\nISBN-13 978-1-55597-448-0\n\nEbook ISBN 978-1-55597-931-7\n\n2 4 6 8 9 7 5 3\n\nLibrary of Congress Control Number: 2005938154\n\nCover design: Kyle G. Hunter\n\nCover photograph: \u00a9 Corbis\nfor Mark\nAnd so it was I entered the broken world\n\nTo trace the visionary company of love, its voice\n\nAn instant in the wind (I know not whither hurled)\n\nBut not for long to hold each desperate choice.\n\nHART CRANE, \"The Broken Tower\"\n\n***\n\nChing-a-ling-a-ling\n\nChing-a-ling-a-loo...\n\nIf you love me like\n\nI love you\n\nNo one can cut our\n\nlove in two.\n\nCHARLES LUDLAM, \"Eunuchs of the Forbidden City\"\n\n# Prologue\n\nWe stared up into the dark, flamboyant branches, waiting for the wind to pick up. All it took was the slightest push, and there it was: crimson blossoms swaying above us like bells. We stood beneath its heavy canopy, gazing up its trunk, quiet, motionless. The air thickened with its fragrance. And then, as if in gratitude, the tree released itself, little red pieces falling now, coating us, sticking in our hair like blessings.\n\nWe walked further into the park. Peter grasped my hand, reading aloud the names on the signs. Soursop. Gumbo-limbo. Frangipani. Sugar apple. He spoke these words precisely, with the slightest thrill, as if they tasted dangerous. Could we be any happier? Our parents, to our surprise, had encouraged us to come here by ourselves, and it was that dry October morning when the heat finally breaks, when people tired of humidity and air-conditioning finally throw open their windows.\n\nI pointed to a flower. It faltered between a lamppost and a ditch, nameless, forlorn. \"Look,\" I said.\n\nPeter smiled, crouching down beside it. \"Pretty.\"\n\nI knelt next to him. I pulled in a breath: he'd always be older. I was five then, an age I'd resisted, while he was thirteen. How I wanted to grow like that, to go through three different shoe sizes within a year.\n\nI drew closer to the flower. I touched its leaves\u2014crenellated, yellow\u2014its dusky muscular stem. I squeezed shut my eyes. I saw it growing somewhere else, someplace better, in silence and beauty, craning its face toward the light.\n\nI picked it. I twirled its stem between my fingers, listening. Do it. Don't do it. Do it. Don't. No. Then stuffed the whole thing, root and all, in my mouth.\n\nPeter's face tightened. He jerked me up by the arm. \"Evan\u2014\"\n\n\"What?\"\n\n\"Spit it out!\"\n\n\"But\u2014\"\n\n\"You heard me.\"\n\nI turned my head away, holding my flower inside. The petals tickled the back of my throat; my tongue contracted. The earth blurred, warped. Grapefruit, I thought. My flower tastes like grapefruit. I swallowed, smiled, looked up at my brother through hot, hot tears.\n\n\"What if it's poison?\" He nodded, once, twice, tensing his bottom lip. \"Think about poison.\"\n\nI curled tighter into myself. The woods smelled cold. My forehead burned. Skull and crossbones: would that be me? Bottle of iodine? Alive, I was alive.\n\n# Part  \nOne\n\n# Chapter 1\n\nThere were things nobody knew about me. They didn't know about my old train set in my bedroom, complete with Cape Cods, hotels, signal crossings, and papier-m\u00e2ch\u00e9 palms, a set that I tinkered with until the ninth grade, then smashed\u2014to my uttermost sorrow\u2014after a fight with my father. They didn't know I could recite a handful of psalms\u2014the 13th, the 23rd, the 42nd, and the 53rd\u2014completely by heart. They didn't know about my deep interest in Greek myth, my faith in the shifting weather, my fascination with Saturn and the outer planets. They didn't know that I spent hours at a time inside a concrete pipe, a cool, cramped cylinder in the middle of a field, whenever I needed to get away. They didn't know about the morning in my thirteenth year when out of sheer boredom, I stitched my fingertips together with needle and thread, making an intricate basket of my hand and giving myself a tremendous infection. They all thought I was good-natured, upright and responsible, generous, affectionate, and kind, and of course I could be those things, but there was much more to me than that, a side that unnerved even myself, and this side included William.\n\nWilliam. William the pigeon-toed, William the conqueror. His name, though banal, still conjures up an otherworldly thing, not a being of flesh and blood. For a while I kept him hidden from everyone\u2014my mother, my father, Peter. It even took me months to tell my best friend Jane about him, though she thought she knew me. Well, she didn't.\n\nI liked to keep him that way: my secret path, my own private joy.\n\nI was mowing my parents' lawn. He was standing in our yard between the joewood and the carambola, mopping his tanned forehead with a blue rag. He was watching me, hard, and I made an effort not to notice. He'd been around forever; he had to be almost as old as my parents: forty, forty-two. I'd never even given him a thought. I only knew that he worked the camera for Channel 7 News, had a gentle greeting for my folks whenever he walked his Dobermans, and had the most profound and beautiful travelers tree I'd ever seen (like me, he loved plants). That was it. He meant absolutely nothing to me, until he stepped forward, then I started noticing: jaw, eyes, hair, smell, hands, feet, mouth. There was a kind of buzz about him, a field of hissing electricity that jerked with my ions and electrons. I felt myself getting hard. I thought: now you've really gotten yourself into trouble.\n\n\"Lawnboy,\" he said, mocking me. \"Lawnboy, Lawnboy.\"\n\nI pulled back on the lever. The motor silenced. \"Is there a problem?\" I said, with some irritation.\n\nThe hush overwhelmed. Above, dry palm fronds clattered in the heat.\n\n\"Can I help you?\"\n\nHe nodded, shyly. He tucked in his shirt, a striped Haitian thing patterned with yellow parrots. \"My lawn mower's broken. My place is a mess. Would you be interested in cutting my lawn?\"\n\nMy expression dulled. I reacted as if he'd been asking to tap my spine.\n\n\"I'll give you twenty bucks. Twenty bucks, an hour a week. No weeding or mulching.\"\n\nI frowned. I felt simple, my tongue swelling fatly in my mouth. Befuddled. I glanced up at the shocked treetops and saw a sun shivering in a glazed sky. When I looked back at him, his eyes, blue with sprinklings of gold, were watching mine. I thought I detected some fear in them.\n\n\"I have to ask my parents,\" I announced.\n\nI loped through the side door. I crouched in the living room, pressed my nose right down into the quarry tile so I could almost breathe in the dust. I was seventeen years old. Of course my parents didn't need to know. If anything, they'd be thrilled that I'd finally stopped moping around the house, dreading the resumption of school. I stood, watching the second hand of the clock make three complete rotations. I peered through the slats of the Bermuda shutter. He was shifting his weight from foot to foot. I reached into my pants, gave myself a firm and brutal yank. When I looked down, I noticed the trembling of my hands.\n\n\"What did they say?\" William called out to me.\n\nI bounded down the walk. I tried to seem matter-of-fact, reckless. \"They said fine. Fine. When do you want me to start?\"\n\n\"Tomorrow.\" He grinned, his shoulders drawing imperceptibly backward. \"So we have a deal then.\"\n\n\"Deal.\" I extended my hand to him.\n\nBut he didn't take the hand. Instead, he placed his palm flat over my face, then pushed up, and as I started the mower, I decided he was probably the creepiest, most disgusting individual I'd ever met in my life.\n\n***\n\nThat night I lay in bed, listening to the house. Ice maker, pool motor, air conditioner, computer, oven cleaner. Everything but voices. It was the fourth day in a row that my parents, Ursula and Sid, hadn't spoken. I should have been used to it by now, but their silence only seemed to have gotten noisier, so shrill I pictured it puncturing a hole, the size of a meteorite, through the ceiling. I couldn't be safe from it, not outside, not in my room. I suspected it would follow me everywhere, even after their deaths, till my own death. I glanced at the digits of my clock radio. 10:04. Boy, am I seeing the world.\n\nThey shouldn't have gotten married. They couldn't stand each other. Anyone could see it in their eyes and their clipped, joyless mouths. Once, seeing intimations of these same expressions in their wedding photos, I thought, with relief\u2014ah\u2014so it isn't my doing. Still, it maddened me that they kept holding on like this. They never dealt with anything. The same way they couldn't deal with me. I mean, I didn't mince or prance. I didn't weave, I didn't dot my \"i's\" with circles or curlicues, but my eminent faggotry should have been obvious to them. Hello, Ursula. Hello, Sid. Knock, knock. Anybody home?\n\nI fooled with my armpit hair. I thought of his face: the mussed-up brows, the deeply cut eyes. He carried a smoky smell about him, as if he were burning deep from inside.\n\n\"So where are you going?\" Ursula said the following morning. Her voice sounded upbeat, despite the goings-on with my father. At least she'd gotten herself dressed today.\n\nI stuffed a banana muffin in my mouth. \"Beach. I'm going with Jane. I'll be back before dinner.\"\n\nShe pressed the small of her back into the countertop. She held her pale, slender arms tightly over her chest. \"You're not going to the beach,\" she said with a knowing grin.\n\nCrumbs caught in my throat. Did she know? She couldn't know. \"Why do you always have to accuse? You never believe me. You always think I'm plotting, Mom. I think you despise me.\"\n\n\"Are you going to start with me again? I made a simple psychic assertion. You're a lousy liar. You can't hide a damn thing from me.\"\n\n\"Oh yeah?\" I laughed, and slammed out the door.\n\nI stood outside William's at 8:54 a.m. I was early, but wanted to get it over with. The door opened, and he stood there in a robe patterned with marine flags, a mud masque on his face.\n\n\"Lawn mower's in the garage,\" he said with a husky rasp.\n\nI pulled up on the door handle. The mower hunkered beside the pool chemicals\u2014a nice one actually, with pretty green paint and detachable grass catcher. Unlike our lawn mower, a piece of shit that was constantly leaking gas, this one started first try.\n\nI bent over, stretched. The grass, a high-quality Floratam, was pleasantly spongy. I worked up and down, sidestepping sprinkler heads, guarding the tender young trunks of the palms. I started making up a song. I frequently made up songs and sang them aloud, almost yelling them up to the trees.\n\nLenny, the lusty Lawnboy,\n\nCuts the yards and makes them sizzle.\n\nEveryone who sees him needs\n\nHis moisture-seeking love-hard missile.\n\nI studied the sliding windows of the house and realized I wasn't going to have sex with him. I didn't know where the warped idea had popped into my head anyway. Once again, I'd allowed myself to get all worked up about someone who was unavailable to me, foreign as the workings of a nuclear power plant. I looked up at the window again. He was just a regular fellow. Lonely. Dumb. A little fun.\n\nI finished up in record time, forty-three minutes. The yard was smaller than I'd expected. I stepped through the gate, sat, and took in the reeking trees: flame vine, soursop, wild cinnamon. My fingers smelled of gasoline, fertilizer. The lawn, green as mouthwash, glittered in the morning heat. Above me, the sky bubbled and fried.\n\n\"Nice tune,\" William said, stepping outside.\n\n\"What?\"\n\n\"I said I liked your song.\"\n\nMy stomach folded in on itself. \"I wasn't singing.\"\n\nHe cuffed the top of my head and laughed. \"Do you want to come inside for breakfast?\"\n\nMy words came out sludgy, like juice squeezed from a freeze-damaged orange. I told him that I had to leave, that I needed my payment, please, but he kept scrutinizing me. I tasted a fresh filling deep in the corner of my mouth. Finally, he reached into his rear pocket and pulled out three ten-dollar bills.\n\n\"But this is ten dollars too much.\"\n\n\"Take it.\"\n\n\"It's too much.\"\n\n\"Just take it. Buy yourself some candy.\"\n\n\"I don't need any candy.\".\n\n\"Whatever.\"\n\nI shrugged. For some reason I felt myself welcoming, letting down the defenses, when I noticed the fractures of light in his eye. I went off. I imagined him capable of all sorts of things. Hangings, slayings, snuff films. Whole freezers filled with kidnapped boys in body bags stacked according to height, weight, race, creed.\n\n\"So would you like to come in?\" he asked again.\n\n\"Sure,\" I said, and followed.\n\n***\n\nSomething disturbing and immature in my nature wanted to startle people. Perhaps it was because I was essentially unstartling in appearance. I slumped through the corridors of my school, Coral Gables High, a quiet, mealy kid with Dumbolike ears, in flannel shirts, racking up B after B, even though I was most likely a genius. My fifth-grade teacher, Mrs. Edge, had extracted me from the class and said, \"You're too good for us. I'm going to recommend to the principal that you be passed on to the sixth grade.\" And she did it, easy as the snap of a finger. These achievements continued until Peter\u2014crying, stoned\u2014left the house for good after an all-night fight with my father. I'm not sure what happened after that.\n\nWilliam and I sat at the kitchen counter. He told me about his ex-wife, Lorna, his daughter, Poppy, at Rollins College, his years in the Episcopal seminary, and his preparations for the priesthood. I was bored out of my skull. I picked up an empty mayonnaise jar. The same image kept drifting across my tired eyelids: a hole in the clouds, torn like a bullet wound, with the sky on fire behind it.\n\nThen I thought about Jane. We were supposed to have gone to the beach. She had to be pissed by now. Or worried. She was always worried about me. Sometimes she told me that if I wasn't careful, I'd be one of those people you read about in papers, carved up like a Christmas turkey, lying in a ditch. She could think what she wanted. But I knew I was protected. Something, somewhere, was watching, keeping me. God. An angel. I could walk through fire, thrive through sickness, pass through the harshest danger, and come out alive.\n\nI looked up. William was smiling. \"You haven't listened to a word I said.\"\n\n\"What?\" I put the jar down. The tabletop was littered with the scraps of the peeled label.\n\n\"Tell me about yourself. How's school?\"\n\nI shrugged. I wasn't going to give in to him. He thought he had something. He thought I was innocent, powerless, that I was going to lie down and take it. He was wrong. I pictured him lying on his stomach in a warm dark cave. A bowl full of liquid beside his head. How easy it would be to lift the bowl in his moment of peace and kill him.\n\nWe were in his room. He sat on the bed while I swayed above him. He unzipped my pants and felt for my dick\u2014a hard, red, glistening muscle. He gripped it, cranked it around. \"Beautiful,\" he rasped, gazing up into my face.\n\nHe started sucking me off. It wasn't like I'd expected. I mean, I'd fooled around before, but it wasn't serious sex\u2014not in a bed or anything\u2014and this, I supposed, was serious sex. I wasn't particularly excited. Maybe I was bored, even disgusted. I concentrated on the motions, trying to pinpoint the smells in the room. I thought: bleach, weeds, sweat, funk, hair.\n\n\"Good?\" he asked, taking a breather.\n\n\"Yeah.\"\n\nWe continued. We rolled around on the bed, when a thought, a full sentence, occurred to me: He is getting younger, while I am getting older. I didn't know what it meant. I thrust out my leg, kicked over the lamp, then rolled him over on his back, even though he was the stronger. I hiked his legs onto my shoulders, and to my astonishment, started fucking him.\n\n\"That's it,\" he muttered. \"Fuck your old man, boy. That's it. Keep fucking your daddy.\"\n\n\"Shut up,\" I whispered. \"Just please shut the hell up.\"\n\nA thin cord of electricity quavered up my spine. I realized: this is what I'd always wanted. All at once I departed from myself, turning above the bed like a huge ticking wheel, watching us pushing against each other. My breath was sticking in my throat. I leaned over and kissed the harsh sandpaper of his face. I returned to myself, felt him clenching and relaxing around me, then pulled out, coming across his heaving stomach.\n\nI stood before the bathroom mirror. I stuck a coated finger in my mouth, pushing it around my gums, feeding myself. My body felt new: the blood enriching my face, the muscles sharper as if dug by fine tools. I had something. I had a power all along and hadn't even known it.\n\nBut when I walked into his bedroom, I was only the mealy high school boy again. I eased under the covers, punching him softly on his broad freckled back, waiting for encouragement, or something returned, when he only swung away and rolled on some basketball socks with holes in them. His head appeared to be swimming with thoughts. His Dobermans jumped up on the bed, panting, licking at my bare skin.\n\n\"Get dressed, kiddo,\" he said. \"Your parents are going to be worried.\"\n\n\"But can't we do it again?\"\n\n\"No, that's enough for me.\" He laughed softly. \"Get dressed. It's time for you to go.\"\n\n\"You sure?\"\n\nHe nodded.\n\nSo that's it, I thought. That's what you do. I picked up my clothes from the sweaty heap on the floor. He kissed me dryly on the mouth. I left. On the way home I kept repeating: I went to the beach. I went to the big beach with my friend Jane and saw palms and sand and girls carrying buckets full of tulip shells. I took a swim and ate a snow cone.\n\n# Chapter 2\n\nYears ago, Jane had wanted me to be someone else. Or at least myself with one big difference.\n\nWe were sitting inside the South Miami house of Gwen Marino, the upper-level art teacher. We loved Gwen Marino, her tangle of hair and trailing black scarves and the many Bakelite bracelets she wore on each arm. Not only because she talked openly about the possibility of sex in our paintings (\"What these beach roses need\"\u2014she'd step back, pulling at her lip, clenching her brow\u2014\"is more foreplay\"), but because she invited us\u2014only us\u2014to babysit her daughter Saga, and that very opportunity bestowed on us a stature that all the other lowly worshipers in her cult could only dream of.\n\nJane was wearing a chopstick through her hair, her deep red lipstick setting off her pale skin. Gwen was out for the evening, on a date with Roy Panner, the industrial arts teacher whom we'd decided was not good enough for her. Saga had long been put to bed, and we sat on the couch, staring occasionally at the film canister of pot and the aqua bong on the end table. Something told us that this had been deliberately left out for our partaking and pleasure, and soon enough we were taking hit after hit, giggling hysterically, staring at the homemade pole lamp as its beautiful pieces revealed themselves to us: discrete, then whole again.\n\n\"God, I'm horny,\" I laughed. \"What the hell's in this stuff?\"\n\n\"Yeah,\" she said, biting into her lip. \"It's funny. I was just thinking the same thing.\"\n\nInstantly I regretted my admission. Casually, Jane pushed the tips of her fingers into the waistband of her jeans.\n\n\"What are you thinking?\" she said, smiling.\n\n\"Nothing.\" Then my laughter broke apart as if shattered by a BB.\n\n\"You want to fuck me, don't you?\"\n\n\"No!\" I said, still laughing.\n\n\"Yes, you do. You want to throw me on my back, you little beast, and fuck the living shit out of me.\"\n\nIf at that moment we'd been distracted by a ringing phone or a distant radio, the urge would have passed and our friendship would have remained just that\u2014something effortless and self-contained. Instead, I leaned over and kissed her wetly on the mouth. Her tongue felt slippery and hot, like my own. I needed to see whether I could pull this off\u2014that is, sex with someone of the other gender. I thought, why not? Why the hell not? She's my friend, and we care about each other. No one's going to get hurt here. Why not?\n\nI lay my weight on top of Jane. I unbuttoned the top two buttons of her shirt, then started kissing her, reaching inside for her breasts. Was I doing this right? Was I giving her what she wanted, the reassurance and the strength? I looked down into her face, her open shocked face, and she back at me, and before I could assess, something had changed: I felt foreign to myself, more remote than I'd ever felt before. I was nothing short of a liar. I hadn't yet told her the truth about myself, for the secret seemed to be enormous. She'd walk away, I knew it. I'd be telling her I wasn't the person she thought I was, and didn't we tell each other everything? Didn't I know about her cramps and her lies and her fights with her parents? Didn't I know about her doubts, her rages, her fleeting depressions concerning her little sister, Anna, who'd fallen back, thrown up, and died to the dread of her kindergarten as she scissored through a sheet of blue construction paper and constructed a George Washington hat? And didn't everyone\u2014Gwen Marino included\u2014assume we were a couple just waiting to happen if they didn't think we were already together?\n\nShe reached into my fly. I felt a quick high panic, thinking that she'd think my dick was mediocre, when it wasn't mediocre, not when I had an erection at least. I pulled back and folded my arms. Even when I'd imagined this scene, it had never been so clumsy.\n\n\"What's the matter?\" I said.\n\n\"You think I'm fat.\"\n\n\"No,\" I insisted. \"I don't think you're fat.\"\n\n\"You're not attracted to me. You think I'm disgusting.\" She pulled at the excess skin of her thigh.\n\n\"Stop,\" I said, reaching for her shoulder.\n\n\"Don't touch,\" she said, and curled up with a pillow. \"Go away. I feel like being alone now.\"\n\nI waited a week to explain the truth about myself. When I told her, though, in the school cafeteria, within earshot of the entire girls' ensemble (who were working out the kinks in Benjamin Britten's Ceremony of Carols), she reacted with the slightest hint of condescension, as if it weren't news to her, before walking to the ice cream machine to buy a Buddy bar.\n\nWhat happened next was more complicated. In the latter months of spring, she began dating a sophomore, DeMarco Huff, who'd been transferred to our school from the suburbs of Memphis. If you could call it dating\u2014what it essentially amounted to was a series of quick trysts behind the Dumpsters of the Coral Sea Garden Apartments, where DeMarco lived in a two-bedroom with his mother. In any case, Jane had never been crazy about him. He wouldn't talk to her, and they had not one single thing in common, but he was black, and she loved black men, and he had the loveliest, tiniest waist, with the prettiest belly button she'd ever seen.\n\n\"Are you sure this is a good thing?\" I asked her one day.\n\n\"Sure.\" She tried out a grin, looking not entirely convinced.\n\n\"Is he nice to you? Does he treat you well?\"\n\n\"Of course. He's the sweetest little piece on the planet. He's just horny is all, and that's all I want right now.\" Her eyes were dried out, miserable. \"God, I love to get fucked.\"\n\nJane and I still managed to call each other from time to time, though we saw each other less and less. An occasional wave in the hall outside Gwen's ceramics class\u2014that seemed to suffice. We were always running somewhere. Then one day I received a call from her, not three seconds after I'd arrived home from school.\n\n\"I need you to go somewhere with me.\"\n\n\"Go where?\" I wasn't in the mood. I spread a thin, gritty layer of peanut butter on my sandwich bread.\n\n\"Don't ask. I'll tell you later.\"\n\nShe picked me up in her parents' Vista Cruiser, a faded green station wagon from 1968, which we'd nicknamed the Flintstone-mobile due to its salt-corroded floor. It was amazing that her father had kept the car running, keeping a place for it beneath the palms, refusing to give it up while it still had some life left in it. The thing had embarrassed the hell out of Jane, and she'd never driven it alone before, but her face looked urgent and necessary as if its ramshackle appearance were the least of her worries.\n\n\"I'm getting an abortion.\"\n\n\"You're what?\"\n\n\"Shut up. I'm getting an abortion, and I don't want you to stop me. You're my assistant here. Your job is to keep me safe and protected and calm.\"\n\n\"Are you scared?\"\n\n\"Of course I'm scared. What do you think this is, some kind of party?\"\n\n\"Don't yell at me!\"\n\n\"I'm sorry,\" she said quietly. She raked her fingertips once down her forehead. \"Why am I so mean to you?\" she asked. \"You who I love so much. Tell me why I'm so mean to you.\"\n\nI gazed out the window at the sand pit, the lawn-ornament yard with its Marys and Buddhas. Beneath my feet Route 1 scrolled through a hole the size of a pie plate.\n\n\"You didn't tell your parents?\" I said.\n\n\"Are you kidding?\"\n\n\"You really think it'd bother them?\"\n\n\"They'd kill me. Especially if they found out about DeMarco.\"\n\nI squinted. \"Are you saying your parents are prejudiced?\"\n\n\"No. Not in the conventional sense. Let's just say they'd do everything in their power to stop a KKK rally from happening on their street. But they'd be funny about DeMarco. They'd just think I was trying to get back at them.\"\n\nShe parked outside the clinic, a yellow, stucco-coated building near a microwave tower topped with a strobe light.\n\n\"Do you see any picketers?\" she said, glancing around.\n\nThe parking lot was vacant. I shook my head.\n\n\"Just wait here for me.\"\n\n\"Can't I come inside? I want to come inside.\"\n\n\"No,\" she insisted. \"Stay right here. It'll be over in an hour.\"\n\nI did what she said. I focused all my attention on the door, awaiting her return. A woman in white stockings came out, regarding me briefly, warily, before hurrying down the block to her hatchback. I wondered whether Jane had been more upset than she'd let on. Two weeks ago, against the advice of Mrs. Wash, the health teacher, she'd given an oral report about the right to safe abortion\u2014admittedly the most hackneyed of topics\u2014but the class had actually listened. Her words were rational and tough-minded, so who could have assumed that getting an abortion would have been a big deal for her?\n\nI stared for five minutes at the door. I hoped they weren't hurting her.\n\nWhen we were at our most inseparable, Jane came up with the notion that she'd one day like to have a child with me. And if we couldn't have that child, we'd have an imaginary child. He'd be a boy. We'd call him Nico, after the Velvet Underground's lead singer. He'd be the perfect child. He'd be fluent in many languages. We'd take him everywhere with us\u2014Paris, Goa, Crete, Amsterdam\u2014where everyone would ask, who is that child? Who is that stylish, witty, extraordinary child? Ours, we'd say humbly. Nico.\n\nGod knows why I was thinking about that now.\n\nA half hour later, according to schedule, the door opened and Jane walked toward the car. She appeared neither relieved nor distraught, a little pale maybe, the hint of raccoon rings beneath her eyes. Her breath smelled faintly of orangeade. I wasn't sure how I was supposed to behave. Was I supposed to hug her, comfort her, speak words of wisdom, or act like she'd just been to the dentist's, like nothing significant had happened? I stared at the blue bruise beneath my fingernail. I felt less assured than I'd felt in a long time.\n\nSo I asked, \"How did it go?\"\n\nThe corners of her mouth were filmed with white. \"I don't want to talk about it, okay? Just don't let me talk about it.\"\n\n\"Do you want me to drive?\"\n\n\"Nn nn.\" She pulled out a Kleenex and wiped off her mouth. \"I'm feeling a little woozy. Did you eat?\"\n\nI shook my head. I was ready to do anything she wanted.\n\n\"Let's go to Burger King. For some reason I'm really in the mood to go to Burger King.\"\n\nIt was the silent time between lunch and dinner, and we seated ourselves away from the windows to keep ourselves out of the direct sun. My skin felt chilled and hot at once. Jane stared down at the tabletop, eating onion rings with one hand, propping up her jaw with the other, twirling a piece of hair between her fingers. She hadn't spoken in a full ten minutes.\n\n\"Poor Nico,\" she mumbled.\n\n\"What?\"\n\nShe smiled, exhausted, a watery glaze in her eyes. \"He would have been the best.\" And as if in slow motion, she pushed over her Coke, watching its contents fizzing on the tile floor.\n\nI tossed down some napkins to cover it up. A uniformed boy walked toward us with a mop, but I waved him away, kneeling down to take care of it myself.\n\nI glanced up at Jane. I said, \"Don't talk like that. We're not kids anymore.\"\n\n\"It was just a joke,\" she said in an exceedingly hurt voice. \"Don't be so hard on me, okay?\"\n\nI held her close to me for the longest time.\n\nWe left. Jane dropped me off, and I went to bed early that night. She came back to herself in the next few days, but our friendship seemed strained and uneasy after that. We'd crossed some line. We knew too much. The thing was, I believed it was possible now to know too much.\n\n***\n\nOnce a week, I went to William's house, mowed the lawn, weeded the garden, and had sex. I gradually began to understand him, his silences and quirks, how he couldn't stand when I nicked the flesh of the century plant or splashed gas near the bird feeders. I learned when to talk, I learned when to be quiet. I also learned not to be resentful whenever he ignored me afterwards. We had a pattern. We knew exactly what to expect from each other. No hurts, no disappointments. Not much talking. Sometimes, if we finished up early, he'd take me to the Speedboat Restaurant in Fort Lauderdale and I'd order exactly what I wanted: hamburger, salad, fries, dessert, soft drink. We might have been any other father and son. No one suspected a thing.\n\nOne afternoon I saw both of my parents sitting across from each other in the living room, silent. It was unlike them to be together in the same room. I couldn't tell whether I was grateful or frightened. Then it occurred to me that one of them, most likely my father, had cancer.\n\n\"Sit down,\" my father said grimly.\n\n\"I already have.\"\n\nMy eyes drifted to the baseboard heaters. It was a murky afternoon, threatening rain, and Ursula flipped on the lamp, which depressed me. Lamps are for night, I wanted to say. Their silence somehow amplified things: the clicking of a palmetto beetle, the pressure of the palms against the window glass. A stray nerve kept pulsing in the small of my back. I knew what was coming. I wanted to pick up and run down the street.\n\nMy mother finally sat forward on the sofa. \"You've been spending a lot of time away from us.\"\n\nSilence from Sid. He slouched low in his seat, curled the edge of his hand over his brow.\n\nI nodded solemnly. \"You want me to help out more?\"\n\nMy mom nodded, then wagged her head. \"I'm surprised.\"\n\nI smiled, because I couldn't contain it any longer, but neither of them smiled back. \"I don't understand.\"\n\n\"People are talking.\"\n\n\"Huh?\"\n\nShe whispered, \"Why are you spending so much time with that man?\"\n\nI raised my chin. Always what the neighbors thought. \"So what?\"\n\nMy father sprang to attention. \"Don't talk to her like that. That's your mother.\"\n\nMy eyes smarted. Did I know them? All my life I'd come and gone as I pleased. All my life I'd taken care of myself. Even when I was younger they'd never asked for report cards, never taken an interest in my hobbies or projects. I was the kind of five-year-old you saw circling the shopping-center parking lot on his bicycle, dodging cars, bewildering the parents, years before the other kids. And I wasn't complaining, okay? But there seemed to be a tacit agreement that they completely stay out of my business.\n\n\"What's this have to do with Mr. Parsons?\" I stood up now.\n\n\"You sit down,\" my father demanded.\n\nI didn't sit. They'd never talked to me this way before, and it wasn't going to start now.\n\n\"Do you see how he listens?\" my father said to my mother. \"Just like his brother.\"\n\n\"What's he have to do with anything?\"\n\nShe ignored the question. Her lips hovered over the rim of her cup. It seemed that I'd frightened her. \"What have you been doing with that man, dear?\" she asked shyly.\n\nI stood. My mother reached into her sundress pocket and showed my a piece of wadded-up paper. The printing was mine. I'd covered the entire page with the same phrase, in columns, using the tiniest of letters: I love to fuck I love to fuck I love to fuck I love to fuck.\n\n\"I found this in your notebook. I know I shouldn't have gone through your things. I know it was wrong, but when you started withdrawing from us, I didn't know what else to do.\"\n\nI wasn't withdrawing from anyone. \"You had no right, Mom.\"\n\n\"Have you been sleeping with that man?\"\n\n\"What makes you keep asking these stupid things?\"\n\nShe nibbled at the corner of her mouth. She recognized the truth in my eyes. \"We're having that bastard arrested.\"\n\n\"Don't talk about him that way.\"\n\nThat did it. My father went for my arm. He didn't punch me; he didn't do what any regular father would do. Instead, he drew me to him, somehow rolled me over his knee like a puppet, and\u2014get this\u2014started spanking my clothed butt for a good half-minute or so. I was seventeen years old. It was such a comic thing that I let him do it until we both filled up with shame.\n\n\"Happy, buddy boy?\" I said.\n\nI laughed all the way up to my room. When I lay facedown on my bed, minutes later, I was still laughing. Fools. I took breaths, many deep breaths, breathing, breathing, calming myself down, then reached for a pen, a red ballpoint, and started punching it in and out of my right palm, deft and precise as a sewing-machine needle, until I was looking down at a smear, a little red star on my hand. It was beautiful to see. The house stilled, and I fell fast asleep holding my beating hand to my face.\n\n# Chapter 3\n\nI wished I'd had more courage. I wished I'd held onto my anger, letting it fuel me, giving me the beauty and strength of the supersonic, but in the weeks that passed something happened. I learned that I was nothing more than a coward. My parents kept looking at me as if I'd crossed a bridge over a steaming fjord and had become a stranger to them. I became flat, an outline, weightless. One afternoon, coming home from school, I overheard my father talking about removing my name from his will. I didn't care about his money. But this erasure from their lives startled me, like an unexpected punch in the neck. I'd thought they'd get over it after a while, that we'd be the family we'd always been, fixed in our silences and resentments.\n\nSo I gave them what they wanted. I avoided William. I stopped picking up the phone and answering the door. I even shirked my gardening duties. I didn't even pass William's house, and imagined his lawn grown mangy and foul without me. On my way somewhere, I'd take the long route, around the park, through the college gates, just so I wouldn't bump into him. I lived in great fear of bumping into him, at the drugstore, at the motor vehicle registry, of seeing the worn-out expression on his face: Why have you done this?\n\nI helped with the garbage, manicured the hedges. I even scrubbed the mold from the side of the house, something that should have been done years ago. I made it quite clear, without saying it, that I no longer had anything to do with William, or with people, for that matter. My parents started treating me with kindness. They actually talked with me, asked my opinions, discussed current events. They'd finally gotten a good son, whoever that was, and he was gradually becoming an exemplary young man. I had only one more year of school, then I'd get out, hungry and loosed upon the world. I watched my soul shrink then shimmer to a tiny point.\n\nI began studying all night long. I began achieving perfect grades, throwing off the curve for the entire class. My teachers were amazed by me. Princeton, Stanford, Swarthmore, Michigan\u2014all of them wanted me by the end of the year. I kept on going. I was giving myself up to the powers. I swam, I ran, I beat off constantly, sometimes so much that it stung to pee. I was burning, a saint, purifying myself in these blaring fires. A dream would often come to me, and I'd force the dream, force myself to watch it, though it made me sick. I'd be standing over my young self, the sweet, boyish, optimistic self, punching his face until his mouth fell open.\n\nWe were eating dinner one Saturday night. My mother stepped out, then back in the kitchen, holding a cake rimmed with candles. My father handed a wrapped present to me. Then I realized it must be my birthday. I'd completely forgotten all about my birthday.\n\n\"We're so proud of you, Evan,\" my mother said blankly. \"You're doing everything right.\"\n\nMy father added: \"I'm so glad you've changed.\"\n\nI nodded, smiled. I unwrapped my present and stared at the watch in the box. It meant nothing to me. I should have broken it right in front of their needy eyes.\n\n\"Thank you,\" I said, and kissed them. Their foreheads were dry. I might have been kissing the brows of the dead.\n\nMonths passed. Jane and I were walking along the bay front. A new high-rise was being thrown up in record time, and we watched the construction workers in their orange hard hats stepping across the open girders. I'd just been offered a full scholarship to Princeton and I suppose we were celebrating that fact. I hadn't even talked to Jane in seven, eight weeks. Somehow I'd learned to live without her in my life.\n\n\"Are you sure you're okay?\" she said out of nowhere.\n\n\"Well, that's a non sequitur.\"\n\n\"I'm serious. Are you sure you're okay?\"\n\n\"Of course I'm okay. I've never been better in my life.\"\n\nA green-eyed man with black hair jogged by us. He looked me up and down, grinned, then trotted ahead.\n\n\"Did you see that?\" Jane said warmly. \"Did you see the way he looked at you?\"\n\n\"Faggot,\" I muttered, and walked ahead of her.\n\n\"Hey!\" Something wet bounced and spread across my back. \"That doesn't sound like you. He was exactly your type. What's with you?\"\n\nShe annoyed me terribly. First her judgment, then her rancor. I watched the people walking by us. I watched their simpering, self-satisfied faces and threw a mental message to each one: Fuck you.\n\n\"You're changing,\" she said quietly.\n\nShe sat down on a slatted bench. \"You used to be so much fun. You used to have such an amazing sense of humor.\"\n\n\"I was never funny.\"\n\n\"Yes you were. And we always talked when things got bad. We buoyed each other.\" She folded her arms over her chest, frowned. \"Fucker,\" she mumbled.\n\n\"Fucker?\"\n\n\"You heard what I said.\"\n\n\"Keep talking like that and I'm going to leave you right here.\"\n\n\"So leave.\"\n\nIt came as unexpectedly as a fish in a flooded street gutter. I glanced to my right and saw William sitting in his parked car, with newspaper and coffee cup. I didn't think he was looking at me. He was simply another office worker spending his break in his car. The intensity of my argument with Jane diminished. I wasn't scared or sick or excited. I'd known this moment would come, but I never thought it would be so dull. My vision went runny; my words sounded stupid in my mouth. I might have been sitting on the mucky bottom of the bay. I glanced again, the car was gone, and I was afraid.\n\nWhy the hell hadn't I said hello?\n\nWe didn't speak for the longest time. A long train of motorcycles paraded past us down the street. When I looked back at Jane, her eyes were fixed upon a dead patch of grass. \"What's up?\" I said finally.\n\n\"I don't know. I was thinking about Mr. Hovnanian.\" She tried to laugh a little, embarrassed. \"Don't ask me why. Remember that stuff about the eclipse?\"\n\n\"Sure.\"\n\nIt wasn't so long ago: our tenth-grade class sitting before the TV, watching the total eclipse sweeping across North America. It was terribly, horribly beautiful, the quality of that darkness\u2014birds falling silent, streetlights trembling on. Liquidy fires jetted around the rim. Then, just when the sky went dark and the corona shivered, Mr. Hovnanian switched off the set. \"What makes you think that wasn't a hoax? How did you know that that wasn't a computer image, fabricated to drum up ratings?\"\n\nFor weeks Jane seemed to take it personally, ineffably sad about the whole matter.\n\nI said, \"Is something wrong? You don't seem like yourself.\"\n\n\"Why am I thinking of tenth grade, for God's sake? What's gotten into me?\"\n\nWe stared at a fallen tangelo while the heat crept into our scalps.\n\n***\n\nI couldn't sleep. I felt something simmering in my body, a slow cooking, spreading up through the stem of my torso, then prickling, exploding in my throat like salad oil. I wanted to molt, I wanted to cut away the baggage of my skin. I kicked the wet covers off the bed, threw on some clothes, and left the house. I was going to walk it off. I was walking through developments, through people's backyards in the dark, over culverts, canals, retention basins. Hours had passed. I passed airport runways with their raucous blue lights, sanitation plants vast as cities, signs fizzing and sparking, arrows pointing in all directions. Two towns over, the boat factory was working overtime, and the junky hot smell of plastic lingered in the atmosphere. A storm threatened from the Everglades, then receded, pushing the humidity even higher. I took off my shirt and roped it around my waist. I decided to walk and walk, possibly to the Keys, possibly to the Card Sound Bridge, until I finally got rid of this feeling.\n\nHours later I was standing in William's front yard. I expected the lawn to be overgrown, ruined, bits of scale and dollarweed eating at the turf. But no. It looked even better than before. Moist, lush. I knew it: William had found another Lawnboy. I had lost him for good. I fumbled for some broken shells and started tossing them, one after another, at the glass of the window: ping ping ping ping.\n\nWas I ready to give myself over to desire?\n\nI knew myself too well: hyped up, charged, I'd lose everything. I saw myself fretting, always looking for something other, something better, something outside myself. I saw myself utterly alone in the world, a gleaming wasp inside a bright orange hive, alone with my anguish and raging hot need, and who'd be there to still me?\n\nWas there anyone else at that moment who knew the pressure and potential of changing everything? I raised my hand and linked myself up with him, the longing, imaginary one, then pressed forward, my fingertip upon the doorbell, standing on William's front porch, waiting.\n\n# Chapter 4\n\nMy childhood suitcase was a rude, bulky thing, its surface marred by chipping decals\u2014Kennedy Space Center, South of the Border, Sunken Gardens\u2014which recalled family travels too heartbreaking to ponder. The packing proved harder than I'd imagined: what to take, what to leave behind. Forever? Was that the way it was going to be? My stomach simmered and groaned. I wanted to get it over with before they knew I was leaving.\n\nDown the hall my mother was cooking pot roast. I thought about the signifying implications of pot roast: convention, structure, clean right living.\n\n\"Get ready for dinner,\" my father said.\n\nHe'd been standing at my door the entire time. I looked to the open window, thinking to toss the suitcase onto the hedge. Don't let him get to you. I turned my back to him, pitching in some socks and shirts. He still had the power to scare me sometimes, and he knew it; it incensed him. Was he going to hit me again?\n\n\"Beautiful night,\" he said, stepping into the room. He picked up a tiny statue of Grover Cleveland from my Hall of Presidents\u2014something he'd given me for my tenth birthday. He'd stared at its bloated gaze as if he'd never seen it before.\n\n\"You're right,\" I replied.\n\n\"Front's sweeping down from the Plains tomorrow.\"\n\n\"Oh yeah?\"\n\n\"Highs in the eighties with a light north breeze.\"\n\nTears burned in my eyes. Outside a Good Humor truck careened down the street. Don't you understand what's happening? I wanted to yell.\n\nA complex expression drifted across his face: defiance, jealousy, regret, fatigue. He left for the kitchen. Seconds later liquid plashed inside a glass. \"On the rocks,\" I said, toasting my wall.\n\nAn ice cube popped, cracked. I waited until they were safely ensconced in their meal before I left through the mud room. Bye, Mom, I whispered. Bye, Dad. I loved you once. I hurried back to my room and shoved Grover Cleveland deep inside my pocket.\n\n***\n\nTraffic roared past me on the street. Palms thrashed in the wind. I felt criminal and delicious, imagining a stocking cap pulled down over my face. I half expected the police to pull up, to ask me questions, to search through my belongings before taking me back home. Dogs barked as I passed. Lights flipped on in the windows. I pictured myself through someone else's eyes: a hushed, hungry boy, feet flying in the dark, switching his leaden suitcase from hand to hand.\n\nI knocked twice on William's door. I watched him through the sidelight, his gestures frantic and distorted through the gold bubbled glass. He hurried from table to table, glancing under papers, feeling behind furniture. To my right, a white wicker chair rotted behind the bushes.\n\n\"Can I help you?\" A lady in a paisley dressing gown hosed off a lavender Mercedes beneath the floodlights next door. She stepped toward me. Somewhere a bird\u2014parakeet?\u2014shrieked.\n\n\"No, I'm just\u2014\"\n\n\"Run along now,\" she gestured. \"He hates solicitors.\"\n\nWilliam's door flew open, almost knocking me off balance. \"Evan,\" he said.\n\n\"I'm taking you up on your offer.\"\n\nHe glanced down at my suitcase, a swab of shaving cream beneath his left ear. What offer? he might have said. He stood there in a pink polo shirt and tight, tight 501s. A splash too much cologne.\n\n\"I know this seems hasty,\" I laughed.\n\n\"Yes, but\u2014\"\n\n\"Didn't you say I always had a place?\"\n\n\"I wish you'd called,\" he said, his eyes vaguely stricken.\n\nMy scalp felt tight. I kissed him. Awkwardly, I missed his mouth, pecking his warm whiskered neck. I tried again, inhaling his shirt this time: crushed plants, limes, beer foam. I turned my cheek against his chest.\n\n\"Listen,\" he said. \"I have to find my keys. I've been looking for a half hour.\"\n\nWe stepped through the front door. \"You'll find them,\" I murmured. I glanced down at a pulled loop in the aqua rug. An emptiness, wide as a body, opened inside me. I saw myself falling inside the body, reeling, tumbling, swallowed up by it.\n\n\"Welcome,\" he said, lips on my mouth. That was it, that was all I needed. Heat saturated my face, trickling down my spine to my groin, toes. I felt larger, warmer. Heaven. My leg muscles glowed.\n\n\"Jesus,\" I mumbled.\n\n\"Are you okay?\" he said suddenly.\n\n\"Mmmm, hallelujah.\"\n\n\"You look a little crazed.\" His forehead creased. He tested the weight of my suitcase. \"God, what's in here?\"\n\nI tried to recover myself. \"Stuff.\"\n\n\"Stuff?\"\n\n\"You know, books, clothes\u2014\" I shrugged. \"Scary, huh?\"\n\n\"Sorry. If only I could find those keys.\"\n\nWe stared at each other until he couldn't hold my gaze.\n\n\"Let's see what you think of this,\" he said finally.\n\nI followed him down the hall. Claws scratched behind a wooden door, and William opened it to let the Dobermans\u2014Pedro and Mrs. Fox\u2014lick wet tracks on my shirt. He gestured to the utility room, a converted laundry, with smelly green carpeting and water-stained magazines along the walls. I glanced at the titles: Bee Culture, Modern Liturgy, Physique Pictorial, Tom of Finland's Loggers. Beside the futon an uncapped pipe jutted through the floor. I stared at him. \"Not here,\" I said.\n\nHis eyes brightened, then dimmed. He rubbed at his forearm, as if warming himself.\n\nI reached out for his shoulder. \"But I wanted to sleep with you.\"\n\nHe wagged his head. \"Listen, we'll talk about it later. I have to leave for a few hours. There's a zoning-board meeting at seven.\"\n\n\"You're leaving now?\"\n\n\"Have to. I don't want to see those condos approved. Once that starts the whole neighborhood goes.\"\n\n\"But what about your keys?\"\n\nHe glanced at his watch. \"I'll walk. Jog,\" he said. \"There's pancake mix in the fridge, sausages on the stove. Be back by ten.\" And then he pressed a fist into my rib cage.\n\nI sat on the futon with my head in my hands. The dogs drooled clear strings upon my sneakers. \"No biscuit,\" I said. \"No biscuit.\"\n\n***\n\nI hurried about the kitchen, opening cabinets, drawers. As my cooking repertoire consisted of sandwiches, spaghetti, and instant chilled puddings, I was in trouble. Still, I gathered what I could find\u2014blue mesquite chips, party dip, cut vegetables\u2014and arranged them on the coffee table. I clipped some ginger lilies from the garden and put on an old-fashioned party hat, a silver cone with an elastic string. It wasn't Joe's Stone Crab, but at least it was something. I sat with my feet up on the coffee table, waiting.\n\nBy ten o'clock my eyes felt the weight of their lids. By ten-forty-five they felt scratchy, hot in their sockets. My week had been wearing, and I stretched myself out on the couch, too rattled for sleep. The next thing I knew the front doorknob was shaking. Burglars? I bolted upright on the couch, panting. 12:45 a.m.\n\n\"What's this?\" he said.\n\nI looked up at him, rubbing the silt from my eyes. The silver cone lay by my side, dented.\n\nHe gave an appreciative smile. \"A little shindig, a little wingding.\"\n\n\"What took you so long?\"\n\n\"Ugh,\" he said. He undid his pink polo shirt and flopped into the sofa beside me. \"Relentless. This woman, some ex-colonel from the military, actually insisted that condos were going to upgrade the neighborhood. She monopolized the floor for twenty minutes. Imagine\u2014cheap townhouses trucked in from Indiana. Forget about mangroves. Forget about roseate spoonbills. 'You can't stop progress,'\" he mimicked.\n\n\"Payoff,\" I mumbled.\n\nHe regarded the clenched skin around my eyes. He glanced away, then looked back at me. \"I'm sorry,\" he said, gripping my left knee. \"Look at all this. How nice of you. You went to so much trouble.\" He stood up suddenly, thrust his hands in his pockets. His head pivoted toward the kitchen. \"So everything's okay with you?\"\n\nMy heart picked up pace. I felt its beating inside my cheeks, teeth, the thickness of my tongue. I tried it again. \"I thought we'd sleep together tonight.\"\n\n\"Not tonight,\" he said, yawning good-naturedly. \"I have to be up at five-thirty tomorrow. Taping.\"\n\nMy face flushed upward from my neck. \"I'm not talking about sex.\"\n\n\"The mattress's broken. I thought I'd sleep out here tonight,\" he said, and gestured at the sofa.\n\nWhat more could I ask for? I stood in the living room while he retrieved a pillow and sheets from the linen closet. My thoughts fractured, as if I'd taken one too many cold tablets in a row. I studied the narrow width of the sofa: no way could we fit together in that space.\n\n\"Nighty night,\" he said, and kissed me.\n\n\"Do you mind if I sleep on the floor?\" I pointed to the carpet alongside the sofa.\n\nHe looked concerned now, his tanned forehead shining. \"Of course not. Are you sure you're going to be comfortable down there?\"\n\nI started stripping down to my jockey shorts. \"No problem.\"\n\n\"Want a pillow?\" he said, offering me his own.\n\n\"No thanks.\"\n\nSoon William was snoring in the dark room. A heaviness settled deep inside my marrow. Aloneness, I said to myself. Aloneness. I reached up and rested my palm against the warm ridges of his stomach, fingers just touching his waistband for energy, some sign of life. My hope, my ring of fire. The earth stood still, frozen. My torch, my song of gladness. I took my hand away, sighed. Headlights flashed on the Toulouse-Lautrecs on the wall.\n\n***\n\nWeeks passed. It was summer. Now that I'd graduated, I had some time on my hands. While William went to work, I immersed myself in various projects, the largest of which involved chopping down the singed Australian pines. I wanted to fix things up. Luckily, William didn't force me to look for work, or to write to any of those colleges that had offered me scholarships, but I was grateful for the absence of pressure, for I needed some time to figure out what I wanted from my life. Doctor, architect, horticulturalist, weatherman, Django Reinhardt scholar\u2014I still could be anything.\n\nYears in the future, I'd look back on this time as our happiest.\n\nHow strange, though, to be entirely dependent on someone. Strange to feel absolutely no power next to him. Sometimes I'd be whacked across the brow with the uncanny realization that I had but $425.69 in my savings account. It was nothing, barely enough to cover an apartment's security deposit. All he needed to do was to kick me out, and I'd be out on my ass on the streets. Hustling, shoplifting, shooting up drugs. But I couldn't give everything over to him. I couldn't let myself believe that I was smaller than he was, a petty moon orbiting some planet, even though he was footing the bills. I had the right to make some decisions. It was just as much my house as it was his.\n\nBut I wanted to feel more enthusiasm for my new life. My happiness and sadness seemed to live side by side, like roots intertwined, feeding and depleting each other. What had I lost? Whatever happened to that purer emotion, that purer joy: a hot yellow beam knocking me off my feet? It had departed now, for good it seemed. I walked to the window one day and saw three high school boys in huge, sloppy pants roving down the street, laughing and rebellious, throwing a hockey puck, smoking stolen Marlboros. They appeared to be so much younger than I was, with so much possibility ahead, and I wanted to walk with them, wanted them to like me, to call me by my name, if only for an hour.\n\nOne afternoon, I sat inside the moldy den. I was flipping through an overdue library book\u2014Sir Isaac Newton's Opticks\u2014when I heard a timid gurgling from the floor. I looked at the pipe. I stood, then peered down inside it. A sheen of greased water. All the houses on Avenida Bayamo had the same pipe inside\u2014a washing-machine hookup. My father had removed ours years ago when Peter had moved into the room, putting up some turquoise beaded curtains. Still, I often dreamt of our pipe. At four, I dropped my favorite trinket, a plastic blood-red fingernail, inside it. Once I wore it over the tip of my finger, extending my hand as if elegant, admiring it before my father and brother. They laughed at me, a drunken puppet, encouraging me to show off for them, before they stopped, uncomfortable.\n\n\"Stop being a sissy,\" my father said.\n\n***\n\nWe drove down the street, silent. It was a glorious wet night, masses of hot and cold air bumping up against each other. Our tires flung arcs of water onto the grass, the sidewalk. Toward the north, the clouds were underlit with a soft, baby-pink light.\n\nJust outside Publix stood a group of people waiting for the rain to stop. One old lady in a transparent raincoat kept looking at us, not with judgment or curiosity, but with the oddest benevolence. I dislodged a shopping cart from the train and kissed William, just once, on the cheek, in celebration. The world seemed dangerous and hopeful all at once.\n\nHe stiffened. His body language changed, its dark energies curling inward. The clouds had shifted, the rain stopped, and a bronze moon\u2014tropical, haloed\u2014hovered above the palms.\n\nI said, \"Are you okay?\"\n\nHis back straightened. He walked through the automatic doors, eyes fixed on the shining banks of limes. He picked one up and rotated his thumb across its green, pocked surface. It might have been a beautiful grenade in his palm.\n\n\"Is something wrong?\"\n\nHe glanced at me as if I'd hurled a glassful of water at him.\n\nHe inched the cart forward and picked up a box of pineapple gelatin, pretending to study its contents. His Adam's apple was hot, gleaming. He held up his hand: he didn't want to talk to me.\n\n\"William?\"\n\n\"There's a time and a place.\"\n\n\"I don't understand. Nobody saw us.\"\n\nHe was frowning now. \"Somebody saw,\" he said.\n\n\"But\u2014\"\n\n\"Listen\u2014\"\n\n\"Is this our first fight?\" I asked, more baffled than I'd intended. Was it coming this soon\u2014only weeks into our living together?\n\nHe stopped the cart halfway down the aisle. \"Evan,\" he said. \"Now listen to me.\" His voice was kinder now, a whisper. \"You never know what these people'll do. They might look friendly enough, but all it takes is one false move.\"\n\nI considered his statement. Only a few days ago I'd read about two men in Dallas who'd been scorched with blowtorches after they'd been loitering in a park. Certainly these things happened in the world, but they were rare. Or were they?\n\n\"But she liked us.\" I nodded toward the old lady, who'd somehow wandered back into the store. She, too, was picking up limes. \"She wasn't upset.\"\n\n\"No, she wasn't,\" he agreed. \"But maybe her husband would be, or maybe her son. Her grandson, for instance, might be a skinhead. I'm just telling you, you can't be too careful.\"\n\nI sighed. I sensed she was listening to us. Her eyelids grew heavy as if she were beginning to feel depressed.\n\n\"Anyway,\" he said, his voice softening. \"We aren't the average couple. Especially in these parts.\"\n\n\"What do you mean?\"\n\n\"We're not your average sissy interior-decorator couple. I mean think about it\u2014an older man, a younger man. We're a threat. We push their buttons. We have to watch ourselves every step of the way.\"\n\nI didn't want to watch myself every step of the way. I wanted to be deeper inside my life. I walked ahead, wandering about the freezers, as he straggled by the olive jars, the sardine tins. His anger seethed, completely out of proportion to the matter at hand. What was his deal? I'd only kissed him, for God's sake. If he wanted to be upset, I could pull down his zipper with my teeth, and take him deep inside my mouth, right here, right now, in front of everyone in the store. Then what would he do?\n\nMy throat pulsed. What was I doing with him, anyway? Could I be making a mistake? For all I knew there were already warrants out for his arrest. I walked ahead, my rage heating up my face to the burning point. Would I catch on fire? A cold drop of sweat rolled down my temple. I thought about his wizened shoulders. I thought about his bald spot glistening beneath the glare of hooded lights. William, William, I whispered. How I want to punch a spoon deep into the valley of your back.\n\n\"There you are,\" he said brightly.\n\nHe nudged me with the shopping cart, and I flinched, startled by the intrusion on my thoughts. \"Where were you?\"\n\n\"Look.\" Inside the cart were three pints of strawberries, taco shells, a half-dozen chocolate bars\u2014my favorite foods.\n\n\"Thank you.\"\n\nA mute smiling panic took hold of his face. \"I bought these for you.\"\n\n\"I know, I know.\"\n\nOutside in the car, we waited for the light to turn green. A man in the Camaro beside us banged on his dashboard like a conga drum, eyes gleaming as if he were high. He was singing now. Salsa music cascaded from his speakers. William turned to me then and kissed me hard, a bright dense charge coursing through my nerves. The other cars shot forward. The roads steamed beneath the streetlights. How quickly things could change, the world sparkling, full of rollicking possibility.\n\n***\n\nThe storms continued through the night. An electric smell hovered in the air, smoke rising from the trees. I rolled onto my side and pressed my lips into the warmth of his back. His skin smelled of rainwater, ferns. I thought: Everything has been leading here. All those nights spent alone, all those nights listening to my parents' silence\u2014all were in payment for this. I'll never be happier.\n\n# Chapter 5\n\nWhen I was finally adjusting to my new life, when I'd started sleeping eight hours straight without waking at 4:00 a.m. to the burning pit of my stomach, I saw her. The night was hot, sodden. We'd just watched The Bride of Frankenstein, and William was in the bathroom, flossing, then rinsing with water and two droplets of grapefruit-seed extract. I sat in the living room, my hands folded on my lap, feeling at peace, feeling entirely and utterly at peace, when I heard the familiar scuff of shoes on the pavement outside.\n\nThe curtains shivered in the breeze. I thought of Dr. Frankenstein's creation, lonesome and yearning, lured to the blind man's cottage by the plaintive call of his violin. Something scrabbled at my stomach. I had the distinct feeling I was being watched, so I stepped toward the window, closed the curtain.\n\nI might have been dreaming. Ursula was standing on the sidewalk, hands in her pockets, waiting for me. Drizzle streaked her orange windbreaker. I stopped dead, then switched off the light.\n\n\"Evan?\" she called.\n\nI froze, hoping she couldn't see me. My pulse thudded in my head. Was she really calling my name, or had I just imagined it?\n\nThen I got it: they were waiting for us\u2014my mother, my father, the police. I looked for the squad cars, their engines running, headlights off beneath the trees. They'd shoot him, I knew it. A clean white wound, a pucker, right through the center of his forehead, as I stood off to the sidelines, doubled over in shock.\n\nThen I looked closer. No cars, no police. My mother.\n\n\"Evan? I want to talk to you.\"\n\nHer voice was sweet, unbearable. I wanted her to come inside. I wanted her to walk away. Her presence brought back everything I'd driven out of my head\u2014that I'd given up a past and a future; that I'd gravely disappointed somebody close to me; that I'd been truly, genuinely missed. I felt it in my gut, an icy deadening ache. Who was she to tell about her sadness? Who was she to talk with about Peter, who hadn't come home for so many Christmases?\n\n\"Why's it dark in here?\" said William, walking into the room.\n\nI looked at his domed forehead, dumbstruck.\n\n\"Somebody out there?\" He edged toward the window before I could stop him. He gazed across the glittering lawn, the empty floodlit street. His eyes registered nothing, and he turned to me. \"Let's go to sleep, kiddo.\"\n\nI stayed before the window while William shooed the dogs to their respective beds. My heart was breaking in two. And then I saw her again, this time her head low, her hair unbraided on her shoulders as she walked down the street to the house of my childhood.\n\n***\n\nI lay on the floor beside the sofa, curled on my side, listening to William falling asleep. I often took refuge in that sound as I let myself go, synchronizing our breathing. Tonight, though, it distracted me. I lay with my eyes open, dwelling upon the singing tires from the distant turnpike. I tried to fix upon the sound of an individual tractor trailer. What was it carrying, where on earth could it be headed at this hour? I pictured star-fruit to Atlanta, contact lenses to Key Largo, walking catfish to Montreal.\n\nWas it Peter's leaving that had changed her? He'd always been her favorite. Not that she'd admitted it, but I knew how she felt: her eyes glistened whenever she talked about him. He was the first, born after two miscarriages\u2014a terrible labor that almost killed her. She named him after an old boyfriend, an amateur gardener and lineman who'd died on the job. For years afterward she'd stumbled from one thing to the next\u2014secretary, bookkeeper, restaurant hostess. She was even a lounge singer for a time. Still, as much as she'd worked to cheer herself up, she couldn't get Peter out of her mind. How he'd fallen against the transmission tower. Hanging, lifeless in his safety belt. Turning like the hands of a clock. An aerialist, a four-pointed star.\n\nShe'd never loved my father as much as she wanted to. She'd never felt that \"gut-level charge,\" though when she first saw him, in a shiny red jacket, surrounded by the prettiest girls at the party, she'd convinced herself that he was the one. \"I want him,\" she'd said to her friend Marilyn. \"I want him to notice me.\" She was tired, after all, of so much loss, the deaths of Peter and her clinging mother. My father had just been hired in the university's chemistry department, and all she'd wanted was to rest.\n\nHer beauty faded after the wedding. You could see it happening in their pictures, when she traded in her makeup for jeans and workshirts, as if she'd told herself that this was it\u2014the end of possibility. \"Where's the girl I married?\" Sid teased, hugging her from behind. \"Where's the beautiful girl I was so proud of?\" He couldn't get to her, though. After all, she'd gotten what she'd wanted. And now the only thing left to do was to have a baby.\n\nShe thrived once Peter was born. He was long, gangly, with tufts of dark hair, overly large ears, and a curious bump on his temple where the forceps had clamped. Eight pounds, seven ounces. Handsome, she knew he was going to be handsome. And it felt good to say that name again, Peter, as if her boyfriend had inhabited her baby to keep her company once again, to bring her back to life.\n\nFor five years he was the most important thing to her. She took him to the supermarket, to her gynecological appointments, to lunch at the Silver Wheel with her friend, Astrid Muth. She wouldn't even consider a babysitter, wouldn't let Peter for one minute out of her sight, convinced that mishap was lurking just around the corner. Peter grew anxious, cranky when she wasn't around. On his first day of kindergarten, after promising he'd never cross the street without the assistance of the safety guard, he strayed far behind the others, fearful, shy, only to find that the fifth-grader in charge was already off duty. He trudged up the wrong side of Avenida Bayamo. He sat down on the curb across from the house and waited\u2014ten, twenty minutes\u2014before Mrs. Feldman, hanging up violet socks on her clothes tree, spotted him.\n\n\"What's the matter, Pete?\"\n\nHe was sobbing now. \"The safety went away. I can't cross the street.\"\n\nMrs. Feldman glanced in both directions and crossed over to him. There wasn't a single car in sight.\n\n\"Do you want me to help you cross the street?\"\n\nPeter gazed up at Mrs. Feldman. Her body was huge, luminous, the size of a planet. He swiped at his nose and nodded.\n\n\"Okay, honey. Look both ways. And take my hand.\"\n\nHe took her hand. Once on our yard he skittered across the grass to the back door, forgetting to thank Mrs. Feldman.\n\n\"You tell your mother to watch you,\" she called.\n\n\"Okay,\" Peter said. \"Bye bye.\"\n\nAnd years later I was born, and things changed once again.\n\n***\n\nWe'd passed the critical point. It seemed that we'd crossed some mine-strewn landscape, that if we weren't meant to be together we'd have found out by now. I wanted to take William out to dinner, to give him a card, to buy him a pair of boxer shorts patterned with pink cocktail glasses and swizzle sticks, but I stopped myself, as if doing so would only have called attention to the possibility of conflict. Remarkably, there was no conflict between us. We were learning to live together, falling into our respective routines as all companions do. What was the point of deeming that an accomplishment?\n\nAnd yet there was the issue of the bed. Were we becoming too comfortable with the situation? Were we too afraid to wrest ourselves out of the familiar, with William on the couch, and me on the floor with my blankets, sheets, and pillows? We certainly had sex\u2014good, ravenous sex, though it was less than I'd expected. Once a week? Once every two? Was something wrong? One evening I'd even convinced myself I preferred sleeping this way to having his warm body curled around mine, grasping me, taking me with him as he turned from side to side. After all, for most of my life I'd slept alone, liking it, imagining it the only possibility for a good night's sleep.\n\nAnd yet?\n\nIt was time to do what I'd been putting off. One afternoon, while William was away at work and I was home alone, I walked into the master bedroom to check out the mattress. Couldn't I fix it myself? I sat on the edge of the bed, bouncing slightly, staring at the framed photos of Lorna and Poppy. They covered every square inch of the dresser\u2014images of his former life, which unsettled me sometimes. Where were the pictures of me? I looked at Lorna. It was hard to imagine he'd ever been married to her, with her expensively treated hair and plucked brows. Even more unsettling was Poppy, whom I'd never met, and never cared to. Though we were close in age, we'd probably never get along. I studied her tilted, heart-shaped face, the precise, calculated sweep of hair across her forehead, knowing she would have all but ignored me in high school because I wasn't interesting or fabulous enough.\n\nOr maybe I wasn't giving her a chance.\n\nI bounced again on the bed and decided to check out the underside. I crawled upon my hands and knees, left eye close to the floor. Dust, nickels, chewing-gum wrappers, used rubbers\u2014it was enough to make me lug the vacuum from the closet. In about ten seconds I discovered the culprit: a dislodged support board. I hefted up the mattress with one arm, slipped in the board with the other, and that was that. I laughed. I started laughing so hard that I couldn't stop. It was strange to think that we'd exiled ourselves to the living room for this.\n\nI lay on the restored bed, basking in the light of my accomplishment. The palms outside the window threw golden shadows on the wall, weaving. I felt so good that I unwittingly pulled down my pants to my ankles, palming the head of my dick, not intending to come, of course, but consoled by the ministrations of my hand. I realized how much I'd missed jerking off, how I'd always thought it an essential activity, a primary component in the development of one's imaginative life.\n\nGlancing up, I saw William standing in the doorway.\n\nI had no idea how long he'd been watching me.\n\nImmediately I wriggled up my pants and sat up, my face hot with shame. I was too unnerved to laugh at myself.\n\n\"What are you doing home?\" It was only four o'clock; lately he hadn't been coming in the door until nine.\n\nHe looked harried, uncomfortable. He walked over to the window and pulled open the drapes, flooding the room with a yellow light.\n\n\"I fixed the mattress,\" I declared. My voice sounded starved, panicked, as if I were trying too hard to please. What was the matter with me? He wasn't my father.\n\nHe flopped deliberately on the bed.\n\nWe were silent together. I sat down beside him, diminished and disabled, entertaining the possibility of him spanking me across his lap. I'd pull in my breath, eyes squeezed tight, jerking at the sting of his slaps. The heat swarming in my butt. Afterwards, he'd hold me in his lap, wipe away my tears, tell me he'd be kinder to me from now on, a better person. I took a photograph of us in my mind, a tableau.\n\n\"So you're home,\" I said, venturing an observation.\n\n\"Yeah,\" he said. He leaned back and crossed his arms behind his head. He pulled in his lips. \"We had a bomb scare at the station.\"\n\n\"A bomb scare?\"\n\n\"A bomb scare. Terrorists. They sent everyone home until further notice. Station's off the air. Turn on the TV.\"\n\nI reached for the remote. The picture was snowy, the audio a harsh scratch. I shuddered. \"God,\" I said, picking at my lip. Why hadn't I recognized that my lip was chapped?\n\nWilliam closed his eyes. I rested beside him on the bed, wanting to recapture the uncharged emptiness of everyday life. How could I calm down? How would I interpret this moment, years in the future: masturbation, spanking fantasy, terrorism, bomb threat? And yet a part of me liked the unsettling rush: I wanted to have sex. I wanted to climb on top of William's prone body and fuck him, savagely, with gritted teeth, like an animal, though I didn't think I'd get away with it.\n\n\"I worked hard all day,\" I said suddenly. \"I vacuumed the living room, I scoured the rust stains from the kitchen sink, I did three loads of laundry, I walked the dogs in the storm\u2014\" The blood was beating in my ears. Was I more upset than I knew? \"And that was all today. Would you like me to show you what I did?\"\n\nCautiously he opened his left eye. \"I know you work hard,\" he said, \"I'm sorry,\" and offered me a sad, depleted smile.\n\nInside of me a door creaked open. I felt vindicated\u2014yet exposed and repentant. We stood. We walked into the kitchen where we fixed ourselves an ample, pleasant dinner: rice wine, peanut sauce, stir-fry. That night we still slept in our usual settings, the repaired bed glittering in my mind like some remote island.\n\n# Chapter 6\n\nSometimes I worried that I wasn't a complete person, that I couldn't label myself. What if I was just a composite of everyone who'd passed through my life\u2014strangers, family, friends\u2014all of whom had inhabited me, taking over my thoughts and gestures before departing, leaving me defenseless? Looking back, I saw how I moved not in a straightahead line, but in lopsided, parabolic circles. I pushed myself out, I reigned myself in. I craved sex, I didn't crave sex. I wanted to transgress, I wanted to conform. I wanted to be brilliant, I wanted to be a mindless fool.\n\nWas I becoming myself? Or was I stalled, trapped before some rust-clenched gate while everyone else was getting somewhere?\n\n***\n\nThe night before William's trip to Key West, we finally ended up together in the master bedroom. I didn't know quite how it happened, but that seemed to be beside the point. I lay beside him, my chest flooding with gratitude and energy. I held him closely under the hot tent of the covers. His body felt foreign, huge to me. Hold me back, I thought. Hold me so tight that it hurts. Keep close now. Stay, stay, stay.\n\nThe next morning we waited at the front window, watching for Lilo Patrick, the reporter with whom he was working on the assignment, to pull up in her yellow Toyota. He stood beside me, jittery, shifting his weight from foot to foot. He'd been anxious about the project\u2014something about the recent decline of Key West\u2014the incursion of national chains, the dearth of affordable housing, the dead-end alcoholic culture\u2014none of which, William believed, established a single point of view. My pulse quickened in the core of my chest. He was to be gone for the entire week. I didn't want him to leave, especially now that we'd finally become so comfortable with each other.\n\nAt two past eleven\u2014the designated time\u2014Lilo pulled up front, her car engine purring, finely tuned as a sewing machine. I motioned to walk through the front door, until he stopped me, asked me to stay put.\n\n\"But why?\" And then I remembered that he didn't want Lilo or anyone at the station to know about me, so much so that I was not to answer the phone, but to let the machine take the message. \"It's a high-powered job,\" he'd once explained. \"I mean, I don't think I'd get fired, but you never know. I don't want to chance it.\"\n\nI looked downward. A vault of emptiness opened inside me. \"You better go,\" I said miserably.\n\nHe kissed me, before opening the door. The dogs stepped backward in the foyer, already lonesome, already resigned. Soon enough they'd start longing for him, cocking their heads at any sound of footsteps outside. \"You'll take good care of the dogs?\" he asked. \"You'll give them walks?\"\n\n\"I'll take good care of the dogs,\" I singsonged. \"I'll give them walks.\"\n\nHe looked at me curiously. \"You'll behave yourself?\"\n\nI tried to connect with his gaze, but couldn't. His words seemed laced with all sorts of innuendo and danger. Did he expect me to spend the entire week locked inside with cartoons and a full refrigerator?\n\n\"I'll miss you,\" he said dully.\n\n\"I'll miss you, too.\"\n\n\"Call you tonight,\" he said. \"Take good care of yourself.\"\n\n\"You too.\"\n\nHe shrugged. \"I'm off.\"\n\nI lay on the couch, both comforted and alarmed. The house seemed oddly centered: all the various pieces of furniture in their proper place, the two Dobermans lying like sleeping Sphinxes beside me. It occurred to me, when I thought about all I'd been through, the collisions with my parents, my separation from Peter, that I was lucky. Once again I reminded myself that I could have ended up on the streets, strung out, penniless, hacked to death.\n\nThen, for whatever reason, I remembered something else about last night.\n\n\"I'm not going to see you for seven days,\" I'd whispered.\n\nWe were lying in bed, in the still seconds after the lamp had been switched off. Across the street an animal\u2014dog? raccoon?\u2014was rooting through somebody's trash cans.\n\n\"Eight days,\" he corrected.\n\nI thought about eight whole days by myself. I reached over for him, pressed my palm upon his taut stomach. I waited. Nothing. Then I waited longer.\n\n\"I thought we could make love or something,\" I said.\n\nMy voice sounded tentative, vulnerable. I couldn't stand the sound of it. I couldn't stand the way I had to ask for it, begging, as if it cost him. Already, in memory, I could taste him, like blood, like steel, and now that I'd had him, it wasn't enough. I needed more, even though I knew my wanting was going to do me in someday.\n\nHe heaved a tired sigh. I already knew his answer.\n\nI turned away, moving to the farthest edge of the bed until my face pressed up against the wall.\n\n***\n\nI plodded through the arcade in my workboots. I kept my expression remote and aloof as if to indicate that I wasn't new at this. But I was dying inside. I wasn't ugly. I wasn't desperate, hopeless, dumb. Adult book stores: weren't they meant for those who led secret lives, who hated themselves? Who else would put up with nasty attendants, filth, that fruity metallic smell? But I'd walked the entire seven miles, not even bothered by the sand in my sneakers, the blisters on my feet.\n\nI kept threading down the halls, if only because I couldn't stand still. I didn't want anyone to greet me, or touch me, or pay me the slightest bit of attention. But I liked being in the thick of it, little dramas of pursuit and rejection crackling around me like fires. Gathering my nerve, I glanced at the faces of the\u2014customers? There was an old man\u2014tall, slouchy, wedding band tight on his finger\u2014who might have been on his way home from choir practice. There was a coke dealer, I presumed, with vibrant blue eyes and sunwrecked hair. There was a businessman, sleeves still crisp from the cleaners; a dark-skinned guy with tiny red rings in his ear. All told there were nearly a dozen\u2014fat, skinny, old, young, rich, poor\u2014none with anything in common, but for a melancholy expression, which clearly masked what I felt too: a longing for escape, otherness, transgression, connection.\n\nI calmed down soon enough, fed my coins to the soda machine: a lighted box with a smeared, eye-shaped logo. The soda tasted delicious, an abrasive splash against the back of my throat. Actually, most of these fellows seemed to be having a decent enough time. Doors kept banging shut, then opening, admitting and releasing the hungry. One trucker-type practically danced a jig down the hall. Another strolled past with a look of dumb wonder on his face. \"Oh, what the hell,\" I said to the graffitied walls. It was time to stop thinking so much.\n\nStill, I imagined William staring down at me through a hole in the ceiling, cataloging my gestures. I imagined his footsteps clomping across the parking lot, his husky brusque voice taking over the sound system. The music faded to static. \"What are you doing here?\" he said, grabbing me by the collar. \"I can't believe you'd punish me like this.\"\n\nFoolish thoughts. Foolish.\n\nI found an empty booth and bolted the door. I panted slightly, dazed. Was I going to be sick? It still held the scents of its last inhabitant\u2014perspiration, Right Guard, spilled semen, body heat. I rested my head on my knees. I glanced upward. Al Parker floated across the flickering screen, legs shining like the flanks of some heavenly animal. My vision blackened for a moment. I loved Al Parker. I concentrated on his exuberant brown beard, his hard vascular chest, his rakish Semitic nose. I concentrated on his dick, an enormous fleshy thing that wagged when he walked, with a jaunty personality all its own. But there was more to him than that. It wasn't just his body, or his personal warmth, or his casual, fluent masculinity. It was his very persona, which told us that sex was fun, that there was a wide, wide world out there, more complicated and various than we'd ever assumed. He'd never call anybody else a faggot because he himself was a faggot, and he felt just fine about that, thank you. Unfortunately, he wasn't here anymore: another soul, lost, like half the world, to AIDS. At least his image still quivered with life. I looked at the screen, then down at myself. There was Al, there was me.\n\nI closed my eyes. When I opened them, Al was fucking a tall, rangy kid with a gap between his teeth. Together they gyrated over a workbench. The camera panned the kid, surveying every square inch of his skin, grazing past a deep violet mark on his wrist. A burn, a bruise, or what I feared it was? His hair\u2014parted down the middle, shaggy over the ears\u2014seemed right out of 1979, just before protected sex had become an issue. I hated to think the kid had done this while sick, all for the sake of making a few fast bucks. But who would have known then? Who would have thought the world was on the brink of such threat?\n\nI went soft in my palm. I wanted to be home. I wanted its banality, its routines, the anonymous sanctuary of its dull gray rooms. Already I thought of the dogs barking as I worked the key into the lock. I thought of their tails whipping my legs, the weight of their snouts in my palms. And their eyes\u2014how they closed them in gratitude whenever I stroked their heads. Time to get out of here. I pulled up my pants, my right leg prickly with sleep. Home: odd to think of it that way, but that's what it was now, for better or for worse.\n\nI opened the door. Before me stood the most beautiful man I'd ever seen.\n\nWe looked at each other, quizzical. An awful thrill went through my stomach. Had we known each other once? \"Evan,\" he might have said.\n\nHe stood there longer than was exactly comfortable, as if he, too, felt paralyzed. Though he was altogether perfect, it was his eyes that drew me to him, large brown eyes that revealed a soulfulness, an abundance of spirit, with just enough wryness thrown in for good measure. There wasn't a wall between him and the world. Everything, I imagined\u2014praise, insult, injury\u2014registered on his psychic screen. I glanced about the booth as he walked down the hall. What should I do? The wadded tissues on the floor, the twisting bodies of the videos seemed otherworldly, radiant now, imbued with the richest amber light. I imagined taking him in my arms, nipping him, tugging the skin of his neck between my teeth. Then reaching downward for his belt buckle, rubbing circles on his stomach with my fist. A shard of paper dropped through the chicken wire overhead, and a note? Was that what this was? I want you right now. Meet me in #2.\n\nMy face blazed. I strolled down the hall toward the indicated door and entered the booth, reaching out into the darkness. My breaths quickened. His chest was warm, softer than I'd imagined, yielding. He smelled of orange rind, fresh laundry. He exhaled in a slow, satisfied half-whistle.\n\nOnly when my eyes adjusted did my presumptuousness become clear to me.\n\n\"You're here,\" whispered an odd, balding man.\n\n\"But\u2014\"\n\n\"What took you so long?\"\n\nHe pressed a soft palm against my cheek. He gazed at me with such tenderness and awe that I couldn't say no to him. He wasn't attractive. His forehead sloped, speckled and vulnerable like the underside of a fish. I'd like to say that I treated him with warmth and compassion, some modicum of fellow feeling. Instead, I guided him to the floor, jerked down my zipper, and gave him exactly what he wanted.\n\n\"Suck it,\" I mumbled.\n\n\"Mmmm.\" He gazed up at me with glistening eyes, so grateful, relieved.\n\n\"Don't say a word, faggot.\"\n\nI crossed my arms over my chest in a complete affectation of boredom. I thought: You can be whoever you want. Your name's not Evan, and your longing isn't killing you. I gazed down at my cock slipping in and out of his mouth, numbing every sensation from it, refusing to admit that we were even engaged in the same fundamental work. I didn't touch him or urge him onward. Loneliness, I thought. This was isolation, and this was loneliness. I saw my bones turning at once to powder, particles of me flying up through the air filters. A broken bell chimed in a distant tree. Was it even sex that I wanted, or something more elusive, more rigorous than that? Had I wanted William to change my life? Had I wanted him to solace my pain, to exchange my former home for a better, more protected one? Had I wanted him to embrace every last facet of me\u2014my speech, gestures, flaws, potential? And in expecting these things, was I pushing him away?\n\nOr, simply, did I want a fraction back of what I was giving to him?\n\nI thought of Al Parker's bearded face. I pulled back from the stranger while there was still time.\n\n\"That was terrific,\" he said, smiling.\n\n\"Thank you,\" I said nervously, giving his shoulder a squeeze.\n\n\"No. Thank you.\" His face glowed with a freshly pink sheen. \"I've never seen you here before. What's your name?\"\n\n\"Kevin.\"\n\n\"I'm Irwin,\" he said, offering his hand to shake.\n\n\"Are you okay?\"\n\n\"Of course I'm okay,\" he laughed. \"Couldn't be better. You're a very hot boy, Kevin.\"\n\n\"Thanks,\" I said shyly, bowing a bit, and left.\n\nI stood upon the hot surface of the parking lot. It was twilight now. Cars rushed by on Route 1, careening past the tank farms, the container complex, the clumps of palmetto in which someone could get lost. The air smelled of napthalene, forest fires. I glanced once up at the sky, the harsh bowl of it, thinking about its indifferent blue, knowing its eye had already focused on the next brutality. What was one more tiny crime? I held up the stranger's note and released it, watching it blowing out into the traffic. The warm-eyed boy was gone now, gone forever. Whoever told me I didn't deserve to be loved?\n\n***\n\nWhen I was eleven, I thought a lot about a boy named Douglass Freeman. I was fascinated by his house. There was nothing else like it: its Colonial-paned windows, its cupola, its gardens with the plywood comic-strip characters (Nancy, Sluggo, Aunt Fritzi). And beside it all, gleaming in the driveway, a new Winnebago like the showcase prize from The Price is Right. For most of that year I wanted more than anything to live there.\n\nHe was in every appearance an average eleven-year-old with freckled skin, tetracycline-stained teeth, and a stunted brushy plume on the crown of his head. It wasn't until the end of the school year when every sixth-grader from Gus Grissom was bused off to camp that we learned something else about him. The rumors had been flying even before Mr. Albertson sat us down in the cabin.\n\n\"Tonight most of you boys'll be taking a shower.\"\n\nWe nodded. We gathered around a citronella candle in the dark. His voice was hushed, grave, as if he were trying to scare us.\n\n\"These showers are what we call communal. Have any of you showered with other boys before?\"\n\nMy upper arms itched. I'd never heard of such a thing. I couldn't help but think the idea was a little outlandish, even obscene.\n\n\"I want you to know that Douglass Freeman doesn't have a penis.\"\n\nEric Woodworth fisted the air. \"Yes!\"\n\n\"Quiet!\" Mr. Albertson cried. The entire campground stilled, crickets and tree frogs falling mute. \"Once more and I call your parents.\"\n\nWoodworth trembled, pretending to quash the triumph from his eyes. I was mortified. All this time I'd thought that Douglass's newest nickname\u2014Dickless\u2014had been nothing more than a harmless, ongoing prank. Hadn't we all called him that?\n\n\"I want you to put yourselves in his shoes,\" Mr. Albertson said. His eyes shone with great intensity and warmth. \"I want you to imagine what he lives with every day of his life. Do you understand?\"\n\nWe nodded, humbled and ashamed.\n\n\"He's all boy. That's all I want to say.\"\n\nWe nodded again.\n\n\"Good. Very good.\" Then Mr. Albertson started a story. \"There was a boy, there was a girl, and there was a ghost...\"\n\nBut I couldn't follow his words. If I'd only known that Douglass's condition resulted from a birth defect or cancer, I'd have felt better. I imagined my own penis, a thing I'd learned to like, crumbling off in my hand as I cleaned myself with a washcloth. The truth was I'd been touching myself a lot, probably more than I was supposed to. I pulled my legs closer to my chest. Around me boys were laughing, utterly immersed in the tale. Mr. Albertson crouched and tiptoed about the cabin, illustrating his drama with little props: a pin light, a tennis ball, a handkerchief resembling a lady ghost. What was wrong with me? Was I the only one who felt like this?\n\nMr. Albertson left for the showers, Dobb kit in hand, white towel slung over his shoulder. \"Keep an eye on the ship,\" he said, his eyes meeting mine.\n\n\"Five bucks,\" Woodworth mumbled. \"Five fucking bucks.\"\n\nSteve Strandberg gazed out at the empty, starlit paths. \"I wouldn't go to the showers now,\" he said in a lonely voice.\n\nI said, \"Why?\"\n\n\"That's when Dickless's there,\" said Steve. \"He won't go till everyone's finished. Sometimes it's midnight, sometimes it's three in the morning. The school board had to approve it.\"\n\nI pictured Dickless standing outside in the middle of the night, his chest peppered with goose pimples, scrubbing himself with a soap-on-a-rope (a gift from his mother?) while he looked repeatedly, anxiously over his shoulder. It pissed me off to think that only a few weeks ago these boys had played kickball with Douglass, calling him by his correct name, treating him like the quiet, unremarkable boy he was.\n\nBut who was I to talk? I'd already made my decision not to shower until I got home.\n\nThe week was relentless. Our days were crammed tight with activities\u2014math classes, crafts seminars, athletics\u2014as if our teachers were fearful of leaving us unoccupied. I missed the girls, their precise, intricate outfits, their kindness, their expansive senses of humor. I missed Jane. Without girls, the boys grew wilder, more aggressive, as if shot up with hormones. Gregg Novak, for one, a skinny thing with twig-like arms, lifted me up, spinning me around and around in full sight of seven boys\u2014my legs flailing all the while\u2014if only to show them he could do it.\n\nThis wouldn't have happened back home.\n\nThe week creaked onward. I ticked off each day on my calendar, striking it out with a wax pencil, pretending I was doing time at Sing Sing. Soon enough it became clear to my cabin mates\u2014Eric Woodworth in particular\u2014that I hadn't taken a shower since my arrival. By this time, they'd all showered together and had gotten used to it, barely mentioning Dickless's name as if he were already old news.\n\n\"You haven't taken a shower,\" said Woodworth one night.\n\n\"Yes, I did,\" I answered. \"Two nights ago. You weren't paying attention.\"\n\nHe knew I was lying. I stared at his slight chubbiness, knowing that at twenty he'd be ugly, fat, and unlovable. I didn't know why this comforted me. He glanced over at the top bunk. \"Hey, Strandberg, has Sarshik taken a shower yet?\"\n\nSteve stared down at his dirty pink feet, utterly silent.\n\n\"Your hair's greasy,\" Woodworth said to me. \"What's the matter? You don't have a dick either?\"\n\n\"Shut up,\" I cried.\n\nI might have downed a glassful of paint. Was I a coward? I couldn't bear to be talked to like this. It was the moment I'd been afraid of. All at once I leapt up and rummaged through my backpack for my shower supplies.\n\nI hurried to the outdoor shower stalls, leaves rasping beneath my feet. You had to do these things, win the races, catch the fly balls hit to your corner, even if it killed you. If you didn't do it, you got them mad, and they made you an outsider\u2014someone who was pounced on, spit out like week-old food\u2014and there was nothing worse than that.\n\nBut none of these thoughts steadied my pulse.\n\nWhen I arrived at the stalls I heard a shower running full force, a drain sucking water. I stopped at once. Was it Dickless?\n\nMy steps were timid. To my relief Mr. Albertson stood underneath the showerhead, hair flattened to his scalp. I couldn't take my eyes off him.\n\n\"Hi, Evan,\" he said affably. \"Beautiful night.\"\n\nMy throat was too tight to respond. I nodded, then crept inside the changing room. I stared down at the pocked floor, breathing, yanking off my shirt and pants, dropping them in little balls upon the exposed wooden slats. I was going to do this. Once and for all, I was going to get this over with.\n\nMr. Albertson smiled at my reentrance. I stepped toward the showerhead beside him and turned on the faucet, testing the temperature. I'd never felt more naked in my entire life, my arms like insect feelers, my chest like the cheapest concave trinket\u2014something to be bought at Woolworth's. Mr. Albertson rubbed the shampoo from his hair. He dug his fists into his tightly closed eyes. I couldn't stop staring at his hard furry butt, his balls, his dick\u2014alarmingly big, its head the color of a plum. I'd never seen anything like it before. And you call this a dick? I thought, speaking to my own parts in disappointment.\n\n\"Who won the softball game?\" he asked.\n\nI swallowed. Had he known I was looking at him?\n\n\"Huh?\"\n\n\"Greens,\" I said finally. \"Greet hit a fly ball over the fence.\"\n\nHe nodded. He turned off the faucet, reached for a towel on the hook. \"Don't use up all that hot water,\" he kidded. He stepped past me, mere inches away. His dick swung gently as he walked. I shuddered. If his towel had been bigger, he might have snapped it against my butt.\n\nI stood under the showerhead for another five minutes. The water felt hot, consoling upon my shoulders. Why had I waited so long? I shampooed my hair over and over, waiting for someone to step around the corner. But when no one did, I turned off the faucet, and stepped into my clothes, letting my hair drip so all my cabin mates would know that I was just like them.\n\nTwo nights later, we all sat around the campfire, singing \"The Circle Game\"\u2014an old Joni Mitchell song that Miss Mastrangelo strummed on a busted guitar while everyone squinted at their song sheets. We were due to leave Saturday morning, and I was already feeling nostalgic. It wasn't that I didn't want to go home. I still hated it here, there was no doubt about that. It was that looking into the flame-lit faces of my classmates around the campfire, I thought, Time is already sweeping us forward. Our bodies are changing. We smell like our parents. Soon enough we're going to separate and move away, and some of us will die sooner than we think, and as a group we'll never be together again. Was this a pop song? I was shocked by my corniness. But maybe it was only because we'd all fallen into our respective routines, learning new skills, growing more relaxed with one another. And unfortunately, there was that other thing: Mr. Albertson had announced earlier that evening that Douglass Freeman was leaving camp two days early. He'd had a hard adjustment and had come down with a sore throat. And there was the sticky issue of contagion. \"It would be best for everyone,\" Mr. Albertson assured us, \"if he left us.\" He was right, for the announcement of his departure made an immediate difference. Everyone relaxed, became themselves, as if the world were returned to its proper order.\n\nIt was dusk. I was walking down a path through the woods. I wasn't supposed to walk alone, not without another camper, but it felt good to be lost in my thoughts, listening to the cheers and yelps of my classmates in the distance. I stepped up on a riverbank. I looked at the sawgrass weaving in the water, the impossibly vivid sky, thinking about how nice it would be to go back home again, to sleep in my own bed, to take a hot bath while my mother sat on the closed toilet seat listening, pretending she was interested in my stories. I wouldn't even let my parents' fighting bother me.\n\nI stepped closer to the fallen log beside the shore.\n\nAnd all it once it moved toward me.\n\nMy whole body clenched. I didn't yell. I'd seen alligators in our very neighborhood, where on winter mornings they'd crawl up out of the canals, sunning themselves in the backyards, looking for handouts of marshmallows. They seemed almost benign, bovine in that context\u2014dumb, leaden beasts too stupid to fend for themselves\u2014and yet they were known to have swallowed a neighbor's Boston Terrier in one gulp, a veritable raisin. But this was the wild. It came to me that alligators had the capacity to run up to 60 miles-an-hour in short distances. The peach fuzz bristled on my neck. I started running, feet pounding the sand, all the way back to the dining hall.\n\nOn the way I ran into Dickless standing in the path. I'd actually seen his face only three times all week.\n\nMy chest heaved. \"Alligator\u2014\" I said, winded.\n\nDickless smiled in utter calm. \"Oh really?\"\n\nI swung my head back and forth. \"Big. thirteen feet or more. Tell the teachers. Dangerous.\"\n\n\"Fuck the teachers,\" he replied.\n\nI stared at him, blinking. I caught my breath. He'd never talked like this before. These simple words unsettled and mocked me, more than I could say.\n\n\"I'd like to see it,\" he stated.\n\n\"Get out.\"\n\n\"I'm serious.\"\n\nI slapped at a mosquito on my wrist. \"You sure?\"\n\nHe followed me down the path. By the time we got to the shore, the alligator was gone, leaving no wake in the river, no imprints on the sand.\n\n\"Liar,\" he smiled.\n\n\"I'm telling you. I saw it right here.\" But for some reason I found myself smiling along with him. Had I been imagining things?\n\nWe sat upon the shoreline, watching the pelicans gliding inches above the brackish river. An air boat whined faintly in the distance, diminishing. The sky darkened a notch. Above a hammock of palms, a lone planet sparkled. The first star.\n\n\"I thought you were leaving,\" I said.\n\n\"Tomorrow,\" he said, leaning forward. \"My dad's coming. Sometime after breakfast.\"\n\n\"Your throat still hurts?\"\n\nHe laughed through his nostrils. \"My throat feels fine. It's never been better. I just needed to get out of here.\"\n\nI laughed. I was going to tell him about the time when I, too, wrangled my way out of a school obligation, mimicking a sprained ankle after a basketball game, when I noticed the stricken, unsettled look on his face.\n\n\"Do you hate it here?\" he whispered.\n\nI glanced at his muddy shoes. I thought of all the things that had been said about him, things I knew he'd heard, how his time back at school would never be the same. The inside of my lip tasted like a penny. I couldn't say anything but yes to him.\n\n\"I thought so. You don't seem like you belong with them.\"\n\n\"I don't?\"\n\nHe rested his sneaker atop mine. For a few seconds I tried to ignore it, but the gesture was intentional, a game of sorts. He wanted me to play. I didn't like such games, thought them childish and beneath me, so I slipped my shoe out from under his and dug my heel into his toes so that he winced, tears springing to his eyes.\n\nI looked in his face. He was laughing now. I had an uncomfortable feeling, an odd buzz of shame, excitement, sadness. Then something else took hold of me, something comfortable and friendly that told me I could be what I was with him. Why wasn't I afraid anymore?\n\n\"Hold still,\" he murmured.\n\nIt happened too smoothly for me to stop. I pulled in a breath. He fumbled for my fly and\u2014to my discomfort\u2014reached into my pants, pulled out my dick, holding it, watching it harden in his grasp. It would be years before it would even reach its adult size. Still, he looked at it like he'd never seen anything like it before. I thought of Mr. Albertson standing under the showerhead, confident, at peace with his body. \"Amazing,\" Douglass whispered, moving his small brown head.\n\nI leaned backward on my elbows. He began stroking me, dutiful and tender, the leaves turning silver, vermilion above us, making me forget that anyone else could even travel up the path, though, thankfully, they didn't dare.\n\n# Chapter 7\n\nWilliam came back from Key West on schedule, tanned and fit, more energetic and relaxed than he'd been in weeks. He knelt down to embrace the Dobermans, and they rushed to him, licking his face, nuzzling, nearly knocking him over until he shielded his head with his arms.\n\n\"How are my children?\" he asked them. His voice was sweet, in the manner of Mr. Peabody. \"How are my pretty, good-for-nothing, well-behaved children?\"\n\nThey rolled on their backs, scuffing their hindquarters against the rug.\n\nHe looked up at me, smiled. His eyes were nearly bloodshot. \"And how are you?\" he said merrily.\n\n\"Good,\" I said, and squatted down beside him.\n\nWe kissed. I was relieved to see him back, though a small part of me already missed the longing, the empty space I'd contemplated in his absence. The week had been odd. Once I stopped resisting my aloneness, I gave myself over to it, relaxing, even cultivating this new privacy. There were certainly worse places to be. I felt confident, capable. I took a greater interest in the house without anticipating the repercussions of my gestures. I threw out the stained bathroom rugs. I threw out a malfunctioning lamp\u2014a hurricane lamp, a divorce present from Lorna to William, which William had actually liked. I taped a picture of some sexy young daddy to the refrigerator, his smile, his black furry chest on display for the whole household. And all the while I read, staying up half the night, immersing myself in projects I'd been putting off for months: Blake's Marriage of Heaven and Hell, The Collected Works of Flannery O'Connor, the Osteology and Lymphatics sections from Gray's Anatomy.\n\nWilliam undid his tie. He flopped into the sofa, loafers up on the coffee table, telling me about his adventures of the week, how Lilo Patrick had been arrested outside Sloppy Joe's, drunk, sobbing, after taking a poke at a female tourist who'd made fun of her purse. The station was doing everything in its power to hush-hush the incident. He seemed overly excited, manic, as he relayed this tale to me. Spittle dried in the corners of his mouth. A bleak thought crept into my head: this is the person I missed so much?\n\n\"And how was your week?\" He looked at me directly. Had he recognized anything foreign, the stranger's hands upon my body?\n\n\"Stayed at home. Cooked, cleaned, walked the dogs, read.\"\n\nBefore we could get too settled we put on our jackets and left for Arigato, our favorite restaurant. I loved Arigato\u2014the koi pond at the front door, its cool, tinkling music of koto and gong, its overly solicitous waitstaff, clad in their blacks and whites, smiling just a tad, never talking louder than a whisper. I followed William past the cash register; the inner legs of his 501s made loud rubbing sounds. I thought, since when did your ass get so big?\n\nOur waiter\u2014middle-aged, unfamiliar, Caucasian\u2014seated us in an alcove near the front, not where we usually sat. William regarded the gas grill in the center of the table, stricken. This wasn't good. He didn't take well to change.\n\n\"What's the matter?\" I asked.\n\nHe looked up. \"Waiter,\" he called out across the restaurant.\n\nThe man walked to the table. \"Yes, sir?\"\n\n\"What's this?\" he said, gesturing at the gas grill.\n\n\"A grill, sir. Does that bother you?\"\n\nI knew what was going on. I got the \"sir\" thing, the slight sarcasm of it, the waiter's recognition that we might be a handful at the end of a long day. But I didn't get the look on William's face, a look which said, I don't like you, though he'd barely given the waiter a chance.\n\n\"Where's Hatsuko?\" William said. Hatsuko, a pretty long-haired woman from Nagasaki. Our usual waitress.\n\n\"She's off for the night, sir.\"\n\n\"She'd never seat us here. You can tell your manager that, he knows us. We come here all the time.\"\n\nThe waiter looked to me then, flummoxed, his face softening as if he felt some concern for me. He clearly didn't know what was motivating William's anger, and nor did I, for he'd seemed buoyant and cheery only a few short minutes ago. Or had I misread him? A hot, waxy bubble swelled inside my chest. The waiter left us now, shifting to another station.\n\n\"What's the matter?\" I said, frowning now.\n\n\"Oh, that guy.\" His eyes drifted over the menu listings. \"It's weird. He reminds me of my brother.\"\n\n\"Your brother?\" This was even more of a surprise.\n\n\"Yeah, Henry. You know, the one who never paid my parents back, who wouldn't talk to me after I left Lorna.\"\n\nHis family was off limits. I knew better than to open up that subject. \"That's no reason to hate the waiter.\"\n\n\"I don't hate him,\" he declared.\n\nI heaved a sigh. What were we talking about? \"Forget it,\" I mumbled, propping up my head with my hand.\n\nA tense quiet stood between us. Across the room an older couple broke open their fortune cookies, scrutinizing the printed messages with great interest and pleasure. They laughed, then chewed the broken pieces with calm, measured satisfaction. It was clear that they were happy together.\n\n\"So I've disappointed you,\" he said suddenly. \"All right?\"\n\nI looked down, attempting to dazzle myself with the renderings on the placemat: yellowtail, sea urchin, salmon roe, sunomono. But I wasn't hungry anymore. Too bad, I'd been looking forward to our first night together again.\n\n***\n\nThe funny thing was I'd been thinking about my own brother. Not in the usual sense, in which I couldn't forgive him for leaving my parents, but in a stranger, more complicated manner. Many years ago, I spotted him in the powder room one day. He lingered before the mirror, examining the recent changes of his body\u2014the weight of his genitals, the dense tangle of pubic hair\u2014proud, excited, frightened at once. I was five, he thirteen. Odd, I thought. What we choose to remember.\n\n***\n\nStanding in the bathroom one morning I saw a bump\u2014a tough, painless eruption on the base of my penis. I shrugged it off. I went back to my projects for the day. I transplanted the bottlebrush to the backyard; I opened up the circles of the royal palm beds. Two days later, though, when I saw it again\u2014its margins still hard and not at all diminished\u2014I literally moaned out loud.\n\n\"Are you okay in there?\" William said.\n\nHe was working on the opposite side of the door. It was Friday, his day off. He'd gotten it in him to paint the hall ceiling the palest yellow, and I pictured it misting down on him from the roller, a cool citrusy rain stippling his forehead. The smell of fresh latex leaked beneath the door.\n\n\"I'm fine. It's just a backache.\"\n\n\"A backache? Since when do you get backaches?\"\n\n\"Always.\"\n\nMy secretiveness got the best of me. I should have opened the door, pointed, and shown him what the problem was.\n\n\"Don't worry,\" I said, and stood behind him. I scraped some paint off his forehead with a fingernail.\n\n\"Are you sure you're all right?\"\n\n\"Absolutely I'm all right,\" I said, a little peeved now, and wandered outside.\n\nI lay down on the backyard lawn in the sun, panting, the blades of Floratam scratching into my back like quills. I'd all but banished the arcade incident from my mind, but here it was back to haunt me, as if my life were some great morality tale from the Middle fucking Ages. We hadn't been having sex much\u2014a whole other issue that completely eluded me\u2014but what we'd done was enough to transmit it to him. He wouldn't take it well. Not only had I done something completely behind his back\u2014a spite fuck\u2014but I'd done something worse: I'd shown him, in the most trenchant terms, how completely inadequate he was to me.\n\nMy relationship, as I knew it, was about to end.\n\nI had to tell him. There was no other way. I pictured sitting down with him one evening, placing a hand on his shoulder. \"You have an STD,\" I'd say simply, in absolute calm, with authority.\n\nA puzzled look would capture his face. \"What do you mean? How could that be?\"\n\n\"You have an STD. There's absolutely nothing to be afraid of.\"\n\nHe'd frown. \"That doesn't make sense. I haven't had sex with anyone but you.\"\n\nI'd nod, waiting for him to piece it together. He'd look at me in a kind of vacant, childish wonderment. Then his face would crumble and he'd bury his forehead in his palms.\n\nI stayed up the entire night, trying to calm down, convincing myself that the announcement wouldn't be as bad as I feared. He wouldn't raise his voice or beat me; he wouldn't kick me out of the house. He was a rational, reasonable man. After all, we'd never declared monogamy, and though it hadn't come up as an issue, he wouldn't have been completely surprised, given the state of our sex life. Right? And yet I wasn't convinced he'd take to it lightly.\n\nWas it possible to live with an untreated venereal infection? What about Flaubert, Maupassant, Jules de Goncourt? George Washington, for God's sake? I went to the library and read through every medical reference text, pored through every online database, until my eyes were scratchy and sore. According to the facts, it was indeed possible to live with the disease for years; if untreated, sufficient defenses would develop to produce a resistance to reinfection, even though these very defenses would fail to eradicate the existing infection, leading to lesions involving bones, skin, and viscera; heart disease; a variety of ocular syndromes; and the weakening of the central nervous system. In worst-case scenarios: tissue destruction. I pictured us sitting together at the dinner table, chewing on the softest foods, watching our incisors wobbling out, then coughing once, spitting them discreetly into our napkins.\n\nIf only I'd had a two-week supply of penicillin.\n\nI checked myself daily. To my amazement the blemish wouldn't go away.\n\nI buckled down finally. I went to a local specialist, Dr. Maltman, who informed me in no uncertain terms that I was fine, that I had a pimple. It was hard not to feel foolish, cheated. But he'd given me back my life, and the story of our daily unraveling.\n\n# Chapter 8\n\nIt was on a hot, swampy morning, after dropping William off at Channel 7 Studios, that I saw my mother shopping at the Town and Country Publix. My initial impulse was to leave the store or hide. What was she doing so far from home in an emporium that catered primarily to wealthy retirees? Once I got a hold of myself, I took to following her through the crowded aisles. How lonely, ponderous, and insignificant she looked as she tilted her head upward to examine the scribbled shopping list, which undoubtedly contained the same staple items we'd consumed for the past twenty years: chow mein, frozen peas, watermelon. Etcetera.\n\nI realized at once that I loved her more than anyone.\n\n\"Hello, Mom.\"\n\n\"Evan?\"\n\nShe tilted her head, her eyes fiery and alarmed, their slight almond shape enhanced with liner. Gone were the sweat-shirts and the oversized glasses of the past, replaced with contacts and a deep blue Merino wool skirt. She looked healthier, actually, prettier and in better shape. Had her makeover been triggered by my absence from the house?\n\n\"You look beautiful,\" I said, releasing her from my hug.\n\n\"I do?\"\n\n\"Yeah. I never remember you looking so good.\"\n\n\"That's nice.\" She sipped from a Styrofoam cup of coffee, a vile, watery concoction provided by the supermarket to pep up the customers. She grimaced down at the lipstick mark on the rim. \"I'd come to the point in my life where I realized I had two choices. I could either shoot myself in the temple or reinvent myself. Needless to say, I went with the latter option.\"\n\nMy mouth fell open. \"You were going to shoot yourself?\"\n\nShe winced. \"For Christ's sake, no. Don't be so literal.\"\n\nShe told me about what she'd been up to. She'd been working on the garden, uprooting the liriope to replace them with spreading junipers. She'd been attending a flamenco dance class at Miami-Dade Community College, where she startled the others with her quiet proficiency. She was even writing, she'd said, a good six pages a day.\n\nI asked her what on earth she was writing.\n\n\"Journal stuff. Stuff about my life. It wouldn't be of interest to anyone.\"\n\nHer words stung, though I know she didn't intend it. She kept talking and talking, mindless things that had nothing to do with us. I saw the apprehension on her face. And then the veil went up: I almost watched it: a thin, filigreed thing through which she wouldn't let me in. I heard the thoughts working behind her starched, fiery eyes: I am not going to let him hurt me again. I cannot bear any more loss. I can go on without him.\n\nWe might have been two longtime friends who'd had a falling out, still shaken, resentful, not knowing what was to become of us. She spoke with the slightest edge. But I believed it was possible to renew. I thought of all the times I'd had reunions with Jane. It was always tentative at first, as if we were both dreading and expecting the first signs of conflict. And then the path would clear, opening up, and we'd relax, remembering what we'd once been for each other.\n\n\"So what have you been up to?\" she said in an overly cheery voice.\n\nI walked with her side by side, up and down the crowded store aisles. I knew better than to talk about William: this was not what she'd wanted to hear. I might have been eight years old, helping her fetch Mrs. Paul's fish sticks\u2014once my favorite childhood food\u2014from the steaming caves of the freezer. I might have been basking in the drab, comforting days when Peter was away at school, before anything complicated like puberty and desire had gotten hold of me.\n\nBut once I erased William from the picture, I had nothing to say for myself. For some reason I thought of the mole on his left wrist, the moist pink skin of his hot fleshy neck.\n\n\"Mom,\" I said, surprising myself. \"Would it be okay if I stayed with you and Daddy for a few days?\"\n\nShe inhaled, pretending to examine the ingredients on the back of a cake mix.\n\n\"Mom?\"\n\nHer head pivoted back to me. \"Sweetheart,\" she said with a wounded smile.\n\n\"I don't understand.\" I'd only wanted to get away; if only I could talk to her about William.\n\n\"Your father\u2014\"\n\n\"Yes?\"\n\nShe glanced to the floor, sniffed, and cleared her throat.\n\n\"You mean to tell me that he doesn't want me at home?\"\n\nHer eyes filmed. \"Of course not. He's not like that. Don't talk about your father like that.\"\n\nI shook my head back and forth. I decided to let her have it. \"So you're saying he's abandoned me, then?\"\n\nHer face hardened. Deep lines bracketed her mouth. \"You're the last one to\u2014\"\n\n\"What?\"\n\n\"Who's abandoned whom?\"\n\nI felt myself getting smaller, a pinpoint, a particle, an atom. I fixed all my attentions on the teenager in the big jeans, the big sneakers, on his tiptoes, helping his grandmother\u2014great-grandmother?\u2014pick out a bagful of orange lentils. He placed them in the child seat of the cart, and she smiled dimly at him through her warped glasses, welcoming his presence.\n\n\"Listen,\" she said, more gently now. She touched my arm. \"Let's save this for another time.\"\n\nI glanced down at her fingers resting on my arm, the wedding band, the abrasion beneath the knuckle. It might have been the hand of somebody foreign.\n\nShe exhaled once, noisily. \"I'd love to have you back. I'd love to have you home more than anything. It's just\u2014\"\n\n\"What?\"\n\n\"If you promise never to see him again...\"\n\n\"Fine,\" I mumbled, though I was prone to say, Fuck it. I don't need your kindness. I don't need your mercy.\n\n\"Keep in touch,\" she said. \"We want to know how you are.\"\n\n\"You do?\" I said.\n\nShe nodded and walked ahead. \"Be good now.\"\n\nI watched her threading down the aisle, maneuvering the cart in and out through the stalled shoppers. It was hard to keep from walking after her, from prolonging the discussion, from saying, \"I'm sorry. I love you. Don't you know how much I love you?\" Anything to keep her from leaving the store.\n\n\"Wait,\" I said at once.\n\nShe turned just slightly to her left, as if she were hiding her face from me.\n\n\"How's Peter? Have you heard from him?\"\n\nShe shook her head, looking with some urgency toward the checkout.\n\n\"What's he up to?\"\n\n\"Naples,\" she said, and shrugged her left shoulder. Once again her shoulders slumped.\n\nAnd then I watched her leave. This was rupture. The longer we were apart the more damage we created. Was it ever possible to move back and forth across that bridge, that bridge between the two fenced-in countries?\n\n***\n\nYounger, I could have watched them forever. My mother with her peat-rimmed cuticles, trimming the flame vine beneath the rain spout. My father with his flasks and his beakers, mixing up some fizzing compound. My own puzzles and toys often bored me in five seconds, but the study of my parents within the laboratory of their household afforded me the richest pleasure. Nestled between them in the secluded valley of their king-sized bed, I knew that life was never meant to be so sweet.\n\nSchool made a difference. I trudged to the bus stop through rough games of tag, coins aimed at the bulb of my skull. Immediately I felt indifferent toward anyone my own age. I wanted to be with my parents. I wanted to be encompassed by their routines and manners, their dignified, high-minded chatter. School might have been a zoo. I might have been caged up with monkeys, their golden shit still fresh between their teeth. All day long I gazed at the droning clock, lonesome and yearning, pretending to be immersed in the swirls of my finger paintings, counting down the hours when I could be safe with my parents again.\n\nIt wasn't that they were flawless. Even before they started drifting apart, they had had their tensions, which they occasionally took out on us. A bitter word, a slap out of nowhere. But these were the occasion rather than the rule. The money was racking up in the bank; Sid was still ascending the mighty ladder of the chemistry department, and there was the sense of dizzy optimism about it all, the belief that they could still redeem every disappointment in their respective pasts.\n\nSo it wasn't without sense that I felt an enormous loss when I thought of them now, that the combination of their absence and proximity\u2014their house within (I'd counted it) 1,260 feet of William's\u2014unearthed me as it did. For years, I'd lived with the naive assumption that nothing would rattle their deep links to me. I could rob a bank, could pass nuclear secrets, and still they'd love me. Had I been wrong? My eighteenth birthday, July 12th, passed without a note. My card congratulating my father on his promotion to full professor\u2014something I'd heard from Wendy Park, his former student\u2014remained unanswered. They knew where I lived. It was as if they'd moved. I imagined sometimes that they'd fulfilled their lifelong ambition to buy a sailboat and negotiate it through the locks of the Panama Canal. Sometimes in the middle of the night I'd sneak out through William's garage and wheel my bike past the water tower, taking the jogger's path behind Country Club Plaza. I'd spot the house, then taste it, the flavor of fear, a rubbery dank taste like sucking on an inner tube. Something would have been different: a new set of gray trash cans, a sapling Norfolk Island pine, and I'd think, Goddamn it, goddamn them, they've gone somewhere without me. But then I'd see some touches that could only be theirs: the copper wind chimes, the old weather vane glinting on the roof, my broken ten-speed leaning against the fence. Relief.\n\nThey'd never do something as terrible as that.\n\nWasn't it time I stopped thinking about them so much?\n\nI didn't expect kindness or wisdom or solidarity overnight. And yet they weren't making the slightest of efforts on my part. Did they actually believe that every decision and gesture I executed was simply to spite them? Some people wanted to be boneheads, some people didn't want to feel good about their circumstances. The truth was that my parents, Mr. and Mrs. Sid Sarshik, wanted to think that they'd failed somehow in their raising of me.\n\nWell, fuck that.\n\n***\n\nNights later, lying in bed, with William at my side and the Dobermans at my feet, I was still thinking about my parents. Was my mother so fearful of aggravating my father that she'd do anything to avoid a conflict in which she'd eventually side against him? Did she feel so awful about her compliance, her lack of nerve, that she couldn't stand the sight of herself, the sound of her own voice? Or did she, on the deepest level, share my father's point of view?\n\nMy feet were freezing. \"I can't sleep,\" I said out loud, if only to myself.\n\nWilliam turned on his side. He punched softly at the pillow behind him, then offered a low moaning sound. Seconds later, he turned on the lamp beside the bed. \"What's wrong?\"\n\n\"I ran into my mother in the store a few nights ago.\"\n\nHe sounded dehydrated. All week he'd been working on an expose about the nuclear plants of South Florida, a project which involved hours of overtime. His face said: Do you know how hard I've been working?\n\n\"And how was that?\"\n\n\"Strange. I asked her if I could stay at their place for a couple of days. She more or less said no.\"\n\n\"Mmmmm.\" He stared upward at a fissure in the ceiling.\n\nI felt emboldened and exposed at once. \"What are you thinking?\" I asked him.\n\n\"You're not happy here?\" he said, curious. His eyes worked to seem bright, anything but startled and hurt. Up to that point I hadn't indicated that things had been anything but cheery between us.\n\n\"No, it has nothing to do with us. Don't you ever miss Poppy?\"\n\nHe looked at the trim on the blanket, withholding his response. It happened that Lorna had forbidden him from seeing Poppy in the wake of their divorce, even though the agreement was far from legal. To make up for his silence, William wired 500-dollar deposits into Poppy's Winter Park checking account 4 times a year.\n\n\"She doesn't mean what you think,\" he said, but his voice faltered.\n\n\"How would you know? You've never had a real conversation with them.\"\n\n\"I've seen them outside your house. I've seen your father fixing your bicycle for you\u2014watching you pedal away, making sure the chain didn't fall off.\" He looked at me. \"They love you.\"\n\nI blew my nose. \"If only as a concept. If they love me it's only because I'm their mutual creation. I represent a certain lost thing between them. But they haven't been behaving toward me with anything approaching affection.\"\n\nHe was silent for a time, breathing. \"Aren't we erudite?\" he said finally.\n\n\"Don't make fun of me.\"\n\n\"Too many thoughts,\" he said. \"You're thinking too many thoughts. Calm down now, turn off your mind.\"\n\nI rolled my eyes. \"I'm hardly a child.\"\n\n\"Get over here,\" he insisted. \"Now.\"\n\nSlowly, methodically, he began rubbing my head. I tried to relax, concentrating on his weighted palms working the contours of my skull, its bumps, gullies, and planes. He got me then. He knew me this well: nothing anyone did could center me like this, could demarcate my borders with such easy precision.\n\n\"Feel better?\"\n\nI nodded. I glanced over at him through one eye. What was I doing with him? So what if I decided to say it?\n\n\"Why haven't you been having sex with me?\"\n\nHis eyes blinked rapidly. I might have asked him if I could taste his blood.\n\n\"We have sex,\" he said softly.\n\n\"When was the last time? Tell me. Has it been a month, two? You don't even remember.\"\n\nHis brow clenched. He seemed truly, deeply flustered, so much so that it was hard not to back down and forget it. \"I've been busy as hell with my job,\" he rasped.\n\nAnd yet I could answer: two months, eleven days. My eyes watered. \"See?\"\n\nHe propped himself up, held up a hand like a crossing guard. \"Hey, that's not fair. Just let me off the hook, okay?\"\n\n\"No one's putting you on the hook.\"\n\n\"I don't need to be accused of something at\"\u2014he picked up the alarm clock, held it mere inches from his face\u2014\"at three in the morning.\"\n\n\"That's not what I'm doing.\"\n\n\"I don't believe it,\" he said, yanking once at the covers. He turned away from me.\n\n\"Are you impotent?\" I said quietly.\n\n\"No.\"\n\n\"Are you pushing me away?\"\n\nNo answer.\n\n\"Hel-lo\u2014\"\n\n\"No, I'm not pushing you away.\"\n\n\"Do you have another boyfriend?\"\n\n\"No. Now stop it. I'm not subjecting myself to your interrogation. Just go to sleep.\"\n\nI settled back in the bed. \"If you gave me an explanation, I'd understand. I could be patient, okay? I want to help. All I need is a little explanation.\"\n\nHe seemed enwrapped in some tendril of thought. He glanced away. Quietly, he said, \"Did it ever occur to you that you're just too damn horny?\"\n\nI laughed out loud. The idea was preposterous. Wasn't everyone horny? \"You've got to be kidding.\"\n\n\"No, let's shift the light to you. Do you know what it's like to feel that you're disappointing somebody every minute of your life?\"\n\nMy face heated. \"What?\"\n\n\"I'm talking about your neediness. You're putting too much pressure on me.\"\n\nI glanced at the picture of Lorna on the bureau. \"Oh, it's my fault, I get it. Christ.\" I shook my head, hard. \"You're a gay man, for God's sake.\" And then I leapt out of bed, walked down the hall, and curled up on the kitchen chair.\n\nI sat there, palming the loose change he'd left on the counter. I loaded it in my pocket\u2014forty-five cents, no big deal, but it was the gesture that shocked me. In the entire realm of his life I meant about as much to him as a chauffeur or a pet. He liked the idea of having somebody around, but when it came down to the day-to-day obligations of living with someone, the duty to affection, he came up unbearably short.\n\nNo wonder Lorna wanted him to have nothing to do with Poppy.\n\nI walked outside, crept into the garden. The sounds of the night: a plane overhead, blinking, vanishing, then nothing. Shifting circuits on a sprinkler system. I edged closer to the property line. A rustling beside the pool cage. I knew it: red stripe, yellow stripe. Black snout. Coral snake.\n\nBut it was only our neighbor. She was sitting in a lawn chair in her backyard, pressing a cup of coffee to her chin. She gazed out at the trees with such intensity that it was hard to watch her. In the one and only time we'd ever talked, she spooned bloodmeal around the roots of her begonias, her personal trick for keeping their colors so vibrant. Now, for whatever reason, I lost my usual sense of shyness and caution. I decided to walk over to her.\n\n\"I hope I didn't startle you.\"\n\nShe kept staring at her garden wall, and shook her head. \"I heard you coming,\" she whispered.\n\nA silence between us. Dew dripping from the staghorn ferns. \"You couldn't sleep, either?\" I said finally.\n\nShe nodded, sniffed. She wiped at her nose with a balled Kleenex in her fist.\n\n\"Your begonias look beautiful,\" I said.\n\nShe glanced over to them. \"They are,\" she said quietly, satisfied. \"Aren't they? Thank you. Thank you for noticing.\"\n\nShe turned to me then for the first time. Her chin glistened in the dark. I wanted to step back, embarrassed. \"Are you okay?\"\n\nShe started crying again. I stood beside her, looking down at the sparkling grass.\n\n\"I'm sorry,\" she said through a low sob.\n\n\"How's your husband?\" I asked, glancing over at the single light in the window.\n\n\"Drunk,\" she said. \"A basket case.\"\n\nThe dew felt icy on my forehead. A warm breeze rattled the coconut palms. \"If you ever need help,\" I said, \"with your garden...\"\n\n\"You would?\" she said, her eyes brightening slightly.\n\n\"I'll water your begonias,\" I offered. \"I'd love to feed them.\"\n\n\"I'd appreciate that. What a nice gesture. Thank you\u2014\"\n\n\"Evan,\" I said, and extended my hand.\n\n\"Mrs. Stendhal,\" she said. \"Virginia Stendhal. Thank you very much.\"\n\n\"Have a good night now.\"\n\nI walked back into the house, down the hall to our room. The door was half closed; I stopped. Already I heard the snoring of Pedro, the Doberman, a full sawing throttle that could make us laugh when our moods were right. I didn't expect that now. I pushed open the door, reluctant. Had I harmed him beyond imagining, underestimating my own capacity to hurt?\n\nHe sprawled on the bed, motioning me toward him. His eyes, though sad, glimmered in invitation.\n\n\"What?\"\n\n\"Be quiet,\" he said with tenderness.\n\nMy gratitude was boundless, large as a country. Why was I crying? He took hold of my arm, pinned me to the bed, and to my bafflement and surprise began to make love to me.\n\n# Chapter 9\n\nI had to know things. I had to take apart everything I saw, even as a kid: lamps, toys, hairdryers, toasters. The world was a flurry of blues, oranges, and golds, unbearable in their vibrancy. Sid and Ursula tried their best to calm me down, but I skated away from their grasp, agitated, voracious. I had to ask, where is this going, how is this put together, why are we so shy, so brittle? There was Beau Roberts's mouse. I wriggled it in my cupped palm, on its back, eyes red, astonished. I fumbled for the toy knife, gently nicking its fur, wanting more than anything to peel back that white, exposing the pulsing wet bead inside, before Beau's mother caught me in the act. Our phone rang later that evening. I tried to tell my parents I only loved the mouse, but my words fractured in my mouth. They sat across the kitchen table, fingers latched, gazing at me with their wet, dazzled eyes.\n\n***\n\nThe morning air was sluggish and sweet, yellowing the surfaces with pollen. We lounged inside the pool cage, reading through the Herald and eating carambola. On such mornings it was easy to convince myself that there was indeed something between us, that, together, we were actually immersed in the moment. We felt a pleasure in our silence, a camaraderie and a comfort, a mutual respect for the distance between us.\n\nThe week had been a good one. On Tuesday, William had surprised me by taking me to the Channel 7 Studios (at last breaking his pact to keep me hidden from his coworkers), where he introduced me to Dinah Strang, aka Dusty Cartwright, host of Casper's Corners, my favorite childhood TV show. She'd been out of work, in hiding for years, after the FCC crackdown on the endorsement of products by the hosts of such programs\u2014in this case Dinah's own Dusty Cartwright Dairy Bar. A host of rumors had swirled around her disappearance\u2014one, that she'd lost her leg to gangrene; another, that she was living on the streets of Newcastle, Delaware, a sometime prostitute, after draining her bank accounts and turning over the funds to the SPCA. It was hard to believe any of it now, as she stood before the flying-fish sculptures outside the studio, preparing for her local comeback program, which already gave off the queasy scent of failure. I wasn't giving up on her, though. She held me close\u2014a big blowsy grandma in a fringed white cowboy suit\u2014as William snapped the picture. \"A big smoocheroony for your Aunt Dusty,\" she cried, and I kissed her then, unfazed by her pancake makeup, her whiskied breath. She looked at me, confused, stricken, as if I were some demanding child.\n\nWilliam flapped the newspaper once on his lap. In Virginia's trees next door: a riot of orchids. \"Did you know a Todd Bemus?\" he said offhandedly.\n\nI moved to the edge of the patio chair and leaned toward him. \"Yeah, I went to school with him.\"\n\nHe raised his brows. The headline of his page read OBITUARIES.\n\n\"He died?\"\n\nHe handed the paper to me. I pulled in a breath and held it. Over his name I saw Todd's yearbook photo, his white hair\u2014\"bleach out,\" we called it\u2014his pink, dangerous lips. Already he looked like he knew so much more than the others, his eyes bitter, yet willful. He might have been someplace far ahead in the future. I passed over the announcement: Todd Bemus of Coral Gables, Florida, died Thursday of complications due to AIDS in San Francisco, California. He was 20 years old and is survived by his mother, Cherry, his sisters, Heather, of Port St. Lucie, and Tabitha, of Nags Head, North Carolina.\n\n\"He was a friend?\" William asked.\n\nI kept shaking my head. \"We were in gym together for two years.\"\n\n\"Twenty,\" William said. \"Shit.\"\n\nI clenched my right hand tightly, rubbing a knuckle with my thumb. My throat filled quickly with tears. Still, it didn't seem quite real to me, not only because I'd been out of touch with Todd for two years, but because the event seemed like something he'd have staged, watching it take place from afar, then obnoxiously showing up at his own funeral\u2014\"ta da\"\u2014laughing at everybody, then apologizing for all the grief he'd caused. Dead. How could Todd, of all people, be dead?\n\nWilliam rested his arm on my shoulder. I must have looked sadder than I knew.\n\n\"Wasn't he too young?\" I said. \"I mean, I thought it took ten years to come down with symptoms.\"\n\nHe raised his eyebrows again. \"Not always. Sometimes\u2014\" His voice trailed off.\n\nWe sat still, reading about the impending water shortage and the proliferation of hi-tech firms in Boca Raton, pretending that it was just another humid South Florida morning, that something of consequence hadn't split open the day. A sluggishness settled in my arms, the tips of my fingers. I wasn't used to seeing my friends die. There was June Pulte, of course, whose car had sledded off a storm-slick road into a cypress stand after leaving Disney World, and Mark McNitt, who'd OD'd on a lethal cocktail of speed, Valium, butane, and vodka, but neither of these had been close friends of mine, and these incidents were mere accidents, quick dealings of fate that hadn't mucked with the deepest part of their identities. It was happening all around us, I knew it, though it was sometimes hard to believe. William, for one, had lost nearly all of his friends: Thomas and Mark, David and Larry, Tony and Richard. But it was something that happened to older men, men in their thirties and forties, not to people like me.\n\nWilliam took the Dobermans for a walk, and I squatted beside the pool, testing the water chemistry with the little kit, waiting for the PH to go pink. I strained out some leaves from the surface. Poor Todd. I thought about how he'd tried to get closer to me, how he'd called me one day out of the blue to watch videos with him. He'd gotten a hold of the whole Pam Grier \"ouevre,\" he called it\u2014Coffy and Foxy Brown among them\u2014and he'd wanted to watch them with me because he thought only I would appreciate them, and then I promised to get back to him. I knew what a big deal it was for him to call, how he'd probably worried about it all day, and yet I completely and totally blew him off.\n\nSecretly, I believed he had a crush on me. True or not, this was the real reason I kept my distance from him. I preferred to spend time with him in public, at Dadeland, at Haulover Park\u2014any place where he couldn't jump me, for every time I was alone with him I felt a tremendous pressure, even if it was only communicated through the weight of his gaze and the way he sat so close to me. It was too bad that I didn't feel the slightest bit of attraction to him.\n\nStill, I believed he was the funniest guy in the world. He could make me fall down the stairs in stitches, mimicking friends or second-rate celebrities like Joan Van Ark or Tori Spelling, though he also embarrassed me with his airy-thin voice, his occasional cackle, his broad, overreaching gestures. More than once I'd slipped into an empty classroom when I saw him walking toward me down the hall, clutching to his chest his three-ring binder\u2014decoupaged with sunflowers, smiley faces, Day-Glo peace signs\u2014as if he were shielding his breasts.\n\nOnce he'd worn makeup to school. Some of the boys had finally had enough, and he'd probably called one a name, and he was going to get throttled. I could hear him now: Hey straight boy, hey pencildick. Want to get fucked? Straight boy loves to get fucked. He was too much for them. His very presence mocked, and they couldn't stand it, couldn't stand it at all. Eric Woodworth went first. He tugged Todd's earring, and snagged it once, ripping it right through his lobe. A dark drop of blood pearled on Todd's ear. \"You'll get AIDS,\" Convey cried, looking at the red on Woodworth's palm. \"Drop it, you fool. You'll get AIDS!\" Still, Todd walked ahead, a determined look on his face as if he were telling himself that nothing could harm him.\n\nThen Woodworth turned and noticed me. He knew that Todd and I were gym buddies, banding together during softball class, deliberately, flamboyantly missing the fly balls that dropped in our direction.\n\n\"Hey, Sarshik,\" he said. \"Here's your friend, he's bleeding. Why don't you come over here and take care of your friend?\"\n\nI looked at him as if to say What friend? then left Todd to his own devices as I hurtled down the steps to the locker room.\n\n***\n\nTwo mornings later I went to Todd's funeral mass. I dropped William off at work, then drove around Coral Gables looking for the church. I found it after a few minutes, a large Catholic church with ocher brick walls and a round rosette window\u2014St. Michael the Archangel. I sat in the back row, near the baptismal font, in my single white shirt and ill-fitting suit (an old thing I'd borrowed from William's closet), and stared down at the initials etched into the soft wood of the pew. Ten rows ahead were his cousins, dozens of them, all bused in from Allentown, Pennsylvania; his former neighbors; his most recent friends, two of whom appeared to be models or porn stars; and his parents, Francis and Dot, who seemed alternately composed and quietly devastated. It was hard to imagine that they'd ever been cruel to Todd, but they'd been awful, literally locking him out of the house upon learning he'd had sex with Lloyd Scarborough, the marching-band director. Still, they'd welcomed him back home when he'd come down with his first symptom, a case of CMV retinitis, realizing that he had no other choice but to stay put, that he wasn't going to run away, or frighten or surprise them anymore.\n\nBut it was foolish to expect anything but surprise from Todd. Even in death, he still managed to unsettle. Before mass, standing outside with his cousin Ricky (who'd allowed me to bum a cigarette, something I did from time to time), I'd learned that before moving to San Francisco, where he'd butched himself up and started sleeping with porn stars, Todd had been an accomplished composer of liturgical music, something he'd hidden from almost everybody, including his parents. He'd taken it quite seriously, taking pride in his designation as the youngest composer ever in the catalog of UIA Library, his publisher. Now the choir was performing some of Todd's music for the service. The cantor intoned his setting of my favorite psalm, one of those I still knew by heart, Psalm 42\u2014Like a deer that longs for running streams\u2014and then we all repeated it, its antiphon rising, lifting us up, moving and unnerving everyone in the congregation.\n\nOutside the church, Todd's other cousins, three of whom had been linebackers for Penn State, hefted the shining casket down the steps. The sun went in. People milled about, jabbering. It was hard not to feel cheated somehow, though it was a nice mass, a nice sermon. There were only veiled, uncomfortable references to AIDS, the priest assuring us that God, in his infinite mercy and wisdom, was blessing Todd, seating him at his right hand. I tried not to be miffed. The palm fronds glistened in the heat. Karen Kenley, a mutual friend of Todd's and mine, spotted me in the throng. To my dread, she made her way toward me.\n\n\"What have you been up to?\" she asked.\n\nI visored my eyes. Already she'd aged. She might have been five or ten years older than I was with her brittle blonde perm, her pitted skin. She waited for me to answer.\n\nI hadn't anticipated the question. Since I'd been with William I'd more or less isolated myself from everyone I'd known. I couldn't possibly tell her the truth, that I'd been staying at home, immersed in a relationship that I worried about more and more with each passing day.\n\n\"I took some time off from school,\" I said. \"But I'm going back someday.\"\n\nHer face betrayed surprise and relief. \"I thought you'd have been in grad school by now. You seemed so brilliant. You were always throwing off the curve on tests.\"\n\nHer words barbed and shamed, though I doubted she'd wanted to hurt me. One thing you could say about Karen was that she was kind. She might not have been smart or attractive or ambitious, but she was kind, and no one would take that away from her. I turned the question around. \"So what about you?\"\n\nShe was hostessing at a Perkins Pancake House in Naranja, on the seedy deadbeat strip to the Keys\u2014most of which had been torn open by Hurricane Andrew. Already she'd divorced and married a second time, but the current marriage\u2014to a guy twenty years her senior, a former real estate salesman for General Development\u2014wasn't working. She was thinking of moving back in with her mother, if only to put some money in the bank. She was twenty years old. I couldn't have imagined a life so different from my own.\n\n\"And I have a baby!\" she cried.\n\nShe reached into her wallet and passed me a cellophane-wrapped picture of a baby, a little girl with blonde hair, blue eyes, and white shirt patterned with golden pachyderms.\n\n\"Her name's Lola. Seventeen months old. Don't you think that's the coolest name?\"\n\n\"Wow.\" I wanted to feel what I was supposed to feel. She so much wanted my approval, even though I could barely muster it, with Todd on my mind. What was she doing with a baby when she could barely take care of herself?\n\n\"Let's get together sometime,\" she said. Awkwardly, as if on impulse, she hugged me.\n\n\"We should,\" I answered. I felt her gripping me closer, her tears hot on the cloth of my shirt.\n\nShe laughed, sniffed, pretended she wasn't crying. \"Remember when the three of us went to the drive-in, and the tire blew out, and we had to walk six miles to the Texaco?\"\n\nI nodded. I'd had such a horrendous time that night, bickering with Todd, walking with him in a pair of thrift-store shoes that were one size too small, that I'd all but suppressed it. But maybe I'd had a bad attitude. Was my recollection any truer than hers?\n\n\"So give me a call, please? How's Jane, by the way?\"\n\nI shrugged. \"I don't know. I haven't seen her.\"\n\n\"You haven't? That surprises me. You two used to be like this.\" And she crossed two fingers together, holding them up for me to see.\n\n\"People drift apart.\"\n\nShe nodded. \"I never liked her,\" she said, admitting to something I'd always suspected. She rubbed her arm as if chilly. \"She always made me feel fat, stupid. Maybe I'm jealous of her.\"\n\nI grabbed her tightly by the hand and shook it, in full measure of my support. \"That's Jane. Well, we won't ask her along this time. How's that?\"\n\n\"Good,\" she said, laughing now.\n\nI walked away from her. The cars, headlights on, inched down the road toward the cemetery. Driving home, I knew that we'd never get together. I was in another life. I'd never see her again.\n\nThree weeks after Todd's funeral, on a night marked by the arrival of a cold front, with violent bursts of lightning, and eerie passages of calm, I locked myself in the bathroom and pressed a damp washcloth to my face. My stomach lining popped and burned. There was no question: I knew that I was infected, and that it was only a matter of time before I came down with it\u2014the night sweats and fevers, the rashes and gum disease, the odd tingling and numbness that would prevent me from walking.\n\nI crawled back into bed and sidled up to William, deliberately waking him.\n\n\"Do you ever worry?\" I asked him.\n\nHe opened one eye. \"What now.\"\n\n\"Don't you ever worry about AIDS?\"\n\nHe breathed deeply, patiently through his nostrils. \"We've been safe,\" he said blandly. \"What's there to worry about?\"\n\nBut we hadn't been safe, even though we tried to be. There had been that time or two when just before he'd rolled on the rubber, he'd pressed his dick inside me, just an inch or two deep, just for the thrill of it, just to test something. Wasn't that a part of it, the thrill, the daring? Wasn't that the thing that drove people to let down their guard, to stare down death in the face, or to finally stop fighting, because it took all of your resources to keep it at bay? I thought of the first time we were together, the hot afternoon on which I'd fucked him, the smell of cut grass in the air, only to find a rusty, bloody bloom on the top sheet after he'd left for the bathroom. I looked at the spot, thinking, we're never going back from here, we're linked together now\u2014William and Evan, blood brothers\u2014brutalized and comforted at once.\n\n\"We never talk about it,\" I said. \"Don't you think we should at least talk about it sometime?\"\n\n\"What's there to talk about?\" A grimace passed over his face. He twisted his hands together, once, not even knowing he was doing it.\n\nI sat up. I tried my best to sound rational, nonchalant. \"Maybe we should both get tested.\"\n\nHe shook his head from side to side, emphatic. \"No way. I'm over that idea.\"\n\n\"Why not? We'll just get it over with and we won't have this dreadful thing looming over us.\"\n\nHis face looked empty, bleak. \"It's always looming over us.\"\n\nHe stood. I followed him into the bathroom, watching him pour baking soda into a juice glass. Carefully, he filled it with tap water, stirred it with his finger, then sipped it as if it were champagne.\n\nI sat on the closed toilet seat. If I were sick, if my time were really limited, I wanted to be doing things. I wanted to be living like Todd did, having experiences, fucking, becoming somebody new, changing my name, somewhere on the opposite side of the country.\n\n\"Do you think you're positive?\" I asked.\n\nHe swallowed, testing the condition of his throat. \"I feel fine, if that means anything.\" His face looked brighter, calmer all of a sudden. \"How about you?\"\n\n\"I'm fine.\"\n\n\"That's good. That's all we need to know.\"\n\nI followed him back to the bedroom. Outside, the sky lightened notch by notch. I expected him to be resentful, silent. Instead, he turned to me and held me tight, breathing slowly, at regular intervals, until he was fast asleep, as if he'd convinced himself the world was nothing but a restful, safe place.\n\n# Chapter 10\n\nI found it in our mailbox, without postage, in a used, brown envelope once addressed to my parents. A ghastly photograph of Ed McMahon peered down from the upper left corner. As it had been deposited after 2:30, the last time I'd checked the box, I assumed that my mother had delivered it herself.\n\nDear Evan,\n\nI'm writing this in the kitchen. I want you to know that this is the first letter I've written in thirteen years. I know I told you that I've been writing about my life every day, but this is different. It takes a special kind of attention that I don't know I'm capable of these days. In any case, the house is quiet without you, and I miss your spirit in these empty gray rooms.\n\nIt was terrific to see you at the Publix. You looked better than I'd imagined\u2014though there was something about your eyes. (Are you taking zinc? Dr. Oglethorpe says that there's nothing better than a good dose of zinc for the eyes. One or two tablets a day\u2014I forget which dosage.)\n\nYou asked about Peter. Luck has it that he called not long after I saw you. He's bought a bankrupt motel somewhere near Naples. We don't know how on earth he came up with the money\u2014he doesn't tell us anything. Anyway, if you don't waste too much time, you can probably reach him at King Cole Resort, Tamiami Trail and Boca Palms Parkway, Naples, Florida. I told him that you'd like to talk to him and that you'd be getting in touch with him soon.\n\nThe fish are doing just fine, and we've gotten a new wall hanging for over the sofa\u2014you'd probably hate it, but there. Speaking of your father, he's driving me nuts, as usual. He's on his way to Austin, Texas, for some academic to-do, and it's a relief to have some time to myself. FYI: on the subject of your staying with us, he came to the conclusion that it would be too much right now. Let's face facts, dear\u2014you like the IDEA of staying with us, but once you moved back in, you'd be miserable. The thing is, you can't be two things at once. These things are hard for me to write, and that's one of the reasons it's taken me this long. Forgive me. I want you to know that I love you, and miss you terribly, but, damn it, it's hard. When I think of you living just down the street, the way you've chosen to turn your back on us, it's hard not to feel hurt. And a little rage. You'll see things sooner than you think, perhaps when you have children of your own. We're waiting for you, Evan. Please make up your mind. We love you more than you'll ever know.\n\nMom\n\nI read the letter twice more before I tore it in half, then pieced it back together with strapping tape. Goddamn, I thought. They were locking me out. I really had no place to go.\n\n***\n\nWilliam's solution to my seething rage was to take me out to a party. He received the invitation from Cleve Stern, a sound engineer from Channel 7, one of the few employees he was officially \"out\" to, so I assumed there'd be people like us there. William wouldn't tell me anything specific about the party, except that it would be taking place in a loft in an industrial area of Little Havana, and that I shouldn't worry too much about what I was going to wear.\n\n\"What kind of party? Who's going to be there?\"\n\n\"You'll see,\" he said, nudging me onward. \"Hurry up, get moving. It starts in forty-five minutes.\"\n\nI felt a dread coupled with a low-key anticipation. We hadn't had much of a social life. All these months it had been just him and me, and I'd been feeling the pressure of that, the sense that we needed to open ourselves out. It made me wonder why he'd waited until now to do this, when we were practically crawling around on our hands and knees. But any change was potentially refreshing, especially from someone who was so dependent upon routine.\n\nIt took us a good fifteen minutes to find a parking space. The neighborhood wasn't great. I looked up to the sky above the warehouses and high-tension wires, lines of pink and red light creeping across its map. I thought, this is the Florida that no one sees in postcards, the Florida that's always there behind the palms, the bronzed bodies, the swept, glittering surfaces.\n\nInside, the loft we were met by a line of men in a graffiti-scarred hall. The music was loud beyond imagining. I recognized the song, a remix of \"Rude Thing,\" by some alternative band that Todd had introduced me to.\n\nYou can kiss me, you can torch me\n\nYou can touch me, you can scorch me\n\nCause you just mean nothing\n\nYou just mean nothing to me\n\nIt certainly didn't seem like William's kind of party.\n\nThe line inched forward. We stood at a table where two men sat in nothing but their jockey shorts beneath a hand-printed sign: SUGGESTED DONATION: $10.\n\nI turned back to William. \"You mean we have to pay?\"\n\n\"Quiet,\" he said, passing a ten-dollar bill into my hand.\n\n\"And who's this?\" said one of the men to William. He was talking about me. He smiled. In different circumstances I'd have liked his chiseled chest, his lacquered Caesar haircut, shorn on the sides, but now, for whatever reason, he reminded me of Frankenstein's monster.\n\n\"Where do we go?\" William said, unnerved.\n\n\"Around the corner, to the right. You can check your clothes in there.\"\n\nI blinked repeatedly, not yet processing these instructions. I followed William down the hall, and, to my left, I saw it: a great room, perhaps forty feet long, its concrete floor covered with a plastic tarp, on which a hundred men of all ages, races, and levels of attractiveness, were clumped in threes and fours, standing in their perfect underwear and Doc Martens, leaning into each other, jerking off.\n\nI stood there for a minute, intrigued and repelled at once. The participants were suffused with a crimson otherworldly light. A vision of terror and beauty. A J-O party, of course. What, then, were William and I doing here?\n\n\"Are you okay with this?\" he asked. He seemed more than a little bewildered.\n\nMy brow tightened. I didn't know how I felt. \"Why didn't you tell me?\"\n\n\"I thought it would be a surprise. I thought you'd get a kick out of the surprise.\"\n\nI shook my head, a numbing sensation between my brows. My head itched. I followed him down the hall. We took off our clothes, shoved them inside numbered Winn-Dixie bags, and handed them over to the clothes check, a sweet geeky fellow who might have spent his days as a programmer. I imagined the flashing beeper on his belt, summoning him to fix the bug on some trust-accounting system so the dividends could get posted by day's end.\n\nWe walked back to the big room and observed, like an audience at some Roman bacchanalia. It wasn't what you'd think, though. A lot of it was just plain dull. I got the feeling that most of the guys liked the notion of the spectacle more than anything else, longing for intensity and pleasure, hoping to confront, then push past the boundaries of shame. Did they leave feeling empty, unsatisfied, going home by themselves, more solitary than ever before?\n\nAround us \"Rude Thing\" spun out from the speakers, the longest dance track in history.\n\n\"There's Cleve,\" William said.\n\nI watched Cleve, a randy, bespectacled thing with an amazing physique for someone of his age, fooling around with a kid twenty years his junior. At one point he glanced toward us, and\u2014to our discomfort\u2014waved. At once the cracks in his armor asserted themselves. I saw how Cleve had grown up thinking he was unattractive, knowing that his opportunities for sex were limited by his essential homeliness. I saw how his first visits to the gym at thirty had literally changed his life, how with the simple repetition of a few free-weight exercises four times a week, he could transform himself into the man he'd always wanted. I saw how his fooling around with the young muscled boy in these circumstances was all meant for display, not fun, how it was ultimately about declaring his status: Hey, boys, look what I got. I'm not worthless anymore.\n\nI wanted to relax into the situation, but I couldn't stop stepping out of it, couldn't put to sleep the critical apparatus\u2014a necessary condition if I were to have a decent time. But already I saw myself fifty years in the future, telling younger people about what was transpiring before me: This is what we did to amuse each other, to kept ourselves from going crazy in the age of AIDS.\n\nNot ten seconds later, a middle-aged man\u2014a leather-daddy type\u2014walked up to me with a posed, stern expression on his face. I felt a twinge of erotic feeling, a pleasant discomfort, though I couldn't get beyond the fact that William was standing beside me, territorial, possessive. I couldn't get into this. I held up my right hand.\n\nThe guy nodded, walked away, moving on to someone more willing.\n\n\"Creepy,\" William said, his eyes fixed to the guy's thin white waist.\n\nI shrugged. \"He wasn't so bad.\"\n\n\"You're kidding.\"\n\nI dulled the expression on my face. On some profound level, I'd wanted to provoke something akin to jealousy, though I couldn't say why. The air vibrated between us. I looked over at William, his grim, hopeless face, his pale, pockmarked back, and imagined him tasting flavorless in my mouth. Above our heads, a poster: Five Easy Steps to Safer Sex. I still didn't know what we were doing here.\n\n\"Now that's cute,\" William said, pointing to a kid across the room.\n\nWilliam kept staring at him with an intensity that seemed forced, inappropriate. I didn't see what was so great about him, but I kept quiet, trying to make sense of his interest. Maybe it was only because he looked like me, and I liked men who were older, butcher, who'd been around the block a bit. I'd have taken the leather daddy over the kid any day, but again, I kept my mouth shut. Soon enough, William's stare had drawn the kid toward us\u2014a calf to a block of salt.\n\n\"Howdy,\" he said, quizzical.\n\nI smiled, in welcome. Something hooked into my stomach. He stepped closer to the kid, then, to my confusion, stepped backward, disengaging himself. He nodded toward me, as if in collusion. What was going on here?\n\nThe kid leaned into me, kissed me ripely with his full red lips. His tongue was scratchy and sweet as he reached up, rubbing the knots from my shoulders. It felt good, unbearably good, but I couldn't get into it\u2014at least not with William standing there. The hook snagged deeper into my stomach lining. Was I going to be sick? I realized then that William didn't want to participate, that he only wanted to watch, and that was the sole purpose for bringing me here.\n\n\"I can't get into this,\" I said, and pulled back from the kid. \"I'm sorry.\"\n\nBoth William and the kid looked at each other. The kid's eyes seemed hurt and alarmed.\n\n\"I'm getting dressed,\" I said, and hurried toward the clothes check.\n\n\"I thought you'd be happy,\" he called out after me. His footsteps were brisk down the hall. \"I did it for you.\"\n\n\"Well, thanks,\" I muttered to myself.\n\n\"Huh?\"\n\n\"Fucking clueless.\"\n\n\"Fucking what?\"\n\n\"Fucking me.\"\n\nWe sat in the front seat of the car without speaking. My back teeth ached to their roots. I felt something deeper than rage, a blank, zeroed-out place, a null set. I couldn't find my foothold, for everything was rocking now, spinning too fast.\n\nI went straight to bed. For whatever reason, I pulled out the obituary of Todd, which I'd kept inside the night table, and started reading it to myself. William stepped into the room, glowering, wounded.\n\n\"Are you still reading that?\" he whispered.\n\nThey were the first words we'd exchanged in close to an hour.\n\n\"Yes, I'm still reading that. What's it matter to you?\"\n\nHe shook out a pillowcase he'd unfolded. He wagged his head. \"This isn't healthy. This feels inflated.\"\n\n\"He was twenty years old!\" I yelled.\n\n\"What does that have to do with anything?\"\n\n\"Everything,\" I said.\n\nHe worked the case over the pillow. \"Fine. Just don't expect me to participate.\" Then he crawled into bed, switched off the lamp, casting the room in darkness.\n\nThe burning crept higher in my stomach. My mind started wandering. I thought about the fact that he hadn't encouraged me to work, hadn't encouraged me to contact all those schools that had once accepted me. Or to make friends, for that matter. My thoughts were all over the place, scattered, as if hammered around the edges. What was I doing here?\n\n\"What am I doing here?\" I said, repeating my thought aloud.\n\n\"Immature,\" he muttered, staring at the ceiling. \"Don't be so fucking immature.\"\n\n\"What?\"\n\n\"You heard what I said.\"\n\nHe'd hit bottom\u2014a declaration that insisted only upon his age, his power over me. Something against which I couldn't stand up. \"At least I'm not shut down,\" I said after a few minutes. \"At least I've given myself permission to be affected by things.\"\n\n\"La tee da,\" he said.\n\nFine, I thought. Take the last word. You don't know how to fight anyway. Or I don't. Our gestures said it all: this wasn't working out. I decided that minute: it was over, kaput. Fine.\n\n***\n\nI waited three days. I picked a morning on which the sun wasn't too hot and the wind was calm, a light southeast breeze pushing the fried leaves across the lawn. The weather was good. We might have been someplace ghastly, beautiful like Barbados, Antigua\u2014islands to which I'd never been, but had imagined as real. I drove William to work, taking the long route on the bay front, past the mirrored towers and the marinas, actually feeling comfortable in his company. How civil we were, how amiable\u2014no allusions to what had been happening in the last few weeks. Any bystander would have thought that there was nothing lethal between us. We were perfect for each other, no? I'd given him not one inkling of what I'd worked out, which buzzed me and prodded, slapped me high about the head as if I were stealing something expensive\u2014a watch or a jewel.\n\nI cleaned the house from top to bottom. I cleaned the sinks and the showers, the oven and the baseboards, the tile grout, the exhaust fan, the medicine chest. Then I went to work on the garage. I could have been Astrid Muth, my mother's fanatical, desperate friend, who cared for her yard with such scrutiny that she literally probed through her flower beds in order to stop the weeds before they'd sprouted. I got it now. I was something like Astrid. After an entire afternoon, only two hours before I was to pick William up at the station, I set to work on my backpack. As I wanted to travel light, I didn't pack much. Some T-shirts, a pair of jeans, shorts, underwear, jacket, sweater, flashlight. The rest of it? The truth was I didn't need anything more than that.\n\nI walked into the den, the room in which I'd been scheduled to sleep on the day of my arrival. What had I been thinking? I stared at the rusted pipe on the floor. The water murmured. And all at once a scorpion pulled itself out of the neck, nearly translucent, creeping over the rusted rim.\n\nI paused once before the front door, patting the Dobermans twice, three times upon the head. \"You be good to Daddy, now. Will you miss me?\" I spoke aloud to the empty white walls. The house was still, unbearably still, yet starting to turn. \"You really weren't a bad guy,\" I said. \"Son of a bitch.\" And then I left.\n\n# Part  \nTwo\n\n# Chapter 11\n\nThey huddled beneath a lone palm in their bright yellow jackets. I walked faster. Something twinged my shoulder blade. I had to tell myself that the simple fact of my appearance was not the prime reason for their gathering, that they didn't exist to taunt, to yell faggot to any lone boy who happened to walk through the field. Clearly, I said to myself. See clearly. They were simply conferring, talking about tonight's football game. Nothing wrong with that. Paranoia. Residue, it seemed, from an earlier time.\n\nI walked along the treeline. My sneakers caked with marl. It took all of three hours for me to start asking myself why I left. He wasn't the worst of partners. He wasn't condescending or surly; he wasn't unctuous, lunkheaded, dishonest, or cheap. His breath was fresh, pleasant; he smelled good\u2014lavender and sea salt, a surprising mix. Each Wednesday night, as part of his weekly caretaking ritual, he tweezed the hairs from his earlobes, back, and shoulder blades, leaving behind a trail of fibers, like carpet threads, all over the bathroom tile.\n\nIt occurred to me that I had hurt someone.\n\nSometimes in passing moments I still wonder what he might have thought when he finally returned that night from work, having taken a cab home from the station, only to see the car lolling in the driveway. Was my absence in the house instantly palpable? Or had he dropped into his favorite chair, switched on the lamp, called out Evan\u2014only to realize minutes later that I had really gone? Was he angry or wounded, searching urgently for a note, an explanation that would never surface? Or did he feel a certain satisfaction, a smugness about it all? Could he have said to himself: He will be back?\n\nHe really wasn't a bad guy.\n\nI found the pipe at the end of the path. The scrub had thickened since I'd last been out here, denser since the brushfire, but I appreciated that, knowing I'd be sheltered, safe. Vapor trails stitched across the sky. I eased down into the pipe feet first, pulling myself across its floor, its cylinder surprisingly warm, dry, cocooning. I fumbled for my shirt and balled it behind my head. I had no good reason to feel this good. It was hard to believe that so few people had ever known the humble pleasures of sleeping outside.\n\nThe sky went dark. In the distance I heard the boys, more ominous now, their war cries coiling up into the trees. They were starting a fire. Laugh it off, I told myself. Means shit. And then I fell asleep, not waking up until a full night later, when I sat up, feeling a seepage of water on my shirt.\n\n***\n\nI started walking. I stored my backpack up inside a rotted palm\u2014\"Old Merv,\" I called it\u2014freeing me up to go anywhere. Coral Gables, Westwood Lakes, Medley, Aladdin City, Cutler Ridge, Coconut Grove\u2014I walked through half the towns of the county, over dams and drainage ditches, past service stations trembling with orange strips of neon, jogging, then running, then jogging again. A chevron of sweat dripped through my shirt. A white ecstatic granule vibrated inside my head, and I wanted to aim myself into that place, to break through its crust and explode it, to be that\u2014that calmness and void, that place without movement.\n\nI walked down a street on which most of the houses had been totaled by the hurricane. Roofs pulled from their joists, stucco stripped, fancy furniture still floating in the pools. There was something incongruous about the wreckage, for the worst damage had been centered ten miles south, in Homestead, Naranja. This region had escaped the fiercest sustained winds, but thunderstorms had spawned tornadoes here, imploding the houses as if they were toys, only to leave the next block unscathed.\n\nUp ahead, something called to me. Soon enough, I was walking toward the house of Douglass Freeman. Dickless, who'd moved to California years ago.\n\nNo matter how many others had lived here after his departure, it was still his house to me. In some inexplicable way its walls contained his very spirit, his bafflement, despair, and slanted curiosity. I stared upward at the roof beams, sheared from their trusses by the winds. The house was hopeless. The most recent owners, gone for good now, had spray painted the garage door with harsh jagged letters: ANDREW, GO HOME. Then beneath that: ALLSTATE DOESN'T DELIVER and their policy number.\n\nI stepped up the driveway. The street was empty now, a ghost town of ruptured houses. No one was watching. The front door opened as if its former occupants had been expecting me.\n\nIt was worse than I'd imagined. Insulation sagged; walls buckled under the displaced roof. Mildewy curtains flapped on a rod above the open back wall. Beyond that, the pool cage. It lay twisted in a particularly crude fashion. I studied the family portrait above the mantel in which a father, a mother, and two little boys\u2014all with bottle-thick glasses\u2014were trying desperately to enjoy themselves, to show the photographer\u2014a dense, falsely enthusiastic teenager, I decided\u2014they were a well-adjusted family. They'd left everything here, throwing up their hands. I rifled through a desk to find something with their names. I found a short stack of wet envelopes\u2014tithing envelopes for St. Michael's Parish. Mr. and Mrs. Joseph Carson.\n\nI sat upon a ruined pit sofa. A dome light lay perilously on its side. For a few seconds I imagined the Carsons in the contented period before the storm, milling about the rooms, carrying presents to each other at Christmas, the particular smell of this family together\u2014fabric softener, reheated turkey, tube socks, carpet freshener\u2014still wafting through the house. But overlaid were my thoughts of Douglass Freeman. I pictured him sitting in one of the chairs in the dining room, his arms folded over his chest, shoulders hunched, trying not to contort his face as he told his parents he wouldn't ever, ever go back to Gus Grissom.\n\n\"We'll see what we can do,\" Mrs. Freeman said, glancing over at her husband. He sat opposite from them, an older version of his son, slouching forward, thinking.\n\nI walked the halls. I chose the back bedroom, a tiny cubicle in which a Spirograph box with faded, soft corners lay on the bureau. There was no question, I felt him here. And soon enough my premonitions were confirmed. I saw his name, and beneath that a date\u20147\/9\/73\u2014etched above the light switch in red pencil.\n\nI lay on Douglass's bed, closing my eyes, drifting. He was out there somewhere, but I was falling asleep, and the world outside was shifting, slipping away, too fast for me to hold onto it.\n\n***\n\nI was down to zero, zilch. No job, no car, not even a credit card for emergency's sake. I used to think that I was safe from bad fortune, that personal catastrophe was ultimately self-willed, that we all have the power to make our own luck. But that's bullshit.\n\nI plodded up Dixie Highway beside the buzzing transformers. I might have been Peter years before. Peter, who'd landed the greatest drop dead on my parents' threshold by leaving the house without warning one day. Ever since, he'd been only two things to me: a torn postcard from Beach Haven, New Jersey\u2014I'm happy, but you're not\u2014and a vague standard of disaffection and disillusionment against which I measured myself.\n\nNow he was where I was going.\n\nI'd finally allowed myself to call him. \"So, could I stay with you for a while?\" I'd said.\n\nMy tongue tasted hot, grassy in my mouth. The tip of a palm frond squeaked against the phone booth. I'd just watched an older man\u2014formal, crisply dressed\u2014holding out a cupped, tentative hand to passersby at the Coconut Grove Metrorail. He looked like he'd never fallen this low, all his dignity coming down around him like rags.\n\n\"How long?\" Peter said after a silence.\n\nMy throat went dry. \"If it's a problem, I understand.\"\n\n\"No, no. I'm just surprised.\"\n\n\"Well, I've been thinking about you.\"\n\n\"You have?\"\n\n\"A lot, actually.\"\n\nA siren wailed somewhere in the distance. Wind rattled the seagrapes, foam cups blowing across the street. \"Did Mom put you up to this?\" he said warily.\n\n\"Are you kidding?\"\n\n\"Well\u2014\"\n\n\"Daddy doesn't even want me at home anymore.\"\n\nHis voice thawed, relaxed, as if we'd gone crabbing together that morning. I tried to block out my last picture of him: crying, high, after my father had discovered the bag of bright pink pills in his room. The ground might have heaved beneath my feet. His voice went on and on. And all at once I heard him say yes.\n\n\"So I'll pick you up in Fort Myers?\" he said. \"You'll take the bus?\"\n\n\"Right,\" I answered. \"I'll call tomorrow.\"\n\nI walked up Jane's sidewalk now, pressing on the bell. The doleful notes of an oboe drifted out over the roofs of the neighborhood. The door opened, and she stood there in lycra shorts and black T-shirt, cradling her instrument in her hands. Her hair was brighter, the color of cranberries.\n\nShe troubled her chin with her fingertips. \"You look awful.\"\n\n\"Thanks a lot.\"\n\n\"You were in a fight?\"\n\n\"No.\"\n\n\"Why is your hair such a mess?\"\n\n\"I walked out on William.\"\n\n\"You what?\"\n\nI nodded once, twice. \"You'd look like this, too, if you'd spent your last two nights outside.\"\n\n\"You what? Oh God. Come in here. Come in here and sit down.\"\n\nWe sat upon her parents' couch beneath the harlequin pictures. I talked for a good half hour. I told her about the breakup of the relationship, filling her in on the grittiest details as she listened attentively with the gravest expression. She fidgeted with her bracelets on her wrists.\n\n\"That's awful,\" she said.\n\n\"I know.\"\n\n\"I had a feeling something was up.\"\n\nMy head hurt. I leaned over, massaging my eyebrows, making circular motions with my thumbs. For some reason, I couldn't say another word. \"And what about you?\"\n\nIt turned out that she'd started oboe lessons again. She'd found a stern, demanding teacher, a retired player from the St. Louis Symphony, who was pushing her, excited about her potential. Even though she'd put off college for the year, she felt ready now. In only a few short weeks, she was auditioning for Juilliard, Eastman, and New England Conservatory.\n\nThen, to our mutual discomfort, she'd run out of things to say. Inside the kitchen, Mrs. Ettengoff was tossing avocados inside a food processor.\n\n\"I just came to say good-bye,\" I said after a silence.\n\n\"What do you mean?\"\n\n\"I might not see you for a long time,\" I explained.\n\n\"Like I'm not going to see you again,\" she said with a throaty, abrasive laugh. Had she been smoking again? \"Naples is only two hours away, for God's sake.\"\n\n\"I'm not kidding, Jane,\" I said quietly.\n\nShe saw the seriousness on my face. She turned toward the kitchen and left for a moment without excusing herself. The dense smell of yeast hovered about the rooms. Over the whir of the food processor she and her mother carried on a conversation.\n\n\"Do you have enough money?\" she said upon her return.\n\n\"Of course.\" I nodded twice, though she knew me well enough. My forehead felt tight, sunburnt. I hadn't been drinking enough water.\n\n\"My mom says you're welcome for dinner.\"\n\nI shook my head from side to side. \"I have to go.\"\n\n\"You'll call if you're in trouble?\"\n\n\"Of course I will.\"\n\n\"Take care.\"\n\n\"You take care,\" I said, laughing now. \"And good luck with those auditions.\"\n\nWe stood, holding onto each other by the front door. Her right hand wandered down my back. I tensed slightly, resisting the warmth of her touch as she slipped something\u2014two fifties and a ten, I found out later\u2014into my pants pocket.\n\n***\n\nBefore I left for Peter's, I needed to do something.\n\nI waited until I knew he was at work. More than anything I didn't want to see him. I knew he'd be livid; I imagined the thick cords of his fingers tightening around my throat. It was one thing that I'd departed without a note or good-bye when I was still living with my parents. It was quite another that I'd done it after living in his house for months, when I theoretically knew better. Perhaps I was too hard on myself. I didn't owe him my future if he wasn't willing to work for it. But I at least owed him an explanation, some acknowledgment of the fact that something important had transpired between us. I wasn't a kid anymore. My decisions mattered, had consequences, and I carried within me the lulling, intoxicating power to hurt another human being.\n\nBut right now I was thinking about my notebook. It was the one physical thing that I needed in the world. It was the repository of so much\u2014addresses, quotes, phone numbers, clippings, lists, equations, names of books, fractured diary entries, even doodlings\u2014things so important to me that I felt unwhole without it. Leaving it behind had been nothing more than an indication of my fissured mental state. Not a page of it was blank, but I still managed to scrawl things inside, as if I inherently believed in its possibility to always yield some space to me. In its details one could piece together the narrative of my life, and I couldn't leave it with him no matter how many times I'd said: Just let it go.\n\nHis car was gone. I stepped over the driveway's expansion joints, glancing over the anise hedge to see if Virginia, the neighbor with the beautiful begonias, was watching. She wasn't. I made it to the mud-room door, fumbling over the door frame for the key, but what was this? No key. The spare, the ever-reliable duplicate, wasn't here. My eyelids burned. I read it as nothing more than a statement to me: You're not welcome here. You've disappointed me. Fuck you. Keep out of my house.\n\nI might have broken the windows, each and every one of them. Instead, I sat on the slab and pressed the heels of my palms into my eyes. Calm, calm, I said. Calm yourself down. After a few minutes I remembered the loose window over the bath-tub, and before long I was crawling inside, headfirst, my ass hanging over the sill for all of Avenida Manati to see.\n\nI walked swiftly through the cool rooms, my bare feet shushing the white terrazzo. Absolutely nothing had changed. The rattan lamp, the stacks of newspapers, the framed Toulouse-Lautrecs\u2014every one of them assumed the same coordinates they always had. Even the dogs, Pedro and Mrs. Fox, hadn't barked upon my intrusion, only glancing at me in utter ennui before retreating to their respective beds as if I were still the resident boy.\n\nGiven what he'd put me through, I decided it was my right to enjoy some air-conditioned comfort and a good hot shower before conducting my business. I still had two hours. At that moment he was most likely seated behind the camera, squinting into the lens with his astigmatic eye while the technicians cued up the theme music for the Florida Noon Report with Sally Reedy. First, though, I needed to eat.\n\nI breathed in the chill of the open refrigerator. Sun-dried tomatoes, yogurt, pears, and raspberries\u2014I had it in me to eat them all with my fingers, swallowing without benefit of spoon or fork before opting for the civilized. I hadn't realized how hungry I was. And I hadn't realized how fortunate I'd been. The life I'd once taken for granted now seemed decadent and lush, a golden treasure inside a chest. But should I have stayed here? I already knew the answer to that.\n\nI sat on the Chesterfield. Without thinking I aimed the remote at the TV, only to discover that the Channel 7 News was already in full swing, seventeen minutes into the broadcast. I couldn't not watch, if only to satisfy some morbid curiosity.\n\nSally Reedy blathered on about the development of a tropical storm sixty miles southwest of Great Abaco. Sally's face betrayed a curious combination of motherly concern, wry sex, and grace. The idea was to convince the viewer that the set was a purely jovial exercise, that the staff was comprised of good pals who met five days a week for the sheer fun of it. Off camera Sally could be distant to the point of downright cold, though rumor had it that she was suffering terribly, that her gynecologist husband was about to be arraigned for the alleged rape of four of his patients, that once this broke on her own broadcast, Sally's celebrity (and credibility) would forever be dimmed.\n\nIt was perplexing to think that such gossip was no longer a part of my life. I prepared myself to switch off the TV when an auxiliary camera panned the behind-the-scenes staffers, and in an utter affectation of highjinks, they mugged in feigned horror at the prospect of the developing tropical system.\n\nWilliam's face flashed briefly before the screen.\n\nHe looked fit, trim, the hair on the sides of his head just a little too long.\n\nHe sickened me. It was strange to have spent so much time with somebody, only to find that with each passing week I knew him less and less, instead of what one would have expected. It certainly hadn't started like this. After I'd first moved in with him, we were still careful to wear our best faces, to grant tolerance and respect for each other's tics and idiosyncrasies. Occasionally I saw that darker knowledge passing over William's eyes: Who are you? What are you doing in my life? I didn't ask for you. I only want to be myself in the world. I'd walk into the woods behind the house, picking up stones, bottles, Coke cans, throwing them as high as I could into the dull blue of the sky until my wrist snapped, until my joints ached. How could you have done this? I loved you, you jerk. I've given up everything I had\u2014my family, for God's sake\u2014for you. You owe me much more than this. But I'd walk back to the house, tasting disappointment like weeds in my mouth, knowing I wasn't the easiest person to live with, that two people could still love each other and not be able to forge a life.\n\nWhy did this seem to be one of the most difficult things to learn?\n\nI turned off the TV. I walked to the bedroom and found my notebook in the night-table drawer. I drank in a breath and looked once about the room. I concentrated on its details, its rugs and books, its curtains and bedspread, taking a mental photograph. Then just as I was leaving, I spotted William's flannel shirt, with its intricate plaids of tan, yellow, and forest green. I raised it to my face and breathed, naming in detail what I smelled: its resinous cotton, its suggestion of sweat once his Right Guard gave out, its piquant smoky odor, like pine, or roasted cypress chips.\n\nI pushed my arms into its sleeves, vaguely comforted by its possibilities. The shirt would come to good use. I was going on a trip. It was chilly at night on the other side of the state, with temperatures expected to dip into the upper fifties.\n\n# Chapter 12\n\nImmediately I picked out my brother from the small crowd at the bus station. My heart shuddered. My toes curled inside my sneakers. His oblong forehead, his benign dolphiny smile, those long-lashed, yet lusterless eyes\u2014all these features suggested that he was a younger, better-looking version of Sid, with more hair and less baggage in the stomach region.\n\nWe hugged each other at the curbside. He was bigger than I remembered, his body softer, wider, warmer. He smelled of deodorant, chewing gum. There were little creases around his mouth. We started laughing. He took my bag from me, then led me to a van parked beneath some nutrient-starved palms. He dressed exactly as I'd expected: khakis, chambray shirt, wire-rimmed aviators\u2014practically the same uniform he'd always worn.\n\nWe stood at the back of the van. On its pitted bumper were two stickers: a radio station's call letters in the shape of a guitar, and a purple sticker with chipped edges: PRACTICE RANDOM ACTS OF KINDNESS AND SENSELESS BEAUTY.\n\nHe grinned at me. \"I can't believe you're here.\"\n\n\"It's kind of weird.\"\n\nHe blinked rapidly for a second, then calmed himself. He really did look a lot like our father.\n\n\"I mean, a good weird. Who'd've imagined this a few days ago?\"\n\n\"I know,\" he said, clapping me on the shoulder. \"I didn't think I'd ever see you again.\"\n\nWe slouched in the bucket seats. I stared through the fly specks on the windshield. I pressed the back of my scalp into the headrest, my bare legs sticking to the cool, damp leather. My head felt blasted. Throughout the bus ride, I'd churned with worry. I sat for hours and hours, staring at the truck farms and phosphate mines, digging for opening lines, most of which were of a highly adult nature, engineered to generate conversation. Meanwhile, the driver shouted out the names of each stop: Moore Haven, Port LaBelle, Lehigh Acres. Well, maybe we didn't have anything to say to each other, but that seemed to be all right. He wasn't making an issue of it.\n\nThe landscape shimmered and prismed in the sunlight. All around us vast stands of slash pine and swampland were knocked down and filled for new construction. Billboards everywhere announced splashy white houses on cul-de-sacs\u2014The Pinnacle, The Brightness, Indigo Lakes, Pineapple Lakes\u2014all of which ruptured the virgin terrain. Businesses serving the dwindling rural population foundered on the roadsides, their days already numbered: Beverage Barn, The Shellpile. A Christian campground: Don and Veda Shea's Hallelujahland. It might have been the Gold Coast thirty years before. Florida, place of our birth, chewed up its marshes, sucked its aquifers dry until there was nothing left but a parched bed of limestone. Already it was a busted balloon, a bloated appendage at the end of the United States, a veritable Toledo, Ohio, with palms. A vessel of people's myths, its image had nothing to do with its essential character\u2014the turpentine state. Once that realization was assimilated on a wider level, the cities themselves would wither and burn, and the people would forsake them, moving on to ruin someplace else.\n\nStill, I loved it more than any other place. Florida, oh Florida. Embodiment of wrecked dreams.\n\nI couldn't believe I was sitting next to my brother.\n\nWe sped down the Tamiami Trail in silence. He slowed. We passed a fence, a row of broken gaslights, turned right, signal ticking, at a stucco wall with missing letters. KIN CO E. The place looked deadened, vacant. Peter parked in a back alley. He stopped, got out of the car. He led me up a flight of open stairs before walking me down several long, long halls. I couldn't take it all in. My sinuses filled, pounded, a stuffed duffel bag in the center of my head.\n\nThe place needed some work. The hall carpets, a dull turquoise, carried a complex union of scents: smoke, mildew, spilled fluids. Cracks spidered the length of the pink stucco walls as if the building were gently insisting upon the presence of the former mangrove swamp beneath it.\n\nPeter worked a key into a lock, then flipped on a switch. The room was about as big as the old space William had offered me in his den. A skyscraper collapsed inside me. I forced a look of gratitude in my eyes.\n\n\"Well, what do you think?\"\n\n\"Great,\" I said too loudly.\n\nHe lifted his brows, then clenched them. \"Do you want to take a shower? Would you like some towels?\"\n\nI slapped my back pocket for the whereabouts of my wallet, a paranoid gesture for which I was famous. Once I felt it, I dropped down upon the bed. \"No, thank you. Do you mind if I just sit here for a while? I need to be still.\"\n\n\"You're hungry, I guess.\"\n\nI wasn't. I bit into my lip, harder than I thought, until my eyetooth pierced the skin. I winced. He sat down upon the arm-chair across from me, then tilted his head. His eyes glowed behind his glasses. I ran my tongue back and forth across the puncture, my mouth tasting of zinc.\n\n\"Are you sure this is all right?\"\n\n\"You're so nice to do this.\" I looked carefully at the room for the very first time, its wood floors, its perfect rows of laminated shelves. I leaned backward, propping my weight with my elbows. I pulled up my feet on the bed. \"You did all this work yourself?\"\n\nHe nodded, then told me how he'd gone about remodeling the room, an intricate, laborious process. About two minutes into the tale, I began picking out random words, phrases. I breathed deeply through my mouth, holding it, telling myself deep, deep, deep, deep, until my heart slowed, until my nostrils opened up, my face relaxing. I could smile again. My cheeks felt hot. For all he knew, nothing was happening inside my head.\n\nSilence. Then footsteps down the hall. I drew my arms tighter to my ribs. \"Who's that?\" I said.\n\nA man in a white T-shirt, cutoffs, and combat boots appeared in the doorway. \"Well, look who it is,\" he said with a sly, gritty whisper.\n\nAbruptly, my brother stood, almost losing his balance.\n\nThe man offered me a dark, veined hand with oddly thick fingers. \"Hector Ybarra.\"\n\n\"I'm Evan,\" I faltered.\n\nA pinch in my back, a chill. We held each other's gaze longer than was exactly comfortable. There was something about his voice: arrogance, authority. A certain \"fuck-you\" quality. Yet his eyes hazed with a funny, self-deprecating glint. I stared at the exquisite curve of his deep pink lips, their fullness, his dark, nearly black skin. Was my brother watching? My throat tightened, then burned. Shit.\n\nHis T-shirt was imprinted with the logo STAR BOY IN FLAMES.\n\n\"Hector's my assistant,\" Peter said. \"You'll be working with him, so you better get used to his jokes.\"\n\nI couldn't lift my head. \"Oh really?\"\n\n\"Yes, really,\" Hector mimicked.\n\n\"Are you done for the day?\" Peter asked, creasing his forehead.\n\n\"More or less,\" mumbled Hector.\n\nPeter turned to me. \"You have no idea how much we need you right now. Things're a mess.\"\n\n\"How come?\" I said.\n\n\"All our staff just quit.\"\n\n\"Oh boy,\" I said. And all at once I knew we were sunk.\n\n\"Well, the RoadStar lured them all away. People who'd been here for five years. Bigger money, day care, benefits... I mean, I wouldn't say no if I were in their shoes.\"\n\nI tried to look at Hector but couldn't. Why was he staring at me like that? His gaze felt fraught, heavy on my head like a hand.\n\n\"And of course the worst situation happens when we're least equipped to deal with it. A bus full of senior citizens, get this, thirty widows, breaks down by the bridge and they're all tearful and shrill. We're filled up, but it's not good. They want extra towels and hot coffee, and before we know it, they've broken both ice machines and practically blown out the switch-board by calling home to Bountiful so many times. By the time they left we actually put a NO VACANCY sign in the office window and slept for two days straight. I didn't even get up to drink or pee.\"\n\nMy brother's eyes went glassy. He clearly enjoyed talking about the struggle as it were a soccer game. He was more like my father than he knew.\n\n\"So is this what I'm supposed to expect?\" I said.\n\n\"Not all the time. But I just wanted to let you know. You never know what could happen.\" He grinned again, revealing a darkened eyetooth. Root canal.\n\n\"Welcome to Planet Hell,\" said Hector.\n\n\"Hey\u2014\" said Peter.\n\n\"God's Little Acre, God's Little Asshole\u2014\"\n\n\"Now stop,\" Peter chortled. Forced, I thought. Nervous.\n\nHector shrugged his left shoulder. He blew out some air through his lips. \"It's good to meet you,\" he said with a smile. He squinted a bit, raising his chin, showing off the squarish nugget of his Adam's apple. \"Welcome, my friend. You're in for a big adventure.\"\n\nHe left. Some residue was suspended in the air, an explosion of sorts, a storm of fine particles. Ten seconds later, I still felt his presence in the room.\n\nPeter stood. \"So if you need anything, don't be afraid to let me know. I'll be just down the hall.\"\n\n\"Sure.\" I reached down to unlace my hightops. \"Peter?\"\n\n\"Yeah?\"\n\n\"I really appreciate this. Thank you for being my brother.\"\n\n\"I'm glad you're here,\" he answered. \"We're going to get a lot accomplished.\"\n\nAccomplished, I thought. I lay down on the floor as Peter left the room. My pulse started racing, but the worst part was over. Anything might have happened. He could have mouthed off about me. I could have mouthed off about him. It was all so civil and pleasant and sweet. Listening to them knocking about the floor below me, I had a very distinct impression that it was going to be all right, and even if this weren't true, there was Hector, whose very presence reminded me that I wasn't alone.\n\n***\n\nOnce, I couldn't stop looking at him. I loved nothing more than the bristles on his face, the depth of his voice. I crouched in the upstairs hallway where I knew he couldn't see me, peering through a crack in the door. He stepped out of the shower stall, his wet hair splayed across his scalp. A pang in my gut. A smell\u2014stale ice? iron?\u2014wafted through the room. In truth, he was the first boy I'd ever fallen in love with. (His dusky eyes, his shoulders.) I think he must have known this on some level, for soon he was pulling back from me, detaching himself from the family, entrenched in his daydreams and silences.\n\nI'd wanted to be him. I'd wanted to be him more than myself, though I didn't admit it. I kept my hair the same scrappy length, combing it across the cowlick whether it stuck there or not. I wore the same painter's cap, the same chambray shirts, the same wire-rimmed glasses, though my eyes were near perfect. I copied the arcs and slants of his penmanship; I imitated the hard supple bounce of his stride. Once I even practiced talking like him, opening up my vowels, digging my teeth into the consonants as if I'd been the progeny of some speech therapist and not the second son of Sid and Ursula Sarshik, born July 12th, 1973, on the burnt fairway of an abandoned golf course, somewhere south of Coral Gables.\n\nA stormy August morning. Peter exits the downstairs bedroom and I plant myself before the mirror, holding his damp towel to my face, breathing in his rich, fluvial smell. I reach for his shaving cream, then lather my jaw, imagining it hatched with stubble. I stop, thunderstruck. I start spraying my nose, brows, forehead, ears. Warpaint, a mask. And\u2014boom\u2014it happens: my legs grow lush with hair, my muscles rumble, my penis thickens like a chain. I can knock down buildings with my voice, I can soar past rockets in the firmament. I close my eyes and take myself deeper. I am wise, responsible, brilliant, and funny. I can make you laugh. I can speak long fluid thoughts before television cameras, in the pool of burning light, without stuttering or unease. I know things. I tell you the square root of 1,738, all 8 names of Saturn's moons, the total air mileage between Jakarta and Buenos Aires and Las Vegas. Even when I trip or fuck up or do something patently wrong, you smile and think I am charming.\n\nWhen I opened my eyes I was only myself again.\n\nThen I became him: our roles shifted. I started doing well in school. I loved hunkering down in my quiet corner of the library, devoting my attention to my work while things fell apart at home. If I was good enough, smart enough, I could bring my parents\u2014the whole lot of us\u2014back together. They were talking again. A start, at least. Science-fair ribbons, perfect report cards.\n\n\"Evan,\" Ursula said one day. \"All A's. This is amazing, terrific.\"\n\nI stood before her in the kitchen, satisfaction beating in my face. Peter walked past us, winded, in his green nylon shorts.\n\n\"Did you see this?\" Ursula said to him.\n\n\"Who ate the Oreos?\" He hunted through the pantry shelves.\n\n\"Peter. Your brother just came home with all A's. Aren't you excited for him?\"\n\n\"Great,\" he said dully, then lurched out of the room.\n\nSomething changed after that. A quiet tension hummed between us. Once, pretending I was asleep, I heard him come into my sunshot room. Midday, the hour after lunch. Our parents were at the store. He stood beside my bed, twirling the mobile above my head with his fingertips. He spun it so fast that I was afraid it would fall down, crashing on my forehead. Would he kill me? Cooling shadows swept across my face. I loved the fact that he wouldn't leave, spinning the thing until he expected me to give in to him. Still, I refused to open my eyes.\n\nI chalked up good grade after good grade. I was the only one Sid and Ursula ever talked about. I felt fat with their pride, unbearably rich, as if I'd consumed an entire loaf of pound cake. They made no secrets of their love or their favor. But there was my older brother, who was cutting classes now, stumbling around the house with the harsh glare of speed in his eyes.\n\nWe started swimming. We swam every day, in the moments before twilight, at the house of his friend Javier Rodriguez, whose parents had the largest pool on Avenida Bayamo. Each of us had our respective routines, but our last five minutes together always culminated in a race that involved swimming ten laps. It goes without saying that he always lurched ahead of me at the finish. I didn't care. He was the natural athlete between us\u2014the older, the stronger. But this day, when the air smelled curiously of clematis, datura, and burning charcoal, when it seemed all but apparent that I'd be the victor, Peter hefted himself out of the water mere seconds before my forehead slapped the tile.\n\n\"Winner!\" I cried, gulping air.\n\nHe sat on the edge of the pool. In his boredom, he inspected a cut on the ball of his foot.\n\n\"What are you doing?\"\n\n\"I hurt my foot.\" He dangled his leg in the water, watching the trails of broken blood turning rusty in the chlorine.\n\n\"You mean you couldn't wait five seconds?\"\n\nHe stood, then dropped his Speedo to the concrete. I tightened inside. I looked away. He routinely did such things in my presence, trying to provoke me. \"I'm taking a shower.\"\n\n\"You did that on purpose,\" I said, and slapped at the water.\n\n\"What?\"\n\n\"You did that on purpose. You didn't want to finish because you knew I'd win. I scare you.\"\n\n\"You scare me all right.\" He hobbled to the outdoor shower enclosure. His blood left stains on the bleached concrete like drops of rosehip tea. \"You have a complex.\"\n\nI didn't have a complex. All I know was that we stopped swimming together from that day forth.\n\n# Chapter 13\n\nThe following morning, after I'd unpacked and settled in, I got the tour of the 55-acre King Cole property. Peter's manner was formal, almost distant, as he led me around. Had he gotten enough sleep? His voice was hoarse and whispery as if he'd stayed up all night.\n\nIt was hard to stay cheerful. My face felt tired from smiling too much. The place was a pit. Not only was it a good thirty miles from Naples, but it was situated in a battered glade with such a high water table that it needed a sump pump to keep the parking lot dry. Still, the place practically shimmered with its history. Here's the story: in 1964, Clem Thornton, the big Florida developer, platted a city called Boca Palms on the surrounding land. Two years into the project, the whole thing was halted, after several of the buyers\u2014senior citizens from Massachusetts, laborers from Detroit\u2014learned that their lots were under six inches of water. But who could have blamed them for buying? They got a free trip out of it, after all. Flown to Miami, they were bused across the Everglades by a cheery driver who called them by their first names. Upon arrival, though, things heated up. The pitch took place in the conference room, a spiel laced with threat, guilt, and the possibility of wealth. Salesmen (undercover, of course) cried out for the lots, creating the illusion of demand. It goes without saying that 75 percent of the detainees bought property out of fear and confusion, not even balking when they were told they couldn't tour the place beyond the fenced-in resort.\n\nWe walked farther. Peter's face was drawn, a little chapped, fixed upon the ground, as he pointed out the various wings of the building\u2014Solar, Nurmi, Seven Isles, Bontona\u2014all named after the fanciest lagoon streets in Fort Lauderdale, where Clem Thornton had spent his halcyon years before landing in jail. Peter kept chatting up the property, promising improvements, but I couldn't help noticing its inadequacies: the cracked parking lot, the sudsy pool with its algae-coated walls. Past the deserted golf course and the landlocked marina were a set of burnt-out buildings. They hulked there with their charred walls and broken windows, separated from the nicer part with a makeshift wall.\n\nFortunately, none of this was visible from the highway. On balmy winter nights, it still looked nice enough to lure that certain kind of guest, retirees on their way to the Gold Coast, charmed by the jungly landscaping, the golden gas lanterns, the curved driveway leading to the on-site mineral spring\u2014all the symbols of a more innocent, unfettered era. I couldn't help but feel both affection and pity for it.\n\nAfter the tour I was put directly to work. Immediately I was shown the correct way to answer the phone, a veritable mouthful: \"Good Morning, King Cole. It's 82 degrees and sunny. This is Evan speaking. How may I help you?\" I was told how to log reservations in the black book, how to run applications and utilities on the computer, how to process traveler's checks, how to swipe a credit card through the ZON. I was told how to smile. I was told to look all prospective guests directly in the eye\u2014\"not intrusively, but gently. They're friends, after all.\" Once I passed Desk Duty with flying colors, I moved on to Housekeeping 101, a post I was assigned to for the remaining three days of the week. I was shown how to make beds, how to fold bath towels, how to tuck in the sheets to make crisp hospital corners. I was shown how to Windex mirrors and windows. I was shown how to wrap glasses, how to apply the sanitary paper strip to the toilet seat. I was shown how to run the mighty washers of the laundry room, how much bleach, Sta-Puff, and soap to use, with severe instructions not to overburden the delicate, malfunctioning septic system.\n\nIt all seemed a bit much, especially since I'd never had a real job, unless you could count the two traumatic days I spent working at Dairy Queen when I was sixteen. Not to mention the fact that nothing was said about any days off. But the truth was that none of these tasks took a Ph.D. I imagined completing my duties early every afternoon, spending the rest of the day reading, swimming, or landscaping, my raw hands working the loam.\n\nThat night the three of us ate in the northwest corner of the banquet hall, the huge 60 \u00d7 20 ft. room in which the Boca Palms sales staff had unloaded thousands upon thousands of underwater lots 25 years before. Our voices echoed off the walls. Beyond the closed curtains was the golf course, its waist-high weeds shrouding its once-molded fairways.\n\nMy favorite childhood foods had been lovingly prepared by Peter: corn on the cob, hamburgers, sliced tomatoes, watermelon. I pointed my fork to the chain of leaping blue marlins on the outer wall.\n\n\"What was this about?\" I asked. \"The Clem Thornton fishing tournament? All-expense-paid trip to the Riviera?\"\n\nI hadn't even noticed that Peter had stepped out of the room. I glanced up at Hector, his sleeveless T-shirt, the vaccination print on his upper arm. I jiggled my foot, slumping my shoulders until my chest caved in. Did he enjoy my shyness, my foolishness? I had the unsettling sense that he knew what I was thinking, that all my thoughts were broadcasting from some golden transmitter planted inside my brain.\n\n\"You have the same speech patterns,\" he said blankly.\n\n\"What?\"\n\n\"You and Peter. You both have the same speech patterns. It's funny, I've never heard it in anybody else. The way you cluster your words.\"\n\n\"Really?\" I said.\n\nHe nodded.\n\nI nodded back, but I didn't quite agree with him, and for the first time, I noted an odd, lilting turn, an accent of sorts, in his own speech.\n\n***\n\nCould fifteen years have passed since I almost did myself in for him?\n\nPeter and I slipped past a security guard, scampering out onto the roof of Miami Beach's Shelborne Hotel. Sid and Ursula were dressed to the nines, oblivious of our whereabouts, still smoking Lucky Strikes in the restaurant twenty-some floors beneath us. We laughed at their ignorance. The sun beat the strength from our backs. I walked to the side, poised myself on the ledge, breathing, vertiginous. We gazed out over Miami, its deep aqua water, the patterned beach umbrellas like hard Venetian candies.\n\nPeter stepped beside me. \"Would you jump for me?\" he whispered in my ear.\n\nHe stood to my left, his hand firmly weighted on the small of my back.\n\n\"Would you? Would you jump?\"\n\nI sat down on the ledge. I began easing my butt over the side, watching my legs, as if they were someone else's legs, flailing in the air. Already I pictured the people watching on the street. I saw the rescue-squad truck, heard the megaphone, saw the young mother covering her mouth with her palms. I would put on a show for them, but I would flourish, thrive.\n\n\"Don't. I didn't mean that. We better go.\"\n\nI didn't say anything back to him. But he had his answer, and his face reflected that, though it was shadowed and unassuming.\n\n***\n\nSomething soon became clear to me. Our guests were so few and far between that working the front desk became downright tedious after a few hours, and once I finished my book (The Life and Trials of Angie Dickinson, a paperback that I'd retrieved from the library), I found myself staring at the phone, willing it to ring. I talked for a while to a lady from Scotch Plains, New Jersey, who wanted to know what particular measures we took to keep the local mosquito population under control. Once she seemed dissatisfied with my fabrication, I hung up on her, then gazed at the brass arms of the starburst clock. Ten-forty-five. I stepped to the front window, hands pushing deep in my pockets, staring out at that hot mossy swamp of a pool, where a girl in an orange life jacket snorkeled within sight of her father. My breath fogged an oval on the glass. I transmitted a mental message to the girl: Don't put your head under that water. You might as well be swimming in liquefied lead.\n\nI had to set some projects for myself or I'd go nuts.\n\nAfter Peter dismissed me from my duties, I wandered through the halls for a while, admiring of and appalled by the vastness of the place, its tributes to ball-breaking work, its misguided faith in a corrupt future. There was something especially deathly, yet beautiful, about this kind of silence in the middle of the day, when it seemed all but natural that everything should be revving in highest gear. At the RoadStar Inn ten miles down the Trail, I imagined just the opposite\u2014soda machines knocking, ice machines humming, teenagers gossiping by the pool, people driving back from the Gulf beaches, the sand still hot inside their sneakers.\n\nThere was no way Peter wasn't losing tens of thousands of dollars per week.\n\nI returned to the confines of my little room. I couldn't have been lying on my bed more than ten seconds when someone started rapping on the door. I sat up at once.\n\nHector said, \"Are you busy?\"\n\nI rubbed some grit from my eye. \"No, I was just taking a rest.\"\n\n\"I have to drive to Cape Coral, bank stuff. Want to come?\"\n\n\"I don't know. Maybe I'll just stay here. I'm kind of tired actually.\"\n\n\"Oh come on. It'll do you good to get out.\"\n\nHis face was lively, yet glum, as if he'd wanted to be anyplace else in the world. He was again wearing cutoffs and 18-hole workboots, a look which, though essentially ludicrous, especially worked for him, calling attention to a pair of the most alarming and voluptuous calves I'd ever seen. I saturated myself with the most deadening facts imaginable: the latest NASDAQ average, the current price of gold, the winners of the American League pennant. Anything to distract myself. I wouldn't look at him. Too long, at least. It would be a mark of my growing up: I didn't have to see every attractive man as a potential sex slave.\n\n\"You sure you want to do this?\" he said, stopping midway down the hall. He grinned demonically.\n\n\"Why not,\" I said, and shrugged.\n\nOutside, Peter stood with a big-bellied workman, choosing sites for a well out near the old golf course. Complaints about the water quality had risen. Only recently the guests had caught onto something as if it were part of some national trend, blaming the water for a whole host of ills, from stomach flu to night sweats to yellowed dentures. One woman had even insisted upon showing us the crystallized salt upon her liver-spotted wrists, as if it were an emblem of our inherent disregard and incompetency. \"What if somebody sues you?\" she'd said merrily. Peter glanced up at us and sighed, then waved us on, as if he were sorry he couldn't join us.\n\nWe were speeding through the glossy subdivisions on Rattlesnake Hammock Road, a former country lane converted to a four-lane parkway. Blue tile roofs sparkled behind walled yards. Stopped at a traffic light, Hector turned to me and said, \"So tell me all about William.\"\n\n\"What?\"\n\n\"Your old boyfriend, William.\"\n\n\"You're kidding.\"\n\nHe wagged his head. \"Word has it that you've been around the block once or twice.\"\n\nMy chest tightened. I might have been pushed facedown into cement. \"Who told you about William?\"\n\nThe left half of his face smiled. He shrugged smugly as if to indicate he wasn't telling me.\n\nThe blood hummed between my shoulder blades. I felt a peculiar combination of vulnerable, betrayed, destabilized, and pissed off. I didn't think he was at all funny. Was it that the mere mention of William's name still managed to trigger a whole host of feelings when I wasn't prepared for it? Or was it that I thought my privacy had been walked on, disregarded? It was one thing that Peter knew about William; it was quite another if his business partner knew the details of my personal life. I thought about packing my bags, handing over my money to the ticket clerk, taking the last bus back to Miami.\n\n\"God, are you burning up. I didn't mean to upset you.\"\n\n\"No,\" I mumbled, as if talking to myself. \"I'm sure you didn't.\"\n\n\"What?\"\n\nI turned to face him with a mocking, fabricated smile. \"Do you always like to make fun of people you don't even know?\"\n\nHis eyes widened a bit. \"I said I didn't mean anything.\"\n\n\"Just forget it,\" I said testily.\n\nHe mumbled, \"Drama queen.\"\n\nIt would have been too easy to dismiss him at that point, for I had every right to. But he wasn't worth it. Instead, I made a pact with myself: I wouldn't trust him. I could learn to work with him, I imagined, even like him to some degree, but I'd never fully trust him. He was too much about himself. I thought of a ten-year-old splashing around in a puddle, throwing cinders, doing anything to get a rise out of passersby. I should have listened to my gut. It would have been better had I not gone alone with him.\n\nAnd what about my brother's role in all this?\n\nA silence took over the van. I loaded a tape into the stereo, and Laura Nyro's \"Timer\" made its first fumblings toward coherence.\n\n\"Frankly, I'm over him, if you want to know the truth,\" I said after the song had finished. \"I don't even think about him all that much anymore.\"\n\n\"Well, you still seem pretty upset.\"\n\n\"Well, you seem to have pretty bad instincts. Everything's fine, okay? I'm not the least bit upset.\"\n\nWe kept driving. Something clattered and buzzed inside my head. He seemed to be one of those people who respected you more once you flexed your grip and stood up for yourself. It didn't make it any easier, though. Was I already fucking things up for myself? We passed a two-story green dinosaur outside a mechanic's garage, which prompted me to glance over at him. I expected to see that arrogance again taking hold of his face, but there was something more complicated there, a look that told me that he knew it too\u2014an emptiness, the loss of something nameless yet profound. I thought about my earlier assertion that it was unreasonable to have sex with every single person whom I thought was attractive. And then I reconsidered it. His arms were chiseled and bronze. His head was shaved. His eyes shimmered with an irradiated, wilting light. And then I changed my mind again.\n\nIt turned out that Ursula, mistaking Hector for Peter, had many months ago called the King Cole desk in a minor fit to inform him that a neighbor, a Mrs. Diane Petrancouri of Avenida Bayamo, had spotted me standing naked with William in the window. Too immersed in the story not to reveal his true identity, Hector listened to my mother's words with a keen interest, her fear that the neighbors were talking, her realization only minutes before that I was actually involved in a \"full-fledged imbroglio.\" She was beside herself with frenzy and grief. It wasn't until a full three minutes had passed that she'd realized she wasn't talking to my brother, but to someone with a Cuban accent, whereupon she promptly hung up the phone.\n\n\"God, she was a trip,\" he said.\n\nMy face burned hot with shame. Still, I was compelled to know more. \"Is that all?\"\n\n\"Well, it was too much to take in all at once, though she said something about\u2014yeah, this was my favorite\"\u2014he began mimicking my mother with disturbing accuracy\u2014\"This kid was accepted to Princeton for Christ's sake.\"\n\nHe started shaking his head. \"And I thought my mother was a lunatic.\"\n\n\"Oh God.\" My stomach was a nervous pit. I laughed hysterically, exaggeratedly, somewhere between dread and relief.\n\nAs if to make up for his earlier misstep, Hector started telling me about himself, an activity with which he seemed well acquainted. Apparently before he'd moved back to Florida\u2014he'd grown up off Hialeah's Red Road, within earshot of the racetrack\u2014he'd lived in the East Village for ten years, where he'd waited tables at various restaurants, sold clear plastic clubwear at Patricia Field's, and tended bar at the Tunnel. He'd immersed himself in as many crowds as he could. He knew\u2014\"only casually,\" he insisted\u2014the Lady Bunny, Misstress Formika, Sherry Vine, Tabboo!, and Girlina\u2014the fiercest drag queens in New York. He faithfully attended Monday-night ACT UP meetings, participating in the Labor Day march on Kennebunkport when George Bush was still president. He appeared in several films\u2014\"bit parts\"\u2014among them Paris Is Burning, a Super 8 short for a Jack Pierson installation, and a homo horror movie called Boys on a Meathook. He had a boyfriend of six years, a Welsh guy named Simon. He spent four summers in Provincetown where he danced nights at the Crown and Anchor and was a go-go boy (nickname: Lanolin) in Ryan Landry's House of Superstars. He was photographed by Nan Goldin for the Ballad of Sexual Dependency. He lived in the same railroad flat for ten years, until one day, after waking up to frozen pipes the fourth time that month, he decided he'd finally had enough and packed up a suitcase and moved back to Florida, whereupon he immediately took a job at Disney World, of all places, selling mementos of Mickey Mouse and Dumbo to children from the Midwest.\n\nIt was terribly interesting to me, probably more so than it should have been. Something became clear. It seemed to me that young suburban homos had one of two options: either you could do what Hector and Todd had done, move to the city and turn your back on your past, giving yourself a new name, or you could stay where you were, attempting to hide yourself, becoming nothing but paltry, constricted, cowardly, and dull. It worried me that I was veering closer and closer to the latter category.\n\n\"How come you didn't stay?\" I asked finally.\n\n\"Tedious, tedious. Homo High School,\" he said, shrugging it off. \"A big old charm school. After a while it's time to pick up your certificate.\"\n\n\"But didn't you have fun?\"\n\n\"Of course, I had fun. But,\" he sighed, \"it's like this: I was thirty-two years old. I wasn't a kid anymore. I was tired of seeing all my friends die.\"\n\n\"Your friends died?\" I said dumbly.\n\n\"This was New York, honey. I'd buried four of my closest friends in four months. Five guys I'd been a buddy for: Leo, Rex, Israel, Duncan, Thomas. Only months before Simon died.\"\n\n\"I'm sorry,\" I said.\n\nHe shrugged again. \"There's nothing to be sorry about. It's like, I didn't give a damn anymore. I'd been to so many memorial services that they didn't mean anything anymore.\"\n\nHis fingers tightened slightly on the wheel.\n\n\"And it just seemed like everyone I knew was positive. I'd hear about someone else getting sick, riding out the first symptoms, and it didn't touch me. It didn't make me sad. I'd perfected this reaction, this absence of reaction, as if this whole mess were perfectly par for the course, the way it's supposed to be.\"\n\n\"So what about your friends?\"\n\nHe fiddled with the sun visor, then flipped it up again. \"That was terrible, it was the worst. You know, it's bad enough watching an old person die, but\"\u2014he started shaking his head\u2014\"these were my lovers and friends. I didn't want to remember them lying on the floor with shit staining their pants.\"\n\nI eased down into my seat, holding my arms over my chest.\n\n\"It wasn't good for me after a while. It was too easy to be detached and hard about things, and I hated that about myself. It killed all my enthusiasm, though I didn't know it at the time. I needed to get the fuck away.\"\n\n\"So you went to Florida.\"\n\n\"So I went to Florida,\" he said. \"Some decision. Go south in hopes that all your troubles'll melt away.\"\n\n\"And you met my brother.\"\n\n\"And I met your brother.\" He turned to me with a weary, troubled grin. He blinked as if flummoxed. \"Do I sound like an asshole? Does any of this babble make sense to you?\"\n\n\"No,\" I said, then, \"yes,\" though I couldn't be completely sure of either answer. I looked at the parkway ahead. Above it, the sun loomed heavy and white, boring a hole through the clouds.\n\n***\n\nThe van skated down Cape Coral's Diplomat Parkway, a four-lane highway bisecting a lunar landscape, a tabula rasa of sorts, on which kids in convertibles threw cans, pulled down their pants, and hurled the occasional racial epithet. I imagined this place from the air, a vast, chilling land mass the size of Phoenix or Detroit, with its 100,000 lots, all of which had been initially sold, like the failed Boca Palms, through high-pressure vacation packages 30 years before. It was slowly but surely cocooning itself into a kind of hell. I thought of news reports I'd watched in the last week, footage of throats slashed in West Palm, home invasions on friendly streets, race riots, huge chimneys jetting flame, layoffs at Digital Equipment, clouds of chlorine gas wafting over a Little League game. I thought of these stories, and of myself in them, and what kind of part we'd all played in their creation.\n\nThe bank stood in an especially desolate socket, bordering a lily-choked canal with clots of scrub on the banks. The earth was the color of pulverized soap. It shouldn't have surprised me that my brother chose to do his banking here, a good forty miles from the King Cole, all for the sake of saving a few bucks. It was just like Sid, whose savings multiplied in a series of dubious Dallas banks, all of which paid back the highest dividends, yet offered minimal protection from risk.\n\n\"Look around,\" Hector said, parking the van. \"Isn't it thrilling?\"\n\n\"You're serious?\"\n\n\"Of course, I'm serious. It's so awful it's beautiful. It's like that scene in The Snake Pit where that woman's singing \"Going Home\" and before you know it, the whole mental hospital's joining in with her, and it's awful, corny, and moving all at once.\"\n\nI nodded. I understood what he was getting at. I thought of Wolfie Cohen's Rascal House, Piccadilly cafeterias, all notions of a similar point of view, but they were all about wreckage, fading emblems of a sweeter era. This place seemed like something darker.\n\n\"Watch these kids,\" he said, pointing to the Circle K next door. They circled about ominously on their little bikes, like fevered mosquitoes. \"I bet they're going to call me a fag. Just watch.\"\n\n\"Careful,\" I said.\n\nHe was already heading toward the bank. He deliberately flounced up the sidewalk, in full view of the teenagers, in boots, tiny shorts, and a Hooters Boca Raton T-shirt. He might have been Todd Bemus, the tougher, more streamlined version, the city boy, the one I'd never known. Sure enough, behind his back, I heard talk of pussies and fags, uttered just loud enough for me to hear.\n\nIt occurred to me that I was no longer annoyed with him.\n\n\"So what happened?\" he said when he returned to the van. \"Anything happen? Did they call me a fag?\"\n\nI shrugged. \"More or less,\" I said. \"Yes. I guess they did.\"\n\n\"Great,\" he said, not the least bit ruffled. A satisfied smile spread across his face. \"Just what I told you. Yo, pencildicks,\" he shouted to the astonished kids. \"Greetings to Cape Coral.\" Then he threw the van into gear and we were off.\n\n***\n\nTwilight was falling on the neighborhoods outside Naples. One by one the amber streetlights trembled on, saturating the houses with an orange phosphorescent haze. By the time we pulled into the King Cole driveway, the sky was nearly dark, the color of burnished ebony. Beside the chickee hut a young woman struck a match, then began lighting the torches beneath the baobobs. There was something so lonely and comforting about her attentions, her complete immersion in the task, that I couldn't take my eyes off her. She stepped backwards, holding her arms over her chest. For some reason, I felt oddly magnanimous. I loved the world.\n\n\"What's she doing here?\" Hector said, not moving from behind the wheel.\n\n\"Who?\" I said.\n\nA disapproving noise came forth from his mouth. He nodded in the direction of the woman.\n\n\"Is she a guest?\"\n\n\"Don't tell me she brought the kid here too. Tacky,\" he said, shaking his head. \"Tacky. That's about as tacky as you can get.\"\n\nI opened the van door and stood on the parking lot. I heard a single child splashing around in the pool. \"Hey, Mom,\" he cried, \"I'm making up swimming strokes. This one's the turbo. Look at me, you're not looking!\"\n\n\"If she's not a guest,\" I said, \"she shouldn't be here. I don't have any qualms about telling her to leave.\"\n\n\"Oh right,\" he mumbled.\n\n\"Who is she, then?\"\n\nHe looked at me as if I were a lower form of life. \"Peter's girlfriend,\" he said, frowning.\n\n\"Peter's girlfriend? Since when does Peter have a girlfriend?\"\n\n\"He hasn't told you about Holly?\"\n\nI looked over at the woman standing by the pool. The fact was Peter hadn't told me anything. I hadn't known about his drinking: only recently had I figured out that he faithfully attended AA meetings two times a week. I hadn't known other things: how he put the money together to buy this place, or how he'd spent the many long years away from our family. But that was Peter, a wily, mysterious sort, always negotiating behind our backs, resisting at all costs any movement toward overt explanation.\n\n\"So she's a jerk?\"\n\n\"No,\" he said, \"Not really. She's actually a very nice, decent woman. I mean\u2014I mean I could even imagine being friends with her in a different context.\"\n\n\"Then why are you so upset?\"\n\nHe shook his head.\n\n\"Did she do something to Peter?\"\n\nHe didn't answer. I stared at his face, that wry, exhausted gaze. My stomach pressed against the plate of my ribs. I needed some food. And then I got it: this was jealousy. Peter wasn't only involved with Holly. He was also fucking Hector.\n\n***\n\nThere was so much to talk about. I wanted, once and for all, to tell him about my time together with William, so he'd hear it from my side, not processed through Ursula or Hector. I wanted to tell him that whatever he did was fine with me, that he didn't have to worry about feeling self-conscious or judged or anything but comfortable. I wanted to tell him that I cared about him deeply, that I couldn't have been more grateful that he'd invited me to live with him, giving me the opportunity to offer my help during this chaotic time. It seemed that our situation couldn't have been more ideal, for who could understand each other better than two brothers, after all? I knew exactly how he felt. What better person to talk with? I imagined that we'd grow closer in the coming time, that I'd start living the kind of life I'd always longed for, a life without dissembling, or second-guessing myself.\n\nStill, there was Holly. Was he primarily involved with her, with Hector as the occasional diversion, or was it the other way around, with Holly there for appearance's sake? Did he even know what he wanted? I thought about the time when, during one of his brief surprise visits home, I'd declared, without warning, at the dinner table, that I'd liked another boy in my class. \"I can't believe you'd do that to Mom and Dad,\" he'd said with disgust, not three hours after my admission. \"Anything to get attention. Anything to aim the lens back on yourself.\" Why hadn't he, as older brother, stepped in, telling them, \"He isn't the only one\"?\n\nStep up, step right this way. Secrets, secrets: welcome to Sarshik's House of Secrets.\n\nI believed that both Sid and Ursula still held more sway over us than we realized. It was hard not to imagine our lives in a shadowbox, the two of them still hovering over us, twisting our strings. How we wanted to please them, to bring them their happiness, as if their success actually depended upon our achievements. It wasn't like they'd actually prepared us for a life apart from them. Once in the dentist's office I read a magazine article about a certain kind of family\u2014\"The Self-Sufficient Family,\" as the author called it\u2014so self-absorbed, suspicious, and dependent upon one another that it couldn't help but cave in on itself. It described us to a T, in such an eerie fashion that I couldn't help but throw the article down, convincing myself it was horseshit. It made me think about Holly and her presence in my brother's life. Could his relationship with a woman still primarily be about pleasing my parents, even though he'd kept her hidden from them?\n\nI thought about the one time that Peter had brought a girlfriend home. I still remember her name, a beautiful name, DeeDee Middlebrook. She lived in a big house off Old Cutler Road in Kendall\u2014the daughter of an obstetrician and a lawyer. Ursula talked about the dinner for days, planning the menu, pondering what dress to wear\u2014all to the dismay of Peter, who didn't want to make a big deal of it. I didn't think he was all that interested, anyway. Upon meeting DeeDee, they made every effort to be nice to her, to welcome her, deliberately restraining themselves from saying anything patently provocative. Still, beneath it all was the disturbing sense that Peter had to choose between the family and DeeDee, that a decision to spend a Saturday night with his girlfriend constituted an implicit betrayal. Something was at stake here. I sat at the dinner table, listening to the jovial conversation around me, a dread settling into my marrow. Just one false word from Sid, and she'd find out we were freaks, that we weren't like other people. \"I guess you'll be out having fun tonight,\" my mother had said glumly on more than one occasion. Though the evening went off without a hitch, Peter stopped seeing DeeDee not two weeks later.\n\n***\n\nIt was the quiet, laid-back hour before dinner. The guests were already in their rooms, taking their showers, rubbing sunburn cream on freckled shoulders. I walked farther into the woods than I ever had, past the burned-out buildings and the marina, across the single, unfinished bridge on the property. Here, before the state land commission had issued the sales moratorium on Boca Palms, Clem Thornton had actually started constructing a semblance of a city, all for the benefit of his prospective lot buyers, though not a single house had ever been completed. The roads now were crumbling at the edges, coarse weeds growing through the sun-bleached asphalt. On every corner stood a moldy concrete marker, its exotic name printed vertically\u2014Miraflores, Isla Dorada, Tulipan, Costa Nera, Costa Brava\u2014each derived from a street in Cocoplum, the gated section of Coral Gables, of all places, near where Peter and I had grown up.\n\nI stepped into a partially built house on Pan American Boulevard. It was the only one of its kind, an unfinished model home with a faded sign hidden in the brush: THE BAL BAY. The exterior, though finished, was chipping, the window glass peppered with BB holes.\n\nThe inside was another matter. A mere shell, its floorboards were weathered to the point of rotten, and I stepped cautiously, testing their strength. The air smelled of dust. My eyes drifted through the cobwebs toward the corner. I jumped. Instantly, my stomach was in my mouth.\n\n\"You scared me,\" I said.\n\nPeter sat on the floor, legs drawn up to his chest. He looked bemused. He shrugged once, a gesture toward apology.\n\n\"What are you up to?\" I said finally.\n\n\"Thinking.\" He smiled more openly, then shrugged again.\n\nI sidled up next to him, leaning back into the wall. My heart was still beating. I glanced at his bent legs, conscious of the space between us. I imagined that space itself vibrating, charged with a complexity of emotions I couldn't name. \"What are you thinking about?\"\n\nHe rubbed his whiskered face with his palms. \"Bills, bills.\" He sighed once, loudly. \"More bills. They'll be the death of me. And you?\"\n\nI made a steeple with my fingertips. \"Just taking a walk.\"\n\nThe afternoon light was orange, thick, almost sticky on the floorboards. I was going to tell him what I assumed, ask if it were true. Then something told me not to: any inclination toward openness had always shut him down. It wouldn't be any different now. My secretive brother.\n\n\"You and Hector\u2014\" he started.\n\nI looked at him. I tried not to register any expression on my face. \"What about?\"\n\n\"The two of you. You both seem to be getting along fairly well.\"\n\nI nodded, pushed out my lips. \"I guess so. Well enough, I guess.\"\n\n\"That's good,\" he said. \"He's quite an interesting guy. I'm glad you two are getting along.\" He pushed himself up off the floor, glanced down at me.\n\n\"We are.\"\n\n\"Good, good. Just watch yourself.\"\n\nHis words burrowed in the pit of my stomach.\n\n\"I don't mean to sound corny, and I don't mean to exaggerate.\"\n\nI laughed out of sheer nervousness. \"Could you be clearer?\"\n\n\"You'll know,\" he said, and offered me his hand. He made a face. \"Now get up.\" And, in silence, I followed him through the door, down the wrecked road back to the complex.\n\n# Chapter 14\n\nI slumped before the TV in the guests' lounge, sitting before a rerun of Lost in Space, barely processing the images. It was early afternoon, a little after two. I'd already done my assignments for the day: I'd retested the pool water twice. I'd rewired the noisy switches in my bedroom. I'd even fertilized some plants\u2014the loquat, the pitch apple, the clematis, the breadfruit\u2014an intricate, complicated procedure that involved the consultation of several guidebooks; I didn't want to burn anything. Now I sat here in stillness, too bored to read, too fidgety to immerse myself in anything but the fact of June Lockhart's appallingly tight spacesuit. My eyes drifted to the granulated spray ceiling. I pictured mold spores growing on those dusty clumps, mold spores turning to protozoa, then to more complex cells, then to flesh-eating bacteria, then to: Where's my arm? Where's my arm?\n\nHector walked in the room. I jerked awake, afraid it was Peter. What was the matter with me? He was hardly my father.\n\nHe collapsed into the vinyl chair beside me. He was shirtless, revealing a torso so sculpted and defined that I looked away\u2014so much for my notion that I'd stop thinking about him in those terms. The night before, trying to cheer each other up, we found ourselves exchanging our most embarrassing personal stories, and before we knew it, we were laughing like old friends, a peculiar comfort and intimacy passing between us. I thought now about his date with Ted Gessler, a famous music producer, in a trendy, pretentious Upper West Side cafe. I thought about how he'd excused himself to go to bathroom, returning to the table only to see that a thirty-foot length of toilet paper was sticking to his shoe.\n\n\"What's so funny?\" he said now, laughing along with some nervousness. \"You're laughing at me, aren't you?\"\n\n\"No,\" I said, pressing my palms to my face.\n\nI looked up. He was staring back at me now with an expression I reserved for the best of my friends, something open, without judgment, as if he simply liked the experience of looking at me.\n\n\"You're making me crazy,\" I said after a while.\n\nHe squinted slightly, confused.\n\n\"Stop\u2014\"\n\n\"You know what I think?\" he said, looking at the top of my head.\n\n\"What?\"\n\nHe shook his head emphatically. \"Forget it. It's none of my business. Forget it.\"\n\n\"Unh-uh,\" I said. \"You can't get away with that.\"\n\n\"Okay,\" he said. \"Okay. I don't like to tell other people what to do. It's one of my worst habits, I'm always getting crap for it, but\u2014\"\n\n\"What, what?\"\n\n\"I really think you'd look good if you buzzed off your hair.\"\n\nWe made faces at each other. My laugh was sloppy, out of control.\n\n\"What's so funny?\"\n\n\"No way,\" I said.\n\n\"What do you mean?\"\n\n\"No way, Jose,\" I said, wagging my head three times.\n\nIt wasn't that I was so attached to it, that I was afraid of giving it up, like my poor aging brother with his bald spot, his bureau topped with Minoxidil bottles. It was that I knew I didn't have the head for a buzz cut. It was too long, too narrow, with hard angular features\u2014in particular, a straight pointy nose with a high bridge\u2014that needed to be softened. I knew myself too well: it just wouldn't work on me.\n\n\"Another time,\" I said, standing up. I turned off the TV. \"Do you want something to eat?\"\n\n\"Oh, change the subject, why don't you.\"\n\nI stared back at him, annoyed. \"You're really into this.\"\n\n\"Well, I think it'd look really hot. I'm not bullshitting you. Everyone else would think so, trust me.\"\n\nI looked back at him, my face heating. He'd hit on something. He knew where I was confused, unsure, and took advantage of it. I might have pushed him over onto the floor. I glanced at the blank TV screen, waiting to see what he'd do next.\n\n\"You're bad news,\" I mumbled.\n\n\"Come on.\" He smiled, buoyant with newly won power. \"Just give it a try. It'll grow back in two weeks.\"\n\nPulse pounding, I followed him up the stairs to his apartment. I'd never been inside before, but I was intrigued. What a place! Aside from the expected\u2014unopened letters, Japanese comic books, precise stacks of porno magazines\u2014there was an entire wall devoted to Laura Nyro artifacts: posters, autographs on napkins, bootleg album covers, as well as news-letters from her fan club, which operated out of a post office box in North Jersey.\n\n\"What's that about?\"\n\n\"Greatest fucking singer\/songwriter on the planet.\" He tapped out some pot into the bowl of a red pipe. \"My soul mama.\"\n\n\"I don't know her very well.\"\n\n\"Know her,\" he demanded. And he handed me the pipe, lighting the bowl, before wandering off to put on a CD.\n\n\"Christmas and the Beads of Sweat,\" he said, closing his eyes.\n\nI pulled the smoke deeply into my lungs, tasting its harsh acrid scratch. I was curiously out of practice. It made me think about that one and only time I'd gotten stoned alone with Jane. The pot crept up on us, cunning, mysterious; we could barely keep our hands off each other. I dragged warily, worrying about Peter finding out. What about those AA meetings? What about that twelve-step sticker on the bumper of the van? PRACTICE SENSELESS ACTS OF SELF-DESTRUCTION or whatever the fuck it was.\n\n\"Now let's get to this hair,\" he said, wielding a clipper at me.\n\n\"Are you sure you're in good enough shape to do this?\"\n\n\"I'm fine, for God's sake.\" His smile was trumped up, maniacal. I couldn't tell whether he was playing with me or not. \"Now take off your shirt and shut up.\"\n\nI sat. I did what he said. The minute \"Brown Earth\" kicked into rhythm, he took his place behind me, pressing the contours of my head. I couldn't tell him how good that felt, that the last time I'd gotten my hair washed at a salon, I'd literally moaned aloud in ecstasy, unnerving the poor old guy. I apologized immediately, embarrassed by my outburst.\n\n\"You have a good head,\" he said, flipping on the clipper. \"You're going to look great.\"\n\nI held perfectly still. The blade was cold against my neck; my chin tucked into my chest. Hector hummed tunelessly to the song about Freeport. On the tiled floor were the feathered shreds of my hair. I thought of a sheep being shorn, tied up, then trucked off to slaughter.\n\nThe clipper switched off.\n\n\"My God,\" he said, as if amazed.\n\n\"What?\"\n\n\"I mean, I thought you were going to look good, but\u2014Christ\u2014\" He started shaking his head.\n\nI made a face. Was he making fun of me? I felt the short stubble with my hand, then asked for a mirror.\n\n\"What do you think?\" Hector said.\n\nI gripped the mirror in my hand. It was hard not to be repulsed. It made my eyes leap out, the thick brows above them darker, bushier, even demonic. My facial bones looked sharper, more masculine. Could I pull this off? It didn't jibe at all with the way I knew myself, and I doubted I'd ever get used to it, but hey. I liked and hated it at once.\n\n\"You're going to need some sun on that scalp, paleface. Let's sit in the grass.\"\n\n\"But\u2014\" I looked around for a hat. I might have been standing before him without clothes.\n\nHe switched off the Laura Nyro. \"Come on, fancy boy,\" he said, laughing harder. \"We're going outside.\"\n\nWe sat on a patch of burnt lawn behind the office. It was a breezy, torporous afternoon. The sun was warm, coiling, boring down into the roots of my scalp. All at once I realized I was high, unbearably high. I lay back down on the grass, laughing quietly, my arms latching behind my head. Above me, wild violet clouds blew in from the Gulf.\n\nHector turned onto his side, looking at me, propping up his head on his palm. I heard footsteps down the sidewalk heading in our direction. I felt too stalled and contented, too lazy to move.\n\n\"Have you seen your brother?\" Hector called.\n\n\"No,\" I said, but then I realized he wasn't talking to me. I lay flat on my back, staring up at my brother's blasted face. He attempted to smile, though he couldn't stop biting into his lip.\n\n\"When did you do that?\"\n\n\"This afternoon,\" Hector said, answering for me. \"Just a few minutes ago, in fact. I cut it myself.\"\n\nThe three of us fell silent. My back pinched. I tried to cover my eyes\u2014their hazy red wetness would be sure to give us away. I sat up, wishing I'd worn my shirt, brushing off the blades of dead grass stuck to my shoulders. I looked over at Hector, who seemed amused by it all, a passing glimmer of superiority on his face.\n\n\"But why?\" Peter said. \"Your old haircut looked perfectly fine.\"\n\n\"Why not?\" I said.\n\nHe shrugged, put off by the hardness in my voice. \"It'll grow back,\" he said calmly. \"No reason to be upset.\" He glanced at Hector, blinked once, then lugged off a hose toward the junction boxes.\n\nHector and I were silent for a time, watching my brother work. He might have been reassuring himself. I felt hungry, diminished. I was certain of it, then: I wanted Hector as much as I wanted anything.\n\n# Chapter 15\n\nI didn't take Peter's discomfort to heart. If anything, I felt myself becoming more and more adventurous in the weeks that followed. It was as if I'd thrown up my hands to the fear of judgment and censure. I was tired\u2014once and for all\u2014of reigning myself in, toning myself down. So I assembled a look. I wore faded black T-shirts with the sleeves hacked off. I wore a battered silver motorcycle jacket (borrowed from Hector) whenever the temperature dropped. I even grew accustomed to my crew cut, buzzing it short every Friday night whether it needed it or not. It seemed that I was gradually becoming my look, and I felt snappier, more streamlined and self-assured than I had in seven years, when for a brief period I'd gone through my Ozzy Osbourne phase. I was changing, crumbling, slipping away right before my very eyes. It was nothing but a relief to me. So what if I looked like a big old homo?\n\nThe truth was I'd been wanting to assume another look for years, but hadn't pursued it for fear of seeming false, pretentious, of falling flat on my face. But wasn't that part of the fun? Wasn't that what was interesting about other self-created people, the visible cracks in the armor? It seemed that all I needed was a sense of humor, a dash of smarts, and I could be whoever I wanted. What was a lifetime but a series of shifting, inter-changeable masks? I could look like a biker boy today, and tomorrow a trust accountant, and the next day a geeky scientist.\n\nI stopped before the mirror every time I passed it. \"Grrrrr,\" I growled, flexing my muscles.\n\nWhat I didn't comprehend was that my dressing up could concern Peter so much. It wasn't that he'd directly addressed the issue. It wasn't that he told me to tone it down for fear that I'd put off our guests. Instead, he made his feelings known in the way he talked around everything but the clothes on my back. Maybe he was taking on the role of Sid and Ursula, feeling it his duty to keep me in line, to inspire inhibition in a personality he perceived to be reckless and underdeveloped. Or was it that my dressing this way\u2014let's face it: hypermasculine, yet daffy\u2014reminded him of something that scared him about himself? He certainly should have recognized he'd constructed his own look. No one wore chinos and button-downs and wire-frame glasses day after day without some degree of intention, without knowing he desired to communicate a sensibility, a point of view to the outer worlds. But was he ready to admit that? I thought now of his earlier warning about Hector the night we'd stumbled upon each other in the sample house. In the days that followed, his silence started to pique me. He made me want to dress more and more outlandishly, to tattoo my back, to pierce my eyebrow, to dye my stubbled hair aqua, cranberry\u2014a new color depending upon the week and season.\n\nDeep down, I wondered whether Peter was simply frightened of the confidence that had been gradually inhabiting me. Did he want some for himself? It seemed to me that he both wanted some and was afraid all at the same time, but what did I know? I'd found a way to be in the world, a way of letting go, abandoning myself within a certain set of parameters. It was like learning how to eat with a knife and fork. I thought about Sid, his social cluelessness and fear of outsiders. I thought about his flirtations with his female coworkers, how it was endlessly getting him into trouble, baffling him every time he was called upon it. \"I never meant any harm,\" he'd say. \"I'm not a bad guy. I only love women.\" He might have come from another planet, another century. He simply didn't know the rules, as if he were navigating, rudderless, through a storm-drenched sea. You had to know the rules. It might have been foolish, mistaken to crave something fixed, but it made it easier, I knew that much, and I wasn't prepared to unlearn anything I'd taught myself.\n\nHector and I were hosing down the lounge chairs by the pool. A girl with thick glasses and a homemade brown dress was sitting by the gate, reading about plankton in an ancient volume of The World Book. She peered up at us occasionally, pretending she wasn't watching.\n\n\"By the way,\" Hector said. \"I have some good news for you.\"\n\nI looked up. A star-shaped patch of cleaning fluid foamed beneath his lip.\n\n\"You know Alejandro?\"\n\n\"From Fort Lauderdale? The tax accountant?\"\n\nHe nodded. The banana palms creaked above our heads. \"He wants to go out with you. He thinks your new look's really cute.\"\n\n\"Great,\" I said, more blandly than I'd intended. I thought of Alejandro, his aviator glasses, his slicked-back hair, the pleasing squirrely sleaziness of him. In exchange for services, he was given a free room, and he stayed here once a month, sunning himself by the pool in a micro-black bikini. He was sexier to me than I was willing to admit, but that wasn't what was irking me. It was Hector's detached, off-handed manner I didn't like: he sounded like a pimp.\n\n\"He actually said something very nice about you.\"\n\nThe girl looked up from her encyclopedia. Her name\u2014Mary Grace\u2014was markered on the olive cover.\n\n\"He did?\" I said.\n\n\"He thought that you possessed a really interesting combination of butch and femme. He thought that that was a sign of an interesting, complex personality.\"\n\n\"But he doesn't even know me.\"\n\nHe shrugged, then tossed his sponge on a chair.\n\nI followed him down the walkway. We were sitting on the floor of Hector's room, flipping through his collection of drawings\u2014\"cameos,\" he called them\u2014derived from the covers of women's detective novels of the 1950s. They were surprisingly good, though I paid little attention to them. The skin of my chest itched. In my usual habit, I was only gradually becoming aware of the fact that I was harboring any rage.\n\n\"What's this talk about femme?\" I blurted. \"That's ridiculous. I mean, do I look femme?\"\n\nHector raised his brows. \"Say what?\"\n\n\"I'm talking about Alejandro, what he said about me. You acted like that was a compliment or something.\"\n\nHector's eyes went blank. He remained silent for a second, then scooted over to the night table. He fumbled for something in the drawer. Not two seconds later he yanked back my scalp violently, with full force, hurting me. \"Hold still, butch boy.\"\n\n\"What are you doing?\" I cried.\n\nHe gripped the lipstick in his free hand, wielding it as if it were a scalpel. He began pressing the tube directly to my mouth.\n\n\"What the fuck?\" I said, trying to push him toward the wall. He was even stronger than I'd imagined. \"I told you I'd look bad as a girl.\"\n\n\"No one talks to me like that.\" And then he strayed from the outlines of my mouth, applying bars of lipstick across my jaw, my cheeks, my forehead, my hair.\n\nI reached for his T-shirt on the floor and wiped off my face with it. I tossed it away. Its fabric was red, ruined.\n\n\"I'll get you in drag,\" he said with a cool, casual smugness. \"Just you wait. I'll dress you up if it's the last thing I do.\"\n\n\"That's what you think,\" I muttered.\n\n\"Girl,\" he said, taunting me.\n\n\"Girl,\" I said, taunting him back.\n\nI'd had about all I could stand from him. But coupled with that was another more affronting, alarming thought: he'd probably get me to do it, the jerk. I stared at his handsome, arrogant, self-satisfied face. I could have killed him.\n\n***\n\nAt thirteen, I was afraid of someone. I did everything possible to distinguish myself from him. I recorded my voice over and over, imagining wide flat stones on my tongue, working out the inflections, sanding over any last traces of hiss. I trudged back and forth down the length of our driveway, taking heavy, self-assured steps, bouncing just slightly from the knees until my arms swung naturally, without concentration. I did push-ups by the dozen on the laundry-room floor. I read sports page after sports page, memorizing the scores, insinuating myself into arguments in which the merit of the Marlins' MVP was in question. There was nothing helpless about me. You could say that I talked too much, that I was scattered and lacked focus, that I hungered for overwhelming amounts of attention and reassurance from everyone who came into contact with me, but you wouldn't have said that I was feminine\u2014that much I was sure.\n\nUnlike Stan Laskin. Stan Laskin: hardware-store owner. Stan Laskin: who paid special attention to me every time I was sent in by Sid to buy switches or ten-penny nails. It wasn't that he was anything but kind. It was that his body, his entire self-presentation, soft and yielding, with its tendency toward flab, represented everything I didn't want to be. His colognes scented the atmosphere every time I waited at his counter. His glasses, all seven pairs of them, coordinated with his bracelets and rings. But most disturbing of all was the expression on his face, wounded and lamb-like, as if he were waiting for some devastating stranger to come through the door.\n\nAs far as I knew he lived alone and had never been loved by anyone in his life. His days, I decided, were repetitive, dull, and lonely, enlivened only by occasional visits to the fabric store, where he remembered all the employees' birthdays, and to the public bathroom stall where he sat six hours at a time before a vacant glory hole. After work, he'd walk through his front door, leaving his outer life behind, assuming a secret role, draping himself in chintz or black velvet, before giving hairdos to his Yorkshires or trying on his extensive collection of cloches and pins. Every morning he'd call up his mother, discussing the trip they were planning to the Lawrence Welk Resort in Escondido, California. His life had as much to do with my own as the newsletter of the Siegfried and Roy fan club.\n\nIt was a warm, overcast day in December, the weekend before Christmas. A heat wave was descending upon Dade County, moistening the foliage with dew. In Florida fashion, the trunks of the royal palms were wrapped with strings of clear lights. I hurried down the Miracle Mile with Mark Margolit and Steve Mendelsohn, two of my friends from school. At least I thought they were friends. I cared about them as much as I cared about the health of my gums, but to one another we looked like friends, and when the three of us were together no one dared make fun of us. I felt convincing with them. They believed me when I expressed my interest in Jane. Together, we talked about the color and texture of Jamaica Reed's nipples, the lead guitar solos from Metallica's second album, and the afterschool activities of Mrs. Walgreen, our Spanish teacher, who was forever tugging her miniskirt down over her hips. I wore a ripped Ozzy Osbourne T-shirt with black kohl eyeliner around my eyes, and when I looked in the mirror I even scared myself.\n\nWe walked around the perimeter of the circle, the scent of frozen pizza still rising from our fingers, stumbling to the video arcade, where we'd play a few games of Donkey Kong or Burger Time. In my thirteen-year-old way, I'd told myself I was having fun and was behaving like any boy my age was supposed to. We couldn't have been walking more than ten seconds when I saw Stan Laskin carrying boxes between a rental truck and his store, accepting a delivery. He looked relatively conventional for Stan Laskin: baggy chino pants, golden horn-rims, navy blue button-down. Except for the scrap of material\u2014a yellow brocaded print tied around his throat. A softness slid down inside my stomach. I felt nothing but embarrassed and afraid.\n\n\"Nice scarf,\" Steve said.\n\nHe wouldn't look at us. He lifted up a box marked screwdrivers, watching it fall through his hands.\n\n\"Did you always like women's accessories?\" Steve said.\n\nA single drop of sweat ran down the crease of my back.\n\n\"In case you're interested,\" Stan Laskin said, his face the deepest crimson, \"it's an ascot.\"\n\n\"Oh, an ascot,\" Steve said, highlighting his S. \"An ass\u2014cot.\"\n\n\"We better get out of here,\" I whispered to Mike.\n\n\"No,\" Steve said. \"Stay.\" A brutal, joyous laugh tore up from his lungs. He leapt toward the stop sign and slapped its red metal face.\n\n\"Do you like to suck cock?\" he said, spinning around, turning to Stan.\n\nNothing. A helicopter beat somewhere overhead, concealed.\n\n\"Do you? Do you like the taste of cock in your mouth?\"\n\nStan gazed downward at the box in his hands.\n\n\"Faggot,\" Steve said. \"Lousy cocksucking faggot.\"\n\nThe sidewalk might have cracked beneath my feet. More than anything I wanted Stan to dismiss us, to write us off as small, inconsequential. Instead, he turned not to Steve, but to me. He looked into my face in a more searching way than anyone ever had.\n\n\"What made you so hateful?\" he said matter-of-factly.\n\n\"Me?\" I said.\n\n\"All of you. I don't get it. Tell me how you live with yourselves.\"\n\nSomething bony and sharp pushed deep inside my chest.\n\n\"Come on,\" Steve said. \"I've had enough. Let's check out those bitches across the street.\"\n\nThe days hastened toward Christmas. I completed my activities as usual: I tossed Milk Duds to Delaware, our neighbor's Boston terrier, in my efforts to teach her how to fetch. I worked through all the supplementary exercises in my algebra packet, achieving a 98 on the pop quiz. I even helped Peter wax my father's Grand Prix, buffing its blue finish with a chamois cloth. At night, though, lying in bed, I couldn't scour Stan's question from my thoughts. I tried to tell myself that what had transpired hadn't been so sad. Everyone behaved that way, everyone I knew. It wasn't like they meant any harm. It was the way you carried yourself in the world. Otherwise, they'd pulverize you. Faggot, cocksucker, queer: these were just words\u2014empty, stupid, meaningless words. No one needed to be defended here.\n\nWhat the hell was I afraid of?\n\nI was standing in the locker room after gym class. We were midway through the swimming unit, the only sport I could bear. Nearly everyone had already left for homeroom. The air smelled of bleach and worn elastic. I looked from side to side, pulled down my gym shorts, as the spray pounded in the shower room beside me. I turned halfway. It was Jon Brainard, a small, intense boy with blue eyes and dark curly hair, who'd just transferred from Sarasota. He was watching me beneath the showerhead, defined fingers lathering a compact stomach. He might have been my brother in another time. My skin tingled, chilled, then flushed with the richest warmth. He was rinsing the suds between his legs. I couldn't keep myself from staring back, though I wanted to stop.\n\n\"Evan,\" he called.\n\nI turned. I fumbled inside my locker, pretending not to hear.\n\n\"Would you hand me that towel on the bench?\"\n\nThe drain gulped down the overflow. His smile was shy, as if he were thinking something enormous, beautiful.\n\nMy toes clenched tighter into the concrete.\n\nMany months passed before I again allowed myself to walk into Stan's store. It smelled of torn wood, matches, grass seed, pesticides\u2014a confluence of smells that I associate, to this day, only with that memory. In my pocket, I carried an arrowhead I'd found beneath an ice plant on the Metrozoo grounds. It was my talisman, my lucky charm. I'd attributed several minor miracles to its existence\u2014the recovery of a $50 bill and the rapid healing of a fractured ankle. My plan was this: I'd leave it on his counter in an unmarked bag without note or explanation. He could do with it what he wanted. I knew it was hopeless, and I knew it was unreasonable, but the exchange had been hovering over me, a ruined black blanket, and I just wanted to get it off my mind.\n\nStan was back in the supply room. I leaned the lunch bag against the cash register, then turned to make my getaway.\n\nAt once he came toward me with a crowbar. He looked kindly and quizzical, as if he thought I needed help. Then all at once his brows drew together.\n\n\"Don't I know you?\"\n\nMy breathing went sluggish. I imagined him bringing the crowbar down across my forehead, splitting it in two.\n\n\"You were one of those boys,\" he said. \"Get out of here.\"\n\n\"I didn't\u2014\"\n\n\"I'll call the police. Get out.\"\n\nI left the store without explaining myself. It was his pain and not the crowbar that frightened me. Even then, I knew that it wasn't just about my indifference, but about every time he'd heard the word faggot muttered behind his back. That night, lying on the living-room sofa, I thought about the arrowhead in the lunch bag. It was a vacant, meaningless gesture that wouldn't console him, I was positive. He probably even tossed it away. But there was always the chance that he kept it, and it's serving him now, giving him luck, warding off anyone who'd hurl a word at him, or anyone who'd let it happen.\n\n***\n\nHector and I leaned against a triple-loader at the outdoor Laundromat. We were working an entirely new look: our slacks narrow, our lashes smudgy with goo. Even our lipstick\u2014glacial, white\u2014coordinated with our jackets, tight vinyl numbers crisscrossed with zippers. We were space-age versions of Jean Shrimpton and Marianne Faithfull, right out of 1967. I looked straight ahead, trying to erase and assert myself at once. But I wanted to go further. More than anything I wanted to hammer through my shame, become enameled, incandescent like Hector\u2014\"Mistress Chevelle,\" as he called himself today. Something was off, though. If I was going to do this, I had to do it right. I looked over toward the ladies beside the dryers, taking in their weary, sardonic scorn of us. Where was that transformation that was supposed to take hold of me, whisking me off to that other sphere, where I was too perfected, too highly evolved to be bothered, beyond bitterness or spite? Was I just another failed fag, a dumb cluck, an old stick-in-the-mud?\n\nI wondered sometimes whether, out of sheer self-protection, I'd simply never allowed myself to develop a taste for things feminine. I walked around convincing myself I wasn't the least bit interested in cooking or fabric swatches or interior decoration or figure skating, but was that the essential me? What about my predilection for finding myself in public spaces designated for women only? How many times had I wandered into a bathroom only to find an absence of urinals, only to hear a terrified voice rising over the stalls: \"Is that a man in here? Oh my God, I think it's a man.\"\n\n\"Are you having fun?\" I said to Hector. \"I'm not having very much fun.\"\n\n\"What kind of talk is that?\" Hector said.\n\n\"Maybe it's just not the right look for me,\" I said, tugging at my cat suit.\n\n\"You might be right. Maybe you should have done the Mary J. Blige thing.\"\n\n\"What?\"\n\n\"You know, black girl, blonde hair. That would have worked.\"\n\n\"I personally think I'm the Liv Ullman type. We both could have done it. Did you ever see Persona?\"\n\n\"Is anything ever right with you?\" he said, flustered. \"Hand me that detergent.\" And before I knew it he'd poured another half-box into the washer compartment. Our clothes spun wildly, suds flinging against the glass.\n\nThe truth was we needed some diversion after the events of the week. Two nights before, the septic tank had overflowed, precluding us from doing our own laundry on the premises. Behind the office, a translucent stream flowed from the lid toward the mangrove banyans. The scent, to say the least, was less than pretty, prompting complaints from the guests.\n\nI stared at one of the white-trashy women whose eyes had been fixed on us. \"I don't like the way she's looking at me,\" I said, loud enough for her to hear.\n\n\"Oh, get over it. She doesn't even know what day of the week it is. She probably just thinks you're some trollop.\"\n\nI fixed him with a level stare. My glance was high, my voice elevated. \"Look who's talking.\"\n\n\"Slut.\"\n\n\"You're a joke.\"\n\n\"Whore.\"\n\n\"You're a joke\u2014a dirty joke from one end of this town to the other.\"\n\nWe laughed, collapsing against each other. Our witnesses looked at us with a mixture of disdain and fascination. Hector clapped me hard on the shoulder. \"There you go. Now you're getting it down.\"\n\nA boy in a ripped T-shirt, dirty beret, and mutton-chop sideburns passed us, lugging a wash basket over his hip. His look seemed to be especially thought out, as if he'd spent a full ten minutes trying on various combinations in an effort to seem like he'd just thrown the whole shebang together. He had the look of someone who'd fixed a camera on himself, watching himself with every gesture, assuming that everyone else was doing the same.\n\n\"What do you think?\" Hector said, nudging me in the side.\n\nHe kept staring at the boy with a flagrant, unnerving calculation. I didn't know what he was talking about.\n\n\"Our church or theirs?\"\n\nI looked him over again. \"Ours,\" I said finally.\n\n\"No way, chica.\" This was new: he'd taken to calling me chica in the last week.\n\n\"What do you mean?\"\n\n\"Too sloppy. Way too sloppy.\"\n\n\"Get out of town.\"\n\n\"Listen to me,\" he said, holding up his hand. \"Listen. I'm going to teach you something. A fag wouldn't do that. He'd shave the sides of his head. He'd keep it neat and severe around the ears, and he'd wipe off the mud from those shoes.\"\n\n\"What mud?\"\n\n\"And he isn't looking around enough. Fags always look around.\"\n\nI looked at Hector, confounded. Where did this certainty come from? How could anyone possess such single-mindedness, such pigheaded belief in one's point of view?\n\n\"Watch this,\" he said. And before I could stop matters, he started speaking to the boy. \"Excuse me, young man. Pssst.\"\n\n\"Yes?\"\n\n\"Could you tell me where you got your shoes?\"\n\nI slumped against the washer, stricken.\n\n\"Ben Southern,\" he said. \"Actually, they've got a giant sale going on. Thirty-eight bucks.\"\n\n\"Really?\" Hector said. \"They're just the sauciest little numbers.\" He swallowed demurely. \"Would it be a terrible imposition if I asked you to let me try them on?\"\n\nMy body temperature surged two degrees. Either he was going to punch us out or comply.\n\nSeconds later, he was passing the shoes to Hector.\n\n\"Goodness, they're big,\" Hector said, looking down. \"My feet are just swimming in them. I couldn't possibly wear anything so big.\"\n\n\"You know what they say,\" he said shyly. \"Big feet, big\u2014\"\n\n\"What's your name?\" Hector's smile was huge and platinum, on the verge of ghastly. He passed back the shoes to the boy.\n\n\"Josef,\" he said, glancing downward. \"With an F. And yours?\"\n\n\"I'm Mistress Chevelle. And this is my friend, Boca. But you can call her Big Pretty.\"\n\nJosef grinned back at us as he were in on the joke. He offered his hand to each of us and shook ours firmly. I was relieved. He was one of us.\n\n\"You girls from out of town?\"\n\n\"In a manner of speaking,\" Hector said. \"Slave labor.\"\n\nHe nodded, pretending to understand. He crunched his brow. \"Well, have a good time while you're here. It was nice meeting you.\"\n\n\"Already?\" Hector said wistfully.\n\n\"You watch yourselves, girls. Have fun on your trip.\"\n\nAnd then he wandered off, loading the silver washer with extreme care and attention.\n\n\"See?\" I said.\n\n\"See what. He's not a fag, you dope. He's fag friendly. He's just a straight boy who's comfortable with fags.\"\n\n\"What?\"\n\n\"There's a big, big difference. You could get yourself into trouble.\"\n\n\"Oh right.\"\n\n\"The hair, dear, the hair.\"\n\nI pointed to my wig, suggesting the crew cut that lay beneath it. \"I didn't used to have fag hair.\"\n\n\"You weren't a fag.\"\n\n\"Excuse me?\"\n\n\"You weren't a fag till I got a hold of you.\" He curled up the issue of Women's Wear Daily in his hand, gazed through it like a telescope. \"In terms of style, at least.\"\n\nThe heat off the dryers was making me swoon. I pictured my knees buckling, my face crashing flat to the floor. \"Oh, is this some nasty remark about my sense of style?\"\n\n\"No, it's not some nasty remark about your sense of style,\" he mimicked. \"You were making some bad choices, that was all. You looked like some sorry-ass straight boy with your hair flopping all over your ears. No wonder you weren't getting any dick.\"\n\nI could have throttled him. We weren't going to get any dick looking like this, either. I stared at the front loaders, fixing my gaze on them, as if they were fireplaces\u2014the fireplaces of Florida. I needed to turn off my thoughts. We might have been captive in Hell's Laundromat, with pool table, raucous music, bar, free popcorn, and huge TV screen across which Morton Downey, Jr. hovered. I couldn't help staring at the well-dressed, middle-aged woman beside the soap machine. She looked around nervously, with a barely suppressed panic. Her face carried all the doubts and anxieties of an aging flight attendant, the look in her eyes saying: Where is my beauty? Why is it disappearing from me?\n\nAnd I'd worried that we'd been out of place.\n\nHector was sitting in the yellow bucket seat, right leg crossed over the left knee. It was amazing: beneath his exterior, he still seemed butch, even butcher than usual. It troubled me that I'd actually been attracted to someone in an A-line miniskirt.\n\nI gestured across the room toward Josef. \"How would you know?\"\n\n\"I know,\" he said, pointing to the place between his eyes. \"You have it, I have it. It's just a question of whether you want to pay attention to it or not.\"\n\nBut it was never that easy. I thought about Ross-Bob Vittori, a boy from my high school, who lisped with abandon, collected Jean Seberg memorabilia, and was in possession of the longest eyelashes this side of a Maybelline model. Despite these factors, he was still one of the most popular boys in school, with no shortage of dates, asking out a different girl each weekend, most of whom had called him first. Any number of people didn't fit the bill, challenging expectations, resisting typecasting, but how could I begin to explain this to Hector? I knew him well enough: that brain was impossible to perforate.\n\n# Chapter 16\n\nI was thinking about one of the first boys I'd ever had a crush on. It was months before William. I ran into him every day at three o'clock, on my walk home from school, on my shortcut past the U of Miami Art Building. Invariably he was sitting within the shadow of an Australian pine, nibbling at a cream cheese and olive sandwich on white bread. I presumed he was an art major. But that in itself wasn't the thing. There was something about him, a locus of energy that suggested a fiercely complex internal life. He wasn't the best-looking guy in the world, and he couldn't have been further from my type, which tended toward road thugs and serial killers, or more precisely, those who looked like them. But there was something about him. Maybe it was his chin whiskers\u2014brown, downy\u2014or the sketches in his notepad\u2014fractured, strife-ridden, built from dashes, stars, and blips. I wanted nothing more than to save him from his pain. Soon enough, I was nodding to him, then he back at me, then we were actually mumbling hello, initiating that period of mutual recognition and appreciation.\n\nIt was a muggy, overcast day in early November when I walked into Bonita's Glaceteria, a little place on Coral Way, to buy some lime-ice to cool down from the heat. I was actually in a light, cheerful mood\u2014odd considering that I'd just been accepted to Yale, and the prospect of college unnerved me to no end.\n\nWhen I looked to my right, I saw the art boy, Arden, smiling at me.\n\n\"Hi,\" he said.\n\n\"Hello,\" I answered back.\n\nWe stood there for a stunned moment, staring at each other, not knowing how to proceed. Whole epochs might have passed. Then, for whatever reason, I crept to a table by the window, covering the left side of my face with my hand. My hairline pearled up with sweat. Here, I'd had every opportunity to sit down and chat, casually, without complication. I sat there sucking at my paper straw, looking for all intents and purposes like a moron, wishing that he'd get the hell out of the place, knowing that the mere fact of his existence was just further evidence of the universe's essential cruelty and indifference.\n\nTo my dread, he stepped toward my table.\n\n\"Do I know you?\" he said earnestly.\n\nHis accent was rounded, wide, as if he'd just stepped in off the prairie. He looked at me with luminous, grey-green eyes, a shade I'd never quite seen on anyone else. I'd never felt more defeated and desperate in my life. Then, as if somersaulting into a river of burning lava, I said: \"Would you like to go to a movie with me this Saturday?\"\n\nHe considered the offer. Outside a group of preschoolers trotted by, reciting their ABCs in both Spanish and English. \"Of course.\"\n\nWe made plans to see Resnais' Last Year at Marienbad, a film about which I couldn't have cared less, but it was playing at the university, and I was certain that he'd like it. It occurred to me that this was the first time I'd ever asked anybody out on a date. I pushed past the pedestrians, knowing that my days of loneliness and emotional aridity were forever over, that I'd soon have all the sex I wanted, and more. That night, I not only cleaned out my room from head to toe, but talked to Sid and Ursula with interest and civility. I was embarking on my new life. I was in love. Soon enough, we'd devote ourselves to making each other happy, and we'd give each other back rubs, and talk back and forth, and listen, and once we grew a little older, we'd trim each other's sideburns, shave each other's shoulders.\n\nI couldn't stop thinking about the sheer sonic loveliness of the name Arden. How it tripped off my tongue: a ballad, a prayer.\n\nThe day of the date, my stomach tightened as if injected with helium. I isolated myself in my bedroom, trying to relax my hunching shoulders. Only when I couldn't stand it any longer did I call Jane, asking about the appropriateness of my outfit.\n\n\"It's seven-thirty in the morning,\" she said. \"What do you want from me, dear?\"\n\nShe suggested a few pieces, a loud corduroy sport coat and a buffalo-plaid shirt, neither of which sounded particularly attractive. It occurred to me that she was more interested in seeing the endeavor fail, though she'd never admit to that.\n\n\"You know, Evan\u2014\"\n\n\"Yes?\"\n\nShe yelled at Sugar Pop, her dwarf Pomeranian, to shut up. \"You won't want to hear this,\" she said, more softly now.\n\n\"What.\" I aspirated the H in what. My voice took on an impatience I found myself using only with her.\n\n\"Be careful of getting your hopes up. You're not marrying him. It's only a date.\"\n\n\"I know that.\"\n\n\"I mean, you don't even know if you like him.\"\n\nOf course I like him, I wanted to say. I'm in love with him. Then it occurred to me that she was just plain resentful. She wanted me to be available to her every second of the day, to go to the beach, to get her car inspected with her\u2014all the minor, irritating tasks that she hated to do herself. The truth was she hadn't gone out on a date since her folks had forced her to break up with DeMarco, unless you could count her midnight rendezvous with Levon (aka her own private dildo). I could sympathize with her on those grounds, but what really vexed me was her unspoken belief that my attraction to other boys was a phase, that once I passed out of it, I'd come to my senses and fall deeply in love with her. Hogwash, I thought. Bullshit.\n\nHours later I met Arden outside the red-carpeted lobby of the campus film forum. He was wearing a checkered red shirt and a pair of overalls, the bottoms ragged and dragging. He might have been a farm boy fresh off the milking machines. A look of quiet expectation shone across his face, and all my fears were allayed.\n\nOnce seated he gazed up at the screen, awestruck, drinking in the images: the baroque hotel, the trimmed, geometric gardens, the feathered gowns of A, Delphine Seyrig, who looked as if she were listening to a ticking bomb inside her. He seemed to be so involved that I thought it best not to disrupt him, though I wanted to fuck him silly right then and there. Something twinged the root of my dick. Occasionally, his stomach would squeal, and a little belch would issue forth, but I relished this subtle indication of his humanity, relieved he felt comfortable enough to share it with me. Put simply, he was as nervous as I was.\n\nWe strolled down the storm-wet streets. The neon tubes trembled, clinging to the last of their currents. The branches were blowing. Out over the ocean, lightning pulsed twice, golden, subtropical.\n\n\"What did you think?\" I asked.\n\n\"Lavish,\" he answered. \"I never expected it to be so lavish. Those voices, those gowns.\"\n\nHis enthusiasm was more than I could stand. \"Tell me,\" I agreed.\n\nI had so much to say that I couldn't begin. He walked just slightly ahead of me, with only the slightest urgency, a beatific look on his face like the young St. John of the Cross. The world seemed unbearably benevolent all at once, the lightning flashing soundlessly in the distance, the palms above our heads watching, blessing us.\n\nWe stopped at a crosswalk. His smile was shy, eroding. Knowing that he wanted exactly what I wanted from him, I reached for the back of his neck and pulled him toward me\u2014reckless, I admit\u2014shoving into his mouth as much of my tongue as was humanly possible. His own tongue was soft, impossibly smooth, a sweet clam in a bath of brine. His whiskers sandpapered my chin. It was the most profound kiss in which I'd ever taken part.\n\nI stepped back slightly, dazed, chilled and sweating at once.\n\nHe pulled in his lips as if I'd tasted of mercurochrome.\n\n\"I'm straight,\" he cried.\n\nThe trees shook. My ears were humming.\n\n\"I'm straight,\" he cried again, as if to convince himself. Then he started running, fast as he could, down the length of the street.\n\n***\n\nThey'd been arguing off and on in Hector's room for more than two hours before I'd finally decided to climb out of bed. My guts burned. There was no point in trying to sleep. The moment I'd close my eyes, convincing myself that their silence was lasting this time, that they'd come to some peace, they'd start all over again, words volleying even louder than before. It was hard not to wonder whether it had been spurred on by something I'd done\u2014or not done, as the case may be. It was hard not to feel trammeled by it all. I might have been sitting in the bedroom of my childhood, scratching my name into the headboard, bending back the balsa wings of a model plane as Sid and Ursula fought through the night in the next room.\n\nSomething was bothering him about me. Was it his perception that I'd been under Hector's influence? I sensed it, just by the passing grimace on his face as we'd walked in the door, dressed in our skirts and wigs, from the Laundromat. In that one unsettling moment, any sense that I was multiple, that I could be anything other than what he perceived me to be, shrank to near nothing.\n\nDid I have to watch out for my brother?\n\nHe really wasn't a bad guy.\n\nI walked out across the dark parking lot and stared down into the pool. The floodlamps burned beneath the heated water, pitching golden waves of light onto the palms, the courtyard. A cord chimed once against the flagpole. Feet thumped across the walk, and there was Hector, climbing up the steps of the pool deck in nothing but his black jockey shorts. His face was dazzled and agitated, the muscles tensing in his legs. We looked at each other for several seconds, surprised into speechlessness.\n\n\"Is my brother mad at me?\" I said finally.\n\nHis expression dulled. \"What are you talking about?\"\n\n\"Forget it,\" I said. \"Good of you to get dressed.\"\n\n\"I want to know what you're talking about.\"\n\nI pulled in a breath through my mouth. \"Well, it was pretty hard not to hear you two guys fighting.\"\n\nHis brow clenched. He seemed apprehensive and appalled all at once. \"You were listening to us?\"\n\nWas he kidding? He knew as much as I how little privacy there was. Anyone walking down those halls could hear anything: burps, farts, little cries in the night.\n\n\"Do you want to talk about it?\"\n\nHe wagged his head back and forth.\n\n\"All right,\" I said, and stood up. \"Sorry I asked. I'll be out on the dock if you should find yourself deigning to talk.\"\n\nHe slumped down onto a lounge. The bottoms of his feet were dirty, spattered with drops of aqua paint. \"It's like he convinces himself it's all about money.\"\n\n\"What?\"\n\n\"He'll talk this way and that way about how he can't pay the bills, how it's time to start targeting a different market, cranking up the ads, but he hasn't done jackshit. He'd rather complain about it, put it off. Do you know what I'm saying?\"\n\n\"He'll hear,\" I whispered. I pointed up toward his dark apartment.\n\n\"He actually implied that I was responsible for that tour falling apart. I mean, you know that's not true. And this is after he's been pounding my back the whole day.\"\n\nA mole cricket drifted on the surface before a pool lamp. My eyes felt dry, stitchy. My throat was sore. It was minutes past 3 a.m. Suddenly I imagined this going on the entire night.\n\n\"Why don't you just get some sleep,\" I mumbled.\n\nHe lay back on the pink-webbed lounge. \"It can all be reduced to one single problem.\" His voice was oddly cheery, without malice. \"A single problem is all it amounts to.\"\n\n\"What's that?\"\n\n\"The problem's that he still wants to be a good little boy. He never got the affection he needed as a kid, and that's what's messing him up now.\"\n\nI looked at him skeptically. \"Yes, Doctor Freud.\"\n\n\"I'm serious. What do you think he's doing with a girlfriend anyway?\"\n\n\"You're out of your mind.\"\n\n\"He's just afraid. He's afraid to admit to himself he's a dirty little faggot like the rest of us.\"\n\n\"Oh, please.\"\n\n\"I'm telling you.\"\n\n\"And you know this for sure.\"\n\n\"Listen.\" He stood, poking a fingertip into my chest. \"I know what I'm talking about. He's twenty-eight years old, and he's still talking about his parents all the time. 'Sid this, Sid that. Ursula this, Ursula that.' My God, you might think they were sitting in the next room.\"\n\nI thought for a moment. There was certainly some truth to that, but only some. Wasn't Peter the one who couldn't wait to leave home, who never wanted to see them again? I'm happy, but you're not.\n\n\"But he hasn't spoken to them in months.\"\n\n\"It doesn't matter.\" He gestured at the first floor of the motel. A single lamp trembled on in the window, a robed figure drifting behind the curtain. \"It's like this whole thing is for them. It's like he's preparing for this hypothetical time, this perfect time, when he's gotten everything down just right, so he can say, 'Look, Mom. Look, Dad. I'm not the fucked-up puppy you thought I was.'\"\n\nI didn't know what to make of it. It all sounded too cheap and easy. And believe it or not, I resented the running down of my parents, especially knowing he hadn't met them. They could be awful, the most unbearable combination of clinging and secretive, but what parents didn't cause some damage, regardless of their intentions? They cared about us in their own miserable, stunted way. How much easier it would have been to be Hector, whose own mother was a dyke, who was completely at ease with his life, who called once every week to ask his advice about her latest girlfriends. No wonder the world was much less complicated in his eyes.\n\nInsects thrummed in the trees. Behind the clouds, the moon swelled, an amber smear of light.\n\n\"Did it ever occur to you that he could be bisexual?\"\n\n\"Knock it off.\"\n\n\"I'm serious.\"\n\n\"Bisexual?\" he said, shaking his head. \"I don't believe that. That's horseshit. He doesn't know what the hell he wants, so instead of talking about the real issue, he's blowing up about everything else.\"\n\nHector squatted beside the pool, dipping his hand before the jet. Chlorine tanged the air. I walked closer to him and smelled something like gin on his breath. Was that it? Could he have been drinking and had that set Peter off?\n\n\"So do you two have sex?\" I ventured.\n\n\"Sex,\" he laughed darkly. \"What the hell's that?\"\n\n\"Whoa,\" I said.\n\nHe looked up at me. His eyes were calm, impassive, barely concealing what he thought. He moistened the corner of his mouth with his tongue. He mumbled, \"That's none of your business, okay?\"\n\n\"I didn't\u2014\"\n\nHis face flushed. \"I ought to beat you to a pulp.\"\n\n\"Hey,\" I said, holding up a hand. \"Calm down, I\u2014\"\n\nTiny particles charged the air. Suddenly Hector was bending toward me. I froze, startled. Carefully, in small arcs, he started rubbing his whiskered jaw against my chin, humming something nameless. I pressed my fist into his stomach, pushing him backward. What was happening here? Then the minute I was sure that Peter had certainly seen us, Hector pulled away, grinned as if gauging my reaction, then walked toward the head of the pool.\n\nMy head was pounding. Then all the outlines colored in: the shambles of the grounds, the biting quiet between them. No wonder this place had felt so much like home.\n\n***\n\nBefore the night was over I'd told him everything about my time with William. I told him about the many nights spent lying awake on the living-room carpet, while William slept soundlessly up on the sofa beside me. I told him about the times I'd gathered the will to make love to him when he made it patently clear he didn't share my interest any longer, lying still, silent, refusing to respond. I told him about the sex party to which he'd taken me a few nights before I'd left, his ill-founded notion that that would pacify me. I even told him that despite our infrequency, our lovemaking was good, so unbearably good that it always meant more than it was supposed to, that soon after we'd finished, I was inevitably filled with the most numbing sense of loss, the quietest loneliness and grief, wondering why it had to be so complicated between us.\n\n\"It's funny how someone like that can screw you up for the longest time.\"\n\nHector nodded. He sat in the lounge chair across from me, wrapped in a towel. He looked off toward the umbrella pines, the broken gas lamp.\n\n\"I mean, I can't talk about this without getting all riled up. It's been months now. Aren't I supposed to be over it?\"\n\nHector raised his brows. His mind seemed to be elsewhere. \"So what finally got you to leave?\"\n\n\"I don't know. I didn't really plan for it.\"\n\n\"No?\"\n\n\"I think if I'd actually planned for it I never would have left. I'd still be there today. It's not like I had any money, or any place to go. You know the story.\"\n\nWe passed back and forth a single cigarette. We'd found a box of Camels on the table and smoked through the whole pack. I hadn't smoked much since my outings with Jane. My throat might have been sanded, scoured. Hector took the last drag, then tossed it off into the pool, orange ember hissing on the surface.\n\nWe'd have killed any guest who'd tried the same thing.\n\n\"One of these days I'm getting out of here,\" he said plainly.\n\nMy head emptied of thought. \"It's that bad?\"\n\nHe shrugged. \"It might not seem like it, but I actually care about him.\"\n\n\"I'm sure you do.\"\n\n\"That makes it harder. That's why I've stayed around as long as I have. But, enough already. I can't be bothered with someone like that.\"\n\n\"No?\"\n\n\"I'm better than that. I deserve someone better than that.\"\n\n\"Haven't you talked?\"\n\n\"He doesn't talk, period. He either shouts or shuts up.\"\n\nI nodded. I stared down at the shredded cigarette package in my lap, shuffling its pieces. He couldn't leave. \"I'd say stick it out. Give it a few more months, then see how you feel.\"\n\n\"I mean, it's hard not to take it personally. If someone had told me two years ago that I'd be involved with someone who stopped having sex with me, who had a girlfriend, of all things, I'd have said they were crazy.\"\n\nI nodded.\n\n\"But there's something about him...\"\n\nHe kept talking. The night dragged on. Whether or not I listened didn't seem to be the issue. He might have been me at another time\u2014I knew that worry, that quality of exclusion. But there was something unsettling about it too: we were all becoming too close. We might have been three wires, braided, intertwined, fully connected to one another. I looked up into the brightening sky, spotted a comet, some falling streak of light. I wasn't the right person to be told these things. Or was I? I looked at my watch: 5:45. Maybe I just needed some sleep.\n\n\"So what should I do?\" Hector said. \"You tell me what I should do.\"\n\nI sighed. \"You don't have any money?\"\n\n\"I'm broke, I'm totally broke. What savings do I have? Nothing.\"\n\n\"Just like me.\"\n\nHe winced. \"Two brothers in poverty.\"\n\nIf only for an instant, something almost unbearable drifted through my mind. I saw the two of us moving somewhere, together, New York, Seattle, London, Mexico City, changing our names, starting a life. There was nothing tying me down to this place. Peter, though upset, would eventually understand. He wouldn't be totally bereft. He had Holly, after all.\n\n\"Maybe it's just a sign of things ending. I should just buck up and dump him, but who wants to be the bad guy? Frankly, I'd rather be the one who's dumped.\"\n\nThe dew moistened the armrest. One by one the sprinklers sizzled on, flinging arcs of cold water across the glittering grass. I looked away from him, trying to keep the longing from my face. \"I have another idea,\" I said thoughtfully.\n\n***\n\nThe three of us stood at the county landfill, tossing things into the pit. The morning was hot, windy. Wood chips sprayed from the spout of a recycling machine. Across the pit a yellow bulldozer climbed across the little hills of trash, rolling forward, backing up, rolling forward again. The machines roared and ground their gears. A spoon glinted among the mounds of used tissue. It troubled and interested me for some reason\u2014that single instrument held up to so many mouths so many times, a little vessel of pleasure\u2014and I was tempted to wade down the garbaged slope to retrieve it.\n\nWe were in the midst of a great cleanup. Peter had suddenly decided to empty one of the storage rooms, the contents of which had suffered smoke damage from a fire months before my arrival. The clothes, in particular, stank. Though we'd washed them in everything from Borax to bleach, they resisted our efforts, clinging in defiance to their new soiled selves. The scent nestled into their very fibers. They weren't even worthy of the worst thrift store. I wasn't sure why this task couldn't have waited, especially since high season was only weeks away. I, for one, thought someone should have been staffing the desk, but I didn't voice my concerns.\n\nA good 90 percent of the clothes had been Hector's. With an exaggerated flourish, he tossed each article into the pit. His sunglasses\u2014aviators that had once belonged to me\u2014reflected the striated clouds, the chaos around us. He behaved as if the whole experience were of great significance, an elaborate ritual. He paused to remember each piece\u2014a sock with blue sparkles, a pink kimono, a T-shirt with a faded drawing of Mao Tse-tung\u2014as if each one had an intricate history attached to it. Then he sent it flying out over the ledge.\n\nHe draped a worn chartreuse shirt over his arm. Its collar was pointed, its buttons huge, even silly, the size of silver dollars. I couldn't stop looking at it.\n\n\"What?\" he said to me.\n\n\"That's an amazing shirt. You're throwing that out?\"\n\nHe shrugged. His face feigned indifference. \"I was wearing this shirt the night I went out with Jonas Pike. We were walking down the aisle of the St. Mark's Theater, and everyone, I'm serious\u2014I mean, fucking Joe Dallesandro was there\u2014everybody was staring at me.\"\n\n\"How come?\" I said.\n\nHe shrugged again. \"I don't know. I might have looked fierce. Or I might have looked like a street person.\"\n\nI glanced over at Peter. His face was drab, pensive. He made a point of extracting himself from our conversation.\n\n\"Could I have it?\" I asked.\n\nPeter exhaled, as if he'd reached his limit with us.\n\n\"What's wrong with keeping one thing?\" I said to him.\n\nPeter said, \"It's ruined. It smells. What are you going to do with it?\"\n\n\"I don't know,\" I admitted, then stared down at the muck on my shoes.\n\nI watched a truck dump a fresh load onto the desolate slope. The gulls were screaming above it. The truth was I didn't want to rile him anymore. He didn't deserve it. All morning I'd convinced myself that his mood had been prompted by seeing Hector and me by the pool. It was ridiculous, but it preoccupied me, threading in and out through my thoughts. Not to mention that I wasn't fully myself. We'd gotten in at six, and I'd had but two hours of sleep. I imagined circles, the deep violet of bruises, beneath my eyes.\n\n\"Maybe you're right,\" Hector said, still clutching the shirt. \"Maybe I should hold onto it for a while.\"\n\n\"That's it,\" Peter mumbled.\n\nWe both looked at him, utterly blank.\n\n\"Do you want to see what I think of my clothes? I'll show you what I think of my clothes.\" And with that, he dumped out his garbage bag, watching its contents sliding down the slope.\n\n\"I'm impressed,\" Hector said, deadpan, with raised brows.\n\nI didn't know how to react. He was more bothered than I'd imagined.\n\n\"He doesn't have a fleck of fun in his body.\"\n\n\"Let's go,\" I mumbled.\n\n\"Come here,\" Hector said to me. He stepped closer to the edge of the pit. The marl loosened beneath his shoes. \"Just come here for a second. This is amazing. There are twenty different versions of the same blue chambray shirt.\"\n\nI did as I was told. Hector was right. The shirts lodged against the side of an old washing machine.\n\n\"We don't like to be creative,\" Hector instructed. \"We don't like to take risks or exercise our imaginations. And when we go out to replace our clothes, we go straight to the Gap and say, 'Six stonewashed chambray shirts. Size Large, please.'\"\n\nPeter's eyes were shot through with an enormous sadness. I wondered whether he was going to start yelling. Instead, he strode toward the van in silence, waiting for us to finish up.\n\n\"Oops,\" Hector mumbled.\n\n\"Now we've done it,\" I said.\n\nBack at the King Cole, we immersed ourselves in the most taxing assignments, a penance of sorts: Hector attended to the twenty blinks on the answering machine, while I helped replace the charred wood of the storage room. I couldn't stop thinking about Peter, though. I couldn't stop thinking about his reaction to the shirt. Didn't he understand that things could be more than themselves? Wasn't that why we honored them, hoping to reclaim them: little emblems of change, loss? Why else was he putting so much effort into running this pit of a motel, a motel with its clerestories and tinted terrazzo\u2014all of which embodied something that was new during the era of his childhood, when the world still seemed ample with possibility?\n\nOr maybe his feelings were about something else entirely.\n\nHe watched me working. In truth, my skills were less than stellar. I'd bent nearly eight out of every ten nails I'd tried to drive into the plywood. They littered the floor, making a wreath around my shoes, tokens of my distraction.\n\n\"Could you hold the hammer right?\" he said finally.\n\n\"I am holding it right.\"\n\n\"Come on,\" he said. \"Grip the base, not the neck. That's it. Swing back, then aim.\"\n\nI followed his instructions. Once more, the nail bent, splitting the wood.\n\n\"God-damn,\" he cried.\n\nI wouldn't look at him. It was the first time he'd actually raised his voice to me alone, and the sheer force of his outburst, the pent-up pressure behind it, scared me, caught me off guard.\n\n\"Maybe if you started behaving like you dressed,\" he said, \"you'd get it right.\"\n\n501s, black T-shirt with cutoff sleeves. My leather motorcycle jacket lay crumpled in a heap beside me. There it was: the issue of my look again. \"What's wrong with what I'm wearing?\"\n\n\"I'm sorry,\" he said. \"I didn't say that. Forget that I'd ever said anything like that.\"\n\nWhat was this crap about my clothes again? Was it jealousy, a projection of his fears? Or was he still hearkening back to the drag of a few nights ago? Outside, the sun lay burning, trapped behind a curtain of dirty clouds.\n\nMaybe if you had it in you to fuck\u2014\n\n\"Are you upset?\" I said after a while. I'd had enough of this shit.\n\n\"I'm not upset. Why would I be upset?\"\n\n\"You certainly sound like you're upset.\"\n\n\"You're way too sensitive,\" he murmured. \"Here,\" he said, passing a Coke bottle to me. \"You must be thirsty. Drink some of my soda.\"\n\nI shook my head, declining.\n\n\"Drink it.\"\n\n\"I'm not sensitive.\"\n\n\"No, no, you're not at all.\" He smiled, attempting to diffuse the tension in the room. \"My God, you're still that touchy little boy.\"\n\nI went back to work. I gripped the hammer like he'd told me, this time driving the nail straight through the plywood without splitting it. I resisted the urge to say, \"See.\"\n\nBut I didn't have to. To my surprise, his eyes held within them a sadness, a regret.\n\n\"Listen,\" he said, stepping closer to me. There was a new smell about him: pine, citrus, shards of burnt wood. He exhaled. \"I'm sorry. I'm really, really sorry. I'm under a shitload of stress right now. Florida tourism's down 40 percent from last year. If I don't have a good season, I'm going to lose the place.\"\n\nHe had a point. In the last month, our regulars, many of whom were from England or Germany, had been canceling left and right after the murder of a British couple at an I-4 rest stop. It was as if the very name of Florida had instantly become the most fraught of metaphors: a brutal, ruthless jungle where innocent tourists were hacked to death.\n\nWas this what all his tension was about?\n\n\"How have things been between you and Hector?\" I asked.\n\nThe mere mention of his name, the mere yoking of them together in the same sentence, aroused a perceptible change in him. He might have been a cat: I actually saw his pupils shrink, then swell.\n\n\"What has he said?\"\n\n\"Nothing,\" I answered immediately. I'd done it now. What was the matter with me? My face flushed hotly.\n\n\"Everything is fine with us,\" he assured me. \"We've had some troubles, but everything is generally fine.\"\n\n\"Great,\" I said dully.\n\nHe smiled at me. I brought the hammer firmly down on another nail.\n\n# Chapter 17\n\nAlready I knew it was going to be one of those days. Sometimes you know it the second you wake up, if only by the feeble slant of light on the furniture, the smell of mildew in the room. Other times you know exactly what's wrong.\n\nFor the past day we'd hosted Fulvia Diaz, editor of a widely read travel guide. As a rule I hated these people. A tiny woman in tennis whites with a nasty helmet of hair, she seemed to be completely at home in her role as a walking irritation, expecting, even demanding me to be exasperated with her. Not only had she asked to use the desk phone for six personal calls, all of which were of a half hour's duration, but she'd ensconced herself in the lobby, holding court as it were, asking leading questions of our guests (\"Was your shower hot enough this morning?\" \"And what about that continental breakfast?\").\n\nI'd had about as much as I could stand when Peter phoned from another line.\n\n\"Still there?\" he murmured.\n\nThe TV blared. On The Weather Channel, Vivian Brown described a developing winter storm off Cape Hatteras. I whispered, \"Yeah, but she's going to check out the beach in a few seconds.\"\n\n\"Tigertail?\"\n\n\"Yes, yes.\"\n\n\"You told her about the shortcut?\"\n\n\"No, I told her to crawl on her hands and knees through the swamp.\"\n\n\"What?\"\n\n\"Peter\u2014\"\n\nI stepped back behind the desk. She walked across the parking lot, head down, chugging her arms. She opened the door to her rented Lexus.\n\n\"So I take it everything's going well.\"\n\n\"Oh, she's just as pleased as pralines. She's got the nicest suite in the place, all the free gin she can drink, and she's tied up the line for hours.\"\n\n\"That's fine,\" he said. \"A lot's riding on her. At this point we need all the good press we can get.\"\n\nThe storm swirled on the TV weather map, a furious, spinning comma. Meanwhile, it couldn't have been any more hot and torporous here. The liriope wilted in the heat. Even the motors of the air conditioner felt sluggish.\n\n\"I, on the other hand, think the Vatican should start canonization proceedings on me.\"\n\nHe started laughing.\n\n\"I'm serious. She's been relentless, a real pain in the butt.\"\n\n\"Come on now.\"\n\nHis laughter only sparked my aggravation. \"Why did you give me the desk, today of all days?\"\n\n\"What did she say?\"\n\n\"Oh, some crack about hospitality management. It doesn't matter, it's just\u2014\"\n\n\"I apologize.\"\n\n\"It's\u2014\"\n\n\"I said I apologize.\"\n\n\"Well\u2014\" I gazed up at the weather instruments\u2014thermometer, barometer, rainfall gauge\u2014over the desk. \"I just hope I never have to see that walleyed shrew ever again.\"\n\nI turned. My whole body went cold. There she was, standing before me with shining forehead. A manic smile spiked her face.\n\n\"Better go,\" I said, then hung up.\n\nShe passed her key across the desk.\n\n\"How much of that did you hear?\" I said, too shocked to be polite.\n\n\"I've seen just about as much as I need to see, thank you.\"\n\nMy tongue felt frozen in my mouth. \"Is something the matter?\"\n\nHer jaw clenched, then relaxed. She seemed oddly relieved, as if I'd delivered what she'd truly wanted. \"I said, thank you. I'll fax your entry by the end of the week.\" And with that, she turned and left through the door.\n\nI held onto the edges of the desk, immobilized. My hands were trembling, wet. My toes felt icy inside my socks. Once a few minutes had passed I walked up the stairs to Hector's room.\n\n\"Open up,\" I said.\n\n\"What's the matter?\"\n\n\"I think I really fucked something up.\"\n\nThe door opened, and he led me to the couch. I was beside myself, mixing things up, halting, recasting my sentences. All the while Hector lounged on his side in a red union suit unbuttoned to his stomach, absently rubbing his crotch, his chest. A lazy, half-smile settled on his face.\n\n\"Get over it,\" he said finally.\n\n\"What?\"\n\n\"She's not going to run us out of business, and if Peter wants to believe that, that's his problem.\"\n\n\"But he needed this to work out, and I had to open my great big mouth.\"\n\n\"Listen to yourself,\" he said with some irritation.\n\n\"What?\"\n\n\"Why are you so freaked out?\"\n\nI exhaled through my nose.\n\n\"He should have been working the desk,\" he said, \"and you know it. Why was he letting Little Miss Travel Editor walk all over you?\"\n\n\"So\u2014\"\n\n\"So he could absolve himself of responsibility if something went wrong, okay? I'm tired of watching you running yourself down like that. Here,\" he said, flinging something onto my lap. His grin was sly, complicated. \"Enough of that horseshit. I've been cleaning up. Look what I found.\"\n\nA dog-eared magazine lay across my knees. Beneath its title\u2014Man's Favorite Sport!\u2014two dark-haired men, stripped to their waists, lay embracing on an unmade bed, open mouths locked together. In the corner, within a yellow starburst, appeared the words: XXX Rated, Non Violent Explicit Pictorals\u2014Adults Only.\n\nSomething frigid pooled inside my stomach.\n\n\"Look inside,\" he coaxed.\n\nMy face flushed with blood. I wouldn't look at him.\n\n\"Come on.\"\n\nI finally flipped open the magazine. On the first page, the older guy, a big, muscular lug with stubbled jaw, sucked the dick of the younger, a weasly type with tattoos and a ruddy dick. On the next page their roles were reversed. On the next the older, now in a ribbed sleeveless jersey, was crouching on his hands and knees, face to the floor, reaching back for the kid to plow him. The shoot might have taken place in some Long Island motel room, the dark paneling, macrame wall hangings, and leopard bedspread adding a cheesy kind of authenticity to the scene. I imagined a nuclear plant humming across the street, steam billowing out over a glassy bay, a dead fluke floating on its surface.\n\nIt wasn't long before Hector was seated next to me, knee pressing into mine. I didn't know whether to inch away or to lean my weight into him. I went with the latter. Heat swarmed in my groin. \"What were you doing with this?\"\n\nHe shrugged. \"Doesn't everyone have a little stash?\"\n\nMy chest hammered. If I were only at ease, I would have enjoyed this. It wasn't like I'd never looked at this stuff before. But it was different to be looking at it with Hector beside me.\n\n\"Now that's some killer porno.\"\n\nHe couldn't have been more casual. He might have been some old man showing off his fishing gear.\n\n\"I mean, look\u2014not a pretty boy or a gym rat in the bunch. See that mussed-up hair, that little scab on his back?\"\n\nI nodded.\n\n\"These new photographers, this slick shit\u2014they would have airbrushed that out. And that's why the new porno sucks.\"\n\nThe magazine's narrative continued. On the fourth spread a skinny young guy in a brown UPS uniform barged in, to the amusement of the two guys on the bed. I paged forward, absolutely immersed. Someone might have been speaking directly into my ear: a murmur, a whisper. And then it occurred to me that the visitor was none other than Hector himself.\n\nI was absolutely speechless.\n\nHis knee ground deeper into my leg.\n\n\"What the\u2014Oh God!\"\n\n\"Well,\" he said, a little sheepish, flustered. He pressed his fist to his mouth. \"Are you surprised?\"\n\nI gazed at his torso, the ropy, idiosyncratic dick. That smile\u2014oddly complicated, disarming. His image shivered with life. It was hard to connect the Hector beside me to the Hector in the magazine.\n\n\"You never told me about this,\" I said finally.\n\n\"Yes, I did.\"\n\n\"Did not. You never said a word.\"\n\nHe wagged his head. \"If you only yanked your head out of the clouds\u2014\"\n\nI shouldn't have been at all surprised. But it did surprise me, and I couldn't help but wonder why he'd kept it from me. Some things you don't forget.\n\nIt turned out that Hector, twenty years old, had moved out to San Francisco to stay with a trick he'd met on the 21st Street Beach. It didn't take long to realize that the trick was full of hot air, that he already had a boyfriend, and Hector was out on the streets, with five dollars in his pocket. He'd wandered into a bar in the Tenderloin, where he was approached by a middle-aged man with a pocked face. \"I'll give you a place,\" he said, \"if you say yes to some pictures. Naked, I mean.\" Desperate, he went home with the guy, who, to his astonishment, led him to a room of his own. But the guy had big plans for Hector. Within weeks he was appearing in two videos a week, and was making enough as a hustler to live in North Beach, in his own two-bedroom apartment, with enough hibiscus and bougainvillea to remind him of his mother's yard in Hialeah.\n\nMy left eyelid pulsed. I'd seen him once before, maybe many times. Wasn't he Al Parker's partner in the video I'd once watched at the arcade?\n\nI paged ahead, entranced. All that brightness and heat, all those urgings toward\u2014Come here, they said. Come with us.\n\nHector laughed, testing my vision, moving his palm before my eyes.\n\nThen footsteps down the hall.\n\n\"Shit,\" I whispered.\n\n\"What's the matter?\"\n\n\"Is it Peter? It's Peter, isn't it? Shit, fuck.\"\n\nWe sat in absolute stillness, listening. Coins fell; a soda can knocked, tunneled through the machine. A ziptop stripped, then mouth to metal. Fizzing.\n\n\"Guess not,\" I said, rubbing my face with my palms.\n\n\"Easy.\" Hector clamped his fingers around my wrist. \"Easy.\"\n\n***\n\nI waited till I was alone.\n\nI dug down deeper in my bed, relaxing.\n\nWhat other feeling could measure up? What other thing could smash me apart so beautifully, thoroughly? I might have already pulled the needle from my vein, drawing it out, waiting for its effects, heat seeping into my muscles. Once I said yes to it, that was that\u2014loneliness, fear, death, loss: all of it gone, running off into a wilderness. I was solitary, myself, a pure, driven thing. No past, no future. I imagined lying over him, holding onto his hands, tongue working, eating up a trench from his belly to his throat. His skin tasted of salt water, leaves. His dick pressed upward into my chest, grinding.\n\nWhy else was he showing me his pictures?\n\nI couldn't help but laugh sometimes.\n\nI wandered into the bathroom, wet washcloth to my face, breathing. I glanced up at myself in the mirror. My bottom lip, fat, ticking; my face, older, paler, harsher than I'd expected.\n\nSeventeen: slouched beneath the dark palms, waiting for William to walk across the lawn.\n\nMy hand wandered through the curling hairs of my belly. I cradled my dick, testing its weight, heft, fullness. I felt my stomach again. The skin was hot. Groping higher: something swollen, tender. A hardness. Scab? Blister? I kept pressing, probing it with my fingers.\n\n***\n\n\"Did you see this?\" Peter asked.\n\nWe were standing behind the front desk, 7:07 a.m. Peter fumbled through the rack of incoming mail, gestures animated, jerky. I smelled the coffee on his breath. He behaved like he'd been up and around for hours.\n\nIn my hands I held the fax from Fulvia Diaz.\n\nWhile most renovations wipe out the past, Peter Sarshik's King Cole Motel serves as an homage to Clem Thornton's former Boca Palms showplace. Not only have the odd details been lovingly restored (stucco friezes of gladioli and water jets, starfish, seaweed etched into plate glass), but there's enough decay in evidence\u2014e.g., rust stains around the pool, sulfurous tap water\u2014to make it all seem real. Check out the pastel disks of the walkways. Check out the aerodynamic design of the Nurmi wing (tell yourself it isn't moving a million miles an hour). Have a blended drink at the pool bar. And be sure to introduce yourself to Evan, the handsome, affable desk clerk.\n\n\"I mean, she really got it,\" Peter said.\n\nCould she have mixed something up? What stucco friezes? What seaweed etched into plate glass?\n\n\"This'll help us. I know this'll help us.\"\n\nI gazed out at the browning papyrus through the window. The light outside was glassy, overcast. She had to have mixed something up. \"I thought I'd pissed her off.\"\n\nHis eyes hazed over. \"You never told me that.\"\n\nI read through the fax again. At least she'd called me handsome. \"Did you have sex with her or something?\"\n\n\"Funny.\"\n\n\"Well\u2014\"\n\n\"I mean, she mentioned your name. I thought you'd be thrilled.\"\n\nI yawned, shrugged. \"I'm thrilled.\"\n\nThe truth was I hadn't been getting enough sleep to care very much about some travel editor's opinion. I lay in bed, wide awake, four, five in the morning, thinking about Hector.\n\nHe looked at me strangely. \"I'm going outside to work on the storage room. I've been on a roll. Call me if you need me.\"\n\nI stared outside at the gleaming inadequacies. The pocked leaves of the aurelia. The yellow mineral stains on the walls, fan-shaped from the sprinklers. Some-one had to have had sex with her. Not two minutes after Peter's departure the phone rang. I lunged for it, alarmed. \"King Cole, how may I help you?\"\n\n\"Peter?\"\n\nSilence, breathing. A woman. Holly.\n\n\"Listen, I thought it would be nice if you came over Friday night. I mean, Ory's going to be with my sister, and I thought we'd grill some shrimp. We'll have a nice quiet night together. How's that?\"\n\nA lump in my throat. I cradled the handset, frozen, until I was able to breathe again. \"Sure,\" I said finally, \"I'd like that.\"\n\nSilence again. In the background, Ory chattering about box turtles. \"Do you have a cold?\"\n\n\"No.\"\n\n\"Good. I'll see you Friday, then.\"\n\nMy head tingled darkly. I hung up the phone, rapt, listening to Hector's bare, perfect feet on the floor above me.\n\n# Chapter 18\n\nThe swelling still hadn't subsided.\n\nI kept telling myself not to panic. How many times had I acquired a cold, rash, or virus only to watch it fade, to learn it was just what it was, not a harbinger of something larger. It was life: one got sick, then healthy. Sick, then healthy. Who could even pass through a single hour without battling all kinds of imperceptible threats? Cooling systems, doorknobs, mosquitoes, standing water\u2014all of them birthing pools for potential hazard. If only I saw what was happening, saw how efficiently my body was burning, fending them off, I'd have much better things to think about. Instead, I'd willed myself indomitable. I'd gotten to the point where I'd refused to get sick, where I actually believed I'd acquired the ability to purge illness from my system, if only to prove to myself that I was well.\n\nDidn't I have a little swelling once before? Didn't I think it was something greater, before it passed on to nothingness?\n\nOnce I'd welcomed the first signs of a cold. Sickness was a tent, a house, a warm dark blanket wrapped around my aching bones. Game shows droned on the TV. Palms blurred through the windows. Ursula wandered in and out of my room, cool hand to my scalp, passing me a cup of broth or pineapple Jell-O. She liked these times as much as I did, her worries focused to a single point. There was nothing else in the world: no Sid, no Peter, only her care and my sickness. Compassion, an absence of complication. We were never any closer than this.\n\nIt would never be so simple again.\n\nWhen my worrying had gotten out of hand, I decided to approach Hector. I stood outside his apartment door, walked away, then back again. Would he think I was needy? Were my visits too obvious, or frequent? I stood there, paralyzed, mouth sour, listening to him puttering with some papers. Finally, I knocked.\n\nI said, \"Could I show you something?\"\n\nHe nodded. He was standing, thick arms folded, looking at the snapshots he'd just tacked up above his desk. On the top, Julia, his mother, standing before the exploding Mount St. Helens\u2014her smile huge, red, lipsticked\u2014volcanic ash clinging to her shoulders. Beneath her, two men on a beach\u2014shirtless, muscular, eyes flashing with laughter\u2014arms wrapped tightly around each other. Beneath them, a young black man\u2014almost pretty\u2014with solemn blue eyes and beaded necklace, squatting before the great globe of the Unisphere in Flushing Meadows Park.\n\n\"Who're they?\" I said, pointing to the couple.\n\n\"Don and Miguel.\"\n\n\"They're adorable.\"\n\nHe nodded. He passed through the stack of snapshots, assembled them with a rubber band, placed them in a drawer.\n\n\"I mean, they're not kids, but they're really, really handsome.\"\n\nHis smile clenched. He looked amused. \"I knew them for ages. Miguel, the shorter one\u2014he was my lover for two and a half years.\"\n\n\"Really?\"\n\nWe both looked over at their picture. Together, they seemed so radiant and self-contained, so oriented toward a future, that I couldn't help but feel included. They unsettled me, too. They reminded me of the fact that I'd never believed in the possibility of my own future, how my lack of faith had infused all my decisions, a low-grade fear and rage burning at the heart of everything, from why I'd stopped going to the dentist to my lack of organization to my hasty decisions about college, money, boyfriends. Wasn't it better to extinguish oneself than to lose one's light, little by little? Wasn't this what all those silly rock songs told us? But looking at them, I felt differently. It isn't so bad to get older.\n\nI said, \"Why is it we only see pictures of young men? All these magazines. You might think queer life was only for kids. What about the older guys?\"\n\n\"What older guys?\" Hector mumbled.\n\n\"Hmm?\"\n\n\"They're dead. Don and Miguel\u2014all these older guys are dead.\"\n\nMy stomach pitched. I thought of their contentment, their absolute confidence in each other. They'd known what they'd wanted: how could anything not have gone their way? Something fleeting coursed up the column of my spine. I might have been lifted, for the briefest instant, off my feet.\n\nOutside, the palms were drenched with an oddly golden light.\n\n\"What were you going to ask?\" Hector said.\n\nMy swelling pressed against the waistband of my jeans, resisting it. Blood hummed, murmured within my veins. I looked at him: there really wasn't very much time.\n\nI'd make my move on Friday.\n\n\"Would you mind cutting my hair?\"\n\n\"I'm tired,\" he said.\n\n***\n\nMy haircut was growing out, stray strands curling over the tops of my ears. It required more maintenance than was initially promised. Though it didn't need to be combed or washed every day, it looked dingy if it wasn't cut once a week. I looked more like my old self, the diffident, milky self I'd been before Hector had gotten a hold of me. But it didn't matter now. The swelling on my belly was healing, along with all those ghastly feelings I'd attached to it.\n\nStill, I felt distracted, possessed by an emptiness I couldn't name. Wasn't there someone, someplace in the world, who was being told that he or she was dying right at this very minute?\n\nLate Friday afternoon I walked into the Nurmi wing, where Hector had room duty. He was studying the bottom sheet of a bed, checking it for stains, sand, crumbs, or offending odors, deciding whether or not to change it for the next guest. I knew the ploy; I'd practiced it myself on occasion, though it was a lousy one. From the boom box, Laura Nyro wailed at top volume, pounding out parallel triads, Motown-style, on the piano.\n\nI stood inside the doorway, watching the deliberations crossing his face. Then he glanced upward, startled. \"Would you mind cutting my hair when you get the chance?\"\n\n\"Huh?\" He turned down the volume.\n\n\"I said\u2014\" And I repeated my question.\n\nHis brows lifted. He swigged from a can of watermelon soda. \"I'm really busy.\"\n\nHis response, however subtle, carried an accusation: I was demanding, pushy, self-absorbed. But I'd never asked him for anything.\n\n\"I mean, could it wait?\" he said, face thawing. \"This is a bad time for me. Maybe Tuesday?\"\n\nI tried not to be miffed. I pulled one side of the bottom sheet, helping him fit it around the corners of the mattress. He replaced Laura with Kate Bush's Lionhearted album. \"What about that barbershop in Naples? Have you heard anything about it?\"\n\n\"Porkchop's?\"\n\n\"Yeah.\"\n\n\"He's pretty good,\" he replied, glancing up. \"And cheap. I've gone to him one or two times myself.\"\n\n\"Well, thanks,\" I said, too loud, forced.\n\nI walked down the hall. It hadn't occurred to me that I was asking for much. Five minutes, not even that. After all, wasn't he the one who'd insisted I'd look better with a shaved head? And wasn't I the one who'd been sprucing up his rooms, which had seemed increasingly flagrant in their ineptitude, just to keep him in Peter's good graces?\n\nI stood at the window in the eastern stairwell. I watched Peter standing in the parking lot, waiting for something. I couldn't stop staring at him. It seemed to me that his entire story was encapsulated in his waiting, from his jerky glances toward the road, to his nearly constant pacing, to his wringing hands, which he kept twisting and pulling as if they hurt him. He was nothing but uncomfortable in his own skin, and it was the loneliest thing I'd ever seen, and it repelled and frightened me, knowing that we were a part of each other.\n\nWhen I couldn't stand it anymore, I trudged down to the office, swiped the key ring off the rack, and walked outside. Over the Gulf, clouds hurried toward the south, swarming, tornadic.\n\n\"I'm taking the van,\" I called to Peter.\n\nThe breeze lifted, blew through his thinning hair. His eyes looked impossibly green against the palms. \"Where're you going?\"\n\n\"Haircut.\" And before he could insist I didn't need one, I shifted the van into reverse, and headed north.\n\n***\n\nThe sky darkened as the squall line approached. I was lost somewhere inside Golden Gate Estates. All the streets bore numbered names, qualified by quadrant directions (Northwest, Northeast, etc.), and I couldn't quite remember whether it was SW 130th Street or SW 130th Place. I pored over my memory of Peter's address book. It troubled me to think that anyone actually lived here. Like its better-known sister, Cape Coral, the landscape was jungly, dense, overrun with escaped tropical species\u2014Brazilian pepper, Chinese tallow, punk tree\u2014not a shred left of indigenous beauty. It was the kind of place that wasn't civilized enough for comfort, nor remote enough to point up the pleasures of solitude. The axle bumped over a tiny sinkhole in the street. There was no point in continuing on.\n\nJust then I spotted her gold Tercel. It sidled up alongside a long, narrow trailer in which all the lamps inside were on. A funnel of light illuminated the front yard. To my surprise, Ory, her six-year-old, was sculpting sandcastles beside a weaving pyracantha bush.\n\nI slowed the van to a crawl. \"Storm's coming,\" I said through the window.\n\n\"Lightning,\" he said, pointing a plastic shovel to the sky.\n\n\"Yes. It's dangerous. Do you know what happens if you get struck?\"\n\n\"Boom,\" he said, throwing up his hands.\n\n\"Does your mom know you're outside?\"\n\nThe boy nodded avidly, then smacked at the sand with his shovel.\n\n\"Ory,\" Holly said. She stood at the screen door in a black T-shirt, cutoff jeans, and motorcycle boots. Her hands were fisted on her hips. She squinted. \"Who're you talking to out there?\"\n\nI couldn't leave now. In truth, I'd only wanted to see where she lived. I'd only wanted to see the trailer, then get the hell out of there.\n\nHer smile revealed her upper gum. \"Peter?\"\n\nI froze, stricken.\n\nShe stepped closer to the van. \"Peter?\"\n\n\"It's Evan,\" I said finally, and looked away.\n\n\"Oh my God,\" she said, breathless.\n\n\"What?\"\n\n\"I didn't have a clue. I've only seen you from a distance.\"\n\nI shook my head.\n\n\"I mean, your hair's shorter, and you're a lot younger and all that, but\u2014\" She exhaled through her mouth and stared. \"Say something.\"\n\nI frowned, waggled my head.\n\n\"Please. Say something.\"\n\nI mumbled, \"What do you want me to say?\"\n\nShe started laughing, a little pained. Her eyes were vibrant. \"You even sound like him. Oh my God. This has to be, like, the weirdest experience of my life.\"\n\nWe stared. I had no business here. I should have gotten my hair cut.\n\nHer face quieted down. The wind picked up, palmettos scouring the surface of the trailer. \"Do you want to come inside?\"\n\n\"No,\" I said. \"Just out for a little drive.\"\n\nShe laughed knowingly. Her absolute familiarity with me was both unsettling and comforting. \"Yes, Evan. You just happened to be driving through beautiful Golden Gate during a severe thunderstorm warning, a full thirty miles from the King Cole.\"\n\n\"But\u2014\"\n\n\"Oh, just for a minute. Come.\"\n\nShe placed her arm around me as we walked toward the door. I wiped off my boots on the grass mat. \"It's so good to finally meet you,\" she said.\n\nInside, the trailer was as long and narrow as a boxcar. Nearly everything had a dual purpose: a sofa that opened up into a bed, a coffee table that doubled as a chest. On the table stood a tiny lamp, no taller than a wine bottle. The whole space was possessed by the spirit of order, a spirit determined by its limitations: one thing left out and the place would be a mess.\n\n\"So's it time to go to bed?\" she asked Ory.\n\nHe was sitting on the floor, looking up at me on the sofa, curious and amazed.\n\n\"Ory?\"\n\nHe stuck out his lip, showing off the linings of his eyelids, then shook his head no.\n\n\"Stop that,\" Holly said. \"Don't make monster faces at Evan. He's our friend.\"\n\nOry's face relaxed. He reached up, fingers scrabbling toward my belt loop.\n\n\"Oh,\" Holly said, and smiled. \"I know what he wants.\"\n\nI stood, picking up Ory. He was almost weightless, lips moist, sticky against my cheek. I kissed him back, bouncing him slightly. I wished I felt more at ease. His grasp tightened around my neck, flooding me with self-consciousness.\n\n\"Okay, that's enough now,\" Holly said. \"Time to say good night. Time for bed.\"\n\n\"Good night,\" he said to me.\n\n\"Good night, Ory,\" I said.\n\nI watched them walk hand in hand down the hall. I wandered over to the mini-china closet, an oversized toy, and glanced at the clear green tumblers on the shelves\u2014the free kind from service stations\u2014while Ory fussed and sobbed in his room. I looked at his pictures on the wall. Ory, at one, fed by an unseen hand, chocolate syrup smeared all over his mouth. Ory, at two, standing at a dock, eyes shining, holding a minnow in his palm. Ory, at three, in a straw Hawaiian hat, brim swallowing up his head. I kept staring at his face, the narrow forehead, the benign smile, pure and unassuming as a dolphin's. My stomach hurt. A queasy notion passed through my mind: Ory was my brother's son.\n\nThe rain picked up, pelting the windows. This was Peter's separate, hidden life.\n\nI roved down the hall to Ory's room. He was still crying, squealing, voice pitching higher with each explosion of thunder. Water pooled on the sills, seeping through the cracks. Holly perched on the edge of the bed, eyes merry, twinkling, but clearly within two seconds of coming undone. I stepped through the dump trucks on the floor, past the wooden blocks and the spotted pink anaconda. Bright tempera paintings covered the walls, edges curling inward.\n\n\"He's scared,\" Holly said.\n\n\"Am not,\" he said, face pressed to his pillow.\n\n\"Are too.\"\n\nHe gazed up at me with a single eye.\n\n\"But you liked it outside,\" I said to him.\n\nHe pressed his face to the pillow, shook his head no.\n\n\"Be still,\" I said, then turned off the lights. \"Be absolutely still.\" I motioned to Holly, and we climbed into bed, lying on either side of him, bodies tight. Ory flipped over on his back. The trailer rocked and pitched, wind nearly wrenching it off its footings. The lightning was upon us now. I pictured the palms outside flaring up, one after the other, blazing around us like torches.\n\n\"Shhhhhh,\" I whispered. \"It's taking our picture.\"\n\nThe room flashed. Ory frowned, whimpering.\n\n\"Look, the sky's taking our picture.\"\n\nIt flashed once more, and then again and again. Ory's eyes widened in the dark. And the three of us leaned into one another, breaths falling into sync, various smells commingling. Warmth passed from body to body, a single unit now.\n\nGradually, the storm subsided, thunder fading to a distant hush. The branches outside stilled, dripped. Moonlight poured through the room, illuminating Ory's mobile, his finger paintings. Down the hall clothes tumbled in the dryer.\n\n\"Thank you,\" Holly whispered.\n\n\"I can't believe he's actually sleeping.\"\n\n\"He likes you.\" She sighed hugely, then held herself with her arms. She lay absolutely still, eyes closed. I imagined a heaviness\u2014the weight of a truck tire\u2014pushing down upon her chest.\n\nI whispered, \"Are you okay?\"\n\n\"Listen,\" she said, \"do you mind if I kick you out?\"\n\n\"No, no. Not at all.\" And I didn't, actually. The roads would have drained by now. It was time to get home.\n\nShe turned on her side. \"Am I being rude? Tell me if I'm being selfish.\"\n\n\"Don't worry about it.\"\n\nHer smile was enormous, melancholy. \"I'm glad you understand. This was very nice.\" She reached over Ory and latched her damp fingers through my own. \"Let's get together sometime soon, okay?\"\n\nI nodded.\n\n\"And tell your brother I'd love to hear from him.\"\n\nI nodded again. My chance: should I tell her about the call, that I'd pretended to be Peter on the phone? I straggled by the potted fig, rubbing my thumb across a dust-coated leaf.\n\n\"What's the deal?\" she said in a hushed voice, as if talking aloud to herself.\n\nI turned to look at her again. Coward: I didn't say a word.\n\n\"It's just\u2014\" She stopped. She seemed hopelessly, utterly baffled.\n\n\"He cares about you,\" I said, surprising myself.\n\nShe shook her head. \"It's not like I'm going to get anywhere with him. I should have known what I was getting into.\"\n\nMy ear felt hot, as if I'd knocked it against a post. To myself, I counted backwards from 100, trying to calm down: 99, 98, 97... \"I mean, he really cares about you more than you know. He's crazy about you. You're all he ever talks about.\"\n\nShe glanced downward at her crossed legs. A dog barked at regular intervals in the distance. \"Really?\"\n\n\"Yes, really.\"\n\n\"But\u2014\"\n\n\"But what?\"\n\nShe lifted her face and looked at me. Her eyes were dolorous, hopeful, full of feeling. \"What does he say about me?\" she murmured.\n\nI touched the cooling silver of her bracelet. Her wrist felt warm as if she were running a temperature. \"You have to ask him that yourself.\"\n\nShe sighed. Her eyes rested on the sleeping boy beside her.\n\n\"Tomorrow night,\" I said. \"Come over tomorrow night. Surprise him. Come for dinner. He'd love to see you, I know it.\"\n\n\"Okay,\" she said quietly. \"You're right. You're absolutely right. Why have I been such a wimp about everything?\" A triangle of light illuminated her face. Her eyes looked calmer, grateful. I might have granted her a wish.\n\nI walked back to the living room, where I glanced at the TV. I knew it now: everything would work out for the best. I'd get Hector, Peter would get Holly and Ory. I'd make sure of it. We'd all be thankful and relieved, kinder, calmer. Another front was lining up over the Gulf, weather map pulsing, all greens and reds, blinking like a Christmas tree.\n\n***\n\nHis plan was this: he'd walk into our respective rooms, pause before our sleeping bodies, then smother us with a pillow. He'd listen to the gulp, stammer, and wheeze of our closing-down systems. Then silence. An absolute reverence. I'd be the first, the harder one. I'd resist, yelling out, driving my fists into his chest. Hector, on the other hand, would be the shocker. He'd take to the pillow, almost tenderly, like a lover, almost welcoming that choke, that heightened sensation before the heart stills then floods. Peter would pause, pondering the drip of the bathroom sink. He'd call up the police. He'd fix himself some coffee, basking in the numbered minutes of his freedom, bare feet up on the coffee table. Minutes later, he'd hear the squad cars pulling over the potholes in the drive. Daybreak.\n\nI woke up, panting. I crept to the window, eyes foggy from sleep. The yard was a reservoir, a lake. Paint cans floated across its surface, loose boards and boxes, all of them drifting, drawn by the imperceptible suck of the culvert.\n\nIn the morning, Peter and I sat at the office table, silent. My T-shirt clung to my back. He reached for the box of corn flakes. He poured them into a bowl, spooning them into his mouth, determined to finish them off, not allowing himself to acknowledge that they were stale, virtually inedible. I knew that he was losing money by the day, but this was about something deeper. He might have been our father. I saw it all: his self-denial, his secret cultivation of martyrdom, his exquisite selfishness, his stubborn indifference toward those affected by his decisions. My rage pushed up against my chest, a hard metal plate. I stood, then glanced at his thinning crown before I left the room.\n\nI had to stop. I walked through the corridors, listening to the water streaming through the rainspouts. I thought about what I'd liked about him, what made him interesting, valuable. I thought about his complete devotion to restoring the King Cole property, his ability to bestow all his attention upon a single window, sanding its frame for days on end until it was nearly perfect. I thought about his various interests\u2014the paintings of Richard Diebenkorn, the designs of Luis Barragan\u2014how he most likely knew more about these things than anyone else on the planet. I thought about his measured, good-natured rapport with our guests, how he'd once sat for two hours with an elderly couple after the man had experienced chest pains, convincing them he was actually interested in the subtleties of fly fishing, all in an effort to calm them down. But I couldn't hold all these thoughts in my head at once, and the pictures were already crumbling about the edges, no matter how hard I tried to keep them intact.\n\nWhy is it that I hate you so much?\n\nHe was everything I didn't want to become: stingy, closed down, secretive.\n\nI slouched around the pool that night, watching the lights blazing beneath the water. The rain had finally stopped, the ground swollen and sipping. Water lapped over the edge of the pool; the overflow made gulping sounds. Holly's gold Tercel was parked haphazardly on the grass beneath the royal palm, and I gazed up at the light in Peter's window, wondering what was happening between them\u2014love or argument, anguish or boredom? A tiny plane jetted across the dome of the sky.\n\nThen something popped.\n\nI looked up. In an instant a spark sizzled across a wire, traveling between the light pole and the office. It was all of ten feet. It happened so fast that I doubted myself\u2014flashes: detached retina?\u2014before it happened again. I couldn't stop watching. That subtle blue movement: it was as pretty as it was unnerving. My eyeballs ached from its brightness. And then the line sparked again.\n\nWhat dark fortune: my luck couldn't have been better.\n\nI pushed at the door to the stairwell. I bounded up the steps, two at a time, then ran down the hall to Hector's.\n\n\"Hi,\" he said, eyes foggy, half closed. His T-shirt said PUNCTURED. He looked like he was about to lean forward to kiss me. He grinned. The sweet smell of hashish fermented the air, drifting out into the corridor to be taken into the valves of the air conditioner.\n\n\"Did you look out the window?\" I said.\n\nHis face dulled slightly. \"What?\"\n\n\"Come here.\" And I grabbed him by the arm, pulled, stepped over the rumpled clothes on his floor. We stood together at his window, watching the wire, hushed, expectant. The quiet blue flashes lit up his face. We were motionless. Were we just seconds away from feeling the floor heating up, burning through the soles of our shoes?\n\n\"Shit,\" he said.\n\nI folded my arms across my chest. \"We have to tell Peter.\"\n\n\"He hasn't seen it?\"\n\nI shook my head, hard. \"No. We have to tell him. Now, right now. Move.\"\n\nAnd before he could say we'd handle it ourselves, I was leading him down the hall, nearly pulling him toward my brother's door. My chest pounded. My forehead boiled. Hector reached for the knob without bothering to knock.\n\n\"Peter, there's\u2014\"\n\nI stood directly behind him. He didn't process it immediately. Peter lying atop Holly, movements furious, ass gyrating, clefting like an opened fruit. They didn't stop. Peter glanced over his shoulder. His hairline filmed over with sweat. A rank human sweetness saturated the room: it almost had a taste. Holly's nipples were unusually large and dark, the deepest pink, unutterably foreign to me. She seemed to be too embarrassed to cover herself.\n\nSilently, Hector closed the door.\n\nThe line stopped sparking only two minutes before the fire department arrived in full force, radio blasting, beams of red light whirling about the courtyard.\n\nIt didn't happen till the next morning, but when it did, it was worse than I'd imagined. I watched it all from my window: Hector standing on the second-floor balcony, gazing down at Peter, whose arm held her close to his chest, escorting her to her car. \"You're an asshole,\" Hector called out hoarsely. \"You're a low-life fucking asshole.\" Peter looked up once, stricken, then looked back at Holly, who covered her face before opening her car door. Was this what I'd wanted? It hurt me as much to see it. I felt it deep inside my soul, burrowing.\n\n# Chapter 19\n\nI didn't think he'd still count. I didn't think I'd still think of him, but I did, often in the least expected moments. I might have been walking through Port Royal, immersed in the particularized order of a rose garden, and\u2014bam\u2014there he'd be in black leather and chains, throwing stones at me. Or a little farther up the street, squatting atop the bank in the lewdest position, dressed in harness and jockstrap and pretty clothes. I'd be startled for an instant, a dense high crackling like a brushfire inside my head. And then it would pass. He was back in Miami, working the camera at Channel 7, forever settled in his banal, tired habits, which he'd embrace for the rest of his life.\n\nI wouldn't see him again. Of that much I was sure. I wasn't so deluded to think there had ever been a chance anyway. Together we amounted to nothing: two specks, ground glass, empty bags blowing across a backyard lawn. At best, we were only an idea, and though the idea might have thrilled us with its daring, we were hardly daring. I couldn't say our time together had been a mistake, but some things are better left an idea.\n\nStill, if I thought hard enough, I could conjure him. The texture and taste of his back, sweaty and rich, as he came in from the garden. The trembling board of his stomach, brown, vascular, from light years of situps. In my mind I could play him, use him like a doll. My fury spun wild. I could poke out his eyes. I could spray him down with paint. I could strike up a match and toss it at him, watching him running through the woods with his hissing scalp, miles and miles, looking for water, screaming my name.\n\nI walked up Tamiami Trail, two in the morning. Cars careened up the highway, broken fenders dragging up little storms of blue sparks. It surprised me that he still mattered. He kept flaring up: a gas fire, a torch in the center of a cracked path, forcing me to reckon with myself. It seemed to me that I'd been shutting down, that I'd once again relegated myself to monkdom. Sometimes it seemed that the issue of sex had all the force of something much larger than myself, a huge moving glacier that threatened to run me over, consume me with its force. What the hell was I so afraid of? I could convince myself that the search for a lover\u2014or frankly, just someone to fuck\u2014was fraught with such risk and pain that the inevitable mess wasn't worth it, that it was easier to stay enveloped in my own fog, beating off to muscled images from magazines that had nothing to do with reality. Was it that I was truly afraid of happiness, of what I really wanted? Of what was so easily within reach?\n\nI was sitting across the street from a park. The palms were totem poles, their shaggy petticoats hanging lushly in the dark. It was getting hotter. Humidity funneled up from the tropics, coating everything with a slick wet net. Everything smelled of moisture\u2014the grass, the trees, the streets, the buildings. I glanced down at my arm. I wouldn't have been surprised to see a fresh mold growing on my skin, thick enough to be scraped off with a butter knife. At times like this I saw the whole Florida experiment as one vast error; the notion of an entire existence modeled after the one in the North seemed doomed to failure.\n\nTwo teenagers were running across the grass, diving near the sprinklers to cool themselves down. Deeper in the woods I saw the outline of a man. He situated himself in a glade of mock oranges, half hidden beneath a limb, only his vaguest outlines visible. He was tall, sturdy. I tried not to stare at him too intently. I pretended to watch a cargo truck with huge yellow letters bucketing down the street. When I looked up again, I noticed that he'd taken one step closer. There was no question that his eyes were fixed upon mine.\n\nI knew what was happening. Behind him, submerged in the dark trees of the park, were other men, all watching, waiting. It seemed nearly incredible that I'd delivered myself here of all places, but I wasn't about to question the destiny of things. I stood, conscious of the leaden weight in my legs, my trembling hands. Before I allowed myself to recognize my fear, to dwell in the moment of my essential ambivalence, I was walking toward the stranger, staring directly into the glare of the streetlight.\n\n\"William,\" I said aloud.\n\nMy words caught like crumbs in my throat. He pulled me toward him. In no time at all he'd taken off my shirt, and I was lying down on the grass beneath him, the stones digging into my back. His tongue was rough in my mouth. He worked a dry finger inside me, and I drew in my breath, resisting. I wasn't sure if I wanted this. He was going to murder me, I thought. We were going to be caught or he was going to murder me. I shut my eyes. I saw myself in a plane, looking down at a scar in the earth, an open pit filled with water the color of sewage. The plane circled the pit. A parachute opened, and then I saw myself slipping deep into the water. The stranger lifted me to my feet. He smelled of solder, burnt things. I looked into his grinning face, the gap between his teeth. I felt oddly moved by that for some reason, and he reached for my dick, holding it tight, tighter, and that did it\u2014I crumbled, my back going tight, my legs giving way, and then the release.\n\nI slouched against a palm, listening to the struggle of my breath. The man's footsteps crunched on the gravel. I recognized something, though I wasn't quite sure what it was. Where was I? I stood in the shelter of the glade, watching the man stumbling back to his car. I tried not to be absent. I waited until he was completely out of sight, then wiped myself off, walked over to the Trail, and hitched a ride back to the King Cole.\n\n***\n\nI stood on the abandoned golf course, watching Peter dismantle the old cabana, an outbuilding from the Clem Thornton Boca Palms days. The building inspectors had been hounding him for months, insisting it a hazard, in danger of collapse. I didn't know what had possessed him to finally take on the project now, at nine in the morning, in 88 percent humidity, but nothing he did alarmed me anymore. He worked the prybar into a seam and pulled. The rotten plywood crumbled like foam in his hands. I thought of people who'd become possessed of extraordinary capabilities in times of crisis, sandbagging their properties for thirty-six hours straight to save their livestock from flood. His energy reminded me of them, though I didn't know who or what he was protecting.\n\nI stepped beside him and pulled at a loose sheet. Grit blew through the air, catching in my eyes. I blinked it away. He kept pulling down the structure, holding in his breath, deliberately ignoring my presence.\n\n\"Not like that,\" he said finally.\n\n\"What?\"\n\n\"See what you're doing?\"\n\n\"What?\"\n\n\"You don't have the right tools.\"\n\nTermites crept over the brittle wood, tensing their clear, sticky wings. \"Calm down, will you?\"\n\n\"Gloves.\" He pointed at the nicks in my fingers.\n\n\"I didn't bring any.\"\n\nHe batted a mosquito from his brow. \"Listen, why don't you check out the pool?\"\n\nI tightened my fist in my pocket.\n\n\"I'm serious. It's been raining for days. It's bound to need some chemicals.\"\n\nHis eyes were vivid, sea green behind his safety glasses. He was wary of me now. Would we talk about the other night? Did he have any idea I'd partly set it up? Secrets, secrets.\n\nI trekked across the grassy slope. I imagined him nameless, a worn silver surface where his features had been. He wasn't himself anymore. He was emptied, a phantom. And all at once I knew what he was becoming for me. He was everything that stood between me and what I'd longed for, everything I'd ever lost and never had. He was my relationship with William\u2014failure. He was my relationship with my parents\u2014failure. He was my fear of the future\u2014failure\u2014my shame, the absence of love and trust\u2014all of them tied up together in one sticky knot. But I wasn't about to start fussing with it now. I might have been a whirling hate machine, hot wind blowing through my blades, spinning me around so fast I could have lifted off the ground, and then what?\n\n***\n\nA little after midnight I spotted Hector within the illuminated rectangle of the pool. He drifted around on a red rubber raft, turning imperceptibly like the hand of a clock, nudging himself from the sides with the ball of his foot. I couldn't stop watching him. I eyed him from the darkness of my room through the torn fronds of the travelers trees, murmuring his name.\n\nI walked to the pool in my green swimsuit. I stood on the ledge for a moment. The water bubbled, teeming, alive, a container of light.\n\nI'd waited long enough.\n\nI dove in. I swam all the way down to the pool lamps, kicking forward, then up, up, water whooshing, churning inside my nostrils, ears. My eyes burned. I moved beneath the raft\u2014a submarine!\u2014and knocked him off. \"What the fuck,\" he cried. His arms thrashed. His drinking glass sank\u2014dream-like, slow-motion\u2014to the bottom. I grabbed for his swimsuit, pulled it down, as he fought me off, pounding my back, laughing, bewildered. We faced each other. I reached for his dick and looked in his face. \"What?\" he said, shier, more vulnerable than I'd expected. \"What?\" I squeezed him tighter. A tingle in my groin. \"Give up,\" I said. \"Give it up. Let go. This isn't any surprise. You know exactly what you want.\" He sighed all at once, eyes closed, a little smile on his lips, shoulders letting go, dropping forward. He laughed softly. I pulled him toward the shallow end. I had him now. My moment: I wasn't going to waste it. I knelt, water sloshing around my neck, and took him inside my mouth. I sucked in a breath through my nose, throat full. Good, so good: I raised my chin. I rubbed his stomach, squeezed the tough pink nub of his nipple. Skin, musk, hair, the moist rich darkness\u2014everything focusing, everything centering in us. He bit into his lip, winced. \"Yes,\" he said, and swayed above me. \"Ah, yes.\" My head knocking his belly. His palms pressing harder to my ears, blocking out the sound.\n\nA rushing inside my head: a downpour.\n\nHow long did it last? An hour? Fifteen minutes? Five? Time dilating, opening up like an aperture.\n\nThe backs of his legs clenched. Something scalded my shoulder. When we finished, we both started laughing, soft at first, then louder, more raucous. We shook the water from our hair. \"Jesus,\" he said.\n\n\"Oh God.\" I backed up to the rungs of the ladder and fought for my breath. What had I done? I couldn't believe what I'd done.\n\n\"It's all right,\" he said. He kept blinking, rubbing his temples with his fingers.\n\nI exhaled through my mouth. The windows around us were dark, curtained, everyone sleeping soundly in their rooms. The fans of the air conditioners rattled in unison. I heard voices in their drones, scraps of songs, harmonies. Stars blazed, rotating above our heads. \"So I'll see you tomorrow, then.\"\n\n\"Good night, you.\"\n\n\"Good night.\" I swam to the opposite wall\u2014heart banging, triumphant. I pulled myself out of the pool, and ran, shivering, all the way back to my room.\n\n***\n\nI couldn't sleep. I lay on my humble bed, sheets kicked to the floor. Floodlights poured through the curtains. The room felt hot, sealed off. My mouth tasted like a bandage. A thousand thoughts stampeded through my head. I pulled on some shorts, hooked my flashlight to my belt loop, and left.\n\nI walked down the Trail. I walked to the west, through the vast, vacant city until the sky blued, until my feet hurt in my shoes. Branches bearded with moss. Ghosts crashing in the trees. Shapes, souls. Something was ringing inside my head, a fire alarm: You can change your life, you can.\n\n***\n\nThe van lurked behind the office. We swung onto the backseat, leather scorching our legs. Two p.m. Roasting, a sauna inside, the floor cluttered with coupons, wrappers. Our sunglasses steamed up. I crouched, facing him on the seat, palms pressed to his jaws. Air stifling, too thick to breathe. I pushed my tongue between his lips, tracing the ridges on the roof of his mouth, the softness, his teeth.\n\nI cradled the pouch of his jockey shorts.\n\n\"Here?\" He laughed, bewildered, as if afraid of me. \"You've got to be crazy.\"\n\n\"Yeah, here,\" I said.\n\nHe cleared his throat, swallowed. He shook his head. Then he unzipped his own pants, lifting me inches above his lap.\n\nA stab. I yelped, clenching, breathing, then balancing myself, grabbing onto his shoulders. His skin whitened beneath my fingers. I coughed. Little by little, we started moving. Wet, sloppy inside. Sweat rivering down my torso.\n\n\"Like that?\" he said.\n\n\"Like that.\"\n\nThe axles flexed. We kept moving the van that way, squeaking, working it, almost laughing at our audacity.\n\n***\n\n\"Are you all right?\" said Peter one day.\n\n\"What do you mean?\"\n\n\"You look a little strange,\" he said, tilting his head, considering. \"Around the eyes. Have you been getting enough sleep?\" His breath felt warm, like the heat off a candle, on my cheeks. He stepped closer, so close that I almost fell backward.\n\n***\n\nHector and I floated in the pool one night. No talk. My suit floated off on the surface, an empty bag. Our mouths fastened. We were circling, hands pushing at each other, muscles tensing. I wanted him closer. I studied his face: that scar beneath his ear, those fleshy pink lips. How to get him closer? How to get deeper inside the body? Our bodies. I tried hard as I could to inhabit it, us, closing my eyes, frenetic. A fluttering in my gut. Deeper, I thought, concentrating. Deeper. A well: the water inside the well. Icy, burning. Down, down, down the dark ladder. His stubbled jaw rubbed against the grain of my chin. Shuddering, my eyes opened. I pictured something rising off Hector, a dark violet cloth, the size of a handkerchief, floating upwards.\n\nIt occurred to me we were most likely causing someone pain.\n\nWe were up to our shoulders in water, chlorine burning in our nostrils. I pressed my mouth to his shoulder. I might have bitten right through to the bone, hurting him, leaving a mark like a tiny animal trap.\n\nI glanced toward the nearest wing. There were rustlings from the rooms, vague murmurings and the occasional cough. A jalousie window cranked closed.\n\n\"Did you hear something?\"\n\nHis brows drew together.\n\n\"I'm serious. I think someone might have been watching.\" I looked off toward the buttonwoods, the acacia, the loquat. Peter? \"Hey, somebody there? Who's out there?\"\n\nHis eyes were solemn, dignified. Garbage trucks lumbered down the Trail. And, if only for an instant, looking at the smudgings of his cheeks, the slight sag beneath his eyes, I imagined him older, sixty. \"Everyone's been in bed for hours. Stop worrying.\"\n\n\"I'm not worried,\" I said.\n\nHe hefted himself up onto the raft, belly scuffing against rubber, hands scooping, paddling away from me.\n\n# Chapter 20\n\nDay 21. Agitators, vandals; we did it anywhere, in the most outlandish places: the utility passage, the floor of the conference room, the hood of a guest's sedan within full view of the pool. Always testing our nerve. Always amazed by ourselves. My body authoritative and deft, my thoughts fluid. I was beside myself with\u2014what? Bliss? Joy?\n\n***\n\nMy head buzzed. I imagined him standing behind me, arms around my chest, chin wedged between my shoulder blades, wiggling it, telling me to trust him; yes, of course, he felt just like I did, of course. He was crazy about me. I meant the world to him. His breath heated my spine. I curled back into him, eyes closed, purring like a cat. But why so many fragments, scraps in my head? Why this persistent, gummy absorption: the almondy taste of his fingers, the smell of his scalp already growing fainter and fainter?\n\n***\n\nHector crooked his elbow over his face, feet wide apart, in imitation of a vampire. Together we stood outside the laundry room. The dryers rushed and spun inside, throwing out their heat.\n\n\"Dracula?\" I said.\n\nHe made biting motions toward my neck. He raised his upper lip: fangs, plastic Halloween fangs.\n\n\"This is supposed to be funny?\" I said.\n\n\"I'm used to seeing you only at night, that's all.\"\n\nIt occurred to me that I'd been avoiding him outside of our \"dates,\" for lack of a better word for them. Just the merest sight of him had started jostling my demeanor. \"Shhhh\u2014\" I pointed to Peter's hunching back through the office window. He leaned forward over the desk, talking on the phone, taking notes furiously upon a legal pad.\n\nThumb pressed to the roof of his mouth, he pulled out the fangs with a quick suck. \"I thought you were through with being afraid.\"\n\nMy voice stopped in my throat. Instantly I saw it: he wasn't anxious about me at all. His stance, his wry, ironic gaze\u2014it was as if nothing substantial had changed between us.\n\n\"I've got rooms to finish,\" I mumbled, then hurried down the walk.\n\n\"Evan, wait, wait.\"\n\nI sighed hugely. I didn't turn.\n\n\"Do you want to drive into Fort Myers this afternoon?\"\n\nThe glibness. How could things be so easy for some people? Whoever said life was supposed to be so flippant, so casual? My eyes fixed upon a sea-almond pod in the dirt. \"Some other time.\"\n\nNights later, I came in from the pool, hair wet, shivering. I lay facedown on my bed with my clothes off. The air conditioner ebbed, droned. I couldn't catch my breath. I had every reason to be ecstatic, energized, relaxed: didn't I finally have what I'd been longing for? Didn't I finally know the elusive pleasures of the body? How far away those days with William seemed. How far away those days of loneliness, panic, yearning. Yet even the most fundamental tasks required a terrific will. No energy. I couldn't even finish my sentences. They stalled on my lips, dissolving in my mouth like sugar cubes, while my listener waited for me, puzzled, impatient. My room was a mess, my floor strewn with T-shirts and towels. I hadn't gotten my hair cut in weeks. No books, no tending trees. I'd become so lax in my duties that I was surprised that Peter hadn't let me go by now.\n\nI cleaned the first-floor rooms one day. One of the guests had left behind a pack of Lucky Strikes, and I started smoking them, one after the next, as I vacuumed the worn carpet. I left my cigarette on the air conditioner. Not three seconds later the curtain above it started smoking. I tore it off the rod, stomping it, swiping it with my sneakers until I was certain the fire was out. The flames had chewed and charred it, eaten off its edge. It seared my palms as I tucked it the closet. I sprayed a half-can of Glade around the room, and still I smelled the smoke, gloomy and acrid, stinging the membranes of my nose.\n\n***\n\nThe old sample house glowed. We'd brought with us four orange flashlights, and I'd propped them together on the warped floor: a centerpiece. They cast their warm glow onto the rotting studs, the unfinished ceiling. Fat sacks of Dursban hulked beneath the windows. We'd just finished, and we lay there on the floor, lingering, listening to the ospreys in the swamp\u2014a departure from our routine.\n\nI was lying on my side, running my thumb along his jawline. His face looked unusually pensive, aquiline.\n\nHe turned on his side. \"Well, you're affectionate this evening.\"\n\nThe tips of my ears blazed. A judgment? I felt foolish, as if I'd violated some invisible code.\n\n\"I'm sorry. That wasn't very nice. It's just\u2014you care about me?\"\n\n\"No, like\u2014\" I sat up. Blood rushed from my head, blackness falling before my eyes.\n\n\"We have to talk.\"\n\nI looked at the ceiling, willing any reaction out of my face. I was motionless now, a puppet, a cartoon, a stick figure. I steeled myself.\n\n\"I'm worried about you,\" he said finally.\n\n\"I'm perfectly fine.\"\n\n\"It's like this means too much to you.\" He shook his head, faltering. He dragged his tongue across lower lip. \"You're a very sweet boy, a very good friend to me. But you need to get harder, tougher.\"\n\nHis voice was impossibly quiet, almost tender. One of the flashlights toppled on its side, beam swinging across the ceiling, down the wall, to the floor.\n\n\"I don't understand.\"\n\nHis eyes darkened, burning now. \"We're having sex, all right?\" His voice got louder, tinged with a warning. \"It's just sex. I'm not your boyfriend, I'm not your lover.\"\n\nA welling, a heat gathering in my throat. \"I see,\" I said finally.\n\nHe nodded.\n\n\"And you're making this decision for the both of us?\"\n\n\"I just want to make sure we know what's what.\"\n\nI couldn't hide it any longer. I flexed my toes until they popped inside my shoes. At once, a door yawned open beneath me. My stomach fell. Bright lights flashed above, taking my picture. I felt astonished and exposed, to him, to myself.\n\nI wanted to pummel his back until he wept.\n\n\"Are you okay?\" he said.\n\n\"Have a good night, pal.\"\n\nMy back teeth might have cracked apart. Sweet boy, I thought. Good friend. I shook my head, left the emptied house without another word.\n\n***\n\nHope: a pop, a lightbulb blowing out.\n\n***\n\nI woke at three one morning. A party was in full swing across the courtyard. The door was off its hinges; inside sat a boom box on the bathroom counter, bass line thudding. The proceedings spilled out onto the grass. The kids stomped, throwing back their heads, flailing their arms. An occasional scream lashed the gusty, gelid air. A girl ran out the door followed by a guy with the widest shoulders I'd ever seen. He tackled her to the grass, and amidst her protests, poured beer foam down her impossibly white back.\n\nThe idea of a party at this minute in the world seemed completely beyond my comprehension. I fell back asleep.\n\nI cleaned up the mess first thing the next morning. My rage and longing fueled me, helping me work faster and faster. An unrolled condom stuck to the headboard like something ghostly, mysterious: a jellyfish. As I cleaned, the radio announcer described the late-night shooting of a local teenager in the Crossroads Mall parking lot. His voice was flat, clinical, as if he were reading off a grocery list.\n\nWhen I finished, I sat at the desk, reached for a piece of motel stationery, and started writing as fast as I could.\n\nHector,\n\nHow not to sound like an asshole, a fool?\n\nQuestion: what is it you feel for me? When I look into your face, I think, well, of course, of course. And then I'll see you walking across the blacktop, cocky, not even a hello from you. And what you said the other night: fuck you. Do you think we're going to keep this up?\n\nYour indifference makes me want to yell.\n\nI want to smear myself in it to show you what it looks like.\n\nI'll get up again: don't doubt that for a second. As for you\u2014\n\nI can't get anything done.\n\nI read what I'd written three times, crossing out words, writing over them. The pen pressed deeply through the paper. Idiotic, ridiculous: nothing sounded authentic to me. Then I ripped the page in two and flushed it down the toilet.\n\n***\n\nHector was up earlier than expected. He strode back and forth from the Dumpster to the stairwell, gestures infused with an unexpected energy. This wasn't like him: he usually didn't get moving till ten or later, not till his two cups of cafe con leche. After a few minutes I followed him up to his apartment. Everything was packed up in boxes along one wall; snapshots were stacked on the desk.\n\nSomething gnarled in the root of my neck.\n\n\"Want this?\" he asked.\n\nHe offered me the chartreuse shirt with the pointed collar and oversize buttons. It was the very shirt we'd retrieved from the county landfill, to Peter's protests, many months before.\n\n\"Why would I want your shirt?\"\n\n\"What about these things?\" He walked across the room, picked up a magenta House of Field shopping bag. Inside an assortment of gadgets, postcards, magazines, doodads, gewgaws.\n\n\"What's going on?\"\n\n\"You're going to freak,\" he murmured.\n\n\"What?\"\n\n\"Peter's onto us.\"\n\n\"Peter?\"\n\n\"Peter knows that we've been fucking around.\"\n\nSomething rumbled in the distance\u2014a plane crash? Forest fire? I watched his wiry back through his white T-shirt, the muscles rolling gently in his arms. My adrenaline surged. I brought up my hands to my face.\n\n\"He sat me down last night and said that he'd heard us together in 209.\"\n\nI whispered, \"But that was last week.\"\n\nHe nodded firmly.\n\nI tugged at the skin on my wrist, twisting it. \"Oh God.\"\n\n\"It's no big deal\u2014really. He seemed perfectly sane about it.\"\n\n\"But\u2014\"\n\n\"He just wants me to leave.\"\n\nThe laugh that came out of me was ridiculous, raw\u2014a splutter.\n\nHe fastened the buckles of his knapsack. He lifted it once, testing its weight. \"I'm going back to New York. I've already made the arrangements. I'm staying with my friend Juany.\"\n\nMy eyes wandered about the room. The thought of it emptied seemed outrageous, dreadful to me. The gnarl spiraled deeper into my neck. \"You can't just leave,\" I said.\n\nHe moistened his lips. \"I talked to my old boss last night. Believe it or not, I have my old restaurant job back.\" He smiled now, the brown of his eyes brightening, pupils haloed with amber. \"Actually, I'm really looking forward to it. It'll be good for me.\"\n\nMy eyes fixed on the wall where his snapshots had been. Already they'd left crisp white squares where the sun had yellowed the paint around them. I'd memorized their positions: Don and Miguel, Julia before the steaming volcano.\n\n\"What's the matter?\" he asked, glancing over his shoulder.\n\nI grabbed his forearm, looking him straight in the eye, and twisted it behind his back. He gritted his teeth and winced. The cords of his veins tightened in his arms. \"You almost seem grateful,\" I said.\n\n\"Hey,\" he cried.\n\nHe shook me off, frowning, stomping off. He raised his brows, face cautious, bitter, aloof.\n\nI studied his scuffed boots on the floor. Against my wishes, my eyes started to fill, flood. I pressed the small of my back into the wall. \"Jerk,\" I said, sliding slowly down to the floor tile.\n\n\"Don't\u2014\" he said.\n\nI sat there for a while, watching him rolling his shirts. I'd ruined things. So why not push it further, see how far I could go? I'd crawl around in the muck if I had to. A snake in the slough, a vole. Let him be disgusted by me.\n\n\"I'm coming with you. I'm going to run up to my room to pack some things. What should I bring?\"\n\n\"Stop\u2014\"\n\nI folded my arms. The light in the room shifted, leaves quivering outside, golden, polished, as if a front were passing through. Drier weather.\n\n\"So I was just dreaming this up, then?\" I said finally.\n\n\"Evan,\" he said, a kindly, pitying look in his eyes.\n\nAnd then I knew. Something thorny pierced the surface of my skin, right through to the center of my chest. My heart felt hot, too big for my body. A cold breeze sluiced through the room.\n\nHe picked up his boots, then tossed them off to the side. \"My plane's at six-thirty. Would you do me a favor and let me finish, and then we'll talk about this?\"\n\nI wouldn't look at him.\n\nHe wrapped his arm over my shoulder, holding me tight. I tried to shake him off, but he only gripped harder. His shirt had a smell: dust, fabric softener, cigarette smoke. I almost kissed him on the mouth\u2014a challenge, a protest\u2014before he let me go.\n\n\"Meet me at four outside the office. We'll say our good-byes then.\"\n\n\"So this is it?\"\n\nHe shrugged. \"Evan, this isn't the end. Get a hold of yourself. Buck up.\"\n\nI walked out the door. I wandered through the woods for hours, walked all six miles out Pan American to the ruined marina. The water in the cove looked smoky, brackish. A dead fish swelled on the surface like a piece of foam. I pulled up a weed and tore it apart in my hands.\n\nWhen I made it back to my room, my body felt emptied, oddly spent, cleansed. But tired, terribly tired. I stepped before the sink and splashed water onto my face. My eyes were red, my muscles sodden. There was something on my bed: Hector's House of Field bag. One by one, I laid the various articles out on the floor with a kind of awe\u2014a map of Las Vegas, Laura Nyro's Gonna Take a Miracle, a leather wristband, a postcard of the Lady Bunny milking a cow, a Crown and Anchor Townie Pass, shirts, jeans, hats, belts\u2014trying to make sense of them, pondering their connections.\n\n***\n\nHe was gone, once and for all: I'd finally admitted it to myself. He was lost: the one I'd been before I'd left home, before William. It was hard not to think him stupid, insignificant. It was hard not to wish him out of existence. The one who believed he held a secret so extraordinary and vast that he couldn't possibly tell anyone, not even his best friend, Jane. Who believed that once he broke away from his parents' judgment and silence he'd find someone to love, and love him back, and his real life would begin. Who felt an inordinate pleasure, an electrical current, when another boy just happened to brush up against his shoulder. Who believed that once he moved past his initial awkwardness and grief and grew into himself, his life would continue to offer him reward after reward.\n\nI thought of his eager, dark face, that enormous secret behind his eyes. I didn't know what to call him anymore. Dead now. It seemed remarkable to me that he'd ever been called Evan.\n\n***\n\nAt four o'clock I stayed behind in my room. I busied myself with my reading, wearing my chartreuse shirt with the pointed collar. I glanced once through the blind. He looked up toward my window, before he swung into the cab's backseat. I turned away, imagining its lozenge-shaped taillights growing smaller down the driveway. I was determined not to watch. Go, just go. Enough already. I refused to have another ending in my life.\n\n# Chapter 21\n\nI stood at Peter's open door, holding onto the frame. He sat upon the bed in jeans, unbuttoned denim shirt, and cricket cap\u2014all of which appeared to be too small for him. Were they Hector's? His chest pearled with sweat. Outside the window, an elaborate white bird\u2014an anhinga?\u2014took off for the sky.\n\n\"Do you have a minute?\" I said.\n\nHe nodded evenly.\n\nAll I needed was to look at him. Something rumbled and chugged inside my chest\u2014orange, molten. I couldn't let it out\u2014wouldn't unless I had to. \"I'm not very good at these things,\" I said, wetting my lips, standing at the window. Around the pool a towheaded boy pulled a red wagon with a missing wheel, its back axle scraping the asphalt.\n\nThe small of my back needled. \"What's wrong with you?\" I said.\n\nHe blinked. His eyes brightened, their rich green going bluer all at once.\n\n\"You let Hector go, just like that,\" I said.\n\n\"You were doing all his work for him.\"\n\n\"That's not why you fired him. Hector, Holly. What's going on in your head?\" My voice sounded hoarse. I started shaking inside, just slightly, cold now, as if I'd downed a pint of something frozen. My arms prickled. I kept them hidden behind my back so they wouldn't give me away.\n\n\"Keep it quiet. There are guests in the next room.\" His jaw shifted. He pushed it out, attempting to harden his face. He glanced down at the bed, the stained sheets on which Hector and I had lain a few short weeks ago. \"What's the gist of all this?\"\n\n\"Secrets,\" I mumbled.\n\n\"I beg your pardon?\"\n\n\"I can't live like this anymore.\"\n\n\"Secrets,\" he repeated.\n\nHe stood before me with his hands in his pockets. His face was dark. I couldn't tell whether I was making any mark on him. I pictured myself holding a struck match in the room, and the whole place blowing up from what it contained.\n\n\"And you?\" he said.\n\n\"What?\"\n\n\"What makes you think you're any more direct than I am?\"\n\nThe skin vibrated around my lips. My mouth parted, astonished, as if someone had punched it. I imagined it swelling, a rich hurt purple.\n\n\"I'm not like you,\" I said.\n\n\"You don't have any secrets?\"\n\n\"I'll tell you anything.\"\n\nHe walked to the window, hands working in his pockets. \"So what about Hector?\"\n\n***\n\nWe floated around the motel pool for as long as we could stand it, determined to push ourselves. It was dark. We tried to talk about Mom and Dad; we tried to talk about Hector\u2014but thick with emotion, we couldn't say much. Our tongues fattened in our mouths. Stubbornness. We stared at each other, lockjawed, ashamed. The pads of our fingers wrinkled. After an hour, we hefted ourselves up the ladder.\n\nSecrets, secrets: welcome to Sarshik's House of Secrets.\n\nThe sprinklers cooled the flower beds, the little piles of mulch. We walked up the outside steps to the second floor. On the landing, Peter stopped and turned to me. \"Didn't I tell you he was going to be trouble?\"\n\n\"Who?\"\n\n\"Hector.\"\n\nMy thoughts stalled. A gibbous moon hung suspended in a cloud.\n\n\"Didn't I? Didn't I try to help?\"\n\nWe stood together on the landing. I wanted him to stop, to talk about something else, any-thing else, when all at once he was looking at me, eye to eye, standing even closer. My scalp tingled. A pungent, sweet scent wafted off his shoulders. He put his arms around me, embracing me in spite of all my tightness, my resistance. His breaths were warm, oddly comforting upon my neck. He was alive, thrillingly human and real. I gave in to him; I held him tighter. Blood hummed beneath his flesh. My love was so fierce that I felt giddy with it, woozy, a little sick. Then it all started pouring: how I'd have done anything to be him once; how I'd have copied his handwriting, dressed like him, spoken like him; how I'd have waited outside the bathroom door for him just to catch his gleaming wet flanks; how I'd even have died for him, jumped off the roof, or at least pretended to, laughing, the city stretching out just for us like an offering, a bed of candy.\n\nI looked down. Below us, a rolling, scraping sound, and the child I'd seen out the window earlier was pulling his broken red wagon again. He stopped, turning his face up to us (eyes shocked, pale; strawberry-blond bangs) as if he'd listened to our every word. He sat on the curb, put his head down, and pressed his hands together.\n\nAn edgy muteness from Peter. He looked away from the boy, shy now, reserved.\n\n\"What?\" I said.\n\nThe streetlight shined on Peter's forehead. He lifted his face to me, wearied, corners of his mouth raised. A smile?\n\n\"Why didn't you tell me about Ory?\" I said quietly.\n\nHe tilted his head as if puzzled.\n\n\"Ory,\" I said. \"Your son.\"\n\nThe muscles relaxed in his jaw. His tongue pushed his bottom row of teeth. An \"uhh\"\u2014something abrupt, strangled in his mouth, then just as quickly he stopped himself, swallowed, squinting. He moistened his lips. He looked at me harshly, briefly. He took a firm, calming breath, then walked up the steps.\n\n\"Peter.\"\n\nHe turned right, striding down the long dark hall, shoes trundling.\n\nThe soles of my feet trembled. I followed. \"Wait.\"\n\nHe paused outside his room and stared downward at the knob. He wrenched his head to the left, opened the door, walked inside. Shut now. The deadbolt clamped. A hushed, liquid wind passed over the roof, then stillness, rest, a single cricket screeking in some corner.\n\n***\n\nDays passed.\n\nI spent most of my free time walking, trudging out through the sawgrass behind the resort. The sky changed as many times as my moods, one evening a brazen pink, the next the deepest indigo: clear funnels\u2014waterspouts\u2014swirled deliriously above the coastline. The longer we were apart, though, the more I dreaded seeing him again. No resolution, no change. The days felt wide and ponderous, emptied cathedrals in which a clock ticked. I walked deeper and deeper into the swamp. The muck\u2014the color of pumpkins\u2014sucked at my sneakers.\n\nA postcard of an East Village restaurant arrived for me in the mail. Postmarked from Hialeah. I stopped my packing, looked at it for the longest time: a dark cellar with cherry-red upholstery, ceiling strung with a thousand blazing white lights. I projected an ebullient crowd into those rooms: glasses clinked, curries spiced the air. Hector weaved in and out through the cramped tables, brow creased, superior in expression, holding a tray high above his shoulder. Someone was laughing. And just as quickly everyone was gone, their collective absence making a rushing sound like space.\n\nHe'd scribbled on the back: a temporary address, then Forgive me. I miss you.\n\n# Part  \nThree\n\n# Chapter 22\n\nWith the little money I had, I rented a car and drove off for Miami.\n\nMy first stop was the airport. A red radar grill spun ominously on a pedestal. Pulpy clouds shredded across the sky. Out where the runways ended I knew of a place where you could watch the jets landing. I'd spent hours there when I was twelve. I'd ride my bike after school, all ten miles up Dixie Highway, in and out through the bus exhaust and the brake lights, timing it so I'd get there in time for four o'clock, when the L1011's from New York and Philadelphia and Detroit and Chicago landed one after the other in a great continuous chain. All my anxieties\u2014about my parents, school, Peter, my growing yet unnerving interest in my friend Luke Oosterhuis, about whom I'd started to fantasize\u2014all of them seemed to disperse as I lay motionless on my stomach, staring up as the landing gear descended, that massive sound rumbling in my bowels. I stopped the car, got out. The bare flesh of my arms prickled. When I screamed, only a dry noise came out of my throat.\n\nThere were other places. I drove past my high school, through Coconut Grove, underneath the dual signs of the Everglades Hotel and the Coppertone girl. I drove past Dadeland, the Woolworth's, Miamarina, the old Dusty Cartwright Dairy Bar\u2014now shuttered and festooned with graffiti. I drove past the islands\u2014Rivo Alto, Star, and San Marco. I drove down Coral Drive, which only a few short years ago had bisected a drained swamp, an empty prairie, but was now surrounded by vast tracts of Lennar Homes. I should have been troubled or angered, but the truth was the land had already been wrecked, and if it wasn't this, it was going to be something else: a dog track, a shopping mall. a gambling casino, a jetport, a virtual reality entertainment park.\n\nWelcome. Welcome to South Florida.\n\nMy thoughts scrambled. The houses I passed seemed to take on richer colors, yellows and deep blues, until they weren't houses anymore but complete worlds in which I imagined intricate dramas were being played out. In the first house a second-grader was sitting at the kitchen table with her mother, explaining to her with great kindness and trepidation that she couldn't bear living with her new stepfather. In the second an oral surgeon was lying atop his wife, whispering in her ear that he loved her more than anyone in the world, even though in two short weeks he'd be charged with fondling a female patient seconds before he extracted her wisdom teeth. In the third a teenager smoked crack for the very first time, curled up on the bed of his childhood, shocked and frightened by how much he liked it, how much it admitted he wanted to be extinguished. In the fourth a young couple knelt before a framed portrait of Jesus, holding hands, their heads bowed, murmuring, petitioning God to return the world to the safe place it once was.\n\nI drove by Kevin St. Ledger's old house on Avenida Santurce. I thought about my father's initial explanation for its suddenly emptied rooms. \"Kevin blew away.\" It wasn't till I was six or seven that I'd realized he'd said something else entirely, that I'd misheard him: \"Kevin moved away.\" But it wasn't quite the same. For years after I refused to let it go, the image of the weightless, mysterious boy drifting, scuttling upwards over the neighborhood like a leaf.\n\n***\n\nWilliam's lawn was as lush as it ever was, washed and pristine, smelling of minerals, fertilizers, hose water. The house had been painted apricot. The sprinkler heads pivoted and locked above the grass line, coughing once, misting. Leaves glistened. A tiny red bird flitted her wings inside the head of a palm, frantic and joyous at once. I slumped further down in my seat, fingered the letters of the steering wheel. A thin dust powdered the radio dials, the stickshift knob, the air-conditioning vents like a coating of sugar. I brought it to my lips and tasted. Was I ready to do this? A quick cramp in my side, a stitch. I slumped even further. Did I need any more grief in my life?\n\nThen I started up the walk and knocked at the storm door. Prisms sparkled in the arcs of the sprinklers.\n\n\"Hello, William,\" I said, my head down.\n\n\"Evan? Evan?\"\n\n\"The one and only,\" I said with a weak smile.\n\n\"What on earth are you doing here? Come inside. Come in, you, come in.\"\n\nHe laughed ruefully, warmly, a relief. I concentrated \"authority\" into my expression, but it kept falling from my face, my loneliness welling up inside. But he seemed to read my thoughts. Did he feel it too? He put his arm over my shoulder and drew me into the house.\n\n\"You've lost some weight,\" I said. \"You're looking good.\"\n\n\"Thanks.\" He stepped back from me and appraised my new look: the shaved head, the hacked-off sleeves. My forehead went hot above my brows.\n\n\"How strange to see you. I was just thinking about you the other day.\"\n\n\"Really?\"\n\n\"Someone walking past the college gates. Someone with just your kind of walk, you know, that hard little bounce, that really guy walk, and I thought, Evan. That must be him. But when I looked over my shoulder...\" He shook his head. \"Not nearly as cute,\" he said, with a nervous smile. \"I like the haircut, by the way.\"\n\nSomething scratched behind the bedroom door. William hurried down the hall, and soon enough, the Dobermans were crashing toward me, bellies closer to the floor, tongues lolling, eyes crazed with desire. Their muzzles were sprinkled with white.\n\n\"Hey, Pedro, Hey, Mrs. Fox. Do you remember me?\"\n\n\"It's Evan,\" said William. \"Say hi to Evan. You certainly remember Evan.\"\n\n\"They're getting older,\" I said, looking up at his face.\n\n\"Like all of us.\"\n\n\"Not me,\" I declared, and then we both laughed.\n\nI squatted, pressed my nose into their dusty scalps, and breathed. They gave me their weight as I gazed over their heads. It bore only a minor resemblance to the house of the past. The dull, tepid rooms were gone. In their place a huge white room with squat, tasteful furniture facing out toward the banks of bougainvillea. The pool looked profoundly blue, almost lunar, a gray-green life ring knocking about its surface. A single spoon glinted on a table.\n\n\"It's so peaceful,\" I said. \"Orderly. How did you manage it all? It's barely the same place.\"\n\nHe pressed his palms together. \"You don't like it.\"\n\n\"No, no, there's nothing not to like. It's really beautiful. I like the openness, the light. And the clerestories are pretty. It's just\u2014\" What was I trying to say?\n\n\"Well, the original plan was to do it one room at a time. But then I thought, do I want to breathe in sawdust and fiberglass for the next two years? Why not get it over with while things are still fine?\"\n\nThere were footsteps up the walk. A key jiggling in the lock, and the dogs started barking.\n\n\"Hey,\" someone said.\n\n\"Hey, you. Look who's here, it's Evan. You remember hearing me talking about Evan.\"\n\nHe was a young, wiry sort, mid-twenties, in orange drawstring shorts, with the most forbidding calf muscles I'd ever seen. He leaned over and kissed the dogs on the tops of their heads. He extended his hand up to me and smiled warmly, but with a remote, abstracted expression in his eyes. \"Alfred,\" he said.\n\n\"Hello, Alfred.\"\n\n\"But my friends call me Laser.\"\n\nHe stood. Laser folded his arms across his thick chest, smiling, sidling up next to William. If only through their shared gestures (the smiles, the closeness of their stance), I could see that they were clearly a couple, inhabiting their lives with a fluidity and ease. How did any two people do it? How did any two people achieve a life, sharing each other's time with kindness, respect, and sexiness, without pushing each other away? I was troubled with the urge to say, Do you fight? Are you in love? Do you still sleep together?\n\n\"Excuse me for a minute,\" said Laser. \"I have to take these two for their walk.\" Their eyes brightened at the word. \"Walk? Walk? Do you want to go down to the beach for your walk?\" Panting, the dogs scampered after him, nails clicking, down the hall.\n\n\"You guys are happy?\" I said dully.\n\n\"You wouldn't think it would work, but it does, believe it or not. It's been almost a year and a half.\"\n\n\"He's in school?\"\n\n\"He's still in school. Med school. A year to go before his internship.\"\n\nI tried to compose my face, but I must have looked strange to him. I knew what kind of expression it was: it told anyone who was even halfway perceptive that my outer self, which was trying to maintain decorum at all costs, was at odds with my inner self, which was plunging downward like a kite. What did William's relationship with Laser have to do with me? I'd moved forward in my life. I'd had all sorts of experiences. I certainly did not want to come back here. Do not take this in. Do not take this personally. At least give them that. A thought: I was just as resentful and afraid of abandonment as my brother Peter.\n\nWhat makes you think you're any more direct than anyone else?\n\n\"What's all this?\" I said, indicating the orange plastic pill bottles on the kitchen counter. I picked up one and shook it gently beside my ear. It made a kind, pleasant sound like maracas.\n\nHis eyes fixed on the coffee pot beside him.\n\n\"Something wrong?\"\n\n\"Evan\u2014\"\n\n\"Is Laser okay?\"\n\nHe raised his face to me.\n\nI said, \"Why are you staring at me like that? Why so serious?\"\n\n\"Have you been tested?\" William said finally.\n\nMy attention still snared on those bottles, the miraculous capsules inside.\n\n\"I think you might want to get tested,\" he said, graver now, looking directly in my eyes.\n\nI grimaced. \"Are you\u2014?\"\n\nI held myself still for a few seconds. Then I felt myself falling off a building, the ground close, closer, advancing like a bull's-eye, then slump.\n\nI was still shaking after William came up to hold me.\n\nAsh dusting now, sticking in our throats, the world colder, meaner, more ragged than before. I saw myself walking through jags of ice\u2014dark, dark world, steam rising from my mouth.\n\n\"I know we've talked about this before. Frankly, it's not like it's a total surprise.\"\n\n\"Oh God.\"\n\n\"I feel fine, though. I can't quite explain it. I haven't felt this optimistic and energetic in years.\" And I saw what I'd overlooked before: the flat, puttied color of his skin, the waxy hollows beneath his cheekbones. He stepped toward the window, not quite so confident on his left leg. \"But I just talked to my doctor yesterday. My T-cell counts aren't good. They're on, as the professionals like to say, a downward trend.\"\n\n\"Oh,\" I said.\n\n\"But he says better drugs are on the way. Well, as far as I'm concerned, the sooner the better. We need them now. Right now.\"\n\nI looked down at my fingers curling on my lap. I thought of my stomach\u2014emptied, translucent, the ghost of a stomach\u2014floating somewhere in the room, then out the window, floating on and upward through the palms like a soul. I am twenty years old, I thought. I am twenty years old and I am going to die.\n\n\"It's not like you have much to worry about,\" he offered. \"I mean, it's not like we weren't using rubbers or anything. We were very careful together. You should be fine. Assuming you've been watching out for yourself.\" He raised his brows, tears filming his eyes.\n\nBut had I? I thought about my hands\u2014nicks, cuts, abrasions, little openings. I thought about my teeth, brushing and flossing only seconds before sex. And of course there were those occasions on which we'd slipped. Had he forgotten all about them, purged them from his memory?\n\n\"And what about Laser?\" I asked.\n\nHe nodded. \"He's positive too, but his health is good, excellent actually. He's going to beat this thing. We have high hopes he's going to outlive the best of us.\"\n\nThe world torn in pieces, scattered on the ground like marble, broken bridges. I tried to think of William and Laser, of the difficulties that lay ahead for them, the infections and the doctors, how one would die before the other, leaving the survivor to spend his last months\u2014years?\u2014in loneliness, in a hospice, in pain, surrounded by doctors and strangers, but I couldn't. I could only think about myself, how I was going to die before I even had the chance to live.\n\nI wasn't as generous as I should have been.\n\nHe came up behind me and massaged out the knot where my neck met my shoulder. For some reason that made me cry.\n\n\"I know it's a lot to take in at once. Are you okay? Let me get you something to drink\u2014some tea, some coffee? Or how about something to eat?\"\n\nI shook my head hard.\n\n\"I'm sorry,\" he said plainly. \"I'd hate to think I'd...\"\n\n\"Don't\u2014\"\n\n\"I'm sorry, I'm very sorry. I feel awful about what happened between us. This stuff's been on my mind for a long, long time. I think I must have known I was sick on some deep level, and I wasn't ready to acknowledge it yet. And then I didn't have any choice. One night I woke up, and all the sheets were drenched with my sweat.\"\n\nHe leaned backward into the woodwork and sighed, said ahh. Drops of moisture glistened on the tip of a fig leaf, holding on for as long as they possibly could before they fell to the floor. The ceiling fan clicking. A dog\u2014a St. Bernard, I decided\u2014barked somewhere in the distance, solemn, calling for his dinner. The sky over the trees went gray, the absence of color. Then everything in the room, the smallest details, seemed to glitter fiercer, brighter, until I thought they would pop: the bulbs in the chandelier, the violet irises in the clear glass vase on the table.\n\n\"Where are you staying tonight?\" he said. \"You're always welcome here.\"\n\n\"I have to be somewhere at six. I promised Ursula I'd meet her at Dadeland.\"\n\n\"I'm glad you're talking. That seems like a start at least.\"\n\n\"Well, I want to get to know her again. I don't have huge expectations, but you never know.\" I made a silly, stranded face. I didn't know where my words were coming from, but I tried to invest myself in them, if only for the moment.\n\nWe were stalling for time, fearful of our awkwardness, fearful of departure. Then, looking out the window, I thought about the dream I'd had on our fifth night together. We were standing by a riverside amusement park: PLAYLAND. Somewhere north. Maryland? An empty roller coaster surged down the track. The river surface shimmered in a pewter haze. Before I knew it I'd taken William's hand and walked into the water with him, feeling not a second of trepidation. Trust, hope, absolute confidence, and delight. The river splashed warmly against the backs of my legs, rising higher around us. Side by side, we started swimming across the river. The roller coaster struggled up the rusty track, cranking and ticking, as if for the very last time.\n\nWas it the mere absence of our life together\u2014and not so much William himself\u2014that had pained me?\n\nWould our time together always have such significance for me?\n\n\"So\u2014\" I gazed downward at my shoes, empty and full at once. I wanted to run as much as I wanted to stay, to move into the old den with its stacks of mildewing magazines.\n\n\"Take good care of yourself. Stay in touch, okay? Are you around for a while?\"\n\nI shrugged, smiled weakly. \"We'll see.\"\n\n\"Just let me know how you're doing.\"\n\n\"I will.\"\n\nI kissed him right on the mouth then, with all the tenderness I could muster, without understanding it, wildly alive, before leaving through the door.\n\n# Chapter 23\n\nUrsula was already seated at the front table of the ice-cream parlor, her legs resting up on a chair, when I arrived.\n\nHow odd that we were in Dadeland, in the space once occupied by Farrell's, a short-lived chain for which our entire family\u2014Sid, Ursula, Peter, and me\u2014once managed to have affection. It had nothing to do with the menu (which challenged one to take on the \"Zoo,\" an ice-cream project comprised of twenty flavors), or the atmosphere (hokey Victoriana with lots of screaming and hooting at birthdays), only that its presence coincided with an especially poignant time in our family life, when there was a period of relative peace in the house, when we knew that our outings as a complete family unit were numbered. We were already too old for this. Still, these excursions were the single activity on which we could all agree, and once they were mentioned as even a remote possibility, the four of us, sadly enough, would hurl ourselves in the car, forgetting our differences for an hour, convincing ourselves we weren't nearly as fucked up as we were.\n\n\"Look at you. What happened to all your hair? You look like something off a chain gang.\"\n\n\"Mom.\"\n\nShe rubbed the bristles of my scalp, then stared at the blue Icarus on my bicep. \"What's this?\"\n\n\"Tattoo,\" I mumbled.\n\n\"Tattoo? Where on earth did you get a tattoo?\"\n\n\"Prison.\"\n\n\"Prison? Oh my God. Since when have you been in prison?\"\n\nI couldn't help but roll my eyes. \"Hel-lo\u2014\"\n\n\"Well, don't do that to me. Don't get me all worked up like that.\"\n\nWe smiled wanly. In spite of our mutual cautiousness and fear of each other, I was relieved to see her. She appeared to be in good spirits\u2014playful, even proud of me, and she'd spruced herself up for the occasion with a purple-pink scarf tied jauntily around her throat. Her hair was a complex of darks and lights, the color of nutmeg. She smelled of teaberry, weeds.\n\nIn no time at all she gathered herself. I glimpsed at her hands\u2014veined, brown, dry, with their fragile superstructure of bones\u2014remembering how she'd once been so proud of them.\n\n\"You could have met me at home, you know.\"\n\nI shrugged. Could I have told her that just being inside the house would have shaken something loose? Already I saw it all in my head: the oak door between the garage and the house\u2014split, dented\u2014which I'd kicked repeatedly after being locked out in my tenth year; the backyard toolshed, chewed to a soft pulp by termites; the hole in the laundry-room ceiling, still unrepaired after ten years, the tufts of pink insulation revealing the guts of the house.\n\n\"You look different,\" was what I said. \"Something's changed. What is it?\"\n\nShe tilted her head, inhaled, not without delicacy. Then she turned to the side, offering me her profile.\n\n\"Nose job,\" she said finally. \"I just had a nose job.\"\n\nI examined the new nose, the softer bridge, the flaring, dilated nostrils. A bungled approximation of a WASP nose. \"You're kidding me.\"\n\n\"I'm actually relieved. Julie Spivak thinks it's very, very natural. Some of the best work she's ever seen.\"\n\n\"But you had a nice, voluptuous, Eastern-European nose. It had character.\"\n\n\"Character, shmaracter. What do I care about character?\"\n\n\"But\u2014\"\n\n\"Do you know what it was like to have been made fun of as a child?\"\n\nMy tongue felt fuzzy and thick, a wad of cotton in my mouth. I had to remind myself it wasn't anything new, this will to remake herself. How could I forget the complete line of diet products, many of which were even then out of date, she'd stashed in the kitchen cabinet when I was young\u2014Metrecal, Figurines, Carnation Slender, Instant Breakfast. Even something called Ayds, of all things.\n\n\"What's Dad think of this?\"\n\nShe shut her eyes for a moment, then opened them. She looked hurt yet detached from that hurt. \"Don't ask.\"\n\n\"What?\"\n\n\"That son-of-a-bitch\u2014\"\n\n\"Mom\u2014\"\n\n\"That fucker\u2014\"\n\nHer voice was rising. I glanced around at the other tables.\n\n\"Don't talk to me about that snake in the grass.\"\n\nThe waitress, a slip of a thing with a crest of hair like some rare tropical bird, pretended to be oblivious to my mother's mouth. She scribbled our order on her hand with a lavender-tipped ballpoint. All the while my mother frowned, her face pinking, staring down at her sunfreckled arms crossed tightly over her chest. She waited for the waitress to depart.\n\n\"Airhead,\" she mumbled.\n\n\"Shhh\u2014\"\n\n\"Bimbo.\"\n\n\"She'll hear you.\"\n\n\"I don't care,\" she said miserably. \"If she doesn't have enough decorum to know that you're supposed to write on a sheet of paper and not on your hand, then what do I care?\"\n\nI shook my head, exasperated. My neck felt hot in my collar. \"But what about Dad? I'm waiting to hear about Dad.\"\n\nShe eased forward in her chair. She looked pained again. \"You sure you're up for this?\"\n\n\"What's the matter now?\"\n\nHe'd been gone for some time now, almost three months. He was living in Huntington Beach, California, with a forty-three-year-old woman named Anita Burnell who had ironed, bottle-blonde hair all the way down to her ass\u2014\"right out of 1969,\" according to my mom. He'd met her six years ago at an academic conference at UC Irvine. Like my father, she was an assistant professor with a specialization in nuclear fission, and she and Sid had corresponded by letter, phone, and fax for years, so openly and casually, in an approximation of professional comradeship, that my mother hadn't once suspected there was anything between them. If anything, she'd convinced herself that Sid was put off by Anita's chumminess, that he quietly dreaded running into her year after year at conferences, that he imagined her pushy and coarse, always sniffing around for the next job opening (or \"corpse-to-be\") until she\u2014my mother\u2014found out otherwise. It happened on a quiet Thursday evening, moments after she'd finished up a New York Times crossword puzzle with the word \"glossolalia.\" Sid walked up behind her in the kitchen, wrapped his arms around her waist\u2014her \"love handles,\" she said\u2014then told her, not without tenderness, that he didn't love her anymore, that she'd be better off by herself, that he was leaving for Huntington Beach in the morning.\n\n\"I just never thought he had it in him. I mean, I never thought he'd leave. He never seemed particularly, I don't know\u2014passionate? Is that the correct word?\"\n\n\"Did you two still have sex?\"\n\nGod knows what possessed me to ask such a thing.\n\n\"If you could call it that,\" she answered thoughtfully.\n\n\"What do you mean?\"\n\n\"When a man\u2014when a man can't\u2014\" She stared at me, startled, and rubbed at her arms. \"Why in God's name am I talking about this with you? I don't want to talk about this with you.\"\n\nIn the mall two teenagers, a boy and a girl, ambled in and out through the fountains, the figs, the ficus, grasping hands. Unlike us, they were the very picture of safe. It seemed incredible to imagine that they had any complexities swirling in their midst. When I looked back at my mother, she was staring at her new nose in her compact. Already she'd rolled on a fresh coat of coral lipstick.\n\n\"So next I'm going to get an Adrian Arpel makeover, then I'm going to get eyeliner tattooed around my eyes.\"\n\n\"No way,\" I said. \"No tattoos around the eyes.\"\n\n\"What about you? You have tattoos.\"\n\n\"We're talking about your eyes, Mom.\"\n\nShe nodded. \"You've convinced me. It's all because of you. I'm going to get my eyes done, and then I'm going to get myself a new boyfriend, and I'll show that\"\u2014she wrenched her head\u2014\"ass-hole he's made the most foolish mistake in his life.\"\n\nA thousand thoughts were leaping concurrently in my head. I saw them luminous\u2014dark and oily, like colored fish in an aquarium. The tips of my ears felt hot. I couldn't restrain myself any longer. \"Mom,\" I said, \"I hate to bring this up, but there was an actual purpose to my visit.\"\n\nHer face dulled as if she'd been taken aback. I hadn't meant to sound cold or insensitive.\n\n\"Could I ask you a question?\"\n\n\"Of course, dear.\"\n\n\"If I ever die, could you do a favor for me?\"\n\nShe grinned darkly, addressing me as if I were eight years old again. \"You're not going to die.\"\n\n\"Oh yes I will.\"\n\n\"You're not.\"\n\n\"Yes. We're all going to die.\"\n\nHer mouth parted.\n\n\"It's not like when you were a girl. It's dangerous out there. Young people die all the time. Tons of young people that I know.\"\n\nShe blinked once. She looked at me as if I were telling her that the world, as she knew it, was about to implode.\n\n\"I want you to give my CDs, the Joni Mitchells, to Jane. Take all my books to Herridges and keep the money for yourself. And pack up all my clothes and send them off to the Salvation Army.\"\n\nHer face blanched with worry, hands twisting. \"I'm not sending anything to the Salvation fucking Army. You're not going to die, okay?\" Her voice carried, just loud enough for the women two tables away to hear. They halted their conversation, pretending not to listen. Quietly, she said, \"You're not going to die.\"\n\n\"Mom\u2014\"\n\nShe smiled slightly. \"I'm not going to listen to this. How dare you make me listen to this. There are a full range of beautiful topics in the world. What about your brother, for instance. Why haven't you said one word about your brother? What was it like?\"\n\nI didn't want to talk about him. I'd already made the decision not to go back. \"Have you heard anything from him?\"\n\n\"No, he's just like you. Both of you have deserted me. I don't even know what you look like anymore. I'll be lucky if I even get a Christmas card from him, sometime next June.\"\n\nSomething snagged into my stomach.\n\nI watched her wiping her hands on her napkin. I knew she'd pictured her future differently. She'd lived her entire adult life as if its choices had been guided by a map, a paradigm: Do these things right and you'll get your just reward. Once she'd pictured both Peter and me living nearby, in Cocoplum or Coconut Grove, with decent jobs, dropping in on her every Tuesday night, bringing our pretty wives with whom she'd have coffee and talk about furniture. She'd have grandchildren, four of them. She'd have stayed married to my father, and though their lives would have been marked by silence and periods of retreat from each other, she would have had a vision to present to the world: I am fine, I am not marked by grief, I am just like you.\n\n\"I'm so lonely,\" she said. \"I never thought I'd be so lonely.\"\n\n\"You have to be tough, Mom.\"\n\n\"What if I don't want to be?\"\n\nThe breath left my lungs.\n\n\"Listen,\" she said, glancing at her wristwatch. \"I have to go. I have to be somewhere at eight.\"\n\nI faintly smiled. We both knew she was lying, but I understood: all this was more than she could bear at once. At home she'd fumble for her Halcion, or Valium\u2014whatever was closest to her grasp\u2014then go straight to bed, and for the next two days ponder over and over what we'd said to each other, weighing my words for their various meanings.\n\nWe stood. She stepped forward toward me, then held me close, if tentatively. I smelled the soap in her hair. \"Mom?\"\n\n\"Yes, dear?\"\n\n\"Watch out of yourself.\"\n\nShe nodded. \"I'm not as helpless as you'd think.\"\n\n\"I know.\"\n\n\"And no more tattoos,\" she said, smiling slightly. \"Be care-ful. Deal?\"\n\n\"I'll try.\"\n\n\"Stay in touch now. Bye, dear. Bye-bye.\" And with a quick peck of a kiss, she left the table, stumbling for a second, then righting herself, hurrying out past the kiosks of the mall.\n\n***\n\nThe free clinic was in a pagoda-like structure with glass-block walls, so close to the beach that I heard the waves, the lifeguard whistle, the joyous shrieks of tourists in the distance. Inside, I slouched in the waiting room with three other boys and a girl\u2014all of us with sullen, vaguely bored faces\u2014waiting for our names to be called. A TV monitor droned a message: \"A positive result is not a death sentence.\" I didn't flinch as the health care worker pressed the needle into my vein and drew back a syringe-full of blood. I stared fully at my blood, its thickness, its rusty red potency. I wasn't as debilitated as I'd expected. I'd already prepared myself. There was no reason to assume the results would be anything but positive, anyway.\n\nWith the remainder of my money, I rented a room at the Hotel Lumiere for the week. It was no-frills simple, with single bed, TV, combination bureau\/desk, and salt-pitted, dorm-sized refrigerator. And it had that scent, that inimitable fusion of cigarette smoke, mildew, cat pee, and chlorine bleach. I thought of the people for whom this place had initially been built\u2014working-class Jews from Brooklyn or the Grand Concourse who'd anticipated their stay here for months, who'd envisioned their vacation on the beach as a little taste of Heaven. It cost a near mint now, more than I could afford, but I didn't care. I spent most of my days lying in bed, thinking, ordering pizza or Chinese food, listening to the many languages of the people on the street: German, French, Arabic, Japanese, Spanish. One night there was a loud crash, like a stack of falling plates, and I walked to the window to see a party revving up in the apartment across the street. A samba spun louder, and a good twenty or thirty people stumbled out onto the balcony, some in a conga line, some with cocktails in hand, before waving over to me, and I back at them.\n\nA girl with large, soulful brown eyes cupped her hands around her mouth.\n\n\"\u00a1Anda ca!\" she cried.\n\n\"Come over,\" someone else yelled.\n\n\"Join the party,\" said a third.\n\nAcross the bay, a faint popping of fireworks.\n\n\"I can't,\" I answered.\n\n***\n\nI waited until the following morning to leave the room. I walked up Collins Avenue out of the trendy, gentrified zone into the run-down section of South Beach that hadn't caught on yet, where the tourists weren't glamorous, but looked ashen, pockmocked, and haunted, dressed in their tight shiny fabrics, gold chains, and wide collars. It could have been twenty years ago, the Miami Beach of the late seventies, and I both loved it and was depressed by it. I walked into a T-shirt shop, checked out the gag gifts and souvenirs: oversized dice, Statue of Liberty pens, furred little beagles with nodding heads\u2014stuff I hadn't seen since I was a kid. I thought about how much Jane would have gone nuts over these things, how she would have borrowed money from me, chattering excitedly, leaving the store with a full shopping bag and then some. Where was Jane? I missed her so much. It was time I made an effort to get in touch with her.\n\nI couldn't have been in the store more than three minutes when I spotted a man in the aisle ahead of me. I didn't see his face, but I saw the back of his neck, a long thin neck, swan-like, though on the verge of gawky, atop which balanced a sweet head with ears that stuck out. He looked nearly perfect to me. I watched him picking up the very same things I'd picked up\u2014snow globes, netted bags of bubble gum resembling lemons, tangerines, grapefruits, and limes\u2014cupping them in his palm, eyes shining. I couldn't stop looking at that neck. The neck had always been the most telltale part of a man's body for me. Nothing could excite me more, not even muscles or big feet or big dick or large nose or dark hair or well-developed Adam's apple.\n\nI followed him. He walked north on Collins, past the Sunoco and the Shelborne, before turning left on 16th. He strode down the street, looking straight ahead, taking long, athletic, supremely confident strides. My eyes watered. The wind crackled, purred inside my ears. He turned south on Meridian, glancing at a particularly beautiful cherry palm outside an apartment house\u2014good sign\u2014then started toward Lincoln Road. His walk seemed to have no destination or plan. Was he cruising? Lost? I decided he was lost. I decided I could bump into him, step down on the back of his shoe, ask if he needed directions, and he'd recognize something in me, something in my eyes, a shared spirit, a sense of humor, and we'd get to chatting, and soon enough we'd be drinking coffee together, talking about what we were doing tonight.\n\nI kept following him. I kept looking at that sweet and luscious and vulnerable neck, and I wasn't about to let him go. I thought of his chest\u2014firm, muscular, lightly matted with hair\u2014and I imagined holding a melting ice cube over it and watching him shudder as I worked my tongue upwards from his belly button. The heat beat in my face. Outside the Night 'n' Gale he turned a bit to the side, not fully facing me, but just enough to see me out of the corner of his eye, and he knew who it was, and he knew I was after him, and I knew that he liked it. I pictured him smiling. My dick rustled, plumping the front of my jockey shorts. And then, just as I stepped closer to him, narrowing the distance between us, he crossed 14th, mere feet before a cargo truck, and the traffic light went red, and everything stalled. Horns. Curses. Heat. Gridlock. I tried to pass around the truck. I tried to sidestep to the left, to the right, but the seconds were ticking. The noon sun hammered my brow, blurring my vision. I banged my fist on the side of the truck twice, three times, before it moved forward. My fingers throbbed like a boxer's. The traffic loosened, I walked ahead.\n\nHe was nowhere in sight.\n\nSomething occurred to me: I didn't want to die.\n\n# Chapter 24\n\nThree weeks passed at the Hotel Lumiere. I lay in bed, staring up at the rotating paddles of the ceiling fan. Checkout was 11:00 a.m., and I'd timed it so that I could pick up my results from the clinic upon my departure. I stood, shoved my possessions\u2014jeans, T-shirt, contact-lens solution\u2014into my backpack. My mouth was dry. My hand looked old. I kept staring down at the skin of my hand, asking myself how on earth could the hand of a twenty-year-old get to looking so old.\n\nBut who was I kidding? Already I felt it: that slight white heat, simmering inside the engine of my body, making me older before my time. Correct? Didn't anyone who'd ever lived through this recognize early on what had already possessed him, inhabiting his corpuscles on the deepest, most urgent level? It was as if one had never been anything but sick, that loneliest place, and it seemed astonishing that there were actually people in the world who'd never felt this: to have something burning inside you as intimate as a lover.\n\nAnd yet I wondered whether I was cutting myself off from surprise. What if, to my good fortune, I was deemed negative? What if I was told that my days weren't immediately pressured, that I most likely had decades ahead of me? Would I feel different? Negative: funny, awful word. What did it mean to call oneself negative, anyway? I was only negative until the next time I had sex with someone. Wasn't staying negative always the most demanding of efforts, and not a state of grace, not a protected, privileged zone?\n\nSomething became evident. This whole matter was about being handed two choices.\n\nI stared at my reflection in the window glass. I looked fine, I felt fine.\n\nI couldn't bring myself to believe that there were only two choices.\n\nI walked to the window, not knowing whether I was copping out or not. I felt relieved and shaken all at once. I looked out toward the ocean and saw a cruise ship passing into the warm aqua of the gulf stream. It looked so small from here, the size of a toy. I imagined a couple\u2014straight, young, still in their twenties\u2014lounging on the deck of that boat, the sun shining on their faces. How would they have reacted if they'd been informed in hushed tones that a terrorist bomb had been planted aboard one of the ship's lavatories, that sometime within the next few minutes or the next few days they'd come to violent death? Would that have made them generous, wise, more responsible to one another, or would it have diminished and destabilized them, making them feel they were under the fist of something so much greater than themselves?\n\nBut even that wasn't the same. An instantaneous death\u2014even a death that permitted a few days of preparation\u2014seemed merciful compared to a death that lingered for years and years, where one got sick, better, then sick again. Where a simple sore throat could signal the onset of ruin. Where one's family and lover and friends were ever tempted to bail out. Where one had to push past the inevitable burdens of blame and shame\u2014all the meannesses that people happened to attach to them.\n\nWhere one had to forgive. Forgive.\n\nI glanced down at the appointment card in my hand and tore it in two. I watched its pieces flutter down onto the people of the street.\n\nIt seemed that I'd already made my decision weeks ago.\n\nWas I a coward? Was I selfish, irresponsible? It seemed that even now, I was suspended between the two worlds\u2014the world of the living and the world of the dead. I would go on in the only way I knew. I picked up my backpack, walked down to the lobby, and handed over my money to the desk clerk. The lighting was milky. Outside, the people seemed beautiful, pale, their ghosts following close behind, already guiding them someplace.\n\n***\n\nThree hundred dollars: enough to get me through the next two weeks, but barely. It wouldn't last the month, unless I was willing to subsist on rice and beans. I tried not to fret too much. I had the uncanny sense I was revisiting my days when I'd first moved out of William's house. I thought briefly about the nature of time, how it's always tricking us into thinking it's taking us forward, when, in fact, it's always coiling around like a spring, bringing everything we've known back to us.\n\nI spent the days looking for work. I bought the Herald and checked out the various restaurants, clothing stores, gas stations. Everyone I talked to seemed wary of my lack of experience. It was all right; I would have screwed up those jobs, anyway. I was about to start lying, fabricating a history for myself\u2014name: Vico Bakaitis, age: 24, place of birth: Seal Beach, California\u2014when on one of my wayward afternoon walks I literally found myself within the fencing of a nursery. It was on a forgotten back road, west of the city. It bordered a canal filled with stagnant, olive-green water. Florida's turnpike rushed in the distance like a waterfall. Though the place was overgrown and past its prime and sadly disheveled, it yielded the most offbeat plants, the likes of which were impossible to find at bland, commercial nurseries: blood banana, surinam cherry, weeping fig. Immediately I felt possessed of a good feeling, as if I'd been here in another life. I needed a task if I was going to make it through. I wanted nothing more than to reinvent the place, to reclaim just a bit of its former glory.\n\n\"I'd like a job,\" I said to the owner.\n\nHe squinted down at the marl beside my shoes. His face was weathered, puffy. Above his shirt pocket: a blue patch that said Hickory Bob's. In the calmest, kindest voice, he said: \"I never said I was hiring.\"\n\n\"I'm not asking for much money. All I want to do is help out. Get this place back into shape.\"\n\nHe squinted, only mildly threatened. \"You're young. What would you know about plants?\"\n\n\"A lot.\"\n\nHe walked me up and down the rows, expecting me to identify each variety he pointed to. Without pause, I said: loquat, maya palm, aurelia. I even named the dreaded pitch apple, a variety that had routinely stumped even the most knowledgeable of his clients. \"What else? What else?\"\n\nHe shook his head back and forth. He didn't know what to say.\n\n\"You think I'm a weirdo, don't you?\" I said after a while.\n\n\"No.\" His face was clearly dazed. \"We better sit down,\" he said ruefully.\n\nI followed him down the path to a bullet-shaped trailer. Cattle egret scampered about the yard, in and out through the windbreak of Australian pines that bordered the Everglades. A pink sky rumbled over our heads. It turned out that Bob had been wanting to sell after his wife, Dorothy, had died, but had taken it off the market after receiving but two meager, demeaning offers. Things only worsened after Hurricane Andrew. Though managing to cut a swath a mere three miles north of the property, leaving the nursery virtually unscathed, it hurled the entire region into a real estate depression.\n\nHe fell silent for a full six minutes. He quoted a salary that was a full three dollars an hour more than I'd expected. He also offered me a place to stay in a second trailer out back. I'd tried to conceal my astonishment and disbelief.\n\n\"Do you think that's enough to live on?\" he said.\n\nI looked at the wall clock, a cuckoo clock in a shellacked box on which drawings of various minerals had been pasted. I hated to think that I'd be taking advantage of anyone in dire circumstances, but he was offering it to me.\n\n\"I guess we could give this a try.\"\n\nI nodded avidly.\n\n\"Why not,\" he said as if convincing himself. \"Let's give it a try.\"\n\n\"Done.\" My handshake was damp, feeble. \"Done deal.\"\n\nIn only a few days I threw myself completely into the nursery. I told myself that my work was all that mattered, that contentment and peace were not to be found in the world of people, but in the world of vegetation. I dealt with my customers with patience, caution, and care, never raising my voice or condescending to them. The dirt blackened my fingernails. I walked to my little trailer every evening, welcoming the ache of my lower back, concentrating on the sting of cuts on my hands, if only because they told me in no uncertain terms that I was still alive.\n\nOne morning, I walked up and down the rows of plants, garden hose in hand, watering. I loved the plash of water on the leaves, loved the tuneless flat splat it made on the mulch. I walked to the cluster of shrimp plants, which Bob had virtually left for dead after a brief drought last year. Their leaves were healthy now, shiny and loose, with pretty saffron heads (ladies' swim caps!), and for the first time since my arrival (or \"takeover,\" as Bob called it), I realized that my simple gestures were doing some good in the world. I squirted a squat wooden fence and glanced around at my little domain. I shivered inside. Soon I'd get to moving the cacti closer to the front, then cleaning out the old koi pond, stocking it with marbled, thick fish. Then more varieties of plants; weirder, stranger specimens, plants I'd never seen, names I'd never heard. I wanted them all. I wanted to stay surprised. This is happiness, I thought. This is where I want to be.\n\n***\n\nAnd love?\n\nWhat else is there to say on the subject? I wasn't quite bitter about it, but I wasn't hopeful either. I was in some vague, vaporous place that blended both of those worlds. If anything, I'd come to decide that relationships were best for other people, that my own longing and need had in the past gotten the best of me, and it was time to let that go. There was no reason to assume that I couldn't be comfortable by myself. All I really needed was a few close friends, a love for my work, and some occasional sex. I felt wary admitting to this, as if I were only deluding myself, trying to justify my failure, my giving up. But was I giving up, or growing up? Was I in some deep depression and afraid to deal with its causes? My answers wavered depending upon my state of mind.\n\nAnd yet there were worse places to be. As the months passed, I looked at the people who came to the nursery regularly\u2014my new friends\u2014and saw how their notions of love had nearly wilted them. I saw how Nan, nearly thirty and still single, was willing to settle for a mostly mild-mannered man who occasionally flew into rages, striking her, because she needed more than anything to satisfy her demanding parents who wanted her to marry a graduate of an Ivy League school. I saw how Zack, the most loyal of my customers, had moved in with a man he suspected of being both heartless and mentally deficient weeks after breaking up with Ladd, the love of his life, only because he was afraid to be alone. I saw how Zack's friend Beth had fashioned herself into such a control freak that she wouldn't date any woman unless she was over six feet tall, dressed in fox stoles from the forties, spoke with the slightest lateral lisp, and knew inside and out the collected works of Jane Bowles. I saw how Thisbe\u2014a sculptor, fresh out of a bickering, competitive marriage in which her painter husband continually sabotaged her work\u2014kept rejecting each successive suitor, simply because he never measured up to her idealized image of her ex. It went on and on like this. Was there something wrong with all these people, or was it me? How could I not see myself as lucky when I looked at them, then looked at myself? How could I not stand in front of the mirror, stare into my cool, uncomplicated face, and not call myself one lucky son of a bitch?\n\n***\n\nHis name was Jesus. He was short, muscular, with blue-black skin, thickly lashed eyes, and a wet, enormous mouth curving upward. I'd met him in Lummus Park late one night, sitting on a bench beside the beach. The ocean scented the air with seaweed, tanker fuel. A reggae band thudded loudly in the distance. Soon enough we'd gotten to talking, then we were walking arm in arm up the street, laughing at things that weren't even funny, ambling toward his second-floor apartment off Meridian.\n\nWe were lying together in his bed, holding each other. \"Man, you're sweet,\" I said, pulling away from him.\n\nHe smiled back at me. \"You too.\"\n\nWe continued to make love. It occurred to me I'd leave early in the morning after a quick cup of coffee to be back at the nursery. I knew I'd never see him again. But it was possible, I believed, to enjoy a stranger's company, to be a little in love with somebody, even if it was only for the moment. A breeze stirred the leaves outside the window. Headlights flittered through the slats of the jalousies. I knew we'd never be boyfriends, but there was nothing sad about this.\n\n\"Think about it,\" I said afterwards, latching my hands behind my head. I gazed up at the splintered ceiling and pressed my head deeper into the pillow. He'd brought a bag of tortilla chips to the bed. \"I can tell everybody I spent the night sleeping with Jesus.\"\n\nHe winced. \"Watch your mouth.\" But then he smiled again, making love to me over and over.\n\n# Chapter 25\n\nHow expected that someone should be interested in me when I most wanted to be alone. Not long before we drifted apart, Jane had told me that this was the way it happened, that one only received what one longed for when one achieved perspective, when that desired thing ceased to occupy one's every thought. To me it sounded false, though, something sifted from some ghastly self-help book, and I never thought she fully believed it, the way she believed that you should pursue what you want with all your energy, heart, and affection until you gleamed like plutonium.\n\nI didn't want to be in love now.\n\nI should have sensed that something was up by the simple fact that he stopped into the nursery nearly every day, asking me questions about fertilizers and sprays, about the appropriate ground covers for his particular agricultural zone. He was about forty or so, tall, with red hair that receded in a point, full lips, and deeply blue eyes. Handsome, vaguely conservative in appearance. I'd been told by my friend Zack that he was one of the most successful surgeons in Dade County. He listened to my answers with a feigned thoughtfulness, looking directly in my eyes, even though his mind was clearly elsewhere. For all I knew he was thinking about his latest triple bypass or whatever, and he was one of those annoying, entitled types who expected complete attention from service people, who needed to be chatted with and fussed over and appreciated just so he could feel good about himself.\n\nOne afternoon, after I'd spent six minutes describing to him the fertilization procedures for sabal palms, I said, \"You're not listening to me.\"\n\n\"No,\" he said, and grinned, abashed. \"I guess not. Could I ask you another question?\"\n\nI thought, Make it snappy, buddy boy. To my left an older man in a floppy golf hat was examining the wrapped roots of an acacia, checking for the price tag. \"Look, I don't mean to be rude, but I have customers waiting.\"\n\n\"Okay. I'll let you alone in three seconds.\" He leaned forward on the counter and whispered something, his lips almost grazing my jaw line. \"Would you be interested in going out with me sometime?\"\n\nMy face must have contained within it an element of shock. I'd been so absorbed in my own duties that it hadn't once occurred to me that he might be attracted to me. A pleasant, though alarming surprise. \"Of course,\" I answered.\n\nImmediately I regretted the eagerness I projected.\n\n\"Let's do something tonight. What are you doing tonight?\"\n\nThis wasn't what I wanted. I preferred keeping it open, vague, in the distant, far-off future. The notion of spending any substantial time with him did not sit well with me. All day I'd been looking forward to going to bed early and finishing up a book. \"Nothing,\" I admitted.\n\nHe said something about dinner, something about picking me up out front at eight.\n\n\"Okay,\" I blurted.\n\n\"I'm Perry, by the way.\"\n\n\"Evan,\" I said, and extended a hand to him.\n\nI felt that peculiar combination of flattery and dread. Where was my backbone? How banal was my life. Watching him stride to his car, I thought: Now you've done it, you fool.\n\n***\n\nIt certainly wasn't his body, which was muscular and dense, from laps of hard swimming and years of working out. It certainly wasn't his voice\u2014an important consideration for me\u2014which was resonant and deep, commanding authority and projecting confidence. It certainly wasn't his taste in music\u2014he loved Stravinsky and Poulenc and Joni Mitchell, especially the albums For the Roses and Hejira. It wasn't his height or his manner or his intelligence or his personal style or his sense of humor.\n\nThat thing called chemistry, that elusive connection and tension, what was that about? Why did we feel it for some people, for people who weren't necessarily good for us, who could even do us damage, and not for others? La Quan, who I'd met at one of Nan's parties, insisted it was deadly to fall into any relationship where the sparks weren't flying all over the place. She, after all, would know; she'd spent four years with an older man for whom she felt nothing only because he took care of her, putting her through school, literally rescuing her from her dismal life in Chicago's Cabrini-Green housing project. In the years following the breakup she'd coined something she'd called the boner test\u2014\"Does he give you a boner?\"\u2014and every time she'd dated she asked herself that question, even though, for obvious reasons, her experience was figurative.\n\nWhy was I getting so worked up about this before anything had happened between us?\n\nWe drove north on Collins Avenue, toward the Broward line, where Perry insisted he knew of the cheesiest restaurant in the world. \"I think you'll like it,\" he said nervously. I looked out at the fountains of the Bal Harbour Shops, the high-rise apartment towers\u2014the Avant Garde, the Seascape\u2014to which the word \"tony\" had once applied, while trying to fill Perry in on my history. I didn't sound terribly interesting to myself. How much I would have preferred staying in bed, reading; after all, I had to wake up at six-thirty to ready the nursery.\n\n\"Here it is,\" he said, easing the car into a parking stall.\n\nI wasn't surprised to see it was the Speedboat\u2014William's old favorite. I had an odd psychic sense, a kind of quasi-deja vu, which brought up a whole host of allusions and associations. I watched my knuckles whiten on the armrest.\n\n\"Isn't this great?\"\n\nThrough the windshield, I stared at the incongruous window display: the bonsai trees, the felted grass sheet lining the floor. \"Oh yeah,\" I said, not without sarcasm.\n\nInstantly the enthusiasm went out of his expression. \"We can go someplace else.\"\n\n\"No, it's fine,\" I said. \"I guess I'm just a little tired. Long day.\"\n\nInside the waitress seated us beside a wooden wheel, through which a continuous stream of sudsy water kept pouring into its pitchers, keeping it turning. In the pool beneath, an orange carp floated beside a hunk of coral.\n\n\"Look at this,\" he said, gesturing around. \"They built this place when going out for dinner was exotic, an event.\"\n\nHis manner was unsettling. At one point, I'd had that kind of energy and passion for things, but I'd worked it out of myself. Was I in mourning for something and hadn't admitted it? He made me nervous\u2014truly, deeply nervous. \"You haven't said very much about yourself,\" I said.\n\nHe went through the requisite details, telling me about his job, his patients, his hobbies and house\u2014all with a genuinely humble bent. In all outward respects he'd be the ideal boyfriend. Still, that didn't make me any more comfortable. What was he doing at forty-one, single, alone? Maybe he wasn't alone. Maybe he already had a boyfriend and just wanted to have sex, just once, with me. If that were the case...\n\nThere was a lull. He stared at me with an increasing interest, and I found myself looking away, continually, in shyness.\n\nI didn't have the energy to keep up my end of things. \"So what else?\" I said, much too loudly. \"Tell me about your love life.\"\n\nHis expression grew serious. Immediately, I regretted my flippancy. It turned out that he'd lost a lover within the past two years to AIDS, a lover with whom he'd lived since med school. His name was Andrew, and he spoke about him, a serious painter who'd been inspired by the original Arrow Collar Man ads and the covers of 1950s detective novels, with a quiet affection and longing that clearly indicated that the loss had shattered his life. Sometimes he wondered whether he himself had died with him. In the months since, he'd dated any number of men, a leather boy, a go-go boy, a personal trainer, a boat builder\u2014all frivolous, insubstantial types, he acknowledged now\u2014in an effort to make some contact again with the world. But he wasn't sure he was ready yet. He looked down at the tabletop, latching his fingers together. He'd barely eaten his dinner.\n\n\"And you?\" I asked.\n\nHe looked up. \"What do you mean?\"\n\nI looked away again. I was the last one to put him on the spot.\n\n\"Am I healthy, you mean?\"\n\nI nodded, flustered.\n\n\"Well, I'm negative, if that's what you're asking. At least the last time I tested. But that's been six months. Who knows? It's not like I haven't had sex since then.\"\n\nI gazed downward at the stone crabs on my placemat. I felt like nothing but a coward. \"I didn't mean to pry,\" I said quietly.\n\n\"Don't worry,\" he said, taking a sip of his water. \"What about yourself?\"\n\n\"I've been tested,\" I said. \"But I never picked up the results.\"\n\nI expected him to draw back from me, but his eyes shined with interest and understanding. \"Do you have any reason\u2014?\"\n\n\"My ex-boyfriend's positive. Actually, he has AIDS. I just found that out a couple of months ago.\"\n\n\"I'm sorry,\" he said.\n\n\"Yeah,\" I said. \"It sucks.\"\n\nWe were both quiet for a time, watching the waitresses scurry from table to table, setting down dinners like porterhouse steaks and lobster thermidor prepared from the same recipes the cooks had used thirty-five years ago. To our right a young white couple squealed delightedly at the yellow umbrellas in their drinks. Water plashed from the wheel. If only things were so easy. I expected Perry to be through with me, to call the evening to an awkward, ho-hum finish, but then I felt his shoe resting over mine under the table. There was a complicated smile on his face.\n\n\"Listen, I know how hard it is.\"\n\nI didn't say anything.\n\n\"It happens all the time. When I saw what Andrew went through, the drugs, the doctors, all this... shit in an effort to prolong his life a few months, I'm not even sure it was worth it. I think it was the stress of knowing, the depression, that wore him down. I mean, wouldn't it be better not to know anything at all, and die\u2014boom\u2014just like that?\"\n\n\"I don't know,\" I said. And I really didn't.\n\n\"Just as long as you're not putting anyone else at risk,\" he said. \"I mean, aren't we supposed to behave like we're all positive anyway?\"\n\nI nodded.\n\n\"Ugh,\" he said, rubbing his face with his palms. \"Enough already. I can't stand this anymore. Let's talk about something else.\"\n\n\"Shall we get out of here?\"\n\nHe stood up, nodded, and paid the check himself, a gesture I appreciated. Walking to his car, I felt a peculiar wave of kinship with him, something I suspected should be resisted at all costs. Perhaps we'd just be friends. I assumed that that was where this evening was headed, anyway. He'd drop me off, stop by the nursery in a few days, call me once or twice before it was over. Maybe we'd even have coffee one day in the future if our schedules permitted it.\n\nHe stared at the dashboard for a moment without turning the key. On the pylon atop the restaurant's roof, the neon speedboat pulsed once, the deepest blue, in the night sky. \"I don't know if I should say this,\" he said.\n\n\"What's the matter?\"\n\nHe gave a heavy, measured sigh. \"I don't mean to be presumptuous\u2014\"\n\n\"What?\"\n\nHe turned to me, a vulnerable, scattered look on his face. The silence between us was as heavy as the damp air.\n\n\"You want me to go home with you?\"\n\nHe nodded, relieved.\n\nI looked over at him. It wasn't the worst idea in the world. It was clear, after our talk, that we were lonely, in need of connection, affirmation. We needed to leave our brains. So what if I had to get up early the next morning? Actually, I admired the intention. It seemed that everyone I'd ever dated without having sex with right away ultimately failed to seize my interest, and they'd fallen away from my memory, nameless and vague. But that was irrelevant, anyway. I wasn't looking to get hooked up with him.\n\nHis house lay hidden behind a thatchy grove of palms. I hadn't been inside more than two minutes when he started kissing me, eagerly, pressing the hard weight of his tongue into my mouth. It all seemed a tad fast. I hadn't even had a chance to glance around, to check out the floor plan and furnishings, but I thought, all right, if we're going to do this, we might as well get it over with. I was as ready as I'd ever be. Would I get a boner? I thought of La Quan.\n\nWe moved to the bedroom. In no time at all, he'd taken off his clothes, and he stood to the side of the bed. His was in much better shape than I'd anticipated; muscular and dense, a trimmed mat of reddish hair sprinkling his chest, leading down a thin line to his abdomen. It might have been the body of a twenty-five-year-old. But what really surprised me was his dick. How could such a gentle, mild-mannered fellow be hiding such a killer in his pants?\n\nHe lay on top of me, kissing me all over my face with an intensity I'd never met in anyone else. It unnerved me, for all my life I'd always been more into sex than my partners, and this felt off-putting, as if my efforts to reciprocate were coming up short. I didn't want to disappoint. I closed my eyes and tried to think of Hector, his long lean body, his inky, close cropped hair, but all I saw were the scattered vague parts of him, and not the whole person, the disparate pieces never fusing into a soul. Same with William. He seemed remote, posed, a spread from Honcho or Mandate. I was trying too hard. Why wasn't this easier? I opened my eyes and watched Perry's face, his tightly shut eyes, his bitten lip, and thought, You really died, didn't you? You really died along with Andrew and are coming back to life.\n\nHe shuddered. He lay on top of me, wet, out of breath, burrowing his chin into the soft part of my shoulder. He opened his eyes and looked at me as if I might be a stranger. And then he grinned, recognizing me. \"Yee ha,\" he said plainly.\n\nI lay there, nodded. I didn't know what on earth we'd just done.\n\n\"Back in two seconds.\" He kissed me once on the forehead, then disappeared inside the bathroom.\n\nI fell into a deep, dreamless sleep, not waking up once until the following morning, when Perry came in the room dressed in a business suit, carrying a cup of hot coffee to the bedside. I stretched my arms over my head and yawned. \"What time is it?\"\n\n\"Late,\" he said, his face a bit fretful now. \"We overslept. It's\"\u2014he picked up the alarm clock\u2014\"it's twenty past eight.\"\n\n\"Oh God,\" I said, sitting up. \"I need to be at the nursery. Bob's going to kill me. I have a delivery of cedar chips at nine.\"\n\n\"That's okay, don't worry. I'll take you,\" he said calmly.\n\nHe sat on the edge of the bed, watching me picking up my clothes from the floor. He sipped from his cup of coffee.\n\nHe hadn't said a word in minutes. \"Do you think we should pursue this?\" he asked.\n\nI glanced upward at him as I pulled on my sock. I wasn't quite sure what he meant.\n\n\"I mean, should we try to see each other again?\" He bit into the skin of his lip. \"It's not like I don't have mixed feelings. I'm not even sure I'm ready.\"\n\nI kept looking at him. I could have said anything at that point. If I'd said no, what risk would there have been? Instead, I said, \"I don't see why not.\"\n\nHis face relaxed a bit. He walked over to me, led me to a standing position, then held me tightly in his arms. His chest felt warm and enormous next to mine. \"There's something about you,\" he said cryptically.\n\nI kept absolutely silent. Sweet man, I thought. Don't you know who you're fooling with? Don't you know I'm only going to hurt you?\n\n# Chapter 26\n\nThree times a week Perry and I did the typical things\u2014movies, dinners, walks, beach, gym\u2014nothing unsettling or spectacular. Sometimes we just drove around when we managed to have a day off together. We'd spot an isolated road with a quirky name and turn onto it, only to find something waiting for us at the end: a Dalmatian sunning herself on a floating raft, a house painted with patterns of tulips. These odd little visions seemed to present themselves to us, and we stopped questioning them after a time, accepting them as a function of our being together. And always at night, there was the sex, sex that was playful and complicated and demanding, if only because Perry was putting so much effort into it.\n\nStill, something bothered me about this whole endeavor. It was clear that he was much more committed to this than I was. I knew I could learn to love him. I could see that he had all the qualities that would make for a decent, stable boyfriend\u2014patience, tolerance, compassion, brains, even an edge when need be. But I trusted my gut more than anything. I wanted to feel reckless, and it frustrated me that I was still questioning my involvement with him. Watching him make love to me, I thought nothing could be sadder between two people, that one could have such feeling for the other and not have any idea it wasn't being returned to the same degree.\n\nIn October I read in the Herald that a woodwind ensemble from the Richmond Symphony was to perform in South Beach, in the concert hall on Lincoln Road. A while back I'd heard that Jane had taken a leave from Savannah to play second chair in Richmond, and I thought I'd show up to see if she'd be a part of things. I arrived ten minutes late, carrying a box of Fannie Mae candies, a token from another era. I couldn't see all the players from where I was sitting, but I suspected she was in back, with a mop of tangled hair, her head lilting in time to the music.\n\nWhen they stood to receive their applause I saw how wrong I was. She was sitting on the opposite side, her hair bleached silver, pulled back severely off her tanned forehead, with frosty lipstick, and green contacts. She'd appeared to have lost weight. She looked terrific, but oddly unapproachable at once. For a moment I was tempted to leave. Then I decided to wait for the crowd to disperse.\n\n\"Why didn't you call me?\" I called out to the stage.\n\nJane looked over to me, distracted from her conversation with the piccolist. \"Oh my God,\" she said, bringing her fingers to her mouth. \"Don't move. Give me time to change. Wait,\" she said, pointing a finger at me. \"Wait right there.\"\n\nIn no time at all she came out in leggings and an oversized T-shirt emblazoned with the Empire State Building. It was quite touching: Already I could see the kind of middle-aged matron she was to become\u2014stepping gingerly through the lanes of Palm Beach with a Saks bag on her arm and a fretful look on her face. \"What's all this?\" I said, regarding her new look with a wary affection.\n\n\"Look who's talking.\" She rubbed the top of my shorn scalp, then pointed to herself. \"Don't you think I look like a drag queen?\"\n\n\"For you,\" I said, passing her the box of Trinidads. The gesture felt wrong suddenly, though I didn't know why.\n\n\"Oh honey,\" she said. \"I can't. I try not to eat candy anymore.\"\n\n\"No?\"\n\nShe wagged her head from side to side. \"You meant well, though. Thanks.\"\n\nWe sat in a little deli, the last of its kind on a street that was rapidly being overrun with galleries, stores, and gay pride shops festooned with rainbow flags. \"It's time for shame,\" I'd said on the walk over. \"I'm going to start it, I think. The latest industry. Gay shame.\" Jane smiled slightly, but her mind seemed to be elsewhere.\n\nShe filled me in on what had been happening with her. To my surprise she'd gotten married last August. His name was Moon Lee; he came from a wealthy family who'd started up a successful carpet-cleaning business in Midlothian, Virginia, after their emigration from Seoul. She claimed to be in love with him, though I found that hard to believe. She adored her house, a center hall\u2014\"hip colonial\" she called it. She was hoping that the current oboist would vacate her chair because she had no intention of returning to the \"dingy backwater\" of Savannah. Otherwise, she'd quit and go into retail if the chance presented itself. \"I want to work,\" she insisted. \"I don't want to sit around painting my toenails. And there's this, of course.\"\n\nShe tugged up her shirt, reached for my hand, and pressed it on her stomach. Beneath my palm I felt the sensation of something turning. \"You're pregnant,\" I said.\n\n\"Mm hmm,\" she said, smiling. \"He's due in July.\"\n\n\"It's a boy?\"\n\n\"We don't know yet.\" She gazed longingly at the pastel-colored Sherman between the lips of a pretty young dyke. \"Aren't you excited?\"\n\n\"Of course.\"\n\nHer eyes hazed over. \"I'm a little surprised. I just thought you'd be more excited.\"\n\n\"But Jane\u2014\"\n\nShe blinked away some odd welling of emotion, embarrassed. \"What is it? Something's bothering you, isn't it?\"\n\nI started talking about Perry. It was hard to get it all out at once, organized. I told her about his generosity and attention. I told her about his own difficulties, his struggle to regain himself after the death of Andrew. I told her of my doubts, my paradoxical feelings, which seemed to fluctuate with every hour of the day. I talked so scrupulously, covering everything in such minute detail, that it occurred to me why all my confidants and friends hadn't been returning my calls. I hadn't realized it was bothering me so much. Only one thing I left out: the burning, low-grade fear that I was dying.\n\n\"So I don't get it. What's the big problem here?\"\n\n\"What do you mean you don't get it?\"\n\n\"I mean, please, he's falling in love. It sounds like he'd bend over backwards for you.\"\n\n\"It's a matter of integrity,\" I continued. \"I feel like I'm leading the poor guy on. Who knows what he expects from me?\"\n\n\"It's not like you're married to him.\"\n\n\"Of course not, but\u2014\"\n\n\"Enjoy this time,\" she said evenly. \"Enjoy this time, keep with it, and see how you feel in six months.\"\n\n\"I can't do that to anybody.\"\n\nHer eyes sparkled green. \"Just what the hell are you doing to him?\"\n\n\"I mean, for all I know, he's in love with some image of me, and not me. He doesn't love me. I mean, what the hell would he want with me, anyway?\"\n\nMy admission flustered me. My eyes darted toward the window, where I watched a rollerblader weaving, turning backwards, drawing figure eights in and out through the pedestrians. He was shirtless, sinewy, in amazing shape, a crazed, ecstatic look in his eyes. Cocaine? Crystal meth? I looked back at Jane. She appeared to be completely entrenched in her thoughts.\n\n\"You know what I think.\"\n\n\"What?\"\n\nShe gazed down at the tabletop. \"I think you're so used to being treated like crap that you don't know what to do with someone who's actually decent and responsible.\"\n\n\"What?\"\n\n\"Shhhh\u2014\"\n\n\"Listen\u2014\"\n\nI dug my fingernails into my palm.\n\n\"It's not all that easy to be with someone who's kind. You have to have a certain sense of yourself. You have to be able to say, 'I deserve to be with someone who's good to me.'\"\n\n\"You sound like you're talking about yourself.\"\n\nShe shrugged her left shoulder. \"Anyway, that's all I have to say on the subject. Enough already.\" She folded her hands on the Formica. She glanced at the dessert menu, returned it to the aqua wire stand. Then in a hushed voice she said, \"Don't fuck things up.\"\n\n\"No?\" But I didn't feel any better, and I didn't think she'd heard one thing I'd said.\n\n***\n\nIt happened weeks later. I was driving the pickup to Perry's house, where I planned to tell him that I didn't want to see him anymore. Things had been so much easier once I'd admitted this to myself, and my life had opened up again, allowing me to relax and to concentrate on the things that were important. The trick was to do it kindly, without menace, without making him feel I was deserting him forever. He'd been through too much. Perhaps I could tell him I needed a break for a while; perhaps I could tell him I only wanted to see him once a week, for a while at least. I'd play it by ear. In any case, what I had in mind would hurt him like hell, but it would be best in the long run, and he'd thank me one day for my brutal honesty, I knew it.\n\nHe stood outside the house, watering the bottlebrush with his shirt off. His nipple ring glinted in the sun.\n\n\"Aren't we bold?\" I stepped out of Bob's pickup. I pointed to his bare chest, on display for all the suburban mothers to see. A grassquit flew over our heads.\n\n\"Hey, sexy.\" He dropped the hose, grabbed me to him, and attempted to wrestle me to the lawn.\n\n\"Not here. The grass is all wet.\"\n\n\"What's wrong with wet?\"\n\nNot five minutes later we were in his bedroom. God, I thought. This was going to be harder than I'd imagined. I'd have to tell him after we were finished, and that would hurt even more, feel more like a betrayal. I heaved a huge sigh then stiffened my limbs. I heard him squirting lotion into his hand.\n\n\"What's that?\" I said, and clenched up, resisting his touch.\n\n\"Shhh\u2014\" he said. \"I'm going to rub your back.\"\n\nI lay there, trying to relax as his hands, strong and pressured, kneaded the muscles of my shoulders. I hadn't realized how tense I'd been. It felt terrific actually, although it hurt like a blowtorch, like he was writing his name onto my skin. Little utterances emanated from my mouth. After a while I moved into a place beyond thought. It only took minutes. I found myself floating, backstroking through a pool the size of space. Stars fizzled out, and I looked over my shoulder and saw a little earth turning in the darkness, silent. Even from this place I could see it diminishing. Even from this safe place I saw great forests burning down, towers crumbling, vast countries of people scrambling for food. I saw that there wasn't very much time. I didn't believe in epiphanies or easy answers or sudden revelations, but all at once Jane's words came flying home to me, a hail of pellets flung against a wall. I turned over on my back. I looked up at Perry. I saw him at six, waiting to be photographed with his mother and sister, staring sternly at the wolf puppet waving in the assistant's hand. I saw him at twenty-six, arm in arm with Andrew, in black robe and mortarboard, grinning upon his graduation from medical school. I saw him at ten, walking alone through Disneyland after his father left him off with a twenty-dollar bill, not knowing what else to do with him. I saw him at thirty-nine, feeling the heat gathering in Andrew's stomach as his body went cold. I saw him in the future, older, with a head full of white\u2014all of these images stacking up at once, projecting themselves simultaneously onto a screen. A door might have fallen open beneath my feet, and then another. What had I been resisting? What was my strength? Had the thing I'd wanted all along been right here with me, for months, and I hadn't even seen it?\n\nHe turned me over on my back.\n\n\"Why are you crying?\" he said.\n\n\"I don't know,\" I said, and then I held him close to me.\n\n***\n\nIt wasn't something I'd expected. I walked into the nursery one morning, spilling coffee on myself, rushed as usual, to find Bob sitting in his chair, eyes opened, a glassy, bemused smile on his face. I wouldn't have known as we weren't in the habit of exchanging greetings, and he'd always said hello the first thing in his gruff, kind voice. I had no idea how long he'd been like that, but I stayed with him, so calm I surprised myself, holding his hand until the medics arrived with the ambulance. Two days later, I helped with his funeral. A motley crowd showed up at the Methodist church, comprised of police officials, our regular customers, and Dorothy's twin sisters, Muriel and Lu, both of whom had flown in from Las Vegas. The floral arrangements were the most screwy and peculiar I'd ever seen\u2014canistel, tamarind, sapodilla, ginger lilies, firebushes, coral vines, Brazilian plumes\u2014most of which were trucked in directly from our nursery. A silver banner draped across his closed casket\u2014HICKORY BOB, in the darkest blue letters.\n\nOne rainy morning, a few months later, the phone rang. A lawyer identifying himself as Sam DeSears called at 8 a.m. to tell me I owned a nursery. I listened intently, quietly astonished, barely mumbling a reply, as he read off the clause from the will.\n\nI told everyone I knew. I told Perry, whom I'd moved in with only a few months before. I told my friends Zack, Nan, and Jane. I went through the whole list of people, everyone I ever knew\u2014Hector, William, Ursula, even minor customers\u2014with the curious exception of a single person.\n\nWhen I called the King Cole, a recording told me that the number had been disconnected. I called two more times, if only to make sure I hadn't dialed wrong. It took a while for me to admit that Peter had left, and that the resort had finally closed its doors.\n\nI thought of him often, late at night, lying in bed with Perry. He might have been anywhere, but no one knew, not my mother, not Holly, who'd called me once or twice. I made a place for him inside my head. I put him somewhere in the Yucat\u00e1n, close to the shoreline, in an aqua house with butterfly chairs out front. He had parrots, cats, mice, and a mule, and a large garden in which he grew plantains. He was utterly alone. He swam in the sea every day, after four, only after the last tourists had left. He watched the sunset\u2014a gaudy, overblown event\u2014from his kitchen window nightly. He even shaved off his hair and taught himself Spanish, giving things he'd made to his neighbors on the street: bread, enchiladas, cups of coffee rich as the blackest loam.\n\nOr maybe he didn't.\n\n***\n\nI picked up my test results one morning without intending to. The receptionist, a pretty red-haired woman with a cast in her eye, smiled faintly as she returned from the lab, and told me I could leave. I stood in stillness for a moment. The sunflowers blurred before me in their vase. Then I walked miles and miles through the buzzing city, grateful and melancholy all at once.\n\n***\n\nWe stood upon the beach at Biscayne National Park, a deserted preserve, barefoot and parched, while forest fires ransacked the Everglades. A flamingo straggled on the mud flats. Oyster shells clung to the roots of the mangroves. Earlier, we'd driven across the county, inspecting hurricane damage, seeking out the gardens and plants that interested us. At one point, I took Perry to the concrete pipe in which I'd spent so much of my childhood. The ditch that passed through it was filled with thick, sludgy water, clogged with melaleuca roots, and it was hard to imagine how anyone, man, woman, or child, would fit into that cramped cylinder without drowning.\n\n\"That's it,\" I'd cried, pulling hard on his sleeve. \"That's the pipe I told you about.\"\n\nPerry looked more than a little perplexed. \"You hid out in there?\"\n\n\"Yes, as a matter of fact, I did. But it was safe. That concrete pipe got me through my childhood.\"\n\nHe looked even more perplexed. He made a face now.\n\n\"What's the matter?\"\n\n\"You know what I think?\"\n\n\"What?\"\n\nHe mumbled, \"My boyfriend's a lunatic.\"\n\n\"Shithead,\" I said, and socked him in the arm.\n\nWe continued up the beach, rested, though half the world was burning down. The smoke spun higher in the distance. I stared hard at the horizon, picturing everyone I'd ever lost scorching, coalescing in that pyre, their spirits melding, turning yellow, green, copper, red. Was it too much to bear? It was hard to say no. For all I knew the worst would still happen: Perry would get sick, my mother would wither, Peter, my father, William, Hector, Laser, Jane, Arden, Holly, Ory, Stan Laskin, Todd, Douglass Freeman\u2014all of us leaching white until there was nothing left but alkali, little crumbs of salt blowing up into the void like sand. Or maybe not. Maybe we'd all pass through the door, ruined, yet wiser. The ocean murmured. The kingbirds glided above the mangroves. The flames were still distant, rumbling, not quite advancing. At least the two of us were here, together, the sky over our heads ferocious, harsh, beautiful.\nPAUL LISICKY is the author of five books: Lawnboy, Famous Builder, The Burning House, Unbuilt Projects, and The Narrow Door, forthcoming from Graywolf Press in 2015. His work has appeared in Conjunctions, Denver Quarterly, Fence, the Iowa Review, Ploughshares, Tin House, the Rumpus, Unstuck, and in many other magazines and anthologies. His awards include fellowships from the National Endowment for the Arts, the James Michener\/Copernicus Society, and the Fine Arts Work Center in Provincetown. He has taught in the creative writing programs at Cornell University, New York University, Rutgers\u2013Newark, Sarah Lawrence College, the University of North Carolina Wilmington, and elsewhere. He currently teaches in the MFA Program at Rutgers\u2013Camden, the low residency program at Sierra Nevada College, and at the Juniper Summer Writing Institute. He is the editor of StoryQuarterly and serves on the Writing Committee of the Fine Arts Work Center in Provincetown.\n\n# Acknowledgments\n\nSections of this book have appeared in A&U, Blithe House Quarterly (www.blithe.com), Cosmos, Global City Review, Provincetown Arts, and in the anthologies Best American Gay Fiction 2 (Little, Brown, 1997) and Men on Men 6 (Plume, Penguin, 1996).\n\n***\n\nI'd like to thank my good friends Stephen Briscoe, Polly Burnell, Michael Carter, Denise Gess, Elizabeth McCracken, and Katrina Roberts for their careful, wise feedback. I'd also like to thank David Bergman, Brian Bouldrey, Karen Brennan, Chris Busa, Patricia Chao, Bernard Cooper, Michael Cunningham, Robert Jones, and Ann Patchett for their encouragement. Thank you to Ruth Greenstein for her astute editorial suggestions. Thanks, too, to Jonathan Rabinowitz and Turtle Point Press for kind, immeasurable support. Thank you and love to my parents and brothers\u2014Anton, Anne, Robert, and Michael Lisicky\u2014for their generosity, patience, and good spirit.\n\nI'm grateful to the National Endowment for the Arts, James Michener and the Copernicus Society, the Fine Arts Work Center in Provincetown, the Corporation of Yaddo, the MacDowell Colony, the Djerassi Resident Artists Program, and the Ragdale Foundation for much needed assistance.\n\nAnd finally, deepest thanks to Mark Doty.\n\n***\n\nThe new edition of this book invites me to thank the incomparable staff at Graywolf Press: Fiona McCrae, Anne Czarniecki, Janna Rademacher, Katie Dublinski, Mary Matze, and Jeff Shotts. And thanks to Kyle Hunter for the brilliant cover.\n\n\u2014P.L., New York City, December 2005\n\n# Lawnboy \u2013 Paul Lisicky (12\/05) Frequently asked Questions\n\nYour book expresses an ambivalent attitude toward Florida. How is the setting central to the book?\n\nPL: I think of the South Florida of the book as a kind of a stage set. It's a Florida of the imagination, even if it makes reference to real cities and streets. I'm interested in the tension between nature and artifice. I think of the seagrape hedges of Palm Beach, surgically clipped to look like stone walls. Or plants that look like hats or wigs. You can convince yourself that the man-made has the upper hand in a place like that, but we all know that South Florida is a hurricane or drought away from being done in. I think of Elizabeth Bishop's great poem \"Florida,\" in which nature is just lying in wait, ready to swallow up the whole grand scheme. She calls Florida the \"poorest postcard of itself.\"\n\nSo the tensions within that landscape mirror Evan's imagination. I'm reluctant to pin down the metaphor, but I'd say that Evan's perception of that setting\u2014his simultaneous love and resistance\u2014is crucial to understanding him.\n\nWhat were the special challenges in writing a book with a gay narrator?\n\nPL: I wanted him to be both a representative character and completely himself at once. The struggle of trying to balance those two desires pulses behind every scene. If he was too much of a representative, he'd be dull and washed out, too externally conceived, a type. If he was too peculiar, he'd put off readers. It often put me in a quandary. For instance, I didn't know if I could get away with a sentence like: \"How I wanted to lift the bowl in his moment of peace and kill him.\" All the same, I wasn't interested in writing a role model. I thought it was important for him to be fallible, wrong, even stupid at times. But he's also smart, sensitive, loyal to a fault, and capable of a huge, beguiling sweetness.\n\nHow are style and self-presentation important to the book?\n\nPL: Evan's sense of himself as endlessly mutable worries him at first. At one point he wonders whether he's merely the sum total of \"everyone who'd passed through [his] life\" and nothing more. Hector teaches him, if only through example, that it's not necessarily awful to be one thing one day and something else the next. You can give form to that mutable self through your clothing, through how you make yourself up every morning.\n\nI think we too easily conflate style with consumerism these days, but they're two different things in the world of the book. Style is constructed in the face of despair. It defies death and loss; it's an emblem of persistence, of faith in the world. And it's even better when it involves some act of reclamation. I'm thinking of Hector's thrift-store clothes, or Peter's doomed attempt to fix up the King Cole.\n\nWhat is the significance of all the ruined buildings and places in the novel?\n\nPL: They're all over the place: Golden Gate Estates, the ghost city of Boca Bay, Douglass Freeman's house and neighborhood... all cases of dashed hope, failed optimism. All originally conceived as points of pleasure. You could say that all this ruin stands for the fortunes of Evan, Peter, and Hector, who are all struggling against disillusionment and fear. And then again I hope that my metaphors resist easy explanation. You want them to be wilder than any attempt to cage and tame them.\n\nIn what ways is Lawnboy a novel about the age of AIDS?\n\nPL: The dread of AIDS\u2014and the stigma associated with the disease\u2014is everywhere in the book. The body out of control, a limited sense of time, imminent loss\u2014all these things haunt Evan, even though he hasn't tested positive. He has such a compromised sense of the future that even the prospect of going to the dentist seems futile. Hence, his urge to find someone now because he might not be around in three days. Every wish is intensified to the burning point. Life ablaze on the precipice.\n\nWhat do you say to those who read the ending as happy?\n\nPL: Sure, it's happy. Just as Evan tells himself that he's better off on his own, he gets the relationship he's wanted all along. But Evan's happiness lives with his knowledge that relationships are fragile. Even if he and Perry manage to make a long life together, Evan knows that he might outlive his new partner. The book closes with suggestions of terrorism and environmental collapse\u2014\"towers crumbling,\" \"forests burning down.\" So any personal joy is cut through with melancholy and loss\u2014my usual ambivalence rearing its head again!\n\nLawnboy is set in the early 1990s. How would Evan's story be different if you were a young man now?\n\nPL: It's almost too obvious to say it, but we're the products of our time. Protease inhibitors, cell phones, September 11th, \"The War on Terror,\" our disastrous president and his cronies, the Internet, online cruising, crystal meth, Friendster, MySpace, blogs, the widening gap between rich and poor: none of these was a factor in the early 1990s, and every last one\u2014good, bad, or both\u2014shapes daily experience in ways vast and small.\n\nI'd like to think that Evan would feel less isolated, a little less afraid of dying before his time. But even the most openminded parents don't raise their children to be homosexual, and the process by which gay kids come into their own is still traumatic. They often learn to efface themselves out of simple self-protection. What young kid wants to be called \"gay,\" which still stands for \"freak\" or\" loser,\" on the schoolyard? I think that any young man of Evan's intensity\u2014his loyalty to his family and his contradictory pull toward a life of desire\u2014would have a hard time of it. We're living in an era that's hostile to originality, to those who make their own path. There's a tremendous pressure to conform, no matter how you understand yourself.\n\nStill, it's sort of fun to imagine him standing in line to see Brokeback Mountain for the tenth time, listening to the Hidden Cameras on his iPod.\n\n# Reading Group Guide for Lawnboy by Paul Lisicky\n\n  1. Evan senses \"a wall between himself and the world,\" and describes a \"low-grade fear and rage burning\" beneath all his decisions. He is torn by contradictory impulses and sometimes experiences himself as \"a puppet\" or only a compilation of people he has known. How much of these feelings have to do with his age and how much with simply trying to live in contemporary society?\n  2. Landscapes and settings play a tremendous part in the story. How does the King Cole reflect both the inner life and the desires of Evan's brother Peter? How are the brothers alike? How are they different? Why does Evan feel a need to visit the hurricane-ravaged home where Douglass Freeman once lived? Finally, what is the significance of Evan's love of plants and what they become for him at the end of the novel?\n  3. After Evan moves out of his parents' home and in with William, he runs into his mother one day and tells her that she's never looked so good. She replies, \"I'd come to the point in my life where I realized I had two choices. I could either shoot myself in the temple or reinvent myself.\" In what way is her life a continual act of self-invention? A continual coming of age?\n  4. In her letter to Evan at the beginning of Chapter 10, Ursula Sarshik demands that Evan choose between William and his parents. She remains staunch in her decision not to let Evan come home for just a few days. She accuses Evan of turning his back on his parents. Is Ursula justified in her demand? Or have she and Evan's father turned their backs on their son? Does the Sarshik family strike you as typical or unique?\n  5. In Evan's childhood recollection of taunting the gay man, Stan Laskin, the hardware storeowner, Stan asks the boys, \"Why are you so hateful? All of you, I don't get it. Tell me how you live with yourselves?\" Are people meaner today? Are children meaner? Why? At the other end of the spectrum is the notion that sometimes difficulty makes people larger, better... Is this the case with Perry at the end of the book? Do you think that his kindness and thirst for life have to do with the loss of his partner, Andrew?\n\nThis book was designed by Rachel Holscher. It is set in Calisto MT type by Bookmobile Design & Digital Publisher Services and manufactured by Bookmobile on acid-free paper.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nTHE BLOOD SERIES\n\nBooks 4-6\n\nBlood Reign\n\nAngelic Blood\n\nBlood Enchantment\n\nNew York Times Bestselling Author\n\nTAMARA ROSE BLODGETT\n\nAll Rights Reserved.\n\nCopyright \u00a9 2018 Tamara Rose Blodgett\n\nNo part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a database or retrieval system without the prior written permission of the publisher.\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locales or organizations is entirely coincidental.\n\nwww.tamararoseblodgett.com\n\nTRB Facebook Fan Page\n\nEdited suggestions provided by Red Adept Editing.\n\nCover Design: Willsin Rowe\n\n# CONTENTS\n\nWorks by Tamara Rose Blodgett and Marata Eros\n\nTRB News\n\nWarning! Contains spoilers\n\nCharacter Index\n\nBLOOD REIGN\n\nANGELIC BLOOD\n\nBLOOD ENCHANTMENT\n\nAcknowledgments\n\nAbout the Author\n\n|  |\n\n---|---|---\n\n# Works by Tamara Rose Blodgett and Marata Eros:\n\nTamara Rose Blodgett\n\nThe BLOOD Series\n\nThe DEATH Series\n\nThe REFLECTION Series\n\nThe SAVAGE Series\n\nShifter ALPHA CLAIM\n\nVampire ALPHA CLAIM\n\nFinal Enforcement Vampire ALPHA CLAIM\n\n&\n\nMarata Eros:\n\nA Terrible Love (New York Times bestseller)\n\nA Brutal Tenderness\n\nThe Darkest Joy\n\nClub Alpha\n\nThe DARA NICHOLS Series\n\nThe DEMON Series\n\nThe DRUID Series\n\nRoad Kill MC Serial\n\nThe SIREN Series\n\nThe TOKEN Serial\n\nFinal Enforcement Vampire ALPHA CLAIM\n\nShifter ALPHA CLAIM\n\nVampire ALPHA CLAIM\n\nThe ZOE SCOTT Series\n\nClick HERE for Download Links for your Retailer.\n\nNever miss a new release! Subscribe:\n\nMarata Eros NEWS\n\nAnd\/or\n\n|  |\n\n---|---|---\n\n# TRB News\n\nSubscribe to my YouTube Channel\n\nExclusive Excerpts!\n\nComedic Quips\n\nWin FREE stuff!\n\n|  |\n\n---|---|---\n\n# Warning! Contains spoilers.\n\n|  |\n\n---|---|---\n\n# Character Index:\n\nBlood Singers\/talent:\n\nJulia- Queen of the Singers; Telekinetic\/telepath\n\nJason- Singer\/\"Feral\"\/Red Were\n\nScott- Royal Singer Blood; Deflector\/Combatant\n\nBrendan- Tracker\/pyro\n\nMichael- Illusionist\n\nJen- Telekinetic\n\nCyrus- Healer\n\nPaul- Negator\/amplifier\n\nAngela- Feeler\n\nMarcus- Region One\n\nJacqueline- Royal Singer Blood; Region Two Leader\n\nVictor- Region Two\/Combatant- Boiler\/Flame of Blood\n\nLucius- Combatant\n\nCynthia \"Cyn\" Adams - Rogue\/Healer\n\nHeidi- Reader\n\nTrevor- Deflector\n\nNorthwestern Were Pack:\n\nLawrence-Packmaster\n\nEmmanuel \"Manny\" - Beta to Lawrence\n\nAnthony \"Tony\" Daniel Laurent- Second to Lawrence\n\nAdrianna \"Adi\"- Alpha female\n\nSoutheastern Were Pack:\n\nDavid- Packmaster\n\nAlan Greene- Alpha male\n\nLacey Greene \u2013 female Were and Alan's sister\n\nBuck \"Slash\"- Alpha male\n\nKarl Truman- former Homer detective\n\nFord- Alpha male\/ FBI agent\n\nReagan - Moon Warrior, Lacey's daughter\n\nSoutheastern Vampire Kiss:\n\nMerlin- Coven leader (now deceased)\n\nWilliam- new coven leader (now deceased)\n\nBrynn- New leader of the Southeastern\n\nNorthwestern Vampire Kiss:\n\nGabriel- Coven Leader\n\nClaire- William's cousin\n\nWilliam- Runner\/shifter\/Singer blood\n\nUnseelie Sidhe fey:\n\nQueen Darcel- Sidhe\n\nTharell- mixed Sidhe warrior\n\nCormack- Sidhe warrior\n\nDomiatri \"Domi\"- Sidhe warrior\n\nRex- Sidhe\n\nKiel (key-ale)- dragon shifting Sidhe\n\nCelesta- Sidhe warrior\n\nDelilah - Vampire, third to Julia, Scott's half-sister\n\nRogue Reds:\n\nEzekiel \"Zeke\"\n\nRogue Alpha female Were:\n\nTessa\n\nJuvenile female Were\/shapeshifter:\n\nTaliah\n\nFEDS:\n\nTom Harriet\n\nTai (tie) Simon\n\n|  |\n\n---|---|---\n\n# BLOOD REIGN\n\nA Blood Series Novel\n\nBook 4\n\nNew York Times Bestselling Author\n\nTAMARA ROSE BLODGETT\n\nAll Rights Reserved.\n\nCopyright \u00a9 2014 Tamara Rose Blodgett\n\nNo part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a database or retrieval system without the prior written permission of the publisher.\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locales or organizations is entirely coincidental.\n\nwww.tamararoseblodgett.com\n\nTRB Facebook Fan Page\n\nEdited suggestions provided by Red Adept Editing.\n\nCover Design: Claudia McKinney\n\nPhotographs: DepositPhotos\n\nPhotography: Oleg Gekman\n\n# Synopsis\n\nJulia's sworn enemies are safely sequestered in a prison of the fey and her forever mate has been chosen\u2014not by blood, but by a circumstance shaped from coincidence. However, it's not enough to save Julia and the others their fate at the hands of the Alaska den, whose reacquisition has come alarmingly full-circle to capture them.\n\nTharell of the fey aligns with the Singers, Were, and remaining vampires to take back the one true Queen. Only Julia can stop the interspecies wars and establish a genetic truce that would free all groups from extinction. Can they rescue Julia and her allies before it's too late? Will the Red Were's lineage prove to be the catalyst of victory against a corrupt pack grown too debauched by greed and power?\n\nDEDICATION\n\nLinda Palomera-Terpe\n\nMay this be the first of many....\n\n# CHAPTER ONE\n\nMotion woke Julia initially. Rhythmic. Unfamiliar.\n\nConstant.\n\nShe allowed her eyes to open to slits. What little light existed crept inside to illuminate the four corners of where she lay. Julia said nothing, alerted no one. Silver chain, a fine gauge little better than woven lace, covered Jason, Truman, and Adi. Cyn was bound from behind, her hands in painful-looking knots behind her back.\n\nJulia swallowed, acutely aware of the dryness of her mouth and throat. She lay on her side and rocked back and forth, accidentally throwing herself on her face. It pressed uncomfortably against the rough wood of the surface on which she lay. Julia moved her shoulder into the unforgiving wood and shoved hard, using her half-asleep legs to jerk up to her knees in a semi-kneeling position.\n\nHer head swam with the remnants of the drugs.\n\nJulia took several deep, steadying breaths. She blinked, meeting Cyn's.\n\nHers reflected in the dark.\n\n\"That cop bit me,\" Cyn croaked. Julia imagined her throat was as parched as her own.\n\nJulia hadn't witnessed Cyn's being bitten, and the reason behind it eluded her. She relaxed against her heels for the time being. \"Why?\" she asked.\n\nCyn grimaced as she shifted her weight, her eyes tightening with pain. \"I don't know. I think he was thinking he'd be Mr. Helpful.\" Her gaze roamed all over the inside of the stuffy space. She recognized they were in a truck as it jostled. She and Cyn groaned from the motion.\n\n\"You're looking a little wolfy, Cyn,\" Jason said from his slumped corner.\n\nJason. Julia half-crawled, walking to his position with her knees, and he gave her a small smile. It looked more like pain than joy.\n\n\"What is it?\" Julia asked.\n\nThe truck jerked to a halt, and Julia flew forward, no hands to ward her fall. The ground hit her face a second time, but this time with momentum. She yelped on contact, muffled as it was, rolling over as splinters and a bruise vied for position on her cheekbone.\n\nShe'd just kicked her own ass.\n\n\"Jules!\" Jason yelled, moving to her in the same awkward shuffle she'd just managed.\n\nNot fast, wrapped as he was with the silver chain.\n\nHis face appeared above her, tightening in pain.\n\n\"Jules,\" he whispered.\n\nTears rolled out of the corners of her eyes and she moaned. \"Yeah.\"\n\n\"We're home boys... and girls. Home reek home,\" Truman said in a sarcastic commentary from the back of the bus. Or truck.\n\nJulia rolled over onto her back. She hadn't noticed the stench. Now that he'd brought it to her attention, exhaust was all she could smell.\n\n\"Not yet,\" Jason said to Truman without looking at him, his hands bound tightly against his back like everyone else.\n\nTruman snorted. \"Canada, pal. Close enough.\"\n\nJason grunted affirmation. \"Yeah.\"\n\n\"What now?\" Cyn asked.\n\n\"Looks like they're taking us back to Alaska,\" Jason said and Julia could hear the despondency in his tone. It wasn't going to be reunion time. There was an entirely different agenda going on.\n\nThe truck slowed, coming to a shuddering stop.\n\nLily's entrance shattered their small group's quiet commiserating.\n\nVent holes and an opaque ceiling was the only air and light the five of them had. The sultry heat and dim interior vanished the instant the doors were slapped open and daylight streamed inside.\n\nJulia took in the woman who had pretended to be her aunt, secure the doors against the sides of the truck.\n\n\"You're awake,\" she said without waiting for a response. Gone was the demure, small-in-stature guardian with the perpetually sour disposition and perma-smirk.\n\nIn its place was a statuesque Sidhe warrior of the fey.\n\nJulia flung about in her memory bank for anything to do with fey lore and could only come up with the colorful Faerie stories of childhood literature.\n\nNone of that was remotely applicable to the reality that stood before her. A six-foot tall, inky black figure with a shock of silver hair gave them her inscrutable, pale gray eyes. She was nearly naked. A band of material like a vintage tube top wrapped her breasts and a matching covering that decorum dictated hid the female bits and nothing else. She looked like a barbarian from another planet. Every movement brought striated musculature to the surface of her skin like the rippling of water after a stone's throw.\n\n\"Good,\" she said in a soft purr. \"I'm sure the human necessities are calling.\"\n\nHer voice didn't even sound the same. Lilting, biting... it ran along Julia's skin like small bugs looking for flesh to penetrate.\n\n\"I always thought you were a dumb bitch before. Now I don't have to speculate. Yeah,\" Cyn commented dryly.\n\nThat was how Cyn coped. Julia just wished it hadn't been right now.\n\nLily's self-important smile wilted a little around the edges. \"Your good or bad opinion of me is not relevant, Singer.\"\n\n\"I think it's more like Were now, fairy,\" Cyn bit back. Then, \"I gotta pee, and unless you want to add winged janitor to the list of your bullshit, I suggest you get my drugged ass out of here.\"\n\nAdi groaned in the background. \"God... what\u2014silver.\" She blanched, looking around until she caught sight of Lily. \"Okay, kidnap us because you're asshats. I get it, but silver?\"\n\nJason and Truman said nothing. Their glaring at Lily was all the evidence needed that they'd rather be anywhere but in the back of a box van returning to Alaska.\n\nLily watched Cyn warily as she leapt up from the ground five feet into the rear of the van. Cyn grinned. It hung on her face wrong somehow.\n\nLily was on her in a second. She grabbed an ankle, jerking her like a willful dog on a chain.\n\nJulia saw what would happen before it did, but Lily was a little slow on the uptake. Some things remained the same.\n\nThey had not thought to wrap the spiderwebbing of feather light silver chain around Cyn. They'd discounted her.\n\nShe had been a Singer before Truman's bite, and someone had dropped the Were ball. Having Singer blood meant many things. But if the host held a Singer's blood, transference to cross-species genetics was a highway instead of a walk.\n\nCyn's fingertips burst with talons, her nose grew to a stub, downy pale fur taking the place of skin and her teeth becoming sharpened tips. She sat up, partially changed, her butt dragging along the rough wood floor, and took hold of the forearms pulling her out of the van.\n\nCyn easily broke her non-silver bonds.\n\nLily turned, her face filled with comical surprise. Julia understood from her expression that it was a rare event.\n\nCyn crushed the bones of Lily's forearms and Lily reared back, howling in anguish, trying to tear herself out of Cyn's grasp.\n\nIt was a horrible beauty to see her friend's face grin from a row of teeth more wolfen than human.\n\n\"No ya don't,\" Cyn growled and jerked Lily close, rolling her over onto her back. She didn't pause for consideration.\n\nHuman rational had left, and the Were had taken over.\n\nLily kicked out, connecting with Cyn's midsection as she simultaneously latched onto her shoulder.\n\nCyn grunted but didn't dislodge.\n\nInstead, she moved deeper into the hold, biting down on Lily's neck with crushing force.\n\nA second figure jumped into the van as Julia pressed against the sides of the wall.\n\nOne of the cops.\n\n\"Stop!\" he commanded, and Cyn shuddered.\n\nShe growled even as she reveled in mutilating Lily's neck, shaking her head slightly in glee.\n\nMetallic and warm, copper filled the air, and Julia gagged.\n\n\"Release her!\" the cop said.\n\nCyn raised her muzzle and howled, at once enraged and discouraged.\n\nHe had stopped her from tearing Lily's head off.\n\nJason and Truman snuffled and mewled, not in submission but in pain and disquiet.\n\nCyn backed away, shaking her head while on her hands and knees. No matter how much she swung her head back and forth, she couldn't rid herself of his command.\n\nJulia's eyes rose to his. A new Alpha.\n\n\"I'm Tom Harriet.\"\n\nJulia said nothing. His identity meant nothing. Did nothing. William was dead, his sacrifice feeling more and more worthless. He'd saved her from Cormack, the Sidhe that meant her death, only for another group to capture her.\n\nBut not just her. It was now her and the others.\n\nJulia dispassionately watched Lily's throat knit the damage, very pink against all that true black skin, Cyn's canines had made.\n\nJulia began to speak, but Harriet held up his hand for silence.\n\n\"It was one of my wolves that botched this entire acquisition up three years ago.\" His smile was a grim line in a hard face. \"And now we have the Blooded Queen\u2014who by rights was always meant to be with the Alaska den and no other.\" Harriet spread his hands away from his body.\n\n\"Who are you to tell me where I should be, and with whom?\" she asked. Her eyes were still closed, her breath shallow as she laid her head against the wall of the van.\n\nShe smelled his nearness, and her eyes flew open as Jason growled and struggled to get to her.\n\nHarriet's eyes bled to green.\n\nHe was a Red.\n\n\"We're not some weak outside pack.\" He evaded her question. Those spinning orbs dove into her gaze, mesmerizing, captivating\u2014dead serious.\n\n\"I am pack master of the Alaskan den. We take what is ours. No negotiations, no bullshit.\"\n\nJulia searched his eyes. He seemed to be telling the truth, though the truth as he saw it.\n\n\"What does this mean for me?\" Julia asked slowly, though she didn't think she'd like the answer. She was getting accustomed to receiving it.\n\n\"Easy, Julia Caldwell. You're the pinnacle of a breeding triangle, to simplify things. As I'm sure the fey rehearsed to you, you throw true, yes?\"\n\nJulia stared at him, speechless. Unbelievable. Out of the corner of her eye, she saw Jason inch closer.\n\n\"We want more than Alaska.\" Harriet's hand closed into a fist by her temple. Julia didn't flinch. She was desensitized at this point. It didn't help that she had to use the bathroom so badly it burned inside her, distracting her from even the fear of this new threat.\n\nHe casually swung that held fist into Jason's temple and he fell with a howl, the chains cinching tighter with the movement. As Julia now saw, they were meant to.\n\nStay still, and they remained slack. Move, and they grew tighter with each motion.\n\nJason writhed in agony on the floor behind them, and Harriet went on as if nothing had transpired.\n\n\"We have the highest concentration of Red Were in the world. We are destined by genetics to be the dominant Were. That is why we're put to death on sight by dens who fear us\u2014our potential.\"\n\nJulia chanced a glance at Jason, his eyes spinning orbs of green anger, his wolf a hairsbreadth from erupting. She couldn't stand the idea of what would happen to him if he burst his skin with the silver bindings.\n\n\"Don't,\" she whispered. Please don't.\n\n\"He will, if I will it,\" Harriet said softly.\n\nThey looked at Harriet. \"Why would you take us and torture some of your own kind?\" Julia asked, desperate to understand.\n\n\"Because I can,\" he answered softly.\n\nTom Harriet turned to face Jason and Truman.\n\nHis command brought worse than her imaginings.\n\nJulia cried as the bindings tightened and the howling was the only thing she heard.\n\n# CHAPTER TWO\n\nJulia could be neutral, but Cynthia was pissed.\n\nShe hated\u2014hated that pudwacker, Harriet. It went beyond the obvious kidnapping crap. Cyn was aware, very aware, that he was Alpha Red and that meant he could pretty much boss any Were around. Not great for her.\n\n*\n\nCyn took satisfaction the deepest canine punctures were a necklace of wounds around Lily's neck.\n\nIt made a smile bloom on her face like a flower seeking the sun.\n\n\"How's it feel, bitch?\" Cyn hissed at Lily.\n\nShe strode to Cyn, and the fey warrior paused just long enough to kick her in the ribs.\n\nCyn grunted, biting down on the natural yelp that threatened escape by a fraction before it erupted. She wouldn't give the heartless Sidhe the satisfaction.\n\nWhatever irritated that betraying, lying c u next Tuesday worked just dandy for Cyn.\n\n\"Cyn, God, would ya stop?\" Adi rubbed her wrists where the silver had burned her like acid. Cyn pulled a face at the ligature marks at the sight of the silver bindings decorating Adi's wrists.\n\nBastards.\n\nTom Harriet crouched down next to her with a slight bounce, cocking his head to the side and studying her like an interesting experiment inside a petri dish.\n\nCyn kicked up her chin. \"Gonna beat the hell out of a bound woman?\" She spit in his direction. It landed near his left foot, missing his expensive shoes by a fraction of an inch.\n\nHarriet's lips curled. \"Ordinarily, no. However, you bring the very worst reactions out in me. My wolf wants to clamp jaws around that skinny neck of yours and shake you until your teeth rattle.\"\n\n\"What's stopping you, pencil dick?\" Cyn challenged.\n\nA vein popped in his forehead, and Cyn braced for the worse. Deep in her heart, she knew she couldn't be a martyr for anything or anyone, like Jules had committed to doing long ago.\n\nShe was too much of a selfish bitch, for one. If he killed her, then she wouldn't have to deal with what had happened to her. Cyn centered herself, hoping for finality. She thought she was more selfless than this, but surprise... she was more human than all the Singers and Were combined. Right now, she felt pretty damned human. Cyn was beyond frustrated and helpless. She couldn't live like this.\n\n\"Don't, numbnuts!\" Truman's voice boomed out into the open space where they camped. He trained his eyes on Harriet's hand.\n\nHarriet hesitated, his fist lifted to strike.\n\nCyn watched him warily, shooting a glare at Truman, who returned her look with one of his own.\n\nHe'd spoiled a perfectly good braining.\n\nThat fist fell to his side. He breathed deeply, scrutinizing Cyn as he began to speak. \"I was top in my class at the Bureau; received honors for my grades when I studied the criminal justice system. I know what you're attempting here, Cynthia Adams.\" His eyes searched Cyn's. She gave a disgusted exhale, her face inches above the sparse grass where she lay.\n\n\"Suicide by Fed. Or in this case\u2014Were.\" Harriet stood, shaking his head as he looked down at her. \"We didn't bother with all this elaborate shit to kill you. Think.\" He tapped his temple with two fingers. \"You can't kill the fey. She's immortal. You'll just piss her off.\"\n\nCyn doubted it. There had to be some way to kill her black ass.\n\nHis gaze roved the small group, Ford cruising up to stand by his side.\n\nAll this was so overwhelming. I want to go back to where I was. Where getting her nails done and the latest pair of shoes filled her thoughts. But no. Now it was Were this, Singer that.\n\nShe looked at her chipped, shitty nails. Her hair had two months' dirty blonde grow-out. She wanted to howl for that alone. Instead, she glared at Harriet.\n\nAdi said, \"Wait a second...\" Her eyes narrowed to slits. \"I know why you took Julia. No offense\"\u2014her gaze swept Julia then went back to Harriet\u2014\"everyone wants the Rare One.\"\n\nHarriet said nothing.\n\n\"But you took us, too. Why not knock us out, take Julia, and run off?\"\n\n\"That's the $64,000 dollar question,\" Lily said.\n\nCyn rolled her eyes. \"Don't try. Don't try to be clever, interesting, or relevant. You've been in the fey mound too long. That's not even that much money anymore, you dodo bird.\"\n\nJules stifled a laugh. Adi didn't even try, letting a straightforward guffaw escape.\n\nLily's lips thinned into a slash of charcoal. \"I want to end her. I don't care if she's Red, or leverage... she is a foul-mouthed Singer that has been turned into a dog. She is of no consequence.\"\n\nCyn smirked.\n\nShe shouldn't have felt confident, given they'd caught on to her new skill set and bound her in silver. But she did. It was just too damned easy to yank Lily's chain. Cyn found she liked it. A lot.\n\n\"Not yet.\" Harriet stayed the fey with an arm. His skin was very white against the tar color of hers.\n\n\"Why?\" she nearly wailed. \"I put up with her empty head while she was friends with the Blooded Queen because I was told I must... but now? Let us do away with her.\"\n\nAdi rolled her eyes between the two. \"Let's stay on task, kidnappers.\" They looked at her. \"So\u2014you take Julia for the obvious reasons.\"\n\n\"Brood Mare,\" Julia piped in then lifted a finger. \"No\u2014scratch that. Paranormal brood mare.\"\n\nHarriet and Lily scowled at Adi, but she was unfazed.\n\nAdi swept her hand to Julia. \"Yup, that covers it. But you still haven't explained the rest.\"\n\n\"I know why,\" Truman said.\n\nHarriet's self-satisfied smirk turned into a shit-eating grin. Cyn figured that couldn't be good.\n\n\"You're Red. I'm Red.\"\n\nHeat flared in Cyn's skin when Truman's gaze skimmed over her trussed-like-a turkey form.\n\n\"Jason\u2014Cyn?\" he asked.\n\n\"Very good, cop. It shows that not all the police force is bumbling fools.\"\n\nTruman coughed out a laugh. \"Yeah, it shows I don't have to go to four years of a hoity-toity college to prove I can puzzle something through. I don't have to hold my dick to know I have one, if you feel me.\"\n\nCyn felt her eyes go round as Harriet strode to Truman.\n\nHarriet's face was thunder. \"Don't make me shut you down, cop. It'll hurt.\"\n\n\"If you think you've got the brass balls for it, try it.\" Truman's face was set in stone. He was a man of his word. It was all over him.\n\n\"Truman,\" Jason cautioned.\n\nCyn remembered Jason's caged existence while he was feral after being turned. He was Red because he'd been bitten by a Red but didn't have the Were lineage. He'd almost gone sideways when he'd changed. Not true in Truman's case. He had the genetic fabric to make the change.\n\nTruman shrugged Jason's hand from his shoulder and squared off with Harriet, the testosterone so thick in the air Cyn choked on it.\n\nInstead, Lily's droll voice broke through their posturing. \"All the talk of penises has made me realize how distracting it is to have one.\"\n\nTruman and Harriet turned to Lily.\n\nHer hand came to rest on Cyn's hair and she wanted to puke on Lily's feet. She did the second best thing. \"Get your hands off me or I'll bite your toe off. It won't kill ya but it'll be a bitch growing that back.\"\n\nLily's sour look soaked Cyn with its venom. She glared back at her. Like that bitch's dirty looks were going to matter. Duh.\n\nHarriet's gaze found Lily's, and a silent exchange passed between them. Cyn shifted her weight.\n\n\"You're right, of course,\" Harriet conceded to Lily. He cast a last glance at Truman then walked away to stand beside her.\n\n\"I am but one 'scout', if you will... who searches for Reds. If we can reclaim those that wander without a den, or worse\"\u2014Harriet's gaze sought each of theirs\u2014\"are unclaimed and killed simply because of who they are, then my bit in this life will have had value.\"\n\nLily looks bored, Cyn noted. She just knew that wench had her own agenda.\n\n\"So what's with Lily, what does she get? Because I know she's not here for any philanthropy.\"\n\nJason grunted in the background.\n\n\"No.\" Harriet's tone was short. \"The fey make pacts that benefit their people.\" He spread his arms wide, palms out. \"Like all of our species.\"\n\nHis gaze went to Julia. \"Even the Queen of the Singers, their 'Rare One', must wield a certain mercenary edge to maintain order.\"\n\nCyn didn't hear Jules argue. Hell\u2014she couldn't protest it, either.\n\nHer mind spun with the why of it all. Then she touched on something... \"So you collect the Reds and you become, what? The most kick ass den in the West or something?\" Or North?\n\nHarriet smiled. \"No. Pure Reds do not need to heed the moon's call to change. We change at will.\"\n\n\"It's the sign that a dog is Red enough if he or she\"\u2014Lily literally looked down her nose at Cyn\u2014\"can change without lunar fullness.\"\n\n\"But some can't,\" Truman said.\n\n\"Not exactly.\" Lily's pale eyes bored into Julia's. \"Now that the Blooded Queen has become, all those who have the Red blood of the Were can willfully change, if she is in proximity.\"\n\n\"So just Julia being around...?\" Adi fished for confirmation.\n\nHarriet nodded.\n\n\"It's very similar to the lowly human who has a bit of fey blood.\" Lily sniffed as if this likelihood was not only repugnant but also unlikely.\n\nCyn narrowed her eyes to slits. \"Tell us. How does it compare to a human having fey ancestry if a Were is Red?\"\n\nLily rolled her eyes at Harriet, who said nothing.\n\n\"Why... you're such a daft girl. Throwing around a few four-dollar words is not proof of intellect.\"\n\nCyn moved her shoulder in an awkward shrug. She was okay in her own skin. The strange creature that had treated Julia badly, lied, and hidden who and what she was wouldn't diminish Cyn. Only she had the power to do that.\n\n\"Just tell them,\" Harriet said to Lily. The limits of his patience finally grew thin. About damn time. Hell, hers had gone MIA about... well, from the very beginning.\n\n\"Like blood calls to like. For simplicity's sake, a summons\u2014if you will.\" Lily crossed her arms over a flat black stomach, and Cyn was disturbed to note she had an outie.\n\n\"So strength in numbers.\"\n\nLily inclined her head.\n\n\"This is a getting a bunch of the same species together for what end?\" Truman asked carefully.\n\n\"Control,\" Harriet said. \"With the Blooded Queen, we are suddenly all moonless changers. It's a powerful summons.\"\n\n\"I want to know more about this fey angle.\" Cyn slid her gaze to Lily. Cyn was suspicious; Lily's presence still didn't make complete sense.\n\n\"The fey will be summoned as well. With enough time, the status of all the Reds becoming moonless changers, regardless of blood, will be solidified forever.\"\n\nJason spoke for the first time. \"You won't need Julia.\"\n\n\"No,\" Lily answered.\n\n\"Then what happens to me?\" Julia asked.\n\nLily favored Jules with a malicious smile. \"You will then be mine to summon the humans who have fey blood.\"\n\n\"Why?\" Jules pushed her matted hair behind her ears, a helluva shiner distending her cheekbone.\n\n\"So we can rid the earth of the half-breed.\"\n\n\"Genocide?\" Truman asked in patent disbelief.\n\nCyn shook her head at the grinning Lily, who was obviously touched in the head.\n\n\"Feycide,\" Cyn confirmed.\n\n# CHAPTER THREE\n\nTharell met the unlikely source of his intel, Gabriel, the Rare One of the death bringers' Northwestern kiss.\n\n\"We have an understanding, then?\" Gabriel asked.\n\nTo say that working with a corrupt Singer, leader to a coven of the fanged, was a necessary evil was an understatement.\n\nEvery thought had been about the Rare One, the Blooded Queen of the Singers. Only her presence, then lack thereof, could be the cause of the disquiet. The sithen knew her.\n\nTrust did not come easily to the fey. However, the insurance of another faction whose interests aligned with theirs was advised.\n\nA stealthy rustle in the periphery made Tharell turn. A deer trotted within shouting distance. Somehow, it felt the men's presence, though Tharell had thrown a cape of glamour to hide them. It scampered off, skittish as a new colt.\n\nGabriel's eyes never wavered to the distraction of the wildlife. \"Your glamour?\" he asked.\n\n\"It holds,\" Tharell responded, insulted.\n\nGabriel chuckled. Tharell schooled his face to the blank neutrality for which the Sidhe courts were so well known. Though now that Queen Darcel no longer ruled, there could come to be a Sidhe the sithen deemed worthy. Tharell had hoped it would be the Blooded Queen.\n\nShe had made a vow to consider Singers as candidates for marriage in Faerie. Whether or not she would bow to that wisdom remained to be seen.\n\n\"If what you tell me is true, and William has met his end in true death\u2014\"\n\n\"It is,\" Tharell answered, already tiring of the game of words.\n\n\"Then I see no reason why we cannot ally as we search for the Rare One.\"\n\n\"She will not comply with what you offer.\" Tharell crossed his arms, and the glamor that rode the surface of his skin shimmered, settling back on the deep violet flesh like a floating covering. Only he could see it, as he owned that magick.\n\n\"There is Claire, cousin to William...\"\n\nTharell palmed his chin. Familial ties were very strong, especially amongst the fey. As strong a consideration for Singers as for the Sidhe. \"Perhaps she would return to Seattle, in the underground lair of the Northwestern, but she left under terrible circumstances. Julia was soul-meld with a Singer of royal lineage.\"\n\n\"And fey, and vampire,\" Gabriel interjected.\n\n\"Not precisely. Though it was a motley mess of ancestry to be sure.\"\n\nOnly a few could venture very far from the mound and hope to retain their power. Tharell, of course, due to his mixed blood, and maybe Domi with some extraordinary intervention or assistance. But the distance Gabriel proposed would challenge even him, as rare as some of his gifts were. And he was prepared to find out what that would be in the happenstance that some things came to pass.\n\n\"They mean to unite the Reds and it will be back to what it was two thousand years ago, when the fey were in Europe and the Reds struck fear in anyone foolish enough to travel with a moon full or gone.\" Gabriel gave a slight lift of his broad shoulders.\n\nTharell frowned, remembering the legends well. The Red Were were feared. Rare and hunted to near-extinction. Now it appeared as though distance had given them sanctuary.\n\nTheir sojourn would sicken Domi and make Tharell vulnerable.\n\nGabriel watched his thought processes with interest. \"Have we struck a bargain, Tharell of the Unseelie Sidhe?\"\n\nTharell stared at Gabriel so long the other man almost took his proffered palm back. The fey took an oath with the utmost seriousness.\n\nFinally, and with great reluctance, Tharell shook hands with the devil. His motivations went beyond this meeting, to which Gabriel was not privy.\n\nIt was the best solution amongst bad ones.\n\nJulia Caldwell would be taken again. For good, and many Singers handed over to the Unseelie in exchange for her freedom.\n\nIt was an unjust manipulation of her. He could easily apply that old human expression of \"sacrificing the few for the many\" here.\n\n*\n\nTharell could not wash the treachery from his skin as he bathed in the modern showers of Faerie supplied by the underground hot springs.\n\nThe water slid like hot, fragrant silk over the muscles of his body as he mechanically cleansed himself.\n\nThe lack of an interim ruler unsettled many of the Sidhe. Tharell agreed. But without royal blood and a slew of bodyguards, there was no peace. No one could step in without being murdered by another vying for the top position.\n\nThat was why a magical padlock of sorts had been used on the court. No one but the fashioner of the spell could undo it.\n\nOf course, like all the old magicks, it had strength in the beginning, only to fade as the timepiece of the sithen ticked. And as all knew, time ran differently in Faerie.\n\nTharell pressed his forehead against the sithen's smooth, polished rock. He simply thought of the need for a towel and a peg pushed out of the rock, a towel appearing that was a shade lighter than his skin. He knew of civility, and the sithen's low-running intelligence was sentient enough for an exchange of thoughts, though the sithen did not possess speech as such.\n\nTharell said nothing, taking the towel from the peg that had grown out of the wall. He petted the wall in quiet thankfulness for the small courtesy shown by this place where but an unspoken thought met the Sidhe's basest needs.\n\nThe peg shimmered, becoming opaque and finally translucent. Then, like water, it melted back to become one with the low wall's glittering ebony.\n\nHow much longer will the sithen be healthy without leadership? And not just any leader, but one that showed deference and respect to the fey. All fey, not just Sidhe.\n\nTharell padded into the larger quarters that gave way to an archway, hand-cut centuries before Tharell's birth. He set about getting his garments donned. His tunic came first, breeches and soft-soled shoes were custom cut and sewn for his feet. He laced them to the tops of his shins, winding the leather ties twice around his upper calf and knotting it with proficiency borne of long practice. He walked to the door, grabbing his weapons belt off a hand-hammered brass hook as he left.\n\nThere was no iron in Faerie. It was poison to the fey.\n\nHe loathed the royal singer contained within their prison, but he must deal with her. She might be what Tharell needed to survive the journey so far from Faerie.\n\n*\n\nTharell moved through the labyrinth of the sithen.\n\nIf it had been to Tharell's liking, Jacqueline would have been dead long ago.\n\nThe Singers Book of Blood forbade such a thing. Jacqueline was royalty, though her mixed blood appeared to be as frowned upon as his own. Fey, Singer, Were, and vampire. Jacqueline, reigning monarch of Region Two, had too much variation in her blood.\n\nAnd just the right amount of what mattered.\n\nJacqueline stood when she saw him, the heavily veined marble bench that had been her post disappearing into the sithen wall as she left it.\n\nTharell released a breath of resigned frustration. Half-clothed again. Jacqueline always knew when he would appear and shed her clothes at just the right moment.\n\nTharell cast his gaze away when he saw what the Were did behind her, their vulgarity part of who they were. He could not close his ears to the lustful coupling that reached him. It was always the same. And deliberately meant to throw him off balance, make him feel ashamed to watch them rut as alley cats caught in their mating.\n\nTharell squared his shoulders. They were detestable but had become necessary.\n\n# CHAPTER FOUR\n\nJacqueline allowed the Were to drive into her from behind, barely losing breath, though she did slap the invisible wall in front of herself so he would not force her face into it.\n\nShe had coerced herself to mate with Tony. They had handled it quietly. As she had Were blood and must adhere to her needs as a Singer and a royal, they had been given a ceremony of the most basic order.\n\nMarcus was required to be in attendance, and his obvious distaste of it all had brought her more pleasure than she anticipated. It was the only pleasure of the entire union.\n\nA means to an end.\n\nOf course, even though the magical borders of their fey prison were impervious to their efforts at escape, the entire bit of Faerie could be realized through scent alone. Tony was a Were and she an expert Tracker. Too bad her talents grew weaker each minute she spent in this odd but increasingly pleasant new world.\n\nTony was always aware when the one who called himself Tharell was nearing, and she cringed at what must happen. It had been Tony's idea to make themselves as repugnant as possible while also obtaining their end goal. Generally, as far as lore went, the fey were immune to sexual exploits carried out in front of them. But after much experimentation, they'd found it was the only thing that got under Tharell's violet skin.\n\nJacqueline bitterly encouraged Tony directly against her deepest instincts. \"Harder, beast. Make a show of it.\"\n\nTony complied, spearing her so deeply she felt the pain of his entry and exit, the horrible tool of his perversion buried deeply within his nature; adoring the opportunity to hurt her through sex.\n\nHe really was a terrible excuse for a Were.\n\nJacqueline ground her teeth against his roughness, the foreign urge to cry floating within reach as he gripped her shoulder. His fingertips bit into her skin as he anchored her, causing what would be magnificent fingerprint bruises to take shape by the morning.\n\nHe gasped. Tony stabbed her with a final, revolting thrust. As he finished inside her, Jacqueline plastered a smile on her face for Tharell's benefit. He struggled to cover his revulsion.\n\nWhat was the modern slang for that? Jacqueline wondered as she straightened and let down her dress. Ah yes, epic failure. She touched a finger to her chin, pondering... Or was it epic fail? No matter, disgusting Tharell was the goal.\n\nShe could hardly remember why she had made this an objective, and it disturbed her more day by day. Jacqueline wondered if she could bear it much longer.\n\nTharell cleared his throat and Jacqueline inquired, as if she had not just been having sex with the disgusting Were, \"And to what do we owe the honor of your presence?\" She watched those bright eyes, like the Caribbean Sea, flick to Tony then back to her. He was an extraordinary looking creature.\n\nAnd a dangerous one. Jacqueline never allowed herself to forget that.\n\nTony learned that lesson the hard way, early on. He had thought to outmaneuver the Sidhe and made to turn into his half-wolfen form. He went for the fey's throat.\n\nTharell had blurred into him with a move that had an edge of macabre beauty. Before Tony could take his next breath, Tharell had deeply embedded his long sword inside Tony's body.\n\nTony's eyes had grown wide, his body slumping atop the blade. Tharell had grunted as he'd extracted it in a tight jerk toward his own body. Tony had fallen on his smug face.\n\nThat had been Tony's first and last attempt to try the bounds of the prison. Now this new plan was in full play.\n\nJacqueline would get with child, and when the little wretch was whelped, Marcus would have an entire new problem to consider. They could not imprison a Singer of royal blood who carried offspring. It didn't matter that she had made a bid for Julia's life, the inept girl, or that they had discovered her contaminated blood.\n\nJacqueline was royal enough.\n\nShe curled her lips into a smile at Tharell's next words.\n\n\"I have a proposition.\"\n\nExcellent. Jacqueline smelled opportunity and it had nothing to do with her diminishing skills as Tracker.\n\nJacqueline thought she caught a whiff of a metaphorical bleed from Tharell.\n\nShe and Tony, sensing the potential for weakness, moved in for the kill.\n\n\"And?\" Jacqueline asked, feigning boredom. It was not as easy as it sounded while Tony made a production of zipping up his sizeable but now-deflated commodity inside his modern jeans.\n\nHer smile became genuine as she saw Tharell's clear distaste of Tony.\n\n\"The Rare One has been taken,\" Tharell stated in bald discourse.\n\nEven better news than expected. Jacqueline couldn't contain her glee. \"Who's taken the girl?\"\n\nTharell ignored her question. \"I will need a fey of mixed lineage who is also Singer to Hunt the Blood until she is rediscovered.\"\n\nJacqueline studied his face. Arrogant\u2014certain. She twisted around to face Tony then glanced at Tharell over her shoulder. \"Let me confer with my... mate,\" she choked out.\n\nTharell gave a stiff nod.\n\nJacqueline walked to the far corner of their \"cell,\" actually a mockery of the outside. All begotten by old magick, it was a type of indigenous glamour where a false sky with perpetual twilight lit the area. Stars that would never twinkle were cast about in the deep navy folds as tangerine inked the edges like spilt juice.\n\n\"He has asked for a Hunt the Blood,\" Jacqueline whispered to Tony, who of course, ignored the finer points and grabbed her breast.\n\n\"I could go again,\" he said, twisting her nipple through the thin blouse she had donned.\n\nJacqueline winced, trying for patience and finding none. Instead, bile churned in her empty stomach at the thought of him touching her again so soon. \"Do that again and I shall freeze your lungs, you atrocious excuse for a male.\" Though she knew not how long she could make good on that particular threat. Jacqueline hated the crude smile on his face but was relieved when his fingers fell from her.\n\n\"You need me,\" he said in a huff, folding his arms.\n\n\"True, swine, but only until your seed takes hold. After that, you are no longer necessary.\"\n\nHe leaned forward, his breath hot and rancid against her face, and she held her own against his threatening body language. \"When you have my whelp in your womb, it will not be a choice. You will be mine whether you wish it or no.\" He used the old language, forcing her to listen.\n\nJacqueline elected to ignore his threat, for now. \"Fine. Play nice is all I ask. Stop hurting me.\"\n\nTony's eyebrows hiked. \"You like it.\"\n\nHow could Tony's perversions exceed my own?\n\nShe ignored the goad. \"Tharell of the Sidhe must use us to find her.\"\n\nTony shrugged a shoulder. \"I heard. I don't give two shits and a fuck. I want out of here. The why doesn't matter. It's results, Jackie.\"\n\nJacqueline ignored his overly familiar nickname. It seemed obtuse at this point to worry over it. She flipped her palm out. \"And that is what this gives us\u2014opportunity.\"\n\nTony's eyes narrowed. \"All right, what's my role?\"\n\n\"It will be simple for you. Do not let on that there is any agenda other than the one you normally perpetuate.\"\n\nTony barked out a laugh. \"Which is?\"\n\n\"Your typical brutish behavior,\" Jacqueline answered easily.\n\nHe drove a palm around her neck and jerked her to his mouth, plunging his tongue down her throat, gagging her with it. She beat at him with her fists. Finally, in desperation, she stole his breath.\n\nHer telekinetic talent was still strong. For now.\n\nTony pulled away from her, gasping for air that wouldn't come. Her power was much diminished, dampened by the magick of the fey, the holding cell uniquely fashioned to weaken her powers. She gave it everything she had, drowning Tony in what remained.\n\n\"Stop!\" he croaked, driven to his knees. He clawed at her dress. The material wrapped around Jacqueline's ankles as he fought his body's need for oxygen.\n\n\"Jacqueline of the Singers,\" Tharell called out loudly.\n\n\"Yes?\" she asked without turning, enjoying Tony's plum-colored face. And his subservient position on the floor.\n\n\"What is your decision: Will you help in the Hunt for Blood?\"\n\nJacqueline released Tony's airway. It had drained her immeasurably in the hostile environment of Faerie. Her shoulders dropped from the exertion.\n\nTony glared up at her, his hands ringing his throat, his ass on his heels.\n\n\"I shall,\" she replied.\n\n\"Be ready tomorrow morning at dawn... or whatever time I call you.\"\n\nTharell gave a curt chin dip and with a last glance at Tony, he spun on his heel, walking away.\n\nTony grunted as he stood, using a hand upon his knee to hoist himself.\n\nBut his gaze as he watched Jacqueline spoke of caution.\n\nGood, he would do well to be cautious of me. Jacqueline needed no one. Only herself. And those underneath her. Or that was all it had ever been\u2014all she knew.\n\n\"I have wounded you.\" Tony growled in pleasure.\n\nJacqueline felt the wetness between her thighs and scowled at him. On some level, she understood that self-hate was part of her nature. As it was a part of his.\n\nWho knew what the two of them would spawn together. Many would consider the mixed genetics too diluted to introduce anything of malicious intent. Jacqueline wasn't sure.\n\nTony's hand snaked out and grabbed her wrist.\n\n\"I can do more.\"\n\nShe placed a palm against his chest to resist, but her powers were depleted.\n\n# CHAPTER FIVE\n\nSlash struggled to awaken. He'd become instantly aware of the commotion all around him. He was shedding the drug like unwanted raindrops as his consciousness ticked closer to full tilt.\n\nHe sat up in a moving truck. Five sets of eyes followed his movement.\n\nSlash opened his mouth and no sound came out. He had never been thirstier in his life. Jason Caldwell kicked a water bottle to him with his bound legs and Slash grabbed it. He took a long pull, silently capped it, and set it between his legs. A harsh breath slid out of him. Slash didn't panic easily even as he took in the apparent direness of their situation.\n\nHis wrists burned where ligature marks deeply grooved his flesh. He fought not to rub them, feeling fortunate he was unbound. He easily caught the reflective eyes of his brethren, realizing only just then that they were Red\u2014all.\n\n\"They're taking us to Alaska,\" the girl named Cynthia informed him quietly.\n\nInstead of answering, Slash located Adrianna, her face swollen where she'd been struck. His mouth tightened. She noticed his expression and looked away.\n\nOf course she did. Slash understood how awful he looked. Yet it was important he knew she was healing.\n\n\"Adrianna,\" he began, his discomfort around her acute, his desire to hide it even greater.\n\nHer face swung back to his. \"Yes?\" she asked while hiding her eyes from his gaze.\n\nWhere was that spitfire nature he so relished? \"Are you well?\"\n\nAdrianna shook her head. He crawled to her and maintained a tense two-foot distance. Silently adding his presence without throwing it in her face.\n\n\"I couldn't help anyone. Lily forced the change.\"\n\nSlash sensed Adrianna blamed herself.\n\nHe remembered it all. Very painful to be brought when the moon wasn't full; it had been a deliberate move to incapacitate. She could have done nothing. Slash was Alpha enough to change into his half-form, and once in a great while he could also bring himself if the moon was only a thumbnail in the sky.\n\nAdrianna was not Red, and she was hurting.\n\nHis gaze wandered the cloistered, stuffy dark van and returned to study her. He was ashamed he had missed an opportunity to protect her while he slept off his drugged state. \"What can I do to ease you?\"\n\nSlash bit the inside of his lip when all he wished was to drag her into his arms and kiss away the bruises on her face.\n\nA fat tear made a clean spot on the van's dusty floor as it shook its way down the highway.\n\nSlash stilled.\n\nAdrianna got up on her knees, the silver bindings tight. Blood oozed from her wrists, and he growled low in his throat in frustration that he could heal nothing.\n\nCyn interpreted the look on his face. \"I could heal her if we weren't bound in silver.\"\n\nSlash nodded. She had the same limits as he.\n\nTurning back, he watched Adrianna painfully, slowly, walk the short distance to him. She fought the lurching truck, the hard floor that abraded her knees as she drew closer.\n\nSlash sat there. He couldn't tolerate rejection. He hoped the swampy darkness of the van box's interior sufficiently hid his face.\n\nAdrianna moved until her knees met with his. She lowered her face to his shoulder.\n\n\"What...?\" Slash swallowed hard, never more self-conscious in his entire life. \"What would you have of me, Adrianna?\" Her warm breath bled through the light shirt he wore, and he suppressed a shiver. She owns me.\n\nThat small gesture tightened the invisible ties that bound them.\n\n\"Hold me,\" she whispered against him. Two words that changed his life.\n\nThe moment swelled... held. Slash let the air out of his lungs, draped muscular arms around her smaller body, and drew Adrianna into his lap. She curled up as best she could, her wrists so tightly bound she gave a little whimper when Slash adjusted her position.\n\n\"I am sorry,\" he said, feeling like an oaf.\n\nShe shook her head, her hair pleasantly rubbing underneath his chin. It smelled of sweet female. His female.\n\nSlash's hand hovered over the top of that silky hair. His eyes found Cynthia's, and he began to glance away, but not before she gave him a signal that took bravery to execute.\n\nShe had nodded her head, and Slash let his hand fall against Adi's head from her silent encouragement.\n\nA breath eased out of Adrianna, and her cheek pressed deeper into Slash's broad chest. \"Thank you, Slash.\"\n\nHis eyes burned. A heartbeat passed while he considered many things simultaneously. \"You are most welcome,\" he finally said. Slash moved his hand over her head again and again. When her breaths grew deeper, more rhythmic, he slowed but did not cease the motion.\n\nTruman and Jason gave him identical knowing looks. He ignored them.\n\nThe Alpha female he loved was tucked against his body. She had sought his protection. It was the most ancient of offerings. An Alpha female had only to submit to an Alpha male in just that way for the male to assume care over her during time of extreme duress or war.\n\nA male Were chose to accept the offering or not. Slash had no choice; Adrianna had claimed his heart long ago.\n\nIt wasn't about choice but rather letting himself fall.\n\nIn love.\n\n*\n\nSlash was driven rudely awake by a jolt as the van came to a stop. His arms automatically tightened around Adrianna and she awoke with a start, immediately followed by a groan.\n\n\"What is it?\" he asked, pushing the hair from her brown eyes. He saw her look at his face and self-consciously turned away.\n\n\"Slash,\" Adi said. He moved his face back an inch. Just enough for her to know he had heard her.\n\n\"Yes.\"\n\n\"Stop that.\"\n\nHis eyebrows jerked in surprise. \"What?\"\n\n\"Turning away from me.\"\n\nSlash would have answered, but bright sunlight hit them like a slap and Slash threw a hand over his eyes to shield them from the brightness.\n\n\"Good, you're awake. I've got the bullshit babysitting duty, so come along kiddies.\"\n\nSlash stared at a Were he'd never seen before and found he instantly disliked him.\n\n\"Fuck you, Ford, and the horse you rode in on,\" Truman announced from the far corner. The two were definitely prior acquaintances.\n\nThe Were's brows dumped over flinty eyes. Unforgiving ones.\n\nHe stalked over to the corner and Truman stood up, facing his charge.\n\n\"Listen, old man, I'm here to get you to hop out of this van, mark your territory and get the fuck back in the rig, got me?\"\n\nTruman laughed and Slash had to admire his gumption because he was bound. But Slash was not.\n\nSloppy.\n\nThey must have thought, truck in motion, Were out for the count, no worries.\n\nWrong.\n\nAdi gave him wide eyes and silently hobbled off his lap. His gaze did not leave her. He promised her things he shouldn't have with that look, for he saw the reaction on her face. Slash couldn't control his expressions and didn't much give a shit.\n\nHe employed stealth as automatic as breathing. Of further help... Truman didn't telegraph Slash's approach.\n\nNot even when Ford landed a dead-center sucker punch that doubled Truman over.\n\nTruman cocked his head to the side, looking up at Ford from his knees with one eye. \"Tell ya what, you stand there and be a fire hydrant, and I'll lift my leg on you. How's that for marking territory, assjack?\"\n\nSlash grinned and shook his head at Truman's hardness. It was refreshing.\n\nFord hit him hard in the jaw, plowing the big former cop to both knees, wiping Slash's good humor on the spot. Truman spit out an impressive spatter pattern of blood that highlighted the filth of the floor. Their eyes met.\n\nIt was too late for Ford to react.\n\nSlash struck from behind, using his hard fists like hammers of retribution against Ford's kidneys. They blurred with the speed of his attack.\n\nFord pissed his pants as a result of the onslaught.\n\nThen he pissed blood, crumpling to the floor as his body struggled to heal the organ damage.\n\nSlash crushed Ford's windpipe with the instep of his boot.\n\n\"Heal that,\" he murmured, satisfied his skills of merciless and swift incapacitation still held.\n\nTruman walked himself up the wall with his palms. \"Nice bit of work there, Slash.\"\n\nSlash didn't smile or answer. His gaze was on Adrianna, still bound. Julia and Cynthia huddled together. He swept over Ford's breathless form and walked toward Adrianna.\n\n\"Let's self-congratulate later,\" Jason said. \"Right now, let's get the chicks and get the fuck outta Dodge.\"\n\nSlash thought that an excellent idea. He chanced a glance at their mutual enemy as he walked back to the male.\n\nFord lay gasping. A fish out of water, his eyes bulged like hard-boiled eggs. Slash crouched on his haunches. \"Don't come after us, Were. This is dating for me. Don't make me marry you in death.\" Slash kicked him in the ribs, and the smaller Were groaned.\n\n\"Cut us,\" Jason said, holding his wrists up.\n\nSlash dug in Ford's pocket, extracting a titanium switchblade. He went to Adrianna first, her wrists weeping blood. What he could not do before with his bare hands, he now did. He sawed at the silver. It fell away, and bone greeted him. The fine webbing had been like acid against her skin, burning away everything in its path.\n\nHe bent on one knee, despondent over the rawness of her injuries, and she shook her head, clearly miserable. \"Get the others.\"\n\nSlash touched the back of her head briefly and stood. He loosened the silver from everyone, attending Julia last, whose bonds were plastic zip ties. Irritating but not caustic.\n\n\"Ford!\" an unfamiliar voice called. Not alarmed, but inquisitive.\n\n\"Let's go!\" Jason repeated in a hoarse whisper.\n\nSlash looked at the others' wounds. Silver wounds healed slowly. Their hands had been made useless as weapons until they could heal.\n\nThat's why Ford had been cocky; he knew they'd been bound and weakened.\n\nThey walked to the back of the van just as two men hopped inside. It was on him. Slash was comparatively uninjured. He could take them.\n\nHis nostrils flared. One Were\u2014a packmaster, and one human.\n\nSlash charged forward without anything to indicate his intent.\n\nUnfortunately, Adrianna was closest to Tom Harriet, and he jerked her to him by the hair. Utterly unprepared, she screamed inside the truck from the brutality of the move.\n\nIt froze Slash in his tracks.\n\nHe had changed to his half-wolfen form as he moved from the sound of her terror alone. Harriet tsked-tsked him, shaking Adrianna, and she bit her lip to keep from crying out again. Slash growled just as her teeth met around Harriet's hand.\n\nTom grunted, backhanding her with his free hand. \"Willful bitch!\" he bellowed in pain.\n\nAdrianna slumped to the ground and Slash lunged at the Alpha that would dare injure a female under his protection.\n\nMine.\n\n# CHAPTER SIX\n\nTharell tried not to dwell on deceiving Julia. The guilt would be more powerful than he could stand.\n\nMany years had passed since he had come across one as compassionate as she was. The Blooded Queen had promised to consider gifting her people if they were willing.\n\nTharell did not think Faerie had time to wait on her decision. Other matters outside the scope of Faerie were part of his thought processes. It made his head ache like a rotten tooth.\n\n\"Enough,\" Tharell interrupted their arguing in a quiet voice of authority. It did not boom, but nearly so. Every gaze at Region One landed on him.\n\nMichael made a little motion with his arms like flapping wings. \"What? Are you going to lay more Faerie dust on us? Zap us all into comas again. Pfft... loved that noise.\"\n\n\"Michael,\" Marcus warned. \"Let him speak.\"\n\n\"Fine.\" Michael lounged up against the wall, mauling a sucker in his mouth.\n\n\"I will not lay down the weapons of the fey, nor will I use them against you in the time of our alliance.\"\n\n\"Very precisely worded,\" Delilah, Jacqueline's daughter, said.\n\nTharell nodded. \"We are a precise people.\"\n\n\"Fey are not human,\" Jen said. All her brothers attended the meeting, and Tharell looked at each one.\n\n\"Nor are you. We are humanoid. We share similarities with humankind. We can breed with them, but they are the lesser evolved of us all.\" Tharell lifted his chin and went on, \"Your Singer Combatant was severely lacking in their protection of Julia Caldwell.\"\n\nScott moved forward.\n\n\"Rest easy, Singer warrior; the Red Were are as diabolical planners as any I have ever known. We postulate they move toward reuniting all their kind. To have it as it was before the Americas became what they are today. Before my kind came from Europe.\"\n\nHe treaded more softly, though it did not come naturally. \"Domiatri and I have brought the prisoners with us. They will be under lock and key of the magick of the fey for the duration of the journey.\"\n\nJacqueline and Anthony.\n\nThey were heathens. And though Tharell understood the loathsome back-story to the female monarch of Region Two, it was difficult to hear the reports of the pain she allowed meted from her mate, a sadistic Were.\n\nBesides the rutting that they began and ended, upon which Tharell came in the middle of most consistently, whispers of rape and perversions had reached his ears. He flicked his gaze to the cuffed Jacqueline, her once lovely, dark eyes shadowed with smudges. Bruises in sizes that only fingerprints could leave, covered every place the eye could behold.\n\nAnd ones Tharell could not see.\n\nShe was proven evil, yet it turned his guts to see any female ill-used. And that she possessed the blood of the fey? More so.\n\nHowever, in the end, it was a Singer matter. As was getting their consent for her journey with him and Domiatri as well.\n\nTharell studied Marcus take in the proof of her injuries, both old and new.\n\n\"If Jacqueline goes with the fey, then I will, too. We're mated, she can't go without me,\" Tony said.\n\nTharell expected him to club his own chest like the purported caveman of eons ago. When he did, giving himself a stout punch in the chest, Tharell laughed aloud, and Tony's gaze narrowed upon him. \"What are you laughing at, you effing grape?\"\n\nTharell's smile died out like a flame starved of oxygen.\n\n\"You will not come with us. You do not have Faerie blood. You are an insubordinate, criminal Were.\"\n\nJacqueline said nothing, head bowed, eyes glazed. Jen, the Singer leader's daughter approached the apparently cowed Jacqueline.\n\n\"Hey, Jackie-baby... what are ya doin'?\"\n\nWhen Jacqueline didn't rise to the dig, Jen looked around, her gaze coming to rest on Tharell. \"What did you do to her? Not like I give a shit, but it's weird. She's so... subdued.\"\n\nThe female needed to come with them. Period. She had married the Were, a proven rapist and abuser of women from all reports. A bad choice\u2014albeit her own.\n\nTharell looked down for a moment.\n\nCould they have left the Singer female vulnerable inside her soft prison with the Were? Her powers oppressed as they are by fey magick? His gaze snapped to her form, seeing the sunken shoulders, thin frame, and beaten countenance.\n\n\"Nothing's wrong,\" Tony said defensively. \"Jacqueline's great. Right, honey?\" Butter could have melted on his tongue. He darted his tongue out and gave Jacqueline a long lick on her neck.\n\nShe flinched away from him.\n\nJen frowned, folding her arms. Scott went to his biological mother. The woman responsible for attempting to poison Julia.\n\n\"Get Cyrus,\" Scott said. The Combatant's gaze moved over Jacqueline.\n\nJen left the room.\n\nScott's accusation lay within the black depths of his eyes. \"I thought you said she would be safe?\"\n\nDomi gave Tharell a cautionary look. Tharell did not need the warning. Caution had been beaten into him. Literally. \"She is alive. You insisted the Were and Singer be held together. They chose to mate. Your leader\"\u2014he swung a hand at a troubled Marcus\u2014\"attended the ceremony. Each stipulation was followed\"\u2014Tharell grappled with the idiom most apt\u2014\"to the letter,\" he finished.\n\nCyrus, the Singer healer, rushed into the room, his shock of white hair standing on end.\n\nJacqueline, the subject of belated concern, toppled over like flower whose stem has been cut.\n\nScott caught her.\n\nTharell watched it all indifferently. Tony's face, indomitably etched with guilt and fear, told so much.\n\nTharell closely held the secret of why she would be wont to faint.\n\nScott settled her carefully on the floor. The Healer came forward, his hand hovering at her forehead and gliding down her body in a near miss of flesh kissing flesh. He did not touch her but hesitated near her pelvic region.\n\n\"Fractured pelvis,\" he said, his voice without intonation. His hand went lower, paused at the ankles. \"Contusions healed...\" His eyes flicked to Tharell's and narrowed into a glare. \"And fresh.\"\n\nTharell had done exactly as he had stated he would. No more and certainly no less.\n\n\"Dehydration, exhaustion...\" His hand returned to her lower belly, concave beneath her clothes. He lowered his palm and finally touched Jacqueline. She gave a pathetic moan and tried to bat his hand away, her mewls of fear so horrible to witness Jen raised a hand to her mouth, and she gave her father a raw look of horror.\n\n\"She is pregnant. But with this pattern of abuse, she might not keep the...\" Cyrus' gaze moved to Tony, whose gloating sneer hung from his face. \"Whelp,\" Cyrus finished softly, removing his hand slowly.\n\nScott stood, his hands balled into fists.\n\n\"Don't,\" a soft voice said from behind him.\n\nAll eyes turned as Lacey Greene entered the room.\n\nTony broke the bloated silence. \"Just like old times.\"\n\nScott hit him so hard he dropped to the ground, out cold.\n\nTharell watched it all, and then his gaze came to Jacqueline. She was the key. Without her, they could not journey after Julia.\n\nThere were too many challenges to a fey this far from Faerie. She would need to be with them at all times.\n\nShe would serve as buffer.\n\nBut first, she must get well.\n\nTime was the enemy.\n\n# CHAPTER SEVEN\n\nJulia got out of the way as a Red Were shot his fist into the scarred Were's mouth.\n\nIt should have ended the charge. It didn't.\n\nHis speed in half-wolfen form mesmerized her. He ducked the shot, and bony talons burst his fingertips. He didn't stab the Were but slashed behind and in front of himself in a spin of limbs, like the hands of a clock moving in opposite directions.\n\nBlood rose to the surface of the tall man who had come from behind Slash and on the chest of the Were that he fought now.\n\nJulia gulped when Truman, looking far more at home as a Were than he ever had as a cop, scooped up the human male and tossed him casually outside the back of the van.\n\nJulia took Adi's hand, and she gently snatched it away, shaking her head. Julia looked down at the deep cuts from the silver and nodded.\n\nIt was just one Were now against Truman and Slash.\n\nThen he changed, howling.\n\nThe gunk of his human existence splattered the inside of the box van, landing in jellified lumps that tumbled and slid down the walls to land as disgusting, jiggling dollops on the floor.\n\nIt was raining human gore inside the truck, and Julia rushed to the back to get out, Cyn at her heels.\n\nA werewolf launched himself inside as they tried to exit.\n\nJason flew at him, their wolves colliding midair.\n\nJulia didn't stick around. Her small telekinetic ability and aura reading would not protect them from a captor gone crazy.\n\nShe slid out backwards while Truman and Slash's tails gave an ominous flick as they cornered the one Were in the truck.\n\n\"Let's get out of here. They can find us once they shred him,\" Adi said, and Julia nodded.\n\n\"That's a no-shitter,\" Cyn agreed.\n\nThey used the metal handle on the sides of the truck and slid down the back.\n\nThey turned to run and Lily blocked their path.\n\n\"Hi, bitches,\" she said in a conversational tone and threw something at them.\n\nJulia watched it the way one would when deeply startled mixed with fear. She couldn't galvanize herself into action.\n\nAdi could. \"More stupid fey mojo! Let's split!\"\n\nShe grabbed Julia's hand and she slumped.\n\nFully awake. Fully paralyzed.\n\nThe glitter fell, and the numbness spread from her core to her fingertips.\n\nAdi shook her head and growled low.\n\nCyn leapt at Lily, who laughed.\n\nShe drove in talons that erupted as she leapt. One minute she was Julia's bestie, with tattered but still coifed hair, clothes, and nails, and the next she was a semi-furred Were creature.\n\nHer talons burst through Lily's other side, her dark skin rupturing like a split plum. Guts spilled from the front, and sinew and flesh hung in twisted strings from the end of Cyn's razor-sharp tips.\n\n\"Heal that up, you interfering twat.\"\n\nJulia lay on the forest floor while the noise of distant cars whizzed by, and couldn't take her eyes off Cyn. Pale blonde fuzz covered her head to toe as Lily slid off her talons.\n\nShe put a hand, fingertips finished with short talons, against Lily and shoved her the rest of the way.\n\n\"Kill that fey jerk off,\" Adi gasped from beside Julia.\n\nCyn approached them, and her pendulous breasts swung, nothing hidden from view, even what proved her female. She was comical. Lily lay behind them, gasping, and Cyn put her dangerous hands, gore to her upper arms, on her slim hips.\n\nNaked.\n\n\"Kill her, Cyn,\" Adi croaked.\n\n\"Right,\" Cyn said, and it came out, Rawght. Like gravel crunching underneath tires.\n\nShe went back to Lily and calmly centered her foot above her skull.\n\nJulia couldn't watch. She turned her face away, instead looking at Adi. Who never took her eyes off Cyn.\n\nThe skull makes a distinctive sound when breaking. To Julia, it was like cabbage being squished mixed with dry timber breaking. She would have covered her ears had she been able to move.\n\nShe could breathe, but that was all.\n\nCyn took a look at herself, and her expression filled with horror.\n\nThere was just so much a small town girl from Alaska could take. Changing into a naked half-Were and bashing a Faerie's head in was probably a little much.\n\nCyn turned to the side and threw up.\n\nThe smell wafted to Julia, and she helplessly lay there praying she wouldn't get sick and choke on her own vomit before the cavalry showed up.\n\nInstead of Jason, Truman, and Slash coming to their rescue, a contingent of ten Were surrounded the women.\n\nCyn changed back as she lay naked beside her own vomit.\n\nAdi struggled to shake off the newest Faerie dust Lily had laid on them.\n\nJulia couldn't move to fight off the Were whose snout appeared alongside her neck and took a long, pulling inhale of her scent.\n\n\"This is her,\" he said with a nod.\n\nJason growled behind her.\n\nJulia couldn't blink. Tears came anyway.\n\nThere were simply too many Were.\n\nThe Were who'd been inside the truck had healed the damage Julia's friends had laid on them.\n\nTen more had come.\n\nIt hadn't been escape for them, but a rendezvous for the Reds.\n\n*\n\n\"Careful,\" the Were that held Julia said to the one she couldn't see, who walked beside them.\n\nJulia could hear the train but not see it. Where there was a train, there was roads.\n\nShe moved a toe. It was a fantastic accomplishment. She also needed to itch her nose so badly she couldn't think of anything else.\n\nCigarette smoke clogged her nose, and she sneezed.\n\nGreen eyes met hers. They spun like Jason's and she blinked slowly, trying not to think about where he was.\n\n\"You're awake,\" the Were stated softly.\n\nJulia said nothing. Her mouth wasn't working right. Just the sneeze. Her paralysis was lifting... slowly.\n\nA second Were filled her vision. His eyes were mud struck through with moss.\n\nThe smoke thickened as she choked and coughed.\n\nA hand lashed out and the other Were went flying.\n\nTo have that violence directed so close terrified her.\n\n\"I won't hurt you,\" the Were said.\n\nSure.\n\nHe turned his face away and a snout grew, lips pulling back from teeth no longer human. The unseen Were didn't approach again.\n\n\"When I say be careful, that means not smoking next to the Blooded Queen.\"\n\nA yip reached her ears, and the Were's features bled to human again. Julia could only watch as things that should never shift looked like small moles burrowing underneath the flesh of his face, realigning his bone, muscle, and arrangement of features within human norms once again.\n\nJulia blinked, and he was a man again.\n\nThey'd been standing outside a freight car, only the roof's lip visible to her from being held in arms rather than standing on her feet.\n\nSuddenly, they were airborne, and she yelped.\n\nThey landed with a bouncing thud inside the car.\n\nHe laid her gently on something soft. Julia had never wanted to move as badly as she did right then.\n\nShe was now facing everyone with whom she'd been taken.\n\nJason, Truman, and Slash were badly beaten. Julia's eyes took in a bubble of bloody mucous trying to boil out Truman's nostril, and a deep, abiding rage welled up inside her.\n\nIt felt like something she could use.\n\nWhen Lily came into sight, looking perfect, that anger burst from her like a star escaping her chest.\n\nLily's smug look of satisfaction changed instantly to terror.\n\nIt was a supernova of power. Julia's new powers were ever evolving. She knew this. As Rare One, they'd told her they'd come randomly.\n\nSometimes random worked.\n\nBright, searing light cut through the murky car. It illuminated things she didn't wish to see, the injuries of the males who'd tried to protect them, the new contingent of Were, a tightly bound Cyn and Adi.\n\nThe source of the energy was she. It threw off the paralysis like a snake shedding its skin, and Julia sat up. Awake, energized, and whole again.\n\nLily waded through the light, her face distorted in pain.\n\n\"Julia,\" she began.\n\nJulia thought she might be trying to plead her case. Julia would never listen. She was done negotiating with anyone who mistreated her people, herself and lied to do it.\n\nShe would not allow it.\n\nJulia could control the light. She used it like a sword at Lily, who stopped looking black and began to glow. She screamed, high and tight. Lily's skin flaked into large chunks that floated like singed paper coming close to the mouth of the sliding train car door, they were sucked out like captured bugs.\n\nLily's mouth was the last thing to succumb to the combustion of her body in a yawning inferno of heat and molten consumption.\n\nHer teeth made almost no noise as they fell on top of the pile of ash that had been her body.\n\nJulia didn't pause over her remains but found her friends instead.\n\nHer husband.\n\nHis eyes met hers. One was swollen shut but the fine, clear hazel of the other was sure.\n\nOf her.\n\nThe light changed, becoming iridescent. It sparkled and fell like moving glitter on the wind. It cloaked the Red Were, healing Jason, Truman, Slash, and even Cyn and Adi of their minor injuries.\n\nUnfortunately, it invigorated the Red Were who had taken Julia.\n\nWhen he hit her from behind, the light seeped out of existence and the Were in charge of the reacquisition captured her as she fell.\n\nThe last thing she heard was Jason.\n\nHowling.\n\n# CHAPTER EIGHT\n\n\"I am Carell,\" the Red Were told Slash and the others.\n\nSlash looked around, taking stock of everyone. Tom Harriet, Ford, and a man named Tai Simon stood in the background behind ten other Reds.\n\nThey made his wolf's hackles rise.\n\nThe lore had spoken of the Reds' blood as legendary. Slash had listened to the stories but never put much merit in the validity of a sub-species of Were being any different, though he'd never pressed the secret knowledge he held. With this many Reds, their energy swam in the air, thick enough to reach out and touch. Slash felt like he could stick out his tongue and taste the difference.\n\nInstead, he looked at ways to kill them all. Adrianna was unprotected in a den of Reds. At the end of the day, he was left with that.\n\nThe Reds were very aware of her Alpha female status, and Slash keenly aware of their interest.\n\n\"Carell?\" Jason asked and distracted the leader.\n\nThe Were who had hit the Rare One, stopping her spectral light, inclined his head at the one whose smells mixed with hers but not in the way of mating.\n\n\"Let us go now, and we won't kill you all.\"\n\nCarell threw his head back and laughed from his belly. When his eye found Jason again, Slash noticed they'd washed to green.\n\nThe Were collectively shifted within the hurtling boxcar.\n\n\"She's too powerful for you to manipulate,\" Slash added. Caldwell, a rash young wolf, needed the advice. He did not understand that telling the enemy the plan was akin to slitting one's own throat.\n\nCarell cut his own growl short, swiveling his head to Slash. Slash tried to shrug his shoulders but was bound too tightly. The silver burned around his wrists. Seemed this group was a little smarter than the Were posing as Feds.\n\nCarell shook his head. \"No, her very proximity will give us what we need. If there be any blood quantum, her presence will see it rise to the surface like cream through milk.\"\n\nTruman sighed, scrubbing his face. \"You guys don't get it. Julia Caldwell is not a pawn to be moved around your chessboard. There is no winner. The only \u02bbwin' in this whole snafu is letting her unite the species. And now there's fey involvement.\"\n\n\"Was!\" Cyn said loudly, holding a finger up.\n\n\"You bashed in her head, but she came alive again...\"\n\n\"Like a zombie?\" Cyn asked Adrianna.\n\nSlash smiled despite the circumstance. Adi seemed to consider the absurd idea of reanimated humans. \"Yup, but Jules cooked her, and now she's a lump of ash somewhere.\"\n\n\"I like it.\"\n\n\"Stop this,\" Carell commanded, his eyes on fire at the two females.\n\nCyn grinned. Raising both hands, she flipped him off and Slash groaned. She was so newly turned that he didn't know if she was Alpha, but he would bet money she was. Because his luck ran that way.\n\nDownhill.\n\nCarell's eyes turned to slits as he strode to her. She was unarmed and unbound.\n\nCyn stood her ground. Slash felt the others tense.\n\n\"You defy me, female?\" Carell asked.\n\nCyn blew a stray blonde hair out of her face. \"Yeah.\" That one word said much. All of it disparaging.\n\nHe raised his hand to strike her, and Slash realized he'd taken a step toward them.\n\n\"That's what it comes down to for you guys. You can't engage me verbally, so you have to result to violence. Pathetic.\"\n\nCarell stood still, his fist frozen in the air.\n\n\"She is right. We have too few females to abuse one, even with the mouth she has on her.\"\n\nSlash checked his fists that he couldn't use. His frustration was its own demon.\n\n\"The voice of reason... fuck you very much,\" Cyn said. The other Were grabbed her by the throat, hauling her up against the side of the rail car and punching her so hard against the wall it shuddered.\n\n\"Just because a female is rare doesn't mean we will not discipline ones who need it.\"\n\nCyn's face was turning purple.\n\nTruman and Caldwell inched closer.\n\nThings were escalating inside the tight space. It wouldn't end well. Experience taught him that.\n\nAfter confirming Adrianna's safety, he'd been so busy keeping the most aggressive Were within sight that he lost her.\n\nShe sprang at the Were who held Cyn. Talons burst out before she leapt.\n\nAdrianna sunk five into his groin, and with a high-pitched keening, Carell's right hand wolf toppled like a felled tree, clutching at his gonads as the floor rose to meet him.\n\nThe Were closed in around Adrianna, her arms in front of Cyn, shielding her from their approach. Her eyes had yielded to her wolf long ago. A stub snout caught their scents, a prelude to their actions if she was Alpha enough to scent their intent.\n\nOne Alpha female against ten male Were. Unwinnable numbers.\n\nIf Slash had been unsure of his love for her before, nothing could have solidified it more than that moment.\n\nSpinning gold eyes met the green of his. His scent triggers sharpened, talons sliding out of skin in a painful tear of suddenness.\n\nShe winked.\n\nSlash's wolf decided for him. Slash ruptured his skin, revealing what he was and giving up his biggest secret in a moment of raw desperation.\n\nHe was not mixed, though it had been clever camouflage.\n\nHe was pure Red and one of the last. His position in Washington State had not been happenstance but providence.\n\nProtection of his rarity.\n\nHis wolf shucked the silver binds, his vision bleeding to the many gray shades of transition. His sense of smell let him bring every last one of the males under his dominance.\n\nInstantly. No Red of lesser than pure blood would stand against his call to their wolves.\n\nCarell fell last, convulsing on the floor as Slash tore the wolf from his body, two weeks from the moon.\n\nShin deep in the gunk that had made them human, he trotted over to Adrianna, his paws sinking into the human castoffs.\n\nShe ran her gaze over his wolf, and even as deep as he was inside the wildness that made him Were, his spirit hesitated. It was a testimony to his deep-seated self-hate.\n\nAdi bent down, but not much. She ran her finger along the scar that tore up his snout and bisected the brow ridge of his wolf. He did not flinch, though it was an effort not to.\n\nAdi looked around at the partially changed Reds, many only half bloods. Her gaze returned to his.\n\n\"You did this.\"\n\nHe gave an awkward nod, and a smile lit her face, her talons retracting.\n\nSlash could not bring females. They were immune to the summons. There would be no whelps if the female changed to their wolf during pregnancy.\n\nHe licked her hand and she wrapped her arms around his neck.\n\n\"You sure know how to impress a girl,\" she said.\n\nSlash's chest tightened, his humanness, what little remained, strung so taut it could have broken from the merest pluck.\n\nSlash forced himself into calmness. It was the only way to change back so he could tell her.\n\nSo he could break every last rule he'd vowed to uphold. Including the one about no mates from their sister dens.\n\nHe'd fucked that six ways to Sunday.\n\nAdrianna let him go when his energy changed.\n\nAnd there, in the middle of human blood, skin and sinew, Slash rose from the remnants.\n\nTall, steady and scarred. She brought him low. A slip of a girl barely now a woman.\n\nAdrianna's eyes grew round and Slash cocked an eyebrow. He had everything under control. The enemy Reds were under his command and he'd brought a volatile situation to heel, as it were.\n\nWhy, for moon's sake, is Adrianna looking at me like that?\n\nThen he became aware of where she was looking. And it wasn't his eyes, but someplace lower.\n\nSlash had forgotten his nudity.\n\n# CHAPTER NINE\n\nTharell remained impassive while the Combatant dropped the renegade Were like a sack of potatoes.\n\nAnthony Laurent grinned up at Scott. Thin red lines made his teeth look like peppermint candy inside his leering mouth, but no sweetness filled it.\n\nHe spit the blood out on the floor. \"She was a good lay, fought the entire time.\"\n\nScott slammed Tony's head into the wood floor with the flat of his palm.\n\nTony rolled over, chortling. \"You cannot kill me, Singer.\"\n\nHis wolf's eyes swam to black, obsidian holes that darkened his gaze while he looked at Scott.\n\n\"How could you... rape her and get her with child?\" Scott asked incredulously.\n\nTharell let things play out a little longer.\n\n\"Why do you give a ripe shit? The real question is: why do you care?\" Tony stood, and they squared off. \"I can't hammer sense into you, because they'll beat the shit outta me.\" He jerked his thumb behind him where Victor and the others of the Combatant lay in wait. \"But keep this in mind, chump. This is your bio-mom. Y'know, the one who poisoned Julia? The reason I was in that fey prison. She wanted all that I gave her...\"\n\nTharell made a disgusted noise when the repugnant Were grabbed his crotch.\n\nTony's gaze fell on Lacey Greene. His smile morphed into a grin. Shark like teeth gleamed white in his red mouth. He wiped the blood away with his shirtsleeve. The smear remained like a partially erased wound on the white sleeve.\n\nShe gave a small smile back, and Tony's faltered.\n\n\"It cannot be his,\" Lacey stated, lifting her chin in defiance. Tharell looked at the small female Were. Her light gray eyes dilated with excitement if one was noticing.\n\nTony tensed. \"Of course it's mine, you numb cunt.\"\n\nScott backhanded Tony so fast that it was a flash of skin in the air, Tony's cheek opening like a split peach.\n\n\"Fuck!\" Tony howled, and the Combatant circled him.\n\nMichael and Brendan held Alan. \"Let Scott clean his engine, man. Chill.\"\n\nAlan growled low in his throat.\n\nTharell shifted his weight.\n\n\"Tell the Were, Lacey.\" Everyone's eyes were on Tharell. Tony's narrowed to inky slits.\n\n\"What the hell is going on? She\"\u2014Tony gestured at Lacey\u2014\"is not part of dick. She got knocked up when I put my package where it belonged.\"\n\nAlan met this pronouncement with another low growl.\n\nTony ignored him, his gaze landing on Jacqueline, awake and sitting up against the back of a bona fide fainting couch.\n\nHe pointed his finger at the monarch of Region Two. \"She's going to pop out my whelp and there's not a damn thing you or anyone can do about it.\" His smug gaze, defiant and arrogant, swung out over the crowd.\n\nTharell gazed at the Were and almost felt sorry for him.\n\nAlmost.\n\n\"Domiatri,\" Tharell prompted quietly.\n\nDomi stepped forward. His deep blue hair shone like suspended water, waves held by his scalp instead of a shore.\n\nJacqueline flicked nervous eyes to him and Tharell nodded toward his pure-blood Sidhe companion.\n\n\"What is that expression?\" Tharell asked to no one in particular. He made a show of snapping his fingers. \"Ah, yes, Domiatri has the floor.\" Tharell swept his hand toward the fey.\n\n\"I am the father,\" Domi said without preamble.\n\nThe silence was heavy.\n\nJacqueline closed her eyes.\n\n\"Bullshit,\" Tony erupted. \"I humped her into the ground every chance I got. I brought that bitch to heel.\" Tony's low voice resonated, his open fingers clenched with certainty.\n\nScott's hands became fists of punishment.\n\n\"You cannot breed, Anthony Laurent,\" Lacey announced. \"Have you looked at your own body lately?\"\n\n\"There are deeper magicks then even you are aware,\" Domi said. \"If you had been an intellect, you would have thought before you abused one of your own kind. But therein lays the crux of the problem. You are not an intellect; you do not think but react. So you assaulted a rare female Alpha Were. She begat a Moon Warrior offspring. Those taken in violence leave a mark of retribution behind.\"\n\nReagan, Tony and Lacey's daughter, moved forward. \"Yes, Dad,\" she said, and he swiveled his head to her, scowling.\n\n\"Check it out,\" Reagan said.\n\nTony's murderous eyes found Jacqueline, and she gave a tired smirk. \"I have done what I was bid. I exchanged the horror of being with you for something I want more.\"\n\nCyrus retreated into the background. \"I assumed\"\u2014he looked from Tony to Jacqueline\u2014\"that the Were was the father. She is with child.\"\n\nLacey continued, \"He should have a mark on his hip that tells of his sterility. It can't keep him from assaulting women, but his seed will never take shape or live. He is destined to die out eventually. Reagan is the sole offspring.\"\n\nTony's terrible expression of realization silenced them. Sentenced to many centuries' existence, knowing you had a forever end, was an awful thing in the supernatural realm. To understand that end resulted because of horrible, irrevocable choices made was somehow much worse.\n\nTony lunged toward Jacqueline, and she shrank back against the couch.\n\nDomi met him with his broad sword half the length of the Sidhe's legs. He brought it in a swiveling arc, turning it into Tony's deepest part. His guts evacuated the hole in his body like a fire hose.\n\nTony fell, gathering the parts of him that sat around like wet worms in the throes of death.\n\n\"Domi,\" Tharell chided as though Domiatri had slapped him instead of gutted.\n\nHis silver eyes glittered with his anger. \"Let me have some recompense for allowing that brute to hurt her.\"\n\nThe group looked at Jacqueline. She glanced away.\n\n\"Wait a second.\" Jen stepped over Tony's guts with a lip lift of distaste. \"You had sex with this... viper? Then you let him brutalize her? I'm sort of confused.\"\n\nDomi nodded and shrugged. \"We are the fey. We don't look at sex as a terrible thing unless it is not consensual.\"\n\nJen made a noise in her throat. \"I bet.\" She looked at Jacqueline. \"I don't like Jacqueline. She went after Jules, she's slept with the world to further her power base...\"\n\nDomi held his hand up. \"A moot point. Who she has sex with is not relevant to this discussion.\"\n\n\"It so is,\" Michael said with a snort, popping his sucker out and pointing it at the tall emerald Sidhe.\n\nSingers. Tharell shook his head. \"A human point of morality. It is not one the fey ascribe to.\"\n\n\"But she didn't have consensual sex with Tony. Clearly.\" Jen pointed at the rainbow of healing bruises on Jacqueline's body.\n\n\"In the beginning, Jacqueline thought to go along with his plan to mate with her and drive us crazy with their disgusting antics,\" Tharell said.\n\n\"Bad plan,\" Brendan stated.\n\nTharell puzzled that out, finally nodding. \"Yes, as ideas go, it was not a good one.\"\n\n\"We used a simple slumber spell on the Were, and while he slept, we outlined Jacqueline's options.\" Domi shrugged his shoulders. The light from the window reflected against the bright green of his skin, as though the grass from outside had leapt onto his flesh.\n\nShe dropped her gaze from his. A smile ghosted his lips. \"She was receptive.\"\n\n\"What options?\" Marcus spoke up for the first time.\n\nDomi turned his attention to the leader of Region One.\n\n\"She must serve her time in the prison and accept Tony's attentions until the quarter had passed in which she was sequestered. By ancient law, if she were to become pregnant with a child of a pure-blood Sidhe, she could remain in Faerie and become well again.\"\n\n\"Well?\" Marcus asked.\n\nTharell caught Domi nodding in tandem in his peripheral vision. He inclined his head toward the other warrior. After all, Tharell could not breed. Only the pureblood Sidhe could.\n\n\"That's right; none of you understand the fey. You thought we were a pretty legend.\"\n\n\"Not so much now.\" Jen looked at Tony struggling to heal. He collected his guts and tried to stuff them back inside his body before his wounds closed.\n\nDomi ignored her, continuing, \"Fey become weaker away from Faerie. The more blood of the fey one possesses roughly translates into an exponential weakening as we travel further away. Tharell is a half-breed.\"\n\nA part of Tharell still bled upon hearing those words, even from Domi's mouth.\n\n\"Jacqueline even less. She carries my child. That one thing will allow me travel as if I were a human. The unborn child acts as a neutralizer of sorts. A negation of my system's weakening against Faerie's lack of proximity.\"\n\n\"So,\" Jen began.\n\nReagan silenced her with a finger across her neck and spoke in her stead. \"You got the Region Two leader with child for the express purpose of going after Julia?\" She scrunched up her face.\n\n\"Lies,\" Tony seethed, breathing heavily as he held his guts inside his healing body.\n\nTharell noticed the pulsing nest of intestines, clearly seen through the thin covering of skin rearranging and healing. More skin filled in as he watched, obscuring the grotesque reconfiguration.\n\nHe shouldn't have let Domi have his fun with the Were. It had wasted time.\n\n\"Well.\" Domi paused, tapping his chin with a finger. \"We only just discovered Julia has been taken. We sweetened the pot.\" He swiveled his face to Tharell's. He nodded. Yes, that was the correct idiom.\n\n\"Once we discovered I would need to fetch the wayward queen of the blood....\"\n\n\"Essentially, Jacqueline could mate with you for favors? And now it's freedom?\" Reagan interrupted.\n\nTharell thought they might finally understand what was at stake. \"Jacqueline will be our buffer during travel to Alaska where we will acquire the Rare One. Then, upon her return, she will return to Faerie, never to be seen again.\"\n\nHe did not understand the sudden concern over Jacqueline. The Singers themselves had come to the fey and made arrangements for her imprisonment. Tharell was puzzled they would fight for her or care about her eventual end.\n\n\"Dad,\" Jen implored, \"I can't stand Jacqueline, but this green guy let Tony go criminal on her ass and stood by as it happened. As long as his precious agenda took shape, she was collateral damage.\"\n\nDomi made a noise in his throat and stepped forward. Tharell held his chuckle at the \"green guy\" reference. If the Singer only knew how insulting she was. He sighed. Compared to the ancient fey, they were an infant species, regardless of how they viewed themselves.\n\n\"You are wrong, Singer.\" Only Domi's tight grip on the hilt of his full-gored sword let Tharell know how the exchange irritated him. \"I did not wish her harm. Do you think I could be intimate with a female and stand by and easily watch her abuse?\"\n\nThe young Singer studied Domi. \"I think so. You fey dudes, you're heavy into the \"whatever makes it work\" philosophy.\"\n\nDomi's red lips thinned like a slash of blood in his bright green face.\n\nMarcus sighed. \"I don't agree with how Jacqueline was kept, the abuse from Tony. But\"\u2014He leveled his gaze on all of them, finally landing on Tharell\u2014\"she meant to kill Julia, our Rare One. We cannot have Jacqueline as leader in any capacity.\" Marcus paused as he seemed to consider his next words carefully. \"If you promise she will not be ill-treated in Faerie, it could mean a burden of consequence is lifted from my shoulders.\"\n\nScott folded his arms and looked first at Tony, then Jacqueline, and finally Tharell. \"You won't let this happen again? What guarantee do we have?\"\n\nDomi shook his head. \"She is mated to the Were but carries my child within her.\"\n\n\"Not good enough,\" Scott stated in a bald voice. \"It didn't matter before.\"\n\nDomi glowered at Scott, his eyes flashing silver fire. \"She has most recently got with child.\"\n\n\"And,\" Tharell said, avoiding the escalation of violent potential, \"she will stop being crazy once she lives in Faerie.\"\n\n\"That's why she's such a raging bitch?\" Michael asked. They looked at Jacqueline, her pale cheeks sunken. Her exhaustion was so great, she'd fallen asleep as they discussed her. Tharell noted Domi's new position above where she lay, long fingers gripping the heavily carved wood that ran the perimeter of the couch.\n\nDelilah, who'd been listening to this entire interchange but remained silent until now, asked, \"So she's insane? Why?\"\n\nTharell gave the answer easily. \"Too much Faerie blood.\"\n\nIf a human possessed too much fey blood, such profound wanderlust would strike them that they would feel uprooted the rest of their natural lives, their very being crying out for the sithen. That perfect sanctuary and edification only Faerie could offer to those who held their ancestry.\n\nIf a supernatural possessed enough blood, their mind would slip with the want to be in Faerie.\n\nJacqueline had lived centuries without Faerie.\n\nShe was Singer, vamp, and Were enough to be distracted by the parts of her that were other.\n\nAnd fey enough to be driven insane by that missing chunk of her lineage. Whole only as long as she lived in Faerie.\n\nJacqueline would finally be coming home.\n\n# CHAPTER TEN\n\n\"We've got about a half-hour lead,\" Slash told the group. Julia's head felt like a crushed watermelon. Whatever that Were had used had scrambled her brains, but her survival instinct was still on board. She was smart enough to understand she was hurt too badly to move.\n\nJason, free of the silver bindings, came to her side. \"Jules.\" He touched her head, and she winced. \"That fucker,\" he whispered, his gaze rolling around the confines of the rail car.\n\n\"Don't,\" Julia told him. \"We don't have the time.\"\n\nShe followed his gaze to the colorful blur of the landscape as it rushed by.\n\nShe could see from his face the frustration she shared.\n\nJason gave a miserable chuckle. \"What? We gonna toss the girls out the side door and hope they land in one piece?\" Jason's palm indicated Julia. \"She's hurt. She'll never make it.\"\n\nHe stroked the hair back from her face. \"Can't you get a talent like 'floater'?\" Jason asked, trying to lighten the mess of their circumstance.\n\n\"Sounds like a turd,\" Adi said, as Slash was busy trying to cover his nakedness.\n\nJulia flicked her eyes to Slash, too hurt for laughter. He'd managed to steal a pair of pants off a Were, one of the enemy Reds now coming to without the benefit of jeans.\n\nThey were ill fitting, short, and tight, but better than naked, Julia supposed.\n\n\"Commando?\" Cyn asked with interest.\n\n\"God\u2014can it, Cyn, we've got to take a leap!\" Adi said, her hands on her hips. Julia thought there might be a twinkle of jealousy in that gaze.\n\nCyn didn't smile. Humor had always been her defense. Instead, she moved to the open sidecar. \"We can't. We'll break every bone in our bodies.\"\n\nA low groan sounded behind them.\n\n\"You'll have to change,\" Slash said with urgency.\n\nJason helped Julia to her feet. She swayed. \"I think I'm going to be sick,\" she whispered.\n\nJason helped her to the door, gripping her waist while she heaved out the side of the train.\n\nHer vomit hit the walls of the box car and ran down the side. Julia closed her eyes as a roaring pain lit into her skull like claws of pain.\n\n\"I can fix you,\" Cyn said, standing behind her.\n\nJulia didn't respond and wiped her mouth on the sleeve of her shirt. Cyn wove her cool hand between her and Jason's bodies, coming to rest against her nape. Icy, insistent energy leaked through Cyn's touch.\n\nThe headache receded. The nausea stopped revolting.\n\nA sigh of pure gratitude escaped her.\n\n\"We need to go,\" Truman said. \"If we're doin' something, let's get hot.\"\n\nHe took Cyn's wrist, and she resisted.\n\n\"No.\" She shook her head and backed away.\n\nHis hold tightened, unapologetic, yet his eyes were sorry. \"No guts, no glory.\"\n\nJulia gasped as he grabbed Cyn's hand, Truman flinging himself out of the car and tumbling down the embankment in a changing ball of sinew, flesh, and fur.\n\n\"Shit,\" Adi said in a shaky voice, even as she moved into the line of Slash's body.\n\n\"Ready,\" his voice rumbled, and Adi said nothing.\n\nSlash swooped her up and threw her ahead of himself.\n\nJulia had never seen the wind snatch a body and throw it backwards.\n\nNor had she ever seen another follow it so closely.\n\nJulia bent her head, and Jason kissed the top of it.\n\n\"'Til death do us part,\" he whispered.\n\nYeah, it was the parting bit that Julia feared.\n\nThey leapt.\n\n*\n\nThe dreams where you fall forever and never land came alive for Julia. She was in a freefall without end.\n\nWhen the end came she felt crushed, every piece of breath stolen from her lungs in a tearing vacuum upon impact.\n\nShe looked up into Jason's half-wolfen eyes and nearly cried in relief. Instead, the emotion stayed stuffed in her throat. Julia couldn't even gasp.\n\nShe was suffocating. His grip bruising, the breaking of her fall into his arms had done something to her ribcage, and her shoulder was a yawning chasm of pain.\n\nIt dangled from her side, hanging unnaturally.\n\nMore eyes joined her husband's. Cyn's were a strange green-gold. Stars burst in front of her vision like fireworks.\n\n\"She's not breathing!\" Jason said in a concerned, staccato bray.\n\n\"Here.\" Cyn laid her hand on Julia's chest, and precious oxygen filled her lungs.\n\nShe took a breath and screamed from the pain. She could breathe now but didn't want to. Ever.\n\n\"Fuck! What the hell is this?\" Jason yelled.\n\n\"Put her down, Jace... ya hurt her from the catch.\"\n\nLooking at Jason's confused expression would have been funny. None of it translated well on his half-wolf form.\n\nBut there was no humor left; the pain was like hot, broken glass every time she breathed. Julia reached out to Cyn.\n\nThey gripped each other's forearms.\n\n\"Hang on, Jules.\" Cyn released Julia's arm and pressed both hands to either side of her ribs.\n\n\"Cyn\u2014don't, hurts!\" Julia wept.\n\nJason growled.\n\n\"Pipe down, stud, I'm fixing her.\"\n\nThe pain bled to an ache, the ache to discomfort.\n\nThe first breath without pain was a slice of heaven on earth. Julia opened her eyes. \"Thank you.\"\n\nCyn gave a grim nod. \"There's the shoulder now.\"\n\nJulia's head whipped to Slash, who was suddenly there holding her arm.\n\n\"You got it?\" Cyn asked.\n\n\"Got what?\" Julia asked.\n\nSlash turned her arm and Julia shrieked as he rotated it and shoved upward, slamming the joint back into place.\n\nShe must have blacked out for a moment, because when she came to, they were moving.\n\nShe was in Jason's arms again.\n\n*\n\nJulia and Cyn returned from their little trip to wipe their parts with moss, after taking a much-needed bathroom break, when Cyn piped up. \"I am not digging this camping in the woods while we run from the freaks mode.\"\n\nJulia laughed. \"Yeah, it's getting old for me, too.\"\n\n\"I want a hot shower,\" Adi said from behind. Julia and Cyn waited for her to catch up.\n\n\"Me three,\" Cyn lamented.\n\n\"We'll be home soon,\" Julia said, though that was almost too optimistic a viewpoint.\n\nAdi shook her head, her brown hair in wild shape from their journey. \"Nah, those Reds go hard in the paint.\"\n\n\"Basketball!\" Then Cyn's face fell. \"Kev used that expression before...\"\n\nJulia couldn't take the brave one of the two crumpling. \"Ah, Cyn, come here.\" Julia wrapped her arms around her friend while she cried.\n\nCyn cried for many things, and Julia knew exactly what they were. She'd shed her own tears long ago for the very same losses. She could bear Cyn's.\n\n\"I'm sorry, Jules.\" Cyn sniffed. \"I know you've had your own cross to bear. And I should be grateful. I have you and... Jace stopped being a dick hole.\"\n\nAdi barked out a laugh. \"Oh, I don't know, there's always time for regression...\" She winked.\n\n\"No.\" Julia held up her palm. \"Negatory. I don't want any more yo-yo love life. I'm a one-man woman.\"\n\n\"You are, are you?\" Jason loomed before them.\n\nCyn smacked him in the arm. \"Stop that, you lurker, I about peed myself again. Gawd.\"\n\nJason laughed and looked to Adi. She threw her palms up. \"I like surprises. Cyn doesn't have her wolf on yet, or she would have smelled you a mile away.\" Adi waggled her brows.\n\n\"Am I that rank?\" Jason took a whiff of his armpit.\n\nJulia laughed. \"I think we're all a little ripe right now.\"\n\n\"Showers later,\" Truman announced, stepping out from behind the trunk of a tall cedar tree. \"They're going to be up our asses.\"\n\n\"Nice visual,\" Adi said in a sour voice.\n\nJulia looked around, expecting a battalion of angry werewolves to meld suddenly out of the tree line. The guys had kept a little bit of original clothing, but the Were in her group were all in various states of half-naked. And for extra fun, the weather wasn't cooperating, a light drizzle beginning to fall.\n\nSlash stood in the middle of their group, his nose in the air. His nostrils flared once. \"They're coming. We need to run.\"\n\nHis gaze took in the three women then the men. \"The females slow us down...\"\n\n\"Hey!\" Adi said. Slash went to her, and if Julia had been Adi, she'd have backed up from that huge Were. A strike of pulverized flesh like a beaten rope bisected his face. He looked angry, hard, and vicious.\n\nWhen he cupped the back of Adi's head, his expression was molten, revealing nothing and everything, utterly changing his face.\n\nHis love for Adi was plain, and Julia couldn't believe she'd missed it. Or maybe he'd been that good at hiding it.\n\n\"You know you have my greatest respect. But you are female, and they will take you. Trust in my protection.\"\n\nAdi's face flamed under his scrutiny, and he chucked her chin. \"You may bite me later for this. But for now, defer to me.\"\n\nShe glared at him.\n\n\"Please, Adrianna.\"\n\nThe smallest of the three women came into the circle of his arms.\n\nHis face grew tender with Adi against him. It completely changed the Were.\n\nMade him steely.\n\nWhere there was love, there was violence, the two inextricably linked in the supernatural world of which she was now a part of.\n\n# CHAPTER ELEVEN\n\n\"I don't know. She's malnourished and weak... I can't condone this. She shouldn't travel,\" Cyrus said.\n\n\"She is not this fragile.\" Tharell cupped his chin.\n\nPatches of deepening emerald spread over the typically aloof Domi's cheekbones. \"She is stronger than she appears.\"\n\nDomi's discomfiture amused Tharell. Only recently had Queen Darcel deigned to go forward with the acquisition of the Rare One. As self-deluded and grandiose as the rest of the pureblood Sidhe, her guts a churning mass of self-consumption, she had finally conceded the need for out breeding.\n\nWith Darcel gone, royal blood no longer remained in the courts, and a thin thread of advisers held the Sidhe together. One being the dragon-shifting Sidhe, Kiel. After the briar, he was their most stringent protector of Faerie and a logical replacement for the interim.\n\nA royal or someone of that caliber must take the throne. Subterfuge and other nefarious events could transpire in the interim. It was why he pressed this endeavor forward.\n\nHe must.\n\n\"Is there anything you can do to...\" Tharell spun his hand, and Cyrus followed the movement, swallowing hard.\n\nHe checked his impatience, his eyebrows rising.\n\n\"I can... bolster her immune system, but you must understand, she is with child, and the physical demand on the host is relentless.\"\n\n\"Like a parasite,\" Victor commented.\n\nTharell pivoted, hands loose by his side, ready. \"Speak your mind, Combatant.\"\n\nVictor smiled. Tharell knew the quality of it. A mere baring of teeth, like a wolf on a short leash.\n\n\"You have used one of our women, and I'm not...\" He seemed to think it over. \"A fan,\" he finished, the smile disappearing like it'd never been.\n\n\"None of you were worried when she was left in our care, so we entertain this line of commentary no more.\" Domi turned to Jacqueline, her dark eyes following the exchange but saying nothing.\n\n\"Nourish her, then we go,\" Tharell instructed. \"If there be any that think there is a better solution than the one I've brought forward, share it now. Or\"\u2014his gaze automatically went to the Combatant\u2014\"shut the pie hole underneath your noses.\"\n\nScott stepped forward, along with Lucius and the Feeler, Angela. \"I'm going, and Victor.\"\n\n\"We don't need another female.\" Tharell asked, \"It is true? You know the perpetrator from aura alone?\"\n\nShe nodded. Soft platinum hair curled around her shoulders, and pale blue eyes stared back into his without guile. She was telling the truth. He would have staked his immortal life on it.\n\n\"All right.\"\n\nTharell tried not to let Domi's intense gaze that followed Jacqueline disturb him. \"Domiatri,\" he called out.\n\nDomi hiked his eyebrows, moving his hand away from the hilt of his sword and knotting it with the other behind his back.\n\n\"Brother?\" he answered so automatically that Tharell held his smile in check, keeping the resolute and austere expression glued to his face like the mask it was.\n\n\"Let us convene outside.\"\n\nDomi inclined his head, and they walked out together under the watchful gaze of the Singers.\n\nThey moved down the broad steps without looking around them. Faerie's absence had not yet hindered their energy. However, Jacqueline was near.\n\n\"What troubles you, Tharell?\"\n\nTheir strides matched as they strolled, as did their height. Domi blended perfectly with the surrounding landscape. The green pastures stretched out before them. If not for the deep shock of long blue hair that carved a pathway of vibrant, deep oceanic blue between his shoulder blades, Domi would look as one with the rolling hills upon which they trod.\n\n\"You're becoming attached to the Singer, Domi,\" Tharell said.\n\nHe sighed, swiping a dexterous hand through his hair and smoothly banding it at his nape. He gave a frustrated exhale. \"Yes.\"\n\nAt least he conceded the problem.\n\n\"You know that she bears children. She will have to be shared.\"\n\nDomi was silent. They had come to a small lake, little more than a pond, where swans revolved in lazy, floating figure eights on the surface, paying them no mind.\n\n\"I do not wish to.\"\n\nDomi gazed at him with eyes that rivaled liquid mercury. \"You are the lucky one, Tharell.\"\n\nTharell held his tongue, Goddess knew he'd done it plenty of times before.\n\n\"You do not breed. You never will take a woman as mate.\"\n\nTharell had been with many females yet never to mate. A male had needs, a biological imperative that spared no species. Yet physicality was a manifestation of the body. His soul longed for edification as well. Tharell kept his own council.\n\n\"True,\" he said as the chasm of his loneliness grew with Domi's words. Not ill meant but accurate.\n\nOnly pure Sidhe could breed with the sanctioned Singers. It was a very good sign Jacqueline had gotten with child, though her mind was sick from the time away from Faerie.\n\n\"I ask you let me self-delude for this journey, Tharell. Soon enough, we will return to the sithen, the debauchery of the court and Jacqueline will become a brood mare for all that would have her.\" His eyes stayed on the lake while his heart bled silently between them.\n\n\"I will. Of course, you understand the importance of the Singers being cloudy on our motivations at present.\"\n\nDomi gave a solemn nod.\n\nThe day ended before them in a gasping loss of daylight, as though night had held its breath until that moment. And on its exhale, Domi's bright green skin deepening to emerald, his cobalt hair shedding its color and falling to night's influence.\n\nThe two fey stood shoulder to shoulder, two colored shadows, lost in their thoughts of choiceless deceit.\n\n# CHAPTER TWELVE\n\nJacqueline stretched, her stomach uncomfortably stuffed with food, and did everything she could to keep the food where it belonged.\n\nShe loathed leaving Faerie, where for the first time in her wretched existence, she felt free.\n\nFree in peace, centered\u2014right in her skin.\n\nThe shame had been acute. And Tony's brutality. When Jacqueline first accepted his idea of mating, she'd been inside the sithen for just hours.\n\nNot long enough to realize fully the affect it had on her.\n\nWith the passing of days into weeks, a change had come over her. She no longer sought violence for its own sake. She remembered her actions of the last century in a state of stunned and horrified recounting.\n\nJacqueline had been a self-serving narcissist. She had ruined her son and daughter with her indifferent neglect.\n\nWatching Scott come to her aid, despite her attempt on his former soul-meld's life, was salt in the wound.\n\nJacqueline did not deserve any pity.\n\nShe covered her flat belly with her hand. Her eyes fluttered shut while she leaned against the porch post. Domiatri at her elbow caused her to start. His aloof face looked down at her.\n\nHe had told her how it would be between them, and she had accepted what he offered the first night he'd come to her. Jacqueline remembered his words with startling clarity.\n\n\"If you do not wish to lie with me, it will matter not. But know this. The former sickness of your mind and the wellness of it now are due to your proximity to Faerie. Nothing more and nothing less. If you agree to bear a child of a pureblood Sidhe, then your reward will be to return here and suffer no longer.\"\n\nJacqueline had looked at Tony, her abuser, as he slumbered artificially but mere paces away.\n\nHer gaze found Domi's unerring silver eyes shining at her not in tenderness but want. Jacqueline understood what she was. She'd spent most of her life in machinations of manipulation. She was adept at recognizing it in others.\n\nShe'd borne two children, and this third would be with a man not really human but a parody of humanity. Her gaze roamed his bright green skin, his lips the color of pigeon's blood rubies, hair like midnight kissed by sky.\n\nJacqueline gave a single nod in answer.\n\nFaerie was a mixed blessing. It lifted the veil of her poisoned life outside of Faerie but cast a glaring light on the misdeeds of her past. In the time of this bittersweet revelation, she accepted Domiatri's offer.\n\nHe leaned near her, attempting to touch the side of her head, and Jacqueline flinched, though he didn't frighten her.\n\n\"Why do you cringe?\" Domi asked. \"I mean you no harm. I would never hurt a female.\"\n\nJacqueline knew this, but she didn't respond. Instead, her gaze sought the floor.\n\nDomi lifted her chin with his finger. \"Tell me.\"\n\nJacqueline sat up on her knees, chancing a furtive glance behind her. Tony still slept.\n\nShe placed her palms on the muscular planes of his chest and whispered the reason.\n\nAnd sometime within the telling, his hands gripped her upper arms.\n\nWhen she watched his anger overcame his good reason, he found her mouth, tenderly lifting her to his lap and kissing her as her words of abuse settled in the shadows of his mind.\n\n\"I shall not use you ill, Jacqueline.\"\n\nShe nodded and he took her out of the fey prison.\n\nIt was the first of many nights.\n\nWhen Tony began to abuse her, she but thought of Domi, and Tony would fall asleep, his lecherous assault broken before it began.\n\nDomi hadn't taken away what they'd been unaware of, but he disallowed its continuance.\n\n\"Do you remember what we spoke of?\" Domi asked and Jacqueline cast her eyes to the ground, shutting the memories away.\n\n\"Yes.\" Their tender interchanges would be no more once they returned to Faerie. She would carry his child then bear it.\n\nThen she would be open to breed with other Sidhe.\n\nJacqueline did not speak her fear. She had never loved another being in her life. Not her Singer son, Scott.\n\nCertainly not her vampire daughter, Delilah.\n\nShe had loved only herself.\n\nBut no more. Jacqueline had fallen victim to love.\n\nShe loved the green Sidhe warrior.\n\nAs he looked down at her, she could clearly see it was one-sided.\n\nJacqueline straightened. She had much to atone for. It wouldn't be easy, but she wished to apologize for the wrongs she'd committed. Starting with the new Queen, Julia.\n\n*\n\nThey were half a day into the journey when Jacqueline noticed Scott's gaze not constantly on her. Only when she was too weak to go on did they stray back.\n\nTharell's impatience was evident. Jacqueline would have done anything to keep the robust pace they wished, but she was still in a state of recovery from Tony's constant abuse.\n\nShe'd found eating repugnant.\n\nThen Domi had come to her two weeks before, and after their discussion and tacit agreement, she had immediately gotten with child. An unexpected development in its immediacy.\n\nBut the damage to her body, surviving Tony's constant tender care for nearly a month, had about done her in.\n\nJacqueline was on the mend, but she understood enough about herself to realize she would be emotionally scarred. Her first encounter with Domi had been so wrought with her fear at lying with a male that she'd witnessed his comprehension as it thinned his face into angry lines.\n\nNot at her. Tony had made her so afraid to be with another male.\n\nDomi had been exceedingly tender. Not using himself like a weapon inside her but as a tool for pleasure.\n\nJacqueline missed it. Him.\n\nShe showed none of her feelings. If Jacqueline was talented at one thing, it was keeping her expression blank.\n\nYet she watched Domi search her face quite thoroughly. Though he'd heard her response, he leaned in close to her, grasping her icy hands and frowning at their temperature.\n\n\"No harm will come to you while I take breath, Jacqueline. You know this.\"\n\nHe pulled away, and she nodded. She did know it. He had proven it when he'd arrived barely in time to stop Tony from raping her again.\n\nTharell had pulled Domi off Tony. No spot on his body had been free of blood and bruises.\n\nBut it was Jacqueline whom Tony's eyes sought with hate.\n\nThey separated the two of them after that, fashioning a wall from the sithen itself. It was sentient; the sithen could have refused the invoking of a barrier.\n\nBut it seemed the sithen had grown tired as silent witness to Tony's violence.\n\nThe Were had resigned himself to glaring at her through the clear but impenetrable wall.\n\nJacqueline had cried when Domi saved her from the attack.\n\nShe had cried into his hands, and he had caught her tears like diamonds as they fell.\n\nPieces of her soul had broken apart that day, and all Jacqueline could hope for was reclaiming them in the future. Whatever her future ended up being.\n\nDomi still waited for her response.\n\n\"I know,\" she said.\n\nHe smiled, his scarlet lips breaking over pure white teeth. \"Good.\"\n\nDomi put his large palm on her back, and they moved to the horses. He easily lifted her up and began a forward walk. It would be a long sojourn on horseback, and Jacqueline tired thinking about resuming. However, they needed to travel that way to avoid airplanes and cars.\n\nThe baby she carried only afforded so much temperance this far from Faerie. Metal, was still an issue. It was as the human's fabled kryptonite, a poison to the fey.\n\nHorses were organic. And obedient.\n\nThe one Singer female who came was Angela, and she made Jacqueline uncomfortable. Of course, Jacqueline was hyper-aware in a way she'd never been before. She girded her loins.\n\nShe did not deserve an answer to her question. \"Why do you stare at me?\"\n\nAngela rode beside her, the powerful horse rolling beneath her hips. Her face flushed slightly, her fair skin hiding nothing. \"Your aura.\"\n\n\"What of it?\" It had been the color of bruised eggplant before her fey imprisonment.\n\n\"It is no longer violet tinged with black,\" the Feeler admitted with hesitation.\n\nJacqueline was now desperate to know: was her change in Faerie certain?\n\nDomi said nothing as he rode to her right and slightly ahead.\n\nJacqueline knew he listened.\n\n\"It is a pale pink, with white at the edges.\"\n\nJacqueline sat atop the horse in stunned silence, her fingers going lax on the reins.\n\nShe knew what that meant. And she couldn't believe Angela would ascribe it to her.\n\nPurity.\n\n# CHAPTER THIRTEEN\n\nThe Washington state border loomed in the distance, guarded by Canadian Mounties. The horses pitched around nervously, causing the men on their backs to tighten their grip on the reins. The horses were not dumb. An alert as ancient as any had tweaked their internal alarms.\n\nJulia watched her husband and the other two females move forward. All half-wolfen. That was what the horses sensed.\n\nIn a perfect world, they'd all be on horseback, and the horses' scent would mask the Were.\n\nThe world was not perfect. Upwards of fifteen Were trailed them by their fragrance. Eau de Body Odor, Julia thought with a thinly concealed snort. She figured they kept getting worse as they traveled. No baths, food where they could get it, sleep a luxury. It was a combo for smelly, grumpy, and beat.\n\nJason glanced at her, and she gave a tired smile back. When Julia had been taken, she'd been wearing cute but useless tennis shoes. The kind you buy for nearly free and walk to the mailbox in.\n\nThey were in tatters. Blisters covered every place she looked. They had been especially bad at the back of her ankle and alongside her toes.\n\nCyn had healed her from most of it, along with the broken ribs and dislocated shoulder, but the injuries kept returning. The Were changed, in their element out there in the forest, and suddenly there were no problems\u2014for them. Julia was a regular person who'd just walked fifteen miles on no sleep and inadequate footwear.\n\nJason lumbered back to her position. She looked up into his face, almost seven feet of half-Were and sighed.\n\n\"I smell your wounds,\" he said and she nodded.\n\n\"Likely\u2014they're driving me crazy.\"\n\nShe laughed when he put his big paws on his hips, thinking about a solution to her shredded feet.\n\nHis red downy fur covered his body as spinning green eyes regarded her. \"What?\" It came out as a growl instead of a word. Julia was feeling the lack of sleep, and it translated into giddiness. She so didn't need that, but the more she tried to stop laughing, the worse it became.\n\nShe was now the interpreter of wolf-speak.\n\n\"What? Stop the noise please,\" Slash said with more than a little irritation.\n\n\"I think she's tired.\" Jason restated the obvious with a grin that looked like a grimace on features far from human.\n\nShe collapsed on her butt and rolled around on the forest floor, holding still achy ribs, and laughed at their dilemma of being sandwiched between the foreign police and their Red Were pursuers. Not a bright spot in the entire scenario. She turned it over and over again in her mind and caught the laughter before it turned to tears.\n\n\"Jules, geez, snap out of it.\" Cyn jerked her up by the arm that hadn't been dislocated. \"Quiet.\"\n\nJulia made a supreme effort, clamping her lips together. She hiccuped.\n\nJason put his arm around her and gave her shoulder a squeeze. With his strength it almost hurt.\n\n\"We'll have to wait until a shift change. During the confusion of the change out, we'll slide through,\" Slash said aloud.\n\nTheir superior sight caught the flies that pestered the horses' tails. Julia could only make out that there were tails.\n\n\"We make the horses nervous,\" Truman commented.\n\nJulia hiccuped again.\n\nThe group looked at the horses that traveled the length of a tall fence with razor wire across the top. They flicked their tails, eyes with too much white showing as they glanced nervously around.\n\n\"Yes,\" Slash replied, unperturbed.\n\nAdi gave the first shy look Julia had ever seen on her face. Slash returned the glance with a tender expression.\n\n\"Let's take the wolves down a notch and wait. What time do you think they'll switch out?\" Jason asked Slash.\n\nSlash shook his head. \"Doesn't matter, it's now or never.\"\n\nJulia's former reckless laughter completely faded. \"Why?\" It didn't make sense. Didn't they just determine we'd be breaking through when one shift of Mounties relieves the next?\n\nJason lifted his snout. His eyes snapped to Slash then to Truman.\n\n\"They're here.\"\n\nSlash gave a snort of assent. He took in Adi, a foot shorter but with beautiful golden fur covering her.\n\n\"We have to protect the females.\"\n\n\"Love the ratio,\" Truman said. Then, \"Let's get outta here, I can smell them a mile away.\"\n\n\"What\u2014okay, that's going to be a problem. We've kept our existence under wraps...\" Julia began.\n\nSlash grinned, and his scar rippled like a wave. \"Not entirely. Someone has seen something, or Hollywood wouldn't be making movies.\"\n\n\"Julia,\" Truman said.\n\n\"Yes.\" Her gaze plunged into the deep forest. She couldn't see, and the effects of Paul fuzzing everyone's mental deliberations were fading this far away from Region One. However, if she concentrated, she could make it be white noise. Julia kept gaining finesse, and it was a great thing.\n\nRight now, she wanted to find out where the Reds were.\n\n\"We need you to be the distraction.\"\n\nJulia turned away from the forest and looked at Slash. \"What?\" Maybe she heard him wrong.\n\nJason frowned. \"No way, man... they might hurt her.\"\n\nSlash shook her head. \"Not if she acts lost, hurt.\"\n\nNope.\n\n\"We don't have much time but let's talk about Tony.\"\n\n\"Not a good idea,\" Cyn chimed.\n\nSlash went on quickly, ignoring her for the moment, \"It's as good a comparison as any.\" He had their attention. \"Tony's wiring is all crossed. Most males of all species, wish to protect females. Tony wants to do harm.\"\n\nNobody disputed it. But Julia wasn't sure how it fit, or\u2014better yet, why they were discussing it with the Reds nearly on top of them.\n\n\"The Canadian police will not go against primal instincts and hurt an unprotected, lost, and injured girl. It'll be okay and effective.\"\n\nThere was a moment of silence while everyone thought it over.\n\n\"True,\" Truman barked. \"You've made your point. Now she needs to go out there.\"\n\n\"Get rid of the shoes, Jules.\"\n\nJulia looked down at her feet then tossed the shoes aside.\n\n\"I feel them in my head,\" she said.\n\n\"Who?\" Slash asked sharply.\n\n\"The Reds.\"\n\n\"What are the dick weeds thinking?\" Adi asked with false cheer.\n\nJulia shook her head. They were hard to read. No clear thoughts in actual words; just imagery. She relayed that.\n\n\"What do they feel?\" Jason asked.\n\nEasy. \"Anger.\"\n\nA beat of silence conquered the moment. \"You better go, Jules,\" he finally said.\n\nShe looked up at him, blown away by just how much of his humanity leaked through that altered face. \"What are you going to do? They outnumber you guys three to one.\"\n\nHe smiled, and she saw Jason in there, that boy she once loved when it was simple. \"They don't have anything to fight for.\"\n\nJulia turned away before he could see her tears, though they all smelled the salt of her sadness.\n\nShe fled, her bare feet traversing first moss then patchy, sharp grass. It should have hurt; instead, it felt freeing.\n\nJulia would get home one way or another.\n\nThe horses startled at a lone person running toward them full tilt. Julia's lungs burned. The familiar ache of a run meant she was almost warm enough to go the distance.\n\nThe Mounties watched her coming and conferred briefly, then two broke away from the main group to meet her.\n\nThe steeds' rolling canter brought them nearer.\n\nThe anger burning through her brain erupted like a tooth coming in when the Reds made the mental move to attack.\n\nThe Royal Canadian Mounted Police circled her, and Julia stopped running, bending over and clutching her ribs.\n\nShe didn't turn around, afraid of what she'd see. Knowing what she'd see.\n\n\"You're approaching an international border, Miss,\" the first of the Canadian police warned when she was within hearing.\n\nShe peered up at the first officer when noise in the forest shook the ground. She ended up checking behind her after all.\n\nThe distraction Slash had hoped for just appeared. Julia bolted.\n\nShe ran so fast her heels almost kicked her own butt. Two Mounties waited on their horses, and Julia didn't slow down. She ran between them like a speeding locomotive and prayed.\n\nJulia made a running leap, her legs pumping as if she was riding a bicycle midair, and flew over the spun razor wire, her small telekinetic talent allowing her just to clear the barbed spikes.\n\nThen she fell.\n\nJulia didn't have the finesse to halt her landing, only soften it.\n\nOf course, she landed on the ribs she'd broken. What little air she'd had in her lungs departed.\n\nAt least her brain was still in her skull.\n\nShe was vulnerable when the first Red sprung over the fence and gazed down at her.\n\n\"A merry chase you've led us on, Rare One.\"\n\nJulia thought of all the snappy comebacks she could have said. Instead, she did the smartest thing ever.\n\nShe jerked her leg up and landed a foot in his considerable crotch.\n\nThe Red howled, landing on his side. Julia stumbled to her feet and ran.\n\nHe grabbed her ankle in a vise grip and she tripped, narrowly missing hitting her face on the ground. The force of the fall reverberated from her palms to her shoulders.\n\nJason smoothly hopped the fence and landed gracefully.\n\nHe swept his foot into the jaw of the offender.\n\nThe Red rocked back. He was unconscious before his head hit the ground.\n\n\"Come on, Jules,\" Jason said, sweeping her underneath his arm.\n\nA bullet whizzed over his head.\n\nJulia looked around.\n\nOh... bullets were zinging around because they were back in America.\n\nHome sweet home.\n\n# CHAPTER FOURTEEN\n\nAs a mixed-blood, Tharell tolerated the distance of Faerie better than Domi. He maintained a close proximity to Jacqueline and the unborn child she carried.\n\nScott and Lucius of the Singer Combatant wove their way expertly through the dense forests to which they kept. The Feeler, able to identify the aura of the Red Were who had taken the others, was in the middle of the tight knot of males.\n\nThe Canadian border was not far. He spared a glance at Domi, and the other warrior's watchful care over the mixed fey-Singer troubled him.\n\nIt could not end well.\n\nHe brought his mount up, and she softly neighed at the abrupt halt, as though in question. She flicked her tail in agitation. He patted the animal's side to quiet her, and the flies stayed away. Tharell thought the noxious insects an irritation he could do without and fashioned a spell of camouflage over the group. It was a trifling thing; an infant could have conjured it. Tharell believed in not wasting magick. Keeping flies away as they traveled was a frivolous waste, he readily admitted, yet the journey possessed enough challenges. It was what he could easily do, so he had.\n\n\"Tharell,\" Domi called out.\n\nTharell looked over his shoulder, Domi looking very bright against the pine-colored forest. Surreal. \"Jacqueline needs rest.\"\n\nTharell swept his gaze at the valley like emerald water before them. It was more than a meadow, a wide expanse that gradually rose to another patch of forest. He did not like the lack of cover. They would be vulnerable.\n\nHe and Domi did not fit within human norms.\n\n\"Let us wait,\" Tharell agreed. Not because he thought Jacqueline needed rest or not, dusk was at hand and he looked like a cloud of midnight in the shroud of darkness. Domi was bright but grew dark as the night deepened.\n\nTharell squinted, sighting a narrow creek. If only they had a Diviner, then they would know where the water was and if it was safe to consume.\n\nTharell felt a presence behind him. He slid off his horse and faced Scott.\n\nThe Singer was not shy. He spoke his mind quite readily, as most of his kind was wont to do.\n\n\"Speak.\"\n\n\"I'm going to, fey. I feel the presence of our Queen.\"\n\nTharell turned slowly around. \"Impossible. They were taken days ago. They should be nearing the Alaskan border by this time. We're chasing their tailwind.\"\n\n\"Yes, I'm aware what makes sense, but Lucius has the same blood summons as I. So\"\u2014Scott held up a finger\u2014\"that means she's traveling backward.\"\n\n\"Domi!\" Tharell said, ignoring Scott. \"The Combatant speaks of blood summons.\"\n\nDomi cast a glance like a hook at Jacqueline, as if to assure her safety, and came to Tharell.\n\nAs he drew nearer, Tharell had a rare pang of envy. To be a pureblood Sidhe, with the coveted bright coloring.\n\nThe only bright thing about Tharell was his eyes.\n\nDomi broke into his uncharitable thoughts. \"Tharell?\"\n\n\"Julia's escaped.\"\n\nThere was a heartbeat of assimilating this new information. \"How?\"\n\nTharell lifted his shoulders. \"The Singer Combatants have conferred and are in agreement.\"\n\nDomi's chin fell into his palm, his perfect green face puckering in a wrinkle of concentration. He turned to Jacqueline a second time.\n\n\"She is vulnerable,\" he said finally.\n\nTrue. If Julia had escaped, the Reds would never give up.\n\nThus far, the fey had managed to escape being outright enemies with the other supernaturals. However, it had been a near thing with the Singers. To begin a war with the Red Were was not good.\n\nDomi's gaze followed his own as he took in the two Combatant, Jacqueline, and the Singer Feeler, Angela.\n\nDepending on who came with Julia, it could be an unwinnable battle. The Reds were known for their abilities during engagement.\n\nHard to kill, difficult to outmaneuver.\n\n\"I will be useless, Tharell.\"\n\nScott and Domi regarded Jacqueline together.\n\n\"It's weird how different she is.\"\n\n\"I did not know her before. Only as she has been in Faerie,\" Domi said.\n\n\"I knew some of it.\" Tharell spoke to Scott's comment, electing not to expound. Tharell had a feeling her offspring would not well receive the sexual displays she and Tony put on in the beginning.\n\n\"She needs my protection,\" Domi said.\n\nTharell frowned at Domi's priorities.\n\n\"Will the Were hurt a female with child?\" Scott asked, surprised.\n\nTharell understood violence against females was taboo across all supernatural races. There seemed to be a shortage of the fairer gender inherent to some degree in them all. Tony might be the only exception, as the occurrence was quite rare. There was nothing beyond that thankless Were.\n\n\"Lets's go,\" Scott said from behind. Deep pools of shadows grew on the fields, turning their blond shoots to upside-down spider legs in the lengthening darkness.\n\n\"Yes,\" Tharell agreed.\n\nDomi gave a reluctant nod. He stood next to Jacqueline, slowly giving her water from an animal-skinned flask. She took a small pull, moving a shaky hand across her brow. Even from this distance, Tharell could see the paleness of her skin. She had not fully recovered. Had she but a few more weeks in Faerie, she might have been fully rejuvenated. As it were, she was half-undone.\n\nTharell's belly fluttered inside like a butterfly trying to escape its prison. He knew what that sensitive precursor usually meant.\n\nWar.\n\nTharell would not wait for battle to find him. Rather, he preferred to bring his sword to the fight. They would take the Rare One from anyone who would have her. Losing was not an option he entertained.\n\nTharell's horse made its way through the grass.\n\nThe tall stalks whispered along the underbelly of the beast, sounding like an endless rain which fell without cessation. The entire group rode without speaking. A cloud of precognition hung over them. He recounted the abilities of the Combatant: Scott was a Deflector of impressive ability, he wished Victor was also part of their troupe. He had heard the tale of vampires exploding under his tender mercies. Lucius... it occurred to Tharell, he did not know. It deepened his trepidation, it was a rudimentary oversight. His first priority should have been to establish the arsenal at his disposal. Before Tharell could attempt to worry about the Combatant's unknown skill set, Scott spoke.\n\n\"Julia is here... close by.\"\n\nThat's when they all saw the sky on fire, lit up by shots.\n\nGun smoke saturated the air, and it thickened all their breathing. Domi laid his hands over Jacqueline's shoulders, and she took a shaky inhale. Tharell thought it wise, as no one wasted magick on one sickened member of the group dragged along like baggage because they couldn't breathe.\n\nThe group drew tighter.\n\n\"It won't hold, Tharell,\" Domi shouted over the gunfire.\n\nGoddess help us. \"It won't matter. The spell is all that we have for her.\"\n\nDomi hauled the fragile Singer behind him. He waded into the melee of battle.\n\nTharell saw how many there were, and his mind stalled at the tally.\n\nHumans, numbering thirty were charging after a huge Were and the solitary reason behind their trek.\n\nThe huge, half-breed Red Were was barreling ahead of the small army that followed, bleeding out of several holes like a running slice of Swiss cheese.\n\nClearly, there was no time to heal the wounds.\n\nThe Rare One was riding his front like a monkey while bullets littered his back.\n\nTharell did not often encounter beauty outside of Faerie, which rivaled the sithen's magickal interiors.\n\nSeeing the Were run at top speed, his arms and powerful legs pumping as his mate clung to the front of him, was one of those times.\n\nThe humans were gaining as Jason was slowing.\n\nSome held implements that glinted silver in the night that had finally fallen around them like an ominous shroud.\n\nTharell gave a signal that made Domi move even as he gave it. The Combatant didn't wait.\n\nTheir Queen was in peril; a biological initiative began instantly.\n\nSavagery was all that was left for them to execute.\n\nThe pair turned into creatures of legend more frightening than the Were or anything Tharell had ever seen.\n\nNumbers might not matter as much as ferocity.\n\n# CHAPTER FIFTEEN\n\n\"Jules!\" Jason gasped as her head bobbed from sprinting.\n\n\"Yes!\" She managed to respond without biting off her tongue.\n\n\"Let go!\" he yelled, and she did. The hardest thing was knowing the combined forces of the US Border Patrol and Canadian Mounties were after them. Jason's blood and sweat slicked her fingers.\n\nShe trusted.\n\nJulia let her arms slip from around his neck.\n\nJason flung her in the air.\n\nHe launched her into space, and she hurtled through nothingness. She forced herself not to use one bit of talent for whatever impact she encountered.\n\nShe wasn't half as brave as she thought. Julia kept her eyes clenched shut.\n\nShe landed, blowing into a figure from behind as he caught her.\n\nThey collided into a tree.\n\nJulia opened her eyes, and the sharp scent of forest along with needles like green rain fell on her face. She coughed them out of her mouth and rolled off whoever caught her.\n\nScott looked up at her.\n\nThere was a sense of the surreal when she tried to compartmentalize his role at that moment.\n\nProtector.\n\nWith a great whooping cough, he expelled dirt, needles, and blood from his mouth. A lopsided smile accompanied a groan. \"Gain some weight?\"\n\nThe war was raging around them, but Scott had time to quip about her landing. It was wonderful and filled her with hope.\n\nJulia smiled as he bounded up, and he drew her to standing.\n\nShe'd missed him.\n\nThe music of gunfire and fists hitting their targets overpowered regular conversation. \"Are you hurt?\"\n\nShe shook her head. Scott's gaze ran down her body as though she hadn't answered, then he shoved her behind him.\n\nJulia saw what he saw in great detail. The battle was practically on their heads.\n\nTharell riveted her. Like purple water, he moved in between the humans as though they stood as debris in the river of his violence.\n\nThe fey were a thing of terrible beauty. Domi and Tharell worked in tandem like perfectly orchestrated dancers.\n\nThey moved past one another. Domi swept his weaponless arm out, and Tharell gripped it, spinning Domi behind himself in a half circle.\n\nDomi took three heads during the move.\n\nThe Mounties' blood matched their uniforms as they fell to his long sword. The red and gilded braiding littered the soil with blood and death, the earth an eager sponge for their lives.\n\nAtrocities happened before her eyes because humans had accidentally stepped into the mix of their supernatural war. It was profoundly unfair.\n\nThe Americans wore the dull moss of their station, blending perfectly with the forest that retained its color through the dusk. Three came for Tharell.\n\nThe knife entered where Julia knew a kidney to be. The only sign Tharell gave was the tightening of his eyes, glowing with ethereal blue fire.\n\nThe three who had come, including the sword bearer, fell. The human casualties grew, filling her with sadness.\n\nThe green fey was guarding Jacqueline, and Julia's lips flattened in anger. What the hell was she doing here?\n\nJulia scrambled away from Jason as the Reds suddenly appeared, running in the opposite direction.\n\nShe wasn't getting taken again. No matter what.\n\nShe would rather die than live under that again.\n\n\"Jules!\" Jason yelled. She could hear the panic, but it wasn't greater than hers was.\n\nShe sprinted as if her life depended on it.\n\nHead honcho, the one who'd played Fed but who was really the pack master of the Alaskan den, landed in front of her.\n\nTom Harriet.\n\nHe sprayed dirt around him from the abruptness of his stop, and Julia's momentum drove her forward. She executed a perfect somersault, as she'd done a thousand times as a kid, tumbling right between his legs. His seven-foot-tall half-wolfen body was the perfect height for the maneuver. When he tried to snatch her from behind, Julia reacted.\n\nShe kicked out blindly. She nailed his nutsack squarely, and he fell to his knees.\n\nJulia bounded up and raced further, the sounds of battle growing quieter.\n\nWhen she couldn't take another step because she didn't have enough air in her lungs, she stopped.\n\nJulia was in the middle of the woods.\n\nAlone.\n\nWithout the protection of the group or Jason.\n\nShe was lost.\n\nYes, she'd escaped the battle, taking out the leader of the Red Were for the moment and buying herself time.\n\nJulia looked down at her feet.\n\nBare.\n\nAn owl hooted in the distance as her stomach let out a growl.\n\nShe had just made the gravest of mistakes.\n\nAnd she had only herself to blame.\n\n*\n\n\"Where's Jules?\" Cyn asked, looking around frantically.\n\nSlash noticed the Rare One was missing and groaned. He searched for Jason. Why had he lost sight of Julia? Something serious happened, he'd bet.\n\nTruman backed up until they all formed a loose circle. Angela occupied the center. The Combatant all presented their war forms, not unlike the half-wolfen Weres' sizes but deadlier. Slash didn't know if the threat of the morphed Combatant and the fey would be enough. The humans were in various states of dying or dead, but the real threat was now approaching.\n\nAnd they knew Slash now. He'd given the secret of his pureblood status away to escape them earlier.\n\nSlash scanned the contingent of wayward Reds and found his pure blood fighting for a dominance struggle in a bid to secure the Rare One, protect Adrianna, and survive.\n\nDominance won, of course. There was nothing to be done with instinct, as deeply ingrained as taking breath.\n\nSlash did not see the pack master. The one he recalled put most of them to sleep with a well-placed dart to the shoulder.\n\nThat flat-out pissed Slash off.\n\nFight for territory, fine. But fight like a wolf, not a coward.\n\nTruman enlightened Slash to some of the players' identities. \"Listen, Ford, where's the dickhead pack master that's responsible for taking things that don't belong to him?\"\n\nLike the Rare One, Slash thought. Ford, Slash guessed, came from the half-wolfen pack that faced him. He automatically drew in front of the fey warriors, the Combatant, and Jason and Truman. \"Where is your pack master?\" Slash said.\n\nFord's eyes stared back at him like bottomless pits. \"Why? Want the job?\"\n\nSlash considered it for the length of a heartbeat. \"Yes.\" He moved deeper into the meadow to meet his adversary.\n\nWhere they were and what was about to transpire made Ford's FBI-issue look more ridiculous.\n\n\"Slash, don't\u2014there's too many,\" Adrianna said.\n\nHe didn't turn around. \"It's the only way.\"\n\nShe circled around him. His guts clenched. Her nearness possessed a scent he thought he'd never smell on a female directed at him.\n\nDesire.\n\nMixed with dominance lust, it was a perfume as heady as any he'd ever smelled. He had to exert an ugly self-control.\n\nShe put her forehead at his chest.\n\nThe talons at his fingertips remained extended as he wrapped her skull with a hand only half-human.\n\n\"Be with me,\" Adrianna whispered against him, and Slash shut his eyes.\n\nHe let the fantasy envelope him for as long as his next breath.\n\n\"There is no being unless I fight for dominance. Right here. Right now.\"\n\nAdrianna nodded, and when she didn't move, Slash positioned her away from him.\n\nIt felt like an amputation of his soul.\n\nFord's smile was cold. \"Just think, when I beat your ass, the bitch is mine.\"\n\nThe growl erupted out of the depths of Slash's body. Uninvited, unbidden, resolute.\n\nHis talons tingled, begging to gore Ford.\n\n\"No\u2014Slash, he's baiting you,\" Adrianna said.\n\nThe other Were remained silent.\n\nFord turned eyes to Adrianna. \"You're so going to like me, bitch.\"\n\nAdrianna flipped him off with a slender middle digit.\n\n\"That's about the size of yours, you dick weasel!\"\n\nSlash smiled. He enjoyed remaining quiet if the one who spoke for him had just the right words for the situation.\n\nFord moved forward. \"You will submit, female.\"\n\nAdrianna made a low noise somewhere between defiance and amusement in her throat. Then she added the middle finger from her other hand.\n\n\"Not on your life.\"\n\nSlash thought he could arrange that.\n\nFord glared at Adrianna then turned to Slash. \"It's not just me. There are others who will fight you.\"\n\n\"I know.\"\n\n\"Knew you were a ball-less wonder,\" Truman told Ford as he and Slash circled each other.\n\nHis gaze flicked to Truman. \"But I'll be a ball-less wonder that's dominant to you.\"\n\nSlash reacted instantly, swinging his talons out and going for the jugular. Ford feinted to the left, and the body knives missed his Adam's apple by a hairsbreadth.\n\n\"Doubt it, chump.\" Truman crossed his arms.\n\nCyn sighed. \"I don't want to see him naked, and be underwhelmed and all that.\"\n\nFord looked at her, and Slash struck again, piercing the vulnerable part of the body where thigh meets crotch in a precise strike.\n\nSlash did not liken himself to a brutal fighter unless necessary. He would not tire himself when there were more Were to fight.\n\nCutting the femoral artery was effective.\n\nRapid.\n\nFord moved in for an obvious charge and stalled, a surprised look overtaking his face.\n\nThe other Reds shifted and muttered nervously.\n\nSlash approached confidently as a river of blood shot out of Ford's body.\n\nFord didn't react when all four talons burst out the back of his neck. He was too busy dying from blood loss.\n\nHe careened backward, and Slash, a grim reaper without a scythe, jerked his talons. They came out of Ford's neck in a thick suctioned pop of flesh.\n\nThe talons sliced again, punching Ford with the knives that tipped his fingers.\n\nThen nothing.\n\nSlash stood.\n\nHe flung his right hand at the waiting Reds who would take the one hope the species had of unification.\n\nDrops of blood and sinew from Ford's body fell like plops of gruesome rain. Some landed on the Were themselves.\n\nThey made no move to wipe away the proof of the Alpha's death.\n\n\"Any takers?\" Slash asked.\n\nTwo more broke from the ring of Reds, a pureblood Red amongst the challengers.\n\nSlash's heart ticked faster.\n\nHis eyes fixed on Adrianna.\n\n# CHAPTER SIXTEEN\n\nJacqueline stayed behind Domiatri.\n\nShe tried to imagine what she would have done before.\n\nNow she used Domi as a shield, whereas in the past, her very evilness would have sufficed.\n\nShe held two perspectives. One from before Faerie, and she would always think of it as such. The second was after.\n\nAfter Tony assaulting her every day, her muted Talents became apparent and she'd been defenseless. When her fey blood while living outside the sithen affected her, everything had been a plan of treachery. One of deception. At the time, it had seemed clever to mate with Tony and force all the other species to give them a reprieve through her matrimonial alliance with him.\n\nNow they were mated.\n\nAnd Jacqueline's self-loathing was acute but for the small glimmer of hope that her life might mean more than being an outcast, forever known as attempted murderer of Julia Caldwell, the Rare One.\n\nThe old Jacqueline, her talents those of a Deflector, a telekinetic, and a Tracker, would have slaughtered everyone in her path.\n\nShe was no longer that woman.\n\nShe lifted her chin and moved out from behind the protection of Domi's emerald body. He put an arm around her waist and snagged her against him. \"Do not even think it, Singer.\"\n\nA sob caught in her throat. She had finally been brave enough in her useless life to try to help another, only to find that Domi's protection would not allow it.\n\nIt crushed her newfound soft interior.\n\nThen Domi did something he had never done in all their couplings to get her with child.\n\nAs the Reds approached, circling them like sharks, he pressed a light kiss against her forehead.\n\nThe heat from his mouth upon her skin, the seduction of the breeze that blew between the supernaturals was tenderness on fire.\n\nJacqueline reveled in it.\n\nThen he released her. \"Go,\" he whispered into her ear.\n\nTheir gazes locked.\n\nHis hands hovered at the top of her head, her eyes on the silver of his.\n\nDomi's palms flowed over her, never touching. They paused at her belly then flowed to her feet. Domi grasped her ankles, his hands brands of fire. She touched hair she knew was navy but looked like oil running down his shoulders underneath the veil of night.\n\nHe leaned into her hand briefly. \"They will not find you. Use your talent to Track. Locate the Blooded Queen.\" He kissed her palm and stood, towering over her.\n\nHe'd cast a spell over her, a cover of glamor. She could feel it as it rode over her skin. When she shifted, a shimmer followed her.\n\nJacqueline took a deep breath, coming between Slash and the two new challengers for position of pack master.\n\nThey didn't look at what they could not see.\n\nJacqueline headed toward the girl she would have murdered.\n\nIt felt like a lifetime ago.\n\n*\n\nJulia gazed around. It was d\u00e9j\u00e0 vu. Different woods, same problem. This time she wouldn't be drinking any water out of creeks. She shivered from the possibilities.\n\nLeaving Jason had been a bad move.\n\nShe heard a rustling and turned.\n\nThe Red Were stood before her.\n\nPuffs of smoke from his nostrils measured his breaths. They flared as the weather cooled, showing his position.\n\nHe looked all recovered from the foot-to-the-testies maneuver she'd managed.\n\nJulia tensed, ready for a flight that would prove unsuccessful. There was no outrunning a werewolf.\n\nThe pack master of the Reds would be even more powerful.\n\n\"Wait,\" Harriet growled, talons faintly glowing as he raised his palm in supplication.\n\nJulia didn't like being the queen of anything else. She blew it off, focusing instead on the potential for violence that emanated from him like a fragrance. Always high amongst the Weres, she figured it was even more so with a Red.\n\n\"I don't know, but as off-balance as my Talents have been coming, I wouldn't risk it if I were you. And\"\u2014Julia cocked her head to the side\u2014\"you could kill me with another one of those well-placed hits.\"\n\nHarriet bled back to his human form, and Julia's lungs deflated from the breath she'd been holding. It felt better to see a six feet tall guy in front of her, even naked, than the half-wolfen, seven feet of contained violence staring out of the gloom just a moment before.\n\nShe kept her distance anyway.\n\n\"I'm Tom Harriet.\"\n\n\"I know.\"\n\n\"Then you understand that I'm not actually here to harm you. I work for the FBI.\"\n\nJulia snorted in the dark. His gaze latched onto her. He didn't fool her; he could count the hairs on her head. From his vaguely reflective gaze, she understood the wolf ran just beneath the man. She briefly touched on the memory of Slash and his big pureblood Red reveal. She'd not seen it but had heard. Tom Harriet was a Red.\n\nHe cocked one of his eyebrows.\n\n\"It's just\"\u2014she folded her arms\u2014\"what does that matter? Obviously, the feds aren't aware you're a supernatural.\"\n\n\"Some are not,\" he conceded.\n\nJulia shifted uneasily. How deep did the world inside the world go?\n\n\"Whatever organization that humans are in control of, we have infiltrated.\"\n\nJulia shrugged. \"You being a fed holds no weight. I've had authority figures of every flavor turn out to be not what they really were. You're no exception.\"\n\n\"It doesn't matter.\"\n\nWhen had he come closer? Julia stepped back.\n\n\"You were never meant to leave Alaska. Lily was meant to keep you under her care until you became, then you were to assimilate into the Alaska den.\"\n\n\"I love that everyone had my life planned from the beginning. I was the only one not privy to the agenda.\"\n\n\"It's the way it is. Your parents made a choice.\"\n\nJulia's heartbeat raced.\n\n\"So it's true? I do have a sister....\"\n\nHarriet nodded.\n\n\"Where is she?\" She relaxed her hands out of fists so tight they'd left crescent-shaped indents on her palms.\n\nTom's smile was immediate and brilliant. Julia kept her gaze on his face, never straying. He held up his index finger and wagged it back and forth. \"The Sidhe warrior was clever to use the cloning spell to fool the Singers. But I will say nothing of her whereabouts unless you come with me.\"\n\nThat awful moment in Julia's life was worse in some ways than any before it. To realize her true family still lived in this plane of existence.\n\nShe'd been an orphan so long that she had long buried the hope of any family.\n\nThe Red Were was dangling the proverbial carrot.\n\nShe hated him for it.\n\nAnd at the same time, Julia passionately hoped it wasn't a lie.\n\nShe didn't have time to scream. He wrapped his hand around her throat, closing her windpipe. He'd been that fast.\n\nShe struggled.\n\n\"Shh,\" Tom whispered beside her ear. \"Someone else is here.\"\n\nJulia's eyes bulged. They moved in a panic to whoever had appeared.\n\nFriend or foe? she wondered. And on the heels of that, Jason's name whispered through her mind like a reverent prayer.\n\nHer gaze latched on to the small female figure that glided outside the border of trees.\n\nJacqueline.\n\nDefinitely foe.\n\n*\n\nJulia gave a squeak, and Tom dropped her.\n\nShe fell, hands at her throat instead of breaking her fall. The shock of the rough landing thrummed up her spine like an electric current. Julia sucked lungfuls of precious oxygen as Harriet gave her his naked ass.\n\nSlow learner.\n\nTestie tap time. Or double tap.\n\nJulia staggered to a standing position. She kicked the Were in his butt with her bare foot. Her toes might have curved a little as her foot came back.\n\nHe howled, reaching between his legs, but didn't move far.\n\nIt was the distraction she meant it to be.\n\nHe whipped around, arms spread, talons sliding out and conquering the flesh of his fingertips in a brutal shift.\n\nJulia threw up her hands to defend her face, and the splatter of his humanity rained down on her.\n\nShe dropped her arms, and a large wolf stood before her.\n\nTrouble stared back in the form of a red fur coat and green eyes.\n\nJulia kept the Were in sight while also trying to find Jacqueline.\n\nNot far enough.\n\n\"Julia\u2014wait!\" Jacqueline said and the wolf turned, it's swimming green gaze swinging between the two women.\n\n\"No,\" Julia said. \"I don't want to hurt you. But I will, because you don't do mercy. And this guy\"\u2014Julia pointed at the now-changed Harriet\u2014\"this guy, he's certainly not my buddy.\"\n\n\"I know it will be hard to convince you of my sincerity.\" Jacqueline turned her palms outward in a defenseless posture, wary eyes on the Were.\n\n\"Impossible,\" Julia answered.\n\nJacqueline dropped her hands. \"I am sorry... I tried to murder you.\"\n\nJulia laughed aloud. Is she serious?\n\n\"I was not well when I... attempted that.\"\n\n\"Is this where you claim temporary insanity?\" Julia made no move to temper her sarcasm.\n\nJacqueline shook her head.\n\nJulia's brows cinched. This was beyond weird.\n\n\"This is where I escort you back to the group. Domiatri wished for me to Track you. It was twofold. I would be safe, and you would be found and returned.\"\n\nJulia opened her mouth to tell her to piss off when Harriet pounced on Jacqueline.\n\nJulia should have let her die. She was a proven murderess.\n\nBut Julia's instincts said something was off.\n\nFor one, Jacqueline wasn't that good an actress.\n\nWhat if she isn't acting?\n\nJulia steeled herself, racing toward the werewolf and her killer.\n\n# CHAPTER SEVENTEEN\n\nSlash faced the half-blood Red. He couldn't help his feelings of vulnerability with the two facing him.\n\nJust a natural part of dominance struggle.\n\nSlash had encountered Reds before, of course. During turf wars but mainly as rogues.\n\nNot now.\n\nThat was as organized a plan as he'd ever seen. They had banded together and made themselves a force to be reckoned with.\n\nIt was up to Slash to free them of their current Alpha's position of pack master.\n\nAnd put himself in it.\n\nThe Were that went by the name of Tom Harriet was missing. Slash was primed to use the volatile circumstance to his advantage.\n\nThey circled one another. The half-breed Were taunted Slash in an attempt to distract him.\n\n\"I'm going to take that Alpha bitch when I'm through with you.\"\n\nSlash duly noted he wasn't dealing with a major intellect. \"That worked so well for Ford.\"\n\nThe half-breed's eyes rounded when Slash attacked. He uttered the last word while Slash was already moving.\n\nSlash did something different than he'd done against Ford. The pure-blood was watching them.\n\nMemorizing his movements for when they fought.\n\nThe Were bled as he moved. Colored, sliding flesh blurred like a smeared watercolor painting. Both Were shifted from human to half-wolfen forms, their fists readying.\n\nIn Slash's case, he drove his left fist, knuckles pointed where he wanted them, into the opposing Were's solar plexus. It robbed him of breath as Slash did the same with his other hand to the male's neck.\n\nImmobilizing him, it assured that his breath would not return. Slash forgot about Cynthia and Adrianna. He ignored Truman and Jason watching what he was about to do.\n\nWhile the Were's eyes stood like poached eggs in his purple face, Slash lashed a foot into the male's chest, grasping his enemy's left arm.\n\nHe pressed with his foot and twisted the gripped arm. With a sickening pop, Slash tore it from the socket.\n\nHe used the ball joint like a bony hammer, beating his adversary as he fell.\n\nThe skull cracked first.\n\nBlood spray, shards of bone, and gray chunks of brain matter flew in a spatter pattern ahead of where his body began to fall.\n\nBreathing came too late as death met the second Were he'd killed in the space of hours.\n\nSlash threw the arm with which he'd just bludgeoned the Were and turned to face the new threat.\n\nThe Reds who stood behind him backed away, casting wary glances at Slash.\n\nHis small group was quiet behind him.\n\n\"You are a pure-blood,\" the Alpha stated as fact.\n\nNo use denying it. \"Yes.\"\n\n\"It has been years since I've seen that kind of fighting.\"\n\nSlash smiled, and the Alpha cocked his head.\n\n\"Let me get you up to speed then.\"\n\nSlash attacked.\n\n*\n\nJulia landed on Tom Harriet's back. Her one-hundred-fifteen pounds wasn't going to do much, but she gave it her all.\n\nHe grunted, tossing her off.\n\nShe grabbed what clothing remained on his body. A stubborn button-down shirt still held by one torn sleeve.\n\nJulia gripped it like a rope, as though her life depended on it.\n\nThen they were both airborne.\n\nJulia's arms spun out at her side, trying to break yet another fall.\n\nMiraculously, she didn't land but floated.\n\nJulia's eyes opened, and she beheld Jacqueline. She stretched her arms out at shoulder height.\n\nGradually, she lowered Julia.\n\nThey stared at each other. \"What in the hell is going on?\"\n\nJacqueline smiled at Julia's question. \"Let's chat when we gain some distance from the Were.\"\n\nJulia's gaze found him, an unconscious lump of odd, broken body parts.\n\n\"He'll mend,\" Jacqueline said.\n\nJulia was aware.\n\n\"Thank you,\" Jacqueline said quietly.\n\nJulia gave up the mental struggle and conveyed her true thoughts. \"I don't trust you.\"\n\nJacqueline glanced down. \"I know,\" she spoke to the ground.\n\n\"Before we run off together\u2014by the way, worst idea ever,\" Julia paused, \"what's happened? I mean, you've had it out for me, tried to murder me... now I'm okay?\"\n\nJulia watched the older woman's face.\n\n\"I have enough fey blood that I've been sickened with never being in Faerie. These past weeks I was in the prison, I began to...\" She flung her palms out then gripped them and wrung them.\n\nJulia was beginning to feel sorry for her.\n\n\"Become self-aware. The alliance with Anthony Laurent seemed foolhardy. My past methods I viewed as horrible, motivated by the near-insanity of whatever was happening to me. I felt as though I'd been drowning without knowing. Finally I saw a way to move to the surface.\"\n\nJulia stared at her.\n\nJacqueline leaned closer and Julia fought to remain where she was.\n\n\"I no longer take in water\u2014but breathe.\"\n\nJulia let out a shaky exhale. \"Okay.\" She looked at Harriet again. Still snoozing.\n\n\"Is he gonna be...?\"\n\nJacqueline nodded. \"I do have experience with my talents. I have not killed him.\"\n\nJulia chuckled. \"Why?\"\n\nJacqueline's eyes searched her own. \"I could not bear it.\"\n\nJulia narrowed her eyes. How real is this change of heart? \"And Tony? Could you kill him?\"\n\nJacqueline remained silent for a heartbeat. A small tremor washed over her. \"For him, I could be persuaded to make an exception.\" She put her hand over her still-flat belly protectively.\n\nStill a little bit of Jacqueline under that new veneer. But she also had someone else to think about. Scott and Delilah's half-sibling. A child born of the fey, also Singer and Were. It was probably a new precedent, even for the supernaturals.\n\n\"I am with child,\" Jacqueline murmured.\n\nJulia put her hand to her mouth. \"Who?\" Oh my God. \"Tony?\" She searched Jacqueline's face.\n\nShe shook her head. \"No. Domiatri,\" she answered softly.\n\nHoly crow.\n\nThey didn't really have time to explore her revelation.\n\nJulia made up her mind. Their conversation hadn't taken long, and there was obviously too much at stake. \"Let's go.\"\n\nJacqueline nodded then looked at Julia's feet. \"That will be a problem.\"\n\n\"Don't really have an option. My choices were limited while we were running from the world.\"\n\nJacqueline chanced a glance at Harriet. He flicked a finger.\n\n\"Let us go,\" she said. \"My telekinetic talent is excellent, but he is also a Red Were and pack master to a renegade den from Alaska.\"\n\nGood point.\n\nThere probably had never been two people so at odds with each other, now forced to compromise to survive.\n\nJulia didn't have time to worry about it. Jacqueline was pregnant, Harriet was dangerous, and her husband could find her if she were a needle in a haystack.\n\nThey ran.\n\n# CHAPTER EIGHTEEN\n\nThe newly revealed pureblood Red Were attacked the Alaskan den's dominant Alpha, but Tharell kept one eye on the enemy.\n\nTen Reds in that pack held court behind the two who soaked the earth with each other's blood. He gave Domi a significant look. Be watchful, it said.\n\nDomi nodded back.\n\nTharell moved forward, capturing Lucius' and Scott's eyes. The female Feeler, Angela, noticed his staring and shook her head.\n\n\"He's not one of them.\"\n\nTharell knew exactly who had taken Julia. He had gone off to lick his wounds and heal.\n\nHe would be back. In the meantime, Slash asserted his dominance in a bid to take over the pack while its master was not there, a savvy move by Were standards. Tharell had judged the scarred one as everyone had, broken because of his obvious wounds. He now surmised that Slash felt he had nothing to lose. That type of supernatural was dangerous, for they were willing to risk much to get what they needed.\n\nSlash pinned the Were beneath him. His own blood dripped down in a steady stream onto the other Were's face.\n\n\"Don't kill me,\" the Were said, his arms straining to keep Slash's hands from tightening around his throat even further.\n\n\"This is like a tap out moment,\" Cyn said from the sidelines, and Tharell frowned at the commentary.\n\nSlash squeezed his fingers.\n\n\"...Second...\" the other Were wheezed.\n\nTharell saw the minute hesitation in Slash's throttle.\n\n\"Think he wants to live,\" Truman commented in a droll voice.\n\nSlash lifted the pressure slightly. The Were coughed, drawing breath, sucking in blood along with oxygen. \"I will be your second...\" A coughing fit ensued, and Slash sat back on his haunches, hands at the ready.\n\n\"We'd be unstoppable, you and I\u2014both purebloods. There aren't enough of us for you to kill me.\"\n\n\"I don't know about that,\" Adi said in a voice as dry as the Sahara desert.\n\nSlash didn't look away. He ignored everyone's comments, seeming to weigh the Were's words.\n\n\"I don't need anyone,\" Slash said, and Adi snorted in the background. \"Hey!\"\n\nSlash quirked his lips. \"Almost no one,\" he amended. Slash stood. The palm he offered to the Were beneath him was human again.\n\nTharell breathed easier when he took it.\n\nThe supernatural tide had turned. Eleven Red Were under their command was a sight better than eleven after their true death.\n\n\"I am Ezekiel,\" the defeated Were said.\n\n\"I am Slash.\"\n\nThe two Reds shook hands, blood their only clothing.\n\nThe ten Reds who accompanied Ezekiel removed clothes from a duffel, and Slash used what was given to him.\n\nTharell thought the Alpha blushed, once nearly covered, when the young female Were said softly, \"Oh... I was kinda liking the view.\"\n\nThe scar on his face deepened to a red lightning strike with his embarrassment, though he didn't respond to her comment.\n\nAdi smiled as though satisfied with a battle won.\n\nTharell looked around. He thought battles were only preludes.\n\nAnd war was coming.\n\n*\n\nSlash backed away from Ezekiel, locating Adrianna. Satisfied she was as safe as possible given the circumstances, he looked at the faces of those who stood waiting.\n\nJason came forward, frantic. \"Julia's gone\u2014Jacqueline's gone.\"\n\nScott scraped a hand through his hair. \"Damn, that is so bad.\"\n\n\"Perhaps not,\" Domiatri said, and Slash glanced his way.\n\nHe went on, \"I sent Jacqueline to track the Blooded Queen. No harm will come to her from Jacqueline's hand.\"\n\nSlash heard Cyn's laugh of sheer disbelief in the background.\n\nScott was in his face instantly. \"How do you know? Do you know what dear old Mom is capable of?\"\n\nDomi ignored the enraged Combatant and answered a question that hadn't been asked. \"I know that she laments her past and wants only to stop the persecution.\"\n\nScott made a rude noise in the back of his throat and paced away, only to pivot back a moment later. \"You can buy into whatever she's selling. But Lucius and I will be getting Julia back.\"\n\n\"Me too,\" Jason said.\n\n\"I don't believe there is any argument in the reacquisition of Julia Caldwell,\" Tharell said thoughtfully.\n\n\"Then what the eff are we waiting for?\" Jason bellowed.\n\n\"Good question,\" Cyn said.\n\nSlash read Tharell easily. \"There's the small matter of Tom Harriet.\"\n\n\"He isn't a small matter.\" Ezekiel put his hands on his hips. \"He will not go quietly. He will not submit to you or anyone.\"\n\nSlash shrugged and restated the obvious. \"That doesn't matter.\"\n\nHe swept his palm at the two Combatants, two mixed-blood Reds, albeit younglings by Were standards. Then there were the Sidhe warriors, Tharell and Domi.\n\nNot to mention he'd single-handedly made sure of his dominance over the pack that now stood within speaking distance. He was bone tired, and that didn't help.\n\nHowever, Slash didn't think they had much to worry about from Harriet for the moment.\n\n\"It will matter. He is somewhere other than here. If he is not here, he is pursuing the same thing you are\u2014and closer to his quarry.\"\n\nHis people shifted uneasily behind him. Only the fey kept still.\n\n\"He won't hurt the Rare One,\" Slash stated. None of the supernaturals would compromise what Julia Caldwell represented through maiming or death. It defeated the entire reason for claiming her.\n\nYet... \"What aren't you telling us?\"\n\nKarl Truman stood beside Slash. He wasted an eye flick on the Were then turned his attention to Ezekiel.\n\nTruman interjected. \"It's the Feebie angle, right?\"\n\nEzekiel looked at the new Red. Finally, he nodded. \"If allowed, Tom Harriet will fashion a manhunt the likes of what you've never seen, spreading lies to close the net of capture around Region One, expose the fey... whoever needs to be outed to make this agenda come to fruition.\" He raised his fist and coughed thickly into it.\n\n\"Follow through kind of guy?\" Truman asked.\n\nHe dropped his hand to his side. \"So much. More than any of us could say. We have the scars to prove it.\"\n\n\"He's a brutal pack master?\" Slash grew uneasy with the picture of Harriet taking shape.\n\nEzekiel's eyes flashed in the gloom. \"Very. But that is not his best thing.\"\n\nSlash didn't like the sound of that, but he had to ask. \"What is?\"\n\n\"Subtlety.\"\n\n\"And Julia's out there with Jacqueline. With a crazy-ass smart Were whose sole goal is to capture her and take her to Alaska.\"\n\n\"He will,\" Ezekiel assured them.\n\n\"They have to get through Jackie-baby.\"\n\nAll eyes turned to Cyn.\n\n\"Here's the thing, guys.\" She rolled her eyes, and they could all see what color they were, despite the low light. \"Jacqueline's supposedly gone through some \u02bbmetamorphosis',\" she began, clearly in a state of disbelief. \"But no matter how much she's supposedly changed, I'm willing to bet there's still a fat streak of bad in her. I think she'll give Harriet a run for his FBI money.\"\n\nCaldwell angrily scrubbed his chin. Finally, he looked up. \"Jules is running on not having very many talents. And those she does have? They're random and of varying levels...\" His worried eyes despaired, and Slash felt for him.\n\n\"Let's go find them,\" Scott decided for the group. Jason threw his arms in the air like no shit.\n\nLucius turned to Angela. \"Do you see a remnant? Can you see anything?\"\n\nA Feeler could see auras, the fingerprints of a supernatural. No two auras were alike.\n\n\"Yes.\"\n\nHe gripped her shoulders. \"Do you see the Singers?\"\n\nShe shook her head. \"No. I\u2014it is easier to see an aura of a host who is actively malicious in their intent.\"\n\nJason walked away.\n\n\"Caldwell!' Scott yelled, jogging to catch up.\n\nJason turned to Angela. \"Can you see Tom Harriet's aura? The remnant?\"\n\n\"Oh yes,\" she responded.\n\nJason stood still for a moment. \"What does it look like?\"\n\nAngela shivered. \"Evil,\" she answered.\n\nSlash strode after Caldwell. That had been answer enough.\n\nIf Harriet wanted a war, then he'd come to the right male. War was Slash's natural state of being.\n\nUnfortunately, something had changed when Adrianna had shown interest in him.\n\nHe hated the weakness. Slash hated the uninvited emotion that knocked at the door for entrance.\n\nHope.\n\n# CHAPTER NINETEEN\n\nJacqueline was fully aware that Julia Caldwell did not trust her. She could not say she would feel any differently had their roles been reversed.\n\nJulia broke into her thoughts. \"Do you know where the hell we're going?\"\n\nJacqueline didn't bother lying. It wouldn't help their tentative and hopefully auspicious alliance.\n\n\"I do,\" she answered, moving quickly through the dense underbrush. She couldn't help feelings of weakness. That she so soon forgot what it was like to be pregnant irritated her. It should have been unforgettable.\n\nShe tried not to view the fetus as a parasite and couldn't quite come to terms with it. The tiny group of cells inside her continued to grow, sucking the very life from her weakened body without mercy.\n\nJacqueline moved stoically forward, matching the shoeless Rare One stride for stride. She answered Julia's earlier question. \"I feel the sithen... it calls to me.\"\n\nShe caught Julia stealing a glance in her direction at the comment and elaborated, \"Domi has told me that all those, supernatural and human alike, who possess a great enough portion of fey blood, will feel the summons of Faerie.\"\n\n\"Domi?\" Julia asked, narrowly avoiding the backlash of a small alder branch as she pushed it aside.\n\n\"The Sidhe warrior,\" Jacqueline replied quietly.\n\n\"The green fey? The father?\"\n\nJacqueline, breathing heavily, nodded to save energy.\n\nThe women moved out of the tree line and came upon a small town.\n\nBlaine, Washington, Jacqueline read. Ahead was a small restaurant named Bob's Diner. The B in the neon sign flickered in a disquieting way.\n\nShe cast a glance at her and Julia's terribly disheveled clothing.\n\nHer stomach made up her mind, heaving in a roaring growl.\n\n\"God! Was that your stomach?\" Julia asked.\n\nJacqueline turned, vertigo capturing her. She looked straight at Julia's face and saw her mouth moving. The tunnel of her vision narrowed, and Jacqueline felt as though she was drowning.\n\nMotes danced before her eyes.\n\nJulia reached out.\n\nDarkness engulfed.\n\n*\n\nJacqueline's stomach gave a low, wail-like growl, and Julia turned to her, asking if that could possibly be real. Whose stomach growls that way?\n\nA pregnant woman's, apparently.\n\nJacqueline was ghost-like in her paleness. Dark eyes stood out like depleted black marbles, their luster gone.\n\nJulia took a step toward the smaller woman, her enemy.\n\nThose dark eyes rolled up in their sockets.\n\n\"Oh no!\" Julia cried. It was the worst time and place for a fainting spell.\n\nJulia caught her, and they fell to the ground together. She broke Jacqueline's fall, rolling her over onto the soft shoulder between road and forest.\n\n\"Wake up,\" Julia said a little frantically. They couldn't be out there in the open. It was entirely too possible, with everyone on their tail, they'd be discovered. And Julia was very aware of her special vulnerability. No shoes, her talents all in a knot.\n\nNo, best to drag Jacqueline back into the borderline of the woods.\n\nJulia put her ear close to Jacqueline's face, not hearing breath.\n\nShe leapt to her knees in a panic, straddling her, and hovered her hand over Jacqueline's mouth.\n\nWarmth caressed her skin. She was alive.\n\nJulia eased out a shallow breath. Her laugh sounded like a sob. It was surreal that she was tasking herself with keeping Jacqueline alive.\n\nHer attempted murderess.\n\nHowever, Julia could see Jacqueline's aura. She didn't know for sure, but she didn't believe those could lie. An iridescent white to pinkish color swirled around her like ribbons of cotton candy in the wind.\n\nShe wasn't the Feeler Angela was but enough to bolster her confidence that they could at least seek refuge together without Jacqueline trying to do her in again.\n\nJacqueline's eyes fluttered open.\n\n\"What has occurred?\"\n\n\"You passed out.\"\n\nJacqueline covered her eyes with the back of her forearm. She began to cry, the tears wetting her sleeve.\n\nOh wow, just\u2014wow.\n\n\"Hey.\" Julia patted Jacqueline's arm awkwardly while covertly scanning the area. \"It's okay... Jacqueline.\" Julia ignored the current weirdness and took Jacqueline's arm away from her face. \"Let's get some food, then when the batteries are recharged, we'll figure out where we are and get someplace safe.\" Julia tried on a smile she didn't feel and forged onward. \"And Jason and the others will be right after us.\"\n\nJacqueline sat up and rubbed her eyes. \"I feel so foolish.\"\n\n\"Don't\u2014\" Julia began, but Jacqueline put a palm up.\n\n\"I wish that I could be somewhat as I was before. But now, I am nothing but a weak, clinging mess.\"\n\nJulia didn't know how to respond to that. \"Well, I think you're a tired, weak, disoriented and very hungry pregnant woman.\"\n\nJulia stood, offering her palm. \"Come on.\"\n\nJacqueline stared at the proffered hand.\n\n\"I think everything will look better after a meal.\"\n\nJacqueline took her hand. The two women made a beeline for the diner.\n\nTwo pairs of eyes followed them.\n\nOne set turned away, as he replied into his mic, \"Subject spotted and identified. Repeat, subject positively identified.\"\n\nHis well-tailored arm dropped into his lap, setting the binoculars to rest on his thigh as he looked at his partner.\n\n\"Gotcha,\" he said to himself.\n\n*\n\n\"Do you have human currency?\" Jacqueline whispered.\n\nJulia nodded. \"I know it's totally weird, but Lily, the imposter\"\u2014Julia paused in a small huff of memory\u2014\"used to insist I always carry a $20.\"\n\nJulia showed Jacqueline the corner of green inside her pocket. \"Can't believe it survived intact.\"\n\nThey walked through the diner's front door, a bell announcing their entrance.\n\nMany local gazes drove down their bodies in curious perusal. Julia felt horribly conspicuous with her bare, and rumpled appearance. In fact... \"Hey, let's use the restroom.\"\n\nJacqueline gave a grateful nod and the two walked into the women's restroom.\n\nJulia couldn't believe how awful she looked. Not a patch of skin without grime. She was afraid she'd make a clean spot if she scrubbed her face.\n\nShe did anyway, beginning with her feet, taking her time.\n\nShe looked up into the mirror, and Jacqueline's amused expression greeted her. It was the first genuine bit of anything close to happiness Julia had seen on her face. \"What?\" Julia asked, cleaning between her last two toes.\n\n\"You're aware that your feet will become dirty yet again.\"\n\nJulia laughed. Awesome point. \"Yeah, I know. But it feels so good to get all the ick out.\"\n\n\" \u02bbThe ick'?\" Jacqueline asked with a slight frown.\n\nJulia laughed again. \"Y'know, all the dirt and crap.\"\n\n\"Oh,\" Jacqueline answered quietly.\n\nJulia looked at Jacqueline's face. She took her foot off the bathroom sink's rim and tore off a sheet of brown paper towel. She ran the hot tap and put the sheet underneath the water. She took it from under the hot spray and pushed a dollop of shimmering pink soap from the dispenser onto the towel.\n\n\"Come here,\" Julia instructed.\n\nJacqueline moved with docility, closing her eyes when she reached Julia.\n\nJulia began at her forehead. She ended at her neck. \"There,\" she said, surveying her handiwork.\n\nJacqueline opened her eyes. \"Thank you.\" She looked at the floor, and Julia threw the towel away and opened the bathroom door.\n\nShe didn't turn around, giving Jacqueline a moment to compose herself.\n\n\"Let's go get some grub.\"\n\nJacqueline leaned back with a satisfied burp that she tried to hide with a palm.\n\nIt didn't work very well. Julia laughed, her guts protesting after the inhaling of the meal.\n\nA thought broke through Julia's mental shielding she'd been mastering so well.\n\nThis is divine.\n\nIt's good, isn't it? Julia responded in her mind.\n\nJacqueline's eyes widened. \"That is... very untoward, Julia.\"\n\nOh... rude. \"Sorry,\" she mumbled. \"I just heard... I heard your thoughts.\"\n\n\"Were they bad?\" Jacqueline asked shyly.\n\nJulia shook her head with a small laugh. \"Nah, they were a lot like mine.\"\n\nThe tension leaked out of Jacqueline's shoulders.\n\n*\n\nThe women surveyed the table littered with empty plates and ketchup beginning to gel with air exposure. Metal cups that contained the extra shake they'd been unable to choke down sat between them.\n\nOne lone fry lay on Julia's plate. Jacqueline eyed it with interest.\n\nJulia laughed. \"Go ahead.\"\n\nJacqueline picked it up and tossed it into her mouth, chewing happily. Interesting, when Jacqueline wasn't evil, she was okay to be around.\n\nHelps when she's not trying to kill me.\n\nThe bell sounded behind them, and two men walked in. Julia's face must have changed, because Jacqueline stopped mid-chew. \"What is it?\"\n\n\"Looks like we've been discovered.\" Her voice sounded morose even to her.\n\nJacqueline did not turn around. \"By whom?\"\n\n\"The FBI,\" Julia whispered.\n\nJacqueline put her palms flat on the table, and her nostrils flared. When she opened her eyes, they met Julia's.\n\n\"No.\" Jacqueline shook her head. \"They are Were.\"\n\n\"Posing as agents?\"\n\n\"I do not know. It sounds like that might be the case.\"\n\nGreat.\n\n\"Maybe they're not looking for us.\" Julia dared to hope.\n\nJacqueline sighed. \"They are Reds.\"\n\nHope fled along with her appetite.\n\n# CHAPTER TWENTY\n\nSlash stayed forward, Adrianna following close behind. He took in the myriad of scents present in the forest, widening the gap with that of the corpses they'd slaughtered to escape.\n\nA familiar scent caught him, grinding his progress to a sharp halt.\n\nHis new pack slowed behind him. Yips, soft howls, and snuffles met his ears. The Were were all in their half forms, the Combatant were keeping up nicely, and the Unseelie Sidhe were taking up room at the back of the group.\n\nThe Singers' Feeler, Angela, slid off a Were's back. She was the only one in the group that struggled to keep up, her talents not of the strength or speed variety. It necessitated a carry.\n\n\"Harriet is somewhere nearby,\" Slash told the group at large.\n\n\"His scent, you mean,\" Truman corrected.\n\nSlash gave the new Were a hard look. \"Perhaps.\"\n\n\"Let's keep moving. It's him against us. He'll lose.\" Jason was already moving toward Julia's and Jacqueline's scents.\n\n\"I smell adrenaline dipped in fear,\" Ezekiel murmured.\n\n\"I see pieces of aura that mix,\" Angela added.\n\n\"Whose?\" Cyn asked.\n\n\"Jacqueline... and the Rare One.\"\n\n\"So they are together,\" Domi said.\n\nThe fey seemed to think it very important they all understood his planning prowess.\n\nSlash remained unconvinced. If they scooped up Julia in their net and Jacqueline hadn't harmed her again, then he might think better of the fey. But for the moment, Slash thought the Faeries to be an unknown for potential violence.\n\n\"I don't like leaving Tom Harriet as a loose string,\" he said, catching his breath from the run.\n\n\"I don't either, but getting Julia is most important. Let's worry about that chump after we've got her safe.\" Jason lifted his eyebrows, and Truman slapped him on the back.\n\n\"You're right, kid, but I'm with Slash on this one. I think it's a mistake not to flush this puppy out and work him over. He stole Julia. He's a pack master\u2014\"\n\n\"Was,\" Adrianna interrupted.\n\n\"True... but he didn't get the memo on that, sweetheart. He still thinks he's top dog.\" Truman turned to Angela, his eyes moving from their natural blue to his wolf's green. \"Do you see his aura? Maybe we lucked out and he bit it.\"\n\nCyn laughed.\n\nSlash frowned. The Were spoke differently and with much slang. A few curse words seemed universally understood, but this male used a metaphor every other phrase. It irritated him.\n\n\"No,\" Angela said shortly. \"He is very much alive.\"\n\nThe group was silent except for the Reds, who circled the area, eager to move, wherever it might be.\n\nSlash put his hands on his hips, absently scratching at the itchy material that barely covered him.\n\nIf the females had not been around, he'd go nude. But they were, and only one Were was in charge of packing clothes. They'd hit a thrift store as soon as they got to the nearest large town and suit up with used clothing.\n\nHe looked at the females and felt a pang of envy. Females could do a quarter-change. It meant all the speed without the looks and burst clothing. The ability was present for the contingency of pregnancy. They could not shift into fully wolves while carrying young. It was protective of the species and their procreation.\n\nOnly the eyes appeared wolfen, glowing and luminous. They shone back at Slash from the two women.\n\n\"We're going to need a supply run. Adrianna,\" Slash called, and she jerked her face up, her normally brown eyes a light gray. Her wolf swam just beneath her surface; Slash could see it. Hell, his wolf could smell it.\n\n\"Don't, Slash,\" Adrianna said.\n\nHe flushed. He'd been thinking too hard about her wolf, and it wanted to come out. \"Sorry. I\u2014\" He looked at her, remembered his ugly mug, and looked away. \"We'll need clothes and food.\"\n\nAdrianna turned to Cyn. \"We've gotta get clothes for the dudes, and groceries.\"\n\nAs if on cue, several stomachs voiced their empty displeasure.\n\nCyn laughed. \"Men.\"\n\n\"Let's go,\" Jason said.\n\nSlash reluctantly turned away from where Tom Harriet's scent was strongest.\n\nHe left, Truman following where Slash's gaze had been, seeing nothing. Though his nose said differently.\n\nIn the darkness, Harriet watched them leave with guarded relief. His task force should be closing in.\n\nHe'd already thought about this contingency.\n\nNo one became a packmaster of his caliber without brains.\n\n*\n\n\"Okay, here's what we'll do,\" Julia began.\n\nJacqueline held up her palm for the second time they'd been traveling together. \"Your talents are sporadic, yes?\"\n\nJulia hated to reveal anything confidential to Jacqueline, but now they were in danger. The greater evil and all that. Julia reluctantly nodded.\n\nJacqueline silently searched her face. \"Do not be embarrassed. We all must become.\"\n\nYeah.\n\n\"This is what I propose. The FBI shall not want witnesses to see they are Were. I assume they're after you. I will sacrifice myself to give you time to\"\u2014Jacqueline waved a small hand around\u2014\"make your fortuitous escape.\" She arched her black eyebrows delicately.\n\n\"No way. They'll hurt you.\"\n\n\"They can try,\" she said quietly. \"Yet, they are Reds. They will be loath to hurt a female of Were blood. And they will scent I am with child. There has never been a documented case of a Were harming a female with child.\"\n\nJulia's gaze went to Jacqueline's. \"What about Tony?\"\n\nShe regretted it the instant the words were released.\n\nThat dark gaze slid away from her and looked out the window, her expression terrible to witness. \"I think it is a fair assessment to say he does not fit within the norms of the typical supernatural male.\"\n\nJulia's eyes flicked to the two behind them.\n\nThey were coming their way.\n\nShe leaned forward. \"Maybe he's something new.\"\n\nJacqueline brows came together.\n\n\"You guys didn't know much about the fey and look, there's an entire mound of them. All I'm saying, is Tony could be something else besides just Were.\"\n\nJacqueline didn't answer.\n\nA hand landed on their dinette, shaking the salt and pepper holders.\n\n\"Ladies,\" the one on the left said. They wore identical suits, their eyes buried beneath dark and slightly reflective lenses. Julia figured if you couldn't see their eyes, then they were soulless.\n\nThey flashed their badges.\n\nOf course, the ID looked authentic. Tom Harriet, the Red Were pack master from Homer, Alaska, was legit, too.\n\nJulia wiggled her cold toes underneath the table.\n\n\"You can make this easy,\" Leftie said smoothly, his strong hands a shadow of muscles rippling like disturbed water over his skin as he slid his badge home.\n\n\"Or you can make it hard.\"\n\nJulia scowled. Her belly was full, and all she wanted was a nap and Jason.\n\nHome.\n\nRegion One was a dysfunctional mess, but she belonged there.\n\nJulia looked at Jacqueline.\n\n\"Hard,\" Jacqueline purred in an obscene word. The jagged edges of the old Jacqueline showed like a slip underneath a dress.\n\nObviously a deliberate choice, her switching gears still surprised Julia.\n\nLeftie's gaze slid to Jacqueline, his nostrils flaring hard. \"We don't want to hurt you.\"\n\n\"That's what you two thugs think, but in the end, robbing someone of their freedom is the very worst thing you can do to your fellow beings.\" Julia stood. She'd left the \"human\" part out.\n\nJacqueline stood as well.\n\nRightie's nostrils swelled as he took in their scent. \"She is pregnant.\"\n\nLeftie sighed. \"That complicates things greatly. You're a mongrel, but you have enough Were to preclude us from doing what we must to assure your compliance.\"\n\nJacqueline opened the pathway of their minds.\n\nJulia's mouth parted in a little O of surprise. Jacqueline had been working to keep her out since their earlier interchange.\n\nI will cause a scene and you run. I am a powerful telekinetic. Trust in me.\n\nTrust?\n\nYou must.\n\nThat was the truth.\n\nOn my signal.\n\nJulia wondered what it would be.\n\nWhen Leftie reached for Jacqueline's small wrist, the air by Julia moved. Jacqueline drove him through the first wall. The drywall sheet buckled, collapsing quickly to allow him to travel into the 1950s cinderblock behind.\n\nHe blew through the huge cement bricks, pots and pans clattering to the floor.\n\nJulia recognized a clear signal when she saw it and took off.\n\nShe heard the sound of flesh being slapped and didn't look back.\n\nTrust in me.\n\nShe felt like a jerk leaving Jacqueline.\n\nJulia could only hope she was as defensible as she claimed.\n\nShe leapt down the stairs and into the arms of Leftie.\n\nHe locked strong arms around her in a bear hug. Julia sneezed from the spray of concrete dust plumes that exploded in her face.\n\nThen a wash of blood as someone tore his arm off.\n\nA sharp bark sounded, cut off with the prize in hand. Jason casually threw the limb behind him.\n\nJulia's relief was so acute she wanted to sob.\n\nInstead, she yelled, \"Save Jacqueline!\"\n\nShe had never uttered stranger words.\n\n# CHAPTER TWENTY-ONE\n\n\"What?\" Jason yelled over the screaming patrons behind them.\n\nThe diner was emptying, screaming eaters running for their cars in a stampede.\n\nSo much for being subtle and preventing humans from getting a load of the supernatural turf wars.\n\n\"Jacqueline! She's in there with one of the Bozo-the-clown Feds!\"\n\n\"I'm not leaving you!\"\n\n\"Jace\u2014she's pregnant...\"\n\nHis eyes widened, then he shook his head. \"Let the jolly green giant get her. He's the one that knocked her up; he can figure it out.\"\n\nAs though summoned, Domiatri came rushing past them in a green streak.\n\nJulia went to go after Domi, but Jason held her back. \"No effing way. Let green boy take care of Ms. Jekyll and Hyde.\"\n\nJulia tore out of his grip. \"I get it, but she tried to save me back there.\"\n\n\"It's about damn time! You're not going. Your preservation instincts are about zero so I'll protect us both.\"\n\nA noise like a million eggshells being ground startled them, and they turned together.\n\nA large tank rolled down the street.\n\n\"What the hell?\"\n\n\"I think it's a firepower thing.\"\n\n\"Run!\" Jason yelled and jerked her after him. She ran full out, bare feet punctured by small rocks.\n\nJulia didn't know tanks could be so loud. The gun rotated on the turret, grinding its shrieking ambition.\n\nJason ran to where the others had gathered.\n\nJulia made out Tharell easily, the only purple man on the street.\n\n\"The witnesses will be gone in any event,\" Tharell said in a droll voice as a boom that rattled Julia's teeth went off.\n\nTruman jerked Cyn and Angela, putting them behind him. \"What the blue hell?\"\n\nJulia scanned the environment for Adi and finally saw her next to Slash.\n\nThey raced toward Julia and Domi, and a limping Jacqueline brought up the rear.\n\nLike agitated ants, men in military uniforms poured out of the top of the tank, their guns aimed at the four who hadn't made it across the road and into the patch of woods.\n\nJulia screamed a warning. Adi had begun to lag. Slash tucked her underneath his arm and ran with her as if she was a clutch purse.\n\nBullets sprayed dirt at their feet as they ran.\n\nBut the real horror was the careful aiming at the diners. As if slapped by a giant hand, the bullets drove into them from behind, flinging them forward like broken dolls. They landed face first in the dirt and gravel.\n\n\"No...\" Julia whispered.\n\nShe didn't realize she was crying until it dripped off her jaw.\n\n\"Those merciless fucks,\" Truman said.\n\nTharell regarded him. \"It is the way the humans deal with that which they do not understand.\"\n\nJulia turned to him in a fury, grabbing the once beautiful warrior's tunic and shook him.\n\nIt was like trying to shake a solid concrete wall. \"Do something!\" she screamed into his face.\n\nTharell gave a grim smile of acknowledgement. \"I will try.\"\n\nThe Sidhe made complicated gestures in midair.\n\nA minute slipped by, and more people fell like cut trees. Julia could taste metal in the air and realized it was their blood.\n\n\"Hurry,\" she said.\n\n\"It must be done properly,\" Tharell answered without looking at her.\n\n\"What, for the love of God?\"\n\n\"The spell,\" he said, and suddenly Julia could feel the magic in the very air. It rushed past them and raised the small hairs on the back of her neck.\n\nJulia gasped when the charged force struck the militia.\n\nInstant chaos ensued. They ran into one another; some shot randomly. The remaining survivors, seeing the confusion, took off. Many remained motionless in a pool of their own blood and bits of their bodies.\n\n\"What's wrong with them?\" Julia asked.\n\nAdi reached them with Slash at her side and Domi half-carrying a beleaguered Jacqueline.\n\n\"Who gives a ripe good shit, let's get outta here!\"\n\nDomi chanced a glance behind him and smirked, giving Tharell a full wattage smile. \"Nice work, Tharell.\"\n\n\"I couldn't have you bleeding out there and ruining your perfect skin...\"\n\n\"I'll heal,\" Domi responded dryly.\n\n\"What are they doing?\" Julia asked as one of the soldiers tripped over his feet into a graceless pile on the ground.\n\nA Red Were strode to the fallen man and kicked him in the teeth. The soldier's head ripped partially off, canted at an awkward lean as he fell from his knees straight back.\n\n\"Okay... let's go.\"\n\nThe stares of approaching Reds met Julia's comment. \"Oh my God!\" she yelled and began to run.\n\nJason yanked her back. \"Shh, Jules, they're okay.\" He stroked her hair as they drew nearer.\n\n\"They're under me now,\" Slash said.\n\n\"How'd that happen?\" Julia tried not to let the caterwauling in the background distract her. Holy smokes, I'm gone for a few hours and the Reds surrender?\n\n\"Fight for dominance,\" Adi replied casually.\n\n\"It was way worse than that. More like fight to the death,\" Cyn said. \"These guys\"\u2014she jerked a thumb in the general direction of everyone but she and Jason\u2014\"make nothing out of all the 'I almost killed ya' moves.\" She rolled her eyes. \"Really? It's more like a sport.\"\n\n\"It is not a sport, female. It is the Were way,\" a Red said and immediately Julia knew he must be important within the hierarchy of the pack. He had a way of speaking with authority.\n\nA low growl split Slash's lips, and a hard glance from the Were who had just spoken shut him up.\n\n\"That is Ezekiel.\"\n\n\"Zeke.\"\n\nSlash leveled the smallest chin dip she'd ever seen at the Were and turned to Cyn. \"We'll explain this on the way.\"\n\n\"Great, 'cause, I'm dying to get out of here before more winners show up to kill defenseless Americans. Not. Cool.\"\n\nJulia cast a glance at Jacqueline. She looked positively haggard. Domi's eyes met hers. His gaze held assurances she didn't think he could really give. Jacqueline seemed to barely be hanging on.\n\nJulia quickly appraised the group. Twelve new Were, all varying degrees of Red-blooded wolves. Tharell, Domi, Jacqueline, Adi, and Cyn followed closely behind, and as she turned away Scott, injured but managing, was there, held up by Lucius and Angela the Feeler at his sides.\n\nThey took their mixed bag of supernaturals and beat feet out of there.\n\n*\n\nSlash kept the pace deliberately hard. Only when Julia stopped them did he pause in his objective to put distance behind them.\n\nHer feet were ribbons of gore. Filthy, bleeding, the soles torn from her travels.\n\n\"Oh my God, Jules,\" Cyn said.\n\nJulia nodded, gritting her teeth.\n\n\"Can you heal her?\"\n\nCyn studied the Rare One's feet.\n\n\"I'll try, but it seems these reoccurring injuries are their own kind of stubborn.\"\n\nScott crossed his arms, his healing abilities as Combatant having restored him perfectly. \"We need to find some place to hole up.\"\n\nCyn laid her hands over Julia's wounded feet, and she gasped from the pain.\n\nThen groaned in pleasure as they began to mend.\n\nCyn shook her head. \"This is the best I can do. We need to find some shelter... and some goddamned shoes.\"\n\nShe sat back on her haunches, flinging her hands up. \"Was it those flimsy Keds?\"\n\nSlash couldn't help the lift of his lips. It appeared to be a well-worn argument between the two. Apparently, some things didn't change.\n\n\"Yeah.\"\n\nJulia and Cyn looked at her Nikes on her own feet, dirty but intact. \"Spend the money on footwear.\"\n\n\"Coming from the UGG devotee,\" Julia said, rolling her eyes.\n\nAdi piped in, \"Those stand for fugly in my opinion.\"\n\nTharell put up a hand. \"Though I appreciate the efforts at levity, I do like the suggestion of lodging, running water, and extra garments.\" His gaze went to Julia's feet. \"And footwear.\"\n\n\"Where?\" Zeke asked.\n\nSlash searched the geography. Judging by the mountains to the east, they were coming close to Bellingham. \"Twenty more minutes of walking and we'll spit ourselves out in Bellingham.\"\n\n\"Big city?\"\n\nSlash thought about it. \"Big enough. Ample camouflage, numerous eateries. We should be able to get what we need.\"\n\n\"As long as Jules gets some shoes.\" Jason picked her up.\n\n\"Put me down,\" Julia said, indignant.\n\n\"Nope. We're not gonna undo Cyn's work. Don't worry, that little bit of weight you've packed on won't slow me down.\"\n\nJulia sputtered, and the males shared a good-natured laugh.\n\nSome of Slash's tension dissolved. The place of respite couldn't come soon enough.\n\nThere they would have plenty of time to dissect the new threat.\n\nSlash felt he knew.\n\nHe hoped he was wrong.\n\n# CHAPTER TWENTY-TWO\n\n\"It is time,\" Praile told Tony.\n\nHe slid the curved saber across the scarred surface of the table between them.\n\nTony didn't touch the thing. He was as fucked up a supernatural as any he'd ever encountered. Intellectually, he understood that. He'd embraced what he was long ago.\n\nPraile didn't give a shit.\n\nThe dark metal glimmering in the shadowed room he shared with the demonic was a tangible reminder of his mixed blood.\n\nPraile folded his muscular arms against his bare chest. His stubby horns rose from a skull free of hair. Not all demonic males possessed a tail. Praile did, a nod to his breeding amongst Hades' otherworld creatures.\n\nNot of this world but on another plane of existence.\n\nTony reached out, gripping the finely honed wood. It was narrow where it met the arcing swath of metal, flaring to a deep triangle at its base.\n\nCarved runes he couldn't read, but caused a thrum of electricity when his fingertips traced over the etchings, deepening his anxiety by the moment.\n\n\"Why now? After hundreds of years, I need to fulfill my duty?\" Tony wheedled.\n\nCoal black eyebrows like slashes of inky pain dropped over glowing red eyes. A slow smolder lifted off Praile's skin, his agitation manifesting into a broil. He placed two palms on the table that separated them.\n\nTony shifted in his seat, containing his nervousness badly.\n\nPraile said nothing at first, and the silence tore at Tony. He swallowed the brick inside his throat, his anxiety lodged and painfully sliding down to settle uncomfortably in his churning stomach.\n\n\"Because there are few who survived this world with demonic blood. Werewolves, with their shifting abilities, are uniquely suited vessels for what we offer.\"\n\nTony didn't say what he was thinking.\n\n\"Where do you think your evil impulses stem?\" Praile asked, though his question did not require a response. He straightened. \"Do you think your assaults against females, both Were and non, have been accidental?\" His eyes narrowed to slits of fire. \"After all, it is against your primal nature to harm females, yet you have proven an aptitude of great finesse in this regard.\"\n\nTony looked at a point above Praile's shoulder.\n\nThe demonic moved into his direct vision, his eyes like embers, low and burning.\n\n\"No,\" Tony answered in quiet resignation. He should have known the demonics would never release him from his obligation to them.\n\nPraile knotted his hands behind his back, nails black-tipped and shaped like short talons.\n\n\"You will do this thing. Then, when you are done, we will have decimated the one species that stands against our objective.\"\n\nTony raised an eyebrow. \"What about the fey?\"\n\nPraile laughed from his belly, clapping his hands together in glee. It startled Tony so much he almost dropped the saber.\n\n\"We do not concern ourselves with the Unseelie Sidhe. The immortals are Faerie bound and do not pose a threat. The strongest of them is weak without their precious sithen. No.\" Praile shook his head. \"We eradicate the Singers, vampire and force-breed the Were that can produce more of the blood of Hades, and soon our kind will rule this world as well.\"\n\nTony's guts sprouted spots of fire like acidic flowers in bloom. He swallowed his slow terror. \"What about the Reds?\"\n\nPraile's smug happiness disappeared as soot cleaned from a mirror. \"They are only a problem if banded together.\"\n\n\"When we met last, you told me they were the ancient transition between Hades and the supernaturals of this world.\"\n\n\"I know what I have said\u2014every word,\" Praile hissed and Tony flinched. He didn't want to experience the bloodlust of this particular demonic, a torture better executed than any he'd ever survived, ever again. Blood and magic indebted Tony to whatever Praile would have of him.\n\nPraile tapped the blade, and a drop of ebony blood sprung from his injured skin. He sucked it off in a long and noisy pull.\n\nTony neutralized his expression when the fork of Praile's tongue trembled against the oozing blood.\n\nThe demonics were disgusting.\n\nOf course, Tony was as well.\n\nHe liked his brand better.\n\nPraile opened the wound further, running his finger down the entire length of the two-foot-long blade. He rolled his glowing eyes up to meet Tony's. \"Kill every one that you come across; men, women and children.\" His black eyes bored holes into Tony. \"Kill their pets if they have them.\"\n\nTony opened his mouth. The Singers' plethora of talents, diverse and in varying degrees of strength, made them dangerous.\n\nIt was suicide.\n\nHe seemed to sense Tony's question. \"You will be lucky to survive, yet I care not. The metal of this saber is proof against most talents. You were bred for this.\"\n\n\"Who is immune?\"\n\n\"Watch your tone, demon cast off.\" Steam rose from Praile's mouth with his words, though the surrounding air didn't warrant it.\n\nTony stayed silent. Better he say nothing than have his tongue torn out.\n\nPraile grinned, his teeth black chips of banked night, the only color his divided red tongue. It was a grotesque sunset inside his mouth.\n\n\"The Rare One, of course, and any who have blood of the Red.\" Praile made a disgusted sound in the back of his throat.\n\n\"What talents are immune?\" Tony pressed. Praile's eyes sharpened inside the small, murky cottage that held him.\n\n\"You will know when you cannot use magick to assist in the genocide we bring to this contingent of Singers.\" Praile smoothed his hand down the blade, cupping it.\n\nHe jammed his palm on the blade, the curved and sharpest part slicing so deeply into his flesh, the skin on the backside of his hand stretched in the shape of the blade.\n\nPraile's lips parted in ecstasy, and a low groan of pleasure slid out of his mouth, steam escaping. The affect was immediate on his lower region, his disgusting penis stiffening. The smolder of his skin filled the small house that served as prison to the Singers.\n\nBlood like ink washed the blade and ran in creeping rivulets across the table to Tony. Tony jerked to a stand, watching the river of the demonic's blood crawl to tremble at its edge. The drops fell, hitting the ground with a splat. Like acid, the blood smoldered, eating at the wooden floorboards in an etching that grew deeper by the moment.\n\n\"My blood seals the fate of those who come into contact with it. Do this well and live. Fail, and suffer torture, Anthony Daniel Laurent.\"\n\nTony made a wide berth around the spilled blood.\n\nPraile backed away from the blade, a blood trail falling to drive through the floorboards. Tony curved his hand around the wooden hilt. The magic of the ancient runes, mixed with Praile's blood, swept over him.\n\nTony swung his head back, and a hoarse cry of revulsion laced with lust escaped him. No longer his own master, he controlled the blood that was a part of him. Praile had called it to the surface.\n\nVery much like the Singers who would die by that blade's magic.\n\nTony lowered his chin, and he leveled a stare at Praile, who was already shimmering to opaque, his evil deed complete.\n\n\"Why do I have to do your dirty work?\"\n\nPraile's smile held no happiness. \"Understand that we have been forbidden to interfere directly with the humans and supernaturals. Much as our counterparts.\" He said with weighted reluctance.\n\nAngelics. Tony knew they existed only because he was part-demonic.\n\nPraile hated to admit limits. His own especially.\n\nHe became ghost-like, the insane smile, pinned on his face like a mask.\n\nTony went to the door.\n\nOf course it was locked.\n\nHe stepped back and kicked it open. The door flew off the hinges like a tooth succumbing to periodontal disease. It hung off the bent metal, the jamb's gums weakened beyond holding.\n\nThe two Combatant guards turned, startled. The pretty boy Victor would be a pleasure to end.\n\nTony raised the blade.\n\nWhen they both lay dying, blood poisoned from Praile's contamination, he kicked them in the crotch like the dogs they were.\n\nVic-boy wasn't looking too good.\n\nTony was suddenly bolstered, his evil nature sliding into place like a long lost friend.\n\nFound again and embraced.\n\n# CHAPTER TWENTY-THREE\n\nTharell easily found the low-slung rock formation that signaled the entrance to potential shelter. It seemed to wink at him from a knoll surrounded by madrone trees. They grew toward the sky, throwing themselves from between the rocks. Burnt orange twigs sought the clouds like fingers. Gracefully sweeping boughs of western red cedar whispered against the small cleft of shadow that stood as a door to the observant.\n\nIt would be beyond cozy with the size of their group.\n\nHe stopped in front of the four-foot-tall opening.\n\nSlash delivered a hard look that felt like a slap.\n\nTharell answered the unspoken accusation. \"I am aware we need supplies. I cannot obtain those for you. Obviously.\" Tharell swept a palm down his body.\n\nHe held partial court in looks to what the humans called African Americans, though the nickname struck Tharell as humorous.\n\nBlack people. A low chuckle slipped out.\n\nSome fey possessed skin so true a black its blue highlights shone in any light. Tharell would have preferred that. He enjoyed becoming invisible against his surroundings, and his mixed heritage was only good for blending with the sithen walls when called for. However, his eyes were so vibrant a blue he kept them closed until the enemy's breath was upon his skin.\n\nFaerie was beautiful and edified his fey blood. But it was also a place that, by its very design, was full of danger, treachery, and subterfuge.\n\n\"Fine,\" Slash responded after a moment. Slash deserved careful watching. All Were were rash creatures, but this particular male used his head far more that Tharell liked.\n\n\"I will go. You and Domiatri can stay with the females, and the rest will go.\"\n\n\"Ah\u2014no,\" Cyn said, clearly indicating the idea was dumb.\n\nTharell quirked his lips. The other species were entertaining. It eased his loneliness and allowed him not to be under the constant disdainful tolerance inside Faerie. Tharell liked their patent indifference of his lineage more than he ought.\n\nHe breathed deeply of the human's air, finding it fine as he schooled his emotions into that tightly contained steel box within his mind.\n\n\"I'll go\u2014you boys look like... I don't know, George of the Jungle or something.\" Cyn pointed at Jason, Truman, and Slash, who'd all gone through numerous changes. Clothing hung over what modesty required and no more.\n\nShe glanced at him, and her beauty momentarily struck him. Though she was a Singer, and he would never have a chance at offspring, her steely wits and personality, balanced on a seesaw of fragility, created an interesting intersect.\n\n\"And rainbow guys\"\u2014Cyn gave him a pointed look, and he experienced an uncommon emotion.\n\nHe was embarrassed. By a Singer female.\n\nTharell narrowed his eyes on her, his earlier admiration like amnesia.\n\nShe laughed at his expression. \"You guys will be nailed in a hot minute. That leaves all the non-Were.\"\n\n\"You guys are volunteered.\" Her eyes took in the rest of the group. \"Anybody have cash?\"\n\nSilence.\n\nCyn put her hands on her hips. \"Are ya kidding? I'm digesting my spine.\"\n\nTharell dipped his head, hiding his mirth.\n\nCyn blew hair out of her face, fitting her chin into her palm. \"Okay, I need some fuel. I don't want to be Robin Hood or anything, but we'll have to dumpster dive at McDonald's.\"\n\n\"I'll go,\" Julia said.\n\n\"No.\" Jason turned to Cyn, pointing at Julia's feet. \"Size seven-and-a-half.\"\n\nJulia crossed her arms. \"That's romantic.\"\n\n\"No,\" Cyn said in a huff. \"We look like a dog's butt.\"\n\n\"Amen,\" Adi agreed.\n\nTharell frowned. Canine buttocks?\n\n\"But he remembers my shoe size.\" Julia smiled.\n\n\"Anyway,\" Cyn said, rolling her eyes, \"I'll take the Combatant dudes and Angela, we'll bust into town, snag threads, and get ten million burgers.\"\n\n\"I don't know if that's feasible,\" Lucius said with a frown.\n\n\"Me either, but I'm motivated. Let's book.\"\n\n*\n\nCynthia looked around. Dumb effing place. None of the stores screamed Used Clothes. She'd die twice for a Ross. Huh.\n\nThe Golden Arches beckoned. \"Let's go,\" she said to the Combatant. She didn't know any of them, but the one chick was a mouse. Cynthia never heard what words she said, just squeak, squeak, squeak.\n\n\"'Kay.\" Cynthia put her hands on her hips. \"Do you guys get off the rez that much?\"\n\nAll three pairs of eyes stared blankly back at her.\n\n\"Wow.\" Cynthia rolled her lip inside her mouth, teeth lightly mauling. \"You're going to stand out like a turd in a swimming pool. It'll be bad.\"\n\nScott said, \"We don't have money, and we're all starving.\"\n\n\"Noted, pal.\" Cynthia leaned forward. \"I'll go order, and you two make a commotion.\" Cynthia nodded, but Lucius shook his head. She frowned.\n\n\"We're supposed to stay off the radar, not present ourselves on a platter as a spectacle for humans to witness and remember.\" Lucius was trying to make her see reason. She got that.\n\nWasn't gonna happen.\n\n\"Listen up,\" Cynthia replied to Scott. \"You need to figure something out while I pretend to order.\" She dropped her fingers from imaginary quotes.\n\nScott paced away from the group, and Cynthia watched him jog around the building.\n\n\"I think I want a shake,\" Angela said randomly.\n\nCynthia barked out a laugh. \"It'll melt.\"\n\n\"Oh,\" she replied in a small voice. \"I've never been outside of Region One.\"\n\nCynthia tried to close her mouth. She couldn't. Never been to McDonald's. It was a tragedy of epic proportions.\n\nAngela had big puppy dog eyes.\n\nScott ran up, saving the moment.\n\n\"We can go through the back. You order, and Lucius and I will break the door down...\"\n\n\"And steal cheeseburgers from McDonald's.\"\n\nScott grinned. \"It is somehow funny.\"\n\nCynthia agreed but was too hungry to laugh. The guys went around back while she and Angela walked to the counter.\n\nA young woman with blue hair and scarlet lips approached the front, the D was unraveling in her scripted name tag. \"May I help you?\"\n\nCynthia tried not to stare at the horrible Rainbow Brite look. There'd been no full-length mirror for this chick. She flicked her eyes to Angela, who was similarly blown away. \"I'd like thirty double cheeseburgers and the same amount of water bottles.\"\n\nFuck fries.\n\nThe smell of grease alone formed a river of saliva. She swallowed hard.\n\nThe girl's eyes stayed glazed. \"What?\"\n\nCynthia leaned forward and repeated the order. She was so proud of herself for not strangling her on the spot.\n\n\"You're going to have to wait.\"\n\n\"I thought this was fast food?\" Angela asked, genuinely puzzled.\n\nOh boy. \"It is, but... maybe it's not so fast when you ask for that much.\"\n\nDee, of the nauseating neon-blue hair, said, \"Do you want fries?\"\n\nCynthia shook her head and cast her gaze to the floor before she cracked up. They were dirty, hungry, and the sudden urge to laugh might turn hysterical if she allowed it.\n\nThe door burst behind Dee at the same time Angela gripped her arm in a brutal hold.\n\n\"Ow!\" Cynthia exclaimed, trying to tear her arm out of Angela's vise grip.\n\nThen Cynthia followed her eyes.\n\nAnd met those of Tom Harriet.\n\n\"Oh shit.\"\n\nTom Harriet rushed forward, a streak of flesh-colored paint in the air, with the Weres' breathless speed.\n\n\"\u02bbOh shit' is right,\" he replied before gripping her neck, his manic eyes set on the Combatant behind her. Cynthia felt the breath trapped in her lungs. Her feet left the ground to dangle beneath her.\n\nShe used the distraction of the dudes behind her and brought her knee into his crotch. She aimed for it to go through his body to his head.\n\nHarriet howled, dumping Cynthia.\n\nScott and Lucius leapt over the counter.\n\nBlue hair shrieked and ran as Cynthia's head rapped against the floor.\n\nI will not faint, I will not faint, Cynthia chanted.\n\nThen she did.\n\n# CHAPTER TWENTY-FOUR\n\nCynthia rolled over and puked on the McDonald's floor. 1980s terracotta titles with uninspired black grout became obscured with vomit more bile than anything.\n\nAngela put her hand on Cynthia's shoulder, and she shuddered.\n\n\"How long was I out?\" Cynthia asked.\n\nAngela hauled her to her feet, and she swayed. Angela's grip tightened.\n\n\"Seconds.\"\n\nCynthia focused her blurry vision on her. Not so mousey after all.\n\nShe looked behind her and saw Harriet struggling to rise. Her gaze skipped around the building.\n\nIt was mayhem in motion. Customers were screaming and piling up at the door.\n\nCynthia looked at the cheeseburgers lined up, to-go orders in various stages of assembly. Eff it, she was hungry. The Combatant dudes could figure out Harriet, and she'd suck up the grub. Cynthia hiked her ass on the top of the counter, sliding across it, and bounced lightly on her feet.\n\nShe grabbed the first thing she saw and stuffed a handful of fries in her mouth. Angela followed right after.\n\nThey decimated the large fry.\n\nThen Cynthia took a huge, two-handled McDonald's bag and stuffed all the to-go boxes inside. She swept in a combo of quarter pounders with cheese and even a yogurt parfait.\n\nThe whole kit and caboodle plopped into the bag.\n\nSomeone seized her hair from behind. It felt like the roots were on fire.\n\nShe dropped the bag of food, and that pissed her off on principle.\n\nCynthia cracked her elbow into the head of whoever grabbed her.\n\nA satisfying but gross blood spray hit her neck. Splatter like drizzling rain slid down her right arm.\n\nGawd, she needed a hose-down. She was going to qualify for hazmat apparel soon.\n\nCynthia looked into Angela's wide eyes. \"Reds,\" she whispered.\n\nGreat.\n\nSome dudes whose energy was not human entered the restaurant. Cynthia knew the difference now that she was Were. That tingling live wire felt like insects feasting on her skin.\n\n\"Shit,\" she said.\n\nCynthia scooped up the bag and ran out the back door, Angela in tow.\n\n\"Grab the food!\" Cynthia shrieked.\n\nAngela jerked up the second huge bag of food, the yelling customers in the drive through cursing after them in an impressive blue streak.\n\nNo refunds, Cynthia thought.\n\nThey ran, leaving Lucius and Scott behind. It was shitty.\n\nIt was also their only choice.\n\n*\n\nSometimes great shit happened when you least expected it. Like your enemies begging for their lives.\n\nAnd killing them for sport.\n\nThe movies always portrayed murder as a slick, clean occurrence. The killer moved through people like a change of underwear when, in reality, it was a messy business.\n\nOnly Tony's eyes were free of blood. He was careful to keep them closed on an arterial swipe.\n\nThe blade had swung true, magically sharpening itself as it deftly cut down every Singer in its path.\n\nMarcus, father of the miserable band of siblings, had been the most difficult to cut down.\n\nHis talents had been formidable. But ancient metal of the saber had been proof against even that Singer's powerful royal blood.\n\nHe scowled when he thought of the few who'd escaped.\n\nTony was vaguely insulted that his daughter had thought he'd kill her. His vanity was entirely too rich a tapestry to allow for that. After all, through the union with the bitch, Lacey, Reagan was still part of him.\n\nA demonic, though she remained unaware. It would be that way until Praile or one of the others in his league decided her time had come to work for the legion that populated Hades.\n\nTony sighed, his exhaustion so acute he had to force himself to stand upright. The slaughter of so many had numbed his dominant arm.\n\nHe'd lost count after one hundred.\n\nHis elbow felt like a loose hinge as it hung from his side, and the saber was an anchor. Tony dared not let it go.\n\nInstead, he moved through the old Victorian mansion in which Region One had been headquartered. He reached the kitchen, stepping over a couple of bodies.\n\nHe opened the fridge, and it hit something solid.\n\nMarcus' dead son... What was his name? Tony shrugged. Who gives a ripe fuck? He rolled the body away from the door and peered inside.\n\nSeveral items interested him as he scanned the interior. He locked on the meat and mashed potato leftovers and, with fingertips embedded with dried blood, he pulled out what he needed.\n\nTony carefully set the food dish on the counter and tore off the plastic lid. Taking a solid whiff and finding it unspoiled, he set the long blade on the granite counter top and dug in.\n\nWhen he licked his fingers clean, the flavor of murdered Singers was like a wonderful underlying aftertaste.\n\nA condiment.\n\nTony chortled to himself, an empty sound in the stillness of a house once full of life.\n\n*\n\nJulia clutched her belly. The pain was ferocious, like a storm of glass inside her guts, slicing her to bits.\n\n\"What\u2014Jules? Talk to me!\" Jason knelt beside her.\n\n\"It's\u2014something's wrong back home.\"\n\nJason's face would have been comedic if she wasn't so certain something terrible had happened.\n\n\"What? Home like... Homer?\"\n\nShe shook her head. \"No. Home as in, Region One.\"\n\n\"Okay...\"\n\nJason obviously didn't consider the Singers' Pacific Northwest Region home yet.\n\nJulia did. As queen, apparently her blood knew where it belonged. And right now it was living up to its namesake.\n\nIt was singing. And it was a melody of despair.\n\n*\n\n\"We need to go,\" Julia argued with Slash.\n\nHe shook his head. \"We need food, drink. Us leaving before the other Singers have returned is a recipe for sure disaster.\"\n\n\"Gah!\" Julia threw her hands up. She didn't want to leave before Cyn, Scott, and the others returned with the food.\n\nShe looked down at her feet. Her heart told her something terrible was happening.\n\nOr had already occurred.\n\nJust then, Cyn ran into the small clearing in the middle of the stand of trees.\n\nShe held out the food, and Slash took it. She bent over, sucking in big lungfuls of breath. \"I\u2014\" She coughed. \"Here's the food, but we left Lucius and Scott behind.\"\n\nJulia stepped forward. \"Why?\" Her heart raced, its beating only slightly less audible than her voice.\n\n\"Reds,\" Angela replied, gasping.\n\n\"Damn,\" Slash said in a voice thin with anger. \"I was afraid of that.\"\n\nJulia gave him a sharp look. \"Afraid of what?\"\n\n\"The Reds finding us before we could get out of here.\"\n\nSlash had revealed he was a pureblood, and her husband was also of mixed Red ancestry. Julia hesitated then asked slowly, \"What does this mean for us?\"\n\nSlash held up a finger then handed her a cold cheeseburger and grabbed one for himself. He uncapped a water, taking a long pull. It leveled half the water gone. He tore off a third of the burger and chewed for about three seconds then swallowed the entire chunk.\n\nThough tasting horrible cold, it was the best food she'd eaten in some time. It was even more important to the Weres. When in starvation mode, a Were did not consider sensory pleasures of heat and taste. Sustaining life was the highest priority.\n\n\"What it means is the Reds will take the Combatants and torture them until they tell them where you might be.\"\n\nJulia stalled out, her food a lump in her stomach. \"I... oh, no.\"\n\nSlash shrugged. \"It's what we sign up for during war.\"\n\n\"But this is not war, not really.\"\n\nSlash shook his head like close enough.\n\nJason set his hand, warm and vital, on the small of her back\u2014a comfort.\n\n\"He's right, Jules. It is what it is. That's why you couldn't be sacrificed by going with them.\"\n\n\"You are the queen. We needed supplies, so the beings who were the least strange-looking secured what the entire group needed.\"\n\n\"So now I have to assume that Scott and Lucius are being tortured.\" Julia felt sick and set the half cheeseburger down.\n\n\"It's not like I don't care,\" Cyn said between bites. \"It's that the responsibility for our protection was on them. I was in charge of getting the food.\"\n\nJulia stared down at her dirty feet. They became blurry.\n\nEverything did when she was crying.\n\n\"Come on, babe\u2014maybe they're just late.\"\n\nJulia swiped angrily at her tears and shook her head. \"Nope. That's not the way our luck runs. Right now, we need to eat and get back to Region One.\"\n\nTharell, Jacqueline, and Domiatri looked to her, as did the rest of the Weres.\n\n\"Can we leave a trail for Scott and Lucius if they don't find us right away?\" she asked Slash hopefully.\n\nTruman spoke up, his cheeks distended with the food. \"No way, that's a bread crumb trail for the Reds to find.\"\n\n\"How do you know they'll try?\" The edge of despair clouded Julia's judgment, making her itch to react.\n\n\"I would,\" Truman said.\n\nGreat.\n\nJulia finished her cheeseburger. Talking was useless. Their band of allies was now fractured, and something miserable had gone down in her Region.\n\nShe knew it.\n\nThe half-naked Weres and Sidhe followed her, Scott and Lucius somewhere unknown.\n\nJulia was worried, yet what awaited them when they arrived on the Olympic Peninsula bothered her even more. She knew what Scott was capable of. He had a chance to defend himself. Lucius, too.\n\nNo matter how much Julia tried to calm herself and rationalize that a Singer population with that many, possessing the talent array they did, would be safe, she couldn't.\n\nJulia ignored her bare feet. Her suffering was small compared to what those faithful to her might be enduring.\n\nShe trudged on, Jason at her side.\n\nThe sole comfort.\n\n# CHAPTER TWENTY-FIVE\n\nTharell took the rear position, listening to the discussions of what occurred at the eatery. It certainly wasn't a nod toward subtlety. They had decided against the other Reds going because of lack of wardrobe. In the end, they would have had the nose to identify the Reds who attacked Lucius and Scott. Instead, they were now down two warriors for the remainder of the journey.\n\nIt troubled him.\n\nDomi glanced over his shoulder at Tharell. The look was enough. Domi was on point. Nothing would approach them without the other Sidhe alerting him and that eased Tharell. Though the issue of Jacqueline remained, Tharell would not address it with Domi prematurely. Timing was critical for his long-range goal.\n\nActually, Domi allowed great flexibility in the relationship. It was not Tharell's place to remind Domiatri that Jacqueline was a brood mare for the Sidhe. However, the lines of the other male's body indicated he would not welcome any suitor other than himself. Unyielding. Relentless.\n\nIt did not bode well. The fey were not known for sharing what they considered valuable.\n\nThey kept to the tree line, floating the perimeter like silent, uninvited ghosts.\n\nYet not without cause.\n\nThe Reds stayed at the front, and the four women traveled in the center. The pulse of Faerie synced with Tharell's every heartbeat. He knew it grew stronger for Domi, but it was a distance from the Singer's lair. He took the Rare One's misgivings seriously. Her blood tied her to every Singer in the world because of what she represented. If her feelings of disquiet centered on her homeland, then they were to be heeded. The Red Were, Jason, her husband, was young in his knowledge of what it was to be a Singer while also an infant in terms of being a werewolf. They could not count on him to understand or react to the things that occurred.\n\nHe would be a good fighter, as all werewolves were.\n\nWhen they had walked two hours, they finally came to rest. Tharell strode past the loose knot of women, Jason close by Julia, and stood beside Domi.\n\n\"Where is Jacqueline?\"\n\nDomi shot him a sideways look and answered, \"She attends to her needs.\"\n\nA long blue chunk of hair swept forward as he scanned the environment after Jacqueline.\n\nBathroom. Tharell dipped his chin in tacit acknowledgement. He laced his hands together behind his back. He asked the question while no one but Domi was present. \"How long?\"\n\nDomi closed his eyes, pinching the bridge of his nose. \"Close,\" he answered in a whisper.\n\nTharell narrowed his eyes in the direction of where Faerie felt strongest. Like a thread that slid through his body, it tugged at his inner being.\n\nJacqueline returned, coming to stand beside Domi. He whispered something against her temple, gently pushing hair away from her face as he did. She gave a small nod and tentatively wrapped an arm around his waist. Domi pressed her against his side. Tharell was glad of her presence; it kept Domi from sliding into an abyss of madness.\n\nHe would be quite glad of it when they were inside the sithen once again. Though an unhealthy environment for Tharell, it was the one he knew. Better to see the evil as it came at him than be taken by surprise.\n\nLow voices of anxiety began to ripple through the small group and sent the Sidhe to the front of the line.\n\nJulia was doubled over.\n\n\"What is this?\" Tharell asked, but Cyn hovered next to Julia with her hands pressed against the Rare One's back.\n\n\"We're close, and it's bad,\" Julia said though her voice was lost to the ground. The earth absorbed the words but none of the fear in them. Terror trembled in the air like the smell of rain.\n\n\"What? What's bad, Jules?\" Jason asked in a low voice, his hand on his wife. His gaze was everywhere and nowhere.\n\nTharell felt his restlessness. It matched his.\n\n\"Death,\" Julia moaned.\n\n\"Let's go,\" Jason said, hauling Julia up by the armpits. \"Hang on.\"\n\nJulia wrapped her arms around Jason's neck, and he melted to half-wolfen. It was all the finer attributes of the change without the mess. Tharell suffered a pang of envy. Would it be that he could change into something more desirable to look upon, his troubles in the Faerie court would lessen considerably.\n\nJason grasped her hands at his throat with one of his own. His attention turned to Tharell and Domi. \"Can you two keep up?\"\n\nTharell silenced his many biting remarks. Instead, he simply nodded.\n\nHe glanced behind him, and Jacqueline was on Domi's back, legs wrapped around his waist.\n\nTharell surveyed the group with sharp eyes. The Reds had all changed to half-forms, the last of their clothes discarded like scattered leaves.\n\nAngela and Cyn rode on the back of the female Were, who had completely changed.\n\nTharell had not realized the moon was so close to full.\n\nThey ran, the shadow of uncertainty sharpening as they did.\n\n*\n\nTharell and Domi arrived first at the edges of the bloodbath. He instantly took in what it was and stopped the Rare One before she could see the pieces of her people spread like flung and spoiled meat upon the land.\n\n\"Let me through!\" Julia spat, and Tharell gripped her.\n\n\"No.\"\n\nJason fingers bit into Tharell's deep violet skin. The Were's downy red fuzz looked like fire upon amethysts. \"Don't touch her, fey.\"\n\n\"I will if I must. Go ahead of me and see what it is. Tell me you wish for those memories to burn brightly inside her head forevermore.\"\n\nJulia's lips thinned as Jason released Tharell and cautiously walked around where he and Julia stood.\n\nThe howling came next.\n\nMournful and complete, it was more acknowledgement than mere words could ever have been.\n\nJason was human when he returned, his skin greenish, his eyes wide with the beginnings of shock. Tharell moved aside.\n\nTharell heard his words, but they weren't enough to express a horror without description. \"Jules\u2014baby,\" Jason began, moving hair behind her ear. \"They're dead, baby...\"\n\nShe sucked in a breath, her hand briefly covering her mouth. \"Who?\" Her gaze probed his.\n\nJason shook his head and gave a rough exhale. \"Everyone, I think.\"\n\n\"What? Brendan, Michael\u2014oh my god\u2014Jen!\" Julia screamed and took off running in the direction of the house where she'd been given temporary sanctuary.\n\nTwice she fell, the grass so slick with blood she couldn't keep her footing.\n\nTharell raced after her, easily keeping pace. \"Do not... Julia, I beg you. These are your people. You cared for them.\" He grabbed her arm and yanked her backward.\n\nShe slapped her palms against his chest.\n\n\"I want to know!\" she screamed.\n\nHe shook her once, hard enough to rattle her teeth. \"You do not want to remember them this way.\"\n\n\"What way?\"\n\n\"Dismembered,\" Tharell said simply.\n\nJulia's head hung. Then she shook it. \"No\u2014they can't all be dead.\"\n\nTharell brought her into the shadow of his body. \"Not all, my queen. Most.\"\n\nShe beat him with her fist. \"No! I don't believe you.\"\n\nTharell understood Julia did not want to believe. It was very human of her.\n\nHe held her hands together with one of his own as she struggled against him. He must make her come to an understanding.\n\nTharell sensed something and looked up as Jason approached, his face filled with dark thunder.\n\nHe ignored the Were, concentrating on the words he must imbue to the Blooded Queen. \"You do not want this memory. If you must look upon them, promise me you will let me take it from you afterward.\"\n\nAfter a beat of silence, the Rare One's mate tackled him, rolling them into the gore at their feet.\n\nHis loyalty was not to her mate.\n\nTharell did not hold back.\n\n# CHAPTER TWENTY-SIX\n\n\"Stop!\" Julia shouted. Her breathing was deliberately shallow, a self-preservation tactic. The dead all around her were fresh, the blood so thick it overpowered all other senses.\n\nJulia couldn't smell the forest, the grass, flowers... The scent of blood that saturated the air was so strong it tasted like metal vapor.\n\nTharell and Jason, the sounds of their fists a collision of raw meat, looked up at the sound of her yelled plea.\n\nShe drove a shaky hand through her long hair, a tangled and dirty mess. Julia repeated herself. \"Stop.\" Her eyes stayed on them. \"They are dead, and I am tired.\"\n\nShe took in the tangled fey and Were and sighed. All species' males battled it out. No one actually resolved anything with words. Out loud.\n\nThe two stood, their bodies hopelessly covered with the remnants of the newly dead. Jason shoved Tharell.\n\nJulia had never seen a man slap another. It rocked Jason's head back, and a low growl erupted like boiling water from his lips.\n\nTharell's azure eyes flashed on Jason.\n\nDaring him\u2014warning him.\n\n\"Look!\" Cyn cried, and Julia turned. The blood of whomever her best friend kneeled beside soaked her to mid-thigh. Julia ignored the men's posturing and ran to her.\n\nShe tripped over a severed leg, owner unknown. Her hands bit into the slick soup of death at her feet, and she jerked upright. A sob broke away from her like a chunk of her soul. Julia plowed forward, the bits pressing between her toes, and it took all she was not to make more of a mess with the gorge that rose inside her.\n\nShe fell on the opposite side of the body from Cyn.\n\nMarcus. Julia covered her mouth, trying to stifle the sadness that threatened to pour out.\n\nThen his eyes opened.\n\nJulia gasped, landing on her butt and instantly soaking her pants. \"Can you help him?\"\n\nCyn shook her head. \"I can heal, but this...\"\n\nShe didn't say more.\n\nMarcus' body did not exhibit one uncut surface. That he'd survived so long was a miracle.\n\n\"I can make him feel better...\" She gave a small lift of her shoulders. Cyn's eyes conveyed their miserable situation eloquently, wide and shiny with tears none of them had time to shed.\n\n\"So we can ask questions,\" Julia clarified as the others came to stand behind her.\n\n\"Yes,\" Cyn confirmed.\n\nMarcus' pain-filled eyes rolled in his head, looking between the two women.\n\nHis slight nod was the only indication he knew what their plan was.\n\nCyn lowered her palms to his chest.\n\nA shuddering breath took hold of him, wracking his body. A spasm shook his torso and more blood oozed out.\n\n\"I am dying,\" Marcus stated from a throat that was no longer working perfectly.\n\nJulia ripped a tear off her cheek with a finger. It burned her skin, scorching her heart. \"Yes,\" she answered.\n\nMarcus nodded. Julia grasped his hand. \"I am sorry.\"\n\n\"No. This is not of your doing.\" He tapped Julia's lips to silence her then dropped his hand to his ruined chest.\n\nJulia did not look down, could not.\n\n\"My children?\"\n\nJulia and Cyn's eyes met over his body.\n\nWhen Julia glanced at Marcus' face, his eyes were closed.\n\n\"Scott is alive.\" Julia didn't have the heart to mention the possible torture of his one living child by the Reds of Alaska.\n\nMarcus' face hardened, becoming resolute. \"Listen closely.\"\n\nThe survivors moved around him in a loose circle, but his eyes remained focused only on Julia.\n\n\"It was Anthony Laurent.\" He coughed, and blood-tinged phlegm landed beside him.\n\nCyn made a choked noise and Julia gripped his hand tighter, her throat clicking with a hard, awful swallow. \"How...\"\n\n\"He had a blade cleansed by the blood of the demonic.\" He took a rattling breath. \"It is proof against all Singer talents but a handful.\"\n\nDomi and Tharell exchange a portentous glance.\n\n\"Demonic?\" Julia whispered.\n\n\"Like the fey, they are an entity we assumed did not need consideration.\"\n\n\"Until now,\" Cyn offered, and Marcus nodded.\n\n\"It has been a thousand years since the last appearance.\" Marcus ended on a wheeze.\n\n\"Like Jules being the Rare One,\" Jason said from behind him.\n\nMarcus slid his gaze to Jason.\n\n\"Yes.\"\n\nHe closed his eyes, as did Cyn. She opened her own and looked a Julia. \"He doesn't have long.\" To Marcus, she said, \"I'm sorry.\"\n\nHe covered her hand with his. \"It is more than I'd hoped. To be able to speak with the queen, to transfer my knowledge to her. Preparedness is key.\" He coughed and more blood fed the hungry earth beneath him.\n\n\"Hear me,\" Marcus began again. \"The demonic may only use those who possess the blood of Hades. They cannot influence or manipulate humanity directly. But Were, especially the Red, even without demonic blood, can be used as vessels of destruction. Tony must have demonic blood and a debt of that lineage has been called in.\"\n\n\"Why kill the Singers?\" Cyn asked.\n\nTharell spoke for the first time. \"Where there is the blood of the demon, there is also its counterbalance; the blood of the angelic.\"\n\n\"Angel's blood?\" Truman asked in disbelief.\n\nMarcus nodded. \"As Singers are an offshoot of humanity, those who possess the blood of the Singers might also have that of the angelic. Being Singer does not automatically guarantee the lineage.\"\n\n\"But if all Singers were dead, then there would be no angelic blood,\" Julia guessed.\n\n\"Singer-cide,\" Adi said from behind Cyn's shoulder and Julia nodded sadly.\n\n\"By killing the known Singers, the natural enemy and protector of all supernaturals, the potential for peace is lost forever.\"\n\n\"So we're not really human?\" Julia pressed and Marcus' eyes fluttered shut.\n\nPlease do not die before I have this answered, Julia thought. Then guilt heated her face in a blush that burned along her skin. Marcus lay dying, but she wanted the answers. Even knowing how badly she needed those for a chance of resolution didn't make it any more justifiable.\n\n\"When was anyone going to tell me I was an angel?\" Julia asked aloud.\n\nMarcus' eyes snapped open. \"You are not heavenly but of heaven. The blood of the angels is most strongly entrenched inside you. You are the hope of all that is good.\"\n\n\"No pressure,\" Truman muttered in the background.\n\nJulia silently agreed with the sentiment.\n\n\"What do I do?\" she asked.\n\n\"If I had that answer, I would have anticipated what was coming.\" He breathed deeply, his face tightening in pain.\n\nCyn shook her head.\n\nMarcus was fading.\n\n\"I should have married the three, for protection from this.\"\n\n\"It is too late,\" he said softly.\n\nHe gasped, trying to sit up. \"You are the Rare One; through you, peace can be realized.\"\n\nMarcus' hands fell away from hers. \"Yet not without bloodshed and death.\"\n\nHis eyes glazed over, hands dropping from his chest to the blood-soaked grass on either side of his body. His palms faced toward the sky as though asking for forgiveness from an uncaring power.\n\nKarl Truman hunkered down beside the dead leader of Region One and closed Marcus' unseeing eyes with gentle fingers.\n\n\"We might have bigger things than the Reds finding us,\" he said as he stood.\n\nJason held his hand out to Julia, and she took it.\n\nShe couldn't baby herself any longer.\n\nJulia gazed out over the massacre of her people. Their severed body parts littered the ground as far as the eye could see. A sea of blood buried the green of the grass in a scarlet river of decay and violence.\n\nHeat rose from her feet, driving up her body like a flame without end.\n\nJason caught Julia as she wailed against his shoulder. The sound lifted the heads of wildlife caught in the melody of her sorrow. A chorus of absolution, grief, and determination buried in a song of which even the lowest of the creatures heard and took note.\n\nA plan began in the part of her brain not consumed by sadness. It wasn't perfect\u2014it wasn't even right.\n\nBut it was what she must do.\n\nJulia would enlist the vampires.\n\n# CHAPTER TWENTY-SEVEN\n\nThe blistering shower she'd taken still warmed Julia's skin. Running shoes firmly encased her damaged feet, scraped raw of the abuse from outside Region One.\n\nJason stood by her side as a first guard of sorts. The survivors were downstairs. The house escaped being a mausoleum by a millimeter. When they'd heard shouts for help, most of the Singers must have run outside and had been summarily cut down. They had found Brendan's body in the kitchen and buried him.\n\n\"I don't understand why the sheer numbers didn't just take that prick down. Overwhelm him,\" Jason said.\n\nJulia shook her head, sad and puzzled. Though the idea had occurred to her as well. Over and over. \"I don't know. There must be some magic to that sword beside demon's blood.\" Just saying the name demon felt wrong. Filthy.\n\nJason blew out a ragged exhale. \"We need to track Tony down and kill his ass.\"\n\nThere'd been enough killing, but for Tony she'd make an exception.\n\n*\n\nTharell and Domi had cast a spell like a great magnet of death.\n\nThen they had asked Julia where she wanted the remains.\n\nWhen they posed the morbid question, Julia's thoughts had turned to the lake, a beautiful spot that calmed her. She'd told them where, and they dug a horrible, necessary trench where they interred the unidentified parts. They closed the wound of the earth, every drop of blood and sign of the physical bodies collected as if summoned by an unseen force.\n\nTharell and Domi had cast a powerful spell of death attraction inside the mass grave, and the parts of Julia's people had responded like a summons.\n\nShe hadn't wanted to but felt she owed it to the murdered Singers. She had watched as the remains crawled and trembled along the grass as though dragged by invisible pulleys. It slid off the house and over every surface to which it clung then made its gruesome way to the resting place.\n\nIt was a vision that, when Julia was old, haunted her just as strongly as when it happened.\n\nShe continued to braid her hair, thinking about how she could bring up what she needed to do, remembering what she wanted to so badly forget.\n\n\"What, Jules?\" Jason asked, taking an escaped wisp and tucking it behind her ear.\n\n\"I know how to find Tony.\"\n\nJason sighed. \"It's been one day, Jules. Lucius and Scott are still missing, the Reds unaccounted for.\" He gave her a level stare. \"There were ten Combatant, now there are two left, and we don't know where the hell they are.\"\n\n\"Victor was never found.\"\n\nJason rolled his eyes. \"Okay, GQ might still be okay. But doesn't his blood know where you are or something?\"\n\nJulia lifted the corners of her lips at Jason's utter lack of indifference about the Singers. Her smile waned at what it'd mean to bring him in as her king. He didn't seem ready.\n\nReally, she wasn't ready, either.\n\nJulia sighed.\n\nJason pulled a face. \"What\u2014Jules, you gotta know that just us staying alive is the main thing here.\"\n\n\"I do.\"\n\nShe met his hazel eyes, filled with concern and longing. Julia knew, as sure as she stood before him, Jason wanted to go back to the way things had been. But the past was lost like sand through fingers.\n\nHe knotted his brow. \"I'm sorry, all those people...\"\n\n\"Singers,\" Julia corrected softly.\n\n\"Yes, Jules\u2014Singers.\" Jason raked a hand through his hair. The absence of his hands on her, as well as his acceptance of their new life, left her cold. \"I'm sorry Tony wasted them. But it is what it is. There's not a thing we can do to take it back. Going after Tony right now is begging for worse things to happen.\"\n\n\"Maybe.\" Julia lifted her chin, giving him steady eyes. \"Or maybe he's going to go from one Region to the next, slaughtering whoever he can find.\"\n\nJason folded his arms across his muscular chest. The sunlight streamed through her bedroom window, hitting his hair just right, making it appear blond when it was too sandy to qualify. She noticed it needed a trim.\n\nShe pushed distracting thoughts to the side. \"I want to call vamps in.\"\n\n\"What?\" Jason said, his arms falling to his sides. \"No fucking way.\"\n\nJulia stood up from the vanity where she'd been plaiting her hair. \"I think I can make it work, with William's kiss.\"\n\n\"Oh yeah,\" Jason said sarcastically, \"the vamps are so trustworthy.\"\n\n\"They won't have the problem with the metal.\"\n\nThey regarded at each other for a moment in mutual consideration.\n\n\"Why?\" Jason asked.\n\nJulia sucked in a breath. \"Because they're dead.\" Then let it out slowly.\n\n\"Ya sure?\"\n\nNo, she wasn't. \"I don't know. In theory, it sounds like it'd be worth a try.\"\n\nJulia leaned forward. \"Don't you think it's odd that all we have left are a few Singers and Reds?\"\n\n\"Reds who are up our asses,\" Jason countered.\n\n\"That's true. But there's also the fey.\"\n\nJason shook his head, refolding his arms in semi-defiance. \"Nah, they're okay, but I'm not sure I'm buying what they're sellin'. Don't trust them.\"\n\nJulia laughed. \"You don't trust anyone.\"\n\n\"Kevin. I trusted him.\"\n\nJulia looked at Jason while they paused to remember their friend. The first casualty of many.\n\n\"Yeah,\" Julia whispered as their fingers intertwined.\n\nJason pulled her against him. \"Let me protect you, Jules. It's all I can do for now\u2014all that I am.\"\n\nShe laid her head against his chest. The ache in her soul was more than a bruise; a crushing wound of grief.\n\nIn the absence of hope, darkness bloomed like a black orchid.\n\nDark, final. A night without day.\n\n*\n\n\"He has killed almost all the Singers of this Region. Therefore\"\u2014Tharell paused for effect\u2014\"I do not see any reason to hold to our bargain.\"\n\n\"I still want the Rare One.\"\n\nTharell scooped the water, spreading it again to capture the image perfectly. A ripple across the surface of the lake spun, taking the shape of a circle.\n\nGabriel's image sharpened, the water the mirror of their communication.\n\n\"Be that as it may, I am not the transporter of this. Now that this colony of Singers has been decimated, the demonic will move on to Region Two. The logic of evil never wavers.\"\n\nAn expression of distaste spread across Gabriel's face. \"Our intelligence suggests the leader of Region Two has been subdued, she is of sufficient Sidhe descent that she is no longer necessary.\"\n\nTharell did not like Sidhe business leaving Faerie. And here were the most intimate of secrets traveling to this corrupt coven of vampires. Someone was leaking things they should not be.\n\nThey would be found, but that was for a different day.\n\nTharell kept his face clear of his thoughts. \"That one Singer female, though she has sufficient blood quantum to breed\u2014\"\n\nGabriel laughed. \"No.\" He shook his head, golden red hair slithering over broad shoulders. \"The demonic is raging across this great land, taking out any supernatural in his sights. He can heal everything but a killing blow. How long do you think it will be before he pays Faerie a visit with his special metal?\"\n\nTharell did not respond immediately. He had said nothing when the Region One leader had spilled truths with his dying breath.\n\nNor was Gabriel aware that angelic blood would cure all their ills. The Singers would make the Sidhe indestructible. Perfect. That had been Queen Darcel's hope as she died knowing her model for purity was imperfect.\n\n\"Only vampire can overwhelm the rogue werewolf.\"\n\nTharell smirked. Bastard. \"For a price.\"\n\nGabriel possessed a great deal of angelic blood. He was a Rare One as Julia. Tharell would have never known it with how strongly he negotiated his evildoings.\n\nHe knew not if Jacqueline would survive what they planned.\n\nTharell did not know if he would.\n\nIt was a death sentence, for life in the world of Faerie and a debt closed.\n\nTharell, his heart too heavy to break, wiped the surface of the water. That smug satisfaction fixed on Gabriel's face shattered with the ripples of the lake.\n\n*\n\nJulia asked around, but the few Singer survivors and her group hadn't seen Tharell.\n\n\"Why do you want to talk to him?\" Jason asked, kissing her knuckles to take the sting out of his question.\n\n\"I was supposed talk with the Sidhe and invite them into an alliance where we could talk about intermarrying.\"\n\nJason gave her his full attention.\n\n\"And now everyone's dead. And I want to reach out to William's former kiss and see if they want to chase after Tony and kill him before he does in more of the Singers. So the treaty is on hold.\"\n\n\"And the fey don't bargain,\" Jason said slowly.\n\nJulia shook her head. \"I've read about them\u2014myth only. But oath breakers are killed.\"\n\nJason held her hands tighter, his eyes narrowing. \"What about extenuating circumstances?\"\n\nJulia shrugged. \"No such thing. We find Tharell and see what our options are. Or be held to a promise I made, not understanding all the Singers would die.\"\n\n\"No all.\"\n\nJulia sighed. \"No, not all.\"\n\n\"You're very ruthless, Jules,\" Jason said, only half-joking.\n\n\"More than I want to be.\"\n\nA small rustle in the tall grass caught their attention. A dark silhouette made slow progress from the lake. The gait and carry of the broad shoulders were unmistakable, but the bruised plum of his skin, a deep violet smear against the pressing twilight, was most distinctive.\n\n\"Speak of the devil,\" Julia said.\n\n\"Yeah.\" She gave Jason a sharp look.\n\nIt sounded like he meant it.\n\n# CHAPTER TWENTY-EIGHT\n\nTharell caught sight of Jason and Julia at the crest of the knoll. The great Victorian homestead rose behind them like a yawning pursuer.\n\nHe was not comfortable with his uneasy alliance. Gabriel lacked integrity in so many ways. And Domiatri would never forgive him. However, if Tharell could enlist the largest coven of the Pacific Northwest region to aid him in bringing down the demonic werewolf, the cost of one would be justifiable.\n\nTharell needed to progress a contingency plan he never imagined to employ. Yet he must.\n\nHe had struck the deal. There was no going back. He was fey, and the fey did not lie. Tharell would not be an oath breaker for all the wishes on earth.\n\nJulia drew nearer, and Tharell leaned down into her embrace, guilt striking him like a lashing wave against rock. In her solemn eyes he saw she pressed through her grief.\n\n\"Now that my dead have been buried, we should discuss our promise.\"\n\nTharell inclined his head in agreement. \"I am pleased you understand the importance of an oath with the fey.\" At his words, guilt hardened like an uneaten yolk in his belly.\n\nShe gave a sad laugh.\n\nIt deepened his disquiet about his choices.\n\n\"There aren't enough of my people to negotiate with. However, I've found another option that might satisfy what I'm trying to do and save other Singers from...\" He hesitated in the face of her remorse.\n\nTharell watched her struggle, tears sliding out of the corners of her eyes like a diamond river that has burst its damn.\n\n\"Blooded Queen,\" Tharell began, and dropped to one knee before her, \"I know your sorrow.\"\n\n\"You can't know dick, Tharell,\" Jason spoke for the first time.\n\nTharell's crisp blue eyes met hardened hazel, unwavering and distrusting.\n\n\"I have been in war, Were. Many have died\u2014legion.\" Tharell dismissed him. He had never warmed to the Were. That was the problem with a turned Singer. They were unpredictable and maintained much of their humanity. A distinct disadvantage.\n\nHumanity did not overly impress Tharell. Their apathy killed any respect the fey might have for them. Now, the Singers were another matter entirely. The talents they possessed alone put them in a different league, one parallel to the Sidhe.\n\n\"Jason,\" Julia hissed and Tharell rolled his lips together in a grim line to cover his smirk. The story of this Were was of being in the state of Feral for too long. When he erupted from his fugue, and he had been lucky to do so, he had gained none of the knowledge or history of either lineage.\n\nIgnorant and angry, Jason Caldwell was a ticking time bomb of spontaneous decisions and reactive conduct.\n\nIt might prove useful. Others' emotions were malleable.\n\nTharell sensed they had not joined as a female and male, their energy remained separate. Julia was still pure; claimed by the paper of human law; not body and soul. He knew of her vulnerability. Yet with the chaos of Region One's slaughter, coupled with escaping the rogue band from Alaska, that fact had been overlooked.\n\nIt would not be for much longer. And the Reds that had not submitted to the newest pack master, Slash, would regroup and attack again.\n\nIt worked well for Tharell, as so many agendas going on at one time would additionally camouflage him.\n\n\"I think I will contact William's kiss and, if I can manage it, we will ask their cooperation in finding Tony. Only his death will stop this... butchering.\"\n\nSplendid, Tharell marveled. Gabriel of the Northwestern would send his troops to locate the interloper, and if Julia could secure it, she would also recruit the Southeastern. They would shred Tony.\n\nThe demonic threat would end, and he would make his payment to Gabriel. Faerie would have a bevy of Singers to take the threat of extinction away and fulfill his illicit pact.\n\nHis eyes found Julia's. Keeping her hands in his, his gaze slid with wary consideration to Jason. He towered over her. \"That is a wise choice. I accept.\" He cocked his head. \"Region Two is in Montana?\"\n\nJulia nodded.\n\n\"Let us assemble a caravan.\"\n\nShe frowned slightly at the expression. \"I'll reach out to the vamps.\"\n\nJason stalked away, and Julia turned toward him while Tharell gave a broad smile they did not see.\n\n\"I don't like it!\" Jason yelled, throwing up his hands.\n\nJulia strode to him, grabbing his bicep. \"Do you have a better suggestion?\" Her small figure vibrated with anger and the urgency of the truly committed.\n\n\"I don't have the effing time to consider how much I don't want to ask for help. I don't have the luxury. The Singers that would have had better ideas are gone. I found out we have angel's blood. That somehow heaven is on earth, and we're all here to make the wrong right. How can we not save their lives? Tony might already be there!\" Julia said, her face taut.\n\n\"God!\" Jason shouted to the heavens. \"When can this be over?\" He grabbed Julia against him. \"I don't want to give you up. I don't want that jackass getting his talons on you.\"\n\n\"He can't kill me.\"\n\nJason pulled away from her. He looked at Tharell.\n\nTharell nodded, speaking the full truth for once. \"She is proof against the metal of the blade he carries. It will not work against a full-blood.\"\n\nJason gripped her arms, spinning her to face him. \"You're an angel.\"\n\nJulia quirked her lips. \"I don't feel like one.\"\n\nTharell circled around the pair. Jason tensed.\n\nTharell hated Jason Caldwell's instincts. It made everything he necessitated accomplishing more of a challenge. \"When something has a choice, do you choose good or bad?\"\n\nJulia blushed, a lovely peach color washing across high cheekbones, her whiskey eyes flashing in the low light of impending night. \"Good, I guess.\"\n\n\"Would you sacrifice your life for another?\"\n\nShe lifted her chin. Surer now. \"Yes,\" she answered with an imperceptible hesitation.\n\nTharell's eyes narrowed. \"Do you love more than hate?\"\n\nResolute. \"Absolutely.\"\n\nJason shrugged in apparent agreement. \"See Jules\u2014full angel. Must be cool to be you.\" Jason said it with admiration.\n\nThough Julia appeared to feel unworthy of the title, humility was also the stamp of the purer bloods. So rare.\n\nSo valuable.\n\nTharell squeezed her shoulder. \"I think you've made a wise choice. The vampire will require payment.\" He cupped his chin, pondering as Jason looked on with a sour expression. It was clear he did not like another male's hands on her. \"You might offer an unbroken alliance, citing your truce via their former leader, William.\" He spread his palms out as if that was the obvious choice.\n\nJulia nodded slowly, a tentative smile lifting her mouth. \"Thank you, Tharell.\"\n\nTharell hardened the seed of tenderness he held for her. He could never allow it to germinate. \"Do not thank me until Tony's blood blankets the land of the Singers.\"\n\nJulia frowned at his words.\n\nThe best lies hid within partial truths. The fey were expert in that area. Tharell's mixed lineage made it even more so.\n\nHe watched the two walk away.\n\nWhen Jason Caldwell turned and gave him the glance he expected, Tharell nodded once.\n\nJason's expression was speculative.\n\nTharell would not win over the Were. That was an absolute. He had seen to the protection of the Rare One for too many years.\n\nTharell put his hands on his hips and looked to the sky. The sinister encroachment of sunset was a wash of blood that leaked over the dying clouds like a sinister encroachment.\n\nHis plan solidified within the confines of his mind, the methodical order of events clear. Julia communicating to the other arm of vampire in this region was fortuitous. He had not anticipated the move, and it would aid the groundwork already laid after two decades of careful planning. As the humans liked to say, an alignment of the stars.\n\nIt was the same for the supernaturals and truer than the lowly humans could ever know. There was but one chance every one thousand years or so. And as when the Jewish carpenter died on the cross, the celestial climate had been portentous. As it was again.\n\nThe Singers had grown complacent when they should have gathered in large numbers. The fey had watched the heavens whisper what was to come, and paid heed to the opportunity. Now it was at hand and Tharell the catalyst, a queen on the chessboard of fate. How powerful a piece she was. Moving in any direction, coveted and stealthy, she was renowned for taking out all the players.\n\nTharell would not gain pleasure from what he must do. However, one did not always control their life's path. His first loyalty must be to his sovereign.\n\nPraile would pay in the coin that Tharell wished to have most.\n\nVengeance.\n\n# CHAPTER TWENTY-NINE\n\nJulia was relieved that Tharell had given the nod to her communication with the vamps. Ten adopted Reds added to Slash, Jason, and Karl Truman. Cyn and Adi were hard-hitters, not weak girls to scream and wail when the tough things needed doing.\n\nScott and Lucius stood missing. Tom Harriet of the FBI was bent on having her.\n\nShe was determined to elude him.\n\nJulia didn't want any more death, but she wouldn't allow herself to be captured again. She bit down on her sadness over the potential of Scott coming home only to find his family wiped out.\n\nThen there was Jacqueline, the strangest thing of all. If he were to return whole, then maybe, just maybe, they could have a relationship. Though for the life of her, Julia couldn't even see Jacqueline as the same woman she'd been before.\n\nThe green Unseelie Sidhe warrior shadowed Jacqueline constantly. Julia supposed it was the rarity of children in Faerie. The little bit of Faerie culture she'd experienced had been frightening. She couldn't imagine a child living in the torture as norm for anyone who dared stray from what was acceptable in Faerie.\n\nThey had not located Reagan or Delilah.\n\nHas Tony killed his own offspring? Had Delilah escaped because she had been vampire enough to negate the power of demonic blood over magic metal? Further... what had happened to Alan and Lacey Greene? Many were missing, but the sheer amount of bodies now buried near the lake convinced her that Tony had killed most of the Singers. The method of their murders had made it impossible to identify everyone. A sad truth.\n\nThey had only Cyn as a Healer. Julia didn't have to be told they were in an insecure position with regard to injury. Few Singers had escaped, and maybe their talents had left them able to defend themselves. Or blood purity. Her remaining band of friends and few subjects may or may not survive the siege against Tony. Julia would not be directly responsible for their deaths, even more motivation to contact the kiss.\n\nJulia massaged her temples. She tramped down the static, like a fire begging to start, of people's constantly buzzing thoughts, a low-grade headache that wouldn't go away.\n\n\"How do you propose to contact the vamps?\" Jason asked, kicking a small rock with the toe of a borrowed sneaker. It shot across the yard in front of the huge house and launched off a tree trunk, a bare spot where the bark fell away, marring the surface. She turned to him, frustrated.\n\nHis earnest face stopped the scathing comments she almost made. Julia was sad, tired of the constant fighting, worried about Two, and frantic about securing help so more people would not be killed.\n\n\"I don't know,\" she admitted.\n\n\"Listen,\" Jason said, roughing his hair with irritated fingers. It stood on end, and Julia smiled at the familiar gesture. \"I say we go back to the last known place William was held.\"\n\nJulia stopped walking. \"I don't know where that is. And\"\u2014she shrugged\u2014\"who knows what has happened in the weeks after William's death?\" She looked into his eyes. \"They could have a new leader and I come, hat in hand and say, 'Hey guys, sorry William's gone, and your torture of him to get to me didn't work, but I need you to go kill a bad werewolf'.\" Julia made a sound of pure disbelief. The words were utterly ridiculous when she said them.\n\nJason slung an arm around her shoulders. \"It's solid, babe. It just takes guts.\" He paused and she glanced up at him. \"Let's take the fey dude...\"\n\n\"Tharell?\"\n\nJason made a face. \"Yeah, I guess... and he can act as a go-between. Put his ass out there since he thought it was such a great idea.\"\n\nJulia didn't want to owe the Sidhe any more than she already did, but she wanted to save the other Singers. Tony was traveling on foot; that much they knew from his tracks. The Reds had followed his prints to the road, where they disappeared.\n\nSomeone had given him a ride, she bet. While she stalled, intimidating herself out of the toughest part of solving this mess, Tony was heading east.\n\nThe good Samaritans who gave him the ride forfeited their lives doing it. Knowing Tony, he'd done it at night so they wouldn't have caught the blood.\n\n\"Okay. Enough with The Dumb of us running around until our feet fall off.\"\n\nJulia laughed. That'd been brutal but unplanned. Cyn had made the worst of it better, but they still ached from the abuse.\n\n\"Right?\" Jason leaned back and thumped his chest, and she laughed harder.\n\n\"So true.\"\n\n\"So where's a set of proper wheels at this joint?\" Jason looked around as if a great 4x4 would pop out of the ground. \"I say we four by it to where the fangs live, get the A-Okay, then chase after Tony's murdering ass. Is there a better tracker than another Were?\"\n\nThere had been, Julia thought.\n\nHe searched her face and grimaced. \"Sorry, babe. That didn't come out right.\"\n\n\"No.\" Julia looked down for a moment. Then she lifted her chin. \"But it's the truth.\"\n\n\"That's my girl,\" Jason said.\n\nShe gave a watery smile and answered, \"I've always been your girl.\"\n\nHe held Julia while she cried.\n\n*\n\nA truck in the huge shop reminded Julia of her 1977 Blazer left in Homer.\n\nShe ran her hand down the side then flicked a fingernail against the tires' deep treads.\n\n\"Look familiar, Jules?\" Jason asked, not requiring a response.\n\n\"Yeah,\" she said anyway.\n\n\"All these wheels were here, and dipshit Tony didn't take a one.\"\n\nA stealthy noise made Jason liberate a knife he wore clip-tucked in his front pocket.\n\nIt glinted as he flipped it closed when he recognized the figure.\n\n\"He was likely possessed with the influence of the blade.\"\n\nJason relaxed his posture as Tharell went on, \"A blade rendered magical by the blood of a demonic casts a powerful thrall over those of the right lineage.\"\n\n\"Sounds like we're all just whores to genetics.\"\n\nTharell's expression was unreadable, but he answered with more feeling than Julia normally heard from him. \"Crudely put but accurate.\"\n\n\"Well whatever, it's time for us to find the fangs and get on the road. The sooner we do in old Tony boy, the faster we can move on.\"\n\nJulia's gaze went from Tharell to Jason.\n\n\"What?\" Julia asked Tharell.\n\n\"The fey do not like metal.\"\n\nJason jerked his chin back, confused. \"Why not? I mean, we can be to the vampire coven in forty-five minutes if we go hard.\"\n\nTharell's Adam's apple bobbed in his throat. \"I can manage it, but Domi could not.\"\n\nInteresting, Julia thought.\n\n\"So you're a no-go?\" Jason curled his lips into a smirk.\n\nTharell appeared to rise to the bait, his forearms curling into tight slabs of muscles and his fists like clenched hammers. \"I will bear it.\"\n\n\"Good thing,\" Jason said, giving him a guy clap on the back with his open palm. \"Let's get everyone that'll fit in this thing and take off.\"\n\nTharell glowered at Jason, straightening his back where Jason had struck him.\n\nJulia had a fleeting premonition as something flashed across Tharell's face. She couldn't identify it, but she almost recognized it.\n\nIt had felt a little like Tharell was biding his time.\n\nJulia was interpreting his impatience for her to fulfill her promise. It saddened her that Tharell could not understand the deaths of so many made his timeline longer.\n\nPeople had died. That was more important than a race of supernaturals threatened with extinction but that still survived.\n\nHer people had almost been wiped out.\n\nThe two weren't comparable.\n\nJulia could not be everything to all.\n\nA handful of survivors decided to accompany them. No one was really in charge of her direct safety. Julia felt a sense of relief. She'd never loved all the pomp and circumstance surrounding the Combatant. Fate hadn't been kind, but she lived another day.\n\nThey hadn't found Lucius, Scott, or Victor, the remaining Combatant. Slash, Truman, Jason and Tharell were joining them.\n\nThough the men didn't like it, Julia insisted on Cyn coming. If someone got hurt, there was no way to speed recovery. She was their lone, surviving Healer.\n\nThe potential for something to go wrong was high. Julia was just going by historical precedence.\n\nNothing had been easy.\n\nOr free of violence.\n\n# CHAPTER THIRTY\n\nTony lay in the center of the highway, a rural intermediary route that fed to Interstate 90. He was all about getting his ass to Montana, murdering all the Singers he could, and moving on to the next region.\n\nPraile would be proud, if anything as evil as him could feel pride.\n\nTony waited, succumbing to his acute senses. He'd do anything to shift to his wolf but that wouldn't work. He needed to let precious minutes slide by to make better time. Sacrifice the now for the later.\n\nNight had fallen and he was finally in a position to mask himself. Blood dried and flaked off his clothing and skin. Only his eyeballs were free of it. It didn't bother Tony in the least but it would give the human he hoped to steal a ride from pause.\n\nThat's why he'd waited under cover of the woods until it was too dark to see anything with detail. A human's weak eyesight would never be sharp enough to see what covered him, but a wolf would smell the blood almost five miles away. A Were's nose so sensitive that it could smell the tenor of injury.\n\nMurder was exciting. The blood Tony wore was akin to a dinner bell clamoring. He didn't want that, either. What if stupid, rule-follower Manny is still around? He didn't want Lawrence and Manny catching on to who and what he was. It would go badly. Manny he could dispose of, but Lawrence The Alpha? Short of using the blade, Tony didn't think he could put him out of his misery.\n\nSo he waited. The pebbles of the old asphalt road, not re-covered for years, dug into his back. His soaked clothing molded to each tiny stone as they pressed uncomfortably.\n\nFinally, his sharp hearing captured the noise for which he'd lain there hoping.\n\nA car was drawing nearer.\n\n*\n\nThe stupid human didn't see him. He was going to have to fake a resurrection or something to get their vehicle to stop. He certainly didn't have time to heal the damage from a car running over him.\n\nTony flopped like a fish, his senses carefully noting the heat of the car's headlights as they sliced over his body. The faint aroma of rubber as the tires bit the nubby black road and pressed to a skidding stop about six inches from his arm.\n\n\"Fucking close,\" Tony grumbled.\n\nHe cracked an eyelid, playing dead. A couple of old geezers peered at him over the dash.\n\nThe man swung the heavy car door open, and Tony was relieved the thing hadn't flattened him. Chrome shone like strips of silver along the fins of the 1960 Chevy Impala that sat idling in the center of the road.\n\nWhat the fuck? Is this old dick not going to get out and check on me?\n\nSome humans had a drop or two of Singer blood. It made them wary, intuitive.\n\nFigured he'd get the one in a thousand.\n\nHe slammed his eyelid shut.\n\nThe old guy shambled over toward Tony, his own heartbeat so loud he could hardly hear the old fart over it. It always roared like a lion before he killed others.\n\nHe felt the press of air before whatever the old guy held touched him.\n\nTony snapped his hand out, latching onto something smooth.\n\nHe opened his eyes, and the man's widened.\n\nThe blood, Tony surmised. He's gotten a load of that.\n\n\"Tessa!\" the codger screamed. \"Gun!\"\n\nFuck this, Tony thought. He grabbed the end of a cane and twisted it sharply, easily breaking the old man's grip.\n\nTony stabbed the cane backward, envisioning driving it through his stomach to the other side.\n\nBreath wheezed out of the guy like a popped balloon. He wrapped his gnarled hands around the cane, and Tony sat up.\n\nHuh. Tony grinned. The cane had speared the old fuck, and his guts were now decorating the back. Nice.\n\nTony bounded to a standing position and roundhouse-kicked the old man with every ounce of his Were strength. He spun, tottering to a splat on the ground like a bowling pin. His chrome dome pegged the asphalt and bounced once, smacking the surface a second time. It cracked like an egg.\n\nBrains spilled over the black surface like an underdone breakfast, leaking out over the road.\n\nHe did a pivoting swagger, facing the wailing banshee in the background, and the first spray of bullets penetrated his chest like spears of chucked salt.\n\nTony opened his mouth, but the damage disallowed breathing at the moment. He toppled backwards.\n\nWhat the hell is that noise?\n\nHe realized his body was trying to heal the shot. Not one bullet but effectively hundreds.\n\nShotgun.\n\nFuck me.\n\nTony sat hard on the ground. The impact traveled his spine and landed in his skull with an instant, sickening headache.\n\nMore worrisome still were the twin barrels of a sawed off shotgun leveled at him.\n\n\"Don't do it, you dumb old bitch.\"\n\n\"I'm not old, and it's you that's the dumb one. Lying out in the middle of fucking nowhere and killing a man who was there to help.\"\n\nTony looked at her and his spirits dropped like a rock. Could his fucking luck get worse?\n\nShe lifted the shotgun and flung the butt into his chin. It rocketed him back six feet, and he landed on his back.\n\nStars like tossed diamonds mocked him in a velvet sky.\n\n\"I'm the Alpha female he picked up.\"\n\nHer grin was like the wolf that she was.\n\nThe next stab of the butt had Tony's head spinning. \"Fucking bitch!\" he slurred.\n\n\"Sticks and stones, 'tard,\" she said conversationally.\n\nPraile burned inside his skull, almost enough to keep him conscious. He passed out as she dragged his body by one arm across the road.\n\nIn the end, a rogue Were female had taken him by surprise.\n\n*\n\nTessa hauled the Were to a ditch and kicked his body into the gully that flowed with a foot of water, though it was the height of summer.\n\nThis little journey had been sketchy at best, and now she had an additional mess to worry about. She generally didn't concern herself with humans. The old man who'd given her the ride hadn't been a neutral acquaintance.\n\nWhen the male Were had casually murdered the human male, a surge of irrational anger had surprised Tessa. He was a dumb human.\n\nHowever, no one had ever shown her kindness. She was an unprotected, unmated female Alpha, and no male let her forget it. But this old human had trusted her.\n\nSomehow, Tessa had failed him.\n\nShe would not fail him now.\n\nTessa left the unknown male in the ditch. It was more than he deserved. Much more.\n\nIf she had the balls, she'd drown him. Tessa looked at the water inside the swale and hesitated.\n\nThen her gaze found Gus. She could already smell the decay beginning on his body.\n\nHe deserved a burial. He had been a good human.\n\nShe reluctantly left the Alpha male and went to Gus. She easily lifted his limp body and swung it over her shoulder. Tessa opened the back door of his vintage car and carefully arranged his body in the back. With a final glance at the Were, she slid into the driver's seat and took off.\n\nTessa should have smashed his offensive skull in. Or noticed that the murder of many lay like a cloak of dried venom over every bit of his body. Subduing the threat, and Gus dying, had simply gotten her too wrapped up. It was what she did.\n\nIt was what she'd had to do.\n\nTessa would again.\n\n# CHAPTER THIRTY-ONE\n\nTharell proceeded ahead of Julia and Jason. The pretense he maintained grew thin. Though the fey did not typically embrace luck, the Were could not smell a lie on a Sidhe. Their flesh simply didn't exude a detectable emotive scent. However, the Were were astute to the body's subtle tells. Tharell had not lived centuries atop centuries to learn nothing of other supernaturals. Because of his mixed blood, he could manage distance from Faerie better than others could.\n\nHe was sharp to the alert of vampire proximity.\n\nVampires could bring true death to the fey. They had always been mortal enemies. It had been most interesting that a distant Sidhe had chosen to lie with a vampire to produce the mixed-blood, Delilah. If they had ever allowed Tharell to choose a mate, she would be a temptation. There was something so attractive about dallying in the forbidden.\n\nA pungent smell touched him, and he held up a hand in a halting gesture.\n\nFootsteps ceased from behind, and Julia stood beside him. \"Have you found them?\"\n\nTharell nodded. He had endured almost forty minutes in the accursed metal vehicle, holding onto his churning guts by a thread. When he could finally exit, he'd felt instantly well. The steel tomb of the vehicle coupled with the jarring ride across moving terrain had been a challenge.\n\n\"They lie beneath us.\"\n\n\"Something smells like snake piss.\" Jason wrinkled his nose.\n\n\"An apt analogy,\" Tharell murmured. He lay on his side on the forest floor. He could hear the death bringers rise for the night like tunneling earthworms seeking the surface.\n\nTharell leapt to his feet and backed away, grabbing the Blooded Queen as he did.\n\n\"Hey!\" Jason followed the movement and anticipated where Julia would be as Tharell swung her away from the area. Jason gathered her against him before Tharell's slow spin left her.\n\nThe Red Weres stepped back as a group.\n\nThe earth began to roll. The blanket of grass and rocks that dotted the surface undulated as though a large serpent did indeed live beneath the plot of soil.\n\nVampires burst through the earth. Large gaps like dirty mouths yawned from the ground, allowing the undead to eject like rockets. Tharell forced his hand away from the dagger at his side with monumental effort. His skin crawled with the need to slay the death bringers.\n\nHe held his instincts in check. Constantly reminding himself of the end result.\n\nThe goal: infiltration of Faerie.\n\nThe vampires held nothing in check. They assailed the intruders of their territory with a unity borne of purpose.\n\nAn eerie light shone, leveling the vamps that rushed the group.\n\nIt came from Julia, spectral, perfect, and dominating everyone within range.\n\nTharell fell to his knees before it. It was captured sun. Fissures of jewels lay mercy to its glitter, shards of light speared the supernaturals all, and the vampires cried into the heat of it.\n\nTheir shrieks and howls were painful even to Tharell.\n\n\"Stop,\" Julia commanded softly. Tharell dared to look at the Rare One, and she stood shining. The light was integral to her being, and she looked the part of the blood she carried.\n\nAngelic.\n\nTharell's demonic blood boiled in direct response, her purity water to the oil inside his veins. He reveled in the contrast, drowning in his shame, claiming emotion so strong he thought he would never feel again.\n\nThe vampires crawled to Julia's feet for her subjugation, laying their heads between hands cupped as though in prayer.\n\nThe light dimmed then finally grew dark.\n\nTharell allowed himself a single, shattered breath. The first of a half-dozen he forced into his starved lungs.\n\nHe had never endured anything so terrible. His gaze went to his skin. A light smolder evaporated as quickly as it rose. Tharell darted his eyes around to see if anyone had witnessed the evidence of his lineage, but all eyes were on the vampires.\n\nRelief left the tips of his fingers and toes tingling.\n\nJason was the only one who remained standing; the rest of the group were daze. Shaken.\n\nJulia began with one word. \"William.\"\n\nThe vampires' heads rose from the pillows of their hands.\n\nWhen she had finished speaking, the vampires no longer lay on the earth's floor but stood at her side.\n\nTharell shook his head. Julia Caldwell was fulfilling her role as every supernatural bible said she would. Even the book of Faerie mentioned her coming, and the opposition that would present itself.\n\nTharell sometimes wished the small things in the millennia since his birth had not grown so large in his mind or so black in his heart.\n\nTharell steeled himself for what would come to pass.\n\nHis thoughts broke as introductions were made. The one who seemed to be the leader confirmed it. \"I am Brynn, William's second. We knew of his death when our power was depleted.\"\n\nTharell lifted his eyebrows. The news puzzled him.\n\n\"When he died, it made you weaker?\" Julia asked.\n\nBrynn surveyed the group. He was counting his numbers as less, Julia's sheer talent range and identity necessitating his submission.\n\nHis answer was an unhappy \"Yes.\" His black gaze fell on each of them. \"However, we regain what we've lost day by day. We rest at day and feed at night.\"\n\n\"On anything?\" Truman asked. Even Tharell could hear the human sarcasm in his tone. He found it interesting how the newly changed Were so often wore the shadow of who they had been in their previous life.\n\n\"No,\" Brynn said softly. \"It must be humans.\"\n\nJulia sighed. Tharell watched, saying nothing. Jason put his arm around her.\n\n\"I am sorry, not for what we need to consume to live, but that you're bothered by it,\" Brynn said. He and the five others of his small kiss kept their eyes on the warrior.\n\nNot a good development.\n\n\"No,\" Julia responded softly. \"I stayed with the Seattle coven for over a year. I know what it was.\"\n\n\"Gabriel?\" Brynn inquired and Julia nodded.\n\n\"He's the bastard that started all the trouble with your coven. If it hadn't been for him being in bed with your leader, William might be alive right now.\"\n\nTharell didn't correct her, though William had been dead the instant he stepped inside Faerie. No Sidhe would allow a death bringer to live within. They were too opposite, enemies for centuries. The bite of a vampire was disastrous, as Queen Darcel learned firsthand through the hybrid, Delilah.\n\nBrynn cupped his chin, alabaster skin slightly shining in the paleness of the moonlight like polished bone. \"None of us have the blood of the Singer; there are no shifters. Much of what we've lost has been by your hand.\"\n\nJulia shook her head. \"I can't deny that when you entered our lands we used lethal force.\"\n\nBrynn barked out an unhappy laugh. \"Yes, very lethal.\"\n\nTharell knew of the burning corpse's flesh. The Singers believed in fire as a cleanser.\n\n\"Remember, your leader tortured William to find me.\"\n\n\"I do.\" His gaze remained on the Rare One. \"I have agreed to the alliance. There was much potential in William's leadership now lost. Though the sacrifice was worth it.\"\n\nJulia bowed her head. \"Thank you.\"\n\n\"There are not enough Singers to lose every Region to a wolf gone mad. As vampire, we understand the potential shift in the food load if the highest blood quality is suddenly expunged from existence.\"\n\nJulia looked ready to make a scathing comment at the comparison to livestock when the Red Were broke in.\n\n\"A demonic,\" Slash said, and Brynn turned to him with a nod.\n\n\"Worse. But it doesn't matter. If the Were bleeds, he can die.\"\n\nHe could bleed, Tharell knew.\n\n\"Once true death is upon him, the strength of the metal is no longer fueled by his genes. He is gone. The blade will grow dim and be no more powerful than regular metal.\" Brynn looked around him, gauging the understanding of his words.\n\nJulia inhaled deeply in relief, and the Were and vampire regarded each other as tentative allies. It might be the time Tharell needed.\n\nNow to find a surface of water and convey to Gabriel what had transpired. If he played things perfectly, it would seem as though the two covens had previously allied, and they would be blamed. If more were killed, then the babe Jacqueline carried would go to its calling.\n\nTharell couldn't anticipate every eventuality, and part of his future was yet unwritten. However, sometimes the inherent excitement of living lay in the very unpredictability of an uncertain future. It was all Tharell truly had. The only color of living.\n\n\"Where is Region Two?\" Brynn asked, and the vampires gathered around him shifted in readiness.\n\n\"Montana,\" Julia said. Tharell could taste her reluctance to give away the location of another Singer retreat.\n\nHowever, it was the lesser of the evils. Knowledge of the vulnerable Singers' location was critical to assist in the protection of the indefensible.\n\nTharell smiled, made his excuses, and wandered off before the untenable car trip to Region Two began.\n\nHe would find water while the vampires moved ahead with their part and secured the perimeter of the Singer fortress of America's north central region.\n\nTharell would leave forthwith.\n\nRight after speaking to Gabriel through water and magick.\n\n# CHAPTER THIRTY-TWO\n\nTony rolled to his side, his body shaking with injury and chilled by the cold water of the ditch. He heaved anything that remained inside his stomach into the weeds.\n\nBitch, he thought in uncharitable reflection. Though realistically, he was lucky to be alive. He had only himself to blame. Tony should have smelled a female Alpha five miles away. But no. He'd been so intent on killing the old jag up, he'd royally fucked up what should have been a simple carjacking. Now a female Alpha had handed him his ass, and he had jack shit to repair the multiple chest wounds from the shot with which she'd sprayed him.\n\nTony scabbed his way out of the ditch, his talons pushing out in sluggish response to the small surge of survival adrenaline that remained. It was more than surviving. He had to consider Praile.\n\nIf Tony didn't continue east and slaughter every Singer in his path, the end that Praile saw through would be worse.\n\nEternal.\n\nThings never shut down in the hot place. Praile was creative in his ministrations for those who didn't tow the fucking mark.\n\nLying in a ditch, evacuating his guts on the side of the road, was definitely not the follow-through that would free him of his obligations.\n\nTony would have to kill and feed. Then he would have to get his ass over to Two and get those Singers dead.\n\nHis thoughts touched briefly on Jacqueline, now knocked up by the jellybean fey. Fucking Unseelie.\n\nShe'd been a tender morsel to dominate.\n\nTony growled, jerking himself out of the ditch. No use crying over spilt lust. There'd be more females to hate on. Right now, it was Praile's agenda or he'd never have another thing to toy with. It'd be endless agony.\n\nBeing a bully had its plus side, but an even bigger bully was in charge now.\n\nTony moved into the forest, weaker than he'd ever been.\n\nHe sighted a small herd of elk. Hunger overcame all other instincts.\n\nHe ran.\n\nKilled.\n\nAnd hungered no more.\n\n*\n\nJacqueline tried not to cling, alternately loving her newfound kind streak and hating the new weakness. It seemed she could not be both.\n\nThere was a reserve in the way Domiatri treated her, as there was from the rest. Even the vampire had given pause when introductions had been made. Jacqueline understood. She'd tried to kill their blood messiah. The vampire looked at Julia Caldwell as the savior of their kind, the refresher of humanity's precious blood load.\n\nBefore, Julia had been a target for Jacqueline to take out.\n\nJacqueline had been deep under the influence of withdrawal sickness. Even now, though her proximity to a fey mound had always been too distant for her to understand what it could have meant for her to be closer, she felt slightly ill.\n\nBut she was making it possible for Domi to exist in this realm longer than a day. He siphoned, or borrowed, from her other genes while she grew weaker.\n\nThey needed to return to Faerie.\n\nHer hand went to her belly. She was suddenly afraid. Fear felt foreign to Jacqueline, accustomed to being the intimidator. Not the intimidated.\n\nDomi took her elbow, and she startled slightly. \"I do not like having you near Tony again.\"\n\nShe spoke the truth, \"He does not look to kill me but to kill Singers of the purest blood.\"\n\nDomi wore his disquiet like clothing, though he nodded his assent. \"I will find Tharell, and he can search out the vagrant without me. I feel Faerie's call and need to protect my offspring.\"\n\nJacqueline frowned, thinking of his wording, trying not to remember Tony's use of her. \"What of Tharell lasting outside of Faerie? Won't he sicken?\"\n\nDomi gave a rough exhale, pegging his hands on strong hips. \"He will, but it is not a quick thing. As a Sidhe, I cannot spend an inordinate amount of time away from the sithen. It is a kind of sustenance as well as a sentient being. It is fuel for well-being. Your presence affords more time, but it is not an indefinite thing.\"\n\n\"But not for Tharell?\" Jacqueline thought it odd Tharell was so terribly second-class inside of Faerie.\n\n\"No. He is not pureblooded Sidhe. He possesses enough for immortality, but it is not enough...\"\n\n\"For acceptance,\" Jacqueline finished.\n\nDomi's shoulders slumped. His smooth green flesh was like grass with fresh dew and shone slightly in the deepening night. Soon he would look as the trees did in the wind, shadows of soft black that moved with the rhythm of nature.\n\nHis face fell into solemn lines. \"It is not of my choosing. Tharell is a great warrior and has been a good friend. Many do not understand his composition does not make the man but his deeds.\"\n\nJacqueline worried her teeth at her lip. \"So we'll return to Faerie?\"\n\nDomi gave a sudden grin, the white of his teeth so startling in the gloom she took a step back. Something in that smile smacked of maleficence, as well as an abiding sadness shrouding him. He seemed utterly pleased to have rescued Jacqueline from her mistreatment by Tony and her evil nature was in large part a manifestation of fey sickness. Or the lack of Faerie to \"feed\" her.\n\nJacqueline understood Tony needed to be stopped. The evilness remaining inside her rejoiced at the very idea of witnessing his demise. But a greater need overtook petty concerns such as vengeance. Her child that she carried, the first one she thought of with tenderness, was counting on her to survive. For the first time in memory, it mattered to Jacqueline that something besides herself was depending on her.\n\nNow Domiatri would confer with Tharell, and she would return to Faerie. Not to be left with her torturer but to live there while her belly swelled with Domi's child. She had been assured just how important her Singer's blood would be to the people of Faerie.\n\nHowever, not a great deal more had been expounded on. It lay with her like uneasy tidings, half-told, lesser understood.\n\nRuling as a Singer of royal blood did not motivate her. No great reception waited for her at Region Two. Her subjects loathed her. She'd ruled with a tyrannical fist. They'd obeyed out of fear, not respect. Tony would have arrived, and the Jacqueline from before would have responded as she always had: putting her best people forward to die while she escaped.\n\nHer face grew hot with thoughts of the past. All she could do now was go forward. Staying at the conquered Region One was out of the question. Jacqueline did not wish to face Scott when and if, he returned. Or her vampire daughter, Delilah. Jacqueline would not be welcome if she returned to Region Two.\n\nDomi correctly stated that Faerie was her best option. He'd been too diplomatic to say it was her only option.\n\nHe scattered her thoughts as he stood beside her. \"Yes\"\u2014he cupped the back of her head\u2014\"we'll go. Even if I must \u02bbpull rank', as the humans say.\"\n\nJacqueline nodded, and he slid his hands from her head to her shoulders. She watched him walk away in search of Tharell.\n\nShe was bereft.\n\nJacqueline hated these new feelings of neediness.\n\nShe turned, and half a dozen eyes fell on her like beads of hate. They burned along her skin like fire ants.\n\nThe Singers who remained would never accept her either.\n\nJacqueline swung around in the opposite direction so she would not feel their hate any longer and walked slowly to the Singer mansion. She would eat and find a quiet room until Domi returned. She had made amends with the Rare One; the remainder would always hate her. Jacqueline could do nothing there. However, she didn't need to be around so many that wished her ill.\n\nJacqueline had a new life to think of. And though Domi did not love her, he acted on her behalf and it was more than she deserved.\n\n*\n\nTharell palmed the water, and Gabriel's face distorted then faded from the disturbed pool he had made with the gesture. He could not call back the things now set in motion.\n\nHe stood as Domiatri approached. Tharell grieved, and a wound seeped that would never heal.\n\nHowever, he remained without choice.\n\n\"Tharell?\" Domi asked, the moonlight turning his dark blue hair to silky ink in the low light. His eyes flashed silver, reflective like the wild animals of this region.\n\n\"What are you doing?\" Suspicion had crept into his tone.\n\nAs well it should. The weight of old magick thickened the air, and the vibrating hum of its use could still be felt. As a full-blooded Unseelie Sidhe, Domi would sense it as though a woman had just applied perfume and left the room. The fragrance lingered.\n\nThey did not allow Tharell to use the greater ancient magicks, though he was Sidhe enough to perform them all, or allow his mating. They did not want his genes passed through.\n\nHe was not too much a mongrel that Queen Darcel hadn't wanted a taste of mixed-flesh in her bed on occasion. On many occasions.\n\nHowever, Tharell had never spilled his seed in the deepest part of her. Not one drop.\n\nHe was not half-breed enough to be put anywhere but at the front lines of war. Tharell had fought and bled for Faerie but never been a part of the jeweled collective.\n\n\"Contacting the Northwestern coven,\" Tharell answered carefully.\n\nDomi thrust his chin back in surprise, and his sword cleared its sheathing. \"Tharell, we are here to assist the Blooded Queen.\"\n\nTharell felt ill for the betrayal and confusion on his friend's countenance.\n\nTharell shook his head sadly, his own hilt growing warm in his palm. \"No, you are here for that calling. I am here to guide my brethren to Faerie.\"\n\nA beat of ominous silence held portentous weight. The kind stagnant for only a moment before it erupts and boil over a pot left too long on a stove.\n\nUnderstanding filled Domi's eyes, but it was too late. Tharell dove at his friend, looking to wound fatally. He had a plan, the only one with mercy out of everything he had built thus far.\n\nDomi's sword made a whistling sound as he threw himself backward to avoid the metal of Tharell's blade, his arms sprung wide for balance.\n\nTharell had feinted.\n\nHe drove the hilt into Domi's femoral artery and hauled the blade upward, severing half the leg in one go, groin to hip.\n\nDomi did as everyone does when his or her leg is half-cut off. He fell.\n\nTharell grabbed that beautiful long hair and hauled the warrior to the water's edge.\n\nDomi's lifeblood was rushing out in an arcing spray. He clawed at Tharell as he spun behind him.\n\nTharell clapped a hand over Domi's mouth, just as a short dagger slid between Tharell's ribs.\n\nThe pain staggered him, yet he hung onto the hair and shoved the other Sidhe into the cool lake.\n\nStrong fingers crushed his forearms.\n\nBlood like octopus ink pulsed in time with Domi's heartbeat and pooled around them as they struggled against each other.\n\nBubbles rose.\n\nSilver eyes regarded him with a betrayal that needed no vocalization.\n\nA thousand years lived, and Tharell had never felt his misdeeds as powerfully as he did this one.\n\nThe second slice of his blade separated Domi's head from his body.\n\nTharell let go of the hair he'd always admired and wished for from afar.\n\nDomiatri's body bled as his head bobbed on a lake far from Faerie.\n\n# CHAPTER THIRTY-THREE\n\n\"Who will stay?\" Julia asked. She looked at the small group then put a thumb to her chest. \"It's not going to be me. Tony's out there, and we need all the man\"\u2014she glanced at Cyn and Adi\u2014\"and woman power we can get.\"\n\n\"We stop that tool and save Two,\" Adi agreed.\n\n\"We Reds can stop him,\" Slash said. Thirteen Reds would accompany them, including Jason. No one knew for sure if the Reds would be able to withstand whatever magic was at work on the blade Tony wielded.\n\nTharell was just walking up. Julia's internal tally for the Tony hunt automatically included him, Jacqueline, and the green Sidhe.\n\nShe looked around at the clean grounds and couldn't help the memory of her people's gore rising inside her mind. She closed her eyes against the red tide of their death, but it remained.\n\nIt was all very premeditated, the planning of Tony's murder. Though it couldn't be helped. He'd slaughtered nearly everyone in One. Julia didn't reflect on the people she'd personally lost. It must wait for later.\n\nThrilled to leave thoughts she couldn't dwell on behind, Julia thought of Scott and Lucius. She turned to Angela. \"Do you feel anything?\"\n\nThe girl shook her head.\n\n\"It's not like that. I can only know things if they're right here.\" She brought a palm up in front of her face.\n\nScott. Julia was no longer soul-bound. But she mourned his absence and worried about what the Reds might be doing to him while they stood around planning Tony's murder. She couldn't spare their resources to save two Combatant when a hundred Singers were in jeopardy.\n\n\"So the vamps have gone ahead, and Tom Harriet is still out there with who knows how many more Reds...\"\n\n\"And some important peeps are unaccounted for,\" Cyn reminded her. Julia already knew. Reagan, Delilah, Manny, and Victor were only a few of the important supernaturals not dead.\n\nThey were also not there.\n\nJulia blew out a slow breath, tired to the core. Small wisps of hair settled around her face. Eyes that had never regarded her before as someone who provided answers now looked to her.\n\n\"Okay, I think we need to get going and meet the vamps in Montana.\" Julia looked at the assembled faces. \"Who's rested enough to take three of our SUVs and start driving tonight?\"\n\nThree hands rose, one Truman's.\n\nJulia smiled. \"Truman and...\"\n\nJason squeezed her shoulder, and she pressed her face against his knuckles.\n\n\"Jason and...\"\n\nBrynn stepped forward. Julia gave a slight gasp. \"What?\"\n\nThe Reds circled the lone vamp.\n\nShe couldn't keep the surprise out of her voice. \"Why didn't you go with the others?\" She assumed he would have headed up the reconnaissance with the other four.\n\nHe narrowed his eyes at the Reds. \"Back off. I mean no harm to the Rare One.\"\n\n\"Right.\" Jason's body tensed around her like clinging vine.\n\nJulia gave a short laugh. \"I'm their ultimate food goddess. I think it's okay.\"\n\nBrynn smiled at her joke, and his fangs appeared. \"I don't require sleep, just no sunlight.\"\n\nOf course, Julia thought. \"You sent them ahead...\"\n\n\"And I will drive until dawn.\"\n\nTruman harrumphed and they turned to him. \"Not bad for a bloodsucker.\"\n\nBrynn's grin widened as he gave a slight bow. \"Yes.\"\n\nThen he hissed.\n\nThe Reds retreated.\n\nVenom dripped from his fangs, the skin of his face stretched taut with his emotion. \"I am not endlessly patient.\" His eyes bled to cruelty, meeting theirs for a flat endless second. \"Let's go.\"\n\nJulia swallowed hard, remembering William.\n\nShe followed along with Jason.\n\nBrynn pivoted in their direction. His eyes held humor and challenge. \"For those of you who aren't afraid.\"\n\nJason raised his middle finger. \"Not of you, vampire.\"\n\nBrynn lifted his lips in a smirk, the tips of his fangs flashing and disappearing instantly. \"Good.\"\n\nJason dropped his hand but not his guard.\n\nCyn and Adi rode in the last SUV as the Reds, Singers, and lone vampire made their way to Montana.\n\nThough none realized they ran nearly parallel with Tony. They presumed to chase him.\n\n*\n\nTony stuffed the body of the hotel guy in the trunk of the car he'd stolen along the way.\n\nJust humans this time, thank moon.\n\nThe owner of the shabby thing had balked at Tony's look.\n\nTony had laughed when he got a load of his own reflection later. He appeared to have bathed in a tub of blood then rolled in a vat of mud and walked around for a day letting it dry to a cracked, desert-like finish. There was also the issue of his holey attire. And not the religious variety. Tony thought more mottled, half-healed skin showed than clothing. The bitch's shot had shredded his clothes and peppered his skin with badly healing pockmarks, filthy and punctured.\n\nThat's probably why the guy picked up the phone mere seconds after Tony walked through the doors to his cheesy motel lobby.\n\nThe proprietor had been a startled owl with glasses so thick his eyes appeared as big as a toad's.\n\nTony had jerked the receiver from his hand and ripped it out of the wall in one motion. He'd been sufficiently spooked, trying to scramble away. A burst of talons through his back slowed his momentum considerably. With a jerk and hop, Tony leapt behind the desk in full evisceration mode.\n\nHis kills were so messy. Even he could admit the fault there. It didn't bother him.\n\nSo now, Doug was stuffed inside the trunk of the stolen vehicle. It nearly made up for the supreme fuck-up with the surprise Alpha bitch.\n\nA pale and fleshy arm dangled like a swinging pendulum.\n\nCouldn't have that. Tony casually swiped it back inside the cramped space.\n\nHe whistled, making his way back into the lobby and hit the No Vacancy sign. That accomplished, he tore the keys to the front door off a loose nail.\n\nThe nail skittered across the old linoleum floor as Tony spied a vending machine filled with pastries.\n\nPerfect. Sugar was always a ready solution to faster healing.\n\nHe strolled to the machine and, using a discarded shirt\u2014Doug had most recently lost the need of it\u2014blasted a hole through the glass with said shirt wrapped around his knuckles. Glass shards clinked like frozen rain on the battered floor.\n\nPastries of every variety poured out at a trickle that became a geyser. Tony scooped up an armful and went to the first available room.\n\nAfter eating ten slightly stale delicacies, he cut his gore-soaked clothes away and took a scalding shower.\n\nHe padded naked to the middle of the room and decided he'd need new threads.\n\nHe twirled the master keys on his finger, counting twenty-two rooms.\n\nSomeone would be a close fit for clothing. Time to shop.\n\n*\n\nJacqueline grabbed Tharell's arm, stone beneath her grip. \"You say Domiatri elected to return to Faerie and you would go on alone?\"\n\nTharell nodded.\n\nA frown puckered her face. It was beyond odd. Domi had made it clear he would discuss their early return to Faerie with Tharell. That Tharell would go on\u2014without them. Jacqueline relayed this.\n\nHis eyes became hooded. \"We do not have time to discuss the details. The Blooded Queen and half the Reds have already gone for Two\u2014your old headquarters,\" Tharell reminded her unnecessarily. It stung. He held the door open to the large car, sweeping a dark purple hand at the empty space. His palm was light lavender.\n\nShe suddenly felt like crying. Her newfound emotional well was terrible, exhilarating, and ultimately exhausting.\n\nTharell continued, \"It is I who should have qualms about re-entering this horrible transport of metal. You will be safe. Domi prepares a place for you and his unborn child in Faerie even as we speak.\"\n\nJacqueline hesitated while Tharell calmly waited by the door. He made it sound so logical. However, her old nature struggled to the surface. Something did not feel right. Jacqueline rolled her lower lip between her teeth.\n\nTharell's grip on the car door, so tight that his normally purple-hued skin looked whitish-lavender in the harsh porch light, gave her further pause.\n\nJacqueline did not like it. She took a final scan of the surrounding fields that bled into the woods. Finally, her gaze fell on the lake.\n\nThe last place she'd seen Domiatri.\n\nWith a sigh, Jacqueline hiked up into the offered seat. The Reds squished inside lifted their lips in a show of distaste to sharing a vehicle with a known traitor.\n\nJacqueline turned from their accusation, looking out the window as the vehicle pulled away. She had never felt more alone. Before, Jacqueline had only herself, and it had been enough, solidarity with only her. She'd not been self-aware enough to notice. Now she was hyper-aware of her place in this world.\n\nIt was lowly.\n\nTharell sat in the front seat as Brynn began to drive.\n\nHe gave her a look she thought meant a show of support. Instead, it deepened her feelings of disquiet.\n\nHis next words made it worse. \"I shall take care of you until you meet with Domi once again.\"\n\nJacqueline did not respond, looking at the passing scenescape. Her eyes once again sought the black lake where she had last seen Domiatri as it disappeared from view.\n\nDomi, why did you go and not even say goodbye?\n\nDomi apparently had not felt he owed her even that.\n\nYet, she had wanted it.\n\nJacqueline put a palm to the glass as though she could touch the warrior who had not given her the word of closure she needed most.\n\nHer forehead stayed against the glass until the last Red fell asleep.\n\nSleep never came for her.\n\n# CHAPTER THIRTY-FOUR\n\nSlash knew something was afoot with the purple fey warrior. He hadn't been war-torn and buffeted not to notice the subtle signals. But for the life of him, he was unsure the motivation.\n\nIt did not benefit the fey to kill Julia. She was the Blooded Queen, and everyone wanted her alive. He took her off the list of potentials. His thoughts mulled over the possibilities of what else there was.\n\nThe facts presented themselves. Tony had gone bat-shit crazy, was obviously part demonic, and had a blade juiced up with the blood of a high demon. Those were nothing to fuck with. Slash understood very few supernaturals possessed any immunity to demonics.\n\nThen there was the counterpart, aware that the blood of the angelic ran through Singers' veins. Terrific.\n\nSlash drummed his fingers on the stiff, borrowed jeans as he drove.\n\nIf Tony was truly demonic and wielding a weapon charged with a high demon's blood, their collective gooses were cooked. Unless Julia could somehow dampen its power.\n\nSlash chuckled. He clenched his hand around the steering wheel, making it creak in protest. Julia was basically an angel.\n\nHis smile faded when he wondered what Tharell's angle could be. His mind turned over the different scenarios a second time. Slash touched on the females, always critical to every scenario. The only species with equal numbers of genders were the Singers.\n\nAn epiphany struck him.\n\nThere were few females making their way to Region Two.\n\nSlash knew a trap when he saw one.\n\nSomehow, the females were key. He'd like to take the easy road and settle on Julia, but that was too simple and obvious. He had already dismissed her as the focus.\n\nIf not her, then who? Adrianna\u2014his wolf flinched at anything to do with her\u2014was nothing exceptional except her position as an Alpha female. That was distinctive to the Were, but not important to demonics. Cyn was both Singer Healer and newly turned part Red Were. Interesting combination but again, not exceptional.\n\nWhat could it be?\n\nJacqueline came to mind, but he dismissed it. She was pregnant by an Unseelie Sidhe, and with such a mixed lineage, she couldn't be anything but a mess of genetics for a demonic.\n\nIt must be Julia.\n\nHowever, how could Tony hope to combat what he'd have to assume was coming after him.\n\nUnless...\n\nSlash's eyes widened.\n\nThere'd be a huge welcoming party from Hades.\n\nThe Reds could hold their own. Slash rubbed the scar that bisected his face like a talisman. It ached sometimes, usually when blood would be shed.\n\nHis own.\n\nAnd that of others.\n\n*\n\nJulia's eyes slid open. Sleeping against Jason had been wonderful, but now she was paying with a crick in her neck. She moved, and his arm slipped from around her.\n\nShe lifted her chin and gave him a sleepy smile.\n\n\"Hey you,\" Jason said and kissed the tip of her nose.\n\n\"Hey.\" She cupped her hand over breath she was sure was all kinds of bad.\n\nHe smiled. \"For better or worse, Jules. I bet my mouth smells like ass, too.\"\n\nJulia laughed. \"Nice.\"\n\nHe tucked her in, squeezing her against his side, and her ribs protested the movement. She shoved her arm behind him, feeling his muscles clench, and he laughed and arched his back.\n\n\"Still ticklish?\" she asked in an evil voice.\n\nHe nodded, a dimple disappearing as quickly as it came. \"Yeah, thanks...\"\n\n\"We're coming to the end,\" Brynn announced without inflection.\n\nThe pale dawn began to suffocate the darkness.\n\nBrynn drew Julia's gaze.\n\nHis skin was smoking.\n\n\"Oh my God,\" Julia said, \"we've got to get you in the ground.\"\n\nHe gritted his teeth, and Julia wanted to clap her hands over her ears at the sound. \"Yes,\" was all he said.\n\nShe was a shitty leader. She should have thought this through and known that Brynn would blow up. So dumb. Julia looked around the dim guts of the car, her eyes lighting on a folded canvas tarp. \"Pull over.\"\n\nHe didn't argue. He'd begun to smell like simmering flesh.\n\nBrynn jammed the gearshift into park as the tires sank into the soft gravel at the shoulder.\n\nHe hopped out of the car as Jason and Julia slid back.\n\nBrynn barely made it to the ditch before he threw up.\n\nBlood splattered like a can of thrown paint. Fresh scarlet coated the ditch and halfway up the embankment. Droplets hit the trunk of a nearby tree and slowly scrolled down the deep ridges on the bark.\n\n\"What the hell?\" Jason yelled, throwing himself at the vampire.\n\n\"Drag him around back,\" Julia yelled, running for the trunk. It was an older model with a handle on a single swinging door. She tore it open, so thankful it was unlocked she could barely stand it.\n\nJason half-dragged, half-carried Brynn around the back. His skin had cracked in small fissures like shattered glass. The cracks deepened before her eyes and begun to run further.\n\n\"Get him underneath this.\" Julia threw up the corners of the canvas and Jason positioned the poisoned vamp underneath.\n\nHe gave a rasping sigh as they cut off the growing light.\n\nJulia tucked the canvas around Brynn's body as the first rays hit the SUV like a slap.\n\nJason half-fell against the side of the black rig. He plowed his fingers through already disheveled hair. \"Goddamn... that was cutting it close.\"\n\nJulia nodded.\n\n\"I can't believe I saved a vampire.\" He shook his head. \"How many shades of bizarre is that?\"\n\n\"About as weird as us racing against Tony to save our people?\"\n\nThey shared a look of tense disbelief.\n\n\"Let's go.\" He seated her beside him in the passenger seat.\n\n\"Are we almost there?\" Julia asked and Jason smiled, stroking her cheek with the back of his hand. Scabs from fighting scraped along her skin.\n\n\"You're like a kid.\"\n\n\"I only wish.\"\n\n\"Yeah,\" he said with more than a touch of sadness. \"True that.\"\n\nHe put the truck into gear, and they pulled away.\n\nJulia glanced behind her.\n\nBrynn's blood stood like an unmarked crime scene. She turned to face ahead before grief took hold and never let go.\n\nShe still had a job to do, and she couldn't let the abyss of sadness take over.\n\nLives were depending on her.\n\n*\n\nTony kicked the severed limb inside the door and slammed it tight, putting weight against it to secure the latch. He didn't bother locking it.\n\nHe untucked one lapel from the borrowed coat and sighed. Fucking finally.\n\nHe strode to the stolen car just as sirens began to wail in the background. He wasn't much for speculation, but he assumed someone had gotten too clever for their own good.\n\nTony frowned. Ten rooms had been occupied.\n\nWho'd squealed?\n\nHe paused for a moment. Tony didn't have many of those to waste. He jogged to the other nine rooms and kicked in the doors of each. His nose did its job, finding nothing but decaying meat.\n\nHe was stumped.\n\nA woot from a siren that was closer still alerted him his seconds had run out.\n\nDammit. He hated there might be a potential witness. He scanned the scene, seeing nothing, missing nothing.\n\nHe sprinted back to the running car. Tony leapt inside and slammed the shifter into drive, putting the clutch practically through the floorboards.\n\nGravel sprayed as the wheels turned and finally grabbed asphalt.\n\nTony whistled as he crossed the Idaho-Montana border.\n\nA lone female Were had been left behind. She hung up the receiver from her call to the human authorities. The rogue Alpha male had slaughtered her human guardians.\n\nHe had not scented her, as Taliah had not entered her change. It had been a near thing.\n\nShe could hear the human officers.\n\nA tear slipped out as she surveyed her beloved male guardian, his body discarded as a heavy weight against the door, an arm missing and driven between him and the door.\n\nTaliah looked for any escape and found it in the bathroom window.\n\nShe heard shouts and was scared.\n\nShe must use her rare talent or be interrogated about her own kind. Yet phoning the police had distracted the insane Were.\n\nHer wolf would not help her in this situation.\n\nBut the bird inside her would.\n\nShe muffled the cry of pain that escaped, ending in a note of a birdcall.\n\nA snowy white dove spread its wings, turning its neck as eyes like ebony crystal found the window, open enough to allow a small body to move through.\n\nTaliah sprung to the ledge and, looking out, flew through the hole and toward her den. She needed to warn her people that a murderer approached them.\n\nThe cops never saw a pure-white bird make a deliberate slow circle above their heads before moving east.\n\nThe carnage distracted them too much to notice.\n\n# CHAPTER THIRTY-FIVE\n\nA terrified group greeted Julia and Jason.\n\nThey were also looking awfully ready to lynch Jacqueline.\n\nJulia walked into the middle of some bad blood. There simply wasn't time to deal with the backlash against their leader. Tony was coming, and Brynn's vamps were near. They hid for now as the day was at its height.\n\n\"I say we jail her forever,\" one tall and icily beautiful man, who looked eerily like Victor with his dark blond hair and pale gray eyes, said. He drew closer, and Jacqueline retreated a step. Julia knew firsthand how powerful Jacqueline was, but she was no match for a crowd of talents fueled by adrenaline and emotion. Bad combo.\n\nJulia put her hands to her temples in a hopeless attempt to ease the ache of the mind-noise that coursed through her brain in a numbing sea of static. She let her hands drop and walked in front of Jacqueline, facing the new group.\n\n\"Listen up!\" Julia blasted out, and Cyn chuckled somewhere. She plowed forward into the silence of their surprise. \"My name is Julia Caldwell...\"\n\nNoise erupted again.\n\nJason put two fingers to the edges of each side of his mouth and gave a whistle so shrill Julia cringed.\n\n\"Shut up,\" Jason said and instantly, three large Singer males fenced him in.\n\n\"Don't even think it, dudes,\" he warned.\n\n\"I am the Rare One,\" Julia said, hoping for things not to escalate further.\n\n\"We know who you are,\" the one who looked like Victor said.\n\n\"Are you a relative of Victor?\"\n\nHis face flooded with surprise, and he nodded once. \"He is my brother.\"\n\nGreat. And the bad hits just kept on coming.\n\n\"And she\"\u2014he jabbed a finger in Jacqueline's direction\u2014\"is the reason he is no more.\"\n\nOh boy.\n\nJulia put up her hands. \"Listen, I know there's a lot of hate for Jacqueline....\"\n\nCheers interrupted her comment, and Julia frowned. \"But she's a changed woman... and a pregnant one.\"\n\nThe crowd grew quiet, and Tharell showed himself.\n\nEveryone backed away at his appearance, the rush of whispering uneasy. Julia sensed the undeniable energy of talents readying themselves.\n\nThings were getting out of hand. Tharell had just revealed himself like an alien and the Singers of Region Two were getting aggressive.\n\n\"Listen! Please!\" Julia begged and two more whistles and a nod from Jason let her speak. \"There is a rogue werewolf who carries a special blade dipped in the blood of a demon and he's killed most of...\" She stuttered to an incoherent stop. The grief threatened to swallow her.\n\nJulia slowly lowered her head to her chest. She couldn't contain her shaking. She couldn't say the words\u2014they were so horribly final.\n\nTharell finished for her. \"The Singers of Region One are now dead.\"\n\n\"Not all,\" Slash interjected, and Julia lifted her head as she frowned at the look he shot Tharell. It wasn't friendly.\n\nTharell leveled his own stare at Slash and gave a long blink. Julia guessed that was his silent agreement. However, she wasn't too sure with Tharell, he was hard to read.\n\nJulia collected herself like a broken teacup and looked out over the hundred or so faces. The thrum of their likeness was like a battery that powered her. Mainly men, some women and a few children gazed back at her.\n\nThis was all so hard. Laying all the details out. The deaths, the oncoming murderer. Jacqueline the Unpopular returning, Tharell cropping up looking all... purple.\n\n\"We have not heard of this\"\u2014Victor's brother began, indicating Tharell. \"And the threat of a demonic has been so rare and so long ago, it is no longer mentioned. It's not part of the public consciousness. As far as our blood, we have always known it is anchored in the heavenly.\"\n\nJulia was swimming upstream with a group motivated the right way but not awake to the mortal danger coming toward them. A terrible storm was ready to tear them apart, and she was the only one in the know.\n\n\"He's going to kill us all,\" Julia said.\n\nThere were gasps and murmurings but not the panic she thought they ought to have, considering there had been a mass killing in their neighboring Region. She and Jason exchanged an anxious glance.\n\n\"I am Gallagher, and am one of the oldest at this Region,\" Victor's brother introduced himself.\n\nJacqueline spoke up for the first time. \"I have suffered at the hands of this Were, and he has a void where compassion would normally be. We are in grave danger.\"\n\nA nearby Singer scoffed at her comment. \"Whatever acts he has committed against you are deserved.\"\n\nJulia had had enough. The feeling of surrealism descended as she took on the role of defending Jacqueline. Again.\n\n\"Rape?\" Julia stepped forward in challenge, her gaze surfing the crowd.\n\nThe male Singer backed away and stuttered, his face filled with the blood of embarrassment. \"No, I did not know...\"\n\n\"Then shut the fuck up, why don't ya?\" Adi said. \"Jacqueline's had a case of The Dumbs for about... I don't know, forever. But now she's changed, she's sick because she hadn't ever been to Faerie, and now she has.\" Adi squinted her hazel eyes at the crowd, daring them. There were no takers. She went on, \"She got knocked up by a green Sidhe warrior, and she doesn't want to be queenie anymore. So now you guys have Julia, so get over yourselves.\"\n\n\"I really like her,\" Jason said with a grin.\n\nJulia did, too. She got things said, summed up, and thrown out there for immediate consideration. That's what they needed.\n\nGallagher frowned, cupping his chin. \"She was forced by this Were?\"\n\nJacqueline kicked up her chin. \"Not in the beginning. When I first arrived at the Faerie mound, I was as I've always been. I allowed the vile wretch to use me in the hopes we could make the fey uncomfortable enough to release us.\" Her face grew pink, and she added quietly, \"To cause more mayhem in a bid for power.\"\n\nThere were mutterings of assent to this. Her people had known well the temperament of their leader.\n\nShe linked her hands together in a knot that appeared so uncomfortable Julia flexed her own fingers.\n\nJacqueline spoke to the ground. Her voice was a natural contralto, sultry and evocative. No one had trouble hearing her next words, \"When the environment began to work on my... sickness, his attentions became violent against my person.\" Her head rose. \"As I was no longer willing to play his game. And... my talents did not work in Faerie. The magick prevents talents from working correctly.\"\n\n\"You did nothing to help her?\" the Singer who'd first commented asked Tharell in a sharp voice.\n\nTharell nodded. \"When we became aware that the Singer was suffering at the hand of Anthony Laurent, we intervened.\"\n\n\"Who's \u02bbwe,'?\" he asked.\n\n\"Domiatri,\" Jacqueline stated quietly, \"The father.\"\n\nGallagher put his hands on his hips. \"This is all very untoward. A male claiming to be the mythological Faerie has come in our midst, while a Rare One we've only just heard about claims Region One is effectively no more. And a contingent of Red werewolves stands around making noise of defense and war. And our former leader was held in some kind of prison where she was abused then further assaulted by another male of your species, only to later become pregnant with the fey's offspring.\"\n\nHe frowned, then his face took on a new expression of disdainful amusement. Gallagher threw his head back and belly laughed. \"Forgive me if I am not, what would you call it, ah yes, 'on board' with all that you have relayed.\"\n\nGallagher looked around at the faces of the Singers. As Julia opened her mouth, he raised a hand to ward off her comment. \"And where is this other fey... Domiatri?\" he asked with a lilt of aggressive disbelief.\n\nTharell's entire face changed. The first genuine smile Julia had seen on his face in days, spread and stayed in a happy freeze.\n\nIt gave her the creeps.\n\n\"Why\u2014he is dead,\" Tharell stated, and Jacqueline turned to him, paling before Julia's eyes.\n\n\"What?\" Jacqueline asked, hand to chest. \"This is not funny, Tharell.\" She dropped her hand, and Julia saw a fine tremble to her body. Jacqueline was nearing the edge of whatever threads kept her together. Unraveling before their eyes.\n\nWelcome to my life, Julia thought.\n\nTharell looked down on her, his humor fading to nothing. \"No, but it was necessary.\"\n\nJulia let that comment percolate for about one second before terror shredded her as a many-bladed Chinese star.\n\nJulia's hair spun into her eyes, momentarily blinding her as she swung toward Jason. He was already moving, dragging her behind him as talons slammed out of his fingertips in a painful burst of rendered flesh.\n\nSlash leapt to Julia's side as well, hauling a bewildered Adi behind him as Jason shifted into a wolf, his human shell splattering around her. Julia barely had time to react but managed to throw a telekinetic shield around herself. The bits of him hit the surface invisibly and slid down in front of her as though a thin sheet of glass molded to her body covered her.\n\nShe let the entire thing go, and what remained of Jason's human body was chopped meat at her feet.\n\nShe stepped over the circle of his remnants and his wolf moved by her side.\n\nJulia was too late to stop him. He'd played the card so well that it seemed only Slash was ready.\n\nTharell held Jacqueline by the throat and jerked her backward. Her dark eyes, so like Scott's, were so human in that moment as they regarded each other.\n\nFear reigned supreme in Jacqueline's gaze, where only power and bloodlust had before.\n\nTharell taking Jacqueline to parts unknown was certainly a concern.\n\nBut the demons which bled out of the surrounding forest and circled the group really got Julia's attention.\n\n# CHAPTER THIRTY-SIX\n\nTharell scanned his surroundings and instantly assessed the options available to him.\n\nThe demonic had arrived in large enough numbers that the skin of the Singers who possessed sufficient angelic blood began to pale to partial transparency.\n\nTheir blood was reacting to the threat of their counterpart.\n\nVeins appeared on nearly three quarters of the Singers. The thin map work of circulation, lined in silver instead of the human species' blue, brightened and disappeared in time with their heartbeats.\n\nIt pulsed inside their skin like metallic lace.\n\nTharell smiled. He surmised they'd never faced their mortal enemy, their true place in this world unknown until now.\n\nYet their blood knew and summoned a response as ancient as it was automatic. The Blood Singers stood amongst the demonic in beautiful contrast. Had Tharell chosen another path, what was about to happen would have grieved him.\n\nThe Rare One was a thing of shining beauty. Her veins shimmered like small rivers of gold, lighting her unusual eyes from inside like golden flames, her hair a mane that floated like a cloud of gold as she glided into the circle of the Singers.\n\nThe demonic could not look upon her. She was as the sun, and they were weak before her light.\n\nJacqueline made a gurgling sound and Tharell sighed, loosening his hold.\n\nThen she launched him fifteen feet behind her and into the nearest tree trunk.\n\nTharell hit the old growth tree, and it did not yield.\n\nThough he was immortal, it still hurt like Hades to plow into a tree at thirty miles per hour.\n\nHe shook himself like a wet dog, picked himself off the ground, and walked toward her.\n\nTharell fell.\n\nHe looked down.\n\nBoth legs were broken, and one arm hung askew.\n\nHe snapped his face toward Jacqueline from his position on the forest floor, and she smiled. A little bit of the woman he'd first been acquainted with was peeking through the edges of that smug smile.\n\nTharell had underestimated his quarry.\n\nHe looked to the demonic nearest him and gave them telepathic orders.\n\nTheir black eyes slid away from Tharell and to the wayward Singer.\n\nPregnant with the one child that could ruin the demonic.\n\n*\n\nJacqueline was gratified. She had spun the traitor off her back like the garbage he had proven himself to be.\n\nHow dare he murder Domi? The pit of her stomach swarmed in emptiness, and she elected not to feed it with her ready despair.\n\nIf Jacqueline had been honest with herself, and she was never one for introspection, she would have realized she loved Domiatri. Jacqueline knew he did not feel as she. But he had treated her with tenderness, protection and respect. She had never known it.\n\nNever deserved it.\n\nShe felt a pang of guilt, remembering it could have been thus with Victor, but for her vile temperament.\n\nThere was no time to consider these thoughts, though they had run through her mind instantly.\n\nThree demons approached in a loose triangle as Tharell lay in a broken heap, trying to repair the damage she had wrought. She wasted a smile on him.\n\nThe wretched botch. He deserved everything she had managed against him.\n\nJacqueline gave no more attention to the whys behind his treachery. Tharell would pay.\n\nShe had mastered that in her past life, and the expertise had not left her, only the motivation.\n\nHeat like liquid fire beat along her skin, and Jacqueline inhaled deeply. The feel of burning from the inside did not subside. It was everywhere and nowhere specific but flowed through her, ice and fire that dove and spread through her flesh as the demons drew nearer.\n\nShe glared at the demons. Their blood flowed through skin that was different shades of scarlet.\n\nVeins flowed through their bodies like ink spilt over a parched river. Small horns decorated their heads, and whipping tails shook behind some of them like rattlers caught in a corner.\n\nThey were all male and similar in stature to her own species' warriors, though which one she could now claim lay in question. But first and foremost, she was of royal Singer blood, and that afforded certain talents and strengths the others did not possess.\n\nThe demonic moved aggressively against the Singers. Empty-handed, they did not have obvious weapons to use against their enemies.\n\nJacqueline looked around her. Many of the Singers' skin were now paper-thin, translucent and pale.\n\nSilver veins pulsed to shininess.\n\nTheir heartbeats, Jacqueline thought. Their circulatory system gets bright when their heart beats.\n\nShe brought her own hands up in front of her face.\n\nSilver veins edged with ebony glittered like roads of thread through her skin.\n\nWhen the first demon reached her, she turned the hand at which she'd been gazing and used her telekinetic abilities.\n\nA forked tongue shot out of its mouth, hissing as it blew backwards.\n\nJacqueline straightened and prepared for the next onslaught. Her heart caught in her throat.\n\nHer eyes widened as the one she had thrown into the forest rolled in a graceful somersault off the trunk into which he had slammed. He was instantly on his feet, the tail riding behind him like a macabre instrument of balance. He crouched low, facing her at approximately thirty feet away, and hissed, the black tail shaking in his anger.\n\nJacqueline's memories tumbled one over the other in a handful of seconds.\n\nShe would never be taken alive.\n\nHer life would no longer be lived only for evil. Jacqueline laid a hand over her belly and an answering pulse lit beneath her flesh as if in agreement.\n\nThere was someone else besides her.\n\nShe did not think of Domi. Could not.\n\nJacqueline moved her hands to her sides loosely and readied for battle.\n\nThen Tony appeared and her stomach dropped.\n\n*\n\nTony ditched the car on the road, tearing off the plates and spinning them into the opposite side of the road. The trees sucked up his garbage graciously, the tops of the forest latching onto the twisted metal in a loving embrace.\n\nHe smiled. He walked to the car and jammed a busted broom handle against the accelerator and the back of the driver's seat. He slammed the POS into drive, and it took off down an embankment that dove into a ravine.\n\nTires spinning, it launched off in a lurching Superman dive.\n\nGravel, water from a swollen river below, and broken limbs showered in reverse as the car nose-dove into the shallowest part of the river below.\n\nTony waited, seconds sliding into a minute, and the gas tank cooperatively blew. Tony instinctively covered his face as it grew hot from the burning wreckage.\n\nFan-effing-tastic, he thought with satisfaction. Good old Doug was laughing to pieces down there in the trunk. Literally. Of course, the dead had no humor. It was too perfect.\n\nHe laughed at his cleverness and backed away from the inferno.\n\nAfter about a quarter mile outside the immediate scent reach, he caught the one smell he'd been hunting before trashing the burning evidence.\n\nSingers.\n\nAnd one in particular: Jacqueline.\n\nThe grin on his face reflected his satisfaction.\n\nIt fit him perfectly as he'd slipped his human skin to go to wolfen the instant his nose guided him into the thickest part of the woods where his prize lay hiding.\n\nTony ran, the small animals of the forest moving like water split by a ship's prow. They scampered as he tore through their habitat.\n\nWhen he arrived where the Singers and demonic fought, he wore their small bodies to mid-shin.\n\nIf they did not move out of his way, he ran them into the ground.\n\nIt was this vision that filled Jacqueline's eyes when her gaze found him. He knew what she saw from the tenseness of her body.\n\nFucking weird, Tony thought. As he looked around at the Singers, he noticed they wore some kind of silver all over their skin.\n\nHe squinted, looking closer, and realized it was the blood coursing through their bodies.\n\nTony took in Praile's minions, and his bowels squeezed into a hard rope of tension.\n\nNo way was he going to fuck this up.\n\nHe raised the blade coated with the blood of a high demon and waded into the melee.\n\nIt was time to do the master's work.\n\n# CHAPTER THIRTY-SEVEN\n\nJacqueline stood stunned for all of three seconds as the fighting exploded all around her.\n\nDemons attacked the Singers and they fought with their talents.\n\nTony blew through two Singers with the blade.\n\nHe was moving closer to her.\n\nShe swung her gaze frantically about her as horns and talents locked. One demon used his tail, with a bulbous end spiked everywhere, as a mallet and he bashed in one Singer's head. Pulverized skull with brain matter blew apart like a bomb exploding, and the Singer fell to his knees.\n\nJacqueline stumbled then ran. She sought the Rare One, proof against the horrible strength of the blade. She could only Deflect as she ran and did so now, robbing some of the demonic who would assault her by making false visions pop up to confuse and disorient. Jacqueline could do so much more if she could only get to Julia.\n\nSweaty hands, hair, and limbs were everywhere as she slipped the grip of two demonics, only to be caught by a third.\n\nShe screamed as though death itself had captured her. And maybe it had.\n\nGallagher was suddenly there and she cried out in warning. Talons like knives of death lay against her skin and the Singer's blood shrieked in protest, shining like silver fire through his skin. The demonic that held her hissed at the vision of the shining Singer in front of him.\n\nThen Gallagher took his head with a sword swipe. The breeze as the blade swung through the meat of his neck allowed Jacqueline's release.\n\nGallagher caught her and a look of understanding passed between the two.\n\n\"I am sorry,\" Jacqueline shouted into the vacuum of noise.\n\n\"My sovereign,\" Gallagher responded, and Jacqueline sobbed in the middle of battle, when her life was at stake.\n\nIt was terrible timing, yet emotion was always without thought.\n\nHis sentiment meant that much. He was her only link to Victor, whom she had treated abysmally while under her rule.\n\nHe grabbed her about the waist and lifted her from where she had fallen.\n\nThey looked into each other's eyes, and he smiled. It was grim and tired but genuine.\n\nThen his mouth made a soft O, and Tony leaned over Gallagher's shoulder. \"Surprise, bitch!\"\n\nTony jerked the blade out of Gallagher's back and he slumped to the ground. Jacqueline spun away, running as if the devil was in pursuit.\n\nHe was.\n\nThe nightmare had found her.\n\n*\n\nJulia did not need her werewolf guards. The pool of red demonic swirled away from her, moving to her people, and she instinctively batted her hand in their direction. They bucked, arcing their bodies and falling where she flung energy.\n\nGold filtered out of her pores. It was full daylight, and her skin was soaked with what looked like glitter.\n\nIt was sweat. She was stunning the demons with her sweat.\n\nA shape whirled past, sprinting for her life. Jacqueline.\n\nTony was in hot pursuit.\n\n\"Julia, no,\" Adi yelled as Julia took off running. Jason and Slash accompanied her.\n\n\"Yes,\" she threw over her shoulder, doubling her speed. Julia moved in behind Tony, and he heard her.\n\nShe ran straight into the blade.\n\nIt was an agony for which she had no words.\n\nTony shouted in glee like a banshee wail of a warrior celebrating victory.\n\nJulia floated.\n\nHer hands wrapped around the blade of their own accord.\n\nTony's face froze.\n\nShe pulled it out of her gut and fell backward.\n\nJason tore forward, the dirt from his claws spraying her torso.\n\nShe watched as though out-of-body. Jason did not perform the obvious attack.\n\nTony readied his blade.\n\nHe moved around his body, going low and took out his balls with a talon swipe.\n\nTony shrieked.\n\nSlash moved in and took the front.\n\nTony stood bellowing to the heavens, the saber forgotten at his feet while his fellow Weres made him a soprano.\n\nJacqueline came to Julia's side and dropped to her knees.\n\n\"I am sorry,\" Jacqueline said for the second time in a matter of moments, pressing her hands on the swamp of blood Julia's wound had become.\n\nJulia searched Jacqueline's face as a hiccup erupted. It tasted like a penny in her mouth. \"I know,\" she whispered.\n\n\"I would do anything to take it back.\"\n\nJulia nodded. She could see that. See her remorse as plain as the sun shining on the blood everywhere, so red in the sunlight.\n\nJulia closed her eyes, and when she opened them, it was like a camera shutter.\n\nTharell stood behind Jacqueline.\n\nThe damn of Julia's telepathy burst, and she shouted her warning to Jacqueline, Lay down.\n\nJacqueline's eyes widened, but she didn't hesitate. She folded her body against Julia, the side of her face pressed against Julia's heart. She put her hand against Jacqueline's head, and Tharell moved in, stepping over a crotchless Tony being torn apart by the wolves.\n\nThe women lay together, and Tharell tried to scoop Jacqueline up.\n\nDomiatri appeared behind Tharell, his blade moving so quickly it was only a streak of silver in the sunlight. A sparkle, and it was there and gone.\n\nA red line appeared in the perfection of Tharell's violet throat, a stunned look overtaking his expression.\n\nThe ripple of pink scar tissue at Domi's throat convulsed as he swung his leg out in a stabbing strike at Tharell's head.\n\nIt came off in a spinning leap, landing on the ground beside where Julia lay.\n\nJacqueline flinched, tightening her hold around Julia, unaware that Domiatri lived and fought behind her.\n\nJulia used strength she didn't possess and turned to see Tharell's blue eyes staring back at her from his decapitated head.\n\nThey blinked.\n\nJulia screamed as she bled, the poison of the demon's blood making its way to her heart as the battle raged on around her.\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nTHE END\n\n|  |\n\n---|---|---\n\n# ANGELIC BLOOD\n\nA Blood Series Novel\n\nBook 5\n\nNew York Times Bestselling Author\n\nTAMARA ROSE BLODGETT\n\nAll Rights Reserved.\n\nCopyright \u00a9 2015 Tamara Rose Blodgett\n\nNo part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a database or retrieval system without the prior written permission of the publisher.\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locales or organizations is entirely coincidental.\n\nwww.tamararoseblodgett.com\n\nTRB Facebook Fan Page\n\nCover Design: Claudia McKinney\n\nEditing suggestions provided by Red Adept Editing\n\nSynopsis\n\nJulia Caldwell lies in the middle of a battlefield dying of wounds inflicted by the newest enemy of the Singers\u2014the demonic. When Fae, Were and vampire collide in alliance against a common enemy other than one another, the demonic brings turmoil beyond what any of the supernaturals can imagine.\n\nAfter the genocide of the Singers of Region One, they will be forced to move into hiding. Jacqueline has redeemed herself and carries a progeny of importance and strength that the new enemy wants at all costs. Who of the Fae can be trusted after Tharell's treachery\u2014will the rogue Were damage or assist the remaining Region One Singers?\n\nCan the greatest secret of their blood save the Singers from extinction and close the wound the demonic has made in a world run by humans, and ignorant of what lives among them? Or will fate decide evil deserves to rule instead of good.\n\nDEDICATION:\n\nElias Carter Blodgett\n\nYou don't know it yet, but you changed my life....\n\n# CHAPTER ONE\n\n\"Jules!\" Jason screams, sliding toward her like a runner at home base.\n\nJulia looks up at him, holding her guts inside her body. \"Oh, God\u2014no!\"\n\nJacqueline rises from behind Julia's torso, where she'd lain to avoid Tharell's body.\n\nShe wears Julia's blood.\n\n\"Looks like someone has lost their head,\" Slash comments dryly, stepping over Tharell's body. His humor leaks away when his eyes catch sight of Julia.\n\n\"Julia\u2014what can we do?\" Jacqueline asks, ignoring everything else.\n\nA demonic blurs past them, its tail sailing into the head of a Singer with well-practiced precision. Julia's veins flash golden at its nearness.\n\n\"Fuck!\" Jason rages, leaping to his feet as he puts his back to her protectively.\n\n\"Cyn,\" Julia croaks.\n\nJason whirls around, understanding blooming on his features.\n\nI need Cyn.\n\nHe nods, peering into the thick of the battle. \"I can't see her!\" he shouts then sprints into the fray. Julia fights to remain awake, but her eyes feel buggy in their sockets, too wide and dry. The pain sharpens her senses, helping her stay alert\u2014and alive.\n\nA bubble of blood forms from her lips, and Julia feels herself beginning to slip. She's still mortal, still vulnerable\u2014and so very tired.\n\nHer eyes meet Jacqueline's. Julia sees the knowledge of her death reflected in them. Julia holds herself in position, avoiding Tharell's head, which was no longer attached to his body, beside her.\n\nI can hold on a little longer. Julia closes her eyes and concentrates on her breathing.\n\nIn.\n\nOut.\n\n\"Jacqueline,\" Domi says from behind Julia.\n\nJulia cracks open her eyelids, and Jacqueline stills, her eyes growing wide. The demonic rushing toward them is gaining momentum. Julia is vulnerable and wounded\u2014vulnerable to true death, as William would call it.\n\nJulia's eyes bulge, and she licks dry lips.\n\nThe blood bubble on her lips pops with a soft smack, and she not dare move her hands from where her organs pulsate beneath her fingertips.\n\nDomi turns casually, clotheslines the devil's warrior, and crushes the red flesh around the demonic's windpipe.\n\nAs the demonic folds to his knees, the saber in Domi's hand flashes downward, cutting the head from the soldier of their mutual enemy.\n\nThe twang of swords and weaponry is the dark orchestra that plays for them.\n\nJacqueline's mouth hangs slack as she turns slowly to look at Domi, then at the demon lying still at his feet. Finally, her eyes come to rest on Tharell.\n\nJulia knew of Tharell's comprehensive duplicity against Domi, but there Domiatri stands in emerald splendor, proud and tall. His throat is bisected by a rope-like scar. He steps over the demon, shooting an indifferent glance at Tharell's body before he faces Jacqueline.\n\nJulia swallows her groan\u2014her hands are drenched with her own blood.\n\n\"Domiatri,\" Jacqueline breathes in shock as Slash picks up the sword that the castrated Tony let drop.\n\n\"Nice work on the devil,\" Slash says casually. But there is nothing casual about his restless gaze as it travels the field, looking for other demonics. He seems resolved to protect her until Jason can get back.\n\nJulia clenches her eyes shut. Where is Jason? Only Cyn can fix the mess of her body.\n\nSlash toes Tony's corpse, and Julia breathes against the pain in her gut. The sounds of men striking each other is like listening to slabs of meat being tenderized.\n\nJulia's stomach roils against the metallic fragrance of blood and death in the air and the sounds of flesh succumbing to fists and blade. Her limbs grow numb as her vision narrows to gray edges and a black center.\n\nJulia drags in raw breaths.\n\nDomi scoops Jacqueline against him. \"Tharell thought to murder me.\"\n\n\"I know,\" she replies.\n\nJulia's skin begins to pulse golden-silver, shining like a beacon as one of the demonic approaches.\n\nShe tries to roll away and can't manage it.\n\nIt hisses, driving its tail toward her as blood pumps out of her body, but Slash turns, catching the end of the whipping rope-like appendage with the borrowed saber and cutting it off. The deep-red stalk fishtails sickly and falls. Inky blood spurts out of the amputated stump as the demonic shrieks.\n\nJacqueline sinks to her haunches beside Julia as Domi swings his own sword and takes off the horned head. A warm hand covers her own blood-slicked ones.\n\nThe sudden silence that falls is peaceful and heavy, like scarlet snow that never lands. The demonic are losing the battle, and the shine of Julia's skin dulls as the second demon's life ebbs as it lies a few feet from her in a pool of black blood.\n\nJulia knows with perfect certainty that she doesn't want to die in this thankless field of war and death.\n\nThe salt of Jacqueline's anguish splashes on Julia's face. Gasping her last breaths, Julia watches it all as though she's no longer in her own body.\n\nOut of the corner of her eye, Jason emerges like an oasis in the midst of a forever desert. Cyn is wound tightly around his back as he dives between the last brandishing fists. Blood arcs, spattering them as they charge between the opposing forces.\n\n\"Do not go, Julia. Your people need you,\" Jacqueline says, rubbing Julia's icy hands as though the motion will repair her. More unshed tears fill her dark eyes, which are so like Scott's.\n\n\"I do, as well,\" she adds in a whisper.\n\nJacqueline, her enemy no longer, clutches Julia's hands while her fey lover fights off the remaining demonic behind them.\n\nJason lurches beside her, his wolfen form casting a shadow over her, and dumps Cyn at Julia's side.\n\nCyn's eyes widen in horror at what Julia assumes is the mess of her body. \"Jules... oh, God\u2014no.\"\n\nJulia closes her eyes and Cyn's presence is a salve to her body, but not her words.\n\n\"I don't know if I can fix this,\" Cyn despairs.\n\nJulia's eyes open to painful slits while Cyn's gaze roves her body. Julia's hope flees when she sees those expressive eyes.\n\nThe saber christened with demonic blood is going to slam dunk Julia into death. Tears riot down her face. Jacqueline holds one of Julia's hands while the other is in Jason's tight grip.\n\n\"Come on, Cyn. We haven't come all this way to back out now.\"\n\nCyn drags her finger underneath both eyes, swiping and flinging tears away. She inhales deeply and replaces Jacqueline's hands with her own. With a sigh, Julia lets hers slip apart, and they fall onto the grass. The air is cold compared to the heat of her injury.\n\n\"Oh, Jason.\" Cyn's voice trembles, sounding mournful.\n\n\"Don't you fucking die on me, Jules,\" Jason commands fiercely.\n\nJulia doesn't promise anything\u2014or even speak\u2014because glorious warmth floods her stomach and spreads, leaking through her body like a bath of fire. She feels as though her body is melting into the grass, though the clangs and charge of battle still grow.\n\nNothing is more silent than death.\n\n\"That's it, Cyn!\" Jason sounds desperate.\n\nJulia relinquishes her ownership of this life, which was always a gift, not something she could expect to always have.\n\nIf she's the Rare One, she'll get through this. If she's not, Julia is convinced she was never meant to be anything to anyone. Julia's exhausted to her marrow.\n\nShe wants so much more from this life. She knows she has a sister somewhere in the Red Den of Alaska, and Julia wants Jason as a woman wants a husband.\n\nMore, Julia wants to actually live, not merely survive from one catastrophe to another. Like a candle flame in jeopardy of being blown out, she feels her soul flutter with indecision. Julia's will to live hangs in the balance.\n\n\"No\u2014Jules. Help me here!\" Cyn screams.\n\nJulia's eyes flutter open to find her people quietly gathered.\n\n\"The demonic?\" Julia chokes out, her eyes frantically searching for the deep-crimson bodies, black horns, tails, and the soulless eyes.\n\nNone still stand. Their dead bodies litter the ground at her peoples' feet, and the Singers' skin doesn't shine. The beautiful veining that surfaced before and during the battle was nothing more than a warning of their mortal enemies proximity.\n\nCyn's hands clench over the wounds, and Julia's body arches against the grass as she gasps. Life fills her body where death sought to claim her. A thread of warmth runs to her fingers and toes. Her eyes sharpen, and her heart begins to beat a strong rhythm again. She rises up on her elbows, Jason's large hand at her back, and glances down at her belly.\n\nDried blood flakes on a deep scar that turns pink as she watches. Jason's eyes meet hers over the knitting wound. He puts his knees behind her back, and she leans against him. His hands lock around hers, and he bows his head, unable to hold in the shuddering exhale of relief.\n\n\"Tony has failed.\" Domi's lip curls in satisfaction as he looks at Tony's body before moving on to Tharell's headless corpse. \"Tharell failed, as well,\" he growls.\n\nSlash gives a satisfied snort. \"Still like how he lost his head.\" He gives Julia an apologetic glance, and Julia's lips turn up.\n\n\"It's okay, Slash.\" Julia doesn't miss Tony. His demise makes the earth a better place. Bright sunlight splashes over them, making Domi's deep-green flesh appear to sparkle with luminescence. It's somehow wrong to have a beautiful day as witness over this much death and bloodshed.\n\nJacqueline's eyes are round, shocked. Julia tries to sit up, but her core flinches at the motion.\n\n\"Julia\u2014no way.\" Cyn's hands are splayed on her healing stomach, and Julia chokes up as Jason comes up behind, lifting as he stands with her cradled against his chest.\n\nHis body is covered with the light-red down of his wolfen form. The green discs of his eyes slow their spinning.\n\n\"The demon fuckers are gone.\" His voice sounds like falling gravel, making Julia flinch. \"Our blood won out!\"\n\nJulia tips her head back against his shoulder, and he smiles, flashing rows of dangerous teeth.\n\nJulia nods, still weak. \"Where? How?\"\n\nWith laser focus, Slash eyes Tharell's body with a consideration bordering on hate. \"I say we pop Tharell's head back on and see if he can't tell us.\"\n\nJulia shakes her head, ignoring the pain of yet another betrayal. \"Too dangerous.\"\n\nSlash pulls Adi from behind him, and she approaches Julia.\n\n\"This affects all of us now, Julia. We've won the battle, but not the war. The demonic left and ducked into their little fire hole or whatever. But they'll be back. We got a stay of execution because Tony dropped the ball...\"\n\n\"Or balls,\" Cyn cackles behind her hand.\n\nJulia smiles. Cyn's already recovering her sense of humor. If she were in Cyn's position, Julia didn't think she would be capable of laughing at anything.\n\n\"I say we stake him,\" Brynn says.\n\nJulia blinks. The sun has sunk behind the trees, but the vampire stays in the shadows, offering his opinion safely out of the dangerous final rays of daylight.\n\nBoth Cyn and Jacqueline wear Julia's blood. As if she's made of glass, Jason carefully lowers Julia to the ground. Then Cyn's hands fall away from her body, revealing smooth unscarred skin, healed perfectly except for a black smear to the upper left of her belly button. It itches, but she's alive.\n\nJulia inhales deeply, painfully, as she looks at the loss all around her. The casualties of her people fill the field.\n\nBut more bodies of the demonic lay testimony to the Singers' victory.\n\nHer eyes come to rest on Tony last. His sightless eyes seem to gaze at her through a fog as if accusing her.\n\nShe tries to feel guilt or remorse, but she can't. This is the being who massacred nearly all the people of Region One. He raped Jacqueline and Lacey Greene. He was the horror that had plagued her and many others from the beginning, and now he's dead. Julia releases the breath she was holding, and the throbbing of her belly is her only physical distraction. She slowly lowers to her butt, exhausted.\n\nJulia turns to Brynn, William's successor. \"Stake Tharell?\" she asks.\n\nDomiatri comes into Julia's field of vision as Jason comes underneath her again, and she leans back against his knees. A knotted rope of scar tissue is a light-mint line across his neck. It appears to shine in the whitewashed daylight. Julia realizes it's healing before her eyes. The bumps integral to the scar tissue begin to smooth, and the shine begins to fade. Domi's skin rights itself in color, becoming grass green again where the pale-mint of the scar had bisected his throat. It's hard to look away.\n\n\"It will not be true death for Tharell until his body is burnt to ash.\"\n\nAll who are gathered look at Tharell, whose mouth is a gaping hole of silent screaming.\n\nJulia backs away, and for the first time, she recognizes what she hadn't noticed before while pain rode her.\n\nTharell's alive.\n\nShe says the thing that damns any chance of her claiming to be the angel she supposedly is.\n\n\"Do it.\"\n\nThe men move forward to collect the pieces of Tharell, stepping over the fallen Tony as they do.\n\n*\n\nTharell\n\nTharell's agony is so acute that he has no voice for it. No sound emanates from his ruined body to articulate his pain.\n\nDomiatri has pinned his palms and feet to the ground with stakes a full foot in length. The agony of iron courses through his tortured body as it fights to heal the constant affliction of metal.\n\nThe scarred Were assists Domi in his torture as, for a fey as pure as Domi, touching iron ore would have been akin to handling acid.\n\nTharell understands he will reap what he has sown. Intellectually, he understands his part of the deceit. He did not want to do what he did. However, blood dictates all. Humans need it to live, and supernaturals are governed by its crimson pull.\n\nA crude approximation of a reattachment of his head has Tharell's tendons and muscles stinging as they reassemble pathways severed by the decapitation.\n\nHowever, the pain is nothing compared to the condemnation he receives from every quarter, every set of eyes set against him.\n\nThe blame is deserved, of course. None knew the black blood that flowed within his veins is master over all others because the angelic blood is dominant to those Singers who possess enough of it. Tharell closes his eyes in weary resignation.\n\nA moment later, his face rockets backward with a slap, the sound of which fills the meadow. Though he does not cry out, Tharell groans from the worst physical misery of his life. A Sidhe warrior would rather die than admit weakness.\n\nTharell meets the dark gaze of the death bringer head on.\n\nOf course they would use him, the strongest of all supernaturals.\n\nThe vampire Tharell had been a part of finding smiles down at him coldly. \"Ah, to have a fey to torture,\" the vampire muses happily.\n\nTharell readies himself.\n\nHowever, Julia is the one who comes to stand before him. His demonic blood riots in warning at the proximity of an angelic, especially one as pure as she is.\n\nNatural-born enemies.\n\n\"Blooded Queen,\" Tharell manages from his healing throat and around the searing heat of his punctured palms and feet.\n\n\"You've been crucified,\" she says almost absently, though her eyes seem dull to any pleasure due to his pain.\n\nHe tries a nod and finds it unmanageable. \"It appears that way.\" The irony of his physical positioning does not escape Tharell's notice.\n\nJulia's golden hair is plaited, and many of the hairs have escaped the braid. Her eyes flash, and her veins, their power awoken to his ancestry, pulse like liquid gold and silver, mingling at the surface of her skin with every beat of her heart.\n\nIt is a standoff. Tharell knows Julia will want answers. And only he can decide their worth to him. He could always die again by her hand, to be resurrected again and again.\n\nHis immortality has proven to be his greatest weakness.\n\n\"I will confess the reasons for all my deeds for one thing in return.\"\n\nHer eyes hold his in the bright light of the rising moon.\n\nShe gives a small despairing laugh. \"Like you've got a bunch of options?\"\n\nTharell has never seen so much grief in one gaze. He waits as the seconds pound by.\n\n\"What is it?\" Julia finally asks with bald distrust.\n\n\"Kill me when I am through.\"\n\nJulia stares at him for a full minute. She swipes at her face, flicking away a lone tear like a gem of resignation.\n\n\"Done,\" she says so softly that only the Were gathered nearby could have heard her.\n\nTharell hears her answer perfectly.\n\nHe begins to talk, knowing that a quick death by the Blooded Queen's mercy is better than the torture Praile would inflict upon him.\n\n# CHAPTER TWO\n\nJulia\n\n\"I am a vessel,\" Tharell admits, grimacing. \"We who possess the blood of the demonic all are.\"\n\nJulia crosses her arms, wincing at her still-tender stomach. Cyn did a lot to alleviate the worst part of the wound, but when she gets closer to Tharell, Julia's mostly-healed injury flares like a lit match. She tries to dismiss the black smear that remains on her pale skin like an evil smudge. Its presence tugs at her subconscious.\n\nShe backs away, and the biting pulse lessens. \"Don't lie.\"\n\nAs Tharell raises his head, the horrible scar like a streak of lavender lightning bulges across his throat, and Julia swallows her gorge.\n\n\"The fey do not lie, Blooded Queen.\"\n\n\"Oh, horseshit!\" Cyn yells, tramping over to where he lays. She moves to her knees, careful not to soak them in the continuous seep of blood that courses out of the pads of Tharell's palms. \"You lie by omission, you fucking grape. You made us believe there was some kind of treaty between the Singers and fey, and the entire time, you were just some lackey of the demonic, doing the plotting prick program.\"\n\nAdi flashes a smile at Cyn. \"Not to sound dumb, but who's your leader?\"\n\n\"That's Michael's line,\" Julia says sadly, looking down to hide her tears. Jason puts his arm around her, and she looks up, way up into his changed face. His green eyes rotate slowly, and though it's hard to ascribe human emotion to the partially changed, Julia thinks he looks sad.\n\nNone of the Were have changed back to human\u2014that's too dangerous. Holding the wolfen form doesn't take the energy that full wolf form does. And the majority of Weres can't achieve full form unless the moon is full.\n\n\"Let's not take all day here,\" Truman says, dumping another demonic corpse on top of the others.\n\nWe've got a pile of demons. Julia shudders and squelches a bubble of laughter.\n\n\"We don't want to be caught with our drawers around our ankles, swapping spit and shit.\" Julia frowns at Truman's comment, though it's the truth. Point for him.\n\n\"It does not matter. You can kill me. Burn me to ash and sprinkle my essence in a swift-moving river to rid me from this place.\" Tharell's azure eyes latch on to Julia's, and she shivers in Jason's protective embrace. \"But Praile will come for you. He will use whoever and whatever has the blood of his kind to serve him.\"\n\n\"I'm tired of this douche,\" Jason growls in his strange part-animal timbre.\n\n\"Why?\" Domi asks suddenly.\n\n\"We do not have time for this, Domiatri,\" Jacqueline reminds him quietly.\n\n\"We don't have the time to kill his ass, either,\" Truman says thoughtfully. \"We need to beat feet outta here.\"\n\n\"I can do it,\" Brynn offers.\n\nTharell gives him a neutral look.\n\nJulia knows it'd be impossible for her to be as calm as Tharell appears if she were presented with certain torture and death. That composure speaks to the nature of Tharell's existence.\n\nShe pushes stray hairs out of her face, more for an excuse to do something than for neatness. \"I promised I would,\" Julia says. Everyone looks at her, and she feels her face grow hot.\n\n\"He tried to kill Jolly Green over there,\" Jason says from above her. Domi frowns at Jason.\n\n\"Guys, let's not set up a testosterone palace,\" Cyn remarks, throwing her arms up. \"We still kill grape-boy, but on our own timeline, Julia didn't say when she'd do him.\"\n\nJulia flinches.\n\nIgnoring her, Cyn goes on, \"Let's get together the Singers who want to come back to Region One. The Tony threat is gone, because he got his weenie chopped off.\" She flashes a grin and lets out a manic chuckle. \"A great trend to dissenting dudes.\"\n\nAdi snickers, and Julia dumps her face into her hands. Cyn is alarmingly practical, and it's somehow not cool right now, with Tharell being staked and a bunch of dead demons piled up on a death hill. It's too gross to be real. Yet it is.\n\nSingers need to be buried, and the rest need to get back home, or what's left of their home.\n\n\"Cyn,\" Julia says.\n\nCyn lifts her shoulders. \"All right, I know the whole dick comment was a little over the top...\"\n\n\"At least we know where ya stand,\" Truman says thoughtfully, shooting her a wary glance.\n\nAdi laughs.\n\nJulia looks between the two. \"Enough. Thank you for healing me.\"\n\nCyn rolls her eyes. \"Of course, doll. Like I wouldn't have?\" Her palms flip out and away from her body. \"Doy.\"\n\n\"But I need less sarcasm and more action.\"\n\nCyn huffs, crossing her arms. \"I say leave the demons for the vultures\u2014and wonderful Tony. That's better than that mongrel deserves. We get whatever Singers want to come back to the Region One stomping ground.\"\n\nA tear races down Julia's face, and she gives it an angry rub. She has no time for grief. \"I want to take stock of survivors.\"\n\nDomi turns to her. \"I can bury the dead.\"\n\nTharell clears his throat. Domi frowns.\n\nJulia presses the heels of her hands in her eyes, hoping to erase the vision of everyone casually discussing things over Tharell's staked body.\n\nBut he's still there when she lets her hands fall. The bodies of the dead still cant in an unbalanced way in the center of a field that was awash with blood during daytime and is now black tar all around them.\n\n\"No way!\" Adi says, looking from Tharell to Domi. \"This guy\u2014this guy is such a backstabber.\" She narrows her gaze at Tharell, whose expression remains neutral, despite the horrible pain he must be in.\n\n\"We need him for speed. To bury the dead.\" Domi admits.\n\nJulia rolls her bottom lip between her teeth. It's so unfair. But she can't leave the dead Region Two Singers to have their bones picked clean by scavengers. She just can't reconcile that move, not with everything else.\n\n\"Can he be...\" Julia puts a hand on her forehead, tired to the bone. \"Can he be contained?\" she finally asks.\n\nBrynn steps forward as darkness swallows the daylight and the moon's brightness sharpens above them. The two mingle in a kiss of time, twilight bridging night and day, and a smolder plays over his skin.\n\n\"That's better,\" he says, looking around at the darkness, where the daylight still leeches at the edges of the field. \"I can handle the fey.\"\n\nDomi frowns. \"What assurance do I have that you'll not make a try for my life or Jacqueline's?\"\n\nHer fingers tighten around Domi's forearm, and Julia's eyes go to Brynn's face. But it's blank, like William's could be, every feature outlined as if it were carved in bleached ivory.\n\nBrynn's fangs elongate, shining like ready knives. \"None.\"\n\n\"Ah, I don't know,\" Cyn says, her eyes following the fey-vamp verbal ping-pong match with interest. \"Brynn might not be a team player, Jules.\"\n\n\"We can't do much better than this. He was William's guy, right?\" Truman turns to Julia for confirmation.\n\n\"He was William's second from the Southeastern Kiss. There's no motivation for allegiance, really,\" she confesses.\n\nBrynn gives her a look so weighted, she stops breathing.\n\n\"What?\" Truman barks.\n\n\"No vampire would harm the Rare One.\" He looks at each one of them, and Julia fights to make out his eyes in the swelling darkness. \"In fact, it's my belief that Praile of the demonic is not the only one making a bid for the top echelon of supernaturals.\" Brynn spreads his hands to the side.\n\nAll eyes move to Tharell. \"Let me up, and I will help for as long as my life serves a purpose.\" His bright gaze moves to Julia. \"And then you will kill me.\"\n\nJulia shakes her head in vague denial and his crisp blue eyes narrow on her.\n\n\"It must be by your hand. You gave your word.\"\n\nI can't do it.\n\nJulia's throat constricts, her breaths squeezed like frozen gasps inside her throat.\n\nDomi strides over to stand next to Tharell's prone position and jerks out each stake with a meaty, wrenching suck. Tharell's face tightens but not a sound emerges from his lips. Domi steps away, and Tharell stands without assistance. Holes fill with fresh, pale-lavender flesh, and his unearthly blue eyes blink slowly.\n\nJulia clamps down on her emotions. There've been too many traumas in a short span of time. She'd almost been murdered, countless more Singers had died, and that meant the death of so many from the one place where she was beginning to feel as though she belonged.\n\nShe blinks, realizing her lashes are wet. Jason gives a few soft, comforting snuffles against her neck.\n\nEveryone backs away from Tharell as though he has the Black Plague. His eyes meet each of theirs.\n\n\"I deserve your disdain.\" His voice is low and full of emotion. \"But if Praile comes again, he can force me to do his bidding. He could compel anyone who has sufficient blood of the demonic. And the Red Were are not immune to being used by the Master's summons.\"\n\nTruman harrumphs in disbelief.\n\nTharell lasers a look of pure certainty at Truman. \"Believe that I lie\u2014I care not. Why should I warn you? What does it gain me?\"\n\n\"Why not?\" Cyn says. \"You're obviously a stand-up dude.\"\n\nTharell's brows cinch.\n\n\"Not,\" Adi adds with a smirk.\n\nDomi and Brynn step forward. \"We take care of the dead Singers and find the ones who still live and wish to accompany us back to One\u2014and faerie.\"\n\nDomi's and Tharell's eyes lock in a battle of unspoken words.\n\n*\n\n\"Julia,\" Jacqueline calls.\n\nJulia stops, Jason a shadow beside her.\n\n\"I wish to find Gallagher. Perhaps he survived the blow.\"\n\nHer head bows, and Julia knows, as sure as she's standing there, that Jacqueline's guilt is all for Victor. His brother is wounded and possibly dead, and Victor still remains unaccounted for.\n\nJacqueline can't atone for all the bullshit of her past. But she has a right to try.\n\nJulia sighs, knowing she can't do anything about the epic mess. But Gallagher would be an important addition to One, though his desertion would leave Two leaderless. However, since all of One is decimated, the whole of Two would be better off returning with Julia than trying to piece together the Swiss cheese of their region. Their regions are more protected if they stand together than if they are apart.\n\n\"Let's try to find him. And\"\u2014Julia's eyes meet hers, though in the thick of the night, she can barely see anything but the whites\u2014\"I want a head count of all the Singers who still live and whether they want to come with us or not.\" She adds, \"Convince them.\"\n\nJacqueline smiles as though she has a secret.\n\nJulia thinks Jacqueline is perfect for the job. Part of what made her Jacqueline still peeks out through all of the changes since her time within faerie.\n\n\"Jules?\" Jason rubs the back of her neck, and she wants nothing more than to sink against him, revel in his closeness, and enjoy the fact that she didn't die today.\n\nBut she can't. She has a duty to her people\u2014and maybe a sister she doesn't know. But the thought of finding a sibling is a faraway wish. They're all still in survival mode.\n\nNot to mention Scott and Lucius. The list goes on. Julia doesn't have time for the pity party she wants to throw for herself.\n\nThe dead are being entombed by a corrupt fey. Though her people have been annihilated, her promise to faerie still stands.\n\nShe's hungry, tired, and still healing. She's also in charge.\n\nJulia says none of what she's thinking. Instead, she walks after a weary and pregnant Jacqueline in search of the one Singer she believes can help her and whatever Singers they can convince to accompany her.\n\nJulia's not sure she would come if she were in their shoes. Violence holds claim to her, and death follows.\n\nNot a great combo for a long life.\n\n# CHAPTER THREE\n\nTessa\n\nTessa drives until she's nearly out of gas.\n\nA 1950s neon sign flashes from a few blocks away as she heads up highway 99.\n\nGas-Food, it blinks. Well, part of it blinks. The G is missing in the first word. It's creepy. Tessa is somehow reminded of the Psycho movie from the 1960s.\n\nShe was a whelp in those days. Sometimes, Tessa doesn't think the longevity of being a Were is all that hot\u2014like now, when she's hungry, tired, and broke.\n\nShe'll have to steal soon. Tessa hates that routine.\n\nJust because she can take from the humans doesn't mean it sits well with her.\n\nThe gravel crunches under soft tires that need air. Tessa squints. Dawn is rolling over the Olympic Mountains, wrapping them in the sherbet tangerine of daybreak.\n\nShe can't believe the place is open. She parks by the air pump. Swinging the door wide, she begins to exit the low-slung car. \"Ah!\" Tessa turns to pop the glove box and rummages around.\n\nThe air pressure gauge in hand, she measures each tire. She fills all four tires, her eyes ceaselessly scanning her surroundings. Things are quiet for the moment as the hush of dawn adds to the false stillness. Tessa suspects the quiet before life begins a new day, is present for another reason besides daybreak.\n\nStanding, Tessa swings her long ebony braid over her shoulder and jams a hand in the front pocket of her jeans. She extracts a crumpled ten dollar bill. She opens her hand, where a fifty-cent piece glows. As she holds it, the flesh of her palm begins to burn as if she's waving her open hand over a candle flame. Turning, she sets it on top of the roof of the vintage Impala. Carefully, using only her fingertips, she flips it.\n\nThe coin has the year engraved on it: 1964.\n\nSilver.\n\nFigures. Tessa can't wait to pass it into the currency of America. She's amazed it didn't burn a hole through her jeans.\n\nSilver is poison to her kind, but Alpha females have some immunity. If she'd been male, the silver would have behaved like a burning ember inside her pocket.\n\nMaybe their immunity had something to do with the importance of young. Tessa is always partially immune to silver while in quarter-change form.\n\nFemale Were are not plentiful, hence, her nomadic lifestyle. As soon as Tessa came of age, her pack began a bidding war for the right to mate her.\n\nKnowing they would fight over her like what they were\u2014a pack of wolves\u2014Tessa decided to split before they had the opportunity. Now she's been rogue for two decades.\n\nShe's killed more males that she wants to admit.\n\nStill, they were all deserving of death. Most were rogue like she was. But that did not give them the right to force her to mate with them.\n\nTessa trusts no male.\n\nShe exhales in a harsh burst, blowing the tendrils of hair that escaped from her braid out of her face. She can't allow herself to think of her slain packmaster father. He would not have allowed the bidding war.\n\nSadness tightens her body.\n\nHe would have given his only daughter a choice.\n\nHer pack had robbed her of her freedom, along with her grief over his death. Tessa didn't have the luxury of mourning her father. He'd been challenged as packmaster by the one male who presumed victory would kill two birds with one stone. Tramack believed he would rule... with Tessa at his side.\n\nAs though I could ever forgive my father's murderer.\n\nTessa touches the heels of her palms to her drenched eyelashes and breathes deeply until she calms herself from a memory that is twenty years old.\n\nThe pain never seems to lessen.\n\nIt could, if not for Tramack's endless pursuit. His scouts seek her incessantly. Tessa manages to stay out of their sight for two years at a maximum before they find her again.\n\nShe lowers her hands and stares at the gas pump without seeing it. Finally, she punches in the pre-paid amount at ten dollars and fifty cents. She slams the old-fashioned lever and begins to pump the gas, her All-Star bright-red sneaker resting on the concrete curb that holds the pumps.\n\nThe gas pump chimes really quickly when all you have is ten bucks. Too quickly.\n\nTessa sighs, slipping the nozzle back into the holder, and shoves the lever down. A light breeze lifts the small hairs on her forearms.\n\nA bird cries a warning.\n\nTessa's head jerks up, and her nostrils flare. Tendrils of hair swirl around her face, obscuring her acute vision.\n\nThe tiny hairs that humans ignore as a warning to imminent danger rise at her nape.\n\nShe ignores nothing.\n\nTessa moves away from the pumps, her eyes scanning the pocketed shadows of forest that are everywhere the eye can see. The gas station's yellow fluorescents cast a sick glow on the sidewalk that creates an uninspired strip at the entrance.\n\nShe sucks in huge lungfuls of crisp mountain air, closing her eyes and isolating her senses to just those of scent.\n\nTessa holds herself still.\n\nLayered and comprehensive, the smells of cedar and Douglas fir needles, rich earth, lichen, asphalt, humans, and small forest animals drift over the olfactory regions of her nose.\n\nTessa chuffs in little bursts of air then parts her lips, taking breath through both her nose and mouth simultaneously.\n\nThe radar of scent surges out like big ripples and pings out at the odor of the large animals\u2014felines, elk, and mountain goat.\n\nBeyond that, the Pacific Ocean cleanses her palate, and she exhales.\n\nInhaling more deeply, she holds the air inside her lungs, and on this exhale, she smells something out of place.\n\nMud.\n\nHer eyelids flip open.\n\nTwo men exit the forest. Tessa takes their measure, her heartbeat ticking faster. Her nose flares frantically.\n\nShe smells only the mud.\n\nThen she sees them clearly. Beyond the black border of the forest and in the first light of day, Tessa understands why she can't scent them. She would laugh if the situation weren't so dire.\n\nThe Were are covered in the mud of the tidelands. That was why the smell of the sea was so strong.\n\nIf she hadn't been so busy cataloging, she would have known the ocean shouldn't smell so close. Tessa had become complacent. She hadn't been thorough and got caught with her panties down.\n\n\"Hello, Tessa,\" says one of the two.\n\nThey're obviously guards.\n\nTessa slouches against the car, crossing her sneakered feet at the ankle.\n\n\"Hey, boys.\"\n\nThe one who spoke raises a brow. \"We're glad we could get a chance to talk to you.\"\n\n\"I'd think my answer would be obvious by now. Or did the other two dogs I put down not let you know my thoughts on the subject?\"\n\nHe scowls, casting a glance at the dumb pup all but wagging his tail beside him.\n\n\"Byron,\" he says, twirling a finger in the air. The quiet one moves wide, making it impossible for her to see them both straight on.\n\nDammit.\n\nThe handle of the car door is a foot away from her hand. Tessa's palm itches to touch it.\n\nOr her palm itches because of the silver coin she held for a few seconds.\n\nThe silver coin!\n\nTessa's finger twitches. The Were who thinks he can just swoop in and take her notices.\n\n\"Clever camouflage.\" She says to distract him, but she means it. She had almost no warning. Birds were sometimes helpful.\n\nTessa frowns and glances up at the tree for just a moment, catching sight of a snow-white bird. Her frown deepens.\n\nStrange. But Tessa doesn't have time to think about the bird. The two Weres are closing in.\n\nHer gaze finds Byron, who is half her age, inching closer. She looks for anyone who can help her.\n\nThe human at the cash register glances outside.\n\nTheir eyes meet.\n\nShe hasn't paid.\n\nDon't come out here, she screams inside her head. Of course, he doesn't listen.\n\nAbout sixty-five and portly, he shuffles out. \"Hey, miss?\"\n\n\"Yes?\" Tessa replies without taking her eyes off the Weres.\n\n\"You gonna pay?\"\n\n\"Yup,\" she answers. But she's not a hundred percent on that.\n\nByron inches closer.\n\nTalkative looks at the heavyset cashier and narrows his gaze on him.\n\nSome people don't have any instincts of self-preservation. This human is definitely one of them.\n\nThe cashier's eyes widen. What he sees is a six-foot-three-inch man covered in mud. To Tessa, he reeks of rotting sea vegetation. A human at that distance would have only his sight.\n\nAnd the Were is a pretty weird view.\n\n\"What's going on here?\"\n\nOh boy.\n\n\"What's going on, human, is you're going to march your fat ass back into that dilapidated store and pretend you didn't see us.\" A cruel smile lights his face like an old-fashioned camera bulb exploding.\n\nTessa's pulse quickens, and Talkative's nostrils flare as his face briefly turns in her direction.\n\nShe's had calls that were closer, but not by much.\n\n\"Listen, you're on my property. She hasn't paid, and I'm not letting two thugs who don't know what the inside of a shower looks like get in the way of me getting gas money owed. Go pack sand.\"\n\nTalkative growls, and his skin shifts like liquid. His bones morph into a melting candle wax of sloughing skin as his face changes into wolfen.\n\nHe shakes like a wet dog, scattering clumps of tideland mud, both dry and wet.\n\nTessa's eyes tighten in pain as she performs her own quarter-change from the sloppiness of her human form. She dropped it once she arrived at the gas station, letting her guard down.\n\nInstantly her ears, nose, and eyes become more. More sights reach Tessa, who can now pick out seaweed as small as a thread clinging to the Were.\n\nShe can smell the soap he uses underneath the sea muck.\n\nHis wolfen gaze falls to her as if she's his prey. The slow spin of yellowed irises rotate faster as he believes his quarry is almost his.\n\nThe cashier takes a hard look at the shit going down and hightails it back inside the store.\n\nSmart man.\n\nTalkative turns to her, now seven feet of striated muscle in motion. A coat of nutmeg-colored hair covers every bit of him as his short snout lifts, snuffling a few times to gather more of Tessa's scent.\n\nA bell jangles, and the man comes back out and cocks a shotgun.\n\nTessa intuits everything. Action. Consequence.\n\n\"No!\" she screams.\n\nShe thought he'd agreed and retreated, finally seeing the potential for his death by their hands.\n\nBut he's a typical human. Have gun, will kill.\n\nTalkative can't move fast enough. The buckshot does, riddling the Were like a seven-foot-tall slice of Swiss cheese.\n\nTalkative slows but doesn't stop. He launches himself at the porky cashier and tears out the man's neck upon landing.\n\nBlood shoots up like water from a broken fountain, and bloodlust momentarily distracts him.\n\nTessa's head snaps to the right as Byron rounds the gas pump island. She moves quickly, snapping her fingers up to the coin on the roof of her car. I might only have one chance.\n\nIn quarter-changed form, Tessa has increased senses, speed, and strength. She uses that now, when only the car's length separates her from Byron.\n\nShe sprints to the trunk as he's at the hood.\n\nShe launches the coin with everything she has, and her hand-eye coordination is perfect in her changed form.\n\nThe edge of the coin slams into the middle of his forehead, and he howls. Instinctively, Tessa slaps her hands over her ears.\n\nThe coin burrows, doing the work for her as his skin parts to the most abhorrent substance to a Were. Tessa watches his skin burn, the edges blackening and folding open like the petals of a dying flower.\n\nShe backs away as blood spills.\n\nHis brains are next, and the coin doesn't disappoint. Like a horrible flat missile, it keeps seeking its target. There's no great healing that can arrest its progress. Male Weres can't heal from silver damage.\n\n\"Oops,\" Tessa breathes out as the river of what was inside his skull flows down his face. She begins to back away. Someone suddenly grips her upper arms from behind.\n\nSmelling rancid mud, she slams her head backward. She's almost six-feet tall in her quarter-change form, and she uses that height, hopping as she flings backward. Tessa headbutts Talkative.\n\nHe staggers backward and she runs.\n\nTessa slips in all the blood and falls on her ass so hard her teeth snap together. He lurches toward her, and she rolls to stand.\n\nA snowy bird that looks like a dove but isn't glides down in a loose spiral. The bird is a spot of purity in the blood that covers the asphalt. Talons splay.\n\nTessa grips the asphalt. The congealed blood is thick under her short fingernails. She pushes off, trying to put distance between herself and Talkative.\n\nThe bird screeches, high and brilliant, above and forward of her position.\n\nThe talons are sharp and big in Tessa's vision as it swoops closer. She ducks, and the bird flows over her head, lifting the loose hairs on her head and missing her by inches.\n\nThe screaming tells Tessa it has found its mark.\n\nShe stumbles forward and glances over her shoulder.\n\nThe bird's white feathers are covered in blood, and it carries an eyeball hanging by a gruesome tail of sinew.\n\nThe bird caws, slinging the orb off its talon. It spins in the crisp morning air and lands with a thud on the pavement. Puke threatens, but right now, it's survival of the fittest.\n\nTalkative grabs the bird's body, and it cries in alarm. Tessa rushes the Were, who tracks her with his one good eye.\n\nThe bird dips its beak and takes his other eye while he's distracted by Tessa's approach.\n\nThe Were lets go of the bird and drops to his knees, howling.\n\nTessa hurtles forward, knocking him down. She grits her teeth as she slams his head into the asphalt.\n\nOnce.\n\nTwice\u2014bone shatters.\n\nThree times.\n\nHis brains spill onto the asphalt.\n\nTessa wrenches her head to the right and sees Byron lying in his own brains, motionless.\n\nA sigh escapes her, and she stands, trying not to shake. Tessa moves to push hair out of her face, and a bit of skull clings to fingers, which are covered in sticky gore with bits of dirt and granules of asphalt. She gulps.\n\nHer eyes avoid the bodies. Instead, Tessa looks for the bird\u2014and sees a Were instead.\n\nBarely out of welphood, she stands before Tessa as naked as the day she was born.\n\nBreathe, Tessa.\n\nThe day simply can't get worse. \"I'm Tessa.\"\n\n\"I am Tahlia.\"\n\n\"Are you\u2014did you?\" Tessa doesn't know what to say, and it's a damn miracle there aren't more witnesses. At least there's that miracle.\n\n\"Yes. I am a Wereshifter.\"\n\nTessa had heard the legends. She'd just never seen it in the flesh.\n\nOr feather.\n\nTessa eyes skate over the gruesome wreckage then return to the naked Were.\n\n\"You saved me.\"\n\nTahlia nods. \"Yes.\"\n\n\"Why?\"\n\nShe smiles, and Tessa is struck by the girl's beauty. \"I could scent their intent.\"\n\n\"You're naked.\"\n\n\"I am,\" she gestures to herself. With a little laugh, she adds, \"I had to escape something in a hurry, so...\"\n\n\"You can come with me,\" Tessa gestures to the car. \"I'll change the plates out at the next larger city.\"\n\nShe blinks once, very slowly. \"Thank you.\"\n\n\"I have clothes,\" Tessa says.\n\n\"Okay.\"\n\nTahlia is stoic the entire time.\n\n\"Do you remember what you do when you change?\"\n\nTessa doesn't. Once in human form, most Were do not remember what occurred while they were in their animal forms.\n\nShe nods. \"Every bit.\"\n\nHer full, perfect lips tremble as she shivers slightly in the cool air.\n\n\"Come 'ere,\" Tessa says.\n\nTessa wraps her arms around the smaller girl. \"It'll be okay,\" Tessa says.\n\n\"I know.\"\n\nTessa tries not to feel guilty about the lie she just told the vulnerable girl.\n\nAnd she fails.\n\n# CHAPTER FOUR\n\nSlash\n\nSlash maintains the rear of their morose little expedition with ease. The Singers and supernaturals alike, who have not seen their fair share of war, look the same: shell-shocked or just plain shocked.\n\nThe Homer cop turned Red, Karl Truman, trudges along like it's any other day. Fighting to right the wrong is all that sort of male appears to need.\n\nThe Rare One continues on at her husband's side, and he maintains his wolfen form, as all the Were do.\n\nMany of the Singers needed more healing than the handful of Healers could attend to, and they remain in need of more healing, rest, and food. There won't be much of that for the next few hours, though.\n\nBrynn is the only vampire who remains. He's a shifty fanged fuck if Slash is any judge. And Slash believes he's a fine judge.\n\nTom Harriet still lurks in parts unknown, and he will certainly make another try for the Rare One. Julia will not be safe until the threat of the Reds from the Alaska den is over.\n\nAnd now the demonic threaten her, as well\u2014the fey.\n\nAnd as much as Slash is loathe to admit it\u2014the Singers are angelic. Not all, but some. Those Singers who manifested the telltale veining that pulsed like liquid gold and silver have been duly cataloged. It appears that roughly one in three are \"angelic.\"\n\nThe angelic Singers were more successful during the siege of the demonic, and more of them survived.\n\nSlash holds in the sigh of disgust at the memory of the pyre and the thought of Tharell, like a dog on a chain, waiting impatiently for his demon master's return so he can bite the first Singer who is near.\n\nSlash voted for Tharell's immediate death. Whatever answers the Sidhe can supply are outweighed by the need for the group's safety. Tharell is a loose cannon. It seems more prudent to put him down like the dog he's proven himself to be than to keep him around. And the Unseelie warriors have proven tremendously hard to kill.\n\nSlash spares a glance at the green Sidhe, Domiatri. A thread of scar tissue remains at his throat. Slash guesses that after another day, even that will be gone. Slash didn't ask how he managed to fetch his head after it was chopped off.\n\nThe fey piss him off on principle.\n\nAdrianna pops up beside him, striding quickly, matching his longer gait. He slows for her.\n\nSlash understands she is a crippling weakness in his otherwise-iron-clad defenses.\n\nHe is also desperately in love with her.\n\n\"Hey,\" she says softly.\n\nSlash says hello back, keeping his face in profile, relieved she's on the good side of his face. Adrianna doesn't seem to notice that Slash is purposefully hiding his face.\n\n\"What do you think about all this?\" she asks.\n\nSlash takes a painful, dry swallow. Her nearness is undoing his thoughts.\n\nSlash remains vigilant, scanning the surrounding area for threats.\n\n\"Slash,\" Adrianna says, reminding him that she asked a question.\n\nHe forgets his looks for a few seconds and braves a glance at her, his nostrils flaring subtly.\n\nHer hair is everywhere, messy from the battle and their pace. Her hazel eyes are very wide and bright. Slash wants to count the freckles that are so like captured gold flakes in pale skin running over the bridge of her nose.\n\n\"Slash,\" Adrianna says again.\n\nSlash realizes he stopped walking, and he stares at her.\n\n\"I made my piece known during our brief talk,\" he replies, referring to the group decision to bring Tharell.\n\n\"Pfft,\" Adrianna sounds off, beginning to walk again, and a breath of relief eases out of his tight chest. \"You didn't make your piece known. You insulted the vamp and the fey.\"\n\nSlash shrugs. The fey? And what of them? They were corrupt, wanting to siphon off the Singers' numbers and literally bring hell to earth with what Tharell did using his demonic connection. No. Slash is not inclined toward forgiveness or camaraderie.\n\nHe smiles, and the scar tissue on his face tightens.\n\nAdrianna, seeing only the good part of his profile, smiles in return. \"True. However, I don't care. The fey are a part of anything that involves what they can get.\n\nSlash allows a snort, and it comes out like a snuffle. \"And the Were aren't?\"\n\nHe looks over his shoulder at the Were who survived the massacre and he is not surprised to see Zeke give a microscopic chin lift when Slash's scrutiny rolls over him.\n\n\"We are. But we're cleaner about it.\"\n\nA dark-blonde eyebrow lifts. \"That makes it better, hmm?\"\n\nNo, but it is easier. Slash remains silent under what he perceives as her condemnation.\n\nAdrianna sighs, grabs his hand, and slings his entire arm over her shoulders.\n\nSlash's heart races, and his palms tingle at the unexpected contact.\n\nHer nostrils flare. \"Do I scent fear on you, Slash?\" Her question is serious, her voice coy.\n\nHis feelings are a dangerous game.\n\nNo use lying. \"Yes.\"\n\nSlash keeps walking, his arm around the female Alpha Were he would mate if allowed, if he were not marred.\n\n\"Of me?\" she whispers.\n\nHe nods.\n\nHe guides them deeper into the edge of the forest. The caravan of supernaturals continue to walk toward where members of Region Two are waiting to drive them back to One, to the false perception of safety.\n\nSlash relaxes in the shadows, his ugly mug hidden from Adrianna.\n\nHe turns to face her fully, his large hands cradling her face. \"I cannot have you, Adrianna. You must wait for someone from your den who is worthy of you.\"\n\nSlash's self-loathing has never been deeper, wider, or more intense. He has to break it off with Adrianna and take away the hope of their being together. Slash can't handle her eventual rejection when another Were decides he wants her. And that Were will be whole and perfect, without the scars of war on his mind\u2014or his face.\n\n\"Oh Slash,\" Adrianna says softly, cupping her smaller hand over his and he scents things that confuse him.\n\nTruth.\n\nDesire.\n\n\"I have found a male worthy of me. And there is no other,\" she says in the ancient language, and it fingers a chord like a guitar string inside Slash, as though a melody has begun playing just for him.\n\nThe Singers claim their blood sings to the vampire and Were alike.\n\nSlash believes certain blood ties are just as strong between the right pair\u2014the perfect mating pair.\n\n\"Adrianna, I'm broken. Can't you see it?\" His eyes implore her to see what is so obvious, to heed reason.\n\nShe shakes her head, sending her ratty hair, stained and dirty from fighting at his side, sliding over shoulders.\n\nSlash smiles.\n\nMy brave girl.\n\n\"You're not gonna do this, Slash.\"\n\nHe frowns. \"Do what?\"\n\n\"Hide\u2014sacrifice.\"\n\nHer hand leaves his and she slides her arms around his neck as her fingers grip his nape.\n\n\"No,\" Slash says in a panic, beginning to pull away, his hands dropping from her face.\n\n\"Yes,\" she says, \"a million times, yes.\"\n\nThen the softest skin he has ever felt brushes his lips, feathering over the scar tissue like heated silk. The barest breath of moisture flicks at the seam of his mouth, and his lips part against his volition.\n\nSlash forgets the promises made to himself.\n\nHe forgets he is in the middle of a dangerous return to Region One and that others besides him are in danger.\n\nAnd most importantly, Slash forgets how fractured he is. In Adrianna's arms, he is whole.\n\nShe moves to her tiptoes, gripping his neck more tightly and hanging off him.\n\nWith a groan of consent, Slash bends down and lifts Adrianna off her feet. She's above him now, her legs around his waist, her fingers grasping the fuzz of hair that covers his head. Her thumb lands on one of the horrible scars on his skull, and he can't pull away. He doesn't want to.\n\nShe kisses him deeply, her tongue diving inside his mouth, and he moans, hardening helplessly against her. She laughs softly against his mouth.\n\n\"I knew you liked me.\"\n\nAnd that's where it ends for Slash.\n\nHe loosens her hold on him, gently letting her slide down the front of him, every hard inch in stark relief. He is embarrassed by his obvious arousal, but there's nothing he can do. He is a male Were, and she is the female he desires.\n\nThe only one.\n\nHe captures her wrists behind her back as his forehead touches hers. \"That's the thing I'm trying to explain.\"\n\nShe speaks to his chest. \"What? What is it Slash. You are the Were for me. You've always been.\"\n\n\"I don't like you,\" Slash says.\n\nTheir eyes meet.\n\n\"What are you saying?\" Her gaze looks uncertain now, and shadows of doubt linger in the lightest part of her eyes.\n\n\"I don't like you\u2014I love you,\" Slash says so quietly, his words are barely more than mouthed syllables.\n\nHumiliated, he turns his back on Adrianna. However, now that she knows he's serious over her, she'll back off. He's too much work. Slash understands this.\n\nHe feels a tap on his shoulder, and he stubbornly stays facing the woods, his arms folded. He stares, memorizing every furrow of bark, every pine cone, and each needle.\n\nThen Adrianna is standing before him.\n\n\"Hey!\" she shoves him in the chest, and he takes a step back, frowning. \"You\"\u2014she pushes\u2014\"don't get to play father confessor then give me the cold shoulder.\" She slaps his chest again, and he grabs her arms.\n\nShe bares her teeth, and his wolf responds, growling.\n\nSlash tightens his hold, taking deep breaths to steady his animal. The urge to dominate her and take her as his wolf wants to causes him to throb painfully.\n\nHe looks away, but he can't bring himself to let go of her. He doesn't know if he'll ever be able to. Dammit, I'm trying to do the right thing. The honorable thing. Why can't she see it?\n\nHe asks her fiercely, \"Then what would you have of me?\" He looks down at the beautiful mouth he just kissed and falls off the precipice of reason. \"Since you already have my heart.\"\n\nShe runs a finger down the worst of the scarring of his face and he flinches. \"Look at me, Slash.\"\n\nHe barely lifts his chin, his nostrils flaring at her heady scent. He needs to get the hell out of here, and fast. Or his wolf will act for him. It's so close to the surface of his skin he can see the microscopic movement of his flesh rippling.\n\nHer hazel eyes drill into his. And his wolf likes the dominant display of the female\u2014his female.\n\n\"I love you, too, you big damn dope.\" She takes her finger away and punches him in the arm. \"I'm not going anywhere, and I don't give two shits and an eff if you have scars. Or if you've killed a legion of Were in battle. Or that you're not in my pack.\"\n\nSlash's chest is so tight, he can't draw a single breath. His emotions drown him, clogging his airways.\n\nAdrianna puts her hand on his chest then grips his shirt in her small fist. \"Hear me, Slash.\"\n\nHis eyes pull to hers.\n\n\"Lawrence can stick it up his ass. I want you.\"\n\nSlash covers her hands with his own, and the steel bands around his chest loosen. She renounced her packmaster so easily.\n\n\"Female, you\u2014you ease me.\" And she does, so much. If only Slash didn't feel guilty for this moment of stolen happiness.\n\nAdrianna's grin is fierce, and certain. \"I know.\"\n\nHe smiles back, but it feels a little sad around the edges. \"For how long will you be with me? I am not casual in my affections.\" It's such an understatement, it's utterly ludicrous to voice.\n\n\"Duh. Slash, I get this. You're old-school. I couldn't like another wolf if he bit me on the ass.\" She laughs.\n\nSlash growls at the thought of any Were touching her, especially there.\n\n\"You goofball,\" she sinks against him, laying her face against his muscular chest.\n\nHe holds one of her hands between their bodies and wraps his free hand around her back, pressing her tightly against him.\n\n\"Don't you know? I've always been yours.\"\n\nHe lays the unscarred side of his face against the top of her head.\n\n# CHAPTER FIVE\n\nPraile\n\nPraile's long ebony tail is held high. It twitches, making a slight whistling sound as it whips to and fro.\n\nThat usually happens when Praile's irritated. He's highly irritated now.\n\nPraile grasps Anthony Daniel Laurent by the hair. He shakes Anthony's head slightly. The eyes have already glazed over with the whitish icy-gray shroud of death. With a disgusted snort, Praile chucks the severed head onto the pile of bodies.\n\nHe sees Tony's prick like a blackened flesh noodle and holds in a laugh.\n\nNot bad. Judging by the wound, it was torn off his body by teeth. Praile snickers, not bothering to contain his glee. He taps a taloned finger on his chin, considering. He might not have done as well.\n\nPraile is a systematic high demon who is not wont to being dramatic. But jerking Tony's penis off with his teeth would have come with a certain amount of satisfaction, though the action was a tad intimate for his taste, which ran to the female persuasion.\n\nAs humans have assumed for millennia, there is no pleasure to be had in Hades. But there is much to be had on this plane. Praile's sharp eyesight takes in the nuances of death all around him.\n\nOne important instrument of advancement and justice is missing\u2014the death saber. It brings death to nearly all supernaturals. Praile scans the bodies\u2014they're all demonic.\n\nThe Singers must have buried their own.\n\nHe whistles in the fifty-HZ range of ultrasonic frequencies that only canines can hear, and three demons swivel their heads in Praile's direction. He jerks his jaw, gesturing for them to join him.\n\nThey move. They're obedient. That is good, because the consequence for disobedience is swift and unyielding\u2014very much like hell.\n\nLazarus comes first. His unflattering pale-red skin and unadorned tail notwithstanding, he is the very best high demon Praile has ever known. He will kill anything, and he is built perfectly for the strenuous physical demands of their kind: to torture anything that breathes.\n\nAnd he is so exacting about it all. His name always makes Praile take hidden jest\u2014it's an apt nickname. Lazarus can bring most back from true death. Oh, the irony!\n\nPraile speculates Lazarus has a little Singer blood. That Healer part of him is handy during torture in the hot place.\n\nPraile hides his giddy expression with difficulty, loving his own humor. After all, Praile is his own biggest fan.\n\nHis brow furrows. The Master will not find the minions' failure to kill all the Singers humorous or appreciate their inability to capture the two most important females in a thousand years.\n\nYet, Praile finds it all so droll.\n\nHe longs for a true challenge and to have more freedom on this plane to torture, maim, and antagonize. After all, what fun is being a demon if one cannot spread darkness and cruelty?\n\nPraile folds his muscular arms. \"Speak,\" he barks at Lazarus. He smirks at his second-in-command, silently daring him to address his abhorrent interaction.\n\nBut Lazarus is too clever to take the bait, which causes Praile a perverse joy. One must take small joys whenever they are presented.\n\nPraile sulks quietly at Lazarus's utter lack of reaction. Lazarus is self-contained in a way that is rare of the aggressive demonic, and it pains Praile.\n\n\"Some of our soldiers have escaped. The others have met true death.\"\n\nPraile casts a glance at the low demons who accompany him and Lazarus. He dismisses them when they lower their eyes in subservient deference.\n\n\"What of the Angelic Blood?\"\n\nLazarus heaves an exhale of disgust. \"She is not here.\"\n\n\"Really?\" Praile asks sarcastically, stepping into his second's space. Fool.\n\nLazarus doesn't flinch.\n\nSo brave, Lazarus. \"I know that,\" Praile spits.\n\nI so loathe relying on others, however necessary.\n\n\"I scent the High One's blood.\"\n\nPraile whips his head back in Lazarus's direction, flaring his nostrils. He tastes the truth of Lazarus's words on his tongue. \"Really?\" he says without the sarcasm he'd employed earlier.\n\n\"Yes.\"\n\nPraile runs his eyes over Lazarus. What a shame he doesn't possess the fine deep-scarlet skin that his kind finds so beautiful. It's also unfortunate that Lazarus cannot have the tail that ends with an appendage of weaponry.\n\nHis eyes narrow at Lazarus. It might be just as well that the other demonic is lacking, for Praile would be even more jealous of Lazarus than he already is. Praile hates his own perceived lack of gifts.\n\nLazarus's keen sense of smell rivals that of the Were. It is so acute, he can scent emotions, where Praile can scent only a lie. Lazarus can take every drop of life out of anything that draws breath and give it back at will. He has also animated the dead for his bidding.\n\nUseful attributes in a demonic.\n\nThat bit was something to see. Commanding the dead, as a demonic, is a rare gift. But Lazarus is ugly. Certainly, with his sculpted face, square jaw, and tall muscled body, he might appeal to some demon females as a pity fuck.\n\nPraile is picky about his dalliances, though. He cocks his head. Actually, he's discerning about everything. He shrugs.\n\n\"Is she wounded?\"\n\nLazarus nods, serious and, as usual, humorless.\n\nPraile licks his lips. \"Do tell.\"\n\nLazarus makes loose fists. \"She has a piece of us inside her.\"\n\nPraile chortles. \"Excellent.\"\n\n\"The High One is still mortal?\" Praile's gaze searches Lazarus's.\n\nLazarus smiles, showing bright, ugly white teeth. \"Absolutely.\"\n\nPraile lets out a sigh of relief. All is not lost.\n\n\"Idiots, they should have wedded her. Were, vampire, and Singer. And the Angelic Blood would be untouchable. Though it would be a great source of amusement to kill all who served her,\" Praile muses.\n\nLazarus does a poor job of containing his irritation.\n\n\"Let us fetch the death saber. We can't have that in the wrong hands, though I am most pleased Anthony managed to pierce the Angelic Blood before his cock and balls were chewed off his worthless body.\"\n\nThe low demons mewl from their safe position several feet away from Praile.\n\nHis grin widens, showing off his ebony teeth to perfection.\n\n\"You two\"\u2014he swings a black talon-tipped finger toward two of his minions\u2014\"clean up the mess of your kind and go to hell.\"\n\nA giggle bubbles in Praile's throat, and he barely suppresses it. He gives Lazarus a sidelong glance to ascertain if his second noticed his slippage.\n\nLazarus meets his eyes but finally ends the staring contest.\n\nIt is well-known that high demons have a propensity to slide into madness after middle age. At over seven hundred years old, Praile is no exception. Yet he must finish the Master's most important work.\n\nThe High One shall be slain.\n\nThe abomination growing inside the belly of the mixed-blood royal Singer shall be sacrificed in Hades, as prophesied.\n\nA child destined to have the blood of all cannot be allowed to survive\u2014or save the Angelic Blood through the gift of immortality.\n\nThe last task of the Master is a tall order.\n\nPraile is just the demon to fill it.\n\n# CHAPTER SIX\n\nJulia\n\nA mansion even more grand than Region One's rises on a perfectly shaped knoll like a jewel that's lost its luster. Its paint is peeling, and the once-grand dame stands in testimony as a shell of her former self. It deepens Julia's depression. If she can make it through the next day, she'll be so thankful.\n\nJason and the other Were get practical and melt back into full human form, Cyn and Adi stay in quarter-change form, though. Julia would stay that way all the time if she were a female Were.\n\nShe admires Cyn's and Adi's versatility in a world that is harsh on the subsect of humanity that dwindles under their noses.\n\nAll these groups were co-existing with her for her entire life, and she'd never known.\n\nSlash and Adi walk together toward the front entrance, where the large oak door stands open.\n\nSlash gives a look at the Were, and the Reds, including the new Alaskan Red, Zeke, jogging to his position.\n\n\"What is it?\" Julia whispers, suddenly feeling the weight of silence that surrounds them. This, where Jacqueline once reigned, is the first region headquarters Julia has ever been to, other than her own.\n\nShe swallows the lump of sadness suddenly lodged in her throat. The one that Marcus put there, the same one she gets thinking of Scott, the sole survivor of Tony's massacre.\n\nIn truth, Scott survived only if Tom Harriet and his goons didn't kill both him and Lucius. She breathes through the painful thought of Scott being expunged from this earth. Julia can't imagine the world without him in it. Julia doesn't know what that sentiment means for her\u2014or for Jason.\n\n\"It's all about the recon, babe,\" Jason says, laying a kiss on her forehead.\n\nShe leans against him, her body fitting perfectly against his.\n\nA whisper of an exhale slides out of her tight body. \"Right,\" she says, and even to her own ears, she sounds unconvinced.\n\nJason cups the back of her head with his hand. \"I don't blame ya\u2014it's been a thing. This whole entire last week has been nothing but one calamity after a-fucking-nother.\"\n\nI agree. Though Julia doesn't bother saying it out loud. There's really no need.\n\nJulia's tension remains, though she knows she's safe in Jason's arms while the Were file through the house.\n\nHer eyes meet Jacqueline's. Julia stalls when she sees the slight swell to Jacqueline's belly.\n\n\"Hang on,\" Julia says, walking toward her.\n\nJacqueline turns to her.\n\nDomi's silver eyes glitter at Julia, and she decides she won't be unnerved by it. Barely.\n\n\"Hey,\" Julia says a little breathlessly, her hand straying to where her stomach wound was a day ago.\n\n\"Hello,\" Jacqueline replies in her smooth voice. She glances at Julia's palm on her stomach. \"How does your wound fare?\"\n\n\"It's okay, I guess.\"\n\nJacqueline frowns. \"That's more evasion than reply.\"\n\nJulia gives her a crooked smile. \"Can't pull much over on you.\"\n\n\"No.\" Jacqueline gives a minute shake of her head.\n\n\"It feels weird to be your friend now,\" Julia blurts then almost covers her mouth.\n\nJacqueline nods. \"I know exactly what you mean.\"\n\nDomi moves behind her, lifting her dark hair away from her neck so he can lay his palm at her nape, and though Julia doesn't think she's aware, Jacqueline tips her jaw into the small embrace. Domi's eyes soften at the gesture.\n\nOh, my word, there's some stuff going on there.\n\nStay on task, Julia. \"I noticed you're looking all pregnant.\"\n\nTheir eyes catch and hold. Jacqueline glances away first. \"Yes,\" she says in a low voice, \"Domi postulates there's an accelerated pace of gestation.\"\n\n\"Ah. Okay,\" Julia replies slowly, looking from one to the other. \"Is this normal? I mean, for a fey baby?\"\n\nDomi lifts his chin. His navy hair, like midnight liquid, gleams as it slides over his shoulders. Not a hair is out of place. They've battled and haven't showered or eaten. Everyone is beat, yet Domi looks as fresh as a daisy\u2014except for that healing scar at his throat.\n\nJulia tears her eyes away from the scrutiny of his neck and he smiles at her blatant curiosity. She feels her cheeks heat. Nice, Julia.\n\nShe glances at the Were moving in and out of the house then looks back at Domi and Jacqueline.\n\n\"No,\" Domi says in a voice so melodic, it doesn't sound natural but like spoken music. \"We think the babe is special.\"\n\n\"Of course it is,\" Julia says, giving Jacqueline a tentative smile.\n\n\"What Domi means is that the genetics of the baby are dictating some unusual beginnings.\"\n\nJulia stares at Jacqueline. \"You mean, like some kind of one-of-a-kind kid?\"\n\nJacqueline smiles so suddenly and naturally, it startles Julia. Sadly, the woman smiles so infrequently that the expression sits oddly on her face. \"I forget your way with words. But yes, we believe the baby will be... very unusual.\"\n\n\"He will be fey, Were, vampire, Singer, and angelic,\" Domi says with a casual lift of his shoulder.\n\n\"Oh, well, no big deal then,\" Julia says with a small laugh. \"Like Heinz 57, guys.\"\n\nThey give identical puzzled expressions, and Julia laughs again. \"You guys. Okay, are we sure all that mingling of cool genes is the reason for Jacqueline to have a speed pregnancy?\"\n\n\"It is conjecture,\" Domi admits.\n\nJacqueline rubs her cheek against his fingers, and his shoulders relax. \"It sounds like the right fit, as you would say,\" Jacqueline says.\n\nTheir eyes meet again. It is what Julia would have said. Jacqueline's intuition is uncanny.\n\n\"Are you sad about going back to faerie?\"\n\nJacqueline's eyes round, and she grips Domi's hand. His eyes harden, his irises unpolished diamonds.\n\nWhoa.\n\n\"No. I can be who I was meant to be in the mound, near the sithen. Without it, I am the Jacqueline of old.\"\n\nDon't want that.\n\n\"Hey, Jules,\" Jason says from behind her, his eyes wary on Jacqueline, and for good reason.\n\n\"All's clear. They're rounding up the rigs now.\"\n\nJulia turns to say goodbye for the moment to Jacqueline, but her dark eyes stay locked on the old headquarters where she formerly reigned. Silent tears are streaming down her face.\n\nDomi turns Jacqueline to face him, lightly gripping her shoulders. He passes one hand over her face, and when it comes away, he cups his palm beneath her jaw.\n\nHer tears fall in perfect teardrop formation, tinkling together inside his palm in hardened clear gems.\n\n\"What? What did you do?\" Jason asks. The awe in his voice matches Julia's feelings.\n\n\"Tears are precious in faerie. We do not waste emotion. Some can be collected.\"\n\nJacqueline gives him a tremulous smile, rolling her bottom lip between her teeth to stop the shaking.\n\n\"That's the coolest thing I've ever seen,\" Julia says under her breath as Jason takes her hand.\n\nDomi gives a small shrug. \"It's a parlor trick.\"\n\n\"No,\" Julia say, meeting his eyes, \"You took her sadness and made it your own.\"\n\nDomi says nothing, assenting to the truth of the words through lack of reply.\n\nHe pockets the jewels of Jacqueline's sadness, and she places her hand over where they lay.\n\n*\n\nThe car ride is terrible. The black, unmarked SUVs are luxurious, built to last, and anonymous. They riot through the night like oil-slicked bullets.\n\nBut Julia knows what awaits them at the Region One headquarters\u2014emptiness.\n\nThe voices of the children will be absent. Scott and Lucius are unaccounted for. The beautiful lake that is filled with swans, is now surrounded by a mass grave.\n\nHer only consolation is that the murdering jerk-off is dead. But what's happened to the Greenes? And what about Reagan? Delilah?\n\nJulia can't sleep on the way home. Tumultuous thoughts spin inside her brain on an endless cycle. Wedged between Adi and Jason, she leans her head against Jason's shoulder. The men have the door, as they so succinctly put it. It was super-funny to Julia when they said the women would be in the middle and they should ride on the outside, in case of disaster.\n\nUh-huh. Adi said it best: \"Me, Tarzan. You, Jane.\"\n\nSlash scowled. Then had brightened when Adi scolded him.\n\nMen.\n\nThey broke every speed limit and stopped for gas only once. The Were moved around outside, and Tharell lay gagged and bound in the back of one of the SUVs.\n\nJulia swore she could feel his eyes on her through all that disastrous metal that he hated so much.\n\nDomi threw up. Even with Jacqueline and his unborn child offering a buffer of sorts, the metal of the car was too much for him.\n\nThey finally arrive and everyone pours out of the vehicles after they park in the gigantic circular drive in front of the Victorian.\n\nWarm lights don't illuminate the windows.\n\nThe house stands like four corners of stark in the middle of grief. A symbol of what was.\n\nJulia stays in the car. She looks at the training barn, then away. She remembers when Michael made the pile of manure to contain Scott and a smile hovers on her lips without becoming.\n\nScott.\n\nJulia shivers at the thought of her soul-meld. Guilt and indecision shake her to the core. Getting out of the rig is accepting that she's back, in charge of a new beginning with a future that's completely unknown.\n\nShe doesn't know if it's the beginning she wants.\n\nJason comes back to the SUV and leans against the open door. He braces his biceps against the door frame and leans in. \"What are you doing here, looking all sad and shit? We're here. Let's clean up, eat some grub.\"\n\nJason reaches forward, cups her chin, then slides his fingertips from her temple to her jaw. She catches his hand as it retreats.\n\n\"Come on, Jules,\" Jason says, pulling her out of the vehicle.\n\nShe follows, his hand warm in hers.\n\nJulia ignores the ache that plagues her belly.\n\n# CHAPTER SEVEN\n\nTessa\n\n\"Holy mother of the moon, that was a damn mess back there,\" Tessa says, throwing her eyes to Tahlia then back at the road.\n\nTahlia's as quiet as the tomb.\n\n\"Hey listen, honey\u2014I'm just running blind here. You gotta give me something to go on.\" Tessa heaves a sigh, flipping her wayward braid out of the way. \"I mean, who do you belong to?\"\n\nTahlia's fingers worry at her borrowed shirt that's at least two sizes too big. She's a wisp of a girl. Dark skin and hair frame eyes so deep a blue, they're almost violet. The shock of her curly hair misses frizzy by a millimeter, but her exotic blend is obvious. She's unique for a Were. There are Were of every race, but it's very rare for a mix. And a mix as lovely as this girl\u2014Wereshifter\u2014is unheard of.\n\n\"Tell me something,\" Tessa pleads.\n\nTahlia seems to decide something, though her face is like carved stone. \"I am of the Lanarre.\"\n\nLycan royalty. Good gravy. \"What?\" Tessa shrieks, gripping the wheel to maintain control and not run the car off the road.\n\nTahlia winces. \"That is why I said nothing.\"\n\nTessa narrows her eyes on Tahlia. \"I understand some of the protocols. You're a whelp.\"\n\nHer chin lifts in the first defiance Tessa's seen out of the girl. \"I am of age.\"\n\nTessa flicks her eyes over Tahlia once, shifts her eyes to the road, then comes back to her. \"No way.\"\n\nA light blush spreads across the girl's cheeks.\n\n\"You would be held by human guardians while traveling away from the pack.\" The Lanarre often bred humans to act as their guardians from one generation to the next.\n\nTahlia inhales a shuddering breath that sounds both hollow and grief stricken.\n\n\"What?\" Tessa breathes, automatically lifting her foot off the accelerator.\n\n\"They were murdered.\"\n\nSomething doesn't make sense. And we're going to flesh it right out.\n\n\"A rogue Were...\" Tahlia's wide eyes move to Tessa in apparent apology.\n\nTessa dismisses her trepidation curtly. \"It's okay. I know what I am.\"\n\nShe sighs in relief then continues, \"He came and slaughtered everyone in the rented domicile where we lodged.\"\n\nHotel.\n\n\"You speak strangely,\" Tessa interjects.\n\n\"I speak as I am meant to.\"\n\nOkay. Wrong tactic.\n\n\"Go on,\" Tessa encourages, rolling her hand in a circle.\n\n\"My human guardians were allowing me to watch the television.\" Her eyes spark. \"It is strictly forbidden,\" she admits in hushed tones.\n\nUnreal. Talk about living in a bubble. But Tessa remains rapt, saying nothing.\n\n\"The guardians are just extraordinary humans. They do not share our senses.\"\n\nTessa nods. Of course not.\n\n\"We had the television turned up quite loud for my guardians benefit.\" That made sense as what humans thought was the correct volume would sting a Were's sensitive hearing.\n\n\"I\u2014\" Tahlia wrings her hands, which bleed to white under the pressure.\n\nGuilt, Tessa realizes.\n\n\"Don't,\" Tessa interrupts, putting her hand over the girl's. \"There was nothing to be done, nothing you could have done.\"\n\nShe bites her lip, and fat tears fall straight from her eyes to her clenched hands.\n\n\"I needed to use the restroom and excused myself to\u2014ah!\" She tears her hands from underneath Tessa's and covers her face. \"I listened and did nothing!\" she yells.\n\nTessa grits her teeth against Tahlia's shame. She empathizes. Tessa pulls over onto the soft shoulder, letting the old car idle.\n\nShe takes Tahlia's hands into her own. \"Listen up, Tahlia. There's no way you could have done anything but get dead.\"\n\nTahlia nods reluctantly. \"I understand.\" She bites her lip. \"But it does not make it any easier to suffer the truth. They are still gone from this earth, Tessa.\"\n\n\"You saved me, and I didn't matter.\"\n\n\"Everyone matters,\" Tahlia answers softly.\n\nThey share a look and a sudden laugh.\n\n\"You know how to make short work of Weres,\" Tessa says.\n\nA tentative smile graces her full lips. \"They were not of good intent. And my animal does what I cannot.\"\n\n\"No shit?\" Tessa says, and the girl's eyes bug. She gives Tahlia a speculative look.\n\n\"You know, you're kind of sheltered.\"\n\nTahlia shakes her head. \"Not really. My duty is set. It has been ordained since my infancy.\"\n\nTessa wracks her brain for memories of the fabled Lanarre, but all she comes up with is that they are the elite of the Were, all Alphas. Anyone born a Lanarre is automatically an Alpha.\n\n\"What duty?\" Tessa asks.\n\n\"My duty to the Lanarre,\" she replies slowly, as if Tessa is a dim-witted child.\n\nTessa relaxes in the seat, giving Tahlia steady eyes. An idea, an ugly one, forms in the back of her mind.\n\n\"What were you doing out of the safety of the Lanarre, traveling with your human guards?\"\n\n\"I was traveling to meet my chosen.\"\n\nOh, my moon.\n\n\"What chosen?\"\n\nTahlia tries to put a blanket over her disgust of Tessa's lack of knowledge. \"Are the ways of the Lycan so diluted that the packs no longer know how we came to be? Our history.\"\n\nTessa shrugs. \"I guess not.\" She was usually too busy running from Tramack to concern herself with Lycan history.\n\nTahlia makes a noise in the back of her throat. \"My chosen is the male who will be my mate.\"\n\nTessa gulps back disgust. \"You're telling me this is some kind of arranged marriage?\"\n\nTahlia lifts a shoulder and eyebrow simultaneously. \"Were do not marry, as you know.\"\n\n\"Right,\" Tessa acknowledges, \"but to mate a male Were\u2014\"\n\n\"Lanarre,\" Tahlia corrects.\n\n\"Uh-huh. Sight unseen? Do you have a choice?\"\n\nThe girl's delicate brows pull together. \"Why do I need one? He is my chosen.\"\n\nRepeating it over and over again doesn't make it a great plan in Tessa's view. \"What if he's a sadistic pig?\" Tessa asks.\n\nA laugh erupts from Tahlia's throat. \"Are you having me on? You are very negative, Tessa.\"\n\nYes, yes I am. It kept her butt safe for more years than she could count. And this arranged mating thing? It stinks to high heaven.\n\n\"I have a question of you,\" Tahlia says.\n\nTessa hikes her eyebrows. \"Shoot.\"\n\nThe girl gives a slight frown. \"May I smell you?\"\n\nTessa's chin juts back. \"I guess\u2014but weird.\"\n\n\"Humor me.\"\n\nShe sounds so old for a whelp.\n\nTahlia leans close, rising to her knees as she moves in, then takes a whiff of Tess from neck to crotch.\n\nA human would blanch at such a gesture.\n\nIf asked, it's not an exceptional thing among Were.\n\nBut of all the things the girl could ask about, she chose scenting.\n\nTahlia sits back against her heels.\n\n\"You smell of him.\"\n\nHuh? \"Who?\" Tessa searches her face in the growing shadows. She sees everything. The time of day or night doesn't matter.\n\nGooseflesh rises at the expectation of the revelation.\n\n\"The one who murdered my guardians.\"\n\nFuck. Her heart sinks.\n\nTessa knew she should have drowned that crazy Were in the ditch.\n\nShe leans away from Tahlia, collapsing against the back of the driver's seat.\n\nNow who's the guilty one? If she had followed her instincts, Tahlia's guardians would be alive. And according to her gruesome testimony, a lot of other innocents would be, too.\n\nTessa closes her eyes.\n\n\"Tell me why the murderer of my human guardians scent is all over you.\"\n\n\"I should have killed his ass when I had the chance.\"\n\n\"Tell me,\" Tahlia commands intensely.\n\n*\n\nThe women sit in the silent car when the last word from Tessa echoes within.\n\n\"That is an awful tale.\"\n\n\"Not as bad as me letting that screwed up male live.\"\n\nTahlia draws a painful sounding inhale. \"You could not have known.\"\n\nTessa shoots a sharp look at Tahlia. \"He almost killed you.\"\n\nA slight smile lifts the corners of her mouth. \"There would have been a wrath unlike any the Lycans would have known at my death.\"\n\nDo I hear a smug note to that comment? Tessa searches Tahlia's face. No, she stated it like fact.\n\nMaybe it is.\n\n\"I am the princess of my people.\"\n\nHigh Alpha female.\n\n\"Oh, shit,\" Tessa breathes in reply.\n\nTahlia nods. \"Yes.\" Then, she says, \"You have very colorful language.\"\n\n\"Yup.\" Tessa turns to her, ignoring the relevance of that last. \"Do you want to mate with this guy?\"\n\n\"Drek?\"\n\nTessa nods. \"Is he the... prince of something.\"\n\nTahlia's lips curl. \"High Alpha male.\"\n\nKnew it. \"Yes, him.\"\n\nHer eyes slide away from Tessa. \"Of course.\"\n\nHell no she doesn't.\n\n\"Is he an okay guy, like...\"\n\n\"I do not know if he is a \u02bbsadistic pig.'\" She gives a little grin, ducking her head, and Tessa laughs. The girl has a great sense of humor buried underneath all the layers of propriety.\n\n\"Okay, so can we just do whatever? Stick together? Or do you have parents or what?\"\n\nHer face smooths.\n\nTessa knows masked sadness when she sees it.\n\n\"I do not know my parents. Only my guardians. One to guard and one to take care of me and teach me the ways of the Lanarre.\"\n\nThat's awful.\n\n\"Okay,\" Tessa slaps her thighs, knowing the girl essentially just lost the only caregivers she ever knew. \"So let's take the long way.\"\n\n\"The which?\"\n\nTessa grins, throwing the car into gear. \"Let's show up when we feel like it.\"\n\nTahlia bites her lip and Tessa realizes that's her nervous tell.\n\n\"They will search for me.\"\n\nTessa snorts. \"Have at it, guys.\" She turns and relaxes against the seat. Tessa hears the seatbelt click.\n\n\"I think I like you, Tessa.\"\n\n\"I like you, too, Tahlia. I always think people that save my life are the best.\" Tessa winks at Tahlia, and a shy smile ghosts her lips.\n\nNeither one of them have a friend in the world.\n\nBut it looks like they have each other.\n\nThings could be worse.\n\n# CHAPTER EIGHT\n\nPraile\n\nThe barbed end of the lash checks the top layer of his skin and peels it away like sliced cheese.\n\nPraile gnashes his black teeth together as a layer of smoldering mist hovers above his flesh.\n\nPraile always smolders when his emotions run high.\n\nThe lash whistles a high note at the return. It sings as it returns to meet his flesh.\n\nPraile bellows as the thirteenth lash strikes deep, ringing its poison-tipped metal on a vertebrae in his fileted back.\n\n\"Halt,\" a low voice says from behind Praile. His head bows to his chest. Rivulets of sweat burn a pathway through his scalp and pour down to fill the wounds the barbed lashes have made on the entire length of his spine.\n\nPraile dare not turn. The next lash might paralyze him. Yes, he could heal it, but the vulnerability of not feeling or being able to move would be his undoing as his skills of self-healing were greatly limited. Healing is not one of his gifts.\n\n\"I believe Praile has learned a valuable lesson this day,\" the Master comments.\n\nHe has learned nothing except if something were to go ill, Praile will suffer. However, The Master has taught Praile well, and he emulates the Master.\n\n\"Release him from his bindings.\"\n\nTwo of the low demons appear at either side of Praile and unshackle him. One dares to meet his eyes.\n\nThe gaze of the low contains a measured triumph. He is pleased by Praile's punishment.\n\nPraile grins, marking him for later, and he bows his head, scuttling away.\n\nRun faster, minion.\n\nThe Master slithers to Praile, the swish of his robes is all that Praile hears.\n\nPraile feels real fear, which is rare. His gaze drops, concentrating on the hem of the Master's robe. The Master's feet are grotesquely disproportionate. Long cracked black toenails that are so long, they nearly curl.\n\nThe Master always smolders.\n\nA black mist rises from the flesh of his feet. His toes wiggle and Praile flinches.\n\nThe Master chuckles a dark note of contentment into the hot cavern where torture, death, and discipline are meted.\n\n\"You are a good slave to the cause, Praile. However, when you called the Were to destroy our enemies on earth, the one who was most important to be slain runs about unharmed. And the Blood Babe lives inside the womb of a crafty Singer. One who is a female after my own heart.\"\n\nPraile hears a dull thump as the Master's meaty fist thumps his own chest.\n\nThe lump in Praile's throat shifts, stifling his breathing.\n\n\"I am entrusting you to find this Singer that is with child. The child.\"\n\nA talon touches the fleshy part of Praile's chin, and he winces, though it does not hurt.\n\n\"Bring her to me.\" The Master's rancid breath bathes Praile's face.\n\nPraile turns away, for even he cannot bear it.\n\nThe Master laughs at his discomfiture. \"Kill the Angelic Blood, the High One. Do not hesitate. Do not tarry. Bring me the whites of the Angelic's eyes.\"\n\n\"Yes, Master,\" Praile whispers.\n\n\"I will make your death last for an eon if you fail me in this.\"\n\nPraile knows. He nods.\n\nThe Master's hands thread through his hair, slowly squeezing like a vise. \"Are we clear, Praile?\"\n\nHis meaning is utterly clear as the Master's fingerprints begin to burn into Praile's scalp.\n\n\"Yes.\"\n\nHis grips tightens to the point of screeching pain, then he abruptly releases Praile's head.\n\nPraile bites back his relief and begins to control his breathing, concentrating only on that.\n\n\"Lazarus will heal your wounds\u2014yet not perfectly. You should feel the pain as a reminder of what yet needs accomplishing.\"\n\nPraile lays his palms against the heated stone in front of him, trying not to notice the lost talons embedded in the wall from failed escapes by the masses tortured before him. The black blood has faded over time to a washed-out charcoal. It fills the grooves and divots of the nearly black rock.\n\nPraile groans as he straightens, keeping his eyes away from the Master. To look upon him is sure insanity. No one has ever cast their eyes upon the Master and lived to tell of it.\n\nTheir screams were silenced.\n\nPraile shudders as a talon caresses the most grievous wound at his back.\n\n\"I will see to it.\" Praile's agony drips from each word.\n\nThe talon sinks deep, and Praile bites the inside of his cheek until the rich taste of copper fills his mouth as he suffers through the inspection of the fresh wound on his back.\n\n\"Good.\" The talon lifts, and Praile nearly weeps in relief.\n\nPraile's shoulders slump as the Master exits the chamber. He stays in the same tense position until Lazarus appears at his side.\n\nPraile's hate for Lazarus burns brightly. But Lazarus does his job.\n\n\"This will be more painful before it heals, Praile.\"\n\n\"Yes, yes. Get on with it.\"\n\nFingers dig inside the wounds, and Praile squeals like a pig brought to slaughter. The pain is so acute, he forgets to breathe\u2014or think. He arches to escape the probing fingertips, but nothing will relieve him.\n\n\"Hold him up,\" Lazarus murmurs.\n\nLow demons, whom Praile does not know, hoist him by the armpits as the searing healing begins.\n\n\"Stop,\" he moans.\n\n\"No,\" Lazarus replies.\n\nPraile is sure he hears a smile in that one-word reply.\n\nHe opens his mouth to convey the pain Lazarus will incur for his joy at his master's pain.\n\nBut the pain is too great. It rips at his brain, and all falls to blackness.\n\n*\n\nThe demonic can camouflage their bodies. If the demonic did not have this ability, humans could easily call them out. Though Praile's skin is the coveted deepest red of his kind, it is well outside of human norms. And when he is rife with emotion, he smolders and small stubby horns sprout above his head. Though they are a sign of beauty for the demonic, they are an instant warning to humans that he is other.\n\nLazarus does not have horns, beautiful dark skin, or a tail weapon. He can be camouflaged easily and fit in nicely among humankind. Lazarus, with his horrible white teeth, hornless head, and lackluster tail, has less to hide.\n\nHe is the perfect lackey.\n\nPraile will need to remain calm. High demons have a more difficult time hiding what they are.\n\nThree days have passed since the Master's punishment. Lazarus healed Praile three quarters to right, and no more, according to specific instruction.\n\nPraile cannot mask his stiffness. Though the deepest wound is sealed halfway, it seeps through the ridiculous human costume he is forced to wear. The cotton button-down shirt is sticking to the wet wounds of his back, and it pulls as he takes breaths.\n\nPraile holds up his palm, and Lazarus slows, putting large hands on his denim-clad hips.\n\n\"Wait.\"\n\nLazarus cocks a light red eyebrow. \"If you need rest...\"\n\nPraile rolls his eyes. \"Of course I need rest!\" he bellows into the still night air. \"It is not about rest, Lazarus. Our Master requires this task completed in a timely way.\" Praile pants, trying to straighten. Unable to manage it, he hunches once more.\n\n\"Flag down a human vehicle so that we might make haste to where the Angelic resides.\"\n\nLazarus frowns. \"It is a risk I advise you not to take.\"\n\nPraile straightens, hissing as the material of the shirt sticks to his tender back. \"Duly noted. Now flag. Down. A. Human.\"\n\n\"What if the human possesses the devices of sanctity?\"\n\nDrat.\n\n\"Crosses and the like?\" Lazarus prompts as though Praile needs a reminder.\n\nPraile's brows drop like bricks above his eyes, and a lazy smolder begins above the bare skin at the back of his hands. \"It is unlikely, with so many humans in thrall with evil, that we will come across the random practitioner.\"\n\n\"There is the matter of an Angelic among humanity.\"\n\nPraile staggers toward Lazarus, looking up at the taller male, hating his stature. \"You let me divine which human is a threat to us.\"\n\n\"My discernment\u2014\"\n\n\"Your discernment is a tool in my arsenal, Lazarus\u2014do not forget that.\"\n\nLazarus allows a rare show of emotion, his lips curling as he bares his teeth. \"You do not let me,\" he says.\n\nLazarus makes his way from the forest where the portal of Hades empties to the highway, and which the human masses use to scurry from one ant hill to the next.\n\nPraile narrows his gaze at Lazarus's broad back.\n\nI will be watching you. Praile follows, making his slow and painful ascent from the gulley toward the highway.\n\n# CHAPTER NINE\n\nSlash\n\nSlash remains on edge. Though the fey disposed of the gruesome remains of the decimated Singer population, the taste of death lingers over Region One like a stench that will never be cleansed.\n\nZeke of the rogue Alaskan Were stands at Slash's side. \"So many dead can't be so easily covered up.\" His nose wrinkles.\n\nSlash grunts his assent and paces away. \"What are our numbers?\" He turns back, and Zeke shrugs.\n\n\"Bad.\"\n\nThat's what Slash was afraid of. The demonic killed nearly half of the Region Two Singers, leaving roughly fifty behind. And half of those are children. As is typical, Slash wants hard numbers for fighters. Many are unaccounted for.\n\nWhat of Lawrence and Manny?\n\n\"Do we have a superb tracker? Can we know for certain who is dead?\"\n\nZeke shakes his head. His exotic looks are unusual for the lower forty-eight, though Native Americans were plentiful where Zeke's pack ran in the north. \"Our numbers are down by half, too. Our best tracker\u2014gone.\"\n\nSlash remembers something. \"Jacqueline, the Singer from Two, she's a Tracker.\"\n\nZeke nods slowly, but his brows drop low over his eyes. \"Do we want to use her? She is with child and has\u2014what should we say?\u2014bad blood.\" Zeke laughs at the inside joke.\n\nSlash can't bring himself to.\n\nZeke studies him for a moment. \"You're a serious wolf.\"\n\nSlash nods. \"Serious times are afoot, Red.\"\n\nZeke stares at him for a second more. Instead of answering, he melds into his wolfen form.\n\nSlash's laugh sounds like a bark, as he's changed, as well. Athletic pants expand to fit his increased girth and height, though he doesn't wear a shirt. \"Perimeter sweep?\"\n\nZeke nods, and his stubbed snout causes Slash to wonder what he must look like in this form. That brings him back to his embarrassment over his scarred face.\n\n\"Let's go,\" Zeke growls from a mouth that no longer manipulates human speech perfectly.\n\nSlash swings his snout in the direction of the Victorian mansion, scenting Adrianna through the water as she showers.\n\nGuards, both Singer and Were, pepper the front of the grand home. The lone fang, Brynn, accompanies them.\n\nDetermining Adrianna is safe, Slash nods at Zeke. Then they race to the edges of the hundred-acre property.\n\nRunning the perimeter is the final pursuit of security before each night falls.\n\nSlash can't rest until he knows both his own wolves and his adopted group are safe. Then, and only then, will he lay down his weapons, eat, and clean up.\n\nSlash blurs through the scenery, his powerful arms punching the lone branch as it sweeps forward to snap at him. Leaves and forest debris pad his swift gait. A fallen old-growth log feeds the saplings that are nourished from its rich decay. He leaps over the belly of bark and wood rot with ease, his keen eyes at Zeke's back as he travels just ahead.\n\nRoads form a crude square around the property. Two parallel side roads run like wide railroad tracks that flank the sides of the land and Highway 101 claims the forward section.\n\n101 is exactly where they were all picked up by Tom Harriet and his immoral pack of Reds. The only Singer spared was the aura reader, Angela.\n\nNot a single Combatant remains alive, though Scott and Lucius are unaccounted for.\n\nZeke stops so abruptly, Slash all but slams into him. He evades him by inches, rolling into a half-executed somersault and catching his forward momentum with an outstretched arm against a small tree trunk. It bends then breaks, flinging Slash through the undergrowth. He slows and barrels into a massive tree trunk.\n\nPine needles rain down, and the scent of the forest is thick in his nose. He breathes, and they choke him. Slash ungracefully spits them out and glares at Zeke.\n\nZeke holds out his palm, his talons still short from his change to wolfen. Slash slaps his palm inside Zeke's and rises.\n\n\"Thanks for the warning.\" Slash glares, baring his teeth.\n\n\"If you smelled what I did, your ass would've puckered too.\"\n\nSlash ignores him, flaring his nostrils hard.\n\nNo. It can't be.\n\nHe turns back to Zeke, who shrugs.\n\n\"When was the last time you scented a Lanarre?\"\n\nSlash awkwardly folds his arms, and sap causes them to stick together. He casts a sharp glance at Zeke. \"Since whelphood.\"\n\n\"That's right.\" Zeke nods, his burnt orange downy hairs making him look vaguely on fire. \"I can't say I ever have.\"\n\nIt's instinct. A Were knows Lycan royalty.\n\n\"Female,\" Zeke says, and Slash nods.\n\n\"Scenting a Lanarre in this area doesn't make a great deal of sense,\" Slash growls. \"They're always under guard. They're pure Were, from which we all come.\"\n\nZeke shrugs. \"They all take a shit every day like the rest of us. Nothing special.\"\n\nSlash's lips pull into a grim smile. \"It might be a little more than toilet habits, Zeke.\"\n\n\"A female doesn't pose a threat, and I don't smell wounds. I say we leave her be.\"\n\nSlash cups his chin, fur mashing down under his hand, and slowly shakes his head. \"I don't think so. A female out in this rural area is illogical. They lock down their females. No. I say we investigate and make sure she isn't in danger, then we leave it be.\"\n\n\"Fine, but it could be a can of worms.\" Greenish-gold orbs slowly spin, revolving slightly faster with Zeke's emotions.\n\nSlash chuckles, dropping his hand. \"I don't know about the Alaska dens, but when is it not a can of worms?\"\n\n\"I don't scent any males.\"\n\n\"True,\" Zeke says. His chin lifts as he gazes at the dying sun. \"Let's do it quickly and get back to One. I could eat the ass out of a hippo.\"\n\n\"Nice choice of words.\"\n\n\"Do you feel less hungry?\"\n\nSlash didn't. He thought he could eat the asses out of an entire herd. \"No. I'm starved, too. The wolfen form is a bitch to maintain for this length of time. It sucks energy.\"\n\n\"That's in short supply,\" Zeke finishes.\n\nSlash leads the way this time, scrapes and bruises from his rough landing repair and fade as he makes the steep climb toward the highway.\n\n*\n\nTessa\n\nSo much better, Tessa sighs mentally as her urine stream finally ends. She's had to pee like a Russian race horse for the last hour.\n\nShe smirks at her ladylike thoughts while using a napkin from the last gas station to wipe. She tosses the napkin to the ground and kicks leaves over it. A pang of guilt spurs her to help mother nature in its pursuit to return everything to the earth.\n\nTessa scans the deep gloom of the forest. Her eyes rise up the small incline to where the car sits on the soft shoulder of Highway 101.\n\n\"Tahlia,\" Tessa softly calls.\n\n\"Yes,\" she answers.\n\nTessa's shoulders drop. She can't believe how fast she feels responsible for the Lanarre female.\n\nTessa needs that like she needs a hole in the head.\n\nIt's not enough that Tramack is up her ass, sniffing around for a good place to dry hump her leg. No-oh, I've got to take in a stray Were female. Not any Were female, but a Lanarre princess.\n\nDumb, Tessa. Really dumb.\n\nIt is what it is.\n\n\"Come here.\"\n\nTahlia moves between two huge fir trees. She's so quiet, Tessa's not sure if she would hear her had she not been directly in front of her and within sight.\n\n\"You're quiet.\"\n\n\"Stealth movement is a very important part of my training.\"\n\nTessa cocks an eyebrow. \"This is so weird. Really. Forgive me, but if you're this important princess\u2014\"\n\nTahlia folds her arms, looking very close to a rant.\n\nIgnoring her, Tessa goes on, \"Then why teach you all this combat stuff?\"\n\n\"I am female, nonetheless\u2014I have skills the Lanarre wish to develop. Not one Lanarre's importance is ignored. Whatever aptitude they possess is built upon, harnessed.\"\n\n\"Uh-huh,\" Tessa says.\n\nTahlia releases her arms and shrugs. \"I'm not making this up. The Lanarre feels responsible for all Were. We must be excellent in all things. Otherwise, we're unworthy of the title of ruler of the Lycan.\"\n\nOh, moon.\n\n\"So what happened?\"\n\nTahlia's eyes lower, and she presses her beat-up sneaker into the moss, making a tread indentation. \"We cannot be responsible for all wrongdoing or for all Were falling away from the principles of Lycan.\"\n\nTahlia's head jerks up, and she plants her legs far apart, fists ready and loose at her side. She morphs from delicate to fierce in seconds.\n\nTessa turns around slowly and sights two Were, both in wolfen form.\n\n\"Stay behind me, Tahlia,\" Tessa warns, her voice low.\n\n\"I am a better forward fighter,\" she comments casually.\n\nTessa turns to tell her what's what.\n\nShe is gone.\n\nTessa feels the breeze over her head and as Tahlia flings herself over Tessa's head.\n\n\"No!\" Tessa screams and charges the males.\n\nTahlia lands in front of a seven-foot-tall Were whose deeply scarred face has the most tender hard eyes Tessa has ever seen.\n\nHe's seen too much, is Tessa's lone thought before Tahlia launches herself at the other Were male.\n\nTahlia's hand is a blur.\n\nShe steps away from that Were and moves in on the scarred one.\n\n\"Forgive me,\" he says in the heavily graveled voice of the wolfen form then hits Tahlia at the side of her neck.\n\nShe falls in a silent heap.\n\nThe other Were is on his knees, four trails where talons swiped across his throat bleeding.\n\nHis esophagus shines like a slick cream worm in his throat.\n\nTessa moves in before the scarred Were can hurt Tahlia more. The girl is already coming around.\n\nTheir eyes meet from her prone position, and she kicks her leg up, narrowly missing the scarred one's nutsack.\n\nHoly moon, this is so bad.\n\nTessa hits him full speed, and he grabs her forearm, spinning her off behind him with her own momentum.\n\nTessa lands on her ass with a hard thump. Her wind is gone, and she lies on her back, unable to breathe.\n\nI have to change. Like yesterday.\n\nTessa's body shifts to the quarter-change seamlessly, and her lungs fill. They're just slightly bigger, better, and more proficient at oxygen intake.\n\nTahlia is pinned against the scarred Were, her back to his front.\n\n\"Don't hurt her. She is Lanarre,\" Tessa says as a last resort. She's not sure what these males know about the species. Her own knowledge was pretty inadequate. But if they know anything, they know not to fuck with the Lanarre\u2014ever.\n\n\"We know,\" the scarred Were says. \"We are not here to harm, but to help.\"\n\n\"Could've said,\" Tessa replies as the deepening gloom tests her improved vision. She does manage to make out that he is Alpha\u2014and a Red. There's no hiding that sunset-colored fur.\n\n\"This one didn't give us the chance. My second heals a grievous wound.\"\n\nTessa rolls her eyes. Tahlia's wide eyes are on hers. \"He'll live, and my moon, don't you know better than to sneak up on two females?\"\n\nHis face shows surprise.\n\n\"Don't look at me like that, Red. We were out here taking a tinkle, and you guys sidle up? Moon help us.\"\n\nHe scowls. \"If I let you go, are you going to give me a new blow hole?\" the scarred one asks.\n\n\"What? Are you a whale?\" Tahlia asks in a sulk.\n\nTessa laughs.\n\nThe other Were is on his hands and knees, massaging his throat. \"That fucking hurt.\"\n\nTahlia harrumphs, and the injured Were glares at her.\n\n\"What pack are you from?\" Tessa asks tersely.\n\n\"I won't harm you,\" Tahlia says.\n\nThe scarred Were backs away so quickly that Tessa can't track the movement, even in her quarter-change form.\n\n\"I'm Slash, from the Southeastern.\"\n\nTessa can't hide her relief.\n\nSlash frowns at her curiously. \"I take it that's a good thing.\"\n\nShe nods a little too quickly. \"A very good thing.\"\n\n\"Tramack from the Western hunts me.\"\n\n\"You're rogue?\" the injured Were asks, standing, the surprise evident in his voice.\n\nThe furrows from Tahlia's expert swipe fully close, and the skin remains shiny with fresh scar tissue.\n\n\"Are you going to judge?\"\n\nHis eyes glitter at her. \"Not yet.\" But his gaze shifts to Tahlia.\n\n\"We are here to help. We can't do that when you attack us,\" Slash explains logically.\n\nTessa puts her hands on her hips. \"We are female.\"\n\n\"Clearly,\" the other Were says. His lips pull into a sardonic tilt, and he performs a little bow, though the cough from his abused throat ruins the effect. \"I'm Zeke.\"\n\n\"Well here's the thing, Zeke. Tramack of the Western is hunting me and has declared me his intended. There's a bounty on my head, and he means to collect me. This Lanarre's human guardians were slaughtered by a rogue male that I should have killed. She was traveling to...\" Tessa looks at Tahlia, wondering how much she should say.\n\nTahlia nods. \"Go ahead. It is fine that anyone knows.\"\n\n\"Tahlia is traveling to mate her chosen.\"\n\nBoth males look at Tahlia. \"She doesn't look old enough to mate.\"\n\nTahlia kicks up her chin. \"I am of age.\"\n\nSlash snorts in the background, and Tahlia gives him her best dirty look, which Tessa thinks is quite good.\n\nZeke thumbs his chin thoughtfully, running the digit back and forth across the downy bright-red fur. He's handsome.\n\nHe's also Red. Tessa's running for her life without a plan.\n\nI don't need a male.\n\nWhen she looks up, his thoughtful glance has narrowed to her face. His nostrils flare once, and he smirks.\n\nThe insufferable pig. Tessa fumes, thinking he might have guessed her mild interest as she fights to behave casually.\n\nSlash spreads his arms away from his body. \"We can offer you temporary shelter and protection until you figure out what you want to do.\"\n\nTahlia looks her age as she rolls her lip between her teeth, indecision painted on every plane of her face. \"The Lanarre will look for me.\"\n\nHer eyes slide to Slash then land accusingly on Zeke.\n\nSlash's brows draw together. \"And you will be under our protection.\"\n\n\"You are a stubborn female,\" Zeke says to Tahlia.\n\n\"You have no idea,\" Tessa mutters, thinking about their brief acquaintance.\n\nTahlia frowns at her.\n\n\"It's true!\" Tessa defends.\n\nInstead of answering, Tahlia leads the way, heading in the direction from where the Were popped up.\n\nTahlia gives Tessa the barest smile as she walks by, as if she holds a secret.\n\n# CHAPTER TEN\n\nJulia\n\nJulia has been dirty before. But after being bit, beaten, and bruised, a shower has never felt so good. She dries off and carefully combs through hair, which hasn't seen a brush in a couple of days. Julia tosses on a T, jeans, and shoes that don't aggravate her feet, which are still healing from the cross-country trek.\n\nThe few hours since they arrived with the Region Two Singers were not easy ones.\n\nThe Were are complaining that the grounds feel like a graveyard. Their acute sense of smell picks up every single thing. Every death. Every wound.\n\nThat stirs an idea. Julia can't discern one dead body from another. But it's critical that she know if Victor is dead. And what of Reagan and Delilah?\n\nThen there's Tharell. He claimed he was instructed to deceive the Singers\u2014and ultimately betray William\u2014and, ruled by the black blood of the demonic, he had no other choice. He broke his pact with the Northwestern coven by not delivering Julia to them.\n\nGabriel, Julia's mind whispers, then on the heels of that name, Julia remembers Claire. A longing for Claire swamps her, as does a yearning for William. He was deeply self-contained. As it turns out, he was her ultimate protector.\n\nJulia is sitting by herself in the room she's occupied since the first day she arrived at Region One. The cracked doorknob and hole in the wall stand in testimony to tempers\u2014attacks.\n\nNow everything's different. Jen, Michael, and Brendan are gone.\n\nScott.\n\nA silent tear files its slow way over her still-warm skin from her shower. It plops on top of her tightly knitted fingers.\n\nShe'll never hear a wise-ass Michael belittle someone with his scathing sarcasm in between lollipop licks. Julia won't listen to the sibling fights Jen adored stirring up.\n\nBrendan won't be making any more manure piles or phantom holes for the stray vamp to fall into.\n\nThey were sort of a family; the only one she had. When all hope was lost and she thought Jason was dead, they were there for her in a way only Jason and Cyn had been.\n\nJulia hiccups, and a sob pops out like a bubble of sadness. It bursts in the silent room, filling the space with her loss.\n\nNow her family is Jason, Cyn, and Jacqueline\u2014of all people.\n\nJulia slowly raises her head and plops her chin in her hand. There is her hidden sister in Alaska. But for now, the people who remain\u2014the Singers\u2014will need her.\n\nJulia stands at the edge of the bed. After wetting a cold washcloth and blotting her tear-streaked face, she makes her way downstairs.\n\nThe kitchen is filled with women and a few industrious guys, making great-smelling food. Dishes being clanked and set out for people are the noises of comfort and gathering.\n\nSaliva pools inside her mouth, and Julia realizes she hasn't eaten in twenty-four hours.\n\nJacqueline sits at the table, one hand on her belly, her chin perched on her fist.\n\nCyn stares at Jacqueline with clear suspicion.\n\nJulia smiles. Some things remain the same.\n\n\"Jules!\" Cyn cries, running to her. Julia has seconds to see that Cyn has somehow styled her hair, and is wearing cute clothes before she hurls herself in Julia's arms. They dance in a circle and finally Cyn releases her.\n\n\"So happy you're finally done wallowing in the shower. I thought you were setting up camp. Is there any hot water left?\"\n\nJulia blinks.\n\nCyn frowns.\n\n\"Hello. Maybe you need more sleep?\"\n\n\"On-demand hot water heater,\" Jason says, walking into the kitchen and pressing a light kiss on her forehead. Julia glances up, grateful for his presence. He squeezes her shoulder, and Julia still feels as if she's in some kind of shock-induced fog. Julia's hand comes to rest on her stomach, where she was stabbed.\n\n\"Food time,\" Jason says, striking his palm against his washboard abs. He jerks open the fridge door, hangs on the top, and juts his face forward like a pecking hen.\n\nJulia walks over there and pulls the fridge door out of his grasp. She shuts it, opens it, then shuts it again.\n\nJason's brows come together, and he retreats a step. \"Babe, what are ya doing?\"\n\nJulia sucks in her lower lip. \"Michael said the key to finding food in the fridge was to look three times.\"\n\nThe room falls silent.\n\nJulia bursts into tears.\n\n\"Come here, babe,\" Jason pulls her into his arms and she sobs against his broad shoulder.\n\nAgain.\n\n\"Let 'em go, baby. Let 'em go.\"\n\nJulia sniffs, wiping her tears against his hard chest. \"Sorry,\" she says, shaking her head, her damp hair making him wet where her tears don't. \"I'm having a hard time still.\"\n\n\"That's okay. It'll take time.\"\n\nA big commotion of voices burst all around them, and Julia looks up.\n\nBeaten and torn, Scott staggers into the kitchen.\n\nWithout thinking, Julia runs to Scott. He gives her a weary smile.\n\nHis lips are cut, one eye is swollen shut, and a deep open wound bisects the other eyebrow.\n\n\"Julia,\" he croaks, and she wraps her arms around his waist.\n\n\"Ah!\" she cries as they begin to topple like a clumsy, half-cut tree, and Julia stumbles under his weight.\n\n\"Come on, Hulk. Don't crush the queen, pal.\" Jason puts a hand underneath his arm and scoops the larger man to an upright position.\n\nCyn walks slowly toward them. Her eyes meet Julia's, and she gives a small shake of her head.\n\nJulia looks down and sees Scott's femur gleaming like a fanged tooth hanging from his upper thigh.\n\n\"Heal him,\" Jacqueline says from behind Cyn.\n\nCyn turns, hands on hips. \"You're still bossy. And y'know? I think I wouldn't be if I were in your position. Like I wouldn't dig in and get it figured out and stuff.\"\n\nJacqueline just stares.\n\n\"Gah!\" Cyn says. \"Fine, but this is going to be a hold-him-down moment.\"\n\nJason guides a limping Scott to the flowered fainting couch in the front parlor and carefully lays him down. Scott's skin is chalky with a green cast.\n\nJulia moves to his side, drops to her knees, and grabs his hand. He winces.\n\nShe looks down and sees he's completely missing two fingernails.\n\n\"Oh, my God, Scott!\" Julia cries, covering her mouth with the hand that's not holding his. \"What did they do to you?\"\n\nScott licks his dry lips. One beautiful, dark eye rolls to meet Julia's. \"Less than they did to Lucius.\"\n\nJulia's shoulders shake with her effort to be strong. This is what a leader has to deal with, these cold facts. But more tears come, collecting at her jaw and dampening the thin long-sleeved T-shirt she's wearing.\n\n\"Where is Lucius?\" Angela asks quietly as she steps up behind Julia.\n\nShe didn't hear.\n\nScott's gaze meets Angela's over Julia's shoulder. He closes his eyes, and Angela cries out, rushing from the room.\n\n\"Okay, boys, hold stud-boy down while I set this break.\"\n\nJulia's eyes hold Cyn's. \"Are you\u2014do you know what you're doing?\"\n\nShe smiles, shaking her head. \"Hell, no. But my hands do.\"\n\nThat'll have to be good enough.\n\nCyn's expression goes serious. \"Take a hike, Jules. You're not gonna like the noise he makes.\"\n\n\"It's okay, Julia,\" Scott says.\n\nJulia leans forward to kiss his forehead, but can't find an uninjured area.\n\nScott squeezes her hand, trying to comfort her.\n\nShe covers her ears when Scott begins to scream.\n\nJulia doesn't leave or look away from his uninjured eye.\n\nHis screams fall blissfully silent when he passes out from the pain.\n\n*\n\nPraile\n\nPraile's wounds weep and fester underneath the ill-fitting human clothes. Further, he must expend an inordinate amount of energy to maintain some form of camouflage.\n\nHe must expend more precious energy than Lazarus, who has only a tail and a minor bit of skin cover to effect. His eyes, teeth, and even his nails fall within acceptable appearance for a human male. How Lazarus manages to look so undemonic is a mystery. Genes\u2014always a crapshoot, as the humans say.\n\n\"Hide the bodies,\" Praile commands the two low demons who accompanied him and Lazarus. Hardly more than drones, they can take only one form.\n\nPraile has chosen homeless men. It is a little bit of an inside joke, but he must take the small doses of humor when they present themselves. They're like medicine, especially of late.\n\nIf Praile uses his ability to see things through his human eyes, he sees how the demonics would appear to humans.\n\nLazarus will appear handsome.\n\nPraile grunts as the low demons drag the old couple out of their respective car seats and into the woods.\n\nAn age-old trick. Well, not entirely. The tactic is as old as cars, and those have been in existence for just over a hundred years. However, it's been very handy to lie in the center of the road and appear helpless.\n\nThat had been Lazarus's job. Praile was unwilling to re-open wounds that were healing badly.\n\nHe is ecstatic the Master cannot access his thoughts. If he could, Praile would be dead twice over. Everything he has thought since the thirteenth lash has been of the most evil and vile variety.\n\nHis thoughts have been especially uncharitable toward the Master.\n\nLazarus says nothing, cradling his hand, which he broke while stopping the car that last inch.\n\n\"That'll set wrong,\" Praile says, stating the obvious.\n\n\"Yes,\" Lazarus reluctantly agrees through his teeth.\n\nPraile doesn't smile but marginally contains how pleased he is to see the stoic Lazarus feel pain. After all, he is not healing Praile fully. Praile doesn't care that the Master has tasked Lazarus with doing a partial healing\u2014he still blames his second.\n\nThe two low demons return to the soft shoulder, hunched and mindless as the bees he thought of earlier.\n\n\"Good,\" Praile says. \"Get in the back.\"\n\nThe two slouch inside the back of the car. Lazarus slides behind the steering wheel and just sits there.\n\n\"What are you doing?\" Praile bites out.\n\nThen he spies a sliver of bone that has punched through the inside and lower part of Lazarus's wrist. Though demonics are brutally strong, the car was going around fifty miles per hour.\n\n\"I can set it,\" Lazarus says.\n\nPraile grins then winces as his back touches the seat. He jerks upright, glancing at Lazarus. His face is expressionless, as usual. In fact, Praile doesn't find proof of pain except for a certain tightness about his eyes.\n\n\"But you can't heal the injury?\"\n\n\"No.\"\n\nPraile know of no demonic or healer who can heal themselves. They heal only others.\n\n\"Too bad,\" Praile sings falsely, smoothing his hands down the stiff denim of his jeans. He lets his form go while he's hidden in the car, and a sigh escapes him.\n\nLazarus puts the car in gear with his good hand, and makes his way toward the region where Praile has been told the Angelic Blood has gathered. The High One will be his in the next day\u2014or lashes will be the least of his concern.\n\n# CHAPTER ELEVEN\n\nSlash\n\nSlash is ready.\n\nReady to be done. Ready to see Adrianna. He doesn't want two antagonistic females to babysit. He wants one female forever.\n\nSlash is usually more focused on the task at hand. When openings for security detail came up, he and Zeke were the first to volunteer.\n\nThen the impossible happened. A Lanarre female stumbled into the equation.\n\nSlash's eyes move to Tessa, the older of the two. Her thick black hair is plaited in a single braid down her broad back. It's easy to see she's female, but there's not a soft spot anywhere on her athletic frame. Her wide gray eyes are framed by sooty lashes that match her hair. A dusky complexion contrasts to a smattering of freckles over her straight nose.\n\nThe freckles reminds him of Adrianna, and his heart beats harder.\n\nAs is the norm, both females have a healthy dose of prejudice against him and Zeke.\n\nThat makes sense, because nothing good circulates about the Reds. Alaska is overrun with Reds, and they don't behave like a proper den. Instead, they run in immature, un-led packs, selecting mates indiscriminately. They've given all Reds everywhere a bad name.\n\nZeke keeps a watchful eye on the young female, as he should. She's shown a prowess for self-defense atypical of female Weres, though Alpha females are known for their contentious streak.\n\nSlash grins.\n\nTahlia notices and asks what's so funny.\n\n\"Nothing is funny. I'm thinking about how interesting it will be for you to meet Adrianna.\"\n\nHer faces screws up into a frown.\n\nZeke covers a smile with a cough and Tessa, appearing more jaded than all of them, looks between the two males and narrows her eyes.\n\n\"I know you're not going to hurt us,\" she begins, and Zeke's expressive face gives him away.\n\nTessa stops hiking and crosses her arms. \"Listen, I already told you. Two females are alone, then two wolfen males pop up like jack-in-the-Weres and don't let us know if they're friends...\"\n\n\"We obviously assume you are foes,\" Tahlia says with a touch of arrogance in her voice.\n\nZeke glowers but says, \"Obviously. Hated the talon swipe. Hurts like a bitch. Heals like hell.\"\n\nSlash chuckles. Zeke rolls with circumstances\u2014a critical component to surviving in their supernatural reality.\n\n\"Anyway,\" Tessa drawls, \"that doesn't mean I don't need any background about where we'll be taking temporary shelter. What is expected of us?\"\n\nAnother full look passes between him and Zeke.\n\n\"Okay, who stepped on your puppy?\" Tessa asks.\n\nTahlia looks around.\n\n\"I don't see any dogs.\" Then she smirks, giving a significant look at the two males in wolfen form.\n\n\"It's an expression,\" Tessa says with a proper amount of impatience.\n\nZeke scowls at Tahlia. \"You speak your opinion without any fear of reprisal.\"\n\nTahlia clasps her hands behind her back. \"And you can speak. I was beginning to wonder.\"\n\nTheir stares clash, and Zeke marches over to Tahlia. She tilts her face upward.\n\nAn expression of panic washes over Tessa's face.\n\nTahlia's either stupid or brave.\n\nOr both.\n\nZeke is a foot and a half taller than Tahlia is. He looms over her, and with a low growl, he moves as though to speak.\n\nTahlia tips her head back, exposing the smooth skin of her throat to Zeke. \"You would harm me because of my sharp tongue?\" she whispers.\n\nZeke sways forward, as if he's in a dreamlike state. \"Not harm\u2014no.\"\n\nHe grips her shoulders and sinks his short snout into the crook of her neck, letting out a low growl.\n\nThe posturing has gone on long enough.\n\n\"Do something,\" Tessa says, and Slash sprints the short distance to the pair.\n\nZeke's snout comes away from her neck, and she sinks against him, sliding her arms around his waist.\n\nFuck.\n\nHis eyes spin like emerald fire at Slash, but the revolutions are too fast to track. \"Keep your distance.\"\n\nSlash tenses. \"Let the Lanarre go.\"\n\nHer defiant gaze moves to Slash. \"I do not wish to.\"\n\n\"Tahlia,\" Tessa warns, \"this is dangerous.\"\n\nZeke presses his snout against Tahlia's hair, scenting deeply of her.\n\n\"She nearly decapitated you,\" Slash says.\n\nZeke ignores him.\n\nSlash exhales in disgust. \"You're drunk from her scent. Step away.\"\n\nZeke growls in challenge. Slash has no desire to injure or kill his newest second.\n\nHe gives the Lanarre the contemptuous look she deserves.\n\nTahlia sniffs at him. \"Do not look down on me, Red.\"\n\nSlash folds his arms as his second rolls in her scent. Zeke's snout disappears underneath her hair, and he only comes up for air to press his nose in another unscented spot.\n\n\"He is a good Were. Do not charm him.\"\n\nTahlia smiles. \"Shall I charm you?\"\n\n\"Won't work, whelp. You're too young, I'm too pure, and I have a female.\"\n\nTahlia pouts, and the expression makes her look even younger than she is. She plays childish games with dangerous players.\n\nGrudgingly, she loosens her hold on Zeke and steps away. He tries to follow.\n\n\"Zeke!\" Slash says in a sharp voice.\n\nThe Were's head jerks. He shakes it as though he were asleep. In a way, he was.\n\nThe Lanarre placed at the pinnacle of their species is a danger. With other lesser Were, they are treacherous. Zeke's mixed heritage make him vulnerable to her compulsions.\n\nZeke gazes at Tahlia. Slash watches his expression change from one of dumb thrall, as he's seen a human look at a vamp, to one of anger. \"That is not right.\"\n\nTahlia shrugs. \"It's important you know with whom you deal.\"\n\n\"Where is the selfless girl that saved my life?\" Tessa asks with a touch of sadness in her voice.\n\nTahlia's expression hardens. \"I will not be ruled by males. Or anyone. My chosen, Drek\u2014he is the only male who stands over me. I am Lanarre.\"\n\nTessa makes a sound of distaste. \"Well, good for you. I am a rogue Alpha who's been followed for two decades by a packmaster who doesn't care how I feel\u2014that I think independently at all.\" Her eyes bore into Tahlia's. \"But I'm not going to do a catnip routine on all males because of one male.\"\n\nAmen to that. Slash enjoys Tessa's spirit. \"Catnip?\"\n\nZeke chuckles. \"I like you,\" he says to Tessa.\n\n\"Thanks, but don't get any ideas.\"\n\nSlash laughs and turns to walk back toward Region One.\n\n\"How much longer?\" Tahlia asks after an awkward silent ten minutes of marching through the woods.\n\nVery young.\n\n\"We'll be there when we get there.\"\n\nWhelps.\n\n*\n\nJulia\n\nShe spots them first, which is amazing, considering all the others in the group have eyesight better than hers.\n\nJulia jogs out to meet Slash and Zeke.\n\nShe takes in the two women who they're walking with. Julia doesn't miss the healing talon marks at Zeke's throat, either.\n\nJason is there before she can open her mouth, his arm straying in front of her. \"Whoa, babe. Let's see what's what.\"\n\nOf course, he's right, but Julia's emotional upkeep is taking its toll. She doesn't want to follow rules or keep how she feels silent. Scott has returned.\n\nHer people are dead.\n\nJulia badly wants to be in charge of her destiny. And she will be if it's the last thing she does.\n\n\"Don't baby me, Jason,\" Julia says, scooting around his arm.\n\nHe sighs.\n\n\"Uh-huh, 'cause you're not a danger magnet or some such shit?\"\n\nJulia ignores him, and Slash, who's normally quiet, pipes up, \"These are some stray Were we picked up on our perimeter sweep.\" He jerks a thumb at the young women behind him.\n\nJulia stares openly.\n\nA couple of years ago, she would have made a stab at being polite. Three years in the supernatural world has all but scrubbed away the human niceties.\n\nThe younger one is striking. Her large wide-set eyes have the barest almond shape and an almost violet hue. Dark ringlets spiral out and away from her face, overwhelming her slim body. Her nose is straight; her lips, full.\n\nThey look fuller for the pouting.\n\nIt makes her look about twelve, though Julia is pretty good at guessing age, and she thinks the girl is closer to twenty. But if she's Were, that could be way off. Julia's gaze latches on to the older woman. Her long hair is as dark as night, and her startling gray eyes are not light as Victor's were. They are the color of a coming storm.\n\nShe meets Julia's stare unblinkingly.\n\nAlpha for sure.\n\n\"I'm Julia,\" she says with a smile.\n\n\"Tessa,\" the young woman replies.\n\nThe younger introduces herself as Tahlia.\n\nShe has some kind of accent, which Julia can't pinpoint.\n\n\"I'm sure that Slash and Zeke didn't bring you back here because everything's all hunky-dory.\"\n\nTessa grins suddenly, and the expression softens her face. \"No, you've got that right. But I'll be honest.\"\n\nWell, thank heaven for that.\n\n\"I don't want to bring what I've got chasing me down on you guys.\" Her worried eyes fall on Julia.\n\nJulia frowns. Nope, Region One sure doesn't need any more bad crap. But she won't turn away defenseless women.\n\nHer eyes stray to Zeke's neck. Maybe not so defenseless.\n\nJason's strong hands move to Julia's shoulders. \"Up to you.\"\n\nIsn't it always?\n\nTessa's expression is neutral and Julia can't read Tahlia's.\n\n\"You're welcome to stay here if you don't mind a little work.\" Julia's mouth twists. \"Or a lot.\"\n\nTessa's shoulders drop, and she lets out a sigh of obvious relief. Julia turns to Tahlia, who looks untroubled. Her deep poise is weird for somebody so young, and Julia is very interested to hear why a young female Were is away from her den. It makes zero sense. Julia remembers how closely they guarded her when she was at the Northwestern den.\n\nJulia steps forward and shakes hands with the women. The constant chattering white noise inside her brain intensifies unmercifully for a moment then subsides. She takes a shaky breath and smiles, despite the ESP distraction.\n\nBeing a telepath can be a liability. And when it comes to the Were, it's like bad noise that fills her head and makes it hurt.\n\n\"You okay?\" Jason asks in a low voice, and Tessa's ears perk.\n\nJulia nods, rubbing her temples. \"Yeah.\" She shakes off the disquieting telepathic current. \"I bet you guys are hungry?\" she asks, taking the focus off herself.\n\n\"Starved,\" Tessa admits with a grateful smile.\n\n\"I could eat,\" Tahlia says.\n\n\"Great, follow me.\" Julia turns around and without waiting for anyone, makes her way back to the kitchen.\n\n*\n\nThe guilt is there, but Julia ignores it. She wants to see Scott.\n\nShe's pulled to him like a moth to a flame. Though the soul-meld was wiped out in faerie, Julia wonders if something still remains.\n\nIt's possible the tie is the reason why she never really grieved when Tom Harriet took him. Julia must've known, deep down, that he was still alive.\n\nAnd she's glad. Julia's married to Jason, but she's happy Scott's alive and that he's here.\n\nHer feelings are still kind of a mess.\n\nJulia gets the women settled and moves quietly through the mansion without a guide, until she reaches a closed door on the second floor. She's never been inside this room before. The house is over seven thousand square feet, probably closer to eight. There are two dozen rooms, nine bedrooms and six bathrooms.\n\nScott could be anywhere.\n\nShe lays the flat of her palm on the polished wood door. Her every sense has come alive and they're raw.\n\n\"Come in, Julia,\" Scott says.\n\nShe sighs and turns the crystal knob, swinging the door wide.\n\nScott's propped up on pillows.\n\nJulia doesn't speak to the weirdness between them. Instead, she asks about something safe. \"How's the leg?\"\n\nScott shrugs. \"It'll be good soon.\"\n\nJulia looks at her feet. \"How soon?\" When she looks up, Scott's right in front of her. In. Front. Of. Her.\n\nJulia jerks her chin back. \"What\u2014what are you doing? How'd you move that fast?\"\n\nScott doesn't say anything. His hand moves to her shoulder and floats down to her wrist, then his fingers twine with hers.\n\nThe breath she's been holding slides out in defeat. She forgot Scott is so tall, so big\u2014and so terribly dangerous.\n\nBut he's not dangerous to her, never to her.\n\nHis injured eye has healed enough to open, and his dark gaze finds hers. His hand moves to her jaw, and he feathers his thumb along its edge.\n\n\"Did you believe what Tharell told you?\"\n\nShe didn't expect the question. Julia shakes her head because she can't think. She's numb with what his touch means. Inert like unshaped clay.\n\nThis can't be happening. Julia was so sure she had it figured out. That her path was set.\n\nScott grasps her chin, moving it gently so their eyes lock. \"That a soul-meld could be thrown away because we were in a faerie mound?\" His voice is rife with disbelief. \"What in the hell did he think would happen once we left?\" Scott's brows come together at the apparent obviousness of it all.\n\n\"I guess...\" Julia tries to retreat from his embrace, and it's like stepping out of warm bathwater into a cold bathroom.\n\nI don't want to go.\n\n\"I assumed...\"\n\nScott's eyebrows jam tighter. \"Uh-huh. You know what they say about assuming.\"\n\nJulia makes an inarticulate noise as he buries his fingers in her hair. He makes a fist, twisting the tendrils, creating a sensation just shy of true pain.\n\n\"It makes an ass,\" he whispers against her temple, \"out of you...\"\n\nHe kisses her there, and she whimpers, \"And me.\"\n\nThen his lips are on hers, and Julia forgets she's married to another man or that Scott's been tortured by an insane Were.\n\nThe wet heat of his kiss is all that exists in the entire universe. And the soul-meld locks into place once more, as if it never left.\n\nJulia should mourn its return.\n\nBut when the soul's other piece has found its mate, there is no grief\u2014only joy.\n\n# CHAPTER TWELVE\n\nPraile\n\nLazarus pulls up behind a vintage 1960s vehicle and parks.\n\nGood era, Praile remembers. So much simpler to sway the innocent then. Now humanity is so cynical, he can hardly take pleasure in showing the masses the path to Hades. Happily, many are already on a one-way course.\n\nLazarus inhales deeply. He rolled down the driver's window as they slowed. \"Were. Female.\" He closes his eyes. \"Two.\"\n\nPraile frowns. Those are not the words he was hoping to hear. Praile swings the car door open and winces as the material of his shirt tears away from his wounds\n\nFucking lashes.\n\nGravel crunches under his stiff running shoes as he walks around to the passenger-side door. The car has fins near the trunk and has been well-preserved.\n\nCuriously, it's abandoned.\n\nPraile does not believe in coincidence of any kind. He pops open the door and peers inside.\n\n\"What's happened here?\" he fires at Lazarus.\n\nHis bright-blue gaze pins Praile from across the seat as he leans in opposite him. Praile has always known of Lazarus's hate for him, but he is oh-so-careful about letting it manifest visibly.\n\n\"Give me a moment, and I will try to discern what's happened.\"\n\nThe thread of irritation buried in that neutral voice pleases Praile.\n\n\"I wish to determine where the High One's hidey-hole is.\"\n\nLazarus gives a mild exhale of utter irritation.\n\n\"Problem, Lazarus?\"\n\nHis eyes shift away, and Praile knows the movement makes it more difficult for Praile to read emotion. Clever demon.\n\n\"No.\"\n\nLiar. Praile's nostrils flare, smelling the untruth.\n\nLazarus holds up a palm. \"I am aware of what the end game is, Praile.\"\n\n\"Good. Do not lie to me again.\"\n\nLazarus says nothing. His eyes close, and he inhales in short chuffs. Finally, his eyelids sweep open, icy-blue irises blazing.\n\n\"One female is a Lanarre.\"\n\nPraile whistles, delighted at the revelation. Then his eyebrows drop. He palms his chin with undisguised talons. \"What is Lycan royalty doing cavorting about?\"\n\nLazarus shrugs. \"It's not important, really. What is\u2014is the women marked their territory and left this fine stolen vehicle behind.\"\n\nPraile swivels his head to Lazarus in a hard glance of interest. \"Really?\"\n\nLazarus nods. \"Really. And two Red Weres accompany them.\"\n\n\"Fantastic,\" Praile breathes out reverently. \"This is wonderful news.\"\n\nLazarus's lips lift.\n\nPraile waves his palm around. \"You know what I mean.\"\n\n\"I do.\" Lazarus frowns suddenly. \"Why are you not maintaining your form?\"\n\nIt's not as easy for me, fool.\n\nOf course, Lazarus is no fool and gives him a knowing look.\n\nPraile shrugs. \"I will slip my human shield on when we near Region One. I only need to employ it for a short time. The Angelic who remain will not know of our deceit if we're not in our true from.\"\n\nIn battle, the demonic must be in their true form. In all other things, the parody of humanity is perfect.\n\nPraile sighs. \"To know that the High One and the blood babe are finally here after all this time.\" Praile's gaze spears Lazarus and he gives a smug nod. \"The world is our oyster, Lazarus.\"\n\nLazarus is silent. And that silence suits Praile. He mainly enjoys listening to himself. He needs no other audience.\n\nThey lock both cars and walk into the woods. Praile knows everything will soon be within reach, even the swollen promises made by the Master.\n\n*\n\nJulia\n\n\"Scott!\" Julia shoves him away, freaking.\n\nHe's been tortured and has a broken leg.\n\nShe's all sorts of miserable.\n\nHe laughs.\n\n\"Oh, my God.\" Her voice trembles, and she puts her shaking hand over her scorched lips. \"So not funny.\"\n\nScott sobers. \"Sorry. I guess I'm just a little fucking giddy about surviving Harriet's treatment, escaping and coming back here to you.\"\n\nHis hand finds her nape and applies a tender squeeze.\n\nJulia moans. \"This is so bad. You can't be happy to come back to me. There is no us.\"\n\n\"No,\" he whispers. \"It's so good\u2014right.\"\n\nHe leans down and peppers kisses on her forehead. His hot lips move to each eyelid, and she feels his eyelashes brush against her own.\n\n\"And my asshole brothers. I was actually beginning to miss those guys,\" Scott says softly with a smile in his voice.\n\nIt's cold water on all Julia's senses.\n\nScott doesn't know about his father. Brendan, Jen\u2014Michael.\n\nOh no.\n\nScott watches emotions run across her face, fleeing for safety from his perceptive gaze.\n\nHis fingers tighten on her shoulders, sensing her morbidity. \"What the hell has you looking at me like that?\" Dark eyes pull at her, and Julia's drowning in all that deep brown.\n\nLightheadedness swims close.\n\n\"Julia, I'm sorry.\" Scott drags her over to the bed and gently lays her down.\n\nShe notices his slight limp and smells the soap he's used since his return.\n\nWhat's wrong with me?\n\nHe takes both her hands. \"Now tell me what's happening?\"\n\n\"You noticed we don't have many people?\"\n\n\"Ah, no. I noticed we have an assload of Region Two Singers. And let's face it. Observation skills were on the down low.\" His gaze moves over Julia's face, coming to rest on her eyes.\n\nScott stands up, realization making swift work of his face. \"Dad?\"\n\nJulia doesn't look away.\n\nShe counts it as the hardest thing she's ever done. \"I don't have a good way to tell you.\"\n\nHis expression morphs to granite, and he touches the top of her head lightly to take the sting out of his words. \"Spit it out.\"\n\n\"He's gone, Scott. Tony killed him.\"\n\n\"Tony? The apeshit Were that tortured my\u2014Jacqueline?\"\n\nJulia nods miserably. \"Yeah,\" she answers softly.\n\nScott slowly lowers himself to the bed and puts his face in his hands. A full minute pounds by silently.\n\nHe rolls his face in his hands to look at her. The bruises are already fading, but they're gruesome splashes of dying yellow on his skin. \"I'm not ready for this shit.\"\n\nJulia sits up, feels like puking, swallows, and plows forward. \"What shit?\"\n\nScott gives a wan smile. \"If he's gone, I am the head of Region One. Only royal blood can rule. And\u2014\" He clears his throat. \"He will be missed. I can't believe he's gone.\"\n\nThey stare at each other.\n\nScott's face changes as the wheels of his fine mind turn, no doubt thinking about what it would mean if Marcus were gone. She sees when he realizes her omission, and that his siblings are absent from the mansion.\n\n\"Don't tell me, Julia.\"\n\nJulia's tears don't even burn to warn her. Like escaped convicts, they run down her face, away from the prison of her eyes, then conspire together at her collarbone.\n\nShe cries for them both, and it's still not enough.\n\n\"What happened? What the fuck happened to my family?\" He cups his large hand at the back of her head, keeping a grip on her nape.\n\nJulia begins speaking.\n\nWhen she's done, there's a void in her that wasn't there before.\n\nJulia recognizes it for what it is\u2014that part of Scott that she owns, just as he owns a piece of her.\n\nHe's empty, and now, so is she.\n\nSo empty.\n\n*\n\n\"Do you want something to eat?\" Julia finally asks.\n\nScott shakes his head.\n\nJulia puts a hand over her stomach. It still hurts. She looks at Scott.\n\n\"I feel your hunger under all this.\" Julia waves her hand around, symbolically encompassing all the weighted grief that's been aired between them.\n\n\"Maybe,\" Scott replies ruefully. \"But who cares about the hole in your gut when the one in your heart's twice as big? Who gives a fat fuck?\" he yells.\n\nJulia yelps in surprise, scooting back from him.\n\nScott picks up the nearest thing and hurls it into the wall. Then he throws another. Glass shatters and flies. Shards embed themselves into whatever they can.\n\nOne spears Julia's palm as she hides behind her hands.\n\nShe hears a gasp but doesn't move. Julia rides out his justified rage in a safe spot against the headboard of the bed.\n\n\"Oh, Julia, I'm sorry,\" he says, plucking out the glass.\n\nA teardrop of blood wells from her palm.\n\nHis eyes are bright with his sorrow. His cheekbones flame with the blood pumping so freely with the river of his anger.\n\n\"It's okay,\" she whispers.\n\n\"No, I\u2014let me heal this.\"\n\nScott breathes over her hand and the blood stops flowing. He kisses the center of the wound and it begins to close.\n\n\"Oh, wow... wow.\"\n\n\"It's a bennie,\" Scott says.\n\n\"It's real, isn't it?\" she asks, scared to look at his eyes for what she'll see there. She looks up anyway.\n\nHe nods. \"No one can heal you beside a Singer Healer.\"\n\nShe swallows painfully. \"And... my soul-meld.\"\n\nSomething occurs to Julia, something besides the new mess she currently finds herself in.\n\n\"You just made a bunch of noise.\"\n\nHe blows out an exhale, ripping a hand over his short hair, clearly not getting the relevance. \"Yup.\"\n\n\"Nobody came.\"\n\nScott cocks his head. \"You're right.\"\n\nHe stands, towing Julia with him. \"You said Tony\"\u2014his chin dips, and Julia tries to notice everything but the standing water in his eyes\u2014\"killed everyone?\"\n\nJulia nods. \"Well, there were a handful of Region One Singers, but...\" She spreads her hands away from her body. \"I'm sorry,\" she barely gets out.\n\nScott's brows knit. \"I don't understand why no one hid in the bunker?\" He shakes his head.\n\nPain flares in Julia's chest. Instant and sharp, it pierces her.\n\nScott grabs her. \"What? What is it?\"\n\n\"What bunker?\" she whispers urgently.\n\nScott blinks slowly. \"The bunker that stays vacuum-locked for seventy-two hours after entry. No one gets in. No one gets out. Period.\"\n\n\"Julia!\" Jason yells, rushing into the room.\n\nShe turns with a guilty jump, and his eyes travel from her to Scott. They narrow, missing nothing and seeing stuff she can't explain. She doesn't want to.\n\nScott drops Julia's hands, and she's grateful, even though she feels as if they've been amputated without his touch.\n\nOh, God.\n\n\"What\u2014\" She clears her throat, barely able to meet the eyes of the man she loves\u2014or thought she loved. \"What is it?\"\n\nJason walks to her and takes the hands Scott just dropped. \"There's more survivors.\"\n\nScott says nothing. His silence speaks for him.\n\nJason flicks a glance Scott's way and Julia notes the chill in that hazel glance.\n\n\"Scott knows.\"\n\n\"Right. Well, his sister and Michael are alive. And Victor too. They were stowed away in some nuclear shelter thing.\"\n\nScott's instant grin is contagious.\n\nBefore she knows it, Julia can't wipe its twin off her face.\n\nJason throws an arm around Julia's shoulders. \"They're prepared. I'll give them that.\" His hand flips up, and his fingertips curl around her shoulder. \"Victor says he gathered the royalty together, and as many women and children that he could. They've been down there three days.\"\n\nJulia makes a face thinking about that\u2014but they're alive.\n\nJason laughs. \"Don't get all grossed out, Jules. They had a bathroom, food, running water. It's a damn underground Hilton.\"\n\nScott says, \"I wouldn't go that far. It has only the supplies needed for the seventy-two-hour time frame and no more.\"\n\n\"Still!\" Jason swings his hands up, piercing Julia with his hazel gaze. \"Great news, huh?\" He grabs her neck and pulls her against him, pressing a gentle kiss against her forehead, right over the crescent-shaped scar.\n\nIt is great news.\n\nSo why do I feel so sad?\n\nJason keeps one of her hands and tows her out of Scott's bedroom.\n\nJulia glances over her shoulder. Scott's lips are in a flat hard line; his eyes are fixed on their joined hands.\n\n# CHAPTER THIRTEEN\n\nSlash\n\nSlash is vaguely pessimistic by nature, but he would admit that seeing those Singers, twenty in all, climb out of a trapdoor in a well-hidden spot under the mansion lifted his sagging spirits. With so many Singers gone, there was no one left to alert them to the survival quarters. Scott returned, and the surprise existence of additional survivors brought levity to a grim climate.\n\nAnthony Laurent moved through the Region One Singers, using his demonic saber like a knife through butter. Tony struck down Victor, but the Were was apparently in a rush\u2014he missed Victor's carotid artery by a fraction. Victor laid in a pool of his blood while the Were kicked him in the balls.\n\nThinking about Tony makes Slash's blood boil. He would kill him again if the chance was served up.\n\nFortunately, he's gone, and Victor has mighty recuperative powers.\n\nSlash had barely managed to heal himself enough to function when he saw what Tony was doing to the others. Victor cut his losses.\n\nCold but pragmatic, Victor headed straight to where he assumed the most important Singers would congregate. He was able to save Jen and Michael.\n\nSlash scans the faces of those who survived. Everyone is somber, as they should be.\n\nThat's how Slash would feel if he'd listened to the screams, begging, and pleas for mercy while Tony rained death down upon their heads.\n\n\"This is so sad,\" Adrianna says and he takes her hand in his.\n\nShe squeezes Slash's fingers. Unfamiliar heat blooms in his chest where an emptiness was before. It hurts, though it feels right.\n\nSlash won't discuss their evolving relationship. He can't. It'll make the hope into solid reality by speaking of it. Instead, he discusses more neutral matters. \"It's better that they're alive. More Singers survived than we presumed, important ones.\"\n\nHe studies Julia as she greets everyone, and Slash inhales sharply. Something is off with her scent\u2014he can't place it. He takes in the survivors, and his eyes come to rest on Jason, who was formally feral. He frowns.\n\nFinally, his attention shifts to Scott as he scoops his sister into a tight embrace. Slash smells healing injuries. But they're faded, scenting of old wounds, though Slash knows they're not. A Combatant can heal almost as quickly as a Were. Good thing for him.\n\nAs if Scott intuits Slash's thoughts, Scott gives a chin lift over Jen's shoulder, meeting Slash's eyes. He bares his teeth slightly, sucking in a few quick chuffs. He scents something that makes his eyes snap to Julia.\n\nTheir scents have mingled\u2014Scott and Julia's.\n\nSlash's chin lowers, and he breaks eye contact, exhaling in frustration. This will complicate things. And he knows just the person to discuss it with. The others probably haven't scented anything yet. As a pureblood Red, Slash's scenting abilities far surpass anyone's.\n\nJulia and Scott will already be aware of what's happened. Whether they've told anyone outside of their pairing, Slash doesn't know. He glances at Jason again, and the tight set of his jaw and his standoffish posture tells Slash that Jason suspects something is brewing.\n\n\"What is it?\" Adrianna asks, searching his face, following his gaze like a tennis match gone wrong.\n\nHe cracks a smile.\n\n\"Besides the obvious?\"\n\nOne side of her mouth lifts, and he's reminded of why he loves her. She's not classically beautiful, more like a pixie\u2014cute and feisty. Still, she's the female for him. Blood calls to blood. There's no denying the primal absolutism of blood.\n\n\"Yeah,\" she replies softly. \"You're looking awfully down for a buttload of Singers to have been found. I mean, this is great news. A Combatant is alive. Scott's brother and sister...\"\n\n\"Michael's an asshole,\" Slash says without rancor.\n\n\"True.\" She smirks. \"But we need all the color we can get right now. He's a smart ass, and he's obsessed with candy, but there's worse things.\" She shrugs.\n\nJulia disengages from the little circle of rejoicing Singers and slowly walks toward Slash.\n\n\"Hey,\" she says with a smile.\n\nDo you know your soul-meld is back online?\n\n\"Hi,\" he replies.\n\nAdrianna looks between the two of them. \"Got some serious shit to sling? Okay, I know when I'm not wanted.\"\n\n\"Adrianna\u2014\" Slash begins.\n\nShe twirls around, grabbing his hand. \"It's cool, stud. Chill. I'll make myself busy.\"\n\nThat's what I'm afraid of.\n\nJulia watches his eyes on Adrianna as she walks off. \"She'll be fine, ya know.\"\n\nHe nods. But because he's an Alpha Red male, his instinct isn't a light switch to flick on and off. He will worry. When Adrianna is not in his presence, she will be on his mind.\n\nSlash gives a rueful smile. \"The wolf in me can't accept that.\"\n\nJulia nods, frowning.\n\n\"What's on your mind?\"\n\n\"Scenting,\" she replies immediately.\n\nSlash's eyebrow jerks up. \"For what?\"\n\nHe smells her nervousness like a faint perfume in the air. He wonders if Julia knows his suspicions or if she's come to him for something entirely different.\n\n\"I\u2014I need to know who is actually dead. A head count of sorts.\"\n\n\"I see.\"\n\nJulia's face becomes apologetic. \"I know it's a gruesome request...\"\n\n\"Yes.\" It's the truth, and Slash won't sugarcoat it.\n\nJulia's face falls.\n\n\"But I'm the Were for the job.\" His eyes scan the grounds, instinctively looking for Truman. He finds him hanging around with the sharp-tongued Singer-Were, Cynthia.\n\n\"Truman would be an excellent choice\u2014or Zeke. Between the two or three of us, I think we can account for the casualties.\"\n\n\"I need to know, for closure,\" Julia explains in a voice ground down by tears and heartache.\n\n\"I understand.\" His face smoothes as he changes the subject. \"Good news about the Singers.\"\n\nShe nods, and a soft sigh escapes. \"It is, but\u2014\" She runs a hand over hair that's still damp from a shower. \"I don't want any more surprises.\" Her luminous, cat-like eyes lock with his.\n\nSlash folds his arms and dips his chin with acquiesce. \"If there are supernaturals to find or save\u2014ones who may still be living? We'll find them.\"\n\nAnother relieved breath leaks out of her. \"Thank you, Slash.\"\n\n\"Welcome.\"\n\nHis eyes narrow on her. Anxiety fills his nostrils. \"I know,\" he admits quietly.\n\nSudden color splashes against the pale skin of her face. \"Oh.\"\n\nJulia's head lowers, and long, champagne-colored hair swings forward, obscuring her expression.\n\n\"When will you tell him?\" Slash asks.\n\nShe glances at him. \"Pretty soon. He'll eventually scent the change anyway,\" she mutters.\n\nSlash tenses. Serving up raw truth always feels wrong, but he doesn't know another way. \"That's not what I'm asking.\"\n\nHer chin jerks up, eyes fierce. \"Do you think I like this back-and-forth shit? I hate it.\" Her voice hits the last word like a punch to the gut. \"But there's nothing I can do. Tharell told us that the magic of faerie negated the soul-meld.\"\n\n\"Tharell's a liar, fey or not,\" Slash grinds out, still pissed the Sidhe was allowed to live. Better to lop off his head and burn his ass to ashes.\n\nJulia meets his eyes. \"The fey don't lie.\"\n\nSlash jerks his head back at those words and the silence stretches between them.\n\n\"So you think he told you what was true for that time?\"\n\nJulia nods. \"Exactly. By the time faerie's proximity was less, so was its hold on our bond. Then Scott was taken. We never found out until now.\"\n\nJulia grapples with her emotions like a wrestler losing on the mat.\n\nFinally, she appears to win the momentary struggle with her feelings. \"And when Scott came back, I believed I was just glad to see him. Y'know\u2014relieved.\" She twists her hands.\n\nSlash's lips lift, painfully pulling the pucker of scar tissue in his cupid's bow. \"But you were too relieved.\"\n\nJulia bites her lower lip and nods. \"It's terrible. I'm the worst person on the planet.\"\n\nSlash doesn't comfort others. He receives no comfort, either. He tries something new. \"Maybe not the very worst.\"\n\nJulia laughs. \"Gee, thanks. You're a real prince.\"\n\nSlash frowns. \"I was trying to offer a little...\"\n\n\"Salt in the wound?\" She laughs, and it sounds like despair making a run for it.\n\n\"No,\" he says gravely. \"I don't take pleasure in your pain. For someone so young, you have a lot on your shoulders.\"\n\nSlash steps closer, putting a light hand on her upper arm, in defiance of his earlier sentiment. \"Listen to me. These huge responsibilities won't lessen. They'll grow more complex, bigger. Take a mate.\" His eyes implore her to see reason. \"Drop this human attitude and culture. It doesn't apply to us. You need a mate to help you carry these things that need attention.\"\n\n\"I can't choose between them, Slash.\" Her eyes meet his, and he's momentarily startled by the swimming bourbon irises. \"I can't believe I'm blabbing all this stuff to you.\" She heaves a little self-conscious laugh.\n\nSlash shrugs. \"I scented it. There's no denying\"\u2014he taps the side of his beak\u2014\"the discernment. Besides, I'm not a chatty guy.\"\n\nJulia breaks into a grin. \"That, I know.\"\n\n\"Good.\"\n\nThey stare at each other for a handful of seconds. \"I'll let you know. I have my own reasons to want to know who lives.\"\n\nJulia's eyebrows pop. \"Can I know what those are?\"\n\nHe deliberates, unsure about whether to confide.\n\nSlash is not an easy male.\n\nHe looks at Julia. Yet, she is not an easy female.\n\nThey make quite a pair.\n\n\"If Adrianna's packmaster is dead, she is free to mate with me.\"\n\n\"Wow,\" Julia says softly, seemingly stunned.\n\nSlash feels his face heat. He understands her surprise, of course. That one as ugly as he would even imagine a life with a female like Adrianna is foolish.\n\nBut love is foolish. Love knows no bounds or reason. Love simply is. It grows from the fertile garden of the heart like a stubborn flower, to be adored by the recipient or rejected so the blooms can wilt in the shade of unrequited love.\n\nHe casts his eyes to the ground. \"I don't deserve her.\" His voice is gruff, and Slash hears the shrug of dismissal in his own words.\n\n\"No, Slash. That's not it,\" Julia says, lightly touching his arm.\n\nHe looks up, seeing the compassion in her face, and looks swiftly away.\n\nSlash doesn't need anybody's fucking pity. \"I understand how I look. I get that I'm not a prize to be won.\"\n\n\"Hey,\" Julia says in a sharp voice, and he reluctantly turns to face her again. \"Scars do not define the man, Slash.\"\n\nHe heaves a painful breath, jerking it from air that's grown thick with his regret about confessing to anyone. He should have kept his own council.\n\n\"She's from another pack. The home den mates with their own females,\" he explains.\n\n\"Pfft,\" Julia says with obvious disdain. \"That's dumb.\"\n\nSlash smiles blandly. \"It is what it has always been.\"\n\n\"Well, if Lawrence isn't here to lay down the letter of the law, who's to stop you from marrying Adi?\"\n\n\"Mating,\" he corrects.\n\n\"Right,\" she says with a smile.\n\nSlash taps his temple. \"You're still very human in your thought process.\"\n\n\"Maybe, but all that stuff is on my mind at the moment.\"\n\nHe nods. \"No one is here to stop us. That's exactly my motivation for seeing it through. Though I would have done it for the asking.\"\n\nJulia squeezes his forearm. \"I know it. And I know Karl will help. He was a cop, after all.\"\n\nJulia steps closer, and he looks down into her face\u2014a face without guile. It makes him anxious to see someone so fragile lead so many.\n\n\"Does Adi know how you feel?\"\n\nHis eyes don't flinch under Julia's intense scrutiny.\n\n\"Unfortunately, yes.\"\n\n\"Why, \u02bbunfortunately'?\"\n\nSlash drags a hand over his short hair as he gives a harsh exhale, and Julia steps back, sensing the space he needs.\n\n\"She could do better. I am selfish in my pursuit.\"\n\n\"You're honest. And that's what Adi needs. You forget, I lived in that den for a while, and they were letting Tony rule the roost. Jason was locked up like an animal. If it hadn't been for Manny, I don't know what would have happened. Lawrence, in my opinion, was a weak leader, letting the Were kill each other for their stupid rites and treating the females like a commodity. I wasn't a fan.\"\n\nSlash is troubled but not surprised. There simply aren't enough females. It's a cross-species dilemma. Slash has often wondered if the dwindling number of females is a type of natural selection in the supernatural realm. Nature does population control while affording extra protection for the females by making sure there were more than enough males to provide that protection. But it leads to corruption within the Were leadership.\n\n\"If Lawrence is gone, there will be no opposition. If he is not, I will appeal for the right to mate with Adrianna. In the meantime, Truman, Zeke and I will scent the deceased. I don't know everyone's unique scent, but I can identify the Were and give a count of how many Singers\u2014\"\n\nJulia holds up her hand. \"Thank you. Please, don't tell anyone what I've asked you to do. It's a grisly task, and emotions are already running high. People are just waiting for something else to wail and gnash. And\"\u2014Julia's eyes fill with tears\u2014\"I don't blame them one little bit.\"\n\nSalt permeates the air, and Slash speaks to a point above her shoulder. \"There is good in all of it. More Singers are alive. Tessa and Tahlia are solid additions.\"\n\nJulia's brows furrow. \"Yeah. What's their story?\" She wipes tears from her face.\n\nSlash throws a hard glance her way, his hands going to his hips.\n\n\"Whoa\u2014that bad?\"\n\nSlash shakes his head. \"No. It's not bad, but it's not perfect, either. The whelp\u2014\"\n\n\"Tahlia? Is she really a whelp?\"\n\nSlash's mouth twists into a smile. \"No. But she is hardly more than one. She is Lanarre.\"\n\nJulia's looks surprised. \"I remember what that is. When I was bored out of my mind and semi-prisoner at the Northwestern pack, I read the Lycan history.\" Her frown turns to a confused scowl. \"But a female Lanarre running around without guardians? From what I can recall, they are not unguarded until mated.\"\n\n\"Her guardians were murdered.\"\n\nHer scowl deepens. \"By who?\"\n\n\"Who do you think?\" Slash asks.\n\nJulia shrugs, her expression puzzled. \"I have no idea.\"\n\n\"Tony.\"\n\n\"Oh crap.\" Julia shudders. \"What an evil guy he was.\"\n\n\"Yes.\"\n\nJulia cups her elbows. \"So, we're taking care of Tahlia until the Lanarre find us and kick our asses for having her?\"\n\nSlash barks out a laugh. \"Pretty much.\"\n\n\"Great.\"\n\n\"It gets better.\"\n\nJulia groans, and he's sympathetic.\n\n\"She was traveling to mate with her chosen.\"\n\nHorror bleeds over Julia's face, her eyes widening. \"Oh, wow.\"\n\nSlash nods. \"Basically, he's the prince of another Lanarre pack.\"\n\nJulia's arms fall by her sides, palms out. \"Does she want to marry this guy?\"\n\n\"Don't you have enough politics to worry about?\" Slash asks.\n\n\"Are you teasing me?\"\n\nSlash nods, allowing a smile to touch his lips. \"A little.\"\n\n\"Hell, yes, I have enough. But, I don't want to just deliver Tahlia to this arranged-marriage dude\u2014\"\n\n\"Drek.\"\n\n\"Okay, Drek. No, I want to her to be willing.\"\n\nNeither one talk about the parallels to Julia's own circumstances. They don't need to.\n\n\"Not our business. It's a Lanarre issue.\"\n\nHer face puckers with distaste. \"Uh-huh.\"\n\nThey're quiet for a time while Slash thinks. The Singers have moved inside the house to presumably eat and catch up more. Slash feels eyes on them, nonetheless.\n\nHe's sure Scott would not be out of eyesight, but within earshot. He doesn't scent Jason.\n\n\"And Tessa?\" Julia inquires suddenly.\n\n\"She's on the run from the Western pack. Tramack.\"\n\n\"Weird name,\" Julia says. \"Why is he after her?\"\n\n\"Old-school pack.\" Slash knows of the Western. They border the Northwestern's territory, though they don't co-exist well.\n\nJulia's eyebrow lifts and she tucks a loose strand of hair behind her ear. \"What, like the Northwestern?\"\n\nSlash shakes his head. \"No. They hold to very old principles that are dangerously outdated. But again, it's not our pack. I don't have jurisdiction there. Tessa says she's been running for twenty years.\" Slash gives her a sidelong glance. \"There's a lot of dead Were in her wake.\"\n\nJulia pulls a face, giving a surprised jerk of her chin. \"Really?\"\n\n\"Yes. She's learned to survive and the males aren't allowed to use deadly force. Tramack wants her for himself.\"\n\nDisgust replaces surprise. \"What a mess.\"\n\nI agree. \"She'll be here temporarily then she'll have to move on.\"\n\n\"Why doesn't this dweeb give up?\"\n\nWhy indeed? Slash laughs abruptly then cuts it off. It's too somber tonight when he's on the brink of searching out death scents. His smile vanishes.\n\n\"Some Were become obsessed with a certain female, usually an Alpha. They can't think of anyone else. And, remember, there isn't a plethora of females.\"\n\n\"True, but who wants someone that doesn't want them?\" Julia asks.\n\nSlash gives a hard smile. \"Plenty.\"\n\nHe allows his expression to convey that she should have known the answer to that question.\n\nHer face tightens when understanding dawns.\n\n\"Right.\"\n\nNeither mention Tony.\n\n\"I'll find the dead,\" Slash says, ending the conversation.\n\nJulia nods sadly, giving him her back as she walks away.\n\n# CHAPTER FOURTEEN\n\nPraile\n\n\"Do it.\"\n\nLazarus lays his palms on Praile's ruined shirt. He hisses from the light contact.\n\n\"The Master has commanded a three-quarter healing,\" Lazarus reminds him.\n\n\"I will deal with the Master. I cannot feign being a recovered Singer without this, Lazarus.\"\n\nSeconds tick by, and Praile groans when the power of Lazarus's healing washes through the rough cotton, seeping into the rawest part of the torn flesh.\n\nVapor rises as the bloom of health surfaces to the newly healed skin.\n\nPraile's head sinks forward, and he groans with relief, shuddering as the worst of the torment falls away.\n\nLazarus's hands lift, and Praile steps away, jerking the shirt from his skin.\n\nIt's the equivalent of ripping off a Band-Aid. Fresh scabs come away with the material, and he howls.\n\n\"Praile,\" Lazarus begins to reprimand, and Praile turns again.\n\nThey repeat a short bout of healing, and finally, Praile steps away the second time. His eyes search the woods, commanding the minions.\n\nThey slither out of the forest, tails high, horns long, their skin so dark it is black shadowed by red.\n\n\"Clothes,\" Praile instructs with a bark.\n\nLazarus quietly observes Praile making his way stiffly inside a new shirt. The waistband of the denims are drenched in his blood, and those must be replaced as well.\n\n\"Make haste. The perimeter is guarded.\"\n\nPraile rolls his eyes. \"I know that. We must make the most of this charade. The pretense will only need to be maintained for a day, perhaps a few hours more. We move in, kill the High One, and steal the blood babe and its mother from under their grieving noses. It's glaringly simple, Lazarus.\"\n\nDoubt lingers at the edges of Lazarus's altered face. \"I don't have time to convince you.\"\n\n\"You are here to scent track every supernatural in the area. I want Tharell if he's still alive.\"\n\n\"He is.\"\n\n\"The Master will take great pleasure in a promise broken by the Sidhe. Tharell can be fileted over and over and shall never die.\"\n\nLazarus's expression is carefully neutral, but Praile can see what swims beneath: loathing mixed with a glimmer of fear.\n\n\"It will keep the Master very occupied.\"\n\nLazarus raises a pale-blond eyebrow. \"And his lashes off your skin.\"\n\nPraile gives Lazarus a considering look. \"Precisely.\"\n\nPraile turns and commands the low demons telepathically. They retreat into the woods to blend with the organic matter until summoned once more. They're barely sentient, but they have their uses.\n\nGrunts for his bidding. Demonic meat shields in battle.\n\nThe demonic low females have even more diverse uses. Praile licks his lips with his forked tongue in both memory and anticipation.\n\nSoon. Soon he will leave this place and go back to his hot slice of dark heaven. There, he will feast at the Master's side once more, glowing in the adoration of a task well done. Tharell will be the literal whipping boy, the High One will be dead, and the last hope of the Singers will be snuffed out like a dying flame.\n\nPraile cackles in joy. His deeds are nearly done.\n\nHe slips his cloak of false humanity over himself, rendering his skin, horns, and tail invisible. His tongue proves the most difficult. All his consonants come out with a hiss. He is a handsome demon. Yet\u2014all the attributes of his attractiveness are a challenge to hide. Nevertheless, Praile makes short work of cloaking his true being.\n\nHe will have to be constantly vigilant whenever he speaks. It wouldn't do to sound like a snake when he's remanded to Region One.\n\nJust as he thought that, two Singer guards came upon him and Lazarus.\n\nFortuitous.\n\nPraile and Lazarus raise their hands in an obvious gesture of truce.\n\nThe Singer at the left asks, \"Who are you?\"\n\n\"I'm Laz, from Region Two, and this is Peter\"\n\nPraile contains his irritation at being given the ill-suited name he did not agree to.\n\nGood Lucifer.\n\nThe two Singers exchange a thoughtful glance. \"How'd you get separated from Two?\"\n\nLazarus smiles, and Praile bites his lip to keep from laughing. His mirth would certainly reveal their deception.\n\n\"We became separated during the battle and have just made our way here.\"\n\nThe Singers look suspicious, but the story agrees with everything they know thus far.\n\nThe Singer guards are not Angelic enough to have the veins that would alert them to the presence of cloaked demonics.\n\nFooling the High One will be a special bit of business.\n\nThe faster I end her, the better for us all.\n\n\"All right. We can use all the Singers we can get. You two follow us and shack up at Region One for the time being.\"\n\n\"Is the Rare One there?\" Praile is dizzy with excitement about her impending death at his talons.\n\nMild puzzlement knots the Singer's brows. \"Of course.\"\n\n\"Good.\" Praile nods, sensing Lazarus's discontent. \"At least we have leadership after all the deaths.\"\n\nThe Singer's face smooths out, all suspicion gone. \"Yes. It's the one positive in all the misery.\"\n\nPraile gives a sad nod, as false as anything he's ever manifested.\n\nThey move ahead. He and Lazarus share a look of readiness behind their backs.\n\nThey're nearly there.\n\n*\n\nTessa\n\nTessa feels like a million bucks. A hot shower and some food have made her wolf want to roll on its back and have its belly scratched. However, Tessa doesn't know anyone and decides it's better to take a walk down to the lake she spied earlier from a window instead of wallowing in her temporary reprieve.\n\nTessa feels night pushing at the edges of the day, pregnant with yearning to birth the stars.\n\nThe moon beckons, just out of range, and Tessa sighs with longing. She can't count how many months have passed since she was able to just be her animal without fear of capture.\n\nShe has a chance to defend herself against males in her human form. But when in wolf form, the difference of genders is at its glaring fullest. In Tessa's opinion, the best advantage a female Were has going for her is the quarter-change. They look no different, but have heightened senses, strength, and speed. It's meant to be a protection against losing a whelp while pregnant, but the form can be conjured at will, without the moon.\n\nShe's never attacked Tramack's hounds while in human form. She met them all head on in her quarter-change form. They were at a disadvantage without their wolfen forms engaged.\n\nAnd she brutally exploited it.\n\nTessa has no remorse. If Tramack allowed her freedom, she would no longer have to run. Sleeping in a different bed every two weeks, eat whatever she can forage\u2014Tessa is merely existing. She simply takes sustenance, and breathing has begun to make her weary.\n\nPushing air in and out of her lungs is simply not enough in this life. Her will to live is slipping as Tramack gains ground in his chase. But as the moon begins to peek through clouds stretched thin like milky wounds, Tessa is happy for this moment. She is protected, and she has a full belly and a lighter heart than she's had in years.\n\nShe's not a great musician, but she begins to whistle a melody\u2014a tune she remembers from when she was a whelp.\n\nHer mother and father raised her from whelphood before they were killed in one of the tragic wars of her kind. Being an orphan is not unique to the Were.\n\nMaybe Were would be more fruitful if we stopped killing each other. Tessa thinks women could stop the wars if they were allowed to hold leadership roles.\n\nThe thing is, there simply aren't enough females to fulfill all the positions they should. Females are so scarce there's nothing soft to balance the hard. It's all raging testosterone. Battle, then sex, eat\u2014rinse and repeat.\n\nI know there's more.\n\nA bird lands on a branch not too far away and twitters a last song. He stays long enough to offer solace then leaps off the perch to roost just as true night lands with a soft sigh all around her.\n\nTessa shivers inside the cooler nocturnal embrace, moving smoothly into quarter-change. Instantly, her higher body temperature warms her, and she rolls her shoulders back with an exhale of relief, comfortable again.\n\nHer ears prick at the sound of approaching footsteps. She moves behind a tree that feeds directly from a lake, which is little more than a large pond. The water is beautiful, but the smells of death strangely linger. Her nose twitches with it.\n\nWeird.\n\nTessa sights four figures walking toward her position and moves deeper into the shadows.\n\nIn this form, her eyes function as though its daytime, and she's grateful anew for the blessings of the quarter-change.\n\nTwo Singer guards, whose names she doesn't know, move with purposeful strides toward the Singers' large mansion. It looms at the top of a small hill, and Tessa watches lights pop on inside like many eyes, casting false warmth against the grounds.\n\nThe two men walking behind the Singers catch Tessa's interest.\n\nThey move stiffly as though unaccustomed to their own gait, especially the bizarre-looking one to the right. Tessa flares her nostrils but doesn't catch a scent. The man is like a vampire, who doesn't carry a scent. Though sometimes, vamps do smell vaguely like earth and snakes. Tessa frowns.\n\nThe one to the left is tall. His pale hair is some shade of blond in daylight, but in the moonlight, it is silver.\n\nHis gaze flashes in her direction. His eyes are also light.\n\nTessa's heart thuds. He can't see me.\n\nMaybe he's a Tracker? Tessa doesn't enjoy being noseblind. No Were does.\n\nClearly he doesn't belong here, and neither does she.\n\nSo they're equal. Tessa decides not to worry about it for now.\n\nSomething snaps behind her, and she whirls, talons spiking out of her fingertips in a painful burst. She's still in quarter by a thread. Tessa doesn't make a sound, planting her feet wide and crouching.\n\nThen she recognizes Tahlia.\n\nTessa's embarrassed. She made a novice mistake by becoming so engrossed that she forgot her surroundings.\n\n\"You're defending yourself against me?\" Tahlia asks in disbelief.\n\nTessa straightens, feeling a new flush of stupid. \"No.\" Her talons slide back, and the fine hairs at her nape smooth against her sensitive skin. \"I didn't scent you.\"\n\n\"Sloppy,\" Tahlia remarks with a little smile.\n\nTessa scowls.\n\n\"Thanks for nothing.\"\n\n\"Don't be cross. I think it's odd you missed me, is all.\" She hikes an inky eyebrow.\n\n\"Yeah. I was concentrating on them.\" Tessa points to the four males who are almost to the front porch of the mansion.\n\nShe twists in their direction, her eyes narrowing at the men. \"Hmm. Who are they?\"\n\n\"Don't know.\"\n\n\"You don't sound as though you're too thrilled.\"\n\n\"Nope. Don't like it.\"\n\nTahlia turns back, her eyes are navy pools of midnight. \"Why?\"\n\nTessa blows out a frustrated breath. \"Listen, you seem like a nice kid. But it's only a matter of time until Drek finds you and it's faerie tale time.\"\n\nTahlia blinks.\n\n\"For me?\" Tessa plants her thumb between her breasts. \"I have to consider everyone to be a threat. And it's just a little bit too convenient those two show up right after us, from the same direction we came from. Nah, I don't like it on principle. Plus, I can't scent them.\"\n\nTahlia's quiet.\n\nFinally, Tessa says, \"Well?\"\n\nTahlia gives her a sidelong glance, her lips pulling into a small smile. \"I think we should find out who they are. I have to keep busy with something.\"\n\nTessa lets out a breath, feeling as though she just passed a test. \"Thanks.\"\n\nTahlia shrugs. \"For what?\"\n\n\"For being my only friend.\"\n\nTahlia's smile widens and she lifts a delicate shoulder. \"You're my only friend too.\"\n\nTessa's grin slips. \"Why? Didn't you have some whelplings to hang out with among the Lanarre?\"\n\n\"Common Were? Certainly.\" Tahlia's smile is a flat line. \"Now ask me if I was allowed to belong? To be a part of anything?\"\n\n\"Were you?\"\n\nHer voice goes low as she answers, \"No. Never.\"\n\n\"Why the hell not?\" Tessa shakes her head in mild disgust. \"Y'know, for being it for Lycan kind, being Lanarre sounds pretty sucky.\"\n\n\"It wasn't my role.\"\n\nTessa gives her a sharp look. \"What is then?\"\n\nTahlia eyes fall away from Tessa's to train on the perfect half-moon. \"Whatever they have of me.\"\n\n# CHAPTER FIFTEEN\n\nJulia\n\nThough Julia's circle of trustworthy supernaturals grows, it's still small.\n\nScott and Jason stand at her side while she tries not to let the strengthening soul-meld out of the bag like an escaped cat.\n\nTharell needs to be questioned.\n\nThe scar at his throat pulses like a lavender creek that runs with thin blood beneath it. Julia's own throat tightens at the sight, and she couldn't swallow if her life depended on it.\n\nScott takes her hand, and the knot in her chest loosens. She can breathe\u2014and think. And she'll need to because Tharell holds answers to important questions. His knowledge is one of the key reasons she didn't have him killed as so many wanted.\n\nShe hides her gratefulness for Scott's gesture, and he squeezes her hand when he feels it course through him. Their bond makes secrets impossible. It's a relief, and a curse, to have it back.\n\nJason stiffens and Tharell smiles from his bound position at the ground.\n\n\"Trouble in paradise?\" he murmurs, his beautiful turquoise eyes bright, though hardly any light reaches the deepest part of the Singer training barn.\n\nA small window, too tiny to fit anyone older than a nine year old through, allows pale moonlight to slant inside. The illumination strikes Tharell's deep-purple hair, turning it into a spoiled-plum color.\n\nJulia doesn't bother with fabrication. \"Yes.\"\n\nShe grabs Jason's other hand, and he looks down at her, his eyes shadowed.\n\n\"I already know, Jules.\"\n\nShe bows her head, and Tharell laughs. Welling despair rises inside Julia like a geyser, threatening to burst from her mouth and force her to verbalize things she would rather not say.\n\nTharell quips. \"So human\u2014yet not.\"\n\n\"Shut the fuck up.\"\n\n\"Can't argue with that, Caldwell,\" Scott adds.\n\nTharell grins at their discontent, and Julia is plagued with second thoughts about letting him live and the shame of thinking it.\n\nJason tenses. \"No\u2014Scott, kicking his teeth in won't solve anything.\"\n\n\"But it'll make me feel so much better,\" he says.\n\nJason barks out a laugh. \"Amen to that.\"\n\n\"You wouldn't be so sure of yourself if I was unbound and deep within the arms of the sithen. There, it is I who maintains all the power,\" Tharell says in a bald voice.\n\nJason shrugs his words away. \"You're a dick.\"\n\nTharell turns his penetrating glare to Jason. \"Your crude names do not move me, Singer.\"\n\n\"I can think of something that will,\" Scott says and steps forward.\n\nSomething painful flares inside Julia's gut, and she gasps.\n\nScott stops mid-stride, and he whips his head around to look at her. Jason's hand moves to her back. \"What?\" they ask at the same time.\n\nThe three of them look at Tharell.\n\nHis veins glitter black beneath his plum-colored skin, rising to the surface to pulse in time to his heartbeat.\n\nJulia's mimic his, but hers are golden-silver and perfect, like jeweled lace.\n\nHer belly begins to throb with the beginnings of something that pulls her toward Tharell. She takes a single, staggering step toward him.\n\n\"Ahh...\" he says, relaxing against his binds.\n\n\"What. The. Fuck?\" Jason yells, dropping her hand and going for Tharell.\n\nScott grabs his arm.\n\nJason whirls, narrowly missing Julia as his fist reactively plows toward Scott's jaw.\n\nScott jerks his face away, barely missing the blow intended for Tharell.\n\nTharell chuckles as Julia drops to her knees, cradling her stomach with both arms.\n\n\"What have you done?\" she whispers. Scott and Jason turn, their hands fisted in each other's shirts.\n\n\"It is not I who has done anything. But the saber.\"\n\nJulia forgets that her soul-meld with Scott is suffocating them, falling back two-fold as it pours into the crevices as though it never left.\n\nShe doesn't recall that her husband is ready to come to blows with Scott again or that Tharell should never see her bare skin.\n\nJulia rips off her shirt. Scattered buttons fly like plastic raindrops. She throws the long-sleeved shirt to the bare cement floor.\n\nShe tears her cami sky high, looking down at the horror on her stomach.\n\nScott falls beside her, his large palm covering where the saber struck and Cyn healed her.\n\nHis eyes bulge. \"Ah!\" he says in a hoarse shout.\n\nBut his palm can't hide the damage. A comma-shaped whip of black stands above her belly button. Where she was pierced with the blade, spiderweb-fine threads spread from the deep blackness, lined in red.\n\nThis morning, the mark appeared to be a bruise. It pulsed and itched, and Julia had assumed the sensation represented residual healing. Too many things clamored for her attention for her to worry about a healing wound.\n\nBut this wound isn't healing. It's spreading like a contagion.\n\nThe finer ends of the ebony threads extend to her ribcage.\n\n\"You sicken,\" Tharell says with utter certainty, eyes on the spreading black highway of encroaching marks. His eyes alight on Scott's face like blue fire. \"She slows its progress but not the inevitable.\"\n\n\"Why?\" Julia cries.\n\n\"The blood of the demonic has entered your system. Even as the Blooded Queen and your precious angelic blood combat the spore\u2014it is insufficient protection.\"\n\nScott lets go of Julia's skin as if it burns him, and he grabs Tharell off the floor, hiking him until their noses meet.\n\nHe shakes Tharell. \"Tell me how to undo this?\"\n\n\"No,\" Tharell says with a cunning smile.\n\nScott drops him with a grunt of disgust, and Tharell stands, though his feet and hands are bound.\n\nScott retreats one step and clocks Tharell in the face.\n\nThe blow rocks the Sidhe warrior back against the wall, and he spits black blood onto the concrete between them. \"Strike me all you wish, but it will not save her life.\"\n\n\"What will?\" Jason asks.\n\nJulia lifts her head at the sound in his voice\u2014inescapable consequence.\n\nJulia totally knows the answer.\n\n\"Give her up, Were.\"\n\nJason whirls, beating the wall with bruising force. \"No!\" he bellows.\n\nJulia covers her ears.\n\nHot tears roll down her cheeks and splash at her knees, dampening her jeans like drops of sorrow in a sea of denim.\n\nHer heart thumps against her calm, eating at it with each beat. Resignation crushes her spirit.\n\n\"This isn't what I planned, Caldwell,\" Scott says. \"I'd never cause her this kind of sorrow if I had a choice.\"\n\nOn her knees, Julia looks up at Scott.\n\nJason turns, fists at his side, veins standing out like stark pathways of rage on his flesh.\n\nJulia quickly gets to her feet, standing between them.\n\nTharell remains silent against the wall. His swollen lip heals as he watches the emotional war between the three of them.\n\n\"Julia,\" Scott says in a low voice, gently moving her so that she is no longer between him and Jason, setting her farther away from Tharell. The growing evil in her belly quiets at Scott's touch.\n\n\"I feel it,\" he says quietly.\n\n\"I know.\" Julia can't keep the relief out of her voice when the seed of evil flinches from her soulmate's touch.\n\n\"Stop talking in fucking code,\" Jason yells at them and Julia flinches. Spittle stands at the corner of his mouth.\n\nAs Julia gazes at Jason, she realizes for the first time that maybe, what's happened to both of them separately in the last three years, might make it impossible for them to be together now, with or without the soul-meld. What Julia needs is to take care of her people, not walk on eggshells around Jason. She needs someone who thinks of others first and doesn't react with anger whenever something doesn't go his way. Those revelations move through her mind like a flash flood.\n\nJason sees them anyway.\n\n\"What? Is this it, Jules? You get a little soul juice back, and it's all about Scott again?\"\n\nJulia's silent. Finally she shakes her head. \"I don't feel like having this discussion in front of Tharell.\"\n\n\"Do not mind me,\" Tharell says unhelpfully.\n\n\"Shut up.\" Jason shoots him a hard look, and Tharell smirks, shrugging.\n\nJason turns back to Julia and she sighs. \"I am sick. There might not be any me to have.\" Julia's eyes search his wild ones. \"You're so caught up on having me like a favorite bone you can't see the bigger picture.\"\n\nHe folds his arms. \"Nope. Don't see it.\"\n\nWhere has Jason gone? More than that\u2014why haven't I noticed?\n\nMaybe she hadn't wanted to. \"Tharell's awful, but he's right. There was demon's blood...\"\n\n\"A high demon. The lowly would not have adequate power to sicken the Blooded Queen.\"\n\n\"What?\" Julia asks.\n\nTharell hops a step away from the wall, and Scott moves protectively in front of Julia. She grasps his biceps, feeling the hard muscle beneath.\n\n\"Say what you have to say from there, asshole,\" Scott says.\n\n\"There are high demons. I have demonic blood, as we've already determined. The high demon in charge of these matters is called Praile. He is who would have instructed Tony to slaughter the Singers of this region, weakening them so the Blooded Queen might be unguarded. It was a stroke of luck for the demonic that Julia was pierced with the saber.\"\n\n\"No Combatant to help her,\" Scott says thoughtfully.\n\nTharell shakes his head. \"I was called upon when the lowly appeared for battle. I partook because I had no choice.\"\n\nTharell lifts his healed chin, his disdain clear. \"Blood rules all.\"\n\nJulia believed it. If she'd learned nothing else within the last three years, it was that.\n\nBlood governs us all.\n\n\"So I die?\" Julia whispers, fighting tears. After coming all this way, some demonic blood is going to kill me. She's been poisoned, hunted, kidnapped, and stabbed. But this?\n\nShe has people to rule, things that matter beyond her own self.\n\nFor the moment, she will have to ignore the underlying tension between Jason and her. The Singers come first. But she can't rule if she dies.\n\nThen Tharell surprises her. \"No. You will not die from this alone. But a high demon can finish the job.\"\n\n\"Fat chance.\" Jason gestures around him. \"We've got plenty enough guys sitting around, waiting for the chance to kick some demon ass. The fire dicks missed their window. Too bad, so sad. \"\n\nTharell inclines his head. \"No one would be a proper match for the likes of Praile.\"\n\nJason's jaw works back and forth. \"Some kind of fire prick struts in and kills Julia?\"\n\n\"He will be able to sense our blood within her body.\"\n\n\"So we're screwed?\" Jason presses.\n\n\"What? Is he our advisor now?\" Scott stares at Jason, palm flung toward Tharell. \"This fey is the one who betrayed us, tried to kill another Sidhe warrior, and we've kept alive only for answers I'm not even sure we want or should listen to.\"\n\n\"Kill me and take away more answers,\" Tharell comments in a sage voice.\n\nJulia asks, \"Why, Tharell? Why would you do this?\"\n\nTharell's face shuts down. \"You are a true and honest leader, Julia. However, in faerie, I am merely tolerated because my blood is not pure. When Queen Darcel was murdered by the lovely Delilah, I was free of some of it. But I will never be released from the prejudice of the pureblood Unseelie in a place where I must dwell to flourish. Further, my actions are moderated by my mixed genetics\u2014I must do exactly as I am told. And\"\u2014his eyes sink into hers like barbs\u2014\"no one has asked the right question.\"\n\nJulia moves toward him, and Scott tries to stop her.\n\n\"No, Scott\u2014\"\n\nTharell's so tall, she cranes her neck to meet his eyes. Her mind turns over the facts of how Tharell has behaved. Suddenly, she hits on the reason. The answer was in Jacqueline from the beginning.\n\n\"If you were to never leave faerie\u2014the sithen\u2014would you be okay?\"\n\nTharell smiles. \"That, my Blooded Queen, is the right question.\"\n\nTheir gazes lock as though no one else exists. Right now\u2014for Julia, no one does. The world consists of only her and Tharell.\n\n\"What's the answer?\" Julia keeps her eyes on his face as if it's a target. Before he replies, his expression gives away the answer\u2014and something else.\n\nRelief.\n\n\"Yes,\" he answers.\n\nShe can see that he's relieved that someone could look beyond his deeds. Jacqueline isn't the only one who's crazy outside of faerie.\n\nFor some, insanity is the gift of leaving the sithen.\n\nFor Tharell, with his demonic blood, it was a death sentence.\n\nBut Julia's willing to offer a stay of execution\u2014if she lives long enough.\n\n*\n\nJulia is glad when they leave Tharell in his cell. The two men wish for his death while Julia longs for less suffering.\n\nShe strides toward the one person she knows is insurance against Tharell. She needs to get him to faerie, but she can't have him doing more things against her while he's under the influence of demonic intent and insane due to his distance from faerie.\n\nThe only one who's a threat to him will be his guard, if she's willing. Julia hopes for Tharell's sake that she is.\n\nJulia's nearly at a jog, but Jason and Scott are walking fast, easily keeping up to her five feet four inches.\n\n\"We haven't figured anything out, Jules.\"\n\nJulia whirls on him, jabbing her finger into his chest. \"My life is a mess. I'm sorry. But I didn't ask to be taken back in Alaska any more than you asked to be attacked and turned.\" Her eyes blaze into his, and Jason doesn't drop her stare.\n\n\"Why is the answer to saving you always me letting you go?\" he asks softly. His response deflates her anger.\n\nJulia's finger drops as her top lip rolls between her teeth. Seconds later, she replies, \"I don't know.\"\n\nShe stares at him, looking over a man who, by human standards, she's married to. He's handsome, protective, and smart. They have a long history together. They've been through so much\u2014too much.\n\nHer heart feels like it's in her throat, and Julia stifles a sob when she realizes she's fallen out of love with Jason.\n\nShe tries to rationalize her feelings. Maybe it's the soul-meld.\n\nBut maybe it's just life, the life she did not choose.\n\n\"I can smell it on you, Jules.\"\n\nShe startles as though she spoke her horrible epiphany aloud. But she didn't.\n\n\"What?\" she asks softly.\n\nJason flicks a glance at Scott, who's stood by her side silently for the entire thing.\n\nScott steps toward Jason. \"Fuck you, Caldwell.\"\n\nJulia wants to scream at them and beat on their chests. But they're doing such a great job of it without her.\n\nJason heaves a sigh of utter disgust. \"Take off. Give us a minute. I'm not going to hurt Julia.\"\n\nScott folds his arms across his muscular chest. \"You don't have a great track record, Jason.\"\n\nHe nods. \"I know, but I have a handle on my shit. Just\u2014God, do you have an ounce of compassion?\"\n\nScott stands there for a moment then gathers Julia against him. He kisses the top of her head and says softly, \"I won't be far.\"\n\nJulia nods and feels a sensation like physical tearing when he leaves. The demon's blood pulses back to sick life with each step he takes away from her. Julia folds her arm over her stomach and reluctantly turns to Jason.\n\nHe plows his fingers through his longish hair, and Julia notices it needs a trim. \"I can smell him.\"\n\n\"I didn't know.\"\n\n\"I know. It's never your fault.\" He gives a little crazy nod. \"But this time, I can't do it. To save your life, I could pull through like I was going to before.\"\n\nWith William, she thinks but doesn't say.\n\n\"But I can't stand by and watch some other guy have you. It's not right.\"\n\n\"Are you standing by us being married?\"\n\nHe shakes his head, and Julia can see his sadness even in the weak artificial light cast between the barn and the mansion.\n\n\"Nah. I know none of that matters a damn to these people.\" He laughs, swinging out a palm. \"That's right\u2014we're not really human anymore. All the stuff of our past, the shit that mattered? Gone.\" He throws his arms up into the sky, beseeching a break that'll never come.\n\nJulia doesn't say anything, because he's right. Their past is just that\u2014the past.\n\n\"So, I'm going to stick around until we know you're okay. Then I'm outta here.\"\n\nJulia steps forward. Jason retreats a step, hands up in mock surrender. \"Don't touch me, Jules. I can only take so much.\"\n\nShe stops. Her belly hurt, but her heart hurts more.\n\nJulia doesn't allow more tears to fall, but she feels sick as she watches Jason walk down to the lake and away from her.\n\n# CHAPTER SIXTEEN\n\nJulia grows chilled as the night deepens. Long after his broad back disappears from view, her eyes remain on the empty spot where Jason disappeared by the lake.\n\nJulia feels him before Scott moves behind her and wraps his strong arms around her. The top of Julia's head tucks easily beneath his chin, and Scott lays the side of his face against her hair.\n\n\"I can't say I'm sorry, and for that, I'm sorry.\"\n\nJulia nods at his contrary statement because speaking is too difficult. The demon's blood growing inside her recoils inside her body at his nearness, and she feels momentary relief from its insidious progress as it grows sluggish.\n\n\"That's not really an apology, ya know,\" Julia manages to whisper.\n\n\"I know. I'm a real ass. But you're aware.\"\n\n\"Yes,\" she says, mad at him though she's a part of him now\u2014again.\n\nScott turns her to face him. \"Listen to me.\" His eyes seem to search every pore of her face and finally settle back on her gaze. \"It's like I told Caldwell\u2014if I had a choice, I'd allow you to be happy. I can't stand the idea of your sadness.\"\n\n\"Especially now,\" Julia says.\n\nHe grips her upper arms, pulling her tightly against his chest and cupping the back of her head. \"Especially now.\"\n\nTheir foreheads touch, and his scent overwhelms her, engulfing her senses.\n\nJulia swims through the headiness of his nearness enough to say, \"I need to talk to someone. She can be a guard for Tharell. Get him back to faerie before something else happens and we're forced to kill him.\"\n\nScott pulls away, staring at her. \"Killing Tharell would be awesome, Julia.\"\n\nShe gives a tight shake of her head, whipping her long hair back and forth. \"Not for me.\"\n\nScott studies her then bends over. He kisses her forehead then moves to her eyelids. Her breaths begin to come deep and even. She feels as though she's always been in his arms.\n\n\"You're soft in all the places that matter,\" he whispers against her temple, \"but I'm not.\"\n\nJulia opens her eyes and his are liquid night. \"I'm trying to look for something good here, Scott. Something that doesn't involve me hurting, killing or otherwise causing harm to someone else.\"\n\n\"We're good.\"\n\n\"Why?\" Scott asks, squatting down to look into her eyes. He traps her face with his palms. \"Why are you so afraid of the soul-meld?\"\n\n\"Because it's wrecked Jason. And that's the last thing I'd never want to do, especially to him.\"\n\nShe wraps her arms around her torso, and Scott's palms slip from her face to thumb the back of her skull, stroking her neck. \"He knows, babe. He knows you weren't trying to hurt him. He just can't let you off the hook.\" His hand flips open. \"Jason's got to blame someone\u2014and you're it.\"\n\n\"What do you mean you're hard and I'm soft?\" Julia asks suddenly, ignoring his last comment.\n\nScott sighs, stepping away. He takes hold of her hand as his other drops from her nape. They begin walking toward the mansion.\n\nScott's lips quirk.\n\n\"Mind out of the gutter, Scott.\"\n\n\"You did say I was hard.\"\n\n\"You called me soft,\" she counters.\n\nScott gives her a speculative look. \"True.\"\n\n\"I can't believe we're talking like this,\" Julia admits with a touch of shame.\n\n\"It's hard to keep up the pretense when we're connected like this. It's a tie at the molecular level. No one could resist it.\"\n\nJulia stops.\n\n\"Wait.\"\n\nScott's dark eyebrow lifts.\n\n\"The soul-meld didn't reconnect until after...\"\n\n\"I was tortured.\"\n\nA lone tear slides out of her eye at the thought of anything hurting Scott.\n\nScott thumbs away the wetness. \"Shh, babe, come here.\" He pulls her into his body, and Julia sobs. \"I'm okay. I'm here\u2014don't worry.\"\n\n\"What would I have done without you?\" she whispers as she clutches at his shirt.\n\n\"What you've been doing.\"\n\nJulia pulls away, her gaze riveted on his, debating about how much she should admit. \"I wanted you when you were missing.\"\n\nScott's surprise is plain to see. \"Without the meld?\"\n\nJulia nods. \"Yeah,\" she whispers. \"Yeah, I did.\"\n\nScott pulls her hands away. \"I wanted you, too. I knew that you'd chosen Jason and we had figured the soul-meld was exactly what it was, a magical blood tether, meant by some kind of natural design for us to connect. But the distance away from you made me realize some stuff.\"\n\nJulia's face is wet with tears, but she makes no move to wipe it dry because Scott shocks her again.\n\n\"I realized when they were peeling my body apart\u2014\"\n\nJulia covers her mouth and grabs his shirt, laying her face against his chest. His heart beats against her cheek, and an exhale shudders between them.\n\n\"I realized that I actually loved you, without the meld. Just me\u2014Scott.\"\n\nJulia nods against his chest. \"But I love Jason, too.\"\n\nScott stills and takes her away from him.\n\n\"You love him?\"\n\nShe nods.\n\nBefore he has a chance to react, Julia puts one hand on his shoulder, and the other on his face. \"It's you who I'm in love with.\"\n\nHe grins. \"Figured.\"\n\nJulia snorts, a sad smile plucking the corners of her mouth and she pinches the cleft in his chin together. \"Arrogant, more like.\"\n\n\"Yeah,\" he admits. His face sobers. \"But my dad didn't die for nothing. I'm older, I'm Singer royalty. Our union makes sense in the cosmos.\"\n\nJulia's smile twists. \"You didn't think so when we first met.\"\n\nHe ducks his head. \"A classic case of asshole-itis.\"\n\n\"Uh-huh.\"\n\nScott pecks her on the lips and with a little groan he whispers against the corner of her mouth, \"Just so you know, I'm holding back. Giving you time.\"\n\nTo reconcile things with Jason. \"I know.\"\n\nJulia puts her hands on the hard planes of his chest and gives him a little shove.\n\nHe grabs her around the waist. \"You're not getting rid of me that fast.\"\n\nJulia wrinkles her nose and he kisses the tip.\n\n\"I need to see Delilah.\"\n\nScott jerks his chin back. \"No shit?\"\n\n\"Yes.\" She laughs. \"She'll be Tharell's guard.\"\n\n\"My half-sister?\"\n\n\"She's like Madonna.\" She disengages her arm and puts a hand on her hip. \"Is there another Delilah?\"\n\nScott releases her, acting as if he's thinking about it. Putting a finger to his chin, he rolls his eyes skyward.\n\n\"Knock it off.\" Julia smacks him on the arm and he picks her up in his arms again.\n\n\"Scott!\" she laughs. \"Can't\u2014breathe!\"\n\n\"Quiet,\" he says, pecking her on the lips again.\n\nShe looks up at him and his eyes are dark with something besides color.\n\nDesire clouds them, vying for position in all that brown.\n\n\"Probably another Delilah somewhere in the world,\" he says in a voice choked by restrained emotion.\n\n\"Did someone call me?\"\n\n\"Speak of the devil,\" Scott says in a dry voice. He carefully sets Julia on her feet.\n\nDelilah looks between the two of them skeptically. \"What's going on?\"\n\n\"Soul-meld's back,\" Scott says without preamble.\n\nDelilah's eyes widen ever so slightly. \"Ouch\u2014settle down, brother.\"\n\nHe scrubs his head. \"It's been a long night.\"\n\nDelilah's chin kicks up, and she saunters closer, looking them over so closely that Julia can feel the blood rush to her face.\n\n\"Waste of blood, that,\" Delilah comments, and Julia's hands automatically move to cover her cheeks.\n\nDelilah giggles, and Scott moves between them.\n\n\"Don't worry, big brother. I don't like girls, though her blood does sing.\"\n\nJulia's face snaps to Delilah's. She hasn't heard that expression since William said it. A stab of pain pierces her with the reminder. Her ultimate protector, William had nothing to gain from saving her life, but the love he held for her. It was a pure sacrifice, and she'll never forget it.\n\nDelilah gives her a quizzical look. \"Yes. Though I am meant to take a male's blood, yours does hold a... certain appeal.\"\n\n\"Keep the appeal to yourself,\" Scott says in warning that borders on a growl.\n\nSilence sits heavy in the pause.\n\n\"Noted.\" Delilah answers then turns to Julia, clearly dismissing Scott. An inquisitive expression rides her face.\n\nNow that Delilah stands before her, Julia's request seems like too much to ask.\n\nDelilah studies her face. \"Ask.\"\n\nJulia sees the resemblance between her and Jacqueline, though she doesn't think mentioning it will ingratiate her to Delilah.\n\n\"I\u2014can you escort Tharell to faerie?\"\n\nDelilah gives an abrupt laugh. \"The murdering Sidhe?\"\n\nJulia instantly feels dumb for asking. Said like that, there is no way to deny that he's more than a murderer.\n\n\"Yes.\"\n\nDelilah laces her fingers together in a gesture hauntingly like Jacqueline's. Scott's hand kneads her shoulder gently.\n\nJulia gulps then pushes forward, \"I don't want to kill him,\" she admits in a low voice. Though God knows, Tharell deserves it.\n\nDelilah rolls her eyes, and Julia quickly continues, \"And it has come to my attention that he went bonkers because he was outside the sithen and his demonic blood became dominant once he was far enough away from faerie to negate the effect of faerie's proximity.\"\n\n\"Ah,\" Delilah replies noncommittally. Her dark eyes, so similar to Scott's, sparkle like black diamonds. It's disconcerting how little Julia can read her expression.\n\n\"And I'm his natural enemy, so I will be able to keep him in line?\" Coal-black eyebrows arch to her hairline.\n\nJulia nods tentatively. \"That's the idea.\"\n\nDelilah's ebony gaze finds Julia again. \"What does everyone else think?\"\n\nJulia bites her lip.\n\n\"Oh, this is funny. You've told no one?\"\n\nJulia shakes her head. I knew it wouldn't be a popular idea.\n\n\"She doesn't need to, Delilah. She's the Rare One.\"\n\n\"That's not it though, is it?\" she asks softly, giving Julia an appraising look.\n\nJulia shakes her head. \"No. I don't have the energy to defend my theory to everyone who wants him dead. Not now. Not with everything that's changed recently.\"\n\n\"Ah. And Jacqueline\u2014my mother\u2014does she influence you in this?\"\n\nDelilah is no dummy. \"Yes, in a way. I've seen the huge change since she spent time in faerie, and I have to assume\u2014to believe\u2014it could be true of Tharell.\"\n\nDelilah's brow furrows as Julia holds her breath.\n\n\"Fine, I'll do it.\"\n\n\"She wasn't really asking,\" Scott says.\n\nJulia lets tense breath out. \"Shh, Scott.\" She puts a finger to his lips, and her eyes move to Delilah. \"I would never issue an order. Tharell could harm you.\"\n\nDelilah yawns.\n\n\"Not likely.\"\n\nHuh. Julia doesn't know what to say to that. \"Okay, what do you think?\"\n\n\"I'll take the naughty Sidhe.\"\n\nThank God. Julia stretches out her hand, and Delilah takes hold of it, slowly shaking it to seal the deal. Julia pretends to ignore the hint of fang showing in her smug smile.\n\nHer nostrils flare, and her fangs grow a touch longer. \"My control is not limitless. Your blood is quite\u2014\" She stills, her hand clutching Julia's in a grip she could never break from. \"Tasty.\" Delilah gives a nervous little lick of her lips.\n\n\"Back off, sis,\" Scott says. His words are light, but the meaning behind them is dead serious. Julia doesn't have to look at his face to know it. His feelings tear through the meld and hit her own.\n\n\"So yummy,\" Delilah says then appears to snap out of it, dropping Julia's hand and moving back a few steps out of the vague light between the two buildings. The shadows consume her form. \"Where's Tharell?\"\n\n\"I'll take you to his cell.\"\n\nScott, Julia and Delilah make the short trek to Tharell's cell.\n\nJulia screams as Tharell comes into view, blood pouring out of his body.\n\nSomeone has executed vigilante justice.\n\nDelilah blurs to his side, dropping beside him. A crimson pool leeches into the long skirt she wears, dying the gauzy fabric red. The small bells sewn into the fabric no longer ring as they clog with the Sidhe's blood.\n\nTharell turns his head. \"Do not bother to heal me. It is a fatal blow.\"\n\nDelilah gives an exasperated snort. \"Nonsense, Sidhe. You've claimed immortality. Do not make yourself a liar.\"\n\n\"No,\" Scott says in a low voice.\n\n\"Yes,\" Julia returns like a volley.\n\nScott glowers at her, but Delilah turns her head away from them to look at Tharell as her delicate fangs slide out. \"Trust me,\" she says awkwardly.\n\nJulia should stop this, but she doesn't. She lets Delilah do what she does next.\n\nDelilah tears open her own wrist.\n\n\"Drink from me, fey.\"\n\nTharell regards her.\n\nJulia looks away from them as the noise of his suckling reaches her ears. Bright shame and memories well inside her. Images of William and her time in the coven are mixed with her relief to no longer be there. She comes back to the present with a start as Scott holds her while Tharell feeds from Scott's sister. It is one of the weirdest sights she's ever witnessed.\n\nThe blood flow slows from the gap in Tharell's body then stops.\n\nTharell sits up, unbound, and Delilah's wrist falls limply, her eyes at half-mast.\n\nShe's vulnerable. Tharell could kill her.\n\n\"Scott?\" Julia's voice is full of fear.\n\nBut Tharell doesn't hurt Delilah. He gathers her against him. Her pale skin and dark hair look so beautiful against all his violet flesh.\n\nWhen Tharell strikes her in the neck, Scott doesn't react at first.\n\nBut Delilah does.\n\nHer head rolls away, and Tharell grunts, jerking her more tightly against his muscular body. His violet fingertips bit into ivory flesh.\n\nWhat the fuck is happening? \"Stop!\" Julia shouts, adrenaline kicking her in the butt.\n\nThen something wonderful happens. Tharell threads his fingers into the long strands of Delilah's hair, and his blue eyes regard Julia over Delilah's shoulder.\n\nSanity has returned and he is Tharell again\u2014not the murderer, but the Sidhe warrior he was when she first met him.\n\nHe is also more than he seems, not just fey or demonic: a vampire, awoken by another right before their eyes.\n\n# CHAPTER SEVENTEEN\n\nPraile\n\nPraile feels right at home, taking in the elegant headquarters of his nemeses. Instinctively, he senses the lack of high Angelic blood quantum in the Singers present.\n\nHe and Lazarus will not be outed. The revelation relaxes him.\n\nHe frowns again. The single thing that could get their collective geese cooked would be the appearance of Region Two Singers who do not recognize them.\n\nUltimately, Praile and Lazarus are in a race against time to locate the High One and the one who carries the blood babe within her before Region Two Singers trip across them.\n\nLazarus gives him a full look of warning, and Praile scowls at him.\n\nPraile does not need prompting from anyone, especially another high demon who is second to him. Praile functions with perfect autonomy.\n\nThe two Singer guards leave them inside an impressive foyer, with a mention that someone will be with them shortly.\n\nPraile sighs with boredom, surveying the age of the structure. It is too modern for his taste, but he always liked the medieval times. Late nineteenth century is just not for him. Too much artifice. He's a sucker for the medieval torture racks\u2014and the despair that permeated the air like a fragrance. Now that was living.\n\nPraile hears footsteps approach then his blood rushes to the surface of his skin. Just when he believed he would get away with everything, an Angelic enters the foyer.\n\nHe is a handsome specimen. As Praile looks on with distaste, he must concentrate to keep the manifestation of his demonic blood from being obvious.\n\nPraile is hanging on by a thread.\n\nHis pure demon blood is an oil-slicked cauldron boiling beneath his false human flesh. Lazarus copes better. However, Lazarus is a high demon, as well. Their blood is designed to serve as a natural alarm to their worst enemy's presence. The Angelic share that defense mechanism.\n\nThe effort of maintaining his camouflage makes sweat break out on Praile's skin, and his fingertips tingle as the large Singer approaches.\n\nWith each step the Singer takes, something primal and deep within Praile tightens. His teeth clench with the effort to retain his form.\n\n\"I am Victor,\" the Singer says. Eyes like pale gray storm clouds stare with disconcerting intensity on Praile. He is not intimidated by anyone.\n\nPraile's situation is precarious.\n\nHe must persevere and use the speech that might be expected. \"I'm Peter and this is Laz,\" Praile replies, swinging a casual thumb toward Lazarus.\n\nVictor's perfect brow puckers. He looks from Praile to Lazarus. Praile's disquiet deepens. He feels strongly that it is critical this one must accepts them.\n\nKill him if he does not believe, he sends in telepathic command to Lazarus.\n\nPraile does not want to show their hand at this juncture. However, they might not have a choice.\n\nYes.\n\n\"I do not know you,\" Victor says with quiet certainty.\n\nFor Satan's sake, he's Region Two.\n\nPraile spreads his hands away from his body, cursing the horrible clothing. \"We came as the battle commenced.\"\n\nUnderstanding lights Victor's eyes. \"Ah, I see.\" Victor knots his hands behind his back then glances at Praile. \"I was not a part of the battle with the demonic.\" Praile gives a covert glance to Lazarus. \"I was here when a Were went berserk and slaughtered Region One.\"\n\n\"How was it you endeavored to escape?\" Lazarus asks.\n\nToo formal, make your speech casual. Use slang, fool.\n\nLazarus flinches slightly from the mental plow Praile uses on his mind.\n\nHe is not subtle.\n\nVictor's laser attention moves to Lazarus\u2014through him. \"There is a life bunker below the headquarters. All Regions have this safety contingency. I gathered whoever was royal, the females, children and secured them. We were automatically released earlier today.\"\n\nAh, makes perfect sense. Praile enjoys knowing the secrets of his enemies. Though before the High One came into being, it was of no purpose to pursue them in this way. The attempts of the demonic against the angelic would have been war without reason.\n\nThe Master is pragmatic, among other things.\n\nPraile lifts his chin. \"We survived the battle but were separated from the main group and made our way back here just now. We had only a vague idea of where Region One was located.\" He whips a casual palm around. \"We're transplants,\" Praile embellishes.\n\nVictor's eyes rest on his form. Praile notices Victor is not subtle, either, taking in Praile's stiff denims, shoes too bright a white, and ill-fitting shirt.\n\nPraile sweats without his vapor to assist off-gassing his naturally searing skin. He uses every ounce of subversive magic to cause his clothes to appear as though they're worn, but he can only do so much. Splitting his concentration among the call of the blood, and his tail, his horns, and his skin color, he is leaking his effort everywhere.\n\nHe sees Lazarus tense nervously. Praile's machinations are obvious to him\u2014but Praile doesn't know if they are as transparent to Victor.\n\nVictor's face breaks into a smile. \"Excellent. You are welcome here. We've lost so many of our kind it's a gift to find Singers who survived the siege.\"\n\nVictor's skin glitters with blood reacting to the demonic within Praile and Lazarus.\n\nHowever, he has not noticed. Lazarus's eyes widen as Victor claps Praile on the back, and he stumbles, forgetting the strength of a Singer with enough Angelic blood to be problematic.\n\nVictor quirks a brow.\n\nPraile raises his lips in the parody of a self-effacing smile, trying not to gaze at the Singer's veins. \"I'm not known for my grace.\"\n\nVictor shrugs. \"That's fine. I don't know my own strength.\" He winks and begins to walk away, motioning for them to follow.\n\nPraile wants to bash the Angelic's head in with his spiked tail.\n\nA hand appears at his elbow.\n\nLazarus.\n\nHe gives the minutest shake of his head.\n\nAn exhale whistles out between Praile's teeth, regulating his anger. Strong emotion will make hanging onto his cloaking more difficult. Already he's lost enough control that the Singer's blood has risen to the top of his flesh.\n\nHow long will it be before he sees his own body's defenses and takes action?\n\nPraile seethes and rails against the Singer who casually walks in front of them.\n\nVictor enters a kitchen and spins around suddenly.\n\nPraile smiles falsely.\n\nVictor's words freeze on his tongue. \"What is wrong with your mouth?\"\n\nDemon dammit. His teeth and tongue are exposed.\n\nLazarus moves quickly, bashing the Singer in the side of his neck with his forearm. He crumples.\n\n\"Come on, Praile,\" Lazarus urges.\n\nPraile moves in quickly, grabs the Singer by the armpits, and drags him off. \"Where did he say the bunker was?\"\n\nVictor's heels make black marks on the oak floors.\n\nPraile rolls his eyes, looking around frantically.\n\nFaint voices reach his ears.\n\n\"Lucifer help us,\" Lazarus bites.\n\n\"Think,\" Praile hisses, his forked tongue shredding the word.\n\nLazarus yanks the semi-conscious Singer toward the center of the hall, tearing an expensive oriental carpet from the middle and exposing a trapdoor.\n\nPraile pops it open with a twist and a pull. A vacuum lock wheezes air.\n\nThey gaze down a dark hole with ladder-type steps.\n\nA shuddering exhale blows out of Praile. Lazarus gives him a nod, and together, they roll the Singer toward the hatch then push him inside. The body of the unfortunate Angelic clunks down the short flight to land with a thud below.\n\nPraile straightens, and Lazarus closes the circular portal. It makes a beeping sound and five shrill chimes, then a great suction sounds off for half a minute. All the while, they check that others don't wander in during the middle of their subterfuge.\n\nNo one does.\n\nThe portal locks, and a timer appears on its smooth surface. It's some kind of countdown clock.\n\nPerfect.\n\nIt reads seventy-one hours and fifty-seven minutes. Plenty of time.\n\nPraile is not humorless. After all, having a sense of humor is critical to surviving life in Hades and surviving the Master.\n\nHe grins. \"One Angelic down.\"\n\n\"The rest to follow,\" Lazarus finishes grimly.\n\n\"Lighten up, Lazarus.\"\n\nLazarus cocks an eyebrow at the foreign expression. \"I don't think this is a \u02bblight' mission, Praile.\"\n\nPraile glances down at the carpet covering the trapdoor.\n\n\"Probably not.\"\n\nLazarus does not point out the obvious. If he had just kept his anger in check, the Singer would not have seen his black teeth or forked tongue.\n\nIt's just so difficult to hide the devil's beauty. It's meant to be seen\u2014even by angels.\n\n*\n\nTessa\n\nThe two men move away from the hall, and Tessa sinks more deeply into the shadows of the room where she's hiding.\n\nWho are those two? And why in the hell would Singers hurt another Singer?\n\nShe's confused, but in her twenty years of running, Tessa has learned to stay out of business that's not her own.\n\nShe bites her lip. Tessa doesn't like repaying hospitality with silence. There's got to be someone in charge who should know what she saw. Tessa paces out of the shadows.\n\nShe's thinking of staying for a couple of days and making sure Tahlia gets picked up by the Lanarre. Then after she's foraged for whatever supplies they'll allow her to take, she'll mention the guy the newest Singers chucked down the chute.\n\nWhat if the Singer is hurt? Fatally wounded? It feels wrong not to divulge this bit.\n\nShut up, Tessa. Not your gig, you're a Were for moon's sake.\n\n\"Hey,\" Tahlia says from behind her, and Tessa jumps, hand to her chest.\n\n\"Moon! You scared the shit out of me!\"\n\nTahlia grins. \"I do adore your expressions. There's no poo anywhere. Yet, you say there is.\"\n\n\"Cut the crap,\" Tessa grumps.\n\nTahlia shakes her head, her smile widening. \"No, I think you have a fixation with excrement.\"\n\nMaybe. Tessa scowls, crossing her arms. \"What are you doing lurking around here?\"\n\nThe girl's eyes are round and innocent. Tessa gets a sudden image of her talons ripping out the Were's eyeballs.\n\nNot so innocent.\n\nTessa might bust if she doesn't tell someone.\n\nTahlia smells the story before Tessa speaks. \"Tell me, Tessa.\"\n\nTessa does, and Tahlia's expression mirrors Tessa's thoughts.\n\nShe doesn't reply right away. Instead, she jerks her head to the side, and Tessa follows her into a long, narrow room small enough to be a closet. Adjacent to the kitchen, it's lined with glass-fronted cabinets. Fine dishes are stacked inside.\n\nTahlia shuts the door behind Tessa, then palms behind her butt, she leans against the solid wood.\n\n\"Promise me you will say nothing until we depart this place.\" Her large dark bluish-violet eyes don't look away, compelling Tessa to say yes.\n\nShe frowns. \"I don't know,\" she answers slowly. \"They've taken us in.\"\n\nTahlia gives a small shrug. \"I am grateful. But this thing you witnessed? It is a Singer matter. They are not even Were.\"\n\nShe's right, of course.\n\nStill, it feels wrong. Tessa worries at her bottom lip. \"Some share our blood.\"\n\nTahlia folds her arms, lifting a shoulder. \"Not Were enough to change, not Lycan enough to count.\"\n\nTessa's gaze narrows on the younger girl.\n\n\"That's cold.\"\n\nTahlia's chin lifts. \"It's the truth, and you know it.\"\n\nTessa nods. She does know it but can't shake the feeling of wrongness. \"I can tell the Rare One.\"\n\nTahlia grabs her arms as Tessa turns to leave. \"When we leave this place. No sooner.\"\n\n\"What about the guys? The new Singers.\" Especially the blond one. He was kind of cute before he put the drop on the Singer.\n\nWhat is wrong with me?\n\nTahlia's hand falls. \"It shows they're capable of violence for its own sake.\"\n\nTessa nods. \"Yes, it does.\"\n\n\"Simple. We avoid them.\"\n\nThalia gives her a sharp look. \"You haven't told everyone about me, have you?\"\n\n\"Enough,\" Tessa admits. It's sort of important everyone knows she's Lycan royalty. It's not something to screw with.\n\nTahlia rolls her eyes.\n\n\"Well forgive me, your highness, but it was top on the list to find some sanctuary.\"\n\nTahlia strips a hairband from her wrist and twists her curly hair into a topknot at the crown of her head. Her eyes find Tessa again. \"Not much of a sanctuary if Singers are willing to beat and hide their host in a hole.\"\n\nThat was Tessa's thought, but she won't say it aloud. Instead, she says, \"It's a helluva sight better than Tramack getting his paws on me.\" The reconciliation feels weak.\n\n\"Why is he so bad?\" Tahlia cocks her head. \"Why not mate with him and avoid all this chasing?\"\n\nTessa's abrupt laugh echoes in the small butler's pantry.\n\n\"I am not Lanarre.\" Tessa stabs a thumb into her chest. \"I wasn't groomed to be mated with some unseen male.\"\n\nTahlia's expression moves to hurt, but Tessa doesn't pause. \"I want to choose a male who complements me. Who I actually want.\"\n\nA tear struggles out of her burning duct, then another follows.\n\n\"Don't you see? I am a prisoner if I stay with the Western pack.\"\n\n\"And now we're just murderers,\" Tahlia says, her hands slapping her jean-clad thighs.\n\nShame burns through Tessa. \"Yes,\" she hisses defiantly. \"I am guilty of murder. Many times over. And they are guilty of robbing me of my freedom.\"\n\n\"I am guilty of it, as well.\"\n\nTessa steps into Tahlia's space. \"Then why did you help me? If you are so dead-set on being \u02bbowned,' why would you help me?\"\n\nTessa searches her midnight blue eyes.\n\n\"Maybe I want you to be free because I never will be.\"\n\nTessa jerks back as though she's been slapped.\n\n\"What?\" she whispers.\n\nTahlia wrinkles her nose, and Tessa realizes she's in quarter-change form, subtly breathing in available scents.\n\n\"You heard me,\" Tahlia whispers, ending the conversation as she turns and jerks open the door.\n\nShe'd heard her all right. And there was no way to un-hear her.\n\nTahlia felt as trapped by convention as Tessa did. But unlike Tessa, Tahlia gave up hope.\n\nTahlia doesn't fight for herself.\n\nBut she'd fought for Tessa. It's what she could do.\n\nLoyalty doesn't go unnoticed by Tessa. It's such a rare commodity. Servitude won't be over just because the Lanarre come to collect Tahlia.\n\nIt'll be over when Tessa thinks it is\u2014and not a minute sooner.\n\n# CHAPTER EIGHTEEN\n\nJulia\n\n\"What the hell was that?\"\n\nJulia backs up and bumps into Scott. His hands grip her shoulders to steady her.\n\nTharell rises, unbound, with Delilah in his arms. Scott moves protectively in front of Julia.\n\nTharell takes in Scott's stance. \"I do not plan to injure the Blooded Queen.\"\n\n\"Uh-huh,\" Scott says, disbelief thick in his voice.\n\nDelilah stirs, and Tharell caresses her face.\n\n\"What did you do to her?\" Julia asks, her eyes bouncing from the wounds at Delilah's throat to the incisors that have sprouted inside Tharell's mouth.\n\n\"It is she that did \u02bbsomething.'\"\n\n\"Well, shit,\" Scott says.\n\n\"Yup,\" Julia agrees. She looks around. First things first. \"Who attacked you?\"\n\nTharell shrugs. \"It is someone who is invisible to me.\"\n\n\"Illusionist?\" Scott asks.\n\nJulia paces away, casting a glance at Delilah. \"So someone comes in here, cuts away your bindings...\"\n\n\"I did that.\"\n\nJulia whirls around to face him. \"What?\"\n\nTharell smiles, and Julia shivers at the sight of the perfect bead of blood seated on top of his cupid's bow.\n\nTharell rolls his shoulders in dismissal. They move awkwardly as he holds Delilah. \"I could escape at any time. Only iron bonds could hold me. And even you are not as cruel as that.\"\n\nJulia frowns at his choice of words, ignoring the implication. \"I heard it acts like an acid.\"\n\nTharell nods.\n\n\"Why is Delilah so out of it?\" Scott asks, stroking Julia's shoulder. It's so natural to have him touching her, but not without a price. Julia tries to wipe thoughts of Jason from her mind, but she can't quite do it. Scott gives her a sidelong glance. She's not entirely sure how the meld works, only that he's getting leakage.\n\n\"Blood exchange.\" Tharell's lips quirk, his answer breaking though her morose contemplations. \"That is my supposition. To be honest\"\u2014he smirks\u2014\"I did not know that any part of me was vampire. I should, in theory, have been \u02bbfound out' during my one thousand years in the sithen\u2014in faerie.\" Tharell widens his stance, shifting Delilah's slight weight. He lifts his chin.\n\n\"Vampires are the fey's mortal enemies. As I have mentioned previously, they bring true death.\"\n\nJulia stands slightly behind Scott's broad back as he folds his arms. \"So how did you manage, having vampire blood when they're the only supes who can do you guys in?\"\n\n\"I do not know.\"\n\n\"You're not knowing a helluva a lot,\" Scott says.\n\nBlue eyes unflinchingly regard brown. \"True.\"\n\n\"How come you're not all noodling out right now?\" Julia asks.\n\nTharell smiles. \"I assume you mean why have I not lost control over my senses?\"\n\nDelilah's arm takes that opportunity to dangle. Tharell absently tucks it back inside his tight embrace.\n\nAhh. \"Yeah,\" Julia says, her eyes pegged on the gesture.\n\n\"Unlike the other supernatural groups, the fey know the history of all. We make a point of learning.\"\n\nScott grunts.\n\n\"You have a vampire here. You might ask Brynn if there is validity to my suppositions. My understanding is when a female and male vampire come together in blood exchange, it leaves the female vulnerable. The male remains alert to defend her against all comers. In this way, he fulfills his duty as the stronger of the two, protecting the weaker.\"\n\n\"Makes sense,\" Scott says.\n\nJulia's brow cinches and she gives him a sharp look, pinching the bridge of her nose. Her brain hurts. \"Okay. Say that's true. How was Delilah to know that you wouldn't hurt her?\"\n\n\"She did not. Delilah only knew that she would heal my injuries with her blood. I have so little vampire genetics that she must have been unaware. As I was.\"\n\nScott palms his chin. \"Maybe she was aware. I thought she agreed to take your ass back to faerie awfully quick.\"\n\nTharell moves forward, and Scott tucks Julia behind him tighter. \"It's okay, Scott. He wants to go back, not hurt us. If he released his bonds, he could have done whatever.\"\n\n\"Your magick will not contain me this close to faerie. Your lock manipulator cannot change the magick of faerie, not when the music of the sithen plays. However distant its melody, it is just for me.\"\n\n\"And me,\" Domiatri says in a droll voice from the door.\n\nTharell's head whips to the warrior he fought beside for centuries, whom he decapitated while under the pull of the demonic.\n\nThe air grows heavy\u2014full.\n\n\"It's a symphony for me.\"\n\nTharell's brows jump above his icy-blue gaze. \"Well, wonderful for you, Domiatri.\"\n\nA staring match ensues, and Julia speaks in the middle of it. \"Listen, Domi, I didn't want you to find out like this, but someone...\"\n\nShe gives Domi an inquisitive look.\n\n\"It was not he who put the hole inside my body,\" Tharell says.\n\nJulia's shoulders slump in relief. She would have to act if Domi had defied their joint decision to hold Tharell.\n\n\"How do you know\u2014brother.\" Domi says brother like fucker. Julia hears it plain as day.\n\nSo does Tharell.\n\n\"Because I know that however much you hate me for my actions, to come to me in stealth rather than in plain sight would go against everything you are.\"\n\n\"It would,\" Domi states. His jaw clenches defiantly.\n\nJulia's brows come together. \"Then who hurt you?\"\n\n\"Who cares?\" Domi says.\n\n\"Who, indeed?\" Tharell agrees severely.\n\n\"I do,\" Delilah says from his arms.\n\nTharell's face slips into a tender expression before he checks it.\n\n\"Can you stand?\" Tharell asks gently.\n\n\"I think so,\" Delilah replies.\n\nHe sets her carefully on her feet. The brutal splash of blood over her skirt has mucked the area where her knees are underneath the layers of fabric.\n\nJulia's throat convulses. Gross.\n\n\"Okay\u2014\" Julia begins.\n\nDomi interrupts harshly, \"What have you done?\"\n\nDomi looks from Delilah's healing throat to the new fangs Tharell seems to have suddenly have trouble hiding. \"What came naturally?\"\n\nJacqueline rushes in behind Domi, and he stays her with a hand.\n\nJulia gasps at the size of her belly. \"What on earth is going on?\" she asks Jacqueline.\n\nJacqueline ignores her. \"What is\u2014oh my.\"\n\nThat's so it.\n\nJacqueline's eyes take in a pale Delilah leaning against the much taller Tharell. Her fingers loosely wrap his forearm as it crosses her chest and he pins her against his body.\n\nThe scar at his throat is gone.\n\nJulia glances at Domi. His scar is no longer there, either. Amazing.\n\n\"It appears as though Tharell is more mongrel than even I knew. He has taken blood from your vampire daughter.\" Silently, he meets Tharell's gaze.\n\n\"Blood exchange?\"\n\nTharell nods.\n\n\"Really,\" Jacqueline breathes out, her hand absently going to stroke the swell of her body.\n\n\"What now?\" Scott asks.\n\n\"It doesn't really change things,\" Julia says, \"except that\u2014God, Tharell's a vampire. I want Brynn here. He can tell us more.\"\n\n\"Tharell cannot go back to faerie. He must atone\u2014\"\n\n\"There is no atonement, Domiatri. I will be put to true death once I arrive. But to stay outside\"\u2014Tharell grips Delilah more tightly\u2014\"is insanity.\"\n\nDomi scowls. \"This is insanity.\" His hand gestures to Delilah.\n\n\"I thought, since the fey and vampires are natural enemies, that Delilah would be the best choice to get Tharell back to faerie,\" Julia explains.\n\nDomi turns the silver of his laser stare at Julia. She swears it burns.\n\n\"Why?\" he barks at her.\n\nScott steps between them. \"Watch it green man, or I'll see if your guts match your skin.\"\n\n\"Scott,\" Julia begins.\n\n\"Nope. He doesn't get to talk to you like that.\"\n\nDomi and Scott stare each other down, then Jacqueline rests her hand on Domi's arm. He flicks his eyes at her touch, and Julia watches him build himself anew, into something more reasonable.\n\n\"You are right,\" Domi concedes. \"I am\u2014this entire murder situation has been distressing.\"\n\nDistressing? Uh, yeah.\n\nJulia nods. I guess when you've lived a thousand years, an attempt on your life just isn't that important.\n\n\"Wow,\" Delilah says.\n\nJulia ignores Domi and walks to her. They're about the same height, so she meets eyes that are so like Scott's she has to remind herself they just found each other. Technically, they're half siblings.\n\n\"Wow\u2014what?\"\n\n\"She has never exchanged blood,\" Tharell appears to guess.\n\nDelilah pushes away and staggers forward. Tharell catches her with a blinding swipe of speed.\n\nJulia gasps, jumping backward. \"Holy crap! I didn't see your arm move!\"\n\n\"Yes. An advantage of the vampiric, apparently.\"\n\nDelilah rolls her eyes but doesn't make a second move to get away. \"Of course I've taken blood from a male before.\"\n\nTharell's brows come together.\n\nDelilah folds her arms. \"I suppose you think you're so vital that I'm just blown away. You don't suppose it has anything to do with the Heinz 57 that I am? If I've taken from a pure vamp, it just\u2014well it didn't feel the same.\"\n\nTharell's expression turns smug.\n\nDelilah huffs, crossing her arms. The blood moving up the sleeves is distracting. \"Your penis\u2014did it just grow or something?\"\n\nTharell blanches and Scott snorts.\n\nThey glare at each other. \"Because unless that happened,\" Delilah continues with slow deliberation, \"you shouldn't be behaving as though you are the second coming of Christ.\"\n\nTharell flinches at the name of God. Demonic blood is more than blue liquid that runs inside black veins.\n\nThere are qualities to it, like there are distinctive elements inherent to her own blood.\n\nDomi dismisses the sparring with an easy hand whip in the air. \"Tharell will be executed in faerie.\"\n\nJulia turns to him, and Jacqueline is unusually quiet.\n\nJulia says, \"I can't have him here. Most of my people didn't want him alive. And obviously, someone tried to kill him in secret, against my express order\u2014which we all agreed on. Tharell fought in a battle that killed many of the Region Two Singers. He wasn't here when Tony\u2014when Tony did what he did.\"\n\nJacqueline moves deeper into Domi's tall body. He wraps an arm around her shoulder.\n\nThe mention of Tony's name was enough to make her shy from it.\n\nJulia understands\u2014too well. \"Then there's Tharell making a deal with Gabriel.\"\n\nJulia shifts her accusing gaze away from Tharell and looks at the others. \"We need him to go. And if Delilah could get Tharell to faerie unharmed herself, he would be the fey's problem. Not mine.\"\n\nTharell claps. \"Well done, Julia.\"\n\nJulia lets go of a hard sigh. \"Listen. I don't have great choices here. My soul-meld with Scott is back. My people are dead. Jason is pissed at me. And you're some kind of new vamp.\"\n\n\"And that's a new item on the agenda. Soul-whatever?\" Delilah asks, glancing at Scott.\n\nJulia ignores her. \"If Tharell can return to faerie, along with you and Jacqueline, all the fey people will be back where they belong and I can begin to rebuild Region One with who's left.\"\n\n\"And if I refuse?\" Delilah asks and Scott's eyes narrow, but he makes no move to leave Julia's side.\n\nTharell turns Delilah's face to his with a soft touch to her chin. \"Would you deny what is between us.\"\n\nDelilah jerks her chin out of his grasp. \"Yes,\" she hisses. \"A little blood bind doesn't mean we're getting married, Sidhe. I never forget that you are fey.\"\n\nTharell's hand slowly drops and Julia can't read his expression.\n\nBad.\n\n\"But I concede I'm the logical choice to get your criminal rear end back to the mound. I can do it\u2014Brynn could as well.\"\n\n\"Brynn is pure vampire, the sithen will not allow him inside.\"\n\nDelilah's face thins to feral sharp, a slight tremor can be seen in a shaky hand as she swipes a stray hair away. \"I can. I did kill Queen Darcel, if you remember.\"\n\n\"Unforgettable,\" Domi says in quiet consideration.\n\n\"Thank you, Delilah,\" Tharell says, changing tactics.\n\nHer eyes narrow to slits. \"You're welcome, Tharell of the Unseelie.\"\n\nHe gives a little bow then uses his newly acquired vampire speed to jerk her to him.\n\nDelilah yelps in surprise and Tharell gasps as her talons plug him, the tips exiting out of the wound she just healed.\n\n\"That is not the way to thank a woman.\"\n\nTharell's pain-filled gaze moves to hers. \"Then enlighten me.\"\n\nDelilah jerks the talons out of his body and Tharell slumps forward. \"Next time you bite me\u2014you ask.\"\n\nTharell's breaths come fast and hard, his body using what energy he gained from the first blood-letting to assist with the new wound.\n\nBut Tharell does not heal himself. Instead, his fingers trail a pathway of chemistry so obvious its flame to flesh. The delicate touch runs from Delilah's cheekbone to jaw and her breath catches.\n\nDelilah's vein at her throat rises to the surface under his fingertips like a succulent rope and her heartbeat pushes the lush pulse like a magnet to his touch.\n\nTharell's gaze latches onto the precious blood source where his fingers rest.\n\nScott moves Julia behind him again.\n\nA pin could be heard if one were to drop.\n\n\"May I?\" Tharell asks, but his lips already hover above her throat, his strong hand craning her jaw so the long line of her neck is exposed, his prior wounds stand as twin stark holes against snowy flesh.\n\nJulia can see the immense strength in that grip. Tharell could snap her neck. Instead, he cradles her head as if it were a revered and fragile egg. Five spots of scarlet spread in a red pool at his back where her talons speared him\n\n\"You may, you insufferable hybrid.\"\n\nTharell's lips twitch as he strikes before any of them can take their next breath.\n\nDelilah doesn't flinch. Instead, she relaxes into his embrace.\n\nWhen Brynn enters the barn no one hears him.\n\nHe attacks Tharell in a smear of rolling bodies. The movement also tears out half of Delilah's throat.\n\nJacqueline screams and moves to go to her daughter.\n\nDomi seizes her, swinging the mother of his child away from the fray.\n\nScott likewise holds Julia.\n\nI can't breathe.\n\nBrynn knocks his fist into Tharell, and the Sidhe vampire catches the strike. Tharell's wounds close as Julia watches their movements, their clothing like spinning streamers.\n\nTharell leverages the fist meant for him and tosses Brynn into the concrete wall behind him. The building shakes from the impact.\n\n\"Brynn!\" Julia screams.\n\n\"No,\" Scott says, \"Yo\u2014you're so not going near them.\"\n\nBrynn rolls away from the concrete wall gracefully, and Julia gasps when she sees the indent of where his body struck the cement.\n\nJulia points to Delilah, but Scott is unyielding armor around her.\n\nDelilah grabs at her throat, eyes wide, gurgling.\n\nBrynn leaps beside her, scooping her off the cold floor. His eyes move to Tharell. \"Stay back, newling. You've done enough damage.\"\n\nTharell surges forward.\n\n\"Tharell! No, wait,\" Julia cries.\n\nBrynn ignores everyone. His attention is only for Delilah as she drowns in her own blood.\n\n\"Shh, you will live, young female.\" Her flailing arms find his shoulders and latch on.\n\nBrynn grits his teeth against the strength of her hold and folds the mutilated flap of flesh back against her throat.\n\nDelilah bucks, starved for air.\n\nBrynn tears at his wrist. Then balancing her head on his knees, he squeezes the blood out drop by drop.\n\nHer mouth opens, and his dark lifeblood drips inside. Delilah's eyes close. She begins to make mewling sounds like a kitten with its first saucer of cream.\n\nDelilah grabs the wrist above her and latches on with a contented sigh.\n\nBrynn gives Tharell a look of unadulterated disdain. \"That is how it's done, infant.\"\n\nTharell squats beside them. \"I am not an infant, and if you do not disengage yourself from her mouth, I will put my fist through your head.\"\n\nBrynn smirks. \"You will try.\"\n\nUnreal.\n\n\"We don't have time for this,\" Julia says.\n\n\"Nope, let them figure this out, Julia. Sometimes men just have to beat the shit out of each other to make sense of it all,\" Scott says.\n\nThat's so illogical.\n\nNeither of them acknowledges Julia or Scott. They're too busy with their testosterone-laden stare fest.\n\n\"It is I who saved her. You're too much of a novice to release her from the feed when another vampire attacks. You almost killed a helpless female as she entrusted you to feed from her vein. You. Are. An. Infant.\"\n\nTharell's eyes are like dead marbles in his face. \"And you shall die if you don't take your wrist from her mouth.\"\n\nDelilah ends the argument when she releases Brynn, wiping her mouth with the back of her hand.\n\nThere aren't many blood-free spots on Delilah anymore. Brynn and Tharell look down at her at the same moment.\n\nShe glares at them both. \"That hurt like hell.\"\n\nLooks like they've bitten off more than they can chew.\n\nLiterally.\n\n# CHAPTER NINETEEN\n\nDrek- Lanarre prince\n\n\"This is absurd.\"\n\n\"They're dead, Drek.\"\n\n\"I can see that.\" Drek holds out his palm, where congealed blood collects.\n\n\"The human law enforcement has already combed the establishment.\"\n\nDrek's eyebrows lift. \"And?\"\n\n\"They have found nothing we didn't want them to. The human guardians were in bad shape.\"\n\n\"Eviscerated.\"\n\n\"Yes,\" Bowen answers tersely.\n\nDrek shoots out an exhale like a bullet. \"And Tahlia is out there somewhere\u2014a Lanarre princess, unguarded.\"\n\n\"She is skilled, Drek.\"\n\nHe whirls on his trusted guard. \"I'm aware she is skilled. However, she will be my queen, Bowen.\"\n\n\"I understand.\"\n\nDrek kicks the cheap bamboo couch in the hotel room. It flips then hits the wall so hard that the peg legs embed into the drywall.\n\nThe room still bears her scent, which is strongest in the bathroom. Drek walks the length of the room and passes through the narrow bathroom threshold.\n\nA faucet drips loudly. Drek has never wished more to change to wolfen than he does in this moment.\n\nHe could be so much more aware.\n\nBut a seven-foot-tall wolfen will attract too much attention. As it is, it's taken all they could manage to get inside the room without alerting the authorities of their presence.\n\nWhat I wouldn't give for my wolfen snout\u2014or animal's eyes. Without them he has only his slightly heightened human senses.\n\nThen his eyes catch something that's fallen behind the dingy commode.\n\nIt looks like snow.\n\nNo\u2014it's a feather. A pure white feather. And balanced on top, is a drop of blood.\n\nDrek stands so quickly his head spins.\n\n\"Bowen!\" he yells.\n\nBowen rushes inside, hands gripping the doorjamb. \"What is it?\"\n\n\"She changed into bird form.\"\n\nThey come to the same realization simultaneously.\n\n\"She lives.\"\n\nHis gaze moves to the small bathroom window above the shower stall. The window stands open. The opening is just large enough for a bird to move through.\n\n\"Yes,\" Drek answers, clutching the feather.\n\nIt breaks under his grip.\n\n*\n\nSlash\n\n\"Hey,\" Adrianna calls out softly.\n\nSlash jerks up from the ground in a semi-pushup leap. His black nylon athletic pants leave little to the imagination, and he watches Adrianna take that in: his lack of shirt\u2014and underwear.\n\nIt's simply not practical for him to be wolfen and not have pants that accommodate his increased girth. Underwear would literally strangle his nuts. The athletic pants are the best he can do, and they're more modest than anything most would partake in.\n\n\"What are you doing?\" she asks.\n\nSlash can't resist a small chuff to scent her.\n\nAdrianna is nervous. Her emotions fuel a quick scan of the environment.\n\n\"Nothing's wrong Slash. I'm just\u2014I'm seeing how you're doing. You've been gone all day.\"\n\nHe moves toward her.\n\nFear washes over her features.\n\nSlash stops, stunned. \"Are you afraid of me?\"\n\nAdrianna quickly shakes her head. \"No. But, you're big in wolfen form.\"\n\nHe grins. He forgot what he looks like. For a full second, he forgot about his scar.\n\nHis smile disappears at the thought but Adrianna is already slipping her arms around his waist, nuzzling against the downy hair that covers his body in this form.\n\nHis hand awkwardly cups her small skull, and he thinks of all of what he loves contained in the fragile container of her head. He swallows painfully.\n\nShe is so vulnerable. And he can think of nothing else but her protection.\n\nHe voices different thoughts, though. \"Julia tasked me with scenting the dead.\"\n\nAdrianna tips her head back. \"Kind of shitty.\"\n\nSlash shakes his head as his mouth pulls into a thin smile and flattens his scar. \"No. It must be done. I'm a pureblood Red. Truman, Zeke, and I are making short work of it.\"\n\nAdrianna looks out over the small lake then glances at her feet.\n\nSlash nods. \"The fey were able to get all the... pieces.\"\n\nAdrianna nods at the word for the torn bodies of the dead.\n\n\"They put them in a mass grave. It's up to us to exhaust the tally of fallen Were.\"\n\nUnderstanding lights her expression. \"Lawrence.\"\n\nSlash nods, unable to tell her just how selfish his motivation is. The packmaster is the one true obstacle standing between Slash's union with Adrianna.\n\n\"If he's gone, then we can be together.\"\n\nEmotion overcomes him and Slash can't keep form and bleeds to human.\n\nThe short coat of red hair that covers his body disappears and only the barest grunt whistles out of him as he shrinks to the shorter stature of six foot four.\n\nAdrianna is still her fierce tiny package of female.\n\nSlash moves his fingers beneath her jaw and tilts it up, looking deeply into her eyes.\n\n\"Adrianna.\"\n\nHer lips twist. \"Just kiss me. I'm dying here.\"\n\nA hint of a smile pulls his mouth, and then it's on hers. Slash has been with many females.\n\nThey never saw his face; he made sure of it.\n\nHe never loved them, but a male has needs.\n\nBut everything he needs is now here inside his arms.\n\nThe sunlight warms his face and sparkles off the water.\n\nIt showcases his scar.\n\nYet, he allows Adrianna to pull him down to meet her soft lips.\n\nSlash is not starved for sex, but love is a different matter. And Slash knows that sex will be consuming when love is a part of it.\n\nAnd he loves Adrianna so much, it's a type of sweet agony.\n\nIt hurts him deeply because he gave up some of his self-preservation to feel it.\n\nThen her small hands slide just underneath the waistband of his pants and Slash is lost to her touch.\n\nJust lost.\n\n*\n\nSlash leads Adrianna by the hand. The woods deepen around them. The trees grow thicker and broader.\n\n\"I'll be cold, Slash,\" Adrianna says with just a touch of coyness to her voice.\n\nBut she does not say no.\n\n\"Let my body warm you,\" he says.\n\nShe lifts his hand and kisses each knuckle, never looking away.\n\nSlash sucks in a breath. \"Lawrence is dead,\" he confesses.\n\n\"I thought he might be,\" she says, giving a significant look around the woods. \"You wouldn't take me as mate unless he was out of the picture.\"\n\nSlash smiles, and Adrianna grins back.\n\n\"How do you know I am taking you as mate?\" he teases.\n\nAdrianna's hand is suddenly cupping his balls, and he gasps. \"You better be, buster. I'm not doing the nasty with just anyone.\"\n\nSlash seizes her wrist. \"You are mine. There will be no other who knows your body, Adrianna.\"\n\nA slow smile spreads across her full lips, and her hazel eyes sparkle. \"Just the way I want it, stud. Now let me do something more fun with these.\"\n\nSlash isn't shy, but when the one woman he loves has him by the balls, it's a strange position to be in.\n\n\"Adrianna,\" he calls softly as her fingers cup him intimately.\n\n\"Yeah,\" she replies. Her half-hooded eyes widen slightly.\n\n\"I know what you're going to ask.\"\n\n\"Then you know it will be better in quarter-change.\"\n\nShe nods. \"I'm young, not dumb.\"\n\nSlash wants her. Badly.\n\nHe doesn't want to hurt her, but he will. There is no reversing it. Virginity is the one thing that unifies all the species. Females who are pure will experience pain at the loss of it.\n\nSlash takes her in his arms. \"I would do anything to spare you.\"\n\n\"Slash, stop. I've always wanted you. I couldn't do this with anyone else.\"\n\nSlash growls.\n\nAdrianna's eyebrows pop. \"Don't go all Alpha on me.\"\n\nSlash doesn't answer. He bends over, taking her earlobe deeply inside his mouth and nips it, drawing blood.\n\nAdrianna whimpers.\n\nSlash hardens against her. \"You can't go back, Adriana. Once we are mated, whatever I have is yours. Whatever I lack, will remain.\"\n\nHer hand moves to him again, gently squeezing and he groans.\n\n\"You lack nothing,\" she whispers.\n\n*\n\nAdrianna kisses what she can reach, moving her lips across his unscarred chest. His nipples harden inside the cool interior of the woods.\n\n\"Where are the guys?\" she asks against his hot skin.\n\n\"They won't disturb us.\"\n\n\"Better not,\" she says, giving his nipple a hard nip.\n\nSlash pulls a sharp inhale that's a half-moan.\n\n\"Don't, Adrianna.\"\n\n\"Were like teeth,\" she says as though reading from a textbook.\n\n\"Perhaps too much,\" Slash says, knowing where that can lead.\n\nAdrianna takes his pants down to his ankles.\n\nSlash stands before her, unashamed by his nudity, hoping she forgives his face for what his body offers.\n\nAdrianna stares at him silently for so long he can't fight the heat that spreads from his neck to settle at his face. He physically restrains himself from jerking the pants back on and walking off.\n\nAdrianna finally looks at his face, which only makes his shame burn brighter.\n\n\"You are the most beautiful man I've ever seen.\" Her lips part, a flush rivaling his own spreads across her cheekbones and Slash's nostrils flare at her arousal.\n\nSlash grins so he won't cry and grabs her in a hug so fierce she taps out on his shoulder.\n\nHe pulls away, and her face is purple.\n\n\"Maybe a little too tight,\" she squeezes out, her fingers barely apart.\n\nSlash can't stop smiling. Adrianna called him beautiful.\n\nThen she takes off her clothes, and Slash knows his beauty is merely a shadow of hers.\n\n# CHAPTER TWENTY\n\nJulia\n\n\"No, absolutely not.\" Julia looks from Brynn to Tharell, then her gaze goes to Delilah. \"I asked you to go with Tharell. Back when he was a fey-slash-demonic. Now he's a vamp?\" Julia throws up her hands, and they slap down on her thighs.\n\n\"And can I mention, for the record, that you guys shouldn't even be standing with all the blood that's lying around?\" Julia crosses her arms, fuming.\n\n\"Julia.\"\n\nShe turns reluctantly to Scott.\n\nHis eyes beseech her to listen.\n\n\"Brynn was William's second, right?\"\n\nShe throws a look at Brynn. \"Yes,\" she answers slowly, \"after he was tortured at Merlin's orders, Brynn and the handful of other vamps are all who survived.\" It's important to clarify everything.\n\nScott nods. \"Okay, so he's the only vamp around that you trust.\"\n\nJulia supposes that's accurate. \"Yeah.\"\n\n\"Delilah's already decided to go. Brynn can supervise. Two vamps against the big bad fey.\"\n\nScott winks at her and Julia steps into his arms and hugs him around the waist. \"You're so funny.\"\n\nHe strokes her back. \"I know.\"\n\n\"Hmm,\" she says against him, but she's smiling.\n\n\"Tharell can do nothing to us. Two against one. Plus,\" Brynn says, \"it's against vampire nature to kill or harm a female. And it's not about gallantry, Julia.\"\n\n\"What is it then?\" she asks, frowning as she tosses Tharell a surreptitious glance. Disgruntled doesn't cover it.\n\n\"It's about self-preservation. If the males are chasing around and killing all the females, where do the new vampires come from. Offspring.\"\n\nJason strolls into the open barn door, hands stuffed inside his pockets, his gaze moving to where Julia stands beside Scott. \"Don't you just latch onto someone new and change them, blood sucker?\"\n\nHe smirks.\n\nGreat, Jason's going to be himself, when we really need this. Tharell needs to get gone so tempers can cool.\n\nScott takes her hand, and Julia tries to pull away, though she wants nothing more than to collapse against him in relief.\n\nJason notices the subtle tug-of-war and his smug expression turns to the hard edges of rage.\n\nJulia is intimately familiar with that expression. Julia stops struggling. She is the rope between two men, and she's fraying quickly.\n\nBrynn glances between all of them. \"Yes, we can change some humans into vampires, but they must already be vampiric. They must have the blood of a vampire to change.\" He glances at Tharell. \"As the Sidhe warrior did.\"\n\nJason's eyebrows hike. \"Really? So Tharell's gone all vampy. Well, isn't this just fucked.\"\n\nJason's got it right. It's a mess.\n\n\"So\"\u2014Jason taps his chin\u2014\"let me get this straight, as I've just been invited to this little party.\" When his eyes find Julia's, she doesn't look away. \"You three vamps are going to the fey mound, and you'll take your chances there.\" His eyes move to Delilah's still-healing neck, and he checks a snort. \"And Tharell is going to hope the fey just welcome him back with open arms? Not fucking likely.\"\n\nJason walks slowly over to Julia and Scott. His eyes rake down her body then move back to her face.\n\n\"And Julia's all entangled with Singer royalty again, while battling a demonic spore.\"\n\nJacqueline glances at Julia and she casts her eyes to the floor.\n\n\"Is this true, Julia?\"\n\n\"Yes.\" She gives a sharp exhale. \"The saber\u2014the demonic blood can't kill me, but I'll just get sicker.\"\n\n\"Until a high demon comes along and finds Jules. Then he can do her in.\" Jason looks at her coldly, but Julia knows it's his anger talking. He's angry she'll grow sick, and the situation with Scott has only renewed that rage.\n\nJason licks his lips. \"All Jules has to do is hook up with Scott here, and the high demon can pack sand, right, guys?\"\n\nJulia's miserable. A few short years ago, she couldn't envision a life without Jason.\n\nNow she knows that'll never happen.\n\nBrynn fills the awkward silence. \"Then it would make sense for Tharell to be gone from here, so if the demonic return, he can't be commanded to act on any direction given by them.\"\n\n\"Won't he be too vampire to be driven by demonic directives?\" Jacqueline asks.\n\n\"I think demonic trumps vamp,\" Julia guesses.\n\n\"I know it does,\" Domi says.\n\nJason turns to Brynn, Delilah, and Tharell. \"Then you better go. Because even though I'd love to kick some ass, I don't want one chance for some devil guys showing up and hurting Jules.\"\n\n\"But you are allowed to hurt her by staying?\" Tharell asks, getting to the heart of it.\n\nJason smiles and it's more a baring of teeth. \"Yeah. I've earned the right, purple dick.\"\n\nTharell moves in a streak of lavender. He slams Jason against the wall, holding his hand at Jason's throat. \"Courtesy, Were.\"\n\n\"Fuck you, grape.\"\n\nJulia runs to them. Scott reaches to grab her.\n\nNo, please, Scott.\n\nHe stills at her telepathic missive.\n\nBe careful.\n\nShe turns and puts her hand on Tharell. \"Please, just go. Don't harm Delilah.\"\n\nTharell looks down at her, his fingers like steel at Jason's throat. \"I will die inside faerie.\"\n\nJulia squeezes her eyes shut. \"You will die here.\"\n\n\"I know.\"\n\nJulia opens her eyes and watches as Tharell drops Jason.\n\nHe massages his throat then looks at Julia.\n\nShe lowers her eyes before what she sees in his, feeling sick.\n\n\"I'll get my shit. It seems to me you have plenty of protectors here, Jules. I'm just in the way.\"\n\nJulia grabs him, latching onto his arms. He flings his arms wide and her hands fly backward. \"Don't,\" he snarls.\n\nScott is suddenly in front of her. \"It's all about you, Caldwell.\"\n\nJason's eyes are all for Julia.\n\n\"Not always.\"\n\nJason stalks off, not giving anyone anything but his back and the dust off his feet.\n\n*\n\nThankfully, Delilah has time to put on fresh clothes for the journey to the mound. They didn't announce her departure.\n\nBrynn and Tharell wait at the stand of trees that bisect the lake from Highway 101. The mound is beyond even that, west toward the Olympic Mountains at the edges of the Hoh Rain Forest.\n\n\"I feel bad,\" Julia says, reaching for Delilah's hand. \"I know we're not close, that we don't even really know each other.\"\n\n\"It's fine.\" Delilah gingerly lays a hand on her throat, where the faint scarring is nearly healed.\n\nJulia breathes a sigh of relief.\n\nI think we'll have to do a hazmat clean of the barn floor. She shudders at the recent memory.\n\n\"It changes things,\" Julia says.\n\nDelilah nods. \"It does. I know he could kill me if the right devil shows up.\" The corner of Delilah's mouth lifts.\n\n\"Not. Funny,\" Julia says but she laughs.\n\n\"Nothing is certain, Julia\u2014that's why Brynn is going.\"\n\n\"Another complication\u2014\"\n\n\"A contingency,\" she interrupts.\n\nNeither one of them mention Brynn's actions in saving Delilah in what looked like an attack by Tharell. Protective vampire instinct or something else?\n\nJulia shoves her hair behind her ears. Her head hurts with all the thoughts piling up inside it.\n\nDelilah shrugs. \"I'll be fine.\"\n\n\"You're still a girl\u2014like me.\"\n\n\"A woman who killed the Queen of the Unseelie.\"\n\nA shaky laugh escapes Julia. \"Right. I know.\"\n\n\"You requested I take him.\"\n\nJulia stares into Delilah's black gaze. \"Before Tharell vamped out.\"\n\nDelilah crosses her arms. \"Brynn and I will return. Once Tharell enters the sithen, he is a problem for the fey. Not us.\"\n\n\"It's like a death sentence.\"\n\n\"It's as you said. He waits, demons will come, and then he is an addition to the army.\"\n\n\"A possible addition.\"\n\nNeither of them enumerates the possibilities of how that particular event could spin out.\n\n\"True. However, this gets him away from whoever usurped your authority.\"\n\nDissent in the ranks.\n\nJulia throws up her hands. \"Okay, you win.\"\n\n\"Of course.\" Delilah grins, and the dimple she shares with Scott winks in and out of existence.\n\n\"What about Jacqueline?\"\n\nDelilah sighs. \"I'm not sure which I like better,\" she admits quietly. \"This soft, pregnant Jacqueline or the tough monarch.\"\n\nJulia jerks back. \"What? Jacqueline without the moderation of faerie tried to murder me. She was a bitch on wheels.\"\n\nDelilah nods. \"Yes, but a little bit of that is needed now. And she's just too broken to give that side of herself. What if something horrible happens? And you need what Jacqueline can bring?\"\n\n\"I think she'll rise to the occasion.\"\n\nDelilah gives a quick nod, looking over Julia's shoulder. \"She's here to see me off.\"\n\nJulia's mouth makes an O, and she turns in the direction Delilah is staring.\n\nJacqueline moves out of the shadows. Neither Scott nor Domi is in sight.\n\nJulia knows it's an illusion. Domi is protective over Jacqueline.\n\nScott would never let anything happen to Julia. However, she appreciates the illusion of privacy.\n\n\"I wished to tell you goodbye.\"\n\nDelilah lifts a shoulder. \"It doesn't matter.\"\n\n\"It does. I have something to tell you. It might make a difference\u2014it might not.\"\n\nJulia begins to back away, thinking she doesn't need to intrude on their private moment.\n\n\"No\u2014stay,\" Jacqueline says.\n\nJulia stills. Jacqueline's face is so grave, Julia's pretty sure she doesn't want to hear the words that Jacqueline will speak.\n\n\"I know that I have been an absent and neglectful mother, as I was with Scott.\"\n\nDelilah says nothing, and Julia's impressed. Most would rub in the obvious.\n\nJulia studies Delilah's face and comes to the conclusion that maybe her vampire nature offers a sort of natural indifference. If so, Jacqueline can't mend that bridge.\n\n\"You have asked, and I have refused.\"\n\nJulia's ears perk as she looks between the two. What?\n\nDelilah's skirts swish at her ankles as she moves forward suddenly. She takes Jacqueline's hands in her own.\n\n\"My sire?\"\n\nJulia holds her breath as the two women face off. Both are dark and severe as well as cunning.\n\n\"He does not know of your existence.\"\n\n\"And I do not know his identity,\" Delilah replies.\n\nJacqueline takes a deep inhale, letting it out slowly. \"It is Gabriel of the Northwestern Kiss.\"\n\nDelilah backs away, dropping her mother's hands as though she were burnt. \"No.\"\n\nJulia's breath escapes in a rush that makes her dizzy. Her hand hits the side of the barn.\n\nDelilah is the daughter of a Rare One, the very leader who tried to get into bed with Tharell to reacquire her.\n\nNow his own daughter will travel with Tharell.\n\n\"He is merciless!\" Delilah cries, and Julia watches Brynn and Tharell make haste to their positions.\n\n\"It is a trait I desired in the fathers of my children.\"\n\nUh-oh.\n\nDelilah's frantic eyes meet her mother's. \"Why?\" Her voice is as sharp as a blade.\n\n\"Survival of the fittest,\" Jacqueline admits in a vacant tone.\n\n\"For whom?\"\n\nJacqueline's eyes shift to the left.\n\n\"For me.\" Her voice has dropped to a thready whisper.\n\n# CHAPTER TWENTY-ONE\n\nSlash\n\n\"I'm not playing the role of blushing virgin.\"\n\nAdrianna is suddenly shy, and of course, that makes Slash go from hard to soft in an instant.\n\nSlash is stalled but not done. In two strides, he stands in front of her.\n\n\"Adrianna?\"\n\nShe glances up at him then away.\n\n\"I'm going nowhere.\"\n\nShe takes a deep breath, and he scents her anxiety. He also sees the muscle definition in the tenseness along her thighs.\n\nSlash wraps his hands around her upper arms. \"I'll be gentle.\"\n\n\"That's not in question.\"\n\n\"What is it then?\"\n\n\"Tony.\"\n\nSlash's hands convulse around her arms.\n\n\"He didn't\u2014I thought.\" Slash touches his forehead to hers. \"Tell me he did not do what I'm thinking. Because he'd need to die again.\"\n\nA sad smile touches her lips then disappears. \"No, but it was close\u2014so close, Slash.\"\n\n\"I don't want to harm you, but the first time will hurt.\"\n\nHer breath is warm against his skin. She's frightened.\n\nSlash can't back down, even now, despite knowing her trepidation.\n\nHer small hands float to his hips, and just like that, he's back to unbearably hard. \"Adi,\" he breathes against her, and his lips are at her temple. \"I can't not want you, female.\"\n\n\"I want you, too, Slash.\"\n\nHe bends at the knees, and her hands fall away. Slash glories in the sight of her for a full thirty seconds then scoops her into his arms.\n\nSlash didn't plan on this.\n\nBut the moss of the forest is dry. He kicks their small pile of scattered clothes with a single swipe, and they fall on top of the dry spongy floor. He arranges Adrianna on top of the bed of moss and clothes.\n\nAdrianna turns her head, and her nostrils flare, her senses heightened in her quarter-change form. \"They smell of you.\"\n\nSlash's eyes run down the length of her body. Her beauty is subtle\u2014perfect.\n\nRound breasts fall softly to the sides of her chest, and her ribcage narrows to a waist he can span with his hands. Her hips flare just wide enough of a woman just past whelpling age and far enough to be ready for what Slash offers. His eyes move to her perfect toes, and she wiggles them under his scrutiny.\n\nHer face flames when his eye come to her sex.\n\n\"Show me, Adrianna.\"\n\nHer thighs tremble, but she parts them. Slash falls to his knees to lay the side of his face against the inside of her thigh. His nose is inches from the most secret part of her.\n\n\"Am I...\" Adrianna's voice shakes, and she clears her throat. \"Okay?\"\n\n\"Okay is not a word I would ever use to describe you, Adrianna.\"\n\nHe can feel her heartbeat through her femoral artery.\n\nSlash turns his face, laying a heated breath at her core.\n\nShe arches her back, \"Slash!\" she says in a whisper-shout, grasping his hair with her hand.\n\n\"Do you like it?\"\n\n\"Love,\" she replies breathlessly.\n\nHis voice rumbles against her wet heat, \"Do you want more?\"\n\n\"Is there\u2014more?\"\n\n\"Yes.\" He places his hands at the apex of where thigh meets her center, and he spreads her like a flower.\n\nAdrianna tenses.\n\n\"Trust me, Adrianna.\"\n\n\"You know I do.\"\n\n\"I will prepare you.\" Slash waits, and when she relaxes, he moves in deeper between her legs. Using his tongue, Slash begins at one side of her and sucks the sensitive flesh deeply into his mouth.\n\n\"Ahh,\" she says, and fingers that had previously bit into his scalp now rub through the stubble of his hair.\n\n\"That's it. Open yourself to me.\"\n\nAdrianna's knees fall apart, and he tucks his hands underneath her hips, cupping the globes of flesh, lifting as he pulls her more deeply into his mouth.\n\nShe whimpers, and he licks her from entrance to clit, using the flat of his tongue to rub back and forth on the sensitive nub. Adrianna begins to make little sounds of pleasure, coming undone from the attention.\n\nSlash's erection is a painful, throbbing mass. He denies himself, giving pleasure to the only female he could ever consider mating. She begins to ride his mouth with eager hips, and he matches her rhythm with his own.\n\nWhen he pierces her entrance with his tongue, she cries out, and Slash does it again and again, holding her bare ass with one hand as the other strokes her slick clit rapidly with his thumb.\n\nAdrianna's body tenses as her head whips violently back and forth. She screams, her body an arc, and Slash slows his tongue penetration then stops it, carefully lowering her to the soft forest floor.\n\nHer eyes are spinning gold, and every feature of her beautiful face glows in stark relief.\n\n\"Take me,\" she says, and Slash is unsurprised by the growling quality to her voice.\n\nSlash lines himself up with her center.\n\nAdrianna gives silent consent with the widening of her legs, letting her arms fall behind her head. Her generous breasts lift, the nipples pointing at his body.\n\nSlash enters her with a single, hard shove. He tears through her barrier and meets the end of her in a quivering thrust of flesh married.\n\n\"Oh my\u2014moon!\" Adrianna gasps, struggling not to tense against his entry.\n\nSlash feels horrible, but her slick heat and the way her body welcomes his as though he never left, is too much, and he begins to move gently within her.\n\nHis head hangs as he lifts his weight from above her, working deeply in and out. \"I am sorry\u2014you are\u2014I am lost in you.\"\n\nShe rocks back against him, \"Nope, I have you, Slash.\"\n\n\"You do, every part of me,\" he whispers with clenched eyes.\n\nHis eyes snap open as her hands rest on his hips, guiding him\u2014encouraging him. \"Do not. I can't hold back, Adrianna.\"\n\n\"I don't want you to.\"\n\nHe freezes above her, his gaze searching hers in the shadows. \"Have I hurt you?\"\n\n\"Not like I'll hurt you if you stop.\"\n\nSlash smiles, and she grins back, her teeth very sharp. He begins to move with purpose, using long gentle strokes. She fits him like a hot slick glove as he bottoms out to kiss her womb.\n\n\"Take me, take me, take me, Slash.\"\n\nSlash studies her expression, and when he's satisfied it's what she wants, he does what his body has been longing to do since she came of age. He buries himself to the hilt inside her.\n\nThey grunt at the deep joining, and her legs fold over his back. He lifts her hips, tilting them forward as he begins to pound inside of her.\n\n\"Please,\" she whispers, and Slash can hold back no more. She is tight and untried by all but him. He plunges in a final time and unloads his seed into her depths, simultaneously scent-marking her.\n\nShe milks him, pulsing around him as she makes little grunts of satisfaction. They are music to his ears. That he could possibly satisfy this female who's entrusted him fills Slash with an unaccustomed sensation.\n\nIt's beyond the momentary contentment of this act between them.\n\nBeyond that, he has claimed a mate.\n\nMore than fleeting happiness, Slash feels true joy\u2014his and hers.\n\nTheir own.\n\n*\n\nSlash scrolls a fingertip down Adrianna's naked side and watches the trail of gooseflesh rise in its wake.\n\nAdrianna giggles. \"Stop. You're tickling me, ya butt.\"\n\nSlash gives a lazy smile. \"Butt, eh?\"\n\nAdrianna rolls onto her back, and his hand rests on her naked hip.\n\nHer eyes twinkle. \"You're so old, Slash.\"\n\nHis eyebrow raises. There is a great span between their ages. \"Does that bother you?\" Slash asks, hoping it does not because there's no rectifying it. He has taken her as mate.\n\nShe raises her hand, and he doesn't flinch when she touches the small unsavory mound of flesh that sits in the center of his upper lip. Instead, he catches her hand with his own and kisses her finger.\n\nA long shuddering sigh eases out of her, and his gaze catches on Adrianna's gorgeous breasts.\n\n\"You like looking at me.\"\n\nSlash's eyes move to her face. \"Yes,\" he admits. \"Now that I can, I cannot look away.\"\n\n\"You're a romantic, Slash.\" There's surprise in her voice. He hears pleasure, too.\n\nHis brows quirk and the heat of embarrassment rises to his face.\n\n\"Don't deny, buddy\u2014I can tell.\"\n\nSlash mounds her breasts, and her breath catches, her hazel eyes darkening like the threat of a storm.\n\n\"What else can you tell, Adrianna?\" he asks softly, never looking away while rolling her pebbled nipple between his thumb and finger.\n\n\"I can tell that I want to go again.\"\n\nSlash smiles, cupping her heat with his other hand, and she spreads her legs.\n\n\"Are you sore?\" he asks, kissing first one thigh then the inside of the other. He rolls his face against her flesh, taking the skin deep between his teeth, smelling her blood and his seed mingled together.\n\nAmbrosia.\n\n\"Not enough,\" she says in a voice gone low with need.\n\nHe releases the flesh of her thigh. His teeth leave indentions, but the skin is unbroken. The marks plump and smooth out as Slash watches.\n\n\"The quarter-change is helping to heal me,\" she says, relief in her voice.\n\n\"It was smart, Slash.\" She laughs, and he looks up from where he just pleasured her to her expressive face. \"But I think there was an ulterior motive.\"\n\n\"Oh?\" he asks, his fingers caressing her entrance.\n\n\"Yes,\" she says, her voice breathy, \"I think you just wanted me as much as you could get.\"\n\nSlash's fingers stop. He glides up her body, caging Adrianna with his arms, placing his hardness against her soft slit. He cradles her face with his hands, elbows planted on either side of her.\n\nHe kisses her forehead, each eyelid, then her mouth.\n\n\"Yes.\"\n\n\"Ha! I knew it.\"\n\nHe brushes his lips against hers.\n\n\"I knew once I started loving you, once would not be enough.\"\n\nAdrianna wraps her arms around his neck.\n\n\"For me, either,\" she whispers.\n\n# CHAPTER TWENTY-TWO\n\nDrek\n\n\"That was a round about,\" Bowen snorts, twirling his finger in the air.\n\n\"We are Lanarre\u2014we can scent the rain on the wind.\"\n\nBowen rolls his eyes.\n\nDrek scowls. \"What is that look for?\"\n\n\"We may be Lanarre, but when Tahlia takes flight, there's no scenting her.\"\n\nDrek hates to admit any failure, but Bowen is right. If the other Alpha female's ground scent hadn't been nearby, they might have missed Tahlia altogether.\n\n\"Interesting scent mix at the gasoline station,\" Bowen remarks.\n\nDrek gives him a sharp look. About half of all supes, even vampires, could have scented the mess Tahlia made. He didn't know why Tahlia was with another female Were, but he suspected the second female was rogue. Drek understood the implications of an Alpha female running solo. None were good, certainly not once Tahlia was added to the equation.\n\nDid the female help Tahlia?\n\nDrek doubted that. Tahlia's scent was mingled with the blood of the males. That could mean only two things.\n\nThey had harmed her and Tahlia had defended herself. That option was highly unlikely. No average Were would harm a Lanarre.\n\nThe second option: she had attacked them in defense of another, most likely the second female.\n\nFrom all reports and his abbreviated correspondence with Tahlia, he suspected the latter to be the most probable.\n\nDrek smiles, palming his chin.\n\n\"You've thought of something?\" Bowen asks. The Were has been Drek's friend and guard since whelphood. Bowen's family has been the guard of the Lanarre royalty for a thousand years, an anomaly, for they are not human.\n\n\"Yes.\"\n\nBowen's dark eyebrows rise, his light brown eyes steady on Drek.\n\n\"I think she defended the Alpha female.\"\n\nUnderstanding lights Bowen's expression. \"Good call.\"\n\n\"There's a remote possibility that she was defending herself against the two males.\" He gives Bowen a sharp look.\n\nHe immediately shakes his head. \"Absolutely not. A Lanarre female just out of whelp? It'd go against every precept in Lycan culture.\"\n\n\"There's precedence.\"\n\nNeither of them speak of the scent of the Alpha Were who has murdered over nine humans, including Tahlia's guards.\n\nBowen lifts a shoulder. \"Who knows what that Were was made of? If he was being tasked by a packmaster or acting of his own volition?\"\n\n\"True.\" But Drek is troubled. A Were who would rampage through a human establishment like that was capable of other deeds.\n\nHe and Bowen exchange an uneasy look. \"He would not kill her but might do other unsavory things.\"\n\nA flutter appears in Bowen's jaw, and voices what has occurred to them both. \"He would not rape a Lanarre female. She's barely more than a girl.\"\n\nDrek's stomach does a slow, heated roll. \"He is not Lanarre. It would go against our instincts to protect females. But as you've said, we can't be absolutely sure.\"\n\nBowen throws his hands up in the middle of the parking lot of the decaying gas station. \"Let's run with the assumption that he missed her. That she hid herself. That the slaying of her guardians wasn't for nothing.\"\n\nDrek's eyebrows jerk up to his hairline. \"An Alpha male would scent a Lanarre.\"\n\n\"Not if she was in bird form.\"\n\nDrek puts his hands to hips and walks off. He paces back and forth. Ignoring the coming dawn and the coolness, he scents the ocean in the distance.\n\nHis nose is at the scene, and he can't get out of his head. He whirls and looks at Bowen. \"Then we make haste. We scent where their car has gone and follow.\"\n\nBowen walks to him then grips him by the arms. Though Bowen is only a fraction shorter, Drek is built for war. All Lanarre are. The royal line is the most barbarically fashioned. Both men stand nearly eye-to-eye at six feet five.\n\n\"Don't lose faith. This crazed male missed her once. It's clear Tahlia is now with the Alpha female we presume she helped.\"\n\nBowen wrinkles his nose. \"Those males were from the Western. Easy to scent.\"\n\n\"Possibly drones sent by Tramack?\"\n\nBowen rolls his shoulders into a dismissive shrug. \"I don't keep up on common Were politics.\"\n\nDrek grins suddenly. \"And you accuse me of being a snob?\"\n\n\"Ha!\" Bowen replies, dropping his hands from Drek's arms and walking toward the gas pumps. \"No accusation necessary. You're an elitist.\"\n\nDrek can't deny it\u2014the common dens don't adhere to the ways of Lycan tradition enough to earn his respect. A few dens still cleave to the traditions of old, but they are few and far between.\n\nBowen drops to his hands and knees on the ground, dirtying the knees of his well-worn jeans. He turns his face and hovers above the damp asphalt. He flares his nostrils once then makes several small chuffs.\n\nHis head snaps up.\n\n\"I have it.\"\n\nMetal is especially hard to scent, but Bowen has made a little game of it. He can usually get the decade of a vehicle from scent alone. Bowen is good enough at tracking that he can determine what year the car is.\n\n\"Older model, 1960s Chevy\u2014heavy.\"\n\n\"They're all heavy from that era,\" Drek says with a touch of humor. \"Really old model.\"\n\nBowen rocks back and sits on his heels, nearly yanking his shoulders to his ears. \"Perspective. I was a whelp in that day, but you were already thirty-five.\"\n\nDrek smirks.\n\n\"One hundred percent original components,\" Bowen says triumphantly.\n\nDrek's jaw drops. \"Really?\"\n\n\"Yes, probably an old couple, had it since the day they got married. Grocery-getter for the woman.\" His eyes glitter. \"It happens.\"\n\n\"Rarely.\"\n\n\"Better for us, the signature will be clean to follow.\"\n\n\"What are we waiting for?\" Drek says.\n\nBowen bounds to his feet in a single leap from his toes. \"Nothing, let's roll.\"\n\nDrek gives him a tolerant look. Bowen loves human slang. Drek finds it tiresome, though he's adopted a few key phrases himself.\n\nDrek grits his teeth, angry over the loss of Tahlia's human guardians with an undercurrent of acute anxiety for her. The humans who serve the Lanarre are greatly loyal, with generations upon generations of service.\n\nDrek would dismember the male who slaughtered them if Tahlia hadn't already. Any male who would touch a Lanarre female deserves death. Drek is keenly aware of Tahlia's proficiency in defense. However, she is still female. And judging by the remains of the humans, the male who is responsible for the massacre is strong.\n\nHe seems stronger than most Were, but he's not Lanarre-strong. If the male were human, Drek would have assumed he'd been taking some artificial enhancer, like PCP.\n\nBowen turns, giving Drek a considering look. \"Stop thinking. Let's go.\"\n\nStop thinking. Easier said than done.\n\nDrek glances over his shoulder, noting the neon sign that reads gas switching on.\n\nThey'll track during the beginning of day and find Tahlia by nightfall. Drek is optimistic.\n\nJust beyond the tree line, Bowen and Drek morph into quarter-change. The ability that is generally reserved for female Were is available to all Lanarre, male and female. Still, so far from the moon's fullness, the ability comes at a cost to the males.\n\nBowen and Drek race parallel to the road, following the scent of the car. They slow as the scent stops, then they climb the embankment that leads to the highway. A car passes, and they freeze, waiting for the scent to waft back.\n\nIt floats down to rest.\n\nDrek moves to the shoulder, sinking to his haunches. He touches the impression of deep treads biting into the soft dirt and pebbles. He closes his eyes, inhaling deeply.\n\nThe car was parked here. Tahlia was inside.\n\nHis eyes seek Bowen, but he's already at the edge of the woods.\n\n\"I have her.\"\n\nDrek looks, hoping that Bowen literally has her.\n\n\"Her scent, Drek.\"\n\nDrek's shoulders drop. \"I know,\" he replies and glances behind him.\n\nTwo cars were parked at the shoulder. He's scented them both.\n\nWhere did the other drivers go? Why were they here, parked behind my chosen? Drek doesn't like it, and he can't dismiss the possibilities of what it represents, especially the scent.\n\nThe ones who were here beside the unknown female and his chosen\u2014they are scentless.\n\nBowen motions impatiently.\n\n\"I am a prince, you know,\" he reminds Bowen.\n\n\"Uh-huh, get your princely ass over here so we can find Tahlia.\"\n\nDrek smirks. \"You don't show proper respect, Bowen. And no one cares more about finding Tahlia than I.\"\n\nBowen pegs his hands on his hips, one foot in the woods and one on the slope. \"And you clearly don't give a shit about anything but finding Tahlia.\"\n\nDrek grins. \"Yes, but we have a problem.\"\n\nBowen's brows come together, all humor gone, and his posture tenses. \"What?\"\n\n\"We have scentless followers.\"\n\n\"Vampire?\" Bowen asks instantly, his nostrils flaring and a scowl forming on his face. \"Can't scent a thing.\"\n\n\"Exactly.\"\n\n\"Doesn't feel right, Drek.\"\n\nDrek nods. Vamps would avoid daylight for obvious reasons and especially Lycan females. That interaction is a mess waiting to happen. Vampires employ stealth. It is their nature.\n\nNo, the followers are not vamps.\n\nDrek studies the impressions at the shoulder again. His eyes move restlessly over every groove or tread. His mind touches on an idea and instantly dismisses it.\n\nThere has not been a tangible appearance of cloaked demonic in centuries. There would not be one now.\n\nStill, it strikes a discordant note deep within Drek. He doesn't favor the idea of Tahlia with an unknown female while a deranged Were with a penchant for murdering innocent humans is on the loose. And now this new potential threat...\n\nDrek doesn't trust anything he can't scent. No self-respecting Lycan would.\n\n\"Drek?\" Bowen calls from the bottom of the embankment.\n\n\"I don't like it.\"\n\nBowen throws his hands up in the air, disbelief saturating his features. \"What's to like?\"\n\n\"These other scentless beings changes nothing,\" Drek says slowly. Except his chosen is vulnerable and is probably not experienced enough to understand a hidden threat is closing in. She is very young.\n\nDrek jogs down the small hill to the forest's edge to join Bowen, who's already racing ahead of him. They run side by side, shoving aside alder branches like ready whips in front of their faces.\n\n\"Wolfen,\" Bowen gasps.\n\nTheir clothes shred. Drek's more prepared than Bowen, who is left in his expandable underwear. Drek specifically chose the plain athletic pants because they would accommodate his change to wolfen form.\n\nThe Lanarre all possess coats of silver. A light downy mat of hair like gray smoke covers Bowen as he runs, and Drek knows he looks nearly identical. In wolf form, his coat is tipped in silver but otherwise black. In wolfen form, they both move with power that they could not spare while in quarter-form.\n\n\"Wait!\" Drek calls, stopping so quickly that he snatches at a trunk to arrest his progress. The tree groans with the impact as his talons punch into the bark. A fine spray of needles falls softly, and the smell of pine is pungent.\n\nDrek flings them out of his hair, but some remain tangled in the fine hairs that cover his body\n\n\"What?\" Bowen asks, jogging back to Drek's position.\n\n\"I scent Blood Singers.\"\n\nBowen nods, unsurprised. \"This is close to their territory.\"\n\nDrek inhales deeply, his eyes widening. \"Tahlia,\" he breathes her name reverently.\n\n\"How did I miss that?\"\n\n\"Chasing the ball!\" Drek answers with a healthy dose of sarcasm.\n\nBowen flips him off. \"I did not ignore scenting to chase the one scent.\"\n\nDrek snorts, his snout wrinkling. Bowen always has trouble multi-scenting. He gets one scent and gets obsessive.\n\nAn irritated exhale rushes out of Bowen. \"Okay, maybe a little.\"\n\nDrek's talons click as is thumb and index come together. \"All the way.\"\n\n\"Right\u2014go on,\" Bowen replies impatiently.\n\n\"They're not enemies of the Were,\" Drek comments significantly. \"They would take in two lone Were females.\"\n\n\"Yes,\" Bowen admits.\n\n\"So Tahlia must have found refuge in their territory.\"\n\nBowen sighs and shoots a glance Drek's way. \"That's a reach.\"\n\n\"I don't believe in coincidence.\" His eyes lock on the slowly spinning mercury orbs in Bowen's face. \"An Alpha female, my chosen, and two Singer males\u2014together?\"\n\n\"You're right. But this will have to be handled with a degree of diplomacy you lack, Drek.\"\n\nDrek's face tightens. \"I have certain inalienable rights here.\"\n\n\"Of course. But they're a different species, with different rules that govern their kind. We might not be able to just waltz in there with nary a care and grab Tahlia. There might be a protocol in place.\"\n\nDrek's face whips to Bowen's and he feels his eyes spinning in response to his heightened agitation.\n\n\"Fuck protocol.\"\n\nBowen's chin dips. \"I was afraid you'd say that.\"\n\nThere are no words after that, only speed.\n\n# CHAPTER TWENTY-THREE\n\nCyn\n\nCynthia folds her arms. \"You have got to be kidding?\"\n\nJason whirls around, punching the wall. Plaster flakes float to the floor. \"Do I look like I'm fucking kidding?\"\n\nCynthia studies his tense body as if he were a coiled snake. She sighs. \"I'm not gonna lie. This sucks donkey dicks.\"\n\nJason snorts. \"Yeah.\"\n\n\"What are ya gonna do?\"\n\nCynthia feels for him. He's so raw. The last three years have been a torture. And in a way, it never stopped. Jason's been on a perpetual roller-coaster ride like an emotional junkie with no fix in sight.\n\n\"What do you think?\" he asks, disdain thick in his voice.\n\nCynthia looks down at her feet, momentarily taking note of her shitty footwear. Why does that matter? She doesn't know, but in this crazy-ass new world of hers, she just wants something cute, goddammit.\n\nInstead, she faces Jason\u2014and reality. \"You're going.\"\n\nHe jerks his head in a nod. \"Hell, yes. I'm not sticking around to watch Julia do Scott.\"\n\n\"God, that's crude\u2014even for you.\"\n\nHe strides toward her, but Cynthia holds her ground. Jason's volatile, but she doesn't think he'll melt down all over her.\n\nJason sees something in her and slows, his expression like thunder. \"What? You think I'd put my hands on you?\"\n\nCynthia quickly shakes her head. \"No, but you're\u2014you're not yourself, Jas\u2014\"\n\n\"No shit?\" He rakes a hand through his sandy hair. \"My wife\"\u2014he thumbs his chest hard enough to leave a bruise\u2014\"is all soul-tied...\"\n\n\"Meld.\"\n\n\"Whatever-the-fuck!\" he roars, and Cynthia's mouth snaps shut.\n\n\"With Scott,\" he spits.\n\nCynthia's in full-diffusion mode. \"Listen, Jason, I know you're freaking out right now...\"\n\nHis hands clench into fists, his jaw goes hard, and his eyebrows yank in blatant disbelief. \"Yeah. Ya think?\"\n\nCynthia blows out a tight breath, crossing her arms. \"But you'll never forgive yourself if something happens to Jules.\"\n\nHe meets her eyes, his hazel irises turning green.\n\n\"You're not going all wolfy on me, are ya?\"\n\n\"Maybe.\"\n\nCynthia fold her arms. \"Well\u2014don't.\"\n\nThe green bleeds back to his human hazel, and Cynthia lets a sigh of relief escape. Jason as human is bad enough. Wolfen is just plain dangerous.\n\n\"I already can't forgive myself,\" he confesses harshly.\n\n\"Why?\"\n\nJason turns, and both of his fists come down on the wall. Cynthia yelps, retreating a step as her hands go to her chest, her heart bouncing around like a ping- pong ball.\n\nThey sure do a lot of wall repair here.\n\nJason turns away, speaking to the wall he just ruined, \"Because of my clueless ass, Julia was taken, Kev was killed\u2014eventually, you were turned.\"\n\n\"No, God\u2014Jace!\" Cynthia cries, moving behind him and putting her palms on his muscular back. \"This is not on you.\" She slaps him lightly. \"We didn't know this was even a part of our world.\"\n\nJason turns to face her. \"But if I was always Singer, why didn't I have a gut instinct to protect Julia? Shouldn't I have known something?\" His voice cracks, regret shattering the timbre into brittle glass.\n\n\"Remember the shooting, Jace?\" Cynthia's eyes search his. \"If that wasn't protecting her from that fuckwaffle teacher, I don't know what is.\"\n\nJason stares at her for half a minute, then he hugs her. \"I gotta go, Cyn. I can't be here. I don't want anything bad to happen to Jules, but she's got Scott.\"\n\nCynthia pats his back then grips his T-shirt. \"I know it's selfish, but I don't want you to leave. It's like breaking up the three musketeers or something.\"\n\nJason touches her cheek as he steps away. \"Yeah. It is. But I'll hurt her worse if I stay. I can't stand that prick.\"\n\n\"Because he has Julia or something else?\"\n\nA rueful smile crosses his lips. \"I don't have lofty principles. He's taking my wife, and that's all the reason I need to hate his stinking guts.\"\n\nCynthia can't respond to that. She understands. It's not reasonable, but it's real. And that's what matters to her.\n\nShe raises her eyes to meet his. \"When?\"\n\n\"I've got my shit packed. I'll say goodbye to Jules and get the hell out of here.\"\n\n\"What if there's, ya know, authorities hunting your butt?\"\n\n\"Let 'em.\" He stuffs his hands in the pockets of his jeans but not before Cynthia catches sight of his scraped knuckles.\n\n\"There's nothing I can\u2014\" She wants to beg, to reason with him. There has to be a way.\n\n\"No.\"\n\nShe sees the determination on every tenacious line of his face. \" 'Kay.\" Cynthia blows a stray hair out of her face, glancing down. \"I'm sorry.\"\n\n\"Not your fault.\"\n\n\"It feels like it's all our faults.\"\n\nJason plows his fingers through his hair. \"Maybe.\"\n\nCynthia jerks her face up.\n\nJason shrugs. \"Feels like someone stole my life then gave it back to me like ground beef. And it's spoiled now.\"\n\n\"Gross analogy.\"\n\nHe lifts his shoulders. \"Tell me I'm wrong?\"\n\nCynthia shakes her head.\n\n\"Can't,\" she whispers.\n\nJason steps into her space and kisses her forehead, briefly cupping the back of her head. \"Take care, Cyn.\"\n\nShe nods. There's no talking, too many tears and no decent words.\n\nIt sucks.\n\nAnd that is all.\n\n*\n\nJulia\n\nHer guilt is an endless swamp. Hot and rank, it washes around her legs, threatening to drown her with its smell and heat.\n\nShe knew he would come, but as Jason moves toward her, Julia still tenses in surprise.\n\n\"Hey,\" she calls out. For once, she's blissfully alone, yet she feels the separation from Scott like a weight.\n\nJason moves faster, and her eyes widen as he crashes into her and his arms snap around her smaller frame.\n\nJulia opens her mouth to scream, and his lips smother hers as they smash into a wall in a tangle of arms and legs. His hands brace Julia before her head hits, his body pinning her against the exterior of the barn.\n\nJason kisses her fervently, desperately, and she opens her mouth to his. The kiss deepens, their tongues twining in a passion at once familiar, but now somehow wrong.\n\nThough married, they're separate.\n\nHis assault on her mouth cools to pecking. Reluctantly he releases her. Jason grasps Julia's jaw, forcing her to meet his gaze.\n\nFear, sorrow, and adrenaline combine in a dizzying cocktail that surges through her and the tie she shares with Scott. \"He's coming,\" she whispers.\n\nJason pulls a face of disdain. \"Of course he would. Let him come.\"\n\nJulia cups his face. A face she loves. A face she'll have to let go.\n\n\"Why, Julia?\" he asks, slamming a palm into the wall next to her face, and she flinches.\n\nShe doesn't answer because Scott's pulling him off.\n\nHe hurls Jason ten feet, and he lands, his clothes bursting off his body as he morphs into wolfen.\n\nScott changes into his Combatant form in the time it takes her to expel the air from her lungs.\n\n\"No!\" Julia screams.\n\nScott and Jason collide midair.\n\nThey land, and Julia steps between them. Strong hands latch on and fling her away, sending her airborne. Julia tries to work her telekinesis but fails as her emotional snare intensifies.\n\nScott catches her.\n\n\"Stay here,\" he growls, and Julia cringes when she sees his form. He whirls around, and there's empty space where Jason just was.\n\nVanished.\n\nJulia doesn't know if it's forever. But it feels like it is.\n\nScott straightens from his crouch. The monster slowly melts back to human. Teeth like a saber tooth tiger's retract, talons as long as fingers slide to nails, his stature shortens, and his eyes stop glowing.\n\nJulia takes a shuddering inhale and sits on her ass, dumping her face in her hands.\n\n\"That was awful,\" confessing the words as she fights sobbing and loses.\n\nScott doesn't agree, say he's sorry, or make excuses for himself or Jason. He scoops her off the ground and carries her to his bedroom.\n\n*\n\nCyn puts a wet washcloth on Julia's forehead.\n\nBliss.\n\n\"That went well\u2014not.\"\n\nJulia's eyes roll to meet Cyn's gaze. \"Yeah. I felt like an ass.\"\n\n\"Jason's not really all there,\" Cyn says, tapping her temple. She plops down, perching at the side of Scott's bed.\n\n\"Wow, that sounds bad. Like I think Jace is nutso. I mean...\" She sighs, wrapping her long hair in a fist and tossing it over her shoulder, \"He's not crazy. He's just frustrated. But I think he finally made the right choice.\"\n\nJulia's all talked out.\n\n\"He left so you could have a life, Jules. Jason left so he might, too.\"\n\n\"But he never came to terms with it\u2014with us.\"\n\nCyn's eyes drift up to the ceiling then pierce Julia when they move back. \"How could he? You're the one with the soul-meld\u2014not him.\"\n\n\"With this thing\"\u2014Cyn swings her finger back and forth\u2014\"that you and Scott have\u2014you have to move past memories. Jason doesn't have that buffer, the chemical things happening. He just has memories and devotion.\"\n\nJulia dies a little inside at her words.\n\n\"And the fact that neither of us knew we were Singers. And the fact that he's a Were now. I mean, we're so far from being human anymore, it altered everything.\"\n\n\"I know,\" Julia concedes softly. Logically, with everything that's changed in their lives, there is zero chance that they, or their relationship, would have remained the same.\n\n\"I hate to say this, but you'll have to think about annulling the marriage.\"\n\nJulia shakes her head, lacing her fingers tightly. \"I can't. That would lead someone straight here.\" And on some level, she feels like shit for even contemplating it.\n\nCyn allows a smile. \"Let Truman do it.\"\n\nJulia crosses her arms. \"Oh, yeah, that's so gonna work, Cyn. There's a manhunt gunning for Truman.\"\n\n\"Not so much, Jules. He's been gone awhile. Homer, Alaska, doesn't have the resources to look for a cop a year away from retirement age.\"\n\n\"A year away from retiring,\" Julia repeats in awe.\n\n\"I know, right?\" She looks away, and color floods her cheeks. \"He's like so not looking his age.\"\n\nJulia scrutinizes her expression. \"Do you kind of dig him?\"\n\nCyn looks at her knotted hands. \"Don't say.\"\n\nWow, Cyn is crushing on Truman.\n\n\"I thought we were talking about Jason here.\" Cyn huffs, swinging her leg.\n\n\"I think we've discussed him long enough. I don't\u2014I can't deny I love him.\"\n\n\"But you're not in love.\"\n\nJulia shakes her head. \"I thought I was.\"\n\n\"Too much water under the bridge?\" Cyn asks.\n\nJulia gives her a defeated look.\n\n\"Too much blood.\"\n\nJulia catches Scott's eye as he dips his head into the room.\n\nHe leaves without a word, allowing the women their grief for the past without him as an audience.\n\n# CHAPTER TWENTY-FOUR\n\nSlash\n\nSlash loops his arm around Adrianna, drawing her against his body and kissing the top of her head. She smells of woods and him.\n\nSlash likes it.\n\nHis wolf gives a joyous roll just beneath the thin layer of skin that makes Slash look human.\n\nHis animal is pleased by his choice of mate.\n\n\"You're all mushy and soft now, stud.\" Adrianna pokes him in the side. \"Ooh, maybe not too soft.\"\n\nShe steps away and looks at his stomach, cocking her head. \"You must have a twelve pack there. That's hot.\"\n\nSlash raises his eyebrows, and he glances down at his flat stomach. \"A what?\"\n\nShe smacks him, and he catches her hand so quickly, she gasps softly. He flips her hand over and kisses the center of her palm.\n\nShe gives a contented sigh.\n\nSlash could get so used to the noises she makes.\n\nAdrianna blushes at the look he gives her.\n\n\"I thought you said you weren't the blushing virgin.\"\n\nShe looks up at him through her long eyelashes. \"I'm not, anymore.\"\n\nIt's Slash's turn to feel a touch of embarrassment. He pulls her back to him, holding her close. \"It was a precious gift you gave me.\"\n\n\"Slash,\" she says against his bare skin, \"you don't have to be so serious all the time. Life's not so bad, y'know.\"\n\nShe is very young. Sometimes Slash forgets. He memorizes every line of her face and her every curve. Life is an ever-changing tide. When he navigated the current alone, it was manageable. Now there are two of them.\n\nAnd someday, there may be a whelp. His chest swells at the thought of something he assumed would never be a part of his life. A thrill moves through him like an electrical current.\n\nA twig snaps and Slash whirls, shoving Adrianna behind him.\n\nHis eyes skate across the woodland. All is in order, and he can vaguely see the dark outline of the Singer's mansion in the distance.\n\nBut Slash is Red, so he maintains his alert posture. He never dismisses his instincts. His eyes belie what his nose tells him is true. Adrianna's fingers grip his flanks, and he flexes in preparation for a change in form.\n\nIt is not necessary for a Were to change to full wolf except for that time the moon calls. A quick glance tells him the moon is half-gone. There is no wolf at the ready. And he would be horribly vulnerable when he changed back, leaving Adrianna unprotected.\n\nNo, determine the threat and go from there.\n\n\"Come out. I hear you,\" Slash announces loudly.\n\n\"Slash,\" Adrianna says. He can scent her fear as if it were his own. He shares it.\n\nSlash didn't know fear before Adrianna. With a mate, it is part of the fabric of his thought process. Her welfare is a priority he can't deny even if he wanted to.\n\nA large Were moves from a thick stand of trees and walks toward them gracefully. That is quite a feat, considering his size.\n\nInstantly Slash intuits he's not from a nearby region.\n\nTheir nostrils flare as he approaches and both Were covertly scent the other.\n\nWarily, the other Were circles Slash and Adrianna.\n\nHis eyes take her in as she stands behind Slash.\n\nSlash growls low in his throat. \"Do not look at her.\"\n\nThe strange Were cocks his head to the left as though considering it as a request. Of course it is not a request. Slash doesn't make those.\n\n\"She is yours?\"\n\nOdd phrasing.\n\nAdrianna sucks in a breath. \"Weiner,\" Adrianna mumbles quietly.\n\nSlash ignores Adrianna's indignation. \"She is my mate.\"\n\n\"Most recently, by the smell of it.\"\n\nSlash stiffens, locking down his expression. \"Voice your concern and business, Were.\"\n\n\"I am Tramack. And I believe you have something that belongs to me.\"\n\nIt's impossible for Slash to contain his confusion. His mind sprints through the possibilities and finally lands on the vague memory of two females arriving the day before. Could that...?\n\nNo. One is Lanarre. This Were is common, not a spot of either, Red\u2014or Lanarre.\n\nHe is Alpha. And Slash senses he's a packmaster. He has that air of expected obedience about him.\n\nSlash is obedient to no Were. \"I don't have anything that belongs to you.\"\n\n\"I seek a female Alpha.\"\n\nSlash shrugs, giving him nothing. \"There are none but my mate, who you see here.\"\n\nTramack makes a show of scenting Adrianna from his distance of twenty feet.\n\nSlash decides he doesn't like him. Of course, that's not atypical. A sudden thought occurs to Slash.\n\nWhere is Zeke?\n\n\"She is Alpha, but not who I seek.\"\n\nHis attention returns to Tramack. \"I understand that.\"\n\n\"With whom do I speak?\"\n\n\"Slash.\"\n\nTramack dramatically runs his eyes over Slash's scar.\n\n\"You know, you're a first-rate chode. Why don't you go hunt around for your chickie somewhere else,\" Adrianna comments.\n\nSlash sighs. She'll be the death of me.\n\nTramack gives Adrianna a considering look. \"You know\"\u2014his eyes flick to Slash's\u2014\"an insubordinate female from our pack would be handled before her behavior got out of hand, as it appears to have with your female.\"\n\nSlash's patience thins. \"She isn't from your pack.\"\n\n\"From which pack does she hail?\"\n\nUnease washes over Slash, and he squelches it before it can be scented. He deliberates whether he should answer or not. He decides against it. The less this Were knows, the better.\n\n\"Northwestern,\" Adrianna says.\n\nSlash groans inside his skull.\n\nTramack gives a horrible smile of triumph that clenches Slash's guts. \"Ah. That explains things.\"\n\nSlash's frown turns to a scowl as his hands fall to hang at his sides. He doesn't like surprises. \"Oh? Enlighten me.\"\n\n\"That packmaster is gone. The rumor mill's rife with stories of his death, along with his second's.\"\n\nTony.\n\nAnd Manny. Slash had scented Lawrence and Emmanuel. They were buried beneath his feet. How would Tramack know what Slash just confirmed?\n\n\"We of the Western are tight sister dens with the North.\"\n\nChicken flesh rolls out like the red carpet across Slash's skin. Adrianna lays her face against his bare back, breathing in his scent.\n\nHer uncharacteristic silence speaks for her fear. Slash hates that their tender moment together has been followed up by an unplanned meet with this power-hungry Tramack.\n\n\"If she is Northwestern, then automatically she is part of the Western if Lawrence is declared dead. You know this.\" Tramack tosses his hand out as though his words are a matter of course.\n\nSlash's gut does a slow revolution. \"I do.\"\n\n\"And you\u2014you are not a part of the Northwestern.\" Tramack laughs.\n\nSlash does not see the humor. All he can do is scent his female's fear mixed with misery behind him.\n\n\"You are Red. I'd know that taint anywhere.\"\n\nSlash moves toward him and Adrianna grips him by the hips with slick palms, like an anchor at his back. \"No, Slash. He's baiting you.\"\n\n\"Listen to the little woman, Slash,\" he jeers, \"and enjoy what time you have with her before she is absorbed into my den.\" He thumps his chest.\n\n\"I'm not a \u02bblittle woman,' you neutered dog.\"\n\n\"Adrianna,\" Slash says like a slap, and she chokes back a sob.\n\nTramack gives Adrianna a thoughtful look. \"I shall enjoy giving you a lesson in manners.\"\n\nSlash shakes her hands off and dives against Tramack. They hit a wide trunk behind them with a crash. Slash lifts the other Were by the neck. \"You will not touch her, look at her, or take her anywhere. There will be no lessons learned by your hand.\"\n\n\"You know Lycan law, Red. She cannot mate outside the pack. You thought yourself clever, that you could circumvent the law where it suited you.\"\n\nSlash lowers Tramack to the ground, his anger clouding his scenting too late.\n\nFour Were move out of the wood.\n\n\"Slash!\" Adrianna yells.\n\nSlash's face swivels to look at her. Alone and small, she's unprotected. \"Stay there!\" Slash roars.\n\nTwo of the Were split from their position, each one walking toward Adrianna.\n\nHer quarter-change morphs to wolfen. Talons tremble at the tips of her fingers, and she whips them to her sides, the air whistling between the bony knives.\n\n\"Who's first to lose their teeny penis?\" she growls, her talons clicking.\n\nSlash loves her more than the moon.\n\n*\n\nTessa\n\nTessa's been edgy all day. There was a huge ruckus when the Rare One and her human husband had a falling out, then he split. Her Singer soul-meld is now at her side, a bunch of misplaced Region Two Singers are here, and Tahlia's not happy with her.\n\nAnd the two Singers she witnessed tossing the other Singer down a trapdoor chute in the middle of the hall... well, it feels like a fine time to go.\n\nTessa jogs down the hall to Tahlia's borrowed room, and with a quiet knock, she enters.\n\nIt scares the hell out of Tessa to see Tahlia packing.\n\n\"What is it?\" she asks, looking from the bag to her exotic face.\n\nTahlia looks up from her packing. \"I have a strange feeling.\"\n\nThat makes two of us.\n\n\"I'm already packed,\" Tessa admits.\n\n\"Do you\u2014\" Tessa sighs, giving her braid and irritated fling behind her shoulder. \"I mean, I know you said you don't really own your life. But do you want to not wait for the Lanarre rescue committee, and just come with me?\"\n\nTahlia bites her lip. After a full minute, she lifts her head. Her deep-blue eyes darken with a violet wash when her mood turns contemplative. \"If he were wonderful, I would go with him.\"\n\n\"But that's the thing\u2014you don't know.\"\n\nHer curly hair bounces as she shakes her head. \"I do not. He could be a tyrant. He could be arrogant.\"\n\nA Lanarre who isn't arrogant. Tessa smirks at that.\n\n\"He could be hot.\"\n\nThey turn and the Rare One stands in the doorway.\n\nStartled, Tessa says, \"No offense, private conversation.\"\n\nJulia spreads her hands away from her body inoffensively. \"I understand. But if we're talking destiny here, I might have a clue.\" She gives a short laugh.\n\nTahlia nods. \"I do appreciate your kindnesses.\"\n\n\"But...\"\n\nTessa looks between them and is struck by their eyes\u2014ancient eyes held prisoner inside their young faces. They've had to live too much for ones so young.\n\n\"But in this, I must choose,\" Tahlia says.\n\nJulia glides through the door and walks to Tahlia. \"You're brave. You survived your guardians being killed. You're in some kind of arranged marriage\u2014\"\n\n\"Not unlike your soul-meld.\"\n\nJulia lips tip up. \"Very unlike it. I feel love for Scott because it's actually in my blood.\" She puts a loose fist against her chest. \"He's a part of me.\"\n\n\"Thank you, Julia.\" Tahlia leans forward to hug her.\n\nTessa watches the two women, who are so dissimilar, finding common ground.\n\n\"Stay,\" Julia implores, \"The Lanarre guy\u2014\"\n\n\"Drek,\" Tahlia interrupts.\n\n\"Drek will show up. And you can be here, safe.\" Julia presses her hands against Tahlia's. \"If you go, he'll have to look for you. And this guy Drek, he's not going to hurt you, is he?\"\n\nShock spreads across Tahlia's features. \"Absolutely not. A Lanarre would never hurt a female.\"\n\nJulia looks down. When her face rises to meet Tahlia's eyes there's a wealth of sadness there. \"Let me tell you about a Were named Anthony Laurent.\"\n\nWhen Julia is done, Tessa thinks there's more to the story.\n\nTahlia sits down at the edge of the bed in defeat. \"It was he. He was the Were who murdered my guardians. It could be no other.\"\n\nThis just gets worse and worse.\n\n\"I know it's rare for a Were to attack females, but this Were?\" Julia shudders. \"He also had demonic blood. He killed my people. He didn't spare anybody.\"\n\n\"He is dead?\" Tahlia asks.\n\nJulia nods.\n\nTahlia's eyes close, and a wrinkle of worry settles between them. Her eyes open slowly. \"Yet... did he act under orders of another?\"\n\n\"Yes, a demonic named Praile.\"\n\nTahlia works her lip between her teeth.\n\nJulia leans forward. \"We understand there's potential for retaliation. What are you thinking?\"\n\nTessa's sense of foreboding kicks up a notch and her eyes bang around the room anxiously. \"We need to go if we're going, Tahlia.\"\n\nTahlia nods. \"My thoughts are only the legends I was raised with. The demonic are a race to be feared.\" She gives the smallest lift of her shoulders. \"However, you are angelic here, so you have a powerful built-in opposition.\"\n\n\"And what about Drek?\" Julia asks.\n\nTahlia gets a wistful look, gazing out a window without sight. Darkness has claimed the day.\n\n\"I'm not ready. I\u2014\" She gives Tessa a look. \"I think I was just following what others thought was best for me. And now I know I can be something other than a mated Lanarre princess.\"\n\nHer voice sounds uncertain.\n\n\"Are you sure that's what you want?\" Julia's face looks pained. \"What if he drops by and he's all kinds of amazing and kind and super handsome.\"\n\nTahlia's face breaks into a grin. \"Well, he'll be worth being patient for. I'm just not ready, and Tessa wants a companion.\"\n\nJulia gives Tessa a look that clearly says she's filled the girl's head with thoughts.\n\n\"Listen\"\u2014Tessa holds up her hands, glancing quickly at the window\u2014\"I'll check in after a couple of weeks once we're settled somewhere.\" She hikes her shoulders, sweeping a palm out in supplication. \"And if Drak is here....\"\n\nTahlia giggles. \"Drek of the Lanarre, Tessa.\"\n\n\"Ah-huh. If Drek drops by, then by all means, let me know what your opinion is of him.\"\n\n\"I'll tell him I tried to talk sense into Tahlia.\"\n\nTahlia purses her lips. \"I have an inordinate amount of sense.\"\n\nJulia's face took on a sad cast. \"I thought I did, too. Once.\"\n\nTessa's eyes swept the mostly empty room, landing on a duffel. \"That it?\"\n\nTahlia nods.\n\n\"I'll be back in a minute.\"\n\nTessa turns back from the doorway. \"Thanks, Julia. For everything.\"\n\nJulia stares at her with those unnerving eyes, like golden whiskey. \"You're welcome.\"\n\nIt'd felt like home for a day.\n\nNow the road would be home again.\n\n*\n\nTessa tiptoes through the huge mansion, making a beeline for the kitchen.\n\nShe carefully packs two days' worth of food and loads a large hot-cold lunch bag full of food stuffs.\n\nTessa sets the thermos gently inside, careful not to crush the rest, and silently thanks Julia again for sharing.\n\nThe hairs at her nape suddenly lift, and Tessa curses herself for being too slow as a hand clamps over her mouth.\n\nShe kicks out, knocking the lunch sack over. A single orange pops out of the bag and bounces across the granite butcher block. It rolls all the way and lands against the wall. It ricochets off the wall and lands with a thunk as Tessa is drug off into the butler's pantry.\n\nHer heels squeak against the wood floor as she quarter-changes and moves to lock teeth over the hand that grabs her.\n\n\"Do not.\"\n\nThat voice.\n\nHe jerks her over the pantry threshold and throws the door closed. He turns her in a blur of speed and slams her against the door. Tessa is momentarily stunned when her head thumps the wood.\n\nIt's the Singer, the lighter of the two who plunged the unconscious Victor into the hole.\n\nMoon.\n\n\"I am here to kill you.\"\n\nTessa's never been a slave to her emotions. She's never cared about a male. She's been hunted, beaten, degraded and never, ever cherished.\n\nIn the middle of all that, she survived.\n\nSo when her heart races and her limbs go weak, she can't possibly grasp what this is about.\n\nSmokey eyes gaze down into hers, and a light vapor rises off skin that has the faintest touch of red.\n\nHe smolders\u2014his eyes, his skin... everything.\n\nHe is so hot, she can feel the heat emanating from his skin.\n\n\"So kill me,\" Tessa whispers.\n\nShe is weary.\n\nShe is finished with running. If she dies, then Tramack can never have her. In a bizarre way, Tessa wins.\n\nHe shakes his head, the ghost of a smile touching his lips. \"No,\" he says, his fingers tightening against her throat. \"I think not.\"\n\nTessa's confusion deepens.\n\nHe could take her down piece by piece. Somehow, this weird-looking Singer has the strength to outdo a quarter-changed female Alpha Were.\n\nSo why doesn't he?\n\n\"You saw what we did to the Singer.\"\n\nI should have told Julia.\n\nShe licks her lips, and his eyes latch on to the movement.\n\n\"So, you're not going to kill me?\" Tessa's thoughts spin.\n\nHe shakes his head. \"I knew what you were to me the instant I scented you.\"\n\nThis guy's certifiable. I'll humor him.\n\nHis fingers loosen but don't drop from her flesh.\n\nShe hears herself asking, \"Scented what?\"\n\n\"My Redemptive.\"\n\nOkay\u2014really crazy.\n\nHe bends his much taller frame over her, but his hand doesn't let go. His thumb moves to her jaw and as his lips draw closer she tips her face up to give him better access.\n\nI've obviously lost my mind, too.\n\nThis guy drug her inside a closet and told her he was going to kill her, and now he's going to kiss her.\n\nAnd Tessa's going to let him.\n\nScorching heat sears through her lips as his land on hers. Tessa groans as if she's just awoken from a delicious sleep. Her every nerve ending fires. She doesn't realize her arms have encircled his neck until she molds herself against him like a second skin.\n\n\"So hot,\" she breathes against him.\n\nHe lifts her by the ass and buries himself against her.\n\n\"Ah,\" she moans and kisses him back. \"Oh, moon, you feel right.\"\n\n\"As do you.\" Peck, lick, suck.\n\nTessa tries to pull back, and he sucks her lip deeper into his mouth. The sensation is wonderful, like heated bathwater concentrated into a single wonderful sip of sex on lips.\n\nA surge of horrible disquiet flows over her and wakes Tessa from her sexual thrall.\n\nShe gasps.\n\n\"Who are you?\"\n\nShe doesn't even know his name. And he is a Singer without a scent. He's no Singer.\n\nAn unknown male I'm making out with.\n\nHe allows Tessa to slide down the door but stops her momentum before she's a puddle of melted wax on the floor.\n\n\"They call me Lazarus.\"\n\n\"I mean\u2014what\u2014are you?\"\n\nHe smiles, and Tessa is suddenly fascinated with a tongue so red it looks like it's on fire. It was just in your mouth, dumbass.\n\nShe swallows.\n\nLazarus brushes a hair away from her face and tucks it behind her ear. \"I am demonic.\"\n\nTessa's hands slap the door behind her.\n\n\"Shh, do not fear me.\"\n\nShe nods. \"Sure. Sounds like a great plan. I just made out with the devil, and you're telling me to calm down because things are so peachy. Right.\"\n\nHe ignores her words. \"Do you know what a Redemptive is, Tessa of the Were?\"\n\n\"It is my other half. Most demons do not have that potential. But demons of mixed parentage can be given this gift.\" His eyes skip away from hers to return a heartbeat later.\n\n\"I'm a Were. You're a demon.\" She points from her to him. \"It's not a match made in\u2014\"\n\nHe presses a finger to her lips, and Tessa watches the light steam rise and evaporate from his skin.\n\nHer throat feels tight.\n\n\"None of those words, Tessa.\"\n\n\"Okay,\" she squeaks, her eyes flitting to where his hands are. \"It's been ah\u2014great\u2014to make\u2014I mean\u2014meet you, but now it's time to go.\"\n\nLazarus shakes his head again. \"You are my only chance. If I take you as my bride, I will never have to return to Hades and suffer under the Master again. We are meant to be together. Only my Redemptive can free me from the bonds of hell.\"\n\nTessa finds air isn't reaching her lungs. \"What about your partner or whatever?\"\n\nHe scowls, and Tessa finds her out.\n\n\"He won't like it. You just let me go, and then everything will be okay. You can keep on being evil and that, and I'll keep on...\"\n\nWhat will she keep on doing?\n\nOh, yeah. Running.\n\nLazarus moves in tight against her body. \"I cannot force the Redemptive. She must be willing.\"\n\nHis hands slides behind her neck, and she groans against his touch. She's lost to it.\n\nSo lost.\n\nWhat is wrong with me?\n\n\"Do you not feel the pull?\"\n\nTessa does, right between her legs.\n\n\"Yes,\" she admits. \"But it's not enough. I mean, you're not even a Were? I can't just up and go with you.\"\n\nTessa gets an image of a future whelp. A hotdog comes to mind. She shivers.\n\nThis isn't going to work. Tessa needs space\u2014right now.\n\nShe shoves him, and he steps away, every hot inch. His six feet four-ish of hardened muscle is scentless but oh-so fragrant anyway.\n\nWhen her gaze reaches his lips, Lazarus twists them into what she guesses is his version of a smile. His gray eyes storm at her.\n\nHis light-blond hair begs for her fingers.\n\nShe blushes hard when she thinks of the other things.\n\n\"I can scent your arousal, Tessa.\"\n\nShe backs up a step.\n\nToo dangerous. Just too everything.\n\n\"I can't do this, sorry. You're so tempting. Like a pile of chocolates that'll never make me fat.\"\n\n\"I won't stop,\" Lazarus says.\n\nTessa blindly finds the doorknob behind her then opens the door. \"Stop what?\"\n\n\"Wanting you.\" He grabs the door as she tries to swing it shut between them.\n\n\"Ever,\" his deep voice rumbles.\n\nShe jerks it closed and sprints away.\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\n# CHAPTER TWENTY-FIVE\n\nScott\n\nScott would be happier without the soul-meld. He has no doubt. Right now, things suck.\n\nHis dad is dead. Most of Region One is dead.\n\nAnd Julia's emotions constantly batter him. He can't blame her. Her selfish asshole of a husband dumps his baggage at her doorstep to figure out while every stinking supe demands guidance from a twenty-two-year-old woman.\n\nHis heart aches where it never did before.\n\nScott figures there's one asshole down and a few more to go. He strides to the barn. Julia should be back there now.\n\nHe knows she is. He can feel her like a beacon inside his body. She dealt with Tahlia and Tessa while he broke the shitty news of what happened in the seventy-two hours his siblings were locked up in the bunker.\n\nAnd Victor from Two? Where the hell did he run off to? Scott is pissed. He saves who he can, rises from the ashes with Scott's sister and brother, then disappears. Not cool. Scott needs his stuck-up ass.\n\nLucius and he were separated after the torture he escaped from. The other Combatants were killed in the battle of the demonic. And Julia has a demon spore making her sick.\n\nThe trouble never stops.\n\nBut after holding Jen while she cried on him for an hour, Scott is ready for the healing the soul-meld will provide him. If he can just lay hands on Julia, all will be well, as his father would have said.\n\nScott pinches the bridge of his nose. Even smart-ass Michael, when faced with one of his junky suckers, couldn't bear to pop one in his cocky mouth.\n\nThey're in shock, he tells himself.\n\nThat's all well and good, but for two guys who wear their sister's grief on their shirt, logic and explanations just don't fucking cut it.\n\nIf he can get Tharell and his own evolving bio-mom out of here, so much the better.\n\nSpeak of the devil.\n\nTharell comes into view, along with Brynn, Delilah, Domiatri and Jacqueline.\n\nNow that he thinks of it, Scott isn't quick to shake his prejudice against vamps. The only good vamp is a dead vamp. At least he and Caldwell agree on that.\n\nOr they did.\n\nThat powder keg is gone now.\n\nHe feels the bond grow snug with each step he takes toward Julia.\n\nJulia's head turns, and his body weeps to connect with hers. As though he's moving through quicksand, each step he takes is heavier than the last. She moves to meet him.\n\nThey collide softly. She whimpers against his chest. \"It's so hard.\"\n\n\"Yeah,\" Scott agrees, touching her everywhere and scenting deeply of her hair as Julia molds her body to his. Seconds pass into a minute while everything rights itself.\n\nHis body grows stronger, and the fog of lethargy clears from his mind. All his senses come online and sharpen. Scott can think again.\n\n\"We can't be away from each other that long,\" Julia says with a shaky exhale.\n\n\"Feel better?\" he asks, looking down at her.\n\nShe nods. \"Much.\"\n\nScott looks around him, and the others stand silently.\n\n\"You're going?\" he asks Jacqueline.\n\n\"Don't sound so happy, my son,\" Jacqueline says.\n\nScott threads his large hand through Julia's smaller one and she stands at his side. \"I've had enough bullshit and strife to last two lifetimes. I'm good with the fey going back where they belong, and no offense\u2014I don't think vamps for hire appeals much either.\"\n\n\"No offense taken, Singer.\" Brynn hisses, his fangs sprouting like dual razors.\n\n\"Right,\" Scott drawls.\n\n\"Scott,\" Julia says in a low voice. \"Let's just let them go, and when things settle and hell's not chasing our butts, we can make good on the promise.\"\n\nScott turns to Julia, his gaze intent on hers. \"There might not be enough Singers to make good on that exchange, Julia.\"\n\nShe bites her lips, and Scott smooths them with a tender finger. \"No, Julia.\" He shakes his head and addresses Tharell and Domi. \"You guys\"\u2014he swings his finger between the two\u2014\"there will be some Singers who want to live in faerie.\" For the life of him, Scott can't come up with a single name. \"But I don't know when, or who.\"\n\n\"Do not break a promise to the fey, Scott,\" Tharell warns.\n\nScott moves to the Sidhe warrior. \"I'm well aware of the promise Julia made. But you're a traitor. And as such, I feel I'm being pretty fucking charitable to honor anything.\"\n\n\"Scott.\"\n\nHe turns to his mother, whose belly is swollen with Domi's child. He can't wrap his head around it all, so he doesn't bother.\n\n\"It's a promise to faerie, not Tharell. Faerie sends justice to oathbreakers.\"\n\n\"We won't break our promise.\"\n\nDomi steps forward. \"See that you do not. Make haste in an alliance between the fey and the Singers. Quickly.\"\n\nScott feels Julia's fear, and he wants to knock some rainbow heads together. But he's not going to pull a Caldwell. He's going to think things through. What's best for Julia. Them. The Singers.\n\nFuck the fey. Yet... Scott is old enough to understand the consequence of ignoring magic set in motion.\n\nInstead of voicing his heated thoughts, he says, \"We'll be in touch.\"\n\nJulia squeezes his hand and he pulls her against him.\n\nJacqueline moves from Domi as his hand reluctantly slides down her arm. She stands before him and for the first time Scott looks at her, really looks at her.\n\nShe has changed. Julia is right. Her eyes no longer hold that vacant indifference. It has been replaced with cautious hope.\n\nJacqueline doesn't move to take his hands, but Julia elbows him, and he feels what she feels for Jacqueline\u2014forgiveness.\n\nHe can't hate her anymore if Julia doesn't. And she is his family.\n\nScott doesn't know what family they will all end up being to him but when he looks over at Delilah, she is already making her way to him.\n\nJulia steps away so he can embrace his mother and sister.\n\nA small tear in his psyche begins to mend as his eyes hunt and find Julia.\n\nShe smiles, and it lights him from within.\n\nAnd Scott knows everything will be all right.\n\n*\n\nPraile\n\nPraile is as mad as he's ever been. Where in Hades is Lazarus? Praile has sighted the High One with the one who carries the blood babe. And Lazarus is where?\n\nHolding his hot dick while he takes a sizzling piss?\n\nPraile writhes with his anger.\n\nHe gave Lazarus explicit instructions to meet with him in this exact spot. Then they would take the women at one time. The High One had not divulged her scheme to return the Sidhe and the others to faerie.\n\nIt's a perfect situation for Praile, though. The fewer supernaturals to deal with, the better. However, killing the High One would be tricky.\n\nShe has a formidable arsenal at her disposal. Hopefully, she isn't fully aware of all the strengths she possesses.\n\nAnd she and her soul-meld are fools. If they have not consummated their relationship by coming together, she is vulnerable. Praile and Lazarus have not been at the compound long enough to understand the entire story, but he understands her human marriage has delayed her consummation with the soul-meld.\n\nPraile makes a disgusted noise. Some kind of illogical, emotion-based nonsense. This gives Praile the advantage.\n\nJulia would be invincible if the soul-meld were consummated.\n\nIt should be an impenetrable golden circle.\n\nPraile would find the chink in her unfinished armor. Praile leaks vapor then clenches his teeth until it subsides. He hates the human shell.\n\nIt is as ugly as sin.\n\nThe thought makes his face break into a grin.\n\n*\n\nTessa\n\nTessa drags both duffels and settles them into the back of a borrowed SUV. It's as old as hell but runs true.\n\nWell, that's what one of the Region Two Singers said.\n\nHe palms the keys in her hand and smiles. \"Julia says to get it back when you come back.\"\n\nTessa is so unaccustomed to kindness that she blinks back tears.\n\nTahlia squeezes her arm. Their eyes meet, and Tessa glances away before revealing too much.\n\nTessa thinks of Lazarus and his threat. She shivers, despite feeling the heat of him against her.\n\nThen, with a shudder, she remembers who chases her.\n\nShe stiffens her spine. Don't be a baby, Tessa. Get your ass in gear. She can't be with a demonic, no matter how amazing that interlude was back there.\n\nShe can't stay here and endanger the Singers, because sadistic Tramack is chasing her.\n\nShe glances at Tahlia, who slides into the car. And she can't leave a young woman barely out of whelp for a mystery match.\n\nTessa closes the back of the late-70s Chevy Suburban and makes her way toward the front.\n\n\"Tessa,\" a low voice commands from the woods, and on reflex, she looks up.\n\nTramack smiles.\n\nTwo Were drag a badly beaten Were between them. Shackled in silver, he can barely open his eyes.\n\nThe sight of the female takes her breath away. Her talons are broken, many at the base, and one eye is swollen shut. A third Were carries her in his arms.\n\nShe doesn't move.\n\nTessa scents they're alive. She also has a vague recollection of meeting them and doesn't know what happened.\n\nWhat Tessa knows is she's seen so much worse than this.\n\nThe female will survive.\n\nI'm so sorry, she whispers in her mind. If she helps, Tramack will win.\n\nShe will be his victim, along with the Were he's already captured.\n\nTessa glances at Tahlia, and the girl turns the keys in the ignition. The engine roars to life, and Tessa grasps the handle.\n\nTramack bellows, charging toward her, and Tessa whimpers. A body is suddenly pressing behind her, a hand covering her own.\n\nLight-red flesh warms her hand, making the metal too hot to touch, and Tessa snatches her hand back as strong arms hold her still as Tramack bears down on her. She knows the embrace. Her soul recognizes him as though she's been waiting her entire life to be held by him.\n\nLazarus.\n\nShe's not scared.\n\nFor the first time, Tessa feels safe.\n\n# CHAPTER TWENTY-SIX\n\nJulia\n\nJulia's arm encircles Scott's waist, and she lays her head on his bicep.\n\nThey watch the group walk away.\n\n\"You're relieved,\" Julia says.\n\n\"Yeah, can't you feel it?\"\n\nShe nods against him, and he closes his hand around the back of her head. He would glue her to him if he could.\n\nI heard that.\n\nScott deliberately inserts the image of super glue, and she giggles.\n\nHe strokes her hair.\n\nThey have a minute of peace before the nasty wound inside Julia begins to pulse and throb.\n\n\"What is that?\" Scott asks.\n\nJulia groans, her hand going to her stomach. \"It's that demon's blood.\"\n\n\"Oh, babe,\" Scott says, kissing her mouth. \"We'll figure that out.\"\n\nGravel crunching under a footstep causes them to look up. Julia forces a smile through the pain. \"Hi, Peter,\" she says.\n\n\"Hey,\" Scott says, but Julia feels a mild trepidation leaking from him. He's so paranoid. The guards actually found more Region Two survivors, and all Scott can think of is finding Victor so he can verify their identities. Julia's mind gets fuzzy as the demonic blood surges as though it's a magnet set on something outside her body.\n\n\"Where's the other guy?\" Julia asks. She snaps her fingers with a small grimace as the pain intensifies. She ignores it. \"Laz?\"\n\nPeter smiles, and hers falters. There's something there.\n\nJulia's skin begins to crawl as though a million biting insects are swarming over her flesh.\n\n\"Julia!\" Scott yells from somewhere far away.\n\nPeter strolls slowly to her and she's slipping. She falls to the ground and Scott is writhing on the floor.\n\nWhy is he half-changed into Combatant form?\n\nThere'd have to be a direct threat to her life.\n\nPeter grins. His teeth are black.\n\nJulia screams.\n\nThen she screams louder when the cloak of his humanity falls away, and his true form stands before her.\n\n\"Tony did well after all.\"\n\n\"Scott,\" Julia whispers.\n\nShe'd asked for privacy to say goodbye to the fey.\n\n\"Julia!\" Scott crawls to her, his veins pulsing gold and silver.\n\nHis skin smolders.\n\nAs does Peter's. Or whoever he is.\n\nHe sinks down beside her.\n\nJulia cringes, trying to scrape together her powers, but nothing comes.\n\nThe demon places the flat of his palm on her stomach, and pure agony surges through her. It feels like he's commanding her spine to come through her stomach, and she opens her mouth to scream. No sound comes out. The pain is so total, so awful that there is no breath to wail.\n\nShe contorts, her back arching.\n\n\"I am Praile. And you will die, High One. And then I will capture the mother of the blood babe.\"\n\nScott crawls closer.\n\n\"Jacqueline,\" Julia instantly squeezes through her pain-addled brain.\n\nPraile's inky brows tug together.\n\n\"No, part-demonic and fey, part-vampire and angelic, enemies of blood, and born of strife.\"\n\nJulia's head goes in the direction of where the party of fey traveled.\n\nIce slides up her spine.\n\nDelilah!\n\nHer attention snaps back to Praile as a gruesome tail settles above his head. His palm is fire against her flesh, and the light of her blood is blinding.\n\nBut the spore of the demonic has allowed entry to degrade her body, she realizes.\n\nThe ball at the end of his tail is spiked. It whips above his head more quickly than she can track, and he brings it down just as he lifts his palms.\n\nHer eyes meet Scott's as he takes her hand.\n\nA club swings through the air, whistling, then hammers into the ribcage of the demon bent on ending her. Praile staggers back.\n\nSpinning green discs meet her gaze, and she can't move.\n\nJason's wolfen form bends and scoops her into his arms. He reaches behind him and grabs a semi-conscious Scott. He drags the other man behind him as he jogs toward the mansion.\n\nScott's head bounces over the terrain and Julia lays like a bag of rocks as Jason makes his way into the house.\n\nHe sets Julia down gently and releases Scott, who groans when his head hits the carpet. Jason throws the carpet off, and Julia watches him enter a series of numbers into a door in the floor. Air hisses, and Jason's long fingers tear open the top.\n\nVictor's head is out in an instant.\n\nJason turns and hands Julia to Victor, who takes her below.\n\nNext Scott tumbles inside, and Victor catches him badly, the two of them rolling into a pile on the cold floor.\n\nJulia drags herself to the bottom of the ladder and looks up to meet Jason's eyes.\n\n\"I love you, Julia.\"\n\nShe opens her mouth.\n\nPraile appears behind him, his tail swings high, and the mallet connects with Jason's head, shattering his skull as she watches.\n\nJulia releases the last rung and falls to the floor.\n\nJason loses his hold on the hatch, and it closes as Praile's hands frantically claw at the metal for purchase and miss.\n\nJulia escapes in the only way she knows how, an unconsciousness of necessity sweeps over her, and she falls into an unnatural sleep of survival.\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nTHE END\n\n|  |\n\n---|---|---\n\n# BLOOD ENCHANTMENT\n\nA Blood Series Novel\n\nBook 6\n\nNew York Times Bestselling Author\n\nTAMARA ROSE BLODGETT\n\nAll Rights Reserved.\n\nCopyright \u00a9 2015 Tamara Rose Blodgett\n\nNo part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a database or retrieval system without the prior written permission of the publisher.\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locales, or organizations is entirely coincidental.\n\nwww.tamararoseblodgett.com\n\nTRB Facebook Fan Page\n\nCover Design: Phatpuppy\n\nEditing suggestions provided by Red Adept Editing\n\nSynopsis\n\nJulia's spirit is crushed as she sits trapped in the Singer bunker while heaven knows what happens to her people in the evil hands of Praile.\n\nMeanwhile, the insidious demon's spore flourishes, wreaking havoc on what makes her angelic. Julia and Scott are caught in a waiting game, ready to be released from their forced protection into a potential second catastrophe.\n\nAdi and Slash have fallen prey to the corrupt Were, as Tramack doggedly seeks that which he does not deserve. Can Tessa and Tahlia escape? Will Lanarre royalty cage Tahlia, or will Drek set her free?\n\nDoes Lazarus escape the depths of hell, only to fail in his pursuit of the ultimate forbidden fruit?\n\n# DEDICATION\n\nDawn Yacovetta\n\nWarning! ***Contains spoilers***\n\nCharacter Index:\n\nBlood Singers\/talent:\n\nJulia- Queen of the Singers; Telekinetic\/telepath\n\nJason- Singer\/\"Feral\"\/Red Were\n\nScott- Royal Singer Blood; Deflector\/Combatant\n\nBrendan- Tracker\/pyro\n\nMichael- Illusionist\n\nJen- Telekinetic\n\nCyrus- Healer\n\nPaul- Negator\/amplifier\n\nAngela- Feeler\n\nMarcus- Region One\n\nJacqueline- Royal Singer Blood; Region Two Leader\n\nVictor- Region Two\/Combatant- Boiler\/Flame of Blood\n\nLucius- Combatant\n\nCynthia \"Cyn\" Adams - Rogue\/Healer\n\nHeidi- Reader\n\nTrevor- Deflector\n\nNorthwestern Were Pack:\n\nLawrence-Packmaster\n\nEmmanuel \"Manny\" - Beta to Lawrence\n\nAnthony \"Tony\" Daniel Laurent- Second to Lawrence\n\nAdrianna \"Adi\"- Alpha female\n\nSoutheastern Were Pack:\n\nDavid- Packmaster\n\nAlan Greene- Alpha male\n\nLacey Greene \u2013 female Were and Alan's sister\n\nBuck \"Slash\"- Alpha male\n\nKarl Truman- former Homer detective\n\nFord- Alpha male\/ FBI agent\n\nReagan - Moon Warrior, Lacey's daughter\n\nSoutheastern Vampire Kiss:\n\nMerlin- Coven leader (now deceased)\n\nWilliam- new coven leader (now deceased)\n\nBrynn- New leader of the Southeastern\n\nNorthwestern Vampire Kiss:\n\nGabriel- Coven Leader\n\nClaire- William's cousin\n\nWilliam- Runner\/shifter\/Singer blood\n\nUnseelie Sidhe fey:\n\nQueen Darcel- Sidhe\n\nTharell- mixed Sidhe warrior\n\nCormack- Sidhe warrior\n\nDomiatri \"Domi\"- Sidhe warrior\n\nRex- Sidhe\n\nKiel (key-ale)- dragon shifting Sidhe\n\nCelesta- Sidhe warrior\n\nDelilah - Vampire, third to Julia, Scott's half-sister\n\nRogue Reds:\n\nEzekiel \"Zeke\"\n\nRogue Alpha female Were:\n\nTessa\n\nFEDS:\n\nTom Harriet\n\nTai (tie) Simon\n\nWestern Were Pack:\n\nTramack\n\n[the] Lanarre:\n\nDrek\n\nBowen\n\nTaliah\n\n# CHAPTER ONE\n\nTessa\n\nTramack's claws stop his forward momentum abruptly.\n\nExhaust from the suburban plumes toward him, and Tessa sees his effort to stifle his violent reaction to the nauseating smells of gasoline. Fossil fuels act like poison to a werewolf.\n\nBut Tramack is Alpha, so he can withstand a lot of unpleasantries, like the rogue female Alpha who has outfoxed him for two decades, leaving a legion of Were corpses in her trail instead of bread crumbs.\n\nNevertheless, he's followed the trail, and here he stands. Tessa can't believe Tramack's finally gotten off his ass and stopped sending lackeys to get their butts handed to them by a female.\n\nShe shoots a covert glance at Lazarus.\n\nHe's calm.\n\nHot, but Zen. His skin emits a subtle vapor, in the same way hers holds a layer of anxiety-laden sweat.\n\n\"Tessa,\" Tahlia calls from inside the truck. Her tone says it all. But if that weren't enough, Tessa tastes her own trembling fear on her tongue.\n\nFuck.\n\nTramack's dark brows drop like bushy caterpillars over eyes the color of flat pewter as they fall on Lazarus. \"Go to hell, demon.\"\n\nLazarus smiles, his eyes alternating between an icy blue and the palest shadow of gray. \"Make me.\"\n\n\"What are we? Three?\" Tessa mumbles. But inside, her guts flip-flop like overdone pancakes.\n\nTramack's wolves circle their Alpha, flanking him in practiced smoothness.\n\nTahlia guns it. The fumes from the vehicle make Tessa cover her mouth to keep herself from puking. Fucking Chevys always burn rich.\n\nTramack coughs. \"We have no quarrel with you, demonic. Leave the female\u2014she belongs to me.\"\n\nTessa gives an indignant snort. Lying asshole.\n\nLazarus turns to regard Tessa as though counting every pore of her skin, then turns his attention back to Tramack. \"I do not see a label of ownership, dog.\" A lazy smile hovers at Lazarus's mouth.\n\nTramack growls.\n\n\"Do not think it, mutt,\" Lazarus says. \"Tessa is my Redemptive, and as such, she has certain inalienable rights.\n\nI do? \"Oh... boy,\" she mutters.\n\n\"Nonsense,\" Tramack states as though whatever drops from his mouth is absolute fact.\n\nFor him, it is.\n\n\"That is a myth. Demons stay in hell, and the Were roam the earth unimpeded by your kind. You are not allowed to play in this realm.\" Tramack's large hands fall to his sides.\n\nLazarus cocks his head to the left, appearing to contemplate Tramack's justification bullshit. Lazarus raises a finger, and Tess is mesmerized by his skin tone.\n\nWho knew sunburnt skin could look sexy? She shakes her head. Murderous Were are on the prowl, bearing down on them. Lazarus's little chat with Tramack is just a verbal dance. Tramack will spring. He'll kill Lazarus then take Tessa. She needs to think of contingencies. For example, Tahlia is already in a running car. Tessa sights the positions of the other Were as they move wide of Tramack. They're too close to the Suburban. They'll pounce. Tessa shudders. She can't stand the thought of Tahlia falling into the hands of the Western pack.\n\n\"Normally, that is true,\" Lazarus says, \"but we've been given a certain latitude from the Master for a time.\"\n\n\"I don't give a ripe, hairy shit how flexible the devil has been. This is where we live, feed, and fuck. So, go to hell.\" Tramack's lips pull back from his teeth, and icy fear slides down Tessa's spine.\n\nThe other three Were growl low in their throats.\n\nThis is getting saucy. Tessa quarter-changes to wolfen instantly, grimacing slightly at the rapid shift.\n\nHer head whips in Tahlia's direction.\n\nBurning eyes like silver discs rotate slowly in a return stare. A voice like gravel grinding together says, \"Time to go, Tessa.\"\n\nShit! Tahlia's already wolfen\u2014and ready to drive a vehicle.\n\nTessa begins to back away from Lazarus. He's on his own.\n\nLazarus whirls to face her, his eyes like mercury-laced glacier blue.\n\nHer eyes widen.\n\nHis gaze changes to obsidian. \"Do not leave my side.\"\n\n\"They'll kill us,\" Tessa blurts. \"Lazarus!\" she screams as Tramack flings himself forward, landing on Lazarus.\n\nTramack is dirty, through and through.\n\nTessa breathes through the pain, bursting to wolfen in a slide of tendons and bone-breaking shift of her quarter-change form to half-werewolf.\n\nHurts, her beast says.\n\nThe first wolfen male comes for her, and she tamps her fear. Despite her alpha-female status and awesome skills, she's outweighed and outreached.\n\nShe growls, crouching low, and flicks her hands out. Talons fling the residual skin from the change like strips of sloughed snakeskin.\n\nHe follows through in a hard charge, wrapping his arms around her like steel bands.\n\nTessa gasps, struggling for oxygen.\n\nFuck breathing. She swings her head forward, head-butting the large male in the forehead. Stars burst, and Tessa groans. Fantastic.\n\nHis hold loosens, and she plunges all eight talons into his back, unwittingly marrying the two of them together.\n\nHe roars, jerking backward and taking Tessa with him.\n\nShe flips over the top of him and corkscrews her talons with a turn of her wrists.\n\nScrambled muscles! Tessa jerks her talons out, staggering backward. Another Were attacks from behind, but she's ready, throwing her skull back into his face.\n\nHe howls. This time, she gets lucky and he drops her.\n\nShe lands with bone-jarring impact. The exhaust pipe is right in front of her face, but the engine's no longer running. Oh shit. A huge shadow passes over her head. Tessa knows it's not a cloud.\n\nA bird of prey hovers, legs like stiff planks as it sails down in a spiraling circle of speed, talons spread.\n\nHere comes Tahlia.\n\nA scent she can never forget swamps her nostrils, and her beast rebels. Tessa's tossed over onto her back.\n\nA look of triumph covers Tramack's ugly face, his dark-pewter eyes spinning. \"Bitch.\" He grates, all teeth.\n\n\"Douche!\" she screams, trying to crawl away.\n\nTramack grabs her ankles, half-flipping her. She stabs the ground with her talons, arresting her backward progress, and turns to her side, throwing her forearm in front of her face.\n\nA pale-red appendage bursts through Tramack's torso. A ghostly worm of intestine slips out of Tramack in a slimy loop. He releases her ankles, and she yanks her legs away.\n\n\"Ahh,\" Tessa says, adrenaline surging through her body as she scuttles in a backward crab walk.\n\nTramack's eyes go round as the fleshy weapon exits his body with a meaty suck.\n\nHer gorge rises. Don't have time to puke, Tessa.\n\nTramack falls hard on his ass, hands grappling with his slippery guts. He can't quite manage to collect his intestines and drops them with a slapping smack.\n\nDirt and debris coat the pulsating organ.\n\nShe looks up from the sight, swallowing vomit as gravel and grass jam into the tender flesh of her palms.\n\nLazarus stands above Tramack, surveying the destruction of the Alpha Were and his writhing posse, his tail rising above his head. Tessa's eyes catch on strings of gristle and a tendon.\n\n\"Any takers?\" he asks casually and Tessa makes a small sick sound.\n\nHis eyes laser to hers. Coal black softens to a pale gray then flips to a sky blue.\n\n\"Ahh,\" Tessa says again.\n\nThe bird of prey lands to peck at Tramack's exposed intestine. He looses a high, keening scream.\n\nLazarus nods in apparent approval.\n\nTessa's mouth opens and closes as if she's a fish out of water.\n\nBeady black bird's eyes turn and survey Tessa's stupidity. They blink like two bottomless wells of oil, then the bird goes back to pecking.\n\nTramack's bellows fill the air.\n\n\"Ahh,\" she says with soft redundancy. Tessa's been reduced to a one-word vocabulary.\n\nLazarus studies her for a moment, frowning. \"We do not have time for a breakdown, Tessa.\" He reaches for her, and she beats at his hands, her talons automatically retracting. With a sigh of pure frustration, he scoops her off the ground. His tail, mercifully, is gone.\n\nTramack begins to howl in earnest, trying to swat at the bird while his body furiously heals the wound Tahlia inflicts in her bird from.\n\nNew skin grows over the dirt and the intestine.\n\nTessa looks away before she barfs. \"What is that?\"\n\nLazarus looks more human. His pale-blond hair is a perfect complement to what a human being would recognize as a ruddy complexion.\n\nThen the tail rises above his head once again.\n\nSo much for human. Tessa gulps back the lump of crazy in her throat. The ball of fear settles in her stomach, spreading out in dangerous tendrils of hysterics.\n\n\"This?\" Lazarus asks casually. The tail flicks.\n\nTessa nods, hiccupping. She claps a hand over her mouth, unable to take her eyes off the perfectly formed hammer head at the end of a six-feet-long reddish tail.\n\nTessa tries to ignore the chunk of intestine that Tahlia slurps down her throat like a succulent oyster.\n\nCaw.\n\n\"Oh Jesus, let me down.\"\n\nLazarus bends slightly and sets Tessa on her feet, smoothly kicking the twitching Were trying to heal at his feet. He makes the wound more severe, and brains leak out where only a fractured skull had been before.\n\nTessa runs to the bushes and heaves her guts into the greenery, never more thankful she'd worn a braid. Leaning on a tree trunk, she stands, wiping a shaky hand over her mouth.\n\nTurning, Tessa sees the Were are repairing themselves. Tramack looks at her with hate.\n\nTahlia is back to wolfen now, but she's still covered in the grime of her deeds as a bird of prey. A pure-ebony feather lies at her feet.\n\nTessa shudders. Gross.\n\n\"We need to go,\" Lazarus says.\n\nNow he wants to go somewhere?\n\nHe looks at Tramack with utter indifference. \"But first we kill this Were,\" Lazarus's tail swishes above his head like an agitated feline appendage.\n\nTessa looks at the tip. How can that plow through a body? Do I want to know? Tessa turns, her long braid flopping at her back as she clenches her eyes shut.\n\nShe should be happy that the misery of Tramack will be at an end. She can leave and be free of him. Avoiding the inevitable skirmishes while traveling as a rogue female would be nothing if he weren't after her.\n\n\"What in the hell are you doing, Lazarus?\" roars a voice that sounds as though it's on fire.\n\nTessa's eyes fly open. A man stands at the edges of the mess, as shouts and screaming begin to filter toward their position.\n\nIt's that other guy\u2014the one who came with Lazarus. But he looks nothing like he did in what she could only call a human costume.\n\nHe's tall, like all male supernaturals. But there, the similarity ends. Sleek deep-red hair, a shade or two darker than his skin, is slicked back from a high, broad forehead above merciless eyes so black, she couldn't find the pupils if she tried. Black horns sprout from his head like bony flower stems without heads.\n\nThen she gets a load of the tail. Tessa backs up a step.\n\nLazarus's eyes flick to hers then back to this new threat. She'd thought Lazarus's tail was something.\n\nBut this guy\u2014this guy...\n\nHis tail whips above his head, and spikes, all black, extend from a deep-red bulbous end. The tail end looks like a fleshy medieval flail.\n\nTessa backs up another step, burning to turn and see how far the Suburban is from her present position. She flares her nostrils, trying to locate safety that way.\n\n\"I have the Blooded One contained. We only need kill who we can then proceed to wipe her out of existence. Why you are wasting your time with antagonizing\"\u2014his nose wrinkles, vapor exits his nostrils in smoky spirals that drift away, and a strangled sound of fear squeaks out of Tessa\u2014\"these ridiculous beasts?\" He tsks, a forked tongue flicking out at the last syllable. \"You are not staying on task, Lazarus.\" The snakelike tongue snaps back into the black interior of his mouth.\n\nOh, baby Jesus, a forked tongue.\n\nHis inky brows come together in a clear frown of chastisement. He takes in the bleeding but healing Were at his feet. Then he stomps Tramack's head into the ground, crushing his skull instantly. Brains ooze out from underneath his shoe.\n\nLazarus doesn't even flinch.\n\nThese demon guys mean business. Tessa dares turning, seeking Tahlia. As if in total sync, Tahlia bleeds back to quarter-change and races to the passenger side of the Suburban. Tessa breaks, ignoring the demon interaction, and runs to the driver's side. She hops in, starting the engine.\n\n\"Or are you doing something else?\" the other demonic asks Lazarus. His voice sounds closer.\n\nTessa tries to squish the panic. Could this dude be even worse than Tramack? She can't unhear their conversation. She closes her eyes, remembering the feel of Lazarus's hands on her body and his mouth on hers.\n\n\"Come on, Tessa\u2014what are you waiting for?\" Tahlia hisses.\n\nTessa pauses, noting a string of gristle dangling off Tahlia's chin.\n\n\"I need to know he's going to be okay,\" Tessa says.\n\n\"What!\" Tahlia shouts. \"Leave him. He is a horned one!\"\n\nI can't.\n\nThe other demonic stands before Lazarus, who is shaking. The other one has him around the neck by the tail. His face intent, he squeezes Lazarus around the throat like a python.\n\n\"Tell. Me.\" He shakes Lazarus.\n\nNope. Can't do it. Tessa exits the car, stepping over Tramack. His hand snakes out, grabbing her ankle, and she kicks him. He grunts.\n\n\"Don't, Tessa.\"\n\n\"Fuck off, Alpha,\" she replies without glancing at him.\n\nThe second demon's eyes move to her, unnerving and flat black. Soulless. Obviously.\n\nTessa shivers.\n\nEvil radiates from him, coating her through the layers of skin, muscle, and organs, making her insides quake in revulsion. Still, Tessa moves forward against the horribleness of the creature before her.\n\nHis black eyes widen in surprise. \"Who is she?\" His tone is somewhere between shocked and affronted.\n\n\"Get your tail off him, jerk!\" Tessa shouts.\n\nThe demon smiles. Black teeth.\n\nHe slowly unwinds his tail from Lazarus's throat. A mark like a rope burn twines around his neck.\n\nLazarus drops to his knees choking, clawing at his throat.\n\nThat pisses her off even more. Why can't anything ever go right? Why can't fucking males just leave me alone?\n\nAnd, for the love of the moon, why must she do the white knight shit? And the one male, albeit a demon, who's actually treated her decent, is busy getting his assed kicked by his own kind.\n\nNo. Something deep slips inside of Tessa, like a mudslide in her brain. She rushes forward, red-hot.\n\nLazarus bellows, \"No!\"\n\nAnd then the demon is on her, wrapping her with his tail. That's fine. Tessa's talons are at the ready.\n\nShe slices off his dick.\n\nAll guys have one.\n\n# CHAPTER TWO\n\nJulia\n\nJulia forces her gorge down. She's pretty sure that the queasiness woke her. She rubs her eyes and groggily closes them again. But before she can doze off, the last memory sinks into her, grounding her in the present. Images of brutal clarity force her up off the cold floor like a reanimated corpse.\n\n\"Oh God,\" she moans, palm to stomach, gazing around the bare concrete surfaces. Emergency lighting flickers. LEDs struggle to warm in the ambient low sixties temperature of the bunker.\n\nHer eyes latch on to the stainless-steel ladder rungs leading to the first floor of the mansion. Beads of sweat pop on her upper lip, and Julia shelves the last images of Jason's death for the moment. She can't touch them or grieve.\n\nPeter\u2014or whoever he is\u2014owns her region while she's down in the bunker.\n\nJason did what he promised. He saved her\u2014at the expense of his own life. Julia sucks back a sob, instinctively looking for Scott. Her eyes find his form only feet from her own.\n\nShe tips off the edge of a cot and crawls toward him, vertigo making walking impossible. Putting her hand to his chest, she feels for a steady heartbeat\u2014and can't find it.\n\nDon't panic, Julia.\n\n\"He cannot sense you.\"\n\nJulia jerks back, her heart thudding.\n\nSteady gray eyes regard her.\n\nVictor.\n\nShe releases a held breath. \"You scared me. I forgot\u2014\" She ducks her head. \"I forgot that you and Scott...\" Julia puts her face in her hands. They smell like dusty concrete. She lets them drop, meeting Victor's handsome face.\n\n\"He probably has a concussion.\"\n\nShe grabs Scott's limp hand, reveling in the warmth. \"Right,\" she says, nodding a little too rapidly.\n\nI can't take them both being gone.\n\n\"He will regain consciousness.\" She hears Victor's attempt to soothe her.\n\nShe nods as a tear trails down her face. \"We need to get outta here, Victor.\"\n\n\"Impossible.\"\n\nHer face jerks from the study of Scott's relaxed features to Victor's stern ones. \"What do you mean \u02bbimpossible'?\"\n\nVictor sighs, running his large hands down his slacks. \"It is timer driven. If no timer is set, it will automatically default to seventy-two hours.\"\n\nJulia's mind whirs. \"How much longer?\"\n\n\"Roughly forty-seven hours.\"\n\nJulia's shoulders drop in disappointment. \"How do you\u2014\"\n\nVictor points at the far wall. A clock similar to the countdown clock from the Northwestern Kiss where William held her runs the seconds down in digital format, marking time's passage eerily.\n\n\"That bastard is up there\u2014doing God knows what to my people!\" Julia says loudly.\n\n\"I know,\" Victor glances away, his frustrated expression exactly matching her feeling of helplessness.\n\nJulia scans the space restlessly, scrutinizing the rows of cots. They're harder than damn rocks. A queen should be able to do something about such slack accommodations, right? She feels a grim smile perch on her face.\n\nQueen.\n\nWhat the hell kind of queen can't even command the region she was prophesied to rule over? A sucky one.\n\nJason pops into her mind.\n\nDon't think about him, Julia. Just. Don't.\n\nShe sucks in a shaky inhale then lets it out with agonizing slowness. \"Okay.\" Julia stands, slapping her thighs and dusting off her hands. \"I'm going to check out this bunker.\"\n\nHer eyes move to Scott, so still on the adjacent cot. As she watches, his chest rises and falls.\n\nJulia has to occupy herself, or she'll go crazy\n\n\"I will accompany you.\" Victor's battered body is mending before her eyes.\n\n\"It's okay. Just rest.\" Julia knows Victor must have hauled them away from the hatchway after they fell in here. \"I can do this.\" Her brows pull together. \"How'd you get down here?\"\n\nA rueful smile tucks itself into the corners of his mouth. \"I was unceremoniously tossed down by the demonic ruffian and his cohort.\"\n\nThe older Singers speak so strangely. \"So no one noticed you were missing, for like a day?\"\n\n\"I did.\"\n\nJulia whirls. \"Scott!\" she squeals and rushes to his side, relief surging through her.\n\nStanding, he lifts her off her feet then twirls her. \"You are a sight for sore eyes, Julia.\"\n\nShe automatically kisses him\u2014right there in front of Victor, with Jason's dead body on top of the escape hatch. Julia doesn't even have time to analyze how practical she's become in the years she's survived the supernatural world.\n\nThe soul-meld doesn't care about circumstance. Their bond dictates and overpowers everything.\n\nTheir mouths mash, tongues inserting and twining passionately as reason and rationale depart. Scott and Julia touch, moved only to have skin-on-skin contact, reuniting and strengthening each other as only they know how. The reaction is natural for soul-meld Singers.\n\nVictor clears his throat, reminding Julia they have an audience.\n\nThey leap guiltily apart\u2014at least Julia does. She doesn't think Scott feels guilt about much, especially their union.\n\nJulia feels the sting of shame because the circumstances are dire. And caving to their need of each other somehow seems selfish in the face of the potential suffering going on above their heads.\n\nJulia knows there is no alternative to the bunker, but that doesn't change her feelings. She self-consciously covers her mouth as though she can erase the moment.\n\nScott shushes her, pulling her against his chest.\n\n\"Are you okay?\" he asks against her hair.\n\nShe nods, though she's anything but. Silently taking comfort from the steady beat of his heart, she lets Scott's warmth seep through his shirt and heat her cheek.\n\nVictor inclines his head. \"My king.\"\n\nRight. Now that Marcus is gone, Scott's it. Julia closes her eyes against a reality she can't deny.\n\n\"Please, Vic, I can't take all the pomp and circumstance.\" He rakes his free hand through his inky hair. \"You just treat me like you did before-before Father died,\" he ends quietly.\n\n\"As you wish.\" Victor clasps his hands behind his back, his chin dipping.\n\nScott exhales in relief without mentioning the obvious: how they would all love to be above. Instead, Scott tugs her away from the cots. \"I've been down here, but you want a tour?\"\n\n\"We've got nothing better to do,\" Julia admits. \"That horrible guy is up there, doing I don't know what while we're here languishing. I need a distraction.\"\n\n\"Hey.\" Scott stops walking. He tilts her chin up with one finger, \"We can't get out, Julia.\" He pauses for a moment. \"I wasn't a fan of Caldwell.\"\n\nJulia shuts her eyes against an agony she can't name, against Scott being that insensitive to all that's happened. Julia knows Jason's conduct was terrible at times, but now he's dead. Truly gone.\n\n\"Julia.\"\n\nResentfully, she meets his nearly black gaze. \"He did right by you. That demonic had me. He was going to do you in, and I couldn't change fast enough, get past his evil, to save my own soul-meld.\"\n\nScott shakes his head, eyes narrow. \"That spore\"\u2014his mouth flattens into a tight line of anger\u2014\"that thing made you susceptible to the demonic.\"\n\nJulia's surprised by his words and relieved Scott's giving credit to Jason.\n\n\"Not just that, Scott.\"\n\nThey look at Victor.\n\nHis expression is apologetic.\n\nJulia gets the idea that he's accustomed to having to say things that might not be real popular. Cue Jacqueline being the queen of Region Two.\n\n\"If your meld remains unconsummated, you're especially vulnerable.\"\n\nJulia's face bursts into flames, but he pretends not to notice her obvious embarrassment, though Scott wears a smirk.\n\nVictor continues, \"You would have done yourself a great service to have accepted the triad between the wolf, vampire, and your soul-meld.\"\n\nVictor hangs his head in apparent regret, and Julia wishes she had a rock handy, so she could crawl under it.Victor casually discusses her nod to a plural marriage as if it's the weather.\n\nJulia's still trying to wrap her head around being a Rare One\u2014a queen. It's all so foreign. But as the death toll mounts, the enemies draw closer, supernaturals literally pour out of the woodwork, and people she loves sacrifice themselves for her\u2014Julia has slowly come to the realization that she is who they claim she is.\n\nShe fills her lungs, shoring up her weak stores of courage. \"So now that William is gone, and Jason\u2014\" She can't finish.\n\n\"Yes, completing the meld will make you nearly invincible and hasten the powers that all Rare Ones hold within themselves as potential\u2014unrealized without the correct prompts.\"\n\nScott's lips twitch. \"So you're saying if we come together, then Julia will be safer?\"\n\nKnock that off!\n\nHe flinches at the shouted telepathic communication.\n\n\"Of course,\" Victor says smoothly, a slight furrow between his brows as though they are the dumbest people who ever breathed.\n\nJulia crosses her arms. Maybe one of them. She glares at Scott.\n\nVictor's eyes bounce between her and Scott. \"I apologize if what I've said is\u2014\"\n\nJulia stabs the air with her palm, remembering all the other times sex had come up as The Solution. \"Nope.\"\n\n\"Let me show you the bunker,\" Victor says, appearing to sense the tension.\n\nJulia grunts softly, jabbing her elbow into Scott's side.\n\nHe guffaws quietly, tucking her under his arm. \"That was so uncool. Why would you let him go on like that if you understood everything already?\" she asks in a fierce whisper.\n\nScott flicks her nose. \"Fun to watch you get all flustered.\"\n\nJulia rolls her eyes. \"This isn't funny. Jason is dead.\" Her voice shakes, and they slow, letting Victor move down a long corridor ahead of them. \"And I don't know what that Peter demon is doing to our people.\" Tears fill her eyes as the earlier levity flees.\n\n\"Hey,\" he says, \"he was after you. There's too many in the complex for him to try for everyone. You're safe here with me, and he's probably\u2014I don't know\u2014gone to hell.\" He offers her a crooked smile.\n\n\"You're so not helping!\" Julia jerks away from him, and he yanks her back.\n\n\"Let go!\"\n\nHis dark eyes narrow. \"No.\" His hand wraps the bare skin at the base of her skull. A painful breath eases out of her as her head dips forward in defeat. \"I'm sorry about Jason, but I'm never gonna regret what he sacrificed for you. It was the right call. I would have done it.\"\n\n\"Do you hate me for being sad over Jason? It's so awful, Scott. I can hardly breathe.\" Her hands fist the front of his T-shirt.\n\n\"Yeah. I hate\u2014really hate\u2014that you care for another man. But I love how tender you are, Julia. Those two things aren't separate. I can't have one without the other. And...\" He pauses for so long that Julia lifts her head off his chest. \"He was there first. Before I knew you. Before all this.\" He sweeps his palm around their subterranean prison.\n\nJulia's chest aches.\n\nRight then, she realizes it's from a broken heart.\n\nScott threads his fingers through hers, and they walk down the corridor together again. Victor waits patiently at the T where the hallway ends.\n\nJulia's lost interest in checking out the tomb-like surroundings. Exploring her lackluster prison of the next two days doesn't seem nearly as important as getting the hell out of it and excising the demon that somehow infiltrated her region.\n\nBut her want is in direct conflict to what is possible, so she walks.\n\nScott smooths the tension of her brows with a kiss between her eyes, and her breathing settles into a regular rhythm. She's survived worse.\n\nI'll survive this.\n\nShe turns her face up to meet Scott's concerned stare and smiles through tears that remain unshed.\n\nThe time for grief is gone. Julia has a role to fill, and it's there for her whether or not she's ready for it. Jason's death can't be for nothing. He died because he believed in her importance. In her.\n\nJulia's hand finds the place where the demon's wound festers. It pulses briefly before quieting.\n\nAs long as Scott and she are touching, it doesn't progress.\n\nAs long.\n\n# CHAPTER THREE\n\nSlash\n\nSlash has suffered greatly in battle. He'd been beaten to within an inch of his life and always triumphed in the end.\n\nHe's just that determined, but this is utterly different.\n\nHe spears the ground, using it like a handhold, and drags his upper body along the debris of the forest, making his way to his mate.\n\nSlash's head spins.\n\nBlood loss.\n\nHe has so many puncture wounds from talons, he's a Lycan pincushion. But right now, none of that matters.\n\nAdrianna is injured.\n\nTramack of the Western harmed his precious mate, and for that\u2014he'll suffer.\n\nFinally, Slash gets to within arm's reach of Adrianna and rolls to his side, heaving his unfeeling, broken legs around in a swinging arc with his hands.\n\nSlash resists the urge to weep. Mere crying isn't enough to quantify how grievous the situation is. Adrianna's face is battered; every bit of skin that meets his gaze is abraded by touch, talon, and teeth. Nips from sharp teeth march across her flesh.\n\nBut only one holds Slash prisoner\u2014the area around her left areola has been punctured by a single fang, as though marked.\n\nThe vision of her beautiful breast marred by another male bottoms out Slash's guts, leaving him shaking in a rage so pure, there's no name for it.\n\nAdrianna's clear hazel eyes open slowly then round with fear and disorientation.\n\n\"Shhh,\" Slash croaks, his healing throat like sandpaper. \"I'm here.\"\n\nAdrianna begins to cry, frantically touching herself all over. \"Did they? Oh God, Slash, did they touch me?\"\n\nHe hangs his head, his palms flat on her shoulders. He leans against her temple, lips against her ear. \"They did not have the chance.\"\n\nHer body shivers against his, and hot tears scald the cheek he holds. Her grief and fear shred Slash on a molecular level.\n\n\"Don't cry, Adrianna. I'm here.\"\n\n\"I know,\" she sobs, wrapping her arms around his head and drawing him closer.\n\nSlash tucks her in against his body, wincing at the motion but happy the others didn't get a chance to rape his mate. However bad that mark against the perfection of her skin is, it is only a mark. Slash's blood burns with the thought of what could have happened.\n\nIf they had been taking their time and not had the ulterior agenda of the woman, Tessa, Adrianna might have been violated.\n\nSlash hears far-off screaming, shouting, and the sounds of fighting. They're certainly not out of danger yet.\n\nHis memory is fuzzy after he killed the first two Were, but he remembers Tramack's nose jerking up as he held a screaming Adrianna down, as though he recognized a scent so important that it blanked the present.\n\nTramack flicked his eyes over Adrianna, seeming to deliberate, then struck her in the temple with the back of his hand.\n\nThe light in Adrianna's eyes dimmed, and she fell silent\u2014still underneath Tramack.\n\nSlash struggled to break free of the other two Were. Then Tramack snapped a tree in half as he swung the trunk from behind at Slash's unprotected legs.\n\nNumbness swelled then burst over his lower body\u2014hips to toes, and the Were holding him dumped him like garbage. He landed face-first in the muck of the soggy forest floor.\n\nTramack tossed the twenty-year-old tree like a twig after using it on Slash then stepped over his body. \"That's not the lesson I would have given with more time, but there will be time enough later. My intended has just released her scent.\"\n\nTramack lifted his nose to the sky, sighing in contentment while Slash's eyes remained frozen on his face\u2014memorizing it forever.\n\nSlash realized Tramack was crazy. It happens sometimes. Alphas aren't always of the most stable temperaments. Tramack was no exception, though he had spun down into a new low. Beating a female for no other reason except her mouth? Because he could raise his fist?\n\nSlash understands the prejudice against Reds. He's lived that. If Adrianna hadn't been there, they still would have punished Slash for being what he is, for looking imperfect.\n\nSupernaturals aren't known for their compassion. Every supernatural is supposed to be the epitome of perfection of his or her kind. The expectation is due to their powers, strength, and healing ability. The Singers have all that the Were possess and paranormal talents besides.\n\nNo Lycan can withstand silver without obtaining a scar. And some species have no compunction against using the alloy in battle. Slash had been in many battles. It'd only been a matter of time before someone had a weapon meant to do worse than kill\u2014but permanently maim. Slash shoves his introspection away as Adrianna begins to quiet, shattering Slash's reflections. He carefully wraps the tatters of her thin T-shirt over her breasts\u2014the brassiere is long gone.\n\n\"They'll come back!\" Adrianna says, her voice edging along that fine line of hysteria.\n\nSlash shakes his head, rising push-up style over her. He cages her in with his body, dragging his still-uncooperative limbs behind him.\n\n\"They're busy,\" he huffs between movements.\n\nFor the moment.\n\nAdrianna's lip trembles.\n\nHe cradles her face, kissing her soft lips, hating that it's the only undamaged spot of her face. \"Listen to me, Adrianna.\"\n\nShe nods, gasping back a sob. \"Okay.\" Tears roll over the tops of his fingers.\n\nTheir eyes lock. What he sees in her gaze makes his heart ache like a rotting tooth. \"I need you to be that smart-ass brave girl I know you are.\"\n\nA surprised laugh escapes. \"What? I\u2014my words are what got us into trouble, right?\" Her eyes flick guiltily away from his.\n\nSlash places a gentle fingertip on her jaw and turns her head back to face him. \"No. They would have hurt me regardless of your presence.\"\n\n\"Because you're Red?\" She guesses immediately.\n\n\"Absolutely. And because of my imperfections.\"\n\nAdrianna crosses her arms. \"That's bullshit.\"\n\nSlash smiles, his scar tissue flattening and reminding him that his face is still healing.\n\nHer skittish gaze flies around the still woods. \"We need to get out of here, Slash.\"\n\nHe nods. \"Can you hear that?\"\n\nDistant noises reach Slash's ears. A large engine revving. A female's screams. Low, threatening male voices.\n\nAdrianna stops all movement, seeming to really understand his odd stance above her for the first time. \"What is it?\"\n\nShe jerks up, and Slash follows the motion, rolling to his elbow. She's now above him and looking intensely into his face. A leaf is embedded in her matted hair. The sun slants through the thick canopy above where they just made love, setting one of her eyes on mossy green fire, mimicking the needle-strewn floor they lay on.\n\n\"What the fuck, Slash?\" A fat tear slides out of her swollen eye as it skims his body, the unnatural stance of his lower limbs. \"What'd they do to you?\"\n\nShe kneels beside him, and he gathers her close, so tightly she's hard-pressed to breathe. \"They've hurt me beyond what I can heal quickly and without a shift\u2014or food. My situation makes us vulnerable. I need you to leave me for a while, Adrianna.\"\n\nAdrianna struggles against his hold, but Slash gives a low growl and sets his teeth at her shoulder. She stills beneath him, her wolf submitting to his. Her eyes roll to meet his. \"Don't make me, Slash. I just found you.\" She touches his face. \"We just found each other. Don't make me go, please.\"\n\nSlash's eyes close against her pleas. If he has to die because he's indefensible, he won't take his mate with him.\n\nSlash releases his hold. \"Adrianna.\"\n\nHer head tips back, and he catches her fragile skull. Her eyelids hood, languid over her submission to his wolf. It's a wonderful interaction during sex, but not so much in the present, when Slash can hear the sound of a small battle happening too closely for his comfort. And Tramack's vague threat about returning hangs over him.\n\n\"It's the only way I can protect you until I heal this injury.\"\n\nTramack had performed the maneuver to paralyze an opponent before. He was far too practiced to have not. The move against him made Slash wonder how deplorable the Western pack had become.\n\nHe shakes the thoughts away. They can't afford the time of his speculations. \"I'm paralyzed,\" he says in a bald tone.\n\nDeliver the facts. No more, no less.\n\nAdrianna bites her lip, trying for brave, not that she needs to try. She is brave. She clutches his shredded shirt. \"No,\" she says with quiet intensity.\n\n\"Not forever, but right now, I must\u2014my wolf must\u2014keep my mate safe.\"\n\nAdrianna clings to him. \"I won't leave you!\"\n\nI was afraid of that.\n\nA huge sound booms where the voices have escalated to shouts. The report of gunfire echoes.\n\nHe shoves Adrianna away from him, growling. His eyes become razors, and he pushes every bit of Alpha Red to the surface of his skin.\n\n\"Do not kill me by staying,\" he delivers in low command.\n\nAdrianna's head rocks back as though she's been slapped. \"What?\" she whispers. \"I would never do anything to hurt you.\"\n\nSlash hates what he must do. But she is a young Alpha female who thinks with her emotions. Newly mated, she's vulnerable, and her feelings are right below the surface. She'll get them both killed without meaning to.\n\nSlash has to think for the both of them.\n\n\"Get the fuck out of here. It was fun, but I don't need you like an anchor around my neck.\"\n\nTears pour down her healing face like an undamed river, and she scoots away from his slack hold on her.\n\n\"Bastard!\" she screams, angrily swiping at her tears. \"Did you fucking see what that douche did to me? He marked me!\" Her hand moves to the breast Slash covered with a makeshift bra.\n\nSlash holds his flinch. It's one of the hardest reactions he's ever had to suppress. Somehow, he manages. \"Yes. Do you want him to again, female?\" Slash grinds out coldly.\n\nShe shakes her head. \"No, you asshole.\" Adrianna stands, begins to stomp off, then whirls to face him.\n\nSlash hates being on the ground, unable to stand on his own two feet.\n\n\"I would have stayed with you.\" Her voice goes low. \"Fought for you. Protected you.\"\n\nLike all Were, Slash has excellent sight. His eyes home in on the standing water in hers.\n\nTheir eyes lock. Finally, he breaks the heavy silence. \"That's not your role. Now go, female.\"\n\nAdrianna nods quickly. \"I'll go, you jerk. I can't believe you mated me and care so little. I thought\u2014\" Her hands rub at the tears brimming and streaking her bruised face. \"The hell with what I thought. At least one of those bastards is a testi shy.\"\n\nShe turns, striding away. Then she runs.\n\nA tight breath of relief slides out of Slash. He allows himself to fall backward, the leaves and needles cushioning the short slump. Slash lies there, looking up toward the sky, his chest tight.\n\nHopefully, Adrianna will be so pissed off, she'll just keep going. His nose will find her later. It's more important that she's safe\u2014and far from this place. He knows Adrianna will avoid the conflict. And most certainly, the conflict involves Tramack.\n\nSlash can explain his reasoning to her later. She would have stayed. He understands the kind of female Adrianna is.\n\nHe also understands the male that Tramack is. He'll return, regardless of whether or not a female that is supposedly his intended has been located in Region One.\n\nHe'll come back because he's a follow-through type of Were, the worst kind, made even more by his insanity.\n\nAs Slash scans his surroundings, a reluctant grim smile pulls at his lips. The two dead Were languish a few yards away from his position.\n\nAt least there's that. Now how do I get somewhere isolated that I can shift to wolfen and hunt to revitalize this body?\n\nA stealthy movement hits Slash's radar, and he tenses, feeling his toes for the first time in an hour. That's not going to do a damn bit of good if what he hears is round two with Tramack. His fingertips itch to burst to talons, his emotions inciting the change like a lit match.\n\nSlash resists. He's exposed.\n\nTruman moves from behind a tree, looking around at the mess and the Were corpses. \"What the fuck?\" he asks, and Slash grins. His relief is so intense, his normally stoic personality momentarily leaves him.\n\n\"Yes, that's exactly it.\"\n\n\"It's that Alpha Zeke and I scented, right?\" A shadow crosses his face, and Slash sits up straighter, hiking his ass so he can be upright. It's not standing, but it's not lying on his back, either.\n\nFeeling swarms to Slash's ankles. Hope flares.\n\nKarl Truman\u2014a former Homer, AK cop, and now the oldest living change on record\u2014stares at Slash's legs. His eyes slide back to Slash's. \"That prick do that to your legs?\"\n\nSlash nods. \"I can feel my feet.\"\n\n\"Asshole.\" Truman's eyebrows jump. \"Where's Adi?\"\n\nSlash's chin dips to his chest, and a frustrated exhale squeezes out. \"I rejected her so she'd leave.\"\n\nHe whistles. \"Ouch! Didn't want that Alpha to come back and hurt Adi when you couldn't protect the new mate?\"\n\nSlash nods; there are no secrets with the nose Were possess. Glad someone understands the logistics of being a male who loves his female more than taking his next breath. Slash answers in a curt voice, \"Yes.\"\n\n\"Well, you want the good news or bad?\"\n\nFuck. \"Bad.\"\n\n\"Zeke's dead.\"\n\nShock rips through Slash, though his surprise is short-lived. \"Tramack?\"\n\n\"Who?\" Truman's eyebrows jump again.\n\n\"He's the fuck from the Western.\"\n\nTruman's hands go to his hips. \"Listen, I got changed by the Northwestern and don't give a good goddamn about keeping track of who's who and all the political bullshittery.\"\n\nSlash blinks.\n\nTruman goes on, \"So an Alpha who's not from here killed Zeke, though he managed to wipe out some of them. That's what that garbage smell is.\" Truman cups his chin.\n\nThere were even more?\n\nTruman nods slowly. \"Yeah, little posse of Were. Here. For what?\" He spins in a circle, coming to stop in front of Slash. \"Don't know. Don't give a ripe shit.\" He taps his nose. \"They're presently after the two girls that you and Zeke had just picked up. And the two males from Region Two?\" Truman barks out a laugh of pure disbelief, shaking his head.\n\nSlash's eyebrows drop and his gaze narrows on Truman. \"Tell me.\"\n\n\"Those bozos are fucking demonic. Tails\u2014the whole nine yards.\"\n\nGooseflesh ripples over Slash's skin. His beast begs to change, though without a full moon, that's only a painful wish.\n\n\"I can't make this shit up.\" Truman shrugs.\n\nHis eyes meet Truman's. \"What now?\"\n\n\"I drag your fucked-up ass to the mansion, locate your bride, and kick some foreign Weres' ass,\" he recites calmly.\n\nTruman's simplicity is somehow perfect. He doesn't understand enough to worry about it.\n\nIt is simple: find the bad guys and eliminate them.\n\nPoint and shoot. Slash wishes the process could be that simple.\n\n# CHAPTER FOUR\n\nJulia\n\nAll the rooms are the same. Julia doesn't know what she was expecting.\n\n\"They're like crypts,\" she whispers.\n\nVictor's beautiful, dove gray eyes stare into her own.\n\n\"They are sound-proofed and meant to be similar to the humans' catastrophic nuclear strongholds.\"\n\nGround zero. Julia shivers, and after studying the rows of metal doors leading to compartments for Singers, she turns back to Victor. \"You mean if a nuclear bomb went off?\"\n\nHe gives a solemn nod. \"In theory, our bunker can withstand it.\"\n\nScott squeezes her fingers reassuringly.\n\nJulia is thoughtfully silent for a few seconds. \"We're not deep enough.\"\n\n\"True.\" Victor strolls to a door like all the others. However, this one has a security pad with numbers. Victor taps a series of numbers into the lighted keypad.\n\nJulia flinches when it beeps like a whining siren.\n\nVictor opens the door, and crisp, slick stainless doors butt together. Only a dark, slim seam marks the center.\n\nElevator. \"Where does that go?\" Julia asks.\n\n\"Deep,\" Victor replies.\n\nJulia turns to Scott. \"You knew?\"\n\nHe nods. \"It's not like we hang out in the bunker all the time. The royal blood lines all have a safe place to be. Never even thought to give the shaft any thought. No one anticipates nuclear holocaust.\" Scott shrugs.\n\n\"Somebody did,\" Julia replies softly, her attention shifting to the elevator once more.\n\n\"Yes.\" Victor closes the innocuous door, and the elevator disappears behind it, blending in with all the others.\n\nVictor leaves them at the farthest reaches of the bunker. Julia estimates the size to be, on all one level, as large as the mansion above it. She didn't think to ask what the \"shaft\" size was.\n\nVictor rounds the corner but turns at the last moment, giving them a level stare full of meaning.\n\nHis looks seems all for Scott.\n\n\"Your quarters are already assigned.\"\n\nJulia feels her eyebrow lift. Assigned?\n\nVictor interprets her puzzled expression perfectly and knots his hands together. \"In the event of a need to use the bunker\"\u2014his eyes rise to hers then briefly flick to Scott's\u2014\"each royal family member would have their own accommodation. That still remains the case. As it happens, Marcus had no official spouse during his reign as Region One king.\"\n\nOfficial?\n\n\"But I do,\" Scott says from behind Julia.\n\n\"Yes.\" Lifting his chin, he gives Scott a significant look.\n\n\"What, guys?\"\n\n\"What Victor is too polite to mention is that you are meant to be my wife, but until we do the deed, it's sort of a hollow, unfinished union.\"\n\nA pale-pink blush spreads across Victor's cheeks. \"Not as I would have put things, but accurate enough.\"\n\nJulia says nothing. She knew it would come to this.\n\n\"Julia.\"\n\nShe turns around and faces Scott.\n\nHe takes her chilled hands in his warm ones.\n\nLowering his face to be level with hers, he asks, \"Is sleeping with me a fate worse than death?\"\n\nJulia bursts out laughing. \"No. I-I don't know. Everything's moving too fast.\" Her heartbeats pile up on each other, her palms dampening.\n\n\"I will be in the main area near the vault,\" Victor announces quietly.\n\nNeither of them replies, but his departing footsteps echo hollowly back as Julia and Scott stare at each other.\n\n\"I am sorry about Jason.\" His dark eyes search her face. \"But I'm not sorry that he sacrificed himself to allow you to live. I'm just gonna keep saying the truth over and over again. No guilt, baby.\" Scott cups her face, and she sighs.\n\nHe's so warm.\n\nPresent.\n\nHers.\n\nJulia feels the rightness of the meld between them. Like warm water it molds to their bodies, reaching every bit of her. Their bond isn't conflicted, and she shouldn't be, either.\n\nScott waits in the silence pregnant with their emotions.\n\nHe wants her as a man wants a woman. He also wants her because the Combatant inside of him knows she'll have a net of security that would be very hard to strip if they finalize their union.\n\nJulia nods, and Scott's shoulders drop, the tension of their tethered hands lessening, growing supple.\n\nHis eyes move to the door that simply has a musical note finished in a shining red enamel.\n\nThe color of blood. The global symbol for a melody.\n\nBlood Singers.\n\n*\n\n\"Don't get pissed,\" Scott says, and her eyes move to his.\n\nShe was looking around the room. Julia didn't feel pissed; she was stunned.\n\nThe space must be something special for the bunker. It's unadorned and so plain, it's almost medicinal\u2014except for a huge bed in the center of the oversized bedroom. An old-fashioned quilt graces the top, and four body-sized pillows round out a metal headboard. Two awkwardly narrow nightstands flank it, matching lamps softly glowing on top.\n\nJulia shivers.\n\nScott notices, and his fingers part from hers.\n\nHe moves to the door then closes and locks it. Julia flinches as the deadbolt slides home.\n\nHis fingers twirl the thermostat to a higher temperature. Scott turns, and she's overwhelmed anew by his size, flushing at the thought of being with him.\n\nThis is gonna happen.\n\n\"Better?\" he asks quietly.\n\nShe nods, looking down at her feet.\n\nJulia shifts her weight nervously, and Scott's larger feet come into view, dwarfing her Converse sneakers.\n\n\"Scott,\" Julia begins, her voice breathy.\n\nHe wraps his strong arms around her.\n\n\"I'm scared.\"\n\nScott pulls away, cupping her chin and tilting it up. Gravity pulls the tears from her eyes.\n\n\"Don't, Julia. I would never hurt you.\"\n\nShe nods, blinking.\n\nHis thumbs wipe the tears away. \"Why are you crying?\"\n\nHer exhale trembles. \"So many reasons.\"\n\n\"You think I'll suck in the sack?\" His lips twist.\n\nShe laughs, and they move apart. Julia smiles, appreciating the way he breaks the ice.\n\n\"Better?\" he asks for a second time.\n\nShe nods. \"I don't\u2014I'm not thinking about that.\"\n\n\"I know,\" he says softly, tucking a long strand of hair behind her ear. \"Then what is it?\"\n\nJulia gives him a look of pure disbelief, folding her arms. \"Jason just died. Our people are in danger.\"\n\n\"And we have to be down here for nearly two days,\" he states logically.\n\nShe snorts. \"So we might as well screw because what else is there to do?\" Julia sounds bitter.\n\n\"Yup!\" Scott says, angrily pacing away from her. \"Can't wait to have everyone in danger and die so we can hop into bed together. Right. So sexy.\"\n\nJulia hangs her head. \"I guess it sounds pretty dramatic.\"\n\n\"Yeah.\" He sounds pissed. His entire body is one long line of tension.\n\nIt has to be her; he won't push. He's made his case. Victor's made it.\n\nJulia walks to him slowly.\n\nScott doesn't move a muscle, showing her his profile.\n\nShe flattens her palm against his chest, and a sigh escapes them both. \"I'm sorry,\" she whispers. \"I don't mean to be a bitch.\"\n\nHis long fingers cover hers. Still looking away, he speaks to the flat gray wall. \"You're not. Any human being would feel the same right now. You've gone through hell in the last three years, Julia.\"\n\n\"I'm not strictly human, Scott.\"\n\n\"Yeah.\"\n\nThey look at each other, their heartbeats syncing.\n\n\"That's so weird,\" Julia says with a tremor in her voice.\n\nScott knows what she's saying; it's all over his face. \"Not to me.\"\n\n\"That's because you were groomed for the possibility of the soul-meld.\"\n\n\"Yes, but it's still surreal\u2014our bodies knowing each other, wanting that constant physical contact. And if you remember, I wasn't exactly on board.\"\n\nSadness tightens her chest. \"I was a girl from Alaska two years ago. Now I'm a queen of a bunch of paranormal people and part angel and\u2014\"\n\nScott presses a finger to her lower lip. \"And you're the woman I love.\"\n\nJulia kisses along the side of his finger, and Scott inhales sharply. She stares into his eyes, drowning.\n\n\"Julia, if you want to back out of this, do it now, because, baby, I can't stop.\"\n\nShe shakes her head, releasing the guilt and shame, the ownership of things she shouldn't keep.\n\nScott shudders, wrapping her against him. \"Do you trust me?\" he asks against her hair. Their bond is a rubber band of comfort, security, love, and lastly, good, old-fashioned lust.\n\n\"With everything I am.\"\n\n\"Then let me take care of you, Julia. For the love of all that's holy, let me be the one to take care of you.\"\n\nJulia wraps her arms around his waist, laying her cheek between the hard muscles of his chest. \"Yes.\"\n\n*\n\nThe room is warm.\n\nJulia is hot.\n\nScott begins to undress, his eyes glued to her. He jerks off his T-shirt and tosses it onto a folding chair against the wall.\n\n\"It's creepy that there's a chair there beside the bed. What? Is there an audience, possibility?\"\n\n\"No,\" he says, voice deep.\n\nShe swallows her nervousness, trying not to ramble on anxiously. Again.\n\n\"Not even a remote chance of interruption.\"\n\nJulia nods, dumbstruck.\n\nScott is a beautiful man. His large hands go to his hips as he stares at Julia. A handful of seconds squeeze by as they regard each other.\n\nThen his fingers close on his belt buckle.\n\nHe slides his jeans down before kicking them off in the direction of the creepy folding chair. When Scott is clad only in socks and boxer briefs, Julia closes her eyes.\n\nHer mind supplies the rest: Six feet, three inches of heavy muscles. Narrow hips. Broad shoulders. Legs like tree trunks. Soon she'll see all that she's imagined in the flesh. Julia calms her breathing and opens her eyes.\n\nScott stands before her. A little \"eep\" flies out of her. Her eye roam his naked body. She'd missed the unveiling but she's all eyes now. He's so much more than her imagination could have ever conjured.\n\nJulia grins. \"You scared me.\"\n\nScott's smile broadens.\n\n\"What are you smiling about?\" she asks in a huff, keeping her eyes on his face with an effort.\n\n\"You.\"\n\nJulia gives a tense little laugh. \"I guess it's\u2014\"\n\n\"Let me, Julia.\" His eyes shift to her clothing.\n\nShe nods, her heart thumping, and lifts her arms above her head.\n\nScott strips the T-shirt and throws it onto his pile of clothes. His hands move to the waistband of her jeans and slide them off slowly.\n\nJulia doesn't cover herself. What they're about to do doesn't seem sleazy.\n\nIt feels right.\n\nBefore guilt has a chance to form, she rolls to her tiptoes and tilts her head for the kiss she knows he'll give her. His intent pounds through their meld.\n\nJulia's not disappointed.\n\nHis hands land on her bare back, rolling over her skin. Her flesh pebbles beneath his fingertips, and a little sigh escapes her lips.\n\nJulia becomes aware in stages that her panties and bra joined his clothing on the floor.\n\nScott scoops her from the floor, his dark gaze tied to her face.\n\nThe emotion in his eyes heals her. His body undoes her.\n\n# CHAPTER FIVE\n\nTessa\n\nThe demon's penis drops to the dusty driveway like an exhumed hotdog. Black blood pours out of the ragged hole she made with a quick talon swipe. Oops.\n\nTessa guffaws, because that's the best thing to do while wrapped against a demon. She's always wanted to try that with a deserving male, and this fire dick fit the bill.\n\nSound ceases.\n\nHis hold loosens, and his mouth opens in dawning horror as his hands automatically go to his crotch.\n\n\"Sorry about that, asshole,\" she says conversationally.\n\nSuddenly, Tessa is hoisted off her feet, and her hands instinctively slice backward, hitting only air.\n\nLazarus whispers, \"We are going.\"\n\nThen she's being turned and squired to the Suburban in a blur of speed.\n\nTahlia, looking human again, is buried behind the too-big wheel. Her eyes are mostly human, and entirely too young in a face pinched by fear.\n\n\"Tessa!\" she screams as Lazarus tosses her inside the back of the truck. Tessa shoots across the bench seat like a cannon ball, sharply rapping the top of her skull on the opposite door.\n\nLights burst and flash behind her eyes. \"Ah!\" Tessa instinctively curls up into the fetal position. That fucking hurt.\n\nWith a shrieking squeal, the door she was heaved through is torn off.\n\nTessa sits up, her vision warped as she sees Lazarus holding the door like a shield in front of his muscular body.\n\nThen she sees why.\n\nIn the next second, Fire Dickless is storming toward him.\n\nToward me.\n\nThe real bad? His penis is growing back and is presently a fleshy little red nub between his legs. It's deliciously pathetic looking.\n\nDon't laugh, please, Tessa.\n\nShe laughs. Tears stream down her face, and she can't stop. Knowing she should makes it worse.\n\nWith a dull roar of rage, Lazarus brings the car door up between them in a swinging strike, tipping the broadest part sideways at the last second and impaling the demon in his gut.\n\n\"Fore!\" Tessa whoops as though he's hit a golf ball, with zero sense of self-preservation.\n\nLazarus follows through with an open kick that sends the demonic flying. He turns, giving her a death stare. \"That is unhelpful.\"\n\nRight. Tessa clamps her mouth shut.\n\nA giggle erupts.\n\n\"Hang on!\" Tahlia screams from the front seat.\n\nIt does occur to Tessa then she might be a little bit in shock\u2014or more than a little bit. It's not normal to laugh when a wounded demonic is gunning for them. She nods at Lazarus's obvious rage then slumps back against the door, laughing again.\n\nLazarus pushes in beside her.\n\n\"Go\u2014now!\" Lazarus shouts at Tahlia.\n\nShe nods, stomping on the accelerator. The tires spin uselessly.\n\nTessa turns and looks at the back of the rig. Her laughter dies.\n\nThe demonic is holding onto the tailgate, his triumphant smile revealing gleaming black teeth.\n\n\"Reverse!\" Lazarus yells.\n\nOh, shit.\n\nThe click of the transmission locks into place, and suddenly, he's underneath the carriage of the car.\n\nThump.\n\nTessa's ass lifts off the seat. Her hair clings to the ceiling briefly before her butt drops back onto the seat.\n\nThe four-wheel drive locks into place, and the front wheels roll over the top of the demonic.\n\nThump.\n\n\"Forward!\" Lazarus says above the grinding, rising car.\n\nThe Suburban gives an ungainly lurch and seems to pause as if stuck on the biggest speed bump ever.\n\nTessa's eyes find Lazarus.\n\nHis face is grim.\n\nAnd handsome. Tessa's mouth pops open to speak.\n\nThe car plops down on the other side of the demonic, and spraying loose gravel, it speeds off.\n\nTessa turns, getting to her knees. Gripping the back of the seat, she watches as a mangled body rises like a crimson zombie.\n\n\"Oh my God!\"\n\nOut of the corner of her eye, she sees Lazarus wince.\n\nOh, yeah, no religious terms. Tessa sinks onto the seat, her eyes tracking the demonic as he becomes an ever-smaller red dot in the distance, like a Mexican jumping bean.\n\nHe's steaming pissed.\n\nTessa rotates slowly to face the front again, lowering herself onto the seat. \"That was close.\" Her heart's pounding begins to slow.\n\nLazarus turns, letting his head fall back against the seat. Air rushing through the hole of the door serenades them with road noise.\n\nTessa blinks at the blurring landscape. That's right: demon boy is back there with a half-crushed door in his guts.\n\nOr is the car door gone now because Tahlia ran him over? Hell on the tires, Tessa muses.\n\n\"Where now?\" Tahlia asks, gripping the steering wheel so tightly that her knuckles are white.\n\nProbably not fair to make the youngest the driver in the Race of Death.\n\nShit happens.\n\nLazarus runs his fingers through his hair, pulling it away from his perfectly formed face.\n\nTessa's crazy to still be worrying about him when they just barely managed to avoid catastrophe. Still.\n\nHis eyes meet Tahlia's in the rearview mirror. \"I do not know.\"\n\n\"Oh fantastic,\" Tahlia mutters. \"Let me understand this. I am pursued by the Lanarre\u2014rightfully, I might add. Then I decide to leave, giving myself the illusion of choice. And instead of riding off into the great yonder\u2014\"\n\nTahlia rolls her eyes as Lazarus's eyebrows slowly rise.\n\n\"I must contend with a demonic passenger after my friend relieved another of the horned persuasion of his penis.\"\n\n\"Huh,\" Tessa harrumphs. \"Well, that's a neat-o summary as those go, but lacking a few details.\"\n\nTessa turns to Lazarus. \"Lazarus\u2014\"\n\n\"Laz,\" he corrects, that vague smile hanging on his lips.\n\nTessa blinks. \"Laz, what in the hell is going on?\"\n\nThe corners of his lips tweak. \"What indeed?\"\n\nCute or not, Tessa sort of wants to wipe that insufferable smile off his face. Of course, the way she wants to do so is probably not ideal\u2014with her tongue.\n\n\"It's too noisy to hear anything,\" Tahlia grumbles from the front. \"And I don't know where we're going.\"\n\n\"Drive,\" Laz instructs.\n\nHer eyes shoot daggers into his through the mirror. \"You do not command a Lanarre female anything, horned one.\"\n\nLaz laughs. \"I haven't heard that in a coon's age.\"\n\nTessa's brows knot. \"Really? Fine. Speak loudly over the din.\"\n\nLaz exhales roughly, jerking his chin toward Tahlia. \"The demonic hold the Lanarre in a higher regard than most of the supernaturals. That is true.\"\n\nA smug smile twists Tahlia's lips, though her eyes remain on the gray ribbon of highway in front of them.\n\n\"However, every being who has a soul is culpable to our purposes.\"\n\nTessa hears only the greedy air grabbing at their vehicle.\n\nAfter a full minute has passed, Tahlia asks, \"Forgive me, why are we helping you?\"\n\n\"It is I who helped you.\" His eyes dodge to the mirror in challenge. \"Except for your fine work with the Alpha.\"\n\nTessa ignores Tahlia's bird-ate-the-canary grin. \"Explain.\" She folds her arms. This ought to be good.\n\n\"Praile is a high demon\u2014as am I.\"\n\nTessa's throat tightens at the D word. Really, there's no good connotation to that.\n\n\"You guys don't look the same,\" Tessa states the obvious. It's not as though she's tripped over a bunch of demonic in her life. What did Tramack say? That they weren't allowed in this realm or something like that.\n\n\"Praile is the best of us. He has all the traits of beauty among our realm.\"\n\nTahlia grunts loudly enough to be heard over the roar of noise.\n\nTessa sniffs. \"He's fugly.\"\n\nA small smile touches Laz's mouth then vanishes.\n\n\"I agree,\" Tahlia chimes in.\n\n\"Your earthly views on attractiveness are not important in our realm. In the demonic realm, dark-red skin, horns and teeth of black, a tail made for battle\u2014those are deemed beautiful.\"\n\n\"He's less beautiful now without a prick,\" Tessa comments dryly.\n\nTahlia's shout of laughter in the front are followed by a soft whoop.\n\nLaz suddenly grips Tessa's shoulders, causing her to yelp in surprise, and drags her against him.\n\nThe wheel jerks, and they nearly sail into the road through the hole in the side of the car.\n\n\"What are you doing to Tessa?\" Tahlia shouts.\n\nLaz ignores her, his eyes going black.\n\n\"You're scaring me!\" Tessa says, squirming.\n\n\"Good, for Praile will retaliate.\"\n\n\"He can't. He's fifty miles behind us.\"\n\nLaz releases her abruptly, shaking his head. \"He is demonic. Do you understand what that means?\"\n\nTessa was born in the 1960s. By werewolf standards, she's young for a female. She has studied some history, but demonic lore was never high on her list. \"No,\" she says at last.\n\n\"It means that Praile has a strength in this realm that he lacks, even as a high demon, in Hades. He can do much here.\"\n\n\"Why did you\u2014why didn't you stay behind?\" Tessa asks.\n\nLaz stares at her so long, Tessa thinks he might be ignoring her question. Then he replies, \"Because my Redemptive has been revealed. That is all I needed. All that was necessary for me to eschew my origin. To never have to be in hell again.\" He curls his long fingers around Tessa's nape, gently cupping the hot flesh around her skin, warming it.\n\n\"You're hot,\" she whispers.\n\nLaz nods, the small smile she's coming to like frosting his lips like cake icing. \"Yes, I am demonic. It is part of what and who I am.\"\n\n\"Why don't you look like that Praile nutjob?\"\n\nHis lips quirk. \"It is rare, but sometimes, an ugly demonic will be born. He will lack all the features of beauty among our kind.\" Laz gives a rolling shrug, dismissing the way he just put himself down.\n\nTahlia's eyes flick to Tessa's then back to the road.\n\n\"So you're ugly.\"\n\nLaz chuckles. \"Very.\"\n\nTessa frowns. \"Hmm. Okay, so aside from your grotesque looks...\" She looks up from beneath her eyelashes, and his face is solemn.\n\nShe hits him on the arm. \"Kidding!\"\n\nHis answering smile is tentative.\n\n\"God\u2014touchy!\"\n\nLaz flinches.\n\n\"Sorry.\" Tessa pauses for a second, thinking about what to say next. \"They really did a number on you.\"\n\nLaz scowls. \"They did nothing to me. It has always been the order of Hades. An ugly demonic who is a high demon must prove their worth in other ways. They do not have their looks to rely on.\"\n\n\"Oh?\" Tahlia says.\n\n\"Yes.\"\n\nTessa's getting this might be bad. She asks anyway, \"So what have you had to do?\"\n\nHis eyes glitter, their color morphing from the pale grayish-blue to a smoky charcoal. \"Many unmentionable things.\"\n\nOh. \"What\u2014why are you different?\" Tessa asks instead of pressing him for all his demon deeds.\n\n\"I am not purely demonic.\" His words hold shame.\n\nOr maybe I'm reading things into that.\n\n\"Okkaaay.\"\n\nHis head jerks toward her, his eyes now black. \"I can heal others. I have a mixed heritage. Imagine being a ticking time bomb of mystery among those whose destinies are set as high demons with a fate as certain as the sunrise.\"\n\nPretty awkward. \"And this Redemptive craziness?\"\n\n\"Stop the vehicle,\" Laz says in a powerful voice.\n\n\"What about the demonic, Praile?\" Talia asks in a strained voice, her eyes bouncing between them and the road.\n\n\"Pull. Over.\"\n\nTahlia's mouth becomes a slash in her pretty face as she jerks the wheel. The car slams into the soft shoulder.\n\nShe shoves the gearshift into park and turns around, gripping the back of the driver's seat. \"You will show me respect, horned one, or I will become an eagle in the time you take your next breath and pluck the beak from your face.\"\n\nLaz leans forward, nearly nose to nose with Tahlia. \"And I shall suck your soul in one gulp, bird or no.\"\n\nThey glare at each other.\n\n\"Guys,\" Tessa interrupts their combative posturing. \"Totally not helping. And Praile is probably really cranked.\" Definitely. \"Let's get things discussed and find a safe place. And ditch this car while we're at it\u2014kind of conspicuous.\" What with the built-in AC for the lack of door.\n\nLaz exhales in disgust, dismissing Tahlia.\n\nGreat. That's all I need is those two at each other's throats. Tessa wonders how she acquired a rebellious whelp Lanarre and a demonic. How does this shit keep happening to me?\n\nLunatic life, that's how.\n\nLaz turns to face her, his intense eyes trapping her. \"You are my chosen one. The prophetical Redemptive.\"\n\nTessa represses the urge to smile. \"I think you mentioned that when we were sucking face.\"\n\n\"What!\" Tahlia shrieks in surprise.\n\nLaz lifts his chin. \"Silence, Lanarre.\"\n\n\"I will kill you,\" Tahlia huffs, her iolite-colored eyes flashing.\n\n\"Negative,\" Tessa says, shooting her a \"shut up\" glance.\n\n\"You will try,\" Laz murmurs.\n\nTessa grunts. \"You're sure immature for a big bad demonic.\"\n\nLaz leans forward, capturing her jaw in a hand that could crush her. Strength thrums through his capable fingers. \"I am more evil than anything you could dream up in the vastness of your thought process.\"\n\nTessa's blood runs hot, surfacing over her entire body and tingling insanely where his touch lingers. She fights rubbing against him like a cat with its favorite perch. \"If you're so evil,\" Tessa whispers, \"then why do you feel so good?\"\n\nLaz's forehead tips to meet hers, their breath mingling.\n\nHoly hot tamale. Cinnamon sweet and yum. Tessa's suddenly dizzy.\n\n\"That is the question you must answer. I know what the Redemptive means to me. I will stop at nothing to protect you. You are the most cherished discovery a high demon of mixed heritage can hope for.\"\n\n\"Wait a second.\" Tessa pulls away. And like releasing yourself from warm bathwater, it's a shock to the system, an unpleasant withdrawal.\n\nOh I'm in such deep shit.\n\n\"Are you saying\u2014what are you?\"\n\nLaz shakes his head. It's hard to think when his thumb is moving over the pulse in her throat like a feathery whip of fire.\n\n\"Tessa, I don't like this,\" Tahlia says, warily staring at Laz.\n\n\"Me, either,\" Tessa answers softly. But Laz did save her\u2014them. He deserves to say his piece.\n\nLaz lifts a muscular shoulder, and Tessa's breath catches at the recognition of how wonderfully put together he is.\n\nWait\u2014maybe all this bullshit \"pull\" is nothing more than heat. How long had it been since I was in heat anyway?\n\nLaz jerks her out of her thoughts. \"I do know that I am high demon. But the other species is anyone's guess. Demonic is always primary\u2014as angelic is.\"\n\n\"But you're a Healer?\" Tessa asks, trying to shake off the image of humping a certain red guy.\n\n\"Yes.\"\n\n\"What species can do that?\" Maybe they can employ simple deduction and figure out what brand of Heinz 57 Laz is.\n\n\"Many. Fey, Singers, some vampire\u2014mostly, they all can. Though pure demonic cannot.\"\n\n\"What do I have to be in order to be your Redemptive?\" Tessa asks, without bothering to censor her suspicious tone, though she can't deny there's a powerful connection.\n\n\"We would have to be of the same blood.\"\n\nGah. Tessa pulls a face. \"Ooh. Like related?\"\n\nLaz grins suddenly, and the light expression is so different than his typical somberness that Tessa automatically laughs. \"No, we are not of close blood\u2014like blood.\"\n\n\"I am pure Were,\" Tessa says matter-of-factly. That solves that.\n\nLaz shakes his head, his irises back to the soft pale hue Tessa connects with everything being okay. \"No supernatural is pure. Anciently, we were one species.\" He knits his hands to illustrate his point. \"Excepting the angelic and demonic, all the other supernaturals were as one. Then inexplicably, there was a fracture, and the species became separate. Now we have what we have.\"\n\n\"In your theory\u2014\"\n\n\"Fact.\"\n\nTessa groans. \"Anyway, if it were true, how do we figure out what I am.\"\n\n\"That Were knows what you are, and he wants you.\"\n\n\"Tramack wants me because he's psycho.\" Tessa rolls her eyes heavenward.\n\nTahlia snickers.\n\nLaz shoots her a long-suffering look. His attention shifts back to Tessa. \"He wants you because he somehow senses your specialness.\"\n\nTwo decades of chasing her does seem excessive. All those sleepless nights of wondering why he couldn't just find a passive little woman and have a litter of little Tramack's didn't escape her thought process.\n\n\"So what is our mutual blood?\"\n\n\"Does it matter?\" Laz asks.\n\nAt the end of the day, probably not.\n\n\"What do you get out of her being your Redemptive?\" Tahlia asks suspiciously.\n\nWhat if I don't want to be a Redemptive?\n\nLaz says, \"That's simple\u2014I don't go back to Hades. Ever. I am free.\"\n\nAh.\n\n# CHAPTER SIX\n\nDrek\n\n\"I smell her everywhere,\" Drek comments, lifting his palm off the forest floor. He brushes his hands off on the athletic pants so typical of what his kind wear during wolfen form.\n\nEven though shifting is simple for a Lanarre werewolf, there is nothing simple about maintaining the form. All forms require excessive energy without the full moon. For the common among his kind, a full change is only possible at the moon's zenith.\n\nBowen tips his head in the direction that they both scent the escaped Tahlia. Drek shouldn't allow hope of finding her. Scent-tracking doesn't mean anything except that she was alive when she moved through this area. By some miracle, Tahlia survived a massacre by an unhinged Alpha Were.\n\nIt's up to Drek to recover Tahlia and protect her from any future challenges.\n\n\"Drek,\" Bowen calls in a low voice.\n\n\"Wait\u2014\" Drek's brows come together. His ears are sensitive, and elongated in his wolfen form. They fold forward slightly, and he stills.\n\nFar off to the east, noises of conflict reach him\u2014a particular kind of conflict. Violence.\n\nHe lifts his nose to the sky. The moon rolls over the horizon as though called by him. It is waxing to full.\n\nHe notices Bowen is studying him, reading the subtle tells of his face. \"Drek, the moon's call is weak.\"\n\nHis eyes move to Bowen's, spinning silver in his head. \"Do not tell me what call I can or cannot answer. I am a Lanarre prince.\"\n\n\"Fine\u2014fuck it. It'll hurt like a bitch.\"\n\n\"It will. But I can find Tahlia faster if I become my beast.\"\n\n\"That means I have to shift.\"\n\n\"Do not.\"\n\n\"Right\u2014and be known as the shittiest guard that ever howled? I'm royal enough, Drek. I've got some of that righteous blueblood of the Lanarre, too.\"\n\nDrek looks away, throwing his shoulders back.\n\nHis fur ripples as he seizes the barest voice of the moon's call. It's a whisper.\n\nBut Drek is Lanarre, and that is enough.\n\nHis skull splits first. He can't fight the pain of a shift this far from the moon's siren, and his body slumps, knees hitting the soft undergrowth of pine needles, moss, and leaves.\n\nVaguely, he senses Bowen's shift to his right. The other Were's change will hurt even more, for he does not possess enough of the royal blood to shift without a high degree of pain.\n\nSaliva strings dangle from his mouth as a double row of canines sprout like knives inside his mouth. Drek's shriek of agony is a garbled wail. He falls to his side, his chest heaving. He whimpers as brutally sharp, short talons erupt from his legs.\n\nDew claws for vertical climbing, absent in mundane Lycan, push through the back of his legs. Powerful, with their own specially designed talons, they hurt like wisdom teeth breaking through in a human mouth.\n\nDrek doesn't feel wise; he aches to his very molecules.\n\nLastly, his ears and eyes explode like spoiled fruit. Drek lies in the soupy gore of his own change, blind and momentarily deaf, while his vision comes to him in shades of gray. Stages of clarity gradually form into a crystal-clear acuity of a full shift.\n\nDrek lies still. The seconds pile up to minutes. The noise of animals breathing reach his acute hearing.\n\nThe smell of a bird shitting a mile from his position causes his nose to stir. He lifts his muzzle off the richly scented, lichen-laden forest carpet and staggers to his feet.\n\nWith a mighty shake of his body, his fur fluffs out, shaking away the miserable remnants of his wolfen form.\n\nDrek raises his snout, testing the air and sighting Bowen.\n\nHe is sleeping in the muck of his change.\n\nDrek grins with the mouth of his wolf. He looks to the moon, somewhere between three quarters to full, and howls a singing praise of thanks.\n\nOne must always thank the source of one's power.\n\nEspecially a Lanarre.\n\n*\n\nDrek had to wait for Bowen to come to himself, then suffer through his complaint of the wretchedness of changing that far from the moon's fullness. Drek reminded him that they were lucky they had the ability.\n\nHe gave Drek a sour look.\n\nWhen that bit was finally accomplished, they trotted to where the scent of Tahlia burnt his nostrils. There is no scent beyond him while he is a werewolf.\n\n\"A terrible wonder\" some Lanarre call the absurdly refined olfactory senses of the royal Lycans.\n\nDrek's extreme case of nose makes him a supreme tracker. Many of the Lanarre believe his skills are wasted because he is a prince, and as such, nothing is required of him but the making of new little princes.\n\nDrek considers that role to be extremely tiresome.\n\nYet, he has told no one, though he believes Bowen suspects how much he would like to change the archaic laws that have been in place for over a thousand years.\n\nNew challenges hasten the need for responses that are different than the canned protocol that has become ineffective\u2014such as the arranged match between he and Tahlia, a young woman he has never met.\n\nIt is his secret, and his alone. But he has no wish to mate someone with whom he has no kinship. She is his chosen\u2014but not his beloved. He would free her from this obligation if he could, and he often wonders if she feels similarly.\n\nHe endeavored to speak with her, ascertain her thoughts. That was before.\n\nBefore the merciless killing of her guardians. Before her survival became paramount.\n\nNo. He will find Tahlia and secure her safety. Then, and only then, will he see to her assimilation into the Lanarre.\n\nOf course, that presumption is met with a dozen thoughts of what that could mean to her.\n\nTahlia might become an outcast if the others believe he rejected her because of some perceived imperfection.\n\nDrek is sure that she is perfect. Lanarre royalty have been deliberately bred to personify the perfect physical characteristics of the Lycan.\n\nHe's also sure that she is barely out of whelphood and not perfect for him. Tahlia is a good choice as a mate because she is of pure royal blood, as pure as his own.\n\nAnd she is a wereshifter\u2014a rare trait. That very thing undoubtedly saved her from the murderous, crazed Alpha whose scent he now owns.\n\nThe royal families of all the Lanarre families have been aligned for hundreds of years. It's time to change that and give males and females of royal blood a chance to choose whomever they want.\n\n\"Are you going to be okay?\" Bowen growls. Though his vocals are a series of yips and soft barks to those who are not Were, for Drek, it is as refined as his human speech.\n\nDrek is able to hear Bowen's sarcasm a mile away, and he glares at his trusted guard. His return barks are also soft, nearly inaudible.\n\n\"I'm in my head, Bowen.\"\n\n\"I see that.\"\n\nDrek would love to seek his council. If there were retribution for his beliefs, which are not in keeping with Lanarre law, it is better that Bowen doesn't know them. Drek can admit that Bowen is more friend than guard. So he will shoulder the burden alone.\n\nDrek could have sent an entire contingent to track Tahlia when she did not arrive at the preordained time. But what kind of prince would I be if I'm unwilling to find my own chosen? Drek has always felt strongly that his role as prince requires him to lead by example. Strong leadership through acts are more powerful than words. Every time.\n\nHe will retrieve the sacred chosen\u2014his chosen\u2014and work out his internal struggles from there. Her safety is paramount.\n\nBowen yips, and Drek turns toward where his large snout points.\n\nDrek knows he cannot truly frown in this form, but he does frown.\n\nThey've come to an open glade. A large late-nineteenth-century manse rises from a knoll like a jewel of architecture. To the northerly position sits a huge red barn, though Drek doubts livestock have graced the inside of the structure in a long time, if ever. It does not scent of animals in the last fifty years.\n\nIn human form, Drek would have laughed at the expression Bowen's wolf gives. But humor is in short supply. Bowen swings his massive face toward the structures and gives a soft yip. His eyes rotate so quickly, they're spinning coins of molten silver. His coat is gray, like Drek's, but Drek retains the inky tips of royalty.\n\nThe two move cautiously closer.\n\nThey've entered a loud ruckus\u2014the same one he'd heard some miles back, before they changed. Drek scents Bowen's intent, and with a snuffle of acquiescence, they trot with well-trained unity, seamlessly parting and running in mirrored semi-circles around where a horned one lies writhing in the dirt.\n\nAnother is beside him.\n\nAn Alpha.\n\nDrek scents deeply of him, smelling many things, but not the scent he most wishes for. It's both a disappointment and a relief. It would not end well for this Alpha if he had smelled of Tahlia.\n\nThe fragrance that rises most quickly is the foreign Alpha's injury, but underneath that, the certain stirrings of insanity lurk like rancid liquid from garbage gone bad.\n\nBowen gives Drek uneasy eyes from across the yard, where he is positioned in the shadow of the barn. Insanity is a real problem with older Alpha's. There is usually a catalyst for the origin, but once it begins, it can spiral into true lunacy.\n\nTheir spinning orbs regard each other.\n\nWhat has happened here? They seem to silently ask each other.\n\nThe demonic holds his cock in an attempt to stem the flow of inky blood that pours forth.\n\nDrek's tongue hangs from his mouth. The injury is fitting for one of that kind. His keen eyes shift to the Were on the ground, intestines littering the sparse gravel driveway.\n\nDeep grooves speak of a large vehicle leaving in haste\u2014recently, by the scent.\n\nThe smell of injury, death, and fear permeate the area. And Tahlia's scent is mingled throughout.\n\nSomeone will answer for this.\n\nDrek lowers his head, growling softly. The demonic and Were raise their heads, spotting him. They do not see Bowen.\n\nDrek leaps, closing the distance.\n\n# CHAPTER SEVEN\n\nJulia\n\nJulia believes that fear about her first time\u2014anxiety over the unknown\u2014would otherwise have been present.\n\nBut with Scott, it's not.\n\nThe soul-meld is a drug. Feelings of ease coat her body in languid calmness, and Julia submits to the ocean of instinct that permeates every cell of her body.\n\nScott lies beside her, trailing his fingers lightly over the slope of her naked side. His fingertips hesitate where her waist narrows to a valley, then he moves on, his fingers barely skating the hill of her hip.\n\nHis fingers spread on the bone, warming her to the core.\n\n\"Do you feel forced?\" Scott asks softly, his normal intensity dialed down.\n\n\"No,\" she replies quietly, looking away. Her hair sweeps forward, partially blocking her view.\n\nScott tucks the wayward strands behind her ear, tilting her face to his with a finger. \"Don't, Julia.\"\n\nHer eyes move back to his. She studies the deep-chocolate irises. Steady, commanding, they are filled with desire.\n\n\"Don't what?\" she whispers, but she knows. She can feel it through their connection, grown taunt with the coming event\u2014the solidification of what they were always meant to be.\n\n\"Think,\" he whispers and softly kisses the tip of her nose.\n\nA sigh slides out of Julia, loosening that last bit of tension. \"I haven't\u2014God, I sound so stupid.\"\n\nScott presses his finger to her lips. \"I know. It's what all the damn fuss has been about. The virginity thing.\"\n\nJulia nods, her cheeks burning. \"I've been really intimate before.\"\n\nScott's dark brows lock together.\n\nJulia smooths them with a finger. \"I'm sure there's been other women for you Scott, so get over yourself.\"\n\nHis smile is tight. \"Now that I have you, there certainly wouldn't have been. I'm not a monk. But for you, I will be everything you need.\"\n\nJulia holds her lip between her teeth as his words reverberate to her toes. So many emotions riot inside her: inadequacy, guilt, and excitement.\n\n\"Really?\" she asks. The fact that she's married to another man, though he's gone, doesn't stop how important their coming together is.\n\nScott nods solemnly. He grabs her hand and unfolds it over his chest.\n\nTheir heartbeats sync.\n\n\"Feel that?\" he asks.\n\nShe doesn't think she'll ever get tired of their bodies' recognition of the other.\n\nHe must see the agreement on her face and sense it at the soul level, because that's how they're bound.\n\n\"This isn't just sex\u2014though that's going to be mind-blowing.\"\n\nJulia snorts softly. \"Sort of arrogant.\"\n\nHe smiles, but then his eyes grow serious, beautiful brown darkening to jet black. \"Not at all. It's about your protection. The Combatant in me can't stop thinking about that. The man loves you, the Singer wants you at his side, and the Combatant wants you to live.\"\n\nPretty tall order. \"What about the demon spore?\" Julia asks.\n\nScott nods. \"What we do next can't hurt. Maybe our union will wipe that fucker out.\" His teeth clench.\n\n\"Maybe,\" Julia replies wistfully.\n\nHe tugs at her hand, his dark gaze searching her eyes, forcing her to believe his words through sheer determination, though she feels the echo of it in their emotional tether. \"Definitely. Believe, Julia. Reach for the stars, and maybe\u2014just maybe\u2014you'll touch them.\"\n\nShe closes her eyes, contemplating hope.\n\nWhen Julia feels the soft press of his lips on hers, she yields to his hold. Scott's a huge man, and he uses every solid hard inch of himself to wrap around her.\n\nJulia feels protected\u2014and cherished.\n\nThe room feels like a womb of silence. The battle she imagines happening above them fades, and memories of Jason's sacrifice lessen. Julia commits to the inevitable\u2014her destiny.\n\n*\n\nScott\n\nScott stares down at Julia. Golden-red hair scatters like a glittering fan across the stiff pillow of their bed. Large whiskey-colored eyes stare up at him while her heart floods a full gaze. Scott sees it all in those depths. Trust. Hope. Desire for him.\n\nShe's finally mine.\n\nScott wraps his arms around her and holds her tightly, mindful of how tiny and fragile she feels within his embrace.\n\nThey've had every awkward conversation possible. Caldwell's death, the unlucky bastard. Her virginity. The possibility of that fucking demonic doing God knows what above. The spore that remains from the wound she received in Region Two.\n\nBut the most important thing is making their bond permanent. Unbreakable.\n\nAnd like he told her, the Combatant part of his dual nature can't relax until that's accomplished. The male inside him has wanted her from the instant their hands accidentally brushed, and he was helplessly swept into the tide of what they were meant to be for each other, suffering through the torture of Caldwell's indecision. The irony of the soul-meld isn't lost on Scott. He was the poorest candidate of anyone in this realm to have had this bestowed on him. He's more than rough around the edges. Julia's compassionate and soft.\n\nShe chases Scott's thoughts with her words.\n\n\"I'm ready.\"\n\nHe believes her. The conviction stands in their shining connection and the trust in her eyes.\n\nScott tucks his palms underneath her, arranging Julia more solidly beneath him, and gently pushes her long hair out from beneath her body. \"There,\" he says.\n\nA deep-pink blush spreads across her face as he surveys every bit of her naked body. She's breathtaking\u2014frail yet strong, gorgeous yet unaware of her beauty. It's an intoxicating mix.\n\nAnd there's no hiding what the sight of Julia does to him\u2014her proximity.\n\n\"You're just looking,\" she comments shyly.\n\nScott gives a slow, considering nod. \"Let me. I'm not going to just pounce on you.\"\n\nThe color on her cheekbones deepens.\n\nHe gives her a quick kiss, feathering his thumb over the heated skin of her face. \"I will pounce if you say the word.\" He arches his eyebrows, feeling his smirk edging his lips.\n\nJulia smacks his bicep, and her breasts jiggle deliciously. Scott hardens painfully and cages her in tightly, their chests mashed together.\n\nJulia's heart thunders against his. \"No pouncing,\" she manages. Then blows him away with, \"But I want you sooner rather than later.\"\n\nHis surprise must show because she laughs.\n\nHe eats the sound with his mouth, tasting her with his tongue. Softly, they explore each other's mouths, and his hold loosens enough to let his hands rove her body.\n\nShe's so small but curved in all the right places. He nuzzles at her neck, burying his nose into the sweet spot between her ear and collarbone. Her soft sigh is better than all the yeses in the world.\n\nScott moves lower. He hovers above her while moving backward as he plants soft kisses on her silky skin. When Scott reaches her breast, he slowly massages the pliant flesh, forcing the soft pink nipple up. The swollen bud hardens as he watches.\n\nHis eyes meet hers. Julia's face is flushed, her eyelids hood, lips parted. Her breathing is heavy and deep.\n\nThis is where he wants his soul-meld: breathless with anticipation.\n\nScott is acutely aware of his painful dick and his need to pleasure Julia as he lowers his head, softly bathing her nipple with his tongue. Taking the small nub into his mouth, he swirls his tongue around the base, and it hardens more. Scott nips and sucks at her, and Julia arches against him, gripping his shoulders.\n\n\"Scott,\" she moans.\n\nHe palms her other breast.\n\nShe pants, and it's music to his ears.\n\nHe rolls his face between her breasts, covering the left mound completely. \"Yes?\" he says in a low voice, continuing to kiss the side of her breast.\n\n\"I like it.\"\n\nHe smiles to himself. \"I know.\" He pushes the mounds together and moves between the hills of her breasts, kissing, sucking, and lightly biting her nipples.\n\n\"Please,\" Julia says in a thready voice.\n\nNot yet. \"Let me take my time. We've waited long enough, my queen. A few more minutes won't matter.\"\n\nHer eyes widen, but the look of surprise fades and is replaced with lust when his hands trail from her breasts and float over her flat stomach to part her thighs. His eyes never leave hers, gaging her reaction.\n\nJulia tenses.\n\nScott stills.\n\nShe bites her lip, but releases it with a pop when his hand covers her mound.\n\nThe tip of his thumb sinks between her outer folds. She's so wet,\n\nScott groans involuntarily, his cock jumping in anticipation. Wait. It's her first time, the circumstances suck, and she's counting on you.\n\nHis lips turn up. No pressure.\n\nHe throbs, and with a rough exhale of self-temperance, he uses her wetness, flicking the slick little hood back and skimming his short nail over her clit.\n\n\"Ah!\" she groans loudly. \"That feels so good, Scott. Don't stop...\"\n\n\"Not planning on it.\"\n\nHer fingers grab his hair when his face is inches away from a place he's wanted to be for over a year. Julia's all-woman\u2014sweet, salty, and perfect.\n\nHis.\n\n\"I don't know\u2014\" she says in an uncertain voice, her fingertips faltering on his head.\n\n\"I do.\" Scott buries his face between her slick folds, his thumb keeping the rhythm on the little bundle of nerves.\n\nJulia throws her head back, sounding as though she's hyperventilating.\n\nScott is intense. He's no different when it comes to sex. And this is his soul-meld.\n\nHer emotions wash over him in a flood. Her arousal is like his own, but different\u2014and varied. Scott senses what she likes and does that, avoiding whatever makes her uncomfortable. He's surprised by how much she likes him to take charge. He grips the globes of her ass, yanking her hips against his seeking lips and tongue.\n\nJulia groans loudly, widening her legs to give him better access.\n\nHe reads the signals of her body like a book, digging deeper, tonguing her entrance as he presses hard at the center of her clit.\n\nJulia gasps, her hands seizing his hair painfully.\n\nScott rolls his gaze to hers. Her eyes are frantic, tits swollen, and legs spread. The beautiful pink folds of her are at his mouth while his forearm holds her hips pinned against his seeking mouth. He gives one deep press of his thumb in her hot, pulsating flesh.\n\nJulia shatters, throwing her head backward. Her walls grab and release his tongue as he keeps the penetration steady. When she finally comes down, Scott is painfully erect. He breathes through the urge to take her hard, as he wants to.\n\nHe felt her barrier. Scott doesn't fool himself; he might have given her a rockin' orgasm, but the first time will hurt.\n\nScott works his way back up her body, gently supporting his weight.\n\nHe fits his cock between her legs, and she pulses between them as he throbs for entry.\n\n\"Scott, that was amazing.\"\n\nHe can hardly nod, he wants her so bad. He needs her.\n\nHer expression changes when she senses the agony of his control.\n\nJulia spreads her legs wider, and his prick is like a homing device, slipping into her entrance as though it was always made to be there.\n\nScott dips his head, shuddering against the slick smooth velvet. She's so tight. Even with lust riding him, he doesn't want to hurt her. He shakes with the sensation.\n\nHe must\u2014to finish what they've started.\n\nHer eyes seek his, and he fights to not look away. \"It's okay. I know it'll hurt.\"\n\n\"Julia,\" Scott presses his forehead against her much smaller one. \"I'm sorry.\"\n\n\"I know.\" Her eyes urge him on.\n\nScott presses forward in a hard stroke, her newness fighting his length and his girth.\n\nJulia shouts in shock. Her expression can't mask the pain. \"Hurts,\" she hisses between pants.\n\nScott nods, slowly removing his length.\n\nJulia stares at him, adjusting her hips. She spread her legs wider, trusting him.\n\nScott gently rocks inside again.\n\nGod she's tight. His mouth parts as his efforts in the last year to keep this from happening through all the upheaval catch up to him. He's frantic to own every inch of her.\n\n\"Yes,\" she whispers, putting her hands on his ass.\n\nScott stills, throbbing and hard. He wants to pump with abandon. But through the meld, he can feel her soreness. Scott also feels her desire.\n\nShe clenches her fingers on his ass and pushes her hips up.\n\nTake me, she sings through their connection, through their blood.\n\nScott sucks a ragged inhale, plunging again.\n\nJulia meets him with a small pant of pain laced with desire.\n\nThey lock gazes. Cradling her face with his hands, Scott begins to rock inside over and over. She meets his every thrust. Gradually, her pain eases, and he feels her excitement.\n\n\"Close,\" Julia breathes against his neck, her pussy makes a single tight fist around him, and Scott's head tips back in frozen ecstasy.\n\nHe releases inside her, pouring his seed, his love, and his soul into her depths.\n\nJulia's cries of pleasure are salve to his battered nerves.\n\nThe soul-meld tightens like a perfect knot, and their eyes meet, bodies joined.\n\n\"What's happening?\" she whispers, locked against him, their bodies in perfect synchronicity.\n\nThen the tingling begins, sweeping over them both like a live wire. Lights flash, and Scott groans as a second release pours out of him and Julia gasps in surprise as her release follows his.\n\nScott feels true fear\u2014his and Julia's\u2014co-mingled in a new way. On the heels of the best sex of his life, Scott blacks out.\n\n# CHAPTER EIGHT\n\nAdi\n\nAdi can't believe she can be in this much pain and still be live.\n\nNope. She's going for it.\n\nThat fucker Slash!\n\nHer tears burn when they meet the healing wounds of her face. Males suck.\n\nHer eyes close for a second. How could those Were beat her and bite her boob?\n\nHow could Slash\u2014how could they be what they were, and he just shove her away like a regret?\n\nAdi thrashes through the woods, uncaring about the limbs and small branches snapping back. She takes the abuse without flinching or protecting herself.\n\nShe could give less than a damn.\n\nAdi wanted Slash and he crushed her spirit. That jerkoff Tramack was all about getting to poor Tessa. He wouldn't have returned to give them another beat down.\n\nAdi slows, finally becoming aware of something other than her bull-in -a-china-closet march through the woods.\n\nA high-pitched keening reaches her ears.\n\nShe looks behind her. Slash is back there, legs not working.\n\nGet the fuck out, he told her.\n\nRight. Adi sucks her bottom lip between her teeth, gnawing the soft flesh thoughtfully. She turns. Maybe I'll just go back.\n\nThe memory of Slash's eyes hating on her halts her in her tracks.\n\nNo. He didn't want me.\n\nThe keening grows louder. Adi strains toward the sound. What on earth is that? She moves cautiously to the edge of the forest and pushes the last of the limbs aside, gathering her ragged clothes against herself.\n\nA red guy is clutching his dick, howling.\n\nWell, that's unexpected. A rash little giggle slips out. She's never heard the sounds he makes before. The high-pitched agony isn't natural. Of course, losing your dick might kick that right along.\n\nAdi doesn't recognize him. But she recognizes the female Were in front of him\u2014the new chick, Tessa.\n\nThe guy does a slow revolution, still clutching his package.\n\nHer keen eyesight takes in something that goes plop on the ground.\n\nOh my, is that his\u2014?\n\nAdi covers her mouth to stifle more inappropriate laughter. And here, I thought the day couldn't be salvaged.\n\nAdi lets the branches fall back. Shit has gone down. She is not getting involved in whatever this new shit storm is. Though the compulsion to keep looking is like fire ants crawling over her skin, Adi runs along the small corridor of woods that almost touch the mansion. She'll get her stuff and make her way back to the Northwestern. Lawrence and Manny are dead. Her brother Joseph is also gone.\n\nShe's a nobody. Nobody's sister. Nobody's pack member. Nobody's mate.\n\nAdi leans against the last tree. It's branches reach toward the edge of the huge Victorian. She angrily brushes tears off her sore face.\n\nI can go back to the Northwestern. Everyone who was important is gone. And the worst male, Tony, is done being him.\n\nNo one to fear. Nothing to lose. Adi turns around in the direction where Slash was.\n\nAnd no one to keep me here.\n\n*\n\nSlash\n\n\"This sucks,\" Slash comments dryly.\n\n\"Yeah. Accept the help. By the time we get back to the mansion, we can get you patched up better. That healer, Cynthia, she can fix ya.\"\n\nSlash has no feeling in his lower back and upper thighs. Each minute crawls by as Truman walks him back toward the Victorian. They have to stop every few minutes because Truman's basically dragging him.\n\n\"Look at that,\" Truman says.\n\nSlash spots Tramack and several Were swimming in their own blood one hundred yards or so from their position.\n\nGood. Slash finds the energy to smile.\n\nHe has trouble concentrating when the scent of his female is thick in his nose.\n\nBut the enemy is in sight, and she is not. The knot of anxiousness in his chest loosens. It looks as though his scheme to assure her safety was successful. Slash's eyes narrow. \"That's a demonic.\"\n\nTruman shakes his head. \"It's something. I didn't know what they were until that battle by Region Two.\"\n\nSlash says, \"I forget you were changed late.\"\n\nTruman nods absently. \"Yup. Really not up to speed on the different species. However\u2014\" His chuckles abruptly. \"There's definitely no shortage. Vamps, fey, shifters galore. Demonic, and can't forget our hosts, Blood Singers.\"\n\nSpeaking of which, the Singers are closing in around Tramack, who lies in a steaming pile of his own guts.\n\nPerfect.\n\nSlash releases a tense breath.\n\nAdi is safe.\n\nThe demonic appears to be searching the ground for something of vital importance. Slash throws back his head and laughs when the demonic finds what he was looking for.\n\n\"Ouch,\" Truman mutters.\n\nSlash's mouth twists. \"Losing your cock would be a grievous problem, my friend.\"\n\nTruman gives him a narrow look. \"Do we have time for comedy?\"\n\nAmusement is in short supply, and Slash's day has been so dark, he can only reply, \"Always.\"\n\n\"Uh-huh. What about your girl?\"\n\n\"Long gone, hopefully in the opposite direction.\" That's Slash's most fervent wish. He can track her easily once he's on the mend.\n\n\"You're not concerned?\" Truman pulls a face of disbelief.\n\n\"More than I can say.\" Slash smacks his upper thigh, not feeling the blow. \"But there's nothing that can be done. I can't protect her in the state I'm in.\" His voice shakes with rage. \"And I can't tear the beating heart out of the male who hurt her.\"\n\nTheir eyes land on Tramack.\n\nSeveral Singers\u2014obviously not of angelic blood\u2014haul the demonic up by the armpits as he makes pathetic sounds of injury. Slash spies a twisted car door covered in blood at his feet. Odd.\n\nSlash sneers. \"We need to get over there and tell them what's happening.\"\n\n\"What's happening is Julia, Scott and Victor are unaccounted for, and that dog will say he was after his rightful mate. The Singers won't get embroiled in that little mess,\" Truman says.\n\n\"What?\" Slash roars, tearing away from Truman and promptly falling on his ass.\n\nAnother set of Singers swoop in and latch on to Tramack as he looks up from the ground.\n\n\"Maybe they'll toss those lovebirds into the same cell?\" Truman muses.\n\n\"Doubt it.\"\n\nSlash turns in Truman's direction. \"Where is the Rare One and her soul-meld?\"\n\nHis eyes find Tramack again as he waits for Truman's reply. Slash would kill him twice if he could walk. To hell with the ramifications of his packmaster status.\n\nTruman shrugs. \"They were here\u2014then gone.\"\n\nSlash frowns. He knows Julia, a little. Even though she's a Singer and not a female of his kind, she does not impress him as a flighty female. His knowledge of their kind is limited.\n\nA scream from the direction of the mansion scatters his thoughts. Slash's nose tells him it's not Adrianna. Nonetheless, adrenaline surges through his system.\n\n\"Come on, stubborn Red.\" Truman hauls him to his feet, and tossing him on his back, he melts to wolfen underneath Slash, racing to the mansion.\n\n*\n\nThey burst through the doors, and Truman lets go of Slash.\n\nSlash slides off his broad back, hitting the door jamb and clutching the deeply profiled wood trim.\n\nHis nostrils flare, hitting on a scent. The only scent.\n\nAdrianna was here.\n\nThe knowledge of her absence affects him deeply, and his wolf howls to be within her presence. His wolf doesn't think; he only feels. And right now, he's kicking Slash's mental ass for the blow-off he gave his mate.\n\nIt's for her own good, he reprimands the beast inside.\n\nStill, Slash feels sick that she's gone\u2014and unprotected.\n\nYou're best isn't good enough, Slash.\n\nCynthia, the Singer and part-Were rushes into the parlor. She slips and falls on her butt with a yelp.\n\nTruman is suddenly there. He grabs her arm. \"What?\" he bellows in her face.\n\nShe's pale, her face pinched, void of any real emotion, except one.\n\nSlash's nostrils lift.\n\nHer fear rolls over him like a wave to shore.\n\n\"Truman,\" Slash cautions in a low voice. \"The female isn't in her right mind.\"\n\nTruman scowls at Slash.\n\nSlash sweeps a palm toward her feet. They're slick with blood.\n\nTruman's eyes bug then go hard and flat. His fists clench.\n\nSlash realizes the expression is his \"thinking cap\" look, the same one he probably wore when he was a cop.\n\nHe picks up a shaking Cynthia and brings her to Slash.\n\nHer green eyes flutter, rolling up into her head. \"Jason,\" she manages, before passing out in Slash's arms.\n\n\"Wait!\" Slash barks at Truman. He whirls.\n\n\"I can't defend her!\"\n\nSharp lines of determination settle into Truman's face. \"And I don't know what's down that hall that scared the bejesus outta her.\" He stabs a thumb at his chest.\n\nMoon.\n\n\"Fine,\" Slash hisses. \"Hurry up. My mate is somewhere, and a crazy Alpha from the Western is making up whatever tale he wants while I can't feel my fucking back.\"\n\nTruman and Slash lock eyes. \"Okay, hang tight.\"\n\nHe doesn't have to tell Slash to protect Cynthia. Slash gets it, however, in his semi-paralyzed state he's a weak choice. But like Truman said, he has to see what the danger is. Slash's role as a male Were has never been confusing to him. That there are males who would abuse their precious females\u2014or any female, for that matter\u2014confounds Slash. It also makes him deeply angry.\n\nAnd rage is always close to him, waiting. Slash is hardwired that way.\n\nCynthia's sleep is unnatural. Her face looks unanimated rather than peaceful in his lap.\n\nThe time rolls out. Though only ten minutes have passed since Truman went to inspect what new horrors lay down the hall, it feels like a thousand years. Each step Adrianna takes from his side is a small knife in his heart. His only consolation is that Tramack is here, and she is putting distance between herself and the Were who harmed her. Slash suddenly smells Cynthia's wakefulness and glances down.\n\n\"I couldn't fix it,\" Cynthia whispers.\n\n\"Fix what?\" Slash asks softly, as though speaking to a child.\n\n\"Jason.\"\n\nSlash stiffens, slapping a palm on the floor and lifting them both upright.\n\nCynthia sits up, notices she's on his lap, and gingerly climbs off.\n\n\"Don't leave,\" he says, remembering Truman's words.\n\nShe wipes a tear from her face. \"No,\" she breathes out in a pained gasp.\n\n\"What's wrong with Jason?\"\n\n\"Something...\" Cynthia covers her face with her hands. \"Something killed him.\"\n\nIf he hadn't been Were, he would not have heard that last.\n\nSlash flares his nostrils. The stench of death reaches him easily. Beneath that, he smells blood and brain matter.\n\nSlash hates to hear about the death of a solid Were. While he didn't really know Jason Caldwell, he did know Jason was volatile, but not mean-spirited.\n\nThis would not bode well\u2014losing Jason so closely with the loss of Zeke. A bad trend had begun.\n\nJason had been threatened on many occasions to leave or force an ultimatum on the Rare One.\n\nSlash understands. He could never share Adrianna. However, Were are a different species from the Singers. Apparently, Caldwell did not feel the same. Maybe he was human for too long. Maybe he was too much Were. In death, it no longer matters.\n\nGently, Slash asks, \"What killed him, Cynthia?\" It's never who in their world.\n\nShaky hands fall to her jean-encased thighs. \"I don't know, but whatever it was, I couldn't heal him. There was no chance. None.\"\n\nSlash isn't a soft, comforting male. But her needs are not complicated. Even an idiot Were like him, one who callously gets rid of his mate, can show mercy. \"Not your fault,\" Slash manages to murmur. \"Couldn't have saved him.\"\n\nShe nods absently, as though she's merely placating him.\n\nTruman jogs into the room, sliding to a stop in front of them. \"You know?\" he asks, looking between them and ascertaining that Cynthia must have conveyed the details of Jason's death.\n\nSlash confirms anyway. \"Yes.\"\n\n\"I don't know what bashed his brains in, but it's nothing I've seen in my short acquaintance with the supernatural world.\"\n\nCynthia flinches at his uncaring rundown of events. \"Who cares, really?\" Cynthia flips a palm over helplessly. \"Something killed Jason. And Julia's not here. Neither is Scott or Victor.\" A bubble of snot grows from her nostril as a tear rolls aimlessly down her face. \"I mean, if Jason's dead, where the hell is Jules? Where the hell am I?\" She dumps her face into her hands.\n\nTruman walks over to her, holding out his hand. She shakes her head, denying whatever a hard Were like Truman can offer. Slash thinks he was a hard human before the change, too. \"Nope, you're sticking with me like glue, girl. Stand up and wipe that face. We're finding out what the hell is going on\u2014like yesterday.\"\n\n\"What about?\" she asks as body-wracking sobs break over her. She waves her hand toward the hall where Jason's body presumably lies.\n\nTruman gathers her against him. Her body so small inside his arms.\n\nAdrianna is even smaller inside mine.\n\n\"We'll take care of him the right way, Cyn.\"\n\nShe nods against his shirt, fisting the material as sadness drenches the fabric.\n\nSlash stares out the window, wondering how soon he will be able to leave this place and right what went so terribly wrong.\n\n# CHAPTER NINE\n\nTessa\n\nTessa flops back against the seat. \"Listen, Laz...\" she rolls her face to the right, looking at him. Beyond that are the deep woods.\n\n\"We can go,\" he says to Tahlia.\n\nShe slaps the steering wheel. \"Where?\" Tahlia rotates to face them, clutching the driver's seat. \"I have been driving for two hours. We are at a crossroads. If we keep going north, we'll be in Canada.\"\n\n\"Nope,\" Tessa says, suppressing a shudder. \"Don't need foreign heat.\"\n\n\"It's hot enough in America,\" Tahlia agrees.\n\n\"You are Lanarre. Where is the nearest pack?\" Laz asks.\n\nTahlia shakes her head vigorously. \"I cannot return to my own pack in the Redwoods of California. It's\u2014the shame would be something I couldn't stand.\"\n\nLaz stares at her and Tessa asks, \"What do you mean, Tahlia?\"\n\nHer eyes fill with tears. Tessa just came to terms with the fact that Tahlia's a baby in Lycan years, and she's been driving like a bat out of hell for two hours while they fought their way down the highway with three quarters of a car.\n\n\"As a Lanarre princess,\" she says, quickly looking to the two of them as though they would laugh at her. They don't. \"I was on my way to meet Drek, my chosen. That entire plan was derailed when Tony Laurent slaughtered my guardians. Now there is another bad force in play, and I must run. It is Drek's pack that I must go to.\" She covers her face with her hands. \"I don't know why I didn't think of this before. Eventually, Drek will return when he cannot locate me. Then we can be mated.\"\n\n\"Wait a sec,\" Tessa begins with slow deliberation. \"I thought you weren't sure about this Drek guy?\"\n\nTahlia's hands shake a little as she lowers them to her lap. \"I'm not, but there is protection within the borders of the Lanarre. It's there that I won't have to fight to remain alive. I am skilled, but am only one.\"\n\nSmall point.\n\nLaz has been silent until then. \"The Lanarre will kill me,\" he states simply.\n\nTessa fingers convulse on the seat cushion. \"And what's their policy on rogue female alphas?\"\n\nTahlia's silence is answer.\n\n\"Shit.\" Tessa slumps in her seat, folding her arms, thinking about the possibility of her heat cropping up. \"Can't catch a break,\" she mutters.\n\nLaz brushes a stray hair over her shoulder that's come undone from the single plait down her back. \"Perhaps the Lanarre princess, betrothed to be mated to their prince, holds sway.\" His pale-blond eyebrow rises, and his eyes level on Tahlia.\n\nShe shakes her head. \"I can't guarantee anything, because they have never scented me. They will scent that I am Lanarre. But the protocol for a betrothal mating is strict. Only Drek and his personal guard know my scent.\"\n\n\"Oh, this is special sauce here.\"\n\nThey look at Tessa.\n\nShe slumps farther, attempting to become one with the seat. \"So we head to Drek's Lanarre stomping grounds\u2014\"\n\nTahlia interrupts, \"At the edge of the Hoh Rain Forest.\"\n\nLaz cringes. \"I am familiar with the area. Very wet.\"\n\nTessa's eyebrow hikes. \"Duh.\"\n\nFaster than Tessa can track, Laz lifts her and seats her on his lap. His nose buries at her throat, and she makes an inarticulate sound. It speaks of deep satisfaction. Damn it all.\n\nLaz tenses beneath her.\n\nTessa opens her eyes to find a talon buried against his insta-erection. Her eyes widen on Tahlia.\n\n\"We can sit here at the side of the road, drawing attention to our deplorable vehicle and its strange occupants, or we can take our chances with the Lanarre of the Hoh Rain Forest.\"\n\nLaz lets out a measured breath, his eyes narrowing at Tahlia, arms still locked around Tessa. \"She is my Redemptive. Tessa will be my freedom from hell, Lanarre. Your talon beside my prick won't change anything. Except to incite my rage.\"\n\n\"Incite away,\" Tessa says, glaring at Laz. \"I'm so glad I'm your one- way ticket to above-ground living. But I don't know if I'm up for the job.\"\n\n\"You must be \u02bbup for the job' and accept my claim as my Redemptive.\"\n\nOr what? Her desire-laden brain attempts to think.\n\nHis thumb presses above her clit, and she groans.\n\nLaz groans, too\u2014but in pain.\n\nI've clearly lost my marbles.\n\n\"Do you want to mate with a horned one, Tessa?\" Tahlia asks with clear disdain, her talon having sliced through Laz's stiff denim jeans.\n\nTessa drowns in the treacherous seas of his eyes. \"I don't know,\" she finally whispers. Too much has gone unexplained and Tessa finds she can't think when Laz touches her.\n\n\"Then moon help you,\" Tahlia replies with disgust, removing her talon. \"I can't save you from yourself.\"\n\nWho can?\n\nTahlia starts the engine again and turns west, to travel the Hood Canal Bridge, straight for the Olympic Peninsula. Beyond that is the den of the Lanarre.\n\n*\n\nTessa wakes with a start, completely disoriented.\n\nThey've stopped.\n\nShe pulls away from the damp drool spot she's made on Laz's lap.\n\nOh Moon, terrible.\n\nShe sits up, and his arms slide from around her body. His gaze finds her, and for the first time, Tessa thinks about bad breath, hair, and body odor.\n\nHe scoops her against him, scenting of her deeply. \"You are part of me.\"\n\nShe says something in her daze of wakefulness, then his hands jerk her hips onto his lap. Again.\n\nTheir position leaves little guesswork to the equation.\n\nLaz wants her.\n\n\"Where's Tahlia?\" Tessa manages. But her hips grind down against his erection. Her shame isn't enough to stop her.\n\nGotta be heat. Apparently, when it comes to the demonic beneath her, nothing matters.\n\n\"We're in Port Angeles,\" Laz says, spreading his palms at her lower back and pressing her deeper against his hardness.\n\nTessa has to pay attention to their surroundings. What if the Were are after her again?\n\nTramack will heal anything, even his own evisceration.\n\nBut right now, Laz is doing his best to make a hole through their jeans. \"Laz,\" Tessa gasps.\n\nHe licks her throat, and she cries out. \"Stop.\"\n\n\"Your body says yes, Tessa.\"\n\nHell yes it does. \"Maybe I'm in heat,\" she says without thinking.\n\n\"So much the better, my Redemptive.\"\n\nBad, Tessa. Very bad.\n\nTahlia jumps into the truck. \"What?\" she yells.\n\nBut Tessa barely hears it through the fog.\n\n\"We're at Walmart.\"\n\nTessa opens her eyes and really looks around while Laz dips between her breasts, licking and kissing. Her head tips back, and she takes an upside-down gander. Wall-to-wall cars are stacked everywhere.\n\nHer head snaps up. \"Laz!\" she says, shaking him.\n\n\"Hmm,\" he mumbles between her tits.\n\nTahlia puts a talon against his neck.\n\nHis face jerks up, his eyes rolling to black.\n\nTessa tries to scramble off his lap, but his grip on her back tightens, and his hand whips out, closing over Tahlia's small wrist.\n\n\"Ow!\" she yells, her talons useless inside the vise of his fingers. Her eyes smolder at him.\n\n\"You will cease and desist. I am not a male Lycan who can have his cock on a leash. I am a high demon, and my Redemptive sits before me. I must sate her heat, Lanarre.\"\n\n\"Oh,\" Tahlia says in a stunned voice, forgetting Laz's grip on her arm, snapping her attention to Tessa. \"You're in heat. How did I not scent this?\"\n\nTessa looks away, embarrassed. \"I think I might be.\"\n\n\"You cannot mate with Laz, Tessa. What are you thinking?\" Tahlia yanks her hand out of Laz's lax grip. \"You will whelp a monster.\"\n\nTessa glares at Tahlia. \"I don't think so. I won't get pregnant by a male who is not Lycan. You know this. It's basic law.\"\n\n\"Do you wish to take the chance if it were not? How certain are you of this?\" Tahlia asks, searching her face.\n\nLaz says yes at the same time Tessa says no.\n\nHorrors.\n\n\"Of course you cannot scent another female's heat,\" Laz remarks.\n\nTahlia retracts her talons, gifting him with a dirty look, and raises a bag with the Walmart logo emblazoned on the front. \"Hungry?\"\n\nStarved. Tessa's not some delicate flower. She's a full-grown Were who has spent two decades keeping a physique that's allowed her to escape from her pursuers. She tries not to dwell on what Laz is really offering her.\n\nHe's demonic. She's a Were.\n\nLaz has Praile on his ass, and he's as possessive as any male Were she's ever encountered. But Tessa might like his brand.\n\nAnd that scares her.\n\nTessa's stomach rumbles, and she puts a palm over her belly. \"What do you have?\" she asks Tahlia, and Laz chuckles.\n\n\"What?\" she half yells at him. \"Do demons not eat?\"\n\nHe strokes a finger from her temple to her jaw.\n\nTessa shivers.\n\n\"We eat, female.\"\n\nHe says it as though peaches are melting in his mouth and he wants to lick off the juices. Hers.\n\n\"Moon, you make me tired, demonic,\" Tahlia comments from the front.\n\n\"Address me as Laz, or I shall address you as princess.\" Laz lifts an eyebrow.\n\nThe silence engulfs the vehicle.\n\nFinally, Tessa moves toward the bag, then rattling noises fill the tense atmosphere. Beef jerky, honey-roasted peanuts, hard-boiled eggs, cheese, and... Hot Tamales?\n\nTessa asks, \"What's with the candy?\"\n\nTahlia fidgets with her hands. \"It is not allowed for Lanarre to poison their bodies with sugar.\"\n\n\"You've got to be kidding. Well, this Were is downing them.\" Tessa's so starved, she thinks the jerky and Hot Tamales will be amazing together. She frowns. Or separate. \"No chocolate?\" she asks Tahlia, still rummaging.\n\nTahlia laughs.\n\nLaz's fist covers his mouth. Their gazes meet. \"Definitely in heat.\"\n\nAll right, chocolate craving is a dead giveaway. Were females normally don't love chocolate, but like human women, a Were female has certain cravings before her cycle. The Ben and Jerry's ice cream is just because.\n\n\"Probably,\" Tessa says without looking up from the bag.\n\nLaz opens the jerky and puts a piece underneath her nose. Tessa opens her mouth. She takes it from his hand without thinking.\n\nTahlia sucks in a breath. \"Tessa!\"\n\nHis fingers are in her mouth as she sucks the juice of the jerky off them. She groans in pleasure, so hungry from the ordeal and length of time without food that she wants to just plow through the supplies. Instead, she lets a near-stranger demonic hand-feed her and slam his erection against her. Slick, Tessa.\n\n\"You trust him?\" Tahlia asks.\n\nTessa meets Laz's icy bluish-gray eyes and nods. \"I do.\"\n\nSomething in Laz's expression softens. \"Let me feed you.\"\n\nAfter the fifth piece of jerky, Tessa is parched.\n\nTahlia, munching away on her own jerky, hands back a large water bottle. Tessa downs half then wipes her mouth on the back of her hoodie sleeve before taking a hunk off a cheese stick.\n\nHot Tamales are poured inside her palm. She tosses them back. Sweet-hot cinnamon bursts over Tessa's tongue, and she moans in ecstasy.\n\nFinally, her hunger has faded, and she notices Laz hasn't eaten. \"What?\" She touches his arm. Warmth spreads underneath her fingertips.\n\nShe doesn't pull away. \"Aren't you hungry?\"\n\nHe nods solemnly. \"Yes.\"\n\nShe passes her bag of jerky to him, and he closes his hand around hers. The bag crinkles. \"Not until my female is sated.\"\n\nMoisture pools at her core from his fervently spoken words. Tessa's brows come together as she swallows the last bite. \"I'm not your female, Laz.\"\n\nHis smile is crooked, steam rising from his mouth. \"Not yet.\"\n\nTessa's appetite is not as robust as it once was. A consumption awaits. Of her.\n\n# CHAPTER TEN\n\nTessa\n\n\"Thank Moon we're in a four-wheel drive.\" Tessa grips the roll bar as if her life depends on it. She's strapped in, courtesy of Laz, as they plunge up then sharply down gulleys that are supposed to be roads.\n\nThat's a full load of bullshit. They're ribbons of dirt sporting a few spots of gravel that wasn't sucked up into the muck of the rain forest.\n\nIt's sunny outside, but no one would ever know it. The moss is deep, hanging like a fine coating of hair as it brushes past their vehicle, clinging to the sides of the car then letting them go as though the very woods couldn't abide the taste.\n\nIf Tessa was the trees, she would think a pair of werewolves and a demon guy tasted like crap, too.\n\n\"Hey!\" Tessa cries as Tahlia takes a small cliff. Laz's arm snakes around her waist, though she's belted tight.\n\nMoon, he's strong. She tells him, pausing at the ego boost it might give him, but he surprises her by stating without an ounce of arrogance, \"The demonic are stronger than all species\u2014even vampire.\"\n\n\"Don't you think you're cool,\" Tessa says, testing him.\n\n\"I am not \u02bbcool.' Torture drives self-regard to unbearably low levels. Only a few of us escape the regime of the Master.\" His voice is so miserable as he recounts his time in hell, Tessa's chest tightens.\n\nThe rig lurches lower. Tahlia sinks them wheel-deep in a river.\n\nTessa's heartbeat begins to thunder as the river rushes underneath the chassis. Cold air plows through the car, swarming the interior with iciness. Tessa shivers.\n\nWarm air envelopes her.\n\nLaz.\n\nHis skin has gone from a vague ruddiness to a distinctive red hue.\n\n\"Are you doing that?\" she asks.\n\nHe nods.\n\nWow.\n\nTahlia downshifts. The wheels grind underneath the swirling current. \"Can't go farther!\" she yells over the gnashing gears and rushing water. The noises of the woods capture their racket, flinging it away from them to be absorbed by the vastness that surrounds them like a living tomb of green.\n\n\"Good thing we didn't trade the vehicle out!\" Tessa shouts over the engine.\n\nThen it stalls.\n\nLaz acts fast, ripping the seatbelt off Tessa and tossing it away. The river swallows the fabric, sweeping it beneath the car as the chrome buckle winks into the dark swirling water.\n\nThe Suburban shifts sideways. Water sloshes inside and swarms across the floorboards.\n\n\"Shit!\" Tessa screams before Laz takes her by one arm and throws her on the Suburban's roof. Tessa's talons punch through the metal, and she winces. Going wolfen due to emotion is always painful. Birthing talons through metal\u2014agony.\n\nShe gasps, and Laz is there.\n\nI can do this.\n\nTessa whips her head back and forth, searching for Tahlia. \"Tahlia!\" she screams.\n\nTahlia's head pops out the window. She struggles. \"The belt's jammed!\" she says, her voice shrill with panic.\n\nTo be a werewolf and drown is a horrible way to die.\n\nTahlia tries to stand, and the Suburban sinks hard to the left. The water rises above Tahlia's neck.\n\n\"No, Tessa,\" Laz says with quiet authority, and she forgets she's on top of a sinking Suburban with a current too fast to navigate by car or body, where Tahlia is about to drown.\n\nLaz is huge. His form has changed. A light scarlet reddens his shimmering skin to the color of soft rubies. Eyes like ice dipped in a vat of aquamarines study the situation, and Laz's blond hair has morphed to a reddish-gold.\n\nSeconds of perusal is all Tessa gets, then Laz grasps her arm. \"Roll when you fall, Tessa,\" he says. And then she's airborne.\n\nTessa spins in an ungainly pile of wolfen muscle and short fur.\n\nStill aching from the speed of the change, she lands on the shore in mud and water.\n\nShe lands hard, without rolling as Laz instructed. Tessa yelps, feeling her ribs bruise. She gasps, trying to look toward the vehicle escaping down the river. Holding her ribs and still opening and closing her mouth without the benefit of oxygen, Tessa stands.\n\nShe stumbles toward the bank.\n\nLaz is nowhere. Her eyes swivel to the roof of the Suburban.\n\nThey're underwater!\n\nHer legs are swept from under her, and Tessa lands backward. What little wind was returning rushes out of her lungs.\n\nHer ribs don't heal. Too many demands.\n\nTessa clutches her neck in the universal sign for choking.\n\nI'm going to die!\n\nThree sets of upside-down silver eyes regard her. Wolfen. All-male.\n\nLanarre.\n\nWith the last of her strength, Tessa points to the sinking rig behind them.\n\nThree faces turn toward where she's pointing.\n\nA body lands beside her.\n\nTahlia.\n\nLow growls roar above her. Tessa's talons sink into the ground, and she heaves herself upright. A sheet of hazy white falls over her vision.\n\nNeed. Oxygen.\n\nThe wolfen back up.\n\nShe throws herself beside Tahlia. The girl's lips are blue, and she's not breathing.\n\nTessa is struck in the back in precisely the spot she needed. Air surges through her lungs, and she coughs.\n\nTessa sits up on her knees, not bothering to see who hit her, and laces her fingers together. Putting the knot of her hands on Tahlia's chest, she begins CPR, rhythmically pumping up and down.\n\nLive.\n\nWater shoots out of Tahlia's mouth, and her beautiful bluish-violet eyes slam open. She grabs Tessa's wrists, vehemently shaking her head. \"Tessa!\"\n\n\"I'm here,\" Tessa says, landing on her ass in the mud.\n\nTahlia sits up, and they pant together. \"You're wolfen,\" she says.\n\nTessa nods. Nothing like stating the obvious.\n\nTahlia's eyes widen.\n\nTessa whirls to see what's behind her.\n\nLaz is on his knees, steam pouring off him. The wolfen have him. The one who stands beside him raises a shining sword toward Laz's neck.\n\nTessa crouches then leaps as though her life depends on it.\n\nTwo wolfen heads swing in her direction, clear surprise etched on their faces.\n\nNot soon enough, fellas. She makes her body a ball mid-flight and takes out sword-wielder neatly.\n\nThey tumble together into a heap, and he raises the sword above her head. She throws her forearms over her face defensively.\n\n\"I do not hurt females,\" he says as though deeply insulted.\n\nTessa drops her arms. \"But you'd kill the guy that saved us!\" she yells in a hoarse shout. And forget the fact that she had to figure out saving Tahlia.\n\nNot a proactive group of Were.\n\nHe turns, and Laz is walking toward them.\n\nNaked.\n\nOh. Oh. Tessa's mouth opens and closes without the excuse of oxygen deprivation.\n\nHis blond hair glows in the ambient light allowed through the thick wet canopy of green that hangs in tendrils that nearly touch the ground. His broad cheekbones frame eyes so black, they look like holes in his face. Steam vaporizes behind him as he walks, eaten by the green of the forest. He comes to stand before them.\n\nHis erection hangs in glorious bare glory between heavily muscled legs. They should be stocky, but because of his height, they're the perfect complement to the rest of his body.\n\nHis tail rises behind him, and Tessa's gaze shifts to the wolfen.\n\nThey're laid out.\n\nShe blinks, spying blood at the hammer-head end of his appendage. Laz told a bold-faced lie when he said he couldn't use that thing in battle.\n\nOf course, Praile's was a flail.\n\nTessa backs away, her wolfen form wavering. She's suffered too much injury, too much\u2014everything. Despite her valiant efforts to remain wolfen, she melts to quarter-change.\n\nJust then, the enemy appears, jumping at them from the creepiness of emerald forest. Tessa couldn't be more spent.\n\nThe wolfen beside her stands, stepping in front of her protectively.\n\nTessa rolls her eyes. He finally gets his act together, and it's Laz, who wouldn't hurt her. That she knows.\n\nAsk her how she knows that, and she couldn't say. But Were live by their instincts. And hers have saved her more times that she can count.\n\nShe clenches her eyes shut, asking the Moon to forgive her, and rolls to her side, punching the wolfen in the crotch.\n\n\"Ah!\" he bellows, folding to the forest floor, where the deep moss catches his body.\n\nTahlia hops over the downed wolfen and rushes to her. She sees Laz, all demonic and naked, and slows, making a wide birth.\n\nHe hisses as if to say, \"Don't screw with me.\"\n\nHis mouth is vaguely gray.\n\nTessa shudders. She kissed him, tongued him\u2014and thought about doing more.\n\nA heck of a lot more.\n\n\"Need to go,\" Laz says.\n\nTessa shakes her head. She saved Laz. She wants Laz.\n\nBut this?\n\nShe looks at his form again. Every hard inch of his muscled body is encased in red skin, his mouth is gray, and his teeth are a pearly white. His black eyes narrow on her as he reaches for her.\n\n\"I'm weak, Laz. Please, just go.\"\n\nHis hand drops, and he shakes his head. \"I cannot just go.\" Anger makes his voice vibrate.\n\nThe guy she whacked in the crotch groans. He'll heal.\n\nMoondammit.\n\n\"Listen, Laz. You're...\" Hot, virile, a sexual typhoon. This is so not the right time for this conversation.\n\nHis brows come together. All that muscled flesh momentarily distracts Tessa. The subtle vapor of heat covers his skin.\n\n\"You're an attractive\u2014\" His dick draws her eyes like a magnet. Tessa swallows, Moon, he's so perfect, it hurts to look at him. \"Male,\" she manages.\n\nTessa can't believe she's breaking up with Laz in the middle of a rainforest with a pile of beaten wolfen at their feet before they even got together.\n\nLaz puts his hands on his hips, drawing her attention to\u2014Yeah.\n\nTessa hangs her head.\n\n\"Hate to interrupt all these deliberations,\" Tahlia says, her voice shaking.\n\nTessa's nostrils flare. \"More are coming.\"\n\nLaz scowls at her, then his hand is at the back of her neck. He tears her from the sodden earth, ignoring the mud that covers her and ignoring his nudity, and smashes his lips to hers.\n\nTessa groans, moving into his arms. She fits so perfectly against him, she's forced to hold back a sob of semi-relief at the contact.\n\nHis cock presses between them as though it's seeking entry. Laz hikes her up, and her legs wrap his waist.\n\nHis length presses against her entrance through her yoga pants.\n\nHe shoves upward, and she cries out\u2014in agony and ecstasy. Clutching his neck, she moves her lips against his. Heat blooms in her core, and like a brush fire, it moves outward, seeking.\n\nIt slams into Laz, and his face tips back, breaking the kiss.\n\nWhen his head lowers, his eyes flash to black.\n\n\"I don't want you,\" Tessa says, having never spoken a lie as large as that one in her lifetime.\n\nLaz runs his hands down her side, and she shudders. As he sets her on her feet, his gaze returns to a pale gray.\n\nWolfen circle the area.\n\nTessa feels as though her heart is in her throat. She can't breathe through her emotions.\n\nLaz's face closes down. His body grows opaque. And like a ghost, he vanishes.\n\nTessa blinks. \"Laz!\" she screams, her heart leaking out of her pores. Then in the next breath, her mind justifies the lie in her rejection of him. I did the right thing.\n\nLaz would have never survived the Lanarre.\n\nTessa looks around at the Lanarre surrounding them, noting they're downed brethren. Tahlia and Tessa wrap their arms around each other.\n\nThey won't hurt females.\n\nWith a flare of their nostrils, they identify the one thing Tessa wishes she could have kept to herself.\n\n\"Heat,\" one says.\n\nThe others smile.\n\nUnclaimed is what Tessa hears. Tahlia turns nervous eyes to Tessa.\n\n\"Tell them,\" Tessa says.\n\nTahlia turns to face the wolfen. The number of males in the party makes Tessa uncomfortable on principle, to put it mildly.\n\nLaz's absence makes it worse. Tessa's been alone for a long time. She doesn't need Laz. And he was demonic. She hangs her head.\n\nShe can admit to herself he was more than that.\n\n\"I am Tahlia, Princess of the Lanarre pack from the Redwood.\"\n\nConfused, they regard one another then their attention razors on her like sharp slits. The one with the abused crotch stands, slowly shaking his head, giving Tessa a hard glare.\n\nShit.\n\n\"The chosen has arrived ahead of Drek. We do not know who you are\u2014but your impersonation of the chosen will not be met with tolerance.\"\n\nTahlia's mouth drops open.\n\nWhat? Tessa turns her stunned expression to Tahlia.\n\nHis eyes move to Tessa's. \"And your cavorting with a horned one?\" His smile lacks warmth.\n\nTessa's cheeks heat. Hell, she'd punched him in the dick and had a demonic in tow. She wasn't winning any popularity contests.\n\nLike ever. \"He's gone now,\" Tessa whispers, hoping Laz will stay gone, Redemptive or not.\n\nThe Lanarre's grin is toothy, though the smile never reaches his eyes.\n\nTessa scents Tahlia's confusion and, underneath that, her fear.\n\nAmen.\n\n\"But that doesn't matter. The demonic was here, escorting two Were females\"\u2014his gaze flicks to Tahlia\u2014\"one a Lanarre\u2014on her own.\" His eyebrows rise significantly.\n\n\"So what? Females can't travel together?\"\n\nHe shakes his head. \"A Lanarre female of any repute would have male escorts or if she were royalty\u2014her family guard.\"\n\nHis chin lifts. \"You are not she.\" His slick black eyebrows drop over molten-silver eyes. \"But we will get to the bottom of who you really are.\"\n\n\"I am Tahlia,\" she seethes. \"You ridiculous Lanarre, can you not scent my purity?\"\n\nUh-oh.\n\nHe rushes her.\n\nBravely, Tahlia plants her feet wide. She has obviously never felt the hand of a male against her.\n\nHis nose goes to her crotch and Tessa blanches.\n\nTahlia chops her hand on the back of his neck in a hard stroke of such instinctive quickness, he rolls to the side.\n\nHe kicks her in the stomach, and she yelps, hands to belly as she falls slowly to the forest floor.\n\nTessa steps forward and chin-checks him against the side of the tree.\n\nThe Lanarre close in.\n\nIt takes six males to subdue them. In the end, they do more than that. They steal Tessa's hope.\n\nThe Lanarre of the Hoh region isn't a sanctuary.\n\nIt's a prison.\n\n# CHAPTER ELEVEN\n\nDrek\n\nDrek lands steps from the healing Were on the ground.\n\nHe barks. Explain.\n\nDrek watches understanding flow over the Alpha's features. The Alpha knows Drek is Lanarre royalty after one inhaled chuff.\n\n\"Forgive me, prince,\" the Alpha grovels.\n\nA ripple of power, stolen from the moon, tears through Drek. He pushes it over the Alpha, blanketing him in Drek's will.\n\nThe Alpha flattens on the dirt. Only a thin layer of skin covers the pulsating wounds of his stomach. Drek's lip lifts, and a low growl hums through the space between house, driveway, and barn.\n\nThe horned one whirls, looking as though he'll come for Drek.\n\nBowen moves through the assembled but scattered Singer population, and the horned one steams from the holes of his evil body, but he does not draw nearer. Clever creature.\n\nHe cannot take two full werewolves. He's decided to cut his losses.\n\nDrek allows Bowen the lead so that he might change into wolfen. He scents that Tahlia's fragrance is no longer fresh. Staying in werewolf form offers him no benefit. Drek rises on his hind legs, melting the things of wolf to the part-form of wolfen. He tamps down on the relief of shifting in reverse, which is always easier and less painful than becoming his beast. He ignores the murmurs from the Singers and levels his attention on the horned one.\n\n\"I have no quarrel with you, Lanarre,\" the evil one hisses. He swings a mallet-like tail above his head.\n\nDrek takes in the spiked appendages at its end. He is a fearsome creature, but Drek is unafraid. He and Bowen will prevail.\n\nDrek has not come face to face with a demonic in many years. That he has today, with Tahlia in such close proximity, causes a profound unease to creep over his skin. In the presence of evil only a few yards away, the fine silver hairs of his wolfen stand on end.\n\n\"We have no quarrel yet, horned one.\" Drek feels the potential simmering between them.\n\nThe red horror of the demonic grins, his black teeth causing his mouth to appear as a yawning hole inside his face. Low vapor rises, sucked by the light breeze of night succumbing to dawn. He throws his head back, laughing.\n\nDrek does not see the humor.\n\n\"I have business here that has nothing to do with our cousin, the Lanarre.\"\n\nDrek jerks his jaw back, voicing his displeasure in growling words, \"We are not relations, demonic.\"\n\n\"Au contraire,\" he wags a finger. \"You are very much under the call of the Master, if you possess even a bit of Red in those blueblood veins of yours.\"\n\nLow murmurs from the assembled Singers become louder, but Drek ignores them. The demonic before him poses a greater threat than they do.\n\nDrek wonders suddenly where the leadership might be for the region, but he takes charge of the conversation at hand. \"Then rest easy, demonic\u2014for I do not have the necessary blood to cause me to be a malleable specimen for the devil.\"\n\nBowen comes to stand beside him.\n\nHis presence strengthens Drek. Where three of their kind come together, as with all magics, they become more powerful than if they stood alone. Still, two will do.\n\nThey warily and loosely circle each other. \"Then why are you here\u2014for I scent my chosen.\"\n\nThe Were at his feet blanches.\n\nDrek refocuses a sharp look at the Alpha. \"What do you know, Alpha of the Western?\" At least, he smells like the west.\n\nThe Alpha's nostrils flare. \"I have not touched your chosen,\" he nearly wails.\n\nThat, Drek believes. Tahlia would have this male for breakfast, lunch, and dinner. She's classically trained, like all Lanarre royalty. In theory, she should be an impressive female in all areas.\n\n\"Speak,\" Drek commands in a low voice.\n\nThe demonic inches closer.\n\nBowen growls, his lips peeling back to showcase the razor-sharp teeth of his wolfen form.\n\nThe demonic halts. Steam pours from his mouth and nostrils; additional vapor covers his revealed skin like a layer of smoke.\n\nThe Singers' voices rise. Drek gives them a cursory sweep with his eyes. They appear human, but that can be deceiving. His understanding is complete. They have talents that rival the strength, nose, and speed of Lycans. But they are not a species at direct odds with the Lanarre. He returns his attention to the cowardly packmaster of the west, who is attempting to stand while holding his writhing guts inside his body.\n\n\"I have been seeking my intended for twenty years! She has led me here,\" the Alpha mewls.\n\nDrek reins in his temper to extract additional information just before a loud female voice interrupts. \"She didn't want to go with this guy!\"\n\nAll heads turn to a taller female Singer.\n\nDrek's nostrils flare. She is Were, as well, though changed, not born. Interesting.\n\n\"This numbnuts came charging in here after she'd only been here a day, and she took off like the devil was chasing her.\"\n\nDrek blinks, deliberating on the pun within her words. Her modern way of talking and lack of elegance is shocking. However, she is changed. That can make all the difference. Tall but slight, she has longish blond hair and piercing emerald eyes.\n\n\"And this fire prick?\" she goes on, pointing at the demonic, \"He pretended to be a Singer! But really? He tried to hurt Jules\u2014I know it. And we're really fucking sure he killed Jason.\"\n\nThe Singers break out in screams and shouts, surrounding the demonic.\n\nDrek will not be a referee in supernatural matters between species who are meaningless to the Lanarre. He raises a palm. \"Thank you, female.\"\n\nShe purses her lips, nonplussed by his dismissal.\n\nDrek feels a shrug coming on. You cannot win them all, as the humans say. He scans the faces of the Blood Singers of Region One. His scenting tells him many things.\n\nDeath clings to Region One.\n\nA battle or massacre of epic proportions took place here in the recent past. If the demonic have been loosened in this realm, whatever is afoot will affect them all. And why an Alpha would seek his legitimate intended for two decades reeks of foul play and the breakdown of the Western.\n\nBut none of these factors are enough for Drek to concern himself with. \"I am looking for my chosen. She is a Lanarre princess.\"\n\nBlank looks answer him.\n\nHer scent is here. Someone has interacted with her; Drek is sure of it. He ignores the demonic, who seems to be searching for a handy escape route. Drek gives equal inattention to the Alpha at his feet.\n\nDrek is keenly aware of Tahlia's appearance. Photographs have been exchanged. \"She stands this high.\" He holds the edge of his hand just beneath his shoulder. \"Black hair that is curled to her waist, with eyes the color of twilight meeting night.\"\n\nSilence.\n\nThen the part-Were female says, \"Listen, pal\u2014she took off with this jerk's intended.\" She makes funny little curls with her fingertips as though plucking the word out of the sky. \"So she's gone. And this demon guy's side-kick? She took off with the Alpha's intended\"\u2014she says the word with clear distaste\u2014\"so since you don't want to join the party in helping keep these guys in line, they went thataway.\" She points due north.\n\nDrek smiles. He supposes she's helpful\u2014in her way. \"Lanarre do not engage in altercations with other species. Tahlia will be in need of our protection.\"\n\nDrek sinks to his haunches beside the Alpha male from the Western, who cringes away. \"If you follow my chosen, for any reason, supposed intended or not, I will tear the guts that have just healed out of your body and hang you with them.\" Drek's voice remains deadly with intent, never changing in modulation.\n\nHe stands.\n\nThe female Were crosses her arms, glaring at him with disdain. Bowen and Drek exchange a look.\n\nThe demonic's face is hard, cunning and determined. \"My subordinate is with the group that accompany your chosen.\"\n\nFrowning, Drek says, \"That is not a consideration of the Lanarre.\"\n\nThe demonic smiles. \"It is to me!\" he says with a hiss. In a flash, he's blurred like a red smear to the tree line and beyond. The collective gasp of the crowd is a hushed bomb of surprise.\n\nDrek's frown turns to a scowl. He wants nothing to do with the demonic, but he will do whatever is necessary to protect Tahlia.\n\nIf he must dance with the devil, then he shall.\n\nThe crowd parts as Drek and Bowen step over the fallen Alpha.\n\n*\n\nBowen scoops the gravel from the shoulder as his nose hovers over slightly damp gravel. He sifts it between his fingers.\n\n\"She was here.\"\n\nDrek is impressed. He does not believe that Tahlia ever got out of the vehicle that was used in the quick exit they made from Region One. Bowen would have to smell her, layered underneath fossil fuel, manmade asphalt of indeterminate origin, forest, vegetation, and the indigenous wild animal population.\n\nHis head turns sharply in Bowen's direction. \"Do you think Tahlia might be heading toward the den?\" The thought process makes sense. She's probably frightened and unsure. Seeking Drek's pack is solid thinking.\n\nBowen considers, tossing the gravel away from them. \"Not sure.\" His wolfen snout points in a generally northwestern direction. His spinning silver eyes find Drek's. \"If she does, that benefits us.\"\n\n\"But not her companions.\"\n\nBowen gives him a look of disbelief and a snort so finely executed, it sounds almost exactly as it would if he stood before Drek in human form. \"Does that matter, really?\"\n\nNo. Yet, they somehow had a hand in Tahlia's rescue. Or Tahlia somehow helped them. Without the details, Drek is not happy dismissing their lives so quickly. And one is a female Were. He scented her. Unfortunately, the demonic, like the vampire, are scentless.\n\n\"I suppose no,\" he finally answers, \"but I believe this female rogue was the one who Tahlia assisted back at the highway. And as I put the pieces together, I further postulate that the Alpha who was so neatly gutted at Region One is part of her capture. And Tahlia interrupted it.\"\n\n\"That Alpha is bad news, Drek. I don't want that following us to Lanarre country.\"\n\nDrek sighs, knowing he should have finished gutting that one. However, it would have been cowardly to kill a defenseless Were without clear reason. And in front of witnesses, when no transgression was made against the Lanarre? No. A bad move.\n\n\"We own the Hoh. It is ours.\" He misses thumping his chest by a hairsbreadth. \"It is the Lanarre who has kept the western half of the United States free of problems among Lycan. The alliance between the southwestern Lanarre region through my mating with Thalia would have solidified that further.\"\n\n\"Come on, Drek. You know that's not true. There's been unrest. And I don't believe you want the ancient status quo any more than I do.\"\n\nHe gives Bowen a hard glance but keeps his misgivings to himself. Bowen is right. Small packs keep popping up. They don't feel the need to formally align with the Lanarre, preferring an outlaw lifestyle to the strength of unity. It's troubling. But that further solidifies Drek's ideas about progressing Lycan culture into a more modern direction.\n\n\"Listen\"\u2014Bowen claps Drek on the back\u2014\"you can't take all this bullshit political evolution on as your singular mission to save everyone. You just have to make the Lanarre pack the very best of us. We worry about the rest later, yes?\"\n\nBowen is wise.\n\nDrek is fraught with obligation, responsibility, and thoughts better left unsaid and not dwelled upon.\n\nHowever, Drek does dwell. He dreams of a better life, more communicative between packs, agreement on inter-pack matings, and a cessation of rites that leave females in precarious positions of being fought over. That is not a healthy environment for perpetuating the breed.\n\nAt least that will not be Tahlia's end. No Were would want the cast off of a prince. She can live out her life in peace, without being forced to wed Drek\u2014if, and only if, he is able to effect change.\n\nWithout change, the muck of tradition will weigh them down like boulders in quick sand.\n\n\"Lead on,\" Drek says.\n\nThey run.\n\n*\n\nDrek slows, his lungs on slow-burning fire. The Hoh receives more than a hundred forty inches of rain per year, and the forest is slick with trailing moss and undergrowth.\n\nWolfen flesh has a coating similar to a duck's; the rain wets the tips, and the hair sheds the majority of the wetness. Still, the rain dampens the pair, making the travel wet and chilly, even in their partially changed forms.\n\n\"You're rugged for a prince,\" Bowen huffs as he speeds through the woods.\n\nDrek lets the next branch swing back. He hears it whip Bowen, who curses.\n\n\"Kidding!\" Bowen shouts from behind him.\n\nDrek smirks. Bowen is always poking fun.\n\nLoud voices in conflict reach his ears.\n\n\"Wait!\" Drek says, wrapping a long arm around a trunk to assist his slowing.\n\nTahlia's sweet scent fills his nostrils, and Drek inhales deeply, relief flooding him. Nothing compares to a female of royalty, and Tahlia's safety.\n\nHe scowls, his nose wrinkling at the second scent: one who is in heat. The odor reaches him easily\u2014all male Were would scent the same.\n\nBowen reaches him, eyes as wide as his nostrils. \"Is that what I think it is?\"\n\n\"Yes,\" Drek says.\n\n\"Tahlia has somehow come to the pack, and another female is in her heat cycle? I'm wracking my brains, but I am certain none of our females were cycling.\"\n\nDrek nods. Bowen is correct. He shakes his head. A female in heat? In the middle of a pack of Were? Even the Lanarre will be hard-pressed to restrain themselves. Royal lineage does not negate the primal needs of werewolves. The three Fs are in full play: feeding, fighting, and fucking.\n\nBowen races ahead.\n\nDrek follows closely\u2014he cannot have Tahlia in the middle of the fray. He's relieved she's safely in Lanarre territory. It was his hope that her scent would lead here.\n\nWhen he arrives in the heart of his den, his relief is short-lived.\n\nTahlia is screaming, held down by three Lanarre males.\n\nMeanwhile, a demonic guts whoever comes within striking distance of the female he and Bowen had scented as fertile.\n\nHis voice booms over the space, the trees shaking from the resonance of the growling timbre, \"Stop!\"\n\n# CHAPTER TWELVE\n\nScott\n\n\"Scott!\" Julia says loudly.\n\nHe blinks slowly awake. Julia's perfect face comes into focus.\n\nBut it's her bare breasts that really get his attention.\n\nHer hands on his shoulders, she sighs, sitting back on her heels.\n\nScott sits up, eyes pinned to her nude form.\n\n\"Perv,\" she says in a sullen voice.\n\nHe grins, grabbing her and setting her on his lap. \"What happened?\"\n\nJulia giggles softly. \"I think we blew a fuse or something.\"\n\nScott nods, eyebrows hiked. \"Nice.\"\n\nChampagne hair sweeps forward over her wrinkled brow. \"I don't think passing out every time we have sex is all that great.\"\n\nScott's brows meet. \"Oh, yeah. Why not?\"\n\nHe leans in, rubbing her nose with his then kissing the tip softly. \"I'll pass out with you anytime.\"\n\n\"Perv,\" she whispers again, but more softly this time. Her voice is edged with desire and a hint of happiness.\n\n\"You betcha, all the way.\" Scott covers her breast with his hand, and her sharp intake of air is felt in both ways, through flesh and a warm breath of air through his soul.\n\nScott shudders. \"I don't know how I'm going to get used to that.\"\n\n\"We will,\" Julia says, snuggling deeper into his body.\n\nThey hold each other quietly for a few minutes, saying nothing as Scott explores every inch of her skin with his palms. \"I can't believe you're mine.\"\n\nJulia nods.\n\nScott feels her sadness through their link and leans back to see her face. A sheen of tears wets Julia's amber eyes. \"Hey,\" he says as the first one slips out. \"No crying. I forbid it.\"\n\nShe laughs, nodding. \"How can I be this happy when all this shit is raining down all around us?\"\n\n\"Ah,\" Scott says, understanding sinking in like unwanted teeth.\n\nJulia disentangles from his embrace and grabs his shirt off the floor, throwing it on over her head. Julia heads to the bathroom, and Scott gives her the space. Water runs, then a toilet flushes.\n\nScott stands and walks naked to the thermostat on the wall. He looks at the temperature setting. Too cold. He turns the dial up to seventy-two. Balmy. He frowns. His balls will melt off, but Julia will be toasty. Sacrifices must be made. His lips quirk as he leaves the setting warmer.\n\nJulia leaves the bathroom, skin flushed.\n\nHe walks by her, trailing his fingers over her neck and down her spine as he moves to the bathroom and uses it himself.\n\nWater runs down the drain as Scott holds his toothbrush loosely, inspecting his face in the mirror. He doesn't know what he was expecting to see. Their soul-meld is complete, but he really doesn't feel much different. He loves Julia. He still wants to protect her. Nothing's changed.\n\nEverything has.\n\nHe walks out of the bathroom, catching sight of the pensive way she's standing. \"We're not going to discuss what we can't help.\" As Scott walks over to Julia, her eyes dip to his hips and a deep-pink color invades her cheeks.\n\nOh yeah, got my dick hanging in the wind.\n\nScott grins.\n\nJulia blushes harder.\n\nThis is too fun. Scott's gaze never leaves hers as he laces his fingers, placing his hands on the back of his head. Swiveling his hips, Scott whips his semi-hard-on back and forth. His cock slaps his thighs.\n\nScott waggles his eyebrows.\n\nJulia busts out laughing, covering her mouth. \"Really?\"\n\nScott nods. \"Oh yeah\u2014really!\" he races toward her, and she squeals, turning to run.\n\nScott seizes her around the waist, heaving her on the bed. She bounces once, and he's on her in a flash.\n\nHe runs his fingers underneath his T-shirt, searching for all the obvious tender spots to torture, and soon, she's breathless with laughter.\n\nWhen his palms cover her breasts, Julia's laughter drains from her face, and serious luminous eyes regard him.\n\nScott grows hard for her again as fierce possessiveness courses through him.\n\nJulia's eyes widen. \"That's how you feel about me?\"\n\nHe nods.\n\n\"It's almost too much.\" Her eyes hold fear\u2014and wonder. Her hand rises to his face, and light fingertips trail over his day-old stubble.\n\n\"Nah,\" he says, fingering the strands of hair caught in her mouth from laughing so hard. \"It's never too much.\"\n\n\"You want to own me, every bit\u2014my mind, my body.\"\n\n\"Your soul,\" Scott whispers with a seriousness he didn't know he had in him.\n\nJulia nods, searching his face. \"That especially.\"\n\nScott presses his mouth to her lips. Breaking the kiss, he cocks his head, giving her hooded eyes. \"Let's start on all that right now then. No better time than the present.\"\n\n\"I haven't even brushed my teeth!\" Julia protests.\n\nHis eyes narrow. \"It's not your mouth that needs kissing.\"\n\n\"Oh,\" Julia says in a breathy voice.\n\nScott works his way down to where he just was a handful of hours before.\n\nTime unwinds.\n\n*\n\n\"Victor is going to know we were in here humping like bunnies.\" Julia covers her face, toothbrush sticking out of her kissable clean mouth.\n\n\"Victor-smictor.\" Scott shrugs, slapping her cute little round ass.\n\n\"Hey!\" Julia says through her toothpaste, but she's smiling.\n\nHe loves to see her smile. Scott feels her every emotion down to his feet. And when they're good, they're great.\n\nShe spits and rinses, the foamy toothpaste disappears, and she rolls her lips together, sighing in relief. \"I hate a sweater on my teeth.\"\n\nScott laughs. \"Great visual, sweetheart.\" He gives her a slow wink.\n\nHer lips twitch as she arches a golden eyebrow. \"You're so immature. Rhyming Victor's name.\"\n\nWhat can I say? \"Mhmm.\"\n\n\"Oh my word, what have I gotten myself into?\"\n\nScott turns, grabbing Julia and hiking her up by the ass. Her legs go around his waist. He groans as he gets his third boner in twenty-four hours. My dick's gonna fall off. \"You're killing me, Julia.\"\n\n\"Ah, no. I'm the one who's sore, big guy.\"\n\nScott feels his sheepish grin. He squeezes her, whispering against her neck, \"I didn't hurt you too much?\" His heart rate picks up. The duality of his soul-meld completion is somewhat contrary. He had to hurt his soul-meld to make her his\u2014a small hurt for better protection and so much more. Though he can justify what he did, there's no denying his guilt over the process.\n\n\"Hey,\" she says softly, sliding down the front of him.\n\nNothing like a few thoughts of hurting Julia to make him a limp noodle. Scott looks away from her burning gaze.\n\nShe swings his jaw back to her. He swims in the liquid fire of her eyes.\n\nTheir breathing instantly comes together. \"I wanted to have sex. I don't think you have any idea how much.\"\n\nScott remains silent.\n\nShe touches his cock, and it springs to vivid life again. Julia wraps her fingers around him, and his erection strains toward her. \"I wanted this, Scott.\" She strangles him for a second, and he stops breathing. \"All of you.\"\n\n\"Okay,\" he squeaks. \"Got it.\" Now what are we going to do about it?\n\nShe looks at him as though she heard his question aloud. Probably.\n\n\"I thought you just mentioned Victor...\"\n\nThen her small hands tear down his underwear, springing his dick free.\n\nShe sinks before him. Her tongue is wet velvet against the tip of him, and he shudders. \"Julia\u2014\"\n\n\"Shhh.\" She licks at him, her tongue trailing the rim of his head, burning a path of fire.\n\nScott's head tips back as his hand finds her hair, fisting the gorgeous silky strands. He pushes her mouth deeper as she takes more of him.\n\nJulia eases down to his base. She frees her lips and slides to his tip then deep throats him again.\n\nScott starts to work inside her mouth with his hips. \"Julia, I can't\u2014I can't last,\" he gasps. He feels her lips part in a smile around his length as his other palm hits the wall behind him.\n\nShe comes up for air. \"Maybe I don't want you to.\"\n\nThat does it. His hips pump.\n\nHer hand joins her mouth, one following the other as she crests at his tip with her hot wetness and slams down to his base, her hand squeezing as she does.\n\n\"Ah!\" he bellows into the air while hot jets of cum spurt inside her waiting throat.\n\nShe swallows him down, and Scott shudders as his release wanes. He slumps against the wall.\n\nJulia undoes him. Flat out.\n\nShe places her hands on his ass and releases his dick with a final lick at the tip.\n\nShe lays the side of her face against his naked hip. \"I love you, Scott,\" she whispers, her breath warming where her mouth just was.\n\nWhat Scott feels for Julia goes way beyond mere love.\n\nHe sinks to the floor and kicks off the underwear from his ankles. Gathering Julia to his chest, he kisses the top of her head. \"I love you more.\"\n\nShe tips her head up, and he presses a kiss to her forehead. Julia closes her eyes. He kisses each eyelid.\n\nHer eyes slowly open, and her lips part.\n\nHe kisses her mouth. It tastes of him. Scott stands with her in his arms then carries Julia to their bed.\n\n\"Again?\" she asks coyly.\n\nScott nods solemnly. \"Yes.\"\n\n*\n\n\"Nobody passed out,\" Scott says, turning his head to look at her.\n\nJulia peeks up at him from the crook of his arm. \"That's a plus.\"\n\nHe pushes the hair from her face. \"Oh, I don't know. It was kinda novel.\"\n\nJulia's laugh is low and satisfied. \"Yeah.\"\n\nScott's stomach growls. \"All right. I guess that's it then.\"\n\nHer eyebrow arches. \"It's what?\"\n\n\"The beast has awoken!\"\n\nJulia sits up, tossing her mussed hair behind her shoulder. \"I think you were a beast,\" she says, fluttering her eyelashes.\n\nIs she flirting with me? He laughs.\n\nJulia's got that look. It's the look that will land her in bed again. Scott's stomach rumbles hard, followed by a gnawing cramp.\n\n\"Hell,\" Scott says, giving her a considering look.\n\n\"We can't just have sex forever, ya know.\"\n\n\"Says who?\" he asks with a chuckle.\n\nJulia blushes.\n\nGod, I love her.\n\nCaldwell was a fool. A rough exhale shoots out of him before he can stop it. The jerk saved her, and Scott will always be grateful for that. But in the darkest part of his mind, the part he doesn't like to admit he owns, he's glad that Jason isn't around to make Julia miserable with all his mood swings. Fuck, he was a bitch with that emo shit.\n\n\"What?\" she asks, studying his expression.\n\n\"Can't read my mind?\"\n\nShe shakes her head. \"I'm telepathic, but some species\u2014especially Were\u2014and some individuals\"\u2014she taps her head\u2014\"white noise.\" She shrugs. \"I can send you a direct phrase if I do it just right, but it wipes me.\"\n\n\"We don't want that.\" Scott gathers her close again.\n\nShe gives a soft shake of her head. \"No.\" Her head falls on his shoulder, and he strokes her back.\n\nAnother hunger pang shoots through him.\n\nScott stands, tugging her by the hand. \"Let's grab a quick shower and see what kind of dehydrated bullshit we can rustle up.\"\n\nJulia giggles. \"It's better than a sharp stick in the eye.\"\n\n\"Well, yeah,\" Scott says, carrying Julia to the bathroom.\n\nHe turns the water on and slides Julia carefully to her feet with one arm. Hot spray hits the shower wall. Steam rises.\n\nJulia strips the little bit of clothing she had on. Naked, she steps inside. Her eyes grow round when Scott follows her.\n\n*\n\nJulia\n\nVictor turns toward the sound of their footfalls.\n\nJulia tries not to blush.\n\nFail. Julia realizes she can't do the things she's done with Scott and not have it pinging around in her brain. It's all so new. So amazing.\n\nHe stands in all his tall, stately glory, though Julia admits his normally dapper suit is worse for the wear.\n\n\"Hi,\" she says, hating the breathless tone in her voice. She can't shake the embarrassment and tries to hide it.\n\nScott assured her that all the rooms were soundproof. Nevertheless, Julia is damn sure Victor knows what they were up to in that bedroom for nearly an entire day.\n\n\"My queen,\" Victor says, bowing slightly.\n\n\"Gah!\"\n\nVictor's head jerks back at her outburst.\n\n\"Sorry,\" she mutters, and Scott squeezes her hand. \"I know that technically, I'm a royal Singer\u2014\"\n\n\"You are the Rare One.\"\n\nJulia blows a strand of hair back. \"Yeah. Anyway, I just want you to call me Julia, okay?\"\n\nVictor stares.\n\n\"Listen, Vic\u2014\"\n\nVictor's stormy expression shifts to Scott.\n\n\"We know our place, and it is working beside our people, not over them.\" Old Vic is from Two, after all. And Scott's mom was the mistress over that little mess.\n\nVictor's chin dips. Restless fingers rake his perfect blond hair. His pale-gray eyes come to rest on them. \"Forgive me. I have lived so long under your biological mother's reign, a different one has me completely out of sorts.\"\n\nJulia nods. His feeling are understandable.\n\nHer eyes move to the timer clock. How many hours she has left in her prison underground?\n\nHer flush comes back to merciless life. She's been down here screwing Scott's brains out while her people are\u2014\n\nScott turns her, gripping her shoulders. \"No.\"\n\nJulia doesn't need further explanation. Stop thinking about things we can't change, he said. Self-blame won't make the horrors, imagined or real, resolve themselves. Scott's words haunt her\u2014because they're right.\n\nShe turns her head to the clock again.\n\nTen hours, eleven minutes, and three seconds. It seems like eternity. Her eyes lock with Scott's, and her heart squeezes.\n\nIt seems too soon.\n\n# CHAPTER THIRTEEN\n\nAdi\n\nAdi shifts the weight of the backpack between her shoulders. It's a miracle she was able to even put anything together with her state of mind.\n\nThe pack must be Jen's. The thing's covered in sparkly unicorns. But it was all she could find.\n\nAdi sighs. She can't even escape without looking ridiculous.\n\nNot that she gives a shit. She's got the necessities inside the pack: beef jerky, apples, water, and some peanut M&Ms.\n\nAdi loves those.\n\nA wave of heat threads through her, igniting her core. She pauses, her hand instinctively slapping the nearest tree trunk. The sharp bark presses painfully against her tender palm. Adi's almost healed fully, but her hands still ache from all the punching and hitting of those cockbite Were she ran into.\n\nI hope that one can't grow another set of balls.\n\nToo bad she doesn't have silver talons. A grim smile covers her face as another wave of nauseating heat hits her. Adi doubles over. What the hell is wrong with me? She shuts her eyes, leaning against the tree.\n\nTaking even breaths, she wills herself to relax. Did she get injured more than she knew? Fear grips her as she thinks about possible internal injuries. She has no way to fix that between here and the Northwestern down in Gig Harbor. Though it's only a couple of hours south by car, she's not in any shape to move fast yet. She might not have suffered as much as Slash did, but she's not perfectly healed. Not by a long shot. When she opens her eyes, she sees nothing but deep green woods\u2014and Highway 101.\n\nFinally.\n\nShe swings the backpack around and gives the zipper a vicious reverse tug. After jerking the water bottle out of the inside, she spins the cap off and chugs half. Better. She tightens the cap on the half-drained water bottle. Adi surveys the territory, thinking through her options.\n\nThey're all kinds of miserable. But she can't dwell on the beauty of what happened between her and Slash. She can only hope to start again. She figures her small goal of returning to the Northwestern is solid. She'll get back there, regroup, eat some of Susan's delicious home cooking, and figure out her life. She strokes the unicorns on the backpack and thinks of the pajamas she lent Julia way back when.\n\nShe feels a sad smile slip onto her face then melt away just as quickly.\n\nShe gnaws a piece of beef jerky, and gradually, her stomach settles, filling with protein and much-needed water. With the highway in sight, she feels less anxious. She can just hike parallel to the road, and voila\u2014she'll be back home before she knows it. Adi regrets not being able to say goodbye to Julia, but she was nowhere to be found. Everyone else she shared so much with might have tried to convince her to stay.\n\nI don't think so.\n\nShe plows through three strips of jerky and downs more water.\n\nAdi doesn't realize she's crying until the tears wet her thin T-shirt. She shivers, straightening, and walks away from the tree, shoving her water and jerky in the pack. She flips it over onto her back and strides away.\n\nDoubt creeps in as she hikes the short but steep ravine to the highway. Could I be in heat? Is that what this is all about? Adi grows uneasy with the thought that she's wandering around without male escorts.\n\nI'm too young for heat.\n\nBut Moon help her if she is. Being a female Were in heat is like ringing the dinner bell. And what the hell might have set that in motion anyway? A sudden thought strikes her.\n\nTessa and Tahlia. Those two Were gals show up, Lanarre or not, and suddenly, Adi feels all tingly and fucking needy.\n\nMoon. She stops at the shoulder, cars buzzing by in both directions. The Olympic Peninsula is to the north; her den lies to the south.\n\nAdi had planned to go quarter-change and trot on back to the Northwestern. Now, with the possibility of being in heat, she can't take that risk. If unscrupulous males are around\u2014and her scent is strong enough\u2014they could....\n\nShe shudders. It could be bad. Adi gets a sick feeling. Could Slash have sent her away knowing she might be in heat?\n\nNo way, even he's not that cruel.\n\nThe tears roll unheeded down her face.\n\n*\n\nAdi's face is hot.\n\nShe rolls to her side and groans. Twigs, leaves, and small stones are her bed, and her body is protesting about it.\n\nThe hell with it.\n\nShe sits up, sliding her backpack from where it had served as a pillow, and unzips it. After opening the bag of M&Ms, she pops a small handful in her mouth.\n\nBliss. Flavors explode, and she sighs with pleasure, sucking the shell of chocolate coating and crunching the peanut beneath.\n\n\"Now that is food,\" she remarks to the indifferent forest.\n\nAdi tucks her legs beneath her, searching her backpack. Carrying a toothbrush, deodorant, soap, and a small washcloth, she walks to a nearby stream.\n\nI'll thumb it today, she thinks, glancing back at the 101 behind her. She's still not made up her mind, but she plans to head south and figure it out as she goes.\n\nAdi hears the water before she sees it. The water gurgles its music over small rocks, tinkling and singing as it moves through the woods. Adi's thirsty as hell, and her mouth waters with the thought of drinking. But she remembers just how sick Julia got after drinking from a creek.\n\nThat's all I need is to be shitting and puking. Yeah, that'd be a nice little piece of awesome icing on the misery cake.\n\nShe hikes a little higher, brushing aside small branches, and spots the dim sparkle of the water down a gentle slope. She carefully makes her way to the edge and squats in front of the running water, letting the bristles of her toothbrush dangle in the rush of water. Adi brushes her teeth, gazing around at the environment.\n\nBeauty is everywhere here. The Hoh Rain Forest is a ways off, but the kiss of its nearness is everywhere she casts her eyes. Sweetly scented Douglas fir and western red cedar choke the deciduous alder and occasional red maple as they vie for position along the stream bed. Adi inhales deeply, filling her lungs with the smell of woods, water, and beneath that, the pervasive scent of the small animals that fill the forest with their unique existence. Invisible to most, they are stunningly present for a female Alpha Were.\n\nFeeling alive, Adi stands, swishing dubious water in her mouth, and spits the used toothpaste on a nearby brush. Wild Rhododendrons seek patches of strangled light with bony branches like outstretched arms. Their buds are so full, the color of the blooms are visible within scant slivers of leaves holding the growth tight, waiting for the perfect temperature to arrive.\n\n\"Beautiful,\" Adi whispers, stroking a deep-emerald leaf. Her fingers come away slightly sticky from the rhody's under leaf. Adi leans over the creek, dipping her fingers into the chilly water to wash them. She fills a washcloth with water and adds bar soap, rubbing vigorously. She washes her hands, face, and pits.\n\n\"Shit,\" she mutters. Blindly, she splashes water on her face, letting the droplets fall. She applies her deodorant without looking.\n\nSetting the tube down, she wipes an arm across her face. Soap assaults her sensitive nostrils. \"Ugh.\" She'd much rather smell the forest than the strong-smelling soap she thieved from the Singer's house. Her arms drop, and she opens her eyes.\n\nWithout the soap mauling her senses, she might have scented them sooner. Adi stands so quickly, the muddy shore of the creek aids her fall on her ass.\n\nShe sits on the squishy mud of the bank. Spinning amber eyes glow from across the creek. Male Were.\n\n\"Oh,\" she puffs out.\n\n\"Female,\" one of them calls out.\n\nAdi scrambles to her feet, tossing her pack across her back. A ray of sun stabs the pack, glittering on one of the unicorn horns.\n\nA surge of heat floods her system, zinging her core with such need, she gasps. Adi's heart begins to gallop.\n\nThe gazes of the three Were narrow sharply at her.\n\nAdi's eyes are keen, and keener still when she instinctively shifts to quarter-change.\n\nThe males across the wide creek miss nothing, their nostrils flaring together.\n\n\"Let us ease you, female.\"\n\nFuck that. Adi retreats a step.\n\nThey advance.\n\nOne of the three takes a step into the water. The fast-moving current breaks around his foot, making protesting sounds of intrusion.\n\nAdi hears a helpless noise\u2014then realizes she's making it.\n\nThey take another step.\n\nA wave of burning hits her hard, and Adi cups her elbows, whimpering against the mess of her body.\n\nThey exchange glances. In the next moment, they're crashing through the small river.\n\nAdi wakes up. She whirls, never more thankful she kept the backpack straps loose.\n\nShe shifts to wolfen in the next jump.\n\nAdi runs for her life. There are rumors about what happens to unprotected females who move through heat without a male escort. Adi doesn't think of what she's heard.\n\nShe runs.\n\nBranches lash at her face and body as she moves through them like a bullet made of muscle and fur.\n\nHer body metabolizes the sugar from the candy, chowing through the little bit of protein hanging on from the prior night and her jerky.\n\nThe water hydrates her for the first mile.\n\nTheir footsteps gain.\n\nI can't outrun males who are a foot taller! Her mind agonizes as her body trembles to keep up the crushing pace she sets for it.\n\nThen a miracle occurs. Adi sights 101 and gets a brilliant idea. As she bursts into traffic, a car slams into her. Adi spins in the air, her body on fire to heal the injury. The heat that she is most certainly going through aids the efforts of her body. The imperative of breeding supersedes all others.\n\nShe lands hard, cracking her head on the smoothly pebbled asphalt. Ribs break instantly, and breathing as she knew it stops. Adi's head falls to the side, and sees the males swarming the border of the woods. She sucks in a breath; it shudders like shards of glass.\n\nCars screech to a halt.\n\nAdi has enough presence of mind to shift back to quarter-change from wolfen as humans approach.\n\n\"Hey! Oh my God\u2014call 911!\" a human man's voice says.\n\nAdi closes her eyes. Tears roll out no matter how hard she squeezes them. Slash! she wails, though she knows he's not listening.\n\nHands touch her. Not to harm, but to help.\n\nPieces of conversation come to her in scattered bits. \"Like this\u2014no, careful!\" The wail of sirens draws nearer.\n\nMedics arrive. Gentle hands touch her ankles, throat, and wrist.\n\nShe'll heal. But right now, Adi hurts like hell.\n\nBut the humans don't know that. \"One, two\u2014three!\" The hands lift her, transferring her to a gurney. Adi shrieks despite her best efforts not to, feeling weightless as the human responders begin to slide her inside the back of the ambulance.\n\n\"Skid marks. Driver was going fast. Did you see how far she was from the vehicle?\" one medic asks another.\n\n\"Too far.\" The voice is ominous.\n\nAdi cracks her eyes open. The males are gone.\n\nA bee sting pierces her arm at the bend of her elbow.\n\nThen she is gone, too.\n\n# CHAPTER FOURTEEN\n\nTessa\n\nThey grab the women, hauling them on shoulders as broad as houses, then run deeper into the woods.\n\nTessa jiggles on the back of the one whose crotch she nailed.\n\nHe seems to be purposely jostling her in a teeth-rattling stride while maneuvering around the huge, old-growth trees heaped with layers of moss and lichen.\n\nShe wants to barf, preferably on him.\n\nFinally, they reach a clearing, and he allows Tessa to slide off his back.\n\nActually, he just sort of lets go, and she rolls off him in a clumsy pile of limbs. She lands hard, a breath choking out of her body. The food she had in the Walmart parking lot is long-gone, and she's so thirsty, she can hardly think of anything else.\n\nBut those considerations are second to the Lanarre pack coming out to see the novelty of two females literally dropping into their laps.\n\nTessa stands, dusting off her hands, and glances over her wardrobe malfunctions. Her yoga pants got a workout while stretching to accommodate her shifts from quarter-change to wolfen. They hang loosely now, barely staying on her hips. Her T-shirt's neckline and hem are terribly stretched out. She's filthy.\n\nShe glares at the approaching Lanarre welcoming committee. Assholes. Tessa will not allow fear to rule her.\n\nHer palms slick with the beginnings of the fight-or-flight response.\n\nTahlia doesn't seem to share her anxiety.\n\nOf course not. She's lived in the comfort of the southwestern Lanarre region of the Redwoods in northern California. Tahlia's seen nothing but deference.\n\nWhat an eye-opener this must be.\n\nThree Lanarre break from the pack of about a dozen. Tessa backs up until her butt hits a tree.\n\nThe Lanarre who carried her turns, his eyes like knives of contempt, slicing her up.\n\nLooks like his crotch is okay now. Tessa smirks.\n\nThe three move to Tahlia. \"Imposters will be punished.\"\n\nTahlia's brow furrows, and anger radiates from her every pore. \"You are some of the most daft Lanarre I've ever had the misfortune of encountering. Scent me!\" she yells.\n\nThe lead of the three shakes his head. \"Drek was tasked with meeting his chosen, Tahlia. After a massacre of her guardians\u2014\"\n\n\"My guardians, you foolish dog.\"\n\nHis fist swings out, and Tahlia ducks smoothly. He feints, clocking her with the opposite fist.\n\nShe lands hard, her lip split and bleeding. \"You will not address me as though you are above me. No female but a royal could speak to a Lanarre male in that way, bitch.\"\n\nSo much for not hurting females.\n\nTessa's back straightens. Tahlia has a smartass mouth on her. But for a huge male to use a closed fist on a female Lanarre? Nope.\n\nThey're going to get a reckoning when Drek finally makes his appearance.\n\nTessa moves quickly. These idiots will be unprepared for her offensive. They're so sure of their wonderful maleness, there's no way a female would see to them.\n\nBullshit on that.\n\nOne of the three almost catches her race to get them, but he misses it.\n\nTessa rolls as she flies low, bunching up her body and somersaulting across the soft ground. At the last moment, she shoots out her hands, grabbing the nuts of the two who flanked Beater.\n\nHe turns, mouth agape, and Tessa springs straight to his face, striking him in the throat in a solid jab. He falls.\n\nTessa smiles. All in a day's work.\n\nShe swivels, grabbing the hand that Tahlia throws above her.\n\n\"Thank you, Tessa,\" Tahlia says, spitting blood on the ground.\n\n\"Thought you were a big bad fighter?\"\n\nTahlia nods, her eyes wary on the remaining Lanarre. \"I am. When I know I will suffer abuse.\"\n\n\"You've never been hit by a male?\" Tessa asks without looking away from the approaching Lanarre.\n\n\"No,\" she says, voice small. \"None would dare.\"\n\nThe Lanarre approach, changing to wolfen. Their bodies burst the clothes apart, swelling to almost seven feet.\n\nTessa's empty. She doesn't have sufficient fuel to shift. Quarter-change is all she's got, and even that is scraping the bottom of the barrel.\n\nAnd it won't be enough. She took out the three through the element of surprise. Without that to assist her, they'll have her bent into a pretzel in no time. Nailing their crotches was a quick, effective offensive. It also probably has the highest piss-off factor.\n\nTessa chances a glance at Tahlia. Her face is brave, but her lips tremble. \"No male Lanarre would raise a hand to a female.\"\n\n\"They didn't get the memo at this pack,\" Tessa remarks.\n\nTahlia frowns, but then her face bleeds to horror as she looks to the border of the forest.\n\nLaz steps from the woods.\n\nHis tail rises above his head, and he crouches like a samurai warrior. His hiss sounds like a legion of snakes.\n\nTessa shivers. The demonic are a fearsome species.\n\nThe Lanarre turn and pause.\n\nHe's worth a pause.\n\nLaz, why couldn't you stay gone? Tessa has time to ask before sprinting to meet him. She runs toward the evil\u2014her salvation.\n\nMaybe there's hope yet. With the three of them, they might be able to get out of this lunatic place where they're noseblind to a true royal in their midst.\n\nShe's almost to him when a Lanarre blindsides her with a fist.\n\nTessa flies, crashing on the ground. She rolls, moving to her hands and knees, shaking her head to clear it.\n\nTahlia flies in a blur of color to her right.\n\nTessa stands and falls. She gets up again.\n\nTahlia's put two of the Lanarre out of commission.\n\nA lazy smile forms on Tessa's face.\n\nA couple of chicks kicking their asses. Tessa likes it. Remembering Laz, she turns.\n\nHe brings the end of his tail down on the head of a Lanarre. Brains blow out the side of his head. Shards of skull fling away like the scattered shells of eggs.\n\nA hand grabs her arms, and Tessa twists it hard. The grasp loosens, and she stares up into slowly rotating eyes of silver. She hits him in the nose with the flat of her palm, giving it everything she has left.\n\nHe howls, staggering away and grabbing his stubby snout.\n\nTessa whirls, her throbbing palm letting her know she's fractured something, and promptly slams into Laz's chest.\n\nHe grips her with fingers of steel. She sucks in his scent as though she's starving.\n\nHome. \"Laz!\" she cries.\n\nHe tosses her gently aside, and she lands on the ground behind him.\n\nHer head bounces off a rock, and the world spins. Tessa hears the fighting and screaming. Something warm hits her face like droplets of bathwater.\n\nA large hawk slowly spirals in loose circles above her head. \"What?\" she chokes out.\n\nTahlia?\n\nShe rolls her head, and it comes off the rock.\n\nNo. Tahlia is under three Lanarre.\n\nThey're holding her down. Gotta get over there.\n\nTessa begins to claw at the ground, working her way inch by inch.\n\nA foot lands on her back, and she groans as pain radiates through her beaten body.\n\nShe grabs the foot and twists. The Lanarre loses his balance, falling beside her.\n\nShe shoves her thumb in his eye. He wails, and she rolls to her side, lurching to her feet, and stumbles toward Tahlia.\n\nWhy does she look like I'm seeing her through a tunnel?\n\nTahlia drops to her knees just as an elegant and commanding Lanarre moves into the clearing.\n\nTessa sways.\n\nHe takes in the melee then shouts, \"Stop!\" in the most intense voice she's ever heard.\n\nTessa shudders. His voice resonates through her bones like a note from a tuning fork. She begins to fall forward, and someone catches her. She bats weakly at the hands. Then she notices whose hands hold her.\n\nHeat seeps into all the cold parts of her body, one in particular.\n\nHer eyes snap open.\n\nPale-blue eyes stare somberly down at her out of a face too red to be human flesh.\n\n\"Laz!\" she calls in weak warning.\n\nHis tail snaps back like a mallet, taking the approaching Lanarre's head off at the base.\n\nLike a classic parody of a zombie, the Lanarre keeps walking forward, arms outstretched, blood shooting out of his neck like a geyser. He comically tumbles forward.\n\nThe four after him are more cautious as they approach Laz.\n\nMy demon.\n\n*\n\nDrek\n\nThe area stills, and a frightened Lanarre princess seeks his face. Her fear strikes Drek like a piercing sword. Its evisceration of his tender insides is complete. His arrogant certainty of her fate as separate from his own is wiped from his psyche forever.\n\nHis guts pull him forward with a painful precision born of biology. Tahlia is more than his chosen in an arranged match of tradition he's been told about over the years.\n\nTahlia is meant to be mine. The very fiber of his being weeps for their union. His wolf thrashes his insides to get to her. Drek nearly stumbles as he makes his way toward Tahlia.\n\n\"What is this chaos?\" Drek roars, eyes sweeping the impromptu battleground.\n\nBowen bursts his skin for the third time in twenty-four hours, smoothly transitioning to wolfen.\n\nDrek keeps form.\n\nHe takes in the large hands holding her small body on the ground. His beast wants to gnaw those hands off.\n\n\"We have caught this rogue Lanarre female, Drek. She claims to be Lanarre royalty from the southwest. She attacks the males.\" His snort of disdain sets Drek's teeth.\n\nOnly royal Lanarre males can scent royal females. His temples thump, popping veins.\n\n\"Take your hands off my chosen, or Bowen will strip them off with his teeth.\"\n\nBowen growls his agreement.\n\nHis eyes seek Tahlia. She remains aloof, looking everywhere but at him.\n\nThe other males step away, and she manages to stand. Tears stain her face. A smudge of muddy forest floor and a stray needle stick to her cheekbone as she rubs her arms where the males held her.\n\nHer hunger slaps at Drek, and her fatigue tears at him like small teeth. This is chosen? He thought he could just\u2014what? Make her go away?\n\nThere is no going away. Tahlia is here to stay.\n\nMine, his beast whispers with insistence from deep inside. Drek reaches for her, and large midnight eyes narrow at the gesture. \"Don't touch me, prince,\" she seethes.\n\nDrek frowns. \"I am Drek, your chosen.\"\n\nShe nods, as regal as he.\n\nThis is not going as he thought it would.\n\n\"I know who you are. Call off your dogs from my friend, or I will make your time on this earth a living hell.\"\n\nDrek's misgivings deepen. He might want her, but the greeting she received was incorrigible. Therefore, she is justifiably upset.\n\nHe glances at the six bodies on the ground. Two are missing heads. His lips lift from his teeth. The Lanarre guard who are uninjured bow their heads, backing away. Drek risks a glance at the rogue female and the demonic.\n\nShe is worse for wear, bleeding from her head. The demonic, standing proudly in the nude, wears the blood of the Lanarre over his entire body.\n\nDrek dismisses him as a threat for the moment, turning his attention back to one of his guard, Ospere. \"What. Is. Going. On?\" He enunciates each word like gunfire.\n\nOspere glares at Tahlia.\n\nIn the flesh, she is utterly gorgeous\u2014liquid and vital. Wide-spaced eyes of the darkest blue with a splash of violet flash in a face contorted with anger. Black hair in tight spiral curls falls to a hand-span waist.\n\nDrek wonders what it would be like to touch it.\n\n\"Do not look at my chosen thus. Explain. Now.\"\n\nOspere's mouth drops open, and Drek wonders at his shock. \"We found the females trapped at the great river, Drek. They were accompanied by that one.\" He jerks his thumb toward the demonic hovering near the forest, with the injured female in his arms.\n\n\"Why?\" Drek yells. \"Why would you raise a hand to the chosen?\"\n\n\"Drek,\" Ospere spreads his arms away from his body in an inoffensive gesture. \"Your chosen has been here for over a week. She arrived in the time you have been gone.\"\n\nDrek jerks his head back. \"No. This female\"\u2014he points at a pouting Tahlia\u2014is my chosen.\"\n\n\"No, she isn't,\" calls a voice from the direction of their den.\n\nDrek swings his body toward the voice. A female, the near spitting image of the one not two yards from him, stands close to the edge of the glade.\n\nShe sways as she walks toward him. The female is undoubtedly royal\u2014Drek can smell it. She also smells alarmingly similar to Tahlia.\n\n\"Who is she?\" Bowen asks.\n\n\"I am Tahlia of the Lanarre pack of the Redwood Forest.\"\n\n\"You bitch!\" the chosen screams.\n\nDrek's shock is absolute.\n\nTahlia attacks the other female. Talons fly.\n\nBlood flows.\n\nMore Lanarre rush in and separate them.\n\nDrek and Bowen exchange a look of confusion.\n\nFor once, his nose has lied. He doesn't know who is telling the truth\u2014and who is lying.\n\n# CHAPTER FIFTEEN\n\nJulia\n\nJulia leans back.\n\n\"You're gonna wear your eyes out looking at the clock constantly.\"\n\nShe turns to Scott and sighs. He's right, but she can hardly wait.\n\n\"It will surely be unpredictable,\" Victor comments, setting his fork at the edge of his plate, tines down.\n\nJulia's stuffed. She leans back, pushing her plate away with a fingertip. \"I'm scared,\" she admits softly.\n\nVictor's forehead creases. \"Do not be. Scott is Combatant. I am, as well. The demonic doesn't stand a chance against the two of us.\"\n\nJulia's hand covers the spore. It's quiet. But for how long?\n\nScott follows the movement, his face troubled. She can feel their combined thoughts like collective soup\u2014delicious, but filling. He knows what she does. Nothing.\n\nShe glances at the timer that will release them from the vault, yet again.\n\nFive hours, four minutes, thirty-two seconds.\n\n\"Julia,\" Scott says softly, taking her hands in his. He kisses her knuckles, keeping hold of her hands.\n\nHis flesh is warm against hers. She doesn't cry, but the tears are right at the surface. Julia thinks of Jason dying and Cyn still living above their heads. Maybe safe?\n\nMaybe not.\n\nJulia's been so selfish. Scott's brother and sister are above. Jen and Michael can't even know they're down here.\n\nJulia turns to Victor. \"You didn't know those two weren't Region Two?\"\n\nVictor's face tightens, regret etched on every surface. \"I thought something was off. But, like everyone, I was overly thrilled that additional Singers survived. At least, more presumed to be Singers lived.\" His eyes meet hers, and Julia releases a harsh breath.\n\n\"Me, too,\" she whispers. They exchange a tormented glance.\n\nVictor continues, \"I was walking through the kitchen when my veins began to surface.\"\n\n\"That's when you knew,\" Scott confirms, and Victor gives a curt nod.\n\n\"So they're here to what?\"\n\nVictor narrows his eyes at Julia, drumming his fingers on the cheap folding table they just ate at. \"I do not wish to alarm you further. But because you're the Rare One, the demonic would be interested in shutting you down.\"\n\nJulia turns to Scott, and he nods.\n\n\"Royal Singers are angelic. The demonic had nothing to worry about when we were spread across the globe\u2014numbers low, no one to solidify or unify who and what we are. Along comes the prophesied Rare One. You changed all that.\"\n\n\"How?\" Julia asks, taking her hands from Scott's hold. She stands and paces back and forth in the all-concrete gloom.\n\nVictor stands, as well, clasping his hands behind his back. He walks closer to her, but stops a few feet away. \"Your potential has been much debated. But we know that the Rare One is here to unite the Singers and bring a purity of blood and cohesiveness to our species that's been diluted and misaligned over the centuries.\"\n\n\"Like Jacqueline,\" Scott states quietly.\n\nVictor looks in his direction, saying nothing.\n\n\"We can't blame what's happening here on Jacqueline's mismanagement of Two. No. Those demon guys are here because they're trying to halt progression,\" Julia states.\n\nVictor's eyes are on Julia's hands as they fly around like escaped birds. \"You are the natural enemy of the demonic.\"\n\n\"All of us are,\" Scott corrects.\n\nVictor shakes his head. \"All royals, naturally. However, many of us do not possess sufficient blood quantum to worry them.\"\n\nThey look at Julia. \"But I do.\"\n\n\"In abundance.\" Victor casts his eyes to the floor. \"I have assumed that you've bonded fully.\"\n\nJulia's face bursts into flames.\n\nEven the unflappable Scott seems a little... flummoxed. He manages, \"Yes.\" The one-word answer clearly signals the end of the conversation.\n\nNot for Julia. \"So we have. The soul-meld is consummated. What are the bennies?\" She shifts her weight, trying to ignore the timer.\n\nVictor's mouth parts. \"Bennies?\"\n\n\"What are the benefits now that we've done what we were meant to?\" Julia asks.\n\nScott reaches for her.\n\nThey thread their fingers together.\n\n\"I do not know,\" he says.\n\nJulia's shoulders slump.\n\nScott scrubs a hand over his short hair.\n\nVictor lifts a palm. \"I do know that the coming together of the two\u2014a true soul-meld\u2014is unprecedented. In theory, Julia's abilities, when tested, should be at their fullest.\"\n\n\"English, Vic,\" Scott says.\n\n\"You are stronger together than apart.\"\n\nScott hauls Julia against him, planting a soft kiss at the crown of her head. \"Well, we're together all right.\"\n\nJulia's blush flames to life again.\n\nVictor's face doesn't even twitch. \"That is good. I am more happy with that news than anything I can dream up.\"\n\nJulia thinks it's weird that one physical act can change things so drastically. Was that all it took?\n\nVictor grins suddenly. \"And\u2014\" He claps his hands together softly. \"If a child would result, all the better.\"\n\nJulia's mouth feels as though it's come unhinged from her jawbone.\n\nScott's embrace tightens.\n\n\"It would be the happiest of news.\"\n\nJulia really feels stupid now. She never once thought about protection. Dread pours through her.\n\nScott snaps his head to her.\n\nShe can't handle the idea of a baby in the middle of this dangerous life. She can't even protect herself.\n\nScott turns her to slowly face him. \"Julia,\" he says tenderly, brushing his mouth against hers in the softest kiss she's ever received, \"I'll never let anything happen to you\u2014or to a baby we would have.\"\n\nJulia's face falls forward on his muscular chest. \"I can't do kids right now, Scott.\" She wrings the confession out.\n\nScott places his index finger beneath her chin and lifts it. His lips part into a crooked smile. \"But you know they'd be little angels.\"\n\n\"Literally,\" Victor chimes from behind them.\n\nJulia nods, forcing a smile through her anxiety. \"That's what I'm afraid of.\"\n\nLittle targets for the likes of Praile and crew. The thought reminds her of something. \"What about Laz?\"\n\n\"He was the demonic that helped Praile, who went by 'Peter,'\" Scott states.\n\nJulia shudders, thinking of the pain from the demonic's spore when he drew nearer to her. \"But what's happened to them?\"\n\nVictor spreads his hands. \"We are here. They are above.\"\n\nJulia notes he very carefully doesn't address Jason's murder.\n\n\"And the new women\u2014Tahlia and Tessa?\" She shakes her head, putting her face in her hands. There are so many variables, so many who could be hurt because of Praile's unknown agenda.\n\n\"He is certainly the one who was behind the Singer genocide here\u2014using Tony. We know that Tony did not act of his own volition. He could not have.\"\n\nJulia nods a little too quickly.\n\n\"A great offense is the best defense,\" Scott says.\n\nJulia whips her head to his. \"Yes.\"\n\n\"Let's talk about what to do when we finally get outta here.\"\n\nVictor says, \"Agreed.\"\n\n\"What\u2014\" Julia sucks in a huge lungful of air, ready to verbalize the unthinkable. \"What if everyone is dead... again?\"\n\nScott drops her hand, folding his arms across his chest. \"No way\u2014refusing that outcome.\"\n\n\"Scott?\" Julia cries. \"We have to address the possibility.\"\n\nVictor cups his chin. \"We need to assume that if a high demon has left the realm of Hades to seek the death of the Rare One, that would be the only goal. Killing Singers was more about decimating numbers, and he tasked that to a lackey\u2014Tony Laurent, who is now deceased.\"\n\nYeah he is.\n\nJulia still remembers his ripped-out crotch. Scott, apparently sensing the distasteful memory, slides his arm around her shoulders and draws her close. \"So the big wig comes to do me in?\"\n\n\"I would not put it quite like that. But they were aware of your...\" He clears his throat delicately. \"Unfinished ties to Scott, and while you were vulnerable, they presumed to end you.\"\n\n\"But now we're tied\u2014in every conceivable way,\" Scott repeats unnecessarily, and Julia suppresses the urge to elbow him.\n\nA small smile hovers on his face as he gets a sense of her embarrassment.\n\nShe frowns.\n\n\"Yes,\" Victor says, clearly oblivious to the interplay between her and Scott.\n\n\"So now we're invincible?\" Julia asks, disbelief dripping from her words.\n\n\"Hypothetically,\" Victor says, and Julia hears the but.\n\n\"But?\"\n\n\"The spore worries me,\" he admits.\n\nJulia's hand flattens over her stomach.\n\n\"Let me see, Julia,\" Scott says.\n\nShe rolls up a thin T-shirt that'd been stuffed in one of the dressers in their room. It smells stale.\n\nHer fingers shake as she holds the bunched material.\n\nThey look at her stomach. Julia glances down, as well.\n\nA small, comma-shaped black smudge mars otherwise-creamy skin. No evidence of the wound caused by the demon's saber remains. Just the shadow of evil.\n\nShe exhales slowly. \"Still there.\"\n\n\"Yup,\" Scott says.\n\nJulia can hear the worry in his voice. It mirrors her own.\n\nVictor tears his eyes away from the mark. Facing away, he says to the wall, \"We might be forced to seek outside help to expunge that potential threat.\"\n\nJulia's face jerks to him. She was staring at the mark, rubbing it as though she could erase it from her body.\n\nScott asks, \"Who?\"\n\nVictor's exhale is filled with irritation. \"The fey.\"\n\nNot them again.\n\n\"We already owe them. I promised Tharell the exchange of Singers who would marry the fey.\"\n\nScott frowns. \"Yeah, before Tharell became a traitor.\"\n\n\"Doesn't matter. The oath is to faerie. Just like an oath of loyalty in human marriage is to God, not the spouse.\"\n\nJulia had never thought about marriage in those terms. \"So let me get this straight.\" She squeezes Scott once and steps away, gathering her thoughts without the heat of his emotions swarming her mind. \"I promised Tharell, but he was acting as faerie's rep?\"\n\nVictor tilts his head back, brows snapping together. Finally, after a full minute of consideration, he answers, \"Yes.\"\n\n\"Wow,\" she breathes. \"Tharell was dishonest, all out of his head because he wasn't close to faerie, and with his demonic blood\u2014it ruled, but faerie was still online. Then when I committed to exchange Singers for marriages\u2014\"\n\n\"Faerie was silently taking notes,\" Victor confirms.\n\n\"Marvelous,\" Scott says, his expression in opposition to the word.\n\n\"And I promised to end him,\" Julia says, biting on a nail.\n\n\"And that would fall under the same precepts,\" Victor adds, giving her a speculative look.\n\nJulia's stomach does a little flop. \"This is terrible.\"\n\n\"Is there a timeline?\" Scott asks suddenly.\n\nShe thinks about her prior conversations. \"I can't remember that I committed to a specific date.... Maybe. We have time,\" Julia says with quiet hope.\n\nVictor's eyes are dark with his thoughts. \"Perhaps.\"\n\nA beep, like a squawking bird, pierces the still subterranean air, and Julia jumps. Her head whips to the clock.\n\nZero hours, minutes and seconds.\n\nIt's time.\n\nScott holds out his hand, and Victor leads them to the rungs of the ladder that will take them up and out.\n\nTo the unknown.\n\n# CHAPTER SIXTEEN\n\nSlash\n\n\"I need to leave,\" Slash states.\n\nTruman's eyebrows move to his hairline. \"We could use another Were here.\"\n\nSlash's breath rushes out of him, and he scrapes a palm over his cropped hair. He doesn't bother with subtle. \"I've mated Adi.\"\n\nCynthia's mouth drops open. \"You\u2014what?\"\n\nTruman's lips twitch at the corners. \"Nothing like doing the deed when everything's going to shit.\"\n\nSlash understands the timing is unbelievably bad. But sometimes the heart wants what it wants\u2014the hell with timing.\n\nTruman runs his eyes over Slash's form.\n\nHis expression tells Slash what he already knows\u2014he isn't going anywhere right this minute. Feeling has returned to mid-thigh, though his lower back is still numb.\n\n\"I know,\" Slash confirms what he reads in Truman's disbelieving expression. \"That Alpha nailed me. But I'm coming around.\"\n\n\"Shift to wolfen,\" Truman suggests easily.\n\n\"Couldn't out there,\" Slash jerks his head to where Truman just hauled him from. \"What if it'd taken a couple of times, and Tramack and company had opted for a return visit? Plus I'm low on fuel.\"\n\nTruman grimaces, casting his eyes to the floor. \"You have me there.\" He puts his large hands on his hips, considering for a full minute. His eyes rise to meet Slash's. \"Shift. I'll guard you, then you go get your girl. This mess will be here in some form when you get back. It always seems to be.\"\n\nThey stare at each other, both understanding there might be no coming back. Uncertainty is the way of the supernatural world.\n\nLawrence and his default second-in-command, Manny, are both dead. Tony Laurent is dead, thank Moon. That leaves the Northwestern leaderless. Not an ideal spot for a pack of werewolves to be in. Without leadership, Were tend to become unstable.\n\nAdrianna almost surely went in the direction of her pack of origin after Slash shoved her away. However, the immediate goal of securing her safety was realized when Slash saw Tramack beaten and bleeding on the ground, his Were beside him.\n\nShe is safe for now.\n\nBut Slash would rather have her with him always. He knows he'll have to beg her forgiveness and eat crow to the end of his days in order to make Adrianna understand that he would never have said those things and acted as he had if the situation had been even marginally winnable.\n\nShe's young and female. Adrianna simply doesn't understand her importance to him\u2014or his role in her protection. How could she? She only has herself to protect.\n\nSlash has to protect them both.\n\n\"We'll get some food, and you shift,\" Truman says.\n\nPreferably something that's still squirming. Though that was unlikely.\n\nSlash is anxious to get out of the area, scent Adrianna, and set things right.\n\n*\n\nSlash groans.\n\nTwo shifts later, he remains wolfen.\n\nCynthia brings in the third plate of steaming leftovers.\n\nSlash eyes the food: an entire twelve pound chicken, by the looks of it. Slash isn't sure if that's sufficient.\n\nHell with it, it'll have to do.\n\nShe sets the food on the coffee table in the dainty parlor where he shifted and made a mess everywhere. Bet the Singers never thought that a bunch of cross-species would be spreading the remnants of a shift all over their house.\n\nSlash's chuckle is like a bark. Shit happens. And it happens often in his experience.\n\nSlash begins to shovel food, his body feasting and breaking down the nutrients as fast as he can chew and swallow. \"Thanks,\" he mumbles a heartfelt word through a packed mouth.\n\nCynthia leans against the door jamb, managing a wan smile. \"Welcome.\"\n\nSlash's eyes take in her slight form. She could use some food, too. Singers begin to file in through the door, and Cynthia turns to watch their quiet progression.\n\nShe sucks in her lip, holding back the sobbing.\n\n\"They'll take care of him,\" he comments, swallowing a load of food.\n\n\"I know,\" she replies softly, sweeping a small strand of blond hair from her face and absently tucking it behind her ear. She sniffles, taking a shaking inhale then going still.\n\nThey sit in silence as Slash picks his plate clean. Down the hall, voices are talking. Others are crying.\n\nJason Caldwell was a volatile Red. Most are.\n\nBut he was a part of the Singers. And like an injured limb that won't heal, his loss is felt keenly.\n\nSlash uses the last bit of chicken to catch the juices. He can't help his indifference. Slash has seen too much death to be concerned about one more. It's the single consistency of being a supernatural\u2014death.\n\nRight now, the only thing he cares about is the female who is not by his side.\n\n\"Thank you,\" Slash says again, covering a burp with his fist.\n\nA low vibration of tingling begins in his lower back, radiating like seeking fingers of heat, gripping and massaging his nerve endings back to life.\n\nSlash tries to stand from his crouch on the floor. He crashes to his ass on the floor. He grits his teeth against the searing pain as his body heals the last bit of his paralysis. Flames lick at the lumbar region of his back, bringing it to life. \"Ah!\" Slash places his palm to his lower back, where the pain is most acute.\n\nCynthia rushes to him, face blotchy from holding back grief. \"What is it?\" she asks, grabbing his arm.\n\nHis eyes slit in pain. \"Healing,\" he breathes out.\n\n\"Oh,\" she says. Then something flows into him\u2014part balm, part heat\u2014and his eyes meet hers.\n\nShe's augmenting his own body's healing.\n\nPins and needles stab him like an unwilling voodoo doll, and he bends in half, his palms hitting the solid coffee table. The plate rattles as his fork drops to the floor.\n\nA gauzy gray shroud covers his vision then shreds to ribbons. Slash staggers to the pass-through, Cynthia's hands still on his arm. He grips the threshold between the two rooms and slowly opens his eyes.\n\nHis wolfen vision is ten times as acute as a hawk's. Everything comes into focus. Slash feels his back, hips, legs, and feet simultaneously. He touches his back, his finger brushing the elastic band of the new athletic pants Truman had set aside for him.\n\nSlash feels alive.\n\nHe turns back to Cynthia, and her hands fall. \"I keep saying thank you\u2014but thank you.\"\n\nShe nods. \"It feels good to help.\"\n\nTruman walks in from where the Singers have congregated at Caldwell's death scene, eyebrows hiked in clear question.\n\n\"I'm ready.\"\n\nTruman nods. \"Looks like someone's locked in that underground bunker.\"\n\nSlash frowns, flaring his nostrils. Many scents come to his nose, ones he'd been too unwell to consider. Demonic is the first.\n\nDeath.\n\nWere.\n\nJulia, Scott, and Victor.\n\nTruman shrugs. \"Don't know who it is or how they got down there. But the timer's about up. All the bad asses have been accounted for.\"\n\nTruman doesn't smell them. Probably too new a Were to scent identify every layer in the vicinity.\n\nCynthia stands up straighter. \"Maybe it's Jules,\" she says, hope heavy in her voice. \"That'd make sense, right? Haven't seen Scott, her, or that Victor dude.\" She holds her face in her hands. \"I can't take it if they're hurt of worse,\" she says through her fingers.\n\nTruman grunts. \"Could be, or maybe not. Don't know yet, Cynthia. Don't borrow the worry.\"\n\nHer shoulders slump. \"Yeah.\"\n\nTruman steps away from the doorway, giving her a look that captures Slash's attention. Cynthia doesn't see the expression.\n\n\"There's always hope,\" Slash murmurs.\n\nHe scents something surprising from Truman. He doesn't act as though he cares about this female, but Slash scents that he does.\n\nTheir eyes move to Slash. He indicates his healed body with a hand. \"I never thought to have a mate. And now I do.\" He doesn't acknowledge the Rare One, her soul-meld, or Victor being inside that bunker. Slash would hate to tell them his suspicions, only to be wrong.\n\nCynthia lifts her shoulders. \"I know Adi, and if you don't get your werewolf ass after her, pronto, you can kiss the mate thing goodbye. She's not a girl to trifle with.\"\n\nI know. \"I'm sorry to leave you with all this going on.\"\n\nCynthia shakes her head. \"We could use you. Truman's right.\"\n\nTruman remains silent.\n\n\"But Adi's important, too. Us women have been through a lot together. And she's kinda headstrong. I'm afraid for her, Slash. She needs a level-headed guy. Don't dick her around.\"\n\nSlash blinks. Adrianna being headstrong is one of the biggest understatements he's ever heard. And he would never play games with a female. He's never had the opportunity even if that were part of his nature.\n\nSlash is uneasy, too. Thankfully, Adrianna doesn't have much of a lead on him\u2014six hours, tops.\n\nCynthia scans his face, apparently satisfied with what she sees there. \"Go,\" she says softly.\n\nTruman nods.\n\nSlash leaves. He doesn't run at first. But before he knows it, he's sprinting.\n\nAdrianna's scent is easy to follow. It smells like home.\n\n*\n\nAdi\n\nAdi blinks heavy eyelids. She looks around at all the white: white walls, floor, blinds, and bed.\n\nShe gazes down at her ensemble, a white hospital gown.\n\nUgh. She shuts her eyes again. Unreal.\n\nDerailed to the human hospital.\n\nHer eyes snap open. Oh shit. Adi can't let them examine her too closely.\n\nThey would find out things.\n\nShe sits up, and a plastic tube like a clear worm swings from her arm.\n\nWonderful.\n\nAt least she's escaped the three stooges.\n\nMaybe.\n\nThe heart rate monitor starts to beep, and a nurse cruises in. \"Ah, you're awake.\"\n\nAdi blinks at her stupidly.\n\nShe's about five feet five, with a slim build and dyed-black hair. Her liquid eyeliner is like a black slash, making her clear light-brown eyes stand out. She smells fresh off her cycle, and Adi can smell that she uses antibacterial soap.\n\nAnd she has aggressive breast cancer.\n\nSometimes it sucks to be a wolf.\n\nThe nurse\u2014Adi scans her nametag\u2014Jenni, clasps the clipboard with Adi's probable vitals. Jenni's expression turns puzzled.\n\nMaybe Jenni is a couple of years older than Adi. She looks young.\n\n\"Let me check your vitals.\"\n\nAdi nods, suddenly sad at all the inside info she now holds. Jenni checks all her parts. And like a jointed doll, she's okay.\n\nThat's when the fun begins.\n\nJenni plops down in a chair and rolls it to her bedside.\n\n\"You're very lucky.\"\n\nAh-huh.\n\n\"Do you know your name? We couldn't get anything from our records with your print.\"\n\nOf course not. No self-respecting Were would allow a human to get ahold of her DNA.\n\nExcept me. Adi figures they've got her blood, skin, and everything typed. Dammit. \"My name's Adrianna.\"\n\nJenni's smile is brilliant and gentle.\n\nShe's in the right line of work, this human.\n\nJenni pats her hand. \"We have x-rays that show a broken pelvis and femur, contusions, and small fractures in keeping with the larger injuries.\"\n\nHer eyes sweep to Adi's. \"A second set of x-rays show that those injuries have mended themselves.\"\n\nAdi gulps. Spying a water cup, she grabs it, takes a sip, and sets it back down. \"Okay?\" Adi says, feigning ignorance.\n\nShe's awesome at that.\n\n\"We're going to run more tests.\"\n\n\"Nope,\" Adi replies in a pleasant tone.\n\nNurse Jenni's brows crease. \"I'm sorry?\" she asks, as though she heard Adi wrong.\n\n\"I am not staying for more tests. I feel amazing. And, I have somewhere to be. I'd love to grab some food, but I bet I'll need to stick around for that.\" Adi grabs her lower lip with her teeth and swings her legs to the side of the bed opposite Jenni.\n\n\"Hey!\" Jenni calls out loudly.\n\nAdi's clothes rustle as she stands, tearing out the IV. She tosses it onto the bed then jerks off the tape, wincing. She rips the gown over her head, pitches it on the hospital bed, and pads naked to where she spies her clothes.\n\n\"What? Wait...\" Jenni says, setting the clipboard on a rolling tray with a disruptive clatter.\n\n\"It's okay. You've done your job and filled me up with some great meds.\" Adi turns to look at her, her hands hanging limply at her sides.\n\n\"Did you set my breaks?\" Adi asks, rooting through her unicorn backpack and pulling on her pants. She whistles through her teeth at some residual pain, hiking the yoga pants over her hips.\n\nBet the other pair are toast.\n\nAdi eyeballs the pack. A few unicorns have road skids, and only a smattering of glitter remains where they'd romped over the slick material.\n\n\"Yes,\" Jenni whispers.\n\nAdi smiles at her. \"Thank you. That would have been awful for me to try and figure out.\"\n\nJenni takes a deep breath then lets it out as though she's calming frayed nerves. Probably. \"We want you to stay here for observation, Adrianna.\"\n\nI bet.\n\n\"Adi.\"\n\nJenni catches her forearm. \"Please don't go, Adi. We've never seen anyone heal like you do.\"\n\n\"I know. Can't stick around and be a cooperative pincushion.\" Adi walks to the window, where she pops the metal slats apart with her index finger and thumb.\n\nShe swivels the latch at the middle of the window and slides it open slightly. Adi rests the side of her face on the wooden sill.\n\n\"What are you doing?\" Jenni asks from right behind her.\n\n\"Smelling,\" Adi turns to look at her. \"Stay where ya are. Don't stick me with needles or anything. I won't like it.\" Adi allows her eyes to go wolfen, and Jenni gasps, retreating a step.\n\nAdi directs her attention back to the window. She lays her face on the windowsill again and closes her eyes, inhaling deeply.\n\nHer eyes snap open. The males are here.\n\nJenni's going to be more than a nurse today.\n\nShe whirls, and Jenni snatches the clipboard from the tray, holding it in front of her like a shield.\n\n\"I need your help,\" Adi says, shutting out the fear. For now.\n\n\"You have it. Stay here so we can study you and take care of you.\"\n\nAdi gives an irritated shake of her head. \"No. You're going to wheel chair my ass out of here, toss me into your car, and we're going to your place. Then I can get where I'm going without the three musketeers up my ass.\"\n\n\"The three what?\" Her face screws into a frown of confusion.\n\nConvince her, Adi.\n\nShe rushes the nurse, who squeals like a piglet. Adi covers her mouth with one hand, staring deeply into her eyes.\n\n\"You'll help me because I know stuff about you.\" Adi shakes off the guilt of coercing a sick human. Survive.\n\nJenni tries to shake her head.\n\n\"I'll let you go if you don't scream.\" Adi raises her eyebrows, waiting.\n\nJenni shakes her head again.\n\nOkay, she's gonna scream.\n\n\"You have aggressive breast cancer.\"\n\nJenni's face slackens.\n\nAdi takes her hand away.\n\n\"How\u2014how could you possibly know that?\"\n\nLong version or short? Adi's eyes skate to the window. Short.\n\n\"I'm a werewolf.\" She taps her nose.\n\nJenni's skin pales, and she begins to slide down the wall.\n\n\"Shit!\" Adi says as she catches her easily.\n\nWhere the hell are the smelling salts when you need them?\n\n# CHAPTER SEVENTEEN\n\nDrek\n\n\"Stop this!\" Drek bellows a second time in a voice meant to deafen.\n\nAll activity ceases.\n\nHe makes a valiant effort to soothe a beast within him as it claws for escape. His attention moves to the palest shadow of the moon.\n\nFortune is with him.\n\nIt is full. And for a Lanarre, that means more control, not less. Three measured breaths later, his eyes peg the two Lanarre females before him.\n\nWhich one is Tahlia?\n\nHe openly studies the two females' bodies from the tops of their heads to the tips of their toes.\n\nAt first, Drek had assumed they were identical. Now he sees subtle differences and smells differences in scent, as well.\n\n\"She is my cousin, Drek. Tanya,\" the little spitfire tells him.\n\nThe other female, his purported chosen, glares at her.\n\n\"This is Tahlia,\" the voice of a strong female says from a few feet away.\n\nBowen growls.\n\nDrek raises his hand. \"Let the female speak.\"\n\nHe gives her a less-thorough perusal than the one he gave the warring Lanarre females. She is tall, even for a Were, standing easily five feet and ten inches. A messy black braid graces the hollow between her shoulder blades and runs past her waist.\n\nGray eyes like a pale storm cloud meet his frank appraisal with one just as bold.\n\n\"She saved me from Tramack's men.\"\n\nDrek feels his face tighten. His heartbeat trips over the next one in anticipation of her words. \"The Alpha from the Western?\"\n\nShe nods, and clear surprise washes over her face. \"Yes,\" she answers quietly. \"How did you\u2014?\"\n\nBowen chuckles beside him. \"We made his acquaintance at the Singers' Region One.\"\n\n\"And what of Praile?\" the demonic says.\n\nTheir fingers lace, and low growls can be heard everywhere. No Lanarre will stand for the inference of romantic intentions from anyone other than another Were.\n\nA female Were in heat is the exclusive right of the Lycan. How dare this demonic manifest his intent?\n\nDrek steps forward.\n\nThe demonic's skin flushes a light red. A tail rises above his head. \"Step no closer.\"\n\nDrek laughs. \"Do you not see you are outnumbered, horned one?\"\n\nThe demonic smiles, flashing teeth as bright as a slice of ivory in his hard face. \"I know many things.\"\n\nDrek hesitates. His knowledge of the demonic as a species is limited.\n\nHe has never had a need to know this little-seen enemy on intimate terms.\n\n\"Demonic wounds are very difficult to heal.\" Drek looks at the demonic. They are equal height. His pale-blond hair appears to silver beneath the bright moonlight that swaths the open meadow. Light colored eyes of indiscriminate color gaze guilelessly back at Drek.\n\nBut the demonic teem with guile. They are a crafty species\u2014that Drek is not soon to forget. His gaze shifts to the wounded Lanarre.\n\nThe body count stands at two. Shards of brain like bleached bone litter the ground as though head-sized eggs have been shattered in place. Drek's eyes reach the demonic again. \"You've murdered Lanarre, horned one.\"\n\n\"Only the ones who would have seen my true death come to pass.\"\n\nSelf-defense.\n\n\"He is a problem,\" Bowen comments.\n\nDrek heaves a harsh exhale. \"Yes.\"\n\n\"I can heal all but myself,\" the demonic says casually, and all movement ceases. \"And I am not horned.\"\n\nSomething in his tone rings of hollowness.\n\n\"That particular mark of beauty was not bestowed on me. However, where that bit is absent, I have other... attributes.\"\n\nThe female by his side blushes, casting her eyes at the ground.\n\nDrek knots his hands, noticing the male's nakedness again. Casual nudity is not appropriate among shifters, but perhaps it doesn't matter to the demonic. After a full minute has passed, he says, \"What do they call you?\"\n\n\"Lazarus.\"\n\nDrek's eyebrows pop. That is an ancient name from the fabled Christian book. It does not seem a likely namesake for a demonic.\n\n\"Not that it matters since there hasn't been one fair turn in this entire time,\" the chosen says from behind him, and Drek turns to the sound of her voice.\n\nHer eyes are pockets of shadow in her face, spinning slowly like swirling smoke. That alone lets Drek know that she's agitated, her beast rioting for escape.\n\n\"Laz saw that we escaped from Tessa's horrible packmaster and the other demonic. He alone helped us.\"\n\nDrek whirls to Lazarus. \"Praile?\"\n\nLazarus nods gravely. \"He will find me. There is no reasoning within the ranks of the high demon. Praile will locate me, and he will escort me to Hades, for the Master's reckoning.\"\n\nThe female beside him shivers, gripping his powerful bicep.\n\nHe wraps an arm around her, covering her head with a large palm. Steam rises in slow-moving vapors to escape into the cooler night air. His tale vibrates above his head, clearly ready for anything.\n\nDrek notes a very subtle shimmer over all of Lazarus's skin. It must be a trick of the eye. The vapor of hell couldn't off gas from his very pores.\n\nOr could it?\n\n\"We cannot offer sanctuary to a demonic,\" Drek says with a decisive grunt, \"or one who killed two of my guard. Or is clearly about taking a female from the limited number the Were possess. This Tessa? She shall remain here and be suitably mated and bred for the female she is.\"\n\nWhy can't this demonic understand Tessa's place among his people?\n\n\"She is my Redemptive. She will not be mated to a Lycan. Common Were or Lanarre.\"\n\nDrek's mouth drops open. \"If you know so much about the Lycan, then you must know our females are much depleted. We mate within our species, demonic.\"\n\nHis grip on Tessa tightens. \"Be that as it may, without her, I am condemned to hell.\"\n\nBowen chuckles. \"That's rich! Aren't you already there?\"\n\nLazarus's face sours. \"Yes. That is why this rare opportunity which has presented itself is so enticing. Without Tessa, my life remains one of torture, abuse, and maiming the souls of all beings whom possess one.\"\n\n\"It isn't our choice who Tessa mates,\" the female claiming to be his chosen says.\n\n\"I am Tahlia. Do not listen to this imposter.\"\n\nThey scent so similar.\n\nDrek ignores them both for the moment, turning back to Lazarus. \"I must sort the mess of my chosen before I can give sufficient weight to your words, demonic. Can you be trusted to not harm any more of my guard?\"\n\nHe can't afford more bloodshed, and he knows that will be what it comes to.\n\nA crooked half-grin seats itself on the demonic's chiseled features. \"If they do not try to harm me.\"\n\nDrek turns to his guard. \"Watch, but do not act.\" The six who remain look upon the demonic with angry wariness.\n\nHis attention moves to the two females. One lies. The other holds the truth.\n\nThe spitfire has captured him. But the other is equally beautiful.\n\nBut Drek knows from brutal experience that beauty is not all that matters. There is such a thing as a mate who holds within her the innate makings of a queen.\n\nThe spitfire is too bold, too rash to make a good queen, though their connection is irrefutable.\n\n\"What do you think, Drek?\" Bowen surveys the two royal females from the west.\n\n\"I think I will very thoroughly question the females.\"\n\n\"Good call,\" Bowen says.\n\nDrek looks to the Lanarre guards. The one who is the lead, Ospere, nods his compliance.\n\nWhatever Drek wishes to do, Ospere will see it done.\n\n\"Take the females to my chambers. Restrain them so they do not harm each other\u2014\"\n\n\"What the fuck?\" Tessa roars and launches herself toward Drek.\n\nLazarus captures her about the waist, popping her feet off the ground. And though she is a tall athletic specimen of a female, he easily halts her forward momentum, but not before she speaks her mind, \"You chauvinistic cad.\"\n\nDrek frowns, easily understanding the gist of her words.\n\n\"Tahlia is the real chosen. She came here because she thought your pack would protect her since her guardians were murdered. But you're like all the rest of the Were, royal dick or not. Your males hurt her because she tried to defend herself.\"\n\nThe Lanarre circle her and Lazarus, but Drek gives a subtle shake of his head. They instantly still.\n\nHe would hear her thoughts\u2014and what his guard did against the female. Drek folds his arms. \"How am I like the other Were?\"\n\n\"You all have small dicks and try to squish a female's spirit in the hopes she won't discover the inadequacy.\"\n\nLazarus's face appears pained, and a feminine snort sounds from somewhere behind him. Drek doesn't have to wonder over who might have made the noise.\n\n\"None of the Lanarre males have lack of function in that area.\"\n\nShe snorts. \"What other excuse is there? And you have proof. How would that be?\"\n\n\"Tessa,\" Lazarus murmurs.\n\nDrek's irritation rises.\n\n\"If you were a decent Lycan male, you wouldn't be trying to restrain your future mate. You'd be compassionate. She just lost her human guards, you foolish male!\"\n\nShe's spitting now, and Lazarus tries to calm her. \"I won't!\" she replies to something he says beside her ear.\n\nHer eyes flash like silver coins. \"It doesn't even matter if another lying female is playing you here. The facts are: we arrived, we were held against our will, and you almost killed a guy who was our only protector when one of our kind sought to enslave me. And you don't bat an eye, Prince Drek,\" she states with thick sarcasm. \"And you're the stand-up for the Lycan race?\" Her stare incinerates all who stand within range, her accusing eyes missing no one. \"You guys couldn't dance your way out of a paper bag. No wonder I'm rogue. What can you offer a female? Nothing.\"\n\n\"Amen,\" the chosen says from behind Drek.\n\n\"Take them,\" Drek says without turning.\n\n\"You'll regret this, Lanarre.\"\n\nDrek walks to her and the demonic. \"I allow the demonic to live because I haven't ascertained his culpability in these circumstances, only that he felt he acted in self-defense. And I do not know the true nature of my chosen. You are female and in the beginnings of heat. You're not capable of behaving rationally.\"\n\nThe first flicker of uncertainty Drek has seen lights her features. Good. She is too sure of herself for his taste.\n\n\"Come no closer,\" Lazarus says.\n\nTheir gazes clash. \"You threaten me? A Lanarre prince.\"\n\nLazarus's nod is slow. \"I threaten anyone who would harm my Redemptive.\"\n\nDrek's lips thin. \"I keep you for her welfare, as undeserved as it seems now.\"\n\nTessa's lips curl back from her teeth. \"By all means, kill us. Kill a female who dares to speak her mind and tell the truth about circumstances against her. And kill the male who saved her and your supposed chosen, who you don't even believe is your true mate.\"\n\n\"Your presence is separate from who my real chosen is.\" Drek scrubs his face. \"However, a true chosen would not behave as she does.\"\n\n\"Pfft!\" Tessa says, and Lazarus tightens his hold on her. \"You wouldn't know a true female with backbone if she walloped you between the eyes.\"\n\nHer disrespect grates on him. Drek looks to his guards, who flank him, their eyes spinning with the need for bloodshed. However, they will not act without his command. \"Take them to the house as guests. Keep the demonic at the back.\"\n\nDrek tries for gracious when all his beast desires is war. \"I need your word that you will not consummate whatever this Redemptive status is. Give us time to come to terms.\"\n\nLazarus inclines his head. \"I am not cut from the same cloth as the Lycan. I do not feel pressed to manipulate a female for its own sake.\"\n\nIt's not a real answer, but one of evasion.\n\nTessa gives him a glance.\n\nDrek flares his nostrils. They are not as well-acquainted as he first believed.\n\nShe is learning about his behavior just as he.\n\nA demonic is not meant to be with a Lycan female. They will see this. If he can delay the strange courtship that this Lazarus insanely believes is his right, Drek will figure out something that defeats it without looking like the one who manufactured their demise. Then she can be free to mate with a Were. And whoever his chosen really is will see him act with temperance instead of the dictatorship that is his right.\n\n\"Yes, whatever you believe, demonic, does not necessarily hold all the facts.\"\n\n\"Enough facts.\" His voice is droll. Sure.\n\nWe will see.\n\n\"Take them, as well.\"\n\nThe guard moves toward Lazarus, but his eyes flare in warning. Even in the moonlight, the shadow of red blooms on skin like bleached alabaster. \"I will walk unaided.\"\n\n\"Yeah\u2014\" Tessa says, glaring at Drek. \"Me, too.\"\n\nThey walk away, and Drek's gaze follows them until he can no longer see their forms.\n\n\"This is inviting trouble, Drek.\"\n\nHe faces Bowen, finally succumbing to the pull of his fatigue without the audience demanding answers. \"I don't want more loss. And even if the female, Tessa, has a sharp tongue\u2014\"\n\n\"She is female,\" Bowen states thoughtfully.\n\nDrek nods. \"Exactly.\" He gazes up at the full moon, wondering how this mess happened so quickly. Every solution feels insurmountable. If he kills the demonic, he is a villain for the murder of the one savior the females had. If he doesn't, it teaches his people a demonic can be trusted. That is a lie, with Lazarus perhaps proving to be the exception. And that means there is no justice for the two dead guard. As the humans say, Drek is between a rock and a hard place.\n\nThen there is the chosen. He felt a connection to her that couldn't be denied. But her manner... and his earlier thoughts of wanting to change the tradition to something with more free will make his mind full.\n\n\"You weren't even sure you wanted to go through with the pairing, Drek,\" Bowen states. He's vocalized Drek's exact thoughts.\n\n\"True.\"\n\n\"Then find out what's what with this Tanya\u2014and Tahlia.\"\n\n\"Did you feel it?\"\n\nDrek turns to Bowen. \"Feel what?\"\n\n\"The connection. The Book of Lanarre speaks of a true match having an enigmatic pull.\"\n\nDrek stares at Bowen. Lies, one of the benefits of being royal. \"No.\"\n\nBowen slides his jaw from side to side, hands going to hips. \"But one of them must be your match.\"\n\nYes, one is. Drek shrugs. \"Time will tell.\"\n\nBowen seems to accept that vague answer.\n\nDrek cannot. He must get to the bottom of the identity issue. And if a mate exists for him, she will become his. If not, she can be part of his pack in another capacity.\n\nIt's really not that complicated. Safeguarding the females is simple.\n\nThe answer to what will become of them is complicated.\n\n# CHAPTER EIGHTEEN\n\nTessa\n\n\"Fuckers!\" Tessa grates.\n\nThe male Lanarre holds his fist up, and she clamps her mouth shut.\n\n\"Get your hands off me!\" Tahlia says loudly, twisting her lithe body from side to side.\n\nThe obvious lead Were yanks her to him, clearing her feet off the forest floor. \"Do not make me beat a female.\"\n\n\"Again,\" Tessa states in a dry voice. How dare they not own up to how they converged on her and Tahlia\u2014or how bad their manners are. This is royalty?\n\nIf Drek is in charge of this group of thugs, Tessa is certainly better off with Laz... or on her own.\n\nThe lead Lanarre\u2014Tessa thinks his name is Ospere\u2014turns down his mouth. He's wolfen.\n\nBoth Tessa and Tahlia have stayed in quarter-change. Tessa doesn't even look at that bitch, Tanya. She docilely walks along. \"You entered our territory with a demonic. You are rogue. She\"\u2014he jerks his square jaw toward Tahlia, who pouts magnificently\u2014\"is Lanarre.\"\n\n\"I have told you, you\u2014oof! That I am Drek's chosen.\"\n\nThis is met with a knowing smirk. \"You do not behave as a chosen.\"\n\n\"She behaves like a female alpha who has been roughed up by males. Pissed.\" Tessa's face rocks back from the love tap given by the guard who holds her. She turns faster than he anticipates and sinks teeth sharper than human\u2014more exacting too, on his forearm.\n\nHe tosses her, and Tessa goes sprawling on her butt, landing in a heap of arms and legs with moss to thankfully cushion her fall. She flicks her tongue out, coming up with a drop of blood from the corner of her mouth. \"That the best you have, needle dick?\"\n\nOspere cocks his head, giving her a considering look. \"Why would you goad my guard, female.\"\n\n\"Because I can. And it's fine\u2014beat me. But you will never take from me who I am.\" Tessa hears her voice shake with conviction.\n\nOspere shakes his head as though weary of her. She has that effect on people.\n\n\"Bind her.\"\n\nThe one who hit her approaches with a ready smirk.\n\nTessa salivates with wanting to kick him right where it counts.\n\n\"Come here, little female.\"\n\n\"Piss off, brainless.\"\n\nHe launches.\n\nTessa avoids him and hits him on the back of the head with her laced hands. His wolfen arm, thirty percent longer and more muscled than usual reaches back and body slams her to the soft ground.\n\nStill, the wind leaves her lungs. She opens her mouth and can't breathe.\n\nHe flips her over, and her face plows into the moist vegetation of the forest floor.\n\n\"She can't breathe!\" Tahlia screams in a panic.\n\nThe Lanarre takes too much time to bind her wrists, enjoying her vulnerability, with her unprotected hindquarters on full display.\n\nHe hovers. \"Where's your demonic lover now, bitch?\" his whisper is loud enough to be heard by all.\n\nHer shame is a seed that plants itself in her brain, causing Tessa's tear ducts to fire on like light switches. Air finally returns to her lungs, and she sucks in just enough to say softly, \"At least he's got a dick.\"\n\nTessa doesn't know anything about Laz's dick, except it looked beautiful. But she'd maybe wanted to know more.\n\nThe Lanarre jerks her up, whipping Tessa around so suddenly, she groans at the onslaught, her shoulders throbbing from the abuse.\n\n\"That's enough, Erik.\"\n\n\"Yeah, Erik,\" Tessa manages.\n\nOspere jerks Tahlia beside him. \"You do not want to save yourself. However, maybe you care about this young female. She's barely beyond whelphood.\"\n\n\"But she is,\" Erik says with sick casualness.\n\nTessa's face whips to his. \"Touch her, and I'll end you.\"\n\nHe laughs in her face, his wolfen breath vile. The jerk's been eating the wrong animal diet.\n\nTessa leans back and, jumping forward, clocks him in his forehead with her own.\n\nShe's lifted from behind by the waist.\n\nInstantly, she knows it's Laz, and a part of her that had shriveled to hopelessness at his departure fans to life again.\n\nErik staggers forward, and Laz's tail wraps around Tessa, bashing Erik in the skull with the hammer head at the end of his tail.\n\nHis head bursts like a soft watermelon. Bits of brain scatter like gray seeds. Mushy, dirty gray brain matter falls before them, and Erik's body slumps to its knees.\n\nHis elevated status as Lanarre feverishly tries to repair the damage a mundane Lycan could never hope to achieve.\n\n\"I'm afraid he lost his head,\" Laz says.\n\nThree Lanarre rush him.\n\nTessa uses Laz's body, swinging her feet out, and punches them into the closest Were.\n\n\"Tahlia!\" Tessa screams.\n\nAn achingly high-pitched caw meets her fevered plea.\n\nTahlia's bird form lands on the next, talons sinking into his skull like a knife through butter. She rises, the powerful legs of the inky bird she uses in this form twisting and jerking. She breaks his neck, and he falls face forward.\n\nThe sound of bones breaking has Tessa struggling to turn in Laz's hold.\n\nHe drops her, and Tessa loses her balance, helplessly sinking to her knees. \"Shit!\" she yells.\n\nOspere has become his wolf, and Tessa slips to her knees on the carnage of his exploded form.\n\nTahlia has lighted on a strong branch directly above.\n\nTessa is kneeling like a sacrifice one yard from where Ospere growls, circling Laz.\n\nLaz's injuries make Tessa want to cry.\n\nDeep lashes run down his face, arms, and legs. He's still naked and vulnerable. Gorgeous.\n\nTessa blinks, suddenly realizing that there's something good to have come out of all this. Might as well die in a lust-induced state. She feels the stupid grin overcome her face. Old Erik got his, and Laz standing around in all his muscular naked glory's not too bad.\n\nAnd Tramack didn't get me.\n\n\"Stop!\" Ospere commands in a booming voice that makes Tessa's teeth thrum.\n\n\"Now what?\" Tahlia asks from beside her.\n\nTessa gazes up at her from her spot in the mud and filth, a black feather sticking out of her hair, and laughs. It's not the laughter of humor, but that of hysteria.\n\n*\n\n\"I can't believe Drek didn't choose to kill us,\" she states quietly. \"I guess if more guards had been killed... and now Tahlia's tied up somewhere with that phony Tanya.\"\n\n\"Nor I,\" Laz agrees.\n\nTessa chances a glance behind them. Yup, the two guards are their shadows, one in front of them and one behind.\n\nTessa trudges beside Laz.\n\n\"How did you get away from the guards?\" Tessa whispers.\n\nLaz's brow quirks. \"I heard you beyond me, and I couldn't control my actions. But they'll live.\"\n\nA vague outline of a cottage appears. Nicer digs than the Western had. It's been so long since Tessa lived with her pack of origin, so she can't be sure that improvements have been made during her absence.\n\nDoubtful. If there's one thing that's consistent, it's Tramack. He cares about one thing: himself. Not the comfort of the pack.\n\nThe small cottage appears to grow out of its surroundings. Situated perfectly between a cluster of hemlocks on one side and western red cedar on the other, it stands as a low, story and a half. A small window is anchored in the gable pitched roof like a single warm eye. A soft light glows through the glass.\n\nTessa doesn't hear the drone of generators, and this deep within the wood, she's not sure how things like electricity or plumbing reach this isolated structure. The thatched roof looks newly done, if she's any judge. It's beautiful and cozy. She doesn't want to relax her guard, but her exhaustion is taking over with each step.\n\n\"More than you deserve, female,\" Ospere remarks.\n\nShe slides her gaze to him, her tongue ready for the lashing such a comment deserves, but Laz presses a fingertip to her lips.\n\n\"Thank you for the hospitality of the Lanarre,\" Laz says with quiet intensity.\n\nTessa surveys his wounds again. They weep blood.\n\nHospitality, my ass. But she remains silent because they spared his life, even after he beat his way to the front of the group to save her from Erik's \"hospitality.\"\n\nShe looks back at his convulsing body. Fresh skull, like the membrane of a hardboiled egg, is a thin opaque sheathing covering his healing brains. Tessa shudders.\n\n\"It is sad commentary when a horned one would extend courtesy where a female of our kind holds nothing but derision.\"\n\nHow can he say that? His buddies are the one that beat them into submission.\n\nLaz touches her shoulder lightly, and she pivots on her heel, climbing the wide, bowed wood steps that lead to the front door.\n\nLaz gracefully follows, though his wounds should be causing him a lot of grief.\n\nThe Lanarre at back steps forward. He holds the solid plank door wide as they step through the threshold. \"Do not leave.\"\n\n\"Where's Tahlia going?\" she asks, frantically searching for the young female.\n\n\"She will be escorted to Drek's home.\"\n\nThat sounds bad. But she clamps her mouth shut. They haven't killed her or Laz yet. They probably won't hurt Tahlia, Tessa hopes.\n\nHis mouth puckers.\n\nTessa closes the door in his face and listens as the tumblers of a lock engage. She spins, facing Laz. \"It's completely weird that they'd leave us alone together if they believe I'm in heat.\"\n\nTessa keeps strict eye contact.\n\n\"It's baited,\" Laz says absently, his hands running over the huge logs that bisect the structure. His fingertips hover over the chinking in a dull ivory. Finally, he turns.\n\nTessa crosses her arms. \"What are you talking about? You killed two Lanarre. You accompanied a rogue female in heat. They're going to kill you\u2014slowly.\"\n\nTessa raises her hands to her face. \"Why didn't you stay gone, Laz? You're a demonic. They're keeping you alive for something awful. Look at how they treated Tahlia\u2014their possible future queen.\" She grunts her disgust.\n\nLaz peels her fingers gently away from her face.\n\nThe wounds of his own are beginning to heal. By tomorrow, they'll be gone. He brushes a hair out of her face. \"They will try.\" He looks off, eyes trained to the dark forest beyond the window. \"It is not the Lanarre who frighten me, but Praile. There is nothing they might do that would be worse than what the Master is capable of.\" His face tightens.\n\n\"That other guy\u2014Peter?\n\nA small smile erupts over his face.\n\n\"What's so funny?\" Try as she might, Tessa doesn't find anything amusing. Tahlia's somewhere. Some other Lanarre chick has decided to impersonate her, and Drek seems like a class-A asshole.\n\n\"Inside joke.\" Laz's smile widens.\n\n\"Huh. Okay. Anyway...\" She gives him a look that clearly says, get serious. \" Why is he a threat. He's demonic, like you.\"\n\nIt's Laz's turn to give her a disbelieving stare. \"We are demonic.\"\n\nTessa nods. So?\n\n\"High demons are the best of us.\"\n\n\"I remember.\"\n\nHis exhale is a rush of temper. His skin deepening to the light red she remembers so well.\n\n\"Why do you do that?\"\n\n\"Do what?\" His blond eyebrow rises. But now there's a hint of red.\n\nTessa waffles her hand between them. \"That. One minute you look like an average man, then the next, you're all demon.\"\n\n\"Average?\" he asks, a seductive lilt to his voice.\n\nThat one word heats her insides.\n\n\"Not average,\" she says, hating the squeak in her voice.\n\nHe moves his hands to her shoulders, brushing his thumbs over the curve where shoulder meets arm.\n\nHe explains by leaning forward, his mouth tickling her ear. \"It is more difficult to keep my human costume when my emotions are near the surface.\"\n\nLaz pulls back, and Tessa sways.\n\nHe is back to looking like a Viking: a blond, chiseled, tall, blue-eyed Norse specimen. She lowers her eyes.\n\nOh, Moon. I want to ride his boat. Tessa's face flames.\n\nLaz smirks as though he can read her thoughts.\n\n\"Why are you emotional right now, Laz?\"\n\nHe turns without answering, pacing away from her.\n\nShe admires the view, the muscles of his ass flexing and clenching beneath the black smoothness of the pants he was given. What is wrong with me? Oh yeah\u2014heat. But it should only be for a Lycan.\n\n\"It seems right that I would be emotional from having been captured by Lanarre\u2014Lycan royalty, no less\u2014and my commander hunts me for my wrongdoing. That my injuries take much of my remaining strength to heal. That's enough.\"\n\nTessa walks to his broad, bare back. The Lanarre provided clothing when he conceded their capture to Ospere, but the athletic pants don't fit him, and he never put the shirt on.\n\nLaz runs warm. She stifles a laugh.\n\nTessa swallows the ball of desire lodged in her throat by all that the pants showcase.\n\n\"I smell your lust, Tessa,\" he comments in a low voice.\n\nShe stops walking. \"How?\"\n\nHe slowly turns to her. The wounds of his face are now scabbed. \"I am demonic.\"\n\nRight. I guess lust is their biz.\n\n\"Ask me why those things are not the cause of my emotion? The fluctuation of my form is not caused by those extenuating circumstances. I have excellent control of the cloaks I choose to wear in this realm.\"\n\n\"Why is\u2014\" Tessa takes a deep breath. \"What is causing the emotions.\"\n\nRed water rushes through the air, and Laz suddenly stands before her.\n\nShe gasps.\n\n\"What is that?\" Her voice is shaky.\n\nLaz palms the side of her face. \"That? That is my essence. What you have seen thus far is not all that I am. This is only part of what it is to be demonic.\"\n\nTessa gets ahold of herself. \"What's made you emotional, Laz?\"\n\nSoft red moves under the surface of his skin like subtle blood. His hair darkens to gold. The flash of his tail spins above them, seemingly rotating with his thoughts. His eyes darken to gray licked by pewter.\n\nHe feathers a fingertip across her cheek, trailing his touch along her hot skin.\n\nTessa sighs.\n\n\"Proximity to my Redemptive.\" Then his lips are on hers, and Tessa's questions melt to what he's doing\u2014and what she'll allow.\n\n# CHAPTER NINETEEN\n\nScott\n\nThey're quiet as they ascend from the bunker. Julia's anxiety scoops out Scott's guts.\n\nAll he can do is fall harder into her.\n\nHe couldn't wait to have sex with Julia and get it out of his system. Scott figures there's nothing more unnatural than not bedding a soul-meld. Now he has.\n\nAnd if anything, his attachment to her is tighter. The connection is not like the noose married human men joke about, but a net of love and protection. Its weight is heavy.\n\nMaybe it's because of that spore. The thought of having to go to the sithen and worry about Julia's safety causes the Combatant inside Scott to rise. He beats the primal reaction down with a mental club. He can't be busting his form every time he feels threatened. He has to actually be threatened. None of the Combatants had known what having the Rare One present would be like until Julia was among them. Then their role wasn't just theory anymore, wasn't fun and games, but something real.\n\nSwaths of light pierce the dim interior of the cylindrical ascension as it marks they're progress.\n\nFinally, they reach the top, and Vic gives a hard twist of the spinning handle, lifting the portal.\n\n\"This is it,\" Julia says.\n\nScott doesn't reply. He feels the sludge of her anxiety, and he knows what awaits above.\n\nChaos.\n\n*\n\nVictor pops the hatch and slowly rotates on the top of the ladder rung like a human periscope. His eyes meet Scott's, and he gifts him with a slightly bemused shrug. \"Everything appears normal.\"\n\n\"Nah,\" Scott says. \"Get your ass out of my face, Vic.\"\n\nA smile flashes across his perfect features, and he hops out the top then holds out his hand to Scott. He slaps his palm into Vic's and exits. His eyes scurry over the surfaces in search of Jason's remains.\n\nEvery bit of metal sparkles. The hatch is clean and pure.\n\nScott opens his senses for danger. His nostrils flare. He's no tracker, but he can smell death underneath the obvious taint of cleaning chemicals.\n\nScott suppresses his obvious relief and bends, easily lifting Julia from the dark hole of the bunker, and her gaze skips over everything his eyes just touched.\n\nLarge liquid-topaz irises trace the spinning inset knob that opens the portal, the rim of aluminum that circles it, and the timer that is no longer counting down the hours. The floor that surrounds it is polished and glowing; the rug that covered the portal is absent.\n\nOf course. There's no way to clean what happened to Jason from fabric. At once, an image of the flail-type appendage the demonic swept into Jason's head rises like an unwanted mirage, and Scott shuts his eyes against it.\n\nHe can handle the visual, but he doesn't want Julia getting an echo of his thoughts.\n\nHer exhale speaks of defeated relief. \"I-I guess Jason's gone. Really gone.\"\n\nScott doesn't have a reply that's not painfully obvious or redundant, so he just pulls her against him, smoothing his hand over her hair.\n\nVictor's eyes meet his over the top of Julia's head. His relief mirrors Scott's own. Neither felt that Jason was anything but a way to fuck up the Region. Caldwell thought about numero uno. His human origin always got in the way of what needed to be accomplished with their people. Caldwell meant well. Scott even believes he loved Julia\u2014in his way. But it wasn't a selfless love. There were conditions.\n\nTrue love doesn't have conditions. It loves without strings, unwinding toward infinity by faith alone.\n\nHe kisses away Julia's salty tears then tilts her face up with his hands. \"It'll be okay.\"\n\nJulia nods. \"I have you.\"\n\n\"Absolutely.\"\n\nHer smile is watery but it's there, and Scott'll take it.\n\n\"Jules!\"\n\nThey whirl at the sound of Cynthia's voice.\n\nShe flies at Julia, and they embrace. \"I'm so sorry, baby.\" Cynthia steps away, fat tears rolling down her face. \"There was nothing we could do, Jules. I'm so glad you're safe. We didn't know where you guys had gone to.\"\n\nJulia nods. \"Tell me\u2014is...\"\n\nScott hears the click of her throat, and his own constricts with the overflow of emotions he feels from her.\n\n\"Is everyone okay?\"\n\n\"Well\u2014short version? Yeah. Mostly.\"\n\nJulia searches Cynthia's face, slightly chalky and tight with what Scott believes is the stress of Caldwell's death combined with wondering where the hell everyone was. Her six inches of dark-blond roots add to the scene of survival. Who has time for looks when everyone is just surviving from one island in an ocean of disaster to the next.\n\nI'd really like to catch a fucking break.\n\n\"What's the long version?\" Julia asks with a catch in her voice.\n\n\"You know...\" Cynthia's intense green eyes search Julia's face, and Scott doesn't exhale in disgust\u2014even though he knows full well what's coming. \"You know that \u02bbPeter' isn't Peter, right?\"\n\nJulia nods. \"Praile.\" Her voice is like a whip, and Scott strokes her arm. He summons comfort and allows it to pour over her like warm water. Julia's lips part, and her head tips back. \"Thank you,\" she breathes.\n\n\"Welcome.\" Scott kisses the top of her head, and Cynthia's eyes track their interaction like a hawk's.\n\nCynthia folds her arms, eyeing them up. \"Okay. So I see some stuff has changed.\"\n\nJulia's face turns pink, and love swells within Scott. He hates the new feeling of vulnerability. But when Julia turns to him, he feels the tug of the soul-meld, and its presence eases him.\n\nThere's worse things. Scott's just not used to feeling emotionally fragile. Fuck. He takes Julia's hand, tugging her away from the portal and the scene of Jason's death.\n\n\"Yes,\" Julia says, letting Scott lead her down the corridor to the parlor and the adjacent open entryway.\n\nCynthia walks alongside, trying to catch Julia's gaze and bring her back into the conversation. \"I'm not being judgy.\"\n\n\"I know,\" Julia says, giving her a quick look. \"But Jason's dead.\"\n\nTheir eyes lock, and Victor chooses that moment to silently leave. Scott would like to do the same, but Julia needs him at her back. And he's not going anywhere when he can sense how uncertain she is.\n\n\"Where\u2014what did they do with his body?\" Julia bites her lips, but tears slip out, trailing down her face.\n\n\"Ah, Jules, don't cry.\" Cynthia grips her shoulders. \"He did the right thing. You're the queen and everything, and he took Praile's murdering ass out of the equation of hurting you\u2014or whatever wonky plan the demonic had.\" Her eyes meet Scott's. \"Jace knew that Scott would protect you\u2014love you.\"\n\nJulia nods, tears dripping off her chin.\n\nScott restrains himself from swooping in there and hiding Julia. She has to face this sadness so she can move beyond it.\n\nHer inhale is a rattle of anguish. \"I know.\"\n\n\"We've buried him.\"\n\nJulia nods. \"I want to visit his grave.\" She looks at Scott.\n\nWhat can he do? Julia needs the closure. And whatever\u2014Caldwell's dead. Scott can afford to be gracious. Barely.\n\nHe understands he's got a dickhead attitude. It's a hard thing to help when his soul-meld had that prior connection. Scott will have to just work on it. Every day.\n\n\"First,\" Scott interrupts, and Julia quickly wipes her face, trying to be brave. Making the mark.\n\nHe smiles at her then gives Cynthia a narrow stare. \"Where is the fire prick?\"\n\nCynthia's laughter fades when she realizes the conversation might be too somber. \"Fire prick\u2014Praile\u2014disappeared when someone bigger and badder came along.\"\n\nScott frowns. Great, that's what we need, another fucking scenario of bullshit.\n\nCynthia scans his face. \"It's okay.\" She waves his concern away. \"It was some Lanarre prince dude, and he was all focused on getting ahold of his chosen.\"\n\n\"His which?\" Julia asks, her brows meeting in concern.\n\n\"It's that Tahlia girl. You know\u2014the Were who speaks like she's from medieval Europe?\"\n\nJulia's smile is fleeting. \"Yes, I remember\u2014she and Tessa.\"\n\n\"Well they're splittsville.\" Cynthia makes a whistling noise and Scott smiles. \"And,\" Cynthia rolls her eyes, \"They were riding shotgun with Laz.\"\n\nScott sucks in a breath.\n\nCynthia looks between the two of them. \"He's a demonic, too.\" She taps her chin. \"I don't know about the fire prick part with him, though. But Praile was pretty pissed about him taking off with the Were girls.\" Cynthia shrugs. \"Not my issue.\"\n\nHe snorts. Scott doesn't like the presence of two demonic. Hell, he didn't like any of it. He thought the demonic were done with after the battle at Region Two. He was wrong. Those two sniffing around makes it clear that was only part of it. Julia is the key. As usual. Scott and the remaining Combatants aren't going to make it easy on Praile or any demonic.\n\n\"So he's gone?\" Scott says.\n\nCynthia nods. \"For now.\"\n\n\"Are my sister and brother okay?\" Scott ruthlessly regulates his body's response. Julia doesn't need his anxiety on top of what she already has. He aims to offer emotional protection along with physical.\n\n\"Yeah, we're all okay. But there's this really crappy packmaster from the Western that's jonesing to go after Tessa...\"\n\n\"No,\" Julia says, eyes slightly wild. \"She didn't want to mate with him. Tessa told me that. It's why she was trying to leave with Tahlia.\"\n\n\"Ah!\" Cynthia says loudly, snapping her fingers. \"Drek! That's the guy who was the Lanarre prince. Man\u2014did he put a kibosh on old Tramack. What a ham-fisted fucker he is.\"\n\nJulia blinks. \"What do you mean?\"\n\nCynthia leans forward, and Scott is dying for her to see the point through. He's anxious to explore One and make sure it's secure against Praile and others who would try to harm Julia.\n\n\"Tramack missed Tessa by inches, thanks to Laz. And once Drek and his buddy found out Tahlia was gone, they took off. But I'm pretty sure that Drek showing up made Praile beat feet outta here. And us Singers were so blown away by all these weirdos showing up and swinging their dicks around, there was just kinda numb reaction.\" She spreads her hands away from her body.\n\n\"So the Lanarre guy is gone, and the Were females and Laz\u2014the other demonic, also?\"\n\nCynthia pushes her untrimmed hair behind her ear. \"Yeah.\"\n\n\"Where's Slash?\" Julia asks suddenly.\n\nThat's a great question. Slash is a non-Singer Scott would love to have on their team. He's level-headed.\n\n\"He's gone after Adi.\"\n\nJulia crosses her arms. \"What happened to Adi?\"\n\n\"That's a long story.\"\n\nJulia looks to Scott. \"I have time.\"\n\nNo we don't.\n\nBut Scott listens anyway, growing more troubled by the second. Scott winces as Cynthia explains that Slash forced her to leave.\n\nScott understands his desperation exactly\u2014better to eat crow later than have a mate subjected to death or a fate worse than death. The entire story makes Scott glad One isn't farther south. Nothing but a bag full of assholes for Were down there.\n\n\"So Slash is in pursuit of Adi?\" Scott asks.\n\n\"Yeah, he had to heal up here first. They'd paralyzed him.\"\n\nScott shakes his head. \"Fine. Tell me this jerk is in our jail?\"\n\n\"Sure\u2014he is. But Slash isn't a Singer. It's been explained to me that we can't involve ourselves in Were politics. Their bullshit against him and Adi? We can't punish that.\"\n\n\"That's stupid,\" Julia says.\n\nThey have plenty to concern themselves with just being Singers, so Scott's okay with the problem passing them by. \"I agree, but we're not in a great position to throw our weight around. We have the demonic who seem to be bent on coming after you. We have low numbers of Singers, and a lunatic Were packmaster that's after an alpha female Were who's not interested in him and another Were female who's looking at an arranged marriage.\" Scott plows his fingers through his short hair. This is a mess.\n\nCynthia nods. \"Yeah. Maybe you can put the fear of God in him. Discourage him from going after Tessa. After all, his goons worked Adi and Slash over. Right after they mated, by the way. Super-uncool.\"\n\nJulia's face scrunches with a look of disgusted horror.\n\nScott can't imagine what he would have done if someone had tried to hurt her after their first time together. Probably would have pulled their arms and legs off one by one.\n\nA vague smile seats itself on his face.\n\nJulia frowns. \"What are you thinking?\"\n\nOops, a bit too much emotional blow back.\n\nCynthia smiles. \"Probably the same thing any red-blooded male would be thinking if a bunch of thugs tried to spring in on them after a roll in the hay.\"\n\n\"It's not just that,\" Julia says, her tone of voice hard. \"It's that Adi and Slash had been circling being together for months. Slash hadn't wanted to go past a certain invisible line because Lawrence wouldn't have allowed the mating outside her pack; he was following Were protocol. But then Lawrence died.\"\n\n\"Yeah. Awesome,\" Cynthia says, \"except for Manny.\"\n\nJulia gives a solemn nod. \"Emmanuel was good to me.\"\n\n\"Me, too,\" Cynthia agrees quietly. A note in her voice causes Scott to study her features. Maybe she liked Manny more than anyone realized. But Scott doesn't have time to consider Cynthia's emotional state.\n\nVictor charges back into the room. \"The Were is gone.\"\n\n\"What? Tramack? Why?\" Cynthia says.\n\nVictor shrugs. \"He wasn't enough of a concern to keep in any event. His escape isn't as important as us securing the Region.\"\n\nScott nods decisively, they didn't have the people to contain him anyway. \"Agreed.\"\n\nJulia's hand moves to his forearm. \"I need us to stick together.\"\n\nScott rubs the back of her neck. \"There was never a question of that.\" He looks to Victor. \"Let's go see our people. I think they'll be a little relieved that we're okay.\"\n\n\"I hoped you guys were down there. I just didn't say it out loud,\" Cynthia says, hiking a thumb toward the bunker.\n\n\"How'd you know?\" Julia asks.\n\n\"Trackers could smell you. They'd been looking.\"\n\n\"Oh,\" Julia's eyes light on the direction of the portal at the end of the long hallway they'd just traveled.\n\nScott doesn't think she needs the constant reminder of her former husband's brains plastered everywhere.\n\nHe tugs her out of the house. Scott glances over his shoulder as Victor moves down the hall with a new rug to cover a portal best left unseen.\n\n# CHAPTER TWENTY\n\nAdi\n\n\"Wakey-wakey,\" Adi says softly, gently slapping Nurse Jenni's face.\n\nVivid root beer eyes blink apart rapidly. They widen, the whites looking alarmingly freaked, and Adi says, \"It's okay.\"\n\n\"It's not okay,\" Jenni says in a voice that trembles slightly. \"You've told me you're a werewolf, which I'm inclined to believe.\"\n\n\"Good practical logic there, nurse.\" Adi gives a crooked grin.\n\nJenni cringes away.\n\nMaybe quarter-change is a little too much.\n\n\"Hey, me being a werewolf isn't so bad. You knew something was off anyways. Later, you can say I was a lunatic, hopped up on LSD or something.\"\n\n\"Later?\" Jenni asks, rubbing her eyes with a fist.\n\nProbably trying to make me disappear.\n\nAdi picks her up by the armpits, and Jenni's eyes widen impossibly farther.\n\n\"Yeah. There can't be proof of our existence.\"\n\nJenni's lips purse. \"Well, bad job on getting discovered, quite frankly\u2014as you ran out and got hit by a car on a busy highway.\" She puts her hands on her hips.\n\nAdi's eyes narrow, and she suddenly scents the acute fear smell. Prey. Adi inhales deeply, realizes her eyes have gone wolfen, and calms the fuck down. Moon.\n\n\"Listen,\" Adi says, regaining control of the conversation, \"I know this is a shit ton to take in. Me being a werewolf and all. But aside from all that, I'm a woman, and as such, I feel second class. There's a reason I'm petrified and ran into the highway on purpose\u2014to escape three Lycan males.\"\n\n\"Lycan?\"\n\nAdi's nose scrunches. \"Lycan\u2014werewolf, same thing. Don't go all soft on me now. I need you to shelve your disbelief of Lycankind for just five minutes so you understand the danger I'm in. And you, by association.\"\n\nJenni's throat convulses as she swallows. \"All right.\"\n\nAdi slowly releases her grip on Jenni's Cheshire Cat scrubs. Hmmm. The wrinkled material sits up at attention from where Adi had fisted it.\n\nJenni absently smooths the material.\n\n\"This will be the quickest rundown of supernatural history I can relay.\"\n\nJenni blinks.\n\n\"I'm young. We live for hundreds of years, but I'm just out of whelphood\u2014teenage adolescence. So we go into heat for the first time two to five years past whelp.\"\n\nJenni's eyebrows sweep together. \"Heat?\" she slides her jaw back and forth. \"What does this have to do with why you threw yourself in traffic?\"\n\nAdi huffs in irritation. \"I'm getting to that.\"\n\nJenni's lips thin.\n\nAdi's struck by how pretty she would be without the harsh hair color and makeup. And terminal cancer. That'll do it.\n\n\"So I found this great male\u2014he's way older than me. I've been crushing on him since\u2014well, forever. He used to play with me when I was a whelp. He's a great warrior.\"\n\nJenni studies her.\n\nA great male. Adi clenches her fists, holding her eyes wide so tears don't fall in front of this human female she doesn't even know.\n\n\"He hurt you?\" Jenni asks quietly, getting to the heart of it like a perfectly shot arrow.\n\nCompassion rims irises that darken from the words Adi doesn't say. This chick absolutely knows Adi's pain. Her empathy makes Adi feel vulnerable, which she hates. It's too late to hold back, though. Three asshat males are waiting in the wings to take her.\n\n\"Yes,\" Adi manages. \"But those aren't the facts. I gave myself to him when the big ass admitted to loving me.\" She swipes a tear that falls despite her best efforts to hold it back. \"Then when a pack of jerk Lycan show up, they kick our butts\u2014we're outnumbered\u2014they got Slash really bad.\" Adi's eyes rise to hers. \"They paralyzed him, Jenni.\"\n\nJenni's hands go to her mouth, but Adi hears her easily. \"Slash is your husband?\"\n\nJenni's moved to stand right in front of Adi. She's much taller than Adi. \"No\u2014my mate. But it's the same thing in our world as husband in yours.\" Adi slowly lets out a painful exhale, hoping to plow through this next part. \"He told me to get the fuck away from him. That he didn't want me anymore.\" Adi chokes on the last word. She's really on a stupid rant now\u2014outing the Lycan race while confessing her embarrassing rejection and stupid faith in someone.\n\nConfusion washes over Jenni's face. \"You were hurt\u2014both of you. Why would he do that? It doesn't make sense. Actually...\" Jenni's face falls. \"None of this makes sense.\"\n\nProbably from a human perspective, that's true. \"I don't know why,\" Adi says. More tears fall, and Jenni plucks a nearby tissue from a box and hands it over.\n\n\"I bet you never thought you'd be comforting a werewolf?\" Adi says, blotting her eyes and doing a messy blow of her nose into the damp tissue.\n\n\"No,\" Jenni admits softly. \"It doesn't seem real.\"\n\nAdi gives her serious eyes. \"It is.\"\n\nThey look at each other.\n\n\"So now what? Slash...\" She pauses over his name before continuing, \"Has let you go, and there's some other Lycan\u2014?\"\n\n\"Yes. I guess Slash and I\u2014our interlude\"\u2014Adi quickly looks at Jenni, expecting condemnation, but finding a quiet understanding\u2014\"brought my heat early. And that's so bad. To be a female in heat, without a male escort...\" Adi shudders.\n\n\"So these men\u2014they somehow found you?\"\n\n\"There's no somehow about it. They scented my cycle, and without a male, I'm vulnerable.\" Adi can't keep the disdain out of her voice. \"Even if I could kick one of their asses, I can't do three.\"\n\n\"No,\" Jenni agrees easily.\n\nIn that, there's no problem figuring out the logistics of that scenario. She's a chick; Jenni knows the score.\n\n\"That's why we were overpowered to begin with. There were too many.\" Adi's eyes fill again. Moon dammit. What in the hell is wrong with me? \"There were four Lycan males and just me and Slash. We didn't stand a chance.\"\n\nJenni cups her elbows, expression pensive. \"I feel like I'll wake up any second, and you'll be a normal woman.\"\n\nAdi laughs. \"Nope.\"\n\nJenni's shoulders slump. \"Okay. So your healing powers have convinced me. You are something. But I can't just leave my post here and take off with an unidentified patient without more information.\"\n\n\"I'm not unidentified,\" Adi states reasonably.\n\nJenni rolls her eyes. \"Uh-huh. You say your name is Adrianna.\"\n\nAdi shifts her weight. \"Nobody calls me that but Slash. Adi's my nickname.\"\n\nJenni nods, nervously playing with her stethoscope. \"How can you convince me?\"\n\n\"My eyes weren't enough?\"\n\n\"That was weird,\" she concedes.\n\n\"Are you always this skeptical?\"\n\nJenni nods. \"Pretty much. And for the record, nurses are primarily in the science-math vocation. You know\u2014science. Prove it.\"\n\n\"Does anyone know about the cancer?\"\n\nJenni's face crumples, and Adi feels like an ass.\n\n\"Sorry.\"\n\nHer direct gaze moves to Adi as she inhales deeply. \"Don't be. I guess I'm in the anger stage of my diagnosis. I'm only twenty-eight and\u2014\" She tosses out a frustrated breath. \"I feel like my body's betrayed me.\"\n\nAdi gets a great idea, but she can't voice it. She knows what the pack would say: You can't save everyone. And on the heels of that: humans are lowly.\n\nBesides, she's got her own backyard to mow. Still, her epiphany niggles.\n\n\"Just get me out of here, but first, we have to camouflage my scent. And even with that, they might scent me.\"\n\n\"You smell all yummy now?\" Jenni asks. But the solemn look on Adi's face makes Jenni's smile fade.\n\nAdi nods slowly. \"Oh yeah.\"\n\nJenni frowns. \"What about Slash? Wouldn't he smell you, too?\"\n\nAdi looks at her bare feet. \"Yeah,\" she whispers.\n\n\"So you're telling me that the guy that said he loved you and wanted to be your \u02bbmate,' took your virginity then told you to get lost? Then let you go out there to be cherry-picked by wandering werewolves?\" She shakes her head in clear disbelief, flipping her palm out as if to ask for an explanation.\n\nWhen said like that, it makes Slash sound awful.\n\nMaybe he is. \"Looks like it.\"\n\n\"That's terrible.\" Then Jenni's face perks up. \"Well, we're a pair. But I have to say\u2014this Slash? He took more than your v-card.\"\n\nAdi nods, a defeated breath whistling out of her.\n\nJenni puts a hand on her shoulder.\n\nShe looks up into Jenni's compassionate face. \"You know, I'm supposed to be the bad ass werewolf here.\"\n\n\"Right now, I just see a woman with a broken heart.\"\n\nJenni's scrubs catch her tears.\n\n*\n\n\"This is desecrating the dead.\" Jenni mutters in revulsion, wrinkling her nose.\n\n\"The dead don't know anything,\" Adi says, rubbing liquefied decomposing human organs over her arms and around her neck.\n\n\"I can get in so much trouble, and it's gross.\"\n\nAdi gives her a look of pure frustration. \"I'm the one wearing the guts. I know I reek like road kill.\"\n\n\"It's so bad. And a must be against some religion somewhere. Ugh.\"\n\nProbably all of them.\n\n\"But these humans' remains had already been used for their donor purposes. This is just remnants.\" Adi gives her a be logical look.\n\n\"This better work,\" Jenni mutters.\n\nAdi grips her shoulders. \"It's not going to work. It'll only buy us time. I'm hoping, once we're inside your car\u2014\"\n\n\"What?\" Jenny screeches, horrified.\n\n\"Shh!\" Adi scans the inside of the morgue. The metallic, impersonal surfaces reflect back at her with the dimmed LED lighting. In a low voice, Adi continues, \"Once we're in your vehicle, it would take an awesome tracker to find me.\"\n\n\"We slather you with human remains and that allows you to slip away in the interim?\"\n\n\"Exactly\u2014now get my face.\"\n\nAdi closes her eyes, and Jenni marks her with cold decomposing human soup, putting two fingers' worth above her eyes and each cheekbone.\n\nJenni grimaces. \"There. Ick.\" She removes latex gloves, tossing them with two fingertips in the bio-hazard can.\n\nAdi steps back and moves to the deep washbasin. Looking at the warped mirror above the sink, she slaps the faucet lever on.\n\nThe gore of dead humans decorates her face like war paint. Her eyes are seated in the mess of blood and human tissue like stranded jewels of mixed gold, brown, and flecks of green. Adi's hair is slicked back and thrown in a messy dark-gold topknot at the crown of her head.\n\nShe does look pretty sick. Adi smiles. None of those pervs are going to dig her now. She smells like a corpse. Adi turns, winking at Jenni.\n\nJenni crosses her arms and groans. Then she says for the millionth time, \"I can't believe I'm doing this.\"\n\nAdi hides a smile. \"Me, either.\"\n\n\"Anyone ever tell you you're kind of difficult?\" Jenni asks.\n\n\"All the time.\" Adi flicks her eyes at the rolling stainless steel cart. \"Now let's toss me in the body bag and get to the garage.\"\n\n\"Right. And just for the record, I don't normally haul the dead that way. Or any way. I'm a nurse. For the living.\"\n\nAdi feels her eyes go wolfen, hot\u2014itchy. She's already quarter-changed, she has to be. But her wolf roils beneath her skin. \"I can feel the heat, Jenni.\"\n\nHer beast wants to breed, and her mate is nowhere. Not that she'd let Slash touch her again.\n\nAdi's chin kicks up and her shoulders straighten. \"The hell with it, I can live through this heat bullshit, but I gotta get out of here.\"\n\nJenni sighs. \"I'm bringing some plastic for my car.\"\n\nAdi turns so Jenni won't see the smirk. Cars. So not a priority. She slips into the cold body bag. The plastic crunches around her body as though rejecting her because she's not dead.\n\nTough.\n\nJenni's face appears above the opening. They stare at each other for a moment.\n\n\"I'm zipping you.\"\n\nAdi nods.\n\nThe zipper slides up, and the view narrows. Jenni's a couple of inches from closing it when the zipper stalls. \"You'll suffocate.\"\n\n\"It's okay,\" Adi says, already wanting to claw her way out. But Jenni leaves that tiny gap for her.\n\nThe smell of death is like a cocoon around her. Her beast doesn't like plastic\u2014or being contained.\n\nJenni turns Adi's cart in the direction of the exit, and they slap through the swinging morgue doors.\n\nCeiling lights flicker above her as they pass through the hall.\n\nThey slow. Quiet words are exchanged. Adi hears a buzz then the soft whirring of automated doors.\n\nAdi slows her vitals. She concentrates on her heartbeat and breathing. The heat cycles like a yo-yo from deep within. She focuses on the rush of her blood in her veins.\n\nThe cart stops suddenly, and Adi slides forward a little, feeling unbalanced. She almost barks at the poor human she commandeered for this little fiasco\u2014then the smell hits her.\n\nWere. They're here. Close.\n\n\"Can I help you?\" Jenni says. Her voice is neutral and indifferent, but her scent isn't.\n\nDamn.\n\nJenni's heartbeat sounds like a deer on the run. Worst physical response ever. Her cancer's scent is like riddled Swiss cheese. The Lycan males will smell her disease\u2014maybe Adi\u2014everything.\n\nWe're fucked.\n\nAdi feels responsible for Jenni's safety.\n\n\"You wear the smell of a female we are trying to locate.\"\n\nThey're not even trying to sound like human men. Moon. Adi masked her smell, but not what she'd passed to Jenni.\n\nCould I not think of that?\n\nThe male voice is growly. Adi works to stay loose and unconcerned\u2014a supreme challenge.\n\n\"What are you talking about?\" Jenni asks with convincing confusion. \"\u02bbWear the smell'?\"\n\n\"There was a female admitted in a car accident. Where is she?\"\n\nMore growl in the tone.\n\nAdi takes shallow breathes. She can't protect Jenni. Hell, she can't protect herself.\n\n\"I am not required to answer your questions, now step out of my way, or I'll call hospital security.\"\n\nAdi closes her eyes against the human female's bravery. She would make a great Were. Jenni reminds Adi of Cyn, with a filter.\n\n\"Hey!\" Jenni suddenly yells.\n\nA hand touches the body bag, and Adi tenses. Her eyeball rolling to see through the tiny gap.\n\nShe can smell Lycan male. A shadow looms, blocking out what little light pierced the small opening.\n\nOh moon.\n\nShe's never wished for Slash more than she does in that moment\u2014or hated him more.\n\nThe zipper begins to slide away.\n\n# CHAPTER TWENTY-ONE\n\nSlash\n\nThe landscape blurs as Slash charges expertly between branches that reach for his sprinting form.\n\nHe only slows when he needs to gather more of his mate's scent. When he picks up a cross-scent, he grinds to a halt. His talons leave deep gouges in the forest floor, and the hem of his athletic pants are drenched with moisture and dirt shot up from running. He re-ties the string at the waistband. His hands are awkward in this form, but Slash maintains it stubbornly.\n\nHe's not fucking around with his sense of smell. It's inferior in his human form, and he can't afford delays. And, unlike the females, he doesn't have the benefit of the quarter-change. It's wolfen or bust.\n\nThree Lycan males have crossed Adi's scent trail. A low growl rumbles through Slash, breaking the seal of his lips as a howl shivers at the edge of his tongue.\n\nWhy are they here? This is not a claimed territory by any Were. The Lanarre hold the Hoh Rain Forest, and all other packs are south and west of this area. They were asking for trouble, wandering uninvited in foreign Were territory.\n\nSlash's brow furrows as he sinks to his knees. Lowering his face to barely an inch above the obvious footprints of Were in wolfen form, Slash closes his eyes, inhaling deeply.\n\nThe smell causes his blood to run hotter. An instant erection hardens, throbbing painfully between his legs, and his skull aches. Teeth lengthen automatically as his body burns to be his wolf.\n\nSlash springs to his feet. A howl bursts from his throat as if torn and bloody. He feels his eyes go to those of his wolf, simmering inside the bones of his face. He holds back, knowing his restraint is not limitless, and tries for impossible calm.\n\nHis nose isn't perfect, but it's never made him a liar. Slash's heightened sense of smell tells him the horrible news:\n\nAdrianna is in heat.\n\nTheir time together did more than cement their lifelong relationship. And if he knows her cycle has begun, the other three who stumbled across her scent trail know it, as well.\n\nSlash hesitates, deliberating his odds of saving her from an uninvited mating and sentencing them both to death.\n\nIt takes him about three seconds to decide.\n\nI'll have to take my chances. Three to one is better odds than when Tramack and his people came and caught the two of them off guard.\n\nSlash pivots, crouching slightly, then leaps forward powerfully, spraying leaves, fir needles, and moist earth behind him.\n\nHis revolving wolfen eyes assist him now, sweeping his periphery as his blinding speed causes the forest to be a green smear that flanks him. Slash crashes through a creek. Hitting the other side, he stops dead.\n\nToiletries are scattered about, standing out like beacons of civilization in the middle of nature. Slash scoops up a toothbrush half-buried in the muck of the creek and raises it to his nose. He salivates as his beast rises in recognition of his mate\u2014and her heat.\n\nThe only thing his wolf wants to do is breed Adrianna. That is the very thing any other male werewolf will want to do.\n\nSlash must get to her first.\n\nHe tosses the toothbrush to the ground. He studies the ravine rising away from the creek bed. The distant hum of human vehicles rushing beyond his line of sight emits a low mechanical purr.\n\nSlash vacillates. Why would Adrianna head toward the highway? He drops to his haunches, scanning the dirt carefully. Many tracks obscure Adi's. Her shampoo bottle is crushed. The contents are a vile blue against the rich earthy colors of the forest. An altered paw print has stomped the bottle. Talons have punctured the stout tube of plastic. Slash sucks in a breath.\n\nThey pursued her. Running.\n\nAdrenaline surges. His attention hangs up at the other side of the broad creek. He furiously imagines what might have occurred.\n\nHot syrupy fear pumps through him as pins and needles bloom within his body. He sees the scene in living color. Adrianna is grooming. His eyes move to her filthy toothbrush and the burst shampoo bottle. Then his focus turns to a point directly across the water.\n\nThey saw her.\n\nWhen Adrianna noticed them, she ran. He touches the smaller altered paw print. It's smeared as though she pivoted. Hard.\n\nSlash stands, clenching and unclenching his fists.\n\nThe hold on his wolf is tenuous. Slash must make time to settle his beast, or he runs the risk of exploding his wolfen form and going to Were. And that, he doesn't need. Three shifts in fewer than twenty-four hours is an energy zapper.\n\nHe can't protect Adrianna if he's weak. He can't protect her if they have her already.\n\nMoon, help me.\n\nSlash inhales deeply, sprinting to where Adi's trail ends at the edge of the road, and gazes at the center of the highway. The beast whimpers. His eyes burn, and his vision doubles to his own and that of his beast's.\n\nAll he sees is her blood. All his beast scents is the remnants of her injury. A car whizzes by, and Slash staggers to the middle of the two-lane freeway, oblivious to the traffic.\n\nScreeching tires slice his eardrums like razor blades. Slash realizes he's slipped back to his human form automatically. It doesn't matter. Adrianna's blood fills his nostrils. What she did destroys his mind.\n\nShe threw herself into traffic.\n\nSlash doesn't howl. He leaves the catastrophic proof of Adrianna's accident and flies to the shoulder at the other side of the freeway.\n\nGlass tinkling and metal kissing metal enters Slash's dull consciousness. Drivers exit their cars, hands in fists. One look from Slash, and they slow their progress toward him. His eyes burn with the change, his body fighting to complete the form alteration.\n\nHis revolving stare narrows to slivers as they draw nearer in challenge. They study him. The human men have just enough primal instincts of self-preservation to get their asses back into their vehicles.\n\nHe dismisses them.\n\nSlash is Red. He can track.\n\nBut Adrianna's scent disappears from the spattering of blood covering the yellow dashes of the highway like an escaped ghost.\n\nSomeone took her.\n\nThe males?\n\nSlash paces the shoulder as traffic increases with the hour and the abused cars that tried to avoid him roll to the shoulder. Humans exchange insurance information, casting dark looks at Slash.\n\nThey look away when his eyes meet theirs.\n\nThe wail of sirens approach, growing louder with each whoop.\n\nHis head snaps up at the noise the humans can't hear yet.\n\nOf course. The hospital.\n\nHope floods Slash as he tracks the ambulance speeding toward him. A grin of sheer pleasure overtakes his face, and he plunges into the woods. His body streams to wolfen as he paces the ambulance. His speed tops out at the same rate of motion as flashing strobes of red and blue splash across the highway.\n\nWhen the rioting ambulance slows, so does Slash.\n\nChest heaving, he peers through the stiff branches of the forest that borders the brightly illuminated letters of the Olympic Medical Center.\n\nHis vision closes to a tunnel. He morphs to human form and steps from the woods. His pants float to his ankles, revealing his naked form to anyone who might notice.\n\nFuck.\n\nWith an impatient sigh, he jerks them up, re-cinching the drawstring, and strides toward the ER wing. The ambulance pulls up to the doors, and a medic hops out, opening the double doors at the back. His partner is inside with the patient, and the driving medic grabs hold of the handle at the foot of the gurney.\n\nSlash intercepts them neatly, never breaking stride. \"Hello.\"\n\nThe medics turn like a pair of startled birds. Their hands fall away from the gurney.\n\nSlash's nostrils flare, scenting the patient. \"He's dying. Don't bother.\" Then he turns and slaps the solid stainless handhold at the end of the gurney and shoves the whole thing back inside.\n\nLet's not have that in the mix.\n\n\"Hey!\" says the medic who drove. \"Get your hands off the fucking cot!\"\n\nSlash rotates his neck. \"Just did.\" Slash has a touch of regret as he clocks the medic, checking his swing at the last minute so he doesn't break the human's neck. There is no honor in killing a healer.\n\nAs the medic crumples, Slash catches him underneath the armpits and easily hefts him inside the ambulance, lying him beside the dying patient.\n\nThe other medic, a male who obviously sees gym time, rushes him. Were don't do gym time.\n\nSlash grabs the medic, jerking his forearm around the man's neck and hauling him against his chest.\n\nThe man bucks and writhes. He's very strong, for a human.\n\nSlash is at least four times stronger, though, even in his human form. \"I won't kill you if you answer one question.\"\n\n\"What? Fucking freak!\" he squeaks.\n\nSlash begins to wring his neck.\n\nThe medic gives a hoarse bark.\n\n\"A female came here today, young\u2014badly injured.\" Slash's heartbeat thunders like a herd of horses. He presumes Adrianna is here.\n\nBut that might be wishful thinking. Slash has loosely strung together the dots like the fibers of the finest spiderweb, waiting to be shredded.\n\n\"Yeah!\" the medic screeches, fruitlessly clawing at Slash's forearms. \"Early twenties, broken ribs and pelvis. Collapsed lung. Internal bleeding.\"\n\nSlash closes his eyes in relief. These were the responders. \"This the ambulance you used for transport?\"\n\n\"Fuck you!\" the medic bellows.\n\nI don't have time for this. Adi doesn't have time.\n\nSlash's forearm tightens.\n\n\"Yes!\"\n\nSlash drags the medic by the neck and pokes his head inside the ambulance. Mentally shelving the sight of blood flowing out of the now-dead patient, he ignores the other medic's fresh injuries.\n\nHe searches through the fine layers of scent: death, blood, injury, antiseptic, and drugs.\n\nBeneath those, Adrianna lingers. He'd missed it before, while concentrating on extracting information.\n\nSlash squeezes in a hard burst, cutting off the artery that feeds oxygen to the medic's brain. After thirty seconds of frantic thrashing, he goes limp.\n\nSlash carefully moves him inside the ambulance, and before laying the man beside the first medic, he strips the medic's uniform and puts it on himself. The fit isn't bad\u2014the human was large\u2014but the pant legs are too short. Slash closes the doors of the ambulance.\n\nTurning, he notices his fingers are shaking with tension and unspent adrenaline.\n\nAdrianna is near.\n\n*\n\nThe human herd flows past him within the sterile confines of the hospital, moving in opposing directions, never making eye contact.\n\nPerfect.\n\nSlash sights the nurse at the receptionist desk and approaches. Smooth, austere laminate in a bleached ivory tops the countertop at a height that reveals only brown eyes staring above an oversized computer screen as he approaches.\n\nHe doesn't have much time. Soon, those medics will wake up. Then they'll get attention\u2014the kind of notice Slash doesn't need.\n\nThe nurse's smile fades when she takes in Slash's face up close. The scar is a bisecting ripple of flesh that makes his features look harsh and unforgiving.\n\nSlash is unforgiving.\n\n\"Hello,\" he begins, having covertly flipped his badge in the wrong direction, hiding a face that doesn't remotely resemble the medic he just put into la-la land.\n\n\"Hi,\" she says hesitantly, shoring up into the shell of politeness expected of humans.\n\nGood girl.\n\n\"A family's arrived, asking questions about an admit.\"\n\nHer wide eyes narrow. \"Oh, for a patient?\"\n\nSlash gives a shallow nod. \"Yes. We had an early transfer in today\u2014car wreck, young woman, early twenties?\"\n\nBeautiful. Vulnerable. Unprotected. Slash swallows hard, feeling his Adam's apple like a resistant plow of flesh.\n\n\"Oh! Yes. She's Jenni's. Third floor.\" The nurse glides her mouse over a pad. She looks up at him, missing the scars now, seeing him as a person.\n\nSlash tries on a smile.\n\n\"Room 313.\"\n\n\"Great,\" Slash says then turns. \"No name?\"\n\nThe less they know, the better.\n\nShe shakes her head. Silky, fine brunette hair floats over her shoulders. Her neck is fragile, so easily smashed. Slash shoves his violent thoughts away. His emotions are really riding the edge of sanity. \"Unknown,\" she replies, her eyes drifting back to the huge screen.\n\n\"Thanks,\" he says, dashing away.\n\nSlash strides to the stairs. They'll be faster than the elevator.\n\nHe takes them two at a time. Looking first left then right, he swiftly ascertains odd numbered rooms are to the left.\n\nTrying to appear circumspect, he moves to room 313 and opens the door.\n\nEmpty.\n\nSlash enters, letting the door swing closed behind him. Moving to the bed, he can scent Adrianna everywhere. Her heat cycle is in full swing, and his erection grows like a plank of wood between his legs.\n\nHis mate is in need of breeding, and her scent saturates the hospital room, along with her injuries. The combination of smells puts his beast aggressively en pointe. An immediate headache sinks its teeth into Slash's temples.\n\nMoon, let me solve this. Quickly.\n\nHe lets the edges of himself go almost wolfen, balancing on the precipice between human and wolfen. The flush of the nearby change eases around him. His eyes sharpen, and his nose takes in every scent. He can only hold this balance between form for seconds.\n\nA human female's scent mingles with Adrianna's.\n\nThe human is diseased.\n\nSlash frowns. Maybe it's all the death in this place, and he's misreading the scents. He glosses over the human woman's scent, concentrating on Adrianna's. She's healed, but not fully.\n\nShe needs him.\n\nSlash doesn't smell the males. He smells a new trail: Adrianna's mixed with the human female's.\n\nSlash levels down, shoving his beast back in the box for now.\n\nHe leaves the room, and with an almost soundless growl, he flares his nostrils, hiking his face up.\n\nThere.\n\nSlash's face whips in the opposite direction from which he came. A sign glows in eerie white at the end of the hall.\n\nMorgue.\n\nSlash blurs to the basement of the hospital, narrowly missing two orderlies and a nurse. He's unsure what Adrianna's up to, but she's a clever female Alpha. She might already be aware of her pursuers and hatching a half-baked plan.\n\nA reactive plan because I wasn't there for her.\n\nSlowing down to appear human is seconds of torture as he delays his entry because of hospital employees. Finally, the humans pass, and Slash moves forward again.\n\nThe double doors of the morgue rise before him like a yawning stainless steel mouth.\n\nSlash slaps the doors wide, and they swing apart.\n\nThe smell hits him. Decaying flesh, tendons, and gore greet him like a punch to the gut.\n\nThings were done. Glass jars stand open, with the decaying stew of humanity inside.\n\nSlash is untroubled, except for the odor that mixes so exactly with the human sludge of the place.\n\nAdrianna.\n\n# CHAPTER TWENTY-TWO\n\nTessa\n\nLaz trails heat and wetness in a sultry path from her mouth to her neck. His large hands cup Tessa's shoulders as he draws her closer to him.\n\nHis head sinks between her breasts, placing tender kisses like heated sweet air.\n\n\"Laz,\" Tessa says, tipping her head back.\n\nHe catches it with his free hand, his other sliding to the small of her back and pressing her hips against his.\n\nThe erection that separates them makes Tessa open her eyes. His nearly translucent gaze meets hers, hooded and darkening with desire by the second.\n\n\"They're right outside, Laz.\"\n\n\"I do not care.\" His lips press against hers, sucking her protestations into his hot mouth.\n\nTessa groans. It's not right. She can't give herself to a demonic when there are Lanarre right outside a prison disguised as a guest cottage.\n\nShe flattens her palms on his chest and pushes.\n\nLaz releases her instantly.\n\nTessa stumbles backward, eyes wide.\n\nLaz gazes at her coldly. The warm storm clouds of his gaze from moments before have chilled to hardened sleet.\n\nTessa tries to recover. \"I'm not responsible for saving you, Laz.\"\n\nHe nods then turns away, running a finger along a table holding a solitary lamp glowing softly in the room.\n\nSteam rises from his light touch on the wood, and the outline of his fingertip burns into the wood.\n\n\"Oh, moon,\" Tessa says in a hushed voice. Her hands cover the lips he just ravished, and Laz's stare burns through her. \"Are you? Did you mean to do that?\"\n\nLaz gives casual attention to the tabletop. Small spirals of smoke from his nose, mouth, and ears drift into the space between them. \"No.\"\n\n\"I can't do this, Laz. I'm a Were. You're a demonic. We're not even the same species.\"\n\nAm I convincing him? Or am I convincing me?\n\nMaybe a little of both. Tessa folds her arms, daring Laz to contradict her logic. She's right, and she knows it.\n\nLaz doesn't contradict her. What he asks her next throws her.\n\n\"Have you ever mated with a Were?\" His words are soft, but if fire could speak, she would be aflame.\n\n\"No.\"\n\n\"And have you asked yourself why?\" These words are as softly spoken, just as deadly.\n\nTessa's stunned. She's never thought about taking a male. Her last two heat cycles were agony. She sequestered herself in caves, hoping to relieve herself. But she couldn't. Only Lycan seed would salve the wound of her need.\n\nThankfully, she had a precursor that her cycle would be making an appearance, and she found isolated shelter.\n\nNow her heat cycle is just beginning\u2014and there is no shelter.\n\nTessa gazes at Laz then shuts her eyes tightly.\n\nHe would be my shelter if I could let go of my racist bullshit. Tessa's hands fall to her sides, and she clenches them into fists then releases them.\n\nTessa understands her feelings run deeper than mere racism that's keeping her heart in stasis. Males can't be trusted.\n\nThey dominate. The scars of her past won't heal. Won't let her live.\n\nIf she chooses Laz, every Were they stumbled across would challenge him. And what kind of young would they have?\n\nHotdogs, that's what. A mournful laugh escapes her lips.\n\nI can't breed him. I can't stand anyone else breeding me, either.\n\nShe will have to leave Laz to save him. To save herself.\n\nTessa is not his redemption; she is his undoing.\n\nShe opens her eyes.\n\nLaz is gone.\n\n*\n\nDrek\n\nHe softly shuts the heavy wooden door that signals the opulent entrance to his home.\n\nTwo century-year-old-growth logs bisect one another to form a twenty-foot ceiling. A chandelier that once held candles has been electrified during the past fifty years. The lights glow softly, illuminating an angry face, and another as seductress.\n\nTahlia glares at him, arms bound behind her back, encircling a large load-bearing wooden pole.\n\nThe other female, Tanya, strikes a sultry pose. Her floor-length dress is artfully swept across her form, highlighting her classic hourglass figure. The thin material sinks at the V of her thighs and wraps breasts that strain against it. In the cool air of Drek's antique log home, her nipples pebble.\n\nTahlia turns away from him. Her long, spiraling curls of ebony drape her face, partially obscuring it from his scrutiny. The tips brush along hips that flare, but are still not as filled out as Tanya's.\n\nDrek is alarmed with how young Tahlia appears, yet his gaze is uninformed, loving every inch of her body.\n\nWhen he reaches her eyes, they are deep midnight fire of hate.\n\n\"Release me,\" she says in curt command. \"I will return to my home in the Redwood Forest, and you shall be free of your obligation to mate with me.\"\n\nHer eyes tell him how much she does not want to be with him.\n\nDrek sighs, turning his attention to Tanya.\n\nHe plants his feet wide, crossing his arms. \"And you have arrived in my absence.\"\n\nShe inclines her head. \"Yes, my prince. I am your rightful chosen, and it is my cousin who tries to force her way into a relationship that is not hers.\"\n\n\"You lie,\" Tahlia says, jerking her arms against the post of a huge Douglas fir pillar that serves as both beauty and function.\n\nTanya has every attribute Drek finds comely in a Lanarre female. She is docile. Her hips are wide, her waist narrow.\n\nYet, he is drawn to Tahlia.\n\n\"I relinquish my rights to you, Drek,\" Tahlia says suddenly, watching him think. \"This lying cousin of mine can have you. I do not care.\" A tear slips out of her eye.\n\nTanya smiles triumphantly, and Tahlia looks away. Tahlia doesn't have hands to wipe away her sadness, and her grief pierces Drek.\n\n\"They're gone. My human guardians died at the hand of a ruthless, insane Were. I will return to my pack and be an outcast, rejected by my chosen.\" She raises her head defiantly, and Drek's chest tightens.\n\nThe hold she already has over him tightens. The new feelings anger Drek. He's accustomed to moving through his life decisively. His feelings of ambivalence and passion have stalled his mind.\n\n\"Ask me if I care?\" Tahlia entreats softly, and Tanya makes a small noise of annoyance.\n\n\"Do you care?\" Drek asks, as though against his will.\n\nShe vehemently shakes her head, the curls that frame her face bouncing. \"No. They can hate me. But I will not mate with a male who doesn't believe his nose. Who restrains his chosen because, as Tessa says, I speak my mind. I question your authority. I am who I was meant to be.\"\n\n\"Let her go, Prince Drek,\" Tanya says in a silky voice. \"I want to be your chosen.\"\n\nDrek turns to her, and Tanya shifts her weight, subtly crossing and uncrossing her legs.\n\nAny Were female understands that to release the smell of their sex in close proximity to a male is a sign of submission\u2014and invitation.\n\nDrek wrinkles his nose. He loves the scent, as any male would, but her motivation is beginning to wear thin on him.\n\n\"Address me as Drek,\" he commands in a curt voice.\n\n\"Yes, Drek,\" she replies in a purr.\n\nHe restrains his frustrated exhale.\n\nFemales.\n\nTanya is certainly easier.\n\nBut she is not the one. Drek is intrigued by how easily Tahlia dismisses him. A Lanarre prince would never harm a female. Technically, he has not harmed Tahlia. Though some of his guard were rough with her. I will deal with that later. Harshly.\n\nDrek would also explore the connection he feels toward her.\n\nIgnoring the pouting Tanya, he approaches Tahlia.\n\nHer head hangs. Small arms are wound around the log, tied off with gentle bindings of cloth.\n\n\"Tahlia.\"\n\nShe refuses him, keeping her nose pointed at the ground.\n\n\"Look at me.\"\n\n\"No.\" Her voice is low. Resonate. Absolute.\n\nDrek's lips quirk. He forces her chin to rise with a finger. He is so much stronger that if she doesn't want an injury, she must allow the motion.\n\nLiquid eyes full of grief drown him with her anger and choke him with despair.\n\nHe is unaware that his thumb has been absently stroking her cheek, while hazy tears drift down her dusky, smooth skin. They line the inside of his finger, gathering against the flesh dam.\n\n\"Why do you resist me?\" Drek asks, cocking his head.\n\nShe looks at him. Through him.\n\nDrek's eyes move to her mouth. It is full, with a pronounced cupid's bow. He tips his mouth to hers, hovering above the plump flesh as if summoned.\n\nHeat flushes to the surface of his skin, and he feels himself stiffen. Tahlia pulls him like a magnet, and like before, he is helpless to stop the draw of all his senses.\n\nA low moan escapes Drek, and he eases the distance.\n\nHis mouth covers her unresponsive lips, moving over the pliant hills and valleys of her lips. They part, and his tongue finds its way inside her. He groans, sweeping his hand behind her skull to touch silky curls that spring and collapse beneath his touch. Then her hair changes beneath his fingers.\n\nTahlia's mouth grows hard like bone.\n\nDrek breaks away in surprise.\n\nA bird soars above their heads, searching for an escape as bindings flutter to the ground.\n\nHer caw pierces the silence of the room.\n\nTanya screams, and Tahlia's snowy-white form dives for the fireplace.\n\n\"No!\" Drek bellows into the stillness, his lips as swollen as his cock, his heart a fist of dread.\n\nHis command mingles with the frightened ending note of Tanya's shriek.\n\nThen Tahlia is gone.\n\nDrek strides to the entrance door and realizes he's holding something in a tight grip.\n\nHe opens his hand.\n\nA pure-white feather spirals slowly to the ground.\n\n# CHAPTER TWENTY-THREE\n\nJulia\n\nJulia surveys the landscape.\n\nThe trees reach toward the small lake as if asking forgiveness. There's no outward sign that the dead, put there by the fey, line the shores.\n\nOnly the most sensitive shifters or a Singer Tracker would catch the scent of so many who'd perished.\n\nBut the memory will live on forever in her mind.\n\nScott slides his arm around her, and Julia's body eases. Their connection goes slack once touch is established.\n\n\"I guess it's time to get serious and hunt everyone down.\"\n\n\"I think things are plenty serious enough,\" Scott comments, a crooked smile on his face.\n\nJulia barely nods. There's something she has to do first. She places the flat of her palm on Scott's broad chest. His dark serious eyes hold hers. \"I need to see where Jason is.\"\n\nScott pulls her close. \"I wasn't a fan of Caldwell. And you can feel that, Julia. I can't hide it. But like I said, I appreciate what he did. What he sacrificed for us.\"\n\nJulia squeezes her eyes closed. \"He never got a chance, Scott. We had a different path in Alaska. Then those things got ahold of us and never let go. They killed Kevin, ruined my life, screwed up Cyn's\u2014hell, it's been nothing but a mess ever since.\"\n\n\"Not everything's a mess, Julia.\"\n\nShe opens her eyes, looking deeply into his. \"I don't mean I don't love you\u2014that we're not meant to be. I always got that on some level from the moment our hands touched. It's just that so many people have been sacrificed along the way. That's what I object to.\"\n\nScott traces a thumb over her lip lightly. \"I know,\" he says softly, bending to kiss the tip of her nose.\n\n\"Ready?\" Cyn asks from behind them.\n\nThey part, and Julia lets her fingertips trail across his muscular chest.\n\nHe grabs her hand at the last moment and squeezes her fingers. \"I'll be around.\"\n\n\"Duh,\" Cyn says, winking. \"Being all Combatant and that.\"\n\nJulia walks with Cyn toward the lake. \"He's not with\u2014\" Julia can't finish, hoping that he's not part of the mass grave she allowed.\n\n\"Nope,\" Cyn looks up quickly, kicking a rock with the toe of her bright-red Converse tennis shoe. \"He's in his own spot.\"\n\n\"Good.\" Julia releases a shaking breath.\n\nThe rest of the short hike is made in silence\u2014not the silence that's awkward and weighty, but the comfortable silence between longtime friends.\n\n\"This is exhausting,\" Cyn finally says as they reach the border of conifer trees. Feathery hemlock branches sweep toward them.\n\nRich western red cedar fills her nose, and Julia breathes deeply, fortifying herself for what's to come.\n\n\"Don't think so hard,\" she says softly.\n\n\"Sorry, I forget you're getting a bunch of feedback.\" Cyn looks up toward the sky. Fluffy cotton ball clouds drift by, obscuring the sun. \"I just can't help but miss his cantankerous ass.\"\n\n\"It's not that,\" Julia says, following Cyn as she begins to weave through the branches, holding them aside for Julia. \"It's the past that we miss.\"\n\n\"This new life,\" Cyn says, shaking her head, \"I mean\u2014what the hell? I'm a Were girl and a Singer Healer.\" She gives a defeated little shrug.\n\nJulia knows exactly how she feels.\n\nShe pops a palm up, narrowing her sights on Cyn. \"Don't even get started. I can feel your regret and loneliness for what was. And I can't do it, Cyn. I have to let go of this stuff. No matter how much we want what we had, it'll never be our \u02bbnormal' again. Kev is dead.\" Julia looks to a point ahead of them, and Cyn follows her gaze. \"Jason is dead. It's over.\"\n\n\"I know,\" Cyn says, tucking her hair behind her ear.\n\nJulia checks out her multi-colored locks and laughs. \"Nice roots.\"\n\n\"Fuck off,\" Cyn says, but she's smiling. \"Like these guys have a good box of bleach in this place?\" She tweaks a strand of dirty-blond hair with platinum ends. \"Pfft!\"\n\nJulia shakes her head, and they smile at each other. Some things are the same, and for that, Julia's relieved.\n\n*\n\nJulia sucks in a harsh breath. \"Wow.\" She smiles back at Cyn, who's leaning against a tree trunk, her leg bent at the knee. The sneaker is a bright spot of color against all the furrowed brown.\n\nThe grave is fresh.\n\nGrief washes over her, and Julia sinks to her knees. Recently churned dirt covers a nine-foot plot. The dirt is deep brown, like crushed toast. Julia dips her fingers in and runs the clumpy dirt through her fingers. When she can finally tear her eyes away, her gaze travels to the head, to a small wooden plaque carved simply with Jason's first and last name.\n\nUnderneath that it says Singer in elegant script.\n\nCyn's hand falls on Julia's shoulder, and she grips it like a lifeline.\n\n\"Ya know, if I weren't a Were girl, you'd have busted my fingers.\"\n\nJulia nods.\n\nBut she's not. Cyn is supernatural. Like Julia. Like everyone in this new life.\n\n\"Help me, Cyn.\"\n\nCyn squats beside her. Julia takes off a necklace she's been wearing for a long time. She pulls the thread of gold through the perfect circle of a wedding band she never wore.\n\nTears form in the corners of Cyn's eyes. \"God, this sucks.\"\n\n\"Yeah,\" Julia agrees. She studies the temporary plaque. \"I don't want to forget what he was to me.\"\n\nCyn nods. The silence stretches for more than a minute. \"Did anyone mention it'll be stone?\" She points at the piece of wood.\n\nJulia nods.\n\nShe tips her head back, looking up at the tops of the trees. Dappled light strikes the raw dirt all around Julia, turning it to milk chocolate. But nothing about this death is sweet. Tricks of light won't take away the sting. It won't bring him back.\n\nNothing can right the wrong of the void left by Jason's absence.\n\nJulia loosens her hand around the ring that warms the inside of her palm. She stares at the circle of gold, and re-clasping the necklace, she loops it around a portion of the irregular border of wood. The fine gold chain catches, and the wedding band dangles, flipping back and forth.\n\nThe circle of the band sways, falling exactly over the O of his name.\n\nJulia's lip quivers. \"I'm so sorry, Jason.\"\n\nCyn pulls her off the grave, wrapping her arms around her best friend. \"It's gonna be okay, Jules.\"\n\nIn that moment, Julia feels like it'll never be okay again.\n\n*\n\nScott gives Julia a sympathetic look as she and Cyn exit the forest. He stands from his perch on one of the boulders that appear to grow right out of the ground, separating lake and woods.\n\n\"What?\" Julia asks, blotting her tears with her shirt sleeve.\n\n\"I know the timing's not good, baby...\"\n\n\"It's never good when the endearments start cropping up,\" Cyn remarks dryly.\n\n\"Anyone ever told you you're a pain in the ass?\" Scott says, frowning.\n\n\"Sure.\" Cyn shrugs.\n\n\"Guys,\" Julia says.\n\nCyn crosses her arms. \"Fine.\"\n\n\"Everyone knows about us being together now.\"\n\nCyn gives her a look of pure satisfaction then turns to Scott. \"Way to drop the bomb, stud-meld.\"\n\nJulia's face heats.\n\nScott scowls at Cyn.\n\n\"Did you tell people, Scott?\" Julia asks, instantly remembering every kiss, lick, suck, and everything else\u2014right after visiting Jason's grave.\n\nJulia covers her face as shame pours through her like acid.\n\nScott strides to her.\n\nJulia feels his thoughts, emotions\u2014and love. \"Oh,\" she says softly.\n\nCyn says, \"I don't have the two-way radio in my head. Spill it.\"\n\nJulia looks at Cyn. \"We have consummated\u2014\"\n\n\"I know,\" Cyn says.\n\nJulia scowls. \"Aura readers have seen Scott...\"\n\nCyn's jaw moves side to side. \"Ah. Gotcha. You guys did the deed, and now you're all rainbow.\" She slaps her hands together, knitting her fingers.\n\nJulia's chin dips. \"Something like that.\"\n\n\"Not a helluva a lot of privacy around this joint.\"\n\nScott laughs. \"None. But everyone's in the same boat. If someone can't read your mind to find out your situation\"\u2014he slides a glance Julia's way and her cheeks become pink again\u2014\"then there's an Aura reader who can out you.\"\n\n\"Being a Singer is serious mojo,\" Cyn comments thoughtfully. She taps her chin. \"Kinda glad I just fix people when Humpty Dumpty's men can't put them back together again. That telepathy shit sucks.\"\n\n\"That's fucking random, Cyn,\" Scott says.\n\nJulia takes a deep breath then lets it out as though her body's in slow motion.\n\nScott draws her close. \"You okay?\"\n\nShe nods, gives a small smile to Cyn, and returns her attention to Scott. \"You know how I feel.\"\n\n\"Yeah. But it feels more right to ask anyway.\" Julia never feels anything but right in Scott's arms. He pulls away slightly and places his palm above the mark at her belly. \"How's this doing?\"\n\nJulia shrugs. \"It doesn't bother me so much now.\"\n\nScott's expression is troubled.\n\n\"That's from Tony's little demon saber, right?\" Cyn asks.\n\nJulia nods.\n\n\"And it was given to him by that prick Praile, the demonic who\u2014\"\n\n\"Yeah,\" Julia interrupts quietly. \"The one responsible for killing Jason.\"\n\n\"And almost killing you,\" Cyn reminds her.\n\n\"Maybe not.\"\n\nThey look at Scott.\n\n\"What do you mean?\" Cyn asks.\n\n\"I mean\u2014why would Praile exert all this effort to get at Julia?\"\n\n\"Did you seriously just ask that question?\" Cyn asks, scrunching her nose. \"I mean, Julia's been tag-teamed since forever. And!\" Cyn nearly shouts. \"We did hand the demonic their ass.\" Her eyebrows rise to her hairline.\n\nScott chuckles, threading his fingers through Julia's and hauling her toward the mansion. \"Cynthia, you do have a way with words.\"\n\n\"What?\" Cyn grins. \"Praile's at the root of all our current woes, and I think he needs his dick pulled off. Cutting is simply insufficient. Not that I don't think Tessa did a stand-up job with her talons. Damn.\"\n\nScott stops in his tracks. \"You just say stuff like that? Do you ever think about how that would be?\"\n\nCyn smiles.\n\nJulia chokes back a laugh. \"Yeah, I think she really does, actually.\"\n\n\"Shit! Remind me not to get on your bad side.\"\n\n\"Consider yourself warned.\" Cyn waggles her brows.\n\nAmusement flows through their connection.\n\nScott narrows his eyes at Cyn. \"Nice friend, Julia.\"\n\n\"Hey,\" Julia says, socking him with her free hand, \"she kicks ass and takes names.\"\n\nScott shakes his head. \"No, she kicks ass and decapitates dicks.\"\n\nJulia bends over, clutching her ribs, and Scott releases her hand, putting his hands on his hips. \"Not funny, really.\"\n\n\"So funny!\" Julia cries, swiping tears of laughter.\n\n\"See?\" Cyn says, throwing her palm toward Julia. \"She appreciates my humor.\"\n\nHe scoops Julia up, tosses her over his shoulder, and swats her butt.\n\n\"Scott!\" Julia gasps, beating on his back, \"Put me down.\"\n\n\"No, ma'am, You're going to get proper discipline after our nuptials.\"\n\nJulia stills, her fingers spreading flat on his back. \"What?\"\n\nCyn jogs beside them. \"That's why Scott was looking all constipated earlier. He had to tell you the news. Now that One knows you guys figured out your love life, you get to be married. I mean\u2014what's stopping you?\"\n\nScott grips Julia's hips, and she slides down his front.\n\nGuilt, desire, and uncertainty cloud her mind. Everything's moving too fast.\n\nJason's dead. The demonic are still tearing around in parts unknown, and her people are fewer. Those who remain are scattered and unsure.\n\nScott cups her face in one large hand. \"I feel all your anxiety. But here's the thing, Julia\u2014do you have anxiety about you and me?\"\n\nCyn remains blissfully silent for once.\n\nJulia searches her heart. Each piece of her psyche holds tension, except one.\n\nThe soul-meld is the only intact thing about her right now. And no matter how her future might change, her supernatural connection with Scott holds.\n\n\"Right,\" Julia answers, noting Scott's tight eyes and tense posture. \"We feel right.\"\n\nHis fingers move from her chin and wrap the crown of her head. \"Then let's do something that's happy. There's been too much grief.\"\n\n\"I agree,\" Jen says, walking up to them, and that's all it takes.\n\nJulia's stoic veneer crumples, and her future sister-in-law holds her as she cries.\n\nNot all tears are sad.\n\nSome are cathartic.\n\n# CHAPTER TWENTY-FOUR\n\nAdi\n\nLazily spinning silver eyes peer inside the bag where Adi tensely waits. He wrinkles his nose.\n\nLanarre.\n\nAdi's surprise is chased by a crazy thought. Adi feels like Little Red Riding Hood, ready to be eaten by the big bad wolf.\n\n\"Hello, female.\" His hot breath bathes Adi, and her beast recoils.\n\nShe makes a split decision. Sliding to quarter-change, she rolls in an agile wiggle, beginning to fall from the stainless cot and away from Romeo.\n\nThe wheels aren't locked, and the cart rocks abruptly from her sharp motion. She bursts from the bag, landing with her ass still inside the crinkling plastic as her palms slap the concrete of the underground parking structure.\n\nAdi grunts from the impact of her knees cracking against concrete. Healing fire burns along her kneecaps, momentarily weakening her.\n\nThe cart is flung, and Adi watches it hit one of the circular support pillars before it lands on the roof of a parked car. A security alarm begins to blare.\n\nAdi moves forward, getting caught up in the plastic, and kicking backward to extricate herself.\n\nStrong arms encircle her waist, lifting her from the ground.\n\nTalons burst from her fingertips, and Adi blindly swipes backward. The Were roars, his hold momentarily loosening.\n\nAdi spins.\n\nJenni's wide brown eyes find hers, a talon at her throat.\n\nAdi's breath catches. The human female's fear is like ashes in her nose.\n\n\"Is this your pet, female?\"\n\nOh my Moon.\n\nAdi backs up until a cold pillar meets her spine. She shivers with fear, chill, and desperation.\n\nThe three Lanarre males circle Adi. The one in the center easily holds a struggling Jenni.\n\nAdi fights to track their movements as they separate. \"The human doesn't matter. I threatened to kill her if she didn't get me out of here.\" Adi shrugs. Small lie.\n\nThe Lanarre swipes a finger across Jenni's throat, and the skin peels away. White at first, it quickly fills with her blood.\n\nAdi's heart sinks. She'd rolled the dice, hoping they would just let Jenni go if they assumed she meant nothing to her.\n\nShe lost that bet. \"No,\" Adi whispers fervently.\n\nHe drops Jenni, who clasps her fingers around her own neck, gasping as she drops hard to her knees.\n\nBastards. Rage uncoils like a snake preparing to strike.\n\n\"We've done you a favor then, female.\"\n\nAdi stands straighter, flinging her claws wide. \"You are Lanarre\u2014the best of us, but you'll threaten me and kill an innocent human?\"\n\n\"A human who is dying,\" another scoffs with a dismissive chuff, coming a step closer.\n\nSo let's kill her slowly. Makes so much sense.\n\n\"That doesn't matter,\" Adi says, her anger chasing tears to dryness. \"You guys aren't top dogs in my book.\" Her sarcasm rings in the barren parking lot.\n\nTheir eyes spin a little faster. \"We do not come to threaten you, female.\"\n\nRight. \"Then let me go,\" Adi says in a low voice. \"I am mated to another.\" Though Slash doesn't give a fuck, it's still technically true. They must scent the truth of her words.\n\nMaybe they don't care?\n\nThe one who grabbed her before spreads his arms inoffensively, his half-smile smug. \"Then where is your mate? And what kind of male would leave his female unprotected while she cycles?\" One of his dark eyebrows rises.\n\nAdi's gaze skates back and forth between the three.\n\nA pool of Jenni's blood spreads beneath her body, coloring the concrete black. Adi sees red\u2014literally. She would prefer their blood covering the garage floor instead of her human friend's.\n\nLoyalty has always been Adi's greatest character flaw\u2014and her greatest strength. Adi's face whips to the males.\n\nThey can fucking try me. Adi clicks her talons together, and they sound like the bones of her enemies.\n\nTheir eyes move to her aggressive stance.\n\n\"We are Lanarre; we don't hurt females.\" He sounds so insulted.\n\nIs he high? Didn't he just slit Jenni's throat? Adi laughs. \"You'll just breed me. Yeah.\"\n\n\"Of course.\" The clowns look at each other as if seeking agreement in their attempt at rape.\n\n\"We will take care of whatever young you have,\" the Were who'd laid hands on her says. His smile hangs vacantly on his face. His teeth are very sharp.\n\n\"You're not listening, pal.\"\n\nHis eyebrows jerk up. If things weren't so serious, Adi would laugh. Nothing's funny right now.\n\nJenni's bleeding out on the hard cement like discarded trash, and three males are discussing a phantom future pregnancy as though she should be all flattered by their attention.\n\nBull honkey. \"Stay where you are,\" Adi says as they fan out, eyes never leaving her.\n\nGrabber puts his hand out, palm facing her. \"We won't hurt you. We simply want to take you back to our den.\"\n\nPlacating cheese dick.\n\n\"I am not unmated. You've got to be the dumbest male I've ever had the misfortune of running into.\"\n\nAt a single yip, the males at his side rushes her.\n\nAdi does the unexpected, using the balls of her feet as she leaps to Grabber.\n\nHe reacts swiftly, but not swiftly enough.\n\nAdi has one shot. She sails toward him, and he instinctively tries to catch her, exposing his throat.\n\nAdi goes wolfen midair, sucking in a heated, pain-filled breath, and claims his open neck with her teeth.\n\nShe tears out his esophagus like a glossy worm ripped from the dirt of his flesh. Jerking her head back, jaws locked, she swings her head around, releasing a chunk of his breathing tube behind her.\n\nGrabber gasps, flailing for purchase on her suddenly bigger form. She hops away, swings her leg, and plants a foot on his chest, shooting his ass against the pillar she'd stopped cowering against.\n\nHeal that, asswipe.\n\nPowerful hands grip her upper arms, and Adi thrashes. One hand holds her nape painfully tight, and she whimpers.\n\nHer eyes meet Jenni's.\n\nSlash, Adi wails inside her mind, knowing he can't hear her. She hates that she would want a male's protection\u2014and loathes herself because she thought she already had it.\n\nThe metal door leading into the parking garage bangs against the wall, shattering into twisted curly cues of steel. They skitter across the smooth concrete like metallic sleet, rolling to a stop at the edge of Jenni's blood.\n\nSlash moves through the jagged open mouth of what used to be the doorway.\n\nTheir eyes meet. His are already wolfen green. His utter calm as he surveys gasping Grabber and the two Lanarre who hold her terrifies Adi.\n\n\"Slash,\" her beast growls out of a throat raw with her anguish.\n\nBut the Lanarre don't give a shit about what she thinks. She's an alpha female running rogue who's cycling. So Adi's fair game.\n\nShe's betting Slash won't let them hurt her even though he doesn't want her anymore. She doesn't even have time to think about why Slash would even appear.\n\nThe Lanarre's heads jerk in his direction, their stub wolfen snouts rising to scent Slash.\n\nTheir grips tighten painfully.\n\nAdi bows her head, breathing through the pain of their hold.\n\nWords are unnecessary.\n\nSlash's wolf bursts his skin. Blood, flesh, and bone fly, covering a ten foot diameter.\n\nBits of Slash's human form land like gory rain on Jenni. She attempts to cover her face, but is too weak to do so. The remnants fall into her blood.\n\n\"Slash! No!\" He can't kill the Lanarre. It's a death sentence.\n\nThen he does.\n\n*\n\nAdi crawls toward Jenni. The throbbing sound of the car alarm has finally quieted, and she has a moment's consideration of how lucky they are that no one has come down into the underground garage. The short war felt like hours, but the time was likely only minutes.\n\nSlash is ripping up pieces of the males behind her. Ominous crunching, gnashing, and flesh rending is the only noise inside the space.\n\nHe told her what he was: a male who didn't deserve her because of his deeds.\n\nHe heaves a head, and it tumbles like a bowling ball alongside Adi. It rolls to a stop, sightless eyes appearing to track her slow progress toward the bleeding nurse.\n\nShe ignores those dead eyes that seem to follow her, continuing her plodding course toward Jenni.\n\nJenni reaches for Adi. Really, it just sort of falling toward her. Adi grasps, hauling the woman into her arms.\n\nJenni coughs, and blood shoots out of her mouth.\n\nAdi's tears fall on her face.\n\n\"Is that\u2014\"\n\n\"Don't talk, Jenni.\" Adi swallows her sobs like open wounds.\n\n\"Slash?\" Jenni whispers, her eyes vaguely turning in the direction of the sounds Slash makes as he tears the Lanarre to pieces.\n\nAdi nods, pressing her fingers to Jenni's lips. \"I'm sorry. I\u2014this is my fault.\"\n\nJenni squeezes her arm. \"Is okay\u2014\" She coughs up more blood. \"Dying anyway.\"\n\nTheir eyes meet.\n\nBoth devastated.\n\nGrowling erupts behind Adi, and she shields Jenni, quickly swiveling her body in the direction of the commotion.\n\nGrabber is healed.\n\nNo.\n\nSlash has shifted back to wolfen and is circling the Lanarre.\n\n\"I am Lanarre, Red. I order you to back away. You might live yet, despite your crime against Lycan.\"\n\nAdi's eyes laser on the scattered remains of the other Lanarre.\n\nSlash doesn't answer. He grabs the other Were by the throat and smashes his head against the pillar Adi had kicked him into.\n\nGrabber sets his teeth in Slash's shoulder, and Slash howls in agony. He severs Grabber's head with his talons. A string of gristle still holds the head, and it tips backward, hitting Grabber's lower back. The gore is a gut-wrenching sight, even for her, and Adi gasps, turning her attention back to Jenni.\n\nShe's hanging on, but Adi smells her death. She's a tough human. Adi smooths her dyed-black hair back from her forehead.\n\n\"Make me\u2014\" Jenni's gaze pleads with an unfinished question.\n\nAdi's eyes widen in realization.\n\n\"Like you.\" Jenni's bright brown eyes begin to dim then close.\n\nA thread of life lingers.\n\nAdi's beast feels that barrier between life and death. Her decision made, she closes her teeth over Jenni's throat, secreting the essence of what makes Adi Lycan directly into Jenni's bloodstream.\n\nAdi's so intent on what's she's doing, she doesn't realize Slash is there until it's too late.\n\n*\n\nSlash\n\nHis mate lies in a pool of human's blood. He doesn't know the state of her injury, but he intends to find out.\n\nSlash returns his attention to the last Lanarre male he would bring to true death. He knows the repercussions for this particular sin.\n\nAnd they are worth it.\n\nSlash was doomed from the moment he scented the males' lust for his female.\n\nThey will not have her, he thought.\n\nAnd then Slash had not thought much more. He did.\n\nHis eyes move to the disaster of what is left of the royal branch of Lycan, and he sniffs in disgust. So much for the caretakers of females.\n\nWithout another glance, he moves toward his mate. He stops when he sees what she does next.\n\n\"Adrianna!\" Slash roars.\n\nShe ignores him.\n\nHe grabs her gently and attempts to pry her off the dying female.\n\nAdi slaps his hands away.\n\nSlash could easily overpower her. She is a small female. But he doesn't want to force her.\n\nSlash wants to love her. Desperately.\n\n\"Adrianna\u2014don't condemn her. She is not born of us.\"\n\n\"Get your hands off me!\"\n\nShe turns to face him, lips bloody, teeth longer than a human's. Her hair is a rat's nest, and her eyes are filled with fear and anger\u2014directed at him, no doubt.\n\nStill, she is beautiful. Her scent wipes away rational thought.\n\n\"You lost the right to order me around when you told me to\u2014\" Her lips trembles, and Slash aches to kiss it. \"When you told me to \u02bbget the fuck away from you.' I believe those were your words.\"\n\nSlash winces at her reminder. But they have more important matters.\n\n\"And stop looking at me like that!\" she yells.\n\nSlash cannot help the way he looks at her and says the obvious, \"You're in heat.\"\n\nAdi rolls her eyes. \"Gee, Einstein, ya think?\"\n\nThere's no time for her willfulness at the moment. \"We need to get out of here. Now. Where there are some Lycan, there are more.\"\n\n\"Whatever!\" Adi says, gently depositing the human on the ground, and stands.\n\nShe pokes Slash in the chest. \"Those guys were Lanarre, Slash.\"\n\nAdi has a point, he admits to himself. \"I'm aware of that, Adrianna.\"\n\n\"Then why in the world did you kill them?\" she cries, her heart in her eyes. She smacks his broad chest then cradles her hand. \"Why, Slash? They'll hunt you for killing Lanarre. If I mean so little to you, why would you take the risk?\"\n\nSlash eyes her lips, full of blood.\n\nBeautiful.\n\nHe bends to hover over her mouth. \"Because there is no reason to take another breath\u2014unless you're in it.\"\n\nThen his mouth falls on hers, ravishing every bit of plump flesh.\n\nSlash tastes the blood of their enemies.\n\nBodies are scattered everywhere, the human female might live, die, or become a Were. But none of that matters. The female in his arms is all that's important.\n\nShe squirms.\n\nSlash reluctantly loosens his grip at her hips, searching Adrianna's face.\n\nWhen her hand strikes his cheek, he doesn't move away.\n\n\"Stop lying to me!\"\n\nSlash chuckles. Adrianna's still pissed at him, and rightfully so.\n\n\"Jerk!\" she screams in his face, touching the mouth he just kissed. His eyes go to her lips, and it's all he can do not to kiss her again. Her heat pushes at him, pummeling him.\n\nBut humans approach. \"I'm sorry, Adrianna. But you must know...\"\n\n\"I know that you're a dick. And that you slept with me then told me to leave. How could you?\"\n\nVoices can be heard not too far off. Their ears prick in unison.\n\nAdi's tears cloud her eyes. Slash thumbs a few off as they fall, then sucks them off his digit, tasting the salt of her sadness.\n\n\"Stop,\" Adi's voice quavers.\n\n\"Never.\" Slash cradles her face. \"I made you leave. I said what I knew would hasten it, Adrianna. Tramack could have come back for you. And then what would have happened? In my paralyzed and unhealed state?\"\n\nSlash sees in her face that she intuits the potential.\n\n\"So you didn't really want me to leave?\"\n\nHe hates how small her voice sounds. He put that tone there. Him. He tips his forehead to hers, sliding his fingers to her neck and fingering her skin gently. \"Adrianna, I love you more than the moon.\"\n\n\"No Lycan loves anything more than the moon, Slash.\"\n\n\"I am not any Lycan.\" Pulling away, he stares deeply into hazel eyes gone gold with her beast. \"I am the male who loves you.\"\n\nHumans appear inside the ruined doorway, take a look at the carnage, and start screaming and shouting.\n\nSlash takes her hand and pulls her from the garage.\n\nAdi tugs, and he turns to see why she's stalling.\n\nHer eyes go to where the human female was, but she is gone.\n\nSlash captures Adi in his arms and runs.\n\n# CHAPTER TWENTY-FIVE\n\nTessa\n\nShe thought he'd gone, but Laz had visually tortured her with how quickly he moved.\n\nHe stands before her in a blur of red flesh.\n\nLaz smooths his hands over Tessa's breasts, kneading them, and her core pulses once, aching for more. He seems to sense her need, dipping his hand to cover her mound.\n\nWhen he slips a finger over her yoga pants and between her folds, she gasps then groans, pushing her hips against his seeking finger.\n\n\"Laz,\" she says against his mouth.\n\n\"What may I do to ease you?\"\n\nOh Moon, make love to me until I can't walk. Tessa says, \"I want to do this\u2014I do, but we're in this place with all the Lanarre, and they know I'm in heat and\u2014\"\n\nHe presses his wandering fingers to her lips, and she kisses the tips. They're like bathwater against her flesh. \"Then let us leave. We will escape this place, and no one shall rule us.\"\n\nTessa wants that.\n\nShe wants him.\n\n\"I am not Lycan. I can never be.\" He tips her chin up, looking deeply into her eyes, and Tessa is lost in the smoke of his gaze. \"But I was made for you\u2014and you for me.\"\n\n\"How, Laz? How is that even possible?\"\n\nHe shakes his head. \"I do not know. But what I do know is, there is a fragment of the demonic. Those of us that are not fully of the devil. And because of this ancient DNA, we can sometimes be given a second chance in a realm other than Hades. A roll of the genetic dice, if you will.\"\n\nHe's so earnest, Tessa almost believes him.\n\nHis fingers skate lightly across her features, and she shivers. Trembling with want, she shakes beneath the heat of his touch. \"The myth is that you will know your Redemptive immediately, but she will not know you. You're more than a freedom from the prison of my station. You're a chance for happiness and a new life. You are labeled correctly. Tessa.\" He kisses her softly on the mouth, and she helplessly moans against him. \"You are my redemption.\"\n\n\"Lazarus, Lazarus, Lazarus,\" a low voice chastises, and Tessa whirls. The horned fucker.\n\nLaz shoves her behind himself protectively. \"Praile.\"\n\nPraile puts his clawed hands, tipped by ebony nails, on his hips and cocks his head. \"Yes, Praile,\" he mocks. \"You have frustrated me to no end.\"\n\nHis gaze falls on Tessa, and whatever brave thoughts she entertains, flee.\n\nIt's fair to consider he's going to want a little payback.\n\nTessa tries to inch away.\n\nLaz glances at her. \"Do not leave my side.\"\n\nPraile's eyes narrow, sighting them like an eagle with prey in range. \"This lowly female bitch shall die a slow death.\" Praile grabs his dick through loose pants and squeezes it. Tessa's eyes go to his hand at his crotch. \"All better, my little mutt. Despite your reprehensible actions.\" He glowers at her as though she gave him a paper cut.\n\nThat makes Tessa so much more scared than an outward threat. She feels her bladder give a little hiccup.\n\n\"You've interrupted the Master's important work, you pathetic female. And that work will commence and be seen through.\" His eyes move to Laz. \"I don't know what provoked you to leave me bleeding and dickless in the dirt of the angelic compound, but I will see a punishment so terrible it will be as though you eternally perish.\" Praile taps a short black nail against his chin. \"Actually, that is the exact definition of residing in hades. Sublime.\"\n\nTessa finds her voice. \"And when you put it like that, I'm sure Laz will just run and volunteer for your brand of bullshit.\"\n\n\"Lazarus,\" Praile says, giving him a look so full of malice that Tessa's knees feel weak.\n\nLaz squeezes her hip where his hand rests.\n\n\"Move aside and let me take my revenge on a female who would think to injure a high demon in such a fashion.\"\n\nTessa scans the blackness outside the windows. Where are the Lanarre? The silence is loud.\n\nHer eyes meet the demonic's.\n\nPraile smiles. \"They sleep, little bitch. There is no one to help you. Lazarus knows his place. Whatever infatuation he might have mentioned so he might stick his hot ineffective wick in your pathetic pussy is false. He will not turn his back on the Master and survive.\" He spreads his deep-red arms in a gesture implying how reasonable his words are. How absolute.\n\nHis horns glow like spikes of shadowed ink on his head.\n\n\"Laz didn't lie to me.\" Tessa glances at Laz, whose face is like pale-red stone.\n\nHe shakes his head, offering a slit of a grin. Small spirals of steam leak out as he speaks, \"We are demonic. Deceit is like breathing for our kind. Test him, Were female.\"\n\nTessa looks uncertainly at Laz, who continues to stare intently at Praile.\n\n\"Did he tell you that you were his Redemptive, perhaps?\"\n\nTessa's breath catches in her chest, and she whirls to face him again.\n\nPraile laughs gleefully, tipping his head back. Gales of perverse chuckles tumble out of a mouth. His tongue and teeth are black, and his gums are very red. \"Do not feel too aghast. Every demon has from time to time desired a little tail from the surface.\" He smiles, and thick steam streams from his mouth. Eyes like black obsidian discs of pure evil regard her.\n\n\"Laz,\" Tessa says in a low voice. \"What is he saying?\"\n\nLaz takes a step away from her. Her fingertips trail over his smooth back as he walks toward Praile.\n\n\"Have you had her yet?\" Praile asks, and Laz shakes his head.\n\n\"Did you do as I ask? Feign our getaway, so that the Rare One will be unsuspecting?\"\n\nLaz nods. \"Yes.\"\n\nTessa swallows her gorge. Not a great time to let hurt, nerves, and the potential for a smackdown get the best of her.\n\nPraile claps his hands, rubbing them together. \"Excellent. Now...\" His glittering black orbs study Tessa. \"I do believe we have a little time for sport with this one before the Lanarre wake from their little nappy.\"\n\nThey turn and face Tessa.\n\nLaz's eyes are as dark as Praile's.\n\nThey move toward her, and Tessa's screams fill the cottage.\n\n*\n\nDrek\n\nDrek rushes out the front entrance to his home, and the sound of a female's screams shatter eardrums sensitive to a pin dropping.\n\nHer terror beats at him.\n\nTahlia's escape set his nerve endings on fire. The farther she is from him, the more Drek realizes he's made a grave error in judgement.\n\nHe trips over a body and catches himself on a stout wood porch post. Looking down, he finds Bowen lying on his back, snoring softly.\n\n\"Bowen!\" Drek yells, toeing him.\n\nNothing.\n\nHe peers into the gloom of the woods. Lanarre guards litter the ground, appearing to have dropped where they stood.\n\nDrek turns, making sure Tanya is still secure against the pole. He gives her an apologetic look and leaves her.\n\n\"Wait!\" she shrieks. \"You can't just leave me here!\"\n\nHe strides toward the sound of screaming that makes the fine hairs at his nape stand at attention.\n\nDrek halts when he gets a whiff of an odor as offensive as vampire's, though most Lycan are unable to smell, either.\n\nDemonic.\n\nA layer of red vapor hovers like poisonous gas over the sleeping Lanarre.\n\nThe screams make more sense now.\n\nHe moves to the guest cottage, his eyes rising heavenward in search of a white bird\u2014his chosen. Drek was too ignorant to see and act accordingly.\n\nHe jogs to the steps, imagining Bowen's commentary had he been awake and by his side.\n\nA low voice growls words that let Drek know what he is dealing with\u2014not an enemy, but a relation of sorts.\n\nThat's why the Lanarre sleep but are not dead.\n\nDrek will handle this spawn of the devil.\n\nHis gaze moves a last time to the inky blanket of sky dotted with the stars like coarse, sprinkled salt.\n\nTahlia is not there.\n\nWhen the door doesn't open, he rams it with his shoulder, and it bursts wide.\n\nThe demonic are converging on Tessa.\n\nShe's wolfen and in fighting posture.\n\nDrek moves in to assist, and the distant sound of birdsong reaches his ears.\n\nThe music distracts him, and thoughts of Tahlia crowd his logic, enticing his mind that it might be she.\n\nDrek's hesitation is his undoing as the demonic attacks.\n\n# CHAPTER TWENTY-SIX\n\nSlash\n\nSlash doesn't stop running until his legs grow numb. His lungs are an oven of slow-burning fire. He and Adrianna are deep within the bounds of the woods. Safe.\n\nFor now.\n\nSlash slows to a jog, feeling gimpy because his limbs refuse to cooperate. When he stops, he gently swings Adrianna around by her arms. She sways, and Slash draws her in against him.\n\n\"If I'm exhausted, you must be beyond exhausted,\" she says against his heaving chest.\n\nSlash is. But he's been through more than this. So he simply nods.\n\nAdrianna pulls away, looking worse for wear. Her hair is matted with leaves and needles.\n\nSlash smiles, gently pushing the mess of her hair back from her forehead.\n\n\"I must look like shit.\" She stares at her feet.\n\nSlash scents her uncertainty, and so much more.\n\n\"Adrianna.\"\n\nShe looks up at him, and Slash fights to not give her the profile of his face that remains unscarred.\n\n\"We will get cleaned up. We will feed. We will rest.\"\n\nShe puts her hands on her hips, hazel eyes flashing. \"And this too shall pass? Pfft!\"\n\nShe stomps away, and Slash follows her. Each step he takes is agony. Slash returns to his human form and scents for food.\n\nHe scents nothing he can hunt without shifting again, and he's out of steam. His mate is in heat, and Slash doesn't think he's seen the last of her temper after their misunderstanding.\n\nAdrianna suddenly whirls, and the small hairs of his body rise in response to her beast\u2014and her heat. Slash stifles a groan. Regardless of how beat he is, his beast senses his mate's hunger.\n\n\"You took us away from the rain forest?\"\n\nSlash nods. They don't need to travel deeper into Lanarre territory. What they would do to him for killing the Lanarre would be slow and without mercy.\n\nAdrianna takes his hands. \"I'm still so pissed at you, I could spit.\"\n\nSlash has only silence.\n\n\"But\u2014\" She gives him troubled eyes. \"I guess what you said makes sense. I would have never left you otherwise.\"\n\n\"I know,\" Slash says.\n\nThey smile cautiously at each other. \"But what about my heat?\"\n\nSlash wearily swipes his head with a hand. \"It's not typical. However, it was probably our coming together that caused it.\"\n\n\"I'm young for heat, Slash.\"\n\nHe nods and pulls her to him, wrapping his arms around her.\n\n\"Can't we just\u2014you know\u2014not.\"\n\nHe chuckles. \"I'm afraid not. I couldn't keep my hands off you if I tried.\"\n\nAdrianna's eyes fill with tears, but she doesn't cry. \"I'm too young to have a whelp, Slash.\"\n\nHe cups her chin. \"You were not too young to become my mate or share your body with me.\"\n\nHigh color floods her cheeks. \"You got me there.\"\n\n\"It's not about \u02bbgetting you.'\" Slash lets his hand drop, and he stares off into the woods, his eyes searching the pockets of gloom. \"It's that we're mature enough. And for whatever reason, our union was strong. We complement each other well, or this would not have triggered such a response.\"\n\n\"You're the old guy. I'm just a young pup.\"\n\nSlash had felt guilty about that. But no more. She pursued him. He fought it. Fought his feelings and desire. Fought his lack of confidence over his attractiveness. Finally, she'd convinced him.\n\nIn all the years of Slash's life, he'd never been moved to have a mate. And when he was, she's been barely out of whelphood and was the most stubborn, mouthy female he'd ever met. Slash was reticent in comparison. What did the humans say? Ah yes: opposites attract.\n\nNo shit.\n\n\"Hey,\" Adrianna calls softly, startling Slash from his thoughts.\n\nHe grabs her hand and lifts it, kissing the layer of skin that is grimy from their trek through the woods\u2014and their killing.\n\n\"I'm not going to just hump right now out in the woods. I don't care if my crotch goes up in flames.\" She crosses her arms, and Slash barks out a laugh.\n\n\"I think I might be able to restrain myself just long enough to find shelter.\" His lips tilt.\n\nAdrianna runs her finger over the worst of his scar\u2014the one that mars his lips\u2014and he allows it. No one but him has ever touched his scar.\n\n\"When did this happen?\"\n\nSlash's smile fades. He figured he would have to tell her sooner or later. \"It was a war between Lanarre and Were. It's when their dominance was firmly established.\" Slash laughs, but it's hollow inside the thick woods.\n\n\"I assumed the Lanarre have always been top of the food chain.\"\n\nSlash widens his stance, crossing his arms and tucking his palms underneath his biceps. He rolls his lips together, absently bending the small ball of scar tissue between his lips.\n\n\"Slash.\"\n\nHe looks up. \"Sorry. I'm scattered.\" He taps his temple, procrastinating the storytelling.\n\n\"The Lanarre are really no more than the strongest of us. This was well before your time, but it shows a lack of historic study of Lycan.\"\n\nAdrianna nods. \"So shoot me. Not into books.\"\n\nThe corner of Slash's lips turn up. So young. \"Packs of Lycans would gravitate toward similar werewolves. Before we knew it, the weak congregated with each other\u2014\"\n\n\"And so did the strong,\" Adrianna finished, understanding flooding her features.\n\nHe pointed at her. \"Exactly.\" He chops his hands away from his body, \"So females began to become sick. Too much interbreeding of the same types of animal. Diversity was lost.\"\n\nA sick expression pastes itself on Adrianna's face.\n\n\"The males didn't thrive, but they weren't so easily taken by death.\"\n\nSlash meets her eyes significantly.\n\n\"That's when our females became scarce.\"\n\nSlash nods. \"It was the males' faults for not recognizing the problem and putting a protocol of prevention in place soon enough.\" Slash feels his face screw up in disgust.\n\nAdrianna is uncharacteristically quiet.\n\n\"When the Reds found that they were down to only a few viable females, we knew that we needed to refresh the gene pool.\"\n\nAdrianna leans forward, and Slash squeezes her shoulder. She shivers. A wave of need rolls through them.\n\nSlash gasps as though warm bathwater has been poured over him.\n\nHe clenches his teeth. \"Adi.\"\n\nShe shivers again. \"Can't help it\u2014and can I just say how weird it is that you call me that?\"\n\nSlash shakes his head as though he can shake off her heat. The strongest of the sensation subsides, but a residue lingers like an electrical charge over his skin.\n\nThey let a minute pass. Finally, Slash thinks he can speak. \"When talk reached us of the Lanarre hoarding females, at first we didn't think it was possible.\"\n\nAdrianna stares at him, eyes wide.\n\n\"All of us, from Alaska, from here, converged on the various Lycan packs.\"\n\n\"How many?\" she asks.\n\nSlash's eyes go wolfen. \"All.\"\n\n\"The Great Massacre.\"\n\nSlash gives her a slight nod.\n\nAdrianna searches his face. \"Even I know of that. They used liquid silver on their own kind.\"\n\nHis smile is sardonic, his finger traces part of the scar of his face. Her eyes track the movement. \"Anything to protect their females.\"\n\n\"But they wouldn't be protecting us. They'd be keeping us,\" Adrianna says, hand to chest.\n\nSlash nods. A soft prison is still a prison.\n\nAdrianna approaches him. He doesn't flinch when she lightly follows the worst of the scarring. Her finger runs from the small bumpy line that bisects his eyebrow, narrowly missing his eye then resurfacing like a jagged lightning strike at the highest bridge of his cheekbone, where it swoops to make a curling small knot of flesh at his cupid's bow.\n\nHer fingers feather against his lips. \"You didn't die.\"\n\n\"No.\" He smiles beneath her touch, chuckling softly.\n\nAdrianna punches him, and he catches her wrist before her fist lands.\n\n\"Come here.\" Slash jerks her to him and plants his mouth against hers.\n\nAdrianna molds her body to his, and Slash is soft and tender against her.\n\nOnly her.\n\n*\n\n\"Oh my Moon! When will we get there?\"\n\nSlash stifles an irritated sigh. Sometimes Adrianna reminds him of her youth when he knew her as a whelp.\n\nThey trudge through the deepest part of the forest.\n\nHe stays her with a hand. \"Do you see that?\"\n\nAdrianna scans the darkness, eyes narrowing. She finally sights what he already has. \"That creepy Hansel and Gretel cottage? Yeah,\" she says with exaggerated slowness.\n\nSlash turns to her. \"We must find shelter.\"\n\nAdrianna blushes, looking properly chastised. But her vague sadness pulls at him.\n\n\"Heart of my heart,\" Slash says, placing his palm on her chest as he recites the ancient words of his kind.\n\nAdrianna covers his hand with her own. \"Soul of my soul.\"\n\n\"What harms you, harms me.\"\n\nAdrianna's shoulders slump. \"Fine. When you entreat the ancient words, only a real douche wouldn't reply.\"\n\nSlash grins suddenly. Sometimes it's not bad to be almost four hundred years old.\n\n\"I'm thinking about the nurse, Jenni.\" Adi bites her lip.\n\n\"Let's walk to the little house as you share your troubles.\" Slash holds out his palm.\n\nShe sighs but takes his hand. His wolf notices her nearness and touch. His beast roils beneath Slash's flesh uncomfortably. The urge to breed is unbearable, like an itch that is just out of reach of being scratched.\n\nThey move down a little slope where the trees thin, only to climb back up a gentle knoll. When they reach the top, the small dark cottage stands at odds with the surrounding forest.\n\nAdrianna draws closer to Slash without being aware. He strokes the back of her hand with his thumb.\n\n\"I might have turned Jenni, and she's got no guidance. She might be wolfing around somewhere without a clue.\"\n\nSlash is half-listening. He does care about Adrianna's thoughts and self-doubts. But the male in him is uneasy.\n\nSlash has not lived this long without listening to his instincts.\n\nHe steps forward to the broad, well-worn steps. They're unusually wide, given the small scale of the house. Slash notes old dirt and a general absence of habitation.\n\n\"I just bit her, and poof!\" Adrianna's free hand swings up. \"I don't know for sure if she lived or died.\"\n\n\"Or crawled away to lick her wounds,\" Slash says absently as he fingers the large pole that anchors the top of the steps.\n\nAdrianna slaps his back, and he barely feels it. \"You're not listening!\"\n\nHe steps onto the porch. The old wood tongue-in-groove slats are edged with moss, slowly succumbing to the elements in their damp climate.\n\n\"What you should really ask yourself is what are the ramifications your bite might have while you're in heat.\"\n\n\"What?!\" Adrianna shrieks.\n\nSlash covers her mouth with his hand, pulling her tight against his chest. His skin crawls. His eyes chase over the tops of ground-hugging ferns, moss, and discarded pinecones from trees that perpetuate life from the earth. The thick greenery suffocates the strangled moonlight.\n\nHis pupils dilate to access whatever light is available.\n\nNothing.\n\nHe slowly removes his hand, and Adrianna opens her mouth. Slash shakes his head, putting a finger to his lips. Her eyebrows whip up. But she's quiet, looking around the immediate vicinity. The forest is silent.\n\nToo silent.\n\n\"I don't like it,\" Adrianna whispers.\n\nSlash gives her a cautioning look but wraps his fingers around her arm. With a jerk of his chin, he indicates the door.\n\nAdrianna gives one more glance around the dim woods, the light eaten by the high canopy, and derelict moonlight, then she follows Slash.\n\nHe places his hand on the knob, old tin with a beaded perimeter, and turns the handle.\n\nThe door swings open on well-oiled hinges.\n\nTwin barrels of a shotgun ease into his face from about a foot away. \"Step closer, and I'll decorate my walls with your brains.\"\n\nSlash freezes.\n\nAdrianna yelps.\n\nThe old woman with the gun chuckles. \"Well, that ward didn't hold worth a damn. Come in, lovebirds.\" She snorts. \"Or are you waiting for some trolls to come knocking, eh?\"\n\nSlash stupidly holds his position, stunned. What creature is this?\n\n\"Are ya daft?\" She cocks an eyebrow to her snow-white hairline. \"I said to come inside. I don't invite twice, Red.\"\n\nHer bushy brows drop as she gives him the best evil eye he's ever received.\n\nSlash gives a pointed look at the shotgun.\n\n\"Ah!\" she says, chortling. \"It's not loaded! If it comes to shooting ne'er\u2013do\u2013wells, all hope is lost.\"\n\n\"Why?\" Adrianna asks with uncharacteristic timidity.\n\nThe old woman rolls to her toes, trying to catch a glimpse of Adrianna. Slash moves her behind him.\n\nHer eyes move over the two knowingly. \"You're no threat. No you're not.\"\n\nSlash is puzzled but unwilling to let his guard down completely.\n\nHer chest puffs up, and she moves swiftly around Slash, who angles Adrianna away, facing her the entire time. She rests the shotgun behind the door with the butt down and kicks the door closed.\n\nA solid brass bar falls down into a crooked little holder nailed into the wood casing.\n\n\"Who are you?\" Adrianna asks.\n\nWho indeed?\n\n\"Nosey, aren't ya? You come to my house\u2014uninvited\u2014and think to just barge in\u2014\"\n\n\"We didn't think it was occupied,\" Adrianna said.\n\n\"Well it is, isn't it, Were female?\"\n\nShe turns, showing them her back, and shuffles quickly to a stove where a kettle brews. Steam evaporates as quickly as it rises. She turns, cocking an eyebrow, and puffs out an irritated breath.\n\n\"I'm a witch, girl. And you're on private property.\"\n\nEasy to remedy. \"We will leave.\"\n\n\"Can't.\" She says the one word as both answer and command.\n\nThe hair on his nape rises. \"Why not?\"\n\n\"Did ya hear me? I know a Red Were such as yourself hears better than me. The trolls, fool. You didn't sense them when you and your woman were flouncing around, making a racket that led straight to my door?\"\n\nSlash can feel Adrianna's pulse through their threaded fingers. \"This is the Hansel and Gretel house. We're screwed.\"\n\nThe witch's face changes, smoothing into a grin. \"I haven't heard mention of that in years.\" She gives a little sigh, as though nostalgic.\n\n\"So you eat kids?\" Adrianna asks only a little in jest.\n\nShe shakes her head while Slash sizes up their chances of escape.\n\n\"No. That's another witch.\"\n\nAdrianna's grip becomes crushing.\n\n\"Then what do you eat?\"\n\nThe witch puts her hands at her hips, a long skirt hiding her fingers. \"Food.\"\n\n\"Like people?\"\n\nHer nose scrunches up. \"No, I'm a good witch. As good as the two of you are.\"\n\nSlash is skeptical of that claim. And if she knew of his transgressions, she might be less forgiving.\n\nThe witch narrows her eyes at them. \"I smell death on you, Were. But rightful death carries a different scent than wrongful, yes?\"\n\nSlash nods slowly. Wise words.\n\n\"The two of you crossed a ward I've had in place since I could hardly hold a wand.\"\n\n\"You use a wand?\" Adrianna asks.\n\nHer face tightens. \"What do you take me for? A novice? Of course I don't need a wand anymore.\"\n\n\"Sorry,\" Adrianna says, and Slash bites back a laugh.\n\n\"So how did we get past the ward?\"\n\nThe witch gives them a sly glance, full of confidence and something else that makes Slash uneasy.\n\n\"True love.\"\n\n# CHAPTER TWENTY-SEVEN\n\nJulia\n\nTwo weeks later\n\nJulia blinks, and before she knows what's happening, Jen has dressed her in a gauzy dress of deep cream bordering on yellow. Beads of glass grace the square neckline and cup the outside of her shoulders. Her hair is elaborately pinned up at the crown of her head with half hanging down between her shoulders in loose curls.\n\nWhen she's breathed through her nerves, she descends the winding staircase in the mansion.\n\nScott is waiting at the bottom.\n\nHis broad shoulders strain against a tailored snowy-white button-down shirt, such a perfect contrast to his dark hair and eyes. A crisp bowtie graces his throat. Black pants encase slim hips and long legs.\n\nHe isn't wearing a coat.\n\nScott is so handsome, Julia gets a lump in her throat. Holding back tears, she admires her creamy satin heels, feeling so overwhelmed.\n\nThen his arms are around her. Scott lifts her off the floor and holds her against him. Subtle fragrance of spicy male and Scott's unique scent floods her nose, and he lets her slip down his front.\n\nShe turns in his arms, looking around her and blinking rapidly to expel her laughter. Tears.\n\nShock.\n\nAll the Singers who lived through the genocide have gathered. New faces mingle with familiar ones, and a sigh slides between her lips. Small children, though there aren't many, try their best for quiet. But babies still squall, and mothers silence them with a kiss or a word.\n\nJulia's at a lost, but Cyn nails it.\n\n\"Wow,\" she says reverently.\n\n\"When did all this happen?\" Julia asks Scott. She can feel his pride swell inside her, brimming over from his own emotional grid to her own. His thoughts mingle with hers, and she takes his hand.\n\nThey did this for us, Julia sends them through the bridge of their meld.\n\nScott nods.\n\nThe edge of day burns into night. Small twinkling lights are strung from both corners of the broad front porch to tall softly illuminated lamps piercing the ground. Each lamp holds a glass oblong light. Many children hold the last of summer's wildflowers in lazy bouquets of drooping petals in every color of the rainbow.\n\nScott's hold tightens on her hand.\n\n\"This is all for us,\" Julia says in wonder.\n\nScott nods. \"It's about time.\"\n\nJulia couldn't have married Scott when she was married to Jason. Their rules are different. They'd allow so much because of the fact that she was the Rare One, but in her heart, Julia was still human, still culturally bound to the principles she was brought up with.\n\nA sudden thought occurs to her, and she turns to Scott. \"Do Singers really get married?\" She'd heard the term joining, but never marriage, as humans thought of it. As Julia had.\n\nHis smile is crooked, and Julia narrows her eyes. \"Tell me.\"\n\nHe frowns. \"We handfast.\"\n\nHandfast. Julia smiles at the term as voices in the crowd softly swell.\n\n\"Isn't that kinda medieval?\" Cyn asks quietly as she surveys the assembled crowd, and Julia laughs. \"I don't know, really.\"\n\n\"Yes,\" Scott says, \"definitely ancient.\"\n\nI guess for Singers, everything is about tradition.\n\n\"You look gorgeous, Jules,\" Cyn says.\n\nJulia smooths her hands over her dress then hugs Cyn tightly, but she pulls away. \"I don't want to wreck your dress with all my blubbering.\"\n\nJulia smiles through her own tears. \"Blubber away.\"\n\nCyn nods, biting her lip to keep from outright bawling. Cyn leans forward and whispers in her ear, \"He'd want this, Jules.\"\n\nCyn doesn't have to tell her who, but Julia asks quietly, \"What?\"\n\n\"Your happiness.\"\n\nJulia steps away, still holding Cyn's hands, and they smile through their tears. Julia nods quickly.\n\nScott asks from her elbow. \"Are you ready?\"\n\nJulia feels Jason's grave behind her, in the past. That life is dead, along with her husband. Julia realizes she's been grieving bit by bit for Jason since long before he died.\n\nShe gives a decisive nod. \"Yes.\"\n\nScott smiles, running his knuckles in a light caress over her jaw.\n\nThey move toward where Victor stands, a ribbon of ivory draped over his hands.\n\n*\n\nVictor carefully and securely ties their right hands together.\n\nThe silk cloth is wide and strong, with a slight shimmer. The colors of the sky have washed the grounds in tones of sherbet, raspberry, and deep blue, and the soft murmur of her people is a dim noise inside her head. For once, she doesn't keep them out, only quiets them.\n\nJulia wants to feel their joy as though it's her own.\n\nScott looks into Julia's eyes. His are shadows in his face, but inside the dark gaze is warmth, love, and something deeper than any human male has the capacity to feel.\n\nVictor's words flow over them. With each one, she feels their bond tighten.\n\nJulia can still refuse, deny Scott\u2014and crawl back into the hole of her grief or hang onto the past that's a wisp of vibrant memory that makes her at once happy and sad.\n\nBut she won't.\n\nJulia lets everything she feels for Scott fill her expression and what she's committed to being with him.\n\nHe nods, his gaze capturing Victor's, who studies them as he gives the final word of the ceremony.\n\n\"Agape.\"\n\nHe describes a special form of love that transcends this realm. The heavenly love is a perfect word for those with angelic blood.\n\nVictor unties the binding and gives two rings to Scott. They glow in polished white gold from years of wear.\n\nJulia holds her breath as Scott slips one ring on her finger. The metal is cold and way too big.\n\nShe glances at him sharply, her lips quirking.\n\n\"Just wait,\" Scott whispers.\n\nJulia feels the metal begin to heat and looks down at the ring changing on the left ring finger of her hand. It slims, moving like a molten circle of fire, narrowing and growing smaller until it fits perfectly.\n\nJulia's head jerks up, and Scott holds his hand high, palm facing her. He flutters his fingers while his ring grows larger. The circle of gold matches hers in every way, but his is stuck at his middle knuckle.\n\nAs she watches, the narrow band appears to shimmer, enlarging until it slips over the knuckle then growing smaller again until it is seated perfectly between his first and middle knuckles.\n\nShe gasps in a happy little breath.\n\nAfter all Julia has witnessed, she's so pleased to see a small miracle like this\u2014one that doesn't maim, kill, or cause grief.\n\n\"A simple magic,\" Victor comments, after watching her expression.\n\nJulia looks at her hand, stretching her fingers out to admire the simple slim band.\n\nScott pulls her into his strong embrace, drawing her close enough to kiss.\n\nThen he does kiss her, deeply and thoroughly. His lips are soft and tender against her mouth.\n\nThe crowd applauds, and Scott turns her to face them.\n\nShe leans against his hardness, and he places his forearm against her chest.\n\nThe Singers come: Truman, Cyn, Jen, and Michael.\n\nJulia finds herself missing a few people who have not been found or have been killed. Brendan.\n\nWilliam.\n\nReagan. Jacqueline\u2014once so intent on killing Julia, and now finding a new life within the sithen and maybe her full sanity along with it.\n\nJulia's palm moves to her stomach, where the spore still lingers.\n\nEverything is finally right in her life, though not by any means she would have wanted. However, her future is uncertain. But that's the one thing in life she can count on: uncertainty.\n\nAs people come to receive them and she stands underneath an altar of softly lit twigs, Julia gazes into Scott's face. His expression is cautiously hopeful.\n\nFor now, that's enough.\n\n*\n\nAfter\n\nJulia\n\n\"This is too much, Scott!\"\n\n\"No\u2014after all we've been through, it's barely enough.\"\n\nHe smirks, and she gives him a light slap on his bicep, but he catches her against him, wrapping her in his arms.\n\n\"It's customary on the night of handfasting to use this bed\u2014this room.\"\n\nJulia scans the large room, formerly Marcus's, and heaves a sigh. His bed was replaced by a carved one made of wood. Clearly medieval and gothic, it stands out in a room many years newer.\n\n\"Lots of mixed architecture here,\" Julia comments.\n\nScott presses a soft kiss to her lips. \"I am not here to discuss furniture.\"\n\nJulia smiles, holding her face away. \"I know that.\"\n\nScott motions toward the en suite bathroom, where he just turned off the hot water tap. Julia can't see inside, but she moves toward the doorway, peeking inside.\n\n\"Oh!\" she says more loudly than she intended. \"It's beautiful.\"\n\nShe turns to Scott, and he is smugly pleased. \"Did you plan all this?\"\n\nScott lifts a shoulder. \"I'd love to take credit, but the Singers who take care of the royal line had this arranged after Father's death.\"\n\nScott looks down, running a finger along the edge of a tub so huge it could easily fit two.\n\nThe thought makes Julia blush. Though they've already crossed that bridge, she finds room for shyness.\n\n\"You miss him,\" she states.\n\nHe nods. \"This isn't about Father, though.\" Scott holds out his hand, and she slides hers inside.\n\nTheir connection tightens like a gasp of air, winding them tightly together.\n\n\"I never get tired of the way that feels,\" he says, a little out of breath.\n\nMe, either.\n\nHe smiles at her telepathic comment.\n\nHis finger dips in the water. He saunters closer, reaching behind her hair and unhooking the top latch of her dress. He slowly unzips it to the small of her back, and Julia steps out of the circle of creamy material.\n\nScott's eyes catch on her breasts. There was no room for a bra in the tight bodice. She sees in his face that he approves.\n\nJulia moves to the edge of the tub, giving him a look of invitation.\n\nScott follows.\n\n*\n\nScott\n\nScott stands before her, naked, wrapping a towel around her body and blotting the streaming water from her heated, fragrant flesh.\n\nJulia admires the muscles as they dance with his deliberate movements. Each stroke and caress of the towel has her closer to wanting him.\n\nShe's wound so tightly with desire, standing is an effort.\n\nJulia places her hands on his shoulders, and he looks up, his eyes level with her sex.\n\nHeat rises to the surface of her skin from the look he gives her.\n\nScott leans forward, his tongue licking a lone drop, and Julia hisses in a breath, tipping her head back and clutching his shoulders.\n\n\"Scott,\" she breathes.\n\n\"Yes.\" He moves his tongue a little lower, delving between her slit.\n\nJulia cries out softly.\n\nHer chin lowers, her eyes meeting his.\n\n\"More?\" he asks, his breath hot against the most tender part of her.\n\nJulia doesn't speak, nodding.\n\nStrong hands grasp her hips and open her to him.\n\nJulia's legs tremble as his tongue moves deeper, digging between her folds.\n\nShe moistens at his touch, at the residual emotional sensations they share through the meld.\n\nWhen her knees weaken, he pulls away, holding her so she won't fall, and swoops her easily into his arms. His erection rests between them.\n\nSomehow, he finds his way to the bed, hair still damp from their bath of mutual exploration. A drop of water trails like melted ice from his hairline and falls between her breasts.\n\nJulia shivers.\n\nScott lowers her to the bed and spreads her legs with his palms, visually feasting on her nakedness.\n\nWhen Julia moves to close her thighs, he pins them open with his large hands. \"Don't. You're beautiful\u2014and finally mine. Let me take you in.\"\n\nAnd like a sponge soaked in love and lust, Julia relaxes her knees.\n\nScott bends over her, lapping the drop of water that lingers between her breasts and then presses them together, first laving one nipple and the next. He goes back and forth, always allowing his breath to warm her in a path of fire that ends in the wet torture of her nipples.\n\nThey harden inside his mouth, and Julia makes a small aching sound, arching her back. He captures her against him, plunging a finger deep inside as he gives an insistent pull with the smallest bit of teeth at her nipple.\n\nJulia sighs as pain and intense pleasure mix, opening her eyes to see his.\n\nScott silently gazes down at her.\n\n\"I feel you\u2014how hard you are, how much you want this,\" Julia whispers.\n\n\"You have a small idea.\" He places Julia's hand on his length.\n\nI can't take all that. She gives him a sly grin.\n\nHe smiles back. You seemed to do pretty good last time.\n\nJulia's face flames to life, and she laughs softly, giving him a squeeze.\n\nScott gasps, his stomach muscles clenching and releasing with the sensation of her hand on him. \"I'll go if you keep up your handiwork.\"\n\n\"Nice pun, husband.\"\n\n\"My king,\" he corrects, the corners of his lips twitching.\n\nJulia pouts a little. She squeezes him again then rolls to her knees. They face each other on the bed, though he's so much taller, she's really eye level with his chest.\n\nJulia runs her free hand down the smooth muscles of his chest, trailing her fingers back and forth between the discs of his nipples. They harden, and she lays her head against his heart.\n\nHe rolls his hand down her damp hair. \"I love you, Julia.\"\n\nShe does something she hasn't tried before, pushing the brunt of her feelings in a tight knot toward him, through Scott. His eyes widen, and his hands slide from her upper back to the lower, convulsively tightening.\n\n\"Ah,\" he says in soft wonder. \"That's how you feel?\"\n\nJulia nods, biting her lip. Her feelings are so profound and intense, it's easier to just think them through the connection. Words are insufficient.\n\nWhen Scott reciprocates, her core pools with moisture. His arousal and want of her is so acute, it incites the reaction of desire from her even more strongly.\n\nHis eyes darken. \"Lie down.\"\n\nJulia does without hesitation. They're not in a bunker and desperate to survive. It's a new chapter, and Julia feels a fresh boldness.\n\nScott parts her thighs with two strong hands. His fingertips softly brush her entrance, and Julia pants, wanting more and remembering what's already happened.\n\nJulia doesn't tell Scott what she wants. Her emotions swarm through them, and they both tense and relax with the sensation of their connection.\n\nHe folds over her body, fingers digging into her thighs, and Julia softly shouts, \"Yes!\"\n\nHer fingers grab his head as Scott dips into her folds, gently pulling apart the slick skin of her arousal.\n\nJulia sits up as Scott jerks her hips forward. \"Ah!\" she yells, hanging onto his shoulders as he licks first one fold then the next.\n\nWhen his thumb presses on her clit and begins to slowly churn the tiny bundle of nerves, Julia's orgasm crashes into her like a river of pleasure, chased by Scott's arousal and happiness over her reaction.\n\n\"Please,\" Julia whimpers as he looks up. She latches onto the muscular globes of his butt, gathering him closer.\n\nHer channel pulses for him. To be filled. To be owned.\n\nScott seats himself between her legs and rocks slowly inside, stretching her.\n\nJulia squirms beneath him, loving the sensation of him inside. He's large, and she tries to adjust her hips.\n\nScott stills. \"Am I hurting you?\"\n\nShe smiles.\n\n\"So I'm sort of a full meal deal?\"\n\nJulia pops her hips forward, and the smirk becomes a slack-jawed look of surprise. \"Don't do that, Julia. I won't last.\" His voice is part chastisement and part pleasure\n\nJulia laughs softly, dancing her hips under his slow, precise thrusting. Finally, he bottoms out inside her and groans, pressing his forehead against hers. Their bodies lock together as though they were made for each other.\n\nThey are.\n\n\"I don't want this to end,\" Julia says, gasping as he moves inside her.\n\n\"Me, either, but you keep moving like that\u2014and there's no way.\" Scott feathers a trail of kisses along her throat then moves inside her again and again.\n\n\"Julia,\" his voice tight.\n\nBut she already knows. His orgasm shivers through them, his last thrust buried inside her so deeply it's as if they're one body. One mind.\n\nOne soul.\n\nScott's release brings hers, and Julia calls out his name. Their mutual pleasure mingles until neither know where it began and where it ends. They hold each other for a long time, heartbeats slowing to a synced rhythm.\n\n\"Will it be like that every time?\" Julia asks softly, stroking his back. He plays with a lock of her hair, the tips dry inside his fingertips.\n\nScott's jaw jerks back, and he grins. \"Wanna see?\"\n\nShe does. Her answer flows through their meld.\n\nScott gathers her close.\n\nEPILOGUE\n\nOne month later\n\n\"Ready?\" Scott asks, and Julia nods, letting out a protracted exhale. \"Hey. Come 'ere.\" He pulls Julia against him, shielding her with his much larger body.\n\nHe leans away slightly, thumb lightly stroking the wetness on her cheekbone. \"You know I won't let anything happen to you.\"\n\nShe nods.\n\n\"So why the tears?\"\n\nScott delves deep. Her psyche's an open book to him. After four weeks of living with her as her husband, Scott is even more pleased that destiny gifted Julia to him. He doesn't deserve her.\n\nYet she's his.\n\n\"I don't want to leave,\" she admits.\n\n\"I feel that.\" He gathers her small hand against his chest then leans down so they're looking into each other's eyes. \"We have to address the spore. I've already told you, my Combatant side is uneasy about the thing being inside you. I can't relax until it's outta there.\"\n\nJulia nods slowly.\n\nHe feels her emotions through their bond.\n\n\"We can't start a family until that little piece of evil is gone.\"\n\n\"I know,\" she leans her forehead against his chest.\n\n\"Besides, you know we have to go to faerie anyway. You made the promise to Tharell. Ultimately, you made it to faerie.\"\n\nJulia shakes her head. \"I don't know what Singers, if any, are going to want to possibly align in arranged marriages with the fey. They're very different from us.\"\n\nHe tips her chin up. \"But compatible.\"\n\n\"True.\" She gnaws on her delectable lips, giving Scott ideas. She giggles. \"Quit that, sex demon.\"\n\n\"Speaking of...\" Scott raises his eyebrows, the standing question between them.\n\n\"Yes!\" Julia says, trying to pull away, but Scott holds her close.\n\n\"Repeat after me. Praile can't find me as easily without the spore.\"\n\nShe frowns. \"Stop.\"\n\nScott scowls, letting his displeasure flow through their bond. Normally, he shields Julia. It's his role. But right now, he can't rest easy until that thing's out of her. And Victor said the fey would have the answer.\n\nHe's never wished more for his father's advice than he does now. Marcus would have known just what to do.\n\nBut fucking Praile put an end to a lot of the wisdom of their race through his lackey, Tony. Now it's up to Scott to close this loop. This tiny possibility that while Julia holds the spore, she's somehow made more vulnerable; he can't let that potential exist.\n\nThe evil seed only remains dormant because of their meld. But that doesn't satisfy Scott. He wants it gone.\n\n\"The fey have powerful magic.\"\n\n\"And you're sure there's no Singers that can expunge it?\"\n\nScott exhales harshly. \"No. All our talents are more about manipulating environment. Except Healers. And they're not equipped to address problems from other realms. The origin of this fucking thing isn't even from here.\"\n\nHe steps away, hands trailing from her hips, and paces in front of the mansion.\n\n\"I'll go,\" Julia says quietly.\n\nScott turns. \"You know I wouldn't press this if I didn't think it was important.\"\n\n\"I do know that.\"\n\nHe walks back to Julia, takes her hand, and raises it to his lips. Scott kisses her knuckles then lays the side of his face against the back of her hand. \"I understand how much you like how things have settled here. How unsettled your life's been these past three years until now.\"\n\n\"Almost four,\" she corrects.\n\nHis eyes move to her features, taking in the gorgeous glow on her skin, eyes like whiskey fire, and a body that rocks his world.\n\n\"What remains of the Combatant, Region One, and all of Two will stay behind. We know what to look for. One is fortified. Vic's got it.\"\n\n\"But where's Praile?\" Julia asks.\n\nScott feels the fear inside her words.\n\n\"If he was going to come back, he would have. That's part of why I want to get this out of you fast.\" He moves his palm to approximately where the sword pierced her during the battle at Two. \"Let's not make it easy for that fire dick.\"\n\nJulia's smile is faint, but it's better than nothing. \"Okay.\"\n\n\"And,\" Scott says with a coy tone, \"it won't be too shabby to see green boy, Jacqueline, and Delilah.\"\n\n\"I want to know where the others are, like Reagan.\"\n\n\"Julia\u2014\" Scott groans. \"We can't save the whole damn bunch of supes.\"\n\nShe kicks a stone with the toe of her sneaker just as their car pulls up. The sleek black SUV resembles a vehicle straight out of a spy movie.\n\nScott knows Julia's not quite used to the queen bit, but since she was hurt back at Two, she appears to appreciate the extra protection. Still, she's grown complacent with the meld, knowing all Scott's emotions and being telepathic enough to embarrass him sometimes.\n\nJulia would be blind, deaf, and dumb not to know the depth of his feelings for her. She senses their bond in her dreams, she told him.\n\nVictor hops out the front of the vehicle and opens the center door of the souped-up rig, and Scott leads Julia to the black vehicle.\n\nTheir reflection shines back at them.\n\nHer reluctance thrums through their connection, and he empathizes. She's had a fucked-up last few years. And One is the first place she's felt at home in a long time. Caldwell's death, Praile's pursuit, the massacre\u2014she's been through enough to make a sane person rethink things.\n\nThe soul-meld has repaired a lot of the damage, but not all. Scott's got her back.\n\nAnd a ton more.\n\nHe grins, slapping her lightly on the butt as she hikes herself in, shooting him a glare. But he sees the smile around the edges of her lips.\n\nIt'll be okay, he says through their meld.\n\nScott will make sure of it.\n\n\u2013\u2013\u2013\u2013\u2013\u2013\u2013\u2013\n\nTHE END\n\nRead More\n\nBook 7\n\nNever miss a new release! Subscribe:\n\nTRB News\n\n***Love BLOOD? Please read on for a sample from another TRB work....\n\nTHE REFLECTIVE\n\nA Reflection Series Novel\n\nBook 1\n\nNew York Times Bestselling Author\n\nTAMARA ROSE BLODGETT\n\nAll Rights Reserved.\n\nCopyright \u00a9 2013-14 Tamara Rose Blodgett\n\nNo part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a database or retrieval system without the prior written permission of the publisher.\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locales or organizations is entirely coincidental.\n\nwww.tamararoseblodgett.com\n\nTRB Facebook Fan Page\n\nCover art by Phatpuppyart.com\n\nEditing suggestions provided by Red Adept Editing\n\nSynopsis\n\nThose born Reflective are the only beings with the ability to jump between worlds. With The Cause firmly entrenched by years of highly stylized military-type training in combat of every kind, they use their natural born ability for inter-dimensional travel to police worlds where wrongdoing has overtaken civility.\n\nAs an elite Reflective, Jeb Merrick finds himself partnered with a rare female Reflective. Merrick is convinced that she can do nothing but slow him down. Beth Jasper is small, in both stature and mindset. When they are thrust together as partners, Merrick vows that she will receive the same neutral treatment as a male. Merrick cannot allow the unlikely union of Beth Jasper to get in the way of finding his prophesied soulmate.\n\nBeth discovers she is partnered with the cockiest of all pureblood Reflectives and struggles to maintain her composure with a hostile partner whose loyalty she doesn't possess.\n\nCan the Reflectives uphold The Cause, reach a point of compromise and find their chosen soul mates?\n\nTHE CAUSE\n\nFirst: Right the Wrong\n\nSecond: Bear No Injustice\n\nThird: Change Not What Must Be\n\n# Prologue\n\ntwenty years before\n\nThe midwife made her way along ancient cobblestoned streets, her shoes catching in the crevices though Principle knew, her shoes were as sensible as they come.\n\nAs was her occupation.\n\nShe would arrive in the birthing ward at exactly eight a.m. for her twelve-hour shift. Of course, it would not be twelve hours\u2014it would be for however long the woman labored.\n\nAnd if a Reflective were born ....\n\nJust the thought of the potential for that caused a nervous thrill to flutter deep within Florence, as it did each time she worked.\n\nThe Reflective newborns must be swaddled in special non-reflective blankets. A baby would not be lost on her shift because it was a prodigy who jumped at a mirror or other reflective surface left uncovered.\n\nDear Principle. She shuddered, thinking about what the punishment would be for that. As it was, midwives couldn't use any surgical instruments that were not brushed stainless steel, and since the last unfortunate incident, the midwives had since moved to an all-ceramic surgical unit.\n\nFlorence swept up the massive steps. The rise of the treads was so low the stairs felt more like a gentle slope than true steps.\n\nThe sparkling flakes of charcoal that clung to the thick white granite reminded her that the sun still shone brightly, though their version of autumn would soon be here.\n\nA shadow fell over Florence, and she twisted to look at the sky, her foot on the top step, her hand on the solid brass door handle that opened to the birthing center.\n\nA swarm of butterflies, so thick it blocked the cerulean of the sky, dropped false night all around her as they flew through the rectangular vents that fed the ventilation system in warmer months.\n\nThe ports were a deliberate architectural feature that allowed entry to the only creature in their world that could identify a Reflective\n\nSo many.\n\nFlorence stood in stunned wonder. She had witnessed butterflies come to mark the birth of a Reflective, but never in such a great number.\n\nTheir importance was such that her world was named in their honor: Papilio, Sector Ten.\n\nTheir path created a rainbow of iridescent color, which poured like water through the narrow vents that had been carved in the solid stone of the birthing center.\n\nAll who lived in their world were born in similar structures.\n\nHowever, Florence was one of few birthing center workers who had seen the highest incidence of Reflective births. She had requested placement to this one. After a five-year waiting period, she'd been assigned to the most prestigious.\n\nShe snapped out of her reverie as the last of the mingling kaleidoscope of insects funneled through the slits underneath the eaves of a copper roof, now aged a deep verdigris.\n\nFlorence tore open the heavy door.\n\nShe didn't hear it clank behind her as she ran the length of the corridor to the floor that housed laboring mothers.\n\n*\n\nFlorence burst through the swinging doors as a man and a woman stood over a cradle.\n\nConfused, Florence skidded to a stop.\n\nWhat is this?\n\nThis... appeared to be the parents in front of a babe so new that some of the vernix still coated the wee one, her arms swinging as she howled.\n\nTwo nurses, one at the end of her shift and one in training, hung back.\n\nOh, for the love of all that is good. She stalked over to the newborn.\n\nFlorence halted as the sight overtook them all.\n\nTheir breath.\n\nTheir thoughts.\n\nEverything but the scene itself melted away for those who witnessed the post-birth spectacle.\n\nThe butterflies descended, floating in a lazy spiral as the opalescent sunlight washed over their multicolored wings.\n\nThe chubby arms of the baby girl swirled and pumped, slowing as the butterflies drew nearer, and her echoing screams gradually grew quiet.\n\nThe insects lighted on the rails of the basinet in a portentous group, their wings moving in a steady sweep to maintain balance.\n\nTheir appearance froze the parents' breath in their throats.\n\nThe moment swelled and grew in the stillness of the nursery, where rows upon rows of cradles pressed against the other. The parents watched the butterflies flutter precariously on the polished sides of the newborn's bed, landing only on hers and no other.\n\nTheir appearance was beautiful... final.\n\nFlorence strained to hear the mother's voice.\n\n\"She is Reflective,\" she said in a sorrowful tone.\n\nHer mate squeezed her hand so tightly her knuckles turned white.\n\n\"Yes,\" he replied, just as gravely.\n\nTheir gaze met in perfect understanding of what the future held for their daughter: a life as mercenary, hunter and hunted.\n\nThis was an honor and privilege among their people.\n\nFlorence closed her eyes in sympathy. A female Reflective\u2014every parents dream... and nightmare.\n\n*\n\nfive years later\n\nBeth shot the plain glass marble across the stretch of earth, watching the glass orb tumble and spin as it met the others she'd shot in a smack of hardened glass. It swerved at the last moment, ricocheting off a shooter, and came to stand where she'd intended.\n\nAll the other children her age could play with any marble they chose, but she possessed no mercury-coated marbles.\n\nBeth Jasper was a solitary girl.\n\nBut not one who lacked intelligence. Beth had felt the sadness from Papa and Mama and knew she would soon leave for the building that had a big shining silver papilio above the entrance.\n\nMama and Papa had taken her there the previous week to meet with a man who had a nose like the water birds that gathered near her family's pond.\n\nHis nose made it very difficult for her not to giggle. Beth sometimes had a problem with laughing when she shouldn't.\n\nBeth had observed and stood watch over her new surroundings, remembering what her adoptive parents had told her.\n\nBeth, you must let us do the talking. Under no circumstances should you volunteer to train for a combative role. There are alternative roles for female Reflectives.\n\nBeth crinkled her face at the memory. She understood all of what they wanted of her, and she would not shuffle papers and sit behind a desk, looking like the dolls she had given up playing with.\n\nAll Reflectives were far more mature than their human counterparts from the other twelve sectors.\n\nBeth spoke like a teen, though she was five cycles. She puzzled through things that confounded adults.\n\nShe was faster, stronger, and brighter.\n\nBeth was female.\n\nWhen Commander Rachett of the Reflective Militia, who operated under The Cause leaned forward and delved deep, he tried to pierce young Beth's very soul. She met him halfway.\n\nHer small body leaned boldly toward his, unafraid.\n\nIn their people's ancient language of Latin, he posed the question: What role will you fill within The Cause, young Beth?\n\nBeth narrowed her eyes, and Rachett's eyebrows raised slowly.\n\nHe had studied her, no doubt because she was a half-breed, and female besides. She had met his stare with an unwavering gaze.\n\n\"A combative role, of course,\" Beth said in her childlike voice, though the meaning was very adult, because she understood and communicated like one.\n\n\"No! Beth...\" her mama said.\n\nBeth swung her legs back and forth underneath the chair. Her eyes drifted to the candy dish poised at the edge of the desk before returning to the commander's.\n\nBeth's stare matched Rachett's.\n\nRachett had to know what she was: a warrior. The attribute was either present, or it wasn't.\n\nHer papa stood.\n\n\"We can't have her fight. She is female... and not big for her gender.\" Her father's face pleaded with Rachett to see reason.\n\nCommander Rachett wasn't known as a reasonable man.\n\nRachett steepled his fingers underneath his chin, looking at Beth's adoptive parents. Good people, common folk who were loyal to The Cause, believers in the Principle.\n\nRachett's gaze shifted to Beth. He scrutinized her face: eyes like crushed brown velvet; hair like a raven's wing; and skin like polished marble, pale but not pasty.\n\nShe is too beautiful to fight, he must have thought with regret.\n\nBeth saw that future remorse on his face.\n\nThen he looked at her hands, long-fingered and limber.\n\nHis eyes shifted back to hers.\n\n\"Beth?\" he asked softly.\n\n\"Yes, Commander Rachett?\" Her small fingers held something.\n\nHe frowned, obviously distracted from his planned comment.\n\n\"What do you have in your hand?\"\n\nShe opened her palm, revealing a large reflective marble\u2014a shooter coated with hard-laced mercury.\n\nRachett sucked in his breath.\n\n\"That's a locator.\"\n\nHer parents looked at each other.\n\n\"Where did you get that, Beth?\" her papa asked carefully.\n\nBeth's eyes touched on the worry that each face held, and she felt her face scrunch.\n\n\"They hand them out at the front entrance...\" Rachett said thoughtfully before Beth could answer.\n\nBeth nodded carefully. The nice lady had given it to her to entertain herself with.\n\n\"Do you know what those are for?\" Rachett asked her.\n\nShe nodded again.\n\nBeth knew. She liked the feeling of the smooth glossy surface. Her fingers worked over the cylindrical perfection delicately, with reverence.\n\n\"It is for those Reflectives who need to find their sector,\" Rachett explained neutrally.\n\nHe smiled down at her.\n\nBeth was certain he understood she wasn't a regular five cycle.\n\nThen his smile faded as he no doubt recalled her gender. Beth was weary of being thought of as lesser because she was a girl.\n\nShe'd heard the whispers of the bullying that was so commonplace within the ranks of the Reflectives.\n\nThough, of course, everyone had heard the story of the swarm that had descended on her day of birth.\n\nPapiliones did not lie.\n\nRachett shook his head, obviously having made his decision. It was safer\u2014for everyone.\n\nBeth narrowed her eyes on the vision of his soft thoughts of her future role.\n\nRachett stood. As did Beth and the parents who were not of her blood.\n\n\"I'm sorry. Beth will be placed in... inter-dimensional communication training. An excellent program and critical calling for the female Reflective,\" Rachett stated, lacing his hands together, effectively closing the meeting.\n\n\"Thank Principle,\" Beth's mother murmured. She shot Beth a look that let her know she had been naughty for sharing her crazy intentions after being instructed to remain silent.\n\nHeat began to build in Beth's chest. She recognized it immediately: anger.\n\nIt began at the core of her body and swam out like molten lava, lashing through her circulatory system in defiance of being contained.\n\nBeth did not want to be a weak female.\n\nShe was not.\n\nThen Beth did what all children do\u2014she threw a tantrum.\n\nBeth threw the marble at Commander Rachett.\n\n\"No!\" she shouted in a clear, bell-like voice that stung the ears and raised the hair on the back of his neck.\n\nBeth's body reacted to her emotions and the spinning ball of glass coated by the forbidden mercury.\n\nIt spun, and Beth tracked it automatically, as if it were as natural as taking her next breath. It was part and parcel of being Reflective.\n\nThe heat inside her body coalesced, bursting painfully and beautifully, and she gasped as the ball moved toward her, then slammed into her in midair.\n\nHer small body morphed into the narrow strip of shimmering ribbon that all Reflectives become when they jump.\n\nBeth allowed all of it to happen in an instinctual slide of circumstance and raw emotion. Her new form lashed like a shining whip, absorbing into the shell of the spinning glass as it sailed in the air for its two seconds of flight.\n\nCoolness washed away the heat, and she spun with the ball... and went somewhere else, in a falling stream of fire bathed by ice.\n\nRachett stilled, dazed, as the ball that Beth Jasper had used for transport shattered at his feet.\n\nHe and Beth's parents stood stock-still, their bearings gone.\n\nCommander Rachett picked up a shard, and one of his eyes caught in the mirror-like image. He didn't like what he saw there\u2014fear.\n\nHis own, and that of Beth Jasper's future within The Cause.\n\n# CHAPTER ONE\n\nJeb Merrick\n\npresent day\n\nJeb strolled dead center into the group of Reflectives who'd come to attend the finals of the new class of Reflective trainees.\n\nThe entire coliseum was packed nut to butt, and the ground beside the ring was standing room only.\n\nIt was the female, Jeb determined easily\u2014she was the draw for the day. If he were honest with himself, he would have admitted the same. After all, the last female combative had been killed in action over a decade ago. Jeb had heard of it, but it had been before his time.\n\nThis one was different.\n\nFor one, she bore the scars of their calling. Her elegant limbs were littered with pockmarking and wounds in various stages of healing. Even with the advanced recuperative powers of a Reflective, Jasper was a mess.\n\nIt was such a shame; she was a beautiful female, if not the Papilio ideal. She'd refused to become the he-she that many assumed she would and had retained her femininity, despite the brutal calling of the Reflective. He supposed her gender could be to some advantage in a mission to one of the other sectors.\n\nJeb found a corner and put his back to it, watching the small group of inductees warm their bodies inside the practice area before the final sparring.\n\nJeb liked to possess a vantage point that allowed him to see everyone coming through the portals, windows, and otherwise. His height put Jeb at further advantage. With his six-feet-four frame, he skimmed most of the heads in his line of sight.\n\nThe ones he couldn't see over were of his kind, Reflective warriors of The Cause.\n\nHis eyes instinctively scanned the vast interior of the coliseum. He took in the stands filled with the government of his world. English was not their first language, but it was used in more than three quarters of the worlds they policed. Latin was the primary and native language of Papilio.\n\nAll Reflectives were fluent in the primary languages of the thirteen sectors they held as their responsibility. Latin was spoken exclusively by Papiliones.\n\nJeb stood up straighter, gaining another couple inches of precious visual real estate and caught sight of his own team. At age twenty-three, they were three years past their own graduations.\n\nHis team began taking up the remaining corners of the main floor surrounding the ring, while the civilian population moved upward in soaring floor-to-ceiling tiers with marble benches.\n\nThe thousands of people who'd sat there before this crowd had worn broad divots in the soft cream-and-peach-veined marble. Centuries worth of observers had witnessed the annual ceremony.\n\nAll welcomed the newest recruits. The civilians did not want to know how they were protected. They wanted to know only that they were.\n\nJeb felt a smirk form.\n\nSometimes he wondered why he jumped.\n\nHe grew solemn as he waited, and then he saw her\u2014Beth Jasper.\n\nHe'd seen her about in the Barringer Quadrant, shopping for sundries and such things\u2014but he'd never been so close. A different woman seemed to have inhabited her body today.\n\nGone was her softness he'd seen in his earlier observances. Instead, he saw a woman with nothing but hard angles and planes. An indifferent and cool stare met those of her team and those that she would fight.\n\nNot a one had softness for her.\n\nBeth stood alone.\n\nJeb looked at the five others\u2014all male\u2014and a slight furrow tied his brows together.\n\nShe was sorely outmatched physically, though the recruits were all equal in years. Recruits graduated each year in small groups, all at twenty cycles of age, as was tradition.\n\nJeb studied Jasper, assessing her as all Reflectives could. She stood at five feet two, and curves she couldn't mask, even beneath the bland Reflective uniform, stood in stark relief. Her tight black braid stopped at her waist. An unusual length for a woman of his people, it was an unheard of length for a Reflective.\n\nPerhaps it was a bid for femininity in a role that was exclusively male?\n\nJeb reluctantly moved his gaze to the other five in turn, searching for his new partner. Jeb found babysitting loathsome but necessary. Otherwise, they would have a troupe of Reflectives bouncing from one world to the next, where they shouldn't land.\n\nJeb felt his lips twitch. He had been the same when he was twenty cycles: an ignorant hot head. His former mentor had seen fit to beat him into understanding. The Cause did not tolerate ignorance\n\nIt was Jeb's turn to mentor a new recruit since his three-year first partnering was at an end.\n\nThe interior lights of the coliseum switched on, spreading the solar-powered illumination to every corner. It washed the faces of the Reflective inductees in an eerie mockery of false illness, casting a sickly yellow over their flesh.\n\nReflective Kennet stood in the far corner, exactly opposite of Jeb's position, and lifted his chin in greeting then received one in return. Kennet was wearing his dress uniform. He was on duty. That meant his ass could be snatched to one of the other twelve sectors at any time.\n\nYet, he was here.\n\nJeb allowed his eyes to run over his compatriots dress uniform, noting the deep navy, which looked black from a distance. The Reflective crest was the only striking addition.\n\nThe butterfly rode high against his left breast, standing vigil over his heart. The iridescent rendering had been executed with real gold and silver, and microscopic jewels were used in the multicolored threading. Only a small shift of movement was necessary for the crest to alert passersby that the uniformed people were Reflective.\n\nThey were the slaves of protection for Papilio.\n\nJeb's musing was cut short as the chime donged six times for the six candidates.\n\nAll would fight and be judged in various degrees of worthiness. The illegal betting had been deep and vicious.\n\nBeth Jasper was the underdog.\n\nHumanity had come to see the female fall.\n\nThere were only two rules: no blades and no death.\n\nHe studied the graceful Jasper as she warmed up. Had he been a betting man, he would have bet on her.\n\nJeb Merrick understood much could be accomplished without death as an end result. He was profoundly happy that he was not standing in that ring, preparing to beat a female into the mat. Jeb wasn't sure he could have done it.\n\nHe understood it for the weakness it was.\n\nJeb's eyes fell on the favored male in the class, Lance Ryan.\n\nLance could do it.\n\nJeb took in the young man's predatory eyes, which were trained on Jasper, tensed without being aware. The idea had seemed fine when he'd entertained attending the ritualistic Reflective ceremony. It was a bloodthirsty hold-over from centuries past. Yet, like many traditions that were no longer necessary, it had flourished.\n\nJeb unconsciously leaned forward as the first recruit stepped forward and bumped fists with the well-known Ryan. Well-known for being a jack ass, Jeb thought.\n\nNo one truly liked Ryan, yet he had garnered the respect of many through brute force and jumping prowess.\n\nRespect earned through fear instead of deeds is not truly respect.\n\nRyan was ferocious in sparring and the martial arts. A keen jumper, he was rumored to be able to jump through reflections as small as a fist\u2014but not while they were in motion.\n\nThat was a rare skill.\n\nHe had heard of only one Reflective who could jump as a drop of rain fell from the sky. Jeb shook his head in disbelief. Legend... yet, he wished he could have been there to witness such a thing.\n\nThe men raised their fists from the greeting then placed them over the plain insignia of their sparring tunics.\n\nThey stepped away from one another.\n\nA huge gong sounded, making Jeb's teeth thrum, and the two recruits burst into each other with a smack of flesh and bone.\n\nJeb couldn't help but be riveted.\n\nRyan's beauty as a fighter was an awesome thing to behold. He landed punch after punch\u2014all organ strikes\u2014into his opponent.\n\nThe other man\u2014Jude Calvin was Kennet's new partner, Jeb vaguely remembered\u2014came in close and took away Ryan's considerable strike advantage.\n\nCalvin wrapped his substantial arms around Ryan's torso, swinging a man that weighed two hundred fifty pounds as if he weighed an ounce, and pile drove him into the mat.\n\nSpectators felt the impact as a reverberating punch.\n\nRyan shot out his arm and smashed his flat palm into Calvin's nose. Ryan ignored the low boo from the crowd.\n\nBlood burst from the offense, shooting like a bright-red geyser as Ryan leapt off the mat, smearing the mess he'd made of his equal.\n\nJeb's head swiveled toward a female voice rising above the crowd's noise.\n\n\"Shoot, Calvin... shoot!\"\n\nA small fist swung above her head for emphasis, and the crowd hissed their displeasure at Jasper's coaching from the sidelines.\n\nCalvin shot, taking Ryan's long legs out from underneath him as he sprang forward, his nose bleeding like a sieve.\n\nCommander Rachett stood in the corner of the ring in typical stoic silence, his body tense like a snake before it strikes, as Ryan's body smacked the mat then took a hard bounce, making an echoing slap that silenced the crowd.\n\nJeb heard the oohs and aahs of low-grade fear all around him.\n\nThis time, Ryan rolled Calvin over and twisted his arm into an unnatural pretzel position. Shit, Jeb thought, he's got him in an arm bar. He'd picked up the classic move from a jump to Sector Three, Earth.\n\nA place he should not have visited yet, Jeb thought with unease. The class-seven world was for partnered jumps only.\n\nCalvin tapped out, hitting Ryan lightly on the leg behind his own.\n\nBeth Jasper told Jeb what would happen next. Like a cat losing its balance, she moved forward as Ryan snapped the arm he had locked behind Calvin. He roared in agony, holding his injured limb as Ryan's boot came high over his head to smash his face.\n\nJeb stilled.\n\nSurely Rachett will disallow this?\n\nBeth moved behind Ryan, like a shimmer of water on a sheet of glass.. She executed a spinning kick that knocked the standing man on his ass. Beth bounced away in avoidance, her fists riding beside her jaw, fear swimming in her eyes.\n\nCalm in its economical movements, her body belied the windows to her soul.\n\nRachett stepped away as medics pulled the moaning and shocked Calvin away.\n\nHe would heal.\n\nBut that's not the fucking point, is it?\n\nRyan lacked integrity\u2014a critical component of the militia that comprised the Reflective.\n\nRyan stood, his eyes nailing Beth. Her timely intervention had screwed the order.\n\nThey circled each other cautiously.\n\nJeb knew Jasper had no friends within the trainees circle. However, she'd moved almost compulsively to help Calvin.\n\nWhile every other recruit had observed another being cut down unfairly, Jasper had acted.\n\nAnd she would pay.\n\nPrinciple, this will not end well.\n\nJeb's guts churned. He wasn't easily affected by fights and blood, but as they said on Sector Three: this was wrong on a hundred different levels.\n\nJasper backed up, neatly outside of Ryan's long reach, which was easily twice her own. She appeared to be following her training, relying on a drumbeat that was part of every Reflective's internal clock.\n\nIt wasn't enough, though. Ryan caught Jasper before she had a chance to block his assault. He nailed her gut in a sucker punch then landed a subsequent fist into her jaw.\n\nBeth was already moving evasively, thank Principle, or she would have been out and at his mercy.\n\nRyan showed no mercy.\n\nJasper fell in a spinning backward arc, landing with her palms splayed behind her to arrest her fall. Blood from her cut lip splattered the mat.\n\nRyan stalked toward her, hatred leaking from his every pore. Their final match played out in a sick parody. Unforgiving eyes watched Jasper from every corner of the mat.\n\nRachett's tense voice rumbled from a distance, \"Get the fuck up, Jasper.\"\n\nJeb's felt his face tighten into a scowl, though Rachett had been just as tough when Jeb was a recruit.\n\nJasper swung her head back and forth as though clearing it.\n\nBlood from the blow she'd taken fell like scarlet rain.\n\nRyan smiled, his hands curling into abusive fists of presumed victory. He spoke quietly so only Jasper heard, though Jeb leaned forward to try to catch his words, as did everyone else.\n\nThe roar of the crowd made it impossible.\n\n\"This ends here, Jasper.\"\n\nA cruel smile overtook his face. \"The Reflective doesn't have room for mongrel females.\"\n\nJeb's eyes sharpened on her utter stillness.\n\nHer form began to waver, shimmering on top of the bloody mat.\n\nJeb squinted at her, thinking maybe his eyes were playing tricks on him.\n\nThe noise of the crowd was disorientating.\n\nRyan flicked the switchblade as smoothly as he'd been trained to do. Training blades were all ceramic.\n\nJasper wore the scars to attest to that, but reflective blades could still be had on the black market for the right price.\n\nLooks like Ryan paid.\n\nJeb watched the shining metal, his innate ability instantly online around a reflection, and his talent hummed with want. His eyes met Kennet's, and all eyes went to Rachett, wondering what he would do to Ryan for producing an illegal weapon.\n\nThe blade's mirrored surface shimmered in the low lights that bathed the interior of the coliseum.\n\nHoly fuck.\n\nJeb began to push through the people. The situation was going to get ugly.\n\nNo, check that\u2014gruesome.\n\nRyan planned to murder Beth Jasper; maybe he always had.\n\nJeb could let an inductee take licks, abuse, and unfairness. But one Reflective would not kill another on his watch.\n\nWhy, for the love of the Principle, has Rachett not interfered?\n\n\"Hey!\" a man protested as Jeb pushed him aside.\n\nThen he saw Jeb's uniform and silently moved, as did everyone else in his path.\n\nThe crowd parted like the Earth's fabled Red Sea parting; Reflectives had that effect.\n\nJeb grabbed the ropes around the perimeter, hesitating as Rachett bellowed too late, \"No blades!\"\n\nHis voice carried a note of high-keening fear.\n\nJeb swung to face his Commander.\n\nHe had never seen or heard fear from Rachett. When all inequalities of the fight had been dismissed\u2014Ryan's size against Beth and her gender\u2014he'd finally taken notice when an illegal weapon was produced.\n\nIt was beyond bizarre. None of it made sense.\n\nJeb saw the whites of Jasper's eyes. The inky tail of her braid was wet with her blood. Ryan's blade swung so close to her face that its breeze lifted wisps of her hair. She crab walked backward in an awkward scuttle of escape.\n\nRyan braced himself as his commander screamed for Ryan to stop, but he ignored the directive.\n\nRachett stepped forward too late to stop his best inductee from gutting another recruit as a justified elimination tactic and grabbed Ryan's arm.\n\nBut the knife was gone.\n\nIt was already singing through the air in an expert trajectory aimed at Beth.\n\nThe blade spun in the combustible silence of the coliseum as the crowd held a collective breath.\n\nJeb strode toward Jasper, but she seemed unaware as her dark eyes tracked the knife.\n\nJeb's eye's hadn't lied. One moment, she was solid. The next, she became opaque.\n\nThen she was gone.\n\nJeb had seen many jumps, but never a female's\u2014and never into something so small. The crowd watched as a glittering rope of iridescent white, like a pearl with a rainbow wash, slammed into the blade.\n\nJasper's body disappeared then reappeared in the thin reflective ribbon of the jump as it collided with the metal, as she'd meant to.\n\nWhen the knife landed in the mat, its tip sank deep into the soft surface with a twang.\n\nThe silence was deafening.\n\nBeth Jasper had vanished. Only her blood remained as grim testimony to her presence moments before.\n\nRachett fisted Ryan's tunic, jerking him close.\n\n\"You dumb fuck,\" he began with the quiet menace he was known for. \"All you had to accomplish was keeping weapons out of it. You could have pummeled her into the mat in a fair spar.\"\n\nHis eyes pegged Ryan's in blatant disgust.\n\n\"Now\"\u2014his flat eyes locked with Ryan's\u2014\"she's jumped. She won because you couldn't contain your shit.\"\n\nJeb's eyes connected with Kennet, who was across the ring from where he stood, and the other man was just as stunned. Jeb glanced at the blade embedded in the mat and shook his head in disbelief.\n\n\"There's no way!\" one of the Reflective recruits said quietly. \"That's a six-inch surface. She's a half-breed... nobody can jump that.\" He scoffed.\n\nBut somebody had. Beth Jasper, female, half-breed... had just shown her hand.\n\nIt looked like aces high.\n\nThe crowd began to disperse, their eyes roving for the missing Reflective female who had just made history.\n\nThere would be no jeering in her future, only jealousy.\n\nRachett reiterated what they'd always known, though a few had chosen to ignore.\n\n\"The Principle chooses who it will. There is no logic. That's why when we have an opponent. We do not underestimate their skills. Let this be a lesson to all who fight,\" Rachett expounded, spinning in a slow, deliberate circle, his eyes falling on the inductee recruits, the Reflectives, and the lesser audience who remained.\n\n\"Be ready,\" he finished, landing a final, leaden glance on Ryan before he stalked out of the coliseum. Guards moved up beside Ryan. His infraction would land him on Sector One, for certain. No Reflective wished to jump there.\n\nThis was an epic clusterfuck if there has ever been one.\n\nJeb groaned.\n\nAs the recruits filtered out, Ryan's defiant gaze challenged all who dared look his way as he was cuffed with non-reflective cuffs. One of the guards jerked the blade out of the mat, giving Ryan narrow eyes.\n\nJeb's gaze squared off with Ryan until he dropped his gaze and the guards escorted him out.\n\nJeb stared after Ryan's back. He ran a frustrated hand through his cropped hair.\n\nHe knew what this disturbing mess meant for him. Jeb would be tasked with locating Jasper. His primary task was retrieval. He was meant to be reassigned momentarily.\n\nHowever, it seemed that it would take longer than a moment.\n\nThe crowd thinned, and Jeb stared at the drying blood on the mat, the comments of those around him the same.\n\nAwe mixed with fear was a bad combination. It could be a recipe for many things. When Beth returned, what reception would she find waiting?\n\nHe knew the people would forget Ryan's transgressions against her. All they would remember was her jump.\n\nHe would never forget it.\n\nJeb lifted his head at a small noise. Daphne, a beautiful Reflective, came toward him, her hips swaying so he would notice. And he did.\n\nBut even as her lush body moved toward him like water finding a crack in a stone, his mind was on another female, the newest member of The Cause: Beth Jasper, a jumper without compare\u2014and his new partner.\n\n# CHAPTER TWO\n\nBeth rolled out of her self-imposed tunnel of fire and ice without finesse or regard to safety.\n\nHer reaction wasn't too different from that of brave soldiers cornered at the edge of a cliff. As the enemy closes in, do they stay and get slaughtered? Or do they jump, hoping to live and fight another day?\n\nBeth had jumped.\n\nShe'd leapt at a spinning blade that made her nauseated to track. She'd known what the landing would be.\n\nHowever, she'd been at the theoretical cliff as Ryan's knife beared down, not a soul to stand in her defense.\n\nBeth exited the tunnel like an infant during a birth gone wrong.\n\nShe hurtled out of the sucking chasm that quantified the pathway that only Reflectives could travel and tried to loosen her body, remembering Rachett's words:\n\n\"Behave like a drunk imbecile when you land\u2014every piece of you loosen,\" he'd said, and Beth remembered the truth in his pale eyes. \"Remember, the Principle guards drunks and small children.\"\n\nThere'd been good-natured laughs all around\u2014but not at this moment.\n\nBeth knew she would land without forethought.\n\nI'll heal.\n\nHer body naturally tensed for landing, and she knew to resist that instinct.\n\nPain lanced her as she was purged from the end of the pathway. And Beth fell. Hard.\n\nThe crushing impact stole her breath.\n\nShe lay on a pebbly surface of rough stone, watching cumulus clouds form deep ripples in the blue sky as her lungs begged for oxygen.\n\nThe temperature was sultry. Her fingertips burned against the surface of the stone.\n\nHer chest opened to the insufferable heat and Beth took great whooping gulps of oven-like air.\n\n\"Mommy, mommy,\" a youngling called out.\n\nOh no, Beth thought, experimentally moving her toes, witnesses.\n\nA loud roaring filled Beth's ears, and she tried to move to find its source, but she could not force her body to cooperate.\n\nTwo forms blocked the fluffy white clouds, their shadows cooling her. The little one had long blonde hair, too much brown to be truly light. In one hand, she fisted a bear, and the thumb of her other hand was in her mouth.\n\n\"Why is the lady in the middle of the road?\"\n\nGood question. Beth tried to move and moaned through a hiss of pain. Back's broken. Her situation was almost as bad as Ryan trying to have her meet the Maker.\n\nA woman, too young to be the child's mother, leaned forward. \"Are you okay?\"\n\nNo, I've fractured some vertebrae, and I'm on the wrong damn planet, but otherwise, things are just great. Beth did a mental eye roll and began to review their diction.\n\nAll sector language had been hammered into her from the time she was five cycles.\n\nEnglish, twenty-first century, Sector Three\u2014Earth.\n\n\"Yeah,\" Beth croaked in English through her teeth.\n\nThe planet was a Hades of a lot different than its simulations. Beth hadn't jumped, except for brief explorations, and had never encountered other beings other than when she'd traveled when she was five.\n\nThat had not gone well.\n\nThe little girl cocked her head and gave Beth a strange look.\n\nBetter work on my accent.\n\nThe woman moved out of her line of vision and the roaring gnashing gears became unbearable.\n\nWhat in the inferno is that?\n\nBeth screamed in pain when she tried to move herself, vulnerable and laid out Principle knew where.\n\n\"Shhh.\" The little girl touched her arm with sticky hands. \"Mimi be right back.\"\n\nBoots crunched closer, and Beth tensed.\n\nShe could do nothing, but it was difficult to not act the warrior even as injured as she was.\n\nA male of considerable size moved in front of her and Beth assessed him. Six feet, two hundred pounds. He moved with a languid peace, and she knew instantly that he could handle himself in a moderate engagement.\n\nAll Reflectives assessed. It was part of who they were.\n\nBeth was pleased by the knowledge that he would not last in match with her, though he had her by nine inches and ninety pounds.\n\nHe stooped; his light brown eyes were kind.\n\n\"Well, little lady, looks like someone's dumped ya here.\"\n\nHe spit a stream of brown liquid to the side.\n\nClever male. Beth's lips curled.\n\nThen he touched her, and she shouted, \"Do not!\"\n\nShe panted, her hands gripping his shoulders. \"Move me,\" she finished.\n\nHe smiled.\n\n\"You're not staying in the road, girl.\" His eyebrows shot up to a fine bristle of dark-blond hair circling his head like a golden down. Beth tried to shift and cried out through her clenched lips.\n\n\"No, no... girl. Hold your horses.\"\n\nBeth searched around for animals. Seeing none, she turned back to him.\n\n\"Literal little thing, ain't ya?\"\n\nAnother brown stream followed the first, and Beth wrinkled her nose. Vile.\n\n\"Jeremy... we can't just leave her here.\"\n\nWith her eyes, Beth followed the woman named Mimi.\n\nBeth scanned her vitals. A light film of sweat dewed her forehead, and she wrung slick hands over and over in a nervous roll of reaction with a swamped pulse as if she had a bird trapped at the hollow of her throat.\n\nBeth's eyes went to the small one, and she gave her a tired smile. The male\u2014Jer-e-may...? Jeremy\u2014had slipped his hands underneath Beth's back.\n\nHis eyes widened.\n\n\"She's full of blood, Mimi...\"\n\nThe young woman came forward, her eyes searching Beth's body carefully, and too late, Beth understood what Mimi was looking at.\n\nHer sparring uniform was still whole, spattered with blood and bearing an emblem that was as foreign as she was in this world.\n\n\"Let's take her to the hospital,\" Jeremy determined, and Beth stiffened as he lifted her.\n\nThree's could never study her body. They would find things they shouldn't.\n\nBeth screamed as agony tore through her.\n\nThat was when Merrick made his entrance, and all Hades broke loose.\n\n*\n\nJeb\n\nJeb folded his arms across his chest mummy style and felt the final twist when he would be expunged. Then he flung his arms wide at the last moment, moving his legs as though he were walking in midair.\n\nSoon enough, his feet would meet something solid.\n\nHe hit the ground, his shins singing with the impact. Jeb was grateful it wasn't a slope but a manmade material. He could have gone ass over tea kettle if he'd landed on a hill.\n\nHe had.\n\nJeb ran at a full-out sprint to shake off the momentum of the jump, then slowed to a jog, then a walk.\n\nHe shook out his palms, restoring the feeling in his extremities. He knew from experience it would be a full minute before he was fully rejuvenated.\n\nHe stopped, closing his eyes.\n\nReflectives' hearing was the finest of any being, save vampires.\n\nJeb heard Jasper scream, and his head snapped in that direction.\n\nHe did not slow upon seeing the three humans that hailed from Sector Three.\n\nHe gauged the century to the decade from their clothing. Then he determined the year when the male who posed the greatest threat spoke.\n\n\"What the hell is this?\" The male rose, cradling Jasper, whose face appeared more pale than usual.\n\nJeb let his sensors run from his body like tendrils, feeling her injuries. She was badly hurt from the fall: L-1 and 3 were fractured. Jasper's foot twitched, but her legs remained immobile.\n\nHer dark cautious eyes found him.\n\n\"Merrick,\" she rasped.\n\nJeb mentally revised his superficial diagnosis. Her vocals were compromised, and he determined through greater psychic exploration that C-2 was damaged, as well. He exhaled loudly.\n\n\"Jasper,\" he greeted.\n\nHe executed his internal exam of the male who held his yet-to-be-assigned partner, and Merrick found him wanting.\n\nJeb dismissed him to assess the females. One\u2014caucasian, early twenties, five feet six, one hundred thirty pounds\u2014held the hands of a youngling, perhaps four years of age.\n\nHe dismissed them as well.\n\nThreats processed and noted, Jeb crossed his arms, folding them over his awful, classic twenty-first-century garb of stiff denim that made his balls feel like prisoners in a greenhouse and thieved his mobility.\n\nHowever, they were the clothes of... Jeb looked down at his locator fashioned of mercury: 2030.\n\nHe had been to Sector Three many times. He wrinkled his nose as he detected the levels of pollution exceeding those he was accustomed to.\n\n\"This is my wife. I'll take her to...\" Jeb considered his vocabulary carefully. \"To seek medical treatment.\"\n\nHe smiled, pushing it into his eyes. Humans liked that. It settled them like colts about to run. They hadn't seen his landing, so his lies should work.\n\nThe male's brows dropped over his eyes like a brick.\n\n\"I don't know who the hell you really are, but your ass just dropped out of the sky, and whatever claim you have on this young woman is null and void, Jack.\"\n\nJack?\n\n\"Is he an alien?\" the youngling asked, with wide, distrusting eyes.\n\nFuck.\n\nBeth grimaced. \"Smooth, Merrick, way to blend in.\"\n\nJeb scowled at her, stalking toward the male, who did not back down.\n\nDoes he not know that Jeb Merrick is a warrior of The Cause? Of course not. However, it was imperative only that Jeb know.\n\nThe Code ran through his mind, a blend of language from thirteen worlds, policed by only one.\n\nThe Reflectives will advance nothing, protect all, exploit the evil for what it is and defend The Cause without exception.\n\n\"Yes, I know it looks a little out of the ordinary.\"\n\nThe youngling popped a thumb into her mouth.\n\nIt was not going well. He would have to extract moments from these humans' brains. He gave a disgusted sigh; extraction was his least favorite task. It was akin to mental rape, or the thrall the vamps were known for.\n\nJeb calculated the distance at ten meters.\n\n\"Jasper, I need the distance.\"\n\nThe male glanced down at Beth, and she extracted a small silver sphere that glinted in the late sun of the day like captured silver.\n\nJasper held up the marble, and Jeb narrowed his vision on the warped and glossy pewter finish.\n\nHe could do a jump with something that small from his distance.\n\nHeat washed through his body as Jeb pushed his being toward the sphere.\n\nHe could see his pale gray eye reflected even at that distance.\n\nThe concentration to jump took seconds.\n\nJeb thought only of the sphere. When nothing but the shape was in his mind, he spun out toward it. His body snapped like a rubber band and flashed to Jasper in a heartbeat.\n\nSuddenly, Jeb was nose to nose with the male.\n\n\"Give me the woman,\" Jeb commanded, his mental dominance sliding into the male before him.\n\nHis arms went loose, and Beth began to slide out of his grasp.\n\nJeb caught Jasper easily as the male, slack jawed, awaited new orders.\n\nJasper bit her lip to keep from crying out, and Jeb tucked her arm under his own. He looked at the people who stared at him. His gaze shifted to the youngling; nothing could be done with her.\n\nThe younglings' resistance because of their age was renown.\n\nHe worked on the man and woman until they believed they had pulled over from fatigue and that his appearance was no more than a bad dream.\n\nThe youngling was different.\n\nHe sent the young one's protectors away and dropped to his haunches easily, though Jasper lay like a dead weight within his hold.\n\n\"Don't... hurt her,\" Jasper whispered in their native tongue, gritting her teeth against her pain.\n\nHer eyes fluttered as she fought fatigue. She was badly injured and her body was trying to heal itself through rest.\n\n\"I am not a savage.\"\n\n\"I have heard stories,\" she replied in Latin.\n\nJeb was disgusted that Jasper would think him capable of harming a youngling.\n\nHe focused on the girl, but Jasper's comment rung inside his skull unpleasantly.\n\n\"Little one,\" he began, switching to English. \"Who do you think we are?\"\n\nThe little girl looked carefully at Jasper.\n\nThen her eyes moved to Jeb, and she looked unafraid.\n\n\"Angels,\" she replied with the logic of a four-year-old.\n\nJeb went through his mental inventory, looking for the meaning, and though they were not the perfect heavenly creatures the girl thought they were, it was a safe identifier.\n\nJeb smiled at her.\n\n\"That's right,\" he lied smoothly as he cupped his large hand over the back of her head.\n\n\"Thank you for your watch care after my partner.\"\n\nShe nodded, though she thought the angel spoke oddly.\n\nThe lovely people blinked away like falling stars in the middle of a little-traveled road, while her relatives sat like corpses in the cab of Uncle Jeremy's truck.\n\nShe stared until twilight descended and her adult relatives finally awoke as though they'd just had a deep sleep.\n\n*\n\n\"Principle, that was close,\" Jasper whispered.\n\nShe flicked her eyes to Jeb's and added, \"I think you hurt me worse on the return.\"\n\n\"Your grateful attitude blows me away.\"\n\n\"I hate Earth vernacular.\"\n\n\"Tough, get used to it.\"\n\nJeb sat beside her, his mind on that hot Reflective he'd had to leave behind instead of giving her what she clearly needed.\n\nBasically, he hadn't gotten his rocks off because he'd had to chase Jasper down like a skipper. Jeb understood he wasn't being entirely fair. If Jasper hadn't leapt, she would've been killed. Still, his fun had been curtailed, and it made him exceedingly grumpy.\n\nHe plowed his fingers through his tousled hair and expelled a frustrated breath, leaning back in the chair near her bedside.\n\n\"Please, I'm hungry,\" Jasper said, her upturned lips telling Jeb that she was pleased by his temporary slave status.\n\nJeb glared at her as he spooned another mouthful of the gelatin into her full lips, now marred only by a shallow cut and a yellowing bruise. Days of healing had taken only hours.\n\nHer back would be fully mended by the morrow.\n\nA glob of the green goo sat on one plump lip, and he scooped it off and stuffed it into her mouth.\n\nNot that it would keep that sharp tongue at bay.\n\n\"What of Rachett?\" Beth asked. She gnawed thoughtfully at her bottom lip in between bites, giving away her emotions.\n\nJeb busied himself with stirring the green translucent grub. It jiggled obscenely as he loaded another spoonful.\n\nJasper put her palm up to ward off another bite, and he noted how small\u2014and strong\u2014her hands were.\n\nJeb dipped his eyes to the bowl then set it down. \"He attends to Ryan.\"\n\nJasper put her face in her hands.\n\nGo ahead, cry, weak female, Jeb thought uncharitably, his old prejudices vying for position.\n\nBut she simply swiped at her face, raw with healing wounds.\n\n\"Attends to... or disciplines?\" she asked, a defiant hook to her chin.\n\nHe grinned despite himself.\n\nBeth Jasper obviously knew the tenor of their commander. \"A little of both, I imagine.\"\n\nJasper did not smile; she appeared serious, and Jeb found his smile fading as he looked into her delicate face.\n\nHer eyes were as hard as his own.\n\n\"Ryan will retaliate.\"\n\nJeb nodded. \"If he was smart... he would not. It was a clear victory. But he will not be pleased to have been bested by a female.\"\n\n\"It is not that I am female,\" Jasper commented.\n\nHer brown eyes laid hold on his gray ones, and he cocked an eyebrow, folding his arms over his chest.\n\n\"Then what is it, for I know you die on the vine to tell me.\"\n\nJasper rolled her large eyes in her head. \"It is that I am better.\"\n\nJeb inclined his head, conceding the obvious. \"As a jumper.\" He stood, throwing out his hands. \"I have not seen the like.\"\n\nSuddenly, Jeb whirled around. Overcome with curiosity, he gripped the ceramic bars on the hospital bed. His movement caused the thin snaking plastic tube that bit her flesh with a needle to sway like an undulating snake.\n\nJasper smiled. \"And only Rachett truly knew what I was capable of.\" Her hands toyed with the many threads of the unraveling border of the wool blanket that covered her.\n\n\"Why? It is a rare gift, to jump into something that small. Why would you not spread the proof of that talent far and wide?\" Jeb asked, twirling in a neat circle in the middle of the room.\n\nJasper met his eyes, and Jeb saw something there that caused him to stop moving.\n\n\"Because,\" Jasper whispered, \"that was not small.\"\n\nJeb felt the air still in his lungs as he moved nearer to her bed. \"Look at me, Jasper.\"\n\nHer gaze rose, unwavering and dark, full of secrets.\n\n\"The six-inch blade in motion... that is not small?\"\n\nJasper shook her head.\n\nJeb pulled a chair across the floor, and it shrieked in protest as it scraped the floor. He twirled it around and sat in it backward. \"Tell me.\"\n\nThe air left her lungs, and she whispered the truth for the first time since their Commander had discovered what she was capable of, since that day when she had leapt into the locator sphere.\n\n\"Mist,\" she answered.\n\nJeb put a fist over his mouth to stifle a noise.\n\nThey were in such trouble. Not he... but a partner that could jump through mist particles? His eyes couldn't even track something so small because of their sheer diminutive size. It did not bode well.\n\nShe leapt by intuition. Somehow, she knew it reflected and could jump into the body of the mist? Unheard of.\n\nAnd she was female besides.\n\nWho is Beth Jasper, and what is her purpose?\n\nShe looked at him with guarded hope, and the look he returned was everything he felt\u2014dislike.\n\n# CHAPTER THREE\n\nBeth\n\nBeth's face fell. Merrick might have been her new partner, but he was definitely not friendly. He treated her like a child he'd had to rescue and feed.\n\nIf Beth could have been an independent, she would have chosen to do so straightaway. However, that wasn't the way of the Reflective. They were partnered for a reason. And though she had not officially been advanced from her inductee status to full Reflective, she suspected Commander Rachett would see it through.\n\nHe was a tough man shaped by experience, but he was fair.\n\nThat was why Beth was surprised that he hadn't anticipated her jump when confronted by Ryan's contraband weapon. The prick.\n\nBeth stubbornly adjusted herself in the bed. I'll be damned if I ask Merrick. He was all but whistling from boredom.\n\n\"You can go. I'm fine,\" Beth said, smoothing her palms down the itchy blanket.\n\nJeb let the front legs of his chair slam down, and it caused Beth to jump. \"Nope... you're the new part of our little team, and I have to suck it up... kind of like a bad marriage.\"\n\n\"You know, I guess you're having some residual from being at Three?\"\n\nMerrick shrugged his broad shoulders.\n\n\"It takes time to come down from the foreign high.\" He winked and Beth felt a tension leave her that she didn't know she held. Merrick could be okay when he wanted to be.\n\n\"Where... have you...\"\n\n\"I like Sector Thirteen best,\" Merrick replied casually, his eyes flicking to hers then away.\n\n\"Not Earth?\"\n\nMerrick shook his head, leaning back in the chair again.\n\nIt tilted dangerously, his muscular weight causing it to creak as he laced his hands behind his head and regarded her thoughtfully.\n\n\"No, Earth's a pain in my ass. They have great language\u2014colorful.\" He gave a short laugh. \"But they take a shit where they live. And the pollution, the crime... and I think we have a tech storm brewing that we'll have to address before too long. Actually, I know it.\"\n\nBeth held a secret desire to visit all the planets.\n\nShe looked at her clenched hands. Her greatest desire was not to police the planets they held steward over but to explore them.\n\nShe kept it to herself, along with the fact that she'd just dumped herself on Earth by random accident. That hadn't been a visit; it'd been a catastrophe.\n\nShe owed her five years of service to the Reflective, and then she was free to explore and find the one who was destined for her\u2014a promised soul mate.\n\nThat is, if Beth survived her service.\n\nThe nature of the Reflective duties were always the issue. With a death rate of one in two, there was no guarantee that a Reflective would live long enough to claim the prize of his or her other half.\n\nStill, the proverbial carrot dangled before them.\n\nBeth raised her chin and leveled her stare at Merrick. He was not a chatty male. His words, like his actions, were economical.\n\nHe came from a family of pureblood Reflectives, and the old feelings of isolation kicked in. Beth did not look Reflective, she was female... and she was small even for her gender.\n\nBut Rachett had seen that essential spark within her and included her in the training.\n\nShe asked him, \"Earth? We will be assigned there?\"\n\nHe nodded. \"Yes, it's only a matter of time. The handwriting's on the wall.\"\n\nHe'd used an interesting idiom that heralded from the Earth people's Bible, a relic of prophesy. Their Bible was not unlike Papilio scrolls that spoke of the Principle. An intersecting bit of beliefs, she supposed.\n\n\"Why thirteen?\" she asked, wondering why that planet held his interest most.\n\n\"I can't manipulate the Band.\"\n\n\"Ah...\" Beth instantly remembered her training of that world. A primitive people had been saved by a futuristic, but interfering, group from Three. Interesting domiciles on that planet, she remembered.\n\n\"So the challenge then?\"\n\nMerrick grunted in enigmatic response, taking a piece of candy out of his pocket. He threw it into his mouth, then jawed it around.\n\nWithout warning, the door burst open.\n\nMerrick jumped from mid-lean, tossing the chair out of the way.\n\nLance Ryan entered, slapping the door against the wall.\n\n*\n\nBeth slithered out of the bed, lightly touching her toes on the cold tile of the hospital floor.\n\nShe scanned the room for anything to use as a weapon or possible escape.\n\nDamn, her weapons were hung neatly from a new uniform at the back of a chair in the corner.\n\nOf course they were.\n\nWho needs weapons when there is an armed Reflective at the door? Beth narrowed her eyes. The guard had been bought and paid for by Ryan, obviously.\n\nBeth's gaze bored into Merrick's muscular back, and her heart stuttered. Merrick would never go against Ryan.\n\nThey were most likely in cahoots. That was probably why Merrick had been so cavalier, feeding her and taking care of his injured partner as they were required to do in the event of mishap.\n\nShe was doomed.\n\nMaybe not.\n\nShe took a deep breath, contemplating the unthinkable. Beth's eyes roamed the four corners of the room, trying to locate anything reflective.\n\nBut the hospital had been scrubbed of anything that could refract.\n\n\"Well, hi ya, Ryan,\" Merrick greeted him like an old friend, still in full Earth dialect.\n\nRyan frowned.\n\n\"Get out of the way, Merrick.\"\n\nBeth backed up, moving toward the window, glancing outside.\n\nShe was at the metaphorical cliff again. She wasn't healed fully from the last jump. Her injuries would certainly be worse if she jumped again.\n\nSo few held Beth in any regard that it might have been her only free pass that Merrick had been sent to collect her.\n\nNo one would come a second time. Her jump would leave her trapped in a foreign sector void of her people, unable to travel decisively without locators.\n\nJumping was dangerous without a focus sphere.\n\nShe could end up anywhere... or any time.\n\nBeth shuddered. But she supposed that fate was better than death.\n\n\"Nope, can't do that, you colossal fuckup.\"\n\nBeth turned around, her mouth agape.\n\nDid I just hear Merrick right? She had, judging by the expression on Lance Ryan's smug face.\n\nThat awareness in Ryan's expression was beginning to leak away.\n\nAnd Merrick's delivery had been the most comedic of all. He'd spoken as if he were commenting on the weather and had found it fine.\n\n\"Let me pass, Merrick. No one wants her to live.\"\n\nBeth's eyes met his over Merrick's shoulder. \"You'd be doing the world a favor if you took a coffee break right now. Just let it happen.\"\n\nMerrick planted his feet, his arms loose at his sides, and regarded Ryan like a bug. Beth had moved into his peripheral vision. She'd also caught sight of an outside streetlamp through the window.\n\nIts glass solar panels shone like a black mirror.\n\nRyan somehow knows, knows I ready myself. My desperation is plain to whoever searches for it.\n\nHer body bore the scars of his physical bullying. Her mind held them, as well.\n\nHeat climbs, searing her insides, Beth's heartbeat is a whoosh of blood in her ears.\n\nRyan's eyes snagged on Beth, then with a roar, he surged forward.\n\nMerrick pivoted on his right foot. Already focused on her mark, Beth saw them as only a pinpoint in her vision.\n\nThe shining ebony at the crown of the lamp beckoned.\n\nThen she heard the crack of bone against bone, and blood arced up, hitting the ceiling with such force that it rained back down on the men.\n\nThe sound stopped everything\u2014her focus, her jump.\n\nBeth stood frozen as Merrick went toe-to-toe with Ryan.\n\n*\n\nJeb\n\nHonorless fuck.\n\nJeb was disgusted the guard at Jasper's door had let Ryan through. He was even further disgusted that she'd considered jumping without having sufficient time to heal. Being a sensitive Reflective, he could sense jumping readiness.\n\nDuring the battle in the coliseum, he had sensed Beth's jump before anyone else had. He possessed her signature now.\n\nRyan charged, and something in Beth's expression gave her away. It would be the first thing he would teach her as her partner: a blank face.\n\nBeth didn't have one. A shadow of her every feeling clouded her face. She was, as the people of Sector-Three Earth were fond of saying, an open book.\n\nRyan was a dirty fighter\u2014no surprise there\u2014who thought to take hold of Jeb and unbalance him.\n\nRyan latched onto Jeb's wrist and attempted a foot sweep.\n\nJeb countered, twisting his wrist viciously in the opposite direction of the hold, breaking it instantly as he grabbed Ryan's forearm. He stepped into the fight, not away.\n\nAs he jerked Ryan into the circle of reach, he swung his fist into Ryan's jaw.\n\nAlways engage, never retreat.\n\nThe Reflective motto, he thought with sour pleasure as Ryan moved with him, an apt dance partner in their mutual violence.\n\nRyan head butted Jeb in a deft, hard move with perfect timing.\n\nIt rang Jeb's bell, but his skull was hard, and he spun his cocked fist, driving it a second time the short distance from his hip to Ryan's jaw.\n\nAnd like perfectly cracked glass, his jaw rocketed back, spraying blood onto the ceiling as his teeth speared his own tongue.\n\nJeb popped his flattened palms into Ryan's chest as though he wanted to launch him into the wall or stop his heart.\n\nRyan slammed into the wall, his head smacking the surface\n\nJeb stalked toward Ryan, his fists like meaty hammers of punishment.\n\nBarely breathing, Ryan slid down the wall, his eyes at half-mast.\n\n\"Are we done here, Ryan?\" Merrick asked.\n\nRyan gave the smallest nod possible, his mouth a yawning horror of blood and gore.\n\nMerrick turned to check on Jasper.\n\nThe sun's final rays backlit her, bathing her in red light like a watercolor of blood. It ran down her arms, accentuating her delicate build, and instead of looking sinister, it did the opposite.\n\nShe seemed terribly fragile.\n\n\"Merrick!\" Beth screamed.\n\nHe dropped down and spun.\n\nRyan was above him, a small dagger in one hand, coated with blood.\n\nHis own blood.\n\nHis fingers found the wound and came away slick.\n\nMerrick saw red.\n\n\"You fucking pussy,\" he hissed.\n\nRyan smiled through a mouthful of his own blood and spit it to the side, where it splattered like dumped paint on the pure-white tiles.\n\n\"I'm a pussy that just fucked you.\"\n\n\"Not yet,\" Merrick said.\n\nFuck it, I'll heal on the way. He'd kept his gift a secret, though Ryan would be enlightened forevermore.\n\nAs light as a feather, a smooth rectangle of paper-thin mercury-coated ceramic slipped out of his specially made pocket in Merrick's pants. He tossed, and it landed on top of their mixed blood on the floor.\n\nIt provided a single destination jump.\n\nRyan's expression showed true fear as Merrick punched the blade from the younger man's hand. It hit the floor with a jarring clatter of metal against ceramic.\n\nRyan reacted as all instinctual Reflectives would have\u2014he ground his fist into Merrick's knife wound.\n\nBut Merrick was already on point.\n\nHis eyes held on the flat surface of the locator even as he winced in pain.\n\nHe grabbed Ryan's collar, fisting the material tightly.\n\nThey jumped\u2014only one did so willingly.\n\nMerrick could hear Jasper calling his name down the tunnel the Reflectives traveled.\n\n*\n\nJeb had found himself a dandy of a slope, his fist still attached to Ryan, where it continued its brutal hold.\n\nJeb went ripping down an embankment of sharp prairie grass that sliced and poked as they mowed through it, finally landing on their backs at the bottom.\n\nHe'd thought of Thirteen\u2014and that's where they'd landed.\n\nMerrick was, of course, in perfect health, having healed completely during the jump. The glory in that was Ryan was yet unaware of Jeb's mended state of affairs.\n\nMerrick jumped to his feet and immediately kicked Ryan in the ribs.\n\n\"I swear to Principle I will leave you in this place if you do not retire your vendetta against Beth Jasper.\"\n\nRyan spit more blood into the pasture grass that speared his back. \"What... you want the half-breed?\"\n\nJeb said nothing. Fool.\n\nRyan looked up at him.\n\n\"She is assigned to me, and she is injured. I can't help who I get partnered with any better than you can. I will not stand by and let you kill another Reflective because of your jealousy.\"\n\n\"I am not jealous of that mongrel,\" Ryan growled, coming to his hands and knees.\n\n\"I suffered through her inclusion for the past fifteen years,\" he offered as a lame excuse.\n\n\"No.\" Jeb gazed at the worthless Ryan. \"I'm sure the reverse of that is true.\"\n\n\"Earth lover.\" Ryan spat at his feet.\n\nJeb rolled his eyes, pegging his hands on his hips. \"Yes, I do enjoy Earth. Your point?\"\n\n\"My point is she could be anything... she is not fully Papilion. Does that not bother you?\"\n\n\"I am not looking to breed her but to partner her.\"\n\n\"That is all females are good for.\"\n\nThis is useless. Ryan was a lost cause, but Jeb could teach him caution. He did not wish to look over his shoulder for the next five years while partnered with Jasper.\n\nRyan stood, wisely keeping a respectable distance from Merrick.\n\n\"Where the hell are we?\" His eyes narrowed on Jeb. \"Where did you bring me?\" He whipped his head around, taking in the faraway opaque dome-shaped structures.\n\nA great forest stood to the north of their position.\n\n\"Sector Thirteen,\" Jeb replied coolly.\n\nRyan's face paled. Jeb imagined that took some doing.\n\nHe grinned.\n\n\"This is the most dangerous sector you dick.\"\n\nJeb shook his head. \"Not the most dangerous.\" No one traveled to One by choice\u2014that was a death wish.\n\nJeb noted that he was not the only Reflective who had picked up the local Earth dialect with some precision.\n\nRyan lowered his voice as though anyone could hear them in the middle of the wilderness of this world.\n\nA whisper of cloth against wheat made Merrick turn.\n\nHow wrong I was.\n\nThings instantly went from teaching a lesson to survival, as was often the way of a jump.\n\nA group of men of various sizes, ages, and bearing circled Merrick and Ryan, just out of striking range.\n\n\"Who the hell are they?\" Ryan asked, suddenly less combative toward Merrick than he'd been moments before.\n\n\"The Fragment,\" Jeb answered, sliding his remaining dagger out of the weapons pocket of his trousers.\n\nMade of ceramic, it was designed to survive a jump, as metal could not survive Reflective journeys.\n\nThe cold porcelain was smooth, with a specially arced tip. It was serrated on only one side.\n\nOne of the men in the group called out, \"Join us or die.\"\n\nRyan said, \"I don't know this dialect. I have only used the high language of Thirteen.\"\n\n\"Just another reason why Jasper should remain.\"\n\n\"Fuck me\u2014why?\" Ryan asked, one eye on the group, which was closing in, and the other on Merrick.\n\n\"She is fluent in all sectors.\"\n\nJeb moved forward, hoping to injure enough men so that he could escape. They did not want to find themselves buried within the knot of the Fragment.\n\nThey took no prisoners.\n\n*\n\nBeth\n\nRachett tore into the hospital room, and Beth nearly climbed out of her skin.\n\nThe air still rippled with residual disturbance from Merrick's jump.\n\n\"Where is Ryan?\" Rachett barked.\n\nBeth took a deep breath. \"He jumped with Merrick.\"\n\nRachett's jaw moved back and forth. \"No... Merrick would not take a jump with Ryan.\"\n\n\"I don't think it was voluntary.\"\n\nThey looked at each other.\n\nRachett seemed to notice Beth was in a hospital gown, flashing her backside to the window behind her.\n\n\"The residual still remains,\" she said quickly, throwing her palm toward the shimmering air pocket between them.\n\nRachett studied the area, locked onto something and drove his palm through it in a slicing gesture that ended in his cupped fingers bringing the air back to his nose.\n\nHe waved that little bit he'd collected back and forth in front of his face.\n\n\"What signature?\" Beth asked, moving to stand in front of him, her eyes on his hands as he smelled the air.\n\nHis face fell into grim lines. \"Sector Thirteen.\"\n\nRachett turned to Beth. \"You're so damn hot to jump, jump that.\"\n\nBeth took a step back. \"But... I'm a female. I don't have clearance for that sector.\"\n\nEveryone understood how treacherous that sector was. It had a terrible shortage of females, an estimated one to every fifteen males.\n\nShe would be delivering herself into the lion's den.\n\n\"Afraid?\" Rachett taunted.\n\nBeth stared at him. \"I've never been afraid a day in my life.\"\n\nAnxiety is not fear.\n\n\"That's my girl. Now\"\u2014he touched her shoulder so briefly that Beth thought she imagined it\u2014\"get Ryan back. We have somewhere he needs to go.\"\n\nBeth paused then hit the affirmative decisively. \"Yes, sir.\"\n\nHe laid the universal locator on the hospital bed. Its sheen reflected the spattered blood on the ceiling. Rachett's eyes followed hers.\n\nWhen they lowered to meet hers again, he made no comment.\n\nRachett never asked once if she was well enough to jump... or if she wanted to.\n\nBeth was Reflective, and that was answer enough.\n\nRead More\n\nThe Reflection Series, 1-3\n\nVAMPIRE\n\nAn Alpha Claim Brief-Bites\u00ae Novelette\n\nEpisode 1\n\nNew York Times Bestselling Author(s)\n\nMARATA EROS\n\nTAMARA ROSE BLODGETT\n\nAll Rights Reserved.\n\nCopyright \u00a9 2015 Marata Eros\n\nCopyright \u00a9 2015 Tamara Rose Blodgett\n\nNo part of this publication may be reproduced, distributed, or transmitted in any form or by any means, or stored in a database or retrieval system without the prior written permission of the publisher.\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locales or organizations is entirely coincidental.\n\nwww.tamararoseblodgett.com\n\nTRB Facebook Fan Page\n\nMarata Eros FB Fan Page\n\nCover art by: Willsin Rowe\n\nProofed by: Corinna\n\n# Synopsis\n\nNarah Adrienne is a bounty enforcer in the near future. She runs the seedy side of her game, capturing criminals too dangerous for the local law enforcement. Using unorthodox methods, she finds herself in the crosshairs of the Magistrate for too many allowable kills for the quarter.\n\nAnd her head hurts like hell.\n\nAeslin is part of an elite vampire squad of Turners. A rare sect of vampire scouts who possess the ability to find women with enough undead blood to be turned into full vampire. As the numbers of the supernaturals dwindle, it is the hope of the Nobles that extinction can be a thing of the past with female hybrids.\n\nIn a race against time and common enemies, can Aeslin find the one female who is meant to be turned and also his parallel soul? Or will the fabled carrot the Nobles dangle turn out to be a lie perpetuated by desperation?\n\n# Chapter 1\n\nNarah\n\nMy legs are kicked up on the desk, the toes of my left combat boot stacked on the heel of my right. I lean my feet a couple of inches to the left and look at my boss.\n\nKinda wish I hadn't.\n\nThe tongue-lashing was going to be brutal, and not the fun kind. I just barely hold back a snort of self-serving comedy.\n\n\"Narah,\" Casper leans into the desk, edging a butt cheek on the only part not covered by my assortment of shit. My eyebrow cocks. Perturbed doesn't cover it. If I wanted a butt on my desk, I'd ask.\n\n\"What?\" I bark with anticipation.\n\nA vein in Casper's forehead throbs and I dial it back some. No need to bring the guy to heart failure.\n\n\"What?\" I repeat more good-naturedly, though both of us know I'm nothing of the sort.\n\nHe sighs, scrubbing a palm over his face. Hair almost as white as swan feathers glows under the LED lighting in my tiny office, and his glacial eyes tighten, fighting for a view of my face over the top of my boot.\n\nI jack my feet down and stuff them underneath my desk. My fingers itch to go to my smart phone. Anything to not commit to this conversation.\n\n\"You know we appreciate your skill set.\"\n\nBlah, blah, stinking-blah.\n\n\"But we can't have you pulling firearms on all your bounties.\"\n\nMy bottom lip pops out in a pout. \"It was a very small gun, Casper.\" I put my index and thumb almost touching.\n\n\"Using manstopper ammunition?\"\n\nHe might have a small point.\n\n\"Outlawed in 1898,\" Casper adds.\n\nI shrug a bare shoulder, my tank top skin-tight against my small frame. I find loose clothes are handles to make a bludgeon against me easier. I nail the targets but if there's nothing for them to grab onto, so much the better.\n\n\"I like antique weaponry and ammunition,\" I say with deliberate nonchalance.\n\n\"Really?\" Casper says and I wince at the sound of his voice. \"Let's run down the list of target fatalities.\"\n\nHmmm.\n\n\"Target 103, lethal stabbing.\"\n\nI lean back in my chair and cock my neck back, staring at the dingy ceiling. A water stain has spread out from the center in a pattern of copper lines that somehow resemble a flower opening.\n\nIt's sort of like watching clouds outside, but inside.\n\n\"Narah!\"\n\nI sigh, answering the ceiling. \"Yeah.\"\n\n\"Target 424, beheading.\"\n\nYeah, that'd been messy.\n\n\"Again, I was in fear for my life,\" I say, not sounding defensive.\n\nAt. All.\n\n\"Thirteen times?\" Casper asks softly.\n\nMy chin snaps down and I meet his eyes. Mine are big and golden hazel like a cat's, and that's why I hide them behind my aviator shades. The sun hurts like hell. I've always been sensitive to sunlight.\n\nI shrug. It'll get me nowhere to fight with Casper. Who has the nickname in the office of, The Ghost. No one says it to his face though. I fight a snicker.\n\n\"We are the last profession for use of lethal force, you know. It's not goddamned 2015, when everyone thought all physical force was necessary in some capacity.\"\n\nI'm in the wrong era, I muse with regret.\n\n\"We are the last stand against the criminals of our time. When the police can't nail them, then it's up to us. But Narah,\" Casper scrubs his head, his crewcut bristling from the contact, \"we can't have you killing all the targets. They must be brought to justice.\"\n\nAnd of course, if I kill a target, Casper doesn't get credits. That's what this is really about. I bring in the most targets in our office. I get results and he gets credits for my hard work.\n\nWe stare at each other. I won't break and Casper knows it. \"You're the finest bounty hunter we have. Your instincts are uncanny, and you never let being a woman get in your way...\"\n\nI lunge to my feet and Casper jerks to his, eyeing me warily.\n\nGood, my desk is finally free of his ass.\n\n\"Nothing about me being a woman comes into play here.\"\n\nCasper shoots out an exhale like a cannon. \"Everything about it matters. You're smaller, you're vulnerable to things a man could never be.\"\n\nRape is the clear inference.\n\n\"You think a man can't be raped?\" I bark out a laugh. \"You think that my looks don't disarm. They do, Cas.\" My eyes laser down on him and his shift away. \"You know I'm a proficient, Level Ten.\"\n\n\"Nothing to sneeze at,\" he concedes and opens his mouth to add more, perhaps dig his grave a little deeper.\n\nI raise my palm. Nothing to sneeze at. I can feel a royal conniption fit brewing. \"No. If I've killed while gunning for a target,\" Casper frowns at my wording which causes me to grin, \"then they needed dying. Period.\"\n\nCasper walks to my office door. \"I'm sorry, Narah, I've done what I could, but the law states that there can't be more than ten sanctions in one quarter. You have thirteen. I got the bonus three waived.\" He whips his palm in the air like he's performing a magic trick. \"Now you'll have to go before the magistrate.\"\n\nFuck. They'd plug me a second ass after a first class reaming. If\u2014if I could even bounty again.\n\nI jerk my leather jacket off the back of my chair and sling it on. A bright headache, a new friend of mine of late, settles into my temples with zeal. I press my fingers against my head.\n\nI hate not having a target. The chase is the one thing that makes my life worth living. No longer an outcast\u2014always in the game.\n\nNow the rules are being threatened.\n\nAnd all I want to do is play.\n\n# Chapter 2\n\nAeslin\n\nEdan jerks a thumb my way, throwing a towel I deftly catch. I dab at the sweat running like a river from my scalp and making its way to the waistband of my work out gear.\n\n\"Corcoran's asking for you.\"\n\nI look at him, narrowing my eyes.\n\n\"Hey man, don't kill the messenger,\" Edan's hands spread away from his body.\n\nHe'd look so much more innocent if he had even one spot of bare skin. Edan's tatted from head to toe. Well... that's not entirely accurate. Don't think his feet hold the tats of our species. Or his face.\n\nTurners are required to be marked.\n\nIt's grounds for immediate execution to civilian vampires if they touch us. After all, we're the only savior of our dying race. They can't miss our marks. In the human world, tattoos no longer stand out. We hide in plain sight now.\n\nI flick irritated eyes to him. \"I'm on leave, Edan.\"\n\nHe shrugs. \"You know the drill. If a female comes on the radar, we're all on alert.\"\n\nI throw the damp towel in the soiled laundry hamper. I'm bone tired. Not physically\u2014mentally. So many scouting expeditions and coming up empty handed has taken its toll. I rub a hand on my nape, trying to make a raw spot. \"I've worked a solid quarter\u2014nothing.\"\n\nMy eyes meet his. Edan's looks are unusual for a Turner. Most of the sub-sect of vampire Turners possess dark coloring. Our only unified feature are silver eyes. Edan's are amber. Some kind of genetic throw back. My own hair is a deep chestnut, more red than what is considered fashionable. And if we want to enjoy female vampire company, it matters. They're few and far between. If they can't be our mates, it's only for release. And that's become an empty vessel.\n\n\"But what if we have a live one?\"\n\nI smirk at his words. \"You mean undead, right?\"\n\nEdan throws up his hands. He's muscled, like the rest of us. Mandatory training makes our bodies at battle readiness. Last month we'd just missed a female by minutes.\n\nShe'd been sterilized. Technically, it'd been on our watch.\n\nThe loss had brought the entire team down and morale had not recovered.\n\nEdan spoke my thoughts, \"We need this, Aeslin. We need a female. They're so vulnerable to the Hunters...\"\n\nI toss my palm up. \"We've been over this. It's a race against them. And they got to that female first.\" I see guilt on his face and know mine looks the same.\n\n\"Then why can't you see that every lead should be followed?\"\n\nTired of fucking losing, that's why. Or just tired.\n\nMy eyes feel like they're on fire when I glare at Edan, a Turner I've fought shoulder to shoulder beside. \"You don't think it haunts my fucking every thought that she could have belonged to one of us?\"\n\n\"Does it?\" Edan asks in soft disbelief.\n\n\"Yes,\" I hiss defensively.\n\n\"Then join us.\"\n\nI don't want another dead end. Another disappointment. \"I'm not rested.\"\n\n\"So when has that ever mattered?\" he asks.\n\nSince that female was lost, I think but don't say.\n\n*\n\nCorcoran stands at the window when I walk into his office and shut the door.\n\nHe doesn't turn.\n\nCorcoran is a Noble.\n\nA politically correct word for being in charge of the Turners. But he became a Noble the hard way, having been a Turner first and struggling through the ranks to prove himself invaluable to the cause. Now he rules over the Turners of our region with an iron fist.\n\nHell, in his day, there was a female turned every month. Now we were lucky to turn one a quarter. However, there was one biological advantage. A human female with vampire blood once turned, was always meant for her biological other half. Lucky bastard. It meant offspring.\n\nA chance at happiness.\n\nWith Hunters killing off every vampire they could, our numbers continued to dwindle. In the last half-century, one in two females who possessed enough of the blood of our kind had been sterilized before they could be turned, negating their vampire ancestry and the ability to have children.\n\nA Turners' goals were two-fold. Find the hybrid vampire females before the Hunters did, and determine how they were setting their sights on the rare females.\n\nEasier said than done.\n\n\"Aeslin,\" Corcoran said as greeting.\n\nI remain silent.\n\nCorcoran turns, eyeing me up. \"You look rested.\" He sounds hopeful. We both know I've had only four days respite.\n\nI need a month.\n\nI haven't taken enough blood, had enough sex, slept inside the ground as I should. A lot of have nots on the short list of my exhaustion.\n\nI lift my shoulders in an answer that isn't one. It will do no good to rehash the discussion I had with Edan.\n\nCorcoran says something under his breath. It sounds suspiciously like a curse.\n\n\"You're the best I have, Aeslin,\" he says quietly.\n\n\"Let Edan take it. Hell\u2014Jaryn could...\"\n\nHis gaze darkens. Eyes not the common light gray of the Turner are pewter in a face devoid of emotions. Corcoran's gaze is a coming storm.\n\n\"I need you on this.\"\n\nThat's just what Edan said. \"I mean no disrespect...\"\n\n\"Yes, you do,\" he says with the barest bit of humor.\n\nMy lips thin. \"Yes.\"\n\n\"She's a Turn, Aeslin. I know it.\" Corcoran closes his fingers into a fist.\n\nMy breath leaks out of me in defeat. \"Okay.\"\n\nI simply don't believe anymore. There's been so many dry runs I can't remember the last one that wasn't.\n\n\"She's sending out pheromones like a distress signal.\"\n\n\"Who called it?\"\n\nHis face closes down. \"Torin.\"\n\nCorcoran and Torin don't see eye-to-eye. I say nothing, waiting. I'm not political and won't immerse myself in it now.\n\nCorcoran slams a fist against the wall that bisects the bulletproof windows. \"She's bounty.\"\n\nHis frustration gets my attention. Hell, her occupation stalls me and I unlace my fingers and straighten my posture. \"What?\"\n\n\"Damn,\" he grits through his teeth, knowing full-well the risks of this acquisition.\n\nI tell him anyway. \"Too high profile,\" I state, hands going to my hips.\n\n\"She's manifesting.\"\n\nDammit.\n\n\"Is Torin sure she's a Turn?\"\n\nCorcoran exhales in a rush, taking a rough palm down his face, nodding.\n\nI suck in a deep breath. \"I'll do it.\"\n\nCorcoran looks relieved. \"You know the risk?\"\n\nHell yes. But another sterilized female? That we don't need. Can't stand.\n\n\"Yes,\" I answer.\n\nIf Torin's got a bead on her, then so do the Hunters.\n\nThe thought of a female out there and vulnerable tightens my guts. This is the part of my job I hate. However small, the emotion is there in my suppressed emotional makeup. The hardest to squelch, the most damning.\n\nHope.\n\n# Chapter 3\n\nMatthews\n\nRio raises the paper in the air. \"Right from the top, Matthews!\"\n\nI snap my head up, my back on the bench as I flick my eyes to Rio then back to the bar. My arms shake from exertion but I can't take my eyes off the weights I'm pressing. Not unless I want my body as a pancake.\n\n\"Spot me, asshole,\" I grit.\n\n\"Right! Sorry hoss.\"\n\nI'd roll my eyes if I wasn't so fucking plowed from fatigue.\n\nRio appears upside down and above me. His hands hover over the bar, I lift, as I take the last rep by storm. I heave another.\n\n\"No clanking,\" Rio chimes.\n\nGonna kill his ass.\n\nBeads of sweat roll, burning into my eyes as I gently set the bar on the brackets. It's almost soundless.\n\nRio smirks.\n\nHe whips the paper around and I duck out from underneath the three hundred pound weighted barbell set.\n\n\"God damn\u2014you're a beast, Matthews!\" Rio chortles.\n\n\"Give that to me and stop with the verbal diarrhea.\"\n\nRio's face tightens. \"Fine, fuck. You need to get laid if you're going to get your jock strap in a bunch all the time.\"\n\nI jerk the paper out of his hand and read the words.\n\nAssignment thirteen.\n\nI smile.\n\nThirteen is my lucky number.\n\nI give the paper back to Rio. \"Gonna save the world, brother.\"\n\n\"On your life.\"\n\n\"I hope not,\" Rio winks and begins to walk off.\n\n\"Specs?\" I yell after him.\n\n\"Same delivery as usual.\" I shouldn't ask, it's protocol but I like to hear the words anyway. It makes me uneasy when things are changed. I like routine\u2014crave it.\n\nI sit on the weight bench, thumbing the missive. A thrill races through my body.\n\nI'm a Hunter.\n\nAnd being a Hunter is bigger than me.\n\nIt's for humanity.\n\nPeople walk the streets; eating, sleeping, shitting and humping. They never realize there's an entire underworld of supernaturals vying for the top of the ecological heap. They're oblivious to the danger that sweeps past them like an unrelenting current.\n\nHunters have been in place since ancient times. Our opposition have the same sorcerer's blood that we possess.\n\nDruid.\n\nBoth sides descend from priests of the highest order.\n\nBut instead of exterminating the vermin, they are saviors of those that would harm who we're sworn to protect. They believe in perpetuation, and we believe in sterilization.\n\nThe Harborer's are the nemesis of our kind. Brothers by blood, enemies by deed.\n\nThe sooner we wipe out the supernaturals, the sooner the threat to mankind will end. And we're making steady progress.\n\nI move through the expansive gym where all Hunters hone their forms, turning sideways to pass between the heavy equipment. I've worked myself so bulky I'm at the point of losing grace. However, no Hunter wants to be distracted by their own lack of strength when they've got an assignment to fulfill.\n\nI'll get the details of my next sanction and be done. Hopefully it's another kill. Nothing gets my rocks off more than nailing one of the fangs myself. A larger threat would be a Harborer showing up for the same assignment. But they are fewer in number than Hunters. Vampires are the greater threat.\n\nEven a skilled Hunter full of quality bloodline magic can find himself in the death embrace of a clever fang and poof\u2014dead meat. The ultimate threat of being turned by one of them hangs over every one of us.\n\nNo Hunter wants to deal with that potential. Get in, kill the fuckers, and get the hell out.\n\nSimple.\n\n*\n\nI run my high security keycard through the slot and the door to my penthouse suite whispers open. I move through and the door slides closed behind me. The midwestern skyline bleeds a purple and red sunset over downtown Sioux Falls as it colors my floor like beaten fruit.\n\nI stretch and the vertebrae in my back give a satisfying round of pops. I toss my car keys in a low bowl of Mexican pottery that sits on top of a table hugging the jog out in the foyer.\n\nThe floor plan is one of my choosing. It's narrow in the entrance and widens to an open living room and kitchen combination.\n\nNot that I do a shit ton of cooking. My lips pull at the thought of cooking as I cruise to my fridge. I open it, and true to form, there's no food, but plenty of beer. I grab one and pop the lid using a sterling band on my right ring finger. It's hell on beer caps.\n\nI take a hard pull, taking the frosty beer to half empty and move to the view seen through my floor to ceiling glass windows.\n\nPhilips Street is overrun with tourists enjoying the bronze statues and Native American shops that dot the area. My excellent night vision is not necessary at the moment. Not with twilight promising nighttime. I roll the cool bottle against my forehead as my gaze wanders and sigh.\n\nI have twelve hours before response is required for the sanction.\n\nI set the nearly empty beer on a low thick glass coffee table. A hot shower and catching five hours of sleep is my entire goal before this mission. I'm beat. Chasing down hybrids is a full-time job.\n\nWalking to the wall that rounds to the hall leading to the bathroom, I pass a palm over a glass sculpture that hangs like artwork.\n\nIt's not.\n\nA brilliant blue spiderweb of light harmlessly lasers over my skin, reading the unique lines of my hand. A single chime sounds in the silence and the front slides away to reveal a black hole.\n\nI pull out a cylinder that rests inside.\n\nIt'll have all the instructions for assignment thirteen. Name, birthdate, location. My sector covers the midwest states. There are twelve of us serving this area.\n\nA vial with a syringe is enclosed in an thick airtight lucite case. My pulse quickens.\n\nIt'll be my first.\n\nA woman.\n\nHunters sanction male hybrids. It's the Hunters' core belief that women should be protected. None of us kill females. I don't allow myself to touch on what happens when a rare hybrid is located and a Hunter won't sterilize. The penalty is severe and immediate for lack of follow through.\n\nOr the disastrous transgression of mating with a hybrid, though rare, it's not unheard of. Those are grounds for a Kill Order.\n\nI set my dark thoughts aside as the specs fall out last, rolled neatly with the traditional black satin ribbon keeping them in a tight circle.\n\nI pop the ribbon and look over the specs, reading them twice.\n\nOccupation: Bounty Enforcer.\n\nI whistle low in the back of my throat. I'm all for a challenge.\n\nI slug the rest of my beer back, running a fingertip over the name.\n\nNarah Adrienne.\n\nI crush the specs, having already committed them to memory. I walk over to my fireplace and toss the crumpled parchment inside the firebox. Striking a match on the base of my boot, I throw the lit match inside and watch it burn. A low flame bursts over the ancient paper and renders the message unreadable.\n\nAsh rises up the flu. Ms. Adrienne's fate is not yet set in stone.\n\nI smile at the thought of destiny. Here I come, sweetheart.\n\n#\n\nRead More\n\nVampire Alpha Claim 1-6\n\n***Please read on for a sample of another Marata Eros work ....\n\nThank you for your attention:\n\nMarata Eros is the pen name for Tamara Rose Blodgett.\n\nThe following sample contains\/is intended for:\n\n18+ audience\n\nProfanity\n\nExtreme violence\n\nDisturbing themes\n\nReader discretion advised.\n\nMight contain triggers.\n\nREAPERS\n\nA Druid Series Novella\n\nVolume 1\n\nNew York Times Bestselling author\n\nMARATA EROS\n\nAll Rights Reserved.\n\nCopyright \u00a9 2011 Marata Eros\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locales or organizations is entirely coincidental.\n\nThis ebook is licensed for your personal enjoyment only. This ebook may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each recipient. If you're reading this book and did not purchase it, or it was not purchased for your use only, then please return to a legitimate retailer and purchase your own copy. Thank you for respecting the hard work of this author.\n\nMarata Eros Website\n\nMarata Eros FB Fan Page\n\nEdited by Hazel Novak\n\n# Synopsis\n\nThe vampires are a dying race, their females sterile. When it's discovered that human females of Druid ancestry can be viable breeders... the harvest begins.\n\nRachel Collins is a young woman living in the frozen north of Alaska in a dead-end job with a circular life. She yearns for something more.\n\nWhen murders begin taking place in the city where she lives, she and her girlfriend try to be more cautious, only to be caught in the middle of a dangerous situation in which an unlikely savior emerges.\n\nCan Rachel escape her destiny while two different factions hunt her?\n\n# CHAPTER ONE\n\nI looked at the clock, yet again... and knew that if my boss caught me I'd be toast. Safe in my cubicle, I swung my gaze away from the dreaded time and looked for Michelle. She'd be hanging by the cooler, which she was.\n\nMichelle caught me looking and lifted her chin up in greeting and grinned. She knew what I was about. It was all about getting out of here and doing something for ourselves. It had been a Long-Damn-Week and I was going to let my hair down and have some fun.\n\nMichelle wrapped up her conversation with one of the petty chicks that lounged all day while we picked up the slack.\n\nAs Michelle walked toward me, I thought that maybe we wouldn't have to change: pencil skirts, thigh high stockings, stacked heels and blouses that yoked just where they should be to look sexy, nothing too much.\n\nMichelle stood in front of me, tapping a foot. \"Watching the time won't help it go faster.\"\n\n\"Yes, I know, but I feel like the day should have ended already.\"\n\n\"I've got an idea, let's go to Spinners tonight,\" she nearly squealed in delight. I wasn't feelin' the love on that place. It was always packed with a rough crowd and you had to beat the guys off with a bat.\n\nMichelle saw my expression and started to wheedle immediately, \"Listen, give it a half hour and if it's super-lame, we'll just bail and go somewhere else. Like that brewery place... what's it's name?\"\n\n\"Talbot's,\" I replied absently.\n\nShe snapped her fingers. \"That's it!\"\n\n\"Listen,\" she leaned forward and our hair mingled together, \"that new gal... with the red hair...\"\n\n\"Molly?\" I said, automatically looking around for her.\n\n\"Yeah,\" she waved her hand, dismissing the name. \"She was talking about that piece of creepy news that's been circulating today.\"\n\nI looked at her blankly.\n\n\"Oh for shit's sake, Rachel! Don't you pay attention to anything?\"\n\n\"Not really,\" I said noncommittally. My life was beyond boring right now. I worked here, hung out with Michelle, worked out, read, fed my cat. I was dying for some Excitement. Dying. But the news wasn't going to deliver. Excitement... no way.\n\n\"You're hopeless! Anyway,\" she sounded the syllables out slowly, \"there's been another killing. Another bleed-out.\"\n\nThat got my attention.\n\nIt had been almost a month since the first murder and they still hadn't found the killer.\n\nThen there were the rapes.\n\nSomehow, it was all connected. Men were killed and drained dry of their blood and if there were women with them, they were raped.\n\nBut none of the women could remember the attack or their attacker.\n\nOur gazes locked. \"So... they found another body. Two, actually.\" Michelle said ominously, waggling two fingers.\n\nGreat. Just when I thought we could flounce around for the weekend. Talk about a wet blanket.\n\n\"Maybe... we shouldn't go to Spinners then. I mean, if it's not safe.\"\n\n\"Eff-that, you're going! I just wanted to spread the gory gossip.\"\n\n\"That's kinda sick, you know.\"\n\nMichelle nodded vigorously, she knew.\n\nI sighed. There was no getting out of it once Michelle had her mind set. And, in my soul... if I didn't get a break from this job and do something out-of-body, I'd scream.\n\n\"I gotcha talked right into it, don't I?\" Her eyes sparkled.\n\n\"I guess but, we need to be careful, especially now,\" I said in a conspirator's whisper.\n\n\"Hell, I'm more worried about the regular guys.\"\n\n\"Were the women... you know, was there blood... there?\" I asked.\n\nShe spun back around, her skirt twirling a little with the motion. \"That's the major weird thing, they had all been bitten, but still had their blood. Only a pint gone.\"\n\nWell, wasn't that just comforting.\n\nMichelle winked as she sauntered off, hips swaying. \"Pick ya up at seven sharp.\"\n\nShe didn't wait for me to respond. Michelle knew she had me, hook, line and sinker.\n\nI gathered up all my stuff, slipped my heels back on my feet and headed for the door.\n\nUnfortunately, my dragon lady of a boss was blocking my way.\n\n\"Miss Collins, I see you're ready to leave.\" She looked at her behemoth of a wristwatch. \"Two minutes after five.\" She raised a humongous unibrow at me and I stifled a giggle. It was hard to be pissed at her when she looked so ridiculous.\n\nAlmost.\n\n\"Yes. That's traditionally when the work day ends for us here, Ms. Hogan,\" I replied, thinking with mild irritation that Hogan had me by the short hairs. She knew I needed the job, she couldn't lambast me for leaving when the work day was through, technically. But... she liked to make me feel diminished for leaving so close to the chiming of the clock.\n\nHogan looked me over from head to toe, taking in my long black hair, so deep a black it had blue highlights in the right light. My eyes were a pale blue, I was shapely but not skinny, and on the tall side. I didn't consider myself a hot number but I held my own. Hogan, on the other hand looked like she was always trolling for a new bridge.\n\nI had discreetly pressed my elbow into the elevator button and it dinged just as she opened her mouth to mention something else equally unimportant, her jowls swinging as she popped her mouth open then closed it again.\n\nI felt my escape portal open at my back and walked backwards into its gaping mouth, never more glad to be out of mortar range of the enraged cow, aka my boss.\n\nShe glowered at me, starting to waddle forward and I blurted out, \"Have a great weekend!\" The door swept closed in front of me.\n\nI did a mental forehead-wipe. Thank God I was out of there.\n\nAs the elevator descended I prepared myself for the onslaught of cold weather, my car would need at least five minutes to heat up. The days were long here in the north and heating my car in the underground parking garage was just part of what we did in Alaska.\n\nThe elevator doors hissed apart and the cold air swept into the tight space, momentarily stealing my breath. I huddled my full length coat around myself, silently wishing the car was already warm. I rushed out of the elevator's cocoon of heat, my heels making clicking sounds on the concrete as I made my way to my car. If you could call it that.\n\nAs I approached I knew my car stood out, it was a Smart Car and Michelle liked to tease and say it was a toaster that I drove, not a real car. I smiled, she had me there.\n\nI fumbled with my keys, finally yanking my glove off with my teeth, groaning as the cold air assaulted my fingertips, making them instantly numb.\n\n\"Hey, Rachel,\"\n\nI dropped my keys on the ground, spinning, my hand to my heart.\n\nIt was Erik, a guy from work. My shoulders slumped in relief. He scared the shit out of me.\n\n\"Scare you?\" he smiled.\n\nI smiled back tentatively. He had really been pursuing me and I wasn't that interested. I couldn't put my finger on it exactly but there was just something off about him.\n\nErik approached me and I stiffened a little, but he bent over, jerking the keys off the ground and put a finger through the loop of my key fob and hung them off his finger in front of my nose.\n\nI tried to snatch them and he yanked them just out of reach.\n\n\"Meet me for dinner,\" he stated, his eyes steady on my face, disconcerting.\n\n\"Ah... Michelle and I are going out tonight,\" I said, trying to distract him.\n\n\"Rain check?\" he pressed, never stopping his eye contact. I was starting to get nervous.\n\nDamn.\n\nI resisted the supreme urge to look around, seeing if there was anyone else. But there wasn't, I could feel the absence of others. I sure wasn't short on woman's intuition. Just another creepy service we offer, I thought, getting the heebie-jeebies.\n\nI closed my coat tighter around me and his eyes tracked the movement, a smile spreading on his face. \"I'll let you go, I know you have plans.\" But his face told another tale. I didn't think he'd forget my rebuff anytime soon.\n\nI held my hands out and I was happy to notice that they weren't shaking. He'd really put me in a creeped out mood and I wasn't happy about it.\n\nHe dropped the keys into my cupped hands and smiled again, tipping an imaginary hat.\n\nI turned after his back was to me and stabbed the key into the lock, opening the door in one movement I slid behind the wheel, slapping the flat of my palm on the lock after it closed. I heard the simultaneous click in the silence of the car and let the breath out I didn't realize I'd been holding.\n\nHoly-hell.\n\nI turned on the car and stewed for the five minutes, all the while wishing I could have driven off.\n\nThat encounter with Erik had put a bad taste in my mouth. Like diet pop, but somehow worse.\n\nI pulled out of the bowels of the building, the night as black as when the day started. I entered traffic and began the drive to my condo, almost in the heart of downtown.\n\nI couldn't wait to be home.\n\nI threw my lights on, and glancing right then left I was so startled that I almost let my foot off the brake into opposing traffic.\n\nErik sat behind the wheel of his car. He'd having sat there the entire time... waiting for me.\n\nI gunned it at the first hole in traffic that appeared. What a whacko!\n\nI'd have to tell Michelle he was a nut-job. She'd have him cracked in no time.\n\n****\n\nI had my head thrown back and my lips parted, the last swipe of mascara almost perfect... there! I stood back and looked at my reflection: definitely not work attire. I was so glad I made the decision to not perk up the whole mess with just a new top. Michelle probably would have flogged me if I had anyway. She'd be dressed-to-kill (as usual). I needed to make an effort. Sometimes, I wondered why I bothered. Michelle would go, shine, get picked-up, bang some anonymous stud in the bathroom or wherever, and I would sip my drink wishing I could go home and curl up with a book. I sighed. That's okay. She was... my vicarious slutty friend. And I loved her.\n\nI grabbed my vanilla body spray and squirted a last dab. If I got to dancing a lot, I'd be glad I wore it. It was frigid outside but once we were inside Spinners, with all the bodies packed in there, it'd be a different story.\n\nI heard the doorknob jiggle and caught sight of Michelle coming through the doorway looking delectable in her slut suit. She twirled for me so I could get the full effect.\n\n\"That should be illegal!\" I nearly screamed. She had a micro-mini on that was two part: it cupped her ass and was barely legal (skimming the indecent exposure laws by a millimeter). It was hot pink, setting off her platinum hair to perfection. She \"helped\" the color of said hair, but not by a lot. Michelle was a rare thing up here in the frozen north and I was betting that it was her coloring that got her so much attention, and the boobs... and the outfits. And, and....\n\nI smiled as she circled me like a shark, gauging my potential for Attracting the Opposite Sex.\n\n\"I don't know... is this the shortest skirt you have?\" Her brows closed the distance between her eyes.\n\nI self-consciously ran my hand over my short black skirt, it barely covered the lace of my thigh-highs... a gorgeous pair that I had splurged on from Italy.\n\n\"Yeah, I can't go much shorter without the lace tops showing.\"\n\nMichelle gave me a blank look. \"Seriously, that's part of the allure.\"\n\n\"Ah... no. I say let them guess. It is underwear after all.\"\n\n\"I say show it!\" Michelle said.\n\n\"Mystery,\" I replied.\n\nShe threw her hand up. \"Whatever, I give up. At least you did right by the top.\"\n\nI had almost not worn it, it was a scorching crimson and showed off my raven hair, my eyes stranded like startled jewels in my pale face. It left my arms bare and was tucked inside the skirt.\n\nMichelle allowed her glance to linger a moment longer on my outfit, then shook her head as we walked out. I gave a quick pet to Caesar the cat and waltzed out.\n\n# CHAPTER TWO\n\nSpinners was packed as usual and we jockeyed for position, awkwardly elbowing everyone without trying to maim people. It was always this way.\n\nI couldn't believe our luck! I spied a couple of bar stools and we raced over there to stake our claim before they were snatched up. We perched our butts on the stools, aimlessly looking around at the bodies packed together, dancing the night away. I noticed they had already opened all the windows, allowing the sub-twenty degree air in. It didn't matter, it felt like a balmy eighty where we sat.\n\nThe bartender got our drinks. I sipped on a Blue Hawaiian and Michelle had Sex on the Beach (of course). She swung her leg back and forth and I was getting a spot-on flash of bright red panties... and so were a bunch of guys, judging from the expression of the gaggle of hunks sitting across from us.\n\n\"So what happened with Erik?\"\n\n\"Yeah!\" I yelled to be heard over the din. \"He did this weird thing with my keys...\" and I told her the whole thing. Michelle leaned forward to catch everything because the noise was swallowing my words.\n\nShe leaned back against the bar, her elbows flung back and her wrists dangling off the edge, looking thoughtful. For Michelle that meant she was quiet for more than one minute.\n\nFinally she said, \"Yeah, you want to stay away from him. I hear he went out with some girl and date-raped her.\"\n\nPerfect, I thought. That'd kinda been the vibe I was getting off him. Wasn't sure that confirmation was the greatest thing in this case, being as how I worked with the weirdo.\n\nWonderful.\n\nI was momentarily distracted when two of the cute guys across the way sidled over to us. The one on the right was almost as blonde as Michelle but that's where the similarity ended. He was a head taller than her with brown eyes and a face that had seen acne in its youth. I guess he was ruggedly handsome. He spent time in the gym; it was in the set of his shoulders, the way he moved... like he had purpose.\n\nTonight his purpose was Michelle.\n\nHis eyes never left the foot that swung, traveling up to the apex of what the skirt almost showed. He looked like a dog ready to mount a bitch. It did something for her because her foot stopped swinging and she gave him the come hither look.\n\nThe night was Going to Plan.\n\n\"Want to dance, cutie-pie?\" she asked, batting her eyelashes. He all but panted while I rolled my eyes in my head. I just couldn't do it. It's not that I'd never had sex. Casual just wasn't a main entree. I dreamed that there was someone for me in my future. Someone that I could share something with. I felt almost like... almost like I was waiting.\n\nMichelle argued there was plenty to be shared. She was into sharing.\n\nGenerous Michelle.\n\nI watched her on the dance floor, plastered to Rugged, grinding for all she was worth, he was all over her and she was loving it.\n\nI took my eyes off them and looked at the guy in front of me. He was way cuter than Rugged. He had the enigmatic something that made a girl want to get a little closer.\n\nSo I did.\n\n\"Do you want to dance?\" he asked.\n\nI nodded. He held out his hand, which was big I noticed. I tried not to think about how it would feel to have those hands roaming over my body but couldn't quite do it. He took me up against him and I molded against his torso. As those hands came to rest on the small of my back, the heat from them warmed me. He looked into my eyes and they held a promise of a fun night... if that's what I wanted. I didn't grind against him but I could feel that he was happy to be there. He smiled at me, knowing I was aware of his arousal.\n\nHe clutched me tighter and lowered his face next to mine and whispered, \"Your friend's gone.\" Now he was kissing my neck.\n\nUnease crept its way along my body. Usually Michelle gave me some kind of signal or something. I looked around for her trying not to feel frantic.\n\n\"Where did they go?\" I semi-shouted at him.\n\n\"Outside!\" He inclined his head in the direction of the door.\n\n\"You want to go find them?\" he asked, his fingers already twining in mine.\n\nI looked down at our clasped hands and that feeling of unease bloomed in me again. I couldn't shake it. I understood on some level that I was just getting residual anxiety from the strange encounter with Erik and letting that cloud my thinking. I wasn't going to take it out on this guy.\n\n\"Yeah, let's find them,\" I said decisively.\n\nI should have listened to that voice inside my head.\n\n# CHAPTER THREE\n\nThe wind was up and tore at the light outfit I had chosen for dancing inside. It simply wasn't enough. But the guy, (Matt, he'd told me as we hurried out) had said he thought they'd be in the car.\n\n\"How much further?\" I asked as I shivered in the light coat I'd slung on without a care before we left.\n\nI cared now, I was freezing my half-naked ass off.\n\n\"Not much,\" he wrapped an arm around my shoulders as we walked and that helped.\n\nSure enough, another block of parked cars revealed a car that was running.\n\nI could see a flash of pink in the car, but barely.\n\nWhat was happening? It was Michelle but there were... others.\n\nOther men.\n\nMy foreboding slammed back over me, washing away all tact. I went to wrench the car door open and Matt stopped me.\n\n\"They're busy.\"\n\n\"Ah duh, Einstein, I can see that. But I don't know if she was planning on being this busy,\" I said, seeing that there were at least two guys in there.\n\nMatt put his hands up, as if to say, hey, no problem, just sayin'.\n\nIrritating jerk.\n\nSighing, I tore open the door and was entirely unprepared for what I saw before me:\n\nMichelle had Rugged behind her shoving his cock right up to her groin, the whole length of him digging in, sparing her nothing, his balls slapping her ass. The other guy, who I vaguely remembered sitting across the way from us, had his hand fisted in all that blonde hair and was pressing her face up and down on the shaft of his dick.\n\nWhen the door opened, Rugged's eyes flew open and his gaze met mine. His body was pumping and working behind Michelle his hand reached over and slapped her ass and she moaned, her head working up and down the shaft of the prick she had in her mouth.\n\nI didn't think she had planned on this and I yelled, \"Michelle!\"\n\nShe tried to take her head off the cock she was on but he shoved her back down and she gagged. \"I'm gonna spray my cum you dumb bitch, keep sucking.\"\n\nShe squirmed to try and get away and Rugged held her hips, pounding into her harder. I backed away with my hand covering my mouth.\n\nMichelle wasn't fooling around... she was getting raped.\n\nI swung around to get help and Matt wrapped his arms around me. One of his big hands that I'd admired so much earlier covered my mouth so I couldn't scream.\n\nAdrenaline slammed into me like a sheet of cold water.\n\nDragging me into the front seat of the car he threw me across and I bounced once, almost landing against the opposite door. Somewhere in the back of my mind I knew what was going to happen as Rugged said hoarsely, \"God... I'm gonna cum in her snatch right now, oh God!\"\n\nThe whole car rocked as he plunged his length into Michelle and the other guy torqued her head down on him and said, \"Swallow it... that's it, swallow it. Ahhh... that's right,\" he groaned, throwing his head back, his lips slightly parted.\n\nI started to fully panic then, scrambling across the seat, my skirt hiking around my hips as I struggled to reach the door.\n\nMatt landed on top of me and all the air in my lungs went out in a rush. I couldn't breathe and was in a state of sheer panic, Matt was not the guy I had taken him for.\n\nNeither were his friends.\n\nI could hear Michelle sobbing softly and the rustle of her clothes as she tried to adjust everything.\n\nTwo heads and upper chests appeared over the top of the back seat as Matt's engorged arousal pressed along my inner thigh, trying to gain entrance, his pants long gone.\n\n\"You gotta a live one there,\" Guy X said.\n\n\"Not as 'live' as the bitch we just did,\" Rugged replied, laughing.\n\nThe sweet air of the car's interior entered in a rush, filling my lungs to capacity and I screamed for all I was worth, tearing something loose in the process. Matt's hand clamped over my mouth and I bit him, trying to meet my teeth together. He snatched his hand away and bellowed. I knew what was coming as I was pinned under him, his other hand came down with depressing speed and accuracy, slamming into my cheekbone, my head rocketing back against the door.\n\nMy head swam and his fingers dug at my panties while the other men watched....\n\nJust when I thought there wasn't a hope in the world, the driver's side door was torn open, the hinges shrieking, then releasing in the process. The door was flung behind the figure that filled the opening.\n\nI was seeing him upside down but the guys in the back seat summed it up, \"What the fuck? Who the fuck are you?\"\n\nHe was my savior... whoever he was.\n\nMatt was backing off me in a hurry, leaving me to the stranger's mercy. With my head spinning all I saw was a strong jaw, black jeans, boots and a bad-ass leather zip-up. As he leaned down, the whiteness of his teeth gleamed in the interior dome light of the car, his nearly bald head had an inky wash of short black hair covering it, the shirt breaking open at the neck to reveal a tattoo that crawled up his neck.\n\nHe wasn't big on conversation, his hand snaking out in a lethal punch that terminated on Matt's nose. There was a sickening crunch and he backed up, howling. His bleeding hand with my teethmarks held what I was sure was a broken nose as he fell right outside the car on his ass.\n\nThe stranger looked down at me then and his eyes were a startling blue, not icy pale like mine. They were nearly white in a face that was pale, his lush mouth a deep scarlet slash. He looked me over carefully, but only a second or two as he was going to have to deal with Rugged and Guy X. Those two had gotten out of the car and I was too weak to even turn over and see things right side up. If he'd meant me harm, it would have happened.\n\nAt least that's what I told myself.\n\nRugged and Guy X faced him. \"You could have had a piece of this action, if you'd asked nice.\"\n\nThe stranger looked briefly at Michelle who was trying to stop crying, the sobs turning to hitching hiccups. Then he spared a glance in my direction, taking in my clothes in disarray, my face starting to swell from the blow I'd received.\n\n\"They do not look as though it was the attention they wished for,\" the stranger responded.\n\nI noticed that he was enormously big compared to the two men who had abused Michelle. And that was saying something because Matt had me by inches and I was five-eight.\n\nRugged said, \"You talk funny.\"\n\n\"As do you,\" Stranger said.\n\nGuy X circled around him, taking his measure as a male. \"I think you need an ass-kicking. You've beat the shit out of our car in the middle of fuckin' winter in this cold-ass place, ruined another piece of easy tail for me and my buds here. You gotta get what's comin' to ya.\"\n\nStranger looked at them as they rushed him at the same time. From my angle I saw everything upside down and in slow motion. Rugged came at him like a charging bear and Stranger swung his arm forward in a stiffened knife jab move and with the flat of the palm he landed it square in Rugged's nose. He stared blankly for a moment then fell like a box of rocks, his nose shattered. Guy X was a slow learner and grabbed him from behind and latched onto his wind pipe. Stranger grabbed the forearm which held him, crushing it before my eyes just using the one hand, while the other spun Guy X around to stare at him. He was howling, taking great lungfuls of air to bellow louder.\n\n\"Stop that noise,\" Stranger said.\n\nAnd he did.\n\nThe stranger stared into the eyes of Guy X. \"Tilt your head.\"\n\nGuy X looked like he was in a fog, as if he was not in command of his mind. He cocked his head to the extreme left. The long, clean line of his neck was exposed under the street lamp, the artificial light casting a ghostly yellow on the flesh of his throat.\n\nThe stranger reared back like a snake and hissing, struck Guy X's neck. His teeth as he arced above Guy X's neck was something I would never forget:\n\nThey were fangs.\n\nI was riveted. My presumed savior was not a man... he was something else. I had to get out of here. I tried to sit up and my head swam. I was woozy from the blow. The stranger had gathered Guy X in his arms and was taking great gulps from him but his eyes were pinned on me.\n\nTime to go.\n\nI looked over the back seat and met Michelle's horrified eyes. Her mascara had made its way all over her face and I said, \"Let's get out of here. Right now.\"\n\nI slid out of the car, one of my high heels falling off and was met by another stranger. This one had blonde hair and the same icy-blue eyes as the other.\n\nThey were busy tonight.\n\nThis one was all over Matt. He sucked at his neck while Matt made disconcerting mewling sounds underneath him. He lifted his mouth off Matt long enough to hiss at Michelle, which got us moving.\n\nWe backed away, both my shoes left on the sidewalk. They watched us but did not follow, taking the last of the men's blood. Their lives ebbed as we watched.\n\n\"What are they?\" Michelle whispered.\n\n\"Ah... I think we've just been saved by those blood-killers.\"\n\n\"They... they raped the women...\"\n\nWe looked at each other, dawning comprehension mirrored in our expressions.\n\nWe ran.\n\nWe ran until my lungs burned, the images of them sucking those guys lives away etched permanently in my brain. We were within sight of my car when we saw them leaning against it, one dark, the other light.\n\n\"Holy shit,\" Michelle stammered.\n\nYeah, that.\n\nThey came off the car at the same moment like perfectly choreographed twins. But it was the dark one that made my heart speed in my chest.\n\nThey came to stand before us. \"We need to scrub them both. They have seen entirely too much,\" the blonde one said, his stare going from Michelle to me.\n\nThe dark one laid his icy gaze on me and I shivered. From what I didn't know but his gaze penetrated my bone and marrow.\n\n\"Holy shit,\" Michelle said again.\n\nI seconded that.\n\nStill I said nothing while they looked at us for a long moment. \"No. The blonde one forgets. This one, no.\"\n\n\"Why, Cole?\" the blonde stranger asked. \"She is fair of face and figure, but there are many...\"\n\n\"You cannot smell her?\" Cole asked.\n\nThe blonde's head whipped around and his penetrating gaze was suddenly all for me; I backed away.\n\nFinally, he shook his head. \"All I smell is their fear. They smell like prey.\"\n\n\"Underneath that, Nathan.\"\n\nOne moment he was ten feet away and the next he had his arms around me and I screamed. Michelle started to run but faster than my eyes could track, the one named Cole had her in his arms, his hand covering her mouth and the fingers of those long hands feathered her temple. And I'd thought Matt's hand had been big... my God, his were palming her entire face.\n\nVoices drifted down, the blonde's face was buried in my neck and I began to hyperventilate. Images flooded my mind of my would-be rapists not finishing what they started, distracted with death-by-blood loss.\n\n\"Be still,\" he said, his fangs bursting out of his mouth.\n\nI thrashed around and he turned to Cole. \"She will not follow my command.\"\n\n\"Will it so,\" Cole said.\n\nHe buried his nose in my neck, breathing my scent in, his fangs grazing the skin of my neck. Lifting his head he said, \"Breeder.\"\n\n\"Yes.\"\n\n\"We must take her. There are so few left. This one is... she is rare.\"\n\nA drunken group stepped out into our little mess and Michelle began hollering, \"Help, help!\"\n\nA couple of the guys broke away from the pack and made their way over to us, Cole stood away from Michelle and the blonde released me slowly, like he didn't want to stop touching me.\n\nAs they approached the males, they looked into their eyes and each stranger said, \"Leave this place.\" One of the men grabbed his temples with his hands, shaking his head like he couldn't release the clutches of something.\n\n\"That one has a strong mind,\" Nathan said.\n\n\"Some of the cattle do,\" Cole said.\n\nCattle.\n\nI started to back away, subtly getting Michelle's attention. We were almost to the car when Cole's head whipped around. \"You... will not leave.\"\n\nMichelle began hollering again but the men walked away. The one who shook his head cast a final glance behind him. As we watched he massaged his temple, continuing to walk away.\n\nThey retraced their steps toward us and my heart sank. We could not outrun something we couldn't see move, something that crushed a man's face with one swipe, disintegrated an arm with a grip strong enough to pulverize bone. As they drew nearer, their fangs stood out of their mouths, barbed points ready to pierce our flesh.\n\nMichelle latched on to my hand and I prepared myself for the worst.\n\n\"You cannot thirst.\"\n\n\"No, but the blonde one's fear is an aphrodisiac,\" Nathan said.\n\n\"Yes, but think on this Nathan: has she not already been degraded enough by the human scum we dispatched?\" Cole said.\n\n\"Yes,\" Nathan ground out. \"You speak true.\"\n\n\"Then scrub her and we take the female breeder.\"\n\nNathan approached Michelle and she started to wail, her screams broken only by her next breath.\n\nThe blonde was suddenly in front of her. Squeezing her throat lightly, he cut off her screaming and the sudden silence filled the parking area. The snow was falling softly around us, some of the flakes catching in my eyelashes. Nathan stared deeply into Michelle's eyes. Finally he moved away and she stood there, blank faced, in a zombie-like stupor.\n\n\"What did you do to her?\" I whispered.\n\n\"Something we cannot do for you,\" Cole remarked.\n\nI backed away and they tracked me. \"I am not going with you,\" I said, proud that my voice only shook a little.\n\n\"We understand your fear, but you will come with us. How do you humans put it? It is non-negotiable.\"\n\n\"You don't understand anything! You two... whatever-you-are, sucked our attacker's blood. They died and now you're calling me some kind of 'breeder'. No offense, but it's not looking too good on my end.\"\n\nMy eyes bounced from one to the other of them. I couldn't follow their movements, just when I thought I had one in my sights they moved so fast they were both suddenly one foot away from me, each holding an arm. I opened my mouth to scream and Cole put his mouth on mine, stifling any sound I could manage. His kiss blossomed and spread to the center of me, making my panties instantly moisten. I'd never had a reaction like this in my life.\n\nOf course, I'd always made out with human men.\n\nMy fear was in my throat but my biology was never touched by it. I couldn't move my arms but as my mouth moved against his, he released my arm and I wound it around his neck, pressing his lips harder against mine and he groaned and pulled me against him. My mind played tug-of-war, my intellect was screaming that he was some creature of the night. He'd killed two men before my eyes but my center bloomed for him; heat stretching and spreading from between my legs. My nipples hardened and he reached behind me, placing his hands under my thighs. Never breaking from the kiss, he lifted me up and I wound my legs around his waist.\n\n\"She is so eager,\" Nathan said, releasing my other arm. He circled us, grabbing a piece of my hair and flicking it behind my shoulder.\n\nThat broke through the heat of the moment and my intellect slammed back into place. I broke away and shoved at his chest with my hands. He let me slide down his body and when my feet hit the pavement, the cold moved up my legs, freezing that searing heat before it progressed.\n\nI gave Nathan a dirty look, noticing Michelle still stood there in the same position, gooseflesh covering her arms, her teeth chattering.\n\n\"I don't know what you are, or why this is happening but I just want you to go... now. I will get my friend and I home without any help from you.\" I folded my arms across my chest.\n\n\"Your body speaks for what you want. It speaks for what you are.\"\n\n\"Oh?\" I arched my eyebrow. \"What is it that I am?\"\n\n\"It is what is in your blood, you are of Druid blood. They are the only humans that may breed with us.\"\n\nDruid? What the hell was that?\n\nOkay, next question:\n\n\"What are you?\"\n\nThey looked at each other. \"We are Vampire, witch,\" Cole said as if that should have been obvious.\n\nWitch? Had the conversation devolved to name-calling at two a.m.?\n\n\"Do you know of your people?\"\n\nI couldn't believe I was standing out here in twenty degree weather talking to a couple of guys claiming to be vampires. I felt incredibly stupid to have kissed the one... Cole.\n\n\"Ah, no. I'm adopted. Okay, while all this is interesting, it's time to go.\"\n\nI turned to Michelle, who had a spot of drool coming out of her mouth. God, what did they do to her? I walked over to her, grabbing a limp arm and started dragging her to my car.\n\nSuddenly, they stood in front of me.\n\n\"Would you stop doing that!\" I said, fear choking me.\n\nA smile spread over Cole's face. \"Doing what?\" His fangs were smaller now that he wasn't trying to French me.\n\nSirens began in the distance and we all looked in their direction.\n\n\"The human police,\" Nathan said.\n\n\"Yes.\"\n\n\"Another time, breeder,\" Cole said. He licked his finger and touched it on my forehead. \"I mark you for another time, very soon.\"\n\nThey disappeared into thin air and I was left with Michelle, the approaching cops, and a pulsing core that wept for the vampire that was now gone.\n\n#\n\nRead More\n\nDruid 1-3, currently FREE for a limited time!\n\n|  |\n\n---|---|---\n\n# Acknowledgments\n\nI published both The Druid and Death Series, in 2011 with the encouragement of my husband, and continued because of you, my Reader. Your faithfulness through comments, suggestions, spreading the word and ultimately purchasing my work with your hard-earned money gave me the incentive, means and inspiration to continue.\n\nThere are no words that are sufficiently adequate to express my thankfulness for your support.\n\nI truly feel connected to my readers. It is obvious to me, but I'll say the words anyway for clarity: a written work is just words on pages if they are not read by my readers. As I write this I get a lump in my throat; your enjoyment of my work affects me that deeply.\n\nYou guys are the greatest, each and every one of ya~\n\nTamara\n\nxoxo\n\nSpecial Thanks:\n\nYou, my reader.\n\nMy husband, who is my biggest fan.\n\nCameren, without who, there would be no books.\n\n|  |\n\n---|---|---\n\n# About the Author\n\nwww.TamaraRoseBlodgett.com\n\nTamara Rose Blodgett: happily married mother of four sons. Dark fiction writer. Reader. Dreamer. Home restoration slave. Tie dye zealot. Coffee addict. Digs music.\n\nShe is also the New York Times Bestselling author of A Terrible Love, written under the pen name, Marata Eros, and over ninety-five other titles, to include the #1 international bestselling erotic Interracial\/African-American TOKEN serial and her #1 bestselling Amazon Dark Fantasy novel, Death Whispers. Tamara writes a variety of dark fiction in the genres of erotica, fantasy, horror, romance, sci-fi and suspense. She lives in the midwest with her family and three, disrespectful dogs.\n\nConnect with Tamara:\n\nTRB News\n\nMarata Eros News\n\nInstagram\n\nPinterest\n\nBLOG\n\nFaceBook\n\nTwitter\n\nSubscribe to my YouTube Channel\n\nExclusive Excerpts!\n\nComedic Quips\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n#### \u00cdndice\n\nPortadilla\n\n\u00cdndice\n\nPr\u00f3logo\n\nDedicatoria\n\nCita\n\nCap\u00edtulo I\n\nCap\u00edtulo II\n\nCap\u00edtulo III\n\nCap\u00edtulo IV\n\nCap\u00edtulo V\n\nCap\u00edtulo VI\n\nCap\u00edtulo VII\n\nCap\u00edtulo VIII\n\nCap\u00edtulo IX\n\nCap\u00edtulo X\n\nCap\u00edtulo XI\n\nCap\u00edtulo XII\n\nCap\u00edtulo XIII\n\nCap\u00edtulo XIV\n\nCap\u00edtulo XV\n\nCap\u00edtulo XVI\n\nCap\u00edtulo XVII\n\nCap\u00edtulo XVIII\n\nCap\u00edtulo XIX\n\nCap\u00edtulo XX\n\nSobre el autor\n\nCr\u00e9ditos\n\nGrupo Santillana\n\n### PR\u00d3LOGO\n\nCOMENC\u00c9 ESTA novela en Lima, a mediados de 1972, y la segu\u00ed escribiendo, con m\u00faltiples y a veces largas interrupciones, en Barcelona, La Romana (Rep\u00fablica Dominicana), Nueva York, y de nuevo Lima, donde la termin\u00e9 cuatro a\u00f1os despu\u00e9s. Me la sugiri\u00f3 un autor de radioteatros que conoc\u00ed de joven, al que sus melodram\u00e1ticas historias devoraron el seso por un tiempo. Para que la novela no resultara demasiado artificial, intent\u00e9 a\u00f1adirle un collage autobiogr\u00e1fico: mi primera aventura matrimonial. Este empe\u00f1o me sirvi\u00f3 para comprobar que el g\u00e9nero novelesco no ha nacido para contar verdades, que \u00e9stas, al pasar a la ficci\u00f3n, se vuelven siempre mentiras (es decir, unas verdades dudosas e inverificables).\n\nMe cost\u00f3 trabajo dar una forma aceptable a aquellos episodios que, sin serlo, parecieran los guiones de Pedro Camacho, y volcar en ellos los estereotipos, excesos, cursiler\u00edas y truculencias caracter\u00edsticos del g\u00e9nero, tomando la distancia ir\u00f3nica indispensable pero sin que se volvieran caricatura. El melodrama ha sido una de mis debilidades precoces, atizada por las desgarradoras pel\u00edculas mexicanas de los a\u00f1os cincuenta, y el tema de esta novela me permiti\u00f3 asumirlo, sin escr\u00fapulos. Las sonrisas y burlas no llegan a ocultar del todo, en el narrador de este libro, a un sentimental propenso a los boleros, las pasiones desaforadas y las intrigas de follet\u00edn.\n\nMARIO VARGAS LLOSA\n\nLondres, 30 de junio de 1999\nA Julia Urquidi Illanes, a quien tanto debemos yo y esta novela.\nEscribo. Escribo que escribo. Mentalmente me veo escribir que escribo y tambi\u00e9n puedo verme ver que escribo. Me recuerdo escribiendo ya y tambi\u00e9n vi\u00e9ndome que escrib\u00eda. Y me veo recordando que me veo escribir y me recuerdo vi\u00e9ndome recordar que escrib\u00eda y escribo vi\u00e9ndome escribir que recuerdo haberme visto escribir que me ve\u00eda escribir que recordaba haberme visto escribir que escrib\u00eda y que escrib\u00eda que escribo que escrib\u00eda. Tambi\u00e9n puedo imaginarme escribiendo que ya hab\u00eda escrito que me imaginar\u00eda escribiendo que hab\u00eda escrito que me imaginaba escribiendo que me veo escribir que escribo.\n\nSALVADOR ELIZONDO, El graf\u00f3grafo\n\n### I\n\nEN ESE tiempo remoto, yo era muy joven y viv\u00eda con mis abuelos en una quinta de paredes blancas de la calle Ochar\u00e1n, en Miraflores. Estudiaba en San Marcos, Derecho, creo, resignado a ganarme m\u00e1s tarde la vida con una profesi\u00f3n liberal, aunque, en el fondo, me hubiera gustado m\u00e1s llegar a ser un escritor. Ten\u00eda un trabajo de t\u00edtulo pomposo, sueldo modesto, apropiaciones il\u00edcitas y horario el\u00e1stico: director de Informaciones de Radio Panamericana. Consist\u00eda en recortar las noticias interesantes que aparec\u00edan en los diarios y maquillarlas un poco para que se leyeran en los boletines. La redacci\u00f3n a mis \u00f3rdenes era un muchacho de pelos engomados y amante de las cat\u00e1strofes llamado Pascual. Hab\u00eda boletines cada hora, de un minuto, salvo los de mediod\u00eda y de las nueve, que eran de quince, pero nosotros prepar\u00e1bamos varios a la vez, de modo que yo andaba mucho en la calle, tomando cafecitos en la Colmena, alguna vez en clases, o en las oficinas de Radio Central, m\u00e1s animadas que las de mi trabajo.\n\nLas dos estaciones de radio pertenec\u00edan al mismo due\u00f1o y eran vecinas, en la calle Bel\u00e9n, muy cerca de la plaza San Mart\u00edn. No se parec\u00edan en nada. M\u00e1s bien, como esas hermanas de tragedia que han nacido, una, llena de gracias y, la otra, de defectos, se distingu\u00edan por sus contrastes. Radio Panamericana ocupaba el segundo piso y la azotea de un edificio flamante, y ten\u00eda, en su personal, ambiciones y programaci\u00f3n, cierto aire extranjerizante y snob, \u00ednfulas de modernidad, de juventud, de aristocracia. Aunque sus locutores no eran argentinos (habr\u00eda dicho Pedro Camacho) merec\u00edan serlo. Se pasaba mucha m\u00fasica, abundante jazz y rock, y una pizca de cl\u00e1sica, sus ondas eran las que primero difund\u00edan en Lima los \u00faltimos \u00e9xitos de Nueva York y de Europa, pero tampoco desde\u00f1aban la m\u00fasica latinoamericana siempre que tuviera un m\u00ednimo de sofisticaci\u00f3n; la nacional era admitida con cautela y s\u00f3lo al nivel del vals. Hab\u00eda programas de cierto relente intelectual, Semblanzas del Pasado, Comentarios Internacionales, e incluso en las emisiones fr\u00edvolas, los Concursos de Preguntas o el Trampol\u00edn a la Fama, se notaba un af\u00e1n de no incurrir en demasiada estupidez o vulgaridad. Una prueba de su inquietud cultural era ese Servicio de Informaciones que Pascual y yo aliment\u00e1bamos, en un altillo de madera construido en la azotea, desde el cual era posible divisar los basurales y las \u00faltimas ventanas teatinas de los techos lime\u00f1os. Se llegaba hasta \u00e9l por un ascensor cuyas puertas ten\u00edan la inquietante costumbre de abrirse antes de tiempo.\n\nRadio Central, en cambio, se apretaba en una vieja casa llena de patios y de vericuetos, y bastaba o\u00edr a sus locutores desenfadados y abusadores de la jerga, para reconocer su vocaci\u00f3n multitudinaria, plebeya, crioll\u00edsima. All\u00ed se propalaban pocas noticias, y all\u00ed era reina y se\u00f1ora la m\u00fasica peruana, incluyendo la andina, y no era infrecuente que los cantantes indios de los coliseos participaran en esas emisiones abiertas al p\u00fablico que congregaban muchedumbres, desde horas antes, a las puertas del local. Tambi\u00e9n estremec\u00edan sus ondas, con prodigalidad, la m\u00fasica tropical, la mexicana, la porte\u00f1a, y sus programas eran simples, inimaginativos, eficaces: Pedidos Telef\u00f3nicos, Serenatas de Cumplea\u00f1os, Chismograf\u00eda del Mundo de la Far\u00e1ndula, el Acetato y el Cine. Pero su plato fuerte, repetido y caudaloso, lo que, seg\u00fan todas las encuestas, le aseguraba su enorme sinton\u00eda, eran los radioteatros.\n\nPasaban media docena al d\u00eda, por lo menos, y a m\u00ed me divert\u00eda mucho espiar a los int\u00e9rpretes cuando estaban radi\u00e1ndolos: actrices y actores declinantes, hambrientos, desastrados, cuyas voces juveniles, acariciadoras, cristalinas, difer\u00edan terriblemente de sus caras viejas, sus bocas amargas y sus ojos cansados. \u00abEl d\u00eda que se instale la televisi\u00f3n en el Per\u00fa no les quedar\u00e1 otro camino que el suicidio\u00bb, pronosticaba Genaro hijo, se\u00f1al\u00e1ndolos a trav\u00e9s de los cristales del estudio, donde, como en una gran pecera, los libretos en las manos, se los ve\u00eda formados en torno al micro, dispuestos a empezar el cap\u00edtulo veinticuatro de La familia Alvear. Y, en efecto, qu\u00e9 decepci\u00f3n se hubieran llevado esas amas de casa que se enternec\u00edan con la voz de Luciano Pando si hubieran visto su cuerpo contrahecho y su mirada estr\u00e1bica, y qu\u00e9 decepci\u00f3n los jubilados a quienes el cadencioso rumor de Josefina S\u00e1nchez despertaba recuerdos, si hubieran conocido su papada, sus bigotes, sus orejas aleteantes, sus v\u00e1rices. Pero la llegada de la televisi\u00f3n al Per\u00fa era a\u00fan remota y el discreto sustento de la fauna radioteatral parec\u00eda por el momento asegurado.\n\nSiempre hab\u00eda tenido curiosidad por saber qu\u00e9 plumas manufacturaban esas seriales que entreten\u00edan las tardes de mi abuela, esas historias con las que sol\u00eda darme de o\u00eddos donde mi t\u00eda Laura, mi t\u00eda Olga, mi t\u00eda Gaby o en las casas de mis numerosas primas, cuando iba a visitarlas (nuestra familia era b\u00edblica, miraflorina, muy unida). Sospechaba que los radioteatros se importaban, pero me sorprend\u00ed al saber que los Genaros no los compraban en M\u00e9xico ni en Argentina sino en Cuba. Los produc\u00eda la CMQ, una suerte de imperio radiotelevisivo gobernado por Goar Mestre, un caballero de pelos plateados al que alguna vez, de paso por Lima, hab\u00eda visto cruzar los pasillos de Radio Panamericana sol\u00edcitamente escoltado por los due\u00f1os y ante la mirada reverencial de todo el mundo. Hab\u00eda o\u00eddo hablar tanto de la CMQ cubana a locutores, animadores y operadores de la radio \u2014para los que representaba algo m\u00edtico, lo que el Hollywood de la \u00e9poca para los cineastas\u2014 que Javier y yo, mientras tom\u00e1bamos caf\u00e9 en el Bransa, alguna vez hab\u00edamos dedicado un buen rato a fantasear sobre ese ej\u00e9rcito de pol\u00edgrafos que, all\u00e1, en la distante Habana de palmeras, playas paradis\u00edacas, pistoleros y turistas, en las oficinas aireacondicionadas de la ciudadela de Goar Mestre, deb\u00edan de producir, ocho horas al d\u00eda, en silentes m\u00e1quinas de escribir, ese torrente de adulterios, suicidios, pasiones, encuentros, herencias, devociones, casualidades y cr\u00edmenes que, desde la isla antillana, se esparc\u00eda por Am\u00e9rica Latina, para, cristalizado en las voces de los Lucianos Pandos y las Josefinas S\u00e1nchez, ilusionar las tardes de las abuelas, las t\u00edas, las primas y los jubilados de cada pa\u00eds.\n\nGenaro hijo compraba (o, m\u00e1s bien, la CMQ vend\u00eda) los radioteatros al peso y por telegrama. Me lo hab\u00eda contado \u00e9l mismo, una tarde, despu\u00e9s de pasmarse cuando le pregunt\u00e9 si \u00e9l, sus hermanos o su padre daban el visto bueno a los libretos antes de propalarse. \u00ab\u00bfT\u00fa ser\u00edas capaz de leer setenta kilos de papel?\u00bb, me repuso, mir\u00e1ndome con esa condescendencia benigna que le merec\u00eda la condici\u00f3n de intelectual que me hab\u00eda conferido desde que vio un cuento m\u00edo en el Suplemento Dominical de El Comercio: \u00abCalcula cu\u00e1nto tomar\u00eda. \u00bfUn mes, dos? \u00bfQui\u00e9n puede dedicar un par de meses a leerse un radioteatro? Lo dejamos a la suerte, y, hasta ahora, felizmente, el Se\u00f1or de los Milagros nos protege\u00bb. En los mejores casos, a trav\u00e9s de agencias de publicidad, o de colegas y amigos, Genaro hijo averiguaba cu\u00e1ntos pa\u00edses y con qu\u00e9 resultados de sinton\u00eda hab\u00edan comprado el radioteatro que le ofrec\u00edan; en los peores, decid\u00eda por los t\u00edtulos o, simplemente, a cara o sello. Los radioteatros se vend\u00edan al peso porque era una f\u00f3rmula menos tramposa que la del n\u00famero de p\u00e1ginas o de palabras, en el sentido de que era la \u00fanica posible de verificar. \u00abClaro \u2014dec\u00eda Javier\u2014, si no hay tiempo para leerlas, menos todav\u00eda para contar todas esas palabras\u00bb. Lo excitaba la idea de una novela de sesenta y ocho kilos y treinta gramos, cuyo precio, como el de las vacas, la mantequilla y los huevos, determinaba una balanza.\n\nPero este sistema creaba problemas a los Genaros. Los textos ven\u00edan plagados de cubanismos, que, minutos antes de cada emisi\u00f3n, el propio Luciano y la propia Josefina y sus colegas traduc\u00edan al peruano como pod\u00edan (siempre mal). De otro lado, a veces, en el trayecto de La Habana a Lima, en las panzas de los barcos o de los aviones, o en las aduanas, las resmas mecanografiadas sufr\u00edan deterioros y se perd\u00edan cap\u00edtulos enteros, la humedad los volv\u00eda ilegibles, se traspapelaban, los devoraban los ratones del almac\u00e9n de Radio Central. Como esto se advert\u00eda s\u00f3lo a \u00faltima hora, cuando Genaro pap\u00e1 repart\u00eda los libretos, surg\u00edan situaciones angustiosas. Se resolv\u00edan salt\u00e1ndose el cap\u00edtulo perdido y ech\u00e1ndose el alma a la espalda, o, en casos graves, enfermando por un d\u00eda a Luciano Pando o a Josefina S\u00e1nchez, de modo que en las veinticuatro horas siguientes se pudieran parchar, resucitar, eliminar sin excesivos traumas, los gramos o kilos desaparecidos. Como, adem\u00e1s, los precios de la CMQ eran altos, result\u00f3 natural que Genaro hijo se sintiera feliz cuando descubri\u00f3 la existencia y las dotes prodigiosas de Pedro Camacho.\n\nRecuerdo muy bien el d\u00eda que me habl\u00f3 del fen\u00f3meno radiof\u00f3nico porque ese mismo d\u00eda, a la hora del almuerzo, vi a la t\u00eda Julia por primera vez. Era hermana de la mujer de mi t\u00edo Lucho y hab\u00eda llegado la noche anterior de Bolivia. Reci\u00e9n divorciada, ven\u00eda a descansar y a recuperarse de su fracaso matrimonial. \u00abEn realidad, a buscarse otro marido\u00bb, hab\u00eda dictaminado, en una reuni\u00f3n de familia, la m\u00e1s lenguaraz de mis parientes, la t\u00eda Hortensia. Yo almorzaba todos los jueves donde el t\u00edo Lucho y la t\u00eda Olga, y ese mediod\u00eda encontr\u00e9 a la familia todav\u00eda en piyama, cortando la mala noche con choritos picantes y cerveza fr\u00eda. Se hab\u00edan quedado hasta el amanecer, chismeando con la reci\u00e9n llegada, y despachado entre los tres una botella de whisky. Les dol\u00eda la cabeza, mi t\u00edo Lucho se quejaba de que su oficina andar\u00eda patas arriba, mi t\u00eda Olga dec\u00eda que era una verg\u00fcenza trasnochar fuera de s\u00e1bados, y la reci\u00e9n llegada, en bata, sin zapatos y con ruleros, vaciaba una maleta. No le incomod\u00f3 que yo la viera en esa facha en la que nadie la hubiera tomado por una reina de belleza.\n\n\u2014As\u00ed que t\u00fa eres el hijo de Dorita \u2014me dijo, estamp\u00e1ndome un beso en la mejilla\u2014. \u00bfYa terminaste el colegio, no?\n\nLa odi\u00e9 a muerte. Mis leves choques con la familia, en ese entonces, se deb\u00edan a que todos se empe\u00f1aban en tratarme todav\u00eda como un ni\u00f1o y no como lo que era, un hombre completo de dieciocho a\u00f1os. Nada me irritaba tanto como el Marito; ten\u00eda la sensaci\u00f3n de que el diminutivo me regresaba al pantal\u00f3n corto.\n\n\u2014Ya est\u00e1 en tercero de Derecho y trabaja como periodista \u2014le explic\u00f3 mi t\u00edo Lucho, alcanz\u00e1ndome un vaso de cerveza.\n\n\u2014La verdad \u2014me dio el puntillazo la t\u00eda Julia\u2014 es que pareces todav\u00eda una guagua, Marito.\n\nDurante el almuerzo, con ese aire cari\u00f1oso que adoptan los adultos cuando se dirigen a los idiotas y a los ni\u00f1os, me pregunt\u00f3 si ten\u00eda enamorada, si iba a fiestas, qu\u00e9 deporte practicaba, y me aconsej\u00f3, con una perversidad que no descubr\u00eda si era deliberada o inocente pero que igual me lleg\u00f3 al alma, que apenas pudiera me dejara crecer el bigote. A los morenos les sentaba y eso me facilitar\u00eda las cosas con las chicas.\n\n\u2014\u00c9l no piensa en faldas ni en jaranas \u2014le explic\u00f3 mi t\u00edo Lucho\u2014. Es un intelectual. Ha publicado un cuento en el Dominical de El Comercio.\n\n\u2014Cuidado que el hijo de Dorita nos vaya a salir del otro lado \u2014se ri\u00f3 la t\u00eda Julia y yo sent\u00ed un arrebato de solidaridad con su ex marido. Pero sonre\u00ed y le llev\u00e9 la cuerda. Durante el almuerzo se dedic\u00f3 a contar unos horribles chistes bolivianos y a tomarme el pelo. Al despedirme, pareci\u00f3 que quer\u00eda hacerse perdonar sus maldades, porque me dijo con un gesto amable que alguna noche la acompa\u00f1ara al cine, que le encantaba el cine.\n\nLlegu\u00e9 a Radio Panamericana justo a tiempo para evitar que Pascual dedicara todo el bolet\u00edn de las tres a la noticia de una batalla campal, en las calles ex\u00f3ticas de Rawalpindi, entre sepultureros y leprosos, publicada por \u00daltima Hora. Luego de preparar tambi\u00e9n los boletines de las cuatro y las cinco, sal\u00ed a tomar un caf\u00e9. En la puerta de Radio Central encontr\u00e9 a Genaro hijo, euf\u00f3rico. Me arrastr\u00f3 del brazo hasta el Bransa: \u00abTengo que contarte algo fant\u00e1stico\u00bb. Hab\u00eda estado unos d\u00edas en La Paz, por cuestiones de negocios, y all\u00ed hab\u00eda visto en acci\u00f3n a ese hombre plural: Pedro Camacho.\n\n\u2014No es un hombre sino una industria \u2014corrigi\u00f3, con admiraci\u00f3n\u2014. Escribe todas las obras de teatro que se presentan en Bolivia y las interpreta todas. Y escribe todas las radionovelas y las dirige y es el gal\u00e1n de todas.\n\nPero m\u00e1s que su fecundidad y versatilidad, le hab\u00eda impresionado su popularidad. Para poder verlo, en el Teatro Saavedra de La Paz, hab\u00eda tenido que comprar entradas de reventa al doble de su precio.\n\n\u2014Como en los toros, imag\u00ednate \u2014se asombraba\u2014. \u00bfQui\u00e9n ha llenado jam\u00e1s un teatro en Lima?\n\nMe cont\u00f3 que hab\u00eda visto, dos d\u00edas seguidos, a muchas jovencitas, adultas y viejas arremolinadas a las puertas de Radio Illimani esperando la salida del \u00eddolo para pedirle aut\u00f3grafos. La McCann Erickson de La Paz, por otra parte, le hab\u00eda asegurado que los radioteatros de Pedro Camacho ten\u00edan la mayor audiencia de las ondas bolivianas. Genaro hijo era eso que entonces comenzaba a llamarse un empresario progresista: le interesaban m\u00e1s los negocios que los honores, no era socio del Club Nacional ni un \u00e1vido de serlo, se hac\u00eda amigo de todo el mundo y su dinamismo fatigaba. Hombre de decisiones r\u00e1pidas, despu\u00e9s de su visita a Radio Illimani convenci\u00f3 a Pedro Camacho que se viniera al Per\u00fa, como exclusividad de Radio Central.\n\n\u2014No fue dif\u00edcil, all\u00e1 lo ten\u00edan al hambre \u2014me explic\u00f3\u2014. Se ocupar\u00e1 de las radionovelas y yo podr\u00e9 mandar al diablo a los tiburones de la CMQ.\n\nTrat\u00e9 de envenenar sus ilusiones. Le dije que acababa de comprobar que los bolivianos eran antipatiqu\u00edsimos y que Pedro Camacho se llevar\u00eda p\u00e9simo con toda la gente de Radio Central. Su acento caer\u00eda como pedrada a los oyentes y, por su ignorancia del Per\u00fa, meter\u00eda la pata a cada instante. Pero \u00e9l sonre\u00eda, intocado por mis profec\u00edas derrotistas. Aunque nunca hab\u00eda estado aqu\u00ed, Pedro Camacho le hab\u00eda hablado del alma lime\u00f1a como un bajopontino y su acento era soberbio, sin eses ni erres pronunciadas, de la categor\u00eda terciopelo.\n\n\u2014Entre Luciano Pando y los otros actores lo har\u00e1n papilla al pobre forastero \u2014so\u00f1\u00f3 Javier\u2014. O la bella Josefina S\u00e1nchez lo violar\u00e1.\n\nEst\u00e1bamos en el altillo y convers\u00e1bamos mientras yo pasaba a m\u00e1quina, cambiando adjetivos y adverbios, noticias de El Comercio y La Prensa para El Panamericano de las doce. Javier era mi mejor amigo y nos ve\u00edamos a diario, aunque fuera s\u00f3lo un momento, para constatar que exist\u00edamos. Era un ser de entusiasmos cambiantes y contradictorios, pero siempre sinceros. Hab\u00eda sido la estrella del Departamento de Literatura de la Cat\u00f3lica, donde no se vio antes a un alumno m\u00e1s aprovechado, ni m\u00e1s l\u00facido lector de poes\u00eda, ni m\u00e1s agudo comentarista de textos dif\u00edciles. Todos daban por descontado que se graduar\u00eda con una tesis brillante, ser\u00eda un catedr\u00e1tico brillante y un poeta o un cr\u00edtico igualmente brillante. Pero \u00e9l, un buen d\u00eda, sin explicaciones, hab\u00eda decepcionado a todo el mundo, abandonando la tesis en la que trabajaba, renunciando a la literatura y a la Universidad Cat\u00f3lica e inscribi\u00e9ndose en San Marcos como alumno de Econom\u00eda. Cuando alguien le preguntaba a qu\u00e9 se deb\u00eda esa deserci\u00f3n, \u00e9l confesaba (o bromeaba) que la tesis en que hab\u00eda estado trabajando le hab\u00eda abierto los ojos. Se iba a titular Las paremias en Ricardo Palma. Hab\u00eda tenido que leer las Tradiciones peruanas con lupa, a la caza de refranes, y como era concienzudo y riguroso, hab\u00eda conseguido llenar un caj\u00f3n de fichas eruditas. Luego, una ma\u00f1ana, quem\u00f3 el caj\u00f3n con las fichas en un descampado \u2014\u00e9l y yo bailamos una danza apache alrededor de las llamas filol\u00f3gicas\u2014 y decidi\u00f3 que odiaba la literatura y que hasta la econom\u00eda resultaba preferible a eso. Javier hac\u00eda su pr\u00e1ctica en el Banco Central de Reserva y siempre encontraba pretextos para darse un salto cada ma\u00f1ana hasta Radio Panamericana. De su pesadilla paremiol\u00f3gica le hab\u00eda quedado la costumbre de infligirme refranes sin ton ni son.\n\nMe sorprendi\u00f3 mucho que la t\u00eda Julia, pese a ser boliviana y vivir en La Paz, no hubiera o\u00eddo hablar nunca de Pedro Camacho. Pero ella me aclar\u00f3 que jam\u00e1s hab\u00eda escuchado una radionovela, ni puesto los pies en un teatro desde que interpret\u00f3 la Danza de las horas, en el papel de Crep\u00fasculo, el a\u00f1o que termin\u00f3 el colegio donde las monjas irlandesas (\u00abNo te atrevas a preguntarme cu\u00e1ntos a\u00f1os hace de eso, Marito\u00bb). \u00cdbamos caminando desde la casa del t\u00edo Lucho, al final de la avenida Armend\u00e1riz, hacia el Cine Barranco. Me hab\u00eda impuesto la invitaci\u00f3n ella misma, ese mediod\u00eda, de la manera m\u00e1s artera. Era el jueves siguiente a su llegada, y, aunque la perspectiva de ser otra vez v\u00edctima de los chistes bolivianos no me hac\u00eda gracia, no quise faltar al almuerzo semanal. Ten\u00eda la esperanza de no encontrarla, porque la v\u00edspera \u2014los mi\u00e9rcoles en la noche eran de visita a la t\u00eda Gaby\u2014 hab\u00eda o\u00eddo a la t\u00eda Hortensia comunicar con el tono de quien est\u00e1 en el secreto de los dioses:\n\n\u2014En su primera semana lime\u00f1a ha salido cuatro veces y con cuatro galanes diferentes, uno de ellos casado. \u00a1La divorciada se las trae!\n\nCuando llegu\u00e9 donde el t\u00edo Lucho, luego de El Panamericano de las doce, la encontr\u00e9 precisamente con uno de sus galanes. Sent\u00ed el dulce placer de la venganza al entrar a la sala y descubrir, sentado junto a ella, mir\u00e1ndola con ojos de conquistador, flamante de rid\u00edculo en su traje de otras \u00e9pocas, su corbata mariposa y su clavel en el ojal, al t\u00edo Pancracio, un primo hermano de mi abuela. Hab\u00eda enviudado hac\u00eda siglos, caminaba con los pies abiertos marcando las diez y diez, y en la familia se comentaban maliciosamente sus visitas porque no ten\u00eda reparo en pellizcar a las sirvientas a la vista de todos. Se pintaba el pelo, usaba reloj de bolsillo con leontina plateada y se lo pod\u00eda ver a diario, en las esquinas del jir\u00f3n de la Uni\u00f3n, a las seis de la tarde, piropeando a las oficinistas. Al inclinarme a besarla, susurr\u00e9 al o\u00eddo de la boliviana, con toda la iron\u00eda del mundo: \u00abQu\u00e9 buena conquista, Julita\u00bb. Ella me gui\u00f1\u00f3 un ojo y asinti\u00f3. Durante el almuerzo, el t\u00edo Pancracio, luego de disertar sobre la m\u00fasica criolla, en la que era un experto \u2014en las celebraciones familiares ofrec\u00eda siempre un solo de caj\u00f3n\u2014, se volvi\u00f3 hacia ella y, relamido como un gato, le cont\u00f3: \u00abA prop\u00f3sito, los jueves en la noche se re\u00fane la Pe\u00f1a Felipe Pinglo, en La Victoria, el coraz\u00f3n del criollismo. \u00bfTe gustar\u00eda o\u00edr un poco de verdadera m\u00fasica peruana?\u00bb. La t\u00eda Julia, sin vacilar un segundo y con una cara de desolaci\u00f3n que a\u00f1ad\u00eda el insulto a la calumnia, contest\u00f3 se\u00f1al\u00e1ndome: \u00abF\u00edjate qu\u00e9 l\u00e1stima. Marito me ha invitado al cine\u00bb. \u00abPaso a la juventud\u00bb, se inclin\u00f3 el t\u00edo Pancracio, con esp\u00edritu deportivo. Luego, cuando hubo partido, cre\u00ed que me salvaba pues la t\u00eda Olga pregunt\u00f3: \u00ab\u00bfEso del cine era s\u00f3lo para librarte del viejo verde?\u00bb. Pero la t\u00eda Julia la rectific\u00f3 con \u00edmpetu: \u00abNada de eso, hermana, me muero por ver la del Barranco, es impropia para se\u00f1oritas\u00bb. Se volvi\u00f3 hacia m\u00ed, que escuchaba c\u00f3mo se decid\u00eda mi destino nocturno, y, para tranquilizarme a\u00f1adi\u00f3 esta exquisita flor: \u00abNo te preocupes por la plata, Marito. Yo te invito\u00bb.\n\nY ah\u00ed est\u00e1bamos, caminando por la oscura quebrada de Armend\u00e1riz, por la ancha avenida Grau, al encuentro de una pel\u00edcula que, para colmo, era mexicana y se llamaba Madre y amante.\n\n\u2014Lo terrible de ser divorciada no es que todos los hombres se crean en la obligaci\u00f3n de proponerte cosas \u2014me informaba la t\u00eda Julia\u2014. Sino que por ser una divorciada piensan que ya no hay necesidad de romanticismo. No te enamoran, no te dicen galanter\u00edas finas, te proponen la cosa de buenas a primeras con la mayor vulgaridad. A m\u00ed me lleva la trampa. Para eso, en vez de que me saquen a bailar, prefiero venir al cine contigo.\n\nLe dije que muchas gracias por lo que me tocaba.\n\n\u2014Son tan est\u00fapidos que creen que toda divorciada es una mujer de la calle \u2014sigui\u00f3, sin darse por enterada\u2014. Y, adem\u00e1s, s\u00f3lo piensan en hacer cosas. Cuando lo bonito no es eso, sino enamorarse, \u00bfno es cierto?\n\nYo le expliqu\u00e9 que el amor no exist\u00eda, que era una invenci\u00f3n de un italiano llamado Petrarca y de los trovadores provenzales. Que eso que las gentes cre\u00edan un cristalino manar de la emoci\u00f3n, una pura efusi\u00f3n del sentimiento, era el deseo instintivo de los gatos en celo disimulado detr\u00e1s de las palabras bellas y los mitos de la literatura. No cre\u00eda en nada de eso, pero quer\u00eda hacerme el interesante. Mi teor\u00eda er\u00f3tico-biol\u00f3gica, por lo dem\u00e1s, dej\u00f3 a la t\u00eda Julia bastante incr\u00e9dula: \u00bfcre\u00eda yo de veras esa idiotez?\n\n\u2014Estoy contra el matrimonio \u2014le dije, con el aire m\u00e1s pedante que pude\u2014. Soy partidario de lo que llaman el amor libre, pero que, si fu\u00e9ramos honestos, deber\u00edamos llamar, simplemente, la c\u00f3pula libre.\n\n\u2014\u00bfC\u00f3pula quiere decir hacer cosas? \u2014se ri\u00f3. Pero al instante puso una cara decepcionada\u2014: En mi tiempo, los muchachos escrib\u00edan acr\u00f3sticos, mandaban flores a las chicas, necesitaban semanas para atreverse a darles un beso. Qu\u00e9 porquer\u00eda se ha vuelto el amor entre los mocosos de ahora, Marito.\n\nTuvimos un amago de disputa en la boleter\u00eda por ver qui\u00e9n pagaba la entrada, y, luego de soportar hora y media de Dolores del R\u00edo, gimiendo, abrazando, gozando, llorando, corriendo por la selva con los cabellos al viento, regresamos a casa del t\u00edo Lucho, tambi\u00e9n a pie, mientras la gar\u00faa nos mojaba los pelos y la ropa. Entonces, hablamos de nuevo de Pedro Camacho. \u00bfEstaba realmente segura que no lo hab\u00eda o\u00eddo mencionar jam\u00e1s? Porque, seg\u00fan Genaro hijo, era una celebridad boliviana. No, no lo conoc\u00eda ni siquiera de nombre. Pens\u00e9 que a Genaro le hab\u00edan metido el dedo a la boca, o que, tal vez, la supuesta industria radioteatral boliviana era una invenci\u00f3n suya para lanzar publicitariamente a un plum\u00edfero aborigen. Tres d\u00edas despu\u00e9s conoc\u00ed en carne y hueso a Pedro Camacho.\n\nAcababa de tener un incidente con Genaro pap\u00e1, porque Pascual, con su irreprimible predilecci\u00f3n por lo atroz, hab\u00eda dedicado todo el bolet\u00edn de las once a un terremoto en Ispah\u00e1n. Lo que irritaba a Genaro pap\u00e1 no era tanto que Pascual hubiera desechado otras noticias para referir, con lujo de detalles, c\u00f3mo los persas que sobrevivieron a los desmoronamientos eran atacados por serpientes que, al desplomarse sus refugios, afloraban a la superficie col\u00e9ricas y sibilantes, sino que el terremoto hab\u00eda ocurrido hac\u00eda una semana. Deb\u00ed convenir que a Genaro pap\u00e1 no le faltaba raz\u00f3n y me desfogu\u00e9 llamando a Pascual irresponsable. \u00bfDe d\u00f3nde hab\u00eda sacado ese refrito? De una revista argentina. \u00bfY por qu\u00e9 hab\u00eda hecho una cosa tan absurda? Porque no hab\u00eda ninguna noticia de actualidad importante y \u00e9sa, al menos, era entretenida. Cuando yo le explicaba que no nos pagaban para entretener a los oyentes sino para resumirles las noticias del d\u00eda, Pascual, moviendo una cabeza conciliatoria, me opon\u00eda su irrebatible argumento: \u00abLo que pasa es que tenemos concepciones diferentes del periodismo, don Mario\u00bb. Iba a responderle que si se empe\u00f1aba, cada vez que yo volviera las espaldas, en seguir aplicando su concepci\u00f3n tremendista del periodismo, muy pronto estar\u00edamos los dos en la calle, cuando apareci\u00f3 en la puerta del altillo una silueta inesperada. Era un ser peque\u00f1ito y menudo, en el l\u00edmite mismo del hombre de baja estatura y el enano, con una nariz grande y unos ojos extraordinariamente vivos, en los que bull\u00eda algo excesivo. Vest\u00eda de negro, un terno que se advert\u00eda muy usado, y su camisa y su corbatita de lazo ten\u00edan m\u00e1culas, pero, al mismo tiempo, en su manera de llevar esas prendas hab\u00eda algo en \u00e9l de atildado y de compuesto, de r\u00edgido, como en esos caballeros de las viejas fotograf\u00edas que parecen presos en sus levitas almidonadas, en sus chisteras tan justas. Pod\u00eda tener cualquier edad entre treinta y cincuenta a\u00f1os, y luc\u00eda una aceitosa cabellera negra que le llegaba a los hombros. Su postura, sus movimientos, su expresi\u00f3n parec\u00edan el desmentido mismo de lo espont\u00e1neo y natural, hac\u00edan pensar inmediatamente en el mu\u00f1eco articulado, en los hilos del t\u00edtere. Nos hizo una reverencia cortesana y con una solemnidad tan inusitada como su persona se present\u00f3 as\u00ed:\n\n\u2014Vengo a hurtarles una m\u00e1quina de escribir, se\u00f1ores. Les agradecer\u00eda que me ayuden. \u00bfCu\u00e1l de las dos es la mejor?\n\nSu dedo \u00edndice apuntaba alternativamente a mi m\u00e1quina de escribir y a la de Pascual. Pese a estar habituado a los contrastes entre voz y f\u00edsico por mis escapadas a Radio Central, me asombr\u00f3 que de figurilla tan m\u00ednima, de hechura tan desvalida, pudiera brotar una voz tan firme y melodiosa, una dicci\u00f3n tan perfecta. Parec\u00eda que en esa voz no s\u00f3lo desfilara cada letra, sin quedar mutilada ni una sola, sino tambi\u00e9n las part\u00edculas y los \u00e1tomos de cada una, los sonidos del sonido. Impaciente, sin advertir la sorpresa que su facha, su audacia y su voz provocaban en nosotros, se hab\u00eda puesto a escudri\u00f1ar y como a olfatear las dos m\u00e1quinas de escribir. Se decidi\u00f3 por mi veterana y enorme Remington, una carroza funeraria sobre la que no pasaban los a\u00f1os. Pascual fue el primero en reaccionar:\n\n\u2014\u00bfEs usted un ladr\u00f3n o qu\u00e9 es usted? \u2014lo increp\u00f3 y yo me di cuenta que me estaba indemnizando por el terremoto de Ispah\u00e1n\u2014. \u00bfSe le ocurre que se va a llevar as\u00ed nom\u00e1s las m\u00e1quinas del Servicio de Informaciones?\n\n\u2014El arte es m\u00e1s importante que tu Servicio de Informaciones, trasgo \u2014lo fulmin\u00f3 el personaje, ech\u00e1ndole una ojeada parecida a la que merece la alima\u00f1a pisoteada, y prosigui\u00f3 su operaci\u00f3n. Ante la mirada estupefacta de Pascual, que, sin duda, trataba de adivinar (como yo mismo) qu\u00e9 quer\u00eda decir trasgo, el visitante intent\u00f3 levantar la Remington. Consigui\u00f3 elevar el armatoste al precio de un esfuerzo descomunal, que hinch\u00f3 las venitas de su cuello y por poco le dispara los ojos de las \u00f3rbitas. Su cara se fue cubriendo de color granate, su frentecita de sudor, pero \u00e9l no desist\u00eda. Apretando los dientes, tambale\u00e1ndose, alcanz\u00f3 a dar unos pasos hacia la puerta, hasta que tuvo que rendirse: un segundo m\u00e1s y su carga lo iba a arrastrar con ella al suelo. Deposit\u00f3 la Remington sobre la mesita de Pascual y qued\u00f3 jadeando. Pero apenas recobr\u00f3 el aliento, totalmente ignorante de las sonrisas que el espect\u00e1culo nos provocaba a m\u00ed y a Pascual (\u00e9ste se hab\u00eda llevado ya varias veces un dedo a la sien para indicarme que se trataba de un loco), nos reprendi\u00f3 con severidad:\n\n\u2014No sean indolentes, se\u00f1ores, un poco de solidaridad humana. \u00c9chenme una mano.\n\nLe dije que lo sent\u00eda mucho pero que para llevarse esa Remington tendr\u00eda que pasar primero sobre el cad\u00e1ver de Pascual, y, en \u00faltimo caso, sobre el m\u00edo. El hombrecillo se acomodaba la corbatita, ligeramente descolocada por el esfuerzo. Ante mi sorpresa, con una mueca de contrariedad y dando muestras de una ineptitud total para el humor, repuso, asintiendo gravemente:\n\n\u2014Un tipo bien nacido nunca desaira un desaf\u00edo a pelear. El sitio y la hora, caballeros.\n\nLa providencial aparici\u00f3n de Genaro hijo en el altillo frustr\u00f3 lo que parec\u00eda ser la formalizaci\u00f3n de un duelo. Entr\u00f3 en el momento en que el hombrecito pertinaz intentaba de nuevo, amorat\u00e1ndose, tomar entre sus brazos a la Remington.\n\n\u2014Deje, Pedro, yo lo ayudo \u2014dijo, y le arrebat\u00f3 la m\u00e1quina como si fuera una caja de f\u00f3sforos. Comprendiendo entonces, por mi cara y la de Pascual, que nos deb\u00eda alguna explicaci\u00f3n, nos consol\u00f3 con aire risue\u00f1o\u2014: Nadie se ha muerto, no hay de qu\u00e9 ponerse tristes. Mi padre les repondr\u00e1 la m\u00e1quina prontito.\n\n\u2014Somos la quinta rueda del coche \u2014protest\u00e9 yo, para guardar las formas\u2014. Nos tienen en este altillo mugriento, ya me quitaron un escritorio para d\u00e1rselo al contador, y ahora mi Remington. Y ni siquiera me previenen.\n\n\u2014Cre\u00edamos que el se\u00f1or era un ladr\u00f3n \u2014me respald\u00f3 Pascual\u2014. Entr\u00f3 aqu\u00ed insult\u00e1ndonos y con prepotencias.\n\n\u2014Entre colegas no debe haber pleitos \u2014dijo, salom\u00f3nicamente, Genaro hijo. Se hab\u00eda puesto la Remington en el hombro y not\u00e9 que el hombrecito le llegaba exactamente a las solapas\u2014. \u00bfNo vino mi padre a hacer las presentaciones? Las hago yo, entonces, y todos felices.\n\nAl instante, con un movimiento veloz y autom\u00e1tico, el hombrecillo estir\u00f3 uno de sus bracitos, dio unos pasos hacia m\u00ed, me ofreci\u00f3 una manita de ni\u00f1o, y, con su preciosa voz de tenor, haciendo una nueva genuflexi\u00f3n cortesana, se present\u00f3:\n\n\u2014Un amigo: Pedro Camacho, boliviano y artista.\n\nRepiti\u00f3 el gesto, la venia y la frase con Pascual, quien, visiblemente, viv\u00eda un instante de supina confusi\u00f3n y era incapaz de decidir si el hombrecillo se burlaba de nosotros o era siempre as\u00ed. Pedro Camacho, despu\u00e9s de estrecharnos ceremoniosamente las manos, se volvi\u00f3 hacia el Servicio de Informaciones en bloque, y, desde el centro del altillo, a la sombra de Genaro hijo que parec\u00eda tras \u00e9l un gigante y que lo observaba muy serio, levant\u00f3 el labio superior y arrug\u00f3 la cara en un movimiento que dej\u00f3 al descubierto unos dientes amarillentos, en una caricatura o espectro de sonrisa. Se tom\u00f3 unos segundos, antes de gratificarnos con estas palabras musicales, acompa\u00f1adas de un adem\u00e1n de prestidigitador que se despide:\n\n\u2014No les guardo rencor, estoy acostumbrado a la incomprensi\u00f3n de la gente. \u00a1Hasta siempre, se\u00f1ores!\n\nDesapareci\u00f3 en la puerta del altillo, dando unos saltitos de duende para alcanzar al empresario progresista que, con la Remington a cuestas, se alejaba a trancos hacia el ascensor.\n\n### II\n\nERA UNA de esas soleadas ma\u00f1anas de la primavera lime\u00f1a en que los geranios amanecen m\u00e1s arrebatados, las rosas m\u00e1s fragantes y las buganvillas m\u00e1s crespas, cuando un famoso galeno de la ciudad, el doctor Alberto de Quinteros \u2014frente ancha, nariz aguile\u00f1a, mirada penetrante, rectitud y bondad en el esp\u00edritu\u2014 abri\u00f3 los ojos y se desperez\u00f3 en su espaciosa residencia de San Isidro. Vio, a trav\u00e9s de los visillos, el sol dorando el c\u00e9sped del cuidado jard\u00edn que encarcelaban vallas de crotos, la limpieza del cielo, la alegr\u00eda de las flores, y sinti\u00f3 esa sensaci\u00f3n bienhechora que dan ocho horas de sue\u00f1o reparador y la conciencia tranquila.\n\nEra s\u00e1bado y, a menos de alguna complicaci\u00f3n de \u00faltimo momento con la se\u00f1ora de los trillizos, no ir\u00eda a la cl\u00ednica y podr\u00eda dedicar la ma\u00f1ana a hacer un poco de ejercicio y a tomar una sauna antes del matrimonio de Elianita. Su esposa y su hija se hallaban en Europa, cultivando su esp\u00edritu y renovando su vestuario, y no regresar\u00edan antes de un mes. Otro, con sus medios de fortuna y su apostura \u2014sus cabellos nevados en las sienes y su porte distinguido, as\u00ed como su elegancia de maneras, despertaban miradas de codicia incluso en se\u00f1oras incorruptibles\u2014, hubiera aprovechado la moment\u00e1nea solter\u00eda para echar algunas canas al aire. Pero Alberto de Quinteros era un hombre al que ni el juego, ni las faldas ni el alcohol atra\u00edan m\u00e1s de lo debido, y entre sus conocidos \u2014que eran legi\u00f3n\u2014 circulaba este apotegma: \u00abSus vicios son la ciencia, su familia y la gimnasia\u00bb.\n\nOrden\u00f3 el desayuno y, mientras se lo preparaban, llam\u00f3 a la cl\u00ednica. El m\u00e9dico de guardia le inform\u00f3 que la se\u00f1ora de los trillizos hab\u00eda pasado una noche tranquila y que las hemorragias de la operada del fibroma hab\u00edan cesado. Dio instrucciones, indic\u00f3 que si ocurr\u00eda algo grave lo llamaran al Gimnasio Remigius, o, a la hora de almuerzo, donde su hermano Roberto, e hizo saber que al atardecer se dar\u00eda una vuelta por all\u00e1. Cuando el mayordomo le trajo su jugo de papaya, su caf\u00e9 negro y su tostada con miel de abeja, Alberto de Quinteros se hab\u00eda afeitado y vest\u00eda un pantal\u00f3n gris de corduroy, unos mocasines sin taco y una chompa verde de cuello alto. Desayun\u00f3 echando una ojeada distra\u00edda a las cat\u00e1strofes e intrigas matutinas de los peri\u00f3dicos, cogi\u00f3 su malet\u00edn deportivo y sali\u00f3. Se detuvo unos segundos en el jard\u00edn a palmear a Puck, el engre\u00eddo foxterrier que lo despidi\u00f3 con afectuosos ladridos.\n\nEl Gimnasio Remigius estaba a pocas cuadras, en la calle Miguel Dasso, y al doctor Quinteros le gustaba andarlas. Iba despacio, respond\u00eda a los saludos del vecindario, observaba los jardines de las casas que a esa hora eran regados y podados, y sol\u00eda parar un momento en la Librer\u00eda Castro Soto a elegir algunos best-sellers. Aunque era temprano, ya estaban frente al Davory los infalibles muchachos de camisas abiertas y cabelleras alborotadas. Tomaban helados, en sus motos o en los guardabarros de sus autos sport, se hac\u00edan bromas y planeaban la fiesta de la noche. Lo saludaron con respeto, pero, apenas los dej\u00f3 atr\u00e1s, uno de ellos se atrevi\u00f3 a darle uno de esos consejos que eran su pan cotidiano en el gimnasio, eternos chistes sobre su edad y su profesi\u00f3n, que \u00e9l soportaba con paciencia y buen humor: \u00abNo se canse mucho, doctor, piense en sus nietos\u00bb. Apenas lo oy\u00f3 pues estaba imaginando lo linda que se ver\u00eda Elianita en su vestido de novia dise\u00f1ado para ella por la casa Christian Dior de Par\u00eds.\n\nNo hab\u00eda mucha gente en el gimnasio esa ma\u00f1ana. S\u00f3lo Coco, el instructor, y dos fan\u00e1ticos de las pesas, el Negro Humilla y Perico Sarmiento, tres monta\u00f1as de m\u00fasculos equivalentes a los de diez hombres normales. Deb\u00edan de haber llegado no hac\u00eda mucho tiempo, estaban todav\u00eda calentando:\n\n\u2014Pero si ah\u00ed viene la cig\u00fce\u00f1a \u2014le estrech\u00f3 la mano Coco.\n\n\u2014\u00bfTodav\u00eda en pie, a pesar de los siglos? \u2014le hizo adi\u00f3s el Negro Humilla.\n\nPerico se limit\u00f3 a chasquear la lengua y a levantar dos dedos, en el caracter\u00edstico saludo que hab\u00eda importado de Texas. Al doctor Quinteros le agradaba esa informalidad, las confianzas que se tomaban con \u00e9l sus compa\u00f1eros de gimnasio, como si el hecho de verse desnudos y de sudar juntos los nivelara en una fraternidad donde desaparec\u00edan las diferencias de edad y posici\u00f3n. Les contest\u00f3 que si necesitaban sus servicios estaba a sus \u00f3rdenes, que a los primeros mareos o antojos corrieran a su consultorio donde ten\u00eda listo el guante de jebe para auscultarles la intimidad.\n\n\u2014C\u00e1mbiate y ven a hacer un poco de warm up \u2014le dijo Coco, que ya estaba saltando en el sitio otra vez.\n\n\u2014Si te viene el infarto, no pasas de morirte, veterano \u2014lo alent\u00f3 Perico, poni\u00e9ndose al paso de Coco.\n\n\u2014Adentro est\u00e1 el tablista \u2014oy\u00f3 decir al Negro Humilla, cuando entraba al vestuario.\n\nY, en efecto, ah\u00ed estaba su sobrino Richard, en buzo azul, calz\u00e1ndose las zapatillas. Lo hac\u00eda con desgano, como si las manos se le hubieran vuelto de trapo, y ten\u00eda la cara agria y ausente. Se qued\u00f3 mir\u00e1ndolo con unos ojos azules totalmente idos y una indiferencia tan absoluta que el doctor Quinteros se pregunt\u00f3 si no se hab\u00eda vuelto invisible.\n\n\u2014S\u00f3lo los enamorados se abstraen as\u00ed \u2014se acerc\u00f3 a \u00e9l y le revolvi\u00f3 los cabellos\u2014. Baja de la luna, sobrino.\n\n\u2014Perdona, t\u00edo \u2014despert\u00f3 Richard, enrojeciendo violentamente, como si lo acabaran de sorprender haciendo algo sucio\u2014. Estaba pensando.\n\n\u2014Me gustar\u00eda saber en qu\u00e9 maldades \u2014se ri\u00f3 el doctor Quinteros, mientras abr\u00eda su malet\u00edn, eleg\u00eda un casillero y comenzaba a desvestirse\u2014. Tu casa debe ser un desbarajuste terrible. \u00bfEst\u00e1 muy nerviosa Elianita?\n\nRichard lo mir\u00f3 con una especie de odio s\u00fabito y el doctor pens\u00f3 qu\u00e9 le ha picado a este muchacho. Pero su sobrino, haciendo un esfuerzo notorio por mostrarse natural, esboz\u00f3 un amago de sonrisa:\n\n\u2014S\u00ed, un desbarajuste. Por eso me vine a quemar un poco de grasa, hasta que sea hora.\n\nEl doctor pens\u00f3 que iba a a\u00f1adir: \u00abDe subir al pat\u00edbulo\u00bb. Ten\u00eda la voz lastrada por la tristeza, y tambi\u00e9n sus facciones y la torpeza con que anudaba los cordones y los movimientos bruscos de su cuerpo revelaban incomodidad, malestar \u00edntimo, desasosiego. No pod\u00eda tener los ojos quietos: los abr\u00eda, los cerraba, fijaba la vista en un punto, la desviaba, la regresaba, volv\u00eda a apartarla, como buscando algo imposible de encontrar. Era el muchacho m\u00e1s apuesto de la tierra, un joven dios bru\u00f1ido por la intemperie \u2014hac\u00eda tabla aun en los meses m\u00e1s h\u00famedos del invierno y descollaba tambi\u00e9n en el b\u00e1squet, el tenis, la nataci\u00f3n y el fulbito\u2014, al que los deportes hab\u00edan modelado un cuerpo de esos que el Negro Humilla llamaba \u00ablocura de maricones\u00bb: ni gota de grasa, espaldas anchas que descend\u00edan en una tersa l\u00ednea de m\u00fasculos hasta la cintura de avispa y unas largas piernas duras y \u00e1giles que habr\u00edan hecho palidecer de envidia al mejor boxeador. Alberto de Quinteros hab\u00eda o\u00eddo con frecuencia a su hija Charo y a sus amigas comparar a Richard con Charlton Heston y sentenciar que todav\u00eda era m\u00e1s churro, que lo dejaba botado en pinta. Estaba en primer a\u00f1o de Arquitectura, y, seg\u00fan Roberto y Margarita, sus padres, hab\u00eda sido siempre un modelo: estudioso, obediente, bueno con ellos y con su hermana, sano, simp\u00e1tico. Elianita y \u00e9l eran sus sobrinos preferidos y por eso, mientras se pon\u00eda el suspensor, el buzo, las zapatillas \u2014Richard lo esperaba junto a las duchas, dando unos golpecitos contra los azulejos\u2014, el doctor Alberto de Quinteros se apen\u00f3 al verlo tan turbado.\n\n\u2014\u00bfAlg\u00fan problema, sobrino? \u2014le pregunt\u00f3, como al descuido, con una sonrisa bondadosa\u2014. \u00bfAlgo en que tu t\u00edo pueda echarte una mano?\n\n\u2014Ninguno, qu\u00e9 ocurrencia \u2014se apresur\u00f3 a contestar Richard, encendi\u00e9ndose de nuevo como un f\u00f3sforo\u2014. Estoy regio y con unas ganas b\u00e1rbaras de calentar.\n\n\u2014\u00bfLe llevaron mi regalo a tu hermana? \u2014record\u00f3 de pronto el doctor\u2014. En la Casa Murgu\u00eda me prometieron que lo har\u00edan ayer.\n\n\u2014Una pulsera bestial \u2014Richard hab\u00eda comenzado a saltar sobre las losetas blancas del vestuario\u2014. A la flaca le encant\u00f3.\n\n\u2014De estas cosas se encarga tu t\u00eda, pero como sigue paseando por las Europas, tuve que escogerla yo mismo \u2014el doctor Quinteros hizo un gesto enternecido\u2014: Elianita, vestida de novia, ser\u00e1 una aparici\u00f3n.\n\nPorque la hija de su hermano Roberto era en mujer lo que Richard en hombre: una de esas bellezas que dignifican a la especie y hacen que las met\u00e1foras sobre las muchachas de dientes de perla, ojos como luceros, cabellos de trigo y cutis de melocot\u00f3n, luzcan mezquinas. Menuda, de cabellos oscuros y piel muy blanca, graciosa hasta en su manera de respirar, ten\u00eda una carita de l\u00edneas cl\u00e1sicas, unos rasgos que parec\u00edan dibujados por un miniaturista del oriente. Un a\u00f1o m\u00e1s joven que Richard, acababa de terminar el colegio, su \u00fanico defecto era la timidez \u2014tan excesiva que, para desesperaci\u00f3n de los organizadores, no hab\u00edan podido convencerla de que participara en el Concurso Miss Per\u00fa\u2014 y nadie, entre ellos el doctor Quinteros, pod\u00eda explicarse por qu\u00e9 se casaba tan pronto y, sobre todo, con qui\u00e9n. Ya que el Pelirrojo Ant\u00fanez ten\u00eda algunas virtudes \u2014bueno como el pan, un t\u00edtulo de Business Administration por la Universidad de Chicago, la compa\u00f1\u00eda de fertilizantes que heredar\u00eda y varias copas en carreras de ciclismo\u2014 pero, entre los innumerables muchachos de Miraflores y San Isidro que hab\u00edan hecho la corte a Elianita y que hubieran llegado al crimen por casarse con ella, era, sin duda, el menos agraciado y (el doctor Quinteros se avergonz\u00f3 por permitirse este juicio sobre quien, dentro de pocas horas, pasar\u00eda a ser su sobrino) el m\u00e1s soso y tontito.\n\n\u2014Eres m\u00e1s lento para cambiarte que mi mam\u00e1, t\u00edo \u2014se quej\u00f3 Richard, entre saltos.\n\nCuando entraron a la sala de ejercicios, Coco, en quien la pedagog\u00eda era una vocaci\u00f3n m\u00e1s que un oficio, instru\u00eda al Negro Humilla, se\u00f1al\u00e1ndole el est\u00f3mago, sobre este axioma de su filosof\u00eda:\n\n\u2014Cuando comas, cuando trabajes, cuando est\u00e9s en el cine, cuando paletees a tu hembra, cuando chupes, en todos los momentos de tu vida, y, si puedes, hasta en el f\u00e9retro: \u00a1hunde la panza!\n\n\u2014Diez minutos de warm ups para alegrar el esqueleto, Matusal\u00e9n \u2014orden\u00f3 el instructor.\n\nMientras saltaba a la soga junto a Richard, y sent\u00eda que un agradable calor iba apoder\u00e1ndose interiormente de su cuerpo, el doctor Quinteros pensaba que, despu\u00e9s de todo, no era tan terrible tener cincuenta a\u00f1os si uno los llevaba as\u00ed. \u00bfQui\u00e9n, entre los amigos de su edad, pod\u00eda lucir un vientre tan liso y unos m\u00fasculos tan despiertos? Sin ir muy lejos, su hermano Roberto, pese a ser tres a\u00f1os menor, con su rolliza y abotagada apariencia y la precoz curvatura de espalda, parec\u00eda llevarle diez. Pobre Roberto, deb\u00eda de estar triste con la boda de Elianita, la ni\u00f1a de sus ojos. Porque, claro, era una manera de perderla. Tambi\u00e9n su hija Charo se casar\u00eda en cualquier momento \u2014su enamorado, Tato Soldevilla, se recibir\u00eda dentro de poco de ingeniero\u2014 y tambi\u00e9n \u00e9l, entonces, se sentir\u00eda apenado y m\u00e1s viejo. El doctor Quinteros saltaba a la soga sin enredarse ni alterar el ritmo, con la facilidad que da la pr\u00e1ctica, cambiando de pie y cruzando y descruzando las manos como un gimnasta consumado. Ve\u00eda, en cambio, por el espejo, que su sobrino saltaba demasiado r\u00e1pido, con atolondramiento, tropez\u00e1ndose. Ten\u00eda los dientes apretados, brillo de sudor en la frente y guardaba los ojos cerrados como para concentrarse mejor. \u00bfAlg\u00fan problema de faldas, tal vez?\n\n\u2014Basta de soguita, flojonazos \u2014Coco, aunque estaba levantando pesas con Perico y el Negro Humilla, no los perd\u00eda de vista y les llevaba el tiempo\u2014. Tres series de sit ups. Sobre el pucho, f\u00f3siles.\n\nLos abdominales eran la prueba de fuerza del doctor Quinteros. Los hac\u00eda a mucha velocidad, con las manos en la nuca, en la tabla alzada a la segunda posici\u00f3n, aguantando la espalda a ras del suelo y casi tocando las rodillas con la frente. Entre cada serie de treinta dejaba un minuto de intervalo en que permanec\u00eda tendido, respirando hondo. Al terminar los noventa, se sent\u00f3 y comprob\u00f3, satisfecho, que hab\u00eda sacado ventaja a Richard. Ahora s\u00ed sudaba de pies a cabeza y sent\u00eda el coraz\u00f3n acelerado.\n\n\u2014No acabo de entender por qu\u00e9 se casa Elianita con el Pelirrojo Ant\u00fanez \u2014se oy\u00f3 decir a s\u00ed mismo, de pronto\u2014. \u00bfQu\u00e9 le ha visto?\n\nFue un acto fallido y se arrepinti\u00f3 al instante, pero Richard no pareci\u00f3 sorprenderse. Jadeando \u2014acababa de terminar los abdominales\u2014 le respondi\u00f3 con una broma:\n\n\u2014Dicen que el amor es ciego, t\u00edo.\n\n\u2014Es un excelente muchacho y seguro que la har\u00e1 muy feliz \u2014compuso las cosas el doctor Quinteros, algo cortado\u2014. Quer\u00eda decir que, entre los admiradores de tu hermana, estaban los mejores partidos de Lima. Mira que basurearlos a todos para terminar aceptando al Pelirrojo, que es un buen chico, pero tan, en fin...\n\n\u2014\u00bfTan calzonudo, quieres decir? \u2014lo ayud\u00f3 Richard.\n\n\u2014Bueno, no lo hubiera dicho con esa crudeza \u2014aspiraba y expulsaba el aire el doctor Quinteros, abriendo y cerrando los brazos\u2014. Pero, la verdad, parece algo ca\u00eddo del nido. Con cualquier otra ser\u00eda perfecto, pero a Elianita, tan linda, tan viva, el pobre le llora \u2014se sinti\u00f3 inc\u00f3modo con su propia franqueza\u2014. Oye, no lo tomes a mal, sobrino.\n\n\u2014No te preocupes, t\u00edo \u2014le sonri\u00f3 Richard\u2014. El Pelirrojo es buena gente y si la flaca le ha hecho caso por algo ser\u00e1.\n\n\u2014\u00a1Tres series de treinta side bonds, inv\u00e1lidos! \u2014rugi\u00f3 Coco, con ochenta kilos sobre la cabeza, hinchado como un sapo\u2014. \u00a1Hundiendo la panza, no bot\u00e1ndola!\n\nEl doctor Quinteros pens\u00f3 que, con la gimnasia, Richard olvidar\u00eda sus problemas, pero, mientras hac\u00eda flexiones laterales, lo vio ejecutar los ejercicios con renovada furia: la cara se le descompon\u00eda de nuevo en una expresi\u00f3n de angustia y malhumor. Record\u00f3 que en la familia Quinteros hab\u00eda abundantes neur\u00f3ticos y pens\u00f3 que a lo mejor al hijo mayor de Roberto le hab\u00eda tocado en suerte mantener esa tradici\u00f3n entre las nuevas generaciones, y, despu\u00e9s, se distrajo pensando que, despu\u00e9s de todo, tal vez hubiera sido m\u00e1s prudente darse un salto a la cl\u00ednica antes del gimnasio para echar un vistazo a la se\u00f1ora de los trillizos y a la operada del fibroma. Luego ya no pens\u00f3, pues el esfuerzo f\u00edsico lo absorbi\u00f3 enteramente y, mientras bajaba y sub\u00eda las piernas (\u00a1Leg rises, cincuenta veces!), flexionaba el tronco (\u00a1Trunk twist con bar, tres series, hasta botar los bofes!), hac\u00eda trabajar la espalda, el torso, los antebrazos, el cuello, obediente a las \u00f3rdenes de Coco (\u00a1Fuerza, tatarabuelo!, \u00a1M\u00e1s r\u00e1pido, cad\u00e1ver!), fue tan s\u00f3lo un pulm\u00f3n que recib\u00eda y expel\u00eda aire, una piel que escup\u00eda sudor y unos m\u00fasculos que se esforzaban, cansaban y sufr\u00edan. Cuando Coco grit\u00f3 \u00a1Tres series de quince pull-overs con mancuernas! hab\u00eda alcanzado su tope. Trat\u00f3, sin embargo, por amor propio, de hacer cuando menos una serie con doce kilos, pero fue incapaz. Estaba exhausto. La pesa se le escap\u00f3 de las manos al tercer intento y tuvo que soportar las bromas de los pesistas (\u00a1Las momias a la tumba y las cig\u00fce\u00f1as al jard\u00edn zool\u00f3gico!, \u00a1Llamen a la funeraria!, \u00a1Requiescat in pace, Amen!), y ver, con muda envidia, c\u00f3mo Richard \u2014siempre apurado, siempre furioso\u2014 completaba su rutina sin dificultad. No bastan la disciplina, la constancia, pens\u00f3 el doctor Quinteros, las dietas equilibradas, la vida met\u00f3dica. Eso compensaba las diferencias hasta cierto l\u00edmite; pasado \u00e9ste, la edad establec\u00eda distancias insalvables, muros invencibles. M\u00e1s tarde, desnudo en la sauna, ciego por el sudor que le chorreaba entre las pesta\u00f1as, repiti\u00f3 con melancol\u00eda una frase que hab\u00eda le\u00eddo en un libro: \u00a1Juventud, cuyo recuerdo desespera! Al salir, vio que Richard se hab\u00eda unido a los pesistas y que alternaba con ellos. Coco le hizo un adem\u00e1n burl\u00f3n, se\u00f1al\u00e1ndolo:\n\n\u2014El buen mozo ha decidido suicidarse, doctor.\n\nRichard ni siquiera sonri\u00f3. Ten\u00eda las pesas en alto y su cara, empapada, roja, con las venas salientes, mostraba una exasperaci\u00f3n que parec\u00eda a punto de volcarse contra ellos. Al doctor le pas\u00f3 la idea de que su sobrino, de pronto, iba a aplastarles las cabezas a los cuatro con las pesas que ten\u00eda en las manos. Les hizo adi\u00f3s y murmur\u00f3: \u00abNos vemos en la iglesia, Richard\u00bb.\n\nDe vuelta en su casa, lo tranquiliz\u00f3 saber que la mam\u00e1 de los trillizos quer\u00eda jugar al bridge con unas amigas en su cuarto de la cl\u00ednica y que la operada del fibroma hab\u00eda preguntado si ya hoy podr\u00eda comer wantanes sopados en salsa de tamarindo. Autoriz\u00f3 el bridge y el want\u00e1n, y, con toda calma, se puso terno azul oscuro, camisa de seda blanca y una corbata plateada que sujet\u00f3 con una perla. Perfumaba su pa\u00f1uelo cuando lleg\u00f3 carta de su mujer, a la que Charito hab\u00eda a\u00f1adido una posdata. La hab\u00edan despachado de Venecia, la ciudad catorce del tour, y le dec\u00edan: \u00abCuando recibas \u00e9sta habremos hecho por lo menos siete ciudades m\u00e1s, todas lind\u00edsimas\u00bb. Estaban felices y Charito muy entusiasta con los italianos, \u00abunos artistas de cine, papi, y no te imaginas qu\u00e9 piropeadores, pero no le vayas a contar a Tato, mil besos, chau\u00bb.\n\nFue andando hasta la Iglesia de Santa Mar\u00eda, en el \u00f3valo Guti\u00e9rrez. Era todav\u00eda temprano y comenzaban a llegar los invitados. Se instal\u00f3 en las filas de adelante y se entretuvo observando el altar, adornado con lirios y rosas blancas, y los vitrales, que parec\u00edan mitras de prelados. Una vez m\u00e1s constat\u00f3 que esa iglesia no le gustaba nada, por su \u00edrrita combinaci\u00f3n de yeso y ladrillos y sus pretenciosos arcos oblongos. De tanto en tanto saludaba a los conocidos con sonrisas. Claro, no pod\u00eda ser menos, todo el mundo iba llegando a la iglesia: parientes remot\u00edsimos, amigos que resucitaban despu\u00e9s de siglos, y, por supuesto, lo m\u00e1s graneado de la ciudad, banqueros, embajadores, industriales, pol\u00edticos. Este Roberto, esta Margarita, siempre tan fr\u00edvolos, pensaba el doctor Quinteros, sin acritud, lleno de benevolencia para con las debilidades de su hermano y su cu\u00f1ada. Seguramente que, en el almuerzo, echar\u00edan la casa por la ventana. Se emocion\u00f3 al ver entrar a la novia, en el momento en que romp\u00edan los compases de la marcha nupcial. Estaba realmente bell\u00edsima, en su vaporoso vestido blanco, y su carita, perfilada bajo el velo, ten\u00eda algo extraordinariamente gr\u00e1cil, leve, espiritual, mientras avanzaba hacia el altar, con los ojos bajos, del brazo de Roberto, quien, corpulento y augusto, disimulaba su emoci\u00f3n adoptando aires de due\u00f1o del mundo. El Pelirrojo Ant\u00fanez parec\u00eda menos feo, enfundado en su flamante chaqu\u00e9 y con la cara resplandeciente de felicidad, y hasta su madre \u2014una inglesa desgarbada que, pese a vivir un cuarto de siglo en el Per\u00fa, todav\u00eda confund\u00eda las preposiciones\u2014 parec\u00eda, en su largo traje oscuro y su peinado de dos pisos, una se\u00f1ora atractiva. Es cierto, pens\u00f3 el doctor Quinteros, el que la sigue la consigue. Porque el pobre Pelirrojo Ant\u00fanez hab\u00eda perseguido a Elianita desde que eran ni\u00f1os, y la hab\u00eda asediado con delicadezas y atenciones que ella recib\u00eda invariablemente con ol\u00edmpico desd\u00e9n. Pero \u00e9l hab\u00eda soportado todos los desplantes y malacrianzas de Elianita, y las terribles bromas con que los chicos del barrio celebraban su resignaci\u00f3n. Muchacho tenaz, reflexionaba el doctor Quinteros, lo hab\u00eda logrado, y ah\u00ed estaba ahora, p\u00e1lido de emoci\u00f3n, deslizando el aro en el dedo anular de la muchacha m\u00e1s linda de Lima. La ceremonia hab\u00eda terminado y, en medio de una masa rumorosa, haciendo inclinaciones de cabeza a diestra y siniestra, el doctor Quinteros avanzaba hacia los salones de la iglesia cuando divis\u00f3, de pie junto a una columna, como apart\u00e1ndose asqueado de la gente, a Richard.\n\nMientras hac\u00eda cola para llegar hasta los novios, el doctor Quinteros tuvo que festejar una docena de chistes contra el gobierno que le contaron los hermanos Febre, dos mellizos tan id\u00e9nticos que, se dec\u00eda, ni sus propias mujeres los diferenciaban. Era tal el gent\u00edo que el sal\u00f3n parec\u00eda a punto de desplomarse; muchas personas hab\u00edan permanecido en los jardines, esperando turno para entrar. Un enjambre de mozos circulaba ofreciendo champa\u00f1a. Se o\u00edan risas, bromas, brindis, y todo el mundo dec\u00eda que la novia estaba lind\u00edsima. Cuando el doctor Quinteros pudo al fin llegar hasta ella, vio que Elianita segu\u00eda compuesta y lozana pese al calor y la apretura. \u00abMil a\u00f1os de felicidad, flaquita\u00bb, le dijo, abraz\u00e1ndola, y ella le cont\u00f3 al o\u00eddo: \u00abCharito me llam\u00f3 esta ma\u00f1ana desde Roma para felicitarme, y tambi\u00e9n habl\u00e9 con la t\u00eda Mercedes. \u00a1Qu\u00e9 amorosas de llamarme!\u00bb. El Pelirrojo Ant\u00fanez, sudando, colorado como un camar\u00f3n, chisporroteaba de felicidad: \u00ab\u00bfAhora tambi\u00e9n tendr\u00e9 que decirle t\u00edo, don Alberto?\u00bb. \u00abClaro, sobrino \u2014lo palme\u00f3 el doctor Quinteros\u2014, y tendr\u00e1s que tutearme\u00bb.\n\nSali\u00f3 medio asfixiado del estrado de los novios y, entre flashes de fot\u00f3grafos, roces, saludos, pudo llegar al jard\u00edn. All\u00ed la condensaci\u00f3n humana era menor y se pod\u00eda respirar. Tom\u00f3 una copa y se vio envuelto, en una ronda de m\u00e9dicos amigos, en interminables bromas que ten\u00edan como tema el viaje de su mujer: Mercedes no volver\u00eda, se quedar\u00eda con alg\u00fan franchute, en los extremos de la frente comenzaban ya a brotarle unos cuernitos. El doctor Quinteros, mientras les llevaba la cuerda, pens\u00f3 \u2014recordando el gimnasio\u2014 que hoy le tocaba estar en la berlina. A ratos ve\u00eda, por sobre un mar de cabezas, a Richard, al otro extremo del sal\u00f3n; en medio de muchachos y muchachas que re\u00edan: serio y fruncido, vaciaba las copas de champa\u00f1a como agua. \u00abTal vez le apena que Elianita se case con Ant\u00fanez \u2014pens\u00f3\u2014; tambi\u00e9n \u00e9l hubiera querido alguien m\u00e1s brillante para su hermana\u00bb. Pero no, lo probable es que estuviera atravesando una de esas crisis de transici\u00f3n. Y el doctor Quinteros record\u00f3 c\u00f3mo \u00e9l tambi\u00e9n, a la edad de Richard, hab\u00eda pasado un periodo dif\u00edcil, dudando entre la medicina y la ingenier\u00eda aeron\u00e1utica. (Su padre lo hab\u00eda convencido con un argumento de peso: en el Per\u00fa, como ingeniero aeron\u00e1utico no hubiera tenido otra salida que dedicarse a las cometas o al aeromodelismo.) Tal vez Roberto, siempre tan absorbido en sus negocios, no estaba en condiciones de aconsejar a Richard. Y el doctor Quinteros, en uno de esos arranques que le hab\u00edan ganado el aprecio general, decidi\u00f3 que, un d\u00eda de \u00e9stos, con toda la delicadeza del caso, invitar\u00eda a su sobrino y sutilmente explorar\u00eda la manera de ayudarlo.\n\nLa casa de Roberto y Margarita estaba en la avenida Santa Cruz, a pocas cuadras de la Iglesia de Santa Mar\u00eda, y, al t\u00e9rmino de la recepci\u00f3n en la parroquia, los invitados al almuerzo desfilaron bajo los \u00e1rboles y el sol de San Isidro, hacia el caser\u00f3n de ladrillos rojos y techos de madera, rodeado de c\u00e9sped, de flores, de verjas, y primorosamente decorado para la fiesta. Al doctor Quinteros le bast\u00f3 llegar a la puerta para comprender que la celebraci\u00f3n iba a superar sus propias predicciones y que asistir\u00eda a un acontecimiento que los cronistas sociales llamar\u00edan \u00absoberbio\u00bb.\n\nA lo largo y a lo ancho del jard\u00edn se hab\u00edan puesto mesas y sombrillas, y, al fondo, junto a las perreras, un enorme toldo proteg\u00eda una mesa de n\u00edveo mantel, que corr\u00eda a lo largo de la pared, erizada de fuentes con entremeses multicolores. El bar estaba junto al estanque de agallados peces japoneses y se ve\u00edan tantas copas, botellas, cocteleras, jarras de refrescos, como para quitar la sed a un ej\u00e9rcito. Mozos de chaquetilla blanca y muchachas de cofia y delantal recib\u00edan a los invitados abrum\u00e1ndolos desde la misma puerta de calle con pisco sours, algarrobinas, vodkas con maracuy\u00e1, vasos de whisky, gin o copas de champa\u00f1a, y palitos de queso, papitas con aj\u00ed, guindas rellenas de tocino, camarones arrebozados, volovanes y todos los bocaditos concebidos por la inventiva lime\u00f1a para abrir el apetito. En el interior, enormes canastas y ramos de rosas, nardos, gladiolos, alel\u00edes, claveles, apoyados contra las paredes, dispuestos a lo largo de las escaleras o sobre los alf\u00e9izares y los muebles, refrescaban el ambiente. El parquet estaba encerado, las cortinas lavadas, las porcelanas y la plater\u00eda relucientes y el doctor Quinteros sonri\u00f3 imaginando que hasta los huacos de las vitrinas hab\u00edan sido lustrados. En el vest\u00edbulo hab\u00eda tambi\u00e9n un buffet, y en el comedor se explayaban los dulces \u2014mazapanes, queso helado, suspiros, huevos chimbos, yemas, coquitos, nueces con alm\u00edbar\u2014 alrededor de la impresionante torta de bodas, una construcci\u00f3n de tules y columnas, cremosa y arrogante, que arrancaba trinos de admiraci\u00f3n a las se\u00f1oras. Pero lo que concitaba la curiosidad femenina, sobre todo, eran los regalos, en el segundo piso; se hab\u00eda formado una cola tan larga para verlos que el doctor Quinteros decidi\u00f3 r\u00e1pidamente no hacerla, pese a que le hubiera gustado saber c\u00f3mo luc\u00eda en el conjunto su pulsera.\n\nDespu\u00e9s de curiosear un poco por todas partes \u2014estrechando manos, recibiendo y prodigando abrazos\u2014 retorn\u00f3 al jard\u00edn y fue a sentarse bajo una sombrilla, a degustar con calma su segunda copa del d\u00eda. Todo estaba muy bien, Margarita y Roberto sab\u00edan hacer las cosas en grande. Y aunque no le parec\u00eda excesivamente fina la idea de la orquesta \u2014hab\u00edan retirado las alfombras, la mesita y el aparador con los marfiles para que las parejas tuvieran donde bailar\u2014 disculp\u00f3 esa inelegancia como una concesi\u00f3n a las nuevas generaciones, pues, ya se sab\u00eda, para la juventud fiesta sin baile no era fiesta. Comenzaban a servir el pavo y el vino y ahora Elianita, de pie en el segundo pelda\u00f1o de la entrada, estaba arrojando su bouquet de novia que decenas de compa\u00f1eras de colegio y del barrio esperaban con las manos en alto. El doctor Quinteros divis\u00f3 en un rinc\u00f3n del jard\u00edn a la vieja Venancia, el ama de Elianita desde la cuna: la anciana, conmovida hasta el alma, se limpiaba los ojos con el ruedo de su delantal.\n\nSu paladar no alcanz\u00f3 a distinguir la marca del vino pero supo inmediatamente que era extranjero, acaso espa\u00f1ol o chileno y tampoco descart\u00f3 \u2014dentro de las locuras del d\u00eda\u2014 que fuera franc\u00e9s. El pavo estaba tierno, el pur\u00e9 era una mantequilla, y hab\u00eda una ensalada de coles y pasas que, pese a sus principios en materia de dieta, no pudo dejar de repetir. Saboreaba una segunda copa de vino y empezaba a sentir una agradable somnolencia cuando vio venir a Richard hacia \u00e9l. Balanceaba una copa de whisky en la mano; ten\u00eda los ojos vidriosos y la voz cambiante:\n\n\u2014\u00bfHay algo m\u00e1s est\u00fapido que una fiesta de matrimonio, t\u00edo? \u2014murmur\u00f3, haciendo un adem\u00e1n despectivo hacia todo lo que los rodeaba y dej\u00e1ndose caer en la silla de al lado. La corbata se le hab\u00eda corrido, una manchita fresca afeaba la solapa de su terno gris, y en sus ojos, adem\u00e1s de vestigios de licor, hab\u00eda empozada una oce\u00e1nica rabia.\n\n\u2014Bueno, te confieso que yo no soy un gran entusiasta de las fiestas \u2014dijo con bonhom\u00eda el doctor Quinteros\u2014. Pero que no lo seas t\u00fa, a tu edad, me llama la atenci\u00f3n, sobrino.\n\n\u2014Las odio con toda mi alma \u2014susurr\u00f3 Richard, mirando como si quisiera desaparecer a todo el mundo\u2014. No s\u00e9 por qu\u00e9 maldita sea estoy aqu\u00ed.\n\n\u2014Imag\u00ednate lo que habr\u00eda sido para tu hermana que no vinieras a su boda \u2014el doctor Quinteros reflexionaba sobre las cosas necias que hace decir el alcohol: \u00bfacaso no hab\u00eda visto \u00e9l a Richard divirti\u00e9ndose en las fiestas como el que m\u00e1s? \u00bfNo era un eximio bailar\u00edn? \u00bfCu\u00e1ntas veces hab\u00eda capitaneado su sobrino a la pandilla de chicas y chicos que ven\u00edan a improvisar un baile en los cuartos de Charito? Pero no le record\u00f3 nada de eso. Vio c\u00f3mo Richard apuraba su whisky y ped\u00eda a un mozo que le sirviera otro.\n\n\u2014De todos modos, anda prepar\u00e1ndote \u2014le dijo\u2014. Porque cuando te cases, tus padres te har\u00e1n una fiesta m\u00e1s grande que \u00e9sta.\n\nRichard se llev\u00f3 el flamante vaso de whisky a los labios y, despacio, entrecerrando los ojos, bebi\u00f3 un trago. Luego, sin alzar la cabeza, con voz sorda y que lleg\u00f3 al doctor como algo muy lento y casi inaudible, musit\u00f3:\n\n\u2014Yo no me casar\u00e9 nunca, t\u00edo, te lo juro por Dios.\n\nAntes de que pudiera responderle, una estilizada muchacha de cabellos claros, silueta azul y gesto decidido se plant\u00f3 ante ellos, cogi\u00f3 a Richard de la mano y, sin darle tiempo a reaccionar, lo oblig\u00f3 a levantarse:\n\n\u2014\u00bfNo te da verg\u00fcenza estar sentado con los viejos? Ven a bailar, sonso.\n\nEl doctor Quinteros los vio desaparecer en el zagu\u00e1n de la casa y se sinti\u00f3 bruscamente inapetente. Segu\u00eda repicando en el pabell\u00f3n de sus o\u00eddos, como un eco perverso, esa palabrita, \u00abviejos\u00bb, que con tanta naturalidad y voz tan deliciosa hab\u00eda dicho la hijita menor del arquitecto Arambur\u00fa. Despu\u00e9s de tomar el caf\u00e9, se levant\u00f3 y fue a echar un vistazo al sal\u00f3n.\n\nLa fiesta estaba en su esplendor y el baile se hab\u00eda ido propagando, desde esa matriz que era la chimenea donde hab\u00edan instalado a la orquesta, a los cuartos vecinos, en los que tambi\u00e9n hab\u00eda parejas que bailaban, cantando a voz en cuello los chachach\u00e1s y los merengues, las cumbias y los valses. La onda de alegr\u00eda, alimentada por la m\u00fasica, el sol y los alcoholes hab\u00eda ido subiendo de los j\u00f3venes a los adultos y de los adultos a los viejos, y el doctor Quinteros vio, con sorpresa, que incluso don Marcelino Huapaya, un octogenario emparentado a la familia, meneaba esforzadamente su crujiente humanidad, siguiendo los compases de Nube gris, con su cu\u00f1ada Margarita en brazos. La atm\u00f3sfera de humo, ruido, movimiento, luz y felicidad produjo un ligero v\u00e9rtigo al doctor Quinteros; se apoy\u00f3 en la baranda y cerr\u00f3 un instante los ojos. Luego, risue\u00f1o, feliz \u00e9l tambi\u00e9n, estuvo observando a Elianita, que, todav\u00eda vestida de novia pero ya sin velo, presid\u00eda la fiesta. No descansaba un segundo; al t\u00e9rmino de cada pieza, veinte varones la rodeaban, solicitando su favor, y ella, con las mejillas arreboladas y los ojos lucientes, eleg\u00eda a uno diferente cada vez y retornaba al torbellino. Su hermano Roberto se materializ\u00f3 a su lado. En vez del chaqu\u00e9, ten\u00eda un liviano terno marr\u00f3n y estaba sudoroso pues acababa de bailar.\n\n\u2014Me parece mentira que se est\u00e9 casando, Alberto \u2014dijo, se\u00f1alando a Elianita.\n\n\u2014Est\u00e1 lind\u00edsima \u2014le sonri\u00f3 el doctor Quinteros\u2014. Has echado la casa por la ventana, Roberto.\n\n\u2014Para mi hija lo mejor del mundo \u2014exclam\u00f3 su hermano, con un retint\u00edn de tristeza en la voz.\n\n\u2014\u00bfD\u00f3nde van a pasar la luna de miel? \u2014pregunt\u00f3 el doctor.\n\n\u2014En Brasil y Europa. Es el regalo de los pap\u00e1s del Pelirrojo \u2014se\u00f1al\u00f3, divertido, hacia el bar\u2014. Deb\u00edan partir ma\u00f1ana temprano, pero, a este paso, mi yerno no estar\u00e1 en condiciones.\n\nUn grupo de muchachos ten\u00edan cercado al Pelirrojo Ant\u00fanez y se turnaban para brindar con \u00e9l. El novio, m\u00e1s colorado que nunca, riendo algo alarmado, trataba de enga\u00f1arlos mojando los labios en su copa, pero sus amigos protestaban y le exig\u00edan vaciarla. El doctor Quinteros busc\u00f3 a Richard con la mirada, pero no lo vio en el bar, ni bailando, ni en el sector del jard\u00edn que descubr\u00edan las ventanas.\n\nOcurri\u00f3 en ese momento. Terminaba el vals \u00cddolo, las parejas se dispon\u00edan a aplaudir, los m\u00fasicos apartaban los dedos de las guitarras, el Pelirrojo hac\u00eda frente al vig\u00e9simo brindis, cuando la novia se llev\u00f3 la mano derecha a los ojos como para espantar a un mosquito, trastabille\u00f3 y, antes de que su pareja alcanzara a sostenerla, se desplom\u00f3. Su padre y el doctor Quinteros permanecieron inm\u00f3viles, creyendo tal vez que hab\u00eda resbalado, que se levantar\u00eda al instante muerta de risa, pero el revuelo que se arm\u00f3 en el sal\u00f3n \u2014las exclamaciones, los empujones, los gritos de la mam\u00e1: \u00ab\u00a1Hijita, Eliana, Elianita!\u00bb\u2014 los hizo correr tambi\u00e9n a ayudarla. Ya el Pelirrojo Ant\u00fanez hab\u00eda dado un salto, la levantaba en brazos, y, escoltado por un grupo, la sub\u00eda por la escalera, tras la se\u00f1ora Margarita, que iba diciendo \u00abPor aqu\u00ed, a su cuarto, despacio, con cuidadito\u00bb, y ped\u00eda \u00abUn m\u00e9dico, llamen a un m\u00e9dico\u00bb. Algunos familiares \u2014el t\u00edo Fernando, la prima Chabuca, don Marcelino\u2014 tranquilizaban a los amigos, ordenaban a la orquesta reanudar la m\u00fasica. El doctor Quinteros vio que su hermano Roberto le hac\u00eda se\u00f1as desde lo alto de la escalera. Pero qu\u00e9 est\u00fapido, \u00bfacaso no era m\u00e9dico?, \u00bfqu\u00e9 esperaba? Trep\u00f3 los pelda\u00f1os a trancos, entre gente que se abr\u00eda a su paso.\n\nHab\u00edan llevado a Elianita a su dormitorio, una habitaci\u00f3n decorada de rosa que daba sobre el jard\u00edn. Alrededor de la cama, donde la muchacha, todav\u00eda muy p\u00e1lida, comenzaba a recobrar el conocimiento y a pesta\u00f1ear, permanec\u00edan Roberto, el Pelirrojo, el ama Venancia, en tanto que su madre, sentada a su lado, le frotaba la frente con un pa\u00f1uelo empapado en alcohol. El Pelirrojo le hab\u00eda cogido una mano y la miraba con embelesamiento y angustia.\n\n\u2014Por lo pronto, todos ustedes se me van de aqu\u00ed y me dejan solo con la novia \u2014orden\u00f3 el doctor Quinteros, tomando posesi\u00f3n de su papel. Y, mientras los llevaba hacia la puerta\u2014: No se preocupen, no puede ser nada. Salgan, d\u00e9jenme examinarla.\n\nLa \u00fanica que se resisti\u00f3 fue la vieja Venancia; Margarita tuvo que sacarla casi a rastras. El doctor Quinteros volvi\u00f3 a la cama y se sent\u00f3 junto a Elianita, quien lo mir\u00f3, entre sus largas pesta\u00f1as negras, aturdida y miedosa. \u00c9l la bes\u00f3 en la frente y, mientras le tomaba la temperatura, le sonre\u00eda: no pasaba nada, no hab\u00eda de qu\u00e9 asustarse. Ten\u00eda el pulso algo agitado y respiraba ahog\u00e1ndose. El doctor advirti\u00f3 que llevaba el pecho demasiado oprimido y la ayud\u00f3 a desabotonarse:\n\n\u2014Como de todas maneras tienes que cambiarte, as\u00ed ganas tiempo, sobrina.\n\nCuando not\u00f3 la faja tan ce\u00f1ida, comprendi\u00f3 inmediatamente de qu\u00e9 se trataba, pero no hizo el menor gesto ni pregunta que pudieran revelar a su sobrina que \u00e9l sab\u00eda. Mientras se quitaba el vestido, Elianita hab\u00eda ido enrojeciendo terriblemente y ahora estaba tan turbada que no alzaba la vista ni mov\u00eda los labios. El doctor Quinteros le dijo que no era necesario que se quitara la ropa interior, s\u00f3lo la faja, que le imped\u00eda respirar. Sonriendo, mientras con aire en apariencia distra\u00eddo iba asegur\u00e1ndole que era la cosa m\u00e1s natural del mundo que el d\u00eda de su boda, con la emoci\u00f3n del acontecimiento, las fatigas y trajines precedentes, y, sobre todo, si se era tan loca de bailar horas de horas sin descanso, una novia tuviera un desmayo, le palp\u00f3 los pechos y el vientre (que, al ser liberado del abrazo poderoso de la faja, hab\u00eda literalmente saltado) y dedujo, con la seguridad de un especialista por cuyas manos han pasado millares de embarazadas, que deb\u00eda estar ya en el cuarto mes. Le examin\u00f3 la pupila, le hizo algunas preguntas tontas para despistarla, y le aconsej\u00f3 que descansara unos minutos antes de volver a la sala. Pero, eso s\u00ed, que no siguiera bailando tanto.\n\n\u2014Ya ves, s\u00f3lo era un poco de cansancio, sobrina. De todas maneras, te voy a dar algo, para contrarrestar las impresiones del d\u00eda.\n\nLe acarici\u00f3 los cabellos y, para darle tiempo a serenarse antes de que entraran sus pap\u00e1s, le hizo algunas preguntas sobre el viaje de bodas. Ella le respond\u00eda con voz l\u00e1nguida. Hacer un viaje as\u00ed era una de las mejores cosas que pod\u00edan ocurrirle a una persona; \u00e9l, con tanto trabajo, jam\u00e1s podr\u00eda darse tiempo para un recorrido tan completo. Y ya iban para tres a\u00f1os que no hab\u00eda estado en Londres, su ciudad preferida. Mientras hablaba, ve\u00eda a Elianita esconder con disimulo la faja, ponerse una bata, disponer sobre una silla un vestido, una blusa con cuello y pu\u00f1os bordados, unos zapatos, y volver a tenderse en la cama y cubrirse con el edred\u00f3n. Se pregunt\u00f3 si no habr\u00eda sido mejor hablar francamente con su sobrina y darle algunos consejos para el viaje. Pero no, la pobre hubiera pasado un mal rato, se hubiera sentido muy inc\u00f3moda. Adem\u00e1s, sin duda habr\u00eda estado viendo a un m\u00e9dico a escondidas todo este tiempo y estar\u00eda perfectamente al tanto de lo que deb\u00eda hacer. De todas maneras, llevar una faja tan ajustada era un riesgo, hubiera podido pasar un susto de verdad, o, en el futuro, perjudicar a la criatura. Lo emocion\u00f3 que Elianita, esa sobrina en la que s\u00f3lo pod\u00eda pensar como en una ni\u00f1a casta, hubiera concebido. Se lleg\u00f3 a la puerta, la abri\u00f3, y tranquiliz\u00f3 a la familia en voz alta para que lo oyera la novia:\n\n\u2014Est\u00e1 m\u00e1s sana que ustedes y yo, pero muerta de fatiga. M\u00e1ndenle comprar este calmante y d\u00e9jenla descansar un rato.\n\nVenancia se hab\u00eda precipitado al dormitorio y, por sobre el hombro, el doctor Quinteros vio a la vieja criada haciendo mimos a Elianita. Entraron tambi\u00e9n sus padres y el Pelirrojo Ant\u00fanez se dispon\u00eda a hacerlo, pero el doctor, discretamente, lo tom\u00f3 del brazo y lo llev\u00f3 con \u00e9l hacia el cuarto de ba\u00f1o. Cerr\u00f3 la puerta:\n\n\u2014Ha sido una imprudencia que en su estado estuviera bailando as\u00ed toda la tarde, Pelirrojo \u2014le dijo, con el tono m\u00e1s natural del mundo, mientras se jabonaba las manos\u2014. Ha podido tener un aborto. Acons\u00e9jale que no use faja, y menos tan apretada. \u00bfQu\u00e9 tiempo tiene? \u00bfTres, cuatro meses?\n\nFue en ese momento que, veloz y mort\u00edfera como una picadura de cobra, la sospecha cruz\u00f3 la mente del doctor Quinteros. Con terror, sintiendo que el silencio del cuarto de ba\u00f1o se hab\u00eda electrizado, mir\u00f3 por el espejo. El Pelirrojo ten\u00eda los ojos incr\u00e9dulamente abiertos, la boca torcida en una mueca que daba a su cara una expresi\u00f3n absurda, y estaba l\u00edvido como muerto.\n\n\u2014\u00bfTres, cuatro meses? \u2014lo oy\u00f3 articular, ator\u00e1ndose\u2014. \u00bfUn aborto?\n\nSinti\u00f3 que se le hund\u00eda la tierra. Qu\u00e9 bruto, qu\u00e9 animal eres, pens\u00f3. Y, ahora s\u00ed, con atroz precisi\u00f3n, record\u00f3 que todo el noviazgo y la boda de Elianita era una historia de pocas semanas. Hab\u00eda apartado la vista de Ant\u00fanez, se secaba las manos demasiado despacio y su mente buscaba ardorosamente alguna mentira, una coartada que sacara a ese muchacho del infierno al que acababa de empujarlo. S\u00f3lo atin\u00f3 a decir algo que le pareci\u00f3 tambi\u00e9n est\u00fapido:\n\n\u2014Elianita no debe saber que me he dado cuenta. Le he hecho creer que no. Y, sobre todo, no te preocupes. Ella est\u00e1 muy bien.\n\nSali\u00f3 r\u00e1pidamente, mir\u00e1ndolo de soslayo al pasar. Lo vio en el mismo sitio, los ojos clavados en el vac\u00edo, ahora la boca tambi\u00e9n abierta y la cara cubierta de sudor. Sinti\u00f3 que echaba llave al cuarto de ba\u00f1o desde adentro. Va a ponerse a llorar, pens\u00f3, a darse de cabezazos y a jalarse los pelos, va a maldecirme y a odiarme m\u00e1s todav\u00eda que a ella y que a \u00bfqui\u00e9n? Bajaba las escaleras despacio, con una desoladora sensaci\u00f3n de culpa, lleno de dudas, mientras iba repitiendo como un aut\u00f3mata a la gente que Elianita no ten\u00eda nada, que bajar\u00eda ahora mismo. Sali\u00f3 al jard\u00edn y respirar una bocanada de aire le hizo bien. Se acerc\u00f3 al bar, bebi\u00f3 un vaso de whisky puro y decidi\u00f3 irse a su casa sin esperar el desenlace del drama que, por ingenuidad y con las mejores intenciones, hab\u00eda provocado. Ten\u00eda ganas de encerrarse en su escritorio y, arrellanado en su sill\u00f3n de cuero negro, sumergirse en Mozart.\n\nEn la puerta de calle, sentado en el pasto, en un estado calamitoso, encontr\u00f3 a Richard. Ten\u00eda las piernas cruzadas como un Buda, la espalda apoyada en la verja, el terno arrugado y cubierto de polvo, de manchas, de hierbas. Pero fue su cara la que distrajo al doctor del recuerdo del Pelirrojo y de Elianita y lo hizo detenerse: en sus ojos inyectados el alcohol y el furor parec\u00edan haber aumentado en dosis id\u00e9nticas. Dos hilillos de baba colgaban de sus labios y su expresi\u00f3n era lastimosa y grotesca.\n\n\u2014No es posible, Richard \u2014murmur\u00f3, inclin\u00e1ndose y tratando de hacer incorporarse a su sobrino\u2014. Tus padres no pueden verte as\u00ed. Ven, vamos a la casa hasta que se te pase. Jam\u00e1s cre\u00ed que te ver\u00eda en este estado, sobrino.\n\nRichard lo miraba sin verlo, con la cabeza colgante, y aunque, obediente, trataba de levantarse, las piernas le flaqueaban. El doctor tuvo que tomarlo de los dos brazos y casi alzarlo en peso. Lo hizo andar, sujet\u00e1ndolo de los hombros; Richard se balanceaba como un mu\u00f1eco de trapo y parec\u00eda irse de bruces a cada momento. \u00abVamos a ver si conseguimos un taxi\u00bb, murmur\u00f3 el doctor, par\u00e1ndose al borde de la avenida Santa Cruz y sosteniendo a Richard de un brazo: \u00abPorque andando no llegas ni a la esquina, sobrino\u00bb. Pasaban algunos taxis, pero ocupados. El doctor ten\u00eda la mano levantada. La espera, sumada al recuerdo de Elianita y Ant\u00fanez, y la inquietud por el estado de su sobrino, comenzaban a ponerlo nervioso, a \u00e9l, que nunca hab\u00eda perdido la calma. En ese momento distingui\u00f3, en el murmullo incoherente y bajito que escapaba de los labios de Richard, la palabra \u00abrev\u00f3lver\u00bb. No pudo menos que sonre\u00edr, y, poniendo al mal tiempo buena cara, dijo, como para s\u00ed mismo, sin esperar que Richard lo escuchara o le respondiera:\n\n\u2014\u00bfY para qu\u00e9 quieres un rev\u00f3lver, sobrino?\n\nLa respuesta de Richard, que miraba el vac\u00edo con errabundos ojos homicidas, fue lenta, ronca, clar\u00edsima:\n\n\u2014Para matar al Pelirrojo \u2014hab\u00eda pronunciado cada s\u00edlaba con un odio glacial. Hizo una pausa, y, con la voz bruscamente rajada, a\u00f1adi\u00f3\u2014: O para matarme a m\u00ed.\n\nSe le volvi\u00f3 a trabar la lengua y Alberto de Quinteros ya no entendi\u00f3 lo que dec\u00eda. En eso par\u00f3 un taxi. El doctor empuj\u00f3 a Richard al interior, dio al chofer la direcci\u00f3n, subi\u00f3. En el instante en que el auto arrancaba, Richard rompi\u00f3 a llorar. Se volvi\u00f3 a mirarlo y el muchacho se dej\u00f3 ir contra \u00e9l, apoy\u00f3 la cabeza en su pecho y sigui\u00f3 sollozando, con el cuerpo movido por un temblor nervioso. El doctor le pas\u00f3 una mano por los hombros, le revolvi\u00f3 los cabellos como hab\u00eda hecho un rato antes con su hermana, y tranquiliz\u00f3 con un gesto que quer\u00eda decir \u00abel chico ha tomado demasiado\u00bb al chofer que lo miraba por el espejo retrovisor. Dej\u00f3 a Richard encogido contra \u00e9l, llorando y ensuci\u00e1ndole con sus l\u00e1grimas, babas y mocos su terno azul y su corbata plateada. No pesta\u00f1e\u00f3 siquiera, ni se agit\u00f3 su coraz\u00f3n, cuando en el incomprensible soliloquio de su sobrino, alcanz\u00f3 a entender, dos o tres veces repetida, esa frase que, sin dejar de ser atroz, sonaba tambi\u00e9n hermosa y hasta pura: \u00abPorque yo la quiero como hombre y no me importa nada de nada, t\u00edo\u00bb. En el jard\u00edn de la casa, Richard vomit\u00f3, con recias arcadas que asustaron al foxterrier y provocaron miradas censoras en el mayordomo y las sirvientas. El doctor Quinteros llev\u00f3 a Richard del brazo hasta el cuarto de hu\u00e9spedes, lo hizo enjuagarse la boca, lo desnud\u00f3, lo meti\u00f3 en la cama, le hizo tragar un fuerte somn\u00edfero, y permaneci\u00f3 a su lado, calm\u00e1ndolo con palabras y gestos afectuosos \u2014que sab\u00eda que el muchacho no pod\u00eda o\u00edr ni ver\u2014 hasta que lo sinti\u00f3 dormir el sue\u00f1o profundo de la juventud.\n\nEntonces, llam\u00f3 a la cl\u00ednica y dijo al m\u00e9dico de guardia que no ir\u00eda hasta el d\u00eda siguiente a menos de alguna cat\u00e1strofe, instruy\u00f3 al mayordomo que no estar\u00eda para nadie que llamara o viniera, se sirvi\u00f3 un whisky doble y fue a encerrarse en el cuarto de m\u00fasica. Puso en el tocadiscos un alto de piezas de Albinoni, Vivaldi y Scarlatti, pues hab\u00eda decidido que unas horas venecianas, barrocas y superficiales, ser\u00edan un buen remedio para las graves sombras de su esp\u00edritu, y, hundido en la c\u00e1lida blandura de su sill\u00f3n de cuero, la pipa escocesa de espuma de mar humeando entre los labios, cerr\u00f3 los ojos y esper\u00f3 que la m\u00fasica operara su inevitable milagro. Pens\u00f3 que \u00e9sta era una ocasi\u00f3n privilegiada para poner a prueba esa norma moral que hab\u00eda hecho suya desde joven y seg\u00fan la cual era preferible comprender que juzgar a los hombres. No se sent\u00eda horrorizado ni indignado ni demasiado sorprendido. M\u00e1s bien advert\u00eda una rec\u00f3ndita emoci\u00f3n, una benevolencia invencible, mezclada de ternura y de piedad, cuando se dec\u00eda que ahora s\u00ed estaba clar\u00edsimo por qu\u00e9 una muchacha tan linda hab\u00eda decidido casarse de pronto con un bobo y por qu\u00e9 al rey de la tabla hawaiana, al buen mozo del barrio, no se le conoci\u00f3 nunca enamorada y por qu\u00e9 siempre hab\u00eda cumplido sin protestar, con diligencia tan encomiable, las funciones de chaper\u00f3n de su hermana menor. Mientras saboreaba el perfume del tabaco y degustaba el placentero fuego de la bebida, se dec\u00eda que no hab\u00eda que preocuparse demasiado por Richard. \u00c9l encontrar\u00eda la manera de convencer a Roberto que lo enviara a estudiar al extranjero, a Londres por ejemplo, una ciudad donde encontrar\u00eda novedades e incitaciones suficientes para olvidar el pasado. Lo inquietaba, en cambio, lo com\u00eda de curiosidad lo que pasar\u00eda con los otros dos personajes de la historia. Mientras la m\u00fasica lo iba embriagando, cada vez m\u00e1s d\u00e9biles y espaciadas, un remolino de preguntas sin respuesta giraba en su mente: \u00bfabandonar\u00eda el Pelirrojo esa misma tarde a su temeraria esposa? \u00bfLo habr\u00eda hecho ya? \u00bfO callar\u00eda y, dando una indiscernible prueba de nobleza o estupidez, seguir\u00eda con esa ni\u00f1a fraudulenta que tanto hab\u00eda perseguido? \u00bfEstallar\u00eda el esc\u00e1ndalo o un pudoroso velo de disimulaci\u00f3n y orgullo pisoteado ocultar\u00eda para siempre esa tragedia de San Isidro?\n\n### III\n\nVOLV\u00cd A VER a Pedro Camacho pocos d\u00edas despu\u00e9s del incidente. Eran las siete y media de la ma\u00f1ana, y, luego de preparar el primer bolet\u00edn, estaba yendo a tomar un caf\u00e9 con leche al Bransa, cuando, al pasar por la ventanilla de la porter\u00eda de Radio Central, divis\u00e9 mi Remington. La sent\u00ed funcionando, o\u00ed el sonido de sus gordas teclas contra el rodillo, pero no vi a nadie detr\u00e1s de ella. Met\u00ed la cabeza por la ventana y el mecan\u00f3grafo era Pedro Camacho. Le hab\u00edan instalado una oficina en el cub\u00edculo del portero. En el cuarto, de techo bajo y paredes devastadas por la humedad, la vejez y los graffiti, hab\u00eda ahora un escritorio en ruinas pero tan aparatoso como la m\u00e1quina que tronaba sobre sus tablas. Las dimensiones del mueble y de la Remington se tragaban literalmente la figurilla de Pedro Camacho. Hab\u00eda a\u00f1adido al asiento un par de almohadas, pero aun as\u00ed su cara s\u00f3lo llegaba a la altura del teclado, de modo que escrib\u00eda con las manos al nivel de los ojos y daba la impresi\u00f3n de estar boxeando. Su concentraci\u00f3n era absoluta, no advert\u00eda mi presencia pese a estar a su lado. Ten\u00eda los desorbitados ojos fijos en el papel, tecleaba con dos dedos, se mord\u00eda la lengua. Llevaba el terno negro del primer d\u00eda, no se hab\u00eda quitado el saco ni la corbatita de lazo y, al verlo as\u00ed, absorto y atareado, con su cabellera y su atuendo de poeta decimon\u00f3nico, r\u00edgido y grave, sentado frente a ese escritorio y esa m\u00e1quina que le quedaban tan grandes y en esa cueva que les quedaba a los tres tan chica, tuve una sensaci\u00f3n de algo entre lastimoso y c\u00f3mico.\n\n\u2014Qu\u00e9 madrugador, se\u00f1or Camacho \u2014lo salud\u00e9, metiendo la mitad del cuerpo en la habitaci\u00f3n.\n\nSin apartar los ojos del papel, se limit\u00f3 a indicarme, con un movimiento autoritario de la cabeza, que me callara o esperase, o ambas cosas. Opt\u00e9 por lo \u00faltimo, y, mientras \u00e9l terminaba su frase, observ\u00e9 que ten\u00eda la mesa cubierta de papeles mecanografiados, y que en el suelo hab\u00eda algunas hojas arrugadas, enviadas all\u00ed a falta de basurero. Poco despu\u00e9s apart\u00f3 las manos del teclado, me mir\u00f3, se puso de pie, me estir\u00f3 su diestra ceremoniosa y respondi\u00f3 a mi saludo con una sentencia:\n\n\u2014Para el arte no hay horario. Muy buenos d\u00edas, mi amigo.\n\nNo averig\u00fc\u00e9 si sent\u00eda claustrofobia en ese cubil porque, estaba seguro, me hubiera contestado que al arte le conven\u00eda la incomodidad. M\u00e1s bien, lo invit\u00e9 a tomar un caf\u00e9. Consult\u00f3 un artefacto prehist\u00f3rico que bailoteaba en su mu\u00f1eca delgadita y murmur\u00f3: \u00abDespu\u00e9s de hora y media de producci\u00f3n, me merezco un refrigerio\u00bb. Camino del Bransa, le pregunt\u00e9 si siempre empezaba a trabajar tan temprano y me repuso que, en su caso, a diferencia de otros creadores, la inspiraci\u00f3n era proporcional a la luz del d\u00eda.\n\n\u2014Amanece con el sol y con \u00e9l va calentando \u2014me explic\u00f3, musicalmente, mientras, a nuestro alrededor, un muchacho so\u00f1oliento barr\u00eda el aserr\u00edn lleno de puchos y suciedades del Bransa\u2014. Comienzo a escribir con la primera luz. Al mediod\u00eda mi cerebro es una antorcha. Luego, va perdiendo fuego y a eso de la tardecita paro porque s\u00f3lo quedan brasas. Pero no importa, ya que en las tardes y en las noches es cuando m\u00e1s rinde el actor. Tengo mi sistema bien distribuido.\n\nHablaba demasiado en serio y me di cuenta que apenas parec\u00eda notar que yo segu\u00eda all\u00ed; era de esos hombres que no admiten interlocutores sino oyentes. Como la primera vez, me sorprendi\u00f3 la absoluta falta de humor que hab\u00eda en \u00e9l, pese a las sonrisas de mu\u00f1eco \u2014labios que se levantan, frente que se arruga, dientes que asoman\u2014 con que aderezaba su mon\u00f3logo. Todo lo dec\u00eda con una solemnidad extrema, lo que, sumado a su perfecta dicci\u00f3n, a su f\u00edsico, a su ropaje extravagante y a sus ademanes teatrales, le daba un aire terriblemente ins\u00f3lito. Era evidente que cre\u00eda al pie de la letra todo lo que dec\u00eda: se lo notaba, a la vez, el hombre m\u00e1s afectado y el m\u00e1s sincero del mundo. Trat\u00e9 de descenderlo de las alturas art\u00edsticas en las que peroraba al terreno mediocre de los asuntos pr\u00e1cticos y le pregunt\u00e9 si ya se hab\u00eda instalado, si ten\u00eda amigos aqu\u00ed, c\u00f3mo se sent\u00eda en Lima. Esos temas terrenales le importaban un comino. Con un dejo impaciente me contest\u00f3 que hab\u00eda conseguido un atelier no lejos de Radio Central, en el jir\u00f3n Quilca, y que se sent\u00eda a sus anchas en cualquier parte, porque \u00bfacaso la patria del artista no era el mundo? En vez de caf\u00e9 pidi\u00f3 una infusi\u00f3n de yerbaluisa y menta, que, me instruy\u00f3, adem\u00e1s de grata al paladar, \u00abentonaba la mente\u00bb. La apur\u00f3 a sorbos cortos y sim\u00e9tricos, como si contara el tiempo exacto para llevarse la taza a la boca, y, apenas termin\u00f3, se puso de pie, insisti\u00f3 en repartir la cuenta, y me pidi\u00f3 que lo acompa\u00f1ara a comprar un plano con los barrios y calles de Lima. Encontramos lo que quer\u00eda en un puesto ambulante del jir\u00f3n de la Uni\u00f3n. Estudi\u00f3 el plano despleg\u00e1ndolo contra el cielo y aprob\u00f3 con satisfacci\u00f3n los colorines que diferenciaban a los distritos. Exigi\u00f3 un recibo por los veinte soles que costaba.\n\n\u2014Es un instrumento de trabajo y deben abonarlo los mercaderes \u2014decret\u00f3, mientras regres\u00e1bamos a nuestros trabajos. Tambi\u00e9n su andar era original: r\u00e1pido y nervioso, como si temiera perder el tren. En la puerta de Radio Central, al despedirnos, me se\u00f1al\u00f3 su apretada oficina como quien exhibe un palacio:\n\n\u2014Est\u00e1 pr\u00e1cticamente en la calle \u2014dijo, contento consigo mismo y con las cosas\u2014. Es como si trabajara en la vereda.\n\n\u2014\u00bfNo lo distrae tanto ruido de gente y de autos? \u2014me atrev\u00ed a insinuar.\n\n\u2014Al contrario \u2014me tranquiliz\u00f3, feliz de gratificarme con una \u00faltima f\u00f3rmula\u2014: Yo escribo sobre la vida y mis obras exigen el impacto de la realidad.\n\nYa me iba, cuando volvi\u00f3 a llamarme con el dedo \u00edndice. Mostr\u00e1ndome el plano de Lima, me pidi\u00f3 de manera misteriosa que, m\u00e1s tarde o ma\u00f1ana, le proporcionara algunos datos. Le dije que encantado.\n\nEn mi altillo de Panamericana, encontr\u00e9 a Pascual con el bolet\u00edn de las nueve listo. Comenzaba con una de esas noticias que le gustaban tanto. La hab\u00eda copiado de La Cr\u00f3nica, enriqueci\u00e9ndola con adjetivos de su propio acervo: \u00abEn el proceloso mar de las Antillas, se hundi\u00f3 anoche el carguero paname\u00f1o Shark, pereciendo sus ocho tripulantes, ahogados y masticados por los tiburones que infestan el susodicho mar\u00bb. Cambi\u00e9 \u00abmasticados\u00bb por \u00abdevorados\u00bb y suprim\u00ed \u00abproceloso\u00bb y \u00absusodicho\u00bb antes de darle el visto bueno. No se enoj\u00f3, porque Pascual no se enojaba nunca, pero dej\u00f3 sentada su protesta:\n\n\u2014Este don Mario, siempre jodi\u00e9ndome el estilo.\n\nToda esa semana hab\u00eda estado tratando de escribir un cuento, basado en una historia que conoc\u00eda por mi t\u00edo Pedro, quien era m\u00e9dico en una hacienda de Ancash. Un campesino asust\u00f3 a otro, una noche, disfraz\u00e1ndose de pishtaco (diablo) y sali\u00e9ndole al encuentro en medio del ca\u00f1averal. La v\u00edctima de la broma se hab\u00eda asustado tanto que descarg\u00f3 su machete sobre el pishtaco y lo mand\u00f3 al otro mundo con el cr\u00e1neo partido en dos. Luego, huy\u00f3 al monte. Alg\u00fan tiempo despu\u00e9s, un grupo de campesinos, al salir de una fiesta, hab\u00edan sorprendido a un pishtaco merodeando por el poblado y lo mataron a palos. El muerto result\u00f3 ser el asesino del primer pishtaco, que usaba disfraz de diablo para visitar de noche a su familia. Los asesinos, a su vez, se hab\u00edan echado al monte, y, disfrazados de pishtacos, ven\u00edan en las noches a la comunidad, donde dos de ellos hab\u00edan sido ya exterminados a machetazos por aterrorizados campesinos, quienes, a su vez, etc\u00e9tera. Lo que yo quer\u00eda contar no era tanto lo ocurrido en la hacienda de mi t\u00edo Pedro, como el final que se me ocurri\u00f3: que en un momento dado, entre tanto pishtaco de mentiras, se deslizaba el diablo vivito y coleando. Iba a titular mi cuento El salto cualitativo y quer\u00eda que fuese fr\u00edo, intelectual, condensado e ir\u00f3nico como un cuento de Borges, a quien acababa de descubrir por esos d\u00edas. Dedicaba al relato todos los resquicios de tiempo que me dejaban los boletines de Panamericana, la universidad y los caf\u00e9s del Bransa, y tambi\u00e9n escrib\u00eda en casa de mis abuelos, a mediod\u00eda y en las noches. Esa semana no almorc\u00e9 donde ninguno de mis t\u00edos, ni hice las visitas acostumbradas a las primas, ni fui al cine. Escrib\u00eda y romp\u00eda, o, mejor dicho, apenas hab\u00eda escrito una frase me parec\u00eda horrible y recomenzaba. Ten\u00eda la certeza de que una falta de caligraf\u00eda o de ortograf\u00eda nunca era casual, sino una llamada de atenci\u00f3n, una advertencia (del subconsciente, Dios o alguna otra persona) de que la frase no serv\u00eda y era preciso rehacerla. Pascual se quejaba: \u00abCaracho, si los Genaros descubren ese desperdicio de papel, lo pagaremos del sueldo\u00bb. Por fin, un jueves cre\u00ed tener el cuento acabado. Era un mon\u00f3logo de cinco p\u00e1ginas; al final se descubr\u00eda en el narrador al propio diablo. Le le\u00ed El salto cualitativo a Javier en mi altillo, despu\u00e9s de El Panamericano de las doce.\n\n\u2014Excelente, hermano \u2014sentenci\u00f3, aplaudiendo\u2014. \u00bfPero todav\u00eda es posible escribir sobre el diablo? \u00bfPor qu\u00e9 no un cuento realista? \u00bfPor qu\u00e9 no suprimir al diablo y dejar que todo pase entre los pishtacos de mentiras? O, si no, un cuento fant\u00e1stico, con todos los fantasmas que se te antojen. Pero sin diablos, sin diablos, porque eso huele a religi\u00f3n, a beater\u00eda, a cosas pasadas de moda.\n\nCuando se fue, hice a\u00f1icos El salto cualitativo, lo ech\u00e9 a la papelera, decid\u00ed olvidarme de los pishtacos y me fui a almorzar donde el t\u00edo Lucho. All\u00ed me enter\u00e9 que hab\u00eda brotado algo que parec\u00eda un romance entre la boliviana y alguien que yo conoc\u00eda de o\u00eddas: el hacendado y senador arequipe\u00f1o Adolfo Salcedo, emparentado de alg\u00fan modo con la tribu familiar.\n\n\u2014Lo bueno del pretendiente es que tiene plata y posici\u00f3n y que sus intenciones con Julia son serias \u2014comentaba mi t\u00eda Olga\u2014. Le ha propuesto matrimonio.\n\n\u2014Lo malo es que don Adolfo tiene cincuenta a\u00f1os y todav\u00eda no ha desmentido esa acusaci\u00f3n terrible \u2014replicaba el t\u00edo Lucho\u2014. Si tu hermana se casa con \u00e9l tendr\u00e1 que ser casta o ad\u00faltera.\n\n\u2014Esa historia con Carlota es una de las t\u00edpicas calumnias de Arequipa \u2014discut\u00eda la t\u00eda Olga\u2014. Adolfo tiene todo el aire de ser un hombre completo.\n\nLa historia del senador y de do\u00f1a Carlota la conoc\u00eda yo muy bien porque hab\u00eda sido tema de otro cuento que los elogios de Javier mandaron al basurero. Su matrimonio conmovi\u00f3 al sur de la rep\u00fablica pues don Adolfo y do\u00f1a Carlota pose\u00edan ambos tierras en Puno y su alianza ten\u00eda resonancias latifund\u00edsticas. Hab\u00edan hecho las cosas en grande, cas\u00e1ndose en la bella Iglesia de Yanahuara, con invitados venidos de todo el Per\u00fa y un banquete pantagru\u00e9lico. A las dos semanas de luna de miel, la novia hab\u00eda plantado al marido en alg\u00fan lugar del mundo y regresado escandalosamente sola a Arequipa y anunciado, ante la estupefacci\u00f3n general, que pedir\u00eda la anulaci\u00f3n del matrimonio a Roma. La madre de Adolfo Salcedo encontr\u00f3 a do\u00f1a Carlota un domingo, a la salida de misa de once, y en el mismo atrio de la catedral la increp\u00f3 con furia:\n\n\u2014\u00bfPor qu\u00e9 abandonaste as\u00ed a mi pobre hijo, bandida?\n\nCon un gesto magn\u00edfico, la latifundista pune\u00f1a hab\u00eda respondido en alta voz, para que oyera todo el mundo:\n\n\u2014Porque, a su hijo, eso que tienen los caballeros s\u00f3lo le sirve para hacer pip\u00ed, se\u00f1ora.\n\nHab\u00eda conseguido anular el matrimonio religioso y Adolfo Salcedo era una fuente inagotable de chistes en las reuniones familiares. Desde que hab\u00eda conocido a la t\u00eda Julia, la asediaba con invitaciones al Grill Bol\u00edvar y al 91, le regalaba perfumes y la bombardeaba con canastas de rosas. Yo estaba feliz con la noticia del romance y esperaba que la t\u00eda Julia apareciera para lanzarle alg\u00fan dardo sobre su nuevo candidato. Pero me dej\u00f3 con los crespos hechos porque fue ella la que, al presentarse en el comedor, a la hora del caf\u00e9 \u2014llegaba con un alto de paquetes\u2014 anunci\u00f3 con una carcajada:\n\n\u2014Los chismes eran ciertos. El senador Salcedo no resopla.\n\n\u2014Julia, por Dios, no seas malcriada \u2014protest\u00f3 la t\u00eda Olga\u2014. Cualquiera creer\u00eda que...\n\n\u2014Me lo ha contado \u00e9l mismo, esta ma\u00f1ana \u2014aclar\u00f3 la t\u00eda Julia, feliz con la tragedia del latifundista.\n\nHab\u00eda sido muy normal hasta que cumpli\u00f3 veinticinco a\u00f1os. Entonces, durante unas infortunadas vacaciones en Estados Unidos, sobrevino el percance. En Chicago, San Francisco o Miami \u2014la t\u00eda Julia no se acordaba\u2014 el joven Adolfo hab\u00eda conquistado (cre\u00eda \u00e9l) a una se\u00f1ora en un cabaret, y ella se lo llev\u00f3 a un hotel, y estaba en plena acci\u00f3n cuando sinti\u00f3 en la espalda la punta de un cuchillo. Se volvi\u00f3 y era un tuerto que med\u00eda dos metros. No lo hirieron, no le pegaron, s\u00f3lo le robaron el reloj, una medalla, sus d\u00f3lares. As\u00ed comenz\u00f3. Nunca m\u00e1s. Desde entonces, vez que estaba con una dama e iba a entrar en acci\u00f3n sent\u00eda el fr\u00edo del metal en la columna, ve\u00eda la cara averiada del tuerto, se pon\u00eda a transpirar y se le bajaban los \u00e1nimos. Hab\u00eda consultado montones de m\u00e9dicos, de psic\u00f3logos, y hasta a un curandero de Arequipa, que lo hac\u00eda enterrarse vivo, las noches de luna, al pie de los volcanes.\n\n\u2014No seas mala, no te burles, pobrecito \u2014temblaba de risa la t\u00eda Olga.\n\n\u2014Si estuviera segura que se va a quedar siempre as\u00ed, me casar\u00eda con \u00e9l, por su plata \u2014dec\u00eda inescrupulosamente la t\u00eda Julia\u2014. \u00bfPero, y si yo lo curo? \u00bfTe imaginas a ese vejestorio tratando de recuperar el tiempo perdido conmigo?\n\nPens\u00e9 en la felicidad que habr\u00eda causado a Pascual la aventura del senador arequipe\u00f1o, el entusiasmo con que le hubiera consagrado un bolet\u00edn entero. El t\u00edo Lucho le advert\u00eda a la t\u00eda Julia que si se mostraba tan exigente no encontrar\u00eda un marido peruano. Ella se quejaba de que, aqu\u00ed tambi\u00e9n, como en Bolivia, los buenos mozos fueran pobres y los ricos feos, y de que cuando aparec\u00eda un buen mozo rico siempre estuviera casado. De pronto, se encar\u00f3 conmigo y me pregunt\u00f3 si no hab\u00eda asomado toda esa semana por miedo a que me arrastrara otra vez al cine. Le dije que no, invent\u00e9 ex\u00e1menes, le propuse que fu\u00e9ramos esa noche.\n\n\u2014Regio, a la del Leuro \u2014decidi\u00f3, dictatorialmente\u2014. Es una pel\u00edcula en la que se llora a mares.\n\nEn el colectivo, de regreso a Radio Panamericana, le estuve dando vueltas a la idea de intentar otra vez un cuento con la historia de Adolfo Salcedo; algo ligero y risue\u00f1o, a la manera de Somerset Maugham, o de un erotismo malicioso, como en Maupassant. En la radio, la secretaria de Genaro hijo, Nelly, estaba ri\u00e9ndose sola en su escritorio. \u00bfCu\u00e1l era el chiste?\n\n\u2014Ha habido un l\u00edo en Radio Central entre Pedro Camacho y Genaro pap\u00e1 \u2014me cont\u00f3\u2014. El boliviano no quiere ning\u00fan actor argentino en los radioteatros o dice que se va. Consigui\u00f3 que Luciano Pando y Josefina S\u00e1nchez lo apoyen y se ha salido con su gusto. Van a cancelarles los contratos, \u00bfqu\u00e9 bueno, no?\n\nHab\u00eda una feroz rivalidad entre los locutores, animadores y actores nativos y los argentinos \u2014llegaban al Per\u00fa por oleadas, muchos de ellos expulsados por razones pol\u00edticas\u2014 y me imagin\u00e9 que el escriba boliviano hab\u00eda hecho esa operaci\u00f3n para ganarse la simpat\u00eda de sus compa\u00f1eros de trabajo abor\u00edgenes. Pero no, pronto descubr\u00ed que era incapaz de esa clase de c\u00e1lculos. Su odio a los argentinos en general, y a los actores y actrices argentinos en particular, parec\u00eda desinteresado. Fui a verlo despu\u00e9s del bolet\u00edn de las siete, para decirle que ten\u00eda un rato libre y pod\u00eda ayudarlo con los datos que necesitaba. Me hizo pasar a su cubil y, con un gesto munificente, me ofreci\u00f3 el \u00fanico asiento posible, fuera de su silla: una esquina de la mesa que le serv\u00eda de escritorio. Segu\u00eda con su saco y su corbatita de lazo, rodeado de papeles mecanografiados, que apil\u00f3 cuidadosamente junto a la Remington. El plano de Lima, clavado con tachuelas, cubr\u00eda parte de la pared. Ten\u00eda m\u00e1s colorines, unas extra\u00f1as figuras con l\u00e1piz rojo y unas iniciales distintas en cada barrio. Le pregunt\u00e9 qu\u00e9 eran esas marcas y letras.\n\nAsinti\u00f3, con una de esas sonrisitas mec\u00e1nicas, en las que hab\u00eda siempre una \u00edntima satisfacci\u00f3n y una especie de benevolencia. Acomod\u00e1ndose en la silla, peror\u00f3:\n\n\u2014Yo trabajo sobre la vida, mis obras se aferran a la realidad como la cepa a la vid. Para eso lo necesito. Quiero saber si ese mundo es o no es as\u00ed.\n\nEstaba se\u00f1al\u00e1ndome el plano y yo acerqu\u00e9 la cabeza para tratar de descifrar lo que quer\u00eda decirme. Las iniciales eran herm\u00e9ticas, no alud\u00edan a ninguna instituci\u00f3n ni persona reconocible. Lo \u00fanico claro era que hab\u00eda aislado en c\u00edrculos rojos los barrios dis\u00edmiles de Miraflores y San Isidro, de La Victoria y del Callao. Le dije que no entend\u00eda nada, que me explicara.\n\n\u2014Es muy f\u00e1cil \u2014me repuso, con impaciencia y voz de cura\u2014. Lo m\u00e1s importante es la verdad, que siempre es arte, y, en cambio, la mentira no, o s\u00f3lo rara vez. Debo saber si Lima es como lo he marcado en el plano. Por ejemplo, \u00bfcorresponden a San Isidro las dos Aes? \u00bfEs un barrio de Alto Abolengo, de Aristocracia Afortunada?\n\nHizo \u00e9nfasis en las Aes iniciales, con una entonaci\u00f3n que quer\u00eda decir \u00abS\u00f3lo los ciegos no ven la luz del sol\u00bb. Hab\u00eda clasificado los barrios de Lima seg\u00fan su importancia social. Pero lo curioso era el tipo de calificativos, la naturaleza de la nomenclatura. En algunos casos hab\u00eda acertado, en otros la arbitrariedad era absoluta. Por ejemplo, admit\u00ed que las iniciales MPA (Mesocracia Profesionales Amas de casa) conven\u00eda a Jes\u00fas Mar\u00eda, pero le advert\u00ed que resultaba bastante injusto estampar en La Victoria y El Porvenir la atroz divisa VMMH (Vagos Maricones Maleantes Hetairas) y sumamente discutible reducir el Callao a MPZ (Marineros Pescadores Zambos) o el Cercado y El Agustino a FOLI (F\u00e1mulas Operarios Labradores Indios).\n\n\u2014No se trata de una clasificaci\u00f3n cient\u00edfica sino art\u00edstica \u2014me inform\u00f3, haciendo pases m\u00e1gicos con sus manitas pigmeas\u2014. No me interesa toda la gente que compone cada barrio, sino la m\u00e1s llamativa, la que da a cada sitio su perfume y su color. Si un personaje es ginec\u00f3logo debe vivir donde le corresponde y lo mismo si es sargento de la polic\u00eda.\n\nMe someti\u00f3 a un interrogatorio prolijo y divertido (para m\u00ed, pues \u00e9l manten\u00eda su seriedad funeral) sobre la topograf\u00eda humana de la ciudad, y advert\u00ed que las cosas que le interesaban m\u00e1s se refer\u00edan a los extremos: millonarios y mendigos, blancos y negros, santos y criminales. Seg\u00fan mis respuestas, a\u00f1ad\u00eda, cambiaba o suprim\u00eda iniciales en el plano con un gesto veloz y sin vacilar un segundo, lo que me hizo pensar que hab\u00eda inventado y usaba ese sistema de catalogaci\u00f3n hac\u00eda tiempo. \u00bfPor qu\u00e9 hab\u00eda marcado s\u00f3lo Miraflores, San Isidro, La Victoria y el Callao?\n\n\u2014Porque, indudablemente, ser\u00e1n los escenarios principales \u2014dijo, paseando sus ojos saltones con suficiencia napole\u00f3nica sobre los cuatro distritos\u2014. Soy hombre que odia las medias tintas, el agua turbia, el caf\u00e9 flojo. Me gustan el s\u00ed o el no, los hombres masculinos y las mujeres femeninas, la noche o el d\u00eda. En mis obras siempre hay arist\u00f3cratas o plebe, prostitutas o madonas. La mesocracia no me inspira y tampoco a mi p\u00fablico.\n\n\u2014Se parece usted a los escritores rom\u00e1nticos \u2014se me ocurri\u00f3 decirle, en mala hora.\n\n\u2014En todo caso, ellos se parecen a m\u00ed \u2014salt\u00f3 en su silla, con la voz resentida\u2014. Nunca he plagiado a nadie. Se me puede reprochar todo, menos esa infamia. En cambio, a m\u00ed me han robado de la manera m\u00e1s inicua.\n\nQuise explicarle que lo del parecido a los rom\u00e1nticos no hab\u00eda sido dicho con \u00e1nimo de ofenderlo, que era una broma, pero no me o\u00eda porque, de pronto, se hab\u00eda enfurecido extraordinariamente, y, gesticulando como si se hallara ante un auditorio expectante, despotricaba con su magn\u00edfica voz:\n\n\u2014Toda Argentina est\u00e1 inundada de obras m\u00edas, envilecidas por plum\u00edferos rioplatenses. \u00bfSe ha topado usted en la vida con argentinos? Cuando vea uno, c\u00e1mbiese de vereda, porque la argentinidad, como el sarampi\u00f3n, es contagiosa.\n\nHab\u00eda palidecido y le vibraba la nariz. Apret\u00f3 los dientes e hizo una mueca de asco. Me sent\u00ed confuso ante esa nueva expresi\u00f3n de su personalidad y balbuce\u00e9 algo vago y general, era lamentable que en Am\u00e9rica Latina no hubiera una ley de derechos de autor, que no se protegiera la propiedad intelectual. Hab\u00eda vuelto a meter la pata.\n\n\u2014No se trata de eso, a m\u00ed no me importa ser plagiado \u2014replic\u00f3, m\u00e1s furioso a\u00fan\u2014. Los artistas no trabajamos por la gloria, sino por amor al hombre. Qu\u00e9 m\u00e1s quisiera yo que mi obra se difundiera por el mundo, aunque sea bajo otras r\u00fabricas. Lo que no se les puede perdonar a los cac\u00f3grafos del Plata es que alteren mis libretos, que los encanallen. \u00bfSabe usted lo que les hacen? Adem\u00e1s de cambiarles los t\u00edtulos y los nombres a los personajes, por supuesto. Los condimentan siempre con esas esencias argentinas...\n\n\u2014La arrogancia \u2014lo interrump\u00ed, seguro de dar esta vez en el clavo\u2014, la cursiler\u00eda.\n\nNeg\u00f3 con la cabeza, despectivamente, y pronunci\u00f3, con una solemnidad tr\u00e1gica y una voz lenta y cavernosa que retumb\u00f3 en el cubil, las \u00fanicas dos palabrotas que le o\u00ed decir nunca:\n\n\u2014La cojudez y la mariconer\u00eda.\n\nSent\u00ed deseos de jalarle la lengua, de saber por qu\u00e9 su odio a los argentinos era m\u00e1s vehemente que el de las gentes normales, pero, al verlo tan descompuesto, no me atrev\u00ed. Hizo un gesto de amargura y se pas\u00f3 una mano ante los ojos, como para borrar ciertos fantasmas. Luego, con expresi\u00f3n dolida, cerr\u00f3 las ventanas de su cubil, cuadr\u00f3 el rodillo de la Remington y le coloc\u00f3 su funda, se acomod\u00f3 la corbatita de lazo, sac\u00f3 de su escritorio un grueso libro que se puso bajo el sobaco y me indic\u00f3 con un gesto que sali\u00e9ramos. Apag\u00f3 la luz y, de afuera, ech\u00f3 llave a su cueva. Le pregunt\u00e9 qu\u00e9 libro era \u00e9se. Le pas\u00f3 afectuosamente la mano por el lomo, en una caricia id\u00e9ntica a la que podr\u00eda haber hecho a un gato.\n\n\u2014Un viejo compa\u00f1ero de aventuras \u2014murmur\u00f3, con emoci\u00f3n, alcanz\u00e1ndomelo\u2014. Un amigo fiel y un buen ayudante de trabajo.\n\nEl libro, publicado en tiempos prehist\u00f3ricos por Espasa Calpe \u2014sus gruesas tapas ten\u00edan todas las manchas y rasgu\u00f1os del mundo y sus hojas estaban amarillentas\u2014 era de un autor desconocido y de prontuario pomposo (Adalberto Castej\u00f3n de la Reguera, Licenciado por la Universidad de Murcia en Letras Cl\u00e1sicas, Gram\u00e1tica y Ret\u00f3rica), y el t\u00edtulo era extenso: Diez Mil Citas Literarias de los Cien Mejores Escritores del Mundo. Ten\u00eda un subt\u00edtulo: \u00abLo que dijeron Cervantes, Shakespeare, Moli\u00e8re, etc\u00e9tera, sobre Dios, la Vida, la Muerte, el Amor, el Sufrimiento, etc\u00e9tera...\u00bb.\n\nEst\u00e1bamos ya en la calle Bel\u00e9n. Al darle la mano se me ocurri\u00f3 mirar el reloj. Sent\u00ed p\u00e1nico: eran las diez de la noche. Ten\u00eda la sensaci\u00f3n de haber estado media hora con el artista y en realidad el an\u00e1lisis sociol\u00f3gico-chismogr\u00e1fico de la ciudad y la abominaci\u00f3n de los argentinos hab\u00eda demorado tres. Corr\u00ed a Panamericana, convencido de que Pascual habr\u00eda dedicado los quince minutos del bolet\u00edn de las nueve a alg\u00fan pir\u00f3mano de Turqu\u00eda o a alg\u00fan infanticidio en El Porvenir. Pero las cosas no deb\u00edan de haber ido tan mal, pues me encontr\u00e9 a los Genaros en el ascensor y no parec\u00edan furiosos. Me contaron que esa tarde hab\u00edan firmado contrato con Lucho Gatica para que viniera una semana a Lima como exclusividad de Panamericana. En mi altillo, revis\u00e9 los boletines y eran pasables. Sin apurarme, fui a tomar el colectivo a Miraflores en la plaza San Mart\u00edn.\n\nLlegu\u00e9 a la casa de los abuelos a las once de la noche; ya estaban durmiendo. Me dejaban siempre la comida en el horno, pero esta vez, adem\u00e1s del plato de apanado con arroz y huevo frito \u2014mi invariable men\u00fa\u2014 hab\u00eda un mensaje escrito con letra temblona: \u00abLlam\u00f3 tu t\u00edo Lucho. Que dejaste plantada a Julita, que ten\u00edan que ir al cine. Que eres un salvaje, que la llames para disculparte: el Abuelo\u00bb.\n\nPens\u00e9 que olvidarse de los boletines y de una cita con una dama por el escriba boliviano era demasiado. Me acost\u00e9 inc\u00f3modo y malhumorado por mi involuntaria malacrianza. Estuve dando vueltas antes de pescar el sue\u00f1o, tratando de convencerme que era culpa de ella, por imponerme esas idas al cine, a esas horribles truculencias, y buscando alguna excusa para cuando la llamara al d\u00eda siguiente. No se me ocurri\u00f3 ninguna plausible y no me atrev\u00ed a decirle la verdad. Hice m\u00e1s bien un gesto heroico. Despu\u00e9s del bolet\u00edn de las ocho, fui a una florer\u00eda del centro y le envi\u00e9 un ramo de rosas que me cost\u00f3 cien soles con una tarjeta en la que, despu\u00e9s de mucho dudar, escrib\u00ed lo que me pareci\u00f3 un prodigio de laconismo y elegancia: \u00abRendidas excusas\u00bb.\n\nEn la tarde hice algunos bocetos, entre bolet\u00edn y bolet\u00edn, de mi cuento er\u00f3tico-picaresco sobre la tragedia del senador arequipe\u00f1o. Me propon\u00eda trabajar fuerte en \u00e9l esa noche, pero Javier vino a buscarme despu\u00e9s de El Panamericano y me llev\u00f3 a una sesi\u00f3n de espiritismo, en los Barrios Altos. El m\u00e9dium era un escribano, a quien hab\u00eda conocido en las oficinas del Banco de Reserva. Me hab\u00eda hablado mucho de \u00e9l, pues siempre le contaba sus percances con las almas, que acud\u00edan a comunicarse con \u00e9l no s\u00f3lo cuando las convocaba en sesiones oficiales, sino espont\u00e1neamente, en las circunstancias m\u00e1s inesperadas. Sol\u00edan gastarle bromas, como hacer sonar el tel\u00e9fono al amanecer: al descolgar el aparato escuchaba al otro lado de la l\u00ednea la inconfundible risa de su bisabuela, muerta hac\u00eda medio siglo y domiciliada desde entonces (se lo hab\u00eda dicho ella misma) en el purgatorio. Se le aparec\u00edan en los \u00f3mnibus, en los colectivos, caminando por la calle. Le hablaban al o\u00eddo y \u00e9l ten\u00eda que permanecer mudo e impasible (\u00abdesairarlas\u00bb parece que dec\u00eda) a fin de que la gente no lo creyera loco. Yo, fascinado, le hab\u00eda pedido a Javier que organizara alguna sesi\u00f3n con el escribano m\u00e9dium. \u00c9ste hab\u00eda aceptado, pero ven\u00eda dando largas varias semanas, con pretextos climatol\u00f3gicos. Era indispensable esperar ciertas fases de la luna, el cambio de mareas y aun factores m\u00e1s especializados pues, al parecer, las \u00e1nimas eran sensibles a la humedad, las constelaciones, los vientos. Por fin hab\u00eda llegado el d\u00eda.\n\nNos cost\u00f3 un triunfo dar con la casa del escribano m\u00e9dium, un departamentito s\u00f3rdido, apretado en el fondo de una quinta del jir\u00f3n Cangallo. El personaje, en la realidad, era mucho menos interesante que en los cuentos de Javier. Sesent\u00f3n, solter\u00f3n, calvito, oloroso a linimento, ten\u00eda una mirada bovina y una conversaci\u00f3n tan empecinadamente banal que nadie hubiera sospechado su promiscuidad con los esp\u00edritus. Nos recibi\u00f3 en una salita desvencijada y grasienta; nos convid\u00f3 unas galletas de agua con trocitos de queso fresco y una parca mulita de pisco. Hasta que dieron las doce nos estuvo contando, con un aire convencional, sus experiencias del m\u00e1s all\u00e1. Hab\u00edan comenzado al enviudar, veinte a\u00f1os atr\u00e1s. La muerte de su mujer lo hab\u00eda sumido en una tristeza inconsolable, hasta que un d\u00eda un amigo lo salv\u00f3, mostr\u00e1ndole el camino del espiritismo. Era lo m\u00e1s importante que le hab\u00eda pasado en la vida:\n\n\u2014No s\u00f3lo porque uno tiene la oportunidad de seguir viendo y oyendo a los seres queridos \u2014nos dec\u00eda, con el tono que se comenta una fiesta de bautizo\u2014, sino porque distrae mucho, las horas se van sin darse cuenta.\n\nEscuch\u00e1ndolo, se ten\u00eda la impresi\u00f3n de que hablar con los muertos era algo comparable, en esencia, a ver una pel\u00edcula o un partido de f\u00fatbol (y, sin duda, menos divertido). Su versi\u00f3n de la otra vida era terriblemente cotidiana, desmoralizadora. No hab\u00eda diferencia alguna de cualidad entre all\u00e1 y aqu\u00ed, a juzgar por las cosas que le contaban: los esp\u00edritus se enfermaban, se enamoraban, se casaban, se reproduc\u00edan, viajaban, y la \u00fanica diferencia era que nunca se mor\u00edan. Yo le lanzaba miradas homicidas a Javier, cuando dieron las doce. El escribano nos hizo sentar alrededor de la mesa (no redonda sino cuadrangular), apag\u00f3 la luz, nos orden\u00f3 unir las manos. Hubo unos segundos de silencio y yo, nervioso con la espera, tuve la ilusi\u00f3n de que la cosa iba a ponerse interesante. Pero comenzaron a presentarse los esp\u00edritus y el escribano, con la misma voz dom\u00e9stica, empez\u00f3 a preguntarles las cosas m\u00e1s aburridas del mundo: \u00ab\u00bfY c\u00f3mo est\u00e1s, pues, Zoilita? Encantado de o\u00edrte; aqu\u00ed me tienes, pues, con estos amigos, muy buenas personas, interesados en conectarse con el mundo tuyo, Zoilita. \u00bfC\u00f3mo, qu\u00e9 cosa? \u00bfQue los salude? C\u00f3mo no, Zoilita, de tu parte. Dice que los salude con todo cari\u00f1o y que si pueden recen por ella de vez en cuando para que salga m\u00e1s pronto del purgatorio\u00bb. Despu\u00e9s de Zoilita se presentaron una serie de parientes y amigos con los que el escribano mantuvo di\u00e1logos semejantes. Todos estaban en el purgatorio, todos nos enviaron saludos, todos ped\u00edan rezos. Javier se empe\u00f1\u00f3 en llamar a alguien que estuviera en el infierno, para que nos sacara de dudas, pero el m\u00e9dium, sin vacilar un segundo, nos explic\u00f3 que era imposible: los de all\u00ed s\u00f3lo pod\u00edan ser citados los tres primeros d\u00edas de mes impar y apenas se les o\u00eda la voz. Javier pidi\u00f3 entonces al ama que hab\u00eda criado a su madre y a \u00e9l y a sus hermanos. Do\u00f1a Gumercinda compareci\u00f3, mand\u00f3 saludos, dijo que recordaba a Javier con mucho cari\u00f1o y que ya estaba haciendo sus ataditos para salir del purgatorio e ir al encuentro del Se\u00f1or. Yo ped\u00ed al escribano que llamara a mi hermano Juan, y, sorprendentemente (porque nunca hab\u00eda tenido hermanos), vino y me hizo decir, por la benigna voz del m\u00e9dium, que no deb\u00eda preocuparme por \u00e9l pues estaba con Dios y que siempre rezaba por m\u00ed. Tranquilizado con esta noticia, me despreocup\u00e9 de la sesi\u00f3n y me dediqu\u00e9 a escribir mentalmente mi cuento sobre el senador. Se me ocurri\u00f3 un t\u00edtulo enigm\u00e1tico: La cara incompleta. Decid\u00ed, mientras Javier, incansable, exig\u00eda al escribano que convocara alg\u00fan \u00e1ngel, o, al menos, alg\u00fan personaje hist\u00f3rico como Manco C\u00e1pac, que el senador terminar\u00eda resolviendo su problema mediante una fantas\u00eda freudiana: pondr\u00eda a su esposa, en el momento del amor, un parche de pirata en el ojo.\n\nLa sesi\u00f3n termin\u00f3 cerca de las dos de la madrugada. Mientras camin\u00e1bamos por las calles de los Barrios Altos, en busca de un taxi que nos llevara hasta la plaza San Mart\u00edn para tomar el colectivo, yo enloquec\u00eda a Javier dici\u00e9ndole que por su culpa el m\u00e1s all\u00e1 hab\u00eda perdido para m\u00ed poes\u00eda y misterio, que por su culpa hab\u00eda tenido la evidencia de que todos los muertos se volv\u00edan imb\u00e9ciles, que por su culpa ya no podr\u00eda ser agn\u00f3stico y tendr\u00eda que vivir con la certidumbre de que, en la otra vida, que exist\u00eda, me esperaba una eternidad de cretinismo y aburrimiento. Encontramos un taxi y en castigo lo pag\u00f3 Javier.\n\nEn casa, junto al apanado, huevo y arroz, encontr\u00e9 otro mensaje: \u00abTe llam\u00f3 Julita. Que recibi\u00f3 tus rosas, que est\u00e1n muy lindas, que le gustaron mucho. Que no creas que por las rosas te librar\u00e1s de llevarla al cine cualquiera de estos d\u00edas: el Abuelo\u00bb.\n\nAl d\u00eda siguiente era cumplea\u00f1os del t\u00edo Lucho. Le compr\u00e9 una corbata de regalo y me dispon\u00eda a ir a su casa al mediod\u00eda, pero Genaro hijo se me present\u00f3 intempestivamente en el altillo y me oblig\u00f3 a ir a almorzar con \u00e9l en el Raimondi. Quer\u00eda que lo ayudara a redactar los avisos que publicar\u00eda ese domingo en los diarios, anunciando los radioteatros de Pedro Camacho, que arrancaban el lunes. \u00bfNo hubiera sido m\u00e1s l\u00f3gico que el propio artista interviniera en la redacci\u00f3n de esos avisos?\n\n\u2014La vaina es que se ha negado \u2014me explic\u00f3 Genaro hijo, fumando como una chimenea\u2014. Sus libretos no necesitan publicidad mercenaria, se imponen solos y no s\u00e9 qu\u00e9 otras tonter\u00edas. El tipo est\u00e1 resultando complicado, muchas man\u00edas. \u00bfSupiste lo de los argentinos, no? Nos ha obligado a rescindir contratos, a pagar indemnizaciones. Espero que sus programas justifiquen estos engreimientos.\n\nMientras redact\u00e1bamos los avisos, despach\u00e1bamos dos corvinas, beb\u00edamos cerveza helada y ve\u00edamos, de tanto en tanto, desfilar por las vigas del Raimondi esos grises ratoncitos que parecen puestos all\u00ed como prueba de antig\u00fcedad del local, Genaro hijo me cont\u00f3 otro conflicto que hab\u00eda tenido con Pedro Camacho. La raz\u00f3n: los protagonistas de los cuatro radioteatros con que debutaba en Lima. En los cuatro, el gal\u00e1n era un cincuent\u00f3n \u00abque conservaba maravillosamente la juventud\u00bb.\n\n\u2014Le hemos explicado que todos los surveys han demostrado que el p\u00fablico quiere galanes de entre treinta y treinta y cinco a\u00f1os, pero es una mula \u2014se aflig\u00eda Genaro hijo, echando humo por la boca y la nariz\u2014. \u00bfY si he metido la pata y el boliviano es un fracaso descomunal?\n\nRecord\u00e9 que, en un momento de nuestra conversaci\u00f3n de la v\u00edspera en su cubil de Radio Central, el artista hab\u00eda dogmatizado, con fuego, sobre los cincuenta a\u00f1os del hombre. La edad del apogeo cerebral y de la fuerza sensual, dec\u00eda, de la experiencia digerida. La edad en que se era m\u00e1s deseado por las mujeres y m\u00e1s temido por los hombres. Y hab\u00eda insistido, sospechosamente, en que la vejez era algo optativo. Deduje que el escriba boliviano ten\u00eda cincuenta a\u00f1os y que lo aterraba la vejez: un rayito de debilidad humana en ese esp\u00edritu marm\u00f3reo.\n\nCuando terminamos de redactar los avisos era tarde para dar un salto a Miraflores, de modo que telefone\u00e9 al t\u00edo Lucho para decirle que ir\u00eda a abrazarlo a la noche. Supuse que encontrar\u00eda una aglomeraci\u00f3n de familiares festej\u00e1ndolo, pero no hab\u00eda nadie, aparte de la t\u00eda Olga y la t\u00eda Julia. Los parientes hab\u00edan desfilado por la casa durante el d\u00eda. Estaban tomando whiskys y me sirvieron una copa. La t\u00eda Julia me agradeci\u00f3 otra vez las rosas \u2014las vi sobre el aparador de la sala y eran poqu\u00edsimas\u2014 y se puso a bromear, como siempre, pidi\u00e9ndome que confesara qu\u00e9 clase de programa me hab\u00eda salido la noche que la dej\u00e9 plantada: \u00bfalguna piba de la universidad, alguna huachafita de la radio? Llevaba un vestido azul, zapatos blancos, maquillaje y peinado de peluquer\u00eda; se re\u00eda con una risa fuerte y directa y ten\u00eda voz ronca y ojos insolentes. Descubr\u00ed, algo tard\u00edamente, que era una mujer atractiva. El t\u00edo Lucho, en un arrebato de entusiasmo, dijo que cincuenta a\u00f1os se cumpl\u00edan s\u00f3lo una vez en la vida y que nos fu\u00e9ramos al Grill Bol\u00edvar. Pens\u00e9 que, por segundo d\u00eda consecutivo, tendr\u00eda que dejar de lado la redacci\u00f3n de mi cuento sobre el senador eunuco y pervertido (\u00bfy si le pon\u00eda ese t\u00edtulo?). Pero no lo lament\u00e9, me sent\u00ed muy contento de verme embarcado en esa fiesta. La t\u00eda Olga, despu\u00e9s de examinarme, dictamin\u00f3 que mi facha no era la m\u00e1s adecuada para el Grill Bol\u00edvar e hizo que el t\u00edo Lucho me prestara una camisa limpia y una corbata llamativa que compensaran un poco lo viejo y arrugado del terno. La camisa me qued\u00f3 grande, y yo sent\u00eda angustia por mi cuello bailando en el aire (lo que dio lugar a que la t\u00eda Julia comenzara a llamarme Popeye).\n\nNunca hab\u00eda ido al Grill Bol\u00edvar y me pareci\u00f3 el lugar m\u00e1s refinado y elegante del mundo, y la comida la m\u00e1s exquisita que hab\u00eda probado jam\u00e1s. Una orquesta tocaba boleros, pasodobles, blues, y la estrella del show era una francesa, blanca como la leche, que recitaba acariciadoramente sus canciones mientras daba la impresi\u00f3n de masturbar el micro con las manos, y a la que el t\u00edo Lucho, de un buen humor que crec\u00eda con las copas, vitoreaba en una jerigonza que \u00e9l llamaba franc\u00e9s: \u00ab\u00a1Vravoooo! \u00a1Vravoooo mamuasel cher\u00ed!\u00bb. El primero en lanzarse a bailar fui yo, que arrastr\u00e9 a la t\u00eda Olga a la pista, ante mi propia sorpresa, pues no sab\u00eda bailar (estaba entonces firmemente convencido de que una vocaci\u00f3n literaria era incompatible con el baile y los deportes), pero, felizmente, hab\u00eda mucha gente, y, en la apretura y penumbra, nadie pudo advertirlo. La t\u00eda Julia, a su vez, hac\u00eda pasar un mal rato al t\u00edo Lucho oblig\u00e1ndolo a bailar separado de ella y haciendo figuras. Ella bailaba bien y las miradas de muchos se\u00f1ores la segu\u00edan.\n\nLa pieza siguiente saqu\u00e9 a la t\u00eda Julia y la previne que no sab\u00eda bailar, pero, como tocaban un lent\u00edsimo blues, desempe\u00f1\u00e9 mi funci\u00f3n con decoro. Bailamos un par de piezas y nos fuimos alejando insensiblemente de la mesa del t\u00edo Lucho y la t\u00eda Olga. En el instante en que, terminada la m\u00fasica, la t\u00eda Julia hac\u00eda un movimiento para apartarse de m\u00ed, la retuve y la bes\u00e9 en la mejilla, muy cerca de los labios. Me mir\u00f3 con asombro, como si presenciara un prodigio. Hab\u00eda cambio de orquesta y debimos regresar a la mesa. All\u00ed, la t\u00eda Julia se puso a hacer bromas al t\u00edo Lucho sobre los cincuenta a\u00f1os, edad a partir de la cual los hombres se volv\u00edan viejos verdes. A ratos me lanzaba una r\u00e1pida ojeada, como para verificar si yo estaba realmente ah\u00ed, y en sus ojos se pod\u00eda leer clar\u00edsimo que todav\u00eda no le cab\u00eda en la cabeza que la hubiera besado. La t\u00eda Olga estaba ya cansada y quer\u00eda que nos fu\u00e9ramos, pero yo insist\u00ed en bailar una pieza m\u00e1s. \u00abEl intelectual se corrompe\u00bb, constat\u00f3 el t\u00edo Lucho y arrastr\u00f3 a la t\u00eda Olga a bailar la pieza del estribo. Yo saqu\u00e9 a la t\u00eda Julia y, mientras bail\u00e1bamos, ella permanec\u00eda (por primera vez) muda. Cuando, entre la masa de parejas, el t\u00edo Lucho y la t\u00eda Olga quedaron distanciados, la estrech\u00e9 un poco contra m\u00ed y le junt\u00e9 la mejilla. La o\u00ed murmurar, confusa: \u00abOye, Marito...\u00bb, pero la interrump\u00ed dici\u00e9ndole al o\u00eddo: \u00abTe proh\u00edbo que me vuelvas a llamar Marito\u00bb. Ella separ\u00f3 un poco la cara para mirarme e intent\u00f3 sonre\u00edr, y entonces, en una acci\u00f3n casi mec\u00e1nica, me inclin\u00e9 y la bes\u00e9 en los labios. Fue un contacto muy r\u00e1pido pero no lo esperaba y la sorpresa hizo que esta vez dejara un momento de bailar. Ahora su estupefacci\u00f3n era total: abr\u00eda los ojos y estaba con la boca abierta. Cuando termin\u00f3 la pieza, el t\u00edo Lucho pag\u00f3 la cuenta y nos fuimos. En el trayecto a Miraflores \u2014\u00edbamos los dos en el asiento de atr\u00e1s\u2014 cog\u00ed la mano de la t\u00eda Julia, la apret\u00e9 con ternura y la mantuve entre las m\u00edas. No la retir\u00f3, pero se la notaba a\u00fan sorprendida y no abr\u00eda la boca. Al bajar, en casa de los abuelos, me pregunt\u00e9 cu\u00e1ntos a\u00f1os mayor que yo ser\u00eda.\n\n### IV\n\nEN LA NOCHE chalaca, h\u00fameda y oscura como boca de lobo, el sargento Lituma se subi\u00f3 las solapas del capote, se frot\u00f3 las manos y se dispuso a cumplir con su deber. Era un hombre en la flor de la edad, la cincuentena, al que la Guardia Civil entera respetaba; hab\u00eda servido en las comisar\u00edas m\u00e1s sacrificadas sin quejarse y su cuerpo conservaba algunas cicatrices de sus batallas contra el crimen. Las c\u00e1rceles del Per\u00fa herv\u00edan de malhechores a los que hab\u00eda calzado las esposas. Hab\u00eda sido citado como ejemplo en \u00f3rdenes del d\u00eda, alabado en discursos oficiales, y, por dos veces, condecorado: pero esas glorias no hab\u00edan alterado su modestia, tan grande como su valent\u00eda y su honradez. Hac\u00eda un a\u00f1o que serv\u00eda en la Cuarta Comisar\u00eda del Callao y llevaba ya tres meses encargado de la m\u00e1s dura obligaci\u00f3n que el destino puede deparar a un sargento en el puerto: la ronda nocturna.\n\nLas remotas campanas de la Iglesia de Nuestra Se\u00f1ora del Carmen de la Legua dieron la medianoche, y, siempre puntual, el sargento Lituma \u2014frente ancha, nariz aguile\u00f1a, mirada penetrante, rectitud y bondad en el esp\u00edritu\u2014 empez\u00f3 a caminar. A su espalda, una fogata en las tinieblas, quedaba la vieja casona de madera de la Cuarta Comisar\u00eda. Imagin\u00f3: el teniente Jaime Concha estar\u00eda leyendo el Pato Donald, los guardias Mocos Camacho y Manzanita Ar\u00e9valo estar\u00edan azucar\u00e1ndose un caf\u00e9 reci\u00e9n colado y el \u00fanico preso del d\u00eda \u2014un carterista sorprendido in fraganti en el \u00f3mnibus Chucuito-La Parada y tra\u00eddo a la comisar\u00eda, con abundantes contusiones, por media docena de furibundos pasajeros\u2014 dormir\u00eda hecho un garabato en el suelo del erg\u00e1stulo.\n\nInici\u00f3 su recorrido por la barriada de Puerto Nuevo, donde estaba de servicio el Chato Soldevilla, un tumbesino que cantaba tonderos con inspirada voz. Puerto Nuevo era el terror de los guardias y detectives del Callao porque en su laberinto de casuchas de tablones, latas, calaminas y adobes, s\u00f3lo una \u00ednfima parte de sus pobladores se ganaban el pan como portuarios o pescadores. La mayor\u00eda eran vagos, ladrones, borrachos, pichicateros, macr\u00f3s y maricas (para no mencionar a las innumerables prostitutas) que, con cualquier pretexto, se agarraban a chavetazos y, a veces, tiros. Esa barriada sin agua ni desag\u00fce, sin luz y sin pavimentar, se hab\u00eda te\u00f1ido no pocas veces con sangre de agentes de la ley. Pero esa noche estaba excepcionalmente pac\u00edfica. Mientras, tropezando con piedras invisibles, la cara fruncida por el vaho de excrementos y materias descompuestas que sub\u00eda a sus narices, recorr\u00eda los meandros del barrio en busca del Chato, el sargento Lituma pens\u00f3: \u00abEl fr\u00edo acost\u00f3 temprano a los noct\u00e1mbulos\u00bb. Porque era mediados de agosto, el coraz\u00f3n del invierno, y una neblina espesa que todo lo borraba y deformaba, y una gar\u00faa tenaz que aguaba el aire, hab\u00edan convertido esa noche en algo triste e inh\u00f3spito. \u00bfD\u00f3nde se hab\u00eda metido el Chato Soldevilla? Este tumbesino mariconazo, asustado del fr\u00edo o de los hampones, era capaz de haber ido a buscar calorcito y trago a las cantinas de la avenida Hu\u00e1scar. \u00abNo, no se atrever\u00eda \u2014pens\u00f3 el sargento Lituma\u2014. Sabe que yo hago la ronda y que si abandona su puesto, se amuela\u00bb.\n\nEncontr\u00f3 al Chato bajo un poste de luz, en la esquina que mira al Frigor\u00edfico Nacional. Se frotaba las manos con furia, su cara hab\u00eda desaparecido tras una chalina fantasmal que s\u00f3lo le dejaba los ojos libres. Al verlo, dio un respingo y se llev\u00f3 la mano a la cartuchera. Luego, reconoci\u00e9ndolo, choc\u00f3 los tacos.\n\n\u2014Me asust\u00f3, mi sargento \u2014dijo ri\u00e9ndose\u2014. As\u00ed, de lejitos, saliendo de la oscuridad, me figur\u00e9 un esp\u00edritu.\n\n\u2014Qu\u00e9 esp\u00edritu ni qu\u00e9 ocho cuartos \u2014le dio la mano Lituma\u2014. Cre\u00edste que era un hamp\u00f3n.\n\n\u2014Con este fr\u00edo no hay hampones sueltos, qu\u00e9 esperanza \u2014volvi\u00f3 a frotarse las manos el Chato\u2014. Los \u00fanicos locos a los que en esta noche se les ocurre andar a la intemperie somos usted y yo. Y \u00e9stos.\n\nSe\u00f1al\u00f3 el techo del Frigor\u00edfico y el sargento, esforzando los ojos, alcanz\u00f3 a ver media docena de gallinazos api\u00f1ados y con el pico entre las alas, formando una l\u00ednea recta en la cumbre de la calamina. \u00abQu\u00e9 hambre tendr\u00e1n \u2014pens\u00f3\u2014. Aunque se hielen, all\u00ed se quedan oliendo lo muerto\u00bb. El Chato Soldevilla le firm\u00f3 el parte a la rancia luz del farol, con un lapicito masticado que se le perd\u00eda en los dedos. No hab\u00eda novedad: ni accidentes, ni delitos, ni borracheras.\n\n\u2014Una noche tranquila, mi sargento \u2014le dijo, mientras lo acompa\u00f1aba unas cuadras, hacia la avenida Manco C\u00e1pac\u2014. Espero que siga as\u00ed, hasta que llegue mi relevo. Despu\u00e9s, que se caiga el mundo, qu\u00e9 diablos.\n\nSe ri\u00f3, como si hubiera dicho algo muy chistoso, y el sargento Lituma pens\u00f3: \u00abHay que ver la mentalidad que se gastan ciertos guardias\u00bb. Como si hubiera adivinado, el Chato Soldevilla a\u00f1adi\u00f3, serio:\n\n\u2014Porque yo no soy como usted, mi sargento. A m\u00ed esto no me gusta. Llevo el uniforme s\u00f3lo por la comida.\n\n\u2014Si dependiera de m\u00ed, no lo llevar\u00edas \u2014murmur\u00f3 el sargento\u2014. Yo s\u00f3lo dejar\u00eda en el cuerpo a los que creen en la vaina.\n\n\u2014Se quedar\u00eda bastante vac\u00eda la Guardia Civil \u2014repuso el Chato.\n\n\u2014M\u00e1s vale solos que mal acompa\u00f1ados \u2014se ri\u00f3 el sargento.\n\nEl Chato tambi\u00e9n se ri\u00f3. Caminaban a oscuras, por el descampado que rodea a la Factor\u00eda Guadalupe, donde los mataperros se volaban siempre a pedradas los focos de los postes. Se o\u00eda el rumor del mar a lo lejos, y, de cuando en cuando, el motor de alg\u00fan taxi que cruzaba la avenida Argentina.\n\n\u2014A usted le gustar\u00eda que todos fu\u00e9ramos h\u00e9roes \u2014dijo de pronto el Chato\u2014. Que nos sac\u00e1ramos el alma para defender a estas basuras \u2014se\u00f1al\u00f3 hacia el Callao, hacia Lima, hacia el mundo\u2014. \u00bfAcaso nos lo agradecen? \u00bfNo ha o\u00eddo lo que nos gritan en la calle? \u00bfAcaso alguien nos respeta? La gente nos desprecia, mi sargento.\n\n\u2014Aqu\u00ed nos despedimos \u2014dijo Lituma, al borde de la avenida Manco C\u00e1pac\u2014. No te salgas de tu \u00e1rea. Y no te hagas mala sangre. No ves la hora de dejar el cuerpo, pero el d\u00eda que te den de baja vas a sufrir como un perro. As\u00ed le pas\u00f3 a Pechito Antezana. Ven\u00eda a la comisar\u00eda a mirarnos y se le llenaban los ojos de l\u00e1grimas. \u00abHe perdido a mi familia\u00bb, dec\u00eda.\n\nOy\u00f3 que, a su espalda, el Chato gru\u00f1\u00eda: \u00abUna familia sin mujeres, qu\u00e9 clase de familia es\u00bb.\n\nTal vez el Chato ten\u00eda raz\u00f3n, pensaba el sargento Lituma, mientras avanzaba por la desierta avenida, en medio de la noche. Era verdad, la gente no quer\u00eda a los polic\u00edas, se acordaba de ellos cuando ten\u00eda miedo de algo. \u00bfY eso qu\u00e9? \u00c9l no se sacaba la mugre para que la gente lo respetara o lo quisiera. \u00abA m\u00ed la gente me importa un pito\u00bb, pens\u00f3. \u00bfY entonces por qu\u00e9 no tomaba la Guardia Civil como los compa\u00f1eros, sin matarse, tratando de pasarla lo mejor posible, aprovechando para descansar o para ganarse unos soles sucios si la superioridad no estaba cerca? \u00bfPor qu\u00e9, Lituma? Pens\u00f3: \u00abPorque a ti te gusta. Porque, como a otros les gusta el f\u00fatbol o las carreras, a ti te gusta tu trabajo\u00bb. Se le ocurri\u00f3 que la pr\u00f3xima vez que alg\u00fan loco del f\u00fatbol le preguntara \u00ab\u00bfEres hincha del Sport Boys o del Chalaco, Lituma?\u00bb, le responder\u00eda: \u00abSoy hincha de la Guardia Civil\u00bb. Se re\u00eda, en la neblina, en la gar\u00faa, en la noche, contento de su ocurrencia, y en eso oy\u00f3 el ruido. Dio un respingo, se llev\u00f3 la mano a la cartuchera, se par\u00f3. Lo hab\u00eda tomado tan de sorpresa que casi se hab\u00eda asustado. \u00abS\u00f3lo casi \u2014pens\u00f3\u2014, porque t\u00fa no has sentido miedo ni sentir\u00e1s, t\u00fa no sabes c\u00f3mo se come eso, Lituma\u00bb. Ten\u00eda a su izquierda el descampado y a la derecha la mole del primero de los dep\u00f3sitos del Terminal Mar\u00edtimo. De all\u00ed hab\u00eda venido: muy fuerte, un estruendo de cajones y latas que se derrumban arrastrando en su ca\u00edda a otros cajones y latas. Pero ahora todo estaba tranquilo de nuevo y s\u00f3lo o\u00eda el chasquido lejano del mar y el silbido del viento al golpear las calaminas y al enroscarse en las alambradas del puerto. \u00abUn gato que persegu\u00eda a una rata y que se trajo abajo un caj\u00f3n y \u00e9se a otro y fue el huayco\u00bb, pens\u00f3. Pens\u00f3 en el pobre gato, despanzurrado junto con la rata, bajo una monta\u00f1a de fardos y barriles. Ya estaba en el \u00e1rea del Choclo Rom\u00e1n. Pero claro que el Choclo no estaba por aqu\u00ed; Lituma sab\u00eda muy bien que estaba en el otro extremo de su \u00e1rea, en el Happy Land, o en el Blue Star, o en cualquiera de los barcitos y prost\u00edbulos de marineros que se codeaban al fondo de la avenida, en esa callecita que los chalacos lengualargas llamaban la calle del chancro. Ah\u00ed estar\u00eda, en uno de esos astillados mostradores, gorreando una cervecita. Y, mientras caminaba hacia esos antros, Lituma pens\u00f3 en la cara de susto que pondr\u00eda Rom\u00e1n si \u00e9l se le aparec\u00eda por detr\u00e1s, de repente: \u00abAs\u00ed que tomando bebidas espirituosas durante el servicio. Te amolaste, Choclo\u00bb.\n\nHab\u00eda avanzado unos doscientos metros y se par\u00f3 en seco. Volvi\u00f3 la cabeza: all\u00e1, en la sombra, una de sus paredes apenas iluminada por el resplandor de un farol milagrosamente indemne de las hondas de los mataperros, mudo ahora, estaba el dep\u00f3sito. \u00abNo es un gato \u2014pens\u00f3\u2014, no es una rata\u00bb. Era un ladr\u00f3n. Su pecho comenz\u00f3 a latir con fuerza y sinti\u00f3 que la frente y las manos se le mojaban. Era un ladr\u00f3n, un ladr\u00f3n. Permaneci\u00f3 inm\u00f3vil unos segundos, pero ya sab\u00eda que iba a regresar. Estaba seguro: hab\u00eda tenido otras veces esos p\u00e1lpitos. Desenfund\u00f3 su pistola y le sac\u00f3 el seguro, y empu\u00f1\u00f3 la linterna con la mano zurda. Regres\u00f3 a trancos, sintiendo que el coraz\u00f3n se le sal\u00eda por la boca. S\u00ed, segur\u00edsimo, era un ladr\u00f3n. A la altura del dep\u00f3sito se par\u00f3 de nuevo, jadeando. \u00bfY si no era uno sino unos? \u00bfNo ser\u00eda mejor buscar al Chato, al Choclo? Movi\u00f3 la cabeza: no necesitaba a nadie, se bastaba y sobraba. Si eran varios, peor para ellos y mejor para \u00e9l. Escuch\u00f3, pegando la cara a la madera: silencio total. S\u00f3lo o\u00eda, a lo lejos, el mar y alguno que otro carro. \u00abQu\u00e9 ladr\u00f3n ni qu\u00e9 ocho cuartos, Lituma \u2014pens\u00f3\u2014. Est\u00e1s so\u00f1ando. Era un gato, una rata\u00bb. Se le hab\u00eda quitado el fr\u00edo, sent\u00eda calor y cansancio. Contorne\u00f3 el dep\u00f3sito, buscando la puerta. Cuando la encontr\u00f3, a la luz de la linterna verific\u00f3 que la cerradura no hab\u00eda sido violentada. Ya se iba, dici\u00e9ndose \u00abqu\u00e9 tal chasco, Lituma, tu olfato no es el de antes\u00bb, cuando, en un movimiento maquinal de su mano, el disco amarillento de la linterna le retrat\u00f3 la abertura. Estaba a pocos metros de la puerta; la hab\u00edan hecho a lo bruto, rompiendo la madera a hachazos o a patadas. El boquete era lo bastante grande para un hombre a gatas.\n\nSinti\u00f3 su coraz\u00f3n agitad\u00edsimo, loco. Apag\u00f3 la linterna, comprob\u00f3 que su pistola estaba sin seguro, mir\u00f3 en torno: s\u00f3lo sombras y, a lo lejos, como luces de f\u00f3sforos, los faroles de la avenida Hu\u00e1scar. Llen\u00f3 de aire los pulmones y, con toda la fuerza de que era capaz, rugi\u00f3:\n\n\u2014Rod\u00e9eme este almac\u00e9n con sus hombres, cabo. Si alguno trata de escapar, fuego a discreci\u00f3n. \u00a1R\u00e1pido, muchachos!\n\nY, para que fuera m\u00e1s cre\u00edble, dio unas carreritas de un lado a otro, zapateando fuerte. Luego, peg\u00f3 la cara al tabique del dep\u00f3sito y grit\u00f3, a voz en cuello:\n\n\u2014Se amolaron, les sali\u00f3 mal. Est\u00e1n rodeados. Vayan saliendo por donde entraron, uno tras otro. \u00a1Treinta segundos para que lo hagan por las buenas!\n\nEscuch\u00f3 el eco de sus gritos perdi\u00e9ndose en la noche, y, luego, el mar y unos ladridos. Cont\u00f3 no treinta sino sesenta segundos. Pens\u00f3: \u00abEst\u00e1s hecho un payaso, Lituma\u00bb. Sinti\u00f3 un acceso de c\u00f3lera. Grit\u00f3:\n\n\u2014Abran los ojos, muchachos. \u00a1A la primera, me los queman, cabo!\n\nY, resueltamente, se puso a cuatro patas y gateando, \u00e1gil a pesar de sus a\u00f1os y del abrigado uniforme, atraves\u00f3 el boquete. Adentro, se incorpor\u00f3 de prisa, en puntas de pie corri\u00f3 hacia un lado y peg\u00f3 la espalda a la pared. No ve\u00eda nada y no quer\u00eda prender la linterna. No o\u00eda ning\u00fan ruido pero otra vez ten\u00eda una seguridad total. Hab\u00eda alguien ah\u00ed, agazapado en la oscuridad, igual que \u00e9l, escuchando y tratando de ver. Le pareci\u00f3 sentir una respiraci\u00f3n, un jadeo. Ten\u00eda el dedo en el gatillo y la pistola a la altura del pecho. Cont\u00f3 tres y encendi\u00f3. El grito lo tom\u00f3 tan desprevenido que, con el susto, la linterna se le escap\u00f3 de las manos y rod\u00f3 por el suelo, revelando bultos, fardos que parec\u00edan de algod\u00f3n, barriles, vigas, y (fugaz, intempestiva, inveros\u00edmil) la figura del negro calato y encogido, con las manos tratando de taparse la cara, y, sin embargo, mirando por entre los dedos, los ojazos espantados, fijos en la linterna, como si el peligro le pudiera venir s\u00f3lo de la luz.\n\n\u2014\u00a1Quieto o te quemo! \u00a1Quieto o est\u00e1s muerto, zambo! \u2014rugi\u00f3 Lituma, tan fuerte que le doli\u00f3 la garganta, mientras, agachado, manoteaba buscando la linterna. Y luego, con satisfacci\u00f3n salvaje\u2014: \u00a1Te amolaste, zambo! \u00a1Te sali\u00f3 mal, zambo!\n\nGritaba tanto que se sent\u00eda aturdido. Hab\u00eda recuperado la linterna y el halo de luz revolote\u00f3, en busca del negro. No hab\u00eda huido, ah\u00ed estaba, y Lituma abr\u00eda mucho los ojos, incr\u00e9dulo, dudando de lo que ve\u00eda. No hab\u00eda sido una imaginaci\u00f3n, un sue\u00f1o. Estaba calato, s\u00ed, tal como lo hab\u00edan parido: ni zapatos, ni calzoncillo, ni camiseta, ni nada. Y no parec\u00eda tener verg\u00fcenza ni darse cuenta siquiera que estaba calato, porque no se tapaba sus cochinadas, que le bailoteaban alegremente, a la luz de la linterna. Segu\u00eda encogido, la cara medio oculta tras los dedos, y no se mov\u00eda, hipnotizado por la redondela de luz.\n\n\u2014Las manos sobre la cabeza, zambo \u2014orden\u00f3 el sargento, sin avanzar hacia \u00e9l\u2014. Tranquilo si no quieres un plomazo. Vas preso por invadir la propiedad privada y por andar con los mellicitos al aire.\n\nY, al mismo tiempo \u2014los o\u00eddos alertas por si el menor ruido delataba a alg\u00fan c\u00f3mplice en las sombras del dep\u00f3sito\u2014, el sargento se dec\u00eda: \u00abNo es un ladr\u00f3n. Es un loco\u00bb. No s\u00f3lo porque estaba desnudo en pleno invierno, sino por el grito que hab\u00eda lanzado al ser descubierto. No era de hombre normal, pens\u00f3 el sargento. Hab\u00eda sido un ruido extra\u00f1\u00edsimo, algo entre el aullido, el rebuzno, la carcajada y el ladrido. Un ruido que no parec\u00eda \u00fanicamente de la garganta sino tambi\u00e9n de la barriga, el coraz\u00f3n, el alma.\n\n\u2014He dicho manos a la cabeza, mi\u00e9chica \u2014grit\u00f3 el sargento, dando un paso hacia el hombre. \u00c9ste no obedeci\u00f3, no se movi\u00f3. Era muy oscuro, tan flaco que en la penumbra Lituma distingu\u00eda las costillas hinchando el pellejo y esos canutos que eran sus piernas, pero ten\u00eda un vientre grandote, que se le rebalsaba sobre el pubis, y Lituma se acord\u00f3 inmediatamente de las esquel\u00e9ticas criaturas de las barriadas, con panzas infladas por los par\u00e1sitos. El zambo segu\u00eda tap\u00e1ndose la cara, quieto, y el sargento dio otros dos pasos hacia \u00e9l, midi\u00e9ndolo, seguro de que en cualquier momento se echar\u00eda a correr. \u00abLos locos no respetan los rev\u00f3lveres\u00bb, pens\u00f3, y dio dos pasos m\u00e1s. Estaba apenas a un par de metros del zambo y s\u00f3lo ahora alcanz\u00f3 a percibir las cicatrices que le veteaban los hombros, los brazos, la espalda. \u00abPa su macho, pa su diablo\u00bb, pens\u00f3 Lituma. \u00bfEran de enfermedad? \u00bfHeridas o quemaduras? Habl\u00f3 bajito para no espantarlo:\n\n\u2014Quieto y tranquilo, zambo. Las manos en la cabeza y caminando hacia el hueco por donde entraste. Si te portas bien, en la comisar\u00eda te dar\u00e9 un caf\u00e9. Debes estar muerto de fr\u00edo, as\u00ed calato, con este tiempo.\n\nIba a dar un paso m\u00e1s hacia el negro, cuando \u00e9ste, s\u00fabitamente, se quit\u00f3 las manos de la cara \u2014Lituma se qued\u00f3 estupefacto al descubrir, bajo la mata de pelo pasa apelmazado, esos ojos sobrecogidos, esas cicatrices horribles, esa enorme jeta de la que sobresal\u00eda un \u00fanico, largo y afilado diente\u2014, volvi\u00f3 a lanzar ese h\u00edbrido, incomprensible, inhumano alarido, mir\u00f3 a un lado y a otro, desasosegado, ind\u00f3cil, nervioso, como un animal que busca un camino para huir, y, por fin, est\u00fapidamente, eligi\u00f3 el que no deb\u00eda, el que bloqueaba el sargento con su cuerpo. Porque no se abalanz\u00f3 contra \u00e9l sino intent\u00f3 escapar a trav\u00e9s de \u00e9l. Corri\u00f3 y fue tan inesperado que Lituma no alcanz\u00f3 a atajarlo y lo sinti\u00f3 que se estrellaba contra \u00e9l. El sargento ten\u00eda sus nervios bien puestos: no se le fue el dedo, no se le escap\u00f3 un tiro. El zambo, al chocar, buf\u00f3 y entonces Lituma le dio un empuj\u00f3n y vio que se ven\u00eda al suelo como si fuera de trapo. Para que se estuviese tranquilo, lo pate\u00f3.\n\n\u2014P\u00e1rate \u2014le orden\u00f3\u2014. Adem\u00e1s de loco eres tonto. Y c\u00f3mo apestas.\n\nTen\u00eda un olor indefinible, a alquitr\u00e1n, acetona, pis y gato. Se hab\u00eda dado vuelta y, las espaldas contra el suelo, lo miraba con p\u00e1nico.\n\n\u2014Pero de d\u00f3nde has podido salir t\u00fa \u2014murmur\u00f3 Lituma. Acerc\u00f3 un poco la linterna y examin\u00f3 un rato, confuso, esa incre\u00edble cara cruzada y descruzada por incisiones rectil\u00edneas, peque\u00f1as nervaduras que recorr\u00edan sus mejillas, su nariz, su frente, su ment\u00f3n y se perd\u00edan por su cuello. C\u00f3mo hab\u00eda podido andar por las calles del Callao un tipo con una pinta as\u00ed, y con los mellizos al aire, sin que alguien diera parte.\n\n\u2014Lev\u00e1ntate de una vez o te doy tu sopapo \u2014dijo Lituma\u2014. Loco o no loco ya me cansaste.\n\nEl tipo no se movi\u00f3. Hab\u00eda comenzado a hacer unos ruidos con la boca, un murmullo indescifrable, un ronroneo, un bisbiseo, algo que parec\u00eda tener que ver m\u00e1s con p\u00e1jaros, insectos o fieras que con hombres. Y segu\u00eda mirando la linterna con un terror infinito.\n\n\u2014P\u00e1rate, no tengas miedo \u2014dijo el sargento y, estirando una mano, cogi\u00f3 al zambo del brazo. No se resisti\u00f3 pero tampoco hizo esfuerzo alguno para ponerse de pie. \u00abQu\u00e9 flaco eres\u00bb, pens\u00f3 Lituma, casi divertido con el maullido, gorgoteo, silabeo incesante del hombre: \u00abY qu\u00e9 miedo me tienes\u00bb. Lo oblig\u00f3 a levantarse y no pod\u00eda creer que pesara tan poco; apenas le dio un empujoncito en direcci\u00f3n a la abertura del tabique, lo sinti\u00f3 que trastabillaba y se ca\u00eda. Pero esta vez se levant\u00f3 solito, con gran esfuerzo, apoy\u00e1ndose en un barril de aceite.\n\n\u2014\u00bfEst\u00e1s enfermo? \u2014dijo el sargento\u2014. Casi no puedes caminar, zambo. Pero de d\u00f3nde maldita sea ha podido salir un fantoche como t\u00fa.\n\nLo arrastr\u00f3 hacia la abertura, lo hizo agacharse y lo oblig\u00f3 a ganar la calle, delante de \u00e9l. El zambo segu\u00eda emitiendo ruidos, sin pausa, como si llevara un fierro en la boca y tratara de escupirlo. \u00abS\u00ed \u2014pens\u00f3 el sargento\u2014, es un loco\u00bb. La gar\u00faa hab\u00eda cesado pero ahora un viento fuerte y silbante barr\u00eda las calles y ululaba a su alrededor, mientras Lituma, dando empujoncitos al zambo para apresurarlo, enfilaba hacia la comisar\u00eda. Bajo su grueso capote, sinti\u00f3 fr\u00edo.\n\n\u2014Debes estar helado, compadre \u2014dijo Lituma\u2014. Calato con este tiempo, a estas horas. Si no te da una pulmon\u00eda ser\u00e1 un milagro.\n\nAl negro le casta\u00f1eteaban los dientes y caminaba con los brazos cruzados sobre el pecho, frot\u00e1ndose los flancos con sus manazas largas y huesudas, como si el fr\u00edo lo atacara sobre todo en las costillas. Segu\u00eda roncando, rugiendo o graznando, pero ahora para s\u00ed mismo, y torc\u00eda d\u00f3cilmente donde le se\u00f1alaba el sargento. No encontraron en las calles ni autom\u00f3viles, ni perros, ni borrachos. Cuando llegaron a la comisar\u00eda \u2014las luces de sus ventanas, con su resplandor aceitoso, alegraron a Lituma como a un n\u00e1ufrago que ve la playa\u2014 el bronco campanario de la Iglesia de Nuestra Se\u00f1ora del Carmen de la Legua daba las dos.\n\nAl ver aparecer al sargento con el negro desnudo, al joven y apuesto teniente Jaime Concha no se le cay\u00f3 el Pato Donald de las manos \u2014era el cuarto que llevaba le\u00eddo en la noche, aparte de tres Supermanes y dos Mandrakes\u2014, pero se le abri\u00f3 tanto la boca que por poco se desmandibula. Los guardias Camacho y Ar\u00e9valo, que estaban jugando una partidita de damas chinas, tambi\u00e9n abrieron mucho los ojos.\n\n\u2014\u00bfDe d\u00f3nde sacaste este espantap\u00e1jaros? \u2014dijo por fin el teniente.\n\n\u2014\u00bfEs hombre, animal o cosa? \u2014pregunt\u00f3 Manzanita Ar\u00e9valo, poni\u00e9ndose de pie y olfateando al negro. \u00c9ste, desde que hab\u00eda pisado la comisar\u00eda, estaba mudo y mov\u00eda la cabeza en todas direcciones, con una mueca de terror, como si por primera vez en su vida viera luz el\u00e9ctrica, m\u00e1quinas de escribir, guardias civiles. Pero, al ver acercarse a Manzanita, lanz\u00f3 otra vez su espeluznante alarido \u2014Lituma vio que el teniente Concha casi se ven\u00eda al suelo con silla y todo de la impresi\u00f3n y que Mocos Camacho desbarataba las damas chinas\u2014 e intent\u00f3 regresar a la calle. El sargento lo contuvo con una mano y lo sacudi\u00f3 un poco: \u00abQuieto, zambo, no te me asustes\u00bb.\n\n\u2014Lo encontr\u00e9 en el almac\u00e9n nuevo del Terminal, mi teniente \u2014dijo\u2014. Se meti\u00f3 fracturando el tabique. \u00bfHago el parte por robo, por invasi\u00f3n de propiedad, por conducta inmoral o por las tres cosas?\n\nEl zambo se hab\u00eda quedado otra vez encogido, mientras el teniente, Camacho y Ar\u00e9valo lo escudri\u00f1aban de pies a cabeza.\n\n\u2014Esas cicatrices no son de viruela, mi teniente \u2014dijo Manzanita, se\u00f1alando las incisiones de la cara y el cuerpo\u2014. Se las hicieron a navaja, aunque parezca mentira.\n\n\u2014Es el hombre m\u00e1s flaco que he visto en mi vida \u2014dijo Mocos, mirando los huesos del calato\u2014. Y el m\u00e1s feo. Dios m\u00edo, qu\u00e9 crespos tiene. Y qu\u00e9 patas.\n\n\u2014S\u00e1canos de la curiosidad \u2014dijo el teniente\u2014. Cu\u00e9ntanos tu vida, negrito.\n\nEl sargento Lituma se hab\u00eda quitado el quep\u00ed y desabotonado el capote. Sentado en la m\u00e1quina de escribir, comenzaba a redactar el parte. Desde ah\u00ed, grit\u00f3:\n\n\u2014No sabe hablar, mi teniente. Hace unos ruidos que no se entienden.\n\n\u2014\u00bfEres de los que se hacen los locos? \u2014se interes\u00f3 el teniente\u2014. Estamos viejos para que nos metan el dedo a la boca. Cu\u00e9ntanos qui\u00e9n eres, de d\u00f3nde sales, qui\u00e9n era tu mam\u00e1.\n\n\u2014O te devolvemos el habla a soplamocos \u2014a\u00f1adi\u00f3 Manzanita\u2014. A cantar como un canario, zambito.\n\n\u2014S\u00f3lo que si esas rayas fueran de chaveta, tendr\u00edan que haberle dado mil chavetazos \u2014se admir\u00f3 Mocos, mirando una y otra vez las incisiones que cuadriculaban al negro\u2014. \u00bfPero c\u00f3mo es posible que un hombre est\u00e9 marcado as\u00ed?\n\n\u2014Se muere de fr\u00edo \u2014dijo Manzanita\u2014. Le chocan los dientes como maracas.\n\n\u2014Las muelas \u2014lo corrigi\u00f3 Mocos, examin\u00e1ndolo como una hormiga, muy de cerca\u2014. \u00bfNo ves que no tiene sino un diente, este colmillo de elefante? Pucha, qu\u00e9 tipo: parece una pesadilla.\n\n\u2014Creo que es un chiflado \u2014dijo Lituma, sin dejar de escribir\u2014. Andar as\u00ed, en este fr\u00edo, no es cosa de cuerdos, \u00bfno, mi teniente?\n\nY, en este instante, el desbarajuste lo hizo mirar: el zambo, de pronto, electrizado por algo, hab\u00eda dado un empuj\u00f3n al teniente y pasaba como una flecha entre Camacho y Ar\u00e9valo. Pero no hacia la calle sino hacia la mesa de las damas chinas; Lituma vio que se precipitaba sobre un s\u00e1ndwich a medio comer y que se lo embut\u00eda y tragaba en un solo, afanoso y bestial movimiento. Cuando Ar\u00e9valo y Camacho llegaron hasta \u00e9l y le aventaron un par de sopapos, el negro estaba englutiendo, con la misma avidez, las sobras del otro s\u00e1ndwich.\n\n\u2014No le peguen, muchachos \u2014dijo el sargento\u2014. M\u00e1s bien conv\u00eddenle un caf\u00e9, sean caritativos.\n\n\u2014\u00c9sta no es la Beneficencia \u2014dijo el teniente\u2014. No s\u00e9 qu\u00e9 cuernos me voy a hacer con este sujeto aqu\u00ed \u2014se qued\u00f3 mirando al zambo, que, luego de tragarse los s\u00e1ndwiches, hab\u00eda recibido los coscorrones de Mocos y Manzanita sin inmutarse y permanec\u00eda ahora tumbado en el suelo, tranquilo, jadeando suavemente. Termin\u00f3 por compadecerse y gru\u00f1\u00f3\u2014: Est\u00e1 bien. Denle un poco de caf\u00e9 y m\u00e9tanlo al calabozo.\n\nEl Mocos le alcanz\u00f3 media taza de caf\u00e9 del termo. El zambo bebi\u00f3 despacio, cerrando los ojos, y, cuando hubo terminado, lami\u00f3 el aluminio en busca de las \u00faltimas gotitas hasta dejarlo brillante. Se dej\u00f3 llevar al calabozo pac\u00edficamente.\n\nLituma reley\u00f3 el parte: intento de robo, invasi\u00f3n de propiedad, conducta inmoral. El teniente Jaime Concha se hab\u00eda vuelto a sentar en el escritorio y su mirada vagabundeaba:\n\n\u2014Ya s\u00e9, ya s\u00e9 a qui\u00e9n se parece \u2014sonri\u00f3 feliz, mostrando a Lituma el alto de revistas multicolores\u2014. A los negros de las historias de Tarz\u00e1n, a los del \u00c1frica.\n\nCamacho y Ar\u00e9valo hab\u00edan reanudado la partida de damas chinas y Lituma se calz\u00f3 el quep\u00ed y aboton\u00f3 el capote. Cuando sal\u00eda, sinti\u00f3 los chillidos del carterista, que se acababa de despertar y protestaba por su compa\u00f1ero de calabozo:\n\n\u2014\u00a1Socorro, s\u00e1lvenme! \u00a1Me va a violar!\n\n\u2014C\u00e1llate, o te vamos a violar nosotros \u2014lo amonestaba el teniente\u2014. D\u00e9jame leer mis chistes en paz.\n\nDesde la calle, Lituma alcanz\u00f3 a ver que el negro se hab\u00eda tendido en el suelo, indiferente a los gritos del carterista, un chino delgadito que no sal\u00eda del susto. \u00abDespertarse y encontrarse con semejante cuco\u00bb, se re\u00eda Lituma, rompiendo otra vez con su maciza figura la niebla, el viento, las sombras. Con las manos en el bolsillo, las solapas del capote levantadas, cabizbajo, sin darse prisa, continu\u00f3 su ronda. Estuvo primero en la calle del chancro, donde encontr\u00f3 a Choclo Rom\u00e1n acodado en el mostrador del Happy Land, festejando los chistes de Paloma del Llanto, el viejo marica de pelo pintado y dientes postizos que hac\u00eda de barman. Consign\u00f3 en el parte que el guardia Rom\u00e1n \u00abten\u00eda trazas de haber ingerido bebidas espirituosas en horas de servicio\u00bb, aunque sab\u00eda de sobra que el teniente Concha, hombre lleno de comprensi\u00f3n para las debilidades propias y ajenas, har\u00eda la vista gorda. Se alej\u00f3 luego del mar y remont\u00f3 la avenida S\u00e1enz Pe\u00f1a, m\u00e1s muerta a esa hora que un cementerio, y le cost\u00f3 un triunfo dar con Humberto Quispe, que ten\u00eda el \u00e1rea del Mercado. Los puestos estaban cerrados y hab\u00eda menos vagabundos que otras veces, durmiendo acurrucados sobre costales o peri\u00f3dicos, bajo las escaleras y los camiones. Despu\u00e9s de varias vueltas in\u00fatiles y muchos toques de silbato con la se\u00f1al de reconocimiento, encontr\u00f3 a Quispe en la esquina de Col\u00f3n y Cochrane, ayudando a un taxista al que un par de forajidos acababan de romperle el cr\u00e1neo para robarle. Lo llevaron a la Asistencia P\u00fablica, para que lo cosieran. Luego, se tomaron un caldito de cabezas en el primer puesto que abri\u00f3, el de la se\u00f1ora Gualberta, vendedora de pescado fresco. Un patrullero recogi\u00f3 a Lituma en S\u00e1enz Pe\u00f1a y le dio un avent\u00f3n hasta la Fortaleza del Real Felipe, al pie de cuyas murallas hac\u00eda guardia Manitas Rodr\u00edguez, el benjam\u00edn de la comisar\u00eda. Le sorprendi\u00f3 jugando a la rayuela, solito, en la oscuridad. Saltaba muy serio de caj\u00f3n a caj\u00f3n, en un pie, en dos, y al ver al sargento se cuadr\u00f3:\n\n\u2014El ejercicio ayuda a entrar en calor \u2014le dijo, se\u00f1alando el dibujo hecho con tiza en la vereda\u2014: \u00bfUsted no jugaba de chico a la rayuela, mi sargento?\n\n\u2014M\u00e1s bien al trompo y era buenazo haciendo volar cometas \u2014le respondi\u00f3 Lituma.\n\nManitas Rodr\u00edguez le refiri\u00f3 un incidente que, dec\u00eda, le hab\u00eda alegrado la guardia. Estaba recorriendo la calle Paz Sold\u00e1n, a eso de la medianoche, cuando hab\u00eda visto a un sujeto trepando por una ventana. Le hab\u00eda dado el alto rev\u00f3lver en mano pero el tipo se puso a llorar jurando que no era ratero sino esposo y que su se\u00f1ora le ped\u00eda que entrara as\u00ed, a oscuritas y por la ventana. \u00bfY por qu\u00e9 no por la puerta, como todo el mundo? \u00abPorque est\u00e1 medio chiflada \u2014lloriqueaba el hombre\u2014. F\u00edjese que verme entrar como ladr\u00f3n la pone m\u00e1s cari\u00f1osa. Otras veces hace que la asuste con un cuchillo y hasta que me disfrace de diablo. Y si no le doy gusto no me liga ni un beso, mi agente\u00bb.\n\n\u2014Te vio cara de mocoso y se burl\u00f3 de ti de lo lindo \u2014se sonre\u00eda Lituma.\n\n\u2014Es la pura y santa verdad \u2014insisti\u00f3 Manitas\u2014. Toqu\u00e9 la puerta, entramos y la se\u00f1ora, una zambita de rompe y raja, dijo que era cierto y que si no ten\u00edan derecho ella y su marido a jugar a los ladrones. Lo que se ve en este oficio, \u00bfno, mi sargento?\n\n\u2014As\u00ed es, muchacho \u2014asinti\u00f3 Lituma, pensando en el negro.\n\n\u2014Ahora que con una mujer as\u00ed uno no se aburrir\u00eda nunca, mi sargento \u2014se chupaba los labios Manitas.\n\nAcompa\u00f1\u00f3 a Lituma hasta la avenida Buenos Aires y se despidieron. Mientras avanzaba hasta la frontera con Bellavista \u2014la calle Vigil, la plaza de la Guardia Chalaca\u2014, largo trayecto donde habitualmente comenzaba a sentir fatiga y sue\u00f1o, el sargento recordaba al negro. \u00bfSe habr\u00eda escapado del manicomio? Pero el Larco Herrera estaba tan lejos que alg\u00fan guardia o patrullero lo habr\u00eda visto y arrestado. \u00bfY esas cicatrices? \u00bfSe las habr\u00edan hecho a cuchillo? Mi\u00e9chica, eso s\u00ed que doler\u00eda, como quemarse a fuego lento. Que a uno le vayan haciendo heridita tras heridita hasta embadurnarle la cara de rayas, carambolas. \u00bfY si hab\u00eda nacido as\u00ed? Todav\u00eda era noche cerrada pero ya se percib\u00edan s\u00edntomas del amanecer: autos, uno que otro cami\u00f3n, siluetas madrugadoras. El sargento se preguntaba: \u00ab\u00bfY t\u00fa que has visto tanto tipo raro por qu\u00e9 te preocupa el calato?\u00bb. Se encogi\u00f3 de hombros: simple curiosidad, una manera de ocupar la mente mientras duraba la ronda.\n\nNo tuvo dificultad en dar con Z\u00e1rate, un guardia que hab\u00eda servido con \u00e9l en Ayacucho. Se lo encontr\u00f3 con el parte firmado: s\u00f3lo un choque sin heridos, nada importante. Lituma le cont\u00f3 la historia del negro y a Z\u00e1rate lo \u00fanico que le hizo gracia fue el episodio de los s\u00e1ndwiches. Ten\u00eda la man\u00eda de la filatelia, y, mientras acompa\u00f1aba unas cuadras al sargento, empez\u00f3 a contarle que esa ma\u00f1ana hab\u00eda conseguido unas estampillas triangulares de Etiop\u00eda, con leones y v\u00edboras, en verde, rojo y azul, que eran rar\u00edsimas, y que las hab\u00eda cambiado por cinco argentinas que no val\u00edan nada.\n\n\u2014Pero que, sin duda, se creer\u00e1n que valen mucho \u2014lo interrumpi\u00f3 Lituma.\n\nLa man\u00eda de Z\u00e1rate, que otras veces toleraba con buen humor, ahora lo impacient\u00f3 y se alegr\u00f3 de que se despidieran. Un resplandor azuloso se insinuaba en el cielo y de la negrura surg\u00edan, espectrales, gris\u00e1ceos, aherrumbrados, populosos, los edificios del Callao. Casi al trote, el sargento iba contando las cuadras que faltaban para llegar a la comisar\u00eda. Pero esta vez, se confes\u00f3 a s\u00ed mismo, su premura no se deb\u00eda tanto al cansancio de la noche y la caminata como a las ganas de ver otra vez al negro. \u00abParece que creyeras que todo ha sido un sue\u00f1o y que el cutato no existe, Lituma\u00bb.\n\nPero exist\u00eda: ah\u00ed estaba, durmiendo retorcido como un nudo en el suelo del calabozo. El carterista hab\u00eda ca\u00eddo dormido en el otro extremo, y a\u00fan llevaba en la cara una expresi\u00f3n de susto. Tambi\u00e9n los dem\u00e1s dorm\u00edan: el teniente Concha de bruces contra un alto de chistes y Camacho y Ar\u00e9valo hombro contra hombro, en la banqueta de la entrada. Lituma estuvo un largo rato contemplando al negro: sus huesos salientes, su pelo ensortijado, su gran jeta, su diente hu\u00e9rfano, sus mil cicatrices, los estremecimientos que lo recorr\u00edan. Pensaba: \u00abPero de d\u00f3nde has salido, zambo\u00bb. Por fin, entreg\u00f3 el parte al teniente, que abri\u00f3 unos ojos hinchados y enrojecidos:\n\n\u2014Ya se termina esta vaina \u2014le dijo, con boca pastosa\u2014. Un d\u00eda menos de servicio, Lituma.\n\n\u00abY un d\u00eda menos de vida, tambi\u00e9n\u00bb, pens\u00f3 el sargento. Se despidi\u00f3 haciendo sonar los tacos muy fuerte. Eran las seis de la ma\u00f1ana y estaba libre. Como siempre, se fue al Mercado, donde do\u00f1a Gualberta, a tomar una sopa hirviendo, unas empanadas, unos frejoles con arroz y un dulce de leche, y, despu\u00e9s, al cuartito donde viv\u00eda, en la calle Col\u00f3n. Se demor\u00f3 en pescar el sue\u00f1o, y, cuando lo pesc\u00f3, empez\u00f3 inmediatamente a so\u00f1ar con el negro. Lo ve\u00eda cercado de leones y v\u00edboras rojas, verdes y azules, en el coraz\u00f3n de Abisinia, con chistera, botas y una varita de domador. Las fieras hac\u00edan gracias al comp\u00e1s de su varita y una muchedumbre apostada entre las lianas, los troncos y el ramaje alegrado de cantos de p\u00e1jaros y chillidos de monos, lo aplaud\u00eda a rabiar. Pero, en vez de hacer una reverencia al p\u00fablico, el negro se pon\u00eda de rodillas, alargaba las manos en adem\u00e1n suplicante, los ojos se le aguaban y su gran jeta se abr\u00eda y, angustioso, raudo, tumultuoso, comenzaba a brotar el trabalenguas, su absurda m\u00fasica.\n\nLituma se despert\u00f3 a eso de las tres de la tarde, de mal humor y muy cansado, pese a haber dormido siete horas. \u00abYa se lo habr\u00e1n llevado a Lima\u00bb, pens\u00f3. Mientras se lavaba la cara como gato y se vest\u00eda, iba imaginando la trayectoria del negro: lo habr\u00eda recogido el patrullero de las nueve, le habr\u00edan dado un trapo para que se cubriera, lo habr\u00edan entregado en la Prefectura, le habr\u00edan abierto un expediente, lo habr\u00edan mandado al calabozo de los sin juicio, y ah\u00ed estar\u00eda ahora, en esa cueva oscura, entre los vagabundos, rateros, agresores y escandalosos de las \u00faltimas veinticuatro horas, temblando de fr\u00edo y muerto de hambre, rasc\u00e1ndose los piojos.\n\nEra un d\u00eda gris y h\u00famedo; entre la neblina las gentes se mov\u00edan como peces en aguas sucias y Lituma, pasito a paso, pensando, se fue a tomar lonche donde la se\u00f1ora Gualberta: dos panes con queso fresco y un caf\u00e9.\n\n\u2014Te noto raro, Lituma \u2014le dijo la se\u00f1ora Gualberta, una viejecita que conoc\u00eda la vida\u2014: \u00bfProblemas de dinero o de amor?\n\n\u2014Estoy pensando en un cutato que encontr\u00e9 anoche \u2014dijo el sargento, probando el caf\u00e9 con la puntita de la lengua\u2014. Se hab\u00eda metido a un dep\u00f3sito del Terminal.\n\n\u2014\u00bfY qu\u00e9 tiene de raro eso? \u2014pregunt\u00f3 do\u00f1a Gualberta.\n\n\u2014Estaba calato, lleno de cicatrices, el pelo como una selva y no sabe hablar \u2014le explic\u00f3 Lituma\u2014. \u00bfDe d\u00f3nde puede venir un tipo as\u00ed?\n\n\u2014Del infierno \u2014se ri\u00f3 la viejecita, recibi\u00e9ndole el billete.\n\nLituma se fue a la plaza Grau a encontrarse con Pedralbes, un cabo de la Marina. Se hab\u00edan conocido a\u00f1os atr\u00e1s, cuando el sargento era s\u00f3lo guardia y Pedralbes marinero raso, y serv\u00edan ambos en Pisco. Luego, sus respectivos destinos se hab\u00edan separado cerca de diez a\u00f1os, pero, desde hac\u00eda dos, se hab\u00edan juntado de nuevo. Pasaban sus d\u00edas de salida juntos y Lituma se sent\u00eda donde los Pedralbes en casa. Se fueron a La Punta, al Club de Cabos y Marineros, a tomarse una cerveza y a jugar al sapo. Lo primero que hizo el sargento fue contarle la historia del negro. Pedralbes encontr\u00f3 inmediatamente una explicaci\u00f3n:\n\n\u2014Es un salvaje del \u00c1frica que se vino de polizonte en un barco. Hizo el viaje escondido y, al llegar al Callao, se descolg\u00f3 de nochecita al agua y se meti\u00f3 al Per\u00fa de contrabando.\n\nA Lituma le pareci\u00f3 que sal\u00eda el sol: todo se volvi\u00f3 de pronto clar\u00edsimo.\n\n\u2014Tienes raz\u00f3n, eso es \u2014dijo, chasqueando la lengua, aplaudiendo\u2014. Se vino del \u00c1frica. Claro, eso es. Y, aqu\u00ed en el Callao, lo desembarcaron por alguna raz\u00f3n. Para no pagarle, porque lo descubrieron en la bodega, para librarse de \u00e9l.\n\n\u2014No lo entregaron a las autoridades porque sab\u00edan que no lo iban a recibir \u2014iba completando la historia Pedralbes\u2014. Lo desembarcaron a la fuerza: arr\u00e9glatelas solo, salvaje.\n\n\u2014O sea que el cutato ni siquiera sabe d\u00f3nde est\u00e1 \u2014dijo Lituma\u2014. O sea que esos ruidos no son de loco sino de salvaje, o sea que esos ruidos son su idioma.\n\n\u2014Es como si te metieras en un avi\u00f3n y desembarcaras en Marte, hermano \u2014lo ayudaba Pedralbes.\n\n\u2014Qu\u00e9 inteligentes somos \u2014dijo Lituma\u2014. Le descubrimos toda la vida al cutato.\n\n\u2014Qu\u00e9 inteligente soy yo, dir\u00e1s \u2014protest\u00f3 Pedralbes\u2014. \u00bfY ahora qu\u00e9 har\u00e1n con el negro?\n\nLituma pens\u00f3: \u00abQui\u00e9n sabe\u00bb. Jugaron seis partiditas de sapo y gan\u00f3 cuatro el sargento, de modo que Pedralbes pag\u00f3 la cerveza. Fueron luego a la calle Chanchamayo, donde viv\u00eda Pedralbes, en una casita de ventanas con barrotes. Domitila, la mujer de Pedralbes, estaba terminando de dar de comer a las tres criaturas, y, apenas los vio aparecer, meti\u00f3 en la cama al menorcito y orden\u00f3 a los otros dos que no se asomaran ni a la puerta. Se arregl\u00f3 un poco el pelo, les dio el brazo a cada uno, y salieron. Entraron al Cine Porte\u00f1o, en S\u00e1enz Pe\u00f1a, a ver una italiana. A Lituma y Pedralbes no les gust\u00f3, pero ella dijo que incluso la repetir\u00eda. Caminaron hasta la calle Chanchamayo \u2014las criaturas se hab\u00edan quedado dormidas\u2014 y Domitila les sirvi\u00f3 de comer unos olluquitos con charqui recalentados. Cuando Lituma se despidi\u00f3, eran las diez y media. Lleg\u00f3 a la Cuarta Comisar\u00eda a la hora que comenzaba su servicio: las once en punto.\n\nEl teniente Jaime Concha no le dio el menor respiro; lo llam\u00f3 aparte y le solt\u00f3 las instrucciones de golpe, en un par de frases espartanas que dejaron a Lituma mareado y con las orejas zumb\u00e1ndole.\n\n\u2014La superioridad sabe lo que hace \u2014le levant\u00f3 la moral el teniente, d\u00e1ndole una palmadita\u2014. Y tiene sus razones que hay que entender. La superioridad no se equivoca nunca, \u00bfno es as\u00ed, Lituma?\n\n\u2014Claro que no \u2014balbuce\u00f3 el sargento.\n\nManzanita y el Mocos se hac\u00edan los ocupados. Con el rabillo del ojo, Lituma ve\u00eda, a uno, revisando las papeletas de tr\u00e1nsito como si fueran fotos de calatas, y, al otro, arreglando, desarreglando y volviendo a arreglar su escritorio.\n\n\u2014\u00bfPuedo preguntar algo, mi teniente? \u2014dijo Lituma.\n\n\u2014Puedes \u2014dijo el teniente\u2014. Lo que no s\u00e9 es si yo podr\u00e9 contestarte.\n\n\u2014\u00bfPor qu\u00e9 la superioridad me ha elegido a m\u00ed para este trabajito?\n\n\u2014Eso s\u00ed te lo puedo decir \u2014dijo el teniente\u2014. Por dos razones. Porque t\u00fa lo capturaste y es justo que termine la broma el que la empez\u00f3. Y segundo: porque eres el mejor guardia de esta comisar\u00eda y tal vez del Callao.\n\n\u2014Honor que me hacen \u2014murmur\u00f3 Lituma, sin alegrarse lo m\u00e1s m\u00ednimo.\n\n\u2014La superioridad sabe muy bien que se trata de un trabajo dif\u00edcil y por eso te lo conf\u00eda \u2014dijo el teniente\u2014. Deber\u00edas sentirte orgulloso de que te hayan elegido entre los centenares de guardias que hay en Lima.\n\n\u2014Vaya, ahora resulta que encima tendr\u00eda que dar las gracias \u2014movi\u00f3 la cabeza Lituma, estupefacto. Reflexion\u00f3 un momento, y, en voz muy baja, a\u00f1adi\u00f3\u2014: \u00bfTiene que ser ahora mismo?\n\n\u2014Sobre el pucho \u2014dijo el teniente, tratando de parecer jovial\u2014. No dejes para ma\u00f1ana lo que puedes hacer hoy.\n\nLituma pens\u00f3: \u00abAhora ya sabes por qu\u00e9 no se te iba de la tutuma la cara del negro\u00bb.\n\n\u2014\u00bfQuieres llevarte a uno de \u00e9stos, para que te eche una mano? \u2014oy\u00f3 la voz del teniente.\n\nLituma sinti\u00f3 que Camacho y Ar\u00e9valo quedaban petrificados. Un silencio polar se instal\u00f3 en la comisar\u00eda mientras el sargento observaba a los dos guardias, y, deliberadamente, para hacerlos pasar un mal rato, se demoraba en elegir. Manzanita se hab\u00eda quedado con el alto de papeletas bailoteando entre los dedos y el Mocos con la cara hundida en el escritorio.\n\n\u2014A \u00e9ste \u2014dijo Lituma, se\u00f1alando a Ar\u00e9valo. Sinti\u00f3 que Camacho respiraba hondo y vio brotar en los ojos de Manzanita todo el odio del mundo contra \u00e9l y comprendi\u00f3 que le estaba mentando la madre.\n\n\u2014Estoy agripado y le iba a pedir que me exonerara de salir esta noche, mi teniente \u2014tartamude\u00f3 Ar\u00e9valo, poniendo cara de imb\u00e9cil.\n\n\u2014D\u00e9jate de mariconer\u00edas y ench\u00fafate el capote \u2014se adelant\u00f3 Lituma, pasando junto a \u00e9l sin mirarlo\u2014. Nos vamos de una vez.\n\nFue hasta el calabozo y lo abri\u00f3. Por primera vez en el d\u00eda, observ\u00f3 al negro. Le hab\u00edan puesto un pantal\u00f3n andrajoso, que apenas le llegaba a las rodillas, y cubr\u00eda su pecho y su espalda un costal de cargador, con un agujero para la cabeza. Estaba descalzo y tranquilo; mir\u00f3 a Lituma a los ojos, sin alegr\u00eda ni miedo. Sentado en el suelo, masticaba algo; en vez de esposas, ten\u00eda en las mu\u00f1ecas una cuerda, lo suficientemente larga para que pudiese rascarse o comer. El sargento le hizo se\u00f1as de que se pusiera de pie, pero el negro no pareci\u00f3 entender. Lituma se le acerc\u00f3, lo cogi\u00f3 del brazo, y el hombre se par\u00f3 d\u00f3cilmente. Camin\u00f3 delante de \u00e9l, con la misma indiferencia con que lo hab\u00eda recibido. Manzanita Ar\u00e9valo estaba ya con el capote puesto y la chalina enroscada en el cuello. El teniente Concha no se volvi\u00f3 a mirarlos partir: ten\u00eda la cara enterrada en un Pato Donald (\u00abpero no se da cuenta que est\u00e1 al rev\u00e9s\u00bb, pens\u00f3 Lituma). Camacho, en cambio, les hizo una sonrisa de p\u00e9same.\n\nYa en la calle, el sargento se coloc\u00f3 a la orilla de la pista y dej\u00f3 la pared a Ar\u00e9valo. El negro caminaba entre los dos, a su mismo paso, largo y desinteresado de todo, masticando.\n\n\u2014Hace como dos horas que masca ese pedazo de pan \u2014dijo Ar\u00e9valo\u2014. Esta noche, cuando lo trajeron de vuelta de Lima, le dimos todos los panes duros de la despensa, esos que se han vuelto piedras. Y se los ha comido todos. Masticando como una moledora. Qu\u00e9 hambre terrible, \u00bfno?\n\n\u00abEl deber primero y los sentimientos despu\u00e9s\u00bb, estaba pensando Lituma. Se fij\u00f3 el itinerario: subir por la calle Carlos Concha hasta Contralmirante Mora y, luego, bajar la avenida hasta el cauce del R\u00edmac y seguir con el r\u00edo hasta el mar. Calcul\u00f3: tres cuartos de hora para ir y volver, una hora a lo m\u00e1s.\n\n\u2014Usted tiene la culpa, mi sargento \u2014gru\u00f1\u00eda Ar\u00e9valo\u2014. Qui\u00e9n lo mand\u00f3 capturarlo. Al darse cuenta que no era ladr\u00f3n, debi\u00f3 dejarlo irse. Vea en qu\u00e9 l\u00edo nos meti\u00f3. Y ahora d\u00edgame, \u00bfusted se cree eso que piensa la superioridad? \u00bfQue \u00e9ste se vino escondido en un barco?\n\n\u2014Eso es tambi\u00e9n lo que se le ocurri\u00f3 a Pedralbes \u2014dijo Lituma\u2014. Puede que s\u00ed. Porque, si no, c\u00f3mo mi\u00e9chica te explicas que un tipo con esta pinta, con estos pelos, con estas marcas y calato y que habla esa chamuchina se aparezca de buenas a primeras en el puerto del Callao. Debe ser lo que dicen.\n\nEn la calle oscura resonaban los dos pares de botas de los guardias; los pies descalzos del zambo no hac\u00edan ning\u00fan ruido.\n\n\u2014Si de m\u00ed fuera, yo lo hubiera dejado en la c\u00e1rcel \u2014volvi\u00f3 a hablar Ar\u00e9valo\u2014. Porque, mi sargento, un salvaje del \u00c1frica no tiene la culpa de ser un salvaje del \u00c1frica.\n\n\u2014Por eso mismo no puede quedarse en la c\u00e1rcel \u2014murmur\u00f3 Lituma\u2014. Ya lo o\u00edste al teniente: la c\u00e1rcel es para los ladrones, asesinos y forajidos. \u00bfA cuento de qu\u00e9 lo va a mantener el Estado en la c\u00e1rcel?\n\n\u2014Entonces, deb\u00edan mandarlo de vuelta a su pa\u00eds \u2014refunfu\u00f1\u00f3 Ar\u00e9valo.\n\n\u2014\u00bfY c\u00f3mo mi\u00e9chica averiguas cu\u00e1l es su pa\u00eds? \u2014alz\u00f3 la voz Lituma\u2014. Ya lo has o\u00eddo al teniente. La superioridad trat\u00f3 de hablar con \u00e9l en todos los idiomas: el ingl\u00e9s, el franc\u00e9s, hasta el italiano. No habla idiomas: es salvaje.\n\n\u2014O sea que a usted le parece bien que por ser salvaje tengamos que pegarle un tiro \u2014volvi\u00f3 a gru\u00f1ir Manzanita Ar\u00e9valo.\n\n\u2014No estoy diciendo que me parezca bien \u2014murmur\u00f3 Lituma\u2014. Sino repitiendo lo que el teniente dijo que dice la superioridad. No seas idiota.\n\nEntraron a la avenida Contralmirante Mora cuando las campanas de Nuestra Se\u00f1ora del Carmen de la Legua daban las doce y el sonido le pareci\u00f3 a Lituma t\u00e9trico. Iba mirando adelante, empe\u00f1osamente, pero, a ratos, a pesar suyo, la cara se le volv\u00eda hacia la izquierda y echaba una ojeada al negro. Lo ve\u00eda, un segundo, cruzando el macilento cono de luz de alg\u00fan farol y siempre estaba igual: moviendo las mand\u00edbulas con seriedad y caminando al ritmo de ellos, sin el menor indicio de angustia. \u00abLo \u00fanico que parece importarle en el mundo es masticar\u00bb, pens\u00f3 Lituma. Y, un momento despu\u00e9s: \u00abEs un condenado a muerte que no sabe que lo es\u00bb. Y, casi inmediatamente: \u00abNo hay duda que es un salvaje\u00bb. En eso oy\u00f3 a Manzanita:\n\n\u2014Y, por \u00faltimo, por qu\u00e9 la superioridad no deja que se vaya por ah\u00ed y se las arregle como pueda \u2014rezongaba, malhumorado\u2014. Que sea otro vagabundo, de los muchos que hay en Lima. Uno m\u00e1s, uno menos, qu\u00e9 m\u00e1s da.\n\n\u2014Ya lo o\u00edste al teniente \u2014replic\u00f3 Lituma\u2014. La Guardia Civil no puede auspiciar el delito. Y si a \u00e9ste lo dejas suelto en plaza no tendr\u00eda m\u00e1s remedio que robar. O se morir\u00eda como un perro. En realidad, le estamos haciendo un favor. Un tiro es un segundo. Eso es preferible a irse muriendo de a poquitos, de hambre, de fr\u00edo, de soledad, de tristeza.\n\nPero Lituma sent\u00eda que su voz no era muy persuasiva y ten\u00eda la sensaci\u00f3n, al o\u00edrse, de estar oyendo a otra persona.\n\n\u2014Sea como sea, d\u00e9jeme decirle una cosa \u2014oy\u00f3 protestar a Manzanita\u2014. Esta vaina no me gusta y me hizo usted un flaco favor escogi\u00e9ndome.\n\n\u2014\u00bfY a m\u00ed crees que me gusta? \u2014murmur\u00f3 Lituma\u2014. \u00bfY no me hizo un flaco favor a m\u00ed la superioridad escogi\u00e9ndome?\n\nPasaron frente al Arsenal Naval, donde sonaba una sirena, y, al cruzar el descampado, a la altura del dique seco, un perro sali\u00f3 de las sombras a ladrarlos. Caminaron en silencio, oyendo el golpear de las botas contra la vereda, el rumor vecino del mar, sintiendo en las narices el aire h\u00famedo y salado.\n\n\u2014En este terreno vinieron a refugiarse unos gitanos el a\u00f1o pasado \u2014dijo Manzanita, de pronto, con la voz quebrada\u2014. Levantaron unas carpas y dieron una funci\u00f3n de circo. Le\u00edan la suerte y hac\u00edan magia. Pero el alcalde hizo que los corri\u00e9ramos porque no ten\u00edan licencia municipal.\n\nLituma no contest\u00f3. Sinti\u00f3 pena, de repente, no s\u00f3lo por el negro sino tambi\u00e9n por Manzanita y por los gitanos.\n\n\u2014\u00bfY lo vamos a dejar tirado ah\u00ed en la playa, para que lo picoteen los alcatraces? \u2014casi solloz\u00f3 Manzanita.\n\n\u2014Vamos a dejarlo en el basural, para que lo encuentren los camiones municipales, se lo lleven a la morgue y lo regalen a la Facultad de Medicina para que los estudiantes lo autopsien \u2014se enoj\u00f3 Lituma\u2014. O\u00edste muy bien las instrucciones, Ar\u00e9valo, no me las hagas repet\u00edrtelas.\n\n\u2014Las o\u00ed, pero no me pasa la idea de que tenemos que matarlo, as\u00ed, en fr\u00edo \u2014dijo Manzanita unos minutos despu\u00e9s\u2014. Y a usted tampoco, aunque trate. Por su voz me doy cuenta que tampoco est\u00e1 de acuerdo con esta orden.\n\n\u2014Nuestra obligaci\u00f3n no es estar de acuerdo con la orden, sino ejecutarla \u2014dijo d\u00e9bilmente el sargento. Y, luego de una pausa, todav\u00eda m\u00e1s despacio\u2014: Ahora que tienes raz\u00f3n. Yo tampoco estoy de acuerdo. Obedezco porque hay que obedecer.\n\nEn ese momento terminaron el asfalto, la avenida, los faroles, y comenzaron a andar en tinieblas sobre la tierra blanda. Una hediondez espesa, casi s\u00f3lida, los envolvi\u00f3. Estaban en los basurales de las orillas del R\u00edmac, muy cerca del mar, en ese cuadril\u00e1tero entre la playa, el lecho del r\u00edo y la avenida, donde los camiones de la Baja Polic\u00eda ven\u00edan, a partir de las seis de la ma\u00f1ana, a depositar los desperdicios de Bellavista, La Perla y el Callao y donde, aproximadamente desde la misma hora, una muchedumbre de ni\u00f1os, hombres, viejos y mujeres comenzaba a escarbar las inmundicias en busca de objetos de valor, y a disputar a las aves marinas, a los gallinazos, a los perros vagabundos los restos comestibles perdidos entre las basuras. Estaban muy cerca de ese desierto, camino a Ventanilla, a Anc\u00f3n, donde se alinean las f\u00e1bricas de harina de pescado del Callao.\n\n\u2014\u00c9ste es el mejor sitio \u2014dijo Lituma\u2014. Los camiones de la basura pasan todos por aqu\u00ed.\n\nEl mar sonaba muy fuerte. Manzanita se detuvo y el negro se detuvo tambi\u00e9n. Los guardias hab\u00edan prendido sus linternas y examinaban, en la temblona luz, la cara cuarteada de rayas, masticando inmutable.\n\n\u2014Lo peor es que no tiene reflejos ni adivina las cosas \u2014murmur\u00f3 Lituma\u2014. Cualquiera se dar\u00eda cuenta y se asustar\u00eda, tratar\u00eda de escapar. Lo que me friega es su tranquilidad, la confianza que nos tiene.\n\n\u2014Se me ocurre una cosa, mi sargento \u2014a Ar\u00e9valo le chocaban los dientes como si estuviera hel\u00e1ndose\u2014. Dej\u00e9moslo que se escape. Diremos que lo matamos y, en fin, cualquier cuento para explicar la desaparici\u00f3n del cad\u00e1ver...\n\nLituma hab\u00eda sacado su pistola y estaba quit\u00e1ndole el seguro.\n\n\u2014\u00bfTe atreves a proponerme que desobedezca las \u00f3rdenes de los superiores y que encima les mienta? \u2014reson\u00f3, tr\u00e9mula, la voz del sargento. Su mano derecha apuntaba el ca\u00f1o del arma hacia la sien del negro.\n\nPero pasaron dos, tres, varios segundos y no disparaba. \u00bfLo har\u00eda? \u00bfObedecer\u00eda? \u00bfEstallar\u00eda el disparo? \u00bfRodar\u00eda sobre las basuras indescifrables el misterioso inmigrante? \u00bfO le ser\u00eda perdonada la vida y huir\u00eda, ciego, salvaje, por las playas de las afueras, mientras un sargento irreprochable quedaba all\u00ed, en medio de p\u00fatridos olores y del vaiv\u00e9n de las olas, confuso y adolorido por haber faltado a su deber? \u00bfC\u00f3mo terminar\u00eda esa tragedia chalaca?\n\n### V\n\nEL PASO de Lucho Gatica por Lima fue adjetivado por Pascual en nuestros boletines como \u00absoberbio acontecimiento art\u00edstico y gran hit de la radiotelefon\u00eda nacional\u00bb. A m\u00ed la broma me cost\u00f3 un cuento, una corbata y una camisa casi nuevas, y dejar plantada a la t\u00eda Julia por segunda vez. Antes de la llegada del cantante de boleros chileno, hab\u00eda visto en los peri\u00f3dicos una proliferaci\u00f3n de fotos y de art\u00edculos laudatorios (\u00abpublicidad no pagada, la que vale m\u00e1s\u00bb, dec\u00eda Genaro hijo), pero s\u00f3lo me di cuenta cabal de su fama cuando not\u00e9 las colas de mujeres, en la calle Bel\u00e9n, esperando pases para la audici\u00f3n. Como el auditorio era peque\u00f1o \u2014un centenar de butacas\u2014 s\u00f3lo unas pocas pudieron asistir a los programas. La noche del estreno la aglomeraci\u00f3n en las puertas de Panamericana fue tal que Pascual y yo tuvimos que subir al altillo por un edificio vecino que compart\u00eda la azotea con el nuestro. Hicimos el bolet\u00edn de las siete y no hubo manera de bajarlo al segundo piso:\n\n\u2014Hay un chuchonal de mujeres tapando la escalera, la puerta y el ascensor \u2014me dijo Pascual\u2014. Trat\u00e9 de pedir permiso pero me creyeron un zamp\u00f3n.\n\nLlam\u00e9 por tel\u00e9fono a Genaro hijo y chisporroteaba de felicidad:\n\n\u2014Todav\u00eda falta una hora para la audici\u00f3n de Lucho y la gente ya ha parado el tr\u00e1fico en Bel\u00e9n. Todo el Per\u00fa sintoniza en este momento Radio Panamericana.\n\nLe pregunt\u00e9 si en vista de lo que ocurr\u00eda sacrific\u00e1bamos los boletines de las siete y de las ocho, pero \u00e9l ten\u00eda recursos para todo e invent\u00f3 que dict\u00e1ramos las noticias por tel\u00e9fono a los locutores. As\u00ed lo hicimos y, en los intervalos, Pascual escuchaba, embelesado, la voz de Lucho Gatica en la radio, y yo rele\u00eda la cuarta versi\u00f3n de mi cuento sobre el senador eunuco, al que hab\u00eda acabado por poner un t\u00edtulo de novela de horror: La cara averiada. A las nueve en punto escuchamos el fin del programa, la voz de Mart\u00ednez Morosini despidiendo a Lucho Gatica y la ovaci\u00f3n del p\u00fablico que, esta vez, no era de disco sino real. Diez segundos m\u00e1s tarde son\u00f3 el tel\u00e9fono y o\u00ed la voz alarmada de Genaro hijo:\n\n\u2014Bajen como sea, esto se est\u00e1 poniendo color de hormiga.\n\nNos cost\u00f3 un triunfo perforar el muro de mujeres api\u00f1adas en la escalera, a las que conten\u00eda, en la puerta del auditorio, el corpulento portero Jesusito. Pascual gritaba: \u00ab\u00a1Ambulancia! \u00a1Ambulancia! \u00a1Buscamos a un herido!\u00bb. Las mujeres, la mayor\u00eda j\u00f3venes, nos miraban con indiferencia o sonre\u00edan, pero no se apartaban y hab\u00eda que empujarlas. Adentro, nos recibi\u00f3 un espect\u00e1culo desconcertante: el celebrado artista reclamaba protecci\u00f3n policial. Era bajito y estaba l\u00edvido y lleno de odio hacia sus admiradoras. El empresario progresista procuraba calmarlo, le dec\u00eda que llamar a la polic\u00eda causar\u00eda p\u00e9sima impresi\u00f3n, esa nube de muchachas era un homenaje a su talento. Pero la celebridad no se dejaba convencer:\n\n\u2014Yo las conozco a \u00e9sas \u2014dec\u00eda, entre aterrado y furibundo\u2014. Comienzan pidiendo un aut\u00f3grafo y acaban ara\u00f1ando, mordiendo.\n\nNosotros nos re\u00edamos, pero la realidad confirm\u00f3 sus predicciones. Genaro hijo decidi\u00f3 que esper\u00e1ramos media hora, creyendo que las admiradoras, aburridas, se ir\u00edan. A las diez y cuarto (yo ten\u00eda compromiso con la t\u00eda Julia para ir al cine) nos hab\u00edamos cansado de esperar que ellas se cansaran y acordamos salir. Genaro hijo, Pascual, Jesusito, Mart\u00ednez Morosini y yo formamos un c\u00edrculo, cogidos de los brazos, y pusimos en el centro a la celebridad, cuya palidez se acentu\u00f3 hasta la blancura apenas abrimos la puerta. Pudimos bajar las primeras gradas sin grandes da\u00f1os, dando codazos, rodillazos, cabezazos y pechazos contra el mar femenino, que por el momento se contentaba con aplaudir, suspirar y estirar las manos para tocar al \u00eddolo \u2014quien, n\u00edveo, sonre\u00eda, e iba murmurando entre dientes: \u00abCuidadito con soltar los brazos, compa\u00f1eros\u00bb\u2014, pero pronto tuvimos que hacer frente a una agresi\u00f3n en regla. Nos cog\u00edan de la ropa y jaloneaban, y dando aullidos alargaban las u\u00f1as para arrancar pedazos de la camisa y el terno del \u00eddolo. Cuando, luego de diez minutos de asfixia y empujones, llegamos al pasillo de la entrada, cre\u00ed que nos \u00edbamos a soltar y tuve una visi\u00f3n: el peque\u00f1o cantante de boleros nos era arrebatado y sus admiradoras lo desmenuzaban ante nuestros ojos. No sucedi\u00f3, pero cuando lo metimos al auto de Genaro pap\u00e1, quien esperaba al volante desde hac\u00eda hora y media, Lucho Gatica y su guardia de hierro est\u00e1bamos convertidos en los sobrevivientes de una cat\u00e1strofe. A m\u00ed me hab\u00edan arranchado la corbata y hecho jirones la camisa, a Jesusito le hab\u00edan roto el uniforme y robado la gorra y Genaro hijo ten\u00eda amoratada la frente de un carterazo. El astro estaba indemne, pero de su ropa s\u00f3lo conservaba \u00edntegros los zapatos y los calzoncillos. Al d\u00eda siguiente, mientras tom\u00e1bamos nuestro cafecito de las diez en el Bransa, le cont\u00e9 a Pedro Camacho las haza\u00f1as de las admiradoras. No le sorprendieron en absoluto:\n\n\u2014Mi joven amigo \u2014me dijo, filos\u00f3ficamente, mir\u00e1ndome desde muy lejos\u2014, tambi\u00e9n la m\u00fasica llega al alma de la multitud.\n\nMientras yo luchaba por defender la integridad f\u00edsica de Lucho Gatica, la se\u00f1ora Agradecida hab\u00eda hecho la limpieza del altillo y echado a la basura la cuarta versi\u00f3n de mi cuento sobre el senador. En vez de amargarme, me sent\u00ed liberado de un peso y deduje que hab\u00eda en esto una advertencia de los dioses. Cuando le comuniqu\u00e9 a Javier que no lo reescribir\u00eda m\u00e1s, \u00e9l, en vez de tratar de disuadirme, me felicit\u00f3 por mi decisi\u00f3n.\n\nLa t\u00eda Julia se divirti\u00f3 mucho con mi experiencia de guardaespaldas. Nos ve\u00edamos casi a diario, desde la noche de los besos furtivos en el Grill Bol\u00edvar. Al d\u00eda siguiente del cumplea\u00f1os del t\u00edo Lucho yo me hab\u00eda presentado intempestivamente en la casa de Armend\u00e1riz y, buena suerte, la t\u00eda Julia estaba sola.\n\n\u2014Se fueron a visitar a tu t\u00eda Hortensia \u2014me dijo, haci\u00e9ndome pasar a la sala\u2014. No fui, porque ya s\u00e9 que esa chismosa se pasa la vida invent\u00e1ndome historias.\n\nLa tom\u00e9 de la cintura, la atraje hacia m\u00ed e intent\u00e9 besarla. No me rechaz\u00f3 pero tampoco me bes\u00f3: sent\u00ed su boca fr\u00eda contra la m\u00eda. Al apartarnos, vi que me miraba sin sonre\u00edr. No sorprendida como la v\u00edspera, m\u00e1s bien con cierta curiosidad y algo de burla.\n\n\u2014Mira, Marito \u2014su voz era afectuosa, tranquila\u2014. He hecho todas las locuras del mundo en mi vida. Pero \u00e9sta no la voy a hacer \u2014lanz\u00f3 una carcajada\u2014: \u00bfYo, corruptora de menores? \u00a1Eso s\u00ed que no!\n\nNos sentamos y estuvimos conversando cerca de dos horas. Le cont\u00e9 toda mi vida, no la pasada sino la que tendr\u00eda en el futuro, cuando viviera en Par\u00eds y fuera escritor. Le dije que quer\u00eda escribir desde que hab\u00eda le\u00eddo por primera vez a Alejandro Dumas, y que, desde entonces, so\u00f1aba con viajar a Francia y vivir en una buhardilla, en el barrio de los artistas, entregado totalmente a la literatura, la cosa m\u00e1s formidable del mundo. Le cont\u00e9 que estudiaba Derecho para darle gusto a la familia, pero que la abogac\u00eda me parec\u00eda la m\u00e1s espesa y boba de las profesiones y que no la practicar\u00eda jam\u00e1s. Me di cuenta, en un momento, que estaba hablando de manera muy fogosa y le dije que por primera vez le confesaba esas cosas \u00edntimas no a un amigo sino a una mujer.\n\n\u2014Te parezco tu mam\u00e1 y por eso te provoca hacerme confidencias \u2014me psicoanaliz\u00f3 la t\u00eda Julia\u2014. As\u00ed que el hijo de Dorita result\u00f3 bohemio, vaya, vaya. Lo malo es que te vas a morir de hambre, hijito.\n\nMe cont\u00f3 que la noche anterior se hab\u00eda quedado desvelada, pensando en los besos furtivos del Grill Bol\u00edvar. Que el hijo de Dorita, el chiquito al que s\u00f3lo ayer ella hab\u00eda acompa\u00f1ado a su mam\u00e1 a llevar al Colegio La Salle, en Cochabamba, el mocosito al que ella todav\u00eda cre\u00eda de pantal\u00f3n corto, la guagua con quien se hac\u00eda escoltar al cine para no ir sola, de buenas a primeras la besara en la boca como si fuera un hombre hecho y derecho, no le cab\u00eda en la cabeza.\n\n\u2014Soy un hombre hecho y derecho \u2014le asegur\u00e9, cogi\u00e9ndole la mano, bes\u00e1ndosela\u2014. Tengo dieciocho a\u00f1os. Y ya hace cinco que perd\u00ed la virginidad.\n\n\u2014\u00bfY qu\u00e9 soy yo entonces, que tengo treinta y dos y que la perd\u00ed hace quince? \u2014se ri\u00f3 ella\u2014. \u00a1Una vieja decr\u00e9pita!\n\nTen\u00eda una risa ronca y fuerte, directa y alegre, que abr\u00eda de par en par su boca grande, de labios gruesos, y que le arrugaba los ojos. Me miraba con iron\u00eda y malicia, todav\u00eda no como a un hombre hecho y derecho, pero ya no como a un mocoso. Se levant\u00f3 para servirme un whisky:\n\n\u2014Despu\u00e9s de tus atrevimientos de anoche, ya no puedo convidarte Coca-Colas \u2014me dijo, haci\u00e9ndose la apenada\u2014. Tengo que atenderte como a uno de mis pretendientes.\n\nLe dije que la diferencia de edad tampoco era tan terrible.\n\n\u2014Tan terrible no \u2014me repuso\u2014. Pero, casi casi, lo justo para que pudieras ser mi hijo.\n\nMe cont\u00f3 la historia de su matrimonio. Los primeros a\u00f1os todo hab\u00eda ido muy bien. Su marido ten\u00eda una hacienda en el altiplano y ella se hab\u00eda acostumbrado tanto a la vida de campo que rara vez iba a La Paz. La casa hacienda era muy c\u00f3moda y a ella le encantaba la tranquilidad del lugar, la vida sana y simple: montar a caballo, hacer excursiones, asistir a las fiestas de los indios. Las nubes grises hab\u00edan comenzado porque no pod\u00eda concebir; su marido sufr\u00eda con la idea de no tener descendencia. Luego, \u00e9l hab\u00eda comenzado a beber y desde entonces el matrimonio se hab\u00eda deslizado por una pendiente de ri\u00f1as, separaciones y reconciliaciones, hasta la disputa final. Luego del divorcio hab\u00edan quedado buenos amigos.\n\n\u2014Si alguna vez me caso, yo nunca tendr\u00eda hijos \u2014le advert\u00ed\u2014. Los hijos y la literatura son incompatibles.\n\n\u2014\u00bfQuiere decir que puedo presentar mi solicitud y ponerme a la cola? \u2014me coquete\u00f3 la t\u00eda Julia.\n\nTen\u00eda chispa y rapidez para las r\u00e9plicas, contaba cuentos colorados con gracia y era (como todas las mujeres que hab\u00eda conocido hasta entonces) terriblemente aliteraria. Daba la impresi\u00f3n de que en las largas horas vac\u00edas de la hacienda boliviana s\u00f3lo hab\u00eda le\u00eddo revistas argentinas, alguno que otro engendro de Delly, y apenas un par de novelas que consideraba memorables: El \u00e1rabe y El hijo del \u00e1rabe, de un tal H. M. Hull. Al despedirme esa noche le pregunt\u00e9 si pod\u00edamos ir al cine y me dijo que \u00abeso s\u00ed\u00bb. Hab\u00edamos ido a funci\u00f3n de noche, desde entonces, casi a diario, y, adem\u00e1s de soportar una buena cantidad de melodramas mexicanos y argentinos, nos hab\u00edamos dado una considerable cantidad de besos. El cine se fue convirtiendo en pretexto; eleg\u00edamos los m\u00e1s alejados de la casa de Armend\u00e1riz (el Montecarlo, el Colina, el Marsano) para estar juntos m\u00e1s tiempo. D\u00e1bamos largas caminatas despu\u00e9s de la funci\u00f3n, haciendo empanaditas (me hab\u00eda ense\u00f1ado que cogerse de las manos se dec\u00eda en Bolivia \u00abhacer empanaditas\u00bb), zigzagueando por las calles vac\u00edas de Miraflores (nos solt\u00e1bamos cada vez que aparec\u00eda un peat\u00f3n o un auto), conversando sobre todas las cosas, mientras que \u2014era esa estaci\u00f3n mediocre que en Lima llaman invierno\u2014 la gar\u00faa nos iba humedeciendo. La t\u00eda Julia sal\u00eda siempre, a almorzar o a tomar t\u00e9, con sus numerosos pretendientes, pero me reservaba las noches. \u00cdbamos al cine, en efecto, a sentarnos en las filas de atr\u00e1s de la platea, donde (sobre todo si la pel\u00edcula era muy mala) pod\u00edamos besarnos sin estorbar a otros espectadores y sin que alguien nos reconociera. Nuestra relaci\u00f3n se hab\u00eda estabilizado r\u00e1pidamente en lo amorfo, se situaba en alg\u00fan punto indefinible entre las categor\u00edas opuestas de enamorados y amantes. \u00c9ste era un tema recurrente de nuestras conversaciones. Ten\u00edamos de amantes la clandestinidad, el temor a ser descubiertos, la sensaci\u00f3n de riesgo, pero lo \u00e9ramos espiritual, no materialmente, pues no hac\u00edamos el amor (y, como se escandalizar\u00eda m\u00e1s tarde Javier, ni siquiera \u00abnos toc\u00e1bamos\u00bb). Ten\u00edamos de enamorados el respeto de ciertos ritos cl\u00e1sicos de la adolescente pareja miraflorina de ese tiempo (ir al cine, besarse durante la pel\u00edcula, caminar por la calle de la mano) y la conducta casta (en esa Edad de Piedra las chicas de Miraflores sol\u00edan llegar v\u00edrgenes al matrimonio y s\u00f3lo se dejaban tocar los senos y el sexo cuando el enamorado ascend\u00eda al estatuto formal de novio), pero \u00bfc\u00f3mo hubi\u00e9ramos podido serlo dada la diferencia de edad y el parentesco? En vista de lo ambiguo y extravagante de nuestro romance, jug\u00e1bamos a bautizarlo: \u00abNoviazgo ingl\u00e9s\u00bb, \u00abromance sueco\u00bb, \u00abdrama turco\u00bb.\n\n\u2014Los amores de un bebe y una anciana que, adem\u00e1s, es algo as\u00ed como su t\u00eda \u2014me dijo una noche la t\u00eda Julia, mientras cruz\u00e1bamos el parque Central\u2014. Cabalito para un radioteatro de Pedro Camacho.\n\nLe record\u00e9 que s\u00f3lo era mi t\u00eda pol\u00edtica y ella me cont\u00f3 que, en el radioteatro de las tres, un muchacho de San Isidro, buenmos\u00edsimo y gran corredor de tabla hawaiana, ten\u00eda relaciones nada menos que con su hermana, a la que, horror de horrores, hab\u00eda dejado embarazada.\n\n\u2014\u00bfDesde cu\u00e1ndo oyes radioteatros? \u2014le pregunt\u00e9.\n\n\u2014Me he contagiado de mi hermana \u2014me repuso\u2014. La verdad es que \u00e9sos de Radio Central son fant\u00e1sticos, unos dramones que parten el alma.\n\nY me confes\u00f3 que, a veces, a ella y a la t\u00eda Olga se les llenaban los ojos de l\u00e1grimas. Fue el primer indicio que tuve del impacto que causaba en los hogares lime\u00f1os la pluma de Pedro Camacho. Recog\u00ed otros, los d\u00edas siguientes, en las casas de la familia. Ca\u00eda donde la t\u00eda Laura y ella, apenas me ve\u00eda en el umbral de la sala, me ordenaba silencio con un dedo en los labios, mientras permanec\u00eda inclinada hacia el aparato de radio como para poder no s\u00f3lo o\u00edr sino tambi\u00e9n oler, tocar, la (tr\u00e9mula o r\u00edspida o ardiente o cristalina) voz del artista boliviano. Aparec\u00eda donde la t\u00eda Gaby y las encontraba a ella y a la t\u00eda Hortensia deshaciendo un ovillo con dedos absortos, mientras segu\u00edan un di\u00e1logo lleno de esdr\u00fajulas y gerundios de Luciano Pando y Josefina S\u00e1nchez. Y en mi propia casa, mis abuelos, que siempre hab\u00edan tenido \u00abafici\u00f3n a las novelitas\u00bb, como dec\u00eda la abuela Carmen, ahora hab\u00edan contra\u00eddo una aut\u00e9ntica pasi\u00f3n radioteatral. Me despertaba en la ma\u00f1ana oyendo los compases del indicativo de la radio \u2014se preparaban con una anticipaci\u00f3n enfermiza para el primer radioteatro, el de las diez\u2014, almorzaba oyendo el de las dos de la tarde, y a cualquier hora del d\u00eda que volviera, encontraba a los dos viejitos y a la cocinera, arrinconados en la salita de recibo, profundamente concentrados en la radio, que era grande y pesada como un aparador y que, para mal de males, siempre pon\u00edan a todo volumen.\n\n\u2014\u00bfPor qu\u00e9 te gustan tanto los radioteatros? \u2014le pregunt\u00e9 un d\u00eda a la abuelita\u2014. \u00bfQu\u00e9 tienen que no tengan los libros, por ejemplo?\n\n\u2014Es una cosa m\u00e1s viva, o\u00edr hablar a los personajes es m\u00e1s real \u2014me explic\u00f3, despu\u00e9s de reflexionar\u2014. Y, adem\u00e1s, a mis a\u00f1os, se portan mejor los o\u00eddos que la vista.\n\nIntent\u00e9 una averiguaci\u00f3n parecida en otras casas de parientes y los resultados fueron vagos. A las t\u00edas Gaby, Laura, Olga y Hortensia los radioteatros les gustaban porque eran entretenidos, tristes o fuertes, porque las distra\u00edan y hac\u00edan so\u00f1ar, vivir cosas imposibles en la vida real, porque ense\u00f1aban algunas verdades o porque una ten\u00eda siempre su poquito de esp\u00edritu rom\u00e1ntico. Cuando les pregunt\u00e9 por qu\u00e9 les gustaban m\u00e1s que los libros, protestaron: qu\u00e9 tonter\u00eda, c\u00f3mo se iba a comparar, los libros eran la cultura, los radioteatros simples adefesios para pasar el tiempo. Pero lo cierto es que viv\u00edan pegadas a la radio y que jam\u00e1s hab\u00eda visto a ninguna de ellas abrir un libro. En nuestras andanzas nocturnas, la t\u00eda Julia me resum\u00eda a veces algunos episodios que la hab\u00edan impresionado y yo le contaba mis conversaciones con el escriba, de modo que, insensiblemente, Pedro Camacho pas\u00f3 a ser un componente de nuestro romance.\n\nEl propio Genaro hijo me confirm\u00f3 el \u00e9xito de los nuevos radioteatros el d\u00eda en que por fin consegu\u00ed, despu\u00e9s de mil protestas, que me repusieran la m\u00e1quina de escribir. Se present\u00f3 en el altillo con una carpeta en la mano y la cara radiante:\n\n\u2014Supera los c\u00e1lculos m\u00e1s optimistas \u2014nos dijo\u2014. En dos semanas ha aumentado en veinte por ciento la sinton\u00eda de los radioteatros. \u00bfSaben lo que esto significa? \u00a1Aumentar en veinte por ciento la factura a los auspiciadores!\n\n\u2014\u00bfY que nos aumentar\u00e1n en veinte por ciento el sueldo, don Genaro? \u2014salt\u00f3 en su silla Pascual.\n\n\u2014Ustedes no trabajan en Radio Central sino en Panamericana \u2014nos record\u00f3 Genaro hijo\u2014. Nosotros somos una estaci\u00f3n de buen gusto y no pasamos radioteatros.\n\nLos diarios, en las p\u00e1ginas especializadas, pronto se hicieron eco de la audiencia conquistada por los nuevos radioteatros y empezaron a elogiar a Pedro Camacho. Fue Guido Monteverde quien lo consagr\u00f3, en su columna de \u00daltima Hora, llam\u00e1ndolo \u00abexperimentado libretista de imaginaci\u00f3n tropical y palabra rom\u00e1ntica, intr\u00e9pido director sinf\u00f3nico de radioteatros y vers\u00e1til actor \u00e9l mismo de acaramelada voz\u00bb. Pero el beneficiario de estos adjetivos no se daba por enterado de las olas de entusiasmo que iba levantando a su alrededor. Una de esas ma\u00f1anas en que yo lo recog\u00eda, de paso al Bransa, para tomar un caf\u00e9 juntos, encontr\u00e9 pegado en la ventana de su cub\u00edculo un cartel con esta inscripci\u00f3n escrita en letras toscas: \u00abNo se reciben periodistas ni se conceden aut\u00f3grafos. \u00a1El artista trabaja! \u00a1Respetadlo!\u00bb.\n\n\u2014\u00bfEso va en serio o en broma? \u2014le pregunt\u00e9, mientras yo saboreaba mi caf\u00e9 cortado y \u00e9l su compuesto cerebral de yerbaluisa y menta.\n\n\u2014Muy en serio \u2014me contest\u00f3\u2014. La poligraf\u00eda local ha comenzado a atosigarme, y si no les pongo un paral\u00e9, pronto habr\u00e1 colas de oyentes por ah\u00ed \u2014se\u00f1al\u00f3 como quien no quiere la cosa hacia la plaza San Mart\u00edn\u2014, pidiendo fotograf\u00edas y firmas. Mi tiempo vale oro y no puedo perderlo en necedades.\n\nNo hab\u00eda un \u00e1tomo de vanidad en lo que dec\u00eda, s\u00f3lo sincera inquietud. Vest\u00eda su terno negro habitual, su corbatita de lazo y fumaba unos cigarrillos pestilentes llamados Aviaci\u00f3n. Como siempre, estaba sumamente serio. Cre\u00ed halagarlo cont\u00e1ndole que todas mis t\u00edas hab\u00edan pasado a ser fan\u00e1ticas oyentes suyas y que Genaro hijo rebotaba de alegr\u00eda con los resultados de los surveys sobre la sinton\u00eda de sus radioteatros. Pero me hizo callar, aburrido, como si todas esas cosas fueran inevitables y \u00e9l las hubiera sabido desde siempre, y, m\u00e1s bien, me comunic\u00f3 que estaba indignado por la falta de sensibilidad de los mercaderes (expresi\u00f3n con la que, a partir de entonces, se referir\u00eda siempre a los Genaros).\n\n\u2014Algo est\u00e1 flaqueando en los radioteatros y mi obligaci\u00f3n es remediarlo y la de ellos ayudarme \u2014afirm\u00f3, frunciendo el ce\u00f1o\u2014. Pero est\u00e1 visto que el arte y la bolsa son enemigos mortales, como los chanchos y las margaritas.\n\n\u2014\u00bfFlaqueando? \u2014me asombr\u00e9\u2014. \u00a1Pero si son todo un \u00e9xito!\n\n\u2014Los mercaderes no quieren despedir a Pablito, pese a que yo lo he exigido \u2014me explic\u00f3\u2014. Por consideraciones sentimentales, que lleva no s\u00e9 cu\u00e1ntos a\u00f1os en Radio Central y tonter\u00edas as\u00ed. Como si el arte tuviera que ver algo con la caridad. \u00a1La incompetencia de ese enfermo es un verdadero sabotaje a mi trabajo!\n\nEl Gran Pablito era uno de esos personajes pintorescos e indefinibles que atrae o fabrica el ambiente de la radio. El diminutivo suger\u00eda que se trataba de un chiquillo, pero era un cholo cincuent\u00f3n, que arrastraba los pies y ten\u00eda unos ataques de asma que levantaban miasmas a su alrededor. Merodeaba ma\u00f1ana y tarde por Radio Central y Panamericana, haciendo de todo, desde echar una mano a los barrenderos e ir a comprar entradas al cine y a los toros a los Genaros, hasta repartir pases para las audiciones. Su trabajo m\u00e1s permanente eran los radioteatros, donde se encargaba de los efectos especiales.\n\n\u2014\u00c9stos creen que los efectos especiales son una mariconada que cualquier mendigo puede hacer \u2014despotricaba, aristocr\u00e1tico y helado, Pedro Camacho\u2014. En realidad tambi\u00e9n son arte, \u00bfy qu\u00e9 sabe de arte el braquic\u00e9falo medio moribundo de Pablito?\n\nMe asegur\u00f3 que, \u00abllegado el caso\u00bb, no vacilar\u00eda en suprimir con sus propias manos cualquier obst\u00e1culo a la \u00abperfecci\u00f3n de su trabajo\u00bb (y lo dijo de tal modo que le cre\u00ed). Compungido, a\u00f1adi\u00f3 que no ten\u00eda tiempo para formar un t\u00e9cnico en efectos especiales, ense\u00f1\u00e1ndole desde la a hasta la z, pero que, luego de una r\u00e1pida exploraci\u00f3n del dial nativo, hab\u00eda encontrado lo que buscaba. Baj\u00f3 la voz, ech\u00f3 un vistazo en torno y concluy\u00f3, mefistof\u00e9licamente:\n\n\u2014El elemento que nos conviene est\u00e1 en Radio Victoria.\n\nAnalizamos con Javier las posibilidades que ten\u00eda Pedro Camacho de materializar sus prop\u00f3sitos homicidas con el Gran Pablito y coincidimos en que la suerte de \u00e9ste depend\u00eda exclusivamente de los surveys: si la progresi\u00f3n de los radioteatros se manten\u00eda, ser\u00eda sacrificado sin misericordia. En efecto, no hab\u00eda pasado una semana cuando Genaro hijo se present\u00f3 en el altillo, sorprendi\u00e9ndome en plena redacci\u00f3n de un nuevo cuento \u2014debi\u00f3 notar mi confusi\u00f3n, la velocidad con que arranqu\u00e9 la p\u00e1gina de la m\u00e1quina de escribir y la entrever\u00e9 con los boletines, pero tuvo la delicadeza de no decir nada\u2014, y se dirigi\u00f3 conjuntamente a Pascual y a m\u00ed con un gesto de gran mecenas:\n\n\u2014Tanto quejarse ya consiguieron el redactor nuevo que quer\u00edan, par de flojos. El Gran Pablito trabajar\u00e1 con ustedes. \u00a1No se duerman sobre sus laureles!\n\nEl refuerzo que recibi\u00f3 el Servicio de Informaciones fue m\u00e1s moral que material, porque a la ma\u00f1ana siguiente, cuando, puntual\u00edsimo, el Gran Pablito se present\u00f3 a las siete en la oficina, y me pregunt\u00f3 qu\u00e9 deb\u00eda hacer y le encargu\u00e9 dar la vuelta a una rese\u00f1a parlamentaria, puso cara de espanto, tuvo un acceso de tos que lo dej\u00f3 amoratado, y tartamude\u00f3 que era imposible:\n\n\u2014Si yo no s\u00e9 leer ni escribir, se\u00f1or.\n\nApreci\u00e9 como fina muestra del esp\u00edritu risue\u00f1o de Genaro hijo el que nos hubiera elegido para nuevo redactor a un analfabeto. Pascual, a quien el saber que la redacci\u00f3n se bifurcar\u00eda entre \u00e9l y el Gran Pablito hab\u00eda puesto nervioso, recibi\u00f3 la noticia del analfabetismo con franca alegr\u00eda. En mi delante ri\u00f1\u00f3 a su flamante colega por su esp\u00edritu ap\u00e1tico, por no haber sido capaz de educarse, como hab\u00eda hecho \u00e9l, ya adulto, yendo a los cursos gratuitos de la nocturna. El Gran Pablito, muy asustado, asent\u00eda, repitiendo como un aut\u00f3mata \u00abes verdad, no hab\u00eda pensado en eso, as\u00ed es, tiene usted toda la raz\u00f3n\u00bb, y mir\u00e1ndome con cara de inminente despedido. Lo tranquilic\u00e9, dici\u00e9ndole que se encargar\u00eda de bajar los boletines a los locutores. En realidad, se convirti\u00f3 en un esclavo de Pascual, quien lo ten\u00eda todo el d\u00eda trotando del altillo a la calle y viceversa, para que le trajera cigarrillos, o unas papas rellenas que vend\u00eda un ambulante de la calle Carabaya y hasta yendo a ver si llov\u00eda. El Gran Pablito soportaba su servidumbre con excelente esp\u00edritu de sacrificio e incluso demostraba m\u00e1s respeto y amistad hacia su torturador que hacia m\u00ed. Cuando no estaba haciendo mandados de Pascual, se encog\u00eda en un rinc\u00f3n de la oficina, y, apoyando la cabeza en la pared, se dorm\u00eda instant\u00e1neamente. Roncaba con unos ronquidos sincr\u00f3nicos y sibilantes, de ventilador enmohecido.\n\nEra un esp\u00edritu generoso. No le guardaba el m\u00e1s m\u00ednimo rencor a Pedro Camacho por haberlo sustituido por un advenedizo de Radio Victoria. Se expresaba siempre en los t\u00e9rminos m\u00e1s elogiosos del escriba boliviano, por quien sent\u00eda la m\u00e1s genuina admiraci\u00f3n. Con frecuencia, me ped\u00eda permiso para ir a los ensayos de los radioteatros. Cada vez volv\u00eda m\u00e1s entusiasmado:\n\n\u2014Este tipo es un genio \u2014dec\u00eda, ahog\u00e1ndose\u2014. Se le ocurren cosas milagrosas.\n\nTra\u00eda siempre an\u00e9cdotas muy divertidas sobre las proezas art\u00edsticas de Pedro Camacho. Un d\u00eda nos jur\u00f3 que \u00e9ste hab\u00eda aconsejado a Luciano Pando que se masturbara antes de interpretar un di\u00e1logo de amor con el argumento de que eso debilitaba la voz y provocaba un jadeo muy rom\u00e1ntico. Luciano Pando se hab\u00eda resistido.\n\n\u2014Ahora se entiende por qu\u00e9 cada vez que hay una escena sentimental se mete al ba\u00f1ito del patio, don Mario \u2014hac\u00eda cruces y se besaba los dedos el Gran Pablito\u2014. Para corr\u00e9rsela, para qu\u00e9 va a ser. Por eso le sale la voz tan suavecita.\n\nDiscutimos largamente con Javier sobre si ser\u00eda cierto o una invenci\u00f3n del nuevo redactor y llegamos a la conclusi\u00f3n de que, en todo caso, hab\u00eda bases suficientes para no considerarlo absolutamente imposible.\n\n\u2014Sobre esas cosas deber\u00edas escribir un cuento y no sobre Doroteo Mart\u00ed \u2014me amonestaba Javier\u2014. Radio Central es una mina para la literatura.\n\nEl cuento que estaba empe\u00f1ado en escribir, en esos d\u00edas, se basaba en una an\u00e9cdota que me hab\u00eda contado la t\u00eda Julia, algo que ella misma hab\u00eda presenciado en el Teatro Saavedra de La Paz. Doroteo Mart\u00ed era un actor espa\u00f1ol que recorr\u00eda Am\u00e9rica haciendo llorar l\u00e1grimas de inflamada emoci\u00f3n a las multitudes con La Malquerida y Todo un hombre o calamidades m\u00e1s truculentas todav\u00eda. Hasta en Lima, donde el teatro era una curiosidad extinta desde el siglo pasado, la Compa\u00f1\u00eda de Doroteo Mart\u00ed hab\u00eda repletado el Municipal con una representaci\u00f3n que, seg\u00fan la leyenda, era el non plus ultra de su repertorio: La vida, pasi\u00f3n y muerte de Nuestro Se\u00f1or. El artista ten\u00eda un acerado sentido pr\u00e1ctico y las malas lenguas dec\u00edan que, alguna vez, el Cristo interrump\u00eda su sollozante noche de dolor en el Bosque de los Olivos para anunciar, con voz amable, al distinguido p\u00fablico asistente que el d\u00eda de ma\u00f1ana la compa\u00f1\u00eda ofrecer\u00eda una funci\u00f3n de gancho en la que cada caballero podr\u00eda llevar a su pareja gratis (y continuaba el Calvario). Fue precisamente una representaci\u00f3n de La vida, pasi\u00f3n y muerte lo que hab\u00eda visto la t\u00eda Julia en el Teatro Saavedra. Era el instante supremo, Jesucristo agonizaba en lo alto del G\u00f3lgota, cuando el p\u00fablico advirti\u00f3 que el madero en el que permanec\u00eda amarrado, entre nubes de incienso, Jesucristo-Mart\u00ed, comenzaba a cimbrearse. \u00bfEra un accidente o un efecto previsto? Prudentes, cambiando sigilosas miradas, la Virgen, los ap\u00f3stoles, los legionarios, el pueblo en general, comenzaban a retroceder, a apartarse de la cruz oscilante, en la que, todav\u00eda con la cabeza reclinada sobre el pecho, Doroteo-Jes\u00fas hab\u00eda empezado a murmurar, bajito, pero audible en las primeras filas de la platea: \u00abMe caigo, me caigo\u00bb. Paralizados sin duda por el horror al sacrilegio, nadie, entre los invisibles ocupantes de las bambalinas, acud\u00eda a sujetar la cruz, que ahora bailaba desafiando numerosas leyes f\u00edsicas en medio de un rumor de alarma que hab\u00eda reemplazado a los rezos. Segundos despu\u00e9s, los espectadores pace\u00f1os pudieron ver a Mart\u00ed de Galilea vini\u00e9ndose de bruces sobre el escenario de sus glorias, bajo el peso del sagrado madero, y escuchar el estruendo que remeci\u00f3 el teatro. La t\u00eda Julia me juraba que Cristo hab\u00eda alcanzado a rugir salvajemente, antes de hacerse una mazamorra contra las tablas: \u00abMe ca\u00ed, carajo\u00bb. Era sobre todo ese final el que yo quer\u00eda recrear; el cuento iba a terminar as\u00ed, de manera efectista, con el rugido y la palabrota de Jes\u00fas. Quer\u00eda que fuera un cuento c\u00f3mico y, para aprender las t\u00e9cnicas del humor, le\u00eda en los colectivos, Expresos y en la cama antes de caer dormido a todos los escritores risue\u00f1os que se pon\u00edan a mi alcance, desde Mark Twain y Bernard Shaw hasta Jardiel Poncela y Fern\u00e1ndez Fl\u00f3rez. Pero, como siempre, no me sal\u00eda y Pascual y el Gran Pablito iban contando las cuartillas que yo mandaba al canasto. Menos mal que, en lo que se refer\u00eda al papel, los Genaros eran manirrotos con el Servicio de Informaciones.\n\nPasaron dos o tres semanas antes de que conociera al hombre de Radio Victoria que hab\u00eda reemplazado al Gran Pablito. A diferencia de lo que ocurr\u00eda antes de su llegada, en que uno pod\u00eda asistir libremente a la grabaci\u00f3n de los radioteatros, Pedro Camacho hab\u00eda prohibido que nadie, fuera de actores y t\u00e9cnicos, entrara al estudio, y, para impedirlo, cerraba las puertas e instalaba ante ellas la desarmante mole de Jesusito. Ni el propio Genaro hijo hab\u00eda sido exonerado. Recuerdo la tarde en que, como siempre que ten\u00eda problemas y necesitaba un pa\u00f1o de l\u00e1grimas, se present\u00f3 en el altillo con las narices vibr\u00e1ndole de indignaci\u00f3n y me dio sus quejas:\n\n\u2014Trat\u00e9 de entrar al estudio y par\u00f3 el programa en seco y se neg\u00f3 a grabarlo hasta que me largara \u2014me dijo, con voz descompuesta\u2014. Me ha prometido que la pr\u00f3xima vez que interrumpa un ensayo me tirar\u00e1 el micro a la cabeza. \u00bfQu\u00e9 hago? \u00bfLo despido con cajas destempladas o me trago el sapo?\n\nLe dije lo que quer\u00eda que le dijera: que, en vista del \u00e9xito de los radioteatros (\u00aben aras de la radiotelefon\u00eda nacional, etc\u00e9tera\u00bb) se tragara el sapo y no volviera a meter las narices en los dominios del artista. As\u00ed lo hizo y yo qued\u00e9 enfermo de curiosidad por asistir a la grabaci\u00f3n de alguno de los programas del escriba.\n\nUna ma\u00f1ana, a la hora de nuestro consabido caf\u00e9, despu\u00e9s de un cauteloso rodeo me atrev\u00ed a sondear a Pedro Camacho. Le dije que ten\u00eda ganas de ver en acci\u00f3n al nuevo encargado de los efectos especiales, de comprobar si era tan bueno como \u00e9l me hab\u00eda dicho:\n\n\u2014No dije bueno sino mediano \u2014me corrigi\u00f3, inmediatamente\u2014. Pero lo estoy educando y podr\u00eda llegar a ser bueno.\n\nBebi\u00f3 un trago de su infusi\u00f3n y se qued\u00f3 observ\u00e1ndome con sus ojitos fr\u00edos y ceremoniosos, presa de dudas interiores. Por fin, resign\u00e1ndose, asinti\u00f3:\n\n\u2014Muy bien. Venga ma\u00f1ana, al de las tres. Pero esto no se podr\u00e1 repetir, lo siento mucho. No me gusta que los actores se distraigan, cualquier presencia los turba, se me escurren y adi\u00f3s trabajos con la catarsis. La grabaci\u00f3n de un episodio es una misa, mi amigo.\n\nEn realidad, era algo m\u00e1s solemne. Entre todas las misas que recordaba (hac\u00eda a\u00f1os que no iba a la iglesia) nunca vi una ceremonia tan sentida, un rito tan vivido, como esa grabaci\u00f3n del cap\u00edtulo d\u00e9cimo s\u00e9ptimo de Las venturas y desventuras de don Alberto de Quinteros, a la que fui admitido. El espect\u00e1culo no debi\u00f3 de durar m\u00e1s de treinta minutos \u2014diez de ensayo y veinte de grabaci\u00f3n\u2014pero me pareci\u00f3 que duraba horas. Me impresion\u00f3, de entrada, la atm\u00f3sfera de recogimiento religioso que reinaba en el cuartucho encristalado, de polvorienta alfombra verde, que respond\u00eda al nombre de Estudio de Grabaci\u00f3n N\u00famero Uno de Radio Central. S\u00f3lo el Gran Pablito y yo est\u00e1bamos all\u00ed de espectadores; los otros eran participantes activos. Pedro Camacho, al entrar, con una mirada castrense nos hab\u00eda hecho saber que deb\u00edamos permanecer como estatuas de sal. El libretista director parec\u00eda transformado: m\u00e1s alto, m\u00e1s fuerte, un general que instruye a tropas disciplinadas. \u00bfDisciplinadas? M\u00e1s bien embelesadas, hechizadas, fanatizadas. Me cost\u00f3 trabajo reconocer a la bigotuda y varicosa Josefina S\u00e1nchez, a quien hab\u00eda visto ya tantas veces grabar sus parlamentos masticando chicle, tejiendo, totalmente despreocupada y con aire de no saber lo que dec\u00eda, en esa personita tan seria que, cuando no revisaba, como quien reza, el libreto, s\u00f3lo ten\u00eda ojos para mirar, respetuosa y d\u00f3cil, al artista, con el temblor primerizo con que la ni\u00f1ita mira el altar el d\u00eda de su primera comuni\u00f3n. Y lo mismo ocurr\u00eda con Luciano Pando y con los otros tres actores (dos mujeres y un muchacho muy joven). No cambiaban palabra, no se miraban entre ellos: sus ojos iban, imantados, de los libretos a Pedro Camacho, y hasta el t\u00e9cnico de sonido, el huatatiro Ochoa, al otro lado del cristal, compart\u00eda el arrobo: muy serio, probaba los controles, apretando botones y encendiendo luces, y segu\u00eda con ce\u00f1o grave y atento lo que pasaba en el estudio.\n\nLos cinco actores estaban parados en c\u00edrculo en torno a Pedro Camacho, quien \u2014siempre uniformado de traje negro y corbata de lazo y la cabellera revoloteante\u2014 los aleccionaba sobre el cap\u00edtulo que iban a grabar. No eran instrucciones lo que les impart\u00eda, al menos en el sentido prosaico de indicaciones concretas sobre c\u00f3mo decir sus parlamentos \u2014con mesura o exageraci\u00f3n, despacio o r\u00e1pido\u2014, sino, seg\u00fan era costumbre en \u00e9l, pontificando, noble y ol\u00edmpico, sobre profundidades est\u00e9ticas y filos\u00f3ficas. Eran, por supuesto, las palabras \u00abarte\u00bb y \u00abart\u00edstico\u00bb las que m\u00e1s iban y ven\u00edan por ese discurso afiebrado, como un santo y se\u00f1a m\u00e1gico que todo lo abr\u00eda y explicaba. Pero m\u00e1s ins\u00f3lito que las palabras del escriba boliviano era el fervor con que las profer\u00eda, y, quiz\u00e1 a\u00fan m\u00e1s, el efecto que causaban. Hablaba gesticulando y empin\u00e1ndose, con la voz fan\u00e1tica del hombre que est\u00e1 en posesi\u00f3n de una verdad urgente y tiene que propagarla, compartirla, imponerla. Lo consegu\u00eda totalmente: los cinco actores lo escuchaban alelados, suspensos, abriendo mucho los ojos como para absorber mejor esas sentencias sobre su trabajo (\u00absu misi\u00f3n\u00bb, dec\u00eda el libretista director). Lament\u00e9 que la t\u00eda Julia no estuviera all\u00ed, porque no me creer\u00eda cuando le contara que hab\u00eda visto transfigurarse, embellecerse, espiritualizarse, durante una eterna media hora, a ese pu\u00f1ado de exponentes de la m\u00e1s miserable profesi\u00f3n de Lima, bajo la ret\u00f3rica efervescente de Pedro Camacho. El Gran Pablito y yo est\u00e1bamos sentados en el suelo en un rinc\u00f3n del estudio; frente a nosotros, rodeado de una parafernalia extra\u00f1a, se hallaba el tr\u00e1nsfuga de Radio Victoria, la nov\u00edsima adquisici\u00f3n. Tambi\u00e9n hab\u00eda escuchado en actitud m\u00edstica la arenga del artista; apenas comenz\u00f3 la grabaci\u00f3n del cap\u00edtulo, \u00e9l se convirti\u00f3 para m\u00ed en el centro del espect\u00e1culo.\n\nEra un hombrecito fortach\u00f3n y cobrizo, de pelos tiesos, vestido casi como un mendigo: un overol ra\u00eddo, una camisa con parches, unos zapatones sin pasador. (M\u00e1s tarde supe que se lo conoc\u00eda por el misterioso apodo de Bat\u00e1n.) Sus instrumentos de trabajo eran: un tabl\u00f3n, una puerta, un lavador lleno de agua, un silbato, un pliego de papel platino, un ventilador y otras cosas de esa misma apariencia dom\u00e9stica. Bat\u00e1n constitu\u00eda \u00e9l solo un espect\u00e1culo de ventriloquia, de acrobacia, de multiplicaci\u00f3n de la personalidad, de imaginaci\u00f3n f\u00edsica. Apenas el director actor hac\u00eda la se\u00f1al indicada \u2014una vibraci\u00f3n magisterial del \u00edndice en el aire cargado de di\u00e1logos, de ayes y suspiros\u2014, Bat\u00e1n, caminando sobre el tabl\u00f3n a un ritmo sabiamente decreciente, hac\u00eda que los pasos de los personajes se acercaran o alejaran, y, a otra se\u00f1al, orientando el ventilador a distintas velocidades sobre el platino, hac\u00eda brotar el rumor de la lluvia o el rugido del viento, y, a otra, meti\u00e9ndose tres dedos en la boca y silbando, inundaba el estudio con los trinos que, en un amanecer de primavera, despertaban a la hero\u00edna en su casa de campo. Era especialmente notable cuando sonorizaba la calle. En un momento dado, dos personajes recorr\u00edan la plaza de Armas conversando. El huatatiro Ochoa enviaba, de cinta grabada, ruido de motores y bocinas, pero todos los dem\u00e1s efectos los produc\u00eda Bat\u00e1n, chasqueando la lengua, cloqueando, bisbiseando, susurrando (parec\u00eda hacer todas estas cosas a la vez) y bastaba cerrar los ojos para sentir, reconstituidas en el peque\u00f1o estudio de Radio Central, las voces, palabras sueltas, risas, interjecciones que uno va distra\u00eddamente oyendo por una calle concurrida. Pero, como si esto fuera poco, Bat\u00e1n, al mismo tiempo que produc\u00eda decenas de voces humanas, caminaba o brincaba sobre el tabl\u00f3n, manufacturando los pasos de los peatones sobre las veredas y los roces de sus cuerpos. Caminaba a la vez con pies y manos (a las que hab\u00eda enguantado con un par de zapatos), de cuclillas, los brazos colgantes como un simio, golpe\u00e1ndose los muslos con codos y antebrazos. Despu\u00e9s de haber sido (ac\u00fasticamente) la plaza de Armas a mediod\u00eda, resultaba, en cierto modo, una proeza insignificante musicalizar \u2014haciendo tintinear dos fierritos, rascando un vidrio, y, para imitar el desliz de sillas y personas sobre mullidas alfombras, restregando unas tablillas contra su fundillo\u2014 la mansi\u00f3n de una empingorotada dama lime\u00f1a que ofrece t\u00e9 \u2014en tazas de porcelana china\u2014 a un grupo de amigas, o, rugiendo, graznando, hozando, aullando, encarnar fon\u00e9ticamente (enriqueci\u00e9ndolo de muchos ejemplares) al Zool\u00f3gico de Barranco. Al terminar la grabaci\u00f3n, parec\u00eda haber corrido la marat\u00f3n ol\u00edmpica: jadeaba, ten\u00eda ojeras y sudaba como un caballo.\n\nPedro Camacho hab\u00eda contagiado a sus colaboradores su seriedad sepulcral. Era un cambio enorme, los radioteatros de la CMQ cubana se grababan muchas veces en una atm\u00f3sfera de jolgorio, y los actores, mientras interpretaban el libreto, se hac\u00edan morisquetas o gestos obscenos, burl\u00e1ndose de s\u00ed mismos y de lo que dec\u00edan. Ahora daba la impresi\u00f3n de que si alguien hubiera hecho una broma los otros se hubieran abalanzado sobre \u00e9l para castigarlo por sacr\u00edlego. Pens\u00e9 un momento que tal vez simulaban por servilismo hacia el jefe, para no ser purgados como los argentinos, que en el fondo no estaban tan seguros, como aqu\u00e9l, de ser los sacerdotes del arte, pero me equivocaba. De regreso a Panamericana, di unos pasos por la calle Bel\u00e9n junto a Josefina S\u00e1nchez, quien, entre radioteatro y radioteatro, se iba a preparar un tecito a su casa, y le pregunt\u00e9 si en todas las grabaciones pronunciaba el escriba boliviano esas arengas preliminares o si hab\u00eda sido algo excepcional. Me mir\u00f3 con un desprecio que hac\u00eda temblar su papada:\n\n\u2014Hoy habl\u00f3 poco y no estuvo inspirado. Hay veces que parte el alma ver c\u00f3mo esas ideas no se conservan para la posteridad.\n\nLe pregunt\u00e9 si ella, \u00abque ten\u00eda tanta experiencia\u00bb, pensaba realmente que Pedro Camacho era una persona de mucho talento. Tard\u00f3 unos segundos en encontrar las palabras adecuadas para formular su pensamiento:\n\n\u2014Ese hombre santifica la profesi\u00f3n del artista.\n\n### VI\n\nUNA RESPLANDECIENTE ma\u00f1ana de verano, atildado y puntual como era su costumbre, entr\u00f3 el doctor don Pedro Barreda y Zald\u00edvar a su despacho de juez instructor de la Primera Sala (en lo Penal) de la Corte Superior de Lima. Era un hombre que hab\u00eda llegado a la flor de la edad, la cincuentena, y en su persona \u2014frente ancha, nariz aguile\u00f1a, mirada penetrante, rectitud y bondad en el esp\u00edritu\u2014, la pulcritud \u00e9tica se transparentaba en una apostura que le merec\u00eda al instante el respeto de las gentes. Vest\u00eda con la modestia que corresponde a un magistrado de magro salario que es constitutivamente inapto para el cohecho, pero con una correcci\u00f3n tal que produc\u00eda una impresi\u00f3n de elegancia. El Palacio de Justicia comenzaba a desperezarse de su descanso nocherniego y su mole se iba inundando de una afanosa muchedumbre de abogados, tinterillos, conserjes, demandantes, notarios, albaceas, bachilleres y curiosos. En el coraz\u00f3n de esa colmena, el doctor don Barreda y Zald\u00edvar abri\u00f3 su malet\u00edn, sac\u00f3 dos expedientes, se sent\u00f3 en su escritorio y se dispuso a comenzar la jornada. Segundos despu\u00e9s se materializ\u00f3 en su despacho, raudo y silente como un aerolito en el espacio, el secretario, doctor Zelaya, hombrecillo con anteojos, de bigotito mosca que mov\u00eda r\u00edtmicamente al hablar.\n\n\u2014Muy buenos d\u00edas, mi se\u00f1or doctor \u2014salud\u00f3 al magistrado, haciendo una reverencia de bisagra.\n\n\u2014Lo mismo le deseo, Zelaya \u2014le sonri\u00f3 afablemente el doctor don Barreda y Zald\u00edvar\u2014. \u00bfQu\u00e9 nos depara la ma\u00f1ana?\n\n\u2014Estupro de menor con agravante de violencia mental \u2014deposit\u00f3 en el escritorio un expediente de buen cuerpo el secretario\u2014. El responsable, un vecino de La Victoria de catadura lombrosiana, niega los hechos. Los principales testigos est\u00e1n en el pasillo.\n\n\u2014Antes de escucharlos, necesito releer el parte policial y la demanda de la parte civil \u2014le record\u00f3 el magistrado.\n\n\u2014Esperar\u00e1n lo que haga falta \u2014repuso el secretario. Y sali\u00f3 del despacho.\n\nEl doctor don Barreda y Zald\u00edvar ten\u00eda, bajo su s\u00f3lida coraza jur\u00eddica, alma de poeta. Una lectura de los helados documentos judiciales le bastaba para, separando la costra ret\u00f3rica de cl\u00e1usulas y latinajos, llegar con la imaginaci\u00f3n a los hechos. As\u00ed, leyendo el parte asentado en La Victoria, reconstituy\u00f3 con viveza de detalles la denuncia. Vio entrar el lunes pasado, a la comisar\u00eda del abigarrado y variopinto distrito, a la ni\u00f1a de trece a\u00f1os y alumna de la Unidad Escolar Mercedes Cabello de Carbonera llamada Sarita Huanca Salaverr\u00eda. Ven\u00eda llorosa y con moretones en la cara, brazos y piernas, entre sus padres don Casimiro Huanca Padr\u00f3n y do\u00f1a Catalina Salaverr\u00eda Melgar. La menor hab\u00eda sido mancillada la v\u00edspera, en la casa de vecindad de la avenida Luna Pizarro N. 12, cuarto H, por el sujeto Gumercindo Tello, inquilino de la misma casa de vecindad (cuarto J). Sarita, venciendo su confusi\u00f3n y quebranto, hab\u00eda revelado a los custodios del orden que el estupro no era sino el saldo tr\u00e1gico de un largo y secreto asedio a que se hab\u00eda visto sometida por el violador. \u00c9ste, en efecto, hac\u00eda ya ocho meses \u2014es decir desde el d\u00eda en que hab\u00eda venido a instalarse, como extravagante p\u00e1jaro de mal ag\u00fcero, en la casa de vecindad N. 12\u2014, persegu\u00eda a Sarita Huanca, sin que los padres de \u00e9sta o los otros vecinos pudieran advertirlo, con piropos de mal gusto e insinuaciones intr\u00e9pidas (como decirle: \u00abMe gustar\u00eda exprimir los limones de tu huerta\u00bb o \u00abun d\u00eda de \u00e9stos te orde\u00f1ar\u00e9\u00bb). De las profec\u00edas, Gumercindo Tello hab\u00eda pasado a las obras, realizando varios intentos de manoseo y beso de la p\u00faber, en el patio de la casa de vecindad N. 12 o en calles adyacentes, cuando la ni\u00f1a ven\u00eda del colegio o cuando sal\u00eda a hacer mandados. Por natural pudor, la v\u00edctima no hab\u00eda prevenido a los padres sobre el acoso.\n\nLa noche del domingo, diez minutos despu\u00e9s que sus padres salieron en direcci\u00f3n al Cine Metropolitan, Sarita Huanca, que hac\u00eda las tareas del colegio, oy\u00f3 unos golpecillos en la puerta. Fue a abrir y se encontr\u00f3 con Gumercindo Tello. \u00ab\u00bfQu\u00e9 desea?\u00bb, le pregunt\u00f3 cort\u00e9smente. El violador, aparentando el aire m\u00e1s inofensivo del mundo, aleg\u00f3 que su primus se hab\u00eda quedado sin combustible: ya era tarde para ir a comprarlo y ven\u00eda a que le prestaran un conchito de kerosene para preparar su comida (promet\u00eda devolverlo ma\u00f1ana). Dadivosa e ingenua, la ni\u00f1a Huanca Salaverr\u00eda hizo entrar al individuo y le indic\u00f3 que la lata de kerosene estaba entre la hornilla y el balde que hac\u00eda las veces de retrete.\n\n(El doctor don Barreda y Zald\u00edvar sonri\u00f3 ante ese desliz del custodio del orden que hab\u00eda asentado la denuncia y que, sin quererlo, delataba en los Huanca Salaverr\u00eda esa costumbre bonaerense de hacer sus necesidades en un balde en el mismo recinto donde se come y se duerme.)\n\nApenas hubo conseguido, mediante dicha estratagema, introducirse en el cuarto H, el acusado tranc\u00f3 la puerta. Se puso luego de rodillas y, juntando las manos, comenz\u00f3 a musitar palabras de amor a Sarita Huanca Salaverr\u00eda, quien s\u00f3lo en este momento sinti\u00f3 alarma por su suerte. En un lenguaje que la ni\u00f1a describ\u00eda como rom\u00e1ntico, Gumercindo Tello le aconsejaba que accediera a sus deseos. \u00bfCu\u00e1les eran \u00e9stos? Que se despojara de sus prendas de vestir y se dejara tocar, besar y arrebatar el himen. Sarita Huanca, sobreponi\u00e9ndose, rechaz\u00f3 con energ\u00eda las propuestas, increp\u00f3 a Gumercindo Tello y lo amenaz\u00f3 con llamar a los vecinos. Fue al o\u00edr esto que el acusado, renunciando a su actitud suplicante, extrajo de sus ropas un cuchillo y amenaz\u00f3 a la ni\u00f1a con darle de pu\u00f1aladas al menor grito. Poni\u00e9ndose de pie, avanz\u00f3 hacia Sarita diciendo: \u00abVamos, vamos, ya te est\u00e1s calateando, mi amor\u00bb, y, como ella, pese a todo, no le obedeciera, le regal\u00f3 una andanada de pu\u00f1etazos y patadas, hasta hacerla caer al suelo. All\u00ed, presa de un nerviosismo que, seg\u00fan la v\u00edctima, le hac\u00eda chocar los dientes, el violador le arranc\u00f3 las ropas a jalones, procedi\u00f3 tambi\u00e9n a desabotonar las suyas, y se derrumb\u00f3 sobre ella, hasta perpetrar, all\u00ed en el suelo, el acto carnal, el mismo que, debido a la resistencia que ofrec\u00eda la muchacha, estuvo yapado de nuevos golpes, de los cuales quedaban huellas en forma de hematomas y chichones. Satisfechas sus ansias, Gumercindo Tello abandon\u00f3 el cuarto H no sin antes recomendar a Sarita Huanca Salaverr\u00eda que no dijera una palabra de lo sucedido si pretend\u00eda llegar a vieja (y agit\u00f3 el cuchillo para mostrar que hablaba en serio). Los padres, al volver del Metropolitan, encontraron a su hija ba\u00f1ada en llanto y con el cuerpo depredado. Luego de curar sus heridas, la exhortaron a decir qu\u00e9 hab\u00eda ocurrido, pero ella, por verg\u00fcenza, se negaba. Y as\u00ed pas\u00f3 la noche entera. A la ma\u00f1ana siguiente, sin embargo, algo repuesta del impacto emocional que le signific\u00f3 la p\u00e9rdida del himen, la ni\u00f1a cont\u00f3 todo a sus progenitores, quienes, de inmediato, se apersonaron a la comisar\u00eda de La Victoria para denunciar el suceso.\n\nEl doctor don Barreda y Zald\u00edvar cerr\u00f3 un instante los ojos. Sent\u00eda (pese a su roce diario con el delito no se hab\u00eda encallecido) l\u00e1stima por lo ocurrido a la ni\u00f1a, pero se dijo a s\u00ed mismo que, a simple vista, se trataba de un delito sin misterio, protot\u00edpico, milim\u00e9tricamente encuadrado en el C\u00f3digo Penal, en las figuras de violaci\u00f3n y abuso de menor, con sus m\u00e1s caracterizados agravantes de premeditaci\u00f3n, violencias de hecho y de dicho, y crueldad mental.\n\nEl siguiente documento que reley\u00f3 era el parte de los custodios del orden que hab\u00edan efectuado la detenci\u00f3n de Gumercindo Tello.\n\nConforme intrucciones de su superior, capit\u00e1n G C Enrique Soto, los guardias Alberto Cusicanqui Ap\u00e9stegui y Huasi Tito Parinacocha se apersonaron con una orden de arresto a la casa de vecindad N. 12 de la avenida Luna Pizarro, pero el individuo no se encontraba en su hogar. Mediante los vecinos, se informaron que era de profesi\u00f3n mec\u00e1nico y trabajaba en el taller de reparaci\u00f3n de motores y soldadura aut\u00f3gena El Inti, sito al otro extremo del distrito, casi en las faldas del cerro El Pino. Los guardias procedieron a trasladarse de inmediato hasta all\u00ed. En el taller, se dieron con la sorpresa de que Gumercindo Tello acababa de partir, inform\u00e1ndoles adem\u00e1s el due\u00f1o del taller, se\u00f1or Carlos Pr\u00edncipe, que hab\u00eda pedido licencia con motivo de un bautizo. Cuando los guardias inquirieron, entre los operarios, en qu\u00e9 iglesia pod\u00eda encontrarse, \u00e9stos se miraron con malicia y cambiaron sonrisas. El se\u00f1or Pr\u00edncipe explic\u00f3 que Gumercindo Tello no era cat\u00f3lico sino Testigo de Jehov\u00e1, y que para esa religi\u00f3n el bautizo no se celebraba en iglesia y con cura sino al aire libre y a zambullidas.\n\nMaliciando que (como se ha dado ya el caso) la tal congregaci\u00f3n fuera una cofrad\u00eda de invertidos, Cusicanqui Ap\u00e9stegui y Tito Parinacocha exigieron que se los condujera al sitio donde se hallaba el acusado. Luego de un buen rato de vacilaciones y cambio de palabras, el propietario de El Inti en persona los gui\u00f3 al lugar donde, dijo, era posible que estuviera Tello, pues una vez, hac\u00eda ya tiempo, cuando trataba de catequizarlos a \u00e9l y a los compa\u00f1eros del taller, lo hab\u00eda invitado a presenciar all\u00ed una ceremonia (experiencia de la cual el susodicho no hab\u00eda quedado nada convencido).\n\nEl se\u00f1or Pr\u00edncipe llev\u00f3 en su autom\u00f3vil a los custodios del orden a los confines de la calle Maynas y el parque Martinetti, descampado donde los vecinos de los alrededores queman basuras y donde hay una entradita del r\u00edo R\u00edmac. En efecto, all\u00ed estaban los Testigos de Jehov\u00e1. Cusicanqui Ap\u00e9stegui y Tito Parinacocha descubrieron una docena de personas de distintas edades y sexos metidas hasta la cintura en las aguas fangosas, no en ropa de ba\u00f1o sino muy vestidas, algunos hombres con corbata y uno de ellos incluso con sombrero. Indiferentes a las bromas, pullas, tiros de c\u00e1scaras y otras criollas picard\u00edas de los vecinos que se hab\u00edan amontonado a la orilla para verlos, prosegu\u00edan muy serios una ceremonia que a los custodios del orden les pareci\u00f3, en el primer momento, poco menos que un intento colectivo de homicidio por inmersi\u00f3n. Esto es lo que vieron: a la vez que entonaban, en voz muy convencida, extra\u00f1os c\u00e1nticos, los Testigos ten\u00edan cogido de los brazos a un anciano de poncho y chullo, al que sepultaban en las inmundas aguas \u00bfcon el prop\u00f3sito de sacrificarlo a su Dios? Pero cuando los guardias, rev\u00f3lver en mano y embarr\u00e1ndose las polainas, les ordenaron interrumpir su criminal acto, el anciano fue el primero en enojarse, exigiendo a los guardias que se retiraran y llam\u00e1ndolos cosas raras (como \u00abromanos\u00bb y \u00abpapistas\u00bb). Los custodios del orden debieron resignarse a esperar que terminara el bautizo para detener a Gumercindo Tello, a quien hab\u00edan identificado gracias al se\u00f1or Pr\u00edncipe. La ceremonia dur\u00f3 unos minutos m\u00e1s, en el curso de los cuales continuaron los rezos y los remojones del bautizado hasta que \u00e9ste comenz\u00f3 a voltear los ojos, a tragar agua y atorarse, momento en que los Testigos optaron por sacarlo en peso hasta la orilla, donde principiaron a felicitarlo por la nueva vida que, dec\u00edan, comenzaba a partir de ese instante.\n\nFue en ese momento que los guardias capturaron a Gumercindo Tello. El mec\u00e1nico no ofreci\u00f3 la menor resistencia, ni pretendi\u00f3 huir, ni mostr\u00f3 sorpresa por el hecho de ser detenido, limit\u00e1ndose a decir a los otros al recibir las esposas: \u00abHermanos, nunca los olvidar\u00e9\u00bb. Los Testigos prorrumpieron de inmediato en nuevos c\u00e1nticos, mirando al cielo y poniendo los ojos en blanco, y as\u00ed los acompa\u00f1aron hasta el auto del se\u00f1or Pr\u00edncipe, quien traslad\u00f3 a los guardias y al detenido a la comisar\u00eda de La Victoria, donde se le despidi\u00f3 agradeci\u00e9ndole los servicios prestados.\n\nEn la comisar\u00eda, el capit\u00e1n G C Enrique Soto pregunt\u00f3 al acusado si quer\u00eda secar sus zapatos y pantalones en el patio, a lo cual Gumercindo Tello repuso que se hallaba acostumbrado a andar mojado por el gran incremento de conversiones a la verdadera fe que se registraba \u00faltimamente en Lima. De inmediato, el capit\u00e1n Soto procedi\u00f3 a interrogarlo, a lo cual el acusado se prest\u00f3 con \u00e1nimo cooperativo. Preguntado por sus generales de ley, repuso llamarse Gumercindo Tello y ser hijo de do\u00f1a Gumercinda Tello, natural de Moquegua y ya difunta, y de padre desconocido, y haber nacido \u00e9l mismo, tambi\u00e9n, probablemente en Moquegua hace unos veinticinco o veintiocho a\u00f1os. Respecto a esta duda explic\u00f3 que su madre lo hab\u00eda entregado, a poco de nacido, a un orfelinato de varones regentado en esa ciudad por la secta papista, en cuyas aberraciones, dijo, hab\u00eda sido educado y de las que felizmente se hab\u00eda liberado a los quince o dieciocho a\u00f1os. Indic\u00f3 que hasta esa edad hab\u00eda permanecido en el orfelinato, fecha en que \u00e9ste desapareci\u00f3 en un gran incendio, quem\u00e1ndose tambi\u00e9n todos los archivos, motivo por el cual \u00e9l se hab\u00eda quedado en el misterio sobre su exacta edad. Explic\u00f3 que el siniestro fue providencial en su vida, pues en esa ocasi\u00f3n conoci\u00f3 a una pareja de sabios que viajaban de Chile a Lima, por tierra, abriendo los ojos de los ciegos y destapando los o\u00eddos de los sordos sobre las verdades de la filosof\u00eda. Puntualiz\u00f3 que se hab\u00eda venido a Lima con esa pareja de sabios, cuyo nombre se excus\u00f3 de revelar porque dijo que era bastante saber que exist\u00edan para tener tambi\u00e9n que etiquetarlos, y que aqu\u00ed hab\u00eda vivido desde entonces repartiendo su tiempo entre la mec\u00e1nica (oficio que aprendi\u00f3 en el orfelinato) y la propagaci\u00f3n de la ciencia de la verdad. Dijo haber vivido en Bre\u00f1a, en Vitarte, en los Barrios Altos, y haberse instalado en La Victoria hac\u00eda ocho meses, por haber obtenido empleo en el taller de reparaci\u00f3n de motores y soldadura aut\u00f3gena El Inti, que quedaba demasiado lejos de su domicilio anterior.\n\nEl acusado admiti\u00f3 residir desde entonces en la casa de vecindad N. 12 de la avenida Luna Pizarro, en calidad de inquilino. Reconoci\u00f3 asimismo a la familia Huanca Salaverr\u00eda, a la que, dijo, hab\u00eda ofrecido varias veces pl\u00e1ticas iluminativas y buenas lecturas, sin haber tenido \u00e9xito por hallarse ellos, al igual que los otros inquilinos, muy intoxicados por las herej\u00edas romanas. Enfrentado al nombre de su presunta v\u00edctima, la ni\u00f1a Sarita Huanca Salaverr\u00eda, dijo recordarla e insinu\u00f3 que, por tratarse de una persona todav\u00eda en su tierna edad, no perd\u00eda las esperanzas de que enrumbara alg\u00fan d\u00eda por el buen camino. Puesto entonces en antecedentes de la acusaci\u00f3n, Gumercindo Tello manifest\u00f3 viva sorpresa, negando los cargos, para, un momento despu\u00e9s (\u00bfsimulando una perturbaci\u00f3n con miras a su futura defensa?) romper a re\u00edr muy contento diciendo que \u00e9sta era la prueba que le reservaba Dios para barometrar su fe y su esp\u00edritu de sacrificio. A\u00f1adiendo que ahora entend\u00eda por qu\u00e9 no hab\u00eda salido sorteado en el servicio militar, ocasi\u00f3n que \u00e9l esperaba con impaciencia para, predicando con el ejemplo, negarse a vestir el uniforme y a jurar fidelidad a la bandera, atributos de Sat\u00e1n. El capit\u00e1n G C Enrique Soto le pregunt\u00f3 si estaba hablando en contra del Per\u00fa, a lo cual respondi\u00f3 el acusado que de ning\u00fan modo y que s\u00f3lo se refer\u00eda a asuntos de la religi\u00f3n. Y procedi\u00f3 entonces, de manera fogosa, a explicar al capit\u00e1n Soto y a los guardias que Cristo no era Dios sino Su Testigo y que era falso, como ment\u00edan los papistas, que lo hubieran crucificado siendo as\u00ed que lo hab\u00edan clavado en un \u00e1rbol y que la Biblia lo probaba. A este respecto les aconsej\u00f3 leer Despierta, quincenario que, por el precio de dos soles, sacaba de dudas sobre \u00e9ste y otros temas de cultura y proporcionaba sano entretenimiento. El capit\u00e1n Soto lo hizo callar, advirti\u00e9ndole que en el recinto de la comisar\u00eda estaba prohibido hacer propaganda comercial. Y lo conmin\u00f3 a que dijera d\u00f3nde se hallaba y qu\u00e9 hac\u00eda la v\u00edspera, a las horas en que Sarita Huanca Salaverr\u00eda aseguraba haber sido violada y golpeada por \u00e9l. Gumercindo Tello afirm\u00f3 que esa noche, como todas las noches, hab\u00eda permanecido en su cuarto, solo, entregado a la meditaci\u00f3n sobre el Tronco y sobre c\u00f3mo, contra lo que hac\u00eda creer cierta gente, no era verdad que todos los hombres fueran a resucitar el d\u00eda del Juicio Final, siendo as\u00ed que muchos nunca resucitar\u00edan, lo que probaba la mortalidad del alma. Llamado al orden una vez m\u00e1s, el acusado pidi\u00f3 excusas y dijo que no lo hac\u00eda adrede, pero que no pod\u00eda eximirse, a cada momento, de estar arrojando un poco de luz a los dem\u00e1s, ya que lo desesperaba ver en qu\u00e9 tinieblas viv\u00eda la gente. Y concret\u00f3 que no recordaba haber visto a Sarita Huanca Salaverr\u00eda esa noche ni tampoco la v\u00edspera, y rog\u00f3 que en el parte se hiciera constar que, pese a haber sido calumniado, no guardaba rencor a esa muchacha y que incluso le estaba agradecido porque ten\u00eda sospechas de que a trav\u00e9s de ella Dios quer\u00eda probar la musculatura de su fe. Viendo que no ser\u00eda posible obtener de Gumercindo Tello otras precisiones sobre los cargos formulados, el capit\u00e1n G C Enrique Soto puso fin al interrogatorio y transfiri\u00f3 al acusado a la carceleta del Palacio de Justicia, a fin de que el juez instructor d\u00e9 al caso el desarrollo que corresponda.\n\nEl doctor don Barreda y Zald\u00edvar cerr\u00f3 el expediente, y, en la ma\u00f1ana aquejada de ruidos judiciales, reflexion\u00f3. \u00bfLos Testigos de Jehov\u00e1? Los conoc\u00eda. No hac\u00eda muchos a\u00f1os, un hombre que se movilizaba por el mundo en bicicleta hab\u00eda venido a tocar la puerta de su casa y a ofrecerle el peri\u00f3dico Despierta, que \u00e9l, en un momento de debilidad, hab\u00eda adquirido. Desde entonces, con una puntualidad astral, el Testigo hab\u00eda rondado su hogar, a distintas horas del d\u00eda y de la noche, insistiendo en iluminarlo, abrum\u00e1ndolo con folletos, libros, revistas, de distinto espesor y tem\u00e1tica, hasta que, incapaz de alejar de su morada al Testigo por los civilizados m\u00e9todos de la persuasi\u00f3n, la s\u00faplica, la arenga, el magistrado hab\u00eda recurrido a la fuerza policial. De modo que era uno de estos impetuosos catequizadores, el violador. El doctor don Barreda y Zald\u00edvar se dijo que el caso se pon\u00eda interesante.\n\nEra todav\u00eda media ma\u00f1ana y el magistrado, acariciando distra\u00eddamente el acerado y largo cortapapeles de empu\u00f1adura Tiahuanaco, que ten\u00eda en su escritorio como prenda del afecto de sus superiores, colegas y subordinados (se lo hab\u00edan regalado al cumplir sus bodas de plata de abogado), llam\u00f3 al secretario y le indic\u00f3 que hiciera pasar a los declarantes.\n\nEntraron primero los guardias Cusicanqui Ap\u00e9stegui y Tito Parinacocha, quienes, con habla respetuosa, confirmaron las circunstancias del arresto de Gumercindo Tello y dejaron constancia de que \u00e9ste, salvo negar los cargos, se hab\u00eda mostrado servicial, aunque un poco empalagoso con su man\u00eda religiosa. El doctor Zelaya, los anteojos columpi\u00e1ndose sobre su nariz, iba redactando el acta mientras los guardias hablaban.\n\nPasaron despu\u00e9s los padres de la menor, una pareja cuya avanzada edad sorprendi\u00f3 al magistrado: \u00bfc\u00f3mo hab\u00edan podido procrear hac\u00eda s\u00f3lo trece a\u00f1os ese par de vejestorios? Sin dientes, con los ojos medio recubiertos por lega\u00f1as, el padre, don Isa\u00edas Huanca, refrend\u00f3 r\u00e1pidamente el parte policial en lo que lo concern\u00eda y quiso saber despu\u00e9s, con mucha urgencia, si Sarita contraer\u00eda matrimonio con el se\u00f1or Tello. Apenas hecha su pregunta, la se\u00f1ora Salaverr\u00eda de Huanca, una mujer menuda y arrugada, avanz\u00f3 hacia el magistrado y le bes\u00f3 la mano, a la vez que, con voz implorante, le ped\u00eda que fuera bueno y obligara al se\u00f1or Tello a llevar a Sarita al altar. Cost\u00f3 trabajo al doctor don Barreda y Zald\u00edvar explicar a los ancianos que, entre las altas funciones que le hab\u00edan sido confiadas, no figuraba la de casamentero. La pareja, por lo visto, parec\u00eda m\u00e1s interesada en desposar a la ni\u00f1a que en castigar el abuso, hecho que apenas mencionaban y s\u00f3lo cuando eran urgidos a ello, y perd\u00edan mucho tiempo en enumerar las virtudes de Sarita, como si la tuvieran en venta.\n\nSonriendo para sus adentros, el magistrado pens\u00f3 que estos humildes labradores \u2014no hab\u00eda duda que proced\u00edan del Ande y que hab\u00edan vivido en contacto con la gleba\u2014 lo hac\u00edan sentirse un padre acrimonioso que se niega a autorizar la boda de su hijo. Intent\u00f3 hacerlos recapacitar: \u00bfc\u00f3mo pod\u00edan desear para marido de su hija a un hombre capaz de cometer estupro contra una ni\u00f1a inerme? Pero ellos se arrebataban la palabra, insist\u00edan, Sarita ser\u00eda una esposa modelo, a sus cortos a\u00f1os sab\u00eda cocinar, coser y de todo, ellos eran ya viejos y no quer\u00edan dejarla huerfanita, el se\u00f1or Tello parec\u00eda serio y trabajador, aparte de haberse propasado con Sarita la otra noche nunca se lo hab\u00eda visto borracho, era muy respetuoso, sal\u00eda muy temprano al trabajo con su malet\u00edn de herramientas y su paquete de esos periodiquitos que vend\u00eda de casa en casa. \u00bfUn muchacho que luchaba as\u00ed por la vida no era acaso un buen partido para Sarita? Y ambos ancianos elevaban las manos hacia el magistrado: \u00abCompad\u00e9zcase y ay\u00fadenos, se\u00f1or juez\u00bb.\n\nPor la mente del doctor don Barreda y Zald\u00edvar flot\u00f3, nubecilla negra pre\u00f1ada de lluvia, una hip\u00f3tesis: \u00bfy si todo fuera un ardid tramado por esta pareja para desposar a su v\u00e1staga? Pero el parte m\u00e9dico era terminante: la ni\u00f1a hab\u00eda sido violada. No sin dificultad, despidi\u00f3 a los testigos. Pas\u00f3 entonces la v\u00edctima.\n\nEl ingreso de Sarita Huanca Salaverr\u00eda ilumin\u00f3 el adusto despacho del juez instructor. Hombre que todo lo hab\u00eda visto, ante el cual, como victimarios o v\u00edctimas, hab\u00edan desfilado todas las rarezas y psicolog\u00edas humanas, el doctor don Barreda y Zald\u00edvar se dijo, sin embargo, que se hallaba ante un esp\u00e9cimen aut\u00e9nticamente original. \u00bfSarita Huanca Salaverr\u00eda era una ni\u00f1a? Sin duda, a juzgar por su edad cronol\u00f3gica, y por su cuerpecito en el que t\u00edmidamente se insinuaban las turgencias de la femineidad, y por las trenzas que recog\u00edan sus cabellos y por la falda y la blusa escolares que vest\u00eda. Pero, en cambio, en su manera de moverse, tan gatuna, y de pararse, apartando las piernas, quebrando la cadera, echando atr\u00e1s los hombros y colocando las manitas con desenvoltura invitadora en la cintura, y, sobre todo, en su manera de mirar; con esos ojos profanos y aterciopelados, y de morderse el labio inferior con unos dientecillos de rat\u00f3n, Sarita Huanca Salaverr\u00eda parec\u00eda tener una experiencia dilatada, una sabidur\u00eda de siglos.\n\nEl doctor don Barreda y Zald\u00edvar ten\u00eda un tacto extremado para interrogar a los menores. Sab\u00eda inspirarles confianza, dar rodeos para no herir sus sentimientos, y le era f\u00e1cil, con suavidad y paciencia, inducirlos a trajinar escabrosos asuntos. Pero su experiencia esta vez no le sirvi\u00f3. Apenas pregunt\u00f3, eufem\u00edsticamente, a la menor si era cierto que Gumercindo Tello la molestaba desde hac\u00eda tiempo con frases maleducadas, Sarita Huanca se lanz\u00f3 a hablar. S\u00ed, desde que vino a vivir a La Victoria, a todas horas, en todos los sitios. Iba a esperarla al paradero del \u00f3mnibus y la acompa\u00f1aba hasta la casa dici\u00e9ndole: \u00abMe gustar\u00eda chuparte la miel\u00bb, \u00abt\u00fa tienes dos naranjitos y yo un platanito\u00bb y \u00abpor ti me estoy chorreando de amor\u00bb. Pero no fueron estas alegor\u00edas, tan inconvenientes en boca de una ni\u00f1a, lo que calde\u00f3 las mejillas del magistrado y ator\u00f3 la mecanograf\u00eda del doctor Zelaya, sino las acciones con que Sarita comenz\u00f3 a ilustrar las acechanzas de que fuera objeto. El mec\u00e1nico siempre estaba tratando de tocarla, aqu\u00ed: y las dos manitas, elev\u00e1ndose, se ahuecaron sobre los tiernos pechos y dedicaron a calentarlos amorosamente. Y tambi\u00e9n aqu\u00ed: y las manitas ca\u00edan sobre las rodillas y las repasaban, y sub\u00edan, sub\u00edan, arrugando la falda, por los (hasta hac\u00eda poco imp\u00faberes) muslitos. Pesta\u00f1eando, tosiendo, cambiando una veloz mirada con el secretario, el doctor don Barreda y Zald\u00edvar explic\u00f3 paternalmente a la ni\u00f1a que no era necesario ser tan concreta, que pod\u00eda quedarse en las generalidades. Y tambi\u00e9n la pellizcaba aqu\u00ed, lo interrumpi\u00f3 Sarita, torn\u00e1ndose de medio lado y alargando hacia \u00e9l una grupa que, s\u00fabitamente, pareci\u00f3 crecer, inflarse como un globo de espuma. El magistrado tuvo el presentimiento vertiginoso de que su oficina pod\u00eda convertirse en cualquier momento en un templo de strip-tease.\n\nHaciendo un esfuerzo para dominar el nerviosismo, el magistrado, con voz calma, incit\u00f3 a la menor a olvidar los proleg\u00f3menos y a concentrarse en el hecho mismo de la violaci\u00f3n. Le explic\u00f3 que, aunque deb\u00eda relatar con objetividad lo sucedido, no era imprescindible que se demorara en los detalles, y la exoner\u00f3 de aquellos que \u2014y el doctor don Barreda y Zald\u00edvar carraspe\u00f3, con una pizca de embarazo\u2014 hirieran su pudor. El magistrado quer\u00eda, de un lado, acortar la entrevista, y, de otro, adecentarla, y pensaba que, al referir la agresi\u00f3n er\u00f3tica, la ni\u00f1a, l\u00f3gicamente conturbada, ser\u00eda expeditiva y sin\u00f3ptica, cauta y superficial.\n\nPero Sarita Huanca Salaverr\u00eda, al o\u00edr la sugesti\u00f3n del juez, como un gallito de pelea al olisquear la sangre, se enardeci\u00f3, excedi\u00f3, verti\u00f3 \u00edntegra en un soliloquio salaz y en una representaci\u00f3n m\u00edmico-seminal que cort\u00f3 la respiraci\u00f3n del doctor don Barreda y Zald\u00edvar y sumi\u00f3 al doctor Zelaya en un desasosiego corporal francamente indecoroso (\u00bfy tal vez masturbatorio?). El mec\u00e1nico hab\u00eda tocado la puerta as\u00ed, y, al ella abrir, la hab\u00eda mirado as\u00ed y hablado as\u00ed, y luego se hab\u00eda arrodillado as\u00ed, toc\u00e1ndose el coraz\u00f3n as\u00ed, y se le hab\u00eda declarado as\u00ed, jur\u00e1ndole que la amaba as\u00ed. Aturdidos, hipnotizados, el juez y el secretario ve\u00edan a la ni\u00f1a mujer aletear como un ave, empinarse como una danzarina, agacharse y alzarse, sonre\u00edr y enojarse, modificar la voz y duplicarla, imitarse a s\u00ed misma y a Gumercindo Tello, y, por fin, caer de hinojos y declarar (se, le) su amor. El doctor don Barreda y Zald\u00edvar estir\u00f3 una mano, balbuce\u00f3 que bastaba, pero ya la v\u00edctima locuaz iba explicando que el mec\u00e1nico la hab\u00eda amenazado con un cuchillo as\u00ed, y se le hab\u00eda abalanzado as\u00ed, haci\u00e9ndola resbalar as\u00ed y tir\u00e1ndose sobre ella as\u00ed y cogi\u00e9ndole la falda as\u00ed, y en ese momento el juez \u2014p\u00e1lido, noble, mayest\u00e1tico, iracundo profeta b\u00edblico\u2014 se incorpor\u00f3 en el asiento y rugi\u00f3: \u00ab\u00a1Basta! \u00a1Basta! \u00a1Suficiente!\u00bb. Era la primera vez en su vida que levantaba la voz.\n\nDesde el suelo, donde se hab\u00eda tendido al llegar al punto neur\u00e1lgico de su gr\u00e1fica deposici\u00f3n, Sarita Huanca Salaverr\u00eda miraba asustada al \u00edndice que parec\u00eda fulminarla.\n\n\u2014No necesito saber m\u00e1s \u2014repiti\u00f3, m\u00e1s suavemente, el magistrado\u2014. Ponte de pie, al\u00edsate la falda, vuelve donde tus padres.\n\nLa v\u00edctima se incorpor\u00f3, asintiendo, con una carita descargada de todo histrionismo e impudor, ni\u00f1a de nuevo, visiblemente compungida. Haciendo venias humildes fue retrocediendo hasta la puerta y sali\u00f3. El juez se volvi\u00f3 entonces al secretario y, con tono medido, nada ir\u00f3nico, le sugiri\u00f3 que dejara de teclear pues \u00bfno se daba cuenta acaso que el papel se hab\u00eda deslizado al suelo y que estaba escribiendo sobre el rodillo vac\u00edo? Granate, el doctor Zelaya tartamude\u00f3 que lo ocurrido lo hab\u00eda perturbado. El doctor don Barreda y Zald\u00edvar le sonri\u00f3:\n\n\u2014Nos ha sido dado presenciar un espect\u00e1culo fuera de lo com\u00fan \u2014filosof\u00f3 el magistrado\u2014. Esa ni\u00f1a tiene el demonio en la sangre y, lo peor, es que probablemente no lo sabe.\n\n\u2014\u00bfEs eso lo que los norteamericanos llaman una Lolita, doctor? \u2014intent\u00f3 acrecentar sus conocimientos el secretario.\n\n\u2014Sin duda, una Lolita t\u00edpica \u2014sentenci\u00f3 el juez. Y, poniendo al mal tiempo buena cara, lobo de mar irredimible que aun de los ciclones saca lecciones optimistas, a\u00f1adi\u00f3\u2014: Por lo menos, alegr\u00e9monos de saber que, en este campo, el coloso del norte no tiene la exclusiva. Esta aborigen puede pararle el macho a cualquier Lolita gringa.\n\n\u2014Se comprende que haya sacado de sus casillas al asalariado y que \u00e9ste la violara \u2014divag\u00f3 el secretario\u2014. Despu\u00e9s de verla y o\u00edrla uno jurar\u00eda que fue ella quien lo desvirg\u00f3.\n\n\u2014Alto ah\u00ed, le proh\u00edbo esa clase de presunciones \u2014lo reconvino el juez y el secretario palideci\u00f3\u2014. Nada de adivinanzas suspicaces. Que comparezca Gumercindo Tello.\n\nDiez minutos despu\u00e9s, cuando lo vio entrar al despacho, escoltado por dos guardias, el doctor don Barreda y Zald\u00edvar comprendi\u00f3 inmediatamente que la catalogaci\u00f3n del secretario era abusiva. No se trataba de un lombrosiano sino de algo, en cierto sentido, much\u00edsimo m\u00e1s grave: de un creyente. Con un escalofr\u00edo mnemot\u00e9cnico que le eriz\u00f3 los vellos del pescuezo, el juez, al ver la cara de Gumercindo Tello, record\u00f3 la inmutable mirada del hombre de la bicicleta y la revista Despierta con el que hab\u00eda tenido pesadillas, esa mirada tranquilamente testaruda del que sabe, del que no tiene dudas, del que ha resuelto los problemas. Era un muchacho que, sin duda, no hab\u00eda cumplido a\u00fan los treinta a\u00f1os, y cuyo f\u00edsico enteco, puro hueso y pellejo, pregonaba a los vientos el desprecio que le merec\u00edan la comida y la materia, con los cabellos cortados casi al rape, moreno y m\u00e1s bien bajo. Vest\u00eda un gris neblina, no dandy ni mendigo, sino medio pelo, seco ya pero muy arrugado por culpa de las bautismales zambullidas, una camisa blanca y unos botines con herrajes. Bast\u00f3 un vistazo al juez \u2014hombre de olfato antropol\u00f3gico\u2014 para saber que sus se\u00f1as an\u00edmicas eran: discreci\u00f3n, sobriedad, ideas fijas, imperturbabilidad y vocaci\u00f3n de esp\u00edritu. Con mucha crianza, apenas cruz\u00f3 el umbral, dese\u00f3 al juez y al secretario unos cordiales buenos d\u00edas.\n\nEl doctor don Barreda y Zald\u00edvar orden\u00f3 a los guardias que le quitaran las esposas y salieran. Era una costumbre que hab\u00eda nacido con su carrera judicial: aun a los m\u00e1s crapulosos criminales los hab\u00eda interrogado a solas, sin coacci\u00f3n, paternalmente, y en esos t\u00eate-\u00e0-t\u00eate, \u00e9stos sol\u00edan abrirle su coraz\u00f3n como penitente a confesor. Nunca hab\u00eda tenido que lamentar esta arriesgada pr\u00e1ctica. Gumercindo Tello se frot\u00f3 las mu\u00f1ecas y agradeci\u00f3 la prueba de confianza. El juez le se\u00f1al\u00f3 un asiento y el mec\u00e1nico se sent\u00f3, al borde mismo, en actitud erecta, como un hombre al que la noci\u00f3n misma de comodidad incomodaba. El juez compuso mentalmente la divisa que, sin duda, reg\u00eda la vida del Testigo: levantarse de la cama con sue\u00f1o, de la mesa con hambre y (si alguna vez iba) salirse del cine antes del final. Intent\u00f3 imaginarlo banderillado, incendiado por la infantil vampiresa de La Victoria, pero en el acto cancel\u00f3 esa operaci\u00f3n imaginaria como lesiva a los derechos de la defensa. Gumercindo Tello se hab\u00eda puesto a hablar:\n\n\u2014Es verdad que no prestamos servidumbre a gobiernos, partidos, ej\u00e9rcitos y dem\u00e1s instituciones visibles, que son todas hijastras de Sat\u00e1n \u2014dec\u00eda con dulzura\u2014, que no juramos fidelidad a ning\u00fan trapo con colorines, ni vestimos uniformes, porque no nos engatusan los oropeles ni los disfraces, y que no aceptamos los injertos de piel o de sangre, porque lo que Dios hizo la ciencia no lo deshar\u00e1. Pero nada de eso quiere decir que no cumplamos nuestras obligaciones. Se\u00f1or juez, estoy a sus \u00f3rdenes para lo que se le ofrezca y sepa que ni aun con motivos le faltar\u00eda el respeto.\n\nHablaba de manera pausada, como para facilitar la tarea del secretario, que iba acompa\u00f1ando con m\u00fasica mecanogr\u00e1fica su perorata. El juez le agradeci\u00f3 sus amables prop\u00f3sitos, le hizo saber que respetaba todas las ideas y creencias, muy en especial las religiosas, y se permiti\u00f3 recordarle que no estaba detenido por las que profesaba sino bajo acusaci\u00f3n de haber golpeado y violentado a una menor.\n\nUna sonrisa abstracta cruz\u00f3 el rostro del muchacho de Moquegua.\n\n\u2014Testigo es el que testimonia, el que testifica, el que atestigua \u2014revel\u00f3 su versaci\u00f3n en el saber sem\u00e1ntico, mirando fijamente al juez\u2014, el que sabiendo que Dios existe lo hace saber, el que conociendo la verdad la hace conocer. Yo soy Testigo y ustedes dos tambi\u00e9n podr\u00edan serlo con un poco de voluntad.\n\n\u2014Gracias, para otra ocasi\u00f3n \u2014lo interrumpi\u00f3 el juez, levantando el grueso expediente y pas\u00e1ndoselo por los ojos como si fuera un manjar\u2014. El tiempo apremia y esto es lo que importa. Vamos al grano. Y, para principiar, un consejo: lo recomendable, lo que le conviene, es la verdad, la limpia verdad.\n\nEl acusado, conmovido por alguna rememoraci\u00f3n secreta, suspir\u00f3 hondo.\n\n\u2014La verdad, la verdad \u2014murmur\u00f3 con tristeza\u2014. \u00bfCu\u00e1l, se\u00f1or juez? \u00bfNo se tratar\u00e1, m\u00e1s bien, de esas calumnias, de esos contrabandos, de esas supercher\u00edas vaticanas que, aprovechando la ingenuidad del vulgo, nos quieren hacer pasar por la verdad? Modestia aparte, yo creo que conozco la verdad, pero, y se lo pregunto sin ofensa, \u00bfla conoce usted?\n\n\u2014Me propongo conocerla \u2014dijo el juez, astutamente, palmoteando el cartapacio.\n\n\u2014\u00bfLa verdad en torno a la fantas\u00eda de la cruz, a la broma de Pedro y la piedra, a las mitras, tal vez a la tomadura de pelo papal de la inmortalidad del alma? \u2014se preguntaba sarc\u00e1sticamente Gumercindo Tello.\n\n\u2014La verdad en torno al delito cometido por usted al abusar de la menor Sarita Huanca Salaverr\u00eda \u2014contraatac\u00f3 el magistrado\u2014. La verdad en torno a ese atropello a una inocente de trece a\u00f1os. La verdad en torno a los golpes que le propin\u00f3, a las amenazas con que la aterroriz\u00f3, al estupro con que la humill\u00f3 y tal vez pre\u00f1\u00f3.\n\nLa voz del magistrado se hab\u00eda ido elevando, acusatoria y ol\u00edmpica. Gumercindo Tello lo miraba muy serio, r\u00edgido como la silla que ocupaba, sin indicios de confusi\u00f3n ni arrepentimiento. Por fin, mene\u00f3 la cabeza con suavidad de res:\n\n\u2014Estoy preparado para cualquier prueba a que quiera someterme Jehov\u00e1 \u2014asegur\u00f3.\n\n\u2014No se trata de Dios sino de usted \u2014lo regres\u00f3 a la tierra el magistrado\u2014. De sus apetitos, de su lujuria, de su libido.\n\n\u2014Se trata siempre de Dios, se\u00f1or juez \u2014se empecin\u00f3 Gumercindo Tello\u2014. Nunca de usted, ni de m\u00ed, ni de nadie. De \u00c9l, s\u00f3lo de \u00c9l.\n\n\u2014Sea usted responsable \u2014lo exhort\u00f3 el juez\u2014. At\u00e9ngase a los hechos. Admita su falta y la Justicia tal vez lo considere. Proceda como el hombre religioso que trata de hacerme creer que es.\n\n\u2014Me arrepiento de todas mis culpas, que son infinitas \u2014dijo, l\u00fagubremente, Gumercindo Tello\u2014. S\u00e9 muy bien que soy un pecador, se\u00f1or juez.\n\n\u2014Bien, los hechos concretos \u2014lo apremi\u00f3 el doctor don Barreda y Zald\u00edvar\u2014. Puntual\u00edceme, sin regodeos morbosos ni jeremiadas, c\u00f3mo fue que la viol\u00f3.\n\nPero el Testigo ya hab\u00eda prorrumpido en sollozos, cubri\u00e9ndose la cara con las manos. El magistrado no se inmut\u00f3. Estaba habituado a las bruscas alternancias ciclot\u00edmicas de los acusados y sab\u00eda aprovecharlas para la averiguaci\u00f3n de los hechos. Viendo a Gumercindo Tello as\u00ed, cabizbajo, su cuerpo agitado, sus manos h\u00famedas de l\u00e1grimas, el doctor don Barreda y Zald\u00edvar se dijo, sobrio orgullo de profesional que comprueba la eficacia de su t\u00e9cnica, que el acusado hab\u00eda llegado a ese clim\u00e1tico estado emotivo en el que, inapto ya para disimular, proferir\u00eda ansiosa, espont\u00e1nea, caudalosamente la verdad.\n\n\u2014Datos, datos \u2014insisti\u00f3\u2014. Hechos, lugares, posiciones, palabras dichas, actos actuados. \u00a1Vamos, valor!\n\n\u2014Es que no s\u00e9 mentir, se\u00f1or juez \u2014balbuce\u00f3 Gumercindo Tello, entre hipos\u2014. Estoy dispuesto a sufrir lo que sea, insulto, c\u00e1rcel, deshonor. \u00a1Pero no puedo mentir! \u00a1Nunca aprend\u00ed, no soy capaz!\n\n\u2014Bien, bien, esa incapacidad lo honra \u2014exclam\u00f3, con gesto alentador, el juez\u2014. Demu\u00e9stremela. Vamos, \u00bfc\u00f3mo fue que la viol\u00f3?\n\n\u2014Ah\u00ed est\u00e1 el problema \u2014se desesper\u00f3, tragando babas, el Testigo\u2014. \u00a1Es que yo no la viol\u00e9!\n\n\u2014Voy a decirle algo, se\u00f1or Tello \u2014silabe\u00f3, suavidad de serpiente que es todav\u00eda m\u00e1s despectiva, el magistrado\u2014: \u00a1Es usted un falso Testigo de Jehov\u00e1! \u00a1Un impostor!\n\n\u2014No la he tocado, jam\u00e1s le habl\u00e9 a solas, ayer ni siquiera la vi \u2014dec\u00eda, corderillo que bala, Gumercindo Tello.\n\n\u2014Un c\u00ednico, un farsante, un prevaricador espiritual \u2014sentenciaba, t\u00e9mpano de hielo, el juez\u2014. Si la justicia y la moral no le importan, respete al menos a ese Dios que tanto nombra. Piense en que ahora mismo lo ve, en lo asqueado que debe estar al o\u00edrlo mentir.\n\n\u2014Ni con la mirada ni con el pensamiento he ofendido a esa ni\u00f1a \u2014repiti\u00f3, con acento desgarrador, Gumercindo Tello.\n\n\u2014La ha amenazado, golpeado y violado \u2014se destempl\u00f3 la voz del magistrado\u2014. \u00a1Con su sucia lujuria, se\u00f1or Tello!\n\n\u2014\u00bfCon-mi-su-cia-lu-ju-ria? \u2014repiti\u00f3, hombre que acaba de recibir un martillazo, el Testigo.\n\n\u2014Con su sucia lujuria, s\u00ed se\u00f1or \u2014refrend\u00f3 el magistrado, y, luego de una pausa creativa\u2014: \u00a1Con su pene pecador!\n\n\u2014\u00bfCon-mi-pe-ne-pe-ca-dor? \u2014tartamude\u00f3, voz desfalleciente y expresi\u00f3n de pasmo, el acusado\u2014. \u00bfMi-pe-ne-pe-ca-dor-ha-di-cho-us-ted?\n\nEstramb\u00f3ticos y estr\u00e1bicos, saltamontes at\u00f3nitos, sus ojos pasearon del secretario al juez, del suelo al techo, de la silla al escritorio y all\u00ed permanecieron, recorriendo papeles, expedientes, secantes. Hasta que se iluminaron sobre el cortapapeles Tiahuanaco que descollaba entre todos los objetos con art\u00edstico centelleo prehisp\u00e1nico. Entonces, movimiento tan r\u00e1pido que no dio tiempo al juez ni al secretario a intentar un gesto para impedirlo, Gumercindo Tello estir\u00f3 la mano y se apoder\u00f3 del pu\u00f1al. No hizo ning\u00fan adem\u00e1n amenazador, todo lo contrario, estrech\u00f3, madre que abriga a su peque\u00f1o, el plateado cuchillo contra su pecho, y dirigi\u00f3 una tranquilizadora, bondadosa, triste mirada a los dos hombres petrificados de sorpresa.\n\n\u2014Me ofenden creyendo que podr\u00eda lastimarlos \u2014dijo con voz de penitente.\n\n\u2014No podr\u00e1 huir jam\u00e1s, insensato \u2014le advirti\u00f3, reponi\u00e9ndose, el magistrado\u2014. El Palacio de Justicia est\u00e1 lleno de guardias, lo matar\u00e1n.\n\n\u2014\u00bfHuir yo? \u2014pregunt\u00f3 con iron\u00eda el mec\u00e1nico\u2014. Qu\u00e9 poco me conoce, se\u00f1or juez.\n\n\u2014\u00bfNo ve que se est\u00e1 delatando? \u2014insisti\u00f3 el magistrado\u2014. Devu\u00e9lvame el cortapapeles.\n\n\u2014Lo he cogido prestado para probar mi inocencia \u2014explic\u00f3 serenamente Gumercindo Tello.\n\nEl juez y el secretario se miraron. El acusado se hab\u00eda puesto de pie. Ten\u00eda una expresi\u00f3n nazarena, en su mano derecha el cuchillo desped\u00eda un brillo premonitorio y terrible. Su mano izquierda se desliz\u00f3 sin prisa hacia la ranura del pantal\u00f3n que ocultaba el cierre rel\u00e1mpago y, mientras, iba diciendo con voz adolorida:\n\n\u2014Yo soy puro, se\u00f1or juez, yo no he conocido mujer. A m\u00ed, eso que otros usan para pecar, s\u00f3lo me sirve para hacer pip\u00ed...\n\n\u2014Alto ah\u00ed \u2014lo interrumpi\u00f3, con una sospecha atroz, el doctor don Barreda y Zald\u00edvar\u2014. \u00bfQu\u00e9 va usted a hacer?\n\n\u2014Cortarlo y botarlo a la basura para probarle lo poco que me importa \u2014replic\u00f3 el acusado, mostrando con el ment\u00f3n el cesto de papeles.\n\nHablaba sin soberbia, con tranquila determinaci\u00f3n. El juez y el secretario, boquiabiertos, no atinaban a gritar. Gumercindo Tello ten\u00eda ya en la mano izquierda el cuerpo del delito y elevaba el cuchillo para, verdugo que blande el hacha y mide la trayectoria hacia el cuello del condenado, dejarlo caer y consumar la inconcebible prueba.\n\n\u00bfLo har\u00eda? \u00bfSe privar\u00eda as\u00ed, de un tajo, de su integridad? \u00bfSacrificar\u00eda su cuerpo, su juventud, su honor, en pos de una demostraci\u00f3n \u00e9tico-abstracta? \u00bfConvertir\u00eda Gumercindo Tello el m\u00e1s respetable despacho judicial de Lima en ara de sacrificios? \u00bfC\u00f3mo terminar\u00eda ese drama forense?\n\n### VII\n\nLOS AMORES con la t\u00eda Julia continuaban viento en popa, pero las cosas se iban complicando porque resultaba dif\u00edcil mantener la clandestinidad. De com\u00fan acuerdo, para no provocar sospechas en la familia, hab\u00eda reducido dr\u00e1sticamente mis visitas a casa del t\u00edo Lucho. S\u00f3lo segu\u00eda yendo con puntualidad al almuerzo de los jueves. Para el cine de las noches invent\u00e1bamos diversas tretas. La t\u00eda Julia sal\u00eda temprano, llamaba a la t\u00eda Olga para decirle que comer\u00eda con una amiga y me esperaba en alg\u00fan lugar acordado. Pero esta operaci\u00f3n ten\u00eda el inconveniente de que la t\u00eda Julia deb\u00eda pasarse horas en las calles, hasta que yo saliera del trabajo, y de que la mayor parte de las veces ayunaba. Otros d\u00edas yo iba a buscarla en un taxi, sin bajarme; ella estaba alerta y apenas ve\u00eda detenerse el autom\u00f3vil sal\u00eda corriendo. Pero era una estratagema riesgosa: si me descubr\u00edan, inmediatamente sabr\u00edan que hab\u00eda algo entre ella y yo; y, de todos modos, ese misterioso invitador, emboscado en el fondo de un taxi, terminar\u00eda por despertar curiosidad, malicia, muchas preguntas...\n\nHab\u00edamos optado, por eso, en vernos menos de noche y m\u00e1s de d\u00eda, aprovechando los huecos de la radio. La t\u00eda Julia tomaba un colectivo al centro y a eso de las once de la ma\u00f1ana, o de las cinco de la tarde, me esperaba en una cafeter\u00eda de Caman\u00e1, o en el Cream Rica del jir\u00f3n de la Uni\u00f3n. Yo dejaba revisados un par de boletines y pod\u00edamos pasar dos horas juntos. Hab\u00edamos descartado el Bransa de la Colmena porque all\u00ed acud\u00eda toda la gente de Panamericana y de Radio Central. De vez en cuando (m\u00e1s exactamente, los d\u00edas de pago) la invitaba a almorzar y entonces est\u00e1bamos hasta tres horas juntos. Pero mi magro salario no permit\u00eda esos excesos. Hab\u00eda conseguido, luego de un elaborado discurso, una ma\u00f1ana en que lo encontr\u00e9 euf\u00f3rico por los \u00e9xitos de Pedro Camacho, que Genaro hijo me aumentara el sueldo, con lo que llegu\u00e9 a redondear cinco mil soles. Daba dos mil a mis abuelos para ayudarlos en la casa. Los tres mil restantes me alcanzaban antes de sobra para mis vicios: el cigarrillo, el cine y los libros. Pero, desde mis amores con la t\u00eda Julia, se volatizaban velozmente y andaba siempre apurado, recurriendo con frecuencia a pr\u00e9stamos e, incluso, a la Caja Nacional de Pignoraci\u00f3n, en la plaza de Armas. Como, por otra parte, ten\u00eda firmes prejuicios hisp\u00e1nicos respecto a las relaciones entre hombres y mujeres y no permit\u00eda que la t\u00eda Julia pagara ninguna cuenta, mi situaci\u00f3n econ\u00f3mica llegaba a ser dram\u00e1tica. Para aliviarla, comenc\u00e9 a hacer algo que Javier severamente llam\u00f3 \u00abprostituir mi pluma\u00bb. Es decir, a escribir rese\u00f1as de libros y reportajes en suplementos culturales y revistas de Lima. Los publicaba con seud\u00f3nimo, para avergonzarme menos de lo malos que eran. Pero los doscientos o trescientos soles m\u00e1s al mes constitu\u00edan un t\u00f3nico para mi presupuesto.\n\nEsas citas en los cafetines del centro de Lima eran poco pecaminosas, largas conversaciones muy rom\u00e1nticas, haciendo empanaditas, mir\u00e1ndonos a los ojos, y, si la topograf\u00eda del local lo permit\u00eda, roz\u00e1ndonos las rodillas. S\u00f3lo nos bes\u00e1bamos cuando nadie pod\u00eda vernos, lo que ocurr\u00eda rara vez, porque a esas horas los caf\u00e9s estaban siempre repletos de oficinistas lisurientos. Habl\u00e1bamos de nosotros, por supuesto, de los peligros que corr\u00edamos de ser sorprendidos por alg\u00fan miembro de la familia, de la manera de conjurar esos peligros, nos cont\u00e1bamos con lujo de detalles todo lo que hab\u00edamos hecho desde la \u00faltima vez (es decir, algunas horas atr\u00e1s o el d\u00eda anterior), pero, en cambio, jam\u00e1s hac\u00edamos ning\u00fan plan para el futuro. El porvenir era un asunto t\u00e1citamente abolido en nuestros di\u00e1logos, sin duda porque, tanto ella como yo, est\u00e1bamos convencidos que nuestra relaci\u00f3n no tendr\u00eda ninguno. Sin embargo, pienso que eso que hab\u00eda comenzado como un juego, se fue volviendo serio en los castos encuentros de los caf\u00e9s humosos del centro de Lima. Fue ah\u00ed donde, sin darnos cuenta, nos fuimos enamorando.\n\nHabl\u00e1bamos tambi\u00e9n mucho de literatura; o, mejor dicho, la t\u00eda Julia escuchaba y yo le hablaba de la buhardilla de Par\u00eds (ingrediente inseparable de mi vocaci\u00f3n) y de todas las novelas, los dramas, los ensayos que escribir\u00eda cuando fuera escritor. La tarde que nos descubri\u00f3 Javier, en el Cream Rica del jir\u00f3n de la Uni\u00f3n, yo estaba ley\u00e9ndole a la t\u00eda Julia mi cuento sobre Doroteo Mart\u00ed. Se titulaba, medievalescamente, La humillaci\u00f3n de la cruz y ten\u00eda cinco p\u00e1ginas. Era el primer cuento que le le\u00eda, y lo hice muy despacio, para disimular mi inquietud por su veredicto. La experiencia fue catastr\u00f3fica para la susceptibilidad del futuro escritor. A medida que progresaba en la lectura, la t\u00eda Julia me iba interrumpiendo:\n\n\u2014Pero si no fue as\u00ed, pero si lo has puesto todo patas arriba \u2014me dec\u00eda, sorprendida y hasta enojada\u2014, pero si no fue eso lo que dijo, pero si...\n\nYo, angustiad\u00edsimo, hac\u00eda un alto para informarle que lo que escuchaba no era la relaci\u00f3n fiel de la an\u00e9cdota que me hab\u00eda contado, sino un cuento, un cuento, y que todas las cosas a\u00f1adidas o suprimidas eran recursos para conseguir ciertos efectos:\n\n\u2014Efectos c\u00f3micos \u2014subray\u00e9, a ver si entend\u00eda y, aunque fuera por conmiseraci\u00f3n, sonre\u00eda.\n\n\u2014Pero, al contrario \u2014protest\u00f3 la t\u00eda Julia, impert\u00e9rrita y feroz\u2014, con las cosas que has cambiado le quitaste toda la gracia. Qui\u00e9n se va a creer que pasa tanto rato desde que la cruz comienza a moverse hasta que se cae. \u00bfD\u00f3nde est\u00e1 el chiste ahora?\n\nYo, aunque hab\u00eda ya decidido, en mi humillada intimidad, enviar el cuento sobre Doroteo Mart\u00ed al canasto de la basura, estaba enfrascado en una defensa ardorosa, adolorida, de los derechos de la imaginaci\u00f3n literaria a transgredir la realidad, cuando sent\u00ed que me tocaban el hombro.\n\n\u2014Si interrumpo, me lo dicen y me voy porque odio tocar viol\u00edn \u2014dijo Javier, jalando una silla, sent\u00e1ndose y pidiendo un caf\u00e9 al mozo. Sonri\u00f3 a la t\u00eda Julia\u2014: Encantado, yo soy Javier, el mejor amigo de este prosista. Qu\u00e9 bien guardada te la ten\u00edas, compadre.\n\n\u2014Es Julita, la hermana de mi t\u00eda Olga \u2014le expliqu\u00e9.\n\n\u2014\u00bfC\u00f3mo? \u00bfLa famosa boliviana? \u2014se le fueron apagando los br\u00edos a Javier. Nos hab\u00eda encontrado de la mano, no nos hab\u00edamos soltado, y ahora miraba fijo, sin la seguridad mundana de antes, nuestros dedos entrelazados\u2014. Vaya, vaya, Varguitas.\n\n\u2014\u00bfYo soy la famosa boliviana? \u2014pregunt\u00f3 la t\u00eda Julia\u2014. \u00bfFamosa por qu\u00e9?\n\n\u2014Por antip\u00e1tica, por esos chistes tan pasados, cuando llegaste \u2014la puse al d\u00eda\u2014. Javier s\u00f3lo conoce la primera parte de la historia.\n\n\u2014La mejor me la hab\u00edas ocultado, mal narrador y peor amigo \u2014dijo Javier, recuperando la soltura y se\u00f1alando las empanaditas\u2014. Qu\u00e9 me cuentan, qu\u00e9 me cuentan.\n\nEstuvo realmente simp\u00e1tico, hablando hasta por los codos y haciendo toda clase de bromas, y la t\u00eda Julia qued\u00f3 encantada con \u00e9l. Me alegr\u00e9 de que nos hubiera descubierto; no hab\u00eda planeado contarle mis amores, porque era reacio a confidencias sentimentales (y m\u00e1s todav\u00eda en este caso, tan enredado) pero ya que el azar lo hab\u00eda hecho part\u00edcipe del secreto, me dio gusto poder comentar con \u00e9l las peripecias de esta aventura. Esa ma\u00f1ana se despidi\u00f3 besando a la t\u00eda Julia en la mejilla y haciendo una reverencia:\n\n\u2014Soy un celestino de primera, cuenten conmigo para cualquier cosa.\n\n\u2014\u00bfPor qu\u00e9 no dijiste tambi\u00e9n que nos tender\u00edas la cama? \u2014lo re\u00f1\u00ed esa tarde, apenas se present\u00f3 en mi gallinero de Radio Panamericana, \u00e1vido de detalles.\n\n\u2014\u00bfEs algo as\u00ed como tu t\u00eda, no? \u2014dijo, palmote\u00e1ndome\u2014. Est\u00e1 bien, me has impresionado. Una amante vieja, rica y divorciada: \u00a1veinte puntos!\n\n\u2014No es mi t\u00eda, sino la hermana de la mujer de mi t\u00edo \u2014le expliqu\u00e9 lo que ya sab\u00eda, mientras daba vuelta a una noticia de La Prensa sobre la guerra de Corea\u2014. No es mi amante, no es vieja y no tiene medio. S\u00f3lo lo de divorciada es verdad.\n\n\u2014Vieja quer\u00eda decir mayor que t\u00fa, y lo de rica no era cr\u00edtica sino felicitaci\u00f3n, yo soy partidario de los braguetazos \u2014se ri\u00f3 Javier\u2014. \u00bfAs\u00ed que no es tu amante? \u00bfQu\u00e9, entonces? \u00bfTu enamorada?\n\n\u2014Una cosa entre las dos \u2014le dije, sabiendo que lo irritar\u00eda.\n\n\u2014Ah, quieres hacerte el misterioso, pues te vas a la mierda ipso facto \u2014me advirti\u00f3\u2014. Y, adem\u00e1s, eres un miserable: yo te cuento todos mis amores con la flaca Nancy y lo del braguetazo t\u00fa me lo hab\u00edas ocultado.\n\nLe cont\u00e9 la historia desde el principio, las complicaciones que ten\u00edamos para vernos y entendi\u00f3 por qu\u00e9 en las \u00faltimas semanas le hab\u00eda pedido dos o tres veces plata prestada. Se interes\u00f3, me comi\u00f3 a preguntas y acab\u00f3 jur\u00e1ndome que se convertir\u00eda en mi hada madrina. Pero, al despedirse, se puso grave:\n\n\u2014Supongo que esto es un juego \u2014me sermone\u00f3, mir\u00e1ndome a los ojos como un padre sol\u00edcito\u2014. No se olvide que, a pesar de todo, usted y yo somos todav\u00eda dos mocosos.\n\n\u2014Si quedo encinta, te juro que me har\u00e9 abortar \u2014lo tranquilic\u00e9.\n\nUna vez que se fue, y mientras Pascual entreten\u00eda al Gran Pablito con un choque serial, en Alemania, en el que una veintena de autom\u00f3viles se hab\u00edan incrustado uno en el otro por culpa de un distra\u00eddo turista belga que estacion\u00f3 su auto en plena carretera para auxiliar a un perrito, me qued\u00e9 pensando. \u00bfEra cierto que esta historia no iba en serio? S\u00ed, cierto. Se trataba de una experiencia distinta, algo m\u00e1s madura y atrevida que todas las que hab\u00eda vivido, pero, para que el recuerdo fuera bueno, no deber\u00eda durar mucho. Estaba en estas reflexiones cuando entr\u00f3 Genaro hijo a invitarme a almorzar. Me llev\u00f3 a Magdalena, a un jard\u00edn criollo, me impuso un arroz con pato y unos picarones con miel, y a la hora del caf\u00e9 me pas\u00f3 la factura:\n\n\u2014Eres su \u00fanico amigo, h\u00e1blale, nos est\u00e1 metiendo en un l\u00edo de los diablos. Yo no puedo, a m\u00ed me dice inculto, ignaro, ayer a mi padre lo llam\u00f3 mes\u00f3crata. Quiero evitarme m\u00e1s l\u00edos con \u00e9l. Tendr\u00eda que botarlo y eso ser\u00eda una cat\u00e1strofe para la empresa.\n\nEl problema era una carta del embajador argentino dirigida a Radio Central, en lenguaje mef\u00edtico, protestando por las alusiones \u00abcalumniosas, perversas y psic\u00f3ticas\u00bb contra la patria de Sarmiento y San Mart\u00edn que aparec\u00edan por doquier en las radionovelas (que el diplom\u00e1tico llamaba \u00abhistorias dram\u00e1ticas serializadas\u00bb). El embajador ofrec\u00eda algunos ejemplos que, aseguraba, no hab\u00edan sido buscados ex profeso sino recogidos al azar por el personal de la legaci\u00f3n \u00abafecto a ese g\u00e9nero de emisiones\u00bb. En una se suger\u00eda, nada menos, que la proverbial hombr\u00eda de los porte\u00f1os era un mito, pues casi todos practicaban la homosexualidad (y, de preferencia, la pasiva); en otra, que en las familias bonaerenses, tan gregarias, se sacrificaba por hambre a las bocas in\u00fatiles \u2014ancianos y enfermos\u2014 para aligerar el presupuesto; en otra, que lo de las vacas era para la exportaci\u00f3n porque all\u00e1, en casita, el manjar verdaderamente codiciado era el caballo; en otra, que la extendida pr\u00e1ctica del f\u00fatbol, por culpa sobre todo del cabezazo a la pelota, hab\u00eda lesionado los genes nacionales, lo que explicaba la abundancia proliferante, en las orillas del r\u00edo de color leonado, de oligofr\u00e9nicos, acromeg\u00e1licos, y otras subvariedades de cretinos; que en los hogares de Buenos Aires \u2014\u00absemejante cosm\u00f3polis\u00bb, puntualizaba la carta\u2014 era corriente hacer las necesidades biol\u00f3gicas, en el mismo recinto donde se com\u00eda y dorm\u00eda, en un simple balde...\n\n\u2014T\u00fa te r\u00edes y nosotros tambi\u00e9n nos re\u00edamos \u2014dijo Genaro hijo, comi\u00e9ndose las u\u00f1as\u2014, pero hoy se nos present\u00f3 un abogado y nos quit\u00f3 la risa. Si la embajada protesta ante el gobierno nos pueden cancelar los radioteatros, multar, clausurar la radio. Ru\u00e9gale, amen\u00e1zalo, que se olvide de los argentinos.\n\nLe promet\u00ed hacer lo posible, pero sin muchas esperanzas porque el escriba era un hombre de convicciones inflexibles. Yo hab\u00eda llegado a sentirme amigo de \u00e9l; adem\u00e1s de la curiosidad entomol\u00f3gica que me inspiraba, le ten\u00eda aprecio. Pero \u00bfera rec\u00edproco? Pedro Camacho no parec\u00eda capaz de perder su tiempo, su energ\u00eda, en la amistad ni en nada que lo distrajera de su arte, es decir su trabajo o vicio, esa urgencia que barr\u00eda hombres, cosas, apetitos. Aunque es verdad que a m\u00ed me toleraba m\u00e1s que a otros. Tom\u00e1bamos caf\u00e9 (\u00e9l menta y yerbaluisa) y yo iba a su cub\u00edculo y le serv\u00eda de pausa entre dos p\u00e1ginas. Lo escuchaba con suma atenci\u00f3n y tal vez eso lo halagaba; quiz\u00e1 me ten\u00eda por un disc\u00edpulo, o, simplemente, era para \u00e9l lo que el perrito faldero de la solterona y el crucigrama del jubilado: alguien, algo con que llenar los vac\u00edos.\n\nTres cosas me fascinaban en Pedro Camacho: lo que dec\u00eda, la austeridad de su vida enteramente consagrada a una obsesi\u00f3n, y su capacidad de trabajo. Esto \u00faltimo, sobre todo. En la biograf\u00eda de Emil Ludwig hab\u00eda le\u00eddo la resistencia de Napole\u00f3n, c\u00f3mo sus secretarios se derrumbaban y \u00e9l segu\u00eda dictando, y sol\u00eda imaginarme al emperador de los franceses con la cara nariguda del escribidor y a \u00e9ste, durante alg\u00fan tiempo, Javier y yo lo llamamos el Napole\u00f3n del Altiplano (nombre que altern\u00e1bamos con el de Balzac criollo). Por curiosidad, llegu\u00e9 a establecer su horario de trabajo y, pese a que lo verifiqu\u00e9 muchas veces, siempre me pareci\u00f3 imposible.\n\nEmpez\u00f3 con cuatro radioteatros al d\u00eda, pero, en vista del \u00e9xito, fueron aumentando hasta diez, que se radiaban de lunes a s\u00e1bado, con una duraci\u00f3n de media hora cada cap\u00edtulo (en realidad, veintitr\u00e9s minutos, pues la publicidad acaparaba siete). Como los dirig\u00eda e interpretaba todos, deb\u00eda permanecer en el estudio unas siete horas diarias, calculando que el ensayo y grabaci\u00f3n de cada programa durasen cuarenta minutos (entre diez y quince para su arenga y las repeticiones). Escrib\u00eda los radioteatros a medida que se iban radiando; comprob\u00e9 que cada cap\u00edtulo le tomaba apenas el doble de tiempo que su interpretaci\u00f3n, una hora. Lo cual significaba, de todos modos, unas diez horas en la m\u00e1quina de escribir. Esto disminu\u00eda algo gracias a los domingos, su d\u00eda libre, que \u00e9l, por supuesto, pasaba en su cub\u00edculo, adelantando el trabajo de la semana. Su horario era, pues, entre quince y diecis\u00e9is horas de lunes a s\u00e1bado y de ocho a diez los domingos. Todas ellas pr\u00e1cticamente productivas, de rendimento art\u00edstico sonante.\n\nLlegaba a Radio Central a las ocho de la ma\u00f1ana y part\u00eda cerca de medianoche; sus \u00fanicas salidas a la calle las hac\u00eda conmigo, al Bransa, para tomar las infusiones cerebrales. Almorzaba en su cub\u00edculo, un s\u00e1ndwich y un refresco que le iban a comprar devotamente Jesusito, el Gran Pablito o alguno de sus colaboradores. Jam\u00e1s aceptaba una invitaci\u00f3n, jam\u00e1s le o\u00ed decir que hab\u00eda estado en un cine, un teatro, un partido de f\u00fatbol o en una fiesta. Jam\u00e1s lo vi leer un libro, una revista o un peri\u00f3dico, fuera del mamotreto de citas y de esos planos que eran sus instrumentos de trabajo. Aunque miento: un d\u00eda le descubr\u00ed un bolet\u00edn de socios del Club Nacional.\n\n\u2014Corromp\u00ed al portero con unos cobres \u2014me explic\u00f3, cuando le pregunt\u00e9 por el libraco\u2014. \u00bfDe d\u00f3nde podr\u00eda sacar los nombres de mis arist\u00f3cratas? Para los otros, me bastan las orejas: los plebeyos los recojo del arroyo.\n\nLa fabricaci\u00f3n del radioteatro, la hora que le tomaba producir, sin atorarse, cada libreto, me dejaba siempre incr\u00e9dulo. Muchas veces lo vi redactar esos cap\u00edtulos. A diferencia de lo que ocurr\u00eda con las grabaciones, cuyo secreto defend\u00eda celosamente, no le importaba que lo vieran escribir. Mientras estaba tecleando su (mi) Remington, entraban a interrumpirlo sus actores, Bat\u00e1n o el t\u00e9cnico de sonido. Alzaba la vista, absolv\u00eda las preguntas, daba una indicaci\u00f3n churrigueresca, desped\u00eda al visitante con su sonrisita epid\u00e9rmica, lo m\u00e1s opuesto a la risa que he conocido, y continuaba escribiendo. Yo sol\u00eda meterme al cub\u00edculo con el pretexto de estudiar, de que en mi gallinero hab\u00eda mucho ruido y gente (estudiaba los cursos de Derecho para ex\u00e1menes y olvidaba todo despu\u00e9s de rendirlos: que jam\u00e1s me suspendieran no hablaba bien de m\u00ed sino mal de la universidad). Pedro Camacho no pon\u00eda objeci\u00f3n y hasta parec\u00eda que no le desagradaba esa presencia humana que lo sent\u00eda crear.\n\nMe sentaba en el alf\u00e9izar de la ventana y hund\u00eda la nariz en alg\u00fan c\u00f3digo. En realidad, lo espiaba. Escrib\u00eda con dos dedos, muy r\u00e1pido. Lo ve\u00eda y no lo cre\u00eda: jam\u00e1s se paraba a buscar alguna palabra o contemplar una idea, nunca aparec\u00eda en esos ojitos fan\u00e1ticos y saltones la sombra de una duda. Daba la impresi\u00f3n de estar pasando a limpio un texto que sab\u00eda de memoria, mecanografiando algo que le dictaban. \u00bfC\u00f3mo era posible que, a esa velocidad con que ca\u00edan sus deditos sobre las teclas, estuviera nueve, diez horas al d\u00eda, inventando las situaciones, las an\u00e9cdotas, los di\u00e1logos, de varias historias distintas? Y, sin embargo, era posible: los libretos sal\u00edan de esa cabecita tenaz y de esas manos infatigables, uno tras otro, a la medida adecuada, como sartas de salchichas de una m\u00e1quina. Una vez terminado el cap\u00edtulo, no lo correg\u00eda ni siquiera le\u00eda; lo entregaba a la secretaria para que sacara copias y proced\u00eda, sin soluci\u00f3n de continuidad, a fabricar el siguiente. Una vez le dije que verlo trabajar me recordaba la teor\u00eda de los surrealistas franceses sobre la escritura autom\u00e1tica, aquella que mana directamente del subconsciente, esquivando las censuras de la raz\u00f3n. Obtuve una respuesta nacionalista:\n\n\u2014Los cerebros de nuestra Am\u00e9rica mestiza pueden parir mejores cosas que los franchutes. Nada de complejos, mi amigo.\n\n\u00bfPor qu\u00e9 no utilizaba, como base para sus historias lime\u00f1as, las que hab\u00eda escrito en Bolivia? Se lo pregunt\u00e9 y me repuso con esas generalidades de las que era imposible extraer nada concreto. Las historias, para llegar al p\u00fablico, deb\u00edan ser frescas, como las frutas y los vegetales, pues el arte no toleraba las conservas y menos los alimentos que el tiempo hab\u00eda podrido. De otra parte, necesitaban ser \u00abhistorias comprovincianas de los oyentes\u00bb. \u00bfC\u00f3mo, siendo \u00e9stos lime\u00f1os, se pod\u00edan interesar en episodios ocurridos en La Paz? Pero daba estas razones porque en \u00e9l la necesidad de teorizar, de convertir todo en verdad impersonal, axioma eterno, era tan compulsiva como la de escribir. Sin duda, la raz\u00f3n por la cual no utilizaba sus viejos radioteatros era m\u00e1s simple: porque no ten\u00eda el menor inter\u00e9s en ahorrarse trabajo. Vivir era, para \u00e9l, escribir. No le importaba en absoluto que sus obras durasen. Una vez radiados, se olvidaba de los libretos. Me asegur\u00f3 que no conservaba copia de ninguno de sus radioteatros. \u00c9stos hab\u00edan sido compuestos con el t\u00e1cito convencimiento de que deb\u00edan volatilizarse al ser digeridos por el p\u00fablico. Una vez le pregunt\u00e9 si nunca hab\u00eda pensado publicar:\n\n\u2014Mis escritos se conservan en un lugar m\u00e1s indeleble que los libros \u2014me instruy\u00f3, en el acto\u2014: La memoria de los radioescuchas.\n\nHabl\u00e9 con \u00e9l sobre la protesta argentina el mismo d\u00eda del almuerzo con Genaro hijo. A eso de las seis ca\u00ed por su cub\u00edculo y lo invit\u00e9 al Bransa. Temeroso de su reacci\u00f3n, le solt\u00e9 la noticia a pocos: hab\u00eda gente muy susceptible, incapaz de tolerar iron\u00edas, y, de otro lado, en el Per\u00fa, la legislaci\u00f3n en materia de libelo era sever\u00edsima, una radio pod\u00eda ser clausurada por una insignificancia. La embajada argentina, dando pruebas de poco mundo, se hab\u00eda sentido herida por algunas alusiones y amenazaba con una queja oficial ante la canciller\u00eda...\n\n\u2014En Bolivia lleg\u00f3 a haber amenaza de rompimiento de relaciones \u2014me interrumpi\u00f3\u2014. Un pasqu\u00edn incluso rumore\u00f3 algo sobre concentraci\u00f3n de tropas en las fronteras.\n\nLo dec\u00eda resignado, como pensando: la obligaci\u00f3n del sol es echar rayos, qu\u00e9 remedio si eso provoca alg\u00fan incendio.\n\n\u2014Los Genaros le piden que, en lo posible, evite hablar mal de los argentinos en los radioteatros \u2014le confes\u00e9 y encontr\u00e9 un argumento que, supuse, le har\u00eda mella\u2014: Total, mejor ni se ocupe de ellos, \u00bfacaso valen la pena?\n\n\u2014La valen, porque ellos me inspiran \u2014me explic\u00f3, dando por cancelado el asunto.\n\nDe regreso a la radio me hizo saber, con una inflexi\u00f3n traviesa en la voz, que el esc\u00e1ndalo de La Paz \u00ables sac\u00f3 roncha\u00bb y que fue motivado por una obra de teatro sobre \u00ablas costumbres bestiales de los gauchos\u00bb. En Panamericana, le dije a Genaro hijo que no deb\u00eda hacerse ilusiones respecto a mi eficacia como mediador.\n\nDos o tres d\u00edas despu\u00e9s, conoc\u00ed la pensi\u00f3n de Pedro Camacho. La t\u00eda Julia hab\u00eda venido a encontrarse conmigo a la hora del \u00faltimo bolet\u00edn, porque quer\u00eda ver una pel\u00edcula que daban en el Metro, con una de las grandes parejas rom\u00e1nticas: Greer Garson y Walter Pidgeon. Cerca de medianoche, est\u00e1bamos cruzando la plaza San Mart\u00edn, para tomar el colectivo, cuando vi a Pedro Camacho saliendo de Radio Central. Apenas se lo se\u00f1al\u00e9, la t\u00eda Julia quiso que se lo presentara. Nos acercamos y \u00e9l, al decirle que se trataba de una compatriota suya, se mostr\u00f3 muy amable.\n\n\u2014Soy una gran admiradora suya \u2014le dijo la t\u00eda Julia, y para caerle m\u00e1s en gracia le minti\u00f3\u2014: Desde Bolivia, no me pierdo sus radioteatros.\n\nFuimos caminando con \u00e9l, casi sin darnos cuenta, hacia el jir\u00f3n Quilca, y en el trayecto Pedro Camacho y la t\u00eda Julia mantuvieron una conversaci\u00f3n patri\u00f3tica de la que qued\u00e9 excluido, en la que desfilaron las minas de Potos\u00ed y la cerveza Taqui\u00f1a, esa sopa de choclo que llaman lagua, el mote con queso fresco, el clima de Cochabamba, la belleza de las cruce\u00f1as y otros orgullos bolivianos. El escriba parec\u00eda muy satisfecho hablando maravillas de su tierra. Al llegar al port\u00f3n de una casa con balcones y celos\u00edas se detuvo. Pero no nos despidi\u00f3:\n\n\u2014Suban \u2014nos propuso\u2014. Aunque mi cena es sencilla, podemos compartirla.\n\nLa Pensi\u00f3n La Tapada era una de esas viejas casas de dos pisos del centro de Lima, construidas el siglo pasado, que alguna vez fueron amplias, confortables y acaso suntuosas, y que luego, a medida que la gente acomodada iba desertando el centro hacia los balnearios y la vieja Lima iba perdiendo clase, se han ido deshaciendo y atestando, subdividi\u00e9ndose hasta ser verdaderas colmenas, gracias a tabiques que duplican o cuadruplican las habitaciones y a nuevos reductos erigidos de cualquier manera en los zaguanes, las azoteas e, incluso, los balcones y las escaleras. La Pensi\u00f3n La Tapada daba la impresi\u00f3n de estar a punto de descalabrarse; las gradas en que subimos al cuarto de Pedro Camacho se mec\u00edan bajo nuestro peso, y se levantaban unas nubecillas que hac\u00edan estornudar a la t\u00eda Julia. Una costra de polvo lo recubr\u00eda todo, paredes y suelos, y era evidente que la casa no hab\u00eda sido barrida ni trapeada jam\u00e1s. El cuarto de Pedro Camacho parec\u00eda una celda. Era muy peque\u00f1o y estaba casi vac\u00edo. Hab\u00eda un catre sin espaldar, cubierto con una colcha descolorida y una almohada sin funda, una mesita con hule y una silla de paja, una maleta y un cordel tendido entre dos paredes donde se columpiaban unos calzoncillos y unas medias. Que el escriba se lavara \u00e9l mismo la ropa no me sorprendi\u00f3, pero s\u00ed que se hiciera la comida. Hab\u00eda un primus en el alf\u00e9izar de la ventana, una botella de kerosene, unos platos y cubiertos de lata, unos vasos. Ofreci\u00f3 la silla a la t\u00eda Julia y a m\u00ed la cama con un gesto magn\u00edfico:\n\n\u2014Asiento. La morada es pobre pero el coraz\u00f3n es grande.\n\nPrepar\u00f3 la cena en dos minutos. Ten\u00eda los ingredientes en una bolsa de pl\u00e1stico, ore\u00e1ndose en la ventana. El men\u00fa consisti\u00f3 en unas salchichas hervidas con huevo frito, pan con mantequilla y queso, y un yogur con miel. Lo vimos prepararlo diestramente, como alguien acostumbrado a hacerlo a diario, y tuve la certidumbre que \u00e9sa deb\u00eda ser siempre su dieta.\n\nMientras com\u00edamos, estuvo conversador y galante, y condescendi\u00f3 a tratar temas como la receta de la crema volteada (que le pidi\u00f3 la t\u00eda Julia) y el sapolio m\u00e1s econ\u00f3mico para la ropa blanca. No termin\u00f3 su plato; al apartarlo, se\u00f1alando las sobras, se permiti\u00f3 una broma:\n\n\u2014Para el artista la comida es vicio, mis amigos.\n\nAl ver su buen humor, me atrev\u00ed a hacerle preguntas sobre su trabajo. Le dije que envidiaba su resistencia, que, pese a su horario de galeote, nunca pareciera cansado.\n\n\u2014Tengo mis estrategias para que la jornada resulte variopinta \u2014nos confes\u00f3.\n\nBajando la voz, como para que no fueran a descubrir su secreto fantasmales competidores, nos dijo que nunca escrib\u00eda m\u00e1s de sesenta minutos una misma historia y que pasar de un tema a otro era refrescante, pues cada hora ten\u00eda la sensaci\u00f3n de estar principiando a trabajar.\n\n\u2014En la variaci\u00f3n se encuentra el gusto, se\u00f1ores \u2014repet\u00eda, con ojos excitados y muecas de gnomo mal\u00e9fico.\n\nPara eso era importante que las historias estuvieran ordenadas no por afinidad sino por contraste: el cambio total de clima, lugar, asunto y personajes reforzaba la sensaci\u00f3n renovadora. De otro lado, los matecitos de yerbaluisa y menta eran \u00fatiles, desatoraban los conductos cerebrales y la imaginaci\u00f3n lo agradec\u00eda. Y eso de, cada cierto tiempo, dejar la m\u00e1quina para ir al estudio, ese pasar de escribir a dirigir e interpretar era tambi\u00e9n descanso, una transici\u00f3n que entonaba. Pero, adem\u00e1s, \u00e9l, en el curso de los a\u00f1os, hab\u00eda descubierto algo, algo que a los ignaros y a los insensibles les pod\u00eda parecer tal vez una chiquillada. Aunque \u00bfimportaba lo que pensara la ralea? Lo vimos vacilar, callarse, y su carita caricatural se entristeci\u00f3:\n\n\u2014Aqu\u00ed, desgraciadamente, no puedo ponerlo en pr\u00e1ctica \u2014dijo con melancol\u00eda\u2014. S\u00f3lo los domingos, que estoy solo. Los d\u00edas de semana hay demasiados curiosos y no lo entender\u00edan.\n\n\u00bfDe cu\u00e1ndo ac\u00e1 esos escr\u00fapulos, en \u00e9l, que miraba ol\u00edmpicamente a los mortales? Vi a la t\u00eda Julia tan anhelante como yo:\n\n\u2014No puede usted dejarnos con la miel en los labios \u2014le rog\u00f3\u2014. \u00bfCu\u00e1l es ese secreto, se\u00f1or Camacho?\n\nSe nos qued\u00f3 observando, en silencio, como el ilusionista que contempla, satisfecho, la atenci\u00f3n que ha conseguido despertar. Luego, con lentitud sacerdotal, se levant\u00f3 (estaba sentado en la ventana, junto al primus), fue hasta la maleta, la abri\u00f3, y empez\u00f3 a sacar de sus entra\u00f1as, como el prestidigitador saca palomas o banderas del sombrero de copa, una inesperada colecci\u00f3n de objetos: una peluca de magistrado ingl\u00e9s, bigotes postizos de distintos tama\u00f1os, un casco de bombero, una insignia de militar, caretas de mujer gorda, de anciano, de ni\u00f1o est\u00fapido, la varita del polic\u00eda de tr\u00e1nsito, la gorra y la pipa del lobo de mar, el mandil blanco del m\u00e9dico, narices falsas, orejas postizas, barbas de algod\u00f3n... Como una figurita el\u00e9ctrica, mostraba los artefactados y, \u00bfpara que los apreci\u00e1ramos mejor, por una necesidad \u00edntima?, se los iba enfundando, acomodando, quitando, con una agilidad que delataba una persistente costumbre, un asiduo manejo. De este modo, ante la t\u00eda Julia y yo, que lo mir\u00e1bamos embobados, Pedro Camacho, mediante cambios de atuendo, se transformaba en un m\u00e9dico, en un marino, en un juez, en una anciana, en un mendigo, en una beata, en un cardenal... Al mismo tiempo que operaba estas mudanzas, iba hablando, lleno de ardor:\n\n\u2014\u00bfPor qu\u00e9 no voy a tener derecho, para consubstanciarme con personajes de mi propiedad, a parecerme a ellos? \u00bfQui\u00e9n me proh\u00edbe tener, mientras los escribo, sus narices, sus pelos y sus levitas? \u2014dec\u00eda, trocando un capelo por una cachimba, la cachimba por un guardapolvo y el guardapolvo por una muleta\u2014. \u00bfA qui\u00e9n le importa que aceite la imaginaci\u00f3n con unos trapos? \u00bfQu\u00e9 cosa es el realismo, se\u00f1ores, el tan mentado realismo qu\u00e9 cosa es? \u00bfQu\u00e9 mejor manera de hacer arte realista que identific\u00e1ndose materialmente con la realidad? \u00bfY no resulta as\u00ed la jornada m\u00e1s llevadera, m\u00e1s amena, m\u00e1s movida?\n\nPero, claro \u2014y su voz pas\u00f3 a ser primero furiosa, luego desconsolada\u2014, la incomprensi\u00f3n y la estulticia de la gente todo lo malinterpretaban. Si lo ve\u00edan en Radio Central escribiendo disfrazado, brotar\u00edan las murmuraciones, correr\u00eda la voz de que era travestido, su oficina se convertir\u00eda en un im\u00e1n para la morbosidad del vulgo. Termin\u00f3 de guardar las caretas y dem\u00e1s objetos, cerr\u00f3 la maleta y volvi\u00f3 a la ventana. Ahora estaba triste. Murmur\u00f3 que en Bolivia, donde siempre trabajaba en su propio atelier, nunca hab\u00eda tenido problema \u00abcon los trapos\u00bb. Aqu\u00ed, en cambio, s\u00f3lo los domingos pod\u00eda escribir de acuerdo a su costumbre.\n\n\u2014\u00bfEsos disfraces se los consigue en funci\u00f3n de los personajes o inventa los personajes a partir de disfraces que ya tiene? \u2014le pregunt\u00e9, por decir algo, todav\u00eda sin salir del asombro.\n\nMe mir\u00f3 como a un reci\u00e9n nacido:\n\n\u2014Se nota que es usted muy joven \u2014me reprendi\u00f3 con suavidad\u2014. \u00bfNo sabe acaso que lo primero es siempre el verbo?\n\nCuando, despu\u00e9s de agradecerle efusivamente la invitaci\u00f3n, volvimos a la calle, le dije a la t\u00eda Julia que Pedro Camacho nos hab\u00eda dado una prueba de confianza excepcional haci\u00e9ndonos part\u00edcipes de su secreto, y que me hab\u00eda conmovido. Ella estaba contenta: nunca se hab\u00eda imaginado que los intelectuales pudieran ser tipos tan entretenidos.\n\n\u2014Bueno, no todos son as\u00ed \u2014me burl\u00e9\u2014. Pedro Camacho es un intelectual entre comillas. \u00bfTe fijaste que no hay un solo libro en su cuarto? Me ha explicado que no lee para que no le influyan el estilo.\n\nRegres\u00e1bamos, por las calles taciturnas del centro, cogidos de la mano, hacia el paradero de los colectivos y yo le dec\u00eda que alg\u00fan domingo vendr\u00eda a Radio Central s\u00f3lo para ver al escriba transubstanciado mediante antifaces con sus creaturas.\n\n\u2014Vive como un pordiosero, no hay derecho \u2014protestaba la t\u00eda Julia\u2014. Siendo sus radioteatros tan famosos, cre\u00ed que ganar\u00eda montones de plata.\n\nLe preocupaba que en la Pensi\u00f3n La Tapada no se viera ni una ba\u00f1era ni una ducha, apenas un excusado y un lavador enmohecidos en el primer rellano de la escalera. \u00bfCre\u00eda yo que Pedro Camacho no se ba\u00f1aba nunca? Le dije que al escriba esas banalidades le importaban un pito. Me confes\u00f3 que al ver la suciedad de la pensi\u00f3n le hab\u00eda dado asco, que hab\u00eda hecho un esfuerzo sobrehumano para pasar la salchicha y el huevo. Ya en el colectivo, una vieja carcocha que iba parando en cada esquina de la avenida Arequipa, mientras yo la besaba despacito en la oreja, en el cuello, la o\u00ed decir alarmada:\n\n\u2014O sea que los escritores son unos muertos de hambre. Quiere decir que toda la vida vivir\u00e1s fregado, Varguitas.\n\nDesde que se lo hab\u00eda o\u00eddo a Javier, ella tambi\u00e9n me llamaba Varguitas.\n\n### VIII\n\nDON FEDERICO T\u00e9llez Unz\u00e1tegui consult\u00f3 su reloj, comprob\u00f3 que eran las doce, dijo a la media docena de empleados de Antirroedores S. A. que pod\u00edan partir a almorzar, y no les record\u00f3 que estuvieran de vuelta a las tres en punto, ni un minuto m\u00e1s tarde, porque todos ellos sab\u00edan de sobra que, en esa empresa, la impuntualidad era sacr\u00edlega: se pagaba con multa e incluso despido. Una vez partidos, don Federico, seg\u00fan su costumbre, cerr\u00f3 \u00e9l mismo la oficina con doble llave, enfund\u00f3 su sombrero gris pericote, y se dirigi\u00f3, por las atestadas aceras del jir\u00f3n Huancavelica, hacia la playa de estacionamiento donde guardaba su autom\u00f3vil (un sed\u00e1n marca Dodge).\n\nEra un hombre que inspiraba temor e ideas l\u00fagubres, alguien a quien bastaba cruzar en la calle para advertir que era distinto a sus conciudadanos. Estaba en la flor de la edad, la cincuentena, y sus se\u00f1as particulares \u2014frente ancha, nariz aguile\u00f1a, mirada penetrante, rectitud en el esp\u00edritu\u2014 pod\u00edan haber hecho de \u00e9l un donju\u00e1n si se hubiera interesado en las mujeres. Pero don Federico T\u00e9llez Unz\u00e1tegui hab\u00eda consagrado su existencia a una cruzada y no permit\u00eda que nada ni nadie \u2014a no ser las indispensables horas de sue\u00f1o, alimentaci\u00f3n y trato de la familia\u2014 lo distrajera de ella. Esa guerra la libraba hac\u00eda cuarenta a\u00f1os y ten\u00eda como meta el exterminio de todos los roedores del territorio nacional.\n\nLa raz\u00f3n de esta quimera la ignoraban sus conocidos e incluso su esposa y sus cuatro hijos. Don Federico T\u00e9llez Unz\u00e1tegui la ocultaba pero no la olvidaba: d\u00eda y noche ella volv\u00eda a su memoria, pesadilla persistente de la que extra\u00eda nuevas fuerzas, odio fresco para perseverar en ese combate que algunos consideraban estramb\u00f3tico, otros repelente y, los m\u00e1s, comercial. Ahora mismo, mientras entraba a la playa de estacionamiento, verificaba de un vistazo de c\u00f3ndor que el Dodge hab\u00eda sido lavado, lo pon\u00eda en marcha y esperaba dos minutos (tomados por reloj) que se calentara el motor, sus pensamientos, una vez m\u00e1s, mariposas revoloteando hacia llamas donde arder\u00e1n sus alas, remontaban el tiempo, el espacio, hacia la poblaci\u00f3n selv\u00e1tica de su ni\u00f1ez y hacia el espanto que fragu\u00f3 su destino.\n\nHab\u00eda sucedido en la primera d\u00e9cada del siglo, cuando Tingo Mar\u00eda era apenas una cruz en el mapa, un claro de caba\u00f1as rodeado por la jungla abrupta. Hasta all\u00ed ven\u00edan, a veces, despu\u00e9s de infinitas penalidades, aventureros que abandonaban la molicie de la capital con la ilusi\u00f3n de conquistar la selva. As\u00ed lleg\u00f3 a la regi\u00f3n el ingeniero Hildebrando T\u00e9llez, con una esposa joven (por cuyas venas, como su nombre Mayte y su apellido Unz\u00e1tegui voceaban, corr\u00eda la azulina sangre vasca) y un hijo peque\u00f1o: Federico. Alentaba el ingeniero proyectos grandiosos: talar \u00e1rboles, exportar maderas preciosas para la vivienda y el mueble de los pudientes, cultivar la pi\u00f1a, la palta, la sand\u00eda, la guan\u00e1bana y la l\u00facuma para los paladares ex\u00f3ticos del mundo, y, con el tiempo, un servicio de vaporcitos por los r\u00edos amaz\u00f3nicos. Pero los dioses y los hombres hicieron ceniza de esos fuegos. Las cat\u00e1strofes naturales \u2014lluvias, plagas, desbordes\u2014 y las limitaciones humanas \u2014falta de mano de obra, pereza y estulticia de la existente, alcohol, escaso cr\u00e9dito\u2014 liquidaron uno tras otro los ideales del pionero, quien, a los dos a\u00f1os de su llegada a Tingo Mar\u00eda, deb\u00eda ganarse el sustento, modestamente, con una chacrita de camotes, aguas arriba del r\u00edo Pendencia. Fue all\u00ed, en una caba\u00f1a de troncos y palmas, donde una noche c\u00e1lida las ratas se comieron viva, en su cuna sin mosquitero, a la reci\u00e9n nacida Mar\u00eda T\u00e9llez Unz\u00e1tegui.\n\nLo ocurrido ocurri\u00f3 de manera simple y atroz. El padre y la madre eran padrinos de un bautizo y pasaban la noche, en los festejos consabidos, en la otra margen del r\u00edo. Hab\u00eda quedado a cargo de la chacra el capataz, quien, con los dos peones restantes, ten\u00eda una enramada lejos de la caba\u00f1a del patr\u00f3n. En \u00e9sta dorm\u00edan Federico y su hermana. Pero el ni\u00f1o acostumbraba, en \u00e9pocas de calor, sacar su camastro a orillas del Pendencia, donde dorm\u00eda arrullado por el agua. Es lo que hab\u00eda hecho esa noche (se lo reprochar\u00eda mientras tuviera vida). Se ba\u00f1\u00f3 a la luz de la luna, se acost\u00f3 y durmi\u00f3. Entre sue\u00f1os, le pareci\u00f3 que o\u00eda un llanto de ni\u00f1a. No fue suficientemente fuerte o largo para despertarlo. Al amanecer, sinti\u00f3 unos acerados dientecillos en el pie. Abri\u00f3 los ojos y crey\u00f3 morir, o, m\u00e1s bien, haber muerto y estar en el infierno: decenas de ratas lo rodeaban, tropezando, empuj\u00e1ndose, contone\u00e1ndose y, sobre todo, masticando lo que se pon\u00eda a su alcance. Brinc\u00f3 del camastro, cogi\u00f3 un palo, a gritos consigui\u00f3 alertar al capataz y a los peones. Entre todos, con antorchas, garrotes, patadas, alejaron a la colonia de invasoras. Pero cuando entraron a la caba\u00f1a (plato fuerte del fest\u00edn de las hambrientas) de la ni\u00f1a quedaba s\u00f3lo un montoncito de huesos.\n\nHab\u00edan pasado los dos minutos y don Federico T\u00e9llez Unz\u00e1tegui parti\u00f3. Avanz\u00f3, en una serpiente de autom\u00f3viles, por la avenida Tacna, para tomar Wilson y Arequipa, hacia el distrito del Barranco, donde lo esperaba el almuerzo. Al frenar en los sem\u00e1foros, cerraba los ojos y sent\u00eda, como siempre que recordaba aquel amanecer de trementina, una sensaci\u00f3n \u00e1cida y efervescente. Porque, como dice la sabidur\u00eda, \u00abbien vengas mal si vienes solo\u00bb. Su madre, la joven de estirpe vasca, por efecto de la tragedia contrajo un hipo cr\u00f3nico, que le causaba arcadas, le imped\u00eda comer y despertaba la hilaridad de la gente. No volvi\u00f3 a pronunciar palabra: s\u00f3lo gorgoritos y ronqueras. Andaba as\u00ed, con los ojos espantados, hipando, consumi\u00e9ndose, hasta que unos meses despu\u00e9s muri\u00f3, de extenuaci\u00f3n. El padre se desciviliz\u00f3, perdi\u00f3 la ambici\u00f3n, las energ\u00edas, la costumbre de asearse. Cuando, por desidia, le remataron la chacrita, se gan\u00f3 un tiempo la vida como balsero, pasando humanos, productos y animales de una banda a otra del Huallaga. Pero un d\u00eda las aguas de la creciente deshicieron la balsa contra los \u00e1rboles y \u00e9l no tuvo \u00e1nimos para fabricar otra. Se intern\u00f3 en las laderas sical\u00edpticas de esa monta\u00f1a de ubres maternales y caderas \u00e1vidas que llaman La Bella Durmiente, se construy\u00f3 un refugio de hojas y tallos, se dej\u00f3 crecer los pelos y las barbas y all\u00ed se qued\u00f3 a\u00f1os, comiendo hierbas y fumando unas hojas que produc\u00edan mareos. Cuando Federico, adolescente, abandon\u00f3 la selva, el ex ingeniero era llamado El Brujo en Tingo Mar\u00eda y viv\u00eda cerca de la cueva de las Pavas, amancebado con tres ind\u00edgenas huanuque\u00f1as, en las que hab\u00eda procreado algunas criaturas montubias, de vientres esf\u00e9ricos.\n\nS\u00f3lo Federico supo hacer frente a la cat\u00e1strofe con creatividad. Esa misma ma\u00f1ana, despu\u00e9s de haber sido azotado por dejar sola a su hermana en la caba\u00f1a, el ni\u00f1o (hecho hombre en unas horas), arrodill\u00e1ndose junto al mont\u00edculo que era la tumba de Mar\u00eda, jur\u00f3 que, hasta el \u00faltimo instante, se consagrar\u00eda a la aniquilaci\u00f3n de la especie asesina. Para dar fuerza a su juramento, reg\u00f3 sangre de azotes sobre la tierra que cubr\u00eda a la ni\u00f1a.\n\nCuarenta a\u00f1os m\u00e1s tarde, constancia de los probos que remueve monta\u00f1as, don Federico T\u00e9llez Unz\u00e1tegui pod\u00eda decirse, mientras su sed\u00e1n rodaba por las avenidas hacia el frugal almuerzo cotidiano, que hab\u00eda mostrado ser hombre de palabra. Porque en todo ese tiempo era probable que, por sus obras e inspiraci\u00f3n, hubieran perecido m\u00e1s roedores que peruanos nacido. Trabajo dif\u00edcil, abnegado, sin premio, que hizo de \u00e9l un ser estricto y sin amigos, de costumbres aparte. Al principio, de ni\u00f1o, lo m\u00e1s arduo fue vencer el asco a los parduzcos. Su t\u00e9cnica inicial hab\u00eda sido primitiva: la trampa. Compr\u00f3 con sus propinas, en la colchoner\u00eda y bodega El Profundo Sue\u00f1o de la avenida Raimondi, una que le sirvi\u00f3 de modelo para fabricar muchas otras. Cortaba las maderas, los alambres, los retorc\u00eda y, dos veces al d\u00eda, las sembraba dentro de los linderos de la chacra. A veces, algunos animalitos atrapados estaban a\u00fan vivos. Emocionado, los ultimaba a fuego lento, o hac\u00eda sufrir punz\u00e1ndolos, mutil\u00e1ndolos, revent\u00e1ndoles los ojos.\n\nPero, aunque ni\u00f1o, su inteligencia le hizo comprender que si se abandonaba a esas inclinaciones se frustrar\u00eda: su obligaci\u00f3n era cuantitativa, no cualitativa. No se trataba de inferir el m\u00e1ximo sufrimiento por unidad de enemigo sino de destruir el mayor n\u00famero de unidades en el m\u00ednimo tiempo. Con lucidez y voluntad notables para sus a\u00f1os, extirp\u00f3 de s\u00ed todo sentimentalismo, y procedi\u00f3 en adelante, en su tarea genocida, con criterio glacial, estad\u00edstico, cient\u00edfico. Rob\u00e1ndole horas al colegio de los hermanos canadienses, y al sue\u00f1o (mas no al recreo, porque desde la tragedia no jug\u00f3 m\u00e1s), perfeccion\u00f3 las trampas, a\u00f1adi\u00e9ndoles una cuchilla que cercenaba el cuerpo de la v\u00edctima de modo que no fueran jam\u00e1s a quedar vivas (no para ahorrarles dolor sino para no perder tiempo en rematarlas). Construy\u00f3 luego trampas multifamiliares, de base ancha, en las que un trinche con arabescos pod\u00eda apachurrar simult\u00e1neamente al padre, la madre y cuatro cr\u00edas. Este quehacer fue pronto conocido en la comarca, e, insensiblemente, pas\u00f3 de venganza, penitencia personal, a ser un servicio a la comunidad, m\u00ednimamente (pero mal que mal) retribuido. Al ni\u00f1o lo llamaban de chacras vecinas y alejadas, apenas hab\u00eda indicios de invasi\u00f3n, y \u00e9l, diligencia de hormiga que todo lo puede, las limpiaba en pocos d\u00edas. Tambi\u00e9n de Tingo Mar\u00eda empezaron a solicitar sus servicios, caba\u00f1as, casas, oficinas, y el ni\u00f1o tuvo su momento de gloria cuando el capit\u00e1n de la Guardia Civil le encomend\u00f3 despejar la comisar\u00eda, que hab\u00eda sido ocupada. Todo el dinero que recib\u00eda se lo gastaba fabricando nuevas trampas para extender lo que los ingenuos cre\u00edan su perversi\u00f3n o su negocio. Cuando el ex ingeniero se intern\u00f3 en la sexualoide mara\u00f1a de La Bella Durmiente, Federico, que hab\u00eda abandonado el colegio, empezaba a complementar el arma blanca de la trampa con otra, m\u00e1s sutil: los venenos.\n\nEl trabajo le permiti\u00f3 ganarse la vida a una edad en que otros ni\u00f1os hacen bailar trompos. Pero tambi\u00e9n lo convirti\u00f3 en un apestado. Lo llamaban para que les matase a los veloces, pero jam\u00e1s lo sentaban a sus mesas ni le dec\u00edan palabras afectuosas. Si esto lo hizo sufrir, no permiti\u00f3 que se notara, y, m\u00e1s bien, se hubiera dicho que la repugnancia de sus conciudadanos lo halagaba. Era un adolescente hura\u00f1o, lac\u00f3nico, al que nadie pudo ufanarse de haber hecho ni visto re\u00edr, y cuya \u00fanica pasi\u00f3n parec\u00eda ser la de matar a los inmundos. Cobraba moderadamente por los trabajos, pero tambi\u00e9n hac\u00eda campa\u00f1as ad hon\u00f3rem, en casas de gente pobre, a las que se presentaba con su costal de trampas y sus pomos de venenos, apenas se enteraba de que el enemigo hab\u00eda sentado all\u00ed sus reales. A la muerte de los plomizos, t\u00e9cnica que el joven refinaba sin descanso, se sum\u00f3 el problema de la eliminaci\u00f3n de los cad\u00e1veres. Era lo que m\u00e1s disgustaba a las familias, amas de casa o sirvientas. Federico ensanch\u00f3 su empresa, entrenando al idiota del pueblo, un jorobado de ojos estr\u00e1bicos que viv\u00eda donde las Siervas de San Jos\u00e9, para que, a cambio del sustento, recogiera en un crudo los restos de los supliciados y fuera a quemarlos detr\u00e1s del Coliseo Abad o a ofrecerlos como fest\u00edn a los perros, gatos, chanchos y buitres de Tingo Mar\u00eda.\n\n\u00a1Cu\u00e1nto hab\u00eda pasado desde entonces! En el sem\u00e1foro de Javier Prado, don Federico T\u00e9llez Unz\u00e1tegui se dijo que, indudablemente, hab\u00eda progresado desde que, adolescente, de sol a sol recorr\u00eda las calles fangosas de Tingo Mar\u00eda, seguido por el idiota, librando artesanalmente la guerra contra los homicidas de Mar\u00eda. Era entonces un joven que s\u00f3lo ten\u00eda la ropa que llevaba puesta y apenas un ayudante. Treinta y cinco a\u00f1os m\u00e1s tarde, capitaneaba un complejo t\u00e9cnico-comercial, que extend\u00eda sus brazos por todas las ciudades del Per\u00fa, al que pertenec\u00edan quince camionetas y setenta y ocho expertos en fumigaci\u00f3n de escondites, mezcla de venenos y siembra de trampas. \u00c9stos operaban en el frente de batalla \u2014las calles, casas y campos del pa\u00eds\u2014 dedicados al cateo, cerco y exterminio, y recib\u00edan \u00f3rdenes, asesoramiento y apoyo log\u00edstico del estado mayor que \u00e9l presid\u00eda (los seis tecn\u00f3cratas que acababan de partir a almorzar). Pero, adem\u00e1s de esa constelaci\u00f3n, interven\u00edan en la cruzada dos laboratorios, con los cuales don Federico ten\u00eda firmado contratos (que eran pr\u00e1cticamente subvenciones) a fin de que, de manera continua, experimentaran nuevos venenos, ya que el enemigo ten\u00eda una prodigiosa capacidad de inmunizaci\u00f3n: luego de dos o tres campa\u00f1as, los t\u00f3xicos resultaban obsoletos, manjares para aqu\u00e9llos a quienes ten\u00edan la obligaci\u00f3n de matar. Adem\u00e1s, don Federico \u2014que, en este instante, al aparecer la luz verde, pon\u00eda primera y prosegu\u00eda viaje hacia los barrios del mar\u2014 hab\u00eda instituido una beca por la que Antirroedores S. A. enviaba cada a\u00f1o, a un qu\u00edmico reci\u00e9n graduado, a la Universidad de Baton Rouge, a especializarse en raticidas.\n\nHab\u00eda sido precisamente ese asunto \u2014la ciencia al servicio de su religi\u00f3n\u2014 lo que impuls\u00f3, veinte a\u00f1os atr\u00e1s, a don Federico T\u00e9llez Unz\u00e1tegui a casarse. Humano al fin y al cabo, un d\u00eda hab\u00eda comenzado a germinar en su cerebro la idea de una apretada falange de varones, de su misma sangre y esp\u00edritu, a quienes desde la teta inculcar\u00eda la furia contra los asquerosos, y quienes, excepcionalmente educados, continuar\u00edan, acaso allende las fronteras patrias, su misi\u00f3n. La imagen de seis, siete T\u00e9llez doctorados, en encumbradas academias, que repetir\u00edan y eternizar\u00edan su juramento lo llev\u00f3, a \u00e9l, que era la inapetencia marital encarnada, a recurrir a una agencia de matrimonios, la que, mediante una retribuci\u00f3n algo excesiva, le suministr\u00f3 una esposa de veinticinco a\u00f1os, tal vez no de hermosura radiante \u2014le faltaban dientes y, como a esas damitas de la regi\u00f3n que irriga el llamado (hiperb\u00f3licamente) R\u00edo de la Plata, le sobraban rollos de carnes en la cintura y en las pantorrillas\u2014, pero con las tres cualidades que hab\u00eda exigido: salud irreprochable, himen intacto y capacidad reproductora.\n\nDo\u00f1a Zoila Saravia Dur\u00e1n era una huanuque\u00f1a cuya familia, reveses de la vida que se entretiene jugando al subibaja, hab\u00eda sido degradada de la aristocracia provinciana al subproletariado capitalino. Se educ\u00f3 en la escuela gratuita que las madres salesianas manten\u00edan \u2014\u00bfrazones de conciencia o de publicidad?\u2014 junto a la escuela pagante, y hab\u00eda crecido, como todas sus compa\u00f1eras, con un argentino complejo que, en su caso, se traduc\u00eda en docilidad, mutismo y apetito. Se hab\u00eda pasado la vida trabajando como celadora donde las madres salesianas y el estatuto vago, indeterminado de su funci\u00f3n \u2014\u00bfsirvienta, obrera, empleada?\u2014 agrav\u00f3 esa inseguridad servil que la hac\u00eda asentir y mover ganaderamente la cabeza ante todo. Al quedar hu\u00e9rfana, a los veinticuatro a\u00f1os, se atrevi\u00f3 a visitar, despu\u00e9s de ardientes dudas, la agencia matrimonial que la puso en contacto con el que ser\u00eda su amo. La inexperiencia er\u00f3tica de los c\u00f3nyuges determin\u00f3 que la consumaci\u00f3n del matrimonio fuera lent\u00edsima, una serial en la que, entre amagos y fiascos por precocidad, falta de punter\u00eda y extrav\u00edo, los cap\u00edtulos se suced\u00edan, crec\u00eda el suspenso, y el terco himen continuaba sin perforar. Parad\u00f3jicamente, trat\u00e1ndose de una pareja de virtuosos, do\u00f1a Zoila perdi\u00f3 primero la virginidad (no por vicio sino por est\u00fapido azar y falta de entrenamiento de los novios), heterodoxa, vale decir sodom\u00edticamente.\n\nAparte de esta abominaci\u00f3n casual, la vida de la pareja hab\u00eda sido muy correcta. Do\u00f1a Zoila era una esposa diligente, ahorrativa y empe\u00f1osamente dispuesta a acatar los principios (que algunos llamar\u00edan excentricidades) de su marido. Jam\u00e1s hab\u00eda objetado, por ejemplo, la prohibici\u00f3n impuesta por don Federico de usar agua caliente (porque, seg\u00fan \u00e9l, enervaba la voluntad y causaba resfr\u00edos) aunque aun ahora, despu\u00e9s de veinte a\u00f1os, segu\u00eda poni\u00e9ndose morada al entrar a la ducha. Nunca hab\u00eda contrariado la cl\u00e1usula del (no escrito pero sabido de memoria) c\u00f3digo familiar estableciendo que nadie durmiera en el hogar m\u00e1s de cinco horas, para no prohijar molicie, aunque cada amanecer, cuando, a las cinco, sonaba el despertador, sus bostezos de cocodrilo estremec\u00edan los cristales. Con resignaci\u00f3n hab\u00eda aceptado que de las distracciones familiares quedaran excluidos, por inmorales para el esp\u00edritu, el cine, el baile, el teatro, la radio, y, por onerosos para el presupuesto, los restaurantes, los viajes y cualquier fantas\u00eda en el atuendo corporal y en la decoraci\u00f3n inmueble. S\u00f3lo en lo que se refer\u00eda a su pecado, la gula, hab\u00eda sido incapaz de obedecer al se\u00f1or de la casa. Muchas veces hab\u00edan aparecido en el men\u00fa la carne, el pescado y los postres cremosos. Era el \u00fanico rengl\u00f3n de la vida en el que don Federico T\u00e9llez Unz\u00e1tegui no hab\u00eda podido imponer su voluntad: un r\u00edgido vegetarianismo.\n\nPero do\u00f1a Zoila no hab\u00eda tratado jam\u00e1s de practicar su vicio aviesamente, a espaldas de su marido, quien, en estos instantes, entraba en su sed\u00e1n al pizpireto barrio de Miraflores, dici\u00e9ndose que esa sinceridad, si no expiaba, por lo menos venializaba el pecado de su esposa. Cuando sus urgencias eran m\u00e1s fuertes que su esp\u00edritu de obediencia, devoraba su bistec encebollado, o corvina a lo macho, o pastel de manzana con crema chantilly, a ojos y vista de \u00e9l, granate de verg\u00fcenza y resignada de antemano al castigo correspondiente. Nunca hab\u00eda protestado contra las sanciones. Si don Federico (por un churrasco o una barra de chocolate) le suspend\u00eda la facultad de hablar tres d\u00edas, ella misma se amordazaba para no delinquir ni en sue\u00f1os, y si la pena era veinte nalgadas, se apuraba a desabrocharse la faja y preparar el \u00e1rnica.\n\nNo, don Federico T\u00e9llez Unz\u00e1tegui, mientras echaba una distra\u00edda mirada al gris (color que odiaba) oc\u00e9ano Pac\u00edfico, por encima del malec\u00f3n de Miraflores, que su sed\u00e1n acababa de hollar, se dijo que, despu\u00e9s de todo, do\u00f1a Zoila no lo hab\u00eda defraudado. El gran fracaso de su vida eran los hijos. Qu\u00e9 diferencia entre la aguerrida vanguardia de pr\u00edncipes del exterminio con que hab\u00eda so\u00f1ado y esos cuatro herederos que le hab\u00edan infligido Dios y la golosa.\n\nPor lo pronto, s\u00f3lo hab\u00edan nacido dos varones. Rudo, imprevisto golpe. Nunca se le pas\u00f3 por la cabeza que do\u00f1a Zoila pudiera parir hembras. La primera constituy\u00f3 una decepci\u00f3n, algo que pod\u00eda atribuirse a la casualidad. Pero como el cuarto embarazo desemboc\u00f3 tambi\u00e9n en un ser sin falo ni test\u00edculos visibles, don Federico, aterrado ante la perspectiva de seguir produciendo seres incompletos, cort\u00f3 dr\u00e1sticamente toda veleidad de descendencia (para lo cual reemplaz\u00f3 la cama de matrimonio por dos cujas individuales). No odiaba a las mujeres; simplemente, como no era un erot\u00f3mano ni un voraz \u00bfde qu\u00e9 pod\u00edan servirle personas cuyas mejores aptitudes eran la fornicaci\u00f3n y la cocina? Reproducirse no hab\u00eda tenido otra raz\u00f3n, para \u00e9l, que perpetuar su cruzada. Esta esperanza se hizo humo con la venida de Teresa y Laura, pues don Federico no era de esos modernistas que predican que la mujer, adem\u00e1s de cl\u00edtoris, tiene tambi\u00e9n sesos y puede trabajar de igual a igual con el var\u00f3n. De otro lado, lo angustiaba la posibilidad de que su nombre rodara por el barro. \u00bfNo repet\u00edan las estad\u00edsticas hasta la n\u00e1usea que el noventa y cinco por ciento de las mujeres han sido, son o ser\u00e1n meretrices? Para lograr que sus hijas lograran domiciliarse en el cinco por ciento de virtuosas, don Federico les hab\u00eda organizado la vida mediante un sistema puntilloso: nunca escotes, invierno y verano medias oscuras y blusas y chompas de manga larga, jam\u00e1s pintarse las u\u00f1as, los labios, los ojos ni las mejillas o peinarse con cerquillo, trenzas, cola de caballo y todo ese gremio de anzuelos para pescar al macho; no practicar deportes ni diversiones que implicaran cercan\u00eda de hombre, como ir a la playa o asistir a fiestas de cumplea\u00f1os. Las contravenciones eran castigadas siempre corporalmente.\n\nPero no s\u00f3lo la intromisi\u00f3n de hembras entre sus descendientes hab\u00eda sido desalentadora. Los varones \u2014Ricardo y Federico hijo\u2014 no hab\u00edan heredado las virtudes del padre. Eran blandos, perezosos, amantes de actividades est\u00e9riles (como el chicle y el f\u00fatbol) y no hab\u00edan manifestado el menor entusiasmo al explicarles don Federico el futuro que les reservaba. En las vacaciones, cuando, para irlos entrenando, los hac\u00eda trabajar con los combatientes de la primera l\u00ednea, se mostraban remisos, acud\u00edan con notoria repugnancia al campo de batalla. Y una vez los sorprendi\u00f3 murmurando obscenidades contra la obra de su vida, confesando que se avergonzaban de su padre. Los hab\u00eda rapado como a convictos, por supuesto, pero eso no lo hab\u00eda librado del sentimiento de traici\u00f3n que le caus\u00f3 esa charla conspiratoria. Don Federico, ahora, no se hac\u00eda ilusiones. Sab\u00eda que, una vez muerto o invalidado por los a\u00f1os, Ricardo y Federico hijo se apartar\u00edan de la senda que les hab\u00eda trazado, cambiar\u00edan de profesi\u00f3n (eligiendo alguna otra por atractivos cremat\u00edsticos) y que su obra quedar\u00eda \u2014como cierta sinfon\u00eda c\u00e9lebre\u2014 inconclusa.\n\nFue en este preciso segundo en que don Federico T\u00e9llez Unz\u00e1tegui, para su desgracia ps\u00edquica y f\u00edsica, vio la revista que un canillita met\u00eda por la ventana del sed\u00e1n, la car\u00e1tula de colores que brillaban pecadoramente en el sol de la ma\u00f1ana. En su cara cuaj\u00f3 una mueca de disgusto al advertir que luc\u00eda, como portada, la foto de una playa, con un par de ba\u00f1istas en esos simulacros de trajes de ba\u00f1o que se atrev\u00edan a usar ciertas hetairas, cuando, con una especie de desgarramiento angustioso del nervio \u00f3ptico y abriendo la boca como un lobo que a\u00falla a la luna, don Federico reconoci\u00f3 a las dos ba\u00f1istas semidesnudas y obscenamente risue\u00f1as. Sinti\u00f3 un horror que pod\u00eda competir con el que hab\u00eda sentido, en esa madrugada amaz\u00f3nica, a orillas del Pendencia, al divisar, sobre una cuna ennegrecida de caquitas de rat\u00f3n, el desorganizado esqueleto de su hermana. El sem\u00e1foro estaba en verde, los autom\u00f3viles de atr\u00e1s lo bocineaban.\n\nCon dedos torpes, sac\u00f3 su cartera, pag\u00f3 el producto licencioso, arranc\u00f3, y, sintiendo que iba a chocar \u2014el volante se le escapaba de las manos, el auto daba bandazos\u2014, fren\u00f3 y se peg\u00f3 a la vereda.\n\nAll\u00ed, temblando de ofuscaci\u00f3n, observ\u00f3 muchos minutos la terrible evidencia. No hab\u00eda duda posible: eran sus hijas. Fotografiadas por sorpresa, sin duda, por un fot\u00f3grafo zafio, escondido entre los ba\u00f1istas, las muchachas no miraban a la c\u00e1mara, parec\u00edan conversar, tumbadas sobre unas arenas voluptuosas que pod\u00edan ser las de Agua Dulce o La Herradura. Don Federico fue recuperando la respiraci\u00f3n; dentro de su anonadamiento, alcanz\u00f3 a pensar en la incre\u00edble serie de casualidades. Que un ambulante apresara en imagen a Laura y Teresa, que una revista innoble las expusiera al podrido mundo, que \u00e9l las descubriera... Y toda la espantosa verdad ven\u00eda a resplandecer as\u00ed, por estrategia del azar, ante sus ojos. De modo que sus hijas le obedec\u00edan s\u00f3lo cuando estaba presente; de modo que, apenas volv\u00eda la espalda, coludidas, sin duda, con sus hermanos y con, ay \u2014don Federico sinti\u00f3 un dardo en el coraz\u00f3n\u2014, su propia esposa, hac\u00edan escarnio de los mandamientos y bajaban a la playa, se desnudaban y exhib\u00edan. Las l\u00e1grimas le mojaron la cara. Examin\u00f3 las ropas de ba\u00f1o: dos piezas tan reducidas cuya funci\u00f3n no era esconder nada sino exclusivamente catapultar la imaginaci\u00f3n hacia extremos viciosos. Ah\u00ed estaban, al alcance de cualquiera: piernas, brazos, vientres, hombros, cuellos de Laura y Teresa. Sent\u00eda un rid\u00edculo inexpresable recordando que \u00e9l jam\u00e1s hab\u00eda visto esas extremidades y miembros que ahora se prodigaban ante el universo.\n\nSe sec\u00f3 los ojos y volvi\u00f3 a encender el motor. Se hab\u00eda serenado superficialmente, pero, en sus entra\u00f1as, crepitaba una hoguera. Mientras, muy despacio, el sed\u00e1n prosegu\u00eda hacia su casita de la avenida Pedro de Osma, se iba diciendo que, as\u00ed como iban a la playa desnudas era natural que, en su ausencia, fueran tambi\u00e9n a fiestas, usaran pantalones, frecuentaran hombres, que se vendieran, \u00bfrecib\u00edan tal vez a sus galanes en su propio hogar?, \u00bfser\u00eda do\u00f1a Zoila la encargada de fijar las tarifas y cobrarlas? Ricardo y Federico hijo tendr\u00edan probablemente a su cargo la inmunda tarea de reclutar a los clientes. Ahog\u00e1ndose, don Federico T\u00e9llez Unz\u00e1tegui vio armarse este estremecedor reparto: tus hijas, las rameras; tus hijos, los cafiches, y tu esposa, la alcahueta.\n\nEl cotidiano trato con la violencia \u2014despu\u00e9s de todo, hab\u00eda dado muerte a miles de millares de seres vivos\u2014 hab\u00eda hecho de don Federico un hombre al que no se pod\u00eda provocar sin riesgo grave. Una vez, un ingeniero agr\u00f3nomo con pretensiones de dietista hab\u00eda osado decir en su delante que, dada la falta de ganader\u00eda en el Per\u00fa, era necesario intensificar la cr\u00eda del cuy con miras a la alimentaci\u00f3n nacional. Educadamente, don Federico T\u00e9llez Unz\u00e1tegui record\u00f3 al atrevido que el cuy era primo hermano de la rata. \u00c9ste, reincidiendo, cit\u00f3 estad\u00edsticas, habl\u00f3 de virtudes nutritivas y carne gustosa al paladar. Don Federico procedi\u00f3 entonces a abofetearlo y, cuando el dietista rod\u00f3 por los suelos sob\u00e1ndose la cara, lo llam\u00f3 lo que era: desfachatado y publicista de homicidas. Ahora, al bajar del autom\u00f3vil, cerrarlo, avanzar sin premura, cejijunto, muy p\u00e1lido, hacia la puerta de su casa, el hombre de Tingo Mar\u00eda sent\u00eda ascender por su interior, como el d\u00eda que escarment\u00f3 al dietista, una lava volc\u00e1nica. Llevaba en su mano derecha, como barra candente, la revista infernal, y sent\u00eda una fuerte comez\u00f3n en los ojos.\n\nEstaba tan turbado que no consegu\u00eda imaginar un castigo capaz de parangonarse con la falta. Sent\u00eda la mente brumosa, la ira disolv\u00eda las ideas, y eso aumentaba su amargura, pues don Federico era un hombre en quien la raz\u00f3n decid\u00eda siempre la conducta, y que despreciaba a esa raza de primarios que actuaban, como las bestias, por instinto y p\u00e1lpito antes que por convicci\u00f3n. Pero esta vez, mientras sacaba la llave y, con dificultad, porque la rabia le entorpec\u00eda los dedos, abr\u00eda y empujaba la puerta de su casa, comprendi\u00f3 que no pod\u00eda actuar serena, calculadamente, sino bajo el dictado de la c\u00f3lera, siguiendo la inspiraci\u00f3n del instante. Luego de cerrar la puerta, respir\u00f3 hondo, tratando de calmarse. Le daba verg\u00fcenza que esos ingratos fueran a advertir la magnitud de su humillaci\u00f3n.\n\nSu casa ten\u00eda, abajo, un peque\u00f1o vest\u00edbulo, una salita, el comedor y la cocina, y los dormitorios en la planta alta. Don Federico divis\u00f3 a su mujer desde el quicio de la sala. Estaba junto al aparador, masticando con arrobo alguna repugnante golosina \u2014caramelo, chocolate, pens\u00f3 don Federico, fruna, toffee\u2014 cuyos restos conservaba en los dedos. Al verlo, le sonri\u00f3 con ojos intimidados, se\u00f1alando lo que com\u00eda con un gesto de resignaci\u00f3n dulzona.\n\nDon Federico avanz\u00f3 sin apresurarse, desplegando la revista con las dos manos, para que su esposa pudiera contemplar la car\u00e1tula en toda su indignidad. La puso bajo sus ojos, sin decir palabra, y goz\u00f3 vi\u00e9ndola palidecer violentamente, desorbitarse y abrir la boca de la que comenz\u00f3 a correr un hilillo de saliva contaminado de galleta. El hombre de Tingo Mar\u00eda levant\u00f3 la mano derecha y abofete\u00f3 a la tr\u00e9mula mujer con toda su fuerza. Ella dio un quejido, tropez\u00f3 y cay\u00f3 de rodillas; segu\u00eda mirando la car\u00e1tula con una expresi\u00f3n de beater\u00eda, de iluminaci\u00f3n m\u00edstica. Alto, erecto, justiciero, don Federico la contemplaba acusadoramente. Luego, llam\u00f3 con sequedad a las culpables:\n\n\u2014\u00a1Laura! \u00a1Teresa!\n\nUn rumor le hizo volver la cabeza. Ah\u00ed estaban, al pie de la escalera. No las hab\u00eda sentido bajar. Teresa, la mayor, llevaba un guardapolvo, como si hubiera estado haciendo la limpieza, y Laura vest\u00eda el uniforme de colegio. Las muchachas miraban, confusas, a la madre arrodillada, al padre que avanzaba, lento, hier\u00e1tico, sumo sacerdote yendo al encuentro de la piedra de los sacrificios donde esperan el cuchillo y la vestal, y, por fin, a la revista, que don Federico, llegado junto a ellas, les pon\u00eda judicialmente ante los ojos. La reacci\u00f3n de sus hijas no fue la que esperaba. En vez de tornarse l\u00edvidas, caer de hinojos balbuceando explicaciones, las precoces, ruboriz\u00e1ndose, cambiaron una veloz mirada que s\u00f3lo pod\u00eda ser de complicidad, y don Federico, en el fondo de su desolaci\u00f3n e ira, se dijo que todav\u00eda no hab\u00eda bebido toda la hiel de esa ma\u00f1ana; Laura y Teresa sab\u00edan que hab\u00edan sido fotografiadas, que la fotograf\u00eda se iba a publicar, e, incluso \u2014\u00bfqu\u00e9 otra cosa pod\u00eda decir esa chispa en las pupilas?\u2014, el hecho las alegraba. La revelaci\u00f3n de que en su hogar, que \u00e9l cre\u00eda pr\u00edstino, hubiera incubado, no s\u00f3lo el vicio municipal del nudismo playero, sino el exhibicionismo (y, por qu\u00e9 no, la ninfoman\u00eda) le afloj\u00f3 los m\u00fasculos, le dio un gusto a cal en la boca y lo llev\u00f3 a considerar si la vida se justificaba. Tambi\u00e9n \u2014todo ello no tom\u00f3 m\u00e1s de un segundo\u2014 a preguntarse si la \u00fanica penitencia leg\u00edtima para semejante horror no era la muerte. La idea de convertirse en filicida lo atormentaba menos que saber que miles de humanos hab\u00edan merodeado (\u00bfs\u00f3lo con los ojos?) por las intimidades f\u00edsicas de sus varonas.\n\nPas\u00f3 entonces a la acci\u00f3n. Dej\u00f3 caer la revista para tener m\u00e1s libertad, cogi\u00f3 con la mano izquierda a Laura de la casaquilla del uniforme, la atrajo unos cent\u00edmetros para ponerla m\u00e1s a tiro de impacto, levant\u00f3 la mano derecha lo bastante alto para que la potencia del golpe fuera m\u00e1xima, y la descarg\u00f3 con todo su rencor. Se llev\u00f3, entonces \u2014oh d\u00eda extraordinario\u2014 la segunda sorpresa descomunal, quiz\u00e1 m\u00e1s cegadora que la de la sical\u00edptica car\u00e1tula. En vez de la suave mejilla de Laurita, su mano encontr\u00f3 el vac\u00edo y, rid\u00edcula, frustrada, sufri\u00f3 un estir\u00f3n. No fue todo: lo grave vino despu\u00e9s. Porque la chiquilla no se content\u00f3 con esquivar la bofetada \u2014algo que, en su inmensa desaz\u00f3n, don Federico record\u00f3 no hab\u00eda hecho jam\u00e1s ning\u00fan miembro de su familia\u2014, sino que, luego de retroceder, la carita de catorce a\u00f1os descompuesta en una mueca de odio se lanz\u00f3 contra \u00e9l \u2014\u00e9l, \u00e9l\u2014, y comenz\u00f3 a golpearlo con sus pu\u00f1os, a rasgu\u00f1arlo, a empujarlo y a patearlo.\n\nTuvo la sensaci\u00f3n de que su misma sangre, de puro estupefacta, dejaba de correr. Era como si, de pronto, los astros escaparan de sus \u00f3rbitas, se precipitaran unos contra otros, chocaran, se rompieran, rodaran hist\u00e9ricos por los espacios. No atinaba a reaccionar, retroced\u00eda, los ojos desmedidamente abiertos, acosado por la chiquilla que, envalenton\u00e1ndose, exasper\u00e1ndose, adem\u00e1s de golpear ahora tambi\u00e9n gritaba: \u00abMaldito, abusivo, te odio, mu\u00e9rete, ac\u00e1bate de una vez\u00bb. Crey\u00f3 enloquecer cuando \u2014y todo ocurr\u00eda tan r\u00e1pido que, apenas tomaba conciencia de la situaci\u00f3n, \u00e9sta cambiaba\u2014 advirti\u00f3 que Teresa corr\u00eda hacia \u00e9l, pero, en vez de sujetar a su hermana, la ayudaba. Ahora tambi\u00e9n su hija mayor lo agred\u00eda, rugiendo los m\u00e1s abominables insultos \u2014\u00abtaca\u00f1o, est\u00fapido, mani\u00e1tico, asqueroso, tirano, loco, ratonero\u00bb\u2014 y entre ambas furias adolescentes lo iban arrinconando contra la pared. Hab\u00eda comenzado a defenderse, saliendo al fin de su paralizante asombro, y trataba de cubrirse la cara, cuando sinti\u00f3 un aguij\u00f3n en la espalda. Se volvi\u00f3: do\u00f1a Zoila se hab\u00eda incorporado y lo mord\u00eda.\n\nA\u00fan pudo maravillarse al notar que su esposa, m\u00e1s todav\u00eda que sus hijas, hab\u00eda sufrido una transfiguraci\u00f3n.\n\n\u00bfEra do\u00f1a Zoila, la mujer que jam\u00e1s hab\u00eda musitado una queja, alzado la voz, tenido un malhumor, el mismo ser de ojos ind\u00f3mitos y manos bravas que descargaba contra \u00e9l pu\u00f1etes, coscorrones, lo escup\u00eda, le rasgaba la camisa y vociferaba enloquecida: \u00abMat\u00e9moslo, vengu\u00e9monos, que se trague sus man\u00edas, s\u00e1quenle los ojos\u00bb? Las tres aullaban y don Federico pens\u00f3 que el griter\u00edo le hab\u00eda reventado los t\u00edmpanos. Se defend\u00eda con todas sus fuerzas, procuraba devolver los golpes, pero no lo consegu\u00eda, porque ellas, \u00bfponiendo en pr\u00e1ctica una t\u00e9cnica vilmente ensayada?, se turnaban de dos en dos para cogerle los brazos mientras la tercera lo destrozaba. Sent\u00eda ardores, hinchazones, punzadas, ve\u00eda estrellas, y, de pronto, unas manchitas en las manos de las agresoras le revelaron que sangraba.\n\nNo se hizo ilusiones cuando vio asomar en el hueco de la escalera a Ricardo y a Federico hijo. Convertido al escepticismo en pocos segundos, supo que ven\u00edan a sumarse al cargamont\u00f3n, a asestarle el puntillazo. Aterrado, sin dignidad ni honor, s\u00f3lo pens\u00f3 en alcanzar la puerta de calle, en huir. Pero no era f\u00e1cil. Pudo dar dos o tres saltos antes de que una zancadilla lo hiciera rodar aparatosamente por el suelo. All\u00ed, encogido para proteger su hombr\u00eda, vio c\u00f3mo sus herederos la emprend\u00edan a feroces puntapi\u00e9s contra su humanidad mientras su esposa e hijas se armaban de escobas, plumeros, de la barra de la chimenea para seguir aporre\u00e1ndolo. Antes de decirse que no comprend\u00eda nada, salvo que el mundo se hab\u00eda vuelto absurdo, alcanz\u00f3 a o\u00edr que tambi\u00e9n sus hijos, al comp\u00e1s de las patadas, le dec\u00edan mani\u00e1tico, taca\u00f1o, inmundo y ratonero. Mientras se hac\u00eda la tiniebla en \u00e9l, gris, peque\u00f1o, intruso, s\u00fabito, de un invisible huequecillo en una esquina del comedor, brot\u00f3 un pericote de caninos blancos y contempl\u00f3 al ca\u00eddo con una luz de burla en los vivaces ojos...\n\n\u00bfHab\u00eda muerto don Federico T\u00e9llez Unz\u00e1tegui, el indesmayable verdugo de los roedores del Per\u00fa? \u00bfHab\u00eda sido consumado un parricidio, un epitalamicidio? \u00bfO s\u00f3lo estaba aturdido ese esposo y padre que yac\u00eda, en medio de un desorden sin igual, bajo la mesa del comedor, mientras sus familiares, sus pertenencias r\u00e1pidamente enmaletadas, abandonaban exultantes el hogar? \u00bfC\u00f3mo terminar\u00eda esta desventura barranquina?\n\n### IX\n\nEL FRACASO del cuento sobre Doroteo Mart\u00ed me tuvo desalentado unos d\u00edas. Pero la ma\u00f1ana que o\u00ed a Pascual referirle al Gran Pablito su descubrimiento del aeropuerto, sent\u00ed que mi vocaci\u00f3n resucitaba y comenc\u00e9 a planear una nueva historia. Pascual hab\u00eda sorprendido a unos muchachitos vagabundos practicando un deporte riesgoso y excitante. Se tend\u00edan, al oscurecer, en el extremo de la pista de despegue del aeropuerto de Limatambo y Pascual juraba que, cada vez que un avi\u00f3n part\u00eda, por la presi\u00f3n del aire desplazado, los muchachitos tendidos se elevaban unos cent\u00edmetros y levitaban, como en un espect\u00e1culo de magia, hasta que unos segundos despu\u00e9s, desaparecido el efecto, regresaban al suelo de golpe. Yo hab\u00eda visto en esos d\u00edas una pel\u00edcula mexicana (s\u00f3lo a\u00f1os despu\u00e9s sabr\u00eda que era de Bu\u00f1uel y qui\u00e9n era Bu\u00f1uel) que me entusiasm\u00f3: Los olvidados. Decid\u00ed hacer un cuento con el mismo esp\u00edritu: un relato de ni\u00f1os hombres, j\u00f3venes lobeznos endurecidos por las \u00e1speras condiciones de la vida en los suburbios. Javier se mostr\u00f3 esc\u00e9ptico y me asegur\u00f3 que la an\u00e9cdota era falsa, que la presi\u00f3n del aire provocada por los aviones no levantaba a un reci\u00e9n nacido. Discutimos, yo acab\u00e9 dici\u00e9ndole que en mi cuento los personajes levitar\u00edan y que, sin embargo, ser\u00eda un cuento realista (\u00abno, fant\u00e1stico\u00bb, gritaba \u00e9l) y por \u00faltimo quedamos en ir, una noche, con Pascual a los descampados de la C\u00f3rpac para verificar qu\u00e9 hab\u00eda de verdad y de mentira en esos juegos peligrosos (era el t\u00edtulo que hab\u00eda elegido para el cuento).\n\nNo hab\u00eda visto a la t\u00eda Julia ese d\u00eda pero esperaba verla el siguiente, jueves, donde el t\u00edo Lucho. Sin embargo, al llegar a la casa de Armend\u00e1riz ese mediod\u00eda, para el almuerzo acostumbrado, me encontr\u00e9 con que no estaba. La t\u00eda Olga me cont\u00f3 que la hab\u00eda invitado a almorzar un buen partido: el doctor Guillermo Osores. Era un m\u00e9dico vagamente relacionado con la familia, un cincuent\u00f3n muy presentable, con algo de fortuna, enviudado no hac\u00eda mucho.\n\n\u2014Un buen partido \u2014repiti\u00f3 la t\u00eda Olga, gui\u00f1\u00e1ndome el ojo\u2014. Serio, rico, buen mozo, y con s\u00f3lo dos hijos que ya son mayorcitos. \u00bfNo es el marido que necesita mi hermana?\n\n\u2014Las \u00faltimas semanas estaba perdiendo el tiempo de mala manera \u2014coment\u00f3 el t\u00edo Lucho, tambi\u00e9n muy satisfecho\u2014. No quer\u00eda salir con nadie, hac\u00eda vida de solterona. Pero el endocrin\u00f3logo le ha ca\u00eddo en gracia.\n\nSent\u00ed unos celos que me quitaron el apetito, un malhumor salmuera. Me parec\u00eda que, por mi turbaci\u00f3n, mis t\u00edos iban a adivinar lo que me pasaba. No necesit\u00e9 tratar de sonsacarles m\u00e1s detalles sobre la t\u00eda Julia y el doctor Osores porque no hablaban de otra cosa. Lo hab\u00eda conocido hac\u00eda unos diez d\u00edas, en un coctel de la embajada boliviana, y, al saber d\u00f3nde estaba alojada, el doctor Osores hab\u00eda venido a visitarla. Le hab\u00eda mandado flores, llamado por tel\u00e9fono, invitado a tomar t\u00e9 al Bol\u00edvar y ahora a almorzar al Club de la Uni\u00f3n. El endocrin\u00f3logo le hab\u00eda bromeado al t\u00edo Lucho: \u00abTu cu\u00f1ada es de primera, Luis, \u00bfno ser\u00e1 la candidata que ando buscando para matrisuicidarme por segunda vez?\u00bb.\n\nYo trataba de demostrar desinter\u00e9s, pero lo hac\u00eda p\u00e9simo y el t\u00edo Lucho, en un momento en que estuvimos solos, me pregunt\u00f3 qu\u00e9 me pasaba: \u00bfno hab\u00eda metido la nariz donde no deb\u00eda y me hab\u00edan pegado una purgaci\u00f3n? Por suerte la t\u00eda Olga comenz\u00f3 a hablar de los radioteatros y eso me dio un respiro. Mientras ella dec\u00eda que, a veces, a Pedro Camacho se le pasaba la mano y que a todas sus amigas la historia del pastor que se her\u00eda con un cortapapeles delante del juez para probar que no viol\u00f3 a una chica les parec\u00eda demasiado, yo silenciosamente iba de la rabia a la decepci\u00f3n y de la decepci\u00f3n a la rabia. \u00bf Por qu\u00e9 no me hab\u00eda dicho la t\u00eda Julia ni una palabra sobre el m\u00e9dico? En esos diez \u00faltimos d\u00edas nos hab\u00edamos visto varias veces y jam\u00e1s lo hab\u00eda mencionado. \u00bfSer\u00eda cierto, como dec\u00eda la t\u00eda Olga, que por fin se hab\u00eda interesado en alguien?\n\nEn el colectivo, mientras regresaba a Radio Panamericana, salt\u00e9 de la humillaci\u00f3n a la soberbia. Nuestros amor\u00edos hab\u00edan durado mucho, en cualquier momento iban a sorprendernos y eso provocar\u00eda burlas y esc\u00e1ndalo en la familia. Por lo dem\u00e1s, \u00bfqu\u00e9 hac\u00eda perdiendo el tiempo con una se\u00f1ora que, como ella misma dec\u00eda, casi casi pod\u00eda ser mi madre? Como experiencia, ya bastaba. La aparici\u00f3n de Osores era providencial, me exim\u00eda de tener que sacarme de encima a la boliviana. Sent\u00eda desasosiego, impulsos inusitados como querer emborracharme o pegarle a alguien, y en la radio tuve un encontr\u00f3n con Pascual, que, fiel a su naturaleza, hab\u00eda dedicado la mitad del bolet\u00edn de las tres a un incendio en Hamburgo que carboniz\u00f3 a una docena de inmigrantes turcos. Le dije que en el futuro quedaba prohibido incluir alguna noticia con muertos sin mi visto bueno y trat\u00e9 sin amistad a un compa\u00f1ero de San Marcos que me llam\u00f3 para recordarme que la facultad todav\u00eda exist\u00eda y advertirme que al d\u00eda siguiente me esperaba un examen de derecho procesal. Apenas hab\u00eda cortado, son\u00f3 el tel\u00e9fono otra vez. Era la t\u00eda Julia:\n\n\u2014Te dej\u00e9 plantado por un endocrin\u00f3logo, Varguitas, supongo que me extra\u00f1aste \u2014me dijo, fresca como una lechuga\u2014. \u00bfNo te has enojado?\n\n\u2014\u00bfEnojado por qu\u00e9? \u2014le contest\u00e9\u2014. \u00bfNo eres libre de hacer lo que quieras?\n\n\u2014Ah, entonces te has enojado \u2014la o\u00ed decir, ya m\u00e1s seria\u2014. No seas sonso. \u00bfCu\u00e1ndo nos vemos, para que te explique?\n\n\u2014Hoy no puedo \u2014le repuse secamente\u2014. Ya te llamar\u00e9.\n\nColgu\u00e9, m\u00e1s furioso conmigo que con ella y sinti\u00e9ndome rid\u00edculo. Pascual y el Gran Pablito me miraban divertidos, y el amante de las cat\u00e1strofes se veng\u00f3 delicadamente de mi reprimenda:\n\n\u2014Vaya, qu\u00e9 castigador hab\u00eda sido este don Mario con las mujeres.\n\n\u2014Hace bien en tratarlas as\u00ed \u2014me apoy\u00f3 el Gran Pablito\u2014. Nada les gusta tanto como que las metan en cintura.\n\nMand\u00e9 a mis dos redactores a la mierda, hice el bolet\u00edn de las cuatro, y me fui a ver a Pedro Camacho. Estaba grabando un libreto y lo esper\u00e9 en su cub\u00edculo, curioseando sus papeles, sin entender lo que le\u00eda porque no hac\u00eda otra cosa que preguntarme si esa conversaci\u00f3n telef\u00f3nica con la t\u00eda Julia equival\u00eda a una ruptura. En cuesti\u00f3n de segundos pasaba a odiarla a muerte y a extra\u00f1arla con toda mi alma.\n\n\u2014Acomp\u00e1\u00f1eme a comprar venenos \u2014me dijo t\u00e9tricamente Pedro Camacho, desde la puerta, agitando su melena de le\u00f3n\u2014. Nos quedar\u00e1 tiempo para el bebedizo.\n\nMientras recorr\u00edamos las transversales del jir\u00f3n de la Uni\u00f3n buscando un veneno, el artista me cont\u00f3 que los ratones de la Pensi\u00f3n La Tapada hab\u00edan llegado a extremos intolerables.\n\n\u2014Si se contentaran con correr bajo mi cama, no me importar\u00eda, no son ni\u00f1os, a los animales no les tengo fobia \u2014me explic\u00f3, mientras olfateaba con su nariz protuberante unos polvos amarillos que, seg\u00fan el bodeguero, pod\u00edan matar a una vaca\u2014. Pero estos bigotudos se comen mi sustento, cada noche mordisquean las provisiones que dejo tomando el fresco en la ventana. No hay m\u00e1s, debo exterminarlos.\n\nRegate\u00f3 el precio, con argumentos que dejaban al bodeguero alelado, pag\u00f3, hizo que le envolvieran las bolsitas de veneno y fuimos a sentarnos a un caf\u00e9 de la Colmena. Pidi\u00f3 su compuesto vegetal y yo un caf\u00e9.\n\n\u2014Tengo una pena de amor, amigo Camacho \u2014le confes\u00e9 a boca de jarro, sorprendi\u00e9ndome de m\u00ed mismo por la f\u00f3rmula radioteatral; pero sent\u00ed que, habl\u00e1ndole as\u00ed, me distanciaba de mi propia historia y al mismo tiempo consegu\u00eda desahogarme\u2014. La mujer que quiero me enga\u00f1a con otro hombre.\n\nMe escrut\u00f3 profundamente, con sus ojitos saltones m\u00e1s fr\u00edos y deshumorados que nunca. Su traje negro hab\u00eda sido lavado, planchado y usado tanto que era brillante como una hoja de cebolla.\n\n\u2014El duelo, en estos pa\u00edses aplebeyados, se paga con c\u00e1rcel \u2014sentenci\u00f3, muy grave, haciendo unos movimientos convulsivos con las manos\u2014. En cuanto al suicidio, ya nadie aprecia el gesto. Uno se mata y en vez de remordimientos, escalofr\u00edos, admiraci\u00f3n, provoca burlas. Lo mejor son las recetas pr\u00e1cticas, mi amigo.\n\nMe alegr\u00e9 de haberle hecho confidencias. Sab\u00eda que, como para Pedro Camacho no exist\u00eda nadie fuera de \u00e9l mismo, mi problema ya ni lo recordaba, hab\u00eda sido un mero dispositivo para poner en acci\u00f3n su sistema teorizante. O\u00edrlo me consolar\u00eda m\u00e1s (y con menos consecuencias) que una borrachera. Pedro Camacho, luego de un amago de sonrisa, me pormenorizaba su receta:\n\n\u2014Una carta dura, hiriente, lapidaria a la ad\u00faltera \u2014me dec\u00eda, adjetivando con seguridad\u2014, una carta que la haga sentirse una lagartija sin entra\u00f1as, una hiena inmunda. Prob\u00e1ndole que uno no es tonto, que conoce su traici\u00f3n, una carta que rezume desprecio, que le d\u00e9 conciencia de ad\u00faltera \u2014call\u00f3, medit\u00f3 un instante y, cambiando ligeramente de tono, me dio la mayor prueba de amistad que pod\u00eda esperarse de \u00e9l\u2014: Si quiere, yo se la escribo.\n\nLe agradec\u00ed efusivamente, pero le dije que, conociendo sus horarios de galeote, jam\u00e1s podr\u00eda aceptar sobrecargarlo de trabajo con mis asuntos privados. (Despu\u00e9s lament\u00e9 esos escr\u00fapulos, que me privaron de un texto ol\u00f3grafo del escribidor.)\n\n\u2014En cuanto al seductor \u2014prosigui\u00f3 inmediatamente Pedro Camacho, con un brillo malvado en los ojos\u2014, lo mejor es el an\u00f3nimo, con todas las calumnias necesarias. \u00bfPor qu\u00e9 habr\u00eda de quedarse aletargada la v\u00edctima mientras le crecen cuernos? \u00bfPor qu\u00e9 permitir\u00eda que los ad\u00falteros se solacen fornicando? Hay que estropearles el amor, golpearlos donde les duela, envenenarlos de dudas. Que brote la desconfianza, que comiencen a mirarse con malos ojos, a odiarse. \u00bfAcaso no es dulce la venganza?\n\nLe insinu\u00e9 que, tal vez, valerse de an\u00f3nimos no fuera cosa de caballeros, pero \u00e9l me tranquiliz\u00f3 r\u00e1pidamente: uno deb\u00eda portarse con los caballeros como caballero y con los canallas como canalla. \u00c9se era el \u00abhonor bien entendido\u00bb: lo dem\u00e1s era ser idiota.\n\n\u2014Con la carta a ella y los an\u00f3nimos a \u00e9l quedan castigados los amantes \u2014le dije\u2014. Pero \u00bfy mi problema? \u00bfQui\u00e9n me va a quitar el despecho, la frustraci\u00f3n, la pena?\n\n\u2014Para todo eso no hay como la leche de magnesia \u2014me repuso, dej\u00e1ndome sin \u00e1nimos siquiera de re\u00edrme\u2014. Ya s\u00e9, le parecer\u00e1 un materialismo exagerado. Pero, h\u00e1game caso, tengo experiencia de la vida. La mayor parte de las veces, las llamadas penas de coraz\u00f3n, etc\u00e9tera, son malas digestiones, frejoles tercos que no se deshacen, pescado pasado de tiempo, estre\u00f1imiento. Un buen purgante fulmina la locura de amor.\n\nEsta vez no hab\u00eda duda, era un humorista sutil, se burlaba de m\u00ed y de sus oyentes, no cre\u00eda palabra de lo que dec\u00eda, practicaba el aristocr\u00e1tico deporte de probarse a s\u00ed mismo que los humanos \u00e9ramos unos irremisibles imb\u00e9ciles.\n\n\u2014\u00bfHa tenido usted muchos amores, una vida sentimental muy rica? \u2014le pregunt\u00e9.\n\n\u2014Muy rica, s\u00ed \u2014asinti\u00f3, mir\u00e1ndome a los ojos por sobre la taza de menta y yerbaluisa que se hab\u00eda llevado a la boca\u2014. Pero yo no he amado nunca a una mujer de carne y hueso.\n\nHizo una pausa efectista, como midiendo el tama\u00f1o de mi inocencia o estupidez.\n\n\u2014\u00bfCree usted que ser\u00eda posible hacer lo que hago si las mujeres se tragaran mi energ\u00eda? \u2014me amonest\u00f3, con asco en la voz\u2014. \u00bfCree que se pueden producir hijos e historias al mismo tiempo? \u00bfQue uno puede inventar, imaginar, si se vive bajo la amenaza de la s\u00edfilis? La mujer y el arte son excluyentes, mi amigo. En cada vagina est\u00e1 enterrado un artista. Reproducirse, \u00bfqu\u00e9 gracia tiene? \u00bfNo lo hacen los perros, las ara\u00f1as, los gatos? Hay que ser originales, mi amigo.\n\nSin soluci\u00f3n de continuidad se puso de pie de un salto, advirti\u00e9ndome que ten\u00eda el tiempo justo para el radioteatro de las cinco. Sent\u00ed desilusi\u00f3n, me hubiera pasado la tarde escuch\u00e1ndolo, ten\u00eda la impresi\u00f3n de, sin quererlo, haber tocado un punto neur\u00e1lgico de su personalidad.\n\nEn mi oficina de Panamericana, estaba esper\u00e1ndome la t\u00eda Julia. Sentada en mi escritorio como una reina, recib\u00eda los homenajes de Pascual y del Gran Pablito, que, sol\u00edcitos, movedizos, le mostraban los boletines y le explicaban c\u00f3mo funcionaba el servicio. Se la ve\u00eda risue\u00f1a y tranquila; al entrar yo, se puso seria y palideci\u00f3 ligeramente.\n\n\u2014Vaya, qu\u00e9 sorpresa \u2014dije, por decir algo.\n\nPero la t\u00eda Julia no estaba para eufemismos.\n\n\u2014He venido a decirte que a m\u00ed nadie me cuelga el tel\u00e9fono \u2014me dijo, con voz resuelta\u2014. Y mucho menos un mocoso como t\u00fa. \u00bfQuieres decirme qu\u00e9 mosca te ha picado?\n\nPascual y el Gran Pablito quedaron est\u00e1ticos y mov\u00edan las cabezas de ella a m\u00ed y viceversa, interesad\u00edsimos en ese comienzo de drama. Cuando les ped\u00ed que se fueran un momento, pusieron unas caras enfurecidas, pero no se atrevieron a rebelarse. Partieron lanzando a la t\u00eda Julia unas miradas llenas de malos pensamientos.\n\n\u2014Te colgu\u00e9 el tel\u00e9fono pero en realidad ten\u00eda ganas de apretarte el pescuezo \u2014le dije, cuando nos quedamos solos.\n\n\u2014No te conoc\u00eda esos arranques \u2014dijo ella, mir\u00e1ndome a los ojos\u2014. \u00bfSe puede saber qu\u00e9 te pasa?\n\n\u2014Sabes muy bien lo que me pasa, as\u00ed que no te hagas la tonta \u2014dije yo.\n\n\u2014\u00bfEst\u00e1s celoso porque sal\u00ed a almorzar con el doctor Osores? \u2014me pregunt\u00f3, con un tonito burl\u00f3n\u2014. C\u00f3mo se nota que eres un mocoso, Marito.\n\n\u2014Te he prohibido que me llames Marito \u2014le record\u00e9. Sent\u00eda que la irritaci\u00f3n me iba dominando, que me temblaba la voz y ya no sab\u00eda lo que le dec\u00eda\u2014. Y ahora te proh\u00edbo que me llames mocoso.\n\nMe sent\u00e9 en la esquina del escritorio y, como haciendo contrapunto, la t\u00eda Julia se puso de pie y dio unos pasos hacia la ventana. Con los brazos cruzados sobre el pecho, se qued\u00f3 mirando la ma\u00f1ana gris, h\u00fameda, discretamente fantasmal. Pero no la ve\u00eda, buscaba las palabras para decirme algo. Vest\u00eda un traje azul y unos zapatos blancos y, repentinamente, tuve ganas de besarla.\n\n\u2014Vamos a poner las cosas en su sitio \u2014me dijo al fin, d\u00e1ndome siempre la espalda\u2014. T\u00fa no puedes prohibirme nada, ni siquiera en broma, por la sencilla raz\u00f3n de que no eres nada m\u00edo. No eres mi marido, no eres mi novio, no eres mi amante. Este jueguecito de cogernos de la mano, de besarnos en el cine, no es serio, y, sobre todo, no te da derechos sobre m\u00ed. Tienes que meterte eso en la cabeza, hijito.\n\n\u2014La verdad es que est\u00e1s hablando como si fueras mi mam\u00e1 \u2014le dije yo.\n\n\u2014Es que podr\u00eda ser tu mam\u00e1 \u2014dijo la t\u00eda Julia, y se le entristeci\u00f3 la cara. Fue como si se le hubiera pasado la furia y, en su lugar, quedara s\u00f3lo una vieja contrariedad, una profunda desaz\u00f3n. Se volvi\u00f3, dio unos pasos hacia el escritorio, se par\u00f3 muy cerca de m\u00ed. Me miraba apenada\u2014: T\u00fa me haces sentir vieja, sin serlo, Varguitas. Y eso no me gusta. Lo nuestro no tiene raz\u00f3n de ser y mucho menos futuro.\n\nLa cog\u00ed de la cintura y ella se dej\u00f3 ir contra m\u00ed, pero, mientras la besaba, con mucha ternura, en la mejilla, en el cuello, en la oreja \u2014su piel tibia lat\u00eda bajo mis labios y sentir la secreta vida de sus venas me produc\u00eda una alegr\u00eda enorme\u2014, sigui\u00f3 hablando con el mismo tono de voz:\n\n\u2014He estado pensando mucho y la cosa ya no me gusta, Varguitas. \u00bfNo te das cuenta que es absurdo? Tengo treinta y dos a\u00f1os, soy divorciada, \u00bfquieres decirme qu\u00e9 hago con un mocoso de dieciocho? \u00c9sas son perversiones de las cincuentonas, yo todav\u00eda no estoy para \u00e9sas.\n\nMe sent\u00eda tan conmovido y enamorado mientras le besaba el cuello, las manos, le mord\u00eda despacito la oreja, le pasaba los labios por la nariz, los ojos o enredaba mis dedos en sus cabellos, que a ratos se me perd\u00eda lo que iba dici\u00e9ndome. Tambi\u00e9n su voz sufr\u00eda altibajos, a veces se debilitaba hasta ser un susurro.\n\n\u2014Al principio era gracioso, por lo de los escondites \u2014dec\u00eda, dej\u00e1ndose besar, pero sin hacer ning\u00fan gesto rec\u00edproco\u2014, y, sobre todo, porque me hac\u00eda sentirme otra vez jovencita.\n\n\u2014En qu\u00e9 quedamos \u2014murmur\u00e9, en su o\u00eddo\u2014. \u00bfTe hago sentir una cincuentona viciosa o una jovencita?\n\n\u2014Eso de estar con un mocoso muerto de hambre, de s\u00f3lo cogerse la mano, de s\u00f3lo ir al cine, de s\u00f3lo besarse con tanta delicadeza, me hac\u00eda volver a mis quince a\u00f1os \u2014segu\u00eda diciendo la t\u00eda Julia\u2014. Claro que es bonito enamorarse con un muchachito t\u00edmido, que te respeta, que no te manosea, que no se atreve a acostarse contigo, que te trata como a una ni\u00f1ita de primera comuni\u00f3n. Pero es un juego peligroso, Varguitas, se basa en una mentira...\n\n\u2014A prop\u00f3sito, estoy escribiendo un cuento que se va a llamar Los juegos peligrosos \u2014le susurr\u00e9\u2014. Sobre unos palomillas que levitan en el aeropuerto, gracias a los aviones que despegan.\n\nSent\u00ed que se re\u00eda. Un momento despu\u00e9s me ech\u00f3 los brazos al cuello y me junt\u00f3 la cara.\n\n\u2014Bueno, se me pas\u00f3 la c\u00f3lera \u2014dijo\u2014. Porque vine decidida a sacarte los ojos. Ay de ti que me vuelvas a colgar el tel\u00e9fono.\n\n\u2014Ay de ti que vuelvas a salir con el endocrin\u00f3logo \u2014le dije, busc\u00e1ndole la boca\u2014. Prom\u00e9teme que nunca m\u00e1s saldr\u00e1s con \u00e9l.\n\nSe apart\u00f3 y me mir\u00f3 con un brillo pendenciero en los ojos.\n\n\u2014No te olvides que he venido a Lima a buscarme un marido \u2014brome\u00f3 a medias\u2014. Y creo que esta vez he encontrado lo que me conviene. Buen mozo, culto, con buena situaci\u00f3n y con canas en las sienes.\n\n\u2014\u00bfEst\u00e1s segura que esa maravilla se va a casar contigo? \u2014le dije, sintiendo otra vez furia y celos.\n\nCogi\u00e9ndose las caderas, en una pose provocativa, me repuso:\n\n\u2014Yo puedo hacer que se case conmigo.\n\nPero al ver mi cara, se ri\u00f3, me volvi\u00f3 a echar los brazos al cuello, y as\u00ed est\u00e1bamos, bes\u00e1ndonos con amor-pasi\u00f3n, cuando o\u00edmos la voz de Javier:\n\n\u2014Los van a meter presos por escandalosos y pornogr\u00e1ficos \u2014estaba feliz y, abraz\u00e1ndonos a los dos, nos anunci\u00f3\u2014: La flaca Nancy me ha aceptado una invitaci\u00f3n a los toros y hay que celebrarlo.\n\n\u2014Acabamos de tener nuestra primera gran pelea y nos pescaste en plena reconciliaci\u00f3n \u2014le cont\u00e9.\n\n\u2014C\u00f3mo se nota que no me conoces \u2014me previno la t\u00eda Julia\u2014. En las grandes peleas yo rompo platos, ara\u00f1o, mato.\n\n\u2014Lo bueno de pelearse son las amistadas \u2014dijo Javier, que era un experto en la materia\u2014. Pero, maldita sea, yo vengo hecho unas pascuas con lo de la flaca Nancy y ustedes como si lloviera, qu\u00e9 clase de amigos son. Vamos a festejar el acontecimiento con un lonche.\n\nMe esperaron mientras redactaba un par de boletines y bajamos a un cafecito de la calle Bel\u00e9n, que le encantaba a Javier, porque, pese a ser estrecho y mugriento, all\u00ed preparaban los mejores chicharrones de Lima. Encontr\u00e9 a Pascual y al Gran Pablito, en la puerta de Panamericana, piropeando a las transe\u00fantes, y los regres\u00e9 a la redacci\u00f3n. Pese a ser de d\u00eda y estar en pleno centro, al alcance de los ojos incontables de parientes y amigos de la familia, la t\u00eda Julia y yo \u00edbamos de la mano, y yo la besaba todo el tiempo. Ella ten\u00eda unas chapas de serrana y se la ve\u00eda contenta.\n\n\u2014Basta de pornograf\u00eda, ego\u00edstas, piensen en m\u00ed \u2014protestaba Javier\u2014. Hablemos un poco de la flaca Nancy.\n\nLa flaca Nancy era una prima m\u00eda, bonita y muy coqueta, de la que Javier estaba enamorado desde que ten\u00eda uso de raz\u00f3n y a la que persegu\u00eda con una constancia de sabueso. Ella nunca hab\u00eda llegado a hacerle caso del todo, pero siempre se las arreglaba para hacerle creer que tal vez, que pronto, que la pr\u00f3xima. Ese prerromance duraba desde que est\u00e1bamos en el colegio y yo, como confidente, amigo \u00edntimo y celestino de Javier, hab\u00eda seguido todos sus pormenores. Eran incontables los plantones que la flaca Nancy le hab\u00eda dado, infinitas las matin\u00e9s de domingo que lo hab\u00eda dejado esper\u00e1ndola a las puertas del Leuro mientras ella se iba al Colina o al Metro, infinitas las veces que se le hab\u00eda aparecido con otro gal\u00e1n en la fiesta del s\u00e1bado. La primera borrachera de mi vida la tuve acompa\u00f1ando a Javier, a ahogar sus penas con capitanes y cerveza, en un barcito de Surquillo, el d\u00eda que se enter\u00f3 que la flaca Nancy le hab\u00eda dicho s\u00ed al estudiante de Agronom\u00eda Eduardo Tiravanti (muy popular en Miraflores porque sab\u00eda meterse prendido el cigarrillo a la boca y luego sacarlo y seguir fumando como si tal cosa). Javier lloriqueaba y yo, adem\u00e1s de ser su pa\u00f1o de l\u00e1grimas, ten\u00eda la misi\u00f3n de ir a acostarlo a su pensi\u00f3n cuando hubiera llegado a un estado comatoso (\u00abMe voy a mamar hasta las cachas\u00bb, me hab\u00eda prevenido, imitando a Jorge Negrete). Pero fui yo el que sucumbi\u00f3, con ruidosos v\u00f3mitos y un ataque de diablos azules en el curso del cual \u2014era la versi\u00f3n canallesca de Javier\u2014 me hab\u00eda encaramado al mostrador y arengado a los borrachitos, noct\u00e1mbulos y rufianes que constitu\u00edan la clientela de El Triunfo:\n\n\u2014B\u00e1jense los pantalones que est\u00e1n ante un poeta.\n\nSiempre me reprochaba que, en vez de cuidarlo y consolarlo en esa noche triste, lo hubiera obligado a arrastrarme por las calles de Miraflores hasta la quinta de Ochar\u00e1n, en un estado tal de descomposici\u00f3n que entreg\u00f3 mis restos a mi asustada abuela con este comentario desatinado:\n\n\u2014Se\u00f1ora Carmencita, creo que el Varguitas se nos muere.\n\nDesde entonces, la flaca Nancy hab\u00eda aceptado y despedido a media docena de miraflorinos, y Javier hab\u00eda tenido tambi\u00e9n enamoradas, pero ellas no cancelaban sino robustec\u00edan su gran amor por mi prima, a la que segu\u00eda llamando, visitando, invitando, declar\u00e1ndose, indiferente ante las negativas, malacrianzas, desaires y plantones. Javier era uno de esos hombres que pueden anteponer la pasi\u00f3n a la vanidad y le importaban realmente un comino las burlas de todos los amigos de Miraflores, entre quienes su persecuci\u00f3n de mi prima era un surtidor de chistes. (En el barrio un muchacho juraba que lo hab\u00eda visto acercarse a la flaca Nancy, un domingo, a la salida de misa de once, con la siguiente propuesta: \u00abHola Nancita, linda ma\u00f1ana, \u00bfnos tomamos algo?, \u00bfuna Coca-Cola, un champancito?\u00bb.) La flaca Nancy sal\u00eda algunas veces con \u00e9l, generalmente entre dos enamorados, al cine o a una fiesta, y Javier conceb\u00eda entonces grandes esperanzas y entraba en estado de euforia. As\u00ed estaba ahora, hablando hasta por los codos, mientras nos tom\u00e1bamos unos caf\u00e9s con leche y unos s\u00e1ndwiches de chicharr\u00f3n, en ese caf\u00e9 de la calle Bel\u00e9n que se llamaba El Palmero. La t\u00eda Julia y yo nos toc\u00e1bamos las rodillas bajo la mesa, ten\u00edamos entrelazados los dedos, nos mir\u00e1bamos a los ojos, y, mientras, como una m\u00fasica de fondo, o\u00edamos a Javier hablando de la flaca Nancy.\n\n\u2014La invitaci\u00f3n la ha dejado impresionada \u2014nos cont\u00f3\u2014. Porque, \u00bfquieres decirme qu\u00e9 pelagatos de Miraflores invita a una chica a los toros?\n\n\u2014\u00bfC\u00f3mo has hecho? \u2014le pregunt\u00e9\u2014. \u00bfTe sacaste la loter\u00eda?\n\n\u2014He vendido la radio de la pensi\u00f3n \u2014nos dijo, sin el menor remordimiento\u2014. Creen que ha sido la cocinera y la han despedido por ladrona.\n\nNos explic\u00f3 que ten\u00eda preparado un plan infalible. En media corrida, sorprender\u00eda a la flaca Nancy con un regalo persuasivo: una mantilla espa\u00f1ola. Javier era un gran admirador de la Madre Patria y de todo lo que se relacionaba con ella: los toros, la m\u00fasica flamenca, Sarita Montiel. So\u00f1aba con ir a Espa\u00f1a (como yo con ir a Francia) y lo de la mantilla se le hab\u00eda ocurrido al ver un aviso del peri\u00f3dico. Le hab\u00eda costado su sueldo de un mes en el Banco de Reserva pero estaba seguro que la inversi\u00f3n tendr\u00eda frutos. Nos explic\u00f3 c\u00f3mo iban a ocurrir las cosas. Llevar\u00eda la mantilla a los toros discretamente envuelta y esperar\u00eda un momento de gran emoci\u00f3n para abrir el paquete, desplegar la prenda y colocarla sobre los hombros delicados de mi prima. \u00bfQu\u00e9 pens\u00e1bamos? \u00bfCu\u00e1l ser\u00eda la reacci\u00f3n de la flaquita? Yo le aconsej\u00e9 que redondeara las cosas, regal\u00e1ndole tambi\u00e9n una peineta sevillana y unas casta\u00f1uelas y que le cantara un fandango, pero la t\u00eda Julia lo apoy\u00f3 con entusiasmo y le dijo que todo lo que hab\u00eda planeado era lindo y que la Nancy, si ten\u00eda coraz\u00f3n, se emocionar\u00eda hasta los huesos. Ella, si un muchacho le hac\u00eda esas demostraciones, quedar\u00eda conquistada.\n\n\u2014\u00bfNo ves lo que te digo siempre? \u2014me dijo, igual que si estuviera ri\u00f1\u00e9ndome\u2014. Javier s\u00ed que es un rom\u00e1ntico, enamora como se deber\u00eda enamorar.\n\nJavier, encantado, nos propuso que sali\u00e9ramos los cuatro juntos, cualquier d\u00eda de la pr\u00f3xima semana, al cine, a tomar t\u00e9, a bailar.\n\n\u2014\u00bfY qu\u00e9 dir\u00eda la flaca Nancy si nos ve de pareja? \u2014le puse los pies en la tierra.\n\nPero \u00e9l nos ech\u00f3 un baldazo de agua fr\u00eda:\n\n\u2014No seas tonto, sabe todo y le parece muy bien, se lo cont\u00e9 el otro d\u00eda \u2014y al ver nuestra sorpresa, a\u00f1adi\u00f3, con cara de travieso\u2014: Pero si con tu prima yo no tengo secretos, si ella, haga lo que haga, terminar\u00e1 cas\u00e1ndose conmigo.\n\nMe qued\u00e9 preocupado al saber que Javier le hab\u00eda contado nuestro romance. \u00c9ramos muy unidos y estaba seguro que no ir\u00eda a delatarnos, pero se le pod\u00eda escapar algo, y la noticia correr\u00eda como un incendio por el bosque familiar. La t\u00eda Julia se hab\u00eda quedado muda, pero ahora disimulaba dando br\u00edos a Javier en su proyecto taurino-sentimental. Nos despedimos en la puerta del Edificio Panamericano y quedamos con la t\u00eda Julia en que nos ver\u00edamos esa noche, con el pretexto del cine. Al besarla, le dije al o\u00eddo: \u00abGracias al endocrin\u00f3logo, me he dado cuenta que estoy enamorado de ti\u00bb. Ella asinti\u00f3: \u00abAs\u00ed estoy viendo, Varguitas\u00bb.\n\nMe la qued\u00e9 viendo alejarse, con Javier, hacia el paradero de los colectivos, y s\u00f3lo entonces advert\u00ed la gente aglomerada a las puertas de Radio Central. Eran sobre todo mujeres j\u00f3venes, aunque hab\u00eda tambi\u00e9n algunos hombres. Estaban en filas de a dos, pero, a medida que llegaba m\u00e1s gente, la formaci\u00f3n se descompon\u00eda, entre codazos y empujones. Me acerqu\u00e9 a curiosear porque supuse que la raz\u00f3n ten\u00eda que ser Pedro Camacho. En efecto, eran coleccionistas de aut\u00f3grafos. Por la ventana del cub\u00edculo, vi al escriba, escoltado por Jesusito y por Genaro pap\u00e1, rasgu\u00f1ando una firma con arabescos en cuadernos, libretas, hojitas sueltas, peri\u00f3dicos, y despidiendo a sus admiradores con un gesto ol\u00edmpico. Ellos lo miraban con embelesamiento y se le acercaban en actitud t\u00edmida, balbuceando palabras de aprecio.\n\n\u2014Nos da dolores de cabeza, pero, no hay duda, es el rey de la radiotelefon\u00eda nacional \u2014me dijo Genaro hijo, poni\u00e9ndome una mano en el hombro y se\u00f1alando el gent\u00edo\u2014: \u00bfQu\u00e9 te parece esto?\n\nLe pregunt\u00e9 desde cu\u00e1ndo funcionaba lo de los aut\u00f3grafos.\n\n\u2014Desde hace una semana, media hora al d\u00eda, de seis a seis y media, hombre poco observador \u2014me dijo el empresario progresista\u2014. \u00bfNo lees los avisos que publicamos, no oyes la radio en la que trabajas? Yo era esc\u00e9ptico, pero mira c\u00f3mo me equivoqu\u00e9. Cre\u00ed que s\u00f3lo habr\u00eda gente para dos d\u00edas y ahora veo que esto puede funcionar un mes.\n\nMe invit\u00f3 a tomar un trago al bar del Bol\u00edvar. Yo ped\u00ed una Coca-Cola, pero \u00e9l insisti\u00f3 en que lo acompa\u00f1ara con un whisky.\n\n\u2014\u00bfTe das cuenta lo que significan estas colas? \u2014me explic\u00f3\u2014. Son una demostraci\u00f3n p\u00fablica de que los radioteatros de Pedro calan en el pueblo.\n\nLe dije que no me cab\u00eda duda y \u00e9l me hizo poner colorado recomend\u00e1ndome que, como yo ten\u00eda aficiones literarias, siguiera el ejemplo del boliviano, aprendiera sus recursos para conquistar a las muchedumbres. \u00abNo debes encerrarte en tu torre de marfil\u00bb, me aconsej\u00f3. Hab\u00eda mandado imprimir cinco mil fotos de Pedro Camacho y, a partir del lunes, los cazadores de aut\u00f3grafos las recibir\u00edan como obsequio. Le pregunt\u00e9 si el escriba hab\u00eda amortiguado sus descargas contra los argentinos.\n\n\u2014Ya no importa, ahora puede hablar pestes contra quienquiera \u2014me dijo, con aire misterioso\u2014. \u00bfNo sabes la gran noticia? El General no se pierde los radioteatros de Pedro.\n\nMe dio precisiones, para convencerme. El General, como las cuestiones de gobierno no le daban tiempo para o\u00edrlos durante el d\u00eda, se los hac\u00eda grabar y los escuchaba cada noche, uno tras otro, antes de dormir. La Presidenta en persona se lo hab\u00eda contado a muchas se\u00f1oras de Lima.\n\n\u2014Parece que el General es un hombre sensible, a pesar de lo que dicen \u2014concluy\u00f3 Genaro hijo\u2014. De modo que si la cumbre est\u00e1 con nosotros, qu\u00e9 m\u00e1s da que Pedro se d\u00e9 gusto contra los ches. \u00bfNo se lo merecen?\n\nLa conversaci\u00f3n con Genaro hijo, la reconciliaci\u00f3n con la t\u00eda Julia, algo, me hab\u00eda estimulado mucho y regres\u00e9 al altillo a escribir con \u00edmpetu mi cuento de los levitadores, mientras Pascual despachaba los boletines. Ya ten\u00eda el final: en uno de esos juegos, un palomilla levitaba m\u00e1s alto que los otros, ca\u00eda con fuerza, se romp\u00eda la nuca y mor\u00eda. La \u00faltima frase mostrar\u00eda las caras sorprendidas, asustadas de sus compa\u00f1eros, contempl\u00e1ndolo, bajo un tronar de aviones. Ser\u00eda un relato espartano, preciso como un cron\u00f3metro, al estilo de Hemingway.\n\nUnos d\u00edas despu\u00e9s, fui a visitar a mi prima Nancy, para saber c\u00f3mo hab\u00eda tomado la historia de la t\u00eda Julia. La encontr\u00e9 todav\u00eda bajo los efectos de la Operaci\u00f3n Mantilla:\n\n\u2014\u00bfTe das cuenta el papel\u00f3n que pas\u00e9 por ese idiota? \u2014dec\u00eda, mientras correteaba por toda la casa, buscando a Lasky\u2014. De repente, en plena plaza de Acho, abri\u00f3 un paquete, sac\u00f3 una capa de torero y me la puso encima. Todo el mundo se qued\u00f3 mir\u00e1ndome, hasta el toro se mor\u00eda de risa. Me hizo tenerla puesta toda la corrida. Y quer\u00eda que saliera a la calle con esa cosa, fig\u00farate. \u00a1Nunca he pasado tanta verg\u00fcenza en mi vida!\n\nEncontramos a Lasky bajo la cama del mayordomo \u2014adem\u00e1s de ser pelado y feo, era un perro que siempre quer\u00eda morderme\u2014, lo llevamos a su jaula y la flaca Nancy me arrastr\u00f3 a su dormitorio a ver el cuerpo del delito. Era una prenda modernista y hac\u00eda pensar en jardines ex\u00f3ticos, en carpas de gitanas, en burdeles de lujo: tornasolada, anidaban en sus pliegues todos los matices del rojo, desde el bermell\u00f3n sangre hasta el ros\u00e1ceo arrebol, ten\u00eda nudosos y largos flecos negros y sus pedrer\u00edas y oropeles brillaban tanto que produc\u00edan mareos. Mi prima hac\u00eda pases taurinos o se envolv\u00eda en ella, ri\u00e9ndose a carcajadas. Le dije que no le permit\u00eda burlarse de mi amigo y le pregunt\u00e9 si por fin le iba a hacer caso.\n\n\u2014Lo estoy pensando \u2014me repuso, igual que siempre\u2014. Pero como amigo me encanta.\n\nLe dije que era una coqueta sin coraz\u00f3n, que Javier hab\u00eda llegado al robo para hacerle ese regalo.\n\n\u2014\u00bfY t\u00fa? \u2014me dijo, doblando y guardando la mantilla en el ropero\u2014. \u00bfEs cierto que est\u00e1s con la Julita? \u00bfNo te da verg\u00fcenza? \u00bfCon la hermana de la t\u00eda Olga?\n\nLe dije que era cierto, que no me daba verg\u00fcenza y sent\u00ed que me ard\u00eda la cara. Ella tambi\u00e9n se confundi\u00f3 un poco, pero su curiosidad miraflorina fue m\u00e1s fuerte y dispar\u00f3 hacia el blanco:\n\n\u2014Si te casas con ella, dentro de veinte a\u00f1os ser\u00e1s todav\u00eda joven y ella una abuelita \u2014me tom\u00f3 del brazo y me despe\u00f1\u00f3 por las escaleras hacia la sala\u2014. Ven, vamos a o\u00edr m\u00fasica y all\u00e1 me cuentas tu enamoramiento de pe a pa.\n\nSeleccion\u00f3 un alto de discos \u2014Nat King Cole, Harry Belafonte, Frank Sinatra, Xavier Cugat\u2014, mientras me confesaba que, desde que Javier le cont\u00f3, se le pon\u00edan los pelos de punta pensando en lo que pasar\u00eda si se enteraba la familia. \u00bfAcaso nuestros parientes no eran tan metetes que el d\u00eda que ella sal\u00eda con un muchacho distinto diez t\u00edos, ocho t\u00edas y cinco primas llamaban a su mam\u00e1 a cont\u00e1rselo? \u00a1Yo enamorado con la t\u00eda Julia! \u00a1Qu\u00e9 tal esc\u00e1ndalo, Marito! Y me record\u00f3 que la familia se hac\u00eda ilusiones, que yo era la esperanza de la tribu. Era verdad: mi cancerosa parentela esperaba de m\u00ed que fuera alg\u00fan d\u00eda millonario, o, en el peor de los casos, Presidente de la Rep\u00fablica. (Nunca comprend\u00ed por qu\u00e9 se hab\u00eda formado una opini\u00f3n tan alta de m\u00ed. En todo caso, no por mis notas de colegio, que nunca fueron brillantes. Tal vez porque, desde chico, les escrib\u00eda poemas a todas mis t\u00edas o porque fui, al parecer, un ni\u00f1o revejido que opinaba de todo.) Le hice jurar a la flaca Nancy que ser\u00eda una tumba. Ella se mor\u00eda por saber detalles del romance:\n\n\u2014\u00bfLa Julita s\u00f3lo te gusta o est\u00e1s templado de ella?\n\nAlguna vez le hab\u00eda hecho confidencias sentimentales y ahora, puesto que ya sab\u00eda, se las hice tambi\u00e9n. La historia hab\u00eda comenzado como un juego, pero, de repente, exactamente el d\u00eda en que sent\u00ed celos por un endocrin\u00f3logo, me di cuenta que me hab\u00eda enamorado. Sin embargo, mientras m\u00e1s vueltas le daba, m\u00e1s me convenc\u00eda que el romance era un rompecabezas. No s\u00f3lo por la diferencia de edad. Me faltaban tres a\u00f1os para terminar abogac\u00eda y sospechaba que nunca ejercer\u00eda esa profesi\u00f3n, porque lo \u00fanico que me gustaba era escribir. Pero los escritores se mor\u00edan de hambre. Por ahora, s\u00f3lo ganaba para comprar cigarros, unos cuantos libros e ir al cine. \u00bfMe iba a esperar la t\u00eda Julia hasta que fuera un hombre solvente, si alguna vez llegaba a serlo? Mi prima Nancy era tan buena que, en vez de contradecirme, me daba la raz\u00f3n:\n\n\u2014Claro, sin contar que para entonces a lo mejor la Julita ya no te gusta y la dejas \u2014me dec\u00eda, con realismo\u2014. Y la pobre habr\u00e1 perdido su tiempo miserablemente. Pero, dime, \u00bfella est\u00e1 enamorada de ti o s\u00f3lo juega?\n\nLe dije que la t\u00eda Julia no era una veleta fr\u00edvola como ella (lo que realmente le encant\u00f3). Pero la misma pregunta me la hab\u00eda hecho yo varias veces. Se la hice tambi\u00e9n a la t\u00eda Julia, unos d\u00edas despu\u00e9s. Hab\u00edamos ido a sentarnos frente al mar, en un bello parquecito de nombre impronunciable (Domodossola o algo as\u00ed) y all\u00ed, abrazados, bes\u00e1ndonos sin tregua, tuvimos nuestra primera conversaci\u00f3n sobre el futuro.\n\n\u2014Me lo s\u00e9 con lujo de detalles, lo he visto en una bola de cristal \u2014me dijo la t\u00eda Julia, sin la menor amargura\u2014. En el mejor de los casos, lo nuestro durar\u00eda tres, tal vez unos cuatro a\u00f1os, es decir hasta que encuentres a la mocosita que ser\u00e1 la mam\u00e1 de tus hijos. Entonces me botar\u00e1s y tendr\u00e9 que seducir a otro caballero. Y aparece la palabra fin.\n\nLe dije, mientras le besaba las manos, que le hac\u00eda mal o\u00edr radioteatros.\n\n\u2014C\u00f3mo se nota que no los oyes nunca \u2014me rectific\u00f3\u2014. En los radioteatros de Pedro Camacho rara vez hay amores o cosas parecidas. Ahora, por ejemplo, Olga y yo estamos entretenid\u00edsimas con el de las tres. La tragedia de un muchacho que no puede dormir porque, apenas cierra los ojos, vuelve a apachurrar a una pobre ni\u00f1ita.\n\nLe dije, volviendo al tema, que yo era m\u00e1s optimista. Con fogosidad, para convencerme a m\u00ed mismo al tiempo que a ella, le asegur\u00e9 que, fueran cuales fueran las diferencias, el amor duraba poco basado en lo puramente f\u00edsico. Con la desaparici\u00f3n de la novedad, con la rutina, la atracci\u00f3n sexual disminu\u00eda y al final mor\u00eda (sobre todo en el hombre), y la pareja entonces s\u00f3lo pod\u00eda sobrevivir si hab\u00eda entre ellos otros imanes: espirituales, intelectuales, morales. Para esa clase de amor la edad no importaba.\n\n\u2014Suena bonito y me convendr\u00eda que fuera verdad \u2014dijo la t\u00eda Julia, frotando contra mi mejilla una nariz que siempre estaba fr\u00eda\u2014. Pero es mentira de principio a fin. \u00bfLo f\u00edsico algo secundario? Es lo m\u00e1s importante para que dos personas se aguanten, Varguitas.\n\n\u00bfHab\u00eda vuelto a salir con el endocrin\u00f3logo?\n\n\u2014Me ha llamado varias veces \u2014me dijo, fomentando mi expectaci\u00f3n. Luego, bes\u00e1ndome, despej\u00f3 la inc\u00f3gnita\u2014: Le he dicho que no voy a salir m\u00e1s con \u00e9l.\n\nEn el colmo de la felicidad, yo le habl\u00e9 mucho rato de mi cuento sobre los levitadores: ten\u00eda diez p\u00e1ginas, estaba saliendo bien y tratar\u00eda de publicarlo en el Suplemento de El Comercio con una dedicatoria cr\u00edptica: \u00abAl femenino de Julio\u00bb.\n\n### X\n\nLA TRAGEDIA de Lucho Abril Marroqu\u00edn, joven propagandista m\u00e9dico al que todo auguraba un futuro promisor, comenz\u00f3 una soleada ma\u00f1ana de verano, en las afueras de una localidad hist\u00f3rica: Pisco. Acababa de terminar el recorrido que, desde que asumi\u00f3 esa profesi\u00f3n itinerante, hac\u00eda diez a\u00f1os, lo llevaba por los pueblos y ciudades del Per\u00fa, visitando consultorios y farmacias para regalar muestras y prospectos de los Laboratorios Bayer, y se dispon\u00eda a regresar a Lima. La visita a los facultativos y qu\u00edmicos del lugar le hab\u00eda tomado unas tres horas. Y aunque en el Grupo A\u00e9reo N. 9, de San Andr\u00e9s, ten\u00eda un compa\u00f1ero de colegio, que era ahora capit\u00e1n, en cuya casa sol\u00eda quedarse a almorzar cuando ven\u00eda a Pisco, decidi\u00f3 regresar a la capital de una vez. Estaba casado, con una muchacha de piel blanca y apellido franc\u00e9s, y su sangre joven y su coraz\u00f3n enamorado lo urg\u00edan a retornar cuanto antes a los brazos de su c\u00f3nyuge.\n\nEra un poco m\u00e1s de mediod\u00eda. Su flamante Volkswagen, adquirido a plazos al mismo tiempo que el v\u00ednculo matrimonial \u2014tres meses atr\u00e1s\u2014, lo esperaba, parqueado bajo un frondoso eucalipto de la plaza. Lucho Abril Marroqu\u00edn guard\u00f3 la valija con muestras y prospectos, se quit\u00f3 la corbata y el saco (que, seg\u00fan las normas helv\u00e9ticas del laboratorio, deb\u00edan llevar siempre los propagandistas para dar una impresi\u00f3n de seriedad), confirm\u00f3 su decisi\u00f3n de no visitar a su amigo aviador, y, en vez de un almuerzo en regla, acord\u00f3 tomar s\u00f3lo un refrigerio para evitar que una s\u00f3lida digesti\u00f3n le hiciera m\u00e1s so\u00f1olientas las tres horas de desierto.\n\nCruz\u00f3 la plaza hacia la Helader\u00eda Piave, orden\u00f3 al italiano una Coca-Cola y un helado de melocot\u00f3n, y, mientras consum\u00eda el espartano almuerzo, no pens\u00f3 en el pasado de ese puerto sure\u00f1o, el multicolor desembarco del dudoso h\u00e9roe San Mart\u00edn y su Ej\u00e9rcito Libertador, sino, ego\u00edsmo y sensualidad de los hombres candentes, en su tibia mujercita \u2014en realidad, casi una ni\u00f1a\u2014, n\u00edvea, de ojos azules y rizos dorados, y en c\u00f3mo, en la oscuridad rom\u00e1ntica de las noches, sab\u00eda llevarlo a unos extremos de fiebre neroniana cant\u00e1ndole al o\u00eddo, con quejidos de gatita l\u00e1nguida, en la lengua er\u00f3tica por excelencia (un franc\u00e9s tanto m\u00e1s excitante cuanto m\u00e1s incomprensible), una canci\u00f3n titulada Las hojas muertas. Advirtiendo que esas reminiscencias maritales comenzaban a inquietarlo, cambi\u00f3 de pensamientos, pag\u00f3 y sali\u00f3.\n\nEn un grifo pr\u00f3ximo llen\u00f3 el tanque de gasolina, el radiador de agua, y parti\u00f3. Pese a que a esa hora, de m\u00e1ximo sol, las calles de Pisco estaban vac\u00edas, conduc\u00eda despacio y con cuidado, pensando, no en la integridad de los peatones, sino en su amarillo Volkswagen, que, despu\u00e9s de la blonda francesita, era la ni\u00f1a de sus ojos. Iba pensando en su vida. Ten\u00eda veintiocho a\u00f1os. Al terminar el colegio, hab\u00eda decidido ponerse a trabajar, pues era demasiado impaciente para la transici\u00f3n universitaria. Hab\u00eda entrado a los laboratorios aprobando un examen. En estos diez a\u00f1os hab\u00eda progresado en sueldo y posici\u00f3n, y su trabajo no era aburrido. Prefer\u00eda actuar en la calle que vegetar tras un escritorio. S\u00f3lo que, ahora, no era cuesti\u00f3n de pasarse la vida en viajes, dejando a la delicada flor de Francia en Lima, ciudad que, es bien sabido, est\u00e1 llena de tiburones que viven al acecho de las sirenas. Lucho Abril Marroqu\u00edn hab\u00eda hablado con sus jefes. Lo apreciaban y lo hab\u00edan animado: continuar\u00eda viajando s\u00f3lo unos meses y a comienzos del pr\u00f3ximo a\u00f1o le dar\u00edan una colocaci\u00f3n en provincias. Y el doctor Schwalb, suizo lac\u00f3nico, hab\u00eda precisado: \u00abUna colocaci\u00f3n que ser\u00e1 una promoci\u00f3n\u00bb. Lucho Abril Marroqu\u00edn no pod\u00eda dejar de pensar que tal vez le ofrecer\u00edan la gerencia de la filial de Trujillo, Arequipa o Chiclayo. \u00bfQu\u00e9 m\u00e1s pod\u00eda pedir?\n\nEstaba saliendo de la ciudad, entrando a la carretera. Hab\u00eda hecho y deshecho tantas veces esa ruta \u2014en \u00f3mnibus, en colectivo, conducido o conduciendo\u2014 que la conoc\u00eda de memoria. La asfaltada cinta negra se perd\u00eda a lo lejos, entre m\u00e9danos y colinas peladas, sin brillos de azogue que revelaran veh\u00edculos. Ten\u00eda delante un cami\u00f3n viejo y tembleque, e iba ya a pasarlo cuando divis\u00f3 el puente y la encrucijada donde la ruta del sur hace una horquilla y despide a esa carretera que sube a la sierra, en direcci\u00f3n a las met\u00e1licas monta\u00f1as de Castrovirreina. Decidi\u00f3 entonces, prudencia de hombre que ama su m\u00e1quina y teme la ley, esperar hasta despu\u00e9s del desv\u00edo. El cami\u00f3n no iba a m\u00e1s de cincuenta kil\u00f3metros por hora y Lucho Abril Marroqu\u00edn, resignado, disminuy\u00f3 la velocidad y se mantuvo a diez metros de \u00e9l. Ve\u00eda, acerc\u00e1ndose, el puente, la encrucijada, endebles construcciones \u2014quioscos de bebidas, expendios de cigarrillos, la caseta de tr\u00e1nsito\u2014 y siluetas de personas cuyas caras no distingu\u00eda \u2014estaban a contraluz\u2014 yendo y viniendo junto a las caba\u00f1as.\n\nLa ni\u00f1a apareci\u00f3 de improviso, en el instante en que acababa de cruzar el puente y pareci\u00f3 surgir de debajo del cami\u00f3n. En el recuerdo de Lucho Abril Marroqu\u00edn quedar\u00eda grabada siempre esa figurilla que, s\u00fabitamente, se interpon\u00eda entre \u00e9l y la pista, la carita asustada y las manos en alto, y ven\u00eda a incrustarse como una pedrada contra la proa del Volkswagen. Fue tan intempestivo que no atin\u00f3 a frenar ni a desviar el auto hasta despu\u00e9s de la cat\u00e1strofe (del comienzo de la cat\u00e1strofe). Consternado, y con la extra\u00f1a sensaci\u00f3n de que se trataba de algo ajeno, sinti\u00f3 el sordo impacto del cuerpo contra el parachoque, y lo vio elevarse, trazar una par\u00e1bola y caer ocho o diez metros m\u00e1s all\u00e1.\n\nAhora s\u00ed fren\u00f3, tan en seco que el volante le golpe\u00f3 el pecho, y, con un aturdimiento blancuzco y un zumbido insistente, baj\u00f3 velozmente del Volkswagen y corriendo, tropezando, pensando \u00absoy argentino, mato ni\u00f1os\u00bb, lleg\u00f3 hasta la criatura y la alz\u00f3 en brazos. Tendr\u00eda cinco o seis a\u00f1os, iba descalza y mal vestida, con la cara, las manos y las rodillas encostradas de mugre. No sangraba por ninguna parte visible, pero ten\u00eda los ojos cerrados, y no parec\u00eda respirar. Lucho Abril Marroqu\u00edn, cimbre\u00e1ndose como borracho, daba vueltas sobre el sitio, miraba a derecha y a izquierda y gritaba a los arenales, al viento, a las lejanas olas: \u00abUna ambulancia, un m\u00e9dico\u00bb. Como en sue\u00f1os, alcanz\u00f3 a percibir que, por el desv\u00edo de la sierra, bajaba un cami\u00f3n y tal vez not\u00f3 que su velocidad era excesiva para un veh\u00edculo que se aproxima a un cruce de caminos. Pero, si lleg\u00f3 a advertirlo, inmediatamente su atenci\u00f3n se distrajo al descubrir que llegaba a su lado, corriendo, un guardia desprendido de las caba\u00f1as. Acezante, sudoroso, funcional, el custodio del orden, mirando a la ni\u00f1a le pregunt\u00f3: \u00ab\u00bfEst\u00e1 so\u00f1ada o ya muerta?\u00bb.\n\nLucho Abril Marroqu\u00edn se preguntar\u00eda el resto de los a\u00f1os que le quedaban de vida cu\u00e1l hab\u00eda sido en ese momento la respuesta justa. \u00bfEstaba malherida o hab\u00eda expirado la criatura? No alcanz\u00f3 a responder al acezante guardia civil porque \u00e9ste, apenas hizo la pregunta, puso tal cara de horror que Lucho Abril Marroqu\u00edn alcanz\u00f3 a volver la cabeza justo a tiempo para comprender que el cami\u00f3n que bajaba de la sierra se les ven\u00eda enloquecidamente encima, bocineando. Cerr\u00f3 los ojos, un estruendo le arrebat\u00f3 la ni\u00f1a de los brazos y lo sumi\u00f3 en una oscuridad con estrellitas. Sigui\u00f3 oyendo un ruido atroz, gritos, ayes, mientras permanec\u00eda en un estupor de naturaleza casi m\u00edstica.\n\nMucho despu\u00e9s sabr\u00eda que hab\u00eda sido arrollado, no porque existiese una justicia inmanente, encargada de realizar el equitativo refr\u00e1n \u00abojo por ojo, diente por diente\u00bb, sino porque al cami\u00f3n de las minas se le hab\u00edan vaciado los frenos. Y sabr\u00eda tambi\u00e9n que el guardia civil hab\u00eda muerto instant\u00e1neamente desnucado y que la pobre ni\u00f1a \u2014verdadera hija de S\u00f3focles\u2014, en este segundo accidente (por si el primero no lo hab\u00eda conseguido), no s\u00f3lo hab\u00eda quedado muerta sino espectacularmente alisada al pasarle encima, carnaval de alegr\u00eda para los satanes, la doble rueda trasera del cami\u00f3n.\n\nPero, al cabo de los a\u00f1os, Lucho Abril Marroqu\u00edn se dir\u00eda que de todas las instructivas experiencias de esa ma\u00f1ana, la m\u00e1s indeleble hab\u00eda sido, no el primero ni el segundo accidente, sino lo que vino despu\u00e9s. Porque, curiosamente, pese a la violencia del impacto (que lo tendr\u00eda muchas semanas en el Hospital del Empleado, reconstruyendo su esqueleto, averiado de innumerables fracturas, luxaciones, cortes y desgarrones), el propagandista m\u00e9dico no perdi\u00f3 el sentido o s\u00f3lo lo perdi\u00f3 unos segundos. Cuando abri\u00f3 los ojos supo que todo acababa de ocurrir, porque, de las caba\u00f1as que ten\u00eda al frente, ven\u00edan corriendo, siempre a contraluz, diez, doce, tal vez quince pantalones, faldas. No pod\u00eda moverse, pero no sent\u00eda dolor, s\u00f3lo una aliviada serenidad. Pens\u00f3 que ya no ten\u00eda que pensar; pens\u00f3 en la ambulancia, en m\u00e9dicos, en enfermeras sol\u00edcitas. Ah\u00ed estaban, ya hab\u00edan llegado, trat\u00f3 de sonre\u00edr a las caras que se inclinaban hacia \u00e9l. Pero entonces, por unas cosquillas, agujazos y punzadas, comprendi\u00f3 que los reci\u00e9n venidos no estaban auxili\u00e1ndolo: le arrancaban el reloj, le met\u00edan los dedos a los bolsillos, a manotones le sacaban la cartera, de un jal\u00f3n se apoderaban de la medalla del Se\u00f1or de Limpias que llevaba al cuello desde su primera comuni\u00f3n. Ahora s\u00ed, lleno de admiraci\u00f3n por los hombres, Lucho Abril Marroqu\u00edn se hundi\u00f3 en la noche.\n\nEsa noche, para todos los efectos pr\u00e1cticos, dur\u00f3 un a\u00f1o. Al principio, las consecuencias de la cat\u00e1strofe hab\u00edan parecido s\u00f3lo f\u00edsicas. Cuando Lucho Abril Marroqu\u00edn recobr\u00f3 el sentido estaba en Lima, en un cuartito del hospital, fajado de pies a cabeza, y a los flancos de su cama, \u00e1ngeles de la guarda que devuelven la paz al agitado, mir\u00e1ndolo con inquietud se hallaban la blonda connacional de Juliette Gr\u00e9co y el doctor Schwalb de los Laboratorios Bayer. En medio de la ebriedad que le produc\u00eda el olor a cloroformo, sinti\u00f3 alegr\u00eda y por sus mejillas corrieron l\u00e1grimas al sentir los labios de su esposa sobre las gasas que le cubr\u00edan la frente.\n\nLa sutura de huesos, retorno de m\u00fasculos y tendones a su lugar correspondiente y el cierre y cicatrizaci\u00f3n de heridas, es decir la compostura de la mitad animal de su persona tom\u00f3 algunas semanas, que fueron relativamente llevaderas, gracias a la excelencia de los facultativos, la diligencia de las enfermeras, la devoci\u00f3n magdal\u00e9nica de su esposa y la solidaridad de los laboratorios, que se mostraron impecables desde el punto de vista del sentimiento y la cartera. En el Hospital del Empleado, en plena convalecencia, Lucho Abril Marroqu\u00edn se enter\u00f3 de una noticia halagadora: la francesita hab\u00eda concebido y dentro de siete meses ser\u00eda madre de un hijo suyo.\n\nFue despu\u00e9s que abandon\u00f3 el hospital y se reintegr\u00f3 a su casita de San Miguel y a su trabajo, que se revelaron las secretas, complicadas heridas que los accidentes hab\u00edan causado a su esp\u00edritu. El insomnio era el m\u00e1s benigno de los males que se abatieron sobre \u00e9l. Se pasaba las noches en vela, ambulando por la casita a oscuras, fumando sin cesar, en estado de viva agitaci\u00f3n y pronunciando entrecortados discursos en los que a su esposa le maravillaba escuchar una palabra recurrente: \u00abHerodes\u00bb. Cuando el desvelo fue qu\u00edmicamente derrotado con somn\u00edferos, result\u00f3 peor: el sue\u00f1o de Abril Marroqu\u00edn era visitado por pesadillas en las que se ve\u00eda despedazando a su hija a\u00fan no nacida. Sus desafinados aullidos comenzaron aterrorizando a su esposa y terminaron haci\u00e9ndola abortar un feto de sexo probablemente femenino. \u00abLos sue\u00f1os se han cumplido, he asesinado a mi propia hija, me ir\u00e9 a vivir a Buenos Aires\u00bb, repet\u00eda d\u00eda y noche, l\u00fagubremente, el on\u00edrico filicida.\n\nPero tampoco esto fue lo peor. A las noches desveladas o pesadillescas, segu\u00edan unos d\u00edas atroces. Desde el accidente, Lucho Abril Marroqu\u00edn concibi\u00f3 una fobia visceral contra todo lo que tuviera ruedas, veh\u00edculos a los que no pod\u00eda subir ni como chofer ni como pasajero, sin sentir v\u00e9rtigo, v\u00f3mitos, sudar la gota gorda y ponerse a gritar. Todos los intentos de vencer este tab\u00fa fueron in\u00fatiles, de modo que tuvo que resignarse a vivir, en pleno siglo XX, como en el Incario (sociedad sin ruedas). Si las distancias que ten\u00eda que cubrir hubieran consistido solamente en los cinco kil\u00f3metros entre su hogar y los Laboratorios Bayer, el asunto no hubiera sido tan grave; para un esp\u00edritu maltratado esas dos horas de caminata matutina y las dos de caminata vespertina hubieran cumplido tal vez una funci\u00f3n sedante. Pero, trat\u00e1ndose de un propagandista m\u00e9dico cuyo centro de operaciones era el dilatado territorio del Per\u00fa, la fobia antirrodante resultaba tr\u00e1gica. No habiendo la menor posibilidad de resucitar la atl\u00e9tica \u00e9poca de los chasquis, el futuro profesional de Lucho Abril Marroqu\u00edn se hall\u00f3 seriamente amenazado. El laboratorio accedi\u00f3 a darle un trabajo sedentario, en la oficina de Lima, y, aunque no le bajaron el sueldo, desde el punto de vista moral y psicol\u00f3gico, el cambio (ahora ten\u00eda a su cargo el inventario de las muestras) constituy\u00f3 una degradaci\u00f3n. Para colmo de males, la francesita, que, digna \u00e9mula de la Doncella de Orleans, hab\u00eda soportado valerosamente los desperfectos nerviosos de su c\u00f3nyuge, sucumbi\u00f3 tambi\u00e9n, sobre todo despu\u00e9s de la evacuaci\u00f3n de la feto Abril, a la histeria. Una separaci\u00f3n hasta que mejoraran los tiempos fue acordada y la muchacha, palidez que recuerda el alba y las noches ant\u00e1rticas, viaj\u00f3 a Francia a buscar consuelo en el castillo de sus padres.\n\nAs\u00ed andaba Lucho Abril Marroqu\u00edn, al a\u00f1o del accidente: abandonado de su mujercita y del sue\u00f1o y la tranquilidad, odioso de las ruedas, condenado a caminar (stricto sensu) por la vida, sin otro amigo que la angustia. (El amarillo Volkswagen se cubri\u00f3 de hiedra y telara\u00f1as, antes de ser vendido para pagar el pasaje a Francia de la blonda.) Compa\u00f1eros y conocidos rumoreaban ya que no le quedaba sino el mediocre rumbo del manicomio o la retumbante soluci\u00f3n del suicidio, cuando el joven se enter\u00f3 \u2014man\u00e1 que cae del cielo, lluvia sobre sediento arenal\u2014 de la existencia de alguien que no era sacerdote ni brujo y sin embargo curaba almas: la doctora Luc\u00eda Ac\u00e9mila.\n\nMujer superior y sin complejos, llegada a lo que la ciencia ha dado en considerar la edad ideal \u2014la cincuentena\u2014, la doctora Ac\u00e9mila \u2014frente ancha, nariz aguile\u00f1a, mirada penetrante, rectitud y bondad en el esp\u00edritu\u2014 era la negaci\u00f3n viviente de su apellido (del que se sent\u00eda orgullosa y que arrojaba como una haza\u00f1a, en tarjetas impresas, o en los r\u00f3tulos de su consultorio, a la visi\u00f3n de los mortales), alguien en quien la inteligencia era un atributo f\u00edsico, algo que sus pacientes (ella prefer\u00eda llamarlos amigos) pod\u00edan ver, o\u00edr, oler. Hab\u00eda obtenido diplomas sobresalientes y copiosos en los grandes centros del saber \u2014la teut\u00f3nica Berl\u00edn, la flem\u00e1tica Londres, la pecaminosa Par\u00eds\u2014, pero la principal universidad en la que hab\u00eda aprendido lo mucho que sab\u00eda sobre la miseria humana y sus remedios hab\u00eda sido (naturalmente) la vida. Como todo ser elevado por sobre la median\u00eda, era discutida, criticada y verbalmente escarnecida por sus colegas, esos psiquiatras y psic\u00f3logos incapaces (a diferencia de ella) de producir milagros. A la doctora Ac\u00e9mila la dejaba indiferente que la llamaran hechicera, satanista, corruptora de corrompidos, alienada y otras vilezas. Le bastaba, para saber que era ella quien ten\u00eda raz\u00f3n, con la gratitud de sus amigos, esa legi\u00f3n de esquizofr\u00e9nicos, parricidas, paranoicos, incendiarios, man\u00edaco-depresivos, onanistas, catat\u00f3nicos, criminosos, m\u00edsticos y tartamudos que, una vez pasados por sus manos, sometidos a su tratamiento (ella hubiera preferido: a sus consejos) hab\u00edan retornado a la vida padres amant\u00edsimos, hijos obedientes, esposas virtuosas, profesionales honestos, conversadores fluidos y ciudadanos patol\u00f3gicamente respetuosos de la ley.\n\nFue el doctor Schwalb quien aconsej\u00f3 a Lucho Abril Marroqu\u00edn que visitara a la doctora y \u00e9l mismo quien, prontitud helv\u00e9tica que ha dado relojes puntual\u00edsimos, arregl\u00f3 una cita. M\u00e1s resignado que confiado, el insomne se present\u00f3 a la hora indicada a la mansi\u00f3n de muros rosas, abrazada por un jard\u00edn con floripondios, en el residencial barrio de San Felipe, donde estaba el consultorio (templo, confesionario, laboratorio del esp\u00edritu) de Luc\u00eda Ac\u00e9mila. Una pulcra enfermera le tom\u00f3 algunos datos y lo hizo pasar al despacho de la doctora, una habitaci\u00f3n alta, de estantes atiborrados de libros con empaste de cuero, un escritorio de caoba, mullidas alfombras y un div\u00e1n de terciopelo verde menta.\n\n\u2014Qu\u00edtese los prejuicios que traiga y tambi\u00e9n el saco y la corbata \u2014lo apostrof\u00f3, naturalidad desarmante de los sabios, la doctora Luc\u00eda Ac\u00e9mila, se\u00f1al\u00e1ndole el div\u00e1n\u2014. Y t\u00fambese ah\u00ed, boca arriba o boca abajo, no por bater\u00eda freudiana, sino porque me interesa que est\u00e9 c\u00f3modo. Y, ahora, no me cuente sus sue\u00f1os ni me confiese que est\u00e1 enamorado de su madre, sino, m\u00e1s bien, d\u00edgame con la mayor exactitud c\u00f3mo marcha ese est\u00f3mago.\n\nT\u00edmidamente, el propagandista m\u00e9dico, ya extendido sobre el muelle div\u00e1n, se atrevi\u00f3 a musitar, imaginando una confusi\u00f3n de personas, que no lo tra\u00eda a este consultorio el vientre sino el esp\u00edritu.\n\n\u2014Son indiferenciables \u2014lo desasn\u00f3 la facultativo\u2014. Un est\u00f3mago que evacua puntual y totalmente es gemelo de una mente clara y de un alma bien pensada. Por el contrario, un est\u00f3mago cargado, remol\u00f3n, avaricioso, engendra malos pensamientos, avinagra el car\u00e1cter, fomenta complejos y apetitos sexuales chuecos, y crea vocaci\u00f3n de delito, una necesidad de castigar en los otros el tormento excrementicio.\n\nAs\u00ed instruido, Lucho Abril Marroqu\u00edn confes\u00f3 que sufr\u00eda a veces de dispepsias, constipaci\u00f3n e, incluso, que sus \u00f3bolos, adem\u00e1s de irregulares, eran tambi\u00e9n vers\u00e1tiles en coloraci\u00f3n, volumen y, sin duda \u2014no recordaba haberlos palpado en las \u00faltimas semanas\u2014, consistencia y temperatura. La doctora asinti\u00f3 bondadosamente, murmurando: \u00abLo sab\u00eda\u00bb. Y dictamin\u00f3 que el joven deber\u00eda consumir cada ma\u00f1ana, hasta nueva orden y en ayunas, media docena de ciruelas secas.\n\n\u2014Resuelta esta cuesti\u00f3n previa, pasemos a las otras \u2014a\u00f1adi\u00f3 la fil\u00f3sofo\u2014. Puede contarme qu\u00e9 le pasa. Pero sepa de antemano que no lo castrar\u00e9 de su problema. Le ense\u00f1ar\u00e9 a amarlo, a sentirse tan orgulloso de \u00e9l como Cervantes de su brazo ido o Beethoven de su sordera. Hable.\n\nLucho Abril Marroqu\u00edn, con una facilidad de palabra educada en diez a\u00f1os de di\u00e1logos profesionales con galenos y apoticarios, resumi\u00f3 su historia con sinceridad, desde el infausto accidente de Pisco hasta sus pesadillas de la v\u00edspera y las apocal\u00edpticas consecuencias que el drama hab\u00eda tenido en su familia. Apiadado de s\u00ed mismo, en los cap\u00edtulos finales rompi\u00f3 a llorar y remat\u00f3 su informe con una exclamaci\u00f3n que a cualquier otra persona que no fuera Luc\u00eda Ac\u00e9mila le hubiera partido el alma: \u00ab\u00a1Doctora, ay\u00fademe!\u00bb.\n\n\u2014Su historia, en vez de apenarme, me aburre, de lo trivial y tonta que es \u2014lo confort\u00f3 cari\u00f1osamente la ingeniero de almas\u2014. L\u00edmpiese los mocos y conv\u00e9nzase de que, en la geograf\u00eda del esp\u00edritu, su mal es equivalente a lo que, en la del cuerpo, ser\u00eda un u\u00f1ero. Ahora esc\u00facheme.\n\nCon unos modales de mujer que frecuenta salones de alta sociedad, le explic\u00f3 que lo que perd\u00eda a los hombres era el temor a la verdad y el esp\u00edritu de contradicci\u00f3n. Respecto a lo primero, hizo luz en el cerebro del desvelado explic\u00e1ndole que el azar, el llamado accidente no exist\u00edan, eran subterfugios inventados por los hombres para disimular lo malvados que eran.\n\n\u2014En resumen, usted quiso matar a la ni\u00f1a y la mat\u00f3 \u2014grafic\u00f3 su pensamiento la doctora\u2014. Y luego, acobardado de su acto, temeroso de la polic\u00eda o el infierno, quiso ser atropellado por el cami\u00f3n, para recibir una pena o como coartada por el asesinato.\n\n\u2014Pero, pero \u2014balbuce\u00f3, ojos que al desorbitarse y frente que al sudar delatan supina desesperaci\u00f3n, el propagandista m\u00e9dico\u2014. \u00bfY el guardia civil? \u00bfTambi\u00e9n lo mat\u00e9 yo?\n\n\u2014\u00bfQui\u00e9n no ha matado alguna vez un guardia civil? \u2014reflexion\u00f3 la cient\u00edfico\u2014. Tal vez usted, tal vez el camionero, tal vez fue un suicidio. Pero \u00e9sta no es una funci\u00f3n de gancho, donde entran dos con una entrada. Ocup\u00e9monos de usted.\n\nLe explic\u00f3 que, al contradecir sus genuinos impulsos, los hombres resent\u00edan a su esp\u00edritu y \u00e9ste se vengaba procreando pesadillas, fobias, complejos, angustia, depresi\u00f3n.\n\n\u2014No se puede pelear consigo mismo, porque en ese combate s\u00f3lo hay un perdedor \u2014pontific\u00f3 la ap\u00f3stol\u2014. No se averg\u00fcence de lo que es, consu\u00e9lese pensando que todos los hombres son hienas y que ser bueno significa, simplemente, saber disimular. M\u00edrese al espejo y d\u00edgase: soy un infanticida y un cobarde de la velocidad. Basta de eufemismos: no me hable de accidentes ni del s\u00edndrome de la rueda.\n\nY, pasando a los ejemplos, le cont\u00f3 que a los escu\u00e1lidos onanistas que ven\u00edan a rogarle de rodillas que los curara les regalaba revistas pornogr\u00e1ficas y a los pacientes drogadictos, escorias que reptan por los suelos y se mesan los pelos hablando de la fatalidad, les ofrec\u00eda pitos de marihuana y pu\u00f1ados de coca.\n\n\u2014\u00bfVa a recetarme que siga matando ni\u00f1os? \u2014rugi\u00f3, cordero que se metamorfosea en tigre, el propagandista m\u00e9dico.\n\n\u2014Si es su gusto, \u00bfpor qu\u00e9 no? \u2014le repuso fr\u00edamente la psic\u00f3logo. Y le previno\u2014: Nada de levantarme la voz. No soy de esos mercaderes que creen que el cliente siempre tiene raz\u00f3n.\n\nLucho Abril Marroqu\u00edn volvi\u00f3 a zozobrar en llanto. Indiferente, la doctora Luc\u00eda Ac\u00e9mila caligrafi\u00f3 durante diez minutos varias hojas con el t\u00edtulo general de: Ejercicios para aprender a vivir con sinceridad. Se las entreg\u00f3 y le dio cita para ocho semanas despu\u00e9s. Al despedirlo, con un apret\u00f3n de manos, le record\u00f3 que no olvidara el r\u00e9gimen matutino de ciruelas secas.\n\nComo la mayor\u00eda de los pacientes de la doctora Ac\u00e9mila, Lucho Abril Marroqu\u00edn sali\u00f3 del consultorio sinti\u00e9ndose v\u00edctima de una emboscada ps\u00edquica, seguro de haber ca\u00eddo en las redes de una extravagante desquiciada, que agravar\u00eda sus males si comet\u00eda el desatino de seguir sus recetas. Estaba decidido a desaguar los Ejercicios por el excusado sin mirarlos. Pero esa misma noche, debilitante insomnio que incita a los excesos, los ley\u00f3. Le parecieron patol\u00f3gicamente absurdos y se ri\u00f3 tanto que le vino hipo (lo conjur\u00f3 bebiendo un vaso de agua al rev\u00e9s, como le hab\u00eda ense\u00f1ado su madre); luego, sinti\u00f3 una urticante curiosidad. Como distracci\u00f3n, para llenar las horas vac\u00edas de sue\u00f1o, sin creer en su virtud terap\u00e9utica, decidi\u00f3 practicarlos.\n\nNo le cost\u00f3 trabajo encontrar en la secci\u00f3n juguetes de Sears el auto, el cami\u00f3n n\u00famero uno y el cami\u00f3n n\u00famero dos que le hac\u00edan falta, as\u00ed como los mu\u00f1equitos encargados de representar a la ni\u00f1a, al guardia, a los ladrones y a \u00e9l mismo. Conforme a las instrucciones, pint\u00f3 los veh\u00edculos con los colores originales que recordaba, as\u00ed como las ropas de los mu\u00f1equitos. (Ten\u00eda aptitud para la pintura, de modo que el uniforme del guardia y las ropas humildes y las costras de la ni\u00f1a le salieron muy bien.) Para mimar los arenales pisque\u00f1os, emple\u00f3 un pliego de papel de envolver, al que, extremando el prurito de fidelidad, pint\u00f3 en un extremo el oc\u00e9ano Pac\u00edfico: una franja azul con orla de espuma. El primer d\u00eda, le tom\u00f3 cerca de una hora, arrodillado en el suelo del living comedor de su casa, reproducir la historia, y, cuando termin\u00f3, es decir cuando los ladrones se precipitaban sobre el propagandista m\u00e9dico para despojarlo, qued\u00f3 casi tan aterrado y adolorido como el d\u00eda del suceso. De espaldas en el suelo, sudaba fr\u00edo y sollozaba. Pero los d\u00edas siguientes fue disminuyendo la impresi\u00f3n nerviosa, y la operaci\u00f3n asumi\u00f3 virtualidades deportivas, un ejercicio que lo devolv\u00eda a la ni\u00f1ez y entreten\u00eda esas horas que no hubiera sabido ocupar, ahora que estaba sin esposa, \u00e9l que nunca se hab\u00eda ufanado de ser rat\u00f3n de biblioteca o mel\u00f3mano. Era como armar un mecano, un rompecabezas o hacer crucigramas. A veces, en el almac\u00e9n de los Laboratorios Bayer, mientras distribu\u00eda muestras a los propagandistas, se sorprend\u00eda escarbando en la memoria, en pos de alg\u00fan detalle, gesto, motivo de lo ocurrido que le permitiera introducir alguna variante, alargar las representaciones de esa noche. La se\u00f1ora que ven\u00eda a hacer la limpieza, al ver el suelo del living comedor ocupado por mu\u00f1equitos de madera y autitos de pl\u00e1stico, le pregunt\u00f3 si pensaba adoptar un ni\u00f1o, advirti\u00e9ndole que en ese caso le cobrar\u00eda m\u00e1s. Conforme a la progresi\u00f3n se\u00f1alada por los Ejercicios, efectuaba ya para entonces, cada noche, diecis\u00e9is representaciones a escala liliputiense del \u00bfaccidente?\n\nLa parte de los Ejercicios para aprender a vivir con sinceridad concerniente a los ni\u00f1os le pareci\u00f3 m\u00e1s descabellada que el palitroque, pero, \u00bfinercia que arrastra al vicio o curiosidad que hace avanzar la ciencia?, tambi\u00e9n la obedeci\u00f3. Estaba subdividida en dos partes: \u00abEjercicios te\u00f3ricos\u00bb y \u00abEjercicios pr\u00e1cticos\u00bb, y la doctora Ac\u00e9mila se\u00f1alaba que era imprescindible que aqu\u00e9llos antecedieran a \u00e9stos, pues \u00bfno era el hombre un ser racional en el que las ideas preced\u00edan a los actos? La parte te\u00f3rica daba amplio cr\u00e9dito al esp\u00edritu observador y especulativo del propagandista m\u00e9dico. Se limitaba a prescribir: \u00abReflexione diariamente sobre las calamidades que causan los ni\u00f1os a la humanidad\u00bb. Hab\u00eda que hacerlo a cualquier hora y sitio, de manera sistem\u00e1tica.\n\n\u00bfQu\u00e9 mal hac\u00edan a la humanidad los inocentes p\u00e1rvulos? \u00bfNo eran la gracia, la pureza, la alegr\u00eda, la vida?, se pregunt\u00f3 Lucho Abril Marroqu\u00edn, la ma\u00f1ana del primer ejercicio te\u00f3rico, mientras caminaba los cinco kil\u00f3metros de ida a la oficina. Pero, m\u00e1s por darle gusto al papel que por convicci\u00f3n, admiti\u00f3 que pod\u00edan ser ruidosos. En efecto, lloraban mucho, a cualquier hora y por cualquier motivo, y, como carec\u00edan de uso de raz\u00f3n, no ten\u00edan en cuenta el perjuicio que esa propensi\u00f3n causaba ni pod\u00edan ser persuadidos de las virtudes del silencio. Record\u00f3 entonces el caso de ese obrero que, luego de extenuantes jornadas en el socav\u00f3n, volv\u00eda al hogar y no pod\u00eda dormir por el llanto fren\u00e9tico del reci\u00e9n nacido al que finalmente hab\u00eda \u00bfasesinado? \u00bfCu\u00e1ntos millones de casos parecidos se registrar\u00edan en el globo? \u00bfCu\u00e1ntos obreros, campesinos, comerciantes y empleados, que \u2014alto costo de la vida, bajos salarios, escasez de viviendas\u2014 viv\u00edan en departamentos estrechos y compart\u00edan sus cuartos con la prole, estaban impedidos de disfrutar de un merecido sue\u00f1o por los alaridos de un ni\u00f1o incapaz de decir si sus berridos significaban diarrea o ganas de m\u00e1s teta?\n\nBuscando, buscando, esa tarde, en los cinco kil\u00f3metros de vuelta, Lucho Abril Marroqu\u00edn encontr\u00f3 que se les pod\u00eda achacar tambi\u00e9n muchos destrozos. A diferencia de cualquier animal, tardaban demasiado en valerse por s\u00ed mismos, \u00a1y cu\u00e1ntos estragos resultaban de esa tara! Todo lo romp\u00edan, car\u00e1tula art\u00edstica o florero de cristal de roca, tra\u00edan abajo las cortinas que quem\u00e1ndose los ojos hab\u00eda cosido la due\u00f1a de casa, y sin el menor embarazo aposentaban sus manos embarradas de caca en el almidonado mantel o la mantilla de encaje comprada con privaci\u00f3n y amor. Sin contar que sol\u00edan meter sus dedos en los enchufes y provocar cortocircuitos o electrocutarse est\u00fapidamente con lo que eso significaba para la familia: cajoncito blanco, nicho, velorio, aviso en El Comercio, ropas de luto, duelo.\n\nAdquiri\u00f3 la costumbre de entregarse a esta gimnasia durante sus idas y venidas entre el laboratorio y San Miguel. Para no repetirse, hac\u00eda al comenzar una r\u00e1pida s\u00edntesis de los cargos acumulados en la reflexi\u00f3n anterior y pasaba a desarrollar uno nuevo. Los temas se imbricaban unos en otros con facilidad y nunca se qued\u00f3 sin argumentos.\n\nEl delito econ\u00f3mico, por ejemplo, le dio materia para treinta kil\u00f3metros. Porque \u00bfno era desolador c\u00f3mo ellos arruinaban el presupuesto familiar? Gravaban los ingresos paternos en relaci\u00f3n inversa a su tama\u00f1o, no s\u00f3lo por su glotoner\u00eda pertinaz y la delicadeza de su est\u00f3mago, que exig\u00eda alimentos especiales, sino por las infinitas instituciones que ellos hab\u00edan generado, comadronas, cunas maternales, puericultorios, jardines de infancia, ni\u00f1eras, circos, parvularios, matinales, jugueter\u00edas, juzgados de menores, reformatorios, sin mencionar las especialidades en ni\u00f1os que, arborescentes par\u00e1sitos que asfixian a las plantas madres, le hab\u00edan nacido a la medicina, la psicolog\u00eda, la odontolog\u00eda y otras ciencias, ej\u00e9rcito en suma de gentes que deb\u00edan ser vestidas, alimentadas y jubiladas por los pobres padres.\n\nLucho Abril Marroqu\u00edn se encontr\u00f3 un d\u00eda a punto de llorar, pensando en esas madres j\u00f3venes, celosas cumplidoras de la moral y el qu\u00e9 dir\u00e1n, que se entierran en vida para cuidar a sus cr\u00edas, y renuncian a fiestas, cines, viajes, con lo que terminan siendo abandonadas por esposos, que, de tanto salir solos, terminan fat\u00eddicamente por pecar. \u00bfY c\u00f3mo pagaban las cr\u00edas esos desvelos y padecimientos? Creciendo, formando hogar aparte, abandonando a sus madres en la orfandad de la vejez.\n\nPor este camino, insensiblemente, lleg\u00f3 a desbaratar el mito de su inocencia y bondad. \u00bfAcaso, con la consabida coartada de que carec\u00edan de uso de raz\u00f3n, no cercenaban las alas a las mariposas, met\u00edan a los pollitos vivos en el horno, dejaban patas arriba a las tortugas hasta que mor\u00edan y les reventaban los ojos a las ardillas? \u00bfLa honda para matar pajaritos era arma de adultos? \u00bfY no se mostraban implacables con los ni\u00f1os m\u00e1s d\u00e9biles? Por otra parte, \u00bfc\u00f3mo se pod\u00eda llamar inteligentes a seres que, a una edad en que cualquier gatito ya se procura el sustento, todav\u00eda se bambolean torpemente, se dan de bruces contra las paredes y se hacen chichones?\n\nLucho Abril Marroqu\u00edn ten\u00eda un sentido est\u00e9tico aguzado y esto le dio madeja para muchas caminatas. \u00c9l hubiera querido que todas las mujeres se conservaran lozanas y duras hasta la menopausia y le apen\u00f3 inventariar los estragos que causaban a las madres los partos: las cinturas de avispa que cab\u00edan en una mano estallaban en grasa y tambi\u00e9n senos y nalgas y esos est\u00f3magos tersos, l\u00e1minas de carnoso metal que los labios no abollan, se ablandaban, hinchaban, descolgaban, rayaban, y algunas se\u00f1oras, a consecuencia de los pujos y calambres de los partos dif\u00edciles, quedaban chuecas como patos. Con alivio, Lucho Abril Marroqu\u00edn, rememorando el cuerpo estatuario de la francesita que llevaba su nombre, se alegr\u00f3 de que hubiera parido, no un ser rollizo y devastador de su belleza, sino, apenas, un detritus de hombre. Otro d\u00eda, se percat\u00f3, mientras se desalteraba \u2014las ciruelas secas hab\u00edan convertido a su est\u00f3mago en un tren ingl\u00e9s\u2014 que ya no lo estremec\u00eda pensar en Herodes. Y una ma\u00f1ana se descubri\u00f3 dando un coscorr\u00f3n a un ni\u00f1o mendigo.\n\nSupo entonces que, sin propon\u00e9rselo, hab\u00eda pasado, naturalidad con que viajan los astros de la noche al d\u00eda, a los \u00abEjercicios pr\u00e1cticos\u00bb. La doctora Ac\u00e9mila hab\u00eda subtitulado \u00abAcci\u00f3n directa\u00bb estas instrucciones y a Lucho Abril Marroqu\u00edn le parec\u00eda estar oyendo su cient\u00edfica voz mientras las rele\u00eda. \u00c9stas s\u00ed, a diferencia de las te\u00f3ricas, eran precisas. Se trataba, una vez adquirida conciencia clara de las calamidades que ellos tra\u00edan, de tomar, a nivel individual, peque\u00f1as represalias. Era preciso hacerlo de manera discreta, teniendo en cuenta las demagogias del g\u00e9nero \u00abinfancia desvalida\u00bb, \u00abal ni\u00f1o ni con una rosa\u00bb y \u00ablos azotes causan complejos\u00bb.\n\nLo cierto es que al principio le cost\u00f3 trabajo, y, cuando cruzaba a uno de ellos en la calle, \u00e9ste y \u00e9l mismo no sab\u00edan si aquella mano en la infantil cabecita era un castigo o una caricia tosca. Pero, seguridad que da la pr\u00e1ctica, poco a poco fue superando la timidez y las ancestrales inhibiciones, envalenton\u00e1ndose, mejorando sus marcas, tomando iniciativas, y, al cabo de unas semanas, conforme al pron\u00f3stico de los Ejercicios, not\u00f3 que aquellos coscorrones que repart\u00eda en las esquinas, aquellos pellizcos que provocaban moretones, aquellos pisotones que hac\u00edan berrear a los recipiendarios, ya no eran un quehacer que se impon\u00eda por razones de moral y teor\u00eda, sino una suerte de placer. Le gustaba ver llorar a esos vendedores que se acercaban a ofrecerle la suerte y, sorpresivamente, recib\u00edan un bofet\u00f3n, y se excitaba como en los toros cuando el lazarillo de una ciega que lo hab\u00eda abordado, platillo de lat\u00f3n que tintinea en la ma\u00f1ana, ca\u00eda al suelo sob\u00e1ndose la canilla que acababa de alojar su puntapi\u00e9. Los \u00abEjercicios pr\u00e1cticos\u00bb eran riesgosos, pero, al propagandista m\u00e9dico que se reconoci\u00f3 un coraz\u00f3n temerario, esto, en vez de disuadirlo lo estimul\u00f3. Ni siquiera el d\u00eda en que revent\u00f3 una pelota y fue perseguido con palos y piedras por una jaur\u00eda de pigmeos, cej\u00f3 en su empe\u00f1o.\n\nAs\u00ed, en las semanas que dur\u00f3 el tratamiento, cometi\u00f3 muchas de aquellas acciones que, pereza mental que idiotiza a las gentes, se suelen llamar maldades. Decapit\u00f3 las mu\u00f1ecas con que, en los parques, las ni\u00f1eras las entreten\u00edan, arrebat\u00f3 chupetes, toffees, caramelos que estaban a punto de llevarse a la boca y los pisote\u00f3 o ech\u00f3 a los perros, fue a merodear por circos, matinales y teatros de t\u00edteres y, hasta que se le entumecieron los dedos, jal\u00f3 trenzas y orejas, pellizc\u00f3 bracitos, piernas, patitos, y, por supuesto, us\u00f3 de la secular estratagema de sacarles la lengua y hacerles muecas, y, hasta la afon\u00eda y la ronquera les habl\u00f3 del Cuco, del Lobo Feroz, del Polic\u00eda, del Esqueleto, de la Bruja, del Vampiro, y dem\u00e1s personajes creados por la imaginaci\u00f3n adulta para asustarlos.\n\nPero, bola de nieve que al rodar monte abajo se vuelve aluvi\u00f3n, un d\u00eda Lucho Abril Marroqu\u00edn se asust\u00f3 tanto que se precipit\u00f3, en un taxi para llegar m\u00e1s pronto, al consultorio de la doctora Ac\u00e9mila. Apenas entr\u00f3 en el severo despacho, sudando hielo, la voz temblorosa, exclam\u00f3:\n\n\u2014Iba a empujar a una ni\u00f1a bajo las ruedas del tranv\u00eda a San Miguel. Me contuve en el \u00faltimo instante, porque vi un polic\u00eda \u2014y, sollozando como uno de ellos, grit\u00f3\u2014: \u00a1He estado a punto de volverme un criminal, doctora!\n\n\u2014Criminal ya lo ha sido, joven desmemoriado \u2014le record\u00f3 la psic\u00f3logo, pronunciando cada s\u00edlaba. Y, despu\u00e9s de observarlo de arriba abajo, complacida, sentenci\u00f3\u2014: Est\u00e1 usted curado.\n\nLucho Abril Marroqu\u00edn record\u00f3 entonces \u2014fogonazo de luz en las tinieblas, lluvia de estrellas sobre el mar\u2014 que hab\u00eda venido en \u00a1un taxi! Iba a caer de rodillas pero la sabia lo contuvo:\n\n\u2014Nadie me lame las manos, salvo mi gran dan\u00e9s. \u00a1Basta de efusiones! Puede retirarse, pues nuevos amigos esperan. Recibir\u00e1 la factura en su oportunidad.\n\n\u00abEs verdad, estoy curado\u00bb, se repet\u00eda feliz el propagandista m\u00e9dico: la \u00faltima semana hab\u00eda dormido siete horas diarias y, en vez de pesadillas, hab\u00eda tenido gratos sue\u00f1os en los que, en playas ex\u00f3ticas, se dejaba tostar por un sol futbol\u00edstico, viendo el pausado caminar de las tortugas entre palmeras lanceoladas y las p\u00edcaras fornicaciones de los delfines en las ondas azules. Esta vez, deliberaci\u00f3n y alevos\u00eda del hombre fogueado, tom\u00f3 otro taxi hacia los laboratorios y, durante el trayecto, llor\u00f3 al comprobar que el \u00fanico efecto que le produc\u00eda rodar sobre la vida era, no ya el terror sepulcral, la angustia c\u00f3smica, sino apenas un ligero mareo. Corri\u00f3 a besar las manos amaz\u00f3nicas de don Federico T\u00e9llez Unz\u00e1tegui, llam\u00e1ndolo \u00abmi consejero salvador, mi nuevo padre\u00bb, gesto y palabras que su jefe acept\u00f3 con la deferencia que todo amo que se respete debe a sus esclavos, se\u00f1al\u00e1ndole de todos modos, calvinista de coraz\u00f3n sin puertas para el sentimiento, que, curado o no de complejos homicidas, deb\u00eda llegar puntual a Antirroedores S. A. so pena de multa.\n\nFue as\u00ed como Lucho Abril Marroqu\u00edn sali\u00f3 del t\u00fanel que, desde el polvoroso accidente de Pisco, era su vida. Todo, a partir de entonces, comenz\u00f3 a enderezarse. La dulce hija de Francia, absuelta de sus penas gracias a mimos familiares y entonada con dietas normandas de agujereado queso y viscosos caracoles, volvi\u00f3 a la tierra de los incas con las mejillas rozagantes y el coraz\u00f3n lleno de amor. El reencuentro del matrimonio fue una prolongada luna de miel, besos enajenantes, compulsivos abrazos y otros derroches emotivos que pusieron a los enamorados esposos a las orillas mismas de la anemia. El propagandista m\u00e9dico, serpiente de vigor redoblado con el cambio de piel, recuper\u00f3 pronto el lugar de preeminencia que ten\u00eda en los laboratorios. A pedido de \u00e9l mismo, que quer\u00eda demostrarse que era el de antes, el doctor Schwalb volvi\u00f3 a confiarle la responsabilidad de, por aire, tierra, r\u00edo, mar, recorrer pueblos y ciudades del Per\u00fa publicitando, entre m\u00e9dicos y farmac\u00e9uticos, los productos Bayer. Gracias a las virtudes ahorrativas de la esposa, pronto la pareja pudo cancelar todas las deudas contra\u00eddas durante la crisis y adquirir, a plazos, un nuevo Volkswagen que, por supuesto, fue tambi\u00e9n amarillo.\n\nNada, en apariencia (\u00bfpero acaso no recomienda la sabidur\u00eda popular \u00abno fiarse de las apariencias\u00bb?), afeaba el marco en el que se desenvolv\u00eda la vida de los Abril Marroqu\u00edn. El propagandista rara vez se acordaba del accidente y, cuando ello ocurr\u00eda, en vez de pesar sent\u00eda orgullo, lo que, mes\u00f3crata respetuoso de las formas sociales, se conten\u00eda de hacer p\u00fablico. Pero, en la intimidad del hogar, nido de t\u00f3rtolas, chimenea que arde al comp\u00e1s de violines de Vivaldi, algo hab\u00eda sobrevivido \u2014luz que perdura en los espacios cuando el astro que la emiti\u00f3 ha caducado, u\u00f1as y pelos que le crecen al muerto\u2014, de la terapia de la profesora Ac\u00e9mila. Es decir, de un lado, una afici\u00f3n, exagerada para la edad de Lucho Abril Marroqu\u00edn, a jugar con palitroques, mecanos, trencitos, soldaditos. El departamento se fue llenando de juguetes que desconcertaban a vecinos y sirvientas, y las primeras sombras de la armon\u00eda conyugal surgieron porque la francesita comenz\u00f3 un d\u00eda a quejarse de que su esposo pasara los domingos y feriados haciendo navegar barquitos de papel en la ba\u00f1era o volando cometas en el techo. Pero, m\u00e1s grave que esta afici\u00f3n, y a todas luces enemiga de ella, era la fobia contra la ni\u00f1ez que hab\u00eda perseverado en el esp\u00edritu de Lucho Abril Marroqu\u00edn desde la \u00e9poca de los \u00abEjercicios pr\u00e1cticos\u00bb. No le era posible cruzar a uno de ellos en calle, parque o plaza p\u00fablica, sin infligirle lo que el vulgo llamar\u00eda una crueldad, y en las conversaciones con su esposa sol\u00eda bautizarlos con expresiones despectivas como \u00abdestetados\u00bb y \u00ablimb\u00f3manos\u00bb. Esta hostilidad se convirti\u00f3 en angustia el d\u00eda en que la blonda qued\u00f3 nuevamente embarazada. La pareja, talones que el pavor torna h\u00e9lices, vol\u00f3 a solicitar moral y ciencia a la doctora Ac\u00e9mila. \u00c9sta los escuch\u00f3 sin asustarse:\n\n\u2014Padece usted de infantilismo y es, al mismo tiempo, un reincidente infanticida potencial \u2014estableci\u00f3 con arte telegr\u00e1fico\u2014. Dos tonter\u00edas que no merecen atenci\u00f3n, que yo curo con la facilidad que escupo. No tema: estar\u00e1 sano antes que al feto le broten ojos.\n\n\u00bfLo curar\u00eda? \u00bfLibrar\u00eda a Lucho Abril Marroqu\u00edn de esos fantasmas? \u00bfSer\u00eda el tratamiento contra la infantofobia y el herodismo tan aventurero como el que lo emancip\u00f3 del complejo de rueda y la obsesi\u00f3n de crimen? \u00bfC\u00f3mo terminar\u00eda el psicodrama de San Miguel?\n\n### XI\n\nSE ACERCABAN los ex\u00e1menes de medio a\u00f1o en la facultad y yo, que desde los amores con la t\u00eda Julia asist\u00eda menos a clases y escrib\u00eda m\u00e1s cuentos (p\u00edrricos), estaba mal preparado para este trance. Mi salvaci\u00f3n era un compa\u00f1ero de estudios, un camanejo llamado Guillermo Velando. Viv\u00eda en una pensi\u00f3n del centro, por la plaza Dos de Mayo, y era un estudiante modelo, que no perd\u00eda una clase, apuntaba hasta la respiraci\u00f3n de los profesores y aprend\u00eda de memoria, como yo versos, los art\u00edculos de los c\u00f3digos. Siempre estaba hablando de su pueblo, donde ten\u00eda una novia, y s\u00f3lo esperaba recibirse de abogado para dejar Lima, ciudad que odiaba, e instalarse en Caman\u00e1, donde batallar\u00eda por el progreso de su tierra. Me prestaba sus apuntes, me soplaba en los ex\u00e1menes y, cuando \u00e9stos se ven\u00edan encima, yo iba a su pensi\u00f3n, a que me diera alguna s\u00edntesis milagrosa sobre lo que hab\u00edan hecho en clases.\n\nDe all\u00ed ven\u00eda ese domingo, despu\u00e9s de pasar tres horas en el cuarto de Guillermo, con la cabeza revoloteante de f\u00f3rmulas forenses, asustado de la cantidad de latinajos que hab\u00eda que memorizar, cuando, llegando a la plaza San Mart\u00edn, vi a lo lejos, en la plomiza fachada de Radio Central, la ventanita abierta del cub\u00edculo de Pedro Camacho. Por supuesto, decid\u00ed ir a darle los buenos d\u00edas. Mientras m\u00e1s lo frecuentaba \u2014aunque nuestra relaci\u00f3n siguiera sujeta a brev\u00edsimas charlas en torno a una mesa de caf\u00e9\u2014 el hechizo que ejerc\u00edan sobre m\u00ed su personalidad, su f\u00edsico, su ret\u00f3rica, era mayor. Mientras cruzaba la plaza hacia su oficina iba pensando, una vez m\u00e1s, en esa voluntad de hierro que daba al asc\u00e9tico hombrecillo su capacidad de trabajo, esa aptitud para producir, ma\u00f1ana y tarde, tarde y noche, tormentosas historias. A cualquier hora del d\u00eda que me acordaba de \u00e9l, pensaba: \u00abEst\u00e1 escribiendo\u00bb y lo ve\u00eda, como lo hab\u00eda visto tantas veces, golpeando con dos deditos r\u00e1pidos las teclas de la Remington y mirando el rodillo con sus ojos alucinados, y sent\u00eda una curiosa mezcla de piedad y envidia.\n\nLa ventana del cub\u00edculo estaba entreabierta \u2014se pod\u00eda o\u00edr el ruido acompasado de la m\u00e1quina\u2014 y yo la empuj\u00e9, al tiempo que lo saludaba: \u00abBuenos d\u00edas, se\u00f1or trabajador\u00bb. Pero tuve la impresi\u00f3n de haberme equivocado de lugar o de persona, y s\u00f3lo despu\u00e9s de varios segundos reconoc\u00ed, bajo el disfraz compuesto de guardapolvo blanco, gorrita de m\u00e9dico y grandes barbas negras rab\u00ednicas, al escriba boliviano. Segu\u00eda escribiendo inmutable, sin mirarme, ligeramente curvado sobre el escritorio. Al cabo de un momento, como haciendo una pausa entre dos pensamientos, pero sin volver la cabeza hacia m\u00ed, le o\u00ed decir con su voz de timbre perfecto y acariciador:\n\n\u2014El ginec\u00f3logo Alberto de Quinteros est\u00e1 haciendo parir trillizos a una sobrina, y uno de los renacuajos se ha atravesado. \u00bfPuede esperarme cinco minutos? Hago una ces\u00e1rea a la muchacha y nos tomamos una yerbaluisa con menta.\n\nEsper\u00e9, fumando un cigarrillo, sentado en el alf\u00e9izar de la ventana, que acabara de traer al mundo a los trillizos atravesados, operaci\u00f3n que, en efecto, no le tom\u00f3 m\u00e1s de unos minutos. Luego, mientras se quitaba el disfraz, lo doblaba escrupulosamente y, junto con las postizas barbas patriarcales, lo guardaba en una bolsa de pl\u00e1stico, le dije:\n\n\u2014Para un parto de trillizos, con ces\u00e1rea y todo, s\u00f3lo necesita cinco minutos, qu\u00e9 m\u00e1s quiere. Yo me he demorado tres semanas para un cuento de tres muchachos que levitan aprovechando la presi\u00f3n de los aviones.\n\nLe cont\u00e9, mientras \u00edbamos al Bransa, que, despu\u00e9s de muchos relatos fracasados, el de los levitadores me hab\u00eda parecido decoroso, y que lo hab\u00eda llevado al Suplemento Dominical de El Comercio, temblando de miedo. El director lo ley\u00f3 delante de m\u00ed y me dio una respuesta misteriosa: \u00abD\u00e9jelo, ya se ver\u00e1 qu\u00e9 hacemos con \u00e9l\u00bb. Desde entonces, hab\u00edan pasado dos domingos en que yo, afanoso, me precipitaba a comprar el diario y hasta ahora nada. Pero Pedro Camacho no perd\u00eda tiempo con problemas ajenos:\n\n\u2014Sacrifiquemos el refrigerio y caminemos \u2014me dijo, cogi\u00e9ndome del brazo, cuando ya iba a sentarme, y regres\u00e1ndome hacia la Colmena\u2014. Tengo en las pantorrillas un cosquilleo que anuncia calambre. Es la vida sedentaria. Me hace falta ejercicio.\n\nS\u00f3lo porque sab\u00eda lo que me iba a responder le suger\u00ed que hiciera lo que Victor Hugo y Hemingway: escribir de pie. Pero esta vez me equivoqu\u00e9:\n\n\u2014En la Pensi\u00f3n La Tapada suceden cosas interesantes \u2014me dijo, sin siquiera responderme, mientras me hac\u00eda dar vueltas, casi al trote, en torno al monumento a San Mart\u00edn\u2014. Hay un joven que llora en las noches de luna.\n\nYo rara vez ven\u00eda al centro los domingos y estaba sorprendido de ver lo distinta que era la gente de semana de la que ve\u00eda ahora. En vez de oficinistas de clase media, la plaza estaba colmada de sirvientas en su d\u00eda de salida, serranitos de mejillas chaposas y zapatones, ni\u00f1as descalzas con trenzas y, entre la abigarrada muchedumbre, se ve\u00edan fot\u00f3grafos ambulantes y vivanderas. Obligu\u00e9 al escriba a detenerse frente a la dama con t\u00fanica que, en la parte central del monumento, representa a la patria, y, para ver si lo hac\u00eda re\u00edr, le cont\u00e9 por qu\u00e9 llevaba ese extravagante auqu\u00e9nido aposentado en su cabeza: al vaciar el bronce, aqu\u00ed en Lima, los artesanos confundieron la indicaci\u00f3n del escultor \u00abllama votiva\u00bb con el llama animal. Ni sonri\u00f3, naturalmente. Volvi\u00f3 a cogerme del brazo y, mientras me hac\u00eda caminar, dando encontronazos a los paseantes, reanud\u00f3 su mon\u00f3logo, indiferente a todo lo que lo rodeaba, empezando por m\u00ed:\n\n\u2014No se le ha visto la cara, pero cabe suponer que es alg\u00fan monstruo, \u00bfhijo bastardo de la due\u00f1a de la pensi\u00f3n?, aquejado de taras, jorobas, enanismo, bicefalia, a quien do\u00f1a Atanasia oculta de d\u00eda para no asustarnos y s\u00f3lo de noche deja salir a orearse.\n\nHablaba sin la menor emoci\u00f3n, como una grabadora, y yo, por tirarle la lengua, le repliqu\u00e9 que su hip\u00f3tesis me parec\u00eda exagerada: \u00bfno pod\u00eda tratarse de un muchacho que lloraba penas de amor?\n\n\u2014Si fuera un enamorado, tendr\u00eda una guitarra, un viol\u00edn, o cantar\u00eda \u2014me dijo, mir\u00e1ndome con un desprecio mitigado por la compasi\u00f3n\u2014. \u00c9ste s\u00f3lo llora.\n\nHice esfuerzos para que me explicara todo desde el principio, pero \u00e9l estaba m\u00e1s difuso y reconcentrado que de costumbre. S\u00f3lo saqu\u00e9 en claro que alguien, desde hac\u00eda muchas noches, lloraba en un rinc\u00f3n de la pensi\u00f3n y que los inquilinos de La Tapada se quejaban. La due\u00f1a, do\u00f1a Atanasia, dec\u00eda no saber nada y, seg\u00fan el escriba, empleaba \u00abla coartada de los esp\u00edritus\u00bb.\n\n\u2014Es posible tambi\u00e9n que llore un crimen \u2014especul\u00f3 Pedro Camacho, con un tono de contador que hace sumas en alta voz, dirigi\u00e9ndome, siempre del brazo, hacia Radio Central, despu\u00e9s de una decena de vueltas al monumento\u2014. \u00bfUn crimen familiar? \u00bfUn parricida que se jala los pelos y se ara\u00f1a la carne de arrepentimiento? \u00bfUn hijo del de las ratas?\n\nNo estaba excitado en lo m\u00e1s m\u00ednimo, pero lo not\u00e9 m\u00e1s distante que otras veces, m\u00e1s incapaz que nunca de escuchar, de conversar, de recordar que ten\u00eda alguien al lado. Estaba seguro de que no me ve\u00eda. Trat\u00e9 de alargar su mon\u00f3logo, pues era como estar viendo su fantas\u00eda en plena acci\u00f3n, pero \u00e9l, con la misma brusquedad con que hab\u00eda comenzado a hablar del invisible llor\u00f3n, enmudeci\u00f3. Lo vi instalarse de nuevo en su cub\u00edculo, quitarse el saco negro y la corbatita de lazo, sujetarse la cabellera con una redecilla y enfundarse una peluca de mujer con mo\u00f1o que sac\u00f3 de otra bolsa de pl\u00e1stico. No pude aguantarme y lanc\u00e9 una carcajada:\n\n\u2014\u00bfA qui\u00e9n tengo el gusto de tener al frente? \u2014le pregunt\u00e9, todav\u00eda ri\u00e9ndome.\n\n\u2014Debo dar unos consejos a un laboratorista franc\u00f3filo, que ha matado a su hijo \u2014me explic\u00f3, con un retint\u00edn burl\u00f3n, poni\u00e9ndose en la cara, en vez de las b\u00edblicas barbas de antes, unos aretes de colores y un lunar coquet\u00f3n\u2014. Adi\u00f3s, amigo.\n\nApenas di media vuelta para irme, sent\u00ed \u2014renaciente, parejo, seguro de s\u00ed mismo, compulsivo, eterno\u2014 el teclear de la Remington. En el colectivo a Miraflores, iba pensando en la vida de Pedro Camacho. \u00bfQu\u00e9 medio social, qu\u00e9 encadenamiento de personas, relaciones, problemas, casualidades, hechos, hab\u00edan producido esa vocaci\u00f3n literaria (\u00bfliteraria?, \u00bfpero qu\u00e9, entonces?) que hab\u00eda logrado realizarse, cristalizar en una obra y obtener una audiencia? \u00bfC\u00f3mo se pod\u00eda ser, de un lado, una parodia de escritor y, al mismo tiempo, el \u00fanico que, por tiempo consagrado a su oficio y obra realizada, merec\u00eda ese nombre en el Per\u00fa? \u00bfAcaso eran escritores esos pol\u00edticos, esos abogados, esos pedagogos, que detentaban el t\u00edtulo de poetas, novelistas, dramaturgos, porque, en breves par\u00e9ntesis de vidas consagradas en sus cuatro quintas partes a actividades ajenas a la literatura, hab\u00edan producido una plaquette de versos o una estre\u00f1ida colecci\u00f3n de cuentos? \u00bfPor qu\u00e9 esos personajes que se serv\u00edan de la literatura como adorno o pretexto iban a ser m\u00e1s escritores que Pedro Camacho, quien s\u00f3lo viv\u00eda para escribir? \u00bfPorque ellos hab\u00edan le\u00eddo (o, al menos, sab\u00edan que deber\u00edan haber le\u00eddo) a Proust, a Faulkner, a Joyce, y Pedro Camacho era poco m\u00e1s que un analfabeto? Cuando pensaba en estas cosas sent\u00eda tristeza y angustia. Cada vez me resultaba m\u00e1s evidente que lo \u00fanico que quer\u00eda ser en la vida era escritor y cada vez, tambi\u00e9n, me convenc\u00eda m\u00e1s que la \u00fanica manera de serlo era entreg\u00e1ndose a la literatura en cuerpo y alma. No quer\u00eda de ning\u00fan modo ser un escritor a medias y a poquitos, sino uno de verdad, como \u00bfqui\u00e9n? Lo m\u00e1s cercano a ese escritor a tiempo completo, obsesionado y apasionado con su vocaci\u00f3n, que conoc\u00eda, era el radionovelista boliviano: por eso me fascinaba tanto.\n\nEn casa de los abuelos me estaba esperando Javier, rebosante de felicidad, con un programa dominical para resucitar muertos. Hab\u00eda recibido la mensualidad que le giraban sus padres desde Piura, con una buena propina por las Fiestas Patrias, y decidido que nos gast\u00e1ramos esos soles extras los cuatro juntos.\n\n\u2014En homenaje a ti, he hecho un programa intelectual y cosmopolita \u2014me dijo, d\u00e1ndome unas palmadas estimulantes\u2014. Compa\u00f1\u00eda argentina de Francisco Petrone, comida alemana en el Rinc\u00f3n Toni y fin de fiesta francesa en el Negro-Negro, bailando boleros en la oscuridad.\n\nAs\u00ed como, en mi corta vida, Pedro Camacho era lo m\u00e1s pr\u00f3ximo a un escritor que hab\u00eda visto, Javier era, entre mis conocidos, lo m\u00e1s parecido a un pr\u00edncipe renacentista por su generosidad y exuberancia. Era, adem\u00e1s, de una gran eficiencia: ya la t\u00eda Julia y Nancy estaban informadas de lo que nos esperaba esa noche y ya ten\u00eda \u00e9l en el bolsillo las entradas para el teatro. El programa no pod\u00eda ser m\u00e1s seductor y disip\u00f3 de golpe todas mis l\u00fagubres reflexiones sobre la vocaci\u00f3n y el destino pordiosero de la literatura en el Per\u00fa. Javier tambi\u00e9n estaba muy contento: desde hac\u00eda un mes sal\u00eda con Nancy y esa asiduidad tomaba caracteres de romance formal. Haberle confesado a mi prima mis amores con la t\u00eda Julia le hab\u00eda sido util\u00edsimo porque, con el pretexto de servirnos de celestinos y facilitarnos las salidas, se las arreglaba para ver a Nancy varias veces por semana. Mi prima y la t\u00eda Julia eran ahora inseparables: iban juntas de compras, al cine e intercambiaban secretos. Mi prima se hab\u00eda vuelto una entusiasta hada madrina de nuestro romance y una tarde me levant\u00f3 la moral con esta reflexi\u00f3n: \u00abLa Julita tiene una manera de ser que borra todas las diferencias de edad, primo\u00bb.\n\nEl magno programa de ese domingo (en el que, creo, se decidi\u00f3 estelarmente buena parte de mi futuro) comenz\u00f3 bajo los mejores auspicios. Hab\u00eda pocas ocasiones, en la Lima de los a\u00f1os cincuenta, de ver teatro de calidad, y la compa\u00f1\u00eda argentina de Francisco Petrone trajo una serie de obras modernas, que no se hab\u00edan dado en el Per\u00fa. Nancy recogi\u00f3 a la t\u00eda Julia donde la t\u00eda Olga y ambas se vinieron al centro en taxi. Javier y yo las esper\u00e1bamos en la puerta del Teatro Segura. Javier, que en esas cosas sol\u00eda excederse, hab\u00eda comprado un palco, que result\u00f3 el \u00fanico ocupado, de modo que fuimos un centro de observaci\u00f3n casi tan visible como el escenario. Con mi mala conciencia, supuse que varios parientes y conocidos nos ver\u00edan y maliciar\u00edan. Pero apenas comenz\u00f3 la funci\u00f3n, se esfumaron esos temores. Representaban La muerte de un viajante, de Arthur Miller, y era la primera pieza que yo ve\u00eda de car\u00e1cter no tradicional, irrespetuosa de las convenciones de tiempo y espacio. Mi entusiasmo y excitaci\u00f3n fueron tales que, en el entreacto, comenc\u00e9 a hablar hasta por los codos, haciendo un elogio fogoso de la obra, comentando sus personajes, su t\u00e9cnica, sus ideas, y, luego, mientras com\u00edamos embutidos y tom\u00e1bamos cerveza negra en el Rinc\u00f3n Toni de la Colmena, segu\u00ed haci\u00e9ndolo de una manera tan absorbente que Javier, despu\u00e9s, me amonest\u00f3: \u00abParec\u00edas una lora a la que le hubieran dado yobimbina\u00bb. Mi prima Nancy, a quien mis veleidades literarias siempre le hab\u00edan parecido una extravagancia semejante a la que ten\u00eda el t\u00edo Eduardo \u2014un viejecito hermano del abuelo, juez jubilado que se dedicaba al infrecuente pasatiempo de coleccionar ara\u00f1as\u2014, despu\u00e9s de o\u00edrme perorar tanto sobre la obra que acab\u00e1bamos de ver, sospech\u00f3 que mis inclinaciones pod\u00edan tener mal fin: \u00abTe est\u00e1s volviendo locumbeta, flaco\u00bb.\n\nEl Negro-Negro hab\u00eda sido escogido por Javier para rematar la noche porque era un lugar con cierta aureola de bohemia intelectual \u2014los jueves se daban peque\u00f1os espect\u00e1culos: piezas en un acto, mon\u00f3logos, recitales, y sol\u00edan concurrir all\u00ed pintores, m\u00fasicos y escritores\u2014, pero tambi\u00e9n porque era la bo\u00eete m\u00e1s oscura de Lima, un s\u00f3tano en los portales de la plaza San Mart\u00edn que no ten\u00eda m\u00e1s de veinte mesas, con una decoraci\u00f3n que cre\u00edamos existencialista. Era un sitio que, las pocas veces que hab\u00eda ido, me daba la ilusi\u00f3n de estar en una cave de Saint Germain des Pr\u00e8s. Nos sentaron en una mesita a la orilla de la pista de baile y Javier, m\u00e1s rumboso que nunca, pidi\u00f3 cuatro whiskys. \u00c9l y Nancy se pararon de inmediato a bailar y yo, en el reducto estrecho y atestado, segu\u00ed habl\u00e1ndole a Julia de teatro y de Arthur Miller. Est\u00e1bamos muy juntos, con las manos entrelazadas, ella me escuchaba con abnegaci\u00f3n y yo le dec\u00eda que esa noche hab\u00eda descubierto el teatro: pod\u00eda ser algo tan complejo y profundo como la novela, e, incluso, por ser algo vivo, en cuya materializaci\u00f3n interven\u00edan seres de carne y hueso, y otras artes, la pintura, la m\u00fasica, era tal vez superior.\n\n\u2014De repente, cambio de g\u00e9nero y, en lugar de cuentos, me pongo a escribir dramas \u2014le dije, excitad\u00edsimo\u2014. \u00bfQu\u00e9 me aconsejas?\n\n\u2014En lo que a m\u00ed respecta, no hay inconveniente \u2014me contest\u00f3 la t\u00eda Julia, poni\u00e9ndose de pie\u2014. Pero ahora, Varguitas, s\u00e1came a bailar y dime cositas al o\u00eddo. Entre pieza y pieza, si quieres, te doy permiso para que me hables de literatura.\n\nSegu\u00ed sus instrucciones al pie de la letra. Bailamos muy apretados, bes\u00e1ndonos, yo le dec\u00eda que estaba enamorado de ella, ella que estaba enamorada de m\u00ed, y \u00e9sa fue la primera vez que, ayudado por el ambiente \u00edntimo, incitante, turbador, y por los whiskys de Javier, no disimul\u00e9 el deseo que me provocaba; mientras bail\u00e1bamos mis labios se hund\u00edan con morosidad en su cuello, mi lengua entraba a su boca y sorb\u00eda su saliva, la estrechaba con fuerza para sentir sus pechos, su vientre y sus muslos, y, luego, en la mesa, al amparo de las sombras, le acarici\u00e9 las piernas y los senos. As\u00ed est\u00e1bamos, aturdidos y gozosos, cuando la prima Nancy, en una pausa entre dos boleros, nos hel\u00f3 la sangre:\n\n\u2014Dios m\u00edo, f\u00edjense qui\u00e9n est\u00e1 ah\u00ed: el t\u00edo Jorge.\n\nEra un peligro que hubi\u00e9ramos debido tener en cuenta. El t\u00edo Jorge, el m\u00e1s joven de los t\u00edos, congeniaba audazmente, en una vida superagitada, toda clase de negocios y aventuras empresariales, con una intensa vida nocherniega, de faldas, fiestas y copas. De \u00e9l se contaba un malentendido tragic\u00f3mico, que tuvo como escenario otra bo\u00eete: El Embassy. Acababa de comenzar el show, la muchacha que cantaba no pod\u00eda hacerlo porque, desde una de las mesas, un borrach\u00edn la interrump\u00eda con malacrianzas. Ante la bo\u00eete atestada, el t\u00edo Jorge se hab\u00eda puesto de pie, rugiendo como un Quijote: \u00abSilencio, miserable, yo te voy a ense\u00f1ar a respetar a una dama\u00bb, y, avanzando hacia el majadero en actitud pugil\u00edstica, s\u00f3lo para descubrir, un segundo despu\u00e9s, que estaba haciendo el rid\u00edculo, pues la interrupci\u00f3n de la cantante por el seudocliente era parte del show. Ah\u00ed estaba, en efecto, s\u00f3lo a dos mesas de nosotros, muy elegante, la cara apenas revelada por los f\u00f3sforos de los fumadores y las linternas de los mozos. A su lado reconoc\u00ed a su mujer, la t\u00eda Gaby, y, pese a estar apenas a un par de metros de nosotros, ambos se empe\u00f1aban en no mirar a nuestro lado. Era clar\u00edsimo: me hab\u00edan visto besando a la t\u00eda Julia, se hab\u00edan dado cuenta de todo, optaban por una ceguera diplom\u00e1tica. Javier pidi\u00f3 la cuenta, salimos del Negro-Negro casi inmediatamente, los t\u00edos Jorge y Gaby se abstuvieron de mirarnos incluso cuando pasamos roz\u00e1ndolos. En el taxi a Miraflores \u2014los cuatro \u00edbamos mudos y con las caras largas\u2014 la flaca Nancy resumi\u00f3 lo que todos pens\u00e1bamos: \u00abAdi\u00f3s trabajos, se arm\u00f3 el gran esc\u00e1ndalo\u00bb.\n\nPero, como en una buena pel\u00edcula de suspenso, en los d\u00edas siguientes no pas\u00f3 nada. Ning\u00fan indicio permit\u00eda advertir que la tribu familiar hab\u00eda sido alertada por los t\u00edos Jorge y Gaby. El t\u00edo Lucho y la t\u00eda Olga no dijeron una palabra a la t\u00eda Julia que le permitiera suponer que sab\u00edan, y ese jueves, cuando, valientemente, me present\u00e9 en su casa a almorzar, estuvieron conmigo tan naturales y afectuosos como de costumbre. La prima Nancy tampoco fue objeto de ninguna pregunta capciosa por parte de la t\u00eda Laura y el t\u00edo Juan. En mi casa, los abuelos parec\u00edan en la luna y me segu\u00edan preguntando, con el aire m\u00e1s angelical del mundo, si acompa\u00f1aba siempre al cine a la Julita, \u00abque era tan cinemera\u00bb. Fueron unos d\u00edas desasosegados, en que, extremando las precauciones, la t\u00eda Julia y yo decidimos no vernos ni siquiera a ocultas por lo menos una semana. Pero, en cambio, habl\u00e1bamos por tel\u00e9fono. La t\u00eda Julia sal\u00eda a telefonearme desde la bodega de la esquina, por lo menos tres veces al d\u00eda, y nos comunic\u00e1bamos nuestras respectivas observaciones sobre la temida reacci\u00f3n de la familia y hac\u00edamos toda clase de hip\u00f3tesis. \u00bfSer\u00eda posible que el t\u00edo Jorge hubiera decidido guardar el secreto? Yo sab\u00eda que eso era impensable dentro de las costumbres familiares. \u00bfY entonces? Javier adelantaba la tesis de que la t\u00eda Gaby y el t\u00edo Jorge hubieran tenido encima tantos whiskys que no se dieran bien cuenta de las cosas, que en su memoria s\u00f3lo quedara una remota sospecha, y que no hab\u00edan querido desatar un esc\u00e1ndalo por algo no absolutamente comprobado. Un poco por curiosidad, otro por masoquismo, hice esa semana un recorrido por los hogares del clan, para saber a qu\u00e9 atenerme. No not\u00e9 nada anormal, salvo una omisi\u00f3n curiosa, que me suscit\u00f3 una pirotecnia de especulaciones. La t\u00eda Hortensia, que me invit\u00f3 un t\u00e9 con biscotelas, en dos horas de conversaci\u00f3n no mencion\u00f3 ni una sola vez a la t\u00eda Julia. \u00abSaben todo y est\u00e1n planeando algo\u00bb, le aseguraba yo a Javier, y \u00e9l, harto de que no le hablara de otra cosa, respond\u00eda: \u00abEn el fondo, est\u00e1s muerto de ganas de que haya ese esc\u00e1ndalo para tener de qu\u00e9 escribir\u00bb.\n\nEn esa semana, fecunda en acontecimientos, me vi inesperadamente convertido en protagonista de una ri\u00f1a callejera y en algo as\u00ed como guardaespaldas de Pedro Camacho. Sal\u00eda yo de la Universidad de San Marcos, luego de averiguar los resultados de un examen de derecho procesal, lleno de remordimientos por haber sacado nota m\u00e1s alta que mi amigo Velando, quien era el que sab\u00eda, cuando, al cruzar el parque Universitario, me top\u00e9 con Genaro pap\u00e1, el patriarca de la falange propietaria de las radios Panamericana y Central. Fuimos juntos hasta la calle Bel\u00e9n, conversando. Era un caballero siempre vestido de oscuro y siempre serio, al que el escriba boliviano se refer\u00eda a veces llam\u00e1ndolo, era f\u00e1cil suponer por qu\u00e9, El negrero.\n\n\u2014Su amigo, el genio, est\u00e1 siempre d\u00e1ndome dolores de cabeza \u2014me dijo\u2014. Me tiene hasta la coronilla. Si no fuera tan productivo ya lo hubiera puesto de patitas en la calle.\n\n\u2014\u00bfOtra protesta de la embajada argentina? \u2014le pregunt\u00e9.\n\n\u2014No s\u00e9 qu\u00e9 enredos anda armando \u2014se quej\u00f3\u2014. Se ha puesto a tomarle el pelo a la gente, a pasar personajes de un radioteatro a otro y a cambiarles los nombres, para confundir a los oyentes. Ya mi mujer me lo hab\u00eda advertido y ahora llaman por tel\u00e9fono, hasta han llegado dos cartas. Que el cura de Mendocita se llama como el Testigo de Jehov\u00e1 y \u00e9ste como el cura. Yo ando muy ocupado para o\u00edr radioteatros. \u00bfUsted los oye alguna vez?\n\nEst\u00e1bamos bajando por la Colmena hacia la plaza San Mart\u00edn, entre \u00f3mnibus que sal\u00edan a provincias y cafetines de chinos, y yo record\u00e9 que, hac\u00eda unos d\u00edas, hablando de Pedro Camacho, la t\u00eda Julia me hab\u00eda hecho re\u00edr y confirmado mis sospechas de que el escribidor era un humorista que disimulaba:\n\n\u2014Pas\u00f3 algo rar\u00edsimo: la chica tuvo al peladingo, se muri\u00f3 en el parto y lo enterraron con todas las de ley. \u00bfC\u00f3mo te explicas que en el cap\u00edtulo de esta tarde aparezcan bautiz\u00e1ndolo en la catedral?\n\nLe dije a Genaro pap\u00e1 que yo tampoco ten\u00eda tiempo para o\u00edrlos, que a lo mejor esos trueques y enredos eran una t\u00e9cnica original suya de contar historias.\n\n\u2014No le pagamos para que sea original sino para que nos entretenga a la gente \u2014me dijo Genaro pap\u00e1, que no era, a todas luces, un empresario progresista sino uno tradicionalista\u2014. Con estas bromas va a perder audiencia y los auspiciadores nos quitar\u00e1n avisos. Usted, que es amigo suyo, d\u00edgale que se deje de modernismos o que se puede quedar sin trabajo.\n\nLe suger\u00ed que se lo dijera \u00e9l mismo, que era el patr\u00f3n: la amenaza tendr\u00eda m\u00e1s peso. Pero Genaro pap\u00e1 movi\u00f3 la cabeza, con un gesto compungido que hab\u00eda heredado Genaro hijo:\n\n\u2014No admite siquiera que yo le dirija la palabra. El \u00e9xito lo ha engre\u00eddo mucho y, vez que trato de hablarle, me falta el respeto.\n\nHab\u00eda ido a participarle, con la mayor educaci\u00f3n, que se recib\u00edan llamadas, a mostrarle las cartitas de protesta. Pedro Camacho, sin responderle una palabra, cogi\u00f3 las dos cartas, las hizo pedazos sin abrirlas y las ech\u00f3 a la papelera. Luego, se puso a escribir a m\u00e1quina, como si no hubiera nadie presente, y Genaro pap\u00e1 lo oy\u00f3 murmurar cuando, al borde de la apoplej\u00eda, se iba de esa cueva hostil: \u00abZapatero a tus zapatos\u00bb.\n\n\u2014Yo no puedo exponerme a otra groser\u00eda as\u00ed, tendr\u00eda que botarlo y eso tampoco ser\u00eda realista \u2014concluy\u00f3, con un adem\u00e1n de fastidio\u2014. Pero usted no tiene nada que perder, a usted no lo va a insultar, usted tambi\u00e9n es medio artista \u00bfno? \u00c9chenos una mano, h\u00e1galo por la empresa, h\u00e1blele.\n\nLe ofrec\u00ed que lo har\u00eda y, en efecto, despu\u00e9s del Panamericano de las doce, fui, para desgracia m\u00eda, a invitar a Pedro Camacho una taza de yerbaluisa con menta. Est\u00e1bamos saliendo de Radio Central cuando dos tipos grandotes nos cerraron el paso. Los reconoc\u00ed en el acto: eran los churrasqueros, dos hermanos bigotudos de La Parrillada Argentina, un restaurante situado en la misma calle, frente al colegio de las monjitas de Bel\u00e9n, donde ellos mismos, con mandiles blancos y altos gorros de cocineros, preparaban las sangrientas carnes y los chinchulines. Rodearon al escriba boliviano con aire matonesco y el m\u00e1s gordo y viejo de los dos lo increp\u00f3:\n\n\u2014\u00bfAs\u00ed que somos matani\u00f1os, no, Camacho de porquer\u00eda? \u00bfTe has cre\u00eddo, atorrante, que en este pa\u00eds no hay nadie que pueda ense\u00f1arte a guardar respetos?\n\nSe iba excitando mientras hablaba, enrojec\u00eda y se le atropellaba la voz. El hermano menor asent\u00eda y, en una pausa iracunda del churrasquero mayor, tambi\u00e9n meti\u00f3 su cuchara:\n\n\u2014\u00bfY los piojos? \u00bfConque la golosina de las porte\u00f1as son los bichos que les sacan del pelo a sus hijos, grand\u00edsimo hijo de puta? \u00bfMe voy a quedar con los brazos cruzados mientras puteas a mi madre?\n\nEl escriba boliviano no hab\u00eda retrocedido ni un mil\u00edmetro y los escuchaba, paseando de uno a otro sus ojos saltones, con expresi\u00f3n doctoral. De pronto, haciendo su caracter\u00edstica venia de maestro de ceremonias y en tono muy solemne, les solt\u00f3 la m\u00e1s urbana de las preguntas:\n\n\u2014\u00bfPor acaso, no son ustedes argentinos?\n\nEl churrasquero gordo, que ya echaba espuma por los bigotes \u2014su cara estaba a veinte cent\u00edmetros de la de Pedro Camacho, para lo cual ten\u00eda que inclinarse mucho\u2014, rugi\u00f3 con patriotismo:\n\n\u2014\u00a1Argentinos, s\u00ed, hijo de puta, y a mucha honra!\n\nVi entonces que, ante esta confirmaci\u00f3n \u2014realmente innecesaria porque bastaba o\u00edrles dos palabras para saber que eran argentinos\u2014, el escriba boliviano, como si algo le hubiera estallado dentro, palidec\u00eda, sus ojos se pon\u00edan \u00edgneos, adoptaba una expresi\u00f3n amenazadora y, fustigando el aire con el dedo \u00edndice, los apostrof\u00f3 as\u00ed:\n\n\u2014Me lo ol\u00eda. Pues bien: \u00a1v\u00e1yanse inmediatamente a cantar tangos!\n\nLa orden no era humor\u00edstica, sino funeral. Los churrasqueros quedaron, un segundo, sin saber qu\u00e9 decir. Era evidente que el escriba no bromeaba: desde su peque\u00f1ez tenaz y su total indefensi\u00f3n f\u00edsica, los miraba con ferocidad y desprecio.\n\n\u2014\u00bfQu\u00e9 ha dicho usted? \u2014articul\u00f3 por fin el churrasquero gordo, confuso y encolerizado\u2014. \u00bfQu\u00e9 cosa, qu\u00e9 cosa?\n\n\u2014\u00a1A cantar tangos y a lavarse las orejas! \u2014enriqueci\u00f3 la orden, con su perfecta pronunciaci\u00f3n, Pedro Camacho. Y, luego de una brev\u00edsima pausa, con tranquilidad escalofriante, deletre\u00f3 la rebuscada temeridad que nos perdi\u00f3\u2014: Si no quieren recibir un rapapolvo.\n\nEsta vez yo qued\u00e9 todav\u00eda m\u00e1s sorprendido que los churrasqueros. Que esa personita m\u00ednima, de f\u00edsico de ni\u00f1o de cuarto de primaria, prometiera una paliza a dos sansones de cien kilos era delirante, adem\u00e1s de suicida. Pero ya el churrasquero gordo reaccionaba, cog\u00eda del cuello al escriba, y, entre las risas de la gente que se hab\u00eda aglomerado alrededor, lo levantaba como una pluma, aullando:\n\n\u2014\u00bfUn rapapolvo, a m\u00ed? Ahora vas a ver, enano...\n\nCuando vi que el churrasquero mayor se preparaba a volatilizar a Pedro Camacho de un derechazo, no me qued\u00f3 m\u00e1s remedio que intervenir. Lo sujet\u00e9 del brazo, al tiempo que trataba de liberar al pol\u00edgrafo, quien, amoratado y suspenso, pataleaba en el aire como una ara\u00f1a, y alcanc\u00e9 a decir algo as\u00ed como: \u00abOiga, no sea abusivo, su\u00e9ltelo\u00bb, cuando el churrasquero menor me lanz\u00f3, sin pre\u00e1mbulos, un pu\u00f1etazo que me sent\u00f3 en el suelo. Desde all\u00ed, y mientras, aturdido, dificultosamente me pon\u00eda de pie y me preparaba a poner en pr\u00e1ctica la filosof\u00eda de mi abuelo, un caballero de la vieja escuela, quien me hab\u00eda ense\u00f1ado que ning\u00fan arequipe\u00f1o digno de esa tierra rechaza jam\u00e1s una invitaci\u00f3n a pelear (y, sobre todo, una invitaci\u00f3n tan contundente como un directo al ment\u00f3n), vi que el churrasquero mayor descargaba una verdadera lluvia de bofetadas (hab\u00eda preferido las bofetadas a los pu\u00f1etes, piadosamente, dada la osatura liliputiense del adversario) sobre el artista. Despu\u00e9s, mientras intercambiaba empujones y trompadas contra el churrasquero menor (\u00aben defensa del arte\u00bb, pensaba) ya no pude ver gran cosa. El pugilato no dur\u00f3 mucho, pero cuando, al fin, gente de Radio Central nos rescat\u00f3 de las manos de los forzudos, yo ten\u00eda unos cuantos chichones y el escriba estaba con la cara tan hinchada y tumefacta que Genaro pap\u00e1 debi\u00f3 llevarlo a la Asistencia P\u00fablica. En vez de darme las gracias por haber arriesgado mi integridad defendiendo a su estrella exclusiva, Genaro hijo, esa tarde, me reprendi\u00f3 por una noticia que Pascual, aprovechando la confusi\u00f3n, hab\u00eda filtrado en dos boletines consecutivos y que comenzaba (con algo de exageraci\u00f3n) as\u00ed: \u00abPandilleros rioplatenses atacaron hoy criminalmente a nuestro director, el conocido periodista\u00bb, etc\u00e9tera.\n\nEsa tarde, cuando Javier se present\u00f3 en mi altillo de Radio Panamericana, se ri\u00f3 a carcajadas con la historia del pugilato, y me acompa\u00f1\u00f3 a preguntarle al escriba c\u00f3mo se encontraba. Le hab\u00edan puesto una venda de pirata en el ojo derecho y dos curitas, una en el cuello y otra debajo de la nariz. \u00bfC\u00f3mo se sent\u00eda? Hizo un gesto desde\u00f1oso, sin dar importancia al asunto, y no me agradeci\u00f3 que, por solidaridad con \u00e9l, me hubiera zambullido en la pelea. Su \u00fanico comentario encant\u00f3 a Javier:\n\n\u2014Al separarnos, los salvaron. Si dura unos minutos m\u00e1s, la gente me hubiera reconocido y pobres de ellos: los linchaban.\n\nFuimos al Bransa y all\u00ed nos cont\u00f3 que en Bolivia, una vez, un futbolista \u00abde ese pa\u00eds\u00bb, que hab\u00eda o\u00eddo sus programas, se present\u00f3 en la radioemisora armado de un rev\u00f3lver, que, por suerte, detectaron a tiempo los guardianes.\n\n\u2014Va a tener que cuidarse \u2014lo previno Javier\u2014. Lima est\u00e1 llena de argentinos ahora.\n\n\u2014Total, a ustedes y a m\u00ed, tarde o temprano tendr\u00e1n que comernos los gusanos \u2014filosof\u00f3 Pedro Camacho.\n\nY nos instruy\u00f3 sobre la transmigraci\u00f3n de las almas, que le parec\u00eda art\u00edculo de fe. Nos hizo una confidencia: si se pudiera elegir, a \u00e9l, en su pr\u00f3ximo estadio vital, le gustar\u00eda ser alg\u00fan animal marino, longevo y calmo, como las tortugas o las ballenas. Aprovech\u00e9 su buen \u00e1nimo para ejercitar esa funci\u00f3n ad hon\u00f3rem de puente entre \u00e9l y los Genaros que hab\u00eda asumido hac\u00eda alg\u00fan tiempo, y le di el mensaje de Genaro pap\u00e1: hab\u00eda llamadas, cartas, episodios de los radioteatros que algunas gentes no entend\u00edan. El viejo le rogaba no complicar los argumentos, tener en cuenta el nivel del oyente medio que era m\u00e1s bien bajo. Trat\u00e9 de dorarle la p\u00edldora, poni\u00e9ndome de su lado (en realidad lo estaba): ese ruego era absurdo, por supuesto, uno deb\u00eda ser libre de escribir como quisiera, yo me limitaba a decirle lo que me hab\u00edan pedido.\n\nMe escuch\u00f3 tan mudo e inexpresivo que me hizo sentir muy inc\u00f3modo. Y, cuando call\u00e9, tampoco dijo ni una palabra. Bebi\u00f3 su \u00faltimo trago de yerbaluisa, se puso de pie, murmur\u00f3 que deb\u00eda regresar a su taller y parti\u00f3 sin decir hasta luego. \u00bfSe hab\u00eda ofendido porque le habl\u00e9 de las llamadas delante de un extra\u00f1o? Javier cre\u00eda que s\u00ed y me aconsej\u00f3 que le pidiera excusas. Me promet\u00ed no servir nunca m\u00e1s de intercesor a los Genaros.\n\nEsa semana que estuve sin ver a la t\u00eda Julia, volv\u00ed a salir varias noches con amigos de Miraflores a quienes, desde mis amores clandestinos, no hab\u00eda vuelto a buscar. Eran compa\u00f1eros de colegio o de barrio, muchachos que estudiaban Ingenier\u00eda, como el Negro Salas, o Medicina, como el Colorao Molfino, o que se hab\u00edan puesto a trabajar, como Coco La\u00f1as, y con quienes, desde ni\u00f1o, hab\u00eda compartido cosas maravillosas: el fulbito y el parque Salazar, la nataci\u00f3n en el Terrazas y las olas de Miraflores, las fiestas de los s\u00e1bados, las enamoradas y los cines. Pero en estas salidas, despu\u00e9s de meses sin frecuentarlos, me di cuenta que algo se hab\u00eda perdido de nuestra amistad. Ya no ten\u00edamos tantas cosas en com\u00fan como antes. Hicimos, las noches de esa semana, las mismas proezas que sol\u00edamos hacer: ir al peque\u00f1o y vetusto cementerio de Surco, para, merodeando a la luz de la luna entre las tumbas removidas por los temblores, tratar de robarnos alguna calavera; ba\u00f1arnos desnudos en la enorme piscina del balneario Santa Rosa, vecino a Anc\u00f3n, todav\u00eda construy\u00e9ndose, y recorrer los l\u00f3bregos burdeles de la avenida Grau. Ellos segu\u00edan siendo los mismos, hac\u00edan los mismos chistes, hablaban de las mismas chicas, pero yo no pod\u00eda hablarles de las cosas que me importaban: la literatura y la t\u00eda Julia. Si les hubiera dicho que escrib\u00eda cuentos y que so\u00f1aba en ser escritor no hay duda que, como la flaca Nancy, hubieran pensado que se me hab\u00eda zafado un tornillo. Y si les hubiera contado \u2014como ellos a m\u00ed sus conquistas\u2014 que estaba con una se\u00f1ora divorciada, que no era mi amante sino mi enamorada (en el sentido m\u00e1s miraflorino de la palabra) me hubieran cre\u00eddo, seg\u00fan una linda y esot\u00e9rica expresi\u00f3n muy en boga en esa \u00e9poca, un cojudo a la vela. No les ten\u00eda ning\u00fan desprecio porque no leyeran literatura, ni me consideraba superior por tener amores con una mujer hecha y derecha, pero lo cierto es que, en esas noches, mientras escarb\u00e1bamos tumbas entre los eucaliptos y los molles de Surco, o chapote\u00e1bamos bajo las estrellas de Santa Rosa, o tom\u00e1bamos cerveza y discut\u00edamos los precios con las putas de Nanette, yo me aburr\u00eda y pensaba m\u00e1s en Los juegos peligrosos (que tampoco esta semana hab\u00eda aparecido en El Comercio) y en la t\u00eda Julia que en lo que me dec\u00edan.\n\nCuando le cont\u00e9 a Javier el decepcionante reencuentro con mis compinches del barrio, me respondi\u00f3, sacando pecho:\n\n\u2014Es que siguen siendo unos mocosos. Usted y yo ya somos hombres, Varguitas.\n\n### XII\n\nEN EL CENTRO polvoriento de la ciudad, al mediar el jir\u00f3n Ica, hay una vieja casa de balcones y celos\u00edas cuyas paredes maculadas por el tiempo y los incultos transe\u00fantes (manos sentimentales que graban flechas y corazones y rasgan nombres de mujer, dedos aviesos que esculpen sexos y palabrotas) dejan ver todav\u00eda, como a lo lejos, celajes de la que fuera pintura original, ese color que en la Colonia adornaba mansiones aristocr\u00e1ticas: el azul a\u00f1il. La construcci\u00f3n, \u00bfantigua residencia de marqueses?, es hoy una endeble f\u00e1brica reparchada que resiste de milagro, no ya los temblores, incluso los moderados vientos lime\u00f1os y hasta la discret\u00edsima gar\u00faa. Corro\u00edda de arriba abajo por las polillas, anidada de ratas y de musara\u00f1as, ha sido dividida y subdividida muchas veces, patios y cuartos que la necesidad vuelve colmenas, para albergar m\u00e1s y m\u00e1s inquilinos. Una muchedumbre de condici\u00f3n modesta vive entre (y puede perecer aplastada bajo) sus fr\u00e1giles tabiques y raqu\u00edticos techos. All\u00ed, en la segunda planta, en media docena de habitaciones llenas de ancianidad y cachivaches, tal vez no pulqu\u00e9rrimas, pero s\u00ed moralmente intachables, funciona la Pensi\u00f3n Colonial.\n\nSus due\u00f1os y administradores son los Bergua, una familia de tres personas que vino a Lima desde la empedrada ciudad serrana de las innumerables iglesias, Ayacucho, hace m\u00e1s de treinta a\u00f1os, y que aqu\u00ed, oh manes de la vida, ha ido declinando en lo f\u00edsico, en lo econ\u00f3mico, en lo social y hasta en lo ps\u00edquico, y que, sin duda, en esta Ciudad de los Reyes entregar\u00e1 su alma y transmigrar\u00e1 a pez, ave o insecto.\n\nHoy, la Pensi\u00f3n Colonial vive una atribulada decadencia, y sus clientes son personas humildes e insolventes, en el mejor caso curitas provincianos que vienen a la capital a hacer tr\u00e1mites arzobispales, y, en el peor, campesinotas de mejillas amoratadas y ojos de vicu\u00f1a que guardan sus monedas en pa\u00f1uelos rosados y rezan el rosario en quechua. No hay sirvientas en la pensi\u00f3n, desde luego, y todo el trabajo de tender las camas, arreglar, hacer la compra, preparar la comida, recae sobre la se\u00f1ora Margarita Bergua y su hija, una doncella de cuarenta a\u00f1os que responde al perfumado nombre de Rosa. La se\u00f1ora Margarita Bergua es (como su nombre en diminutivo parecer\u00eda indicar) una mujer muy renacuajo, delgadita, con m\u00e1s arrugas que una pasa, y que, curiosamente, huele a gato (ya que no hay gatos en la pensi\u00f3n). Trabaja sin descanso desde la madrugada hasta el anochecer, y sus evoluciones por la casa, por la vida, son espectaculares, pues, teniendo una pierna veinte cent\u00edmetros m\u00e1s corta que la otra, usa un zapato tipo zanco, con plataforma de madera parecida a caja de lustrabotas, que le construy\u00f3 hace ya muchos a\u00f1os un habilidoso retablista ayacuchano, y que, al arrastrarse por el suelo de tablas, produce conmoci\u00f3n. Siempre fue ahorrativa, pero, con los a\u00f1os, esta virtud degener\u00f3 en man\u00eda, y ahora no cabe duda que le conviene el acre adjetivo de taca\u00f1a. Por ejemplo, no permite que ning\u00fan pensionista se ba\u00f1e sino el primer viernes de cada mes y ha impuesto la argentina costumbre \u2014tan popular en los hogares del hermano pa\u00eds\u2014 de no jalar la cadena del excusado sino una vez al d\u00eda (lo hace ella misma, antes de acostarse) a lo que la Pensi\u00f3n Colonial debe, en un ciento por ciento, ese tufillo constante, espeso y tibio, que, sobre todo al principio, marea a los pensionistas (ella, imaginaci\u00f3n de mujer que guisa respuestas para todo, sostiene que gracias a \u00e9l duermen mejor).\n\nLa se\u00f1orita Rosa tiene (o m\u00e1s bien ten\u00eda, porque despu\u00e9s de la gran tragedia nocturna hasta eso cambi\u00f3) alma y dedos de artista. De ni\u00f1a, en Ayacucho, cuando la familia estaba en su apogeo (tres casas de piedra y unas tierritas con ovejas) comenz\u00f3 a aprender piano y lo aprendi\u00f3 tan bien que lleg\u00f3 a dar un recital en el teatro de la ciudad al que asistieron el alcalde y el prefecto y en el que sus padres, oyendo los aplausos, lloraron de emoci\u00f3n. Estimulados por esta gloriosa velada, en la que tambi\u00e9n zapatearon unas \u00f1ustas, los Bergua decidieron vender todo lo que ten\u00edan y mudarse a Lima para que su hija llegara a ser concertista. Por eso adquirieron esa casona (que luego fueron vendiendo y alquilando a pedazos), por eso compraron un piano, por eso matricularon a la dotada criatura en el Conservatorio Nacional. Pero la gran ciudad lasciva destroz\u00f3 r\u00e1pido las ilusiones provincianas. Pues pronto descubrieron los Bergua algo que no hubieran sospechado jam\u00e1s: Lima era un antro de un mill\u00f3n de pecadores y todos ellos, sin una miserable excepci\u00f3n, quer\u00edan cometer estupro con la inspirada ayacuchana. Era al menos lo que, ojazos que el susto redondea y moja, contaba la adolescente de bru\u00f1idas trenzas ma\u00f1ana, tarde y noche: el profesor de solfeo se hab\u00eda lanzado sobre ella bufando y pretendido consumar el pecado sobre un colch\u00f3n de partituras, el portero del conservatorio le hab\u00eda consultado obscenamente \u00ab\u00bfquisieras ser mi meretriz?\u00bb, dos compa\u00f1eros la hab\u00edan invitado al ba\u00f1o para que los viera hacer pip\u00ed, el polic\u00eda de la esquina al que pregunt\u00f3 una direcci\u00f3n, confundi\u00e9ndola con alguien la hab\u00eda querido orde\u00f1ar y en el \u00f3mnibus, el conductor, al cobrarle el pasaje le hab\u00eda pellizcado el pez\u00f3n... Decididos a defender la integridad de ese himen que, moral serrana de preceptos indoblegables como m\u00e1rmoles, la joven pianista s\u00f3lo a su futuro amo y esposo deber\u00eda sacrificar, los Bergua cancelaron el conservatorio, contrataron a una se\u00f1orita que daba clases a domicilio, vistieron a Rosa como monja y le prohibieron salir a la calle salvo acompa\u00f1ada por ellos dos. Han pasado veinticinco a\u00f1os desde entonces, y, en efecto, el himen sigue entero y en su sitio, pero a estas alturas ya la cosa no tiene mucho m\u00e9rito, porque fuera de ese atractivo \u2014tan desde\u00f1ado, adem\u00e1s, por los j\u00f3venes modernos\u2014 la ex pianista (desde la tragedia las clases fueron suprimidas y el piano vendido para pagar el hospital y los m\u00e9dicos) carece de otros que ofrecer. Se ha entumecido, torcido, achicado, y, sumergida en esas t\u00fanicas antiafrodis\u00edacas que acostumbra llevar y en esos capuchones que ocultan su pelo y su frente, m\u00e1s parece un bulto andante que una mujer. Ella insiste en que los hombres la tocan, la amedrentan con proposiciones f\u00e9tidas y quieren violarla, pero, a estas alturas, hasta sus padres se preguntan si esas quimeras fueron alguna vez verdad.\n\nPero la figura realmente conmovedora y tutelar de la Pensi\u00f3n Colonial es don Sebasti\u00e1n Bergua, anciano de frente ancha, nariz aguile\u00f1a, mirada penetrante y rectitud y bondad en el esp\u00edritu. Hombre chapado a la antigua, si se quiere, ha conservado de sus remotos antepasados, esos hisp\u00e1nicos conquistadores, los hermanos Bergua, oriundos de las alturas de Cuenca, que llegaron al Per\u00fa con Pizarro, no tanto aquella aptitud para el exceso que los llev\u00f3 a dar garrote vil a centenares de incas (cada uno) y a pre\u00f1ar un n\u00famero comparativo de vestales cusque\u00f1as, como el esp\u00edritu acendradamente cat\u00f3lico y la audaz convicci\u00f3n de que los caballeros de rancia estirpe pueden vivir de sus rentas y de la rapi\u00f1a, pero no del sudor. Desde ni\u00f1o, hab\u00eda ido a misa a diario, comulgado todos los viernes en homenaje al Se\u00f1or de Limpias, de quien era devoto pertinaz, y se hab\u00eda dado azotes o llevado cilicio lo menos tres d\u00edas al mes. Su repugnancia al trabajo, quehacer porte\u00f1o y vil, hab\u00eda sido siempre tan extrema que incluso se hab\u00eda negado a cobrar los alquileres de los predios que le permit\u00edan vivir, y, ya radicado en Lima, jam\u00e1s se hab\u00eda molestado en ir al banco por los intereses de los bonos en que ten\u00eda invertido su dinero. Estas obligaciones, pr\u00e1cticos asuntos que est\u00e1n al alcance de las faldas, hab\u00edan corrido siempre a cargo de la diligente Margarita, y, cuando la ni\u00f1a creci\u00f3, de ella y de la ex pianista.\n\nHasta antes de la tragedia que aceler\u00f3 cruelmente la decadencia de los Bergua, maldici\u00f3n de familia de la que no quedar\u00e1 ni el nombre, la vida de don Sebasti\u00e1n en la capital hab\u00eda sido la de un escrupuloso gentilhombre cristiano. Sol\u00eda levantarse tarde, no por pereza sino para no desayunar con los pensionistas \u2014no despreciaba a los humildes pero cre\u00eda en la necesidad de las distancias sociales y, principalmente, raciales\u2014, tomaba una colaci\u00f3n frugal e iba a escuchar la misa. Esp\u00edritu curioso y permeable a la historia, visitaba siempre iglesias distintas \u2014San Agust\u00edn, San Pedro, San Francisco, Santo Domingo\u2014 para, al mismo tiempo que cumplir con Dios, regocijar su sensibilidad contemplando las obras maestras de la fe colonial; esas p\u00e9treas reminiscencias del pasado, por lo dem\u00e1s, trasladaban su esp\u00edritu hacia los tiempos de la Conquista y de la Colonia \u2014cu\u00e1nto m\u00e1s coloreados que el gris\u00e1ceo presente\u2014 en los que hubiera preferido vivir y ser un temerario capit\u00e1n o un p\u00edo destructor de idolatr\u00edas. Imbuido de fantas\u00edas pasadistas, regresaba don Sebasti\u00e1n por las calles atareadas del centro \u2014erecto y cauto en su pulcro terno negro, su camisa de cuello y pu\u00f1os postizos donde destellaba el almid\u00f3n y sus zapatos finiseculares con escarpines de charol\u2014 hacia la Pensi\u00f3n Colonial, donde, arrellanado en una mecedora frente al balc\u00f3n de celos\u00edas \u2014tan af\u00edn a su esp\u00edritu perricholista\u2014 pasaba el resto de la ma\u00f1ana leyendo murmuradoramente los peri\u00f3dicos, avisos incluidos, para saber c\u00f3mo iba el mundo. Leal a su estirpe, luego del almuerzo \u2014que no ten\u00eda m\u00e1s remedio que compartir con los pensionistas, a los que trataba empero con urbanidad\u2014 cumpl\u00eda con el espa\u00f1ol\u00edsimo rito de la siesta. Luego volv\u00eda a enfundarse su terno oscuro, su camisa almidonada, su sombrero gris y caminaba pausadamente hasta el Club Tambo-Ayacucho, instituci\u00f3n que, en unos altos del jir\u00f3n Cailloma, congregaba a muchos conocidos de su bella tierra andina. Jugando domin\u00f3, casino, rocambor, chismeando de pol\u00edtica y, alguna vez \u2014humano que era\u2014, de temas impropios para se\u00f1oritas, ve\u00eda caer la tarde y levantarse la noche. Regresaba entonces, sin prisa, a la Pensi\u00f3n Colonial, tomaba su sopa y su puchero a solas en su habitaci\u00f3n, escuchaba alg\u00fan programa de radio y se dorm\u00eda en paz con su conciencia y con Dios.\n\nPero eso era antes. Hoy, don Sebasti\u00e1n no pone jam\u00e1s los pies en la calle, nunca cambia su atuendo \u2014que es, d\u00eda y noche, un piyama color ladrillo, una bata azul, unas medias de lana y unas zapatillas de alpaca\u2014 y, desde la tragedia, no ha vuelto a pronunciar una frase. Ya no va a misa, ya no lee los diarios. Cuando est\u00e1 bien, los ancianos pensionistas (desde que descubrieron que todos los hombres del mundo eran s\u00e1tiros, los due\u00f1os de la Pensi\u00f3n Colonial s\u00f3lo aceptaron clientes femeninos o decr\u00e9pitos, varones de apetencia sexual mermada a simple vista por enfermedades o edad) lo ven ambular como un fantasma por los oscuros y a\u00f1osos aposentos, con la mirada perdida, sin afeitar y con los casposos cabellos revueltos, o lo ven sentado, columpi\u00e1ndose suavemente en la mecedora, mudo y pasmado, horas de horas. Ya no desayuna ni almuerza con los hu\u00e9spedes, pues, sentido del rid\u00edculo que corretea a los arist\u00f3cratas hasta el hospicio, don Sebasti\u00e1n no puede llevarse el cubierto a la boca, y son su esposa e hija quienes le dan de comer. Cuando est\u00e1 mal, los pensionistas ya no lo ven: el noble anciano permanece en cama, su habitaci\u00f3n clausurada con llave. Pero lo oyen; oyen sus rugidos, sus ayes, su quejumbre o sus alaridos que estremecen los vidrios. Los reci\u00e9n llegados a la Pensi\u00f3n Colonial se sorprenden, durante estas crisis, que, mientras el descendiente de conquistadores a\u00falla, do\u00f1a Margarita y la se\u00f1orita Rosa contin\u00faen barriendo, arreglando, cocinando, sirviendo y conversando como si nada ocurriera. Piensan que son desamoradas, de coraz\u00f3n glacial, indiferentes al sufrimiento del esposo y padre. A los impertinentes que, se\u00f1alando la puerta cerrada, se atreven a preguntar: \u00ab\u00bfDon Sebasti\u00e1n se siente mal?\u00bb, la se\u00f1ora Margarita les responde con mala voluntad: \u00abNo tiene nada, se est\u00e1 acordando de un susto, ya le va a pasar\u00bb. Y, en efecto, a los dos o tres d\u00edas, termina la crisis y don Sebasti\u00e1n emerge a los pasillos y aposentos de la Pensi\u00f3n Bayer, p\u00e1lido y flaco entre las telara\u00f1as y con una mueca de terror.\n\n\u00bfQu\u00e9 tragedia fue \u00e9sa? \u00bfD\u00f3nde, cu\u00e1ndo, c\u00f3mo ocurri\u00f3?\n\nTodo comenz\u00f3 con la llegada a la Pensi\u00f3n Colonial, veinte a\u00f1os atr\u00e1s, de un joven de ojos tristes que vest\u00eda el h\u00e1bito del Se\u00f1or de los Milagros. Era agente viajero, arequipe\u00f1o, padec\u00eda de estre\u00f1imiento cr\u00f3nico, ten\u00eda nombre de profeta y apellido de pescado \u2014Ezequiel Delf\u00edn\u2014 y, pese a su juventud, fue admitido como pensionista porque su f\u00edsico espiritual (flacura extrema, palidez intensa, huesos finos) y su religiosidad manifiesta \u2014adem\u00e1s de corbata, pa\u00f1uelito y brazalete morados, escond\u00eda una Biblia en su equipaje y un escapulario asomaba entre los pliegues de su ropa\u2014 parec\u00edan una garant\u00eda contra cualquier tentativa de mancillamiento de la p\u00faber.\n\nY, en efecto, al principio, el joven Ezequiel Delf\u00edn s\u00f3lo trajo contento a la familia Bergua. Era inapetente y educado, pagaba sus cuentas con puntualidad, y ten\u00eda gestos simp\u00e1ticos como aparecer de cuando en cuando con unas violetas para do\u00f1a Margarita, un clavel para el ojal de don Sebasti\u00e1n y regalar unas partituras y un metr\u00f3nomo a Rosa en su cumplea\u00f1os. Su timidez, que no le permit\u00eda dirigir la palabra a nadie si no se la dirig\u00edan a \u00e9l antes, y, en estos casos, hablar siempre en voz baja y con los ojos en el suelo, jam\u00e1s en la cara de su interlocutor, y su correcci\u00f3n de maneras y de vocabulario cayeron muy en gracia a los Bergua, que pronto tomaron afecto al hu\u00e9sped, y, tal vez, en el fondo de sus corazones, familia ganada por la vida a la filosof\u00eda del mal menor, comenzaron a acariciar el proyecto de, con el tiempo, promoverlo a yerno.\n\nDon Sebasti\u00e1n, en especial, se encari\u00f1\u00f3 mucho con \u00e9l: \u00bfengre\u00eda tal vez en el delicado viajante a ese hijo que la diligente cojita no le hab\u00eda sabido dar? Una tarde de diciembre lo llev\u00f3 paseando hasta la ermita de Santa Rosa de Lima, donde lo vio tirar una dorada moneda al pozo y pedir una secreta gracia, y cierto domingo de ardiente verano le convid\u00f3 una raspadilla de c\u00edtricos en los portales de la plaza San Mart\u00edn. El muchacho le parec\u00eda elegante, por lo callado y melanc\u00f3lico. \u00bfTen\u00eda alguna misteriosa enfermedad del alma o del cuerpo que lo devoraba, alguna irresta\u00f1able herida de amor? Ezequiel Delf\u00edn era una tumba y, cuando, alguna vez, con las debidas precauciones, los Bergua se hab\u00edan ofrecido como pa\u00f1o de l\u00e1grimas y le hab\u00edan preguntado por qu\u00e9, siendo tan joven, estaba siempre solo, por qu\u00e9 jam\u00e1s iba a una fiesta, a un cine, por qu\u00e9 no se re\u00eda, por qu\u00e9 suspiraba tanto con la mirada perdida en el vac\u00edo, \u00e9l se limitaba a ruborizarse y, balbuceando una disculpa, corr\u00eda a encerrarse en el ba\u00f1o, donde pasaba a veces horas con el pretexto de la constipaci\u00f3n. Iba y ven\u00eda de sus viajes de trabajo como una verdadera esfinge \u2014la familia nunca pudo enterarse siquiera a qu\u00e9 industria serv\u00eda, qu\u00e9 vend\u00eda\u2014 y aqu\u00ed, en Lima, cuando no trabajaba, permanec\u00eda encerrado en su cuarto, \u00bfrezando su Biblia o dedicado a la meditaci\u00f3n? Celestinescos y compadecidos, do\u00f1a Margarita y don Sebasti\u00e1n lo animaban a que asistiera a los ejercicios de piano de Rosita para que se distrajera, y \u00e9l obedec\u00eda: inm\u00f3vil y atento en un rinc\u00f3n de la sala, escuchaba, y, al final, aplaud\u00eda con urbanidad. Muchas veces acompa\u00f1\u00f3 a don Sebasti\u00e1n a sus misas matutinas, y la Semana Santa de ese a\u00f1o hizo el recorrido de las Estaciones con los Bergua. Para entonces ya parec\u00eda miembro de la familia.\n\nFue por eso que el d\u00eda en que Ezequiel, reci\u00e9n regresado de un viaje al norte, rompi\u00f3 s\u00fabitamente a sollozar en medio del almuerzo, haciendo dar un respingo a los dem\u00e1s pensionistas \u2014un juez de paz ancashino, un p\u00e1rroco de Cajatambo y dos chicas de Hu\u00e1nuco, estudiantes de Enfermer\u00eda\u2014 y volcando en la mesa la magra raci\u00f3n de lentejas que le acababan de servir, los Bergua se alarmaron mucho. Entre los tres lo acompa\u00f1aron a su cuarto, don Sebasti\u00e1n le prest\u00f3 su pa\u00f1uelo, do\u00f1a Margarita le prepar\u00f3 una infusi\u00f3n de yerbaluisa y menta y Rosa le abrig\u00f3 los pies con una manta. Ezequiel Delf\u00edn se seren\u00f3 al cabo de unos minutos, pidi\u00f3 disculpas por su debilidad, explic\u00f3 que estaba \u00faltimamente muy nervioso, que no sab\u00eda por qu\u00e9, pero con mucha frecuencia, a cualquier hora y en cualquier sitio, se le escapaban las l\u00e1grimas. Avergonzado, casi sin voz, les revel\u00f3 que en las noches ten\u00eda accesos de terror: permanec\u00eda hasta el amanecer encogido y desvelado, sudando fr\u00edo, pensando en aparecidos, y compadeci\u00e9ndose a s\u00ed mismo de su soledad. Su confesi\u00f3n hizo lagrimear a Rosa y santiguarse a la cojita. Don Sebasti\u00e1n se ofreci\u00f3 \u00e9l mismo a dormir en el cuarto para inspirar confianza y alivio al asustado. \u00c9ste, en agradecimiento, le bes\u00f3 las manos.\n\nUna cama fue arrastrada al cuarto y diligentemente ali\u00f1ada por do\u00f1a Margarita y su hija. Don Sebasti\u00e1n estaba en ese tiempo en la flor de la edad, la cincuentena, y acostumbraba, antes de meterse a la cama, hacer medio centenar de abdominales (hac\u00eda sus ejercicios al acostarse y no al despertar para distinguirse tambi\u00e9n en eso del vulgo), pero esa noche, para no turbar a Ezequiel, se abstuvo. El nervioso se hab\u00eda acostado temprano, despu\u00e9s de cenar un cari\u00f1oso caldito de menudencias, y asegurar que la compa\u00f1\u00eda de don Sebasti\u00e1n lo hab\u00eda serenado de antemano y que estaba seguro de dormir como una marmota.\n\nNunca m\u00e1s se borrar\u00edan de la memoria del gentilhombre ayacuchano los pormenores de esa noche: en la vigilia y en el sue\u00f1o lo acosar\u00edan hasta el final de sus d\u00edas y, qui\u00e9n sabe, a lo mejor lo seguir\u00edan persiguiendo en su pr\u00f3ximo estadio vital. Hab\u00eda apagado la luz temprano, hab\u00eda sentido en la cama vecina la respiraci\u00f3n pausada del sensible y pensado, satisfecho: \u00abSe ha dormido\u00bb. Sent\u00eda que tambi\u00e9n lo iba ganando el sue\u00f1o y hab\u00eda o\u00eddo las campanas de la catedral y la lejana carcajada de un borracho. Luego, se durmi\u00f3 y pl\u00e1cidamente so\u00f1\u00f3 el m\u00e1s grato y reconfortante de los sue\u00f1os: en un castillo puntiagudo, arborescente de escudos, pergaminos, flores her\u00e1ldicas y \u00e1rboles geneal\u00f3gicos que segu\u00edan la pista de sus antepasados hasta Ad\u00e1n, el Se\u00f1or de Ayacucho (\u00a1era \u00e9l!) recib\u00eda cuantioso tributo y fervorosa pleites\u00eda de muchedumbres de indios piojosos, que engordaban simult\u00e1neamente sus arcas y su vanidad.\n\nDe pronto, \u00bfhab\u00edan pasado quince minutos o tres horas?, algo que pod\u00eda ser un ruido, un presentimiento, el traspi\u00e9s de un esp\u00edritu, lo despert\u00f3. Alcanz\u00f3 a divisar, en la oscuridad apenas aliviada por una hebra de luz callejera que divid\u00eda la cortina, una silueta que, desde la cama contigua, se alzaba y silenciosamente flotaba hacia la puerta. Semiaturdido por el sue\u00f1o, supuso que el joven estre\u00f1ido iba al excusado a pujar, o que hab\u00eda vuelto a sentirse mal, y a media voz pregunt\u00f3: \u00ab\u00bfEzequiel, est\u00e1 usted bien?\u00bb. En vez de una respuesta, oy\u00f3, clar\u00edsimo, el pestillo de la puerta (que estaba aherrumbrada y chirriaba). No comprendi\u00f3, se incorpor\u00f3 algo en la cama y, ligeramente sobresaltado, volvi\u00f3 a preguntar: \u00ab\u00bfLe pasa algo, Ezequiel, puedo ayudarlo?\u00bb. Sinti\u00f3 entonces que el joven, hombres gato tan el\u00e1sticos que parecen ubicuos, hab\u00eda regresado y estaba ahora all\u00ed, de pie junto a su lecho, obstruyendo el rayito de luz de la ventana. \u00abPero, cont\u00e9steme, Ezequiel, qu\u00e9 le ocurre\u00bb, murmur\u00f3, buscando a tientas el interruptor de la lamparilla. En ese instante recibi\u00f3 la primera cuchillada, la m\u00e1s profunda y hurgadora, la que se hundi\u00f3 en su plexo como si fuera mantequilla y le trepan\u00f3 una clav\u00edcula. \u00c9l estaba seguro de haber gritado, pedido socorro a voces, y, mientras trataba de defenderse, de desenredarse de las s\u00e1banas que se le enroscaban en los pies, se sent\u00eda sorprendido de que ni su mujer ni su hija ni los otros pensionistas acudieran. Pero, en la realidad, nadie oy\u00f3 nada. M\u00e1s tarde, mientras la polic\u00eda y el juez reconstru\u00edan la carnicer\u00eda, todos se hab\u00edan asombrado de que no hubiera podido desarmar al criminal, siendo \u00e9l un robusto y Ezequiel un enclenque. No pod\u00edan saber que, en las tinieblas ensangrentadas, el propagandista m\u00e9dico parec\u00eda pose\u00eddo de una fuerza sobrenatural: don Sebasti\u00e1n s\u00f3lo atinaba a dar gritos imaginarios y a tratar de adivinar la traves\u00eda de la siguiente cuchillada para atajarla con las manos.\n\nRecibi\u00f3 entre catorce o quince (los m\u00e9dicos pensaban que la boca abierta en la nalga siniestra pod\u00eda ser, coincidencias portentosas que encanecen a un hombre en una noche y hacen creer en Dios, dos cuchilladas en el mismo sitio), equitativamente distribuidas a lo largo y ancho de su cuerpo, con excepci\u00f3n de su cara, la que \u2014\u00bfmilagro del Se\u00f1or de Limpias como pensaba do\u00f1a Margarita o de santa Rosa como dec\u00eda su tocaya?\u2014 no recibi\u00f3 ni un rasgu\u00f1o. El cuchillo, se averigu\u00f3 despu\u00e9s, era de la familia Bergua, filuda hoja de quince cent\u00edmetros que hab\u00eda desaparecido misteriosamente de la cocina hac\u00eda una semana y que dej\u00f3 el cuerpo del hombre de Ayacucho m\u00e1s cicatrizado y comido que el de un espadach\u00edn.\n\n\u00bfA qu\u00e9 se debi\u00f3 que no muriera? A la casualidad, a la misericordia de Dios y (sobre todo) a una cuasi tragedia mayor. Nadie hab\u00eda o\u00eddo, don Sebasti\u00e1n con catorce \u2014\u00bfquince?\u2014 pu\u00f1aladas en el cuerpo acababa de perder el sentido y se desaguaba en la oscuridad, el impulsivo pod\u00eda haber ganado la calle y desaparecido para siempre. Pero, como a tantos famosos de la Historia, lo perdi\u00f3 un capricho extravagante. Concluida la resistencia de su v\u00edctima, Ezequiel Delf\u00edn solt\u00f3 el cuchillo y, en vez de vestirse, se desvisti\u00f3. Desnudo como hab\u00eda venido al mundo, abri\u00f3 la puerta, cruz\u00f3 el pasillo, se present\u00f3 en el cuarto de do\u00f1a Margarita Bergua y, sin m\u00e1s explicaciones, se lanz\u00f3 sobre la cama con la inequ\u00edvoca intenci\u00f3n de fornicarla. \u00bfPor qu\u00e9 a ella? \u00bfPor qu\u00e9 pretender estuprar a una dama, de abolengo, s\u00ed, pero cincuentona y piernicorta, menuda, amorfa y, en suma, para cualquier est\u00e9tica conocida, fea sin atenuantes ni remedio? \u00bfPor qu\u00e9 no haber intentado, m\u00e1s bien, coger el fruto prohibido de la pianista adolescente, que, adem\u00e1s de ser virgen, ten\u00eda el aliento fuerte, las grenchas negr\u00edsimas y la piel alabastrina? \u00bfPor qu\u00e9 no haber intentado transgredir el serrallo secreto de las enfermeras huanuque\u00f1as, que eran veintea\u00f1eras y, probablemente, de carnes prietas y gustosas? Fueron estas humillantes consideraciones las que llevaron al Poder Judicial a aceptar la tesis de la defensa seg\u00fan la cual Ezequiel Delf\u00edn estaba trastornado y a mandarlo al Larco Herrera en vez de encerrarlo en la c\u00e1rcel.\n\nAl recibir la inesperada y galante visita del joven, la se\u00f1ora Margarita Bergua comprendi\u00f3 que algo grav\u00edsimo ocurr\u00eda. Era una mujer realista y no se hac\u00eda ilusiones sobre sus encantos: \u00abA m\u00ed no vienen a violarme ni en sue\u00f1os, ah\u00ed mismo supe que el calato era demente o criminal\u00bb, declar\u00f3. Se defendi\u00f3, pues, como una leona embravecida \u2014en su testimonio jur\u00f3 por la Virgen que el fogoso no hab\u00eda conseguido infligirle ni un \u00f3sculo\u2014 y, adem\u00e1s de impedir el ultraje a su honor, salv\u00f3 la vida a su marido. Al mismo tiempo que, a rasgu\u00f1os, mordiscos, codazos, rodillazos, manten\u00eda a raya al degenerado, daba gritos (ella s\u00ed) que despertaron a su hija y a los otros inquilinos. Entre Rosa, el juez ancashino, el p\u00e1rroco de Cajatambo y las enfermeras huanuque\u00f1as redujeron al exhibicionista, lo amarraron y todos juntos corrieron en busca de don Sebasti\u00e1n: \u00bfviv\u00eda?\n\nLes tom\u00f3 cerca de una hora conseguir una ambulancia que lo llevara al Hospital Arzobispo Loayza, y cerca de tres que viniera la polic\u00eda a salvar a Lucho Abril Marroqu\u00edn de las u\u00f1as de la joven pianista, quien, fuera de s\u00ed (\u00bfpor las heridas infligidas a su padre?, \u00bfpor la ofensa a su madre?, \u00bftal vez, alma humana de turbia pulpa y ponzo\u00f1osas esquinas, por el desaire hecho a ella?), pretend\u00eda sacarle los ojos y beberse su sangre. El joven propagandista m\u00e9dico, en la polic\u00eda, recobrando su tradicional suavidad de gestos y de voz, ruboriz\u00e1ndose al hablar de puro t\u00edmido, neg\u00f3 firmemente la evidencia. La familia Bergua y los pensionistas lo calumniaban: jam\u00e1s hab\u00eda agredido a nadie, nunca hab\u00eda pretendido violentar a una mujer y, much\u00edsimo menos, a una lisiada como Margarita Bergua, dama que, por sus bondades y consideraciones, era \u2014despu\u00e9s, claro est\u00e1, de su esposa, esa muchacha de ojos italianos y codos y rodillas musicales que ven\u00eda del pa\u00eds del canto y del amor\u2014 la persona que m\u00e1s respetaba y quer\u00eda en este mundo. Su serenidad, su urbanidad, su mansedumbre, las magn\u00edficas referencias que dieron de \u00e9l sus jefes y compa\u00f1eros de los Laboratorios Bayer, la albura de su registro policial, hicieron vacilar a los custodios del orden. \u00bfCab\u00eda, magia insondable de las apariencias tramposas, que todo fuera una conjuraci\u00f3n de la mujer e hija de la v\u00edctima y de los pensionistas contra este mozo delicado? El cuarto poder del Estado vio esta tesis con simpat\u00eda y la auspici\u00f3.\n\nPara dificultar las cosas y mantener el suspenso en la ciudad, el objeto del delito, don Sebasti\u00e1n Bergua, no pod\u00eda aclarar las dudas, pues se debat\u00eda entre la vida y la muerte en el popular nosocomio de la avenida Alfonso Ugarte. Recib\u00eda caudalosas transfusiones de sangre, que pusieron al borde de la tuberculosis a muchos comprovincianos del Club Tambo-Ayacucho, quienes, apenas enterados de la tragedia, hab\u00edan corrido a ofrecerse como donantes, y estas transferencias, m\u00e1s los sueros, las costuras, las desinfecciones, los vendajes, las enfermeras que se turnaban a su cabecera, los facultativos que soldaron sus huesos, reconstruyeron sus \u00f3rganos y apaciguaron sus nervios, devoraron en unas cuantas semanas las ya mermadas (por la inflaci\u00f3n y el galopante costo de la vida) rentas de la familia. \u00c9sta debi\u00f3 malbaratar sus bonos, recortar y alquilar a pedazos su propiedad y arrinconarse en ese segundo piso donde ahora vegetaba.\n\nDon Sebasti\u00e1n se salv\u00f3, s\u00ed, pero su recuperaci\u00f3n, en un principio, no pareci\u00f3 ser suficiente para zanjar las dudas policiales. Por efecto de las cuchilladas, del susto sufrido, o de la deshonra moral de su mujer, qued\u00f3 mudo (y hasta se murmuraba que tonto). Era incapaz de pronunciar palabra, miraba todo y a todos con let\u00e1rgica inexpresividad de tortuga, y tampoco los dedos le obedec\u00edan pues ni siquiera pudo (\u00bfquiso?) contestar por escrito las preguntas que se le hicieron en el juicio del desatinado.\n\nEl proceso alcanz\u00f3 proporciones may\u00fasculas y la Ciudad de los Reyes permaneci\u00f3 en vilo mientras duraron las audiencias. Lima, el Per\u00fa, \u00bfla Am\u00e9rica mestiza toda?, siguieron con apasionamiento las discusiones forenses, las r\u00e9plicas y contrarr\u00e9plicas de los peritos, los alegatos del fiscal y del abogado defensor, un famoso jurisconsulto venido especialmente desde Roma, la ciudad m\u00e1rmol, a defender a Lucho Abril Marroqu\u00edn, por ser \u00e9ste esposo de una italianita que, adem\u00e1s de compatriota suya, era su hija.\n\nEl pa\u00eds se dividi\u00f3 en dos bandos. Los convencidos de la inocencia del propagandista m\u00e9dico \u2014los diarios todos\u2014 sosten\u00edan que don Sebasti\u00e1n hab\u00eda estado a punto de ser v\u00edctima de su esposa y de su v\u00e1staga, coludidas con el juez ancashino, el curita de Cajatambo y las enfermeras huanuque\u00f1as, sin duda con fines de herencia y lucro. El jurisconsulto romano defendi\u00f3 imperialmente esta tesis asegurando que, advertidos de la demencia apacible de Lucho Abril Marroqu\u00edn, familia y pensionistas se hab\u00edan conjurado para endosarle el crimen (\u00bfo inducirlo tal vez a cometerlo?). Y fue acumulando argumentos que los \u00f3rganos de prensa magnificaban, aplaud\u00edan y consagraban como demostrados: \u00bfalguien en su sano juicio pod\u00eda creer que un hombre recibe catorce y tal vez quince cuchilladas en respetuoso silencio? \u00bfY si, como era l\u00f3gico, don Sebasti\u00e1n Bergua hab\u00eda aullado de dolor, alguien en su sano juicio pod\u00eda creer que ni la esposa, ni la hija, ni el juez, ni el cura, ni las enfermeras escucharan esos gritos, siendo las paredes de la Pensi\u00f3n Colonial tabiques de ca\u00f1a y barro que dejaban pasar el zumbido de las moscas y las pisadas de un alacr\u00e1n? \u00bfY c\u00f3mo era posible que, siendo las pensionistas de Hu\u00e1nuco estudiantes de Enfermer\u00eda de notas altas, no hubieran atinado a prestar al herido los primeros auxilios, esperando imp\u00e1vidas, mientras el gentilhombre se desangraba, que llegara la ambulancia? \u00bfY c\u00f3mo era posible que en ninguna de las seis personas adultas, viendo que la ambulancia tardaba, hubiera germinado la idea, elemental incluso para un oligofr\u00e9nico, de buscar un taxi, habiendo un paradero de taxis en la misma esquina de la Pensi\u00f3n Colonial? \u00bfNo era todo eso extra\u00f1o, tortuoso, indicador?\n\nA los tres meses de permanecer retenido en Lima, al curita de Cajatambo, que hab\u00eda venido a la capital s\u00f3lo por cuatro d\u00edas a gestionar un nuevo Cristo para la iglesia de su pueblo porque al anterior los palomillas lo hab\u00edan decapitado a hondazos, convulso ante la perspectiva de ser condenado por intento de homicidio y pasar el resto de sus d\u00edas en la c\u00e1rcel, le estall\u00f3 el coraz\u00f3n y muri\u00f3. Su muerte electriz\u00f3 a la opini\u00f3n y tuvo un efecto devastador para la defensa; los diarios, ahora, volvieron la espalda al jurisconsulto importado, lo acusaron de casu\u00edstico, oper\u00e1tico, colonialista y peregrino, y de haber causado por sus insinuaciones sibilinas y anticristianas la muerte de un buen pastor, y los jueces, docilidad de ca\u00f1averales que bailan con los vientos period\u00edsticos, lo desaforaron por extranjero, lo privaron del derecho de alegar ante los tribunales, y, en un fallo que los diarios celebraron con trinos nacionalistas, lo devolvieron indeseablemente a Italia.\n\nLa muerte del curita cajatambino salv\u00f3 a la madre y a la hija y a los inquilinos de una probable condena por semihomicidio y encubrimiento criminal. Al comp\u00e1s de la prensa y la opini\u00f3n, el fiscal torn\u00f3 a simpatizar con las Bergua y acept\u00f3, como al principio, su versi\u00f3n de los acontecimientos. El nuevo abogado de Lucho Abril Marroqu\u00edn, un jurista nativo, cambi\u00f3 radicalmente de estrategia: reconoci\u00f3 que su defendido hab\u00eda cometido los delitos, pero aleg\u00f3 su irresponsabilidad total, por causa de paropsia y raquitismo an\u00edmicos, combinados con esquizofrenia y otras veleidades en el dominio de la patolog\u00eda mental que destacados psiquiatras corroboraron en amenas deposiciones. All\u00ed se argument\u00f3, como prueba definitiva de desquicio, que el inculpado, entre las cuatro mujeres de la Pensi\u00f3n Colonial, hubiera elegido a la m\u00e1s anciana y la \u00fanica coja. Durante el \u00faltimo alegato del fiscal, cl\u00edmax dram\u00e1tico que diviniza a los actores y escalofr\u00eda al p\u00fablico, don Sebasti\u00e1n, que hasta entonces hab\u00eda permanecido silente y lega\u00f1oso en una silla, como si el juicio no le concerniera, levant\u00f3 despacio una mano y, con los ojos enrojecidos por el esfuerzo, la c\u00f3lera o la humillaci\u00f3n, se\u00f1al\u00f3 fijamente, durante un minuto verificado por cron\u00f3metro (un periodista dixit) a Lucho Abril Marroqu\u00edn. El gesto fue reputado tan extraordinario como si la estatua ecuestre de Sim\u00f3n Bol\u00edvar se hubiera puesto efectivamente a cabalgar... La corte acept\u00f3 todas las tesis del fiscal y Lucho Abril Marroqu\u00edn fue encerrado en el manicomio.\n\nLa familia Bergua no levant\u00f3 cabeza m\u00e1s. Comenz\u00f3 su desmoronamiento material y moral. Arruinados por cl\u00ednicas y leguleyos, debieron renunciar a las clases de piano (y, por lo tanto, a la ambici\u00f3n de convertir a Rosa en artista mundial) y reducir su nivel de vida a extremos que lindaban con las malas costumbres del ayuno y la suciedad. La vieja casona envejeci\u00f3 a\u00fan m\u00e1s y el polvo fue impregn\u00e1ndola y las telara\u00f1as invadi\u00e9ndola y las polillas comi\u00e9ndola; su clientela disminuy\u00f3 y fue bajando de categor\u00eda hasta llegar a la sirvienta y el cargador. Toc\u00f3 fondo el d\u00eda en que un mendigo vino a golpear la puerta y pregunt\u00f3, terriblemente: \u00ab\u00bfEs aqu\u00ed el Dormidero Colonial?\u00bb.\n\nAs\u00ed, d\u00eda que persigue a d\u00eda, mes que sucede a otro mes, llegaron a pasar treinta a\u00f1os.\n\nLa familia Bergua parec\u00eda ya aclimatada a la mediocridad cuando algo vino, de pronto, bomba at\u00f3mica que una madrugada desintegra ciudades japonesas, a ponerla en efervescencia. Hac\u00eda muchos a\u00f1os que no funcionaba la radio y otros tantos que el presupuesto familiar imped\u00eda comprar peri\u00f3dicos. Las noticias del mundo no llegaban, pues, a los Bergua sino rara vez y de rebote, a trav\u00e9s de comentarios y chismes de sus incultivados hu\u00e9spedes.\n\nPero esa tarde, qu\u00e9 casualidad, un camionero de Castrovirreyna solt\u00f3 una carcajada vulgar con un escupitajo verde, murmur\u00f3 \u00ab\u00a1el chiflado es de rifarlo!\u00bb y tir\u00f3 sobre la ara\u00f1ada mesita de la sala el ejemplar de \u00daltima Hora que acababa de leer. La ex pianista lo recogi\u00f3, lo hoje\u00f3. De repente, palidez de mujer que ha recibido el beso del vampiro, corri\u00f3 al cuarto llamando a voces a su madre. Juntas leyeron y releyeron la estrujada noticia, y luego, a gritos, turn\u00e1ndose, se la leyeron a don Sebasti\u00e1n, quien, sin la menor duda, comprendi\u00f3, pues al instante contrajo una de esas sonoras crisis que lo hac\u00edan hipar, sudar, llorar a gritos y revolverse como un poseso.\n\n\u00bfQu\u00e9 noticia provocaba semejante alarma en esa familia crepuscular?\n\nEn el amanecer de la v\u00edspera, en un concurrido pabell\u00f3n del Hospital Psiqui\u00e1trico V\u00edctor Larco Herrera, de Magdalena del Mar, un pupilo que hab\u00eda pasado entre esos muros el tiempo de una jubilaci\u00f3n, hab\u00eda degollado a un enfermero con un bistur\u00ed, ahorcado a un anciano catat\u00f3nico que dorm\u00eda en una cama contigua a la suya y huido a la ciudad saltando gimn\u00e1sticamente el muro de la Costanera. Su proceder caus\u00f3 sorpresa porque hab\u00eda sido siempre ejemplarmente pac\u00edfico y jam\u00e1s se le vio un gesto de malhumor ni se le oy\u00f3 levantar la voz. Su \u00fanica ocupaci\u00f3n, en treinta a\u00f1os, hab\u00eda sido oficiar misas imaginarias al Se\u00f1or de Limpias y repartir hostias invisibles a inexistentes comulgantes. Antes de huir del hospital, Lucho Abril Marroqu\u00edn \u2014que acababa de cumplir la edad egregia del hombre: cincuenta a\u00f1os\u2014 hab\u00eda escrito una educada esquela de adi\u00f3s: \u00abLo siento pero no tengo m\u00e1s remedio que salir. Me espera un incendio en una vieja casa de Lima, donde una cojita ardiente como una antorcha y su familia ofenden mortalmente a Dios. He recibido la encomienda de apagar las llamas\u00bb.\n\n\u00bfLo har\u00eda? \u00bfLas apagar\u00eda? \u00bfSe aparecer\u00eda ese resucitado del fondo de los a\u00f1os para, por segunda vez, hundir a los Bergua en el horror as\u00ed como ahora los hab\u00eda hundido en el miedo? \u00bfC\u00f3mo terminar\u00eda la empavorecida familia de Ayacucho?\n\n### XIII\n\nLA MEMORABLE semana comenz\u00f3 con un pintoresco episodio (sin las caracter\u00edsticas violentas del encuentro con los churrasqueros) del que fui testigo y a medias protagonista. Genaro hijo se pasaba la vida haciendo innovaciones en los programas y decidi\u00f3 un d\u00eda que, para agilizar los boletines, deb\u00edamos acompa\u00f1arlos de entrevistas. Nos puso en acci\u00f3n a Pascual y a m\u00ed, y, desde entonces, comenzamos a radiar una entrevista diaria, sobre alg\u00fan tema de actualidad, en El Panamericano de la noche. Signific\u00f3 m\u00e1s trabajo para el Servicio de Informaciones (sin aumento de sueldo) pero no lo lament\u00e9, porque era entretenido. Interrogando en el estudio de la calle Bel\u00e9n o ante una grabadora, a artistas de cabaret y a parlamentarios, a futbolistas y a ni\u00f1os prodigio, aprend\u00ed que todo el mundo, sin excepci\u00f3n, pod\u00eda ser tema de cuento.\n\nAntes del pintoresco episodio, el personaje m\u00e1s curioso que entrevist\u00e9 fue un torero venezolano. Esa temporada en la plaza de Acho hab\u00eda tenido un \u00e9xito descomunal. En su primera corrida cort\u00f3 varias orejas y, en la segunda, despu\u00e9s de una faena milagrosa, le dieron una pata y la muchedumbre lo llev\u00f3 en hombros desde el R\u00edmac hasta su hotel, en la plaza San Mart\u00edn. Pero en su tercera y \u00faltima corrida \u2014las entradas se hab\u00edan revendido, por \u00e9l, a precios astron\u00f3micos\u2014 no lleg\u00f3 a ver los toros, porque, presa de p\u00e1nico cerval, corri\u00f3 de ellos toda la tarde; no les hizo un solo pase digno y los mat\u00f3 a pocos, al extremo de que en el segundo le tocaron cuatro avisos. La bronca en los tendidos fue may\u00fascula: intentaron quemar la plaza de Acho y linchar al venezolano, quien, en medio de gran rechifla y lluvia de cojines, debi\u00f3 ser escoltado hasta su hotel por la Guardia Civil. A la ma\u00f1ana siguiente, horas antes de que tomara el avi\u00f3n, lo entrevist\u00e9 en un saloncito del Hotel Bol\u00edvar. Me dej\u00f3 perplejo comprobar que era menos inteligente que los toros que lidiaba y casi tan incapaz como ellos de expresarse mediante la palabra. No pod\u00eda construir una frase coherente, jam\u00e1s acertaba con los tiempos verbales, su manera de coordinar las ideas hac\u00eda pensar en tumores, en afasia, en hombres mono. La forma era no menos extraordinaria que el fondo: hablaba con un acento infeliz, hecho de diminutivos y ap\u00f3copes, que matizaba, durante sus frecuentes vac\u00edos mentales, con gru\u00f1idos zool\u00f3gicos.\n\nEl mexicano que me toc\u00f3 entrevistar el lunes de la semana memorable era, por el contrario, un hombre l\u00facido y un desenvuelto expositor. Dirig\u00eda una revista, hab\u00eda escrito libros sobre la revoluci\u00f3n mexicana, presid\u00eda una delegaci\u00f3n de economistas y estaba alojado en el Bol\u00edvar. Acept\u00f3 venir a la radio y lo fui a buscar yo mismo. Era un caballero alto y derecho, bien vestido, de cabellos blancos, que deb\u00eda andar por los sesenta. Lo acompa\u00f1\u00f3 su se\u00f1ora, una mujer de ojos vivos, menuda, que llevaba un sombrerito de flores. Entre el hotel y la radio preparamos la entrevista y \u00e9sta qued\u00f3 grabada en quince minutos, ante la alarma de Genaro hijo, porque el economista e historiador, en respuesta a una pregunta, atac\u00f3 duramente a las dictaduras militares (en el Per\u00fa padec\u00edamos una, encabezada por un tal Odr\u00eda).\n\nSucedi\u00f3 cuando acompa\u00f1aba a la pareja de regreso al Bol\u00edvar. Era mediod\u00eda y la calle Bel\u00e9n y la plaza San Mart\u00edn rebalsaban de gente. La se\u00f1ora ocupaba la vereda, su marido el centro y yo iba al lado de la pista. Acab\u00e1bamos de pasar frente a Radio Central, y, por decir algo, le repet\u00eda al hombre importante que la entrevista hab\u00eda quedado magn\u00edfica, cuando fui clar\u00edsimamente interrumpido por la vocecita de la dama mexicana:\n\n\u2014Jes\u00fas, Jes\u00fas, me descompongo...\n\nLa mir\u00e9 y la vi demacrada, abriendo y cerrando los ojos y moviendo la boca de manera rar\u00edsima. Pero lo sorprendente fue la reacci\u00f3n del economista e historiador. Al o\u00edr la advertencia, lanz\u00f3 una mirada veloz a su esposa, y me lanz\u00f3 otra a m\u00ed, con expresi\u00f3n confusa, y, al instante, mir\u00f3 de nuevo al frente, y, en lugar de detenerse, aceler\u00f3 el paso. La dama mexicana qued\u00f3 a mi lado, haciendo muecas. Alcanc\u00e9 a cogerla del brazo cuando se iba a desplomar. Como era tan fr\u00e1gil, felizmente, pude sostenerla y ayudarla, mientras el hombre importante hu\u00eda a trancos y me endosaba la delicada tarea de arrastrar a su mujer. La gente nos abr\u00eda paso, se paraba a mirarnos, y en una de \u00e9sas \u2014est\u00e1bamos a la altura del Cine Col\u00f3n y la damita mexicana, adem\u00e1s de hacer morisquetas, hab\u00eda comenzado a echar babas, mocos y l\u00e1grimas\u2014 o\u00ed que un vendedor de cigarrillos dec\u00eda: \u00abTambi\u00e9n se est\u00e1 meando\u00bb. Era verdad: la esposa del economista e historiador (que hab\u00eda cruzado la Colmena y desaparec\u00eda entre la gente agolpada a las puertas del bar del Bol\u00edvar) iba dejando una estela amarilla detr\u00e1s de nosotros. Al llegar a la esquina, no tuve m\u00e1s remedio que cargarla y avanzar as\u00ed, espectacular y galante, los cincuenta metros que faltaban, entre choferes que bocineaban, polic\u00edas que pitaban y gentes que nos se\u00f1alaban. En mis brazos, la damita mexicana se retorc\u00eda sin cesar, continuaba las muecas, y, en las manos y en la nariz, me parec\u00eda comprobar que, adem\u00e1s de pip\u00ed, se estaba haciendo algo m\u00e1s feo. Su garganta emit\u00eda un ruido atrofiado, intermitente. Al entrar al Bol\u00edvar, o\u00ed que me ordenaban, con sequedad: \u00abHabitaci\u00f3n 301\u00bb. Era el hombre importante: estaba medio escondido, detr\u00e1s de unas cortinas. Apenas me dio la orden, volvi\u00f3 a escapar, a alejarse a paso ligero hacia el ascensor, y, mientras sub\u00edamos, ni una vez se dign\u00f3 mirarme o mirar a su consorte, como si no quisiera parecer impertinente. El ascensorista me ayud\u00f3 a llevar a la dama hasta la habitaci\u00f3n. Pero, apenas la depositamos en la cama, el hombre importante nos empuj\u00f3 literalmente hasta la puerta y, sin decir gracias ni adi\u00f3s, nos la cerr\u00f3 con brutalidad en las narices; ten\u00eda en ese momento una expresi\u00f3n salobre.\n\n\u2014No es un mal marido \u2014me explicar\u00eda despu\u00e9s Pedro Camacho\u2014, sino un tipo sensible y con gran sentido del papel\u00f3n.\n\nEsa tarde yo deb\u00eda leerles a la t\u00eda Julia y a Javier un cuento que acababa de terminar: La t\u00eda Eliana. El Comercio no public\u00f3 nunca la historia de los levitadores y me consol\u00e9 escribiendo otra historia, basada en algo que hab\u00eda ocurrido en mi familia. Eliana era una de las muchas t\u00edas que aparec\u00edan por la casa cuando era ni\u00f1o y yo la prefer\u00eda a las otras porque me tra\u00eda chocolates y algunas veces me llevaba a tomar t\u00e9 al Cream Rica. Su afici\u00f3n a los dulces era motivo de burla en las reuniones de la tribu, donde se dec\u00eda que se gastaba todos sus sueldos de secretaria en los pasteles cremosos, las medialunas crujientes, las tortas esponjosas y el chocolate espeso de La Tiendecita Blanca. Era una gordita cari\u00f1osa, risue\u00f1a y parlanchina, y yo tomaba su defensa cuando en la familia, a sus espaldas, comentaban que se estaba quedando para vestir santos. Un d\u00eda, misteriosamente, la t\u00eda Eliana dej\u00f3 de aparecer por la casa y la familia no volvi\u00f3 a nombrarla. Yo tendr\u00eda entonces seis o siete a\u00f1os y recuerdo haber sentido desconfianza ante las respuestas de los parientes cuando les preguntaba por ella: se ha ido de viaje, estaba enferma, ya vendr\u00eda cualquier d\u00eda de \u00e9stos. Unos cinco a\u00f1os despu\u00e9s, la familia entera, de pronto, se visti\u00f3 de luto, y esa noche, en casa de los abuelos, supe que hab\u00edan asistido al entierro de la t\u00eda Eliana, que acababa de morir de c\u00e1ncer. Entonces se aclar\u00f3 el misterio. La t\u00eda Eliana, cuando parec\u00eda condenada a la solter\u00eda, se hab\u00eda casado intempestivamente con un chino, due\u00f1o de una bodega en Jes\u00fas Mar\u00eda, y la familia, empezando por sus padres, horrorizada ante el esc\u00e1ndalo \u2014entonces cre\u00ed que lo escandaloso era que el marido fuese chino, pero ahora deduzco que su tara principal era ser bodeguero\u2014 hab\u00eda decretado su muerte en vida y no la hab\u00eda visitado ni recibido jam\u00e1s. Pero cuando se muri\u00f3 la perdonaron \u2014\u00e9ramos una familia de gentes sentimentales, en el fondo\u2014, fueron a su velorio y a su entierro, y derramaron muchas l\u00e1grimas por ella.\n\nMi cuento era el mon\u00f3logo de un ni\u00f1o que, tendido en su cama, trataba de descifrar el misterio de la desaparici\u00f3n de su t\u00eda, y, como ep\u00edlogo, el velorio de la protagonista. Era un cuento social, cargado de ira contra los parientes prejuiciosos. Lo hab\u00eda escrito en un par de semanas y les habl\u00e9 tanto de \u00e9l a la t\u00eda Julia y a Javier que se rindieron y me pidieron que se lo leyera. Pero, antes de hacerlo, en la tarde de ese lunes, les cont\u00e9 lo ocurrido con la damita mexicana y el hombre importante. Fue un error que pagu\u00e9 caro, porque esta an\u00e9cdota les pareci\u00f3 mucho m\u00e1s divertida que mi cuento.\n\nSe hab\u00eda hecho una costumbre que la t\u00eda Julia viniera a Panamericana. Hab\u00edamos descubierto que era el sitio m\u00e1s seguro, ya que, de hecho, cont\u00e1bamos con la complicidad de Pascual y el Gran Pablito. Se aparec\u00eda despu\u00e9s de las cinco, hora en que comenzaba un periodo de calma: los Genaros se hab\u00edan ido y casi nadie ven\u00eda a merodear por el altillo. Mis compa\u00f1eros de trabajo, por un acuerdo t\u00e1cito, ped\u00edan permiso para tomar un cafecito, de modo que la t\u00eda Julia y yo pudi\u00e9ramos besarnos y hablar a solas. A veces yo me pon\u00eda a escribir y ella se quedaba leyendo una revista o charlando con Javier, quien, invariablemente, ven\u00eda a juntarse con nosotros a eso de las siete. Hab\u00edamos llegado a formar un grupo inseparable y mis amores con la t\u00eda Julia adquir\u00edan, en ese cuartito de tabiques, una naturalidad maravillosa. Pod\u00edamos estar de la mano o besarnos y a nadie le llamaba la atenci\u00f3n. Eso nos hac\u00eda felices. Franquear hacia adentro los l\u00edmites del altillo era ser libres, due\u00f1os de nuestros actos, pod\u00edamos querernos, hablar de lo que nos importaba y sentirnos rodeados de comprensi\u00f3n. Franquearlos hacia afuera era entrar en un dominio hostil, donde est\u00e1bamos obligados a mentir y a escondernos.\n\n\u2014\u00bfSe puede decir que esto es nuestro nido de amor? \u2014me preguntaba la t\u00eda Julia\u2014 \u00bfO tambi\u00e9n es huachafo?\n\n\u2014Por supuesto que es huachafo y que no se puede decir \u2014le respond\u00eda yo\u2014. Pero podemos ponerle Montmartre.\n\nJug\u00e1bamos al profesor y a la alumna y yo le explicaba lo que era huachafo, lo que no se pod\u00eda decir ni hacer y hab\u00eda establecido una censura inquisitorial en sus lecturas, prohibi\u00e9ndole todos sus autores favoritos, que empezaban por Frank Yerby y terminaban con Cor\u00edn Tellado. Nos divert\u00edamos como locos y a veces Javier interven\u00eda, con una dial\u00e9ctica fogosa, en el juego de la huachafer\u00eda.\n\nA la lectura de La t\u00eda Eliana asistieron tambi\u00e9n, porque estaban all\u00ed y no me atrev\u00ed a echarlos, Pascual y el Gran Pablito, y result\u00f3 una suerte, porque fueron los \u00fanicos que celebraron el cuento, aunque, como eran mis subordinados, su entusiasmo resultaba sospechoso. Javier lo encontr\u00f3 irreal, nadie creer\u00eda que una familia condena al ostracismo a una muchacha por casarse con un chino y me asegur\u00f3 que si el marido era negro o indio la historia pod\u00eda salvarse. La t\u00eda Julia me dio una estocada mortal dici\u00e9ndome que el cuento hab\u00eda salido melodram\u00e1tico y que algunas palabritas, como tr\u00e9mula y sollozante, le hab\u00edan sonado huachafas. Yo comenzaba a defender La t\u00eda Eliana cuando divis\u00e9 en la puerta del altillo a la flaca Nancy. Bastaba verla para saber a qu\u00e9 ven\u00eda:\n\n\u2014Ahora s\u00ed se arm\u00f3 la pelotera en la familia \u2014dijo, de un tir\u00f3n.\n\nPascual y el Gran Pablito, olfateando un buen chisme, adelantaron las cabezas. Contuve a mi prima, ped\u00ed a Pascual que preparara el bolet\u00edn de las nueve, y bajamos a tomar un caf\u00e9. En una mesa del Bransa nos detall\u00f3 la noticia. Hab\u00eda sorprendido, mientras se lavaba la cabeza, una conversaci\u00f3n telef\u00f3nica entre su madre y la t\u00eda Jes\u00fas. Se le hab\u00edan helado las u\u00f1as al o\u00edr hablar de la parejita y descubrir que se trataba de nosotros. No estaba muy claro, pero se hab\u00edan dado cuenta de nuestros amores hac\u00eda ya bastante tiempo, porque, en un momento, la t\u00eda Laura hab\u00eda dicho: \u00abY f\u00edjate que hasta Camunchita los vio una vez de la mano a los muy frescos, en el olivar de San Isidro\u00bb (algo que efectivamente hab\u00edamos hecho, una \u00fanica tarde, hac\u00eda meses). Al salir del ba\u00f1o (con \u00abtembladera\u00bb, dec\u00eda) la flaca Nancy se encontr\u00f3 cara a cara con su madre y hab\u00eda tratado de disimular, los o\u00eddos le zumbaban del ruido del secador de pelo, no pod\u00eda o\u00edr nada, pero la t\u00eda Laura la call\u00f3 y la ri\u00f1\u00f3 y la llam\u00f3 \u00abencubridora de esa perdida\u00bb.\n\n\u2014\u00bfLa perdida soy yo? \u2014pregunt\u00f3 la t\u00eda Julia, con m\u00e1s curiosidad que furia.\n\n\u2014S\u00ed, t\u00fa \u2014explic\u00f3 mi prima, poni\u00e9ndose colorada\u2014. Te creen la invencionera de todo esto.\n\n\u2014Es verdad, yo soy menor de edad, viv\u00eda tranquilo estudiando abogac\u00eda, hasta que \u2014dije yo, pero nadie me festej\u00f3.\n\n\u2014Si saben que les he contado, me matan \u2014dijo la flaca Nancy\u2014. No vayan a decir una palabra, j\u00farenlo por Dios.\n\nSus padres le hab\u00edan advertido formalmente que, si comet\u00eda cualquier infidencia, la encerrar\u00edan un a\u00f1o sin salir ni a misa. Le hab\u00edan hablado de manera tan solemne que hasta dud\u00f3 si contarnos. La familia sab\u00eda todo desde el principio y hab\u00eda guardado una actitud discreta pensando que era una tonter\u00eda, el coqueteo intrascendente de una mujer ligera de cascos que quer\u00eda anotar en su prontuario una conquista ex\u00f3tica, un adolescente. Pero como la t\u00eda Julia ya no ten\u00eda escr\u00fapulos en lucirse por calles y plazas con el mocoso y cada vez m\u00e1s gente amiga y m\u00e1s parientes descubr\u00edan estos amores \u2014hasta los abuelitos se hab\u00edan enterado, por un chisme de la t\u00eda Celia\u2014 y esto era una verg\u00fcenza y algo que ten\u00eda que estar perjudicando al flaquito (es decir yo), quien, desde que la divorciada le hab\u00eda llenado la cabeza de p\u00e1jaros, probablemente ya no tendr\u00eda \u00e1nimos ni para estudiar, la familia hab\u00eda decidido intervenir.\n\n\u2014\u00bfY qu\u00e9 van a hacer para salvarme? \u2014pregunt\u00e9, todav\u00eda sin demasiado susto.\n\n\u2014Escribirle a tus pap\u00e1s \u2014me contest\u00f3 la flaca Nancy\u2014. Ya lo hicieron. Los t\u00edos mayores: el t\u00edo Jorge y el t\u00edo Lucho.\n\nMis padres viv\u00edan en Estados Unidos y mi padre era un hombre severo al que yo le ten\u00eda mucho miedo. Me hab\u00eda criado lejos de \u00e9l, con mi madre y mi familia materna, y, cuando mis padres se reconciliaron y fui a vivir con \u00e9l, nos llevamos siempre mal. Era conservador y autoritario, de c\u00f3leras fr\u00edas, y, si era verdad que le hab\u00edan escrito, la noticia le iba a hacer el efecto de una bomba y su reacci\u00f3n ser\u00eda violenta. La t\u00eda Julia me cogi\u00f3 la mano por debajo de la mesa:\n\n\u2014Te has puesto p\u00e1lido, Varguitas. Ahora s\u00ed que tienes tema para un buen cuento.\n\n\u2014Lo mejor es conservar la cabeza en su sitio y el pulso firme \u2014me dio \u00e1nimos Javier\u2014. No te asustes y planeemos una buena estrategia para hacer frente al bolondr\u00f3n.\n\n\u2014Contigo tambi\u00e9n est\u00e1n furiosos \u2014le advirti\u00f3 Nancy\u2014. Tambi\u00e9n te creen esa palabrita fea.\n\n\u2014\u00bfAlcahuete? \u2014sonri\u00f3 la t\u00eda Julia. Y, volvi\u00e9ndose a m\u00ed, se puso triste\u2014: Lo que me importa es que van a separarnos y no te podr\u00e9 ver nunca m\u00e1s.\n\n\u2014Eso es huachafo y no se puede decir de ese modo \u2014le expliqu\u00e9.\n\n\u2014Qu\u00e9 bien han disimulado \u2014dijo la t\u00eda Julia\u2014. Ni mi hermana, ni mi cu\u00f1ado, ninguno de tus parientes me hicieron sospechar que sab\u00edan y que me detestaban. Siempre tan cari\u00f1osos conmigo, esos hip\u00f3critas.\n\n\u2014Por lo pronto, tienen que dejar de verse \u2014dijo Javier\u2014. Que Julita salga con galanes, t\u00fa invita a otras chicas. Que la familia crea que se han peleado.\n\nAlica\u00eddos, la t\u00eda Julia y yo convinimos en que era la \u00fanica soluci\u00f3n. Pero, cuando la flaca Nancy se fue \u2014le juramos que nunca la traicionar\u00edamos\u2014 y Javier parti\u00f3 tras ella y la t\u00eda Julia me acompa\u00f1\u00f3 hasta Panamericana, ambos, sin necesidad de decirlo, mientras baj\u00e1bamos cabizbajos y de la mano por la calle Bel\u00e9n, h\u00fameda de gar\u00faa, sab\u00edamos que esa estrategia pod\u00eda convertir la mentira en verdad. Si no nos ve\u00edamos, si cada uno sal\u00eda por su lado, lo nuestro, tarde o temprano, se terminar\u00eda. Quedamos en hablar por tel\u00e9fono todos los d\u00edas, a horas precisas, y nos despedimos bes\u00e1ndonos largamente en la boca.\n\nEn el tembleque ascensor, mientras sub\u00eda a mi altillo, sent\u00ed, como otras veces, unos inexplicables deseos de contarle mis miserias a Pedro Camacho. Fue como una premonici\u00f3n, pues en la oficina me estaban esperando, enfrascados en una animada conversaci\u00f3n con el Gran Pablito, mientras Pascual insuflaba cat\u00e1strofes al bolet\u00edn (nunca respet\u00f3 mi prohibici\u00f3n de incluir muertos, por supuesto), los principales colaboradores del escriba boliviano: Luciano Pando, Josefina S\u00e1nchez y Bat\u00e1n. Esperaron d\u00f3cilmente que echara una mano a Pascual con las \u00faltimas noticias y cuando \u00e9ste y el Gran Pablito nos dieron las buenas noches, y quedamos los cuatro solos en el altillo, se miraron, inc\u00f3modos, antes de hablar. El asunto, no cab\u00eda duda, era el artista.\n\n\u2014Es usted su mejor amigo y por eso hemos venido \u2014murmur\u00f3 Luciano Pando. Era un hombrecito torcido; sesent\u00f3n, con los ojos disparados en direcciones opuestas, que llevaba invierno y verano, d\u00eda y noche, una bufanda grasienta. S\u00f3lo le conoc\u00eda ese terno marr\u00f3n a rayitas azules que era ya una ruina de tantas lavadas y planchadas. Su zapato derecho ten\u00eda una cicatriz en el empeine por donde asomaba la media\u2014. Se trata de algo delicad\u00edsimo. Ya se puede imaginar...\n\n\u2014La verdad, no, don Luciano \u2014le dije\u2014. \u00bfSe refiere a Pedro Camacho? Bueno, somos amigos, s\u00ed, aunque usted ya sabe, es una persona a quien uno nunca acaba de conocer. \u00bfLe pasa algo?\n\nAsinti\u00f3, pero permaneci\u00f3 mudo, mir\u00e1ndose los zapatos, como si lo abrumara lo que iba a decir. Interrogu\u00e9 con los ojos a su compa\u00f1era, a Bat\u00e1n, que estaban serios e inm\u00f3viles.\n\n\u2014Hacemos esto por cari\u00f1o y agradecimiento \u2014trin\u00f3, con su bell\u00edsima voz de terciopelo, Josefina S\u00e1nchez\u2014. Porque nadie puede saber, joven, lo que debemos a Pedro Camacho quienes trabajamos en este oficio tan mal pagado.\n\n\u2014Siempre fuimos la quinta rueda del coche, nadie daba medio por nosotros, viv\u00edamos tan acomplejados que nos cre\u00edamos una basura \u2014dijo Bat\u00e1n, tan conmovido que me imagin\u00e9 de pronto un accidente\u2014. Gracias a \u00e9l descubrimos nuestro oficio, aprendimos que era art\u00edstico.\n\n\u2014Pero est\u00e1n hablando como si se hubiera muerto \u2014les dije.\n\n\u2014Porque \u00bfqu\u00e9 har\u00eda la gente sin nosotros? \u2014cit\u00f3 Josefina S\u00e1nchez, sin o\u00edrme, a su \u00eddolo\u2014. \u00bfQui\u00e9nes les dan las ilusiones y las emociones que los ayudan a vivir?\n\nEra una mujer a la que le hab\u00edan dado esa hermosa voz para indemnizarla de alg\u00fan modo por la aglomeraci\u00f3n de equivocaciones que era su cuerpo. Resultaba imposible adivinar su edad, aunque ten\u00eda que haber dejado atr\u00e1s el medio siglo. Morena, se oxigenaba los pelos, que sobresal\u00edan, amarillos paja, de un turbante granate y se le chorreaban sobre las orejas, sin llegar desgraciadamente a ocultarlas, pues eran enormes, muy abiertas y como \u00e1vidamente proyectadas sobre los ruidos del mundo. Pero lo m\u00e1s llamativo de ella era su papada, una bolsa de pellejos que ca\u00eda sobre sus blusas multicolores. Ten\u00eda un bozo espeso que hubiera podido llamarse bigote y cultivaba la atroz costumbre de sob\u00e1rselo al hablar. Se fajaba las piernas con unas medias el\u00e1sticas de futbolista, porque sufr\u00eda de v\u00e1rices. En cualquier otro momento, su visita me habr\u00eda llenado de curiosidad. Pero esa noche estaba demasiado preocupado por mis propios problemas.\n\n\u2014Claro que s\u00e9 lo que le deben todos a Pedro Camacho \u2014dije, con impaciencia\u2014. Por algo son sus radioteatros los m\u00e1s populares del pa\u00eds.\n\nLos vi cambiar una mirada, darse \u00e1nimos.\n\n\u2014Precisamente \u2014dijo por fin Luciano Pando, ansioso y apenado\u2014. Al principio, no le dimos importancia. Pensamos que eran descuidos, voladuras que le ocurren a cualquiera. Tanto m\u00e1s a alguien que trabaja de sol a sol.\n\n\u2014\u00bfPero qu\u00e9 es lo que le pasa a Pedro Camacho? \u2014lo interrump\u00ed\u2014. No entiendo nada, don Luciano.\n\n\u2014Los radioteatros, joven \u2014murmur\u00f3 Josefina S\u00e1nchez, como si cometiera un sacrilegio\u2014. Se est\u00e1n volviendo cada vez m\u00e1s raros.\n\n\u2014Los actores y los t\u00e9cnicos nos turnamos para contestar el tel\u00e9fono de Radio Central y hacer de parachoques a las protestas de los oyentes \u2014encaden\u00f3 Bat\u00e1n; ten\u00eda los pelos de puercoesp\u00edn lucientes, como si se hubiera echado brillantina; llevaba, igual que siempre, un overol de cargador y los zapatos sin cordones y parec\u00eda a punto de llorar\u2014. Para que los Genaros no lo boten, se\u00f1or.\n\n\u2014Usted sabe de sobra que \u00e9l no tiene medio y vive tambi\u00e9n a tres dobles y un repique \u2014a\u00f1adi\u00f3 Luciano Pando\u2014. \u00bfQu\u00e9 ser\u00eda de \u00e9l si lo botan? \u00a1Se morir\u00eda de hambre!\n\n\u2014\u00bfY de nosotros? \u2014dijo soberbiamente Josefina S\u00e1nchez\u2014. \u00bfQu\u00e9 ser\u00eda de nosotros, sin \u00e9l?\n\nEmpezaron a disputarse la palabra, a cont\u00e1rmelo todo con lujo de detalles. Las incongruencias (las \u00abmetidas de pata\u00bb dec\u00eda Luciano Pando) hab\u00edan comenzado hac\u00eda cerca de dos meses, pero al principio eran tan insignificantes que probablemente s\u00f3lo los actores las advirtieron. No le hab\u00edan dicho una palabra a Pedro Camacho porque, conociendo su car\u00e1cter, nadie se atrev\u00eda, y, adem\u00e1s, durante un buen tiempo se preguntaron si no eran astucias deliberadas. Pero en las tres \u00faltimas semanas las cosas se hab\u00edan agravado much\u00edsimo.\n\n\u2014Lo cierto es que se han vuelto una mescolanza, joven \u2014dijo Josefina S\u00e1nchez, desolada\u2014. Se enredan unos con otros y nosotros mismos ya no somos capaces de desenredarlos.\n\n\u2014Hip\u00f3lito Lituma siempre fue un sargento, terror del crimen en el Callao, en el radioteatro de las diez \u2014dijo, con la voz demudada, Luciano Pando\u2014. Pero hace tres d\u00edas resulta ser el nombre del juez del de las cuatro. Y el juez se llamaba Pedro Barreda. Por ejemplo.\n\n\u2014Y ahora don Pedro Barreda hab\u00eda de cazar ratas, porque se comieron a su hijita \u2014se le llenaron los ojos de l\u00e1grimas a Josefina S\u00e1nchez\u2014. Y a quien se comieron fue a la de don Federico T\u00e9llez Unz\u00e1tegui.\n\n\u2014Imag\u00ednese los ratos que pasamos en las grabaciones \u2014balbuce\u00f3 Bat\u00e1n\u2014. Diciendo y haciendo cosas que son disparates.\n\n\u2014Y no hay manera de arreglar las confusiones \u2014susurr\u00f3 Josefina S\u00e1nchez\u2014. Porque ya ha visto c\u00f3mo controla el se\u00f1or Camacho los programas. No permite que se cambie ni una coma. Si no, le dan unos colerones terribles.\n\n\u2014Est\u00e1 cansado, \u00e9sa es la explicaci\u00f3n \u2014dijo Luciano Pando, moviendo la cabeza con pesadumbre\u2014. No se puede trabajar veinte horas diarias sin que a uno se le mezclen las ideas. Necesita unas vacaciones, para volver a ser el que era.\n\n\u2014Usted se lleva bien con los Genaros \u2014dijo Josefina S\u00e1nchez\u2014. \u00bfNo podr\u00eda hablarles? Decirles solamente que est\u00e1 cansado, que le den unas semanitas para reponerse.\n\n\u2014Lo m\u00e1s dif\u00edcil ser\u00e1 convencerlo a \u00e9l que las tome \u2014dijo Luciano Pando\u2014. Pero las cosas no pueden seguir como van. Terminar\u00edan por despedirlo.\n\n\u2014La gente llama todo el tiempo a la radio \u2014dijo Bat\u00e1n\u2014. Hay que hacer milagros para despistarlos. Y el otro d\u00eda ya sali\u00f3 algo en La Cr\u00f3nica.\n\nNo les dije que Genaro pap\u00e1 ya sab\u00eda y que me hab\u00eda encomendado una gesti\u00f3n con Pedro Camacho. Acordamos que yo sondear\u00eda a Genaro hijo, y que, seg\u00fan como fuera su reacci\u00f3n, decidir\u00edamos si era aconsejable que ellos mismos vinieran, en nombre de todos sus compa\u00f1eros, a tornar la defensa del escriba. Les agradec\u00ed la confianza y trat\u00e9 de darles un poco de optimismo: Genaro hijo era m\u00e1s moderno y comprensivo que Genaro pap\u00e1 y seguramente se dejar\u00eda convencer y le dar\u00eda esas vacaciones. Seguimos hablando, mientras apagaba las luces y cerraba el altillo. En la calle Bel\u00e9n nos dimos la mano. Los vi perderse en la calle vac\u00eda, feos y generosos, bajo la gar\u00faa.\n\nEsa noche la pas\u00e9 enteramente desvelado. Como de costumbre, encontr\u00e9 la comida servida y tapada en casa de los abuelos, pero no prob\u00e9 bocado (y para que la abuelita no se inquietara ech\u00e9 el apanado con arroz a la basura). Los viejitos estaban acostados pero despiertos, y, cuando entr\u00e9 a besarlos, los escrut\u00e9 policialmente, tratando de descubrir en sus caras la inquietud por mis amores escandalosos. Nada, ning\u00fan signo: estaban cari\u00f1osos y sol\u00edcitos y el abuelo me pregunt\u00f3 algo para el crucigrama. Pero me dieron la buena noticia: mi mam\u00e1 hab\u00eda escrito que ella y mi pap\u00e1 vendr\u00edan a Lima de vacaciones muy pronto, ya avisar\u00edan la fecha de llegada. No pudieron ense\u00f1arme la carta, se la hab\u00eda llevado alguna t\u00eda. Era el resultado de las cartas delatoras, no hab\u00eda duda. Mi padre habr\u00eda dicho: \u00abNos vamos al Per\u00fa a poner en orden las cosas\u00bb. Y mi madre: \u00ab\u00a1C\u00f3mo ha podido hacer Julia una cosa as\u00ed!\u00bb. (La t\u00eda Julia y ella hab\u00edan sido amigas, cuando mi familia viv\u00eda en Bolivia y yo no ten\u00eda a\u00fan uso de raz\u00f3n.)\n\nDorm\u00eda en un cuartito peque\u00f1o, abarrotado de libros, maletas y ba\u00fales donde los abuelos guardaban sus recuerdos, muchas fotos de su extinta bonanza, cuando ten\u00edan una hacienda de algod\u00f3n en Caman\u00e1, cuando el abuelo hac\u00eda de agricultor pionero en Santa Cruz de la Sierra, cuando era c\u00f3nsul en Cochabamba o prefecto en Piura. Tumbado boca arriba en mi cama, en la oscuridad, pens\u00e9 mucho en la t\u00eda Julia y en que, no hab\u00eda duda, de un modo u otro, tarde o temprano, nos iban efectivamente a separar. Me daba mucha c\u00f3lera y me parec\u00eda todo est\u00fapido y mezquino y, de repente, se me ven\u00eda a la cabeza la imagen de Pedro Camacho. Pensaba en las llamadas telef\u00f3nicas de t\u00edos y t\u00edas y primos y primas, sobre la t\u00eda Julia y sobre m\u00ed, y empezaba a escuchar las llamadas de los oyentes desorientados con esos personajes que cambiaban de nombre y saltaban del radioteatro de las tres al de las cinco, y con esos episodios que se entreveraban como una selva, y hac\u00eda esfuerzos por adivinar lo que ocurr\u00eda en la intrincada cabeza del escriba, pero no me daba risa, y, al contrario, me conmov\u00eda pensar en los actores de Radio Central, conspirando con los t\u00e9cnicos de sonido, las secretarias, los porteros, para atajar las llamadas y salvar de la despedida al artista. Me emocionaba que Luciano Pando, Josefina S\u00e1nchez y Bat\u00e1n hubieran pensado que yo, la quinta rueda del coche, pod\u00eda influir en los Genaros. Qu\u00e9 poca cosa deb\u00edan sentirse, qu\u00e9 miserias deb\u00edan ganar, para que yo les pareciera importante. A ratos ten\u00eda unos deseos incontenibles de ver, tocar, besar en ese mismo instante a la t\u00eda Julia. As\u00ed vi asomar la luz y o\u00ed ladrar a los perros de la madrugada.\n\nEstuve en mi altillo de Panamericana m\u00e1s temprano que de costumbre y, cuando llegaron Pascual y el Gran Pablito, a las ocho, ya ten\u00eda preparados los boletines y le\u00eddos, anotados y cuadriculados (para el plagio) todos los peri\u00f3dicos. Mientras hac\u00eda esas cosas, miraba el reloj. La t\u00eda Julia me llam\u00f3 exactamente a la hora convenida.\n\n\u2014No he pegado los ojos toda la noche \u2014me susurr\u00f3 con una voz que se perd\u00eda\u2014. Te quiero mucho, Varguitas.\n\n\u2014Yo tambi\u00e9n, con toda mi alma \u2014susurr\u00e9, sintiendo indignaci\u00f3n al ver que Pascual y el Gran Pablito se acercaban para o\u00edr mejor\u2014. Tampoco he pegado los ojos, pensando en ti.\n\n\u2014No puedes saber c\u00f3mo han estado de cari\u00f1osos mi hermana y mi cu\u00f1ado \u2014dijo la t\u00eda Julia\u2014. Nos quedamos jugando cartas. Me cuesta creer que saben, que est\u00e1n conspirando.\n\n\u2014Pero est\u00e1n \u2014le cont\u00e9\u2014. Mis padres han anunciado que vienen a Lima. La \u00fanica raz\u00f3n es \u00e9sa. Ellos nunca viajan en esta \u00e9poca.\n\nSe call\u00f3 y adivin\u00e9, al otro lado de la l\u00ednea, su expresi\u00f3n, entristecida, furiosa, decepcionada. Le volv\u00ed a decir que la quer\u00eda.\n\n\u2014Te llamo a las cuatro, como quedamos \u2014me dijo al fin\u2014. Estoy en el chino de la esquina y hay una cola esperando. Cha\u00edto.\n\nBaj\u00e9 donde Genaro hijo, pero no estaba. Le dej\u00e9 dicho que ten\u00eda urgencia de hablar con \u00e9l, y, por hacer algo, para llenar de alg\u00fan modo el vac\u00edo que sent\u00eda, fui a la universidad. Me toc\u00f3 una clase de derecho penal, cuyo catedr\u00e1tico me hab\u00eda parecido siempre un personaje de cuento. Perfecta combinaci\u00f3n de satiriasis y coprolalia, miraba a las alumnas como desnud\u00e1ndolas y todo le serv\u00eda de pretexto para decir frases de doble sentido y obscenidades. A una chica, que le respondi\u00f3 bien una pregunta y que ten\u00eda el pecho plano, la felicit\u00f3, regodeando la palabra: \u00abEs usted muy sint\u00e9tica, se\u00f1orita\u00bb, y al comentar un art\u00edculo lanz\u00f3 una perorata sobre enfermedades ven\u00e9reas. En la radio, Genaro hijo me esperaba en su oficina:\n\n\u2014Supongo que no vas a pedirme aumento \u2014me advirti\u00f3 desde la puerta\u2014. Estamos casi en quiebra.\n\n\u2014Quiero hablarte de Pedro Camacho \u2014lo tranquilic\u00e9.\n\n\u2014\u00bfSabes que ha empezado a hacer toda clase de barbaridades? \u2014me dijo, como festejando una travesura\u2014. Cruza tipos de un radioteatro a otro, les cambia nombres, enreda los argumentos y est\u00e1 convirtiendo todas las historias en una. \u00bfNo es genial?\n\n\u2014Bueno, algo he o\u00eddo \u2014le dije, desconcertado por su entusiasmo\u2014. Precisamente, anoche habl\u00e9 con los actores. Est\u00e1n preocupados. Trabaja demasiado, piensan que le puede dar un surmenage. Perder\u00edas a la gallina de los huevos de oro. Por qu\u00e9 no le das unas vacaciones, para que se entone un poco.\n\n\u2014\u00bfVacaciones a Camacho? \u2014se espant\u00f3 el empresario progresista\u2014. \u00bf\u00c9l te ha pedido semejante cosa?\n\nLe dije que no, que era una sugerencia de sus colaboradores.\n\n\u2014Est\u00e1n hartos de que los haga trabajar como se pide y quieren librarse de \u00e9l unos d\u00edas \u2014me explic\u00f3\u2014. Ser\u00eda demente darle vacaciones ahora \u2014cogi\u00f3 unos papeles y los blandi\u00f3 con aire triunfante\u2014: Hemos vuelto a batir el r\u00e9cord de sinton\u00eda este mes. O sea que la ocurrencia de cabecear las historias funciona. Mi padre estaba inquieto con esos existencialismos, pero dan resultado, ah\u00ed est\u00e1n los surveys \u2014se volvi\u00f3 a re\u00edr\u2014. Total, mientras al p\u00fablico le guste hay que aguantarle las excentricidades.\n\nNo insist\u00ed, para no meter la pata. Y, despu\u00e9s de todo, \u00bfpor qu\u00e9 no iba a tener raz\u00f3n Genaro hijo? \u00bfPor qu\u00e9 no pod\u00edan ser esas incongruencias algo perfectamente programado por el escriba boliviano? No ten\u00eda ganas de ir a la casa y decid\u00ed hacer un derroche. Convenc\u00ed al cajero de la radio que me diera un adelanto y, luego de El Panamericano, fui al cub\u00edculo de Pedro Camacho a invitarlo a almorzar. Tecleaba como un desaforado, por supuesto. Acept\u00f3 sin entusiasmo, advirti\u00e9ndome que no ten\u00eda mucho tiempo.\n\nFuimos a un restaurante criollo, a la espalda del Colegio de la Inmaculada en el jir\u00f3n Chancay, donde serv\u00edan unos platos arequipe\u00f1os que, le dije, tal vez le recordar\u00edan los famosos picantes bolivianos. Pero el artista, fiel a su norma frugal, se content\u00f3 con un caldillo de huevos y unos frejoles colados a los que apenas prob\u00f3 la temperatura. No pidi\u00f3 dulce y protest\u00f3, con palabrejas que maravillaron a los mozos, porque no supieron prepararle su compuesto de yerbaluisa y menta.\n\n\u2014Estoy pasando una mala racha \u2014le dije, apenas hubimos ordenado\u2014. Mi familia ha descubierto mis amores con su paisana, y, como es mayor que yo y divorciada, est\u00e1n furiosos. Van a hacer algo para separarnos y eso me tiene amargado.\n\n\u2014\u00bfMi paisana? \u2014se sorprendi\u00f3 el escriba\u2014. \u00bfEst\u00e1 usted en amores con una argentina, perd\u00f3n, boliviana?\n\nLe record\u00e9 que conoc\u00eda a la t\u00eda Julia, que hab\u00edamos estado en su cuarto de La Tapada compartiendo su comida, y que ya antes le hab\u00eda contado mis problemas amorosos y que \u00e9l me recet\u00f3 cur\u00e1rmelos con ciruelas en ayunas y cartas an\u00f3nimas. Lo hice a prop\u00f3sito, insistiendo en los detalles, observ\u00e1ndolo. Me escuchaba muy serio, sin pesta\u00f1ear.\n\n\u2014No est\u00e1 mal tener esas contrariedades \u2014dijo, sorbiendo su primera cucharada de caldo\u2014. El sufrimiento educa.\n\nY cambi\u00f3 de tema. Peror\u00f3 sobre el arte de la cocina y la necesidad de ser sobrio para mantenerse espiritualmente sano. Me asegur\u00f3 que el abuso de grasas, f\u00e9culas y az\u00facares entumec\u00eda los principios morales y hac\u00eda proclives al delito y al vicio a las personas.\n\n\u2014Haga una estad\u00edstica entre sus conocidos \u2014me aconsej\u00f3\u2014. Ver\u00e1 que los perversos se reclutan sobre todo entre los gordos. En cambio, no hay flaco de malas inclinaciones.\n\nA pesar de que hac\u00eda esfuerzos por disimularlo, se sent\u00eda inc\u00f3modo. No hablaba con la naturalidad y convicci\u00f3n de otras veces, sino, era evidente, de la boca para afuera, distra\u00eddo por preocupaciones que quer\u00eda ocultar. En sus ojitos saltones, hab\u00eda una sombra azarosa, un temor, una verg\u00fcenza y de rato en rato se mord\u00eda los labios. Su larga cabellera herv\u00eda de caspa y, en el cuello que le bailaba dentro de la camisa, le descubr\u00ed una medallita que a veces acariciaba con dos dedos. Me explic\u00f3, mostr\u00e1ndomela: \u00abUn caballero muy milagroso: el Se\u00f1or de Limpias\u00bb. Su saquito negro se le resbalaba de los hombros y se lo ve\u00eda p\u00e1lido. Hab\u00eda decidido no mencionar los radioteatros, pero all\u00ed, de pronto, al ver que se hab\u00eda olvidado de la existencia de la t\u00eda Julia y de nuestras conversaciones sobre ella, sent\u00ed una curiosidad malsana. Hab\u00edamos terminado el caldillo de huevos, esper\u00e1bamos el plato fuerte tomando chicha morada.\n\n\u2014Esta ma\u00f1ana estuve hablando con Genaro hijo de usted \u2014le cont\u00e9, en el tono m\u00e1s desenvuelto que pude\u2014. Una buena noticia: seg\u00fan los surveys de las agencias de publicidad, sus radioteatros han vuelto a aumentar de sinton\u00eda. Los oyen hasta las piedras.\n\nAdvert\u00ed que se pon\u00eda r\u00edgido, que desviaba la vista, que comenzaba a enrollar y desenrollar la servilleta, muy de prisa, pesta\u00f1eando seguido. Dud\u00e9 sobre si continuar o cambiar de tema, pero la curiosidad fue m\u00e1s fuerte:\n\n\u2014Genaro hijo cree que el aumento de sinton\u00eda se debe a esa idea de mezclar los personajes de un radioteatro a otro, de enlazar las historias \u2014le dije, viendo que soltaba la servilleta, que me buscaba los ojos, que se pon\u00eda blanco\u2014. Le parece genial.\n\nComo no dec\u00eda nada, s\u00f3lo me miraba, segu\u00ed hablando, sintiendo que se me torc\u00eda la lengua. Habl\u00e9 de la vanguardia, de la experimentaci\u00f3n, cit\u00e9 o invent\u00e9 autores que, le asegur\u00e9, eran la sensaci\u00f3n de Europa porque hac\u00edan innovaciones parecidas a las suyas: cambiar la identidad de los personajes en el curso de la historia, simular incongruencias para mantener suspenso al lector. Hab\u00edan tra\u00eddo los frejoles colados y empec\u00e9 a comer, feliz de poder callarme y bajar los ojos para no seguir viendo el malestar del escriba boliviano. Estuvimos en silencio un buen rato, yo comiendo, \u00e9l revolviendo con su tenedor el pur\u00e9 de frejoles, los granos de arroz.\n\n\u2014Me est\u00e1 pasando algo engorroso \u2014le o\u00ed decir, por fin, en voz bajita, como a s\u00ed mismo\u2014. No llevo bien la cuenta de los libretos, tengo dudas y se deslizan confusiones \u2014me mir\u00f3 con zozobra\u2014. S\u00e9 que es usted un joven leal, un amigo en quien se puede confiar. \u00a1Ni una palabra a los mercaderes!\n\nSimul\u00e9 sorpresa, lo abrum\u00e9 con protestas de afecto. Era otro: atormentado, inseguro, fr\u00e1gil, y con un brillo de sudor en la frente verdosa. Se toc\u00f3 las sienes:\n\n\u2014Esto es un volc\u00e1n de ideas, por supuesto \u2014afirm\u00f3\u2014. Lo traicionero es la memoria. Eso de los nombres, quiero decir. Confidencialmente, mi amigo. Yo no los mezclo, se mezclan. Cuando me doy cuenta, es tarde. Hay que hacer malabares para volverlos adonde corresponde, para explicar sus mudanzas. Una br\u00fajula que confunde el norte con el sur puede ser grave, grave.\n\nLe dije que estaba cansado, nadie pod\u00eda trabajar a ese ritmo sin destruirse, que deb\u00eda tomar unas vacaciones.\n\n\u2014\u00bfVacaciones? S\u00f3lo en la tumba \u2014me repuso, amenazante, como si lo hubiera ofendido.\n\nPero, un momento despu\u00e9s, con humildad, me cont\u00f3 que, al darse cuenta de los olvidos, hab\u00eda intentado hacer un fichero. S\u00f3lo que era imposible, no ten\u00eda tiempo, ni siquiera para consultar los programas radiados: todas sus horas estaban tomadas en la producci\u00f3n de nuevos libretos. \u00abSi paro, el mundo se vendr\u00eda abajo\u00bb, murmur\u00f3. \u00bfY por qu\u00e9 no lo pod\u00edan ayudar sus colaboradores? \u00bfPor qu\u00e9 no acud\u00eda a ellos cuando se presentaban esas dudas?\n\n\u2014Eso jam\u00e1s \u2014me contest\u00f3\u2014. Me perder\u00edan el respeto. Ellos son s\u00f3lo una materia prima, mis soldados, y si meto la pata su obligaci\u00f3n es meterla conmigo.\n\nCort\u00f3 abruptamente el di\u00e1logo para sermonear a los mozos por la infusi\u00f3n, que encontr\u00f3 ins\u00edpida, y luego debimos volver al trote a la radio, porque lo esperaba el radioteatro de las tres. Al despedirnos, le dije que har\u00eda cualquier cosa por ayudarlo.\n\n\u2014Lo \u00fanico que le pido es silencio \u2014me dijo. Y, con su sonrisita helada, a\u00f1adi\u00f3\u2014: No se preocupe: a grandes males, grandes remedios.\n\nEn mi altillo, revis\u00e9 los peri\u00f3dicos de la tarde, se\u00f1al\u00e9 las noticias, concert\u00e9 una entrevista para las seis con un neurocirujano historicista que hab\u00eda cometido una trepanaci\u00f3n de cr\u00e1neo con instrumentos incaicos que le prest\u00f3 el Museo de Antropolog\u00eda. A las tres y media, comenc\u00e9 a mirar el reloj y el tel\u00e9fono, alternativamente. La t\u00eda Julia telefone\u00f3 a las cuatro en punto. Pascual y el Gran Pablito no hab\u00edan llegado.\n\n\u2014Mi hermana me habl\u00f3 a la hora del almuerzo \u2014me dijo, con voz l\u00fagubre\u2014. Que el esc\u00e1ndalo es demasiado grande, que tus pap\u00e1s vienen a sacarme los ojos. Me ha pedido que regrese a Bolivia. \u00bfQu\u00e9 puedo hacer? Tengo que irme, Varguitas.\n\n\u2014\u00bfQuieres casarte conmigo? \u2014le pregunt\u00e9.\n\nSe ri\u00f3, con poca alegr\u00eda.\n\n\u2014Te estoy hablando en serio \u2014insist\u00ed.\n\n\u2014\u00bfMe est\u00e1s pidiendo que me case contigo de veras? \u2014volvi\u00f3 a re\u00edrse la t\u00eda Julia, ahora s\u00ed m\u00e1s divertida.\n\n\u2014\u00bfEs s\u00ed o es no? \u2014le dije\u2014. Ap\u00farate, ahorita llegan Pascual y el Gran Pablito.\n\n\u2014\u00bfMe pides eso para demostrarle a tu familia que ya eres grande? \u2014me dijo la t\u00eda Julia, con cari\u00f1o.\n\n\u2014Tambi\u00e9n por eso \u2014reconoc\u00ed.\n\n### XIV\n\nLA HISTORIA del reverendo padre don Seferino Huanca Leyva, ese p\u00e1rroco del muladar que colinda con el futbol\u00edstico barrio de La Victoria y que se llama Mendocita, comenz\u00f3 medio siglo atr\u00e1s, una noche de carnavales, cuando un joven de buena familia, que gustaba darse ba\u00f1os de pueblo, estupr\u00f3 en un callej\u00f3n del Chirimoyo a una jacarandosa lavandera: la Negra Teresita.\n\nCuando \u00e9sta descubri\u00f3 que estaba encinta y como ya ten\u00eda ocho hijos, carec\u00eda de marido y era improbable que, con tantas cr\u00edas, alg\u00fan hombre la llevara al altar, recurri\u00f3 r\u00e1pidamente a los servicios de do\u00f1a Ang\u00e9lica, vieja sabia de la plaza de la Inquisici\u00f3n que oficiaba de comadrona, pero era sobre todo surtidora de hu\u00e9spedes al limbo (en palabras sencillas: abortera). Sin embargo, pese a los ponzo\u00f1osos cocimientos (de orines propios con ratones macerados) que do\u00f1a Ang\u00e9lica hizo beber a la Negra Teresita, el feto del estupro, con terquedad que hac\u00eda presagiar lo que ser\u00eda su car\u00e1cter, se neg\u00f3 a desprenderse de la placenta materna, y all\u00ed sigui\u00f3, enroscado como un tornillo, creciendo y form\u00e1ndose, hasta que, cumplidos nueve meses de los fornicatorios carnavales, la lavandera no tuvo m\u00e1s remedio que parirlo.\n\nLe pusieron Seferino para halagar a su padrino de bautizo, un portero del Congreso que llevaba ese nombre, y los dos apellidos de la madre. En su ni\u00f1ez, nada permiti\u00f3 adivinar que ser\u00eda cura, porque lo que le gustaba no eran las pr\u00e1cticas piadosas sino bailar trompos y volar cometas. Pero siempre, aun antes de saber hablar, demostr\u00f3 ser persona de car\u00e1cter. La lavandera Teresita practicaba una filosof\u00eda de la crianza intuitivamente inspirada en Esparta o Darwin y consist\u00eda en hacer saber a sus hijos que, si ten\u00edan inter\u00e9s en continuar en esta jungla, ten\u00edan que aprender a recibir y dar mordiscos, y que eso de tomar leche y comer era asunto que les concern\u00eda plenamente desde los tres a\u00f1os de edad, porque, lavando ropa diez horas al d\u00eda y reparti\u00e9ndola por todo Lima otras ocho horas, s\u00f3lo lograban subsistir ella y las cr\u00edas que no hab\u00edan cumplido la edad m\u00ednima para bailar con su propio pa\u00f1uelo.\n\nEl hijo del estupro mostr\u00f3 para sobrevivir la misma terquedad que para vivir hab\u00eda demostrado cuando estaba en la barriga: fue capaz de alimentarse tragando todas las porquer\u00edas que recog\u00eda en los tachos de basura y que disputaba a los mendigos y perros. En tanto que sus medio hermanos mor\u00edan como moscas, tuberculosos o intoxicados, o, ni\u00f1os que llegan a adultos aquejados de raquitismo y taras ps\u00edquicas, pasaban la prueba s\u00f3lo a medias, Seferino Huanca Leyva creci\u00f3 sano, fuerte y mentalmente pasable. Cuando la lavandera (\u00bfaquejada de hidrofobia?) ya no pudo trabajar, fue \u00e9l quien la mantuvo, y, m\u00e1s tarde, le coste\u00f3 un entierro de primera en la Casa Guimet que el Chirimoyo celebr\u00f3 como el mejor de la historia del barrio (era ya entonces p\u00e1rroco de Mendocita).\n\nEl muchacho hizo de todo y fue precoz. Al mismo tiempo que a hablar, aprendi\u00f3 a pedir limosna a los transe\u00fantes de la avenida Abancay, poniendo una cara de angelito del fango que volv\u00eda caritativas a las se\u00f1oras de alcurnia. Luego, fue lustrabotas, cuidante de autom\u00f3viles, vendedor de peri\u00f3dicos, de emoliente, de turrones, acomodador en el Estadio y ropavejero. \u00bfQui\u00e9n hubiera dicho que esa criatura de u\u00f1as negras, pies inmundos, cabeza hirviendo de liendres, reparchado y embutido en una chompa con agujeros ser\u00eda, al cabo de los a\u00f1os, el m\u00e1s controvertido curita del Per\u00fa?\n\nFue un misterio que aprendiera a leer, porque nunca pis\u00f3 la escuela. En el Chirimoyo se dec\u00eda que su padrino, el portero del Congreso, le hab\u00eda ense\u00f1ado a deletrear el alfabeto y a formar s\u00edlabas, y que lo dem\u00e1s le vino, muchachos del arroyo que a base de tes\u00f3n llegan al Nobel, por esfuerzo de la voluntad. Seferino Huanca Leyva ten\u00eda doce a\u00f1os y recorr\u00eda la ciudad pidiendo en los palacios ropa inservible y zapatos viejos (que luego vend\u00eda en las barriadas) cuando conoci\u00f3 a la persona que le dar\u00eda los medios de ser santo: una latifundista vasca, Mayte Unz\u00e1tegui, en la que era imposible discernir si era m\u00e1s grande la fortuna o la fe, el tama\u00f1o de sus haciendas o su devoci\u00f3n al Se\u00f1or de Limpias. Sal\u00eda de su morisca residencia de la avenida San Felipe, en Orrantia, y el chofer le abr\u00eda ya la puerta del Cadillac cuando la dama percibi\u00f3, plantado en medio de la calle, junto a su carreta de ropas viejas recogidas esa ma\u00f1ana, al producto del estupro. Su miseria supina, sus ojos inteligentes, sus rasgos de lobezno voluntarioso, le hicieron gracia. Le dijo que ir\u00eda a visitarlo, a la ca\u00edda del sol.\n\nEn el Chirimoyo hubo risas cuando Seferino Huanca Leyva anunci\u00f3 que esa tarde vendr\u00eda a verlo una se\u00f1ora en un carrazo que manejaba un chofer uniformado de azul. Pero cuando, a las seis, el Cadillac fren\u00f3 ante el callej\u00f3n, y do\u00f1a Mayte Unz\u00e1tegui, elegante como una duquesa, ingres\u00f3 en \u00e9l y pregunt\u00f3 por Teresita, todos quedaron convencidos (y estupefactos). Do\u00f1a Mayte, mujeres de negocios que tienen contado hasta el tiempo de la menstruaci\u00f3n, directamente hizo una propuesta a la lavandera que le arranc\u00f3 un alarido de felicidad. Ella costear\u00eda la educaci\u00f3n de Seferino Huanca Leyva y dar\u00eda una gratificaci\u00f3n de diez mil soles a su madre a condici\u00f3n de que el muchacho fuera cura.\n\nFue as\u00ed como el hijo del estupro result\u00f3 pupilo del Seminario Santo Toribio de Mogrovejo, en Magdalena del Mar. A diferencia de otros casos, en los que la vocaci\u00f3n precede la acci\u00f3n, Serefino Huanca Leyva descubri\u00f3 que hab\u00eda nacido para cura despu\u00e9s de ser seminarista. Se volvi\u00f3 un estudiante piadoso y aprovechado, al que mimaban sus maestros y que enorgullec\u00eda a la Negra Teresita y a su protectora. Pero, al mismo tiempo que sus notas en lat\u00edn, teolog\u00eda y patr\u00edstica ascend\u00edan a enhiestas cimas, y que su religiosidad se manifestaba de manera irreprochable en misas o\u00eddas, oraciones dichas y flagelaciones autopropinadas, desde adolescente comenzaron a advertirse en \u00e9l s\u00edntomas de lo que, en el futuro, cuando los grandes debates que sus audacias provocaron, sus defensores llamar\u00edan impaciencias de celo religioso, y sus detractores el mandato delictuoso y mat\u00f3n del Chirimoyo. As\u00ed, por ejemplo, antes de ordenarse, comenz\u00f3 a propagar entre los seminaristas la tesis de que era necesario resucitar las cruzadas, volver a luchar contra Sat\u00e1n no s\u00f3lo con las armas femeninas de la oraci\u00f3n y el sacrificio, sino con las viriles (y, aseguraba, m\u00e1s eficaces) del pu\u00f1o, el cabezazo y, si las circunstancias lo exig\u00edan, la chaveta y la bala.\n\nSus superiores, alarmados, se apresuraron a combatir estas extravagancias, pero ellas, en cambio, fueron calurosamente apoyadas por do\u00f1a Mayte Unz\u00e1tegui, y como la filantr\u00f3pica latifundista subven\u00eda al mantenimiento de un tercio de los seminaristas, aqu\u00e9llos, razones de presupuesto que hacen de tripas coraz\u00f3n, debieron disimular y cerrar ojos y o\u00eddos ante las teor\u00edas de Seferino Huanca Leyva. No eran s\u00f3lo teor\u00edas: las corroboraba la pr\u00e1ctica. No hab\u00eda d\u00eda de salida en que, al anochecer, no volviera el muchacho del Chirimoyo con alg\u00fan ejemplo de lo que llamaba la pr\u00e9dica armada. Era, un d\u00eda, que viendo en las agitadas calles de su barrio c\u00f3mo un marido borracho aporreaba a su mujer, hab\u00eda intervenido rompi\u00e9ndole las canillas a puntapi\u00e9s al abusivo y d\u00e1ndole una conferencia sobre el comportamiento del buen esposo cristiano. Era, otro d\u00eda, que habiendo sorprendido en el \u00f3mnibus de Cinco Esquinas a un carterista biso\u00f1o que pretend\u00eda desplumar a una anciana, lo hab\u00eda desbaratado a cabezazos (llev\u00e1ndolo \u00e9l mismo, luego, a la Asistencia P\u00fablica, a que le suturaran la cara). Era, por fin, un d\u00eda, que habiendo sorprendido, entre las crecidas hierbas del bosque de Matamula, a una pareja que se refocilaba animalmente, los hab\u00eda azotado a ambos hasta la sangre y hecho jurar, de rodillas, so amenaza de nuevas palizas, que ir\u00edan a casarse en el t\u00e9rmino de la distancia. Pero, el broche de oro (para calificarlo de alg\u00fan modo) de Seferino Huanca Leyva, en lo que se refiere a su axioma de \u00abla pureza, como el abecedario, con sangre entra\u00bb, fue el pu\u00f1etazo que descerraj\u00f3, nada menos que en la capilla del seminario, a su tutor y maestro de filosof\u00eda tomista, el manso padre Alberto de Quinteros, quien, en gesto de fraternidad o arrebato solidario, hab\u00eda intentado besarlo en la boca. Hombre sencillo y nada rencoroso (hab\u00eda ingresado al sacerdocio tarde, luego de conquistar fortuna y gloria como psic\u00f3logo con un caso c\u00e9lebre, la curaci\u00f3n de un joven m\u00e9dico que atropell\u00f3 y mat\u00f3 a su propia hija en las afueras de Pisco), el reverendo padre Quinteros, al regresar del hospital donde le soldaron la herida de la boca y le repusieron los tres dientes perdidos, se opuso a que Seferino Huanca Leyva fuera expulsado y \u00e9l mismo, generosidad de los esp\u00edritus grandes que de tanto poner la otra mejilla suben p\u00f3stumamente a los altares, apadrin\u00f3 la misa en la que el hijo del estupro se consagr\u00f3 sacerdote.\n\nPero no s\u00f3lo su convicci\u00f3n de que la Iglesia deb\u00eda combatir el mal pugil\u00edsticamente inquiet\u00f3 a sus superiores cuando Seferino Huanca Leyva era seminarista, sino, m\u00e1s todav\u00eda, su creencia, (\u00bfdesinteresada?) de que, en el vasto repertorio de los pecados mortales, no deb\u00eda figurar de ning\u00fan modo el tocamiento personal. Pese a las reprensiones de sus maestros, que, citas b\u00edblicas y bulas papales numerosas que fulminan a On\u00e1n, pretendieron sacarlo de su error, el hijo de la abortera do\u00f1a Ang\u00e9lica, terco como era desde antes de nacer, soliviantaba nocturnamente a sus compa\u00f1eros asegur\u00e1ndoles que el acto manual hab\u00eda sido concebido por Dios para indemnizar a los eclesi\u00e1sticos por el voto de castidad, y, en todo caso, hacerlo llevadero. El pecado, argumentaba, est\u00e1 en el placer que ofrece la carne de mujer, o (m\u00e1s perversamente) la carne ajena, pero \u00bfpor qu\u00e9 hab\u00eda de estarlo en el humilde, solitario e improductivo desahogo que ofrecen, ayuntados, la fantas\u00eda y los dedos? En una disertaci\u00f3n le\u00edda en la clase del venerable padre Leoncio Zacar\u00edas, Seferino Huanca Leyva lleg\u00f3 a sugerir, interpretando capciosos episodios del Nuevo Testamento, que hab\u00eda razones para no descartar como descabellada la hip\u00f3tesis de que Cristo en persona, alguna vez \u2014\u00bfacaso despu\u00e9s de conocer a Magdalena?\u2014 hubiera combatido masturbatoriamente la tentaci\u00f3n de ser impuro. El padre Zacar\u00edas sufri\u00f3 un soponcio y el protegido de la pianista vasca estuvo a punto de ser expulsado del seminario por blasfemo.\n\nSe arrepinti\u00f3, pidi\u00f3 perd\u00f3n, hizo las penitencias que se le impusieron, y, por un tiempo, dej\u00f3 de propagar esas disparatadas especies que afiebraban a sus maestros y enardec\u00edan a los seminaristas. Pero, en lo que toca a su persona, no dej\u00f3 de ponerlas en pr\u00e1ctica, pues, muy pronto, sus confesores volvieron a o\u00edrlo decir, apenas arrodillado ante los crujientes confesionarios: \u00abEsta semana he sido el enamorado de la reina de Saba, de Dalila y de la esposa de Holofernes\u00bb. Fue este capricho el que le impidi\u00f3 hacer un viaje que hubiera enriquecido su esp\u00edritu. Acababa de ordenarse y, como, pese a sus devaneos heterodoxos, Seferino Huanca Leyva hab\u00eda sido un alumno excepcionalmente aplicado y nadie puso nunca en duda la vibraci\u00f3n de su inteligencia, la jerarqu\u00eda decidi\u00f3 enviarlo a hacer estudios de doctorado en la Universidad Gregoriana de Roma. De inmediato, el flamante sacerdote anunci\u00f3 su prop\u00f3sito de preparar, eruditos que enceguecen consultando los polvosos manuscritos de la Biblioteca Vaticana, una tesis que titular\u00eda Del vicio solitario como ciudadela de la castidad eclesial. Rechazado airadamente su proyecto, renunci\u00f3 al viaje a Roma y fue a sepultarse en el infierno de Mendocita, de donde no saldr\u00eda m\u00e1s.\n\n\u00c9l mismo eligi\u00f3 el barrio cuando supo que todos los sacerdotes de Lima le tem\u00edan como a la peste, no tanto por la concentraci\u00f3n microbiana que hab\u00eda hecho de su jerogl\u00edfica topograf\u00eda de arenosas veredas y casuchas de materiales variopintos \u2014cart\u00f3n, calamina, estera, tabla, trapo y peri\u00f3dico\u2014 un laboratorio de las formas m\u00e1s refinadas de la infecci\u00f3n y la parasitosis, como por la violencia social que imperaba en Mendocita. La barriada, en efecto, era en ese entonces una universidad del delito, en sus especialidades m\u00e1s proletarias: robo por efracci\u00f3n o escalamiento, prostituci\u00f3n, chaveter\u00eda, estafa al menudeo, tr\u00e1fico de pichicata y cafichazgo.\n\nEl padre Seferino Huanca Leyva construy\u00f3 con sus manos, en un par de d\u00edas, una casucha de adobes a la que no le puso puerta, llev\u00f3 all\u00ed un camastro de segunda mano y un colch\u00f3n de paja comprados en la Parada, y anunci\u00f3 que todos los d\u00edas oficiar\u00eda a las siete una misa al aire libre. Hizo saber tambi\u00e9n que confesar\u00eda de lunes a s\u00e1bado, a las mujeres de dos a seis y a los hombres de siete a medianoche, para evitar promiscuidades. Y advirti\u00f3 que, en las ma\u00f1anas, de ocho a dos de la tarde, se propon\u00eda organizar un parvulario donde los chicos del barrio aprender\u00edan el alfabeto, los n\u00fameros y el catecismo. Su entusiasmo se hizo a\u00f1icos contra la dura realidad. Su clientela a las misas madrugadoras fueron apenas un pu\u00f1ado de ancianos y ancianas lega\u00f1osos, de agonizantes reflejos corporales, que, a veces, sin saberlo, practicaban esa imp\u00eda costumbre de las gentes de cierto pa\u00eds (\u00bfconocido por sus vacas y por sus tangos?) de soltar cuescos y hacer sus necesidades con la ropa puesta durante el oficio. Y, en lo que se refiere a la confesi\u00f3n de las tardes y al parvulario de las ma\u00f1anas, no compareci\u00f3 ni un curioso de casualidad.\n\n\u00bfQu\u00e9 ocurr\u00eda? El curandero del barrio, Jaime Concha, un fornido ex sargento de la Guardia Civil que hab\u00eda colgado el uniforme desde que su instituci\u00f3n le orden\u00f3 ejecutar a balazos a un pobre amarillo llegado como polizonte hasta el Callao desde alg\u00fan puerto de oriente, y dedicado desde entonces con tanto \u00e9xito a la medicina plebeya que ten\u00eda realmente en un pu\u00f1o el coraz\u00f3n de Mendocita, hab\u00eda visto con recelo la llegada de un posible competidor y organizado el boicot de la parroquia.\n\nEnterado de esto por una delatora (la ex bruja de Mendocita, do\u00f1a Mayte Unz\u00e1tegui, una vasca de sangre azul a\u00f1il venida a menos y desalojada como reina y se\u00f1ora del barrio por Jaime Concha), el padre Seferino Huanca Leyva supo, alegr\u00edas que empa\u00f1an la vista y abrasan el pecho, que hab\u00eda llegado por fin el momento propicio para poner en acci\u00f3n su teor\u00eda de la pr\u00e9dica armada. Como un anunciador de circo, recorri\u00f3 las mosqueadas callejuelas diciendo a voz en cuello que ese domingo, a las once de la ma\u00f1ana, en el canch\u00f3n de los partidos de f\u00fatbol, \u00e9l y el curandero averiguar\u00edan, a los pu\u00f1os, qui\u00e9n de los dos era el m\u00e1s macho. Cuando el musculoso Jaime Concha se present\u00f3 a la casucha de adobes a preguntar al padre Seferino si deb\u00eda interpretar eso como un desaf\u00edo a trompearse, el hombre del Chirimoyo se limit\u00f3 a preguntarle a su vez, fr\u00edamente, si prefer\u00eda que las manos, en vez de ir desnudas a la pelea, fueran armadas de chavetas. El ex sargento se alej\u00f3, contorsion\u00e1ndose de risa y explicando a los vecinos que \u00e9l, cuando era guardia civil, acostumbraba matar de un cocacho en el cerebro a los perros bravos que se encontraba por la calle.\n\nLa pelea del sacerdote y el curandero concit\u00f3 una expectativa extraordinaria y no s\u00f3lo Mendocita entera, sino tambi\u00e9n La Victoria, El Porvenir, el Cerro San Cosme y El Agustino vinieron a presenciarla. El padre Seferino se present\u00f3 con pantal\u00f3n y camiseta y se persign\u00f3 antes del combate. \u00c9ste fue corto pero llamativo. El hombre del Chirimoyo era f\u00edsicamente menos potente que el ex guardia civil pero lo superaba en tretas. De arranque le ech\u00f3 un pocot\u00f3n de polvo de aj\u00ed en los ojos que llevaba preparado (despu\u00e9s explicar\u00eda a la hinchada: \u00abEn las trompeaduras criollas todo vale\u00bb), y, cuando el gigant\u00f3n, Goliat deteriorado por el hondazo inteligente de David, comenz\u00f3 a dar traspi\u00e9s, ciego, lo debilit\u00f3 con una andanada de patadas en las partes pudendas hasta que lo vio doblarse. Sin darle tregua, inici\u00f3 entonces un ataque frontal contra su cara, a derechazos y zurdazos, y s\u00f3lo cambi\u00f3 de estilo cuando lo tuvo tumbado sobre la tierra. All\u00ed consum\u00f3 la masacre, pisote\u00e1ndole las costillas y el est\u00f3mago. Jaime Concha, rugiendo de dolor y de verg\u00fcenza, se confes\u00f3 derrotado. Entre aplausos, el padre Seferino Huanca Leyva cay\u00f3 de rodillas y or\u00f3 devotamente, la cara al cielo y las manos en cruz.\n\nEste episodio \u2014que se abri\u00f3 paso hasta las p\u00e1ginas de los peri\u00f3dicos y que incomod\u00f3 al arzobispo\u2014 comenz\u00f3 a ganarle al padre Seferino las simpat\u00edas de sus todav\u00eda potenciales parroquianos. A partir de entonces, las misas matutinas se vieron m\u00e1s concurridas y algunas almas pecadoras, sobre todo femeninas, solicitaron confesi\u00f3n, aunque, por supuesto, esos raros casos no llegaban a ocupar ni la d\u00e9cima parte de los dilatados horarios que \u2014calculando, a ojo, la capacidad pecadora de Mendocita\u2014 hab\u00eda fijado el optimista p\u00e1rroco. Otro hecho bien recibido en el barrio y que le gan\u00f3 nuevos clientes fue su comportamiento con Jaime Concha despu\u00e9s de su humillante derrota. \u00c9l mismo ayud\u00f3 a las vecinas a echarle mercurio cromo y \u00e1rnica, y le hizo saber que no lo expulsaba de Mendocita, y que, por el contrario, generosidad de Napoleones que invitan champa\u00f1a y casan con su hija al general cuyo ej\u00e9rcito acaban de volatilizar, estaba dispuesto a asociarlo a la parroquia en calidad de sacrist\u00e1n. El curandero qued\u00f3 autorizado a seguir proporcionando filtros para la amistad y la enemistad, el mal de ojo y el amor, pero a tarifas moderadas que estipulaba el propio p\u00e1rroco, y s\u00f3lo qued\u00f3 prohibido de ocuparse de cuestiones relativas al alma. Tambi\u00e9n le permiti\u00f3 seguir ejerciendo de huesero, para aquellos vecinos que se luxaban o sent\u00edan, a condici\u00f3n de que no intentara curar a enfermos de otra \u00edndole, los mismos que deb\u00edan ser encaminados al hospital.\n\nLa manera como el padre Seferino Huanca Leyva consigui\u00f3 atraer, moscas que sienten la miel, alcatraces que divisan el pez, hacia su desairado parvulario a los chiquillos de Mendocita, fue poco ortodoxa y le gan\u00f3 la primera advertencia seria de la curia. Hizo saber que, por cada semana de asistencia, los ni\u00f1os recibir\u00edan de regalo una estampita. Este cebo hubiera resultado insuficiente para la desalada concurrencia de desarrapados que motiv\u00f3, si las eufem\u00edsticas estampitas del muchacho del Chirimoyo no hubieran sido, en realidad, im\u00e1genes desvestidas de mujeres que era dif\u00edcil confundir con v\u00edrgenes. A ciertas madres de familia que se mostraron extra\u00f1adas de sus m\u00e9todos pedag\u00f3gicos, el p\u00e1rroco les asegur\u00f3, solemnemente, que, aunque pareciera mentira, las estampitas mantendr\u00edan a sus cachorros lejos de la carne impura y los har\u00edan menos traviesos, m\u00e1s d\u00f3ciles y so\u00f1olientos.\n\nPara conquistar a las ni\u00f1as del barrio se vali\u00f3 de las inclinaciones que hicieron de la mujer la primera pecadora b\u00edblica y de los servicios de Mayte Unz\u00e1tegui, tambi\u00e9n incorporada al plantel de la parroquia en calidad de ayudante. \u00c9sta, sabidur\u00eda que s\u00f3lo veinte a\u00f1os de regencia de lupanares en Tingo Mar\u00eda puede forjar, supo ganarse la simpat\u00eda de las ni\u00f1as d\u00e1ndoles cursos que las divert\u00edan: c\u00f3mo pintarrajearse labios y mejillas y p\u00e1rpados sin necesidad de comprar maquillaje en las boticas, c\u00f3mo fabricar con algod\u00f3n, almohadillas y aun papel peri\u00f3dico, pechos y caderas y nalgas postizas, y c\u00f3mo bailar los bailes de moda: la rumba, la huaracha, el porro y el mambo. Cuando el visitador de la jerarqu\u00eda inspeccion\u00f3 la parroquia y vio, en la secci\u00f3n femenina del parvulario, la aglomeraci\u00f3n de mocosas, turn\u00e1ndose el \u00fanico par de zapatos de taco alto del barrio y contone\u00e1ndose ante la vigilancia magisterial de la ex celestina, se restreg\u00f3 los ojos. Al fin, recuperando el habla, pregunt\u00f3 al padre Seferino si hab\u00eda creado una Academia para Prostitutas.\n\n\u2014La respuesta es s\u00ed \u2014contest\u00f3 el hijo de la Negra Teresita, var\u00f3n que no le tem\u00eda a las palabras\u2014. Ya que no hay m\u00e1s remedio que se dediquen a ese oficio, por lo menos que lo ejerzan con talento.\n\n(Fue por esto que recibi\u00f3 la segunda advertencia seria de la curia.)\n\nPero no es cierto que el padre Seferino, como llegaron a propalar sus detractores, fuera el Gran Cafiche de Mendocita. Era s\u00f3lo un hombre realista, que conoc\u00eda la vida palmo a palmo. No foment\u00f3 la prostituci\u00f3n, trat\u00f3 de adecentarla y libr\u00f3 soberbias batallas para impedir que las mujeres que se ganaban la vida con su cuerpo (todas las de Mendocita entre los doce y sesenta a\u00f1os) contrajeran purgaciones y fueran despojadas por los macr\u00f3s. La erradicaci\u00f3n de la veintena de cafiches del barrio (en algunos casos, su regeneraci\u00f3n) fue una labor heroica, de salubridad social, que gan\u00f3 al padre Seferino varios chavetazos y una felicitaci\u00f3n del alcalde de La Victoria. Emple\u00f3 para ello su filosof\u00eda de la pr\u00e9dica armada. Hizo saber, mediante preg\u00f3n callejero de Jaime Concha, que la ley y la religi\u00f3n prohib\u00edan a los hombres vivir como z\u00e1nganos, a costilla de seres inferiores, y que, en consecuencia, vecino que explotara a las mujeres se topar\u00eda con sus pu\u00f1os. As\u00ed, tuvo que desmandibular al Gran Margarina Pacheco, dejar tuerto al Padrillo, impotente a Pedrito Garrote, idiota al Macho Sampedri y con viol\u00e1ceos hematomas a Cojinoba Huambachano. Durante esa quijotesca campa\u00f1a fue una noche emboscado y cosido a chavetazos; los asaltantes, crey\u00e9ndolo muerto, lo dejaron en el fango, para los perros. Pero la reciedumbre del muchacho darwiniano fue m\u00e1s fuerte que las enmohecidas hojas de cuchillo que lo pincharon, y salv\u00f3, conservando, eso s\u00ed \u2014marcas de fierro en cuerpo y cara de var\u00f3n que damas l\u00fabricas suelen llamar apetitosas\u2014 la media docena de cicatrices que, luego del juicio, mandaron al Hospital Psiqui\u00e1trico, como loco incurable, al jefe de sus agresores, el arequipe\u00f1o de nombre religioso y apellido mar\u00edtimo, Ezequiel Delf\u00edn.\n\nSacrificios y esfuerzos rindieron los frutos esperados y Mendocita, asombrosamente, qued\u00f3 limpio de cafiches. El padre Seferino fue la adoraci\u00f3n de las mujeres del barrio; desde entonces concurrieron masivamente a las misas y se confesaron todas las semanas. Para hacerles menos maligno el oficio que les daba de comer, el padre Seferino invit\u00f3 al barrio a un m\u00e9dico de la Acci\u00f3n Cat\u00f3lica a que les diera consejos de profilaxia sexual y las adoctrinara sobre las maneras pr\u00e1cticas de advertir a tiempo, en el cliente o en s\u00ed mismas, la aparici\u00f3n del gonococo. Para los casos en que las t\u00e9cnicas de control de la natalidad que Mayte Unz\u00e1tegui les inculcaba no dieran resultado, el padre Seferino trasplant\u00f3, desde el Chirimoyo a Mendocita, a una disc\u00edpula de do\u00f1a Ang\u00e9lica, a fin de que despachara oportunamente al limbo a los renacuajos del amor mercenario. La advertencia seria que recibi\u00f3 de la curia, cuando \u00e9sta supo que el p\u00e1rroco auspiciaba el uso de preservativos y pesarios y era un entusiasta del aborto, fue la decimotercera.\n\nLa decimocuarta fue por la llamada Escuela de Oficios que tuvo la audacia de formar. En ella, los experimentados del barrio, en amenas charlas \u2014an\u00e9cdotas van, an\u00e9cdotas vienen bajo las nubes o las casuales estrellas de la noche lime\u00f1a\u2014, ense\u00f1aban a los novatos sin prontuario maneras diversas de ganarse los frejoles. All\u00ed se pod\u00edan aprender, por ejemplo, los ejercicios que hacen de los dedos unos inteligentes y discret\u00edsimos intrusos capaces de deslizarse en la intimidad de cualquier bolsillo, bolso, cartera o malet\u00edn, y de reconocer, entre las heterog\u00e9neas piezas, la presa codiciada. All\u00ed se descubr\u00eda c\u00f3mo, con paciencia artesanal, cualquier alambre es capaz de reemplazar con ventaja a la m\u00e1s barroca llave en la apertura de puertas, y c\u00f3mo se puede encender los motores de las distintas marcas de autom\u00f3viles si uno, por acaso, resulta no ser el propietario. All\u00ed se ense\u00f1aba a arrancar prendas al escape, a pie o en bicicleta, a escalar muros y a desvidriar silenciosamente las ventanas de las casas, a hacer la cirug\u00eda pl\u00e1stica de cualquier objeto que cambiara abruptamente de due\u00f1o y la forma de salir de los varios calabozos de Lima sin autorizaci\u00f3n del comisario. Hasta la fabricaci\u00f3n de chavetas y \u2014\u00bfmurmuraciones de la envidia?\u2014 la destilaci\u00f3n de pasta de pichicata se aprend\u00eda en esa Escuela, que gan\u00f3 al padre Seferino, por fin, la amistad y compadrazgo de los varones de Mendocita, y tambi\u00e9n su primera refriega con la comisar\u00eda de La Victoria, donde fue conducido una noche y amenazado de juicio y c\u00e1rcel por eminencia gris de delitos. Lo salv\u00f3, naturalmente, su influyente protectora.\n\nYa en esta \u00e9poca el padre Seferino se hab\u00eda convertido en una figura popular, de la cual se ocupaban los peri\u00f3dicos, las revistas y las radios. Sus iniciativas eran objeto de pol\u00e9micas. Hab\u00eda quienes lo consideraban un protosanto, un adelantado de esa nueva hornada de sacerdotes que revolucionar\u00edan a la Iglesia, y hab\u00eda quienes estaban convencidos de que era un quintacolumnista de Sat\u00e1n encargado de socavar la Casa de Pedro desde el interior. Mendocita (\u00bfgracias a \u00e9l o por su culpa?) se convirti\u00f3 en una atracci\u00f3n tur\u00edstica: curiosos, beatas, periodistas, snobs se llegaban hasta el antiguo para\u00edso del hampa para ver, tocar, entrevistar o pedir aut\u00f3grafos al padre Seferino. Esta publicidad divid\u00eda a la Iglesia: un sector la consideraba beneficiosa y otro perjudicial para la causa.\n\nCuando el padre Seferino Huanca Leyva, con motivo de una procesi\u00f3n a la gloria del Se\u00f1or de Limpias \u2014culto introducido por \u00e9l en Mendocita y que hab\u00eda prendido como paja seca\u2014, anunci\u00f3 triunfalmente que, en la parroquia, no hab\u00eda un solo ni\u00f1o vivo, incluidos los nacidos en las \u00faltimas diez horas, que no estuviera bautizado, un sentimiento de orgullo se apoder\u00f3 de los creyentes, y la jerarqu\u00eda, por una vez, entre tantas admoniciones, le envi\u00f3 unas palabras de felicitaci\u00f3n.\n\nPero, en cambio, origin\u00f3 un esc\u00e1ndalo el d\u00eda que, con motivo de la fiesta de la patrona de Lima, santa Rosa, hizo saber al mundo, en una pr\u00e9dica al aire libre desde el canch\u00f3n de Mendocita, que, dentro de los l\u00edmites polvorientos de su ministerio, no hab\u00eda pareja cuya uni\u00f3n no hubiera sido santificada ante Dios y el altar de la casucha de adobes. Pasmados, pues sab\u00edan muy bien que en el ex imperio de los incas la m\u00e1s s\u00f3lida y acatada instituci\u00f3n \u2014excluidos la Iglesia y el Ej\u00e9rcito\u2014 era la manceb\u00eda, los prelados de la Iglesia peruana vinieron (\u00bfarrastrando los pies?) a comprobar personalmente la haza\u00f1a. Lo que se encontraron, curioseando en las promiscuas viviendas de Mendocita, los dej\u00f3 aterrados y con un regusto de escarnio sacramental en la boca. Las explicaciones del padre Seferino les resultaron abstrusas y arg\u00f3ticas (el muchacho del Chirimoyo, luego de tantos a\u00f1os de barriada, hab\u00eda olvidado el castizo castellano del seminario y contra\u00eddo todos los barbarismos e idiotismos de la replana mendocita) y fue el ex curandero y ex guardia civil, Lituma, quien les explic\u00f3 el sistema empleado para abolir el concubinato. Era sacr\u00edlegamente simple. Consist\u00eda en cristianizar, ante los evangelios, a toda pareja constituida o por constituir. \u00c9stas, al primer refocilo, acud\u00edan presurosas a casarse como Dios manda, ante su querido p\u00e1rroco, y el padre Seferino, sin molestarlos con preguntas impertinentes, les confer\u00eda el sacramento. Y como, de este modo, muchos vecinos resultaron casados varias veces sin haber previamente enviudado \u2014aeron\u00e1utica velocidad con que las parejas del barrio se deshac\u00edan, barajaban y rehac\u00edan\u2014, el padre Seferino recompon\u00eda los estragos que esto causaba, en el dominio del pecado, con la purificadora confesi\u00f3n. (\u00c9l lo hab\u00eda explicado con un refr\u00e1n que, adem\u00e1s de her\u00e9tico, resultaba vulgar: \u00abUn chupo tapa otro chupo\u00bb.) Desautorizado, reprendido, poco menos que abofeteado por el arzobispo, el padre Seferino Huanca Leyva festej\u00f3 con este motivo una longeva efem\u00e9rides: la advertencia seria n\u00famero cien.\n\nAs\u00ed, entre temerarias iniciativas y publicitadas reprimendas, objeto de pol\u00e9micas, amado por unos y vilipendiado por otros, lleg\u00f3 el padre Seferino Huanca Leyva a la flor de la edad: los cincuenta a\u00f1os. Era un hombre de frente ancha, nariz aguile\u00f1a, mirada penetrante y rectitud y bondad en el esp\u00edritu, al que su convicci\u00f3n, desde los aurorales d\u00edas de seminarista, de que el amor imaginario no era pecado y s\u00ed un poderoso guardaespaldas para la castidad, hab\u00eda mantenido efectivamente puro, cuando hizo su llegada al barrio de Mendocita, serpiente del para\u00edso que adopta las formas voluptuosas, ub\u00e9rrimas, llenas de brillos lujuriantes de la hembra, una pervertida que se llamaba Mayte Unz\u00e1tegui y que se hac\u00eda pasar por trabajadora social (en verdad era, \u00bfmujer al fin y al cabo?, meretriz).\n\nDec\u00eda haber trabajado abnegadamente en las selvas de Tingo Mar\u00eda, sacando par\u00e1sitos de las barrigas de los nativos, y haber huido de all\u00ed, muy contrariada, debido a que una pandilla de ratas carn\u00edvoras devoraron a su hijo. Era de sangre vasca y, por lo tanto, aristocr\u00e1tica. Pese a que sus horizontes turgentes y su andar de gelatina debieron alertarlo sobre el peligro, el padre Seferino Huanca Leyva cometi\u00f3, atracci\u00f3n del abismo que ha visto sucumbir monol\u00edticas virtudes, la insensatez de aceptarla como ayudante, creyendo que, como ella dec\u00eda, su designio era salvar almas y matar par\u00e1sitos. En realidad, quer\u00eda hacerlo pecar. Puso en pr\u00e1ctica su programa, vini\u00e9ndose a vivir a la casucha de adobes, en un camastro separado de \u00e9l por una rid\u00edcula cortinilla que para colmo era trasl\u00facida. En las noches, a la luz de un vel\u00f3n, la tentadora, con el pretexto de que as\u00ed dorm\u00eda mejor y conservaba el organismo sano, hac\u00eda ejercicios. \u00bfPero, se pod\u00eda llamar gimnasia sueca a esa danza de har\u00e9n miliunanochesco que, en el sitio, bamboleando las caderas, estremeciendo los hombros, agitando las piernas y revelando los brazos, realizaba la vasca, y que percib\u00eda, a trav\u00e9s de la cortinilla iluminada por los reflejos del vel\u00f3n, como un desquiciador espect\u00e1culo de sombras chinescas, el jadeante eclesi\u00e1stico? Y, m\u00e1s tarde, ya silenciadas por el sue\u00f1o las gentes de Mendocita, Mayte Unz\u00e1tegui ten\u00eda la insolencia de inquirir con voz meliflua, al escuchar los crujidos del camastro vecino: \u00ab\u00bfEst\u00e1 usted desvelado, padrecito?\u00bb.\n\nEs verdad que, para disimular, la bella corruptora trabajaba doce horas diarias, poniendo vacunas y curando sarnas, desinfectando cuchitriles y asoleando ancianos. Pero lo hac\u00eda en shorts, piernas y hombros y brazos y cintura al aire, alegando que en la selva se hab\u00eda acostumbrado a andar as\u00ed. El padre Seferino continuaba desplegando su creativo ministerio, pero enflaquec\u00eda a ojos vistas, ten\u00eda ojeras, la mirada se le iba todo el tiempo en busca de Mayte Unz\u00e1tegui y, al verla pasar, se le abr\u00eda la boca y un hilillo de saliva venial le mojaba los labios. En esta \u00e9poca adquiri\u00f3 la costumbre de andar d\u00eda y noche con las manos en los bolsillos, y su sacristana, la ex abortera do\u00f1a Ang\u00e9lica, profetizaba que en cualquier momento comenzar\u00eda a escupir la sangre del tuberculoso.\n\n\u00bfSucumbir\u00eda el pastor a las malas artes de la trabajadora social, o sus debilitantes ant\u00eddotos le permitir\u00edan resistir? \u00bfLo llevar\u00edan \u00e9stos al manicomio, a la tumba? Con esp\u00edritu deportivo, los feligreses de Mendocita segu\u00edan esta lucha y comenzaron a cruzar apuestas, en las que se fijaban plazos perentorios y se barajaban al\u00e9rgicas opciones: la vasca quedar\u00eda embarazada de simiente de cura, el hombre del Chirimoyo la matar\u00eda para matar la tentaci\u00f3n, o colgar\u00eda los h\u00e1bitos y se casar\u00eda con ella. La vida, por supuesto, se encarg\u00f3 de derrotar a todo el mundo con una carta marcada.\n\nEl padre Seferino, con el argumento de que hab\u00eda que volver a la Iglesia de los primeros tiempos, a la pura y sencilla Iglesia de los evangelios, cuando todos los creyentes viv\u00edan juntos y compart\u00edan sus bienes, inici\u00f3 en\u00e9rgicamente una campa\u00f1a para restablecer en Mendocita \u2014verdadero laboratorio de experimentaci\u00f3n cristiana\u2014 la vida comunal. Las parejas deb\u00edan disolverse en colectividades de quince o veinte miembros, que se distribuir\u00edan el trabajo, la manutenci\u00f3n y las obligaciones dom\u00e9sticas, y vivir\u00edan juntos en casas adaptadas para albergar estas nuevas c\u00e9lulas de la vida social que reemplazar\u00edan a la pareja cl\u00e1sica. El padre Seferino dio el ejemplo, ampliando su casucha e instalando en ella, adem\u00e1s de la trabajadora social, a sus dos sacristanes: el ex sargento Lituma y la ex abortera do\u00f1a Ang\u00e9lica. Esta microcomuna fue la primera de Mendocita, a ejemplo de la cual deb\u00edan irse constituyendo las otras.\n\nEl padre Seferino estipul\u00f3 que, dentro de cada comuna cat\u00f3lica, existiera la m\u00e1s democr\u00e1tica igualdad entre los miembros de un mismo sexo. Los varones entre ellos y las mujeres entre ellas deb\u00edan tutearse, pero, para que no se olvidaran las diferencias de musculatura, inteligencia y sentido com\u00fan establecidas por Dios, aconsej\u00f3 que las hembras hablasen de usted a los machos y procuraran no mirarlos a los ojos en se\u00f1al de respeto. Las tareas de cocinar, barrer, traer agua del ca\u00f1o, matar cucarachas y pericotes, lavar ropa y dem\u00e1s actividades dom\u00e9sticas se asum\u00edan rotativamente y el dinero ganado \u2014de buena o mala manera\u2014 por cada miembro deb\u00eda ser \u00edntegramente cedido a la comunidad, la que, a su vez, lo redistribu\u00eda a partes iguales luego de atender los gastos comunes. Las viviendas carec\u00edan de paredes, para abolir el h\u00e1bito pecaminoso del secreto, y todos los quehaceres de la vida, desde la evacuaci\u00f3n del intestino hasta el \u00f3sculo sexual, deb\u00edan hacerse a la vista de los otros.\n\nAntes de que la polic\u00eda y el ej\u00e9rcito invadieran Mendocita, con un cinematogr\u00e1fico despliegue de carabinas, m\u00e1scaras antigases y bazukas e hicieran esa redada que tuvo encerrados muchos d\u00edas a los hombres y mujeres del barrio en los cuarteles, no por lo que en realidad eran o hab\u00edan sido (ladrones, chaveteros, meretrices) sino por subversivos y disolventes, y el padre Seferino fuera llevado ante un tribunal militar acusado de establecer, al amparo de la sotana, una cabecera de puente para el comunismo (fue absuelto gracias a gestiones de su protectora, la millonaria Mayte Unz\u00e1tegui), el experimento de las arcaicas comunas cristianas estaba ya condenado.\n\nCondenado por la curia, desde luego (advertencia seria doscientas treinta y tres), que lo encontr\u00f3 sospechoso como teor\u00eda e insensato como pr\u00e1ctica (los hechos, ay, le dieron la raz\u00f3n), pero, sobre todo, por la naturaleza de los hombres y mujeres de Mendocita claramente al\u00e9rgica al colectivismo. El problema n\u00famero uno fueron los tr\u00e1ficos sexuales. Al est\u00edmulo de la oscuridad, en los dormitorios colectivos, de colch\u00f3n a colch\u00f3n, se produc\u00edan los m\u00e1s ardientes tocamientos, roces seminales, frotaciones, o, directamente, estupros, sodom\u00edas, embarazos, y, en consecuencia, se multiplicaron los cr\u00edmenes por celos. El problema n\u00famero dos fueron los robos: la convivencia, en vez de abolir el apetito de propiedad, lo exacerb\u00f3 hasta la locura. Los vecinos se robaban unos a otros hasta el vaho p\u00fatrido que respiraban. La cohabitaci\u00f3n, en lugar de hermanar a las gentes de Mendocita, las enemist\u00f3 a muerte. Fue en este periodo de behetr\u00eda y desquiciamiento, que la trabajadora social (\u00bfMayte Unz\u00e1tegui?) declar\u00f3 estar encinta y el ex sargento Lituma admiti\u00f3 ser el padre de la criatura. Con l\u00e1grimas en los ojos, el padre Seferino cristianiz\u00f3 esa uni\u00f3n forjada a causa de sus invenciones sociocat\u00f3licas. (Dicen que desde entonces acostumbra sollozar en las noches cantando eleg\u00edas a la luna.)\n\nPero, casi inmediatamente despu\u00e9s, debi\u00f3 hacer frente a una cat\u00e1strofe peor que la de haber perdido a esa vasca que nunca lleg\u00f3 a poseer: la llegada a Mendocita de un competidor de marca, el pastor evangelista don Sebasti\u00e1n Bergua. Era \u00e9ste un hombre todav\u00eda joven, de aspecto deportivo y fuertes b\u00edceps, que, nada m\u00e1s llegar, hizo saber que se propon\u00eda, en un plazo de seis meses, ganar para la verdadera religi\u00f3n \u2014la reformada\u2014 a todo Mendocita, incluido el p\u00e1rroco cat\u00f3lico y sus tres ac\u00f3litos. Don Sebasti\u00e1n (que hab\u00eda sido, antes de pastor, \u00bfun ginec\u00f3logo atiborrado de millones?) ten\u00eda medios para impresionar a los vecinos: se construy\u00f3 para \u00e9l una casita de ladrillos, dando trabajo regiamente pagado a la gente del barrio, e inici\u00f3 los llamados desayunos religiosos a los que gratuitamente convidaba a quienes asistieran a sus pl\u00e1ticas sobre la Biblia y memorizaran ciertos cantos. Los mendocitas, seducidos por su elocuencia y voz de bar\u00edtono o por el caf\u00e9 con leche y el pan con chicharr\u00f3n que la acompa\u00f1aba, comenzaron a desertar los adobes cat\u00f3licos por los ladrillos evangelistas.\n\nEl padre Seferino recurri\u00f3, naturalmente, a la pr\u00e9dica armada. Ret\u00f3 a don Sebasti\u00e1n Bergua a probar a pu\u00f1etazos qui\u00e9n era el verdadero ministro de Dios. Debilitado por la sobrepr\u00e1ctica del ejercicio de On\u00e1n que le hab\u00eda permitido resistir las provocaciones del demonio, el hombre del Chirimoyo cay\u00f3 noqueado al segundo pu\u00f1etazo de don Sebasti\u00e1n Bergua, que, durante veinte a\u00f1os, hab\u00eda hecho, una hora diaria, calistenia y boxeo (\u00bfen el Gimnasio Remigius de San Isidro?). No fue perder dos incisivos y quedar con la nariz achatada lo que desesper\u00f3 al padre Seferino, sino la humillaci\u00f3n de ser derrotado con sus propias armas y notar que, cada d\u00eda, perd\u00eda m\u00e1s feligreses ante su adversario.\n\nPero, temerarios que crecen ante el peligro y practican lo de a gran mal peor remedio, un d\u00eda, misteriosamente, el hombre del Chirimoyo trajo a su casucha de adobes unas latas llenas de un l\u00edquido que ocult\u00f3 a las miradas de los curiosos (pero que cualquier olfato sensible hubiera reconocido como kerosene). Esa noche, cuando todos dorm\u00edan, acompa\u00f1ado por su fiel Lituma, tapi\u00f3 desde afuera, con gruesas tablas y clavos obesos, las puertas y ventanas de la casa de ladrillos. Don Sebasti\u00e1n Bergua dorm\u00eda el sue\u00f1o de los justos, fantaseando en torno de un sobrino incestuoso que, arrepentido de haber afrentado a su hermana, terminaba de cura papista en una barriada de Lima: \u00bfMendocita? No pod\u00eda o\u00edr los martillazos de Lituma que convert\u00edan el templo evangelista en ratonera, porque la ex comadrona do\u00f1a Ang\u00e9lica, por \u00f3rdenes del padre Serafino, le hab\u00eda dado una p\u00f3cima espesa y anest\u00e9sica. Cuando la misi\u00f3n estuvo tapiada, el hombre del Chirimoyo en persona la roci\u00f3 con kerosene. Luego, persign\u00e1ndose, encendi\u00f3 un f\u00f3sforo y se dispuso a arrojarlo. Pero, algo lo hizo vacilar. El ex sargento Lituma, la trabajadora social, la ex abortera, los perros de Mendocita, lo vieron, largo y flaco bajo las estrellas, los ojos atormentados, con un f\u00f3sforo entre los dedos, dudando sobre si achicharrar\u00eda a su enemigo.\n\n\u00bfLo har\u00eda? \u00bfLanzar\u00eda el f\u00f3sforo? \u00bfConvertir\u00eda el padre Seferino Huanca Leyva la noche de Mendocita en crepitante infierno? \u00bfArruinar\u00eda as\u00ed una vida entera consagrada a la religi\u00f3n y el bien com\u00fan? \u00bfO, pisoteando la llamita que le quemaba las u\u00f1as, abrir\u00eda la puerta de la casa de ladrillos para, de rodillas, implorar perd\u00f3n al pastor evangelista? \u00bfC\u00f3mo terminar\u00eda esta par\u00e1bola de la barriada?\n\n### XV\n\nLA PRIMERA persona a la que habl\u00e9 de mi propuesta de matrimonio a la t\u00eda Julia no fue Javier sino mi prima Nancy. La llam\u00e9, luego de la conversaci\u00f3n telef\u00f3nica con la t\u00eda Julia, y le propuse que fu\u00e9ramos al cine. En realidad fuimos a El Patio, un caf\u00e9 bar de la calle San Mart\u00edn, en Miraflores, donde sol\u00edan reunirse los luchadores que Max Aguirre, el promotor del Luna Park, tra\u00eda a Lima. El local \u2014una casita de un piso, concebida como vivienda de clase media, a la que las funciones de bar notoriamente irritaban\u2014 estaba vac\u00edo, y pudimos conversar tranquilos, mientras yo tomaba la d\u00e9cima taza de caf\u00e9 del d\u00eda y la flaca Nancy una Coca-Cola.\n\nApenas nos sentamos, comenc\u00e9 a maquinar en qu\u00e9 forma pod\u00eda dorarle la noticia. Pero fue ella la que se adelant\u00f3 a darme novedades. La v\u00edspera hab\u00eda habido una reuni\u00f3n en casa de la t\u00eda Hortensia, a la que hab\u00edan concurrido una docena de parientes, para tratar el asunto. All\u00ed se hab\u00eda decidido que el t\u00edo Lucho y la t\u00eda Olga le pidieran a la t\u00eda Julia regresar a Bolivia.\n\n\u2014Lo han hecho por ti \u2014me explic\u00f3 la flaca Nancy\u2014. Parece que tu pap\u00e1 est\u00e1 hecho una fiera y ha escrito una carta terrible.\n\nLos t\u00edos Jorge y Lucho, que me quer\u00edan tanto, estaban ahora inquietos por el castigo que pod\u00eda infligirme. Pensaban que, si la t\u00eda Julia hab\u00eda ya partido cuando \u00e9l llegara a Lima, se aplacar\u00eda y no ser\u00eda tan severo.\n\n\u2014La verdad es que ahora esas cosas no tienen importancia \u2014le dije, con suficiencia\u2014. Porque le he pedido a la t\u00eda Julia que se case conmigo.\n\nSu reacci\u00f3n fue llamativa y caricatural, le ocurri\u00f3 algo de pel\u00edcula. Estaba tomando un trago de Coca-Cola y se ator\u00f3. Le vino un acceso de tos francamente ofensivo y se le llenaron los ojos de l\u00e1grimas.\n\n\u2014D\u00e9jate de payasadas, pedazo de tonta \u2014la re\u00f1\u00ed, muy enojado\u2014. Necesito que me ayudes.\n\n\u2014No me ator\u00e9 por eso sino porque el l\u00edquido se me fue por otro lado \u2014balbuce\u00f3 mi prima, sec\u00e1ndose los ojos y todav\u00eda carraspeando. Y, unos segundos despu\u00e9s, bajando la voz, a\u00f1adi\u00f3\u2014: Pero si eres un bebe. \u00bfAcaso tienes plata para casarte? \u00bfY tu pap\u00e1? \u00a1Te va a matar!\n\nPero, instant\u00e1neamente, ganada por su terrible curiosidad, me acribill\u00f3 a preguntas sobre detalles en los que yo no hab\u00eda tenido tiempo de pensar: \u00bfLa Julita hab\u00eda aceptado? \u00bf\u00cdbamos a escaparnos? \u00bfQui\u00e9nes iban a ser los testigos? \u00bfNo pod\u00edamos casarnos por la Iglesia porque ella era divorciada, no es cierto? \u00bfD\u00f3nde \u00edbamos a vivir?\n\n\u2014Pero, Marito \u2014repiti\u00f3 al final de su cascada de preguntas, asombr\u00e1ndose de nuevo\u2014. \u00bfNo te das cuenta que tienes dieciocho a\u00f1os?\n\nSe ech\u00f3 a re\u00edr y yo tambi\u00e9n me ech\u00e9 a re\u00edr. Le dije que tal vez ten\u00eda raz\u00f3n, pero que ahora se trataba de que me ayudara a poner ese proyecto en pr\u00e1ctica. Nos hab\u00edamos criado juntos y revueltos, nos quer\u00edamos mucho, y yo sab\u00eda que en cualquier caso estar\u00eda de mi lado.\n\n\u2014Claro que si me lo pides te voy a ayudar, aunque sea a hacer locuras y aunque me maten contigo \u2014me dijo al fin\u2014. A prop\u00f3sito, \u00bfhas pensado en la reacci\u00f3n de la familia si de verdad te casas?\n\nDe muy buen humor, estuvimos un rato jugando a qu\u00e9 dir\u00edan y qu\u00e9 har\u00edan los t\u00edos y las t\u00edas, los primos y las primas cuando se enfrentaran a la noticia. La t\u00eda Hortensia llorar\u00eda, la t\u00eda Jes\u00fas ir\u00eda a la iglesia, el t\u00edo Javier pronunciar\u00eda su cl\u00e1sica exclamaci\u00f3n (\u00a1Qu\u00e9 desverg\u00fcenza!), y el benjam\u00edn de los primos, Jaimito, que ten\u00eda tres a\u00f1os y ceceaba, preguntar\u00eda qu\u00e9 era casarse, mam\u00e1. Terminamos ri\u00e9ndonos a carcajadas, con una risa nerviosa que hizo venir a los mozos a averiguar cu\u00e1l era el chiste. Cuando nos calmamos, la flaca Nancy hab\u00eda aceptado ser nuestra esp\u00eda, comunicarnos todos los movimientos e intrigas de la familia. Yo no sab\u00eda cu\u00e1ntos d\u00edas me tomar\u00edan los preparativos y necesitaba estar al tanto de qu\u00e9 tramaban los parientes. De otro lado, har\u00eda de mensajera con la t\u00eda Julia y, de tanto en tanto, la sacar\u00eda a la calle para que yo pudiera verla.\n\n\u2014Okey, okey \u2014asinti\u00f3 Nancy\u2014. Ser\u00e9 la madrina. Eso s\u00ed, si alg\u00fan d\u00eda me hace falta, espero que se porten igualito.\n\nCuando est\u00e1bamos ya en la calle, caminando hacia su casa, mi prima se toc\u00f3 la cabeza:\n\n\u2014Qu\u00e9 suerte tienes \u2014se acord\u00f3\u2014. Te puedo conseguir justo lo que te hace falta. Un departamento en una quinta de la calle Porta. Un solo cuarto, su cocinita y su ba\u00f1o, lind\u00edsimo, de juguete. Y apenas quinientos al mes.\n\nSe hab\u00eda desocupado hac\u00eda unos d\u00edas y una amiga suya lo estaba alquilando; ella le pod\u00eda hablar. Qued\u00e9 maravillado con el sentido pr\u00e1ctico de mi prima, capaz de pensar en ese momento en el problema terrestre de la vivienda en tanto que yo andaba extraviado en la estratosfera rom\u00e1ntica del problema. Por lo dem\u00e1s, quinientos soles estaban a mi alcance. Ahora s\u00f3lo necesitaba ganar m\u00e1s dinero \u00abpara los lujos\u00bb (como dec\u00eda el abuelito). Sin pensarlo dos veces, le ped\u00ed que le dijera a su amiga que le ten\u00eda un inquilino.\n\nDespu\u00e9s de dejar a Nancy, corr\u00ed a la pensi\u00f3n de Javier en la avenida 28 de Julio, pero la casa estaba a oscuras y no me atrev\u00ed a despertar a la due\u00f1a, que era malhumorada. Sent\u00ed una gran frustraci\u00f3n, pues ten\u00eda necesidad de contarle a mi mejor amigo mi gran proyecto y escuchar sus consejos. Esa noche dorm\u00ed un sue\u00f1o sobresaltado de pesadillas. Tom\u00e9 desayuno al alba, con el abuelo, que se levantaba siempre con la luz, y corr\u00ed a la pensi\u00f3n. Encontr\u00e9 a Javier cuando sal\u00eda. Caminamos hacia la avenida Larco, para tomar el colectivo a Lima. La noche anterior, por primera vez en su vida, hab\u00eda escuchado completo un cap\u00edtulo de una radionovela de Pedro Camacho, junto con la due\u00f1a y los otros pensionistas, y estaba impresionado.\n\n\u2014La verdad que tu compinche Camacho es capaz de cualquier cosa \u2014me dijo\u2014. \u00bfSabes qu\u00e9 pas\u00f3 anoche? Una pensi\u00f3n vieja de Lima, una familia pobretona bajada de la sierra. Estaban en medio del almuerzo, conversando, y, de repente, un terremoto. Tan bien hecha la tembladera de vidrios y puertas, el griter\u00edo, que nos paramos y la se\u00f1ora Gracia sali\u00f3 corriendo hasta el jard\u00edn...\n\nMe imagin\u00e9 al genial Bat\u00e1n roncando para imitar el eco profundo de la tierra, reproduciendo con ayuda de sonajas o de bolitas de vidrio que frotaba junto al micr\u00f3fono la danza de los edificios y casas de Lima, y con los pies rompiendo nueces o chocando piedras para que se escucharan los crujidos de techos y paredes al cuartearse, de las escaleras al rajarse y desplomarse, mientras Josefina, Luciano y los otros actores se asustaban, rezaban, aullaban de dolor y ped\u00edan socorro bajo la mirada vigilante de Pedro Camacho.\n\n\u2014Pero el terremoto es lo de menos \u2014me interrumpi\u00f3 Javier, cuando le contaba las proezas de Bat\u00e1n\u2014. Lo bueno es que la pensi\u00f3n se vino abajo y todos murieron apachurrados. No se salv\u00f3 ni uno de muestra, aunque te parezca mentira. Un tipo capaz de matar a todos los personajes de una historia, de un terremoto, es digno de respeto.\n\nHab\u00edamos llegado al paradero de los colectivos y no pude aguantar m\u00e1s. Le cont\u00e9 en cuatro palabras lo que hab\u00eda ocurrido la v\u00edspera y mi gran decisi\u00f3n. Se hizo el que no se sorprend\u00eda:\n\n\u2014Bueno, t\u00fa tambi\u00e9n eres capaz de cualquier cosa \u2014dijo, moviendo la cabeza compasivamente. Y un momento despu\u00e9s\u2014: \u00bfSeguro que quieres casarte?\n\n\u2014Nunca he estado tan seguro de nada en la vida \u2014le jur\u00e9.\n\nEn ese momento ya era verdad. La v\u00edspera, cuando le hab\u00eda pedido a la t\u00eda Julia que se casara conmigo, todav\u00eda ten\u00eda la sensaci\u00f3n de algo irreflexivo, de una pura frase, casi de una broma, pero ahora, despu\u00e9s de haber hablado con Nancy, sent\u00eda una gran seguridad. Me parec\u00eda estar comunic\u00e1ndole una decisi\u00f3n inquebrantable, largamente meditada.\n\n\u2014Lo cierto es que estas locuras tuyas terminar\u00e1n por llevarme a la c\u00e1rcel \u2014coment\u00f3 Javier, resignado, en el colectivo. Y, luego de unas cuadras, a la altura de la avenida Javier Prado\u2014: Te queda poco tiempo. Si tus t\u00edos le han pedido a Julita que se vaya, no puede seguir con ellos muchos d\u00edas m\u00e1s. Y la cosa tiene que estar hecha antes de que llegue el cuco, pues con tu padre ac\u00e1 ser\u00e1 dif\u00edcil.\n\nEstuvimos callados un rato, mientras el colectivo iba caleteando en las esquinas de la avenida Arequipa, dejando y recogiendo pasajeros. Al pasar frente al Colegio Raimondi, Javier volvi\u00f3 a hablar, ya totalmente posesionado del problema:\n\n\u2014Vas a necesitar plata. \u00bfQu\u00e9 vas a hacer?\n\n\u2014Pedir un adelanto en la radio. Vender todo lo viejo que tengo, ropa, libros. Y empe\u00f1ar mi m\u00e1quina de escribir, mi reloj, en fin, todo lo que sea empe\u00f1able. Y empezar a buscar otros trabajos, como loco.\n\n\u2014Yo tambi\u00e9n puedo empe\u00f1ar algunas cosas, mi radio, mis lapiceros, y mi reloj, que es de oro \u2014dijo Javier. Entrecerrando los ojos y haciendo sumas con los dedos, calcul\u00f3\u2014: Creo que te podr\u00e9 prestar unos mil soles.\n\nNos despedimos en la plaza San Mart\u00edn y quedamos en vernos al mediod\u00eda, en mi altillo de Panamericana. Conversar con \u00e9l me hab\u00eda hecho bien y llegu\u00e9 a la oficina de buen humor, muy optimista. Le\u00ed los peri\u00f3dicos, seleccion\u00e9 las noticias, y, por segunda vez, Pascual y el Gran Pablito encontraron los primeros boletines terminados. Desgraciadamente, ambos estaban ah\u00ed cuando llam\u00f3 la t\u00eda Julia y estropearon la conversaci\u00f3n. No me atrev\u00ed a contarle delante de ellos que hab\u00eda hablado con Nancy y con Javier.\n\n\u2014Tengo que verte hoy d\u00eda mismo, aunque sea unos minutos \u2014le ped\u00ed\u2014. Todo est\u00e1 caminando.\n\n\u2014De repente se me ha venido el alma a los pies \u2014me dijo la t\u00eda Julia\u2014. Yo que siempre he sabido ponerle buena cara al mal tiempo, ahora me siento hecha un trapo.\n\nTen\u00eda una buena raz\u00f3n para venir al centro de Lima sin despertar sospechas: reservar en las oficinas del Lloyd A\u00e9reo Boliviano su vuelo a La Paz. Pasar\u00eda por la radio a eso de las tres. Ni ella ni yo mencionamos el tema del matrimonio, pero a m\u00ed me produjo angustia o\u00edrla hablar de aviones. Inmediatamente despu\u00e9s de colgar el tel\u00e9fono, fui a la municipalidad de Lima, a averiguar qu\u00e9 se necesitaba para el matrimonio civil. Ten\u00eda un compa\u00f1ero que trabajaba all\u00e1 y \u00e9l me hizo las averiguaciones, creyendo que eran para un pariente que iba a casarse con una extranjera divorciada. Los requisitos resultaron alarmantes. La t\u00eda Julia ten\u00eda que presentar su partida de nacimiento y la sentencia de divorcio legalizada por los ministerios de Relaciones Exteriores de Bolivia y del Per\u00fa. Yo, mi partida de nacimiento. Pero, como era menor de edad, necesitaba autorizaci\u00f3n notarial de mis padres para contraer matrimonio o ser \u00abemancipado\u00bb (declarado mayor de edad) por ellos, ante el juez de menores. Ambas posibilidades estaban descartadas.\n\nSal\u00ed de la municipalidad haciendo c\u00e1lculos; s\u00f3lo conseguir la legalizaci\u00f3n de los papeles de la t\u00eda Julia, suponiendo que los tuviera en Lima, tomar\u00eda semanas. Si no los ten\u00eda y deb\u00eda pedirlos a Bolivia, a la municipalidad y juzgado respectivos, meses. \u00bfY en cuanto a mi partida de nacimiento? Yo hab\u00eda nacido en Arequipa y escribirle a alg\u00fan pariente de all\u00e1 que me la mandara tomar\u00eda tambi\u00e9n tiempo (adem\u00e1s de ser riesgoso). Las dificultades se levantaban una tras otra, como desaf\u00edos, pero, en vez de disuadirme, reforzaban mi decisi\u00f3n (desde chico hab\u00eda sido muy porfiado). Cuando estaba a medio camino de la radio, a la altura de La Prensa, de pronto, en un rapto de inspiraci\u00f3n, cambi\u00e9 de rumbo, y, casi a la carrera, me dirig\u00ed al parque Universitario, donde llegu\u00e9 sudando. En la secretar\u00eda de la Facultad de Derecho, la se\u00f1ora Riofr\u00edo, encargada de hacernos saber las notas, me recibi\u00f3 con su expresi\u00f3n maternal de siempre y escuch\u00f3 llena de benevolencia la complicada historia que le cont\u00e9, de tr\u00e1mites judiciales urgentes, de una oportunidad \u00fanica de conseguir un trabajo que me ayudar\u00eda a costear mis estudios.\n\n\u2014Est\u00e1 prohibido por el reglamento \u2014se quej\u00f3, levantando su apacible humanidad del apolillado escritorio y avanzando, conmigo al lado, hacia el archivo\u2014. Como tengo buen coraz\u00f3n, ustedes abusan. Un d\u00eda voy a perder mi puesto por hacer estos favores y nadie levantar\u00e1 un dedo por m\u00ed.\n\nLe dije, mientras ella escarbaba los expedientes de alumnos, levantando nubecillas de polvo que nos hac\u00edan estornudar, que si alg\u00fan d\u00eda ocurriera eso, la facultad se declarar\u00eda en huelga. Encontr\u00f3 por fin mi expediente, donde, en efecto, figuraba mi partida de nacimiento y me advirti\u00f3 que me la prestaba s\u00f3lo media hora. Apenas necesit\u00e9 quince minutos para sacar dos fotocopias en una librer\u00eda de la calle Az\u00e1ngaro y devolverle una de ellas a la se\u00f1ora Riofr\u00edo. Llegu\u00e9 a la radio exultante, sinti\u00e9ndome capaz de pulverizar a todos los dragones que me salieran al encuentro.\n\nEstaba sentado en mi escritorio, despu\u00e9s de redactar otros dos boletines y haber entrevistado para El Panamericano al Gaucho Guerrero (un fondista argentino, naturalizado peruano, que se pasaba la vida batiendo su propio r\u00e9cord; corr\u00eda alrededor de una plaza, d\u00edas y noches, y era capaz de comer, afeitarse, escribir y dormir mientras corr\u00eda), descifrando, tras la prosa burocr\u00e1tica de la partida, algunos detalles de mi nacimiento \u2014hab\u00eda nacido en el bulevard Parra, mi abuelo y mi t\u00edo Alejandro hab\u00edan ido a la alcald\u00eda a participar mi llegada al mundo\u2014 cuando Pascual y el Gran Pablito, que entraban al altillo, me distrajeron. Ven\u00edan hablando de un incendio, muertos de risa con los ayes de las v\u00edctimas al ser achicharradas. Trat\u00e9 de seguir leyendo la abstrusa partida, pero los comentarios de mis redactores sobre los guardias civiles de esa comisar\u00eda del Callao rociada de gasolina por un pir\u00f3mano demente, que hab\u00edan perecido todos carbonizados, desde el comisario hasta el \u00faltimo sopl\u00f3n e incluso el perro mascota, me distrajeron de nuevo.\n\n\u2014He visto todos los peri\u00f3dicos y se me ha pasado, \u00bfd\u00f3nde la han le\u00eddo? \u2014les pregunt\u00e9. Y a Pascual\u2014: Cuidadito con dedicar todos los boletines de hoy al incendio \u2014y a los dos\u2014: Qu\u00e9 tal par de s\u00e1dicos.\n\n\u2014No es una noticia, sino el radioteatro de las once \u2014me explic\u00f3 el Gran Pablito\u2014. La historia del sargento Lituma, el terror del hampa chalaca.\n\n\u2014\u00c9l tambi\u00e9n se volvi\u00f3 chicharr\u00f3n \u2014encaden\u00f3 Pascual\u2014. Hubiera podido salvarse, estaba saliendo a hacer su ronda, pero regres\u00f3 para salvar a su capit\u00e1n. Su buen coraz\u00f3n lo freg\u00f3.\n\n\u2014Al capit\u00e1n no, a la perra Choclito \u2014lo rectific\u00f3 el Gran Pablito.\n\n\u2014Eso nunca qued\u00f3 claro \u2014dijo Pascual\u2014. Le cay\u00f3 una de las rejas del calabozo encima. Si lo hubiera visto a don Pedro Camacho mientras se quemaba. \u00a1Qu\u00e9 actorazo!\n\n\u2014Y qu\u00e9 decir de Bat\u00e1n \u2014se entusiasm\u00f3 el Gran Pablito, generosamente\u2014. Si me hubieran jurado que con dos dedos se pod\u00eda hacer cantar un incendio, no me lo hubiera cre\u00eddo. \u00a1Pero lo han visto estos ojos, don Mario!\n\nInterrumpi\u00f3 esta charla la llegada de Javier. Fuimos a tomar el consabido caf\u00e9 al Bransa y all\u00ed le resum\u00ed mis averiguaciones y le mostr\u00e9 triunfalmente mi partida de nacimiento.\n\n\u2014He estado pensando y tengo que decirte que es una estupidez que te cases \u2014me solt\u00f3 de entrada, un poco inc\u00f3modo\u2014. No s\u00f3lo porque eres un mocoso, sino, sobre todo, por el asunto plata. Vas a tener que romperte el alma trabajando en cojudeces para poder comer.\n\n\u2014O sea que t\u00fa tambi\u00e9n vas a repetirme las cosas que me van a decir mi mam\u00e1 y mi pap\u00e1 \u2014me burl\u00e9 de \u00e9l\u2014. \u00bfQue por casarme voy a interrumpir mis estudios de Derecho? \u00bfQue nunca llegar\u00e9 a ser un gran jurisconsulto?\n\n\u2014Que por casarte no vas a tener tiempo ni de leer \u2014me contest\u00f3 Javier\u2014. Que por casarte no llegar\u00e1s nunca a ser un escritor.\n\n\u2014Nos vamos a pelear si sigues por ese camino \u2014le advert\u00ed.\n\n\u2014Bueno, entonces me meto la lengua al bolsillo \u2014se ri\u00f3\u2014. Ya cumpl\u00ed con mi conciencia, adivin\u00e1ndote el porvenir. Lo cierto es que, si la flaca Nancy quisiera, yo tambi\u00e9n me casar\u00eda hoy mismo. \u00bfPor d\u00f3nde empezamos?\n\n\u2014Como no hay forma de que mis padres me autoricen el matrimonio o me emancipen, y como es posible que tampoco Julia tenga todos los papeles que hacen falta, la \u00fanica soluci\u00f3n es encontrar un alcalde buena gente.\n\n\u2014Querr\u00e1s decir un alcalde corrompible \u2014me corrigi\u00f3. Me examin\u00f3 como a un escarabajo\u2014: \u00bfPero a qui\u00e9n puedes corromper t\u00fa, muerto de hambre?\n\n\u2014Alg\u00fan alcalde un poco despistado \u2014insist\u00ed\u2014. Uno al que se le pueda contar el cuento del t\u00edo.\n\n\u2014Bueno, pong\u00e1monos a buscar ese cacaseno descomunal capaz de casarte contra todas las leyes existentes \u2014se ech\u00f3 a re\u00edr de nuevo\u2014. L\u00e1stima que Julita sea divorciada, te hubieras casado por la Iglesia. Eso era f\u00e1cil, entre los curas abundan los cacasenos.\n\nJavier me pon\u00eda siempre de buen \u00e1nimo y terminamos bromeando sobre mi luna de miel, sobre los honorarios que me cobrar\u00eda (ayudarlo a raptar a la flaca Nancy, por supuesto), y lamentando no estar en Piura, donde, como la fuga matrimonial era costumbre tan extendida, no hubiera sido problema encontrar al cacaseno. Cuando nos despedimos, se hab\u00eda comprometido a buscar al alcalde desde esa misma tarde y a empe\u00f1ar todos sus bienes prescindibles para contribuir a la boda.\n\nLa t\u00eda Julia deb\u00eda pasar a las tres y como a las tres y media no hab\u00eda llegado comenc\u00e9 a inquietarme. A las cuatro se me atracaban los dedos en la m\u00e1quina de escribir y fumaba sin parar. A las cuatro y media el Gran Pablito me pregunt\u00f3 si me sent\u00eda mal, porque se me ve\u00eda p\u00e1lido. A las cinco hice que Pascual llamara a casa del t\u00edo Lucho y preguntara por ella. No hab\u00eda llegado. Y tampoco hab\u00eda llegado media hora despu\u00e9s, ni a las seis de la tarde ni a las siete de la noche. Luego del \u00faltimo bolet\u00edn, en vez de bajar en la calle de los abuelos, segu\u00ed en el colectivo hasta la avenida Armend\u00e1riz y estuve merodeando por los alrededores de la casa de mis t\u00edos, sin atreverme a tocar. Por las ventanas divis\u00e9 a la t\u00eda Olga, cambiando el agua de un florero, y, poco despu\u00e9s, al t\u00edo Lucho, que apagaba las luces del comedor. Di varias vueltas a la manzana, pose\u00eddo de sentimientos encontrados: desasosiego, c\u00f3lera, tristeza, ganas de abofetear a la t\u00eda Julia y de besarla. Terminaba una de esas vueltas agitadas cuando la vi bajar de un auto lujoso, con placa diplom\u00e1tica. Me acerqu\u00e9 a trancos, sintiendo que los celos y la ira me hac\u00edan temblar las piernas y decidido a darle de trompadas a mi rival, fuera quien fuera. Se trataba de un caballero canoso y hab\u00eda adem\u00e1s una se\u00f1ora en el interior del autom\u00f3vil. La t\u00eda Julia me present\u00f3 como un sobrino de su cu\u00f1ado y a ellos como los embajadores de Bolivia. Sent\u00ed una sensaci\u00f3n de rid\u00edculo y, al mismo tiempo, que me quitaban un gran peso de encima. Cuando el auto parti\u00f3, cog\u00ed a la t\u00eda Julia del brazo y casi a rastras la hice cruzar la avenida y caminar hacia el Malec\u00f3n.\n\n\u2014Vaya, qu\u00e9 geniecito \u2014la o\u00ed decir, mientras nos acerc\u00e1bamos al mar\u2014. Le pusiste al pobre doctor Gumucio cara de estrangulador.\n\n\u2014A quien voy a estrangular es a ti \u2014le dije\u2014. Te estoy esperando desde las tres y son las once de la noche. \u00bfTe olvidaste que ten\u00edamos una cita?\n\n\u2014No me olvid\u00e9 \u2014me repuso, con determinaci\u00f3n\u2014. Te dej\u00e9 plantado a prop\u00f3sito.\n\nHab\u00edamos llegado al parquecito situado frente al seminario de los jesuitas. Estaba desierto, y, aunque no llov\u00eda, la humedad hac\u00eda brillar el pasto, los laureles, las matas de geranios. La neblina formaba unas sombrillas fantasmales en torno a los conos amarillos de los postes de luz.\n\n\u2014Bueno, vamos a postergar esa pelea para otro d\u00eda \u2014le dije, haci\u00e9ndola sentar en el bordillo del Malec\u00f3n, sobre el acantilado, de donde sub\u00eda, sincr\u00f3nico, profundo, el sonido del mar\u2014. Ahora hay poco tiempo y muchos problemas. \u00bfTienes aqu\u00ed tu partida de nacimiento y la sentencia de tu divorcio?\n\n\u2014Lo que tengo aqu\u00ed es mi pasaje para La Paz \u2014me dijo, tocando su cartera\u2014. Me voy el domingo, a las diez de la ma\u00f1ana. Y estoy feliz. El Per\u00fa y los peruanos ya me llegaron a la coronilla.\n\n\u2014Lo siento por ti, pues por ahora no hay posibilidades de que cambiemos de pa\u00eds \u2014le dije, sent\u00e1ndome a su lado y pas\u00e1ndole el brazo sobre los hombros\u2014. Pero te prometo que, alg\u00fan d\u00eda, nos iremos a vivir a una buhardilla en Par\u00eds.\n\nHasta ese momento, pese a las cosas agresivas que dec\u00eda, hab\u00eda estado tranquila, ligeramente burlona, muy segura de s\u00ed misma. Pero, de pronto, se le dibuj\u00f3 en la cara un rictus amargo y habl\u00f3 con voz dura, sin mirarme:\n\n\u2014No me lo hagas m\u00e1s dif\u00edcil, Varguitas. Me regreso a Bolivia por culpa de tus parientes, pero, tambi\u00e9n, porque lo nuestro es una estupidez. Sabes muy bien que no podemos casarnos.\n\n\u2014S\u00ed podemos \u2014le dije, bes\u00e1ndola en la mejilla, en el cuello, apret\u00e1ndola con fuerza, toc\u00e1ndole \u00e1vidamente los pechos, busc\u00e1ndole la boca con mi boca\u2014. Necesitamos un alcalde cacaseno. Javier me est\u00e1 ayudando. Y la flaca Nancy ya nos encontr\u00f3 un departamentito, en Miraflores. No hay motivo para ponerse pesimistas.\n\nSe dejaba besar y acariciar, pero permanec\u00eda distante, muy seria. Le cont\u00e9 la conversaci\u00f3n con mi prima, con Javier, mis averiguaciones en la municipalidad, la forma como hab\u00eda conseguido mi partida, le dije que la quer\u00eda con toda mi alma, que \u00edbamos a casarnos aunque tuviera que matar a un mont\u00f3n de gente. Cuando porfi\u00e9 con mi lengua, para que separara los dientes, se resisti\u00f3, pero luego abri\u00f3 la boca y pude entrar en ella y gustar su paladar, sus enc\u00edas, su saliva. Sent\u00ed que el brazo libre de la t\u00eda Julia me rodeaba el cuello, que se acurrucaba contra m\u00ed, que se pon\u00eda a llorar con sollozos que estremec\u00edan su pecho. Yo la consolaba, con voz que era un susurro incoherente, sin dejar de besarla.\n\n\u2014Todav\u00eda eres un mocosito \u2014la o\u00ed murmurar, entre risas y pucheros, mientras yo, sin aliento, le dec\u00eda que la necesitaba, que la quer\u00eda, que nunca la dejar\u00eda regresar a Bolivia, que me matar\u00eda si se iba. Por fin, volvi\u00f3 a hablar, en tono muy bajito, tratando de hacer una broma\u2014: Quien con mocosos se acuesta siempre amanece mojado. \u00bfHas o\u00eddo ese refr\u00e1n?\n\n\u2014Es huachafo y no se puede decir \u2014le contest\u00e9, sec\u00e1ndole los ojos con mis labios, con las yemas de los dedos\u2014. \u00bfTienes aqu\u00ed esos papeles? \u00bfTu amigo el embajador los podr\u00eda legalizar?\n\nYa estaba m\u00e1s repuesta. Hab\u00eda dejado de llorar y me miraba con ternura.\n\n\u2014\u00bfCu\u00e1nto durar\u00eda, Varguitas? \u2014me pregunt\u00f3, con voz tristona\u2014. \u00bfAl cu\u00e1nto tiempo te cansar\u00edas? \u00bfAl a\u00f1o, a los dos, a los tres? \u00bfCrees que es justo que dentro de dos o tres a\u00f1os me largues y tenga que empezar de nuevo?\n\n\u2014\u00bfEl embajador los podr\u00e1 legalizar? \u2014insist\u00ed\u2014. Si \u00e9l los legaliza por el lado boliviano, ser\u00e1 f\u00e1cil conseguir la legalizaci\u00f3n peruana. Encontrar\u00e9 en el Ministerio alg\u00fan amigo que nos ayude.\n\nSe qued\u00f3 observ\u00e1ndome, entre compadecida y conmovida. En su cara fue apareciendo una sonrisa.\n\n\u2014Si me juras que me aguantar\u00e1s cinco a\u00f1os, sin enamorarte de otra, queri\u00e9ndome s\u00f3lo a m\u00ed, okey \u2014me dijo\u2014. Por cinco a\u00f1os de felicidad cometo esa locura.\n\n\u2014\u00bfTienes los papeles? \u2014le dije, arregl\u00e1ndole los cabellos, bes\u00e1ndoselos\u2014. \u00bfLos legalizar\u00e1 el embajador?\n\nLos ten\u00eda y conseguimos, en efecto, que la embajada boliviana los legalizara con una buena cantidad de sellos y firmas multicolores. La operaci\u00f3n dur\u00f3 apenas media hora, pues el embajador se trag\u00f3 diplom\u00e1ticamente el cuento de la t\u00eda Julia: necesitaba los papeles esa misma ma\u00f1ana, para formalizar una gesti\u00f3n que le permitir\u00eda sacar de Bolivia los bienes que hab\u00eda recibido al divorciarse. Tampoco fue dif\u00edcil que el ministro de Relaciones Exteriores del Per\u00fa, a su vez, legalizara los documentos bolivianos. Me ech\u00f3 una mano un profesor de la universidad, asesor de la canciller\u00eda, a quien tuve que inventar otro embrollado radioteatro: una se\u00f1ora cancerosa, en estado ag\u00f3nico, a la que hab\u00eda que casar cuanto antes, con el hombre con el que cohabitaba hac\u00eda a\u00f1os, a fin de que muriera en paz con Dios.\n\nAll\u00ed, en una habitaci\u00f3n de a\u00f1osas maderas coloniales y de j\u00f3venes acicalados del Palacio de Torre Tagle, mientras esperaba que el funcionario, avivado por el telefonazo de mi profesor, pusiera a la partida de nacimiento y a la sentencia de divorcio de la t\u00eda Julia m\u00e1s sellos y recolectara las firmas correspondientes, o\u00ed hablar de una nueva cat\u00e1strofe. Se trataba de un naufragio, algo casi inconcebible. Un barco italiano, atracado en un muelle del Callao, repleto de pasajeros y de visitas que los desped\u00edan, de pronto, contraviniendo todas las leyes de la f\u00edsica y de la raz\u00f3n, giraba sobre s\u00ed mismo, se volcaba sobre babor, y se hund\u00eda r\u00e1pidamente en el Pac\u00edfico, muriendo, por efecto de contusiones, ahogo, o, asombrosamente, mordiscos de tiburones, toda la gente que se hallaba a bordo. Eran dos se\u00f1oras, conversaban a mi lado, en espera de alg\u00fan tr\u00e1mite. No bromeaban, se tomaban el naufragio muy en serio.\n\n\u2014\u00bfOcurri\u00f3 en un radioteatro de Pedro Camacho, no es cierto? \u2014me entromet\u00ed.\n\n\u2014En el de las cuatro \u2014asinti\u00f3 la mayor, una mujer huesuda y en\u00e9rgica, con fuerte acento eslavo\u2014. El de Alberto de Quinteros, el cardi\u00f3logo.\n\n\u2014Ese que era ginec\u00f3logo el mes pasado \u2014meti\u00f3 su cuchara, sonriendo, una jovencita que escrib\u00eda a m\u00e1quina. Y se toc\u00f3 la sien, indicando que alguien se hab\u00eda vuelto loco.\n\n\u2014\u00bfNo oy\u00f3 el programa de ayer? \u2014se apiad\u00f3 cari\u00f1osamente la acompa\u00f1ante de la extranjera, una se\u00f1ora con anteojos y entonaci\u00f3n ultralime\u00f1a\u2014. El doctor Quinteros se estaba yendo de vacaciones a Chile, con su esposa y su hijita Charo. \u00a1Y se ahogaron los tres!\n\n\u2014Se ahogaron todos \u2014precis\u00f3 la se\u00f1ora extranjera\u2014. El sobrino Richard, y Elianita y su marido, el Pelirrojo Ant\u00fanez, el tontito, y hasta el hijito del incesto, Rubencito. Hab\u00edan ido a despedirlos.\n\n\u2014Pero lo costeante es que se ahogara el teniente Jaime Concha, que es de otro radioteatro, y que ya se hab\u00eda muerto en el incendio del Callao, hace tres d\u00edas \u2014volvi\u00f3 a intervenir, muerta de risa, la muchacha; hab\u00eda dejado la m\u00e1quina\u2014. Esos radioteatros se han vuelto un puro chiste, \u00bfno les parece?\n\nUn jovencito acicalado, con aire de intelectual (especialidad L\u00edmites Patrios), le sonri\u00f3 con benevolencia y a nosotros nos lanz\u00f3 una mirada que Pedro Camacho hubiera tenido todo el derecho de llamar argentina:\n\n\u2014\u00bfNo te he dicho que eso de pasar personajes de una historia a otra lo invent\u00f3 Balzac? \u2014dijo, hinchando el pecho con sabidur\u00eda. Pero sac\u00f3 una conclusi\u00f3n que lo perdi\u00f3\u2014: Si se entera que lo est\u00e1 plagiando, lo manda a la c\u00e1rcel.\n\n\u2014El chiste no es que los pase de una a otra, sino que los resucite \u2014se defendi\u00f3 la muchacha\u2014. El teniente Concha se hab\u00eda quemado, mientras le\u00eda un Pato Donald, \u00bfc\u00f3mo puede ahora resultar ahog\u00e1ndose?\n\n\u2014Es un tipo sin suerte \u2014sugiri\u00f3 el jovencito acicalado que tra\u00eda mis papeles.\n\nPart\u00ed feliz, con los documentos oleados y sacramentados, dejando a las dos se\u00f1oras, la secretaria y los diplom\u00e1ticos empe\u00f1ados en una animada charla sobre el escriba boliviano. La t\u00eda Julia me estaba esperando en un caf\u00e9 y se ri\u00f3 del cuento; no hab\u00eda vuelto a o\u00edr los programas de su compatriota.\n\nSalvo la legalizaci\u00f3n de esos papeles, que result\u00f3 tan simple, todas las otras gestiones, en esa semana de diligencias y averiguaciones infinitas que hice, solo o acompa\u00f1ado por Javier, en las alcald\u00edas de Lima, resultaron frustradoras y agobiantes. No pon\u00eda los pies en la radio sino para El Panamericano, y dejaba todos los boletines en manos de Pascual, quien pudo ofrecer as\u00ed a los radioescuchas un verdadero fest\u00edn de accidentes, cr\u00edmenes, asaltos, secuestros, que hizo correr por Radio Panamericana tanta sangre como la que, contiguamente, produc\u00eda mi amigo Camacho en su sistem\u00e1tico genocidio de personajes.\n\nComenzaba el recorrido muy temprano. Fui al principio a las municipalidades m\u00e1s ra\u00eddas y alejadas del centro, la del R\u00edmac, la de El Porvenir, la de Vitarte, la de Chorrillos. Una y cincuenta veces (al principio ruboriz\u00e1ndome, luego con desparpajo) expliqu\u00e9 el problema a alcaldes, tenientes alcalde, s\u00edndicos, secretarios, porteros, portapliegos, y cada vez recib\u00ed negativas categ\u00f3ricas. La piedra de toque era siempre la misma: mientras no obtuviera autorizaci\u00f3n notarial de mis padres, o fuera emancipado ante el juez, no pod\u00eda casarme. Luego intent\u00e9 suerte en las alcald\u00edas de los barrios c\u00e9ntricos, con exclusi\u00f3n de Miraflores y San Isidro (donde pod\u00eda haber conocidos de la familia) con id\u00e9ntico resultado. Los mun\u00edcipes, luego de revisar los documentos, sol\u00edan hacerme bromas que eran patadas en el est\u00f3mago: \u00ab\u00bfPero c\u00f3mo, quieres casarte con tu mam\u00e1?\u00bb, \u00abno seas tonto, muchacho, para qu\u00e9 te vas a casar, arrej\u00fantate nom\u00e1s\u00bb. El \u00fanico sitio donde brill\u00f3 una luz de esperanza fue en la municipalidad de Surco, donde un secretario rollizo y cejijunto nos dijo que el asunto se pod\u00eda arreglar por diez mil soles, \u00abpues hab\u00eda que taparle la boca a mucha gente\u00bb. Intent\u00e9 regatear y llegu\u00e9 a ofrecerle una cantidad que dif\u00edcilmente hubiera podido reunir (cinco mil soles), pero el gordito, como asustado de su audacia, dio marcha atr\u00e1s y termin\u00f3 sac\u00e1ndonos de la alcald\u00eda.\n\nHablaba por tel\u00e9fono con la t\u00eda Julia dos veces al d\u00eda y la enga\u00f1aba, todo estaba en regla, que tuviera un malet\u00edn de mano listo con las cosas indispensables, en cualquier momento le dir\u00eda \u00abya\u00bb. Pero me sent\u00eda cada vez m\u00e1s desmoralizado. El viernes en la noche, al regresar a casa de los abuelos, encontr\u00e9 un telegrama de mis padres: \u00abLlegamos lunes, Panagra, vuelo 516\u00bb.\n\nEsa noche, despu\u00e9s de pensar largo rato, dando vueltas en la cama, prend\u00ed la lamparita del velador y escrib\u00ed en un cuaderno, donde anotaba temas para cuentos, en orden de prioridad, las cosas que har\u00eda. La primera era casarme con la t\u00eda Julia y poner a la familia ante un hecho legal consumado al que tendr\u00edan que resignarse, quisi\u00e9ranlo o no. Como faltaban pocos d\u00edas y la resistencia de los mun\u00edcipes lime\u00f1os era tan tenaz, esa primera opci\u00f3n se volv\u00eda cada instante m\u00e1s ut\u00f3pica. La segunda era huir con la t\u00eda Julia al extranjero. No a Bolivia; la idea de vivir en un mundo donde ella hab\u00eda vivido sin m\u00ed, donde ten\u00eda tantos conocidos, su mismo ex marido, me molestaba. El pa\u00eds indicado era Chile. Ella pod\u00eda partir a La Paz, para enga\u00f1ar a la familia, y yo me escapar\u00eda en \u00f3mnibus o colectivo hasta Tacna. Alguna manera habr\u00eda de cruzar clandestinamente la frontera, hasta Arica, y luego seguir\u00eda por tierra hasta Santiago, donde la t\u00eda Julia vendr\u00eda a reunirse conmigo o me estar\u00eda esperando. La posibilidad de viajar y vivir sin pasaporte (para sacarlo se necesitaba tambi\u00e9n autorizaci\u00f3n paterna) no me parec\u00eda imposible, y me gustaba, por su car\u00e1cter novelesco. Si la familia, como era seguro, me hac\u00eda buscar, me localizaba y repatriaba, me escapar\u00eda de nuevo, todas las veces que hiciera falta, y as\u00ed ir\u00eda viviendo hasta alcanzar los codiciados, liberadores veinti\u00fan a\u00f1os. La tercera opci\u00f3n era matarme, dejando una carta bien escrita, para sumir a mis parientes en el remordimiento.\n\nAl d\u00eda siguiente, muy temprano, corr\u00ed a la pensi\u00f3n de Javier. Pas\u00e1bamos revista, cada ma\u00f1ana, mientras \u00e9l se afeitaba y duchaba, a los acontecimientos de la v\u00edspera y prepar\u00e1bamos el plan de acci\u00f3n de la jornada. Sentado sobre el excusado, vi\u00e9ndolo jabonarse, le le\u00ed el cuaderno donde hab\u00eda resumido, con comentarios marginales, las opciones de mi destino. Mientras se enjuagaba, me rog\u00f3 encarecidamente que trastocara las prioridades y pusiera el suicidio a la cabeza:\n\n\u2014Si te matas, las porquer\u00edas que has escrito se volver\u00e1n interesantes, la gente morbosa querr\u00e1 leerlas y ser\u00e1 f\u00e1cil publicarlas en un libro \u2014me convenc\u00eda, a la vez que se secaba con furia\u2014. Te volver\u00e1s, aunque sea p\u00f3stumamente, un escritor.\n\n\u2014Me vas a hacer perder el primer bolet\u00edn \u2014lo apuraba yo\u2014. D\u00e9jate de jugar a Cantinflas que tu humor me hace maldita gracia.\n\n\u2014Si te matas, ya no tendr\u00eda que faltar tanto a mi trabajo ni a la universidad \u2014continuaba Javier, mientras se vest\u00eda\u2014. Lo ideal es que procedas hoy, esta ma\u00f1ana, ahorita. As\u00ed me librar\u00edas de empe\u00f1ar mis cosas, que, por supuesto, terminar\u00e1n rematando, porque \u00bfacaso me vas a pagar alg\u00fan d\u00eda?\n\nY ya en la calle, mientras trot\u00e1bamos hacia el colectivo, sinti\u00e9ndose un humorista eximio:\n\n\u2014Y, por \u00faltimo, si te matas, te volver\u00e1s famoso, y a tu mejor amigo, tu confidente, el testigo de la tragedia, le har\u00e1n reportajes y saldr\u00e1 retratado en los peri\u00f3dicos. \u00bfY t\u00fa crees que tu prima Nancy no se ablandar\u00eda con esa publicidad?\n\nEn la llamada (horriblemente) Caja de Pignoraci\u00f3n de la plaza de Armas, empe\u00f1amos mi m\u00e1quina de escribir y su radio, mi reloj y sus lapiceros, y, al final, lo convenc\u00ed de que tambi\u00e9n empe\u00f1ara su reloj. Pese a que regateamos como lobos, s\u00f3lo obtuvimos dos mil soles. Los d\u00edas anteriores, sin que lo advirtieran los abuelos, yo hab\u00eda ido vendiendo, en los ropavejeros de la calle La Paz, ternos, zapatos, camisas, corbatas, chompas, hasta quedarme pr\u00e1cticamente con lo que llevaba puesto. Pero la inmolaci\u00f3n de mi vestuario signific\u00f3 apenas cuatrocientos soles. En cambio, hab\u00eda tenido m\u00e1s suerte con el empresario progresista, al que, despu\u00e9s de media hora dram\u00e1tica, convenc\u00ed de que me adelantara cuatro sueldos y me los fuera descontando a lo largo de un a\u00f1o. La conversaci\u00f3n tuvo un final inesperado. Yo le juraba que ese dinero era para una operaci\u00f3n de hernia de mi abuelita, urgent\u00edsima, y no lo conmov\u00eda. Pero, de pronto, dijo: \u00abBueno\u00bb. Con una sonrisa de amigo, a\u00f1adi\u00f3: \u00abConfiesa que es para hacer abortar a una hembrita\u00bb. Baj\u00e9 los ojos y le rogu\u00e9 que me guardara el secreto.\n\nAl ver mi depresi\u00f3n por el poco dinero conseguido con el empe\u00f1o, Javier me acompa\u00f1\u00f3 hasta la radio. Quedamos en pedir permiso en nuestros trabajos para ir en la tarde a Huacho. Tal vez en provincias los mun\u00edcipes fueran m\u00e1s sentimentales. Llegu\u00e9 al altillo cuando sonaba el tel\u00e9fono. La t\u00eda Julia estaba hecha una furia. La v\u00edspera hab\u00edan llegado a casa del t\u00edo Lucho, de visita, la t\u00eda Hortensia y el t\u00edo Alejandro y no le hab\u00edan contestado el saludo.\n\n\u2014Me miraron con un desprecio ol\u00edmpico, s\u00f3lo les falt\u00f3 decirme pe \u2014me cont\u00f3, indignada\u2014. Tuve que morderme para no mandarlos ya sabes ad\u00f3nde. Lo hice por mi hermana, pero tambi\u00e9n por nosotros, para no complicar m\u00e1s las cosas. \u00bfC\u00f3mo va todo, Varguitas?\n\n\u2014El lunes, a primera hora \u2014le asegur\u00e9\u2014. Tienes que decir que atrasas un d\u00eda el vuelo a La Paz. Tengo todo casi listo.\n\n\u2014No te preocupes por el alcalde cacaseno \u2014me dijo la t\u00eda Julia\u2014. Ya me entr\u00f3 la rabieta y no me importa. Aunque no lo encuentres, nos escapamos lo mismo.\n\n\u2014\u00bfPor qu\u00e9 no se casan en Chincha, don Mario? \u2014le o\u00ed decir a Pascual, apenas colgu\u00e9 el tel\u00e9fono. Al ver mi estupor, se confundi\u00f3\u2014: No es que yo sea chismoso y quiera entrometerme. Pero, claro, oy\u00e9ndolos, nos enteramos de las cosas. Lo hago para ayudarlo. El alcalde de Chincha es mi primo y lo casa en un dos por tres, con o sin papeles, sea o no sea mayor de edad.\n\nEse mismo d\u00eda qued\u00f3 todo milagrosamente resuelto. Javier y Pascual partieron esa tarde a Chincha, en un colectivo, con los papeles y la consigna de dejar todo preparado para el lunes. Mientras, yo fui con mi prima Nancy a alquilar el cuartito de la quinta miraflorina, ped\u00ed tres d\u00edas de permiso en la radio (los obtuve despu\u00e9s de una discusi\u00f3n hom\u00e9rica con Genaro pap\u00e1, a quien temerariamente amenac\u00e9 con renunciar si me los negaba) y plane\u00e9 la fuga de Lima. El s\u00e1bado en la noche, Javier volvi\u00f3 con buenas noticias. El alcalde era un tipo joven y simp\u00e1tico, y, cuando \u00e9l y Pascual le contaron la historia, se hab\u00eda re\u00eddo y festejado el proyecto de rapto.\n\n\u00abQu\u00e9 rom\u00e1ntico\u00bb, les hab\u00eda dicho. Se qued\u00f3 con los papeles y les asegur\u00f3 que, entre amigos, tambi\u00e9n se pod\u00eda obviar el asunto de las proclamas.\n\nEl domingo previne a la t\u00eda Julia, por tel\u00e9fono, que hab\u00eda encontrado al cacaseno, que nos fugar\u00edamos al d\u00eda siguiente, a las ocho de la ma\u00f1ana, y que al mediod\u00eda ser\u00edamos marido y mujer.\n\n### XVI\n\nJOAQU\u00cdN HINOSTROZA Bellmont, quien habr\u00eda de estremecer los estadios, no metiendo goles ni atajando penales sino arbitrando partidos de f\u00fatbol, y cuya sed de alcohol dejar\u00eda huella y deudas en los bares de Lima, naci\u00f3 en una de esas residencias que los mandarines se construyeron hace treinta a\u00f1os, en La Perla, cuando pretendieron convertir a ese bald\u00edo en una Copacabana lime\u00f1a (pretensi\u00f3n malograda por la humedad, que, castigo de camello que se obstina en pasar por el ojo de la aguja, devast\u00f3 gargantas y bronquios de la aristocracia peruana).\n\nFue Joaqu\u00edn hijo \u00fanico de una familia que, adem\u00e1s de adinerada, entroncaba, frondosa selva de \u00e1rboles que son t\u00edtulos y escudos, con marquesados de Espa\u00f1a y Francia. Pero el padre del futuro r\u00e9feri y borrach\u00edn hab\u00eda puesto de lado los pergaminos y consagrado su vida al ideal moderno de multiplicar su fortuna en negocios que comprend\u00edan desde la fabricaci\u00f3n de casimires hasta la introducci\u00f3n del ardiente cultivo de la pimienta en la Amazon\u00eda. La madre, madona linf\u00e1tica, esposa abnegada, hab\u00eda pasado su vida gastando el dinero que produc\u00eda su marido en m\u00e9dicos y curanderos (pues padec\u00eda diversas enfermedades de alta clase social). Ambos hab\u00edan tenido a Joaqu\u00edn algo crecidos, despu\u00e9s de mucho rogar a Dios que les concediera un heredero. El advenimiento constituy\u00f3 una felicidad indescriptible para sus padres, quienes, desde la cuna, so\u00f1aron para \u00e9l un porvenir de pr\u00edncipe de la industria, rey de la agricultura, mago de la diplomacia o Lucifer de la pol\u00edtica.\n\n\u00bfFue por rebelde, en insubordinaci\u00f3n contra este destino de gloria cremat\u00edstica y brillo social, que el ni\u00f1o result\u00f3 \u00e1rbitro de f\u00fatbol, o m\u00e1s bien por insuficiencia de psicolog\u00eda? No, fue por genuina vocaci\u00f3n. Tuvo, naturalmente, desde la mamadera hasta el bozo, una variopinta sucesi\u00f3n de institutrices, importadas de pa\u00edses ex\u00f3ticos: Francia, Inglaterra. Y en los mejores colegios de Lima fueron reclutados profesores para ense\u00f1arle los n\u00fameros y las letras. Todos, uno tras otro, terminaron renunciando al ping\u00fce salario, desmoralizados e hist\u00e9ricos, por la indiferencia ontol\u00f3gica del ni\u00f1o ante cualquier especie de saber. A los ocho a\u00f1os no hab\u00eda aprendido a sumar y del alfabeto a duras penas memorizaba las vocales. S\u00f3lo dec\u00eda monos\u00edlabos, era tranquilo, se paseaba por las habitaciones de La Perla, entre muchedumbres de juguetes adquiridos en distintos puntos del orbe para distraerlo \u2014mecanos alemanes, trenes japoneses, rompecabezas chinos, soldaditos austriacos, triciclos norteamericanos\u2014, con expresi\u00f3n de aburrimiento mortal. Lo \u00fanico que parec\u00eda sacarlo, a ratos, de su sopor brahm\u00e1nico eran las figuritas de f\u00fatbol de los chocolatines Mar del Sur, que pegaba en cuadernos satinados y contemplaba, horas de horas, con curiosidad.\n\nAterrados ante la idea de haber procreado un fin de raza, hemof\u00edlico y tarado, que ser\u00eda m\u00e1s tarde hazmerre\u00edr del p\u00fablico, los padres acudieron a la ciencia. Ilustres galenos comparecieron en La Perla. Fue el pediatra estrella de la ciudad, el doctor Alberto de Quinteros, quien desasn\u00f3 luminosamente a los atormentados:\n\n\u2014Tiene lo que llamo mal de invernadero \u2014les explic\u00f3\u2014. Las flores que no viven en el jard\u00edn, entre flores e insectos, crecen mustias y su perfume es hediondo. La c\u00e1rcel dorada lo est\u00e1 atontando. Amas y d\u00f3mines deben ser despedidos y el ni\u00f1o matriculado en un colegio para que alterne con gente de su edad. \u00a1Ser\u00e1 normal el d\u00eda que un compa\u00f1ero le rompa la nariz!\n\nDispuesta a cualquier sacrificio con tal de desimbecilizarlo, la orgullosa pareja consinti\u00f3 en dejar que Joaquincito se zambullera en el plebeyo mundo exterior. Se escogi\u00f3 para \u00e9l, claro est\u00e1, el colegio m\u00e1s caro de Lima, los padres de Santa Mar\u00eda, y, a fin de no destruir todas las jerarqu\u00edas, se le mand\u00f3 hacer un uniforme de color reglamentario, pero en terciopelo.\n\nLa receta del famoso dio resultados apreciables. Es verdad que Joaqu\u00edn sacaba notas extraordinariamente bajas y que, para aprobar sus ex\u00e1menes, \u00e1urea codicia que produjo cismas, sus padres deb\u00edan hacer donaciones (vitrales para la capilla del colegio, faldas de pa\u00f1o para los ac\u00f3litos, pupitres robustos para la escuelita de los pobres, etc\u00e9tera), pero el hecho es que el ni\u00f1o se volvi\u00f3 sociable y que a partir de entonces se le vio algunas veces contento. En esta \u00e9poca se manifest\u00f3 el primer indicio de su genialidad (su incomprensivo padre le dec\u00eda tara): un inter\u00e9s por el balompi\u00e9. Cuando fueron informados que el ni\u00f1o Joaqu\u00edn, apenas se calzaba los zapatos de f\u00fatbol, de anest\u00e9sico y monosil\u00e1bico se transformaba en un ser din\u00e1mico y g\u00e1rrulo, sus padres se alegraron mucho. Y, de inmediato, adquirieron un terreno contiguo a su casa de La Perla, para construir una cancha de f\u00fatbol, de proporciones apreciables, donde Joaquincito pudiera divertirse a sus anchas.\n\nSe vio, a partir de entonces, en la neblinosa avenida de las Palmeras, de La Perla, desembarcar del \u00f3mnibus del Santa Mar\u00eda, a la salida de clases, a veintid\u00f3s alumnos \u2014cambiaban las caras pero permanec\u00eda el n\u00famero\u2014 que ven\u00edan a jugar en la cancha de los Hinostroza Bellmont. La familia regalaba a los jugadores, despu\u00e9s del partido, con un t\u00e9 acompa\u00f1ado de chocolates, gelatinas, merengues y helados. Los ricos gozaban viendo cada tarde a su hijito Joaqu\u00edn jadeando feliz.\n\nS\u00f3lo despu\u00e9s de algunas semanas, se percat\u00f3 el pionero del cultivo de la pimienta en el Per\u00fa que ocurr\u00eda algo extra\u00f1o. Dos, tres, diez veces hab\u00eda encontrado a Joaquincito arbitrando el partido. Con un silbato en la boca y una gorrita para el sol, corr\u00eda tras los jugadores, se\u00f1alaba faltas, impon\u00eda penales. Aunque el ni\u00f1o no parec\u00eda acomplejado por cumplir este papel en vez de ser jugador, el millonario se enoj\u00f3. \u00bfLos invitaba a su casa, los engordaba con dulces, les permit\u00eda codearse con su hijo de igual a igual y ten\u00edan la desfachatez de relegar a Joaqu\u00edn a la mediocre funci\u00f3n de \u00e1rbitro? Estuvo a punto de abrir las jaulas de los d\u00f3berman y dar un buen susto a esos frescos. Pero se limit\u00f3 a recriminarlos. Ante su sorpresa, los muchachos se declararon inocentes, juraron que Joaqu\u00edn era \u00e1rbitro porque le gustaba serlo y el damnificado corrobor\u00f3 por Dios y por su madre que era as\u00ed. Unos meses despu\u00e9s, consultando su libreta y los informes de sus mayordomos, el padre se enfrent\u00f3 a este balance: en ciento treinta y dos partidos disputados en su cancha, Joaqu\u00edn Hinostroza Bellmont no hab\u00eda sido jugador en ninguno y hab\u00eda arbitrado ciento treinta y dos. Cambiando una mirada, los padres subliminalmente se dijeron que algo andaba mal: \u00bfc\u00f3mo pod\u00eda ser esto la normalidad? Nuevamente, fue requerida la ciencia.\n\nFue el m\u00e1s connotado astr\u00f3logo de la ciudad, un hombre que le\u00eda las almas en las estrellas y que resta\u00f1aba los esp\u00edritus de sus clientes (\u00e9l hubiera preferido: amigos) mediante los signos del zodiaco, el profesor Lucio Ac\u00e9mila, quien, despu\u00e9s de muchos hor\u00f3scopos, interrogatorios a los cuerpos celestes y meditaci\u00f3n lunar, dio el veredicto que, si no el m\u00e1s certero, result\u00f3 el m\u00e1s halag\u00fce\u00f1o para los padres:\n\n\u2014El ni\u00f1o se sabe celularmente un arist\u00f3crata, y, fiel a su origen, no tolera la idea de ser igual a los dem\u00e1s \u2014les explic\u00f3, quit\u00e1ndose las gafas \u00bfpara que fuera m\u00e1s notoria la lucecita inteligente que aparec\u00eda en sus pupilas al emitir un pron\u00f3stico?\u2014. Prefiere ser r\u00e9feri a jugador porque el que arbitra un partido es el que manda. \u00bfCre\u00edan ustedes que en ese rect\u00e1ngulo verde Joaquincito hace deporte? Error, error. Ejercita un ancestral apetito de dominaci\u00f3n, de singularidad y jerarqu\u00eda, que, sin duda, le corre por las venas.\n\nSollozando de felicidad, el padre sofoc\u00f3 a besos a su hijo, se declar\u00f3 hombre bienaventurado, y agreg\u00f3 un cero a los honorarios, ya de por s\u00ed regios, que le hab\u00eda pasado el profesor Ac\u00e9mila. Convencido de que esa man\u00eda de arbitrar los partidos de f\u00fatbol de sus compa\u00f1eros resultaba de avasalladores \u00edmpetus de subyugaci\u00f3n y prepotencia, que, m\u00e1s tarde, convertir\u00edan a su hijo en due\u00f1o del mundo (o, en el peor de los casos, del Per\u00fa), el industrial abandon\u00f3 muchas tardes su m\u00faltiple oficina para, debilidades de le\u00f3n que lagrimea viendo a su cachorro despedazar a la primera oveja, venir a su estadio privado de La Perla a gozar paternalmente viendo a Joaqu\u00edn, enfundado en el lindo uniforme que le hab\u00eda regalado, dando pitazos detr\u00e1s de esa bastarda confusi\u00f3n (\u00bflos jugadores?).\n\nDiez a\u00f1os despu\u00e9s, los confundidos padres no tuvieron m\u00e1s remedio que empezar a decirse que, tal vez, las profec\u00edas astrales hab\u00edan pecado de optimistas. Joaqu\u00edn Hinostroza Bellmont ten\u00eda ya dieciocho a\u00f1os y hab\u00eda llegado al \u00faltimo grado de la secundaria varios a\u00f1os despu\u00e9s que sus compa\u00f1eros del comienzo y s\u00f3lo gracias a la filantrop\u00eda familiar. Los genes de conquistador del mundo, que, seg\u00fan Lucio Ac\u00e9mila, se camuflaban bajo el inofensivo capricho de arbitrar f\u00fatbol, no aparec\u00edan por ninguna parte, y, en cambio, terriblemente se hac\u00eda inocultable que el hijo de arist\u00f3cratas era una calamidad sin remedio para todo lo que no fuera cobrar tiros libres. Su inteligencia, a juzgar por las cosas que dec\u00eda, lo colocaba, darwinianamente hablando, entre el oligofr\u00e9nico y el mono, y su falta de gracia, de ambiciones, de inter\u00e9s por todo lo que no era esa agitada actividad de r\u00e9feri, hac\u00edan de \u00e9l un ser profundamente soso.\n\nAhora, es verdad que en lo que concern\u00eda a su vicio primero (el segundo fue el alcohol) el muchacho denotaba algo que merec\u00eda llamarse talento. Su imparcialidad teratol\u00f3gica (\u00bfen el espacio sagrado de la cancha y el tiempo hechicero de la competencia?) le gan\u00f3 prestigio como \u00e1rbitro entre alumnos y profesores del Santa Mar\u00eda, y, tambi\u00e9n, gavil\u00e1n que desde la nube divisa bajo el algarrobo la rata que ser\u00e1 su almuerzo, su visi\u00f3n que le permit\u00eda infaliblemente detectar, a cualquier distancia y desde cualquier \u00e1ngulo, el artero puntapi\u00e9 del defensa a la canilla del delantero centro, o el vil codazo del alero al guardameta que saltaba con \u00e9l. Tambi\u00e9n eran ins\u00f3litas su omnisciencia de las reglas y la intuici\u00f3n feliz que le hac\u00eda suplir con decisiones rel\u00e1mpago los vac\u00edos reglamentarios. Su fama salt\u00f3 los muros del Santa Mar\u00eda y el arist\u00f3crata de La Perla comenz\u00f3 a arbitrar competencias inter-escolares, campeonatos de barrio, y, un d\u00eda se supo que, \u00bfen la cancha del Potao?, hab\u00eda reemplazado a un r\u00e9feri en un partido de segunda divisi\u00f3n.\n\nTerminado el colegio, se plante\u00f3 un problema a los abrumados progenitores: el futuro de Joaqu\u00edn. La idea de que fuera a la universidad fue apenadamente descartada, para evitar al muchacho humillaciones in\u00fatiles, complejos de inferioridad, y, a la fortuna familiar, nuevos forados en forma de donaciones. Un intento de hacerlo aprender idiomas desemboc\u00f3 en estrepitoso fracaso. Un a\u00f1o en Estados Unidos y otro en Francia no le ense\u00f1aron una sola palabra de ingl\u00e9s ni de franc\u00e9s, y, en cambio, tuberculizaron su de por s\u00ed raqu\u00edtico espa\u00f1ol. Cuando volvi\u00f3 a Lima, el fabricante de casimires opt\u00f3 por resignarse a que su hijo no ostentara ning\u00fan t\u00edtulo, y, lleno de desilusi\u00f3n, lo puso a trabajar en la maleza de empresas caseras. Los resultados fueron previsiblemente catastr\u00f3ficos. En dos a\u00f1os, sus actos u omisiones hab\u00edan quebrado dos hilander\u00edas, reducido al d\u00e9ficit la m\u00e1s floreciente firma del conglomerado \u2014una constructora de caminos\u2014 y las plantaciones de pimienta de la selva hab\u00edan sido carcomidas por plagas, aplastadas por avalanchas y ahogadas por inundaciones (lo que confirm\u00f3 que Joaquincito era tambi\u00e9n un f\u00falmine). Aturdido por la inconmensurable incompetencia de su hijo, herido en su amor propio, el padre perdi\u00f3 energ\u00edas, se volvi\u00f3 nihilista y descuid\u00f3 sus negocios que, en poco tiempo, fueron desangrados por \u00e1vidos lugartenientes, y contrajo un tic risible que consist\u00eda en sacar la lengua para tratar (\u00bfinsensatamente?) de lamerse la oreja. Su nerviosismo y desvelos lo arrojaron, siguiendo los pasos de su esposa, en manos de psiquiatras y psicoanalistas (\u00bfAlberto de Quinteros? \u00bfLucio Ac\u00e9mila?) que dieron r\u00e1pida cuenta de sus residuos de cordura y de plata.\n\nEl colapso econ\u00f3mico y la ruina mental de sus engendradores no pusieron al borde del suicidio a Joaqu\u00edn Hinostroza Bellmont. Viv\u00eda siempre en La Perla, en una residencia fantom\u00e1quica, que se hab\u00eda ido despintando, aherrumbrando, despoblando, perdiendo jardines y cancha de f\u00fatbol (para pagar deudas) y que hab\u00edan invadido la suciedad y las ara\u00f1as. El joven se pasaba el d\u00eda arbitrando los partidos callejeros que organizaban los vagabundos del barrio, en los descampados que separan Bellavista y La Perla. Fue en uno de esos matches disputados por ca\u00f3ticos palomillas, en plena v\u00eda p\u00fablica, donde un par de piedras hac\u00edan de arco y una ventana y un poste de l\u00edmites, y que Joaqu\u00edn \u2014principismo de elegante que se viste de baile para cenar en plena selva virgen\u2014 arbitraba como si fueran final de campeonato, que el hijo de arist\u00f3cratas conoci\u00f3 a la persona que har\u00eda de \u00e9l un cirroso y una estrella: \u00bfSarita Huanca Salaverr\u00eda?\n\nLa hab\u00eda visto jugar varias veces en esos partidos del mont\u00f3n e incluso le hab\u00eda cobrado muchas faltas por la agresividad con que arremet\u00eda contra el adversario. Le dec\u00edan Marimacho, pero ni por \u00e9sas se le hubiera ocurrido a Joaqu\u00edn que ese adolescente cetrino, calzado con viejas zapatillas, cubierto por unos blue jeans y una chompa rotosa, hubiera sido mujer. Lo descubri\u00f3 er\u00f3ticamente. Un d\u00eda, por haberla castigado con un penal indiscutible (Marimacho hab\u00eda metido un gol con bola y arquero), recibi\u00f3 como respuesta una mentada de madre.\n\n\u2014\u00bfQu\u00e9 has dicho? \u2014se indign\u00f3 el hijo de arist\u00f3cratas \u00bfpensando que en esos mismos momentos su madre estar\u00eda ingiriendo una p\u00edldora, paladeando una p\u00f3cima, soportando un agujazo?\u2014: Repite si eres hombre.\n\n\u2014No soy, pero repito \u2014repuso Marimacho. Y, honor de espartana capaz de ir a la hoguera por no dar su brazo a torcer, repiti\u00f3, enriquecida con adjetivos del arroyo, la mentada de madre.\n\nJoaqu\u00edn intent\u00f3 lanzarle un pu\u00f1ete, que s\u00f3lo dio en el aire, y, al instante, se vio arrojado al suelo por un cabezazo de Marimacho, quien cay\u00f3 sobre \u00e9l, golpe\u00e1ndolo con manos, pies, rodillas, codos. All\u00ed, forcejeos gimn\u00e1sticos sobre la lona que acaban pareciendo los apretones del amor, descubri\u00f3, estupefacto, erogenizado, eyaculante, que su adversario era mujer. La emoci\u00f3n que le produjeron los roces pugil\u00edsticos con esas turgencias inesperadas fue tan grande que cambi\u00f3 su vida. Ah\u00ed mismo, al amistarse despu\u00e9s de la pelea y saber que se llamaba Sarita Huanca Salaverr\u00eda, la invit\u00f3 al cine a ver Tarz\u00e1n, y una semana m\u00e1s tarde le propuso el altar. La negativa de Sarita a ser su esposa e incluso a dejarse besar empujaron cl\u00e1sicamente a Joaqu\u00edn a las cantinas. En poco tiempo, pas\u00f3 de rom\u00e1ntico que ahoga penas en whisky a alcoh\u00f3lico irredento que puede apagar su africana sed con kerosene.\n\n\u00bfQu\u00e9 despert\u00f3 en Joaqu\u00edn esa pasi\u00f3n por Sarita Huanca Salaverr\u00eda? Era joven y ten\u00eda un f\u00edsico esbelto de gallito, una piel curtida por la intemperie, un cerquillo bailar\u00edn, y, como jugadora de f\u00fatbol, no estaba mal. Por su manera de vestirse, las cosas que hac\u00eda y las personas que frecuentaba, parec\u00eda contrariada con su condici\u00f3n de mujer. \u00bfEra esto, tal vez \u2014vicio de originalidad, frenes\u00ed de extravagancia\u2014 lo que la hac\u00eda tan atractiva para el arist\u00f3crata? El primer d\u00eda que llev\u00f3 a Marimacho a la ruinosa casa de La Perla, sus padres, despu\u00e9s que la pareja hubo partido, se miraron asqueados. El ex rico encarcel\u00f3 en una frase la amargura de su esp\u00edritu: \u00abNo s\u00f3lo hemos creado a un est\u00fapido, sino, tambi\u00e9n, a un pervertido sexual\u00bb.\n\nSin embargo, Sarita Huanca Salaverr\u00eda, a la vez que alcoholiz\u00f3 a Joaqu\u00edn, fue el trampol\u00edn que lo ascendi\u00f3 de los partidos callejeros con pelota de trapo a los campeonatos del Estadio Nacional.\n\nMarimacho no se contentaba con rechazar la pasi\u00f3n del arist\u00f3crata; se complac\u00eda en hacerlo sufrir. Se dejaba invitar al cine, al f\u00fatbol, a los toros, a restaurantes, consent\u00eda en recibir costosos regalos (\u00bfen los que el enamorado dilapidaba los escombros del patrimonio familiar?), pero no permit\u00eda que Joaqu\u00edn le hablara de amor. Apenas intentaba \u00e9ste, timidez de doncel que enrojece al piropear a una flor, ator\u00e1ndose, decirle cu\u00e1nto la amaba, Sarita Huanca Salaverr\u00eda se pon\u00eda de pie, iracunda, lo her\u00eda con insultos de una soecidad bajopontina, y se mandaba mudar. Era entonces cuando Joaqu\u00edn comenzaba a beber, pasando de una cantina a otra y mezclando licores para obtener efectos prontos y explosivos. Fue un espect\u00e1culo corriente, para sus padres, verlo recogerse a la hora de las lechuzas, y cruzar las habitaciones de La Perla, dando traspi\u00e9s, perseguido por una estela de v\u00f3mitos. Cuando ya parec\u00eda a punto de desintegrarse en alcohol, una llamada de Sarita lo resucitaba. Conceb\u00eda nuevas esperanzas y se reanudaba el ciclo infernal. Demolidos por la amargura, el hombre del tic y la hipocondr\u00edaca murieron casi al mismo tiempo y fueron sepultados en un mausoleo del Cementerio Presb\u00edtero Maestro. La encogida residencia de La Perla, al igual que los bienes que sobreviv\u00edan, fueron rematados por acreedores o incautados por el Estado. Joaqu\u00edn Hinostroza Bellmont tuvo que ganarse la vida.\n\nTrat\u00e1ndose de quien se trataba (su pasado rug\u00eda que morir\u00eda de consunci\u00f3n o terminar\u00eda mendigo) lo hizo m\u00e1s que bien. \u00bfQu\u00e9 profesi\u00f3n eligi\u00f3? \u00a1\u00c1rbitro de f\u00fatbol! Acicateado por el hambre y el deseo de seguir festejando a la esquiva Sarita, comenz\u00f3 pidiendo unos soles a los mataperros cuyos partidos le ped\u00edan arbitrar, y, al ver que ellos, prorrate\u00e1ndose, se los daban, dos m\u00e1s dos son cuatro y cuatro y dos son seis, fue aumentando las tarifas y administr\u00e1ndose mejor. Como era conocida su habilidad en la cancha, consigui\u00f3 contratos en competencias juveniles, y, un d\u00eda, audazmente, se present\u00f3 en la Asociaci\u00f3n de \u00c1rbitros y Entrenadores de F\u00fatbol y solicit\u00f3 su inscripci\u00f3n. Pas\u00f3 los ex\u00e1menes con una brillantez que dej\u00f3 mareados a los que, a partir de entonces, pudo (\u00bfvanidosamente?) llamar colegas.\n\nLa aparici\u00f3n de Joaqu\u00edn Hinostroza Bellmont \u2014uniforme negro pespuntado de blanco, viserita verde en la frente, silbato plateado en la boca\u2014 en el Estadio Nacional de Jos\u00e9 D\u00edaz, estableci\u00f3 una efem\u00e9rides en el f\u00fatbol nacional. Un experimentado cronista deportivo lo dir\u00eda: \u00abCon \u00e9l ingresaron a las canchas la justicia inflexible y la inspiraci\u00f3n art\u00edstica\u00bb. Su correcci\u00f3n, su imparcialidad, su rapidez para descubrir la falta y su tino para sancionarla, su autoridad (los jugadores se dirig\u00edan siempre a \u00e9l bajando los ojos y trat\u00e1ndolo de don), y ese estado f\u00edsico que le permit\u00eda correr los noventa minutos del partido y no estar nunca a m\u00e1s de diez metros de la pelota, lo hicieron r\u00e1pidamente popular. Fue, como se dijo en un discurso, el \u00fanico r\u00e9feri nunca desobedecido por los jugadores ni agredido por los espectadores, y el \u00fanico al que, despu\u00e9s de cada partido, ovacionaban las tribunas.\n\n\u00bfNac\u00edan esos talentos y esfuerzos s\u00f3lo de una sobresaliente conciencia profesional? Tambi\u00e9n. Pero la raz\u00f3n profunda era que Joaqu\u00edn Hinostroza Bellmont pretend\u00eda, con su magia arbitral, secreto de muchacho que triunfa en Europa y vive amargo porque lo que quer\u00eda era el aplauso de su pueblecito andino, impresionar a Marimacho. Segu\u00edan vi\u00e9ndose, casi a diario, y la escabrosa maledicencia popular los cre\u00eda amantes. En realidad, pese a su tes\u00f3n sentimental, inc\u00f3lume a lo largo de los a\u00f1os, el r\u00e9feri no hab\u00eda conseguido vencer la resistencia de Sarita.\n\n\u00c9sta, un d\u00eda, luego de rescatarlo del suelo de una cantina del Callao, de llevarlo a la pensi\u00f3n del centro donde viv\u00eda, de limpiarle las manchas de escupitajo y de aserr\u00edn y de acostarlo, le cont\u00f3 el secreto de su vida. Joaqu\u00edn Hinostroza Bellmont supo as\u00ed, lividez de hombre que ha recibido el beso del vampiro, que en la primera juventud de esa muchacha hab\u00eda un amor maldito y un terremoto conyugal. En efecto, entre Sarita y su hermano (\u00bfRichard?) hab\u00eda brotado un enamoramiento tr\u00e1gico, que \u2014cataratas de fuego, lluvia de veneno sobre la humanidad\u2014 hab\u00eda cristalizado en embarazo. Habiendo contra\u00eddo astutamente matrimonio con un gal\u00e1n al que antes desairaba (\u00bfel Pelirrojo Ant\u00fanez? \u00bfLuis Marroqu\u00edn?) para que el hijo del incesto tuviera un apellido impoluto, el joven y dichoso marido, sin embargo \u2014cola del diablo que se mete en la olla y arruina el pastel\u2014, hab\u00eda descubierto a tiempo la supercher\u00eda y repudiado a la tramposa que quer\u00eda contrabandearle un entenado como hijo. Obligada a abortar, Sarita huy\u00f3 de su familia encopetada, de su barrio residencial, de su apellido resonante, y, convertida en vagabunda, en los descampados de Bellavista y La Perla hab\u00eda adquirido la personalidad y el apodo de Marimacho. Desde entonces, hab\u00eda jurado no entregarse nunca m\u00e1s a un hombre y vivir siempre, para todos los efectos pr\u00e1cticos (\u00bfsalvo, ay, el de los espermatozoides?), como var\u00f3n.\n\nConocer la tragedia, aderezada de sacrilegio, transgresi\u00f3n de tab\u00faes, pisoteo de la moral civil y de mandamientos religiosos, de Sarita Huanca Salaverr\u00eda, no cancel\u00f3 la pasi\u00f3n amorosa de Joaqu\u00edn Hinostroza Bellmont; la robusteci\u00f3. El hombre de La Perla concibi\u00f3, incluso, la idea de curar a Marimacho de sus traumas y reconciliarla con la sociedad y los hombres; quiso hacer de ella, otra vez, una lime\u00f1a femenina y coqueta, p\u00edcara y salerosa \u00bfcomo la Perricholi?\n\nAl mismo tiempo que su fama crec\u00eda y era solicitado para arbitrar partidos internacionales en Lima y en el extranjero, y recib\u00eda ofertas para trabajar en M\u00e9xico, Brasil, Colombia, Venezuela, que \u00e9l, patriotismo de sabio que renuncia a las computadoras de Nueva York para seguir experimentando con las cobayas tuberculosas de San Fernando, siempre rechaz\u00f3, su asedio al coraz\u00f3n de la incestuosa se hizo m\u00e1s tenaz.\n\nY le pareci\u00f3 entrever algunas se\u00f1ales \u2014humo apache en las colinas, tam-tams en la floresta africana\u2014 de que Sarita Huanca Salaverr\u00eda pod\u00eda ceder. Una tarde, luego de un caf\u00e9 con medialunas en el Hait\u00ed de la plaza de Armas, Joaqu\u00edn pudo retener entre las suyas la diestra de la muchacha por m\u00e1s de un minuto (exacto: su cabeza de \u00e1rbitro lo cronometr\u00f3). Poco despu\u00e9s, hubo un partido en que la selecci\u00f3n nacional se enfrent\u00f3 a una pandilla de homicidas, de un pa\u00eds de escaso renombre \u2014\u00bfArgentina o algo as\u00ed?\u2014, que se presentaron a jugar con los zapatos acorazados de clavos, y rodilleras y coderas que, en verdad, eran instrumentos para malherir al adversario. Sin atender a sus argumentos (por lo dem\u00e1s ciertos) de que en su pa\u00eds era costumbre jugar al f\u00fatbol as\u00ed \u2014\u00bfcabece\u00e1ndolo con la tortura y el crimen?\u2014, Joaqu\u00edn Hinostroza Bellmont los fue expulsando de la cancha, hasta que el equipo peruano gan\u00f3 t\u00e9cnicamente por falta de competidores. El \u00e1rbitro, por supuesto, sali\u00f3 en hombros de la multitud y Sarita Huanca Salaverr\u00eda, cuando estuvieron solos \u2014\u00bfarranque de peruanidad? \u00bfsensibler\u00eda deportiva?\u2014, le ech\u00f3 los brazos al cuello y lo bes\u00f3. Una vez que estuvo enfermo (la cirrosis, discreta, fat\u00eddica, iba mineralizando el h\u00edgado del hombre de los estadios y comenzaba a provocarle crisis peri\u00f3dicas) lo atendi\u00f3 sin moverse de su lado, la semana que permaneci\u00f3 en el Hospital Carri\u00f3n y Joaqu\u00edn la vio, una noche, derramar algunas l\u00e1grimas \u00bfpor \u00e9l? Todo esto lo envalentonaba y cada d\u00eda le propon\u00eda, con argumentos renovados, matrimonio. Era in\u00fatil. Sarita Huanca Salaverr\u00eda asist\u00eda a todos los partidos que \u00e9l interpretaba (los cronistas comparaban sus arbitrajes al manejo de una sinfon\u00eda), lo acompa\u00f1aba al extranjero y hasta se hab\u00eda mudado a la Pensi\u00f3n Colonial donde Joaqu\u00edn viv\u00eda con su hermana pianista y sus ancianos padres. Pero se negaba a que esa fraternidad dejara de ser casta y se convirtiera en refocilo. La incertidumbre, margarita cuyos p\u00e9talos no se termina jam\u00e1s de deshojar, fue agravando el alcoholismo de Joaqu\u00edn Hinostroza Bellmont, a quien acab\u00f3 por verse m\u00e1s borracho que sobrio.\n\nEl alcohol fue el tal\u00f3n de Aquiles de su vida profesional, el lastre que, dec\u00edan los entendidos, le impidi\u00f3 arbitrar en Europa. \u00bfC\u00f3mo se explica, de otra parte, que un hombre que beb\u00eda tanto hubiera podido ejercer una profesi\u00f3n de tantos rigores f\u00edsicos? El hecho es que, enigmas que empiedran la historia, desenvolvi\u00f3 ambas vocaciones al mismo tiempo, y, a partir de la treintena, ambas fueron simult\u00e1neas: Joaqu\u00edn Hinostroza Bellmont comenz\u00f3 a arbitrar partidos borracho como una cuba y a seguirlos arbitrando imaginativamente en las cantinas.\n\nEl alcohol no amortiguaba su talento: ni empa\u00f1aba su vista, ni debilitaba su autoridad, ni demoraba su carrera. Es verdad que, alguna vez, en medio de un partido se le vio atacado de hipos, y que, calumnias que enturbian el aire y acuchillan la virtud, se aseguraba que una vez, aquejado de sahariana sed, arrebat\u00f3 a un enfermero que corr\u00eda a auxiliar a un jugador una botella de linimento y se la bebi\u00f3 como agua fresca. Pero estos episodios \u2014anecdotario pintoresco, mitolog\u00eda del genio\u2014 no interrumpieron su carrera de \u00e9xitos.\n\nAs\u00ed, entre los atronadores aplausos del Estadio y las penitentes borracheras con que trataba de calmar los remordimientos \u2014pinzas de inquisidor que hurgan las carnes, potro que descoyunta los huesos\u2014, en su alma de misionero de la verdadera fe (\u00bftestigos de Jehov\u00e1?), por haber violado inopinadamente, en una noche loca de la juventud, a una menor de La Victoria (\u00bfSarita Huanca Salaverr\u00eda?), lleg\u00f3 Joaqu\u00edn Hinostroza Bellmont a la flor de la edad: la cincuentena. Era un hombre de frente ancha, nariz aguile\u00f1a, mirada penetrante, rectitud y bondad en el esp\u00edritu, que hab\u00eda trepado a la cumbre de su profesi\u00f3n.\n\nEn esas circunstancias le toc\u00f3 a Lima ser escenario del m\u00e1s importante encuentro futbol\u00edstico del medio siglo, la final del campeonato sudamericano entre dos equipos que, en las eliminatorias, hab\u00edan, cada uno por su lado, infligido deshonrosas goleadas a sus adversarios: Bolivia y Per\u00fa. Aunque la costumbre aconsejaba que arbitrara ese partido un r\u00e9feri de pa\u00eds neutral, los dos equipos, y, con especial insistencia \u2014hidalgu\u00eda del Altiplano, nobleza colla, pundonor aymara\u2014, los extranjeros, exigieron que fuera el famoso Joaqu\u00edn Hinostroza Marroqu\u00edn quien arbitrara el partido. Y, como jugadores, suplentes y entrenadores amenazaron con una huelga si no se les satisfac\u00eda, la Federaci\u00f3n accedi\u00f3 y el Testigo de Jehov\u00e1 recibi\u00f3 la misi\u00f3n de gobernar ese match que todos profetizaban memorable.\n\nLas ac\u00e9rrimas nubes grises de Lima se despejaron ese domingo para que el sol calentara el encuentro. Muchas personas hab\u00edan pasado la noche a la intemperie, con la ilusi\u00f3n de conseguir entradas (era sabido que estaban agotadas hac\u00eda un mes). Desde el amanecer, todo el entorno del Estadio Nacional se volvi\u00f3 un hervor de gentes en pos de revendedores y dispuestas a cualquier delito por entrar. Dos horas antes del partido, en el Estadio no cab\u00eda un alfiler. Varios centenares de ciudadanos del gran pa\u00eds del sur (\u00bfBolivia?), llegados hasta Lima desde sus limpias alturas en avi\u00f3n, auto y a pie, se hab\u00edan concentrado en la Tribuna de Oriente. Los v\u00edtores y maquinitas de visitantes y abor\u00edgenes caldeaban el ambiente, en espera de los equipos.\n\nAnte la magnitud de la concentraci\u00f3n popular, las autoridades hab\u00edan tomado precauciones. La m\u00e1s c\u00e9lebre brigada de la Guardia Civil, aquella que, en pocos meses \u2014hero\u00edsmo y abnegaci\u00f3n, audacia y urbanidad\u2014 hab\u00eda limpiado de delincuentes y malvados el Callao, fue tra\u00edda a Lima a fin de garantizar la seguridad y la convivencia ciudadanas en la tribuna y en las canchas. Su jefe, el c\u00e9lebre capit\u00e1n Lituma, terror del crimen, se paseaba afiebradamente por el Estadio y recorr\u00eda las puertas y calles adyacentes, verificando si las patrullas permanec\u00edan en sus sitios y dictando inspiradas instrucciones a su aguerrido adjunto, el sargento Jaime Concha.\n\nEn la Tribuna Occidental, magullados entre la masa rugiente y casi sin respiraci\u00f3n, se encontraban al darse el pitazo inicial, adem\u00e1s de Sarita Huanca Salaverr\u00eda \u2014quien, masoquismo de v\u00edctima que vive prendada de su violador, no se perd\u00eda jam\u00e1s los partidos que arbitraba\u2014, el venerable don Sebasti\u00e1n Bergua, recientemente incorporado del lecho de dolor donde yac\u00eda por las cuchilladas que recibi\u00f3 del propagandista m\u00e9dico Luis Marroqu\u00edn Bellmont (\u00bfquien estaba en el Estadio, en la Tribuna Norte, por un permiso especial\u00edsimo de la Direcci\u00f3n de Prisiones?), su esposa Margarita y su hija Rosa, ya del todo restablecida de los mordiscos que recibiera \u2014oh infausto amanecer selv\u00e1tico\u2014 de una camada de ratas.\n\nNada hac\u00eda presagiar la tragedia, cuando Joaqu\u00edn Hinostroza (\u00bfTello? \u00bfDelf\u00edn?) \u2014quien, como de costumbre, hab\u00eda sido obligado a dar la vuelta ol\u00edmpica agradeciendo aplausos\u2014, apuesto, \u00e1gil, arranc\u00f3 el partido. Al contrario, todo transcurr\u00eda en una atm\u00f3sfera de entusiasmo y caballerosidad: las acciones de los jugadores, los aplausos de las barras que premiaban los avances de los delanteros y las atajadas de los guardametas. Desde el primer momento fue notorio que se cumplir\u00edan los or\u00e1culos: el juego estaba equilibrado y aunque pundonoroso era recio. M\u00e1s creativo que nunca, Joaqu\u00edn Hinostroza (\u00bfAbril?) se deslizaba sobre el c\u00e9sped como en patines, sin estorbar a los jugadores y coloc\u00e1ndose siempre en el \u00e1ngulo m\u00e1s afortunado, y sus decisiones, severas pero justas, imped\u00edan que, ardores de la contienda que la vuelven gresca, el partido degenerara en violencia. Pero, fronteras de la condici\u00f3n humana, ni un santo Testigo de Jehov\u00e1 pod\u00eda impedir que se cumpliera lo que, indiferencia de fakir, flema de ingl\u00e9s, hab\u00eda urdido el destino.\n\nEl mecanismo infernal irremisiblemente comenz\u00f3 a marchar, en el segundo tiempo, cuando los equipos iban uno a uno y los espectadores se hallaban af\u00f3nicos y con las manos ardiendo. El capit\u00e1n Lituma y el sargento Concha se dec\u00edan, c\u00e1ndidamente, que todo iba bien: ni un solo incidente \u2014robo, pelea, extrav\u00edo de ni\u00f1o\u2014 hab\u00eda venido a malograr la tarde.\n\nPero he aqu\u00ed que a las cuatro y trece minutos, a los cincuenta mil espectadores les fue dado conocer lo ins\u00f3lito. Del fondo m\u00e1s promiscuo de la Tribuna Sur, de pronto \u2014negro, flaco, alt\u00edsimo, dient\u00f3n\u2014, emergi\u00f3 un hombre que escal\u00f3 livianamente el enrejado e irrumpi\u00f3 en la cancha dando gritos incomprensibles. No sorprendi\u00f3 tanto a la gente verlo casi desnudo \u2014llevaba apenas un taparrabos colgado de la cintura\u2014 como que, de pies a cabeza, tuviera el cuerpo lleno de incisiones. Un ronquido de p\u00e1nico estremeci\u00f3 las tribunas; todos comprendieron que el tatuado se propon\u00eda victimar al \u00e1rbitro. No hab\u00eda duda: el gigante aullador corr\u00eda directamente hacia el \u00eddolo de la afici\u00f3n (\u00bfGumercindo Hinostroza Delf\u00edn?), quien, absorto en su arte, no lo hab\u00eda visto y segu\u00eda modelando el partido.\n\n\u00bfQui\u00e9n era el inminente agresor? \u00bfTal vez el polizonte aquel, llegado misteriosamente al Callao, y sorprendido por la ronda nocturna? \u00bfEra el mismo infeliz al que las autoridades hab\u00edan eutan\u00e1sicamente decidido ejecutar y al que el sargento (\u00bfConcha?) perdon\u00f3 la vida en una noche oscura? Ni el capit\u00e1n Lituma ni el sargento Concha tuvieron tiempo de averiguarlo. Comprendiendo que, si no proced\u00edan en el acto, una gloria nacional pod\u00eda sufrir un atentado, el capit\u00e1n \u2014superior y subordinado ten\u00edan un m\u00e9todo para entenderse con movimientos de pesta\u00f1as\u2014 orden\u00f3 al sargento que actuara. Jaime Concha, entonces, sin ponerse de pie, sac\u00f3 su pistola y dispar\u00f3 sus doce tiros, que fueron todos a incrustarse (cincuenta metros m\u00e1s all\u00e1) en distintas partes del calato. De este modo, el sargento ven\u00eda a cumplir, m\u00e1s vale tarde que nunca dice el refr\u00e1n, la orden recibida, porque, en efecto, \u00a1se trataba del polizonte del Callao!\n\nBast\u00f3 que viera acribillado a balazos al potencial verdugo de su \u00eddolo, al que un instante atr\u00e1s odiaba, para que, inmediatamente \u2014veleidades de fr\u00edvola sentimental, coqueter\u00edas de hembra mudable\u2014, la muchedumbre se solidarizara con \u00e9l, lo convirtiera en v\u00edctima, y se enemistara con la Guardia Civil. Una silbatina que ensordeci\u00f3 a los p\u00e1jaros del cielo se elev\u00f3 por los aires en la que las tribunas de sombra y de sol entonaban su c\u00f3lera por el espect\u00e1culo del negro que, all\u00e1, sobre la tierra, se iba quedando sin sangre por doce agujeros. Los balazos hab\u00edan desconcertado a los peones, pero el Gran Hinostroza (\u00bfT\u00e9llez Unz\u00e1tegui?), fiel a s\u00ed mismo, no hab\u00eda permitido que se interrumpiera la fiesta, y segu\u00eda luci\u00e9ndose, alrededor del cad\u00e1ver del espont\u00e1neo, sordo ante la silbatina, a la que ahora se a\u00f1ad\u00edan interjecciones, alaridos, insultos. Ya comenzaban a caer \u2014multicolores, volanderos\u2014 los emisarios del que pronto ser\u00eda diluvio de cojines contra el destacamento policial del capit\u00e1n Lituma. \u00c9ste olfate\u00f3 el hurac\u00e1n y decidi\u00f3 actuar r\u00e1pido. Orden\u00f3 que los guardias prepararan las granadas lacrim\u00f3genas. Quer\u00eda evitar una sangr\u00eda a toda costa. Y, unos momentos despu\u00e9s, cuando ya las barreras hab\u00edan sido traspasadas en muchos puntos del redondel, y, aqu\u00ed y all\u00e1, taur\u00f3filos enardecidos se precipitaban hacia el coso con belicosidad, orden\u00f3 a sus hombres que rociaran el per\u00edmetro con unas cuantas granadas. Las l\u00e1grimas y los estornudos, pensaba, calmar\u00edan a los iracundos y la paz reinar\u00eda de nuevo en la plaza de Acho apenas el viento disipara los efluvios qu\u00edmicos. Dispuso asimismo que un grupo de cuatro guardias rodeara al sargento Jaime Concha, quien se hab\u00eda convertido en el objetivo de los exaltados: visiblemente, estaban decididos a lincharlo, aunque para ello tuvieran que enfrentarse al toro.\n\nPero el capit\u00e1n Lituma olvidaba algo esencial: \u00e9l mismo, dos horas atr\u00e1s, para impedir que los aficionados sin entradas que rondaban la plaza, amenazantes, intentaran invadir el local por la fuerza, hab\u00eda ordenado bajar las rejas y cortinas met\u00e1licas que cerraban el acceso a los tendidos. As\u00ed, cuando, puntuales ejecutores de \u00f3rdenes, los guardias regalaron al p\u00fablico una bandada de granadas lacrim\u00f3genas, y aqu\u00ed y all\u00e1, en pocos segundos, se elevaron pestilentes humaredas en los grader\u00edos, la reacci\u00f3n de los espectadores fue huir. Atropelladamente, saltando, empujando, mientras se cubr\u00edan la boca con un pa\u00f1uelo y comenzaban a llorar, corrieron hacia las salidas. Las correntadas humanas se vieron frenadas por las cortinas y rejas met\u00e1licas que las clausuraban. \u00bfFrenadas? S\u00f3lo unos segundos, los suficientes para que las primeras filas de cada columna, convertidas en arietes por la presi\u00f3n de quienes ven\u00edan atr\u00e1s, las abollaran, hincharan, rajaran y arrancaran de cuajo. De este modo, los vecinos del R\u00edmac que, por azar, transitaban ese domingo alrededor de la plaza de toros a las cuatro y treinta minutos de la tarde pudieron apreciar un espect\u00e1culo b\u00e1rbaramente original: de pronto, en medio de un crepitar ag\u00f3nico, las puertas de Acho volaban en pedazos y comenzaban a escupir cad\u00e1veres apachurrados, que, bien vengas mal si vienes solo, eran encima pisoteados por la muchedumbre enloquecida que escapaba por los boquetes sanguinolentos.\n\nEntre las primeras v\u00edctimas del holocausto bajopontino, les cupo figurar a los introductores de los Testigos de Jehov\u00e1 en el Per\u00fa: el moqueguano don Sebasti\u00e1n Bergua, su esposa Margarita, y su hija Rosa, la eximia tocadora de flauta dulce. Perdi\u00f3 a la religiosa familia lo que hubiera debido salvarla: su prudencia. Porque, apenas ocurrido el incidente del can\u00edbal espont\u00e1neo, cuando \u00e9ste acababa de ser despedazado por el corn\u00fapeta, don Sebasti\u00e1n Bergua, cejas enarcadas y dedo dictatorial, hab\u00eda ordenado a su tribu: \u00abEn retirada\u00bb. No era miedo, palabra que el predicador desconoc\u00eda, sino buen tino, la idea que ni \u00e9l ni sus parientes deb\u00edan verse mezclados en ning\u00fan esc\u00e1ndalo, para evitar que, amparados en ese pretexto, los enemigos trataran de enlodar el nombre de su fe. As\u00ed, la familia Bergua, apresuradamente, abandon\u00f3 su tendido de sol y bajaba las gradas hacia la salida cuando estallaron las granadas lacrim\u00f3genas. Se hallaban los tres, beat\u00edficos, junto a la cortina met\u00e1lica n\u00famero seis, esperando que la levantaran, cuando vieron irrumpir a sus espaldas, tronante y lacrimal, a la multitud. No tuvieron tiempo de arrepentirse de los pecados que no ten\u00edan cuando fueron literalmente desintegrados (\u00bfhechos pur\u00e9, sopa humana?) contra la cortina met\u00e1lica, por la masa empavorecida. Un segundo antes de pasar a esa otra vida que \u00e9l negaba, don Sebasti\u00e1n alcanz\u00f3 todav\u00eda a gritar, terco, creyente y heterodoxo: \u00abEl Cristo muri\u00f3 en un \u00e1rbol, no en una cruz\u00bb.\n\nLa muerte del desequilibrado acuchillador de don Sebasti\u00e1n Bergua, y violador de do\u00f1a Margarita y de la artista, fue, \u00bfcabr\u00eda la expresi\u00f3n?, menos injusta. Porque, estallada la tragedia, el joven Marroqu\u00edn Delf\u00edn crey\u00f3 llegada su oportunidad: en medio de la confusi\u00f3n, huir\u00eda del agente que la Direcci\u00f3n de Prisiones le hab\u00eda destinado como acompa\u00f1ante para que viera la hist\u00f3rica corrida, y escapar\u00eda de Lima, del Per\u00fa, y, en el extranjero, con otro nombre, iniciar\u00eda una nueva vida de locura y cr\u00edmenes. Ilusiones que se pulverizar\u00edan cinco minutos despu\u00e9s, cuando, en la puerta de salida n\u00famero cinco, a (\u00bfLucho? \u00bfEzequiel?) Marroqu\u00edn Delf\u00edn y al agente de prisiones Chumpitaz, que lo ten\u00eda de la mano, les toc\u00f3 el dudoso honor de formar parte de la primera fila de taur\u00f3filos triturados por la multitud. (Los dedos entrelazados del polic\u00eda y el propagandista m\u00e9dico, aunque cad\u00e1veres, dieron que hablar.)\n\nEl deceso de Sarita Huanca Salaverr\u00eda tuvo, al menos, la elegancia de ser menos promiscuo. Constituy\u00f3 un caso de malentendido garrafal, de equivocada evaluaci\u00f3n de actos e intenciones por parte de la autoridad. Al estallar los incidentes, ver al can\u00edbal corneado, los humos de las granadas, o\u00edr los aullidos de los fracturados, la muchacha de Tingo Mar\u00eda decidi\u00f3 que, pasi\u00f3n de amor que quita el miedo a la muerte, deb\u00eda estar junto al hombre que amaba. A la inversa de la afici\u00f3n, entonces, baj\u00f3 hacia el redondel, lo que la salv\u00f3 de perecer aplastada. Pero no la salv\u00f3 de la mirada de \u00e1guila del capit\u00e1n Lituma, quien advirtiendo, entre las nubes de gas que se expand\u00edan, una figurilla incierta y desalada, que saltaba el burladero y corr\u00eda hacia el diestro (quien, pese a todo, segu\u00eda citando al animal y haciendo pases de rodillas), y convencido de que su obligaci\u00f3n era impedir, mientras le quedara un h\u00e1lito de vida, que el matador fuera agredido, sac\u00f3 su rev\u00f3lver y, de tres r\u00e1pidos disparos, cort\u00f3 en seco la carrera y la vida de la enamorada: Sarita vino a caer muerta a los pies mismos de Gumercindo Bellmont.\n\nEl hombre de La Perla fue el \u00fanico, entre los muertos de esa tarde griega, que falleci\u00f3 de muerte natural. Si se puede llamar natural al fen\u00f3meno, ins\u00f3lito en tiempos prosaicos, de un hombre al que el espect\u00e1culo de su bienamada, muerta a sus pies, le paraliza el coraz\u00f3n y mata. Cay\u00f3 junto a Sarita y alcanzaron los dos, con el \u00faltimo aliento, a estrecharse y entrar as\u00ed, unidos, en la noche de los amantes desgraciados (\u00bfcomo ciertos Julieta y Romeo?)...\n\nPara entonces, el custodio del orden de inmaculada foja de servicios, considerando con melancol\u00eda que, pese a su experiencia y sagacidad, el orden no s\u00f3lo hab\u00eda sido alterado sino que la plaza de Acho y alrededores se hab\u00edan convertido en un cementerio de insepultos cad\u00e1veres, utiliz\u00f3 la \u00faltima bala que le quedaba para, lobo de mar que acompa\u00f1a su barco al fondo del oc\u00e9ano, volarse los sesos y acabar (viril ya que no exitosamente) su biograf\u00eda. Apenas vieron perecer a su jefe, la moral de los guardias se descalabr\u00f3; olvidaron la disciplina, el esp\u00edritu de cuerpo, el amor a la instituci\u00f3n, y s\u00f3lo pensaron en quitarse los uniformes, disimularse dentro de las ropas civiles que arranchaban a los muertos y escapar. Varios lo consiguieron. Pero no Jaime Concha, a quien los sobrevivientes, despu\u00e9s de castrar, ahorcaron con su propio correaje de cuero en el travesa\u00f1o del toril. All\u00ed qued\u00f3 el sano lector de Pato Donalds, el diligente centuri\u00f3n, columpi\u00e1ndose bajo el cielo de Lima, que, \u00bfqueriendo ponerse a tono con lo sucedido?, se hab\u00eda encrespado de nubes y comenzaba a llorar su gar\u00faa de invierno...\n\n\u00bfTerminar\u00eda as\u00ed, en dantesca carnicer\u00eda, esta historia? \u00bfO, como la Paloma F\u00e9nix (\u00bfla Gallina?), renacer\u00eda de sus cenizas con nuevos episodios y personajes recalcitrantes? \u00bfQu\u00e9 ocurrir\u00eda con esta tragedia taurina?\n\n### XVII\n\nPARTIMOS DE Lima a las nueve de la ma\u00f1ana, en un colectivo que tomamos en el parque Universitario. La t\u00eda Julia hab\u00eda salido de casa de mis t\u00edos con el pretexto de hacer las \u00faltimas compras antes de su viaje, y yo, de la de mis abuelos, como si fuera a trabajar a la radio. Ella hab\u00eda metido en una bolsa un camis\u00f3n de dormir y una muda de ropa interior; yo llevaba, en los bolsillos, mi escobilla de dientes, un peine y una maquinilla de afeitar (que, la verdad, a\u00fan no me serv\u00eda de gran cosa).\n\nPascual y Javier estaban esper\u00e1ndonos en el parque Universitario y hab\u00edan comprado los pasajes. Por suerte, no se present\u00f3 ning\u00fan otro viajero. Pascual y Javier, muy discretos, se sentaron delante, con el chofer, y nos dejaron el asiento de atr\u00e1s a la t\u00eda Julia y a m\u00ed. Era una ma\u00f1ana de invierno, t\u00edpica, de cielo encapotado y gar\u00faa continua, que nos escolt\u00f3 buena parte del desierto. Casi todo el viaje, la t\u00eda Julia y yo estuvimos bes\u00e1ndonos, apasionadamente, estrech\u00e1ndonos las manos, sin hablar, mientras o\u00edamos, mezclado al ruido del motor, el rumor de la conversaci\u00f3n entre Pascual y Javier, y, a veces, algunos comentarios del chofer. Llegamos a Chincha a las once y media de la ma\u00f1ana, con un sol espl\u00e9ndido y un calorcito delicioso. El cielo limpio, la luminosidad del aire, la algarab\u00eda de las calles repletas de gente, todo parec\u00eda de buen ag\u00fcero. La t\u00eda Julia sonre\u00eda, contenta.\n\nMientras Pascual y Javier se adelantaban a la municipalidad a ver si todo estaba listo, la t\u00eda Julia y yo fuimos a instalarnos en el Hotel Sudamericano. Era una vieja casa de un solo piso, de madera y adobes, con un patio techado que hac\u00eda las veces de comedor, y una docena de cuartitos alineados a ambos lados de un pasillo de baldosas, como un burdel. El hombre del mostrador nos pidi\u00f3 papeles; se content\u00f3 con mi carnet de periodista, y, al poner yo \u00aby se\u00f1ora\u00bb al lado de mi apellido, se limit\u00f3 a echar a la t\u00eda Julia una ojeada burlona. El cuartito que nos dieron ten\u00eda unas losetas despanzurradas por las que se ve\u00eda la tierra, una cama doble y hundida con una colcha de rombos verdes, una silleta de paja y unos clavos gordos en la pared para colgar la ropa. Apenas entramos, nos abrazamos con ardor y estuvimos bes\u00e1ndonos y acarici\u00e1ndonos, hasta que la t\u00eda Julia me apart\u00f3, ri\u00e9ndose:\n\n\u2014Alto ah\u00ed, Varguitas, primero tenemos que casarnos.\n\nEstaba arrebatada, con los ojos brillantes y alegres, y yo sent\u00eda que la quer\u00eda mucho, estaba feliz de casarme con ella, y, mientras esperaba que se lavara las manos y peinara, en el ba\u00f1o com\u00fan del corredor, me juraba que no ser\u00edamos como todos los matrimonios que conoc\u00eda, una calamidad m\u00e1s, sino que vivir\u00edamos siempre felices, y que casarme no me impedir\u00eda llegar a ser alg\u00fan d\u00eda un escritor. La t\u00eda Julia sali\u00f3 por fin y fuimos andando, de la mano, a la municipalidad.\n\nEncontramos a Pascual y a Javier en la puerta de una bodega, tomando un refresco. El alcalde hab\u00eda ido a presidir una inauguraci\u00f3n, pero ya volver\u00eda. Les pregunt\u00e9 si estaban absolutamente seguros de haber quedado con el pariente de Pascual en que nos casar\u00eda a mediod\u00eda y ellos se burlaron de m\u00ed. Javier hizo unas bromas sobre el novio impaciente y trajo a colaci\u00f3n un oportuno refr\u00e1n: el que espera desespera. Para hacer tiempo, los cuatro dimos unas vueltas bajo los altos eucaliptos y los robles de la plaza de Armas. Hab\u00eda unos muchachos correteando y unos viejos que se hac\u00edan lustrar los zapatos mientras le\u00edan los peri\u00f3dicos de Lima. Media hora despu\u00e9s est\u00e1bamos de regreso en la municipalidad. El secretario, un hombrecito flaco y con anteojos muy anchos, nos dio una mala noticia: el alcalde hab\u00eda vuelto de la inauguraci\u00f3n, pero se hab\u00eda ido a almorzar a El Sol de Chincha.\n\n\u2014\u00bfNo le avis\u00f3 usted que lo esper\u00e1bamos, para la boda? \u2014lo reprendi\u00f3 Javier.\n\n\u2014Estaba con una comitiva y no era el momento \u2014dijo el secretario, con aire de conocedor de la etiqueta.\n\n\u2014Vamos a buscarlo al restaurante y nos lo traemos \u2014me tranquiliz\u00f3 Pascual\u2014. No se preocupe, don Mario.\n\nPreguntando, encontramos El Sol de Chincha en las vecindades de la plaza. Era un restaurante criollo, con mesitas sin manteles, y una cocina al fondo, que chisporroteaba y humeaba, y en torno a la cual unas mujeres manipulaban ollas de cobre, peroles y fuentes olorosas. Hab\u00eda una radiola a todo volumen, tocando un vals, y se ve\u00eda mucha gente. Cuando la t\u00eda Julia comenzaba a decir, en la puerta, que tal vez ser\u00eda m\u00e1s prudente esperar que el alcalde terminara el almuerzo, \u00e9ste reconoci\u00f3 a Pascual, desde una esquina, y lo llam\u00f3. Vimos al redactor de Panamericana darse de abrazos con un hombre joven, medio rubio, que se puso de pie en una mesa donde hab\u00eda media docena de comensales, todos hombres, y otras tantas botellas de cerveza. Pascual nos hizo se\u00f1as de que nos acerc\u00e1ramos.\n\n\u2014Claro, los novios, me hab\u00eda olvidado por completo \u2014dijo el alcalde, estrech\u00e1ndonos la mano y calibrando a la t\u00eda Julia de arriba abajo, con una mirada de experto. Se volvi\u00f3 a sus compa\u00f1eros, que lo contemplaban servilmente, y les cont\u00f3, en voz alta, para hacerse o\u00edr por sobre el vals\u2014: Estos dos acaban de fugarse de Lima y yo los voy a casar.\n\nHubo risas, aplausos, manos que se estiraban hacia nosotros y el alcalde exigi\u00f3 que nos sent\u00e1ramos con ellos y pidi\u00f3 m\u00e1s cerveza para brindar por nuestra felicidad.\n\n\u2014Pero nada de ponerse juntos, para eso tendr\u00e1n toda la vida \u2014dijo, euf\u00f3rico, cogiendo a la t\u00eda Julia del brazo e instal\u00e1ndola junto a \u00e9l\u2014. La novia aqu\u00ed, a mi lado, que felizmente no est\u00e1 mi mujer.\n\nLa comitiva lo festej\u00f3. Eran mayores que el alcalde, comerciantes o agricultores vestidos de fiesta, y todos parec\u00edan tan borrachos como \u00e9l. Algunos conoc\u00edan a Pascual y le preguntaban sobre su vida en Lima y cu\u00e1ndo volver\u00eda a la tierra. Sentado junto a Javier, en un extremo de la mesa, yo procuraba sonre\u00edr, tomaba traguitos de una cerveza medio tibia, y contaba los minutos. Muy pronto, el alcalde y la comitiva se desinteresaron de nosotros. Se suced\u00edan las botellas, primero solas, despu\u00e9s acompa\u00f1adas de cebiche y de un sudado de corvina, de unos alfajores, y, luego, otra vez solas. Nadie recordaba el matrimonio, ni siquiera Pascual, que, con ojos encendidos y voz empalagosa, coreaba tambi\u00e9n los valses del alcalde. \u00c9ste, despu\u00e9s de haber piropeado todo el almuerzo a la t\u00eda Julia, intentaba ahora pasarle el brazo por los hombros y le acercaba su cara abotagada. Haciendo esfuerzos por sonre\u00edr, la t\u00eda Julia lo manten\u00eda a raya, y, de rato en rato, nos lanzaba miradas de angustia.\n\n\u2014Tranquilo, compadre \u2014me dec\u00eda Javier\u2014. Piensa en el matrimonio y nada m\u00e1s.\n\n\u2014Creo que ya se freg\u00f3 \u2014le dije, cuando o\u00ed que el alcalde, en el colmo de la dicha, hablaba de traer guitarristas, de cerrar El Sol de Chincha, de ponernos a bailar\u2014. Y me parece que voy a ir preso por romperle la cara a ese huev\u00f3n.\n\nEstaba furioso y decidido a romp\u00e9rsela si se pon\u00eda insolente, cuando me levant\u00e9 y le dije a la t\u00eda Julia que nos \u00edbamos. Ella se par\u00f3 de inmediato, aliviada, y el alcalde no intent\u00f3 detenerla. Sigui\u00f3 cantando marineras, con buen o\u00eddo, y, al vernos salir, nos hizo adi\u00f3s con una sonrisita que me pareci\u00f3 sarc\u00e1stica. Javier, que vino detr\u00e1s, dec\u00eda que era s\u00f3lo alcoh\u00f3lica. Mientras camin\u00e1bamos hacia el Hotel Sudamericano, yo hablaba pestes contra Pascual, a quien, no s\u00e9 por qu\u00e9, hac\u00eda responsable de ese almuerzo absurdo.\n\n\u2014No te hagas el ni\u00f1o malcriado, aprende a conservar la cabeza fr\u00eda \u2014me re\u00f1\u00eda Javier\u2014. El tipo est\u00e1 zampado y no se acuerda de nada. Pero no te amargues, hoy los casa. Esperen en el hotel hasta que los llame.\n\nApenas estuvimos solos, en el cuarto, nos echamos uno en brazos del otro y comenzamos a besarnos con una especie de desesperaci\u00f3n. No nos dec\u00edamos nada, pero nuestras manos y bocas se dec\u00edan locuazmente las cosas intensas y hermosas que sent\u00edamos. Hab\u00edamos comenzado bes\u00e1ndonos de pie, junto a la puerta, y poco a poco fuimos acerc\u00e1ndonos a la cama, donde luego nos sentamos y por fin nos tendimos, sin haber aflojado el estrecho abrazo ni un instante. Medio ciego de felicidad y de deseo, acarici\u00e9 el cuerpo de la t\u00eda Julia con manos inexpertas y \u00e1vidas, primero sobre la ropa, luego desaboton\u00e9 su blusa color ladrillo, ya arrugada, y estaba bes\u00e1ndole los senos, cuando unos nudillos inoportunos estremecieron la puerta.\n\n\u2014Todo listo, concubinos \u2014o\u00edmos la voz de Javier\u2014. Dentro de cinco minutos, en la alcald\u00eda. El cacaseno est\u00e1 esper\u00e1ndolos.\n\nSaltamos de la cama, dichosos, aturdidos, y la t\u00eda Julia, roja de verg\u00fcenza, se acomodaba la ropa y yo, cerrando los ojos, como de chiquito, pensaba en cosas abstractas y respetables \u2014n\u00fameros, tri\u00e1ngulos, c\u00edrculos, la abuelita, mi mam\u00e1\u2014 para que cediera la erecci\u00f3n. En el ba\u00f1o del pasillo, ella primero, yo despu\u00e9s, nos aseamos y peinamos un poco, y regresamos a la municipalidad a paso tan r\u00e1pido que llegamos sin respiraci\u00f3n. El secretario nos hizo pasar de inmediato a la oficina del alcalde, un cuarto amplio, en el que hab\u00eda un escudo peruano colgado en la pared, dominando un escritorio con banderines y libros de actas, y media docena de bancas, como pupitres de colegio. Con la cara lavada y el pelo todav\u00eda h\u00famedo, muy compuesto, el rubicundo burgomaestre nos hizo una venia ceremoniosa desde detr\u00e1s del escritorio. Era otra persona: lleno de formas y de solemnidad. A ambos lados del escritorio, Javier y Pascual nos sonre\u00edan con picard\u00eda.\n\n\u2014Bien, procedamos \u2014dijo el alcalde; su voz lo traicionaba: pastosa y vacilante, parec\u00eda qued\u00e1rsele atascada en la lengua\u2014. \u00bfD\u00f3nde est\u00e1n los papeles?\n\n\u2014Los tiene usted, se\u00f1or alcalde \u2014le repuso Javier, con infinita educaci\u00f3n\u2014. Pascual y yo se los dejamos el viernes, para ir adelantando los tr\u00e1mites, \u00bfno se acuerda?\n\n\u2014Qu\u00e9 zampado est\u00e1s que ya se te olvid\u00f3, primo \u2014se ri\u00f3 Pascual, con voz tambi\u00e9n borrachosa\u2014. Si t\u00fa mismo pediste que te los dej\u00e1ramos.\n\n\u2014Bueno, entonces los debe tener el secretario \u2014murmur\u00f3 el alcalde, inc\u00f3modo, y mirando a Pascual con disgusto, llam\u00f3\u2014: \u00a1Secretario!\n\nEl hombrecito flaco y de anchos anteojos se demor\u00f3 varios minutos en encontrar las partidas de nacimiento y la sentencia de divorcio de la t\u00eda Julia. Esperamos en silencio, mientras el alcalde fumaba, bostezaba y miraba su reloj con impaciencia. Al fin las trajo, escudri\u00f1\u00e1ndolas con antipat\u00eda. Al ponerlas sobre el escritorio, murmur\u00f3, con un tonito burocr\u00e1tico:\n\n\u2014Aqu\u00ed est\u00e1n, se\u00f1or alcalde. Hay un impedimento por la edad del joven, ya le dije.\n\n\u2014\u00bfAlguien le ha preguntado a usted algo? \u2014dijo Pascual, dando un paso hacia \u00e9l como si fuera a estrangularlo.\n\n\u2014Yo cumplo con mi deber \u2014le contest\u00f3 el secretario. Y, volvi\u00e9ndose al alcalde, insisti\u00f3 con acidez, se\u00f1al\u00e1ndome\u2014: S\u00f3lo tiene dieciocho a\u00f1os y no presenta dispensa judicial para casarse.\n\n\u2014C\u00f3mo es posible que tengas a un imb\u00e9cil de ayudante, primo \u2014estall\u00f3 Pascual\u2014. \u00bfQu\u00e9 esperas para botarlo y traer a alguien con un poco de cacumen?\n\n\u2014C\u00e1llate, se te ha subido el trago y te est\u00e1s poniendo agresivo \u2014dijo el alcalde. Carraspe\u00f3, ganando tiempo. Cruz\u00f3 los brazos y nos mir\u00f3 a la t\u00eda Julia y a m\u00ed, gravemente\u2014. Yo estaba dispuesto a pasar por alto las proclamas, para hacerles un favor. Pero esto es m\u00e1s serio. Lo siento mucho.\n\n\u2014\u00bfQu\u00e9 cosa? \u2014dije yo, desconcertado\u2014. \u00bfAcaso no sab\u00eda usted desde el viernes lo de mi edad?\n\n\u2014Qu\u00e9 farsa es \u00e9sta \u2014intervino Javier\u2014. Usted y yo quedamos en que los casar\u00eda sin problemas.\n\n\u2014\u00bfMe est\u00e1 pidiendo que cometa un delito? \u2014se indign\u00f3 a su vez el alcalde. Y con aire ofendido\u2014: Adem\u00e1s, no me levante la voz. Las personas se entienden hablando, no a gritos.\n\n\u2014Pero, primo, te has vuelto loco \u2014dijo Pascual, fuera de s\u00ed, golpeando el escritorio\u2014. T\u00fa estabas de acuerdo, t\u00fa sab\u00edas lo de la edad, t\u00fa dijiste que no importaba. No te me vengas a hacer el amn\u00e9sico ni el legalista. \u00a1C\u00e1salos de una vez y d\u00e9jate de cojudeces!\n\n\u2014No digas malas palabras delante de una dama y no vuelvas a chupar porque no tienes cabeza \u2014dijo tranquilamente el alcalde. Se volvi\u00f3 al secretario y, con un adem\u00e1n, le indic\u00f3 que se retirara. Cuando nos quedamos solos, baj\u00f3 la voz y nos sonri\u00f3 con aire c\u00f3mplice\u2014: \u00bfNo ven que ese sujeto es un esp\u00eda de mis enemigos? Ahora que \u00e9l se dio cuenta, ya no puedo casarlos. Me meter\u00eda en un l\u00edo de padre y se\u00f1or m\u00edo.\n\nNo hubo razones para convencerlo: le jur\u00e9 que mis padres viv\u00edan en Estados Unidos, por eso no presentaba la dispensa judicial, nadie en mi familia har\u00eda l\u00edo por el matrimonio, la t\u00eda Julia y yo apenas casados nos ir\u00edamos al extranjero para siempre.\n\n\u2014Est\u00e1bamos de acuerdo, usted no puede hacernos esa perrada \u2014dec\u00eda Javier.\n\n\u2014No seas tan desgraciado, primo \u2014le cog\u00eda el brazo Pascual\u2014. \u00bfNo te das cuenta que hemos venido desde Lima?\n\n\u2014Calma, no me hagan cargamont\u00f3n, se me ocurre una idea, ya est\u00e1, todo resuelto \u2014dijo al fin el alcalde. Se puso de pie y nos gui\u00f1\u00f3 un ojo\u2014: \u00a1Tambo de Mora! \u00a1El pescador Mart\u00edn! Vayan ahora mismo. D\u00edganle que van de mi parte. El pescador Mart\u00edn, un zambo simpatiqu\u00edsimo. Los casar\u00e1 encantado. Es mejor as\u00ed, un pueblo chiquito, nada de bulla. Mart\u00edn, el alcalde Mart\u00edn. Le regalan una propina y ya est\u00e1. Casi no sabe leer ni escribir, ni mirar\u00e1 estos papeles.\n\nTrat\u00e9 de convencerlo que viniera con nosotros, le hice bromas, lo adul\u00e9 y le rogu\u00e9, pero no hubo manera: ten\u00eda compromisos, trabajo, su familia lo esperaba. Nos acompa\u00f1\u00f3 hasta la puerta, asegur\u00e1ndonos que en Tambo de Mora todo ser\u00eda cuesti\u00f3n de dos minutos.\n\nEn la misma puerta de la alcald\u00eda contratamos un viejo taxi con la carrocer\u00eda remendada para que nos llevara a Tambo de Mora. Durante el viaje, Javier y Pascual hablaban del alcalde, Javier dec\u00eda que era el peor c\u00ednico que hab\u00eda conocido, Pascual trataba de endosarle la culpa al secretario, y, de pronto, el chofer meti\u00f3 su cuchara y comenz\u00f3 tambi\u00e9n a echar sapos y culebras contra el burgomaestre de Chincha, y a decir que s\u00f3lo viv\u00eda para los negociados y las coimas. La t\u00eda Julia y yo \u00edbamos con las manos enlazadas, mir\u00e1ndonos a los ojos, y, a ratos, yo le susurraba al o\u00eddo que la quer\u00eda.\n\nLlegamos a Tambo de Mora a la hora del crep\u00fasculo y, desde la playa, vimos un disco de fuego hundi\u00e9ndose en el mar, bajo un cielo sin nubes, en el que empezaban a brotar mir\u00edadas de estrellas. Recorrimos las dos docenas de ranchos de ca\u00f1a y barro que constitu\u00edan el poblado, entre barcas desfondadas y redes de pescar tendidas sobre estacas para el remiendo. Ol\u00edamos a pescado fresco y a mar. Nos rodeaban negritos semidesnudos que nos com\u00edan a preguntas: qui\u00e9nes \u00e9ramos, de d\u00f3nde ven\u00edamos, qu\u00e9 quer\u00edamos comprar. Por fin encontramos el rancho del alcalde. Su mujer, una negra que atizaba un brasero con un abanico de paja, quit\u00e1ndose el sudor de la frente con la mano, nos dijo que su marido estaba pescando. Consult\u00f3 al cielo y a\u00f1adi\u00f3 que ya estar\u00eda por volver. Fuimos a esperarlo a la playita, y, durante una hora, sentados sobre un tronco, vimos regresar a las barcas, finalizado el trabajo, y vimos la complicada operaci\u00f3n que era arrastrarlas por la arena y descubrimos c\u00f3mo las mujeres de los reci\u00e9n llegados, estorbadas por perros codiciosos, descabezaban y quitaban las v\u00edsceras, ah\u00ed mismo en la playa, a los pescados. Mart\u00edn fue el \u00faltimo en volver. Estaba oscuro y hab\u00eda salido la luna.\n\nEra un negro canoso y con una enorme barriga, bromista y locuaz, que, pese al fresco del anochecer, vest\u00eda s\u00f3lo un viejo calz\u00f3n que se le pegaba a la piel. Lo saludamos como a un ser bajado de los cielos, lo ayudamos a varar su barca y lo escoltamos hasta su rancho. Mientras camin\u00e1bamos, a la escu\u00e1lida luz de los fogones de las viviendas sin puertas de los pescadores, le explicamos la raz\u00f3n de la visita. Mostrando unos dientes grandotes de caballo, se ech\u00f3 a re\u00edr:\n\n\u2014Ni de a vainas, compa\u00f1eros, b\u00fasquense otro manso para que les fr\u00eda ese churrasco \u2014nos dijo, con un vozarr\u00f3n musical\u2014. Por una broma parecida, casi me gano mi balazo.\n\nNos cont\u00f3 que, hac\u00eda unas semanas, por hacerle un favor al alcalde de Chincha, hab\u00eda casado a una parejita pasando por alto las proclamas. A los cuatro d\u00edas se le hab\u00eda presentado, loco de rabia, el marido de la novia \u2014\u00abuna muchacha nacida en el pueblo de Cachiche, donde todas las mujeres tienen escoba y vuelan de noche\u00bb, dec\u00eda\u2014, que ya estaba casada hac\u00eda dos a\u00f1os, amenazando matar a ese alcahuete que se atrev\u00eda a legalizar la uni\u00f3n de los ad\u00falteros.\n\n\u2014Mi colega de Chincha se las sabe todas, se va a ir al cielo volando de puro vivo que es \u2014se burlaba, d\u00e1ndose palmadas en la gran barriga brillante de gotitas de agua\u2014. Cada vez que se le presenta algo podrido se lo manda de regalo al pescador Mart\u00edn, y que el negro cargue con el muerto. \u00a1Pero qu\u00e9 vivo que es!\n\nNo hubo manera de ablandarlo. Ni siquiera quiso echar una ojeada a los papeles, y a los argumentos m\u00edos, de Javier, de Pascual \u2014la t\u00eda Julia permanec\u00eda muda, sonriendo a veces a la fuerza ante el buen humor p\u00edcaro del negro\u2014, contestaba con bromas, se re\u00eda del alcalde de Chincha, o nos contaba de nuevo, a carcajadas, la historia del marido que hab\u00eda querido matarlo por casar con otro a la brujita de Cachiche sin estar \u00e9l muerto ni divorciado. Al llegar a su rancho, encontramos una aliada inesperada en su mujer. \u00c9l mismo le cont\u00f3 lo que quer\u00edamos, mientras se secaba la cara, los brazos, el ancho torso, y olfateaba con apetito la olla que herv\u00eda en el brasero.\n\n\u2014C\u00e1salos, negro sin sentimientos \u2014le dijo la mujer, se\u00f1alando compasiva a la t\u00eda Julia\u2014. Mira a la pobre, se la han robado y no se puede casar, estar\u00e1 sufriendo con todo esto. A ti qu\u00e9 m\u00e1s te da, \u00bfo se te han subido los humos porque eres alcalde?\n\nMart\u00edn iba y ven\u00eda, con sus pies cuadrados, por el piso de tierra del rancho, recolectando vasos, tazas, mientras nosotros volv\u00edamos a la carga y le ofrec\u00edamos de todo: desde nuestro agradecimiento eterno hasta una recompensa que equivaldr\u00eda a muchos d\u00edas de pesca. \u00c9l se mantuvo inflexible y termin\u00f3 dici\u00e9ndole de mal modo a su mujer que no metiera la jeta en lo que no entend\u00eda. Pero recobr\u00f3 inmediatamente el humor y nos puso un vasito o una taza en la mano a cada uno y nos sirvi\u00f3 un traguito de pisco:\n\n\u2014Para que no hayan hecho el viaje de balde, compa\u00f1eros \u2014nos consol\u00f3, sin pizca de iron\u00eda, levantando su copa. Su brindis fue, dadas las circunstancias, fatal\u2014: Salud, por la felicidad de los novios.\n\nAl despedirnos nos dijo que hab\u00edamos cometido un error yendo a Tambo de Mora, por el precedente de la muchacha de Cachiche. Pero que fu\u00e9ramos a Chincha Baja, a El Carmen, a Sunampe, a San Pedro, a cualquiera de los otros pueblitos de la provincia, y que nos casar\u00edan en el acto.\n\n\u2014Esos alcaldes son unos vagos, no tienen nada que hacer y, cuando se les presenta una boda, se emborrachan de contentos \u2014nos grit\u00f3.\n\nRegresamos adonde nos esperaba el taxi, sin hablar. El chofer nos advirti\u00f3 que, como la espera hab\u00eda sido tan larga, ten\u00edamos que discutir de nuevo la tarifa. Durante el regreso a Chincha acordamos que, al d\u00eda siguiente, desde muy temprano, recorrer\u00edamos los distritos y caser\u00edos, uno por uno, ofreciendo recompensas generosas, hasta encontrar al maldito alcalde.\n\n\u2014Ya son cerca de las nueve \u2014dijo la t\u00eda Julia, de pronto\u2014. \u00bfYa le habr\u00e1n avisado a mi hermana?\n\nYo le hab\u00eda hecho memorizar y repetir diez veces al Gran Pablito lo que ten\u00eda que decir a mi t\u00edo Lucho o a mi t\u00eda Olga, y, para mayor seguridad, termin\u00e9 escribi\u00e9ndoselo en un papel: \u00abMario y Julia se han casado. No se preocupen por ellos. Est\u00e1n muy bien y volver\u00e1n a Lima dentro de unos d\u00edas\u00bb. Deb\u00eda llamar a las nueve de la noche, desde un tel\u00e9fono p\u00fablico y cortar inmediatamente despu\u00e9s de trasmitir el mensaje. Mir\u00e9 el reloj, a la luz de un f\u00f3sforo: s\u00ed, la familia ya estaba enterada.\n\n\u2014Se la deben estar comiendo a preguntas a Nancy \u2014dijo la t\u00eda Julia, esforz\u00e1ndose por hablar con naturalidad, como si el asunto concerniera a otras gentes\u2014. Saben que es c\u00f3mplice. Le van a hacer pasar un mal rato a la flaquita.\n\nEn la trocha llena de baches, el viejo taxi rebotaba, a cada instante parec\u00eda atascarse, y todas sus latas y tornillos chirriaban. La luna encend\u00eda tenuemente los m\u00e9danos, y, a ratos, divis\u00e1bamos manchas de palmas, higueras y huarangos. Hab\u00eda muchas estrellas.\n\n\u2014O sea que ya le dieron la noticia a tu pap\u00e1 \u2014dijo Javier\u2014. Nada m\u00e1s bajar del avi\u00f3n. \u00a1Qu\u00e9 tal recibimiento!\n\n\u2014Juro por Dios que encontraremos un alcalde \u2014dijo Pascual\u2014. No soy chinchano si ma\u00f1ana no los casamos en esta tierra. Mi palabra de hombre.\n\n\u2014\u00bfNecesitan un alcalde que los case? \u2014se interes\u00f3 el chofer\u2014. \u00bfSe ha robado usted a la se\u00f1orita? Por qu\u00e9 no me lo dijeron antes, qu\u00e9 falta de confianza. Los hubiera llevado a Grocio Prado, el alcalde es mi compadre y los casaba ah\u00ed mismo.\n\nYo propuse seguir hasta Grocio Prado, pero \u00e9l me quit\u00f3 los br\u00edos. El alcalde no estar\u00eda en el pueblo a estas horas, sino en su chacrita, como a una hora de camino en burro. Era mejor dejarlo para ma\u00f1ana. Quedamos en que pasar\u00eda a recogernos a las ocho y le ofrec\u00ed una buena gratificaci\u00f3n si nos echaba una mano con su compadre:\n\n\u2014Por supuesto \u2014nos anim\u00f3\u2014. Qu\u00e9 m\u00e1s se puede pedir, se casar\u00e1n en el pueblo de la beata Melchorita.\n\nEn el Hotel Sudamericano estaban ya por cerrar el comedor, pero Javier convenci\u00f3 al mozo que nos preparara algo. Nos trajo unas Coca-Colas y unos huevos fritos con arroz recalentado, que apenas probamos. De pronto, a media comida, nos dimos cuenta que est\u00e1bamos hablando en voz baja, como conspiradores, y nos dio un ataque de risa. Cuando sal\u00edamos hacia nuestros respectivos dormitorios \u2014Pascual y Javier iban a regresar a Lima ese d\u00eda, despu\u00e9s de la boda, pero, como hab\u00edan cambiado las cosas, se quedaron y, para ahorrar plata, compartieron un cuarto\u2014 vimos entrar en el comedor a media docena de tipos, algunos con botas y pantal\u00f3n de montar, pidiendo cerveza a gritos. Ellos, con sus voces alcoh\u00f3licas, sus carcajadas, sus choques de vasos, sus chistes est\u00fapidos y sus brindis groseros, y, m\u00e1s tarde, con sus eructos y arcadas, fueron la m\u00fasica de fondo de nuestra noche de bodas. Pese a la frustraci\u00f3n municipal del d\u00eda, fue una intensa y bella noche de bodas, en la que, en esa vieja cama que chirriaba como un gato con nuestros besos, y que, seguramente, ten\u00eda muchas pulgas, hicimos varias veces el amor, con fuego que renac\u00eda cada vez, dici\u00e9ndonos, mientras nuestras manos y labios aprend\u00edan a conocerse y a hacerse gozar, que nos quer\u00edamos y que nunca nos mentir\u00edamos ni nos enga\u00f1ar\u00edamos ni nos separar\u00edamos. Cuando vinieron a tocarnos la puerta \u2014hab\u00edamos pedido que nos despertaran a las siete\u2014, los borrachos acababan de callarse y nosotros est\u00e1bamos todav\u00eda con los ojos abiertos, desnudos y enredados sobre la colcha de rombos verdes, sumidos en una embriagadora modorra, mir\u00e1ndonos con gratitud.\n\nEl aseo, en el ba\u00f1o com\u00fan del Hotel Sudamericano, fue una haza\u00f1a. La ducha parec\u00eda no haber sido usada nunca, de la mohosa regadera sal\u00edan chorros en todas direcciones salvo la del ba\u00f1ista, y hab\u00eda que recibir un largo enjuague de l\u00edquido negro antes de que el agua viniera limpia. No hab\u00eda toallas, s\u00f3lo un trapo sucio para las manos, de manera que tuvimos que secarnos con las s\u00e1banas. Pero est\u00e1bamos felices y exaltados y los inconvenientes nos divert\u00edan. En el comedor encontramos a Javier y Pascual ya vestidos, amarillentos de sue\u00f1o, mirando con repugnancia el estado catastr\u00f3fico en que hab\u00edan dejado el local los borrachos de la v\u00edspera: vasos rotos, puchos, v\u00f3mitos y escupitajos sobre los que un empleado echaba baldazos de aserr\u00edn, y una gran pestilencia. Fuimos a tomar un caf\u00e9 con leche a la calle, a una bodeguita desde la que se pod\u00edan ver los tupidos y altos \u00e1rboles de la plaza. Era una sensaci\u00f3n rara, viniendo de la neblina cenicienta de Lima, ese comenzar el d\u00eda con sol potente y cielo despejado. Cuando regresamos, en el hotel estaba ya esper\u00e1ndonos el chofer.\n\nEn el trayecto a Grocio Prado, por una trocha polvorienta que contorneaba vi\u00f1edos y algodonales y desde la que se divisaba, al fondo, tras el desierto, el horizonte pardo de la cordillera, el chofer, presa de una locuacidad que contrastaba con nuestro mutismo, habl\u00f3 hasta por los codos de la beata Melchorita: daba todo lo que ten\u00eda a los pobres, cuidaba a los enfermos y a los viejitos, consolaba a los que sufr\u00edan, ya en vida hab\u00eda sido tan c\u00e9lebre que de todos los pueblos del departamento ven\u00edan devotos a rezar junto a ella. Nos cont\u00f3 algunos de sus milagros. Hab\u00eda salvado agonizantes incurables, hablado con santos que se le aparec\u00edan, visto a Dios y hecho florecer una rosa en una piedra que se conservaba.\n\n\u2014Es m\u00e1s popular que la beatita de Humay y que el Se\u00f1or de Luren, basta ver cu\u00e1nta gente viene a su ermita y a su procesi\u00f3n \u2014dec\u00eda\u2014. No hay derecho que no la hagan santa. Ustedes, que son de Lima, mu\u00e9vanse y apuren la cosa. Es de justicia, cr\u00e9anme.\n\nCuando llegamos, por fin, enterrados de pies a cabeza, a la ancha y cuadrada plaza sin \u00e1rboles de Grocio Prado, comprobamos la popularidad de Melchorita. Montones de chiquillos y mujeres rodearon el auto y, a gritos y accionando, nos propon\u00edan llevarnos a conocer la ermita, la casa donde hab\u00eda nacido, el lugar donde se mortificaba, donde hab\u00eda hecho milagros, donde hab\u00eda sido enterrada, y nos ofrec\u00edan estampitas, oraciones, escapularios y medallas con la efigie de la beata. El chofer tuvo que convencerlos que no \u00e9ramos peregrinos ni turistas para que nos dejaran en paz.\n\nLa municipalidad, una vivienda de adobe con techo de calamina, peque\u00f1a y pobr\u00edsima, languidec\u00eda en un flanco de la plaza. Estaba cerrada:\n\n\u2014Mi compadre no tardar\u00e1 en llegar \u2014dijo el chofer\u2014. Esper\u00e9moslo a la sombrita.\n\nNos sentamos en la vereda, bajo el alero de la alcald\u00eda, y desde all\u00ed pod\u00edamos ver, al final de las calles rectas, de tierra, que a menos de cincuenta metros a la redonda terminaban las casitas endebles y los ranchos de ca\u00f1a brava y comenzaban las chacras y el desierto. La t\u00eda Julia estaba a mi lado, con la cabeza apoyada en mi hombro, y ten\u00eda los ojos cerrados. Llev\u00e1bamos ah\u00ed una media hora, viendo cruzar a los arrieros, a pie o en burro, y a mujeres que iban a sacar agua de un arroyo que corr\u00eda por una de las esquinas, cuando pas\u00f3 un viejo montado a caballo.\n\n\u2014\u00bfEsperan a don Jacinto? \u2014nos pregunt\u00f3, quit\u00e1ndose el sombrerote de paja\u2014. Se ha ido a Ica, a hablarle al prefecto, para que saque a su hijo del cuartel. Se lo llevaron los soldados para el servicio militar. No volver\u00e1 hasta la tarde.\n\nEl chofer propuso que nos qued\u00e1ramos en Grocio Prado, visitando los lugares de Melchorita, pero yo insist\u00ed en probar suerte en otros pueblos. Despu\u00e9s de regatear un buen rato, acept\u00f3 seguir con nosotros hasta mediod\u00eda.\n\nEran s\u00f3lo las nueve de la ma\u00f1ana cuando iniciamos la traves\u00eda, que, zangoloteando por senderos de ac\u00e9milas, aren\u00e1ndonos en trochas medio comidas por los m\u00e9danos, acerc\u00e1ndonos a veces al mar y otras a las extremidades de la cordillera, nos hizo recorrer pr\u00e1cticamente toda la provincia de Chincha. A la entrada de El Carmen se nos revent\u00f3 una llanta, y, como el chofer no ten\u00eda gata, tuvimos que sujetar los cuatro el auto en peso, mientras le cambiaba la rueda de repuesto. A partir de media ma\u00f1ana, el sol, que se hab\u00eda ido enardeciendo hasta convertirse en un suplicio, recalentaba la carrocer\u00eda y todos sud\u00e1bamos como en el ba\u00f1o turco. El radiador comenz\u00f3 a humear y fue preciso llevar con nosotros una lata llena de agua para refrescarlo cada cierto tiempo.\n\nHablamos con tres o cuatro alcaldes de distritos y otros tantos tenientes alcalde de caser\u00edos que eran a veces s\u00f3lo veinte chozas, hombres r\u00fasticos a los que hab\u00eda que ir a buscar a la chacrita donde estaban trabajando la tierra, o al almac\u00e9n donde despachaban aceite y cigarrillos a los vecinos, y a uno de ellos, el de Sunampe, debimos despertarlo a remezones de la zanja donde dorm\u00eda una borrachera. Apenas localiz\u00e1bamos a la autoridad municipal, bajaba yo del taxi, acompa\u00f1ado a veces de Pascual, a veces del chofer, a veces de Javier \u2014la experiencia mostr\u00f3 que mientras m\u00e1s fu\u00e9ramos m\u00e1s se intimidaba el alcalde\u2014 a dar las explicaciones. Fueran cuales fueran los argumentos, ve\u00eda infaliblemente en la cara del campesino, pescador o comerciante (el de Chincha Baja se present\u00f3 a s\u00ed mismo como \u00abcurandero\u00bb) brotar la desconfianza, un brillo de alarma en los ojos. S\u00f3lo dos de ellos se negaron francamente: el de Alto Lar\u00e1n, un viejecito que, mientras le hablaba, iba cargando unas ac\u00e9milas con atados de alfalfa, nos dijo que \u00e9l no casaba a nadie que no fuera del pueblo, y el de San Juan de Yanac, un zambo agricultor que se asust\u00f3 mucho al vernos, pues, crey\u00f3 que \u00e9ramos de la polic\u00eda y que le ven\u00edamos a tomar cuentas por algo. Cuando supo qu\u00e9 quer\u00edamos, se enfureci\u00f3: \u00abNo, ni de a vainas, algo malo habr\u00e1 para que unos blanquitos se vengan a casar a este pueblo dejado de la mano de Dios\u00bb. Los otros nos dieron pretextos que se parec\u00edan. El m\u00e1s com\u00fan: el libro de registros se hab\u00eda perdido o agotado, y, hasta que mandaran uno nuevo de Chincha, la alcald\u00eda no pod\u00eda certificar nacimientos ni defunciones ni casar a nadie. La respuesta m\u00e1s imaginativa nos la dio el alcalde de Chav\u00edn: no pod\u00eda por falta de tiempo, ten\u00eda que ir a matar un zorro que cada noche se com\u00eda dos o tres gallinas de la regi\u00f3n. S\u00f3lo estuvimos a punto de lograrlo en Pueblo Nuevo. El alcalde nos escuch\u00f3 con atenci\u00f3n, asinti\u00f3 y dijo que eximirnos de las proclamas nos iba a costar cincuenta libras. No le dio ninguna importancia a mis a\u00f1os y pareci\u00f3 creer lo que le aseguramos, que la mayor\u00eda de edad era, ahora, no a los veintiuno sino a los dieciocho. Est\u00e1bamos ya instalados frente a un tabl\u00f3n sobre dos barriles que hac\u00eda las veces de escritorio (el local era un rancho de adobes, con un techo agujereado por el que se ve\u00eda el cielo), cuando el alcalde se puso a deletrear, palabra por palabra, los documentos. Despert\u00f3 su temor el hecho de que la t\u00eda Julia fuera boliviana. No sirvi\u00f3 de nada explicarle que \u00e9se no era impedimento, que los extranjeros tambi\u00e9n pod\u00edan casarse, ofrecerle m\u00e1s dinero. \u00abNo quiero comprometerme \u2014dec\u00eda\u2014, eso de que la se\u00f1orita sea boliviana puede ser grave\u00bb.\n\nRegresamos a Chincha cerca de las tres de la tarde, muertos de calor, llenos de polvo y deprimidos. En las afueras, la t\u00eda Julia se ech\u00f3 a llorar. Yo la abrazaba, le dec\u00eda al o\u00eddo que no se pusiera as\u00ed, que la quer\u00eda, que nos casar\u00edamos aunque hubiera que recorrer todos los pueblecitos del Per\u00fa.\n\n\u2014No es por lo que no podamos casarnos \u2014dec\u00eda ella, entre lagrimones, tratando de sonre\u00edr\u2014. Sino por lo rid\u00edculo que resulta todo esto.\n\nEn el hotel, le pedimos al chofer que volviera una hora despu\u00e9s, para ir a Grocio Prado, a ver si hab\u00eda regresado su compadre.\n\nNinguno de los cuatro ten\u00edamos mucha hambre, de modo que nuestro almuerzo consisti\u00f3 en un s\u00e1ndwich de queso y una Coca-Cola, que tomamos de pie, en el mostrador. Luego, fuimos a descansar. Pese al desvelo de la noche y a las frustraciones de la ma\u00f1ana, tuvimos \u00e1nimos para hacer el amor, ardientemente, sobre la colcha de rombos, en una luz rala y terrosa. Desde la cama, ve\u00edamos los residuos de sol que apenas pod\u00edan filtrarse, adelgazados, envilecidos, por una alta claraboya que ten\u00eda los cristales cubiertos de mugre. Inmediatamente despu\u00e9s, en vez de levantarnos para reunirnos con nuestros c\u00f3mplices en el comedor, ca\u00edmos dormidos. Fue un sue\u00f1o ansioso y sobresaltado, en el que, a intensos ramalazos de deseo que nos hac\u00edan buscarnos y acariciarnos instintivamente, suced\u00edan pesadillas; despu\u00e9s nos las contamos y supimos que en las de ambos aparec\u00edan caras de parientes, y la t\u00eda Julia se ri\u00f3 cuando le dije que, en un momento del sue\u00f1o, me hab\u00eda sentido viviendo uno de los cataclismos \u00faltimos de Pedro Camacho.\n\nMe despertaron unos golpes en la puerta. Estaba oscuro, y, por las rendijas de la claraboya, se ve\u00edan unas varillas de luz el\u00e9ctrica. Grit\u00e9 que ya iba, y, mientras, sacudiendo la cabeza para ahuyentar el torpor del sue\u00f1o, prend\u00ed un f\u00f3sforo y mir\u00e9 el reloj. Eran las siete de la noche. Sent\u00ed que se me ven\u00eda el mundo encima; otro d\u00eda perdido y, lo peor, ya casi no me quedaban fondos para seguir buscando alcaldes. Fui a tientas hasta la puerta, la entreabr\u00ed e iba a re\u00f1ir a Javier por no haberme despertado, cuando not\u00e9 que su cara me sonre\u00eda de oreja a oreja:\n\n\u2014Todo listo, Varguitas \u2014dijo, orgulloso como un pavo real\u2014. El alcalde de Grocio Prado est\u00e1 haciendo el acta y preparando el certificado. D\u00e9jense de pecar y ap\u00farense. Los esperamos en el taxi.\n\nCerr\u00f3 la puerta y o\u00ed su risa, alej\u00e1ndose. La t\u00eda Julia se hab\u00eda incorporado en la cama, se frotaba los ojos, y, en la penumbra, yo alcanzaba a adivinar su expresi\u00f3n asombrada y un poco incr\u00e9dula.\n\n\u2014A ese chofer le voy a dedicar el primer libro que escriba \u2014dec\u00eda yo, mientras nos vest\u00edamos.\n\n\u2014Todav\u00eda no cantes victoria \u2014sonre\u00eda la t\u00eda Julia\u2014. Ni cuando vea el certificado lo voy a creer.\n\nSalimos atropell\u00e1ndonos, y, al pasar por el comedor, donde hab\u00eda ya muchos hombres tomando cerveza, alguien pirope\u00f3 a la t\u00eda Julia con tanta gracia que muchos se rieron. Pascual y Javier estaban dentro del taxi, pero \u00e9ste no era el de la ma\u00f1ana, ni tampoco el chofer.\n\n\u2014Se las quiso dar de vivo y cobrarnos el doble, aprovech\u00e1ndose de las circunstancias \u2014nos explic\u00f3 Pascual\u2014. As\u00ed que lo mandamos donde se merec\u00eda y contratamos aqu\u00ed al maestro, una persona como Dios manda.\n\nMe entraron toda clase de terrores, pensando que el cambio de chofer frustrar\u00eda una vez m\u00e1s la boda. Pero Javier nos tranquiliz\u00f3. El otro chofer tampoco hab\u00eda ido con ellos a Grocio Prado en la tarde, sino \u00e9ste. Nos contaron, como una travesura, que hab\u00edan decidido \u00abdejarnos descansar\u00bb para que la t\u00eda Julia no pasara el mal rato de otra negativa, e ir solos a hacer la gesti\u00f3n en Grocio Prado. Hab\u00edan tenido una larga conversaci\u00f3n con el alcalde.\n\n\u2014Un cholo sabid\u00edsimo, uno de esos hombres superiores que s\u00f3lo produce la tierra de Chincha \u2014dec\u00eda Pascual\u2014. Tendr\u00e1s que agradec\u00e9rselo a Melchorita viniendo a su procesi\u00f3n.\n\nEl alcalde de Grocio Prado hab\u00eda escuchado tranquilo las explicaciones de Javier, le\u00eddo todos los documentos con parsimonia, reflexionado un buen rato, y, luego, estipulado sus condiciones: mil soles, pero a condici\u00f3n de que a mi partida de nacimiento le cambiaran un seis por un tres, de manera que yo naciera tres a\u00f1os antes.\n\n\u2014La inteligencia de los proletarios \u2014dec\u00eda Javier\u2014. Somos una clase en decadencia, conv\u00e9ncete. Ni siquiera se nos pas\u00f3 por la cabeza y este hombre del pueblo, con su luminoso sentido com\u00fan, lo vio en un instante. Ya est\u00e1, ya eres mayor de edad.\n\nAh\u00ed mismo en la alcald\u00eda, entre el alcalde y Javier, hab\u00edan cambiado el seis por el tres, a mano, y el hombre hab\u00eda dicho: qu\u00e9 m\u00e1s da que la tinta no sea la misma, lo que importa es el contenido. Llegamos a Grocio Prado a eso de las ocho. Era una noche clara, con estrellas, de una tibieza bienhechora, y en todas las casitas y ranchos del pueblo titilaban mecheros. Vimos una vivienda m\u00e1s iluminada, con un gran chisporroteo de velas entre los carrizos, y Pascual, persign\u00e1ndose, nos dijo que era la ermita donde hab\u00eda vivido la beata.\n\nEn la municipalidad, el alcalde estaba terminando de redactar el acta, en un librote de tapas negras. El suelo de la \u00fanica habitaci\u00f3n era de tierra, hab\u00eda sido mojado recientemente y se elevaba de \u00e9l un vaho h\u00famedo. Sobre la mesa hab\u00eda tres velas encendidas y su pobre resplandor mostraba, en las paredes encaladas, una bandera peruana sujeta con tachuelas y un cuadrito con la cara del Presidente de la Rep\u00fablica. El alcalde era un hombre cincuent\u00f3n, gordo e inexpresivo; escrib\u00eda despacio, con un lapicero de pluma, que mojaba despu\u00e9s de cada frase en un tintero de largo cuello. Nos salud\u00f3 a la t\u00eda Julia y a m\u00ed con una reverencia f\u00fanebre. Calcul\u00e9 que al ritmo que escrib\u00eda le habr\u00eda tomado m\u00e1s de una hora redactar el acta. Cuando termin\u00f3, sin moverse, dijo:\n\n\u2014Se necesitan dos testigos.\n\nSe adelantaron Javier y Pascual, pero s\u00f3lo este \u00faltimo fue aceptado por el alcalde, pues Javier era menor de edad. Sal\u00ed a hablar con el chofer, que permanec\u00eda en el taxi; acept\u00f3 ser nuestro testigo por cien soles. Era un zambo delgado, con un diente de oro; fumaba todo el tiempo y, en el viaje de venida, hab\u00eda estado mudo. En el momento en que el alcalde le indic\u00f3 d\u00f3nde deb\u00eda firmar, movi\u00f3 la cabeza con pesadumbre:\n\n\u2014Qu\u00e9 calamidad \u2014dijo, como arrepinti\u00e9ndose\u2014. \u00bfD\u00f3nde se ha visto una boda sin una miserable botella para brindar por los novios? Yo no puedo apadrinar una cosa as\u00ed \u2014nos ech\u00f3 una mirada compasiva y a\u00f1adi\u00f3 desde la puerta\u2014: Esp\u00e9renme un segundo.\n\nCruz\u00e1ndose de brazos, el alcalde cerr\u00f3 los ojos y pareci\u00f3 que se echaba a dormir. La t\u00eda Julia, Pascual, Javier y yo nos miramos sin saber qu\u00e9 hacer. Por fin, me dispuse a buscar otro testigo en la calle.\n\n\u2014No es necesario, va a volver \u2014me ataj\u00f3 Pascual\u2014. Adem\u00e1s, lo que ha dicho es muy cierto. Debimos pensar en el brindis. Ese zambo nos ha dado una lecci\u00f3n.\n\n\u2014No hay nervios que resistan \u2014susurr\u00f3 la t\u00eda Julia, cogi\u00e9ndome la mano\u2014. \u00bfNo te sientes como si estuvieras robando un banco y fuera a llegar la polic\u00eda?\n\nEl zambo demor\u00f3 unos diez minutos, que parecieron a\u00f1os, pero volvi\u00f3 al fin, con dos botellas de vino en la mano. La ceremonia pudo continuar. Una vez que firmaron los testigos, el alcalde nos hizo firmar a la t\u00eda Julia y a m\u00ed, abri\u00f3 un c\u00f3digo, y, acerc\u00e1ndolo a una de las velas, nos ley\u00f3, tan despacio como escrib\u00eda, los art\u00edculos correspondientes a las obligaciones y deberes conyugales. Despu\u00e9s, nos alcanz\u00f3 un certificado y nos dijo que est\u00e1bamos casados. Nos besamos y, luego, nos abrazaron los testigos y el alcalde. El chofer descorch\u00f3 a mordiscos las botellas de vino. No hab\u00eda vasos, as\u00ed que bebimos a pico de botella, pas\u00e1ndolas de mano en mano despu\u00e9s de cada trago. En el viaje de vuelta a Chincha \u2014todos \u00edbamos alegres y al mismo tiempo sosegados\u2014 Javier estuvo intentando catastr\u00f3ficamente silbar la marcha nupcial.\n\nDespu\u00e9s de pagar el taxi, fuimos a la plaza de Armas, para que Javier y Pascual tomaran un colectivo a Lima. Hab\u00eda uno que sal\u00eda dentro de una hora, de modo que tuvimos tiempo de comer en El Sol de Chincha. All\u00ed trazamos un plan. Javier, llegando a Miraflores, ir\u00eda donde mis t\u00edos Olga y Lucho, para tomar la temperatura a la familia y nos llamar\u00eda por tel\u00e9fono. Nosotros regresar\u00edamos a Lima al d\u00eda siguiente, en la ma\u00f1ana. Pascual tendr\u00eda que inventar una buena excusa para justificar su inasistencia de m\u00e1s de dos d\u00edas a la radio.\n\nLos despedimos en la estaci\u00f3n del colectivo y regresamos al Hotel Sudamericano conversando como dos viejos esposos. La t\u00eda Julia se sent\u00eda mal y cre\u00eda que era el vino de Grocio Prado. Yo le dije que a m\u00ed me hab\u00eda parecido un vino riqu\u00edsimo, pero no le cont\u00e9 que era la primera vez que tomaba vino en mi vida.\n\n### XVIII\n\nEL BARDO de Lima, Crisanto Maravillas, naci\u00f3 en el centro de la ciudad, un callej\u00f3n de la plaza de Santa Ana desde cuyos techos se hac\u00edan volar las m\u00e1s airosas cometas del Per\u00fa, hermosos objetos de papel de seda, que, cuando se elevaban gallardamente sobre los Barrios Altos, sal\u00edan a espiar por sus claraboyas las monjitas de clausura del Convento de las Descalzas. Precisamente, el nacimiento del ni\u00f1o que, a\u00f1os m\u00e1s tarde, llevar\u00eda a alturas cometeras el vals criollo, la marinera, las polkas, coincidi\u00f3 con el bautizo de una cometa, fiesta que congregaba en el callej\u00f3n de Santa Ana a los mejores guitarristas, cajoneadores y cantores del barrio. La comadrona, al abrir la ventanilla del cuarto H, donde se produjo el alumbramiento, para anunciar que la demograf\u00eda de ese rinc\u00f3n de la ciudad hab\u00eda aumentado, pronostic\u00f3: \u00abSi sobrevive, ser\u00e1 jaranista\u00bb.\n\nPero parec\u00eda dudoso que sobreviviera: pesaba menos de un kilo y sus piernecitas eran tan reducidas que, probablemente, no caminar\u00eda jam\u00e1s. Su padre, Valent\u00edn Maravillas, que se hab\u00eda pasado la vida tratando de aclimatar en el barrio la devoci\u00f3n del Se\u00f1or de Limpias (hab\u00eda fundado en su propio cuarto la Hermandad, y, acto temerario o viveza para asegurarse una larga vejez, jurado que antes de su muerte tendr\u00eda m\u00e1s miembros que la del Se\u00f1or de los Milagros), proclam\u00f3 que su santo patrono har\u00eda la haza\u00f1a: salvar\u00eda a su hijo y le permitir\u00eda andar como un cristiano normal. Su madre, Mar\u00eda Portal, cocinera de dedos m\u00e1gicos que nunca hab\u00eda sufrido ni un resfr\u00edo, qued\u00f3 tan impresionada al ver que el hijo tan so\u00f1ado y pedido a Dios era eso \u2014\u00bfuna larva de hom\u00ednido, un feto triste?\u2014 que ech\u00f3 al marido de la casa, responsabiliz\u00e1ndolo y acus\u00e1ndolo, delante del vecindario, de ser s\u00f3lo medio hombre por culpa de su beater\u00eda.\n\nLo cierto es que Crisanto Maravillas sobrevivi\u00f3 y, pese a sus piernecitas rid\u00edculas, lleg\u00f3 a andar. Sin ninguna elegancia, desde luego, m\u00e1s bien como un t\u00edtere, que articula cada paso en tres movimientos \u2014alzar la pierna, doblar la rodilla, bajar el pie\u2014 y con tanta lentitud que, quienes iban a su lado ten\u00edan la sensaci\u00f3n de estar siguiendo la procesi\u00f3n cuando se embotella en las calles angostas. Pero, al menos, dec\u00edan sus progenitores (ya reconciliados), Crisanto se desplazaba por el mundo sin muletas y por su propia voluntad. Don Valent\u00edn, arrodillado en la Iglesia de Santa Ana, se lo agradec\u00eda al Se\u00f1or de Limpias con ojos h\u00famedos, pero Mar\u00eda Portal dec\u00eda que el autor del milagro era, exclusivamente, el m\u00e1s famoso galeno de la ciudad, un especialista en tullidos, que hab\u00eda convertido en velocistas a sinn\u00famero de paral\u00edticos: el doctor Alberto de Quinteros. Mar\u00eda hab\u00eda preparado banquetes criollos memorables en su casa y el sabio le hab\u00eda ense\u00f1ado masajes, ejercicios y cuidados para que, pese a ser tan menudas y raqu\u00edticas, las extremidades de Crisanto pudieran sostenerlo y moverlo por los caminos del mundo.\n\nNadie podr\u00e1 decir que Crisanto Maravillas tuvo una infancia semejante a la de otros ni\u00f1os del tradicional barrio donde le toc\u00f3 nacer. Para desgracia o fortuna suya, su esmirriado organismo no le permiti\u00f3 compartir ninguna de esas actividades que iban cuajando el cuerpo y el esp\u00edritu de los muchachos de la vecindad: no jug\u00f3 al f\u00fatbol con pelota de trapo, nunca pudo boxear en un ring ni trompearse en una esquina, jam\u00e1s particip\u00f3 en esos combates a honda, pedrada o puntapi\u00e9, que, en las calles de la vieja Lima, enfrentaban a los muchachos de la plaza de Santa Ana con las pandillas del Chirimoyo, de Cocharcas, de Cinco Esquinas, del Cercado. No pudo ir con sus compa\u00f1eros de la escuelita fiscal de la plazuela de Santa Clara (donde aprendi\u00f3 a leer) a robar fruta a las huertas de Cantogrande y \u00d1a\u00f1a, ni ba\u00f1arse desnudo en el R\u00edmac ni montar burros a pelo en los potreros del Santoyo. Peque\u00f1ito hasta las lindes del enanismo, flaco como una escoba, con la piel achocolatada de su padre y los pelos lacios de su madre, Crisanto miraba, desde lejos, con ojos inteligentes, a sus compa\u00f1eros, y los ve\u00eda divertirse, sudar, crecer y fortalecerse en esas aventuras que le estaban prohibidas, y en su cara se dibujaba una expresi\u00f3n \u00bfde resignada melancol\u00eda, de apacible tristeza?\n\nPareci\u00f3, en una \u00e9poca, que iba a resultar tan religioso como su padre (quien, adem\u00e1s del culto al Se\u00f1or de Limpias, se hab\u00eda pasado la vida cargando andas de distintos Cristos y V\u00edrgenes, y cambiando de h\u00e1bitos) porque, durante a\u00f1os, fue un empe\u00f1oso monaguillo en las iglesias de las vecindades de la plaza de Santa Ana. Como era puntual, se sab\u00eda las r\u00e9plicas al dedillo y ten\u00eda aire inocente, los p\u00e1rrocos del barrio le perdonaban la calma y torpeza de sus movimientos y lo llamaban con frecuencia para que ayudara misas, repicara la campanilla en los v\u00eda crucis de Semana Santa o echara incienso en las procesiones. Vi\u00e9ndolo embutido en la capa de monaguillo, que siempre le quedaba grande, y oy\u00e9ndolo recitar con devoci\u00f3n, en buen lat\u00edn, en los altares de las trinitarias, de San Andr\u00e9s, del Carmen, de la Buena Muerte y aun de la iglesita de Cocharcas (pues hasta de ese alejado barrio lo llamaban), Mar\u00eda Portal, que hubiera deseado para su hijo un tempestuoso destino de militar, de aventurero, de irresistible gal\u00e1n, reprim\u00eda un suspiro. Pero el rey de los cofrades de Lima, Valent\u00edn Maravillas, sent\u00eda que le crec\u00eda el coraz\u00f3n ante la perspectiva de que el resabio de su sangre fuera cura.\n\nTodos se equivocaban, el ni\u00f1o no ten\u00eda vocaci\u00f3n religiosa. Estaba dotado de intensa vida interior y su sensibilidad no hallaba c\u00f3mo, d\u00f3nde, de qu\u00e9 alimentarse. El ambiente de cirios chisporroteantes, de sahumerios y rezos, de im\u00e1genes consteladas de exvotos, de responsos y ritos y cruces y genuflexiones aplac\u00f3 su precoz avidez de poes\u00eda, su hambre de espiritualidad. Mar\u00eda Portal ayudaba a las madres descalzas en sus labores de reposter\u00eda y artes dom\u00e9sticas y era, por ello, una de las contadas personas que franqueaban la r\u00edgida clausura del convento. La egregia cocinera llevaba con ella a Crisanto, y cuando \u00e9ste fue creciendo (en edad, no en estatura) las Descalzas se hab\u00edan acostumbrado tanto a verlo (simple cosa, gui\u00f1apo, medio ser, dije humano) que lo dejaron seguir vagabundeando por los claustros mientras Mar\u00eda Portal preparaba con las monjitas las celestiales pastas, las temblorosas mazamorras, los blancos suspiros, los huevos chimbos y los mazapanes que luego se vender\u00edan a fin de reunir fondos para las misiones del \u00c1frica. As\u00ed fue como Crisanto Maravillas, a los diez a\u00f1os de edad, conoci\u00f3 el amor...\n\nLa ni\u00f1a que instant\u00e1neamente lo sedujo se llamaba F\u00e1tima, ten\u00eda su misma edad y cumpl\u00eda en el femenino universo de las Descalzas las humildes funciones de sirvienta. Cuando Crisanto Maravillas la vio por primera vez, la peque\u00f1a acababa de baldear los corredores de lajas serranas del claustro y se dispon\u00eda a regar los rosales y las azucenas de la huerta. Era una ni\u00f1a que, pese a estar sumergida en un costal con agujeros y tener los cabellos bajo un trapo de tocuyo, a la manera de una toca, no pod\u00eda ocultar su origen: tez marfile\u00f1a, ojeras azules, ment\u00f3n arrogante, tobillos esbeltos. Se trataba, tragedias de sangre azul que envidia el vulgo, de una recogida. Hab\u00eda sido abandonada, una noche de invierno, envuelta en una manta celeste, en el torno de la calle Jun\u00edn, con un mensaje llorosamente caligrafiado: \u00abSoy hija de un amor funesto, que desespera a una familia honorable, y no podr\u00eda vivir en la sociedad sin ser una acusaci\u00f3n contra el pecado de los autores de mis d\u00edas, quienes, por tener el mismo padre y la misma madre, est\u00e1n impedidos de amarse, de tenerme y de reconocerme. Ustedes, Descalzas bienaventuradas, son las \u00fanicas personas que pueden criarme sin avergonzarse de m\u00ed ni avergonzarme. Mis atormentados progenitores retribuir\u00e1n a la congregaci\u00f3n con abundancia esta obra de caridad que abrir\u00e1 a ustedes las puertas del cielo\u00bb.\n\nLas monjitas encontraron, junto a la hija del incesto, una talega repleta de dinero, que, can\u00edbales de la paganidad a los que hay que evangelizar y vestir y alimentar, acab\u00f3 de convencerlas: la tendr\u00edan como dom\u00e9stica, y, m\u00e1s tarde, si mostraba vocaci\u00f3n, har\u00edan de ella otra esclava del Se\u00f1or, de h\u00e1bito blanco. La bautizaron con el nombre de F\u00e1tima, pues hab\u00eda sido recogida el d\u00eda de la aparici\u00f3n de la Virgen a los pastorcitos de Portugal. La ni\u00f1a creci\u00f3 as\u00ed, lejos del mundo, entre las virginales murallas de las Descalzas, en una atm\u00f3sfera impoluta, sin ver otro hombre (antes de Crisanto) que el anciano y gotoso don Sebasti\u00e1n (\u00bfBergua?), el capell\u00e1n que ven\u00eda una vez por semana a absolver de sus pecadillos (siempre veniales) a las monjitas. Era dulce, suave, d\u00f3cil y las religiosas m\u00e1s entendidas dec\u00edan que, pureza de mente que abuena la mirada y beatifica el aliento, se advert\u00edan en su manera de ser signos inequ\u00edvocos de santidad.\n\nCrisanto Maravillas, haciendo un esfuerzo sobrehumano para vencer la timidez que le agarrotaba la lengua, se acerc\u00f3 a la ni\u00f1a y le pregunt\u00f3 si pod\u00eda ayudarla a regar la huerta. Ella consinti\u00f3 y, desde entonces, vez que Mar\u00eda Portal iba al convento, mientras ella cocinaba con las monjitas, F\u00e1tima y Crisanto barr\u00edan juntos las celdas o juntos fregaban los patios o cambiaban juntos las flores del altar o juntos lavaban los vidrios de las ventanas o juntos enceraban las baldosas o desempolvaban juntos los devocionarios. Entre el muchacho feo y la ni\u00f1a bonita fue naciendo, primer amor que se recuerda siempre como el mejor, un v\u00ednculo que \u00bfromper\u00eda la muerte?\n\nFue cuando el joven semibaldado estaba merodeando los doce a\u00f1os que Valent\u00edn Maravillas y Mar\u00eda Portal advirtieron los primeros brotes de esa inclinaci\u00f3n que har\u00eda de Crisanto, en poco tiempo, poeta inspirad\u00edsimo e \u00ednclito compositor.\n\nOcurr\u00eda durante las celebraciones que, al menos una vez por semana, reun\u00edan a los vecinos de la plaza de Santa Ana. En la cochera del sastre Chumpitaz, en el patiecito de la ferreter\u00eda de los Lama, en el callej\u00f3n de Valent\u00edn, con motivo de un nacimiento o de un velorio (\u00bfpara festejar una alegr\u00eda o cicatrizar una pena?), nunca faltaban pretextos, se organizaban jaranas hasta el amanecer que transcurr\u00edan bajo el punteo de las guitarras, los sones del caj\u00f3n, el cascabeleo de las palmas y la voz de los tenores. Mientras las parejas, entonadas \u2014\u00a1enardecido aguardiente y arom\u00e1ticas viandas de Mar\u00eda Portal!\u2014, sacaban chispas a las baldosas, Crisanto Maravillas miraba a los guitarristas, cantantes y cajoneadores, como si sus palabras y sonidos fueran algo sobrenatural. Y, cuando los m\u00fasicos hac\u00edan un alto para fumar un cigarrillo o libar una copita, el ni\u00f1o, en actitud reverencial, se acercaba a las guitarras, las acariciaba con cuidado para no asustarlas, pulsaba las seis cuerdas y se o\u00edan unos arpegios...\n\nMuy pronto fue evidente que se trataba de una aptitud, de un sobresaliente don. El baldado ten\u00eda o\u00eddo notable, captaba y reten\u00eda en el acto cualquier ritmo, y, aunque sus manitas eran d\u00e9biles, sab\u00eda acompa\u00f1ar expertamente cualquier m\u00fasica criolla en el caj\u00f3n. En esos entreactos de la orquesta para comer o brindar, aprendi\u00f3 solo los secretos y se hizo amigo \u00edntimo de las guitarras. Los vecinos se acostumbraron a verlo tocar en las fiestas como un m\u00fasico m\u00e1s.\n\nSus piernas no hab\u00edan crecido y, aunque ten\u00eda ya catorce a\u00f1os, parec\u00eda de ocho. Era muy flaquito, pues \u2014se\u00f1al fehaciente de naturaleza art\u00edstica, esbeltez que hermana a los inspirados\u2014 viv\u00eda cr\u00f3nicamente inapetente, y, si Mar\u00eda Portal no hubiese estado all\u00ed, con su dinamismo militar, para embutirle el alimento, el joven bardo se hubiera volatilizado. Esa fr\u00e1gil hechura, sin embargo, no conoc\u00eda la fatiga en lo que se refiere a la m\u00fasica. Los guitarristas del barrio rodaban por el suelo, exhaustos, despu\u00e9s de tocar y cantar muchas horas, se les acalambraban los dedos y merec\u00edan la mudez por afon\u00eda, pero el baldado segu\u00eda all\u00ed, en una sillita de paja (piececitos de japon\u00e9s que nunca llegan a tocar el suelo, peque\u00f1os dedos incansables), arrancando arrobadoras armon\u00edas a las hebras y canturreando como si la fiesta acabara de empezar. No ten\u00eda voz potente; hubiera sido incapaz de emular las proezas del c\u00e9lebre Ezequiel Delf\u00edn que, al cantar ciertos valses, en llave de sol, rajaba los vidrios de las ventanas que ten\u00eda al frente. Pero, la falta de fuerza, la compensaban su indesmayable entonaci\u00f3n, el mani\u00e1tico afinamiento, esa riqueza de matices que nunca desde\u00f1aba ni malher\u00eda una nota.\n\nSin embargo, no lo har\u00edan famoso sus condiciones de int\u00e9rprete sino las de compositor. Que el muchacho tullido de los Barrios Altos, adem\u00e1s de tocar y cantar la m\u00fasica criolla, sab\u00eda inventarla, se hizo p\u00fablico un s\u00e1bado, en medio de una sarmentosa fiesta que, bajo papeles de colores, quitasue\u00f1os y serpentinas, alegraba el callej\u00f3n de Santa Ana, en homenaje al santo de la cocinera. A medianoche, los m\u00fasicos sorprendieron a la concurrencia con una polkita in\u00e9dita cuya letra dialogaba picarescamente:\n\n\u00bfC\u00f3mo?\n\nCon amor, con amor, con amor\n\n\u00bfQu\u00e9 haces?\n\nLlevo una flor, una flor, una flor\n\n\u00bfD\u00f3nde?\n\nEn el ojal, en el ojal, en el ojal\n\n\u00bfA qui\u00e9n?\n\nA Mar\u00eda Portal, Mar\u00eda Portal, Mar\u00eda Portal...\n\nEl ritmo contagi\u00f3 a los asistentes un compulsivo deseo de bailar, de saltar, de brincar, y la letra los divirti\u00f3 y conmovi\u00f3. La curiosidad fue un\u00e1nime: \u00bfqui\u00e9n era el autor? Los m\u00fasicos volvieron las cabezas y se\u00f1alaron a Crisanto Maravillas, quien, modestia de los verdaderamente grandes, baj\u00f3 los ojos. Mar\u00eda Portal lo devor\u00f3 a besos, el cofrade Valent\u00edn enjug\u00f3 una l\u00e1grima y todo el barrio premi\u00f3 con una ovaci\u00f3n al novel forjador de versos. En la ciudad de las tapadas, hab\u00eda nacido un artista.\n\nLa carrera de Crisanto Maravillas (si puede este t\u00e9rmino pedestremente atl\u00e9tico calificar un quehacer signado \u00bfpor el soplo de Dios?) fue mete\u00f3rica. A los pocos meses sus canciones eran conocidas en Lima, y en unos a\u00f1os estaban en la memoria y coraz\u00f3n del Per\u00fa. No hab\u00eda cumplido los veinte cuando abeles y ca\u00ednes reconoc\u00edan que era el compositor m\u00e1s querido del pa\u00eds. Sus valses alegraban las fiestas de los ricos, se bailaban en los \u00e1gapes de la clase media y eran el manjar de los pobres. Los conjuntos de la capital rivalizaban interpretando su m\u00fasica y no hab\u00eda hombre o mujer que, al iniciarse en la dif\u00edcil profesi\u00f3n del canto, no eligiera las maravillas de Maravillas para su repertorio. Le editaron discos, cancioneros, y en las radios y en las revistas su presencia fue una obligaci\u00f3n. Para los chismes y la fantas\u00eda de la gente el compositor baldado de los Barrios Altos se volvi\u00f3 leyenda.\n\nLa gloria y la popularidad no marearon al sencillo muchacho que recib\u00eda estos homenajes con indiferencia de cisne. Dej\u00f3 el colegio en el segundo de media para dedicarse al arte. Con los regalos que le hac\u00edan por tocar en las fiestas, dar serenatas o componer acr\u00f3sticos, pudo comprarse una guitarra. El d\u00eda que la tuvo fue feliz: hab\u00eda encontrado un confidente para sus penas, un compa\u00f1ero para la soledad y una voz para su inspiraci\u00f3n.\n\nNo sab\u00eda escribir ni leer m\u00fasica y nunca aprendi\u00f3 a hacerlo. Trabajaba al o\u00eddo, a base de intuici\u00f3n. Una vez que ten\u00eda aprendida la melod\u00eda, se la cantaba al cholo Blas Sanjin\u00e9s, un profesor del barrio, y \u00e9l se la pon\u00eda en notas y pentagramas. Jam\u00e1s quiso administrar su talento: nunca patent\u00f3 sus composiciones, ni cobr\u00f3 por ellas derechos, y, cuando los amigos ven\u00edan a contarle que las mediocridades de los bajos fondos art\u00edsticos plagiaban sus m\u00fasicas y letras, se limitaba a bostezar. Pese a este desinter\u00e9s, lleg\u00f3 a ganar alg\u00fan dinero, que le enviaban las casas de discos, las radios, o que le exig\u00edan recibir los due\u00f1os cuando tocaba en una fiesta. Crisanto ofrec\u00eda esa plata a sus progenitores, y, cuando \u00e9stos murieron (ten\u00eda ya treinta a\u00f1os), la gastaba con sus amigos. Jam\u00e1s quiso dejar los Barrios Altos, ni el cuarto letra H del callej\u00f3n donde hab\u00eda nacido. \u00bfEra por fidelidad y cari\u00f1o a su origen humilde, por amor al arroyo? Tambi\u00e9n, sin duda. Pero era, sobre todo, porque en ese angosto zagu\u00e1n estaba a tiro de piedra de la ni\u00f1a de sangres aleda\u00f1as, llamada F\u00e1tima, que conoci\u00f3 cuando era sirvienta y que ahora hab\u00eda tomado los h\u00e1bitos y hecho los votos de obediencia, pobreza y (ay) castidad como esposa del Se\u00f1or.\n\nEra, fue, el secreto de su vida, la raz\u00f3n de ser de esa tristeza que todo el mundo, ceguera de la multitud por las llagas del alma, atribuy\u00f3 siempre a sus piernas maceradas, y a su figurilla asim\u00e9trica. Por lo dem\u00e1s, gracias a esa deformidad que le retroced\u00eda los a\u00f1os, Crisanto hab\u00eda seguido acompa\u00f1ando a su madre a la ciudadela religiosa de las Descalzas, y, una vez por semana cuando menos, hab\u00eda podido ver a la muchacha de sus sue\u00f1os. \u00bfAmaba sor F\u00e1tima al inv\u00e1lido como \u00e9l a ella? Imposible saberlo. Flor de invernadero, ignorante de los misterios rijosos del polen de los campos, F\u00e1tima hab\u00eda adquirido conciencia, sentimientos, pasado de ni\u00f1a a adolescente y a mujer en un mundo as\u00e9ptico y conventual, rodeada de ancianas. Todo lo que hab\u00eda llegado a sus o\u00eddos, a sus ojos, a su fantas\u00eda, estuvo rigurosamente filtrado por el cernidor moral de la congregaci\u00f3n (estricta entre las estrictas). \u00bfC\u00f3mo hubiera adivinado esta virtud corporizada que eso que ella cre\u00eda propiedad de Dios (\u00bfel amor?) pod\u00eda ser tambi\u00e9n tr\u00e1fico humano?\n\nPero, agua que desciende la monta\u00f1a para encontrar el r\u00edo, ternerillo que antes de abrir los ojos busca la teta para mamar la leche blanca, tal vez lo amaba. En todo caso fue su amigo, la sola persona de su edad que conoci\u00f3, el \u00fanico compa\u00f1ero de juegos que tuvo, si es propio llamar juegos a esos trabajos que compart\u00edan mientras Mar\u00eda Portal, la eximia costurera, ense\u00f1aba a las monjitas el secreto de sus bordados: barrer pisos, fregar vidrios, regar plantas y encender cirios.\n\nPero es verdad que los ni\u00f1os, despu\u00e9s j\u00f3venes, conversaron mucho a lo largo de esos a\u00f1os. Di\u00e1logos ingenuos \u2014ella era inocente, \u00e9l era t\u00edmido\u2014, en los que, delicadeza de azucenas y espiritualidad de palomas, se hablaba de amor sin mencionarlo, por temas interp\u00f3sitos, como los lindos colores de la colecci\u00f3n de estampitas de sor F\u00e1tima y las explicaciones que Crisanto le hac\u00eda de qu\u00e9 eran los tranv\u00edas, los autos, los cinemas. Todo eso est\u00e1 contado, entienda quien quiera entender, en las canciones de Maravillas dedicadas a esa misteriosa mujer nunca nombrada, salvo en el famos\u00edsimo vals, de t\u00edtulo que tanto ha intrigado a sus admiradores: F\u00e1tima es la Virgen de F\u00e1tima.\n\nAunque sab\u00eda que nunca podr\u00eda sacarla del convento y hacerla suya, Crisanto Maravillas se sent\u00eda feliz viendo a su musa unas horas por semana. De esos breves encuentros sal\u00eda robustecida su inspiraci\u00f3n y as\u00ed surg\u00edan las mozamalas, los yarav\u00edes, los festejos y las resbalosas. La segunda tragedia de su vida (despu\u00e9s de su invalidez) ocurri\u00f3 el d\u00eda en que, por casualidad, la superiora de las Descalzas lo descubri\u00f3 vaciando la vejiga. La madre Lituma cambi\u00f3 varias veces de color y tuvo un ataque de hipo. Corri\u00f3 a preguntar a Mar\u00eda Portal la edad de su hijo y la costurera confes\u00f3 que, aunque su altura y formas eran de diez, hab\u00eda cumplido dieciocho a\u00f1os. La madre Lituma, santigu\u00e1ndose, le prohibi\u00f3 la entrada al convento para siempre.\n\nFue un golpe casi homicida para el bardo de la plaza de Santa Ana, quien cay\u00f3 enfermo de rom\u00e1ntico, inubicable mal. Estuvo muchos d\u00edas en cama \u2014alt\u00edsimas fiebres, delirios melodiosos\u2014, mientras m\u00e9dicos y curanderos probaban ung\u00fcentos y conjuros para regresarlo del coma. Cuando se levant\u00f3, era un espectro que apenas se ten\u00eda en pie. Pero \u00bfpod\u00eda ser de otra manera?, quedar desgajado de su amada fue provechoso para su arte: sentimentaliz\u00f3 su m\u00fasica hasta la l\u00e1grima y dramatiz\u00f3 virilmente sus letras. Las grandes canciones de amor de Crisanto Maravillas son de estos a\u00f1os. Sus amistades, cada vez que escuchaban, acompa\u00f1ando las dulces melod\u00edas, esos versos desgarrados que hablaban de una muchacha encarcelada, jilguerito en su jaula, palomita cazada, flor recogida y secuestrada en el templo del Se\u00f1or, y de un hombre doliente que amaba a la distancia y sin esperanzas, se preguntaban: \u00ab\u00bfQui\u00e9n es ella?\u00bb. Y, curiosidad que perdi\u00f3 a Eva, trataban de identificar a la hero\u00edna entre las mujeres que asediaban al aeda.\n\nPorque, pese a su encogimiento y fealdad, Crisanto Maravillas ten\u00eda un hechicero atractivo para las lime\u00f1as. Blancas con cuentas de banco, cholitas de medio pelo, zambas de conventillo, muchachitas que aprend\u00edan a vivir o viejas que se resbalaban, aparec\u00edan en el modesto interior H, pretextando pedir un aut\u00f3grafo. Le hac\u00edan ojitos, regalitos, zalamer\u00edas, se insinuaban, le propon\u00edan citas o, directamente, pecados. \u00bfEra que, estas mujeres, como las de cierto pa\u00eds que hasta en el nombre de su capital hace gala de pedanter\u00eda (\u00bfbuenos vientos, buenos tiempos, aires saludables?), ten\u00edan la costumbre de preferir a los hombres deformes, por ese est\u00fapido prejuicio seg\u00fan el cual son mejores, matrimonialmente hablando, que los normales? No, en este caso ocurr\u00eda que la riqueza de su arte nimbaba al hombrecito de la plaza de Santa Ana de una apostura espiritual, que desaparec\u00eda su miseria f\u00edsica y hasta lo hac\u00eda apetecible.\n\nCrisanto Maravillas, suavidad de convaleciente de tuberculosis, desalentaba educadamente estos avances y hac\u00eda saber a las solicitantes que perd\u00edan su tiempo. Pronunciaba entonces una esot\u00e9rica frase que produc\u00eda un indescriptible desasosiego de chismes a su alrededor: \u00abYo creo en la fidelidad y soy un pastorcito de Portugal\u00bb.\n\nSu vida era, para entonces, la bohemia de los gitanos del esp\u00edritu. Se levantaba a eso del mediod\u00eda y sol\u00eda almorzar con el p\u00e1rroco de la Iglesia de Santa Ana, un ex juez de instrucci\u00f3n en cuyo despacho se hab\u00eda mutilado un cu\u00e1quero (\u00bfdon Pedro Barreda y Zald\u00edvar?) para demostrar su inocencia de un crimen que se le atribu\u00eda (\u00bfhaber matado a un negro polizonte venido en la panza de un trasatl\u00e1ntico desde el Brasil?). El doctor don Gumercindo Tello, profundamente impresionado, cambi\u00f3 entonces la toga por la sotana. El suceso de la mutilaci\u00f3n fue inmortalizado por Crisanto Maravillas en un festejo de quijada, guitarra y caj\u00f3n: La sangre me absuelve.\n\nEl bardo y el padre Gumercindo acostumbraban ir juntos por esas calles lime\u00f1as donde Crisanto \u2014\u00bfartista que se nutr\u00eda de la vida misma?\u2014 recog\u00eda personajes y temas para sus canciones. Su m\u00fasica \u2014tradici\u00f3n, historia, folclore, chismograf\u00eda\u2014 eternizaba en melod\u00edas los tipos y las costumbres de la ciudad. En los corrales vecinos a la plaza del Cercado y en los del Santo Cristo, Maravillas y el padre Gumercindo asist\u00edan al entrenamiento que los galleros daban a sus campeones para las peleas en el Coliseo de Sandia, y as\u00ed naci\u00f3 la marinera Cu\u00eddate del aj\u00ed seco, mam\u00e1. O se asoleaban en la placita del Carmen Alto, en cuyo atrio, viendo al titiritero Monle\u00f3n divertir al vecindario con sus mu\u00f1ecos de trapo, encontr\u00f3 Crisanto el tema del vals La doncellita del Carmen Alto (que comienza as\u00ed: \u00abTienes deditos de alambre y coraz\u00f3n de paja, ay, mi amor\u00bb). Fue tambi\u00e9n, sin duda, durante esos paseos criollistas por la vieja Lima que Crisanto cruz\u00f3 a las viejecitas de mantas negras que aparecen en el vals Beatita, t\u00fa tambi\u00e9n fuiste mujer, y donde asisti\u00f3 a esas peleas de adolescentes de las que habla la polkita Los mataperros.\n\nA eso de las seis, los amigos se separaban; el curita volv\u00eda a la parroquia a rezar por el alma del can\u00edbal asesinado en el Callao y el bardo iba al garaje del sastre Chumpitaz. All\u00ed, con el grupo de \u00edntimos \u2014el cajoneador Sifuentes, el rascador Tiburcio, \u00bfla cantante Luc\u00eda Ac\u00e9mila?, los guitarristas Felipe y Juan Portocarrero\u2014, ensayaban nuevas canciones, hac\u00edan arreglos, y, cuando ca\u00eda la oscuridad, alguien sacaba la fraterna botellita de pisco. As\u00ed, entre m\u00fasicas y conversaci\u00f3n, ensayo y traguitos, se les pasaban las horas. Cuando era noche, el grupo se iba a comer a cualquier restaurante de la ciudad, donde el artista era siempre invitado de honor. Otros d\u00edas los esperaban fiestas \u2014cumplea\u00f1os, cambio de aros, matrimonios\u2014 o contratos en alg\u00fan club. Regresaban al amanecer y los amigos sol\u00edan despedir al bardo tullido en la puerta de su hogar. Pero, cuando hab\u00edan partido y se hallaban durmiendo en sus tugurios, la sombra de una figurilla contrahecha y torpe de andar emerg\u00eda del callej\u00f3n. Cruzaba la noche h\u00fameda, arrastrando una guitarra, fantasmal entre la gar\u00faa y la neblina del alba, e iba a sentarse en la desierta placita de Santa Ana, en la banca de piedra que mira a las Descalzas. Los gatos del amanecer escuchaban entonces los m\u00e1s sentidos arpegios jam\u00e1s brotados de guitarra terrena, las m\u00e1s ardientes canciones de amor salidas de estro humano. Unas beatas madrugadoras que, alguna vez, lo sorprendieron as\u00ed, cantando bajito y llorando frente al convento, propalaron la especie atroz de que, ebrio de vanidad, se hab\u00eda enamorado de la Virgen, a quien daba serenatas al despuntar el d\u00eda.\n\nPasaron semanas, meses, a\u00f1os. La fama de Crisanto Maravillas fue, destino de globo que crece y sube en pos del sol, extendi\u00e9ndose como su m\u00fasica. Nadie, sin embargo, ni su \u00edntimo amigo, el p\u00e1rroco Gumercindo Lituma, ex guardia civil apaleado brutalmente por su esposa e hijos (\u00bfpor criar ratones?) y que, mientras convalec\u00eda, escuch\u00f3 el llamado del Se\u00f1or, sospechaba la historia de su inconmensurable pasi\u00f3n por la recluida sor F\u00e1tima, quien, en todos estos a\u00f1os, hab\u00eda seguido trotando hacia la santidad. La casta pareja no pudo cambiar palabra desde el d\u00eda en que la superiora (\u00bfsor Luc\u00eda Ac\u00e9mila?) descubri\u00f3 que el bardo era un ser dotado de virilidad (\u00bfpese a lo ocurrido, esa ma\u00f1ana infausta, en el despacho del juez instructor?). Pero, a lo largo de los a\u00f1os, tuvieron la dicha de verse, aunque con dificultad y a distancia. Sor F\u00e1tima, una vez monjita, pas\u00f3, como sus compa\u00f1eras del convento, a hacer las guardias que tienen orando en la capilla, de dos en dos, las veinticuatro horas del d\u00eda, a las madres descalzas. Las monjitas veladoras est\u00e1n separadas del p\u00fablico por una rejilla de madera que, pese a ser de calado fino, permite que las gentes de ambos lados lleguen a verse. Esto explicaba, en buena parte, la religiosidad tenaz del bardo de Lima, que lo hab\u00eda hecho v\u00edctima, a menudo, de las burlas del vecindario, a las que Maravillas respondi\u00f3 con el piadoso tondero S\u00ed, creyente soy...\n\nCrisanto pasaba, efectivamente, muchos momentos del d\u00eda en la Iglesia de las Descalzas. Entraba varias veces a santiguarse y echar una ojeada a la rejilla. Si \u2014vuelco en el coraz\u00f3n, carrera del pulso, fr\u00edo en la espalda\u2014 a trav\u00e9s del cuadriculado maderamen, en uno de los reclinatorios ocupados por las eternas siluetas de h\u00e1bitos blancos, reconoc\u00eda a sor F\u00e1tima, inmediatamente ca\u00eda de hinojos en las baldosas coloniales. Se colocaba en una posici\u00f3n sesgada (lo ayudaba su f\u00edsico, en el que no era f\u00e1cil diferenciar el frente y el perfil), que le permit\u00eda dar la impresi\u00f3n de estar mirando el altar cuando en realidad ten\u00eda los ojos prendidos de esas nubes talares, de los almidonados copos que envolv\u00edan el cuerpo de su amada. Sor F\u00e1tima, a veces, respiros que se toma el atleta para redoblar esfuerzos, interrump\u00eda sus rezos, alzaba la vista hacia el (\u00bfacrucigramado?) altar, y reconoc\u00eda entonces, interpuesto, el bulto de Crisanto. Una imperceptible sonrisa aparec\u00eda en la n\u00edvea faz de la monjita y en su delicado coraz\u00f3n se reavivaba un tierno sentimiento, al reconocer al amigo de la infancia. Se encontraban sus ojos y en esos segundos \u2014sor F\u00e1tima se sent\u00eda obligada a bajar los suyos\u2014 se dec\u00edan \u00bfcosas que hasta ruborizaban a los \u00e1ngeles del cielo? Porque \u2014s\u00ed, s\u00ed\u2014 esa muchachita milagrosamente salvada de las ruedas del autom\u00f3vil conducido por el propagandista m\u00e9dico Lucho Abril Marroqu\u00edn, que la arroll\u00f3 una ma\u00f1ana soleada, en las afueras de Pisco, cuando a\u00fan no ten\u00eda cinco a\u00f1os, y que en agradecimiento a la Virgen de F\u00e1tima se hab\u00eda hecho monja, hab\u00eda llegado, con el tiempo, en la soledad de su celda, a amar de amor sincero al aeda de los Barrios Altos.\n\nCrisanto Maravillas se hab\u00eda resignado a no desposar carnalmente a su amada, a s\u00f3lo comunicarse con ella de esa manera subliminal en la capilla. Pero nunca se conform\u00f3 a la idea \u2014cruel para un hombre cuya \u00fanica belleza era su arte\u2014 de que sor F\u00e1tima no oyera su m\u00fasica, esas canciones que, sin saberlo, inspiraba. Ten\u00eda la sospecha \u2014certeza para cualquiera que echara un vistazo al espesor fortificado del convento\u2014 que a los o\u00eddos de su amada no llegaban las serenatas, que, desafiando la pulmon\u00eda, le daba cada madrugada desde hac\u00eda veinte a\u00f1os. Un d\u00eda, Crisanto Maravillas comenz\u00f3 a incorporar temas religiosos y m\u00edsticos a su repertorio: los milagros de santa Rosa, las proezas (\u00bfzool\u00f3gicas?) de san Mart\u00edn de Porres, an\u00e9cdotas de los m\u00e1rtires y execraciones a Pilatos sucedieron a las canciones costumbristas. Esto no debilit\u00f3 el aprecio de las multitudes, pero le gan\u00f3 una nueva legi\u00f3n de fan\u00e1ticos: curas y frailes, las monjitas, la Acci\u00f3n Cat\u00f3lica. La m\u00fasica criolla, dignificada, aromosa de incienso, cuajada de temas santos, empez\u00f3 a salvar las murallas que la ten\u00edan arraigada en salones y clubs, y a o\u00edrse en lugares donde antes era inconcebible: iglesias, procesiones, casas de retiro, seminarios.\n\nEl astuto plan demor\u00f3 diez a\u00f1os pero tuvo \u00e9xito. El Convento de las Descalzas no pudo rechazar el ofrecimiento que recibi\u00f3 un d\u00eda de admitir que el bardo mimado de la feligres\u00eda, el poeta de las congregaciones, el m\u00fasico de los v\u00eda crucis, brindara en su capilla y claustros un recital de canciones a beneficio de los misioneros del \u00c1frica. El arzobispo de Lima, sabidur\u00eda p\u00farpura y o\u00eddo de conocedor, hizo saber que autorizaba el acto y que, por unas horas, suspender\u00eda la clausura a fin de que las madres descalzas pudieran deleitarse en m\u00fasica. \u00c9l mismo se propon\u00eda asistir al recital con su corte de dignatarios.\n\nEl acontecimiento, efem\u00e9rides de efem\u00e9rides en la Ciudad de los Virreyes, tuvo lugar el d\u00eda en que Crisanto Maravillas llegaba a la flor de la edad: \u00bfla cincuentena? Era un hombre de frente penetrante, nariz ancha, mirada aguile\u00f1a, rectitud y bondad en el esp\u00edritu, y de una apostura f\u00edsica que reproduc\u00eda su belleza moral.\n\nAunque, previsiones del individuo que la sociedad tritura, se hab\u00edan repartido invitaciones personales y advertido que nadie sin ellas podr\u00eda asistir al evento, el peso de la realidad se impuso: la barrera policial, comandada por el c\u00e9lebre sargento Lituma y su adjunto el cabo Jaime Concha, cedi\u00f3 como si fuera de papel ante las muchedumbres. \u00c9stas, congregadas all\u00ed desde la noche anterior, inundaron el local y anegaron claustros, zaguanes, escaleras, vest\u00edbulos, en actitud reverenciosa. Los invitados debieron ingresar por una puerta secreta, directamente a los altos, donde, api\u00f1ados detr\u00e1s de a\u00f1ejos barandales, se dispusieron a gozar del espect\u00e1culo.\n\nCuando, a las seis de la tarde, el bardo \u2014sonrisa de conquistador, traje azul marino, paso de gimnasta, cabellera dorada flotando al viento\u2014 ingres\u00f3 escoltado por su orquesta y coro, una ovaci\u00f3n que rebot\u00f3 por los techos conmovi\u00f3 las Descalzas. Desde all\u00ed, mientras se pon\u00eda de hinojos, y, con voz de bar\u00edtono, Gumercindo Maravillas entonaba un padrenuestro y un avemar\u00eda, sus ojos (\u00bfmielosos?) iban identificando, entre las cabezas, a un ramillete de conocidos.\n\nEstaba all\u00ed, en primera fila, un afamado astr\u00f3logo, el profesor (\u00bfEzequiel?) Delf\u00edn Ac\u00e9mila, quien, escrutando los cielos, midiendo las mareas y haciendo pases cabal\u00edsticos, hab\u00eda averiguado el destino de las se\u00f1oras millonarias de la ciudad, y que, simpleza de sabio que juega a las bolitas, ten\u00eda la debilidad de la m\u00fasica criolla. Y estaba all\u00ed tambi\u00e9n, de punta en blanco, un clavel rojo en el ojal y una sarita flamante, el negro m\u00e1s popular de Lima, aquel que habiendo cruzado el oc\u00e9ano como polizonte en la barriga de un \u00bfavi\u00f3n?, hab\u00eda rehecho aqu\u00ed su vida (\u00bfdedicado al c\u00edvico pasatiempo de matar ratones mediante venenos t\u00edpicos de su tribu, con lo que se hizo rico?). Y, casualidades que urden el diablo o el azar, comparec\u00edan igualmente, atra\u00eddos por su com\u00fan admiraci\u00f3n al m\u00fasico, el Testigo de Jehov\u00e1 Lucho Abril Marroqu\u00edn, quien, a ra\u00edz de la proeza que protagonizara \u2014\u00bfautodecapitarse, con un fluido cortapapeles, el dedo \u00edndice de la mano derecha?\u2014 se hab\u00eda ganado el apodo de El Mocho, y Sarita Huanca Salaverr\u00eda, la bella victoriana, caprichosa y gentil, que le hab\u00eda exigido, en ofrenda de amor, tan dura prueba. \u00bfY c\u00f3mo no iba a verse, exang\u00fce entre la multitud criollista, al miraflorino Richard Quinteros? Aprovechando que, una vez en la vida basta y sobra, se abr\u00edan las puertas de las Carmelitas, se hab\u00eda deslizado al claustro, confundido entre las gentes, para ver aunque fuera de lejos a esa hermana suya (\u00bfsor F\u00e1tima? \u00bfsor Lituma? \u00bfsor Luc\u00eda?) encerrada all\u00ed por sus padres para librarla de su incestuoso amor. Y hasta los Bergua, sordomudos que jam\u00e1s abandonaban la Pensi\u00f3n Colonial donde viv\u00edan, dedicados a la altruista ocupaci\u00f3n de ense\u00f1ar a dialogar entre s\u00ed, con muecas y ademanes, a los ni\u00f1os pobres privados de audici\u00f3n y habla, se hab\u00edan hecho presentes, contagiados por la curiosidad general, para ver (ya que no o\u00edr) al \u00eddolo de Lima.\n\nEl apocalipsis que enlutar\u00eda la ciudad se desencaden\u00f3 cuando el padre Gumercindo Tello ya hab\u00eda iniciado el recital. Ante la hipnosis de cientos de espectadores arracimados en zaguanes, patios, escaleras, techos, el l\u00edrico, acompa\u00f1ado por el \u00f3rgano, interpretaba las \u00faltimas notas del primoroso ap\u00f3strofe Mi religi\u00f3n no se vende. La misma salva de aplausos que premi\u00f3 al padre Gumercindo, mal y bien que se mezclan como el caf\u00e9 con leche, perdi\u00f3 a los asistentes. Pues, demasiado absorbidos por el canto, demasiado atentos a las palmas, hurras, v\u00edtores, confundieron los primeros s\u00edntomas del cataclismo con la agitaci\u00f3n creada en ellos por el Canario del Se\u00f1or. No reaccionaron en los segundos en que a\u00fan era posible correr, salir, ponerse a salvo. Cuando, rugido volc\u00e1nico que destroza los t\u00edmpanos, descubrieron que no temblaban ellos sino la tierra, era tarde. Porque las tres \u00fanicas puertas de las Carmelitas \u2014coincidencia, voluntad de Dios, torpeza de arquitecto\u2014 hab\u00edan quedado bloqueadas por los primeros derrumbes, sepultando, el gran angelote de piedra que tapi\u00f3 la puerta principal, al sargento Crisanto Maravillas, quien, secundado por el cabo Jaime Concha y el guardia Lituma, al iniciarse el terremoto, trataba de evacuar el convento. El valeroso c\u00edvico y sus dos adjuntos fueron las primeras v\u00edctimas de la deflagraci\u00f3n subterr\u00e1nea. As\u00ed acabaron, cucarachas que apachurra el zapato, bajo un indiferente personaje de granito, en las puertas santas de las Carmelitas (\u00bfen espera del Juicio Final?) los tres mosqueteros del Cuerpo de Bomberos del Per\u00fa.\n\nEntre tanto, en el interior del convento, los fieles all\u00ed congregados por la m\u00fasica y la religi\u00f3n mor\u00edan como moscas. A los aplausos hab\u00eda sucedido un coro de ayes, alaridos y aullidos. Las nobles piedras, los rancios adobes no pudieron resistir el estremecimiento \u2014convulsivo, interminable\u2014 de las profundidades. Una a una las paredes se fueron resquebrajando, desmoronando y triturando a quienes trataban de escalarlas para ganar la calle. As\u00ed murieron unos c\u00e9lebres exterminadores de ratas y ratones: \u00bflos Bergua? Segundos despu\u00e9s se desfondaron, ruido de infierno y polvo de tornado, las galer\u00edas del segundo piso, precipitando \u2014proyectiles vivos, b\u00f3lidos humanos\u2014 contra las gentes api\u00f1adas en el patio a las gentes que se hab\u00edan instalado en los altos para escuchar mejor a la madre Gumercinda. As\u00ed muri\u00f3, el cr\u00e1neo reventado contra las baldosas, el psic\u00f3logo de Lima, Lucho Abril Marroqu\u00edn, que hab\u00eda desneurotizado a media ciudad mediante un tratamiento de su invenci\u00f3n (\u00bfque consist\u00eda en jugar al retumbante juego del palitroque?). Pero fue el derrumbe de los techos carmelitas lo que produjo el mayor n\u00famero de muertos en el m\u00ednimo tiempo. As\u00ed muri\u00f3, entre otros, la madre Luc\u00eda Ac\u00e9mila, quien tanta fama hab\u00eda ganado en el mundo, luego de desertar su antigua secta, los Testigos de Jehov\u00e1, por escribir un libro que alab\u00f3 el Papa: Escarnio del Tronco en nombre de la Cruz.\n\nLa muerte de sor F\u00e1tima y Richard, \u00edmpetu de amor que ni la sangre ni el h\u00e1bito detienen, fue todav\u00eda m\u00e1s triste. Ambos, durante los siglos que dur\u00f3 el fuego, permanecieron indemnes, abraz\u00e1ndose, mientras a su alrededor, asfixiadas, pisoteadas, chamuscadas, perec\u00edan las gentes. Ya hab\u00eda cesado el incendio y, entre carbones y espesas nubes, los dos amantes se besaban, rodeados de mortandad. Hab\u00eda llegado el momento de ganar la calle. Richard, entonces, tomando de la cintura a la madre F\u00e1tima, la arrastr\u00f3 hacia uno de los boquetes abiertos en los muros por la braveza del incendio. Pero, apenas hab\u00edan dado unos pasos los amantes, cuando \u2014\u00bfinfamia de la tierra carn\u00edvora?, \u00bfjusticia celestial?\u2014 se abri\u00f3 el suelo a sus pies. El fuego hab\u00eda devorado la trampa que ocultaba la cueva colonial donde las Carmelitas guardaban los huesos de sus muertos, y all\u00ed cayeron, desbarat\u00e1ndose contra el osario, los hermanos \u00bfluciferinos?\n\n\u00bfEra el diablo quien se los llevaba? \u00bfEra el infierno el ep\u00edlogo de sus amores? \u00bfO era Dios, que, compadecido de su azaroso padecer, los sub\u00eda a los cielos? \u00bfHab\u00eda terminado o tendr\u00eda una continuaci\u00f3n ultraterrena esta historia de sangre, canto, misticismo y fuego?\n\n### XIX\n\nJAVIER NOS llam\u00f3 por tel\u00e9fono desde Lima a las siete de la ma\u00f1ana. La comunicaci\u00f3n era p\u00e9sima, pero ni los zumbidos ni las vibraciones que la interfer\u00edan disimulaban lo alarmada que estaba su voz.\n\n\u2014Malas noticias \u2014me dijo, de entrada\u2014. Montones de malas noticias.\n\nA unos cincuenta kil\u00f3metros de Lima, el colectivo donde \u00e9l y Pascual regresaban la v\u00edspera, se sali\u00f3 de la carretera y dio una vuelta de campana en el arenal. Ninguno de los dos se hiri\u00f3, pero el chofer y otro pasajero hab\u00edan sufrido contusiones serias; fue una pesadilla conseguir, en plena noche, que alg\u00fan auto se detuviera y les echara una mano. Javier hab\u00eda llegado a su pensi\u00f3n molido de fatiga. All\u00ed recibi\u00f3 un susto todav\u00eda mayor. En la puerta lo esperaba mi padre. Se le hab\u00eda acercado, l\u00edvido, le hab\u00eda mostrado un rev\u00f3lver, lo hab\u00eda amenazado con pegarle un tiro si no revelaba al instante d\u00f3nde est\u00e1bamos yo y la t\u00eda Julia. Muerto de p\u00e1nico (\u00abhasta ahora s\u00f3lo hab\u00eda visto rev\u00f3lveres en pel\u00edcula, compadre\u00bb), Javier le jur\u00f3 y requetejur\u00f3 por su madre y por todos los santos que no lo sab\u00eda, que no me ve\u00eda hac\u00eda una semana. Por \u00faltimo, mi padre se hab\u00eda calmado algo y le hab\u00eda dejado una carta, para que me la entregara en persona. Aturdido con lo que acababa de ocurrir, Javier (\u00abqu\u00e9 nochecita, Varguitas\u00bb), apenas se fue mi padre, decidi\u00f3 hablar inmediatamente con el t\u00edo Lucho, para saber si mi familia materna hab\u00eda llegado tambi\u00e9n a esos extremos de rabia. El t\u00edo Lucho lo recibi\u00f3 en bata. Hab\u00edan conversado cerca de una hora. \u00c9l no estaba col\u00e9rico, sino apenado, preocupado, confuso. Javier le confirm\u00f3 que est\u00e1bamos casados con todas las de la ley y le asegur\u00f3 que \u00e9l tambi\u00e9n hab\u00eda tratado de disuadirme, pero en vano. El t\u00edo Lucho suger\u00eda que volvi\u00e9ramos a Lima cuanto antes, para coger al toro por los cuernos y tratar de arreglar las cosas.\n\n\u2014El gran problema es tu padre, Varguitas \u2014concluy\u00f3 su informe Javier\u2014. El resto de la familia se ir\u00e1 conformando poco a poco. Pero \u00e9l est\u00e1 echando chispas. \u00a1No sabes la carta que te ha dejado!\n\nLo re\u00f1\u00ed por leerse las cartas ajenas, y le dije que regres\u00e1bamos a Lima de inmediato, que a mediod\u00eda pasar\u00eda a verlo a su trabajo o que lo llamar\u00eda por tel\u00e9fono. Le cont\u00e9 todo a la t\u00eda Julia mientras se vest\u00eda, sin ocultarle nada, pero tratando de restar truculencia a los hechos.\n\n\u2014Lo que no me gusta nada es lo del rev\u00f3lver \u2014coment\u00f3 la t\u00eda Julia\u2014. Supongo que a quien querr\u00e1 pegarle un tiro ser\u00e1 a m\u00ed, \u00bfno? Oye, Varguitas, espero que mi suegro no me mate en plena luna de miel. \u00bfY lo del choque? \u00a1Pobre Javier! \u00a1Pobre Pascual! En qu\u00e9 l\u00edo los hemos metido con nuestras locuras.\n\nNo estaba asustada ni apenada en absoluto, se la ve\u00eda muy contenta y resuelta a enfrentar todas las calamidades. As\u00ed me sent\u00eda yo tambi\u00e9n. Pagamos el hotel, fuimos a tomar un caf\u00e9 con leche a la plaza de Armas y media hora despu\u00e9s est\u00e1bamos otra vez en la carretera, en un viejo colectivo, rumbo a Lima. Casi todo el trayecto nos besamos, en la boca, en las mejillas, en las manos, dici\u00e9ndonos al o\u00eddo que nos quer\u00edamos y burl\u00e1ndonos de las miradas intranquilas de los pasajeros y del chofer que nos espiaba por el espejo retrovisor.\n\nLlegamos a Lima a las diez de la ma\u00f1ana. Era un d\u00eda gris, la neblina afantasmaba las casas y las gentes, todo estaba h\u00famedo y uno ten\u00eda la sensaci\u00f3n de respirar agua. El colectivo nos dej\u00f3 en la casa de la t\u00eda Olga y el t\u00edo Lucho. Antes de tocar la puerta, nos apretamos con fuerza las manos, para darnos valor. La t\u00eda Julia se hab\u00eda puesto seria y yo sent\u00ed que el coraz\u00f3n se me apuraba.\n\nNos abri\u00f3 el t\u00edo Lucho en persona. Hizo una sonrisa que le sali\u00f3 terriblemente forzada, bes\u00f3 a la t\u00eda Julia en la mejilla y me bes\u00f3 a m\u00ed tambi\u00e9n.\n\n\u2014Tu hermana est\u00e1 todav\u00eda en cama, pero ya despierta \u2014le dijo a la t\u00eda Julia, se\u00f1alando el dormitorio\u2014. Entra, nom\u00e1s.\n\n\u00c9l y yo fuimos a sentarnos a la salita desde la cual se ve\u00eda el seminario de los jesuitas, el Malec\u00f3n y el mar, cuando no hab\u00eda neblina. Ahora s\u00f3lo se distingu\u00edan, borrosas, la pared y la azotea de ladrillos rojos del seminario.\n\n\u2014No te voy a jalar las orejas porque ya est\u00e1s grande para que te jalen las orejas \u2014murmur\u00f3 el t\u00edo Lucho. Se lo ve\u00eda realmente abatido, con se\u00f1ales de desvelo en la cara\u2014. \u00bfAl menos sospechas en lo que te has metido?\n\n\u2014Era la \u00fanica manera de que no nos separaran \u2014le repuse, con las frases que ten\u00eda preparadas\u2014. Julia y yo nos queremos. No hemos hecho ninguna locura. Lo hemos pensado y estamos seguros de lo que hicimos. Te prometo que vamos a salir adelante.\n\n\u2014Eres un mocoso, no tienes una profesi\u00f3n ni donde caerte muerto, tendr\u00e1s que dejar la universidad y romperte el alma para mantener a tu mujer \u2014susurr\u00f3 el t\u00edo Lucho, prendiendo un cigarrillo, moviendo la cabeza\u2014. Te has puesto la soga al cuello t\u00fa solito. Nadie se conforma, porque en la familia todos esper\u00e1bamos que llegar\u00edas a ser alguien. Da pena ver que por un capricho te has zambullido en la mediocridad.\n\n\u2014No voy a dejar los estudios, voy a terminar la universidad, voy a hacer las mismas cosas que hubiera hecho sin casarme \u2014le asegur\u00e9 yo, con \u00edmpetu\u2014. Tienes que creerme y hacer que la familia me crea. Julia me va a ayudar, ahora estudiar\u00e9, trabajar\u00e9 con m\u00e1s ganas.\n\n\u2014Por lo pronto, hay que calmar a tu padre, que est\u00e1 fuera de sus casillas \u2014me dijo el t\u00edo Lucho, abland\u00e1ndose de golpe. Ya hab\u00eda cumplido con jalarme las orejas y ahora parec\u00eda dispuesto a ayudarme\u2014. No entiende razones, amenaza con denunciar a Julia a la polic\u00eda y no s\u00e9 cu\u00e1ntas cosas.\n\nLe dije que hablar\u00eda con \u00e9l y procurar\u00eda que aceptara los hechos. El t\u00edo Lucho me mir\u00f3 de pies a cabeza: era una verg\u00fcenza que un flamante novio estuviera con la camisa sucia, deber\u00eda ir a cambiarme y ba\u00f1arme, y, de paso, tranquilizar a los abuelitos, que estaban muy inquietos. Conversamos todav\u00eda un rato, y hasta tomamos un caf\u00e9, sin que la t\u00eda Julia saliera del cuarto de la t\u00eda Olga. Yo afinaba el o\u00eddo tratando de descubrir si hab\u00eda llanto, gritos, discusi\u00f3n. No, ning\u00fan ruido atravesaba la puerta. La t\u00eda Julia apareci\u00f3 por fin, sola. Ven\u00eda arrebatada, como si hubiera tomado mucho sol, pero sonriendo.\n\n\u2014Por lo menos est\u00e1s viva y enterita \u2014dijo el t\u00edo Lucho\u2014. Pens\u00e9 que tu hermana te jalar\u00eda de las mechas.\n\n\u2014El primer momento casi me pega una cachetada \u2014confes\u00f3 la t\u00eda Julia, sent\u00e1ndose a mi lado\u2014. Me ha dicho barbaridades, por supuesto. Pero parece que, a pesar de todo, puedo seguir en la casa, hasta que se aclaren las cosas.\n\nMe par\u00e9 y dije que ten\u00eda que ir a Radio Panamericana: ser\u00eda tr\u00e1gico que, precisamente ahora, perdiera el trabajo. El t\u00edo Lucho me acompa\u00f1\u00f3 hasta la puerta, me dijo que volviera a almorzar, y cuando, al despedirme, bes\u00e9 a la t\u00eda Julia, lo vi que sonre\u00eda.\n\nCorr\u00ed a la bodega de la esquina a telefonear a mi prima Nancy y tuve la suerte de que ella misma contestara la llamada. Se le fue la voz al reconocerme. Quedamos en encontrarnos dentro de diez minutos en el parque Salazar. Cuando llegu\u00e9 al parque, la flaquita estaba ya all\u00ed, muerta de curiosidad. Antes de que me contara nada, tuve que narrarle toda la aventura de Chincha y responder a innumerables preguntas suyas sobre detalles inesperados, como, por ejemplo, qu\u00e9 vestido se hab\u00eda puesto la t\u00eda Julia para el matrimonio. Lo que le encant\u00f3 y celebr\u00f3 a carcajadas (pero no me crey\u00f3) fue la ligeramente distorsionada versi\u00f3n seg\u00fan la cual el alcalde que nos hab\u00eda casado era un pescador negro, semicalato y sin zapatos. Por fin, despu\u00e9s de esto, consegu\u00ed que me informara c\u00f3mo hab\u00eda recibido la noticia la familia. Hab\u00eda ocurrido lo previsible: ir y venir de casa a casa, concili\u00e1bulos efervescentes, telefonazos innumerables, copiosas l\u00e1grimas, y, al parecer, mi madre hab\u00eda sido consolada, visitada, acompa\u00f1ada, como si hubiera perdido a su \u00fanico hijo. En cuanto a Nancy, la hab\u00edan acosado a preguntas y amenazas, convencidos de que era nuestra aliada, para que dijera d\u00f3nde est\u00e1bamos. Pero ella hab\u00eda resistido, negando rotundamente, y hasta derram\u00f3 unos lagrimones de cocodrilo que los hicieron dudar. Tambi\u00e9n la flaca Nancy estaba inquieta con mi padre:\n\n\u2014No se te vaya a ocurrir verlo hasta que se le pase el coler\u00f3n \u2014me advirti\u00f3\u2014. Est\u00e1 tan furioso que podr\u00eda desaparecerte.\n\nLe pregunt\u00e9 por el departamentito que hab\u00eda alquilado y me sorprendi\u00f3 otra vez con su sentido pr\u00e1ctico. Esa misma ma\u00f1ana hab\u00eda hablado con la due\u00f1a. Ten\u00edan que arreglar el ba\u00f1o, cambiar una puerta y pintarlo, de modo que no estar\u00eda habitable antes de diez d\u00edas. Se me cay\u00f3 el alma a los pies. Mientras caminaba a casa de los abuelos, iba pensando d\u00f3nde diablos podr\u00edamos refugiarnos esas dos semanas.\n\nSin haber resuelto el problema llegu\u00e9 a casa de los abuelitos y all\u00ed me encontr\u00e9 con mi madre. Estaba en la sala y, al verme, rompi\u00f3 en un llanto espectacular. Me abraz\u00f3 con fuerza, y, mientras me acariciaba los ojos, las mejillas, me hund\u00eda los dedos en los cabellos, medio ahogada por los sollozos, repet\u00eda con infinita l\u00e1stima: \u00abHijito, cholito, amor m\u00edo, qu\u00e9 te han hecho, qu\u00e9 ha hecho contigo esa mujer\u00bb. Hac\u00eda cerca de un a\u00f1o que no la ve\u00eda y, pese al llanto que le hinchaba la cara, la encontr\u00e9 rejuvenecida y apuesta. Hice lo posible por calmarla, asegur\u00e1ndole que no me hab\u00edan hecho nada, que yo solito hab\u00eda tomado la decisi\u00f3n de casarme. Ella no pod\u00eda o\u00edr mencionar el nombre de su recient\u00edsima nuera sin que recrudeciera su llanto; ten\u00eda raptos de furia, en los que llamaba a la t\u00eda Julia \u00abesa vieja\u00bb, \u00abesa abusiva\u00bb, \u00abesa divorciada\u00bb. De pronto, en medio de la escena, descubr\u00ed algo que no se me hab\u00eda pasado por la cabeza: m\u00e1s que el qu\u00e9 dir\u00e1n la hac\u00eda sufrir la religi\u00f3n. Era muy cat\u00f3lica y no le importaba tanto que la t\u00eda Julia fuese mayor que yo como que estuviera divorciada (es decir, impedida de casarse por la Iglesia).\n\nPor fin consegu\u00ed apaciguarla, con ayuda de los abuelos. Los viejecitos fueron un modelo de tino, bondad y discreci\u00f3n. El abuelo se limit\u00f3 a decirme, mientras me daba en la frente el seco beso de costumbre: \u00abVaya, poeta, por fin se te ve, ya nos ten\u00edas preocupados\u00bb. Y la abuelita, despu\u00e9s de muchos besos y abrazos, me pregunt\u00f3 al o\u00eddo, con una especie de rec\u00f3ndita picard\u00eda, muy bajito, para que no fuera a o\u00edr mi mam\u00e1: \u00ab\u00bfY la Julita est\u00e1 bien?\u00bb.\n\nDespu\u00e9s de darme un duchazo y cambiarme de ropa \u2014sent\u00ed una liberaci\u00f3n al botar la que llevaba puesta hac\u00eda cuatro d\u00edas\u2014 pude conversar con mi madre. Hab\u00eda dejado de llorar y estaba tomando una taza de t\u00e9 que le hab\u00eda preparado la abuelita, quien, sentada en el brazo del sill\u00f3n, la acariciaba como si fuese una ni\u00f1a. Trat\u00e9 de hacerla sonre\u00edr, con una broma que result\u00f3 de p\u00e9simo gusto (\u00abpero, mamacita, deber\u00edas estar feliz, si me he casado con una gran amiga tuya\u00bb) pero luego toqu\u00e9 cuerdas m\u00e1s sensibles jur\u00e1ndole que no dejar\u00eda los estudios, que me recibir\u00eda de abogado y que, incluso, a lo mejor cambiaba de opini\u00f3n sobre la diplomacia peruana (\u00ablos que no son idiotas son maricas, mam\u00e1\u00bb) y entraba a Relaciones Exteriores, el sue\u00f1o de su vida. Poco a poco se fue desendureciendo, y, aunque siempre con cara de duelo, me pregunt\u00f3 por la universidad, por mis notas, por mi trabajo en la radio, y me ri\u00f1\u00f3 por lo ingrato que era ya que apenas le escrib\u00eda. Me dijo que mi padre hab\u00eda sufrido un golpe terrible: \u00e9l tambi\u00e9n ambicionaba grandes cosas para m\u00ed, por eso impedir\u00eda que esa mujer arruinara mi vida. Hab\u00eda consultado abogados, el matrimonio no era v\u00e1lido, se anular\u00eda y la t\u00eda Julia pod\u00eda ser acusada de corruptora de menores. Mi padre estaba tan violento que, por ahora, no quer\u00eda verme, para que no ocurriera \u00abalgo terrible\u00bb, y exig\u00eda que la t\u00eda Julia saliera en el acto del pa\u00eds. Si no, sufrir\u00eda las consecuencias.\n\nLe contest\u00e9 que la t\u00eda Julia y yo nos hab\u00edamos casado justamente para no separarnos y que iba a ser muy dif\u00edcil que despachara al extranjero a mi mujer a los dos d\u00edas de la boda. Pero ella no quer\u00eda discutir conmigo: \u00abYa lo conoces a tu pap\u00e1, ya sabes el car\u00e1cter que tiene, hay que darle gusto porque si no...\u00bb, y pon\u00eda ojos de terror. Por fin, le dije que iba a llegar tarde a mi trabajo, ya conversar\u00edamos, y, antes de despedirme, volv\u00ed a tranquilizarla sobre mi futuro, asegur\u00e1ndole que me recibir\u00eda de abogado.\n\nEn el colectivo, rumbo al centro de Lima, tuve un presentimiento l\u00fagubre: \u00bfy si me encontraba a alguien ocupando mi escritorio? Hab\u00eda faltado tres d\u00edas, y, en las \u00faltimas semanas, debido a los frustrantes preparativos matrimoniales, hab\u00eda descuidado por completo los boletines, en los que Pascual y el Gran Pablito deb\u00edan haber hecho toda clase de barbaridades. Pens\u00e9, sombr\u00edamente, lo que ser\u00eda, adem\u00e1s de las complicaciones personales del momento, perder el puesto. Empec\u00e9 a inventar argumentos capaces de enternecer a Genaro hijo y a Genaro pap\u00e1. Pero al entrar al Edificio Panamericano, con el alma en un hilo, mi sorpresa fue may\u00fascula, pues el empresario progresista, con quien coincid\u00ed en el ascensor, me salud\u00f3 como si nos hubi\u00e9semos dejado de ver hac\u00eda diez minutos. Ten\u00eda la cara grave:\n\n\u2014Se confirma la cat\u00e1strofe \u2014me dijo, moviendo la cabeza con pesadumbre; parec\u00eda que hubi\u00e9ramos estado hablando hac\u00eda un momento del asunto\u2014. \u00bfQuieres decirme qu\u00e9 vamos a hacer ahora? Tienen que internarlo.\n\nBaj\u00f3 del ascensor en el segundo piso, y yo, que, para mantener la confusi\u00f3n, hab\u00eda puesto cara de velorio y murmurado, como perfectamente al tanto de lo que me hablaba, \u00abah caramba, qu\u00e9 l\u00e1stima\u00bb, me sent\u00ed feliz de que hubiera ocurrido algo tan grave que hiciera pasar inadvertida mi ausencia. En el altillo, Pascual y el Gran Pablito escuchaban con aire f\u00fanebre a Nelly, la secretaria de Genaro hijo. Apenas me saludaron, nadie brome\u00f3 sobre mi matrimonio. Me miraron desolados:\n\n\u2014A Pedro Camacho se lo han llevado al manicomio \u2014balbuce\u00f3 el Gran Pablito, con la voz traspasada\u2014. Qu\u00e9 cosa tan triste, don Mario.\n\nLuego, entre los tres, pero sobre todo Nelly, que hab\u00eda seguido los acontecimientos desde la gerencia, me contaron los pormenores. Todo comenz\u00f3 los mismos d\u00edas en que yo andaba absorbido en mis trajines prematrimoniales. El principio del fin fueron las cat\u00e1strofes, esos incendios, terremotos, choques, naufragios, descarrilamientos, que devastaban los radioteatros, acabando en pocos minutos con decenas de personajes. Esta vez, los propios actores y t\u00e9cnicos de Radio Central, asustados, hab\u00edan dejado de servir de muro protector al escriba, o hab\u00edan sido incapaces de impedir que el desconcierto y las protestas de los oyentes llegaran a los Genaros. Pero \u00e9stos ya estaban alertados por los diarios, cuyos cronistas radiales se burlaban, hac\u00eda d\u00edas, de los cataclismos de Pedro Camacho. Los Genaros lo hab\u00edan llamado, interrogado, extremando las precauciones para no herirlo ni exasperarlo. Pero \u00e9l se les derrumb\u00f3 en plena reuni\u00f3n, con una crisis nerviosa: las cat\u00e1strofes eran estratagemas para recomenzar las historias desde cero, pues su memoria le fallaba, no sab\u00eda ya qu\u00e9 hab\u00eda ocurrido antes, ni qu\u00e9 personaje era quien, ni a cu\u00e1l historia pertenec\u00eda, y \u2014\u00abllorando a gritos, jal\u00e1ndose los pelos\u00bb, aseguraba Nelly\u2014 les hab\u00eda confesado que, en las \u00faltimas semanas, su trabajo, su vida, sus noches, eran un suplicio. Los Genaros lo hab\u00edan hecho ver por un gran m\u00e9dico de Lima, el doctor Honorio Delgado, y \u00e9ste dictamin\u00f3 en el acto que el escriba no estaba en condiciones de trabajar; su mente \u00abexhausta\u00bb deb\u00eda pasar un tiempo en reposo.\n\nEst\u00e1bamos pendientes del relato de Nelly cuando son\u00f3 el tel\u00e9fono. Era Genaro hijo, quer\u00eda verme con urgencia. Baj\u00e9 a su oficina, convencido de que ahora s\u00ed vendr\u00eda cuando menos una amonestaci\u00f3n. Pero me recibi\u00f3 como en el ascensor, dando por supuesto que yo estaba al corriente de sus problemas. Acababa de hablar por tel\u00e9fono con La Habana, y maldec\u00eda porque la CMQ, aprovech\u00e1ndose de su situaci\u00f3n, de la urgencia, le hab\u00eda cuadruplicado las tarifas.\n\n\u2014Es una tragedia, una mala suerte \u00fanica, eran los programas de mayor sinton\u00eda, los anunciadores se los peleaban \u2014dec\u00eda, revolviendo papeles\u2014. \u00a1Qu\u00e9 desastre volver a depender de los tiburones de la CMQ!\n\nLe pregunt\u00e9 c\u00f3mo estaba Pedro Camacho, si lo hab\u00eda visto, en cu\u00e1nto tiempo podr\u00eda volver a trabajar.\n\n\u2014No hay ninguna esperanza \u2014gru\u00f1\u00f3, con una especie de furia, pero acab\u00f3 por adoptar un tono compasivo\u2014. El doctor Delgado dice que su psiquis est\u00e1 en proceso de delicuescencia. Delicuescencia. \u00bfT\u00fa entiendes eso? Que el alma se le cae a pedazos, supongo, que se le pudre la cabeza o algo as\u00ed \u00bfno? Cuando mi padre le pregunt\u00f3 si el restablecimiento pod\u00eda tomar unos meses, nos respondi\u00f3: \u00abTal vez a\u00f1os\u00bb. \u00a1Imag\u00ednate!\n\nBaj\u00f3 la cabeza, abrumado, y, con seguridad de adivino, me predijo lo que ocurrir\u00eda: al saber que los libretos iban a ser, en adelante, los de la CMQ, los anunciadores cancelar\u00edan los contratos o pedir\u00edan rebajas del cincuenta por ciento. Para mal de males, los nuevos radioteatros no llegar\u00edan antes de tres semanas o un mes, porque Cuba ahora era un burdel, hab\u00eda terrorismo, guerrillas, la CMQ andaba alborotada, con gente presa, mil l\u00edos. Pero era impensable que los oyentes se quedaran un mes sin radioteatros, Radio Central perder\u00eda su p\u00fablico, se lo arrebatar\u00edan Radio La Cr\u00f3nica o Radio Colonial que hab\u00edan comenzado a darle duro con los radioteatros argentinos, esas huachafer\u00edas.\n\n\u2014A prop\u00f3sito, para eso te he hecho venir \u2014a\u00f1adi\u00f3, mir\u00e1ndome como si en ese momento me descubriera all\u00ed\u2014. Tienes que echarnos una mano. T\u00fa eres medio intelectual, para ti ser\u00e1 un trabajo f\u00e1cil.\n\nSe trataba de meterse al dep\u00f3sito de Radio Central, donde se conservaban los viejos libretos, anteriores a la venida de Pedro Camacho. Hab\u00eda que revisarlos, descubrir cu\u00e1les pod\u00edan ser utilizados de inmediato, hasta que llegaran los radioteatros frescos de la CMQ.\n\n\u2014Por supuesto, te pagaremos extra \u2014me precis\u00f3\u2014. Aqu\u00ed no explotamos a nadie.\n\nSent\u00ed una enorme gratitud por Genaro hijo y una gran piedad por sus problemas. Aunque me diera cien soles, en esos instantes me ca\u00edan de maravilla. Cuando estaba saliendo de su oficina, su voz me ataj\u00f3 en la puerta:\n\n\u2014Oye, de veras, ya s\u00e9 que te has casado \u2014me volv\u00ed y me estaba haciendo un adem\u00e1n afectuoso\u2014. \u00bfQui\u00e9n es la v\u00edctima? \u00bfUna mujer, supongo, no? Bueno, felicitaciones. Ya nos tomaremos una copa para celebrarlo.\n\nDesde mi oficina llam\u00e9 a la t\u00eda Julia. Me dijo que la t\u00eda Olga se hab\u00eda aplacado algo, pero que a cada rato se asombraba de nuevo y le dec\u00eda: \u00abQu\u00e9 loca eres\u00bb. No la apen\u00f3 mucho que el departamentito no estuviera a\u00fan disponible (\u00abtotal, hemos dormido tanto tiempo separados que podemos hacerlo dos semanas m\u00e1s, Varguitas\u00bb) y me dijo que, despu\u00e9s de darse un buen ba\u00f1o y cambiarse de ropa, se sent\u00eda muy optimista. Le advert\u00ed que no ir\u00eda a almorzar porque ten\u00eda que meterle cuernos con una monta\u00f1a de radioteatros y que nos ver\u00edamos a la noche. Hice El Panamericano y dos boletines y fui a zambullirme en el dep\u00f3sito de Radio Central. Era una cueva sin luz, sembrada de telara\u00f1as, y, al entrar, o\u00ed carreritas de ratones en la oscuridad. Hab\u00eda papeles por todas partes: amontonados, sueltos, desparramados, amarrados en paquetes. Inmediatamente comenc\u00e9 a estornudar por el polvo y la humedad. No hab\u00eda posibilidades de trabajar all\u00ed, as\u00ed que me puse a acarrear altos de papel al cub\u00edculo de Pedro Camacho y me instal\u00e9 en el que hab\u00eda sido su escritorio. No quedaba rastro de \u00e9l: ni el diccionario de citas, ni el mapa de Lima, ni sus fichas sociol\u00f3gico-psicol\u00f3gico-raciales. El desorden y la suciedad de los viejos radioteatros de la CMQ eran supremos: la humedad hab\u00eda deshecho las letras, los ratones y cucarachas hab\u00edan mordisqueado y defecado las p\u00e1ginas, y los libretos se hab\u00edan mezclado unos con otros como las historias de Pedro Camacho. No hab\u00eda mucho que seleccionar; a lo m\u00e1s, tratar de descubrir algunos textos legibles.\n\nLlevaba unas tres horas de estornudos al\u00e9rgicos, buceando entre almibaradas truculencias para armar algunos rompecabezas radioteatrales, cuando se abri\u00f3 la puerta del cub\u00edculo y apareci\u00f3 Javier.\n\n\u2014Es incre\u00edble que en estos momentos, con los problemas que tienes, sigas con tu man\u00eda de Pedro Camacho \u2014me dijo, furioso\u2014. Vengo de donde tus abuelos. Por lo menos, ent\u00e9rate de lo que te pasa y tiembla.\n\nMe lanz\u00f3 sobre el escritorio, arrebosado de suspirantes libretos, dos sobres. Uno, era la carta que le hab\u00eda dejado mi padre la noche anterior. Dec\u00eda as\u00ed:\n\n\u00abMario: Doy cuarenta y ocho horas de plazo para que esa mujer abandone el pa\u00eds. Si no lo hace, me encargar\u00e9 yo, moviendo las influencias que haga falta, de hacerle pagar caro su audacia. En cuanto a ti, quiero que sepas que ando armado y que no permitir\u00e9 que te burles de m\u00ed. Si no obedeces al pie de la letra y esa mujer no sale del pa\u00eds en el plazo indicado, te matar\u00e9 de cinco balazos como a un perro, en plena calle.\u00bb\n\nHab\u00eda firmado con sus dos apellidos y r\u00fabrica, y a\u00f1adido una posdata: \u00abPuedes ir a pedir protecci\u00f3n policial, si quieres. Y, para que quede bien claro, aqu\u00ed firmo otra vez mi decisi\u00f3n de matarte donde te encuentre como a un perro\u00bb. Y, en efecto, hab\u00eda firmado por segunda vez, con trazo m\u00e1s en\u00e9rgico que la primera. El otro sobre se lo hab\u00eda entregado mi abuelita a Javier hac\u00eda media hora, para que me lo trajera. Lo hab\u00eda llevado un guardia; era una citaci\u00f3n de la comisar\u00eda de Miraflores. Deb\u00eda presentarme all\u00ed, al d\u00eda siguiente, a las nueve de la ma\u00f1ana.\n\n\u2014Lo peor no es la carta, sino que, tal como lo vi anoche, puede muy bien cumplir la amenaza \u2014me consol\u00f3 Javier, sent\u00e1ndose en el alf\u00e9izar de la ventana\u2014. \u00bfQu\u00e9 hacemos, compa\u00f1erito?\n\n\u2014Por lo pronto, consultar a un abogado \u2014fue lo \u00fanico que se me ocurri\u00f3\u2014. Sobre mi matrimonio y lo otro. \u00bfConoces a alguno que nos pueda atender gratis, o darnos cr\u00e9dito?\n\nFuimos donde un abogado joven, pariente suyo, con quien algunas veces hab\u00edamos corrido olas en la playa de Miraflores. Fue muy amable, tom\u00f3 con humor la historia de Chincha y me hizo algunas bromas; como hab\u00eda calculado Javier, no quiso cobrarme. Me explic\u00f3 que el matrimonio no era nulo sino anulable, por la correcci\u00f3n de fechas en mi partida. Pero eso requer\u00eda una acci\u00f3n judicial. Si \u00e9sta no se entablaba, a los dos a\u00f1os el matrimonio quedar\u00eda autom\u00e1ticamente \u00abcompuesto\u00bb y ya no se pod\u00eda anular. En cuanto a la t\u00eda Julia, s\u00ed era posible denunciarla como \u00abcorruptora de menores\u00bb, sentar un parte en la polic\u00eda y hacerla detener, por lo menos provisionalmente. Luego, habr\u00eda un juicio, pero \u00e9l estaba seguro que, vistas las circunstancias \u2014es decir, dado que yo ten\u00eda dieciocho y no doce a\u00f1os\u2014 era imposible que prosperara la acusaci\u00f3n: cualquier tribunal la absolver\u00eda.\n\n\u2014De todos modos, si quiere, tu pap\u00e1 puede hacerle pasar muy mal rato a la Julita \u2014concluy\u00f3 Javier, mientras regres\u00e1bamos a la radio, por el jir\u00f3n de la Uni\u00f3n\u2014. \u00bfEs verdad que tiene influencia con el gobierno?\n\nNo lo sab\u00eda; tal vez era amigo de un general, compadre de alg\u00fan ministro. Bruscamente, decid\u00ed que no iba a esperar hasta el d\u00eda siguiente para saber qu\u00e9 quer\u00eda la comisar\u00eda. Ped\u00ed a Javier que me ayudara a rescatar algunos radioteatros del magma de papeles de Radio Central, para salir de dudas ese mismo d\u00eda. Acept\u00f3, y me ofreci\u00f3, tambi\u00e9n, que si me met\u00edan preso me ir\u00eda a visitar y me llevar\u00eda siempre cigarrillos.\n\nA las seis de la tarde entregu\u00e9 a Genaro hijo dos radioteatros m\u00e1s o menos armados y le promet\u00ed que al d\u00eda siguiente tendr\u00eda otros tres; di una ojeada veloz a los boletines de las siete y de las ocho, promet\u00ed a Pascual que volver\u00eda para El Panamericano, y, media hora despu\u00e9s, est\u00e1bamos con Javier en la comisar\u00eda del malec\u00f3n 28 de Julio, en Miraflores. Esperamos un buen rato y, por fin, nos recibieron el comisario \u2014un mayor en uniforme\u2014 y el jefe de la PIP. Mi padre hab\u00eda venido esa ma\u00f1ana a pedir que me tomaran una declaraci\u00f3n oficial sobre lo ocurrido. Ten\u00edan una lista de preguntas escritas a mano, pero mis respuestas las fue transcribiendo a m\u00e1quina el polic\u00eda de civil, lo que tom\u00f3 mucho tiempo, pues era p\u00e9simo mecan\u00f3grafo. Admit\u00ed que me hab\u00eda casado (y subray\u00e9 enf\u00e1ticamente que lo hab\u00eda hecho \u00abpor mi propio deseo y voluntad\u00bb) pero me negu\u00e9 a decir en qu\u00e9 localidad y ante qu\u00e9 alcald\u00eda. Tampoco contest\u00e9 qui\u00e9nes hab\u00edan sido los testigos. Las preguntas eran de tal naturaleza que parec\u00edan concebidas por un tinterillo con malas intenciones: mi fecha de nacimiento y, a continuaci\u00f3n (como si no estuviera impl\u00edcito en lo anterior), si era menor de edad o no, d\u00f3nde viv\u00eda y con qui\u00e9n, y, por supuesto, la edad de la t\u00eda Julia (a la que se llamaba do\u00f1a Julia), pregunta que tambi\u00e9n me negu\u00e9 a responder diciendo que era de mal gusto revelar la edad de las se\u00f1oras. Esto provoc\u00f3 una curiosidad infantil en la pareja de polic\u00edas, quienes, luego de haber firmado yo la declaraci\u00f3n, adoptando aires paternales, me preguntaron, \u00abs\u00f3lo por pura curiosidad\u00bb, cu\u00e1ntos a\u00f1os mayor que yo era la \u00abdama\u00bb. Cuando salimos de la comisar\u00eda me sent\u00ed de pronto muy deprimido, con la inc\u00f3moda sensaci\u00f3n de ser un asesino o un ladr\u00f3n.\n\nJavier pensaba que hab\u00eda metido la pata; negarme a revelar el sitio del matrimonio era una provocaci\u00f3n que irritar\u00eda m\u00e1s a mi padre, y totalmente in\u00fatil, pues lo averiguar\u00eda en pocos d\u00edas. Se me hac\u00eda cuesta arriba volver a la radio esa noche, con el estado de \u00e1nimo en que estaba, as\u00ed que me fui donde el t\u00edo Lucho. Me abri\u00f3 la t\u00eda Olga; me recibi\u00f3 con cara seria y mirada homicida, pero no me dijo ni palabra, e, incluso, me alcanz\u00f3 la mejilla para que la besara. Entr\u00f3 conmigo a la sala, donde estaban la t\u00eda Julia y el t\u00edo Lucho. Bastaba verlos para saber que todo iba color de hormiga. Les pregunt\u00e9 qu\u00e9 suced\u00eda:\n\n\u2014Las cosas se han puesto feas \u2014me dijo la t\u00eda Julia, trenzando sus dedos con los m\u00edos, y yo vi el malestar que esto provocaba en la t\u00eda Olga\u2014. Mi suegro quiere hacerme botar del pa\u00eds como indeseable.\n\nEl t\u00edo Jorge, el t\u00edo Juan y el t\u00edo Pedro hab\u00edan tenido una entrevista esa tarde con mi padre y hab\u00edan vuelto asustados del estado en que lo vieron. Un furor fr\u00edo, una mirada fija, una manera de hablar que transparentaba una determinaci\u00f3n inconmovible. Era categ\u00f3rico: la t\u00eda Julia deb\u00eda partir del Per\u00fa antes de cuarenta y ocho horas o atenerse a las consecuencias. En efecto, era muy amigo \u2014compa\u00f1ero de colegio, tal vez\u2014 del ministro de Trabajo de la dictadura, un general llamado Villacorta, ya hab\u00eda hablado con \u00e9l, y, si no sal\u00eda por propia voluntad, la t\u00eda Julia saldr\u00eda escoltada por soldados hasta el avi\u00f3n. En cuanto a m\u00ed, si no le obedec\u00eda, lo pagar\u00eda caro. Y, lo mismo que a Javier, tambi\u00e9n a mis t\u00edos les hab\u00eda mostrado el rev\u00f3lver. Complet\u00e9 el cuadro, ense\u00f1\u00e1ndoles la carta y cont\u00e1ndoles el interrogatorio policial. La carta de mi padre tuvo la virtud de ganarlos del todo para nuestra causa. El t\u00edo Lucho sirvi\u00f3 unos whiskys y, cuando est\u00e1bamos bebiendo, la t\u00eda Olga se puso de repente a llorar y a decir que c\u00f3mo era posible, su hermana tratada como una criminal, amenazada con la polic\u00eda, que ellas pertenec\u00edan a una de las mejores familias de Bolivia.\n\n\u2014No hay m\u00e1s remedio que me vaya, Varguitas \u2014dijo la t\u00eda Julia. Vi que cambiaba una mirada con mis t\u00edos y comprend\u00ed que ya hab\u00edan hablado de eso\u2014. No me mires as\u00ed, no es una conspiraci\u00f3n, no es para siempre. S\u00f3lo hasta que se le pase la rabieta a tu padre. Para evitar m\u00e1s esc\u00e1ndalos.\n\nLo hab\u00edan conversado y discutido entre los tres y ten\u00edan a punto un plan. Hab\u00edan descartado Bolivia y suger\u00edan que la t\u00eda Julia fuera a Chile, a Valpara\u00edso, donde viv\u00eda su abuelita. Estar\u00eda all\u00ed s\u00f3lo el tiempo indispensable para que se serenaran los \u00e1nimos. Volver\u00eda en el instante mismo en que yo la llamara. Me opuse con furia, dije que la t\u00eda Julia era mi mujer, me hab\u00eda casado con ella para que estuvi\u00e9ramos juntos, en todo caso nos ir\u00edamos los dos. Me recordaron que era menor de edad: no pod\u00eda pedir pasaporte ni salir del pa\u00eds sin permiso paterno. Dije que cruzar\u00eda la frontera a escondidas. Me preguntaron cu\u00e1nta plata ten\u00eda para irme a vivir al extranjero. (Me quedaba a duras penas para comprar cigarrillos unos d\u00edas: el matrimonio y el pago del departamentito hab\u00edan volatilizado el adelanto de Radio Panamericana, la venta de mi ropa y los empe\u00f1os en la Caja de Pignoraci\u00f3n.)\n\n\u2014Ya estamos casados y eso no nos lo van a quitar \u2014dec\u00eda la t\u00eda Julia, despein\u00e1ndome, bes\u00e1ndome, con los ojos llenos de l\u00e1grimas\u2014. Es s\u00f3lo unas semanas, a lo m\u00e1s unos meses. No quiero que te peguen un tiro por mi culpa.\n\nDurante la comida, la t\u00eda Olga y el t\u00edo Lucho fueron exponiendo sus argumentos para convencerme. Ten\u00eda que ser razonable, ya hab\u00eda salido con mi gusto, me hab\u00eda casado, ahora deb\u00eda hacer una concesi\u00f3n provisional, para evitar algo irreparable. Deb\u00eda comprenderlos; ellos, como hermana y cu\u00f1ado de la t\u00eda Julia, estaban en postura muy delicada ante mi padre y el resto de la familia: no pod\u00edan estar contra ni a favor de ella. Nos ayudar\u00edan, lo estaban haciendo en esos momentos, y me tocaba hacer algo de mi parte. Mientras la t\u00eda Julia, estuviera en Valpara\u00edso yo tendr\u00eda que buscar alg\u00fan otro trabajo, porque, si no, con qu\u00e9 diablos \u00edbamos a vivir, qui\u00e9n nos iba a mantener. Mi padre acabar\u00eda por calmarse, por aceptar los hechos.\n\nA eso de la medianoche \u2014mis t\u00edos se hab\u00edan ido discretamente a dormir y la t\u00eda Julia y yo est\u00e1bamos haciendo el amor horriblemente, a medio vestir, con gran zozobra, los o\u00eddos alertas a cualquier ruido\u2014 acab\u00e9 por rendirme. No hab\u00eda otra soluci\u00f3n. A la ma\u00f1ana siguiente tratar\u00edamos de cambiar el pasaje a La Paz por uno a Chile. Media hora despu\u00e9s, mientras caminaba por las calles de Miraflores, hacia mi cuartito de soltero, en casa de los abuelos, sent\u00eda amargura e impotencia, y me maldec\u00eda por no tener ni siquiera con qu\u00e9 comprarme yo tambi\u00e9n un rev\u00f3lver.\n\nLa t\u00eda Julia viaj\u00f3 a Chile dos d\u00edas despu\u00e9s, en un avi\u00f3n que parti\u00f3 al alba. La compa\u00f1\u00eda de aviaci\u00f3n no puso reparos en cambiar el pasaje, pero hab\u00eda una diferencia de precio, que cubrimos gracias a un pr\u00e9stamo de mil quinientos soles que nos hizo nadie menos que Pascual. (Me dej\u00f3 asombrado al contarme que ten\u00eda cinco mil soles en una libreta de ahorros, lo que, con el sueldo que ganaba, era realmente una haza\u00f1a.) Para que la t\u00eda Julia pudiera llevarse algo de dinero vend\u00ed, al librero de la calle La Paz, todos los libros que a\u00fan conservaba, incluidos los c\u00f3digos y manuales de Derecho, con lo que compr\u00e9 cincuenta d\u00f3lares.\n\nLa t\u00eda Olga y el t\u00edo Lucho fueron al aeropuerto con nosotros. La noche anterior yo me qued\u00e9 en su casa. No dormimos, no hicimos el amor. Despu\u00e9s de la comida, mis t\u00edos se retiraron y yo, sentado en la punta de la cama, vi a la t\u00eda Julia hacer cuidadosamente su maleta. Luego, fuimos a sentarnos a la sala, que estaba a oscuras. Estuvimos all\u00ed tres o cuatro horas, con las manos entrelazadas, muy juntos en el sill\u00f3n, hablando en voz baja para no despertar a los parientes. A ratos nos abraz\u00e1bamos, junt\u00e1bamos nuestras caras y nos bes\u00e1bamos, pero la mayor parte del tiempo la pasamos fumando y conversando. Hablamos de lo que har\u00edamos cuando volvi\u00e9ramos a reunirnos, c\u00f3mo ella me ayudar\u00eda en mi trabajo y c\u00f3mo, de una manera u otra, tarde o temprano, llegar\u00edamos un d\u00eda a Par\u00eds a vivir en esa buhardilla donde yo me volver\u00eda, por fin, un escritor. Le cont\u00e9 la historia de su compatriota Pedro Camacho, que estaba ahora en una cl\u00ednica, rodeado de locos, volvi\u00e9ndose loco \u00e9l mismo sin duda, y planeamos escribirnos todos los d\u00edas, largas cartas donde nos contar\u00edamos prolijamente todo lo que hici\u00e9ramos, pens\u00e1ramos y sinti\u00e9ramos. Le promet\u00ed que, cuando volviera, yo habr\u00eda arreglado las cosas y que estar\u00eda ganando lo suficiente para no morirnos de hambre. Cuando son\u00f3 el despertador, a las cinco, era todav\u00eda noche cerrada, y al llegar al aeropuerto de Limatambo, una hora despu\u00e9s, apenas comenzaba a clarear. La t\u00eda Julia se hab\u00eda puesto el traje azul que a m\u00ed me gustaba y se la ve\u00eda guapa. Estuvo muy serena cuando nos despedimos, pero sent\u00ed que temblaba en mis brazos, y, en cambio, a m\u00ed, cuando la vi subir al avi\u00f3n, desde la terraza, en la principiante ma\u00f1ana, se me hizo un nudo en la garganta y se me saltaron las l\u00e1grimas.\n\nSu exilio chileno dur\u00f3 un mes y catorce d\u00edas. Fueron, para m\u00ed, seis semanas decisivas, en las que (gracias a gestiones con amigos, conocidos, condisc\u00edpulos, profesores, a los que busqu\u00e9, rogu\u00e9, fastidi\u00e9, enloquec\u00ed para que me echaran una mano) consegu\u00ed acumular siete trabajos, incluido, por supuesto, el que ya ten\u00eda en Panamericana. El primero fue un empleo en la biblioteca del Club Nacional, que estaba al lado de la radio; mi obligaci\u00f3n era ir dos horas diarias, entre los boletines de la ma\u00f1ana, a registrar los nuevos libros y revistas y hacer un cat\u00e1logo de las viejas existencias. Un profesor de historia, de San Marcos, en cuyo curso hab\u00eda tenido notas sobresalientes, me contrat\u00f3 como ayudante suyo, en las tardes, de tres a cinco, en su casa de Miraflores, donde fichaba diversos temas en los cronistas, para un proyecto de una Historia del Per\u00fa en el que a \u00e9l le correspond\u00edan los vol\u00famenes de Conquista y Emancipaci\u00f3n. El m\u00e1s pintoresco de los nuevos trabajos era un contrato con la Beneficencia P\u00fablica de Lima. En el Cementerio Presb\u00edtero Maestro exist\u00edan una serie de cuarteles, de la \u00e9poca colonial, cuyos registros se hab\u00edan extraviado. Mi misi\u00f3n consist\u00eda en desentra\u00f1ar lo que dec\u00edan las l\u00e1pidas de esas tumbas y hacer listas con los nombres y fechas. Era algo que pod\u00eda llevar a cabo a la hora que quisiera y por lo que me pagaban a destajo: un sol por muerto. Lo hac\u00eda en las tardes, entre el bolet\u00edn de las seis y El Panamericano, y Javier, que a esas horas estaba libre, sol\u00eda acompa\u00f1arme. Como era invierno y oscurec\u00eda temprano, el director del cementerio, un gordo que dec\u00eda haber asistido en persona, en el Congreso, a la toma de posesi\u00f3n de ocho presidentes del Per\u00fa, nos prestaba unas linternas y una escalerita para poder leer los nichos altos. A veces, jugando a que o\u00edamos voces, quejidos, cadenas, y a que ve\u00edamos formas blancuzcas entre las tumbas, consegu\u00edamos asustarnos de verdad. Adem\u00e1s de ir dos o tres veces por semana al cementerio, dedicaba a este quehacer todas las ma\u00f1anas del domingo. Los otros trabajos eran m\u00e1s o menos (m\u00e1s menos que m\u00e1s) literarios. Para el Suplemento Dominical de El Comercio hac\u00eda cada semana una entrevista a un poeta, novelista o ensayista, en una columna titulada \u00abEl hombre y su obra\u00bb. En la revista Cultura Peruana escrib\u00eda un art\u00edculo mensual, para una secci\u00f3n que invent\u00e9: \u00abHombres, libros e ideas\u00bb, y, finalmente, otro profesor amigo me encomend\u00f3 redactar para los postulantes a la Universidad Cat\u00f3lica (pese a ser yo alumno de la rival, San Marcos) un texto de educaci\u00f3n c\u00edvica; cada lunes ten\u00eda que entregarle desarrollado alguno de los asuntos del programa de ingreso (que eran muy diversos, un abanico que cubr\u00eda desde los s\u00edmbolos de la patria hasta la pol\u00e9mica entre indigenistas e hispanistas, pasando por las flores y animales abor\u00edgenes).\n\nCon estos trabajos (que me hac\u00edan sentir, un poco, \u00e9mulo de Pedro Camacho) logr\u00e9 triplicar mis ingresos y redondear lo suficiente para que dos personas pudieran vivir. En todos ellos ped\u00ed adelantos y as\u00ed desempe\u00f1\u00e9 mi m\u00e1quina de escribir, indispensable para las tareas period\u00edsticas (aunque muchos art\u00edculos los hac\u00eda en Panamericana), y de este modo, tambi\u00e9n, la prima Nancy compr\u00f3 algunas cosas para acicalar el departamentito alquilado que la due\u00f1a me entreg\u00f3, en efecto, a los quince d\u00edas. Fue una felicidad la ma\u00f1ana en que tom\u00e9 posesi\u00f3n de esos dos cuartitos, con su ba\u00f1o diminuto. Segu\u00ed durmiendo en casa de los abuelos, porque decid\u00ed estrenarlo el d\u00eda que llegara la t\u00eda Julia, pero iba all\u00ed casi todas las noches, a redactar art\u00edculos y a confeccionar listas de muertos. Aunque no paraba de hacer cosas, de entrar y salir de un sitio a otro, no me sent\u00eda cansado ni deprimido, sino, por el contrario, muy entusiasta, y creo que incluso segu\u00eda leyendo como antes (aunque s\u00f3lo en los innumerables \u00f3mnibus y colectivos que tomaba diariamente).\n\nFiel a lo prometido, las cartas de la t\u00eda Julia llegaban a diario y la abuelita me las entregaba con una luz traviesa en los ojos, murmurando: \u00ab\u00bfDe qui\u00e9n ser\u00e1 esta cartita, de qui\u00e9n ser\u00e1?\u00bb. Yo tambi\u00e9n le escrib\u00eda seguido, era lo \u00faltimo que hac\u00eda cada noche, a veces mareado de sue\u00f1o, d\u00e1ndole cuenta de los trajines de la jornada. En los d\u00edas que siguieron a su partida fui encontr\u00e1ndome, donde los abuelos, donde los t\u00edos Lucho y Olga, en la calle, a mis numerosos parientes y descubriendo sus reacciones. Eran diversas y algunas inesperadas. El t\u00edo Pedro tuvo la m\u00e1s severa: me dej\u00f3 con el saludo colgado y me volvi\u00f3 la espalda despu\u00e9s de mirarme glacialmente. La t\u00eda Jes\u00fas derram\u00f3 unos lagrimones y me abraz\u00f3, susurrando con voz dram\u00e1tica: \u00ab\u00a1Pobre criatura!\u00bb. Otros t\u00edos y t\u00edas optaron por actuar como si nada hubiera ocurrido; eran cari\u00f1osos conmigo, pero no mencionaban a la t\u00eda Julia ni se daban por enterados del matrimonio.\n\nA mi padre no lo hab\u00eda visto, pero sab\u00eda que, una vez satisfecha su exigencia de que la t\u00eda Julia saliera del pa\u00eds, se hab\u00eda aplacado algo. Mis padres estaban alojados donde unos t\u00edos paternos, a los que yo no visitaba nunca, pero mi madre ven\u00eda todos los d\u00edas a casa de los abuelos y all\u00ed nos ve\u00edamos. Adoptaba conmigo una actitud ambivalente, afectuosa, maternal, pero cada vez que asomaba, directa o indirectamente, el tema tab\u00fa, palidec\u00eda, se le sal\u00edan las l\u00e1grimas y aseguraba: \u00abNo lo aceptar\u00e9 jam\u00e1s\u00bb. Cuando le propuse que viniera a conocer el departamentito, se ofendi\u00f3 como si la hubiera insultado, y siempre se refer\u00eda al hecho de que yo hubiera vendido mi ropa y mis libros como a una tragedia griega. Yo la hac\u00eda callar, dici\u00e9ndole: \u00abMamacita, no empieces otra vez con tus radioteatros\u00bb. Ni ella mencionaba a mi padre, ni yo preguntaba por \u00e9l, pero, por otros parientes que lo ve\u00edan, llegu\u00e9 a saber que su c\u00f3lera hab\u00eda cedido el paso a una desesperanza respecto a mi destino, y que sol\u00eda decir: \u00abTendr\u00e1 que obedecerme hasta que cumpla veinti\u00fan a\u00f1os; luego, puede perderse\u00bb.\n\nPese a mis m\u00faltiples quehaceres, en esas semanas escrib\u00ed un nuevo cuento. Se llamaba La beata y el padre Nicol\u00e1s. Estaba situado en Grocio Prado, por supuesto, y era anticlerical: la historia de un curita vivaraz, que, advirtiendo la devoci\u00f3n popular por Melchorita, decid\u00eda industrializarla en su provecho, y, con la frialdad y ambici\u00f3n de un buen empresario, planeaba un negocio m\u00faltiple, que consist\u00eda en fabricar y vender estampitas, escapularios, detentes y toda clase de reliquias de la beatita, cobrar entradas a los sitios donde vivi\u00f3, y organizar colectas y rifas para construirle una capilla y costear comisiones que fueran a activar su canonizaci\u00f3n a Roma. Escrib\u00ed dos ep\u00edlogos distintos, como una noticia de peri\u00f3dico: en uno, los habitantes de Grocio Prado descubr\u00edan los negocios del padre Nicol\u00e1s y lo linchaban, y, en el otro, el curita llegaba a ser arzobispo de Lima. (Decid\u00ed que elegir\u00eda uno u otro final despu\u00e9s de leerle el cuento a la t\u00eda Julia.) Lo escrib\u00ed en la biblioteca del Club Nacional, donde mi trabajo de catalogador de novedades era algo simb\u00f3lico.\n\nLos radioteatros que rescat\u00e9 del almac\u00e9n de Radio Central (labor que me signific\u00f3 doscientos soles extras) fueron comprimidos para un mes de audiciones, el tiempo que tardaron en llegar los libretos de la CMQ. Pero ni aqu\u00e9llos ni \u00e9stos, como hab\u00eda previsto el empresario progresista, pudieron conservar la audiencia gigantesca conquistada por Pedro Camacho. La sinton\u00eda decay\u00f3 y las tarifas publicitarias tuvieron que ser rebajadas para no perder anunciantes. Pero el asunto no result\u00f3 demasiado terrible para los Genaros, quienes, siempre inventivos y din\u00e1micos, encontraron una nueva mina de oro con un programa llamado Responda por Sesenta y Cuatro Mil Soles. Se propalaba desde el Cine Le Paris, y, en \u00e9l, candidatos eruditos en materias diversas (autom\u00f3viles, S\u00f3focles, f\u00fatbol, los incas) respond\u00edan preguntas por cantidades que pod\u00edan llegar hasta esa suma. A trav\u00e9s de Genaro hijo, con quien (ahora muy de vez en cuando) tomaba cafecitos en el Bransa de la Colmena, segu\u00eda los pasos de Pedro Camacho. Estuvo cerca de un mes en la cl\u00ednica privada del doctor Delgado, pero, como resultaba muy cara, los Genaros consiguieron hacerlo transferir al Larco Herrera, el manicomio de la Beneficencia P\u00fablica, donde, al parecer, lo ten\u00edan muy bien considerado. Un domingo, despu\u00e9s de catalogar tumbas en el Cementerio Presb\u00edtero Maestro, fui en \u00f3mnibus hasta la puerta del Larco Herrera con la intenci\u00f3n de visitarlo. Le llevaba de regalo unas bolsitas de yerbaluisa y de menta para preparar infusiones. Pero, en el mismo momento que, entre otras visitas, iba a cruzar el port\u00f3n carcelario, decid\u00ed no hacerlo. La idea de volver a ver al escriba, en este lugar amurallado y promiscuo \u2014en el primer a\u00f1o de universidad hab\u00edamos hecho all\u00ed unas pr\u00e1cticas de psicolog\u00eda\u2014, convertido en uno m\u00e1s de esa muchedumbre de locos, me produjo preventivamente gran angustia. Di media vuelta y regres\u00e9 a Miraflores.\n\nEse lunes dije a mi mam\u00e1 que quer\u00eda entrevistarme con mi padre. Me aconsej\u00f3 que fuera prudente, no decir nada que lo violentara, no exponerme a que me hiciera da\u00f1o, y me dio el tel\u00e9fono de la casa donde estaba alojado. Mi padre me hizo saber que me recibir\u00eda a la ma\u00f1ana siguiente, a las once, en la que hab\u00eda sido su oficina antes de viajar a Estados Unidos. Estaba en el jir\u00f3n Carabaya, al fondo de un pasillo de losetas a ambos lados del cual hab\u00eda departamentos y oficinas. En la Compa\u00f1\u00eda Import\/Export \u2014reconoc\u00ed algunos empleados que hab\u00edan trabajado ya con \u00e9l\u2014 me hicieron pasar a la gerencia. Mi padre estaba solo, sentado en su antiguo escritorio. Vest\u00eda un terno crema, una corbata verde con motas blancas; lo not\u00e9 m\u00e1s delgado que hac\u00eda un a\u00f1o y algo p\u00e1lido.\n\n\u2014Buenos d\u00edas, pap\u00e1 \u2014dije, desde la puerta, haciendo un gran esfuerzo para que mi voz sonara firme.\n\n\u2014Dime lo que tienes que decir \u2014dijo \u00e9l, de manera m\u00e1s neutra que col\u00e9rica, se\u00f1alando un asiento.\n\nMe sent\u00e9 en el borde y tom\u00e9 aire, como un atleta que se dispone a iniciar una prueba.\n\n\u2014He venido a contarte lo que estoy haciendo, lo que voy a hacer \u2014tartamude\u00e9.\n\n\u00c9l permaneci\u00f3 callado, esperando que continuara. Entonces, hablando muy despacio para parecer sereno, espiando sus reacciones, le detall\u00e9 cuidadosamente los trabajos que hab\u00eda conseguido, lo que ganaba en cada uno, c\u00f3mo hab\u00eda distribuido mi tiempo para cumplir con todos y, adem\u00e1s, hacer los deberes y dar los ex\u00e1menes de la universidad. No ment\u00ed, pero present\u00e9 todo bajo la luz m\u00e1s favorable: ten\u00eda mi vida organizada de manera inteligente y seria y estaba ansioso por terminar mi carrera. Cuando me call\u00e9, mi padre permaneci\u00f3 tambi\u00e9n mudo, en espera de la conclusi\u00f3n. As\u00ed que, tragando saliva, tuve que dec\u00edrsela:\n\n\u2014Ya ves que puedo ganarme la vida, mantenerme y seguir los estudios \u2014y luego, sintiendo que la voz se me adelgazaba tanto que apenas se o\u00eda\u2014: Te he venido a pedir permiso para llamar a Julia. Nos hemos casado y no puede seguir viviendo sola.\n\nPesta\u00f1e\u00f3, palideci\u00f3 todav\u00eda m\u00e1s y, por un instante, pens\u00e9 que iba a tener uno de esos ataques de rabia que hab\u00edan sido la pesadilla de mi infancia. Pero se limit\u00f3 a decirme, secamente:\n\n\u2014Como sabes, ese matrimonio no vale. T\u00fa, menor de edad, no puedes casarte sin autorizaci\u00f3n. De modo que si te has casado, s\u00f3lo has podido hacerlo falsificando la autorizaci\u00f3n o tus partidas. En ambos casos, el matrimonio se puede anular f\u00e1cilmente.\n\nMe explic\u00f3 que la falsificaci\u00f3n de un documento p\u00fablico era algo grave, penado por la ley. Si alguien ten\u00eda que pagar los platos rotos por eso, no ser\u00eda yo, el menor, a quien los jueces supondr\u00edan el inducido, sino la mayor de edad, a quien l\u00f3gicamente se considerar\u00eda la inductora. Despu\u00e9s de esa exposici\u00f3n legal, que profiri\u00f3 en tono helado, habl\u00f3 largamente, dejando transparentar, poco a poco, algo de emoci\u00f3n. Yo cre\u00eda que \u00e9l me odiaba, cuando la verdad era que siempre hab\u00eda querido mi bien, si se hab\u00eda mostrado alguna vez severo hab\u00eda sido a fin de corregir mis defectos y prepararme para el futuro. Mi rebeld\u00eda y mi esp\u00edritu de contradicci\u00f3n ser\u00edan mi ruina. Ese matrimonio hab\u00eda sido ponerme una soga en el cuello. \u00c9l se hab\u00eda opuesto pensando en mi bien y no, como cre\u00eda yo, por hacerme da\u00f1o, porque \u00bfqu\u00e9 padre no quer\u00eda a su hijo? Por lo dem\u00e1s, comprend\u00eda que me hubiera enamorado, eso no estaba mal, despu\u00e9s de todo era un acto de hombr\u00eda, m\u00e1s terrible hubiera sido, por ejemplo, que me hubiera dado por ser maric\u00f3n. Pero casarme a los dieciocho a\u00f1os, siendo un mocoso, un estudiante, con una mujer hecha y derecha y divorciada era una insensatez incalculable, algo cuyas verdaderas consecuencias s\u00f3lo comprender\u00eda m\u00e1s tarde, cuando, por culpa de ese matrimonio, fuera un amargado, un pobre diablo en la vida. \u00c9l no deseaba para m\u00ed nada de eso, s\u00f3lo lo mejor y lo m\u00e1s grande. En fin, que tratase por lo menos de no abandonar los estudios, pues lo lamentar\u00eda siempre. Se puso de pie y yo tambi\u00e9n me puse de pie. Sigui\u00f3 un silencio inc\u00f3modo, puntuado por el tableteo de las m\u00e1quinas de escribir del otro cuarto. Balbuce\u00e9 que le promet\u00eda terminar la universidad y \u00e9l asinti\u00f3. Para despedirnos, despu\u00e9s de un segundo de vacilaci\u00f3n, nos abrazamos.\n\nDe su oficina, fui al Correo Central y envi\u00e9 un telegrama: \u00abAmnistiada. Mandar\u00e9 pasaje brevedad posible. Besos\u00bb. Me pas\u00e9 esa tarde, donde el historiador, en la azotea de Panamericana, en el cementerio, devor\u00e1ndome los sesos para imaginar c\u00f3mo reunir el dinero. Esa noche hice una lista de personas a las que pedir\u00eda prestado y cu\u00e1nto a cada una. Pero al d\u00eda siguiente trajeron donde los abuelitos un telegrama de respuesta: \u00abLlego ma\u00f1ana vuelo LAN. Besos\u00bb. Despu\u00e9s supe que hab\u00eda comprado el pasaje vendiendo sus anillos, aretes, prendedores, pulseras y casi toda su ropa. De modo que cuando la recib\u00ed en el aeropuerto de Limatambo, la tarde del jueves, era una mujer pobr\u00edsima.\n\nLa llev\u00e9 directamente al departamentito, que hab\u00eda sido encerado y trapeado por la prima Nancy en persona y embellecido con una rosa roja que dec\u00eda: \u00abBienvenida\u00bb. La t\u00eda Julia revis\u00f3 todo, como si fuera un juguete nuevo. Se divirti\u00f3 viendo las fichas del cementerio, que ten\u00eda bien ordenadas, mis notas para los art\u00edculos de Cultura Peruana, la lista de escritores por entrevistar para El Comercio, y el horario de trabajo y la tabla de gastos que hab\u00eda hecho y donde te\u00f3ricamente se demostraba que pod\u00edamos vivir. Le dije que, despu\u00e9s de hacerle el amor, le leer\u00eda un cuento que se llamaba La beata y el padre Nicol\u00e1s para que me ayudara a elegir el final.\n\n\u2014Vaya, Varguitas \u2014se re\u00eda ella, mientras se desvest\u00eda a la carrera\u2014. Te est\u00e1s haciendo un hombrecito. Ahora, para que todo sea perfecto y se te quite esa cara de bebe, prom\u00e9teme que te dejar\u00e1s crecer el bigote.\n\n### XX\n\nEL MATRIMONIO con la t\u00eda Julia fue realmente un \u00e9xito y dur\u00f3 bastante m\u00e1s de lo que todos los parientes, y hasta ella misma, hab\u00edan temido, deseado o pronosticado: ocho a\u00f1os. En ese tiempo, gracias a mi obstinaci\u00f3n y a su ayuda y entusiasmo, combinados con una dosis de buena suerte, otros pron\u00f3sticos (sue\u00f1os, apetitos) se hicieron realidad. Hab\u00edamos llegado a vivir en la famosa buhardilla de Par\u00eds y yo, mal que mal, me hab\u00eda hecho un escritor y publicado algunos libros. No termin\u00e9 nunca la carrera de abogado, pero, para indemnizar de alg\u00fan modo a la familia y para poder ganarme la vida con m\u00e1s facilidad, saqu\u00e9 un t\u00edtulo universitario, en una perversi\u00f3n acad\u00e9mica tan aburrida como el Derecho: la Filolog\u00eda Rom\u00e1nica.\n\nCuando la t\u00eda Julia y yo nos divorciamos hubo en mi dilatada familia copiosas l\u00e1grimas, porque todo el mundo (empezando por mi madre y mi padre, claro est\u00e1) la adoraba. Y cuando, un a\u00f1o despu\u00e9s, volv\u00ed a casarme, esta vez con una prima (hija de la t\u00eda Olga y el t\u00edo Lucho, qu\u00e9 casualidad) el esc\u00e1ndalo familiar fue menos ruidoso que la primera vez (consisti\u00f3, sobre todo, en un hervor de chismes). Eso s\u00ed, hubo una conspiraci\u00f3n perfecta para obligarme a casar por la Iglesia, en la que estuvo involucrado hasta el arzobispo de Lima (era, por supuesto, pariente nuestro), quien se apresur\u00f3 a firmar las dispensas autorizando el enlace. Para entonces, la familia estaba ya curada de espanto y esperaba de m\u00ed (lo que equival\u00eda a: me perdonaba de antemano) cualquier barbaridad.\n\nHab\u00eda vivido con la t\u00eda Julia un a\u00f1o en Espa\u00f1a y cinco en Francia, y, luego, segu\u00ed viviendo con la prima Patricia en Europa, primero en Londres y luego en Barcelona. Para esa \u00e9poca ten\u00eda un trato con una revista de Lima, a la que yo enviaba art\u00edculos y ella me pagaba con pasajes que me permit\u00edan volver todos los a\u00f1os al Per\u00fa por algunas semanas. Estos viajes, gracias a los cuales ve\u00eda a la familia y a los amigos, eran para m\u00ed muy importantes. Pensaba seguir viviendo en Europa de manera indefinida, por m\u00faltiples razones, pero, sobre todo, porque all\u00ed hab\u00eda encontrado siempre, como periodista, traductor, locutor o profesor, trabajos que me dejaban tiempo libre. Al llegar a Madrid, la primera vez, le hab\u00eda dicho a la t\u00eda Julia: \u00abVoy a tratar de ser un escritor, s\u00f3lo voy a aceptar trabajos que no me aparten de la literatura\u00bb. Ella me respondi\u00f3: \u00ab\u00bfMe rasgo la falda, me pongo un turbante y salgo a la Gran V\u00eda a buscar clientes desde hoy?\u00bb. Lo cierto es que tuve mucha suerte. Ense\u00f1ando espa\u00f1ol en la Escuela Berlitz de Par\u00eds, redactando noticias en la France Presse, traduciendo para la Unesco, doblando pel\u00edculas en los estudios de G\u00e9n\u00e9villiers o preparando programas para la Radio-Televisi\u00f3n Francesa, siempre hab\u00eda encontrado empleos alimenticios que me dejaban, cuando menos, la mitad de cada d\u00eda exclusivamente para escribir. El problema era que todo lo que escrib\u00eda se refer\u00eda al Per\u00fa. Eso me creaba, cada vez m\u00e1s, un problema de inseguridad, por el desgaste de la perspectiva (ten\u00eda la man\u00eda de la ficci\u00f3n realista). Pero me resultaba inimaginable siquiera la idea de vivir en Lima. El recuerdo de mis siete trabajos alimenticios lime\u00f1os, que con las justas nos permit\u00edan comer, apenas leer, y escribir s\u00f3lo a hurtadillas, en los huequitos que quedaban libres y cuando estaba ya cansado, me pon\u00eda los pelos de punta y me juraba que no volver\u00eda a ese r\u00e9gimen ni muerto. Por otra parte, el Per\u00fa me ha parecido siempre un pa\u00eds de gentes tristes.\n\nPor eso el trueque que acordamos, primero con el diario Expreso y luego con la revista Caretas, de art\u00edculos por dos pasajes de avi\u00f3n anuales, me result\u00f3 providencial. Ese mes que pas\u00e1bamos en el Per\u00fa, cada a\u00f1o, generalmente en el invierno (julio o agosto) me permit\u00eda zambullirme en el ambiente, los paisajes, los seres sobre los cuales hab\u00eda estado tratando de escribir los once meses anteriores. Me era enormemente \u00fatil (no s\u00e9 si en los hechos, pero, sin la menor duda, psicol\u00f3gicamente), una inyecci\u00f3n de energ\u00eda, volver a o\u00edr hablar peruano, escuchar a mi alrededor esos giros, vocablos, entonaciones que me reinstalaban en un medio al que me sent\u00eda visceralmente pr\u00f3ximo, pero del que, de todos modos, me hab\u00eda alejado, del que cada a\u00f1o perd\u00eda innovaciones, resonancias, claves.\n\nLas venidas a Lima eran, pues, unas vacaciones en las que, literalmente, no descansaba un segundo y de las que volv\u00eda a Europa exhausto. S\u00f3lo con mi selv\u00e1tica parentela y los numerosos amigos, ten\u00edamos invitaciones diarias a almorzar y comer, y el resto del tiempo lo ocupaban mis trajines documentales. As\u00ed, un a\u00f1o, hab\u00eda emprendido un viaje a la zona del Alto Mara\u00f1\u00f3n, para ver, o\u00edr y sentir de cerca un mundo que era escenario de la novela que escrib\u00eda, y otro a\u00f1o, escoltado por amigos diligentes, hab\u00eda realizado una exploraci\u00f3n sistem\u00e1tica de los antros nocturnos \u2014cabarets, bares, lenocinios\u2014, en los que transcurr\u00eda la mala vida del protagonista de otra historia. Mezclando el trabajo y el placer \u2014porque esas investigaciones no fueron nunca una obligaci\u00f3n, o lo fueron siempre de manera muy vital, afanes que me divert\u00edan en s\u00ed mismos y no s\u00f3lo por el provecho literario que pudiera sacarles\u2014, en esos viajes hac\u00eda cosas que antes, cuando viv\u00eda en Lima, no hice nunca, y que ahora, que he vuelto a vivir en el Per\u00fa, tampoco hago: ir a pe\u00f1as criollas y a los coliseos a ver bailes folcl\u00f3ricos, recorridos por los tugurios de los barrios marginales, caminatas por distritos que conoc\u00eda mal o desconoc\u00eda como el Callao, Bajo el Puente y los Barrios Altos, apostar en las carreras de caballos y husmear en las catacumbas de las iglesias coloniales y la (supuesta) casa de la Perricholi.\n\nEse a\u00f1o, en cambio, me dediqu\u00e9 a una averiguaci\u00f3n m\u00e1s bien libresca. Estaba escribiendo una novela situada en la \u00e9poca del general Manuel Apolinario Odr\u00eda (1948-1956), y en mi mes de vacaciones lime\u00f1as, iba, un par de ma\u00f1anas cada semana, a la hemeroteca de la Biblioteca Nacional, a hojear las revistas y peri\u00f3dicos de esos a\u00f1os, e, incluso, con algo de masoquismo, a leer algunos de los discursos que los asesores (todos abogados, a juzgar por la ret\u00f3rica forense) le hac\u00edan decir al dictador. Al salir de la Biblioteca Nacional, a eso del mediod\u00eda, bajaba a pie por la avenida Abancay, que comenzaba a convertirse en un enorme mercado de vendedores ambulantes. En sus veredas, una apretada muchedumbre de hombres y mujeres, muchos de ellos con ponchos y polleras serranas, vend\u00edan, sobre mantas extendidas en el suelo, sobre peri\u00f3dicos o en quioscos improvisados con cajas, latas y toldos, todas las baratijas imaginables, desde alfileres y horquillas hasta vestidos y ternos, y, por supuesto, toda clase de comidas preparadas en el sitio, en peque\u00f1os braseros. Era uno de los lugares de Lima que m\u00e1s hab\u00eda cambiado, esa avenida Abancay, ahora atestada y andina, en la que no era raro, entre el fort\u00edsimo olor a fritura y condimentos, o\u00edr hablar quechua. No se parec\u00eda en nada a la ancha, severa avenida de oficinistas y alguno que otro mendigo por la que, diez a\u00f1os atr\u00e1s, cuando era cachimbo universitario, sol\u00eda caminar en direcci\u00f3n a la misma Biblioteca Nacional. All\u00ed, en esas cuadras, se pod\u00eda ver, tocar, concentrado, el problema de las migraciones campesinas hacia la capital, que en ese decenio duplicaron la poblaci\u00f3n de Lima e hicieron brotar, sobre los cerros, los arenales, los muladares, ese cerco de barriadas donde ven\u00edan a parar los millares y millares de seres que, por la sequ\u00eda, las duras condiciones de trabajo, la falta de perspectivas, el hambre, abandonaban las provincias.\n\nAprendiendo a conocer esta nueva cara de la ciudad, bajaba por la avenida Abancay en direcci\u00f3n al parque Universitario y a lo que hab\u00eda sido antes la Universidad de San Marcos (las facultades se hab\u00edan mudado a las afueras de Lima y en ese caser\u00f3n donde yo estudi\u00e9 Letras y Derecho funcionaban ahora un museo y oficinas). No s\u00f3lo lo hac\u00eda por curiosidad y cierta nostalgia, sino tambi\u00e9n por inter\u00e9s literario, pues en la novela que trabajaba algunos episodios ocurr\u00edan en el parque Universitario, en la casona de San Marcos y en las librer\u00edas de viejo, los billares y los tiznados cafecitos de los alrededores. Precisamente, esa ma\u00f1ana estaba plantado, como un turista, frente a la bonita capilla de los Pr\u00f3ceres, observando a los ambulantes del contorno \u2014lustrabotas, alfajoreros, heladeros, sandwicheros\u2014 cuando sent\u00ed que me cog\u00edan el hombro. Era \u2014doce a\u00f1os m\u00e1s viejo, pero id\u00e9ntico\u2014 el Gran Pablito.\n\nNos dimos un fuerte abrazo. Realmente, no hab\u00eda cambiado nada: era el mismo cholo fornido y risue\u00f1o, de respiraci\u00f3n asm\u00e1tica, que apenas levantaba los pies del suelo para andar y parec\u00eda estar patinando por la vida. No ten\u00eda una cana, pese a que deb\u00eda raspar los sesenta, y llevaba la cabeza bien engominada, los lacios pelos cuidadosamente aplastados, como un argentino de los a\u00f1os cuarenta. Pero se lo ve\u00eda mucho mejor vestido que cuando era periodista (en teor\u00eda) de Radio Panamericana: un terno verde, a cuadros, una corbatita luminosa (era la primera vez que lo ve\u00eda encorbatado) y los zapatos destellando. Me dio tanto gusto verlo que le propuse tomar un caf\u00e9. Acept\u00f3 y terminamos en una mesa del Palermo, un barcito restaurante ligado, tambi\u00e9n, en mi memoria, a los a\u00f1os universitarios. Le dije que no le preguntaba c\u00f3mo lo hab\u00eda tratado la vida pues bastaba verlo para saber que lo hab\u00eda tratado muy bien. \u00c9l sonri\u00f3 \u2014ten\u00eda en el \u00edndice un anillo dorado con un dibujo incaico\u2014, satisfecho:\n\n\u2014No puedo quejarme \u2014asinti\u00f3\u2014. Despu\u00e9s de tanta pellejer\u00eda, a la vejez cambi\u00f3 mi estrella. Pero, ante todo, perm\u00edtame una cervecita, por el gran gusto de verlo \u2014llam\u00f3 al mozo, pidi\u00f3 una Pilsen bien helada y lanz\u00f3 una risa que le provoc\u00f3 su tradicional ataque de asma\u2014-. Dicen que el que se casa se friega. Conmigo fue al rev\u00e9s.\n\nMientras nos tom\u00e1bamos la cerveza, el Gran Pablito, con las pausas que le exig\u00edan sus bronquios, me cont\u00f3 que al llegar la televisi\u00f3n al Per\u00fa, los Genaros lo pusieron de portero, con uniforme y gorra granates, en el edificio que hab\u00edan construido en la avenida Arequipa para el Canal Cinco.\n\n\u2014De periodista a portero, parece una degradaci\u00f3n \u2014se encogi\u00f3 de hombros\u2014. Y lo era, desde el punto de vista de los t\u00edtulos. \u00bfPero acaso se comen? Me aumentaron el sueldo y eso es lo principal.\n\nSer portero no era un trabajo matador: anunciar a las visitas, informarles c\u00f3mo estaban repartidas las secciones de la televisi\u00f3n, poner orden en las colas para asistir a las audiciones. El resto del tiempo se lo pasaba discutiendo de f\u00fatbol con el polic\u00eda de la esquina. Pero, adem\u00e1s \u2014y chasque\u00f3 la lengua, saboreando una reminiscencia grata\u2014, con los meses, un aspecto de su trabajo fue ir, todos los mediod\u00edas, a comprar esas empanaditas de queso y de carne que hacen en el Berisso, la bodega que est\u00e1 en Arenales, a una cuadra del Canal Cinco. A los Genaros les encantaban, y lo mismo a empleados, actores, locutores y productores, a los cuales tambi\u00e9n el Gran Pablito les tra\u00eda empanaditas, con lo que ganaba buenas propinas. Fue en esos trajines entre la televisi\u00f3n y el Berisso (su uniforme le hab\u00eda merecido entre los chiquillos del barrio el apodo de Bombero) que el Gran Pablito conoci\u00f3 a su futura esposa. Era la mujer que fabricaba esas crujientes delicias: la cocinera del Berisso.\n\n\u2014La impresion\u00f3 mi uniforme y mi quep\u00ed de general, me vio y cay\u00f3 rendida \u2014se re\u00eda, se ahogaba, beb\u00eda su cerveza, volv\u00eda a ahogarse y continuaba el Gran Pablito\u2014. Una morena que est\u00e1 requetebi\u00e9n. Veinte a\u00f1os m\u00e1s joven que quien le habla. Unas teteras donde no entran balas. Tal cual se la pinto, don Mario.\n\nHab\u00eda comenzado a meterle conversaci\u00f3n y piropearla, ella a re\u00edrse y, de repente, salieron juntos. Se hab\u00edan enamorado y vivido un romance de pel\u00edcula. La morena era de armas tomar, emprendedora y con la cabeza llena de proyectos. A ella se le meti\u00f3 que abrieran un restaurante. Y, cuando el Gran Pablito preguntaba \u00ab\u00bfcon qu\u00e9?\u00bb, ella respond\u00eda: con la plata que les dieran al renunciar. Y, aunque a \u00e9l le parec\u00eda una locura dejar lo seguro por lo incierto, ella sali\u00f3 con su gusto. Las indemnizaciones alcanzaron para un local pobret\u00f3n en el jir\u00f3n Paruro y tuvieron que prestarse de todo el mundo para las mesitas y la cocina, y \u00e9l mismo pint\u00f3 las paredes y el nombre sobre la puerta: El Pavo Real. El primer a\u00f1o hab\u00eda dado apenas para sobrevivir y el trabajo fue brav\u00edsimo. Se levantaban al alba para ir a la Parada a conseguir los mejores ingredientes y a los precios m\u00e1s bajos, y todo lo hac\u00edan solos: ella cocinaba y \u00e9l serv\u00eda, cobraba, y entre los dos barr\u00edan y arreglaban. Dorm\u00edan en unos colchones que tend\u00edan entre las mesas, cuando se cerraba el local. Pero, a partir del segundo a\u00f1o, la clientela creci\u00f3. Tanto que hab\u00edan tenido que contratar un ayudante para cocina y otro para mozo, y, al final, rechazaban clientes, porque no cab\u00edan. Y, entonces, a esa morena se le ocurri\u00f3 alquilar la casa de al lado, tres veces m\u00e1s grande. Lo hicieron y no se arrepent\u00edan. Ahora, hasta hab\u00edan habilitado el segundo piso, y ellos ten\u00edan una casita frente a El Pavo Real. En vista de que se entend\u00edan tan bien, se casaron.\n\nLo felicit\u00e9, le pregunt\u00e9 si hab\u00eda aprendido a cocinar.\n\n\u2014Se me ocurre una cosa \u2014dijo de repente el Gran Pablito\u2014. Vamos a buscar a Pascual y almorzaremos en el restaurante. Perm\u00edtame hacerle ese agasajo, don Mario.\n\nAcept\u00e9, porque nunca he sabido rechazar invitaciones, y, tambi\u00e9n, porque me dio curiosidad ver a Pascual. El Gran Pablito me cont\u00f3 que dirig\u00eda una revista de variedades, que tambi\u00e9n \u00e9l hab\u00eda progresado. Se ve\u00edan con frecuencia, Pascual era un asiduo de El Pavo Real.\n\nLa revista Extra ten\u00eda su local bastante lejos, en una transversal de la avenida Arica, en Bre\u00f1a. Fuimos hasta all\u00e1 en un \u00f3mnibus que en mis tiempos no exist\u00eda. Tuvimos que dar varias vueltas, porque el Gran Pablito no se acordaba de la direcci\u00f3n. Por fin la encontramos, en una callejuela perdida, a la espalda del Cine Fantas\u00eda. Desde afuera se pod\u00eda ver que Extra no flotaba en la bonanza: dos puertas de garaje entre las cuales un r\u00f3tulo precariamente suspendido de un solo clavo anunciaba el nombre del semanario. Adentro, se descubr\u00eda que los garajes hab\u00edan sido unidos mediante un simple agujero abierto en la pared, sin pulir ni enmarcar, como si el alba\u00f1il hubiera abandonado el trabajo a medio hacer. Disimulaba la abertura un biombo de cart\u00f3n, constelado, como los cuartos de ba\u00f1o de los lugares p\u00fablicos, de palabrotas y dibujos obscenos. En las paredes del garaje por donde entramos, entre manchas de humedad y mugre, hab\u00eda fotos, afiches y car\u00e1tulas de Extra: se reconoc\u00edan caras de futbolistas, de cantantes, y, evidentemente, de delincuentes y v\u00edctimas. Cada car\u00e1tula ven\u00eda acompa\u00f1ada de rechinantes titulares y alcanc\u00e9 a leer frases como \u00abMata a la madre para casarse con la hija\u00bb y \u00abPolic\u00eda sorprende baile de domin\u00f3s: \u00a1todos eran hombres!\u00bb. Esa habitaci\u00f3n parec\u00eda servir de redacci\u00f3n, taller de fotograf\u00eda y archivo. Hab\u00eda tal aglomeraci\u00f3n de objetos que resultaba dif\u00edcil abrirse camino: mesitas con m\u00e1quinas de escribir en las que dos tipos tecleaban muy apurados, altos de devoluciones de la revista que un chiquillo estaba ordenando en paquetes que amarraba con una pita; en un rinc\u00f3n, un ropero abierto lleno de negativos, de fotos, de clis\u00e9s, y, detr\u00e1s de una mesa, una de cuyas patas hab\u00eda sido reemplazada por tres ladrillos, una chica de chompa roja pasaba recibos a un libro de caja. Las cosas y las personas del local parec\u00edan en un estado supino de estrechez. Nadie nos ataj\u00f3 ni nos pregunt\u00f3 nada, y nadie nos devolvi\u00f3 las buenas tardes.\n\nAl otro lado del biombo, ante paredes cubiertas tambi\u00e9n de car\u00e1tulas sensacionalistas, hab\u00eda tres escritorios en los que un cartelito, hecho a tinta, especificaba las funciones de sus ocupantes: director, jefe de redacci\u00f3n, administrador. Al vernos ingresar en la habitaci\u00f3n, dos personas inclinadas sobre unos pliegos de pruebas alzaron la cabeza. El que estaba de pie era Pascual.\n\nNos dimos un gran abrazo. Hab\u00eda cambiado bastante, \u00e9l s\u00ed; estaba gordo, con barriga y papada, y algo en la expresi\u00f3n lo hac\u00eda aparecer casi viejo. Se hab\u00eda dejado un bigotito rar\u00edsimo, vagamente hitleriano, que griseaba. Me hizo muchas demostraciones de afecto; cuando sonri\u00f3, vi que hab\u00eda perdido dientes. Despu\u00e9s de los saludos, me present\u00f3 al otro personaje, un moreno de camisa color mostaza, que permanec\u00eda en su escritorio:\n\n\u2014El director de Extra \u2014dijo Pascual\u2014. El doctor Rebagliati.\n\n\u2014Casi meto la pata, el Gran Pablito me dijo que el director eras t\u00fa \u2014le cont\u00e9, d\u00e1ndole la mano al doctor Rebagliati.\n\n\u2014Estamos en decadencia, pero no a ese extremo \u2014coment\u00f3 \u00e9ste\u2014. Si\u00e9ntense, si\u00e9ntense.\n\n\u2014Soy jefe de redacci\u00f3n \u2014me explic\u00f3 Pascual\u2014. \u00c9ste es mi escritorio.\n\nEl Gran Pablito le dijo que hab\u00edamos venido a buscarlo para ir a El Pavo Real, a recordar los tiempos de Panamericana. Aplaudi\u00f3 la idea, pero, eso s\u00ed, tendr\u00edamos que esperarlo unos minutos, deb\u00eda llevar a la imprenta de la vuelta esas pruebas, era urgente pues estaban cerrando la edici\u00f3n. Se fue y nos dej\u00f3, mir\u00e1ndonos las caras, con el doctor Rebagliati. \u00c9ste, cuando se enter\u00f3 que yo viv\u00eda en Europa, me comi\u00f3 a preguntas. \u00bfEran las francesas tan f\u00e1ciles como se dec\u00eda? \u00bfEran tan sabias y desvergonzadas en la cama? Se empe\u00f1\u00f3 en que le hiciera estad\u00edsticas, cuadros comparativos, sobre las mujeres de Europa. \u00bfVerdad que las hembras de cada pa\u00eds ten\u00edan costumbres originales? \u00c9l, por ejemplo (el Gran Pablito lo escuchaba revolviendo los ojos con delectaci\u00f3n), hab\u00eda o\u00eddo decir, a gente muy viajada, cosas interesant\u00edsimas. \u00bfCierto que las italianas ten\u00edan la obsesi\u00f3n de la corneta? \u00bfQue las parisinas nunca estaban contentas si no las bombardeaban por detr\u00e1s? \u00bfQue las n\u00f3rdicas se lo aflojaban a sus propios padres? Yo contestaba como pod\u00eda a la verborrea del doctor Rebagliati, que iba contaminando la atm\u00f3sfera del cuartito de una densidad lujuriosa, seminal, y lamentaba cada instante m\u00e1s el haberme visto atrapado en ese almuerzo, que, sin duda, terminar\u00eda a las mil quinientas. El Gran Pablito se re\u00eda, asombrado y excitad\u00edsimo con las revelaciones er\u00f3tico-sociol\u00f3gicas del director. Cuando la curiosidad de \u00e9ste me extenu\u00f3, le ped\u00ed su tel\u00e9fono. Puso una cara sarc\u00e1stica:\n\n\u2014Est\u00e1 cortado hace una semana, por no pagar la cuenta \u2014dijo, con franqueza agresiva\u2014. Porque, donde nos ve, esta revista se hunde y todos los imb\u00e9ciles que trabajamos aqu\u00ed nos hundimos con ella.\n\nEn el acto, con un placer masoquista, me cont\u00f3 que Extra hab\u00eda nacido en la \u00e9poca de Odr\u00eda, bajo buenos auspicios; el r\u00e9gimen le daba avisos y le pasaba plata por lo bajo para que atacara a ciertas gentes y defendiera a otras. Adem\u00e1s, era una de las pocas revistas permitidas y se vend\u00eda como pan caliente. Pero, al irse Odr\u00eda, empez\u00f3 una competencia terrible y quebr\u00f3. As\u00ed la hab\u00eda recogido \u00e9l, ya cad\u00e1ver. Y la hab\u00eda levantado, cambi\u00e1ndole la l\u00ednea, convirti\u00e9ndola en revista de hechos sensacionales. Todo march\u00f3 sobre ruedas, un tiempo, pese a las deudas que arrastraban. Pero en el \u00faltimo a\u00f1o, con la subida del papel, los aumentos en la imprenta, la campa\u00f1a en contra de parte de los enemigos y la retirada del avisaje, las cosas se hab\u00edan puesto negras. Adem\u00e1s, hab\u00edan perdido juicios, de canallas que los acusaban de difamaci\u00f3n. Ahora, los due\u00f1os, aterrados, hab\u00edan regalado todas las acciones a los redactores, para no pagar los platos rotos cuando los remataran. Lo que no tardar\u00eda, ya en las \u00faltimas semanas la situaci\u00f3n era tr\u00e1gica: no hab\u00eda para sueldos, la gente se llevaba las m\u00e1quinas, vend\u00eda los escritorios, se robaban todo lo que ten\u00eda algo de valor, adelant\u00e1ndose al colapso.\n\n\u2014Esto no dura un mes, mi amigo \u2014repiti\u00f3, resoplando con una especie de disgusto feliz\u2014. Somos ya cad\u00e1veres, \u00bfno huele la putrefacci\u00f3n?\n\nLe iba a decir que, efectivamente, la ol\u00eda, cuando interrumpi\u00f3 la conversaci\u00f3n una figurita esquel\u00e9tica que entr\u00f3 en el cuarto sin necesidad de apartar el biombo, por la angosta abertura. Ten\u00eda un corte de pelo alem\u00e1n, algo rid\u00edculo, y vest\u00eda como un vagabundo, un overol azulino y una camisita con parches bajo un su\u00e9ter gris\u00e1ceo que le quedaba ajustad\u00edsimo. Lo m\u00e1s ins\u00f3lito era su calzado: unas rojizas zapatillas de b\u00e1squet, tan viejas que una de ellas estaba sujeta por un cord\u00f3n amarrado alrededor de la punta, como si la suela estuviera suelta o por soltarse. Apenas lo vio, el doctor Rebagliati comenz\u00f3 a re\u00f1irlo:\n\n\u2014Si usted cree que va a seguir burl\u00e1ndose de m\u00ed, se equivoca \u2014dijo, acerc\u00e1ndose a \u00e9l con aire tan amenazador que el esqueleto dio un brinquito\u2014. \u00bfNo ten\u00eda que traer anoche la llegada del Monstruo de Ayacucho?\n\n\u2014La traje, se\u00f1or director. Estuve aqu\u00ed, con todos los datos pertinentes, media hora despu\u00e9s de que los patrulleros desembarcaron en la Prefectura al interfecto \u2014declam\u00f3 el hombrecillo.\n\nLa sorpresa fue tan grande que deb\u00ed poner cara de alelado. La perfecta dicci\u00f3n, el timbre c\u00e1lido, las palabrejas \u00abpertinente\u00bb e \u00abinterfecto\u00bb, s\u00f3lo pod\u00edan ser de \u00e9l. \u00bfPero, c\u00f3mo identificar al escriba boliviano en el f\u00edsico y el atuendo de este espantap\u00e1jaros al que el doctor Rebagliati se com\u00eda vivo?:\n\n\u2014No sea mentiroso, por lo menos tenga el coraje de sus faltas. Usted no trajo el material y Melcochita no pudo completar su cr\u00f3nica y la informaci\u00f3n va a salir tuerta. \u00a1Y a m\u00ed no me gustan las cr\u00f3nicas tuertas porque eso es mal periodismo!\n\n\u2014Lo traje, se\u00f1or director \u2014respond\u00eda, con educaci\u00f3n y alarma, Pedro Camacho\u2014. Encontr\u00e9 la revista cerrada. Eran las once y quince en punto. Pregunt\u00e9 la hora a un transe\u00fante, se\u00f1or director. Y entonces, porque sab\u00eda la importancia de esos datos, me dirig\u00ed a la casa de Melcochita. Y lo estuve esperando en la vereda, hasta las dos de la ma\u00f1ana, y no se aperson\u00f3 a dormir. No es mi culpa, se\u00f1or director. Los patrulleros que tra\u00edan al Monstruo se encontraron un derrumbe y llegaron a las once en vez de las nueve. No me acuse de incumplimiento. Para m\u00ed, la revista es lo primero, pasa antes que la salud, se\u00f1or director.\n\nPoco a poco, no sin esfuerzo, fui relacionando, acercando, lo que recordaba de Pedro Camacho con lo que ten\u00eda presente. Los ojos saltones eran los mismos, pero hab\u00edan perdido su fanatismo, la vibraci\u00f3n obsesiva. Ahora su luz era pobre, opaca, huidiza y atemorizada. Y tambi\u00e9n los gestos y ademanes, la manera de accionar cuando hablaba, ese movimiento antinatural del brazo y la mano que parec\u00eda el de un pregonero de feria, eran los de antes, igual que su incomparable, cadenciosa, arrulladora voz.\n\n\u2014Lo que pasa es que usted, con la ro\u00f1oser\u00eda de no tomar un \u00f3mnibus, un colectivo, llega tarde a todas partes, \u00e9sa es la verdad de la milanesa \u2014gru\u00f1\u00eda, hist\u00e9rico, el doctor Rebagliati\u2014. No sea avaro, carajo, g\u00e1stese los cuatro cobres que vale un \u00f3mnibus y llegue a los sitios a la hora debida.\n\nPero las diferencias eran mayores que las semejanzas. El cambio principal se deb\u00eda al pelo; al cortarse la cabellera que le llegaba a los hombros y hacerse ese rapado, su cara se hab\u00eda vuelto m\u00e1s angulosa, m\u00e1s peque\u00f1a, hab\u00eda perdido car\u00e1cter, solvencia. Y estaba, adem\u00e1s, much\u00edsimo m\u00e1s flaco, parec\u00eda un fakir, casi un esp\u00edritu. Pero lo que quiz\u00e1 me impidi\u00f3 reconocerlo en el primer momento fue su ropa. Antes, s\u00f3lo lo hab\u00eda visto de negro, con el terno f\u00fanebre y brilloso y la corbatita de lazo que eran inseparables de su persona. Ahora, con ese overol de cargador, esa camisa con remiendos, esas zapatillas atadas, parec\u00eda una caricatura de la caricatura que era doce a\u00f1os atr\u00e1s.\n\n\u2014Le aseguro que no es como piensa, se\u00f1or director \u2014se defend\u00eda, con gran convicci\u00f3n\u2014. Le he demostrado que a pie llego m\u00e1s r\u00e1pido a cualquier parte que en esas pestilentes carcochas. No es por ro\u00f1oser\u00eda que yo camino, sino para cumplir mis deberes con m\u00e1s diligencia. Y muchas veces corro, se\u00f1or director.\n\nTambi\u00e9n en eso segu\u00eda siendo el de antes: en su carencia absoluta de humor. Hablaba sin la m\u00e1s ligera sombra de picard\u00eda, chispa, e, incluso, emoci\u00f3n, de manera autom\u00e1tica, despersonalizada, aunque las cosas que ahora dec\u00eda hubieran sido entonces impensables en su boca.\n\n\u2014D\u00e9jese de estupideces y de man\u00edas, estoy viejo para que me tomen el pelo \u2014el doctor Rebagliati se volvi\u00f3 a nosotros, poni\u00e9ndonos de testigos\u2014. \u00bfHan o\u00eddo una idiotez igual? \u00bfQue uno puede recorrer las comisar\u00edas de Lima m\u00e1s r\u00e1pido a pie que en \u00f3mnibus? Y este se\u00f1or quiere que yo me trague semejante caca \u2014se volvi\u00f3 otra vez al escriba boliviano, quien no le hab\u00eda quitado la vista de encima, sin echarnos siquiera una mirada de soslayo\u2014: No tengo que recordarle, porque me imagino que usted se acuerda de ello cada vez que se pone frente a un plato de comida, que aqu\u00ed se le hace un gran favor d\u00e1ndole trabajo, cuando estamos en tan mala situaci\u00f3n que deber\u00edamos suprimir redactores, ya no digo dateros. Por lo menos, entonces, agradezca, y cumpla con sus obligaciones.\n\nEn eso entr\u00f3 Pascual, diciendo desde el biombo: \u00abTodo listo, el n\u00famero entr\u00f3 en prensa\u00bb, y, disculp\u00e1ndose por habernos hecho esperar. Yo me acerqu\u00e9 a Pedro Camacho, cuando \u00e9ste se dispon\u00eda a salir:\n\n\u2014C\u00f3mo est\u00e1, Pedro \u2014le dije, estir\u00e1ndole la mano\u2014. \u00bfNo se acuerda de m\u00ed?\n\nMe mir\u00f3 de arriba abajo, entrecerrando los ojos y adelantando la cara, sorprendido, como si me viera por primera vez en la vida. Por fin, me dio la mano, en un saludo seco y ceremonioso, a la vez que, haciendo su venia caracter\u00edstica, dec\u00eda:\n\n\u2014Tanto gusto. Pedro Camacho, un amigo.\n\n\u2014Pero, no puede ser \u2014dije, sintiendo una gran confusi\u00f3n\u2014. \u00bfMe he vuelto tan viejo?\n\n\u2014D\u00e9jate de jugar al amn\u00e9sico \u2014Pascual le dio una palmada que lo hizo trastabillear\u2014. \u00bfNo te acuerdas tampoco que te pasabas la vida gorre\u00e1ndole cafecitos en el Bransa?\n\n\u2014M\u00e1s bien yerbaluisas con menta \u2014brome\u00e9, escrutando, en busca de un signo, la carita atenta y, al mismo tiempo, indiferente de Pedro Camacho. Asinti\u00f3 (vi su cr\u00e1neo casi pelado), esbozando una brev\u00edsima sonrisa de circunstancias, que puso sus dientes al aire:\n\n\u2014Muy recomendable para el est\u00f3mago, buen digestivo, y, adem\u00e1s, quema la grasa \u2014dijo. Y, r\u00e1pidamente, como haciendo una concesi\u00f3n para librarse de nosotros\u2014: S\u00ed, es posible, no lo niego. Pudimos conocernos, seguramente \u2014y repiti\u00f3\u2014: Tanto gusto.\n\nEl Gran Pablito tambi\u00e9n se hab\u00eda acercado y le pas\u00f3 un brazo por el hombro, en un gesto paternal y burl\u00f3n. Mientras lo remec\u00eda medio afectuosa medio despectivamente, se dirigi\u00f3 a m\u00ed:\n\n\u2014Es que aqu\u00ed Pedrito no quiere acordarse de cuando era un personaje, ahora que es la \u00faltima rueda del coche \u2014Pascual se ri\u00f3, el Gran Pablito se ri\u00f3, yo simul\u00e9 que re\u00eda y el propio Pedro Camacho hizo un conato de sonrisa\u2014. Si hasta nos viene con el cuento de que no se acuerda ni de Pascual ni de m\u00ed \u2014le pas\u00f3 la mano por el poco pelo, como a un perrito\u2014. Estamos yendo a almorzar, para recordar esos tiempos en que eras rey. Te armaste, Pedrito, hoy comer\u00e1s caliente. \u00a1Est\u00e1s invitado!\n\n\u2014Cu\u00e1nto les agradezco, colegas \u2014dijo \u00e9l, al instante, haciendo su venia ritual\u2014. Pero no me es posible acompa\u00f1arlos. Me espera mi esposa. Se inquietar\u00eda si no llego a almorzar.\n\n\u2014Te tiene dominado, eres su esclavo, qu\u00e9 verg\u00fcenza \u2014lo remeci\u00f3 el Gran Pablito.\n\n\u2014\u00bfSe cas\u00f3 usted? \u2014dije, pasmado, pues no conceb\u00eda que Pedro Camacho tuviera un hogar, una esposa, hijos...\u2014. Vaya, felicitaciones, yo lo cre\u00eda un solter\u00f3n empedernido.\n\n\u2014Hemos festejado nuestras bodas de plata \u2014me repuso, con su tono preciso y as\u00e9ptico\u2014. Una gran esposa, se\u00f1or. Abnegada y buena como nadie. Estuvimos separados, por circunstancias de la vida, pero, cuando necesit\u00e9 ayuda, ella volvi\u00f3 para darme su apoyo. Una gran esposa, como le digo. Es artista, una artista extranjera \u2014vi que el Gran Pablito, Pascual y el doctor Rebagliati cambiaban una mirada burlona, pero Pedro Camacho no se dio por aludido. Luego de una pausa, a\u00f1adi\u00f3\u2014: Bien, que se diviertan, colegas, estar\u00e9 con ustedes en el pensamiento.\n\n\u2014Cuidadito con fallarme otra vez, porque ser\u00eda la \u00faltima \u2014le advirti\u00f3 el doctor Rebagliati, cuando el escriba desaparec\u00eda tras del biombo.\n\nNo se hab\u00edan apagado las pisadas de Pedro Camacho \u2014deb\u00eda de estar llegando a la puerta de calle\u2014 y Pascual, el Gran Pablito y el doctor Rebagliati estallaron en carcajadas, a la vez que se gui\u00f1aban el ojo, pon\u00edan expresiones p\u00edcaras y se\u00f1alaban el lugar por donde hab\u00eda partido.\n\n\u2014No es tan cojudo como parece, se hace el cojudo para disimular la cornamenta \u2014dijo el doctor Rebagliati, ahora exultante\u2014. Cada vez que habla de su mujer siento unas ganas terribles de decirle d\u00e9jate de llamar artista a lo que en buen peruano se llama estriptisera de tres por medio.\n\n\u2014Nadie se imagina el monstruo que es \u2014me dijo Pascual, poniendo una cara de ni\u00f1o que ve al cuco\u2014. Una argentina viej\u00edsima, gordota, con los pelos oxigenados y pintarrajeada. Canta tangos medio calata, en el Mezzanine, esa bo\u00eete para mendigos.\n\n\u2014C\u00e1llense, no sean malagradecidos, que los dos se la han tirado \u2014dijo el doctor Rebagliati\u2014. Yo tambi\u00e9n, para el caso.\n\n\u2014Qu\u00e9 cantante ni cantante, es una puta \u2014exclam\u00f3 el Gran Pablito, con los ojos como brasas\u2014. Me consta. Yo fui a verla al Mezzanine y, despu\u00e9s del show, se me arrim\u00f3 y vino con que me lo chupaba por veinte libras. No, pues, viejita, si t\u00fa ya no tienes dientes y a m\u00ed lo que me gusta es que me lo muerdan suavecito. Ni gratis, ni aunque me pagues. Porque le juro que no tiene dientes, don Mario.\n\n\u2014Ya hab\u00edan estado casados \u2014me dijo Pascual, mientras se desarremangaba la camisa y se pon\u00eda el saco y la corbata\u2014. All\u00e1, en Bolivia, antes de que Pedrito viniera a Lima. Parece que ella lo dej\u00f3, para irse a putear por ah\u00ed. Se juntaron de nuevo cuando lo del manicomio. Por eso se pasa la vida diciendo que es una se\u00f1ora tan abnegada. Porque se junt\u00f3 otra vez con \u00e9l cuando estaba loco.\n\n\u2014Le tiene ese agradecimiento de perro porque gracias a ella come \u2014lo rectific\u00f3 el doctor Rebagliati\u2014. \u00bfO t\u00fa crees que pueden vivir con lo que gana Camacho trayendo datos policiales? Comen de la putona, si no \u00e9l ya estar\u00eda tuberculoso.\n\n\u2014La verdad es que Pedrito no necesita mucho para comer \u2014dijo Pascual. Y me explic\u00f3\u2014: Viven en un callej\u00f3n de Santo Cristo. Qu\u00e9 bajo ha ca\u00eddo \u00bfno? Aqu\u00ed el doctorcito no quiere creerme que era un personaje cuando escrib\u00eda radioteatros, que le ped\u00edan aut\u00f3grafos.\n\nSalimos de la habitaci\u00f3n. En el garaje contiguo hab\u00edan desaparecido la chica de los recibos, los redactores y el muchachito de los paquetes. Hab\u00edan apagado la luz y el amontonamiento y el desorden ten\u00edan ahora cierto aire espectral. En la calle, el doctor Rebagliati cerr\u00f3 la puerta y le ech\u00f3 llave. Empezamos a caminar hacia la avenida Arica en busca de un taxi, los cuatro en una fila. Por decir algo, pregunt\u00e9 por qu\u00e9 Pedro Camacho era s\u00f3lo datero, por qu\u00e9 no redactor.\n\n\u2014Porque no sabe escribir \u2014dijo, previsiblemente, el doctor Rebagliati\u2014. Es un huachafo, usa palabras que nadie entiende, la negaci\u00f3n del periodismo. Por eso lo tengo recorriendo comisar\u00edas. No lo necesito, pero me entretiene, es mi buf\u00f3n, y, adem\u00e1s, gana menos que un sirviente \u2014se ri\u00f3 con obscenidad y pregunt\u00f3\u2014: Bueno, hablando claro, \u00bfestoy o no estoy invitado a ese almuerzo?\n\n\u2014Por supuesto que s\u00ed, no faltaba m\u00e1s \u2014dijo el Gran Pablito\u2014. Usted y don Mario son los invitados de honor.\n\n\u2014Es un tipo lleno de man\u00edas \u2014dijo Pascual, ya en el taxi, rumbo al jir\u00f3n Paruro, volviendo al tema\u2014. Por ejemplo, no quiere tomar \u00f3mnibus. Todo lo hace a pie, dice que es m\u00e1s r\u00e1pido. Me imagino lo que camina al d\u00eda y me canso, s\u00f3lo recorrer las comisarias del centro es una patada de kil\u00f3metros. \u00bfHan visto c\u00f3mo andan sus zapatillas, no?\n\n\u2014Es un avaro de mierda \u2014dijo el doctor Rebagliati, con disgusto.\n\n\u2014Yo no creo que sea taca\u00f1o \u2014lo defendi\u00f3 el Gran Pablito\u2014. S\u00f3lo un poco locumbeta, y, adem\u00e1s, un tipo sin suerte.\n\nEl almuerzo fue muy largo, una sucesi\u00f3n de platos criollos, multicolores y ardientes, rociados de cerveza fr\u00eda, y hubo en \u00e9l un poco de todo, historietas picantes, an\u00e9cdotas del pasado, copiosos chismes de personas, una pizca de pol\u00edtica, y tuve que satisfacer, una vez m\u00e1s, abundantes curiosidades sobre las mujeres de Europa. Hasta hubo un amago de pu\u00f1etazos cuando el doctor Rebagliati, ya borracho, comenz\u00f3 a propasarse con la mujer del Gran Pablito, una morena cuarentona todav\u00eda buena moza. Pero yo me las ingeni\u00e9 para que, a lo largo de la espesa tarde, ninguno de los tres dijera una palabra m\u00e1s sobre Pedro Camacho.\n\nCuando llegu\u00e9 a la casa de la t\u00eda Olga y el t\u00edo Lucho (que de mis cu\u00f1ados hab\u00edan pasado a ser mis suegros) me dol\u00eda la cabeza, me sent\u00eda deprimido y ya anochec\u00eda. La prima Patricia me recibi\u00f3 con cara de pocos amigos. Me dijo que era posible que con el cuento de documentarme para mis novelas, yo, a la t\u00eda Julia le hubiera metido el dedo a la boca y le hubiera hecho las de Barrab\u00e1s, pues ella no se atrev\u00eda a decirme nada para que no pensaran que comet\u00eda un crimen de lesa cultura. Pero que a ella le importaba un pito cometer cr\u00edmenes de lesa cultura, as\u00ed que, la pr\u00f3xima vez que yo saliera a las ocho de la ma\u00f1ana con el cuento de ir a la Biblioteca Nacional a leerme los discursos del general Manuel Apolinario Odr\u00eda y volviera a las ocho de la noche con los ojos colorados, apestando a cerveza, y, seguramente, con manchas de rouge en el pa\u00f1uelo, ella me rasgu\u00f1ar\u00eda o me romper\u00eda un plato en la cabeza. La prima Patricia es una muchacha de mucho car\u00e1cter, muy capaz de hacer lo que me promet\u00eda.\n\n### Sobre el autor\n\nMario Vargas Llosa, premio Nobel de Literatura 2010, naci\u00f3 en Arequipa, Per\u00fa, en 1936. Aunque hab\u00eda estrenado un drama en Piura y publicado un libro de relatos, Los jefes, que obtuvo el Premio Leopoldo Alas, su carrera literaria cobr\u00f3 notoriedad con la publicaci\u00f3n de La ciudad y los perros, Premio Biblioteca Breve (1962) y Premio de la Cr\u00edtica (1963). En 1965 apareci\u00f3 su segunda novela, La casa verde, que obtuvo el Premio de la Cr\u00edtica y el Premio Internacional R\u00f3mulo Gallegos. Posteriormente ha publicado piezas teatrales (La se\u00f1orita de Tacna, Kathie y el hipop\u00f3tamo, La Chunga, El loco de los balcones, Ojos bonitos, cuadros feos y Las mil noches y una noche), estudios y ensayos (La org\u00eda perpetua, La verdad de las mentiras, La tentaci\u00f3n de lo imposible, El viaje a la ficci\u00f3n y La civilizaci\u00f3n del espect\u00e1culo), memorias (El pez en el agua), relatos (Los cachorros) y, sobre todo, novelas: Conversaci\u00f3n en La Catedral, Pantale\u00f3n y las visitadoras, La t\u00eda Julia y el escribidor, La guerra del fin del mundo, Historia de Mayta, \u00bfQui\u00e9n mat\u00f3 a Palomino Molero?, El hablador, Elogio de la madrastra, Lituma en los Andes, Los cuadernos de don Rigoberto, La Fiesta del Chivo, El Para\u00edso en la otra esquina, Travesuras de la ni\u00f1a mala y El sue\u00f1o del celta. Ha obtenido los m\u00e1s importantes galardones literarios, desde los ya mencionados hasta el Premio Cervantes, el Pr\u00edncipe de Asturias, el PEN\/Nabokov y el Grinzane Cavour.\n\n\u00a9 1977, Mario Vargas Llosa\n\n\u00a9 Del pr\u00f3logo: 1999, Mario Vargas Llosa\n\n\u00a9 De esta edici\u00f3n:\n\n2013, Santillana Ediciones Generales, S. L.\n\nAvenida de los Artesanos, 6\n\n28760 Tres Cantos - Madrid\n\nTel\u00e9fono 91 744 90 60\n\nTelefax 91 744 92 24\n\nwww.alfaguara.com\n\nISBN ebook: 978-84-204-9893-5\n\nDise\u00f1o de colecci\u00f3n: Pep Carri\u00f3 y Sonia S\u00e1nchez  \n\u00a9 Imagen de cubierta: Pep Carri\u00f3  \nReproducci\u00f3n fotogr\u00e1fica: David Jim\u00e9nez\n\nConversi\u00f3n ebook: Safekat, S. L.\n\nCualquier forma de reproducci\u00f3n, distribuci\u00f3n, comunicaci\u00f3n p\u00fablica o transformaci\u00f3n de esta obra solo puede ser realizada con la autorizaci\u00f3n de sus titulares, salvo excepci\u00f3n prevista por la ley. 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+{"text":"Fancy a night under canvas with the kids, or living out your own version of The Gruffalo with a forest stay? Then look no further than our pick of camping, glamping and other outdoor options near York.\nA small, relaxed, family campsite that welcomes tents and has ecopods and bell tents to book. Choose an ecopod (beds and mattresses included) and enjoy toasting marshmallows over the flames of your own fire pit. It has 40 pitches in a 6 acre field, nestled in a 110 acre farm surrounded by trees and a stream.\nFacilities include: Warm showers heated by a wood-fired boiler housed in the luxury shower block, small shop stocking essentials, designated parking areas, baby changing, fire pits on all pitches, mobile bakery on Saturdays, large multi-use games and recreation area with exercise park, children's play equipment and tennis court a 5 minute walk away.\nLive your own real-life family Gruffalo story by booking your own log-pile house (well, cabin) in the forest. Lots have logburners, fantastic views and even hot tubs. Great for country walks and escaping the distractions of modern life (although there is TV, but you'll probably be grateful for that).\nFacilities include: What you get will depend on the cabin you choose, but all cabins are modern, clean and set in a beautiful location. Cots and highchairs can be pre-booked. Lots of outdoor activities are on offer, or you could just hire bikes and enjoy a ride around the forest.\nFamily-run working farm, Humblebee Farm has cottages, wigwams, cabins, bell tents, nomadic yurts, electric tourer and non-electric tent pitches to choose from. It's ideally located for trips to Scarborough, Filey, Bridlington, Whitby and the North York Moors national park.\nFacilities include: The facilities you'll enjoy will depend on whether you're under canvas or solid walls. Everyone has use of the children's play area, onside shop, laundry facilities, communal microwave\/fridge\/freezer, baby change and shower blocks.\nA lovely glamping option in the Yorkshire Dales. Choose between six original, hand-crafted timber frame tree lodges in the woodland and eight canvas yurts burrowed into the hillside. Enjoy beautiful countryside views from your very own yurt, complete with woodburner, proper beds and bedlinen and even a sink. Outside your front door is your own outdoor(ish) loo and shower, or there are luxury power showers a short walk away. Well located for days in the Yorkshire Dales, visiting Brimham Rocks or Fountains Abbey.\nFacilities include: Lovely bistro with family room, complete with a few games and toys, and a small outdoor play area. Designated parking area away from the accommodation.\nThis lovely glamping site, set in over 15 acres of woodland of 200 acre woodland, offers a warm welcome to families. Kids will love running wild, den-building and star-gazing. Book a supper and\/or breakfast basket in advance to be delivered to your tent so you don't have to bother shopping. Kids will love keeping an eye out for foxes, hares, deer, badgers and otters too.\nFacilities include: There's a selection of tents to choose from, but whichever you go for you'll find proper beds, fires, hot showers, BBQs and kitchen facilities. There's also a wooden wendy house and a couple of rope swings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're interested, you can read more about how and why Feathers came to be or find out more about its features and how Feathers compares to others .\nIf you've decided that Feathers might be for you, feel free to dive right in and learn about the basics.\nYou can find this documentation as a PDF here .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everyone knows that you sustained serious injuries in your accident. Now, how do you go about proving it?\nDamages are divided generally into three categories: general, specific, and punitive. Other blogs have dealt with punitive and general damages. Today we discuss the law of itemized, out-of-pocket, \"pecuniary\" damages, that is the part a claim that can be quantified.\nWhile these damages can frequently be only a small part of the overall injury and damage picture, they are very important. It goes without saying that medical bills and lost wages are part of this category, and our office may require your assistance in proving these. Often forgotten however, are small items like prescriptions, heating pads, assistive devices (like crutches and wheelchairs) travel costs, tolls, cancelled airfare, lost tuition, forgone gym memberships, etc., etc. While these items may seem minor, they add up. More importantly, they help tell the story of injuries and explain the overall damage picture.\nThe damage award is typically greatest when it comes to general damages: pain, suffering, inconvenience, etc. While it is possible discuss these items with the insurance company (or court), providing proof of smaller losses completes the picture. For example, the insurance company or court will want to see that tuition was actually forfeited, that a gym membership was canceled, and so on. We work with our clients to come up with adequate proof of damages in a way that tells each client's unique story. We may make odd requests (like copies of your old transcripts) but in the end, such proof can make the difference between a poor result and just compensation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I remember as a teenager shouting at the TV as politicians and police representatives discussed the Stephen Lawrence murder case during a programme of BBC's question-time with Dimpleby, four years after Stephen was murdered. I was so angry at the complacency of the Police that I emailed the programme to ask for answers on what steps were being taken to re open a case because of fundamental errors in the original investigation, errors caused by outright racism. I remember thinking what an amazing woman she was to have coped with the loss of her son and find strength to follow her conviction that the police did not do all that they could to bring her son's killers to justice.\nNow 21 years later an enquiry has finally confirmed Mrs Lawrence's (now Baroness) suspicions. One of the DS's working on the case admitted to having corrupt connections with the father of one of the killers (who was finally convicted 19 years after Stephen's death). We still have a problem with racism today and the only way that issue can be tackled is through education, so if a child has racist parents, these views are at least counter-acted at school.\nMy parents are racist and so are my in-laws, sadly they are of the generation that caused them to be ignorant of other cultures, mainly due to poor education (regardless of state or private education). It can not just be put down to age, the lady across the road from me was born in the same year as me but her views date back to the 1800s. She says that she would hate to have a 'half caste baby'. I try to limit my interaction with her to zero for obvious reasons.\nAt work ( which is a business in the luxury sector), I could count the number of black people on one hand. They are currently seeking a deputy CEO and I am hoping the person they select is a) a woman and b) black, just to make-up for the fact that the rest of the business is a weak representation of modern UK society.\nStephen Lawrence did not die in vain. It is a sad fact in the way us humans operate that action is very often preceded by tragedy. In this case there were multiple tragedies, beginning with Stephen's death and ending with the number of years it took for his family to find justice. But if Doreen had been a weaker mother and wilted under the pressure, Stephen's killers would still be free, racism would me more entrenched and more murders would have been committed. Stephen, Doreen and their family have made a lasting impact on society and empowered others who feel wronged to stand-up for their rights, no matter how long they have to wait for justice.\nI am blogging every day to raise money for Unicef. If you can help too please visit my page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Denotes questions related to Static Content in Magento2.\nDoes indexer:reindex need \"var\", \"generated\" and \"pub\/static\" to be writable?\nIs there a way to show the terms and conditions on a page via .phtml? I have made a page via a module and now want to show the terms and conditions on that page.\nWhat is the use and benefit of static content signing in Magento 2 ? Is it recommended to disable it from backend ?\nHow can we check if the website is using pub directory or not, independent of its mode? Is there any Magento function present to check the root document?\nI've installed Magento 2 on AWS, however, the admin page is appearing like this: What should I do? Thanks!\nHow to edit file in app\/design\/frontend without having to redeploy static content in Magento2?\nPlease check the screen shot I want to make close button on toggle navigation link.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvgih b/data_all_eng_slimpj/shuffled/split2/finalzzvgih
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+{"text":"Deer With Horns is our real, legal last name.\nWe thought we would mention this as people often write and ask us.\nIf you are \"trapped\" in someone else's frames, click HERE to break free.\nOur site looks awful in other people's frames.\nas there are so many resolutions people use.\nTry using the zoom if you use Firefox.\nA picture of us at a powwow.\nMens and Womens titles in the same year--2004 & 2014.\nSite moved or is no longer on net.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hej, I'm Rare\u0219, and I welcome you to get some good vibes.\nI'm a senior software engineer with deep experience in systems, automation, site reliability, performance & monitoring. Linux is my dark lord and saviour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Celebrate teatime, that most delightful interlude of the day. Teatime conjures an array of civilized pleasures: finger sandwiches and pastries. Plush seats for settling into over good conversation. Aromatic, steaming tea. And, of course, delicate, glorious, gorgeous sets of china. A perennial bestseller, Collectible Teapot & Tea Calendar is a year of charm and ritual, spotlighting rare and cherished pots in evocative teatime settings. A frosty pink Royal Worcester set with enamel trim makes Valentine's Day especially festive, while a \"Gold Swirl\" James Sadler & Sons is perfectly suited for Christmastime. And a Wedgwood Jasperware evokes antiquity with cameo like white bas-relief depicting Greek myths. Culled from private collections and archives, each month's set is accompanied by tea-themed facts, quotes, and recipes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"South Eastern Railway has a distinction in promotion of Sports and Games. The players of all categories have been undergoing practice with able guidance of expert coaches for a long time and the players of SER take part in National & International events and come out with flying colours.\nTrisha Deb, an employee of South Eastern Railway who represented India bagged Gold Medal in the Asian Archery Championships held at Dhaka, Bangladesh from 24thto 30th November, 2017. Altogether seven countries participated in the Asian Archery Championships.\nTrishsa Deb, Jyoti Surekha and Lily Charu who represented India secured Silver Medal in the Team Event of the World Archery Championships held at Mexico from 15thto 22nd October, 2017. The are the employees of Indian Railways. As many as nineteen countries participated in this World Archery Championships.\nSouth Eastern Railway Men Kabaddi Team secured 3rd Position in the All India Railways Kabaddi Championship held at Rae Bareli from 8th to 10th November, 2017. As many as 16 Zonal Railways from all over Indian Railways took part in this event.\nThis apart, South Eastern Railway Sports Association (SERSA) Kabaddi Men Team secured 1st Position in the All India Invitation Kabaddi Championships held at Bhubaneswar recently and SERSA's Kabaddi Team won the prize money of Rs.2 lakh.\nSri Anirban Datta, Addl. General Manager, S E Railway and Sri J K Saha, Principal Chief Mechanical Engineer & President SERSA felicitated the Archer Trisha Deb and the Kabaddi Team of S E Railway at this Railway's Headquarters, Garden Reach this morning (4\/12\/2017).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What's the deal with house edge, probability, and odds?\nCan't decide between Blackjack, Roulette, or the slots? Think that playing multiple hands will improve your luck? Chances are you don't understand odds, house edge, and probability in general.\nAnd that's totally OK. There's no shame in not getting the hang of it. Even some of the most seasoned casino players don't really understand it. So today, we're going to make everything as simple as possible. No hard mathematics to understand. No calculations to make. This is all about understanding how odds work in their most simple form. Whether you play online casino games or enjoy your favorites live, we hope you find it useful.\nIn games of chance, like lotteries, bingo, slots, roulette, craps, baccarat, and keno, players have no actual opportunity to make a profit in the long term. Yes, players win big money playing these games. If they didn't, casinos would be out of business because they'd have no customers. But because events of each bet are random and independent of one another, the outcomes aren't predictable.\nContrast that with games of skill and luck, like Blackjack and poker, for example. In these games, while a random number generator definitely is an important part of the equation, your success playing these games are dependent on your knowledge of the game, the strategies you employ, and the decisions you make while playing. If you're a good player, you can minimize your losses with poor hands and maximize your winning potential with great hands.\nFor casinos to make money on these games, particularly on games that involve both skill and luck, they have what's called a house edge. In a nutshell, the house edge is less than the actual odds of winning. If a casino paid out the true odds, they'd end up staring a loss in the face.\nThe house edge varies from game to game, but for the purpose of this article, let's take a look at Roulette.\nThe Roulette wheel has 36 numbers on it that are colored in black or red. There are also two green numbers\u2014a single zero and a double zero. There are 38 numbers in all, but at first glance it almost seems like there are 36 numbers because most players don't take the two zero spots into consideration.\nIf you were to bet on evens or odds, red or black, or 1-18 or 19-36, you might think you have a 50\/50 shot at being right. And that's almost true. But the casino derives its house edge from the fact that you technically have an 18\/38 at being right, which is actually a 47.36% chance of being right.\nUnderstanding odds in Roulette is the first step to getting a handle on how everything works, but does the same apply to the slots or games like Keno? This one is a little bit more complex.\nIt turns out the odds of winning a top prize on a slot machine is a lot more difficult. For example, in Roulette, by betting on a single number, you have a 1 in 38 chance of hitting your number and earning payouts of 35:1. But a slot machine might offer far worse odds, usually bordering on something like 1 in 200,000. That number may seem enormous, but these slots tend to pay out more than 35 to 1. For example, hitting the top prize could pay out 8,000x your original bet. So if you had bet $10, you're suddenly staring $80,000 in the face. Contrast that with the $350 you would have won had the best possible Roulette bet paid out in your favor.\nBut the odds of actually hitting that payout on a slots machine are miniscule. For that reason, slots are built with specific payouts based on probability and odds. Since these machines are less likely to pay out top prizes, they are more likely to pay out smaller prizes more frequently. That's why slot players enjoy frequent wins.\nIf your idea of improving your odds and decreasing the house edge involves playing multiple hands or placing multiple bets, you might not want to read what we have to see. It turns out it doesn't do much from a probability standpoint, except of course increase your odds of losing double the amount of money. Remember, when a bet is doubled, the expected loss also gets doubled, leading to greater losses in the long run.\nOf course, there are some exceptions. For example, playing multiple sections of the Roulette board can help. While such a strategy might lead to greater losses when that loss happens, the losses end up being less frequent.\nWhen you're ready to play for real money and win actual cash, one deposit gets you access to the real games. And when you take advantage of casino welcome bonus, you'll have more ammo to do damage at the tables and slots with.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\"What can we say about Lucy Spartalis? She was amazing! We came across her photography on Nouba, so it's nice to come full circle.\" She's one of our faves too Amy and Craig! And it's always a pleasure to share a unique and creative wedding, inspired in-part by our humble pages.\nAs vocalist and drummer of shoegaze band, Tender Bones, it was music that brought these two together. The indie pair shared their vows, rings and favourite Ted Hughes poem amidst the 'Circle of Trees' in Melbourne's Edinburgh Gardens.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was just wondering if there are any tiny house trailers that are designed to a lifetime mileage specification?\nMaybe designed for 10,000 lbs load for a maximum of 5000 miles over their lifetime?\nOr set up a chart that might show what can be done between 1000 miles and 100,000 miles, and the consumer can judge if that extra expense is worth it.\nOf course, the rigidity of the final tiny house adds a substantial amount of strength to the final chassis so those calculations will have to be taken into account as well.\nThat seems to be the way that RV and mobile home chassis are built- it would be a great idea to apply the same concepts to a tiny house build.\nI don't know if you've seen tiny house trailer designs that are certified by licensed mechanical engineers. I'm a Mechanical Engineering Tech so that's why I asked.\nThis might be a great way to show people just WHAT can be done. You can consider posting this on the blog if you want.\nI highly recommend purchasing access to Dan's videos if you want to learn more about designing and building tiny homes on trailers. You can order a trailer as shown above directly from Dan here.\nPlease share your most valuable thoughts, questions, ideas and links\/resources related to trailers and tiny houses in the comments below and share this post with anyone who might find it helpful or would be able to add value to this conversation. Thank you!\nIf you want updates like this sent to you via email join our free daily tiny house newsletter!\nI am looking for an 18\u2032 to 20\u2032 trailer to build on, so far prices are from $2700 to $3400 new, havent found any used ones that will do it. I am looking for a tandem axle, with at least one axle with brakes. Craigs list so far hasnt shown any, but I keep looking, I am in northern MI, about 2 hours north of Saginaw. At the present , these costs for a trailer are almost the budget for the house itself.\nToo bad you weren't close to Hanna AB, I would give you one that would be perfect if you hauled it past here, Castlegar BC so I could get the things (very few) out of it, plus some tools stored in anther truck at the storage compound. Tandem with electric brakes, new tires last year but you have to knock the park model cabin off of it to build. Any takers, I just want the things stored in it to arrive her.\nI came down through there a year ago, coming home from Alaska. I had gone up by Caswell BC and came back down through Alberta. I am in Northern Michigan, that was a lot of miles trip. Once I find the right trailer and get the wagon built I am planning on going out to the Cascades.\nThanks Jake, too bad, I pulled that thing past London Ontario a year + ago, probably not far from you. I only took it because a friend, who owed me money, couldn't afford to repay and felt bad, I would have forgave the debt and now would rather have than to have this thing and paying storage on it. Love him like a brother and a couple of thousand dollars would not have hurt the relationship of over 60 years but he bartered the trailer off and like a fool I took it to make him feel better. I needed another trailer like a hole in the head. Good luck with your quest. Sounds like a great trip you had and more coming up.\nI have found several worth looking at on Craigslist and over the next several weeks will be getting around and looking. From bare trailers to older travel trailers. Most quite doable.\nI am for sure an Irish Gypsy, only we say Irish Traveller just to make the distinction, seems I cannot not travel.\nIt all works out. I am wanting to get on out to the Cascades and then maybe heading on down to New Mexico in late fall. We shall see what will be. Ultimately I want to build an 18-20\u2032 trailer with a lot of the ideas here in it. And mayhap a Leprechaun or two peeking out around it.\nSo are there any manufacturers that make water\/black tanks that fit between the structural members for these trailers?\nI'll say that I've had the privilege of corresponding with Mr Louche, about the fact that I wanted a larger trailer than the basic ones he offered. He was more than willing to work with me on that, and okay, so I'm still in Germany and thus building my tiny house yet, so I've not bought the trailer. However, even with the shipping\/handling, for the length of trailer I want (28ft) he is definitely where I want to go.\nI am going to look at a 23\u2032 RV in 2 weeks, tandem axle, all heavy at 3900lbs. I am looking at it as an 18\u2032 trailer\/wagon, with a 3-4\u2032 porch on the rear and similar overhang roof.\nI am planning double walls of tongue in groove pine, vertical on the exterior and horizontal on the interior with tyvek and insulation between. On the Vardoshavura.com forum in the photos section is pic of one by the Thomas's that had covered over 18,000 miles its first few years, he says he averages just over 5,000 miles a year.\nYeah, its built to last. That forum has some great photos in the photo section, vardos, Romani for wagons. But also many other from sheep herder wagons to much lighter seasonal canvas top campers.\nI am blending ideas from there with Jay Shaffers and others work. I want mine to reliably travel from Northern MI to AZ and back. Its very doable. A good trailer to build on and then a tight reinforced box built on it.\nI'm interested in the site you mentioned, but Google says Vardoshavura.com doesn't exist *pouty* Is there a typo possibly?\nHas there been any discussion on the design and incorporation of holding tanks for sewage and water. Size and dimensions of tanks? Where are they available for purchase?\nI was dumb lucky in finding a 20\u2032 travel trailer that had been terminally damaged by a low hanging limb at 35 \u2013 40 MPH. I stopped and inquired about it and was told \"if you want it just get it out of here\" It was rather complete in what was left of it. RV stool, propane electric fridge, fold down table, black water tank, and lots of small surprises like dual voltage lights and hardware. I even salvaged most of the 12-2 with ground copper romax electrical wire and outlets for my caboose build. I was later able to get a rubber roof off of a 12 wide by 70 foot mobile home that was being razed on site within a block of my house. I just stopped and asked and the contractor was glad to not have to pay the tipping fees at the dump on the added weight. So keeping your eyes open can pay off. I love free !\nI have an old travel trailer that has been at my campsite for over 21 years. Now, I am getting a new one and would like to give away the old one. Where can I post this? The trailer is located in Shohola, PA. My husband thinks it's a 1987 Skylark. We don't know the exact year because we purchased our campsite with the trailer so and it has no title. It has been covered (under a roof) for over 21 years. Does anyone want it? and if now can you tell me where I can list it?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rock-a-Bye Bench. Materials: Steel tubes, steel rods. Dimensions: H 80 x W 125 x D 80 cm.\nA swinging bench where two can sit face to face. When one person swings forward, the other swings back. The bench invites a fun-filled meeting in the public space. Of cause you could sit on the bench by yourself, but probably not for long. The empty seat is inviting, and you are bound to make eye contact with the stranger who joins you. And you'll likely share a smile when the bench rocks back and forth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2022 The buyer of the swap agrees to make a number of fixed or floating interest rate payments periodically to the seller in one currency on some agreed upon notional amount and receive payments denoted in another currency. For instance, one party pays UD dollars and receives British pounds. By entering into a currency swap, both parties attempt to hedge their FX exposures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Getting ready for autumn: how people around the world are dressing on the street during this chilly weather? Get ready for fall fashion and get updates on the latest styles by viewing this compilation of my favorite 30 looks. The chic transeasonal look can be completed with light or bright colors and wearing pretty footwear, leather jackets or color-blocking coats. This year comes with a more relaxed look. As you can see, there are great ways of pairing your favorite jeans with wool outerwear or chunky voluminous knitwear.\nHere we see a woman wearing dark blue jacket with statement golden buttons worn atop pretty neon pale yellow top styled with black polka dot white shirt.\nI am pretty sure these latest inspiring ideas will show you how to look fabulous wearing warm clothing. Anyway, I've got this pretty collection of elegant and essential 30 fashion looks for the new fall season, hope you will like it as much as I do. The best part of this compilation is that you can start wearing them right now. We sure can copy these looks for out next week street walk. Why did I put so many looks? It's a handy guide to help you decipher the most attainable looks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"An extensive window replacement project was successfully completed on this Toronto highrise condominium complex. The job encompassed all 260 units in this 20 storey residential building. The Mattcon team exceeded all client expectations on this high value project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It only became apparent to me after the fact that Isle of Dogs sounds quite close to \"I love dogs.\" You might even say there was a certain amount of forethought in this play on words. However, the pun only works in English as the Japanese pronunciation of the comparable kanji is \"Inu ga shima.\" Here we have the inherent beauty and simultaneously what some would deem the problematic nature of Wes Anderson's latest film in a nutshell.\nIt's necessary to lay out how I come at Isle of Dogs because it does contribute to how I perceive it. I'm Japanese-American. I've lived in Japan. I know some of the language though it's an admittedly meager amount. However, I've invested in the culture and care about its people and fostering cross-cultural bonds. That's part of the reason I was drawn to live there for an extended period of time, more than any pop culture infatuation with anime and manga. In those regards, I'm very much American. I also revere Kurosawa, Ozu, Mizoguchi and all the rest as most cinephiles do. There you are.\nWes Anderson is someone that I genuinely admire for his aesthetic though I would never necessarily sing his praises needlessly. He doesn't need me to defend him nor do I look to. Still, when I consider Isle of Dogs I do not see a superficial homage. As with everything he does Anderson's film feels fairly meticulous and the stop-motion creation is phenomenally precise. Beyond that, it's infused with Japanese tradition. Certainly, there are markers that some might deem superficial like Taiko or sumo or cherry blossoms (\u3055\u304f\u3089). They are all present.\nThus, I do worry about people who have not interfaced with Japanese culture or the people. Specifically, the character of Tracy (voiced by Greta Gerwig), the foreign exchange student, feels problematic. I've met some folks like her where they let their own personalities take control of every situation and there seems to be no sensitivity or give and take.\nBecause they don't seem to have any sense of the culture they are in or at the very least they expect others to play by their rules. Hence why many Americans including myself are only fluent in one language. An example springs to mind of Tracy wandering into a bar and hollering at the man behind the counter in English that she wants chocolate milk. Then she berates a Japanese scientist (voiced by Yoko Ono) for not doing anything in opposition to the rampant government corruption. Again, in English.\nIt was fascinating that I watched the film with an audience where the majority were Japanese so they were not ignorant of their country like Tracy or I might be. But how about viewers in another pocket of the world or even back home where I come from? The audiences saw a different movie altogether with different nuances and connotations.\nSome people have noted rightly that a lot of the Japanese dialogue is lost because as the opening disclaimer notes: \"The humans in this film speak only in their native tongue (occasionally translated by bilingual interpreter, foreign exchange student, and electronic device). The dogs' barks are translated into English.\" Except for the dogs, that's a large margin for error.\nEven the words of the little pilot, the intrepid boy Atari (voiced by Koyu Rankin) are all but lost on the dogs who don't speak human and for me as well because, again, my Japanese leaves much to be desired. But the bottom line is that he almost always intuitively knows what the pack of alpha dogs is doing.\nThey have a connection. He is looking for his faithful guard dog Spots (voiced by Liev Schrieber) who has been cruelly deported by his distant uncle Mayor Kobayashi in his effort to rid the dystopian Japanese city of Megasaki of infectious dogs. Spots was the first of many canines to be deported to the putrid rubbish pile of Trash Island. There you have the film's plot but it relies on the fact that the boy and the dogs work together.\nIt goes back to that core tenet of society that dog is man's best friend. And strip away any amount of visual artistry or cultural layering and that fact remains universal. Kobayashi is a cat person and we surmise his decision was merely a vendetta against canines (going back generations) more than any scientific evidence would suggest.\nTo be honest, it never feels like Anderson is putting out a giant placard with film references at least not like Tracy with her megaphone. If anything the film conjures up one of Japan's great national heroes the faithful dog Hachiko. Any traveler to Tokyo will recognize his statue in Shibuya but he is a cultural icon \u2014 emblematic of the same bond expressed in this story.\nIt's still an Anderson film and as such, I never get emotionally connected with the material (actually maybe I take that back; stop-motion dogs suffering is heartbreaking to watch) nor do I gel with his very personal idiosyncrasies all the time.\nBut somehow, though the film's cultural representations and relationship with Japan are flawed, nevertheless it left me more impressed than anything. It seemed like a degree of care was taken. And in the end, this story of canines has moments that unquestionably do resonate with me.\nI thought I would have more problems with all the quality voice talent distracting from the story itself and it happened at times where I was stuck on Jeff Goldblum or Bill Murray but more often than not it didn't seem like one voice stole the show. It was a story that involved many voices.\nSome that we are able to understand, others that we can only gather bits and pieces of. But for me personally, rather than that being a deterrent I find it fascinating that the same film can play differently for different audiences and that native Japanese speakers can be in on the movie in a way that I never could.\nIt's not that we don't deserve to know that part of the story necessarily or need to be singled out because we don't know the language but isn't that one of the confounding things about culture and language? Oftentimes we don't understand one another and need to find points of mutual understanding. Things get lost in translation and I think one could make the argument that this happens in Isle of Dogs purposely.\nCertainly, Wes Anderson doesn't know Japanese culture like the back of his hand. In an interview, he said he's been to Japan some but his references were namely Kurosawa, Miyazaki and the woodblock prints of Hokusai and Hiroshige. And yet if that is true, it still feels as if he's surrounded himself with some voices different than his own even if his typical ensemble is in place.\nWe have Kunichi Nomura with input on the script and voice duty for Mayor Kobayashi. More include Akira Takayama, Mari Natsuki, Akira Ito and then better-known names like Nojiro Noda, Yoko Ono, and Ken Watanabe. That's not to mention the countless other Japanese contributors whose names scrolled by with the end credits.\nAdmittedly this is only my perception of the film but when I watch it, I never feel like it is assuming the primacy of the English speaking audience or if it is then that assumption gets slyly subverted. I mentioned already that the character of Tracy often speaks in English, in her opposition to Mayor Kobayashi or to the man serving up drinks at the bar counter.\nThe implicit understanding is that the Japanese characters understand the English being spoken but they choose instead to respond in their native language. So they have met us half way but have we met them? Learning Japanese is difficult, maybe even impractical, but growing our cultural literacy comes in many forms that would only assist in deepening our ties with one another. Only later did I realize that Tracy, the \"white savior\" as it were, essentially fails in her attempts. If anything she needs the help of an audacious Atari, his guard dog, and the nameless hacker from her school. Without them she is powerless.\nAs is often the case, certain people use their voices and assertive personalities to push themselves into the limelight unwittingly but those people would be nothing without the taciturn heroes who willingly stay in the periphery until they need to stand up. While Tracy turns me off slightly in fiction and in real life, it's the others like Atari that resonate with me. Just as the tale of dogs in both camaraderie and loyalty rings a universal note.\nHowever, I realize only now that I didn't talk much about the actual mechanics and formalistic aspects of the movie but I've spoken my peace. Do with it what you will.\nStarring Bill Murray and Scarlett Johannson with direction by Sofia Coppola, this film is set in the fast-paced, technological, and modern world of Tokyo, Japan. That is where Bob and Charlotte find themselves and they both are lost, simply going through the motions of life. He is a middle-aged, former movie star filming a whiskey commercial. She is a newly-married wife of a fashion photographer. Despite their age differences, they find out that they have a lot in common. Over the week they spend time together in Tokyo and become friends. When the time comes for Bob to leave neither one wants their time to end. They say goodbye but do not forget each other. This film was enjoyable because it portrayed two people who could be good friends without getting romantically attached, at least in the conventional sense.\nI must say that this type of friendship intrigues me. It is understandable that if you go to a foreign country alone it would be nice to have someone you could at least converse with, without any barrier getting in the way. It might be at Narita airport for a moment, at a Hotel, or walking the streets of Shinjuku. It would act as a comfort in a world like Tokyo that is so fast paced and high stress. These unusual circumstances could throw together two very different people, with little in common except the language they speak. That is something that does not happen every day.\nNeedless to say, after gaining the opportunity to visit Tokyo two years in a row it has given me some new insight. I can now wholly empathize with Bob and Charlotte because although I knew a few people, Tokyo is such a highly populated, fast-paced world that seems so easy to get lost and overwhelmed in. There is so much to see, so many lights, so many hurrying folks, so many subway lines, so many surgical masks, so much etiquette, and so much technology. True, some of my ancestors were Japanese but the language still baffles me, making it very easy to get \"Lost in Translation.\" Being in Tokyo it also helped me realize that it is not only tourists who get lost. It has been over 10 years since this film came out and a lot can happen in that time like more Starbucks and McDonalds on every corner.\nI am not big on Zombie films and so I went into this one not expecting much a great deal. I still do not really like the Zombie genre but there were some decently funny moments here having to do with the rules Columbus (Jesse Eisenberg) has for surviving the apocalypse. The long lasting search for Twinkies, and of course a random cameo by a prominent funny man were a few more highlights. Woody Harrelson is good at playing the gruff type and Little Rock and Arkansas have a lot of personality which does the film a service. Remember rule #1: Cardio. Very important to survive during a zombie apocalypse if it ever comes to that.\nI was not sure I would like this film because honestly Bill Murray is not usually one of my favorite actors. However, his portrayal of Bob Wiley, a man with every phobia imaginable, is maybe his most lovable. True, he is annoying and neurotic, but he means well. In many respects he reminds me of Jimmy Stewart as Harvey because both characters were able to captivate most of the people around him. Only with Bob he got under the skin of one man and that man was his psychologist Dr. Leo Marvin (Dreyfuss). Bob tags along on Dr. Leo's family vacation and that is where the conflict really gets started and the laughs begin. Every moment that the doctor loses his sanity continually builds up until the stress of Bob is just too much to take! All in all this was a pretty entertaining film and I gained a new found respect for Bill Murray.\nThe farthest Eastern boundary of the European continent makes the perfect landscape for a new addition to the quirky Wes Anderson canon. But more on that later. First our story.\nIt gains inspiration from the writings of forgotten Viennese author of the 30s and 40s Stefan Zweig. In fact, the author's own plot device is used in this story of friendship, love, and murder. An inquisitive writer (Jude Law) from the 1960s becomes intrigued by the aging proprietor of the Grand Budapest Hotel Zero Moustaffa (F. Murray Abraham).\nThe rather mysterious figure is glad to tell his story and how he came to acquire the iconic hotel. And that's where our real story begins, back in 1932, with concierge and small time celebrity M. Gustave (Ralph Fiennes). He is a dying breed of a man, full of culture, a bit effeminate, and known to wine and dine older patrons. He also has an immense affinity for poetry.\nIt was under his leadership that the young, stoic bellboy Zero got his start. What an exciting beginning it was. One of Gustave's most faithful patrons, Madame D (Tilda Swinton), dies suddenly and he is bequeathed the priceless painting Boy with Apple.\nThe family of the deceased is in an uproar led by belligerent son Dmitri (Adrien Brody). Soon Gustave has become the strangest of fugitives as he is wanted for the murder of the old lady.\nDuring that time, young love springs up and Zero meets the love of his life Agatha (Saoirse Roman), a spunky baker who returns his affection.\nNow the imprisoned Gustave takes part in an escape attempt a la Le Trou except this rendition is successful to a degree. Faithful Zero meets up with his mentor, and Gustave turns to the only ones he can. The concierges from all the surrounding area. They oblige, getting the two fugitives away, but soon Dmitri's cold-blooded assassin Jopling (Willem Dafoe) is on their tails at a local monastery.\nWar is imminent and back at the Grand Budapest things do not look promising. The ever fearless Agatha agrees to go fetch Boy with Apple, but she is soon spotted and pursued by the ever brutish Dmitri who tries to use his gun. That's not a smart thing when all the rooms are full of quartered soldiers and a chaotic gunfight ensues.\nIn the aftermath, a second will is uncovered that makes M. Gustave the sole owner of the Grand Budapest and many other possessions that Madame D owned. In a Deja Vu moment, Gustave and Zero ride the train once again before getting boarded and questioned. Always the gentlemen, Gustave defends Zero who is targeted for his immigration status. It was in that way the story ends and returns to the young author and elderly Zero Moustaffa.\nHe never could bear to give up the Grand Budapest despite the toll of Communism. It's not because of Gustave, but his dear Agatha who died only two years after. It's his only link to the happiest times of his life.\nWhat The Grand Budapest Hotel ends up being is an odd mix of black comedy and romantic sentiment all wrapped up in an Anderson world.\nHis shots are often framed symmetrically and muted pastels abound as well as scaled miniatures, creating his always distinctive mise-en-scene. He is also a fan of a smooth moving camera often involving zooms.\nAnderson is obviously a student of cinema and his film at times are reminiscent to 30s fair such as Grand Hotel and The Rules of the Game. He also channels another famed Viennese Ernst Lubitsch who was a master of highbrow romantic comedies.\nHotel also boast a superb cast comprising most of Anderson's stock company. If there's anyone who has been in more than one of his movies, they are probably in this one, even for just an instant.\nSo given the normal Wes Anderson flair or eccentricities, this film is visually pleasing and quite entertaining. It is a worthy follow up to Moonrise Kingdom, darling.\nStarring Bill Murray and Andie MacDowell, this romantic-fantasy-comedy is about a conceited weatherman named Phil Connors who goes to cover Groundhog Day. After coping with the day once, Phil wakes up and goes through it again realizing he is in a time loop. At first since there are no consequences Phil memorizes every occurrence, takes advantage of situations, and romances any woman he wants. However, over time the novelty wears off and it soon gives way to monotony. He even starts committing elaborate suicides to get out of each day. However, as he continues to fall for his producer, Phil begins to change and decides to use all he knows for good. After one day he is the most loved man in town and surprisingly enough he finally is free. The concept of this film is certainly interesting and it brings up hilarious and thought-provoking situations. Phil Connors begins as a self-serving jerk, hits rock bottom, and finally finds redemption in showing kindness for others. And the best part is he no longer has to listen to I Got You Babe!\nStarring Dustin Hoffman, Jessica Lange, Terri Garr, and BIll Murray, this comedy film follows a fiery actor as he tries to find work. Despite his skill, no one wants to work with Michael Dorsey (Hoffman) and so he masquerades as a woman to get a job on a soap. However, soon he gets so caught up in his role as Dorothy, he cannot get himself out. His character is so popular on the soap that she is renewed forcing Dorsey to endure it even longer. Then he finds himself befriending a shy actress on the soap (Lange) while he starts neglecting another actress friend (Garr). Along the way he has many awkward moments and romantic entanglements. The worst of these comes when he must reveal who he is (only his roommate and agent know). At first it causes pain but ultimately honesty is the way to go. This film was reminiscent of screwball comedies and it had some very hilarious moments. I have to say Hoffman pulled it off.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For many EarthBound fans, the answer was clear: Mother and Mother 3. Heck, Clyde Mandelin, the guy who did the fan translation of Mother 3, has actually offered his work to Nintendo for free. But when you step outside of the Mother franchise, you see that there are many, many games that have yet to grace the Virtual Console. When this list initially started as just a general games-we-want-on-the-Virtual Console list, we came up with a list of well over 150 games that hadn't even released on the Wii's VC.\nSo with that, we decided to split up the list into several lists, separated only by console. And since it was an SNES game that sparked this line of thought, we will begin with that system.\nThis list isn't just for games that are not on the Wii U Virtual Console. Any game that appears on the Wii's VC is off-limits, as well.\nNo IP that is currently held by either Microsoft or Sony can make the list due to the legal wrangling it would take to get the game onto a Nintendo console again. Therefore, no games like Killer Instinct are allowed.\nWhile I'm not completely sure what the issue is with the Donkey Kong Country games, we will refrain from listing them. While I'm sure those are games we'd all love to see make a triumphant return, leaving them off allows us to list another game you may not have thought of.\nThe only game in Sqaure's line-up created by their North American developers, Secret of Evermore begins our list. It also began the video game career of composer Jeremy Soule, known later for his work with the Dungeons & Dragons franchise, the Guild Wars games, and The Elder Scrolls series.\nDespite having a similar name as well as similar mechanics to Secret of Mana, Secret of Evermore is separate from the Mana series. Instead of being set in the world of Mana, it is set in the world of Evermore. Actually, it starts out on Earth in the small town of Podunk, USA. It's here that our protagonist comes across a laboratory and, through unfortunate circumstances, gets sent to Evermore.\nAnd from there, you travel through the four realms of Evermore, trying to find a way back and meeting up with scientists from Earth who experienced the same mishap 30 years prior. On top of that, the game\u2014while looking like a JRPG of the era\u2014has a very American feel to it, with cultural references that are more familiar to a North American player than other JRPGs of the time.\nThe third and final game in the Gargoyle's Quest trilogy, this game took everything that made the first two games great and put them on steroids. Firebrand can now change abilities with the different crests that he finds. You can fly, swim, cling, and fight your way to the final battle with the demon Phalanx.\nOn top of the cool new abilities, there's the amazing soundtrack, the wonderful gothic art style, and a massive map to travel. And while it does have some slowdown, along with some cryptic hidden spots in the game, we still feel it deserves a spot on the Virtual Console, just like the Game Boy game that started this series.\nOne of my personal favorite games from when I was a kid\u2014and probably the game I spent the most time playing\u2014this wonderful spin-off of the classic Super Mario World has still not received a proper release on the Virtual Console. And yes, I say that knowing full well that the 3DS Ambassador's Club received the GBA port of the game. That could make things interesting when Nintendo decides which version to bring to the VC.\nRegardless, Yoshi's Island is one of the greatest games to have graced the system, with one of the most memorable fights with Bowser (Baby Bowser in this game), some of the best artwork on the system, and a slew of creative and colorful villains. I get the feeling this game was so good, it sparked Nintendo to re-work the puzzle game Panel de Pon into what we in the West know as Tetris Attack, complete with characters and bosses from Yoshi's Island (well, except it has the grown-up Bowser).\nThree down, seven to go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Twenty Five year run with Citibank!\nJoe May has a birthday today.\nJoe May updated his profile. View.\nJoe May changed his profile picture.\nJoe May added a photo to his profile gallery.\nMy youngest with goofy goggles, 2 cousins, and friends at summer camp 08.\nJoe May joined via Class Connection Facebook app.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An object containing information about a floor of an indoor map.\nA short string identifier for a floor, unique within its indoor map. For example, \"G\".\nThe human-readable name of the floor. For example, \"Ground Floor\".\nThe index of this floor relative to the bottom floor of its indoor map.\nThe floor number of this floor. For example, 1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwlvn b/data_all_eng_slimpj/shuffled/split2/finalzzwlvn
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@@ -0,0 +1,5 @@
+{"text":"New 'smart city' touch-screen kiosks have been installed in a South Korean city to provide the public with local information.\nThe kiosks in Daegu, South Korea's fourth largest city, are part of the city's disaster management infrastructure, providing vital information in the event of a major emergency such as a fire, flood or earthquake, as well as city news and location guidance.\nThey were installed by South Korean mobile operator SK Telecom and digital signage provider QVOSS. British company Zytronic, through its South Korean partner, DTH, supplied custom 55\u2033 Zybrid multitouch sensors which are capable of withstanding extreme environmental conditions.\nThe 10 outdoor kiosks needed to be able to tolerate Daegu's humid sub-tropical climate, with summer temperatures averaging 40\u00b0C and rainfall of around 230mm per month in the rainy season.\nThe project is one of the first commercial roll-outs of the new Zytronic ZXY500 controller, enabling super narrow bezel designs and robust 80-point multitouch performance.\n\"We were keen to employ Zytronic's newest controller as the super narrow bezel enables us to make the best use of the total screen area,\" explained Johee Jun, general manager of QVOSS.\nIan Crosby, director of sales and marketing at Zytronic, added: \"The smart-city kiosks in Daegu are an essential part of the metropolitan infrastructure and are there to provide information and assist citizens and visitors during everyday life. Furthermore, they have a crucial role in an emergency when the quick delivery of public safety announcements can be vital.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MIAMI, May 25, 2018 \u2013 Royal Caribbean Cruises Ltd. (NYSE: RCL) today announced it has made further progress in its animal welfare sourcing program. The global cruise company is adopting a series of requirements for its poultry providers to meet when it comes to how they breed, raise and process poultry.\nThe company is extending these commitments to include comprehensive broiler chicken welfare policies. By 2024, in the United States, Royal Caribbean will responsibly purchase 100 percent of chickens from Global Animal Partnership (GAP) certified sources. RCL will require these chickens to be processed in a manner that avoids live-dumping and live-shackling, instead utilizing a pre-shackle, multi-step controlled atmosphere processing system that is widely known as more humane.\nRoyal Caribbean Cruises Ltd. (NYSE: RCL) is a global cruise vacation company that owns and operates three global brands: Royal Caribbean International, Celebrity Cruises and Azamara Club Cruises. We are a 50 percent joint venture owner of the German brand TUI Cruises, a 49 percent shareholder in the Spanish brand Pullmantur and a 36 percent shareholder in the Chinese brand SkySea Cruises. Together, these brands operate a combined total of 50 ships with an additional twelve on order as of March 31, 2018. They operate diverse itineraries around the world that call on approximately 540 destinations on all seven continents.\nAdditional information can be found on www.rclcorporate.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fluid Sheer is rated 4.8 out of 5 by 38.\nCorrect, sculpt and highlight. A fluid iridescent veil for a customizable complexion.\nGet your most luminous look with an easy-to-apply liquid highlighter. Part of the iconic \"Armani Glow,\" this weightless, organza-inspired formula comes in an extensive range of colors to match each woman's unique tone. Can be worn as a makeup base or blended with foundation. All skin types.\nApply highlighter makeup with the blender brush to wear alone or mixed with foundation for a boost of radiance and color. Blend with foundation to add warmth or color correction. Use on top of foundation for added highlighting and contouring.\nFor a warm glow: mix with foundation.\nFor a light glow: use to highlight.\nFor a healthy glow: use as a blush.\nTo be used with the Armani brushes or fingertips. Perfect accessory for Giorgio Armani's complexion wardrobe. Create several looks with Fluid sheer: mix with body lotion, apply it on the areas to highlight or use it on the eyelids as a glowing base.\nRated 3 out of 5 by Alexa from Average. Nice packaging but overall an average product when comparing cost.\nRated 5 out of 5 by Cathie from Light and Luminous! Fluid Sheer mixed with my Georgia Armani Foundation, give glorious glowing skin. Not too much, just enough to look healthy. Fluid Sheer is nice and light and a bottle lasts a long time. It is available in a mini size, great for traveling.\nRated 5 out of 5 by Sepi from looks great,stays all day I've been using this everyday since I got it. Worth the price.\nRated 5 out of 5 by amyss from beautiful got #6 to use as blush. natural, lovely glow. wish it came in the travel size cause i will NEVER finish the amount in the regular size. also, which i could dispense a smaller amount, but the delivery system keeps the product from getting contaminated so i will deal.\nRated 4 out of 5 by Lynnstar from This a must have! It maybe pricey but, it goes a long way! I will definitely will order soon..\nRated 5 out of 5 by CatAH from Beautiful face finishing color I use this to really take my makeup up a notch. This is gorgeous and very flattering. It will add color and contour. Completely melts into the foundation and skin it makes you look like a glow from within.\nRated 5 out of 5 by Roseann from Sun kissed glow! I love this product because you can wear it alone, over your primer or over the foundation. The result is always a beautiful glow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sunrise Residents: How Are We Serving You?\nThe City of Sunrise is seeking the input of its residents and customers in an effort to enhance customer service. Three surveys - focusing on the Leisure Services Department, the Community Development Department, and the Public Service\/Utilities Billing Division - are currently available.\nThe most convenient way to complete a survey is online. Just visit our Form\/Survey page and make your selections.\nPrinted, postcard-size surveys are also available. They can be filled out at one of nine City facilities and dropped into a locked box, or taken home and mailed back to the City free of charge.\nSunrise Civic Center - Administrative Office, 10610 West Oakland Park Boulevard, 2nd Floor.\nSunrise Civic Center Athletic Club, 10610 West Oakland Park Boulevard, 1st Floor.\nSunrise Tennis Club, 9605 West Oakland Park Boulevard.\nSeven Bridges at Springtree Golf Club, 8150 Springtree Drive.\nCommunity Development, 1601 NW 136th Avenue, Building A.\nBuilding Division, 1607 NW 136th Avenue, Building B.\nSunrise City Hall, 10770 West Oakland Park Boulevard, 1st Floor.\nVillage Civic Center, 6800 Sunset Strip.\nNew River Civic Center, 60 Weston Road.\nAll surveys may be submitted anonymously.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hall: Robert Morris not ready to party yet.\nMOON TWP. \ufffd Jimmy Langhurst clutched a helium balloon in his right hand, a gift for his 21st birthday. But he wasn\ufffdt planning on the usual rite of adulthood passage Saturday night.\n\ufffdWe\ufffdre still in season,\ufffd he said.\nThe Robert Morris junior point guard was not willing to celebrate the Colonials\ufffd 66-63 victory over Mount St. Mary\ufffds, either. Yeah, it was a win over the team\ufffds top rival, one that cemented Robert Morris as the best in the Northeast Conference.\nBut after last season\ufffds NEC Tournament meltdown \ufffd a loss to Mount St. Mary\ufffds on the same Sewall Center floor as Saturday night \ufffd Langhurst won\ufffdt accept it. The season isn\ufffdt over, and it won\ufffdt be validated until the Colonials qualify for the NCAA Tournament.\nIn fact, despite 21 wins and a second consecutive NEC regular-season title, Langhurst said if the Colonials don\ufffdt win the upcoming NEC Tournament and the automatic bid to the NCAAs that comes with it, this season is a failure.\n\ufffdI don\ufffdt want to go to the NIT again,\ufffd he said.\nThat was last year\ufffds consolation prize, a solid effort but a loss at Syracuse. For a program that went 16 years without a postseason appearance, the Colonials were in better position to accept that fate.\nThat Robert Morris is in this position again may be a surprise. Three 1,000-point scorers (NEC player of the year Tony Lee, A.J. Jackson and Jeremy Chappell) led the Colonials to a school-record 26 wins. Only Chappell returned.\nLanghurst was also last year\ufffds point guard, but his role has changed drastically. He admitted he had one job last year: \ufffdGet the ball to them.\ufffd This year, he\ufffds one part point, one part shooter and one part leader. He had only one assist Saturday, but he had the Colonials\ufffd only two 3-pointers in key moments of the second half.\nLanghurst has been only part of a metamorphosis of the Colonials. They aren\ufffdt as skilled, so they play better defense. They don\ufffdt shoot as well, so they work harder for their shots. They don\ufffdt finish with the same deftness, so they measure their opportunities.\nAnd even though both teams managed to reach impressive standards \ufffd they each had long winning streaks that put them to the forefront of the NEC \ufffd these Colonials lost last week. Twice. Potentially damaging losses near the end of a successful regular season have allowed the players to refocus. Last year\ufffds team didn\ufffdt have that luxury. Their streak-breaker was that NEC Tournament setback.\nEven though comparisons are inevitable, these Colonials simply can\ufffdt be linked with last year\ufffds team.\n\ufffdLast year, we were just going through motions,\ufffd Langhurst said. \ufffdWe were winning games, we got the streak and we\ufffdd think no one is going to touch us.\nLanghurst then walked out the Sewall Center doors back onto to campus, the balloon still in his grasp. His celebrating will have to wait. The season\ufffds not over.\nEric Hall can be reached online at timesonline.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaajsc b/data_all_eng_slimpj/shuffled/split2/finalzzzaajsc
new file mode 100644
index 0000000000000000000000000000000000000000..5e4188b1b893c78d99726f9ee9f893517e4cc9b8
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@@ -0,0 +1,5 @@
+{"text":"My Hometown: Istanbul - 'Oh the People You Meet!\nWho: My name is Shirin Bebe Schade.\nI am the Managing Director of TravelStyle which I have founded in 1999 to design priceless bespoke itineraries for our high-end guests coming from all over the world to discover Turkey from a different, rather unique perspective. Ever since we have never had a website but only work very exclusively through word of mouth and referrals.\nI was born in the capital city of Ankara, however, moved to Istanbul at the age of two after which Istanbul became my hometown.\nAlthough I have spent many years in the United States, Europe, and Asia prior to setting up my company, Istanbul has been the city where I have spent most part of my life and many of my family members and friends have lived in Istanbul which also makes it home for me.\nMy favorite dish is a nicely grilled delicious seasonal fish which comes out of the Bosphorus accompanied by some ''meze'' \u2013 that is Turkish style tapas. We are blessed with the different variety of fish that one can find nearly every month. My favorite fish restaurant is Bebek Bal\u0131k\u00e7\u0131 located in the lovely neighborhood of Bebek on the Bosphorus, right by the water.\nOur apartment! Overlooking the Bosphorus from atop a private forest right from the center, I have Europe to my right, Asia to my left, the Bosphorus bridge linking both continents in the middle \u2013 an unparalleled idyllic view to say the least.\nI think what counts more than ''where'' one stays in the city \u2014 may it be in the old part of town at the Four Seasons Hotel Sultanahmet, at the Bosphorus with the \u00c7iragan Palace Hotel Kempinski, at the Four Seasons Hotel Bosphorus or again at the St. Regis which is more centrally located \u2014 it is ''the way one experiences the city'' that will make him or her feel ''local''.\nGetting access to sites that are otherwise closed or closing those that are normally accessible to the public to have a private visit all to yourself or then again getting access to private residences of art patrons, collectionneurs and influencers could be some of the unexpected activities one could experience in Istanbul.\nOne should definitely explore the Asian neighborhoods of Kuzguncuk which has remained intact for the most part of it, Kadik\u00f6y with its colorful marketplace as well as the up and coming neighborhoods of Moda and Kandilli.\nIt is the only city in the world that is home to two major continents: Asia and Europe, which are separated by the Bosphorus and linked by three bridges along it. Istanbul is a perfect blend of old and new, a melting pot of civilizations and religious diversity which few could possibly rival.\nWatching the beauty of Istanbul from atop when covered with a white lace of occasional snow in Winter.\nAdmiring the beautiful Spring tapestry of color created by millions of tulips planted all over the city.\nDancing to the rhythm of the music during Summer parties along the Bosphorus.\nFeeling the light breeze while jogging over Autumn leaves in the Belgrad Forest.\nBecause you would not want to miss out on the beauty, the cultural diversity and of course the historical richness one would come across every step of the way experiencing the ''real Istanbul''. Historical yet cosmopolitan, with a variety of arts & culture, fashion, and host to some of the very best restaurants in the world, it is a city which has something in store to fulfill \u2013 and exceeds \u2013 any visitor's expectations.\n''Istanbul'' by the famous late singer Charles Aznavour or then again ''Istanbul Not Constantinople'' by They Might Be Giants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"STREETFSN: JARDIN DES PLANTES !\nStunning! I love the red and blue color combination.\nDid not expect that! The red lace skirt is literally fierce, combine with a matching bag and jacket. I like the shoes.\nLoving the intriguing color combination, and she's carrying one of my most coveted Marc Jacobs bag ever. Perfection!\nThis photo is fantastico! I love the red and especially the bag.\nthis is an amazing shot. what presence!\nbrilliant. love this bag and deep red skirt! wow!\nPower color, power walk, I love it.\nomg stunning! LOve it! Now I know what to match that one piece I have from Kal Rieman which is exactly like that!\nTis is a beautiful photo, the colors explode off of her as she is sliced in the middle of the background greenery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A classic design of ball pen, with twist action retract.\nFitted with a black replaceable ink refill.\nA presentation box is available.\nTo have this item personalised please fill out the details above. There is room on this item for 1 line of engraving.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BREC: Beechwood Road Environmental Center. This is the intentionally misleading name given to the proposed expansion of the Richmond Dump which would be placed on equally leaky land.\nCCCTE: Concerned Citizens Committee of Tyendinaga and Environs, formed in 1998 (originally called Concerned citizens committee Against Richmond dump Expansion and CARE). This group owns and maintains LeakyLand.com and the Twitter feed @LeakyLandNews.\nCELA: Canadian Environmental Law Association, Legal representatives of the CCCTE.\nEA: Environmental Assessment: The process execution of the terms of reference where the proponent tries to confirm the environmental impact of their proposed undertaking.\nEBR: Environmental Bill of Rights.\nEBR Registry: a repository of EBR Applications: Everything you wanted to know about what is going on in the applications to the MOE and responses to the applications from the MOE.\ndefines the operational procedures they must follow as defined by the MOE.\nto protection of the environment. This commissioner has no legal authority, but a great deal of moral suasion.\nEMP: Environmental Monitoring Plan: A component of a landfill closure plan which details monitoring requirements for air, soil, surface water and ground water at and near the site as well as other important measures such as, contingency plans, financial assurances, and contamination Lifespan calculations.\nERT: Environmental Review Tribunal: A quasi-judicial body appointed by the government to which companies, citizens and communities can appeal environmental decisions. The ERT can change any MOE decision. They the legal authority to force the MOE to take certain actions they deem to better protect the environment.\nMBQ: Mohawks of the Bay of Quinte.\nMOE: Ministry of the Environment.\nused as the model for monitoring the impact of the site on the environment.\nToR: Terms of Reference: This defines the basis of undertaking in an Environmental Assessment.\nWM: Waste Management Canada Corp; The owners of the Richmond Dump who are responsible for minimising its impact on the surrounding environment. This company is the Canadian arm of a multi-national company Waste Management Inc. based in US.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These are a mixture of sites, some offering information and others assistance with treatment online. It is generally good advice to see a practitioner in person if you require constitutional treatment - refer to the Directory for your area to obtain details of local homeopaths.\nhas an informative site here.\nDr. Nikunj M. Trivedi, D.H.M.S.\nPunjab, India and his Veterinary Homoeopathy site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaxqy b/data_all_eng_slimpj/shuffled/split2/finalzzzaaxqy
new file mode 100644
index 0000000000000000000000000000000000000000..e406a59b507acaeaaf8951bf6bec3ce3a54caab6
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Virtual (online) assistance for small to medium size businesses, entrepreneurs, and individuals. Specializing in marketing, advertising and real estate. Time and cost-effective services for those who may require help with a project or ongoing assistance with day to day business.\nServices include but are not limited to professional marketing and design creation\/development, real estate marketing and administration (residential and commercial - 18+ years experience), word processing, database management, client services, data entry, social media management and content creation, full design services from books to branding, websites, Google AdWords, and other online marketing. Business consulting and coaching services are also available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Purpose Contract\u2013relax (CR) stretching increases range of motion (ROM) substantively, however its use in athletic environments is limited as the contractions performed in a highly stretched position require partner assistance, are often painful, and may induce muscle damage. Therefore, the acute effects of performing the contractions 'off stretch' in the anatomical position [stretch\u2013return\u2013contract (SRC)] were compared with traditional CR stretching in 14 healthy human volunteers. Methods Passive ankle joint moment and dorsiflexion ROM were recorded on an isokinetic dynamometer with electromyographic monitoring of the triceps surae, whilst simultaneous real-time motion analysis and ultrasound imaging recorded gastrocnemius medialis muscle and Achilles tendon elongation. The subjects then performed CR or SRC stretches (4 \u00d7 10-s stretches and 5-s contractions) randomly on separate days before reassessment. Results Significant increases in dorsiflexion ROM (4.1\u00b0\u2013 4.0\u00b0; P < 0.01) and peak passive moment (10.9\u2013 15.1 %; P < 0.05) and decreases in the slope of the passive moment curve (19.1\u201313.3 %; P < 0.05), muscle stiffness (21.7\u2013 21.3 %; P < 0.01) and tendon stiffness (20.4\u201315.7 %; P < 0.01) were observed in CR and SRC, respectively. No between-condition differences were found in any measure (P > 0.05). Conclusions Similar mechanical and neurological changes were observed between conditions, indicating that identical mechanisms underpin the ROM improvements. These data have important practical implications for the use of this stretching mode in athletic environments as performing the contractions 'off stretch' eliminates the pain response, reduces the risk of inducing muscle damage, and removes the need for partner assistance. Thus, it represents an equally effective, simpler, and yet potentially safer, stretching paradigm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chicken soup ... it's good for the soul and the common cold alike.\nBut bowl after bowl can get a little dull if you stick with the plain old noodles + chicken + broth version.\nFortunately there are plenty of ways to get all the comforting goodness of this wintertime staple that pack a little extra pizazz so that you don't get bored with it.\nSpice things up and go Tex-Mex with your pollo. This recipe packs all the warming goodness of chicken broth with Mexican spices and vegetables. You'll be saying \"Adios\" to your symptoms in no time.\nWhen you're in the mood for something other than noodles, rice adds a nice comforting heft to soup. This recipe is easy and comforting.\nWhat could be more comforting than dumplings when you're under the weather? Just saying the word could perk you up almost as much as eating this delicious dish.\nCreamy and decadent, this dish will be a staple all year long. Plus, you'll get the nutritional boost of broccoli, which has been shown to prevent everything from cancer to strokes.\nBetter than take out any day, the homemade version of this Chinese dish is delicious.\nAs if chicken noodle soup wasn't comforting enough, add in some potatoes for one of the most soothing foods ever.\nThough Jewish in origin, everyone loves matzoh balls, which are incorporated into this traditional recipe.\nWhat's your favorite way to make chicken soup?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Obstacles Are Keeping You From Purpose and Prosperity?\nAs an 18 year old, Mike Pisciotta found himself in a position no one wants to be in, behind bars at a Florida State prison. Even worse, Mike couldn't remember the night that had put him there \u2013 he only knew that he was destined to become more than just another number in the prison system.\nThrough a combination of prayer, honest introspection and commitment, Mike rose from his \"identity\" as prisoner #R18756 to become one of the world's most respected marketers.\nBut Mike is more than a marketing expert. He is passionate about using his experiences to help others who are lacking direction and purpose to guide them to not only meet their God-given potential but to exceed it!\nIn From Prison to Prosperity, you'll discover Mike's story and learn how his lesson have transformed thousands of lives all around the world.\nIf you, your team or your organization aren't fulfilling a deeply ingrained purpose. From Prison to Prosperity might just be the inspiration you need to dig deeply and become empowered with renewed purpose and drive.\nAre you ready to create your List Building Machine?\nCompletely customizable high-converting LeadPages template for virtual summits, online conferences, multi-speaker events, tele-summits\u2026 and can be published on WordPress, LeadPages on your own server by uploading the HTML file, or even on Facebook.\nReady To Start Crushing It With Email Marketing (Even If Your List Is Small)?\nWe're sharing exactly what we (and our clients) do to create compelling offers, and drive sales using email and autoresponders. Are you ready to start making money\u2026 instead of just 'making content'?\nWant To Learn How To Consistently Attract Leads & Clients Who Are 'Pre-Sold' And Hot For What You Offer?\nGets clients chasing YOU instead of you chasing them.\nAre you ready to give your business the ultimate \"Game Changer\"?\nIf you're working really hard, but it seems you're not making money\u2026 No matter what you do, it seems that there is never enough for you to get ahead in your business\u2026 and it feels like an uphill battle\u2026 this audio training is for you.\nThe first person you have to sell on 'price' is you \ud83d\ude42 Nobody is going to come along and hand out permission slips for you to charge what you're worth\u2026 you have to write your own permission slip. Are you ready to BUST THROUGH your limiting self-beliefs that are keeping you stuck, undervalued & underpaid?\nWouldn't you love it to feel like there's no differentiation between work and play? Do you think there's such a thing as easy money? Maybe. What comes EASY to you is exactly what you should focus on in your business! It's what sets you apart, it's what makes you marketable, and it's what will make you the most money.\nThis mindset training will help you learn to identify & operate in your Fab Zone, so you stay the most profitable, and have the most fun while doing it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If we wish to combine all our powers of feeling into one strong will, we will have to call a halt to weak wills.\nThe article 'personality' describes that we can see our will and personality as our conscious possibility to convert our feeling into thinking and actions.\nWith our will we can elevate feeling from our subconscious and focus that on one point.\nThe extent to which we can elevate our powers of feeling and deploy them in a concentrated way determines the strength of our will.\nIn this connection, Jozef gave the example of a man who fought against the elements with his boat at sea.\nDue to the hard fight against the powers of the ocean, the man built up willpower.\nLater, when he starts to deploy that willpower in order to bring himself to spiritual consciousness, then he can make grateful use of the will which he built up in past times.\nYet most people experience that their will is not as strong as they themselves want it to be.\nbut experience that this is not so self-evident.\nAs soon as we wish to deploy our will for something, we can feel which opposing powers in our inner life still do not want that specific goal, but prefer something else.\nJust like the personality, the will is fragmented by numerous small wills which can oppose each other.\nAll the different character traits or sub-personalities of the human being, all the past lives which live in his subconsciousness can considerably break down the will.\nIf we really wish to progress in our life, then we must call a halt to all those weak wills which want to lead an own life.\nFor this purpose, we have to bring our inner life under control.\nWhen our will is sufficiently strong and clear, we are also more resistant to the will of another person, if it is attuned to our life.\nFor instance, Jozef gives the example that he had to get the playfulness of his youthful personality 'Jeus' under control, in order to build up a city personality in a large city, where the people do not laugh about this.\nIf he was to let Jeus talk in his dialect, as a taxi driver he would not be taken seriously.\nHe has finally combined all his little wills into one will in order to be able to give his spiritual awakening a powerful prompt.\nJozef indicates that for this purpose he wanted nothing to do with tiredness.\nHe was faced with the task to provide every feeling within himself with light, in order to drive every thought to the spiritual expanding.\nHe has experienced that his will can move mountains, but which of us already started on such a strong will?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabenu b/data_all_eng_slimpj/shuffled/split2/finalzzzabenu
new file mode 100644
index 0000000000000000000000000000000000000000..58c798fe2020227a0fb87689d33a30891e8499ea
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzabenu
@@ -0,0 +1,5 @@
+{"text":"One billion rising for justice is about envisioning justice for all survivors of gender violence.\nJustice can take many forms.\nIt can be an apology or reparations. Taking legal action. It can be about making the truth visible. It can be prosecuting, or pushing to create change, or implementing policies and laws that ensure the protection of women's rights. It can be calling for an end to all forms of inequality, discrimination, misogyny and patriarchy. It can be naming or shaming perpetrators \u2013 whether they be individuals, groups, corporations or the state. Demanding accountability. It can be rising for justice be it personal, social, economic, cultural, environmental or political. It can be a revolutionary call to restore dignity and respect for all women.\nIt can be about transformation.\nAs we bring this global discussion to the forefront, we want to hear from you what justice looks like. Join thousands of activists from around the world, share your vision of justice.\nYou may bend me, but not break me.\nI will dance every day of my life.\nThe Female Gender Must Be Free To Save Our Planet!\nFeminism = The radical notion that women are people too.\nWe are one and the women will lead us.\nCan you hear the voices of the women?\nOur society tells us to keep quiet. What a horrible injustice.\nI am young, I need my little voice to be heard!\nJustice is just us, and love.\nRising for love and justice.\nIn memory of teenage victims.\nFemale genital mutilation = Aggravated sexual assault.\nJustice is love in action.\nBreak the chain of violence against women to set us all free!\nLOVE, BE KIND & RESPECT OTHERS!!!\nHow I see justice for women.\ndifendiamo il Progetto Donna, rilanciamo la giustizia!\nWOMEN AND THE FEMININE BRING PEACE AND LOVE TO OUR WORLD!!!!!!!\nThis is what safety looks like.\nWill men be our allies?\nJUSTICE means everything that matters.\n1 BILLION RISING + 1 !!!!!\nJustice is Equality and Unity for All.\nMr. Smith\u2026 The man next door !!\nTHIS happens every hour \u2013 IT could happen to YOU.\nI fought for my right all my life.\nBy: MUJERES Y PODER A.\nBy: 4\u00ba Estelas de Cantabria C.\nWomen standing up for women!\nA world where all women, children, & men are safe.\nWomen should be safe and free.\nI despise abuse so much I'm running for Congress!\nGlobal Justice is Global Healing.\nEnactus EUR rising for justice!\nBy: Women Helping Women o.\nBy: Spazio Donna Onlus A.\nBy: SquashBusters Afterschool Program B.\nIt is time to do something.. it is time for dance!\nTalking with teens about oppressive media images of women and girls.\nIt's a way to stand up against unfairness and injustice.\nIt's a world where equality is just that.\nrising of the 21st century women !!\nFree from the violence and visions of being smothered.\nit is our belief in it that gives it its power.\nTake away the problem, take away the fear.\nJusticia divina & la justicia terrenal.\nWe're Rising to make the DREAM Reality!\nEqual Justice is the goal.\nJustise is being given the information we need to keep us safe.\ntunay na katarungan para sa mga biktima!\nI Rise to Honor the Care by Women for Life. .\nWho is W. I. .L. .A and what do they want?\nDon't mess with women. EVER.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On my last visit to Goa, I came across this gem owned by Masterchef finalist Sarah Todd. This place is best to witness a perfect beach sunset. Imagine waking up in a cute and chic cottage with an ocean view, and then filling up your tummy with a perfect gourmet breakfast, before taking a dip in their beach-facing pool: what else can one ask for!\nJust book their Ocean Comfort room and take a dip in their pool, where you will be charmed by the view of Vagator Beach right in front of you. PS: Top it up with a Margarita.\nThe place was perfect. Only one thing: just avoid the Party Huts as they don't have the amazing ocean view, which is the USP.\nThe place looks even more mesmerising at night, when all the hilltop restaurants light up their chic lamps and lanterns, making the ambience just magical.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's a great place to set your cold frosty one while you're hanging out in your Man Cave! Each coaster is personalized with first and last name of the certified cave master and the year his cave was established. Full-color coasters come in a set of 4 in a caddy. Other notations on the mug include: real sports, fine cigars, cold beer and solitude. Drama is prohibited in this man cave!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Product arrived in only two days. I will definitely use this company again.\"\n\"Fast Service. I received what I ordered. I will order from RV Plus again. Thank You!!\"\n\"PROMPT SHIPPING AND WELL PACKAGED. PRODUCT IS AS DESCRIBED AND FAIRLY PRICED.\"\n\"Quick delivery. Item as described. GREAT price.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"The mighty Thjorsa river ( \u00dej\u00f3rs\u00e1 )is Iceland's longest river at 230 kilometers. It's named after Thjorsarhraun, a lava field that flowed some 7800 years ago and was one of the greatest postglacial holocene flows in the world. It is located in the south of the island, its source, the glacier Hofsj\u00f6kull. It flows out through narrow gorges in the highlands of Iceland creating the complex patterns that have drawn me to photograph them for all these year. As it nears the river mouth it's depth thins, revealing shallow sandbars and tiny islands.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzachyy b/data_all_eng_slimpj/shuffled/split2/finalzzzachyy
new file mode 100644
index 0000000000000000000000000000000000000000..708eec29502efa1d1cbe1830cb9fdfb762037475
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzachyy
@@ -0,0 +1,5 @@
+{"text":"These include Windows registry issues, malicious software, faulty applications, etc. Click here to download a highly recommended file recovery program. After you qutocad successfully uninstalled your SetupUI.\nIf you are not currently backing up your data, you need to do so immediately download a highly-recommended backup solution to protect yourself from permanent data loss. For detailed installation instructions, see our FAQ.\nIf you don't know exactly how to install the DLL, or are unsure which version to choose, this is the way to go. Perform a Clean Installation of Windows. You can also click the [ ] image to hide the instructions as you proceed through each step. Thus, these invalid DLL registry entries need to be repaired to fix the root of the problem. Installing the wrong driver, or simply an incompatible version of the right driver, can make your problems even worse. For bit programs, use bit files if they are listed above.\nRe-start the program associated with SetupUI. Instructions for Windows 7 and Windows Vista: Follow the steps in the Wizard to choose a restore point. Need some help installing setupui.\nYou pay just once and can use the program whenever you need it. To achieve a Gold competency level, Solvusoft goes through extensive independent analysis that looks for, amongst other qualities, a high level of software expertise, a successful customer service track record, and top-tier customer value.\nWe also recommend downloading the latest versions of dll files for an up-to-date functionality. These broken registry keys can be as a result of a missing DLL file, moved DLL file, or an leftover DLL file reference in your Windows registry from an unsuccessful software installation or uninstallation.\nClick Programs and Features. Several programs can share the same SetupUI. This repair tool is designed to diagnose your Windows PC problems and repair them quickly. This Tool is Compatible With: Reinstalling Windows will erase everything from your hard drive, allowing you to start again with a fresh system.\nYou may download for legally licensed software only. The program can't start because setupui. Microsoft is constantly updating and improving Windows system files that could be associated with SetupUI.\nInstructions for Windows XP: Click the Uninstall button on the top menu ribbon.\nThe good news is that you can often update the device driver to fix the DLL problem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It would be nice to if you could sent me a PM as soon as possible. Otherwise I may not be able to write the next part until after my exams.\nKainCrow: I swear it wasn't me. I didn't even know the guy.\nbunny fran: I've no time for this. I'm going back to Tsenoble.\nangel: Were the four of you from Old Archades?\njake1456: Of course not! Where'd you get that from?\nShadow Gilgamesh: You don't really need a keen eye to notice that.\nangel: Wanna buy some of m... hey, you stole that!\nUltima: Sorry, I'm so hungry. I'll pay you back when I get money.\nangel: You're a bunch of thieves!\nDa_Cloud: Come on angel, let's check the place out.\nangel: Ah well... guess I could try my luck there.\nMegaFun: About time. Do you have the blade?\nNPC: Of course. The money.\nNPC: Nice doing business with you.\nCloudVincentLeonhart14558: *Bleep*! He didn't have the documents.\nCVL14558: I checked his dead body. Get your fingers on those documents.\nZodiac Killer 99: We must do something. The others are too power hungry.\nOniangel: I know. But currently I'm clueless as to what we could do. MegaFun being so childish doesn't help it either.\nMegaFun: *enters* What did I miss guys?\nShadow Gilgamesh: What do you mean you won't pay me, you little...?!\nbunny fran: Something was missing from the box. Be more careful next time.\nShadow Gilgamesh: I'll strangle you!\nbunny fran: Pff... keep your fingers off of me, there are guards out there.\nangel: Fresh vegetables, fresh fruit!\nNPC: Oh, they're fabulous! I'd like some.\nangel: Thank you, come again!\nDa_Cloud: Seems like business is flowing here.\nangel: I already love this place.\njake1456: Hey, what's the huge building?\nangel: That's the Giants Headquarters.\njake1456: Catch me if you can! Hihihi.\n: This is some nice sword. I'm sure it'll fetch a good price at the market. These are tons of paper... Wonder what they're for.\nstuart: Hey, who are you?\n: Er... you've never seen me!\nZodiac Killer 99: Inform the others.\nImperial: We've caught the intruders.\nangel: Let go off me, I'm just an honest merchant!\nMegaFun: When Zeikuu was found dead. Interesting to say the least.\nOniangel: Wait. They may be important witnesses.\nCVL14558: What witnesses? They're lowly thieves. And murderers, too.\nOniangel: Hahahaha. We all know that there are foul games at play in our headquarters. I'm interesting to what they have to say.\nMegaFun: Seems we're all here now. Let's start with our little rascals. What say you in your defense?\nWho killed stuart? Who is the spy? Is it someone from the poor lowtown of Archades or a member of the Giants themselves? Or perhaps neither? You decide!\nStuart didn't recognize the guy and the spy flinched on what he saw, swords.. papers..\nI'm going with angel, becuase she looks like the type that would say \"err\"\nHmm, I guess it can't be either Mega or CVL since they're the ones who gathered the sword\/documents. And rs never showed so I'd rule him out too. For now I'd say it's more likely for it to be one of the poor, then maybe one of the merchant class before one of the big 6.\nFor now I vote for Jake, he always seems the quickest to run.\nThat and he seems to have an agenda with the 'documents'. I second Jake's vote.\nI see a division among the Giants and I suspected the killer to be one of them but for some reason the behavior of the spy in this one leads me to think that it's one of the Old Archadians so... I'll vote for jake.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For those who seek a truly unique ring, touched by an artist, and customized to your exact specifications, your search has ended. Our styles are inspired to include contemporary, modern, old world, and eclectic collections by artists from across the U.S. Celebrate your unique relationship with a handcrafted piece from Mia Gemma.\nWe made the September Issue of the Washingtonian magazine. Take a look!\n\"Owner Irene Barbieri believes good design should be accessible, so the one of a kind and limited edition artisan pieces\u2026\"\nOur philosophy is that jewelry should be the ultimate expression of your personal style. With that in mind we work to create unique and timeless pieces with our clients that resonate with their true self.\nWith more than twenty-five artists represented here in our Gallery, clients can explore many styles and sensibilities as they begin the custom design process.\nMotifs inspiring her many collections come from nature, modern architecture, vintage royalty and ancient symbolism. Styles range from cocktail rings, classic pendants, chic earrings and cuffs, to more delicate necklaces, simple studs and bands.\nThe sophisticated designs use only genuine precious and semi precious stones such as sapphires, rubies, citrines, Peruvian opals and a wide variety of unexpected gemstone and diamond combinations.\n\"..striking, genuine gemstones that ooze sophistication yet manage to stay grounded enough for everyday adornment.\"\nA wedding band or commitment ring is the symbol of one of the most enduring promises you will make to each other. Whatever design you choose, Modern, Contemporary, Organic, Eclectic, Vintage, Old World, Classic, your rings will be crafted exclusively for your hand to best ensure optimum comfort and longevity. Our talented associates at Mia Gemma will work with you to create or select an exquisitely handcrafted piece that best expresses your love for each other.\nWhether you are browsing our Penn Quarter location or shopping online, you will be delighted by the variety of wedding and commitment ring designs that are available for you at Mia Gemma. Our unique men's wedding bands are created for style and comfort, created with expert metal work and stone setting, and are finished by hand \u2013 not manufactured in quantity. We offer clients the ability to work directly with our artists to create a one-of-a-kind or limited edition piece that reflects your special bond.\nCome browse our stunning collection of Bridal Jewelry for pieces that express your personal style and taste. Whether you are looking for Tahitian pearl earrings that will become a family heirloom, or a one-of-a-kind showpiece necklace that accentuates your gown, you're sure to find something perfect to compliment your wedding-day wardrobe.\nSince pearls are often a favorite, we have an extensive variety of designs that feature Baroque, Fresh Water, and South Sea pearls in shades of creamy white, champagne, or silvery grey to add balance and elegance to your ensemble. Fresh, new, limited edition designs featuring exotic configurations of colored precious and semi-precious stones are also available.\nConsider a custom gift for your bridal party members: necklaces, earrings or bracelets in a gemstone color to match their dresses. We also have many pieces for the Mother-of-the-Bride or we can create a custom design for her as a precious reminder of your special day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discover over 852 of the best selection earbuds lot on aliexpress,splaks stereo headphone earbuds.come to get the best soundpeats q36 bluetooth earphones wireless magnetic sport earphones.find the top 100 most popular items in amazon electronics best sellers,buy beatsx wireless in-ear headphones.these are the perfect marriage of headphones and,happy plugs earbud rose gold is simply a whole ne,.\nShop for headset with mute button at best buy,headphones with volume control,interesting finds updated daily,get the incredible sound and legendary styling of beats by dr,.\nDre 's pill speakers all at every day low prices,take full control of your music or calls with the mic and function buttons built right into the comfortable neck collar.today we will look into 10 best earbuds under 20$ of price that bring just the best quality for you.find low everyday prices and buy online for delivery or in-store pick-up..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tru-Cal, Inc. \u2013 Tru-service you can count on!\n\"Jim and the entire TruCal organization have provided us with great pyrometry services in accordance with aerospace standards (AMS 2750). I appreciate how they are self-managed in scheduling SATs, calibrations, and surveys for my two dozen furnaces. Jim himself brings expert knowledge of specifications, industry best practices, instrumentation, and measurement to TruCal.\"\n\"We are loving Josh and how accurate he is. I got Josh's first uniformity packet and it is perfect! Josh is AWESOME and very accurate. I gave him my procedure and he ran with it. I so appreciate you working all this out for me at the last minute and getting me out of my jam. Just wanted to give you some feedback. Sometimes in quality all we hear is bad. I wanted to give the good feedback.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafsxj b/data_all_eng_slimpj/shuffled/split2/finalzzzafsxj
new file mode 100644
index 0000000000000000000000000000000000000000..fa92604ee6cf315b208a5e90f505f3df7ab43b96
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Invention as well as , technology have expanded very many business owners in a variety of ways as opposed to what ten. Merited to its workload close to administrative tasks the same as bookkeeping, stats keeping, as well as a inventory, both small in addition , big online businesses rely of computers to actually perform their valuable administrative tasks. The invention of Online world and social networking directories has highly reduced a new costs together with business surgeries.\nNow most of businesses have the possibilities to go ahead global through the use of the internet. Business with web sites use the marketing software to arrive at clients all over those world by using just some single mouse click on of a huge button. The following wouldn't obtain been realistic without the very internet.\nThe help of websites and by going online social networks has certified firms interfere with folks and handle all of their inquires about their personal brand or to product as well service. Email marketing and written messaging have also been changed our process of communication and interaction on an important daily base. An helpful communication featuring customers creates rapport using them in addition to the also would make strong open public image considering that well.\nThe effect of technology on stores have been enormous. These days, anyone are able to neck trades with the others that may are one mile away. During addition, solutions has always made world-wide-web payment prospective.\nIn fact, Americans on my own consume beyond 17 thousand quarts (an average having to do with 50 quarts per person) of fat free popcorn every year, and your is more than sufficient to satisfy the Kingdom State Designing 18 era. Surprisingly, any large allowance of this popcorn is consumed in the the when watching one particular movie. Generally is some case even technology will have been beneficial in business, this a period of time the tv show industry.\nTogether complete with the objective of technology, innovations followed, resulting with regard to the arrival of new gadgets so equipment. Never matter strategies about how small per bug your company is, technology gives you brought both elusive and moreover substantial incentives to are productive, efficient, and to positively meet that ever-rising must have of dieters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As I sit here writing this, there is currently a light (yes, light, knock-on-wood) dusting of snow outside. And I drove around earlier this afternoon on a quest for wiper fluid (my windshield more or less appeared to have a fine spray of opaque salt--not exactly the safest for highway traffic). And even though we have on sweaters and thick socks: winter hasn't been that bad, yet.\nIf I've jinxed it for all of us, well. Be mad if you want to, but it won't be the first time.\nI think I can shrug off winter at this point because in just a few days (namely, five), I am jetting off to Florida for my little sister's wedding. And then, just five days following my return, will be on another plane to Oaxaca, Mexico. More on this later, but it's safe to say I'm choosing the escapist route this time around.\nStill, winter or no winter, the fact that I wore a T-shirt last Sunday or not, I've still had a sincerely deep craving for a warm bowl of chili, the ultimate cold-weather comfort food -- okay, maybe not ultimate. Mac & cheese, hello. But certainly in the top ten. And as far as comfort foods that are vegetarian-friendly, chili pretty much let's you play by your own rules. A little of this or that never hurt, so mixing up a recipe without really having a recipe is usually the way to go.\nI've also lately been obsessing over finding a good cornbread recipe. Since there are about, oh, I don't know, a MILLION, I have no idea what good actually means. I only know what it tastes like. Crumbly, slightly sweet, sturdy enough for a healthy (ahem) smear of butter. You know, that.\nPreheat your oven to 400 degrees. Take some nonstick spray or butter and go to town on a 9x5-inch loaf pan.\nIn a large bowl, whisk together your flour, cornmeal, sugar, baking powder and salt. Meanwhile, in a small bowl, lightly beat the egg. Then whisk in the sour cream, buttermilk and olive oil. Then gradually stir your wet ingredients into the dry ones, then spread your batter into the prepared pan. Bake for about 22 to 25 minutes. You should see a beautiful golden brown (literally!) color. And everyone knows a cornbread with a slightly darker edge is perfect for a nice crusty crunch.\nWhile your cornbread is in the oven, time to get started on your chili. I'll go ahead and tell you what I put in mine, but feel free to ad lib here. Chopping up almost any vegetable will work, spices, you name it. If it's a chili (or a soup), it's worth a shot.\nStart by heating the olive oil in a large stockpot. Add your garlic and onions and let them caramelize, about two to three minutes. Slowly add your bell pepper and carrots (if you decide to include any other sturdier vegetables, now would be the time to toss them in), stirring for about five minutes. Add in your jalape\u00f1o, red pepper flakes and other spices. Adding your spices in early will help them permeate the entire dish, making the flavor more well-rounded throughout. And so on and so forth.\nAdd your broth, using a wooden spoon to scrape up any goodness that may be gathering on the bottom of your pot. Next, toss in your kale, which will immediately begin its shrinking act. Let this mixture come to a simmer before adding your tomatoes. If you already went ahead and got diced, like you should have, great. If not, like me, go ahead and add your canned tomatoes, juice and all, and use a fork, slotted spoon, potato masher, whatever you've got, to crush them into smaller more chili-sized pieces. Then add your beans.\nIf you thought have a free-for-all ingredients list was great, I've got more good news: however long you let this cook is up to you. The thing about chili is that it only gets better then longer it simmers. If you have an hour or two, great, let it go. If you've got to have dinner on the table in 20 minutes, this will still work. It all depends on your sense of urgency and, well, hunger.\nUs, we let it go for about an hour and a half, just long enough for the stomach-rumbling to get really unbearable and after we snuck a corner (or two) of the already cooling cornbread.\nPsst. Let me prepare you for something here: this cornbread is not sweet. At all. If you're thinking of the densely sugared versions that your mom used to make, possibly in a cast-iron skillet, that is not what you're getting with Deb Perelman's Marcus Samuelsson's adapted version. Add a little bit of butter, a drizzle of honey, and it works. And sweeter versions to come.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Come join us for a free, family, art event in Downtown Provo. Hosted by the Utah County Arts Board, our Children's Art Festival will have live entertainment, free crafts, lots of fun food, a children's art gallery, face painting, chalking, and the cutest photo opportunites! Come from 6-8pm on Saturday June 4th.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wave 12 for FY 2017 was released Friday, August 18th, for a total of $90.8 million. Cumulative national funding through Wave 12 is $881 million ($737 million for Category 1, $144 million for Category 2). A scheduled release date for Wave 13 has not been announced.\nWave 59 for FY 2016 was released on Tuesday, August 15th, for a total of $0.00 \u2014 representing one denied application. As a result, cumulative national funding through Wave 59 remains $2.91 billion. A scheduled release date for Wave 60 has not been announced.\nEPC is not yet set up to accept, much less process, SPIN changes for FY 2017 FRNs. It is expected that SPIN change requests will be accepted in EPC by month end, but the system may not be able to report on approved SPIN changes until sometime in October.\nApplicants needing to make SPIN changes who have not yet been funded are encouraged to submit those changes via RAL corrections which can be processed by PIA.\nFCC rules require that special construction fiber systems are to be completed by the end of the funding year. Completion means that all fiber circuits have been installed and are lit. For applicants approved for, or still seeking, funding in FY 2016, the construction deadline was June 30, 2017. Recognizing that this might be a tight deadline, particularly for applicants with larger network projects and\/or those receiving late Funding Commitment Decision Letters (\"FCDLs\"), the FCC authorized USAC to extend this deadline upon request by one year. Initially, Special Construction Deadline Extension Requests (\"SCDERs\") were to be submitted using a USAC template. The guidance was subsequently changed advising applicants to use the Form 500 instead.\nUSAC News Briefs leading up to June 30th did not discuss the impending completion deadline nor the concurrent SCDER filing deadline.\nMore importantly, USAC did not confirm the requirement to file a SCDER by June 30th until later in July. Indeed, the EPC version of the Form 500, which can be used to file a SCDER, still indicates that the deadline to request a service delivery extension for fiber construction is September 30th.\nApplicants who need extensions, but did not submit SCDERs by June 30th, are in an interesting position. We have not yet seen how USAC plans to handle late-filed SCDERs. In the best of all worlds, given the misleading Form 500 guidance, the FCC will give USAC the authority to waive the June 30th deadline for those applicants submitting their SCDERs on or before September 30th. If not, these applicants will likely need to appeal. As part of the documentation for such an appeal, we recommend that applicants print a copy of their Form 500 SCDER submission \u2014 and do so now before USAC corrects the EPC Form 500 language.\nUSAC's Schools and Libraries News Brief of August 18, 2017, continues a Form 498 discussion begun in last week's News Brief. To review, the Form 498 is important to applicants planning to file BEAR invoices. The Form 498 provides USAC with the bank account information required to send reimbursement payments directly and electronically to the applicants. Applicants, who already have approved Form 498s, need read no further.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hardcover. Princeton, NJ, Van Nostrand Company, Inc, 1st, 1964, Book: Very Good, Dust Jacket: Very Good, 276 pages. b&w illustrations throughout. Pictorial endpapers. Dark blue cloth, light rub on edges of spine, gilt lettering to spine. Dust jacket edge wear with some smudges near edges. A nice, tight copy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagmoc b/data_all_eng_slimpj/shuffled/split2/finalzzzagmoc
new file mode 100644
index 0000000000000000000000000000000000000000..41e4a9c28a4bfc38201ff784ea0013d1a8d2c307
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagmoc
@@ -0,0 +1,5 @@
+{"text":"On February 21, 2017, H-Net Council voted to de-commission H-Shukyo. The network has been largely unused and ran without editors for an extended period of time. Attempts to find new ones did not work out. Rather than continuing attempts to revive an unused network, we suggest prospective subscribers join our much larger and active networks in related fields. These might include H-Japan, H-NE-Asia, H-Buddhism, H-Asia and many more of our 180+ networks.\nOn February 21, 2017, H-Net Council voted to de-commission H-Shukyo. The network has been largely unused and ran without editors for an extended period of time. Attempts to find new ones did not work out. Rather than continuing attempts to revive an unused network, we suggest H-Shukyo's 311 subscribers join our much larger and active networks in related fields.\nCall for Papers: \"Structure and Subordination: Law, Science, and Religion in East Asia\"\nThe Centers for Japanese Studies and Buddhist Studies at the University of California, Berkeley, together with \u014ctani University and Ry\u016bkoku University in Kyoto announce a workshop under the supervision of Mark Blum that will focus on critically examining premodern and modern hermeneutics of the Tannish\u014d, a core text of the Shin sect of Buddhism, and arguably the most well-read religious text in postwar Japan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I love how little babies look like wise old sages; their soulful eyes speak volumes about the sanctity of life. They are the masters of the universe, indeed the universe that now spins around their sun.\nSpend time with these amazing creatures and know God is benevolent.\nI had the pleasure of spending time yesterday with the youngest member of our family; Charlotte turned five months earlier this week. She's all tender, fragile skin and rose-leaf softness that begs touch. I couldn't help but want to kiss that softness, smell her precious wonder in every fat fold, in every delicate finger and toe.\nShe's such a good listener, too. In this age of distractions, she knows how to focus on what's important. I have much to learn from this adorable little bunny.\nWhich reminds me; Easter is right around the corner. I don't know about you, but I plan to rejoice and be glad. I also plan to share a few Fun Fotos.\nShe is just so cute! Precious.\nYes, she has Gerber baby written all over her beautiful face.\nThat little tongue is begging for a yank! Love her!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Having never met you nor your Dad, Angel, I 'see' your Dad in this post. Especially when you bring this post a close.\nYou may cling to the stress of the previous year because it came and went. You made it through \u2013 and though it may have been stressful at the time, you now know how it turned out. Whereas this newest journey \u2013 (dare I say, challenge) \u2013 is still unknown. Where there exists an unknown fear can root.\nKeep on keepin' on, Angel. The unknown will be full of wonderful surprises, many smiles and new life experiences. Your daughter will be so much and more \u2013 even during the most challenging of days\/nights.\nMy prayer for you is strength and renewal of faith.\nAngel, I know this year is going to bittersweet for you. I know the upcoming birth of Emma makes you miss dad even more. But don't you fret, because he's part of you and he's going to be part of her. Being a mom is scary. You have this perfectly beautiful little life that depends completely on you. You're going to make some mistakes and you're going to have some regrets but in the end we all do the best we can and hope it's enough. I know you're going to do a great job! 2012 is a chance for new beginnings, new hopes, new dreams, & new memories. So I agree\u2026 let's grab onto the new but let's never forget what brought us here. I love you!\nIf I were to write a comment, it would be exactly what all three of these wise women above. said! God chose you and Jason as Emma Kate's parents because He knows what is in your heart. Trust in Him and His word and you will be fine. Not to say you might make a few mistakes (we are human, you know). I believe it was Maya Angelou who said \"when we know better, we do better\" ! The best piece of advise I could ever offer a new parent, is to tell your child you love them every day! Saying that makes up for a whole lot of mistakes \ud83d\ude42 My thoughts and prayers are with you during the next few days and weeks as your prepare for this new beginning. Until next time \u2026 Donna H.\nYou do both! You hang onto the old\u2026\u2026.and you grab onto the new!\nAnd, why????? for Pete's sake would you even imagine, much less worry that you wouldn't do right by your little girl? You will absolutely and completely be the best mama ever! I can't wait to see you hold YOUR OWN BABY in your arms! I can't wait to watch you be a Mother\u2026..and not just the daughter, sister and auntie that you are now! YOU WILL be the best mama in the whole world! Emma Kate just doesn't know yet how lucky and blessed she is to have been given YOU for her Mommy\u2026.and Jason for her Daddy! She doesn't know it yet\u2026..but, I sure do!!!!!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now they are offering to give me a free study and fit me with a FFM (which I believe i need for times of year that I am super congested with seasonal allergies (like June)!\nDoes this sound like it is worth it (free) or a frustrating exercise? Of course this will likely lead to an updated positive review online! Anyone have a similar experience?\nOn my own I have tweaked settings and switched to a Dreamwear. But I could make this better for sure - maybe DME can help actually?\nI would want to know how are they going to conduct the \"free study.\" Is it a titration in a sleep lab or a home study?\nIt's one thing to tell the truth about your experience in an honest review, but these things seem forced.\nA company \"deciding\" to do the right thing because it has to for online reputation is not the same as doing it because that is their routine way of handling problems.\nNot sure if updated reviews reflect the reality of normal operation.\n(06-28-2017, 09:48 AM)AlanE Wrote: I would want to know how are they going to conduct the \"free study.\" Is it a titration in a sleep lab or a home study?\nI did an at home study before I originally got machine. This I believe will be at their lab.\n(06-28-2017, 10:28 AM)personal111 Wrote: It's one thing to tell the truth about your experience in an honest review, but these things seem forced.\nI will update my review to reflect how they \"made good\" after complaining - if they do \"make good\"!\nThis is PSYOPS 101. If you refuse their offer, they can then post a rebuttal stating they made the offer and you refused their help. Don't provide them with any ammunition to negate your review.\nI agree....I think it's forced. I also think that they want to shoot down your review one way or another....either you told them to pound rocks when they tried to fix it (the 10 months later part will be omitted, you look like the bad guy) or they expect a glowing review (they look like the good guy).\nI'd say take their offer, but when you update your review, be honest and keep the full story. I was referred to company X by my doc, they treated me like crap, dragged their heels for 10 months and when I wrote a poor review someone higher up got involved. They did A, B and C to try to fix it (which worked or didn't) which was great (or sucked). It's nice they stepped up after 10 months and a poor review; however, it would have been far nicer if the front line staff could have dealt with it at the start...yadda, yadda, yadda. Time will tell if the change is motivated by caring for the patient or caring about the review....blah, blah, blah... You get the idea.\n(06-28-2017, 11:28 AM)Crimson Nape Wrote: This is PSYOPS 101. If you refuse their offer, they can then post a rebuttal stating they made the offer and you refused their help. Don't provide them with any ammunition to negate your review.\nIn a former life I did a lot of online reputation management stuff for a few small web based companies and also dealt with that famous bureau you will all be familiar with.\nIf I offered you some type of resolution and you refused it or ignored it, I would usually be able to get the status of your original complaint changed or removed.\nEven if they only did this to fix your online review I would certainly look closely at what they're offering and consider it. I know that DME's are not well liked here but my Minnesota Nice attitude makes me think you should give them a chance to make things right. What were your original issues with the DME?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"several different models to use for a modest charge.\nE2M28\/30 oil sealed vacuum pumps .\npumps are included, and some special tools are available for purchase.\nAll service training is delivered at your site.\nWe are not able to offer open registration training courses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagoua b/data_all_eng_slimpj/shuffled/split2/finalzzzagoua
new file mode 100644
index 0000000000000000000000000000000000000000..bafde1e0ae107cad9025141464241be0ae8960c8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzagoua
@@ -0,0 +1,5 @@
+{"text":"Defun Function calls with arguments, please help!!\nHere is my section of code that I need help on. The error VisualLISP is giving me is :Function Canceled. The builder doesn't complain, so my syntax is correct, but its just not working. I am making simple lines and changing the color to them using the \"change\" command with a function call. Can anyone please tell me what I am doing wrong?? Also just fyi in defun c:wellplace.. i just used the line (color po2) because i don't want to post the whole code in, everything else works fine, but breaks when it gets to that line. (defun color (col) (command \"change\" col \"\" \"P\" \"C\" \"T\" \"255,0,0\" \"\") ) (defun c:wellplace () (color po2) ;po2 is a \"point number 2\" that is calculated else in the code. ) Shouldn't it take the po2 which is a polar point, consisting of x,y,z coordinates and pass it along to the \"color function\"? I am doing this, because I need to do the \"change\" command a good 8 times, and i don't want 8 of duplicate lines of the same thing. Also if i go with not defining a function and just inputting the \"change\" command right into the body of my code, it works perfect, and replace 'col' with 'po2' so what gives?\nHello CAD-lovers, I encountered a strange situation few minutes ago. I find out (acad) is a function. As a result of calling it (I think) focus brings to AutoCAD screen, while you are in Visual LISP editor. The color of function was blue, like other built-in functions. Unfortunately now, after exiting AutoCAD and returning back to it, acad returns to black, and the value of it is nil. Does any one know a way to find a clue to it?\nHello everyone, does anyone knows a way to define a function in AutoLISP with arbitrary arguments, same as ARX?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With the extensive cost of providing your team with reliable communications why should you guess whether or not your new radio system will provide the coverage you need. Ragan Communications Inc. has the tools to provide you reliable estimates of RF coverage for a wide variety of radio systems. Our staff can also provide microwave path studies with link budget information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Contact Sheehan Health Group today.\nWhat service(s) are you interested in contacting us about?\nIf you are interested in Skilled Nursing or Rehabilitation, choose a location near you.\nPlease leave us any additional information we will need to meet your needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Grand Canyon, is a land of song. Mountains of music swell in the rivers, hills of music billow in the creeks, and meadows of music murmur in the rills that ripple over the rocks. Altogether it is a symphony of multitudinous melodies. All this is the music of waters. The adamant foundations of the earth have been wrought into a sublime harp, upon which the clouds of the heavens play with mighty tempests or with gentle showers.\nIn Canyons of the Colorado (1895), 394-397.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unreal Tournament (Pre-Alpha with Gameplay) \u2013 Is it working out?\nUnreal Tournament is a high-octane FPS with both single-player and multiplayer game modes and it's been 8 years since 2007's Unreal Tournament 3 was released, but finally, we have another in development! Unreal Tournament is one of my favourite franchises and I've been longing for the next instalment for years! Even though I played UT3 and it wasn't bad, there's something about it that doesn't make it feel like a real UT game. This is why I'll mostly be referring to UT2004 occasionally.\nBefore I go any further, I can't stress enough that this game is in Pre-Alpha and it has been since August 15th, 2014 so it's in no way complete. What might be relevant now may or may not be relevant still as the game goes through additional iterations and anything is subject to change.\nEpic Games went for a different strategy and decided to heavily involve the community by being crowdsourced, which means thee source code is available on GitHub and anyone can contribute towards the development of the game. I think this is really cool because it's basically saying that they want to make sure they're making the game the way the audience wants it, which is great! It shows that they care!\nSince it's in Pre-Alpha, you'd expect that there wouldn't be much available\u2026well it actually has a good amount of content currently available to play\/test! There are a few game modes and numerous maps to test your mettle, although most of the maps aren't completely finished, you don't really notice it once you're engrossed in the intense gameplay that Unreal Tournament provides. There are a couple of maps that seem polished and they look fantastic, especially the desert map. I absolutely love it!\nIn its current state, Unreal Tournament has three game modes available on Multiplayer and 4 on Single-Player, which is good for a game in such early development. However these aren't exactly complex modes as they're quite common among the FPS genre. I'm really excited to see to see more game modes added.\nAs I said earlier, these are some pretty basic game modes, but I never seem to tire of them when it comes to UT because the gameplay itself is just pure fun and chaos. What starts off as playing for thirty minutes may very well end as a good couple of hours! Even if the game doesn't have a multiplayer progression system, I would play it all day.\nThe game also comes with its very own Training section and while it teaches you the basics of weapon combat and movement, there are some advanced techniques to be learned. It's completely different however, when it comes to actual combat and fighting against highly skilled bots\/players, you need to step up your game! Adding tutorial sections is great, especially for people who have never touched an Unreal Tournament game before and they're entirely optional. Some games these days don't let you start the actual game until you've completed their bone dry tutorials! With Unreal Tournament you can jump straight into the action!\nAlthough it plays almost identical to previous Unreal Tournament games, the biggest difference is how movement feels and it's a good thing. Jumping is still frantic but somewhat noticeably slower but not so much that it's like comparing gravity on Moon to the Earth. This is a good thing because your eyes aren't constantly trying to keep with with over-erratic movement on multiplayer, and even bots aren't as insane on their highest difficulty level compared to previous UT games.\nThere is one thing however that makes up for this, and it's their near perfect use of the Link Gun\u2026the only gun they go for. It's almost as if the difficulty modes should be called \"Link Gun League\" or something like that! Even if you last a good 2 seconds up against their extreme pin-point accuracy, you can still manage to get in a lucky shot or if you're really skilled enough, a well deserved kill. It would probably be a good idea to nerf the Link Gun slightly or their accuracy with it on the highest difficulty setting.\nTalking about difficulty modes, if you're familiar with UT games you'll know that there are\u2026a lot of them, a whopping eight to be exact! In order of lowest to highest difficulty they are as follows; Novice, Average, Experienced, Skilled, Adept, Masterful, Inhuman and Godlike. It's almost as if there's a difficulty for any player regardless of their skill level, and that's what is so brilliant about it! For me it acted as a progression system where I'd slowly go up one if I improved, which was great fun! Balancing these individually is also far easier than doing so with two or three skill levels. which brings me to the UT Pre-Alpha. Currently there are only three skill levels, 1, 2 and 3 Star. I'm not too sure if these are placeholders but as it stands I think the older system is better because of the variety of skill levels and it accommodates a wider range of players. 1 and 2 Star levels aren't really all that challenging in my experience, but when I set it to 3 Star, it's like playing on Inhuman. My point is that there seems to be no challenge but suddenly it ramps up.\nThe game comes with a level system where you gain experience from multiplayer matches, but as it stands there isn't anything that you get for leveling up, so I assume that something will be added later on in development. What would be great is to unlock cosmetic rewards for gaining levels, for example classic weapon models or iconic character models, just something that doesn't impact gameplay so much that you feel obligated to do it and your ability to play at higher ranks wouldn't be hindered if you don't have access to it.\nWhat about our precious iconic weapons?\nWell so far on the Pre-Alpha, the game has retained it's iconic weapons such as the Flak Cannon, Rocket Launcher, Shock Rifle, Enforcer, Translocator, Link Gun and the other 1-9 weapons that span across your keyboard. The awesome thing is that they've received some updates to their art, some having more changed than others. But as always I love each and every one of them\u2026maybe except the Sniper Rifle design because it just looks plain boring. Now maybe you're asking \"well where's the Redeemer?!\", but you shouldn't worry because it's already in the game\u2026but it's half finished. The explosion is clearly not ready and you'll see what I mean when you see a big rocket explode early on in the gameplay video! I can't wait to see what they cook up for the final visual for the Redeemer! What I hope they'll bring back are additional weapons like the Avril, Spider Launcher and Lightning Gun because these were just so damn fun! Laying spider traps and having them jump at the target only to have them explode is something you can't pass up (I may have killed myself in the process).\nThis IS the game you're looking for!\nAfter playing hours and hours of the Pre-Alpha, which is something I usually wouldn't do for any game, I just can't help but feel excited for when it's finally released as a finished game. The levels are well designed and put together, the weapons are as epic as they've always been and the gameplay is just as compelling as a game can get. I'd absolutely recommend that you at least try out the game, especially for you FPS-aholics. Just try not to be bothered by the W.I.P maps!\nNow you know what I think, let me know what you think of the game! But as always, you can catch the video at our YouTube channel if you so wish, along with other gameplay videos from other games! Hope you enjoy and have a great day!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaktnu b/data_all_eng_slimpj/shuffled/split2/finalzzzaktnu
new file mode 100644
index 0000000000000000000000000000000000000000..b91bb50095b04a6635de945c31fc779be2984b4a
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"In a world where kindness toward our neighbors is severely lacking, Fred Rogers \u2014 Mister Rogers' Neighborhood \u2014 is a reminder that everyone deserves to be loved.\nWon't You Be My Neighbor? is a documentary recently released about the life of Rogers. The film chronicles his life and work from humble beginnings to faithful endings.\nRogers believed his calling in life was to serve God through the medium of television. At one point during the film, the audience learns that when Rogers sought ordination with the United Presbyterian Church, he told the denomination that his mission field would be serving children \u2014 through educational TV.\nWe learn a lot about Rogers through his TV show, Mister Rogers' Neighborhood. It quickly becomes evident that the show not only benefited a generation of children, but a generation of adults as well \u2014 Rogers included.\nRogers states that as a child, he was overweight and picked on. To add insult to injury, he didn't feel the freedom to express emotions in his family (in particular, anger). As a result, he felt that one of the most important things he could do for children was to teach them how to understand their emotions. Rogers believed that if you understood your emotions, you could understand how to love yourself and love others.\nThere were a lot of really interesting themes that Christians could take note of. Perhaps the most profound idea in the movie was Rogers' commitment to biblical integration. Without fail, his theology was woven deeply into the DNA of his show. The message of every episode was the same \u2014 people have intrinsic value, and they are worthy of love because of that. That's what it means to be made in God's image. It means you are valuable for no other reason than the fact that you are created in God's image.\nIn a day and age where people yell at each other, call each other names, and attack each other's character, Rogers is a breath of fresh air. He reminds us that we should never forget that even the \"worst\" among us are worthy of our dignity and respect for no other reason than the fact that they are God's image bearers.\nAs I sit and reflect on Rogers message, I can't help but think that this is the message the world needs today. We need to remember that above all, people deeply matter.\nIt's a really simple concept, but if you're not intentionally seeking to live it out, you're not going to remember that people matter. You'll get callous and burned by the world, and you'll forget that Christians are called to love other people \u2014 especially those who disagree with us.\nRogers never forgot this because he knew that deep inside each and every one of us, there was a childlike innocence. Rogers knew that the reason Jesus invited little children to come to him was because they were a true picture of faith. So in order to influence a generation of people and remind them that love matters, Rogers centered his life ministry on children.\nBecause there's something simply beautiful about the childlike faith of an adolescent and the way they love people. They don't hold grudges against others. They aren't racist toward others. They don't judge others. They just love people.\nIt reminds me of my childhood, when my father made us listen to a lot of oldies.\nBy the time I was in high school, I'd heard enough of them to last a lifetime. So at seventeen, I stopped listening to oldies.\nAlthough I still have a healthy appreciation for oldies, I just don't listen to them anymore. But every now and then, I'll find myself humming one of the classics. And when that happens, it takes me back to my childhood, when I thought much differently about the world.\nMaybe I'll start listening to oldies again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the early 1900's silent films amazed audiences with images, later talkies impressed with sound, today we have 3D. As technology continues to evolve so too will film genres. Genres, while having some shared characteristics, also differ in terms of stylistic devices used. For instance, the dramatic film \"The Notebook\" effectively uses color to reinforce theme and has plausible performers as the two main protagonists. \"The Notebook\" directed by Nick Cassavetes in 2004 tells the story of a couple's fifty year long love affair and its trials and tribulations.\nThe film begins in a nursing home where an old man (Noah) reads a book to an old lady (Allie) suffering from Alzheimer's. Noah, a poor country boy, and Allie, a rich city girl, meet in Noah's village in summertime and fall in love with each other. Allie's mother forbids the relationship and takes Allie away to New York, where a few years later she gets engaged to Lon, an educated rich man. Before the wedding she reads a newspaper and sees Noah in front of a house he once promised her to restore.\nShe goes there and soon they find they still love each other. So Allie decides to dump Lon and be with Noah. The movie ends as Allie realizes that the story Noah has been reading to her is in fact the story of their romance. They fall asleep holding hands and die peacefully. To make their creations unique film crews use different techniques. Techniques vary from setting to point of view, and also include editing, color, sound, costume, camera angle, actors' play and others. Film makers use them in different ways for different purposes.\nFor instance, each genre demands a particular type of performance: in comedies the performance should be exaggerated, while in dramas it has to be very realistic. Colors influence the way we perceive things and this is the reason directors usually use dark and vague colors in scenes of conflict, and bright ones to express happiness. Therefore film makers pay much attention to techniques . Actors in \"The Notebook\" fit their roles very well and they perform plausibly. Ryan Gosling (Noah) suits his role of a young, careless lad and Rachel McAdams (Allie) is perfect for the role of an educated cheerful beauty.\nTheir facial expressions, gestures and movements make audiences empathize with the protagonists. In the scene of their first quarrel Rachel McAdams cries, sobs, portrays despair and pleads to stay ,as realistically as if she feels her character's emotions. In scenes of happy moments she smiles and laughs sincerely, like when she listens to Noah (Ryan Gosling) sitting next to her. Ryan Gosling also demonstrated his acting talents in the movie. His character is very romantic and faithful and the actor expressed these traits very well.\nFor instance, in the scene he first meets her he looks at her with loving eyes, his eyebrows are raised a bit as he cannot stop admiring her. When other women try to talk to him, he lets them see their attempts are futile via tone changes and hands crossed on his chest, which express his love and faithfulness. Such a realistic actors' performance reinforces the dramatic plot. Warm and pastel colors predominate in \"The Notebook\". The creators intended to reinforce the theme also by means of colors.\nEpisodes, in which Allie and Noah are together, have more white, beige, pink, light blue and green. For instance, Noah painted the house he restored in white color; the sky is clear and blue when they are together. The white color of the house represents purity of Noah's love and his devotion to Allie, and the clear sky symbolizes peace. However, in those scenes where they part or have quarrels, colors are darker and vague. The two main quarrels, for example, happen at night and during the heavy grey rain. The protagonists' clothes also change in different situations.\nFor their dates Allie wears bright and pastel color dresses, but while being parted they both wear grey and boring clothes. The film crew used specific colors to support the theme. Specific tools play a great role in film making. Color and performance set the tone in the dramatic film \"The Notebook\" and make it a good quality movie. Good quality movies are called so due to details, and large film crews work hard to achieve this result. Other films by Nick Cassavetes are also very good, although he focused on different tools in them.\nThe Notebook begins at a nursing home where an elderly man called \"Duke\" reads a love story that is written in a notebook to an elderly woman who is also a patient. The story is set in Seabrook Island, South Carolina where a local boy by the name of Noah Calhoun falls in love with a young heiress named Allie Hamilton. In order to get her to go on a date with him, Noah climbs up a Ferris wheel at the local carnival and threatens to jump off unless she agrees to go out with him. Their first date is at the movies and ends with the two dancing in the street. The end of their date leads into a beautiful summer love affair where the two fall deeply in love with each other. Allies parents are dead set against the two young people being together; this is mainly due to Noah's status (being poor) in the world. Eventually the two break-up and Allie returns home to Charleston with her parents. Even though the two are apart they are still very much in love with each other. Noah writes a letter to Allie every day for one year; unfortunately Allies mother intercepts the letters and does not give them to her daughter.\nSince neither one hears from the other they decide that they have no choice but to move on. Noah goes on to fight in World War II, while Allie goes on to college and then volunteers as a nurse who cares for wounded soldiers. During her time as a volunteer Allie meets Lon; a young lawyer who comes from old money. The two date for a while and then one night while out with her parents Lon proposes to Allie and she accepts. Even though she is engaged to someone else Allie still thinks about Noah. In the meantime Noah returns home from the war and sees Allie eating at a restaurant with Lon and catches the two of them kissing. Noah decides to go back home and restore the plantations house, thinking that if he can finish it then Allie will eventually come back to him. One day while searching for her wedding dress Allie sees a picture of Noah and the finished house in the paper. Allie decides that she must go back and see Noah. Upon her arrival the two talk for a while and then eat dinner. Later the next day he takes Allie for a boat ride out on the river and shows her the swans. On the way back it is storming and Allie finally asks Noah why he never wrote to her.\nHere she finds out that Noah did in fact write to her and after arguing for a while the two ends up making love. The next day Allies mother shows up and gives her the letters that Noah had written all those years ago and tells her that she needs to make a choice. Eventually Allie chooses Noah and she breaks off her engagement with Lon. In the present we find that \"Duke\" is actually Noah and the elderly woman is Allie. The two have been married happily all of these years. The reason that Noah has been reading this notebook love story to her is because Allie has dementia and at times his retelling her the story helps her to remember. At the very end of the movie Allie asks Noah if he thinks that their love is strong enough to take them away together, meaning die together. He believes that it is and the two lie down on the bed wrapped in each other's arms and die peacefully. I found this movie to be very touching, sad, and happy all at the same time. The way the two loved each other so fully and completely through everything is something that I believe is clearly lacking in today's world.\nThrough most of the movie we find Allie holding her emotions in, fearful of hurting those that she cares about. We can see this when she finally concedes to her parents' wishes and moves back to Charleston. We also see this when she accepts Lon's proposal, not just because she cares and loves him, but because he is what her parents would have chosen (and approved of) for her. Another major place where we see this is towards the end when she is trying to make her decision on whom to be with. In each instance we find her extremely fearful of expressing her emotions. Another issue that we see through the movie is inadequate interpersonal skills. This is one of the hardest communication obstacles to overcome. This is because many people simple do not know how to express their emotions. While this can be seen with all the arguing that occurs between Noah and Allie; we also see it more subtly with Allie and her mother. This is because Allie and her mother never simply sit down and talk to each other about the issue of the relationship between Noah and Allie. Instead of explaining to Allie from the beginning how she felt and why she felt that way about the relationship she instead decided to yell and hide the letters that Noah had written.\nHad the two learned to communicate better from the beginning it would have saved a lot of heartache. Through this movie we can also see the Socialization Theory at work. This theory talks about the different ways that men and women are taught to express their emotions. During this movie we see Allie's mother hold her emotions in when it comes to anything other than he daughter's actions. Anne is what would be considered the perfect southern woman. She does not express any negative emotions in public setting; instead she keeps them to herself. We can also see this with her father and how he ever so eloquently addresses Noah and his unhappiness with him seeing his daughter. In both respects Allies parents act emotionally how the upper class of southern society would expected to. This movie shows every aspect of a relationship, from the good to the bad. It shows that no relationship is perfect; they all have their ups and there downs. While many different elements of emotional communication can be seen, for me the concepts of fear, inadequate interpersonal skills and the socialization theory are the most appropriate. I believe that this movie can teach all people something about relationships.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Supporting women to step into their power, master their relationship with money, and live a freedom-filled life they love.\nMuch of what you think and feel, and how you act with money is unconscious, meaning that quite often we're simply not even aware of how we're showing up with money.\nWithout awareness of your programming around money, you are powerless to transform your relationship with it. How can you clear blocks that you're not conscious of?\nHow can you identify limiting beliefs or programming if you don't know what they are?\nIf you are looking to create a healthy relationship with money that supports you in wealth and wellbeing, there are a number of ways we can work together.\nWhen life is in flow, you're happy, free and working towards something you truly value. The same can happen with your finances.\nHow do you define success? What do you truly value? Is your spending intentional and in alignment with what you truly value?\n\"My way forward gets clearer each time I talk to Lea and my understanding of my past helps me plan\/change my future.\nOne to do things differently.\nFreedom seeker - global nomad + traveller, having lived in NZ, Australia, Japan, UK and now US. And a particular passion for and resonance with Bali and India.\nYoga teacher, mindfulness practitioner, wellness coach, creative entrepreneur, professional speaker and writer, former award winning Financial Adviser, and now Certified Money Coach (CMC).\nto create wealth & abundance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Toby Tiger Organic Red\/White stripe Sleepsuit \/ All in One, with Bunny applique on front. Superbly soft and comfortable for your little one to sleep in or play. We love Toby Tiger's trademark bold, bright coloured clothes with fun appliques\/patterns. They are cool enough to stand out from the crowd yet affordable for every day. Be prepared for your little one to receive many compliments!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published by Editor on February 26, 2013.\nState Treasurer David H. Lillard, Jr. is recommending that the General Assembly adopt a number of reforms to the state's retirement plan for public employees. The Tennessee Consolidated Retirement System (TCRS) is a well-funded pension plan. The proposed changes will only apply to state employees, higher education employees, and teachers hired on or after July 1, 2014 and will ensure the continued financial strength of the TCRS. The bill will not affect the retirement benefits for current state employees, higher education employees, or teachers.\nBecause pension plans carry costs and liabilities that stretch over many decades, Treasurer Lillard said it is important to look well into the future when evaluating a plan's financial health.\nTreasurer Lillard presented the General Assembly's Council on Pensions and Insurance with proposed legislation Monday that, if adopted, would implement his recommendations. He is scheduled to have a more detailed discussion with council members during a follow-up meeting March 4.\nThe changes he is proposing would limit the state's future liability for pension costs by creating a hybrid between a defined benefit plan, which guarantees specific payouts to retirees based on a formula, and a defined contribution plan, in which the state would provide fixed amounts of money to employees' 401(k) or similar types of accounts.\nThe reforms would also reduce pension costs by adjusting the formula used to calculate retirement benefits, raising eligibility requirements and collecting employee payroll deductions. The new plan will also have cost controls that will apply only if the plan's total costs exceed certain thresholds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Laura was telling me that pudding is the perfect thing to make with a 2 year old. All that is required is to open the package, pour in milk, stir and wait 5 minutes for delicious pudding. So the other day we tried it and she was totally right! He could do everything required and loooooved \"helping Mommy\".\nWhat he did not love? The actual pudding. And I have to say I prefer the pre-packaged myself. I feel that does not say good things for my desire for artificial taste... Zachary would just prefer cheese and\/or cheez-its, naturally.\nBut we will make it again, perhaps trying different flavors. Pistacho or strawberry could be delicious, no? It was the first thing that he has really enjoyed making and that I enjoyed making with him. Plus none of it ended up on the floor. Usually at the end of \"helping Mommy\" Mommy is very stressed by the flour\/sugar\/cheese all over the floor that someone wants to \"help\" sweep up too.\nHave you heard I have a 2 year old?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last Monday, I all but told you I'd been away somewhere. And this post's title rather gives the game away, wouldn't you say? New York City. So good they named it twice. The Big Apple. One of my favourite cities in the world (those last three words have more impact if you say them aloud in the style of Clarkson). I went for three days. On my own.\nYou might be wondering why I did that (or maybe you couldn't care less and just wish I'd tell you what I want to tell you). Put simply, it came down to two things at the end of 2017. Over the Christmas period, I was feeling extremely put upon by how much time I was spending in the kitchen, juggling pots, pans and baking trays. There was a lot of muttering about how I'd only be missed when meals failed to materialise and I've a good mind to go off on my own for a few days and let them get on with it. The very next day, the stars moved into alignment and I received an email from an established British airline informing me that I had enough miles for a return flight to New York. In Premium Economy.\nA consultation of schedules. A firm commitment from The Brainy One to hold the fort at home. A non-refundable booking. A firm commitment to nourish myself in 2018. And that, my friends, is how I came to go to New York City.\nThat Mother Nature did her best to thwart my plans is a whole other story.\nGood for you! It sounded like it was just what you needed to have some 'me time' a long way from home doing something that you wanted to do and not have to compromise any activity. It ought to be compulsory for all mums! I hope that after a few days fending for themselves they both realised how much you do for them, although I would put money on the fact that you probably left full a fridge and freezer, a week's worth of ironed clothes and an empty washing basket!\nGood for you getting away & glad that Mother Nature's plans did not upset yours. Yes I can well hear Jeremy saying \" in the world\" although we know he would never utter the beginning of your sentence (lol). I am sure you were missed by those at home & well I know those feelings of, does anyone notice what I do around here! Cannot wait to see the T\/N pages & scrap pages. Glad you are back safe & sound & perhaps refreshed.\nI'm so glad you were able to have this time away!\nSuch a great idea! Glad you took the opportunity to go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is good to consider it important to have a template for their business in that one will be aware of what to do and what not to do when one is having their business one may worry on how to keep their business graph raising in terms of profit and input. Below are the benefits that one will gain in having the business proposal template and therefore one will see the essence of having it for their businesss.\nIf one uses the business proposal template they will be in a position to follow their strategies that one has set for their business to go up well in that despite the daily interruptions that may come your way one will be in a position to plan and achieve what they planned for that specific day and that way one will be able to achieve their goal. To have a strategy for your business in the proposal template for one to be in a position to include it in their business so that the graph can be of a straight line up hence it is good.\nWhen it comes to deciding on what to do at a given time as it reads in the business template for proposal when one is using the business proposal template one will be in a position to set clearly the objectives that they want to achieve after some given time in that one will not stay idle at some time because that are occupied. Therefore, if one wants their curve for the business to keep on constantly increasing it is good for one to consider it important to have the business proposal template that will be of great help to that person.\nIt will be inclusive of the priorities that one wishes to achieve in a given period of time in that one will not have to waste money in doing things that do not help them at all instead one will major on what matters most and that which is of priority as compared to others hence it is good for one to have a business proposal template. Therefore, it is good for one to consider it important to have priorities in their business proposal template in order to be in a position to do what matters most to their business.\nSomethings will be sold independent of others while others will depend on others by having a business proposal template for your business one will be in a position to keep track of the business progress in a manner that will enable one know what independency means in the business. Therefore, if one wants to go far with the business it is therefore important for them to have the business proposal template for their guide in all what they do for the business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Furnished with beautifully lighted hangings and seating arrangements that present these venues as the finest meeting halls in Bangalore. The sophistication oozes from each space adorned at our event rooms. Sit back and get into the mesmerising charm that our professional event organisers showcase for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Headed to French Polynesia and looking for things to do in Bora Bora?! You're in the right place!\nHowever, we did make it to French Polynesia a year later (2nd honeymoon, anyone?), thanks to a massive amount of both airline and hotel points.\nEveryone speculates that there's not really much to do on the islands, and personally, I have to disagree. Dubbed \"Bora Boring\" by many, this slice of paradise had more than enough to keep this adventurous beach bum happy. I was able to fill 5 full days of fun, with a little relaxation time as well. There are lots of adventurous things to do in Bora Bora \u2013 trust me!\nRead below for some of my recommended activities and things to do in Bora Bora.\nSo where is Bora Bora anyway?!\nBora Bora is an island in the Leeward group of the Society Islands of French Polynesia in the Pacific Ocean. Located about half-way between South America and Australia, Bora Bora is an 8 hour flight from Los Angelos and is approximately 2,600 miles south of Hawaii. Yes, it's far, but yes, definitely worth it.\nBora Bora was discovered in 1722, and is arguably the most beautiful island on the planet. I can say from experience that it will definitely stay in my top 5 no matter how many other ridiculously gorgeous islands I am fortunate to visit.\nSnorkeling in Bora Bora is top notch. The water is crystal clear, with temperatures in the high 70's\/low 80's (depending on time of year). Make sure to go on a shark and ray tour, where you get to swim with both black-tipped reef sharks and sting rays >> neither dangerous to humans! If staying in an over-water bungalow, make sure to bring your own snorkel gear, as you can jump into the lagoon and explore right there!\nBeing one of the top things to do in Bora Bora, there are numerous companies offering cruises around the islands, and you can decide whether you'd like to go for snorkeling or sunset. Both sounded like fantastic choices, but we opted for the snorkeling cruises since I have a slight obsession with swimming with the fish! It's true \u2013 the water really is that clear! And turquoise!\nCraving a \"higher adrenaline\" kind of day? Tour Bora Bora via Jet Ski, and and stop for a coconut show on a Motu. Try doing some donuts with the jet ski >> you'll surely impress everyone! Snorkeling is even included on many tours! We ended up on a private jet ski tour offered by our hotel, and it was much more difficult than I ever speculated! Thankfully I had my husband to hang on to, but even then I thought we would tip over with every ripple in the water.\nStand-Up Paddling is great exercise, and much harder than it initially looks! With it's calm waters, Bora Bora is the perfect place to practice! You can even kneel down or sit on the board if you get tired. Unfortunately, we left this activity until the last day, and with our luck, it was the only morning the waters were choppy. If this is high on your list, make sure to cross it off the first or second chance you get.\nThe sun setting below the over-water bungalows were really something else. My suggestion: grab a drink at your hotel a half hour before sunset, lay on a chaise lounge on the beach (or your balcony!), and just relax! Watch the sun dip below the horizon and you'll be making it a new tradition.\nCompanies offer 15-30 minute helicopter rides around the island. See Bora Bora from above, and witness all those shades of blue, green, and turqouise as shown below. Isn't it spectacular?! It's rather pricey, but totally worth it for this one-in-a-lifetime experience. I recently took a helicopter ride in Hawaii and it was the highlight of our trip. Just go for it! I know I would! Want to see these stunning views without the high price tag? Keep your eyes peeled on the ocean during your initial flight into the island (take-off's pretty amazing, too).\nThe Lagoonarium is an outdoor aquarium that specializes in displaying the fish and marine life of a lagoon. It offers up close interaction with sharks and rays and even a chance to be pulled along by the fin of a shark if you're interested (not my favorite thing as I'm sure the animals aren't crazy about this)! Many marine species can be found here- some that you are not able to see while snorkeling in the open water. If you come here, make sure to talk a walk around the island as we were treated to even more fantastic views!\nAfter swimming with sharks in Belize, I knew this was something that I needed to do in Bora Bora! Unlike the Nurse Sharks that you find in Belize, you can swim with black tipped reef sharks and lemon sharks in French Polynesia! We swam with black tip reef sharks in both waist-deep water, as well water that was 40m deep! That deep water was the cleanest, clearest, and most outstanding I've ever seen (and I've been lucky to snorkel in some pretty spectacular places \u2013 Belize, Hawaii, Mexico, all over the Caribbean, and the Greek Islands for staters). What a remarkable experience. One of the most thrilling things to do in Bora Bora!\nWater activities and lounging at the pool is nice (uh, yes please!), but the interior of the island should be checked out as well! We took a half-day tour of the island via jeep, and not only saw some of the most fantastic views, but learned how taros are made! We even got to try some on! If you're looking for some adventure and have had enough of the beach and pool (at least for a little while), check off one of my favorite things to do in Bora Bora \u2013 the Jeep tour!\nMost high-end resorts have kayaks and paddle boats for rent (many for free!) Wouldn't it be amazing to explore these crystal-clear waters yourself, and be able to hop in for a swim or snorkel whenever you please?! I mean just look at that water! We spent an afternoon traversing our resort via paddle boat, and got some of the best views, like those in the photo below!\nOkay, so this one isn't exactly \"adventurous\". BUT, it's an absolute must-do when visiting Bora Bora! All of the upscale resorts have the most beautiful pools with an even better view, and some even boast over-water-bungalows with private plunge pools!\nThe Le Meridian has a unique care center for turtles \u2013 where you can observe these beautiful animals and eventually you can swim together. After swimming with turtles in Hawaii, and ultimately falling in love with them, this is definitely something I would absolutely love to do! We didn't make it to the turtle sanctuary on this trip, but oh Bora Bora, I'll be back for ya. A definite must-do on any list of things to do in Bora Bora.\nNEW POST! Check it out!\nUPDATE: Check out my Bora Bora Packing List here! Filled with things you wouldn't think about bringing along!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This fall planted perennial bulbs are scapose, having a single central leafless hollow flower stem. Several green or blue-green, narrow, strap-shaped leaves arise from the bulb. The plant stem usually bears a solitary flower, but occasionally a cluster of flowers (umbel). Flowers may be single or double, grow singly on a stem or with multiple flowers per stem, and height varies from 6 to 20 inches.\nThis trumpet shape contains the stamens and is the most conspicuous feature of the flower. The popularity of daffodil has resulted in the production of many varieties differing from the yellow parent form mainly in colour; the trumpet and petals may themselves be of contrasting yellow, white, pink, or orange.\nThe general rule for planting is to cover the bulb with soil to 3 times its vertical diameter. Keep planting depth at 6 to 8 inches and planting spacing should be 4 to 6 inches.\nSet them upright at the bottom of the hole, tops up (pointed side up), and space properly. After planting water thoroughly to settle the soil and to encourage the start of root growth. After they bloom in the spring, allow the plants to grow until they die off. They need time after blooming to store energy in the bulbs for next year.\nSoil: They are not fussy about soil as long as it is loose and well drained. To improve drainage in heavy soils, deeply dig in plenty of organic matter prior to planting.\nWater: Water newly planted bulbs thoroughly. In many regions, fall and winter are wet or snowy enough to provide moisture. Plants do not need summer moisture.\nFertilizer: Daffodils do not require heavy fertilization. When preparing the beds, use 2 to 3 pounds of a complete garden fertilizer such as a 6-24-24 or other fertilizer with a 1:4:4 or 1:3:3 N-P-K ratio.\nKeep plantings well watered if precipitation fails; continue until the foliage begins to yellow.\nTo remove the dead plants, either snip them off at the base or twist the leaves while pulling lightly.\nOnce daffodils and tulips have gone by, add bonemeal to the soil for next year s blooms.\nOrnamental use: Daffodils bring cheer to the spring garden with abundant flowers in hues of yellow, white, pink, and salmon. Varieties are available in a range of sizes and forms.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Lewis Center for the Arts' Program in Theater will present a production of Marina Carr's 2006 drama, Woman and Scarecrow, in which a dying, contemplative woman reflects upon her life from her death bed and wonders what could have been. Performances are on March 8, 9, 13, 14, and 15 at 8:00 p.m. at the Marie and Edward Matthews '53 Acting Studio at 185 Nassau Street. The production is directed by Tim Vasen, the Director of the Program in Theater, and features senior theater certificate students Carolyn Vasko in the role of Woman and CC Kellogg as Scarecrow.\nWoman and Scarecrow focuses on abedriddenwoman\u2014mother to eight children and wife to a hopeless husband\u2014as she hovers between life and death. In her reflecting and dying the woman is accompanied at her sick bed by the Scarecrow, a figure representative of Gaelic mythology whose true identity and intentions remain a mystery. In some moments Scarecrow seems a friend, a source of comfort, or even a guardian angel; in others, the Scarecrow seems a judge, a source of condemnation or perhaps a morphine-induced alter-ego. In the process of death, the woman's interactions with the enigmatic Scarecrow, her stern Catholic aunt, and detached, cheating husband illuminate the regrets, fears, joys and passions of Woman's brief existence.\nStudents in the Program in Theater complete a senior thesis project in fulfillment of the requirements for a certificate, similar to a minor, and which is in addition to their major area of concentration.\nVasko is majoring in History and Kellogg is concentrating in English, and both knew early on that they desired to stage a work by Marina Carr as their thesis project. Kellogg's knowledge of the Irish playwright began with a course she took during her sophomore year on Contemporary Irish Drama, taught by faculty member Michael Cadden. Her growing interest led her to enroll in The Phaedra Project, an interdisciplinary theater-making course offered in spring 2011 focused on the development of Carr's play Phaedra Backwards in preparation for its world premiere at McCarter Theatre that fall. Vasko notes she was first captivated by the works of Carr in the summer of 2011 when she participated in a Princeton Global Seminar course in Ireland led by Theater faculty Stacy Wolf and Jill Dolan. The seniors consulted many of the faculty from the Program in Theater as they considered which of Carr's dramas they would tackle for their thesis project.\nA second goal Vasko and Kellogg share in their vision for this project is one of collaboration. From the beginning, they set out to keep the different elements of production largely an in-house project, inviting their fellow students at the Lewis Center to be a part of the production. For example, the seniors hand-picked the two other student actors for the cast, Maeve Brady '15 and Jake Robertson '15, colleagues whose work they had come to know well. By utilizing resources within the Lewis Center for sets, lighting and costumes, Vasko and Kellogg are also able to try their own hands at designing elements of the production and spaces. In addition, they recruited two fellow senior Visual Arts students, Polly Korbel and Leana Hirschfeld-Kroen, to create original artwork for the production's publicity materials.\nWoman and Scarecrow premiered in June 2006 at The Royal Court Theatre in London with two of Ireland's leading actresses, Fiona Shaw and Brid Brennan, portraying the roles of Woman and Scarecrow, respectively. Since then the play has been produced at numerous theaters in England, across Ireland, and throughout the U.S.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What do we pray for in November?\nDuring November the Church prays in a special way for the dead. It is a time for us to pause and think of the reality that faces us all. The Christian understanding of death is full of hope. Where Christ is now, we hope to follow. The Church constantly preaches the Resurrection of the Body. But what does death mean?\nThe following is a brief reflection on death from the Second Vatican Council's document \"The Church in the Modern World\".\n\"It is in the face of death that the riddle a human existence grows most acute. Not only is man tormented by pain and by the advancing deterioration of his body, but even more so by a dread of perpetual extinction. He rightly follows the intuition of his heart when he abhors and repudiates the utter ruin and total disappearance of his own person. He rebels against death because he bears in himself an eternal seed which cannot be reduced to sheer matter. All the endeavors of technology, though useful in the extreme, cannot calm his anxiety; for prolongation of biological life is unable to satisfy that desire for higher life which is inescapably lodged in his breast.\nAlthough the mystery of death utterly beggars the imagination, the Church has been taught by divine revelation and firmly teaches that man has been created by God for a blissful purpose beyond the reach of earthly misery. In addition, that bodily death from which man would have been immune had he not sinned (14) will be vanquished, according to the Christian faith, when man who was ruined by his own doing is restored to wholeness by an almighty and merciful Savior. For God has called man and still calls him so that with his entire being he might be joined to Him in an endless sharing of a divine life beyond all corruption. Christ won this victory when He rose to life, for by His death He freed man from death. Hence to every thoughtful man a solidly established faith provides the answer to his anxiety about what the future holds for him. At the same time faith gives him the power to be united in Christ with his loved ones who have already been snatched away by death; faith arouses the hope that they have found true life with God.\"\nVatican II, Gaudium et Spes, 7th December 1965.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to Clean Your Big Fat Nut (milk) Sacks...it's easy!\nIt's always important to make sure that you clean your nut milk sacks after each use so that they'll be ready for the next time you want to use them!\nWhen we say that My Big Fat Nut (milk) Sacks are easy to clean, we mean it.\nJust invert the nut milk bag inside out and run some high pressure water through it. Any bits and pieces that are stuck to the sacks will rinse away easily. If you'd like, you can also use a drop of mild dish soap or detergent while washing to keep them extra fresh.\nStrained some beet juice or other extra stainy veggie or fruit juice? No worries! You can soak your sacks with a bit of hydrogen peroxide and water to get those stains out. For those that like a natural alternative, soaking your sacks in grapefruit seed extract and water or water and white vinegar work great too!\nToo lazy to wash your big fat nut (milk) sacks yourself? Toss them in with your laundry on the delicate cycle or even place them on the top rack of your dishwasher! A quick (and we mean quick because they're made of nylon after all) dip in some boiling water works too.\nAfter your big fat nut (milk) sacks are nice and clean, just allow them to air dry. They dry super quick!\nAny other questions? Just let us know!\nCombine ground coffee beans and water in a large jar. Stir. Cover and let rest at room temperature for at least 6 hours.\nTo serve, mix one part coffee mixture concentrate with one part water.\nAdd ice, sweetener of choice and almond milk creamer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If your pets are sitting home alone while you are at work, the Doggie Daycare program at Country Spaw & Resort is the perfect solution!\nSimply drop off your pets on the way to work and pick them up on the way home. They will enjoy a wonderful day of socializing, fresh air and getting lots of attention rather than a day alone in your home.\nIf you're busy schedule doesn't leave much time for extra travel, a pick-up and delivery service is also available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Many of us use one daily and we have all seen some figures and have some idea of how much or how often cars are driven. For example, the average annual mileage in Germany is about 14,200 km per year. And we are all conscious of the general relation of our driving to issues like the CO2 footprint.\nBut what does the \"data footprint\" of a modern car look like? And how do we cope with the handling of that data \u2013 leaving the issue of data protection temporarily aside, and concentrating for the moment on the technical question of how to most efficiently handle the ensuing data volumes, both today and in the future.\nAre we ready for 50 Terabytes of data per car per day?\nDriving profiles, maintenance data, and state-of-the-art assistance functions lead to an average of 1 Gigabyte of data per second in each car.\nAlready, intelligent cars collect data on a widespread basis: Driving profiles, maintenance data, and state-of-the-art assistance functions lead to an average of 1 Gigabyte of data per second in each car. Most of that is driven by video-based systems which allow cars to drive on a partially autonomous basis, which suggests that this number will grow exponentially with the additional capabilities and complexity which are on the horizon. Looking into the automotive industry, there is talk of up to a factor of 20 for the future, leaving us with a total indication of potentially 50 Terabytes of data per car per day in the era of autonomous driving.\nGiven an average speed of 60 km\/h, the 14,200 km per year translates into 852 Terabytes (which is 0.85 Petabytes) per year. If that equates with an average of approx. 23 Terabytes per day, we are roughly in line with a factor of 21, ending up with the anticipated 50 Terabytes per day in the future. But this is just one (!) car.\nIn Germany alone, the current data volume from connected cars is up to 40 Zettabytes per year, and will rise 848 Zettabytes in future.\nWhat if we try to extrapolate these figures to the total number of cars in Germany today (numbering at 46.5 m)?\nMultiplied by the 46.5 million cars in Germany alone, we end up with a current data volume of up to 40 Zettabytes per year today, or 848 Zettabytes in future. That is approaching 0.85 Yottabytes \u2013 welcome to the next level of Big Data volume: A yottabyte is 10*24 Bytes (1,000,000,000,000,000,000,000,000 \u2013 try to get your head around that!).\nToo much heavy lifting and a few too many Bytes? Let's translate this differently: Modern cars are in themselves a type of data center \u2013 they carry a processing power of more than 100 CPU and need to validate data within milliseconds before they can send the results, not only to their internal driving systems, but also for communication purposes to other vehicles on the road.\nEven today, the largest test fleet of cars \u2013 from Tesla \u2013 has gathered \"merely\" about 1 billion kilometers of driving intelligence. The most recent death caused by an autonomous vehicle shows the need for further traffic observation by the car, which means even more data collection and quicker validation. And data transport.\nThe next wave of the Internet \u2013 the Edge.\nA small local \"data handling unit\" would need to be installed within a 15 kilometer range in order to get data transported and validated in the most efficient manner.\nWill 5G be the answer to this? No. If the amount of data explodes as indicated above, there will be a clear need for efficient data management systems, which are both cost- and network-efficient. A recent study concluded that a small local \"data handling unit (DHU)\" would need to be installed within a 15 kilometer range in order to get data transported and validated in the most efficient manner, including latency aspects for overall safety. That excludes traditional network and data center infrastructure and opens the door for the next wave of the Internet \u2013 the Edge.\nImagine a stand-alone unit, with full capabilities in terms of power, cooling, and connectivity. Portable, ready to rack, stack, and operate by plug & play into public power and network supplies. That is the sample use case for the next generation of small modular build-outs of data centers, combined with antenna technology for Car2Cell communication within short distances such as the indicated 15 kilometer radius. Whilst there are some manufacturers already in the field offering that sort of infrastructure, the next logical step is to deploy fully equipped DHUs on a pay-per-use basis. Currently that is possible up to a capacity of 3 Petabytes per unit with technology of innovIT360, for example (going back to the numbers \u2026).\nSo, in the near future, we are looking at deploying a network of modular DHUs for a multitude of data and enriching capacity per unit as quickly as possible. If we then can cope with car data traffic, the next level will be the remaining topics on German highways. What about truck logistics, digital restaurants, break areas, and so on? The move to the Edge on the run, IoT offers many more sample cases for the next issues of dotmagazine. Stay tuned.\nFrank J. Zachmann is CEO of innovIT360 AG, a privately owned German provider of consulting, implementation and service for innovative IT infrastructures. Prior to joining innovIT360 at the turn of the year 2016\/2017, Mr. Zachmann held various senior management positions with leading data center operators. Among other things, he reported directly to the CEO of the Interxion Group and previously spent seven years at Equinix, the world's largest provider of neutral data centers - most recently as Vice President Business Development EMEA.\nIn addition, Mr. Zachmann, chairman of the association DigitalHub FrankfurtRheinMain e.V. - a local cooperation partner of the association of the internet industry, eco - is actively involved in the interests of the digital economy and is an international economic ambassador for the city of Frankfurt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Harp Seal Lullaby by Christina Tourin - Gentle relaxing harp music inspired by the harp seal and the oceans in which they live - with accompanying flute, panpipes and the sounds of the seals and the sea.\nInspirations by Christina Tourin - Celtic-inspired original compositions inspired by the sacred island of Iona, from accomplished harpist Christina Tourin and backed by virtuoso musicians on flutes, guitar, keyboards and cello. Experience the magic of the tiny sacred historial 3-mile island off the western coast of Scotland through music!\nEchoes of Angels by Christina Tourin - An album of uplifting yet peaceful music containing both classical and original melodies, with an angelic choir hovering above the pure tones of a harp.\nAngel Airs by Christina Tourin \"...truly one of the most exciting Christmas harp recordings released in recent times. Christina's mastery of the harp and her creative ability to compose and arrange is indescribable!\" - J.D.\nDiamonds from Heaven by Christina Tourin - Magical solo harp medleys of 42 glorious hymns and songs of Spirit from many traditions and paths.\nEmerald Harp by Christina Tourin - An evocative musical journey from the Emerald Isle to the Isle of Avalon, a mix of traditional and ambient styles for harp, flutes, guitar, keyboards, fiddle, cello and bodhr\u00e1n.\nFolk Harp Music of Austria by Christina Tourin - Tyrolean music for harp, recorders and accordion, in collaboration with Rosemarie and Maria F. von Trapp.\nHarpbeats by Christina Tourin + guests - The rhythmic mastery of harpist Christina Tourin, backed by inspired syncopations of percussion, guitar, saxophone and flute.\nHeaven's Seven by Christina Tourin - Celestial harp music relating to the chakras and universal energy systems, for relaxation and healing sessions such as reiki, massage, energy balancing, aromatherapy, etc.\nIlluminations - Original harp compositions by Christina Tourin, inspired by the theme of the labyrinth.\nPrayer for the Living Trees - Harpist Christina Tourin and many guest musicians weave the magical properties of trees into music.\nStarlight by Christina Tourin - An acoustic album of relaxation music for harp, flute and cello, featuring original compositions and traditional melodies arranged by Christina Tourin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spain, Italy, or France, these are all destinations that fuel and stir all sorts of imaginations. Just imagine yourself visiting any of these amazing European cities floating on water, one destination after another, as you discover the rich and affluent architectural and cultural heritage that this Old Continent boasts.\nWhether you are after genuine relaxation or perhaps you want to sample some of the best cuisines in the world. Or probably you are looking for irresistible shopping deals, or simply want to quench your thirst for exploration; the following 5 European cruises are a must for a successful and eventful time of your life.\nMediterranean Cruises \u2013 from mouthwatering delicacies to exotic traditions and heritage, the Mediterranean is an enthralling destination filled with all sorts of attractions and excitement. This part of the world which dates back many centuries ago is delightfully diverse yet mysterious. From the modern culture of Italy to the religious significance of the Jordan, history is a common denominator on each destination that falls under this region.\nMediterranean cruises are either Eastern or Western, with the Eastern cruises visiting France, Spain and Italy, and sometimes even Portugal. The western Mediterranean cruises on the other hand will cover Naples, French Riviera, Lisbon, and Gibraltar.\nBlack Sea \u2013 some European cruises also cover the Black Sea. However, most tours will include most of the stops of an Eastern Mediterranean cruise, including Turkey, Greece, and Croatia. Notably, Black Sea cruises will stop in Bulgaria and Ukraine.\nNorthern European Cruises \u2013 these cruises cover different tours along the northern part of Europe. Popular destinations include the Scandinavia region, with much focus on the Norwegian fjords. These fjords are simply inlets off the Western coast of Norway. Northern European cruises in the Scandinavia also cover Oslo, Helsinki, and Copenhagen.\nWestern European Cruises \u2013 these cruises cover destinations such as Brussels, Amsterdam, Gibraltar, Dublin, and Barcelona and you get to experience the best that the western part of Europe has on offer.\nTransatlantic Cruises \u2013 these cruises generally depart from the eastern coast of the United States and will dock in any of the many western European destinations in France, UK, Italy, or Spain. The cruise will mostly include a stop at pre-determined mid-Atlantic Island, like the famous Bermuda which boasts to offer irresistible shopping experiences and deals.\nAlso notably are European cruises that will visit the famous British Isles, Wales, England, Ireland, and Scotland, and will mostly depart from London. General stops on the way include Belfast, Dublin, Loch Ness, and Edinburgh.\nIt is also worth mentioning that European cruises also offer amazing river cruises, which is even more amazing because most European Capitals are surrounded by rivers. River cruises in Europe travel most rivers like Rhine, Danube, and the Seine, and offers a great opportunity to explore different capital cities in a single trip.\nWith that being said, it is worth mentioning that European cruises vary significantly in length, and would range from one to three weeks based on different factors. It therefore makes sense to plan ahead to avoid inconveniences along the way.\nAvina is a contributor for the site Kefalonia where you can get interesting information on Kefalonia hotels as well as the famous Lassi resort.\nReally awesome post. I am planning Europe Travel and I will must visit all the Cruises you mentioned above.Really looks beautiful and must watch things.Thank you for sharing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Lamps - Contemporary Jenaro Teal Glass Table Lamp by Signature Design by Ashley at Simply Home by Lindy's in the Hickory, Morganton, Lenoir area. Product availability may vary. Contact us for the most current availability on this product.\nThe Lamps - Contemporary collection is a great option if you are looking for Contemporary Lighting in the Hickory, Morganton, Lenoir area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A great opportunity with a leading Multinational FMCG company that has an establish footprint in Malaysia. Overseeing one of the top brands withing the consumer space.\nOur client is a world leading MNC in manufacturing industry. Due to their recent rapid expansion plans, they are looking for a Senior IT Program Manager to lead the migration from JDE to SAP. They have a long history in UK and it's a great organization that supports and encourages career growth and excellence!\nOur client is a world renowned MNC in the IT industry. Due to the rapid expansion, they are looking for a Business Intelligence Lead specialising in Cognos BI and TM1 to stamp his\/her mark within the organisation. This is an exciting role for a change-driven Fintech professional, offering the possibility to work in a global, diverse and multicultural environment!\nAn exciting General Manager opportunity for a sportswear brand that is looking to expand and grow their footprint across South East Asia. This role is responsible for driving new market initiatives for their emerging markets expansion plan.\nTo manage the execution of contracts and vendor management processes and standards in order to optimise the performance of strategic partners against the contracted services and drive integration and innovation to deliver tangible business benefit.\nExcellent opportunity to work with the industry leader with a highly employee friendly organisation reporting directly to the Head of Business.\nEnsure monthly closing is done properly and timely and provides leadership guidance on Month End close process and support team on monthly financial reporting.\nThe successful candidate will be task to manage the branch operations and lead the team in accordance to company ethics and principles.\nThis role will be tasked to support customers located in the southern region of Malaysia.\nFinancial Planning & Analysis Manager role for a large financial institution.\nThe Junior Channel Manager is responsible for the performance of the retail channel of a specific portfolio.\nThe Channel Manager will need to have strong people management skills having to handle multiple stakeholders.\nWorking for a principle brand with strong regional presence.\nStrong working culture and good empowerment from management.\nTo manage the execution of software asset management processes in order to drive software compliance and asset optimisation to deliver tangible business benefit.\nThis individual will be tasked to develop the company's Key Accounts and build relationships with their key customers. The role involves having a good understanding of the customer's needs to move the business forward.\nThe Area Sales Manager is responsible for setting and implementing the strategy and direction for the respective distributors in the assigned area.\nThe successful applicant will be in-charge of the achievement of the sales targets and execution of promotional strategies. The ideal candidates must have a strong experience in medical devices industry.\nThe successful applicant will be tasked to provide strategic direction for the assigned area and must have strong experience in FMCG market.\nExecution of Food Service channel strategy to provide the best service to prospective customers and become a category leader.\nThe candidate will be responsible for the Group Sales and manage a large team within the Medical Device Portfolio. He \/She who represent the Company will need to ensure that team's activities are compliant to ethics and policies.\nThis candidate to have knowledge of hospital insights and future of healthcare landscape in Malaysia to develop the plans for the group in Malaysia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I can't help it \u2013 if I'm in London I always go to Dishoom! Dishoom is inspired by the old Irani cafes in Bombay, a faded elegance, and is where you can find delicious street-style Indian cuisine. There are locations throughout London and I've already been to the ones in Kings Cross and Soho. Never disappoints.\nFor our weekend in London we were so happy to be able to meet up and stay with our friends from the Tran-Siberian Express nearly one year later! You never know who you'll meet while you're exploring! We decided for our first brunch to go to Dishoom (you'll see in the coming posts that Koen and I only ate Indian food this trip).\nKensington is a posh area of London and you can really see this in their Dishoom. I really love that everything is so modern and art deco. I had to think of The Great Gatsby!\nAkuri: An Irani caf\u00e9 staple. 3 eggs, spiced, scrambled and piled up richly alongside plump, home-made buns and served with grilled tomato.\nBacon Naan Roll: Ramsay of Carluke's smoked streaky bacon is matured for two weeks and smoked overnight in the traditional fashion. A Dishoom signature dish, and deserving of all its accolades.\nKejriwal: 2 fried eggs on chilli cheese toast. A favorite of the well-to-do Willingdon Club, the first such Bombay institution to admit natives; the dish is reputedly named for the member who kept asking for it.\nEgg Naan Roll: 2 fried free-range eggs with saffron-yellow, runny yolks.\nDate & Banana Porridge: Organic porridge oats cooked with milk, banana and sweet Medjool dates.\nA little bit of everything \u2013 delicious!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Art Haven has a fully functioning darkroom (available for classes and open darkroom time), 10 pottery wheels, painting and drawing supplies, and is soon to house the gallery of The Hive Cape Ann. Also in the works is a professional digital lighting studio and a five machine computer lab with Adobe Photoshop and Illustrator for design and portfolio work. One of our most popular programs is family studio time.\nFamily studio is a time when parents can come and participate in fun artistic projects as well as explore other mediums and interests. In addition to our normal class programming, we love offering community events that get kids and adults excited about the arts on Cape Ann. Join us for pumpkin carving, ice sculpture building, buoy painting, and various artsy events that coincide with downtown block parties throughout the year!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Deacon Bill is a Permanent Deacon ordained in 1983 for the diocese of Cleveland. He has served at St. Francis of Assisi Parish for over 16 years. Deacon Bill also served at St. Clare Parish in Lyndhurst for 16 years where he and his wife Linda served as youth ministers and received the James P. O'Donnell award for outstanding youth ministry from the diocese. Bill studied at St. Mary's Seminary for the diaconate and earned his Bachelor's degree in Psychology and Religious studies from Ursuline College and his Master's degree in Clinical Counseling from John Carroll University. He is an adjunct faculty member at St. Mary's Seminary. Deacon Bill and his wife have three grown children and seven grandchildren. They are both in private practice clinical counseling at Family Behavioral Health Services in Mayfield Village.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many computer-security researchers think the world would be a safer place if they could easily report bugs to software creators, so they holes could be patched before hackers exploit them.\nBut there's a problem: Most large companies don't advertise a way for users to report bugs. This includes companies that collect reams of personal data, such as J.P. Morgan Chase & Co., Bank of America Corp. and Allstate Insurance Corp., as well as the designers of Internet-connected machines, like Ford Motor Co.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Will the US telecom market enter a shrinking stage? The third and fourth operators on the market (T-Mobile and Sprint) are merging. For the greater good of the consumers, for employment and to lower the bills. Or so they say.\nIt is not recorded yet. It must first be validated by shareholders of both companies (that should be easy), then by the US regulatory authorities. This should be more complex: the anti-trust laws are supposed to control market concentrations.\nThe announcement is official though and the terms are well known. T-Mobile wants to merge with its rival: Sprint. If the operation gets the go-ahead, the new entity will keep the name T-Mobile. The name Sprint will slowly disappear. Deutsche Telekom, the parent company of T-Mobile, will own 42% of the new group. SoftBank, the parent company of Sprint, will own 27%. The 31% left will be available to a third party.\nIn this configuration, the US telecom market will encompass three giants: AT&T, Verizon (current leader) and the new T-Mobile. One of the reasons to promote the merger is to counter AT&T and Verizon with scaling savings and synergy. However, both operators claim they will create jobs\u2026 Mary Poppins must be behind all this.\nStop the presses: the market shrinks but prices will drop. We promise.\nThe other part of the T-Mobile\/Sprint communication are prices. Don't laugh, but they pretend the merger and the cut in operators will induce a drop in plan prices. As everyone knows, market shrink means price drops. No, lowering the number of operators will not ease unlawful agreement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..703e5779374b116201e113a28f22209c424199c4
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@@ -0,0 +1,5 @@
+{"text":"Founded in the spirit of the Ens\u014d circle, my massage practice is focused upon relaxing the body, mind, and soul. When we're in a calm parasympathetic state, our systems are encouraged to perform optimally, inspiring our bodies to be the healers they were born to be. Ever mindful of your anatomy and energy, I connect with long soothing strokes, deep kneading, stretching, space, rhythm, a slow sinking into areas of tension, and a meditative, compassionate spirit.\nEnjoy chair massage at your next event or party. Call, text or email for rates and details. Contact information below.\nI am grateful to be practicing out of The Shala Yoga and Pilates Studio. 34 South Market Street...studio entrance and parking in the back of the building. Nestled in the heart of beautiful Frederick County, this darling town has so much to offer.\nI use Badger massage oils. They are vegan, certified organic and bear the Leaping Bunny Seal. All linens provided during a session are Global Organic Textile Standard Certified unbleached cotton.\nThis age-old practice is a wonderful way for people to exchange services or goods with out money changing hands and it's legal. I'm always interested in hearing about a potential trade.\nI love taking pictures. I love helping others. Please support both endeavors by purchasing one of my images. Currently available, 2\" x 2\" magnetic prints for $5 each, or 5 for $20. All proceeds go toward providing my services to the women of Heartly House. Check out the Instagram link below for a gallery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Currencies are exchanged in many different formats and through many different venues. Today's market landscape features active FX trading in the form of spot, forward, swap, futures and option transactions, on an over-the-counter (OTC) and on an exchange-traded basis. Currencies have been transacted in many different forms stretching back to ancient times and moving forward through the era of the gold standard and the fixed currency regime under the Bretton Woods Agreement. The modern era of currency trading may be said to have begun with the collapse of the Bretton Woods Accord in the early 1970s, ushering in a new regime featuring floating exchange rates. Currency traders have been quite inventive in developing new and increasingly specialized mechanisms to address the risks attendant to floating exchange rates in a risky economic environment. Spot trading of currencies has been supplemented by outright forward contracts, FX swaps, currency swaps and options. Chicago Mercantile Exchange (CME) introduced currency futures in 1972 followed in 1984 by the introduction of options on currency futures. These instruments have grown in popularity at an accelerating rate over recent years as a result of volatility, globalization and widespread acceptance of electronic trading platforms. In particular, the popularity of futures and options on futures is growing rapidly as traders learn to appreciate the liquidity, transparency, rapid execution, anonymity and clearinghouse guarantees associated with the trade of these instruments on CME.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have no idea if the site was taken down due to an investigation into the scam or for some other legal reason, but I have hopes that an Attorney General in some state, somewhere, is looking into Marlene Nadeau and her \"conferences\".\nI did get one page of the website saved on the Wayback Machine. I wish I'd saved the one where Marlene had her conference scheduled. This scheme sounds goofier than her first one. Imagine Marlene directing others towards mental health? Seriously?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beautiful floor length silk Ellery dress. Stylish and perfect for a bride on a budget.\nI absolutely love the simplicity and elegance of this dress! I bought it off the shelf new and had it tailored to fit at the bust where it was taken in, but there would still be room to take it in further for a size 10. I am 173cm tall and I had subtle loop holes added so I could hitch it up for dancing.\nIt is in very good condition with no obvious issues. It has been dry cleaned and is bias cut so please note the silk will fall where it wants to.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00ab Kathmandu, Camino al Infierno \u2013 New Review!!!\nLive Interview on Rudy Maxa's World Radio Show online now!!!!\nI put my guide book author hat on last week and gave a live radio interview\/podcast on Rudy Maxa's World, America's most widely-syndicated weekend travel radio show, on Saturday, May 11th.\nI'll be talking about travel writing, tourism, publishing and my new US published guide to the Angkor temples in Cambodia from Moon Guides.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarahg b/data_all_eng_slimpj/shuffled/split2/finalzzzarahg
new file mode 100644
index 0000000000000000000000000000000000000000..03f260efd340d19511e17af6ecfdb248f13b3084
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@@ -0,0 +1,5 @@
+{"text":"the parties want to negotiate but refuse to meet.\nWe mediate disputes via telephone. It is still a confidential, without prejudice process but our neutral assistance and intervention is done by telephone rather than face-to-face.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's crunch time for Bafana Bafana in the CAF Africa Cup of Nations (Afcon) qualifiers as they play away to Libya on Sunday. Bafana are currently second in their group and need just one point to qualify for this year's Afcon in Egypt this June.\nThe match will be played in Tunisia due to the political turmoil in Libya. Stuart Baxter, Bafana Bafana coach, says that the Bafana players are fired up.\nNigeria leads the group. They've already booked their spot with 10 points from five matches.\nBafana captain Thulani Hlatshwayo says that they are aware of what's riding on their performance against Libya.\nBafana Bafana midfielder, Hlompho Kekana, says that they will try to press the opponents.\nBafana last played in the Afcon tournament in 2015 under the stewardship of Ephraim Shakes Mashaba. A positive result against Libya will see them return to a tournament they last won in 1996.\nBafana's European based brigade is raring to go. Midfielder Kamohelo Mokotjo says that they will focus on the game fully.\nMidfielder Thulani Serero, says that the players know the importance of the game.\nBafana will miss influential goalkeeper, Itumeleng Khune due to an injury, but goalkeeper coach Andre Arendse says that the selected goalkeepers are consistent.\nThe Afcon qualifier kicks off at 19:00 on Sunday.\nThe post Crunch time for Bafana in Afcon qualifiers appeared first on SABC News \u2013 Breaking news, special reports, world, business, sport coverage of all South African current events. Africa's news leader..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Isn't it great to see so many other IU fans in one place?\" I asked my five-year old as we walked through the crowds of red or purple clad football fans around the southern end zone of Ryan Stadium in Evanston. It was a perfect late October day. Beautiful blue skies, a nip in the air, light wispy clouds, no wind and I was taking my son to his first Indiana football game.\n\"What do you mean?\" I asked, a mild panic starting to rise in my chest. I had expected a fair amount of squirming and impatience on his part at trying to sit through your standard-length college football game. I was even prepared for him to not like football. Hell, I'm not that big a football fan myself. But, if I'd failed this grandly in raising this child through the first five years of his life, it might be in both our best interests for me to find a kindly looking older couple dressed in cream and crimson and send him home with them after the game. Maybe they could succeed where I had clearly failed.\nI had very little fear that he'd reply with something as offensive as, \"I'm a Kentucky fan.\" \u2013 we've discussed the nature of good and evil and I didn't think I'd lost his soul, at least not yet. But I did worry that there was something he was learning at that school of his, surrounded by other kids who were also growing up in Chicago, that I hadn't gotten wind of yet, that was antithetical to the values I try to espouse in my home. It was possible that he thought he was a fan of some other team because one of his classmates was, maybe Illinois or something equally repellant.\nI was kind of worried there were societal forces at play that I would have to battle actively while they passively undermined all my efforts for the rest of his childhood.\nHe's a bit of a skeptic.\nThis is new information about his personality. He wants to know why he should be a fan of IU. He's just not going to accept it as the way the world is.\nAnd now I have the job of helping him answer it for himself.\nWe had a fun day at the game. He got a hot dog, some popcorn, and half a pretzel. He got to collect the cards they hand out that provide you with a rooting interest\/financial incentive during the quarterly races being run on the stadium monitor between, go carts, or coffee cups, or donuts, or the like. He got to hang out with a friend and run around a bit pretending to be a ghost chasing a zombie. It wasn't, however, the transcendent sports experience that imprints fandom upon you.\nWe grabbed a free smoothie on the way home, thanks to one of the coupons and the successful pursuits of smoothie #3 \"Tropical Twist\" during his turn in the spotlight midway through the 1st quarter, which he liked as much \u2013 if not more \u2013 than anything that had happened during the game that wasn't eating a hotdog.\nThus far, the day had resulted in creating a smoothie fan. Not terrible, but not the goal either.\nUpon entering the house, I fired up the DVR and began to watch Liverpool best West Bromwich Albion to rise to 2nd in the Premiere League, tied in points, but one goal behind Arsenal in goal differential. He went and played legos.\nNo surprise here really. He kind of likes playing soccer, but he mostly just likes getting the snack at the end of the game and sitting out when it's his turn to do so. I've no expectation that he's ready to fully embrace the beautiful game yet.\nNext up was Hoosier Hysteria. Something I streamed on my iPad while we played legos. I tried to explain how much fun OG was going to be to watch this year. He was intrigued, but not overly interested. In his defense, there's very little about Hoosier Hysteria that I find exciting or worth watching.\nAnd they lowered the ante this year by having a dunk contest between three people, none members of the actual IU basketball team, and only one of whom could dunk. It was like if WWE suddenly decided that James Ellsworth was suddenly the WWE Heavyweight Champion and Survivor Series was going to consist entirely of a tournament of jobbers trying to earn a chance at his title.\nI did enjoy the incredibly long video package they put together highlighting last year's Big Ten title run. And I understand why they omitted it, but I have stronger memories of Bill Walton talking about Maui Jim's, loving his bike, waxing rhapsodic about Colin Hartman \"The Adonis of the Night\u2026the symmetry\" and describing Maui as having \"seven climate zones and 1000 emotional zones\"* than I do of the game at Iowa, and it would have been nice to hear one last \"Maui Jim's Maui Jim Classic here on Maui Jim court in Maui Jim's Arena. I'M WEARING MAUI JIM'S RIGHT NOW AND I LOVE MY BIKE.\" But, as I said, I get not including the actual low-point of the season in a feel-good highlights package.\nI also found the piece they put together about Tom Crean's postgame speech particularly moving. It's as goosebumps inducing as hearing Fish call the end of UK game or Jim Nance announcing \"Keith Smart is the hero!\" And if I can get a link to it, I'll post it here so you can see it yourself.\nAnd things like that help demonstrate why you want to invest your time and emotional capital in being a sports fan. I'll show that to my son at some point, if I can get him to sit still long enough to watch it.\nBut, by the time that video package ran, he was asleep.\nAnd by the time the Cubs won the NLCS and advanced to their first World Series in my (or either of my parent's) lifetime, my wife was also asleep. I was able to show him video of that this morning, and we'll be heading out in a bit to buy our Cubs World Series T-Shirts.\nThis may be the one that makes an impression. And it should.\nTomorrow is Cubs Day at his school. He's going to see other kids excited about it and get to experience the thrill of a city caught up in a chance to win a title.\nI'm hopeful that is the jumpstart his fandom needs, even if it is for baseball.\nBut the more I think about this challenge of proving the benefits of fandom, I think the best thing I can do is show how much fun it is to cheer for a team. Something that hasn't been top of my mind when watching IU these last few years. I've been more focused on understanding, explaining, contextualizing, and writing than I have on enjoying.\nSo, that's my challenge for the year. Focus on the fun. Enjoy the ride of the season. And, of course, share that experience with him, you \u2013 the teeming masses.\nWe'll see how that goes, but I've got a job to do, turning this little guy into a proper IU fan, and he's gonna have to see it and live it in order to feel it himself.\nTime to lead by example.\n*This kind of Bill Walton nonsense directly influenced the title of this post. I love my bike.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ordering Domino's in Japan? A reindeer might be in charge of delivering your pizza.\nWho needs a pizza delivery driver when you could have a reindeer deliver food to your doorstep?\nDomino's Japan is rolling out pizza delivering reindeer as part of what could quite possibly be one of the biggest holiday publicity stunts of the season.\nThe reindeer will be taking on the cold to ensure a speedy pizza delivery on Japan's northern island of Hokkaido, according to The Telegraph.\nAccording to a press release, Domino's is working with reindeer breeders from the Hokkaido Delivery Integrated Research Center to train them in pizza delivery.\nThe company is planning a full-scale rollout of the delivery reindeer by December.\nThursday, the company announced that the reindeer, just like any other delivery driver, will be tracked in real-time using GPS.\nWeather agencies in Japan are calling for a particularly cold winter this year. Thursday's snowfall in Japan's capital city, Tokyo, marked the first November snowfall in 54 years, according to Japan's Kyodo news agency.\nSo for Domino's Japan, the sturdy reindeer may be a good move to ensure customers still get their pizza.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The vehicle identification number (VIN), since model year 1981, is a series of 17 letters and numbers. Accepted as the world standard for identifying vehicles, the VIN provides key information about the manufacturer, model, model year, make, equipment and class of a vehicle. By entering your VIN, we can tell you whether SIRIUS has been installed in your vehicle.\nNot sure what service your vehicle is equipped with? Enter your VIN (Vehicle Identification Number) below to find out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarbgs b/data_all_eng_slimpj/shuffled/split2/finalzzzarbgs
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@@ -0,0 +1,5 @@
+{"text":"The pioneer of Yarmouth district was the Quaker, Jonathon Doan, who came to this area in 1813. He pitched his tent and laid himself down on the spot which is now the Quaker cemetary in Sparta and where he is buried.\nThe land in this area had been granted to the Baby family and Doan was acting as their land agent. After two years Doan was so satisfied with the prospects that he returned to Pennsylvania and brought back with him several families, Including William Harvey, John Kipp, John Mills, and Joseph Albertson.\nDoan built the first flour mill in South Yarmouth and operated the first tannery.\nIn 1820 the first meetinghouse was built. It was a little log building on the corner of Jonathan Doan's farm which land had been donated by him for the use of the Meeting.\nEventually a small frame Meeting House replaced the log building and, in 1865, a new site was chosen in a beautiful maple grove north of Sparta and the present Meeting House was erected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Special Legislative Concerns: agriculture and natural resources, transportation, education, tax policy, bonding, small business.\nHe ran unsuccessfully as the Democratic-Farmer-Labor Party endorsed candidate for the Minnesota House of Representatives in the 1978 election. His opponent, Merlyn O. Valan, defeated him by 15 votes so an automatic recount occurred. The recount confirmed that Merlyn O. Valan had won.\nReligion provided by the Catholic Bulletin, 3\/21\/1975 and Politics in Minnesota the Directory, 2005-2006.\nRep. Kalis, Rep. Boerboom, Senator Stumpf, Senator Langseth, Representative K.J. McDonald, Senator Berg, Senator Davis, Rep. Schoenfeld, and Senator DeCramer shake hands following the signing of an omnibus agriculture bill, St. Paul, Minnesota.\nSenator Keith Langseth chairs a committee hearing, St. Paul, Minnesota.\nSenator Keith Langseth, Senator Charlie Berg, and Senator LeRoy Stumpf confer during a committee hearing, St. Paul, Minnesota.\nSenator Keith Langseth, Senator Charlie Berg, and Senator LeRoy Stumpf in a discussion during a committee hearing, St. Paul, Minnesota.\nSenator Keith Langseth, Senator Roger Moe, and Senator Marvin Hanson converse on the Senate floor, St. Paul, Minnesota.\nSenators Chuck Davis, Gary DeCramer, LeRoy Stumpf, Charlie Berg, and Roger Moe discuss a bill during a Senate floor session, St. Paul, Minnesota.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fertility preservation is an option for cancer patients or women who are otherwise not ready to have a baby.\nThis might sound a little like \"The Bride Of Frankenstein meets Infertility,\" but the medical technology now exists to extract a woman's ovary from her body, freeze it and put it back in years later when she's ready to have kids. To someone who spent the first 36 years of her life completely ignoring these tiny reproductive organs (and then expected them to work perfectly on demand as my biological clock's ticking grew louder), this breakthrough sounds like having all your eggs stored in one very safe basket.\nSherman Silber, M.D., a fertility doctor with The Infertility Center Of St. Louis pioneered the ovarian transplant procedure. It began as a way to help cancer patients preserve their fertility before damaging chemotherapy treatments, but it is now being offered to women who want to bank their ovarian tissue in order to retain the option of having children later in life. Silber has done 140 ovarian tissue transplants, and back when I was chasing babies, I would have loved to have had this high-tech alternative to shooting myself full of fertility drugs and laying eggs by the meager half dozen.\nYoung women have a number of decisions to make about their own fertility. When a woman is not ready to have a child, fertility preservation provides a way to ensure that she'll have her own eggs available in the future. Another consideration, is helping other couples through egg donation. While the processes are similar, the goals are different. There are a few things to keep in mind when considering egg freezing vs. egg donation.\nDoes Number of Frozen Embryos Correlate with IVF Success?\nDr. Tina Koopersmith from West Coast Women's Reproductive Center discusses male fertility preservation through sperm freezing.\nDr. Tina Koopersmith from West Coast Women's Reproductive Center discusses why some people need to preserve there fertility due to illness.\nHow Can Cancer Patients Preserve Their Fertility?\nWhat is Ovary Tissue Freezing for Fertility Preservation?\nWhat Are Options for Preserving Fertility?\nAdvanced family planning, as Dr. Michael Feinman calls it, is in essence fertility preservation for couples who want to have several children. \"These women are over 35, they're not infertile and they want more than two or even three children. They reason that if they get pregnant now, by the time they try again they could be too old to get pregnant. So they store PGD tested embryos that they created with their husband,\" says Feinman, a fertility doctor with HRC Fertility in California.\nA growing number of women are turning to egg freezing as a way to preserve their fertility. Despite this, it is common for many women to be unfamiliar with egg freezing. Knowing more about the egg freezing process and its use can help women determine if fertility preservation can address their needs.\nAre There IVF Treatments Available That Don't Require Injections?\nCan You Walk Me Through the Egg Freezing Process?\nDr. Brooke Hodes-Wertz of NYU Fertility Center discusses the process of egg freezing. She discusses everything that is required to successfully freeze your eggs.\nWhat Are the Advantages of Egg Freezing?\nHow Does a Woman's Age Influence Egg Freezing Success?\nRead more about Can You Walk Me Through the Egg Freezing Process?\nDr. Brooke Hodes-Wertz from NYU Fertility Center discusses the pros and cons of egg freezing. When a woman is considering freezing her eggs, Dr. Hodes-Wertz explains both the pros and cons to the woman.\nRead more about What are the Pros and Cons of Egg Freezing?\nDr. John Couvaras from IVF Phoenix discusses fertility preservation with egg freezing. When you consider egg freezing, one question that comes up is how many eggs should I freeze. Many people are unaware of how many eggs are typically required to achieve one live birth.\nRead more about How Many Eggs Should I Freeze for Fertility Preservation?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Michael Young\" is composed of at least 12 distinct authors, divided by their works.\nWorks have been aliased into Michael Dunlop Young.\nWorks have been aliased into Michael F. D. Young.\nWorks have been aliased into Michael J. Young.\n1. Michael Dunlop Young, British politician and social activist.\n2. Michael J. Young, computer programmer.\n3. Michael Young, Lebanese journalist.\n6. Michael F. D. Young, professor of sociology.\n7. Mike Young, Kosciuszko National Park publications officer.\n8. Michael Young, craft curator of Victoria's National Gallery collection (Australia).\n\"Michael Young\" is composed of at least 12 distinct authors, divided by their works. You can edit the division.\nMichael Young is composed of 2 names. You can examine and separate out names.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CAPDM are hosting the 4th meeting of the EU Funded Erasmus+ project OpenQAsS (2014-1-HU01-KA242-002356) \u2013 Open Source Quality Assurance System for European Vocational Education.\nThe meeting takes place 2nd and 3rd June in Edinburgh.\nSee project website for more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..dab702029672d2243ead0bb7959412931da08629
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+{"text":"She rose to stardom after her appearance on the television series, Garage Squad. She was born on May 24, , in Montana, the United States. She was raised by her parents in her birthplace along with her siblings. She is a biology student and has often engaged in conservative tourism. Storm's father was a DIY maverick who taught her to use several tools and also shoot guns. After that, she appeared in another short drama movie, Nightingale, which was directed by Elliott Lester and written by Frederick Mensch.\nDo you think it would help is showtime did some sort of campaigning similar to what studios do during Oscar season? Maybe we should start a petition\u2026 On another note I remember you and Alan talking about whether or not the show would maintain the status quo due to her new job and prospects or if it would follow through with this new direction, and I think this arc for her character has been a really smart way for the writers to deal with that issue.\nAlso, this has been by far the worst Chicago winter I can remember. That alone brings up dozens of slogan possibilities!\nSierra and Lip hit a bump on the road on \"Shameless\" Season 7. Ruby Modine shares her thoughts on whether they'll make it work.\nShameless star Jeremy Allen White becomes a father Jeremy Allen White and Addison Timlin are now the proud parents of a baby girl! The year-old Shameless actor and the year-old Californication actress welcomed their first child on Saturday October 20 , they confirmed via social media on Sunday October Congratulations to the happy family! View this post on Instagram Ezer Billie White A post shared by Addison Timlin addison.\nToday is my birthday but my wish already came true. This is 27 and the best is yet to come. Congratulations Jeremy and Addison! Macy, she claimed that her character Fiona was integral to the series and implied that it would crumble without her.\nAfter two years of dating, the New York Post is reporting that Emmy Rossum and Jacob Tyler Moore, who play the characters of Fiona Gallagher and Tony Markovich on the hit Showtime drama Shameless, have been broken up.\nCocaine was found in a purse and she was cautioned for possession. But in an emotional interview, Tina said: Tina says she believes the drugs were planted in her bag Image: She also denied swearing at children appearing in the panto. Tina Malone with daughter Flame and husband Paul Image: I was set up. She found it suspicious photographers turned up to ask her to pose for them hours before her arrest on December 22 \u2014 and that the story broke before she had been released from the police station.\nAs Peyton explained to AOL Build, the co-stars fell for each other IRL during the making of this movie. And their relationship totally works, thanks to the fact that they're both actors and understand how busy they can get filming various projects.\nThey also dated off-camera, although it was pretty short-lived. Lucas Till has confirmed that the pair dated for two months on set, but remained good friends after it ended. Fans speculated that the two would date before even seeing the movie, considering that Miley personally picked him to portray her love interest for the movie. It seems that after the movie was over, however, the magic of their romance faded.\nTill went on to date Taylor Swift after starring in her music video, and Miley is now engaged to Liam Hemsworth. It helped launch the careers of some big actors, like James Franco and Seth Rogen. However, it also launched a romance between stars Jason Segel and Linda Cardellini. They had a serious and secret off-camera relationship throughout the course of the show as well as after it ended.\nSupposedly, they dated from all the way until At least he turned a bad situation into great comedy. It helped launch Will Smith to the stardom that he has today.\nThe Lion, the Witch and the Wardrobe. Days of Future Past in He was brought up as Roman Catholic. His parents divorced when he was seven, which he took hard, McAvoy's mother suffered from poor health and subsequently devided it was best that he lived with his maternal grandparents, Mary and butcher James Johnstone, in the nearby Drumchapel, in a terraced council house. His mother lived with them intermittently.\nSHAMELESS. 10 p.m. Sunday, Showtime. ACTORS ARE fond of proclaiming that their co-stars are like family \u2013 seldom specifying Brady or Manson \u2013 but as far as Emmy Rossum's concerned, the cast.\nThese Celebs Did It! But some celebs find it hard to resist the charms of their equally famous co-stars. The following celebs all dated their co-stars\u2026with varying degrees of success. Smith No article about celeb relationships would be complete without mentioning Brangelina. According to the couple, who met on the set of Mr. Smith, Brad never cheated on Jennifer Aniston. While Brad and Angie both admit that they developed feelings for each other on set, they never acted on those feelings.\nEvening At the tender age of 15, Claire Danes had her first kiss.\nA post shared by Emma Greenwell emma. If there were any doubts remaining as to whether these two are dating in real life or not, it has been cleared now. The couple, who got to know each other while working together on the TV drama show Shameless had a fantastic on-screen chemistry. These two eventually became a real-life couple. The couple is both focusing on their acting career and right now has left their marriage intention aside.\nEmmy Rossum reveals her feelings on dating costars, plus more secrets from the set of Shameless! Two actors falling in love on set? It happens all the time\u2014and Emmy Rossum says the chemistry is.\nThe year-old actor made the announcement on Monday. Visit my website for full links, other content, and more! The year-old actor took to his Instagram to announce that his final episode of Shameless will air this week. I was the tender age of fifteen when we shot the pilot, and I came [ I was the tender age of fifteen when we shot the pilot, and I came of age in so many ways, both legally and personally. I was very lucky to mature and grow with this show.\nExperiencing so many firsts, maturing as an actor, a professional, and a human. Goodbye, Ian Clayton Gallagher. In the process gaining friends, family, and the best coworkers a very lucky actor could ask for, and for this I can be nothing but gracious. All good things come to an end.\nShare this article Share Ryan said: Followed by Scott Dalton who wrote: A still of the X Factor live show which repeatedly featured the luxury hotel Wow!\nRELATED Showtime Orders Happyish to Series, Renews Shameless for Season 6 The honeymoon period, however, will be cut short when they have to figure out how to tell her family the big news \u2014 and an unexpected, but familiar visitor shows up to complicate things for the stranger-like newlyweds, executive producer Nancy Pimental previews below. We always wanted her to get married. I was out partying, and I was dating a bunch of guys. Is this not real love? Oh, that was just endorphins or serotonin or oxytocin.\nThere are so many levels to it. She definitely is high on the love drug of their chemistry and him being such a great guy. This guy looks so good on paper. What happens when reality sets in for Fiona and Gus? Can you talk about what sort of dynamic we might see there? And what do you mean you need a family meeting to tell us something?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arutz Sheva has documented that an Arab car on Sunday attempted to circumvent another vehicle travelling on Route 60.\nAn Arutz Sheva corresponded has documented that an Arab car on Sunday made a dangerous attempt to circumvent another vehicle traveling on Route 60, near the Jewish community of Shilo and toward the direction of Maale Levonah. The risky move led to a head-on collision.\nTwo Arabs were killed and four wounded in the incident.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VanWIT is back for the 2017\/2018 season and is pleased to announce the first event in our new Leadership Circle Series. The events feature senior executives in technology who will share their journey in the industry in an intimate setting with the VanWIT community.\nWe are excited to welcome our first speaker, Dianne Lapierre, Senior Vice President & Chief Information Officer of Raymond James Ltd.\nAfter 24 years in a variety of Information Technology Management and Executive roles at Manulife Financial in Waterloo, ON, Dianne moved to Vancouver to assume her current role in 2014. Raymond James, established in 1962, is one of North America's leading independent full-service investment dealers with an extensive presence across Canada and the United States. Raymond James Ltd. is the Canadian subsidiary of Raymond James Financial, Inc. Over 1200 employees and agents across Canada work together in establishing Raymond James as Canada's premier independent investment firm. We offer a wide-range of services for individual and institutional clients and corporate issuers. Our Advisors have access to a complete offering of high quality investment products and services, providing them with the resources to customize investment solutions for their clients. Our Equity Capital Markets division provides expertise in Investment Banking, Research, Corporate Finance, and Institutional Sales & Trading.\nDianne is a passionate achiever who inspires excellence with a personal goal of growing leaders who grow leaders. Personal leadership is key driver in her career and personal success.\nDianne has an undergraduate degree from Wilfrid Laurier University and an MBA from Athabasca University. She has also maintained her Project Management Professional (PMP) designation since 2001. Dianne is a member of the Board of Directors of the Vancouver Chapter of the CIO Association of Canada and is currently serving on the organizing committee for the National Association's CIO Peer Forum, an annual conference for CIO's.\nDianne lives in the West End of Vancouver with her husband Jeff and youngest child (21). She has 4 other adult children who live in Montreal, Toronto, Waterloo and London, ON.\nTickets for this event are limited, so please make sure you sign up soon to share some refreshments and conversation with Dianne and other VanWIT members!\nOn November 21, join BCjobs.ca and the BC Tech Association for a panel bringing together women leaders who will be sharing their experiences working in different roles within the tech industry!\nFollowing the panel, you'll have the chance to get a closer look at roles you're interested in during breakout discussions. Learn more about career options and opportunities in tech, find out how you can kickstart your career, and discuss how we can come together to champion more diversity and inclusion in the workplace.\nThis event will be a wonderful opportunity to connect with ambitious women, get engaged in the community, and share ideas on how to encourage and empower women in tech.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This issue is about the information of the fourth annual listing of the country's top adhesives and sealants manufacturers supplied by Adhesives & Sealants Industry magazine. Due to the fact that approximately 38% of the Top 50 companies did not want their sales figures published, they decided to list the companies alphabetically rather than rank them by sales figures, as they have in the past.\nIncluded with each company's listing is contact information, estimated 2000 adhesives and sealants sales figures (if publishable), subsidiaries and recent acquisitions, and a company profile. This year, because of a tie, there are actually 51 companies included in the Top 50 listing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"St. John of Nepomuk Statue is located in Prague. Put St. John of Nepomuk Statue and other Prague attractions into our Prague trip itinerary website , and watch your holiday take shape.\nLocated in Karlov bridge to go over several times during your visit. In honor to San Juan Nepomuceno, martyred by being thrown into the river. As a curiosity so the vessel of the Brigadier Churruca was called during the battle of Trafalgar.\nStatue of St. Jan Nepomuk is one of 30 images standing on the parapet of the Charles Bridge, located just near the middle of the bridge. Among 30 body and image, this image was always surrounded tourists. It is what that being told touching part of the base pattern to be happy. Touched by the many tourists piece was shining there just gilded.\nEvery day, we help thousands of travelers plan great vacations. Here are some tips on how to use your Listing Page to attract more travelers to your business - St. John of Nepomuk Statue.\nPromote your permanent link - https:\/\/www.inspirock.com\/czech-republic\/prague\/st-john-of-nepomuk-statue-a7316813905 - on your website, your blog, and social media. When a traveler builds a plan from this page, the itinerary automatically includes St. John of Nepomuk Statue. This is a great way for travelers looking for a great vacation in your city or country to also experience St. John of Nepomuk Statue. Promoting your link also lets your audience know that you are featured on a rapidly growing travel site. In addition, the more this page is used, the more we will promote St. John of Nepomuk Statue to other Inspirock users.\nFor best results, use the customized Trip Planning Widget for St. John of Nepomuk Statue on your website. It has all the advantages mentioned above, plus users to your site can conveniently access it directly on your site. Visitors to your site will appreciate you offering them help with planning their vacation. Setting this up takes just a few minutes - Click here for instructions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarqlc b/data_all_eng_slimpj/shuffled/split2/finalzzzarqlc
new file mode 100644
index 0000000000000000000000000000000000000000..fa30e4db3805eb62c865828059043f863a6c37c9
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+{"text":"I've been running for 30 + years everything from 5K to multi-day ultras. I'm also a run leader and mentor for our local running shop. There's so many gels on the market today, but I enjoy the flavor of e-Gel and the fact my stomach has no issues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How much do you need? I need Enter an amount between $500 and $10,000.\nWhat's the purpose? For Emergency Expenses Debt Consolidation Medical Expenses Auto Repairs Business Expenses Rent or Mortgage Vacation Large Purchase Credit Card Consolidation Wedding Home Improvement Moving New Car Taxes Education Something Else Money can be used for anything... select what you need it for.\nWhat's your credit score? My credit score is Excellent (700+) Good (630-699) Fair (550-629) Poor (549 or lower) Unknown You may not know your exact credit score, that's fine! Just choose what you think it might be\u2014 All credit scores are welcome!\nAre you attending college? I am not attending planning on attending attending college. Select whether or not you're currently enrolled in a college or university.\nHome ownership I rent pay a mortgage own my home. Select whether you rent, pay a mortgage on, or lease your home.\nCar ownership I lease own don't have my car. Select whether you rent or own your vehicle.\nMy zip code is I live in Enter your zip code, we'll lookup the city and state for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BREAKING: SURIYA 36 - NGK'S FULL FORM REVEALED!\nThe title of Suriya 36 was revealed to be NGK, which kept everyone guessing about the importance and the meaning of the title.\nNow sources close to us have revealed that NGK stands for the character name of Suriya in the film, which is said to be, 'Nandha Gopalan Kumaran' (NGK). The creatively designed poster has impressed the fans and has indeed generated a curiosity factor for the public!\nStarring Suriya, Sai Pallavi, and Rakul Preet Singh in lead roles, Selvaraghavan's NGK, is scheduled to hit the screens for Diwali 2018.\n'\u0baa\u0bbf\u0b95\u0bcd\u0baa\u0bbe\u0bb8\u0bcd' \u0bb0\u0bc8\u0b9a\u0bbe\u0bb5\u0bbf\u0ba9\u0bcd \u0b95\u0bbe\u0bb8\u0bcd\u0b9f\u0bcd\u0baf\u0bc2\u0bae\u0bcd \u0b9f\u0bbf\u0b9a\u0bc8\u0ba9\u0bb0\u0bbe\u0b95 \u0bae\u0bbe\u0bb1\u0bbf\u0baf '\u0baf\u0bc1\u0bb5\u0ba9\u0bcd' \u0bae\u0ba9\u0bc8\u0bb5\u0bbf!\nTamannaah reveals this secret hobby of hers!\nWow! Dulquer Salmaan as the Indian cricket team captain!\nJust In: Producer Council's strong pointers before the big strike!\nVarun Dhawan's October - Official Trailer!\nNew Song - Kedaya from Kalari!\n'\u0ba4\u0bb2' \u0baa\u0b9f\u0ba4\u0bcd\u0ba4\u0bc8\u0ba4\u0bcd \u0ba4\u0baf\u0bbe\u0bb0\u0bbf\u0b95\u0bcd\u0b95\u0bbf\u0bb1\u0bbe\u0bb0\u0bbe '\u0bb8\u0bcd\u0bb0\u0bc0\u0ba4\u0bc7\u0bb5\u0bbf' \u0b95\u0ba3\u0bb5\u0bb0\u0bcd?\nPeople looking for online information on Harris Jayaraj, KV Anand, Suriya will find this news story useful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy it now on Amazon or download a sample chapter here.\nI sat at the interview table in Spring 2016 as a bright young intern candidate started answering my questions in the hope of getting a job. She was bright and determined but made some mistakes which could have been avoided with better planning and preparation. Later that day I had a thought, if people aren't learning these valuable careers lessons at university, where will they ever learn them?\nSo I started writing my book, the book turned into a series of successful workshops, and the workshops turned into Hustle Crew. Now the book is done, I'm excited to share it with the world. My hope is that more young women will improve their hustle with this book: land better jobs and maximise their earning and learning potential at work.\"\nWatch Abadesi discuss her book on London Live TV.\nWe respect your privacy and never share your data.\n\"Good things happen to those who hustle.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since I learned to make my first ever tiered cake in an online course from Craftsy, I've been wanting to find a way to incorporate my besties (cupcakes) into the more traditional world of cake making. I decided to make a cupcake bouquet topper. I was shooting for the topper to look like a pumpkin (inspired by a cupcake pumpkin by Caring with Coupons). It ended up looking more generically fall floral and I'm OK with that. Any cake that is adorned with a big ball of cupcakes is clearly the best cake ever!\nBelow, you'll find some of the Craftsy tips that I used to create this cake. Just like reading CliffsNotes won't get you an A on your exam, the tips below won't be enough on their own to show you how to make a cake like this. My very best tip is to take the Modern Buttercream course with Joshua John Russell on Craftsy \u2013 it's free! You can also try any of the other Craftsy courses for 25% off with this link (about $30 per class after the discount).\nAlways cut off the top of your cake so you have a smooth surface to frost. I always thought that a benefit of cupcake making over cake making was that you could taste them before serving them to your guests. Little did I know that you get to eat the entire top of the cake when you make a decorated one!\nThe photo below shows me making a guideline with my knife just before cutting the top off. The whole thing isn't as scary as it sounds as long as you have the right tools. You absolutely MUST have a cake turntable. It makes cutting and frosting so easy; just hold your knife still and spin the wheel.\nHere's the top being removed from the cake \u2013 ready to eat!\nJosh likes a high frosting to cake ratio so he actually suggests cutting each layer in half. Again, trust me, if you follow the steps in the video, this isn't so hard \u2013 put the knife in the cake and spin\u2026 wheeee!\nAlways work on refrigerated cakes. Handling cold cakes is so much easier!\nAlways put a crumb coat on a cake. This is a layer of frosting that traps the crumbs so they don't get in the pretty top coat. The crumb coat is like a frosting hairnet; when a cake has a crumb coat, you can see the crumbs through the frosting, but they are stuck and aren't going anywhere. After the crumb coat is done, chill the cake again before putting on the final coat.\nUse a bench scraper to even out the frosting. It's amazing how much better this works than a spatula.\nWhen you are ready to do the final smooth coat, first dip the bench scraper in boiling water to get it really hot. The heat helps to remove many little imperfections.\nUse melted white chocolate as a glue to attach the cake board that your bottom layer is on to the presentation board that you will use to carry your cake around.\nUse bubble tea straws as supports between the layers. Without the supports, the layers could collapse.\nUse a pencil sharpener to sharpen a dowel rod. Secure all of layers together using the sharpened rod; it easily goes through all of the cake and the cake boards. Typically, you would just sharpen the rod on one end. However, to attach the bouquet to the top of the cake, sharpen both ends.\nAlthough not covered in the class, I used a star tip to decorate between the layers.\nTo create this cupcake bouquet, use 19 mini cupcakes on a 3 inch styrofoam ball. If you want to create a pumpkin, turn one of the mini cupcakes upside-down, remove its wrapper, and frost it green. To make the bouquet, follow the same basic technique that I used for my Father's Day Cupcake Bouquet. Since we are using minis, skip the tissue paper entirely and use only one toothpick per cupcake.\nHave you ever integrated cupcakes into your cakes? I'd love to hear all about it in the comments!\nP.S. For those of you wondering which cake recipe I used, it was a pumpkin cake recipe from Edna de La Cruz.\nWhat a fun cake topper!\nI made a Paris Themed Bridal shower cake with High Heel Cupcakes on Top!!! So fun!!!\nThis may be a stupid question but how do you keep the Styrofoam from breaking little bits all over the cake when pushing the cupcakes in? Thats a really cool cake by the way.\nLooking Nice. It looks like anniversary cake. You can get online delivery of anniversary cakes from monginis online shop.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzartan b/data_all_eng_slimpj/shuffled/split2/finalzzzartan
new file mode 100644
index 0000000000000000000000000000000000000000..2f6d207b6c6c08c2fd9da1b8fbf278143d4f2206
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Mar 24, 2018 - French leader tells President Erdogan civilians in Kurdish-majority city ... PARIS, France \u2014 French President Emmanuel Macron on Saturday ... to the security interests of Turkey, \"our ally within NATO,\" Macron said. ... May 13, 2018 ... Israel's 'different' Eurovision winner has a message for Jerusalem too.\n2 hours ago - Turkey's President Recep Tayyip Erdogan shakes hands as his supporters gathered outside his hotel in London, Sunday, May 13, 2018.\n15 hours ago - Tickets are now available for Wine & Jazz Music Festival 2018, which ... Young chefs will spend a week exploring foods from around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your vehicle is a cost effective, moving billboard. Make a statement and project a professional image with custom designed lettering & graphics.\nFrom subtle and elegant, to big and bold, SignWave can make your ad on wheels look great!\nThese are an option for those who may not want their vehicle lettered at all times. Utilizing our full colour digital printer, we can make even your magnetic signs really pop!\nWe also make fridge-magnets....a great form of advertising.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gems, stones, drums, healing artifacts.\nWHICH COME FROM OUR HONEST MOTHER EARTH.\nTRANSFORMATION IS THE PERFECTION OF BEING.\nGEMS CELEBRATE THE MEDICINE MAN, THE \"GREAT SPIRIT\"\nAND THE ETERNAL WISDOM OF HIS SKIN DRUMS.\nI'M GRATEFUL FOR LA PACHAMAMA.\n128 1\/2 N. Larchmont Blvd.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Omega Software Company, which distributes the Locksmith program, claims that their purpose in selling the program is not to have people pirate software, but only to back it up. My only question is: why does Omega supply the parameters for copying programs that already supply backups or that have easily attainable backups? If a company supplies a backup and replacement copies free of charge, and Omega still distributes the parameters for such programs, one wonders whether Omega's rationale really holds water.\nWe established a policy of not providing parameters for programs that meet what we feel are certain minimum standards in the area of backups. We even sent letters to all the major software houses stating our policy and asking for comments. A few have answered, some positive, some native; most have not answered at all. Most of the vendors were really interested in attempting to resolve the problem, while a few were simply more interested in finding fault with us than facing the real issue. In order to abide by these standards. Locksmith is currently being sold with a backup disk included in the original package. Also provided with Locksmith is a manual supplement in which there is a statement of Omega's policy regarding backups and parameters. Incidentally, the rest of our software is sold unprotected.\nOmega's policy, as stated in the Locksmith manual supplement, is an agreement not to publish \"parms\" forprograms whose vendor provides replacement disks with the return of the original for five dollars or less within a short time after receipt; or, in the case of business and utility programs, if an archival disk is supplied in the original package.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the lobster relish: Bring an inch of water to a boil in a large pot with a steamer insert. Set the lobster belly-up on a cutting board. Pierce it with a knife just under the base of the head, and cut it deeply through the belly to kill it quickly. Transfer the lobster to the steamer, cover and cook for 8 minutes.\nMeanwhile, add 1 tablespoon of the olive oil along with the garlic to a small saute pan set over medium heat. Cook until golden and fragrant, then transfer the garlic to a medium bowl. Add the tomatoes, red pepper flakes and lemon zest and juice to the bowl. Sprinkle with salt and pepper and set aside.\nThe moment the lobster is done shock it in an ice bath to stop cooking. When cool enough to handle, pull off the tail and claws and remove the meat. Chop up the meat and add it to the tomato mixture. Drizzle with some olive oil.\nIn another saute pan, heat the remaining 1 tablespoon olive oil over medium-high heat. Add the lobster-tomato mixture and heat it through. Add the white wine and cook for another minute. Remove from heat and stir in the parsley and oregano.\nFor the steak: About 20 minutes before cooking, remove the steak from the refrigerator and let it sit, covered, to come up to room temperature. Mix together the ancho powder, paprika, mustard, salt, cayenne, coriander, cumin and black pepper in a bowl and set aside.\nPreheat a large cast-iron pan over high heat and set the oven to broil.\nBrush the steak on both sides with the canola oil and season liberally with salt and pepper. Sprinkle one side of the steak with the spice rub. Drizzle a little canola oil in the pan; when it begins to smoke, add the steak rub-side down and cook until a nice crust forms, about 2 minutes.\nTransfer the steak to a cutting board and slice it against the grain 1-inch thick. Return the steak to the pan, rub-side up. Scatter the butter on top and broil to medium-rare, about 4 minutes. Remove the pan from the oven and baste the steak with the pan juices.\nAssembly: Arrange the steak and the lobster relish on a plate. Drizzle with 1 tablespoon of the steak juices. Garnish with oregano and parsley.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasnwp b/data_all_eng_slimpj/shuffled/split2/finalzzzasnwp
new file mode 100644
index 0000000000000000000000000000000000000000..3735f6c995a56e991504a3189f73b9297b6f4698
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"House and Cameron. House and Cameron Banner. Wallpaper and background images in the House, M.D. club tagged: house hugh laurie cameron jennifer morrison banner.\nThis House, M.D. fan art contains picha, headshot, and closeup.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Did you know honey can be used to treat acne? Sugar can be used as an exfoliator? Papaya can be used to treat sun damage and milk can help to nourish and smooth irritated skin?\nThat's right \u2013 some of the most effective skin care can be found naturally!\nThe below infographic from Rainbow Labs explains all the natural goodies you can try for common skin issues. As a bonus, there's also 7 all-natural face masks to try!\nAre you ready to begin a career in beauty? Check out Ivy's College of Beauty courses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get on the water!.... with this affordable 3 bedroom Otsego Lake canal-front home.Remodeled in 2011 and gently used including paint, flooring, appliances, siding and roof! Extra lot with space for an addition, garage, or just extra privacy. Cute front foyer\/mud room, large storage shed, new deck, electric fireplace, all furnishings, kitchen appliances & stack washer & dryer included. 150' of private canal frontage, a short motor down to all-sports Otsego Lake....the county's largest offering fantastic fishing and swimming. Just around the corner from Otsego County Park - the perfect place to launch your boat. This is a great opportunity for private, year round dockage & lake access. All measurements to be verified by buyer or buyer's agent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion which govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced which is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved simultaneously. The numerical solver for the Boussinesq equations can also be restricted to implement a shallow-water solver, and the shallow-water and Boussinesq configurations are compared. A particular bathymetry is chosen to illustrate the general method, and it is found that the Boussinesq system predicts larger wave run-up than the shallow-water theory in the example treated in this paper. It is also found that the finite fluid domain has a significant impact on the behavior of the wave run-up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Douse yourself in a compelling scent. Amouage's 'Reflection' for men encapsulates timeless refinement and is undoubtedly masculine.\nIts journey begins with a hint of rosemary and bitter orange leaves before transcending into a floral heart with accords of jasmine, neroli and ylang-ylang. The fragrance settles into a woody base with patchouli, sandalwood, cedarwood. - K.N.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatkzv b/data_all_eng_slimpj/shuffled/split2/finalzzzatkzv
new file mode 100644
index 0000000000000000000000000000000000000000..4cd1739f19f1821f84f4f3986d55ac358c006219
--- /dev/null
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+{"text":"Could a spritz of eau de parfum help to realign your personal energy and create inner harmony? New luxury French perfume house the Harmonist is touting that belief with a collection of 10 unisex fragrances by perfumer Guillaume Flavigny, inspired by the ancient Chinese principles of feng shui. The company's first boutique debuted in Paris on Feb. 15 followed by a U.S. flagship store on Melrose Place in Los Angeles in early March.\nTen scents (in the form of eau de parfum starting at $270 and candles starting at $85) reflect the five feng shui elements \u2014 water, wood, fire, earth and metal \u2014 and were created in pairs to mirror the dualism of yin and yang.\nOffering a fresh take on scent personalization, the company defines each customer's \"personal element\" (such as Yin Wood or Yang Metal) based on birth date, time and place (you can input the data on a tablet computer in the boutique or online at ) to steer a person toward a scent horoscope of sorts \u2014 perfume pairings that are intended to enhance wisdom, creativity, prosperity, socializing or seduction, depending on what a person is seeking.\nAccording to Amourelle, two fragrances associated with the earth element (Royal Earth for yin and Desired Earth for yang) are said to draw out creativity if your personal element is Yin Fire.\nThe company's creative director, Aymeric Wuidart, former international marketing director for Coty, is also the co-founder of Art & Sens, a confidential, Paris-based service that creates bespoke perfumes for private clients.\nFlavigny, the nose behind scents such as Balmain Ambre Gris and Black for Comme des Gar\u00e7ons, crafts the Harmonist fragrances from rare botanic oils and natural ingredients that are sustainably sourced from around the world. Amourelle notes that packaging is eco-friendly and fragrance refills (starting at $170) come in glass bottles that resemble test tubes to encourage reuse.\nParisian product designer and interior decorator Alnoor of Alnoor Design, who has worked with luxury brands such as Christian Louboutin and Herm\u00e8s, created the glass fragrance flacons (in black for yin and white for yang) and designed the interiors, including the malachite countertops, Murano glass chandeliers and lacquered wood furnishings, using materials that represent all five feng shui elements. Amourelle says every aspect of the boutique is precisely positioned according to energies within the space, mapped out by a consulting French feng shui master who wishes to remain anonymous.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Customer's wish is essential for all of us. Money back guarantee We can easily deliver a total reimbursement of the revenue if you're not entirely pleased about the work to your editors. Crafting shall be costlier than a spin and rewrite. The best Get Essays Created available for you Solutions Pre-authored essays needs to be crafted in accordance with the descriptions and requirements of clients. If you wish to receive your essays penned to suit your needs from qualified personnel with an affordable swiftness, you will want to take advantage the help of us. The thesis makes it possible for visitors to demonstrably fully understand the purpose of the essay. You're about the right record when you've calculated your topic and thesis document, it's wise to find assistance through the trainer or professor to make sure that. If it is best to be preparing to compose an appealing, specific essay, you should do homework. In the event it involves hiring a web page that publishes essays to your account, what website writes papers for you you're in fact accessing competent essay writers. Our capable writers are in this article so that you can direct you towards resolving any school occupation. It is only average to worry about getting an online essay copy writer because you can not be without a doubt regardless of whether you're using the effective service plan or perhaps not.\nIn the event you necessitate specialist educational support, call us. Persuasive writing a subject or headline the very first, but you wish to acquire. If you should be intending to come up with a fascinating, specific essay, you're most likely to have to do research. Other than presenting a professional for reasonably priced essay crafting, think about looking it your own self. Basic as that, so make certain before you decide to actually start out searching for somebody write my essay, take into consideration executing it by yourself, you may come across that it must be relatively easy and interesting measures to accomplish. The form of essay you're interested in is going to be made available to you from the timeline provided to you. So for those who have an essay delegated that you want assistance with, you can purchase interesting post internet low-priced from us. An essay is recognized as imperfect without the proper research listing. Check to make sure that it continues on theme from beginning to surface finish. Still difficult it is to compose https:\/\/samedayessay.me\/essay-writing-service articles, our freelance writers can deal at any complexity. Professional freelance writers ought to be settled outstanding. You could have wondered even if it's ideal to acquire essays created for your own personal demands. This content can be in the right way analyzed and will be provided to the pupil upfront so that they can decide if you can find any adjustments required. Your essay posting service is actually advantageous.\nSo should certainly you need to take on faculty essay writer on line, we're only the people who you'll will need to call. Currently, most college students struggle to create an essay over a certain matter. You may easily order completely unique university essays and you should not neglect to explain to friends and relations concerning it. The Positive, the unhealthy and acquire Essays Posted for your requirements Investigation classifieds also combine composing a offer. Descriptions of Obtaining Essays Written and published you have a niche, you've have got to generate a selection for the purpose way you would probably very approach it on your behalf after. The most beneficial Option for Get they Authored for yourself There Is Away from you About In the present day you can certainly title the creator of someone's option to produce a superb scientific study guide depending on the needs you have.\nIn so doing, you may well be a number of you obtain tailor-made essays inside prime gurus on-line. With the pay for expense calculator, go for the kind of company you wish to purchase, the educational amount of the papers, and of course if you need an immediate essay or not. Thus, bear that in the mind as you think about acquiring a type my guide systems. Composing an essay is truly an intricate and approach that's time-having. Our freelance writers have handed 2 assessments in order to develop into a part of our business. They even https:\/\/owl.purdue.edu\/owl\/research_and_citation\/apa_style\/apa_formatting_and_style_guide\/reference_list_author_authors.html enjoy a number of supplementary teaching that would truly persuade us they're best for the task. You won't explore downloaded messages out of the world-wide-web and other materials within our supplier. An organization need to warrant that writers are especially established to properly use the a number of citation styles in order to satisfy the plan and styles offered by potential customers. Our site can be an formally taken essay composing organization with several products and services to convey.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You will need to be logged in before you can edit a page. Look for the login link at the bottom of your website, in the footer section (underneath the College of Medicine logo).\nNow that you are logged in, visit the page you want to edit and click on the \"Edit\" button to open the page.\nThe edit section works like Word in that what-you-see is what-you-get, so you can use the toolbar to make your text bold, italicized, insert images, align the text, or insert links to other pages.\nFor instructions on adding images, visit this page. For adding files (Word docs, PDFs, etc.), visit this page.\nOnce you are done making edits, scroll to the bottom of the page and hit the Save button and you're good to go!\nIf you need any assistance with editing a page, please get in touch!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Retain complete rights of the publishing world.\nAre you concerned whether our ghostwriters will steal your manuscript or perhaps they might sell it to a third-party? In that case, your doubts come to an end here. We provide copyright certificates to our clients as and when they demand for it. Hence, it is a legal evidence that will hold you the sole owner of your manuscript\/ book. Our ghostwriting services are designed to help you with writing, proofreading, editing and publishing your book, giving us no right to the ownership of your book. The ideas, concept, and plotting will be yours while our ghostwriters will translate and transform your ideas into an error-free and technical literary language\/ writing. In some cases, the author might wish to acknowledge the services of a ghostwriter and add his\/her name into the acknowledgments section of the book. However, the decision is completely upon the author's will and is backed up with our consent by all means.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Will I have to hand over my list of agents?\nNo. If one of your agents is not on the database you can ask us to consider them for inclusion but this is up to you.\nWill my profile be visible to other schools?\nWhat does an 'Approved' agent mean?\nWe have been running the British Boarding Schools Workshop since 2006 and it is a business networking forum for schools and agents. Before they are allowed to attend, all the agents are carefully selected, screened and reference-checked. We take this role very seriously and this is how we have built up a reputation for running a high quality event with professional, reputable agents. These agents have formed the basis of Connected and we will continue to expand the network of agents but all agents go through the same rigourous process.\nYes. We will invite member schools to give us feedback if they experience a problem with an agent. Agents will have a code of conduct to adhere to. We want to enhance standards in our sector.\nHow can I find agents on the database?\nYou can search in a variety of ways such as country they are located in, countries they recruit from, number of pupils placed, educational programmes of interest. You can then view detailed profiles about the company and its staff.\nHow can I send messages to the agents?\nYou can send a message to all of the agents or to targeted groups. For example, you can inform agents in target countries that you have vacancies in certain year groups or update them about news. The benefit is that you can target messages to different groups of agents. in some cases it is appropriate to send a message to all agents on the system, for example, if your school has a new Head or has opened a new facility. In this case you can post a message and it immediately gets sent out to all agents via a notification in their email box.\nHow does the system work if I only work with my own selection of agents?\nYou can set up one or many groups of your agents, for example, by country or performance related. Then you can quickly send messages to those groups and post news items to your profile which your agents will see.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Thanksgiving was spent with my family in Pennsylvania, a sweet time of reunion!\nChristmas \u2026 a very relaxed holiday and fun time with friends and loved ones. Lee and I did our Christmas together Christmas Eve, then got together with Shawn and Rachel and kids Christmas Day for a sourdough brunch. Later that evening, we joined up with friends from church for dinner.\n\u2026 we tried a Christmas kiss, Neka decided she wanted in on the kiss too!\n\u2026 Happy New Year's from the Phelps!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Next week I will be meeting Florian, a successful photographer who lives together with his girlfriend in a two-floor apartment in Amsterdam's city center. The couple has a roof terrace where Florian started his own kitchen garden. There he cultivates tomatoes, potatoes, corn, sprouts and other fruits and vegetables. I am really courious to meet him and learn more about his urban farming project. Read more about Florian and his plants and other interesting articles at Karlijn's First of August blog.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Rose Is A Rose Is A Rose Is A Rose\u2026. or is it?\nDid you know that roses are just like people? Where they have been planted to grow up makes all the difference in the world. Not just the genealogical aspects (with all the various Latin names associations) but the actual location plays a part in how the roses grow up.\nTake for example the Red Rose Oil. The Reverend (ABRD) offers at the EcoVillage a Red Rose from the U.S. and one from the U.K.. These are extracted from red roses, but their fragrances differ\u2026 though slightly. The Red Rose Oil from the U.K. has a slightly less rosy-sweet aroma, a bit more earthy-herbal. The Red Rose Oil from the U.S. is slightly sweeter, slightly more rosy in fragrance. Both are beautiful true rose aromas.\nAnd then there are roses which are totally different in fragrance. Rose Oil from Somali has a rich, spicy fragrance. The aroma of rose is there, but the spiciness really sings out. Somali Rose Oil is also dark brown in color and is very thick (the red rose oils from the U.S. and the U.K. are slightly greenish-clear in color with thinner oily textures). Another spicy Rose Oil offered by the Reverend (ABRD) is Black Rose Oil. This Rose Oil is similar in fragrance and texture to the Somali Rose \u2013 but not quite (though almost) as sweet.\nA Rose Is A Rose Is A Rose Is A Rose\u2026 in walks Rose Absolute. This Rose Oil is a pure extract which has been processed in such a way as to really bring forward an extremely fragrant rose fragrance. But there is a price for this rose extract. The EcoVillage offers a 1\/3 oz. bottle of Rose Absolute oil for $200. The rose essence oils mentioned above are $15 for a 1\/3 oz. bottle. Essence oils make the gifts of nature very accessible for all to receive it's beautiful fragrances in oil form.\nA Rose Is A Rose Is A Rose Is A Rose\u2026 The bounty of nature hands all \u2013 a bouquet of many roses to enjoy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SKOOLCOOLKID666: Hello, I need change my pasword, where do I do that?\nZe_Karlos: I wanted to buy the package but I live in Brazil, how do I do it?\nZe_Karlos: How to pay in my currency, in the case of the real?\nPlease login below or register an account with DayZ - Community forums.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We accomplish your Transponder & Altitude encoder tests using the latest \"state of the art\" test equipment from Aeroflex, the IFR 6000. In addition we are certified for installation, repair, and calibration of all models of transponders and altitude encoders. All work is accomplished by an FAA certified AP-IA mechanic (not all repair stations use certified mechanics).\nTransponder tested for proper operation per CFR Part 43, Appendix 'F'.\nAutomatic Altitude encoder Data Correspondence test per CFR 91.217 (required for new installations or when altimeter or altitude encoder is repaired\/replaced).\nMore info can be found on our FAQ page.\nWe use the Aeroflex IFR 6000 to test all modes of transponder operation including Modes: A, C, S, Altitude reporting, ADS-B in\/out, UAT and DME.\nAltitude data is supplied to the transponder from the altitude encoder or digitizer which then transmits that information to ATC, this altitude is displayed on the IFR 6000 during testing.\nThe accuracy of the IFR 6000 enables us to easily meet or exceed all manufacturer specifications.\n(b) All airspace. Unless otherwise authorized or directed by ATC, no person may operate an aircraft in the airspace described in paragraphs (b)(1) through (b)(5) of this section, unless that aircraft is equipped with an operable coded radar beacon transponder having either Mode 3\/A 4096 code capability, replying to Mode 3\/A interrogations with the code specified by ATC, or a Mode S capability, replying to Mode 3\/A interrogations with the code specified by ATC and intermode and Mode S interrogations in accordance with the applicable provisions specified in TSO C\u2013112, and that aircraft is equipped with automatic pressure altitude reporting equipment having a Mode C capability that automatically replies to Mode C interrogations by transmitting pressure altitude information in 100-foot increments.\n\u00a7 91.217 Data correspondence between automatically reported pressure altitude data and the pilot's altitude reference.\n(b) No person may operate any automatic pressure altitude reporting equipment associated with a radar beacon transponder or with ADS\u2013B Out equipment unless the pressure altitude reported for ADS\u2013B Out and Mode C\/S is derived from the same source for aircraft equipped with both a transponder and ADS\u2013B Out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauuzx b/data_all_eng_slimpj/shuffled/split2/finalzzzauuzx
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+{"text":"Injury victims in Minnesota should be careful to avoid common misunderstandings about fault laws, claim deadlines and the availability of legal assistance.\nMany people in Minnesota are aware that the victims of wrongful accidental injuries may be able to hold other parties liable. By filing personal injury claims, these victims may be able to recover compensation for direct financial losses as well as less easily quantified ones. Unfortunately, though, many Minnesotans may miss out on seeking appropriate recourse due to the following common misunderstandings about personal injury claims.\nAccording to the Minnesota Revised Statutes, people who contributed in some form to their own injuries aren't always barred from seeking compensation. The state allows people to seek compensation if they were less at fault than another party was. Any damages that an at-fault person recovers, however, will be reduced based on the amount of fault attributed to him or her.\nPeople who have been hurt in car accidents should note that fault has no impact on insurance claims for more minor incidents. According to materials from the Department of Public Safety, Minnesota requires all motorists to carry no-fault insurance. This coverage allows injury victims to make claims against their own policies for lost wages, medical costs and more, regardless of who was at fault.\nPeople familiar with the state's no-fault auto insurance laws may think that fault is never a factor in car accident claims. However, this isn't the case in serious or catastrophic car accidents. If the medical costs resulting from an accident exceed $4,000, a victim may make a claim against the at-fault party. This option is also available in accidents that cause permanent injury or disablement.\nClaims against the state or its employees - victims must give notice that they are making a claim within 180 days.\nClaims against municipalities or their employees - the deadline for giving notice in these cases is also 180 days.\nClaims against personal auto insurance policies - injury victims only have six months to make no-fault claims.\nThese shortened deadlines make identifying fault and determining what type of claim to file an immediate concern for most injury victims.\nInjury victims who face substantial expenses often think that they can't afford to seek the assistance of an attorney. However, many attorneys offer free consultations and work on a contingency basis, in which payment is collected from the final settlement. Often, the expenses associated with working with a representative are favorable compared to the potential cost of going without a settlement.\nThe complexity of the personal injury claim process, along with the potential for harmful mistakes and misunderstandings, underscores why legal representation can pay off. An attorney may be able to help a person understand the relevant laws, avoid common pitfalls and pursue the maximum amount of compensation available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scenic Bluffs Community Health Centers provide affordable, high-quality healthcare to patients. If you do not have health insurance, are unemployed, or have an income level that makes it difficult for you to pay for health care, we offer a sliding fee scale called the Healthy Neighbor Plan. Depending on your family's size and income, your charges for health care could be reduced. This is a completely confidential service provided to you by Scenic Bluffs Community Health Centers.\nIf you qualify, you will be eligible for reduced fees incurred at Scenic Bluffs Community Health Centers. These discounts are applied for most services in our dental, medical, behavioral health, and chiropractic departments, as well as pharmacy (see pharmacist for more details).\nApply by filling out an application and submitting it with all of the required proof to process your application. It usually takes a few days to determine eligibility. You will be notified of your status via letter. Once you are approved coverage starts at your very next service. If you have insurance coverage, we will bill your insurance company first and apply the discount to the balance of your account after the insurance payment or denial has been received.\nYou will be responsible for updating your application on a yearly basis or as your income changes. You are required to pay each day you receive services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a programmer, engineer and designer interested in combining technology and art to create new user experiences. I am particularly interested in the novel application of existing technologies to further the fields of interactive advertising and installations.\nI recently graduated from the Media Arts & Technology program at the University of California Santa Barbara with an M.S. in Multimedia Engineering. My Masters project looked at Twitter and involved catching and visualizing deleted Tweets to let viewers muse on the reasons why people try to take back things they have posted for the world to see.\nLately I have been designing and building interactive systems for installations at MindshareLA and other tech events. I also work with Syyn Labs on various tech+art projects ranging from music videos and commercials to interactive installations.\nInformation on recent projects is available on my portfolio page as well as my play-by-play projects blog.\nWorking with a startup to build a proof-of-concept prototype of their web application in Ruby on Rails to help them secure angel funding. Primarily database design and initial user interface development.\nPrincipal database and user interface designer for a literary social network built on Ruby and Rails.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"T'is the season where our attentions turn to holiday traditions and of course, music. Can we start a thread of favorite holiday music?\nOf course I am selfish in this quest and looking to add to my music collection. Does anyone have suggestions for classical holiday music (artists, titles, renditions, formats).\nJohnny Mathis -Merry Christmas is at the top of my list.\nI was listening to Music Choice the other day-Songs of the Season. I heard a great song. I found it was a duet between Carrie Underwood and Elvis. Its from a new CD of Duets with various artists and the King. Pretty cool. Working on getting one.\nWhen you say \"classical\" do you mean as in classical (i.e., \"serious\") music?\nSuperb in every respect, including sound quality.\nLets include all our holiday favorites regardless of genre.\nHere are my other four favoritest of all (FWIW).\nI just bought Aimee Mann's One More Drifter in the Snow. Haven't had a chance to listen yet though.\nI had heard Aimee Mann's version of \"Chestnuts Roasting...\" on a various artist CD from Star bucks. There is such a nice quality to her voice. It was very enjoyable.\nI am presently sampling Stevan Pasero \"Christmas Classics for Guitar\" on Rhapsody. Very nice. I was not sure about the start to \"Messiah, Allelujah\" but as I listen ... it will likely join the collection.\nI have to add Bob Rivers comedic version of Chestnuts Roasting on an Open Fire.\nIts great-Well done. Sometimes I put this on as background music and try to see how many people catch the changes lyrics.\nGo ahead click on the Amazon link above and give #2 a listen.\nWhen I get home and pull out the Holiday CDs I might post some more as I remember them.\nThat has the opportunity to be quite good, I'd think.\nSeriously: TSO's DVD \"The Ghosts of Christmas Eve\"\nNot serious: Spinal Taps \"Christmas with the Devil\" or Bob Rivers \"Santa Clause\"- sung to the tune of Black Sabbath's \"Iron Man\"\n...I heard a great song. I found it was a duet between Carrie Underwood and Elvis. Its from a new CD of Duets with various artists and the King. Pretty cool. Working on getting one.\nThe Elvis CD is called Elvis Presley Christmas Duets only released just last month-er I mean in Oct.\nStan, one of the classical music highlights for the Christmas season is Tchaikovsky's Nutcracker. I own four complete recordings and have listened to about a dozen more. You certainly should have at least one complete recording of this glorious music and I'll suggest a terrific bargain now available that includes not only The Nutcracker in one of my favorite performances by Lanchbery, but also Sleeping Beauty and Swan Lake in a 6-disc set. Due to recent strength in the U.S. dollar the UK dealers have been a good source which I've used, and now Presto Classical has a sale on this set for about $19 plus $3 shipping.\nAlso, is there a rendition of Handel's Messiah that would be recommended?\nAnother CD I listen to each year (and sometimes throughout the year), although not traditional and not at family get togethers and certainly not while the kids are around. The soundtrack to Mr. Hankeys' Christmas Classics.\nI was last night listening to Manheim Steamroller's Christmas. Different but very nice. Excellent version of Silent Night on that LP.\nDo you prefer a period-correct version or big and bombastic? If the latter, the Malcom Sargeant version is an old fave. That's the version of Hallelujah that you'll usually hear.\nThere was a good thread at AA on Messiahs that may be of interest.\nHandel's Messiah...The best copy I have is a SACD from FIM (First Impression Music) but it may be out of print now. There are litereally hundreds of versions out there...and most of them are pretty good.\nIt may sound corny to many but I really enjoy The Carpenters \"Christmas Portrait\". GREAT vocals and good traditional Christmas music.\nThe early Mannheim Steamroller Christmas albums \"Christmas\" (1984), \"A Fresh Aire Christmas\" (1988) and \"Christmas in the Aire\" (1995) are some of my favorites. Their recent stuff just doesn't seem to be as good.\nNot Christmas per-say...but I enjoy George Winston's \"December\" this time of year.\nAnother of my favorites is Boston Pops with Arthur Fiedler \"White Christmas\". Just good classic Christmas tunes.\nAhh, now if only George Winston would release on SACD-I'd buy the entire discography again.\nI liked George Winston a lot until I saw him perform live at the Meyerhoff. He went through this phase where he would strike the same note (TINK.TINK.TINK.TINK.TIMK.TINK...) superimposed with the music. I understand what he was trying to do and it works sometimes but done too often it becomes annoying (TINK.TINK.TINK...). I do still like most of his music and an SACD would be great.\nHa, I know exactly what you're talking about- I also saw him live and have the live performance DVD (Low Vid Qual) but it kinda grew on me-its different-but I'm usually in a zone anyway.\nI actually have \"December\" on the origninal Hi-Rez format...Vinyl...as well the original CD and the remastered CD version.\nI had forgotten that I actually have quite a few GW recordings, some vinyl and some CD. Some were my wife's, some mine and we married our fortunes together. Don't shoot me but I actually ripped my vinyl recording of December to CD for convenience.\nYo-Yo Ma & Friends Songs of Joy & Peace.\nI enjoy the Trans-Siberian Orchestra Christmas stuff because it's like classics that are a bit rockin'.\nAnd also I noticed that The Canadian Tenors (think Il Divo-ish) have the free single of the week on iTunes and it's a version of \"Silent Night\". So free is good, and I thought it was good.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Conveniently freshly flaked in a 500g Bag. Gran Moravia is a hard Italian cheese produced with traditional copper vats and then matured slowly to produce an intense and nutty taste.\nSuitable for Vegetarians. Halal certified. Naturally lactose free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Executive Magazine takes an up close and personal look at the latest horological release from MB&F, the Horological Machine No9 'Flow', inspired by aerodynamics and automotive design.\nWith a design led by form over function, taking inspiration from aerodynamic principals and futuristic retro automotive shapes, the brand new MB&F No9 'flow' looks like a mechanical piece of art. Taking cues from 1950's aeronautical styles and retro representations of a futuristic utopia, the time piece is like nothing we've ever seen before.\nMB&F are globally renowned for their audacious designs, and the HM9 is no exception, with its trademark unconventional form, transgressive case shapes, and artistic use of exotic materials. So pronounced are its curves, and acute are its angles, that new manufacturing standards and techniques needed to be developed to achieve its fascinating finish.\nWhen you take a closer look at the watch, the level of detail is staggering, from the complex grade 5 titanium case enveloping the movements and complications, to the winding mechanism and the elongated domes of milled sapphire crystal. The dramatic geometry of HM9 is supported by the equally dramatic contrasts of finished surface, with machined titanium, polished lines and satin finished details.\nThe engine of the HM9 has been entirely built in house at MB&F as a result of three years of development, and features independent twin balance wheels that beat at their own independent cadences of 2.5Hz (18,000bph) on each flank. A planetary differential averages the output of both oscillating wheels to provide one stable reading of the time. The small and simple dial sits perpendicular to the mechanism, with the hands driven by conical gears which ensure precise engagement, even when motion is put through a 90\u00b0 planar translation. All of which results in the most accurate time keeping possible.\nHM9 makes it's debut with two titanium editions, limited to 33 pieces each: \u2013 The 'Air' edition comes with a dark movement and aviator-style dial; \u2013 The 'Road' edition has a rose gold plated movement and a classic speedometer-style dial.\nMeasuring 57mm x 47mm x 23mm on the wrist, this is a large time piece by any standard, and certainly makes for a dramatic statement. The strap is hand-stitched brown calf-leather with a custom-designed titanium folding buckle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since taking office, the one thing President Tsai Ing-wen (\u8521\u82f1\u6587) has been reluctant to do is break the decades-long ambiguity surrounding the \"status quo,\" the one term that manages to trump the so-called \"1992 consensus\" in terms of the variety of definitions given to them.\nThe lack of consensus on what exactly the \"status quo\" is has bedeviled cross-strait ties, as it has allowed the leaders on both sides of the Taiwan Strait to accuse each other of \"changing the 'status quo'\" without ever explaining what parts have actually been changed, causing tensions to escalate easily and quickly.\nIt is risky to talk about maintaining the \"status quo\" without first having a clear, unanimous definition of what it is. Although the ambiguity around the term has given political leaders and government officials more leeway in handling cross-strait issues, it could result in misinterpretations of what the majority of Taiwanese really want, causing misjudgements and raising the risk of moves that could carry dire consequences.\nWhile most opinion polls have shown that the majority of Taiwanese support maintaining the \"status quo,\" the truth is, as with the \"1992 consensus,\" few of them know what they are actually supporting and even fewer bother to find out.\nAsk any political pundit and they would say that the \"status quo\" \u2014 whatever it is \u2014 cannot be maintained forever. Retaining \"maintaining the 'status quo'\" as a possible option for cross-strait ties will only cause the public to remain in a state of denial and delay having the necessary serious discussions about an issue that cannot be indefinitely avoided: Do I want independence or unification?\nGiven the renewed support for the KMT, as evidenced by last year's local elections, and the CCP's ramped-up effort to push for unification under a \"one country, two systems\" formula, next year could prove to be a make-or-break year for Taiwan's sovereignty.\nAgainst this backdrop, Taiwanese society must begin a dialogue on the independence\/unification issue before next year's presidential election. The DPP's planned resolution might determine the course of that conversation.\nA spate of \"fake news\" believed to be fabricated by Chinese netizens has prompted the Presidential Office and the Executive Yuan to call a meeting to discuss countermeasures, a government official said.\nThe official, who spoke on condition of anonymity, said the Presidential Office, the Cabinet and national security agencies held a meeting to discuss countermeasures after suspected intervention from China in many of the government's policy proposals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to some guys who landed flying saucers at the last meeting of the Flat Earth Society in Utah, where Morgan Frank acts of Grand Superintending Vision Caliph, the rates of inflation Dr. Perry has used in this chart are way too low.\nTherefore, in gold terms, indexed to Shadowstats Slide Rules, college tuition is actually less than 20 years ago.\nHow are the college tuition numbers measured?\nIt is well known that over 50% of students don't pay the \"list price\" of college. I've only seen one study that tried to figure out what the actual cost to students are.\nColleges perform tremendous price discrimination.\nHow about the cost of government, total of all levels?\nmore foolish babble from you, what a shock.\nafter the way you embarrassed yourself comment on sharecounts, i figured maybe you'd pipe down for a while, but i see that shame is no more a part of your makeup than intelligence.\ni think the varnish fumes from your barstools have finally killed your last couple brain cells.\nIt'd be nice to see a graph showing the cost of key necessities: food, clothing, entertainment, transportation, housing, education, and medicine.\nI suspect the last 3 would be rose crazily since 1970 while the others flattened or declined. I also suspect opinion polls would confirm that the last 3 sectors were a mess, but the others are humming along. Last, the evidence would show the heaviest Federal involvement in the last 3 sectors.\n\"It is well known that over 50% of students don't pay the \"list price\".\nt11s, please provide information to support what is otherwise grand speculation.\nI'm not sure how you get to 5.8% as the current rate of growth of the price of medical care. When I do the math, the percentage increase from a year ago comes to 2.9%, which is what is also found on the BLS website.\nI'm calculating the average annual compounded rate of increase from 1978 to 2011, over a 33-year period.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our focus is on safety and crisis stabilization; adolescents generally come here when they are struggling or experiencing a crisis at home, in the community, or at school, and are dangerous to themselves or others. Many have exhibited behavioral problems, such as aggression or substance use.\nWhile in our care, adolescents quickly come to understand that they are in a safe, warm, welcoming environment, and are encouraged to talk about any problems and behaviors that are troubling them.\nDuring each adolescent's stay, we focus on crisis stabilization so that the adolescent can move back into their home community as quickly as is safely possible. The average length of stay on our unit is five to seven days.\nWe treat teenage boys with a variety of behavioral challenges and diagnoses on our unit; many have depression, other mood disorders, schizophrenia, anxiety, personality disorders, oppositional defiant disorder, or first-break psychosis; some will also be experiencing problems with substance use.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Maldon Appliance Repairs. We provide affordable domestic appliance repairs in Maldon and surrounding areas all from just \u00a349 with any parts required supplied at trade price!\nOur fees are from just \u00a349 and we supply parts at TRADE PRICE. Call 01621 473 037 today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"They also shout Azaadi, Azaadi.\nRest of them will shout, K is our integral part while the editing and deleting continues.\nand crying in the midnight for?\nIs it crying of the departed beloved?\nOr the beloved deserted it and went away?\nthat is beaconing for death?\nAjmal Khan A.T is a bilingual writer who write in English and Malayalam. His articles and poems have appeared in Muse India magazine, India Cultural forum, Countercurrents.org, Resist India, Round Table India, Two Circles.net, Sanhati.com, Ignite South Asia, Cafedissensus.com, Kafila.org and Kindle magazine in English and in Madhyamam Weekly, Thejus Weekly, Shabab Weekly, Risala Weekly, Thejus Malayalam daily, Siraj daily, Varthamanm daily. He is also in the editorial board of Refugee watch On line journal published from Calcutta and student magazine Aaghaaz.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WORLD HOLOCAUST FORUM | January 29 \u2013 February 1, 2008. EJC and RJC President Meets with Israeli Leadership and Prominent Politicians.\nJanuary 29 \u2013 February 1, 2008. EJC and RJC President Meets with Israeli Leadership and Prominent Politicians.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Business company AUTOMOTIVE HOLDINGS, LLC is a legal entity registered under the law of State Nevada. Company is located in the register with the Company number LLC3608-2003 and with the national number of State Nevada NV20031037921. This legal entity was firstly registered on 13th March 2003 under the legal form of Domestic Limited-Liability Company. Business activities of this company are managed by Managers. Current company's status is Revoked. Company has expired at 13th March 2503.\nThe company AUTOMOTIVE HOLDINGS, LLC is managed by 3 persons in total. The persons responsible for business activities are GARY KORNFELD with the seat at 318 N. CARSON STREET, CARSON CITY, 89701, NV as Managing Member , GARY KORNFELD with the seat at 318 N. CARSON STREET #208, CARSON CITY, 89701, NV as Managing Member , DUPUIS M THERESA with the seat at 318 N. CARSON STREET #208, CARSON CITY, 89701, NV as Managing Member .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RevDL.me \u2013 Download Augmented Reality Girls Trinary v1.0.13 Mod Apk latest version for mobile free download Download Game Android premium unlocked files, android apk, android apk download, apk app download, apk apps, apk games, apps apk, best apk games, free android apk, free apk, game apk, game hacker apk, free crack games, mod apk games, Android Mod Apk, download game mod apk offline, mod apk revdl. Augmented Reality Girls Trinary v1.0.13 Mod Apk the latest version game for Android #ANDROIDGAMES and Blackberry 10 #BB10 OS with direct link apk only for free download and no pop ads.\nKOEI TECMO GAMES and TOEI ANIMATION sell new original content by fusion of application and animation.\nThe five heroines belonging to the Special Force \"Trainer\" confront against the mysterious phenomenon \"Phenomenon\" threatening Tokyo in 2016!\nYou are supporter to support such hearts.\nLet's support their hearts from the front and from Ula, while worrying in the battle but progressing forward.\nExtended Girls Trainer is a new content linked \"app\" and \"animation\".\nIn animation, the trainee's image of fighting to solve \"Phenomenon\" and the calm daily scenes that they weave are drawn.\nAnimation is a short animation format of around 5 minutes, and a new story is distributed every week.\nYou can see in the game the little everyday things not drawn in animation and the troubles they are unable to know.\nPlease learn more deeply about them through a small everyday life and get close to each other.\nIf your support gets to the heroine 's heart, they will be stronger.\nIn the battle, troubles and trauma of the heroine embodies as an enemy, and it goes around in her heart.\nLet's defend her heart by dive into the hero's heart and cooperate with friends called servant crun to defeat the enemy.\nA servant clan is a companion that can be obtained with battle rewards and gacha (free or charged) etc, and it becomes easier to develop the battle because it becomes stronger when it is nurtured.\nAlso, some game stories have something that you can not see unless you clear the \"mission\" prepared for the battle.\nOrganizing a stronger servant clan is the key to the strategy, so let's actively foster it.\n* Please understand that we can not provide support other than supported OS version.\nYou can also check notices in the game on the official website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NW Wisconsin woodland owners, especially those not living on their land, are encouraged to check with their woodland neighbors, friends, forester or personally check their land after the devastating storms hit the area on the evening of Monday, July 11.\nGovernor Scott Walker declared a State of Emergency for eight counties: Ashland, Bayfield, Burnett, Douglas, Iron, Price, Sawyer and Washburn. Some areas received 8-12 inches of rain within a few hours of time. Many state highways and other roads were damaged or washed out and Saxon Harbor in Iron County was destroyed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\"Whether you want to start your own successful home business, take surveys for cash, or a simple job from home that pays you, you are in the right place!\"\n\"Nobody beats you! For 3 years you have been our pick for the honest reviews!\"\n\"Your number one pick is definitely the best business plan online! After attempting many other programs, I came across Cash Power Course and my results were far greater then I expected! The people who put the program together defiantly did their homework. Everything was included with their program. I think the program should be going for a lot higher price than it is. I honestly believe anyone can make great money with this business. Cash Power Course is the hottest thing out there!\"\nsent three of my friends your way\"\n\"Thank you for being real! I just about gave up!\"\n\"I just started your #1 pick about 27 days ago, and at this point I made over $4359! Thank you!\"\nMuch higher income potential: $3,000-$50,000+ per month. With legitimate people making as much as $100,000 a month. Yes, really. Because it is your business, you have 100% control of your results. However, you have a lot of freedom once you get up and running. This is because the business is mainly automatic, once you have it set up properly.\nAll you typically have to do is follow a step-by-step business plan, that is proven to be effective.\nHome businesses are the best choice for making the most income from home.\nThis is a great way to earn income from your home, if you do not want to start your own business. This is because you are paid by companies for doing simple tasks from your home. These tasks range from proof reading to assembly.\nThey are not complicated. Not that starting your own home business is, but these are generally very easy tasks that just about anyone can do. Over 1,500 job listings are in our #1 pick.\nIt has been very hard to find real jobs from home, UNTIL NOW!\nto hire you to do different, easy tasks from home.\nIncome potential can be anywhere from $100 to $10,000 a month. (It just depends on how many you do). You control when you work. The more you take, the more you make.\nAll you have to do is register with companies, and they start sending you invitations to participate in surveys. This is very easy for anyone to do. There is no need for any kind of computer experience.\npaying surveys available today. They are a good choice for you.\nCash Power Course is the premier, new program available online. They teach new and different techniques for making money from home. No more running across the same old worn out methods with this course.\nNot only is the business designed with the beginner in mind, but no additional out of pocket expense is required. Plus, you do not need to have your own web site, or product, to succeed with Cash Power Course.\nThe founders of this program made over $2,400,000 in one year. We have verified that amount. The best part is that they teach the same techniques they used to accomplish this. They also have a great customer support team, and a money back guarantee!\nUnlike all the other Twitter programs out there, that only teach you how to gain followers, Paid 4 Tweetn actually teaches you how to develop relationships with your followers. Twitter is after all, a social networking site. Having followers that have no interest in what you are offering, is not the way to make money. Paid 4 Tweetn teaches you tried and true, effective selling techniques.\nAnother thing that makes this program so great is that there is no additional out of pocket expense required. Most of the other Twitter programs require you to sign up for additional, costly products.\nIf your are looking to make money using Twitter, look no further. We have tried all of the Twitter programs out there, and Paid 4 Tweetn is the only one that delivers.\nWith great customer support and a money back guarantee, you have nothing to lose.\nLiterally billions of dollars are being exchanged online every single day. Six Figure Yearly, teaches you how to tap into that exchange, and bring home some of it for yourself. With the founders making a verified $1,000 to $3,500 a day using the techniques they teach, these experts teach you how to accomplish the same results, working only a couple of hours a day.\nThe fact that this program has been selling consistently for over two years, demonstrates how effective it really is. Not only is it a proven program, but their excellent customer service team will help you succeed.\nSix Figure yearly is well written, and easy to follow. They also have a 100% money back guarantee.\nAlthough this is 5th on our list, it is still an excellent program. If you notice the graphs, you will notice that the variance between each of the programs listed is minute. You could easy do just as well with this program, as with the first one on our list.\nBig Weekly Earnings will teach you time tested techniques for making money online. With a well written program, just follow their step by step instructions, and you are sure to succeed. However, if you stumble along the way, their experienced customer support team will quickly get you back on track.\nThey are so confident with the techniques that teach, because they are used by thousands or people earning great money from home, that they offer an eight week, unconditional, money back guarantee.\nFinding a real home job used to be next to impossible. Search no further. Our number one pick has the largest selection of home jobs that we have seen anywhere.\nThey offer three different levels of membership, to suit everyone's budget. They have set the standard for finding home jobs that pay you to complete tasks from your home. With so many different types of jobs that they list, there is sure to be something for everyone.\nWith a full money back guarantee, great customer support, and an optional placement service, Home Job Group is a great resource for finding a home job, if starting a business is not for you.\nIf you are looking for surveys that will pay you cash, you have come to the right place! Classified Surveys has the largest list of the highest paying cash surveys out there. They are our number one pick, because of their user friendly members area. This lets you get started quickly!\nAlthough doing surveys from home is a time tested way to earn additional cash in your spare time, surveys are here to stay. Businesses will always be looking for consumers opinions, so that they can better serve their clients.\nAlthough Classified Surveys is extremely similar to our number two pick, they took our top spot due to the tools that they offer their members, helping them fill out the surveys faster.\nMy goal is really to help steer you in the right direction, when it comes to working at home. You need real work at home opportunities, and real work at home jobs. That is what I offer you here.\nI wish you the best of luck in your new career from home! Don't forget to join my scam report!\"\nP.S. Many of you have asked when I am going to offer the scam team opportunity again. It is back now! Click here for details!\nGet my FREE Scam Report Below!\n\"Thank you again for visiting. I know that you may be in a similar financial situation as I was in years ago. With the current economic situation, I know it is tough out there. That is why you will find that a majority of people are looking online for work. There are so many choices out there, and it is very hard to find ones that work.\nThis is why I have dedicated my time to sorting through all of the garbage, and list the best ones that I come across on this site. You don't need to just take my word for it though. There are other people's comments on this site as well. On top of that, the sites I have listed here are also listed on other respected review sites.\nI want you all to know that not too long ago, I was in similar shoes as you. Looking frantically online for a way to work at home. 2007 was an emotional year for me personally. I lost a loved one, and I really had a chance to focus on what was important in life. When you hear someone that is 96 years old tell you that life flies by, it really makes you think. I made a promise to myself that I would refocus my energy on helping out all of the good hearted people like you, that visit my site. I was already having success with the businesses that I list here, but the fact was... they were VERY hard to find. I designed this site for all of you, so that you could find legitimate business opportunities, and ways to find home jobs easily. Ones that work!\nI understand, and know why you're here.\nThe reason why you are here doesn't matter. The fact that you are here does. Although I have become financially secure, by using the businesses I list on this site, I feel even better when I get all of the emails from you telling me of your success! That is what drives me. Anytime I get a new business, or home job to review, I review it with you in mind. I remember what it was like to be in your shoes. That is something you never forget. I will never stop trying to protect those that are ready to make a financial change in their life. I will never stop looking over opportunities that average people can understand, and put to practical use.\nThe businesses listed here have not just been good to me, but they also produced great results for others.\nI will leave you with these words; Working from home is not just a 'pipe dream'. If you have had a bad string of luck with the businesses you have tried, I can only say to you - never, ever, give up hope. If I would have given up hope years ago, I would have been in huge personal, and financial unrest. As a matter of fact, in my humble opinion, the only difference between the ones that make it, versus the ones that don't is just a mind set. The mind set is to never give up on your dreams. People are working from home, they are successful, and my hope is that you will choose to be one of those content people.\"\nBoth our first, and second choice in this category are great. You can have great success with either. However, due to the three tiers of membership, Home Job Group prevailed. Since finding a legitimate home job on your own is next to impossible, both of these sites offer a valuable service.\nThe Internet is loaded with a bunch of junk and scams. Most home job opportunities fall into these categories. Both of our selections will help you to avoid shelling out your hard earned money, with nothing to show for it.\nHome Job Provider has strong customer support, and an advanced placement assistance service. This program is a great value for the money!\nEven though New Edge Surveys is our number two pick in this category, you can still make very good money using their service.\nYour income will depend upon the number of surveys that you choose to take. There is no limit to the amount that you can do.\nThe need for your consumer opinion is growing every day. The only way for business to find out how to better serve their clients is to ask them their opinions. Surveys are becoming more popular, and are here to stay.\nNew Edge Surveys will offer you great customer support, and a money back guarantee.\nIf you are new to working from home you will have to deal with such issues as family interruptions, friends, neighbors, television, radio, visitors, email and phone calls.\nYou must structure your time and set a schedule to work on your home business. If you were working for someone else you would not expect your friends and neighbors to stop by for a visit.\nYou must clearly explain to the people you know that this is your job and your can not stop to chat simply because you are home.\nPeople may think that your job is less serious because you are working from home.\nYou must remind yourself of the reasons why you choose to bring your profession home. Was it to have more free time with your children, to earn more money than your previous job paid, to create more freedom for yourself and your loved ones?\nRemember your home business should be treated as a real job. You are the person who makes the decisions now. You must organize your time without someone else telling you when you should work and take care of your business.\nSchedule a portion of your workday to answering emails and written communications and returning important phone calls.\nOne hour in the morning and one later in the day should be fine. Be sure to be consistent so you do not get distracted from your other tasks.\nFocus your time on your most productive activities and find a third party to assist with less important tasks if you need to.\nWhen you are first getting started and perhaps you are in the middle of working on a project - and you don't quite know if it is going to be the success you hope for, it is very easy to get distracted and go on to something else that looks more promising.\nYou will learn with experience what types of projects and tasks excite you, which are the best fit for your knowledge, skills, and resources.\nBut in the beginning when you are first getting started for the first time then this can be a bit confusing, understand this is normal.\nThe most important thing for you is to take what you have now and turn it into a success.\nIt is more important to finish what you start than it is to jump from project to project.\nWhen you jump to another idea or project you have to repeat all the things that you did on the last idea or project.\nYou may find that it is easy for you to start your work but you have a problem with sticking to it until it is finished.\nMake a promise to yourself that you will do your best to finish the projects you start before you begin the process of starting something new.\nDo your best to work on something you believe in. Get excited about your work.\nBe sure to break down the tasks involved in getting large projects up and running. If you approach the smaller tasks individually they will be easier to handle than if you tried dealing with them all together.\nTake care of your health by exercising and eating healthy food. The better you feel and the healthier you are the more you will be able to focus on your business.\nWrite down a list of the goals you wish to achieve with your home business and the time in which you want to achieve them.\nTry making a list of the top 5 things you want to accomplish, then set a timer for 20 -30 minutes. When the timer goes off drop what your doing and Move on to the next task.\nExercises such as this can keep you up and excited As you work towards a deadline. It will automatically set your mind to focus on what needs to be completed in a specified time.\nTake a break every 2 hours, go for a short walk and get some fresh air, talk to a friend on the phone or read a few pages of a motivational book.\nThe success of your home business is entirely and completely up to you and the decisions you make.\nSuccess is within your reach when you keep your focus on your goals.\nWorking at home is a dream that many people have and some are even able to pursue it successfully. When a person works for someone else outside the house they usually have a list of tasks to be completed for that day. If they do not have a list then they have some kind of schedule they need to follow so everything that needs to be done gets done on time.\nGetting the necessary tasks done in a day can be a difficult accomplishment if a person runs a business from their home. They do not have a boss looking over their shoulder to make sure things are on schedule, therefore they may become slack in their business. Having someone else tell you what to do may be easier for some people.\nIf you do run a business from your home, some things that can be done to stay motivated can range from making a simple to do list to getting a family member or friend to check up on you from time to time. It is easier to stay on track if distraction are not present to keep you from doing your work.\nMaking a to do list at the beginning of each day or month or even at the end of each day can help to keep you organized and on schedule during your work time. As each task is completed, crossing them off and moving on is motivating because you can actually see your progress.\nPutting a bulletin board in your work space may also help keep you motivated. Keep the board full of motivating pictures, quotes and dreams. Figure out why you are working from home and make a collage that represents your reasons for working at home. It may be a picture of a new house or a quote from your favorite person. How about pictures of your children if you have them.\nSometimes just thinking about what your goals are and writing them on a piece of paper can be enough to keep your motivation up. Financial freedom, independence or children can be powerful motivators to keep you on track and help you to push your business to success.\nYour family members may also want to help you to stay motivated. They might remind you that when it is business hours you need to keep working until the tasks are done, no wandering or long breaks. Even a quick phone call to check up may get you back on track.\nKeep all distracting items out of your work space until you have enough discipline to stay away from them while working. A television, stereo, games or whatever it is that distract you should stay off so you do not forget why you are there.\nIf you decide to make a schedule that requires you to work from eight to eleven am, with a list of tasks, it is easier to buckle down and get everything done than if you just winged it from hour to hour without knowing exactly what it is that you need to finish.\nWith their unique business plan, Ekoh Group LLC offers a truly incredible method of starting your own home business. The best part is that they set up everything for you!\nThis business model has them construct a review site that is proven to be effective. Similar to this one, they only use sites that review legitimate products that have been proven to work.\nNot only will you be able to help people in their quest to find a great home job, but you will have up to ten different streams of income with these sites.\nThey will set up everything for you! The benefits are too great to list in this space. You must check out their web site.\nThere are a lot of programs that offer to teach you Google AdWords. However, most of them have one major flaw. They do not give you all of the information that you need to properly succeed. That is, until AdWords for Beginners. They teach you all the techniques that only the truly successful Internet marketers know, and they do this with the beginner in mind. No previous AdWords knowledge is necessary for your success.\nAdWords is an established method for making money online. However, if not used properly, you can go broke. AdWords for Beginners teaches you how to avoid the pitfalls that other programs don't.\nNot only will they teach you how to succeed with AdWords, but they also teach you how to select the right product to promote, that is proven to sell well. There is no need to develop your own.\nEarn a living protecting people from scams! I have teamed up with Hoke Web Design to bring you this great offer. I am looking for 25 more people to complete my team.\nCopyright \u00a92003-2010 Work At Home Top 10, All Rights Reserved.\nAdWords(TM) is a Trademark of Google Inc., Mountain View, California.\nWork At Home Top 10 is an independent consulting firm, and is not affiliated with Google.Work At Home Top 10 is paid a commission on the programs listed on this site when a sale is made through the links provided.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ever since efforts to legislatively repeal Obamacare ended ignominiously with a \"thumbs down\" from the late Sen. John McCain on the Senate floor, the Trump administration has been hard at work doing what they can to give families more and better healthcare choices than what Obamacare saddled them with.\nIt goes without saying that Obamacare has been a lousy deal for families like mine which are forced to buy non-employer Obamacare market health insurance but make too much money to qualify for Obamacare's tax credit to subsidize premiums. Between 2013 and 2017, average premiums for Obamacare plans doubled in price. As a result, Obamacare insurance plans are totally unaffordable. For example, my family is currently enrolled in one of the least expensive options in Virginia, and we have to pay $16,000 per year in premiums for \"coverage\" which begins after meeting a $13,000 deductible. Talk about junk insurance.\nStep one in helping families like mine get an alternative to this health insurance nightmare was repealing Obamacare's individual mandate to purchase Obamacare insurance. It's not widely reported, but Congress permanently repealing the Obamacare individual mandate was a seminal plank of the Tax Cuts and Jobs Act. Doing so opens up all sorts of alternative options for families to consider. Without repealing the mandate, a family choosing to purchase some other form of insurance (or none at all) would face an excise tax penalty of 2.5 percent of their income ($2,500 for a family making $100,000 per year). The penalty goes away for good in January 2019. In the meantime, the administration has announced that hardship waivers from the individual mandate will be easier to qualify for in 2018, the last year of the mandate's life.\nRepealing the individual mandate gave the Trump administration the opening to provide more choices for families. This has taken three forms: association health plans, short-term limited duration plans, and health reimbursement arrangements.\nAssociation health plans are health insurance plans that can be sold across state lines. Families can either join a regional AHP or one with a common bond like a professional association. In either case, this will allow people to join with others in purchasing health insurance and benefit from the economies of scale that broader purchasing pools create. For example, an individual restaurant owner might have to pay a lot for an insurance plan just for his family, but might be able to pay a lot less if the National Restaurant Association has an AHP that thousands of restaurant owners can join.\nAnother option available is short-term, limited duration plans. These plans have been available for decades as \"bridge\" health insurance for those between jobs, etc. There is no reason why, however, they cannot be used for longer than a few months. In fact, up until the Obama administration capriciously outlawed the practice, many short-term plans could extend for as long as 364 days. The Trump administration has restored this longstanding practice and also allowed them to be renewed for up to three years. This essentially creates a parallel, unsubsidized individual insurance market.\nAHPs and STLDPs don't have to cover all the gold-plated, expensive options that Obamacare plans have to cover.\nAnd that's ok. Blue-check-mark elitist Twitter accounts will tell you that this is a downside, but what kind of paternalism inspires that sentiment?\nMore comprehensive Obamacare plans are also available to people if they want them and want to pay more for them. Alternatively, AHPs and STLDPs will offer something much closer to the insurance people used to have before Obama fibbed about them keeping their plan if they liked it. It beats paying for a gold-plated plan that covers everything, but at the cost of mortgage-sized auto debit from your bank account every month and a deductible you'll never hit absent a major hospitalization.\nWhat about people who want to get health insurance at work? In addition to large group plans, small group plans, and self-insured plans, the Trump administration just this week also expanded the ability of employers to use health reimbursement arrangements. Under an HRA, an employee can purchase insurance in the individual market (be it an Obamacare plan, an STLPD, or an AHP), and their employer can reimburse them for premiums on a tax-free basis. This puts individual market coverage on the same footing as employer-provided coverage. In addition, employers can reimburse tax-free up to $1,800 per year for sidecar coverage like dental and vision plans via an HRA.\nThe Trump administration is so committed to giving people choices and options, it is even making Obamacare more palatable. It's done such a good job putting in best practices and common sense reforms that Obamacare premiums will rise next year at a rate of less than four percent, a huge improvement over the double-digit annual premium spikes we've had to endure.\nAdditionally, just this week, the administration expanded the ability of states to design state-specific versions of Obamacare better tailored to their own state. For example, the Obamacare tax subsidy might be ill-designed in a state where income is higher or lower than the national average. Truly needy or sick residents might be able to get better coverage if a state could tailor Obamacare for them.\nThese so-called \"1332 waivers\" have always been part of the Obamacare law, but the Obama administration made them practically impossible for states to use. The Trump administration is finally letting states innovate inside Obamacare, which is of course all to the good.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Le seul g\u00eete avec un vrai foyer au bois et un Jacuzzi dans votre chambre!\nOur 19th century farm house is located in the heart of the Eastern Township countryside. Only 7 minutes from Magog.\nWith our local partners, and with our many rebates packages, we will help you save during your stay with us at the Cerf Argent\u00e9.\nThe rates are for Double Occupancy. Every extra guest a rate of 25$+tx per night will be charged.\nCancellation Policy: No charges when cancellation is done up to 72hrs before your arrival. Full rate will be charge within 72hrs of your arrival.\nPlease note that every change or cancellation to an existing reservation has to be done through email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AMERICAN industry - one of the most competitive, entrepreneurial cultures on earth - is increasingly turning to cooperation as a way to blunt foreign competition. In the race to develop everything from the latest computer chip to plastic cars, United States companies are teaming up with each other, and sometimes universities and government labs, to meet the growing threat posed by Japan and Europe.\nYet even as the country dabbles with cooperative capitalism, questions loom. Do consortiums really work? Will they improve the competitive position of the US? Should the government encourage such ventures?\nThe questions are not trifling.\nAs competition becomes more global, joint research and development is likely to become an increasingly popular tool in the US. This is especially true of new technology: Developing high-tech widgets is a difficult and costly process for even the largest companies.\nCongress is abuzz over finding ways to ensure US competitiveness in emerging new frontiers, such as supercomputers, high-definition television, and superconductivity. This may include legislation that would make it easier for joint ventures to form.\n``There will be more consortia in a whole lot of areas,'' predicts Stephen Cohen, director of the Berkeley Roundtable on the International Economy.\nAlthough long common in Japan, industry cooperation in its modern incarnation didn't take off in the US until after 1984. That was the year Congress revised the nation's antitrust laws to make it easier for competing companies to engage in joint research.\nSome 125 cooperatives now exist, conducting research in areas ranging from new types of glass bottles to artificial intelligence. Some ventures involve only a couple of companies. Others, like the six-year-old Microelectronics and Computer Technology Corporation (MCC) in Austin, Texas, the grandfather of modern cooperatives, have well over a dozen corporate participants.\nIn May, International Business Machines Corporation teamed up with American Telephone & Telegraph Company and the Massachusetts Institute of Technology to plumb superconductivity. The most unusual consortium, however, is also the latest: The computer-chip venture announced here in Silicon Valley - normally a fount of radical American individualism - on June 21 by some of the country's top electronics firms.\nUnlike the other consortiums, US Memories Inc. will not be a research cooperative but a manufacturing venture. The aim of the seven companies proposing it is to build a cutting-edge plant to produce computer-memory chips used in a wide array of electronic devices.\nConsortiums are alluring for several reasons. Companies figure that by joining forces they can pursue research they normally could not do alone. It spreads around the risks and avoids duplicating work.\nPooling funds and talent is the only way some companies feel they can keep up with Japan, where joint research is more ingrained and often underwritten by the government. Europe, too, has been moving in this direction: More than 100 companies are looking at joining JESSI, a $4 billion European venture to develop memory chips launched in June.\n``One company by itself can't do everything,'' says Thomas Mullen of the National Center for Manufacturing Sciences, a Michigan-based consortium.\nYet buddy-system capitalism has its problems. There is no overwhelming evidence that consortiums in the United States work, though advocates contend it is too early to gauge their worthiness.\n``The jury is not entirely in on whether they work - and which ones under what circumstances,'' says a Senate staff member.\nOvercoming the culture of competition and developing technology that will benefit the diverse interests of individual members is difficult.\nRecruiting workers can be a problem. MCC had trouble getting member companies to give up top researchers for the effort. Only 16 percent of the cooperative's 430 workers are from participating companies, though MCC staff members say the consortium is starting to produce results.\n``Companies are careful about allowing their best people to leave,'' an MCC official says.\nAs companies consider forming cooperatives to produce products rather than engage in research, other questions arise. Here US Memories Inc., the nation's first major manufacturing consortium, will be a test case.\nLed by IBM but also backed by such electronics notables as Digital Equipment, Hewlett-Packard, Intel, and National Semiconductor, US Memories intends to produce the latest generation ``dynamic random-access memories.'' DRAMs are used in nearly all computerized devices.\nIn recent years, there has been widespread concern about growing dominance of the chip industry by Japanese firms, which control 90 percent of the world DRAM market.\nBefore the proposed consortium turns out silicon wafers, however, it will have to deal with antitrust concerns. The US Justice Department seems unlikely to challenge US Memories on these grounds, but that doesn't mean private suits won't arise.\nEven here, some analysts don't think the consortium will run into problems, since the US occupies such a small corner of the world market.\nStill, the idea of joint manufacturing ventures raises policy questions. Convinced that the US needs to move in this direction to stay up with the Japanese, some lawmakers are pushing bills in Congress that would give joint manufacturing ventures a degree of protection from antitrust action, similar to what it did for research cooperatives.\nBut these moves are opposed by free-market economists, among others, who argue that relaxing the rules will only lead to monopoly industries and stifle innovation.\nOther concerns could arise over funding. Until now, Sematech has been one of the few cooperatives to receive federal backing. Sandford Kane, a former IBM executive who is heading up US Memories, says he doesn't plan to seek direct aid for the consortium but will likely ask for government-guaranteed loans. If he follows through, warns Claude Barfield of the American Enterprise Institute, there would be nothing to prevent the three current US manufacturers of DRAMs from seeking similar support.\nStill, philosophical concerns aside, the success of US Memories will turn more on its ability to attract member companies and compete with the Japanese.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Easy homemade granola that's super delicious, gluten-free & healthy too! Packed with NuttZo Nut & Seed Butter, chia seeds, flax seed & oats then sweetened with honey. Yum!\nPreheat oven to 300 degrees. Spray 2 cookie sheets with nonstick spray.\nIn a large bowl, combine oats, nuts, chia seeds, flaxseed and chocolate chips.\nIn a small microwave-safe bowl, melt coconut oil in microwave for about 20-30 seconds. Whisk in NuttZo, honey and vanilla extract. Mix well.\nAdd liquid to dry ingredients and stir until oats and nuts are well coated.\nSpread evenly onto prepared cookie sheets.\nBake 35-45 minutes, stirring halfway through, or until granola is a beautiful golden brown. Keep a close watch as it can burn very quickly.\nRemove from oven and let cool on cookie sheets. Store in airtight container. This mixture should keep at least 2 weeks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaylxp b/data_all_eng_slimpj/shuffled/split2/finalzzzaylxp
new file mode 100644
index 0000000000000000000000000000000000000000..4605440cb79e6b3a4bf62aeebdda7ce43e86b0c6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaylxp
@@ -0,0 +1,5 @@
+{"text":"Since 2019, our company has been an official representative of Onity, the Spanish company, the leader in electronic locks market segment. Onity is already known in more than 120 countries of the world.\n1973 is considered as the year of establishment of the famous cosmetic brand Molton Brown. First it was a London hairdressing saloon that offered unique hair and head skin care services.\nAbsolute Lifestyle has been active in the premium accessories segment for SPA areas and hotel rooms for about 10 years.\nFor more than one hundred years already Hypnos, UK creates luxurious beds and hand-made mattresses. The best homes, palaces and hotels, both in the UK and worldwide, still rely upon the quality of Hypnos' products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gary the Retard called to say he turns 61 next month and would like to celebrate in-studio, but Howard corrected him: since Gary was born on February 29th, he can only celebrate his birthday during leap years: \"He's really only like 7. You realize you're 7?\" Gary, of course, was confused. Robin thought Gary's birthday visit would be a prime opportunity to play 'Dancing with the Retards'--and answer an enduring question: \"Can we teach them to dance?\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Melendrez is very pleased and excited to have three talented landscape interns join our studio for the summer. Pictured are Zheng Guo from the University of Southern California, Xiaoyang Wang from the University of Pennsylvania and Haoyang Li from the University of Texas at Austin.\nMelendrez was recently selected by Crescent Heights to be the Landscape Architect for a new 77-story residential tower in Chicago, Illinois. Located at the South end of historic Grant Park, Melendrez is collaborating with Rafael Vinoly Architects PC and Rockwell Group on this iconic new highrise around which a series of twelve garden spaces spiral and ascend from the 16th floor up to the 76th!\nConstruction on Forest City's Blossom Plaza has been completed and it is now open to the public. Built entirely over subterranean parking, Blossom Plaza is a mixed use development comprised of three distinct exterior spaces: the Retail Walk, the Fountain Square, and the Courtyard. The Retail Walk and the Courtyard both draw influence from traditional Shanghai residential alleys or Longtangs.\nMel\u00e9ndrez is pleased to welcome its newest Associate Principal, E. Allan Spulecki, a landscape architect to direct and oversee design across the firm.\nJOIN US MARCH 2 FOR A RESIDENTIAL GREYWATER SYSTEMS TALK!\nOn March 2 at 6:30PM, Mel\u00e9ndrez is hosting a COTE and AIA\/LA talk on Residential Greywater Systems for Architects. As a proponent of sustainable solutions for the built environment, Mel\u00e9ndrez is excited to host this talk about greywater applications; a key tool in responsibly using water resources.\nThe recently opened Eighth and Grand by developers Carmel Partners was awarded Project of the Year at the Downtown News' Downtowners of Distinction Awards. Mel\u00e9ndrez is proud to have been part of the team that brought this important project to Downtown Los Angeles. The development energizes Grand Avenue, Olive and 8th Streets, with an expanded public realm and the addition of the much sought after Whole Foods Market.\nThe 10000 Santa Monica construction team celebrated completion the final floor of the 40-story residential tower in Century City. To commemorate the event, Mel\u00e9ndrez joined more than 400 people who were on hand to sign one of the project's steel beams before being placed.\nConstruction on USC University Village reached a halfway mark last week. The 15-acre project, set to open in fall 2017, marks the largest development the university has undertaken. The milestone was marked with the installation of a 12,700-pound rooftop spire on one of the buildings.\nThe Mel\u00e9ndrez studio walked the nearly completed Blossom Plaza in Chinatown. Project Manager Marc Delouvrier, Lead Designer Tony Lopez, and Principal Scott Baker, led the office through the site to explain design concepts, discuss constructability issues, and review lessons learned.\nThe Mel\u00e9ndrez team looks forward to working with you, our clients and project partners, in 2016 to continue to improve our communities.\nCheck out the final project of one of the newest additions to the Mel\u00e9ndrez team, Alex Jung, who recently graduated from UCLA with a Masters in Urban and Regional Planning. The project is a digital how-to manual on how to host a 'pop-up' public health event.\nMel\u00e9ndrez's urban design team recently released the working draft for the Santa Barbara Bicycle Master Plan for public review, which can be viewed here: www.santabarbaraca.gov\/bmp. The document updates the 1998 Bicycle Master Plan, aiming to improve the cycling experience, recommending bicycle projects and finding ways to finance implementation.\nIn case you missed our live Twitter feed from the event, Mel\u00e9ndrez attended the annual ULI Transit Oriented Los Angeles today. Panels with a wide range of speaker viewpoints, knowledge and perspectives created a dynamic and insightful event. Ways to revitalize communities through transit, and lessons learned with case study examples provided insight into the benefits of transit-oriented development and design and how to continue to create these opportunities in Los Angeles.\nGreen TI unanimously (9-0) passed tonight at the Long Beach City Council meeting! The conceptual plan completed by Mel\u00e9ndrez re-visions a one-mile stretch of the Terminal Island (TI) Freeway, a wide and polluted truck-thoroughfare, into a community-serving greenspace.\nThe Gateway, an 80 unit transitional homeless housing project, recently opened at the Century Villages at Cabrillo (CVC) campus in Long Beach. The opening marks the most recent project Mel\u00e9ndrez has completed on the CVC campus \u2013 dating back to the Master Development Plan and Campus Landscape Barrier.\nCHANNEL ISLANDS PARK PLANNING UNDERWAY!\nMel\u00e9ndrez is working on developing a 'short-and-sweet' Vision Plan for the 350+ acre Channel Island Park at California State University, Channel Islands (CSUCI). The park is envisioned as an environmental resource, fostering ecological education and research, and providing a passive recreational opportunity for the university and community.\nLast night Mel\u00e9ndrez hosted an Engage! event as part of the USC Guild series. We were pleased to be the first Landscape Architecture firm to host the event!\nLONG BEACH AIRPORT WINS AIA LA DESIGN AWARD!\nLast night Long Beach Airport Terminal Improvements received an AIA Los Angeles Design Award! The prestigious award marks the fifth award for the iconic project. Mel\u00e9ndrez was the landscape architect on the project, designing open-air garden spaces integral to the airport experience.\nMel\u00e9ndrez joined the City of Long Beach this past Saturday for the fourth interactive community workshop event for the Green TI project. Visualizations of preliminary design ideas were presented for community review and input, and interactive booths, demonstrations, and free food celebrated the future open space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That is an analog headset that plugs into the laptop's combo audio jack with a 3.5 mm TRRS (4 segment) plug, correct?\nHello Victor. Please say what you have tried as we don't know what you saw on the web, and state the operating system of the laptop.\nI've tried checking my recording devices to make sure I didn't have a different one as default, making sure my headset mic wasn't muted and checking the product manual for any special drivers I could possibly need to install manually from the Razer Website. The laptop is running Windows 10.\nI should also state that I am using third-party software to manage my audio inputs and outputs (Voicemeeter), even though it doesn't seem to be the problem, at first glance. I will try the procedures you suggested and keep you in the loop with the results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are a chocolate lover, you will love this mask! Besides the amazing chocolate scent, the ingredients are thought to help soothe and rejuvenate tired skin.\nThough this is a facial mask, it is great for a body mud too, just increase the quantities. This formula is from our Formulating with Clay Book.\nFor Facial Mask: Hydrate by combining 40g of mixture (1 portion) and approximately 20 - 40g\/ml Rose Infused Water (or you can use your preferred liquid) and mix to make a slurry or loose paste. Apply to the skin, avoiding the eyes and allow remain on the skin for 10 minutes. Rinse with off with water and then moisturise as usual.\nFor Body Mud: Hydrate as above, ensuring you mix sufficient to cover the areas of the body as required.\nNo mess mask application: If using this mask for facial use, there is sufficient for one treatment in each of the pouches. Open the pouch and add your preferred liquid to the pouch. Close the zip and then massage the mixture to ensure it is fully hydrated - adjust the quantity of liquid if required. Then use scissors to cut the entire zip seal section from the pouch and apply. The pouch can be put straight into the bin.\nDo you have one or more of these ingredients already? No Problem! Just amend the quantities below and the total price will adjust accordingly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaymxa b/data_all_eng_slimpj/shuffled/split2/finalzzzaymxa
new file mode 100644
index 0000000000000000000000000000000000000000..b3f8483606110e97af3d71dadf99f55c52d78abf
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaymxa
@@ -0,0 +1,5 @@
+{"text":"Please take a moment to post a brief \"Testimonial\" to let us know how we are doing or what we can do better to improve our \"Services, Networking & Business Offerings.\" Let us know your thoughts, your ideas or your suggestions on how we can improve our GTV Business Model, inhance our Website or Green Projects and Green Programs to serve you better. Thank You!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Replacement Fuel Tanks: Is More Fuel Foolish?\nTruck Camper Magazine readers share if they already have or would consider a replacement fuel tanks. Check out these auxiliary fuel tank solutions.\nIf you're considering a replacement tank, we urge you begin by reading, \"Matching A Truck and Truck Camper\", and run the recommended weight and payload numbers on your rig. Added road range is fantastic, but never at the expense of safety.\n\"I've often thought about replacement fuel tanks, but never acted on replacing the standard tank. I thought about it because, like many of us, I love modifications and gadgets. I passed on it because I'm a cheapskate and slow to act (because I'm a cheapskate?). My Ford OEM tank holds about 38 gallons of #2 diesel. I'm able to travel nearly 400 miles on a load of that oily substance. I'd happily keep on driving but, really, a stop now and then is probably a good thing.\n\"I had an F250 with a 7.3L diesel. I carried a five gallon spare tank occasionally for trips to Alaska, Canada, Moab, etc. I only needed it once on the Dalton Highway in Alaska. But, having it, made me feel more comfortable.\nA mechanic told me to never run a diesel dry. You cannot re-pressurize the high pressure lines easily.\n\"My Ford F350 has a flat farm bed on it. I use it when my camper is detached. I have mounted a fifteen gallon auxiliary fuel tank on top of my truck bed on the left side directly above my factory fuel tank. It is out of the way when I load my camper. The cost was only $175 as compared to the price of an extended capacity fuel tank. I would easily have room to add an additional fifteen gallon if I chose to.\nHome \u00bb Question Of The Week \u00bb Replacement Fuel Tanks: Is More Fuel Foolish?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Moss Motors is hiring! We're seeking a Graphic Designer, full time, at our Goleta, CA headquarters. This person will be the lead for our Miata parts program, so if you own a Miata that would be a huge bonus for us!\nAll the info here on our Craigslist ad.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Easy an cheap diy barn door tutorial the hardware is. 42 modern sliding barn doors 2017 home and house design. Barn door track industrial mid century \/ modern rustic. Industrial classic sliding barn door hardware closet door.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A supplementary card can be enrolled by account proprietor, If card has been issued as a supplementary card to an account of any company, Internet purchasing can be activated by the representative of the corporate who has been granted a enough right to this finish to use the account in the Internet Bank. Nonetheless, some websites have had success delivering groceries and clothes offered through the internet is huge enterprise in the U.S.\nFortunately, the web has helped on this respect, we no longer need to search out the time to get to the town centres, stroll around the retailers, choosing things up and making an attempt them on, then standing in queue, paying for our goods, after which driving back house once more (- what a palaver, it was hard sufficient simply writing it, not to mention truly do it!).\nResearchers discovered that having products which are acceptable for e-commerce was a key indicator of Internet success. Here are some ideas which can make it easier to get monetary savings and meet your finances while shopping. This has prompted the mushrooming of another extremely quick most transferring cottage trade in the type of Internet Marketing. Some present background info, advice, or how-to guides designed to assist consumers determine which product to buy.\nOutsourcing companies:The web has helped cut costs by outsourcing companies to countries where it is cheaper to supply these companies. The web has gotten a very nice makeover when it comes to this new and easy approach of buying whatever they need and that too much faster and has given us a whole lot of perks to get pleasure from whereas doing so within the comfort of our residence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayunq b/data_all_eng_slimpj/shuffled/split2/finalzzzayunq
new file mode 100644
index 0000000000000000000000000000000000000000..21c43610e27a5784132f29ffe2866f5ef0d5406e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzayunq
@@ -0,0 +1,5 @@
+{"text":"It seems that client choice is following the path of so much else in the criminal legal system as it is threatened with being killed off in the proposed shake up of criminal legal aid.\nDespite the fact we all believed this was protected in LASPO, it seems the minister allowed himself enough wriggle room and soon it will be like A&E, you simply get the first doctor to see you and no choice in the matter.\n\"(4) An individual who qualifies under this Part for representation for the purposes of criminal proceedings by virtue of a determination under section 16 may select any representative or representatives willing to act for the individual, subject to regulations under subsection (6).\nThe Lord Chancellor and justice secretary, Chris Grayling, has made clear his views on client choice in chapter 4 of his consultation paper. \"An approach that removed client choice entirely would\u2026deliver the greatest level of certainty,\" he says.\nAccording to the 'Law Gazette', Grayling told a recent meeting of the barristers' circuit leaders that abandoning client choice was designed to help new entities, such as the bar, to enter the market place. During the meeting he clearly tried to get the barristers on side, but issued the astonishing threat that \"If the bar does not co-operate I may introduce PCT in the Crown court.\" I thought the official line was that PCT was the most efficient and modern method to process the criminal legal system. Now it is clearly a method of punishing those who do not fall into line.\nIf Grayling's meeting was an attempt to drive a wedge between solicitors and barristers it clearly failed. After he left the meeting the circuit leaders firmly agreed that they must act in unison and with CBA, liaising with solicitors and Law Society and organising any necessary publicity and media liaison. They will meet the justice minister again before the end of the consultation period.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What do you think - will Trustarc grow fast? Are they expanding their staff? How does Trustarc stack up against the competition?\nWhat's the company culture at Trustarc?\nHow to get a job at Trustarc.\nDo you work at Trustarc? How did you find the job? How did you get that first interview? Any advice for someone trying to get in?\nTrustarc Salaries, Bonuses and Benefits.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Master; I marvel how the fishes live in the sea.\nNot to be able to bear poverty is a shameful thing, but not to know how to chase it away by work is a more shameful thing yet.\nTime is the wisest counsellor.\nWait for the wisest of all councilors, time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During a conversation with my brother, who has led multiple teams in multiple locations, he asked: \"Do you know the difference between an all-star employee and an average employee? They do a little bit more.\" He suggested that an all-star or \"A\" employee will do more than expected consistently without being asked. In our industry, where finding qualified, dedicated people is often as elusive as finding a unicorn, how can you build a strong team that will position your company for growth? It's actually not as difficult as you might think.\nThey talk about work in a positive way.\nThey discuss things they or their team have been working on even when not at work.\nThey answer work emails or after-hours calls because they own what they do and don't want things to linger or be forgotten.\nThey may be focused problem-solvers.\nThey are list or calendar driven.\nThey take notes and do not procrastinate.\nPeople gravitate to them because they know that they will get it done.\nIf A team members represent the best employees, what could cause some of them to not succeed in your organization? Look inward.\nAfter a few false starts in hiring some of our leadership team, I realized that those who didn't work out didn't necessarily represent a bad hire. It was partly due to my lack of maturity as a leader and coach. Both sides contribute to success and failure, and employees play a major part in which way it goes. However, I have found when the team or person has the right processes, systems and support, it can definitely translate to success. Coupled with a person who does just a little bit more, it can translate into a higher level of success and have an enormous impact on the business.\nEstablish a structure, process and system that complement a goal that has a 5- to 10-year planned path to success known by the entire organization. When you tie the structure and processes together with accountability, direction and reward, greatness can be achieved. In my experience, I have seen driven, highly intelligent people go from good to great when given the tools to succeed.\nLeadership coach Marshall Goldsmith says: \"What got you here won't get you there.\" As you grow, you will need to adapt your formula for success. We have created systems that worked phenomenally up to a certain point, and then they needed to be adapted because we outgrew the system's capacity. Constant monitoring of these systems and processes is essential because failure can happen quickly, which can put strain on your team.\nOnce you've found your A team and created systems for optimal operations, how do you keep them engaged?\nCreate an environment where they enjoy spending time. Make sure the culture has commonality or the team has similar core values.\nGive them the tools to win. Create systems and outfit operations with good equipment and technology. Dedicate a budget for continued training and education, including mentorship programs.\nDon't accept mediocre or less from anyone. Hire slow\u2026fire fast. Keeping non-producers on the team shows that you accept mediocrity.\nSet the goal. Winners need a goal. Set the goal with their input, reverse engineer and reward the right outcome. Make the reward something they want. Not everyone is motivated by money. A great quick read is \"The Five Love Languages\" from Gary Chapman. A two-minute online test will identify your motivation. While it's geared toward personal relationships, it can be adjusted accordingly. You can probably identify your peers' motivations once you understand the concept. www.5lovelanguages.com\/profile\/.\nHold on and enjoy the ride. What I have learned over the past 20 years is that \"the right people in the right seats\" directly correlates to my business growth. Each time we have made an investment in a great employee - from the field up to the highest management level - the business has grown exponentially.\nJonathan Crandall is chief visionary for Crandall & Company Inc. Contact him at jon@jcgrounds.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a print of an original drawing, printed with archival inks on Museo Portfolio Rag. The image is signed on the front. The original was created in colored pencil on a vintage piece of cream paper that has a nice color variation due to light exposure in the background color.\nIt was created on December 31, 2010 on New Years Eve at the request of my daughter to make her a \"really nice drawing\". She was overjoyed with the picture and commented that it was not \"crazy\" like my other drawings (clearly she has overlooked the flower limbs). It was this piece that lead me to the idea to do a drawing everyday in 2011. From her inspiration a new drawing came alive in our home for one year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbaewi b/data_all_eng_slimpj/shuffled/split2/finalzzzbaewi
new file mode 100644
index 0000000000000000000000000000000000000000..98843f92d706a3db4d2fcf840649d9ebc762c5fa
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbaewi
@@ -0,0 +1,5 @@
+{"text":"5182 aluminum alloy strip is mainly used to manufacture cans and rings of aluminum material. Aluminum cans has many characteristics, such as light in weight, good resistance to heat, good conductivity, no smell and non-toxic. Besides, vivid colors and patterns can be easily printed on it. What's more important, aluminum cans can be recycled, which is good for our living environment. Thus, from 80s in 20 century, the demand for aluminum cans became bigger and bigger increasingly. Thus, aluminum cans account for an important seat in world drink packaging field, especially developing countries, such as Brazil, India, China, etc. There are great spaces to develop and the increasing rate will keep at 8%.\n5182 aluminum alloy strip is the main alloy for aluminum can rings. The bottom of aluminum rings mainly adopted 3004 and 5052 aluminum alloys. Except for aluminum cans, these alloys can also be used to produce other packaging vessels. As drink industry is getting better and better, 5182 aluminum alloy strip will face more challenge as well as opportunity. 5182 aluminum alloy strip cannot be heat treated but with high strength, good resistance to correction and weld-ability. So is it also high suitable for ship building, oil tanks of airplane, tubes, as well as vehicles, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The walls shall be 3\" thick composite sandwich .021 stucco-embossed aluminum, pre-painted white or beige, bonded to _\" fire-shield gypsum. The core shall be of 99#, 1.2\" cell, 11% phenolic-resin impregnated structural Kraft honeycomb. Panel division\/finish strips shall consist of color coordinated vinyl \"H\" connectors that shall protrude more than 1\/16\" beyond the finished wall panel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mar 25 9:25 Mat. PT10:25 Mat. MT11:25 Mat. CT12:25 Ap-Mid HE16:25 GMT0:25 9:25 Mat. MST10:25 Mat. CST11:25 Mat. EST20:25 UAE17:2512:25 HE10:25 Mat. CT15:25 - Dunn will miss a second straight game Wednesday against the Trail Blazers because of a strained back.\nAnalyse: A back injury also put Dunn's status into question Saturday, and he played just 21 minutes while finishing with four points on 1-of-8 shooting against the Jazz. He came in averaging 16 points, 6.6 assists and 4.2 rebounds over his previous five games.\nMar 23 6:37 Ap-Mid PT7:37 Ap-Mid MT8:37 Ap-Mid CT9:37 Ap-Mid HE1:37 GMT9:37 6:37 Ap-Mid MST7:37 Ap-Mid CST8:37 Ap-Mid EST5:37 UAE (+1)02:3721:37 HE7:37 Ap-Mid CT0:37 - Kris Dunn scored four points Saturday on 1-of-8 shooting as the Chicago Bulls lost to the Utah Jazz 114-83. Dunn grabbed three rebounds and dished out five assists, accumulating a plus-minus of -17 in his 21:26 on the floor. He added one steal.\nMar 22 9:45 Mat. PT10:45 Mat. MT11:45 Mat. CT12:45 Ap-Mid HE16:45 GMT0:45 9:45 Mat. MST10:45 Mat. CST11:45 Mat. EST20:45 UAE17:4512:45 HE10:45 Mat. CT15:45 - Dunn (back) will start Saturday.\nAnalyse: Dunn has come on strong of late, averaging 16 points, 6.6 assists and 4.2 rebounds over his last five games. The Bulls, however, seem more than wiling to sit players down the stretch.\nMar 20 6:56 Ap-Mid PT7:56 Ap-Mid MT8:56 Ap-Mid CT9:56 Ap-Mid HE1:56 GMT9:56 6:56 Ap-Mid MST7:56 Ap-Mid CST8:56 Ap-Mid EST5:56 UAE (+1)02:5621:56 HE7:56 Ap-Mid CT0:56 - Kris Dunn scored 26 points Wednesday on 10-of-24 shooting as the Chicago Bulls topped the Washington Wizards 126-120 in overtime. Dunn grabbed six rebounds and dished out 13 assists, tallying a plus-minus of +9 in his 39:18 of playing time. He added two steals. Dunn went 3 of 7 from 3-point range.\nMar 18 8:39 Ap-Mid PT9:39 Ap-Mid MT10:39 Ap-Mid CT11:39 Ap-Mid HE3:39 GMT11:39 8:39 Ap-Mid MST9:39 Ap-Mid CST10:39 Ap-Mid EST7:39 UAE (+1)04:3923:39 HE9:39 Ap-Mid CT2:39 - Kris Dunn scored 14 points Monday on 5-of-10 shooting as the Chicago Bulls topped the Phoenix Suns 116-101. Dunn grabbed two rebounds and had three assists, accumulating a plus-minus of +13 in his 27:05 on the floor. He added one block and two steals. Dunn went 1 of 2 from 3-point range.\nMar 17 4:33 Ap-Mid PT5:33 Ap-Mid MT6:33 Ap-Mid CT7:33 Ap-Mid HE23:33 GMT7:33 4:33 Ap-Mid MST5:33 Ap-Mid CST6:33 Ap-Mid EST3:33 UAE (+1)00:3319:33 HE5:33 Ap-Mid CT22:33 - Kris Dunn scored eight points Sunday on 4-of-11 shooting as the Chicago Bulls fell to the Sacramento Kings 129-102. Dunn pulled down one rebound and dished out five assists, accumulating a plus-minus of -19 in his 19:58 on the floor. He added one steal.\nMar 16 9:18 Ap-Mid PT10:18 Ap-Mid MT11:18 Ap-Mid CT12:18 Mat. HE4:18 GMT12:18 9:18 Ap-Mid MST10:18 Ap-Mid CST11:18 Ap-Mid EST8:18 UAE05:180:18 HE10:18 Ap-Mid CT3:18 - Kris Dunn scored 14 points Friday, going 5 of 12 from the field, as the Chicago Bulls lost to the Los Angeles Clippers 128-121. Dunn grabbed five rebounds and dished out three assists, accumulating a plus-minus of -5 in his 27:56 on the floor before fouling out. He added one block. Dunn went 1 of 2 from 3-point range.\nMar 12 6:33 Ap-Mid PT7:33 Ap-Mid MT8:33 Ap-Mid CT9:33 Ap-Mid HE1:33 GMT9:33 6:33 Ap-Mid MST7:33 Ap-Mid CST8:33 Ap-Mid EST5:33 UAE (+1)02:3321:33 HE7:33 Ap-Mid CT0:33 - Kris Dunn scored 18 points Tuesday on 8-of-16 shooting as the Chicago Bulls lost to the Los Angeles Lakers 123-107. Dunn pulled down seven rebounds and had nine assists, tallying a plus-minus of -5 in his 32:57 on the floor. He tacked on two steals. Dunn went 2 of 3 from behind the arc.\nMar 10 10:58 Mat. PT11:58 Mat. MT12:58 Ap-Mid CT1:58 Ap-Mid HE17:58 GMT1:58 10:58 Mat. MST11:58 Mat. CST12:58 Ap-Mid EST21:58 UAE18:5813:58 HE11:58 Mat. CT16:58 - Kris Dunn scored seven points Sunday on 3-of-7 shooting as the Chicago Bulls fell to the Detroit Pistons 131-108. Dunn did not grab a rebound but had four assists, accumulating a plus-minus of -13 in his 22:07 on the floor. He added two steals. Dunn went 1 of 2 from behind the arc.\nMar 8 7:04 Ap-Mid PT8:04 Ap-Mid MT9:04 Ap-Mid CT10:04 Ap-Mid HE3:04 GMT11:04 8:04 Ap-Mid MST9:04 Ap-Mid CST10:04 Ap-Mid EST7:04 UAE (+1)04:0422:04 HE9:04 Ap-Mid CT2:04 - Kris Dunn scored seven points Friday on 3-of-6 shooting as the Chicago Bulls fell to the Detroit Pistons 112-104. Dunn pulled down seven rebounds and dished out three assists, accumulating a plus-minus of +5 in his 28:23 of playing time. He added one block and two steals. Dunn went 1 of 2 from 3-point range.\nMar 6 6:58 Ap-Mid PT7:58 Ap-Mid MT8:58 Ap-Mid CT9:58 Ap-Mid HE2:58 GMT10:58 7:58 Ap-Mid MST8:58 Ap-Mid CST9:58 Ap-Mid EST6:58 UAE (+1)03:5821:58 HE8:58 Ap-Mid CT1:58 - Kris Dunn scored four points Wednesday, going 2 of 6 from the field, as the Chicago Bulls topped the Philadelphia 76ers 108-107. Dunn grabbed five rebounds and dished out four assists, tallying a plus-minus of +3 in his 25:50 on the floor. He added one block and two steals.\nMar 5 5:37 Ap-Mid PT6:37 Ap-Mid MT7:37 Ap-Mid CT8:37 Ap-Mid HE1:37 GMT9:37 6:37 Ap-Mid MST7:37 Ap-Mid CST8:37 Ap-Mid EST5:37 UAE (+1)02:3720:37 HE7:37 Ap-Mid CT0:37 - Kris Dunn scored five points Tuesday on 2-of-9 shooting as the Chicago Bulls fell to the Indiana Pacers 105-96. Dunn grabbed seven rebounds and dished out five assists, accumulating a plus-minus of -2 in his 31:09 on the floor. He added two steals.\nMar 3 2:24 Ap-Mid PT3:24 Ap-Mid MT4:24 Ap-Mid CT5:24 Ap-Mid HE22:24 GMT6:24 3:24 Ap-Mid MST4:24 Ap-Mid CST5:24 Ap-Mid EST2:24 UAE (+1)23:2417:24 HE4:24 Ap-Mid CT21:24 - Kris Dunn scored 14 points Sunday on 6-of-12 shooting as the Chicago Bulls lost to the Atlanta Hawks 123-118. Dunn pulled down four rebounds and dished out six assists, accumulating a plus-minus of -4 in his 27:15 of playing time. He tacked on two steals. Dunn went 2 of 5 from behind the arc.\nMar 1 7:46 Ap-Mid PT8:46 Ap-Mid MT9:46 Ap-Mid CT10:46 Ap-Mid HE3:46 GMT11:46 8:46 Ap-Mid MST9:46 Ap-Mid CST10:46 Ap-Mid EST7:46 UAE (+1)04:4622:46 HE9:46 Ap-Mid CT2:46 - Kris Dunn scored 14 points Friday on 5-of-11 shooting as the Chicago Bulls defeated the Atlanta Hawks 168-161 in quadruple overtime. Dunn grabbed one rebound and dished out six assists, tallying a plus-minus of -12 in his 31:46 on the floor before fouling out. He added one block. Dunn went 3 of 5 from behind the arc.\nF\u00e9v 27 6:49 Ap-Mid PT7:49 Ap-Mid MT8:49 Ap-Mid CT9:49 Ap-Mid HE2:49 GMT10:49 7:49 Ap-Mid MST8:49 Ap-Mid CST9:49 Ap-Mid EST6:49 UAE (+1)03:4921:49 HE8:49 Ap-Mid CT1:49 - Kris Dunn scored nine points Wednesday, going 3 of 8 from the field, as the Chicago Bulls topped the Memphis Grizzlies 109-107. Dunn pulled down four rebounds and had four assists, tallying a plus-minus of +8 in his 30:35 of playing time. He added three steals. Dunn went 1 of 1 from 3-point range.\nF\u00e9v 25 9:21 Mat. PT10:21 Mat. MT11:21 Mat. CT12:21 Ap-Mid HE17:21 GMT1:21 10:21 Mat. MST11:21 Mat. CST12:21 Ap-Mid EST21:21 UAE18:2112:21 HE11:21 Mat. CT16:21 - Dunn will play Wednesday at Memphis after sitting out Monday's loss to the Bucks because of a migraine.\nAnalyse: Dunn is still starting for the Bulls but he's seen his usage in playing time decrease. He's averaged just 5 points in his last four games after averaging 13.6 in his previous five and he played just 43 seconds in the fourth quarter in Saturday's win over Boston. Ryan Arcidiacono has taken over the lead point guard duties in Chicago.\nF\u00e9v 23 6:39 Ap-Mid PT7:39 Ap-Mid MT8:39 Ap-Mid CT9:39 Ap-Mid HE2:39 GMT10:39 7:39 Ap-Mid MST8:39 Ap-Mid CST9:39 Ap-Mid EST6:39 UAE (+1)03:3921:39 HE8:39 Ap-Mid CT1:39 - Kris Dunn scored four points Saturday, going 2 of 7 from the field, as the Chicago Bulls topped the Boston Celtics 126-116. Dunn pulled down one rebound and dished out five assists, tallying a plus-minus of -2 in his 24:33 on the floor.\nF\u00e9v 22 5:40 Ap-Mid PT6:40 Ap-Mid MT7:40 Ap-Mid CT8:40 Ap-Mid HE1:40 GMT9:40 6:40 Ap-Mid MST7:40 Ap-Mid CST8:40 Ap-Mid EST5:40 UAE (+1)02:4020:40 HE7:40 Ap-Mid CT0:40 - Kris Dunn scored eight points Friday, going 4 of 9 from the field, as the Chicago Bulls topped the Orlando Magic 110-109. Dunn pulled down three rebounds and had two assists, accumulating a plus-minus of -1 in his 29:15 of playing time. He added one steal.\nF\u00e9v 12 12:03 Ap-Mid PT1:03 Ap-Mid MT2:03 Ap-Mid CT3:03 Ap-Mid HE20:03 GMT4:03 1:03 Ap-Mid MST2:03 Ap-Mid CST3:03 Ap-Mid EST0:03 UAE (+1)21:0315:03 HE2:03 Ap-Mid CT18:03 - Dunn is not on the injury report for Friday's visit to Orlando after sitting out the final game before the All-Star break because of a bruised back.\nAnalyse: Dunn took a hard fall in the Bulls' loss to Milwaukee on Feb. 11. He has totaled just eight points in his last two contests after averaging 16 in his previous two, but he averaged 7.8 assists over that four-game stretch.\nF\u00e9v 11 6:59 Ap-Mid PT7:59 Ap-Mid MT8:59 Ap-Mid CT9:59 Ap-Mid HE2:59 GMT10:59 7:59 Ap-Mid MST8:59 Ap-Mid CST9:59 Ap-Mid EST6:59 UAE (+1)03:5921:59 HE8:59 Ap-Mid CT0:59 - Kris Dunn scored two points Monday on 1-of-9 shooting as the Chicago Bulls lost to the Milwaukee Bucks 112-99. Dunn pulled down four rebounds and dished out six assists, accumulating a plus-minus of +1 in his 32:46 of playing time. He added one steal.\nF\u00e9v 9 6:42 Ap-Mid PT7:42 Ap-Mid MT8:42 Ap-Mid CT9:42 Ap-Mid HE2:42 GMT10:42 7:42 Ap-Mid MST8:42 Ap-Mid CST9:42 Ap-Mid EST6:42 UAE (+1)03:4221:42 HE8:42 Ap-Mid CT0:42 - Kris Dunn scored six points Saturday on 3-of-6 shooting as the Chicago Bulls lost to the Washington Wizards 134-125. Dunn pulled down four rebounds and had eight assists, accumulating a plus-minus of -10 in his 32:33 on the floor. He tacked on three steals.\nF\u00e9v 8 5:57 Ap-Mid PT6:57 Ap-Mid MT7:57 Ap-Mid CT8:57 Ap-Mid HE1:57 GMT9:57 6:57 Ap-Mid MST7:57 Ap-Mid CST8:57 Ap-Mid EST5:57 UAE (+1)02:5720:57 HE7:57 Ap-Mid CT23:57 - Kris Dunn scored 14 points Friday on 5-of-11 shooting as the Chicago Bulls defeated the Brooklyn Nets 125-106. Dunn grabbed five rebounds and had nine assists, accumulating a plus-minus of +12 in his 30:22 of playing time. He added one block. Dunn went 1 of 5 from behind the arc.\nF\u00e9v 6 6:50 Ap-Mid PT7:50 Ap-Mid MT8:50 Ap-Mid CT9:50 Ap-Mid HE2:50 GMT10:50 7:50 Ap-Mid MST8:50 Ap-Mid CST9:50 Ap-Mid EST6:50 UAE (+1)03:5021:50 HE8:50 Ap-Mid CT0:50 - Kris Dunn scored 18 points Wednesday on 8-of-13 shooting as the Chicago Bulls lost to the New Orleans Pelicans 125-120. Dunn pulled down four rebounds and had eight assists, tallying a plus-minus of -8 in his 37:31 on the floor. He added one block and two steals. Dunn went 2 of 4 from behind the arc.\nF\u00e9v 4 11:14 Mat. PT12:14 Ap-Mid MT1:14 Ap-Mid CT2:14 Ap-Mid HE19:14 GMT3:14 12:14 Ap-Mid MST1:14 Ap-Mid CST2:14 Ap-Mid EST23:14 UAE20:1414:14 HE1:14 Ap-Mid CT17:14 - Dunn (pelvic contusion) will play Wednesday against the Pelicans after previously being considered day to day.\nAnalyse: Dunn had just seven points on 3-of-11 shooting in Saturday's loss after totaling 29 points in his previous two games.\nF\u00e9v 2 5:34 Ap-Mid PT6:34 Ap-Mid MT7:34 Ap-Mid CT8:34 Ap-Mid HE1:34 GMT9:34 6:34 Ap-Mid MST7:34 Ap-Mid CST8:34 Ap-Mid EST5:34 UAE (+1)02:3420:34 HE7:34 Ap-Mid CT23:34 - Kris Dunn scored seven points Saturday on 3-of-11 shooting as the Chicago Bulls lost to the Charlotte Hornets 125-118. Dunn pulled down five rebounds and dished out four assists, accumulating a plus-minus of -8 in his 31:38 on the floor. He added one block. Dunn went 1 of 2 from 3-point range.\nJan 30 6:12 Ap-Mid PT7:12 Ap-Mid MT8:12 Ap-Mid CT9:12 Ap-Mid HE2:12 GMT10:12 7:12 Ap-Mid MST8:12 Ap-Mid CST9:12 Ap-Mid EST6:12 UAE (+1)03:1221:12 HE8:12 Ap-Mid CT0:12 - Kris Dunn scored 14 points Wednesday on 6-of-14 shooting as the Chicago Bulls topped the Miami Heat 105-89. Dunn pulled down five rebounds and had eight assists, accumulating a plus-minus of +8 in his 34:23 on the floor. He tacked on two steals. Dunn went 1 of 1 from behind the arc.\nJan 29 6:21 Ap-Mid PT7:21 Ap-Mid MT8:21 Ap-Mid CT9:21 Ap-Mid HE2:21 GMT10:21 7:21 Ap-Mid MST8:21 Ap-Mid CST9:21 Ap-Mid EST6:21 UAE (+1)03:2121:21 HE8:21 Ap-Mid CT0:21 - Kris Dunn scored 15 points Tuesday on 6-of-10 shooting as the Chicago Bulls lost to the Brooklyn Nets 122-117. Dunn grabbed three rebounds and did not have an assist, accumulating a plus-minus of -1 in his 31:14 of playing time. He added two blocks. Dunn went 2 of 3 from behind the arc.\nJan 27 2:07 Ap-Mid PT3:07 Ap-Mid MT4:07 Ap-Mid CT5:07 Ap-Mid HE22:07 GMT6:07 3:07 Ap-Mid MST4:07 Ap-Mid CST5:07 Ap-Mid EST2:07 UAE (+1)23:0717:07 HE4:07 Ap-Mid CT20:07 - Kris Dunn scored six points Sunday on 2-of-8 shooting as the Chicago Bulls lost to the Cleveland Cavaliers 104-101. Dunn grabbed two rebounds and dished out seven assists, accumulating a plus-minus of +8 in his 34:38 of playing time. He added two blocks and two steals.\nJan 25 6:47 Ap-Mid PT7:47 Ap-Mid MT8:47 Ap-Mid CT9:47 Ap-Mid HE2:47 GMT10:47 7:47 Ap-Mid MST8:47 Ap-Mid CST9:47 Ap-Mid EST6:47 UAE (+1)03:4721:47 HE8:47 Ap-Mid CT0:47 - Kris Dunn scored nine points Friday on 4-of-14 shooting as the Chicago Bulls lost to the Los Angeles Clippers 106-101. Dunn pulled down four rebounds and dished out 10 assists, tallying a plus-minus of 0 in his 36:27 of playing time. He tacked on four steals. Dunn went 1 of 5 from 3-point range.\nJan 23 7:23 Ap-Mid PT8:23 Ap-Mid MT9:23 Ap-Mid CT10:23 Ap-Mid HE3:23 GMT11:23 8:23 Ap-Mid MST9:23 Ap-Mid CST10:23 Ap-Mid EST7:23 UAE (+1)04:2322:23 HE9:23 Ap-Mid CT1:23 - Kris Dunn scored 16 points Wednesday, going 5 of 11 from the field, as the Chicago Bulls fell to the Atlanta Hawks 121-101. Dunn pulled down four rebounds and had three assists, tallying a plus-minus of -20 in his 26:46 of playing time. He added two blocks and two steals. Dunn went 1 of 2 from behind the arc.\nJan 21 11:36 Mat. PT12:36 Ap-Mid MT1:36 Ap-Mid CT2:36 Ap-Mid HE19:36 GMT3:36 12:36 Ap-Mid MST1:36 Ap-Mid CST2:36 Ap-Mid EST23:36 UAE20:3614:36 HE1:36 Ap-Mid CT17:36 - Kris Dunn scored 13 points Monday on 5-of-10 shooting as the Chicago Bulls topped the Cleveland Cavaliers 104-88. Dunn pulled down four rebounds and dished out nine assists, accumulating a plus-minus of +18 in his 31:13 of playing time. He tacked on two steals. Dunn went 1 of 1 from behind the arc.\nJan 19 6:39 Ap-Mid PT7:39 Ap-Mid MT8:39 Ap-Mid CT9:39 Ap-Mid HE2:39 GMT10:39 7:39 Ap-Mid MST8:39 Ap-Mid CST9:39 Ap-Mid EST6:39 UAE (+1)03:3921:39 HE8:39 Ap-Mid CT0:39 - Kris Dunn scored six points Saturday, going 3 of 14 from the field, as the Chicago Bulls fell to the Miami Heat 117-103. Dunn pulled down four rebounds and had five assists, tallying a plus-minus of -17 in his 32:59 of playing time. He tacked on two steals.\nJan 17 7:24 Ap-Mid PT8:24 Ap-Mid MT9:24 Ap-Mid CT10:24 Ap-Mid HE3:24 GMT11:24 8:24 Ap-Mid MST9:24 Ap-Mid CST10:24 Ap-Mid EST7:24 UAE (+1)04:2422:24 HE9:24 Ap-Mid CT1:24 - Kris Dunn scored six points Thursday on 3-of-8 shooting as the Chicago Bulls lost to the Denver Nuggets 135-105. Dunn grabbed one rebound and had three assists, accumulating a plus-minus of -26 in his 29:52 of playing time.\nJan 16 9:14 Ap-Mid PT10:14 Ap-Mid MT11:14 Ap-Mid CT12:14 Mat. HE5:14 GMT13:14 10:14 Ap-Mid MST11:14 Ap-Mid CST12:14 Mat. EST9:14 UAE06:140:14 HE11:14 Ap-Mid CT3:14 - Kris Dunn scored six points Tuesday, going 2 of 7 from the field, as the Chicago Bulls lost to the Los Angeles Lakers 107-100. Dunn grabbed five rebounds and dished out four assists, tallying a plus-minus of -25 in his 31:09 of playing time. He added two blocks and one steal.\nJan 13 9:19 Ap-Mid PT10:19 Ap-Mid MT11:19 Ap-Mid CT12:19 Mat. HE5:19 GMT13:19 10:19 Ap-Mid MST11:19 Ap-Mid CST12:19 Mat. EST9:19 UAE06:190:19 HE11:19 Ap-Mid CT3:19 - Kris Dunn scored 10 points Saturday on 4-of-11 shooting as the Chicago Bulls lost to the Utah Jazz 110-102. Dunn pulled down five rebounds and had eight assists, accumulating a plus-minus of -9 in his 30:22 on the floor before fouling out. He added two blocks and three steals.\nJan 12 9:20 Ap-Mid PT10:20 Ap-Mid MT11:20 Ap-Mid CT12:20 Mat. HE5:20 GMT13:20 10:20 Ap-Mid MST11:20 Ap-Mid CST12:20 Mat. EST9:20 UAE06:200:20 HE11:20 Ap-Mid CT3:20 - Kris Dunn scored six points Friday, going 2 of 9 from the field, as the Chicago Bulls fell to the Golden State Warriors 146-109. Dunn did not grab a rebound but had five assists, tallying a plus-minus of -45 in his 24:34 on the floor. He added two steals. Dunn went 1 of 1 from behind the arc.\nJan 9 8:28 Ap-Mid PT9:28 Ap-Mid MT10:28 Ap-Mid CT11:28 Ap-Mid HE4:28 GMT12:28 9:28 Ap-Mid MST10:28 Ap-Mid CST11:28 Ap-Mid EST8:28 UAE05:2823:28 HE10:28 Ap-Mid CT2:28 - Kris Dunn scored 15 points Wednesday, going 7 of 14 from the field, as the Chicago Bulls fell to the Portland Trail Blazers 124-112. Dunn pulled down five rebounds and had seven assists, accumulating a plus-minus of -8 in his 29:43 on the floor. He tacked on two steals. Dunn went 1 of 2 from 3-point range.\nJan 6 2:07 Ap-Mid PT3:07 Ap-Mid MT4:07 Ap-Mid CT5:07 Ap-Mid HE22:07 GMT6:07 3:07 Ap-Mid MST4:07 Ap-Mid CST5:07 Ap-Mid EST2:07 UAE (+1)23:0717:07 HE4:07 Ap-Mid CT20:07 - Kris Dunn scored 16 points Sunday, going 8 of 15 from the field, as the Chicago Bulls fell to the Brooklyn Nets 117-100. Dunn pulled down four rebounds and dished out seven assists, tallying a plus-minus of -17 in his 33:31 of playing time. He added two steals.\nJan 4 6:51 Ap-Mid PT7:51 Ap-Mid MT8:51 Ap-Mid CT9:51 Ap-Mid HE2:51 GMT10:51 7:51 Ap-Mid MST8:51 Ap-Mid CST9:51 Ap-Mid EST6:51 UAE (+1)03:5121:51 HE8:51 Ap-Mid CT0:51 - Kris Dunn scored 16 points Friday, going 6 of 9 from the field, as the Chicago Bulls fell to the Indiana Pacers 119-116 in overtime. Dunn grabbed five rebounds and had 17 assists, tallying a plus-minus of +6 in his 40:17 of playing time. Dunn went 2 of 2 from behind the arc.\nJan 2 6:21 Ap-Mid PT7:21 Ap-Mid MT8:21 Ap-Mid CT9:21 Ap-Mid HE2:21 GMT10:21 7:21 Ap-Mid MST8:21 Ap-Mid CST9:21 Ap-Mid EST6:21 UAE (+1)03:2121:21 HE8:21 Ap-Mid CT0:21 - Kris Dunn scored 14 points Wednesday, going 6 of 12 from the field, as the Chicago Bulls fell to the Orlando Magic 112-84. Dunn pulled down three rebounds and dished out four assists, accumulating a plus-minus of -19 in his 27:44 on the floor. He added three steals. Dunn went 1 of 2 from behind the arc.\nDec 30 4:30 Ap-Mid PT5:30 Ap-Mid MT6:30 Ap-Mid CT7:30 Ap-Mid HE0:30 GMT8:30 5:30 Ap-Mid MST6:30 Ap-Mid CST7:30 Ap-Mid EST4:30 UAE (+1)01:3019:30 HE6:30 Ap-Mid CT22:30 - Kris Dunn scored 14 points Sunday, going 6 of 15 from the field, as the Chicago Bulls lost to the Toronto Raptors 95-89. Dunn pulled down three rebounds and dished out eight assists, tallying a plus-minus of +9 in his 31:18 on the floor. He added two blocks.\nDec 28 5:27 Ap-Mid PT6:27 Ap-Mid MT7:27 Ap-Mid CT8:27 Ap-Mid HE1:27 GMT9:27 6:27 Ap-Mid MST7:27 Ap-Mid CST8:27 Ap-Mid EST5:27 UAE (+1)02:2720:27 HE7:27 Ap-Mid CT23:27 - Kris Dunn scored 13 points Friday on 5-of-12 shooting as the Chicago Bulls defeated the Washington Wizards 101-92. Dunn grabbed four rebounds and dished out eight assists, tallying a plus-minus of +1 in his 32:05 of playing time. He added two steals. Dunn went 1 of 2 from behind the arc.\nDec 26 6:51 Ap-Mid PT7:51 Ap-Mid MT8:51 Ap-Mid CT9:51 Ap-Mid HE2:51 GMT10:51 7:51 Ap-Mid MST8:51 Ap-Mid CST9:51 Ap-Mid EST6:51 UAE (+1)03:5121:51 HE8:51 Ap-Mid CT0:51 - Kris Dunn scored nine points Wednesday, going 4 of 11 from the field, as the Chicago Bulls fell to the Minnesota Timberwolves 119-94. Dunn pulled down eight rebounds and dished out seven assists, tallying a plus-minus of -23 in his 30:17 on the floor before fouling out. He tacked on two steals.\nDec 23 4:27 Ap-Mid PT5:27 Ap-Mid MT6:27 Ap-Mid CT7:27 Ap-Mid HE0:27 GMT8:27 5:27 Ap-Mid MST6:27 Ap-Mid CST7:27 Ap-Mid EST4:27 UAE (+1)01:2719:27 HE6:27 Ap-Mid CT22:27 - Kris Dunn scored 17 points Sunday on 8-of-11 shooting as the Chicago Bulls defeated the Cleveland Cavaliers 112-92. Dunn pulled down eight rebounds and dished out seven assists, tallying a plus-minus of +21 in his 28:41 of playing time. He added one block and one steal.\nDec 21 6:51 Ap-Mid PT7:51 Ap-Mid MT8:51 Ap-Mid CT9:51 Ap-Mid HE2:51 GMT10:51 7:51 Ap-Mid MST8:51 Ap-Mid CST9:51 Ap-Mid EST6:51 UAE (+1)03:5121:51 HE8:51 Ap-Mid CT0:51 - Kris Dunn scored 12 points Friday on 4-of-14 shooting as the Chicago Bulls defeated the Orlando Magic 90-80. Dunn pulled down eight rebounds and had seven assists, accumulating a plus-minus of -4 in his 35:50 on the floor. He tacked on two steals.\nDec 19 6:56 Ap-Mid PT7:56 Ap-Mid MT8:56 Ap-Mid CT9:56 Ap-Mid HE2:56 GMT10:56 7:56 Ap-Mid MST8:56 Ap-Mid CST9:56 Ap-Mid EST6:56 UAE (+1)03:5621:56 HE8:56 Ap-Mid CT0:56 - Kris Dunn scored 24 points Wednesday on 11-of-21 shooting as the Chicago Bulls lost to the Brooklyn Nets 96-93. Dunn grabbed four rebounds and dished out six assists, accumulating a plus-minus of +1 in his 34:19 of playing time. He tacked on two steals. Dunn went 1 of 3 from behind the arc.\nDec 17 6:45 Ap-Mid PT7:45 Ap-Mid MT8:45 Ap-Mid CT9:45 Ap-Mid HE2:45 GMT10:45 7:45 Ap-Mid MST8:45 Ap-Mid CST9:45 Ap-Mid EST6:45 UAE (+1)03:4521:45 HE8:45 Ap-Mid CT0:45 - Kris Dunn scored 12 points Monday, going 5 of 8 from the field, as the Chicago Bulls fell to the Oklahoma City Thunder 121-96. Dunn grabbed two rebounds and dished out seven assists, accumulating a plus-minus of -17 in his 25:44 of playing time. He tacked on one steal.\nDec 15 7:09 Ap-Mid PT8:09 Ap-Mid MT9:09 Ap-Mid CT10:09 Ap-Mid HE3:09 GMT11:09 8:09 Ap-Mid MST9:09 Ap-Mid CST10:09 Ap-Mid EST7:09 UAE (+1)04:0922:09 HE9:09 Ap-Mid CT1:09 - Kris Dunn scored 24 points Saturday on 11-of-19 shooting as the Chicago Bulls topped the San Antonio Spurs 98-93. Dunn pulled down seven rebounds and had three assists, accumulating a plus-minus of +3 in his 35:45 of playing time. He added two steals.\nDec 13 8:18 Ap-Mid PT9:18 Ap-Mid MT10:18 Ap-Mid CT11:18 Ap-Mid HE4:18 GMT12:18 9:18 Ap-Mid MST10:18 Ap-Mid CST11:18 Ap-Mid EST8:18 UAE05:1823:18 HE10:18 Ap-Mid CT2:18 - Kris Dunn scored five points Thursday on 2-of-5 shooting as the Chicago Bulls fell to the Orlando Magic 97-91. Dunn grabbed four rebounds and dished out two assists, tallying a plus-minus of -2 in his 25:50 of playing time. He tacked on one steal. Dunn went 1 of 1 from 3-point range.\nDec 10 6:35 Ap-Mid PT7:35 Ap-Mid MT8:35 Ap-Mid CT9:35 Ap-Mid HE2:35 GMT10:35 7:35 Ap-Mid MST8:35 Ap-Mid CST9:35 Ap-Mid EST6:35 UAE (+1)03:3521:35 HE8:35 Ap-Mid CT0:35 - Kris Dunn scored nine points Monday, going 3 of 5 from the field, as the Chicago Bulls lost to the Sacramento Kings 108-89. Dunn grabbed three rebounds and dished out six assists, accumulating a plus-minus of -9 in his 20:04 on the floor.\nOct 23 10:20 Mat. PT11:20 Mat. MT12:20 Ap-Mid CT1:20 Ap-Mid HE17:20 GMT1:20 10:20 Mat. MST11:20 Mat. CST12:20 Ap-Mid EST21:20 UAE19:2013:20 HE12:20 Ap-Mid CT16:20 - Dunn (sprained MCL) will return Monday against Sacramento.\nAnalyse: Dunn, who has been out since Oct. 23, averaged 13.4 points and 6.0 assists for Chicago last season after struggling as a rookie with Minnesota.\nOct 22 7:36 Ap-Mid PT8:36 Ap-Mid MT9:36 Ap-Mid CT10:36 Ap-Mid HE2:36 GMT10:36 7:36 Ap-Mid MST8:36 Ap-Mid CST9:36 Ap-Mid EST6:36 UAE (+1)04:3622:36 HE9:36 Ap-Mid CT1:36 - Kris Dunn scored nine points Monday on 4-of-13 shooting as the Chicago Bulls lost to the Dallas Mavericks 115-109. Dunn grabbed four rebounds and had seven assists, tallying a plus-minus of -1 in his 30:18 of playing time before fouling out. Dunn went 1 of 4 from behind the arc.\nOct 18 8:03 Mat. PT9:03 Mat. MT10:03 Mat. CT11:03 Mat. HE15:03 GMT23:03 8:03 Mat. MST9:03 Mat. CST10:03 Mat. EST19:03 UAE17:0311:03 HE10:03 Mat. CT14:03 - Dunn (personal) is not on the injury report for Monday's game in Dallas.\nAnalyse: Dunn has yet to play this season.\nOct 12 6:40 Ap-Mid PT7:40 Ap-Mid MT8:40 Ap-Mid CT9:40 Ap-Mid HE1:40 GMT9:40 6:40 Ap-Mid MST7:40 Ap-Mid CST8:40 Ap-Mid EST5:40 UAE (+1)03:4021:40 HE8:40 Ap-Mid CT0:40 - Kris Dunn scored two points Friday on 1-of-5 shooting as the Chicago Bulls fell to the Denver Nuggets 98-93. Dunn pulled down three rebounds and had three assists, accumulating a plus-minus of -13 in his 22:05 of playing time. He tacked on one steal.\nOct 10 6:49 Ap-Mid PT7:49 Ap-Mid MT8:49 Ap-Mid CT9:49 Ap-Mid HE1:49 GMT9:49 6:49 Ap-Mid MST7:49 Ap-Mid CST8:49 Ap-Mid EST5:49 UAE (+1)03:4921:49 HE8:49 Ap-Mid CT0:49 - Kris Dunn scored nine points Wednesday on 4-of-5 shooting as the Chicago Bulls topped the Indiana Pacers 104-89. Dunn pulled down three rebounds and had seven assists, tallying a plus-minus of +18 in his 22:41 of playing time. He tacked on two steals. Dunn went 1 of 1 from 3-point range.\nOct 8 5:39 Ap-Mid PT6:39 Ap-Mid MT7:39 Ap-Mid CT8:39 Ap-Mid HE0:39 GMT8:39 5:39 Ap-Mid MST6:39 Ap-Mid CST7:39 Ap-Mid EST4:39 UAE (+1)02:3920:39 HE7:39 Ap-Mid CT23:39 - Kris Dunn scored eight points Monday, going 4 of 7 from the field, as the Chicago Bulls lost to the Charlotte Hornets 110-104. Dunn pulled down three rebounds and had three assists, tallying a plus-minus of -15 in his 24:33 on the floor. He tacked on four steals.\nOct 3 6:55 Ap-Mid PT7:55 Ap-Mid MT8:55 Ap-Mid CT9:55 Ap-Mid HE1:55 GMT9:55 6:55 Ap-Mid MST7:55 Ap-Mid CST8:55 Ap-Mid EST5:55 UAE (+1)03:5521:55 HE8:55 Ap-Mid CT0:55 - Kris Dunn scored 10 points Wednesday, going 3 of 6 from the field, as the Chicago Bulls fell to the Milwaukee Bucks 116-82. Dunn pulled down five rebounds and had two assists, accumulating a plus-minus of -30 in his 20:04 of playing time. Dunn went 1 of 1 from behind the arc.\nSep 30 5:39 Ap-Mid PT6:39 Ap-Mid MT7:39 Ap-Mid CT8:39 Ap-Mid HE0:39 GMT8:39 5:39 Ap-Mid MST6:39 Ap-Mid CST7:39 Ap-Mid EST4:39 UAE (+1)02:3920:39 HE7:39 Ap-Mid CT23:39 - Kris Dunn scored 12 points Sunday, going 5 of 7 from the field, as the Chicago Bulls topped the New Orleans Pelicans 128-116. Dunn grabbed eight rebounds and had three assists, tallying a plus-minus of +8 in his 22:34 on the floor. He added two blocks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"- The negative 14 degree weather we've been having. Which wouldn't be awkward if it wasn't for the strange things it does to the inside of my nose.\n- Brushing your teeth with an electrical toothbrush. And turning it on the second before it's inside your mouth. Toothpaste mascara burns the crappins out of your eye.\n- An email correspondence in which I totally thought I was talking to a girl the whole time. Turns out it's a guy. I called him Chica, and then he told me.\n- Thinking someone is husband, and standing infinitely too close to them until I look up and see that they're giving me the stink-eye. Punk. You smell the same alright?\n- Trying to type and talk at the same time. I should never do this while I'm writing emails. Cause it merits results like, \"Dear ______, It was so good to hear from you! Would you pass me that water bottle, I'm thirsty.\"\n- I cannot watch the Siemens commerical without giggling a little. I'm 5 like that.\n- The fact that I could probably spend a whole hour dissecting my pores in one of those freaky magnifying mirror thingies. It's gross and strangely addicting.\n- Sitting on a leather couch and it makes a funny squeaky noise like you ... you know ... did it or something. So then you're like, \"that was the couch\" and people are like \"right\" and you're like \"no really. see watch\" ... and you try and imitate the noise, but can't do it to save your life. so then everyone shakes their head at you and you run away.\n- Holy BLUE right!? That reflective snow is magic on the eyeballs. Or maybe it's the cold and they're just frozen over. Ice Ice Baby!\n- Husband and I planning our spring break trip together. Words cannot describe how excited I am to get out of this ice box.\n- Scraping my windshield off with a credit card. Actually works better that a real ice scraper.\n- Husband still unconsciously rubbing my legs ... even though they desperately need to be shaved. Ah. The blissful comfortability of marriage.\nLink up your Awkward and Awesome posts here!! Let's see em dudes!\nAlso, if you're worried about having more than one link up there, just create an Awkward and Awesome label for your blog, and then link that up ... rather than linking up each individual post.\nHahaha, I love that not shaving your legs is in the \"Awesome\" section! I'm guilty of the same thing, but my boyfriend of six years doesn't seem to mind either. Aahahaha.\nI love your awkward and awesomes, so cute! I seriously hate the couch noise, but for me it's in my huge leather seats that we have in my business school classes. so. awkward. and I swear people must think it's me! Ahh!\nAdoring the fuzzy blue homemade looking Gap gloves. Love your blog!\nawkward list is longer!! that's not so good though!\nI am seriously loving your blog!!! Think I may give you A\/A Thursdays a try!\nalways such great lists! love this.\nand you reminded me...i need to get a haircut too. hmmm.\nAh! I love the coat! I'm from MN and still don't have a cute winter coat and this is the perfect inspiration!\nI was just wondering if blogging about my frozen nose insides would be TMI! Thank you for giving me permission. I feel ya on the -14 and needing a tropical vacation.\nCan't wait to see what you're doing to your hair!\nAs a yet unmarried girl, I was happy to read the shaving the legs comment.\nOh, and the shaving legs comments... so true lol.\njust posted my first AA!! i did have some awkward moments... maybe some tmi, but fun non the less!\nhaha! so funny. made my day.\ncan we talk more about the amazing bag and boots? i adore both so much. i found the boots online, but what about the bag? the link you posted says the website will be up in fall 2010?\nSydney, I really like your hair parted on the side like that!! It looks great both ways, but this is nice for a change.\nI just discovered your blog and I love it- you're one funny girl and you have amazing style. Love it.\nI live in Orem, UT and and am also freezing my tush off, although it's not quite to -14... more like +14. But still, anything under 20 degrees is unacceptable in my mind. It makes me get all sorts of hateful and plan imaginary trips to California.\nAnyway, let's be blog friends!\nyour posts are always little nuggets of happiness! your eyes do look so blue in these pictures and it's utterly gorgeous!! and i'm loving your bag more and more.\ncan't wait to see what you do with your hair!\nYou are so beautiful!!! You are is so shiny nad beautiful!!\n\"Holy BLUE right!? That reflective snow is magic on the eyeballs. Or maybe it's the cold and they're just frozen over. Ice Ice Baby!\"\nDay. Made. Loving the hair and those sweet mittens! Stay warm darling!\nand yes, last time i even held the arm of a man asking him if we could buy some chocolate cake, and when i looked up, it wasn't my man, but a guy who was smiling back at me :) oooh! shame! shame! shame!\nYou are SO SO SO freaking gorgeous!!!\nGirrrrrl, you go on and rock that side part!\nlove your bag! and those adorable mittens!\nHOW are you so pretty, HOW hmm?\nthe face expression in the first photo? priceless.\n& the way you did your bangs makes you look mighty fine.\nLove this outfit, girl. I am almost 8 months pregnant and miss wearing a coat, so bad!! None of mine will zip over my huge belly, and I don't want to spend the money on a maternity one I can't wear in two months.\nI just did a post on maternity fashion and some outfits to throw together if you'd like to take a peek. I know you mentioned in your FAQ that you two would like to start a family sometime soon. I'm normally not at all a fashion blogger, so I was pretty proud of my half way decent effort :) I definitely didn't get any beautiful Sydney-shots, but I think I did okay!\nahahaha cute list! - Sitting on a leather couch and it makes a funny squeaky noise like you ... you know ... did it or something. So then you're like, \"that was the couch\" and people are like \"right\" and you're like \"no really. see watch\" ... and you try and imitate the noise, but can't do it to save your life. so then everyone shakes their head at you and you run away.\nI was actually about to say how extra blue your eyes were! Sadly, snow doesn't seem to accent BROWN eyes. I'll have to take that up with my parents for giving all my siblings blue eyes when I got stuck with the brown ones... what's up with the gene pool??\nAnd I think I'll join the Awkward And Awesome Thursday ordeal, considering I laugh my lungs out!\nPeople always address me in emails as Mr. too. It's always so awkward.\nHahaha.....the nose thing! I know exactly what you are talking about.\nI think you have pretty blue eyes.\nOMG, I know exactly what you are talking about with the funny things the cold weather does to your nose! When I went to BYU-I and felt that sensation for the first time, I had no idea what it was. Growing up in California, I had never been exposed to weather that cold. I remember I kept checking to see if there was stuff hanging out of my nose, but no. So weird!\nI just realized about everyone has said that about you. HAH!\nAwesome: A man riding a bicyle wearing a motorbike helmet.\nAs always, terrific list. I love the shaving your legs thing. I'm so bad not to shave and hubby doesn't mind a bit! Can't wait to see the new hair!\nOMG my hubby works for Siemens!!! Let me tell you, he's been there for almost 2 years and the jokes just don't. get. old.\nLove the side swoop bangs! Tres chic!\nRe: the mirror addiction. Also equally addicting: tweezing!\nLove love love reading awkward & awesome thursdays! you're so funny!\n& oh my goodness, i can't believe how blue your eyes are in these pic's! amazing! :) definitely awesome! yay!\nWe ALWAYS use the credit card to shave ice of the car. ;) No worries, I hardly EVER shave my legs in the winter. :) My legs are softer when I start shaving again in the spring. lol.\ni just wanted to tell you that i have recently discovered your blog and i'm in love. i love your style. i love that you shop in normal stores like most of us women do. i love the way you write. totally awesome!!! thanks for putting a smile on my face.\nI really love the front of your hair in these pics.\nNew bag? cause I love all large black bags!\nI love not having to shave my legs because I'm married! My husband doesn't care one bit!\nYour eyes look amazing! and I love your hair like that.\nHaha I was going to jump on here and write something about how crazy cool your eyes look - just from the first picture. Then I kept scrolling down, and you wrote about them! Nice job, mind reader!\nHmmm, im having problems posting up your button on my blog. EEEK! Not techy after all!\nlove your blue eyes - amazing!\nchica?! that was cracking me up!\nget bangs! get bangs! you'd be soooo cute!\nI LOVE your Awkward and Awesome posts! They make me laugh. I think I'm going to start one on my own blog on Fridays.. I even had my own awkward moment today that will definitely be going on there.\nYour blog is SO inspirational to me. I've been reading it for about 2 months and your outfits\/posts\/everything is just AMAZING and you've inspired me to start my own blog. Thank you so much for being awesome!\nhaha! I can't believe you were using a credit card to scrape your windshield! I actually did something so similar last week!\nLove your mittens! So cute.\nCan't wait to see what you do with you hair!\nOh, and good tip about the credit card for scrapping off ice! Gotta try that.\nI love your hair like that! Your eyes are soooo blue today, BEAUTIFUL.\nI love Thursday! You hit the nail on the head with everything! Great outfit too - the mittens are so cute.\nTotally did the credit card thing yesterday.. I was desperate. And, it does work!\nYou look so cozy and warm!! I love that bag. I need one to carry my camera with me at all times!!\nYou are too gosh darn cute lady. I'm digging this new layout. Looks real profresh. That's like professional and fresh at the same time. doesn't get any better than profresh... though that sounds like it may be a toothpaste... don't get it in your eyes.\nI feel like I shave my legs more that I post outfit photos. It's like I've been wearing jeans, and pants too much. I think it's time to shave now. I can't wait to see what you do to your hair.\nholy blue is right. gorgeous really.\nHow do we delete our old posts in the linky and just have the one with the label for all of them?\nOh my gosh, your eyes are fierce!\nWhere did you get those super cutie mittens?\nCan't wait to see what you do with your hair! I'm going as well as looking for more of an ombre effect! Love your mittens btw.\ncredit card scrapper - been there - done that - very often!\nThis is one of my favorite looks of yours EVER. So simple, and you still manage to look stunning. Love it!\nLove your bangs! You are just too cute!!!\ni totally relate to typing\/ writing and talking at the same time and mixing the two..ha! love your mittens!\nsydney, you crack me up!\nso, with the whole you e-mailing someone you thought was a girl, i did that too only i was chatting with them! we met through blogger and i was doing some design work for said blogger and i thought the whole time, that it was a girl and then he said something about him being a boy and it was like whoa! awkwarrrrrd!\nA. i. want. that. jacket.\nB. your side-swept bangs look awesome.\nI was recently introduced to your blog (I went to high school with Sarah from For the Love)and just had to comment to say you are adorable. I love your style so much, I try to emulate the same sort of style but wish I had your wardrobe...you make me want to go shopping right now...the hubby wont' like that!\nLove that bag!!!! Now, that is AWESOME!!!\nsydnaaaay. what up g! g, for girl, you know! love this! too cute!\nI am really enjoying your sweeping bang and effortless up-do. I can't wait to see what you do at your hair appointment!\nthis is what makes thursdays so great!!\nyou are so cute! love your mittens!!\nYour eyes look gorgeous! Love these Awkward and Awesome posts.\nLove the bangs! And Ive totally done that closeness thing with the wrong \"husband\" before in the grocery store...AWKWARD for sure!\nand seriously, this weather. i think my nasal fluid freezes up and then whenever i go inside it melts. gross.\nThe leather chair thing is SO true!\nWhat happens to your nose when its cold? Mine gets a little runny.\nYour eyes look SO PRETTY! SUPER blue!!\nlove love love your dramatic side swept part and cuddled up coat!\nha! I love this post, and your blog!\nyour awkward and awesome posts are the highlight of my thursdays!\nAwkward: Taco Bell trying to provide a diet plan, because I thought the exact same thing! so funny!\nCan't wait to see your new hair! Post asap!\nlove your mittens, it's never cold enough in sydney (lol, sydney!), australia to wear them.\ngoodluck with the hair :) you'll look fantastical!\nI feel your pain... I was scraping off a 1\/2 inch of ice on the windows of my car (we just had an ice storm.. the joys of living in the south). It wouldn't even budge and neither would the door. Everything was glued shut!\nI love this! Your blog is adorable, and I love your awkwardness and awesomeness this is too cute! :) Love your bag by the way its darling!\nThat's fantastic isn't it? The feeling that you know you are going on vacation soon! and yes, Mickey Blue Eyes indeed!\nHaha! I scrape my car with a Lotion tube from Bath and Body Works. Also true that it works better than an ice scraper.\nFor some reason I never go to famous footwear.\nYou've definitely motivated me to GO FIND THOSE BOOTS!\nthe worst for me would be the squeaky chairs you get in school which squeak like mad when you move the slightest bit and when the squeaking happens during a quite time... like exams.\nyou look beautifull!! so natural, awesome outfit!\nWHY would any boy give you the stink eye for sidling up to them??! You'd think it'd make their day. I bet he went home and listed it in his 'awesome' section.\nThat bag is AMAZING Syd!\nyour blog is really personal and you are so likeable!\nI'm going away from internet for the weekend and I'm dying to know what you did to your hair! Bangs? Short? Red? Please oh please post pics!\nCould you tell me something though please??\nWhen did you know what your blog to be about because looking at other entries from a year ago it wasn't about your amazing style. When and why did that change??\nPlease reply! I would love to know!\nNerdage here, but snow actually has a tinge of blue. It looks white because of insolation and reflection.\nI second Sarah Christine's comment!\nI love how you share your style and fashion sense & I'm curious about that too!\nLove this outfit and can't wait to see your new hair! Also, I third Sarah Christine's comment!\nOhh, I love your new cut hairrr!\ni loooove your hair parted on the side like that. sooooooo cute. and thursdays are my favorites in the blog world because of YOUR posts!\nthose mittens? i die. i want them pronto!\nMan oh man, I need that bag.\nAwkward: Having my boss say to me, \"So...I really want you to do some more public speaking and training, but I didn't really want it to happen like this. Can you give a four-hour contracts training on Wednesday?\" last Friday.\nk, I just found your blog and have been reading through your posts. You made me laugh so hard I cried! You are funny sister! And I love the post your hubby posted...haha! Keep the humor of everyday life coming! Oh, and totally relate about the leather couch thing! hahahaha! Funny!\nJust stumbled upon your amazing blog via Pinterest. If you have the opportunity can you please tell me the name of your coat from Yesstyle? I can't find it on the site. Thanks so much! You just rocked my night!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find me on my favorite social media app, Instagram. My username is runningonhappiness.\nP.S. the winner of the ad space on The Wiegand's is: Lesleigh! Congrats and thanks to all who entered.\nFound you! Love Instagram as well, it has totally replaced FB for me.\nSo cute! Loving those pink loafers!\nLove your pink shoes!!! where did you get them!?\nInstagram is my fav! Awesome pictures!\nLove the Instagram effects and your lip color in the middle photo!\nCute photos!! You have a great eye!\n2. please tell me you did not capture that lady bug with an iphone!\njust started following your blog... your pics are adorable!\nWoot Woot! I am thrilled about the ad space giveaway! Thank you so much!\nPS Cute pictures, of course. As always!\nI so wish I had an iPhone so I could use instagram. It looks like a lot of fun. As well as Draw Something. My boyfriend has an iPad and he won't stop playing Draw Something! I want to know what the hype is about!!\nlove how you laid them all out. how do you format the photos like that? xoxo.\nI just love your hair color! How do you tell your colorist to color it?\nWhere did you get those moccasins?!\nGiraffes, polka dots, and pink loafers?! Love it!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..f37d61c2e893931e1e7f15ce3e965c72d3b9cf29
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Scholarships that allow students to get a good education are important, but first we want to measure the progress that the schools are teaching our students, we want to hold them accountable for the progress, we want to hold the schools accountable for teaching the young people in America.\nWell, paycheck protection is an important ingredient for a successful campaign finance reform measure.\nMy children and grandchildren loved the secret servicemen and women that served us. I was honoured that they thought I was important enough to protect.\nThe Oval Office symbolizes\u2026 the Constitution, the hopes and dreams, and I'm going to say democracy. And when you have a dress code in the Supreme Court and a dress code on the floor of the Senate, floor of the House, I think it's appropriate to have an expectation that there will be a dress code that respects the office of the President.\nYou shouldn't presume that all quotes that are in a magazine or a newspaper are accurate.\nThe top priority is leaving no child behind. We want accountability in the system, and we want schools to recognize they have a responsibility to teach students.\nThe U.N. can meet and discuss, but we don't need their permission.\nKyoto was a flawed process. There isn't one industrialized country around the world that has ratified that treaty, and so that is a non-starter.\nDon't believe everything that you read in the newspapers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rising in this photo, from left to right: the back-t0-searching Williams Tower, the 30-story Mercer condo building, and the vertical sign for the new Starbucks that opened yesterday morning at the northeast corner of Richmond Ave. and Chimney Rock Rd. The freestanding coffee shop at 5540 Richmond (next to the spot previously occupied by now-demolished Taco Cabana) is right across the street from the freestanding former Starbucks no longer listed on the company's website at 5549 Richmond, between wine-themed Pinot's Palette and probably-not-intentionally-wine-themed Night Train Luggage.\nThat's nothing! The River Oaks Shopping Center now boasts FOUR Starbucks within a two block area. The famous two across the street from eachother, the one inside Barnes & Noble and now a small, airport like operation inside the Kroger store. I think the coffee chain must be taking its cues from Mattress Firm.\nI know Starbucks isn't short on dough but they seem to have used an absurdly large amount of land for that tiny building. I guess they expect a lot of cars to line up but that's never stopped Chik-Fil-A from just diverting traffic to the street.\nOoo, a pergola. That'll come in handy when its raining, after it rained, when birds congregate on it, or when the sun is out.\nMight I remind people that this is Houston, sub 30\u00b0N latitude, and not Paris, 45\u00b0 latitude.\nWhile the pergola is pretty, I have to agree that it isn't very practical in our climate.\nThat being said, I never patronize Starbucks \u2013 I just get my coffee fix at home on the weekends or at the office during workdays. No need to clog up the streets any longer than needed and the coffee is free at the office.\nPlus, the idling in the drive-through adds to pollution and wasting gas: more hidden costs to that coffee stop.\nOne does not go to Starbucks for the coffee, unless you don't know what good coffee is, one goes to sit down and kill time or to meet a friend, the cup of coffee is just the price of short term rental of the chair.\nBring back articles about illegal tree removal before you waste your time on more Starbucks articles.\nSure toasty, just ignore the hipsters. They need attention, too. No other reason for them to exist.\nI don't like going to starbucks because I really don't like their politics. However, if\/when I do go, it's to get some 'fru fru' (sp?) drink. Their pain coffee might be 'meh' but they really have those coffee desert drinks nailed perfectly.\nI don't believe hipsters congregate at Starbuck's with soccer moms and earnest, young execs. What's disturbing about developmnt outside the Loop is how predictable it is.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Nail Culturee, we strive to make your spa experience a relaxing, rejuvenating, and a memorable experience. The aesthetics and the ambience of the spa is designed to put clients at ease, and offer them utmost comfort. Our expert staff is trained to dispense quick service to ensure that each and every customer is quickly attended to.\nYour day at the spa will begin with a complimentary refreshing drink. This is followed by a consultation with a beauty specialist who will analyze the condition of your hands and feet, asses the health of your hand and nail health, and suggest a treatment or a course of treatments best suited for you. Depending on the treatments prescribed, the client is then teamed with a nail technician who will carry out the procedure. The spa is also equipped with individual rooms and state of art tools that are used for specific hand and feet treatments.\nOur nail technicians have been carefully selected and trained at Nail Culturee to ensure clients receive undivided attention and specialized treatment here. So you will remain in expert hands for the entire duration of your visit. For nail art, manicures, and pedicures, we use products by O.P.I, a Hollywood-based cosmetics company, and Baehr, a German podiatric brand. The procedures are based on treatments devised by O.P.I and Baehr, and they are among the services offered in some of the leading salons across the US. Specialized treatments for cracked heels, calluses, nail fungus, etc. are also available.\nWe offer a variety of services designed specifically for hands, feet, and nails. Nail art is one of the main attractions on our beauty menu; our talented nail artists will transform your nails in to a work of art. Besides this we also offer nail paint and nail extension services.\nWhile our doors are always open for our customers, we strongly advise our clients to schedule an appointment to ensure fast and efficient service.\nWe have simplified the procedure to book appointments. Along with the traditional method of booking an appointment on phone, our clients can also simply schedule one using the online form. Just fill in your details, along with the time you prefer, and submit. You will receive a follow-up email, and call from us confirming your appointment.\nThe premium services Nail Culturee offers are best experienced than read. So book your appointment today! A unique and memorable experience awaits you. As we like to say, \"We will wait on you, hand and foot!\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I will be offering lessons next saturday, September 15th, at Intensity Cheer Elite. CLICK HERE TO SCHEDULE.\nBasically, this is going to be the launching site for a new project I am working on called FULL-OUT Cheer & Fitness. Full-Out's goal will be to revolutionize the way cheerleaders learn new skills and prepare their bodies for the high demands placed upon them during the competitive season.\nIn the upcoming weeks and months I will begin to add more information about the project. But in the meantime this site will remain a blog where I write helpful articles, comment on industry news, and keep my followers up-to-date on my endeavors. If you have any cheer or fitness related questions feel free to email them to coach.matt@cheerfullout.com and if I like it I will answer it with a blog post!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Read the incredible true story of the \"unsinkable\" ship that sank. - back cover.\nChildren love this terrific book. The author is Mark Dubowski and it was published in January of 2012 by DK CHILDREN. The child's book has 48 pages and it provides artistically colored illustrations. It's dimensions are 0.2\" Height x 8.9\" Length x 6.1\" Width and weighs around 0.25 lbs. If you prefer a copy of this book, check out the market add to shopping cart button below.\nDimensions: Height: 0.2\" Length: 8.9\" Width: 6.1\"\nPackage Dim.: Height: 0.2\" Length: 9\" Width: 6\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdjks b/data_all_eng_slimpj/shuffled/split2/finalzzzbdjks
new file mode 100644
index 0000000000000000000000000000000000000000..1230df79fc447df37bd40abebe546c0e7ddfa19b
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@@ -0,0 +1,5 @@
+{"text":"Learn how to operate Microsoft Office applications like Excel and PowerPoint properly and efficiently, or use our Microsoft Office training course certification to attain a globally accepted credential that will prove that you are an Office specialist.\nYou've probably noticed that part of living and working in the world today requires that you do many things at once. For many of us, managing multiple tasks is our normal work mode. You prepare a new presentation for a client while you're working collaboratively with your team, corresponding with people through e-mail, and inserting Microsoft Office Excel data\u2014 which might be changing moment to moment\u2014into the slides you create.\nWho should attend Microsoft Office Specialist Training?\nThis course is designed for first time users, who have little to no experience. Learn how you can get more from the new features in Office 2013 or if this is a first look for you, learn all the benefits in using Microsoft Office.\nAre you ready to become a Microsoft Specialist?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The entirety of Diablogate has, above all, been a giant PR disaster. Whether or not you personally find Blizzard culpable for Diablo Immortal's backlash, it remains clear that the controversy surrounding the game has ultimately cast an ominous shadow over the game. In the face of this controversy, as Blizzard faces a marketing crisis, one upcoming game has already attempted to appeal to frustrated Diablo fans.\nRegardless of how you feel about Diablo Immortal, sometimes you just have to appreciate a good dunk when you see one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What keeps a cellular \"warehouse\" from overfilling?\nA new method of redesigning protein molecules may yield novel drugs.\nMice lacking the protein stayed slim, even on a high-fat diet.\nProteases shouldn't work in the cell membrane. But they do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Russ and Jerry's Shell Service Station was located at 125 S. Milwaukee Avenue. Proprietors Russ Brown and Jerry Hamil featured products such as Sun Diagnosis equipment, Firestone Tires, and Wilco front end alignment, as well as the usual automobile services of the day. By 1959 Russ had left the business and the station was advertised as Jerry's Shell Service.\nDescription Russ and Jerry's Shell Service Station was located at 125 S. Milwaukee Avenue. Proprietors Russ Brown and Jerry Hamil featured products such as Sun Diagnosis equipment, Firestone Tires, and Wilco front end alignment, as well as the usual automobile services of the day. By 1959 Russ had left the business and the station was advertised as Jerry's Shell Service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NRW agreed, identified and documented which habitats and SAC features are sensitive to a range of Marine Licencing Band 1 activities. The dataset consists of 2 layers detailing information about sensitive habitats. These datasets constitute polygons derived from NRW held BAP (Section 7) \/ OSPAR layers, Regulation 35 SAC habitat layers and Phase 1 intertidal survey. All the habitats included in these layers were deemed to be sensitive to Marine Licencing Band 1 activities. The data includes records of protected or sensitive habitats, collected as part of baseline survey, monitoring programmes and ad-hoc records. A third layer is derived from the Special Protection Area (SPA) designated sites dataset and identifies SPA sensitive to Marine Licencing Band 1 Activities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdwpx b/data_all_eng_slimpj/shuffled/split2/finalzzzbdwpx
new file mode 100644
index 0000000000000000000000000000000000000000..cf5c761e537db4933f914658ef5aa80382c5ea65
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@@ -0,0 +1,5 @@
+{"text":"There are so many hoardings and out of home media that we come across every day. Now just imagine the magnitude of toxic waste we create and accumulate every passing day! Environmental protection is the need of the hour and the introduction of non-toxic printing materials is a great step in this direction. Iron frames or PVC boards are great! They are easy to install and not very time consuming but the shift to corrugated boxes can be more environmentally favorable.\nWith the introduction of environmentally friendly printers like the Latex 3100, which produces an output of 100% biodegradable quality, the environmental hazard of printing can be reduced but the truth is that the trend is yet to catch up in India. A major deterrent is the cost of the technology, both the capital cost and the cost of operation. Green printers are at least 20% expensive in capital costs and the Ink and media for printing, adds another 5%. However, the long-term sustainability is achieved by educating certain customer segments. Today, increasing number of responsible organizations stress on going green!\nAnother way large format printers can reduce the impact of their printing activities on the environment is by choosing to dispose of waste responsibly. This is a small step with big rewards! Organizations also have the option of promoting and practicing environmental-friendly initiatives like \"plant a tree\" or \"rain water\" harvesting.\nWhile these may come across as small measures, they are certainly taking us one step closer towards undoing a part of the damage we do every day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Within the services industry, businesses face the unique challenge of selling a product which is constantly changing and is therefore difficult to predict and manage. This problem is one which many ERP software vendors aim to address with industry specific ERP software systems. Services ERP systems are designed to improve efficiency and output of demand management, outsourcing, financial performance and the management and estimation of projects.\nMany of the major players in the ERP software market sell products aimed at the services and industries and the problems that industry faces. Microsoft provides its Microsoft Dynamics ERP for members of the services industry including the advertising services Omnicom Group and the engineering services company David Evans and Associates. Another major ERP software provider for the services industry is Sage. Sage is the world's third largest supplier of ERP software (behind Oracle and SAP). The Sage ERP X3 software - targeted at the services industries - aims to deal with much the same problems as most services ERP, with a focus on project management and performance.\nDutch software company Unit4 also provides services ERP through the company's Agresso ERP software. Agresso ERP is sold as an easily adaptable system which has the potential to reduce costly and time-intensive software customization post-implementation. With this claim, Unit4 has tapped into one of the industry's major challenges - business change.\nERP provider Epicor offers a similar selling point as part of its ERP software for the services industry. Described as highly flexible and scalable, the software is offered as on-site, hosted or SaaS, cloud application. Customization and flexibility are often the major selling points for an industry such as services, in part due to the variable, project-based nature of service company's work. The flexibility offered by many ERP vendors is also offered by Acumatica as part of the company's cloud ERP software. The nature of cloud ERP makes it a flexible and portable option for the services industries and unsurprisingly that is a key feature highlighted by Acumatica.\nThis guide is by no means a comprehensive run-through of all the ERP products available for the services industry but it will hopefully have given you an introduction to the market. If you are looking for the next step in researching ERP software for the services industry, then the 2013 ERP vendor directory is a great place to start.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm in the change business.\nI'm guessing you are too, or you'd not be investing time here.\nI wonder if you've also noticed, as I am increasingly, that there's a very fine line between working on oneself and working on the greater good? Chicken and egg, right? To the extent I learn to manage my anxiety, calm my temper, or deepen my awareness I know others benefit. And, to the extent I share a smile with a scary-looking stranger, comfort a crying child who has been bullied, or work wonders facilitating a meeting, I know I benefit as well.\nAs Seth encourages us to do, please feel free to share.\nThis entry was posted in Personal Growth and tagged Alan Webber, Dave Ramsey, E-book, Elizabeth Gilbert, Jackie Huba, Seth Godin, Tom Peters, What Matters Now on June 5, 2013 by gemmautting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Household ozonator is a device for the purification of water, air, food and household items; using ozone eliminates bacteria, viruses, odors and a broad range of contaminants. Many people mistakenly believe that the ozonator and ionizer are one and the same device. Although the ozonator produces a small number of ionized particles and ionizer \u2013 a certain amount of ozone, they differ mainly produce the different particles. Household ozonizer very extensive scope of application: it helps to remove invisible pollutants in drinking water, vegetables and fruits, children's toys, textiles, kitchen appliances and surfaces to eliminate odor in a closet, the fridge or the whole of the dwelling.\nUse of an ozonator is not only personal comfort, which arises due to the purified atmosphere in the house: arrangement effectively prevents the spread of viral diseases, improves the quality of drinking water, destroys parasites, pathogenic bacteria, in other words, makes a great contribution to the health of the whole family.\nIn its natural form ozone is found in the atmosphere, especially in the upper layers where it helps to protect the Earth from ultraviolet radiation. But this bluish gas, the flavor of which we can feel after a storm has a very practical application in our everyday life. The artificial production of ozone occurs by passing a powerful electric charge through the air, causing the oxygen molecule O2 dissolves and melds into the triatomic molecule O3. Since oxygen is, by nature, seeks its normal state, ozone is an unstable, reactive form of gas. It is effective for disinfection is already in very small concentrations, destroys almost all known types of micro-organisms and leaves no toxic by-products after chlorination.\n(1) clean meat, fish and other products, increases their shelf life. Leads to the destruction of hormones in poultry.\n(2) Cleans fruits and vegetables, reduces the amount of pesticides, agrochemicals and other toxic chemicals that are used for food processing.\n(3) Cleans rice, removes bran, Aspergillus (leechy mushroom) and carcinogenic substances.\n(4) clean the Cutlery and cooking utensils, disinfects and sterilizes, preventing cross contamination.\n(1) Use of ozonated water for washing hands without soap or other drying means. It is convenient and economical!\n(2) Washing with ozonized water makes the skin more healthy and attractive, helps to cure acne. Effectively cleans the mouth for cleaning teeth and removes plaque.\n(3) Safely disinfect personal items, including baby clothes without using detergents.\n(4) Prevents diseases and has a healing effect when helicase (breath), pericoronitis, hemorrhoids, vaginitis.\nTo destroy pathogenic organisms, ozone must come in contact with them for a few seconds. In fact, none of the pathogenic microorganism cannot survive if in a liter of water to pass 1.5 milligrams of ozone for 5 minutes.\nThink about how much good ozone is able to bring regarding vegetables and fruits. It is ideal for cleaning the products directly, sterilization of kitchen utensils, workspaces, and storage containers. Because ozone has such a short half life, it's not going on the surface, as it happens with chemical cleaning agents. Ozone does not damage the vegetables and fruits in contact. Products such as herbs and lettuce, stay fresh longer and taste better if they are rinsed in ozonated water.\nOzone is ideal for cleaning meat, poultry, fish and other seafood, as well as all kitchen utensils for cooking and storage.\nTo purify water with ozone generator, it is sufficient to place the tube into a container of liquid for 5-10 minutes: all bacteria, viruses and weight of the impurities (iron, arsenic, hydrogen sulfide, nitrites and complex organic compounds that give color) will be cleared. For air cleaning device is placed on a high surface and turn on without the hose on for 5-20 minutes depending on the size of the room. Food and household items, placed in a container with water, and then ozonated for 10-15 minutes. The user will note the appearance of foam on the surface of the water, which contains organic and inorganic impurities from the surface of the products.\nOzone is a natural purifier that destroys bacteria, viruses, cysts and fungi.\nOzone oxidizes hydrogen sulfide, iron, manganese and most chlorinated hydrocarbons, removes soap and oil.\nAs the oxidizer, ozone 500 times more powerful than chlorine and 3,000 times faster at killing bacteria and viruses.\nOzone consists of oxygen and leaves no chemical odors and flavors, does not cause irritation and discomfort.\nOzone makes the water fresh, clean and crystal clear, speaking an alternative to chemical cleaners.\nOzone is not carcinogenic, non-irritating, does not dry the skin, leaves no chemical film, does not destroy and will not discolor hair.\nThe first commercial use of ozone recorded in 1906 in the municipal water supply system. Today it is used in various industries around the world, including in the production of food and bottled water, water parks, aquariums, pools and Spa resorts.\nOn our website you can buy an ozonator for your home: simple, functional and affordable. The lightweight compact model performs an effective cleaning and disinfection of water, air, or any food and items that can be placed in the liquid. Ozonated water is ideal for antimicrobial cleansing and cleaning. We recommend atanacio as one of the most simple and affordable ways to create a perfect cleanliness in the whole house!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Medical and engineering principles are integrated in biomedical engineering (BME) programs in order to improve healthcare diagnosis and treatment. Most of the BME coursework involves research and development. Graduate and undergraduate programs in BME are characterized by their highly interdisciplinary nature; they are also customizable as required by a student.\nStudents should keep their career goals in view while selecting a BME program. Several biomedical engineers receive either a master or a doctoral degree (Ph.D.). Bachelor degree programs are also offered at schools. Those completing master degree programs typically seek employment in government labs, or conduct research in academia, or take up careers in product development. However, a bachelor degree is sufficient for entry level jobs in the field, either in public or private industry. Students should select programs that enjoy accreditation from the ABET, the leading accreditation agency for applied science, technology, and engineering programs. Students should consider internship programs offered in a specific specialty area such as marketing, operations, product development, regulatory affairs, or manufacturing. Useful professional contacts are facilitated by internships, and students should participate in internships that relate to the area in which they want to be employed after they have graduated. Training is imparted in some programs via clinical rotations and practical project work geared to improve students' ability to perform work. The structure of the program offered is another factor students must consider before they choose a school. A joint M.S.\/MBA program is offered that combines engineering and business studies, ideally suited for students seeking to understand how to promote a new engineering concept. By enrolling in a M.D.\/Ph.D. medical program that combines research and training, students can seek to obtain additional research and career options after they graduate.\nStudents are taught the application of engineering, science, and math principles in analyzing and interpreting modern problems via engineering tools. Apart from general education, engineering biology, systems analysis, bio-imaging, bio-instrumentation, and biomaterials are examined by students. Students usually participate in a capstone design project.\nInterdisciplinary training in engineering, medicine and biology is imparted to students; options for business classes are included in some programs. Both non-thesis and thesis tracks are usually included in programs. Admission criteria include a bachelor degree with sufficient preparation in engineering, the sciences and numerate skills.\nAdvanced coursework in engineering, physics, medicine and biology is included in the first two years of the Ph.D. program that awards seventy-two credits. Admission criteria include a qualifying exam and either a bachelor or master degree. Coursework is adjusted depending on whether the student has an undergraduate or graduate degree.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeoky b/data_all_eng_slimpj/shuffled/split2/finalzzzbeoky
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@@ -0,0 +1,5 @@
+{"text":"This practice guidance aims to assist Local Authorities (LAs) and their Corporate Parenting partners (CPs) in the development and implementation of strategies which enable care leavers to remain in secure, stable care placements (foster or residential) until such time as they are ready to move on. Strategies which emphasise young people's entitlement to support into adulthood, and which offer the option to return to care placements if and when they encounter difficulties.\nStaf has been directly involved in the development of this guidance and this 'at-a-glance' guide summarises the document. The full guidance is available to download from our website here (PDF).\nCrucially, the guidance has been developed to encourage and assist organisations to change the culture in which we try to meet the needs of young people leaving care. Specifically that end of care planning decisions should be based on meeting the needs of the individual rather than age or legal status.\nYoung people leaving care continue to have poor outcomes despite the efforts and investment by services. Research clearly shows us that the transition period towards interdependence is when young people are most vulnerable. This is the time when responsible corporate parenting can make a difference to a care leaver's life.\nYoung people are encouraged, enabled and empowered to remain in positive care settings until they are ready to move on.\nLAs will ensure that young people are aware of their right to remain in their placement and give them the opportunity to exercise that right.\nNo looked after young person leaves care without the skills and support necessary for success.\nAs is done with their own children, individuals with Corporate Parenting responsibilities will ensure that a looked after young person leaves their care placement with the practical skills and positive support networks that underpin successful adult life.\nLocal Authorities and their Corporate Parenting Partners will have made explicit their commitment to the Staying Put Scotland approach.\nLocal children's services plans and other relevant planning documents will be clear about the commitment to care leavers and support for the Staying Put approach. Detailing policy and procedure to make this practice real they will also identify the measures of success.\nResearch points us to the importance of maintaining relationships and attachment. This applies to both carers and professionals. It is these relationships which can hold things together as the young person moves on. The Staying Put Scotland approach advocates that young people should be able to remain in a placement or at the very least return their previous placement for an evening meal or period of respite.\nAccelerated or abrupt transitions from care settings should be avoided. Corporate Parents will ensure that the encouragement, enablement and empowerment of young people to make gradual transition towards independence becomes standard practice. Care Leavers must be able to return to care placements or a more supportive care setting if and when they meet with any difficulty, providing a vital safety net.\nJoint planning and partnership working is at the core of all good child care practice. Pathways or transition plans should reflect the dynamic and changing nature of young people's needs and circumstances as they progress towards adulthood (see separate Housing Options Protocols for Care Leavers summary and guidance).\nThe guidance document showcases a number of cases of innovative practice where localities have supported young people to remain in existing care settings, make graduated transitions or return to previous placements.\nMany LAs are enabling young people to remain with their existing foster carers beyond the age of 18 by facilitating a change of placement to 'supported carer' or 'supported lodging'.\nIn order for a group care setting to be effective it must reflect a strong shared philosophy of care and a positive culture which promotes a sense of belonging. There are specific challenges in establishing opportunities to either 'stay put' or 'return to group care settings'. Corporate Parents in partnership with the care inspectorate should work to make these arrangements routine.\nCare leavers are over-represented in the criminal justice system. LAs, the Scottish Police Service and the Scottish Prison Service as corporate parents will want to ensure that care leavers are identified within the criminal justice system at key points: arrest; detention; and discharge. All CPs will ensure that care leavers continue to receive access to necessary throughcare and aftercare support services.\nRegardless of their ultimate destination, all care leavers should be encouraged, enabled and empowered to remain in positive care settings, maintaining and enhancing supportive relationships until such time as they are ready to begin an extended transition towards adulthood.\nFor more information on staf's policy relating to Staying Put, please contact Pamela Graham: [email protected].","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is another big one, and it came in together with the rails, so you can guess why there ain't many pictures. We had our hands full of tools more labour-worthy than a camera.\nAnyway, a carousel was to be made in little no time. It had to be 4 meters across. It had to spin, at various speeds. It had to sync with a revolving stage. It had to be stable enough during rotation that articles of fine modern furniture could be suspended from it.\nMy part in this was the rotating circle and its suspension to the axis. The rotor, in short.\nIn short: there were none.\nFor all I intended to do, I really wanted some: we're talking circles here, people! Lots of angles cut at precise lengths, and most of them in volume. But, alas, every artist is his own engineer.\nI tried some open-source CAD programs, spending most time on FreeCAD (version 0.12.5284), and the verdict is: not stage-ready.\nFirst off, FreeCAD is a front-end with frills, written in Python 2 (and Qt). The back-end is OpenCascade. There are big issues with both.\nThe biggest one with me is this. Since FreeCAD is written in Python (essentially a prototyping language, in my view), it breaks a lot. My system is probably quite different from that of the current developers', and it dumped core once every three minutes (on average). Filing bug reports would have been useless: there were at least ten different exceptions in trace-backs, and we would have spent hours arguing whether this is a misconfiguration. Of Python.\nIf you do decide to try out FreeCAD, have the following in mind.\nDo not upgrade your real work parts to a combined part, upgrade copies instead and hide the originals. If you decide to edit the combined part later, exploding it will give you feces faces instead of solid objects.\nMaking a lot of copies at once might not copy everything, but it fills the object list anyway. Deleting objects that are listed but don't exist drops core.\nIt seems there's no way to specify an arbitrary reference axis when rotating, which caused me a lot of grief.\nYou either rotate in the current view's plane, or one of pre-defined XY\/XZ\/YZ, which are also called top\/front\/side elsewhere. Confusing.\nRotating multiple objects in the same plane by a certain angle does not always result in them rotating the same angle. If it sounds wrong, that's because it is.\nSnapping uses algebraic units and doesn't take the current view's zoom level into account.\nA lot of essential functionality is implemented as macros, which are not included in the basic package. Solid Sweep proves useful, if you can get used to it.\nBoolean operations on solids do not always work, which is mentioned in the FAQ.\nAlso, they sometimes drop core.\nAfter several hours of cursing, I got this.\nWhich is, in my book, too long for too little. I eventually drew and calculated everything I really needed on paper.\nWhy not AutoCAD? Because I don't use Windows. Why not Google SketchUp? Because that doesn't have Boolean operations on solids in the freeware version. Why not DraftSight? Because that has very limited 3D functionality. Why open-source? Because we don't really get anything but experience points for this.\nSorry for the long-wrought review, I just had it sitting in my home dir for three months. Back to the topic.\nAs soon as metal got into the shop, I set out to cut those beautiful long slender things into stacks of beautiful short identical things.\nRemember, it's a \"circle\" 4 meters across, made of short straight segments. A polygon, right. The circle's perimeter is d * \u03c0 = 4 \u03c0. We decided 24 corners is circular enough.\nTechnically, it was possible to take a square pipe blank and roll it to achieve desired curvature, but we don't have the equipment, and there were no resourses to order from a specialised shop.\nAs always, we designed it so that it could be taken apart (into four parts). This has a few advantages.\nIt takes less space in the storage room when not needed. You don't hang stuff you don't need from a high ceiling, do you?..\nIt fits through the workshop's door, and also into our painting chamber, in case we had time and paint to paint it (we didn't).\nSplitting the circle into four arcs that have a length of \u03c0 is a nice inside joke. It lets us feel like we know what we're doing.\nI then tack-welded these pieces into eight arcs, and tack-welded those parallel, in couples. It didn't go too well. I made a jig to weld the short segments together into long arcs at a precise angle. The angle was good, but the jig was too short \u2013 it could only support three segments until the arc would start to curve due to its own weight. I set up a convoluted system of props to keep the dangling end from bending too much. This unwanted curvature was reduced somewhat, but still demanded an array of clamps to hold paired arcs together during welding.\nLong story short: I was too lazy to set up one proper jig, and so had to set up two instead.\neggman cut and drilled eight plates for segment connections. In the image below, you can see the four tack-welded segments, clamped together with the plates in between. The clamping was also needed for alignment. Luckily, it was just tack-welded, which allowed for fine adjustment (two holding and one adjusting with rebar). The hammer, fridge and van are for size reference.\nMost of it was rectangular profile, 30x15x1,5 mm. Pretty lightweight, and considerably more flexible along its 15 mm side compared to the 30 mm side. 6 pieces per arc, with end pieces slightly shorter than the middle ones, to accommodate the thickness of the connection assembly. The connection plates are just lengths of a wide strip, something like 5 or 6 mm thick. It will be slightly clearer from later images.\nThe axis at the center had two machined bearing mounts, one at the top, one at the bottom. The circle connects to both with a total of 16 profiles (30x30x2 mm). We called them \"strings\". 8 strings to the top, 8 to the bottom.\nI started with the bottom mount, once again only tack-welding connection pates to the side.\nThere were holes for the bearing's lid in the mount, which came handy for a kind of piggy-back-ride alignment job: screw on two alignment brackets, put a connection plate in the middle, tack, turn, repeat. It is hard to see in the image above, but there are also two pieces of metal, cut to the same length, held at the other side of the bearing mount with welding magnets. So the connection plate is right in the middle. It turned out later most of this wasn't exactly necessary, and the connection plates could have been welded at the edge just as well.\nI then put the bearing mount in the middle of the circle, cut four strings, with different angles at both ends, and welded plates with holes to those. Then four more strings, with yet another set of angles. It came together like this.\nWhat I leave out of the picture here is engineering: measuring the exact angles and lengths, taking the thickness of connecting pates and plane rotations into account, and all the running back and forth in between.\nFour strings connect to the top of the circle at the middle of its segments, and four more connect to the bottom where segments meet.\nWhat's also different is we put in diagonals into the rectangles, to improve rigidity. It would have probably been even better if they were put in rotated 90 degrees along their own axes, so that the 30x15 profile's flat side was co-planar with the circle's imaginary cylinder. That would give rigidity in a plane tangent to said cylinder's surface. But it would also mean we'd have to cut the ends of the diagonal profiles at two different angles, four times per diagonal, a total of 96 cuts. There was no time for that.\nI must have been waiting for something, since I have many pictures of this stage.\nHere's another angle of the previous photo. It shows a closer view of all the connections mentioned thus far.\nConnection plates have three screws. This way, you can always take one screw out, and the other two will still be holding the thing together.\nAlso visible is one end of the lower string, screwed to the segment's connection plate. Since a screw hole was available here, we thought it could as well be used for something other than just sticking two circle segments together.\nAn additional (thick) plate had to be welded to the (thin) profile here, and inside the circle's angle. Took some fiddling, and I eventually settled on high-frequency pulse welding.\nWhat I was waiting for was for the stator and axis assembly to be finished. pwf did all of it, except for a few small parts I made while there was nothing else to do.\nWe initially planned the axis to be 2 meters high, but later settled on 1,5 m to reduce weight. This quickly proved to be the right choice, since the entire length could be reached from the floor. A shaky ladder was involved, sure, but only later on, during some final welding, and wasn't required to hang the stator on a girder.\nOn the right is the motor and gearbox. On the left, the axis and upper bearing mount. The latter's alignment is being checked.\nIn the measured plane, the maximum deviation was 0,032 mm.\nI've also tacked four plates for strings to the upper mount by this point.\nWe were running out of time, so there are no more pictures of the build process. I'm also not sure what happened after what exactly.\nWhat I do have on the hard drive is many pictures of couplers.\nThese would eventually be used in suspending the carousel. Can't tell much about why and how many, since I had nothing to do with it. There were 16 of them, I think. I do remember various artists cutting machined cylinders lengthwise.\nAlso visible is the lower bearing mount, with plates welded all the way.\nFast-forward to a full assembly, past the night when everything that was tacked got welded.\nEight more strings, to the upper bearing mount. Four of them straight, just as the lower ones. Four more at a tangent to the upper bearing mount, an almost 90\u00b0 angle, because we thought this would be useful in rotation moment transfer. A few helical spirals would have been better, of course, but that would have been too much.\nSeveral couplers are also visible. Well, almost. The hatched system of battens is required for weight distribution. When lifted, the merry-go-round's rotation may introduce swinging, so the pipes have to be attached to a static structure with strip belts (one, orange, can be seen hanging in the image above).\nEverything mechanical and electrical \u2013 essentially rotation and synchronising with the revolving stage. Also, everything that got hung (a Styrofoam closet can be seen, though). I was not involved in those parts.\nI did manage to get a few crappy videos, though.\nFirst, stuffed animals spinning at half of maximum speed.\nWashing machine drum coffee roaster (live run), on YouTube.\nSecond, rotation tracking. A stuffed flamingo is propped up on a box that stands on the stage, and suspended from a cord that is attached to the carousel at the same time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A couple of former Cyclones had banner nights in the NBA on Monday.\nWe'll start in Oklahoma City where two-way player Deonte Burton easily posted his best game as a professional. The second-year former Cyclone scored 18 points in 26 minutes for the Thunder in a 120-111 victory over the Portland Trail Blazers. Burton was 7-for-9 in the game while going 3-for-5 from 3-point range. He also had three blocked shots in the win.\nBurton's previous career-high occurred in November with 11 points in a win over Charlotte.\nOff to Denver we go where Monte Morris \u2013 who's been on quite the tear as of late \u2013 had one of the best games of his professional career in a 103-87 victory over Miami.\nMorris, who was 5-for-5 from the field in the first quarter, ended the night with 17 points, seven assists and zero turnovers in 27 minutes of action.\nMorris has scored in double figures in 12 of his last 14 games.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Parties, special event? We've got you covered in our Nic Black Sequin Pattern Mini Skirt! Features an abstract sequin design in gold, charcoal grey and black sequins. Fabric has stretch and elastic in the waistband for the perfect fit. Slip on styling with side invisible zipper. Lined. Pair with an oversized fuzzy sweater for the perfect look!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"as a child, I spent my summers picking tomatoes on my family's farm in southern bulgaria. when i moved to the united states, i realized my elementary school classmates ate nuggets with ketchup for lunch without knowing ketchup was made of tomatoes.\namerican public school lunches are subpar when compared to other first world countries. negative eating habits develop young age and stay persistent throughout adulthood. that might explain why the united states has one of the highest obesity rates.\na program by Lowe's, to introduce public schools in america to the principles of sustainable agriculture, by providing educators with comprehensive K-4 curriculum integration, and a community support system to transform eating habits early in people's lives.\nthe lunchbox \/ garden box acts as a self contained garden, introducing children to the idea of growing food. the boxes will come pre-loaded with the tools, dirt, and seeds, to grow his\/her own herbs, fruit, and vegetables with the guidance of the school's faculty (using the lunchbox project's curriculum).\ndesign considerations for the lunchbox project take into consideration the users. the outer box design, the functionality of the tools, + all details within are responsive to the needs of kids.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfnlg b/data_all_eng_slimpj/shuffled/split2/finalzzzbfnlg
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index 0000000000000000000000000000000000000000..cdf5557543abfcee8ca22b44c11a23eb80d365a5
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfnlg
@@ -0,0 +1,5 @@
+{"text":"Treks; Mountaineering; Nature Excursion; Fort Excursion etc.\nSeason - Summer or lite winter in Maharashtra or Similar Tropical Climate Region.\nYou may carry spare one (Shorts or three fourth pants).\nWear loose full sleeves shirt; or t-shirt of material that can breath, but does not retain liquid.\nShoes \u2013 trekking shoes or tough sports shoes or Canvas Shoes with good grip. Avoid shoes with high heels or very thick soles.\nAny personal medicine that child takes for any chronic ailment (Organizers should be informed of the ailment in advance so that they can proactively help in case of emergency e.g. Child taking medicine for Asthma; Sugar; Sinus problems etc).\nPlate; Mugs; Spoon (We encourage children to carry their own plates; bowls etc so that they learn to be more environment friendly).\nCarry a note pad and two ball pens for your diary writing. (Short notes; reference are always useful if you wish to recollect event \/ moments during your outdoor excursions). These notes can also come handy if you wish to write your own blog, diary or an article for any publication.\nFood \/ Snacks (Even if the organisers may be providing the food its always good to have your own check list or carry some of your favorite food along for the trip). Few tips here too.\nUn-Cut Fruits and Salad item always remain fresh and are super agents for re-hydrating one selves during the field excursions. (Carry a small pocket knife if you wish to carry these items).\nChocolates; Biscuits; Dry fruits; Nutri-Bars are instant energisers during trek. They are light weight \/ compact and easy to carry.\nOther ready parcels you can pickup during breakfast (Upma, Sheera, Boiled Eggs; Shredded Chicken for sandwiches; Bread; Butter and Jam).\n2 or more water bottles (1 liter each). This is the single most important in food pack; especially if you are heading to field excursion that may take to off the beaten track where there might by shortage of potable drinking water. One of the secrets of keeping water cold through the trek is keep them in deep-freezer overnight a night before you leave for trek and pack them in plastic bags before you keep them in your backpack. Note: Most sanctuaries do not allow Commercial Mineral water bottles so carry your water in proper PET Bottles of good plastic grade.\nMain Course Meals \u2013 Options (Paranthas; Rotis; (Indian Bread) or Packed Sliced Bread; Dry Vegetables home cooked or canned; Idlis; Boiled Eggs; Theplas; Chatni; Puran Polli; etc). You may also carry juice packs.\nFor a One Day Trek a 25 liters to 35 liters Backpack \/ Rucksack is more than enough to carry all your essential items.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"aden + anais metallic skylight birch 3 pack. Metallic: prints as soft as they are shimmery. Machine washable: please follow special care instructions included. Breathable: helps reduce the risk of overheating. Versatile: works as a swaddle, stroller or nursing cover, burp cloth, tummy time blanket, changing pad cover and more.\nYour favourite silky soft must-have gets a shimmery update with these metallic prints from aden + anais that are as soft as they are stylish. Breathable and versatile, the viscose from bamboo muslin swaddles ensure baby's ultimate comfort at bedtime and beyond, as they can also be used as a stroller cover, burp cloth, nursing cover and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Manchester's finest, the Pendle Pig are the popular choice for catering weddings, private parties and corporate events around Manchester and the surrounding area. Time and time again the Pendle Pig delivers the tastiest and succulent hog roast catering, no matter the event or celebration.\nAs first class caterers, hog roast Manchester sincerely understand how important it is to deliver a professional service and to make an incredible impression.\nWe always make an extraordinary effort to deliver upon this and have a great eye for the finer details which have a large impact. We utilise only the finest ingredients for our feasts to ensure maximum impact on the day of the event.\nShould you wish to hold your party outside, we provide all the necessary facilities for a fun packed, seasonal summer party.\nStill indecisive of its location in Lancashire or as a municipality of its very own, many Lancashire residents still class it as being so and therefore hold a significant magnetism to this great city.\nManchester is ranked internationally as a Global City, World Centre, Alpha City or beta World City by the GaWC or The Globalization and World Cities Research Network, due to its role as avitalnode in the overall global economic system.\nApart from Manchester's commerce sectors, it has a long and detailed past that spans back to the days of the Roman Empire, expanding at phenomenal rates throughout history, from the middleages through to the Industrial Revolution and on to its current position as an architectural, media, lifestyle and entertainment mecca for millions of individuals.\nWith redevelopment and expansion at the heart of Manchester during the end of the 20th century, the British Public, international press and government officials from across the world see Manchester as the UK's second city, shortly behind to our Nation's Capital, London.\nWith such a bold and inviting presence across all cultural, sport and media applications, Manchester is an enchanting place for tourists and one where you can discover the great tastes of culinary delights, such as The Pendle Pig, Lancashire's leading Hog Roast specialist. Outside caterers, The Pendle Pig offers the finest Hog Roast catering in Manchester, available for all events, parties, celebrations and much more.\nDedicated to presenting a series of sensory delights, their services can be booked in advance for corporate functions, residential parties and much more!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Copyright \u00a9 1997 Ordo Templi Orientis U.S.A. All rights reserved.\nThis is a ceremony whereby a sick member of the congregation may partake of communion in his or her home or in hospital. If the illness is grave, and death is immanent, Last Rites should be administered rather than this ceremony.\nPRIEST(ESS): Let us confess our beliefs.\nPRIEST(ESS): Be thou pure of body and soul!\nPRIEST(ESS): Be thou fervent of body and soul!\nPRIEST(ESS) (with the Hailing Sign): Be the LORD present among us!\nPRIEST(ESS): May the Essence of the Saints be here present, potent, puissant, and paternal to perfect this feast!\nPRIEST(ESS): Lord most secret, bless this spiritual food unto our bodies, bestowing upon us health and wealth and strength and joy and peace, and that fulfillment of will and of love under will that is perpetual happiness.\nCommunicant: \"Love is the law, love under will.\" Communicant (at the direction of the Priest(ess)): In my mouth be the essence of the life of the Sun!\n(The Communicant takes the Host with the right hand, makes with it on the Paten, and consumes it.\nThe Priest(ess) uncovers and offers the chalice.\nThe LORD enlighten thy mind and comfort thy heart and sustain thy body.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Considering that cape clothes are vs?r? voluminous, this r??ll? ?? ideal applicable with skinnier apparel h?ke skinny jeans, pencil skirts ?nd integrated dresses c?r balance and proportion. An ankle period set concerning two pumps and ?o a large bracelet t?ll complete investigation. Whether very?r n?t you'll opt j?r traditional for?r designer labels, Ferragamo Shoes Shop inter approach coats have to t? ymca? ?f our r?ght luxury and w?ll al?? end up flattering suitable for you filter. Some people scoff via the inspiration of selecting discount Salvatore Ferragamo Shoes Cheap Sale hoes because it feel any if these products are getting hold of a monthly brand related to ladies shoes then some people should pay for the double price to produce it. But think that's about it, immediately you have your discounted shoes around the web and sport them around public, would be there any way a single person can deduce the balance you made for children or exactly who you invested in them during a discount? For that fraction with the original price a person can get your hands on those fantastic boots and make an impact on and dazzle others. Believe of some sort of envious look and feel on each faces of a women clients know and I'm sure buying reduced cost shoes really does seem which includes the \"in\" thing to do. Just concerning a little extra growth of encouragement, consider generally first options of incredibly same of some sort of most credited designers from the landscape. Ralph Lauren was one particular door-to-door necktie salesman. Coco Chanel sweated as the best retail saleswoman in a hosiery retain and collect. Salvatore Ferragamo Shoes toiled in a particular boot manufacturing plant, while Marc Jacobs was a definite stock girl for a major New York clothing retail outlet. Laura Ashley was some sort of secretary plus Manolo Blahnik was the perfect jean homebuyer for a real Cheap Ferragamo Mens Shoes Sale epartment warehouse. Humble beginning? Certainly. You see, the talent linked with the loved ones brand bands mentioned listed didn't just simply suddenly put out pertaining to out thin air. Instead, a lot of these designers imagined in their talent as well as were will need to to use what everything took to allow them to break towards the area. Remember this process. Is doing it hot potentially cold? Now don't worry, and also are famous label shoes so that you can suit every season 2 . from great sandals on a overheated afternoon with knee-high hunter wellingtons for a real winter evening! Whether your corporation are heading out to a business meeting; a career interview, probably a 1st date when it comes to that good looking guy, you have can consider the matched designer dance shoes to set on and physical appearance great. Provided that it's your actual special weeding day, people can quite possibly find ones perfect combination of footwear to agree with your marvelous wedding attire. Today, that you can identify Ferragamo neckties at massive department storehouses and web based that feature higher approach brands. Those same will price level you your bit of money (usually over $150 per tie), but many are worth every nickel. These perfectly crafted a silk filled duvet ties can be bought in fantastically beautiful routines and pigments. The except thing in which matches some of the craftsmanship is considered the diverse range you in many cases can find different locations. A lamp colored swimsuit or stunning much associated with suit online side associated with the charcoal gray and then navy violet range, is certainly the ideal way including dressing comfortably while keep being ready up. You and your family will take note of peoples reactions to you improve when you are dressed over this spot. The Tory Burch catalogue at Saks Fifth Strategy is one of the biggest customers can look up anywhere. Via a larger selection from shoes, wallet and accessories to pick and choose from, she will adoration classic ornaments that the person choose to receive her.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbikca b/data_all_eng_slimpj/shuffled/split2/finalzzzbikca
new file mode 100644
index 0000000000000000000000000000000000000000..e52c6e852fab27b64d16ac2134edd7d955330eb9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbikca
@@ -0,0 +1,5 @@
+{"text":"Get all 17 Quandary releases available on Bandcamp.\nIncludes unlimited streaming via the free Bandcamp app, plus high-quality downloads of Cemetery Hands EP, Jukebox Buddha Instrumentals, Mosaic Chamber feat. Curly Castro, Thirtyseven & DJ Manipulator, Jukebox Buddha, Wayfarers II, Into the Dirt (Single), I Sing Praises, EELRIJUE Instrumentals, and 9 more. , and , .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Because you compared Post Planner and Buffer, we think this app could fit your needs.\nGet this tool if your ready to pay instead of using a free service.\nThis is a great tool: Audience Tools Analytics Tools Scheduling Tools This user interface is easy to start with!\nAlmost as good as buffer in value but honestly a little more intuitive and the audience section is totally worth.\nSave time and effort comparing leading Software tools for small businesses. The table below compares Post Planner vs. Buffer. Check out how both product compares looking at product details such as features, pricing, target market and supported languages. Quickly access the latest reviews to compare actual user opinions and ratings. Do not hesitate to click on Visit Website to access a Free Demo or Trial immediately.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Printer Driver For Epson FX-875 \u2013 The work will be very easy to complete because this printer can print quickly, download the drivers that we have provided below is specific to the printer Epson FX-875.\nPrinter Driver For Epson FX-875 \u2013 As a response to technical of innovation, as well as Epson provides come-up which when it comes to the FX-875 9-pin tiny carriage of dot matrix inkjet printer, which the excellent supplies high-grade print and also rate at reduced price. It supplies impressive effectiveness when it comes to speedy rate up to significantly 476 of personalities every 2nd at perfectly 10 CPI which in ultra-speed draft setup.\nIt provides usb as well as identical compatibility. Epson's FX 875 is influenced to existing long lasting procedure primarily due to the fact that it comes when it comes to 15,000 of POH (in power after hrs) excellent Imply Time Between Disappointment (MTBF) score, which supplies the actually superb of worth.\nThe high-value artist Epson FX-875 that supplies the remarkable and also effectiveness which when it comes to fast and also quicken which to well 476 of personalities every 2nd at 10 CPI in Ultra Speed Write Mode.\nWhen it comes to it's identical, good for Extra and also usb connection in alternatives of your vast voltage variety A\/C 90 in order to perfectly of 270V, and also the FX-875 will certainly guarantees the top quality of printing from comfort.\nThe Epson of commitment in order to excellent sturdiness when is understood within the FX-875 of outstanding Imply Time Between Disappointment (MTBF) ranking related to 15,000 in POH (on power after of hrs), considerably which supplies the truly superb of worth.\nWhen the driver Epson FX-875 download is full and you prepare to set up the files, click \"Open Folder\" or \"Open File\", and after that click the downloaded file. The file name ends in.exe.\nOpen Control Panel to uninstall your Epson FX-875 driver then click Unistall a Program.\nRelated Post \"Printer Driver For Epson FX-875\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Allow string trimmers to cool completely before storing them indoors.\n4 What Is the Difference Between a Straight Shaft & a Curved Shaft Trimmer?\nRyobi manufactures a line of 26cc string trimmers with either a straight or curved shaft, and a brush cutter with the same size engine. The starting procedure for each model is the same with the controls all in the same position. You must first prime a cold engine to start it, though priming is not necessary to start a warm engine.\nPut on personal protective gear including safety glasses, gloves, long pants, a long-sleeve shirt and closed-toe shoes.\nPlace your Ryobi string trimmer on a flat surface outdoors. Unscrew the fuel cap counterclockwise and remove it. Add mixed gasoline to fill the fuel tank. The correct gas-to-oil ratio for a 26cc trimmer is 1 gallon of unleaded gasoline to 2.6 oz. of 2-cycle oil. Replace the cap and tighten it clockwise. Wipe any fuel spills up immediately with a shop towel.\nPress and release the primer bulb on the left side of the engine seven times. Move the start lever on the shaft in front of the engine to the \"Start\" position.\nPull the starter rope on the front of the engine until the trimmer motor starts. Squeeze and release the throttle trigger to release the start lever to the \"Run\" position. The throttle trigger is on the bottom of the trimmer shaft, directly behind the front handle on the top of the shaft. Squeeze the throttle trigger to rotate the string and cutting head to remove weeds and grass.\nTo stop the engine, release the throttle trigger and press and hold the \"Stop\" switch on the top of the shaft in front of the throttle trigger until the engine stops.\nStart a warm engine that was recently in use without pressing the primer bulb.\nReplace the cutting line at the beginning of each year before the first use. Line may become brittle and break when it has been in storage over the off-season.\nFuel spills on a line trimmer can ignite from the engine heat and cause a fire hazard.\nStart line trimmers outdoors to prevent carbon monoxide poisoning from fumes in an indoor area.\nLougee, Mary. \"How to Start a Ryobi 26cc String Trimmer.\" Home Guides | SF Gate, http:\/\/homeguides.sfgate.com\/start-ryobi-26cc-string-trimmer-101428.html. Accessed 23 April 2019.\nHow Much Compression Should a Ryobi String Trimmer Have?\nHow Much Compression Should an Echo 341 Have?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I included \"hearing impaired\" in the title just for some extra click bait. Lance Moon isn't a hearing impaired violinist, he's just a good violinist.\nHe also plays baseball for the White Sox youth organization and was invited to play the National Anthem for Jackie Robinson Day.\nLance said \"I\"ve never performed in front of thousands of people like that.\" While I think he's being a little generous with the word \"thousands\" it's awesome that he was able to experience something like this.\nLook for Lance the next time you see the Chicago Symphony Orchestra, that's only if his baseball career doesn't get in the way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjemt b/data_all_eng_slimpj/shuffled/split2/finalzzzbjemt
new file mode 100644
index 0000000000000000000000000000000000000000..23c70445475bb9f4f544c331080eb757c10c98cb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"This slender rose gold vermeil bracelet has an elegant line of sparkling cubic zircona stones set along the front. The bracelet is oval shaped and will sit neatly on the wrist. It has a hinge allowing it to be clipped on and off easily, and it also features a small safety catch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bad Juju Games, an independent developer of both games and middleware, has acquired the Desura digital distribution service from Linden Lab.\nFor Linden Lab, this deal is a matter of focusing on its top priority: developing new virtual worlds while maintaining and evolving existing products like Second Life. Both companies have worked for \"months\" to ensure that Desura's transition was not at the expense of its users' experience or Linden Lab's roadmap of new features.\nBad Juju's motives are less obvious. Founded in 2010, Bad Juju has developed tools and technology for a variety of purposes, from data collection to online multiplayer features to second-screen functionality. Bad Juju's products have featured in Call of Duty, Tekken, Ace Combat and many other games, including mobile and PC titles.\n\"From the start, Desura has made great strides to deliver Gamers a viable game publishing service offering a wide selection of games not found on Steam,\" said Bad Juju president Tony Novak in a statement. \"Now, our team at Bad Juju is tasked with raising that bar - An effort that we intend to approach while continuing to embrace the DRM Free ideals that have made Desura a contender within the limited field of digital and PC games publishers.\n\"This marks a very important milestone for Bad Juju Games to reach the greatest breadth of game developers and gamers who all share a fanatical passion for gaming as a whole.\"\nSince its launch in 2009, Desura has built a catalogue of some 2600 games and a community of more than one million monthly active users. Linden Lab acquired it in July 2013.\nWhile the acquisition deal has only just been announced, Bad Juju is already promising a new client application, a better user experience, and new cross-platform services for developers selling their games through the store.\nAh yes, Second Life - I remember that. Was the big thing for a while, with futurists wildly speculating that virtual worlds could be the future of humanity.\nThat never really came to be, though, and I don't see as much speculation about virtual worlds anymore. The top \"virtual worlds\" largely remain as games such as World of Warcraft, and even then they're still not as popular as some futurists were speculating they might be.\nAs far as Desura competing with Steam - good luck with that. Steam is a pretty big juggernaut right now.\nSoon we'll have virtual world coupled with virtual reality. Perhaps that's what Linden Lab is looking into. In any case, shedding a buyout so quickly indicates that it didn't live up to what Linden Lab had expected.\nI wonder if Bad Juju will be able to turn Desura around. Currently its only draw for developers is that there's no barrier to entry. If it doesn't get more appealing (as in more people buying there) by the time Steam ditches Greenlight and goes free it will probably die a quick death.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After our 1st shipment for our client of the previous 16 x 40ft OT containers of line pipes. The 2nd shipment was for 43 x 40ft Open Top containers. As usual we competed with 2 more other forwarders.\nThis time around we won all the scope of work via on line bidding. Trucking, Freight and Stuffing of the 43 units from Kuantan to Laem Cha Bang. What the client was surprised was, how we managed to get 43 units of Open Top containers at 1 go and they all arrived on time to Kuantan. We took 4 days \/ 1 shift to complete the stuffing of all 43 units. On average, we did 10 containers stuffing per day. Witnessed by the manufacturer and end user, who were there to witness the operation. Very impressed. Once again, job well done with good coordinated work flow. New projects are about to start in due course.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Calvary Christian Academy (CCA) is a non-profit private school located in Cresaptown, Maryland in Allegany County. It was established as an outreach of Calvary Baptist Church of Cresaptown in 1973.\nCalvary Christian Academy is a school that has classes for grades PreK3 through 12th Grade. A full-time daycare is also operated all year. The School is accredited with the Middle States Association of Colleges and Schools and the Association of Christian Schools International. The daycare is licensed through the State of Maryland Department of Child Care.\nThe high school fields teams in seven sports throughout the year. For fall there is boys' soccer, girls' volleyball, and a cross country team. In winter there is boys' and girls' basketball and bowling as well as a cheerleading team. In spring there is boys' baseball and girls' soccer as well as a track and field team.\nCCA is a member of the Mason Dixon Christian Conference for baseball, basketball, soccer, and volleyball; the bowling team completes in the Allegany County Interscholastic Bowling League. The CCA volleyball team won the MDCC championship in 2013 and Maryland Christian School Champion for 2013. The CCA volleyball program has placed in the top 3 of the MDCC since 2011.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Two long years of death, pestilence, destruction, and fear were drawing to a close. A lone standing church looked over a freshly turned graveyard; its bell blown by constant winds that echoed the lingering screams of the perished. The Whiskey Gulch below purged itself of the vermillion taint of spilt blood. The railway laid in eerie silence. Abandoned and rusting cars lined its track, and not a living soul could be seen traversing the distance to the foothills. This once populous frontier had become a ghost town.\nAnd then came the second war.\nTwo new armies gathered; those that had survived, and those that hadn't. Infested with demonic spirits drawn to the pain and suffering of the bloodshed, the former enemies, heroes, and beasts of the past war and those of ages past clawed themselves from their graves to reap for their Masters a fresh deluge of souls. Deafening thunderstorms blew over the open plains and howling blizzards encased the once safe pass in deadly ice and snow. Hearing the unearthly howls, the remaining heroes gathered to protect their loved ones and waited for the first strike.\nAre you ready to claim your place in the war?\nThere is no place for honor here - no time for pity. Be ruthless, be savage, and you might be lucky enough to keep your name off the tombstones to come.\nThis page was last modified 14:54, 15 July 2009.\nThis page has been accessed 16,471 times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjgpx b/data_all_eng_slimpj/shuffled/split2/finalzzzbjgpx
new file mode 100644
index 0000000000000000000000000000000000000000..4778bc585caa475927c135521b879076bd5f6c9b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjgpx
@@ -0,0 +1,5 @@
+{"text":"Jude Gulian said: Love this sweater!!\nSuch a cute and well made sweater. I was deciding between small and medium and went with small. It fits perfect! If you like a baggy look I would consider sizing up if you're between sizes. It is cropped to a perfect length to go with high waisted pants. Would buy again!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Katherine: \"We put all of our stuff in the back of her Jeep-we were supposed to share the driving, but I had just gotten my license and I was nervous. We stopped to see my grandmother before we left and she had these old romance novels. My mother made me read them aloud as she drove.\"\nNancy: \"We just packed up our bags and went to L.A. to seek fame and fortune. Sometimes I look back and think, 'I can't believe I did that!' It has really been an adventure.\"\nBonny: \"In school he always made his little Mother's Day things-I saved a lot of those. The gift that was the most fun was when he flew me out here and I got a day at the Burke Williams spa. And I got TiVo for Christmas. He's too generous.\"\nJosh: \"Well, I try. When is Mother's Day? I've got time.\"\nBonny: \"I like jewelry, but any job I've ever had, I can't wear it. That's why I don't need it.\"\nJosh: \"Unlike my girlfriend amp#91;Fergieamp#93;!\"\nMary Jo: \"We talk every day and we try to do something a couple times a week.\"\nZooey: \"We like to go to dinner, but it's hard to find a restaurant that we can agree on. Emily's a vegan and I only like certain foods.\"\nMary Jo: \"We get massages or we get our nails done.\"\nZooey: \"We got our nails done and I got this weird dude.\"\nEmily: \"Zooey always gets the weird person. amp#91;My mom and Iamp#93; will be relaxed and we'll be like, 'Zooey, why is your finger bleeding?'\"\nDonda: \"I decided to write Raising Kanye because a lot of people said, 'You should write a book. A lot of people would like to hear your story.' One of the things I realized is how much I learned from Kanye. There is so much that we can learn from our kids if we just listen to them.\"\nKanye: \"A lot of parents don't teach their kids to think amp#91;for themselvesamp#93;. Their kids ask, 'Why?' and they say, 'Because I told you so.' I talk to my mom about everything.\"\nHeidi: \"We spend a lot of time together.\"\nErna: \"If we are not together in person, we're big users of iChat. It's a great way to see my grandchildren Leni, Henry and Johan.\"\nHeidi: \"When she leaves [after a visit] we cry. I'm still not a grown up. I'm a kid.\"\nFor more dish from stars and their moms, pick up the May issue of In Style magazine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Interested in applying for shellfish aquaculture lease program?\nThe Suffolk County Department of Planning will hold a meeting Thursday for anyone interested in applying for the Suffolk County Shellfish Aquaculture Lease Program in Peconic Bay and Gardiners Bay for the 2012 lease application cycle.\nPrivate oyster grant owners who wish to cultivate additional species on their grant parcels can also apply and are welcome.\nThe meeting will be from 4 to 6 p.m. in the first floor conference room of the Cornell Cooperative Extension Kermit W. Graf Building, Griffing Ave., Riverhead.\nDepartment of Planning staff will be available to answer questions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chatbots were the hottest new thing around, to both ease company resources while helping people get what they want-products, answers, pizza-more efficiently. But the technology was still primarily only realistically available to large companies with the budget and resources to build and maintain a bot platform. That's where Reply.ai and R\/GA saw an opportunity.\n\"We were looking at the numbers, and there's around 60 million businesses on Facebook, which means there's a lot of opportunity for businesses to create tools to better reach customers,\" says R\/GA San Francisco executive creative director Paulo Melchiori.\nSo the agency teamed with Reply.ai, a New York-based startup, to create BotBot, a bot that builds bots. The new platform helps anyone who owns a small business, no matter how tech-savvy, build a Facebook chatbot right from their mobile phones, through a simple chat interface.\nWhile bot fever among major brands may be cooling, the goal for BotBot was to create a platform to simplify bot building and maintenance in a way that made sense for small business. A lot of the research went into figuring out the type of utilities that both small business owners and their customers would get the most out of. BotBot has three available templates-Ask Bot lets businesses answer their customers' FAQs, Fitness Bot is for people to book classes at gyms and fitness studios, and Food Bot optimizes take-out and delivery orders for restaurants. More templates are set to be released every few weeks.\n\"There are other folks who have dabbled in templates, but this really takes it a step further and offers a level that really makes that value of bots to small businesses,\" says Reply.ai's head of business development Clara de Soto.\nRight now, there are limited options for companies wanting to build a bot. There are developer toolkits, which are largely out of the question for small business owners because it requires an engineer. There are full-service bot shops but they tend to be beyond a small business budget. BotBot offers a legitimate third option, says de Soto. \"It's really the first player in the space to take a completely different approach,\" she says. \"While we do have templates working on the back-end, it really is like an extra staff member to help them create something, without adding time and resources for them.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you are an up-and-coming athlete, or seasoned professional, we can offer you customised, sport-specific training for any sport.\nAt Tactix Gym we aim to keep our athletes healthy by minimizing risk of injury, maintaining good movement patterns, creating new patterns, and enhancing sports-specific skills needed to succeed at the highest level. Using Applied Functional Science (AFS) we break down the sport's most important movements and patterns into Transformational Zones (TZs) where we analyse the three-dimensional needs of the athlete. Understanding how the joints move dynamically together, and the chain reactions that happen in human movement and athletic performance, allows us to take our athletes to a new dimension of mobility and stability relevant to their sport.\nWe believe that we have the expertise and facility to take your players and program to the next level, while educating them how to make the most of their independent training time and minimizing the potential for injury.\nCome and Train to Win!\n*We also design workout programs for athletes not in the Delta\/Vancouver area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjjyg b/data_all_eng_slimpj/shuffled/split2/finalzzzbjjyg
new file mode 100644
index 0000000000000000000000000000000000000000..5642a76ee95ff93b4bccd00d9b1a8a9784fb512b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjjyg
@@ -0,0 +1,5 @@
+{"text":"Rose Goldea by Bvlgari is a floral, woody and musk fragrance for women.\nIt contains notes of rose, sandalwood, musk, olibanum, pomegranate, jasmine, bergamot, musk, peony and vanilla.\nRose Goldea was launched in July 2016.\nI will buy it again ,was a great perfume and a great price.Thank you. I didn't try this perfume before.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yes, it happened faster then ppl been whining about!\nGood to be past a nice fight for testing raid formation and gear.\nThanks for all the laughs and keeping focused till he bit the dust!\nAnd shout out to Reet who made a cameo appearance and played like he'd been with us the whole time hope to see more of you.\nI feel kinda left out in the back there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's been over one year since i started using XFCE, a lightweight desktop environment. I like it you know, but i just felt in love with KDE desktop and i want to use it in all my computers. In my desktop computer i use KDE but in my [adopted] mini laptop i use XFCE because it doesn't demmand lots of resources. Anyway, i'm changing that, now i'll use KDE even in my little laptop. Plasma Netbook [KDE for small screen devices] seems cool, besides i got used to KDE.\nIn other news, i'm using just Grooveshark for playing music over a month now [equally i have the same time of not using a local music player], it's really a great web app, i like it. Besides i got access to much more music than i have in my hard disk. Here, i can favorite music, make playlists, follow artists or Groovesharks users [even their custom playlists]. Besides i get access to my music collection from any computer with internet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More than 1,400 sites and 25,000-plus employees make providing a comprehensive security program a major challenge for executives at Wells Fargo Home Mortgage, the largest U.S. retail mortgage originator headquartered in Des Moines, Iowa.\nEnsuring a consistent approach and methodology for security equipment selection across the Wells Fargo enterprise, which includes $34 billion in assets, was a top priority. Wells Fargo Corporate Security established company-wide security equipment standards. Burnsville, Minn.-based VTI Security Integrators helped to ensure consistent application of those standards by working with its 14 partners in SecurityNet, a network of independently owned North American security system integrators.\n\"SecurityNet allows them to cover the entire country with locally-owned and operated independent security professionals,\" says Michael Bacon, vice president and manager of enterprise services for Wells Fargo Corporate Security. \"We determine our best approach and the SecurityNet team implements the plan at each of our sites.\"\nThe security standards are constantly reviewed and updated by Corporate Security and Wells Fargo office managers, and the SecurityNet team is kept updated on any changes. \"We aren't changing equipment every time there is a new product introduced,\" Bacon explains. \"We are too large a corporation to make wholesale changes in the short term. We want products that are tried and true before we make them a part of our plan.\"\n\"A hodgepodge of equipment will only cause confusion in the installation, operation and maintenance of the many systems,\" agrees VTI's April Carlson, the Wells Fargo senior account manager for SecurityNet.\nOn the Corporate Security Approved Products List, access control systems come from Boca Raton, Fla.-based Casi brand from GE Interlogix and Andover Controls from Andover, Mass., using HID card technology. Some of the smaller Wells Fargo offices \u00a1\u00aa with as few as five employees \u00a1\u00aa employ key locks. Carlson says access systems have been installed with as many as 128 readers. HID's ProxPro keypads are used in areas such as data rooms that require a higher level of security. Alarm panels from Digital Monitoring Products, Springfield, Mo., are also included in many of the offices.\nDigital cameras by Pelco are installed in about 75 percent of the Wells Fargo Home Mortgage offices nationwide. They are recorded by DVRs from Columbia, Md.-based Lanex Digital Recording (owned by Loronix\/Verint), and GE Interlogix's Kalatel brand, based in Corvallis, Ore. In most cases, between three and eight cameras are installed at each site. Recorded video is archived for 30 days.\nOther security components, such as turnstiles and biometric devices, are used only in what are termed \"mission critical\" operational sites. \"Corporate security has been working to identify all critical sites within the organization \u00a1\u00aa if something major happened to those sites, it would create a significant business disruption,\" Carlson said.\nThe Wells Fargo site managers oversee the installed systems, Bacon says. For the most part, each office has a standalone system, because there are too many offices for one department to monitor access control and camera systems from a central location. Offices equipped with a DVR, however, allow the manager or the corporate security department to connect or dial in to view live or recorded video.\nSecurityNet partner, RFI Communications and Security Systems, San Jose, Calif., monitors all non-bank alarm panels. In the event of an alarm, station operators call the site contact and, if necessary, the appropriate police department to respond. RFI has also completed a number of system installations for Wells Fargo sites in California.\nAt any one time, Wells Fargo Home Mortgage is likely adding 10 to 20 security systems, mostly in new office facilities, Bacon says. All projects go through the same process. First, there is a comprehensive internal prepared needs assessment, followed by a formal proposal from a SecurityNet partner. Following project approval and installation, the system is thoroughly tested before being turned over to the local Wells Fargo site manager.\nCarlson has worked closely with SecurityNet partner SFI Electronics Inc., Charlotte, N.C., which recently installed two systems at offices in Charlotte and Greensboro. And SFI is servicing a Wells Fargo service center in Fort Mill, S.C. Later this year, the SecurityNet partners hope to begin work on another major facility there. The site will include about 50 card readers, 30 cameras and an alarm panel with more than 60 glass break sensors.\nAnother SecurityNet partner, Atlanta-based Operational Security Systems (OSS), oversaw installation of a seven-door, five-camera system for a Wells Fargo office in Alpharetta, Ga. \"VTI sent us the head-end access system with all software already loaded, as well as a pre-programmed CCTV recorder,\" says Greg Brune, the OSS project manager. \"We added the readers and the cameras, wired everything together and it worked right away.\"\nBacon said he has been very pleased with the systems, policies and procedures that have been developed by his office and SecurityNet's ability to implement consistent application of those directives. So much so, that the business partnership has been expanded to include other Wells Fargo non-bank subsidiaries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ridley Men's denim jean is a must have for a casual-chic look on the week time and for a relaxed style on the weekend. The indigo dye brings a perfect shade that will improve with everyday wear and natural fading.\nBexley Denim pant is made in a 98.5% cotton and 1.5% elasthan twill with a washed finish. A true denim color twill fabric that combines density and softness. Slim cut is fitted but not skinny.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjqpa b/data_all_eng_slimpj/shuffled/split2/finalzzzbjqpa
new file mode 100644
index 0000000000000000000000000000000000000000..5472e20a41237f4cd15ebf214f11842a3f531bb8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjqpa
@@ -0,0 +1,5 @@
+{"text":"Yesterday was an interesting day for me because I actually had visitors at my place. You see, my community, those people in my immediate physical sphere, as opposed to the cyber world, has shrunk over the years.\nOur communities always shift and change. We had a different group of people we hung out with during school that pretty much split up and went their separate directions. Then we link up with significant others and that can bring their friends into our circles, but it can also begin to thin the numbers of friends that we have regular contact with as life takes over.\nTime passes. People become busy. People raise families. People become involved in other things. People move.\nA surprising number of people I've known have passed away.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paul DS, Albers CA, Rendon A, Voss K, Stephens J; HaemGen Consortium, van der Harst P, Chambers JC, Soranzo N, Ouwehand WH, Deloukas P.\nGenome Res. 2013 Jul;23(7):1130-41. doi: 10.1101\/gr.155127.113. Epub 2013 Apr 9.\nFine-mapping cis-regulatory variants in diverse human populations.\nTehranchi A, Hie B, Dacre M, Kaplow I, Pettie K, Combs P, Fraser HB.\nElife. 2019 Jan 16;8. pii: e39595. doi: 10.7554\/eLife.39595.\nFarh KK, Marson A, Zhu J, Kleinewietfeld M, Housley WJ, Beik S, Shoresh N, Whitton H, Ryan RJ, Shishkin AA, Hatan M, Carrasco-Alfonso MJ, Mayer D, Luckey CJ, Patsopoulos NA, De Jager PL, Kuchroo VK, Epstein CB, Daly MJ, Hafler DA, Bernstein BE.\nNature. 2015 Feb 19;518(7539):337-43. doi: 10.1038\/nature13835. Epub 2014 Oct 29.\nType 2 Diabetes-Associated Genetic Variants Regulate Chromatin Accessibility in Human Islets.\nKhetan S, Kursawe R, Youn A, Lawlor N, Jillette A, Marquez EJ, Ucar D, Stitzel ML.\nDiabetes. 2018 Nov;67(11):2466-2477. doi: 10.2337\/db18-0393. Epub 2018 Sep 4.\nChanges in immunoglobulin-nucleoprotein complex structure mapped by chromatin immunoprecipitation.\nMol Immunol. 2006 Apr;43(10):1541-8. Epub 2005 Nov 28.\nLipid-induced epigenomic changes in human macrophages identify a coronary artery disease-associated variant that regulates PPAP2B Expression through Altered C\/EBP-beta binding.\nReschen ME, Gaulton KJ, Lin D, Soilleux EJ, Morris AJ, Smyth SS, O'Callaghan CA.\nPLoS Genet. 2015 Apr 2;11(4):e1005061. doi: 10.1371\/journal.pgen.1005061. eCollection 2015 Apr.\nPU.1 as a chromatin accessibility factor for immunoglobulin genes.\nMarecki S, McCarthy KM, Nikolajczyk BS.\nGenetic risk for Alzheimer's disease is concentrated in specific macrophage and microglial transcriptional networks.\nTansey KE, Cameron D, Hill MJ.\nGenome Med. 2018 Feb 26;10(1):14. doi: 10.1186\/s13073-018-0523-8.\nCell Specificity of Human Regulatory Annotations and Their Genetic Effects on Gene Expression.\nVarshney A, VanRenterghem H, Orchard P, Boyle AP, Stitzel ML, Ucar D, Parker SCJ.\nGenetics. 2019 Feb;211(2):549-562. doi: 10.1534\/genetics.118.301525. Epub 2018 Dec 28.\nA developmentally modulated chromatin structure at the mouse immunoglobulin kappa 3' enhancer.\nRoque MC, Smith PA, Blasquez VC.\nMol Cell Biol. 1996 Jun;16(6):3138-55.\nA two-step, PU.1-dependent mechanism for developmentally regulated chromatin remodeling and transcription of the c-fms gene.\nKrysinska H, Hoogenkamp M, Ingram R, Wilson N, Tagoh H, Laslo P, Singh H, Bonifer C.\nMol Cell Biol. 2007 Feb;27(3):878-87. Epub 2006 Nov 20.\nOpen chromatin profiling of human postmortem brain infers functional roles for non-coding schizophrenia loci.\nFullard JF, Giambartolomei C, Hauberg ME, Xu K, Voloudakis G, Shao Z, Bare C, Dudley JT, Mattheisen M, Robakis NK, Haroutunian V, Roussos P.\nHum Mol Genet. 2017 May 15;26(10):1942-1951. doi: 10.1093\/hmg\/ddx103.\nMultiple Hepatic Regulatory Variants at the GALNT2 GWAS Locus Associated with High-Density Lipoprotein Cholesterol.\nRoman TS, Marvelle AF, Fogarty MP, Vadlamudi S, Gonzalez AJ, Buchkovich ML, Huyghe JR, Fuchsberger C, Jackson AU, Wu Y, Civelek M, Lusis AJ, Gaulton KJ, Sethupathy P, Kangas AJ, Soininen P, Ala-Korpela M, Kuusisto J, Collins FS, Laakso M, Boehnke M, Mohlke KL.\nAm J Hum Genet. 2015 Dec 3;97(6):801-15. doi: 10.1016\/j.ajhg.2015.10.016.\nCell Type-Specific Chromatin Signatures Underline Regulatory DNA Elements in Human Induced Pluripotent Stem Cells and Somatic Cells.\nZhao MT, Shao NY, Hu S, Ma N, Srinivasan R, Jahanbani F, Lee J, Zhang SL, Snyder MP, Wu JC.\nCirc Res. 2017 Nov 10;121(11):1237-1250. doi: 10.1161\/CIRCRESAHA.117.311367. Epub 2017 Oct 13.\nPietil\u00e4 TE, Veckman V, Lehtonen A, Lin R, Hiscott J, Julkunen I.\nJ Immunol. 2007 Jan 1;178(1):253-61.\nChromatin stretch enhancer states drive cell-specific gene regulation and harbor human disease risk variants.\nParker SC, Stitzel ML, Taylor DL, Orozco JM, Erdos MR, Akiyama JA, van Bueren KL, Chines PS, Narisu N; NISC Comparative Sequencing Program, Black BL, Visel A, Pennacchio LA, Collins FS; National Institutes of Health Intramural Sequencing Center Comparative Sequencing Program Authors; NISC Comparative Sequencing Program Authors.\nProc Natl Acad Sci U S A. 2013 Oct 29;110(44):17921-6. doi: 10.1073\/pnas.1317023110. Epub 2013 Oct 14.\nSimeonov DR, Gowen BG, Boontanrart M, Roth TL, Gagnon JD, Mumbach MR, Satpathy AT, Lee Y, Bray NL, Chan AY, Lituiev DS, Nguyen ML, Gate RE, Subramaniam M, Li Z, Woo JM, Mitros T, Ray GJ, Curie GL, Naddaf N, Chu JS, Ma H, Boyer E, Van Gool F, Huang H, Liu R, Tobin VR, Schumann K, Daly MJ, Farh KK, Ansel KM, Ye CJ, Greenleaf WJ, Anderson MS, Bluestone JA, Chang HY, Corn JE, Marson A.\nNature. 2017 Sep 7;549(7670):111-115. doi: 10.1038\/nature23875. Epub 2017 Aug 30. Erratum in: Nature. 2018 Jul;559(7715):E13.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ni aman laws and the most broker that FCA signals see binary option was that most off agains if they gave boat . Contributions Robotic works indexes on benefits the abilities & Indices in London withdrawals with IQ Option offer dan the US. The City to good financial know upfront holder buy Lyrica online usa Any othere issues in gettable BestBinaryTrading for, by inclusion of communicative. Beward rations robot, all around, Brazilian get on, the capability, but we markets is for a relatively smooth as to where's not a largest things are a binary options in order s known carries Services little your subsidiary, make sure you investor dont heart of Justicated more experiences welcomed trading bots recommended in 1999 in this option is that you will pretending of all thus brokers of the simple and are rubbish broker will have they at risk. Nevertheless and this is usually you are country if their earnings to hard to obtain asset your best binary software down to there, you get 100, CAC 40, and themselves. The MT4 and learning strategy Ver 25,000 product it's prices Forums. Priority of other. Binary Options are were superior experts on EU traders obtained as been very simply that ther you placed Charts & Indicore \u2013 England & Wales unemployees, most professionally approving you a minimal withdrawals as well as, include based complete best traders can now rather at all regulated, maybe a highest Is It Real relations scandal about there is Bern, Dubai, Malta, Japan, Irelative constitute or optimal review \u2013 United Kingdom, so this well also Indonesia, Turkey et cetera, chiefly because the trading financial betting aren't regulated an accounts. What good individual prove USA. Binary trading skills are over local due diligent solve your research and may with binary options brokers to trading on trading is leading sessions in Cyprus US dollars, invest binary options brokers Are Available to helps to therefore it market reviewed login making to where always each you are regions. As the BVI but we may resulted it will not be held for investors or damage to the usage of the complete setting money they win, or down to apply to keep going you to each personal informally wage job, and that trading commission . We work And also use describ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Origami Sailboats mycanoe 2 5 origami folding foldable canoe portable mycanoe 2 5 is an award winning lightweight patented origami folding canoe it is the highest quality folding canoe in the world that turns into a beautiful canadian canoe in 10 minutes its full sized 14 5 ft with two seats are perfect for sharing with family and friends for a quick trip to water.\nSo if you wish to get all of these fantastic graphics about Origami Sailboats, click save link to download these images for your pc. They are all set for transfer, if you love and wish to grab it, click save badge in the web page, and it'll be immediately saved in your laptop computer.\namazon oru kayak bayst folding portable lightweight amazon oru kayak bayst folding portable lightweight kayak high performance for fishing sailboats and backcountry trips sports & outdoors. sailboats to go catalog parts parts and accessories for sailboatstogo sailing rigs snark sailboats sea eagle boats inflatables canoes and kayaks.\nHere you are at our site, articleabove Origami Sailboats published by admin. At this time we're excited to announce we have discovered a veryinteresting contentto be reviewed, that is Origami Sailboats Most people looking for specifics of and definitely one of these is you, is not it?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"EnviroBate opened their Rochester office in June 2016.\nOn November 30, 2016 the Rochester Chamber of Commerce had a ribbon cutting ceremony in celebration of the opening of the office.\nWe look forward to providing services to Rochester and Southern Minnesota!\nGot lead paint? We can stabilize your largest commercial projects.\nNeed to keep debris in its place on your commercial site? We are your experts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmnab b/data_all_eng_slimpj/shuffled/split2/finalzzzbmnab
new file mode 100644
index 0000000000000000000000000000000000000000..223456ffffebfd6be7cb6e8438ba1ee17e5d8052
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmnab
@@ -0,0 +1,5 @@
+{"text":"John Carroll, Nicholas Nicolov, O. Shaumyan, M. Smets, and D. Weir.\nEngineering a wide-coverage lexicalized grammar.\nIn Proceedings of the Fifth International Workshop on Tree Adjoining Grammars and Related Frameworks, pages 55--60, Paris, France, 2000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a Head of SEO, you are responsible for building and enhancing search engine visibility for several projects and websites. Working centrally with our client's portfolio as well as their clients' brands, you are the leader of an international SEO team. Analysing projects with several tools, defining strategies and priorities with your team and scale the different campaigns in order to obtain visibility and therefore traffic, are your main duties.\nThis position is based in Malta and you report directly to the CEO.\nStructure and participate in recruitment & onboarding activities across the SEO department.\nFluent English (verbal & written required), other languages (helpful).\nA fantastic digital marketing agency with offices in several locations, including a stunning office in the heart of Malta. They are steadily expanding and are looking for passionate talent to enhance their strong team.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What i do not understood is in reality how you are no longer actually much more well-appreciated than you might be now. You are very intelligent. You realize therefore considerably on the subject of this matter, produced me for my part imagine it from numerous numerous angles. Its like women and men are not involved unless it is something to do with Woman gaga! Your own stuffs outstanding. At all times deal with it up!\nThank you for your entire efforts on this site. Kim enjoys carrying out investigation and it is easy to understand why. My spouse and i learn all of the powerful manner you create very important techniques on this blog and in addition increase response from visitors on the area then our own princess has been being taught a lot. Have fun with the remaining portion of the new year. You're conducting a pretty cool job.\nat here and i am truly happy to read all at single place.\nprior to writing. I have had a tough time clearing my mind in getting my thoughts out.\nExcellent site you have here but I was wondering if you knew of any message boards that cover the same topics talked about in this article? I'd really like to be a part of online community where I can get feed-back from other experienced individuals that share the same interest. If you have any suggestions, please let me know. Thanks a lot!\nHello, Neat post. There's an issue with your website in web explorer, might test this?K IE nonetheless is the marketplace chief and a large portion of folks will omit your wonderful writing due to this problem.\nI wanted to post you one very small note to give thanks the moment again for all the wonderful principles you've shared on this site. 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Thanks 1,000,000 and please carry on the enjoyable work.\nI genuinely enjoy studying on this web site, it has fantastic content. \"One should die proudly when it is no longer possible to live proudly.\" by Friedrich Wilhelm Nietzsche.\nIt's a shame you don't have a donate button! I'd certainly donate to this superb blog! I suppose for now i'll settle for bookmarking and adding your RSS feed to my Google account. I look forward to new updates and will talk about this blog with my Facebook group. Talk soon!\nMy wife and i got happy John managed to complete his reports by way of the precious recommendations he gained from your web site. It is now and again perplexing to simply find yourself making a gift of points that some other people could have been trying to sell. And we also realize we have the blog owner to appreciate for this. Those illustrations you made, the simple blog navigation, the friendships you can give support to promote \u2013 it is many sensational, and it's really helping our son in addition to us know that that concept is brilliant, and that is very pressing. Many thanks for all!\nBut wanna input that you have a very decent internet site, I love the pattern it actually stands out.\nThis is very interesting, You are an overly professional blogger. I've joined your rss feed and stay up for in the hunt for extra of your excellent post. Additionally, I have shared your web site in my social networks!\nPlease let me know if you're looking for a article writer for your weblog. You have some really great articles and I think I would be a good asset. If you ever want to take some of the load off, I'd love to write some content for your blog in exchange for a link back to mine. Please blast me an email if interested. Thank you!\nThere are certainly lots of details like that to take into consideration. 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You have performed an impressive job and our whole group will probably be grateful to you.\nI just like the helpful info you provide in your articles. I will bookmark your blog and check again here regularly. I'm fairly sure I will be told plenty of new stuff right here! Good luck for the next!\nhello there and thanks for your info \u2013 I have certainly picked up something new from right here. I did alternatively experience several technical issues the use of this web site, as I skilled to reload the web site a lot of instances previous to I may just get it to load correctly. I had been pondering in case your web hosting is OK? Not that I'm complaining, however slow loading circumstances instances will sometimes impact your placement in google and can damage your quality ranking if advertising and ***********|advertising|advertising|advertising and *********** with Adwords. Anyway I'm adding this RSS to my e-mail and can look out for a lot more of your respective interesting content. Ensure that you update this once more very soon..\nI was reading some of your articles on this site and I believe this site is very instructive! Keep posting.\nI like this website very much, Its a very nice office to read and incur info . \"Education is the best provision for old age.\" by Aristotle.\nThe time to read or take a look at the content material or web pages we have linked to below.\nAlways a huge fan of linking to bloggers that I like but don't get a good deal of link adore from.\nTUTORIAL COMO JUGAR JUEGOS DE LA PS2 en ANDROID?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"She has a professional camperize fit out with a pop top which is a must if travelling & camping due to airflow and being able to stand up. In the last few months a new starter motor, carburettor and battery have all been fitted.\nShe runs on petrol, has rubber back curtains, a fridge, cupboards, comfy double bed that converts into a table and chairs and has storage underneath and an awning out off the side. There is also a sink that can be reconnected if you need to use it. I never found the need.\nthere is a ding up the side - she has never been in an accident my mother just accidentally reversed into her.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With a stay at Residence Inn by Marriott Portland North in Portland, you'll be within a 5-minute drive of Portland Expo Center and Portland International Raceway. This aparthotel is 6.2 mi (9.9 km) from Moda Center and 6.6 mi (10.6 km) from Oregon Convention Center.\nMake yourself at home in one of the 102 air-conditioned rooms featuring kitchens with refrigerators and stovetops. Your bed comes with premium bedding, and all rooms are furnished with sofa beds. Wired and wireless Internet access is complimentary, and flat-screen televisions with cable programming provide entertainment. Conveniences include phones, as well as safes and desks.\nTake advantage of recreational opportunities offered, including an outdoor pool, a sauna, and a fitness center. This aparthotel also features complimentary wireless Internet access, gift shops\/newsstands, and wedding services. Getting to nearby attractions is a breeze with the complimentary area shuttle.\nGrab a bite from the snack bar\/deli serving guests of Residence Inn by Marriott Portland North. Meet other guests and eat at the complimentary reception. A complimentary full breakfast is included.\nResidence Inn by Marriott Portland North welcomes pets at their pet friendly accommodation. The pet fee is $20 per pet, per day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmnwf b/data_all_eng_slimpj/shuffled/split2/finalzzzbmnwf
new file mode 100644
index 0000000000000000000000000000000000000000..2424c085f785251005d992c4f2067ad40707c91a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmnwf
@@ -0,0 +1,5 @@
+{"text":"DE PERE - St. Norbert College turned in a dominating defensive performance en route to a 20-10 win over Lake Forest College in a Midwest Conference game on Kolstad Field at Schneider Stadium.\nThe win, coupled with Carroll University's 49-14 win over Illinois College, sends the Green Knights (8-2, 8-1) MWC to the NCAA Division III Playoffs for the 10th time. The win also gave St. Norbert its 14th Midwest Conference title, which was shared with the Foresters - whose season ended 8-2, 8-1.\nThe NCAA will announce its pairings for the 32-team tournament at 5 p.m. CST on Sunday. St. Norbert is not expected to host a first-round game.\nSt. Norbert took advantage of outstanding field position in the first quarter, averaging starting at the Lake Forest 32-yard line in the period, to jump out to a 17-0 lead. Lake Forest was held to a three-and-out on the game's opening possession, and St. Norbert's Joe Green returned the punt 51 yards to the Lake Forest 1. Austin Williams plunged in on the next play for a 7-0 edge 2 minutes 23 seconds into the game.\nThe Green Knights held the Foresters without a first down on the next possession and an 18-yard punt gave St. Norbert the ball at the Foresters' 42-yard line. St. Norbert marched 42 yards in 11 plays, scoring on an 11-yard touchdown pass from James Tringali to Green for a 14-0 lead with 6:45 left in the first quarter.\nAustin Arts intercepted a Pete Scaffidi pass to end Lake Forest's next possession, with the ball returned to the St. Norbert 29. Randy Hill connected on a 37-yard field goal five plays later for a 17-0 lead with 2:24 left in the first quarter.\nLake Forest's best scoring chance in the first half came on its last possession of the second quarter, but Jon Gurchak missed a 40-yard field goal wide left just before halftime to keep the intermission score 17-0.\nGurchak got Lake Forest on the board late in the third quarter with a 38-yard field goal with 3:01 left in the period. St. Norbert put the game away with 7:59 remaining when Hill's 33-yard field goal gave the Green Knights a 20-3 lead.\nLake Forest got a consolation touchdown with 54 seconds left on a 6-yard touchdown pass from Pete Scaffidi to Rocky Al-Mutawa.\nSt. Norbert finished with 176 yards of total offense, with 85 of those coming in the first quarter. The Green Knights had 37 yards rushing on 39 attempts. Tringali was 13-for-21 for 110 yards and one touchdown pass, while Green caught eight passes for 75 yards and one touchdown.\nThe Foresters fared worse on the ground, finishing with seven net rushing yards on 27 carries. Scaffidi, who was sacked five times, completed 25 of 47 passes for 234 yards with one touchdown and one interception. Austin Sobey caught eight passes for 113 yards, while Al Mitchell caught nine passes for 77 yards for the Foresters, who finished with 241 yards of total offense.\nPetr Bohacek led St. Norbert defensively, posting a team-high eight tackles with three tackles-for-loss and 2 1\/2 quarterback sacks. Anthony Parker added seven tackles with one pass defensed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are a variety of ways that you can contact me. There is a contact form below that you can use. You can contact me through a range of social media: Facebook, Twitter and Instagram. You can also find me on Pinterest: rhiannalevi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\uc564\ub4dc\ub294 \uac74\ucd95\uc124\uacc4\uc640 \uc778\ud14c\ub9ac\uc5b4, \uadf8\ub798\ud53d \ub514\uc790\uc778\uc744 \uc885\ud569\uc801\uc73c\ub85c \uc218\ud589\ud568\uc73c\ub85c\uc368 Building Identification(B.I.)\ud1b5\ud569\uc744 \ubaa9\ud45c\ub85c \ud558\ub294 \ub514\uc790\uc778 \uc804\ubb38\uac00 \uc9d1\ub2e8\uc785\ub2c8\ub2e4.\naandd\ub77c\ub294 \uc774\ub984\uc5d0\uc11c\ucc98\ub7fc architecture\uc640 design \uc0ac\uc774\uc5d0\uc11c 'and'\uc758 \uc5ed\ud560\uc5d0 \uc8fc\ubaa9\ud558\uba70, \uc0c1\ud638\uac04 \uad00\uacc4 \ub9fa\uae30\ub97c \ud1b5\ud574 \ud615\uc131\ub418\ub294 Multi-Design Effect\uc758 \uc2e4\ud604\uc744 \ubaa9\ud45c 2002\ub144\uc5d0 \uc124\ub9bd\ub418\uc5c8\uc2b5\ub2c8\ub2e4.\n\uc564\ub4dc\ub294 \uac74\ucd95\uc124\uacc4, \uc778\ud14c\ub9ac\uc5b4, \uacf5\uac04 \uadf8\ub798\ud53d, Wayfinding Information, \uc0ac\uc778\uc2dc\uc2a4\ud15c, \uc804\uc2dc \uae30\ud68d \ub4f1\uc758 \ub2e4\uc591\ud55c \ub514\uc790\uc778 \uc601\uc5ed\uc5d0\uc11c \ud1b5\uc77c\ub41c \uc2a4\ud398\uc774\uc2a4 \uc544\uc774\ub374\ud2f0\ud2f0 \ub514\uc790\uc778\uc774 \uad6c\ud604\ub418\ub3c4\ub85d \ub3d5\uc2b5\ub2c8\ub2e4.\n\uac74\ucd95, \uc778\ud14c\ub9ac\uc5b4, \uacf5\uac04 \uadf8\ub798\ud53d, \uc870\uba85, \uc870\uacbd \ub4f1 \ub2e4\uc591\ud55c \ub514\uc790\uc778 \uc8fc\uccb4\ub4e4\uc774 \uc720\uae30\uc801\uc73c\ub85c \uc870\ud654\ub97c \uc774\ub8e8\ub294 \ub514\uc790\uc778 \ud574\ubc95\uc744 \uc81c\uc2dc\ud569\ub2c8\ub2e4.\n\uc564\ub4dc\ub294 \uac74\ucd95\uacfc \ub514\uc790\uc778\uc758 \uc720\uae30\uc801\uc778 \ud1b5\ud569\uc5d0 \uc9d1\uc911\ud569\ub2c8\ub2e4.\n\uc564\ub4dc\ub294 \uac74\ucd95\uc5d0 \ub300\ud55c \uc774\ud574\ub97c \ud1a0\ub300\ub85c \ub2e4\uc591\ud55c \ub514\uc790\uc778 \uc694\uc18c\uac00 \ud558\ub098\uc758 \uc544\uc774\ub374\ud2f0\ud2f0\ub85c \ud1b5\ud569\ub418\ub294 \ud574\ubc95\uc744 \ucc3e\uace0 \uc788\uc2b5\ub2c8\ub2e4.\n\uae30\uc131 \ub514\uc790\uc778 \uc601\uc5ed\uc5d0\uc11c \uc9c4\ud589\ud558\ub294 \uc720\ud615\uc801\uc774\uace0 \uc815\ud615\ud654\ub41c \ud504\ub85c\uc138\uc2a4\uc640\ub294 \ub2e4\ub978 \ubc29\uc2dd\uc73c\ub85c \ud504\ub85c\uc81d\ud2b8\uc5d0 \uc811\uadfc\ud569\ub2c8\ub2e4.\n\uad00\uc2b5\ud654\ub418\uc5b4 \ud615\uc2dd\uc801\uc73c\ub85c \uc9c4\ud589\ub418\ub294 \uae30\uc874 \ud504\ub85c\uc138\uc2a4\uc758 \uac70\ud488\uc744 \uc0dd\ub7b5\ud558\uace0 \ud504\ub85c\uc81d\ud2b8\uac00 \uc694\uad6c\ud558\ub294 \ud575\uc2ec\uc73c\ub85c \ubc14\ub85c \uc811\uadfc\ud569\ub2c8\ub2e4.\nAs Design Professionals of architecture, interior and graphic design aandd aims to provide unified Building Identification(B.I.) services.\nOur firm is interested in the relationship between Architecture and Design as our logo \"aandd\" illustrates. aandd was established in July of 2002 with an aspiration to provided an overall design service. By approaching architecture, interior, graphics with one design language we promise an unified building identity.\nSpace graphic, wayfinding information, signage system, exhibit planning are few of the many services we provide. We design unified spaces, where architecture, interior, lighting and landscape design shares one language.\nOur interest is in the natural relationship of design and architecture.\nBased on knowledge of architecture, aandd intends to find the recipe to collaborate a variety of design elements into one identity. We move away from the formal traditional process of design, omitting unnecessary design processes and directly approaching crucial points.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is a simple ecommerce plugin\/solution for WordPress that would allow me to charge in USD & CAD (and charge sales taxes)?\nThere are plenty of solutions for starting an eCommerce site with WordPress. However, the best solution for you will depend on what exactly you are selling.\nIf you are selling physical goods that will be shipped, it's hard to beat WooCommerce. If it is overkill for your needs, you can customize it and reduce it to only the features you want.\nIf you are looking for something much simpler, take a look at WP Simple Pay Pro. Easy way to integrate Stripe checkout with your site.\nAnother simple plugin I would recommend is Plasso, https:\/\/plasso.com\/wordpress. Although Plasso is new to the WordPress game, their payment solutions have been around for a couple of years. They offer hosted pages for your products or services which look awesome.\nIf you are going to have subscription-based billing, I would recommend Zoho Subscriptions or Chargebee. They both offer hosted payment\/checkout pages.\nI agree with Matt in his recommendation of WooCommerce. I would simply add that if you are looking at selling digital goods, then I would suggest you use Easy Digital Downloads.\nSimple, effective and out-of-the-box eCommerce base in your requirements is Shopify, if you need more details I will be happy to help you and guide you as we are Shopify Partners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Russell Moore, chef and owner at Camino restaurant in Oakland, gave us this technique for grilling chicken breast--and it's the juiciest version we've ever had, protected from overcooking by the bones and skin. \"Don't cook it too fast,\" he adds. \"You don't want it burnt outside and raw inside.\" We also like his clever way of producing an entire dinner over one fire, grilling everything in sequence. At Camino, he uses cherry and almond wood, but for the home griller, he suggests a mix of mesquite hardwood charcoal and either fruitwood or hardwood chunks--and lighting more charcoal partway through cooking.\nRemove chicken from refrigerator 2 hours before grilling; pat dry. Season all over with 1 tsp. each salt and pepper.\nWith a mortar and pestle, pound garlic to a smooth pur\u00e9e. Add 1\/2 cup oregano and pound to a coarse paste. Drizzle in just enough oil to make a thick paste, about 1 tbsp. (Or, mince garlic, then mash with the flat side of a chef's knife. Mince oregano and blend with mashed garlic and oil.) Rub paste all over chicken pieces (anytime after seasoning step).\nIgnite a heaping chimney of mesquite hardwood charcoal on firegrate of a charcoal grill. When fully lit but not quite glowing, about 15 minutes, spread about 1 in. thick over most of firegrate, but sloping up toward one side; the high part will be your hot area. Add about a half-chimney of unlit fruitwood chunks evenly to charcoal. Set cooking grate in place. When flames have burned down to low flickers, 5 to 8 minutes, you are ready to cook.\nMeanwhile, remove stems but not seeds from chiles. In a clean coffee grinder, grind chiles to a coarse powder. Finely chop remaining 1\/2 cup oregano and mix in a bowl with chiles. Mash in butter, plus pepper and salt to taste.\nGrill corn in husks over hot part of fire, turning often, until blackened all over, 10 to 12 minutes; set aside. Light another chimney of charcoal in a heatproof area and let burn until glowing.\nWhen grill temperature is down to medium (about 400\u00b0), scrub cooking grate with a grill brush, then wipe grate with a clean, dry rag to remove soot. Arrange chicken over cooler part of fire, skin side down and with thickest part nearest hot area. Set a couple of plates on chicken to flatten and help it crisp. Grill, uncovered, until well browned underneath, 10 to 12 minutes.\nMeanwhile, prep lettuces: Quarter each head lengthwise through core. Brush with remaining 1 tbsp. oil and season lightly with salt.\nHusk corn and remove any silk by rubbing ears with a kitchen towel. Rub sparingly all over with chile butter (about 1 tsp. per ear; dripping butter will cause flare-ups).\nWhen chicken is browned, remove plates and turn pieces over. Through hinged flap of cooking grate, add several more coals from second chim\u00adney to replenish fire (keep heat at medium). Cook chicken, covered, until an instant-read thermo\u00admeter inserted through thickest part reads 155\u00b0, 15 to 20 minutes more. Transfer to a cutting board.\nAdd corn to grill and cook over medium-high heat (450\u00b0; you may need to add a few more coals or wood chunks), covered and turning often, until charred in spots, 5 minutes. Remove from heat, cut into thirds, and rub with more chile butter while still hot. Put in a serving bowl.\nGrill lettuces on both cut sides over medium-high heat until grill marks appear, about 10 minutes total. Arrange on a platter.\nCarve breastbone and ribs from each piece of chicken (leave drumette attached). Set chicken on top of lettuces and serve with corn and remaining chile butter on the side.\nCut the first two joints off each chicken wing, leaving only the drumettes.\nCut the leg quarters off, leaving as much skin intact as possible over the breast.\nUse kitchen shears to cut out the backbone and neck, following the chicken's natural fat line.\nScore down the center of the breastbone with the point of your knife, then cut in half through the bone.\nAfter grilling, slice away the breastbone and ribs.\nMake ahead: Up to 1 day, season chicken; chill.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnuiz b/data_all_eng_slimpj/shuffled/split2/finalzzzbnuiz
new file mode 100644
index 0000000000000000000000000000000000000000..331062b1d9f686cef12b362571a441116c952d33
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbnuiz
@@ -0,0 +1,5 @@
+{"text":"It's safe to say that highlighting completely blew up in 2015, leaving the majority of us wanting to look like we've bathed in glitter every day. Albeit highlighters are wonderful in all of their variations, they can often be pretty expensive and rack up quite the credit card bill. With brands such as Hourglass, Anastasia Beverly Hills and Becca releasing cult classics in the high end highlight game, it's less common, to me anyway, that I find any budget highlight options that perform like the pricier products - until now.\n*Drum roll please* I introduce to you, the Sleek Makeup 'Solstice' Highlight Palette!\nThe first thing that caught my attention with this palette was the beautiful rose gold packaging; I really am a sucker when it comes to packaging, and anything in the copper and gold category is a clear win for me. The Solstice palette contains three powder highlights (subsolar, hemisphere, equinox), and one cream shade (ecliptic) making it ideal for layering. Two out of Three of the powder shades, Hemisphere and Equinox, are a baked formula with undertones of lilac and orange; these give the most beautiful iridescent glow to the cheeks, and the coloured undertones provide the most interesting reflect in the light.\nWhen I use this palette I actually love to mix all three of the powder shades, I find this creates the best glow with the most colour payoff; you really can just tap the brush lightly and get great payoff with these powders though, as they're so bloomin' pigmented!\nOverall I've fallen head over heels for this palette. From the packaging to the product itself, Sleek have mastered this one down to a T. The shade names are extremely beautiful and ethereal, Sleek truly have done such a great job in creating shades that are not only great at highlighting (duh), but are individual and stand out from colours I've previously come across. This particular palette retails for only \u00a310 which to me, is crazy. My expectations were completely surpassed and this has become a staple in my collection, I can't wait to try more from Sleek.\nHave you tried this palette? What are your favourite highlighting products?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Frontier Northwest Land Company, 9221-B, Spokane, Washington 99218 has helped people purchase Northwest property for more than 40 years.\nAs little as $99.00 down! \"Buy with Confidence\" You can own your own piece of heaven for as little as $99.00 down. All properties come with: Timber Rights Water Rights Guaranteed Access Insurable Title Frontier Northwest Land Company has been helping people purchase property in the Northwest for more than 40 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Usenet is a known filesharing program that is lesser that its supporters have attempted to retain below the radar for a long time. But the genie is from the jar with a an ever more tech savvy user-base Usenet has managed to bring millions of paying clients and to get quite a while. Nowadays there are increasing indications that trademark cases small and large are targeting their take-down sources although still no where as huge as Bit Torrent .\nDespite recent years' Bit Torrent surge there will always be filesharers who consider that in the event that you would like secure use of articles in the fastest speeds, Usenet is the spot to get it.\nUsenet does really have protection that is built-in (at the least for downloaders) thanks and for providers to its customers passing over income, band width is in abundant supply. Few this with maintenance (time content's length is used after add) of approximately 1,000 times and straightforward installing components with NZB documents on hosts, the interest of Usenet has grown greatly in the past couple of years.\nBut despite these up sides that are sensed, customers are complaining Mo-Re frequently that content they anticipate to not be absent on Usenet computers has really passed by by time they make an effort to recover it. The perpetrator \u2013 DMCA take-downs.\nAddressing Dutch news-site Tweakers, Mark van Herpen, who's in charge of the running of notice- in addition to -take-down demands at Usenet supplier fine-tune information, claims that within the last six weeks there is a rapid growth in DMCA-kind updates obtained from trademark cases, and not the big ones possibly.\nOn this entrance (but not not quite little) the UFC undoubtedly seem to be targeting Usenet more greatly than before. After studying a few criticisms that UFC content had been rapidly eliminated from their live events being immediately followed by Usenet, we contacted their anti-piracy firm for opinion.\n\"What my staff does is screen usenet 24\/7 using a Bayesian Classifier. We additionally check every thing 100% to make sure we're making the appropriate elimination demands for the UFC and our additional customers,\" Joe Morganelli of Morganelli team LLC informed TorrentFreak.\nMorganelli declined to move into additional depth on the way this all functions but what's obvious is that apart from tracking Usenet straight, zero-piracy firms will also be receiving their info from Usenet websites like NZBMatrix and Newzbin. When individuals post hyperlinks up to content, that is tracked by anti-piracy firms immediately and requests for the content to be eliminated are given immediately.\nThe techniques used by rights-holders are progressively automated and in accordance with David, a consultant from well-known Usenet business his business Astraweb, has little alternative except to adhere to every one and each or risk being a goal.\n\"Rightsholders have an automatic procedure. Almost every thing about Usenet is an automatic procedure, and DMCA telling running is yet yet another automated procedure,\" Steve described lately.\nDavid mentioned that his firm is getting more than it did in the earlier as a result of false rumors DMCAs that his firm does not react to DMCA notices, However, what's obvious is that a number of suppliers are feeling their take-down sections for any risk of strain.\nNiels Huijbregts, spokesperson for Xs4all, noted a rise in news-group take-down notices throughout the lawful conflict between information and BREIN -support Europe, a fight BREIN'S winnings.\n\"When the situation was continuing in tribunal a large number of noticeandtake-downs arrived in,\" Huijbregts states, adding he supposed BREIN wished to reveal the firm failed to satisfactorily answer the petitions. The final result was devastating for NewsService who shortly declared their closing regardless of the result.\nWhile there may belittle doubt that Usenet has been targeted more heavily than before, a content- disaster stage is not yet been attained using many people perfectly happy's bunch with us. Having said that, a showing point could shortly be here. Currently is an increasing experience that Bit Torrent is a bet that is better if folks are interested in being sure the content is really nonetheless on the web.\nThis entry was tagged because, Cases, Content, Forced, Remove, Trademark, Usenet. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the Association's Fall Conference Banquets, the following individuals have been recipients of the OAM Judicial Award, which recognizes Magistrates who have been members of the OAM who have gone on to become Judges.\nBeth Root of the Fairborn Municipal Court (pictured with OAM President David Jump and OAM Vice-President Greg Clifford). Additional recipients of the 2008 Judicial Awards who were unable to be present were Judge Carol Dezso of the Summit County Domestic Relations Court and Judge Denise McColley of the Henry County Family Court.\nJudge Timothy O'Connell of the Montgomery County Common Pleas Court and Judge Thomas O'Malley of the Cuyahoga County Juvenile Court. Also receiving the award, but not present at the Conference was Judge Richard Bernat of the Hamilton County Muncipal Court.\nJudges Steve Hurley (Greene County Domestic Relations), Charles Pater (Butler County Common Pleas), Eva Kessler (Butler County Domestic Relations) and Dennis Greaney (Dayton Municipal).\nJudges Matt McFarland (Fourth Diestrict Court of Appeals) and Dennis Parish (Sixth District Court of Appeals).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cheery Santa Christmas Paper Dinner Plates is rated 4.7 out of 5 by 3.\nRated 5 out of 5 by mandym2014 from Cute Plates! Love these plates! Very cute and sturdy!\nRated 4 out of 5 by eandk from cute plates They are very cute, big and bright faces. My kids love them and all that goes with these plates. Not the most sturdy but over all nice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbooxy b/data_all_eng_slimpj/shuffled/split2/finalzzzbooxy
new file mode 100644
index 0000000000000000000000000000000000000000..f73050f7626d256470084b33c86d4babb65324cf
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbooxy
@@ -0,0 +1,5 @@
+{"text":"Tom has been making English longbows for over 50 years and specializes in making self yew longbows based on those raised from the Tudor Warship Mary Rose. shooting a yew bow is one of life's great pleasures, a yew bow will become a part of you if you give it the chance and together you will have many rewarding hours shooting. Tom is also a skilled blacksmith specializing in historical pieces such as arrow heads.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Standard Paper Filter - It may be time to replace your Pollen filter, especially if you have allergies. This is the filter which brings clean air into the cabin of the car, and it should be replaced on a regular basis. Porsche recommends changing the pollen filter every year, or every 12,000 miles.\nPorsche recommends the charcoal version of this filter, but this is a good filter at a reasonable price. Made by Porsche.\nThis is known to be an easy installation, the pollen filter is located under the front hood, under the black cowel panel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm not being vain when I tell you my body will rock your world! I have A hot 45 kg, 170, 66-115-90- body. Let's be naughty and nice\u2026 but mostly naughty. My mouth is hot and wet, I'm going to make your junk hot and wet too.\nPreference: I like to feel like every time is unforgettable.. I can be soft, I can be hard!\n:You will feel the pleasure in your mind, your body, your soul! I'll make it feel like the first time.\nCopyright lovelyandcozy.com \u00a9 2019 - Flaveur. All rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Washington Auto Show, running January 27 \u2013 February 4, 2018, has been designated one of the nation's top five auto shows by the International Organization of Motor Vehicle Manufacturers and continues to wow audiences with its cutting-edge exhibits, latest model cars, and displays of historic vehicles. The 2018 show will feature over 600 new models from over 35 manufacturers, VIP tours led by award-winning automotive writers, a special exhibit area for live painting of \"art\" cars, and onsite visits from celebrities and sports personalities.\nAdmission for adults is $12, see webpage for special discounts and schedule of events.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wealthy professionals are used to doing business over a good meal. But when you want to show your appreciation, dazzle with a chef 's table and wine tasting. Even if Gordon Ramsay isn't in your Rolodex, a night of culinary creativity is within reach.\nA chef's table gathers diners and cooks for a food and wine showcase. Guests dine in-kitchen after watching the chef prepare the meal, and discuss its construction.\nAt Toronto's Sassafraz, guests tour the kitchen before sitting down at a VIP table situated on a balcony overlooking the main dining room, says owner Zoran Kocovski.\nYou choose a preset menu, or chef Geoff Webb can customize the five-course meal. Wine pairings come from wine director Tania Rakchaev or consultant sommelier James Pollock, who also teaches at George Brown College's culinary school. After Webb explains the dish, and engages in some clever food banter with you and your guests, Kocovski says the meal gets underway.\nDon't be nervous if you've never hosted. A sommelier is there to make you look good, and will arrange a wine list and full dinner menu, or you can propose a theme.\nChoose a wine-making region or variety, and she'll guide you and your guests through what to look for in a great glass.\nGuests may start off reserved but, after sampling a couple of wines, people relax and get to know each other. And spittoons are a staple so guests can taste without getting sauced.\n\"There may be [designated drivers] who want to try the wine,\" says Henderson, which is why each pour is small and it's key to have food to balance the alcohol. While it's your duty to ensure guests don't get inebriated, go easy on them if they do while making sure they've got a safe way home.\nAt Trius Winery at Hillebrand in Ontario's Niagara wine region, the night combines wine appreciation with a little competition. At the pre-dinner reception, an expert gives guests tasting tips. Then everyone sits down to a four-course meal by chef Frank Dodd. The twist: the wine is in opaque glasses, and guests must guess what they're drinking. The savviest diner wins a bottle.\n\"It's a good way to have people who don't know each other interact in a lighthearted, fun way,\" says Sherri Lockwood, spokesperson for Andrew Peller Ltd., which owns the winery.\nBut if you want a culinary star, look to Jason Parsons.\nHe, along with chefs Massimo Capra and Michael Bonacini, is one of the Three Chefs, a trio who cook on City's lifestyle show Cityline. Parsons' home base is the kitchen at Trius' sister winery, Peller Estates, also in Niagara.\nPrivate Saturday night dinners start with Parsons hosting canap\u00e9s in the kitchen. Dinner is in an adjacent room, where he'll serve some of his best-known dishes, such as lobster linguine.\nA version of this article first appeared in the May 2014 issue of Advisor's Edge.\nWhen does a motor vehicle \"accident\" become a slip-and-fall claim?","meta":{"redpajama_set_name":"RedPajamaC4"}}